wpmath0000004_2

Calkin_algebra.html

  1. 0 K ( H ) B ( H ) B ( H ) / K ( H ) 0 0\rightarrow K(H)\rightarrow B(H)\rightarrow B(H)/K(H)\rightarrow 0

Canonical_basis.html

  1. ( X i ) i (X^{i})_{i}
  2. A A
  3. 𝒵 := [ v , v - 1 ] \mathcal{Z}:=\mathbb{Z}[v,v^{-1}]
  4. 𝒵 ± := [ v ± 1 ] \mathcal{Z}^{\pm}:=\mathbb{Z}[v^{\pm 1}]
  5. ¯ \overline{\cdot}
  6. v ¯ := v - 1 \overline{v}:=v^{-1}
  7. 𝒵 \mathcal{Z}
  8. F F
  9. ( t i ) i I (t_{i})_{i\in I}
  10. F F
  11. I I
  12. ( - , i ] := { j I | j i } (-\infty,i]:=\{j\in I|j\leq i\}
  13. i I i\in I
  14. F F F\to F
  15. ¯ \overline{\cdot}
  16. ¯ \overline{\cdot}
  17. 𝒵 ± \mathcal{Z}^{\pm}
  18. F ± := 𝒵 ± t j F^{\pm}:=\sum\mathcal{Z}^{\pm}t_{j}
  19. F F
  20. v = 0 v=0
  21. 𝒵 \mathcal{Z}
  22. ( c i ) i I (c_{i})_{i\in I}
  23. F F
  24. c i ¯ = c i \overline{c_{i}}=c_{i}
  25. c i j i 𝒵 + t j c_{i}\in\sum_{j\leq i}\mathcal{Z}^{+}t_{j}
  26. c i t i mod v F + c_{i}\equiv t_{i}\mod vF^{+}
  27. i I i\in I
  28. v = v=\infty
  29. ( c ~ i ) i I (\widetilde{c}_{i})_{i\in I}
  30. c ~ i ¯ = c ~ i \overline{\widetilde{c}_{i}}=\widetilde{c}_{i}
  31. c ~ i j i 𝒵 - t j \widetilde{c}_{i}\in\sum_{j\leq i}\mathcal{Z}^{-}t_{j}
  32. c ~ i t i mod v - 1 F - \widetilde{c}_{i}\equiv t_{i}\mod v^{-1}F^{-}
  33. i I i\in I
  34. v = v=\infty
  35. lim v v - 1 = 0 \lim_{v\to\infty}v^{-1}=0
  36. v v\mapsto\infty
  37. v - 1 = 0 v^{-1}=0
  38. v = v=\infty
  39. r i j 𝒵 r_{ij}\in\mathcal{Z}
  40. t j ¯ = i r i j t i \overline{t_{j}}=\sum_{i}r_{ij}t_{i}
  41. r i i = 1 r_{ii}=1
  42. r i j 0 i j r_{ij}\neq 0\implies i\leq j
  43. v = v=\infty
  44. F + F + ¯ = i c i \textstyle F^{+}\cap\overline{F^{+}}=\sum_{i}\mathbb{Z}c_{i}
  45. F + / v F + F^{+}/vF^{+}
  46. F - F - ¯ = i c ~ i F - / v - 1 F - \textstyle F^{-}\cap\overline{F^{-}}=\sum_{i}\mathbb{Z}\widetilde{c}_{i}\to F^% {-}/v^{-1}F^{-}
  47. v = 0 v=0
  48. ( W , S ) (W,S)
  49. H H
  50. ( T w ) w W (T_{w})_{w\in W}
  51. T w ¯ := T w - 1 - 1 \overline{T_{w}}:=T_{w^{-1}}^{-1}
  52. H H
  53. H H
  54. v = 0 v=0
  55. C w = y w P y , w ( v 2 ) T w C_{w}^{\prime}=\sum_{y\leq w}P_{y,w}(v^{2})T_{w}
  56. P y , w P_{y,w}
  57. A A
  58. J J
  59. A A
  60. D D
  61. A A
  62. λ \lambda
  63. A A
  64. μ \mu
  65. A A
  66. μ \mu
  67. λ \lambda
  68. A A
  69. A A
  70. x m - 1 , x m - 2 , , x 1 x_{m-1},x_{m-2},\ldots,x_{1}
  71. x m x_{m}
  72. λ i \lambda_{i}
  73. A A
  74. μ i \mu_{i}
  75. ( A - λ i I ) , ( A - λ i I ) 2 , , ( A - λ i I ) m i (A-\lambda_{i}I),(A-\lambda_{i}I)^{2},\ldots,(A-\lambda_{i}I)^{m_{i}}
  76. m i m_{i}
  77. ( A - λ i I ) m i (A-\lambda_{i}I)^{m_{i}}
  78. n - μ i n-\mu_{i}
  79. A A
  80. A A
  81. ρ k = r a n k ( A - λ i I ) k - 1 - r a n k ( A - λ i I ) k ( k = 1 , 2 , , m i ) . \rho_{k}=rank(A-\lambda_{i}I)^{k-1}-rank(A-\lambda_{i}I)^{k}\qquad(k=1,2,% \ldots,m_{i}).
  82. ρ k \rho_{k}
  83. λ i \lambda_{i}
  84. A A
  85. r a n k ( A - λ i I ) 0 = r a n k ( I ) = n . rank(A-\lambda_{i}I)^{0}=rank(I)=n.
  86. A = ( 4 1 1 0 0 - 1 0 4 2 0 0 1 0 0 4 1 0 0 0 0 0 5 1 0 0 0 0 0 5 2 0 0 0 0 0 4 ) A=\begin{pmatrix}4&1&1&0&0&-1\\ 0&4&2&0&0&1\\ 0&0&4&1&0&0\\ 0&0&0&5&1&0\\ 0&0&0&0&5&2\\ 0&0&0&0&0&4\end{pmatrix}
  87. λ 1 = 4 \lambda_{1}=4
  88. λ 2 = 5 \lambda_{2}=5
  89. μ 1 = 4 \mu_{1}=4
  90. μ 2 = 2 \mu_{2}=2
  91. γ 1 = 1 \gamma_{1}=1
  92. γ 2 = 1 \gamma_{2}=1
  93. λ 1 = 4 , \lambda_{1}=4,
  94. n - μ 1 = 6 - 4 = 2 , n-\mu_{1}=6-4=2,
  95. ( A - 4 I ) (A-4I)
  96. ( A - 4 I ) 2 (A-4I)^{2}
  97. ( A - 4 I ) 3 (A-4I)^{3}
  98. ( A - 4 I ) 4 (A-4I)^{4}
  99. m 1 = 4. m_{1}=4.
  100. ρ 4 = r a n k ( A - 4 I ) 3 - r a n k ( A - 4 I ) 4 = 3 - 2 = 1 , \rho_{4}=rank(A-4I)^{3}-rank(A-4I)^{4}=3-2=1,
  101. ρ 3 = r a n k ( A - 4 I ) 2 - r a n k ( A - 4 I ) 3 = 4 - 3 = 1 , \rho_{3}=rank(A-4I)^{2}-rank(A-4I)^{3}=4-3=1,
  102. ρ 2 = r a n k ( A - 4 I ) 1 - r a n k ( A - 4 I ) 2 = 5 - 4 = 1 , \rho_{2}=rank(A-4I)^{1}-rank(A-4I)^{2}=5-4=1,
  103. ρ 1 = r a n k ( A - 4 I ) 0 - r a n k ( A - 4 I ) 1 = 6 - 5 = 1. \rho_{1}=rank(A-4I)^{0}-rank(A-4I)^{1}=6-5=1.
  104. A A
  105. λ 1 = 4 , \lambda_{1}=4,
  106. λ 2 = 5 , \lambda_{2}=5,
  107. n - μ 2 = 6 - 2 = 4 , n-\mu_{2}=6-2=4,
  108. ( A - 5 I ) (A-5I)
  109. ( A - 5 I ) 2 (A-5I)^{2}
  110. m 2 = 2. m_{2}=2.
  111. ρ 2 = r a n k ( A - 5 I ) 1 - r a n k ( A - 5 I ) 2 = 5 - 4 = 1 , \rho_{2}=rank(A-5I)^{1}-rank(A-5I)^{2}=5-4=1,
  112. ρ 1 = r a n k ( A - 5 I ) 0 - r a n k ( A - 5 I ) 1 = 6 - 5 = 1. \rho_{1}=rank(A-5I)^{0}-rank(A-5I)^{1}=6-5=1.
  113. A A
  114. λ 2 = 5 , \lambda_{2}=5,
  115. A A
  116. { x 1 , x 2 , x 3 , x 4 , y 1 , y 2 } = { ( - 4 0 0 0 0 0 ) ( - 27 - 4 0 0 0 0 ) ( 25 - 25 - 2 0 0 0 ) ( 0 36 - 12 - 2 2 - 1 ) ( 3 2 1 1 0 0 ) ( - 8 - 4 - 1 0 1 0 ) } . \left\{x_{1},x_{2},x_{3},x_{4},y_{1},y_{2}\right\}=\left\{\begin{pmatrix}-4\\ 0\\ 0\\ 0\\ 0\\ 0\end{pmatrix}\begin{pmatrix}-27\\ -4\\ 0\\ 0\\ 0\\ 0\end{pmatrix}\begin{pmatrix}25\\ -25\\ -2\\ 0\\ 0\\ 0\end{pmatrix}\begin{pmatrix}0\\ 36\\ -12\\ -2\\ 2\\ -1\end{pmatrix}\begin{pmatrix}3\\ 2\\ 1\\ 1\\ 0\\ 0\end{pmatrix}\begin{pmatrix}-8\\ -4\\ -1\\ 0\\ 1\\ 0\end{pmatrix}\right\}.
  117. x 1 x_{1}
  118. λ 1 \lambda_{1}
  119. x 2 , x 3 x_{2},x_{3}
  120. x 4 x_{4}
  121. λ 1 \lambda_{1}
  122. y 1 y_{1}
  123. λ 2 \lambda_{2}
  124. y 2 y_{2}
  125. λ 2 \lambda_{2}
  126. J J
  127. A A
  128. M = ( x 1 x 2 x 3 x 4 y 1 y 2 ) = ( - 4 - 27 25 0 3 - 8 0 - 4 - 25 36 2 - 4 0 0 - 2 - 12 1 - 1 0 0 0 - 2 1 0 0 0 0 2 0 1 0 0 0 - 1 0 0 ) , M=\begin{pmatrix}x_{1}&x_{2}&x_{3}&x_{4}&y_{1}&y_{2}\end{pmatrix}=\begin{% pmatrix}-4&-27&25&0&3&-8\\ 0&-4&-25&36&2&-4\\ 0&0&-2&-12&1&-1\\ 0&0&0&-2&1&0\\ 0&0&0&2&0&1\\ 0&0&0&-1&0&0\end{pmatrix},
  129. J = ( 4 1 0 0 0 0 0 4 1 0 0 0 0 0 4 1 0 0 0 0 0 4 0 0 0 0 0 0 5 1 0 0 0 0 0 5 ) , J=\begin{pmatrix}4&1&0&0&0&0\\ 0&4&1&0&0&0\\ 0&0&4&1&0&0\\ 0&0&0&4&0&0\\ 0&0&0&0&5&1\\ 0&0&0&0&0&5\end{pmatrix},
  130. M M
  131. A A
  132. A M = M J AM=MJ

Canonical_bundle.html

  1. V V
  2. n n
  3. Ω n = ω \,\!\Omega^{n}=\omega
  4. ω D = i * ( ω X 𝒪 ( D ) ) . \omega_{D}=i^{*}(\omega_{X}\otimes\mathcal{O}(D)).
  5. K D = ( K X + D ) | D . K_{D}=(K_{X}+D)|_{D}.
  6. X X
  7. X X
  8. K X K_{X}
  9. X . X.
  10. X X
  11. h - d ( ω X . ) h^{-d}(\omega^{.}_{X})
  12. - d -d
  13. X X
  14. K X K_{X}
  15. X X
  16. R = d = 0 H 0 ( V , K V d ) . R=\bigoplus_{d=0}^{\infty}H^{0}(V,K_{V}^{d}).

Canonical_commutation_relation.html

  1. [ x ^ , p ^ x ] = i [\hat{x},\hat{p}_{x}]=i\hbar
  2. x x
  3. x x
  4. x x
  5. i i
  6. h / 2 π h/2π
  7. [ r ^ i , p ^ j ] = i δ i j [\hat{r}_{i},\hat{p}_{j}]=i\hbar\delta_{ij}
  8. δ i j \delta_{ij}
  9. i iℏ
  10. { x , p } = 1 . \{x,p\}=1\,.
  11. f f
  12. g g
  13. [ f ^ , g ^ ] = i { f , g } ^ . [\hat{f},\hat{g}]=i\hbar\widehat{\{f,g\}}\,.
  14. x 𝐱 , 𝐩 = i x\mathbf{x},\mathbf{p}=iℏ
  15. 𝐱 \mathbf{x}
  16. 𝐩 \mathbf{p}
  17. T r a c e ( A B ) = T r a c e ( B A ) Trace(A B)=Trace(B A)
  18. 2 p x n 2 ‖p‖ ‖x‖≥nℏ
  19. n n
  20. e x p ( i t 𝐱 ) exp(i t\mathbf{x})
  21. e x p ( i s 𝐩 ) exp(i s\mathbf{p})
  22. s i z e = 120 % e x p ( i t 𝐱 ) e x p ( i s 𝐩 ) = e x p ( i s t ) e x p ( i s 𝐩 ) e x p ( i t 𝐱 ) size=120\%exp(i t\mathbf{x}) exp(i s\mathbf{p})=exp(−iℏs t) exp(i s\mathbf{p})%  exp(i t\mathbf{x})
  23. s i z e = 120 % e x p ( i t 𝐱 ) e x p ( i s 𝐩 ) e x p ( i t 𝐱 ) e x p ( i s 𝐩 ) = e x p ( i s t ) size=120\%exp(i t\mathbf{x}) exp(i s\mathbf{p}) exp(−i t\mathbf{x}) exp(−i s% \mathbf{p})=exp(−iℏs t)
  24. [ x , p ] = i , [x,p]=i\hbar,\,
  25. {\mathcal{L}}
  26. x x
  27. Φ ( x ) Φ(x)
  28. p p
  29. π i = def ( x i / t ) . \pi_{i}\ \stackrel{\mathrm{def}}{=}\ \frac{\partial{\mathcal{L}}}{\partial(% \partial x_{i}/\partial t)}.
  30. t π i = x i . \frac{\partial}{\partial t}\pi_{i}=\frac{\partial{\mathcal{L}}}{\partial x_{i}}.
  31. [ x i , π j ] = i δ i j , [x_{i},\pi_{j}]=i\hbar\delta_{ij},\,
  32. [ p i , F ( x ) ] = - i F ( x ) x i ; [ x i , F ( p ) ] = i F ( p ) p i . [p_{i},F(\vec{x})]=-i\hbar\frac{\partial F(\vec{x})}{\partial x_{i}};\qquad[x_% {i},F(\vec{p})]=i\hbar\frac{\partial F(\vec{p})}{\partial p_{i}}.
  33. p p
  34. p kin = p - q A p_{\textrm{kin}}=p-qA\,\!
  35. p kin = p - q A c p_{\textrm{kin}}=p-\frac{qA}{c}\,\!
  36. q q
  37. A A
  38. c c
  39. m m
  40. H = 1 2 m ( p - q A c ) 2 + q ϕ H=\frac{1}{2m}\left(p-\frac{qA}{c}\right)^{2}+q\phi
  41. A A
  42. φ φ
  43. H ψ = i ħ ψ / t Hψ=iħ∂ψ/∂t
  44. A A = A + Λ A\to A^{\prime}=A+\nabla\Lambda
  45. ϕ ϕ = ϕ - 1 c Λ t \phi\to\phi^{\prime}=\phi-\frac{1}{c}\frac{\partial\Lambda}{\partial t}
  46. ψ ψ = U ψ \psi\to\psi^{\prime}=U\psi
  47. H H = U H U , H\to H^{\prime}=UHU^{\dagger},
  48. U = exp ( i q Λ c ) U=\exp\left(\frac{iq\Lambda}{\hbar c}\right)
  49. L = r × p L=r\times p\,\!
  50. [ L i , L j ] = i ϵ i j k L k [L_{i},L_{j}]=i\hbar{\epsilon_{ijk}}L_{k}
  51. ϵ i j k \epsilon_{ijk}
  52. ψ | L | ψ ψ | L | ψ = ψ | L | ψ + q c ψ | r × Λ | ψ . \langle\psi|L|\psi\rangle\to\langle\psi^{\prime}|L^{\prime}|\psi^{\prime}% \rangle=\langle\psi|L|\psi\rangle+\frac{q}{\hbar c}\langle\psi|r\times\nabla% \Lambda|\psi\rangle\,.
  53. K = r × ( p - q A c ) , K=r\times\left(p-\frac{qA}{c}\right),
  54. [ K i , K j ] = i ϵ i j k ( K k + q c x k ( x B ) ) [K_{i},K_{j}]=i\hbar{\epsilon_{ij}}^{\,k}\left(K_{k}+\frac{q\hbar}{c}x_{k}% \left(x\cdot B\right)\right)
  55. B = × A B=\nabla\times A
  56. A A
  57. B B
  58. ψ ψ
  59. Δ A Δ B 1 2 | [ A , B ] | 2 + | { A - A , B - B } | 2 , \Delta A\,\Delta B\geq\frac{1}{2}\sqrt{\left|\left\langle\left[{A},{B}\right]% \right\rangle\right|^{2}+\left|\left\langle\left\{A-\langle A\rangle,B-\langle B% \rangle\right\}\right\rangle\right|^{2}},
  60. A A , B A B B A AA,B≡A B−B A
  61. A A
  62. B B
  63. A A A−\langle A\rangle
  64. B B B−\langle B\rangle
  65. A A
  66. B B
  67. x x
  68. p p
  69. [ L x , L y ] = i ϵ x y z L z , [{L_{x}},{L_{y}}]=i\hbar\epsilon_{xyz}{L_{z}},
  70. ϵ x y z \epsilon_{xyz}
  71. ψ = | [ u e l l ] , m ψ=|[u^{\prime}ell^{\prime}],m\rangle
  72. z z
  73. Δ L x Δ L y 1 2 2 | L z | 2 , \Delta L_{x}\Delta L_{y}\geq\frac{1}{2}\sqrt{\hbar^{2}|\langle L_{z}\rangle|^{% 2}}~{},
  74. | L x 2 L y 2 | 2 2 m \sqrt{|\langle L_{x}^{2}\rangle\langle L_{y}^{2}\rangle|}\geq\frac{\hbar^{2}}{% 2}m
  75. l ( l + 1 ) - m 2 m , l(l+1)-m^{2}\geq m~{},
  76. [ u e l l ] ( [ u e l l ] + 1 ) m ( m + 1 ) [u^{\prime}ell^{\prime}]([u^{\prime}ell^{\prime}]+1)≥m(m+1)
  77. [ u e l l ] m [u^{\prime}ell^{\prime}]≥m

Canonical_coordinates.html

  1. q i q_{i}\,
  2. p i p_{i}\,
  3. { q i , q j } = 0 { p i , p j } = 0 { q i , p j } = δ i j \{q_{i},q_{j}\}=0\qquad\{p_{i},p_{j}\}=0\qquad\{q_{i},p_{j}\}=\delta_{ij}
  4. q i q_{i}
  5. p i p_{i}
  6. p i p_{i}
  7. ( q i , p j ) (q^{i},p_{j})
  8. ( x i , p j ) (x^{i},p_{j})
  9. i p i d q i \sum_{i}p_{i}\,\mathrm{d}q^{i}
  10. P X : T * Q P_{X}:T^{*}Q\to\mathbb{R}
  11. P X ( q , p ) = p ( X q ) P_{X}(q,p)=p(X_{q})
  12. T q * Q T_{q}^{*}Q
  13. X q X_{q}
  14. T q Q T_{q}Q
  15. P X P_{X}
  16. X q = i X i ( q ) q i X_{q}=\sum_{i}X^{i}(q)\frac{\partial}{\partial q^{i}}
  17. / q i \partial/\partial q^{i}
  18. P X ( q , p ) = i X i ( q ) p i P_{X}(q,p)=\sum_{i}X^{i}(q)\;p_{i}
  19. p i p_{i}
  20. / q i \partial/\partial q^{i}
  21. p i = P / q i p_{i}=P_{\partial/\partial q^{i}}
  22. q i q^{i}
  23. p j p_{j}
  24. T * Q T^{*}Q
  25. ( q i , q ˙ i ) (q^{i},\dot{q}^{i})
  26. q i q^{i}
  27. q ˙ i \dot{q}^{i}

Canonical_ensemble.html

  1. T T
  2. N N
  3. V V
  4. N V T NVT
  5. P P
  6. P = e F - E k T , P=e^{\frac{F-E}{kT}},
  7. E E
  8. k k
  9. F F
  10. F F
  11. F F
  12. F ( N , V , T ) F(N,V,T)
  13. P = 1 Z e - E / ( k T ) \textstyle P=\frac{1}{Z}e^{-E/(kT)}
  14. Z = e - F / ( k T ) \textstyle Z=e^{-F/(kT)}
  15. ρ̂ ρ̂
  16. ρ ^ = exp ( 1 k T ( F - H ^ ) ) , \hat{\rho}=\exp\big(\tfrac{1}{kT}(F-\hat{H})\big),
  17. Ĥ Ĥ
  18. e x p ( ) exp()
  19. F F
  20. T r ρ̂ = 1 Trρ̂=1
  21. e - F k T = Tr exp ( - 1 k T H ^ ) . e^{-\frac{F}{kT}}=\operatorname{Tr}\exp\big(-\tfrac{1}{kT}\hat{H}\big).
  22. i i
  23. ρ ^ = i e F - E i k T | ψ i ψ i | \hat{\rho}=\sum_{i}e^{\frac{F-E_{i}}{kT}}|\psi_{i}\rangle\langle\psi_{i}|
  24. e - F k T = i e - E i k T . e^{-\frac{F}{kT}}=\sum_{i}e^{\frac{-E_{i}}{kT}}.
  25. n n
  26. N N
  27. n = 3 N n=3N
  28. ρ = 1 h n C e F - E k T , \rho=\frac{1}{h^{n}C}e^{\frac{F-E}{kT}},
  29. E E
  30. h h
  31. e n e r g y × t i m e energy×time
  32. ρ ρ
  33. C C
  34. F F
  35. F F
  36. ρ ρ
  37. e - F k T = 1 h n C e - E k T d p 1 d q n e^{-\frac{F}{kT}}=\int\ldots\int\frac{1}{h^{n}C}e^{\frac{-E}{kT}}\,dp_{1}% \ldots dq_{n}
  38. h < s u p > n C h<sup>nC
  39. h = 1 e n e r g y u n i t t × × t i m e u n i t t h=1energyunitt××timeunitt
  40. h h
  41. N N
  42. C = N ! C=N!
  43. N N

Canonical_normal_form.html

  1. n n
  2. x 1 , , x n {x_{1},\dots,x_{n}}
  3. n n
  4. a b c abc
  5. a b c ab^{\prime}c
  6. a b c abc^{\prime}
  7. a a
  8. b b
  9. c c
  10. x i x_{i}
  11. x i x^{\prime}_{i}
  12. 2 i value ( x i ) 2^{i}\operatorname{value}(x_{i})
  13. a b c abc^{\prime}
  14. m 6 m_{6}
  15. m 1 , m 2 , m 4 , m_{1},m_{2},m_{4},
  16. m 7 m_{7}
  17. u ( c i , x , y ) = m 1 + m 2 + m 4 + m 7 = ( c i , x , y ) + ( c i , x , y ) + ( c i , x , y ) + ( c i , x , y ) u(ci,x,y)=m_{1}+m_{2}+m_{4}+m_{7}=(ci^{\prime},x^{\prime},y)+(ci^{\prime},x,y^% {\prime})+(ci,x^{\prime},y^{\prime})+(ci,x,y)
  18. n n
  19. x 1 , , x n {x_{1},\dots,x_{n}}
  20. n n
  21. ( x i ) (x_{i})
  22. ( x i ) (x^{\prime}_{i})
  23. a + b + c a^{\prime}+b^{\prime}+c
  24. a + b + c a+b+c
  25. a + b + c a^{\prime}+b^{\prime}+c^{\prime}
  26. M 0 , M 1 , M 2 , M_{0},M_{1},M_{2},
  27. M 4 M_{4}
  28. M 0 M 1 M 2 M 4 M_{0}M_{1}M_{2}M_{4}
  29. M 5 = a + b + c = ( a b c ) = m 5 M_{5}=a^{\prime}+b+c^{\prime}=(ab^{\prime}c)^{\prime}=m_{5}^{\prime}
  30. f = a b c + a b c f=a^{\prime}bc+abc
  31. f = b c f=bc
  32. b c = a b c + a b c bc=a^{\prime}bc+abc
  33. f = ( a + a ) b c f=(a^{\prime}+a)bc
  34. u ( c i , x , y ) = m 1 + m 2 + m 4 + m 7 u(ci,x,y)=m_{1}+m_{2}+m_{4}+m_{7}
  35. u ( c i , x , y ) u(ci,x,y)
  36. M 0 , M 3 , M 5 , M 6 M_{0},M_{3},M_{5},M_{6}
  37. m 0 , m 3 , m 5 , m 6 m_{0},m_{3},m_{5},m_{6}
  38. c o ( c i , x , y ) = M 0 M 1 M 2 M 4 co(ci,x,y)=M_{0}M_{1}M_{2}M_{4}
  39. c o ( c i , x , y ) co(ci,x,y)
  40. M 0 , M 1 , M 2 , M 4 M_{0},M_{1},M_{2},M_{4}
  41. m 0 , m 1 , m 2 , m 4 m_{0},m_{1},m_{2},m_{4}
  42. c o ( c i , x , y ) = AND ( M 3 , M 5 , M 6 , M 7 ) = NOR ( m 3 , m 5 , m 6 , m 7 ) . co^{\prime}(ci,x,y)=\mathrm{AND}(M_{3},M_{5},M_{6},M_{7})=\mathrm{NOR}(m_{3},m% _{5},m_{6},m_{7}).
  43. m 7 m_{7}

Canonical_quantization.html

  1. x x
  2. p p
  3. | ψ |\psi\rangle
  4. | ψ n |\psi_{n}\rangle
  5. H ^ | ψ n = E n | ψ n \hat{H}|\psi_{n}\rangle=E_{n}|\psi_{n}\rangle
  6. | ψ n |\psi_{n}\rangle
  7. | ψ = n = 0 a n | ψ n |\psi\rangle=\sum_{n=0}^{\infty}a_{n}|\psi_{n}\rangle
  8. x x
  9. | x |x\rangle
  10. X ^ | x = x | x \hat{X}|x\rangle=x|x\rangle
  11. ψ ( x ) = x | ψ \psi(x)=\langle x|\psi\rangle
  12. | p |p\rangle
  13. P ^ \hat{P}
  14. ψ ( p ) = p | ψ \psi(p)=\langle p|\psi\rangle
  15. [ X ^ , P ^ ] = X ^ P ^ - P ^ X ^ = i [\hat{X},\hat{P}]=\hat{X}\hat{P}-\hat{P}\hat{X}=i\hbar
  16. Δ x Δ p ħ / 2 ΔxΔp≥ħ/2
  17. ψ ( r ) \psi({r})
  18. ψ ( r 1 , r 2 , , r N ) \psi({r}_{1},{r}_{2},...,{r}_{N})
  19. ψ ( r 1 , , r j , , r k , , r N ) = + ψ ( r 1 , , r k , , r j , , r N ) \psi({r}_{1},...,{r}_{j},...,{r}_{k},...,{r_{N}})=+\psi({r}_{1},...,{r}_{k},..% .,{r}_{j},...,{r}_{N})
  20. ψ ( r 1 , , r j , , r k , , r N ) = - ψ ( r 1 , , r k , , r j , , r N ) \psi({r}_{1},...,{r}_{j},...,{r}_{k},...,{r_{N}})=-\psi({r}_{1},...,{r}_{k},..% .,{r}_{j},...,{r}_{N})
  21. ( r j , r k ) ({r}_{j},{r}_{k})
  22. , , x̂,p̂,p̂
  23. i ħ
  24. { x 3 , p 3 } + 1 12 { { p 2 , x 3 } , { x 2 , p 3 } } \displaystyle\{x^{3},p^{3}\}+\tfrac{1}{12}\{\{p^{2},x^{3}\},\{x^{2},p^{3}\}\}
  25. Q Q
  26. f f
  27. Q x ψ = x ψ Q_{x}\psi=x\psi
  28. Q p ψ = - i x ψ Q_{p}\psi=-i\hbar\partial_{x}\psi~{}~{}
  29. f Q f f\longmapsto Q_{f}~{}~{}
  30. [ Q f , Q g ] = i Q { f , g } [Q_{f},Q_{g}]=i\hbar Q_{\{f,g\}}~{}~{}
  31. Q g f = g ( Q f ) Q_{g\circ f}=g(Q_{f})~{}~{}
  32. ħ 0 ħ→0
  33. | 0 |0\rangle
  34. π π
  35. ( ϕ ) = 1 2 ( t ϕ ) 2 - 1 2 ( x ϕ ) 2 - 1 2 m 2 ϕ 2 - V ( ϕ ) , \mathcal{L}(\phi)=\frac{1}{2}(\partial_{t}\phi)^{2}-\frac{1}{2}(\partial_{x}% \phi)^{2}-\frac{1}{2}m^{2}\phi^{2}-V(\phi),
  36. V ( φ ) V(φ)
  37. S ( ϕ ) = ( ϕ ) d x d t = L ( ϕ , t ϕ ) d t S(\phi)=\int\mathcal{L}(\phi)dxdt=\int L(\phi,\partial_{t}\phi)dt
  38. L L
  39. π = t ϕ \pi=\partial_{t}\phi
  40. H ( ϕ , π ) = d x [ 1 2 π 2 + 1 2 ( x ϕ ) 2 + 1 2 m 2 ϕ 2 + V ( ϕ ) ] . H(\phi,\pi)=\int dx\left[\frac{1}{2}\pi^{2}+\frac{1}{2}(\partial_{x}\phi)^{2}+% \frac{1}{2}m^{2}\phi^{2}+V(\phi)\right].
  41. ϕ ( x ) \phi(x)
  42. π ( x ) \pi(x)
  43. [ ϕ ( x ) , ϕ ( y ) ] = 0 , [ π ( x ) , π ( y ) ] = 0 , [ ϕ ( x ) , π ( y ) ] = i δ ( x - y ) . [\phi(x),\phi(y)]=0,\ \ [\pi(x),\pi(y)]=0,\ \ [\phi(x),\pi(y)]=i\hbar\delta(x-% y).
  44. ϕ \phi
  45. π \pi
  46. 𝒪 ( t ) = e i t H 𝒪 e - i t H . \mathcal{O}(t)=e^{itH}\mathcal{O}e^{-itH}.
  47. φ φ
  48. π π
  49. 𝒪 \mathcal{O}
  50. \mathcal{H}
  51. H H
  52. 𝒪 \mathcal{O}
  53. \mathcal{H}
  54. H H
  55. V ( φ ) V(φ)
  56. V ( φ ) V(φ)
  57. ϕ k = ϕ ( x ) e - i k x d x , π k = π ( x ) e - i k x d x . \phi_{k}=\int\phi(x)e^{-ikx}dx,\ \ \pi_{k}=\int\pi(x)e^{-ikx}dx.
  58. ϕ - k = ϕ k , π - k = π k \phi_{-k}=\phi_{k}^{\dagger},~{}~{}~{}\pi_{-k}=\pi_{k}^{\dagger}
  59. H = 1 2 k = - [ π k π k + ω k 2 ϕ k ϕ k ] , H=\frac{1}{2}\sum_{k=-\infty}^{\infty}\left[\pi_{k}\pi_{k}^{\dagger}+\omega_{k% }^{2}\phi_{k}\phi_{k}^{\dagger}\right],
  60. ω k = k 2 + m 2 \omega_{k}=\sqrt{k^{2}+m^{2}}
  61. a k = 1 2 ω k ( ω k ϕ k + i π k ) , a k = 1 2 ω k ( ω k ϕ k - i π k ) , a_{k}=\frac{1}{\sqrt{2\hbar\omega_{k}}}\left(\omega_{k}\phi_{k}+i\pi_{k}\right% ),\ \ a_{k}^{\dagger}=\frac{1}{\sqrt{2\hbar\omega_{k}}}\left(\omega_{k}\phi_{k% }^{\dagger}-i\pi_{k}^{\dagger}\right),
  62. k k
  63. \mathcal{H}
  64. | 0 |0\rangle
  65. k k
  66. H = k = - ω k a k a k = k = - ω k N k H=\sum_{k=-\infty}^{\infty}\hbar\omega_{k}a_{k}^{\dagger}a_{k}=\sum_{k=-\infty% }^{\infty}\hbar\omega_{k}N_{k}
  67. k k
  68. H H
  69. { θ k , θ l } = δ k l , { θ k , θ l } = 0 , { θ k , θ l } = 0. \{\theta_{k},\theta_{l}^{\dagger}\}=\delta_{kl},\ \ \{\theta_{k},\theta_{l}\}=% 0,\ \ \{\theta_{k}^{\dagger},\theta_{l}^{\dagger}\}=0.
  70. ( θ k ) 2 | 0 = 0 (\theta_{k}^{\dagger})^{2}|0\rangle=0
  71. φ φ
  72. φ φ
  73. φ φ
  74. v v
  75. V ( φ ) = g φ < s u p > 4 V(φ)=gφ<sup>4
  76. v v
  77. φ φ
  78. φ ( x , t ) v φ(x,t)−v
  79. ħ ħ
  80. ħ ħ
  81. A A , B AA,B
  82. ħ 0 ħ→0

Capillary_wave.html

  1. ω 2 = σ ρ + ρ | k | 3 , \omega^{2}=\frac{\sigma}{\rho+\rho^{\prime}}\,|k|^{3},
  2. λ = 2 π k . \lambda=\frac{2\pi}{k}.
  3. ω 2 = σ ρ | k | 3 . \omega^{2}=\frac{\sigma}{\rho}\,|k|^{3}.
  4. g σ / ρ 4 \scriptstyle\sqrt[4]{g\sigma/\rho}
  5. 1 λ σ / ( ρ g ) \scriptstyle\frac{1}{\lambda}\sqrt{\sigma/(\rho g)}
  6. ω 2 = | k | ( ρ - ρ ρ + ρ g + σ ρ + ρ k 2 ) , \omega^{2}=|k|\left(\frac{\rho-\rho^{\prime}}{\rho+\rho^{\prime}}g+\frac{% \sigma}{\rho+\rho^{\prime}}k^{2}\right),
  7. ( ρ - ρ ) / ( ρ + ρ ) (\rho-\rho^{\prime})/(\rho+\rho^{\prime})
  8. λ c = 2 π σ ( ρ - ρ ) g . \lambda_{c}=2\pi\sqrt{\frac{\sigma}{(\rho-\rho^{\prime})g}}.
  9. V g = d x d y 0 η d z ( ρ - ρ ) g z = 1 2 ( ρ - ρ ) g d x d y η 2 , V_{\mathrm{g}}=\iint dx\,dy\;\int_{0}^{\eta}dz\;(\rho-\rho^{\prime})gz=\frac{1% }{2}(\rho-\rho^{\prime})g\iint dx\,dy\;\eta^{2},
  10. V st = σ d x d y [ 1 + ( η x ) 2 + ( η y ) 2 - 1 ] 1 2 σ d x d y [ ( η x ) 2 + ( η y ) 2 ] , V_{\mathrm{st}}=\sigma\iint dx\,dy\;\left[\sqrt{1+\left(\frac{\partial\eta}{% \partial x}\right)^{2}+\left(\frac{\partial\eta}{\partial y}\right)^{2}}-1% \right]\approx\frac{1}{2}\sigma\iint dx\,dy\;\left[\left(\frac{\partial\eta}{% \partial x}\right)^{2}+\left(\frac{\partial\eta}{\partial y}\right)^{2}\right],
  11. T = 1 2 d x d y [ - η d z ρ | Φ | 2 + η + d z ρ | Φ | 2 ] . T=\frac{1}{2}\iint dx\,dy\;\left[\int_{-\infty}^{\eta}dz\;\rho\,\left|\nabla% \Phi\right|^{2}+\int_{\eta}^{+\infty}dz\;\rho^{\prime}\,\left|\nabla\Phi^{% \prime}\right|^{2}\right].
  12. 2 Φ = 0 \nabla^{2}\Phi=0
  13. 2 Φ = 0. \nabla^{2}\Phi^{\prime}=0.
  14. T 1 2 d x d y [ ρ Φ Φ z - ρ Φ Φ z ] at z = 0 . T\approx\frac{1}{2}\iint dx\,dy\;\left[\rho\,\Phi\,\frac{\partial\Phi}{% \partial z}\;-\;\rho^{\prime}\,\Phi^{\prime}\,\frac{\partial\Phi^{\prime}}{% \partial z}\right]_{\,\text{at }z=0}.
  15. η = a cos ( k x - ω t ) = a cos θ , \eta=a\,\cos\,(kx-\omega t)=a\,\cos\,\theta,
  16. Φ z = η t \frac{\partial\Phi}{\partial z}=\frac{\partial\eta}{\partial t}
  17. Φ z = η t \frac{\partial\Phi^{\prime}}{\partial z}=\frac{\partial\eta}{\partial t}
  18. Φ ( x , y , z , t ) = + 1 | k | e + | k | z ω a sin θ , Φ ( x , y , z , t ) = - 1 | k | e - | k | z ω a sin θ . \begin{aligned}\displaystyle\Phi(x,y,z,t)&\displaystyle=+\frac{1}{|k|}\,\text{% e}^{+|k|z}\,\omega a\,\sin\,\theta,\\ \displaystyle\Phi^{\prime}(x,y,z,t)&\displaystyle=-\frac{1}{|k|}\,\text{e}^{-|% k|z}\,\omega a\,\sin\,\theta.\end{aligned}
  19. V g = 1 4 ( ρ - ρ ) g a 2 λ , V st = 1 4 σ k 2 a 2 λ , T = 1 4 ( ρ + ρ ) ω 2 | k | a 2 λ . \begin{aligned}\displaystyle V\text{g}&\displaystyle=\frac{1}{4}(\rho-\rho^{% \prime})ga^{2}\lambda,\\ \displaystyle V\text{st}&\displaystyle=\frac{1}{4}\sigma k^{2}a^{2}\lambda,\\ \displaystyle T&\displaystyle=\frac{1}{4}(\rho+\rho^{\prime})\frac{\omega^{2}}% {|k|}a^{2}\lambda.\end{aligned}
  20. L = 1 4 [ ( ρ + ρ ) ω 2 | k | - ( ρ - ρ ) g - σ k 2 ] a 2 λ . L=\frac{1}{4}\left[(\rho+\rho^{\prime})\frac{\omega^{2}}{|k|}-(\rho-\rho^{% \prime})g-\sigma k^{2}\right]a^{2}\lambda.
  21. ω 2 = | k | ( ρ - ρ ρ + ρ g + σ ρ + ρ k 2 ) , \omega^{2}=|k|\left(\frac{\rho-\rho^{\prime}}{\rho+\rho^{\prime}}\,g+\frac{% \sigma}{\rho+\rho^{\prime}}\,k^{2}\right),
  22. E ¯ = 1 2 [ ( ρ - ρ ) g + σ k 2 ] a 2 . \bar{E}=\frac{1}{2}\,\left[(\rho-\rho^{\prime})\,g+\sigma k^{2}\right]\,a^{2}.

Capital_accumulation.html

  1. s s
  2. k k
  3. K K
  4. Δ K K = Δ K Y K Y = s k {{\Delta K}\over K}={{{\Delta K}\over Y}\over{K\over Y}}={s\over k}
  5. Y Y
  6. k = K Y k={K\over Y}
  7. Y Y
  8. K K
  9. s s
  10. Y Y
  11. k k

Capital_Consumption_Allowance.html

  1. G D P = N D P + C C A + I n d i r e c t T a x e s GDP=NDP+CCA+IndirectTaxes

Carathéodory's_theorem_(convex_hull).html

  1. r d r\leq d
  2. 𝐱 = j = 1 k λ j 𝐱 j \mathbf{x}=\sum_{j=1}^{k}\lambda_{j}\mathbf{x}_{j}
  3. j = 1 k λ j = 1 \sum_{j=1}^{k}\lambda_{j}=1
  4. j = 2 k μ j ( 𝐱 j - 𝐱 1 ) = 0. \sum_{j=2}^{k}\mu_{j}(\mathbf{x}_{j}-\mathbf{x}_{1})=\mathbf{0}.
  5. μ 1 := - j = 2 k μ j \mu_{1}:=-\sum_{j=2}^{k}\mu_{j}
  6. j = 1 k μ j 𝐱 j = 𝟎 \sum_{j=1}^{k}\mu_{j}\mathbf{x}_{j}=\mathbf{0}
  7. j = 1 k μ j = 0 \sum_{j=1}^{k}\mu_{j}=0
  8. 𝐱 = j = 1 k λ j 𝐱 j - α j = 1 k μ j 𝐱 j = j = 1 k ( λ j - α μ j ) 𝐱 j \mathbf{x}=\sum_{j=1}^{k}\lambda_{j}\mathbf{x}_{j}-\alpha\sum_{j=1}^{k}\mu_{j}% \mathbf{x}_{j}=\sum_{j=1}^{k}(\lambda_{j}-\alpha\mu_{j})\mathbf{x}_{j}
  9. α := min 1 j k { λ j μ j : μ j > 0 } = λ i μ i . \alpha:=\min_{1\leq j\leq k}\left\{\tfrac{\lambda_{j}}{\mu_{j}}:\mu_{j}>0% \right\}=\tfrac{\lambda_{i}}{\mu_{i}}.
  10. λ j - α μ j 0. \lambda_{j}-\alpha\mu_{j}\geq 0.
  11. 𝐱 = j = 1 k ( λ j - α μ j ) 𝐱 j \mathbf{x}=\sum_{j=1}^{k}(\lambda_{j}-\alpha\mu_{j})\mathbf{x}_{j}
  12. λ j - α μ j \lambda_{j}-\alpha\mu_{j}
  13. λ i - α μ i = 0 \lambda_{i}-\alpha\mu_{i}=0

Carbene.html

  1. \hbar

Carbon-13.html

  1. C = 100 Y 1.1 X C=\frac{100Y}{1.1X}
  2. δ 13 C Sample = ( C 13 / 12 C Sample C 13 / 12 C PDB - 1 ) 1000 \delta^{13}C\text{Sample}=\left(\frac{{}^{13}C/^{12}C\text{Sample}}{{}^{13}C/^% {12}C_{\mathrm{PDB}}}-1\right)\cdot 1000

Cardioid.html

  1. x = a ( 2 cos t - cos 2 t ) , x=a(2\cos t-\cos 2t),\,
  2. y = a ( 2 sin t - sin 2 t ) . y=a(2\sin t-\sin 2t).\,
  3. z = a ( 2 e i t - e 2 i t ) . z=a(2e^{it}-e^{2it}).\,
  4. ( z z ¯ - a 2 ) 2 - 4 a 2 ( z - a ) ( z ¯ - a ) = 0 (z\bar{z}-a^{2})^{2}-4a^{2}(z-a)(\bar{z}-a)=0
  5. ( x 2 + y 2 - a 2 ) 2 - 4 a 2 ( ( x - a ) 2 + y 2 ) = 0. (x^{2}+y^{2}-a^{2})^{2}-4a^{2}((x-a)^{2}+y^{2})=0.\,
  6. x = a ( 1 + 2 cos t + cos 2 t ) , x=a(1+2\cos t+\cos 2t),\,
  7. y = a ( 2 sin t + sin 2 t ) , y=a(2\sin t+\sin 2t),\,
  8. z = a ( 1 + 2 e i t + e 2 i t ) = a ( 1 + e i t ) 2 . z=a(1+2e^{it}+e^{2it})=a(1+e^{it})^{2}.\,
  9. e i t = 1 + i u 1 - i u , e^{it}=\frac{1+iu}{1-iu},
  10. z = 4 a ( 1 - i u ) 2 , z=\frac{4a}{(1-iu)^{2}},
  11. x = 4 a ( 1 - u 2 ) ( 1 + u 2 ) 2 , x=\frac{4a(1-u^{2})}{(1+u^{2})^{2}},
  12. y = 8 a u ( 1 + u 2 ) 2 . y=\frac{8au}{(1+u^{2})^{2}}.
  13. z = e i t 2 a ( 1 + cos t ) , z=e^{it}2a(1+\cos t),\,
  14. r = 2 a ( 1 + cos θ ) r=2a(1+\cos\theta)\,
  15. r = 4 a cos 2 θ 2 r=4a\cos^{2}\frac{\theta}{2}\,
  16. ( x 2 + y 2 - 2 a x ) 2 = 4 a 2 ( x 2 + y 2 ) . \left(x^{2}+y^{2}-2ax\right)^{2}=4a^{2}\left(x^{2}+y^{2}\right).\,
  17. A = 6 π a 2 . A=6\pi a^{2}.
  18. L = 16 a . L=16a.
  19. ρ ( θ ) = 1 1 - cos θ . \rho(\theta)\,=\,\frac{1}{1-\cos\theta}.\,
  20. y 2 = 2 x + 1 y^{2}=2x+1
  21. ρ ( θ ) = 1 - cos θ . \rho(\theta)\,=\,1-\cos\theta.\,
  22. z z 2 z\to z^{2}
  23. c = 1 - ( e i t - 1 ) 2 4 . c\,=\,\frac{1-\left(e^{it}-1\right)^{2}}{4}.\,

Caret.html

  1. 3 5 3^{5}

Carl_Neumann.html

  1. 1 1 - x = 1 + x + x 2 + \frac{1}{1-x}=1+x+x^{2}+\cdots

Carmichael_function.html

  1. λ ( n ) \lambda(n)
  2. a m 1 ( mod n ) a^{m}\equiv 1\;\;(\mathop{{\rm mod}}n)
  3. ψ ( n ) \psi(n)
  4. λ ( n ) \lambda(n)
  5. φ \varphi
  6. λ ( n ) \lambda(n)
  7. φ ( n ) \varphi(n)
  8. λ ( n ) = { φ ( n ) if n = 2 , 3 , 4 , 5 , 6 , 7 , 9 , 10 , 11 , 13 , 14 , 17 , 19 , 22 , 23 , 25 , 26 , 27 , 29 1 2 φ ( n ) if n = 8 , 16 , 32 , 64 , 128 , 256 \lambda(n)=\begin{cases}\;\;\varphi(n)&\mbox{if }~{}n=2,3,4,5,6,7,9,10,11,13,1% 4,17,19,22,23,25,26,27,29\dots\\ \tfrac{1}{2}\varphi(n)&\,\text{if }n=8,16,32,64,128,256\dots\end{cases}
  9. φ ( p k ) = p k - 1 ( p - 1 ) . \varphi(p^{k})=p^{k-1}(p-1).\;
  10. n = p 1 a 1 p 2 a 2 p ω ( n ) a ω ( n ) n=p_{1}^{a_{1}}p_{2}^{a_{2}}\dots p_{\omega(n)}^{a_{\omega(n)}}
  11. λ ( n ) = lcm [ λ ( p 1 a 1 ) , λ ( p 2 a 2 ) , , λ ( p ω ( n ) a ω ( n ) ) ] . \lambda(n)=\operatorname{lcm}[\lambda(p_{1}^{a_{1}}),\;\lambda(p_{2}^{a_{2}}),% \dots,\lambda(p_{\omega(n)}^{a_{\omega(n)}})].
  12. a λ ( n ) 1 ( mod n ) , a^{\lambda(n)}\equiv 1\;\;(\mathop{{\rm mod}}n),
  13. λ \lambda
  14. a | b λ ( a ) | λ ( b ) a|b\Rightarrow\lambda(a)|\lambda(b)
  15. a a
  16. b b
  17. λ ( lcm ( a , b ) ) = lcm ( λ ( a ) , λ ( b ) ) \lambda(\mathrm{lcm}(a,b))=\mathrm{lcm}(\lambda(a),\lambda(b))
  18. a a
  19. n n
  20. m m
  21. a m 1 ( mod n ) a^{m}\equiv 1\;\;(\mathop{{\rm mod}}n)
  22. m | λ ( n ) m|\lambda(n)
  23. n n
  24. λ ( n ) \lambda(n)
  25. n n
  26. x m a x x_{max}
  27. a a
  28. n n
  29. k x m a x k\geq x_{max}
  30. a k a k + λ ( n ) ( mod n ) a^{k}\equiv a^{k+\lambda(n)}\;\;(\mathop{{\rm mod}}n)
  31. n n
  32. x m a x = 1 x_{max}=1
  33. a a
  34. a a λ ( n ) + 1 ( mod n ) a\equiv a^{\lambda(n)+1}\;\;(\mathop{{\rm mod}}n)
  35. 1 x n x λ ( n ) = x ln x e B ( 1 + o ( 1 ) ) ln ln x / ( ln ln ln x ) \frac{1}{x}\sum_{n\leq x}\lambda(n)=\frac{x}{\ln x}e^{B(1+o(1))\ln\ln x/(\ln% \ln\ln x)}
  36. B = e - γ p ( 1 - 1 ( p - 1 ) 2 ( p + 1 ) ) 0.34537 . B=e^{-\gamma}\prod_{p}\left({1-\frac{1}{(p-1)^{2}(p+1)}}\right)\approx 0.34537\ .
  37. λ ( n ) = n / ( ln n ) ln ln ln n + A + o ( 1 ) \lambda(n)=n/(\ln n)^{\ln\ln\ln n+A+o(1)}\,
  38. A = - 1 + p log p ( p - 1 ) 2 0.2269688 . A=-1+\sum_{p}\frac{\log p}{(p-1)^{2}}\approx 0.2269688\ .
  39. Δ ( ln ln N ) 3 \Delta\geq(\ln\ln N)^{3}
  40. N e - 0.69 ( Δ ln Δ ) 1 / 3 Ne^{-0.69(\Delta\ln\Delta)^{1/3}}
  41. n N n\leq N
  42. λ ( n ) n e - Δ \lambda(n)\leq ne^{-\Delta}
  43. n 1 < n 2 < n 3 < n_{1}<n_{2}<n_{3}<\cdots
  44. 0 < c < 1 / ln 2 0<c<1/\ln 2
  45. λ ( n i ) > ( ln n i ) c ln ln ln n i \lambda(n_{i})>(\ln n_{i})^{c\ln\ln\ln n_{i}}
  46. n > A n>A
  47. λ ( n ) < ( ln A ) c ln ln ln A \lambda(n)<(\ln A)^{c\ln\ln\ln A}
  48. n = ( q - 1 ) | m and q is prime q n=\prod_{(q-1)|m\,\text{ and }q\,\text{ is prime}}q
  49. m < ( ln A ) c ln ln ln A m<(\ln A)^{c\ln\ln\ln A}
  50. x ( log x ) η + o ( 1 ) , \frac{x}{(\log x)^{\eta+o(1)}}\ ,

Cartan_decomposition.html

  1. 𝔤 \mathfrak{g}
  2. B ( , ) B(\cdot,\cdot)
  3. 𝔤 \mathfrak{g}
  4. θ \theta
  5. 𝔤 \mathfrak{g}
  6. 𝔤 \mathfrak{g}
  7. B θ ( X , Y ) := - B ( X , θ Y ) B_{\theta}(X,Y):=-B(X,\theta Y)
  8. θ 1 \theta_{1}
  9. θ 2 \theta_{2}
  10. 𝔰 𝔩 n ( ) \mathfrak{sl}_{n}(\mathbb{R})
  11. θ ( X ) = - X T \theta(X)=-X^{T}
  12. X T X^{T}
  13. X X
  14. 𝔤 \mathfrak{g}
  15. 𝔤 \mathfrak{g}
  16. 𝔤 \mathfrak{g}
  17. 𝔤 \mathfrak{g}
  18. 𝔤 \mathfrak{g}
  19. 𝔤 0 \mathfrak{g}_{0}
  20. 𝔤 \mathfrak{g}
  21. 𝔤 \mathfrak{g}
  22. 𝔤 \mathfrak{g}
  23. 𝔤 0 \mathfrak{g}_{0}
  24. 𝔰 𝔲 ( n ) \mathfrak{su}(n)
  25. θ 0 ( X ) = X \theta_{0}(X)=X
  26. θ 1 ( X ) = - X T \theta_{1}(X)=-X^{T}
  27. 𝔰 𝔲 ( n ) \mathfrak{su}(n)
  28. n = p + q n=p+q
  29. θ 2 ( X ) = ( I p 0 0 - I q ) X ( I p 0 0 - I q ) \theta_{2}(X)=\begin{pmatrix}I_{p}&0\\ 0&-I_{q}\end{pmatrix}X\begin{pmatrix}I_{p}&0\\ 0&-I_{q}\end{pmatrix}
  30. ( I p 0 0 - I q ) \begin{pmatrix}I_{p}&0\\ 0&-I_{q}\end{pmatrix}
  31. 𝔰 𝔲 ( n ) \mathfrak{su}(n)
  32. n = 2 m n=2m
  33. θ 3 ( X ) = ( 0 I m - I m 0 ) X T ( 0 I m - I m 0 ) . \theta_{3}(X)=\begin{pmatrix}0&I_{m}\\ -I_{m}&0\end{pmatrix}X^{T}\begin{pmatrix}0&I_{m}\\ -I_{m}&0\end{pmatrix}.
  34. θ \theta
  35. 𝔤 \mathfrak{g}
  36. θ 2 = 1 \theta^{2}=1
  37. θ \theta
  38. ± 1 \pm 1
  39. 𝔨 \mathfrak{k}
  40. 𝔭 \mathfrak{p}
  41. 𝔤 = 𝔨 + 𝔭 \mathfrak{g}=\mathfrak{k}+\mathfrak{p}
  42. θ \theta
  43. [ 𝔨 , 𝔨 ] 𝔨 [\mathfrak{k},\mathfrak{k}]\subseteq\mathfrak{k}
  44. [ 𝔨 , 𝔭 ] 𝔭 [\mathfrak{k},\mathfrak{p}]\subseteq\mathfrak{p}
  45. [ 𝔭 , 𝔭 ] 𝔨 [\mathfrak{p},\mathfrak{p}]\subseteq\mathfrak{k}
  46. 𝔨 \mathfrak{k}
  47. 𝔭 \mathfrak{p}
  48. 𝔤 = 𝔨 + 𝔭 \mathfrak{g}=\mathfrak{k}+\mathfrak{p}
  49. θ \theta
  50. 𝔤 \mathfrak{g}
  51. + 1 +1
  52. 𝔨 \mathfrak{k}
  53. - 1 -1
  54. 𝔭 \mathfrak{p}
  55. ( 𝔨 , 𝔭 ) (\mathfrak{k},\mathfrak{p})
  56. 𝔤 \mathfrak{g}
  57. ( 𝔤 , 𝔨 ) (\mathfrak{g},\mathfrak{k})
  58. H * ( 𝔤 , 𝔨 ) H^{*}(\mathfrak{g},\mathfrak{k})
  59. 𝔤 = 𝔨 + 𝔭 \mathfrak{g}=\mathfrak{k}+\mathfrak{p}
  60. 𝔤 \mathfrak{g}
  61. 𝔨 \mathfrak{k}
  62. 𝔭 \mathfrak{p}
  63. 𝔨 \mathfrak{k}
  64. 𝔭 \mathfrak{p}
  65. 𝔤 \mathfrak{g}
  66. G G
  67. 𝔤 \mathfrak{g}
  68. θ \theta
  69. 𝔤 \mathfrak{g}
  70. ( 𝔨 , 𝔭 ) (\mathfrak{k},\mathfrak{p})
  71. K K
  72. G G
  73. 𝔨 \mathfrak{k}
  74. Θ \Theta
  75. θ \theta
  76. Θ 2 = 1 \Theta^{2}=1
  77. Θ \Theta
  78. K K
  79. K K
  80. K × 𝔭 G K\times\mathfrak{p}\rightarrow G
  81. ( k , X ) k exp ( X ) (k,X)\mapsto k\cdot\mathrm{exp}(X)
  82. K K
  83. Z Z
  84. G G
  85. K K
  86. K / Z K/Z
  87. K K
  88. G G
  89. Θ \Theta
  90. K × 𝔭 G K\times\mathfrak{p}\rightarrow G
  91. X ( X - 1 ) T X\mapsto(X^{-1})^{T}
  92. 𝔞 \mathfrak{a}
  93. 𝔭 \mathfrak{p}
  94. 𝔭 = k K Ad k 𝔞 . \displaystyle{\mathfrak{p}=\bigcup_{k\in K}\mathrm{Ad}\,k\cdot\mathfrak{a}.}
  95. G = K A K , \displaystyle{G=KAK,}
  96. 𝔞 \mathfrak{a}
  97. 𝔤 𝔩 n ( ) \mathfrak{gl}_{n}(\mathbb{R})
  98. θ ( X ) = - X T \theta(X)=-X^{T}
  99. 𝔨 = 𝔰 𝔬 n ( ) \mathfrak{k}=\mathfrak{so}_{n}(\mathbb{R})
  100. K = SO ( n ) K=\mathrm{SO}(n)
  101. 𝔭 \mathfrak{p}
  102. 𝔭 \mathfrak{p}

Cartan_formalism_(physics).html

  1. e : T M V e\colon{\rm T}M\to V
  2. f a = f 1 f n f_{a}=f_{1}\ldots f_{n}
  3. η a b = η ( f a , f b ) = diag ( 1 , 1 , - 1 , , - 1 ) \eta_{ab}=\eta(f_{a},f_{b})={\rm diag}(1,\ldots 1,-1,\ldots,-1)
  4. x μ = x - 1 , , x - n x^{\mu}=x^{-1},\ldots,x^{-n}
  5. f a f_{a}
  6. μ = x μ \partial_{\mu}=\frac{\partial}{\partial x^{\mu}}
  7. e a := e ( f a ) := e a μ μ . e_{a}:=e(f_{a}):=e^{\mu}_{a}\partial_{\mu}.
  8. e a μ , μ = - 1 , , - n , a = 1 , , n e^{\mu}_{a},\mu=-1,\dots,-n,a=1,\dots,n
  9. V T M V\cong{\rm T}M
  10. B Fr ( M ) B\to{\rm Fr}(M)
  11. g α β g_{\alpha\beta}\!
  12. 𝐱 , 𝐲 = g α β x α y β . \langle\mathbf{x},\mathbf{y}\rangle=g_{\alpha\beta}\,x^{\alpha}\,y^{\beta}.\,
  13. e α i e_{\alpha}^{i}
  14. 𝐱 , 𝐲 = η i j ( e α i x α ) ( e β j y β ) . \langle\mathbf{x},\mathbf{y}\rangle=\eta_{ij}(e_{\alpha}^{i}\,x^{\alpha})(e_{% \beta}^{j}\,y^{\beta}).\,
  15. α \alpha
  16. β \beta
  17. i i
  18. j j
  19. e α i ( 𝐱 ) e_{\alpha}^{i}(\mathbf{x})
  20. g α β ( 𝐱 ) = η i j e α i ( 𝐱 ) e β j ( 𝐱 ) g_{\alpha\beta}(\mathbf{x})=\eta_{ij}\,e_{\alpha}^{i}(\mathbf{x})\,e_{\beta}^{% j}(\mathbf{x})
  21. \nabla
  22. F = def d A + A A {F}\ \stackrel{\mathrm{def}}{=}\ d{A}+{A}\wedge{A}
  23. d A d_{A}
  24. ω \omega
  25. Ω = D ω = d ω + ω ω \Omega=D\omega=d\omega+\omega\wedge\omega
  26. S = def M p l 2 M ϵ a b c d ( e a e b Ω c d ) = M p l 2 M d 4 x ϵ μ ν ρ σ ϵ a b c d e μ a e ν b R ρ σ c d [ ω ] S\ \stackrel{\mathrm{def}}{=}\ M^{2}_{pl}\int_{M}\epsilon_{abcd}(e^{a}\wedge e% ^{b}\wedge\Omega^{cd})=M^{2}_{pl}\int_{M}d^{4}x\epsilon^{\mu\nu\rho\sigma}% \epsilon_{abcd}e^{a}_{\mu}e^{b}_{\nu}R^{cd}_{\rho\sigma}[\omega]
  27. = M p l 2 | e | d 4 x 1 2 e a μ e b ν R μ ν a b =M^{2}_{pl}\int|e|d^{4}x\frac{1}{2}e^{\mu}_{a}e^{\nu}_{b}R^{ab}_{\mu\nu}
  28. = c 4 16 π G d 4 x - g R [ g ] =\frac{c^{4}}{16\pi G}\int d^{4}x\sqrt{-g}R[g]
  29. Ω μ ν a b = R μ ν a b \Omega_{\mu\nu}^{ab}=R_{\mu\nu}^{ab}
  30. ϵ a b c d \epsilon_{abcd}
  31. | e | = ϵ μ ν ρ σ ϵ a b c d e μ a e ν b e ρ c e σ d |e|=\epsilon^{\mu\nu\rho\sigma}\epsilon_{abcd}e^{a}_{\mu}e^{b}_{\nu}e^{c}_{% \rho}e^{d}_{\sigma}
  32. e μ a e_{\mu}^{a}
  33. | e | = - g |e|=\sqrt{-g}
  34. R μ ν λ σ = e a λ e b σ R μ ν a b R^{\lambda\sigma}_{\mu\nu}=e^{\lambda}_{a}e^{\sigma}_{b}R^{ab}_{\mu\nu}
  35. = c = 1 \hbar=c=1
  36. d ω d\omega
  37. ω ^ μ a b = ω μ a b + K μ a b \hat{\omega}^{ab}_{\mu}=\omega^{ab}_{\mu}+K^{ab}_{\mu}

Cartan_matrix.html

  1. A = ( a i j ) A=(a_{ij})
  2. a i j 0 a_{ij}\leq 0
  3. a i j = 0 a_{ij}=0
  4. a j i = 0 a_{ji}=0
  5. [ 2 - 3 - 1 2 ] = [ 3 0 0 1 ] [ 2 / 3 - 1 - 1 2 ] . \left[\begin{smallmatrix}\;\,\,2&-3\\ -1&\;\,\,2\end{smallmatrix}\right]=\left[\begin{smallmatrix}3&0\\ 0&1\end{smallmatrix}\right]\left[\begin{smallmatrix}2/3&-1\\ -1&\;2\end{smallmatrix}\right].
  6. a i j = 2 ( r i , r j ) ( r i , r i ) a_{ij}=2{(r_{i},r_{j})\over(r_{i},r_{i})}
  7. i j , r j - 2 ( r i , r j ) ( r i , r i ) r i i\neq j,r_{j}-{2(r_{i},r_{j})\over(r_{i},r_{i})}r_{i}
  8. D i j = δ i j ( r i , r i ) D_{ij}={\delta_{ij}\over(r_{i},r_{i})}
  9. S i j = 2 ( r i , r j ) S_{ij}=2(r_{i},r_{j})
  10. n × n n\times n
  11. I { 1 , , n } I\subset\{1,\dots,n\}
  12. a i j = 0 a_{ij}=0
  13. i I i\in I
  14. j I j\notin I
  15. A n , B n , C n , D n , E 6 , E 7 , E 8 , F 4 , G 2 A_{n},B_{n},C_{n},D_{n},E_{6},E_{7},E_{8},F_{4},G_{2}
  16. A n A_{n}
  17. B n B_{n}
  18. n 2 n\geq 2
  19. C n C_{n}
  20. n 2 n\geq 2
  21. D n D_{n}
  22. n 4 n\geq 4
  23. E n E_{n}
  24. n = 6 , 7 , 8 n=6,7,8
  25. F 4 F_{4}
  26. G 2 G_{2}
  27. | P / Q | |P/Q|
  28. P , Q P,Q

Categorial_grammar.html

  1. A B A\rightarrow B
  2. B / A B/A\,\!
  3. A \ B A\backslash B
  4. B / A B/A\,\!
  5. B B\,\!
  6. A A\,\!
  7. A \ B A\backslash B\,\!
  8. B B\,\!
  9. A A\,\!
  10. Prim \,\text{Prim}\,\!
  11. Tp ( Prim ) \,\text{Tp}(\,\text{Prim})\,\!
  12. Prim Tp ( Prim ) \,\text{Prim}\subseteq\,\text{Tp}(\,\text{Prim})
  13. X , Y Tp ( Prim ) X,Y\in\,\text{Tp}(\,\text{Prim})
  14. ( X / Y ) , ( Y \ X ) Tp ( Prim ) (X/Y),(Y\backslash X)\in\,\text{Tp}(\,\text{Prim})
  15. ( Σ , Prim , S , ) (\Sigma,\,\text{Prim},S,\triangleleft)
  16. Σ \Sigma\,\!
  17. Prim \,\text{Prim}\,\!
  18. S Tp ( Prim ) S\in\,\text{Tp}(\,\text{Prim})
  19. \triangleleft
  20. ( ) Tp ( Prim ) × Σ (\triangleleft)\subseteq\,\text{Tp}(\,\text{Prim})\times\Sigma
  21. T Y P E symbol TYPE\triangleleft\,\text{symbol}
  22. ( N , N P , and S ) (N,NP,\,\text{ and }S)\,\!
  23. N N\,\!
  24. N P NP\,\!
  25. S S\,\!
  26. N / N N/N\,\!
  27. N P / N NP/N\,\!
  28. N P \ S NP\backslash S
  29. ( N P \ S ) / N P (NP\backslash S)/NP
  30. S S\,\!
  31. N P / N the NP/N\triangleleft\,\text{the}
  32. N P / N that NP/N\triangleleft\,\text{that}
  33. N boy N\triangleleft\,\text{boy}
  34. N mess N\triangleleft\,\text{mess}
  35. N / N bad N/N\triangleleft\,\text{bad}
  36. ( N P \ S ) / N P made (NP\backslash S)/NP\triangleleft\,\text{made}
  37. the N P / N , bad N / N , boy N , made ( N P \ S ) / N P , that N P / N , mess N {\,\text{the}\atop{NP/N,}}{\,\text{bad}\atop{N/N,}}{\,\text{boy}\atop{N,}}{\,% \text{made}\atop{(NP\backslash S)/NP,}}{\,\text{that}\atop{NP/N,}}{\,\text{% mess}\atop{N}}
  38. X X / Y , Y X\leftarrow X/Y,\;Y
  39. X Y , Y \ X X\leftarrow Y,\;Y\backslash X
  40. . N P / N , N / N , N , ( N P \ S ) / N P , N P / N , N .\qquad NP/N,\;N/N,\;N,\;(NP\backslash S)/NP,\;\underbrace{NP/N,\;N}
  41. . N P / N , N / N , N , ( N P \ S ) / N P , N P .\qquad NP/N,\;N/N,\;N,\;\underbrace{(NP\backslash S)/NP,\quad NP}
  42. . N P / N , N / N , N , ( N P \ S ) .\qquad NP/N,\;\underbrace{N/N,\;N},\qquad(NP\backslash S)
  43. . N P / N , N , ( N P \ S ) .\qquad\underbrace{NP/N,\;\quad N},\;\qquad(NP\backslash S)
  44. . N P , ( N P \ S ) .\qquad\qquad\underbrace{NP,\;\qquad(NP\backslash S)}
  45. . S .\qquad\qquad\qquad\quad\;\;\;S
  46. S S\,\!
  47. \star\,\!
  48. Prim Tp ( Prim ) \,\text{Prim}\subseteq\,\text{Tp}(\,\text{Prim})
  49. X , Y Tp ( Prim ) X,Y\in\,\text{Tp}(\,\text{Prim})
  50. ( X / Y ) , ( X \ Y ) , ( X Y ) Tp ( Prim ) (X/Y),(X\backslash Y),(X\star Y)\in\,\text{Tp}(\,\text{Prim})
  51. X Γ X\leftarrow\Gamma
  52. X X\,\!
  53. Γ \Gamma\,\!
  54. \leftarrow
  55. \supseteq\,\!
  56. ( A x i o m ) X X (Axiom)\quad{{}\over X\leftarrow X}
  57. ( C u t ) Z Δ X Δ X Γ Z Δ Γ Δ (Cut)\quad{Z\leftarrow\Delta X\Delta^{\prime}\qquad X\leftarrow\Gamma\over Z% \leftarrow\Delta\Gamma\Delta^{\prime}}
  58. ( \ ) Y X Γ X \ Y Γ (\backslash\leftarrow)\quad{Y\leftarrow X\Gamma\over X\backslash Y\leftarrow\Gamma}
  59. ( \ ) Z Δ Y Δ X Γ Z Δ Γ ( X \ Y ) Δ (\leftarrow\backslash)\quad{Z\leftarrow\Delta Y\Delta^{\prime}\qquad X% \leftarrow\Gamma\over Z\leftarrow\Delta\Gamma(X\backslash Y)\Delta^{\prime}}
  60. ( / ) Y Γ X Y / X Γ (/\leftarrow)\quad{Y\leftarrow\Gamma X\over Y/X\leftarrow\Gamma}
  61. ( / ) Z Δ Y Δ X Γ Z Δ ( Y / X ) Γ Δ (\leftarrow/)\quad{Z\leftarrow\Delta Y\Delta^{\prime}\qquad X\leftarrow\Gamma% \over Z\leftarrow\Delta(Y/X)\Gamma\Delta^{\prime}}
  62. ( ) X Γ Y Γ X Y Γ Γ (\star\leftarrow)\quad{X\leftarrow\Gamma\qquad Y\leftarrow\Gamma^{\prime}\over X% \star Y\leftarrow\Gamma\Gamma^{\prime}}
  63. ( ) Z Δ X Y Δ Z Δ ( X Y ) Δ (\leftarrow\star)\quad{Z\leftarrow\Delta XY\Delta^{\prime}\over Z\leftarrow% \Delta(X\star Y)\Delta^{\prime}}
  64. ( B / A ) \ B A (B/A)\backslash B\leftarrow A
  65. B B A A B ( B / A ) , A ( B / A ) \ B A (Axioms) ( / ) [ Z = Y = B , X = A , Γ = ( A ) , Δ = Δ = ( ) ] ( \ ) [ Y = B , X = ( B / A ) , Γ = ( A ) ] \dfrac{\dfrac{}{B\leftarrow B}\qquad\dfrac{}{A\leftarrow A}}{\dfrac{B% \leftarrow(B/A),\;\;A}{(B/A)\backslash B\leftarrow A}}\qquad\begin{matrix}% \mbox{(Axioms)}\\ {(\leftarrow/)\,\,[Z=Y=B,X=A,\Gamma=(A),\Delta=\Delta^{\prime}=()]}\\ {(\backslash\leftarrow)\,\,[Y=B,X=(B/A),\Gamma=(A)]}\\ \end{matrix}
  66. G = ( V , Σ , : := , S ) G=(V,\,\Sigma,\,::=,\,S)
  67. V V\,
  68. Σ \Sigma\,
  69. : := ::=\,
  70. ( : := ) V × ( V Σ ) * (::=)\subseteq V\times(V\cup\Sigma)^{*}
  71. S S\,
  72. ( Prim , Σ , , S ) (\,\text{Prim},\,\Sigma,\,\triangleleft,\,S)
  73. Prim = V Σ \,\text{Prim}=V\cup\Sigma
  74. Tp ( Prim ) = Prim \,\text{Tp}(\,\text{Prim})=\,\text{Prim}\,\!
  75. x x {x\leftarrow x}
  76. x V Σ x\in V\cup\Sigma
  77. X Γ {X\leftarrow\Gamma}
  78. X : := Γ X::=\Gamma\,\!
  79. s s {s\triangleleft s}
  80. s Σ s\in\Sigma
  81. A : := s A 0 A N - 1 A::=sA_{0}\ldots A_{N-1}
  82. s Σ s\in\Sigma
  83. N 0 N\geq 0
  84. Prim = V \,\text{Prim}=V\,\!
  85. A / A N - 1 / / A 0 s A/A_{N-1}/\ldots/A_{0}\triangleleft s
  86. A : := s A 0 A N - 1 A::=sA_{0}\ldots A_{N-1}
  87. ( Prim , Σ , , S ) (\,\text{Prim},\,\Sigma,\,\triangleleft,\,S)
  88. ( V , Σ , : := , S ) (V,\,\Sigma,\,::=,\,S)
  89. V Tp ( Prim ) V\subset\,\text{Tp}(\,\text{Prim})\,\!
  90. T : := s T::=\,\text{s}\,\!
  91. T s T\triangleleft\,\text{s}
  92. T : := Γ T::=\Gamma\,\!
  93. T Γ T\leftarrow\Gamma
  94. : := ::=
  95. B \ A B\backslash A
  96. A \ B A\backslash B

Category:FFT_algorithms.html

  1. N N

Category:Integers.html

  1. \mathbb{Z}

Category_of_groups.html

  1. E = Hom ( S 3 , S 3 ) E=\operatorname{Hom}(S_{3},S_{3})

Cauchy_horizon.html

  1. B ν 1 + v 1 - v B_{\nu}\propto\sqrt{\frac{1+v}{1-v}\ }

Cauchy_product.html

  1. ( a n ) n 0 \textstyle(a_{n})_{n\geq 0}
  2. ( b n ) n 0 \textstyle(b_{n})_{n\geq 0}
  3. ( c n ) n 0 \textstyle(c_{n})_{n\geq 0}
  4. c n = k = 0 n a k b n - k . c_{n}=\sum_{k=0}^{n}a_{k}b_{n-k}.
  5. n = 0 c n X n \textstyle\sum_{n=0}^{\infty}c_{n}X^{n}
  6. ( a n ) n 0 (a_{n})_{n\geq 0}
  7. ( b n ) n 0 (b_{n})_{n\geq 0}
  8. a n , b n \textstyle a_{n},b_{n}
  9. n = 0 a n , n = 0 b n , \sum_{n=0}^{\infty}a_{n},\qquad\sum_{n=0}^{\infty}b_{n},
  10. ( n = 0 a n ) ( m = 0 b m ) = j = 0 c j , where c j = k = 0 j a k b j - k \left(\sum_{n=0}^{\infty}a_{n}\right)\cdot\left(\sum_{m=0}^{\infty}b_{m}\right% )=\sum_{j=0}^{\infty}c_{j},\qquad\mathrm{where}\ c_{j}=\sum_{k=0}^{j}a_{k}b_{j% -k}
  11. j = 0 c j \sum_{j=0}^{\infty}c_{j}
  12. ( n = 0 a n ) ( m = 0 b m ) \left(\sum_{n=0}^{\infty}a_{n}\right)\left(\sum_{m=0}^{\infty}b_{m}\right)
  13. ( k = 0 n a k ) ( k = 0 n b k ) = k = 0 2 n i = 0 k a i b k - i - k = 0 n - 1 ( a k i = n + 1 2 n - k b i + b k i = n + 1 2 n - k a i ) \left(\sum_{k=0}^{n}a_{k}\right)\cdot\left(\sum_{k=0}^{n}b_{k}\right)=\sum_{k=% 0}^{2n}\sum_{i=0}^{k}a_{i}b_{k-i}-\sum_{k=0}^{n-1}\left(a_{k}\sum_{i=n+1}^{2n-% k}b_{i}+b_{k}\sum_{i=n+1}^{2n-k}a_{i}\right)
  14. n = 0 a n \textstyle\sum_{n=0}^{\infty}a_{n}
  15. A A
  16. n = 0 b n \textstyle\sum_{n=0}^{\infty}b_{n}
  17. B B
  18. A B AB
  19. a n = b n = ( - 1 ) n n + 1 , a_{n}=b_{n}=\frac{(-1)^{n}}{\sqrt{n+1}}\,,
  20. c n = k = 0 n ( - 1 ) k k + 1 ( - 1 ) n - k n - k + 1 = ( - 1 ) n k = 0 n 1 ( k + 1 ) ( n - k + 1 ) c_{n}=\sum_{k=0}^{n}\frac{(-1)^{k}}{\sqrt{k+1}}\cdot\frac{(-1)^{n-k}}{\sqrt{n-% k+1}}=(-1)^{n}\sum_{k=0}^{n}\frac{1}{\sqrt{(k+1)(n-k+1)}}
  21. n 0 n≥0
  22. k 0 , 1 , , n k∈{0,1,...,n}
  23. k + 1 n + 1 k+1≤n+1
  24. n k + 1 n + 1 n–k+1≤n+1
  25. ( k + 1 ) ( n k + 1 ) n + 1 \sqrt{(}{k}{+1)(}{n}{−}{k}{+1)}≤n+1
  26. n + 1 n+1
  27. | c n | k = 0 n 1 n + 1 1 |c_{n}|\geq\sum_{k=0}^{n}\frac{1}{n+1}\geq 1
  28. n 0 n≥0
  29. n n→∞
  30. n = 0 a n \textstyle\sum_{n=0}^{\infty}a_{n}
  31. A n = i = 0 n a i , B n = i = 0 n b i and C n = i = 0 n c i A_{n}=\sum_{i=0}^{n}a_{i},\quad B_{n}=\sum_{i=0}^{n}b_{i}\quad\,\text{and}% \quad C_{n}=\sum_{i=0}^{n}c_{i}
  32. c i = k = 0 i a k b i - k . c_{i}=\sum_{k=0}^{i}a_{k}b_{i-k}\,.
  33. C n = i = 0 n a n - i B i C_{n}=\sum_{i=0}^{n}a_{n-i}B_{i}
  34. C n = i = 0 n a n - i ( B i - B ) + A n B . C_{n}=\sum_{i=0}^{n}a_{n-i}(B_{i}-B)+A_{n}B\,.
  35. ε > 0 ε>0
  36. k | a k | < \textstyle\sum_{k\in{\mathbb{N}}}|a_{k}|<\infty
  37. B B
  38. n n→∞
  39. N N
  40. n N n≥N
  41. | B n - B | ε / 3 k | a k | + 1 |B_{n}-B|\leq\frac{\varepsilon/3}{\sum_{k\in{\mathbb{N}}}|a_{k}|+1}
  42. M M
  43. n M n≥M
  44. | a n | ε 3 N ( sup i { 0 , , N - 1 } | B i - B | + 1 ) . |a_{n}|\leq\frac{\varepsilon}{3N(\sup_{i\in\{0,\dots,N-1\}}|B_{i}-B|+1)}\,.
  45. A A
  46. n n→∞
  47. L L
  48. n L n≥L
  49. | A n - A | ε / 3 | B | + 1 . |A_{n}-A|\leq\frac{\varepsilon/3}{|B|+1}\,.
  50. n m a x L , M + N n≥max{L,M+N}
  51. | C n - A B | = | i = 0 n a n - i ( B i - B ) + ( A n - A ) B | i = 0 N - 1 | a n - i M | | B i - B | ε / ( 3 N ) by (3) + i = N n | a n - i | | B i - B | ε / 3 by (2) + | A n - A | | B | ε / 3 by (4) ε . \begin{aligned}\displaystyle|C_{n}-AB|&\displaystyle=\biggl|\sum_{i=0}^{n}a_{n% -i}(B_{i}-B)+(A_{n}-A)B\biggr|\\ &\displaystyle\leq\sum_{i=0}^{N-1}\underbrace{|a_{\underbrace{\scriptstyle n-i% }_{\scriptscriptstyle\geq M}}|\,|B_{i}-B|}_{\leq\,\varepsilon/(3N)\,\text{ by % (3)}}+{}\underbrace{\sum_{i=N}^{n}|a_{n-i}|\,|B_{i}-B|}_{\leq\,\varepsilon/3\,% \text{ by (2)}}+{}\underbrace{|A_{n}-A|\,|B|}_{\leq\,\varepsilon/3\,\text{ by % (4)}}\leq\varepsilon\,.\end{aligned}
  52. a i = 0 \textstyle a_{i}=0
  53. i > n i>n
  54. b i = 0 \textstyle b_{i}=0
  55. i > m \textstyle i>m
  56. a n \textstyle\sum a_{n}
  57. b n \textstyle\sum b_{n}
  58. ( a 0 + + a n ) ( b 0 + + b m ) \textstyle(a_{0}+\cdots+a_{n})(b_{0}+\cdots+b_{m})
  59. x , y \textstyle x,y\in\mathbb{R}
  60. a n = x n / n ! \textstyle a_{n}=x^{n}/n!\,
  61. b n = y n / n ! \textstyle b_{n}=y^{n}/n!\,
  62. c n = i = 0 n x i i ! y n - i ( n - i ) ! = 1 n ! i = 0 n ( n i ) x i y n - i = ( x + y ) n n ! c_{n}=\sum_{i=0}^{n}\frac{x^{i}}{i!}\frac{y^{n-i}}{(n-i)!}=\frac{1}{n!}\sum_{i% =0}^{n}{\left({{n}\atop{i}}\right)}x^{i}y^{n-i}=\frac{(x+y)^{n}}{n!}
  63. exp ( x ) = a n \textstyle\exp(x)=\sum a_{n}
  64. exp ( y ) = b n \textstyle\exp(y)=\sum b_{n}
  65. exp ( x + y ) = c n \textstyle\exp(x+y)=\sum c_{n}
  66. exp ( x + y ) = exp ( x ) exp ( y ) \textstyle\exp(x+y)=\exp(x)\exp(y)
  67. x , y \textstyle x,y\in\mathbb{R}
  68. a n = b n = 1 \textstyle a_{n}=b_{n}=1
  69. n \textstyle n\in\mathbb{N}
  70. c n = n + 1 \textstyle c_{n}=n+1
  71. n n\in\mathbb{N}
  72. c n = ( 1 , 1 + 2 , 1 + 2 + 3 , 1 + 2 + 3 + 4 , ) \textstyle\sum c_{n}=(1,1+2,1+2+3,1+2+3+4,\dots)
  73. ( a n ) n 0 \textstyle(a_{n})_{n\geq 0}
  74. ( b n ) n 0 \textstyle(b_{n})_{n\geq 0}
  75. a n A \textstyle\sum a_{n}\to A
  76. b n B \textstyle\sum b_{n}\to B
  77. 1 N ( n = 1 N i = 1 n k = 0 i a k b i - k ) A B . \frac{1}{N}\left(\sum_{n=1}^{N}\sum_{i=1}^{n}\sum_{k=0}^{i}a_{k}b_{i-k}\right)% \to AB.
  78. r > - 1 \textstyle r>-1
  79. s > - 1 \textstyle s>-1
  80. ( a n ) n 0 \textstyle(a_{n})_{n\geq 0}
  81. ( C , r ) \textstyle(C,\;r)
  82. ( b n ) n 0 \textstyle(b_{n})_{n\geq 0}
  83. ( C , s ) \textstyle(C,\;s)
  84. ( C , r + s + 1 ) \textstyle(C,\;r+s+1)
  85. \textstyle\mathbb{C}
  86. n \textstyle\mathbb{R}^{n}
  87. n n\in\mathbb{N}
  88. n 2 n\geq 2
  89. n = 1 n=1
  90. k 1 = 0 a 1 , k 1 , , k n = 0 a n , k n \sum_{k_{1}=0}^{\infty}a_{1,k_{1}},\ldots,\sum_{k_{n}=0}^{\infty}a_{n,k_{n}}
  91. n n
  92. n n
  93. k 1 = 0 k 2 = 0 k 1 k n = 0 k n - 1 a 1 , k n a 2 , k n - 1 - k n a n , k 1 - k 2 \sum_{k_{1}=0}^{\infty}\sum_{k_{2}=0}^{k_{1}}\cdots\sum_{k_{n}=0}^{k_{n-1}}a_{% 1,k_{n}}a_{2,k_{n-1}-k_{n}}\cdots a_{n,k_{1}-k_{2}}
  94. k 1 = 0 k 2 = 0 k 1 k n = 0 k n - 1 a 1 , k n a 2 , k n - 1 - k n a n , k 1 - k 2 = j = 1 n ( k j = 0 a j , k j ) \sum_{k_{1}=0}^{\infty}\sum_{k_{2}=0}^{k_{1}}\cdots\sum_{k_{n}=0}^{k_{n-1}}a_{% 1,k_{n}}a_{2,k_{n-1}-k_{n}}\cdots a_{n,k_{1}-k_{2}}=\prod_{j=1}^{n}\left(\sum_% {k_{j}=0}^{\infty}a_{j,k_{j}}\right)
  95. n n
  96. n = 2 n=2
  97. n n\in\mathbb{N}
  98. n 2 n\geq 2
  99. k 1 = 0 a 1 , k 1 , , k n + 1 = 0 a n + 1 , k n + 1 \sum_{k_{1}=0}^{\infty}a_{1,k_{1}},\ldots,\sum_{k_{n+1}=0}^{\infty}a_{n+1,k_{n% +1}}
  100. n + 1 n+1
  101. n + 1 n+1
  102. k 1 = 0 | a 1 , k 1 | , , k n = 0 | a n , k n | \sum_{k_{1}=0}^{\infty}|a_{1,k_{1}}|,\ldots,\sum_{k_{n}=0}^{\infty}|a_{n,k_{n}}|
  103. k 1 = 0 k 2 = 0 k 1 k n = 0 k n - 1 | a 1 , k n a 2 , k n - 1 - k n a n , k 1 - k 2 | \sum_{k_{1}=0}^{\infty}\sum_{k_{2}=0}^{k_{1}}\cdots\sum_{k_{n}=0}^{k_{n-1}}|a_% {1,k_{n}}a_{2,k_{n-1}-k_{n}}\cdots a_{n,k_{1}-k_{2}}|
  104. k 1 = 0 | k 2 = 0 k 1 k n = 0 k n - 1 a 1 , k n a 2 , k n - 1 - k n a n , k 1 - k 2 | \sum_{k_{1}=0}^{\infty}\left|\sum_{k_{2}=0}^{k_{1}}\cdots\sum_{k_{n}=0}^{k_{n-% 1}}a_{1,k_{n}}a_{2,k_{n-1}-k_{n}}\cdots a_{n,k_{1}-k_{2}}\right|
  105. k 1 = 0 k 2 = 0 k 1 k n = 0 k n - 1 a 1 , k n a 2 , k n - 1 - k n a n , k 1 - k 2 \sum_{k_{1}=0}^{\infty}\sum_{k_{2}=0}^{k_{1}}\cdots\sum_{k_{n}=0}^{k_{n-1}}a_{% 1,k_{n}}a_{2,k_{n-1}-k_{n}}\cdots a_{n,k_{1}-k_{2}}
  106. j = 1 n + 1 ( k j = 0 a j , k j ) \displaystyle\prod_{j=1}^{n+1}\left(\sum_{k_{j}=0}^{\infty}a_{j,k_{j}}\right)
  107. n + 1 n+1
  108. \Z \textstyle\Z
  109. ( , 1 , ) \textstyle(\dots,1,\dots)
  110. 1 × \textstyle\ell^{1}\times\ell^{\infty}

Cauchy–Euler_equation.html

  1. a n x n y ( n ) ( x ) + a n - 1 x n - 1 y ( n - 1 ) ( x ) + + a 0 y ( x ) = 0. a_{n}x^{n}y^{(n)}(x)+a_{n-1}x^{n-1}y^{(n-1)}(x)+\cdots+a_{0}y(x)=0.
  2. x = e u x=e^{u}
  3. y = x m y=x^{m}
  4. x 2 d 2 y d x 2 + a x d y d x + b y = 0. x^{2}\frac{d^{2}y}{dx^{2}}+ax\frac{dy}{dx}+by=0.\,
  5. y = x m . y=x^{m}.\,
  6. d y d x = m x m - 1 \frac{dy}{dx}=mx^{m-1}\,
  7. d 2 y d x 2 = m ( m - 1 ) x m - 2 . \frac{d^{2}y}{dx^{2}}=m(m-1)x^{m-2}.\,
  8. x 2 ( m ( m - 1 ) x m - 2 ) + a x ( m x m - 1 ) + b ( x m ) = 0 x^{2}(m(m-1)x^{m-2})+ax(mx^{m-1})+b(x^{m})=0\,
  9. m 2 + ( a - 1 ) m + b = 0. m^{2}+(a-1)m+b=0.\,
  10. y = c 1 x m 1 + c 2 x m 2 y=c_{1}x^{m_{1}}+c_{2}x^{m_{2}}\,
  11. y = c 1 x m ln ( x ) + c 2 x m y=c_{1}x^{m}\ln(x)+c_{2}x^{m}\,
  12. y = c 1 x α cos ( β ln ( x ) ) + c 2 x α sin ( β ln ( x ) ) y=c_{1}x^{\alpha}\cos(\beta\ln(x))+c_{2}x^{\alpha}\sin(\beta\ln(x))\,
  13. α = Re ( m ) \alpha=\mathop{\rm Re}(m)\,
  14. β = Im ( m ) \beta=\mathop{\rm Im}(m)\,
  15. c 1 c_{1}\,
  16. c 2 c_{2}\,
  17. x 2 d 2 y d x 2 + a x d y d x + b y = 0 x^{2}\frac{d^{2}y}{dx^{2}}+ax\frac{dy}{dx}+by=0\,
  18. t = ln ( x ) . t=\ln(x).\,
  19. y ( x ) = ϕ ( ln ( x ) ) = ϕ ( t ) . y(x)=\phi(\ln(x))=\phi(t).\,
  20. d y d x = 1 x d ϕ d t \frac{dy}{dx}=\frac{1}{x}\frac{d\phi}{dt}
  21. d 2 y d x 2 = 1 x 2 ( d 2 ϕ d t 2 - d ϕ d t ) . \frac{d^{2}y}{dx^{2}}=\frac{1}{x^{2}}\bigg(\frac{d^{2}\phi}{dt^{2}}-\frac{d% \phi}{dt}\bigg).
  22. ϕ ( t ) \phi(t)
  23. d 2 ϕ d t 2 + ( a - 1 ) d ϕ d t + b ϕ = 0. \frac{d^{2}\phi}{dt^{2}}+(a-1)\frac{d\phi}{dt}+b\phi=0.\,
  24. ϕ ( t ) \phi(t)
  25. λ 2 + ( a - 1 ) λ + b = 0. \lambda^{2}+(a-1)\lambda+b=0.
  26. λ 1 \lambda_{1}
  27. λ 2 \lambda_{2}
  28. ϕ ( t ) = c 1 e λ 1 t + c 2 e λ 2 t \phi(t)=c_{1}e^{\lambda_{1}t}+c_{2}e^{\lambda_{2}t}
  29. ϕ ( t ) = c 1 e λ 1 t + c 2 t e λ 1 t . \phi(t)=c_{1}e^{\lambda_{1}t}+c_{2}te^{\lambda_{1}t}.
  30. y ( x ) y(x)
  31. t = ln ( x ) t=\ln(x)
  32. ϕ ( ln ( x ) ) = y ( x ) \phi(\ln(x))=y(x)
  33. y ( x ) = c 1 x λ 1 + c 2 x λ 2 y(x)=c_{1}x^{\lambda_{1}}+c_{2}x^{\lambda_{2}}
  34. y ( x ) = c 1 x λ 1 + c 2 ln ( x ) x λ 1 . y(x)=c_{1}x^{\lambda_{1}}+c_{2}\ln(x)x^{\lambda_{1}}.
  35. x 2 u ′′ - 3 x u + 3 u = 0 , x^{2}u^{\prime\prime}-3xu^{\prime}+3u=0\,,
  36. x 2 ( α ( α - 1 ) x α - 2 ) - 3 x ( α x α - 1 ) + 3 x α = α ( α - 1 ) x α - 3 α x α + 3 x α = ( α 2 - 4 α + 3 ) x α = 0 . x^{2}(\alpha(\alpha-1)x^{\alpha-2})-3x(\alpha x^{\alpha-1})+3x^{\alpha}=\alpha% (\alpha-1)x^{\alpha}-3\alpha x^{\alpha}+3x^{\alpha}=(\alpha^{2}-4\alpha+3)x^{% \alpha}=0\,.
  37. u = c 1 x + c 2 x 3 . u=c_{1}x+c_{2}x^{3}\,.
  38. f m ( n ) := n ( n + 1 ) ( n + m - 1 ) . f_{m}(n):=n(n+1)\cdots(n+m-1).
  39. f m f_{m}
  40. D f m ( n ) \displaystyle Df_{m}(n)
  41. f m ( k ) ( n ) \displaystyle f_{m}^{(k)}(n)
  42. m ( m - 1 ) ( m - k + 1 ) x m x k m(m-1)\cdots(m-k+1)\frac{x^{m}}{x^{k}}
  43. f N ( n ) y ( N ) ( n ) + a N - 1 f N - 1 ( n ) y ( N - 1 ) ( n ) + + a 0 y ( n ) = 0 , f_{N}(n)y^{(N)}(n)+a_{N-1}f_{N-1}(n)y^{(N-1)}(n)+\cdots+a_{0}y(n)=0,
  44. y ( n ) = f m ( n ) y(n)=f_{m}(n)\,
  45. m ( m - 1 ) ( m - N + 1 ) + a N - 1 m ( m - 1 ) ( m - N + 2 ) + + a 1 m + a 0 = 0. m(m-1)\cdots(m-N+1)+a_{N-1}m(m-1)\cdots(m-N+2)+\cdots+a_{1}m+a_{0}=0.
  46. φ ( n ) = k = 1 n 1 k - m 1 . \varphi(n)=\sum_{k=1}^{n}\frac{1}{k-m_{1}}.
  47. ln ( x - m 1 ) = 1 + m 1 x 1 t - m 1 d t . \ln(x-m_{1})=\int_{1+m_{1}}^{x}\frac{1}{t-m_{1}}\,dt.
  48. f m ( n ) := Γ ( n + m ) Γ ( n ) f_{m}(n):=\frac{\Gamma(n+m)}{\Gamma(n)}

Causal_contact.html

  1. p p
  2. t 0 t_{0}
  3. p p
  4. p p
  5. p p
  6. p p
  7. E E
  8. E E
  9. E E
  10. E E
  11. E E
  12. E E
  13. E E
  14. E E
  15. E E

Caustic_(mathematics).html

  1. ( a , b ) (a,b)
  2. ( u ( t ) , v ( t ) ) (u(t),v(t))
  3. ( - v ( t ) , u ( t ) ) (-v^{\prime}(t),u^{\prime}(t))
  4. 2 proj d n - d = 2 n n n n d n n - d = 2 n n d n n - d = ( a v 2 - 2 b u v - a u 2 , b u 2 - 2 a u v - b v 2 ) v 2 + u 2 2\mbox{proj}~{}_{n}d-d=\frac{2n}{\sqrt{n\cdot n}}\frac{n\cdot d}{\sqrt{n\cdot n% }}-d=2n\frac{n\cdot d}{n\cdot n}-d=\frac{(av^{\prime 2}-2bu^{\prime}v^{\prime}% -au^{\prime 2},bu^{\prime 2}-2au^{\prime}v^{\prime}-bv^{\prime 2})}{v^{\prime 2% }+u^{\prime 2}}
  5. ( x - u ) ( b u 2 - 2 a u v - b v 2 ) = ( y - v ) ( a v 2 - 2 b u v - a u 2 ) . (x-u)(bu^{\prime 2}-2au^{\prime}v^{\prime}-bv^{\prime 2})=(y-v)(av^{\prime 2}-% 2bu^{\prime}v^{\prime}-au^{\prime 2}).
  6. F ( x , y , t ) = ( x - u ) ( b u 2 - 2 a u v - b v 2 ) - ( y - v ) ( a v 2 - 2 b u v - a u 2 ) F(x,y,t)=(x-u)(bu^{\prime 2}-2au^{\prime}v^{\prime}-bv^{\prime 2})-(y-v)(av^{% \prime 2}-2bu^{\prime}v^{\prime}-au^{\prime 2})
  7. = x ( b u 2 - 2 a u v - b v 2 ) - y ( a v 2 - 2 b u v - a u 2 ) + b ( u v 2 - u u 2 - 2 v u v ) + a ( - v u 2 + v v 2 + 2 u u v ) =x(bu^{\prime 2}-2au^{\prime}v^{\prime}-bv^{\prime 2})-y(av^{\prime 2}-2bu^{% \prime}v^{\prime}-au^{\prime 2})+b(uv^{\prime 2}-uu^{\prime 2}-2vu^{\prime}v^{% \prime})+a(-vu^{\prime 2}+vv^{\prime 2}+2uu^{\prime}v^{\prime})
  8. F t ( x , y , t ) = 2 x ( b u u ′′ - a ( u v ′′ + u ′′ v ) - b v v ′′ ) - 2 y ( a v v ′′ - b ( u ′′ v + u v ′′ ) - a u u ′′ ) + b ( u v 2 + 2 u v v ′′ - u 3 - 2 u u u ′′ - 2 u v 2 - 2 u ′′ v v - 2 u v v ′′ ) + a ( - v u 2 - 2 v u u ′′ + v 3 + 2 v v v ′′ + 2 v u 2 + 2 v ′′ u u + 2 v u u ′′ ) F_{t}(x,y,t)=2x(bu^{\prime}u^{\prime\prime}-a(u^{\prime}v^{\prime\prime}+u^{% \prime\prime}v^{\prime})-bv^{\prime}v^{\prime\prime})-2y(av^{\prime}v^{\prime% \prime}-b(u^{\prime\prime}v^{\prime}+u^{\prime}v^{\prime\prime})-au^{\prime}u^% {\prime\prime})+b(u^{\prime}v^{\prime 2}+2uv^{\prime}v^{\prime\prime}-u^{% \prime 3}-2uu^{\prime}u^{\prime\prime}-2u^{\prime}v^{\prime 2}-2u^{\prime% \prime}vv^{\prime}-2u^{\prime}vv^{\prime\prime})+a(-v^{\prime}u^{\prime 2}-2vu% ^{\prime}u^{\prime\prime}+v^{\prime 3}+2vv^{\prime}v^{\prime\prime}+2v^{\prime% }u^{\prime 2}+2v^{\prime\prime}uu^{\prime}+2v^{\prime}uu^{\prime\prime})
  9. F = F t = 0 F=F_{t}=0
  10. ( x , y ) (x,y)
  11. ( t , t 2 ) . (t,t^{2}).
  12. u = 1 u^{\prime}=1
  13. u ′′ = 0 u^{\prime\prime}=0
  14. v = 2 t v^{\prime}=2t
  15. v ′′ = 2 v^{\prime\prime}=2
  16. a = 0 a=0
  17. b = 1 b=1
  18. F ( x , y , t ) = ( x - t ) ( 1 - 4 t 2 ) + 4 t ( y - t 2 ) = x ( 1 - 4 t 2 ) + 4 t y - t F(x,y,t)=(x-t)(1-4t^{2})+4t(y-t^{2})=x(1-4t^{2})+4ty-t
  19. F t ( x , y , t ) = - 8 t x + 4 y - 1 F_{t}(x,y,t)=-8tx+4y-1
  20. F = F t = 0 F=F_{t}=0
  21. ( 0 , 1 / 4 ) (0,1/4)

Cavity_ring-down_spectroscopy.html

  1. I ( t ) = I 0 exp ( - t / τ ) I(t)=I_{0}\exp\left(-t/\tau\right)
  2. τ 0 = n c l 1 - R + X \tau_{0}=\frac{n}{c}\cdot\frac{l}{1-R+X}
  3. τ = n c l 1 - R + X + α l \tau=\frac{n}{c}\cdot\frac{l}{1-R+X+\alpha l}
  4. A = n c l 2.303 ( 1 τ - 1 τ 0 ) A=\frac{n}{c}\cdot\frac{l}{2.303}\cdot\left(\frac{1}{\tau}-\frac{1}{\tau_{0}}\right)
  5. τ 0 τ = 1 + α l 1 - R = 1 + 2.303 ϵ l C ( 1 - R ) \frac{\tau_{0}}{\tau}=1+\frac{\alpha l}{1-R}=1+\frac{2.303\epsilon lC}{(1-R)}

Cayley_transform.html

  1. Q = ( I - A ) ( I + A ) - 1 Q=(I-A)(I+A)^{-1}\,\!
  2. Q = ( I + A ) - 1 ( I - A ) Q=(I+A)^{-1}(I-A)
  3. A = ( I - Q ) ( I + Q ) - 1 A=(I-Q)(I+Q)^{-1}\,\!
  4. Q \displaystyle Q
  5. [ 0 tan θ 2 - tan θ 2 0 ] [ cos θ - sin θ sin θ cos θ ] . \begin{bmatrix}0&\tan\frac{\theta}{2}\\ -\tan\frac{\theta}{2}&0\end{bmatrix}\leftrightarrow\begin{bmatrix}\cos\theta&-% \sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}.
  6. [ 0 z - y - z 0 x y - x 0 ] 1 K [ w 2 + x 2 - y 2 - z 2 2 ( x y - w z ) 2 ( w y + x z ) 2 ( x y + w z ) w 2 - x 2 + y 2 - z 2 2 ( y z - w x ) 2 ( x z - w y ) 2 ( w x + y z ) w 2 - x 2 - y 2 + z 2 ] , \begin{bmatrix}0&z&-y\\ -z&0&x\\ y&-x&0\end{bmatrix}\leftrightarrow\frac{1}{K}\begin{bmatrix}w^{2}+x^{2}-y^{2}-% z^{2}&2(xy-wz)&2(wy+xz)\\ 2(xy+wz)&w^{2}-x^{2}+y^{2}-z^{2}&2(yz-wx)\\ 2(xz-wy)&2(wx+yz)&w^{2}-x^{2}-y^{2}+z^{2}\end{bmatrix},
  7. w + i x + j y + k z w+{i}x+{j}y+{k}z\,\!
  8. [ 0 - a a b - c 0 0 - b 0 0 0 ] [ 1 2 a 2 c 0 1 2 b 0 0 1 ] . \begin{bmatrix}0&-a&ab-c\\ 0&0&-b\\ 0&0&0\end{bmatrix}\leftrightarrow\begin{bmatrix}1&2a&2c\\ 0&1&2b\\ 0&0&1\end{bmatrix}.
  9. W : z z - i z + i . \operatorname{W}\colon z\mapsto\frac{z-{i}}{z+{i}}.
  10. U = ( A - i I ) ( A + i I ) - 1 A = i ( I + U ) ( I - U ) - 1 \begin{aligned}\displaystyle U&\displaystyle{}=(A-{i}I)(A+{i}I)^{-1}\\ \displaystyle A&\displaystyle{}={i}(I+U)(I-U)^{-1}\end{aligned}

Cayley–Purser_algorithm.html

  1. [ 0 1 2 3 ] + [ 1 2 3 4 ] = [ 1 3 5 7 ] = [ 1 3 0 2 ] \left[\begin{matrix}0&1\\ 2&3\end{matrix}\right]+\left[\begin{matrix}1&2\\ 3&4\end{matrix}\right]=\left[\begin{matrix}1&3\\ 5&7\end{matrix}\right]=\left[\begin{matrix}1&3\\ 0&2\end{matrix}\right]
  2. [ 0 1 2 3 ] [ 1 2 3 4 ] = [ 3 4 11 16 ] = [ 3 4 1 1 ] \left[\begin{matrix}0&1\\ 2&3\end{matrix}\right]\left[\begin{matrix}1&2\\ 3&4\end{matrix}\right]=\left[\begin{matrix}3&4\\ 11&16\end{matrix}\right]=\left[\begin{matrix}3&4\\ 1&1\end{matrix}\right]
  3. χ \chi
  4. α \alpha
  5. χ α - 1 α χ \chi\alpha^{-1}\not=\alpha\chi
  6. β = χ - 1 α - 1 χ , \beta=\chi^{-1}\alpha^{-1}\chi,
  7. γ = χ r . \gamma=\chi^{r}.
  8. n n
  9. α \alpha
  10. β \beta
  11. γ \gamma
  12. χ \chi
  13. δ = γ s \delta=\gamma^{s}
  14. ϵ = δ - 1 α δ \epsilon=\delta^{-1}\alpha\delta
  15. κ = δ - 1 β δ \kappa=\delta^{-1}\beta\delta
  16. μ \mu
  17. μ \mu
  18. μ = κ μ κ . \mu^{\prime}=\kappa\mu\kappa.
  19. μ \mu^{\prime}
  20. ϵ \epsilon
  21. μ \mu
  22. λ = χ - 1 ϵ χ , \lambda=\chi^{-1}\epsilon\chi,
  23. μ = λ μ λ . \mu=\lambda\mu^{\prime}\lambda.
  24. χ \chi
  25. γ \gamma
  26. α \alpha
  27. β \beta
  28. χ β = α - 1 χ \chi\beta=\alpha^{-1}\chi
  29. χ \chi^{\prime}
  30. χ \chi
  31. d d
  32. d ( β - α - 1 ) ( α - 1 γ - γ β ) ( mod n ) d\left(\beta-\alpha^{-1}\right)\equiv\left(\alpha^{-1}\gamma-\gamma\beta\right% )\;\;(\mathop{{\rm mod}}n)
  33. i , j | γ | i,j\in\left|\gamma\right|
  34. x , y n x,y\in\mathbb{Z}_{n}
  35. x ( β i j - 1 - α i j ) y ( mod n ) . x\left(\beta_{ij}^{-1}-\alpha_{ij}\right)\equiv y\;\;(\mathop{{\rm mod}}n).
  36. d d
  37. d I + γ = χ d\mathrm{I}+\gamma=\chi^{\prime}
  38. χ \chi
  39. χ \chi
  40. λ = κ - 1 = v - 1 χ - 1 ϵ v χ \lambda=\kappa^{-1}=v^{-1}\chi^{-1}\epsilon v\chi
  41. ϵ \epsilon

Cellular_network.html

  1. D = R 3 N , D=R\sqrt{3N},\,

Centered_cube_number.html

  1. n 3 + ( n + 1 ) 3 = ( 2 n + 1 ) ( n 2 + n + 1 ) . n^{3}+(n+1)^{3}=(2n+1)(n^{2}+n+1).
  2. ( ( n + 1 ) 2 + 1 2 ) - ( n 2 + 1 2 ) = ( n 2 + 1 ) + ( n 2 + 2 ) + + ( n + 1 ) 2 . {\left({{(n+1)^{2}+1}\atop{2}}\right)}-{\left({{n^{2}+1}\atop{2}}\right)}=(n^{% 2}+1)+(n^{2}+2)+\cdots+(n+1)^{2}.
  3. ( 2 n + 1 ) ( n 2 + n + 1 ) (2n+1)(n^{2}+n+1)

Centered_pentagonal_number.html

  1. 5 n 2 + 5 n + 2 2 . {{5n^{2}+5n+2}\over 2}.

Centered_polygonal_number.html

  1. C k , n = k n 2 ( n - 1 ) + 1. C_{k,n}=\frac{kn}{2}(n-1)+1.
  2. k 2 2 ( k - 1 ) + 1 \frac{k^{2}}{2}(k-1)+1

Centered_square_number.html

  1. C 4 , 1 = 1 C_{4,1}=1
  2. C 4 , 2 = 5 C_{4,2}=5
  3. C 4 , 3 = 13 C_{4,3}=13
  4. C 4 , 4 = 25 C_{4,4}=25
  5. C 4 , n = n 2 + ( n - 1 ) 2 . C_{4,n}=n^{2}+(n-1)^{2}.\,
  6. C 4 , 1 = 0 + 1 C_{4,1}=0+1
  7. C 4 , 2 = 1 + 4 C_{4,2}=1+4
  8. C 4 , 3 = 4 + 9 C_{4,3}=4+9
  9. C 4 , 4 = 9 + 16 C_{4,4}=9+16
  10. C 4 , n = ( 2 n - 1 ) 2 + 1 2 ; C_{4,n}={(2n-1)^{2}+1\over 2};
  11. C 4 , 1 = ( 1 + 1 ) / 2 C_{4,1}=(1+1)/2
  12. C 4 , 2 = ( 9 + 1 ) / 2 C_{4,2}=(9+1)/2
  13. C 4 , 3 = ( 25 + 1 ) / 2 C_{4,3}=(25+1)/2
  14. C 4 , 4 = ( 49 + 1 ) / 2 C_{4,4}=(49+1)/2
  15. C 4 , n = 1 + 4 T n - 1 , C_{4,n}=1+4\,T_{n-1},\,
  16. T n = n ( n + 1 ) 2 = n 2 + n 2 = ( n + 1 2 ) T_{n}={n(n+1)\over 2}={n^{2}+n\over 2}={n+1\choose 2}
  17. C 4 , 1 = 1 C_{4,1}=1
  18. C 4 , 2 = 1 + 4 × 1 C_{4,2}=1+4\times 1
  19. C 4 , 3 = 1 + 4 × 3 C_{4,3}=1+4\times 3
  20. C 4 , 4 = 1 + 4 × 6. C_{4,4}=1+4\times 6.

Centered_triangular_number.html

  1. 3 n 2 + 3 n + 2 2 . {{3n^{2}+3n+2}\over 2}.

Cesàro_summation.html

  1. s k = a 1 + + a k s_{k}=a_{1}+\cdots+a_{k}
  2. n = 1 a n . \sum_{n=1}^{\infty}a_{n}.
  3. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  4. A \R A\in\R
  5. s k s_{k}
  6. A A
  7. lim n 1 n k = 1 n s k = A . \lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}s_{k}=A.
  8. 1 , - 1 , 1 , - 1 , . 1,-1,1,-1,\ldots.\,
  9. n = 1 a n = 1 - 1 + 1 - 1 + 1 - \sum_{n=1}^{\infty}a_{n}=1-1+1-1+1-\cdots
  10. = k = 1 n a k =\sum_{k=1}^{n}a_{k}
  11. 1 , 0 , 1 , 0 , , 1,0,1,0,\ldots,\,
  12. t n = 1 n k = 1 n s k t_{n}=\frac{1}{n}\sum_{k=1}^{n}s_{k}
  13. 1 1 , 1 2 , 2 3 , 2 4 , 3 5 , 3 6 , 4 7 , 4 8 , , \frac{1}{1},\,\frac{1}{2},\,\frac{2}{3},\,\frac{2}{4},\,\frac{3}{5},\,\frac{3}% {6},\,\frac{4}{7},\,\frac{4}{8},\,\ldots,
  14. lim n t n = 1 / 2. \lim_{n\to\infty}t_{n}=1/2.
  15. 1 , 2 , 3 , 4 , . 1,2,3,4,\ldots.\,
  16. n = 1 a n = 1 + 2 + 3 + 4 + 5 + \sum_{n=1}^{\infty}a_{n}=1+2+3+4+5+\cdots
  17. 1 , 3 , 6 , 10 , , 1,3,6,10,\ldots,\,
  18. 1 1 , 4 2 , 10 3 , 20 4 , . \frac{1}{1},\,\frac{4}{2},\,\frac{10}{3},\,\frac{20}{4},\,\ldots.
  19. A n - 1 = a n ; A n α = k = 0 n A k α - 1 A_{n}^{-1}=a_{n};\quad A_{n}^{\alpha}=\sum_{k=0}^{n}A_{k}^{\alpha-1}
  20. ( C , α ) (C,\alpha)
  21. j = 0 a j = lim n A n α E n α \sum_{j=0}^{\infty}a_{j}=\lim_{n\to\infty}\frac{A_{n}^{\alpha}}{E_{n}^{\alpha}}
  22. α \alpha
  23. ( C , α ) (C,\alpha)
  24. j = 0 a j = lim n j = 0 n < m t p l > ( n j ) ( n + α j ) a j . \sum_{j=0}^{\infty}a_{j}=\lim_{n\to\infty}\sum_{j=0}^{n}\frac{<}{m}tpl>{{n% \choose j}}{{n+\alpha\choose j}}a_{j}.
  25. α ( - ) \alpha\in\mathbb{R}\setminus(-\mathbb{N})
  26. n = 0 A n α x n = n = 0 a n x n ( 1 - x ) 1 + α , \sum_{n=0}^{\infty}A_{n}^{\alpha}x^{n}=\frac{\displaystyle{\sum_{n=0}^{\infty}% a_{n}x^{n}}}{(1-x)^{1+\alpha}},
  27. 0 f ( x ) d x \scriptstyle{\int_{0}^{\infty}f(x)\,dx}
  28. lim λ 0 λ ( 1 - x λ ) α f ( x ) d x \lim_{\lambda\to\infty}\int_{0}^{\lambda}\left(1-\frac{x}{\lambda}\right)^{% \alpha}f(x)\,dx
  29. lim λ 1 λ 0 λ { 0 x f ( y ) d y } d x \lim_{\lambda\to\infty}\frac{1}{\lambda}\int_{0}^{\lambda}\left\{\int_{0}^{x}f% (y)\,dy\right\}\,dx

Change_of_basis.html

  1. M = ( 1 / 2 - 1 / 2 1 / 2 1 / 2 ) M=\begin{pmatrix}1/\sqrt{2}&-1/\sqrt{2}\\ 1/\sqrt{2}&1/\sqrt{2}\end{pmatrix}
  2. 𝐑 = [ c α c γ - s α c β s γ - c α s γ - s α c β c γ s β s α s α c γ + c α c β s γ - s α s γ + c α c β c γ - s β c α s β s γ s β c γ c β ] . \mathbf{R}=\begin{bmatrix}\mathrm{c}_{\alpha}\,\mathrm{c}_{\gamma}-\mathrm{s}_% {\alpha}\,\mathrm{c}_{\beta}\,\mathrm{s}_{\gamma}&-\mathrm{c}_{\alpha}\,% \mathrm{s}_{\gamma}-\mathrm{s}_{\alpha}\,\mathrm{c}_{\beta}\,\mathrm{c}_{% \gamma}&\mathrm{s}_{\beta}\,\mathrm{s}_{\alpha}\\ \mathrm{s}_{\alpha}\,\mathrm{c}_{\gamma}+\mathrm{c}_{\alpha}\,\mathrm{c}_{% \beta}\,\mathrm{s}_{\gamma}&-\mathrm{s}_{\alpha}\,\mathrm{s}_{\gamma}+\mathrm{% c}_{\alpha}\,\mathrm{c}_{\beta}\,\mathrm{c}_{\gamma}&-\mathrm{s}_{\beta}\,% \mathrm{c}_{\alpha}\\ \mathrm{s}_{\beta}\,\mathrm{s}_{\gamma}&\mathrm{s}_{\beta}\,\mathrm{c}_{\gamma% }&\mathrm{c}_{\beta}\end{bmatrix}.
  3. v B ( v , w ) v\mapsto B(v,w)
  4. v B ( w , v ) v\mapsto B(w,v)
  5. α 1 , , α n \alpha_{1},\dots,\alpha_{n}
  6. G i , j = B ( α i , α j ) . G_{i,j}=B(\alpha_{i},\alpha_{j}).
  7. v = i x i α i v=\sum_{i}x_{i}\alpha_{i}
  8. w = i y i α i w=\sum_{i}y_{i}\alpha_{i}
  9. B ( v , w ) = v 𝖳 G w . B(v,w)=v^{\mathsf{T}}Gw.
  10. α 1 , , α n \alpha_{1},\dots,\alpha_{n}
  11. α 1 , , α n \alpha^{\prime}_{1},\dots,\alpha^{\prime}_{n}
  12. G = P 𝖳 G P . G^{\prime}=P^{\mathsf{T}}GP.

Chaplygin's_equation.html

  1. f θ θ + v 2 1 - v 2 c 2 f v v + v f v = 0. f_{\theta\theta}+\frac{v^{2}}{1-\frac{v^{2}}{c^{2}}}f_{vv}+vf_{v}=0.
  2. c = c ( v ) c=c(v)

Character_table.html

  1. α , β := 1 | G | g G α ( g ) β ( g ) ¯ \left\langle\alpha,\beta\right\rangle:=\frac{1}{\left|G\right|}\sum_{g\in G}% \alpha(g)\overline{\beta(g)}
  2. β ( g ) ¯ \overline{\beta(g)}
  3. β \beta
  4. χ i , χ j = { 0 if i j , 1 if i = j . \left\langle\chi_{i},\chi_{j}\right\rangle=\begin{cases}0&\mbox{ if }~{}i\neq j% ,\\ 1&\mbox{ if }~{}i=j.\end{cases}
  5. g , h G g,h\in G
  6. χ i χ i ( g ) χ i ( h ) ¯ = { | C G ( g ) | , if g , h are conjugate 0 otherwise. \sum_{\chi_{i}}\chi_{i}(g)\overline{\chi_{i}(h)}=\begin{cases}\left|C_{G}(g)% \right|,&\mbox{ if }~{}g,h\mbox{ are conjugate }\\ 0&\mbox{ otherwise.}\end{cases}
  7. χ i \chi_{i}
  8. | C G ( g ) | \left|C_{G}(g)\right|
  9. g g
  10. g - g , g\mapsto-g,
  11. C 3 C_{3}
  12. u u 2 , u 2 u , u\mapsto u^{2},u^{2}\mapsto u,
  13. χ 1 \chi_{1}
  14. χ 2 \chi_{2}
  15. ω \omega
  16. ω 2 \omega^{2}
  17. ϕ : G G \phi\colon G\to G
  18. ρ : G GL \rho\colon G\to\operatorname{GL}
  19. ρ ϕ := g ρ ( ϕ ( g ) ) \rho^{\phi}:=g\mapsto\rho(\phi(g))
  20. ϕ = ϕ a \phi=\phi_{a}

Character_theory.html

  1. 2 2
  2. V V
  3. F F
  4. ρ : G G L ( V ) ρ:G→GL(V)
  5. G G
  6. V V
  7. ρ ρ
  8. χ ρ ( g ) = Tr ( ρ ( g ) ) \chi_{\rho}(g)=\mathrm{Tr}(\rho(g))
  9. T r Tr
  10. ρ ρ
  11. χ χ
  12. ρ ρ
  13. χ ( 1 ) χ(1)
  14. G G
  15. F F
  16. ker χ ρ := { g G χ ρ ( g ) = χ ρ ( 1 ) } , \ker\chi_{\rho}:=\left\{g\in G\mid\chi_{\rho}(g)=\chi_{\rho}(1)\right\},
  17. ρ ρ
  18. G G
  19. 𝐊 \mathbf{K}
  20. 𝐊 \mathbf{K}
  21. G 𝐊 G→\mathbf{K}
  22. 0
  23. G G
  24. H H
  25. H H
  26. χ ( g ) χ(g)
  27. n n
  28. m m
  29. n n
  30. χ χ
  31. m m
  32. g g
  33. F = 𝐂 F=\mathbf{C}
  34. F = 𝐂 F=\mathbf{C}
  35. χ χ
  36. [ G : C G ( x ) ] χ ( x ) χ ( 1 ) [G:C_{G}(x)]\frac{\chi(x)}{\chi(1)}
  37. x x
  38. G G
  39. F F
  40. c h a r ( F ) char(F)
  41. | G | |G|
  42. G G
  43. G G
  44. G G
  45. G G : Z ( G ) GG:Z(G)
  46. F = 𝐂 F=\mathbf{C}
  47. G G
  48. χ ρ σ = χ ρ + χ σ \chi_{\rho\oplus\sigma}=\chi_{\rho}+\chi_{\sigma}
  49. χ ρ σ = χ ρ χ σ \chi_{\rho\otimes\sigma}=\chi_{\rho}\cdot\chi_{\sigma}
  50. χ ρ * = χ ρ ¯ \chi_{\rho^{*}}=\overline{\chi_{\rho}}
  51. χ Alt 2 ρ ( g ) = 1 2 [ ( χ ρ ( g ) ) 2 - χ ρ ( g 2 ) ] \chi_{{\scriptscriptstyle\rm{Alt}^{2}}\rho}(g)=\tfrac{1}{2}\left[\left(\chi_{% \rho}(g)\right)^{2}-\chi_{\rho}(g^{2})\right]
  52. χ Sym 2 ρ ( g ) = 1 2 [ ( χ ρ ( g ) ) 2 + χ ρ ( g 2 ) ] \chi_{{\scriptscriptstyle\rm{Sym}^{2}}\rho}(g)=\tfrac{1}{2}\left[\left(\chi_{% \rho}(g)\right)^{2}+\chi_{\rho}(g^{2})\right]
  53. ρ σ ρ⊕σ
  54. ρ σ ρ⊗σ
  55. ρ ρ
  56. ρ ρ = ( ρ ρ ) Sym 2 ρ \rho\otimes\rho=\left(\rho\wedge\rho\right)\oplus\textrm{Sym}^{2}\rho
  57. G G
  58. G G
  59. G G
  60. 1 1
  61. C 3 = u u 3 = 1 , C_{3}=\langle u\mid u^{3}=1\rangle,
  62. ( 1 ) (1)
  63. ( u ) (u)
  64. ( u < s u p > 2 ) (u<sup>2)
  65. ω ω
  66. 1 1
  67. 1 1
  68. 1 × 1 1×1
  69. 1 1
  70. G G
  71. α , β := 1 | G | g G α ( g ) β ( g ) ¯ \left\langle\alpha,\beta\right\rangle:=\frac{1}{|G|}\sum_{g\in G}\alpha(g)% \overline{\beta(g)}
  72. β ¯ ( g ) \overline{β}{(}{g}{)}
  73. β ( g ) β(g)
  74. χ i , χ j = { 0 if i j , 1 if i = j . \left\langle\chi_{i},\chi_{j}\right\rangle=\begin{cases}0&\mbox{ if }~{}i\neq j% ,\\ 1&\mbox{ if }~{}i=j.\end{cases}
  75. g , h g,h
  76. G G
  77. χ i χ i ( g ) χ i ( h ) ¯ = { | C G ( g ) | , if g , h are conjugate 0 otherwise. \sum_{\chi_{i}}\chi_{i}(g)\overline{\chi_{i}(h)}=\begin{cases}\left|C_{G}(g)% \right|,&\mbox{ if }~{}g,h\mbox{ are conjugate }\\ 0&\mbox{ otherwise.}\end{cases}
  78. G G
  79. g g
  80. G G
  81. G G
  82. G G
  83. G G
  84. χ χ
  85. g g
  86. G G
  87. χ ( g ) = χ ( 1 ) χ(g)=χ(1)
  88. G G
  89. G G
  90. G G
  91. G G
  92. G G
  93. G G
  94. Q Q
  95. 8 8
  96. H H
  97. G G
  98. χ χ
  99. G G
  100. H H
  101. θ θ
  102. H H
  103. G G
  104. θ θ
  105. G G
  106. G G
  107. G G
  108. θ G , χ G = θ , χ H H \langle\theta^{G},\chi\rangle_{G}=\langle\theta,\chi_{H}\rangle_{H}
  109. χ χ
  110. G G
  111. G G
  112. H H
  113. G G
  114. H H
  115. H H
  116. G G
  117. G G
  118. G G
  119. θ θ
  120. ρ ρ
  121. H H
  122. G G
  123. ρ ρ
  124. G G
  125. H H
  126. G G
  127. H H
  128. G G
  129. H H
  130. G = H t 1 H t n , G=Ht_{1}\cup\ldots\cup Ht_{n},
  131. h h
  132. H H
  133. θ G ( h ) = i : t i h t i - 1 H θ ( t i h t i - 1 ) . \theta^{G}(h)=\sum_{i\ :\ t_{i}ht_{i}^{-1}\in H}\theta\left(t_{i}ht_{i}^{-1}% \right).
  134. θ θ
  135. H H
  136. H H
  137. G G
  138. θ θ
  139. H H
  140. G G
  141. H H
  142. H H
  143. G G
  144. K K
  145. G G
  146. G G
  147. ( H , K ) (H,K)
  148. G = t T H t K G=\bigcup_{t\in T}HtK
  149. θ θ
  150. H H
  151. ( θ G ) K = t T ( [ θ t ] t - 1 H t K ) K , \left(\theta^{G}\right)_{K}=\sum_{t\in T}\left(\left[\theta^{t}\right]_{t^{-1}% Ht\cap K}\right)^{K},
  152. h h
  153. H H
  154. θ θ
  155. ψ ψ
  156. H H
  157. K K
  158. H H
  159. K K
  160. θ G , ψ G \displaystyle\left\langle\theta^{G},\psi^{G}\right\rangle
  161. T T
  162. ( H , K ) (H,K)
  163. θ θ
  164. ψ ψ
  165. 1 1
  166. 0
  167. ψ ψ
  168. θ θ
  169. ψ ψ
  170. | T | |T|
  171. χ ( g ) χ(g)
  172. j j
  173. G G
  174. 𝔤 \mathfrak{g}
  175. H H
  176. 𝔥 \mathfrak{h}
  177. V V
  178. G G
  179. V V
  180. χ V = dim V λ e λ \chi_{V}=\sum\dim V_{\lambda}e^{\lambda}
  181. H H
  182. G G
  183. X X
  184. 𝔥 \mathfrak{h}
  185. Tr ( e X ) = dim V λ e λ ( X ) . \,\text{Tr}(e^{X})=\sum\dim V_{\lambda}e^{\lambda(X)}.
  186. G G
  187. N N

Characteristic_(algebra).html

  1. 1 + + 1 n summands = 0 \underbrace{1+\cdots+1}_{n\,\text{ summands}}=0
  2. a + + a n summands = 0 \underbrace{a+\cdots+a}_{n\,\text{ summands}}=0

Characteristic_energy.html

  1. C 3 C_{3}\,\!
  2. ϵ \epsilon
  3. 1 2 v 2 - μ / r = c o n s t a n t = 1 2 C 3 \tfrac{1}{2}v^{2}-\mu/r=constant=\tfrac{1}{2}C_{3}
  4. μ = G M \mu=GM
  5. M M
  6. r r
  7. C 3 = v 2 C_{3}=v_{\infty}^{2}\,\!
  8. v v_{\infty}
  9. 1 2 m v 2 \tfrac{1}{2}mv^{2}
  10. ϵ \epsilon
  11. C 3 < 0 C_{3}<0\,
  12. C 3 = 0 C_{3}=0\,
  13. C 3 = μ a C_{3}={\mu\over{a}}\,
  14. μ = G M \mu\,=GM
  15. a a\,

Characterizations_of_the_exponential_function.html

  1. e x = lim n ( 1 + x n ) n . e^{x}=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}.
  2. e x = n = 0 x n n ! = 1 + x + x 2 2 ! + x 3 3 ! + x 4 4 ! + e^{x}=\sum_{n=0}^{\infty}{x^{n}\over n!}=1+x+\frac{x^{2}}{2!}+\frac{x^{3}}{3!}% +\frac{x^{4}}{4!}+\cdots
  3. 1 y d t t = x . \int_{1}^{y}\frac{dt}{t}=x.
  4. y = y , y ( 0 ) = 1. y^{\prime}=y,\quad y(0)=1.
  5. f ( x + y ) = f ( x ) f ( y ) for all x and y f(x+y)=f(x)f(y)\,\text{ for all }x\,\text{ and }y\,
  6. x x
  7. f ( x ) f(x)
  8. f ( x + y ) = f ( x ) f ( y ) f(x+y)=f(x)f(y)
  9. x x
  10. y y
  11. f ( x ) f(x)
  12. x x
  13. f ( x ) f(x)
  14. f ( 1 ) = e f(1)=e
  15. f f
  16. f ( 0 ) = 1 f^{\prime}(0)=1
  17. f ( x + y ) = f ( x ) f ( y ) f(x+y)=f(x)f(y)
  18. f ( 0 ) = 1 f(0)=1
  19. f ( 0 ) = f ( 0 + 0 ) = f ( 0 ) f ( 0 ) f(0)=f(0+0)=f(0)f(0)
  20. f ( 0 ) f(0)
  21. f ( x ) = f ( x ) f^{\prime}(x)=f(x)
  22. f ( 0 ) = 1 f^{\prime}(0)=1
  23. f ( x ) \displaystyle f^{\prime}(x)
  24. f ( x + i y ) = e x ( cos ( 2 y ) + i sin ( 2 y ) ) = e x + 2 i y f(x+iy)=e^{x}(\cos(2y)+i\sin(2y))=e^{x+2iy}\,
  25. f ( i ) = cos ( 1 ) + i sin ( 1 ) . f(i)=\cos(1)+i\sin(1).\,
  26. f ( 0 ) = 1 f^{\prime}(0)=1
  27. lim n | x n + 1 / ( n + 1 ) ! x n / n ! | = lim n | x n + 1 | = 0 < 1 . \lim_{n\to\infty}\left|\frac{x^{n+1}/(n+1)!}{x^{n}/n!}\right|=\lim_{n\to\infty% }\left|\frac{x}{n+1}\right|=0<1\mbox{.}~{}
  28. n = 0 x n n ! \sum_{n=0}^{\infty}\frac{x^{n}}{n!}
  29. 1 y d t t = x \int_{1}^{y}\frac{dt}{t}=x
  30. ( 0 , + ) (0,+\infty)
  31. ( - , + ) , (-\infty,+\infty),
  32. 1 d t t = \int_{1}^{\infty}\frac{dt}{t}=\infty
  33. 1 0 d t t = - \int_{1}^{0}\frac{dt}{t}=-\infty
  34. t > 0 t>0
  35. 1 / t > 0 1/t>0
  36. x 0 x\geq 0
  37. s n = k = 0 n x k k ! , t n = ( 1 + x n ) n . s_{n}=\sum_{k=0}^{n}\frac{x^{k}}{k!},\ t_{n}=\left(1+\frac{x}{n}\right)^{n}.
  38. t n = k = 0 n ( n k ) x k n k = 1 + x + k = 2 n n ( n - 1 ) ( n - 2 ) ( n - ( k - 1 ) ) x k k ! n k = 1 + x + x 2 2 ! ( 1 - 1 n ) + x 3 3 ! ( 1 - 1 n ) ( 1 - 2 n ) + + x n n ! ( 1 - 1 n ) ( 1 - n - 1 n ) s n \begin{aligned}\displaystyle t_{n}&\displaystyle=\sum_{k=0}^{n}{n\choose k}% \frac{x^{k}}{n^{k}}=1+x+\sum_{k=2}^{n}\frac{n(n-1)(n-2)\cdots(n-(k-1))x^{k}}{k% !\,n^{k}}\\ &\displaystyle=1+x+\frac{x^{2}}{2!}\left(1-\frac{1}{n}\right)+\frac{x^{3}}{3!}% \left(1-\frac{1}{n}\right)\left(1-\frac{2}{n}\right)+\cdots\\ &\displaystyle{}\qquad\cdots+\frac{x^{n}}{n!}\left(1-\frac{1}{n}\right)\cdots% \left(1-\frac{n-1}{n}\right)\leq s_{n}\end{aligned}
  39. lim sup n t n lim sup n s n = e x \limsup_{n\to\infty}t_{n}\leq\limsup_{n\to\infty}s_{n}=e^{x}
  40. 1 + x + x 2 2 ! ( 1 - 1 n ) + + x m m ! ( 1 - 1 n ) ( 1 - 2 n ) ( 1 - m - 1 n ) t n . 1+x+\frac{x^{2}}{2!}\left(1-\frac{1}{n}\right)+\cdots+\frac{x^{m}}{m!}\left(1-% \frac{1}{n}\right)\left(1-\frac{2}{n}\right)\cdots\left(1-\frac{m-1}{n}\right)% \leq t_{n}.
  41. s m = 1 + x + x 2 2 ! + + x m m ! lim inf n t n s_{m}=1+x+\frac{x^{2}}{2!}+\cdots+\frac{x^{m}}{m!}\leq\liminf_{n\to\infty}t_{n}
  42. lim sup n t n e x lim inf n t n \limsup_{n\to\infty}t_{n}\leq e^{x}\leq\liminf_{n\to\infty}t_{n}\,
  43. lim n t n = e x . \lim_{n\to\infty}t_{n}=e^{x}.\,
  44. ( 1 - r n ) n ( 1 + r n ) n = ( 1 - r 2 n 2 ) n \left(1-\frac{r}{n}\right)^{n}\left(1+\frac{r}{n}\right)^{n}=\left(1-\frac{r^{% 2}}{n^{2}}\right)^{n}
  45. ( 1 + x n ) n = e x ( 1 - x 2 2 n + x 3 ( 8 + 3 x ) 24 n 2 + ) , \left(1+\frac{x}{n}\right)^{n}=e^{x}\left(1-\frac{x^{2}}{2n}+\frac{x^{3}(8+3x)% }{24n^{2}}+\cdots\right),
  46. d d x ( ln x ) = 1 x . \frac{d}{dx}\left(\ln x\right)=\frac{1}{x}.
  47. y = lim n ( 1 + x n ) n . y=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}.
  48. ln y = ln lim n ( 1 + x n ) n = lim n ln ( 1 + x n ) n . \ln y=\ln\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}=\lim_{n\to\infty}\ln% \left(1+\frac{x}{n}\right)^{n}.
  49. ln y = lim n n ln ( 1 + x n ) = lim n x ln ( 1 + ( x / n ) ) ( x / n ) . \ln y=\lim_{n\to\infty}n\ln\left(1+\frac{x}{n}\right)=\lim_{n\to\infty}\frac{x% \ln\left(1+(x/n)\right)}{(x/n)}.
  50. = x lim h 0 ln ( 1 + h ) h where h = x n =x\cdot\lim_{h\to 0}\frac{\ln\left(1+h\right)}{h}\quad\mbox{ where }~{}h=\frac% {x}{n}
  51. = x d d t ( ln t ) at t = 1 =x\cdot\frac{d}{dt}\left(\ln t\right)\quad\mbox{ at }~{}t=1
  52. = x 1 t at t = 1 =x\cdot\frac{1}{t}\quad\mbox{ at }~{}t=1
  53. = x . \!\,=x.
  54. f ( x ) f(x)
  55. f ( x + y ) = f ( x ) f ( y ) f(x+y)=f(x)f(y)
  56. f ( x ) = e k x f(x)=e^{kx}
  57. f ( 1 ) = e f(1)=e
  58. f ( x ) f(x)
  59. f ( x + y ) = f ( x ) f ( y ) f(x+y)=f(x)f(y)
  60. f ( x ) f(x)
  61. f ( x ) f(x)
  62. f ( y ) = f ( x ) f ( y - x ) 0 f(y)=f(x)f(y-x)\neq 0
  63. f ( x ) 0 f(x)\neq 0
  64. f ( 0 ) = 1 f(0)=1
  65. f ( x ) = f ( x + 0 ) = f ( x ) f ( 0 ) f(x)=f(x+0)=f(x)f(0)
  66. f ( x ) f(x)
  67. f ( - x ) = 1 / f ( x ) f(-x)=1/f(x)
  68. 1 = f ( 0 ) = f ( x - x ) = f ( x ) f ( - x ) 1=f(0)=f(x-x)=f(x)f(-x)
  69. f ( x ) f(x)
  70. f ( x + δ ) - f ( x ) = f ( x - y ) [ f ( y + δ ) - f ( y ) ] 0 f(x+\delta)-f(x)=f(x-y)[f(y+\delta)-f(y)]\rightarrow 0
  71. δ 0 \delta\rightarrow 0
  72. f ( x ) = e x f(x)=e^{x}
  73. f ( x ) f(x)
  74. g ( x ) = 0 x f ( x ) d x . g(x)=\int_{0}^{x}f(x^{\prime})\,dx^{\prime}.
  75. g ( x + y ) - g ( x ) = x x + y f ( x ) d x = 0 y f ( x + x ) d x = f ( x ) g ( y ) . g(x+y)-g(x)=\int_{x}^{x+y}f(x^{\prime})\,dx^{\prime}=\int_{0}^{y}f(x+x^{\prime% })\,dx^{\prime}=f(x)g(y).
  76. f ( x ) f(x)
  77. g ( y ) 0 g(y)\neq 0
  78. f ( x ) f(x)
  79. f ( x + δ ) - f ( x ) = [ g ( x + δ + y ) - g ( x + δ ) ] - [ g ( x + y ) - g ( x ) ] g ( y ) f(x+\delta)-f(x)=\frac{[g(x+\delta+y)-g(x+\delta)]-[g(x+y)-g(x)]}{g(y)}
  80. = [ g ( x + y + δ ) - g ( x + y ) ] - [ g ( x + δ ) - g ( x ) ] g ( y ) =\frac{[g(x+y+\delta)-g(x+y)]-[g(x+\delta)-g(x)]}{g(y)}
  81. = f ( x + y ) g ( δ ) - f ( x ) g ( δ ) g ( y ) = g ( δ ) f ( x + y ) - f ( x ) g ( y ) . =\frac{f(x+y)g(\delta)-f(x)g(\delta)}{g(y)}=g(\delta)\frac{f(x+y)-f(x)}{g(y)}.
  82. δ 0 \delta\rightarrow 0
  83. g ( 0 ) = 0 g(0)=0
  84. g ( x ) g(x)
  85. f ( x ) f(x)
  86. f ( q ) = e k q f(q)=e^{kq}
  87. f ( n m ) = f ( 1 m + + 1 m ) = f ( 1 m ) n f\left(\frac{n}{m}\right)=f\left(\frac{1}{m}+\cdots+\frac{1}{m}\right)=f\left(% \frac{1}{m}\right)^{n}
  88. f ( 1 / m ) m = f ( 1 ) f(1/m)^{m}=f(1)
  89. f ( n m ) = f ( 1 ) n / m = e k ( n / m ) . f\left(\frac{n}{m}\right)=f(1)^{n/m}=e^{k(n/m)}.
  90. k = ln [ f ( 1 ) ] k=\ln[f(1)]\,
  91. f ( x ) f(x)
  92. f ( x ) = f ( x / 2 ) 2 f(x)=f(x/2)^{2}
  93. f ( x ) = e k x f(x)=e^{kx}
  94. f ( 1 ) = e f(1)=e

Charge-transfer_complex.html

  1. D + A D A D+A\rightleftharpoons DA
  2. Δ E = E A - E I + J {\Delta}E=E_{A}-E_{I}+J\,

Chargino.html

  1. χ ~ 1 ± \tilde{\chi}_{1}^{\pm}
  2. χ ~ 2 ± \tilde{\chi}_{2}^{\pm}
  3. χ ~ i 0 \tilde{\chi}_{i}^{0}

Charles_Jean_de_la_Vallée-Poussin.html

  1. V n = S n + S n + 1 + + S 2 n - 1 n V_{n}=\frac{S_{n}+S_{n+1}+\ldots+S_{2n-1}}{n}
  2. S n = 1 2 c 0 ( f ) + i = 1 n c i ( f ) T i S_{n}=\frac{1}{2}c_{0}(f)+\sum_{i=1}^{n}c_{i}(f)T_{i}
  3. c i ( f ) c_{i}(f)\,
  4. ( T 0 / 2 , T 1 , , T n ) (T_{0}/2,T_{1},\cdots,T_{n})
  5. S n S_{n}
  6. 2 π 2\pi
  7. F ( θ ) = f ( cos θ ) F(\theta)=f(\cos\theta)\,
  8. F n F_{n}
  9. V n = 2 F 2 n - 1 - F n - 1 V_{n}=2F_{2n-1}-F_{n-1}\,
  10. V n 3 V_{n}\leq 3
  11. f * V n = f f*V_{n}=f\,
  12. f ( x ) = j = - n n a j e i j x . f(x)=\sum_{j=-n}^{n}a_{j}e^{ijx}.\,

Chebyshev's_sum_inequality.html

  1. a 1 a 2 a n a_{1}\geq a_{2}\geq\cdots\geq a_{n}
  2. b 1 b 2 b n , b_{1}\geq b_{2}\geq\cdots\geq b_{n},
  3. 1 n k = 1 n a k b k ( 1 n k = 1 n a k ) ( 1 n k = 1 n b k ) . {1\over n}\sum_{k=1}^{n}a_{k}\cdot b_{k}\geq\left({1\over n}\sum_{k=1}^{n}a_{k% }\right)\left({1\over n}\sum_{k=1}^{n}b_{k}\right).
  4. a 1 a 2 a n a_{1}\leq a_{2}\leq\cdots\leq a_{n}
  5. b 1 b 2 b n , b_{1}\geq b_{2}\geq\cdots\geq b_{n},
  6. 1 n k = 1 n a k b k ( 1 n k = 1 n a k ) ( 1 n k = 1 n b k ) . {1\over n}\sum_{k=1}^{n}a_{k}b_{k}\leq\left({1\over n}\sum_{k=1}^{n}a_{k}% \right)\left({1\over n}\sum_{k=1}^{n}b_{k}\right).
  7. S = j = 1 n k = 1 n ( a j - a k ) ( b j - b k ) . S=\sum_{j=1}^{n}\sum_{k=1}^{n}(a_{j}-a_{k})(b_{j}-b_{k}).
  8. j , k j,k
  9. S 0 S≥ 0
  10. 0 2 n j = 1 n a j b j - 2 j = 1 n a j k = 1 n b k , 0\leq 2n\sum_{j=1}^{n}a_{j}b_{j}-2\sum_{j=1}^{n}a_{j}\,\sum_{k=1}^{n}b_{k},
  11. 1 n j = 1 n a j b j ( 1 n j = 1 n a j ) ( 1 n j = 1 n b k ) . \frac{1}{n}\sum_{j=1}^{n}a_{j}b_{j}\geq\left(\frac{1}{n}\sum_{j=1}^{n}a_{j}% \right)\,\left(\frac{1}{n}\sum_{j=1}^{n}b_{k}\right).
  12. 0 1 f ( x ) g ( x ) d x 0 1 f ( x ) d x 0 1 g ( x ) d x , \int_{0}^{1}f(x)g(x)dx\geq\int_{0}^{1}f(x)dx\int_{0}^{1}g(x)dx,\,

Chebyshev_nodes.html

  1. x k = cos ( 2 k - 1 2 n π ) , k = 1 , , n . x_{k}=\cos\left(\frac{2k-1}{2n}\pi\right)\mbox{ , }~{}k=1,\ldots,n.
  2. x k = 1 2 ( a + b ) + 1 2 ( b - a ) cos ( 2 k - 1 2 n π ) , k = 1 , , n . {x}_{k}=\frac{1}{2}(a+b)+\frac{1}{2}(b-a)\cos\left(\frac{2k-1}{2n}\pi\right)% \mbox{ , }~{}k=1,\ldots,n.
  3. [ - 1 , + 1 ] [-1,+1]
  4. n n
  5. x 1 , x 2 , , x n , x_{1},x_{2},\ldots,x_{n},
  6. P n - 1 P_{n-1}
  7. n - 1 n-1
  8. f ( x i ) f(x_{i})
  9. x i x_{i}
  10. x x
  11. f ( x ) - P n - 1 ( x ) = f ( n ) ( ξ ) n ! i = 1 n ( x - x i ) f(x)-P_{n-1}(x)=\frac{f^{(n)}(\xi)}{n!}\prod_{i=1}^{n}(x-x_{i})
  12. ξ \xi
  13. max x [ - 1 , 1 ] | i = 1 n ( x - x i ) | . \max_{x\in[-1,1]}\left|\prod_{i=1}^{n}(x-x_{i})\right|.
  14. | f ( x ) - P n - 1 ( x ) | 1 2 n - 1 n ! max ξ [ - 1 , 1 ] | f ( n ) ( ξ ) | . \left|f(x)-P_{n-1}(x)\right|\leq\frac{1}{2^{n-1}n!}\max_{\xi\in[-1,1]}\left|f^% {(n)}(\xi)\right|.

Chemiosmosis.html

  1. Δ G ( k J m o l - 1 ) = - m F Δ ψ + 2.3 R T log 10 ( [ X m + ] B [ X m + ] A ) \Delta G(kJ\cdot mol^{-1})=-mF\Delta\psi+2.3RT\log_{10}\left({[X^{m+}]_{B}% \over[X^{m+}]_{A}}\right)
  2. Δ μ X m + ( k J m o l - 1 ) = Δ G ( k J m o l - 1 ) \Delta\mu_{Xm+}(kJ\cdot mol^{-1})=\Delta G(kJ\cdot mol^{-1})
  3. Δ μ H + = - F Δ ψ + 2.3 R T Δ p H \Delta\mu_{H+}=-F\Delta\psi+2.3RT\Delta pH
  4. Δ p H = p H A - p H B \Delta pH=pH_{A}-pH_{B}
  5. Δ p ( m V ) = - Δ μ H + F \Delta p(mV)=-{\Delta\mu_{H+}\over F}
  6. Δ p = Δ ψ - 59 Δ p H \Delta p=\Delta\psi-59\Delta pH

Chernoff_bound.html

  1. S S
  2. S = i = n 2 + 1 n ( n i ) p i ( 1 - p ) n - i . S=\sum_{i=\lfloor\tfrac{n}{2}\rfloor+1}^{n}{\left({{n}\atop{i}}\right)}p^{i}(1% -p)^{n-i}.
  3. S S
  4. S 1 - e - 1 2 p n ( p - 1 2 ) 2 . S\geq 1-e^{-\frac{1}{2p}n\left(p-\frac{1}{2}\right)^{2}}.
  5. μ = n p μ=np
  6. Pr ( k = 1 n X k n 2 ) = Pr ( k = 1 n X k ( 1 - ( 1 - 1 2 p ) ) μ ) e - μ 2 ( 1 - 1 2 p ) 2 = e - n 2 p ( p - 1 2 ) 2 \begin{aligned}\displaystyle\Pr\left(\sum_{k=1}^{n}X_{k}\leq\left\lfloor\tfrac% {n}{2}\right\rfloor\right)&\displaystyle=\Pr\left(\sum_{k=1}^{n}X_{k}\leq\left% (1-\left(1-\tfrac{1}{2p}\right)\right)\mu\right)\\ &\displaystyle\leq e^{-\frac{\mu}{2}\left(1-\frac{1}{2p}\right)^{2}}\\ &\displaystyle=e^{-\frac{n}{2p}\left(p-\frac{1}{2}\right)^{2}}\end{aligned}
  7. n n
  8. ε ε
  9. n 1 ( p - 1 2 ) 2 ln 1 ε . n\geq\frac{1}{(p-\frac{1}{2})^{2}}\ln\frac{1}{\sqrt{\varepsilon}}.
  10. n n
  11. n n
  12. n 2 2 m ln 1 ε . n\geq 2^{2m}\ln\frac{1}{\sqrt{\varepsilon}}.
  13. X X
  14. n n
  15. Pr ( X a ) = Pr ( e t X e t a ) E [ e t X ] e t a = e - t a E [ i e t X i ] . \Pr(X\geq a)=\Pr\left(e^{tX}\geq e^{ta}\right)\leq\frac{E\left[e^{tX}\right]}{% e^{ta}}=e^{-ta}\mathrm{E}\left[\prod_{i}e^{tX_{i}}\right].
  16. Pr ( X a ) min t > 0 e - t a i E [ e t X i ] . \Pr(X\geq a)\leq\min_{t>0}e^{-ta}\prod_{i}E\left[e^{tX_{i}}\right].
  17. Pr ( X a ) = Pr ( e - t X e - t a ) \Pr(X\leq a)=\Pr\left(e^{-tX}\geq e^{-ta}\right)
  18. Pr ( X a ) min t > 0 e t a i E [ e - t X i ] \Pr(X\leq a)\leq\min_{t>0}e^{ta}\prod_{i}\mathrm{E}\left[e^{-tX_{i}}\right]
  19. ε > 0 ε>0
  20. Pr ( 1 n X i p + ε ) ( ( p p + ε ) p + ε ( 1 - p 1 - p - ε ) 1 - p - ε ) n = e - D ( p + ε p ) n Pr ( 1 n X i p - ε ) ( ( p p - ε ) p - ε ( 1 - p 1 - p + ε ) 1 - p + ε ) n = e - D ( p - ε p ) n \begin{aligned}\displaystyle\Pr\left(\frac{1}{n}\sum X_{i}\geq p+\varepsilon% \right)\leq\left(\left(\frac{p}{p+\varepsilon}\right)^{p+\varepsilon}{\left(% \frac{1-p}{1-p-\varepsilon}\right)}^{1-p-\varepsilon}\right)^{n}&\displaystyle% =e^{-D(p+\varepsilon\|p)n}\\ \displaystyle\Pr\left(\frac{1}{n}\sum X_{i}\leq p-\varepsilon\right)\leq\left(% \left(\frac{p}{p-\varepsilon}\right)^{p-\varepsilon}{\left(\frac{1-p}{1-p+% \varepsilon}\right)}^{1-p+\varepsilon}\right)^{n}&\displaystyle=e^{-D(p-% \varepsilon\|p)n}\end{aligned}
  21. D ( x y ) = x ln x y + ( 1 - x ) ln ( 1 - x 1 - y ) D(x\|y)=x\ln\frac{x}{y}+(1-x)\ln\left(\frac{1-x}{1-y}\right)
  22. p 1 2 , p≥\frac{1}{2},
  23. Pr ( X > n p + x ) exp ( - x 2 2 n p ( 1 - p ) ) . \Pr\left(X>np+x\right)\leq\exp\left(-\frac{x^{2}}{2np(1-p)}\right).
  24. q = p + ε q=p+ε
  25. a = n q a=nq
  26. Pr ( 1 n X i q ) inf t > 0 E [ e t X i ] e t n q = inf t > 0 ( E [ e t X i ] e t q ) n . \Pr\left(\frac{1}{n}\sum X_{i}\geq q\right)\leq\inf_{t>0}\frac{E\left[\prod e^% {tX_{i}}\right]}{e^{tnq}}=\inf_{t>0}\left(\frac{E\left[e^{tX_{i}}\right]}{e^{% tq}}\right)^{n}.
  27. ( E [ e t X i ] e t q ) n = ( p e t + ( 1 - p ) e t q ) n = ( p e ( 1 - q ) t + ( 1 - p ) e - q t ) n . \left(\frac{\mathrm{E}\left[e^{tX_{i}}\right]}{e^{tq}}\right)^{n}=\left(\frac{% pe^{t}+(1-p)}{e^{tq}}\right)^{n}=\left(pe^{(1-q)t}+(1-p)e^{-qt}\right)^{n}.
  28. d d t ( p e ( 1 - q ) t + ( 1 - p ) e - q t ) = ( 1 - q ) p e ( 1 - q ) t - q ( 1 - p ) e - q t \frac{d}{dt}\left(pe^{(1-q)t}+(1-p)e^{-qt}\right)=(1-q)pe^{(1-q)t}-q(1-p)e^{-qt}
  29. ( 1 - q ) p e ( 1 - q ) t = q ( 1 - p ) e - q t ( 1 - q ) p e t = q ( 1 - p ) \begin{aligned}\displaystyle(1-q)pe^{(1-q)t}&\displaystyle=q(1-p)e^{-qt}\\ \displaystyle(1-q)pe^{t}&\displaystyle=q(1-p)\end{aligned}
  30. e t = ( 1 - p ) q ( 1 - q ) p . e^{t}=\frac{(1-p)q}{(1-q)p}.
  31. t = log ( ( 1 - p ) q ( 1 - q ) p ) . t=\log\left(\frac{(1-p)q}{(1-q)p}\right).
  32. q = p + ε > p q=p+ε>p
  33. t > 0 t>0
  34. t t
  35. t t
  36. log ( p e ( 1 - q ) t + ( 1 - p ) e - q t ) = log ( e - q t ( 1 - p + p e t ) ) = log ( e - q log ( ( 1 - p ) q ( 1 - q ) p ) ) + log ( 1 - p + p e log ( 1 - p 1 - q ) e log q p ) = - q log 1 - p 1 - q - q log q p + log ( 1 - p + p ( 1 - p 1 - q ) q p ) = - q log 1 - p 1 - q - q log q p + log ( ( 1 - p ) ( 1 - q ) 1 - q + ( 1 - p ) q 1 - q ) = - q log q p + ( - q log 1 - p 1 - q + log 1 - p 1 - q ) = - q log q p + ( 1 - q ) log 1 - p 1 - q = - D ( q p ) . \begin{aligned}\displaystyle\log\left(pe^{(1-q)t}+(1-p)e^{-qt}\right)&% \displaystyle=\log\left(e^{-qt}(1-p+pe^{t})\right)\\ &\displaystyle=\log\left(e^{-q\log\left(\frac{(1-p)q}{(1-q)p}\right)}\right)+% \log\left(1-p+pe^{\log\left(\frac{1-p}{1-q}\right)}e^{\log\frac{q}{p}}\right)% \\ &\displaystyle=-q\log\frac{1-p}{1-q}-q\log\frac{q}{p}+\log\left(1-p+p\left(% \frac{1-p}{1-q}\right)\frac{q}{p}\right)\\ &\displaystyle=-q\log\frac{1-p}{1-q}-q\log\frac{q}{p}+\log\left(\frac{(1-p)(1-% q)}{1-q}+\frac{(1-p)q}{1-q}\right)\\ &\displaystyle=-q\log\frac{q}{p}+\left(-q\log\frac{1-p}{1-q}+\log\frac{1-p}{1-% q}\right)\\ &\displaystyle=-q\log\frac{q}{p}+(1-q)\log\frac{1-p}{1-q}\\ &\displaystyle=-D(q\|p).\end{aligned}
  37. Pr ( 1 n X i p + ε ) e - D ( p + ε p ) n . \Pr\left(\tfrac{1}{n}\sum X_{i}\geq p+\varepsilon\right)\leq e^{-D(p+% \varepsilon\|p)n}.
  38. D ( p + x [ u ! ! ] p ) D(p+x[u^{\prime}!!^{\prime}]p)
  39. d 2 d x 2 D ( p + x p ) = 1 ( p + x ) ( 1 - p - x ) 4 = d 2 d x 2 ( 2 x 2 ) . \frac{d^{2}}{dx^{2}}D(p+x\|p)=\frac{1}{(p+x)(1-p-x)}\geq 4=\frac{d^{2}}{dx^{2}% }(2x^{2}).
  40. D ( ( 1 + x ) p p ) 1 4 x 2 p , - 1 2 x 1 2 , D((1+x)p\|p)\geq\tfrac{1}{4}x^{2}p,\qquad-\tfrac{1}{2}\leq x\leq\tfrac{1}{2},
  41. X X
  42. μ = E X X μ=EXX
  43. δ > 0 δ>0
  44. Pr ( X > ( 1 + δ ) μ ) < ( e δ ( 1 + δ ) ( 1 + δ ) ) μ . \Pr(X>(1+\delta)\mu)<\left(\frac{e^{\delta}}{(1+\delta)^{(1+\delta)}}\right)^{% \mu}.
  45. Pr ( X > ( 1 + δ ) μ ) inf t > 0 E [ i = 1 n exp ( t X i ) ] exp ( t ( 1 + δ ) μ ) = inf t > 0 i = 1 n E [ e t X i ] exp ( t ( 1 + δ ) μ ) = inf t > 0 i = 1 n [ p i e t + ( 1 - p i ) ] exp ( t ( 1 + δ ) μ ) \begin{aligned}\displaystyle\Pr(X>(1+\delta)\mu)&\displaystyle\leq\inf_{t>0}% \frac{\mathrm{E}\left[\prod_{i=1}^{n}\exp(tX_{i})\right]}{\exp(t(1+\delta)\mu)% }\\ &\displaystyle=\inf_{t>0}\frac{\prod_{i=1}^{n}\mathrm{E}\left[e^{tX_{i}}\right% ]}{\exp(t(1+\delta)\mu)}\\ &\displaystyle=\inf_{t>0}\frac{\prod_{i=1}^{n}\left[p_{i}e^{t}+(1-p_{i})\right% ]}{\exp(t(1+\delta)\mu)}\end{aligned}
  46. e t X i e^{tX_{i}}
  47. p i e t + ( 1 - p i ) p_{i}e^{t}+(1-p_{i})
  48. p i ( e t - 1 ) + 1 p_{i}(e^{t}-1)+1
  49. 1 + x e x 1+x\leq e^{x}
  50. x > 0 x>0
  51. x = p i ( e t - 1 ) x=p_{i}(e^{t}-1)
  52. a a
  53. ( 1 + δ ) μ (1+δ)μ
  54. Pr ( X > ( 1 + δ ) μ ) < i = 1 n exp ( p i ( e t - 1 ) ) exp ( t ( 1 + δ ) μ ) = exp ( ( e t - 1 ) i = 1 n p i ) exp ( t ( 1 + δ ) μ ) = exp ( ( e t - 1 ) μ ) exp ( t ( 1 + δ ) μ ) . \Pr(X>(1+\delta)\mu)<\frac{\prod_{i=1}^{n}\exp(p_{i}(e^{t}-1))}{\exp(t(1+% \delta)\mu)}=\frac{\exp\left((e^{t}-1)\sum_{i=1}^{n}p_{i}\right)}{\exp(t(1+% \delta)\mu)}=\frac{\exp((e^{t}-1)\mu)}{\exp(t(1+\delta)\mu)}.
  55. t = l o g ( 1 + δ ) t=log(1+δ)
  56. t > 0 t>0
  57. δ > 0 δ>0
  58. exp ( ( e t - 1 ) μ ) exp ( t ( 1 + δ ) μ ) = exp ( ( 1 + δ - 1 ) μ ) ( 1 + δ ) ( 1 + δ ) μ = [ e δ ( 1 + δ ) ( 1 + δ ) ] μ \frac{\exp((e^{t}-1)\mu)}{\exp(t(1+\delta)\mu)}=\frac{\exp((1+\delta-1)\mu)}{(% 1+\delta)^{(1+\delta)\mu}}=\left[\frac{e^{\delta}}{(1+\delta)^{(1+\delta)}}% \right]^{\mu}
  59. Pr ( X < ( 1 - δ ) μ ) < [ exp ( - δ ) ( 1 - δ ) ( 1 - δ ) ] μ . \Pr(X<(1-\delta)\mu)<\left[\frac{\exp(-\delta)}{(1-\delta)^{(1-\delta)}}\right% ]^{\mu}.
  60. Pr ( X ( 1 + δ ) μ ) e - δ 2 μ 3 , 0 < δ < 1 , \Pr(X\geq(1+\delta)\mu)\leq e^{-\frac{\delta^{2}\mu}{3}},\qquad 0<\delta<1,
  61. Pr ( X ( 1 - δ ) μ ) e - δ 2 μ 2 , 0 < δ < 1. \Pr(X\leq(1-\delta)\mu)\leq e^{-\frac{\delta^{2}\mu}{2}},\qquad 0<\delta<1.
  62. X X
  63. Pr ( X i = 1 ) = Pr ( X i = - 1 ) = 1 2 \Pr(X_{i}=1)=\Pr(X_{i}=-1)=\tfrac{1}{2}
  64. Pr ( X a ) e - a 2 2 n , a > 0 , \Pr(X\geq a)\leq e^{\frac{-a^{2}}{2n}},\qquad a>0,
  65. Pr ( | X | a ) 2 e - a 2 2 n , a > 0. \Pr(|X|\geq a)\leq 2e^{\frac{-a^{2}}{2n}},\qquad a>0.
  66. Pr ( X i = 1 ) = Pr ( X i = 0 ) = 1 2 , E [ X ] = μ = n 2 \Pr(X_{i}=1)=\Pr(X_{i}=0)=\tfrac{1}{2},\mathrm{E}[X]=\mu=\frac{n}{2}
  67. Pr ( X μ + a ) e - 2 a 2 n , a > 0 , \Pr(X\geq\mu+a)\leq e^{\frac{-2a^{2}}{n}},\qquad a>0,
  68. Pr ( X μ - a ) e - 2 a 2 n , 0 < a < μ , \Pr(X\leq\mu-a)\leq e^{\frac{-2a^{2}}{n}},\qquad 0<a<\mu,
  69. d × d d×d
  70. ε > 0 ε>0
  71. Pr ( 1 t i = 1 t M i - E [ M ] 2 > ε ) d exp ( - C ε 2 t γ 2 ) . \Pr\left(\left\|\frac{1}{t}\sum_{i=1}^{t}M_{i}-\mathrm{E}[M]\right\|_{2}>% \varepsilon\right)\leq d\exp\left(-C\frac{\varepsilon^{2}t}{\gamma^{2}}\right).
  72. M 2 γ \lVert M\rVert_{2}\leq\gamma
  73. E [ M ] 2 1 \|\mathrm{E}[M]\|_{2}\leq 1
  74. M 2 γ \|M\|_{2}\leq\gamma
  75. t = Ω ( γ log ( γ / ε 2 ) ε 2 ) . t=\Omega\left(\frac{\gamma\log(\gamma/\varepsilon^{2})}{\varepsilon^{2}}\right).
  76. r t r\leq t
  77. Pr ( 1 t i = 1 t M i - E [ M ] 2 > ε ) 1 𝐩𝐨𝐥𝐲 ( t ) \Pr\left(\left\|\frac{1}{t}\sum_{i=1}^{t}M_{i}-\mathrm{E}[M]\right\|_{2}>% \varepsilon\right)\leq\frac{1}{\mathbf{poly}(t)}
  78. Pr ( r S < ( 1 - d ) r ) < exp ( - r d 2 k / 2 ) \mathrm{Pr}\left(r_{S}<(1-d)\cdot r\right)<\exp\left(-r\cdot d^{2}\cdot k/2\right)
  79. Pr ( r S > 0.5 ) > 1 - exp ( - r ( 1 - 1 2 r ) 2 k / 2 ) \mathrm{Pr}\left(r_{S}>0.5\right)>1-\exp\left(-r\cdot\left(1-\frac{1}{2r}% \right)^{2}\cdot k/2\right)

Chern–Weil_homomorphism.html

  1. 𝔤 \mathfrak{g}
  2. [ 𝔤 ] \mathbb{C}[\mathfrak{g}]
  3. \mathbb{C}
  4. 𝔤 \mathfrak{g}
  5. \mathbb{R}
  6. \mathbb{C}
  7. [ 𝔤 ] G \mathbb{C}[\mathfrak{g}]^{G}
  8. [ 𝔤 ] \mathbb{C}[\mathfrak{g}]
  9. 𝔤 \mathfrak{g}
  10. f ( Ad g x ) = f ( x ) . f(\operatorname{Ad}_{g}x)=f(x).
  11. \mathbb{C}
  12. [ 𝔤 ] G H * ( M , ) \mathbb{C}[\mathfrak{g}]^{G}\to H^{*}(M,\mathbb{C})
  13. [ 𝔤 ] G \mathbb{C}[\mathfrak{g}]^{G}
  14. H * ( B G , ) [ 𝔤 ] G . H^{*}(BG,\mathbb{C})\cong\mathbb{C}[\mathfrak{g}]^{G}.
  15. H k ( B G , ) = lim ker ( d : Ω k ( B j G ) Ω k + 1 ( B j G ) ) / im d . H^{k}(BG,\mathbb{C})=\underrightarrow{\lim}\operatorname{ker}(d:\Omega^{k}(B_{% j}G)\to\Omega^{k+1}(B_{j}G))/\operatorname{im}d.
  16. B G = lim B j G BG=\underrightarrow{\lim}B_{j}G
  17. B j G B_{j}G
  18. f [ 𝔤 ] G f\in\mathbb{C}[\mathfrak{g}]^{G}
  19. f ( a x ) = a k f ( x ) f(ax)=a^{k}f(x)
  20. 𝔤 \mathfrak{g}
  21. 1 k 𝔤 \prod_{1}^{k}\mathfrak{g}
  22. f ( Ω ) f(\Omega)
  23. f ( Ω ) ( v 1 , , v 2 k ) = 1 ( 2 k ) ! σ 𝔖 2 k ϵ σ f ( Ω ( v σ ( 1 ) , v σ ( 2 ) ) , , Ω ( v σ ( 2 k - 1 ) , v σ ( 2 k ) ) ) f(\Omega)(v_{1},\dots,v_{2k})=\frac{1}{(2k)!}\sum_{\sigma\in\mathfrak{S}_{2k}}% \epsilon_{\sigma}f(\Omega(v_{\sigma(1)},v_{\sigma(2)}),\dots,\Omega(v_{\sigma(% 2k-1)},v_{\sigma(2k)}))
  24. ϵ σ \epsilon_{\sigma}
  25. σ \sigma
  26. 𝔖 2 k \mathfrak{S}_{2k}
  27. f ( Ad g x ) = f ( x ) f(\operatorname{Ad}_{g}x)=f(x)
  28. f ( Ω ) f(\Omega)
  29. f ( Ω ) f(\Omega)
  30. f ( Ω ) f(\Omega)
  31. f ¯ ( Ω ) \overline{f}(\Omega)
  32. f ( Ω ) f(\Omega)
  33. D Ω = 0 D\Omega=0
  34. D f ( Ω ) = 0. Df(\Omega)=0.
  35. f ( Ω ) f(\Omega)
  36. π : P M \pi:P\to M
  37. T u P T_{u}P
  38. d π ( h v ) = d π ( v ) d\pi(hv)=d\pi(v)
  39. d π d\pi
  40. f ( Ω ) ( d R g ( v 1 ) , , d R g ( v 2 k ) ) = f ( Ω ) ( v 1 , , v 2 k ) , R g ( u ) = u g ; f(\Omega)(dR_{g}(v_{1}),\dots,dR_{g}(v_{2k}))=f(\Omega)(v_{1},\dots,v_{2k}),\,% R_{g}(u)=ug;
  41. R g * Ω = Ad g - 1 Ω R_{g}^{*}\Omega=\operatorname{Ad}_{g^{-1}}\Omega
  42. f ¯ ( Ω ) \overline{f}(\Omega)
  43. f ¯ ( Ω ) ( v 1 ¯ , , v 2 k ¯ ) = f ( Ω ) ( v 1 , , v 2 k ) \overline{f}(\Omega)(\overline{v_{1}},\dots,\overline{v_{2k}})=f(\Omega)(v_{1}% ,\dots,v_{2k})
  44. v i v_{i}
  45. v i ¯ \overline{v_{i}}
  46. d π ( v i ) = v ¯ i d\pi(v_{i})=\overline{v}_{i}
  47. f ¯ ( Ω ) \overline{f}(\Omega)
  48. ω 0 , ω 1 \omega_{0},\omega_{1}
  49. p : P × P p:P\times\mathbb{R}\to P
  50. ω = t p * ω 1 + ( 1 - t ) p * ω 0 \omega^{\prime}=t\,p^{*}\omega_{1}+(1-t)\,p^{*}\omega_{0}
  51. P × P\times\mathbb{R}
  52. ( x , s ) s (x,s)\mapsto s
  53. Ω , Ω 0 , Ω 1 \Omega^{\prime},\Omega_{0},\Omega_{1}
  54. ω , ω 0 , ω 1 \omega^{\prime},\omega_{0},\omega_{1}
  55. i s : M M × , x ( x , s ) i_{s}:M\to M\times\mathbb{R},\,x\mapsto(x,s)
  56. i 0 i_{0}
  57. i 1 i_{1}
  58. i 0 * f ¯ ( Ω ) i_{0}^{*}\overline{f}(\Omega^{\prime})
  59. i 1 * f ¯ ( Ω ) i_{1}^{*}\overline{f}(\Omega^{\prime})
  60. i 0 * f ¯ ( Ω ) = f ¯ ( Ω 0 ) i_{0}^{*}\overline{f}(\Omega^{\prime})=\overline{f}(\Omega_{0})
  61. Ω 1 \Omega_{1}
  62. f ¯ ( Ω 0 ) , f ¯ ( Ω 1 ) \overline{f}(\Omega_{0}),\overline{f}(\Omega_{1})
  63. [ 𝔤 ] k G H 2 k ( M , ) , f [ f ¯ ( Ω ) ] . \mathbb{C}[\mathfrak{g}]^{G}_{k}\rightarrow H^{2k}(M,\mathbb{C}),\,f\mapsto% \left[\overline{f}(\Omega)\right].
  64. [ 𝔤 ] G H * ( M , ) \mathbb{C}[\mathfrak{g}]^{G}\rightarrow H^{*}(M,\mathbb{C})
  65. G = G L n ( ) G=GL_{n}(\mathbb{C})
  66. 𝔤 = 𝔤 𝔩 n ( ) \mathfrak{g}=\mathfrak{gl}_{n}(\mathbb{C})
  67. 𝔤 \mathfrak{g}
  68. det ( I - t x 2 π i ) = k = 0 n f k ( x ) t k , \det\left(I-t{x\over 2\pi i}\right)=\sum_{k=0}^{n}f_{k}(x)t^{k},
  69. f k f_{k}
  70. 𝔤 \mathfrak{g}
  71. c k ( E ) H 2 k ( M , ) c_{k}(E)\in H^{2k}(M,\mathbb{Z})
  72. det ( I - x 2 π i ) = 1 + f 1 ( x ) + + f n ( x ) \det\left(I-{x\over 2\pi i}\right)=1+f_{1}(x)+\cdots+f_{n}(x)
  73. c ( E ) = 1 + c 1 ( E ) + + c n ( E ) . c(E)=1+c_{1}(E)+\cdots+c_{n}(E).
  74. c t ( E ) = [ det ( I - t Ω / 2 π i ) ] c_{t}(E)=[\det\left(I-t{\Omega/2\pi i}\right)]
  75. det ( I - t Ω / 2 π i ) = det ( I - t Ω 1 / 2 π i ) det ( I - t Ω m / 2 π i ) \det(I-t\Omega/2\pi i)=\det(I-t\Omega_{1}/2\pi i)\wedge\dots\wedge\det(I-t% \Omega_{m}/2\pi i)
  76. c t ( E ) = c t ( E 1 ) c t ( E m ) c_{t}(E)=c_{t}(E_{1})\cdots c_{t}(E_{m})
  77. Ω E E = Ω E I E + I E Ω E \Omega_{E\otimes E^{\prime}}=\Omega_{E}\otimes I_{E^{\prime}}+I_{E}\otimes% \Omega_{E^{\prime}}
  78. c 1 ( E E ) = c 1 ( E ) rk ( E ) + rk ( E ) c 1 ( E ) . c_{1}(E\otimes E^{\prime})=c_{1}(E)\operatorname{rk}(E^{\prime})+\operatorname% {rk}(E)c_{1}(E^{\prime}).
  79. ch ( E ) = [ tr ( e - Ω / 2 π i ) ] H * ( M , ) \operatorname{ch}(E)=[\operatorname{tr}(e^{-\Omega/2\pi i})]\in H^{*}(M,% \mathbb{Q})
  80. ch ( E F ) = ch ( E ) + ch ( F ) , ch ( E F ) = ch ( E ) ch ( F ) . \operatorname{ch}(E\oplus F)=\operatorname{ch}(E)+\operatorname{ch}(F),\,% \operatorname{ch}(E\otimes F)=\operatorname{ch}(E)\operatorname{ch}(F).
  81. c t ( E ) = j = 0 n ( 1 + λ j t ) c_{t}(E)=\prod_{j=0}^{n}(1+\lambda_{j}t)
  82. ch ( E ) = e λ j \operatorname{ch}(E)=e^{\lambda_{j}}
  83. p k ( E ) = ( - 1 ) k c 2 k ( E ) H 4 k ( M , ) p_{k}(E)=(-1)^{k}c_{2k}(E\otimes\mathbb{C})\in H^{4k}(M,\mathbb{Z})
  84. E E\otimes\mathbb{C}
  85. g 2 k g_{2k}
  86. 𝔤 𝔩 n ( ) \mathfrak{gl}_{n}(\mathbb{R})
  87. det ( I - t x 2 π ) = k = 0 n g k ( x ) t k . \operatorname{det}\left(I-t{x\over 2\pi}\right)=\sum_{k=0}^{n}g_{k}(x)t^{k}.
  88. G = G L n ( ) G=GL_{n}(\mathbb{C})
  89. [ 𝔤 ] k H k , k ( M , ) , f [ f ( Ω ) ] . \mathbb{C}[\mathfrak{g}]_{k}\to H^{k,k}(M,\mathbb{C}),f\mapsto[f(\Omega)].
  90. E E ( s s ) = E s s + s E s \nabla^{E\otimes E^{\prime}}(s\otimes s^{\prime})=\nabla^{E}s\otimes s^{\prime% }+s\otimes\nabla^{E^{\prime}}s^{\prime}
  91. E E \nabla^{E\otimes E^{\prime}}

Chiller.html

  1. = Cooling power Input power =\frac{\,\text{Cooling power}}{\,\text{Input power}}

Chirality_(mathematics).html

  1. v A v + b v\mapsto Av+b
  2. A A
  3. b b
  4. A A
  5. F F
  6. P P
  7. F F
  8. ( x , y , z ) ( x , y , - z ) (x,y,z)\mapsto(x,y,-z)
  9. P P
  10. x x
  11. y y
  12. F F
  13. C C
  14. F F
  15. ( x , y , z ) ( - x , - y , - z ) (x,y,z)\mapsto(-x,-y,-z)
  16. C C
  17. F 0 = { ( 1 , 0 , 0 ) , ( 0 , 1 , 0 ) , ( - 1 , 0 , 0 ) , ( 0 , - 1 , 0 ) , ( 2 , 1 , 1 ) , ( - 1 , 2 , - 1 ) , ( - 2 , - 1 , 1 ) , ( 1 , - 2 , - 1 ) } F_{0}=\left\{(1,0,0),(0,1,0),(-1,0,0),(0,-1,0),(2,1,1),(-1,2,-1),(-2,-1,1),(1,% -2,-1)\right\}
  18. ( x , y , z ) ( - y , x , - z ) (x,y,z)\mapsto(-y,x,-z)
  19. F 1 = { ( 1 , 0 , 0 ) , ( - 1 , 0 , 0 ) , ( 0 , 2 , 0 ) , ( 0 , - 2 , 0 ) , ( 1 , 1 , 1 ) , ( - 1 , - 1 , - 1 ) } F_{1}=\left\{(1,0,0),(-1,0,0),(0,2,0),(0,-2,0),(1,1,1),(-1,-1,-1)\right\}
  20. F F
  21. L L
  22. F F
  23. ( x , y ) ( x , - y ) (x,y)\mapsto(x,-y)
  24. L L
  25. x x

Chirality_(physics).html

  1. ψ \psi
  2. γ 5 \gamma^{5}
  3. ( 1 - γ 5 ) / 2 (1-\gamma^{5})/2
  4. ( 1 + γ 5 ) / 2 (1+\gamma^{5})/2
  5. ψ \psi
  6. ψ \psi
  7. ψ L e i θ L ψ L \psi_{L}\rightarrow e^{i\theta_{L}}\psi_{L}
  8. ψ R ψ R \psi_{R}\rightarrow\psi_{R}
  9. ψ L ψ L \psi_{L}\rightarrow\psi_{L}
  10. ψ R e i θ R ψ R . \psi_{R}\rightarrow e^{i\theta_{R}}\psi_{R}.
  11. P R = 1 + γ 5 2 P_{R}=\frac{1+\gamma^{5}}{2}
  12. P L = 1 - γ 5 2 P_{L}=\frac{1-\gamma^{5}}{2}
  13. m ψ ¯ ψ m\overline{\psi}\psi

Chronon.html

  1. θ 0 = 1 6 π ϵ 0 e 2 m 0 c 3 \theta_{0}=\frac{1}{6\pi\epsilon_{0}}\frac{e^{2}}{m_{0}c^{3}}

Circulant_matrix.html

  1. / n \mathbb{Z}/n\mathbb{Z}
  2. n × n n\times n
  3. C \ C
  4. C = [ c 0 c n - 1 c 2 c 1 c 1 c 0 c n - 1 c 2 c 1 c 0 c n - 2 c n - 1 c n - 1 c n - 2 c 1 c 0 ] . C=\begin{bmatrix}c_{0}&c_{n-1}&\dots&c_{2}&c_{1}\\ c_{1}&c_{0}&c_{n-1}&&c_{2}\\ \vdots&c_{1}&c_{0}&\ddots&\vdots\\ c_{n-2}&&\ddots&\ddots&c_{n-1}\\ c_{n-1}&c_{n-2}&\dots&c_{1}&c_{0}\\ \end{bmatrix}.
  5. c \ c
  6. C \ C
  7. C \ C
  8. c \ c
  9. C \ C
  10. c \ c
  11. f ( x ) = c 0 + c 1 x + + c n - 1 x n - 1 f(x)=c_{0}+c_{1}x+\dots+c_{n-1}x^{n-1}
  12. C C
  13. v j = 1 n ( 1 , ω j , ω j 2 , , ω j n - 1 ) T , j = 0 , 1 , , n - 1 , v_{j}=\frac{1}{\sqrt{n}}(1,~{}\omega_{j},~{}\omega_{j}^{2},~{}\ldots,~{}\omega% _{j}^{n-1})^{T},\quad j=0,1,\ldots,n-1,
  14. ω j = exp ( 2 π i j n ) \omega_{j}=\exp\left(\tfrac{2\pi ij}{n}\right)
  15. i i
  16. λ j = c 0 + c n - 1 ω j + c n - 2 ω j 2 + + c 1 ω j n - 1 , j = 0 n - 1. \lambda_{j}=c_{0}+c_{n-1}\omega_{j}+c_{n-2}\omega_{j}^{2}+\ldots+c_{1}\omega_{% j}^{n-1},\qquad j=0\ldots n-1.
  17. det ( C ) = j = 0 n - 1 ( c 0 + c n - 1 ω j + c n - 2 ω j 2 + + c 1 ω j n - 1 ) . \mathrm{det}(C)=\prod_{j=0}^{n-1}(c_{0}+c_{n-1}\omega_{j}+c_{n-2}\omega_{j}^{2% }+\dots+c_{1}\omega_{j}^{n-1}).
  18. det ( C ) = j = 0 n - 1 ( c 0 + c 1 ω j + c 2 ω j 2 + + c n - 1 ω j n - 1 ) = j = 0 n - 1 f ( ω j ) . \mathrm{det}(C)=\prod_{j=0}^{n-1}(c_{0}+c_{1}\omega_{j}+c_{2}\omega_{j}^{2}+% \dots+c_{n-1}\omega_{j}^{n-1})=\prod_{j=0}^{n-1}f(\omega_{j}).
  19. C C
  20. n - d n-d
  21. d d
  22. gcd ( f ( x ) , x n - 1 ) \gcd(f(x),x^{n}-1)
  23. C = c 0 I + c 1 P + c 2 P 2 + + c n - 1 P n - 1 = f ( P ) . C=c_{0}I+c_{1}P+c_{2}P^{2}+\ldots+c_{n-1}P^{n-1}=f(P).
  24. P = [ 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0 ] . P=\begin{bmatrix}0&0&\ldots&0&1\\ 1&0&\ldots&0&0\\ 0&\ddots&\ddots&\vdots&\vdots\\ \vdots&\ddots&\ddots&0&0\\ 0&\ldots&0&1&0\end{bmatrix}.
  25. n × n n\times n
  26. 𝐙 / n 𝐙 , \mathbf{Z}/n\mathbf{Z},
  27. A \ A
  28. B \ B
  29. A + B \ A+B
  30. A B \ AB
  31. A B = B A \ AB=BA
  32. U n * = 1 n F n , and U n = n F n - 1 , where F n = ( f j k ) with f j k = e - 2 j k π i / n , for 0 j , k < n . U_{n}^{*}=\frac{1}{\sqrt{n}}F_{n},\quad\,\text{and}\quad U_{n}=\sqrt{n}F_{n}^{% -1},\quad\,\text{where}\quad F_{n}=(f_{jk})\quad\,\text{with}\quad f_{jk}=% \mathrm{e}^{-2jk\pi\mathrm{i}/n},\quad\,\text{for}\quad 0\leq j,k<n.
  33. U n U_{n}
  34. C = U n diag ( F n c ) U n * = F n - 1 diag ( F n c ) F n , C=U_{n}\operatorname{diag}(F_{n}c)U_{n}^{*}=F_{n}^{-1}\operatorname{diag}(F_{n% }c)F_{n},
  35. c c\!\,
  36. C C\,\!
  37. C C
  38. F n c \ F_{n}c
  39. 𝐑 n \mathbf{R}^{n}
  40. , a 0 , a 1 , , a n - 1 , a 0 , a 1 , \dots,a_{0},a_{1},\dots,a_{n-1},a_{0},a_{1},\dots
  41. C n C_{n}
  42. 𝐙 / n 𝐙 \mathbf{Z}/n\mathbf{Z}
  43. ( c 0 , c 1 , , c n - 1 ) ; (c_{0},c_{1},\dots,c_{n-1});
  44. ( b i ) := ( c i ) * ( a i ) (b_{i}):=(c_{i})*(a_{i})
  45. b k = i = 0 n - 1 a i c k - i b_{k}=\sum_{i=0}^{n-1}a_{i}c_{k-i}
  46. a i a_{i}
  47. 𝐂𝐱 = 𝐛 , \ \mathbf{C}\mathbf{x}=\mathbf{b},
  48. C \ C
  49. n \ n
  50. 𝐜 𝐱 = 𝐛 , \ \mathbf{c}\star\mathbf{x}=\mathbf{b},
  51. c \ c
  52. C \ C
  53. c \ c
  54. x \ x
  55. b \ b
  56. n ( 𝐜 𝐱 ) = n ( 𝐜 ) n ( 𝐱 ) = n ( 𝐛 ) \ \mathcal{F}_{n}(\mathbf{c}\star\mathbf{x})=\mathcal{F}_{n}(\mathbf{c})% \mathcal{F}_{n}(\mathbf{x})=\mathcal{F}_{n}(\mathbf{b})
  57. 𝐱 = n - 1 [ ( ( n ( 𝐛 ) ) ν ( n ( 𝐜 ) ) ν ) ν 𝐙 ] T . \ \mathbf{x}=\mathcal{F}_{n}^{-1}\left[\left(\frac{(\mathcal{F}_{n}(\mathbf{b}% ))_{\nu}}{(\mathcal{F}_{n}(\mathbf{c}))_{\nu}}\right)_{\nu\in\mathbf{Z}}\right% ]^{T}.

Circular_orbit.html

  1. a = v 2 r = ω 2 r a\,=\frac{v^{2}}{r}\,={\omega^{2}}{r}
  2. v v\,
  3. r r\,
  4. ω \omega
  5. 𝐚 \mathbf{a}
  6. v v\,
  7. r r\,
  8. ω \omega
  9. v = G ( M + m ) r = μ r v=\sqrt{G(M\!+\!m)\over{r}}=\sqrt{\mu\over{r}}
  10. μ = G ( M + m ) \scriptstyle\mu=G(M\!+\!m)\,
  11. r = h 2 μ r={{h^{2}}\over{\mu}}
  12. h = r v h=rv
  13. μ = r v 2 \mu=rv^{2}
  14. ω 2 r 3 = μ \omega^{2}r^{3}=\mu
  15. T T\,\!
  16. T = 2 π r 3 μ T=2\pi\sqrt{r^{3}\over{\mu}}
  17. T f f = π 2 2 r 3 μ T_{ff}=\frac{\pi}{2\sqrt{2}}\sqrt{r^{3}\over{\mu}}
  18. T p a r = 2 3 r 3 μ T_{par}=\frac{\sqrt{2}}{3}\sqrt{r^{3}\over{\mu}}
  19. ϵ \epsilon\,
  20. v 2 2 = - ϵ {v^{2}\over{2}}=-\epsilon
  21. - μ r = 2 ϵ -{\mu\over{r}}=2\epsilon
  22. R R
  23. v = G M r - r S v=\sqrt{\frac{GM}{r-r_{S}}}
  24. r S = 2 G M c 2 \scriptstyle r_{S}=\frac{2GM}{c^{2}}
  25. c = G = 1 \scriptstyle c=G=1
  26. u μ = ( t ˙ , 0 , 0 , ϕ ˙ ) u^{\mu}=(\dot{t},0,0,\dot{\phi})
  27. r \scriptstyle r
  28. θ = π 2 \scriptstyle\theta=\frac{\pi}{2}
  29. τ \scriptstyle\tau
  30. ( 1 - 2 M r ) t ˙ 2 - r 2 ϕ ˙ 2 = 1 \left(1-\frac{2M}{r}\right)\dot{t}^{2}-r^{2}\dot{\phi}^{2}=1
  31. x ¨ μ + Γ ν σ μ x ˙ ν x ˙ σ = 0 \ddot{x}^{\mu}+\Gamma^{\mu}_{\nu\sigma}\dot{x}^{\nu}\dot{x}^{\sigma}=0
  32. μ = r \scriptstyle\mu=r
  33. M r 2 ( 1 - 2 M r ) t ˙ 2 - r ( 1 - 2 M r ) ϕ ˙ 2 = 0 \frac{M}{r^{2}}\left(1-\frac{2M}{r}\right)\dot{t}^{2}-r\left(1-\frac{2M}{r}% \right)\dot{\phi}^{2}=0
  34. ϕ ˙ 2 = M r 3 t ˙ 2 \dot{\phi}^{2}=\frac{M}{r^{3}}\dot{t}^{2}
  35. ( 1 - 2 M r ) t ˙ 2 - M r t ˙ 2 = 1 \left(1-\frac{2M}{r}\right)\dot{t}^{2}-\frac{M}{r}\dot{t}^{2}=1
  36. t ˙ 2 = r r - 3 M \dot{t}^{2}=\frac{r}{r-3M}
  37. r \scriptstyle r
  38. t \scriptstyle\partial_{t}
  39. v μ = ( r r - 2 M , 0 , 0 , 0 ) v^{\mu}=\left(\sqrt{\frac{r}{r-2M}},0,0,0\right)
  40. γ = g μ ν u μ v ν = ( 1 - 2 M r ) r r - 3 M r r - 2 M = r - 2 M r - 3 M \gamma=g_{\mu\nu}u^{\mu}v^{\nu}=\left(1-\frac{2M}{r}\right)\sqrt{\frac{r}{r-3M% }}\sqrt{\frac{r}{r-2M}}=\sqrt{\frac{r-2M}{r-3M}}
  41. v = M r - 2 M v=\sqrt{\frac{M}{r-2M}}
  42. v = G M r - r S v=\sqrt{\frac{GM}{r-r_{S}}}

Circulation_(fluid_dynamics).html

  1. d Γ = 𝐕 𝐝𝐥 = | 𝐕 | | d 𝐥 | cos θ d\Gamma=\mathbf{V}\cdot\mathbf{dl}=|\mathbf{V}||d\mathbf{l}|\cos\theta
  2. Γ = C 𝐕 d 𝐥 \Gamma=\oint_{C}\mathbf{V}\cdot d\mathbf{l}
  3. l = ρ V Γ l=\rho V\Gamma\!
  4. ω = × 𝐕 \mathbf{\omega}=\nabla\times\mathbf{V}
  5. Γ = S 𝐕 d 𝐥 = S ω d 𝐒 \Gamma=\oint_{\partial S}\mathbf{V}\cdot d\mathbf{l}=\int\!\!\!\int_{S}\mathbf% {\omega}\cdot d\mathbf{S}

Cissoid.html

  1. r = f 1 ( θ ) r=f_{1}(\theta)
  2. r = f 2 ( θ ) r=f_{2}(\theta)
  3. r = f 2 ( θ ) - f 1 ( θ ) r=f_{2}(\theta)-f_{1}(\theta)
  4. r = - f 1 ( θ + π ) , r = - f 1 ( θ - π ) , r = f 1 ( θ + 2 π ) , r = f 1 ( θ - 2 π ) , r=-f_{1}(\theta+\pi),\ r=-f_{1}(\theta-\pi),\ r=f_{1}(\theta+2\pi),\ r=f_{1}(% \theta-2\pi),\ \dots
  5. r = f 2 ( θ ) - f 1 ( θ ) , r = f 2 ( θ ) + f 1 ( θ + π ) , r = f 2 ( θ ) + f 1 ( θ - π ) , r=f_{2}(\theta)-f_{1}(\theta),\ r=f_{2}(\theta)+f_{1}(\theta+\pi),\ r=f_{2}(% \theta)+f_{1}(\theta-\pi),
  6. r = f 2 ( θ ) - f 1 ( θ + 2 π ) , r = f 2 ( θ ) - f 1 ( θ - 2 π ) , r=f_{2}(\theta)-f_{1}(\theta+2\pi),\ r=f_{2}(\theta)-f_{1}(\theta-2\pi),\ \dots
  7. r = 1 2 - cos θ r=\frac{1}{2-\cos\theta}
  8. r = 1 2 - cos θ - 1 2 - cos θ = 0 r=\frac{1}{2-\cos\theta}-\frac{1}{2-\cos\theta}=0
  9. r = - 1 2 + cos θ r=\frac{-1}{2+\cos\theta}
  10. r = 1 2 - cos θ + 1 2 + cos θ r=\frac{1}{2-\cos\theta}+\frac{1}{2+\cos\theta}
  11. x = f 1 ( p ) , y = p x x=f_{1}(p),\ y=px
  12. x = f 2 ( p ) , y = p x x=f_{2}(p),\ y=px
  13. x = f 2 ( p ) - f 1 ( p ) , y = p x x=f_{2}(p)-f_{1}(p),\ y=px
  14. r = a 1 cos ( θ - α 1 ) r=\frac{a_{1}}{\cos(\theta-\alpha_{1})}
  15. r = a 2 cos ( θ - α 2 ) r=\frac{a_{2}}{\cos(\theta-\alpha_{2})}
  16. ( α 1 - α 2 ) / 2 (\alpha_{1}-\alpha 2)/2
  17. α 1 = α , α 2 = - α \alpha_{1}=\alpha,\ \alpha_{2}=-\alpha
  18. r = a 2 cos ( θ + α ) - a 1 cos ( θ - α ) r=\frac{a_{2}}{\cos(\theta+\alpha)}-\frac{a_{1}}{\cos(\theta-\alpha)}
  19. = a 2 cos ( θ - α ) - a 1 cos ( θ + α ) cos ( θ + α ) cos ( θ - α ) =\frac{a_{2}\cos(\theta-\alpha)-a_{1}\cos(\theta+\alpha)}{\cos(\theta+\alpha)% \cos(\theta-\alpha)}
  20. = ( a 2 cos α - a 1 cos α ) cos θ - ( a 2 sin α + a 1 sin α ) sin θ cos 2 α cos 2 θ - sin 2 α sin 2 θ =\frac{(a_{2}\cos\alpha-a_{1}\cos\alpha)\cos\theta-(a_{2}\sin\alpha+a_{1}\sin% \alpha)\sin\theta}{\cos^{2}\alpha\ \cos^{2}\theta-\sin^{2}\alpha\ \sin^{2}\theta}
  21. r = b cos θ + c sin θ cos 2 θ - m 2 sin 2 θ r=\frac{b\cos\theta+c\sin\theta}{\cos^{2}\theta-m^{2}\sin^{2}\theta}
  22. x 2 - m 2 y 2 = b x + c y x^{2}-m^{2}y^{2}=bx+cy
  23. 2 x ( x 2 + y 2 ) = a ( 3 x 2 - y 2 ) 2x(x^{2}+y^{2})=a(3x^{2}-y^{2})
  24. ( x + a ) 2 + y 2 = a 2 (x+a)^{2}+y^{2}=a^{2}
  25. x = - a 2 x={-{a\over 2}}
  26. y 2 ( a + x ) = x 2 ( a - x ) y^{2}(a+x)=x^{2}(a-x)
  27. ( x + a ) 2 + y 2 = a 2 (x+a)^{2}+y^{2}=a^{2}
  28. x = - a x=-a
  29. x ( x 2 + y 2 ) + 2 a y 2 = 0 x(x^{2}+y^{2})+2ay^{2}=0
  30. ( x + a ) 2 + y 2 = a 2 (x+a)^{2}+y^{2}=a^{2}
  31. x = - 2 a x=-2a
  32. ( x + a ) 2 + y 2 = a 2 (x+a)^{2}+y^{2}=a^{2}
  33. x = k a x=ka
  34. x 3 + y 3 = 3 a x y x^{3}+y^{3}=3axy
  35. x 2 - x y + y 2 = - a ( x + y ) x^{2}-xy+y^{2}=-a(x+y)
  36. x + y = - a x+y=-a
  37. x = - a 1 + p , y = p x x=-\frac{a}{1+p},\ y=px
  38. x = - a ( 1 + p ) 1 - p + p 2 , y = p x x=-\frac{a(1+p)}{1-p+p^{2}},\ y=px
  39. x = - a 1 + p + a ( 1 + p ) 1 - p + p 2 = 3 a p 1 + p 3 , y = p x x=-\frac{a}{1+p}+\frac{a(1+p)}{1-p+p^{2}}=\frac{3ap}{1+p^{3}},\ y=px

Cissoid_of_Diocles.html

  1. r = 2 a sec θ r=2a\sec\theta
  2. r = 2 a cos θ r=2a\cos\theta
  3. r = 2 a sec θ - 2 a cos θ = 2 a ( sec θ - cos θ ) r=2a\sec\theta-2a\cos\theta=2a(\sec\theta-\cos\theta)
  4. r = 2 a sin 2 θ / cos θ = 2 a sin θ tan θ r=2a\sin^{2}\theta/\cos\theta=2a\sin\theta\tan\theta
  5. t = tan θ t=\tan\theta
  6. x = r cos θ = 2 a sin 2 θ = 2 a tan 2 θ sec 2 θ = 2 a t 2 1 + t 2 x=r\cos\theta=2a\sin^{2}\theta=\frac{2a\tan^{2}\theta}{\sec^{2}\theta}=\frac{2% at^{2}}{1+t^{2}}
  7. y = t x = 2 a t 3 1 + t 2 y=tx=\frac{2at^{3}}{1+t^{2}}
  8. ( x 2 + y 2 ) x = 2 a y 2 (x^{2}+y^{2})x=2ay^{2}
  9. t ( y - 2 a t ) + x = 0 t(y-2at)+x=0
  10. t ( t x - 2 a t ) + x = 0 , x ( t 2 + 1 ) = 2 a t 2 , x = 2 a t 2 t 2 + 1 t(tx-2at)+x=0,\ x(t^{2}+1)=2at^{2},\ x=\frac{2at^{2}}{t^{2}+1}
  11. y = t x = 2 a t 3 t 2 + 1 y=tx=\frac{2at^{3}}{t^{2}+1}
  12. x = a ( 1 - sin ψ ) , y = a ( 1 - sin ψ ) 2 cos ψ . x=a(1-\sin\psi),\,y=a\frac{(1-\sin\psi)^{2}}{\cos\psi}.
  13. x = a ( 1 - cos ψ ) , y = a ( 1 - cos ψ ) 2 sin ψ x=a(1-\cos\psi),\,y=a\frac{(1-\cos\psi)^{2}}{\sin\psi}
  14. x = 2 a sin 2 ψ 2 , y = a 4 sin 4 ψ 2 2 sin ψ 2 cos ψ 2 = 2 a sin 3 ψ 2 cos ψ 2 . x=2a\sin^{2}{\psi\over 2},\,y=a\frac{4\sin^{4}{\psi\over 2}}{2\sin{\psi\over 2% }\cos{\psi\over 2}}=2a\frac{\sin^{3}{\psi\over 2}}{\cos{\psi\over 2}}.
  15. r = 2 a sin 2 θ / cos θ r=2a\sin^{2}\theta/\cos\theta
  16. u 3 = a 3 ( u a ) 3 = a 3 ( u a ) ( v u ) ( b v ) = a 3 ( b a ) = 2 a 3 u^{3}=a^{3}(\tfrac{u}{a})^{3}=a^{3}(\tfrac{u}{a})(\tfrac{v}{u})(\tfrac{b}{v})=% a^{3}(\tfrac{b}{a})=2a^{3}
  17. ( x 2 + y 2 ) x = 2 a y 2 (x^{2}+y^{2})x=2ay^{2}
  18. y 2 = x 3 2 a - x y^{2}=\frac{x^{3}}{2a-x}
  19. P N 2 = O N 3 N A . PN^{2}=\frac{ON^{3}}{NA}.
  20. P N 3 O N 3 = P N N A . \frac{PN^{3}}{ON^{3}}=\frac{PN}{NA}.
  21. U C 3 O C 3 = B C C A , \frac{UC^{3}}{OC^{3}}=\frac{BC}{CA},
  22. u 3 a 3 = b a , u 3 = a 2 b \frac{u^{3}}{a^{3}}=\frac{b}{a},\,u^{3}=a^{2}b
  23. r = cos θ sin 2 θ r=\frac{\cos\theta}{\sin^{2}\theta}
  24. r = sin 2 θ cos θ = sin θ tan θ r=\frac{\sin^{2}\theta}{\cos\theta}=\sin\theta\tan\theta

Class_function_(algebra).html

  1. g G f ( g ) g \sum_{g\in G}f(g)g
  2. ϕ , ψ = 1 | G | g G ϕ ( g ) ψ ( g - 1 ) \langle\phi,\psi\rangle=\frac{1}{|G|}\sum_{g\in G}\phi(g)\psi(g^{-1})
  3. ϕ , ψ = G ϕ ( t ) ψ ( t - 1 ) d t . \langle\phi,\psi\rangle=\int_{G}\phi(t)\psi(t^{-1})\,dt.

Classical_unified_field_theories.html

  1. T μ ν T_{\mu\nu}

Classical_XY_model.html

  1. n = 2 n=2
  2. D D
  3. Λ Λ
  4. j Λ j∈Λ
  5. 𝐡 j = ( h j , 0 ) \mathbf{h}_{j}=(h_{j},0)
  6. H ( 𝐬 ) = - i j J i j 𝐬 i 𝐬 j - j 𝐡 j 𝐬 j = - i j J i j cos ( θ i - θ j ) - j h j cos θ j H(\mathbf{s})=-\sum_{i\neq j}J_{ij}\;\mathbf{s}_{i}\cdot\mathbf{s}_{j}-\sum_{j% }\mathbf{h}_{j}\cdot\mathbf{s}_{j}=-\sum_{i\neq j}J_{ij}\;\cos(\theta_{i}-% \theta_{j})-\sum_{j}h_{j}\cos\theta_{j}
  7. i j ij
  8. β 0 β≥0
  9. P ( 𝐬 ) = e - β H ( 𝐬 ) Z Z = [ - π , π ] Λ j Λ d θ j e - β H ( 𝐬 ) . P(\mathbf{s})=\frac{e^{-\beta H(\mathbf{s})}}{Z}\qquad Z=\int_{[-\pi,\pi]^{% \Lambda}}\prod_{j\in\Lambda}d\theta_{j}\;e^{-\beta H(\mathbf{s})}.
  10. Z Z
  11. A ( 𝐬 ) \langle A(\mathbf{s})\rangle
  12. A ( 𝐬 ) A(\mathbf{s})
  13. D D
  14. J > 0 J>0
  15. β β
  16. D D
  17. J > 0 J>0
  18. β 2 \frac{β}{2}
  19. 𝐬 i 𝐬 j J , 2 β σ i σ j J , β \langle\mathbf{s}_{i}\cdot\mathbf{s}_{j}\rangle_{J,2\beta}\leq\langle\sigma_{i% }\sigma_{j}\rangle_{J,\beta}
  20. β β
  21. β c X Y 2 β c Is \beta_{c}^{XY}\geq 2\beta_{c}^{\rm Is}
  22. J x , y | x - y | - α , J_{x,y}\sim|x-y|^{-\alpha},
  23. α > 1 α>1
  24. α 2 α≥2
  25. M ( β ) = 0 M(β)=0
  26. M ( β ) := | 𝐬 i | = 0 M(\beta):=|\langle\mathbf{s}_{i}\rangle|=0
  27. | 𝐬 i 𝐬 j | C ( β ) e - c ( β ) | i - j | |\langle\mathbf{s}_{i}\cdot\mathbf{s}_{j}\rangle|\leq C(\beta)e^{-c(\beta)|i-j|}
  28. M ( β ) := | 𝐬 i | > 0 M(\beta):=|\langle\mathbf{s}_{i}\rangle|>0
  29. θ \langle\;\cdot\;\rangle^{\theta}
  30. 𝐬 i θ = M ( β ) ( cos θ , sin θ ) \langle\mathbf{s}_{i}\rangle^{\theta}=M(\beta)(\cos\theta,\sin\theta)

Classification_of_Clifford_algebras.html

  1. v 2 = Q ( v ) v^{2}=Q(v)\,
  2. Q ( u ) = u 1 2 + u 2 2 + + u n 2 Q(u)=u_{1}^{2}+u_{2}^{2}+\cdots+u_{n}^{2}
  3. P ± = 1 2 ( 1 ± ω ) . P_{\pm}=\frac{1}{2}(1\pm\omega).
  4. C n ( 𝐂 ) = C n + ( 𝐂 ) C n - ( 𝐂 ) C\!\ell_{n}(\mathbf{C})=C\!\ell_{n}^{+}(\mathbf{C})\oplus C\!\ell_{n}^{-}(% \mathbf{C})
  5. C n ± ( 𝐂 ) = P ± C n ( 𝐂 ) C\!\ell_{n}^{\pm}(\mathbf{C})=P_{\pm}C\!\ell_{n}(\mathbf{C})
  6. α ( C n ± ( 𝐂 ) ) = C n ( 𝐂 ) \alpha(C\!\ell_{n}^{\pm}(\mathbf{C}))=C\!\ell_{n}^{\mp}(\mathbf{C})
  7. Q ( u ) = u 1 2 + + u p 2 - u p + 1 2 - - u p + q 2 Q(u)=u_{1}^{2}+\cdots+u_{p}^{2}-u_{p+1}^{2}-\cdots-u_{p+q}^{2}
  8. ω = e 1 e 2 e n . \omega=e_{1}e_{2}\cdots e_{n}.
  9. ω 2 = ( e 1 e 2 e n ) ( e 1 e 2 e n ) \omega^{2}=(e_{1}e_{2}\cdots e_{n})(e_{1}e_{2}\cdots e_{n})
  10. sgn ( σ ) e 1 e 2 e n e n e 2 e 1 \mbox{sgn}~{}(\sigma)e_{1}e_{2}\cdots e_{n}e_{n}\cdots e_{2}e_{1}
  11. sgn ( σ ) ( e 1 e 1 e 2 e 2 e n e n ) \mbox{sgn}~{}(\sigma)(e_{1}e_{1}e_{2}e_{2}\cdots e_{n}e_{n})
  12. ( - 1 ) n / 2 = ( - 1 ) n ( n - 1 ) / 2 (-1)^{\lfloor n/2\rfloor}=(-1)^{n(n-1)/2}
  13. e i e i = ± 1 e_{i}e_{i}=\pm 1
  14. ω 2 = ( - 1 ) n ( n - 1 ) / 2 ( - 1 ) q = ( - 1 ) ( p - q ) ( p - q - 1 ) / 2 = { + 1 p - q 0 , 1 mod 4 - 1 p - q 2 , 3 mod 4. \omega^{2}=(-1)^{n(n-1)/2}(-1)^{q}=(-1)^{(p-q)(p-q-1)/2}=\begin{cases}+1&p-q% \equiv 0,1\mod{4}\\ -1&p-q\equiv 2,3\mod{4}.\end{cases}
  15. C p , q ( 𝐑 ) = C p , q + ( 𝐑 ) C p , q - ( 𝐑 ) C\ell_{p,q}(\mathbf{R})=C\ell_{p,q}^{+}(\mathbf{R})\oplus C\ell_{p,q}^{-}(% \mathbf{R})
  16. C p + 1 , q + 1 ( 𝐑 ) = M 2 ( C p , q ( 𝐑 ) ) C\ell_{p+1,q+1}(\mathbf{R})=\mathrm{M}_{2}(C\ell_{p,q}(\mathbf{R}))
  17. C p + 4 , q ( 𝐑 ) = C p , q + 4 ( 𝐑 ) C\ell_{p+4,q}(\mathbf{R})=C\ell_{p,q+4}(\mathbf{R})
  18. C p + 8 , q ( 𝐑 ) = C p + 4 , q + 4 ( 𝐑 ) = M 2 4 ( C p , q ( 𝐑 ) ) . C\ell_{p+8,q}(\mathbf{R})=C\ell_{p+4,q+4}(\mathbf{R})=M_{2^{4}}(C\ell_{p,q}(% \mathbf{R})).
  19. C p + k , q ( 𝐑 ) = C p , q + k ( 𝐑 ) . C\ell_{p+k,q}(\mathbf{R})=C\ell_{p,q+k}(\mathbf{R}).
  20. C p + k , q ( 𝐑 ) = C p , q + k ( 𝐑 ) = C p - k + k , q + k ( 𝐑 ) = M 2 k ( C p - k , q ( 𝐑 ) ) = M 2 k ( C p , q - k ( 𝐑 ) ) . C\ell_{p+k,q}(\mathbf{R})=C\ell_{p,q+k}(\mathbf{R})=C\ell_{p-k+k,q+k}(\mathbf{% R})=\mathrm{M}_{2^{k}}(C\ell_{p-k,q}(\mathbf{R}))=\mathrm{M}_{2^{k}}(C\ell_{p,% q-k}(\mathbf{R})).

Classifying_space.html

  1. π : Y X \pi:Y\longrightarrow X
  2. π : E G B G . \pi:EG\longrightarrow BG.
  3. γ : Y Z \gamma:Y\longrightarrow Z
  4. 𝕋 n \mathbb{T}^{n}
  5. π 1 ( S ) \pi_{1}(S)
  6. \mathbb{RP}^{\infty}
  7. π 1 ( M ) \pi_{1}(M)
  8. \mathbb{CP}^{\infty}

Clausius–Mossotti_relation.html

  1. ϵ - ϵ 0 ϵ + 2 ϵ 0 M d = ϵ r - 1 ϵ r + 2 M d = 4 π N A α 3 = N A α 3 ϵ 0 \frac{\epsilon-\epsilon_{0}}{\epsilon+2\epsilon_{0}}\cdot\frac{M}{d}=\frac{% \epsilon_{\mathrm{r}}-1}{\epsilon_{\mathrm{r}}+2}\cdot\frac{M}{d}=\frac{4\pi N% _{A}\alpha^{\prime}}{3}=\frac{N_{A}\alpha}{3\epsilon_{0}}
  2. ϵ = ϵ r ϵ 0 \epsilon=\epsilon_{r}\epsilon_{0}
  3. ϵ 0 \epsilon_{0}
  4. M M
  5. d d
  6. N A N_{A}
  7. α \alpha
  8. α = α / ( 4 π ε 0 ) \alpha^{\prime}=\alpha/(4\pi\varepsilon_{0})
  9. K ( ω ) = ϵ p * - ϵ m * ϵ p * + 2 ϵ m * K(\omega)=\frac{\epsilon^{*}_{p}-\epsilon^{*}_{m}}{\epsilon^{*}_{p}+2\epsilon^% {*}_{m}}
  10. ϵ * = ϵ + σ i ω = ϵ - i σ ω \epsilon^{*}=\epsilon+\frac{\sigma}{i\omega}=\epsilon-\frac{i\sigma}{\omega}
  11. ϵ \epsilon
  12. σ \sigma
  13. ω \omega
  14. R e ( K ( ω ) ) Re(K(\omega))
  15. I m ( K ( ω ) ) Im(K(\omega))
  16. α \alpha
  17. E e x t E_{ext}
  18. R i R_{i}
  19. E l o c a l = E ( R i ) E_{local}=E(R_{i})
  20. E = E e x t + i E ( p i , r - R i ) E=E_{ext}+\sum_{i}E(p_{i},r-R_{i})
  21. E ( p i , r - R i ) = 3 ( p r ^ ) r ^ - p r 3 E(p_{i},r-R_{i})=\frac{3(p\cdot\hat{r})\hat{r}-p}{r^{3}}
  22. 𝒩 α = 3 n 2 - 1 n 2 + 2 \mathcal{N}\alpha=3\,\frac{n^{2}-1}{n^{2}+2}
  23. 𝒩 \mathcal{N}
  24. α \ \alpha
  25. n \ n
  26. α \alpha
  27. α \alpha
  28. α = ρ / E l o c a l \ \alpha=\ \rho/E_{local}
  29. ρ \ \rho
  30. E l o c a l \ E_{local}
  31. α \alpha
  32. P = j N j ρ j = j N j α j E l o c a l ( j ) P=\sum_{j}N_{j}\ \rho_{j}=\sum_{j}N_{j}\alpha_{j}E_{local}(j)
  33. N j \ N_{j}
  34. α j \ \alpha_{j}
  35. E l o c a l ( j ) \ E_{local}(j)
  36. j \ j

Clebsch–Gordan_coefficients.html

  1. [ j k , j l ] j k j l - j l j k = i m ε k , l , m j m k , l , m { x , y , z } \begin{aligned}&\displaystyle[\mathrm{j}_{k},\mathrm{j}_{l}]\equiv\mathrm{j}_{% k}\mathrm{j}_{l}-\mathrm{j}_{l}\mathrm{j}_{k}=i\hbar\sum_{m}\varepsilon_{k,l,m% }\mathrm{j}_{m}&\displaystyle k,l,m&\displaystyle\in\{\mathrm{x},\mathrm{y},% \mathrm{z}\}\end{aligned}
  2. 𝐣 = ( j x , j y , j z ) \mathbf{j}=(\mathrm{j}_{\mathrm{x}},\mathrm{j}_{\mathrm{y}},\mathrm{j}_{% \mathrm{z}})
  3. 𝐣 \mathbf{j}
  4. 𝐣 2 = j x 2 + j y 2 + j z 2 \mathbf{j}^{2}=\mathrm{j}_{\mathrm{x}}^{2}+\mathrm{j}_{\mathrm{y}}^{2}+\mathrm% {j}_{\mathrm{z}}^{2}
  5. j ± = j x ± j y i \mathrm{j}_{\pm}=\mathrm{j}_{\mathrm{x}}\pm\mathrm{j}_{\mathrm{y}}i
  6. [ 𝐣 2 , j k ] = 0 \displaystyle[\mathbf{j}^{2},\mathrm{j}_{k}]=0
  7. | j m |jm⟩
  8. j j
  9. m m
  10. 𝐣 2 | j m \displaystyle\mathbf{j}^{2}|j\,m\rangle
  11. m m
  12. j ± | j m = C ± ( j , m ) | j ( m ± 1 ) \mathrm{j}_{\pm}|j\,m\rangle=\hbar C_{\pm}(j,m)|j\,(m\pm 1)\rangle
  13. C ± ( j , m ) = j ( j + 1 ) - m ( m ± 1 ) = ( j m ) ( j ± m + 1 ) . C_{\pm}(j,m)=\sqrt{j(j+1)-m(m\pm 1)}=\sqrt{(j\mp m)(j\pm m+1)}.
  14. C ± ( j , m ) C_{\pm}(j,m)
  15. j 1 m 1 | j 2 m 2 = δ j 1 , j 2 δ m 1 , m 2 . \langle j_{1}\,m_{1}|j_{2}\,m_{2}\rangle=\delta_{j_{1},j_{2}}\delta_{m_{1},m_{% 2}}.
  16. j j
  17. j = 1 / 2 j=1/2
  18. j = 1 j=1
  19. | j 1 m 1 m 1 { - j 1 , - j 1 + 1 , , j 1 } \begin{aligned}&\displaystyle|j_{1}\,m_{1}\rangle&\displaystyle m_{1}&% \displaystyle\in\{-j_{1},-j_{1}+1,\ldots,j_{1}\}\end{aligned}
  20. | j 2 m 2 m 2 { - j 2 , - j 2 + 1 , , j 2 } \begin{aligned}&\displaystyle|j_{2}\,m_{2}\rangle&\displaystyle m_{2}&% \displaystyle\in\{-j_{2},-j_{2}+1,\ldots,j_{2}\}\end{aligned}
  21. | j 1 m 1 j 2 m 2 | j 1 m 1 | j 2 m 2 ( m 1 { - j 1 , - j 1 + 1 , , j 1 } ) ( m 2 { - j 2 , - j 2 + 1 , , j 2 } |j_{1}\,m_{1}\,j_{2}\,m_{2}\rangle\equiv|j_{1}\,m_{1}\rangle\otimes|j_{2}\,m_{% 2}\rangle\quad(m_{1}\in\{-j_{1},-j_{1}+1,\ldots,j_{1}\})\quad(m_{2}\in\{-j_{2}% ,-j_{2}+1,\ldots,j_{2}\}
  22. ( 𝐣 1 ) | j 1 m 1 j 2 m 2 𝐣 | j 1 m 1 | j 2 m 2 (\mathbf{j}\otimes 1)|j_{1}\,m_{1}\,j_{2}\,m_{2}\rangle\equiv\mathbf{j}|j_{1}% \,m_{1}\rangle\otimes|j_{2}\,m_{2}\rangle
  23. ( 1 j ) | j 1 m 1 j 2 m 2 | j 1 m 1 𝐣 | j 2 m 2 (1\otimes\mathrm{\mathbf{}}j)|j_{1}\,m_{1}\,j_{2}\,m_{2}\rangle\equiv|j_{1}\,m% _{1}\rangle\otimes\mathbf{j}|j_{2}\,m_{2}\rangle
  24. 1 1
  25. 𝐉 𝐣 1 + 1 𝐣 \mathbf{J}\equiv\mathbf{j}\otimes 1+1\otimes\mathbf{j}
  26. [ J k , J l ] = i ε k , l , m J m [\mathrm{J}_{k},\mathrm{J}_{l}]=i\hbar\varepsilon_{k,l,m}\mathrm{J}_{m}
  27. 𝐉 2 | [ j 1 j 2 ] J M \displaystyle\mathbf{J}^{2}|[j_{1}\,j_{2}]\,J\,M\rangle
  28. J J
  29. | j 1 - j 2 | J j 1 + j 2 |j_{1}-j_{2}|\leq J\leq j_{1}+j_{2}
  30. J = | j 1 - j 2 | j 1 + j 2 ( 2 J + 1 ) = ( 2 j 1 + 1 ) ( 2 j 2 + 1 ) \sum_{J=|j_{1}-j_{2}|}^{j_{1}+j_{2}}(2J+1)=(2j_{1}+1)(2j_{2}+1)
  31. J 1 M 1 | J 2 M 2 = δ J 1 , J 2 δ M 1 , M 2 \langle J_{1}\,M_{1}|J_{2}\,M_{2}\rangle=\delta_{J_{1},J_{2}}\delta_{M_{1},M_{% 2}}
  32. | [ j 1 j 2 ] J M = m 1 = - j 1 j 1 m 2 = - j 2 j 2 | j 1 m 1 j 2 m 2 j 1 m 1 j 2 m 2 | J M |[j_{1}\,j_{2}]\,J\,M\rangle=\sum_{m_{1}=-j_{1}}^{j_{1}}\sum_{m_{2}=-j_{2}}^{j% _{2}}|j_{1}\,m_{1}\,j_{2}\,m_{2}\rangle\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|J\,M\rangle
  33. j 1 m 1 j 2 m 2 | J M \langle j_{1}\,m_{1}\,j_{2}\,m_{2}|J\,M\rangle
  34. J z = j z 1 + 1 j z \mathrm{J}_{\mathrm{z}}=\mathrm{j}_{\mathrm{z}}\otimes 1+1\otimes\mathrm{j}_{% \mathrm{z}}
  35. M = m 1 + m 2 M=m_{1}+m_{2}
  36. J ± = j ± 1 + 1 j ± \mathrm{J}_{\pm}=\mathrm{j}_{\pm}\otimes 1+1\otimes\mathrm{j}_{\pm}
  37. J ± | [ j 1 j 2 ] J M = C ± ( J , M ) | [ j 1 j 2 ] J ( M ± 1 ) = C ± ( J , M ) m 1 , m 2 | j 1 m 1 j 2 m 2 j 1 m 1 j 2 m 2 | J ( M ± 1 ) \begin{aligned}\displaystyle\mathrm{J}_{\pm}|[j_{1}\,j_{2}]\,J\,M\rangle&% \displaystyle=\hbar C_{\pm}(J,M)|[j_{1}\,j_{2}]\,J\,(M\pm 1)\rangle\\ &\displaystyle=\hbar C_{\pm}(J,M)\sum_{m_{1},m_{2}}|j_{1}\,m_{1}\,j_{2}\,m_{2}% \rangle\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|J\,(M\pm 1)\rangle\end{aligned}
  38. J ± m 1 , m 2 | j 1 m 1 j 2 m 2 j 1 m 1 j 2 m 2 | J M = m 1 , m 2 ( C ± ( j 1 , m 1 ) | j 1 ( m 1 ± 1 ) j 2 m 2 + C ± ( j 2 , m 2 ) | j 1 m 1 j 2 ( m 2 ± 1 ) ) j 1 m 1 j 2 m 2 | J M = m 1 , m 2 | j 1 m 1 j 2 m 2 ( C ± ( j 1 , m 1 1 ) j 1 ( m 1 1 ) j 2 m 2 | J M + C ± ( j 2 , m 2 1 ) j 1 m 1 j 2 ( m 2 1 ) | J M ) . \begin{aligned}&\displaystyle\mathrm{J}_{\pm}\sum_{m_{1},m_{2}}|j_{1}\,m_{1}\,% j_{2}\,m_{2}\rangle\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|J\,M\rangle\\ &\displaystyle=\hbar\sum_{m_{1},m_{2}}\Bigl(C_{\pm}(j_{1},m_{1})|j_{1}\,(m_{1}% \pm 1)\,j_{2}\,m_{2}\rangle+C_{\pm}(j_{2},m_{2})|j_{1}\,m_{1}\,j_{2}\,(m_{2}% \pm 1)\rangle\Bigr)\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|J\,M\rangle\\ &\displaystyle=\hbar\sum_{m_{1},m_{2}}|j_{1}\,m_{1}\,j_{2}\,m_{2}\rangle\Bigl(% C_{\pm}(j_{1},m_{1}\mp 1)\langle j_{1}\,(m_{1}\mp 1)\,j_{2}\,m_{2}|J\,M\rangle% +C_{\pm}(j_{2},m_{2}\mp 1)\langle j_{1}\,m_{1}\,j_{2}\,(m_{2}\mp 1)|J\,M% \rangle\Bigr)\,\text{.}\end{aligned}
  39. C ± ( J , M ) j 1 m 1 j 2 m 2 | J ( M ± 1 ) = C ± ( j 1 , m 1 1 ) j 1 ( m 1 1 ) j 2 m 2 | J M + C ± ( j 2 , m 2 1 ) j 1 m 1 j 2 ( m 2 1 ) | J M C_{\pm}(J,M)\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|J\,(M\pm 1)\rangle=C_{\pm}(j_{1% },m_{1}\mp 1)\langle j_{1}\,(m_{1}\mp 1)\,j_{2}\,m_{2}|J\,M\rangle+C_{\pm}(j_{% 2},m_{2}\mp 1)\langle j_{1}\,m_{1}\,j_{2}\,(m_{2}\mp 1)|J\,M\rangle
  40. M = J M=J
  41. 0 = C + ( j 1 , m 1 - 1 ) j 1 ( m 1 - 1 ) j 2 m 2 | J J + C + ( j 2 , m 2 - 1 ) j 1 m 1 j 2 ( m 2 - 1 ) | J J 0=C_{+}(j_{1},m_{1}-1)\langle j_{1}\,(m_{1}-1)\,j_{2}\,m_{2}|J\,J\rangle+C_{+}% (j_{2},m_{2}-1)\langle j_{1}\,m_{1}\,j_{2}\,(m_{2}-1)|J\,J\rangle
  42. j 1 j 1 j 2 ( J - j 1 ) | J J > 0 \langle j_{1}\,j_{1}\,j_{2}\,(J-j_{1})|J\,J\rangle>0
  43. M = J 1 M=J−1
  44. J M | j 1 m 1 j 2 m 2 j 1 m 1 j 2 m 2 | J M \langle J\,M|j_{1}\,m_{1}\,j_{2}\,m_{2}\rangle\equiv\langle j_{1}\,m_{1}\,j_{2% }\,m_{2}|J\,M\rangle
  45. J = | j 1 - j 2 | j 1 + j 2 M = - J J j 1 m 1 j 2 m 2 | J M J M | j 1 m 1 j 2 m 2 = j 1 m 1 j 2 m 2 | j 1 m 1 j 2 m 2 = δ m 1 , m 1 δ m 2 , m 2 \sum_{J=|j_{1}-j_{2}|}^{j_{1}+j_{2}}\sum_{M=-J}^{J}\langle j_{1}\,m_{1}\,j_{2}% \,m_{2}|J\,M\rangle\langle J\,M|j_{1}\,m_{1}^{\prime}\,j_{2}\,m_{2}^{\prime}% \rangle=\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|j_{1}\,m_{1}^{\prime}\,j_{2}\,m_{2}% ^{\prime}\rangle=\delta_{m_{1},m_{1}^{\prime}}\delta_{m_{2},m_{2}^{\prime}}
  46. m 1 , m 2 J M | j 1 m 1 j 2 m 2 j 1 m 1 j 2 m 2 | J M = J M | J M = δ J , J δ M , M \sum_{m_{1},m_{2}}\langle J\,M|j_{1}\,m_{1}\,j_{2}\,m_{2}\rangle\langle j_{1}% \,m_{1}\,j_{2}\,m_{2}|J^{\prime}\,M^{\prime}\rangle=\langle J\,M|J^{\prime}\,M% ^{\prime}\rangle=\delta_{J,J^{\prime}}\delta_{M,M^{\prime}}
  47. J = 0 J=0
  48. j 1 m 1 j 2 m 2 | 0 0 = δ j 1 , j 2 δ m 1 , - m 2 ( - 1 ) j 1 - m 1 2 j 2 + 1 \langle j_{1}\,m_{1}\,j_{2}\,m_{2}|0\,0\rangle=\delta_{j_{1},j_{2}}\delta_{m_{% 1},-m_{2}}\frac{(-1)^{j_{1}-m_{1}}}{\sqrt{2j_{2}+1}}
  49. M = J M=J
  50. j 1 j 1 j 2 j 2 | ( j 1 + j 2 ) ( j 1 + j 2 ) = 1 \langle j_{1}\,j_{1}\,j_{2}\,j_{2}|(j_{1}+j_{2})\,(j_{1}+j_{2})\rangle=1
  51. j 1 m 1 j 1 ( - m 1 ) | ( 2 j 1 ) 0 = ( 2 j 1 ) ! 2 ( j 1 - m 1 ) ! ( j 1 + m 1 ) ! ( 4 j 1 ) ! \langle j_{1}\,m_{1}\,j_{1}\,(-m_{1})|(2j_{1})\,0\rangle=\frac{(2j_{1})!^{2}}{% (j_{1}-m_{1})!(j_{1}+m_{1})!\sqrt{(4j_{1})!}}
  52. j 1 j 1 j 1 ( - j 1 ) | J 0 = ( 2 j 1 ) ! 2 J + 1 ( J + 2 j 1 + 1 ) ! ( 2 j 1 - J ) ! . \langle j_{1}\,j_{1}\,j_{1}\,(-j_{1})|J\,0\rangle=(2j_{1})!\sqrt{\frac{2J+1}{(% J+2j_{1}+1)!(2j_{1}-J)!}}.
  53. j 1 m 1 0 | ( j 1 + 1 ) m = ( j 1 - m + 1 ) ( j 1 + m + 1 ) ( 2 j 1 + 1 ) ( j 1 + 1 ) j 1 m 1 0 | j 1 m = m j 1 ( j 1 + 1 ) j 1 m 1 0 | ( j 1 - 1 ) m = - ( j 1 - m ) ( j 1 + m ) j 1 ( 2 j 1 + 1 ) \begin{aligned}\displaystyle\langle j_{1}\,m\,1\,0|(j_{1}+1)\,m\rangle&% \displaystyle=\sqrt{\frac{(j_{1}-m+1)(j_{1}+m+1)}{(2j_{1}+1)(j_{1}+1)}}\\ \displaystyle\langle j_{1}\,m\,1\,0|j_{1}\,m\rangle&\displaystyle=\frac{m}{% \sqrt{j_{1}(j_{1}+1)}}\\ \displaystyle\langle j_{1}\,m\,1\,0|(j_{1}-1)\,m\rangle&\displaystyle=-\sqrt{% \frac{(j_{1}-m)(j_{1}+m)}{j_{1}(2j_{1}+1)}}\end{aligned}
  54. j 1 m 1 j 2 m 2 | J M = ( - 1 ) j 1 + j 2 - J j 1 ( - m 1 ) j 2 ( - m 2 ) | J ( - M ) = ( - 1 ) j 1 + j 2 - J j 2 m 2 j 1 m 1 | J M = ( - 1 ) j 1 - m 1 2 J + 1 2 j 2 + 1 j 1 m 1 J ( - M ) | j 2 ( - m 2 ) = ( - 1 ) j 2 + m 2 2 J + 1 2 j 1 + 1 J ( - M ) j 2 m 2 | j 1 ( - m 1 ) = ( - 1 ) j 1 - m 1 2 J + 1 2 j 2 + 1 J M j 1 ( - m 1 ) | j 2 m 2 = ( - 1 ) j 2 + m 2 2 J + 1 2 j 1 + 1 j 2 ( - m 2 ) J M | j 1 m 1 \begin{aligned}\displaystyle\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|J\,M\rangle&% \displaystyle=(-1)^{j_{1}+j_{2}-J}\langle j_{1}\,(-m_{1})\,j_{2}\,(-m_{2})|J\,% (-M)\rangle\\ &\displaystyle=(-1)^{j_{1}+j_{2}-J}\langle j_{2}\,m_{2}\,j_{1}\,m_{1}|J\,M% \rangle\\ &\displaystyle=(-1)^{j_{1}-m_{1}}\sqrt{\frac{2J+1}{2j_{2}+1}}\langle j_{1}\,m_% {1}\,J\,(-M)|j_{2}\,(-m_{2})\rangle\\ &\displaystyle=(-1)^{j_{2}+m_{2}}\sqrt{\frac{2J+1}{2j_{1}+1}}\langle J\,(-M)\,% j_{2}\,m_{2}|j_{1}\,(-m_{1})\rangle\\ &\displaystyle=(-1)^{j_{1}-m_{1}}\sqrt{\frac{2J+1}{2j_{2}+1}}\langle J\,M\,j_{% 1}\,(-m_{1})|j_{2}\,m_{2}\rangle\\ &\displaystyle=(-1)^{j_{2}+m_{2}}\sqrt{\frac{2J+1}{2j_{1}+1}}\langle j_{2}\,(-% m_{2})\,J\,M|j_{1}\,m_{1}\rangle\end{aligned}
  55. j j
  56. j j
  57. ( - 1 ) 4 j = ( - 1 ) 2 ( j - m ) = 1 (-1)^{4j}=(-1)^{2(j-m)}=1
  58. J J
  59. ( - 1 ) 2 ( j 1 + j 2 + J ) = ( - 1 ) 2 ( m 1 + m 2 + M ) = 1 (-1)^{2(j_{1}+j_{2}+J)}=(-1)^{2(m_{1}+m_{2}+M)}=1
  60. j 1 m 1 j 2 m 2 | J M = ( - 1 ) j 1 - j 2 + M 2 J + 1 ( j 1 j 2 J m 1 m 2 - M ) \langle j_{1}\,m_{1}\,j_{2}\,m_{2}|J\,M\rangle=(-1)^{j_{1}-j_{2}+M}\sqrt{2J+1}% \begin{pmatrix}j_{1}&j_{2}&J\\ m_{1}&m_{2}&-M\end{pmatrix}
  61. 0 2 π d α 0 π sin β d β 0 2 π d γ D M , K J ( α , β , γ ) * D m 1 , k 1 j 1 ( α , β , γ ) D m 2 , k 2 j 2 ( α , β , γ ) = 8 π 2 2 J + 1 j 1 m 1 j 2 m 2 | J M j 1 k 1 j 2 k 2 | J K \begin{aligned}&\displaystyle\int_{0}^{2\pi}d\alpha\int_{0}^{\pi}\sin\beta\,d% \beta\int_{0}^{2\pi}d\gamma\,D^{J}_{M,K}(\alpha,\beta,\gamma)^{*}D^{j_{1}}_{m_% {1},k_{1}}(\alpha,\beta,\gamma)D^{j_{2}}_{m_{2},k_{2}}(\alpha,\beta,\gamma)\\ &\displaystyle=\frac{8\pi^{2}}{2J+1}\langle j_{1}\,m_{1}\,j_{2}\,m_{2}|J\,M% \rangle\langle j_{1}\,k_{1}\,j_{2}\,k_{2}|J\,K\rangle\end{aligned}
  62. 4 π Y 1 m 1 ( Ω ) * Y 2 m 2 ( Ω ) * Y L M ( Ω ) d Ω = ( 2 1 + 1 ) ( 2 2 + 1 ) 4 π ( 2 L + 1 ) 1 0 2 0 | L 0 1 m 1 2 m 2 | L M \int_{4\pi}Y_{\ell_{1}}^{m_{1}}{}^{*}(\Omega)Y_{\ell_{2}}^{m_{2}}{}^{*}(\Omega% )Y_{L}^{M}(\Omega)\,d\Omega=\sqrt{\frac{(2\ell_{1}+1)(2\ell_{2}+1)}{4\pi(2L+1)% }}\langle\ell_{1}\,0\,\ell_{2}\,0|L\,0\rangle\langle\ell_{1}\,m_{1}\,\ell_{2}% \,m_{2}|L\,M\rangle
  63. Y 1 m 1 ( Ω ) Y 2 m 2 ( Ω ) = L , M ( 2 1 + 1 ) ( 2 2 + 1 ) 4 π ( 2 L + 1 ) 1 0 2 0 | L 0 1 m 1 2 m 2 | L M Y L M ( Ω ) Y_{\ell_{1}}^{m_{1}}(\Omega)Y_{\ell_{2}}^{m_{2}}(\Omega)=\sum_{L,M}\sqrt{\frac% {(2\ell_{1}+1)(2\ell_{2}+1)}{4\pi(2L+1)}}\langle\ell_{1}\,0\,\ell_{2}\,0|L\,0% \rangle\langle\ell_{1}\,m_{1}\,\ell_{2}\,m_{2}|L\,M\rangle Y_{L}^{M}(\Omega)
  64. m ( - 1 ) j - m j m j ( - m ) | J 0 = δ J , 0 2 j + 1 \sum_{m}(-1)^{j-m}\langle j\,m\,j\,(-m)|J\,0\rangle=\delta_{J,0}\sqrt{2j+1}
  65. 𝐣 < s u b > 1 \mathbf{j}<sub>1

Clenshaw_algorithm.html

  1. ϕ k ( x ) \phi_{k}(x)
  2. S ( x ) = k = 0 n a k ϕ k ( x ) S(x)=\sum_{k=0}^{n}a_{k}\phi_{k}(x)
  3. ϕ k , k = 0 , 1 , \phi_{k},\;k=0,1,\ldots
  4. ϕ k + 1 ( x ) = α k ( x ) ϕ k ( x ) + β k ( x ) ϕ k - 1 ( x ) , \phi_{k+1}(x)=\alpha_{k}(x)\,\phi_{k}(x)+\beta_{k}(x)\,\phi_{k-1}(x),
  5. α k ( x ) \alpha_{k}(x)
  6. β k ( x ) \beta_{k}(x)
  7. ϕ k ( x ) \phi_{k}(x)
  8. α k ( x ) \alpha_{k}(x)
  9. β k ( x ) \beta_{k}(x)
  10. α ( x ) \alpha(x)
  11. k k
  12. β \beta
  13. x x
  14. k k
  15. a 0 , , a n a_{0},\ldots,a_{n}
  16. b k ( x ) b_{k}(x)
  17. b n + 1 ( x ) \displaystyle b_{n+1}(x)
  18. ϕ k ( x ) \phi_{k}(x)
  19. b 2 ( x ) b_{2}(x)
  20. b 1 ( x ) b_{1}(x)
  21. ϕ 0 ( x ) \phi_{0}(x)
  22. ϕ 1 ( x ) \phi_{1}(x)
  23. S ( x ) = ϕ 0 ( x ) a 0 + ϕ 1 ( x ) b 1 ( x ) + β 1 ( x ) ϕ 0 ( x ) b 2 ( x ) . S(x)=\phi_{0}(x)\,a_{0}+\phi_{1}(x)\,b_{1}(x)+\beta_{1}(x)\,\phi_{0}(x)\,b_{2}% (x).
  24. S ( x ) = k = 0 n a k x k S(x)=\sum_{k=0}^{n}a_{k}x^{k}
  25. ϕ 0 ( x ) = 1 , ϕ k ( x ) = x k = x ϕ k - 1 ( x ) \begin{aligned}\displaystyle\phi_{0}(x)&\displaystyle=1,\\ \displaystyle\phi_{k}(x)&\displaystyle=x^{k}=x\phi_{k-1}(x)\end{aligned}
  26. α ( x ) = x \alpha(x)=x
  27. β = 0 \beta=0
  28. b k ( x ) = a k + x b k + 1 ( x ) b_{k}(x)=a_{k}+xb_{k+1}(x)
  29. S ( x ) = a 0 + x b 1 ( x ) = b 0 ( x ) S(x)=a_{0}+xb_{1}(x)=b_{0}(x)
  30. p n ( x ) = a 0 + a 1 T 1 ( x ) + a 2 T 2 ( x ) + + a n T n ( x ) . p_{n}(x)=a_{0}+a_{1}T_{1}(x)+a_{2}T_{2}(x)+\cdots+a_{n}T_{n}(x).
  31. α ( x ) = 2 x , β = - 1 , \alpha(x)=2x,\quad\beta=-1,
  32. T 0 ( x ) = 1 , T 1 ( x ) = x . T_{0}(x)=1,\quad T_{1}(x)=x.
  33. b k ( x ) = a k + 2 x b k + 1 ( x ) - b k + 2 ( x ) b_{k}(x)=a_{k}+2xb_{k+1}(x)-b_{k+2}(x)
  34. p n ( x ) = a 0 + x b 1 ( x ) - b 2 ( x ) . p_{n}(x)=a_{0}+xb_{1}(x)-b_{2}(x).
  35. b 0 ( x ) = 2 a 0 + 2 x b 1 ( x ) - b 2 ( x ) , b_{0}(x)=2a_{0}+2xb_{1}(x)-b_{2}(x),
  36. p n ( x ) = 1 2 [ b 0 ( x ) - b 2 ( x ) ] . p_{n}(x)=\frac{1}{2}\left[b_{0}(x)-b_{2}(x)\right].
  37. m ( θ ) = C 0 θ + C 1 sin θ + C 2 sin 2 θ + + C n sin n θ . m(\theta)=C_{0}\,\theta+C_{1}\sin\theta+C_{2}\sin 2\theta+\cdots+C_{n}\sin n\theta.
  38. C 0 θ C_{0}\,\theta
  39. ϕ 0 ( θ ) = sin 0 θ = sin 0 = 0 \phi_{0}(\theta)=\sin 0\theta=\sin 0=0
  40. sin k θ \sin k\theta
  41. sin ( k + 1 ) θ = 2 cos θ sin k θ - sin ( k - 1 ) θ \sin(k+1)\theta=2\cos\theta\sin k\theta-\sin(k-1)\theta
  42. α k ( θ ) = 2 cos θ , β k = - 1. \alpha_{k}(\theta)=2\cos\theta,\quad\beta_{k}=-1.
  43. b n + 1 ( θ ) = b n + 2 ( θ ) = 0 , b k ( θ ) = C k + 2 cos θ b k + 1 ( θ ) - b k + 2 ( θ ) , for n k 1. \begin{aligned}\displaystyle b_{n+1}(\theta)&\displaystyle=b_{n+2}(\theta)=0,% \\ \displaystyle b_{k}(\theta)&\displaystyle=C_{k}+2\cos\theta\,b_{k+1}(\theta)-b% _{k+2}(\theta),\quad\mathrm{for\ }n\geq k\geq 1.\end{aligned}
  44. ϕ 0 ( θ ) = sin 0 = 0 \phi_{0}(\theta)=\sin 0=0
  45. b 1 ( θ ) sin ( θ ) b_{1}(\theta)\sin(\theta)
  46. C 0 θ C_{0}\,\theta
  47. m ( θ ) = C 0 θ + b 1 ( θ ) sin θ . m(\theta)=C_{0}\,\theta+b_{1}(\theta)\sin\theta.
  48. cos θ \cos\theta
  49. sin θ \sin\theta
  50. m ( θ 1 ) - m ( θ 2 ) = k = 1 n 2 C k sin ( 1 2 k ( θ 1 - θ 2 ) ) cos ( 1 2 k ( θ 1 + θ 2 ) ) . m(\theta_{1})-m(\theta_{2})=\sum_{k=1}^{n}2C_{k}\sin\bigl({\textstyle\frac{1}{% 2}}k(\theta_{1}-\theta_{2})\bigr)\cos\bigl({\textstyle\frac{1}{2}}k(\theta_{1}% +\theta_{2})\bigr).
  51. m ( θ 1 ) + m ( θ 2 ) m(\theta_{1})+m(\theta_{2})
  52. 𝖬 ( θ 1 , θ 2 ) = [ ( m ( θ 1 ) + m ( θ 2 ) ) / 2 ( m ( θ 1 ) - m ( θ 2 ) ) / ( θ 1 - θ 2 ) ] = C 0 [ μ 1 ] + k = 1 n C k 𝖥 k ( θ 1 , θ 2 ) , \mathsf{M}(\theta_{1},\theta_{2})=\begin{bmatrix}(m(\theta_{1})+m(\theta_{2}))% /2\\ (m(\theta_{1})-m(\theta_{2}))/(\theta_{1}-\theta_{2})\end{bmatrix}=C_{0}\begin% {bmatrix}\mu\\ 1\end{bmatrix}+\sum_{k=1}^{n}C_{k}\mathsf{F}_{k}(\theta_{1},\theta_{2}),
  53. δ = 1 2 ( θ 1 - θ 2 ) , μ = 1 2 ( θ 1 + θ 2 ) , 𝖥 k ( θ 1 , θ 2 ) = [ cos k δ sin k μ sin k δ δ cos k μ ] . \begin{aligned}\displaystyle\delta&\displaystyle={\textstyle\frac{1}{2}}(% \theta_{1}-\theta_{2}),\\ \displaystyle\mu&\displaystyle={\textstyle\frac{1}{2}}(\theta_{1}+\theta_{2}),% \\ \displaystyle\mathsf{F}_{k}(\theta_{1},\theta_{2})&\displaystyle=\begin{% bmatrix}\cos k\delta\sin k\mu\\ \displaystyle\frac{\sin k\delta}{\delta}\cos k\mu\end{bmatrix}.\end{aligned}
  54. 𝖬 ( θ 1 , θ 2 ) \mathsf{M}(\theta_{1},\theta_{2})
  55. m m
  56. 𝖥 k ( θ 1 , θ 2 ) \mathsf{F}_{k}(\theta_{1},\theta_{2})
  57. 𝖥 k + 1 ( θ 1 , θ 2 ) = 𝖠 ( θ 1 , θ 2 ) 𝖥 k ( θ 1 , θ 2 ) - 𝖥 k - 1 ( θ 1 , θ 2 ) , \mathsf{F}_{k+1}(\theta_{1},\theta_{2})=\mathsf{A}(\theta_{1},\theta_{2})% \mathsf{F}_{k}(\theta_{1},\theta_{2})-\mathsf{F}_{k-1}(\theta_{1},\theta_{2}),
  58. 𝖠 ( θ 1 , θ 2 ) = 2 [ cos δ cos μ - δ sin δ sin μ - sin δ δ sin μ cos δ cos μ ] \mathsf{A}(\theta_{1},\theta_{2})=2\begin{bmatrix}\cos\delta\cos\mu&-\delta% \sin\delta\sin\mu\\ -\displaystyle\frac{\sin\delta}{\delta}\sin\mu&\cos\delta\cos\mu\end{bmatrix}
  59. α \alpha
  60. β = - 1 \beta=-1
  61. 𝖡 n + 1 = 𝖡 n + 2 = 𝟢 , 𝖡 k = C k 𝖨 + 𝖠𝖡 k + 1 - 𝖡 k + 2 , for n k 1 , 𝖬 ( θ 1 , θ 2 ) = C 0 [ μ 1 ] + 𝖡 1 𝖥 1 ( θ 1 , θ 2 ) , \begin{aligned}\displaystyle\mathsf{B}_{n+1}&\displaystyle=\mathsf{B}_{n+2}=% \mathsf{0},\\ \displaystyle\mathsf{B}_{k}&\displaystyle=C_{k}\mathsf{I}+\mathsf{A}\mathsf{B}% _{k+1}-\mathsf{B}_{k+2},\qquad\mathrm{for\ }n\geq k\geq 1,\\ \displaystyle\mathsf{M}(\theta_{1},\theta_{2})&\displaystyle=C_{0}\begin{% bmatrix}\mu\\ 1\end{bmatrix}+\mathsf{B}_{1}\mathsf{F}_{1}(\theta_{1},\theta_{2}),\end{aligned}
  62. 𝖡 k \mathsf{B}_{k}
  63. m ( θ 1 ) - m ( θ 2 ) θ 1 - θ 2 = 𝖬 2 ( θ 1 , θ 2 ) . \frac{m(\theta_{1})-m(\theta_{2})}{\theta_{1}-\theta_{2}}=\mathsf{M}_{2}(% \theta_{1},\theta_{2}).
  64. θ 2 = θ 1 = μ \theta_{2}=\theta_{1}=\mu
  65. δ = 0 \delta=0\,
  66. d m ( μ ) / d μ dm(\mu)/d\mu
  67. 𝖥 1 \mathsf{F}_{1}
  68. 𝖠 \mathsf{A}
  69. lim δ 0 ( sin δ ) / δ = 1 \lim_{\delta\rightarrow 0}(\sin\delta)/\delta=1
  70. T n * ( x ) = T n ( 2 x - 1 ) T^{*}_{n}(x)=T_{n}(2x-1)

Clifford_module.html

  1. A B = 1 2 ( A B + B A ) = 0. A\cdot B=\frac{1}{2}(AB+BA)=0.
  2. p , q \mathbb{R}_{p,q}\,
  3. γ a 2 = + 1 if 1 a p γ a 2 = - 1 if p + 1 a p + q γ a γ b = - γ b γ a if a b . \begin{matrix}\gamma_{a}^{2}&=&+1&\mbox{if}&1\leq a\leq p\\ \gamma_{a}^{2}&=&-1&\mbox{if}&p+1\leq a\leq p+q\\ \gamma_{a}\gamma_{b}&=&-\gamma_{b}\gamma_{a}&\mbox{if}&a\neq b.\\ \end{matrix}
  4. γ a = S γ a S - 1 \begin{matrix}\gamma_{a^{\prime}}&=&S&\gamma_{a}&S^{-1}\end{matrix}

Clinton_Davisson.html

  1. λ \lambda
  2. λ = h / p \lambda=h/p
  3. h h
  4. p p

Close-packing_of_equal_spheres.html

  1. π 3 2 0.74048. \frac{\pi}{3\sqrt{2}}\simeq 0.74048.
  2. 3 2 \scriptstyle\sqrt{\frac{3}{2}}
  3. 2 \scriptstyle\sqrt{2}
  4. pitch Z = 6 d 3 0.81649658 d , \,\text{pitch}_{Z}=\sqrt{6}\cdot{d\over 3}\approx 0.81649658d,
  5. 3 r \scriptstyle\sqrt{3}r
  6. ( r , r + 3 r , r ) , ( 3 r , r + 3 r , r ) , ( 5 r , r + 3 r , r ) , ( 7 r , r + 3 r , r ) , . \left(r,r+\sqrt{3}r,r\right),\ \left(3r,r+\sqrt{3}r,r\right),\ \left(5r,r+% \sqrt{3}r,r\right),\ \left(7r,r+\sqrt{3}r,r\right),\dots.
  7. 3 \scriptstyle\sqrt{3}
  8. 6 r 2 / 3 \scriptstyle\sqrt{6}r2/3
  9. ( r , r + 3 r 3 , r + 6 r 2 3 ) , ( 3 r , r + 3 r 3 , r + 6 r 2 3 ) , ( 5 r , r + 3 r 3 , r + 6 r 2 3 ) , ( 7 r , r + 3 r 3 , r + 6 r 2 3 ) , . \left(r,r+\frac{\sqrt{3}r}{3},r+\frac{\sqrt{6}r2}{3}\right),\ \left(3r,r+\frac% {\sqrt{3}r}{3},r+\frac{\sqrt{6}r2}{3}\right),\ \left(5r,r+\frac{\sqrt{3}r}{3},% r+\frac{\sqrt{6}r2}{3}\right),\ \left(7r,r+\frac{\sqrt{3}r}{3},r+\frac{\sqrt{6% }r2}{3}\right),\dots.
  10. ( 2 r , r + 4 3 r 3 , r + 6 r 2 3 ) , ( 4 r , r + 4 3 r 3 , r + 6 r 2 3 ) , ( 6 r , r + 4 3 r 3 , r + 6 r 2 3 ) , ( 8 r , r + 4 3 r 3 , r + 6 r 2 3 ) , . \left(2r,r+\frac{4\sqrt{3}r}{3},r+\frac{\sqrt{6}r2}{3}\right),\ \left(4r,r+% \frac{4\sqrt{3}r}{3},r+\frac{\sqrt{6}r2}{3}\right),\ \left(6r,r+\frac{4\sqrt{3% }r}{3},r+\frac{\sqrt{6}r2}{3}\right),\ \left(8r,r+\frac{4\sqrt{3}r}{3},r+\frac% {\sqrt{6}r2}{3}\right),\dots.
  11. 6 r 2 / 3 \scriptstyle\sqrt{6}r2/3
  12. [ 2 i + ( ( j + k ) mod 2 ) 3 [ j + 1 3 ( k mod 2 ) ] 2 6 3 k ] r \begin{bmatrix}2i+((j\ +\ k)\ \bmod{2})\\ \sqrt{3}\left[j+\frac{1}{3}(k\ \bmod{2})\right]\\ \frac{2\sqrt{6}}{3}k\\ \end{bmatrix}r
  13. i i
  14. j j
  15. k k
  16. 0
  17. x x
  18. y y
  19. z z

Closed_convex_function.html

  1. f : n f:\mathbb{R}^{n}\rightarrow\mathbb{R}
  2. α \alpha\in\mathbb{R}
  3. { x dom f | f ( x ) α } \{x\in\mbox{dom}~{}f|f(x)\leq\alpha\}
  4. epi f = { ( x , t ) n + 1 | x dom f , f ( x ) t } \mbox{epi}~{}f=\{(x,t)\in\mathbb{R}^{n+1}|x\in\mbox{dom}~{}f,\;f(x)\leq t\}
  5. f ( x ) f(x)
  6. f : n f:\mathbb{R}^{n}\rightarrow\mathbb{R}
  7. dom f \mbox{dom}~{}f
  8. f f

Closed_manifold.html

  1. 𝐑 n \mathbf{R}^{n}

Closing_(morphology).html

  1. A B = ( A B ) B , A\bullet B=(A\oplus B)\ominus B,\,
  2. \oplus
  3. \ominus
  4. ( A B ) B = A B (A\bullet B)\bullet B=A\bullet B
  5. A C A\subseteq C
  6. A B C B A\bullet B\subseteq C\bullet B
  7. A A B A\subseteq A\bullet B

Clubsuit.html

  1. κ \kappa
  2. S κ S\subseteq\kappa
  3. S \clubsuit_{S}
  4. A δ : δ S \left\langle A_{\delta}:\delta\in S\right\rangle
  5. A κ A\subseteq\kappa
  6. δ \delta
  7. A δ A A_{\delta}\subseteq A
  8. ω 1 \clubsuit_{\omega_{1}}
  9. \clubsuit

Cluster_analysis.html

  1. 𝒪 ( n 3 ) \mathcal{O}(n^{3})
  2. 𝒪 ( 2 n - 1 ) \mathcal{O}(2^{n-1})
  3. 𝒪 ( n 2 ) \mathcal{O}(n^{2})
  4. k k
  5. k k
  6. ε \varepsilon
  7. ε \varepsilon
  8. D B = 1 n i = 1 n max j i ( σ i + σ j d ( c i , c j ) ) DB=\frac{1}{n}\sum_{i=1}^{n}\max_{j\neq i}\left(\frac{\sigma_{i}+\sigma_{j}}{d% (c_{i},c_{j})}\right)
  9. c x c_{x}
  10. x x
  11. σ x \sigma_{x}
  12. x x
  13. c x c_{x}
  14. d ( c i , c j ) d(c_{i},c_{j})
  15. c i c_{i}
  16. c j c_{j}
  17. D = min 1 i < j n d ( i , j ) max 1 k n d ( k ) , D=\frac{\min_{1\leq i<j\leq n}d(i,j)}{\max_{1\leq k\leq n}d^{\prime}(k)}\,,
  18. R I = T P + T N T P + F P + F N + T N RI=\frac{TP+TN}{TP+FP+FN+TN}
  19. T P TP
  20. T N TN
  21. F P FP
  22. F N FN
  23. β 0 \beta\geq 0
  24. P = T P T P + F P P=\frac{TP}{TP+FP}
  25. R = T P T P + F N R=\frac{TP}{TP+FN}
  26. P P
  27. R R
  28. F β = ( β 2 + 1 ) P R β 2 P + R F_{\beta}=\frac{(\beta^{2}+1)\cdot P\cdot R}{\beta^{2}\cdot P+R}
  29. β = 0 \beta=0
  30. F 0 = P F_{0}=P
  31. β = 0 \beta=0
  32. β \beta
  33. J ( A , B ) = | A B | | A B | = T P T P + F P + F N J(A,B)=\frac{|A\cap B|}{|A\cup B|}=\frac{TP}{TP+FP+FN}
  34. F M = T P T P + F P T P T P + F N FM=\sqrt{\frac{TP}{TP+FP}\cdot\frac{TP}{TP+FN}}
  35. T P TP
  36. F P FP
  37. F N FN
  38. F M FM
  39. P P
  40. R R
  41. B I B^{I}
  42. B I I B^{II}

Cluster_decomposition_theorem.html

  1. U ( a ) U(a)
  2. lim | a | Ω | A 1 A n | Ω - Ω | unshifted i A i | Ω Ω | shifted i U ( a ) A i U - 1 ( a ) | Ω = 0 \lim_{|a|\rightarrow\infty}\langle\Omega|A^{\prime}_{1}\cdots A^{\prime}_{n}|% \Omega\rangle-\langle\Omega|\prod_{\mbox{unshifted i}~{}}A_{i}|\Omega\rangle% \langle\Omega|\prod_{\mbox{shifted i}~{}}U(a)A_{i}U^{-1}(a)|\Omega\rangle=0
  3. | Ω |\Omega\rangle
  4. A i = { A i if it is one of the unshifted operators U ( a ) A i U - 1 ( a ) if it is one of the shifted operators A^{\prime}_{i}=\left\{\begin{matrix}A_{i}&\mbox{ if it is one of the unshifted% operators}\\ U(a)A_{i}U^{-1}(a)&\mbox{ if it is one of the shifted operators}\end{matrix}\right.
  5. m > 0 m>0
  6. C e - m | a | Ce^{-m|a|}
  7. C C
  8. | a | |a|
  9. a a
  10. | a | > a 0 |a|>a_{0}

Code_93.html

  1. ( 8 3 ) {\textstyle\left({{8}\atop{3}}\right)}

Code_page_437.html

  1. \varnothing
  2. ϕ \phi\,\!
  3. \varnothing

Coefficient_of_variation.html

  1. σ \ \sigma
  2. μ \ \mu
  3. | μ | |\mu|
  4. σ \ \sigma
  5. μ \ \mu
  6. c v = σ μ c_{v}=\frac{\sigma}{\mu}
  7. Q 3 - Q 1 {Q_{3}-Q_{1}}
  8. Q 2 {Q_{2}}
  9. s s\,
  10. x ¯ \bar{x}
  11. c v ^ = s x ¯ \widehat{c_{v}}=\frac{s}{\bar{x}}
  12. c v ^ * = ( 1 + 1 4 n ) c v ^ \widehat{c_{v}}^{*}=\bigg(1+\frac{1}{4n}\bigg)\widehat{c_{v}}
  13. c v ^ l n = e s l n 2 - 1 \widehat{c_{v}}_{ln}=\sqrt{e^{{s_{ln}}^{2}}\!\!-1}
  14. s l n {s_{ln}}\,
  15. s b s_{b}\,
  16. s l n = s b l n ( b ) s_{ln}=s_{b}ln(b)\,
  17. c v ^ l n \widehat{c_{v}}_{ln}\,
  18. G C V K = e s l n - 1 GCV_{K}={e^{s_{ln}}\!\!-1}
  19. c v c_{v}\,
  20. s l n s_{ln}\,
  21. c v c_{v}\,
  22. d F c v = 2 π 1 / 2 Γ ( n - 1 2 ) e - n 2 ( σ μ ) 2 c v 2 1 + c v 2 c v n - 2 ( 1 + c v 2 ) n / 2 i = 0 n - 1 ( n - 1 ) ! Γ ( n - i 2 ) ( n - 1 - i ) ! i ! n i / 2 2 i / 2 ( σ μ ) i 1 ( 1 + c v 2 ) i / 2 d c v , dF_{c_{v}}=\frac{2}{\pi^{1/2}\Gamma\Big(\frac{n-1}{2}\Big)}e^{-\frac{n}{2(% \frac{\sigma}{\mu})^{2}}\frac{{c_{v}}^{2}}{1+{c_{v}}^{2}}}\frac{{c_{v}}^{n-2}}% {(1+{c_{v}}^{2})^{n/2}}\sideset{}{{}^{\prime}}{\sum}_{i=0}^{n-1}\frac{(n-1)!% \Gamma\Big(\frac{n-i}{2}\Big)}{(n-1-i)!i!}\frac{n^{i/2}}{2^{i/2}(\frac{\sigma}% {\mu})^{i}}\frac{1}{(1+{c_{v}}^{2})^{i/2}}dc_{v},
  23. \sideset{}{{}^{\prime}}{\sum}
  24. μ k / σ k {\mu_{k}}/{\sigma^{k}}
  25. μ k \mu_{k}
  26. σ 2 / μ \sigma^{2}/\mu
  27. μ / σ \mu/\sigma
  28. σ 2 / μ 2 \sigma^{2}/\mu^{2}
  29. μ k / σ k \mu_{k}/\sigma^{k}
  30. σ 2 / μ \sigma^{2}/\mu
  31. σ W 2 / μ W \sigma^{2}_{W}/\mu_{W}

Cogeneration.html

  1. η t h W o u t Q i n Electrical Power Output + Heat Output + Cooling Output Total Heat Input \eta_{th}\equiv\frac{W_{out}}{Q_{in}}\equiv\frac{\,\text{Electrical Power % Output + Heat Output + Cooling Output}}{\,\text{Total Heat Input}}
  2. η t h \eta_{th}
  3. W o u t W_{out}
  4. Q i n Q_{in}

Cograph.html

  1. G G
  2. G ¯ \overline{G}
  3. G G
  4. H H
  5. G H G\cup H
  6. v 1 , v 2 , v 3 , v 4 v_{1},v_{2},v_{3},v_{4}
  7. { v 1 , v 2 } , { v 2 , v 3 } \{v_{1},v_{2}\},\{v_{2},v_{3}\}
  8. { v 3 , v 4 } \{v_{3},v_{4}\}
  9. { v 1 , v 3 } , { v 1 , v 4 } \{v_{1},v_{3}\},\{v_{1},v_{4}\}
  10. { v 2 , v 4 } \{v_{2},v_{4}\}

Coherent_information.html

  1. 𝒩 \mathcal{N}
  2. I ( ρ , 𝒩 ) I(\rho,\mathcal{N})
  3. I ( ρ , 𝒩 ) = def S ( 𝒩 ρ ) - S ( 𝒩 , ρ ) I(\rho,\mathcal{N})\ \stackrel{\mathrm{def}}{=}\ S(\mathcal{N}\rho)-S(\mathcal% {N},\rho)
  4. S ( 𝒩 ρ ) S(\mathcal{N}\rho)
  5. S ( 𝒩 , ρ ) S({\mathcal{N}},\rho)

Cohomology_ring.html

  1. H k ( X ; R ) × H ( X ; R ) H k + ( X ; R ) . H^{k}(X;R)\times H^{\ell}(X;R)\to H^{k+\ell}(X;R).
  2. H ( X ; R ) = k H k ( X ; R ) . H^{\bullet}(X;R)=\bigoplus_{k\in\mathbb{N}}H^{k}(X;R).
  3. ( α k β ) = ( - 1 ) k ( β α k ) . (\alpha^{k}\smile\beta^{\ell})=(-1)^{k\ell}(\beta^{\ell}\smile\alpha^{k}).
  4. H * ( P n ; 𝔽 2 ) = 𝔽 2 [ α ] / ( α n + 1 ) \operatorname{H}^{*}(\mathbb{R}P^{n};\mathbb{F}_{2})=\mathbb{F}_{2}[\alpha]/(% \alpha^{n+1})
  5. | α | = 1 |\alpha|=1
  6. H * ( P ; 𝔽 2 ) = 𝔽 2 [ α ] \operatorname{H}^{*}(\mathbb{R}P^{\infty};\mathbb{F}_{2})=\mathbb{F}_{2}[\alpha]
  7. | α | = 1 |\alpha|=1
  8. P \mathbb{R}P^{\infty}
  9. 𝔽 2 \mathbb{F}_{2}

Cohomotopy_group.html

  1. Σ Y \Sigma Y
  2. \geq
  3. π s p ( X ) = lim k [ Σ k X , S p + k ] \pi^{p}_{s}(X)=\underrightarrow{\lim}_{k}{[\Sigma^{k}X,S^{p+k}]}

Colorfulness.html

  1. s u v = C u v * L * = 13 ( u - u n ) 2 + ( v - v n ) 2 s_{uv}=\frac{C^{*}_{uv}}{L^{*}}=13\sqrt{(u^{\prime}-u^{\prime}_{n})^{2}+(v^{% \prime}-v^{\prime}_{n})^{2}}
  2. s a b = C a b * L * = a * 2 + b * 2 L * s_{ab}=\frac{C^{*}_{ab}}{L^{*}}=\frac{\sqrt{{a^{*}}^{2}+{b^{*}}^{2}}}{L^{*}}
  3. S a b = C a b * C a b * 2 + L * 2 100 % S_{ab}=\frac{C^{*}_{ab}}{\sqrt{{C^{*}_{ab}}^{2}+{L^{*}}^{2}}}100\%
  4. s = M / Q s=\sqrt{M/Q}
  5. p e = ( x - x n ) 2 + ( y - y n ) 2 ( x I - x n ) 2 + ( y I - y n ) 2 p_{e}=\sqrt{\frac{(x-x_{n})^{2}+(y-y_{n})^{2}}{(x_{I}-x_{n})^{2}+(y_{I}-y_{n})% ^{2}}}
  6. C a b * = a * 2 + b * 2 C_{ab}^{*}=\sqrt{a^{*2}+b^{*2}}
  7. h a b = arctan b * a * h_{ab}=\arctan\frac{b^{*}}{a^{*}}

Colpitts_oscillator.html

  1. f 0 = 1 2 π L ( C 1 C 2 C 1 + C 2 ) f_{0}={1\over 2\pi\sqrt{L\left({C_{1}C_{2}\over C_{1}+C_{2}}\right)}}
  2. Z i n = v 1 i 1 Z_{in}=\frac{v_{1}}{i_{1}}
  3. v 1 v_{1}
  4. i 1 i_{1}
  5. v 2 v_{2}
  6. v 2 = i 2 Z 2 v_{2}=i_{2}Z_{2}
  7. Z 2 Z_{2}
  8. C 2 C_{2}
  9. C 2 C_{2}
  10. i 2 i_{2}
  11. i 2 = i 1 + i s i_{2}=i_{1}+i_{s}
  12. i s i_{s}
  13. i s i_{s}
  14. i s = g m ( v 1 - v 2 ) i_{s}=g_{m}\left(v_{1}-v_{2}\right)
  15. g m g_{m}
  16. i 1 i_{1}
  17. i 1 = v 1 - v 2 Z 1 i_{1}=\frac{v_{1}-v_{2}}{Z_{1}}
  18. Z 1 Z_{1}
  19. C 1 C_{1}
  20. v 2 v_{2}
  21. Z i n = Z 1 + Z 2 + g m Z 1 Z 2 Z_{in}=Z_{1}+Z_{2}+g_{m}Z_{1}Z_{2}
  22. R i n R_{in}
  23. R i n = g m Z 1 Z 2 R_{in}=g_{m}\cdot Z_{1}\cdot Z_{2}
  24. Z 1 Z_{1}
  25. Z 2 Z_{2}
  26. R i n R_{in}
  27. Z 1 Z_{1}
  28. Z 2 Z_{2}
  29. R i n R_{in}
  30. R i n = - g m ω 2 C 1 C 2 R_{in}=\frac{-g_{m}}{\omega^{2}C_{1}C_{2}}
  31. R i n = - 30 Ω R_{in}=-30\ \Omega
  32. A v = g m R p 4 A_{v}=g_{m}\cdot R_{p}\geq 4
  33. R i n = - g m ω 2 L 1 L 2 R_{in}=-g_{m}\omega^{2}L_{1}L_{2}
  34. V C = 2 I C R L C 2 C 1 + C 2 V_{C}=2I_{C}R_{L}\frac{C_{2}}{C_{1}+C_{2}}
  35. I C I_{C}
  36. R L R_{L}

Comb_filter.html

  1. y [ n ] = x [ n ] + α x [ n - K ] \ y[n]=x[n]+\alpha x[n-K]\,
  2. K K
  3. α \alpha
  4. Y ( z ) = ( 1 + α z - K ) X ( z ) \ Y(z)=(1+\alpha z^{-K})X(z)\,
  5. H ( z ) = Y ( z ) X ( z ) = 1 + α z - K = z K + α z K \ H(z)=\frac{Y(z)}{X(z)}=1+\alpha z^{-K}=\frac{z^{K}+\alpha}{z^{K}}\,
  6. z = e j Ω z=e^{j\Omega}
  7. H ( e j Ω ) = 1 + α e - j Ω K \ H(e^{j\Omega})=1+\alpha e^{-j\Omega K}\,
  8. H ( e j Ω ) = [ 1 + α cos ( Ω K ) ] - j α sin ( Ω K ) \ H(e^{j\Omega})=\left[1+\alpha\cos(\Omega K)\right]-j\alpha\sin(\Omega K)\,
  9. | H ( e j Ω ) | = { H ( e j Ω ) } 2 + { H ( e j Ω ) } 2 \ |H(e^{j\Omega})|=\sqrt{\Re\{H(e^{j\Omega})\}^{2}+\Im\{H(e^{j\Omega})\}^{2}}\,
  10. | H ( e j Ω ) | = ( 1 + α 2 ) + 2 α cos ( Ω K ) \ |H(e^{j\Omega})|=\sqrt{(1+\alpha^{2})+2\alpha\cos(\Omega K)}\,
  11. ( 1 + α 2 ) (1+\alpha^{2})
  12. 2 α cos ( Ω K ) 2\alpha\cos(\Omega K)
  13. α \alpha
  14. α \alpha
  15. f = 1 2 K , 3 2 K , 5 2 K f=\frac{1}{2K},\frac{3}{2K},\frac{5}{2K}...
  16. α = ± 1 \alpha=\pm 1
  17. α \alpha
  18. α \alpha
  19. H ( z ) = z K + α z K \ H(z)=\frac{z^{K}+\alpha}{z^{K}}\,
  20. z K = - α z^{K}=-\alpha
  21. K K
  22. z K = 0 z^{K}=0
  23. K K
  24. z = 0 z=0
  25. K = 8 K=8
  26. α = 0.5 \alpha=0.5
  27. K = 8 K=8
  28. α = - 0.5 \alpha=-0.5
  29. y [ n ] = x [ n ] + α y [ n - K ] \ y[n]=x[n]+\alpha y[n-K]\,
  30. y y
  31. ( 1 - α z - K ) Y ( z ) = X ( z ) \ (1-\alpha z^{-K})Y(z)=X(z)\,
  32. H ( z ) = Y ( z ) X ( z ) = 1 1 - α z - K = z K z K - α \ H(z)=\frac{Y(z)}{X(z)}=\frac{1}{1-\alpha z^{-K}}=\frac{z^{K}}{z^{K}-\alpha}\,
  33. z = e j Ω z=e^{j\Omega}
  34. H ( e j Ω ) = 1 1 - α e - j Ω K \ H(e^{j\Omega})=\frac{1}{1-\alpha e^{-j\Omega K}}\,
  35. | H ( e j Ω ) | = 1 ( 1 + α 2 ) - 2 α cos ( Ω K ) \ |H(e^{j\Omega})|=\frac{1}{\sqrt{(1+\alpha^{2})-2\alpha\cos(\Omega K)}}\,
  36. α \alpha
  37. α \alpha
  38. α \alpha
  39. f = 0 , 1 K , 2 K f=0,\frac{1}{K},\frac{2}{K}...
  40. 1 1 - α 1\over 1-\alpha
  41. | α | |\alpha|
  42. | α | |\alpha|
  43. H ( z ) = z K z K - α \ H(z)=\frac{z^{K}}{z^{K}-\alpha}\,
  44. z K = 0 z^{K}=0
  45. K K
  46. z = 0 z=0
  47. z K = α z^{K}=\alpha
  48. K K
  49. K = 8 K=8
  50. α = 0.5 \alpha=0.5
  51. K = 8 K=8
  52. α = - 0.5 \alpha=-0.5
  53. y ( t ) = x ( t ) + α x ( t - τ ) \ y(t)=x(t)+\alpha x(t-\tau)\,
  54. τ \tau
  55. H ( s ) = 1 + α e - s τ \ H(s)=1+\alpha e^{-s\tau}\,
  56. y ( t ) = x ( t ) + α y ( t - τ ) \ y(t)=x(t)+\alpha y(t-\tau)\,
  57. H ( s ) = 1 1 - α e - s τ \ H(s)=\frac{1}{1-\alpha e^{-s\tau}}\,

Combinatorial_number_system.html

  1. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  2. N = ( c k k ) + + ( c 2 2 ) + ( c 1 1 ) . N={\left({{c_{k}}\atop{k}}\right)}+\cdots+{\left({{c_{2}}\atop{2}}\right)}+{% \left({{c_{1}}\atop{1}}\right)}.
  3. ( c k k ) N {\textstyle\left({{c_{k}}\atop{k}}\right)}\leq N
  4. ( c k - 1 k - 1 ) N - ( c k k ) {\textstyle\left({{c_{k-1}}\atop{k-1}}\right)}\leq N-{\textstyle\left({{c_{k}}% \atop{k}}\right)}
  5. ( c i i ) {\textstyle\left({{c_{i}}\atop{i}}\right)}
  6. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  7. 2 c 1 + 2 c 2 + + 2 c k 2^{c_{1}}+2^{c_{2}}+\cdots+2^{c_{k}}
  8. ( c i i ) {\textstyle\left({{c_{i}}\atop{i}}\right)}
  9. ( c 1 1 ) + ( c 2 2 ) + + ( c k k ) , {\left({{c_{1}}\atop{1}}\right)}+{\left({{c_{2}}\atop{2}}\right)}+\cdots+{% \left({{c_{k}}\atop{k}}\right)},
  10. ( c k k ) {\textstyle\left({{c_{k}}\atop{k}}\right)}
  11. ( c k k ) {\textstyle\left({{c_{k}}\atop{k}}\right)}
  12. ( c k k ) N {\textstyle\left({{c_{k}}\atop{k}}\right)}\leq N
  13. N - ( c k k ) N-{\textstyle\left({{c_{k}}\atop{k}}\right)}
  14. N - ( c k k ) N-{\textstyle\left({{c_{k}}\atop{k}}\right)}
  15. ( n 5 ) {\textstyle\left({{n}\atop{5}}\right)}
  16. ( n 4 ) {\textstyle\left({{n}\atop{4}}\right)}
  17. 72 = ( 8 5 ) + ( 6 4 ) + ( 3 3 ) 72={\textstyle\left({{8}\atop{5}}\right)}+{\textstyle\left({{6}\atop{4}}\right% )}+{\textstyle\left({{3}\atop{3}}\right)}
  18. ( c i i ) = 0 {\textstyle\left({{c_{i}}\atop{i}}\right)}=0
  19. 0 N < ( n k ) 0\leq N<{\textstyle\left({{n}\atop{k}}\right)}

Combinatorial_proof.html

  1. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  2. ( n k ) = n ( n - 1 ) ( n - k + 1 ) k ( k - 1 ) 1 . {\left({{n}\atop{k}}\right)}=\frac{n(n-1)\cdots(n-k+1)}{k(k-1)\cdots 1}.
  3. n ( n - 1 ) ( n - k + 1 ) n(n-1)\cdots(n-k+1)
  4. n ( n - 1 ) ( n - k + 1 ) = ( n k ) k ! , n(n-1)\cdots(n-k+1)={\left({{n}\atop{k}}\right)}k!,
  5. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}

Combs_method.html

  1. ( ( p q ) r ) ( ( p r ) ( q r ) ) ((p\land q)\Rightarrow r)\iff((p\Rightarrow r)\lor(q\Rightarrow r))
  2. p p
  3. q q
  4. r r
  5. p q r p\land q\Rightarrow r
  6. p r p\Rightarrow r
  7. q r q\Rightarrow r
  8. ( p r ) ( q r ) (p\Rightarrow r)\lor(q\Rightarrow r)
  9. S N S^{N}
  10. S × N S\times N

Comma_category.html

  1. 𝒜 \mathcal{A}
  2. \mathcal{B}
  3. 𝒞 \mathcal{C}
  4. S S
  5. T T
  6. 𝒜 𝑆 𝒞 𝑇 \mathcal{A}\xrightarrow{\;\;S\;\;}\mathcal{C}\xleftarrow{\;\;T\;\;}\mathcal{B}
  7. ( S T ) (S\downarrow T)
  8. ( α , β , f ) (\alpha,\beta,f)
  9. α \alpha
  10. 𝒜 \mathcal{A}
  11. β \beta
  12. \mathcal{B}
  13. f : S ( α ) T ( β ) f:S(\alpha)\rightarrow T(\beta)
  14. 𝒞 \mathcal{C}
  15. ( α , β , f ) (\alpha,\beta,f)
  16. ( α , β , f ) (\alpha^{\prime},\beta^{\prime},f^{\prime})
  17. ( g , h ) (g,h)
  18. g : α α g:\alpha\rightarrow\alpha^{\prime}
  19. h : β β h:\beta\rightarrow\beta^{\prime}
  20. 𝒜 \mathcal{A}
  21. \mathcal{B}
  22. S ( α ) S ( g ) S ( α ) f f T ( β ) T ( h ) T ( β ) \begin{matrix}S(\alpha)&\xrightarrow{S(g)}&S(\alpha^{\prime})\\ f\Bigg\downarrow&&\Bigg\downarrow f^{\prime}\\ T(\beta)&\xrightarrow[T(h)]{}&T(\beta^{\prime})\end{matrix}
  23. ( g , h ) ( g , h ) (g,h)\circ(g^{\prime},h^{\prime})
  24. ( g g , h h ) (g\circ g^{\prime},h\circ h^{\prime})
  25. ( α , β , f ) (\alpha,\beta,f)
  26. ( id α , id β ) (\mathrm{id}_{\alpha},\mathrm{id}_{\beta})
  27. 𝒜 = 𝒞 \mathcal{A}=\mathcal{C}
  28. S S
  29. = 𝟏 \mathcal{B}=\,\textbf{1}
  30. * *
  31. T ( * ) = A T(*)=A
  32. A A
  33. 𝒞 \mathcal{C}
  34. ( 𝒞 A ) (\mathcal{C}\downarrow A)
  35. A A
  36. A A
  37. ( α , * , f ) (\alpha,*,f)
  38. ( α , f ) (\alpha,f)
  39. f : α A f:\alpha\rightarrow A
  40. f f
  41. π α \pi_{\alpha}
  42. ( B , π B ) (B,\pi_{B})
  43. ( B , π B ) (B^{\prime},\pi_{B^{\prime}})
  44. g : B B g:B\rightarrow B^{\prime}
  45. S S
  46. T T
  47. ( A 𝒞 ) (A\downarrow\mathcal{C})
  48. A A
  49. 𝒞 \mathcal{C}
  50. S S
  51. A A
  52. A A
  53. ( B , i B ) (B,i_{B})
  54. i B : A B i_{B}:A\rightarrow B
  55. ( B , i B ) (B,i_{B})
  56. ( B , i B ) (B^{\prime},i_{B^{\prime}})
  57. h : B B h:B\rightarrow B^{\prime}
  58. S S
  59. T T
  60. 𝒞 \mathcal{C}
  61. 𝒜 = = 𝒞 \mathcal{A}=\mathcal{B}=\mathcal{C}
  62. 𝒞 \mathcal{C}^{\rightarrow}
  63. 𝒞 \mathcal{C}
  64. 𝒞 \mathcal{C}
  65. T T
  66. s s
  67. ( s T ) (s\downarrow T)
  68. s s
  69. T T
  70. ( s T ) (s\downarrow T)
  71. s T ( G ) s\rightarrow T(G)
  72. G G
  73. s s
  74. ( s T ) (s\downarrow T)
  75. s s
  76. T T
  77. T T
  78. s s
  79. ( S t ) (S\downarrow t)
  80. S S
  81. t t
  82. S S
  83. t t
  84. S S
  85. T T
  86. S ( * ) = A S(*)=A
  87. T ( * ) = B T(*)=B
  88. ( S T ) (S\downarrow T)
  89. ( A B ) (A\downarrow B)
  90. A A
  91. B B
  92. S T 𝒜 S\downarrow T\to\mathcal{A}
  93. ( α , β , f ) α (\alpha,\beta,f)\mapsto\alpha
  94. ( g , h ) g (g,h)\mapsto g
  95. S T S\downarrow T\to\mathcal{B}
  96. ( α , β , f ) β (\alpha,\beta,f)\mapsto\beta
  97. ( g , h ) h (g,h)\mapsto h
  98. S T C S\downarrow T\to C^{\downarrow}
  99. ( α , β , f ) f (\alpha,\beta,f)\mapsto f
  100. ( g , h ) ( S g , T h ) (g,h)\mapsto(Sg,Th)
  101. ( \bull 𝐒𝐞𝐭 ) \scriptstyle{(\bull\downarrow\mathbf{Set})}
  102. \bull \scriptstyle{\bull}
  103. 𝐒𝐞𝐭 \scriptstyle{\mathbf{Set}}
  104. ( \bull 𝐓𝐨𝐩 ) \scriptstyle{(\bull\downarrow\mathbf{Top})}
  105. ( 𝐒𝐞𝐭 D ) \scriptstyle{(\mathbf{Set}\downarrow D)}
  106. D : 𝐒𝐞𝐭 𝐒𝐞𝐭 \scriptstyle{D:\mathbf{Set}\rightarrow\mathbf{Set}}
  107. s s
  108. s × s s\times s
  109. ( a , b , f ) (a,b,f)
  110. a a
  111. b b
  112. f : a ( b × b ) f:a\rightarrow(b\times b)
  113. b b
  114. a a
  115. f f
  116. b × b b\times b
  117. ( g , h ) : ( a , b , f ) ( a , b , f ) (g,h):(a,b,f)\rightarrow(a^{\prime},b^{\prime},f^{\prime})
  118. f g = T ( h ) f f^{\prime}\circ g=T(h)\circ f
  119. S S
  120. A A
  121. ( S A ) (S\downarrow A)
  122. A A
  123. S S
  124. A A
  125. A A
  126. ( B , π B ) (B,\pi_{B})
  127. B B
  128. π B \pi_{B}
  129. B B
  130. A A
  131. 𝒜 \mathcal{A}
  132. \mathcal{B}
  133. S : 𝒜 𝒞 S:\mathcal{A}\rightarrow\mathcal{C}
  134. T : 𝒞 T:\mathcal{B}\rightarrow\mathcal{C}
  135. ( S T ) (S\downarrow T)
  136. 𝒜 \mathcal{A}
  137. \mathcal{B}
  138. S : 𝒜 𝒞 S:\mathcal{A}\rightarrow\mathcal{C}
  139. T : 𝒞 T:\mathcal{B}\rightarrow\mathcal{C}
  140. ( S T ) (S\downarrow T)
  141. ( S T ) 𝒜 (S\downarrow T)\rightarrow\mathcal{A}
  142. ( S T ) (S\downarrow T)\rightarrow\mathcal{B}
  143. 𝒞 \mathcal{C}
  144. F : 𝒞 𝒞 × 𝒞 F:\mathcal{C}\rightarrow\mathcal{C}\times\mathcal{C}
  145. c c
  146. ( c , c ) (c,c)
  147. f f
  148. ( f , f ) (f,f)
  149. ( a , b ) (a,b)
  150. F F
  151. ( c , c ) (c,c)
  152. ρ : ( a , b ) ( c , c ) \rho:(a,b)\rightarrow(c,c)
  153. ρ : ( a , b ) ( d , d ) \rho^{\prime}:(a,b)\rightarrow(d,d)
  154. σ : c d \sigma:c\rightarrow d
  155. F ( σ ) ρ = ρ F(\sigma)\circ\rho=\rho^{\prime}
  156. ( ( a , b ) F ) ((a,b)\downarrow F)
  157. 𝒞 \mathcal{C}
  158. F : 𝒞 𝒟 F:\mathcal{C}\rightarrow\mathcal{D}
  159. G : 𝒟 𝒞 G:\mathcal{D}\rightarrow\mathcal{C}
  160. ( F i d 𝒟 ) (F\downarrow id_{\mathcal{D}})
  161. ( i d 𝒞 G ) (id_{\mathcal{C}}\downarrow G)
  162. i d 𝒟 id_{\mathcal{D}}
  163. i d 𝒞 id_{\mathcal{C}}
  164. 𝒞 \mathcal{C}
  165. 𝒟 \mathcal{D}
  166. 𝒞 × 𝒟 \mathcal{C}\times\mathcal{D}
  167. S , T S,T
  168. S T S\downarrow T
  169. α = β , α = β , g = h \alpha=\beta,\alpha^{\prime}=\beta^{\prime},g=h
  170. S T S\to T
  171. S ( α ) T ( α ) S(\alpha)\to T(\alpha)
  172. η : S T \eta:S\to T
  173. S , T : 𝒜 𝒞 S,T:\mathcal{A}\to\mathcal{C}
  174. 𝒜 ( S T ) \mathcal{A}\to(S\downarrow T)
  175. α \alpha
  176. ( α , α , η α ) (\alpha,\alpha,\eta_{\alpha})
  177. g g
  178. ( g , g ) (g,g)
  179. S T S\to T
  180. 𝒜 ( S T ) \mathcal{A}\to(S\downarrow T)
  181. S T S\downarrow T