wpmath0000013_6

Indexed_family.html

  1. x : I X i x i = x ( i ) \begin{aligned}\displaystyle x\colon I&\displaystyle\to X\\ \displaystyle i&\displaystyle\mapsto x_{i}=x(i)\end{aligned}
  2. 𝒳 := { x i : i I } \mathcal{X}:=\{x_{i}:i\in I\}
  3. i , j I i,j\in I
  4. i j i\neq j
  5. x i = x j x_{i}=x_{j}
  6. 𝒳 \mathcal{X}
  7. { C i } i I \{C_{i}\}_{i\in I}
  8. C i S C_{i}\subset S
  9. A = [ 1 1 1 1 ] . A=\begin{bmatrix}1&1\\ 1&1\end{bmatrix}.
  10. i I a i . \sum_{i\in I}a_{i}.
  11. i I A i . \bigcup_{i\in I}A_{i}.

Indirect_Fourier_transform.html

  1. I ( 𝐪 ) = 1 V V V ρ ( 𝐫 ) ρ ( 𝐫 ) e - i 𝐪 ( 𝐫 - 𝐫 ) d 𝐫 d 𝐫 , I(\mathbf{q})=\frac{1}{V}\int_{V}\int_{V}\rho(\mathbf{r})\rho(\mathbf{r}^{% \prime})e^{-i\mathbf{q}(\mathbf{r}-\mathbf{r}^{\prime})}\,\text{d}\mathbf{r}\,% \text{d}\mathbf{r}^{\prime},
  2. ρ ( 𝐫 ) \rho(\mathbf{r})
  3. γ ( 𝐫 ) \gamma(\mathbf{r})
  4. I ( 𝐪 ) = V γ ( 𝐫 ) e - i 𝐪 𝐫 d 𝐫 I(\mathbf{q})=\int_{V}\gamma(\mathbf{r})e^{-i\mathbf{q}\cdot\mathbf{r}}\,\text% {d}\mathbf{r}
  5. i i
  6. r r
  7. j j
  8. γ 0 ( r ) \gamma_{0}(r)
  9. γ ( r ) \gamma(r)
  10. γ 0 ( r ) \gamma_{0}(r)
  11. γ ( r ) = b i b j γ 0 ( r ) V \gamma(r)=b_{i}\cdot bj\gamma_{0}(r)V
  12. b k b_{k}
  13. k k
  14. b b
  15. ρ e \rho_{e}
  16. p ( r ) p(r)
  17. p ( r ) = γ ( r ) r 2 . p(r)=\gamma(r)\cdot r^{2}.
  18. p ( r ) p(r)
  19. ρ ( 𝐫 ) \rho(\mathbf{r})
  20. p ( r ) p(r)
  21. Δ ρ ( 𝐫 ) \Delta\rho(\mathbf{r})
  22. r r
  23. p ( r ) p(r)
  24. p i ( r ) p_{i}(r)
  25. ϕ i ( r ) \phi_{i}(r)
  26. c i c_{i}
  27. I ( q ) = 4 π i = 1 N c i ψ i ( r ) , I(q)=4\pi\sum_{i=1}^{N}c_{i}\psi_{i}(r),
  28. ψ i ( r ) \psi_{i}(r)
  29. ψ i ( r ) = 0 ϕ i ( r ) sin ( q r ) q r \psi_{i}(r)=\int_{0}^{\infty}\phi_{i}(r)\frac{\sin(qr)}{qr}
  30. c i c_{i}
  31. c i f i t c_{i}^{fit}
  32. p i ( r ) p_{i}(r)
  33. p f ( r ) p_{f}(r)
  34. c i f i t c_{i}^{fit}
  35. χ 2 \chi^{2}
  36. χ 2 = 1 M - P k = 1 M [ I e x p e r i m e n t ( q k ) - I f i t ( q k ) ] 2 σ 2 ( q k ) \chi^{2}=\frac{1}{M-P}\sum_{k=1}^{M}\frac{[I_{experiment}(q_{k})-I_{fit}(q_{k}% )]^{2}}{\sigma^{2}(q_{k})}
  37. M M
  38. P P
  39. σ ( q k ) \sigma(q_{k})
  40. k k
  41. χ 2 \chi^{2}
  42. S S
  43. S = i = 1 N - 1 ( c i + 1 - c i ) 2 S=\sum_{i=1}^{N-1}(c_{i+1}-c_{i})^{2}
  44. S S
  45. χ 2 \chi^{2}
  46. L = χ 2 + α S L=\chi^{2}+\alpha S
  47. α \alpha
  48. p i ( r ) fitting p f ( r ) p_{i}(r)\rightarrow\,\text{fitting}\rightarrow p_{f}(r)
  49. p ( r ) p(r)

Indistinguishability_quotient.html

  1. A A
  2. G G
  3. H H
  4. A A
  5. G + H G+H
  6. A A
  7. G G
  8. A A
  9. H H
  10. G G
  11. H H
  12. A A
  13. A A
  14. G G
  15. H H
  16. X X
  17. A A
  18. G + X G+X
  19. H + X H+X
  20. A A
  21. A A
  22. A A
  23. A A

Induction_plasma_technology.html

  1. ϕ B = ( μ 0 I c N ) ( π r 0 2 ) ( 1 ) \phi_{B}=(\mu_{0}\,I_{c}\,N)(\pi\,r_{0}^{2})\quad(1)
  2. ϕ B \phi_{B}
  3. μ 0 \mu_{0}
  4. 4 π 10 - 7 Wb/A.m 4\pi\,10^{-7}\,\textrm{Wb/A.m}
  5. I c I_{c}
  6. N N
  7. r 0 r_{0}
  8. E = - N ( Δ ϕ B / Δ t ) ( 2 ) E=-N(\Delta\phi_{B}/\Delta t)\quad(2)
  9. N N

Inductive_type.html

  1. A A
  2. B : A T y p e B:A\to Type
  3. c i r c l e circle
  4. b a s e : c i r c l e base:circle
  5. l o o p : b a s e = b a s e . loop:base=base.
  6. c i r c l e circle

Inertial_navigation_system.html

  1. ω \omega
  2. R ˙ \dot{R}
  3. Ω \Omega
  4. Φ , λ \Phi,\lambda

Infinite_product.html

  1. n = 1 a n = a 1 a 2 a 3 \prod_{n=1}^{\infty}a_{n}=a_{1}\;a_{2}\;a_{3}\cdots
  2. 2 π = 2 2 2 + 2 2 2 + 2 + 2 2 \frac{2}{\pi}=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{% \sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdots
  3. π 2 = 2 1 2 3 4 3 4 5 6 5 6 7 8 7 8 9 = n = 1 ( 4 n 2 4 n 2 - 1 ) . \frac{\pi}{2}=\frac{2}{1}\cdot\frac{2}{3}\cdot\frac{4}{3}\cdot\frac{4}{5}\cdot% \frac{6}{5}\cdot\frac{6}{7}\cdot\frac{8}{7}\cdot\frac{8}{9}\cdots=\prod_{n=1}^% {\infty}\left(\frac{4\cdot n^{2}}{4\cdot n^{2}-1}\right).
  4. n = 1 a n \prod_{n=1}^{\infty}a_{n}
  5. n = 1 log ( a n ) \sum_{n=1}^{\infty}\log(a_{n})
  6. a n 1 a_{n}\geq 1
  7. a n = 1 + p n a_{n}=1+p_{n}
  8. p n 0 p_{n}\geq 0
  9. 1 + n = 1 N p n n = 1 N ( 1 + p n ) exp ( n = 1 N p n ) 1+\sum_{n=1}^{N}p_{n}\leq\prod_{n=1}^{N}\left(1+p_{n}\right)\leq\exp\left(\sum% _{n=1}^{N}p_{n}\right)
  10. n = 1 ( 1 + p n ) \prod_{n=1}^{\infty}(1+p_{n})
  11. n = 1 p n \sum_{n=1}^{\infty}p_{n}
  12. n = 1 | p n | 2 < + \sum_{n=1}^{\infty}|p_{n}|^{2}<+\infty
  13. log ( 1 + x ) = x + O ( x 2 ) \log(1+x)=x+O(x^{2})
  14. f ( z ) = z m e ϕ ( z ) n = 1 ( 1 - z u n ) exp { z u n + 1 2 ( z u n ) 2 + + 1 λ n ( z u n ) λ n } f(z)=z^{m}e^{\phi(z)}\prod_{n=1}^{\infty}\left(1-\frac{z}{u_{n}}\right)\exp% \left\{\frac{z}{u_{n}}+\frac{1}{2}\left(\frac{z}{u_{n}}\right)^{2}+\cdots+% \frac{1}{\lambda_{n}}\left(\frac{z}{u_{n}}\right)^{\lambda_{n}}\right\}
  15. f ( z ) = z m e ϕ ( z ) n = 1 ( 1 - z u n ) . f(z)=z^{m}e^{\phi(z)}\prod_{n=1}^{\infty}\left(1-\frac{z}{u_{n}}\right).
  16. sin ( π z ) = π z n = 1 ( 1 - z 2 n 2 ) \sin(\pi z)=\pi z\prod_{n=1}^{\infty}\left(1-\frac{z^{2}}{n^{2}}\right)
  17. 1 Γ ( z ) = z e γ z n = 1 ( 1 + z n ) e - z n \frac{1}{\Gamma(z)}=ze^{\gamma z}\prod_{n=1}^{\infty}\left(1+\frac{z}{n}\right% )e^{-\frac{z}{n}}
  18. σ ( z ) = z ω Λ * ( 1 - z ω ) e z 2 2 ω 2 + z ω \sigma(z)=z\prod_{\omega\in\Lambda_{*}}\left(1-\frac{z}{\omega}\right)e^{\frac% {z^{2}}{2\omega^{2}}+\frac{z}{\omega}}
  19. Λ * \Lambda_{*}
  20. ( z ; q ) = n = 0 ( 1 - z q n ) (z;q)_{\infty}=\prod_{n=0}^{\infty}(1-zq^{n})
  21. f ( a , b ) = n = - a n ( n + 1 ) 2 b n ( n - 1 ) 2 = n = 0 ( 1 + a n + 1 b n ) ( 1 + a n b n + 1 ) ( 1 - a n + 1 b n + 1 ) \begin{aligned}\displaystyle f(a,b)&\displaystyle=\sum_{n=-\infty}^{\infty}a^{% \frac{n(n+1)}{2}}b^{\frac{n(n-1)}{2}}\\ &\displaystyle=\prod_{n=0}^{\infty}(1+a^{n+1}b^{n})(1+a^{n}b^{n+1})(1-a^{n+1}b% ^{n+1})\end{aligned}
  22. ζ ( z ) = n = 1 1 1 - p n - z \zeta(z)=\prod_{n=1}^{\infty}\frac{1}{1-p_{n}^{-z}}

Infinite_tree_automaton.html

  1. Σ \Sigma
  2. A = ( Σ , D , Q , δ , q 0 , F ) A=(\Sigma,D,Q,\delta,q_{0},F)
  3. Σ \Sigma
  4. D D\subset\mathbb{N}
  5. D D
  6. Q Q
  7. δ : Q × Σ × D 2 Q * \delta:Q\times\Sigma\times D\rightarrow 2^{Q^{*}}
  8. q Q q\in Q
  9. σ Σ \sigma\in\Sigma
  10. d D d\in D
  11. q 0 Q q_{0}\in Q
  12. F Σ ω F\subseteq\Sigma^{\omega}
  13. A A
  14. Σ \Sigma
  15. ( T , V ) (T,V)
  16. Q Q
  17. ( T r , r ) (T_{r},r)
  18. q q
  19. σ Σ \sigma\in\Sigma
  20. d ( t ) d(t)
  21. ( q 1 , , q d ( t ) ) (q_{1},...,q_{d(t)})
  22. δ ( q , σ , d ( t ) ) \delta(q,\sigma,d(t))
  23. d ( t ) d(t)
  24. 0 < i d ( t ) 0<i\leq d(t)
  25. q i q_{i}
  26. t . i t.i
  27. ( T r , r ) (T_{r},r)
  28. r ( ϵ ) = q 0 r(\epsilon)=q_{0}
  29. t T r t\in T_{r}
  30. r ( t ) = q r(t)=q
  31. ( q 1 , , q d ( t ) ) δ ( q , V ( t ) , d ( t ) ) (q_{1},...,q_{d(t)})\in\delta(q,V(t),d(t))
  32. 0 < i d ( t ) 0<i\leq d(t)
  33. t . i T r t.i\in T_{r}
  34. r ( t . i ) = q i r(t.i)=q_{i}
  35. ( T r , r ) (T_{r},r)
  36. F F
  37. Σ \Sigma
  38. ( T , V ) (T,V)
  39. Σ \Sigma
  40. ( A ) \mathcal{L}(A)
  41. A A
  42. q Q q\in Q
  43. σ Σ \sigma\in\Sigma
  44. d D d\in D
  45. δ ( q , σ , d ) \delta(q,\sigma,d)

Inflationism.html

  1. 300 % × 10 % = 30 % , 300\%\times 10\%=30\%,
  2. 1.01 / 1.04 - 1 - 2.88 % , 1.01/1.04-1\approx-2.88\%,
  3. 1 % - 4 % = - 3 % 1\%-4\%=-3\%

Infrastructure_(number_theory).html

  1. 𝒪 ( D 1 / 4 + ε ) \mathcal{O}(D^{1/4+\varepsilon})
  2. ε > 0 \varepsilon>0
  3. D D
  4. 𝒪 ( D 1 / 2 + ε ) \mathcal{O}(D^{1/2+\varepsilon})
  5. ( X , d ) (X,d)
  6. R > 0 R>0
  7. X X\neq\emptyset
  8. d : X / R d:X\to\mathbb{R}/R\mathbb{Z}
  9. d d
  10. / R \mathbb{R}/R\mathbb{Z}
  11. R R
  12. X X
  13. d ( X ) d(X)
  14. b s : X X bs:X\to X
  15. ( X , d ) (X,d)
  16. d ( X ) d(X)
  17. x X x\in X
  18. f x := inf { f > 0 d ( x ) + f d ( X ) } f_{x}:=\inf\{f^{\prime}>0\mid d(x)+f^{\prime}\in d(X)\}
  19. b s ( x ) := d - 1 ( d ( x ) + f x ) bs(x):=d^{-1}(d(x)+f_{x})
  20. / R \mathbb{R}/R\mathbb{Z}
  21. d ( x ) + d ( y ) / R d(x)+d(y)\in\mathbb{R}/R\mathbb{Z}
  22. x , y X x,y\in X
  23. d ( X ) d(X)
  24. d ( X ) d(X)
  25. r e d : / R X red:\mathbb{R}/R\mathbb{Z}\to X
  26. g s ( x , y ) := r e d ( d ( x ) + d ( y ) ) gs(x,y):=red(d(x)+d(y))
  27. g s : X × X X gs:X\times X\to X
  28. r e d red
  29. r e d d = id X red\circ d=\mathrm{id}_{X}
  30. v / R v\in\mathbb{R}/R\mathbb{Z}
  31. f v := inf { f 0 v - f d ( X ) } f_{v}:=\inf\{f\geq 0\mid v-f\in d(X)\}
  32. r e d ( v ) := d - 1 ( v - f v ) red(v):=d^{-1}(v-f_{v})
  33. x d ( X ) x\in d(X)
  34. | d ( x ) - v | |d(x)-v|
  35. | d ( x ) - v | |d(x)-v|
  36. inf { | f - v | f d ( x ) } \inf\{|f-v|\mid f\in d(x)\}
  37. d ( x ) d(x)
  38. v + R v+R\mathbb{Z}
  39. / R \mathbb{R}/R\mathbb{Z}
  40. / R \mathbb{R}/R\mathbb{Z}
  41. / R \mathbb{R}/R\mathbb{Z}
  42. / R \mathbb{R}/R\mathbb{Z}
  43. / R \mathbb{R}/R\mathbb{Z}
  44. X X
  45. ( f , p ) (f,p)
  46. f f
  47. f f
  48. f R e p fRep
  49. X × / R X\times\mathbb{R}/R\mathbb{Z}
  50. Ψ f R e p : f R e p / R , ( x , f ) d ( x ) + f \Psi_{fRep}:fRep\to\mathbb{R}/R\mathbb{Z},\;(x,f)\mapsto d(x)+f
  51. ( x , 0 ) f R e p (x,0)\in fRep
  52. x X x\in X
  53. r e d : / R X red:\mathbb{R}/R\mathbb{Z}\to X
  54. f R e p r e d := { ( x , f ) X × / R r e d ( d ( x ) + f ) = x } fRep_{red}:=\{(x,f)\in X\times\mathbb{R}/R\mathbb{Z}\mid red(d(x)+f)=x\}
  55. f f
  56. f R e p fRep
  57. f f
  58. r e d ( f ) = π 1 ( Ψ f R e p - 1 ( f ) ) red(f)=\pi_{1}(\Psi_{fRep}^{-1}(f))
  59. π 1 : X × / R X , ( x , f ) x \pi_{1}:X\times\mathbb{R}/R\mathbb{Z}\to X,\;(x,f)\mapsto x
  60. f f
  61. Ψ f R e p : f R e p / R \Psi_{fRep}:fRep\to\mathbb{R}/R\mathbb{Z}
  62. / R \mathbb{R}/R\mathbb{Z}
  63. f R e p fRep
  64. f R e p fRep
  65. ( f R e p , + ) (fRep,+)
  66. x + y := Ψ f R e p - 1 ( Ψ f R e p ( x ) + Ψ f R e p ( y ) ) x+y:=\Psi_{fRep}^{-1}(\Psi_{fRep}(x)+\Psi_{fRep}(y))
  67. x , y f R e p x,y\in fRep
  68. Ψ f R e p \Psi_{fRep}
  69. d d
  70. r e d : / R X , v d - 1 ( v - inf { f 0 v - f d ( X ) } ) red:\mathbb{R}/R\mathbb{Z}\to X,\;v\mapsto d^{-1}(v-\inf\{f\geq 0\mid v-f\in d% (X)\})
  71. f R e p r e d = { ( x , f ) f 0 , f [ 0 , f ) : d ( x ) + f d ( X ) } fRep_{red}=\{(x,f)\mid f\geq 0,\;\forall f^{\prime}\in[0,f):d(x)+f^{\prime}% \not\in d(X)\}
  72. ( x , f ) , ( x , f ) f R e p r e d (x,f),(x^{\prime},f^{\prime})\in fRep_{red}
  73. ( x ′′ , f ′′ ) (x^{\prime\prime},f^{\prime\prime})
  74. x ′′ = g s ( x , x ) x^{\prime\prime}=gs(x,x^{\prime})
  75. f ′′ = f + f + ( d ( x ) + d ( x ) - d ( g s ( x , x ) ) ) 0 f^{\prime\prime}=f+f^{\prime}+(d(x)+d(x^{\prime})-d(gs(x,x^{\prime})))\geq 0
  76. f R e p r e d fRep_{red}
  77. b s - 1 ( x ′′ ) bs^{-1}(x^{\prime\prime})
  78. f ′′ - ( d ( x ′′ ) - d ( b s - 1 ( x ′′ ) ) ) f^{\prime\prime}-(d(x^{\prime\prime})-d(bs^{-1}(x^{\prime\prime})))
  79. ( x ′′ , f ′′ ) (x^{\prime\prime},f^{\prime\prime})
  80. ( b s - 1 ( x ′′ ) , f ′′ - ( d ( x ′′ ) - d ( b s - 1 ( x ′′ ) ) ) ) (bs^{-1}(x^{\prime\prime}),f^{\prime\prime}-(d(x^{\prime\prime})-d(bs^{-1}(x^{% \prime\prime}))))
  81. ( x ′′ , f ′′ ) f R e p r e d (x^{\prime\prime},f^{\prime\prime})\in fRep_{red}
  82. Ψ f R e p r e d ( x , f ) + Ψ f R e p r e d ( x , f ) = Ψ f R e p r e d ( x ′′ , f ′′ ) \Psi_{fRep_{red}}(x,f)+\Psi_{fRep_{red}}(x^{\prime},f^{\prime})=\Psi_{fRep_{% red}}(x^{\prime\prime},f^{\prime\prime})
  83. ( x , f ) + ( x , f ) = ( x ′′ , f ′′ ) (x,f)+(x^{\prime},f^{\prime})=(x^{\prime\prime},f^{\prime\prime})

Inglis–Teller_equation.html

  1. log ( N i + N e ) = 23.491 - 7.5 log ( n m ) , \log(N_{i}+N_{e})=23.491-7.5\log(n_{m})\,,

Inoue_surface.html

  1. × H {\mathbb{C}}\times H
  2. × H {\mathbb{C}}\times H
  3. × H {\mathbb{C}}\times H
  4. b 2 = 0 b_{2}=0
  5. b 1 = 1 b_{1}=1
  6. - -\infty
  7. α , α ¯ \alpha,\bar{\alpha}
  8. | α | 2 c = 1 |\alpha|^{2}c=1
  9. {\mathbb{Z}}
  10. 3 {\mathbb{Z}}^{3}
  11. Γ := 3 \Gamma:={\mathbb{Z}}^{3}\ltimes{\mathbb{Z}}
  12. 3 = ( × ) {\mathbb{R}}^{3}\ltimes{\mathbb{R}}=({\mathbb{C}}\times{\mathbb{R}})\ltimes{% \mathbb{R}}
  13. × {\mathbb{C}}\times{\mathbb{R}}
  14. ( × ) ({\mathbb{C}}\times{\mathbb{R}})
  15. \ltimes{\mathbb{R}}
  16. ( z , r ) ( α t z , c t r ) (z,r)\mapsto(\alpha^{t}z,c^{t}r)
  17. × H = × × > 0 {\mathbb{C}}\times H={\mathbb{C}}\times{\mathbb{R}}\times{\mathbb{R}}^{>0}
  18. v e log c t v v\mapsto e^{\log ct}v
  19. \ltimes{\mathbb{R}}
  20. 3 {\mathbb{R}}^{3}\ltimes{\mathbb{R}}
  21. 3 {\mathbb{R}}^{3}
  22. > 0 {\mathbb{R}}^{>0}
  23. × H / Γ {\mathbb{C}}\times H/\Gamma
  24. Λ n \Lambda_{n}
  25. [ 1 x z n 0 1 y 0 0 1 ] , \begin{bmatrix}1&x&\frac{z}{n}\\ 0&1&y\\ 0&0&1\end{bmatrix},
  26. Λ n \Lambda_{n}
  27. Λ n \Lambda_{n}
  28. 2 {\mathbb{Z}}^{2}
  29. Λ n / C = 2 \Lambda_{n}/C={\mathbb{Z}}^{2}
  30. Γ n := Λ n \Gamma_{n}:=\Lambda_{n}\ltimes{\mathbb{Z}}
  31. {\mathbb{Z}}
  32. Λ n \Lambda_{n}
  33. 3 {\mathbb{R}}^{3}
  34. Γ n \Gamma_{n}
  35. 3 = × {\mathbb{R}}^{3}={\mathbb{C}}\times{\mathbb{R}}
  36. Γ n \Gamma_{n}
  37. × H = × × > 0 {\mathbb{C}}\times H={\mathbb{C}}\times{\mathbb{R}}\times{\mathbb{R}}^{>0}
  38. Λ n \Lambda_{n}
  39. > 0 {\mathbb{R}}^{>0}
  40. {\mathbb{Z}}
  41. v e t log b v v\mapsto e^{t\log b}v
  42. S 0 S^{0}
  43. × H / Γ n {\mathbb{C}}\times H/\Gamma_{n}
  44. S + S^{+}
  45. S - S^{-}
  46. 2 {\mathbb{Z}}^{2}

Input_offset_voltage.html

  1. V o s V_{os}
  2. V o s V_{os}

Integer_points_in_convex_polyhedra.html

  1. \mathbb{Z}

Integral_graph.html

  1. K ¯ n \bar{K}_{n}

Integral_sliding_mode.html

  1. x = f ( x ) + B ( x ) u + h ( x ) \overset{\cdot}{x}=f(x)+B(x)u+h(x)
  2. σ ( t ) = G x ( t ) - G x ( 0 ) + 0 t [ G B u 0 ( τ ) + G f ( x ( τ ) ) ] d τ \sigma(t)=Gx(t)-Gx(0)+\int_{0}^{t}[GBu_{0}(\tau)+Gf(x(\tau))]d\tau
  3. σ ( 0 ) = 0 \sigma(0)=0

Integrating_ADC.html

  1. V o u t = - V i n R C t i n t + V i n i t i a l V_{out}=-\dfrac{V_{in}}{RC}t_{int}+V_{initial}
  2. V o u t - u p = - V i n R C t u V_{out-up}=-\dfrac{V_{in}}{RC}t_{u}
  3. V o u t - d o w n = - V r e f R C t d + V o u t - u p = 0 V_{out-down}=-\dfrac{V_{ref}}{RC}t_{d}+V_{out-up}=0
  4. V i n V_{in}
  5. V i n = - V r e f t d t u V_{in}=-V_{ref}\dfrac{t_{d}}{t_{u}}
  6. V i n V_{in}
  7. V i n = - V r e f V_{in}=-V_{ref}
  8. t d = 2 r f c l k t_{d}=\dfrac{2^{r}}{f_{clk}}
  9. t u = t d t_{u}=t_{d}
  10. 2 t d 2t_{d}
  11. t m = 2 2 r f c l k t_{m}=2\dfrac{2^{r}}{f_{clk}}
  12. V i n V_{in}
  13. V i n = - V r e f R a R b t d t u V_{in}=-V_{ref}\dfrac{R_{a}}{R_{b}}\dfrac{t_{d}}{t_{u}}
  14. V i n V_{in}
  15. R p R_{p}
  16. R n R_{n}
  17. R i R_{i}
  18. V o u t = - N V i n t Δ R i + N p V r e f t Δ R p - N n V r e f t Δ R n C V_{out}=-\dfrac{\dfrac{NV_{in}t_{\Delta}}{R_{i}}+\dfrac{N_{p}V_{ref}t_{\Delta}% }{R_{p}}-\dfrac{N_{n}V_{ref}t_{\Delta}}{R_{n}}}{C}
  19. t Δ t_{\Delta}
  20. N p N_{p}
  21. N n N_{n}
  22. N N
  23. V i n V_{in}
  24. N N
  25. N V i n R i = - ( N p V r e f R p - N n V r e f R n ) \dfrac{NV_{in}}{R_{i}}=-\left(\dfrac{N_{p}V_{ref}}{R_{p}}-\dfrac{N_{n}V_{ref}}% {R_{n}}\right)
  26. V i n V_{in}
  27. N p = 0 , N n = N N_{p}=0,N_{n}=N
  28. N p = 1 , N n = N - 1 N_{p}=1,N_{n}=N-1
  29. N p N_{p}
  30. N n N_{n}
  31. N N
  32. r = l o g 2 R i ( R p + R n ) N R n R p r=log_{2}\dfrac{R_{i}(R_{p}+R_{n})}{NR_{n}R_{p}}
  33. R p R_{p}
  34. R n R_{n}
  35. R d / 1000 R_{d}/1000
  36. V i n V_{in}
  37. R d / 1000 R_{d}/1000
  38. R d / 1000 R_{d}/1000
  39. R d / 100 R_{d}/100
  40. R d R_{d}
  41. B B
  42. V Δ = V r e f R C 1 f c l k V_{\Delta}=\dfrac{V_{ref}}{RC}\dfrac{1}{f_{clk}}
  43. B B
  44. T f i r s t = V m a x C R s 1 f c l k V r e f T_{first}=\left\lceil\dfrac{V_{max}CR_{s1}f_{clk}}{V_{ref}}\right\rceil
  45. T f i r s t T_{first}
  46. V m a x V_{max}
  47. R s 1 R_{s1}
  48. T d B ( N - 1 ) T_{d}\leq B(N-1)
  49. N N
  50. C s l o p e = ± V r e f T s l o p e R s l o p e f c l k C_{slope}=\pm\dfrac{V_{ref}T_{slope}}{R_{slope}f_{clk}}
  51. T c l o c k T_{clock}
  52. B B
  53. R d / 100 R_{d}/100
  54. C s l o p e 2 C_{slope2}
  55. 100 V r e f R d f c l k C s l o p e 2 1000 V r e f R d f c l k \dfrac{100V_{ref}}{R_{d}f_{clk}}\leq C_{slope2}\leq\dfrac{1000V_{ref}}{R_{d}f_% {clk}}
  56. 100 V r e f R d f c l k \dfrac{100V_{ref}}{R_{d}f_{clk}}
  57. B B
  58. N N
  59. M M
  60. N = l o g B M N=log_{B}M
  61. T d B ( l o g B ( M ) - 1 ) T_{d}\leq B(log_{B}(M)-1)
  62. N p N_{p}
  63. N n N_{n}
  64. V o u t V_{out}
  65. V i n = R i ( - N p V r e f t Δ R p + N n V r e f t Δ R n - C V o u t ) N t Δ V_{in}=\dfrac{R_{i}\left(-\dfrac{N_{p}V_{ref}t_{\Delta}}{R_{p}}+\dfrac{N_{n}V_% {ref}t_{\Delta}}{R_{n}}-CV_{out}\right)}{Nt_{\Delta}}
  66. C V o u t CV_{out}
  67. V o u t 1 V_{out1}
  68. V o u t 2 V_{out2}
  69. V i n = R i ( - N p V r e f t Δ R p + N n V r e f t Δ R n - C ( V o u t 2 - V o u t 1 ) ) N t Δ V_{in}=\dfrac{R_{i}\left(-\dfrac{N_{p}V_{ref}t_{\Delta}}{R_{p}}+\dfrac{N_{n}V_% {ref}t_{\Delta}}{R_{n}}-C(V_{out2}-V_{out1})\right)}{Nt_{\Delta}}

Interfacial_thermal_resistance.html

  1. Q = Δ T R = G Δ T Q=\frac{\Delta T}{R}=G\Delta T
  2. Q Q
  3. Δ T \Delta T
  4. R R
  5. G G
  6. Q 1 , 2 = k n ( k , T 1 ) E ( k ) α ( k , T 1 , T 2 ) Q_{1,2}=\sum_{k}n\left(k,T_{1}\right)E\left(k\right)\alpha\left(k,T_{1},T_{2}\right)
  7. Q n e t = Q 1 , 2 - Q 2 , 1 Q_{net}\ =\ Q_{1,2}\ -\ Q_{2,1}
  8. R t h = Δ T Q / A R_{th}\ =\ \frac{\Delta T}{Q/A}
  9. E = ω ( k ) ν E\ =\ \hbar\ \omega\left(k\right)\ \nu

Interference_reflection_microscopy.html

  1. r 12 r_{12}\!
  2. R R\!
  3. I r I_{r}\!
  4. I i I_{i}\!

Intermembral_index.html

  1. ( h u m e r u s + r a d i u s ) ( f e m u r + t i b i a ) × 100 \tfrac{(humerus+radius)}{(femur+tibia)}\times 100

Internal_pressure.html

  1. π T \pi_{T}
  2. π T = ( U V ) T \pi_{T}=\left(\frac{\partial U}{\partial V}\right)_{T}
  3. π T = T ( p T ) V - p \pi_{T}=T\left(\frac{\partial p}{\partial T}\right)_{V}-p
  4. d U = ( U S ) V d S + ( U V ) S d V \operatorname{d}U=\left(\frac{\partial U}{\partial S}\right)_{V}\operatorname{% d}S+\left(\frac{\partial U}{\partial V}\right)_{S}\operatorname{d}V
  5. d V \operatorname{d}V
  6. ( U V ) T = ( U S ) V ( S V ) T + ( U V ) S \left(\frac{\partial U}{\partial V}\right)_{T}=\left(\frac{\partial U}{% \partial S}\right)_{V}\left(\frac{\partial S}{\partial V}\right)_{T}+\left(% \frac{\partial U}{\partial V}\right)_{S}
  7. d U = T d S - p d V \operatorname{d}U=T\operatorname{d}S-p\operatorname{d}V
  8. ( U S ) V = T \left(\frac{\partial U}{\partial S}\right)_{V}=T
  9. ( U V ) S = - p \left(\frac{\partial U}{\partial V}\right)_{S}=-p
  10. π T = T ( S V ) T - p \pi_{T}=T\left(\frac{\partial S}{\partial V}\right)_{T}-p
  11. ( S V ) T = ( p T ) V \left(\frac{\partial S}{\partial V}\right)_{T}=\left(\frac{\partial p}{% \partial T}\right)_{V}
  12. d U d T \operatorname{d}U\propto\operatorname{d}T
  13. d T = 0 dT=0
  14. d U = 0 dU=0
  15. π T = 0 \pi_{T}=0
  16. π T = 0 \pi_{T}=0
  17. p V = n R T pV=nRT
  18. π T > 0 \pi_{T}>0
  19. π T < 0 \pi_{T}<0
  20. lim V π T = 0 \lim_{V\to\infty}\pi_{T}=0
  21. p = n R T V - n b - a n 2 V 2 p=\frac{nRT}{V-nb}-a\frac{n^{2}}{V^{2}}
  22. π T = a n 2 V 2 \pi_{T}=a\frac{n^{2}}{V^{2}}
  23. a a

Internal_tide.html

  1. 2 {}^{2}

Interplanetary_scintillation.html

  1. m = Δ I 2 1 / 2 I . m=\frac{\langle\Delta I^{2}\rangle^{1/2}}{\langle I\rangle}.
  2. m 2 Δ ϕ . m\approx\sqrt{2}\Delta\phi.
  3. ϕ R M S = λ r e ( a L ) 1 / 2 [ δ N 2 ] 1 / 2 , \phi_{RMS}=\lambda r_{e}\left(aL\right)^{1/2}\left[\langle\delta N^{2}\rangle% \right]^{1/2},
  4. λ \lambda
  5. r e r_{e}
  6. L L
  7. a a
  8. δ N 2 \delta N^{2}

Intersection_(set_theory).html

  1. A B = { x : x A x B } A\cap B=\{x:x\in A\,\land\,x\in B\}
  2. A B = A\cap B=\varnothing
  3. ( x 𝐌 ) ( A 𝐌 , x A ) . \left(x\in\bigcap\mathbf{M}\right)\Leftrightarrow\left(\forall A\in\mathbf{M},% \ x\in A\right).
  4. i = 1 A i . \bigcap_{i=1}^{\infty}A_{i}.
  5. 𝐌 = { x : A 𝐌 , x A } . \bigcap\mathbf{M}=\{x:\forall A\in\mathbf{M},x\in A\}.
  6. 𝐌 = { x U : A 𝐌 , x A } . \bigcap\mathbf{M}=\{x\in U:\forall A\in\mathbf{M},x\in A\}.

Inventory_theory.html

  1. k k
  2. x k x_{k}
  3. u k u_{k}
  4. w k w_{k}
  5. w w
  6. x k + 1 = x k + u k - w k x_{k+1}=x_{k}+u_{k}-w_{k}
  7. u k 0 u_{k}\geq 0
  8. x k x_{k}
  9. c k = c ( x k , u k ) c_{k}=c(x_{k},u_{k})
  10. c k = p ( x k ) + h ( u k ) c_{k}=p(x_{k})+h(u_{k})
  11. u k u_{k}
  12. k = 0 c k \sum_{k=0}^{\infty}c_{k}
  13. x i k x_{ik}

Inverse_bundle.html

  1. E M E\rightarrow M
  2. E M E^{\prime}\rightarrow M
  3. E E
  4. E E M × n . E\oplus E^{\prime}\cong M\times\mathbb{R}^{n}.\,

Inverse_gas_chromatography.html

  1. V R = j m F ( t R - t o ) T 273.15 V_{R}^{\circ}=\frac{j}{m}F(t_{R}-t_{o})\frac{T}{273.15}
  2. W adh = 2 ( γ 1 γ 2 ) 1 / 2 W_{\mathrm{adh}}=2(\gamma_{1}\gamma_{2})^{1/2}

Inverse_magnetostrictive_effect.html

  1. λ \lambda
  2. M M
  3. H H
  4. σ \sigma
  5. σ \sigma
  6. B B
  7. H H
  8. λ s \lambda_{s}
  9. σ \sigma
  10. ( d λ d H ) σ = ( d B d σ ) H \left(\frac{d\lambda}{dH}\right)_{\sigma}=\left(\frac{dB}{d\sigma}\right)_{H}
  11. σ λ s \sigma\lambda_{s}
  12. B B
  13. σ λ s \sigma\lambda_{s}
  14. B B
  15. σ \sigma
  16. E σ E_{\sigma}
  17. E σ = 3 2 λ s σ sin 2 ( θ ) E_{\sigma}=\frac{3}{2}\lambda_{s}\sigma\sin^{2}(\theta)
  18. λ s \lambda_{s}
  19. θ \theta
  20. λ s \lambda_{s}
  21. σ \sigma
  22. θ \theta
  23. λ 100 \lambda_{100}
  24. λ 111 \lambda_{111}
  25. λ 100 \lambda_{100}
  26. λ 111 \lambda_{111}
  27. λ 100 = λ 111 = λ \lambda_{100}=\lambda_{111}=\lambda
  28. E σ = 3 2 λ σ ( α 1 γ 1 + α 2 γ 2 + α 3 γ 3 ) 2 E_{\sigma}=\frac{3}{2}\lambda\sigma(\alpha_{1}\gamma_{1}+\alpha_{2}\gamma_{2}+% \alpha_{3}\gamma_{3})^{2}
  29. α 1 \alpha_{1}
  30. α 2 \alpha_{2}
  31. α 3 \alpha_{3}
  32. γ 1 \gamma_{1}
  33. γ 2 \gamma_{2}
  34. γ 3 \gamma_{3}

Inversion_(discrete_mathematics).html

  1. ( A ( 1 ) , , A ( n ) ) (A(1),\ldots,A(n))
  2. i < j i<j
  3. A ( i ) > A ( j ) A(i)>A(j)
  4. ( i , j ) (i,j)
  5. A A
  6. inv ( A ) = | { ( A ( i ) , A ( j ) ) i < j and A ( i ) > A ( j ) } | \,\text{inv}(A)=|\{(A(i),A(j))\mid i<j\,\text{ and }A(i)>A(j)\}|
  7. O ( n l o g n ) O(nlogn)
  8. V [ i ] = | { k k < i and A ( k ) > A ( i ) } | V[i]=\left|\{k\mid k<i\,\text{ and }A(k)>A(i)\}\right|

Invertible_module.html

  1. M P R P M_{P}\cong R_{P}
  2. M R M * R M\otimes_{R}M^{*}\cong R

Invex_function.html

  1. f ( x ) - f ( u ) g ( x , u ) f ( u ) , f(x)-f(u)\geqq g(x,u)\cdot\nabla f(u),\,

Ion-association.html

  1. F = q 1 q 2 ϵ r 2 , F=\frac{q_{1}q_{2}}{\epsilon r^{2}},

IPhone_4.html

  1. arctan ( 1 / 300 12 ) \arctan\big(\tfrac{1/300}{12}\big)

Iron_oxide_nanoparticles.html

  1. T C ′′ ′′ {}^{\prime\prime}T^{\prime\prime}_{C}
  2. T N T_{N}

Irreducible_ideal.html

  1. \mathbb{Z}
  2. 4 4\mathbb{Z}

Irving_Stringham.html

  1. ln ( x ) \ln(x)
  2. x x
  3. ln ( x ) \ln(x)
  4. log e ( x ) \log_{e}(x)
  5. log e {}^{e}\log
  6. ln \ln

Isbell_conjugacy.html

  1. 𝒱 \mathcal{V}
  2. 𝒜 \mathcal{A}
  3. 𝒱 \mathcal{V}
  4. 𝒱 𝒜 o p \mathcal{V}^{\mathcal{A}^{op}}
  5. ( 𝒱 𝒜 ) o p (\mathcal{V}^{\mathcal{A}})^{op}
  6. Y : 𝒜 𝒱 𝒜 o p Y:\mathcal{A}\rightarrow\mathcal{V}^{\mathcal{A}^{op}}
  7. Z : 𝒜 ( 𝒱 𝒜 ) o p Z:\mathcal{A}\rightarrow(\mathcal{V}^{\mathcal{A}})^{op}

Iso-damping.html

  1. ω c {\omega}_{c}
  2. d G ( s ) d s | s = j ω c = 0 \frac{d\angle G(s)}{ds}{|}_{s=j\omega_{c}}=0
  3. d G ( s ) d s | s = j ω c = G ( s ) | s = j ω , \angle\frac{dG(s)}{ds}{|}_{s=j\omega_{c}}=\angle G(s){|}_{s=j\omega},
  4. ω c \omega_{c}
  5. G ( s ) G(s)
  6. L ( s ) = ( s ω g c ) α L(s)=\left(\frac{s}{\omega_{gc}}\right)^{\alpha}
  7. ω g c {\omega}_{gc}
  8. α < 0 \alpha<0
  9. L ( s ) L(s)
  10. - 2 < α < - 1 -2<\alpha<-1
  11. 20 α 20\alpha
  12. α π 2 \frac{\alpha\pi}{2}
  13. arg ( L ( j ω ) ) = α π 2 \arg(L(j\omega))=\frac{\alpha\pi}{2}
  14. s s

Isobar_(nuclide).html

  1. m ( A , Z ) = Z m p + N m n - a V A + a S A 2 / 3 + a C Z 2 A 1 / 3 + a A ( N - Z ) 2 A - δ ( A , Z ) m(A,Z)=Zm_{p}+Nm_{n}-a_{V}A+a_{S}A^{2/3}+a_{C}\frac{Z^{2}}{A^{1/3}}+a_{A}\frac% {(N-Z)^{2}}{A}-\delta(A,Z)
  2. A A
  3. Z Z
  4. N N
  5. Z Z
  6. N N
  7. A A
  8. δ = 0 δ=0
  9. Z Z
  10. N N
  11. N Z N−Z
  12. A A
  13. A A
  14. δ δ
  15. δ ( A , Z ) = ( - 1 ) Z a P A - 1 2 \delta(A,Z)=(-1)^{Z}a_{P}A^{-\frac{1}{2}}
  16. a < s u b > P a<sub>P

Isolating_neighborhood.html

  1. F t : X X , t , . F_{t}:X\to X,\quad t\in\mathbb{Z},\mathbb{R}.
  2. Inv ( N , F ) := { x N : F t ( x ) N for all t } Int N , \operatorname{Inv}(N,F):=\{x\in N:F_{t}(x)\in N{\ }\,\text{for all }t\}% \subseteq\operatorname{Int}\,N,
  3. f : X X f:X\to X
  4. A = n 0 f n ( N ) , A Int N . A=\bigcap_{n\geq 0}f^{n}(N),\quad A\subseteq\operatorname{Int}\,N.

Isolation_lemma.html

  1. n n
  2. N N
  3. \mathcal{F}
  4. { 1 , , n } \{1,\dots,n\}
  5. x { 1 , , n } x\in\{1,\dots,n\}
  6. w ( x ) w(x)
  7. { 1 , , N } \{1,\dots,N\}
  8. \mathcal{F}
  9. w ( S ) = x S w ( x ) . w(S)=\sum_{x\in S}w(x)\,.
  10. 1 - n / N 1-n/N
  11. \mathcal{F}
  12. \mathcal{F}
  13. \mathcal{F}
  14. \mathcal{F}
  15. 2 n - 1 2^{n}-1
  16. \mathcal{F}
  17. 1 1
  18. n N nN
  19. ( 2 n - 1 ) / ( n N ) (2^{n}-1)/(nN)
  20. w ( x ) α w(x)\leq\alpha
  21. w ( x ) < α w(x)<\alpha
  22. Pr [ x is singular ] = Pr [ w ( x ) = α ] 1 / N \Pr[x\,\text{ is singular}]=\Pr[w(x)=\alpha]\leq 1/N
  23. \leq
  24. α ( x ) = min S , x S w ( S ) - min S , x S w ( S { x } ) . \alpha(x)=\min_{S\in\mathcal{F},x\not\in S}w(S)-\min_{S\in\mathcal{F},x\in S}w% (S\setminus\{x\}).\,
  25. α ( x ) \alpha(x)
  26. α ( x ) \alpha(x)
  27. α ( x ) \alpha(x)
  28. w ( x ) = α ( x ) w(x)=\alpha(x)
  29. \mathcal{F}
  30. α ( x ) = min S , x S w ( S ) - min S , x S w ( S { x } ) = w ( B ) - ( w ( A ) - w ( x ) ) = w ( x ) , \begin{aligned}\displaystyle\alpha(x)&\displaystyle=\min_{S\in\mathcal{F},x% \not\in S}w(S)-\min_{S\in\mathcal{F},x\in S}w(S\setminus\{x\})\\ &\displaystyle=w(B)-(w(A)-w(x))\\ &\displaystyle=w(x),\end{aligned}
  31. \mathcal{F}
  32. x i j x_{ij}
  33. 2 w i j 2^{w_{ij}}
  34. w i j w_{ij}
  35. ( S , ) (S,\mathcal{F})
  36. \mathcal{F}
  37. \mathcal{F}

Isolator_(microwave).html

  1. S = ( 0 0 1 0 ) S=\begin{pmatrix}0&0\\ 1&0\end{pmatrix}
  2. S = ( 0 0 1 1 0 0 0 1 0 ) S=\begin{pmatrix}0&0&1\\ 1&0&0\\ 0&1&0\end{pmatrix}

Isserlis'_theorem.html

  1. E [ x 1 x 2 x 2 n ] = E [ x i x j ] , \displaystyle\operatorname{E}[\,x_{1}x_{2}\cdots x_{2n}\,]=\sum\prod% \operatorname{E}[\,x_{i}x_{j}\,],
  2. ( 2 n ) ! / ( 2 n n ! ) (2n)!/(2^{n}n!)
  3. E [ x 1 x 2 x 3 x 4 ] = E [ x 1 x 2 ] E [ x 3 x 4 ] + E [ x 1 x 3 ] E [ x 2 x 4 ] + E [ x 1 x 4 ] E [ x 2 x 3 ] . \operatorname{E}[\,x_{1}x_{2}x_{3}x_{4}\,]=\operatorname{E}[x_{1}x_{2}]\,% \operatorname{E}[x_{3}x_{4}]+\operatorname{E}[x_{1}x_{3}]\,\operatorname{E}[x_% {2}x_{4}]+\operatorname{E}[x_{1}x_{4}]\,\operatorname{E}[x_{2}x_{3}].
  4. E [ X 1 X 2 X 3 X 4 X 5 X 6 ] \displaystyle{}\operatorname{E}[X_{1}X_{2}X_{3}X_{4}X_{5}X_{6}]

Iterative_proportional_fitting.html

  1. L 1 L_{1}
  2. ( x i j ) (x_{ij})
  3. m ^ i j = a i b j \hat{m}_{ij}=a_{i}b_{j}
  4. ( m ^ i j ) (\hat{m}_{ij})
  5. ( m i j ) (m_{ij})
  6. x i + = j x i j \textstyle x_{i+}=\sum_{j}x_{ij}\,
  7. x + j = i x i j \textstyle x_{+j}=\sum_{i}x_{ij}\,
  8. log m i j = u + v i + w j + z i j \log\ m_{ij}=u+v_{i}+w_{j}+z_{ij}
  9. m i j := 𝔼 ( x i j ) m_{ij}:=\mathbb{E}(x_{ij})
  10. i v i = j w j = 0 \sum_{i}v_{i}=\sum_{j}w_{j}=0
  11. z i j = 0 z_{ij}=0
  12. m ^ i j ( 0 ) := 1 \hat{m}_{ij}^{(0)}:=1
  13. η 1 \eta\geq 1
  14. m ^ i j ( 2 η - 1 ) = m ^ i j ( 2 η - 2 ) x i + k = 1 J m ^ i k ( 2 η - 2 ) \hat{m}_{ij}^{(2\eta-1)}=\frac{\hat{m}_{ij}^{(2\eta-2)}x_{i+}}{\sum_{k=1}^{J}% \hat{m}_{ik}^{(2\eta-2)}}
  15. m ^ i j ( 2 η ) = m ^ i j ( 2 η - 1 ) x + j k = 1 I m ^ k j ( 2 η - 1 ) . \hat{m}_{ij}^{(2\eta)}=\frac{\hat{m}_{ij}^{(2\eta-1)}x_{+j}}{\sum_{k=1}^{I}% \hat{m}_{kj}^{(2\eta-1)}}.
  16. ( m ^ i j ) := lim η ( m ^ i j ( η ) ) (\hat{m}_{ij}):=\lim_{\eta\rightarrow\infty}(\hat{m}^{(\eta)}_{ij})
  17. x i + x_{i+}
  18. u i u_{i}
  19. m i j = 0 m_{ij}=0
  20. ( i , j ) S (i,j)\in S
  21. m ^ i j ( 0 ) = 0 \hat{m}_{ij}^{(0)}=0
  22. ( i , j ) S (i,j)\in S
  23. b ^ j ( 0 ) := 1 \hat{b}_{j}^{(0)}:=1
  24. η 1 \eta\geq 1
  25. a ^ i ( η ) = x i + j b ^ j ( η - 1 ) , \hat{a}_{i}^{(\eta)}=\frac{x_{i+}}{\sum_{j}\hat{b}_{j}^{(\eta-1)}},
  26. b ^ j ( η ) = x + j i a ^ i ( η ) \hat{b}_{j}^{(\eta)}=\frac{x_{+j}}{\sum_{i}\hat{a}_{i}^{(\eta)}}
  27. m ^ i j ( 2 η ) = a ^ i ( η ) b ^ j ( η ) \hat{m}_{ij}^{(2\eta)}=\hat{a}_{i}^{(\eta)}\hat{b}_{j}^{(\eta)}
  28. ( m ^ i j ) = a ^ b ^ T (\hat{m}_{ij})=\hat{a}\hat{b}^{T}
  29. a ^ = ( a ^ 1 , , a ^ I ) T = lim η a ^ ( η ) \hat{a}=(\hat{a}_{1},\ldots,\hat{a}_{I})^{T}=\lim_{\eta\rightarrow\infty}\hat{% a}^{(\eta)}
  30. b ^ = ( b ^ 1 , , b ^ J ) T = lim η b ^ ( η ) \hat{b}=(\hat{b}_{1},\ldots,\hat{b}_{J})^{T}=\lim_{\eta\rightarrow\infty}\hat{% b}^{(\eta)}
  31. m i j = a i b j = ( γ a i ) ( 1 γ b j ) m_{ij}=a_{i}b_{j}=(\gamma a_{i})(\frac{1}{\gamma}b_{j})
  32. γ > 0 \gamma>0
  33. i a ^ i ( η ) = i a ^ i ( 1 ) \sum_{i}\hat{a}_{i}^{(\eta)}=\sum_{i}\hat{a}_{i}^{(1)}
  34. η 1 \eta\geq 1
  35. j b ^ j ( η ) = j b ^ j ( 0 ) \sum_{j}\hat{b}_{j}^{(\eta)}=\sum_{j}\hat{b}_{j}^{(0)}
  36. η 0 \eta\geq 0
  37. m i j = 0 m_{ij}=0
  38. ( i , j ) S (i,j)\in S
  39. δ i j = 0 \delta_{ij}=0
  40. i , j ) S i,j)\in S
  41. δ i j = 1 \delta_{ij}=1
  42. a ^ i ( η ) = x i + j δ i j b ^ j ( η - 1 ) , \hat{a}_{i}^{(\eta)}=\frac{x_{i+}}{\sum_{j}\delta_{ij}\hat{b}_{j}^{(\eta-1)}},
  43. b ^ j ( η ) = x + j i δ i j a ^ i ( η ) \hat{b}_{j}^{(\eta)}=\frac{x_{+j}}{\sum_{i}\delta_{ij}\hat{a}_{i}^{(\eta)}}
  44. m ^ i j ( 2 η ) = δ i j a ^ i ( η ) b ^ j ( η ) \hat{m}_{ij}^{(2\eta)}=\delta_{ij}\hat{a}_{i}^{(\eta)}\hat{b}_{j}^{(\eta)}
  45. M := ( m i j ( 0 ) ) I × J M:=(m^{(0)}_{ij})\in\mathbb{R}^{I\times J}
  46. u I u\in\mathbb{R}^{I}
  47. v J v\in\mathbb{R}^{J}
  48. M ^ = ( m ^ i j ) I × J \hat{M}=(\hat{m}_{ij})\in\mathbb{R}^{I\times J}
  49. m ^ i + = j = 1 n m ^ i j = u i \hat{m}_{i+}=\sum_{j=1}^{n}\hat{m}_{ij}=u_{i}
  50. m ^ + j = i = 1 m m ^ i j = v j \hat{m}_{+j}=\sum_{i=1}^{m}\hat{m}_{ij}=v_{j}
  51. d i a g : k k × k diag:\mathbb{R}^{k}\longrightarrow\mathbb{R}^{k\times k}
  52. η 0 \eta\geq 0
  53. M ( 2 η + 1 ) = diag ( r ( η + 1 ) ) M ( 2 η ) M^{(2\eta+1)}=\,\text{diag}(r^{(\eta+1)})M^{(2\eta)}
  54. M ( 2 η + 2 ) = M ( 2 η + 1 ) diag ( s ( η + 1 ) ) M^{(2\eta+2)}=M^{(2\eta+1)}\,\text{diag}(s^{(\eta+1)})
  55. r i η + 1 = u i j m i j ( 2 η ) r_{i}^{\eta+1}=\frac{u_{i}}{\sum_{j}m_{ij}^{(2\eta)}}
  56. s j η + 1 = v j i m i j ( 2 η + 1 ) s_{j}^{\eta+1}=\frac{v_{j}}{\sum_{i}m_{ij}^{(2\eta+1)}}
  57. M ^ = lim η M ( η ) . \hat{M}=\lim_{\eta\rightarrow\infty}M^{(\eta)}.
  58. m i j ( 0 ) m h k ( 0 ) m i k ( 0 ) m h j ( 0 ) = m i j ( η ) m h k ( η ) m i k ( η ) m h j ( η ) η 0 and i h , j k \frac{m^{(0)}_{ij}m^{(0)}_{hk}}{m^{(0)}_{ik}m^{(0)}_{hj}}=\frac{m^{(\eta)}_{ij% }m^{(\eta)}_{hk}}{m^{(\eta)}_{ik}m^{(\eta)}_{hj}}\ \forall\ \eta\geq 0\,\text{% and }i\neq h,\quad j\neq k
  59. m i j ( η ) = a i ( η ) b j ( η ) . m^{(\eta)}_{ij}=a_{i}^{(\eta)}b_{j}^{(\eta)}.
  60. I J ( 2 + J ) + I J ( 2 + I ) = I 2 J + I J 2 + 4 I J IJ(2+J)+IJ(2+I)=I^{2}J+IJ^{2}+4IJ\,
  61. I ( 1 + J ) + J ( 1 + I ) = 2 I J + I + J I(1+J)+J(1+I)=2IJ+I+J\,
  62. x i + > 0 , x + j > 0 x_{i+}>0,\ x_{+j}>0
  63. ( m ^ i j ) (\hat{m}_{ij})
  64. X 2 = i , j ( x i j - m i j ^ ) 2 m i j ^ X^{2}=\sum_{i,j}\frac{(x_{ij}-\hat{m_{ij}})^{2}}{\hat{m_{ij}}}
  65. G = 2 i , j x i j log x i j m ^ i j G=2\sum_{i,j}x_{ij}\log\ \frac{x_{ij}}{\hat{m}_{ij}}
  66. \Chi r 2 \Chi^{2}_{r}
  67. r = ( I - 1 ) ( J - 1 ) r=(I-1)(J-1)
  68. 1 - \Chi r 2 ( X 2 ) 1-\Chi^{2}_{r}(X^{2})
  69. 1 - \Chi r 2 ( G ) 1-\Chi^{2}_{r}(G)
  70. p ( X 2 ) 0.1824671 p(X^{2})\approx 0.1824671

Iterative_refinement.html

  1. m m
  2. m m
  3. \lVertsymbol x m - s y m b o l x \lVertsymbol x ( σ κ ( s y m b o l A ) ε 1 ) m + μ 1 ε 1 + μ 2 n κ ( s y m b o l A ) ε 2 \frac{\lVertsymbol{x}_{m}-symbol{x}^{\ast}\rVert_{\infty}}{\lVertsymbol{x}^{% \ast}\rVert_{\infty}}\leq\bigl(\sigma\kappa_{\infty}(symbol{A})\varepsilon_{1}% \bigr)^{m}+\mu_{1}\varepsilon_{1}+\mu_{2}n\kappa_{\infty}(symbol{A})% \varepsilon_{2}
  4. n n
  5. < 𝐯𝐚𝐫 > 𝐀 \mathbf{<var>A}

ITIES.html

  1. Δ wo ϕ = ϕ w - ϕ o = Δ wo ϕ i + R T z i F ln ( a o i a w i ) \Delta\text{w}\text{o}\phi=\phi\text{w}-\phi\text{o}=\Delta\text{w}\text{o}% \phi^{\ominus}_{i}+\frac{RT}{z_{i}F}\ln\left(\frac{a\text{o}_{i}}{a\text{w}_{i% }}\right)
  2. Δ wo ϕ i \Delta\text{w}\text{o}\phi^{\ominus}_{i}
  3. Δ wo ϕ i = Δ G t r , i , w o z i F \Delta\text{w}\text{o}\phi^{\ominus}_{i}=\frac{\Delta G^{\ominus,\,\text{w}% \rightarrow\,\text{o}}_{tr,i}}{z_{i}F}
  4. Δ wo ϕ = Δ wo ϕ ET + R T F l n ( a w R 1 a o O 2 a w O 1 a o R 2 ) \Delta\text{w}\text{o}\phi=\Delta\text{w}\text{o}\phi^{\ominus}\text{ET}+\frac% {RT}{F}ln\left(\frac{a\text{w}_{\,\text{R}_{1}}a\text{o}_{\,\text{O}_{2}}}{a% \text{w}_{\,\text{O}_{1}}a\text{o}_{\,\text{R}_{2}}}\right)
  5. Δ w o ϕ ET \Delta\text{w}_{o}\phi^{\ominus}\text{ET}
  6. Δ wo ϕ ET = [ E O 2 / R 2 ] oSHE - [ E O 1 / R 1 ] wSHE \Delta\text{w}\text{o}\phi^{\ominus}\text{ET}=\left[E^{\ominus}_{\,\text{O}_{2% }/\,\text{R}_{2}}\right]\text{o}\text{SHE}-\left[E^{\ominus}_{\,\text{O}_{1}/% \,\text{R}_{1}}\right]\text{w}\text{SHE}
  7. P i = a o i a w i = exp [ z i F R T ( Δ wo ϕ - Δ wo ϕ i ) ] = P i exp [ z i F R T Δ wo ϕ ] P_{i}=\frac{a\text{o}_{i}}{a\text{w}_{i}}=\exp\left[\frac{z_{i}F}{RT}(\Delta% \text{w}\text{o}\phi-\Delta\text{w}\text{o}\phi^{\ominus}_{i})\right]=P^{% \ominus}_{i}\exp\left[\frac{z_{i}F}{RT}\Delta\text{w}\text{o}\phi\right]
  8. Δ wo ϕ = Δ wo ϕ C+ + Δ wo ϕ A- 2 + R T 2 F ln ( γ oC+ γ wA- γ wC+ γ oA- ) \Delta\text{w}\text{o}\phi=\frac{\Delta\text{w}\text{o}\phi^{\ominus}\text{C+}% +\Delta\text{w}\text{o}\phi^{\ominus}\text{A-}}{2}+\frac{RT}{2F}\ln{\left(% \frac{\gamma\text{o}\text{C+}\gamma\text{w}\text{A-}}{\gamma\text{w}\text{C+}% \gamma\text{o}\text{A-}}\right)}

J1_J2_model.html

  1. H ^ = J 1 i j S i S j + J 2 i j S i S j \hat{H}=J_{1}\sum_{\langle ij\rangle}\vec{S}_{i}\cdot\vec{S}_{j}+J_{2}\sum_{% \langle\langle ij\rangle\rangle}\vec{S}_{i}\cdot\vec{S}_{j}

Jacobi_coordinates.html

  1. s y m b o l r 1 = x 1 - x 2 , symbol{r_{1}=x_{1}-x_{2}}\ ,
  2. s y m b o l r j = 1 m 0 j k = 1 j m k s y m b o l x k - s y m b o l x j + 1 , symbol{r_{j}}=\frac{1}{m_{0j}}\sum_{k=1}^{j}m_{k}symbol{x_{k}}\ -\ symbol{x_{j% +1}}\ ,
  3. m 0 j = k = 1 j m k . m_{0j}=\sum_{k=1}^{j}\ m_{k}\ .
  4. s y m b o l R = 1 m 0 k = 1 N m k s y m b o l x k ; symbolR=\frac{1}{m_{0}}\sum_{k=1}^{N}\ m_{k}symbol{x_{k}}\ ;
  5. m 0 = k = 1 N m k . m_{0}=\sum_{k=1}^{N}\ m_{k}\ .

Jacobi_form.html

  1. H R ( n , h ) H^{(n,h)}_{R}
  2. ϕ ( a τ + b c τ + d , z c τ + d ) = ( c τ + d ) k e 2 π i m c z 2 c τ + d ϕ ( τ , z ) for ( a b c d ) S L 2 ( Z ) \phi\left(\frac{a\tau+b}{c\tau+d},\frac{z}{c\tau+d}\right)=(c\tau+d)^{k}e^{% \frac{2\pi imcz^{2}}{c\tau+d}}\phi(\tau,z)\,\text{ for }{a\ b\choose c\ d}\in SL% _{2}(Z)
  3. ϕ ( τ , z + λ τ + μ ) = e - 2 π i m ( λ 2 τ + 2 λ z ) ϕ ( τ , z ) \phi(\tau,z+\lambda\tau+\mu)=e^{-2\pi im(\lambda^{2}\tau+2\lambda z)}\phi(\tau% ,z)
  4. ϕ \phi
  5. ϕ ( τ , z ) = n 0 r 2 4 m n c ( n , r ) e 2 π i ( n τ + r z ) . \phi(\tau,z)=\sum_{n\geq 0}\sum_{r^{2}\leq 4mn}c(n,r)e^{2\pi i(n\tau+rz)}.

Jacobian_curve.html

  1. Q : { Q 1 ( X 0 , X 1 , X 2 , X 3 ) = 0 } { Q 2 ( X 0 , X 1 , X 2 , X 3 ) = 0 } Q:\{Q_{1}(X_{0},X_{1},X_{2},X_{3})=0\}\cap\{Q_{2}(X_{0},X_{1},X_{2},X_{3})=0\}
  2. Φ : ( x , y ) ( X , Y , Z , T ) = ( x , y , 1 , x 2 ) \Phi:(x,y)\mapsto(X,Y,Z,T)=(x,y,1,x^{2})
  3. 𝐒 : { X 2 - T Z = 0 Y 2 - a X Z - b Z 2 - T X = 0 \mathbf{S}:\begin{cases}X^{2}-TZ=0\\ Y^{2}-aXZ-bZ^{2}-TX=0\end{cases}
  4. 𝐒 1 : { X 2 + Y 2 - T 2 = 0 k X 2 + Z 2 - T 2 = 0 \mathbf{S}1:\begin{cases}X^{2}+Y^{2}-T^{2}=0\\ kX^{2}+Z^{2}-T^{2}=0\end{cases}
  5. ( x , y ) ( X , Y , Z , T ) = ( - 2 y , x 2 - j , x 2 + 2 j x + j , x 2 + 2 x + j ) (x,y)\mapsto(X,Y,Z,T)=(-2y,x^{2}-j,x^{2}+2jx+j,x^{2}+2x+j)
  6. O = ( 0 : 1 : 0 ) ( 0 , 1 , 1 , 1 ) O=(0:1:0)\mapsto(0,1,1,1)
  7. X 3 = T 1 Y 2 X 1 Z 2 + Z 1 X 2 Y 1 T 2 X_{3}=T_{1}Y_{2}X_{1}Z_{2}+Z_{1}X_{2}Y_{1}T_{2}
  8. Y 3 = T 1 Y 2 Y 1 T 2 - Z 1 X 2 X 1 Z 2 Y_{3}=T_{1}Y_{2}Y_{1}T_{2}-Z_{1}X_{2}X_{1}Z_{2}
  9. Z 3 = T 1 Z 1 T 2 Z 2 - k X 1 Y 1 X 2 Y 2 Z_{3}=T_{1}Z_{1}T_{2}Z_{2}-kX_{1}Y_{1}X_{2}Y_{2}
  10. T 3 = ( T 1 Y 2 ) 2 + ( Z 1 X 2 ) 2 T_{3}=(T_{1}Y_{2})^{2}+(Z_{1}X_{2})^{2}
  11. X 3 = 2 Y 1 T 1 Z 1 X 1 X_{3}=2Y_{1}T_{1}Z_{1}X_{1}
  12. Y 3 = ( T 1 Y 1 ) 2 - ( T 1 Z 1 ) 2 + ( Z 1 Y 1 ) 2 Y_{3}=(T_{1}Y_{1})^{2}-(T_{1}Z_{1})^{2}+(Z_{1}Y_{1})^{2}
  13. Z 3 = ( T 1 Z 1 ) 2 - ( T 1 Y 1 ) 2 + ( Z 1 Y 1 ) 2 Z_{3}=(T_{1}Z_{1})^{2}-(T_{1}Y_{1})^{2}+(Z_{1}Y_{1})^{2}
  14. T 3 = ( T 1 Z 1 ) 2 + ( T 1 Y 1 ) 2 - ( Z 1 Y 1 ) 2 T_{3}=(T_{1}Z_{1})^{2}+(T_{1}Y_{1})^{2}-(Z_{1}Y_{1})^{2}
  15. 𝐒 1 : { X 2 + Y 2 - T 2 = 0 4 X 2 + Z 2 - T 2 = 0 \mathbf{S}1:\begin{cases}X^{2}+Y^{2}-T^{2}=0\\ 4X^{2}+Z^{2}-T^{2}=0\end{cases}
  16. P 1 = ( 1 , 3 , 0 , 2 ) P_{1}=(1,\sqrt{3},0,2)
  17. P 2 = ( 1 , 2 , 1 , 5 ) P_{2}=(1,2,1,\sqrt{5})
  18. X 3 = T 1 Y 2 X 1 Z 2 + Z 1 X 2 Y 1 T 2 = 4 X_{3}=T_{1}Y_{2}X_{1}Z_{2}+Z_{1}X_{2}Y_{1}T_{2}=4
  19. Y 3 = T 1 Y 2 Y 1 T 2 - Z 1 X 2 X 1 Z 2 = 4 15 Y_{3}=T_{1}Y_{2}Y_{1}T_{2}-Z_{1}X_{2}X_{1}Z_{2}=4\sqrt{15}
  20. Z 3 = T 1 Z 1 T 2 Z 2 - k X 1 Y 1 X 2 Y 2 = - 8 3 Z_{3}=T_{1}Z_{1}T_{2}Z_{2}-kX_{1}Y_{1}X_{2}Y_{2}=-8\sqrt{3}
  21. T 3 = ( T 1 Y 2 ) 2 + ( Z 1 X 2 ) 2 = 16 T_{3}=(T_{1}Y_{2})^{2}+(Z_{1}X_{2})^{2}=16
  22. P 3 = ( 4 , 4 15 , - 8 3 , 16 ) P_{3}=(4,4\sqrt{15},-8\sqrt{3},16)
  23. X 3 = 2 Y 1 T 1 Z 1 X 1 = 0 X_{3}=2Y_{1}T_{1}Z_{1}X_{1}=0
  24. Y 3 = ( T 1 Y 1 ) 2 - ( T 1 Z 1 ) 2 + ( Z 1 Y 1 ) 2 = 12 Y_{3}=(T_{1}Y_{1})^{2}-(T_{1}Z_{1})^{2}+(Z_{1}Y_{1})^{2}=12
  25. Z 3 = ( T 1 Z 1 ) 2 - ( T 1 Y 1 ) 2 + ( Z 1 Y 1 ) 2 = - 12 Z_{3}=(T_{1}Z_{1})^{2}-(T_{1}Y_{1})^{2}+(Z_{1}Y_{1})^{2}=-12
  26. T 3 = ( T 1 Z 1 ) 2 + ( T 1 Y 1 ) 2 - ( Z 1 Y 1 ) 2 = 12 T_{3}=(T_{1}Z_{1})^{2}+(T_{1}Y_{1})^{2}-(Z_{1}Y_{1})^{2}=12
  27. x 3 = y 2 x 1 z 2 + z 1 x 2 y 1 ( y 2 2 + ( z 1 x 2 ) 2 ) x_{3}=\frac{y_{2}x_{1}z_{2}+z_{1}x_{2}y_{1}}{(y_{2}^{2}+(z_{1}x_{2})^{2})}
  28. y 3 = y 2 y 1 - z 1 x 2 x 1 z 2 ( y 2 2 + ( z 1 x 2 ) 2 ) y_{3}=\frac{y_{2}y_{1}-z_{1}x_{2}x_{1}z_{2}}{(y_{2}^{2}+(z_{1}x_{2})^{2})}
  29. z 3 = z 1 z 2 - a x 1 y 1 x 2 y 2 ( y 2 2 + ( z 1 x 2 ) 2 ) z_{3}=\frac{z_{1}z_{2}-ax_{1}y_{1}x_{2}y_{2}}{(y_{2}^{2}+(z_{1}x_{2})^{2})}
  30. 𝐒 1 : { x 2 + y 2 = 1 k x 2 + z 2 = 1 \mathbf{S}1:\begin{cases}x^{2}+y^{2}=1\\ kx^{2}+z^{2}=1\end{cases}
  31. x = X / T x=X/T
  32. y = Y / T y=Y/T
  33. z = Z / T z=Z/T
  34. X Y = X Y XY=X\cdot Y
  35. Z T = Z T ZT=Z\cdot T
  36. ( p , 0 ) ( 0 : - 1 : 1 ) (p,0)\mapsto(0:-1:1)
  37. ( x , y ) ( p , 0 ) ( 2 ( x - p ) : ( 2 x + p ) ( x - p ) 2 - y 2 : y ) (x,y)\neq(p,0)\mapsto(2(x-p):(2x+p)(x-p)^{2}-y^{2}:y)
  38. O ( 0 : 1 : 1 ) O\mapsto(0:1:1)
  39. C : Y 2 = e X 4 - 2 d X 2 Z 2 + Z 4 C:\ Y^{2}=eX^{4}-2dX^{2}Z^{2}+Z^{4}
  40. e = - ( 3 p 2 + 4 a ) 16 , d = 3 p 4 e=\frac{-(3p^{2}+4a)}{16},\ \ d=\frac{3p}{4}
  41. y 2 = e x 4 + 2 a x 2 + 1 y^{2}=ex^{4}+2ax^{2}+1
  42. P 1 = ( x 1 , y 1 ) P_{1}=(x_{1},y_{1})
  43. P 2 = ( x 2 , y 2 ) P_{2}=(x_{2},y_{2})
  44. P 3 = ( x 3 , y 3 ) P_{3}=(x_{3},y_{3})
  45. x 3 = x 1 y 2 + y 1 x 2 1 - ( x 1 x 2 ) 2 x_{3}=\frac{x_{1}y_{2}+y_{1}x_{2}}{1-(x_{1}x_{2})^{2}}
  46. y 3 = ( ( 1 + ( x 1 x 2 ) 2 ) ( y 1 y 2 + 2 a x 1 x 2 ) + 2 x 1 x 2 ( x 1 2 + x 2 2 ) ) ( 1 - ( x 1 x 2 ) 2 ) 2 y_{3}=\frac{((1+(x_{1}x_{2})^{2})(y_{1}y_{2}+2ax_{1}x_{2})+2x_{1}x_{2}({x_{1}}% ^{2}+{x_{2}}^{2}))}{(1-(x_{1}x_{2})^{2})^{2}}
  47. X 3 = X 1 Z 1 Y 2 + Y 1 X 2 Z 2 X_{3}=X_{1}Z_{1}Y_{2}+Y_{1}X_{2}Z_{2}
  48. Y 3 = ( Z 1 2 Z 2 2 + e X 1 2 X 2 2 ) ( Y 1 Y 2 - 2 d X 1 X 2 Z 1 Z 2 ) + 2 e X 1 X 2 Z 1 Z 2 ( X 1 2 Z 2 2 + Z 1 2 X 2 2 ) Y_{3}=\left(Z_{1}^{2}Z_{2}^{2}+eX_{1}^{2}X_{2}^{2}\right)\left(Y_{1}Y_{2}-2dX_% {1}X_{2}Z_{1}Z_{2}\right)\ +\ 2eX_{1}X_{2}Z_{1}Z_{2}\left(X_{1}^{2}Z_{2}^{2}+Z% _{1}^{2}X_{2}^{2}\right)
  49. Z 3 = Z 1 2 Z 2 2 - e X 1 2 X 2 2 Z_{3}=Z_{1}^{2}Z_{2}^{2}-eX_{1}^{2}X_{2}^{2}
  50. C : Y 2 = X 4 + Z 4 C:\ Y^{2}=X^{4}+Z^{4}
  51. P 1 = ( 1 : 2 : 1 ) P_{1}=(1:\sqrt{2}:1)
  52. P 2 = ( 2 : 17 : 1 ) P_{2}=(2:\sqrt{17}:1)
  53. X 3 = 1 1 17 + 2 2 1 = 17 + 2 2 X_{3}=1\cdot 1\cdot\sqrt{17}+\sqrt{2}\cdot 2\cdot 1=\sqrt{17}+2\sqrt{2}
  54. Y 3 = ( 1 2 1 2 + 1 1 2 2 2 ) ( 2 17 - 2 0 1 2 1 1 ) + 2 1 1 2 1 1 ( 1 2 1 2 + 1 2 2 2 ) = 5 34 + 20 Y_{3}=\left(1^{2}\cdot 1^{2}+1\cdot 1^{2}\cdot 2^{2}\right)\left(\sqrt{2}\cdot% \sqrt{17}-2\cdot 0\cdot 1\cdot 2\cdot 1\cdot 1\right)+2\cdot 1\cdot 1\cdot 2% \cdot 1\cdot 1\left(1^{2}\cdot 1^{2}+1^{2}\cdot 2^{2}\right)=5\sqrt{34}+20
  55. Z 3 = 1 2 1 2 - 1 1 2 2 2 = - 3 Z_{3}=1^{2}\cdot 1^{2}-1\cdot 1^{2}\cdot 2^{2}=-3
  56. P 3 = ( 17 + 2 2 : 5 34 + 20 : - 3 ) P_{3}=(\sqrt{17}+2\sqrt{2}:5\sqrt{34}+20:-3)
  57. X 3 = 1 1 2 + 2 1 1 = 2 2 X_{3}=1\cdot 1\cdot\sqrt{2}+\sqrt{2}\cdot 1\cdot 1=2\sqrt{2}
  58. Y 3 = ( 1 + 1 1 ) ( 2 2 - 2 0 1 1 1 1 ) + 2 1 ( 1 2 1 2 + 1 2 1 2 ) = 8 Y_{3}=\left(1+1\cdot 1\right)\left(\sqrt{2}\cdot\sqrt{2}-2\cdot 0\cdot 1\cdot 1% \cdot 1\cdot 1\right)+2\cdot 1\left(1^{2}\cdot 1^{2}+1^{2}\cdot 1^{2}\right)=8
  59. Z 3 = 1 2 1 2 - 1 1 2 1 2 = 0 Z_{3}=1^{2}\cdot 1^{2}-1\cdot 1^{2}\cdot 1^{2}=0
  60. P 4 = ( 2 2 : 8 : 0 ) P_{4}=(2\sqrt{2}:8:0)
  61. y 2 = x 4 + 2 a x 2 + 1 y^{2}=x^{4}+2ax^{2}+1
  62. x = X / Z x=X/Z
  63. y = Y / Z Z y=Y/ZZ
  64. X X = X 2 XX=X^{2}
  65. Z Z = Z 2 ZZ=Z^{2}
  66. R = 2 X Z R=2\cdot X\cdot Z
  67. x = X / Z x=X/Z
  68. y = Y / Z 2 y=Y/Z^{2}
  69. x = X / Z x=X/Z
  70. y = Y / Z Z y=Y/ZZ
  71. X X = X 2 XX=X^{2}
  72. Z Z = Z 2 ZZ=Z^{2}
  73. x = X / Z x=X/Z
  74. y = Y / Z Z y=Y/ZZ
  75. X X = X 2 XX=X^{2}
  76. Z Z = Z 2 ZZ=Z^{2}
  77. R = 2 X Z R=2\cdot X\cdot Z
  78. x = X / Z x=X/Z
  79. y = Y / Z 2 y=Y/Z^{2}

János_Komlós_(mathematician).html

  1. 1 2 n log n + 1 2 n log log n + c n \frac{1}{2}n\log n+\frac{1}{2}n\log\log n+cn
  2. e - e - 2 c . e^{-e^{-2c}}.

János_Pintz.html

  1. lim inf n p n + 1 - p n log p n = 0 \liminf_{n\to\infty}\frac{p_{n+1}-p_{n}}{\log p_{n}}=0
  2. p n p_{n}

Jean-Marc_Fontaine.html

  1. p p

Job_scheduling_game.html

  1. M M
  2. N N
  3. i i
  4. p i = ( p i 1 , , p i m ) p_{i}=(p_{i}^{1},...,p_{i}^{m})
  5. p i j p_{i}^{j}
  6. i i
  7. j j
  8. M M
  9. x x
  10. M i M_{i}
  11. M d M_{d}
  12. x x
  13. M d M_{d}
  14. x x
  15. M - 1 M-1
  16. M d M_{d}
  17. M M
  18. M M
  19. \infty
  20. M 1 M_{1}
  21. M 2 M_{2}
  22. J 1 = 1 , K J_{1}=1,K
  23. J 2 = K , 1 J_{2}=K,1
  24. K > 1 K>1
  25. J 1 J_{1}
  26. M 1 M_{1}
  27. K K
  28. M 2 M_{2}
  29. J 2 J_{2}
  30. K K
  31. M 1 M_{1}
  32. M 2 M_{2}
  33. J 1 J_{1}
  34. M 1 M_{1}
  35. J 2 J_{2}
  36. M 2 M_{2}
  37. J 1 J_{1}
  38. M 2 M_{2}
  39. J 2 J_{2}
  40. M 1 M_{1}
  41. K K
  42. J 1 J_{1}
  43. M 2 M_{2}
  44. K K
  45. K + 1 K+1
  46. J 2 J_{2}
  47. K K
  48. K K

Johansen_test.html

  1. X t = μ + Φ D t + Π p X t - p + + Π 1 X t - 1 + e t , t = 1 , , T X_{t}=\mu+\Phi D_{t}+\Pi_{p}X_{t-p}+\cdots+\Pi_{1}X_{t-1}+e_{t},\quad t=1,% \dots,T
  2. Δ X t = μ + Φ D t + Π X t - p + Γ p - 1 Δ X t - p + 1 + + Γ 1 Δ X t - 1 + ε t , t = 1 , , T \Delta X_{t}=\mu+\Phi D_{t}+\Pi X_{t-p}+\Gamma_{p-1}\Delta X_{t-p+1}+\cdots+% \Gamma_{1}\Delta X_{t-1}+\varepsilon_{t},\quad t=1,\dots,T
  3. Γ i = Π 1 + + Π i - I , i = 1 , , p - 1. \Gamma_{i}=\Pi_{1}+\cdots+\Pi_{i}-I,\quad i=1,\dots,p-1.\,
  4. Δ X t = μ + Φ D t - Γ p - 1 Δ X t - p + 1 - - Γ 1 Δ X t - 1 + Π X t - 1 + ε t , t = 1 , , T \Delta X_{t}=\mu+\Phi D_{t}-\Gamma_{p-1}\Delta X_{t-p+1}-\cdots-\Gamma_{1}% \Delta X_{t-1}+\Pi X_{t-1}+\varepsilon_{t},\quad t=1,\cdots,T
  5. Γ i = ( Π i + 1 + + Π p ) , i = 1 , , p - 1. \Gamma_{i}=\left(\Pi_{i+1}+\cdots+\Pi_{p}\right),\quad i=1,\dots,p-1.\,
  6. Π = Π 1 + + Π p - I . \Pi=\Pi_{1}+\cdots+\Pi_{p}-I.\,

Johnson_scheme.html

  1. v = | X | = ( n ) v=\left|X\right|={\left({{\ell}\atop{n}}\right)}
  2. p i ( k ) = E i ( k ) , p_{i}\left(k\right)=E_{i}\left(k\right),
  3. q k ( i ) = μ k v i E i ( k ) , q_{k}\left(i\right)=\frac{\mu_{k}}{v_{i}}E_{i}\left(k\right),
  4. μ i = - 2 i + 1 - i + 1 ( i ) , \mu_{i}=\frac{\ell-2i+1}{\ell-i+1}{\left({{\ell}\atop{i}}\right)},
  5. E k ( x ) = j = 0 k ( - 1 ) j ( x j ) ( n - x k - j ) ( - n - x k - j ) , k = 0 , , n . E_{k}\left(x\right)=\sum_{j=0}^{k}(-1)^{j}{\left({{x}\atop{j}}\right)}{\left({% {n-x}\atop{k-j}}\right)}{\left({{\ell-n-x}\atop{k-j}}\right)},\qquad k=0,% \ldots,n.

Johnson–Holmquist_damage_model.html

  1. σ i j = - p ( ϵ k k ) δ i j + 2 μ ϵ i j \sigma_{ij}=-p(\epsilon_{kk})~{}\delta_{ij}+2~{}\mu~{}\epsilon_{ij}
  2. σ i j \sigma_{ij}
  3. p ( ϵ k k ) p(\epsilon_{kk})
  4. δ i j \delta_{ij}
  5. ϵ i j \epsilon_{ij}
  6. σ i j \sigma_{ij}
  7. μ \mu
  8. ϵ k k \epsilon_{kk}
  9. ξ \xi
  10. p ( ξ ) = p ( ξ ( ϵ k k ) ) = p ( ρ ρ 0 - 1 ) ; ξ := ρ ρ 0 - 1 p(\xi)=p(\xi(\epsilon_{kk}))=p\left(\cfrac{\rho}{\rho_{0}}-1\right)~{};~{}~{}% \xi:=\cfrac{\rho}{\rho_{0}}-1
  11. ρ \rho
  12. ρ 0 \rho_{0}
  13. σ h = ( ρ , μ ) = p HEL ( ρ ) + 2 3 σ HEL ( ρ , μ ) \sigma_{h}=\mathcal{H}(\rho,\mu)=p_{\rm HEL}(\rho)+\cfrac{2}{3}~{}\sigma_{\rm HEL% }(\rho,\mu)
  14. p HEL p_{\rm HEL}
  15. σ HEL \sigma_{\rm HEL}
  16. σ intact * = A ( p * + T * ) n [ 1 + C ln ( d ϵ p d t ) ] \sigma^{*}_{\rm intact}=A~{}(p^{*}+T^{*})^{n}~{}\left[1+C~{}\ln\left(\cfrac{d% \epsilon_{p}}{dt}\right)\right]
  17. A , C , n A,C,n
  18. t t
  19. ϵ p \epsilon_{p}
  20. σ * \sigma^{*}
  21. p * p^{*}
  22. T * T^{*}
  23. σ * = σ σ HEL ; p * = p p HEL ; T * = T σ h \sigma^{*}=\cfrac{\sigma}{\sigma_{\rm HEL}}~{};~{}p^{*}=\cfrac{p}{p_{\rm HEL}}% ~{};~{}~{}T^{*}=\cfrac{T}{\sigma_{h}}
  24. σ fracture * = B ( p * ) m [ 1 + C ln ( d ϵ p d t ) ] \sigma^{*}_{\rm fracture}=B~{}(p^{*})^{m}~{}\left[1+C~{}\ln\left(\cfrac{d% \epsilon_{p}}{dt}\right)\right]
  25. B , C , M B,C,M
  26. σ * = σ initial * - D ( σ initial * - σ fracture * ) \sigma^{*}=\sigma^{*}_{\rm initial}-D~{}\left(\sigma^{*}_{\rm initial}-\sigma^% {*}_{\rm fracture}\right)
  27. D D
  28. D D
  29. d D d t = 1 ϵ f d ϵ p d t \cfrac{dD}{dt}=\cfrac{1}{\epsilon_{f}}~{}\cfrac{d\epsilon_{p}}{dt}
  30. ϵ f \epsilon_{f}
  31. ϵ f = D 1 ( p * + T * ) D 2 \epsilon_{f}=D_{1}~{}(p^{*}+T^{*})^{D_{2}}
  32. D 1 , D 2 D_{1},D_{2}
  33. B 4 C B_{4}C
  34. S i C SiC
  35. A l N AlN
  36. A l 2 O 3 Al_{2}O_{3}
  37. p ( ξ ) p(\xi)
  38. p ( ξ ) = { k 1 ξ + k 2 ξ 2 + k 3 ξ 3 + Δ p Compression k 1 ξ Tension p(\xi)=\begin{cases}k_{1}~{}\xi+k_{2}~{}\xi^{2}+k_{3}~{}\xi^{3}+\Delta p&% \qquad\,\text{Compression}\\ k_{1}~{}\xi&\qquad\,\text{Tension}\end{cases}
  39. Δ p \Delta p
  40. k 1 , k 2 , k 3 k_{1},k_{2},k_{3}

Join_dependency.html

  1. R R
  2. R 1 R_{1}
  3. R n R_{n}
  4. R R
  5. R R
  6. * ( R 1 , R 2 , , R n ) *(R_{1},R_{2},\ldots,R_{n})
  7. X Y X\twoheadrightarrow Y
  8. * ( X Y , X ( U - Y ) ) *(X\cup Y,X\cup(U-Y))

Joint_spectral_radius.html

  1. = { A 1 , , A m } n × n , \mathcal{M}=\{A_{1},\dots,A_{m}\}\subset\mathbb{R}^{n\times n},
  2. ρ ( ) = lim k max { A i 1 A i k 1 / k : A i } . \rho(\mathcal{M})=\lim_{k\to\infty}\max{\{\|A_{i_{1}}\cdots A_{i_{k}}\|^{1/k}:% A_{i}\in\mathcal{M}\}}.\,
  3. \mathcal{M}
  4. ρ 1 ? \rho\leq 1?
  5. n × n , \mathcal{M}\subset\mathbb{R}^{n\times n},
  6. A 1 A t A_{1}\dots A_{t}
  7. ρ ( ) = ρ ( A 1 A t ) 1 / t . \rho(\mathcal{M})=\rho(A_{1}\dots A_{t})^{1/t}.
  8. ρ ( A 1 A t ) \rho(A_{1}\dots A_{t})
  9. A 1 A t . A_{1}\dots A_{t}.
  10. x t + 1 = A t x t , A t t x_{t+1}=A_{t}x_{t},\quad A_{t}\in\mathcal{M}\,\forall t
  11. ρ ( ) < 1. \rho(\mathcal{M})<1.
  12. \mathcal{M}
  13. L p L_{p}

Jucys–Murphy_element.html

  1. [ S n ] \mathbb{C}[S_{n}]
  2. X 1 = 0 , X k = ( 1 k ) + ( 2 k ) + + ( k - 1 k ) , k = 2 , , n . X_{1}=0,~{}~{}~{}X_{k}=(1k)+(2k)+\cdots+(k-1\ k),~{}~{}~{}k=2,\dots,n.
  3. [ S n ] \mathbb{C}[S_{n}]
  4. [ S n - 1 ] \mathbb{C}[S_{n-1}]
  5. X k v U = c k ( U ) v U , k = 1 , , n , X_{k}v_{U}=c_{k}(U)v_{U},~{}~{}~{}k=1,\dots,n,
  6. Z ( [ S n ] ) Z(\mathbb{C}[S_{n}])
  7. [ S n ] \mathbb{C}[S_{n}]
  8. [ S n ] \mathbb{C}[S_{n}]
  9. ( t + X 1 ) ( t + X 2 ) ( t + X n ) = σ S n σ t number of cycles of σ . (t+X_{1})(t+X_{2})\cdots(t+X_{n})=\sum_{\sigma\in S_{n}}\sigma t^{\,\text{% number of cycles of }\sigma}.
  10. [ S n ] \mathbb{C}[S_{n}]
  11. Z ( [ S 1 ] ) , Z ( [ S 2 ] ) , , Z ( [ S n - 1 ] ) , Z ( [ S n ] ) Z(\mathbb{C}[S_{1}]),Z(\mathbb{C}[S_{2}]),\ldots,Z(\mathbb{C}[S_{n-1}]),Z(% \mathbb{C}[S_{n}])

Jurkat–Richert_theorem.html

  1. r A ( d ) = | A d | - ω ( d ) d X . r_{A}(d)=\left|A_{d}\right|-\frac{\omega(d)}{d}X.
  2. V ( z ) = p P , p < z ( 1 - ω ( p ) p ) . V(z)=\prod_{p\in P,p<z}\left(1-\frac{\omega(p)}{p}\right).
  3. S ( A , P , z ) X V ( z ) ( F 1 ( log y log z ) + O ( ( log log y ) 3 / 4 ( log y ) 1 / 4 ) ) + m | P ( z ) , m < y 4 ν ( m ) | r A ( m ) | S(A,P,z)\leq XV(z)\left(F_{1}\left(\frac{\log y}{\log z}\right)+O\left(\frac{(% \log\log y)^{3/4}}{(\log y)^{1/4}}\right)\right)+\sum_{m|P(z),m<y}4^{\nu(m)}% \left|r_{A}(m)\right|
  4. S ( A , P , z ) X V ( z ) ( f 1 ( log y log z ) - O ( ( log log y ) 3 / 4 ( log y ) 1 / 4 ) ) - m | P ( z ) , m < y 4 ν ( m ) | r A ( m ) | . S(A,P,z)\geq XV(z)\left(f_{1}\left(\frac{\log y}{\log z}\right)-O\left(\frac{(% \log\log y)^{3/4}}{(\log y)^{1/4}}\right)\right)-\sum_{m|P(z),m<y}4^{\nu(m)}% \left|r_{A}(m)\right|.

K-distribution.html

  1. X X
  2. X X
  3. σ \sigma
  4. L L
  5. σ \sigma
  6. μ \mu
  7. ν \nu
  8. X X
  9. x > 0 x>0
  10. f X ( x ; μ , ν , L ) = 2 x ( L ν x μ ) L + ν 2 1 Γ ( L ) Γ ( ν ) K ν - L ( 2 L ν x μ ) f_{X}(x;\mu,\nu,L)=\frac{2}{x}\left(\frac{L\nu x}{\mu}\right)^{\frac{L+\nu}{2}% }\frac{1}{\Gamma(L)\Gamma(\nu)}K_{\nu-L}\left(2\sqrt{\frac{L\nu x}{\mu}}\right)
  11. K K
  12. L L
  13. μ \mu
  14. ν \nu
  15. E ( X ) = μ \operatorname{E}(X)=\mu
  16. var ( X ) = μ 2 ν + L + 1 L ν . \operatorname{var}(X)=\mu^{2}\frac{\nu+L+1}{L\nu}.
  17. L L
  18. ν \nu
  19. { μ x 2 f ′′ ( x ) - μ x ( L + ν - 3 ) f ( x ) + f ( x ) ( μ ( L - 1 ) ( ν - 1 ) - L ν x ) = 0 , f ( 1 ) = 2 ( L ν μ ) L 2 + ν 2 K ν - L ( 2 L ν μ ) Γ ( L ) Γ ( ν ) , f ( 1 ) = 2 ( L ν μ ) L + ν 2 ( ( L - 1 ) K L - ν ( 2 L ν μ ) - L ν μ K L - ν + 1 ( 2 L ν μ ) ) Γ ( L ) Γ ( ν ) } \left\{\begin{array}[]{l}\mu x^{2}f^{\prime\prime}(x)-\mu x(L+\nu-3)f^{\prime}% (x)+f(x)(\mu(L-1)(\nu-1)-L\nu x)=0,\\ f(1)=\frac{2\left(\frac{L\nu}{\mu}\right)^{\frac{L}{2}+\frac{\nu}{2}}K_{\nu-L}% \left(2\sqrt{\frac{L\nu}{\mu}}\right)}{\Gamma(L)\Gamma(\nu)},\\ f^{\prime}(1)=\frac{2\left(\frac{L\nu}{\mu}\right)^{\frac{L+\nu}{2}}\left((L-1% )K_{L-\nu}\left(2\sqrt{\frac{L\nu}{\mu}}\right)-\sqrt{\frac{L\nu}{\mu}}K_{L-% \nu+1}\left(2\sqrt{\frac{L\nu}{\mu}}\right)\right)}{\Gamma(L)\Gamma(\nu)}\end{% array}\right\}

K-factor_(marketing).html

  1. i = number of invites sent by each customer i=\,\text{number of invites sent by each customer }
  2. c = percent conversion of each invite c=\,\text{percent conversion of each invite }
  3. k = i * c k=i*c

K-means++.html

  1. k k
  2. < v a r > k <var>k

Kanade–Lucas–Tomasi_feature_tracker.html

  1. F ( x ) F(x)
  2. G ( x ) G(x)
  3. x x
  4. x x
  5. h h
  6. F ( x + h ) F(x+h)
  7. G ( x ) G(x)
  8. x x
  9. R R
  10. F ( x + h ) F(x+h)
  11. G ( x ) G(x)
  12. x R | F ( x + h ) - G ( x ) | \sum_{x\in R}\left|F(x+h)-G(x)\right|
  13. x R [ F ( x + h ) - G ( x ) ] 2 \sqrt{\sum_{x\in R}\left[F(x+h)-G(x)\right]^{2}}
  14. - x R F ( x + h ) G ( x ) x R F ( x + h ) 2 x R G ( x ) 2 \dfrac{-\sum_{x\in R}F(x+h)G(x)}{\sqrt{\sum_{x\in R}F(x+h)^{2}}\sqrt{\sum_{x% \in R}G(x)^{2}}}
  15. h h
  16. F ( x ) F(x)
  17. G ( x ) = F ( x + h ) G(x)=F(x+h)
  18. F ( x ) F ( x + h ) - F ( x ) h = G ( x ) - F ( x ) h F^{\prime}(x)\approx\dfrac{F(x+h)-F(x)}{h}=\dfrac{G(x)-F(x)}{h}\,
  19. h G ( x ) - F ( x ) F ( x ) h\approx\dfrac{G(x)-F(x)}{F^{\prime}(x)}\,
  20. h h
  21. x x
  22. h h
  23. x x
  24. h x G ( x ) - F ( x ) F ( x ) x 1 . h\approx\dfrac{\sum_{x}\dfrac{G(x)-F(x)}{F^{\prime}(x)}}{\sum_{x}1}.
  25. | F ′′ ( x ) | \left|F^{\prime\prime}(x)\right|
  26. F ′′ ( x ) G ( x ) - F ( x ) h . F^{\prime\prime}(x)\approx\dfrac{G^{\prime}(x)-F^{\prime}(x)}{h}.
  27. w ( x ) = 1 | G ( x ) - F ( x ) | . w(x)=\dfrac{1}{\left|G^{\prime}(x)-F^{\prime}(x)\right|}.
  28. h = x w ( x ) [ G ( x ) - F ( x ) ] F ( x ) x w ( x ) . h=\dfrac{\sum_{x}\dfrac{w(x)\left[G(x)-F(x)\right]}{F^{\prime}(x)}}{\sum_{x}w(% x)}.
  29. F ( x ) F(x)
  30. h h
  31. h h
  32. { h 0 = 0 h k + 1 = h k + x w ( x ) [ G ( x ) - F ( x + h k ) ] F ( x + h k ) x w ( x ) \begin{cases}h_{0}=0\\ h_{k+1}=h_{k}+\dfrac{\sum_{x}\dfrac{w(x)\left[G(x)-F(x+h_{k})\right]}{F^{% \prime}(x+h_{k})}}{\sum_{x}w(x)}\end{cases}
  33. F ( x + h ) F ( x ) + h F ( x ) , F(x+h)\approx F(x)+hF^{\prime}(x),
  34. h h
  35. E = x [ F ( x + h ) - G ( x ) ] 2 . E=\sum_{x}\left[F(x+h)-G(x)\right]^{2}.
  36. h h
  37. E E
  38. 0 \displaystyle 0
  39. h x F ( x ) [ G ( x ) - F ( x ) ] x F ( x ) 2 \Rightarrow h\approx\dfrac{\sum_{x}F^{\prime}(x)[G(x)-F(x)]}{\sum_{x}F^{\prime% }(x)^{2}}\,
  40. w ( x ) = F ( x ) 2 . w(x)=F^{\prime}(x)^{2}.
  41. { h 0 = 0 h k + 1 = h k + x w ( x ) F ( x + h k ) [ G ( x ) - F ( x + h k ) ] x w ( x ) F ( x + h k ) 2 \begin{cases}h_{0}=0\\ h_{k+1}=h_{k}+\dfrac{\sum_{x}w(x)F^{\prime}(x+h_{k})\left[G(x)-F(x+h_{k})% \right]}{\sum_{x}w(x)F^{\prime}(x+h_{k})^{2}}\end{cases}
  42. h k h_{k}
  43. h h
  44. F ( x ) = sin x , F(x)=\sin x,
  45. G ( x ) = F ( x + h ) = sin ( x + h ) . G(x)=F(x+h)=\sin(x+h).
  46. h h
  47. | h | < π \left|h\right|<\pi
  48. h 1 h_{1}
  49. F ( x ) = sin x F(x)=\sin x
  50. F G , F^{\prime}G,
  51. F F , F^{\prime}F,
  52. ( F ) 2 (F^{\prime})^{2}
  53. R . R.
  54. F ( x ) F^{\prime}(x)
  55. F ( x ) F ( x + Δ x ) - F ( x ) Δ x , F^{\prime}(x)\approx\dfrac{F(x+\Delta x)-F(x)}{\Delta x},
  56. Δ x \Delta x
  57. E = 𝐱 R [ F ( 𝐱 + 𝐡 ) - G ( 𝐱 ) ] 2 , E=\sum_{\mathbf{x}\in R}\left[F(\mathbf{x}+\mathbf{h})-G(\mathbf{x})\right]^{2},
  58. 𝐱 \mathbf{x}
  59. 𝐡 \mathbf{h}
  60. F ( 𝐱 + 𝐡 ) F ( 𝐱 ) + 𝐡 ( 𝐱 F ( 𝐱 ) ) T . F(\mathbf{x}+\mathbf{h})\approx F(\mathbf{x})+\mathbf{h}\left(\dfrac{\partial}% {\partial\mathbf{x}}F(\mathbf{x})\right)^{T}.
  61. E E
  62. 𝐡 \mathbf{h}
  63. 0 \displaystyle 0
  64. 𝐡 [ 𝐱 [ G ( 𝐱 ) - F ( 𝐱 ) ] ( F 𝐱 ) ] [ 𝐱 ( F 𝐱 ) T ( F 𝐱 ) ] - 1 , \Rightarrow\mathbf{h}\approx\left[\sum_{\mathbf{x}}\left[G(\mathbf{x})-F(% \mathbf{x})\right]\left(\dfrac{\partial F}{\partial\mathbf{x}}\right)\right]% \left[\sum_{\mathbf{x}}\left(\dfrac{\partial F}{\partial\mathbf{x}}\right)^{T}% \left(\dfrac{\partial F}{\partial\mathbf{x}}\right)\right]^{-1},
  65. G ( x ) = F ( A x + h ) , G(x)=F(Ax+h),
  66. A A
  67. E = x [ F ( A x + h ) - G ( x ) ] 2 . E=\sum_{x}\left[F(Ax+h)-G(x)\right]^{2}.
  68. Δ A \Delta A
  69. A A
  70. Δ h \Delta h
  71. h h
  72. F ( x ( A + Δ A ) + ( h + Δ h ) ) F(x(A+\Delta A)+(h+\Delta h))
  73. F ( A x + h ) + ( Δ A x + Δ h ) x F ( x ) . \approx F(Ax+h)+(\Delta Ax+\Delta h)\dfrac{\partial}{\partial x}F(x).
  74. F ( x ) = α G ( x ) + β , F(x)=\alpha G(x)+\beta,
  75. α \alpha
  76. β \beta
  77. E = x [ F ( A x + h ) - ( α G ( x ) + β ) ] 2 E=\sum_{x}\left[F(Ax+h)-(\alpha G(x)+\beta)\right]^{2}
  78. α , \alpha,
  79. β , \beta,
  80. A , A,
  81. h . h.
  82. d = e \nabla d=e\,
  83. \nabla
  84. λ 1 \lambda_{1}
  85. λ 2 \lambda_{2}
  86. \nabla
  87. T z = a Tz=a\,
  88. T T
  89. z z
  90. a a
  91. d = e \nabla d=e

Karlsruhe_metric.html

  1. d k ( p 1 , p 2 ) d_{k}(p_{1},p_{2})
  2. d k ( p 1 , p 2 ) = { min ( r 1 , r 2 ) δ ( p 1 , p 2 ) + | r 1 - r 2 | , if 0 δ ( p 1 , p 2 ) 2 r 1 + r 2 , otherwise d_{k}(p_{1},p_{2})=\begin{cases}\min(r_{1},r_{2})\cdot\delta(p_{1},p_{2})+|r_{% 1}-r_{2}|,&\,\text{if }0\leq\delta(p_{1},p_{2})\leq 2\\ r_{1}+r_{2},&\,\text{otherwise}\end{cases}
  3. ( r i , φ i ) (r_{i},\varphi_{i})
  4. p i p_{i}
  5. δ ( p 1 , p 2 ) = min ( | φ 1 - φ 2 | , 2 π - | φ 1 - φ 2 | ) \delta(p_{1},p_{2})=\min(|\varphi_{1}-\varphi_{2}|,2\pi-|\varphi_{1}-\varphi_{% 2}|)

Katapayadi_system.html

  1. \leftrightarrow
  2. \leftrightarrow
  3. \leq
  4. \leftrightarrow
  5. \leftrightarrow
  6. \leftrightarrow
  7. \leftrightarrow
  8. \leftrightarrow
  9. \leftrightarrow

Kato_theorem.html

  1. Z k = - a o 2 n ( 𝐫 ) d n ( 𝐫 ) d r | r 𝐑 𝐤 Z_{k}=-\frac{a_{o}}{2n(\mathbf{r})}\frac{dn(\mathbf{r})}{dr}|_{r\rightarrow% \mathbf{R_{k}}}
  2. 𝐑 𝐤 \mathbf{R_{k}}
  3. Z k Z_{k}
  4. a o = ( h 2 π m e ) 2 a_{o}=\left(\frac{h}{2\pi me}\right)^{2}

Kazuhiko_Nishijima.html

  1. Q = I 3 + B + S 2 Q=I\text{3}+\frac{B+S}{2}

Kelvin–Stokes_theorem.html

  1. γ γ
  2. D D
  3. γ γ
  4. S := ψ ( D ) S:=ψ(D)
  5. Γ Γ
  6. Γ ( t ) = ψ ( γ ( t ) ) Γ(t)=ψ(γ(t))
  7. 𝐅 \mathbf{F}
  8. Γ 𝐅 d Γ = S × 𝐅 d S \oint_{\Gamma}\mathbf{F}\,d\Gamma=\iint_{S}\nabla\times\mathbf{F}\,dS
  9. 𝐀 = ω 𝐀 = a 1 d x 1 + a 2 d x 2 + a 3 d x 3 \mathbf{A}=\omega_{\mathbf{A}}=a_{1}dx_{1}+a_{2}dx_{2}+a_{3}dx_{3}
  10. 𝐀 = ω 𝐀 * = a 1 d x 2 d x 3 + a 2 d x 3 d x 1 + a 3 d x 1 d x 2 \mathbf{A}={}^{*}\omega_{\mathbf{A}}=a_{1}dx_{2}\wedge dx_{3}+a_{2}dx_{3}% \wedge dx_{1}+a_{3}dx_{1}\wedge dx_{2}
  11. × 𝐅 = d ω 𝐅 ψ * ω 𝐅 = P 1 d u + P 2 d v ψ * ( d ω 𝐅 ) = ( P 2 u - P 1 v ) d u d v \begin{aligned}\displaystyle\nabla\times\mathbf{F}&\displaystyle=d\omega_{% \mathbf{F}}\\ \displaystyle\psi^{*}\omega_{\mathbf{F}}&\displaystyle=P_{1}du+P_{2}dv\\ \displaystyle\psi^{*}(d\omega_{\mathbf{F}})&\displaystyle=\left(\frac{\partial P% _{2}}{\partial u}-\frac{\partial P_{1}}{\partial v}\right)du\wedge dv\end{aligned}
  12. d d
  13. ψ < s u p > ψ<sup>∗
  14. 𝐏 ( u , v ) = ( P 1 ( u , v ) , P 2 ( u , v ) ) \mathbf{P}(u,v)=(P_{1}(u,v),P_{2}(u,v))
  15. 𝐏 \mathbf{P}
  16. 𝐅 \mathbf{F}
  17. 𝐏 ( u , v ) \mathbf{P}(u,v)
  18. u , v u,v
  19. P 1 ( u , v ) = 𝐅 ( ψ ( u , v ) ) | ψ u , P 2 ( u , v ) = 𝐅 ( ψ ( u , v ) ) | ψ v P_{1}(u,v)=\left\langle\mathbf{F}(\psi(u,v))\bigg|\frac{\partial\psi}{\partial u% }\right\rangle,\qquad P_{2}(u,v)=\left\langle\mathbf{F}(\psi(u,v))\bigg|\frac{% \partial\psi}{\partial v}\right\rangle
  20. | \langle\ |\ \rangle
  21. | A | \langle\ |A|\ \rangle
  22. A A
  23. n × m n×m
  24. A A
  25. 𝐱 𝐑 m , 𝐲 𝐑 n : 𝐱 | A | 𝐲 = 𝐱 t A 𝐲 \mathbf{x}\in\mathbf{R}^{m},\mathbf{y}\in\mathbf{R}^{n}\ :\qquad\left\langle% \mathbf{x}|A|\mathbf{y}\right\rangle={}^{t}\mathbf{x}A\mathbf{y}
  26. A A
  27. 𝐱 | A | 𝐲 = 𝐲 | A t | 𝐱 . \left\langle\mathbf{x}|A|\mathbf{y}\right\rangle=\left\langle\mathbf{y}\left|{% }^{t}A\right|\mathbf{x}\right\rangle.
  28. 𝐱 | A | 𝐲 + 𝐱 | B | 𝐲 = 𝐱 | A + B | 𝐲 \left\langle\mathbf{x}|A|\mathbf{y}\right\rangle+\left\langle\mathbf{x}|B|% \mathbf{y}\right\rangle=\left\langle\mathbf{x}|A+B|\mathbf{y}\right\rangle
  29. Γ 𝐅 d Γ = a b ( 𝐅 ψ ( t ) ) | d Γ d t ( t ) d t = a b ( 𝐅 ψ ( t ) ) | d ( ψ γ ) d t ( t ) d t = a b ( 𝐅 ψ ( t ) ) | ( J ψ ) γ ( t ) d γ d t ( t ) d t \begin{aligned}\displaystyle\oint_{\Gamma}\mathbf{F}d\Gamma&\displaystyle=\int% _{a}^{b}\left\langle(\mathbf{F}\circ\psi(t))\bigg|\frac{d\Gamma}{dt}(t)\right% \rangle\,dt\\ &\displaystyle=\int_{a}^{b}\left\langle(\mathbf{F}\circ\psi(t))\bigg|\frac{d(% \psi\circ\gamma)}{dt}(t)\right\rangle\,dt\\ &\displaystyle=\int_{a}^{b}\left\langle(\mathbf{F}\circ\psi(t))\bigg|(J\psi)_{% \gamma(t)}\cdot\frac{d\gamma}{dt}(t)\right\rangle\,dt\end{aligned}
  30. J ψ
  31. ψ ψ
  32. ( 𝐅 Γ ( t ) ) | ( J ψ ) γ ( t ) d γ d t ( t ) = ( 𝐅 Γ ( t ) ) | ( J ψ ) γ ( t ) | d γ d t ( t ) = ( 𝐅 t Γ ( t ) ) ( J ψ ) γ ( t ) | d γ d t ( t ) = ( ( 𝐅 ( ψ ( γ ( t ) ) ) ) | ψ u ( γ ( t ) ) , ( 𝐅 ( ψ ( γ ( t ) ) ) ) | ψ v ( γ ( t ) ) ) | d γ d t ( t ) = ( P 1 ( u , v ) , P 2 ( u , v ) ) | d γ d t ( t ) = 𝐏 ( u , v ) | d γ d t ( t ) \begin{aligned}\displaystyle\left\langle(\mathbf{F}\circ\Gamma(t))\bigg|(J\psi% )_{\gamma(t)}\frac{d\gamma}{dt}(t)\right\rangle&\displaystyle=\left\langle(% \mathbf{F}\circ\Gamma(t))\bigg|(J\psi)_{\gamma(t)}\bigg|\frac{d\gamma}{dt}(t)% \right\rangle\\ &\displaystyle=\left\langle({}^{t}\mathbf{F}\circ\Gamma(t))\cdot(J\psi)_{% \gamma(t)}\ \bigg|\ \frac{d\gamma}{dt}(t)\right\rangle\\ &\displaystyle=\left\langle\left(\left\langle(\mathbf{F}(\psi(\gamma(t))))% \bigg|\frac{\partial\psi}{\partial u}(\gamma(t))\right\rangle,\left\langle(% \mathbf{F}(\psi(\gamma(t))))\bigg|\frac{\partial\psi}{\partial v}(\gamma(t))% \right\rangle\right)\bigg|\frac{d\gamma}{dt}(t)\right\rangle\\ &\displaystyle=\left\langle(P_{1}(u,v),P_{2}(u,v))\bigg|\frac{d\gamma}{dt}(t)% \right\rangle\\ &\displaystyle=\left\langle\mathbf{P}(u,v)\bigg|\frac{d\gamma}{dt}(t)\right% \rangle\end{aligned}
  33. Γ 𝐅 d Γ = γ 𝐏 d γ \oint_{\Gamma}\mathbf{F}d\Gamma=\oint_{\gamma}\mathbf{P}d\gamma
  34. P 1 v = ( 𝐅 ψ ) v | ψ u + 𝐅 ψ | 2 ψ v u P 2 u = ( 𝐅 ψ ) u | ψ v + 𝐅 ψ | 2 ψ u v \begin{aligned}\displaystyle\frac{\partial P_{1}}{\partial v}&\displaystyle=% \left\langle\frac{\partial(\mathbf{F}\circ\psi)}{\partial v}\bigg|\frac{% \partial\psi}{\partial u}\right\rangle+\left\langle\mathbf{F}\circ\psi\bigg|% \frac{\partial^{2}\psi}{\partial v\partial u}\right\rangle\\ \displaystyle\frac{\partial P_{2}}{\partial u}&\displaystyle=\left\langle\frac% {\partial(\mathbf{F}\circ\psi)}{\partial u}\bigg|\frac{\partial\psi}{\partial v% }\right\rangle+\left\langle\mathbf{F}\circ\psi\bigg|\frac{\partial^{2}\psi}{% \partial u\partial v}\right\rangle\end{aligned}
  35. ( ( J 𝐅 ) ψ ( u , v ) - ( J 𝐅 ) ψ ( u , v ) t ) 𝐱 = ( × 𝐅 ) × 𝐱 , for all 𝐱 𝐑 3 ({(J\mathbf{F})}_{\psi(u,v)}-{}^{t}{(J\mathbf{F})}_{\psi(u,v)})\mathbf{x}=(% \nabla\times\mathbf{F})\times\mathbf{x},\quad\,\text{for all}\,\mathbf{x}\in% \mathbf{R}^{3}
  36. 𝐚 × \mathbf{a}×
  37. 𝐚 \mathbf{a}
  38. 𝐱 \mathbf{x}
  39. 𝐚 = ( a 1 a 2 a 3 ) , 𝐱 = ( x 1 x 2 x 3 ) \mathbf{a}=\begin{pmatrix}a_{1}\\ a_{2}\\ a_{3}\end{pmatrix},\quad\mathbf{x}=\begin{pmatrix}x_{1}\\ x_{2}\\ x_{3}\end{pmatrix}
  40. 𝐚 × 𝐱 \mathbf{a}×\mathbf{x}
  41. 𝐚 × 𝐱 = ( a 2 x 3 - a 3 x 2 a 3 x 1 - a 1 x 3 a 1 x 2 - a 2 x 1 ) \mathbf{a}\times\mathbf{x}=\begin{pmatrix}a_{2}x_{3}-a_{3}x_{2}\\ a_{3}x_{1}-a_{1}x_{3}\\ a_{1}x_{2}-a_{2}x_{1}\end{pmatrix}
  42. 𝐚 × \mathbf{a}×
  43. 𝐚 × \mathbf{a}×
  44. 𝐚 × 𝐞 1 = ( 0 a 3 - a 2 ) , 𝐚 × 𝐞 2 = ( - a 3 0 a 1 ) , 𝐚 × 𝐞 3 = ( a 2 - a 1 0 ) \mathbf{a}\times\mathbf{e}_{1}=\begin{pmatrix}0\\ a_{3}\\ -a_{2}\end{pmatrix},\quad\mathbf{a}\times\mathbf{e}_{2}=\begin{pmatrix}-a_{3}% \\ 0\\ a_{1}\end{pmatrix},\quad\mathbf{a}\times\mathbf{e}_{3}=\begin{pmatrix}a_{2}\\ -a_{1}\\ 0\end{pmatrix}
  45. 𝐚 × 𝐱 = ( 0 - a 3 a 2 a 3 0 - a 1 - a 2 a 1 0 ) ( x 1 x 2 x 3 ) \mathbf{a}\times\mathbf{x}=\begin{pmatrix}0&-a_{3}&a_{2}\\ a_{3}&0&-a_{1}\\ -a_{2}&a_{1}&0\end{pmatrix}\begin{pmatrix}x_{1}\\ x_{2}\\ x_{3}\end{pmatrix}
  46. A = ( a i j ) A=(a_{ij})
  47. 3 × 3 3×3
  48. 𝐚 \mathbf{a}
  49. a 1 = a 32 - a 23 {a}_{1}={a}_{32}-{a}_{23}
  50. a 2 = a 13 - a 31 {a}_{2}={a}_{13}-{a}_{31}
  51. a 3 = a 21 - a 12 {a}_{3}={a}_{21}-{a}_{12}
  52. ( A - A t ) 𝐱 = 𝐚 × 𝐱 (A-{}^{t}A)\mathbf{x}=\mathbf{a}\times\mathbf{x}
  53. J 𝐅 J\mathbf{F}
  54. 𝐚 = ( × 𝐅 ) \mathbf{a}=(\nabla\times\mathbf{F})
  55. P 1 v - P 2 u = ( 𝐅 ψ ) v | ψ u - ( 𝐅 ψ ) u | ψ v = ( J 𝐅 ) ψ ( u , v ) ψ v | ψ u - ( J 𝐅 ) ψ ( u , v ) ψ u | ψ v chain rule = ψ u | ( J 𝐅 ) ψ ( u , v ) | ψ v - ψ u | ( J 𝐅 ) ψ ( u , v ) t | ψ v = ψ u | ( J 𝐅 ) ψ ( u , v ) - ( J 𝐅 ) ψ ( u , v ) t | ψ v = ψ u | ( ( J 𝐅 ) ψ ( u , v ) - ( J 𝐅 ) ψ ( u , v ) t ) ψ v = ψ u | ( × 𝐅 ) × ψ v ( ( J 𝐅 ) ψ ( u , v ) - ( J 𝐅 ) ψ ( u , v ) t ) 𝐱 = ( × 𝐅 ) × 𝐱 = det [ ( × 𝐅 ) ( ψ ( u , v ) ) ψ u ( u , v ) ψ v ( u , v ) ] Scalar Triple Product \begin{aligned}\displaystyle\frac{\partial P_{1}}{\partial v}-\frac{\partial P% _{2}}{\partial u}&\displaystyle=\left\langle\frac{\partial(\mathbf{F}\circ\psi% )}{\partial v}\bigg|\frac{\partial\psi}{\partial u}\right\rangle-\left\langle% \frac{\partial(\mathbf{F}\circ\psi)}{\partial u}\bigg|\frac{\partial\psi}{% \partial v}\right\rangle\\ &\displaystyle=\left\langle(J\mathbf{F})_{\psi(u,v)}\cdot\frac{\partial\psi}{% \partial v}\bigg|\frac{\partial\psi}{\partial u}\right\rangle-\left\langle(J% \mathbf{F})_{\psi(u,v)}\cdot\frac{\partial\psi}{\partial u}\bigg|\frac{% \partial\psi}{\partial v}\right\rangle&&\displaystyle\,\text{ chain rule}\\ &\displaystyle=\left\langle\frac{\partial\psi}{\partial u}\bigg|(J\mathbf{F})_% {\psi(u,v)}\bigg|\frac{\partial\psi}{\partial v}\right\rangle-\left\langle% \frac{\partial\psi}{\partial u}\bigg|{}^{t}(J\mathbf{F})_{\psi(u,v)}\bigg|% \frac{\partial\psi}{\partial v}\right\rangle\\ &\displaystyle=\left\langle\frac{\partial\psi}{\partial u}\bigg|(J\mathbf{F})_% {\psi(u,v)}-{}^{t}{(J\mathbf{F})}_{\psi(u,v)}\bigg|\frac{\partial\psi}{% \partial v}\right\rangle\\ &\displaystyle=\left\langle\frac{\partial\psi}{\partial u}\bigg|\left((J% \mathbf{F})_{\psi(u,v)}-{}^{t}(J\mathbf{F})_{\psi(u,v)}\right)\cdot\frac{% \partial\psi}{\partial v}\right\rangle\\ &\displaystyle=\left\langle\frac{\partial\psi}{\partial u}\bigg|(\nabla\times% \mathbf{F})\times\frac{\partial\psi}{\partial v}\right\rangle&&\displaystyle% \left((J\mathbf{F})_{\psi(u,v)}-{}^{t}(J\mathbf{F})_{\psi(u,v)}\right)\cdot% \mathbf{x}=(\nabla\times\mathbf{F})\times\mathbf{x}\\ &\displaystyle=\det\left[(\nabla\times\mathbf{F})(\psi(u,v))\quad\frac{% \partial\psi}{\partial u}(u,v)\quad\frac{\partial\psi}{\partial v}(u,v)\right]% &&\displaystyle\,\text{ Scalar Triple Product}\end{aligned}
  56. S ( × 𝐅 ) d S = D ( × 𝐅 ) ( ψ ( u , v ) ) | ψ u ( u , v ) × ψ v ( u , v ) d u d v = D det [ ( × 𝐅 ) ( ψ ( u , v ) ) ψ u ( u , v ) ψ v ( u , v ) ] d u d v scalar triple product \begin{aligned}\displaystyle\iint_{S}(\nabla\times\mathbf{F})\,dS&% \displaystyle=\iint_{D}\left\langle(\nabla\times\mathbf{F})(\psi(u,v))\bigg|% \frac{\partial\psi}{\partial u}(u,v)\times\frac{\partial\psi}{\partial v}(u,v)% \right\rangle\,du\,dv\\ &\displaystyle=\iint_{D}\det\left[(\nabla\times\mathbf{F})(\psi(u,v))\quad% \frac{\partial\psi}{\partial u}(u,v)\quad\frac{\partial\psi}{\partial v}(u,v)% \right]\,du\,dv&&\displaystyle\,\text{ scalar triple product}\end{aligned}
  57. S ( × 𝐅 ) d S = D ( P 2 u - P 1 v ) d u d v \iint_{S}(\nabla\times\mathbf{F})\,dS=\iint_{D}\left(\frac{\partial P_{2}}{% \partial u}-\frac{\partial P_{1}}{\partial v}\right)\,du\,dv
  58. { θ [ a , b ] : [ 0 , 1 ] [ a , b ] θ [ a , b ] = s ( b - a ) + a \begin{cases}\theta_{[a,b]}:[0,1]\to[a,b]\\ \theta_{[a,b]}=s(b-a)+a\end{cases}
  59. θ [ a , b ] \theta_{[a,b]}
  60. 𝐅 \mathbf{F}
  61. c ( a a , b ) ) c(aa,b))
  62. c 𝐅 d c = c θ [ a , b ] 𝐅 d ( c θ [ a , b ] ) \int_{c}\mathbf{F}dc=\int_{c\circ\theta_{[a,b]}}\ \mathbf{F}\ d(c\circ\theta_{% [a,b]})
  63. 0 , 11 0,11
  64. 𝐅 \mathbf{F}
  65. U 𝐑 < s u p > 3 U⊆\mathbf{R}<sup>3
  66. 𝐅 \mathbf{F}
  67. H : 0 , 11 × 0 , 11 U H:0,11×0,11→U
  68. H H
  69. t 0 , 11 t∈0,11
  70. t 0 , 11 t∈0,11
  71. H ( 0 , s ) = H ( 1 , s ) H(0,s)=H(1,s)
  72. s 0 , 11 s∈0,11
  73. c 0 𝐅 d c 0 = c 1 𝐅 d c 1 \int_{c_{0}}\mathbf{F}\,dc_{0}=\int_{c_{1}}\mathbf{F}\,dc_{1}
  74. H : 0 , 11 × 0 , 11 U H:0,11×0,11→U
  75. H : Z × 0 , 11 W H:Z×0,11→W
  76. c < s u b > 0 , c 1 c<sub>0,c_{1}
  77. α β α⊕β
  78. 𝐅 \mathbf{F}
  79. α β α⊕β
  80. α β 𝐅 d ( α β ) = α 𝐅 d α + β 𝐅 d β \int_{\alpha\oplus\beta}\mathbf{F}\,d(\alpha\oplus\beta)=\int_{\alpha}\mathbf{% F}\,d\alpha+\int_{\beta}\mathbf{F}\,d\beta
  81. M M
  82. M M
  83. ( α β ) ( t ) = { α ( t ) a 1 t b 1 , β ( t + ( a 2 - b 1 ) ) b 1 < t b 1 + ( b 2 - a 2 ) (\alpha\oplus\beta)(t)=\begin{cases}\alpha(t)&a_{1}\leq t\leq\ b_{1},\\ \beta(t+(a_{2}-b_{1}))&b_{1}<t\leq b_{1}+(b_{2}-a_{2})\end{cases}
  84. \ominus
  85. M M
  86. \ominus
  87. 𝐅 \mathbf{F}
  88. α 𝐅 d ( α ) = - α 𝐅 d α , \int_{\ominus\alpha}\mathbf{F}\,d(\ominus\alpha)=-\int_{\alpha}\mathbf{F}\,d\alpha,
  89. M M
  90. M M
  91. \ominus
  92. α ( t ) = α ( b 1 + a 1 - t ) \ominus\alpha(t)=\alpha(b_{1}+a_{1}-t)
  93. M M
  94. \ominus
  95. α β \alpha\ominus\beta
  96. α β := α ( β ) \alpha\ominus\beta:=\alpha\oplus(\ominus\beta)
  97. D = 0 , 11 × 0 , 11 D=0,11×0,11
  98. H : D M H:D→M
  99. { γ 1 : [ 0 , 1 ] D γ 1 ( t ) := ( t , 0 ) { γ 2 : [ 0 , 1 ] D γ 2 ( s ) := ( 1 , s ) { γ 3 : [ 0 , 1 ] D γ 3 ( t ) := ( - t + 0 + 1 , 1 ) { γ 4 : [ 0 , 1 ] D γ 4 ( s ) := ( 0 , 1 - s ) [ 6 p t ] γ ( t ) : = ( γ 1 γ 2 γ 3 γ 4 ) ( t ) Γ i ( t ) : = H ( γ i ( t ) ) i = 1 , 2 , 3 , 4 Γ ( t ) : = H ( γ ( t ) ) = ( Γ 1 Γ 2 Γ 3 Γ 4 ) ( t ) \begin{aligned}&\displaystyle\begin{cases}\gamma_{1}:[0,1]\to D\\ \gamma_{1}(t):=(t,0)\end{cases}\\ &\displaystyle\begin{cases}\gamma_{2}:[0,1]\to D\\ \gamma_{2}(s):=(1,s)\end{cases}\\ &\displaystyle\begin{cases}\gamma_{3}:[0,1]\to D\\ \gamma_{3}(t):=(-t+0+1,1)\end{cases}\\ &\displaystyle\begin{cases}\gamma_{4}:[0,1]\to D\\ \gamma_{4}(s):=(0,1-s)\end{cases}\\ \displaystyle[6pt]\gamma(t)&\displaystyle:=(\gamma_{1}\oplus\gamma_{2}\oplus% \gamma_{3}\oplus\gamma_{4})(t)\\ \displaystyle\Gamma_{i}(t)&\displaystyle:=H(\gamma_{i}(t))&&\displaystyle i=1,% 2,3,4\\ \displaystyle\Gamma(t)&\displaystyle:=H(\gamma(t))=(\Gamma_{1}\oplus\Gamma_{2}% \oplus\Gamma_{3}\oplus\Gamma_{4})(t)\\ \end{aligned}
  100. D D
  101. Γ 𝐅 d Γ = S × 𝐅 d S \oint_{\Gamma}\mathbf{F}\,d\Gamma=\iint_{S}\nabla\times\mathbf{F}\,dS
  102. 𝐅 \mathbf{F}
  103. Γ 𝐅 d Γ = 0 \oint_{\Gamma}\mathbf{F}\,d\Gamma=0
  104. Γ 𝐅 d Γ = i = 1 4 Γ i 𝐅 d Γ \oint_{\Gamma}\mathbf{F}\,d\Gamma=\sum_{i=1}^{4}\oint_{\Gamma_{i}}\mathbf{F}d\Gamma
  105. Γ 2 ( s ) = Γ 4 ( 1 - s ) = Γ 4 ( s ) \Gamma_{2}(s)={\Gamma}_{4}(1-s)=\ominus{\Gamma}_{4}(s)
  106. Γ 1 𝐅 d Γ + Γ 3 𝐅 d Γ = 0 \oint_{{\Gamma}_{1}}\mathbf{F}d\Gamma+\oint_{\Gamma_{3}}\mathbf{F}d\Gamma=0
  107. c 1 ( t ) = H ( t , 0 ) = H ( γ 1 ( t ) ) = Γ 1 ( t ) c_{1}(t)=H(t,0)=H({\gamma}_{1}(t))={\Gamma}_{1}(t)
  108. c 2 ( t ) = H ( t , 1 ) = H ( γ 3 ( t ) ) = Γ 3 ( t ) c_{2}(t)=H(t,1)=H(\ominus{\gamma}_{3}(t))=\ominus{\Gamma}_{3}(t)
  109. 𝐅 \mathbf{F}
  110. 𝐩 U \mathbf{p}∈U
  111. H : 0 , 11 × 0 , 11 U H:0,11×0,11→U
  112. c 0 𝐅 d c 0 = 0 \int_{c_{0}}\mathbf{F}\,dc_{0}=0
  113. H H
  114. M 𝐑 < s u p > n M⊆\mathbf{R}<sup>n
  115. H H
  116. H H
  117. 𝐅 \mathbf{F}
  118. c : 0 , 11 U c:0,11→U
  119. c 0 𝐅 d c 0 = 0 \int_{c_{0}}\mathbf{F}\,dc_{0}=0
  120. U U
  121. D D
  122. ψ : D U ψ:D→U
  123. D D
  124. { θ D : I × I D θ D ( u 1 , u 2 ) = ( u 1 ( b 1 - a 1 ) + a 1 u 2 ( b 2 - a 2 ) + a 2 ) \begin{cases}\theta_{D}:I\times I\to D\\ \theta_{D}({u}_{1},{u}_{2})=\begin{pmatrix}{u}_{1}(b_{1}-a_{1})+{a}_{1}\\ {u}_{2}(b_{2}-a_{2})+{a}_{2}\end{pmatrix}\end{cases}
  125. I = 0 , 11 I=0,11
  126. ( u 1 , u 2 ) 𝐑 3 det ( J ( θ D ) ( u 1 , u 2 ) ) > 0. \forall(u_{1},u_{2})\in\mathbf{R}^{3}\qquad\det(J(\theta_{D})_{(u_{1},u_{2})})% >0.
  127. D D
  128. ψ ψ
  129. I × I I×I
  130. φ θ D , S ~ := φ θ D ( I × I ) \varphi\circ\theta_{D},\widetilde{S}:=\varphi\circ\theta_{D}(I\times I)
  131. 𝐅 \mathbf{F}
  132. U U
  133. S 𝐅 d S = S ~ 𝐅 d S ~ \int_{S}\mathbf{F}\,dS=\int_{\widetilde{S}}\mathbf{F}\,d\widetilde{S}
  134. I × I I×I
  135. { δ [ k , j , c ] : 𝐑 k 𝐑 k + 1 δ [ k , j , c ] ( t 1 , , t k ) := ( t 1 , , t j - 1 , c , t j + 1 , , t k ) \begin{cases}\delta_{[k,j,c]}:\mathbf{R}^{k}\to\mathbf{R}^{k+1}\\ \delta_{[k,j,c]}(t_{1},\cdots,t_{k}):=\left(t_{1},\cdots,t_{j-1},c,t_{j+1},% \cdots,t_{k}\right)\end{cases}
  136. I × I I×I
  137. \ominus
  138. { γ 1 : [ 0 , 1 ] I 2 γ 1 ( t ) := δ [ 1 , 2 , 0 ] ( t ) = ( t , 0 ) { γ 2 : [ 0 , 1 ] I 2 γ 2 ( t ) := δ [ 1 , 1 , 1 ] ( t ) = ( 1 , t ) { γ 3 : [ 0 , 1 ] I 2 γ 3 ( t ) := δ [ 1 , 2 , 1 ] ( t ) = ( - t + 0 + 1 , 1 ) { γ 4 : [ 0 , 1 ] I 2 γ 4 ( t ) := δ [ 1 , 1 , 0 ] ( t ) = ( 0 , 1 - t ) [ 6 p t ] γ ( t ) : = ( γ 1 γ 2 γ 3 γ 4 ) ( t ) Γ i ( t ) : = φ ( γ i ( t ) ) i = 1 , 2 , 3 , 4 Γ ( t ) : = φ ( γ ( t ) ) = ( Γ 1 Γ 2 Γ 3 Γ 4 ) ( t ) \begin{aligned}&\displaystyle\begin{cases}\gamma_{1}:[0,1]\to I^{2}\\ \gamma_{1}(t):={\delta}_{[1,2,0]}(t)=(t,0)\end{cases}\\ &\displaystyle\begin{cases}\gamma_{2}:[0,1]\to I^{2}\\ \gamma_{2}(t):={\delta}_{[1,1,1]}(t)=(1,t)\end{cases}\\ &\displaystyle\begin{cases}\gamma_{3}:[0,1]\to I^{2}\\ \gamma_{3}(t):=\ominus{\delta}_{[1,2,1]}(t)=(-t+0+1,1)\end{cases}\\ &\displaystyle\begin{cases}\gamma_{4}:[0,1]\to I^{2}\\ \gamma_{4}(t):=\ominus{\delta}_{[1,1,0]}(t)=(0,1-t)\end{cases}\\ \displaystyle[6pt]\gamma(t)&\displaystyle:=(\gamma_{1}\oplus\gamma_{2}\oplus% \gamma_{3}\oplus\gamma_{4})(t)\\ \displaystyle\Gamma_{i}(t)&\displaystyle:=\varphi(\gamma_{i}(t))&&% \displaystyle i=1,2,3,4\\ \displaystyle\Gamma(t)&\displaystyle:=\varphi(\gamma(t))=(\Gamma_{1}\oplus% \Gamma_{2}\oplus\Gamma_{3}\oplus\Gamma_{4})(t)\end{aligned}
  139. { ( I 2 , φ λ , S λ ) } λ Λ \left\{(I^{2},\varphi_{\lambda},S_{\lambda})\right\}_{\lambda\in\Lambda}
  140. Λ Λ
  141. λ Λ λ∈Λ
  142. I × I I×I
  143. det ( ( J φ ) ( u 1 , u 2 ) ) 0 \det((J\varphi)_{(u_{1},u_{2})})\neq 0
  144. S S
  145. λ λ
  146. Λ Λ
  147. φ λ 1 ( I n t ( I 2 ) ) φ λ 2 ( I n t ( I 2 ) ) = \varphi_{\lambda_{1}}(Int(I^{2}))\cap\varphi_{\lambda_{2}}(Int(I^{2}))=\varnothing
  148. 1 1
  149. 2 2
  150. φ λ 1 δ [ 1 , j 1 , c 1 ] ( I ) φ λ 2 δ [ 1 , j 2 , c 2 ] ( I ) \varphi_{\lambda_{1}}\circ\delta_{[1,j_{1},c_{1}]}(I)\cap\varphi_{\lambda_{2}}% \circ\delta_{[1,j_{2},c_{2}]}(I)\neq\varnothing
  151. φ λ 1 δ [ 1 , j 1 , c 1 ] ( I ) = φ λ 2 δ [ 1 , j 2 , c 2 ] ( I ) \varphi_{\lambda_{1}}\circ\delta_{[1,j_{1},c_{1}]}(I)=\varphi_{\lambda_{2}}% \circ\delta_{[1,j_{2},c_{2}]}(I)
  152. { ( I 2 , φ λ , S λ ) } λ Λ \left\{(I^{2},\varphi_{\lambda},S_{\lambda})\right\}_{\lambda\in\Lambda}
  153. S S
  154. { ( I 2 , φ λ , S λ ) } λ Λ . \left\{(I^{2},\varphi_{\lambda},S_{\lambda})\right\}_{\lambda\in\Lambda}.
  155. φ λ 1 δ [ 1 , j 1 , c 1 ] \varphi_{\lambda_{1}}\circ\delta_{[1,j_{1},c_{1}]}
  156. φ λ 1 δ [ 1 , j 1 , c 1 ] ( I ) = φ λ 2 δ [ 1 , j 2 , c 2 ] ( I ) \varphi_{\lambda_{1}}\circ\delta_{[1,j_{1},c_{1}]}(I)=\varphi_{\lambda_{2}}% \circ\delta_{[1,j_{2},c_{2}]}(I)
  157. ( λ 1 , j 1 , c 1 ) = ( λ 2 , j 2 , c 2 ) (\lambda_{1},j_{1},c_{1})=(\lambda_{2},j_{2},c_{2})
  158. l l
  159. { ( I 2 , φ λ , S λ ) } λ Λ \{(I^{2},\varphi_{\lambda},S_{\lambda})\}_{\lambda\in\Lambda}
  160. λ , c λ,c
  161. j j
  162. l = φ λ δ [ 1 , j , c ] l=\varphi_{\lambda}\circ\delta_{[1,j,c]}
  163. { ( I 2 , φ λ , S λ ) } λ Λ \partial\{(I^{2},\varphi_{\lambda},S_{\lambda})\}_{\lambda\in\Lambda}
  164. { ( I 2 , φ λ , S λ ) } λ Λ \{(I^{2},\varphi_{\lambda},S_{\lambda})\}_{\lambda\in\Lambda}
  165. l l
  166. l { ( I 2 , φ λ , S λ ) } λ Λ l\prec\partial\left\{(I^{2},\varphi_{\lambda},S_{\lambda})\right\}_{\lambda\in\Lambda}
  167. { ( I 2 , φ λ , S λ ) } λ Λ , { ( I 2 , ψ μ , L μ ) } μ M . \left\{(I^{2},\varphi_{\lambda},S_{\lambda})\right\}_{\lambda\in\Lambda},\quad% \left\{(I^{2},\psi_{\mu},L_{\mu})\right\}_{\mu\in M}.
  168. { ( I 2 , φ λ , S λ ) } λ Λ = { ( I 2 , ψ μ , L μ ) } μ M \partial\left\{(I^{2},\varphi_{\lambda},S_{\lambda})\right\}_{\lambda\in% \Lambda}=\partial\left\{(I^{2},\psi_{\mu},L_{\mu})\right\}_{\mu\in M}
  169. { ( I 2 , φ λ , S λ ) } λ Λ \{(I^{2},\varphi_{\lambda},S_{\lambda})\}_{\lambda\in\Lambda}
  170. S := { ( I 2 , φ λ , S λ ) } λ Λ \partial S:=\partial\left\{(I^{2},\varphi_{\lambda},S_{\lambda})\right\}_{% \lambda\in\Lambda}
  171. l { ( I 2 , φ λ , S λ ) } λ Λ l\prec\partial\left\{(I^{2},\varphi_{\lambda},S_{\lambda})\right\}_{\lambda\in\Lambda}
  172. l S l\prec\partial S
  173. l l
  174. S S
  175. γ γ
  176. Γ Γ
  177. Γ Γ

Kemnitz's_conjecture.html

  1. n n
  2. S S
  3. S 1 S S_{1}\subseteq S
  4. n n
  5. S 1 S_{1}

Kendrick_mass.html

  1. Kendrick mass = IUPAC mass × 14.00000 14.01565 \textrm{Kendrick~{}mass}=\textrm{IUPAC~{}mass}\times\frac{14.00000}{14.01565}
  2. Kendrick mass (F) = (observed mass) × nominal mass F exact mass F \textrm{Kendrick~{}mass~{}(F)}=\textrm{(observed~{}mass)}\times\frac{\textrm{% nominal~{}mass~{}F}}{\textrm{exact~{}mass~{}F}}
  3. Kendrick mass defect = nominal Kendrick mass - Kendrick mass \textrm{Kendrick~{}mass~{}defect}=\textrm{nominal~{}Kendrick~{}mass}-\textrm{% Kendrick~{}mass}
  4. Kendrick mass defect = nominal mass - Kendrick exact mass \textrm{Kendrick~{}mass~{}defect}=\textrm{nominal~{}mass}-\textrm{Kendrick~{}% exact~{}mass}
  5. Kendrick mass defect = ( nominal Kendrick mass - Kendrick mass ) × 1 , 000 \textrm{Kendrick~{}mass~{}defect}=(\textrm{nominal~{}Kendrick~{}mass}-\textrm{% Kendrick~{}mass})\times 1{,}000

Kepler-4b.html

  1. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}
  2. R \begin{smallmatrix}R_{\odot}\end{smallmatrix}

Kernighan–Lin_algorithm.html

  1. G ( V , E ) G(V,E)
  2. V V
  3. E E
  4. V V
  5. A A
  6. B B
  7. T T
  8. A A
  9. B B
  10. I a I_{a}
  11. E a E_{a}
  12. D a = E a - I a D_{a}=E_{a}-I_{a}
  13. T o l d - T n e w = D a + D b - 2 c a , b T_{old}-T_{new}=D_{a}+D_{b}-2c_{a,b}
  14. c a , b c_{a,b}
  15. A A
  16. B B
  17. T o l d - T n e w T_{old}-T_{new}

Khabibullin's_conjecture_on_integral_inequalities.html

  1. S \displaystyle S
  2. [ 0 , + ) [0,+\infty)
  3. S ( 0 ) = 0 \displaystyle S(0)=0
  4. S ( e x ) \displaystyle S(e^{x})
  5. x [ - , + ) x\in[-\infty,+\infty)
  6. λ 1 / 2 \lambda\geq 1/2
  7. n 2 n\geq 2
  8. n n\in\mathbb{N}
  9. 0 1 S ( t x ) ( 1 - x 2 ) n - 2 x d x t λ for all t [ 0 , + ) , \int^{1}_{0}S(tx)\,(1-x^{2})^{n-2}\,x\,dx\leq t^{\lambda}\,\text{ for all }t% \in[0,+\infty),
  10. 0 + S ( t ) t 2 λ - 1 ( 1 + t 2 λ ) 2 d t π ( n - 1 ) 2 λ k = 1 n - 1 ( 1 + λ 2 k ) . \int^{+\infty}_{0}S(t)\,\frac{t^{2\lambda-1}}{(1+t^{2\lambda})^{2}}\,dt\leq% \frac{\pi\,(n-1)}{2\lambda}\prod_{k=1}^{n-1}\Bigl(1+\frac{\lambda}{2k}\Bigr).
  11. B B
  12. π ( n - 1 ) 2 λ k = 1 n - 1 ( 1 + λ 2 k ) = π ( n - 1 ) λ 2 1 B ( λ / 2 , n ) \frac{\pi\,(n-1)}{2\lambda}\prod_{k=1}^{n-1}\Bigl(1+\frac{\lambda}{2k}\Bigr)=% \frac{\pi\,(n-1)}{\lambda^{2}}\cdot\frac{1}{B(\lambda/2,n)}
  13. λ 1 / 2 \lambda\geq 1/2
  14. S ( t ) = 2 ( n - 1 ) k = 1 n - 1 ( 1 + λ 2 k ) t λ , S(t)=2(n-1)\prod_{k=1}^{n-1}\Bigl(1+\frac{\lambda}{2k}\Bigr)\,t^{\lambda},
  15. λ 1 \lambda\leq 1
  16. S ( e x ) S(e^{x})
  17. S S
  18. n = 2 n=2
  19. λ > 1 \lambda>1
  20. h \displaystyle h
  21. [ 0 , + ) [0,+\infty)
  22. α > 1 / 2 \alpha>1/2
  23. 0 1 h ( t x ) x ( 1 - x ) n - 1 d x t α for all t [ 0 , + ) , \int_{0}^{1}\frac{h(tx)}{x}\,(1-x)^{n-1}\,dx\leq t^{\alpha}\,\text{ for all }t% \in[0,+\infty),
  24. 0 + h ( t ) t d t 1 + t 2 α π 2 k = 1 n - 1 ( 1 + α k ) = π 2 α 1 B ( α , n ) . \int_{0}^{+\infty}\frac{h(t)}{t}\,\frac{dt}{1+t^{2\alpha}}\leq\frac{\pi}{2}% \prod_{k=1}^{n-1}\Bigl(1+\frac{\alpha}{k}\Bigr)=\frac{\pi}{2\alpha}\cdot\frac{% 1}{\mathrm{B}(\alpha,n)}.\,
  25. q \displaystyle q
  26. [ 0 , + ) [0,+\infty)
  27. α > 1 / 2 \alpha>1/2
  28. 0 1 ( x 1 ( 1 - y ) n - 1 d y y ) q ( t x ) d x t α - 1 for all t [ 0 , + ) , \int_{0}^{1}\Bigl(\,\int_{x}^{1}(1-y)^{n-1}\frac{dy}{y}\Bigr)q(tx)\,dx\leq t^{% \alpha-1}\,\text{ for all }t\in[0,+\infty),
  29. 0 + q ( t ) log ( 1 + 1 t 2 α ) d t π α k = 1 n - 1 ( 1 + α k ) = π B ( α , n ) . \int_{0}^{+\infty}q(t)\log\Bigl(1+\frac{1}{t^{2\alpha}}\Bigr)\,dt\leq\pi\alpha% \prod_{k=1}^{n-1}\Bigl(1+\frac{\alpha}{k}\Bigr)=\frac{\pi}{\mathrm{B}(\alpha,n% )}.\,

Kirchhoff_integral_theorem.html

  1. U ( 𝐫 ) = 1 4 π S [ U 𝐧 ^ ( e i k s s ) - e i k s s U 𝐧 ^ ] d S , U(\mathbf{r})=\frac{1}{4\pi}\int_{S}\left[U\frac{\partial}{\partial\hat{% \mathbf{n}}}\left(\frac{e^{iks}}{s}\right)-\frac{e^{iks}}{s}\frac{\partial U}{% \partial\hat{\mathbf{n}}}\right]dS,
  2. V ( r , t ) = 1 2 π U ω ( r ) e - i ω t d ω , V(r,t)=\frac{1}{\sqrt{2\pi}}\int U_{\omega}(r)e^{-i\omega t}\,d\omega,
  3. U ω ( r ) = 1 2 π V ( r , t ) e i ω t d t . U_{\omega}(r)=\frac{1}{\sqrt{2\pi}}\int V(r,t)e^{i\omega t}\,dt.
  4. V ( r , t ) = 1 4 π S { [ V ] n ( 1 s ) - 1 c s s n [ V t ] - 1 s [ V n ] } d S , V(r,t)=\frac{1}{4\pi}\int_{S}\left\{[V]\frac{\partial}{\partial n}\left(\frac{% 1}{s}\right)-\frac{1}{cs}\frac{\partial s}{\partial n}\left[\frac{\partial V}{% \partial t}\right]-\frac{1}{s}\left[\frac{\partial V}{\partial n}\right]\right% \}dS,

Kirsch_operator.html

  1. h n , m = max z = 1 , , 8 i = - 1 1 j = - 1 1 g ij ( z ) f n + i , m + j h_{n,m}=\rm{max}_{z=1,\ldots,8}\sum_{i=-1}^{1}\sum_{j=-1}^{1}g_{ij}^{(z)}\cdot f% _{n+i,m+j}
  2. 𝐠 ( 𝟏 ) = [ + 5 + 5 + 5 - 3 0 - 3 - 3 - 3 - 3 ] , 𝐠 ( 𝟐 ) = [ + 5 + 5 - 3 + 5 0 - 3 - 3 - 3 - 3 ] , 𝐠 ( 𝟑 ) = [ + 5 - 3 - 3 + 5 0 - 3 + 5 - 3 - 3 ] , 𝐠 ( 𝟒 ) = [ - 3 - 3 - 3 + 5 0 - 3 + 5 + 5 - 3 ] \mathbf{g^{(1)}}=\begin{bmatrix}+5&+5&+5\\ -3&0&-3\\ -3&-3&-3\end{bmatrix},\ \mathbf{g^{(2)}}=\begin{bmatrix}+5&+5&-3\\ +5&0&-3\\ -3&-3&-3\end{bmatrix},\ \mathbf{g^{(3)}}=\begin{bmatrix}+5&-3&-3\\ +5&0&-3\\ +5&-3&-3\end{bmatrix},\ \mathbf{g^{(4)}}=\begin{bmatrix}-3&-3&-3\\ +5&0&-3\\ +5&+5&-3\end{bmatrix}

Klein_cubic_threefold.html

  1. V 2 W + W 2 X + X 2 Y + Y 2 Z + Z 2 V = 0 V^{2}W+W^{2}X+X^{2}Y+Y^{2}Z+Z^{2}V=0\,

Kolmogorov's_three-series_theorem.html

  1. n = 1 ± 1 n . \sum_{n=1}^{\infty}\pm\frac{1}{n}.
  2. ± \pm
  3. 1 / n 1/n
  4. 1 1
  5. - 1 -1
  6. 1 / 2 , 1 / 2 1/2,\ 1/2
  7. X n X_{n}
  8. 1 / n 1/n
  9. - 1 / n -1/n
  10. n = 1 ± 1 n , \sum_{n=1}^{\infty}\pm\frac{1}{\sqrt{n}},
  11. n = 1 ( - 1 ) n / n \sum_{n=1}^{\infty}(-1)^{n}/\sqrt{n}
  12. n = 1 ( - 1 ) n n = ( 2 - 1 ) ζ ( 1 2 ) - 0.604899. \sum_{n=1}^{\infty}\frac{(-1)^{n}}{\sqrt{n}}=(\sqrt{2}-1)\zeta(\frac{1}{2})% \approx-0.604899.

Komornik–Loreti_constant.html

  1. x = n = 0 a n q - n x=\sum_{n=0}^{\infty}a_{n}q^{-n}
  2. β \beta
  3. n 0 n\geq 0
  4. 0 a n q 0\leq a_{n}\leq\lfloor q\rfloor
  5. q \lfloor q\rfloor
  6. a n a_{n}
  7. x x
  8. 0 x q q / ( q - 1 ) 0\leq x\leq q\lfloor q\rfloor/(q-1)
  9. x = 1 x=1
  10. a 0 = 0 a_{0}=0
  11. a n = 0 a_{n}=0
  12. q q
  13. a n = 1 a_{n}=1
  14. 1 < q < 2 1<q<2
  15. q q
  16. q ( 1 , 2 ) q\in(1,2)
  17. q q
  18. 1 < q < 2 1<q<2
  19. q q
  20. q q
  21. 1 = n = 1 t k q k 1=\sum_{n=1}^{\infty}\frac{t_{k}}{q^{k}}
  22. t k t_{k}
  23. t k t_{k}
  24. k k
  25. q = 1.787231650 . q=1.787231650\ldots.\,
  26. q q
  27. k = 0 ( 1 - 1 q 2 k ) = ( 1 - 1 q ) - 1 - 2. \prod_{k=0}^{\infty}\left(1-\frac{1}{q^{2^{k}}}\right)=\left(1-\frac{1}{q}% \right)^{-1}-2.

Kontsevich_quantization_formula.html

  1. f * g = f g + 𝒪 ( ) [ f , g ] = f * g - g * f = i { f , g } + 𝒪 ( 2 ) \begin{aligned}\displaystyle f*g&\displaystyle=fg+\mathcal{O}(\hbar)\\ \displaystyle{}[f,g]&\displaystyle=f*g-g*f=i\hbar\{f,g\}+\mathcal{O}(\hbar^{2}% )\end{aligned}
  2. f * g = f g + k = 1 k B k ( f g ) , f*g=fg+\sum_{k=1}^{\infty}\hbar^{k}B_{k}(f\otimes g),
  3. k k
  4. { D : A [ [ ] ] A [ [ ] ] k = 0 k f k k = 0 k f k + n 1 , k 0 D n ( f k ) n + k \begin{cases}D:A[[\hbar]]\to A[[\hbar]]\\ \sum_{k=0}^{\infty}\hbar^{k}f_{k}\mapsto\sum_{k=0}^{\infty}\hbar^{k}f_{k}+\sum% _{n\geq 1,k\geq 0}D_{n}(f_{k})\hbar^{n+k}\end{cases}
  5. n n
  6. f * g = D ( ( D - 1 f ) * ( D - 1 g ) ) . f\,{*}^{\prime}\,g=D\left(\left(D^{-1}f\right)*\left(D^{-1}g\right)\right).
  7. n n
  8. Π Π
  9. n n
  10. Γ Γ
  11. Γ Γ
  12. Π Π
  13. Π i 2 j 2 i 2 Π i 1 j 1 i 1 f j 1 j 2 g . \Pi^{i_{2}j_{2}}\partial_{i_{2}}\Pi^{i_{1}j_{1}}\partial_{i_{1}}f\,\partial_{j% _{1}}\partial_{j_{2}}g.
  14. Γ Γ
  15. m ( Γ ) m(Γ)
  16. n n
  17. m ( Γ ) = 8 m(Γ)=8
  18. n n
  19. H H⊂ℂ
  20. d s 2 = d x 2 + d y 2 y 2 ; ds^{2}=\frac{dx^{2}+dy^{2}}{y^{2}};
  21. z , w H z,w∈H
  22. z w z≠w
  23. φ φ
  24. z z
  25. i i∞
  26. z z
  27. w w
  28. ϕ ( z , w ) = 1 2 i log ( z - w ) ( z - w ¯ ) ( z ¯ - w ) ( z ¯ - w ¯ ) . \phi(z,w)=\frac{1}{2i}\log\frac{(z-w)(z-\bar{w})}{(\bar{z}-w)(\bar{z}-\bar{w})}.
  29. C n ( H ) := { ( u 1 , , u n ) H n : u i u j i j } . C_{n}(H):=\{(u_{1},\dots,u_{n})\in H^{n}:u_{i}\neq u_{j}\forall i\neq j\}.
  30. w Γ := m ( Γ ) ( 2 π ) 2 n n ! C n ( H ) j = 1 n d ϕ ( u j , u t 1 ( j ) ) d ϕ ( u j , u t 2 ( j ) ) w_{\Gamma}:=\frac{m(\Gamma)}{(2\pi)^{2n}n!}\int_{C_{n}(H)}\bigwedge_{j=1}^{n}% \mathrm{d}\phi(u_{j},u_{t1(j)})\wedge\mathrm{d}\phi(u_{j},u_{t2(j)})
  31. j j
  32. H H
  33. f * g = f g + n = 1 ( i 2 ) n Γ G n ( 2 ) w Γ B Γ ( f g ) . f*g=fg+\sum_{n=1}^{\infty}\left(\frac{i\hbar}{2}\right)^{n}\sum_{\Gamma\in G_{% n}(2)}w_{\Gamma}B_{\Gamma}(f\otimes g).
  34. ħ ħ
  35. f * g = f g + i 2 Π i j i f j g - 2 8 Π i 1 j 1 Π i 2 j 2 i 1 i 2 f j 1 j 2 g - 2 12 Π i 1 j 1 j 1 Π i 2 j 2 ( i 1 i 2 f j 2 g - i 2 f i 1 j 2 g ) + 𝒪 ( 3 ) f*g=fg+\tfrac{i\hbar}{2}\Pi^{ij}\partial_{i}f\,\partial_{j}g-\tfrac{\hbar^{2}}% {8}\Pi^{i_{1}j_{1}}\Pi^{i_{2}j_{2}}\partial_{i_{1}}\,\partial_{i_{2}}f\partial% _{j_{1}}\,\partial_{j_{2}}g-\tfrac{\hbar^{2}}{12}\Pi^{i_{1}j_{1}}\partial_{j_{% 1}}\Pi^{i_{2}j_{2}}(\partial_{i_{1}}\partial_{i_{2}}f\,\partial_{j_{2}}g-% \partial_{i_{2}}f\,\partial_{i_{1}}\partial_{j_{2}}g)+\mathcal{O}(\hbar^{3})

Koras–Russell_cubic_threefold.html

  1. x + x 2 y + z 2 + t 3 = 0 x+x^{2}y+z^{2}+t^{3}=0
  2. 𝐂 * \mathbf{C}^{*}
  3. 𝐀 3 \mathbf{A}^{3}

Kostant_partition_function.html

  1. Δ \Delta
  2. Δ + Δ \Delta^{+}\subset\Delta
  3. 1 α > 0 ( 1 - e - α ) \frac{1}{\prod_{\alpha>0}(1-e^{-\alpha})}
  4. w W ( - 1 ) ( w ) w ( e ρ ) = e ρ α > 0 ( 1 - e - α ) , {\sum_{w\in W}(-1)^{\ell(w)}w(e^{\rho})=e^{\rho}\prod_{\alpha>0}(1-e^{-\alpha}% )},
  5. ch ( V ) = w W ( - 1 ) ( w ) w ( e λ + ρ ) w W ( - 1 ) ( w ) w ( e ρ ) \operatorname{ch}(V)={\sum_{w\in W}(-1)^{\ell(w)}w(e^{\lambda+\rho})\over\sum_% {w\in W}(-1)^{\ell(w)}w(e^{\rho})}
  6. ch ( V ) = w W ( - 1 ) ( w ) w ( e λ + ρ ) e ρ α > 0 ( 1 - e - α ) . \operatorname{ch}(V)={\sum_{w\in W}(-1)^{\ell(w)}w(e^{\lambda+\rho})\over e^{% \rho}\prod_{\alpha>0}(1-e^{-\alpha})}.

Kostant_polynomial.html

  1. ( s ) \ell(s)
  2. s i x = x - 2 ( x , α i ) ( α i , α i ) α i , s_{i}x=x-2{(x,\alpha_{i})\over(\alpha_{i},\alpha_{i})}\alpha_{i},
  3. δ i f = f - f s i α i . \delta_{i}f={f-f\circ s_{i}\over\alpha_{i}}.
  4. ( s ) = m \ell(s)=m
  5. s = s i 1 s i m , s=s_{i_{1}}\cdots s_{i_{m}},
  6. δ s = δ i 1 δ i m \delta_{s}=\delta_{i_{1}}\cdots\delta_{i_{m}}
  7. δ s δ t = δ s t \displaystyle\delta_{s}\delta_{t}=\delta_{st}
  8. ( s t ) = ( s ) + ( t ) \ell(st)=\ell(s)+\ell(t)
  9. δ w 0 f = s W det s f s α > 0 α . \delta_{w_{0}}f={\sum_{s\in W}{\rm det}\,s\,f\circ s\over\prod_{\alpha>0}% \alpha}.
  10. δ s f = det s f s + t < s a s , t f t α > 0 , s - 1 α < 0 α \delta_{s}f={{\rm det}\,s\,f\circ s+\sum_{t<s}a_{s,t}\,f\circ t\over\prod_{% \alpha>0,\,s^{-1}\alpha<0}\alpha}
  11. d = | W | - 1 α > 0 α . \displaystyle d=|W|^{-1}\prod_{\alpha>0}\alpha.
  12. P s = δ s - 1 w 0 d . \displaystyle P_{s}=\delta_{s^{-1}w_{0}}d.
  13. ( s ) \ell(s)
  14. N s t = δ s ( P t ) \displaystyle N_{st}=\delta_{s}(P_{t})
  15. ( s ) ( t ) \ell(s)\geq\ell(t)
  16. det N = 1. \displaystyle{\rm det}\,N=1.
  17. f = s a s P s \displaystyle f=\sum_{s}a_{s}P_{s}
  18. δ t ( f ) = s δ t ( P s ) a s . \displaystyle\delta_{t}(f)=\sum_{s}\delta_{t}(P_{s})a_{s}.
  19. a s = t M s , t δ t ( f ) , \displaystyle a_{s}=\sum_{t}M_{s,t}\delta_{t}(f),
  20. M = N - 1 \displaystyle M=N^{-1}
  21. δ i ( f g ) = δ i ( f ) g + ( f s i ) δ i ( g ) . \delta_{i}(fg)=\delta_{i}(f)g+(f\circ s_{i})\delta_{i}(g).
  22. δ i δ s ( f ) = t δ i ( δ s ( P t ) ) a t ) = t ( δ s ( P t ) s i ) δ i ( a t ) + t δ i δ s ( P t ) a t . \delta_{i}\delta_{s}(f)=\sum_{t}\delta_{i}(\delta_{s}(P_{t}))a_{t})=\sum_{t}(% \delta_{s}(P_{t})\circ s_{i})\delta_{i}(a_{t})+\sum_{t}\delta_{i}\delta_{s}(P_% {t})a_{t}.
  23. δ i δ s = δ s i s \delta_{i}\delta_{s}=\delta_{s_{i}s}
  24. t δ s ( P t ) δ i ( a t ) s i = 0 \sum_{t}\delta_{s}(P_{t})\,\delta_{i}(a_{t})\circ s_{i}=0
  25. δ i ( a t ) = 0 \displaystyle\delta_{i}(a_{t})=0
  26. α i = 2 ( α i , α i ) - 1 α i , \displaystyle\alpha_{i}^{\vee}=2(\alpha_{i},\alpha_{i})^{-1}\alpha_{i},
  27. A ( ψ ) = s W ( - 1 ) ( s ) s ψ . A(\psi)=\sum_{s\in W}(-1)^{\ell(s)}s\cdot\psi.
  28. M t s = t ( λ s ) , N s t = ( - 1 ) ( t ) t ( ψ s ) , M_{ts}=t(\lambda_{s}),\,\,N_{st}=(-1)^{\ell(t)}\cdot t(\psi_{s}),
  29. B s , s = Ω - 1 ( N M ) s , s = A ( ψ s λ s ) Ω B_{s,s^{\prime}}=\Omega^{-1}(NM)_{s,s^{\prime}}={A(\psi_{s}\lambda_{s^{\prime}% })\over\Omega}
  30. ( s ) ( t ) \ell(s)\geq\ell(t)
  31. φ s = C s , t ψ t . \varphi_{s}=\sum C_{s,t}\psi_{t}.
  32. χ = s W a s λ s \chi=\sum_{s\in W}a_{s}\lambda_{s}
  33. a s = A ( φ s χ ) Ω . a_{s}={A(\varphi_{s}\chi)\over\Omega}.
  34. t χ = s W t ( λ s ) a s = s M t , s a s . t\chi=\sum_{s\in W}t(\lambda_{s})\,\,a_{s}=\sum_{s}M_{t,s}a_{s}.

Kostka_number.html

  1. s λ = μ K λ μ m μ . s_{\lambda}=\sum_{\mu}K_{\lambda\mu}m_{\mu}.
  2. M μ = λ K λ μ V λ . M_{\mu}=\bigoplus_{\lambda}K_{\lambda\mu}V_{\lambda}.
  3. GL n ( ) \mathrm{GL}_{n}(\mathbb{C})
  4. K λ μ = K λ μ ( 1 ) = K λ μ ( 0 , 1 ) . K_{\lambda\mu}=K_{\lambda\mu}(1)=K_{\lambda\mu}(0,1).

Kostka_polynomial.html

  1. s λ ( x 1 , , x n ) = μ K λ μ ( t ) P μ ( x 1 , , x n ; t ) . s_{\lambda}(x_{1},\ldots,x_{n})=\sum_{\mu}K_{\lambda\mu}(t)P_{\mu}(x_{1},% \ldots,x_{n};t).
  2. s λ ( x 1 , , x n ) = μ K λ μ ( q , t ) J μ ( x 1 , , x n ; q , t ) s_{\lambda}(x_{1},\ldots,x_{n})=\sum_{\mu}K_{\lambda\mu}(q,t)J_{\mu}(x_{1},% \ldots,x_{n};q,t)
  3. J μ ( x 1 , , x n ; q , t ) = P μ ( x 1 , , x n ; q , t ) s μ ( 1 - q a r m ( s ) t l e g ( s ) + 1 ) . J_{\mu}(x_{1},\ldots,x_{n};q,t)=P_{\mu}(x_{1},\ldots,x_{n};q,t)\prod_{s\in\mu}% (1-q^{arm(s)}t^{leg(s)+1}).
  4. K λ μ = K λ μ ( 1 ) = K λ μ ( 0 , 1 ) . K_{\lambda\mu}=K_{\lambda\mu}(1)=K_{\lambda\mu}(0,1).

Kramers'_law.html

  1. I I
  2. λ \lambda
  3. I ( λ ) d λ = K ( λ λ m i n - 1 ) 1 λ 2 d λ I(\lambda)d\lambda=K\left(\frac{\lambda}{\lambda_{min}}-1\right)\frac{1}{% \lambda^{2}}d\lambda
  4. λ m i n \lambda_{min}
  5. λ m i n \lambda_{min}
  6. \infty
  7. λ \lambda
  8. ω \omega
  9. λ = 2 π c / ω \lambda=2\pi c/\omega
  10. I ~ ( ω ) = I ( λ ) - d λ d ω \tilde{I}(\omega)=I(\lambda)\frac{-d\lambda}{d\omega}
  11. I ~ ( ω ) \tilde{I}(\omega)
  12. ω \omega
  13. ω m a x \omega_{max}
  14. ω m a x = 2 π c / λ m i n \omega_{max}=2\pi c/\lambda_{min}
  15. I ~ ( ω ) = K 2 π c ( ω m a x ω - 1 ) \tilde{I}(\omega)=\frac{K}{2\pi c}\left(\frac{\omega_{max}}{\omega}-1\right)
  16. ψ ( ω ) \psi(\omega)
  17. I ( λ ) I(\lambda)
  18. I ~ \tilde{I}
  19. ω \hbar\omega
  20. ψ ( ω ) = K 2 π c ( ω m a x - ω ) \psi(\omega)=\frac{K}{2\pi c}(\hbar\omega_{max}-\hbar\omega)
  21. ω ω m a x \omega\leq\omega_{max}
  22. ψ ( ω ) = 0 \psi(\omega)=0
  23. ω ω m a x \omega\geq\omega_{max}
  24. ω m a x \hbar\omega_{max}

Krasner's_lemma.html

  1. K ¯ \overline{K}
  2. K ¯ \overline{K}
  3. K ¯ \overline{K}
  4. | α - β | < | α - α i | for i = 2 , , n \left|\alpha-\beta\right|<\left|\alpha-\alpha_{i}\right|\,\text{ for }i=2,% \dots,n\,
  5. 𝔭 \mathfrak{p}
  6. 𝔭 \mathfrak{p}
  7. 𝔭 \mathfrak{p}
  8. 𝔭 ¯ \overline{\mathfrak{p}}
  9. 𝔭 ¯ \overline{\mathfrak{p}}
  10. L ¯ \overline{L}
  11. 𝔭 \mathfrak{p}
  12. f * = k = 1 n ( X - α k * ) f^{*}=\prod_{k=1}^{n}(X-\alpha_{k}^{*})
  13. K ¯ \overline{K}
  14. g = i I ( X - α i ) g=\prod_{i\in I}(X-\alpha_{i})
  15. K ¯ \overline{K}
  16. i I j J : v ( α i - α i * ) > v ( α i * - α j * ) . \forall i\in I\forall j\in J:v(\alpha_{i}-\alpha_{i}^{*})>v(\alpha_{i}^{*}-% \alpha_{j}^{*}).
  17. g * := i I ( X - α i * ) , h * := j J ( X - α j * ) g^{*}:=\prod_{i\in I}(X-\alpha_{i}^{*}),\ h^{*}:=\prod_{j\in J}(X-\alpha_{j}^{% *})

Krichevsky–Trofimov_estimator.html

  1. P ( 0 , 0 ) = 1 , P ( m , n + 1 ) = P ( m , n ) n + 1 / 2 m + n + 1 , P ( m + 1 , n ) = P ( m , n ) m + 1 / 2 m + n + 1 . \begin{array}[]{lcl}P(0,0)&=&1,\\ P(m,n+1)&=&P(m,n)\dfrac{n+1/2}{m+n+1},\\ P(m+1,n)&=&P(m,n)\dfrac{m+1/2}{m+n+1}.\end{array}

Krippendorff's_alpha.html

  1. α = 1 - D o D e = 1 - u = 1 N m u n D u D e \alpha=1-\frac{D_{o}}{D_{e}}=1-\frac{\textstyle\sum_{u=1}^{N}\frac{m_{u}}{n}D_% {u}}{D_{e}}
  2. D u = 1 m u ( m u - 1 ) i = 1 , i i m δ ( c i u , c i u ) D_{u}=\frac{1}{m_{u}(m_{u}-1)}\sum_{i=1,i^{\prime}\neq i}^{m}\delta(c_{iu},c_{% i^{\prime}u})
  3. δ ( c i u c i u ) \delta(c_{iu}c_{i^{\prime}u})
  4. δ c k 2 m e t r i c {{}_{metric}}\delta_{ck}^{2}
  5. D o = u = 1 N m u n D u = 1 n u = 1 N 1 m u - 1 i = 1 , i i m δ ( c i u , c i u ) D_{o}=\sum_{u=1}^{N}\frac{m_{u}}{n}D_{u}=\frac{1}{n}\sum_{u=1}^{N}\frac{1}{m_{% u}-1}\sum_{i=1,i^{\prime}\neq i}^{m}\delta(c_{iu},c_{i^{\prime}u})
  6. D e = 1 n ( n - 1 ) u = 1 , u = 1 N i = 1 , i = 1 m δ ( c i u , c i u ) , [ ( i , u ) ( i , u ) ] D_{e}=\frac{1}{n(n-1)}\sum_{u=1,u^{\prime}=1}^{N}\sum_{i=1,i^{\prime}=1}^{m}% \delta(c_{iu},c_{i^{\prime}u^{\prime}}),[(i,u)\neq(i^{\prime},u^{\prime})]
  7. α = 1 - D w i t h i n u n i t s = i n e r r o r D w i t h i n a n d b e t w e e n u n i t s = i n t o t a l \alpha=1-\frac{D_{within~{}units~{}=~{}in~{}error}}{D_{within~{}and~{}between~% {}units~{}=~{}in~{}total}}
  8. o v v = u = 1 N i i m I ( v i u = v ) * I ( v i u = v ) m u - 1 = o v v o_{vv^{\prime}}=\sum_{u=1}^{N}\frac{\sum_{i\neq i^{\prime}}^{m}I(v_{iu}=v)*I(v% _{i^{\prime}u}=v^{\prime})}{m_{u}-1}=o_{v^{\prime}v}
  9. n v = l = 1 V o v l = v i j m , N I ( v i j = v ) n_{v}=\sum_{l=1}^{V}o_{vl}=\sum_{v_{ij}}^{m,N}I(v_{ij}=v)
  10. n v = l = 1 L o l v = v i j m , N I ( v i j = v ) n_{v}=\sum_{l=1}^{L}o_{lv}=\sum_{v_{ij}}^{m,N}I(v_{ij}=v)
  11. n = l = 1 , p = 1 V o l p n=\sum_{l=1,p=1}^{V}o_{lp}
  12. e v v = i i m I ( v i u = v ) * I ( v i u = v ) n - 1 = 1 n - 1 { n v ( n v - 1 ) iff v = v n v n v iff v v = e k c \ e_{vv^{\prime}}=\frac{\sum_{i\neq i^{\prime}}^{m}I(v_{iu}=v)*I(v_{i^{\prime}% u}=v^{\prime})}{n-1}=\frac{1}{n-1}\begin{cases}n_{v}(n_{v}-1)&\mbox{iff }~{}v% \mbox{ = }~{}v^{\prime}\\ n_{v}n_{v^{\prime}}&\mbox{iff }~{}v\mbox{ ≠ }~{}v^{\prime}\end{cases}=e_{kc}
  13. α = 1 - D o D e = 1 - v = 1 , v = 1 V o v v δ ( v , v ) v = 1 , v = 1 V e v v δ ( v , v ) = 1 - v = 1 , v = 1 V o v v δ ( v , v ) 1 n - 1 v = 1 , v = 1 V n v n v δ ( v , v ) \alpha=1-\frac{D_{o}}{D_{e}}=1-\frac{\sum_{v=1,v^{\prime}=1}^{V}o_{vv^{\prime}% }\delta(v,v^{\prime})}{\sum_{v=1,v^{\prime}=1}^{V}e_{vv^{\prime}}\delta(v,v^{% \prime})}=1-\frac{\sum_{v=1,v^{\prime}=1}^{V}o_{vv^{\prime}}\delta(v,v^{\prime% })}{\frac{1}{n-1}\sum_{v=1,v^{\prime}=1}^{V}n_{v}n_{v^{\prime}}~{}\delta(v,v^{% \prime})}
  14. δ ( v , v ) \delta(v,v^{\prime})
  15. δ ( v , v ) 0 \delta(v,v^{\prime})\geq 0
  16. δ ( v , v ) = 0 \delta(v,v)=0
  17. δ ( v , v ) = δ ( v , v ) \delta(v,v^{\prime})=\delta(v^{\prime},v)
  18. δ n o m i n a l ( v , v ) = { 0 iff v = v 1 iff v v \delta_{nominal}(v,v^{\prime})=\begin{cases}0&\mbox{iff }~{}v\mbox{ = }~{}v^{% \prime}\\ 1&\mbox{iff }~{}v\mbox{ ≠ }~{}v^{\prime}\end{cases}
  19. δ o r d i n a l ( v , v ) = ( g = v g = v n g - n v + n v 2 ) 2 \delta_{ordinal}(v,v^{\prime})=\left(\sum_{g=v}^{g=v^{\prime}}n_{g}-\frac{n_{v% }+n_{v^{\prime}}}{2}\right)^{2}
  20. δ i n t e r v a l ( v , v ) = ( v - v ) 2 \delta_{interval}(v,v^{\prime})=(v-v^{\prime})^{2}
  21. δ r a t i o ( v , v ) = ( v - v v + v ) 2 \delta_{ratio}(v,v^{\prime})=\left(\frac{v-v^{\prime}}{v+v^{\prime}}\right)^{2}
  22. δ p o l a r ( v , v ) = ( v - v ) 2 ( v + v - 2 v m i n ) ( 2 v m a x - v - v ) \delta_{polar}(v,v^{\prime})=\frac{(v-v^{\prime})^{2}}{(v+v^{\prime}-2v_{min})% (2v_{max}-v-v^{\prime})}
  23. δ c i r c u l a r ( v , v ) = ( sin [ 180 v - v U ] ) 2 \delta_{circular}(v,v^{\prime})=\left(\sin\left[180\frac{v-v^{\prime}}{U}% \right]\right)^{2}
  24. 2 2 - 1 + \textstyle\frac{2}{2-1}+
  25. 2 2 - 1 + \textstyle\frac{2}{2-1}+
  26. 2 2 - 1 = 6 \textstyle\frac{2}{2-1}=6
  27. 1 2 - 1 = 1 = \textstyle\frac{1}{2-1}=1=
  28. 2 2 - 1 + \textstyle\frac{2}{2-1}+
  29. 2 2 - 1 = 4 \textstyle\frac{2}{2-1}=4
  30. 2 2 - 1 + \textstyle\frac{2}{2-1}+
  31. 2 3 - 1 + \textstyle\frac{2}{3-1}+
  32. 2 2 - 1 + \textstyle\frac{2}{2-1}+
  33. 2 2 - 1 = 7 \textstyle\frac{2}{2-1}=7
  34. 2 3 - 1 + \textstyle\frac{2}{3-1}+
  35. 1 2 - 1 = 2 = \textstyle\frac{1}{2-1}=2=
  36. 6 3 - 1 = 3 \textstyle\frac{6}{3-1}=3
  37. α m e t r i c = 1 - D o D e = 1 - v = 1 , v = 1 V o v v δ m e t r i c ( v , v ) 1 n - 1 v = 1 , v = 1 V n v n v δ m e t r i c ( v , v ) \alpha_{metric}=1-\frac{D_{o}}{D_{e}}=1-\frac{\sum_{v=1,v^{\prime}=1}^{V}o_{vv% ^{\prime}}\delta_{metric}(v,v^{\prime})}{\frac{1}{n-1}\sum_{v=1,v^{\prime}=1}^% {V}n_{v}n_{v^{\prime}}~{}\delta_{metric}(v,v^{\prime})}
  38. δ ( v , v ) = 0 \delta(v,v)=0
  39. δ ( v , v ) = δ ( v , v ) \delta(v,v^{\prime})=\delta(v^{\prime},v)
  40. α m e t r i c = 1 - 1 δ m e t r i c ( 1 , 3 ) + 2 δ m e t r i c ( 3 , 4 ) 1 26 - 1 ( 4 7 δ m e t r i c ( 1 , 2 ) + 10 7 δ m e t r i c ( 1 , 3 ) + 5 7 δ m e t r i c ( 1 , 4 ) + 10 4 δ m e t r i c ( 2 , 3 ) + 5 4 δ m e t r i c ( 2 , 4 ) + 5 10 δ m e t r i c ( 3 , 4 ) ) \alpha_{metric}=1-\frac{1\delta_{metric}(1,3)+2\delta_{metric}(3,4)}{\frac{1}{% 26-1}(4\cdot 7\delta_{metric}(1,2)+10\cdot 7\delta_{metric}(1,3)+5\cdot 7% \delta_{metric}(1,4)+10\cdot 4\delta_{metric}(2,3)+5\cdot 4\delta_{metric}(2,4% )+5\cdot 10\delta_{metric}(3,4))}
  41. δ n o m i n a l ( v , v ) = 1 \delta_{nominal}(v,v^{\prime})=1
  42. v v v{\neq}v^{\prime}
  43. α n o m i n a l = 1 - 1 + 2 1 26 - 1 ( 4 7 + 10 7 + 5 7 + 10 4 + 5 4 + 5 10 ) = 0.691 \alpha_{nominal}=1-\frac{1+2}{\frac{1}{26-1}(4\cdot 7+10\cdot 7+5\cdot 7+10% \cdot 4+5\cdot 4+5\cdot 10)}=0.691
  44. δ i n t e r v a l ( 1 , 2 ) = δ i n t e r v a l ( 2 , 3 ) = δ i n t e r v a l ( 3 , 4 ) = 1 2 \delta_{interval}(1,2)=\delta_{interval}(2,3)=\delta_{interval}(3,4)=1^{2}
  45. δ i n t e r v a l ( 1 , 3 ) = δ i n t e r v a l ( 2 , 4 ) = 2 2 \delta_{interval}(1,3)=\delta_{interval}(2,4)=2^{2}
  46. δ i n t e r v a l ( 1 , 4 ) = 3 2 \delta_{interval}(1,4)=3^{2}
  47. α i n t e r v a l = 1 - 1 2 2 + 2 1 2 1 26 - 1 ( 4 7 1 2 + 10 7 2 2 + 5 7 3 2 + 10 4 1 2 + 5 4 2 2 + 5 10 1 2 ) = 0.811 \alpha_{interval}=1-\frac{1\cdot 2^{2}+2\cdot 1^{2}}{\frac{1}{26-1}(4\cdot 7% \cdot 1^{2}+10\cdot 7\cdot 2^{2}+5\cdot 7\cdot 3^{2}+10\cdot 4\cdot 1^{2}+5% \cdot 4\cdot 2^{2}+5\cdot 10\cdot 1^{2})}=0.811
  48. α i n t e r v a l > α n o m i n a l \alpha_{interval}>\alpha_{nominal}
  49. α i n t e r v a l \alpha_{interval}
  50. α n o m i n a l \alpha_{nominal}
  51. α i n t e r v a l \alpha_{interval}
  52. α n o m i n a l \alpha_{nominal}
  53. π = P o - P e 1 - P e \pi=\frac{P_{o}-P_{e}}{1-P_{e}}
  54. P o = c o c c n P_{o}=\sum_{c}\frac{o_{cc}}{n}
  55. P e = c n c 2 n 2 P_{e}=\sum_{c}\frac{n_{c}^{2}}{n^{2}}
  56. α n o m i n a l = 1 - D o D e = c o c c - c e c c n - c e c c = c O c c n - c n c ( n c - 1 ) n ( n - 1 ) 1 - c n c ( n c - 1 ) n ( n - 1 ) {}_{nominal}\alpha=1-\frac{D_{o}}{D_{e}}=\frac{\textstyle\sum_{c}o_{cc}-% \textstyle\sum_{c}e_{cc}}{n-\textstyle\sum_{c}e_{cc}}=\frac{\textstyle\sum_{c}% \frac{O_{cc}}{n}-\textstyle\sum_{c}\frac{n_{c}(n_{c}-1)}{n(n-1)}}{1-\textstyle% \sum_{c}\frac{n_{c}(n_{c}-1)}{n(n-1)}}
  57. P o \ P_{o}
  58. P e = c n c 2 n 2 \ P_{e}=\textstyle\sum_{c}\frac{n_{c}^{2}}{n^{2}}
  59. c n c ( n c - 1 ) n ( n - 1 ) \textstyle\sum_{c}\frac{n_{c}(n_{c}-1)}{n(n-1)}
  60. α n o m i n a l = 1 - n - 1 n ( 1 - π ) π {}_{nominal}\alpha=1-\textstyle\frac{n-1}{n}(1-\pi)\geq\pi
  61. lim n α n o m i n a l = π \lim_{n\to\infty}\ {}_{nominal}\alpha=\pi
  62. K = P ¯ - P ¯ e 1 - P ¯ e K=\frac{\bar{P}-\bar{P}_{e}}{1-\bar{P}_{e}}
  63. P ¯ = 1 N u = 1 N c n c u ( n c u - 1 ) m ( m - 1 ) = c o c c m N \bar{P}=\frac{1}{N}\sum_{u=1}^{N}\sum_{c}\frac{n_{cu}(n_{cu}-1)}{m(m-1)}=\sum_% {c}\frac{o_{cc}}{mN}
  64. P ¯ e = c n c 2 ( m N ) 2 \bar{P}_{e}=\sum_{c}\frac{n_{c}^{2}}{(mN)^{2}}
  65. P ¯ \bar{P}
  66. P ¯ e \bar{P}_{e}
  67. c n c ( n c - 1 ) n ( n - 1 ) \textstyle\sum_{c}\frac{n_{c}(n_{c}-1)}{n(n-1)}
  68. P ¯ e \bar{P}_{e}
  69. ρ = 1 - 6 D 2 N ( N 2 - 1 ) \rho=1-\frac{6\sum D^{2}}{N(N^{2}-1)}
  70. D 2 = u = 1 N δ c k u u 2 o r d i n a l \textstyle\sum D^{2}=\textstyle\sum_{u=1}^{N}~{}_{ordinal}\delta_{c{{}_{u}}k{{% }_{u}}}^{2}
  71. ρ \rho
  72. D 2 = N D o \textstyle\sum D^{2}=ND_{o}
  73. ρ \rho
  74. N ( N 2 - 1 ) 6 = n n - 1 N D e \textstyle\frac{N(N^{2}-1)}{6}=\frac{n}{n-1}ND_{e}
  75. N D e \ ND_{e}
  76. α o r d i n a l ρ {}_{ordinal}\alpha\geq\rho
  77. lim n α o r d i n a l = ρ \lim_{n\to\infty}\ {}_{ordinal}\alpha=\rho
  78. α i n t e r v a l r i i {}_{interval}\alpha\geq r_{ii}
  79. lim n α i n t e r v a l = r i i \lim_{n\to\infty}\ {}_{interval}\alpha=r_{ii}

Krogh_length.html

  1. λ K \lambda_{K}
  2. λ K = D s c o / R \lambda_{K}=\sqrt{D_{s}c_{o}/R}
  3. D s D_{s}
  4. c o c_{o}
  5. R R

Kronecker_sum_of_discrete_Laplacians.html

  1. L = 𝐃 𝐱𝐱 𝐃 𝐲𝐲 = 𝐃 𝐱𝐱 𝐈 + 𝐈 𝐃 𝐲𝐲 , L=\mathbf{D_{xx}}\oplus\mathbf{D_{yy}}=\mathbf{D_{xx}}\otimes\mathbf{I}+% \mathbf{I}\otimes\mathbf{D_{yy}},\,
  2. 𝐃 𝐱𝐱 \mathbf{D_{xx}}
  3. 𝐃 𝐲𝐲 \mathbf{D_{yy}}
  4. 𝐈 \mathbf{I}
  5. 𝐃 𝐱𝐱 \mathbf{D_{xx}}
  6. 𝐃 𝐲𝐲 \mathbf{D_{yy}}
  7. L = 𝐃 𝐱𝐱 𝐈 𝐈 + 𝐈 𝐃 𝐲𝐲 𝐈 + 𝐈 𝐈 𝐃 𝐳𝐳 , L=\mathbf{D_{xx}}\otimes\mathbf{I}\otimes\mathbf{I}+\mathbf{I}\otimes\mathbf{D% _{yy}}\otimes\mathbf{I}+\mathbf{I}\otimes\mathbf{I}\otimes\mathbf{D_{zz}},\,
  8. 𝐃 𝐱𝐱 , 𝐃 𝐲𝐲 \mathbf{D_{xx}},\,\mathbf{D_{yy}}
  9. 𝐃 𝐳𝐳 \mathbf{D_{zz}}
  10. 𝐈 \mathbf{I}
  11. λ j x , j y , j z = - 4 h x 2 sin ( π j x 2 ( n x + 1 ) ) 2 - 4 h y 2 sin ( π j y 2 ( n y + 1 ) ) 2 - 4 h z 2 sin ( π j z 2 ( n z + 1 ) ) 2 \lambda_{jx,jy,jz}=-\frac{4}{h_{x}^{2}}\sin\left(\frac{\pi j_{x}}{2(n_{x}+1)}% \right)^{2}-\frac{4}{h_{y}^{2}}\sin\left(\frac{\pi j_{y}}{2(n_{y}+1)}\right)^{% 2}-\frac{4}{h_{z}^{2}}\sin\left(\frac{\pi j_{z}}{2(n_{z}+1)}\right)^{2}
  12. j x = 1 , , n x , j y = 1 , , n y , j z = 1 , , n z , j_{x}=1,\ldots,n_{x},\,j_{y}=1,\ldots,n_{y},\,j_{z}=1,\ldots,n_{z},\,
  13. v i x , i y , i z , j x , j y , j z = 2 n x + 1 sin ( i x j x π n x + 1 ) 2 n y + 1 sin ( i y j y π n y + 1 ) 2 n z + 1 sin ( i z j z π n z + 1 ) v_{ix,iy,iz,jx,jy,jz}=\sqrt{\frac{2}{n_{x}+1}}\sin\left(\frac{i_{x}j_{x}\pi}{n% _{x}+1}\right)\sqrt{\frac{2}{n_{y}+1}}\sin\left(\frac{i_{y}j_{y}\pi}{n_{y}+1}% \right)\sqrt{\frac{2}{n_{z}+1}}\sin\left(\frac{i_{z}j_{z}\pi}{n_{z}+1}\right)
  14. j x , j y , j z {jx,jy,jz}
  15. i x , i y , i z {ix,iy,iz}

Kunen's_inconsistency_theorem.html

  1. j ( x ) = y J ( x , y , p ) . j(x)=y\leftrightarrow J(x,y,p)\,.

Kushner_equation.html

  1. d x = f ( x , t ) d t + σ d w dx=f(x,t)\,dt+\sigma dw
  2. d z = h ( x , t ) d t + η d v dz=h(x,t)\,dt+\eta dv
  3. d p ( x , t ) = L [ p ( x , t ) ] d t + p ( x , t ) [ h ( x , t ) - E t h ( x , t ) ] T η - η - 1 [ d z - E t h ( x , t ) d t ] . dp(x,t)=L[p(x,t)]dt+p(x,t)[h(x,t)-E_{t}h(x,t)]^{T}\eta^{-\top}\eta^{-1}[dz-E_{% t}h(x,t)dt].
  4. L p = - ( f i p ) x i + 1 2 ( σ σ ) i , j 2 p x i x j Lp=-\sum\frac{\partial(f_{i}p)}{\partial x_{i}}+\frac{1}{2}\sum(\sigma\sigma^{% \top})_{i,j}\frac{\partial^{2}p}{\partial x_{i}\partial x_{j}}
  5. d p ( x , t ) = p ( x , t + d t ) - p ( x , t ) dp(x,t)=p(x,t+dt)-p(x,t)
  6. d z - E t h ( x , t ) d t dz-E_{t}h(x,t)dt
  7. f ( x , t ) = a x f(x,t)=ax
  8. h ( x , t ) = c x h(x,t)=cx
  9. d p ( x , t ) = L [ p ( x , t ) ] d t + p ( x , t ) [ c x - c μ ( t ) ] T η - η - 1 [ d z - c μ ( t ) d t ] , dp(x,t)=L[p(x,t)]dt+p(x,t)[cx-c\mu(t)]^{T}\eta^{-\top}\eta^{-1}[dz-c\mu(t)dt],
  10. μ ( t ) \mu(t)
  11. t t
  12. x x
  13. d μ ( t ) = a μ ( t ) d t + Σ ( t ) c η - η - 1 ( d z - c μ ( t ) d t ) . d\mu(t)=a\mu(t)dt+\Sigma(t)c^{\top}\eta^{-\top}\eta^{-1}\left(dz-c\mu(t)dt% \right).
  14. Σ ( t ) \Sigma(t)
  15. d Σ ( t ) d t = a Σ ( t ) + Σ ( t ) a + σ σ - Σ ( t ) c η - η - 1 c Σ ( t ) . \frac{d\Sigma(t)}{dt}=a\Sigma(t)+\Sigma(t)a^{\top}+\sigma^{\top}\sigma-\Sigma(% t)c^{\top}\eta^{-\top}\eta^{-1}c\Sigma(t).
  16. 𝒩 ( μ ( t ) , Σ ( t ) ) \mathcal{N}(\mu(t),\Sigma(t))

L-moment.html

  1. λ r = r - 1 k = 0 r - 1 ( - 1 ) k ( r - 1 k ) E X r - k : r , \lambda_{r}=r^{-1}\sum_{k=0}^{r-1}{(-1)^{k}{\left({{r-1}\atop{k}}\right)}% \mathrm{E}X_{r-k:r}},
  2. E \mathrm{E}
  3. λ 1 = E X \lambda_{1}=\mathrm{E}X
  4. λ 2 = ( E X 2 : 2 - E X 1 : 2 ) / 2 \lambda_{2}=(\mathrm{E}X_{2:2}-\mathrm{E}X_{1:2})/2
  5. λ 3 = ( E X 3 : 3 - 2 E X 2 : 3 + E X 1 : 3 ) / 3 \lambda_{3}=(\mathrm{E}X_{3:3}-2\mathrm{E}X_{2:3}+\mathrm{E}X_{1:3})/3
  6. λ 4 = ( E X 4 : 4 - 3 E X 3 : 4 + 3 E X 2 : 4 - E X 1 : 4 ) / 4. \lambda_{4}=(\mathrm{E}X_{4:4}-3\mathrm{E}X_{3:4}+3\mathrm{E}X_{2:4}-\mathrm{E% }X_{1:4})/4.
  7. λ 1 = mean, L-mean or L-location , \lambda_{1}=\,\text{mean, L-mean or L-location},
  8. λ 2 = L-scale . \lambda_{2}=\,\text{L-scale}.
  9. { x 1 < < x j < < x r } , \left\{x_{1}<\cdots<x_{j}<\cdots<x_{r}\right\},
  10. λ r = r - 1 ( n r ) - 1 x 1 < < x j < < x r ( - 1 ) r - j ( r - 1 j ) x j . \lambda_{r}=r^{-1}{{\textstyle\left({{n}\atop{r}}\right)}}^{-1}\sum_{x_{1}<% \cdots<x_{j}<\cdots<x_{r}}{(-1)^{r-j}{\left({{r-1}\atop{j}}\right)}x_{j}}.
  11. 1 = ( n 1 ) - 1 i = 1 n x ( i ) \ell_{1}={{\textstyle\left({{n}\atop{1}}\right)}}^{-1}\sum_{i=1}^{n}x_{(i)}
  12. 2 = 1 2 ( n 2 ) - 1 i = 1 n { ( i - 1 1 ) - ( n - i 1 ) } x ( i ) \ell_{2}=\tfrac{1}{2}{{\textstyle\left({{n}\atop{2}}\right)}}^{-1}\sum_{i=1}^{% n}\left\{{\textstyle\left({{i-1}\atop{1}}\right)}-{\textstyle\left({{n-i}\atop% {1}}\right)}\right\}x_{(i)}
  13. 3 = 1 3 ( n 3 ) - 1 i = 1 n { ( i - 1 2 ) - 2 ( i - 1 1 ) ( n - i 1 ) + ( n - i 2 ) } x ( i ) \ell_{3}=\tfrac{1}{3}{{\textstyle\left({{n}\atop{3}}\right)}}^{-1}\sum_{i=1}^{% n}\left\{{\textstyle\left({{i-1}\atop{2}}\right)}-2{\textstyle\left({{i-1}% \atop{1}}\right)}{\textstyle\left({{n-i}\atop{1}}\right)}+{\textstyle\left({{n% -i}\atop{2}}\right)}\right\}x_{(i)}
  14. 4 = 1 4 ( n 4 ) - 1 i = 1 n { ( i - 1 3 ) - 3 ( i - 1 2 ) ( n - i 1 ) + 3 ( i - 1 1 ) ( n - i 2 ) - ( n - i 3 ) } x ( i ) \ell_{4}=\tfrac{1}{4}{{\textstyle\left({{n}\atop{4}}\right)}}^{-1}\sum_{i=1}^{% n}\left\{{\textstyle\left({{i-1}\atop{3}}\right)}-3{\textstyle\left({{i-1}% \atop{2}}\right)}{\textstyle\left({{n-i}\atop{1}}\right)}+3{\textstyle\left({{% i-1}\atop{1}}\right)}{\textstyle\left({{n-i}\atop{2}}\right)}-{\textstyle\left% ({{n-i}\atop{3}}\right)}\right\}x_{(i)}
  15. i i
  16. ( ) {\textstyle\left({{\cdot}\atop{\cdot}}\right)}
  17. τ r = λ r / λ 2 , r = 3 , 4 , . \tau_{r}=\lambda_{r}/\lambda_{2},\qquad r=3,4,\dots.
  18. τ 3 \tau_{3}
  19. τ 4 \tau_{4}
  20. τ 4 \tau_{4}
  21. 1 4 ( 5 τ 3 2 - 1 ) τ 4 < 1. \tfrac{1}{4}(5\tau_{3}^{2}-1)\leq\tau_{4}<1.
  22. τ = λ 2 / λ 1 , \tau=\lambda_{2}/\lambda_{1},
  23. λ < s u b > 1 λ<sub>1

Ladder_operator.html

  1. j j
  2. l l
  3. m l m_{l}
  4. [ N , X ] = c X , [N,X]=cX,\quad
  5. | n \scriptstyle{|n\rangle}
  6. N | n = n | n , N|n\rangle=n|n\rangle,\,
  7. | n \scriptstyle{|n\rangle}
  8. N X | n \displaystyle NX|n\rangle
  9. | n \scriptstyle{|n\rangle}
  10. X | n \scriptstyle{X|n\rangle}
  11. [ N , X ] = - c X . [N,X^{\dagger}]=-cX^{\dagger}.\quad
  12. J + = J x + i J y , J_{+}=J_{x}+iJ_{y},\quad
  13. J - = J x - i J y , J_{-}=J_{x}-iJ_{y},\quad
  14. [ J i , J j ] = i ϵ i j k J k , [J_{i},J_{j}]=i\hbar\epsilon_{ijk}J_{k},
  15. [ J z , J ± ] = ± J ± . \left[J_{z},J_{\pm}\right]=\pm\hbar J_{\pm}.\quad
  16. [ J + , J - ] = 2 J z . \left[J_{+},J_{-}\right]=2\hbar J_{z}.\quad
  17. J z J ± | j m \displaystyle J_{z}J_{\pm}|j\,m\rangle
  18. J z | j m ± 1 = ( m ± 1 ) | j m ± 1 . J_{z}|j\,m\pm 1\rangle=\hbar(m\pm 1)|j\,m\pm 1\rangle.\quad
  19. J ± | j m \scriptstyle{J_{\pm}|j\,m\rangle}
  20. | j m ± 1 \scriptstyle{|j\,m\pm 1\rangle}
  21. J + | j m = α | j m + 1 , J_{+}|j\,m\rangle=\alpha|j\,m+1\rangle,\quad
  22. J - | j m = β | j m - 1 . J_{-}|j\,m\rangle=\beta|j\,m-1\rangle.\quad
  23. J ± = J \scriptstyle{J_{\pm}=J_{\mp}^{\dagger}}
  24. j m | J + J + | j m = j m | J - J + | j m = j m + 1 | α * α | j m + 1 = | α | 2 \langle j\,m|J_{+}^{\dagger}J_{+}|j\,m\rangle=\langle j\,m|J_{-}J_{+}|j\,m% \rangle=\langle j\,m+1|\alpha^{*}\alpha|j\,m+1\rangle=|\alpha|^{2}
  25. j m | J - J - | j m = j m | J + J - | j m = j m - 1 | β * β | j m - 1 = | β | 2 \langle j\,m|J_{-}^{\dagger}J_{-}|j\,m\rangle=\langle j\,m|J_{+}J_{-}|j\,m% \rangle=\langle j\,m-1|\beta^{*}\beta|j\,m-1\rangle=|\beta|^{2}
  26. J - J + = ( J x - i J y ) ( J x + i J y ) = J x 2 + J y 2 + i [ J x , J y ] = J 2 - J z 2 - J z , J_{-}J_{+}=(J_{x}-iJ_{y})(J_{x}+iJ_{y})=J_{x}^{2}+J_{y}^{2}+i[J_{x},J_{y}]=J^{% 2}-J_{z}^{2}-\hbar J_{z},
  27. J + J - = ( J x + i J y ) ( J x - i J y ) = J x 2 + J y 2 - i [ J x , J y ] = J 2 - J z 2 + J z . J_{+}J_{-}=(J_{x}+iJ_{y})(J_{x}-iJ_{y})=J_{x}^{2}+J_{y}^{2}-i[J_{x},J_{y}]=J^{% 2}-J_{z}^{2}+\hbar J_{z}.
  28. | α | 2 = 2 j ( j + 1 ) - 2 m 2 - 2 m = 2 ( j - m ) ( j + m + 1 ) , |\alpha|^{2}=\hbar^{2}j(j+1)-\hbar^{2}m^{2}-\hbar^{2}m=\hbar^{2}(j-m)(j+m+1),
  29. | β | 2 = 2 j ( j + 1 ) - 2 m 2 + 2 m = 2 ( j + m ) ( j - m + 1 ) . |\beta|^{2}=\hbar^{2}j(j+1)-\hbar^{2}m^{2}+\hbar^{2}m=\hbar^{2}(j+m)(j-m+1).
  30. J + | j m = ( j - m ) ( j + m + 1 ) | j m + 1 = j ( j + 1 ) - m ( m + 1 ) | j m + 1 , J_{+}|j\,m\rangle=\hbar\sqrt{(j-m)(j+m+1)}|j\,m+1\rangle=\hbar\sqrt{j(j+1)-m(m% +1)}|j\,m+1\rangle,
  31. J - | j m = ( j + m ) ( j - m + 1 ) | j m - 1 = j ( j + 1 ) - m ( m - 1 ) | j m - 1 . J_{-}|j\,m\rangle=\hbar\sqrt{(j+m)(j-m+1)}|j\,m-1\rangle=\hbar\sqrt{j(j+1)-m(m% -1)}|j\,m-1\rangle.
  32. - j m j \scriptstyle{-j\leq m\leq j}
  33. J + | j j = 0 , J_{+}|j\,j\rangle=0,\,
  34. J - | j - j = 0. J_{-}|j\,-j\rangle=0.\,
  35. H ^ D = A ^ 𝐈 𝐉 , \hat{H}\text{D}=\hat{A}\mathbf{I}\cdot\mathbf{J},\quad
  36. J - 1 ( 1 ) = 1 2 ( J x - i J y ) = J - 2 J 0 ( 1 ) = J z J + 1 ( 1 ) = - 1 2 ( J x + i J y ) = - J + 2 . \begin{aligned}\displaystyle J_{-1}^{(1)}&\displaystyle=\dfrac{1}{\sqrt{2}}(J_% {x}-iJ_{y})=\dfrac{J_{-}}{\sqrt{2}}\\ \displaystyle J_{0}^{(1)}&\displaystyle=J_{z}\\ \displaystyle J_{+1}^{(1)}&\displaystyle=-\frac{1}{\sqrt{2}}(J_{x}+iJ_{y})=-% \frac{J_{+}}{\sqrt{2}}.\end{aligned}
  37. 𝐈 ( 1 ) 𝐉 ( 1 ) = n = - 1 + 1 ( - 1 ) n I n ( 1 ) J - n ( 1 ) = I 0 ( 1 ) J 0 ( 1 ) - I - 1 ( 1 ) J + 1 ( 1 ) - I + 1 ( 1 ) J - 1 ( 1 ) , \mathbf{I}^{(1)}\cdot\mathbf{J}^{(1)}=\sum_{n=-1}^{+1}(-1)^{n}I_{n}^{(1)}J_{-n% }^{(1)}=I_{0}^{(1)}J_{0}^{(1)}-I_{-1}^{(1)}J_{+1}^{(1)}-I_{+1}^{(1)}J_{-1}^{(1% )},
  38. a = m ω 2 ( x ^ + i m ω p ^ ) a = m ω 2 ( x ^ - i m ω p ^ ) \begin{aligned}\displaystyle a&\displaystyle=\sqrt{m\omega\over 2\hbar}\left(% \hat{x}+{i\over m\omega}\hat{p}\right)\\ \displaystyle a^{\dagger}&\displaystyle=\sqrt{m\omega\over 2\hbar}\left(\hat{x% }-{i\over m\omega}\hat{p}\right)\end{aligned}

Lady_tasting_tea.html

  1. n = 8 n=8
  2. k = 4 k=4
  3. 8 ! 4 ! ( 8 - 4 ) ! = 70 \frac{8!}{4!(8-4)!}=70

Lagrange's_identity_(boundary_value_problem).html

  1. v L [ u ] - u L * [ v ] = s y m b o l M , vL[u]-uL^{*}[v]=\nabla\cdot symbolM,
  2. M i = j = 1 n a i j ( v u x j - u v x j ) + u v ( b i - j = 1 n a i j x j ) , M_{i}=\sum_{j=1}^{n}a_{ij}\left(v\frac{\partial u}{\partial x_{j}}-u\frac{% \partial v}{\partial x_{j}}\right)+uv\left(b_{i}-\sum_{j=1}^{n}\frac{\partial a% _{ij}}{\partial x_{j}}\right),
  3. s y m b o l M = i = 1 n x i M i , \nabla\cdot symbolM=\sum_{i=1}^{n}\frac{\partial}{\partial x_{i}}M_{i},
  4. L [ u ] = i , j = 1 n a i , j 2 u x i x j + i = 1 n b i u x i + c u L[u]=\sum_{i,\ j=1}^{n}a_{i,j}\frac{\partial^{2}u}{\partial x_{i}\partial x_{j% }}+\sum_{i=1}^{n}b_{i}\frac{\partial u}{\partial x_{i}}+cu
  5. L * [ v ] = i , j = 1 n 2 ( a i , j v ) x i x j - i = 1 n ( b i v ) x i + c v . L^{*}[v]=\sum_{i,\ j=1}^{n}\frac{\partial^{2}(a_{i,j}v)}{\partial x_{i}% \partial x_{j}}-\sum_{i=1}^{n}\frac{\partial(b_{i}v)}{\partial x_{i}}+cv.\,
  6. Ω v L [ u ] d Ω = Ω u L * [ v ] d Ω + S s y m b o l M n d S , \int_{\Omega}vL[u]\ d\Omega=\int_{\Omega}uL^{*}[v]\ d\Omega+\int_{S}symbol{M% \cdot n}\,dS,
  7. a ( x ) d 2 y d x 2 + b ( x ) d y d x + c ( x ) y + λ w ( x ) y = 0 , a(x)\frac{d^{2}y}{dx^{2}}+b(x)\frac{dy}{dx}+c(x)y+\lambda w(x)y=0,
  8. d d x ( p ( x ) d y d x ) + ( q ( x ) + λ w ( x ) ) y ( x ) = 0. \frac{d}{dx}\left(p(x)\frac{dy}{dx}\right)+\left(q(x)+\lambda w(x)\right)y(x)=0.
  9. L f = d d x ( p ( x ) d f d x ) + q ( x ) f . Lf=\frac{d}{dx}\left(p(x)\frac{df}{dx}\right)+q(x)f.
  10. u L v - v L u = - d d x [ p ( x ) ( v d u d x - u d v d x ) ] . uLv-vLu=-\frac{d}{dx}\left[p(x)\left(v\frac{du}{dx}-u\frac{dv}{dx}\right)% \right].
  11. 0 1 d x ( u L v - v L u ) = [ p ( x ) ( u d v d x - v d u d x ) ] 0 1 , \int_{0}^{1}\ dx\ (uLv-vLu)=\left[p(x)\left(u\frac{dv}{dx}-v\frac{du}{dx}% \right)\right]_{0}^{1},
  12. p = P ( x ) \ p=P(x)
  13. q = Q ( x ) \ q=Q(x)
  14. u = U ( x ) \ u=U(x)
  15. v = V ( x ) \ v=V(x)
  16. x \ x
  17. u \ u
  18. v \ v
  19. u L v = u [ d d x ( p ( x ) d v d x ) + q ( x ) v ] , uLv=u\left[\frac{d}{dx}\left(p(x)\frac{dv}{dx}\right)+q(x)v\right],
  20. v L u = v [ d d x ( p ( x ) d u d x ) + q ( x ) u ] . vLu=v\left[\frac{d}{dx}\left(p(x)\frac{du}{dx}\right)+q(x)u\right].
  21. u L v - v L u = u d d x ( p ( x ) d v d x ) - v d d x ( p ( x ) d u d x ) . uLv-vLu=u\frac{d}{dx}\left(p(x)\frac{dv}{dx}\right)-v\frac{d}{dx}\left(p(x)% \frac{du}{dx}\right).
  22. u L v - v L u = d d x ( p ( x ) u d v d x ) - d d x ( v p ( x ) d u d x ) , uLv-vLu=\frac{d}{dx}\left(p(x)u\frac{dv}{dx}\right)-\frac{d}{dx}\left(vp(x)% \frac{du}{dx}\right),
  23. = d d x [ p ( x ) ( u d v d x - v d u d x ) ] , =\frac{d}{dx}\left[p(x)\left(u\frac{dv}{dx}-v\frac{du}{dx}\right)\right],
  24. 0 1 d x ( u L v - v L u ) = [ p ( x ) ( u d v d x - v d u d x ) ] 0 1 , \int_{0}^{1}\ dx\ (uLv-vLu)=\left[p(x)\left(u\frac{dv}{dx}-v\frac{du}{dx}% \right)\right]_{0}^{1},

Lagrangian_coherent_structure.html

  1. 𝒫 {\mathcal{P}}
  2. = [ t 0 , t 1 ] {\mathcal{I}}=[t_{0},t_{1}]
  3. F t 0 t : x 0 x ( t , t 0 , x 0 ) F^{t}_{t_{0}}\colon x_{0}\mapsto x(t,t_{0},x_{0})
  4. x 0 𝒫 x_{0}\in{\mathcal{P}}
  5. x ( t , t 0 , x 0 ) 𝒫 x(t,t_{0},x_{0})\in{\mathcal{P}}
  6. t t\in{\mathcal{I}}
  7. F t 0 t F^{t}_{t_{0}}
  8. t t\in{\mathcal{I}}
  9. ( t 0 ) {\mathcal{M}}(t_{0})
  10. 𝒫 {\mathcal{P}}
  11. = { ( x , t ) 𝒫 × : [ F t 0 t ] - 1 ( x ) ( t 0 ) } {\mathcal{M}}=\{(x,t)\in{\mathcal{P}}\times{\mathcal{I}}\,\colon[F^{t}_{t_{0}}% ]^{-1}(x)\in{\mathcal{M}}(t_{0})\}
  12. 𝒫 × {\mathcal{P}}\times{\mathcal{I}}
  13. ( t ) = F t 0 t ( ( t 0 ) ) {\mathcal{M}}(t)=F^{t}_{t_{0}}({\mathcal{M}}(t_{0}))
  14. {\mathcal{M}}
  15. ( t 0 ) {\mathcal{M}}(t_{0})
  16. 𝒫 × {\mathcal{M}}\in{\mathcal{P}}\times{\mathcal{I}}
  17. ( t ) {\mathcal{M}}(t)
  18. {\mathcal{I}}
  19. ( t ) {\mathcal{M}}(t)
  20. {\mathcal{I}}
  21. ( t 0 ) {\mathcal{M}}(t_{0})
  22. ( t 1 ) {\mathcal{M}}(t_{1})
  23. {\mathcal{I}}
  24. F t 0 t F^{t}_{t_{0}}
  25. {\mathcal{I}}
  26. ( t ) {\mathcal{M}}(t)
  27. {\mathcal{I}}
  28. v = v ( x , t ) , x U 3 , v=v(x,t),\qquad x\in U\subset{\mathbb{R}}^{3},
  29. U U
  30. x = Q ( t ) y + b ( t ) , x=Q(t)y+b(t),
  31. y 3 y\in{\mathbb{R}}^{3}
  32. Q ( t ) Q(t)
  33. 3 × 3 3\times 3
  34. b ( t ) b(t)
  35. 3 3
  36. S ( x , t ) S(x,t)
  37. W ( x , t ) W(x,t)
  38. S ( x , t ) = 1 2 ( v ( x , t ) + ( v ( x , t ) ) T ) , W ( x , t ) = 1 2 ( v ( x , t ) - ( v ( x , t ) ) T ) , S(x,t)=\frac{1}{2}\left(\nabla v(x,t)+(\nabla v(x,t))^{T}\right),\qquad W(x,t)% =\frac{1}{2}\left(\nabla v(x,t)-(\nabla v(x,t))^{T}\right),
  39. S ~ ( y , t ) = Q ( t ) T S ( x , t ) Q ( t ) , W ~ ( y , t ) = Q ( t ) T S ( x , t ) Q ( t ) - Q ( t ) T Q ˙ ( t ) . {\tilde{S}}(y,t)=Q(t)^{T}S(x,t)Q(t),\qquad{\tilde{W}}(y,t)=Q(t)^{T}S(x,t)Q(t)-% Q(t)^{T}{\dot{Q}}(t).
  40. S ( x , t ) S(x,t)
  41. S ( x , t ) S(x,t)
  42. W ( x , t ) W(x,t)
  43. v ( x , t ) \nabla v(x,t)
  44. W ( y , t ) {W}(y,t)
  45. F t 0 t \nabla F^{t}_{t_{0}}
  46. v ( x , t ) v(x,t)
  47. x ˙ = v ( x , t ) = ( sin 4 t 2 + cos 4 t - 2 + cos 4 t - sin 4 t ) x , {\dot{x}}=v(x,t)=\begin{pmatrix}\sin{4t}&2+\cos{4t}\\ -2+\cos{4t}&-\sin{4t}\end{pmatrix}x,
  48. q = 1 2 ( | S | 2 - | W | 2 ) < 0 q=\frac{1}{2}({|S|}^{2}-{|W|}^{2})<0
  49. | | |\,\cdot\,|
  50. x x
  51. y y
  52. x = Q ( t ) y + b ( t ) x=Q(t)y+b(t)
  53. x ( t ) = Q ( t ) y ( t ) + b ( t ) x(t)=Q(t)y(t)+b(t)
  54. x x
  55. y y
  56. y y
  57. 𝒫 × {\mathcal{P}}\times{\mathcal{I}}
  58. ( t 0 ) {\mathcal{M}}(t_{0})
  59. ( t 0 ) {\mathcal{M}}(t_{0})
  60. ( t 0 ) {\mathcal{M}}(t_{0})
  61. ξ ( t ) {\xi}(t)
  62. x ( t , t 0 , x 0 ) x(t,t_{0},x_{0})
  63. F t 0 t \nabla F^{t}_{t_{0}}
  64. ϵ ξ ( t 0 ) \epsilon{\xi}(t_{0})
  65. x 0 x_{0}
  66. 0 < ϵ 1 0<\epsilon\ll 1
  67. ξ ( t 0 ) {\xi}(t_{0})
  68. n {\mathbb{R}}^{n}
  69. x ( t , t 0 , x 0 ) x(t,t_{0},x_{0})
  70. ξ ϵ ( t 1 ; x 0 ) = F t 0 t 1 ( x 0 ) ϵ ξ ( t 0 ) {\xi}_{\epsilon}(t_{1};x_{0})=\nabla F^{t_{1}}_{t_{0}}(x_{0})\epsilon{\xi}(t_{% 0})
  71. x 0 x_{0}
  72. δ t 0 t 1 ( x 0 ) = lim ϵ 0 1 ϵ max | ξ ( t 0 ) | = 1 | ξ ϵ ( t 1 ; x 0 ) | = max | ξ ( t 0 ) | = 1 F t 0 t 1 ( x 0 ) ξ ( t 0 ) , F t 0 t 1 ( x 0 ) ξ ( t 0 ) = max | ξ ( t 0 ) | = 1 ξ ( t 0 ) , C t 0 t 1 ( x 0 ) ξ ( t 0 ) , \delta^{t_{1}}_{t_{0}}(x_{0})=\lim_{\epsilon\to 0}\frac{1}{\epsilon}\max_{% \left|\xi(t_{0})\right|=1}\left|\xi_{\epsilon}(t_{1};x_{0})\right|=\max_{\left% |\xi(t_{0})\right|=1}\sqrt{\left\langle\nabla F_{t_{0}}^{t_{1}}(x_{0})\xi(t_{0% }),\nabla F_{t_{0}}^{t_{1}}(x_{0})\xi(t_{0})\right\rangle}=\max_{\left|\xi(t_{% 0})\right|=1}\sqrt{\left\langle\xi(t_{0}),C_{t_{0}}^{t_{1}}(x_{0})\xi(t_{0})% \right\rangle},
  73. C t 0 t 1 = [ F t 0 t 1 ] T F t 0 t 1 C^{t_{1}}_{t_{0}}=\left[\nabla F_{t_{0}}^{t_{1}}\right]^{T}\nabla F_{t_{0}}^{t% _{1}}
  74. x 0 x_{0}
  75. δ t 0 t 1 ( x 0 ) = λ n ( x 0 ) \delta^{t_{1}}_{t_{0}}(x_{0})=\sqrt{\lambda_{n}(x_{0})}
  76. ( log δ t 0 t 1 ) / ( t 1 - t 0 ) (\log{\delta^{t_{1}}_{t_{0}}})/(t_{1}-t_{0})
  77. FTLE t 0 t 1 ( x 0 ) = 1 2 ( t 1 - t 0 ) log λ n ( x 0 ) . \mathrm{FTLE}_{t_{0}}^{t_{1}}(x_{0})=\frac{1}{2(t_{1}-t_{0})}\log\lambda_{n}(x% _{0}).
  78. t 0 t_{0}
  79. FTLE t 0 t 1 ( x 0 ) \mathrm{FTLE}_{t_{0}}^{t_{1}}(x_{0})
  80. t 1 t_{1}
  81. FTLE t 1 t 0 \mathrm{FTLE}_{t_{1}}^{t_{0}}
  82. F t 0 t ( x 0 ) \nabla F^{t}_{t_{0}}(x_{0})
  83. x ( t ; t 0 , x 0 ) x(t;t_{0},x_{0})
  84. x 0 x_{0}
  85. x = ( x 1 , x 2 , x 3 ) x=(x^{1},x^{2},x^{3})
  86. x ( t ; t 0 , x 0 ) x(t;t_{0},x_{0})
  87. F t 0 t ( x 0 ) ( x 1 ( t ; t 0 , x 0 + δ 1 ) - x 1 ( t ; t 0 , x 0 - δ 1 ) | 2 δ 1 | x 1 ( t ; t 0 , x 0 + δ 2 ) - x 1 ( t ; t 0 , x 0 - δ 2 ) | 2 δ 2 | x 1 ( t ; t 0 , x 0 + δ 3 ) - x 1 ( t ; t 0 , x 0 - δ 3 ) | 2 δ 3 | x 2 ( t ; t 0 , x 0 + δ 1 ) - x 2 ( t ; t 0 , x 0 - δ 1 ) | 2 δ 1 | x 2 ( t ; t 0 , x 0 + δ 2 ) - x 2 ( t ; t 0 , x 0 - δ 2 ) | 2 δ 2 | x 2 ( t ; t 0 , x 0 + δ 3 ) - x 2 ( t ; t 0 , x 0 - δ 3 ) | 2 δ 3 | x 3 ( t ; t 0 , x 0 + δ 1 ) - x 3 ( t ; t 0 , x 0 - δ 1 ) | 2 δ 1 | x 3 ( t ; t 0 , x 0 + δ 2 ) - x 3 ( t ; t 0 , x 0 - δ 2 ) | 2 δ 2 | x 3 ( t ; t 0 , x 0 + δ 3 ) - x 3 ( t ; t 0 , x 0 - δ 3 ) | 2 δ 3 | ) , \nabla F_{t_{0}}^{t}(x_{0})\approx\begin{pmatrix}\frac{x^{1}(t;t_{0},x_{0}+% \delta_{1})-x^{1}(t;t_{0},x_{0}-\delta_{1})}{\left|2\delta_{1}\right|}&\frac{x% ^{1}(t;t_{0},x_{0}+\delta_{2})-x^{1}(t;t_{0},x_{0}-\delta_{2})}{\left|2\delta_% {2}\right|}&\frac{x^{1}(t;t_{0},x_{0}+\delta_{3})-x^{1}(t;t_{0},x_{0}-\delta_{% 3})}{\left|2\delta_{3}\right|}\\ \frac{x^{2}(t;t_{0},x_{0}+\delta_{1})-x^{2}(t;t_{0},x_{0}-\delta_{1})}{\left|2% \delta_{1}\right|}&\frac{x^{2}(t;t_{0},x_{0}+\delta_{2})-x^{2}(t;t_{0},x_{0}-% \delta_{2})}{\left|2\delta_{2}\right|}&\frac{x^{2}(t;t_{0},x_{0}+\delta_{3})-x% ^{2}(t;t_{0},x_{0}-\delta_{3})}{\left|2\delta_{3}\right|}\\ \frac{x^{3}(t;t_{0},x_{0}+\delta_{1})-x^{3}(t;t_{0},x_{0}-\delta_{1})}{\left|2% \delta_{1}\right|}&\frac{x^{3}(t;t_{0},x_{0}+\delta_{2})-x^{3}(t;t_{0},x_{0}-% \delta_{2})}{\left|2\delta_{2}\right|}&\frac{x^{3}(t;t_{0},x_{0}+\delta_{3})-x% ^{3}(t;t_{0},x_{0}-\delta_{3})}{\left|2\delta_{3}\right|}\end{pmatrix},
  88. δ i \delta_{i}
  89. x i x^{i}
  90. 2 × 2 2\times 2
  91. [ t 0 + T , t 1 + T ] [t_{0}+T,t_{1}+T]
  92. FTLE t 0 + T t 1 + T ( x 0 ) \mathrm{FTLE}_{t_{0}+T}^{t_{1}+T}(x_{0})
  93. T T
  94. [ t 0 , t 1 ] [t_{0},t_{1}]
  95. [ t 0 + T , t 1 + T ] [t_{0}+T,t_{1}+T]
  96. t 0 + T t_{0}+T
  97. FTLE t 0 + T t 1 + T \mathrm{FTLE}_{t_{0}+T}^{t_{1}+T}
  98. [ t 0 + T , t 1 + T ] [t_{0}+T,t_{1}+T]
  99. [ t 0 + T , t 1 + T ] [t_{0}+T,t_{1}+T]
  100. T T
  101. [ t 0 , t 1 ] [t_{0},t_{1}]
  102. x 0 x_{0}
  103. n 0 n_{0}
  104. ( t 0 ) \mathcal{M}(t_{0})
  105. T x 0 ( t 0 ) T_{x_{0}}\mathcal{M}(t_{0})
  106. T x 1 ( t 1 ) T_{x_{1}}\mathcal{M}(t_{1})
  107. F t 0 t 1 ( x 0 ) \nabla F_{t_{0}}^{t_{1}}(x_{0})
  108. n 0 n_{0}
  109. F t 0 t 1 ( x 0 ) \nabla F_{t_{0}}^{t_{1}}(x_{0})
  110. ( t 1 ) \mathcal{M}(t_{1})
  111. ρ t 0 t 1 ( x 0 , n 0 ) \rho^{t_{1}}_{t_{0}}(x_{0,}n_{0})
  112. σ t 0 t 1 ( x 0 , n 0 ) \sigma^{t_{1}}_{t_{0}}(x_{0},n_{0})
  113. ρ t 0 t 1 ( x 0 , n 0 ) > 1 \rho^{t_{1}}_{t_{0}}(x_{0},n_{0})>1
  114. ( t ) \mathcal{M}(t)
  115. [ t 0 , t 1 ] [t_{0},t_{1}]
  116. ρ t 0 t 1 ( x 0 , n 0 ) < 1 \rho^{t_{1}}_{t_{0}}(x_{0},n_{0})<1
  117. ( t ) \mathcal{M}(t)
  118. [ t 0 , t 1 ] [t_{0},t_{1}]
  119. ( t ) \mathcal{M}(t)
  120. ρ t 0 t 1 ( x 0 , n 0 ) \rho^{t_{1}}_{t_{0}}(x_{0},n_{0})
  121. n 0 n_{0}
  122. ( t 0 ) {\mathcal{M}}(t_{0})
  123. n = 2 , 3 n=2,3
  124. ( t 0 ) {\mathcal{M}}(t_{0})
  125. ( t 0 ) {\mathcal{M}}(t_{0})
  126. ξ 1 ( x 0 ) \xi_{1}(x_{0})
  127. x 0 = ξ 2 ( x 0 ) x^{\prime}_{0}=\xi_{2}(x_{0})
  128. × ξ 1 ( x 0 ) , ξ 1 ( x 0 ) = 0 \left\langle\nabla\times\xi_{1}(x_{0}),\xi_{1}(x_{0})\right\rangle=0
  129. ξ n ( x 0 ) \xi_{n}(x_{0})
  130. x 0 = ξ 1 ( x 0 ) x^{\prime}_{0}=\xi_{1}(x_{0})
  131. × ξ 3 ( x 0 ) , ξ 3 ( x 0 ) = 0 \left\langle\nabla\times\xi_{3}(x_{0}),\xi_{3}(x_{0})\right\rangle=0
  132. λ 2 ( x 0 ) \lambda_{2}(x_{0})
  133. λ 1 ( x 0 ) \lambda_{1}(x_{0})
  134. F t 0 t F^{t}_{t_{0}}
  135. 𝒪 ( ϵ ) {\mathcal{O}}(\epsilon)
  136. 𝒪 ( ϵ ) {\mathcal{O}}(\epsilon)
  137. 𝒪 ( ϵ ) {\mathcal{O}}(\epsilon)
  138. 𝒪 ( ϵ ) {\mathcal{O}}(\epsilon)
  139. 𝒪 ( ϵ ) {\mathcal{O}}(\epsilon)
  140. D t 0 t 1 D_{t_{0}}^{t_{1}}
  141. D t 0 t 1 ( x 0 ) = 1 2 [ C t 0 t 1 ( x 0 ) Ω - Ω C t 0 t 1 ( x 0 ) ] , Ω = ( 0 - 1 1 0 ) . D_{t_{0}}^{t_{1}}(x_{0})=\frac{1}{2}\left[C_{t_{0}}^{t_{1}}(x_{0})\Omega-% \Omega C_{t_{0}}^{t_{1}}(x_{0})\right],\qquad\Omega=\begin{pmatrix}0&-1\\ 1&0\\ \end{pmatrix}.
  142. ξ i ( x 0 ) \xi_{i}(x_{0})
  143. x 0 = ξ 1 ( x 0 ) x_{0}^{\prime}=\xi_{1}(x_{0})
  144. λ 2 ( x 0 ) \lambda_{2}(x_{0})
  145. x 0 = ξ 2 ( x 0 ) x_{0}^{\prime}=\xi_{2}(x_{0})
  146. λ 1 ( x 0 ) \lambda_{1}(x_{0})
  147. [ t 0 , t 1 ] [t_{0},t_{1}]
  148. F t 0 t 1 = R t 0 t 1 U t 0 t 1 = V t 0 t 1 R t 0 t 1 , \nabla F^{t_{1}}_{t_{0}}=R^{t_{1}}_{t_{0}}U^{t_{1}}_{t_{0}}=V^{t_{1}}_{t_{0}}R% ^{t_{1}}_{t_{0}},
  149. R t 0 t 1 R^{t_{1}}_{t_{0}}
  150. U t 0 t 1 , V t 0 t 1 U^{t_{1}}_{t_{0}},V^{t_{1}}_{t_{0}}
  151. C t 0 t 1 = [ F t 0 t 1 ] T F t 0 t 1 = U t 0 t 1 U t 0 t 1 = V t 0 t 1 V t 0 t 1 , C^{t_{1}}_{t_{0}}=[\nabla F^{t_{1}}_{t_{0}}]^{T}\nabla F^{t_{1}}_{t_{0}}=U^{t_% {1}}_{t_{0}}U^{t_{1}}_{t_{0}}=V^{t_{1}}_{t_{0}}V^{t_{1}}_{t_{0}},
  152. C t 0 t 1 C^{t_{1}}_{t_{0}}
  153. R t 0 t 1 R^{t_{1}}_{t_{0}}
  154. R t 0 t 1 R^{t_{1}}_{t_{0}}
  155. θ t 0 t 1 ( x 0 ) \theta_{t_{0}}^{t_{1}}(x_{0})
  156. R t 0 t 1 R^{t_{1}}_{t_{0}}
  157. x 0 x_{0}
  158. 2 π 2\pi
  159. C t 0 t 1 C^{t_{1}}_{t_{0}}
  160. cos θ t 0 t 1 = ξ i , F t 0 t 1 ξ i λ i , i = 1 o r 2 , sin θ t 0 t 1 = ( - 1 ) j ξ i , F t 0 t 1 ξ j λ j , ( i , j ) = ( 1 , 2 ) o r ( 2 , 1 ) , \begin{aligned}\displaystyle\cos\theta_{t_{0}}^{t_{1}}&\displaystyle=\frac{% \langle\xi_{i},\nabla F^{t_{1}}_{t_{0}}\xi_{i}\rangle}{\sqrt{\lambda_{i}}},% \quad i=1\,\,or\,\,\,2,\\ \displaystyle\sin\theta_{t_{0}}^{t_{1}}&\displaystyle=\left(-1\right)^{j}\frac% {\langle\xi_{i},\nabla F^{t_{1}}_{t_{0}}\xi_{j}\rangle}{\sqrt{\lambda_{j}}},% \qquad(i,j)=(1,2)\,\,or\,\,(2,1),\\ \end{aligned}
  161. θ t 0 t = [ 1 - sign ( sin θ t 0 t ) ] π + sign ( sin θ t 0 t ) cos - 1 ( cos θ t 0 t ) . \theta_{t_{0}}^{t}=\left[1-{\rm sign\,}\left(\sin\theta_{t_{0}}^{t}\right)% \right]\pi+{\rm sign\,}\left(\sin\theta_{t_{0}}^{t}\right)\cos^{-1}\left(\cos% \theta_{t_{0}}^{t}\right).
  162. C t 0 t 1 C^{t_{1}}_{t_{0}}
  163. cos θ t 0 t = 1 2 ( i = 1 3 ξ i , F t 0 t 1 ξ i λ i - 1 ) , sin θ t 0 t = ξ i , F t 0 t 1 ξ j - ξ j , F t 0 t 1 ξ i 2 ϵ i j k e k , i j , \begin{aligned}\displaystyle\cos\theta_{t_{0}}^{t}&\displaystyle=\frac{1}{2}% \left(\sum_{i=1}^{3}\frac{\left\langle\xi_{i},\nabla F^{t_{1}}_{t_{0}}\xi_{i}% \right\rangle}{\sqrt{\lambda_{i}}}-1\right),\\ \displaystyle\sin\theta_{t_{0}}^{t}&\displaystyle=\frac{\left\langle\xi_{i},% \nabla F^{t_{1}}_{t_{0}}\xi_{j}\right\rangle-\left\langle\xi_{j},\nabla F^{t_{% 1}}_{t_{0}}\xi_{i}\right\rangle}{2\epsilon_{ijk}e_{k}},\qquad i\neq j,\end{aligned}
  164. ϵ i j k \epsilon_{ijk}
  165. 𝐞 = { e k } \mathbf{e}=\left\{e_{k}\right\}
  166. [ K t 0 t ] j k = ξ j , F t 0 t 1 ξ k / λ k \left[K_{t_{0}}^{t}\right]_{jk}=\left\langle\xi_{j},\nabla F^{t_{1}}_{t_{0}}% \xi_{k}\right\rangle/{\sqrt{\lambda_{k}}}
  167. t 0 t_{0}
  168. θ t 0 t \theta_{t_{0}}^{t}
  169. [ t 0 , t 1 ] [t_{0},t_{1}]
  170. x 0 ( t 0 ) x_{0}\in{\mathcal{M}}(t_{0})
  171. ( t ) {\mathcal{M}}(t)
  172. T x 0 ( t 0 ) T_{x_{0}}{\mathcal{M}}(t_{0})
  173. σ t 0 t 1 ( x 0 ) \sigma^{t_{1}}_{t_{0}}(x_{0})
  174. η ± ( x 0 ) := λ 2 ( x 0 ) - 1 λ 2 ( x 0 ) - λ 1 ( x 0 ) ξ 1 ( x 0 ) ± 1 - λ 1 ( x 0 ) λ 2 ( x 0 ) - λ 1 ( x 0 ) ξ 2 ( x 0 ) , \eta^{\pm}(x_{0}):=\sqrt{\frac{\lambda_{2}(x_{0})-1}{\lambda_{2}(x_{0})-% \lambda_{1}(x_{0})}}\xi_{1}(x_{0})\pm\sqrt{\frac{1-\lambda_{1}(x_{0})}{\lambda% _{2}(x_{0})-\lambda_{1}(x_{0})}}\xi_{2}(x_{0}),
  175. n ± ( x 0 ) = λ 1 ( x 0 ) λ 1 ( x 0 ) + λ 3 ( x 0 ) ξ 1 ( x 0 ) ± λ 3 ( x 0 ) λ 1 ( x 0 ) + λ 3 ( x 0 ) ξ 3 ( x 0 ) , n_{\pm}(x_{0})=\sqrt{\frac{\sqrt{\lambda_{1}(x_{0})}}{\sqrt{\lambda_{1}(x_{0})% }+\sqrt{\lambda_{3}(x_{0})}}}\xi_{1}(x_{0})\pm\sqrt{\frac{\sqrt{\lambda_{3}(x_% {0})}}{\sqrt{\lambda_{1}(x_{0})}+\sqrt{\lambda_{3}(x_{0})}}}\xi_{3}(x_{0}),
  176. ( t 0 ) {\mathcal{M}}(t_{0})
  177. ( t 0 ) {\mathcal{M}}(t_{0})
  178. ( t 0 ) {\mathcal{M}}(t_{0})
  179. n ± ( x 0 ) n_{\pm}(x_{0})
  180. x 0 = η ± ( x 0 ) x^{\prime}_{0}=\eta^{\pm}(x_{0})
  181. × n ± ( x 0 ) , n ± ( x 0 ) = 0 \langle\nabla\times n_{\pm}(x_{0}),n_{\pm}(x_{0})\rangle=0
  182. λ \lambda
  183. λ \lambda
  184. λ = 1 \lambda=1
  185. 1 2 ( C t 0 t 1 - λ I ) \frac{1}{2}(C^{t_{1}}_{t_{0}}-\lambda I)
  186. λ > 0 \lambda>0
  187. η λ ± ( x 0 ) := λ 2 ( x 0 ) - λ 2 λ 2 ( x 0 ) - λ 1 ( x 0 ) ξ 1 ( x 0 ) ± λ 2 - λ 1 ( x 0 ) λ 2 ( x 0 ) - λ 1 ( x 0 ) ξ 2 ( x 0 ) , \eta^{\pm}_{\lambda}(x_{0}):=\sqrt{\frac{\lambda_{2}(x_{0})-\lambda^{2}}{% \lambda_{2}(x_{0})-\lambda_{1}(x_{0})}}\xi_{1}(x_{0})\pm\sqrt{\frac{\lambda^{2% }-\lambda_{1}(x_{0})}{\lambda_{2}(x_{0})-\lambda_{1}(x_{0})}}\xi_{2}(x_{0}),
  188. λ = 1 \lambda=1
  189. η λ ± ( x 0 \eta^{\pm}_{\lambda}(x_{0}
  190. η ± ( x 0 \eta^{\pm}(x_{0}
  191. η λ ± \eta^{\pm}_{\lambda}
  192. λ \lambda
  193. F t 0 t 1 F^{t_{1}}_{t_{0}}
  194. λ \lambda
  195. λ \lambda
  196. t 0 t_{0}
  197. t 1 t_{1}
  198. λ \lambda
  199. F T L E t 0 t 1 ( x 0 ) FTLE^{t_{1}}_{t_{0}}(x_{0})
  200. D t 0 t 1 D^{t_{1}}_{t_{0}}
  201. λ 1 ( x 0 ) = λ 2 ( x 0 ) \lambda_{1}(x_{0})=\lambda_{2}(x_{0})
  202. t 0 t_{0}
  203. t 1 t_{1}
  204. 𝒫 {\mathcal{P}}
  205. I I

Lagrangian_mechanics.html

  1. 𝐫 ˙ j = ( r ˙ 1 , r ˙ 2 , r ˙ 3 ) \mathbf{\dot{r}}_{j}=(\dot{r}_{1},\dot{r}_{2},\dot{r}_{3})
  2. q j q_{j}
  3. q j ˙ \dot{q_{j}}
  4. r = r ( q i , t ) . {r}={r}(q_{i},t).\,
  5. r ( θ ( t ) ) = ( sin θ , - cos θ ) {r}(\theta(t))=(\ell\sin\theta,-\ell\cos\theta)
  6. r ˙ ( θ ( t ) , θ ˙ ( t ) ) = ( θ ˙ cos θ , θ ˙ sin θ ) {\dot{r}}(\theta(t),\dot{\theta}(t))=(\ell\,\dot{\theta}\cos\theta,\ell\,\dot{% \theta}\sin\theta)
  7. 𝐫 1 = 𝐫 1 ( q 1 , q 2 , , q m , t ) 𝐫 2 = 𝐫 2 ( q 1 , q 2 , , q m , t ) 𝐫 n = 𝐫 n ( q 1 , q 2 , , q m , t ) \begin{array}[]{r c l}\mathbf{r}_{1}&=&\mathbf{r}_{1}(q_{1},q_{2},\cdots,q_{m}% ,t)\\ \mathbf{r}_{2}&=&\mathbf{r}_{2}(q_{1},q_{2},\cdots,q_{m},t)\\ &\vdots&\\ \mathbf{r}_{n}&=&\mathbf{r}_{n}(q_{1},q_{2},\cdots,q_{m},t)\end{array}
  8. δ 𝐫 i = j = 1 m 𝐫 i q j δ q j , \delta\mathbf{r}_{i}=\sum_{j=1}^{m}\frac{\partial\mathbf{r}_{i}}{\partial q_{j% }}\delta q_{j},
  9. δ W = j = 1 m i = 1 n ( 𝐅 i - m i 𝐚 i ) 𝐫 i q j δ q j = 0. \delta W=\sum_{j=1}^{m}\sum_{i=1}^{n}(\mathbf{F}_{i}-m_{i}\mathbf{a}_{i})\cdot% \frac{\partial\mathbf{r}_{i}}{\partial q_{j}}\delta q_{j}=0.
  10. Q j = δ W δ q j = i = 1 n 𝐅 i 𝐫 i q j . Q_{j}=\frac{\delta W}{\delta q_{j}}=\sum_{i=1}^{n}\mathbf{F}_{i}\cdot\frac{% \partial\mathbf{r}_{i}}{\partial q_{j}}.
  11. 𝐅 i = - V Q j = - i = 1 n V 𝐫 i q j = - V q j . \mathbf{F}_{i}=-\nabla V\Rightarrow Q_{j}=-\sum_{i=1}^{n}\nabla V\cdot\frac{% \partial\mathbf{r}_{i}}{\partial q_{j}}=-\frac{\partial V}{\partial q_{j}}.
  12. V V
  13. T = 1 2 i = 1 n m i 𝐫 ˙ i 𝐫 ˙ i . T=\frac{1}{2}\sum_{i=1}^{n}m_{i}\mathbf{\dot{r}}_{i}\cdot\mathbf{\dot{r}}_{i}.
  14. q ˙ j \dot{q}_{j}
  15. T q j = i = 1 n m i 𝐫 ˙ i 𝐫 ˙ i q j \frac{\partial T}{\partial q_{j}}=\sum_{i=1}^{n}m_{i}\mathbf{\dot{r}}_{i}\cdot% \frac{\partial\mathbf{\dot{r}}_{i}}{\partial q_{j}}
  16. T q ˙ j = i = 1 n m i 𝐫 ˙ i 𝐫 ˙ i q ˙ j . \quad\frac{\partial T}{\partial\dot{q}_{j}}=\sum_{i=1}^{n}m_{i}\mathbf{\dot{r}% }_{i}\cdot\frac{\partial\mathbf{\dot{r}}_{i}}{\partial\dot{q}_{j}}.
  17. 𝐫 ˙ i q j ˙ = 𝐫 i q j . \frac{\partial\mathbf{\dot{r}}_{i}}{\partial\dot{q_{j}}}=\frac{\partial\mathbf% {r}_{i}}{\partial q_{j}}.
  18. T q ˙ j = i = 1 n m i 𝐫 ˙ i 𝐫 i q j . \quad\frac{\partial T}{\partial\dot{q}_{j}}=\sum_{i=1}^{n}m_{i}\mathbf{\dot{r}% }_{i}\cdot\frac{\partial\mathbf{r}_{i}}{\partial q_{j}}\ .
  19. d d t T q ˙ j = i = 1 n m i 𝐫 ¨ i 𝐫 i q j + i = 1 n m i 𝐫 ˙ i 𝐫 ˙ i q j = Q j + T q j . \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T}{\partial\dot{q}_{j}}=\sum_{i=1% }^{n}m_{i}\mathbf{\ddot{r}}_{i}\cdot\frac{\partial\mathbf{r}_{i}}{\partial q_{% j}}+\sum_{i=1}^{n}m_{i}\mathbf{\dot{r}}_{i}\cdot\frac{\partial\mathbf{\dot{r}}% _{i}}{\partial q_{j}}=Q_{j}+\frac{\partial T}{\partial q_{j}}\ .
  20. L = T - V L=T-V\,
  21. 𝒮 \mathcal{S}
  22. 𝒮 = t 1 t 2 L d t . \mathcal{S}=\int_{t_{1}}^{t_{2}}L\,\mathrm{d}t.
  23. ( 𝐫 , t ) \mathcal{L}(\mathbf{r},t)
  24. 𝐫 \mathbf{r}
  25. L ( t ) = ( 𝐫 , t ) d 3 𝐫 L(t)=\int\mathcal{L}(\mathbf{r},t)\mathrm{d}^{3}\mathbf{r}\,
  26. 𝒮 = t 1 t 2 ( 𝐫 , t ) d 3 𝐫 d t . \mathcal{S}=\int_{t_{1}}^{t_{2}}\int\mathcal{L}(\mathbf{r},t)\mathrm{d}^{3}% \mathbf{r}\mathrm{d}t.
  27. δ 𝒮 = 0. \delta\mathcal{S}=0.\,\!
  28. F ( r 1 , r 2 , r 3 ) = A F(r_{1},r_{2},r_{3})=A
  29. [ L r j - d d t ( L r ˙ j ) ] + λ F r j = 0 \left[\frac{\partial L}{\partial r_{j}}-\frac{\mathrm{d}}{\mathrm{d}t}\left(% \frac{\partial L}{\partial\dot{r}_{j}}\right)\right]+\lambda\frac{\partial F}{% \partial r_{j}}=0
  30. δ L δ r j + λ F r j = 0 \frac{\delta L}{\delta r_{j}}+\lambda\frac{\partial F}{\partial r_{j}}=0
  31. δ L δ r j = L r j - d d t ( L r ˙ j ) \frac{\delta L}{\delta r_{j}}=\frac{\partial L}{\partial r_{j}}-\frac{\mathrm{% d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial\dot{r}_{j}}\right)
  32. N j = i = 1 e λ i F i r j N_{j}=\sum_{i=1}^{e}\lambda_{i}\frac{\partial F_{i}}{\partial r_{j}}
  33. N j = i = 1 n 𝐍 i 𝐫 i q j N_{j}=\sum_{i=1}^{n}\mathbf{N}_{i}\cdot\frac{\partial\mathbf{r}_{i}}{\partial q% _{j}}
  34. Q j = d d t ( T q ˙ j ) - T q j = - δ T δ q j = i = 1 n 𝐅 i 𝐫 i q j , Q_{j}=\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial T}{\partial\dot{q}_{j% }}\right)-\frac{\partial T}{\partial q_{j}}=-\frac{\delta T}{\delta q_{j}}=% \sum_{i=1}^{n}\mathbf{F}_{i}\cdot\frac{\partial\mathbf{r}_{i}}{\partial q_{j}},
  35. 𝐅 i = - V i + 𝐍 i , \mathbf{F}_{i}=-\nabla V_{i}+\mathbf{N}_{i},
  36. δ T δ q j = i = 1 n 𝐅 i 𝐫 i q j = i = 1 n ( - V i + 𝐍 i ) 𝐫 i q j = - i = 1 n V i 𝐫 i q j + i = 1 n 𝐍 i 𝐫 i q j = - V q j + N j \frac{\delta T}{\delta q_{j}}=\sum_{i=1}^{n}\mathbf{F}_{i}\cdot\frac{\partial% \mathbf{r}_{i}}{\partial q_{j}}=\sum_{i=1}^{n}(-\nabla V_{i}+\mathbf{N}_{i})% \cdot\frac{\partial\mathbf{r}_{i}}{\partial q_{j}}=-\sum_{i=1}^{n}\nabla V_{i}% \cdot\frac{\partial\mathbf{r}_{i}}{\partial q_{j}}+\sum_{i=1}^{n}\mathbf{N}_{i% }\cdot\frac{\partial\mathbf{r}_{i}}{\partial q_{j}}=-\frac{\partial V}{% \partial q_{j}}+N_{j}
  37. δ T δ q j = d d t ( ( L + V ) q ˙ j ) - ( L + V ) q j = - δ L δ q ˙ j - V q j \frac{\delta T}{\delta q_{j}}=\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{% \partial(L+V)}{\partial\dot{q}_{j}}\right)-\frac{\partial(L+V)}{\partial q_{j}% }=-\frac{\delta L}{\delta\dot{q}_{j}}-\frac{\partial V}{\partial q_{j}}
  38. N j = - δ L δ q ˙ j N_{j}=-\frac{\delta L}{\delta\dot{q}_{j}}
  39. N j = i = 1 e λ i F i r j N_{j}=\sum_{i=1}^{e}\lambda_{i}\frac{\partial F_{i}}{\partial r_{j}}
  40. L = L + d f ( q , t ) d t L^{\prime}=L+\frac{\mathrm{d}f(q,t)}{\mathrm{d}t}
  41. L ( q 1 ( t ) , q 2 ( t ) , , q 1 ( t ) , q 2 ( t ) , , t ) L(q_{1}(t),q_{2}(t),\ldots,\dots{q}_{1}(t),\dots{q}_{2}(t),\ldots,t)
  42. q ˙ j \dot{q}_{j}
  43. D = 1 2 j = 1 m k = 1 m C j k q ˙ j q ˙ k . D=\frac{1}{2}\sum_{j=1}^{m}\sum_{k=1}^{m}C_{jk}\dot{q}_{j}\dot{q}_{k}.
  44. Q j = - V q j - D q ˙ j Q_{j}=-\frac{\partial V}{\partial q_{j}}-\frac{\partial D}{\partial\dot{q}_{j}}
  45. 0 = d d t ( L q ˙ j ) - L q j + D q ˙ j . 0=\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial\dot{q}_{j}}% \right)-\frac{\partial L}{\partial q_{j}}+\frac{\partial D}{\partial\dot{q}_{j% }}.
  46. i ( 𝐅 i - m i 𝐚 i ) δ 𝐫 i = 0 \sum_{i}(\mathbf{F}_{i}-m_{i}\mathbf{a}_{i})\cdot\delta\mathbf{r}_{i}=0
  47. t 1 t 2 i ( 𝐅 i - m i 𝐚 i ) δ 𝐫 i d t = t 1 t 2 δ ( - V + 1 2 i m i v i 2 ) d t - [ i m i 𝐯 i δ 𝐫 i ] t 1 t 2 = δ t 1 t 2 L d t - [ i m i 𝐯 i δ 𝐫 i ] t 1 t 2 \begin{aligned}\displaystyle\int_{t_{1}}^{t_{2}}\sum_{i}(\mathbf{F}_{i}-m_{i}% \mathbf{a}_{i})\cdot\delta\mathbf{r}_{i}\,dt&\displaystyle=\int_{t_{1}}^{t_{2}% }\delta\left(-V+\frac{1}{2}\sum_{i}m_{i}v_{i}^{2}\right)\,dt-\left[\sum_{i}m_{% i}\mathbf{v}_{i}\cdot\delta\mathbf{r}_{i}\right]_{t_{1}}^{t_{2}}\\ &\displaystyle=\delta\int_{t_{1}}^{t_{2}}L\,dt-\left[\sum_{i}m_{i}\mathbf{v}_{% i}\cdot\delta\mathbf{r}_{i}\right]_{t_{1}}^{t_{2}}\end{aligned}
  48. δ t 1 t 2 L d t = 0. \delta\int_{t_{1}}^{t_{2}}L\,dt=0.
  49. J = x 1 x 2 F ( x , y , y ) d x J=\int_{x_{1}}^{x_{2}}F(x,y,y^{\prime})\mathrm{d}x
  50. d d x F y = F y \frac{\mathrm{d}}{\mathrm{d}x}\frac{\partial F}{\partial y^{\prime}}=\frac{% \partial F}{\partial y}
  51. x t , y q , y q ˙ , F L , J 𝒮 x\rightarrow t,\quad y\rightarrow q,\quad y^{\prime}\rightarrow\dot{q},\quad F% \rightarrow L,\quad J\rightarrow\mathcal{S}
  52. Q j = d d t ( ( L + V ) q ˙ j ) - ( L + V ) q j = [ d d t ( L q ˙ j ) + 0 ] - [ L q j + V q j ] = d d t ( L q ˙ j ) - L q j + Q j . Q_{j}=\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial\mathcal{(}L+V)}{% \partial\dot{q}_{j}}\right)-\frac{\partial\mathcal{(}L+V)}{\partial q_{j}}=% \left[\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial\dot{q}_{j% }}\right)+0\right]-\left[\frac{\partial L}{\partial q_{j}}+\frac{\partial V}{% \partial q_{j}}\right]=\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{% \partial\dot{q}_{j}}\right)-\frac{\partial L}{\partial q_{j}}+Q_{j}.
  53. d d t ( L q ˙ j ) = L q j \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial\dot{q}_{j}}% \right)=\frac{\partial L}{\partial q_{j}}
  54. q ˙ j \dot{q}_{j}
  55. F = - V . {F}=-{\nabla}V.\,
  56. δ W = F δ r . \delta W={F}\cdot\delta{r}.
  57. F δ r = m r ¨ δ r . {F}\cdot\delta{r}=m\ddot{{r}}\cdot\delta{r}.
  58. F δ r = - V i r q i δ q i = - i , j V r j r j q i δ q i = - i V q i δ q i . {F}\cdot{\delta}{r}=-{\nabla}V\cdot\displaystyle\sum_{i}{\partial{r}\over% \partial q_{i}}\delta q_{i}=-\displaystyle\sum_{i,j}{\partial V\over\partial r% _{j}}{\partial r_{j}\over\partial q_{i}}\delta q_{i}=-\displaystyle\sum_{i}{% \partial V\over\partial q_{i}}\delta q_{i}.
  59. m r ¨ δ r = m j [ i r i ¨ r i q j ] δ q j m\ddot{{r}}\cdot\delta{r}=m\sum_{j}\left[\sum_{i}\ddot{r_{i}}{\partial r_{i}% \over\partial q_{j}}\right]\delta q_{j}
  60. d d t r i ¨ r i q j d t = d d t ( r i q j r ˙ i ) - d d t d d t ( r i q j ) r ˙ i d t = d d t ( r ˙ i r i q j ) - r ˙ i d d t ( r i q j ) \frac{\mathrm{d}}{\mathrm{d}t}\int\ddot{r_{i}}{\partial r_{i}\over\partial q_{% j}}\mathrm{d}t=\frac{\mathrm{d}}{\mathrm{d}t}\left({\partial r_{i}\over% \partial q_{j}}\dot{r}_{i}\right)-\frac{\mathrm{d}}{\mathrm{d}t}\int\frac{% \mathrm{d}}{\mathrm{d}t}\left({\partial r_{i}\over\partial q_{j}}\right)\dot{r% }_{i}\mathrm{d}t=\frac{\mathrm{d}}{\mathrm{d}t}\left(\dot{r}_{i}{\partial r_{i% }\over\partial q_{j}}\right)-\dot{r}_{i}\frac{\mathrm{d}}{\mathrm{d}t}\left({% \partial r_{i}\over\partial q_{j}}\right)
  61. m r ¨ δ r = m j [ i [ d d t ( r i ˙ r i q j ) - r i ˙ d d t ( r i q j ) ] ] δ q j m\ddot{{r}}\cdot\delta{r}=m\sum_{j}\left[\sum_{i}\left[{\mathrm{d}\over\mathrm% {d}t}\left(\dot{r_{i}}{\partial r_{i}\over\partial q_{j}}\right)-\dot{r_{i}}{% \mathrm{d}\over\mathrm{d}t}\left({\partial r_{i}\over\partial q_{j}}\right)% \right]\right]\delta q_{j}
  62. d d t r j q i = r j ˙ q i , r j q i = r j ˙ q i ˙ , {\mathrm{d}\over\mathrm{d}t}{\partial r_{j}\over\partial q_{i}}={\partial\dot{% r_{j}}\over\partial q_{i}},\quad{\partial r_{j}\over\partial q_{i}}={\partial% \dot{r_{j}}\over\partial\dot{q_{i}}},
  63. m r ¨ δ r = m j [ i [ d d t ( r i ˙ r i ˙ q j ˙ ) - r i ˙ r i ˙ q j ] ] δ q j m\ddot{{r}}\cdot\delta{r}=m\sum_{j}\left[\sum_{i}\left[{\mathrm{d}\over\mathrm% {d}t}\left(\dot{r_{i}}{\partial\dot{r_{i}}\over\partial\dot{q_{j}}}\right)-% \dot{r_{i}}{\partial\dot{r_{i}}\over\partial q_{j}}\right]\right]\delta q_{j}
  64. m r ¨ δ r = m j [ i [ d d t q j ˙ ( 1 2 r i ˙ 2 ) - q j ( 1 2 r i ˙ 2 ) ] ] δ q j m\ddot{{r}}\cdot\delta{r}=m\sum_{j}\left[\sum_{i}\left[{\mathrm{d}\over\mathrm% {d}t}{\partial\over\partial\dot{q_{j}}}\left(\frac{1}{2}\dot{r_{i}}^{2}\right)% -{\partial\over\partial q_{j}}\left(\frac{1}{2}\dot{r_{i}}^{2}\right)\right]% \right]\delta q_{j}
  65. m r ¨ δ r = j [ d d t q j ˙ ( i 1 2 m r i ˙ 2 ) - q j ( i 1 2 m r i ˙ 2 ) ] δ q j m\ddot{{r}}\cdot\delta{r}=\sum_{j}\left[{\mathrm{d}\over\mathrm{d}t}{\partial% \over\partial\dot{q_{j}}}\left(\sum_{i}\frac{1}{2}m\dot{r_{i}}^{2}\right)-{% \partial\over\partial q_{j}}\left(\sum_{i}\frac{1}{2}m\dot{r_{i}}^{2}\right)% \right]\delta q_{j}
  66. m r ¨ δ r = i [ d d t T q i ˙ - T q i ] δ q i m\ddot{{r}}\cdot\delta{r}=\sum_{i}\left[{\mathrm{d}\over\mathrm{d}t}{\partial T% \over\partial\dot{q_{i}}}-{\partial T\over\partial q_{i}}\right]\delta q_{i}
  67. m 𝐫 ¨ δ 𝐫 - 𝐅 δ 𝐫 = i [ d d t T q i ˙ - ( T - V ) q i ] δ q i = 0. m\mathbf{\ddot{r}}\cdot\delta\mathbf{r}-\mathbf{F}\cdot\delta\mathbf{r}=\sum_{% i}\left[{\mathrm{d}\over\mathrm{d}t}{\partial{T}\over\partial{\dot{q_{i}}}}-{% \partial{(T-V)}\over\partial q_{i}}\right]\delta q_{i}=0.
  68. [ d d t T q i ˙ - ( T - V ) q i ] = 0 \left[{\mathrm{d}\over\mathrm{d}t}{\partial{T}\over\partial{\dot{q_{i}}}}-{% \partial{(T-V)}\over\partial q_{i}}\right]=0
  69. d d t V q i ˙ = 0. {\mathrm{d}\over\mathrm{d}t}{\partial{V}\over\partial{\dot{q_{i}}}}=0.
  70. L q i = d d t L q i ˙ . {\partial{L}\over\partial q_{i}}={\mathrm{d}\over\mathrm{d}t}{\partial{L}\over% \partial{\dot{q_{i}}}}.
  71. L \scriptstyle L
  72. q i \scriptstyle q_{i}
  73. q i \scriptstyle q_{i}
  74. p i \scriptstyle p_{i}
  75. p i = L q ˙ i , p_{i}=\frac{\partial L}{\partial\dot{q}_{i}},
  76. p ˙ i = d d t L q ˙ i = L q i = 0. \dot{p}_{i}=\frac{d}{dt}\frac{\partial L}{\partial\dot{q}_{i}}=\frac{\partial L% }{\partial q_{i}}=0.
  77. L \scriptstyle L
  78. q ˙ i \scriptstyle\dot{q}_{i}
  79. p 2 = L q ˙ 2 , p_{2}=\frac{\partial L}{\partial\dot{q}_{2}},
  80. L ( q 1 , q 3 , q 4 , ; q ˙ 1 , q ˙ 2 , q ˙ 3 , q ˙ 4 , ; t ) . L(q_{1},q_{3},q_{4},\dots;\dot{q}_{1},\dot{q}_{2},\dot{q}_{3},\dot{q}_{4},% \dots;t)\,.
  81. x ¨ = g \ddot{x}=g
  82. x ( t ) = 1 2 g t 2 x(t)=\frac{1}{2}gt^{2}
  83. 1 / 2 {1}/{2}
  84. L = T - V = 1 2 m x ˙ 2 + m g x . L=T-V=\frac{1}{2}m\dot{x}^{2}+mgx.
  85. L x - d d t L x ˙ = 0 = m g - m d x ˙ d t \frac{\partial L}{\partial x}-\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{% \partial\dot{x}}=0=mg-m\frac{\mathrm{d}\dot{x}}{\mathrm{d}t}
  86. x ¨ = g \ddot{x}=g
  87. T = 1 2 m x ˙ 2 , V = 1 2 k x 2 , T=\frac{1}{2}m\dot{x}^{2}\,,\quad V=\frac{1}{2}kx^{2}\,,
  88. L = T - V = 1 2 m x ˙ 2 - 1 2 k x 2 L=T-V=\frac{1}{2}m\dot{x}^{2}-\frac{1}{2}kx^{2}
  89. d d t L x ˙ = L x m x ¨ = - k x , \frac{d}{dt}\frac{\partial L}{\partial\dot{x}}=\frac{\partial L}{\partial x}% \quad\Rightarrow\quad m\ddot{x}=-kx\,,
  90. x ¨ = - ω 2 x , ω = k / m , \ddot{x}=-\omega^{2}x\,,\quad\omega=\sqrt{k/m}\,,
  91. x ( t ) = A sin ( ω t ) + B cos ( ω t ) x(t)=A\sin(\omega t)+B\cos(\omega t)
  92. T = 1 2 M x ˙ 2 + 1 2 m ( x ˙ pend 2 + y ˙ pend 2 ) = 1 2 M x ˙ 2 + 1 2 m [ ( x ˙ + θ ˙ cos θ ) 2 + ( θ ˙ sin θ ) 2 ] , \begin{array}[]{rcl}T&=&\frac{1}{2}M\dot{x}^{2}+\frac{1}{2}m\left(\dot{x}_{% \mathrm{pend}}^{2}+\dot{y}_{\mathrm{pend}}^{2}\right)\\ &=&\frac{1}{2}M\dot{x}^{2}+\frac{1}{2}m\left[\left(\dot{x}+\ell\dot{\theta}% \cos\theta\right)^{2}+\left(\ell\dot{\theta}\sin\theta\right)^{2}\right],\end{array}
  93. V = m g y pend = - m g cos θ . V=mgy_{\mathrm{pend}}=-mg\ell\cos\theta.
  94. L = T - V = 1 2 M x ˙ 2 + 1 2 m [ ( x ˙ + θ ˙ cos θ ) 2 + ( θ ˙ sin θ ) 2 ] + m g cos θ = 1 2 ( M + m ) x ˙ 2 + m x ˙ θ ˙ cos θ + 1 2 m 2 θ ˙ 2 + m g cos θ \begin{array}[]{rcl}L&=&T-V\\ &=&\frac{1}{2}M\dot{x}^{2}+\frac{1}{2}m\left[\left(\dot{x}+\ell\dot{\theta}% \cos\theta\right)^{2}+\left(\ell\dot{\theta}\sin\theta\right)^{2}\right]+mg% \ell\cos\theta\\ &=&\frac{1}{2}\left(M+m\right)\dot{x}^{2}+m\dot{x}\ell\dot{\theta}\cos\theta+% \frac{1}{2}m\ell^{2}\dot{\theta}^{2}+mg\ell\cos\theta\end{array}
  95. d d t [ ( M + m ) x ˙ + m θ ˙ cos θ ] = 0 , \frac{\mathrm{d}}{\mathrm{d}t}\left[(M+m)\dot{x}+m\ell\dot{\theta}\cos\theta% \right]=0,
  96. ( M + m ) x ¨ + m θ ¨ cos θ - m θ ˙ 2 sin θ = 0 (M+m)\ddot{x}+m\ell\ddot{\theta}\cos\theta-m\ell\dot{\theta}^{2}\sin\theta=0
  97. θ \theta
  98. d d t [ m ( x ˙ cos θ + 2 θ ˙ ) ] + m ( x ˙ θ ˙ + g ) sin θ = 0 ; \frac{\mathrm{d}}{\mathrm{d}t}\left[m(\dot{x}\ell\cos\theta+\ell^{2}\dot{% \theta})\right]+m\ell(\dot{x}\dot{\theta}+g)\sin\theta=0;
  99. θ ¨ + x ¨ cos θ + g sin θ = 0. \ddot{\theta}+\frac{\ddot{x}}{\ell}\cos\theta+\frac{g}{\ell}\sin\theta=0.\,
  100. x ¨ 0 \ddot{x}\to 0
  101. θ ¨ 0 \ddot{\theta}\to 0
  102. L = T - U = 1 2 M 𝐑 ˙ 2 + ( 1 2 μ 𝐫 ˙ 2 - U ( r ) ) = L cm + L rel \begin{aligned}\displaystyle L&\displaystyle=T-U=\frac{1}{2}M\dot{\mathbf{R}}^% {2}+\left(\frac{1}{2}\mu\dot{\mathbf{r}}^{2}-U(r)\right)\\ &\displaystyle=L_{\mathrm{cm}}+L_{\mathrm{rel}}\end{aligned}
  103. M 𝐑 ¨ = 0 , M\ddot{\mathbf{R}}=0,\,
  104. L = 1 2 μ ( r ˙ 2 + r 2 θ ˙ 2 ) - U ( r ) , L=\frac{1}{2}\mu\left(\dot{r}^{2}+r^{2}\dot{\theta}^{2}\right)-U(r),
  105. L θ ˙ = μ r 2 θ ˙ = constant = , \frac{\partial L}{\partial\dot{\theta}}=\mu r^{2}\dot{\theta}=\mathrm{constant% }=\ell,\,
  106. L r = d d t L r ˙ , \frac{\partial L}{\partial r}=\frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{% \partial\dot{r}},\,
  107. μ r θ ˙ 2 - d U d r = μ r ¨ . \mu r\dot{\theta}^{2}-\frac{dU}{dr}=\mu\ddot{r}.\,
  108. θ ˙ = μ r 2 , \dot{\theta}=\frac{\ell}{\mu r^{2}},\,
  109. μ r ¨ = - d U d r + 2 μ r 3 . \mu\ddot{r}=-\frac{dU}{dr}+\frac{\ell^{2}}{\mu r^{3}}.\,
  110. F cf = μ r θ ˙ 2 = 2 μ r 3 . F_{\mathrm{cf}}=\mu r\dot{\theta}^{2}=\frac{\ell^{2}}{\mu r^{3}}.\,
  111. 𝐅 \mathbf{F}
  112. V ( 𝐱 ) V(\mathbf{x})
  113. 𝐅 = - V ( 𝐱 ) \mathbf{F}=-\nabla V(\mathbf{x})
  114. L ( 𝐱 , 𝐱 ˙ ) = 1 2 m 𝐱 ˙ 2 - V ( 𝐱 ) . L(\mathbf{x},\dot{\mathbf{x}})=\frac{1}{2}m\dot{\mathbf{x}}^{2}-V(\mathbf{x}).
  115. d d t ( L x ˙ i ) - L x i = 0 , \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial\dot{x}_{i}}% \right)-\frac{\partial L}{\partial x_{i}}=0,
  116. L x i = - V x i , \frac{\partial L}{\partial x_{i}}=-\frac{\partial V}{\partial x_{i}},
  117. L x ˙ i = x ˙ i ( 1 2 m 𝐱 ˙ 2 ) = 1 2 m x ˙ i ( x ˙ i 2 ) = m x ˙ i , \frac{\partial L}{\partial\dot{x}_{i}}=\frac{\partial~{}}{\partial\dot{x}_{i}}% \left(\frac{1}{2}m\dot{\mathbf{x}}^{2}\right)=\frac{1}{2}m\frac{\partial~{}}{% \partial\dot{x}_{i}}\left(\dot{x}_{i}^{2}\right)=m\dot{x}_{i},
  118. d d t ( L x ˙ i ) = m x ¨ i . \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{\partial L}{\partial\dot{x}_{i}}% \right)=m\ddot{x}_{i}.
  119. m 𝐱 ¨ + V = 0 , m\ddot{\mathbf{x}}+\nabla V=0,
  120. L = m 2 ( r ˙ 2 + r 2 θ ˙ 2 + r 2 sin 2 θ φ ˙ 2 ) - V ( r ) . L=\frac{m}{2}(\dot{r}^{2}+r^{2}\dot{\theta}^{2}+r^{2}\sin^{2}\theta\,\dot{% \varphi}^{2})-V(r).
  121. m r ¨ - m r ( θ ˙ 2 + sin 2 θ φ ˙ 2 ) + V = 0 , m\ddot{r}-mr(\dot{\theta}^{2}+\sin^{2}\theta\,\dot{\varphi}^{2})+V^{\prime}=0,
  122. d d t ( m r 2 θ ˙ ) - m r 2 sin θ cos θ φ ˙ 2 = 0 , \frac{\mathrm{d}}{\mathrm{d}t}(mr^{2}\dot{\theta})-mr^{2}\sin\theta\cos\theta% \,\dot{\varphi}^{2}=0,
  123. d d t ( m r 2 sin 2 θ φ ˙ ) = 0. \frac{\mathrm{d}}{\mathrm{d}t}(mr^{2}\sin^{2}\theta\,\dot{\varphi})=0.
  124. x ( t ) \scriptstyle\vec{x}(t)
  125. 𝐱 \mathbf{x}
  126. T ( t ) = 1 2 m | 𝐱 ˙ ( t ) | 2 T(t)={\tfrac{1}{2}}m\left|\dot{\mathbf{x}}(t)\right|^{2}
  127. V ( t ) = m Φ ( 𝐱 ( t ) , t ) . V(t)=m\Phi(\mathbf{x}(t),t).
  128. L ( t ) = T ( t ) - V ( t ) = 1 2 m | 𝐱 ˙ ( t ) | 2 - m Φ ( 𝐱 ( t ) , t ) . L(t)=T(t)-V(t)={\tfrac{1}{2}}m\left|\dot{\mathbf{x}}(t)\right|^{2}-m\Phi(% \mathbf{x}(t),t).
  129. 𝐱 \mathbf{x}
  130. 0 = δ L ( t ) d t = δ L ( t ) d t 0=\delta\int{L(t)\,\mathrm{d}t}=\int{\delta L(t)\,\mathrm{d}t}
  131. = ( m 𝐱 ˙ ( t ) δ 𝐱 ˙ ( t ) - m Φ ( 𝐱 ( t ) , t ) δ 𝐱 ( t ) ) d t . =\int{(m\dot{\mathbf{x}}(t)\cdot\dot{\delta\mathbf{x}}(t)-m\nabla\Phi(\mathbf{% x}(t),t)\cdot\delta\mathbf{x}(t))\,\mathrm{d}t}.
  132. 0 = - m 𝐱 ¨ ( t ) - m Φ ( 𝐱 ( t ) , t ) 0=-m\ddot{\mathbf{x}}(t)-m\nabla\Phi(\mathbf{x}(t),t)
  133. m c 2 d t d τ = m c 2 1 - v 2 ( t ) c 2 = + m c 2 + 1 2 m v 2 ( t ) + 3 8 m v 4 ( t ) c 2 + . mc^{2}\frac{dt}{d\tau}=\frac{mc^{2}}{\sqrt{1-\frac{v^{2}(t)}{c^{2}}}}=+mc^{2}+% {1\over 2}mv^{2}(t)+{3\over 8}m\frac{v^{4}(t)}{c^{2}}+\dots\,.
  134. - m c 2 d τ ( t ) d t = - m c 2 1 - v 2 ( t ) c 2 = - m c 2 + 1 2 m v 2 ( t ) + 1 8 m v 4 ( t ) c 2 + -mc^{2}\frac{d\tau(t)}{dt}=-mc^{2}\sqrt{1-\frac{v^{2}(t)}{c^{2}}}=-mc^{2}+{1% \over 2}mv^{2}(t)+{1\over 8}m\frac{v^{4}(t)}{c^{2}}+\dots
  135. v 2 ( t ) = 𝐱 ˙ ( t ) 𝐱 ˙ ( t ) . \scriptstyle v^{2}(t)=\dot{\mathbf{x}}(t)\cdot\dot{\mathbf{x}}(t).
  136. 𝐀 \scriptstyle\mathbf{A}
  137. L ( t ) = - m c 2 1 - v 2 ( t ) c 2 - q ϕ ( 𝐱 ( t ) , t ) + q 𝐱 ˙ ( t ) 𝐀 ( 𝐱 ( t ) , t ) . L(t)=-mc^{2}\sqrt{1-\frac{v^{2}(t)}{c^{2}}}-q\phi(\mathbf{x}(t),t)+q\dot{% \mathbf{x}}(t)\cdot\mathbf{A}(\mathbf{x}(t),t).
  138. 𝐱 \mathbf{x}
  139. 0 = - d d t ( m 𝐱 ˙ ( t ) 1 - v 2 ( t ) c 2 ) - q ϕ ( 𝐱 ( t ) , t ) - q 𝐀 ( 𝐱 ( t ) , t ) t - q 𝐱 ˙ ( t ) 𝐀 ( 𝐱 ( t ) , t ) + q 𝐀 ( 𝐱 ( t ) , t ) 𝐱 ˙ ( t ) 0=-\frac{d}{dt}\left(\frac{m\dot{\mathbf{x}}(t)}{\sqrt{1-\frac{v^{2}(t)}{c^{2}% }}}\right)-q\nabla\phi(\mathbf{x}(t),t)-q\frac{\partial\mathbf{A}(\mathbf{x}(t% ),t)}{\partial t}-q\dot{\mathbf{x}}(t)\cdot\nabla\mathbf{A}(\mathbf{x}(t),t)+q% \nabla{\mathbf{A}}(\mathbf{x}(t),t)\cdot\dot{\mathbf{x}}(t)
  140. d d t ( m 𝐱 ˙ ( t ) 1 - v 2 ( t ) c 2 ) = q 𝐄 ( 𝐱 ( t ) , t ) + q 𝐱 ˙ ( t ) × 𝐁 ( 𝐱 ( t ) , t ) \frac{d}{dt}\left(\frac{m\dot{\mathbf{x}}(t)}{\sqrt{1-\frac{v^{2}(t)}{c^{2}}}}% \right)=q\mathbf{E}(\mathbf{x}(t),t)+q\dot{\mathbf{x}}(t)\times\mathbf{B}(% \mathbf{x}(t),t)
  141. 𝐄 ( 𝐱 , t ) = - ϕ ( 𝐱 , t ) - 𝐀 ( 𝐱 , t ) t \mathbf{E}(\mathbf{x},t)=-\nabla\phi(\mathbf{x},t)-\frac{\partial\mathbf{A}(% \mathbf{x},t)}{\partial t}
  142. 𝐁 ( 𝐱 , t ) = × 𝐀 ( 𝐱 , t ) \mathbf{B}(\mathbf{x},t)=\nabla\times\mathbf{A}(\mathbf{x},t)
  143. L ( τ ) = 1 2 m u μ ( τ ) u μ ( τ ) + q u μ ( τ ) A μ ( x ) L(\tau)=\frac{1}{2}mu^{\mu}(\tau)u_{\mu}(\tau)+qu^{\mu}(\tau)A_{\mu}(x)
  144. L x ν - d d τ L u ν = 0 \frac{\partial L}{\partial x^{\nu}}-\frac{d}{d\tau}\frac{\partial L}{\partial u% ^{\nu}}=0
  145. q u μ A μ x ν = d d τ ( m u ν + q A ν ) qu^{\mu}\frac{\partial A_{\mu}}{\partial x^{\nu}}=\frac{d}{d\tau}(mu_{\nu}+qA_% {\nu})
  146. - m c 2 d τ ( t ) d t = - m c 2 - c - 2 g μ ν ( x ( t ) ) d x μ ( t ) d t d x ν ( t ) d t . -mc^{2}\frac{d\tau(t)}{dt}=-mc^{2}\sqrt{-c^{-2}g_{\mu\nu}(x(t))\frac{dx^{\mu}(% t)}{dt}\frac{dx^{\nu}(t)}{dt}}.
  147. L ( t ) = - m c 2 - c - 2 g μ ν ( x ( t ) ) d x μ ( t ) d t d x ν ( t ) d t + q d x μ ( t ) d t A μ ( x ( t ) ) . L(t)=-mc^{2}\sqrt{-c^{-2}g_{\mu\nu}(x(t))\frac{dx^{\mu}(t)}{dt}\frac{dx^{\nu}(% t)}{dt}}+q\frac{dx^{\mu}(t)}{dt}A_{\mu}(x(t)).
  148. L = - m c 2 d τ d t + L I . L=-mc^{2}\frac{d\tau}{dt}+L_{I}\,.
  149. δ L = m d t 2 d τ δ ( g μ ν d x μ d t d x ν d t ) + δ L I . \delta L=m\frac{dt}{2d\tau}\delta\left(g_{\mu\nu}\frac{dx^{\mu}}{dt}\frac{dx^{% \nu}}{dt}\right)+\delta L_{I}\,.
  150. δ L = m d t 2 d τ ( g μ ν , α δ x α d x μ d t d x ν d t + 2 g α ν d δ x α d t d x ν d t ) + L I x α δ x α + L I d x α d t d δ x α d t . \delta L=m\frac{dt}{2d\tau}\left(g_{\mu\nu,\alpha}\delta x^{\alpha}\frac{dx^{% \mu}}{dt}\frac{dx^{\nu}}{dt}+2g_{\alpha\nu}\frac{d\delta x^{\alpha}}{dt}\frac{% dx^{\nu}}{dt}\right)+\frac{\partial L_{I}}{\partial x^{\alpha}}\delta x^{% \alpha}+\frac{\partial L_{I}}{\partial\frac{dx^{\alpha}}{dt}}\frac{d\delta x^{% \alpha}}{dt}\,.
  151. δ L = 1 2 m g μ ν , α δ x α d x μ d τ d x ν d t - d d t ( m g α ν d x ν d τ ) δ x α + L I x α δ x α - d d t ( L I d x α d t ) δ x α + d ( ) d t . \delta L=\frac{1}{2}mg_{\mu\nu,\alpha}\delta x^{\alpha}\frac{dx^{\mu}}{d\tau}% \frac{dx^{\nu}}{dt}-\frac{d}{dt}\left(mg_{\alpha\nu}\frac{dx^{\nu}}{d\tau}% \right)\delta x^{\alpha}+\frac{\partial L_{I}}{\partial x^{\alpha}}\delta x^{% \alpha}-\frac{d}{dt}\left(\frac{\partial L_{I}}{\partial\frac{dx^{\alpha}}{dt}% }\right)\delta x^{\alpha}+\frac{d(...)}{dt}\,.
  152. 0 = 1 2 m g μ ν , α d x μ d τ d x ν d t - d d t ( m g α ν d x ν d τ ) + f α 0=\frac{1}{2}mg_{\mu\nu,\alpha}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{dt}-\frac% {d}{dt}\left(mg_{\alpha\nu}\frac{dx^{\nu}}{d\tau}\right)+f_{\alpha}
  153. f α = L I x α - d d t ( L I d x α d t ) f_{\alpha}=\frac{\partial L_{I}}{\partial x^{\alpha}}-\frac{d}{dt}\left(\frac{% \partial L_{I}}{\partial\frac{dx^{\alpha}}{dt}}\right)
  154. f α d x α d t = 0 f_{\alpha}\tfrac{dx^{\alpha}}{dt}=0
  155. d d t ( m d x ν d τ ) = - m Γ μ σ ν d x μ d τ d x σ d t + g ν α f α \frac{d}{dt}\left(m\frac{dx^{\nu}}{d\tau}\right)=-m\Gamma^{\nu}_{\mu\sigma}% \frac{dx^{\mu}}{d\tau}\frac{dx^{\sigma}}{dt}+g^{\nu\alpha}f_{\alpha}
  156. p ν = m d x ν d τ p^{\nu}=m\frac{dx^{\nu}}{d\tau}
  157. d p ν d t = - Γ μ σ ν p μ d x σ d t + g ν α f α \frac{dp^{\nu}}{dt}=-\Gamma^{\nu}_{\mu\sigma}p^{\mu}\frac{dx^{\sigma}}{dt}+g^{% \nu\alpha}f_{\alpha}
  158. d x ν d t = p ν p 0 \frac{dx^{\nu}}{dt}=\frac{p^{\nu}}{p^{0}}

Lagrangian_system.html

  1. ( Y , L ) (Y,L)
  2. Y X Y→X
  3. L L
  4. Y X Y→X
  5. Q Q→ℝ
  6. Q = × M Q=ℝ×M
  7. L L
  8. r r
  9. n n
  10. n = d i m X n=dimX
  11. r r
  12. Y Y
  13. L L
  14. Y X Y→X
  15. δ δ
  16. L L
  17. δ L δL
  18. Y Y
  19. | Λ | = k r |Λ|=k≤r
  20. L L
  21. L = ( x λ , y i , y Λ i ) d n x , L=\mathcal{L}(x^{\lambda},y^{i},y^{i}_{\Lambda})\,d^{n}x,
  22. δ L = δ i d y i d n x , δ i = i + | Λ | ( - 1 ) | Λ | d Λ i Λ , \delta L=\delta_{i}\mathcal{L}\,dy^{i}\wedge d^{n}x,\qquad\delta_{i}\mathcal{L% }=\partial_{i}\mathcal{L}+\sum_{|\Lambda|}(-1)^{|\Lambda|}\,d_{\Lambda}\,% \partial_{i}^{\Lambda}\mathcal{L},
  23. d Λ = d λ 1 d λ k , d λ = λ + y λ i i + , d_{\Lambda}=d_{\lambda_{1}}\cdots d_{\lambda_{k}},\qquad d_{\lambda}=\partial_% {\lambda}+y^{i}_{\lambda}\partial_{i}+\cdots,
  24. L = ( x λ , y i , y λ i ) d n x , δ i L = i - d λ i λ . L=\mathcal{L}(x^{\lambda},y^{i},y^{i}_{\lambda})\,d^{n}x,\qquad\delta_{i}L=% \partial_{i}\mathcal{L}-d_{\lambda}\partial_{i}^{\lambda}\mathcal{L}.
  25. δ L = 0 δL=0
  26. d L = δ L + d H Θ L , dL=\delta L+d_{H}\Theta_{L},
  27. d H ϕ = d x λ d λ ϕ , ϕ O * ( Y ) d_{H}\phi=dx^{\lambda}\wedge d_{\lambda}\phi,\qquad\phi\in O^{*}_{\infty}(Y)
  28. θ < s u b > L θ<sub>L

Lame's_stress_ellipsoid.html

  1. x 1 , x 2 , x 3 x_{1},x_{2},x_{3}\,\!
  2. 𝐓 ( 𝐧 ) \mathbf{T}^{(\mathbf{n})}\,\!
  3. 𝐧 \mathbf{n}\,\!
  4. P P\,\!
  5. T 1 ( 𝐧 ) = σ 1 n 1 , T 2 ( 𝐧 ) = σ 2 n 2 , T 3 ( 𝐧 ) = σ 3 n 3 T_{1}^{(\mathbf{n})}=\sigma_{1}n_{1},\qquad T_{2}^{(\mathbf{n})}=\sigma_{2}n_{% 2},\qquad T_{3}^{(\mathbf{n})}=\sigma_{3}n_{3}\,\!
  6. 𝐧 \mathbf{n}\,\!
  7. n 1 2 + n 2 2 + n 3 2 = T 1 σ 1 2 2 + T 2 σ 2 2 2 + T 3 σ 3 2 2 = 1 n_{1}^{2}+n_{2}^{2}+n_{3}^{2}=\frac{T_{1}}{{\sigma_{1}}^{2}}^{2}+\frac{T_{2}}{% {\sigma_{2}}^{2}}^{2}+\frac{T_{3}}{{\sigma_{3}}^{2}}^{2}=1\,\!
  8. ± σ 1 , ± σ 2 , ± σ 3 \pm\sigma_{1},\pm\sigma_{2},\pm\sigma_{3}\,\!
  9. I 1 I_{1}\,\!
  10. I 2 I_{2}\,\!
  11. I 3 I_{3}\,\!
  12. n 1 2 + n 2 2 + n 3 2 = T 1 2 σ 1 2 + T 2 2 σ 2 2 + T 3 2 σ 3 2 = 1 n_{1}^{2}+n_{2}^{2}+n_{3}^{2}=\frac{T_{1}^{2}}{{\sigma_{1}}^{2}}+\frac{T_{2}^{% 2}}{{\sigma_{2}}^{2}}+\frac{T_{3}^{2}}{{\sigma_{3}}^{2}}=1\,\!

Laplace–Carson_transform.html

  1. V ( j , t ) V(j,t)
  2. p p
  3. V ( j , p ) = p 0 V ( j , t ) e - p t d t V^{\ast}(j,p)=p\int^{\infty}_{0}V(j,t)e^{-pt}\,dt
  4. V ( j , t ) = 1 2 π i a 0 - i a 0 + i e t p V ( j , p ) p d p V(j,t)=\frac{1}{2\pi i}\int^{a_{0}+i\infty}_{a_{0}-i\infty}e^{tp}\frac{V^{\ast% }(j,p)}{p}\,dp
  5. a 0 a_{0}
  6. i i\infty
  7. e t p V ( j , t ) p e^{tp}\frac{V(j,t)}{p}

Largest_empty_rectangle.html

  1. L 1 L_{1}
  2. Θ ( n log n ) \Theta(n\log n)
  3. O ( min ( n 2 , s log n ) ) O(\min(n^{2},s\log n))
  4. s = O ( n 2 ) s=O(n^{2})

Largest_empty_sphere.html

  1. Θ ( n log n ) \Theta(n\,\log\,n)

Laser_drilling.html

  1. I a b s + k ( T z + r T r ) + ρ l ν i L v - ρ v ν v ( c p T i + E v ) = 0 I_{abs}+k\left(\frac{\partial T}{\partial z}+r\frac{\partial T}{\partial r}% \right)+\rho_{l}\nu_{i}L_{v}-\rho_{v}\nu_{v}(c_{p}T_{i}+E_{v})=0
  2. I a b s = I ( t ) - β z I_{abs}=I(t)^{-\beta z}
  3. ρ u ( r , t ) t = P ( t ) + μ 2 u ( r , t ) r 2 \rho\frac{\partial u(r,t)}{\partial t}=P(t)+\mu\frac{\partial^{2}u(r,t)}{% \partial r^{2}}
  4. 2 σ δ ¯ 2\sigma\over\bar{\delta}

Laves_phase.html

  1. 3 / 2 1.225 \sqrt{3/2}\approx 1.225

Law_of_the_unconscious_statistician.html

  1. E [ g ( X ) ] = x g ( x ) f X ( x ) , \operatorname{E}[g(X)]=\sum_{x}g(x)f_{X}(x),\,
  2. E [ g ( X ) ] = - g ( x ) f X ( x ) d x \operatorname{E}[g(X)]=\int_{-\infty}^{\infty}g(x)f_{X}(x)\,dx
  3. E [ g ( X ) ] = - g ( x ) d F X ( x ) \operatorname{E}[g(X)]=\int_{-\infty}^{\infty}g(x)\,dF_{X}(x)
  4. Ω g X d P = g d ( X * P ) \int_{\Omega}g\circ XdP=\int_{\mathbb{R}}gd(X_{*}P)
  5. X X
  6. d ( X * P ) d(X_{*}P)
  7. μ \mu
  8. d ( X * P ) = f d μ d(X_{*}P)=fd_{\mu}
  9. f : f:\mathbb{R}\to\mathbb{R}
  10. E [ g ( X ) ] = Ω g X d P = g ( x ) f ( x ) d x \operatorname{E}[g(X)]=\int_{\Omega}g\circ XdP=\int_{\mathbb{R}}g(x)f(x)dx

Law_of_total_covariance.html

  1. cov ( X , Y ) = E ( cov ( X , Y Z ) ) + cov ( E ( X Z ) , E ( Y Z ) ) . \operatorname{cov}(X,Y)=\operatorname{E}(\operatorname{cov}(X,Y\mid Z))+% \operatorname{cov}(\operatorname{E}(X\mid Z),\operatorname{E}(Y\mid Z)).\,
  2. cov [ X , Y ] = E [ X Y ] - E [ X ] E [ Y ] \operatorname{cov}[X,Y]=\operatorname{E}[XY]-\operatorname{E}[X]\operatorname{% E}[Y]
  3. = E [ E [ X Y Z ] ] - E [ E [ X Z ] ] E [ E [ Y Z ] ] =\operatorname{E}[\operatorname{E}[XY\mid Z]]-\operatorname{E}[\operatorname{E% }[X\mid Z]]\operatorname{E}[\operatorname{E}[Y\mid Z]]
  4. = E [ cov [ X , Y Z ] + E [ X Z ] E [ Y Z ] ] - E [ E [ X Z ] ] E [ E [ Y Z ] ] =\operatorname{E}\!\left[\operatorname{cov}[X,Y\mid Z]+\operatorname{E}[X\mid Z% ]\operatorname{E}[Y\mid Z]\right]-\operatorname{E}[\operatorname{E}[X\mid Z]]% \operatorname{E}[\operatorname{E}[Y\mid Z]]
  5. = E [ cov [ X , Y Z ] ] + E [ E [ X Z ] E [ Y Z ] ] - E [ E [ X Z ] ] E [ E [ Y Z ] ] =\operatorname{E}\!\left[\operatorname{cov}[X,Y\mid Z]]+\operatorname{E}[% \operatorname{E}[X\mid Z]\operatorname{E}[Y\mid Z]\right]-\operatorname{E}[% \operatorname{E}[X\mid Z]]\operatorname{E}[\operatorname{E}[Y\mid Z]]
  6. = E ( cov ( X , Y Z ) ) + cov ( E ( X Z ) , E ( Y Z ) ) =\operatorname{E}(\operatorname{cov}(X,Y\mid Z))+\operatorname{cov}(% \operatorname{E}(X\mid Z),\operatorname{E}(Y\mid Z))

Lawvere_theory.html

  1. 0 \aleph_{0}
  2. I : 0 op L I:\aleph_{0}\text{op}\rightarrow L

László_Pyber.html

  1. n ( 2 27 + o ( 1 ) ) μ 2 . n^{(\frac{2}{27}+o(1))\mu^{2}}.
  2. n g ( n ) n^{g(n)}

Learning_to_rank.html

  1. k k

Learning_with_errors.html

  1. ( x , y ) (x,y)
  2. x q n x\in\mathbb{Z}_{q}^{n}
  3. y q y\in\mathbb{Z}_{q}
  4. f : q n q , f:\mathbb{Z}_{q}^{n}\rightarrow\mathbb{Z}_{q},
  5. y = f ( x ) y=f(x)
  6. f f
  7. 𝕋 = / \mathbb{T}=\mathbb{R}/\mathbb{Z}
  8. A 𝐬 , ϕ A_{\mathbf{s},\phi}
  9. q n × 𝕋 \mathbb{Z}_{q}^{n}\times\mathbb{T}
  10. 𝐚 q n \mathbf{a}\in\mathbb{Z}_{q}^{n}
  11. e e
  12. ϕ \phi
  13. 𝕋 \mathbb{T}
  14. ( 𝐚 , 𝐚 , 𝐬 / q + e ) (\mathbf{a},\langle\mathbf{a},\mathbf{s}\rangle/q+e)
  15. 𝐬 q n \mathbf{s}\in\mathbb{Z}_{q}^{n}
  16. 𝐚 , 𝐬 = i = 1 n a i s i \textstyle\langle\mathbf{a},\mathbf{s}\rangle=\sum_{i=1}^{n}a_{i}s_{i}
  17. q n × q n q \mathbb{Z}_{q}^{n}\times\mathbb{Z}_{q}^{n}\longrightarrow\mathbb{Z}_{q}
  18. q q
  19. q 𝕋 \mathbb{Z}_{q}\longrightarrow\mathbb{T}
  20. 1 q 1\in\mathbb{Z}_{q}
  21. 1 / q + 𝕋 1/q+\mathbb{Z}\in\mathbb{T}
  22. 𝕋 \mathbb{T}
  23. LWE q , ϕ \mathrm{LWE}_{q,\phi}
  24. 𝐬 q n \mathbf{s}\in\mathbb{Z}_{q}^{n}
  25. A 𝐬 , ϕ A_{\mathbf{s},\phi}
  26. α > 0 \alpha>0
  27. D α D_{\alpha}
  28. D α ( x ) = ρ α ( x ) / α D_{\alpha}(x)=\rho_{\alpha}(x)/\alpha
  29. ρ α ( x ) = e - π ( | x | / α ) 2 \rho_{\alpha}(x)=e^{-\pi(|x|/\alpha)^{2}}
  30. Ψ α \Psi_{\alpha}
  31. 𝕋 \mathbb{T}
  32. D α D_{\alpha}
  33. LWE q , Ψ α \mathrm{LWE}_{q,\Psi_{\alpha}}
  34. q n × 𝕋 \mathbb{Z}_{q}^{n}\times\mathbb{T}
  35. q q
  36. n n
  37. { ( 𝐚 𝐢 , 𝐛 𝐢 ) } q n × 𝕋 \{(\mathbf{a_{i}},\mathbf{b_{i}})\}\subset\mathbb{Z}^{n}_{q}\times\mathbb{T}
  38. 𝐬 \mathbf{s}
  39. i i
  40. { 𝐚 𝐢 , 𝐬 - 𝐛 𝐢 } \{\langle\mathbf{a_{i}},\mathbf{s}\rangle-\mathbf{b_{i}}\}
  41. χ \chi
  42. 𝐬 \mathbf{s}
  43. 𝐬 1 \mathbf{s}_{1}
  44. k Z q k\in Z_{q}
  45. r q r\in\mathbb{Z}_{q}
  46. { ( 𝐚 𝐢 , 𝐛 𝐢 ) } q n × 𝕋 \{(\mathbf{a_{i}},\mathbf{b_{i}})\}\subset\mathbb{Z}^{n}_{q}\times\mathbb{T}
  47. { ( 𝐚 𝐢 + ( r , 0 , , 0 ) , 𝐛 𝐢 + ( r k ) / q ) } \{(\mathbf{a_{i}}+(r,0,\ldots,0),\mathbf{b_{i}}+(rk)/q)\}
  48. k k
  49. A 𝐬 , χ A_{\mathbf{s},\chi}
  50. q q
  51. q q
  52. n n
  53. k k
  54. 𝐬 1 \mathbf{s}_{1}
  55. 𝐬 j \mathbf{s}_{j}
  56. 𝐛 𝐢 \mathbf{b_{i}}
  57. 𝐚 𝐢 \mathbf{a_{i}}
  58. 𝐚 𝐢 + ( 0 , , r , , 0 ) \mathbf{a_{i}}+(0,\ldots,r,\ldots,0)
  59. r r
  60. j t h j^{th}
  61. q q
  62. n n
  63. q = q 1 q 2 q t q=q_{1}q_{2}\cdots q_{t}
  64. q q_{\ell}
  65. 𝐬 j \mathbf{s}_{j}
  66. 0 mod q 0\mod q_{\ell}
  67. 𝐬 j \mathbf{s}_{j}
  68. q q
  69. χ \chi
  70. { ( 𝐚 𝐢 , 𝐛 𝐢 ) } \{(\mathbf{a_{i}},\mathbf{b_{i}})\}
  71. A 𝐬 , χ A_{\mathbf{s},\chi}
  72. { ( 𝐚 𝐢 , 𝐛 𝐢 + 𝐚 𝐢 , 𝐭 ) / q } \{(\mathbf{a_{i}},\mathbf{b_{i}}+\langle\mathbf{a_{i}},\mathbf{t}\rangle)/q\}
  73. A 𝐬 + 𝐭 , χ A_{\mathbf{s}+\mathbf{t},\chi}
  74. 𝒮 q n \mathcal{S}\subset\mathbb{Z}_{q}^{n}
  75. | 𝒮 | / | q n | = 1 / p o l y ( n ) |\mathcal{S}|/|\mathbb{Z}_{q}^{n}|=1/poly(n)
  76. A 𝐬 , χ A_{\mathbf{s^{\prime}},\chi}
  77. 𝐬 𝒮 \mathbf{s^{\prime}}\leftarrow\mathcal{S}
  78. 𝒜 \mathcal{A}
  79. { ( 𝐚 𝐢 , 𝐛 𝐢 ) } \{(\mathbf{a_{i}},\mathbf{b_{i}})\}
  80. A 𝐬 , χ A_{\mathbf{s^{\prime}},\chi}
  81. A 𝐬 , χ A_{\mathbf{s},\chi}
  82. 𝐬 \mathbf{s}
  83. q n \mathbb{Z}_{q}^{n}
  84. 𝐭 \mathbf{t}
  85. q n \mathbb{Z}_{q}^{n}
  86. { ( 𝐚 𝐢 , 𝐛 𝐢 + 𝐚 𝐢 , 𝐭 ) / q } \{(\mathbf{a_{i}},\mathbf{b_{i}}+\langle\mathbf{a_{i}},\mathbf{t}\rangle)/q\}
  87. 𝒜 \mathcal{A}
  88. 𝒮 \mathcal{S}
  89. q n \mathbb{Z}_{q}^{n}
  90. 𝐭 \mathbf{t}
  91. 𝐬 + 𝐭 𝒮 \mathbf{s}+\mathbf{t}\in\mathcal{S}
  92. 𝒜 \mathcal{A}
  93. 𝒮 \mathcal{S}
  94. L L
  95. η ϵ ( L ) \eta_{\epsilon}(L)
  96. s s
  97. ρ 1 / s ( L * { 𝟎 } ) ϵ \rho_{1/s}(L^{*}\setminus\{\mathbf{0}\})\leq\epsilon
  98. L * L^{*}
  99. L L
  100. ρ α ( x ) = e - π ( | x | / α ) 2 \rho_{\alpha}(x)=e^{-\pi(|x|/\alpha)^{2}}
  101. D L , r D_{L,r}
  102. L L
  103. r r
  104. L L
  105. r > 0 r>0
  106. x L x\in L
  107. ρ r ( x ) \rho_{r}(x)
  108. D G S ϕ DGS_{\phi}
  109. n n
  110. L L
  111. r ϕ ( L ) r\geq\phi(L)
  112. D L , r D_{L,r}
  113. G a p S V P 100 n γ ( n ) GapSVP_{100\sqrt{n}\gamma(n)}
  114. D G S n γ ( n ) / λ ( L * ) DGS_{\sqrt{n}\gamma(n)/\lambda(L^{*})}
  115. γ ( n ) \gamma(n)
  116. D G S 2 n η ϵ ( L ) / α DGS_{\sqrt{2n}\eta_{\epsilon}(L)/\alpha}
  117. L W E q , Ψ α LWE_{q,\Psi_{\alpha}}
  118. q q
  119. α ( 0 , 1 ) \alpha\in(0,1)
  120. α q > 2 n \alpha q>2\sqrt{n}
  121. L W E LWE
  122. q q
  123. q q
  124. n n
  125. G a p S V P ζ , γ GapSVP_{\zeta,\gamma}
  126. L W E q , Ψ α LWE_{q,\Psi_{\alpha}}
  127. p o l y ( n ) poly(n)
  128. α ( 0 , 1 ) \alpha\in(0,1)
  129. γ ( n ) n / ( α log n ) \gamma(n)\geq n/(\alpha\sqrt{\log{n}})
  130. ζ ( n ) γ ( n ) \zeta(n)\geq\gamma(n)
  131. q ( ζ / n ) ω log n ) q\geq(\zeta/\sqrt{n})\omega\sqrt{\log{n}})
  132. G a p S V P γ GapSVP_{\gamma}
  133. γ \gamma
  134. S I V P t SIVP_{t}
  135. α \alpha
  136. G a p S V P ζ , γ GapSVP_{\zeta,\gamma}
  137. m , q m,q
  138. χ \chi
  139. 𝕋 \mathbb{T}
  140. q 2 q\geq 2
  141. n 2 n^{2}
  142. 2 n 2 2n^{2}
  143. m = ( 1 + ϵ ) ( n + 1 ) log q m=(1+\epsilon)(n+1)\log{q}
  144. ϵ \epsilon
  145. χ = Ψ α ( n ) \chi=\Psi_{\alpha(n)}
  146. α ( n ) o ( 1 / n log n ) \alpha(n)\in o(1/\sqrt{n}\log{n})
  147. 𝐬 q n \mathbf{s}\in\mathbb{Z}^{n}_{q}
  148. m m
  149. a 1 , , a m q n a_{1},\ldots,a_{m}\in\mathbb{Z}^{n}_{q}
  150. e 1 , , e m 𝕋 e_{1},\ldots,e_{m}\in\mathbb{T}
  151. χ \chi
  152. ( a i , b i = a i , 𝐬 / q + e i ) i = 1 m (a_{i},b_{i}=\langle a_{i},\mathbf{s}\rangle/q+e_{i})^{m}_{i=1}
  153. x { 0 , 1 } x\in\{0,1\}
  154. S S
  155. [ m ] [m]
  156. E n c ( x ) Enc(x)
  157. ( i S a i , x / 2 + i S b i ) (\sum_{i\in S}a_{i},x/2+\sum_{i\in S}b_{i})
  158. ( a , b ) (a,b)
  159. 0
  160. b - a , 𝐬 / q b-\langle a,\mathbf{s}\rangle/q
  161. 0
  162. 1 2 \frac{1}{2}
  163. 1 1
  164. 0
  165. 1 1
  166. A s , χ A_{s,\chi}
  167. q n × q \mathbb{Z}^{n}_{q}\times\mathbb{Z}_{q}

Least-upper-bound_property.html

  1. S S
  2. x x
  3. S S
  4. x s x≥s
  5. s S s∈S
  6. x x
  7. S S
  8. x x
  9. S S
  10. x y x≤y
  11. y y
  12. S S
  13. X X
  14. X X
  15. X X
  16. X X
  17. 𝐐 \mathbf{Q}
  18. { x 𝐐 : x 2 2 } = 𝐐 ( - 2 , 2 ) \left\{x\in\mathbf{Q}:x^{2}\leq 2\right\}=\mathbf{Q}\cap\left(-\sqrt{2},\sqrt{% 2}\right)\,
  19. 𝐐 \mathbf{Q}
  20. 𝐐 \mathbf{Q}
  21. S S
  22. S S
  23. B < s u b > 1 B<sub>1
  24. S S
  25. a a , b aa,b
  26. f f
  27. b b
  28. S S
  29. f f
  30. 𝐑 \mathbf{R}
  31. x < s u b > n x<sub>n

Least_squares_support_vector_machine.html

  1. { x i , y i } i = 1 N \{x_{i},y_{i}\}_{i=1}^{N}
  2. x i n x_{i}\in\mathbb{R}^{n}
  3. y i { - 1 , + 1 } y_{i}\in\{-1,+1\}
  4. y i = 1 y_{i}=1
  5. y i = - 1 y_{i}=-1
  6. { w T ϕ ( x i ) + b 1 , if y i = + 1 , w T ϕ ( x i ) + b - 1 , if y i = - 1. \begin{cases}w^{T}\phi(x_{i})+b\geq 1,&\,\text{if }\quad y_{i}=+1,\\ w^{T}\phi(x_{i})+b\leq-1,&\,\text{if }\quad y_{i}=-1.\end{cases}
  7. y i [ w T ϕ ( x i ) + b ] 1 , i = 1 , , N , y_{i}\left[{w^{T}\phi(x_{i})+b}\right]\geq 1,\quad i=1,\ldots,N\,,
  8. ϕ ( x ) \phi(x)
  9. ξ i \xi_{i}
  10. { y i [ w T ϕ ( x i ) + b ] 1 - ξ i , i = 1 , , N , ξ i 0 , i = 1 , , N . \begin{cases}y_{i}\left[{w^{T}\phi(x_{i})+b}\right]\geq 1-\xi_{i},&i=1,\ldots,% N,\\ \xi_{i}\geq 0,&i=1,\ldots,N.\end{cases}
  11. min J 1 ( w , ξ ) = 1 2 w T w + c i = 1 N ξ i , \min J_{1}(w,\xi)=\frac{1}{2}w^{T}w+c\sum\limits_{i=1}^{N}{\xi_{i}},
  12. Subject to { y i [ w T ϕ ( x i ) + b ] 1 - ξ i , i = 1 , , N , ξ i 0 , i = 1 , , N , \,\text{Subject to }\begin{cases}y_{i}\left[{w^{T}\phi(x_{i})+b}\right]\geq 1-% \xi_{i},&i=1,\ldots,N,\\ \xi_{i}\geq 0,&i=1,\ldots,N,\end{cases}
  13. L 1 ( w , b , ξ , α , β ) = 1 2 w T w + c i = 1 N ξ i + i = 1 N α i { y i [ w T ϕ ( x i ) + b ] - 1 + ξ i } + i = 1 N β i ξ i , L_{1}(w,b,\xi,\alpha,\beta)=\frac{1}{2}w^{T}w+c\sum\limits_{i=1}^{N}{\xi_{i}}+% \sum\limits_{i=1}^{N}\alpha_{i}\left\{y_{i}\left[{w^{T}\phi(x_{i})+b}\right]-1% +\xi_{i}\right\}+\sum\limits_{i=1}^{N}\beta_{i}\xi_{i},
  14. α i 0 , β i 0 ( i = 1 , , N ) \alpha_{i}\geq 0,{\rm}\beta_{i}\geq 0\;(i=1,\ldots,N)
  15. { L 1 w = 0 w = i = 1 N α i y i ϕ ( x i ) , L 1 b = 0 i = 1 N α i y i = 0 , L 1 ξ i = 0 0 α i c , i = 1 , , N . \begin{cases}\frac{\partial L_{1}}{\partial w}=0\quad\to\quad w=\sum\limits_{i% =1}^{N}\alpha_{i}y_{i}\phi(x_{i}),\\ \frac{\partial L_{1}}{\partial b}=0\quad\to\quad\sum\limits_{i=1}^{N}\alpha_{i% }y_{i}=0,\\ \frac{\partial L_{1}}{\partial\xi_{i}}=0\quad\to\quad 0\leq\alpha_{i}\leq c,\;% i=1,\ldots,N.\end{cases}
  16. w w
  17. max Q 1 ( α ) = - 1 2 i , j = 1 N α i α j y i y j K ( x i , x j ) + i = 1 N α i \max\;Q_{1}(\alpha)\;=-\frac{1}{2}\sum\limits_{i,j=1}^{N}{\alpha_{i}\alpha_{j}% y_{i}y_{j}K(x_{i},x_{j})}+\sum\limits_{i=1}^{N}{\alpha_{i}}
  18. K ( x i , x j ) = ϕ ( x i ) , ϕ ( x j ) K(x_{i},x_{j})=\left\langle{\phi(x_{i}),\phi(x_{j})}\right\rangle
  19. min J 2 ( w , b , e ) = μ 2 w T w + ζ 2 i = 1 N e c , i 2 , \min J_{2}(w,b,e)=\frac{\mu}{2}w^{T}w+\frac{\zeta}{2}\sum\limits_{i=1}^{N}{e_{% c,i}^{2}},
  20. y i [ w T ϕ ( x i ) + b ] = 1 - e c , i , i = 1 , , N . y_{i}\left[{w^{T}\phi(x_{i})+b}\right]=1-e_{c,i},\quad i=1,\ldots,N.
  21. y i = ± 1 y_{i}=\pm 1
  22. y i 2 = 1 y_{i}^{2}=1
  23. i = 1 N e c , i 2 = i = 1 N ( y i e c , i ) 2 = i = 1 N e i 2 = i = 1 N ( y i - ( w T ϕ ( x i ) + b ) ) 2 , \sum\limits_{i=1}^{N}{e_{c,i}^{2}}=\sum\limits_{i=1}^{N}{(y_{i}e_{c,i})^{2}}=% \sum\limits_{i=1}^{N}{e_{i}^{2}}=\sum\limits_{i=1}^{N}{\left({y_{i}-(w^{T}\phi% (x_{i})+b)}\right)}^{2},
  24. e i = y i - ( w T ϕ ( x i ) + b ) . e_{i}=y_{i}-(w^{T}\phi(x_{i})+b).
  25. J 2 ( w , b , e ) = μ E W + ζ E D \;J_{2}(w,b,e)=\mu E_{W}+\zeta E_{D}
  26. E W = 1 2 w T w E_{W}=\frac{1}{2}w^{T}w
  27. E D = 1 2 i = 1 N e i 2 = 1 2 i = 1 N ( y i - ( w T ϕ ( x i ) + b ) ) 2 . E_{D}=\frac{1}{2}\sum\limits_{i=1}^{N}{e_{i}^{2}}=\frac{1}{2}\sum\limits_{i=1}% ^{N}{\left({y_{i}-(w^{T}\phi(x_{i})+b)}\right)}^{2}.
  28. μ \mu
  29. ζ \zeta
  30. γ = ζ / μ \gamma=\zeta/\mu
  31. γ \gamma
  32. μ \mu
  33. ζ \zeta
  34. { L 2 ( w , b , e , α ) = J 2 ( w , e ) - i = 1 N α i { [ w T ϕ ( x i ) + b ] + e i - y i } , = 1 2 w T w + γ 2 i = 1 N e i 2 - i = 1 N α i { [ w T ϕ ( x i ) + b ] + e i - y i } , \begin{cases}L_{2}(w,b,e,\alpha)\;=J_{2}(w,e)-\sum\limits_{i=1}^{N}\alpha_{i}% \left\{{\left[{w^{T}\phi(x_{i})+b}\right]+e_{i}-y_{i}}\right\},\\ \quad\quad\quad\quad\quad\;=\frac{1}{2}w^{T}w+\frac{\gamma}{2}\sum\limits_{i=1% }^{N}e_{i}^{2}-\sum\limits_{i=1}^{N}\alpha_{i}\left\{\left[w^{T}\phi(x_{i})+b% \right]+e_{i}-y_{i}\right\},\end{cases}
  35. α i \alpha_{i}\in\mathbb{R}
  36. { L 2 w = 0 w = i = 1 N α i ϕ ( x i ) , L 2 b = 0 i = 1 N α i = 0 , L 2 e i = 0 α i = γ e i , i = 1 , , N , L 2 α i = 0 y i = w T ϕ ( x i ) + b + e i , i = 1 , , N . \begin{cases}\frac{\partial L_{2}}{\partial w}=0\quad\to\quad w=\sum\limits_{i% =1}^{N}\alpha_{i}\phi(x_{i}),\\ \frac{\partial L_{2}}{\partial b}=0\quad\to\quad\sum\limits_{i=1}^{N}\alpha_{i% }=0,\\ \frac{\partial L_{2}}{\partial e_{i}}=0\quad\to\quad\alpha_{i}=\gamma e_{i},\;% i=1,\ldots,N,\\ \frac{\partial L_{2}}{\partial\alpha_{i}}=0\quad\to\quad y_{i}=w^{T}\phi(x_{i}% )+b+e_{i},\,i=1,\ldots,N.\end{cases}
  37. w w
  38. e e
  39. [ 0 1 N T 1 N Ω + γ - 1 I N ] [ b α ] = [ 0 Y ] , \left[\begin{matrix}0&1_{N}^{T}\\ 1_{N}&\Omega+\gamma^{-1}I_{N}\end{matrix}\right]\left[\begin{matrix}b\\ \alpha\end{matrix}\right]=\left[\begin{matrix}0\\ Y\end{matrix}\right],
  40. Y = [ y 1 , , y N ] T Y=[y_{1},\ldots,y_{N}]^{T}
  41. 1 N = [ 1 , , 1 ] T 1_{N}=[1,\ldots,1]^{T}
  42. α = [ α 1 , , α N ] T \alpha=[\alpha_{1},\ldots,\alpha_{N}]^{T}
  43. I N I_{N}
  44. N × N N\times N
  45. Ω N × N \Omega\in\mathbb{R}^{N\times N}
  46. Ω i j = ϕ ( x i ) T ϕ ( x j ) = K ( x i , x j ) \Omega_{ij}=\phi(x_{i})^{T}\phi(x_{j})=K(x_{i},x_{j})
  47. K ( x , x i ) = x i T x , K(x,x_{i})=x_{i}^{T}x,
  48. d d
  49. K ( x , x i ) = ( 1 + x i T x / c ) d , K(x,x_{i})=\left({1+x_{i}^{T}x/c}\right)^{d},
  50. K ( x , x i ) = exp ( - x - x i 2 / σ 2 ) , K(x,x_{i})=\exp\left({-\left\|{x-x_{i}}\right\|^{2}/\sigma^{2}}\right),
  51. K ( x , x i ) = tanh ( k x i T x + θ ) , K(x,x_{i})=\tanh\left({k\,x_{i}^{T}x+\theta}\right),
  52. d d
  53. c c
  54. σ \sigma
  55. k k
  56. θ \theta
  57. c , σ + c,\sigma\in\mathbb{R}^{+}
  58. d N d\in N
  59. k k
  60. θ \theta
  61. c c
  62. σ \sigma
  63. k k
  64. P [ f ] exp ( - β P ^ f 2 ) P[f]\propto\exp\left({-\beta\left\|{\hat{P}f}\right\|^{2}}\right)
  65. β > 0 \beta>0
  66. P ^ \hat{P}
  67. D D
  68. 𝕄 \mathbb{M}
  69. w w
  70. λ \lambda
  71. λ \lambda
  72. w w
  73. p ( w | D , λ , 𝕄 ) p ( D | w , 𝕄 ) p ( w | λ , 𝕄 ) p(w|D,\lambda,\mathbb{M})\propto p(D|w,\mathbb{M})p(w|\lambda,\mathbb{M})
  74. λ \lambda
  75. p ( λ | D , 𝕄 ) p ( D | λ , 𝕄 ) p ( λ | 𝕄 ) p(\lambda|D,\mathbb{M})\propto p(D|\lambda,\mathbb{M})p(\lambda|\mathbb{M})
  76. p ( 𝕄 | D ) p ( D | 𝕄 ) p ( 𝕄 ) . p(\mathbb{M}|D)\propto p(D|\mathbb{M})p(\mathbb{M}).
  77. { x i , y i } i = 1 N \{x_{i},y_{i}\}_{i=1}^{N}
  78. μ \mu
  79. ζ \zeta
  80. 𝕄 \mathbb{M}
  81. w w
  82. b b
  83. p ( w , b | D , log μ , log ζ , 𝕄 ) p(w,b|D,\log\mu,\log\zeta,\mathbb{M})
  84. p ( w , b | D , log μ , log ζ , 𝕄 ) = p ( D | w , b , log μ , log ζ , 𝕄 ) p ( w , b | log μ , log ζ , 𝕄 ) p ( D | log μ , log ζ , 𝕄 ) . p(w,b|D,\log\mu,\log\zeta,\mathbb{M})=\frac{{p(D|w,b,\log\mu,\log\zeta,\mathbb% {M})p(w,b|\log\mu,\log\zeta,\mathbb{M})}}{{p(D|\log\mu,\log\zeta,\mathbb{M})}}.
  85. p ( D | log μ , log ζ , 𝕄 ) p(D|\log\mu,\log\zeta,\mathbb{M})
  86. w w
  87. b b
  88. w w
  89. b b
  90. ζ \zeta
  91. p ( w , b | log μ , log ζ , 𝕄 ) = p ( w | log μ , 𝕄 ) p ( b | log σ b , 𝕄 ) . p(w,b|\log\mu,\log\zeta,\mathbb{M})=p(w|\log\mu,\mathbb{M})p(b|\log\sigma_{b},% \mathbb{M}).
  92. σ b \sigma_{b}\to\infty
  93. b b
  94. w w
  95. b b
  96. w w
  97. b b
  98. σ b \sigma_{b}\to\infty
  99. p ( w , b | log μ , ) = ( μ < m t p l > 2 π ) n f 2 exp ( - μ 2 w T w ) 1 2 π σ b exp ( - b 2 2 σ b ) ( μ 2 π ) n f 2 exp ( - μ 2 w T w ) . \begin{array}[]{l}p(w,b|\log\mu,)=\left({\frac{\mu}{<}mtpl>{{2\pi}}}\right)^{% \frac{{n_{f}}}{2}}\exp\left({-\frac{\mu}{2}w^{T}w}\right)\frac{1}{{\sqrt{2\pi% \sigma_{b}}}}\exp\left({-\frac{{b^{2}}}{{2\sigma_{b}}}}\right)\\ \quad\quad\quad\quad\quad\quad\quad\propto\left({\frac{\mu}{{2\pi}}}\right)^{% \frac{{n_{f}}}{2}}\exp\left({-\frac{\mu}{2}w^{T}w}\right)\end{array}.
  100. n f n_{f}
  101. w w
  102. p ( D | w , b , log μ , log ζ , 𝕄 ) p(D|w,b,\log\mu,\log\zeta,\mathbb{M})
  103. w , b , ζ w,b,\zeta
  104. 𝕄 \mathbb{M}
  105. p ( D | w , b , log ζ , 𝕄 ) = i = 1 N p ( x i , y i | w , b , log ζ , 𝕄 ) . p(D|w,b,\log\zeta,\mathbb{M})=\prod\limits_{i=1}^{N}{p(x_{i},y_{i}|w,b,\log% \zeta,\mathbb{M})}.
  106. p ( x i , y i | w , b , log ζ , 𝕄 ) p ( e i | w , b , log ζ , 𝕄 ) . p(x_{i},y_{i}|w,b,\log\zeta,\mathbb{M})\propto p(e_{i}|w,b,\log\zeta,\mathbb{M% }).
  107. e i = y i - ( w T ϕ ( x i ) + b ) e_{i}=y_{i}-(w^{T}\phi(x_{i})+b)
  108. p ( e i | w , b , log ζ , 𝕄 ) = ζ < m t p l > 2 π exp ( - ζ e i 2 2 ) . p(e_{i}|w,b,\log\zeta,\mathbb{M})=\sqrt{\frac{\zeta}{<}mtpl>{{2\pi}}}\exp\left% ({-\frac{{\zeta e_{i}^{2}}}{2}}\right).
  109. w w
  110. b b
  111. m ^ - \hat{m}_{-}
  112. m ^ + \hat{m}_{+}
  113. w T ϕ ( x ) + b w^{T}\phi(x)+b
  114. ϕ ( x ) \phi(x)
  115. 1 / ζ 1/\zeta
  116. p ( w , b | D , log μ , log ζ , 𝕄 ) exp ( - μ 2 w T w - ζ 2 i = 1 N e i 2 ) = exp ( - J 2 ( w , b ) ) . p(w,b|D,\log\mu,\log\zeta,\mathbb{M})\propto\exp(-\frac{\mu}{2}w^{T}w-\frac{% \zeta}{2}\sum\limits_{i=1}^{N}{e_{i}^{2}})=\exp(-J_{2}(w,b)).
  117. w M P w_{MP}
  118. b M P b_{MP}

Least_trimmed_squares.html

  1. S = i = 1 n r i ( β ) 2 S=\sum_{i=1}^{n}{r_{i}(\beta)}^{2}
  2. r i ( β ) = y i - f ( x i , β ) , r_{i}(\beta)=y_{i}-f(x_{i},\beta),
  3. r ( j ) ( β ) r_{(j)}(\beta)
  4. S ( β ) = j = 1 n ( r ( j ) ( β ) ) 2 , S(\beta)=\sum_{j=1}^{n}(r_{(j)}(\beta))^{2},
  5. S k ( β ) = j = 1 k ( r ( j ) ( β ) ) 2 . S_{k}(\beta)=\sum_{j=1}^{k}(r_{(j)}(\beta))^{2}.

Lebesgue_integration.html

  1. x x
  2. f f
  3. a a
  4. b b
  5. f f
  6. f f
  7. a a , b aa,b
  8. f f
  9. A A
  10. μ ( A ) μ(A)
  11. A A
  12. f f
  13. f f
  14. t t
  15. y = t a n d y = t + d t y=tandy=t+dt
  16. μ ( { x f ( x ) > t } ) d t . \mu\left(\{x\mid f(x)>t\}\right)\,dt.
  17. f * ( t ) = μ ( { x f ( x ) > t } ) . f^{*}(t)=\mu\left(\{x\mid f(x)>t\}\right).
  18. f f
  19. f d μ = 0 f * ( t ) d t \int f\,d\mu=\int_{0}^{\infty}f^{*}(t)\,dt
  20. f * f*
  21. f f
  22. f f
  23. x x
  24. | f | d μ < + . \int|f|\,d\mu<+\infty.
  25. x x
  26. x x
  27. f d μ = f + d μ - f - d μ \int f\,d\mu=\int f^{+}\,d\mu-\int f^{-}\,d\mu
  28. f = f + - f - f=f_{+}-f_{-}
  29. f + ( x ) = max { f ( x ) , 0 } = { f ( x ) , if f ( x ) > 0 , 0 , otherwise f - ( x ) = max { - f ( x ) , 0 } = { - f ( x ) , if f ( x ) < 0 , 0 , otherwise. \begin{aligned}\displaystyle f^{+}(x)&\displaystyle=\max\{f(x),0\}&% \displaystyle=&\displaystyle\begin{cases}f(x),&\,\text{if }f(x)>0,\\ 0,&\,\text{otherwise}\end{cases}\\ \displaystyle f^{-}(x)&\displaystyle=\max\{-f(x),0\}&\displaystyle=&% \displaystyle\begin{cases}-f(x),&\,\text{if }f(x)<0,\\ 0,&\,\text{otherwise.}\end{cases}\end{aligned}
  30. a a , b × c c , d aa,b×cc,d
  31. ( b a ) ( d c ) (b−a)(d−c)
  32. b a b−a
  33. d c d−c
  34. X X
  35. E E
  36. ( E , X , μ ) (E,X,μ)
  37. E E
  38. X X
  39. E E
  40. E E
  41. X X
  42. E E
  43. n n
  44. X X
  45. E E
  46. μ μ
  47. μ ( E ) = 1 μ(E)=1
  48. f f
  49. E E
  50. ( t , ) (t,∞)
  51. X X
  52. { x f ( x ) > t } X for all t . \{x\,\mid\,f(x)>t\}\in X\quad\,\text{for all}\ t\in\mathbb{R}.
  53. X X
  54. sup k f k , lim inf k f k , lim sup k f k \sup_{k\in\mathbb{N}}f_{k},\quad\liminf_{k\in\mathbb{N}}f_{k},\quad\limsup_{k% \in\mathbb{N}}f_{k}
  55. k k∈ℕ
  56. E f d μ = E f ( x ) μ ( d x ) \int_{E}f\,d\mu=\int_{E}f\left(x\right)\,\mu\left(dx\right)
  57. f f
  58. E E
  59. S S
  60. 1 S d μ = μ ( S ) . \int 1_{S}\,d\mu=\mu(S).
  61. + +∞
  62. μ μ
  63. k a k 1 S k \sum_{k}a_{k}1_{S_{k}}
  64. ( k a k 1 S k ) d μ = k a k 1 S k d μ = k a k μ ( S k ) . \int\left(\sum_{k}a_{k}1_{S_{k}}\right)\,d\mu=\sum_{k}a_{k}\int 1_{S_{k}}\,d% \mu=\sum_{k}a_{k}\,\mu(S_{k}).
  65. 0 × = 0 0×∞=0
  66. ∞−∞
  67. f = k a k 1 S k f=\sum_{k}a_{k}1_{S_{k}}
  68. k k
  69. f f
  70. lim k f k d μ = f d μ . \lim_{k}\int f_{k}\,d\mu=\int f\,d\mu.
  71. f = sup k f k = lim k f k . f=\sup_{k\in\mathbb{N}}f_{k}=\lim_{k\in\mathbb{N}}f_{k}.
  72. f d μ lim k f k d μ \int f\,d\mu\geq\lim_{k}\int f_{k}\,d\mu
  73. lim n g n d μ = f d μ . \lim_{n}\int g_{n}\,d\mu=\int f\,d\mu.
  74. n n∈ℕ
  75. g n d μ lim k f k d μ . \int g_{n}\,d\mu\leq\lim_{k}\int f_{k}\,d\mu.
  76. g g
  77. lim k f k ( x ) g ( x ) \lim_{k}f_{k}(x)\geq g(x)
  78. lim k f k d μ g d μ . \lim_{k}\int f_{k}\,d\mu\geq\int g\,d\mu.
  79. g g
  80. g g
  81. A A
  82. E E
  83. lim k f k ( x ) 1 \lim_{k}f_{k}(x)\geq 1
  84. x A x∈A
  85. lim k f k d μ μ ( A ) . \lim_{k}\int f_{k}\,d\mu\geq\mu(A).
  86. ε > 0 ε>0
  87. B k = { x A : f k ( x ) 1 - ε } . B_{k}=\{x\in A:f_{k}(x)\geq 1-\varepsilon\}.
  88. k k∈ℕ
  89. ( 1 - ε ) μ ( B k ) = ( 1 - ε ) 1 B k d μ f k d μ (1-\varepsilon)\mu(B_{k})=\int(1-\varepsilon)1_{B_{k}}\,d\mu\leq\int f_{k}\,d\mu
  90. x x
  91. k k
  92. k B k = A , \bigcup_{k}B_{k}=A,
  93. 0
  94. μ μ
  95. k k
  96. μ ( A ) = lim k μ ( B k ) lim k ( 1 - ε ) - 1 f k d μ . \mu(A)=\lim_{k}\mu(B_{k})\leq\lim_{k}(1-\varepsilon)^{-1}\int f_{k}\,d\mu.
  97. ε ε
  98. ( X , M , μ ) (X,M,μ)
  99. f n f \int f_{n}\leq\int f
  100. n n
  101. lim n f n f \lim\limits_{n\rightarrow\infty}\int f_{n}\leq\int f
  102. α ( 0 , 1 ) α∈(0,1)
  103. φ φ
  104. 0 φ f 0≤φ≤f
  105. E n = { x : f n ( x ) α ϕ ( x ) } E_{n}=\{x:f_{n}(x)\geq\alpha\phi(x)\}
  106. E n = X \bigcup\limits^{\infty}E_{n}=X
  107. f n E n f n α E n ϕ \int f_{n}\geq\int\limits_{E_{n}}f_{n}\geq\alpha\int\limits_{E_{n}}\phi
  108. lim E n ϕ = ϕ \lim\int\limits_{E_{n}}\phi=\int\phi
  109. lim f n α ϕ \lim\int f_{n}\geq\alpha\int\phi
  110. α ( 0 , 1 ) α∈(0,1)
  111. α = 1 α=1
  112. φ f φ≤f
  113. lim f n f \lim\int f_{n}\geq\int f
  114. lim f n = f \lim\int f_{n}=\int f
  115. X 0 , X→0,∞∞
  116. f f
  117. f = | f ( x ) | d x . \|f\|=\int|f(x)|\,dx.
  118. sin ( x ) x \frac{\sin(x)}{x}
  119. - | sin ( x ) x | d μ = . \int_{-\infty}^{\infty}\left|\frac{\sin(x)}{x}\right|d\mu=\infty.
  120. - sin ( x ) x d μ \int_{-\infty}^{\infty}\frac{\sin(x)}{x}d\mu