wpmath0000012_6

Immersed_boundary_method.html

  1. ρ ( u ( x , t ) t + u u ) = - p + μ Δ u ( x , t ) + f ( x , t ) \rho\left(\frac{\partial{u}({x},t)}{\partial{t}}+{u}\cdot\nabla{u}\right)=-% \nabla p+\mu\,\Delta u(x,t)+f(x,t)
  2. u = 0. \nabla\cdot u=0.\,
  3. Γ \Gamma
  4. X ( s , t ) X(s,t)
  5. s s
  6. t t
  7. F ( s , t ) F(s,t)
  8. f ( x , t ) = Γ F ( s , t ) δ ( x - X ( s , t ) ) d s , f(x,t)=\int_{\Gamma}F(s,t)\,\delta\big(x-X(s,t)\big)\,ds,
  9. δ \delta
  10. δ δ
  11. X ( s , t ) t = u ( X , t ) = Ω u ( x , t ) δ ( x - X ( s , t ) ) d x , \frac{\partial X(s,t)}{\partial t}=u(X,t)=\int_{\Omega}u(x,t)\,\delta\big(x-X(% s,t)\big)\,dx,
  12. Ω \Omega

Implied_repo_rate.html

  1. I R R = ( I n v o i c e P r i c e PurchasePriceOfBond - 1 ) ( d a y B a s e daysToDelivery ) IRR=\left(\frac{\,}{\text{}}{InvoicePrice}\,\text{PurchasePriceOfBond}-1\right% )\left(\frac{\,}{\text{}}{dayBase}\,\text{daysToDelivery}\right)

Impossiball.html

  1. 20 ! × 3 19 120 2.36 × 10 25 \frac{20!\times 3^{19}}{120}\approx 2.36\times 10^{25}

Income_elasticity_of_demand.html

  1. ϵ Y = % change in demand % change in income \epsilon_{Y}=\frac{\%\ \mbox{change in demand}~{}}{\%\ \mbox{change in income}% ~{}}
  2. ϵ d \ \epsilon_{d}
  3. Q ( I , P ) Q(I,\vec{P})
  4. ϵ d = Q I I Q \epsilon_{d}=\frac{\partial Q}{\partial I}\frac{I}{Q}
  5. ϵ d = Y 1 + Y 2 Q 1 + Q 2 × Δ Q Δ Y \epsilon_{d}={Y_{1}+Y_{2}\over Q_{1}+Q_{2}}\times{\Delta Q\over\Delta Y}
  6. ϵ d = d ln Q d ln I \epsilon_{d}=\frac{d\ln Q}{d\ln I}
  7. I I
  8. P \vec{P}

Index_of_dispersion.html

  1. σ 2 \sigma^{2}
  2. μ \mu
  3. D = σ 2 μ . D={\sigma^{2}\over\mu}.
  4. μ \mu
  5. σ / μ \sigma/\mu
  6. μ k / σ k \mu_{k}/\sigma^{k}
  7. σ W 2 / μ W \sigma^{2}_{W}/\mu_{W}
  8. μ / σ \mu/\sigma

Index_set_(recursion_theory).html

  1. W e W_{e}
  2. W e W_{e}
  3. ϕ e \phi_{e}
  4. 𝒜 \mathcal{A}
  5. A = { x : ϕ x 𝒜 } A=\{x:\phi_{x}\in\mathcal{A}\}
  6. A A
  7. 𝒜 \mathcal{A}
  8. A A
  9. x , y x,y\in\mathbb{N}
  10. ϕ x ϕ y \phi_{x}\simeq\phi_{y}
  11. x A y A x\in A\leftrightarrow y\in A
  12. 𝒞 \mathcal{C}
  13. C C
  14. C C
  15. C C
  16. C C
  17. ω \omega
  18. ω \omega

India-based_Neutrino_Observatory.html

  1. θ 13 \theta_{13}

Indian_Contract_Act,_1872.html

  1. a g r e e m e n t = o f f e r + a c c e p t a n c e . agreement=offer+acceptance.
  2. C o n t r a c t = A g r e e m e n t + E n f o r c e - a b i l i t y Contract=Agreement+Enforce-ability

Induced_homomorphism_(algebraic_topology).html

  1. f : X Y f:X\to Y
  2. X X
  3. Y Y

Inequalities_in_information_theory.html

  1. H ( X 1 ) , H(X_{1}),
  2. H ( X 2 ) , H(X_{2}),
  3. H ( X 1 , X 2 ) , H(X_{1},X_{2}),
  4. H ( X 1 ) 0 H(X_{1})\geq 0
  5. H ( X 2 ) 0 H(X_{2})\geq 0
  6. H ( X 1 ) H ( X 1 , X 2 ) H(X_{1})\leq H(X_{1},X_{2})
  7. H ( X 2 ) H ( X 1 , X 2 ) H(X_{2})\leq H(X_{1},X_{2})
  8. H ( X 1 , X 2 ) H ( X 1 ) + H ( X 2 ) . H(X_{1},X_{2})\leq H(X_{1})+H(X_{2}).
  9. I ( A ; B | C ) 0 , I(A;B|C)\geq 0,
  10. A A
  11. B B
  12. C C
  13. Γ n * \Gamma^{*}_{n}
  14. 2 n - 1 , \mathbb{R}^{2^{n}-1},
  15. Γ n * \Gamma^{*}_{n}
  16. Γ n * ¯ . \overline{\Gamma^{*}_{n}}.
  17. Γ n * Γ n * ¯ Γ n . \Gamma^{*}_{n}\subseteq\overline{\Gamma^{*}_{n}}\subseteq\Gamma_{n}.
  18. 2 n - 1 \mathbb{R}^{2^{n}-1}
  19. Γ n . \Gamma_{n}.
  20. Γ n , \Gamma_{n},
  21. Γ n * Γ n . \Gamma^{*}_{n}\subseteq\Gamma_{n}.
  22. Γ n * ¯ Γ n , \overline{\Gamma^{*}_{n}}\subset\Gamma_{n},
  23. n 4. n\geq 4.
  24. D K L ( P Q ) Ψ Q * ( μ 1 ( P ) ) , D_{KL}(P\|Q)\geq\Psi_{Q}^{*}(\mu^{\prime}_{1}(P)),
  25. Ψ Q * \Psi_{Q}^{*}
  26. μ 1 ( P ) \mu^{\prime}_{1}(P)
  27. 1 2 D K L ( e ) ( P Q ) sup { | P ( A ) - Q ( A ) | : A is an event to which probabilities are assigned. } . \sqrt{\frac{1}{2}D_{KL}^{(e)}(P\|Q)}\geq\sup\{|P(A)-Q(A)|:A\,\text{ is an % event to which probabilities are assigned.}\}.
  28. D K L ( e ) ( P | | Q ) D_{KL}^{(e)}(P||Q)
  29. sup A | P ( A ) - Q ( A ) | \sup_{A}|P(A)-Q(A)|\,
  30. f : f:\mathbb{R}\rightarrow\mathbb{C}
  31. - | f ( x ) | 2 d x = 1 , \int_{-\infty}^{\infty}|f(x)|^{2}\,dx=1,
  32. g ( y ) = - f ( x ) e - 2 π i x y d x , g(y)=\int_{-\infty}^{\infty}f(x)e^{-2\pi ixy}\,dx,
  33. | f | 2 |f|^{2}
  34. | g | 2 |g|^{2}
  35. - - | f ( x ) | 2 log | f ( x ) | 2 d x - - | g ( y ) | 2 log | g ( y ) | 2 d y 0. -\int_{-\infty}^{\infty}|f(x)|^{2}\log|f(x)|^{2}\,dx-\int_{-\infty}^{\infty}|g% (y)|^{2}\log|g(y)|^{2}\,dy\geq 0.
  36. log ( e / 2 ) , \log(e/2),
  37. X X
  38. Y Y
  39. Y Y^{\prime}
  40. X X
  41. Y Y^{\prime}
  42. Y Y
  43. H ( Y | Y ) = 0 H(Y^{\prime}|Y)=0
  44. 𝔼 ( | 𝔼 ( X | Y ) - 𝔼 ( X | Y ) | ) 2 log 2 I ( X ; Y | Y ) , \mathbb{E}\big(\big|\mathbb{E}(X|Y^{\prime})-\mathbb{E}(X|Y)\big|\big)\leq% \sqrt{2\log 2\,I(X;Y|Y^{\prime})},

Inertance.html

  1. I = ρ A I={\rho\cdot\ell\over A}\,
  2. ρ \rho
  3. \ell
  4. A A
  5. Δ p = I Q ˙ = I d Q d t \Delta p=I\cdot{\dot{Q}}=I\cdot{\mathrm{d}Q\over{\mathrm{d}t}}
  6. p p
  7. Q Q
  8. X = j ω I X=j\omega I
  9. ω = 2 π f \omega=2\pi f
  10. f f

Inertia_tensor_of_triangle.html

  1. 𝐉 \mathbf{J}
  2. 𝐂 \mathbf{C}
  3. 𝐉 = tr ( 𝐂 ) 𝐈 - 𝐂 \mathbf{J}=\mathrm{tr}(\mathbf{C})\mathbf{I}-\mathbf{C}
  4. 𝐂 Δ ρ 𝐱𝐱 T d A \mathbf{C}\triangleq\int_{\Delta}\rho\mathbf{x}\mathbf{x}^{\mathrm{T}}\,dA
  5. 𝐂 = a 𝐕 T 𝐒𝐕 \mathbf{C}=a\mathbf{V}^{\mathrm{T}}\mathbf{S}\mathbf{V}
  6. 𝐕 \mathbf{V}
  7. ( 𝐯 0 , 𝐯 1 , 𝐯 2 ) (\mathbf{v}_{0},\mathbf{v}_{1},\mathbf{v}_{2})
  8. a = | ( 𝐯 1 - 𝐯 0 ) × ( 𝐯 2 - 𝐯 0 ) | a=|(\mathbf{v}_{1}-\mathbf{v}_{0})\times(\mathbf{v}_{2}-\mathbf{v}_{0})|
  9. 𝐒 = 1 24 [ 2 1 1 1 2 1 1 1 2 ] \mathbf{S}=\frac{1}{24}\begin{bmatrix}2&1&1\\ 1&2&1\\ 1&1&2\\ \end{bmatrix}
  10. 𝐉 = a 24 ( 𝐯 0 2 + 𝐯 1 2 + 𝐯 2 2 + ( 𝐯 0 + 𝐯 1 + 𝐯 2 ) 2 ) 𝐈 - a 𝐕 T 𝐒𝐕 \mathbf{J}=\frac{a}{24}(\mathbf{v}^{2}_{0}+\mathbf{v}^{2}_{1}+\mathbf{v}^{2}_{% 2}+(\mathbf{v}_{0}+\mathbf{v}_{1}+\mathbf{v}_{2})^{2})\mathbf{I}-a\mathbf{V}^{% \mathrm{T}}\mathbf{S}\mathbf{V}
  11. 𝐂 x x 0 = Δ x 2 d A = x = 0 1 x 2 y = 0 1 - x d y d x = 0 1 x 2 ( 1 - x ) d x = 1 12 \mathbf{C}^{0}_{xx}=\int_{\Delta}x^{2}\,dA=\int_{x=0}^{1}x^{2}\int_{y=0}^{1-x}% \,dy\,dx=\int_{0}^{1}x^{2}(1-x)\,dx=\frac{1}{12}
  12. 𝐂 x y 0 = Δ x y d A = x = 0 1 x y = 0 1 - x y d y d x = 0 1 x ( 1 - x ) 2 2 d x = 1 24 \mathbf{C}^{0}_{xy}=\int_{\Delta}xy\,dA=\int_{x=0}^{1}x\int_{y=0}^{1-x}y\,dy\,% dx=\int_{0}^{1}x\frac{(1-x)^{2}}{2}\,dx=\frac{1}{24}
  13. 𝐂 y y 0 = 𝐂 x x 0 \mathbf{C}^{0}_{yy}=\mathbf{C}^{0}_{xx}
  14. C C
  15. z = 0 z=0
  16. 𝐂 0 = 1 24 [ 2 1 0 1 2 0 0 0 0 ] = 1 48 [ 1 - 1 0 ] [ 1 - 1 0 ] T + 1 16 [ 1 1 0 ] [ 1 1 0 ] T \mathbf{C}^{0}=\frac{1}{24}\begin{bmatrix}2&1&0\\ 1&2&0\\ 0&0&0\\ \end{bmatrix}=\frac{1}{48}\begin{bmatrix}1\\ -1\\ 0\end{bmatrix}\begin{bmatrix}1&-1&0\end{bmatrix}^{\mathrm{T}}+\frac{1}{16}% \begin{bmatrix}1\\ 1\\ 0\end{bmatrix}\begin{bmatrix}1&1&0\end{bmatrix}^{\mathrm{T}}
  17. 𝐱 = 𝐀𝐱 0 \mathbf{x}^{\prime}=\mathbf{A}\mathbf{x}^{0}
  18. 𝐯 0 = 𝟎 \mathbf{v}^{\prime}_{0}=\mathbf{0}
  19. 𝐯 1 = 𝐯 1 - 𝐯 0 \mathbf{v}^{\prime}_{1}=\mathbf{v}_{1}-\mathbf{v}_{0}
  20. 𝐯 2 = 𝐯 2 - 𝐯 0 \mathbf{v}^{\prime}_{2}=\mathbf{v}_{2}-\mathbf{v}_{0}
  21. 𝐀 \mathbf{A}
  22. 𝐯 1 \mathbf{v}^{\prime}_{1}
  23. 𝐯 2 \mathbf{v}^{\prime}_{2}
  24. 𝐂 = Δ 𝐱 𝐱 T d A = Δ 0 𝐀𝐱 0 𝐱 0 T 𝐀 T a d A 0 = a 𝐀𝐂 0 𝐀 T \mathbf{C}^{\prime}=\int_{\Delta^{\prime}}\mathbf{x}^{\prime}\mathbf{x}^{% \prime\mathrm{T}}\,dA^{\prime}=\int_{\Delta^{0}}\mathbf{A}\mathbf{x}^{0}% \mathbf{x}^{0\mathrm{T}}\mathbf{A}^{\mathrm{T}}a\,dA^{0}=a\mathbf{A}\mathbf{C}% ^{0}\mathbf{A}^{\mathrm{T}}
  25. 𝐂 = a 48 ( 𝐯 1 - 𝐯 2 ) ( 𝐯 1 - 𝐯 2 ) T + a 16 ( 𝐯 1 + 𝐯 2 - 2 𝐯 0 ) ( 𝐯 1 + 𝐯 2 - 2 𝐯 0 ) T \mathbf{C}^{\prime}=\frac{a}{48}(\mathbf{v}_{1}-\mathbf{v}_{2})(\mathbf{v}_{1}% -\mathbf{v}_{2})^{\mathrm{T}}+\frac{a}{16}(\mathbf{v}_{1}+\mathbf{v}_{2}-2% \mathbf{v}_{0})(\mathbf{v}_{1}+\mathbf{v}_{2}-2\mathbf{v}_{0})^{\mathrm{T}}
  26. 𝐯 0 \mathbf{v}_{0}
  27. 𝐂 = Δ ( 𝐱 + 𝐯 0 ) ( 𝐱 + 𝐯 0 ) T d A = 𝐂 + a 2 ( 𝐯 0 𝐯 0 T + 𝐯 0 𝐱 ¯ T + 𝐱 ¯ 𝐯 0 T ) \mathbf{C}=\int_{\Delta}(\mathbf{x^{\prime}}+\mathbf{v}_{0})(\mathbf{x^{\prime% }}+\mathbf{v}_{0})^{\mathrm{T}}\,dA=\mathbf{C}^{\prime}+\frac{a}{2}(\mathbf{v}% _{0}\mathbf{v}_{0}^{\mathrm{T}}+\mathbf{v}_{0}\overline{\mathbf{x}}^{\prime% \mathrm{T}}+\overline{\mathbf{x}}^{\prime}\mathbf{v}_{0}^{\mathrm{T}})
  28. 𝐱 ¯ = Δ 𝐱 d A = 1 3 ( 𝐯 1 + 𝐯 2 ) = 1 3 ( 𝐯 1 + 𝐯 2 - 2 𝐯 0 ) \overline{\mathbf{x}}^{\prime}=\int_{\Delta^{\prime}}\mathbf{x}^{\prime}\,dA^{% \prime}=\frac{1}{3}(\mathbf{v}^{\prime}_{1}+\mathbf{v}^{\prime}_{2})=\frac{1}{% 3}(\mathbf{v}_{1}+\mathbf{v}_{2}-2\mathbf{v}_{0})
  29. 𝐂 \mathbf{C}
  30. 𝐯 i 𝐯 i T \mathbf{v}_{i}\mathbf{v}_{i}^{\mathrm{T}}
  31. a 12 \frac{a}{12}
  32. 𝐯 i 𝐯 j T ( i j ) \mathbf{v}_{i}\mathbf{v}_{j}^{\mathrm{T}}\;(i\neq j)
  33. a 24 \frac{a}{24}
  34. 𝐒 \mathbf{S}

Infinitary_combinatorics.html

  1. κ ( λ ) m n \kappa\rightarrow(\lambda)^{n}_{m}
  2. κ \kappa
  3. κ ( λ ) m < ω \kappa\rightarrow(\lambda)^{<\omega}_{m}
  4. κ ( λ , μ ) n \kappa\rightarrow(\lambda,\mu)^{n}
  5. κ \kappa
  6. \alef 0 ( \alef 0 ) k n \alef_{0}\rightarrow(\alef_{0})^{n}_{k}
  7. n + ( \alef 1 ) \alef 0 n + 1 \beth_{n}^{+}\rightarrow(\alef_{1})_{\alef_{0}}^{n+1}
  8. 2 κ ↛ ( κ + ) 2 2^{\kappa}\not\rightarrow(\kappa^{+})^{2}
  9. 2 κ ↛ ( 3 ) κ 2 2^{\kappa}\not\rightarrow(3)^{2}_{\kappa}
  10. κ ( κ , \alef 0 ) 2 \kappa\rightarrow(\kappa,\alef_{0})^{2}
  11. \alef 1 ( \alef 1 ) 2 \alef 1 \alef_{1}\rightarrow(\alef_{1})^{\alef_{1}}_{2}

Infinity.html

  1. s i z e = 150 % ∞size=150\%
  2. \infty
  3. 1 \frac{1}{\infty}
  4. i i
  5. i i
  6. i i
  7. \infty
  8. \infty
  9. x x\rightarrow\infty
  10. x - x\to-\infty
  11. a b f ( t ) d t = \int_{a}^{b}\,f(t)\ dt\ =\infty
  12. a a
  13. b b
  14. - f ( t ) d t = \int_{-\infty}^{\infty}\,f(t)\ dt\ =\infty
  15. - f ( t ) d t = a \int_{-\infty}^{\infty}\,f(t)\ dt\ =a
  16. a a
  17. i = 0 f ( i ) = a \sum_{i=0}^{\infty}\,f(i)=a
  18. a a
  19. i = 0 f ( i ) = \sum_{i=0}^{\infty}\,f(i)=\infty
  20. + +\infty
  21. - -\infty
  22. + +\infty
  23. - -\infty
  24. \infty
  25. x x\rightarrow\infty
  26. | x | |x|
  27. \infty
  28. z / 0 = z/0=\infty
  29. \infty
  30. 𝐜 \mathbf{c}
  31. 0 {\aleph_{0}}
  32. 𝐜 = 2 0 > 0 \mathbf{c}=2^{\aleph_{0}}>{\aleph_{0}}
  33. 𝐜 = 1 = 1 \mathbf{c}=\aleph_{1}=\beth_{1}

Information_gain_ratio.html

  1. A t t r Attr
  2. E x Ex
  3. v a l u e ( x , a ) value(x,a)
  4. x E x x\in Ex
  5. x x
  6. a A t t r a\in Attr
  7. H H
  8. v a l u e s ( a ) values(a)
  9. a A t t r a\in Attr
  10. a A t t r a\in Attr
  11. I G ( E x , a ) = H ( E x ) - v v a l u e s ( a ) ( | { x E x | v a l u e ( x , a ) = v } | | E x | H ( { x E x | v a l u e ( x , a ) = v } ) ) IG(Ex,a)=H(Ex)-\sum_{v\in values(a)}\left(\frac{|\{x\in Ex|value(x,a)=v\}|}{|% Ex|}\cdot H(\{x\in Ex|value(x,a)=v\})\right)
  12. I V ( E x , a ) = - v v a l u e s ( a ) | { x E x | v a l u e ( x , a ) = v } | | E x | * log 2 ( | { x E x | v a l u e ( x , a ) = v } | | E x | ) IV(Ex,a)=-\sum_{v\in values(a)}\frac{|\{x\in Ex|value(x,a)=v\}|}{|Ex|}*\log_{2% }\left(\frac{|\{x\in Ex|value(x,a)=v\}|}{|Ex|}\right)
  13. I G R ( E x , a ) = I G / I V IGR(Ex,a)=IG/IV

Information_source_(mathematics).html

  1. H { X } = lim n H ( X n | X 0 , X 1 , , X n - 1 ) H\{{X}\}=\lim_{n\to\infty}H(X_{n}|X_{0},X_{1},\dots,X_{n-1})
  2. X 0 , X 1 , , X n X_{0},X_{1},\dots,X_{n}\,
  3. H ( X n | X 0 , X 1 , , X n - 1 ) H(X_{n}|X_{0},X_{1},\dots,X_{n-1})
  4. H { X } = lim n H ( X 0 , X 1 , , X n - 1 , X n ) n + 1 . H\{{X}\}=\lim_{n\to\infty}\frac{H(X_{0},X_{1},\dots,X_{n-1},X_{n})}{n+1}.

Initial_value_theorem.html

  1. F ( s ) = 0 f ( t ) e - s t d t F(s)=\int_{0}^{\infty}f(t)e^{-st}\,dt
  2. lim t 0 f ( t ) = lim s s F ( s ) . \lim_{t\to 0}f(t)=\lim_{s\to\infty}{sF(s)}.\,
  3. s F ( s ) = f ( 0 - ) + t = 0 - e - s t f ( t ) d t sF(s)=f(0^{-})+\int_{t=0^{-}}^{\infty}e^{-st}f^{{}^{\prime}}(t)dt
  4. lim s s F ( s ) = lim s [ f ( 0 - ) + t = 0 - e - s t f ( t ) d t ] \lim_{s\to\infty}sF(s)=\lim_{s\to\infty}[f(0^{-})+\int_{t=0^{-}}^{\infty}e^{-% st}f^{{}^{\prime}}(t)dt]
  5. lim s e - s t \lim_{s\to\infty}e^{-st}
  6. lim s [ t = 0 - e - s t f ( t ) d t ] = lim s { lim ϵ 0 + [ t = 0 - ϵ e - s t f ( t ) d t ] + lim ϵ 0 + [ t = ϵ e - s t f ( t ) d t ] } \lim_{s\to\infty}[\int_{t=0^{-}}^{\infty}e^{-st}f^{{}^{\prime}}(t)dt]=\lim_{s% \to\infty}\{\lim_{\epsilon\to 0^{+}}[\int_{t=0^{-}}^{\epsilon}e^{-st}f^{{}^{% \prime}}(t)dt]+\lim_{\epsilon\to 0^{+}}[\int_{t=\epsilon}^{\infty}e^{-st}f^{{}% ^{\prime}}(t)dt]\}
  7. lim s e - s t ( t ) \lim_{s\to\infty}e^{-st}(t)
  8. lim s [ t = 0 - e - s t f ( t ) d t ] = lim s { lim ϵ 0 + [ t = 0 - ϵ f ( t ) d t ] } + lim ϵ 0 + { t = ϵ lim s [ e - s t f ( t ) d t ] } = f ( t ) | t = 0 - t = 0 + + 0 = f ( 0 + ) - f ( 0 - ) + 0 \begin{aligned}\displaystyle\lim_{s\to\infty}[\int_{t=0^{-}}^{\infty}e^{-st}f^% {{}^{\prime}}(t)dt]&\displaystyle=\lim_{s\to\infty}\{\lim_{\epsilon\to 0^{+}}[% \int_{t=0^{-}}^{\epsilon}f^{{}^{\prime}}(t)dt]\}+\lim_{\epsilon\to 0^{+}}\{% \int_{t=\epsilon}^{\infty}\lim_{s\to\infty}[e^{-st}f^{{}^{\prime}}(t)dt]\}\\ &\displaystyle=f(t)|_{t=0^{-}}^{t=0^{+}}+0\\ &\displaystyle=f(0^{+})-f(0^{-})+0\\ \end{aligned}
  9. lim s s F ( s ) = f ( 0 - ) + f ( 0 + ) - f ( 0 - ) = f ( 0 + ) \lim_{s\to\infty}sF(s)=f(0^{-})+f(0^{+})-f(0^{-})=f(0^{+})

Inosine-5′-monophosphate_dehydrogenase.html

  1. \rightleftharpoons

Instant_centre_of_rotation.html

  1. γ = A D B D \gamma=\frac{AD}{BD}

Insulin_(medication).html

  1. I n s u l i n = g l u c o s e - T R C F + c a r b o h y d r a t e s K F = K F ( g l u c o s e - T R ) + c a r b o h y d r a t e s C F C F × K F Insulin=\frac{glucose-TR}{CF}+\frac{carbohydrates}{KF}=\frac{KF(glucose-TR)+% carbohydratesCF}{{CF}\times{KF}}

Integral_nonlinearity.html

  1. INL = max 0 c c max | V out [ c ] - V out [ 0 ] - c m | \mathrm{INL}=\max_{0\leq c\leq c_{\max}}\left|V_{\mathrm{out}}[c]-V_{\mathrm{% out}}[0]-c\cdot m\right|
  2. m = V out [ c max ] - V out [ 0 ] c max m=\frac{V_{\mathrm{out}}[c_{\max}]-V_{\mathrm{out}}[0]}{c_{\max}}
  3. V out [ c ] V_{\mathrm{out}}[c]

Integrally_closed_domain.html

  1. A = k [ t 2 , t 3 ] B = k [ t ] A=k[t^{2},t^{3}]\subset B=k[t]
  2. Y 2 = X 3 Y^{2}=X^{3}
  3. A = [ 5 ] . A=\mathbb{Z}[\sqrt{5}].
  4. A 𝔭 A_{\mathfrak{p}}
  5. 𝔭 \mathfrak{p}
  6. A 𝔭 A_{\mathfrak{p}}
  7. 𝔭 \mathfrak{p}
  8. Spec ( A ) \operatorname{Spec}(A)
  9. 𝔭 \mathfrak{p}
  10. 𝔭 \mathfrak{p}
  11. 1 \leq 1
  12. A 𝔭 A_{\mathfrak{p}}
  13. A 𝔭 A_{\mathfrak{p}}
  14. 𝔭 \mathfrak{p}
  15. 2 \geq 2
  16. A 𝔭 A_{\mathfrak{p}}
  17. 2 \geq 2
  18. A s s ( A ) Ass(A)
  19. A s s ( A / f A ) Ass(A/fA)
  20. 𝒪 p \mathcal{O}_{p}
  21. d 0 d\neq 0
  22. d x n A dx^{n}\in A
  23. n 0 n\geq 0
  24. A [ [ X ] ] A[[X]]
  25. R [ [ X ] ] R[[X]]
  26. χ ( T ) = p P length p ( T ) p \chi(T)=\sum_{p\in P}\operatorname{length}_{p}(T)p
  27. c ( d ) c(d)
  28. F , F F,F^{\prime}
  29. c ( χ ( M / F ) ) = c ( χ ( M / F ) ) c(\chi(M/F))=c(\chi(M/F^{\prime}))
  30. c ( χ ( M / F ) ) c(\chi(M/F))
  31. c ( M ) c(M)
  32. 𝔪 \mathfrak{m}
  33. 𝔪 \mathfrak{m}
  34. x = 0 x=0
  35. 𝔪 \mathfrak{m}
  36. x s = 0 xs=0
  37. s 𝔪 s\not\in\mathfrak{m}
  38. s s
  39. 𝔪 \mathfrak{m}

Integration_by_reduction_formulae.html

  1. I n = f ( x , n ) d x , I_{n}=\int f(x,n)\,dx,
  2. I k = f ( x , k ) d x , I_{k}=\int f(x,k)\,dx,
  3. k < n . k<n.
  4. cos n x d x , \int\cos^{n}xdx,\,\!
  5. cos n ( x ) d x \int\cos^{n}(x)dx\!
  6. I n = cos n x d x . I_{n}=\int\cos^{n}xdx.\,\!
  7. I n = cos n - 1 x cos x d x , I_{n}=\int\cos^{n-1}x\cos xdx,\,\!
  8. cos x d x = d ( sin x ) , \cos xdx=d(\sin x),\,\!
  9. I n = cos n - 1 x d ( sin x ) . I_{n}=\int\cos^{n-1}xd(\sin x).\!
  10. cos n x d x = cos n - 1 x sin x - sin x d ( cos n - 1 x ) = cos n - 1 x sin x + ( n - 1 ) sin x cos n - 2 x sin x d x = cos n - 1 x sin x + ( n - 1 ) cos n - 2 x sin 2 x d x = cos n - 1 x sin x + ( n - 1 ) cos n - 2 x ( 1 - cos 2 x ) d x = cos n - 1 x sin x + ( n - 1 ) cos n - 2 x d x - ( n - 1 ) cos n x d x = cos n - 1 x sin x + ( n - 1 ) I n - 2 - ( n - 1 ) I n , \begin{aligned}\displaystyle\int\cos^{n}xdx&\displaystyle=\cos^{n-1}x\sin x-% \int\sin xd(\cos^{n-1}x)\\ &\displaystyle=\cos^{n-1}x\sin x+(n-1)\int\sin x\cos^{n-2}x\sin xdx\\ &\displaystyle=\cos^{n-1}x\sin x+(n-1)\int\cos^{n-2}x\sin^{2}xdx\\ &\displaystyle=\cos^{n-1}x\sin x+(n-1)\int\cos^{n-2}x(1-\cos^{2}x)dx\\ &\displaystyle=\cos^{n-1}x\sin x+(n-1)\int\cos^{n-2}xdx-(n-1)\int\cos^{n}xdx\\ &\displaystyle=\cos^{n-1}x\sin x+(n-1)I_{n-2}-(n-1)I_{n},\end{aligned}\,
  11. I n + ( n - 1 ) I n = cos n - 1 x sin x + ( n - 1 ) I n - 2 , I_{n}\ +(n-1)I_{n}\ =\cos^{n-1}x\sin x\ +\ (n-1)I_{n-2},\,
  12. n I n = cos n - 1 ( x ) sin x + ( n - 1 ) I n - 2 , nI_{n}\ =\cos^{n-1}(x)\sin x\ +(n-1)I_{n-2},\,
  13. I n = 1 n cos n - 1 x sin x + n - 1 n I n - 2 , I_{n}\ =\frac{1}{n}\cos^{n-1}x\sin x\ +\frac{n-1}{n}I_{n-2},\,
  14. cos n x d x = 1 n cos n - 1 x sin x + n - 1 n cos n - 2 x d x . \int\cos^{n}xdx\ =\frac{1}{n}\cos^{n-1}x\sin x+\frac{n-1}{n}\int\cos^{n-2}xdx.\!
  15. I 5 = cos 5 x d x . I_{5}=\int\cos^{5}xdx.\,\!
  16. n = 5 , I 5 = 1 5 cos 4 x sin x + 4 5 I 3 , n=5,\quad I_{5}=\tfrac{1}{5}\cos^{4}x\sin x+\tfrac{4}{5}I_{3},\,
  17. n = 3 , I 3 = 1 3 cos 2 x sin x + 2 3 I 1 , n=3,\quad I_{3}=\tfrac{1}{3}\cos^{2}x\sin x+\tfrac{2}{3}I_{1},\,
  18. I 1 = cos x d x = sin x + C 1 , \because I_{1}\ =\int\cos xdx=\sin x+C_{1},\,
  19. I 3 = 1 3 cos 2 x sin x + 2 3 sin x + C 2 , C 2 = 2 3 C 1 , \therefore I_{3}\ =\tfrac{1}{3}\cos^{2}x\sin x+\tfrac{2}{3}\sin x+C_{2},\quad C% _{2}\ =\tfrac{2}{3}C_{1},\,
  20. I 5 = 1 5 cos 4 x sin x + 4 5 [ 1 3 cos 2 x sin x + 2 3 sin x ] + C , I_{5}\ =\frac{1}{5}\cos^{4}x\sin x+\frac{4}{5}\left[\frac{1}{3}\cos^{2}x\sin x% +\frac{2}{3}\sin x\right]+C,\,
  21. x n e a x d x . \int x^{n}e^{ax}dx.\,\!
  22. I n = x n e a x d x . I_{n}=\int x^{n}e^{ax}dx.\,\!
  23. x n d x = d ( x n + 1 ) n + 1 , x^{n}dx=\frac{d(x^{n+1})}{n+1},\,\!
  24. I n = 1 n + 1 e a x d ( x n + 1 ) , I_{n}=\frac{1}{n+1}\int e^{ax}d(x^{n+1}),\!
  25. e a x d ( x n + 1 ) = x n + 1 e a x - x n + 1 d ( e a x ) = x n + 1 e a x - a x n + 1 e a x d x , \begin{aligned}\displaystyle\int e^{ax}d(x^{n+1})&\displaystyle=x^{n+1}e^{ax}-% \int x^{n+1}d(e^{ax})\\ &\displaystyle=x^{n+1}e^{ax}-a\int x^{n+1}e^{ax}dx,\end{aligned}\!
  26. ( n + 1 ) I n = x n + 1 e a x - a I n + 1 , (n+1)I_{n}=x^{n+1}e^{ax}-aI_{n+1},\!
  27. n I n - 1 = x n e a x - a I n , nI_{n-1}=x^{n}e^{ax}-aI_{n},\!
  28. I n = 1 a ( x n e a x - n I n - 1 ) , I_{n}=\frac{1}{a}\left(x^{n}e^{ax}-nI_{n-1}\right),\,\!
  29. x n e a x d x = 1 a ( x n e a x - n x n - 1 e a x d x ) . \int x^{n}e^{ax}dx=\frac{1}{a}\left(x^{n}e^{ax}-n\int x^{n-1}e^{ax}dx\right).\!
  30. a x + b \sqrt{ax+b}\,\!
  31. p x + q {px+q}\,\!
  32. a x + b \sqrt{ax+b}\,\!
  33. x 2 + a 2 x^{2}+a^{2}\,\!
  34. x 2 - a 2 x^{2}-a^{2}\,\!
  35. x > a x>a\,\!
  36. a 2 - x 2 a^{2}-x^{2}\,\!
  37. x < a x<a\,\!
  38. a x 2 + b x + c ax^{2}+bx+c\,\!
  39. a x 2 + b x + c \sqrt{ax^{2}+bx+c}\,\!
  40. I n = x n a x + b d x I_{n}=\int\frac{x^{n}}{\sqrt{ax+b}}dx\,\!
  41. I n = 2 x n a x + b a ( 2 n + 1 ) - 2 n b a ( 2 n + 1 ) I n - 1 I_{n}=\frac{2x^{n}\sqrt{ax+b}}{a(2n+1)}-\frac{2nb}{a(2n+1)}I_{n-1}\,\!
  42. I n = d x x n a x + b I_{n}=\int\frac{dx}{x^{n}\sqrt{ax+b}}\,\!
  43. I n = - a x + b ( n - 1 ) b x n - 1 - a ( 2 n - 3 ) 2 b ( n - 1 ) I n - 1 I_{n}=-\frac{\sqrt{ax+b}}{(n-1)bx^{n-1}}-\frac{a(2n-3)}{2b(n-1)}I_{n-1}\,\!
  44. I n = x n a x + b d x I_{n}=\int x^{n}\sqrt{ax+b}dx\,\!
  45. I n = 2 x n ( a x + b ) 3 a ( 2 n + 3 ) - 2 n b a ( 2 n + 3 ) I n - 1 I_{n}=\frac{2x^{n}\sqrt{(ax+b)^{3}}}{a(2n+3)}-\frac{2nb}{a(2n+3)}I_{n-1}\,\!
  46. I n , m = d x ( a x + b ) n ( p x + q ) m I_{n,m}=\int\frac{dx}{(ax+b)^{n}(px+q)^{m}}\,\!
  47. I n , m = - 1 ( n - 1 ) ( b p - a q ) [ 1 ( a x + b ) m - 1 ( p x + q ) n - 1 + a ( n + m - 2 ) I m , n - 1 ] I_{n,m}=-\frac{1}{(n-1)(bp-aq)}\left[\frac{1}{(ax+b)^{m-1}(px+q)^{n-1}}+a(n+m-% 2)I_{m,n-1}\right]\,\!
  48. I n , m = ( a x + b ) m ( p x + q ) n d x I_{n,m}=\int\frac{(ax+b)^{m}}{(px+q)^{n}}dx\,\!
  49. I n , m = { - 1 ( n - 1 ) ( b p - a q ) [ ( a x + b ) m + 1 ( p x + q ) n - 1 + a ( n + m - 2 ) I m - 1 , n - 1 ] - 1 ( n - m - 1 ) p [ ( a x + b ) m ( p x + q ) n - 1 + m ( b p - a q ) I m - 1 , n ] - 1 ( n - 1 ) p [ ( a x + b ) m ( p x + q ) n - 1 - a m I m - 1 , n - 1 ] I_{n,m}=\begin{cases}-\frac{1}{(n-1)(bp-aq)}\left[\frac{(ax+b)^{m+1}}{(px+q)^{% n-1}}+a(n+m-2)I_{m-1,n-1}\right]\\ -\frac{1}{(n-m-1)p}\left[\frac{(ax+b)^{m}}{(px+q)^{n-1}}+m(bp-aq)I_{m-1,n}% \right]\\ -\frac{1}{(n-1)p}\left[\frac{(ax+b)^{m}}{(px+q)^{n-1}}-amI_{m-1,n-1}\right]% \end{cases}\,\!
  50. I n = ( p x + q ) n a x + b d x I_{n}=\int\frac{(px+q)^{n}}{\sqrt{ax+b}}dx\,\!
  51. ( p x + q ) n a x + b d x = 2 ( p x + q ) n + 1 a x + b p ( 2 n + 3 ) + b p - a q p ( 2 n + 3 ) I n \int(px+q)^{n}\sqrt{ax+b}dx=\frac{2(px+q)^{n+1}\sqrt{ax+b}}{p(2n+3)}+\frac{bp-% aq}{p(2n+3)}I_{n}\,\!
  52. I n = 2 ( p x + q ) n a x + b a ( 2 n + 1 ) + 2 n ( a q - b p ) a ( 2 n + 1 ) I n - 1 I_{n}=\frac{2(px+q)^{n}\sqrt{ax+b}}{a(2n+1)}+\frac{2n(aq-bp)}{a(2n+1)}I_{n-1}\,\!
  53. I n = d x ( p x + q ) n a x + b I_{n}=\int\frac{dx}{(px+q)^{n}\sqrt{ax+b}}\,\!
  54. a x + b ( p x + q ) n d x = - a x + b p ( n - 1 ) ( p x + q ) n - 1 + a 2 p ( n - 1 ) I n \int\frac{\sqrt{ax+b}}{(px+q)^{n}}dx=-\frac{\sqrt{ax+b}}{p(n-1)(px+q)^{n-1}}+% \frac{a}{2p(n-1)}I_{n}\,\!
  55. I n = - a x + b ( n - 1 ) ( a q - b p ) ( p x + q ) n - 1 + a ( 2 n - 3 ) 2 ( n - 1 ) ( a q - b p ) I n - 1 I_{n}=-\frac{\sqrt{ax+b}}{(n-1)(aq-bp)(px+q)^{n-1}}+\frac{a(2n-3)}{2(n-1)(aq-% bp)}I_{n-1}\,\!
  56. I n = d x ( x 2 + a 2 ) n I_{n}=\int\frac{dx}{(x^{2}+a^{2})^{n}}\,\!
  57. I n = x 2 a 2 ( n - 1 ) ( x 2 + a 2 ) n - 1 + 2 n - 3 2 a 2 ( n - 1 ) I n - 1 I_{n}=\frac{x}{2a^{2}(n-1)(x^{2}+a^{2})^{n-1}}+\frac{2n-3}{2a^{2}(n-1)}I_{n-1}\,\!
  58. I n , m = d x x m ( x 2 + a 2 ) n I_{n,m}=\int\frac{dx}{x^{m}(x^{2}+a^{2})^{n}}\,\!
  59. a 2 I n , m = I m , n - 1 - I m - 2 , n a^{2}I_{n,m}=I_{m,n-1}-I_{m-2,n}\,\!
  60. I n , m = x m ( x 2 + a 2 ) n d x I_{n,m}=\int\frac{x^{m}}{(x^{2}+a^{2})^{n}}dx\,\!
  61. I n , m = I m - 2 , n - 1 - a 2 I m - 2 , n I_{n,m}=I_{m-2,n-1}-a^{2}I_{m-2,n}\,\!
  62. I n = d x ( x 2 - a 2 ) n I_{n}=\int\frac{dx}{(x^{2}-a^{2})^{n}}\,\!
  63. I n = - x 2 a 2 ( n - 1 ) ( x 2 - a 2 ) n - 1 - 2 n - 3 2 a 2 ( n - 1 ) I n - 1 I_{n}=-\frac{x}{2a^{2}(n-1)(x^{2}-a^{2})^{n-1}}-\frac{2n-3}{2a^{2}(n-1)}I_{n-1% }\,\!
  64. I n , m = d x x m ( x 2 - a 2 ) n I_{n,m}=\int\frac{dx}{x^{m}(x^{2}-a^{2})^{n}}\,\!
  65. a 2 I n , m = I m - 2 , n - I m , n - 1 {a^{2}}I_{n,m}=I_{m-2,n}-I_{m,n-1}\,\!
  66. I n , m = x m ( x 2 - a 2 ) n d x I_{n,m}=\int\frac{x^{m}}{(x^{2}-a^{2})^{n}}dx\,\!
  67. I n , m = I m - 2 , n - 1 + a 2 I m - 2 , n I_{n,m}=I_{m-2,n-1}+a^{2}I_{m-2,n}\,\!
  68. I n = d x ( a 2 - x 2 ) n I_{n}=\int\frac{dx}{(a^{2}-x^{2})^{n}}\,\!
  69. I n = x 2 a 2 ( n - 1 ) ( a 2 - x 2 ) n - 1 + 2 n - 3 2 a 2 ( n - 1 ) I n - 1 I_{n}=\frac{x}{2a^{2}(n-1)(a^{2}-x^{2})^{n-1}}+\frac{2n-3}{2a^{2}(n-1)}I_{n-1}\,\!
  70. I n , m = d x x m ( a 2 - x 2 ) n I_{n,m}=\int\frac{dx}{x^{m}(a^{2}-x^{2})^{n}}\,\!
  71. a 2 I n , m = I m , n - 1 + I m - 2 , n {a^{2}}I_{n,m}=I_{m,n-1}+I_{m-2,n}\,\!
  72. I n , m = x m ( a 2 - x 2 ) n d x I_{n,m}=\int\frac{x^{m}}{(a^{2}-x^{2})^{n}}dx\,\!
  73. I n , m = a 2 I m - 2 , n - I m - 2 , n - 1 I_{n,m}=a^{2}I_{m-2,n}-I_{m-2,n-1}\,\!
  74. I n = d x x n ( a x 2 + b x + c ) I_{n}=\int\frac{dx}{{x^{n}}(ax^{2}+bx+c)}\,\!
  75. - c I n = 1 x n - 1 ( n - 1 ) + b I n - 1 + a I n - 2 -cI_{n}=\frac{1}{x^{n-1}(n-1)}+bI_{n-1}+aI_{n-2}\,\!
  76. I m , n = x m d x ( a x 2 + b x + c ) n I_{m,n}=\int\frac{x^{m}dx}{(ax^{2}+bx+c)^{n}}\,\!
  77. I m , n = - x m - 1 a ( 2 n - m - 1 ) ( a x 2 + b x + c ) n - 1 - b ( n - m ) a ( 2 n - m - 1 ) I m - 1 , n + c ( m - 1 ) a ( 2 n - m - 1 ) I m - 2 , n I_{m,n}=-\frac{x^{m-1}}{a(2n-m-1)(ax^{2}+bx+c)^{n-1}}-\frac{b(n-m)}{a(2n-m-1)}% I_{m-1,n}+\frac{c(m-1)}{a(2n-m-1)}I_{m-2,n}\,\!
  78. I m , n = d x x m ( a x 2 + b x + c ) n I_{m,n}=\int\frac{dx}{x^{m}(ax^{2}+bx+c)^{n}}\,\!
  79. - c ( m - 1 ) I m , n = 1 x m - 1 ( a x 2 + b x + c ) n - 1 + a ( m + 2 n - 3 ) I m - 2 , n + b ( m + n - 2 ) I m - 1 , n -c(m-1)I_{m,n}=\frac{1}{x^{m-1}(ax^{2}+bx+c)^{n-1}}+{a(m+2n-3)}I_{m-2,n}+{b(m+% n-2)}I_{m-1,n}\,\!
  80. I n = ( a x 2 + b x + c ) n d x I_{n}=\int(ax^{2}+bx+c)^{n}dx\,\!
  81. 8 a ( n + 1 ) I n + 1 2 = 2 ( 2 a x + b ) ( a x 2 + b x + c ) n + 1 2 + ( 2 n + 1 ) ( 4 a c - b 2 ) I n - 1 2 8a(n+1)I_{n+\frac{1}{2}}=2(2ax+b)(ax^{2}+bx+c)^{n+\frac{1}{2}}+(2n+1)(4ac-b^{2% })I_{n-\frac{1}{2}}\,\!
  82. I n = 1 ( a x 2 + b x + c ) n d x I_{n}=\int\frac{1}{(ax^{2}+bx+c)^{n}}dx\,\!
  83. ( 2 n - 1 ) ( 4 a c - b 2 ) I n + 1 2 = 2 ( 2 a x + b ) ( a x 2 + b x + c ) n - 1 2 + 8 a ( n - 1 ) I n - 1 2 (2n-1)(4ac-b^{2})I_{n+\frac{1}{2}}=\frac{2(2ax+b)}{(ax^{2}+bx+c)^{n-\frac{1}{2% }}}+{8a(n-1)}I_{n-\frac{1}{2}}\,\!
  84. I n + 1 2 = I 2 n + 1 2 = 1 ( a x 2 + b x + c ) 2 n + 1 2 d x = 1 ( a x 2 + b x + c ) 2 n + 1 d x I_{n+\frac{1}{2}}=I_{\frac{2n+1}{2}}=\int\frac{1}{(ax^{2}+bx+c)^{\frac{2n+1}{2% }}}dx=\int\frac{1}{\sqrt{(ax^{2}+bx+c)^{2n+1}}}dx\,\!
  85. I n = x n sin a x d x I_{n}=\int x^{n}\sin{ax}dx\,\!
  86. a 2 I n = - a x n cos a x + n x n - 1 sin a x - n ( n - 1 ) I n - 2 a^{2}I_{n}=-ax^{n}\cos{ax}+nx^{n-1}\sin{ax}-n(n-1)I_{n-2}\,\!
  87. J n = x n cos a x d x J_{n}=\int x^{n}\cos{ax}dx\,\!
  88. a 2 J n = a x n sin a x + n x n - 1 cos a x - n ( n - 1 ) J n - 2 a^{2}J_{n}=ax^{n}\sin{ax}+nx^{n-1}\cos{ax}-n(n-1)J_{n-2}\,\!
  89. I n = sin a x x n d x I_{n}=\int\frac{\sin{ax}}{x^{n}}dx\,\!
  90. J n = cos a x x n d x J_{n}=\int\frac{\cos{ax}}{x^{n}}dx\,\!
  91. I n = - sin a x ( n - 1 ) x n - 1 + a n - 1 J n - 1 I_{n}=-\frac{\sin{ax}}{(n-1)x^{n-1}}+\frac{a}{n-1}J_{n-1}\,\!
  92. J n = - cos a x ( n - 1 ) x n - 1 - a n - 1 I n - 1 J_{n}=-\frac{\cos{ax}}{(n-1)x^{n-1}}-\frac{a}{n-1}I_{n-1}\,\!
  93. J n - 1 = - cos a x ( n - 2 ) x n - 2 - a n - 2 I n - 2 J_{n-1}=-\frac{\cos{ax}}{(n-2)x^{n-2}}-\frac{a}{n-2}I_{n-2}\,\!
  94. I n = - sin a x ( n - 1 ) x n - 1 - a n - 1 [ cos a x ( n - 2 ) x n - 2 + a n - 2 I n - 2 ] I_{n}=-\frac{\sin{ax}}{(n-1)x^{n-1}}-\frac{a}{n-1}\left[\frac{\cos{ax}}{(n-2)x% ^{n-2}}+\frac{a}{n-2}I_{n-2}\right]\,\!
  95. I n = - sin a x ( n - 1 ) x n - 1 - a ( n - 1 ) ( n - 2 ) ( cos a x x n - 2 + a I n - 2 ) \therefore I_{n}=-\frac{\sin{ax}}{(n-1)x^{n-1}}-\frac{a}{(n-1)(n-2)}\left(% \frac{\cos{ax}}{x^{n-2}}+aI_{n-2}\right)\,\!
  96. I n - 1 = - sin a x ( n - 2 ) x n - 2 + a n - 2 J n - 2 I_{n-1}=-\frac{\sin{ax}}{(n-2)x^{n-2}}+\frac{a}{n-2}J_{n-2}\,\!
  97. J n = - cos a x ( n - 1 ) x n - 1 - a n - 1 [ - sin a x ( n - 2 ) x n - 2 + a n - 2 J n - 2 ] J_{n}=-\frac{\cos{ax}}{(n-1)x^{n-1}}-\frac{a}{n-1}\left[-\frac{\sin{ax}}{(n-2)% x^{n-2}}+\frac{a}{n-2}J_{n-2}\right]\,\!
  98. J n = - cos a x ( n - 1 ) x n - 1 - a ( n - 1 ) ( n - 2 ) ( - sin a x x n - 2 + a J n - 2 ) \therefore J_{n}=-\frac{\cos{ax}}{(n-1)x^{n-1}}-\frac{a}{(n-1)(n-2)}\left(-% \frac{\sin{ax}}{x^{n-2}}+aJ_{n-2}\right)\,\!
  99. I n = sin n a x d x I_{n}=\int\sin^{n}{ax}dx\,\!
  100. a n I n = - sin n - 1 a x cos a x + a ( n - 1 ) I n - 2 anI_{n}=-\sin^{n-1}{ax}\cos{ax}+a(n-1)I_{n-2}\,\!
  101. J n = cos n a x d x J_{n}=\int\cos^{n}{ax}dx\,\!
  102. a n J n = sin a x cos n - 1 a x + a ( n - 1 ) J n - 2 anJ_{n}=\sin{ax}\cos^{n-1}{ax}+a(n-1)J_{n-2}\,\!
  103. I n = d x sin n a x I_{n}=\int\frac{dx}{\sin^{n}{ax}}\,\!
  104. ( n - 1 ) I n = - cos a x a sin n - 1 a x + ( n - 2 ) I n - 2 (n-1)I_{n}=-\frac{\cos{ax}}{a\sin^{n-1}{ax}}+(n-2)I_{n-2}\,\!
  105. J n = d x cos n a x J_{n}=\int\frac{dx}{\cos^{n}{ax}}\,\!
  106. ( n - 1 ) J n = sin a x a cos n - 1 a x + ( n - 2 ) I n - 2 (n-1)J_{n}=\frac{\sin{ax}}{a\cos^{n-1}{ax}}+(n-2)I_{n-2}\,\!
  107. I m , n = sin m a x cos n a x d x I_{m,n}=\int\sin^{m}{ax}\cos^{n}{ax}dx\,\!
  108. I m , n = { - sin m - 1 a x cos n + 1 a x a ( m + n ) + m - 1 m + n I m - 2 , n sin m + 1 a x cos n - 1 a x a ( m + n ) + n - 1 m + n I m , n - 2 I_{m,n}=\begin{cases}-\frac{\sin^{m-1}{ax}\cos^{n+1}{ax}}{a(m+n)}+\frac{m-1}{m% +n}I_{m-2,n}\\ \frac{\sin^{m+1}{ax}\cos^{n-1}{ax}}{a(m+n)}+\frac{n-1}{m+n}I_{m,n-2}\\ \end{cases}\,\!
  109. I m , n = d x sin m a x cos n a x I_{m,n}=\int\frac{dx}{\sin^{m}{ax}\cos^{n}{ax}}\,\!
  110. I m , n = { 1 a ( n - 1 ) sin m - 1 a x cos n - 1 a x + m + n - 2 n - 1 I m , n - 2 - 1 a ( m - 1 ) sin m - 1 a x cos n - 1 a x + m + n - 2 m - 1 I m - 2 , n I_{m,n}=\begin{cases}\frac{1}{a(n-1)\sin^{m-1}{ax}\cos^{n-1}{ax}}+\frac{m+n-2}% {n-1}I_{m,n-2}\\ -\frac{1}{a(m-1)\sin^{m-1}{ax}\cos^{n-1}{ax}}+\frac{m+n-2}{m-1}I_{m-2,n}\\ \end{cases}\,\!
  111. I m , n = sin m a x cos n a x d x I_{m,n}=\int\frac{\sin^{m}{ax}}{\cos^{n}{ax}}dx\,\!
  112. I m , n = { sin m - 1 a x a ( n - 1 ) cos n - 1 a x - m - 1 n - 1 I m - 2 , n - 2 sin m + 1 a x a ( n - 1 ) cos n - 1 a x - m - n + 2 n - 1 I m , n - 2 sin m - 1 a x a ( m - n ) cos n - 1 a x + m - 1 m - n I m - 2 , n I_{m,n}=\begin{cases}\frac{\sin^{m-1}{ax}}{a(n-1)\cos^{n-1}{ax}}-\frac{m-1}{n-% 1}I_{m-2,n-2}\\ \frac{\sin^{m+1}{ax}}{a(n-1)\cos^{n-1}{ax}}-\frac{m-n+2}{n-1}I_{m,n-2}\\ \frac{\sin^{m-1}{ax}}{a(m-n)\cos^{n-1}{ax}}+\frac{m-1}{m-n}I_{m-2,n}\\ \end{cases}\,\!
  113. I m , n = cos m a x sin n a x d x I_{m,n}=\int\frac{\cos^{m}{ax}}{\sin^{n}{ax}}dx\,\!
  114. I m , n = { - cos m - 1 a x a ( n - 1 ) sin n - 1 a x - m - 1 n - 1 I m - 2 , n - 2 - cos m + 1 a x a ( n - 1 ) sin n - 1 a x - m - n + 2 n - 1 I m , n - 2 cos m - 1 a x a ( m - n ) sin n - 1 a x + m - 1 m - n I m - 2 , n I_{m,n}=\begin{cases}-\frac{\cos^{m-1}{ax}}{a(n-1)\sin^{n-1}{ax}}-\frac{m-1}{n% -1}I_{m-2,n-2}\\ -\frac{\cos^{m+1}{ax}}{a(n-1)\sin^{n-1}{ax}}-\frac{m-n+2}{n-1}I_{m,n-2}\\ \frac{\cos^{m-1}{ax}}{a(m-n)\sin^{n-1}{ax}}+\frac{m-1}{m-n}I_{m-2,n}\\ \end{cases}\,\!
  115. I n = x n e a x d x I_{n}=\int x^{n}e^{ax}dx\,\!
  116. n > 0 n>0\,\!
  117. I n = x n e a x a - n a I n - 1 I_{n}=\frac{x^{n}e^{ax}}{a}-\frac{n}{a}I_{n-1}\,\!
  118. I n = x - n e a x d x I_{n}=\int x^{-n}e^{ax}dx\,\!
  119. n > 0 n>0\,\!
  120. n 1 n\neq 1\,\!
  121. I n = - e a x ( n - 1 ) x n - 1 + a n - 1 I n - 1 I_{n}=\frac{-e^{ax}}{(n-1)x^{n-1}}+\frac{a}{n-1}I_{n-1}\,\!
  122. I n = e a x sin n b x d x I_{n}=\int e^{ax}\sin^{n}{bx}dx\,\!
  123. I n = e a x sin n - 1 b x a 2 + ( b n ) 2 ( a sin b x - b n cos b x ) + n ( n - 1 ) b 2 a 2 + ( b n ) 2 I n - 2 I_{n}=\frac{e^{ax}\sin^{n-1}{bx}}{a^{2}+(bn)^{2}}\left(a\sin bx-bn\cos bx% \right)+\frac{n(n-1)b^{2}}{a^{2}+(bn)^{2}}I_{n-2}\,\!
  124. I n = e a x cos n b x d x I_{n}=\int e^{ax}\cos^{n}{bx}dx\,\!
  125. I n = e a x cos n - 1 b x a 2 + ( b n ) 2 ( a cos b x + b n sin b x ) + n ( n - 1 ) b 2 a 2 + ( b n ) 2 I n - 2 I_{n}=\frac{e^{ax}\cos^{n-1}{bx}}{a^{2}+(bn)^{2}}\left(a\cos bx+bn\sin bx% \right)+\frac{n(n-1)b^{2}}{a^{2}+(bn)^{2}}I_{n-2}\,\!

Integration_using_Euler's_formula.html

  1. e i x = cos x + i sin x . e^{ix}=\cos x+i\,\sin x.
  2. e - i x = cos x - i sin x . e^{-ix}=\cos x-i\,\sin x.
  3. cos x = e i x + e - i x 2 and sin x = e i x - e - i x 2 i . \cos x=\frac{e^{ix}+e^{-ix}}{2}\quad\,\text{and}\quad\sin x=\frac{e^{ix}-e^{-% ix}}{2i}.
  4. cos 2 x d x . \int\cos^{2}x\,dx.
  5. cos 2 x d x \displaystyle\int\cos^{2}x\,dx
  6. 1 4 ( e 2 i x + 2 + e - 2 i x ) d x \displaystyle\frac{1}{4}\int\left(e^{2ix}+2+e^{-2ix}\right)dx
  7. sin 2 x cos 4 x d x . \int\sin^{2}x\cos 4x\,dx.
  8. sin 2 x cos 4 x d x \displaystyle\int\sin^{2}x\cos 4x\,dx
  9. sin 2 x cos 4 x d x = - 1 24 sin 6 x + 1 8 sin 4 x - 1 8 sin 2 x + C . \int\sin^{2}x\cos 4x\,dx\,=\,-\frac{1}{24}\sin 6x+\frac{1}{8}\sin 4x-\frac{1}{% 8}\sin 2x+C.
  10. e x cos x d x . \int e^{x}\cos x\,dx.
  11. e x cos x d x = Re e x e i x d x . \int e^{x}\cos x\,dx\,=\,\operatorname{Re}\int e^{x}e^{ix}\,dx.
  12. e x e i x d x = e ( 1 + i ) x d x = e ( 1 + i ) x 1 + i + C . \int e^{x}e^{ix}\,dx\,=\,\int e^{(1+i)x}\,dx\,=\,\frac{e^{(1+i)x}}{1+i}+C.
  13. e x cos x d x \displaystyle\int e^{x}\cos x\,dx
  14. 1 + cos 2 x cos x + cos 3 x d x . \int\frac{1+\cos^{2}x}{\cos x+\cos 3x}\,dx.
  15. 1 2 6 + e 2 i x + e - 2 i x e i x + e - i x + e 3 i x + e - 3 i x d x . \frac{1}{2}\int\frac{6+e^{2ix}+e^{-2ix}}{e^{ix}+e^{-ix}+e^{3ix}+e^{-3ix}}\,dx.
  16. 1 2 i 1 + 6 u 2 + u 4 1 + u 2 + u 4 + u 6 d u . \frac{1}{2i}\int\frac{1+6u^{2}+u^{4}}{1+u^{2}+u^{4}+u^{6}}\,du.

Integration_using_parametric_derivatives.html

  1. 0 x 2 e - 3 x d x . \int_{0}^{\infty}x^{2}e^{-3x}\,dx.
  2. 0 e - t x d x = [ e - t x - t ] 0 = ( lim x e - t x - t ) - ( e - t 0 - t ) \displaystyle\int_{0}^{\infty}e^{-tx}\,dx=\left[\frac{e^{-tx}}{-t}\right]_{0}^% {\infty}=\left(\lim_{x\to\infty}\frac{e^{-tx}}{-t}\right)-\left(\frac{e^{-t0}}% {-t}\right)
  3. 0 e - t x d x = 1 t , \int_{0}^{\infty}e^{-tx}\,dx=\frac{1}{t},
  4. d 2 d t 2 0 e - t x d x = d 2 d t 2 1 t \displaystyle\frac{d^{2}}{dt^{2}}\int_{0}^{\infty}e^{-tx}\,dx=\frac{d^{2}}{dt^% {2}}\frac{1}{t}
  5. 0 x 2 e - 3 x d x = 2 3 3 = 2 27 . \int_{0}^{\infty}x^{2}e^{-3x}\,dx=\frac{2}{3^{3}}=\frac{2}{27}.

Intensity_(heat_transfer).html

  1. I I
  2. d q = I d ω cos θ d A dq=I\,d\omega\,\cos\theta\,dA
  3. d A dA
  4. d q dq
  5. d A dA
  6. d ω d\omega
  7. d A a dA_{a}
  8. θ \theta
  9. q = ϕ = 0 2 π θ = 0 π / 2 I cos θ sin θ d θ d ϕ q=\int_{\phi=0}^{2\pi}\int_{\theta=0}^{\pi/2}I\cos\theta\sin\theta d\theta d\phi
  10. E = π I E=\pi I
  11. π \pi
  12. θ \theta
  13. 2 π 2\pi
  14. I λ I_{\lambda}

Interbilayer_forces_in_membrane_fusion.html

  1. V R = C R exp [ - z λ R ] V_{R}=C_{R}\cdot\exp\!\left[{-z\over\lambda_{R}}\right]
  2. V A = C A exp [ - z λ A ] V_{A}=C_{A}\cdot\exp\!\left[{-z\over\lambda_{A}}\right]
  3. V V D W = - H 12 π * ( 1 z 2 - 2 ( z + D ) 2 + 1 ( z + 2 D ) 2 ) V_{VDW}=-{H\over 12\pi}*\left({1\over z^{2}}-{2\over(z+D)^{2}}+{1\over(z+2D)^{% 2}}\right)

International_Geomagnetic_Reference_Field.html

  1. V ( r , ϕ , θ ) = a = 1 L m = 0 ( a r ) + 1 ( g m cos m ϕ + h m sin m ϕ ) P m ( cos θ ) V(r,\phi,\theta)=a\sum_{\ell=1}^{L}\sum_{m=0}^{\ell}\left(\frac{a}{r}\right)^{% \ell+1}\left(g_{\ell}^{m}\cos m\phi+h_{\ell}^{m}\sin m\phi\right)P_{\ell}^{m}% \left(\cos\theta\right)
  2. r r
  3. L L
  4. ϕ \phi
  5. θ \theta
  6. a a
  7. g m g_{\ell}^{m}
  8. h m h_{\ell}^{m}
  9. P m ( cos θ ) P_{\ell}^{m}\left(\cos\theta\right)
  10. l l
  11. m m

Interpolation_space.html

  1. r r
  2. p p
  3. q q
  4. X X
  5. Z Z
  6. X X
  7. Z Z
  8. X X
  9. Z Z
  10. Z Z
  11. Z Z
  12. X 0 X 1 X_{0}\cap X_{1}
  13. X 0 + X 1 = { z Z : z = x 0 + x 1 , x 0 X 0 , x 1 X 1 } . X_{0}+X_{1}=\left\{z\in Z:z=x_{0}+x_{1},\ x_{0}\in X_{0},\,x_{1}\in X_{1}% \right\}.
  14. Z Z
  15. x X 0 X 1 := max ( x X 0 , x X 1 ) , \|x\|_{X_{0}\cap X_{1}}:=\max\left(\left\|x\right\|_{X_{0}},\left\|x\right\|_{% X_{1}}\right),
  16. x X 0 + X 1 := inf { x 0 X 0 + x 1 X 1 : x = x 0 + x 1 , x 0 X 0 , x 1 X 1 } . \|x\|_{X_{0}+X_{1}}:=\inf\left\{\left\|x_{0}\right\|_{X_{0}}+\left\|x_{1}% \right\|_{X_{1}}\ :\ x=x_{0}+x_{1},\;x_{0}\in X_{0},\;x_{1}\in X_{1}\right\}.
  17. X 0 X 1 X 0 , X 1 X 0 + X 1 . X_{0}\cap X_{1}\subset X_{0},\ X_{1}\subset X_{0}+X_{1}.
  18. X X
  19. X 0 X 1 X X 0 + X 1 , X_{0}\cap X_{1}\subset X\subset X_{0}+X_{1},
  20. Z Z
  21. 1 p 1≤p≤∞
  22. L p 0 ( 𝐑 ) L p 1 ( 𝐑 ) L p ( 𝐑 ) L p 0 ( 𝐑 ) + L p 1 ( 𝐑 ) , when 1 p 0 p p 1 , L^{p_{0}}(\mathbf{R})\cap L^{p_{1}}(\mathbf{R})\subset L^{p}(\mathbf{R})% \subset L^{p_{0}}(\mathbf{R})+L^{p_{1}}(\mathbf{R}),\ \ \,\text{when}\ \ 1\leq p% _{0}\leq p\leq p_{1}\leq\infty,
  23. ( X , Y ) (X,Y)
  24. ( X , Y ) (X,Y)
  25. θ θ
  26. ( X , Y ) (X,Y)
  27. θ θ
  28. ( X 0 , X 1 ) \mathcal{F}(X_{0},X_{1})
  29. f : 𝐂 X < s u b > 0 + X 1 f:\mathbf{C}→X<sub>0+X_{1}
  30. = =
  31. ( X 0 , X 1 ) \mathcal{F}(X_{0},X_{1})
  32. f ( X 0 , X 1 ) = max { sup t 𝐑 f ( i t ) X 0 , sup t 𝐑 f ( 1 + i t ) X 1 } . \|f\|_{\mathcal{F}(X_{0},X_{1})}=\max\left\{\sup_{t\in\mathbf{R}}\|f(it)\|_{X_% {0}},\;\sup_{t\in\mathbf{R}}\|f(1+it)\|_{X_{1}}\right\}.
  33. T T
  34. T θ T 0 1 - θ T 1 θ . \|T\|_{\theta}\leq\|T\|_{0}^{1-\theta}\|T\|_{1}^{\theta}.
  35. ( R , Σ , μ ) (R,Σ,μ)
  36. x x
  37. μ μ
  38. σ σ
  39. s > 0 s>0
  40. t t
  41. x 𝐑 x∈\mathbf{R}
  42. f 1 ( x ) = { f ( x ) | f ( x ) | < s , s f ( x ) | f ( x ) | otherwise f_{1}(x)=\begin{cases}f(x)&|f(x)|<s,\\ \frac{sf(x)}{|f(x)|}&\,\text{otherwise}\end{cases}
  43. s s
  44. K ( f , t ; L 1 , L ) = 0 t f * ( u ) d u , K\left(f,t;L^{1},L^{\infty}\right)=\int_{0}^{t}f^{*}(u)\,du,
  45. f f
  46. t > 0 t>0
  47. J ( x , t ; X 0 , X 1 ) = max ( x X 0 , t x X 1 ) . J(x,t;X_{0},X_{1})=\max\left(\|x\|_{X_{0}},t\|x\|_{X_{1}}\right).
  48. x x
  49. x = 0 v ( t ) d t t , x=\int_{0}^{\infty}v(t)\,\frac{dt}{t},
  50. v ( t ) v(t)
  51. Φ ( v ) = ( 0 ( t - θ J ( v ( t ) , t ; X 0 , X 1 ) ) q d t t ) 1 q < . \Phi(v)=\left(\int_{0}^{\infty}\left(t^{-\theta}J(v(t),t;X_{0},X_{1})\right)^{% q}\,\tfrac{dt}{t}\right)^{\frac{1}{q}}<\infty.
  52. x x
  53. x θ , q ; J := inf v { Φ ( v ) : x = 0 v ( t ) d t t } . \|x\|_{\theta,q;J}:=\inf_{v}\left\{\Phi(v)\ :\ x=\int_{0}^{\infty}v(t)\,\tfrac% {dt}{t}\right\}.
  54. J θ , q ( X 0 , X 1 ) = K θ , q ( X 0 , X 1 ) , J_{\theta,q}(X_{0},X_{1})=K_{\theta,q}(X_{0},X_{1}),
  55. q q
  56. ( X 0 , X 1 ) θ , q = ( X 1 , X 0 ) 1 - θ , q , 0 < θ < 1 , 1 q . (X_{0},X_{1})_{\theta,q}=(X_{1},X_{0})_{1-\theta,q},\qquad 0<\theta<1,1\leq q% \leq\infty.
  57. θ θ
  58. q q
  59. ( X 0 , X 1 ) θ , q ( X 0 , X 1 ) θ , r . (X_{0},X_{1})_{\theta,q}\subset(X_{0},X_{1})_{\theta,r}.
  60. θ θ
  61. 0 , 11 0,11
  62. ( θ , ) (θ,∞)
  63. sup { | f ( u ) | , | f ( u ) - f ( v ) | 1 + t - 1 | u - v | : u , v [ 0 , 1 ] } . \sup\left\{|f(u)|,\,\frac{|f(u)-f(v)|}{1+t^{-1}|u-v|}\ :\ u,v\in[0,1]\right\}.
  64. 0 θ 1 0≤θ≤1
  65. ( ( X 0 , X 1 ) θ 0 , ( X 0 , X 1 ) θ 1 ) θ = ( X 0 , X 1 ) η , η = ( 1 - θ ) θ 0 + θ θ 1 . \left(\left(X_{0},X_{1}\right)_{\theta_{0}},\left(X_{0},X_{1}\right)_{\theta_{% 1}}\right)_{\theta}=(X_{0},X_{1})_{\eta},\qquad\eta=(1-\theta)\theta_{0}+% \theta\theta_{1}.
  66. j = 0 , 1 j=0,1
  67. ( ( X 0 , X 1 ) θ ) = ( X 0 , X 1 ) θ , 0 < θ < 1. ((X_{0},X_{1})_{\theta})^{\prime}=(X^{\prime}_{0},X^{\prime}_{1})_{\theta},% \quad 0<\theta<1.
  68. n n
  69. q q
  70. 1 p + 1 q = 1 \frac{1}{p}+\frac{1}{q}=1
  71. x θ , p ; J inf { ( n 𝐙 ( 2 θ n max ( x n X 0 , 2 - n x n X 1 ) ) p ) 1 p : x = n 𝐙 x n } . \|x^{\prime}\|_{\theta,p;J}\simeq\inf\left\{\left(\sum_{n\in\mathbf{Z}}\left(2% ^{\theta n}\max\left(\left\|x^{\prime}_{n}\right\|_{X^{\prime}_{0}},2^{-n}% \left\|x^{\prime}_{n}\right\|_{X^{\prime}_{1}}\right)\right)^{p}\right)^{\frac% {1}{p}}\ :\ x^{\prime}=\sum_{n\in\mathbf{Z}}x^{\prime}_{n}\right\}.
  72. n n
  73. n −n
  74. T T
  75. Y Y
  76. Y Y
  77. T T
  78. Y Y
  79. a n , b n > 0 , n = 1 min ( a n , b n ) < . a_{n},b_{n}>0,\ \ \sum_{n=1}^{\infty}\min(a_{n},b_{n})<\infty.
  80. x x
  81. x K ( X 0 , X 1 ) = sup m 1 n = 1 m a n K ( x , b n a n ; X 0 , X 1 ) y n Y < , \|x\|_{K(X_{0},X_{1})}=\sup_{m\geq 1}\left\|\sum_{n=1}^{m}a_{n}K\left(x,\tfrac% {b_{n}}{a_{n}};X_{0},X_{1}\right)\,y_{n}\right\|_{Y}<\infty,
  82. Y Y
  83. H p s , s 𝐑 , 1 p ; B p , q s , s 𝐑 , 1 p , q . H^{s}_{p},\ \ s\in\mathbf{R},\ 1\leq p\leq\infty\,;\quad B^{s}_{p,q},\ \ s\in% \mathbf{R},\ 1\leq p,q\leq\infty.
  84. s 0 s≥0
  85. 0 < θ < 1 , 1 p , p 0 , p 1 , q , q 0 , q 1 , s , s 0 , s 1 𝐑 , 0<\theta<1,\ \ 1\leq p,p_{0},p_{1},q,q_{0},q_{1}\leq\infty,\ \ s,s_{0},s_{1}% \in\mathbf{R},
  86. 1 p θ = 1 - θ p 0 + θ p 1 , 1 q θ = 1 - θ q 0 + θ q 1 , s θ = ( 1 - θ ) s 0 + θ s 1 . \frac{1}{p_{\theta}}=\frac{1-\theta}{p_{0}}+\frac{\theta}{p_{1}},\ \frac{1}{q_% {\theta}}=\frac{1-\theta}{q_{0}}+\frac{\theta}{q_{1}},\ \ s_{\theta}=(1-\theta% )s_{0}+\theta s_{1}.
  87. H p s H^{s}_{p}
  88. ( H p 0 s 0 , H p 1 s 1 ) θ = H p θ s θ , s 0 s 1 , 1 < p 0 , p 1 < , \left(H^{s_{0}}_{p_{0}},H^{s_{1}}_{p_{1}}\right)_{\theta}=H^{s_{\theta}}_{p_{% \theta}},\qquad s_{0}\neq s_{1},\ 1<p_{0},p_{1}<\infty,
  89. ( B p 0 , q 0 s 0 , B p 1 , q 1 s 1 ) θ = B p θ , q θ s θ , s 0 s 1 , 1 p 0 , p 1 , q 0 , q 1 . \left(B^{s_{0}}_{p_{0},q_{0}},B^{s_{1}}_{p_{1},q_{1}}\right)_{\theta}=B^{s_{% \theta}}_{p_{\theta},q_{\theta}},\qquad s_{0}\neq s_{1},\ 1\leq p_{0},p_{1},q_% {0},q_{1}\leq\infty.
  90. ( H p 0 s , H p 1 s ) θ , p θ = H p θ s , 1 p 0 , p 1 . \left(H^{s}_{p_{0}},H^{s}_{p_{1}}\right)_{\theta,p_{\theta}}=H^{s}_{p_{\theta}% },\qquad 1\leq p_{0},p_{1}\leq\infty.
  91. ( H p s 0 , H p s 1 ) θ , q = B p , q s θ , s 0 s 1 , 1 p , q . \left(H^{s_{0}}_{p},H^{s_{1}}_{p}\right)_{\theta,q}=B^{s_{\theta}}_{p,q},% \qquad s_{0}\neq s_{1},\ 1\leq p,q\leq\infty.
  92. ( B p , q 0 s 0 , B p , q 1 s 1 ) θ , q = B p , q s θ , s 0 s 1 , 1 p , q , q 0 , q 1 , \left(B^{s_{0}}_{p,q_{0}},B^{s_{1}}_{p,q_{1}}\right)_{\theta,q}=B^{s_{\theta}}% _{p,q},\qquad s_{0}\neq s_{1},\ 1\leq p,q,q_{0},q_{1}\leq\infty,
  93. ( B p , q 0 s , B p , q 1 s ) θ , q = B p , q θ s , 1 p , q 0 , q 1 , \left(B^{s}_{p,q_{0}},B^{s}_{p,q_{1}}\right)_{\theta,q}=B^{s}_{p,q_{\theta}},% \qquad 1\leq p,q_{0},q_{1}\leq\infty,
  94. ( B p 0 , q 0 s 0 , B p 1 , q 1 s 1 ) θ , q θ = B p θ , q θ s θ , s 0 s 1 , p θ = q θ , 1 p 0 , p 1 , q 0 , q 1 . \left(B^{s_{0}}_{p_{0},q_{0}},B^{s_{1}}_{p_{1},q_{1}}\right)_{\theta,q_{\theta% }}=B^{s_{\theta}}_{p_{\theta},q_{\theta}},\qquad s_{0}\neq s_{1},\ p_{\theta}=% q_{\theta},\ 1\leq p_{0},p_{1},q_{0},q_{1}\leq\infty.

Interpretation_(model_theory).html

  1. ( n , f ) (n,f)
  2. f f
  3. f f
  4. f k f^{k}
  5. ( n , f ) (n,f)
  6. f f

Intersection_form_(4-manifold).html

  1. Q M : H 2 ( M ; ) × H 2 ( M ; ) Q_{M}\colon H^{2}(M;\mathbb{Z})\times H^{2}(M;\mathbb{Z})\to\mathbb{Z}
  2. Q M ( a , b ) = a b , [ M ] . Q_{M}(a,b)=\langle a\smile b,[M]\rangle.
  3. Q ( a , b ) = M α β Q(a,b)=\int_{M}\alpha\wedge\beta
  4. \wedge

Interval_boundary_element_method.html

  1. c u = Ω ( G u n - G n u ) d S c\cdot u=\int\limits_{\partial\Omega}\left(G\frac{\partial u}{\partial n}-% \frac{\partial G}{\partial n}u\right)dS
  2. u ~ ( x ) = { u ( x , p ) : c ( p ) u ( p ) = Ω ( G ( p ) u ( p ) n - G ( p ) n u ( p ) ) d S , p p ^ } \tilde{u}(x)=\{u(x,p):c(p)\cdot u(p)=\int\limits_{\partial\Omega}\left(G(p)% \frac{\partial u(p)}{\partial n}-\frac{\partial G(p)}{\partial n}u(p)\right)dS% ,p\in\hat{p}\}
  3. u ^ ( x ) = h u l l u ~ ( x ) = h u l l { u ( x , p ) : c ( p ) u ( p ) = Ω ( G ( p ) u ( p ) n - G ( p ) n u ( p ) ) d S , p p ^ } \hat{u}(x)=hull\ \tilde{u}(x)=hull\{u(x,p):c(p)\cdot u(p)=\int\limits_{% \partial\Omega}\left(G(p)\frac{\partial u(p)}{\partial n}-\frac{\partial G(p)}% {\partial n}u(p)\right)dS,p\in\hat{p}\}
  4. Ω \Omega

Interval_finite_element.html

  1. a x = b ax=b\,
  2. x = b a x=\frac{b}{a}
  3. a [ 1 , 2 ] = 𝐚 a\in[1,2]=\mathbf{a}
  4. b [ 1 , 4 ] = 𝐛 b\in[1,4]=\mathbf{b}
  5. [ 1 , 2 ] x = [ 1 , 4 ] [1,2]x=[1,4]
  6. 𝐱 = { x : a x = b , a 𝐚 , b 𝐛 } = 𝐛 𝐚 = [ 1 , 4 ] [ 1 , 2 ] = [ 0.5 , 4 ] \mathbf{x}=\left\{x:ax=b,a\in\mathbf{a},b\in\mathbf{b}\right\}=\frac{\mathbf{b% }}{\mathbf{a}}=\frac{[1,4]}{[1,2]}=[0.5,4]
  7. [ [ - 4 , - 3 ] [ - 2 , 2 ] [ - 2 , 2 ] [ - 4 , - 3 ] ] [ x 1 x 2 ] = [ [ - 8 , 8 ] [ - 8 , 8 ] ] \left[\begin{array}[]{cc}{[-4,-3]}&{[-2,2]}\\ {[-2,2]}&{[-4,-3]}\end{array}\right]\left[\begin{array}[]{c}x_{1}\\ x_{2}\end{array}\right]=\left[\begin{array}[]{c}{[-8,8]}\\ {[-8,8]}\end{array}\right]
  8. ( 𝐀 , 𝐛 ) = { x : A x = b , A 𝐀 , b 𝐛 } \sum{{}_{\exists\exists}}(\mathbf{A},\mathbf{b})=\{x:Ax=b,A\in\mathbf{A},b\in% \mathbf{b}\}
  9. ( ( 𝐀 , 𝐛 ) ) = { x : A x = b , A 𝐀 , b 𝐛 } \diamondsuit\left(\sum{{}_{\exists\exists}}(\mathbf{A},\mathbf{b})\right)=% \diamondsuit\{x:Ax=b,A\in\mathbf{A},b\in\mathbf{b}\}
  10. ( ( 𝐀 , 𝐛 ) ) = [ x ¯ 1 , x ¯ 1 ] × [ x ¯ 2 , x ¯ 2 ] × × [ x ¯ n , x ¯ n ] \diamondsuit\left(\sum{{}_{\exists\exists}}(\mathbf{A},\mathbf{b})\right)=[% \underline{x}_{1},\overline{x}_{1}]\times[\underline{x}_{2},\overline{x}_{2}]% \times...\times[\underline{x}_{n},\overline{x}_{n}]
  11. x ¯ i = m i n { x i : A x = b , A 𝐀 , b 𝐛 } , x ¯ i = m a x { x i : A x = b , A 𝐀 , b 𝐛 } \underline{x}_{i}=min\{x_{i}:Ax=b,A\in\mathbf{A},b\in\mathbf{b}\},\ \ % \overline{x}_{i}=max\{x_{i}:Ax=b,A\in\mathbf{A},b\in\mathbf{b}\}
  12. x i { x i : A x = b , A 𝐀 , b 𝐛 } = [ x ¯ i , x ¯ i ] x_{i}\in\{x_{i}:Ax=b,A\in\mathbf{A},b\in\mathbf{b}\}=[\underline{x}_{i},% \overline{x}_{i}]
  13. [ p 1 p 2 p 2 + 1 p 1 ] [ u 1 u 2 ] = [ p 1 + 6 p 2 5.0 2 p 1 - 6 ] , f o r p 1 [ 2 , 4 ] , p 2 [ - 2 , 1 ] . \left[\begin{array}[]{cc}p_{1}&p_{2}\\ p_{2}+1&p_{1}\end{array}\right]\left[\begin{array}[]{cc}u_{1}\\ u_{2}\end{array}\right]=\left[\begin{array}[]{c}\frac{p_{1}+6p_{2}}{5.0}\\ 2p_{1}-6\end{array}\right],\ \ \ for\ \ p_{1}\in[2,4],p_{2}\in[-2,1].
  14. [ 1 , 2 ] x = [ 1 , 4 ] [1,2]x=[1,4]
  15. x = [ 1 , 2 ] x=[1,2]
  16. a x = [ 1 , 2 ] [ 1 , 2 ] = [ 1 , 4 ] ax=[1,2][1,2]=[1,4]
  17. a = [ 1 , 4 ] a=[1,4]
  18. x = [ 1 , 1 ] x=[1,1]
  19. a x = [ 1 , 4 ] [ 1 , 1 ] = [ 1 , 4 ] ax=[1,4][1,1]=[1,4]
  20. a = [ 1 , 8 ] a=[1,8]
  21. ( 1 ) G ( x , u , p ) = 0 (1)\ \ \ G(x,u,p)=0
  22. p = ( p 1 , , p m ) 𝐩 p=(p_{1},\dots,p_{m})\in{\mathbf{p}}
  23. p i [ p ¯ i , p ¯ i ] = 𝐩 i , p_{i}\in[\underline{p}_{i},\overline{p}_{i}]={\mathbf{p}}_{i},
  24. 𝐩 = 𝐩 1 × 𝐩 2 × × 𝐩 m . {\mathbf{p}}={\mathbf{p}}_{1}\times{\mathbf{p}}_{2}\times\cdots\times{\mathbf{% p}}_{m}.
  25. k x 2 u x 2 + k y 2 u y 2 + q = 0 for x Ω k_{x}\frac{\partial^{2}u}{\partial x^{2}}+k_{y}\frac{\partial^{2}u}{\partial y% ^{2}}+q=0\,\text{ for }x\in\Omega
  26. u ( x ) = u * ( x ) for x Ω u(x)=u^{*}(x)\,\text{ for }x\in\partial\Omega
  27. k x , k y k_{x},k_{y}
  28. k x 𝐤 x , k y 𝐤 y k_{x}\in{\mathbf{k}}_{x},\ k_{y}\in{\mathbf{k}}_{y}
  29. u ~ ( x ) := { u ( x ) : G ( x , u , p ) = 0 , p 𝐩 } \tilde{u}(x):=\{u(x):G(x,u,p)=0,p\in{\mathbf{p}}\}
  30. u ~ ( x ) = { u ( x ) : k x 2 u x 2 + k y 2 u y 2 + q = 0 for x Ω , u ( x ) = u * ( x ) for x Ω , k x 𝐤 x , k y 𝐤 y } \tilde{u}(x)=\{u(x):k_{x}\frac{\partial^{2}u}{\partial x^{2}}+k_{y}\frac{% \partial^{2}u}{\partial y^{2}}+q=0\,\text{ for }x\in\Omega,u(x)=u^{*}(x)\,% \text{ for }x\in\partial\Omega,k_{x}\in{\mathbf{k}}_{x},\ k_{y}\in{\mathbf{k}}% _{y}\}
  31. u ~ \tilde{u}
  32. u ~ \tilde{u}
  33. 𝐮 ( x ) = u ~ ( x ) = { u ( x ) : G ( x , u , p ) = 0 , p 𝐩 } {\mathbf{u}}(x)=\lozenge\tilde{u}(x)=\lozenge\{u(x):G(x,u,p)=0,p\in{\mathbf{p}}\}
  34. 𝐮 ( x ) = { u ( x ) : k x 2 u x 2 + k y 2 u y 2 + q = 0 for x Ω , u ( x ) = u * ( x ) for x Ω , k x 𝐤 x , k y 𝐤 y } {\mathbf{u}}(x)=\lozenge\{u(x):k_{x}\frac{\partial^{2}u}{\partial x^{2}}+k_{y}% \frac{\partial^{2}u}{\partial y^{2}}+q=0\,\text{ for }x\in\Omega,u(x)=u^{*}(x)% \,\text{ for }x\in\partial\Omega,k_{x}\in{\mathbf{k}}_{x},\ k_{y}\in{\mathbf{k% }}_{y}\}
  35. K ( p ) u = Q ( p ) , p 𝐩 K(p)u=Q(p),\ \ \ p\in{\mathbf{p}}
  36. K K
  37. Q Q
  38. 𝐮 = { u : K ( p ) u = Q ( p ) , p 𝐩 } {\mathbf{u}}=\lozenge\{u:K(p)u=Q(p),p\in{\mathbf{p}}\}
  39. u ¯ i = min { u i : K ( p ) u = Q ( p ) , p 𝐩 } \underline{u}_{i}=\min\{u_{i}:K(p)u=Q(p),p\in{\mathbf{p}}\}
  40. u ¯ i = max { u i : K ( p ) u = Q ( p ) , p 𝐩 } \overline{u}_{i}=\max\{u_{i}:K(p)u=Q(p),p\in{\mathbf{p}}\}
  41. 𝐮 = 𝐮 1 × × 𝐮 n = [ u ¯ 1 , u ¯ 1 ] × × [ u ¯ n , u ¯ n ] \mathbf{u}=\mathbf{u}_{1}\times\cdots\times\mathbf{u}_{n}=[\underline{u}_{1},% \overline{u}_{1}]\times\cdots\times[\underline{u}_{n},\overline{u}_{n}]
  42. 𝐩 = [ p ¯ , p ¯ ] \mathbf{p}=[\underline{p},\overline{p}]
  43. p [ p ¯ , p ¯ ] p\in[\underline{p},\overline{p}]
  44. p ¯ \overline{p}
  45. p ¯ \underline{p}
  46. n \sqrt{n}
  47. 𝐩 = [ p ¯ , p ¯ ] \mathbf{p}=[\underline{p},\overline{p}]
  48. n 𝐩 = [ n p ¯ , n p ¯ ] n\mathbf{p}=[n\underline{p},n\overline{p}]
  49. n p ¯ - n p ¯ = n ( p ¯ - p ¯ ) = n Δ p n\overline{p}-n\underline{p}=n(\overline{p}-\underline{p})=n\Delta p
  50. m X = E [ X ] = p ¯ + p ¯ 2 , σ X = V a r [ X ] = Δ p 6 m_{X}=E[X]=\frac{\overline{p}+\underline{p}}{2},\sigma_{X}=\sqrt{Var[X]}=\frac% {\Delta p}{6}
  51. E [ n X ] = n p ¯ + p ¯ 2 , σ n X = n V a r [ X ] = n σ = n Δ p 6 E[nX]=n\frac{\overline{p}+\underline{p}}{2},\sigma_{nX}=\sqrt{nVar[X]}=\sqrt{n% }\sigma=\sqrt{n}\frac{\Delta p}{6}
  52. 6 σ n X = 6 n Δ p 6 = n Δ p 6\sigma_{nX}=6\sqrt{n}\frac{\Delta p}{6}=\sqrt{n}\Delta p
  53. w i d t h o f n i n t e r v a l s w i d t h o f n r a n d o m v a r i a b l e s = n Δ p n Δ p = n \frac{width\ of\ n\ intervals}{width\ of\ n\ random\ variables}=\frac{n\Delta p% }{\sqrt{n}\Delta p}=\sqrt{n}
  54. u u
  55. P P
  56. E A L u = P \frac{EA}{L}u=P
  57. L L
  58. A A
  59. E E
  60. E [ E ¯ , E ¯ ] , P [ P ¯ , P ¯ ] E\in[\underline{E},\overline{E}],P\in[\underline{P},\overline{P}]
  61. u u
  62. u E = - P L E 2 A < 0 \frac{\partial u}{\partial E}=\frac{-PL}{E^{2}A}<0
  63. u P = L E A > 0 \frac{\partial u}{\partial P}=\frac{L}{EA}>0
  64. u ¯ = u ( E ¯ , P ¯ ) = P ¯ L E ¯ A \underline{u}=u(\overline{E},\underline{P})=\frac{\underline{P}L}{\overline{E}A}
  65. u ¯ = u ( E ¯ , P ¯ ) = P ¯ L E ¯ A \overline{u}=u(\underline{E},\overline{P})=\frac{\overline{P}L}{\underline{E}A}
  66. ε = 1 L u \varepsilon=\frac{1}{L}u
  67. ε E = 1 L u E = - P E 2 A < 0 \frac{\partial\varepsilon}{\partial E}=\frac{1}{L}\frac{\partial u}{\partial E% }=\frac{-P}{E^{2}A}<0
  68. ε P = 1 L u P = 1 E A > 0 \frac{\partial\varepsilon}{\partial P}=\frac{1}{L}\frac{\partial u}{\partial P% }=\frac{1}{EA}>0
  69. ε ¯ = ε ( E ¯ , P ¯ ) = P ¯ E ¯ A \underline{\varepsilon}=\varepsilon(\overline{E},\underline{P})=\frac{% \underline{P}}{\overline{E}A}
  70. ε ¯ = ε ( E ¯ , P ¯ ) = P ¯ E ¯ A \overline{\varepsilon}=\varepsilon(\underline{E},\overline{P})=\frac{\overline% {P}}{\underline{E}A}
  71. ε u = 1 L > 0 \frac{\partial\varepsilon}{\partial u}=\frac{1}{L}>0
  72. ε ¯ = ε ( u ¯ ) = P ¯ E ¯ A \underline{\varepsilon}=\varepsilon(\underline{u})=\frac{\underline{P}}{% \overline{E}A}
  73. ε ¯ = ε ( u ¯ ) = P ¯ E ¯ A \overline{\varepsilon}=\varepsilon(\overline{u})=\frac{\overline{P}}{% \underline{E}A}
  74. σ = E ε \sigma=E\varepsilon
  75. σ E = ε + E ε E = ε + E 1 L u E = P E A - P E A = 0 \frac{\partial\sigma}{\partial E}=\varepsilon+E\frac{\partial\varepsilon}{% \partial E}=\varepsilon+E\frac{1}{L}\frac{\partial u}{\partial E}=\frac{P}{EA}% -\frac{P}{EA}=0
  76. σ P = E ε P = E 1 L u P = 1 A > 0 \frac{\partial\sigma}{\partial P}=E\frac{\partial\varepsilon}{\partial P}=E% \frac{1}{L}\frac{\partial u}{\partial P}=\frac{1}{A}>0
  77. σ ¯ = σ ( P ¯ ) = P ¯ A \underline{\sigma}=\sigma(\underline{P})=\frac{\underline{P}}{A}
  78. σ ¯ = σ ( P ¯ ) = P ¯ A \overline{\sigma}=\sigma(\overline{P})=\frac{\overline{P}}{A}
  79. σ ε = ε ( E ε ) = E > 0 \frac{\partial\sigma}{\partial\varepsilon}=\frac{\partial}{\partial\varepsilon% }(E\varepsilon)=E>0
  80. σ ¯ = σ ( ε ¯ ) = E ε ¯ = P ¯ A \underline{\sigma}=\sigma(\underline{\varepsilon})=E\underline{\varepsilon}=% \frac{\underline{P}}{A}
  81. σ ¯ = σ ( ε ¯ ) = E ε ¯ = P ¯ A \overline{\sigma}=\sigma(\overline{\varepsilon})=E\overline{\varepsilon}=\frac% {\overline{P}}{A}
  82. σ \sigma
  83. σ 0 \sigma_{0}
  84. σ < σ 0 \sigma<\sigma_{0}
  85. σ ¯ < σ 0 \overline{\sigma}<\sigma_{0}
  86. P ¯ A < σ 0 \frac{\overline{P}}{A}<\sigma_{0}
  87. d d x ( E A d u d x ) + n = 0 \frac{d}{dx}\left(EA\frac{du}{dx}\right)+n=0
  88. u u
  89. E E
  90. A A
  91. n n
  92. u ( 0 ) = 0 u(0)=0
  93. d u ( 0 ) d x E A = P \frac{du(0)}{dx}EA=P
  94. E E
  95. n n
  96. 𝐮 ( x ) = { u ( x ) : d d x ( E A d u d x ) + n = 0 , u ( 0 ) = 0 , d u ( 0 ) d x E A = P , E [ E ¯ , E ¯ ] , P [ P ¯ , P ¯ ] } {\mathbf{u}}(x)=\left\{u(x):\frac{d}{dx}\left(EA\frac{du}{dx}\right)+n=0,u(0)=% 0,\frac{du(0)}{dx}EA=P,E\in[\underline{E},\overline{E}],P\in[\underline{P},% \overline{P}]\right\}
  97. v v
  98. 0 L e ( d d x ( E A d u d x ) + n ) v = 0 \int\limits_{0}^{L^{e}}\left(\frac{d}{dx}\left(EA\frac{du}{dx}\right)+n\right)% v=0
  99. x [ 0 , L ( e ) ] . x\in[0,L^{(e)}].
  100. 0 L ( e ) E A d u d x d v d x d x = 0 L ( e ) n v d x \int\limits_{0}^{L^{(e)}}EA\frac{du}{dx}\frac{dv}{dx}dx=\int\limits_{0}^{L^{(e% )}}nvdx
  101. x [ 0 , L ( e ) ] . x\in[0,L^{(e)}].
  102. x 0 , x 1 , , x N e x_{0},x_{1},...,x_{Ne}
  103. N e Ne
  104. N 1 ( e ) ( x ) = 1 - 1 - x 0 ( e ) x 1 ( e ) - x 0 ( e ) , N 2 ( e ) ( x ) = 1 - x 0 ( e ) x 1 ( e ) - x 0 ( e ) . N_{1}^{(e)}(x)=1-\frac{1-x_{0}^{(e)}}{x_{1}^{(e)}-x_{0}^{(e)}},\ \ N_{2}^{(e)}% (x)=\frac{1-x_{0}^{(e)}}{x_{1}^{(e)}-x_{0}^{(e)}}.
  105. x [ x 0 ( e ) , x 1 ( e ) ] . x\in[x_{0}^{(e)},x_{1}^{(e)}].
  106. x 1 ( e ) x_{1}^{(e)}
  107. x 1 ( e ) x_{1}^{(e)}
  108. u h ( e ) ( x ) = u 1 e N 1 ( e ) ( x ) + u 2 e N 2 ( e ) ( x ) , v h ( e ) ( x ) = u 1 e N 1 ( e ) ( x ) + u 2 e N 2 ( e ) ( x ) u^{(e)}_{h}(x)=u^{e}_{1}N_{1}^{(e)}(x)+u^{e}_{2}N_{2}^{(e)}(x),\ \ v^{(e)}_{h}% (x)=u^{e}_{1}N_{1}^{(e)}(x)+u^{e}_{2}N_{2}^{(e)}(x)
  109. [ E ( e ) A ( e ) L ( e ) - E ( e ) A ( e ) L ( e ) - E ( e ) A ( e ) L ( e ) E ( e ) A ( e ) L ( e ) ] [ u 1 ( e ) u 2 ( e ) ] = [ 0 L ( e ) n N 1 ( e ) ( x ) d x 0 L ( e ) n N 2 ( e ) ( x ) d x ] \left[\begin{array}[]{cc}\frac{E^{(e)}A^{(e)}}{L^{(e)}}&-\frac{E^{(e)}A^{(e)}}% {L^{(e)}}\\ -\frac{E^{(e)}A^{(e)}}{L^{(e)}}&\frac{E^{(e)}A^{(e)}}{L^{(e)}}\\ \end{array}\right]\left[\begin{array}[]{c}u^{(e)}_{1}\\ u^{(e)}_{2}\end{array}\right]=\left[\begin{array}[]{c}\int\limits_{0}^{L^{(e)}% }nN_{1}^{(e)}(x)dx\\ \int\limits_{0}^{L^{(e)}}nN_{2}^{(e)}(x)dx\end{array}\right]
  110. K ( e ) u ( e ) = Q ( e ) K^{(e)}u^{(e)}=Q^{(e)}
  111. K u = Q Ku=Q
  112. K = [ K 11 ( 1 ) K 12 ( 1 ) 0 0 K 21 ( 1 ) K 22 ( 1 ) + K 11 ( 2 ) K 12 ( 2 ) 0 0 K 21 ( 2 ) K 22 ( 2 ) + K 11 ( 3 ) 0 0 0 K 22 ( N e - 1 ) + K 11 ( N e ) K 11 ( N e ) 0 0 K 21 ( N e ) K 22 ( N e ) ] K=\left[\begin{array}[]{ccccc}K_{11}^{(1)}&K_{12}^{(1)}&0&...&0\\ K_{21}^{(1)}&K_{22}^{(1)}+K_{11}^{(2)}&K_{12}^{(2)}&...&0\\ 0&K_{21}^{(2)}&K_{22}^{(2)}+K_{11}^{(3)}&...&0\\ ...&...&...&...&...\\ 0&0&...&K_{22}^{(Ne-1)}+K_{11}^{(Ne)}&K_{11}^{(Ne)}\\ 0&0&...&K_{21}^{(Ne)}&K_{22}^{(Ne)}\end{array}\right]
  113. u = [ u 0 u 1 u N e ] u=\left[\begin{array}[]{c}u_{0}\\ u_{1}\\ ...\\ u_{Ne}\\ \end{array}\right]
  114. Q = [ Q 0 Q 1 Q N e ] Q=\left[\begin{array}[]{c}Q_{0}\\ Q_{1}\\ ...\\ Q_{Ne}\\ \end{array}\right]
  115. K = [ E ( 1 ) A ( 1 ) L ( 1 ) - E ( 1 ) A ( 1 ) L ( 1 ) 0 0 - E ( 1 ) A ( 1 ) L ( 1 ) E ( 1 ) A ( 1 ) L ( 1 ) + E ( 2 ) A ( 2 ) L ( 2 ) - E ( 2 ) A ( 2 ) L ( 2 ) 0 0 - E ( 2 ) A ( 2 ) L ( 2 ) E ( 2 ) A ( 2 ) L ( 2 ) + E ( 3 ) A ( 3 ) L ( 3 ) 0 0 0 E ( N e - 1 ) A ( N e - 1 ) L ( N e - 1 ) + E ( N e ) A ( N e ) L ( N e ) - E ( N e ) A ( N e ) L ( N e ) 0 0 - E ( N e ) A ( N e ) L ( N e ) E ( N e ) A ( N e ) L ( N e ) ] K=\left[\begin{array}[]{ccccc}\frac{E^{(1)}A^{(1)}}{L^{(1)}}&-\frac{E^{(1)}A^{% (1)}}{L^{(1)}}&0&...&0\\ -\frac{E^{(1)}A^{(1)}}{L^{(1)}}&\frac{E^{(1)}A^{(1)}}{L^{(1)}}+\frac{E^{(2)}A^% {(2)}}{L^{(2)}}&-\frac{E^{(2)}A^{(2)}}{L^{(2)}}&...&0\\ 0&-\frac{E^{(2)}A^{(2)}}{L^{(2)}}&\frac{E^{(2)}A^{(2)}}{L^{(2)}}+\frac{E^{(3)}% A^{(3)}}{L^{(3)}}&...&0\\ ...&...&...&...&...\\ 0&0&...&\frac{E^{(Ne-1)}A^{(Ne-1)}}{L^{(Ne-1)}}+\frac{E^{(Ne)}A^{(Ne)}}{L^{(Ne% )}}&-\frac{E^{(Ne)}A^{(Ne)}}{L^{(Ne)}}\\ 0&0&...&-\frac{E^{(Ne)}A^{(Ne)}}{L^{(Ne)}}&\frac{E^{(Ne)}A^{(Ne)}}{L^{(Ne)}}% \end{array}\right]
  116. n n
  117. Q = [ R 0 0 P ] Q=\left[\begin{array}[]{c}R\\ 0\\ ...\\ 0\\ P\\ \end{array}\right]
  118. K = [ 1 0 0 0 0 E ( 1 ) A ( 1 ) L ( 1 ) + E ( 2 ) A ( 2 ) L ( 2 ) - E ( 2 ) A ( 2 ) L ( 2 ) 0 0 - E ( 2 ) A ( 2 ) L ( 2 ) E ( 2 ) A ( 2 ) L ( 2 ) + E ( 3 ) A ( 3 ) L ( 3 ) 0 0 0 E ( e - 1 ) A ( e - 1 ) L ( e - 1 ) + E ( e ) A ( e ) L ( e ) - E ( e ) A ( e ) L ( e ) 0 0 - E ( e ) A ( e ) L ( e ) E ( e ) A ( e ) L ( e ) ] = K ( E , A ) = K ( E ( 1 ) , , E ( N e ) , A ( 1 ) , , A ( N e ) ) K=\left[\begin{array}[]{ccccc}1&0&0&...&0\\ 0&\frac{E^{(1)}A^{(1)}}{L^{(1)}}+\frac{E^{(2)}A^{(2)}}{L^{(2)}}&-\frac{E^{(2)}% A^{(2)}}{L^{(2)}}&...&0\\ 0&-\frac{E^{(2)}A^{(2)}}{L^{(2)}}&\frac{E^{(2)}A^{(2)}}{L^{(2)}}+\frac{E^{(3)}% A^{(3)}}{L^{(3)}}&...&0\\ ...&...&...&...&...\\ 0&0&...&\frac{E^{(e-1)}A^{(e-1)}}{L^{(e-1)}}+\frac{E^{(e)}A^{(e)}}{L^{(e)}}&-% \frac{E^{(e)}A^{(e)}}{L^{(e)}}\\ 0&0&...&-\frac{E^{(e)}A^{(e)}}{L^{(e)}}&\frac{E^{(e)}A^{(e)}}{L^{(e)}}\end{% array}\right]=K(E,A)=K(E^{(1)},...,E^{(Ne)},A^{(1)},...,A^{(Ne)})
  119. Q = [ 0 0 0 P ] = Q ( P ) Q=\left[\begin{array}[]{c}0\\ 0\\ ...\\ 0\\ P\\ \end{array}\right]=Q(P)
  120. E E
  121. A A
  122. P P
  123. E ( e ) [ E ¯ ( e ) , E ¯ ( e ) ] E^{(e)}\in[\underline{E}^{(e)},\overline{E}^{(e)}]
  124. A ( e ) [ A ¯ ( e ) , A ¯ ( e ) ] A^{(e)}\in[\underline{A}^{(e)},\overline{A}^{(e)}]
  125. P [ P ¯ , P ¯ ] P\in[\underline{P},\overline{P}]
  126. 𝐮 = { u : K ( E , A ) u = Q ( P ) , E ( e ) [ E ¯ ( e ) , E ¯ ( e ) ] , A ( e ) [ A ¯ ( e ) , A ¯ ( e ) ] , P [ P ¯ , P ¯ ] } {\mathbf{u}}=\lozenge\{u:K(E,A)u=Q(P),E^{(e)}\in[\underline{E}^{(e)},\overline% {E}^{(e)}],A^{(e)}\in[\underline{A}^{(e)},\overline{A}^{(e)}],P\in[\underline{% P},\overline{P}]\}
  127. 𝐮 {\mathbf{u}}
  128. u i [ u ¯ i , u ¯ i ] u_{i}\in[\underline{u}_{i},\overline{u}_{i}]
  129. u i < u i m a x u_{i}<u^{max}_{i}
  130. u i < u i m a x , u i [ u ¯ i , u ¯ i ] u_{i}<u^{max}_{i},\ \ \ u_{i}\in[\underline{u}_{i},\overline{u}_{i}]
  131. u ¯ i < u i m a x \overline{u}_{i}<u^{max}_{i}
  132. K ( p ) u ( p ) = Q ( p ) K(p)u(p)=Q(p)
  133. p ^ \hat{p}
  134. p ^ \hat{p}
  135. L = { p 1 * , , p n * } L=\{p^{*}_{1},...,p^{*}_{n}\}
  136. u ¯ i = m i n { u i ( p k * ) : K ( p k * ) u ( p k * ) = Q ( p k * ) , p k * L } \underline{u}_{i}=min\{u_{i}(p_{k}^{*}):K(p_{k}^{*})u(p_{k}^{*})=Q(p_{k}^{*}),% p_{k}^{*}\in L\}
  137. u ¯ i = m a x { u i ( p k * ) : K ( p k * ) u ( p k * ) = Q ( p k * ) , p k * L } \overline{u}_{i}=max\{u_{i}(p_{k}^{*}):K(p_{k}^{*})u(p_{k}^{*})=Q(p_{k}^{*}),p% _{k}^{*}\in L\}
  138. u = u ( p ) u=u(p)
  139. u i ( p ) u i ( p 0 ) + j u ( p 0 ) p j Δ p j u_{i}(p)\approx u_{i}(p_{0})+\sum_{j}\frac{\partial u(p_{0})}{\partial p_{j}}% \Delta p_{j}
  140. u ¯ i u i ( p 0 ) - | j u ( p 0 ) p j | Δ p j \underline{u}_{i}\approx u_{i}(p_{0})-\left|\sum_{j}\frac{\partial u(p_{0})}{% \partial p_{j}}\right|\Delta p_{j}
  141. u ¯ i u i ( p 0 ) + | j u ( p 0 ) p j | Δ p j \overline{u}_{i}\approx u_{i}(p_{0})+\left|\sum_{j}\frac{\partial u(p_{0})}{% \partial p_{j}}\right|\Delta p_{j}
  142. u i p j \frac{\partial u_{i}}{\partial p_{j}}
  143. u i = u i ( p ) u_{i}=u_{i}(p)
  144. u i p j 0 \frac{\partial u_{i}}{\partial p_{j}}\geq 0
  145. p i m i n = p ¯ i , p i m a x = p ¯ i p_{i}^{min}=\underline{p}_{i},\ p_{i}^{max}=\overline{p}_{i}
  146. u i p j < 0 \frac{\partial u_{i}}{\partial p_{j}}<0
  147. p i m i n = p ¯ i , p i m a x = p ¯ i p_{i}^{min}=\overline{p}_{i},\ p_{i}^{max}=\underline{p}_{i}
  148. u ¯ i = u i ( p m i n ) , u ¯ i = u i ( p m a x ) \underline{u}_{i}=u_{i}(p^{min}),\ \overline{u}_{i}=u_{i}(p^{max})
  149. u = u ( p ) u=u(p)
  150. K = K ( p ) K=K(p)
  151. Q = Q ( p ) Q=Q(p)
  152. u = u ( p ) u=u(p)

Introduction_to_systolic_geometry.html

  1. L L
  2. A A
  3. 4 π A L 2 , 4\pi A\leq L^{2},\,
  4. x - x . x\mapsto-x.\,
  5. A A
  6. π A \sqrt{\pi A}
  7. P P
  8. 3 {\mathbb{R}}^{3}
  9. L L
  10. P \partial P
  11. P P
  12. L 2 π 4 area ( P ) . L^{2}\leq\frac{\pi}{4}\mathrm{area}(\partial P).
  13. X X
  14. X X
  15. X X
  16. sys ( X ) . \mathrm{sys}(X).\,
  17. X X
  18. 2 \mathbb{RP}^{2}
  19. 3 \mathbb{R}^{3}
  20. 2 \mathbb{RP}^{2}
  21. 2 \mathbb{RP}^{2}
  22. 2 \mathbb{RP}^{2}
  23. S 2 S^{2}
  24. 2 \mathbb{RP}^{2}
  25. 2 \mathbb{RP}^{2}
  26. g g
  27. 2 \mathbb{RP}^{2}
  28. sys ( g ) 2 π 2 area ( g ) , \mathrm{sys}(g)^{2}\leq\frac{\pi}{2}\mathrm{area}(g),
  29. sys \mathrm{sys}
  30. area ( g ) - 2 π sys ( g ) 2 0. \mathrm{area}(g)-\frac{2}{\pi}\mathrm{sys}(g)^{2}\geq 0.
  31. ( T 2 , g ) (T^{2},g)
  32. area ( g ) - 3 2 sys ( g ) 2 0. \mathrm{area}(g)-\tfrac{\sqrt{3}}{2}\mathrm{sys}(g)^{2}\geq 0.
  33. 2 {\mathbb{R}}^{2}
  34. L 2 - 4 π A π 2 ( R - r ) 2 . L^{2}-4\pi A\geq\pi^{2}(R-r)^{2}.\,
  35. A A
  36. L L
  37. R R
  38. r r
  39. π 2 ( R - r ) 2 \pi^{2}(R-r)^{2}
  40. area ( g ) - 3 2 sys ( g ) 2 Var ( f ) , \mathrm{area}(g)-\tfrac{\sqrt{3}}{2}\mathrm{sys}(g)^{2}\geq\mathrm{Var}(f),

Invariant_estimator.html

  1. x x
  2. θ \theta
  3. x x
  4. f ( x | θ ) f(x|\theta)
  5. θ \theta
  6. θ \theta
  7. x x
  8. a a
  9. A A
  10. L = L ( a , θ ) L=L(a,\theta)
  11. R = R ( a , θ ) = E [ L ( a , θ ) | θ ] R=R(a,\theta)=E[L(a,\theta)|\theta]
  12. x x
  13. θ \theta
  14. a a
  15. X X
  16. Θ \Theta
  17. A A
  18. X X
  19. Θ \Theta
  20. f ( x | θ ) f(x|\theta)
  21. L L
  22. X X
  23. X X
  24. G G
  25. X X
  26. g 1 G g_{1}\in G
  27. g 2 G g_{2}\in G
  28. g 1 g 2 G g_{1}g_{2}\in G\,
  29. g G g\in G
  30. g - 1 G g^{-1}\in G
  31. g - 1 ( g ( x ) ) = x . g^{-1}(g(x))=x\,.
  32. e G e\in G
  33. e ( x ) = x e(x)=x\,
  34. x 1 x_{1}
  35. x 2 x_{2}
  36. X X
  37. x 1 = g ( x 2 ) x_{1}=g(x_{2})
  38. g G g\in G
  39. X X
  40. x 0 x_{0}
  41. X ( x 0 ) X(x_{0})
  42. X ( x 0 ) = { g ( x 0 ) : g G } X(x_{0})=\{g(x_{0}):g\in G\}
  43. X X
  44. g g
  45. F F
  46. G G
  47. g G g\in G
  48. θ Θ \theta\in\Theta
  49. θ * Θ \theta^{*}\in\Theta
  50. Y = g ( x ) Y=g(x)
  51. f ( y | θ * ) f(y|\theta^{*})
  52. θ * \theta^{*}
  53. g ¯ ( θ ) \bar{g}(\theta)
  54. F F
  55. G G
  56. L ( θ , a ) L(\theta,a)
  57. G G
  58. g G g\in G
  59. a A a\in A
  60. a * A a^{*}\in A
  61. L ( θ , a ) = L ( g ¯ ( θ ) , a * ) L(\theta,a)=L(\bar{g}(\theta),a^{*})
  62. θ Θ \theta\in\Theta
  63. a * a^{*}
  64. g ~ ( a ) \tilde{g}(a)
  65. G ¯ = { g ¯ : g G } \bar{G}=\{\bar{g}:g\in G\}
  66. Θ \Theta
  67. G ~ = { g ~ : g G } \tilde{G}=\{\tilde{g}:g\in G\}
  68. A A
  69. G G
  70. G , G ¯ , G ~ G,\bar{G},\tilde{G}
  71. G G
  72. δ ( x ) \delta(x)
  73. G G
  74. x X x\in X
  75. g G g\in G
  76. δ ( g ( x ) ) = g ~ ( δ ( x ) ) . \delta(g(x))=\tilde{g}(\delta(x)).
  77. δ \delta
  78. Θ \Theta
  79. R ( θ , δ ) = R ( g ¯ ( θ ) , δ ) R(\theta,\delta)=R(\bar{g}(\theta),\delta)
  80. θ Θ \theta\in\Theta
  81. g ¯ G ¯ \bar{g}\in\bar{G}
  82. g ¯ \bar{g}
  83. g ¯ \bar{g}
  84. θ \theta
  85. X X
  86. f ( x - θ ) f(x-\theta)
  87. Θ = A = 1 \Theta=A=\mathbb{R}^{1}
  88. L = L ( a - θ ) L=L(a-\theta)
  89. g = g ¯ = g ~ = { g c : g c ( x ) = x + c , c } g=\bar{g}=\tilde{g}=\{g_{c}:g_{c}(x)=x+c,c\in\mathbb{R}\}
  90. δ ( x + c ) = δ ( x ) + c , for all c , \delta(x+c)=\delta(x)+c,\,\text{ for all }c\in\mathbb{R},
  91. δ ( x ) = x + K \delta(x)=x+K
  92. K K\in\mathbb{R}
  93. g ¯ \bar{g}
  94. Θ \Theta
  95. θ \theta
  96. R ( θ , δ ) = R ( 0 , δ ) = E [ L ( X + K ) | θ = 0 ] R(\theta,\delta)=R(0,\delta)=\operatorname{E}[L(X+K)|\theta=0]
  97. R ( θ , δ ) R(\theta,\delta)
  98. δ ( x ) = x - E [ X | θ = 0 ] . \delta(x)=x-\operatorname{E}[X|\theta=0].
  99. X = ( X 1 , , X n ) X=(X_{1},\dots,X_{n})
  100. f ( x 1 - θ , , x n - θ ) f(x_{1}-\theta,\dots,x_{n}-\theta)
  101. L ( | a - θ | ) L(|a-\theta|)
  102. G = { g c : g c ( x ) = ( x 1 + c , , x n + c ) , c 1 } , G=\{g_{c}:g_{c}(x)=(x_{1}+c,\dots,x_{n}+c),c\in\mathbb{R}^{1}\},
  103. G ¯ = { g c : g c ( θ ) = θ + c , c 1 } , \bar{G}=\{g_{c}:g_{c}(\theta)=\theta+c,c\in\mathbb{R}^{1}\},
  104. G ~ = { g c : g c ( a ) = a + c , c 1 } . \tilde{G}=\{g_{c}:g_{c}(a)=a+c,c\in\mathbb{R}^{1}\}.
  105. δ ( x ) \delta(x)
  106. - L ( δ ( x ) - θ ) f ( x 1 - θ , , x n - θ ) d θ - f ( x 1 - θ , , x n - θ ) d θ , \frac{\int_{-\infty}^{\infty}{L(\delta(x)-\theta)f(x_{1}-\theta,\dots,x_{n}-% \theta)d\theta}}{\int_{-\infty}^{\infty}{f(x_{1}-\theta,\dots,x_{n}-\theta)d% \theta}},
  107. δ ( x ) = - θ f ( x 1 - θ , , x n - θ ) d θ - f ( x 1 - θ , , x n - θ ) d θ . \delta(x)=\frac{\int_{-\infty}^{\infty}{\theta f(x_{1}-\theta,\dots,x_{n}-% \theta)d\theta}}{\int_{-\infty}^{\infty}{f(x_{1}-\theta,\dots,x_{n}-\theta)d% \theta}}.
  108. x N ( θ 1 n , I ) x\sim N(\theta 1_{n},I)\,\!
  109. δ p i t m a n = δ M L = x i n . \delta_{pitman}=\delta_{ML}=\frac{\sum{x_{i}}}{n}.
  110. x C ( θ 1 n , I σ 2 ) x\sim C(\theta 1_{n},I\sigma^{2})\,\!
  111. δ p i t m a n δ M L \delta_{pitman}\neq\delta_{ML}
  112. δ p i t m a n = k = 1 n x k [ R e { w k } m = 1 n R e { w k } ] , n > 1 , \delta_{pitman}=\sum_{k=1}^{n}{x_{k}\left[\frac{Re\{w_{k}\}}{\sum_{m=1}^{n}{Re% \{w_{k}\}}}\right]},\qquad n>1,
  113. w k = j k [ 1 ( x k - x j ) 2 + 4 σ 2 ] [ 1 - 2 σ ( x k - x j ) i ] . w_{k}=\prod_{j\neq k}\left[\frac{1}{(x_{k}-x_{j})^{2}+4\sigma^{2}}\right]\left% [1-\frac{2\sigma}{(x_{k}-x_{j})}i\right].

Invariant_factorization_of_LPDOs.html

  1. n n
  2. 𝒜 2 = a 20 x 2 + a 11 x y + a 02 y 2 + a 10 x + a 01 y + a 00 . \mathcal{A}_{2}=a_{20}\partial_{x}^{2}+a_{11}\partial_{x}\partial_{y}+a_{02}% \partial_{y}^{2}+a_{10}\partial_{x}+a_{01}\partial_{y}+a_{00}.
  3. 𝒜 2 = ( p 1 x + p 2 y + p 3 ) ( p 4 x + p 5 y + p 6 ) . \mathcal{A}_{2}=(p_{1}\partial_{x}+p_{2}\partial_{y}+p_{3})(p_{4}\partial_{x}+% p_{5}\partial_{y}+p_{6}).
  4. p i p_{i}
  5. x ( α y ) = x ( α ) y + α x y . \partial_{x}(\alpha\partial_{y})=\partial_{x}(\alpha)\partial_{y}+\alpha% \partial_{xy}.
  6. a 20 = p 1 p 4 , a_{20}=p_{1}p_{4},
  7. a 11 = p 2 p 4 + p 1 p 5 , a_{11}=p_{2}p_{4}+p_{1}p_{5},
  8. a 02 = p 2 p 5 , a_{02}=p_{2}p_{5},
  9. a 10 = ( p 4 ) + p 3 p 4 + p 1 p 6 , a_{10}=\mathcal{L}(p_{4})+p_{3}p_{4}+p_{1}p_{6},
  10. a 01 = ( p 5 ) + p 3 p 5 + p 2 p 6 , a_{01}=\mathcal{L}(p_{5})+p_{3}p_{5}+p_{2}p_{6},
  11. a 00 = ( p 6 ) + p 3 p 6 , a_{00}=\mathcal{L}(p_{6})+p_{3}p_{6},
  12. = p 1 x + p 2 y \mathcal{L}=p_{1}\partial_{x}+p_{2}\partial_{y}
  13. a 20 0 , a_{20}\neq 0,
  14. p 1 0 , p_{1}\neq 0,
  15. p 1 = 1. p_{1}=1.
  16. p 2 , p_{2},
  17. ...
  18. p 6 p_{6}
  19. p 2 , p_{2},
  20. p 4 , p_{4},
  21. p 5 p_{5}
  22. a 20 = p 1 p 4 , a_{20}=p_{1}p_{4},
  23. a 11 = p 2 p 4 + p 1 p 5 , a_{11}=p_{2}p_{4}+p_{1}p_{5},
  24. a 02 = p 2 p 5 . a_{02}=p_{2}p_{5}.
  25. 𝒫 2 ( - p 2 ) = a 20 ( - p 2 ) 2 + a 11 ( - p 2 ) + a 02 = 0 \mathcal{P}_{2}(-p_{2})=a_{20}(-p_{2})^{2}+a_{11}(-p_{2})+a_{02}=0
  26. ω \omega
  27. 𝒫 2 , \mathcal{P}_{2},
  28. p 1 = 1 , p_{1}=1,
  29. p 2 = - ω , p_{2}=-\omega,
  30. p 4 = a 20 , p_{4}=a_{20},
  31. p 5 = a 20 ω + a 11 , p_{5}=a_{20}\omega+a_{11},
  32. a 10 = ( p 4 ) + p 3 p 4 + p 1 p 6 , a_{10}=\mathcal{L}(p_{4})+p_{3}p_{4}+p_{1}p_{6},
  33. a 01 = ( p 5 ) + p 3 p 5 + p 2 p 6 , a_{01}=\mathcal{L}(p_{5})+p_{3}p_{5}+p_{2}p_{6},
  34. a 10 = a 20 + p 3 a 20 + p 6 , a_{10}=\mathcal{L}a_{20}+p_{3}a_{20}+p_{6},
  35. a 01 = ( a 11 + a 20 ω ) + p 3 ( a 11 + a 20 ω ) - ω p 6 . , a_{01}=\mathcal{L}(a_{11}+a_{20}\omega)+p_{3}(a_{11}+a_{20}\omega)-\omega p_{6% }.,
  36. ω \omega
  37. 𝒫 2 ( ω ) = 2 a 20 ω + a 11 0 , \mathcal{P}_{2}^{\prime}(\omega)=2a_{20}\omega+a_{11}\neq 0,
  38. p 3 = ω a 10 + a 01 - ω a 20 - ( a 20 ω + a 11 ) 2 a 20 ω + a 11 , p_{3}=\frac{\omega a_{10}+a_{01}-\omega\mathcal{L}a_{20}-\mathcal{L}(a_{20}% \omega+a_{11})}{2a_{20}\omega+a_{11}},
  39. p 6 = ( a 20 ω + a 11 ) ( a 10 - a 20 ) - a 20 ( a 01 - ( a 20 ω + a 11 ) ) 2 a 20 ω + a 11 . p_{6}=\frac{(a_{20}\omega+a_{11})(a_{10}-\mathcal{L}a_{20})-a_{20}(a_{01}-% \mathcal{L}(a_{20}\omega+a_{11}))}{2a_{20}\omega+a_{11}}.
  40. 𝒫 2 \mathcal{P}_{2}
  41. p j p_{j}
  42. a 00 = ( p 6 ) + p 3 p 6 , a_{00}=\mathcal{L}(p_{6})+p_{3}p_{6},
  43. p j p_{j}
  44. ω \omega
  45. a 00 = { ω a 10 + a 01 - ( 2 a 20 ω + a 11 ) 2 a 20 ω + a 11 } + ω a 10 + a 01 - ( 2 a 20 ω + a 11 ) 2 a 20 ω + a 11 × a 20 ( a 01 - ( a 20 ω + a 11 ) ) + ( a 20 ω + a 11 ) ( a 10 - a 20 ) 2 a 20 ω + a 11 a_{00}=\mathcal{L}\left\{\frac{\omega a_{10}+a_{01}-\mathcal{L}(2a_{20}\omega+% a_{11})}{2a_{20}\omega+a_{11}}\right\}+\frac{\omega a_{10}+a_{01}-\mathcal{L}(% 2a_{20}\omega+a_{11})}{2a_{20}\omega+a_{11}}\times\frac{a_{20}(a_{01}-\mathcal% {L}(a_{20}\omega+a_{11}))+(a_{20}\omega+a_{11})(a_{10}-\mathcal{L}a_{20})}{2a_% {20}\omega+a_{11}}
  46. l 2 = a 00 - { ω a 10 + a 01 - ( 2 a 20 ω + a 11 ) 2 a 20 ω + a 11 } + ω a 10 + a 01 - ( 2 a 20 ω + a 11 ) 2 a 20 ω + a 11 × a 20 ( a 01 - ( a 20 ω + a 11 ) ) + ( a 20 ω + a 11 ) ( a 10 - a 20 ) 2 a 20 ω + a 11 = 0 , l_{2}=a_{00}-\mathcal{L}\left\{\frac{\omega a_{10}+a_{01}-\mathcal{L}(2a_{20}% \omega+a_{11})}{2a_{20}\omega+a_{11}}\right\}+\frac{\omega a_{10}+a_{01}-% \mathcal{L}(2a_{20}\omega+a_{11})}{2a_{20}\omega+a_{11}}\times\frac{a_{20}(a_{% 01}-\mathcal{L}(a_{20}\omega+a_{11}))+(a_{20}\omega+a_{11})(a_{10}-\mathcal{L}% a_{20})}{2a_{20}\omega+a_{11}}=0,
  47. 𝒜 2 \mathcal{A}_{2}
  48. p j p_{j}
  49. 𝒜 3 = j + k 3 a j k x j y k = a 30 x 3 + a 21 x 2 y + a 12 x y 2 + a 03 y 3 + a 20 x 2 + a 11 x y + a 02 y 2 + a 10 x + a 01 y + a 00 . \mathcal{A}_{3}=\sum_{j+k\leq 3}a_{jk}\partial_{x}^{j}\partial_{y}^{k}=a_{30}% \partial_{x}^{3}+a_{21}\partial_{x}^{2}\partial_{y}+a_{12}\partial_{x}\partial% _{y}^{2}+a_{03}\partial_{y}^{3}+a_{20}\partial_{x}^{2}+a_{11}\partial_{x}% \partial_{y}+a_{02}\partial_{y}^{2}+a_{10}\partial_{x}+a_{01}\partial_{y}+a_{0% 0}.
  50. 𝒜 3 = ( p 1 x + p 2 y + p 3 ) ( p 4 x 2 + p 5 x y + p 6 y 2 + p 7 x + p 8 y + p 9 ) . \mathcal{A}_{3}=(p_{1}\partial_{x}+p_{2}\partial_{y}+p_{3})(p_{4}\partial_{x}^% {2}+p_{5}\partial_{x}\partial_{y}+p_{6}\partial_{y}^{2}+p_{7}\partial_{x}+p_{8% }\partial_{y}+p_{9}).
  51. 𝒜 2 , \mathcal{A}_{2},
  52. a 30 = p 1 p 4 , a_{30}=p_{1}p_{4},
  53. a 21 = p 2 p 4 + p 1 p 5 , a_{21}=p_{2}p_{4}+p_{1}p_{5},
  54. a 12 = p 2 p 5 + p 1 p 6 , a_{12}=p_{2}p_{5}+p_{1}p_{6},
  55. a 03 = p 2 p 6 , a_{03}=p_{2}p_{6},
  56. a 20 = ( p 4 ) + p 3 p 4 + p 1 p 7 , a_{20}=\mathcal{L}(p_{4})+p_{3}p_{4}+p_{1}p_{7},
  57. a 11 = ( p 5 ) + p 3 p 5 + p 2 p 7 + p 1 p 8 , a_{11}=\mathcal{L}(p_{5})+p_{3}p_{5}+p_{2}p_{7}+p_{1}p_{8},
  58. a 02 = ( p 6 ) + p 3 p 6 + p 2 p 8 , a_{02}=\mathcal{L}(p_{6})+p_{3}p_{6}+p_{2}p_{8},
  59. a 10 = ( p 7 ) + p 3 p 7 + p 1 p 9 , a_{10}=\mathcal{L}(p_{7})+p_{3}p_{7}+p_{1}p_{9},
  60. a 01 = ( p 8 ) + p 3 p 8 + p 2 p 9 , a_{01}=\mathcal{L}(p_{8})+p_{3}p_{8}+p_{2}p_{9},
  61. a 00 = ( p 9 ) + p 3 p 9 , a_{00}=\mathcal{L}(p_{9})+p_{3}p_{9},
  62. = p 1 x + p 2 y , \mathcal{L}=p_{1}\partial_{x}+p_{2}\partial_{y},
  63. a 30 0 , a_{30}\neq 0,
  64. p 1 = 1 , p_{1}=1,
  65. 𝒫 3 ( - p 2 ) := a 30 ( - p 2 ) 3 + a 21 ( - p 2 ) 2 + a 12 ( - p 2 ) + a 03 = 0. \mathcal{P}_{3}(-p_{2}):=a_{30}(-p_{2})^{3}+a_{21}(-p_{2})^{2}+a_{12}(-p_{2})+% a_{03}=0.
  66. ω \omega
  67. p 1 = 1 , p_{1}=1,
  68. p 2 = - ω , p_{2}=-\omega,
  69. p 4 = a 30 , p_{4}=a_{30},
  70. p 5 = a 30 ω + a 21 , p_{5}=a_{30}\omega+a_{21},
  71. p 6 = a 30 ω 2 + a 21 ω + a 12 . p_{6}=a_{30}\omega^{2}+a_{21}\omega+a_{12}.
  72. a 20 - a 30 = p 3 a 30 + p 7 , a_{20}-\mathcal{L}a_{30}=p_{3}a_{30}+p_{7},
  73. a 11 - ( a 30 ω + a 21 ) = p 3 ( a 30 ω + a 21 ) - ω p 7 + p 8 , a_{11}-\mathcal{L}(a_{30}\omega+a_{21})=p_{3}(a_{30}\omega+a_{21})-\omega p_{7% }+p_{8},
  74. a 02 - ( a 30 ω 2 + a 21 ω + a 12 ) = p 3 ( a 30 ω 2 + a 21 ω + a 12 ) - ω p 8 . a_{02}-\mathcal{L}(a_{30}\omega^{2}+a_{21}\omega+a_{12})=p_{3}(a_{30}\omega^{2% }+a_{21}\omega+a_{12})-\omega p_{8}.
  75. n n
  76. 𝒜 \mathcal{A}
  77. 𝒜 ~ \tilde{\mathcal{A}}
  78. 𝒜 ~ g = e - φ 𝒜 ( e φ g ) . \tilde{\mathcal{A}}g=e^{-\varphi}\mathcal{A}(e^{\varphi}g).
  79. 𝒜 ~ \tilde{\mathcal{A}}
  80. 𝒜 = j + k n a j k x j y k = j + k ( n - 1 ) p j k x j y k \mathcal{A}=\sum_{j+k\leq n}a_{jk}\partial_{x}^{j}\partial_{y}^{k}=\mathcal{L}% \circ\sum_{j+k\leq(n-1)}p_{jk}\partial_{x}^{j}\partial_{y}^{k}
  81. = x - ω y + p \mathcal{L}=\partial_{x}-\omega\partial_{y}+p
  82. ω \omega
  83. 𝒫 ( t ) = k = 0 n a n - k , k t n - k , 𝒫 ( ω ) = 0. \mathcal{P}(t)=\sum^{n}_{k=0}a_{n-k,k}t^{n-k},\quad\mathcal{P}(\omega)=0.
  84. ω ~ \tilde{\omega}
  85. n = 2 l 2 = 0 , n=2\ \ \rightarrow l_{2}=0,
  86. n = 3 l 3 = 0 , l 31 = 0 , n=3\ \ \rightarrow l_{3}=0,l_{31}=0,
  87. n = 4 l 4 = 0 , l 41 = 0 , l 42 = 0 , n=4\ \ \rightarrow l_{4}=0,l_{41}=0,l_{42}=0,
  88. l 2 , l 3 , l 31 , l 4 , l 41 , l 42 , l_{2},l_{3},l_{31},l_{4},l_{41},\ \ l_{42},...
  89. l 2 = a 00 - ( p 6 ) + p 3 p 6 , l_{2}=a_{00}-\mathcal{L}(p_{6})+p_{3}p_{6},
  90. l 3 = a 00 - ( p 9 ) + p 3 p 9 , l_{3}=a_{00}-\mathcal{L}(p_{9})+p_{3}p_{9},
  91. l 31 = a 01 - ( p 8 ) + p 3 p 8 + p 2 p 9 , l_{31}=a_{01}-\mathcal{L}(p_{8})+p_{3}p_{8}+p_{2}p_{9},
  92. l 2 = a 00 - ( p 6 ) + p 3 p 6 , l 3 = a 00 - ( p 9 ) + p 3 p 9 , l 31 , . l_{2}=a_{00}-\mathcal{L}(p_{6})+p_{3}p_{6},l_{3}=a_{00}-\mathcal{L}(p_{9})+p_{% 3}p_{9},l_{31},....
  93. l 2 = a 00 - ( p 6 ) + p 3 p 6 , l 3 = a 00 - ( p 9 ) + p 3 p 9 , l 31 , . . l_{2}=a_{00}-\mathcal{L}(p_{6})+p_{3}p_{6},l_{3}=a_{00}-\mathcal{L}(p_{9})+p_{% 3}p_{9},l_{31},.....
  94. 𝒜 ~ \tilde{\mathcal{A}}
  95. e - φ x e φ = x + φ x , e - φ y e φ = y + φ y , e^{-\varphi}\partial_{x}e^{\varphi}=\partial_{x}+\varphi_{x},\quad e^{-\varphi% }\partial_{y}e^{\varphi}=\partial_{y}+\varphi_{y},
  96. e - φ x y e φ = e - φ x e φ e - φ y e φ = ( x + φ x ) ( y + φ y ) e^{-\varphi}\partial_{x}\partial_{y}e^{\varphi}=e^{-\varphi}\partial_{x}e^{% \varphi}e^{-\varphi}\partial_{y}e^{\varphi}=(\partial_{x}+\varphi_{x})\circ(% \partial_{y}+\varphi_{y})
  97. A 1 = x y + x x + 1 = x ( y + x ) , l 2 ( A 1 ) = 1 - 1 - 0 = 0 ; A_{1}=\partial_{x}\partial_{y}+x\partial_{x}+1=\partial_{x}(\partial_{y}+x),% \quad l_{2}(A_{1})=1-1-0=0;
  98. A 2 = x y + x x + y + x + 1 , A 2 = e - x A 1 e x ; l 2 ( A 2 ) = ( x + 1 ) - 1 - x = 0 ; A_{2}=\partial_{x}\partial_{y}+x\partial_{x}+\partial_{y}+x+1,\quad A_{2}=e^{-% x}A_{1}e^{x};\quad l_{2}(A_{2})=(x+1)-1-x=0;
  99. A 3 = x y + 2 x x + ( y + 1 ) y + 2 ( x y + x + 1 ) , A 3 = e - x y A 2 e x y ; l 2 ( A 3 ) = 2 ( x + 1 + x y ) - 2 - 2 x ( y + 1 ) = 0 ; A_{3}=\partial_{x}\partial_{y}+2x\partial_{x}+(y+1)\partial_{y}+2(xy+x+1),% \quad A_{3}=e^{-xy}A_{2}e^{xy};\quad l_{2}(A_{3})=2(x+1+xy)-2-2x(y+1)=0;
  100. A 4 = x y + x x + ( cos x + 1 ) y + x cos x + x + 1 , A 4 = e - sin x A 2 e sin x ; l 2 ( A 4 ) = 0. A_{4}=\partial_{x}\partial_{y}+x\partial_{x}+(\cos x+1)\partial_{y}+x\cos x+x+% 1,\quad A_{4}=e^{-\sin x}A_{2}e^{\sin x};\quad l_{2}(A_{4})=0.
  101. 𝒜 t \mathcal{A}^{t}
  102. 𝒜 = a α α , α = 1 α 1 n α n . \mathcal{A}=\sum a_{\alpha}\partial^{\alpha},\qquad\partial^{\alpha}=\partial_% {1}^{\alpha_{1}}\cdots\partial_{n}^{\alpha_{n}}.
  103. 𝒜 t u = ( - 1 ) | α | α ( a α u ) . \mathcal{A}^{t}u=\sum(-1)^{|\alpha|}\partial^{\alpha}(a_{\alpha}u).
  104. γ ( u v ) = ( γ α ) α u , γ - α v \partial^{\gamma}(uv)=\sum{\left({{\gamma}\atop{\alpha}}\right)}\partial^{% \alpha}u,\partial^{\gamma-\alpha}v
  105. 𝒜 t = ( - 1 ) | α + β | ( α + β α ) ( β a α + β ) α . \mathcal{A}^{t}=\sum(-1)^{|\alpha+\beta|}{\left({{\alpha+\beta}\atop{\alpha}}% \right)}(\partial^{\beta}a_{\alpha+\beta})\partial^{\alpha}.
  106. 𝒜 t = a ~ α α , \mathcal{A}^{t}=\sum\tilde{a}_{\alpha}\partial^{\alpha},
  107. a ~ α = ( - 1 ) | α + β | ( α + β α ) β ( a α + β ) . \tilde{a}_{\alpha}=\sum(-1)^{|\alpha+\beta|}{\left({{\alpha+\beta}\atop{\alpha% }}\right)}\partial^{\beta}(a_{\alpha+\beta}).
  108. ( α β ) = ( ( α 1 , α 2 ) ( β 1 , β 2 ) ) = ( α 1 β 1 ) ( α 2 β 2 ) . {\left({{\alpha}\atop{\beta}}\right)}={\left({{(\alpha_{1},\alpha_{2})}\atop{(% \beta_{1},\beta_{2})}}\right)}={\left({{\alpha_{1}}\atop{\beta_{1}}}\right)}\,% {\left({{\alpha_{2}}\atop{\beta_{2}}}\right)}.
  109. 𝒜 2 \mathcal{A}_{2}
  110. a ~ j k = a j k , j + k = 2 ; a ~ 10 = - a 10 + 2 x a 20 + y a 11 , a ~ 01 = - a 01 + x a 11 + 2 y a 02 , \tilde{a}_{jk}=a_{jk},\quad j+k=2;\tilde{a}_{10}=-a_{10}+2\partial_{x}a_{20}+% \partial_{y}a_{11},\tilde{a}_{01}=-a_{01}+\partial_{x}a_{11}+2\partial_{y}a_{0% 2},
  111. a ~ 00 = a 00 - x a 10 - y a 01 + x 2 a 20 + x x a 11 + y 2 a 02 . \tilde{a}_{00}=a_{00}-\partial_{x}a_{10}-\partial_{y}a_{01}+\partial_{x}^{2}a_% {20}+\partial_{x}\partial_{x}a_{11}+\partial_{y}^{2}a_{02}.
  112. x x - y y + y x + x y + 1 4 ( y 2 - x 2 ) - 1 \partial_{xx}-\partial_{yy}+y\partial_{x}+x\partial_{y}+\frac{1}{4}(y^{2}-x^{2% })-1
  113. [ x + y + 1 2 ( y - x ) ] [ ] \big[\partial_{x}+\partial_{y}+\tfrac{1}{2}(y-x)\big]\,\big[...\big]
  114. 𝒜 1 t \mathcal{A}_{1}^{t}
  115. [ ] [ x - y + 1 2 ( y + x ) ] . \big[...\big]\,\big[\partial_{x}-\partial_{y}+\tfrac{1}{2}(y+x)\big].

Inverse_floating_rate_note.html

  1. 100 ( collateral price ) = 20 ( inverse price ) + 80 ( floater price ) \ 100(\mbox{collateral price}~{})=20(\mbox{inverse price}~{})+80(\mbox{floater% price}~{})

Inversive_distance.html

  1. δ \delta
  2. r r
  3. R R
  4. d d
  5. I = | d 2 - r 2 - R 2 2 r R | . I=\left|\frac{d^{2}-r^{2}-R^{2}}{2rR}\right|.
  6. I I
  7. δ \delta
  8. δ = arcosh ( I ) . \delta=\operatorname{arcosh}(I).
  9. δ \delta
  10. I I
  11. p p
  12. p p
  13. δ \delta
  14. p p
  15. p = π sin - 1 tanh ( δ / 2 ) . p=\frac{\pi}{\sin^{-1}\tanh(\delta/2)}.
  16. p p

Irrational_number.html

  1. π \pi
  2. 3 1 5 3 \frac{3\quad 1}{5\quad 3}
  3. log 2 3 = m n . \log_{2}3=\frac{m}{n}.
  4. 2 m / n = 3 2^{m/n}=3\,
  5. ( 2 m / n ) n = 3 n (2^{m/n})^{n}=3^{n}\,
  6. 2 m = 3 n . 2^{m}=3^{n}.\,
  7. p ( x ) = a n x n + a n - 1 x n - 1 + + a 1 x + a 0 = 0 p(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}=0\,
  8. A = 0.7 162 162 162 . A=0.7\,162\,162\,162\,\cdots.
  9. 10 A = 7.162 162 162 . 10A=7.162\,162\,162\,\cdots.
  10. 10 , 000 A = 7 162.162 162 . 10,000A=7\,162.162\,162\,\cdots.
  11. 9990 A = 7155. 9990A=7155.
  12. A = 7155 9990 = 135 × 53 135 × 74 = 53 74 , A=\frac{7155}{9990}=\frac{135\times 53}{135\times 74}=\frac{53}{74},
  13. ( 2 ) log 2 3 = 3. \left(\sqrt{2}\right)^{\log_{\sqrt{2}}3}=3.
  14. log 2 3 \log_{\sqrt{2}}3
  15. log 2 3 = log 2 3 log 2 2 = log 2 3 1 / 2 = 2 log 2 3 \log_{\sqrt{2}}3=\frac{\log_{2}3}{\log_{2}\sqrt{2}}=\frac{\log_{2}3}{1/2}=2% \log_{2}3
  16. log 2 3 = m / 2 n \log_{2}3=m/2n
  17. 2 log 2 3 = 2 m / 2 n 2^{\log_{2}3}=2^{m/2n}
  18. 3 = 2 m / 2 n 3=2^{m/2n}
  19. 3 2 n = 2 m 3^{2n}=2^{m}
  20. ( ( 1 / e ) 1 / e , ) ((1/e)^{1/e},\infty)
  21. a a a a^{a^{a}}
  22. n n n n^{n^{n}}
  23. π \pi
  24. π \pi
  25. π \pi
  26. π \pi
  27. π \pi
  28. π \pi
  29. π \pi
  30. π \pi
  31. π \pi
  32. π \pi

Ishimori_equation.html

  1. 𝐒 t = 𝐒 ( 2 𝐒 x 2 + 2 𝐒 y 2 ) + u x 𝐒 y + u y 𝐒 x , ( 1 a ) \frac{\partial\mathbf{S}}{\partial t}=\mathbf{S}\wedge\left(\frac{\partial^{2}% \mathbf{S}}{\partial x^{2}}+\frac{\partial^{2}\mathbf{S}}{\partial y^{2}}% \right)+\frac{\partial u}{\partial x}\frac{\partial\mathbf{S}}{\partial y}+% \frac{\partial u}{\partial y}\frac{\partial\mathbf{S}}{\partial x},\qquad(1a)
  2. 2 u x 2 - α 2 2 u y 2 = - 2 α 2 𝐒 ( 𝐒 x 𝐒 y ) . ( 1 b ) \frac{\partial^{2}u}{\partial x^{2}}-\alpha^{2}\frac{\partial^{2}u}{\partial y% ^{2}}=-2\alpha^{2}\mathbf{S}\cdot\left(\frac{\partial\mathbf{S}}{\partial x}% \wedge\frac{\partial\mathbf{S}}{\partial y}\right).\qquad(1b)
  3. L t = A L - L A ( 2 ) L_{t}=AL-LA\qquad(2)
  4. L = Σ x + α I y , ( 3 a ) L=\Sigma\partial_{x}+\alpha I\partial_{y},\qquad(3a)
  5. A = - 2 i Σ x 2 + ( - i Σ x - i α Σ y Σ + u y I - α 3 u x Σ ) x . ( 3 b ) A=-2i\Sigma\partial_{x}^{2}+(-i\Sigma_{x}-i\alpha\Sigma_{y}\Sigma+u_{y}I-% \alpha^{3}u_{x}\Sigma)\partial_{x}.\qquad(3b)
  6. Σ = j = 1 3 S j σ j , ( 4 ) \Sigma=\sum_{j=1}^{3}S_{j}\sigma_{j},\qquad(4)
  7. σ i \sigma_{i}
  8. I I

Isoelastic_utility.html

  1. u ( c ) = { c 1 - η - 1 1 - η η 1 ln ( c ) η = 1 u(c)=\begin{cases}\frac{c^{1-\eta}-1}{1-\eta}&\eta\neq 1\\ \ln(c)&\eta=1\end{cases}
  2. c c
  3. u ( c ) u(c)
  4. η \eta
  5. ln ( c ) \ln(c)
  6. η \eta
  7. η \eta
  8. η \eta
  9. η \eta
  10. - c u ′′ ( c ) / u ( c ) -c\cdot u^{\prime\prime}(c)/u^{\prime}(c)
  11. η \eta
  12. η = 0 \eta=0
  13. η = 1 \eta=1
  14. u ( c ) u(c)
  15. log c \log c
  16. η \eta
  17. lim η 1 c 1 - η - 1 1 - η = ln ( c ) \lim_{\eta\rightarrow 1}\frac{c^{1-\eta}-1}{1-\eta}=\ln(c)
  18. η = 1 \eta=1
  19. η \eta
  20. \infty

Isomonodromic_deformation.html

  1. d Y d x = A Y = i = 1 n A i x - λ i Y \frac{dY}{dx}=AY=\sum_{i=1}^{n}\frac{A_{i}}{x-\lambda_{i}}Y
  2. i = 1 n A i = 0. \sum_{i=1}^{n}A_{i}=0.
  3. Y = Y M i . Y^{\prime}=YM_{i}.
  4. π 1 ( 𝐏 1 ( 𝐂 ) - { λ 1 , , λ n } ) G L ( n , 𝐂 ) . \pi_{1}\left(\mathbf{P}^{1}(\mathbf{C})-\{\lambda_{1},\dots,\lambda_{n}\}% \right)\to GL(n,\mathbf{C}).
  5. x ^ \hat{x}
  6. g - 1 ( x ) A g ( x ) - g - 1 ( x ) d g ( x ) d x g^{-1}(x)Ag(x)-g^{-1}(x)\frac{dg(x)}{dx}
  7. A i λ j \displaystyle\frac{\partial A_{i}}{\partial\lambda_{j}}
  8. 𝔰 𝔩 ( 2 , 𝐂 ) \mathfrak{sl}(2,\mathbf{C})
  9. d Y d x = A Y = i = 1 n j = 1 r i + 1 A j ( i ) ( x - λ i ) j Y , \frac{dY}{dx}=AY=\sum_{i=1}^{n}\sum_{j=1}^{r_{i}+1}\frac{A^{(i)}_{j}}{(x-% \lambda_{i})^{j}}Y,
  10. ( r i + 1 ) (r_{i}+1)
  11. A j ( i ) A^{(i)}_{j}
  12. x i = x - λ i x_{i}=x-\lambda_{i}
  13. r i + 1 r_{i}+1
  14. d ( g i - 1 Z i ) d x i = ( j = 1 r i ( - j ) T j ( i ) x i j + 1 + M ( i ) x i ) ( g i - 1 Z i ) \frac{d(g_{i}^{-1}Z_{i})}{dx_{i}}=\left(\sum_{j=1}^{r_{i}}\frac{(-j)T^{(i)}_{j% }}{x_{i}^{j+1}}+\frac{M^{(i)}}{x_{i}}\right)(g_{i}^{-1}Z_{i})
  15. M ( i ) M^{(i)}
  16. T j ( i ) T^{(i)}_{j}
  17. Z i = g i exp ( M ( i ) log ( x i ) + j = 1 r i T j ( i ) x i j ) . Z_{i}=g_{i}\exp\left(M^{(i)}\log(x_{i})+\sum_{j=1}^{r_{i}}\frac{T^{(i)}_{j}}{x% _{i}^{j}}\right).
  18. Y = G i exp ( M ( i ) log ( x i ) + j = 1 r i T j ( i ) x i j ) . Y=G_{i}\exp\left(M^{(i)}\log(x_{i})+\sum_{j=1}^{r_{i}}\frac{T^{(i)}_{j}}{x_{i}% ^{j}}\right).
  19. A j ( i ) A^{(i)}_{j}
  20. T j ( i ) T^{(i)}_{j}
  21. Ω = i = 1 n ( A d λ i - g i D ( j = 1 r i T j ( i ) ) g i - 1 ) \Omega=\sum_{i=1}^{n}\left(Ad\lambda_{i}-g_{i}D\left(\sum_{j=1}^{r_{i}}T^{(i)}% _{j}\right)g_{i}^{-1}\right)
  22. T j ( i ) T^{(i)}_{j}
  23. d A + [ Ω , A ] + d Ω d x = 0. dA+[\Omega,A]+\frac{d\Omega}{dx}=0.

Isoparametric_manifold.html

  1. 𝐠 = 𝐡 𝐩 \mathbf{g}=\mathbf{h}\oplus\mathbf{p}

Isotope.html

  1. × 10 1 5 \times 10^{1}5
  2. × 10 1 4 \times 10^{1}4
  3. × 10 1 1 \times 10^{1}1
  4. × 10 8 \times 10^{8}
  5. m ¯ a \overline{m}_{a}
  6. m ¯ a = m 1 x 1 + m 2 x 2 + + m N x N \overline{m}_{a}=m_{1}x_{1}+m_{2}x_{2}+...+m_{N}x_{N}
  7. N = A - Z N=A-Z
  8. A A
  9. D = N - Z D=N-Z

Item_tree_analysis.html

  1. i j i\rightarrow j
  2. i j i\rightarrow j
  3. i j i\rightarrow j
  4. i j k i\wedge j\rightarrow k
  5. i j k i\vee j\rightarrow k
  6. I I T A \leq_{IITA}
  7. C I T A \leq_{CITA}

IVX.html

  1. IVX Call ( 30 ) = IVX Call April + ( 30 - 12 ) ( IVX Call May - IVX Call April ) 47 - 12 \,\text{IVX Call}(30)=\frac{\,\text{IVX Call April}+(\sqrt{30}-\sqrt{12})\ast(% \,\text{IVX Call May}-\,\text{IVX Call April})}{\sqrt{47}-\sqrt{12}}

J0.html

  1. j 0 j_{0}

Jacobi's_four-square_theorem.html

  1. 1 2 + 0 2 + 0 2 + 0 2 \displaystyle 1^{2}+0^{2}+0^{2}+0^{2}
  2. r 4 ( n ) = { 8 m | n m if n is odd 24 m | n m odd m if n is even . r_{4}(n)=\begin{cases}8\sum\limits_{m|n}m&\,\text{if }n\,\text{ is odd}\\ 24\sum\limits_{\begin{smallmatrix}m|n\\ m\,\text{ odd}\end{smallmatrix}}m&\,\text{if }n\,\text{ is even}.\end{cases}
  3. r 4 ( n ) = 8 m : 4 m | n m . r_{4}(n)=8\sum_{m\,:\,4\nmid m|n}m.

Jacobi–Madden_equation.html

  1. a 4 + b 4 + c 4 + d 4 = ( a + b + c + d ) 4 a^{4}+b^{4}+c^{4}+d^{4}=(a+b+c+d)^{4}\,
  2. a 4 + b 4 + c 4 + d 4 = e 4 a^{4}+b^{4}+c^{4}+d^{4}=e^{4}\,
  3. 5400 4 + 1770 4 + ( - 2634 ) 4 + 955 4 = ( 5400 + 1770 - 2634 + 955 ) 4 . 5400^{4}+1770^{4}+(-2634)^{4}+955^{4}=(5400+1770-2634+955)^{4}.\,
  4. a 4 + b 4 + c 4 + d 4 = ( a + b + c + d ) 4 a^{4}+b^{4}+c^{4}+d^{4}=(a+b+c+d)^{4}
  5. a 4 + b 4 + ( a + b ) 4 = 2 ( a 2 + a b + b 2 ) 2 a^{4}+b^{4}+(a+b)^{4}=2(a^{2}+ab+b^{2})^{2}
  6. ( a + b ) 4 + ( c + d ) 4 (a+b)^{4}+(c+d)^{4}
  7. a 4 + b 4 + ( a + b ) 4 + c 4 + d 4 + ( c + d ) 4 = ( a + b ) 4 + ( c + d ) 4 + ( a + b + c + d ) 4 a^{4}+b^{4}+(a+b)^{4}+c^{4}+d^{4}+(c+d)^{4}=(a+b)^{4}+(c+d)^{4}+(a+b+c+d)^{4}
  8. ( a 2 + a b + b 2 ) 2 + ( c 2 + c d + d 2 ) 2 = ( ( a + b ) 2 + ( a + b ) ( c + d ) + ( c + d ) 2 ) 2 = 1 4 ( ( a + b ) 2 + ( c + d ) 2 + ( a + b + c + d ) 2 ) 2 (a^{2}+ab+b^{2})^{2}+(c^{2}+cd+d^{2})^{2}=\big((a+b)^{2}+(a+b)(c+d)+(c+d)^{2}% \big)^{2}=\tfrac{1}{4}\big((a+b)^{2}+(c+d)^{2}+(a+b+c+d)^{2}\big)^{2}
  9. ( - 31764 ) 4 + 27385 4 + 48150 4 + 7590 4 = ( - 31764 + 27385 + 48150 + 7590 ) 4 (-31764)^{4}+27385^{4}+48150^{4}+7590^{4}=(-31764+27385+48150+7590)^{4}\,

Jaffe_profile.html

  1. ρ ( r ) = ρ 0 4 π ( r r 0 ) - 2 ( 1 + r r 0 ) - 2 . \rho(r)={\rho_{0}\over 4\pi}\left({r\over r_{0}}\right)^{-2}\left(1+{r\over r_% {0}}\right)^{-2}.
  2. ρ 0 \rho_{0}
  3. r 0 r_{0}
  4. r - 3 r^{-3}
  5. r - 1 r^{-1}
  6. r - 4 r^{-4}
  7. r - 2 r^{-2}

Jeep_problem.html

  1. 1 + 1 2 + 1 3 + + 1 n = k = 1 n 1 k 1+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}=\sum_{k=1}^{n}\frac{1}{k}
  2. 1 + 1 3 + 1 5 + + 1 2 n - 1 = k = 1 n 1 2 k - 1 = H 2 n - 1 - 1 2 H n - 1 1+\frac{1}{3}+\frac{1}{5}+\cdots+\frac{1}{2n-1}=\sum_{k=1}^{n}\frac{1}{2k-1}=H% _{2n-1}-\frac{1}{2}H_{n-1}
  3. k = 1 n 1 2 k - 1 > k = 1 n 1 2 k = 1 2 H n \sum_{k=1}^{n}\frac{1}{2k-1}>\sum_{k=1}^{n}\frac{1}{2k}=\frac{1}{2}H_{n}

Jeu_de_taquin.html

  1. λ / μ \lambda/\mu
  2. λ μ \lambda\setminus\mu
  3. { c } λ μ \{c\}\cup\lambda\setminus\mu
  4. λ / μ \lambda/\mu
  5. λ μ \lambda\setminus\mu
  6. ( 2 , 1 ) / ( 2 ) (2,1)/(2)
  7. ( 1 , 1 ) / ( 1 ) (1,1)/(1)
  8. λ / μ \lambda/\mu
  9. μ \mu
  10. μ { c } \mu\setminus\left\{c\right\}
  11. ( λ { d } ) / ( μ { c } ) (\lambda\setminus\left\{d\right\})/(\mu\setminus\left\{c\right\})
  12. λ / μ \lambda/\mu
  13. λ \lambda
  14. λ { c } \lambda\cup\left\{c\right\}
  15. ( λ { c } ) / ( μ { d } ) (\lambda\cup\left\{c\right\})/(\mu\cup\left\{d\right\})

Job_shop_scheduling.html

  1. m = 1 m=1
  2. M = { M 1 , M 2 , , M m } M=\{M_{1},M_{2},\dots,M_{m}\}
  3. J = { J 1 , J 2 , , J n } J=\{J_{1},J_{2},\dots,J_{n}\}
  4. M i \displaystyle M_{i}
  5. J j \displaystyle J_{j}
  6. 𝒳 \displaystyle\ \mathcal{X}
  7. x 𝒳 x\in\mathcal{X}
  8. n × m n\times m
  9. i \displaystyle i
  10. M i \displaystyle M_{i}
  11. x = ( 1 2 2 3 3 1 ) x=\begin{pmatrix}1&2\\ 2&3\\ 3&1\end{pmatrix}
  12. M 1 \displaystyle M_{1}
  13. J 1 , J 2 , J 3 \displaystyle J_{1},J_{2},J_{3}
  14. J 1 , J 2 , J 3 \displaystyle J_{1},J_{2},J_{3}
  15. M 2 \displaystyle M_{2}
  16. J 2 , J 3 , J 1 \displaystyle J_{2},J_{3},J_{1}
  17. C : 𝒳 [ 0 , + ] C:\mathcal{X}\to[0,+\infty]
  18. C i j : M × J [ 0 , + ] C_{ij}:M\times J\to[0,+\infty]
  19. M i \displaystyle M_{i}
  20. J j \displaystyle J_{j}
  21. x 𝒳 x\in\mathcal{X}
  22. C ( x ) \displaystyle C(x)
  23. y 𝒳 y\in\mathcal{X}
  24. C ( x ) > C ( y ) \displaystyle C(x)>C(y)
  25. x 𝒳 x_{\infty}\in\mathcal{X}
  26. C ( x ) = + C(x_{\infty})=+\infty
  27. x x_{\infty}
  28. ( 2 1 / m ) (2−1/m)
  29. ( 2 2 / m ) (2−2/m)

Joback_method.html

  1. T b [ K ] = 198 + T b , i T_{b}[K]\,=\,198+\sum{T_{b,i}}
  2. T m [ K ] = 122.5 + T m , i T_{m}[K]\,=\,122.5+\sum{T_{m,i}}
  3. T c [ K ] = T b [ 0.584 + 0.965 T c , i - ( T c , i ) 2 ] - 1 T_{c}[K]\,=\,T_{b}\left[0.584+0.965\sum{T_{c,i}}-\left(\sum{T_{c,i}}\right)^{2% }\right]^{-1}
  4. P c [ b a r ] = [ 0.113 + 0.0032 * N A - P c , i ] - 2 P_{c}[bar]\,=\,\left[{0.113+0.0032*N_{A}-\sum{P_{c,i}}}\right]^{-2}
  5. V c [ c m 3 / m o l ] = 17.5 + V c , i V_{c}[cm^{3}/mol]\,=\,17.5+\sum{V_{c,i}}
  6. H f o r m a t i o n [ k J / m o l ] = 68.29 + H f o r m , i H_{formation}[kJ/mol]\,=\,68.29+\sum{H_{form,i}}
  7. G f o r m a t i o n [ k J / m o l ] = 53.88 + G f o r m , i G_{formation}[kJ/mol]\,=\,53.88+\sum{G_{form,i}}
  8. C P [ J / ( m o l . K ) ] = a i - 37.93 + [ b i + 0.210 ] T + [ c i - 3.91 10 - 4 ] T 2 + [ d i + 2.06 10 - 7 ] T 3 C_{P}[J/(mol.K)]\,=\,\sum a_{i}-37.93+\left[\sum b_{i}+0.210\right]T+\left[% \sum c_{i}-3.91\cdot 10^{-4}\right]T^{2}+\left[\sum d_{i}+2.06\cdot 10^{-7}% \right]T^{3}
  9. Δ H v a p [ k J / m o l ] = 15.30 + H v a p , i \Delta H_{vap}[kJ/mol]\,=\,15.30+\sum H_{vap,i}
  10. Δ H f u s [ k J / m o l ] = - 0.88 + H f u s , i \Delta H_{fus}[kJ/mol]\,=\,-0.88+\sum H_{fus,i}
  11. η L [ P a . s ] = M w e [ η a - 597.82 ] / T + η b - 11.202 \eta_{L}[Pa.s]\,=\,M_{w}e^{\left[\sum\eta_{a}-597.82\right]/T+\sum\eta_{b}-11.% 202}
  12. G i \sum G_{i}

John_Knox_(meteorologist).html

  1. e e 1 / 2 / e - 1 / 2 \scriptstyle e\,\equiv\,e^{1/2}/e^{-1/2}

John_R._Stallings.html

  1. 3 \mathcal{F}_{3}
  2. n \mathcal{F}_{n}
  3. n + 1 \mathcal{F}_{n+1}
  4. G = A C B \scriptstyle G=A\ast_{C}B
  5. G = H , t | t - 1 K t = L \scriptstyle G=\langle H,t|t^{-1}Kt=L\rangle

Jordan's_inequality.html

  1. | E G | = sin ( x ) |EG|=\sin(x)
  2. | D E | | D C ^ | | D G ^ | sin ( x ) x π 2 sin ( x ) 2 π x sin ( x ) x \begin{aligned}&\displaystyle|DE|\leq|\widehat{DC}|\leq|\widehat{DG}|\\ \displaystyle\Leftrightarrow&\displaystyle\sin(x)\leq x\leq\tfrac{\pi}{2}\sin(% x)\\ \displaystyle\Rightarrow&\displaystyle\tfrac{2}{\pi}x\leq\sin(x)\leq x\end{aligned}
  3. 2 π x sin x x for x [ 0 , π 2 ] . \frac{2}{\pi}x\leq\sin{x}\leq x\,\text{ for }x\in\left[0,\frac{\pi}{2}\right].

Jordan's_totient_function.html

  1. J k ( n ) J_{k}(n)
  2. J k ( n ) = n k p | n ( 1 - 1 p k ) . J_{k}(n)=n^{k}\prod_{p|n}\left(1-\frac{1}{p^{k}}\right).\,
  3. d | n J k ( d ) = n k . \sum_{d|n}J_{k}(d)=n^{k}.\,
  4. J k ( n ) 1 = n k J_{k}(n)\star 1=n^{k}\,
  5. J k ( n ) = μ ( n ) n k J_{k}(n)=\mu(n)\star n^{k}
  6. n 1 J k ( n ) n s = ζ ( s - k ) ζ ( s ) \sum_{n\geq 1}\frac{J_{k}(n)}{n^{s}}=\frac{\zeta(s-k)}{\zeta(s)}
  7. n k ζ ( k + 1 ) \frac{n^{k}}{\zeta(k+1)}
  8. ψ ( n ) = J 2 ( n ) J 1 ( n ) \psi(n)=\frac{J_{2}(n)}{J_{1}(n)}
  9. J k ( n ) J 1 ( n ) \frac{J_{k}(n)}{J_{1}(n)}
  10. J 2 k ( n ) J k ( n ) \frac{J_{2k}(n)}{J_{k}(n)}
  11. δ n δ s J r ( δ ) J s ( n δ ) = J r + s ( n ) \sum_{\delta\mid n}\delta^{s}J_{r}(\delta)J_{s}\left(\frac{n}{\delta}\right)=J% _{r+s}(n)
  12. | GL ( m , 𝐙 n ) | = n m ( m - 1 ) 2 k = 1 m J k ( n ) . |\operatorname{GL}(m,\mathbf{Z}_{n})|=n^{\frac{m(m-1)}{2}}\prod_{k=1}^{m}J_{k}% (n).
  13. | SL ( m , 𝐙 n ) | = n m ( m - 1 ) 2 k = 2 m J k ( n ) . |\operatorname{SL}(m,\mathbf{Z}_{n})|=n^{\frac{m(m-1)}{2}}\prod_{k=2}^{m}J_{k}% (n).
  14. | Sp ( 2 m , 𝐙 n ) | = n m 2 k = 1 m J 2 k ( n ) . |\operatorname{Sp}(2m,\mathbf{Z}_{n})|=n^{m^{2}}\prod_{k=1}^{m}J_{2k}(n).

Jordan–Wigner_transformation.html

  1. j j
  2. σ j + , σ j - , σ j z \sigma_{j}^{+},\sigma_{j}^{-},\sigma_{j}^{z}
  3. σ j + \sigma_{j}^{+}
  4. σ j - \sigma_{j}^{-}
  5. { σ j + , σ j - } = 1 \{\sigma_{j}^{+},\sigma_{j}^{-}\}=1
  6. σ j + = ( σ j x + i σ j y ) / 2 = f j \sigma_{j}^{+}=(\sigma_{j}^{x}+i\sigma_{j}^{y})/2=f_{j}^{\dagger}
  7. σ j - = ( σ j x - i σ j y ) / 2 = f j \sigma_{j}^{-}=(\sigma_{j}^{x}-i\sigma_{j}^{y})/2=f_{j}
  8. σ j z = 2 f j f j - 1. \sigma_{j}^{z}=2f_{j}^{\dagger}f_{j}-1.
  9. { f j , f j } = 1 \{f_{j}^{\dagger},f_{j}\}=1
  10. [ f j , f k ] = 0 [f_{j}^{\dagger},f_{k}]=0
  11. j k j\neq k
  12. a j = e + i π k = 1 j - 1 f k f k f j a_{j}^{\dagger}=e^{+i\pi\sum_{k=1}^{j-1}f^{\dagger}_{k}f_{k}}\cdot f^{\dagger}% _{j}
  13. a j = e - i π k = 1 j - 1 f k f k f j a_{j}=e^{-i\pi\sum_{k=1}^{j-1}f^{\dagger}_{k}f_{k}}\cdot f_{j}
  14. a j a j - 1 2 = f j f j - 1 2 . a_{j}^{\dagger}a_{j}-\frac{1}{2}=f^{\dagger}_{j}f_{j}-\frac{1}{2}.
  15. e ± i π k = 1 j - 1 f k f k e^{\pm i\pi\sum_{k=1}^{j-1}f^{\dagger}_{k}f_{k}}
  16. k = 1 , , j - 1 k=1,\ldots,j-1
  17. + 1 +1
  18. - 1 -1
  19. e ± i π k = 1 j - 1 f k f k = k = 1 j - 1 e ± i π f k f k = k = 1 j - 1 ( 1 - 2 f k f k ) . e^{\pm i\pi\sum_{k=1}^{j-1}f^{\dagger}_{k}f_{k}}=\prod_{k=1}^{j-1}e^{\pm i\pi f% ^{\dagger}_{k}f_{k}}=\prod_{k=1}^{j-1}(1-2f^{\dagger}_{k}f_{k}).
  20. f k f k { 0 , 1 } . f^{\dagger}_{k}f_{k}\in\{0,1\}.
  21. { a i , a j } = δ i , j , { a i , a j } = 0 , { a i , a j } = 0. \{a_{i}^{\dagger},a_{j}\}=\delta_{i,j},\,\{a_{i}^{\dagger},a_{j}^{\dagger}\}=0% ,\,\{a_{i},a_{j}\}=0.
  22. a j = e + i π k = 1 j - 1 a k a k σ j + a^{\dagger}_{j}=e^{+i\pi\sum_{k=1}^{j-1}a^{\dagger}_{k}a_{k}}\cdot\sigma_{j}^{+}
  23. a j = e - i π k = 1 j - 1 a k a k σ j - a_{j}=e^{-i\pi\sum_{k=1}^{j-1}a^{\dagger}_{k}a_{k}}\cdot\sigma_{j}^{-}
  24. a j a j = σ j z + 1 2 . a^{\dagger}_{j}a_{j}=\sigma_{j}^{z}+\frac{1}{2}.

Josef_Mattauch.html

  1. π / 4 2 \pi/4\sqrt{2}
  2. π / 2 \pi/2

Jónsson_function.html

  1. f : [ x ] ω x f:[x]^{\omega}\to x
  2. f f
  3. [ y ] ω [y]^{\omega}
  4. x x
  5. [ x ] ω [x]^{\omega}
  6. x x
  7. x x
  8. ω \omega

Juggler_sequence.html

  1. a k + 1 = { a k 1 2 , if a k is even a k 3 2 , if a k is odd . a_{k+1}=\begin{cases}\left\lfloor a_{k}^{\frac{1}{2}}\right\rfloor,&\mbox{if }% ~{}a_{k}\mbox{ is even}\\ \\ \left\lfloor a_{k}^{\frac{3}{2}}\right\rfloor,&\mbox{if }~{}a_{k}\mbox{ is odd% }~{}.\end{cases}
  2. a 1 = 3 3 2 = 5.196 = 5 , a_{1}=\lfloor 3^{\frac{3}{2}}\rfloor=\lfloor 5.196\dots\rfloor=5,
  3. a 2 = 5 3 2 = 11.180 = 11 , a_{2}=\lfloor 5^{\frac{3}{2}}\rfloor=\lfloor 11.180\dots\rfloor=11,
  4. a 3 = 11 3 2 = 36.482 = 36 , a_{3}=\lfloor 11^{\frac{3}{2}}\rfloor=\lfloor 36.482\dots\rfloor=36,
  5. a 4 = 36 1 2 = 6 = 6 , a_{4}=\lfloor 36^{\frac{1}{2}}\rfloor=\lfloor 6\rfloor=6,
  6. a 5 = 6 1 2 = 2.449 = 2 , a_{5}=\lfloor 6^{\frac{1}{2}}\rfloor=\lfloor 2.449\dots\rfloor=2,
  7. a 6 = 2 1 2 = 1.414 = 1. a_{6}=\lfloor 2^{\frac{1}{2}}\rfloor=\lfloor 1.414\dots\rfloor=1.

Justesen_code.html

  1. ( N , K , D ) q k (N,K,D)_{q^{k}}
  2. C o u t C_{out}
  3. ( n , k , d ) q (n,k,d)_{q}
  4. C i n i C_{in}^{i}
  5. 1 i N 1\leq i\leq N
  6. C o u t ( C i n 1 , , C i n N ) C_{out}\circ(C_{in}^{1},...,C_{in}^{N})
  7. m [ q k ] K m\in[q^{k}]^{K}
  8. C o u t C_{out}
  9. C o u t ( m ) = ( c 1 , c 2 , . . , c N ) C_{out}(m)=(c_{1},c_{2},..,c_{N})
  10. C o u t ( C i n 1 , . . , C i n N ) ( m ) = ( C i n 1 ( c 1 ) , C i n 2 ( c 2 ) , . . , C i n N ( c N ) ) C_{out}\circ(C_{in}^{1},..,C_{in}^{N})(m)=(C_{in}^{1}(c_{1}),C_{in}^{2}(c_{2})% ,..,C_{in}^{N}(c_{N}))
  11. N N
  12. N N
  13. N N
  14. C o u t C_{out}
  15. 𝔽 q k \mathbb{F}_{q^{k}}
  16. 𝔽 q k - { 0 } \mathbb{F}_{q^{k}}-\{0\}
  17. R R
  18. 0
  19. C o u t C_{out}
  20. δ o u t = 1 - R \delta_{out}=1-R
  21. N = q k - 1 N=q^{k}-1
  22. { C i n α } α 𝔽 q k * \{C_{in}^{\alpha}\}_{\alpha\in\mathbb{F}_{q^{k}}^{*}}
  23. 1 2 \frac{1}{2}
  24. C * = C o u t ( C i n 1 , C i n 2 , . . , C i n N ) C^{*}=C_{out}\circ(C_{in}^{1},C_{in}^{2},..,C_{in}^{N})
  25. R 2 \frac{R}{2}
  26. C * C^{*}
  27. ε \varepsilon
  28. C * C^{*}
  29. ( 1 - R - ε ) H q - 1 ( 1 2 - ε ) (1-R-\varepsilon)H_{q}^{-1}(\frac{1}{2}-\varepsilon)
  30. C * C^{*}
  31. ( 1 - R - ε ) H q - 1 ( 1 2 - ε ) 2 k N (1-R-\varepsilon)H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2k\cdot N
  32. ( 1 - R - ε ) H q - 1 ( 1 2 - ε ) 2 k N (1-R-\varepsilon)H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2k\cdot N
  33. Δ ( c 1 , c 2 ) \Delta(c^{1},c^{2})
  34. c 1 c^{1}
  35. c 2 c^{2}
  36. m 1 m_{1}
  37. m 2 m_{2}
  38. ( 𝔽 q k ) K (\mathbb{F}_{q^{k}})^{K}
  39. m 1 m 2 m_{1}\neq m_{2}
  40. Δ ( C * ( m 1 ) , C * ( m 2 ) ) \Delta(C^{*}(m_{1}),C^{*}(m_{2}))
  41. C o u t ( m ) = ( c 1 , c 2 , . . , c N ) C_{out}(m)=(c_{1},c_{2},..,c_{N})
  42. C * ( m ) = ( C i n 1 ( c 1 ) , C i n 2 ( c 2 ) , . . , C i n N ( c N ) ) C^{*}(m)=(C_{in}^{1}(c_{1}),C_{in}^{2}(c_{2}),..,C_{in}^{N}(c_{N}))
  43. Δ ( C * ( m 1 ) , C * ( m 2 ) ) \Delta(C^{*}(m_{1}),C^{*}(m_{2}))
  44. C i n i C_{in}^{i}
  45. C o u t ( m 1 ) = ( c 1 1 , c 2 1 , . . , c N 1 ) C_{out}(m_{1})=(c_{1}^{1},c_{2}^{1},..,c_{N}^{1})
  46. C o u t ( m 2 ) = ( c 1 2 , c 2 2 , . . , c N 2 ) C_{out}(m_{2})=(c_{1}^{2},c_{2}^{2},..,c_{N}^{2})
  47. { C i n i } 1 i N \{C_{in}^{i}\}_{1\leq i\leq N}
  48. ( 1 - ε ) N (1-\varepsilon)N
  49. C i n i C_{in}^{i}
  50. H q - 1 ( 1 2 - ε ) 2 k H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2k
  51. 1 i N 1\leq i\leq N
  52. c i 1 c i 2 c_{i}^{1}\neq c_{i}^{2}
  53. C i n i C_{in}^{i}
  54. H q - 1 ( 1 2 - ε ) 2 k \geq H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2k
  55. Δ ( C i n i ( c i 1 ) , C i n i ( c i 2 ) ) H q - 1 ( 1 2 - ε ) 2 k \Delta(C_{in}^{i}(c_{i}^{1}),C_{in}^{i}(c_{i}^{2}))\geq H_{q}^{-1}(\frac{1}{2}% -\varepsilon)\cdot 2k
  56. T T
  57. 1 i N 1\leq i\leq N
  58. c i 1 c i 2 c_{i}^{1}\neq c_{i}^{2}
  59. C i n i C_{in}^{i}
  60. H q - 1 ( 1 2 - ε ) 2 k \geq H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2k
  61. Δ ( C * ( m 1 ) , C * ( m 2 ) ) H q - 1 ( 1 2 - ε ) 2 k T \Delta(C^{*}(m_{1}),C^{*}(m_{2}))\geq H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2% k\cdot T
  62. T T
  63. i i
  64. 1 i N 1\leq i\leq N
  65. c i 1 c i 2 c_{i}^{1}\neq c_{i}^{2}
  66. T T
  67. C i n i C_{in}^{i}
  68. i S i\in S
  69. H q - 1 ( 1 2 - ε ) 2 k H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2k
  70. | S | \left|S\right|
  71. | S | = Δ ( C o u t ( m 1 ) , C o u t ( m 2 ) ) ( 1 - R ) N \left|S\right|=\Delta(C_{out}(m_{1}),C_{out}(m_{2}))\geq(1-R)N
  72. ε N \varepsilon N
  73. H q - 1 ( 1 2 - ε ) 2 k H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2k
  74. T | S | - ε N ( 1 - R ) N - ε N = ( 1 - R - ε ) N T\geq\left|S\right|-\varepsilon N\geq(1-R)N-\varepsilon N=(1-R-\varepsilon)N
  75. Δ ( C * ( m 1 ) , C * ( m 2 ) ) H q - 1 ( 1 2 - ε ) 2 k T H q - 1 ( 1 2 - ε ) 2 k ( 1 - R - ε ) N \Delta(C^{*}(m_{1}),C^{*}(m_{2}))\geq H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2% k\cdot T\geq H_{q}^{-1}(\frac{1}{2}-\varepsilon)\cdot 2k\cdot(1-R-\varepsilon)\cdot N
  76. m 1 m 2 m_{1}\neq m_{2}
  77. C * C^{*}
  78. ( 1 - R - ε ) H q - 1 ( 1 2 - ε ) (1-R-\varepsilon)H_{q}^{-1}(\frac{1}{2}-\varepsilon)
  79. C * C^{*}
  80. R R
  81. δ \delta
  82. 𝐛 = ( a 1 , a 1 , a 2 , α 1 a 2 , , a N , α N - 1 a N ) \mathbf{b}=\left(a_{1},a_{1},a_{2},\alpha^{1}a_{2},\ldots,a_{N},\alpha^{N-1}a_% {N}\right)
  83. i = 1 i ( 2 m i ) , \sum_{i=1}^{\ell}i{\left({{2m}\atop{i}}\right)},
  84. \ell
  85. i = 1 ( 2 m i ) N - K + 1 \sum_{i=1}^{\ell}{\left({{2m}\atop{i}}\right)}\leq N-K+1

Jürgen_Ehlers.html

  1. \to
  2. \to
  3. S L ( 2 ) SL(2)
  4. λ \lambda
  5. 1 / c 2 1/c^{2}

K-equivalence.html

  1. 𝒦 \mathcal{K}
  2. 𝒦 \scriptstyle\mathcal{K}
  3. f , g : X ( Y , 0 ) f,g:X\to(Y,0)
  4. 𝒦 \scriptstyle\mathcal{K}
  5. Ψ : X × Y X × Y \Psi:X\times Y\to X\times Y
  6. Ψ ( x , 0 ) = ( ϕ ( x ) , 0 ) \Psi(x,0)=(\phi(x),0)
  7. Ψ ( x , f ( x ) ) = ( ϕ ( x ) , g ( ϕ ( x ) ) ) \Psi(x,f(x))=(\phi(x),g(\phi(x)))
  8. 𝒜 \scriptstyle\mathcal{A}
  9. 𝒦 \scriptstyle\mathcal{K}
  10. 𝒦 \scriptstyle\mathcal{K}
  11. X × { 0 } X\times\{0\}
  12. X × V X\times V
  13. y V ψ ( x , y ) V y\in V\Rightarrow\psi(x,y)\in V
  14. 𝒜 \scriptstyle{\mathcal{A}}
  15. 𝒦 \scriptstyle\mathcal{K}

K-factor_(fire_protection).html

  1. q = K p q=K\sqrt{p}
  2. L P M / bar LPM/\sqrt{\,}\text{bar}

K-Poincaré_algebra.html

  1. [ P μ , P ν ] = 0 [P_{\mu},P_{\nu}]=0\,
  2. [ R j , P 0 ] = 0 , [ R j , P k ] = i ε j k l P l , [ R j , N k ] = i ε j k l N l , [ R j , R k ] = i ε j k l R l [R_{j},P_{0}]=0,\;[R_{j},P_{k}]=i\varepsilon_{jkl}P_{l},\;[R_{j},N_{k}]=i% \varepsilon_{jkl}N_{l},\;[R_{j},R_{k}]=i\varepsilon_{jkl}R_{l}\,
  3. [ N j , P 0 ] = i P j , [ N j , P k ] = i δ j k ( 1 - e - 2 λ P 0 2 λ + λ 2 | P | 2 ) - i λ P j P k , [ N j , N k ] = - i ε j k l R l [N_{j},P_{0}]=iP_{j},\;[N_{j},P_{k}]=i\delta_{jk}\left(\frac{1-e^{-2\lambda P_% {0}}}{2\lambda}+\frac{\lambda}{2}|\vec{P}|^{2}\right)-i\lambda P_{j}P_{k},\;[N% _{j},N_{k}]=-i\varepsilon_{jkl}R_{l}\,
  4. P μ P_{\mu}
  5. R j R_{j}
  6. N j N_{j}
  7. Δ P j = P j 1 + e - λ P 0 P j , Δ P 0 = P 0 1 + 1 P 0 \Delta P_{j}=P_{j}\otimes 1+e^{-\lambda P_{0}}\otimes P_{j}~{},\qquad\Delta P_% {0}=P_{0}\otimes 1+1\otimes P_{0}\,
  8. Δ R j = R j 1 + 1 R j \Delta R_{j}=R_{j}\otimes 1+1\otimes R_{j}\,
  9. Δ N k = N k 1 + e - λ P 0 N k + i λ ε k l m P l R m . \Delta N_{k}=N_{k}\otimes 1+e^{-\lambda P_{0}}\otimes N_{k}+i\lambda% \varepsilon_{klm}P_{l}\otimes R_{m}.
  10. S ( P 0 ) = - P 0 S(P_{0})=-P_{0}\,
  11. S ( P j ) = - e λ P 0 P j S(P_{j})=-e^{\lambda P_{0}}P_{j}\,
  12. S ( R j ) = - R j S(R_{j})=-R_{j}\,
  13. S ( N j ) = - e λ P 0 N j + i λ ε j k l e λ P 0 P k R l S(N_{j})=-e^{\lambda P_{0}}N_{j}+i\lambda\varepsilon_{jkl}e^{\lambda P_{0}}P_{% k}R_{l}\,
  14. ε ( P 0 ) = 0 \varepsilon(P_{0})=0\,
  15. ε ( P j ) = 0 \varepsilon(P_{j})=0\,
  16. ε ( R j ) = 0 \varepsilon(R_{j})=0\,
  17. ε ( N j ) = 0 \varepsilon(N_{j})=0\,

K-Poincaré_group.html

  1. a μ a^{\mu}
  2. Λ μ ν {\Lambda^{\mu}}_{\nu}
  3. η ρ σ Λ μ ρ Λ ν σ = η μ ν , \eta^{\rho\sigma}{\Lambda^{\mu}}_{\rho}{\Lambda^{\nu}}_{\sigma}=\eta^{\mu\nu}~% {},
  4. η μ ν \eta^{\mu\nu}
  5. η μ ν = ( - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) . \eta^{\mu\nu}=\left(\begin{array}[]{cccc}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{array}\right)~{}.
  6. [ a j , a 0 ] = i λ a j , [ a j , a k ] = 0 [a_{j},a_{0}]=i\lambda a_{j}~{},\;[a_{j},a_{k}]=0\,
  7. [ a μ , Λ ρ σ ] = i λ { ( Λ ρ 0 - δ ρ 0 ) Λ μ σ - ( Λ α σ η α 0 + η σ 0 ) η ρ μ } [a^{\mu},{\Lambda^{\rho}}_{\sigma}]=i\lambda\left\{\left({\Lambda^{\rho}}_{0}-% {\delta^{\rho}}_{0}\right){\Lambda^{\mu}}_{\sigma}-\left({\Lambda^{\alpha}}_{% \sigma}\eta_{\alpha 0}+\eta_{\sigma 0}\right)\eta^{\rho\mu}\right\}\,
  8. a μ a^{\mu}
  9. Λ μ ν {\Lambda^{\mu}}_{\nu}
  10. Λ μ ν = ( cosh τ sinh τ sinh τ cosh τ ) {\Lambda^{\mu}}_{\nu}=\left(\begin{array}[]{cc}\cosh\tau&\sinh\tau\\ \sinh\tau&\cosh\tau\end{array}\right)\,
  11. [ a 0 , ( cosh τ sinh τ ) ] = i λ sinh τ ( sinh τ cosh τ ) [a_{0},\left(\begin{array}[]{c}\cosh\tau\\ \sinh\tau\end{array}\right)]=i\lambda~{}\sinh\tau\left(\begin{array}[]{c}\sinh% \tau\\ \cosh\tau\end{array}\right)\,
  12. [ a 1 , ( cosh τ sinh τ ) ] = i λ ( 1 - cosh τ ) ( sinh τ cosh τ ) [a_{1},\left(\begin{array}[]{c}\cosh\tau\\ \sinh\tau\end{array}\right)]=i\lambda\left(1-\cosh\tau\right)\left(\begin{% array}[]{c}\sinh\tau\\ \cosh\tau\end{array}\right)\,
  13. Δ a μ = Λ μ ν a ν + a μ 1 \Delta a^{\mu}={\Lambda^{\mu}}_{\nu}\otimes a^{\nu}+a^{\mu}\otimes 1\,
  14. Δ Λ μ ν = Λ μ ρ Λ ρ ν \Delta{\Lambda^{\mu}}_{\nu}={\Lambda^{\mu}}_{\rho}\otimes{\Lambda^{\rho}}_{\nu}\,
  15. S ( a μ ) = - ( Λ - 1 ) μ ν a ν S(a^{\mu})=-{(\Lambda^{-1})^{\mu}}_{\nu}a^{\nu}\,
  16. S ( Λ μ ν ) = ( Λ - 1 ) μ ν = Λ ν μ S({\Lambda^{\mu}}_{\nu})={(\Lambda^{-1})^{\mu}}_{\nu}={\Lambda_{\nu}}^{\mu}\,
  17. ε ( a μ ) = 0 \varepsilon(a^{\mu})=0
  18. ε ( Λ μ ν ) = δ μ ν \varepsilon({\Lambda^{\mu}}_{\nu})={\delta^{\mu}}_{\nu}\,

Kalman_decomposition.html

  1. x ˙ ( t ) = A x ( t ) + B u ( t ) \dot{x}(t)=Ax(t)+Bu(t)
  2. y ( t ) = C x ( t ) + D u ( t ) \,y(t)=Cx(t)+Du(t)
  3. x \,x
  4. y \,y
  5. u \,u
  6. A \,A
  7. B \,B
  8. C \,C
  9. D \,D
  10. x ( k + 1 ) = A x ( k ) + B u ( k ) \,x(k+1)=Ax(k)+Bu(k)
  11. y ( k ) = C x ( k ) + D u ( k ) \,y(k)=Cx(k)+Du(k)
  12. ( A , B , C , D ) \,(A,B,C,D)
  13. n \,n
  14. ( A , B , C , D ) \,(A,B,C,D)
  15. ( A ^ , B ^ , C ^ , D ^ ) \,(\hat{A},\hat{B},\hat{C},\hat{D})
  16. A ^ = T - 1 A T \,{\hat{A}}={T^{-1}}AT
  17. B ^ = T - 1 B \,{\hat{B}}={T^{-1}}B
  18. C ^ = C T \,{\hat{C}}=CT
  19. D ^ = D \,{\hat{D}}=D
  20. T \,T
  21. n × n \,n\times n
  22. T = [ T r o ¯ T r o T r o ¯ T r ¯ o ] \,T=\begin{bmatrix}T_{r\overline{o}}&T_{ro}&T_{\overline{ro}}&T_{\overline{r}o% }\end{bmatrix}
  23. T r o ¯ \,T_{r\overline{o}}
  24. T r o \,T_{ro}
  25. [ T r o ¯ T r o ] \,\begin{bmatrix}T_{r\overline{o}}&T_{ro}\end{bmatrix}
  26. T r o ¯ \,T_{\overline{ro}}
  27. [ T r o ¯ T r o ¯ ] \,\begin{bmatrix}T_{r\overline{o}}&T_{\overline{ro}}\end{bmatrix}
  28. T r ¯ o \,T_{\overline{r}o}
  29. [ T r o ¯ T r o T r o ¯ T r ¯ o ] \,\begin{bmatrix}T_{r\overline{o}}&T_{ro}&T_{\overline{ro}}&T_{\overline{r}o}% \end{bmatrix}
  30. T \,T
  31. T = T r o \,T=T_{ro}
  32. ( A ^ , B ^ , C ^ , D ^ ) \,(\hat{A},\hat{B},\hat{C},\hat{D})
  33. A ^ = [ A r o ¯ A 12 A 13 A 14 0 A r o 0 A 24 0 0 A r o ¯ A 34 0 0 0 A r ¯ o ] \,\hat{A}=\begin{bmatrix}A_{r\overline{o}}&A_{12}&A_{13}&A_{14}\\ 0&A_{ro}&0&A_{24}\\ 0&0&A_{\overline{ro}}&A_{34}\\ 0&0&0&A_{\overline{r}o}\end{bmatrix}
  34. B ^ = [ B r o ¯ B r o 0 0 ] \,\hat{B}=\begin{bmatrix}B_{r\overline{o}}\\ B_{ro}\\ 0\\ 0\end{bmatrix}
  35. C ^ = [ 0 C r o 0 C r ¯ o ] \,\hat{C}=\begin{bmatrix}0&C_{ro}&0&C_{\overline{r}o}\end{bmatrix}
  36. D ^ = D \,\hat{D}=D
  37. ( A r o , B r o , C r o , D ) \,(A_{ro},B_{ro},C_{ro},D)
  38. ( [ A r o ¯ A 12 0 A r o ] , [ B r o ¯ B r o ] , [ 0 C r o ] , D ) \,\left(\begin{bmatrix}A_{r\overline{o}}&A_{12}\\ 0&A_{ro}\end{bmatrix},\begin{bmatrix}B_{r\overline{o}}\\ B_{ro}\end{bmatrix},\begin{bmatrix}0&C_{ro}\end{bmatrix},D\right)
  39. ( [ A r o A 24 0 A r ¯ o ] , [ B r o 0 ] , [ C r o C r ¯ o ] , D ) \,\left(\begin{bmatrix}A_{ro}&A_{24}\\ 0&A_{\overline{r}o}\end{bmatrix},\begin{bmatrix}B_{ro}\\ 0\end{bmatrix},\begin{bmatrix}C_{ro}&C_{\overline{r}o}\end{bmatrix},D\right)

Kapitsa–Dirac_effect.html

  1. λ = h p , \lambda=\frac{h}{p},
  2. τ 1 / ω r e c \tau<<1/\omega_{rec}
  3. n λ = 2 d sin Θ , n\lambda=2d\sin\Theta,
  4. ω r e c = k 2 2 m \omega_{rec}=\frac{\hbar k^{2}}{2m}
  5. E r e c = ω r e c . E_{rec}=\hbar\omega_{rec}.
  6. U ( z , t ) = ω r e c 2 δ f 2 ( t ) sin 2 ( k z ) , U(z,t)=\frac{\hbar\omega^{2}_{rec}}{\delta}f^{2}(t)\sin^{2}(kz),
  7. δ Γ 2 / 4 \delta<<\Gamma^{2}/4
  8. Γ \Gamma
  9. | ψ = | ψ 0 e - i d t U ( z , t ) = | ψ 0 e - i 2 δ ω r e c 2 τ e i 2 δ ω r e c 2 τ cos ( 2 k z ) , \left|\psi\right\rangle=\left|\psi_{0}\right\rangle e^{-\frac{i}{\hbar}\int dt% ^{\prime}U(z,t^{\prime})}=\left|\psi_{0}\right\rangle e^{-\frac{i}{2\delta}% \omega^{2}_{rec}\tau}e^{\frac{i}{2\delta}\omega^{2}_{rec}\tau\cos(2kz)},
  10. τ = d t f 2 ( t ) \tau=\int dt^{\prime}f^{2}(t^{\prime})
  11. e i α cos ( β ) = n = - inf inf i n J n ( α ) e i n β e^{i\alpha\cos(\beta)}=\sum^{\inf}_{n=-\inf}i^{n}J_{n}(\alpha)e^{in\beta}
  12. | ψ = | ψ 0 e - i 2 δ ω r e c 2 τ n = - inf inf i n J n ( ω r e c 2 2 δ τ ) e i 2 n k z = e - i 2 δ ω r e c 2 τ n = - inf inf i n J n ( ω r e c 2 2 δ τ ) | g , 2 n k . \left|\psi\right\rangle=\left|\psi_{0}\right\rangle e^{-\frac{i}{2\delta}% \omega^{2}_{rec}\tau}\sum^{\inf}_{n=-\inf}i^{n}J_{n}(\frac{\omega^{2}_{rec}}{2% \delta}\tau)e^{i2nkz}=e^{-\frac{i}{2\delta}\omega^{2}_{rec}\tau}\sum^{\inf}_{n% =-\inf}i^{n}J_{n}(\frac{\omega^{2}_{rec}}{2\delta}\tau)\left|g,2n\hbar k\right\rangle.
  13. 2 n k 2n\hbar k
  14. P n = J n 2 ( θ ) P_{n}=J^{2}_{n}(\theta)
  15. n = 0 , ± 1 , ± 2... n=0,\pm 1,\pm 2...
  16. θ = ω r e c 2 2 δ τ = ω r e c ( 2 ) τ \theta=\frac{\omega^{2}_{rec}}{2\delta}\tau=\omega^{(2)}_{rec}\tau
  17. p r m s = n = - inf inf ( n k ) 2 P n = 2 θ k . p_{rms}=\sum^{\inf}_{n=-\inf}(n\hbar k)^{2}P_{n}=\sqrt{2}\theta\hbar k.

Karel_Lambert.html

  1. x ϕ x ϕ y \forall x\,\phi x\rightarrow\phi y
  2. ( x ϕ x and E ! y ϕ y ) ϕ z (\forall x\,\phi x\and E!y\,\phi y)\rightarrow\phi z
  3. x ( M x L x ) \forall x\,(Mx\rightarrow Lx)
  4. x U x \forall x\,Ux
  5. x ( M x and L x ) \exists x\,(Mx\and Lx)
  6. x U x \exists x\,Ux
  7. ( λ x ) ( ϕ x and ¬ ϕ x ) (\lambda x)(\phi x\and\neg\phi x)

Karen_Vogtmann.html

  1. O n , n O_{n,n}

Kayles.html

  1. n n
  2. K n K_{n}
  3. K n K_{n}
  4. K 5 = mex { K 0 + K 4 , K 1 + K 3 , K 2 + K 2 , K 0 + K 3 , K 1 + K 2 } , K_{5}=\mbox{mex}~{}\{K_{0}+K_{4},K_{1}+K_{3},K_{2}+K_{2},K_{0}+K_{3},K_{1}+K_{% 2}\},\,
  5. K 0 + K 4 , K 1 + K 3 , K 2 + K 2 , K 0 + K 3 , and K 1 + K 2 . K_{0}+K_{4},\quad K_{1}+K_{3},\quad K_{2}+K_{2},\quad K_{0}+K_{3},\,\text{ and% }K_{1}+K_{2}.\,
  6. K 0 = 0 K_{0}=0
  7. K n K_{n}
  8. n n
  9. 12 a + b 12a+b
  10. K 83 K_{83}
  11. K n K_{n}

Kempner_series.html

  1. n = 1 1 n {\sum_{n=1}^{\infty}\ }^{\prime}\frac{1}{n}
  2. 8 n = 1 ( 9 10 ) n - 1 = 80. 8\sum_{n=1}^{\infty}\left(\frac{9}{10}\right)^{n-1}=80.

Kenneth_L._Clarkson.html

  1. k k
  2. n n
  3. O ( k + n log n ) O(k+n\log n)
  4. n n
  5. O ( n log n ) O(n\log n)
  6. n n
  7. d d
  8. O ( n d / 2 + n log n ) O(n^{\lfloor d/2\rfloor}+n\log n)

Kepler_orbit.html

  1. r ( ν ) = a ( 1 - e 2 ) 1 + e cos ( ν ) r(\nu)=\frac{a(1-e^{2})}{1+e\cos(\nu)}
  2. r r
  3. a a
  4. e e
  5. ν \nu
  6. r ( ν ) = p 1 + e cos ( ν ) r(\nu)=\frac{p}{1+e\cos(\nu)}
  7. p p
  8. 𝐅 = m 𝐚 = m d 2 𝐫 d t 2 \mathbf{F}=m\mathbf{a}=m\frac{d^{2}\mathbf{r}}{dt^{2}}
  9. 𝐅 \mathbf{F}
  10. m m
  11. 𝐚 \mathbf{a}
  12. 𝐫 \mathbf{r}
  13. F = G m 1 m 2 r 2 F=G\frac{m_{1}m_{2}}{r^{2}}
  14. F F
  15. G G
  16. m 1 m_{1}
  17. m 2 m_{2}
  18. r r
  19. m 1 m_{1}
  20. m 2 m_{2}
  21. 𝐫 1 \mathbf{r}_{1}
  22. 𝐫 2 \mathbf{r}_{2}
  23. m 1 𝐫 ¨ 1 = - G m 1 m 2 r 2 𝐫 ^ m_{1}\ddot{\mathbf{r}}_{1}=\frac{-Gm_{1}m_{2}}{r^{2}}\mathbf{\hat{r}}
  24. m 2 𝐫 ¨ 2 = G m 1 m 2 r 2 𝐫 ^ m_{2}\ddot{\mathbf{r}}_{2}=\frac{Gm_{1}m_{2}}{r^{2}}\mathbf{\hat{r}}
  25. 𝐫 \mathbf{r}
  26. 𝐫 = 𝐫 1 - 𝐫 2 \mathbf{r}=\mathbf{r}_{1}-\mathbf{r}_{2}
  27. 𝐫 ^ \mathbf{\hat{r}}
  28. r r
  29. 𝐫 ¨ = - μ r 2 𝐫 ^ \ddot{\mathbf{r}}=-\frac{\mu}{r^{2}}\mathbf{\hat{r}}
  30. μ \mu
  31. μ = G ( m 1 + m 2 ) \mu=G(m_{1}+m_{2})
  32. a a\,\!
  33. e e\,\!
  34. i i\,\!
  35. Ω \Omega\,\!
  36. ω \omega\,\!
  37. ν \nu
  38. M M
  39. T T
  40. i i
  41. Ω \Omega
  42. ω \omega
  43. H = r × r ˙ {H}={r}\times{\dot{{r}}}
  44. H ˙ = d d t ( r × r ˙ ) = r ˙ × r ˙ + r × r ¨ = 0 + 0 = 0 \dot{{H}}=\frac{d}{dt}\left({r}\times{\dot{{r}}}\right)=\dot{{r}}\times{\dot{{% r}}}+{r}\times{\ddot{{r}}}={0}+{0}={0}
  45. H {H}
  46. ( r ¨ ) \left(\ddot{{r}}\right)
  47. ( θ ¨ ) \left(\ddot{\theta}\right)
  48. ( r ¨ ) \left(\ddot{r}\right)
  49. ( x ^ , y ^ ) (\hat{{x}}\ ,\ \hat{{y}})
  50. ( r ^ , s y m b o l θ ^ ) (\hat{{r}}\ ,\ \hat{symbol\theta})
  51. H {H}
  52. r ^ = cos ( θ ) x ^ + sin ( θ ) y ^ \hat{{r}}=\cos(\theta)\hat{{x}}+\sin(\theta)\hat{{y}}
  53. s y m b o l θ ^ = - sin ( θ ) x ^ + cos ( θ ) y ^ \hat{symbol\theta}=-\sin(\theta)\hat{{x}}+\cos(\theta)\hat{{y}}
  54. r {r}
  55. r = r ( cos θ x ^ + sin θ y ^ ) = r 𝐫 ^ {r}=r(\cos\theta\hat{{x}}+\sin\theta\hat{{y}})=r\hat{\mathbf{r}}
  56. r ˙ = r ˙ 𝐫 ^ + r θ ˙ s y m b o l θ ^ \dot{{r}}=\dot{r}\hat{\mathbf{r}}+r\dot{\theta}\hat{symbol{\theta}}
  57. r ¨ = ( r ¨ - r θ ˙ 2 ) 𝐫 ^ + ( r θ ¨ + 2 r ˙ θ ˙ ) s y m b o l θ ^ \ddot{{r}}=(\ddot{r}-r\dot{\theta}^{2})\hat{\mathbf{r}}+(r\ddot{\theta}+2\dot{% r}\dot{\theta})\hat{symbol\theta}
  58. ( r ¨ - r θ ˙ 2 ) 𝐫 ^ + ( r θ ¨ + 2 r ˙ θ ˙ ) s y m b o l θ ^ = ( - μ r 2 ) 𝐫 ^ + ( 0 ) s y m b o l θ ^ (\ddot{r}-r\dot{\theta}^{2})\hat{\mathbf{r}}+(r\ddot{\theta}+2\dot{r}\dot{% \theta})\hat{symbol\theta}=\left(-\frac{\mu}{r^{2}}\right)\hat{\mathbf{r}}+(0)% \hat{symbol\theta}
  59. r ¨ - r θ ˙ 2 = - μ r 2 \ddot{r}-r{\dot{\theta}}^{2}=-\frac{\mu}{r^{2}}
  60. H = | r × r ˙ | = | ( r cos ( θ ) , r sin ( θ ) , 0 ) × ( r ˙ cos ( θ ) - r sin ( θ ) θ ˙ , r ˙ sin ( θ ) + r cos ( θ ) θ ˙ , 0 ) | = | ( 0 , 0 , r 2 θ ˙ ) | = r 2 θ ˙ H=|{r}\times{\dot{{r}}}|=|(r\cos(\theta),r\sin(\theta),0)\times(\dot{r}\cos(% \theta)-r\sin(\theta)\dot{\theta},\dot{r}\sin(\theta)+r\cos(\theta)\dot{\theta% },0)|=|(0,0,r^{2}\dot{\theta})|=r^{2}\dot{\theta}
  61. θ ˙ = H r 2 \dot{\theta}=\frac{H}{r^{2}}
  62. θ ¨ = - 2 H r ˙ r 3 \ddot{\theta}=-\frac{2\cdot H\cdot\dot{r}}{r^{3}}
  63. θ \theta
  64. r r
  65. r ˙ = d r d θ θ ˙ \dot{r}=\frac{dr}{d\theta}\cdot\dot{\theta}
  66. r ¨ = d 2 r d θ 2 θ ˙ 2 + d r d θ θ ¨ \ddot{r}=\frac{d^{2}r}{d\theta^{2}}\cdot{\dot{\theta}}^{2}+\frac{dr}{d\theta}% \cdot\ddot{\theta}
  67. r r
  68. θ \theta\,
  69. r ¨ - r θ ˙ 2 = - μ r 2 \ddot{r}-r{\dot{\theta}}^{2}=-\frac{\mu}{r^{2}}
  70. d 2 r d θ 2 θ ˙ 2 + d r d θ θ ¨ - r θ ˙ 2 = - μ r 2 \frac{d^{2}r}{d\theta^{2}}\cdot{\dot{\theta}}^{2}+\frac{dr}{d\theta}\cdot\ddot% {\theta}-r{\dot{\theta}}^{2}=-\frac{\mu}{r^{2}}
  71. d 2 r d θ 2 ( H r 2 ) 2 + d r d θ ( - 2 H r ˙ r 3 ) - r ( H r 2 ) 2 = - μ r 2 \frac{d^{2}r}{d\theta^{2}}\cdot\left(\frac{H}{r^{2}}\right)^{2}+\frac{dr}{d% \theta}\cdot\left(-\frac{2\cdot H\cdot\dot{r}}{r^{3}}\right)-r\left(\frac{H}{r% ^{2}}\right)^{2}=-\frac{\mu}{r^{2}}
  72. H 2 r 4 ( d 2 r d θ 2 - 2 ( d r d θ ) 2 r - r ) = - μ r 2 \frac{H^{2}}{r^{4}}\cdot\left(\frac{d^{2}r}{d\theta^{2}}-2\cdot\frac{\left(% \frac{dr}{d\theta}\right)^{2}}{r}-r\right)=-\frac{\mu}{r^{2}}
  73. r = 1 s r=\frac{1}{s}
  74. d r d θ = - 1 s 2 d s d θ \frac{dr}{d\theta}=-\frac{1}{s^{2}}\cdot\frac{ds}{d\theta}
  75. d 2 r d θ 2 = 2 s 3 ( d s d θ ) 2 - 1 s 2 d 2 s d θ 2 \frac{d^{2}r}{d\theta^{2}}=\frac{2}{s^{3}}\cdot\left(\frac{ds}{d\theta}\right)% ^{2}-\frac{1}{s^{2}}\cdot\frac{d^{2}s}{d\theta^{2}}
  76. d 2 r d θ 2 \frac{d^{2}r}{d\theta^{2}}
  77. d r d θ \frac{dr}{d\theta}
  78. H 2 ( d 2 s d θ 2 + s ) = μ H^{2}\cdot\left(\frac{d^{2}s}{d\theta^{2}}+s\right)=\mu
  79. s = μ H 2 ( 1 + e cos ( θ - θ 0 ) ) s=\frac{\mu}{H^{2}}\cdot\left(1+e\cdot\cos(\theta-\theta_{0})\right)
  80. θ 0 \theta_{0}\,
  81. d s d θ \frac{ds}{d\theta}
  82. θ 0 \theta_{0}\,
  83. x ^ , y ^ \hat{x}\ ,\ \hat{y}
  84. θ 0 \theta_{0}\,
  85. θ \theta\,
  86. s s
  87. r = 1 s r=\frac{1}{s}
  88. H 2 μ \frac{H^{2}}{\mu}
  89. r = 1 s = p 1 + e cos θ r=\frac{1}{s}=\frac{p}{1+e\cdot\cos\theta}
  90. u {u}
  91. r = r u {r}=r{u}
  92. r ¨ = - μ r 2 u \ddot{{r}}=-\frac{\mu}{r^{2}}{u}
  93. H = r × r ˙ = r u × d d t ( r u ) = r u × ( r u ˙ + r ˙ u ) = r 2 ( u × u ˙ ) + r r ˙ ( u × u ) = r 2 u × u ˙ {H}={r}\times\dot{{r}}=r{u}\times\frac{d}{dt}(r{u})=r{u}\times(r\dot{{u}}+\dot% {r}{u})=r^{2}({u}\times\dot{{u}})+r\dot{r}({u}\times{u})=r^{2}{u}\times\dot{{u}}
  94. r ¨ × H = - μ r 2 u × ( r 2 u × u ˙ ) = - μ u × ( u × u ˙ ) = - μ [ ( u u ˙ ) u - ( u u ) u ˙ ] \ddot{{r}}\times{H}=-\frac{\mu}{r^{2}}{u}\times(r^{2}{u}\times\dot{{u}})=-\mu{% u}\times({u}\times\dot{{u}})=-\mu[({u}\cdot\dot{{u}}){u}-({u}\cdot{u})\dot{{u}}]
  95. u u = | u | 2 = 1 {u}\cdot{u}=|{u}|^{2}=1
  96. u u ˙ = 1 2 ( u u ˙ + u ˙ u ) = 1 2 d d t ( u u ) = 0 {u}\cdot\dot{{u}}=\frac{1}{2}({u}\cdot\dot{{u}}+\dot{{u}}\cdot{u})=\frac{1}{2}% \frac{d}{dt}({u}\cdot{u})=0
  97. r ¨ × H = μ u ˙ \ddot{{r}}\times{H}=\mu\dot{{u}}
  98. r ˙ × H = μ u + c \dot{{r}}\times{H}=\mu{u}+{c}
  99. r ( r ˙ × H ) = r ( μ u + c ) = μ r u + r c = μ r ( u u ) + r c cos ( θ ) = r ( μ + c cos ( θ ) ) {r}\cdot(\dot{{r}}\times{H})={r}\cdot(\mu{u}+{c})=\mu{r}\cdot{u}+{r}\cdot{c}=% \mu r({u}\cdot{u})+rc\cos(\theta)=r(\mu+c\cos(\theta))
  100. θ \theta
  101. r ¯ \bar{r}
  102. c ¯ \bar{c}
  103. r = r ( r ˙ × H ) μ + c cos ( θ ) = ( r × r ˙ ) H μ + c cos ( θ ) = | H | 2 μ + c cos ( θ ) r=\frac{{r}\cdot(\dot{{r}}\times{H})}{\mu+c\cos(\theta)}=\frac{({r}\times\dot{% {r}})\cdot{H}}{\mu+c\cos(\theta)}=\frac{|{H}|^{2}}{\mu+c\cos(\theta)}
  104. ( r , θ ) (r,\theta)
  105. p = | H | 2 μ p=\frac{|{H}|^{2}}{\mu}
  106. e = c μ e=\frac{c}{\mu}
  107. r = p 1 + e cos θ r=\frac{p}{1+e\cdot\cos\theta}
  108. θ \theta\,
  109. e = 0 e\ =\ 0\,
  110. 0 < e < 1 0\ <e\ <\ 1\,
  111. a = p 1 - e 2 a=\frac{p}{1-e^{2}}
  112. b = p 1 - e 2 = a 1 - e 2 b=\frac{p}{\sqrt{1-e^{2}}}=a\cdot\sqrt{1-e^{2}}
  113. e = 1 e\ =\ 1\,
  114. p 2 \frac{p}{2}
  115. e > 1 e\ >\ 1\,
  116. a = p e 2 - 1 a=\frac{p}{e^{2}-1}
  117. b = p e 2 - 1 = a e 2 - 1 b=\frac{p}{\sqrt{e^{2}-1}}=a\cdot\sqrt{e^{2}-1}
  118. θ = 0 \theta=0\,
  119. p 1 + e \frac{p}{1+e}
  120. p 1 - e \frac{p}{1-e}
  121. θ \theta\,
  122. [ - cos - 1 ( - 1 e ) < θ < cos - 1 ( - 1 e ) ] \left[-\cos^{-1}\left(-\frac{1}{e}\right)<\theta<\cos^{-1}\left(-\frac{1}{e}% \right)\right]
  123. [ - π < θ < π ] \left[-\pi<\theta<\pi\right]
  124. H 2 μ \frac{H^{2}}{\mu}
  125. V r = r ˙ = H p e sin θ = μ p e sin θ V_{r}=\dot{r}=\frac{H}{p}\cdot e\cdot\sin\theta=\sqrt{\frac{\mu}{p}}\cdot e% \cdot\sin\theta
  126. V r V_{r}
  127. V t = r θ ˙ = H r = μ p ( 1 + e cos θ ) V_{t}=r\cdot\dot{\theta}=\frac{H}{r}=\sqrt{\frac{\mu}{p}}\cdot(1+e\cdot\cos\theta)
  128. θ \theta\,
  129. x ˙ = - a sin E E ˙ \dot{x}=-a\cdot\sin E\cdot\dot{E}
  130. y ˙ = b cos E E ˙ \dot{y}=b\cdot\cos E\cdot\dot{E}
  131. H = x y ˙ - y x ˙ = a b ( 1 - e cos E ) E ˙ H=x\cdot\dot{y}-y\cdot\dot{x}=a\cdot b\cdot(1-e\cdot\cos E)\cdot\dot{E}
  132. t = 0 t=0
  133. H = μ p H=\sqrt{\mu\cdot p}
  134. t = a a μ ( E - e sin E ) t=a\cdot\sqrt{\frac{a}{\mu}}(E-e\cdot\sin E)
  135. x ˙ = - a sinh E E ˙ \dot{x}=-a\cdot\sinh E\cdot\dot{E}
  136. y ˙ = b cosh E E ˙ \dot{y}=b\cdot\cosh E\cdot\dot{E}
  137. H = x y ˙ - y x ˙ = a b ( e cosh E - 1 ) E ˙ H=x\cdot\dot{y}-y\cdot\dot{x}=a\cdot b\cdot(e\cdot\cosh E-1)\cdot\dot{E}
  138. t = a a μ ( e sinh E - E ) t=a\cdot\sqrt{\frac{a}{\mu}}(e\cdot\sinh E-E)
  139. θ \theta\,
  140. [ - < t < ] [ - < E < ] \left[-\infty<t<\infty\right]\longleftrightarrow\left[-\infty<E<\infty\right]
  141. cos θ = x r = cos E - e 1 - e cos E \cos\theta=\frac{x}{r}=\frac{\cos E-e}{1-e\cdot\cos E}
  142. tan 2 θ 2 = 1 - cos θ 1 + cos θ = 1 - cos E - e 1 - e cos E 1 + cos E - e 1 - e cos E = 1 - e cos E - cos E + e 1 - e cos E + cos E - e = 1 + e 1 - e 1 - cos E 1 + cos E = 1 + e 1 - e tan 2 E 2 \tan^{2}\frac{\theta}{2}=\frac{1-\cos\theta}{1+\cos\theta}=\frac{1-\frac{\cos E% -e}{1-e\cdot\cos E}}{1+\frac{\cos E-e}{1-e\cdot\cos E}}=\frac{1-e\cdot\cos E-% \cos E+e}{1-e\cdot\cos E+\cos E-e}=\frac{1+e}{1-e}\ \cdot\ \frac{1-\cos E}{1+% \cos E}=\frac{1+e}{1-e}\ \cdot\ \tan^{2}\frac{E}{2}
  143. ( cos E , sin E ) (\ \cos E\ ,\ \sin E\ )
  144. ( cos θ , sin θ ) (\ \cos\theta\ ,\ \sin\theta\ )
  145. ( cos E 2 , sin E 2 ) \left(\cos\frac{E}{2}\ ,\ \sin\frac{E}{2}\right)
  146. ( cos θ 2 , sin θ 2 ) \left(\cos\frac{\theta}{2}\ ,\ \sin\frac{\theta}{2}\right)
  147. tan θ 2 = 1 + e 1 - e tan E 2 \tan\frac{\theta}{2}=\sqrt{\frac{1+e}{1-e}}\cdot\tan\frac{E}{2}
  148. θ = 2 arg ( 1 - e cos E 2 , 1 + e sin E 2 ) + n 2 π \theta=2\cdot\operatorname{arg}\left(\sqrt{1-e}\ \cdot\ \cos\frac{E}{2}\ ,\ % \sqrt{1+e}\ \cdot\ \sin\frac{E}{2}\right)+n\cdot 2\pi
  149. E = 2 arg ( 1 + e cos θ 2 , 1 - e sin θ 2 ) + n 2 π E=2\cdot\operatorname{arg}\left(\sqrt{1+e}\ \cdot\ \cos\frac{\theta}{2}\ ,\ % \sqrt{1-e}\ \cdot\ \sin\frac{\theta}{2}\right)+n\cdot 2\pi
  150. arg ( x , y ) \operatorname{arg}(x\ ,\ y)
  151. ( x , y ) (\ x\ ,\ y\ )
  152. | E - θ | < π \left|E-\theta\right|<\pi
  153. arg ( x , y ) \operatorname{arg}(x\ ,\ y)
  154. [ - < θ < ] [ - < E < ] \left[-\infty<\theta<\infty\right]\longleftrightarrow\left[-\infty<E<\infty\right]
  155. cos θ = x r = e - cosh E e cosh E - 1 \cos\theta=\frac{x}{r}=\frac{e-\cosh E}{e\cdot\cosh E-1}
  156. tan 2 θ 2 = 1 - cos θ 1 + cos θ = 1 - e - cosh E e cosh E - 1 1 + e - cosh E e cosh E - 1 = e cosh E - e + cosh E e cosh E + e - cosh E = e + 1 e - 1 cosh E - 1 cosh E + 1 = e + 1 e - 1 tanh 2 E 2 \tan^{2}\frac{\theta}{2}=\frac{1-\cos\theta}{1+\cos\theta}=\frac{1-\frac{e-% \cosh E}{e\cdot\cosh E-1}}{1+\frac{e-\cosh E}{e\cdot\cosh E-1}}=\frac{e\cdot% \cosh E-e+\cosh E}{e\cdot\cosh E+e-\cosh E}=\frac{e+1}{e-1}\ \cdot\ \frac{% \cosh E-1}{\cosh E+1}=\frac{e+1}{e-1}\ \cdot\ \tanh^{2}\frac{E}{2}
  157. tan θ 2 \tan\frac{\theta}{2}
  158. tanh E 2 \tanh\frac{E}{2}
  159. tan θ 2 = e + 1 e - 1 tanh E 2 \tan\frac{\theta}{2}=\sqrt{\frac{e+1}{e-1}}\cdot\tanh\frac{E}{2}
  160. [ - cos - 1 ( - 1 e ) < θ < cos - 1 ( - 1 e ) ] [ - < E < ] \left[-\cos^{-1}\left(-\frac{1}{e}\right)<\theta<\cos^{-1}\left(-\frac{1}{e}% \right)\right]\longleftrightarrow\left[-\infty<E<\infty\right]
  161. E 2 \frac{E}{2}
  162. tanh - 1 x = 1 2 ln ( 1 + x 1 - x ) \tanh^{-1}x=\frac{1}{2}\ln\left(\frac{1+x}{1-x}\right)
  163. P = 2 π a a μ P=2\pi\cdot a\cdot\sqrt{\frac{a}{\mu}}
  164. - μ r -\frac{\mu}{r}
  165. V r 2 + V t 2 2 - μ r \frac{{V_{r}}^{2}+{V_{t}}^{2}}{2}-\frac{\mu}{r}
  166. - μ 2 a -\frac{\mu}{2\cdot a}
  167. μ 2 a \frac{\mu}{2\cdot a}
  168. x ^ , y ^ \hat{x}\ ,\ \hat{y}
  169. x ^ \hat{x}
  170. V x = cos θ V r - sin θ V t = - μ p sin θ V_{x}=\cos\theta\cdot V_{r}-\sin\theta\cdot V_{t}=-\sqrt{\frac{\mu}{p}}\cdot\sin\theta
  171. V y = sin θ V r + cos θ V t = μ p ( e + cos θ ) V_{y}=\sin\theta\cdot V_{r}+\cos\theta\cdot V_{t}=\sqrt{\frac{\mu}{p}}\cdot(e+% \cos\theta)
  172. ( r ¯ , v ¯ ) (\ \bar{r}\ ,\bar{v}\ )
  173. v ¯ ˙ = - μ r ^ r 2 \dot{\bar{v}}=-\mu\cdot\frac{\hat{r}}{r^{2}}
  174. r ¯ ˙ = v ¯ \dot{\bar{r}}=\bar{v}
  175. ( r 0 ¯ , v 0 ¯ ) (\ \bar{r_{0}}\ ,\bar{v_{0}}\ )
  176. ( r ^ , t ^ ) (\hat{r}\ ,\ \hat{t})
  177. r 0 ¯ = r r ^ \bar{r_{0}}=r\cdot\hat{r}
  178. v 0 ¯ = V r r ^ + V t t ^ \bar{v_{0}}=V_{r}\cdot\hat{r}+V_{t}\cdot\hat{t}
  179. r > 0 r>0
  180. V t > 0 V_{t}>0
  181. p = ( r V t ) 2 μ p=\frac{{(r\cdot V_{t})}^{2}}{\mu}
  182. e 0 e\geq 0
  183. θ \theta
  184. e cos θ = V t V 0 - 1 e\cdot\cos\theta=\frac{V_{t}}{V_{0}}-1
  185. e sin θ = V r V 0 e\cdot\sin\theta=\frac{V_{r}}{V_{0}}
  186. V 0 = μ p V_{0}=\sqrt{\frac{\mu}{p}}
  187. θ \theta
  188. V r V_{r}
  189. V t V_{t}
  190. ( r ^ , t ^ ) (\hat{r}\ ,\ \hat{t})
  191. θ \theta
  192. ( r ¯ , v ¯ ) (\bar{r}\ ,\ \bar{v})
  193. ( r 0 ¯ , v 0 ¯ ) (\ \bar{r_{0}}\ ,\bar{v_{0}}\ )
  194. θ \theta
  195. ( x ^ , y ^ ) (\hat{x}\ ,\ \hat{y})
  196. x ^ \hat{x}
  197. x ^ = cos θ r ^ - sin θ t ^ \hat{x}=\cos\theta\cdot\hat{r}-\sin\theta\cdot\hat{t}
  198. y ^ = sin θ r ^ + cos θ t ^ \hat{y}=\sin\theta\cdot\hat{r}+\cos\theta\cdot\hat{t}
  199. V r = 0 V_{r}=0
  200. V t = V 0 = μ p = μ ( r V t ) 2 μ V_{t}=V_{0}=\sqrt{\frac{\mu}{p}}=\sqrt{\frac{\mu}{\frac{{(r\cdot V_{t})}^{2}}{% \mu}}}
  201. V t = μ r V_{t}=\sqrt{\frac{\mu}{r}}
  202. ( r 0 ¯ , v 0 ¯ ) (\ \bar{r_{0}}\ ,\bar{v_{0}}\ )
  203. ( r ¯ , v ¯ ) (\bar{r},\bar{v})
  204. p , e , θ p,e,\theta
  205. r , V r , V t r,V_{r},V_{t}
  206. x ^ , y ^ \hat{x},\hat{y}
  207. r ¯ ¨ = F ¯ ( r ¯ , r ¯ ˙ , t ) \ddot{\bar{r}}=\operatorname{\bar{F}}(\bar{r},\dot{\bar{r}},t)
  208. F ¯ ( r ¯ , r ¯ ˙ , t ) \operatorname{\bar{F}}(\bar{r},\dot{\bar{r}},t)
  209. - μ r ^ r 2 -\mu\cdot\frac{\hat{r}}{r^{2}}
  210. p , e , θ , x ^ , y ^ p,\,e,\,\theta,\,\hat{x},\,\hat{y}
  211. r ¯ , r ¯ ˙ \bar{r},\dot{\bar{r}}
  212. θ \theta
  213. F ¯ ( r ¯ , r ¯ ˙ , t ) = - μ r ^ r 2 + f ¯ ( r ¯ , r ¯ ˙ , t ) \operatorname{\bar{F}}(\bar{r},\dot{\bar{r}},t)=-\mu\cdot\frac{\hat{r}}{r^{2}}% +\operatorname{\bar{f}}(\bar{r},\dot{\bar{r}},t)
  214. f ¯ ( r ¯ , r ¯ ˙ , t ) \operatorname{\bar{f}}(\bar{r},\dot{\bar{r}},t)
  215. e ¯ = e x ^ \bar{e}=e\cdot\hat{x}
  216. e ¯ = ( V t - V 0 ) r ^ - V r t ^ V 0 \bar{e}=\frac{(V_{t}-V_{0})\cdot\hat{r}-V_{r}\cdot\hat{t}}{V_{0}}
  217. e ¯ \bar{e}
  218. ( r ¯ , v ¯ ) (\bar{r},\bar{v})

Kernel_smoother.html

  1. f ( X ) ( X p ) f(X)\,\,\left(X\in\mathbb{R}^{p}\right)
  2. K h λ ( X 0 , X ) = D ( X - X 0 h λ ( X 0 ) ) K_{h_{\lambda}}(X_{0},X)=D\left(\frac{\left\|X-X_{0}\right\|}{h_{\lambda}(X_{0% })}\right)
  3. X , X 0 p X,X_{0}\in\mathbb{R}^{p}
  4. \left\|\cdot\right\|
  5. h λ ( X 0 ) h_{\lambda}(X_{0})
  6. Y ^ ( X ) : p \hat{Y}(X):\mathbb{R}^{p}\to\mathbb{R}
  7. X 0 p X_{0}\in\mathbb{R}^{p}
  8. Y ^ ( X 0 ) = i = 1 N K h λ ( X 0 , X i ) Y ( X i ) i = 1 N K h λ ( X 0 , X i ) \hat{Y}(X_{0})=\frac{\sum\limits_{i=1}^{N}{K_{h_{\lambda}}(X_{0},X_{i})Y(X_{i}% )}}{\sum\limits_{i=1}^{N}{K_{h_{\lambda}}(X_{0},X_{i})}}
  9. K ( x * , x i ) = exp ( - ( x * - x i ) 2 2 b 2 ) K(x^{*},x_{i})=\exp\left(-\frac{(x^{*}-x_{i})^{2}}{2b^{2}}\right)
  10. h m ( X 0 ) = X 0 - X [ m ] h_{m}(X_{0})=\left\|X_{0}-X_{[m]}\right\|
  11. X [ m ] X_{[m]}
  12. D ( t ) = { 1 / m if | t | 1 0 otherwise D(t)=\begin{cases}1/m&\,\text{if }|t|\leq 1\\ 0&\,\text{otherwise}\end{cases}
  13. Y ^ ( X 0 ) \hat{Y}(X_{0})
  14. λ \lambda
  15. h λ ( X 0 ) = λ = constant , h_{\lambda}(X_{0})=\lambda=\,\text{constant},
  16. Y ^ ( X 0 ) \hat{Y}(X_{0})
  17. Y ^ ( X ) \hat{Y}(X)
  18. h λ ( X 0 ) = λ = constant . h_{\lambda}(X_{0})=\lambda=\,\text{constant}.
  19. min α ( X 0 ) , β ( X 0 ) i = 1 N K h λ ( X 0 , X i ) ( Y ( X i ) - α ( X 0 ) - β ( X 0 ) X i ) 2 \displaystyle\min_{\alpha(X_{0}),\beta(X_{0})}\sum\limits_{i=1}^{N}{K_{h_{% \lambda}}(X_{0},X_{i})\left(Y(X_{i})-\alpha(X_{0})-\beta(X_{0})X_{i}\right)^{2}}
  20. Y ^ ( X 0 ) = ( 1 , X 0 ) ( B T W ( X 0 ) B ) - 1 B T W ( X 0 ) y \hat{Y}(X_{0})=\left(1,X_{0}\right)\left(B^{T}W(X_{0})B\right)^{-1}B^{T}W(X_{0% })y
  21. y = ( Y ( X 1 ) , , Y ( X N ) ) T y=\left(Y(X_{1}),\dots,Y(X_{N})\right)^{T}
  22. W ( X 0 ) = diag ( K h λ ( X 0 , X i ) ) N × N W(X_{0})=\operatorname{diag}\left(K_{h_{\lambda}}(X_{0},X_{i})\right)_{N\times N}
  23. B T = ( 1 1 1 X 1 X 2 X N ) B^{T}=\left(\begin{matrix}1&1&\dots&1\\ X_{1}&X_{2}&\dots&X_{N}\\ \end{matrix}\right)
  24. min α ( X 0 ) , β j ( X 0 ) , j = 1 , , d i = 1 N K h λ ( X 0 , X i ) ( Y ( X i ) - α ( X 0 ) - j = 1 d β j ( X 0 ) X i j ) 2 \underset{\alpha(X_{0}),\beta_{j}(X_{0}),j=1,...,d}{\mathop{\min}}\,\sum% \limits_{i=1}^{N}{K_{h_{\lambda}}(X_{0},X_{i})\left(Y(X_{i})-\alpha(X_{0})-% \sum\limits_{j=1}^{d}{\beta_{j}(X_{0})X_{i}^{j}}\right)^{2}}
  25. Y ^ ( X 0 ) = α ( X 0 ) + j = 1 d β j ( X 0 ) X 0 j \hat{Y}(X_{0})=\alpha(X_{0})+\sum\limits_{j=1}^{d}{\beta_{j}(X_{0})X_{0}^{j}}
  26. β ^ ( X 0 ) = arg min β ( X 0 ) i = 1 N K h λ ( X 0 , X i ) ( Y ( X i ) - b ( X i ) T β ( X 0 ) ) 2 \displaystyle\hat{\beta}(X_{0})=\underset{\beta(X_{0})}{\mathop{\arg\min}}\,% \sum\limits_{i=1}^{N}{K_{h_{\lambda}}(X_{0},X_{i})\left(Y(X_{i})-b(X_{i})^{T}% \beta(X_{0})\right)}^{2}

Keulegan–Carpenter_number.html

  1. K C = V T L , K_{C}=\frac{V\,T}{L},
  2. δ = A L , \delta=\frac{A}{L},
  3. K C = 2 π δ . K_{C}=2\pi\,\delta.\,
  4. ( 𝐮 ) 𝐮 V 2 L , (\mathbf{u}\cdot\nabla)\mathbf{u}\sim\frac{V^{2}}{L},
  5. 𝐮 t V T . \frac{\partial\mathbf{u}}{\partial t}\sim\frac{V}{T}.

Khmaladze_transformation.html

  1. X 1 , , X n X_{1},\ldots,X_{n}
  2. F F
  3. { F θ : θ Θ } \{F_{\theta}:\theta\in\Theta\}
  4. F n F_{n}
  5. X 1 , , X n X_{1},\ldots,X_{n}
  6. F F
  7. X i X_{i}
  8. F F
  9. v n ( x ) = n [ F n ( x ) - F ( x ) ] . v_{n}(x)=\sqrt{n}[F_{n}(x)-F(x)].
  10. v n v_{n}
  11. x x
  12. v n v_{n}
  13. v n ( x ) = u n ( t ) v_{n}(x)=u_{n}(t)
  14. t = F ( x ) t=F(x)
  15. u n u_{n}
  16. U i = F ( X i ) U_{i}=F(X_{i})
  17. [ 0 , 1 ] [0,1]
  18. X i X_{i}
  19. F F
  20. v n v_{n}
  21. ψ ( v n , F ) \psi(v_{n},F)
  22. F F
  23. φ ( u n ) \varphi(u_{n})
  24. ψ ( v n , F ) = φ ( u n ) \psi(v_{n},F)=\varphi(u_{n})
  25. sup x | v n ( x ) | = sup t | u n ( t ) | , sup x | v n ( x ) | a ( F ( x ) ) = sup t | u n ( t ) | a ( t ) \sup_{x}|v_{n}(x)|=\sup_{t}|u_{n}(t)|,\quad\sup_{x}\frac{|v_{n}(x)|}{a(F(x))}=% \sup_{t}\frac{|u_{n}(t)|}{a(t)}
  26. - v n 2 ( x ) d F ( x ) = 0 1 u n 2 ( t ) d t . \int_{-\infty}^{\infty}v_{n}^{2}(x)\,dF(x)=\int_{0}^{1}u_{n}^{2}(t)\,dt.
  27. F F
  28. F F
  29. F F
  30. F F
  31. F = F θ n F=F_{\theta_{n}}
  32. θ n \theta_{n}
  33. X 1 , , X n X_{1},\ldots,X_{n}
  34. θ ^ n \hat{\theta}_{n}
  35. θ \theta
  36. v ^ n ( x ) = n [ F n ( x ) - F θ ^ n ( x ) ] \hat{v}_{n}(x)=\sqrt{n}[F_{n}(x)-F_{\hat{\theta}_{n}}(x)]
  37. v n v_{n}
  38. u ^ n ( t ) = v ^ n ( x ) \hat{u}_{n}(t)=\hat{v}_{n}(x)
  39. t = F θ ^ n ( x ) t=F_{\hat{\theta}_{n}}(x)
  40. n n\to\infty
  41. F θ F_{\theta}
  42. θ ^ n \hat{\theta}_{n}
  43. θ \theta
  44. v ^ n \hat{v}_{n}
  45. v ^ n \hat{v}_{n}
  46. w n w_{n}
  47. v ^ n ( x ) - K n ( x ) = w n ( x ) \hat{v}_{n}(x)-K_{n}(x)=w_{n}(x)
  48. K n ( x ) K_{n}(x)
  49. v ^ n ( x ) \hat{v}_{n}(x)
  50. w n w_{n}
  51. K n K_{n}
  52. w n w_{n}
  53. F θ ^ n ( x ) F_{\hat{\theta}_{n}}(x)
  54. x x
  55. θ n \theta_{n}
  56. ω n ( t ) = w n ( x ) , t = F θ ^ n ( x ) \omega_{n}(t)=w_{n}(x),t=F_{\hat{\theta}_{n}}(x)
  57. [ 0 , 1 ] [0,1]
  58. F θ ^ n F_{\hat{\theta}_{n}}
  59. v ^ n \hat{v}_{n}
  60. w n w_{n}
  61. v ^ n \hat{v}_{n}
  62. w n w_{n}
  63. w n w_{n}
  64. v ^ n \hat{v}_{n}
  65. w n w_{n}
  66. X 1 , , X n X_{1},\ldots,X_{n}
  67. d \mathbb{R}^{d}

Kind_(type_theory).html

  1. * *
  2. * *
  3. * *
  4. * * *\rightarrow*
  5. * * * *\rightarrow*\rightarrow*
  6. * *
  7. ( * * ) * (*\rightarrow*)\rightarrow*
  8. * *
  9. k 1 k 2 k_{1}\rightarrow k_{2}
  10. k 1 k_{1}
  11. k 2 k_{2}
  12. * *
  13. * * *\rightarrow*
  14. * *
  15. * *
  16. * * * *\rightarrow*\rightarrow*
  17. * * * * *\rightarrow*\rightarrow*\rightarrow*
  18. * *
  19. * * *\rightarrow*
  20. * *
  21. * * *\rightarrow*
  22. * *
  23. ( * * ) * * (*\rightarrow*)\rightarrow*\rightarrow*

Kingman's_formula.html

  1. 𝔼 ( W q ) ( ρ 1 - ρ ) ( c a 2 + c s 2 2 ) τ \mathbb{E}(W_{q})\approx\left(\frac{\rho}{1-\rho}\right)\left(\frac{c_{a}^{2}+% c_{s}^{2}}{2}\right)\tau

Kiowa_phonology.html

  1. ( C ) ( j ) V ( { C : } ) + T o n e \left(C\right)\left(j\right)V\left(\begin{Bmatrix}C\\ :\end{Bmatrix}\right)+Tone

Kirillov_model.html

  1. π ( ( a b 01 ) ) f ( x ) = τ ( b x ) f ( a x ) . \pi({ab\choose 01})f(x)=\tau(bx)f(ax).

KiteGen.html

  1. 2 {}^{2}

Kleene's_O.html

  1. 𝒪 \mathcal{O}
  2. ω 1 C K \omega_{1}^{CK}
  3. ω 1 C K \omega_{1}^{CK}
  4. 𝒪 \mathcal{O}
  5. 𝒪 \mathcal{O}
  6. 𝒪 \mathcal{O}
  7. p p
  8. 𝒪 \mathcal{O}
  9. p p
  10. | p | |p|
  11. < 𝒪 <_{\mathcal{O}}
  12. 𝒪 \mathcal{O}
  13. 𝒪 \mathcal{O}
  14. | 0 | = 0 |0|=0
  15. i i
  16. 𝒪 \mathcal{O}
  17. | i | = α |i|=\alpha
  18. 2 i 2^{i}
  19. 𝒪 \mathcal{O}
  20. | 2 i | = α + 1 |2^{i}|=\alpha+1
  21. i < 𝒪 2 i i<_{\mathcal{O}}2^{i}
  22. { e } \{e\}
  23. e e
  24. e e
  25. 𝒪 \mathcal{O}
  26. n n
  27. { e } ( n ) < 𝒪 { e } ( n + 1 ) \{e\}(n)<_{\mathcal{O}}\{e\}(n+1)
  28. 3 5 e 3\cdot 5^{e}
  29. 𝒪 \mathcal{O}
  30. { e } ( n ) < 𝒪 3 5 e \{e\}(n)<_{\mathcal{O}}3\cdot 5^{e}
  31. n n
  32. | 3 5 e | = lim k | { e } ( k ) | |3\cdot 5^{e}|=\lim_{k}|\{e\}(k)|
  33. 3 5 e 3\cdot 5^{e}
  34. γ k \gamma_{k}
  35. | { e } ( k ) | = γ k |\{e\}(k)|=\gamma_{k}
  36. k k
  37. p < 𝒪 q p<_{\mathcal{O}}q
  38. q < 𝒪 r q<_{\mathcal{O}}r
  39. p < 𝒪 r p<_{\mathcal{O}}r
  40. < 𝒪 <_{\mathcal{O}}
  41. < 𝒪 <_{\mathcal{O}}
  42. γ \gamma
  43. α < γ \alpha<\gamma
  44. α \alpha
  45. < 𝒪 <_{\mathcal{O}}
  46. | i | = α |i|=\alpha
  47. | j | = β |j|=\beta
  48. i < 𝒪 j , i<_{\mathcal{O}}j\,,
  49. α < β \alpha<\beta
  50. < 𝒪 <_{\mathcal{O}}
  51. 𝒪 \mathcal{O}
  52. 𝒪 \mathcal{O}
  53. 𝒪 \mathcal{O}
  54. 𝒪 \mathcal{O}
  55. n n
  56. n 𝒪 n_{\mathcal{O}}
  57. ω 1 C K \omega^{CK}_{1}
  58. ω 1 C K \omega^{CK}_{1}
  59. 𝒪 \mathcal{O}
  60. < 𝒪 <_{\mathcal{O}}
  61. < 𝒪 <_{\mathcal{O}}
  62. 𝒪 \mathcal{O}
  63. p p
  64. { q q < 𝒪 p } \{q\mid q<_{\mathcal{O}}p\}
  65. p p
  66. 𝒪 \mathcal{O}
  67. Π 1 1 \Pi^{1}_{1}
  68. 𝒪 \mathcal{O}
  69. Π 1 1 \Pi^{1}_{1}
  70. Σ 1 1 \Sigma^{1}_{1}
  71. 𝒪 \mathcal{O}
  72. 𝒪 \mathcal{O}
  73. 𝒪 \mathcal{O}
  74. f f
  75. e e
  76. f ( e ) f(e)
  77. 𝒪 \mathcal{O}
  78. < e <_{e}
  79. { p p < 𝒪 f ( e ) } \{p\mid p<_{\mathcal{O}}f(e)\}
  80. + 𝒪 +_{\mathcal{O}}
  81. 𝒪 \mathcal{O}
  82. max { p , q } p + 𝒪 q \max\{p,q\}\leq p+_{\mathcal{O}}q
  83. 𝒪 \mathcal{O}
  84. 𝒪 \mathcal{O}
  85. 𝒫 \mathcal{P}
  86. 𝒪 \mathcal{O}
  87. < 𝒪 <_{\mathcal{O}}
  88. p p
  89. 𝒫 \mathcal{P}
  90. q < 𝒪 p q<_{\mathcal{O}}p
  91. q q
  92. 𝒫 \mathcal{P}
  93. 𝒫 \mathcal{P}
  94. 𝒪 \mathcal{O}
  95. < 𝒪 <_{\mathcal{O}}
  96. 𝒫 \mathcal{P}
  97. 𝒫 \mathcal{P}
  98. 𝒫 \mathcal{P}
  99. 𝒫 \mathcal{P}
  100. p p
  101. 𝒪 \mathcal{O}
  102. 𝒫 \mathcal{P}
  103. 2 0 2^{\aleph_{0}}
  104. 𝒪 \mathcal{O}
  105. 2 0 2^{\aleph_{0}}
  106. 𝒪 \mathcal{O}
  107. ω 2 \omega^{2}
  108. λ < ω 1 C K \lambda<\omega_{1}^{CK}
  109. 2 0 2^{\aleph_{0}}
  110. 𝒪 \mathcal{O}
  111. ω 2 λ \omega^{2}\cdot\lambda
  112. 𝒫 \mathcal{P}
  113. ω 2 \omega^{2}
  114. 𝒫 \mathcal{P}
  115. d d
  116. e d e_{d}
  117. 𝒪 \mathcal{O}
  118. 𝒫 = { p p < 𝒪 e d } \mathcal{P}=\{p\mid p<_{\mathcal{O}}e_{d}\}
  119. d d
  120. α ω 2 \alpha\geq\omega^{2}
  121. e d e_{d}
  122. | e d | = α |e_{d}|=\alpha
  123. 0 \aleph_{0}
  124. 𝒪 \mathcal{O}
  125. Π 1 1 \Pi_{1}^{1}
  126. Π 1 0 \Pi_{1}^{0}
  127. Π 1 1 \Pi^{1}_{1}
  128. 𝒪 \mathcal{O}
  129. 𝒪 \mathcal{O}

Klincewicz_method.html

  1. T c = 45.40 - 0.77 * M W + 1.55 * T b + j = 1 35 n j Δ j T_{c}\,=\,45.40-0.77*MW+1.55*T_{b}+\sum_{j=1}^{35}n_{j}\Delta_{j}
  2. ( M W / P c ) 1 / 2 = 0.348 + 0.0159 * M W + j = 1 35 n j Δ j (MW/P_{c})^{1/2}\,=\,0.348+0.0159*MW+\sum_{j=1}^{35}n_{j}\Delta_{j}
  3. V c = 25.2 + 2.80 * M W + j = 1 35 n j Δ j V_{c}\,=\,25.2+2.80*MW+\sum_{j=1}^{35}n_{j}\Delta_{j}
  4. T c = 50.2 - 0.16 * M W + 1.41 * T b T_{c}\,=\,50.2-0.16*MW+1.41*T_{b}
  5. ( M W / P c ) 1 / 2 = 0.335 + 0.009 * M W + 0.019 A (MW/P_{c})^{1/2}\,=\,0.335+0.009*MW+0.019A
  6. V c = 20.1 + 0.88 * M W + 13.4 * A V_{c}\,=\,20.1+0.88*MW+13.4*A
  7. n j Δ j \sum n_{j}\Delta_{j}

Klinkenberg_correction.html

  1. K g = C q 1 μ g P 1 a G ( P 1 2 - P 2 2 ) 2 K_{g}=Cq_{1}\mu_{g}\frac{P_{1}}{a}G(P_{1}^{2}-P_{2}^{2})^{2}
  2. K g K_{g}
  3. q 1 q_{1}
  4. P 1 P_{1}
  5. P 2 P_{2}
  6. μ g \mu_{g}
  7. a a
  8. G G
  9. C C

Knizhnik–Zamolodchikov_equations.html

  1. 𝔤 ^ k \hat{\mathfrak{g}}_{k}
  2. k k
  3. h h
  4. v v
  5. 𝔤 ^ k \hat{\mathfrak{g}}_{k}
  6. Φ ( v , z ) \Phi(v,z)
  7. t a t^{a}
  8. 𝔤 \mathfrak{g}
  9. t i a t^{a}_{i}
  10. Φ ( v i , z ) \Phi(v_{i},z)
  11. η η
  12. i , j = 1 , 2 , , N i,j=1,2,\ldots,N
  13. ( ( k + h ) z i + j i a , b η a b t i a t j b z i - z j ) Φ ( v N , z N ) Φ ( v 1 , z 1 ) = 0. \left((k+h)\partial_{z_{i}}+\sum_{j\neq i}\frac{\sum_{a,b}\eta_{ab}t^{a}_{i}% \otimes t^{b}_{j}}{z_{i}-z_{j}}\right)\left\langle\Phi(v_{N},z_{N})\dots\Phi(v% _{1},z_{1})\right\rangle=0.
  14. 𝔤 ^ k \hat{\mathfrak{g}}_{k}
  15. 𝔤 ^ k \hat{\mathfrak{g}}_{k}
  16. ( L - 1 - 1 2 ( k + h ) k 𝐙 a , b η a b J - k a J k - 1 b ) v = 0 , \left(L_{-1}-\frac{1}{2(k+h)}\sum_{k\in\mathbf{Z}}\sum_{a,b}\eta_{ab}J^{a}_{-k% }J^{b}_{k-1}\right)v=0,
  17. v v
  18. J k a J^{a}_{k}
  19. t a t^{a}
  20. v v
  21. J k a J^{a}_{k}
  22. J - 1 a J 0 b J^{a}_{-1}J^{b}_{0}
  23. d d
  24. V ( a , 0 ) Ω = a . V(a,0)\Omega=a.
  25. X ( z ) = X ( n ) z - n - 1 X(z)=\sum X(n)z^{-n-1}
  26. 𝔤 \mathfrak{g}
  27. T ( z ) = L n z - n - 2 . T(z)=\sum L_{n}z^{-n-2}.
  28. α α
  29. V ( a , z ) = V ( a , n ) z - n - α . V(a,z)=\sum V(a,n)z^{-n-\alpha}.
  30. d d z V ( a , z ) = [ L - 1 , V ( a , z ) ] = V ( L - 1 a , z ) [ L 0 , V ( a , z ) ] = ( z - 1 d d z + α ) V ( a , z ) \begin{aligned}\displaystyle\frac{d}{dz}V(a,z)&\displaystyle=\left[L_{-1},V(a,% z)\right]=V\left(L_{-1}a,z\right)\\ \displaystyle\left[L_{0},V(a,z)\right]&\displaystyle=\left(z^{-1}\frac{d}{dz}+% \alpha\right)V(a,z)\end{aligned}
  31. V ( a , z ) V ( b , w ) = V ( b , w ) V ( a , z ) = V ( V ( a , z - w ) b , w ) . V(a,z)V(b,w)=V(b,w)V(a,z)=V(V(a,z-w)b,w).
  32. V i ( a , z ) V i ( b , w ) = V i ( b , w ) V i ( a , z ) = V i ( V ( a , z - w ) b , w ) . V_{i}(a,z)V_{i}(b,w)=V_{i}(b,w)V_{i}(a,z)=V_{i}(V(a,z-w)b,w).
  33. Φ ( v , z ) Φ(v,z)
  34. Φ ( v , z ) = Φ ( v , n ) z - n - δ \Phi(v,z)=\sum\Phi(v,n)z^{-n-\delta}
  35. V j ( a , z ) Φ ( v , w ) = Φ ( v , w ) V i ( a , w ) = Φ ( V k ( a , z - w ) v , w ) V_{j}(a,z)\Phi(v,w)=\Phi(v,w)V_{i}(a,w)=\Phi\left(V_{k}(a,z-w)v,w\right)
  36. 𝔤 \mathfrak{g}
  37. Φ ( v , w ) Φ(v,w)
  38. Φ ( v 1 , z 1 ) Φ ( v 2 , z 2 ) Φ ( v n , z n ) = ( Φ ( v 1 , z 1 ) Φ ( v 2 , z 2 ) Φ ( v n , z n ) Ω , Ω ) . \left\langle\Phi(v_{1},z_{1})\Phi(v_{2},z_{2})\cdots\Phi(v_{n},z_{n})\right% \rangle=\left(\Phi\left(v_{1},z_{1}\right)\Phi\left(v_{2},z_{2}\right)\cdots% \Phi\left(v_{n},z_{n}\right)\Omega,\Omega\right).
  39. 𝔤 \mathfrak{g}
  40. 𝔤 \mathfrak{g}
  41. s X s ( w ) X s ( z ) Φ ( v 1 , z 1 ) Φ ( v n , z n ) ( w - z ) - 1 \sum_{s}\left\langle X_{s}(w)X_{s}(z)\Phi(v_{1},z_{1})\cdots\Phi(v_{n},z_{n})% \right\rangle(w-z)^{-1}
  42. 1 2 ( k + h ) T ( z ) Φ ( v 1 , z 1 ) Φ ( v n , z n ) = - j , s X s ( z ) Φ ( v 1 , z 1 ) Φ ( X s v j , z j ) Φ ( X n , z n ) ( z - z j ) - 1 . {1\over 2}(k+h)\left\langle T(z)\Phi(v_{1},z_{1})\cdots\Phi(v_{n},z_{n})\right% \rangle=-\sum_{j,s}\left\langle X_{s}(z)\Phi(v_{1},z_{1})\cdots\Phi(X_{s}v_{j}% ,z_{j})\Phi(X_{n},z_{n})\right\rangle(z-z_{j})^{-1}.
  43. L 0 \displaystyle L_{0}
  44. [ X ( m ) , Φ ( a , n ) ] = Φ ( X a , m + n ) . [X(m),\Phi(a,n)]=\Phi(Xa,m+n).
  45. 𝔤 \mathfrak{g}
  46. X ( z ) Φ ( v 1 , z 1 ) Φ ( v n , z n ) = j Φ ( v 1 , z 1 ) Φ ( X v j , z j ) Φ ( v n , z n ) ( z - z j ) - 1 . \left\langle X(z)\Phi(v_{1},z_{1})\cdots\Phi(v_{n},z_{n})\right\rangle=\sum_{j% }\left\langle\Phi(v_{1},z_{1})\cdots\Phi(Xv_{j},z_{j})\cdots\Phi(v_{n},z_{n})% \right\rangle(z-z_{j})^{-1}.
  47. s X s ( z ) Φ ( z 1 , v 1 ) Φ ( X s v i , z i ) Φ ( v n , z n ) = j s Φ ( X s v j , z j ) Φ ( X s v i , z i ) ( z - z j ) - 1 . \sum_{s}\langle X_{s}(z)\Phi(z_{1},v_{1})\cdots\Phi(X_{s}v_{i},z_{i})\cdots% \Phi(v_{n},z_{n})\rangle=\sum_{j}\sum_{s}\langle\cdots\Phi(X_{s}v_{j},z_{j})% \cdots\Phi(X_{s}v_{i},z_{i})\cdots\rangle(z-z_{j})^{-1}.
  48. s X s ( z ) Φ ( X s v i , z i ) = ( z - z i ) - 1 Φ ( s X s 2 v i , z i ) + ( k + g ) z i Φ ( v i , z i ) + O ( z - z i ) \sum_{s}X_{s}(z)\Phi\left(X_{s}v_{i},z_{i}\right)=(z-z_{i})^{-1}\Phi\left(\sum% _{s}X_{s}^{2}v_{i},z_{i}\right)+(k+g){\partial\over\partial z_{i}}\Phi(v_{i},z% _{i})+O(z-z_{i})
  49. ( k + g ) z i Φ ( v i , z i ) = lim z z i [ s X s ( z ) Φ ( X s v i , z i ) - ( z - z i ) - 1 Φ ( s X s 2 v i , z i ) ] . (k+g)\frac{\partial}{\partial z_{i}}\Phi(v_{i},z_{i})=\lim_{z\to z_{i}}\left[% \sum_{s}X_{s}(z)\Phi\left(X_{s}v_{i},z_{i}\right)-(z-z_{i})^{-1}\Phi\left(\sum% _{s}X_{s}^{2}v_{i},z_{i}\right)\right].
  50. 𝔰 𝔩 2 \mathfrak{sl}_{2}
  51. Φ j ( z j ) \Phi_{j}(z_{j})
  52. Ψ ( z 1 , , z n ) \Psi(z_{1},\dots,z_{n})
  53. Φ j ( z j ) \Phi_{j}(z_{j})
  54. ( ρ , V i ) (\rho,V_{i})
  55. V 1 V n V_{1}\otimes\cdots\otimes V_{n}
  56. ( k + 2 ) z i Ψ = i , j i Ω i j z i - z j Ψ , (k+2)\frac{\partial}{\partial z_{i}}\Psi=\sum_{i,j\neq i}\frac{\Omega_{ij}}{z_% {i}-z_{j}}\Psi\ ,
  57. Ω i j = a ρ i ( J a ) ρ ( J a ) \Omega_{ij}=\sum_{a}\rho_{i}(J^{a})\otimes\rho(J_{a})
  58. X n n X_{n}\subset\mathbb{C}^{n}
  59. z i z j , i j z_{i}\neq z_{j},i\neq j
  60. B n B_{n}
  61. 𝔤 \mathfrak{g}
  62. ( ρ , V i ) (\rho,V_{i})
  63. θ : B n V 1 V n \theta:B_{n}\rightarrow V_{1}\otimes\cdots\otimes V_{n}
  64. V 1 V n V_{1}\otimes\dots\otimes V_{n}
  65. V i V_{i}

Knudsen_layer.html

  1. l c l_{c}
  2. l c = k T s π d 2 p s l_{c}=\frac{kT_{s}}{\pi d^{2}p_{s}}
  3. k k
  4. T s T_{s}
  5. d d
  6. p s p_{s}

Koebe_quarter_theorem.html

  1. g ( z ) = z + b 1 z - 1 + b 2 z - 2 + g(z)=z+b_{1}z^{-1}+b_{2}z^{-2}+\cdots
  2. n 1 n | b n | 2 1. \sum_{n\geq 1}n|b_{n}|^{2}\leq 1.
  3. X ( r ) d x d y = 1 2 i X ( r ) z ¯ d z = - 1 2 i | z | = r g ¯ d g = 1 2 π r 2 - 1 2 π n | b n | 2 r 2 n . \int_{X(r)}dx\,dy={1\over 2i}\int_{\partial X(r)}\overline{z}\,dz=-{1\over 2i}% \int_{|z|=r}\overline{g}\,dg={1\over 2\pi r^{2}}-{1\over 2\pi}\sum n|b_{n}|^{2% }r^{2n}.
  4. f ( z ) = z ( 1 - z ) 2 = n = 1 n z n f(z)=\frac{z}{(1-z)^{2}}=\sum_{n=1}^{\infty}nz^{n}
  5. f α ( z ) = z ( 1 - α z ) 2 = n = 1 n α n - 1 z n f_{\alpha}(z)=\frac{z}{(1-\alpha z)^{2}}=\sum_{n=1}^{\infty}n\alpha^{n-1}z^{n}
  6. g ( z ) = z + a 2 z 2 + a 3 z 3 + g(z)=z+a_{2}z^{2}+a_{3}z^{3}+\cdots
  7. g ( z 2 ) - 1 / 2 = z - 1 - 1 2 a 2 z + . g(z^{2})^{-1/2}=z^{-1}-{1\over 2}a_{2}z+\cdots.
  8. f ( 0 ) = 0 , f ( 0 ) = 1 , f(0)=0,\,\,\,f^{\prime}(0)=1,
  9. f ( z ) = z + a 2 z 2 + . f(z)=z+a_{2}z^{2}+\cdots.
  10. h ( z ) = w f ( z ) w - f ( z ) = z + ( a 2 + w - 1 ) z 2 + h(z)={wf(z)\over w-f(z)}=z+(a_{2}+w^{-1})z^{2}+\cdots
  11. | w | 1 4 . |w|\geq{1\over 4}.
  12. 1 - r ( 1 + r ) 3 | f ( z ) | 1 + r ( 1 - r ) 3 {1-r\over(1+r)^{3}}\leq|f^{\prime}(z)|\leq{1+r\over(1-r)^{3}}
  13. 1 - r 1 + r | z f ( z ) f ( z ) | 1 + r 1 - r {1-r\over 1+r}\leq\left|z{f^{\prime}(z)\over f(z)}\right|\leq{1+r\over 1-r}
  14. f ( z ) = z ( 1 - e i θ z ) 2 . f(z)={z\over(1-e^{i\theta}z)^{2}}.

Kolmogorov's_criterion.html

  1. p j 1 j 2 p j 2 j 3 p j n - 1 j n p j n j 1 = p j 1 j n p j n j n - 1 p j 3 j 2 p j 2 j 1 p_{j_{1}j_{2}}p_{j_{2}j_{3}}\cdots p_{j_{n-1}j_{n}}p_{j_{n}j_{1}}=p_{j_{1}j_{n% }}p_{j_{n}j_{n-1}}\cdots p_{j_{3}j_{2}}p_{j_{2}j_{1}}
  2. j 1 , j 2 , , j n S . j_{1},j_{2},\ldots,j_{n}\in S.
  3. p i j p j l p l k p k i = p i k p k l p l j p j i . p_{ij}p_{jl}p_{lk}p_{ki}=p_{ik}p_{kl}p_{lj}p_{ji}.
  4. q j 1 j 2 q j 2 j 3 q j n - 1 j n q j n j 1 = q j 1 j n q j n j n - 1 q j 3 j 2 q j 2 j 1 q_{j_{1}j_{2}}q_{j_{2}j_{3}}\cdots q_{j_{n-1}j_{n}}q_{j_{n}j_{1}}=q_{j_{1}j_{n% }}q_{j_{n}j_{n-1}}\cdots q_{j_{3}j_{2}}q_{j_{2}j_{1}}
  5. j 1 , j 2 , , j n S . j_{1},j_{2},\ldots,j_{n}\in S.

Kolmogorov_structure_function.html

  1. h x ( α ) = min S { log | S | : x S , K ( S ) α } h_{x}(\alpha)=\min_{S}\{\log|S|:x\in S,K(S)\leq\alpha\}
  2. x x
  3. n n
  4. x S x\in S
  5. S S
  6. x x
  7. K ( S ) K(S)
  8. S S
  9. α \alpha
  10. S S
  11. log | { x } | = 0 \log|\{x\}|=0
  12. α = K ( x ) + c \alpha=K(x)+c
  13. c c
  14. x x
  15. { x } \{x\}
  16. K ( x ) K(x)
  17. x x
  18. S S
  19. x x
  20. K ( S ) + K ( x | S ) = K ( x ) + O ( 1 ) K(S)+K(x|S)=K(x)+O(1)
  21. h x ( α ) h_{x}(\alpha)
  22. L ( α ) + α = K ( x ) L(\alpha)+\alpha=K(x)
  23. h x h_{x}
  24. α = K ( x ) + c \alpha=K(x)+c
  25. α \alpha
  26. α + h x ( α ) = K ( x ) + O ( 1 ) \alpha+h_{x}(\alpha)=K(x)+O(1)
  27. S S
  28. h x ( α ) h_{x}(\alpha)
  29. x x
  30. K ( S ) α K(S)\leq\alpha
  31. S S
  32. x x
  33. K ( S ) + log | S | = K ( x ) + O ( 1 ) K(S)+\log|S|=K(x)+O(1)
  34. x x
  35. S S
  36. x x
  37. S S
  38. log | S | \log|S|
  39. x x
  40. K ( x ) K(x)
  41. K ( x ) K ( x , S ) + O ( 1 ) K ( S ) + K ( x | S ) + O ( 1 ) K ( S ) + log | S | + O ( 1 ) K ( x ) + O ( 1 ) K(x)\leq K(x,S)+O(1)\leq K(S)+K(x|S)+O(1)\leq K(S)+\log|S|+O(1)\leq K(x)+O(1)
  42. K ( x | S ) = log | S | + O ( 1 ) K(x|S)=\log|S|+O(1)
  43. S x S\ni x
  44. x x
  45. log | S | + O ( 1 ) \log|S|+O(1)
  46. log | S | - K ( x | S ) \log|S|-K(x|S)
  47. x x
  48. S S
  49. x x
  50. S S
  51. x x
  52. S S
  53. x x
  54. K ( x , S ) = K ( x ) + O ( 1 ) K(x,S)=K(x)+O(1)
  55. x x
  56. S S
  57. S S
  58. K ( S | x * ) = O ( 1 ) K(S|x^{*})=O(1)
  59. S S
  60. x x
  61. x * x^{*}
  62. x x
  63. α \alpha
  64. λ x ( α ) \lambda_{x}(\alpha)
  65. β x ( α ) \beta_{x}(\alpha)
  66. S x S\ni x
  67. x x
  68. K ( S ) α K(S)\leq\alpha
  69. S S
  70. β x ( α ) = 0 \beta_{x}(\alpha)=0
  71. α \alpha
  72. S x S\ni x
  73. x x
  74. K ( S ) α K(S)\leq\alpha
  75. x x
  76. S S
  77. K ( x ) K(x)
  78. O ( log n ) O(\log n)
  79. α \alpha
  80. S S
  81. O ( log n ) O(\log n)
  82. α \alpha
  83. α \alpha
  84. λ x ( α ) = min S { Λ ( S ) : S x , K ( S ) α } , \lambda_{x}(\alpha)=\min_{S}\{\Lambda(S):S\ni x,\;K(S)\leq\alpha\},
  85. Λ ( S ) = log | S | + K ( S ) K ( x ) - O ( 1 ) \Lambda(S)=\log|S|+K(S)\geq K(x)-O(1)
  86. α \alpha
  87. O ( log n ) O(\log n)
  88. P P
  89. - log P ( x ) = log | S | + O ( log n ) -\log P(x)=\log|S|+O(\log n)
  90. P P
  91. S S
  92. n n
  93. x S x\in S
  94. P ( x ) = 1 / | S | P(x)=1/|S|
  95. h x ( α ) = min P { - log P ( x ) : P ( x ) > 0 , K ( P ) α } h^{\prime}_{x}(\alpha)=\min_{P}\{-\log P(x):P(x)>0,K(P)\leq\alpha\}
  96. - log P ( x ) > 0 -\log P(x)>0
  97. P P
  98. n n
  99. x x
  100. K ( P ) K(P)
  101. P P
  102. α \alpha
  103. P P
  104. log | { x } | = 0 \log|\{x\}|=0
  105. α = K ( x ) + c \alpha=K(x)+c
  106. x x
  107. { x } \{x\}
  108. K ( x ) K(x)
  109. x x
  110. h x ( α ) = h x ( α ) + O ( log n ) h^{\prime}_{x}(\alpha)=h_{x}(\alpha)+O(\log n)
  111. α \alpha
  112. h x ( α ) h^{\prime}_{x}(\alpha)
  113. α \alpha
  114. S S
  115. x x
  116. O ( log n ) O(\log n)
  117. - log P ( x ) -\log P(x)
  118. α \alpha
  119. α \alpha
  120. λ x ( α ) = min P { Λ ( P ) : P ( x ) > 0 , K ( P ) α } , \lambda^{\prime}_{x}(\alpha)=\min_{P}\{\Lambda(P):P(x)>0,\;K(P)\leq\alpha\},
  121. Λ ( P ) = - log P ( x ) + K ( P ) K ( x ) - O ( 1 ) \Lambda(P)=-\log P(x)+K(P)\geq K(x)-O(1)
  122. α \alpha
  123. O ( log n ) O(\log n)

Konrad_Osterwalder.html

  1. 𝒟 ϕ F [ ϕ ( x ) ] F [ ϕ ( x ¯ ) ] * e - S [ ϕ ] = 𝒟 ϕ 0 ϕ + ( τ = 0 ) = ϕ 0 𝒟 ϕ + F [ ϕ + ] e - S + [ ϕ + ] ϕ - ( τ = 0 ) = ϕ 0 𝒟 ϕ - F [ ϕ ¯ - ] * e - S - [ ϕ - ] . \int\mathcal{D}\phi F[\phi(x)]F[\phi(\bar{x})]^{*}e^{-S[\phi]}=\int\mathcal{D}% \phi_{0}\int_{\phi_{+}(\tau=0)=\phi_{0}}\mathcal{D}\phi_{+}F[\phi_{+}]e^{-S_{+% }[\phi_{+}]}\int_{\phi_{-}(\tau=0)=\phi_{0}}\mathcal{D}\phi_{-}F[\bar{\phi}_{-% }]^{*}e^{-S_{-}[\phi_{-}]}.

Koornwinder_polynomials.html

  1. 1 i < j n ( x i x j , x i / x j , x j / x i , 1 / x i x j ; q ) ( t x i x j , t x i / x j , t x j / x i , t / x i x j ; q ) 1 i n ( x i 2 , 1 / x i 2 ; q ) ( a x i , a / x i , b x i , b / x i , c x i , c / x i , d x i , d / x i ; q ) \prod_{1\leq i<j\leq n}\frac{(x_{i}x_{j},x_{i}/x_{j},x_{j}/x_{i},1/x_{i}x_{j};% q)_{\infty}}{(tx_{i}x_{j},tx_{i}/x_{j},tx_{j}/x_{i},t/x_{i}x_{j};q)_{\infty}}% \prod_{1\leq i\leq n}\frac{(x_{i}^{2},1/x_{i}^{2};q)_{\infty}}{(ax_{i},a/x_{i}% ,bx_{i},b/x_{i},cx_{i},c/x_{i},dx_{i},d/x_{i};q)_{\infty}}
  2. | x 1 | = | x 2 | = | x n | = 1 |x_{1}|=|x_{2}|=\cdots|x_{n}|=1
  3. | a | , | b | , | c | , | d | , | q | , | t | < 1 , |a|,|b|,|c|,|d|,|q|,|t|<1,

Korringa–Kohn–Rostoker_approximation.html

  1. [ 2 + E ] ψ ( r ) = V ( r ) ψ ( r ) [\nabla^{2}+E]\psi({r})=V({r})\psi({r})
  2. V ( r ) V({r})
  3. ψ ( r ) \psi({r})
  4. [ 2 + E ] G ( r , r ) = δ ( r - r ) [\nabla^{2}+E]G({r},{r}^{\prime})=\delta({r}-{r}^{\prime})
  5. e i k r = l ( 2 l + 1 ) i l j l ( k r ) P l ( cos θ ) e^{i{k}\cdot{r}}=\sum_{l}(2l+1)i^{l}j_{l}(kr)P_{l}(\cos\theta)
  6. j l ( k r ) j_{l}(kr)
  7. P l ( cos θ ) P_{l}(\cos\theta)

Köhler_theory.html

  1. ln ( p w ( D p ) p 0 ) = 4 M w σ w R T ρ w D p - 6 n s M w π ρ w D p 3 \ln\left(\frac{p_{w}(D_{p})}{p^{0}}\right)=\frac{4M_{w}\sigma_{w}}{RT\rho_{w}D% _{p}}-\frac{6n_{s}M_{w}}{\pi\rho_{w}D_{p}^{3}}
  2. p w p_{w}
  3. p 0 p^{0}
  4. σ w \sigma_{w}
  5. ρ w \rho_{w}
  6. n s n_{s}
  7. M w M_{w}
  8. D p D_{p}

Krawtchouk_matrices.html

  1. K ( 0 ) = [ 1 ] K ( 1 ) = [ 1 1 1 - 1 ] K ( 2 ) = [ 1 1 1 2 0 - 2 1 - 1 1 ] K ( 3 ) = [ 1 1 1 1 3 1 - 1 - 3 3 - 1 - 1 3 1 - 1 1 - 1 ] K^{(0)}=\begin{bmatrix}1\end{bmatrix}\qquad K^{(1)}=\left[\begin{array}[]{rr}1% &1\\ 1&-1\end{array}\right]\qquad K^{(2)}=\left[\begin{array}[]{rrr}1&1&1\\ 2&0&-2\\ 1&-1&1\end{array}\right]\qquad K^{(3)}=\left[\begin{array}[]{rrrr}1&1&1&1\\ 3&1&-1&-3\\ 3&-1&-1&3\\ 1&-1&1&-1\end{array}\right]
  2. K ( 4 ) = [ 1 1 1 1 1 4 2 0 - 2 - 4 6 0 - 2 0 6 4 - 2 0 2 - 4 1 - 1 1 - 1 1 ] K ( 5 ) = [ 1 1 1 1 1 1 5 3 1 - 1 - 3 - 5 10 2 - 2 - 2 2 10 10 - 2 - 2 2 2 - 10 5 - 3 1 1 - 3 5 1 - 1 1 - 1 1 - 1 ] . K^{(4)}=\left[\begin{array}[]{rrrrr}1&1&1&1&1\\ 4&2&0&-2&-4\\ 6&0&-2&0&6\\ 4&-2&0&2&-4\\ 1&-1&1&-1&1\end{array}\right]\qquad K^{(5)}=\left[\begin{array}[]{rrrrrr}1&1&1% &1&1&1\\ 5&3&1&-1&-3&-5\\ 10&2&-2&-2&2&10\\ 10&-2&-2&2&2&-10\\ 5&-3&1&1&-3&5\\ 1&-1&1&-1&1&-1\end{array}\right].
  3. N N
  4. K i j ( N ) K^{(N)}_{ij}
  5. ( 1 + v ) N - j ( 1 - v ) j = i v i K i j ( N ) (1+v)^{N-j}\,(1-v)^{j}=\sum_{i}v^{i}K^{(N)}_{ij}
  6. i i
  7. j j
  8. 0
  9. N N
  10. p = 1 / 2 p=1/2

Krull_ring.html

  1. A A
  2. P P
  3. A A
  4. A A
  5. A 𝔭 A_{\mathfrak{p}}
  6. 𝔭 P \mathfrak{p}\in P
  7. A A
  8. A A
  9. A A
  10. A ^ \widehat{A}
  11. A A
  12. A [ x ] A[x]
  13. A [ [ x ] ] A[[x]]
  14. R [ x 1 , x 2 , x 3 , ] R[x_{1},x_{2},x_{3},\ldots]
  15. R R
  16. A A
  17. K K
  18. L L
  19. K K
  20. A A
  21. L L

Krull–Schmidt_category.html

  1. R R
  2. R R
  3. X 1 X 2 X r Y 1 Y 2 Y s X_{1}\oplus X_{2}\oplus\cdots\oplus X_{r}\cong Y_{1}\oplus Y_{2}\oplus\cdots% \oplus Y_{s}
  4. X i X_{i}
  5. Y j Y_{j}
  6. r = s r=s
  7. π \pi
  8. X π ( i ) Y i X_{\pi(i)}\cong Y_{i}
  9. i i
  10. R R
  11. R R
  12. R R
  13. R R
  14. R R

Krull–Schmidt_theorem.html

  1. 1 = G 0 G 1 G 2 1=G_{0}\leq G_{1}\leq G_{2}\leq\cdots\,
  2. G = G 0 G 1 G 2 . G=G_{0}\geq G_{1}\geq G_{2}\geq\cdots.\,
  3. 𝐙 \mathbf{Z}
  4. p p^{\infty}
  5. 𝐐 / 𝐙 \mathbf{Q}/\mathbf{Z}
  6. G G
  7. G G
  8. G 1 × G 2 × × G k G_{1}\times G_{2}\times\cdots\times G_{k}\,
  9. G G
  10. G = H 1 × H 2 × × H l G=H_{1}\times H_{2}\times\cdots\times H_{l}\,
  11. G G
  12. k = l k=l
  13. H i H_{i}
  14. G i G_{i}
  15. H i H_{i}
  16. i i
  17. G = G 1 × × G r × H r + 1 × × H l G=G_{1}\times\cdots\times G_{r}\times H_{r+1}\times\cdots\times H_{l}\,
  18. r r
  19. E 0 E\neq 0
  20. E E

Krylov–Bogoliubov_averaging_method.html

  1. d 2 u d t 2 + k 2 u = a + ε f ( u , d u d t ) \frac{d^{2}u}{dt^{2}}+k^{2}u=a+\varepsilon f\left(u,\frac{du}{dt}\right)
  2. 0 < ε k . 0<\varepsilon\ll k.
  3. u ( t ) = a k 2 + A sin ( k t + B ) , u(t)=\frac{a}{k^{2}}+A\sin(kt+B),
  4. d u d t = k A ( t ) cos ( k t + B ( t ) ) , \frac{du}{dt}=kA(t)\cos(kt+B(t)),
  5. d d t [ A B ] = ε k f ( a k 2 + A sin ( ϕ ) , k A cos ( ϕ ) ) [ cos ( ϕ ) - 1 A sin ( ϕ ) ] , \frac{d}{dt}\begin{bmatrix}A\\ B\end{bmatrix}=\frac{\varepsilon}{k}f\left(\frac{a}{k^{2}}+A\sin(\phi),kA\cos(% \phi)\right)\begin{bmatrix}\cos(\phi)\\ -\frac{1}{A}\sin(\phi)\end{bmatrix},
  6. ϕ = k t + B \phi=kt+B
  7. d d t [ A 0 B 0 ] = ε 2 π k 0 2 π f ( a k 2 + A sin ( θ ) , k A cos ( θ ) ) [ cos ( θ ) - 1 A 0 sin ( θ ) ] d θ , \frac{d}{dt}\begin{bmatrix}A_{0}\\ B_{0}\end{bmatrix}=\frac{\varepsilon}{2\pi k}\int_{0}^{2\pi}f(\frac{a}{k^{2}}+% A\sin(\theta),kA\cos(\theta))\begin{bmatrix}\cos(\theta)\\ -\frac{1}{A_{0}}\sin(\theta)\end{bmatrix}d\theta,
  8. A 0 A_{0}
  9. B 0 B_{0}
  10. u 0 ( t , ε ) := a k 2 + A 0 ( t , ε ) sin ( k t + B 0 ( t , ε ) ) . u_{0}(t,\varepsilon):=\frac{a}{k^{2}}+A_{0}(t,\varepsilon)\sin(kt+B_{0}(t,% \varepsilon)).
  11. | u ( t , ε ) - u 0 ( t , ε ) | C 1 ε , \left|u(t,\varepsilon)-u_{0}(t,\varepsilon)\right|\leq C_{1}\varepsilon,
  12. 0 t C 2 ε 0\leq t\leq\frac{C_{2}}{\varepsilon}
  13. C 1 C_{1}
  14. C 2 C_{2}

Kugel–Khomskii_coupling.html

  1. H = t 2 U i , j [ 4 ( S i S j ) ( τ i α - 1 2 ) ( τ j α - 1 2 ) + ( τ i α + 1 2 ) ( τ j α + 1 2 ) - 1 ] H=\frac{t^{2}}{U}\sum_{\langle i,j\rangle}\left[4\left(\overrightarrow{S_{i}}% \cdot\overrightarrow{S_{j}}\right)(\tau_{i}^{\alpha}-\frac{1}{2})(\tau_{j}^{% \alpha}-\frac{1}{2})+(\tau_{i}^{\alpha}+\frac{1}{2})(\tau_{j}^{\alpha}+\frac{1% }{2})-1\right]

Kullback's_inequality.html

  1. Ψ Q * \Psi_{Q}^{*}
  2. Q Q
  3. μ 1 ( P ) \mu^{\prime}_{1}(P)
  4. P . P.
  5. M Q M_{Q}
  6. D K L ( P Q ) = D K L ( P Q θ ) + supp P ( log d Q θ d Q ) d P . D_{KL}(P\|Q)=D_{KL}(P\|Q_{\theta})+\int_{\mathrm{supp}P}\left(\log\frac{% \mathrm{d}Q_{\theta}}{\mathrm{d}Q}\right)\mathrm{d}P.
  7. D K L ( P Q θ ) 0 D_{KL}(P\|Q_{\theta})\geq 0
  8. D K L ( P Q ) supp P ( log d Q θ d Q ) d P = supp P ( log e θ x M Q ( θ ) ) P ( d x ) D_{KL}(P\|Q)\geq\int_{\mathrm{supp}P}\left(\log\frac{\mathrm{d}Q_{\theta}}{% \mathrm{d}Q}\right)\mathrm{d}P=\int_{\mathrm{supp}P}\left(\log\frac{e^{\theta x% }}{M_{Q}(\theta)}\right)P(dx)
  9. M Q ( θ ) < : M_{Q}(\theta)<\infty:
  10. D K L ( P Q ) μ 1 ( P ) θ - Ψ Q ( θ ) , D_{KL}(P\|Q)\geq\mu^{\prime}_{1}(P)\theta-\Psi_{Q}(\theta),
  11. μ 1 ( P ) \mu^{\prime}_{1}(P)
  12. Ψ Q = log M Q \Psi_{Q}=\log M_{Q}
  13. D K L ( P Q ) sup θ { μ 1 ( P ) θ - Ψ Q ( θ ) } = Ψ Q * ( μ 1 ( P ) ) . D_{KL}(P\|Q)\geq\sup_{\theta}\left\{\mu^{\prime}_{1}(P)\theta-\Psi_{Q}(\theta)% \right\}=\Psi_{Q}^{*}(\mu^{\prime}_{1}(P)).
  14. lim h 0 D K L ( X θ + h X θ ) h 2 lim h 0 Ψ θ * ( μ θ + h ) h 2 , \lim_{h\rightarrow 0}\frac{D_{KL}(X_{\theta+h}\|X_{\theta})}{h^{2}}\geq\lim_{h% \rightarrow 0}\frac{\Psi^{*}_{\theta}(\mu_{\theta+h})}{h^{2}},
  15. Ψ θ * \Psi^{*}_{\theta}
  16. X θ X_{\theta}
  17. μ θ + h \mu_{\theta+h}
  18. X θ + h . X_{\theta+h}.
  19. lim h 0 D K L ( X θ + h X θ ) h 2 = lim h 0 1 h 2 - ( log d X θ + h d X θ ) d X θ + h \lim_{h\rightarrow 0}\frac{D_{KL}(X_{\theta+h}\|X_{\theta})}{h^{2}}=\lim_{h% \rightarrow 0}\frac{1}{h^{2}}\int_{-\infty}^{\infty}\left(\log\frac{\mathrm{d}% X_{\theta+h}}{\mathrm{d}X_{\theta}}\right)\mathrm{d}X_{\theta+h}
  20. = lim h 0 1 h 2 - [ ( 1 - d X θ d X θ + h ) + 1 2 ( 1 - d X θ d X θ + h ) 2 + o ( ( 1 - d X θ d X θ + h ) 2 ) ] d X θ + h , =\lim_{h\rightarrow 0}\frac{1}{h^{2}}\int_{-\infty}^{\infty}\left[\left(1-% \frac{\mathrm{d}X_{\theta}}{\mathrm{d}X_{\theta+h}}\right)+\frac{1}{2}\left(1-% \frac{\mathrm{d}X_{\theta}}{\mathrm{d}X_{\theta+h}}\right)^{2}+o\left(\left(1-% \frac{\mathrm{d}X_{\theta}}{\mathrm{d}X_{\theta+h}}\right)^{2}\right)\right]% \mathrm{d}X_{\theta+h},
  21. log x \log x
  22. 1 - 1 / x 1-1/x
  23. = lim h 0 1 h 2 - [ 1 2 ( 1 - d X θ d X θ + h ) 2 ] d X θ + h =\lim_{h\rightarrow 0}\frac{1}{h^{2}}\int_{-\infty}^{\infty}\left[\frac{1}{2}% \left(1-\frac{\mathrm{d}X_{\theta}}{\mathrm{d}X_{\theta+h}}\right)^{2}\right]% \mathrm{d}X_{\theta+h}
  24. = lim h 0 1 h 2 - [ 1 2 ( d X θ + h - d X θ d X θ + h ) 2 ] d X θ + h = 1 2 X ( θ ) , =\lim_{h\rightarrow 0}\frac{1}{h^{2}}\int_{-\infty}^{\infty}\left[\frac{1}{2}% \left(\frac{\mathrm{d}X_{\theta+h}-\mathrm{d}X_{\theta}}{\mathrm{d}X_{\theta+h% }}\right)^{2}\right]\mathrm{d}X_{\theta+h}=\frac{1}{2}\mathcal{I}_{X}(\theta),
  25. lim h 0 Ψ θ * ( μ θ + h ) h 2 = lim h 0 1 h 2 sup t { μ θ + h t - Ψ θ ( t ) } . \lim_{h\rightarrow 0}\frac{\Psi^{*}_{\theta}(\mu_{\theta+h})}{h^{2}}=\lim_{h% \rightarrow 0}\frac{1}{h^{2}}{\sup_{t}\{\mu_{\theta+h}t-\Psi_{\theta}(t)\}}.
  26. Ψ θ ( τ ) = μ θ + h , \Psi^{\prime}_{\theta}(\tau)=\mu_{\theta+h},
  27. Ψ θ ( 0 ) = μ θ , \Psi^{\prime}_{\theta}(0)=\mu_{\theta},
  28. Ψ θ ′′ ( 0 ) = d μ θ d θ lim h 0 h τ . \Psi^{\prime\prime}_{\theta}(0)=\frac{d\mu_{\theta}}{d\theta}\lim_{h% \rightarrow 0}\frac{h}{\tau}.
  29. lim h 0 Ψ θ * ( μ θ + h ) h 2 = 1 2 Ψ θ ′′ ( 0 ) ( d μ θ d θ ) 2 = 1 2 V a r ( X θ ) ( d μ θ d θ ) 2 . \lim_{h\rightarrow 0}\frac{\Psi^{*}_{\theta}(\mu_{\theta+h})}{h^{2}}=\frac{1}{% 2\Psi^{\prime\prime}_{\theta}(0)}\left(\frac{d\mu_{\theta}}{d\theta}\right)^{2% }=\frac{1}{2\mathrm{Var}(X_{\theta})}\left(\frac{d\mu_{\theta}}{d\theta}\right% )^{2}.
  30. 1 2 X ( θ ) 1 2 V a r ( X θ ) ( d μ θ d θ ) 2 , \frac{1}{2}\mathcal{I}_{X}(\theta)\geq\frac{1}{2\mathrm{Var}(X_{\theta})}\left% (\frac{d\mu_{\theta}}{d\theta}\right)^{2},
  31. Var ( X θ ) ( d μ θ / d θ ) 2 X ( θ ) . \mathrm{Var}(X_{\theta})\geq\frac{(d\mu_{\theta}/d\theta)^{2}}{\mathcal{I}_{X}% (\theta)}.

Kuratowski_embedding.html

  1. Φ : X C b ( X ) \Phi:X\rightarrow C_{b}(X)
  2. Φ ( x ) ( y ) = d ( x , y ) - d ( x 0 , y ) for all x , y X \Phi(x)(y)=d(x,y)-d(x_{0},y)\quad\mbox{for all}~{}\quad x,y\in X
  3. Ψ : X C b ( X ) \Psi:X\rightarrow C_{b}(X)
  4. Ψ ( x ) ( y ) = d ( x , y ) for all x , y X \Psi(x)(y)=d(x,y)\quad\mbox{for all}~{}\quad x,y\in X

Kurosh_subgroup_theorem.html

  1. H = F ( X ) * ( * i I g i A i g i - 1 ) * ( * j J f j B j f j - 1 ) . H=F(X)*(*_{i\in I}g_{i}A_{i}g_{i}^{-1})*(*_{j\in J}f_{j}B_{j}f_{j}^{-1}).
  2. H = F ( X ) * ( * j J g j H j g j - 1 ) , H=F(X)*(*_{j\in J}g_{j}H_{j}g_{j}^{-1}),

Kuznetsov_trace_formula.html

  1. g : g:\mathbb{R}\rightarrow\mathbb{R}
  2. c 0 mod N c - r K ( m , n , c ) g ( 4 π m n c ) = Integral transform + Spectral terms . \sum_{c\equiv 0\,\,\text{mod}\ N}c^{-r}K(m,n,c)g\left(\frac{4\pi\sqrt{mn}}{c}% \right)=\,\text{Integral transform}\ +\ \,\text{Spectral terms}.
  3. G G
  4. H G H\subset G
  5. G / H G/H

Kværner-process.html

  1. C n H m nC + m 2 H 2 \mathrm{C_{n}H_{m}\rightarrow nC+\frac{m}{2}H_{2}}

K·p_perturbation_theory.html

  1. ( p 2 2 m + V ) ψ = E ψ \left(\frac{p^{2}}{2m}+V\right)\psi=E\psi
  2. ψ n , 𝐤 ( 𝐱 ) = e i 𝐤 𝐱 u n , 𝐤 ( 𝐱 ) \psi_{n,\mathbf{k}}(\mathbf{x})=e^{i\mathbf{k}\cdot\mathbf{x}}u_{n,\mathbf{k}}% (\mathbf{x})
  3. H 𝐤 u n , 𝐤 = E n , 𝐤 u n , 𝐤 H_{\mathbf{k}}u_{n,\mathbf{k}}=E_{n,\mathbf{k}}u_{n,\mathbf{k}}
  4. H 𝐤 = p 2 2 m + 𝐤 𝐩 m + 2 k 2 2 m + V H_{\mathbf{k}}=\frac{p^{2}}{2m}+\frac{\hbar\mathbf{k}\cdot\mathbf{p}}{m}+\frac% {\hbar^{2}k^{2}}{2m}+V
  5. 𝐤 𝐩 = k x ( - i x ) + k y ( - i y ) + k z ( - i z ) \mathbf{k}\cdot\mathbf{p}=k_{x}(-i\hbar\frac{\partial}{\partial x})+k_{y}(-i% \hbar\frac{\partial}{\partial y})+k_{z}(-i\hbar\frac{\partial}{\partial z})
  6. H = H 0 + H 𝐤 , H 0 = p 2 2 m + V , H 𝐤 = 2 k 2 2 m + 𝐤 𝐩 m H=H_{0}+H_{\mathbf{k}}^{\prime},\;\;H_{0}=\frac{p^{2}}{2m}+V,\;\;H_{\mathbf{k}% }^{\prime}=\frac{\hbar^{2}k^{2}}{2m}+\frac{\hbar\mathbf{k}\cdot\mathbf{p}}{m}
  7. H 𝐤 H_{\mathbf{k}}^{\prime}
  8. H 𝐤 H_{\mathbf{k}}^{\prime}
  9. u n , 𝐤 = u n , 0 + m n n u n , 0 | 𝐤 𝐩 | u n , 0 E n , 0 - E n , 0 u n , 0 u_{n,\mathbf{k}}=u_{n,0}+\frac{\hbar}{m}\sum_{n^{\prime}\neq n}\frac{\langle u% _{n,0}|\mathbf{k}\cdot\mathbf{p}|u_{n^{\prime},0}\rangle}{E_{n,0}-E_{n^{\prime% },0}}u_{n^{\prime},0}
  10. E n , 𝐤 = E n , 0 + 2 k 2 2 m + 2 m 2 n n | u n , 0 | 𝐤 𝐩 | u n , 0 | 2 E n , 0 - E n , 0 E_{n,\mathbf{k}}=E_{n,0}+\frac{\hbar^{2}k^{2}}{2m}+\frac{\hbar^{2}}{m^{2}}\sum% _{n^{\prime}\neq n}\frac{|\langle u_{n,0}|\mathbf{k}\cdot\mathbf{p}|u_{n^{% \prime},0}\rangle|^{2}}{E_{n,0}-E_{n^{\prime},0}}
  11. u n , 0 | 𝐤 𝐩 | u n , 0 = 𝐤 u n , 0 | 𝐩 | u n , 0 \langle u_{n,0}|\mathbf{k}\cdot\mathbf{p}|u_{n^{\prime},0}\rangle=\mathbf{k}% \cdot\langle u_{n,0}|\mathbf{p}|u_{n^{\prime},0}\rangle
  12. u n , 0 | 𝐩 | u n , 0 \langle u_{n,0}|\mathbf{p}|u_{n^{\prime},0}\rangle
  13. E c ( s y m b o l k ) E c 0 + ( k ) 2 2 m + 2 E g m 2 n | u c , 0 | 𝐤 𝐩 | u n , 0 | 2 E_{c}(symbolk)\approx E_{c0}+\frac{(\hbar k)^{2}}{2m}+\frac{\hbar^{2}}{{E_{g}}% m^{2}}\sum_{n}{|\langle u_{c,0}|\mathbf{k}\cdot\mathbf{p}|u_{n,0}\rangle|^{2}}
  14. 1 m = 1 2 m 2 E c ( s y m b o l k ) k k m 1 m + 2 E g m 2 m , n u c , 0 | p | u n , 0 u n , 0 | p m | u c , 0 \frac{1}{m}_{\ell}={{1}\over{\hbar^{2}}}\sum_{m}\cdot{{\partial^{\ 2}E_{c}(% symbol{k})}\over{\partial k_{\ell}\partial k_{m}}}\approx\frac{1}{m}+\frac{2}{% E_{g}m^{2}}\sum_{m,\ n}{\langle u_{c,0}|p_{\ell}|u_{n,0}\rangle}{\langle u_{n,% 0}|p_{m}|u_{c,0}\rangle}
  15. 2 E g m 2 m , n | u c , 0 | p | u n , 0 | | u c , 0 | p m | u n , 0 | 20 e V 1 m E g , \frac{2}{E_{g}m^{2}}\sum_{m,\ n}{|\langle u_{c,0}|p_{\ell}|u_{n,0}\rangle|}{|% \langle u_{c,0}|p_{m}|u_{n,0}\rangle|}\approx 20\mathrm{eV}\frac{1}{mE_{g}}\ ,
  16. H 𝐤 u n , 𝐤 = E n , 𝐤 u n , 𝐤 H_{\mathbf{k}}u_{n,\mathbf{k}}=E_{n,\mathbf{k}}u_{n,\mathbf{k}}
  17. H 𝐤 = p 2 2 m + m 𝐤 𝐩 + 2 k 2 2 m + V + 4 m 2 c 2 ( V × ( 𝐩 + 𝐤 ) ) σ H_{\mathbf{k}}=\frac{p^{2}}{2m}+\frac{\hbar}{m}\mathbf{k}\cdot\mathbf{p}+\frac% {\hbar^{2}k^{2}}{2m}+V+\frac{\hbar}{4m^{2}c^{2}}(\nabla V\times(\mathbf{p}+% \hbar\mathbf{k}))\cdot\vec{\sigma}
  18. σ = ( σ x , σ y , σ z ) \vec{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})

L-estimator.html

  1. x 1 , , x n x_{1},\ldots,x_{n}
  2. n = 2 k + 1 n=2k+1
  3. x ( k + 1 ) x_{(k+1)}
  4. ( n + 1 ) / 2 (n+1)/2
  5. n = 2 k n=2k
  6. ( x ( k ) + x ( k + 1 ) ) / 2 (x_{(k)}+x_{(k+1)})/2
  7. 2 2 erf - 1 ( 1 / 2 ) 1.349 2\sqrt{2}\operatorname{erf}^{-1}(1/2)\approx 1.349
  8. n \sqrt{n}

Lack-of-fit_sum_of_squares.html

  1. y = α x + β y=\alpha x+\beta\,
  2. ( observed value - fitted value ) 2 \displaystyle\sum(\,\text{observed value}-\,\text{fitted value})^{2}
  3. Y i j = α x i + β + ε i j , i = 1 , , n , j = 1 , , n i . Y_{ij}=\alpha x_{i}+\beta+\varepsilon_{ij},\qquad i=1,\dots,n,\quad j=1,\dots,% n_{i}.
  4. α ^ , β ^ \widehat{\alpha},\widehat{\beta}\,
  5. Y ^ i = α ^ x i + β ^ \widehat{Y}_{i}=\widehat{\alpha}x_{i}+\widehat{\beta}\,
  6. ε ^ i j = Y i j - Y ^ i \widehat{\varepsilon}_{ij}=Y_{ij}-\widehat{Y}_{i}\,
  7. N = i = 1 n n i N=\sum_{i=1}^{n}n_{i}
  8. i = 1 n j = 1 n i ε ^ i j = 0 \sum_{i=1}^{n}\sum_{j=1}^{n_{i}}\widehat{\varepsilon}_{ij}=0\,
  9. i = 1 n ( x i j = 1 n i ε ^ i j ) = 0. \sum_{i=1}^{n}\left(x_{i}\sum_{j=1}^{n_{i}}\widehat{\varepsilon}_{ij}\right)=0.\,
  10. Y ¯ i = 1 n i j = 1 n i Y i j \overline{Y}_{i\bullet}=\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}Y_{ij}
  11. i = 1 n j = 1 n i ε ^ i j 2 = i = 1 n j = 1 n i ( Y i j - Y ^ i ) 2 \displaystyle\sum_{i=1}^{n}\sum_{j=1}^{n_{i}}\widehat{\varepsilon}_{ij}^{\,2}=% \sum_{i=1}^{n}\sum_{j=1}^{n_{i}}\left(Y_{ij}-\widehat{Y}_{i}\right)^{2}
  12. 1 σ 2 i = 1 n j = 1 n i ε ^ i j 2 \frac{1}{\sigma^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n_{i}}\widehat{\varepsilon}_{ij}% ^{\,2}
  13. F \displaystyle F

Lacunarity.html

  1. Λ \Lambda
  2. λ \lambda
  3. \Epsilon \Epsilon
  4. ϵ \epsilon
  5. ϵ \epsilon
  6. ϵ \epsilon
  7. \Epsilon \Epsilon
  8. ϵ \epsilon
  9. λ ϵ \lambda_{\epsilon}
  10. 𝐶𝑉 \mathit{CV}
  11. σ \sigma
  12. μ \mu
  13. ϵ \epsilon
  14. G \mathit{G}
  15. g \mathit{g}
  16. λ ϵ , g \lambda_{\epsilon,g}
  17. λ ϵ , g = ( C V ϵ , g ) 2 = ( σ ϵ , g μ ϵ , g ) 2 \lambda_{\epsilon,g}=(CV_{\epsilon,g})^{2}=\left(\frac{\sigma_{\epsilon,g}}{% \mu_{\epsilon,g}}\right)^{2}
  18. B B
  19. m m
  20. p p
  21. λ ϵ , g \lambda_{\epsilon,g}
  22. μ ϵ = i = 1 B m i , ϵ p i , ϵ \mu_{\epsilon}=\sum_{i=1}^{B}{m_{i,\epsilon}p_{i,\epsilon}}
  23. v ϵ = i - 1 B ( m i , ϵ - μ ϵ ) 2 p i , ϵ \mathit{v}_{\epsilon}=\sum_{i-1}^{B}{\left(m_{i,\epsilon}-\mu_{\epsilon}\right% )^{2}p_{i,\epsilon}}
  24. σ ϵ = v ϵ = i = 1 B m i , ϵ 2 p i , ϵ - μ ϵ 2 \sigma_{\epsilon}=\sqrt{\mathit{v}_{\epsilon}}=\sum_{i=1}^{B}{m_{i,\epsilon}^{% 2}p_{i,\epsilon}-\mu_{\epsilon}^{2}}
  25. λ ϵ = i = 1 B m i , ϵ 2 p i , ϵ - μ ϵ 2 μ ϵ 2 = σ ϵ 2 μ ϵ 2 \lambda_{\epsilon}=\frac{\sum_{i=1}^{B}{m_{i,\epsilon}^{2}p_{i,\epsilon}-\mu_{% \epsilon}^{2}}}{\mu_{\epsilon}^{2}}=\frac{\sigma_{\epsilon}^{2}}{\mu_{\epsilon% }^{2}}
  26. λ \lambda
  27. λ ϵ , g \lambda_{\epsilon,g}
  28. λ ϵ , g \lambda_{\epsilon,g}
  29. ϵ \epsilon
  30. λ ϵ , g ¯ = ϵ = 1 \Epsilon λ ϵ , g \Epsilon \overline{\lambda_{\epsilon,g}}=\frac{\sum_{\epsilon=1}^{\Epsilon}\lambda_{% \epsilon,g}}{\Epsilon}
  31. Λ g = g = 1 G λ ϵ , g ¯ G \Lambda_{\mathit{g}}=\frac{\sum_{\mathit{g}=1}^{\mathit{G}}\overline{\lambda_{% \epsilon,g}}}{\mathit{G}}
  32. λ ϵ , g \lambda_{\epsilon,g}
  33. ϵ \epsilon
  34. l n ln
  35. l n ln
  36. g \mathit{g}
  37. l n ln
  38. l n ln
  39. l n ln
  40. l n ln
  41. f λ ϵ , g = λ ϵ , g + 1 f\lambda_{\epsilon,g}=\lambda_{\epsilon,g}+\mathit{1}
  42. σ \sigma
  43. ϵ \epsilon
  44. l n ln
  45. l n ln
  46. f λ ϵ , g f\lambda_{\epsilon,g}
  47. ( r ) = i = 1 r 2 S i 2 Q ( S i , r ) ( i = 1 r 2 S i Q ( S i , r ) ) 2 . \mathcal{L}(r)=\frac{\sum_{i=1}^{r^{2}}S_{i}^{2}Q(S_{i},r)}{\left(\sum_{i=1}^{% r^{2}}S_{i}Q(S_{i},r)\right)^{2}}.
  48. S i S_{i}
  49. Q ( S i , r ) Q(S_{i},r)
  50. S i S_{i}
  51. D B D_{B}
  52. A A
  53. N = A ϵ D B N=A\epsilon^{D_{B}}
  54. A A
  55. y \mathit{y}
  56. ϵ \epsilon
  57. N N
  58. m m
  59. g g
  60. A A
  61. ϵ \epsilon
  62. A g = 1 e y g A_{g}=\frac{1}{e^{\mathit{y}_{g}}}
  63. A ¯ = g = 1 G A g G \overline{A}=\frac{\sum_{g=1}^{G}A_{g}}{G}
  64. P Λ = g = 1 G ( A g A ¯ - 1 ) 2 G P\Lambda=\frac{\sum_{g=1}^{G}{\left({\frac{A_{g}}{\overline{A}}}-1\right)^{2}}% }{G}

Ladder_graph.html

  1. ( x - 1 ) x ( x 2 - 3 x + 3 ) ( n - 1 ) (x-1)x(x^{2}-3x+3)^{(n-1)}

Lagrange_number.html

  1. | α - p q | < 1 5 q 2 . \left|\alpha-\frac{p}{q}\right|<\frac{1}{\sqrt{5}q^{2}}.
  2. L n = 9 - 4 m n 2 L_{n}=\sqrt{9-\frac{4}{{m_{n}}^{2}}}
  3. m 2 + x 2 + y 2 = 3 m x y m^{2}+x^{2}+y^{2}=3mxy\,

Lambert's_problem.html

  1. r ¯ ¨ = - μ r ^ r 2 \ddot{\bar{r}}=-\mu\cdot\frac{\hat{r}}{r^{2}}\
  2. t 1 , t 2 \ t_{1}\ ,\ t_{2}
  3. r ¯ 1 = r 1 r ^ 1 , r ¯ 2 = r 2 r ^ 2 \bar{r}_{1}=r_{1}{\hat{r}}_{1},\ \bar{r}_{2}=r_{2}{\hat{r}}_{2}
  4. r ¯ ( t ) \bar{r}(t)
  5. r ¯ ( t 1 ) = r ¯ 1 \bar{r}(t_{1})=\bar{r}_{1}
  6. r ¯ ( t 2 ) = r ¯ 2 . \bar{r}(t_{2})=\bar{r}_{2}.
  7. F 1 F_{1}
  8. P 1 P_{1}
  9. r ¯ 1 \bar{r}_{1}
  10. P 2 P_{2}
  11. r ¯ 2 \bar{r}_{2}
  12. P 1 P_{1}
  13. P 2 P_{2}
  14. F 1 F_{1}
  15. F 1 F_{1}
  16. F 2 F_{2}
  17. P 1 P_{1}
  18. P 2 P_{2}
  19. F 1 F_{1}
  20. P 1 P_{1}
  21. r ¯ 1 \bar{r}_{1}
  22. P 2 P_{2}
  23. r ¯ 2 \bar{r}_{2}
  24. r ¯ 1 \bar{r}_{1}
  25. r ¯ 2 \bar{r}_{2}
  26. P 1 P_{1}
  27. P 2 P_{2}
  28. 2 d 2d
  29. P 1 P_{1}
  30. F 1 F_{1}
  31. r 1 = r m - A r_{1}=r_{m}-A
  32. P 2 P_{2}
  33. F 1 F_{1}
  34. r 2 = r m + A r_{2}=r_{m}+A
  35. A A
  36. P 1 P_{1}
  37. P 2 P_{2}
  38. F 1 F_{1}
  39. P 1 P_{1}
  40. P 2 P_{2}
  41. F 1 F_{1}
  42. F 1 F_{1}
  43. P 1 P_{1}
  44. P 2 P_{2}
  45. F 1 F_{1}
  46. P 1 P_{1}
  47. P 2 P_{2}
  48. F 1 F_{1}
  49. A A
  50. | A | |A|
  51. E E
  52. d | A | \frac{d}{|A|}
  53. x 2 A 2 - y 2 B 2 = 1 ( 1 ) \frac{x^{2}}{A^{2}}-\frac{y^{2}}{B^{2}}=1\quad(1)
  54. B = | A | E 2 - 1 = d 2 - A 2 ( 2 ) B=|A|\sqrt{E^{2}-1}=\sqrt{d^{2}-A^{2}}\quad(2)
  55. F 1 F_{1}
  56. r 2 r_{2}
  57. P 2 P_{2}
  58. r 1 r_{1}
  59. P 1 P_{1}
  60. r 2 - r 1 = 2 A ( 3 ) r_{2}-r_{1}=2A\quad(3)
  61. F 2 F_{2}
  62. s 1 - s 2 = 2 A ( 4 ) s_{1}-s_{2}=2A\quad(4)
  63. r 1 + s 1 = r 2 + s 2 ( 5 ) r_{1}+s_{1}=r_{2}+s_{2}\quad(5)
  64. P 1 P_{1}
  65. P 2 P_{2}
  66. F 1 F_{1}
  67. F 2 F_{2}
  68. a = r 1 + s 1 2 = r 2 + s 2 2 ( 6 ) a=\frac{r_{1}+s_{1}}{2}=\frac{r_{2}+s_{2}}{2}\quad(6)
  69. F 2 F_{2}
  70. r ¯ 1 × r ¯ 2 \bar{r}_{1}\times\bar{r}_{2}
  71. - r ¯ 1 × r ¯ 2 -\bar{r}_{1}\times\bar{r}_{2}
  72. α \alpha
  73. r ¯ 2 \bar{r}_{2}
  74. 0 < α < 180 \ 0<\alpha<180^{\circ}
  75. 180 < α < 360 180^{\circ}<\alpha<360^{\circ}
  76. r ¯ ( t ) \bar{r}(t)
  77. r ¯ 2 \bar{r}_{2}
  78. r ¯ 1 × r ¯ 2 \bar{r}_{1}\times\bar{r}_{2}
  79. r ¯ 1 \bar{r}_{1}
  80. r ¯ 2 \bar{r}_{2}
  81. α \alpha
  82. r ¯ 2 \bar{r}_{2}
  83. 180 180^{\circ}
  84. α \alpha
  85. 0 < α < \ 0<\alpha<\infty
  86. P 1 P_{1}
  87. P 2 P_{2}
  88. F 1 F_{1}
  89. d = r 1 2 + r 2 2 - 2 r 1 r 2 cos α 2 ( 7 ) d=\frac{\sqrt{{r_{1}}^{2}+{r_{2}}^{2}-2r_{1}r_{2}\cos\alpha}}{2}\quad(7)
  90. A = r 2 - r 1 2 ( 8 ) A=\frac{r_{2}-r_{1}}{2}\quad(8)
  91. E = d A ( 9 ) E=\frac{d}{A}\quad(9)
  92. B = | A | E 2 - 1 = d 2 - A 2 ( 10 ) B=|A|\sqrt{E^{2}-1}=\sqrt{d^{2}-A^{2}}\quad(10)
  93. F 1 F_{1}
  94. E E
  95. r 2 - r 1 r_{2}-r_{1}
  96. x 0 = - r m E ( 11 ) x_{0}=-\frac{r_{m}}{E}\quad(11)
  97. y 0 = B ( x 0 A ) 2 - 1 ( 12 ) y_{0}=B\sqrt{{\left(\frac{x_{0}}{A}\right)}^{2}-1}\quad(12)
  98. r m = r 2 + r 1 2 ( 13 ) r_{m}=\frac{r_{2}+r_{1}}{2}\quad(13)
  99. F 2 F_{2}
  100. F 2 F_{2}
  101. A A
  102. r 2 - r 1 r_{2}-r_{1}
  103. x = A 1 + ( y B ) 2 ( 14 ) x=A\sqrt{1+{\left(\frac{y}{B}\right)}^{2}}\quad(14)
  104. P 1 P_{1}
  105. P 2 P_{2}
  106. F 1 F_{1}
  107. F 2 F_{2}
  108. a = r 1 + s 1 2 = r 2 + s 2 2 = r m + E x 2 ( 15 ) a=\frac{r_{1}+s_{1}}{2}=\frac{r_{2}+s_{2}}{2}\ =\frac{r_{m}+Ex}{2}\quad(15)
  109. ( x 0 - x ) 2 + ( y 0 - y ) 2 ( 16 ) \sqrt{{(x_{0}-x)}^{2}+{(y_{0}-y)}^{2}}\quad(16)
  110. e = ( x 0 - x ) 2 + ( y 0 - y ) 2 2 a ( 17 ) e=\frac{\sqrt{{(x_{0}-x)}^{2}+{(y_{0}-y)}^{2}}}{2a}\quad(17)
  111. θ 1 \theta_{1}
  112. P 1 P_{1}
  113. sin α \sin\alpha
  114. cos θ 1 = - ( x 0 + d ) f x + y 0 f y r 1 ( 18 ) \cos\theta_{1}=-\frac{(x_{0}+d)f_{x}+y_{0}f_{y}}{r_{1}}\quad(18)
  115. f x = x 0 - x ( x 0 - x ) 2 + ( y 0 - y ) 2 ( 19 ) f_{x}=\frac{x_{0}-x}{\sqrt{{(x_{0}-x)}^{2}+{(y_{0}-y)}^{2}}}\quad(19)
  116. f y = y 0 - y ( x 0 - x ) 2 + ( y 0 - y ) 2 ( 20 ) f_{y}=\frac{y_{0}-y}{\sqrt{{(x_{0}-x)}^{2}+{(y_{0}-y)}^{2}}}\quad(20)
  117. P 2 P_{2}
  118. P 1 P_{1}
  119. sin α \sin\alpha
  120. sin θ 1 = ( x 0 + d ) f y - y 0 f x r 1 ( 21 ) \sin\theta_{1}=\frac{(x_{0}+d)f_{y}-y_{0}f_{x}}{r_{1}}\quad(21)
  121. sin α \sin\alpha
  122. sin θ 1 = - ( x 0 + d ) f y - y 0 f x r 1 ( 22 ) \sin\theta_{1}=-\frac{(x_{0}+d)f_{y}-y_{0}f_{x}}{r_{1}}\quad(22)
  123. α \alpha
  124. t 2 - t 1 t_{2}-t_{1}
  125. r 1 = r 2 r_{1}=r_{2}
  126. A = 0 A=0
  127. P 1 P_{1}
  128. P 2 P_{2}
  129. x = 0 ( 1 ) x=0\quad(1^{\prime})
  130. x 0 = 0 ( 11 ) x_{0}=0\quad(11^{\prime})
  131. y 0 = r m 2 - d 2 ( 12 ) y_{0}=\sqrt{{r_{m}}^{2}-d^{2}}\quad(12^{\prime})
  132. x = 0 ( 14 ) x=0\quad(14^{\prime})
  133. a = r m + d 2 + y 2 2 ( 15 ) a=\frac{r_{m}+\sqrt{d^{2}+y^{2}}}{2}\quad(15^{\prime})
  134. μ \mu
  135. V r = μ p e sin θ V_{r}=\sqrt{\frac{\mu}{p}}\cdot e\cdot\sin\theta
  136. V t = μ p ( 1 + e cos θ ) . V_{t}=\sqrt{\frac{\mu}{p}}\cdot(1+e\cdot\cos\theta).