wpmath0000007_1

Banach_bundle.html

  1. { U i | i I } \{U_{i}|i\in I\}
  2. τ i : π - 1 ( U i ) U i × X i \tau_{i}:\pi^{-1}(U_{i})\to U_{i}\times X_{i}
  3. τ i x : π - 1 ( x ) X i \tau_{ix}:\pi^{-1}(x)\to X_{i}
  4. U i U j Lin ( X i ; X j ) U_{i}\cap U_{j}\to\mathrm{Lin}(X_{i};X_{j})
  5. x ( τ j τ i - 1 ) x x\mapsto(\tau_{j}\circ\tau_{i}^{-1})_{x}
  6. τ i x : π - 1 ( x ) X \tau_{ix}:\pi^{-1}(x)\to X
  7. π : T V V ; \pi:\mathrm{T}V\to V;
  8. ( x , v ) x . (x,v)\mapsto x.
  9. τ = id : π - 1 ( V ) = T V V × V ; \tau=\mathrm{id}:\pi^{-1}(V)=\mathrm{T}V\to V\times V;
  10. ( x , v ) ( x , v ) . (x,v)\mapsto(x,v).
  11. π - 1 ( x ) = T x * M = ( T x M ) * ; \pi^{-1}(x)=\mathrm{T}_{x}^{*}M=(\mathrm{T}_{x}M)^{*};
  12. f 0 : M M ; f_{0}:M\to M^{\prime};
  13. f : E E . f:E\to E^{\prime}.
  14. f x : E x E f 0 ( x ) f_{x}:E_{x}\to E^{\prime}_{f_{0}(x)}
  15. τ : π - 1 ( U ) U × X \tau:\pi^{-1}(U)\to U\times X
  16. τ : π - 1 ( U ) U × X \tau^{\prime}:\pi^{\prime-1}(U^{\prime})\to U^{\prime}\times X^{\prime}
  17. f 0 ( U ) U f_{0}(U)\subseteq U^{\prime}
  18. U Lin ( X ; X ) U\to\mathrm{Lin}(X;X^{\prime})
  19. x τ f 0 ( x ) f x τ - 1 x\mapsto\tau^{\prime}_{f_{0}(x)}\circ f_{x}\circ\tau^{-1}
  20. f * π : f * E = M × X M f^{*}\pi:f^{*}E=M\times X\to M
  21. f * ( E V ) = ( f * E ) U f^{*}(E_{V})=(f^{*}E)_{U}

Banach_function_algebra.html

  1. ( f A ) (f\in A)
  2. ( p , q X ) (p,q\in X)
  3. ( f A ) (f\in A)
  4. f ( p ) f ( q ) f(p)\neq f(q)
  5. x X x\in X
  6. ε x ( f ) = f ( x ) ( f A ) \varepsilon_{x}(f)=f(x)\ (f\in A)
  7. ε x \varepsilon_{x}
  8. A A
  9. A A
  10. X X
  11. A A

Bandwidth_expansion.html

  1. γ \gamma
  2. A ( z ) A^{\prime}(z)
  3. A ( z ) A(z)
  4. A ( z ) = A ( z / γ ) A^{\prime}(z)=A(z/\gamma)
  5. A ( z ) A(z)
  6. A ( z ) = k = 0 N a k z - k A(z)=\sum_{k=0}^{N}a_{k}z^{-k}
  7. A ( z ) = k = 0 N a k γ k z - k A^{\prime}(z)=\sum_{k=0}^{N}a_{k}\gamma^{k}z^{-k}
  8. a k a_{k}
  9. γ k \gamma^{k}

Banked_turn.html

  1. μ m g > m v 2 r . \mu mg>{mv^{2}\over r}.
  2. v < r μ g . v<{\sqrt{r\mu g}}.
  3. N sin θ = m v 2 r N\sin\theta={mv^{2}\over r}
  4. N cos θ = m g N\cos\theta=mg
  5. m v 2 r = m g tan θ {mv^{2}\over r}={mg\tan\theta}
  6. v 2 r = g tan θ {v^{2}\over r}={g\tan\theta}
  7. v = r g tan θ v={\sqrt{rg\tan\theta}}
  8. m v 2 r = μ s N cos θ + N sin θ {mv^{2}\over r}=\mu_{s}N\cos\theta+N\sin\theta
  9. N cos θ = μ s N sin θ + m g N\cos\theta=\mu_{s}N\sin\theta+mg
  10. v 2 ( N cos θ - μ s N sin θ ) r g = μ s N cos θ + N sin θ {v^{2}\left(N\cos\theta-\mu_{s}N\sin\theta\right)\over rg}=\mu_{s}N\cos\theta+% N\sin\theta
  11. v = r g ( sin θ + μ s cos θ ) cos θ - μ s sin θ = r g ( tan θ + μ s ) 1 - μ s tan θ v={\sqrt{rg\left(\sin\theta+\mu_{s}\cos\theta\right)\over\cos\theta-\mu_{s}% \sin\theta}}={\sqrt{rg\left(\tan\theta+\mu_{s}\right)\over 1-\mu_{s}\tan\theta}}
  12. v = r g ( sin θ - μ s cos θ ) cos θ + μ s sin θ = r g ( tan θ - μ s ) 1 + μ s tan θ v={\sqrt{rg\left(\sin\theta-\mu_{s}\cos\theta\right)\over\cos\theta+\mu_{s}% \sin\theta}}={\sqrt{rg\left(\tan\theta-\mu_{s}\right)\over 1+\mu_{s}\tan\theta}}
  13. a = v 2 r a={v^{2}\over r}
  14. L sin θ = m v 2 r L\sin\theta={mv^{2}\over r}
  15. L = m g cos θ L={mg\over{\cos\theta}}
  16. r = v 2 g tan θ r={v^{2}\over{g\tan\theta}}

Barker_code.html

  1. a j a_{j}
  2. j = 1 , 2 , , N j=1,2,\dots,N
  3. c v = j = 1 N - v a j a j + v c_{v}=\sum_{j=1}^{N-v}a_{j}a_{j+v}
  4. | c v | 1 |c_{v}|\leq 1\,
  5. 1 v < N 1\leq v<N
  6. c v { - 1 , 0 } c_{v}\in\{-1,0\}
  7. 2 n - 1 2^{n}-1

Barrier_function.html

  1. x x
  2. b b
  3. f ( x ) + g ( x , b ) f(x)+g(x,b)
  4. g ( x , b ) g(x,b)
  5. g ( x , b ) g(x,b)
  6. - log ( b - x ) -\log(b-x)
  7. x < b x<b
  8. \infty
  9. log ( t ) \log(t)
  10. t t
  11. x x
  12. b b
  13. x i x_{i}
  14. b i b_{i}
  15. - log ( b i - x i ) -\log(b_{i}-x_{i})
  16. c T x c^{T}x
  17. a i T x b i , i = 1 , , m a_{i}^{T}x\leq b_{i},i=1,\ldots,m
  18. { x | A x < b } \{x|Ax<b\}\neq\emptyset
  19. Φ ( x ) = { i = 1 m - log ( b i - a i T x ) for A x < b + otherwise \Phi(x)=\begin{cases}\sum_{i=1}^{m}-\log(b_{i}-a_{i}^{T}x)&\,\text{for }Ax<b\\ +\infty&\,\text{otherwise}\end{cases}

Base_(group_theory).html

  1. G G
  2. Ω \Omega
  3. B = [ β 1 , β 2 , , β k ] B=[\beta_{1},\beta_{2},...,\beta_{k}]
  4. Ω \Omega
  5. G G
  6. β i B \beta_{i}\in B
  7. G G

Base_excess.html

  1. B a s e e x c e s s = 0.93 × ( [ H C O 3 - ] - 24.4 + 14.8 × ( p H - 7.4 ) ) Base~{}excess=0.93\times\left(\left[HCO_{3}^{-}\right]-24.4+14.8\times\left(pH% -7.4\right)\right)
  2. B a s e e x c e s s = 0.93 × [ H C O 3 - ] + 13.77 × p H - 124.58 Base~{}excess=0.93\times[HCO_{3}^{-}]+13.77\times pH-124.58
  3. p H = p K + l o g [ H C O 3 - ] [ C O 2 ] pH=pK+log\frac{[HCO_{3}^{-}]}{[CO_{2}]}
  4. B E = 0.02786 × P a C O 2 × 10 ( p H - 6.1 ) + 13.77 × p H - 124.58 BE=0.02786\times PaCO_{2}\times 10^{(pH-6.1)}+13.77\times pH-124.58

Beam_parameter_product.html

  1. λ / π \lambda/\pi
  2. λ \lambda
  3. σ 2 ( z ) = σ 0 2 + M 4 ( λ π σ 0 ) 2 ( z - z 0 ) 2 \sigma^{2}(z)=\sigma_{0}^{2}+M^{4}\left(\frac{\lambda}{\pi\sigma_{0}}\right)^{% 2}(z-z_{0})^{2}
  4. σ 2 ( z ) \sigma^{2}(z)
  5. z 0 z_{0}
  6. σ 0 \sigma_{0}
  7. z 0 z_{0}
  8. σ 0 \sigma_{0}

Beap.html

  1. n \sqrt{n}
  2. n \sqrt{n}
  3. O ( n ) O(\sqrt{n})
  4. O ( n ) O(n)
  5. O ( n ) O(\sqrt{n})
  6. O ( n ) O(\sqrt{n})

Beeman's_algorithm.html

  1. x ¨ = A ( x ) \ddot{x}=A(x)
  2. t + Δ t t+\Delta t
  3. x ( t + Δ t ) x(t+\Delta t)
  4. t and t - Δ t t\,\text{ and }t-\Delta t
  5. x ( t + Δ t ) = x ( t ) + v ( t ) Δ t + 1 6 ( 4 a ( t ) - a ( t - Δ t ) ) Δ t 2 + O ( Δ t 4 ) x(t+\Delta t)=x(t)+v(t)\Delta t+\frac{1}{6}\Bigl(4a(t)-a(t-\Delta t)\Bigr)% \Delta t^{2}+O(\Delta t^{4})
  6. t + Δ t t+\Delta t
  7. t and t + Δ t t\,\text{ and }t+\Delta t
  8. a ( t + Δ t ) a(t+\Delta t)
  9. x ( t + Δ t ) \displaystyle x(t+\Delta t)
  10. x ( t + Δ t ) = x ( t ) + v ( t ) Δ t + 1 6 ( 4 a ( t ) - a ( t - Δ t ) ) Δ t 2 + O ( Δ t 4 ) v ( t + Δ t ) = v ( t ) + 1 6 ( 2 a ( t + Δ t ) + 5 a ( t ) - a ( t - Δ t ) ) Δ t + O ( Δ t 3 ) ; \begin{aligned}\displaystyle x(t+\Delta t)&\displaystyle=x(t)+v(t)\Delta t+% \frac{1}{6}\Bigl(4a(t)-a(t-\Delta t)\Bigr)\Delta t^{2}+O(\Delta t^{4})\\ \displaystyle v(t+\Delta t)&\displaystyle=v(t)+\frac{1}{6}\Bigl(2a(t+\Delta t)% +5a(t)-a(t-\Delta t)\Bigr)\Delta t+O(\Delta t^{3});\end{aligned}
  11. v ( t + Δ t ) = v ( t ) + 1 12 ( 5 a ( t + Δ t ) + 8 a ( t ) - a ( t - Δ t ) ) Δ t + O ( Δ t 3 ) v(t+\Delta t)=v(t)+\frac{1}{12}\Bigl(5a(t+\Delta t)+8a(t)-a(t-\Delta t)\Bigr)% \Delta t+O(\Delta t^{3})
  12. t t
  13. Δ t \Delta t
  14. x ( t ) x(t)
  15. v ( t ) v(t)
  16. a ( t ) a(t)
  17. x ( t ) x(t)
  18. t + Δ t t+\Delta t
  19. x ( t + Δ t ) = x ( t ) + v ( t ) Δ t + 2 3 a ( t ) Δ t 2 - 1 6 a ( t - Δ t ) Δ t 2 + O ( Δ t 4 ) . x(t+\Delta t)=x(t)+v(t)\Delta t+\frac{2}{3}a(t)\Delta t^{2}-\frac{1}{6}a(t-% \Delta t)\Delta t^{2}+O(\Delta t^{4}).
  20. t + Δ t t+\Delta t
  21. v ( t + Δ t ) ( p r e d i c t e d ) = v ( t ) + 3 2 a ( t ) Δ t - 1 2 a ( t - Δ t ) Δ t + O ( Δ t 3 ) v(t+\Delta t)(predicted)=v(t)+\frac{3}{2}a(t)\Delta t-\frac{1}{2}a(t-\Delta t)% \Delta t+O(\Delta t^{3})
  22. t + Δ t t+\Delta t
  23. v ( t + Δ t ) ( c o r r e c t e d ) = v ( t ) + 5 12 a ( t + Δ t ) Δ t + 2 3 a ( t ) Δ t - 1 12 a ( t - Δ t ) Δ t + O ( Δ t 3 ) v(t+\Delta t)(corrected)=v(t)+\frac{5}{12}a(t+\Delta t)\Delta t+\frac{2}{3}a(t% )\Delta t-\frac{1}{12}a(t-\Delta t)\Delta t+O(\Delta t^{3})
  24. O ( Δ t 4 ) O(\Delta t^{4})
  25. O ( Δ t 3 ) O(\Delta t^{3})
  26. O ( Δ t 3 ) O(\Delta t^{3})
  27. O ( Δ t 4 ) O(\Delta t^{4})
  28. O ( Δ t 2 ) O(\Delta t^{2})
  29. O ( Δ t 2 ) O(\Delta t^{2})

Behavioral_modeling.html

  1. Σ = ( 𝕋 , 𝕎 , ) \Sigma=(\mathbb{T},\mathbb{W},\mathcal{B})
  2. 𝕋 \mathbb{T}\subseteq\mathbb{R}
  3. 𝕎 \mathbb{W}
  4. 𝕎 𝕋 \mathcal{B}\subseteq\mathbb{W}^{\mathbb{T}}
  5. 𝕎 𝕋 \mathbb{W}^{\mathbb{T}}
  6. 𝕋 \mathbb{T}
  7. 𝕎 \mathbb{W}
  8. w w\in\mathcal{B}
  9. w w
  10. w w\notin\mathcal{B}
  11. w w
  12. 𝕎 𝕋 \mathbb{W}^{\mathbb{T}}
  13. \mathcal{B}
  14. 𝕋 = \mathbb{T}=\mathbb{R}
  15. 𝕋 = \mathbb{T}=\mathbb{Z}
  16. 𝕎 = q \mathbb{W}=\mathbb{R}^{q}
  17. 𝕎 \mathbb{W}
  18. Σ = ( 𝕋 , 𝕎 , ) \Sigma=(\mathbb{T},\mathbb{W},\mathcal{B})
  19. 𝕎 \mathbb{W}
  20. \mathcal{B}
  21. 𝕎 𝕋 \mathbb{W}^{\mathbb{T}}
  22. σ t \sigma^{t}\mathcal{B}\subseteq\mathcal{B}
  23. t 𝕋 t\in\mathbb{T}
  24. σ t \sigma^{t}
  25. t t
  26. σ t ( f ) ( t ) := f ( t + t ) \sigma^{t}(f)(t^{\prime}):=f(t^{\prime}+t)
  27. Σ = ( , q , ) \Sigma=(\mathbb{R},\mathbb{R}^{q},\mathcal{B})
  28. \mathcal{B}\,
  29. R ( d / d t ) w = 0 R(d/dt)w=0
  30. R R
  31. R R
  32. w : q w:\mathbb{R}\rightarrow\mathbb{R}^{q}
  33. R ( d / d t ) w = 0 R(d/dt)w=0
  34. local ( , q ) \mathcal{L}^{\rm local}(\mathbb{R},\mathbb{R}^{q})
  35. = { w 𝒞 ( , q ) | R ( d / d t ) w ( t ) = 0 for all t } . \mathcal{B}=\{w\in\mathcal{C}^{\infty}(\mathbb{R},\mathbb{R}^{q})~{}|~{}R(d/dt% )w(t)=0\,\text{ for all }t\in\mathbb{R}\}.

Beltrami–Klein_model.html

  1. d ( p , q ) = 1 2 log | a q | | p b | | a p | | q b | d(p,q)=\frac{1}{2}\log\frac{\left|aq\right|\,\left|pb\right|}{\left|ap\right|% \,\left|qb\right|}
  2. A B AB
  3. A B AB
  4. A B AB
  5. A B AB
  6. B A C \angle BAC
  7. s s
  8. u = s 1 + 1 - s 2 = ( 1 - 1 - s 2 ) s . u=\frac{s}{1+\sqrt{1-s^{2}}}=\frac{\left(1-\sqrt{1-s^{2}}\right)}{s}.
  9. u u
  10. s = 2 u 1 + u 2 . s=\frac{2u}{1+u^{2}}.
  11. d ( 𝐮 , 𝐯 ) = 1 2 log 𝐯 - 𝐚 𝐛 - 𝐮 𝐮 - 𝐚 𝐛 - 𝐯 , d(\mathbf{u},\mathbf{v})=\frac{1}{2}\log\frac{\left\|\mathbf{v}-\mathbf{a}% \right\|\,\left\|\mathbf{b}-\mathbf{u}\right\|}{\left\|\mathbf{u}-\mathbf{a}% \right\|\,\left\|\mathbf{b}-\mathbf{v}\right\|},
  12. d s 2 = g ( 𝐱 , d 𝐱 ) = d 𝐱 2 1 - 𝐱 2 + ( 𝐱 d 𝐱 ) 2 ( 1 - 𝐱 2 ) 2 ds^{2}=g(\mathbf{x},d\mathbf{x})=\frac{\left\|d\mathbf{x}\right\|^{2}}{1-\left% \|\mathbf{x}\right\|^{2}}+\frac{(\mathbf{x}\cdot d\mathbf{x})^{2}}{\bigl(1-% \left\|\mathbf{x}\right\|^{2}\bigr)^{2}}
  13. 𝐱 𝐲 = x 0 y 0 - x 1 y 1 - - x n y n \mathbf{x}\cdot\mathbf{y}=x_{0}y_{0}-x_{1}y_{1}-\cdots-x_{n}y_{n}\,
  14. 𝐱 = 𝐱 𝐱 \left\|\mathbf{x}\right\|=\sqrt{\mathbf{x}\cdot\mathbf{x}}
  15. d ( 𝐮 , 𝐯 ) = arcosh ( 𝐮 𝐯 ) . d(\mathbf{u},\mathbf{v})=\operatorname{arcosh}(\mathbf{u}\cdot\mathbf{v}).
  16. d ( 𝐮 , 𝐯 ) = arcosh ( 𝐮 𝐮 𝐯 𝐯 ) , d(\mathbf{u},\mathbf{v})=\operatorname{arcosh}\left(\frac{\mathbf{u}}{\left\|% \mathbf{u}\right\|}\cdot\frac{\mathbf{v}}{\left\|\mathbf{v}\right\|}\right),
  17. u u
  18. s = 2 u 1 + u u . s=\frac{2u}{1+u\cdot u}.
  19. s s
  20. u = s 1 + 1 - s s = ( 1 - 1 - s s ) s s s . u=\frac{s}{1+\sqrt{1-s\cdot s}}=\frac{\left(1-\sqrt{1-s\cdot s}\right)s}{s% \cdot s}.

Bent's_rule.html

  1. n + 1 n+1
  2. s + λ i p i | s + λ j p j = s | s + λ i s | p i + λ j s | p j + λ i λ j p i | p j = 1 + 0 + 0 + λ i λ j cos ω i j = 1 + cos ω i j \langle s+\sqrt{\lambda_{i}}p_{i}|s+\sqrt{\lambda_{j}}p_{j}\rangle=\langle s|s% \rangle+\sqrt{\lambda_{i}}\langle s|p_{i}\rangle+\sqrt{\lambda_{j}}\langle s|p% _{j}\rangle+\sqrt{\lambda_{i}\lambda_{j}}\langle p_{i}|p_{j}\rangle=1+0+0+% \sqrt{\lambda_{i}\lambda_{j}}\cos{\omega_{ij}}=1+\cos{\omega_{ij}}
  3. s | s = 1 \langle s|s\rangle=1
  4. 1 + λ i λ j cos ω i j = 0 1+\sqrt{\lambda_{i}\lambda_{j}}\cos{\omega_{ij}}=0
  5. cos ω i j = - 1 λ i λ j \cos{\omega_{ij}}=-\frac{1}{\sqrt{\lambda_{i}\lambda_{j}}}
  6. s + [ u r a d i c a l , u 3 b b l e s s t h a n s u b > i ] p i s+[u^{\prime}radical^{\prime},u^{\prime}\u{0}3bb\\ lessthansub>i^{\prime}]p_{i}

Bergman_kernel.html

  1. L 2 , h ( D ) = L 2 ( D ) H ( D ) L^{2,h}(D)=L^{2}(D)\cap H(D)
  2. sup z K | f ( z ) | C K f L 2 ( D ) \sup_{z\in K}|f(z)|\leq C_{K}\|f\|_{L^{2}(D)}
  3. ev z : f f ( z ) \operatorname{ev}_{z}:f\mapsto f(z)
  4. ev z f = D f ( ζ ) η z ( ζ ) ¯ d μ ( ζ ) . \operatorname{ev}_{z}f=\int_{D}f(\zeta)\overline{\eta_{z}(\zeta)}\,d\mu(\zeta).
  5. K ( z , ζ ) = η z ( ζ ) ¯ . K(z,\zeta)=\overline{\eta_{z}(\zeta)}.
  6. f ( z ) = D K ( z , ζ ) f ( ζ ) d μ ( ζ ) . f(z)=\int_{D}K(z,\zeta)f(\zeta)\,d\mu(\zeta).
  7. L 2 L^{2}
  8. d z 1 d z n dz^{1}\wedge\cdots\wedge dz^{n}
  9. L 2 L^{2}

Bergman_space.html

  1. F A 2 ( + ) 2 := 1 π + | F ( z ) | 2 d z = 0 | f ( t ) | 2 t , \|F\|^{2}_{A^{2}(\mathbb{C}_{+})}:=\frac{1}{\pi}\int_{\mathbb{C}_{+}}|F(z)|^{2% }\,dz=\int_{0}^{\infty}\frac{|f(t)|^{2}}{t},
  2. F ( z ) = 0 f ( t ) e - t z d t F(z)=\int_{0}^{\infty}f(t)e^{-tz}\,dt
  3. f A w p ( D ) := ( D | f ( x + i y ) | 2 w ( x + i y ) d x d y ) 1 / p , \|f\|_{A^{p}_{w}(D)}:=\left(\int_{D}|f(x+iy)|^{2}\,w(x+iy)\,dx\,dy\right)^{1/p},
  4. A w p ( D ) A^{p}_{w}(D)
  5. p = 2 p=2
  6. D = 𝔻 D=\mathbb{D}
  7. A α p A^{p}_{\alpha}
  8. f f
  9. f A α p := ( 1 π 𝔻 | f ( z ) | p ( 1 - | z | p ) α d z ) 1 / p < , \|f\|_{A^{p}_{\alpha}}:=\left(\frac{1}{\pi}\int_{\mathbb{D}}|f(z)|^{p}\,(1-|z|% ^{p})^{\alpha}dz\right)^{1/p}<\infty,
  10. A α p ( + ) A^{p}_{\alpha}(\mathbb{C}_{+})
  11. f A α p ( + ) := ( 1 π + | f ( x + i y ) | p x α d x d y ) 1 / p , \|f\|_{A^{p}_{\alpha}(\mathbb{C}_{+})}:=\left(\frac{1}{\pi}\int_{\mathbb{C}_{+% }}|f(x+iy)|^{p}x^{\alpha}\,dx\,dy\right)^{1/p},
  12. L 2 ( + , d μ α ) L^{2}(\mathbb{R}_{+},\,d\mu_{\alpha})
  13. d μ α := Γ ( α + 1 ) 2 α t α + 1 d t d\mu_{\alpha}:=\frac{\Gamma(\alpha+1)}{2^{\alpha}t^{\alpha+1}}\,dt
  14. Γ Γ
  15. A ν 2 A^{2}_{\nu}
  16. ν \nu
  17. + ¯ \overline{\mathbb{C}_{+}}
  18. A ν p := { f : + analytic : f A ν p := ( sup ϵ > 0 + ¯ | f ( z + ϵ ) | p d ν ( z ) ) 1 / p < } . A^{p}_{\nu}:=\left\{f:\mathbb{C}_{+}\longrightarrow\mathbb{C}\;\,\text{% analytic}\;:\;\|f\|_{A^{p}_{\nu}}:=\left(\sup_{\epsilon>0}\int_{\overline{% \mathbb{C}_{+}}}|f(z+\epsilon)|^{p}\,d\nu(z)\right)^{1/p}<\infty\right\}.
  19. k z A 2 k_{z}^{A^{2}}
  20. z 𝔻 z\in\mathbb{D}
  21. k z A 2 ( ζ ) = 1 ( 1 - z ¯ ζ ) 2 ( ζ 𝔻 ) , k_{z}^{A^{2}}(\zeta)=\frac{1}{(1-\overline{z}\zeta)^{2}}\;\;\;\;\;(\zeta\in% \mathbb{D}),
  22. A 2 ( + ) A^{2}(\mathbb{C}_{+})
  23. k z A 2 ( + ) ( ζ ) = 1 ( z ¯ + ζ ) 2 ( ζ + ) , k_{z}^{A^{2}(\mathbb{C}_{+})}(\zeta)=\frac{1}{(\overline{z}+\zeta)^{2}}\;\;\;% \;\;(\zeta\in\mathbb{C}_{+}),
  24. φ \varphi
  25. Ω \Omega
  26. D D
  27. k z A 2 ( Ω ) ( ζ ) = k φ ( z ) 𝒜 2 ( D ) ( φ ( ζ ) ) φ ( z ) ¯ φ ( ζ ) ( z , ζ Ω ) . k^{A^{2}(\Omega)}_{z}(\zeta)=k^{\mathcal{A}^{2}(D)}_{\varphi(z)}(\varphi(\zeta% ))\,\overline{\varphi^{\prime}(z)}\varphi^{\prime}(\zeta)\;\;\;\;\;(z,\zeta\in% \Omega).
  28. k z A α 2 ( ζ ) = α + 1 ( 1 - z ¯ ζ ) α + 2 ( z , ζ 𝔻 ) , k_{z}^{A^{2}_{\alpha}}(\zeta)=\frac{\alpha+1}{(1-\overline{z}\zeta)^{\alpha+2}% }\;\;\;\;\;(z,\zeta\in\mathbb{D}),
  29. k z A α 2 ( + ) ( ζ ) = 2 α ( α + 1 ) ( z ¯ + ζ ) α + 2 ( z , ζ + ) . k_{z}^{A^{2}_{\alpha}(\mathbb{C}_{+})}(\zeta)=\frac{2^{\alpha}(\alpha+1)}{(% \overline{z}+\zeta)^{\alpha+2}}\;\;\;\;\;(z,\zeta\in\mathbb{C}_{+}).

Berndt–Hall–Hall–Hausman_algorithm.html

  1. β k + 1 = β k - λ k A k Q β ( β k ) , \beta_{k+1}=\beta_{k}-\lambda_{k}A_{k}\frac{\partial Q}{\partial\beta}(\beta_{% k}),
  2. β k \beta_{k}
  3. λ k \lambda_{k}
  4. Q = i = 1 N Q i Q=\sum_{i=1}^{N}Q_{i}
  5. A k = [ i = 1 N ln Q i β ( β k ) ln Q i β ( β k ) ] - 1 . A_{k}=\left[\sum_{i=1}^{N}\frac{\partial\ln Q_{i}}{\partial\beta}(\beta_{k})% \frac{\partial\ln Q_{i}}{\partial\beta}(\beta_{k})^{\prime}\right]^{-1}.
  6. A k A_{k}

Bernstein's_theorem_on_monotone_functions.html

  1. ( - 1 ) n d n d t n f ( t ) 0 (-1)^{n}{d^{n}\over dt^{n}}f(t)\geq 0
  2. f ( t ) = 0 e - t x d g ( x ) , f(t)=\int_{0}^{\infty}e^{-tx}\,dg(x),
  3. f ( t ) = a + b t + 0 ( 1 - e - t x ) μ ( d x ) f(t)=a+bt+\int_{0}^{\infty}(1-e^{-tx})\mu(dx)
  4. a , b 0 a,b\geq 0
  5. μ \mu
  6. 0 ( 1 x ) μ ( d x ) < . \int_{0}^{\infty}(1\wedge x)\mu(dx)<\infty.

Bevel_gear.html

  1. 2 π 2\pi

Bi-elliptic_transfer.html

  1. r b r_{b}
  2. v 2 = μ ( 2 r - 1 a ) v^{2}=\mu\left(\frac{2}{r}-\frac{1}{a}\right)
  3. v v\,\!
  4. μ = G M \mu=GM\,\!
  5. r r\,\!
  6. a a\,\!
  7. r b r_{b}
  8. a 1 a_{1}
  9. a 2 a_{2}
  10. a 1 = r 0 + r b 2 a_{1}=\frac{r_{0}+r_{b}}{2}
  11. a 2 = r f + r b 2 a_{2}=\frac{r_{f}+r_{b}}{2}
  12. r 0 r_{0}
  13. Δ v 1 = 2 μ r 0 - μ a 1 - μ r 0 \Delta v_{1}=\sqrt{\frac{2\mu}{r_{0}}-\frac{\mu}{a_{1}}}-\sqrt{\frac{\mu}{r_{0% }}}
  14. r b r_{b}
  15. Δ v 2 = 2 μ r b - μ a 2 - 2 μ r b - μ a 1 \Delta v_{2}=\sqrt{\frac{2\mu}{r_{b}}-\frac{\mu}{a_{2}}}-\sqrt{\frac{2\mu}{r_{% b}}-\frac{\mu}{a_{1}}}
  16. r f r_{f}
  17. Δ v 3 = 2 μ r f - μ a 2 - μ r f \Delta v_{3}=\sqrt{\frac{2\mu}{r_{f}}-\frac{\mu}{a_{2}}}-\sqrt{\frac{\mu}{r_{f% }}}
  18. r b = r f r_{b}=r_{f}
  19. Δ v 3 \Delta v_{3}
  20. r b = r_{b}=\infty
  21. Δ v \Delta v
  22. ( 2 - 1 ) ( μ / r 0 + μ / r f ) \left(\sqrt{2}-1\right)\left(\sqrt{\mu/r_{0}}+\sqrt{\mu/r_{f}}\right)
  23. T = 2 π a 3 μ T=2\pi\sqrt{\frac{a^{3}}{\mu}}
  24. t t
  25. t 1 = π a 1 3 μ a n d t 2 = π a 2 3 μ t_{1}=\pi\sqrt{\frac{a_{1}^{3}}{\mu}}\quad and\quad t_{2}=\pi\sqrt{\frac{a_{2}% ^{3}}{\mu}}
  26. t = t 1 + t 2 t=t_{1}+t_{2}\;

Bianchi_group.html

  1. P S L 2 ( 𝒪 d ) PSL_{2}(\mathcal{O}_{d})
  2. 𝒪 d \mathcal{O}_{d}
  3. ( - d ) \mathbb{Q}(\sqrt{-d})
  4. P S L 2 ( ) PSL_{2}(\mathbb{C})
  5. P S L 2 ( ) PSL_{2}(\mathbb{C})
  6. 3 \mathbb{H}^{3}
  7. M d = P S L 2 ( 𝒪 d ) \ 3 M_{d}=PSL_{2}(\mathcal{O}_{d})\backslash\mathbb{H}^{3}
  8. ( - d ) \mathbb{Q}(\sqrt{-d})
  9. D D
  10. ( - d ) \mathbb{Q}(\sqrt{-d})
  11. Γ = S L 2 ( 𝒪 d ) \Gamma=SL_{2}(\mathcal{O}_{d})
  12. \mathcal{H}
  13. v o l ( Γ \ ) = | D | 3 2 4 π 2 ζ ( - d ) ( 2 ) . vol(\Gamma\backslash\mathbb{H})=\frac{|D|^{\frac{3}{2}}}{4\pi^{2}}\zeta_{% \mathbb{Q}(\sqrt{-d})}(2)\ .
  14. M d M_{d}
  15. ( - d ) \mathbb{Q}(\sqrt{-d})

Bicomplex_number.html

  1. t = w + x i + y j + z k , w , x , y , z t=w+xi+yj+zk,\quad w,x,y,z\in\mathbb{R}
  2. i j = j i = k , i 2 = - 1 , j 2 = + 1. ij=ji=k,\quad i^{2}=-1,\quad j^{2}=+1.
  3. t = w + y j t=w+yj
  4. t = w + x i + y j + z k , t=w+xi+yj+zk,
  5. t = ( w + x i ) + ( y + z i ) j t=(w+xi)+(y+zi)j
  6. t ( p q q p ) , p = w + x i , q = y + z i t\mapsto\begin{pmatrix}p&q\\ q&p\end{pmatrix},\quad p=w+xi,\quad q=y+zi
  7. ( i 0 0 i ) ( 0 i i 0 ) = ( 0 - 1 - 1 0 ) . \begin{pmatrix}i&0\\ 0&i\end{pmatrix}\begin{pmatrix}0&i\\ i&0\end{pmatrix}=\begin{pmatrix}0&-1\\ -1&0\end{pmatrix}.
  8. g = ( 1 - h i ) / 2 , g = ( 1 + h i ) / 2 g=(1-hi)/2,\quad g^{\prime}=(1+hi)/2
  9. ( a b ) (a\oplus b)
  10. ( c d ) (c\oplus d)
  11. a c b d ac\oplus bd
  12. t = u + v j t=u+vj
  13. s = w + z j s=w+zj
  14. t s = ( u w + v z ) + ( u z + v w ) j . ts=(uw+vz)+(uz+vw)j.
  15. t ( u + v ) ( u - v ) , s ( w + z ) ( w - z ) . t\mapsto(u+v)\oplus(u-v),\quad s\mapsto(w+z)\oplus(w-z).
  16. ( u + v ) ( w + z ) ( u - v ) ( w - z ) . (u+v)(w+z)\oplus(u-v)(w-z).
  17. w := a + i b w:=~{}a+ib
  18. z := c + i d z:=~{}c+id
  19. a + b i + c ε + d i 0 a+bi+c\varepsilon+di_{0}
  20. { 1 , i , ε , i 0 } \{1,~{}i,~{}\varepsilon,~{}i_{0}\}
  21. 1 ( 1 0 0 1 ) i ( i 0 0 i ) ε ( 0 1 1 0 ) i 0 ( 0 i i 0 ) . 1\equiv\begin{pmatrix}1&0\\ 0&1\end{pmatrix}\qquad i\equiv\begin{pmatrix}i&0\\ 0&i\end{pmatrix}\qquad\varepsilon\equiv\begin{pmatrix}0&1\\ 1&0\end{pmatrix}\qquad i_{0}\equiv\begin{pmatrix}0&i\\ i&0\end{pmatrix}.
  22. { 1 , i 1 , i 2 , j } \{1,~{}i_{1},i_{2},j\}
  23. 1 ( 1 0 0 1 ) i 1 ( i 0 0 i ) i 2 ( 0 i i 0 ) j ( 0 - 1 - 1 0 ) . 1\equiv\begin{pmatrix}1&0\\ 0&1\end{pmatrix}\qquad i_{1}\equiv\begin{pmatrix}i&0\\ 0&i\end{pmatrix}\qquad i_{2}\equiv\begin{pmatrix}0&i\\ i&0\end{pmatrix}\qquad j\equiv\begin{pmatrix}0&-1\\ -1&0\end{pmatrix}.
  24. { 1 , i 1 , i 2 , i 3 } \{1,~{}i_{1},~{}i_{2},~{}i_{3}\}
  25. { 1 , i 1 , , i 7 } \{1,~{}i_{1},\dots,~{}i_{7}\}
  26. A = ( X 2 + 1 , Y 2 - 1 ) A=(X^{2}+1,\ Y^{2}-1)
  27. B = ( X 2 + 1 , Y 2 + 1 ) B=(X^{2}+1,\ Y^{2}+1)
  28. X 2 + 1 , Y 2 - 1 , X Y - Y X X^{2}+1,\ Y^{2}-1,\ XY-YX
  29. ( X Y ) 2 + 1 A (XY)^{2}+1\in A
  30. X Y 2 X = X ( Y 2 - 1 ) X + ( X 2 + 1 ) - 1 , XY^{2}X=X(Y^{2}-1)X+(X^{2}+1)-1,
  31. X Y 2 X + 1 = X ( Y 2 - 1 ) X + ( X 2 + 1 ) A . XY^{2}X+1=X(Y^{2}-1)X+(X^{2}+1)\in A.
  32. X Y ( X Y - Y X ) + X Y 2 X + 1 A . XY(XY-YX)+XY^{2}X+1\in A.
  33. X 2 + 1 , Y 2 + 1 , X Y - Y X X^{2}+1,\ Y^{2}+1,\ XY-YX
  34. ( X Y ) 2 - 1 B (XY)^{2}-1\in B
  35. Y X Y Y\leftrightarrow XY
  36. j ε j\equiv\varepsilon
  37. ( z ± z ± z z ) z ( 1 ± j ) z ( 1 ± ε ) \begin{pmatrix}z&\pm z\\ \pm z&z\end{pmatrix}\equiv z(1\pm j)\equiv z(1\pm\varepsilon)
  38. ( z z z z ) ( z - z - z z ) z 2 ( 1 + j ) ( 1 - j ) z 2 ( 1 + ε ) ( 1 - ε ) = 0. \begin{pmatrix}z&z\\ z&z\end{pmatrix}\begin{pmatrix}z&-z\\ -z&z\end{pmatrix}\equiv z^{2}(1+j)(1-j)\equiv z^{2}(1+\varepsilon)(1-% \varepsilon)=0.
  39. k = 1 n ( a k , b k ) ( u , v ) k = ( k = 1 n a i u k , k = 1 n b k v k ) . \sum_{k=1}^{n}(a_{k},b_{k})(u,v)^{k}\quad=\quad\left({\sum_{k=1}^{n}a_{i}u^{k}% },\quad\sum_{k=1}^{n}b_{k}v^{k}\right).
  40. f ( u , v ) = ( 0 , 0 ) f(u,v)=(0,0)
  41. u 1 , u 2 , , u n , v 1 , v 2 , , v n . u_{1},u_{2},\dots,u_{n},\ v_{1},v_{2},\dots,v_{n}.
  42. ( u i , v j ) (u_{i},v_{j})\!

Biconjugate_gradient_method.html

  1. A x = b . Ax=b.\,
  2. A A
  3. x 0 x_{0}\,
  4. x 0 * x_{0}^{*}
  5. b * b^{*}\,
  6. M M\,
  7. r 0 b - A x 0 r_{0}\leftarrow b-A\,x_{0}\,
  8. r 0 * b * - x 0 * A T r_{0}^{*}\leftarrow b^{*}-x_{0}^{*}\,A^{T}
  9. p 0 M - 1 r 0 p_{0}\leftarrow M^{-1}r_{0}\,
  10. p 0 * r 0 * M - 1 p_{0}^{*}\leftarrow r_{0}^{*}M^{-1}\,
  11. k = 0 , 1 , k=0,1,\ldots
  12. α k r k * M - 1 r k p k * A p k \alpha_{k}\leftarrow{r_{k}^{*}M^{-1}r_{k}\over p_{k}^{*}Ap_{k}}\,
  13. x k + 1 x k + α k p k x_{k+1}\leftarrow x_{k}+\alpha_{k}\cdot p_{k}\,
  14. x k + 1 * x k * + α k ¯ p k * x_{k+1}^{*}\leftarrow x_{k}^{*}+\overline{\alpha_{k}}\cdot p_{k}^{*}\,
  15. r k + 1 r k - α k A p k r_{k+1}\leftarrow r_{k}-\alpha_{k}\cdot Ap_{k}\,
  16. r k + 1 * r k * - α k ¯ p k * A r_{k+1}^{*}\leftarrow r_{k}^{*}-\overline{\alpha_{k}}\cdot p_{k}^{*}\,A
  17. β k r k + 1 * M - 1 r k + 1 r k * M - 1 r k \beta_{k}\leftarrow{r_{k+1}^{*}M^{-1}r_{k+1}\over r_{k}^{*}M^{-1}r_{k}}\,
  18. p k + 1 M - 1 r k + 1 + β k p k p_{k+1}\leftarrow M^{-1}r_{k+1}+\beta_{k}\cdot p_{k}\,
  19. p k + 1 * r k + 1 * M - 1 + β k ¯ p k * p_{k+1}^{*}\leftarrow r_{k+1}^{*}M^{-1}+\overline{\beta_{k}}\cdot p_{k}^{*}\,
  20. r k r_{k}\,
  21. r k * r_{k}^{*}
  22. r k = b - A x k , r_{k}=b-Ax_{k},\,
  23. r k * = b * - x k * A r_{k}^{*}=b^{*}-x_{k}^{*}\,A
  24. x k x_{k}\,
  25. x k * x_{k}^{*}
  26. A x = b , Ax=b,\,
  27. x * A = b * ; x^{*}\,A=b^{*}\,;
  28. x * x^{*}
  29. α ¯ \overline{\alpha}
  30. x 0 x_{0}\,
  31. r 0 b - A x 0 r_{0}\leftarrow b-A\,x_{0}\,
  32. r ^ 0 b ^ - x ^ 0 A T \hat{r}_{0}\leftarrow\hat{b}-\hat{x}_{0}A^{T}
  33. p 0 r 0 p_{0}\leftarrow r_{0}\,
  34. p ^ 0 r ^ 0 \hat{p}_{0}\leftarrow\hat{r}_{0}\,
  35. k = 0 , 1 , k=0,1,\ldots
  36. α k r ^ k r k p ^ k A p k \alpha_{k}\leftarrow{\hat{r}_{k}r_{k}\over\hat{p}_{k}Ap_{k}}\,
  37. x k + 1 x k + α k p k x_{k+1}\leftarrow x_{k}+\alpha_{k}\cdot p_{k}\,
  38. x ^ k + 1 x ^ k + α k p ^ k \hat{x}_{k+1}\leftarrow\hat{x}_{k}+\alpha_{k}\cdot\hat{p}_{k}\,
  39. r k + 1 r k - α k A p k r_{k+1}\leftarrow r_{k}-\alpha_{k}\cdot Ap_{k}\,
  40. r ^ k + 1 r ^ k - α k p ^ k A T \hat{r}_{k+1}\leftarrow\hat{r}_{k}-\alpha_{k}\cdot\hat{p}_{k}A^{T}
  41. β k r ^ k + 1 r k + 1 r ^ k r k \beta_{k}\leftarrow{\hat{r}_{k+1}r_{k+1}\over\hat{r}_{k}r_{k}}\,
  42. p k + 1 r k + 1 + β k p k p_{k+1}\leftarrow r_{k+1}+\beta_{k}\cdot p_{k}\,
  43. p ^ k + 1 r ^ k + 1 + β k p ^ k \hat{p}_{k+1}\leftarrow\hat{r}_{k+1}+\beta_{k}\cdot\hat{p}_{k}\,
  44. x k := x j + P k A - 1 ( b - A x j ) , x_{k}:=x_{j}+P_{k}A^{-1}\left(b-Ax_{j}\right),
  45. x k * := x j * + ( b * - x j * A ) P k A - 1 , x_{k}^{*}:=x_{j}^{*}+\left(b^{*}-x_{j}^{*}A\right)P_{k}A^{-1},
  46. j < k j<k
  47. P k := 𝐮 k ( 𝐯 k * A 𝐮 k ) - 1 𝐯 k * A , P_{k}:=\mathbf{u}_{k}\left(\mathbf{v}_{k}^{*}A\mathbf{u}_{k}\right)^{-1}% \mathbf{v}_{k}^{*}A,
  48. 𝐮 k = [ u 0 , u 1 , , u k - 1 ] , \mathbf{u}_{k}=\left[u_{0},u_{1},\dots,u_{k-1}\right],
  49. 𝐯 k = [ v 0 , v 1 , , v k - 1 ] . \mathbf{v}_{k}=\left[v_{0},v_{1},\dots,v_{k-1}\right].
  50. P k + 1 = P k + ( 1 - P k ) u k v k * A ( 1 - P k ) v k * A ( 1 - P k ) u k . P_{k+1}=P_{k}+\left(1-P_{k}\right)u_{k}\otimes{v_{k}^{*}A\left(1-P_{k}\right)% \over v_{k}^{*}A\left(1-P_{k}\right)u_{k}}.
  51. P k = A k - 1 A P_{k}=A_{k}^{-1}A
  52. x k + 1 = x k - A k + 1 - 1 ( A x k - b ) x_{k+1}=x_{k}-A_{k+1}^{-1}\left(Ax_{k}-b\right)
  53. A k + 1 - 1 = A k - 1 + ( 1 - A k - 1 A ) u k v k * ( 1 - A A k - 1 ) v k * A ( 1 - A k - 1 A ) u k . A_{k+1}^{-1}=A_{k}^{-1}+\left(1-A_{k}^{-1}A\right)u_{k}\otimes{v_{k}^{*}\left(% 1-AA_{k}^{-1}\right)\over v_{k}^{*}A\left(1-A_{k}^{-1}A\right)u_{k}}.
  54. p k = ( 1 - P k ) u k , p_{k}=\left(1-P_{k}\right)u_{k},
  55. p k * = v k * A ( 1 - P k ) A - 1 p_{k}^{*}=v_{k}^{*}A\left(1-P_{k}\right)A^{-1}
  56. v i * r k = p i * r k = 0 , v_{i}^{*}r_{k}=p_{i}^{*}r_{k}=0,
  57. r k * u j = r k * p j = 0 , r_{k}^{*}u_{j}=r_{k}^{*}p_{j}=0,
  58. r k = A ( 1 - P k ) A - 1 r j , r_{k}=A\left(1-P_{k}\right)A^{-1}r_{j},
  59. r k * = r j * ( 1 - P k ) r_{k}^{*}=r_{j}^{*}\left(1-P_{k}\right)
  60. i , j < k i,j<k
  61. u k = M - 1 r k , u_{k}=M^{-1}r_{k},\,
  62. v k * = r k * M - 1 . v_{k}^{*}=r_{k}^{*}\,M^{-1}.\,
  63. P k P_{k}
  64. A = A * A=A^{*}\,
  65. x 0 * = x 0 x_{0}^{*}=x_{0}
  66. b * = b b^{*}=b
  67. r k = r k * r_{k}=r_{k}^{*}
  68. p k = p k * p_{k}=p_{k}^{*}
  69. x k = x k * x_{k}=x_{k}^{*}
  70. p i * A p j = r i * M - 1 r j = 0 p_{i}^{*}Ap_{j}=r_{i}^{*}M^{-1}r_{j}=0
  71. i j i\neq j
  72. P j P_{j^{\prime}}\,
  73. deg ( P j ) + j < k \mathrm{deg}\left(P_{j^{\prime}}\right)+j<k
  74. r k * P j ( M - 1 A ) u j = 0 r_{k}^{*}P_{j^{\prime}}\left(M^{-1}A\right)u_{j}=0
  75. P i P_{i^{\prime}}\,
  76. i + deg ( P i ) < k i+\mathrm{deg}\left(P_{i^{\prime}}\right)<k
  77. v i * P i ( A M - 1 ) r k = 0 v_{i}^{*}P_{i^{\prime}}\left(AM^{-1}\right)r_{k}=0

Bicycle_gearing.html

  1. 0.0254 π 0.0254\pi

Big_Omega_function.html

  1. Ω ( ) \Omega(\,\text{ })\,\!
  2. f = Ω ( g ) f=\Omega(g)\,\!
  3. f f\,\!
  4. g g\,\!
  5. Ω \Omega
  6. Ω ( n ) \Omega(n)\,\!
  7. n n\,\!
  8. Ω ( x ) \Omega(x)\,\!
  9. y = x e x y=x\cdot e^{x}\,\!
  10. W ( x ) W(x)\,\!
  11. ω ( x ) \omega(x)\,\!

Bigraph.html

  1. ( V , E , c t r l , p r n t , l i n k ) : k , X m , Y , (V,E,ctrl,prnt,link):\langle k,X\rangle\to\langle m,Y\rangle,
  2. V V
  3. E E
  4. c t r l ctrl
  5. p r n t prnt
  6. l i n k link
  7. k , X m , Y \langle k,X\rangle\to\langle m,Y\rangle
  8. k k
  9. X X
  10. m m
  11. Y Y

Binary_icosahedral_group.html

  1. Spin ( 3 ) SO ( 3 ) \operatorname{Spin}(3)\to\operatorname{SO}(3)\,
  2. Spin ( 3 ) Sp ( 1 ) \operatorname{Spin}(3)\cong\operatorname{Sp}(1)
  3. 1 { ± 1 } 2 I I 1. 1\to\{\pm 1\}\to 2I\to I\to 1.\,
  4. I A 5 I\cong A_{5}
  5. - 1 -1
  6. H 1 ( 2 I ; 𝐙 ) H 2 ( 2 I ; 𝐙 ) 0. H_{1}(2I;\mathbf{Z})\cong H_{2}(2I;\mathbf{Z})\cong 0.
  7. S 5 S_{5}
  8. ( n - 1 ) (n-1)
  9. S 5 , S_{5},
  10. A 5 , A_{5},
  11. 2 A 5 2 I ; 2\cdot A_{5}\cong 2I;
  12. A 5 I , A_{5}\cong I,
  13. S 5 S_{5}
  14. 2 S 5 ± , 2\cdot S_{5}^{\pm},
  15. Pin ± ( 4 ) O ( 4 ) \operatorname{Pin}^{\pm}(4)\to\operatorname{O}(4)
  16. I A 5 I\cong A_{5}
  17. P G L ( 2 , 5 ) S 5 , PGL(2,5)\cong S_{5},
  18. 2 A 5 , 2 S 5 , A 5 , S 5 , 2\cdot A_{5},2\cdot S_{5},A_{5},S_{5},
  19. r , s , t r 2 = s 3 = t 5 = r s t \langle r,s,t\mid r^{2}=s^{3}=t^{5}=rst\rangle
  20. s , t ( s t ) 2 = s 3 = t 5 . \langle s,t\mid(st)^{2}=s^{3}=t^{5}\rangle.
  21. s = 1 2 ( 1 + i + j + k ) t = 1 2 ( φ + φ - 1 i + j ) . s=\tfrac{1}{2}(1+i+j+k)\qquad t=\tfrac{1}{2}(\varphi+\varphi^{-1}i+j).
  22. - 1 -1
  23. 2 I S 5 2I\to S_{5}
  24. A 5 A_{5}

Binary_moment_diagram.html

  1. { if x { if y , 3 if ¬ y , 2 if ¬ x { if y , 1 if ¬ y , 0 \begin{cases}\,\text{if }x\begin{cases}\,\text{if }y,3\\ \,\text{if }\neg y,2\end{cases}\\ \,\text{if }\neg x\begin{cases}\,\text{if }y\,\text{ , }1\\ \,\text{if }\neg y\,\text{ , }0\end{cases}\end{cases}
  2. { always { always 0 if y , + 1 if x , + 2 \begin{cases}\,\text{always}\begin{cases}\,\text{always }0\\ \,\text{if }y,+1\end{cases}\\ \,\text{if }x,+2\end{cases}
  3. ( 4 x 2 + 2 x 1 + x 0 ) ( 4 y 2 + 2 y 1 + y 0 ) (4x_{2}+2x_{1}+x_{0})(4y_{2}+2y_{1}+y_{0})
  4. x 2 x_{2}
  5. x 1 x_{1}
  6. x 0 x_{0}
  7. y 2 y_{2}
  8. y 1 y_{1}
  9. y 0 y_{0}
  10. x 2 x_{2}
  11. x 1 x_{1}
  12. x 0 x_{0}
  13. y 2 y_{2}
  14. y 1 y_{1}
  15. y 0 y_{0}
  16. y 2 y_{2}
  17. y 1 y_{1}
  18. y 0 y_{0}
  19. y 2 y_{2}
  20. y 1 y_{1}
  21. y 0 y_{0}

Binary_splitting.html

  1. S ( a , b ) = n = a b p n q n S(a,b)=\sum_{n=a}^{b}\frac{p_{n}}{q_{n}}
  2. S ( a , b ) = P ( a , b ) Q ( a , b ) . S(a,b)=\frac{P(a,b)}{Q(a,b)}.

Binet–Cauchy_identity.html

  1. ( i = 1 n a i c i ) ( j = 1 n b j d j ) = ( i = 1 n a i d i ) ( j = 1 n b j c j ) + 1 i < j n ( a i b j - a j b i ) ( c i d j - c j d i ) \biggl(\sum_{i=1}^{n}a_{i}c_{i}\biggr)\biggl(\sum_{j=1}^{n}b_{j}d_{j}\biggr)=% \biggl(\sum_{i=1}^{n}a_{i}d_{i}\biggr)\biggl(\sum_{j=1}^{n}b_{j}c_{j}\biggr)+% \sum_{1\leq i<j\leq n}(a_{i}b_{j}-a_{j}b_{i})(c_{i}d_{j}-c_{j}d_{i})
  2. n \scriptstyle\mathbb{R}^{n}
  3. ( a c ) ( b d ) = ( a d ) ( b c ) + ( a b ) ( c d ) (a\cdot c)(b\cdot d)=(a\cdot d)(b\cdot c)+(a\wedge b)\cdot(c\wedge d)\,
  4. ( a b ) ( c d ) = ( a c ) ( b d ) - ( a d ) ( b c ) . (a\wedge b)\cdot(c\wedge d)=(a\cdot c)(b\cdot d)-(a\cdot d)(b\cdot c).\,
  5. | a b | 2 = | a | 2 | b | 2 - | a b | 2 . |a\wedge b|^{2}=|a|^{2}|b|^{2}-|a\cdot b|^{2}.\,
  6. 1 = cos 2 ( ϕ ) + sin 2 ( ϕ ) 1=\cos^{2}(\phi)+\sin^{2}(\phi)
  7. 1 i < j n ( a i b j - a j b i ) ( c i d j - c j d i ) \sum_{1\leq i<j\leq n}(a_{i}b_{j}-a_{j}b_{i})(c_{i}d_{j}-c_{j}d_{i})
  8. = 1 i < j n ( a i c i b j d j + a j c j b i d i ) + i = 1 n a i c i b i d i - 1 i < j n ( a i d i b j c j + a j d j b i c i ) - i = 1 n a i d i b i c i =\sum_{1\leq i<j\leq n}(a_{i}c_{i}b_{j}d_{j}+a_{j}c_{j}b_{i}d_{i})+\sum_{i=1}^% {n}a_{i}c_{i}b_{i}d_{i}-\sum_{1\leq i<j\leq n}(a_{i}d_{i}b_{j}c_{j}+a_{j}d_{j}% b_{i}c_{i})-\sum_{i=1}^{n}a_{i}d_{i}b_{i}c_{i}
  9. = i = 1 n j = 1 n a i c i b j d j - i = 1 n j = 1 n a i d i b j c j . =\sum_{i=1}^{n}\sum_{j=1}^{n}a_{i}c_{i}b_{j}d_{j}-\sum_{i=1}^{n}\sum_{j=1}^{n}% a_{i}d_{i}b_{j}c_{j}.
  10. det ( A B ) = S { 1 , , n } | S | = m det ( A S ) det ( B S ) , \det(AB)=\sum_{\scriptstyle S\subset\{1,\ldots,n\}\atop\scriptstyle|S|=m}\det(% A_{S})\det(B_{S}),
  11. A = ( a 1 a n b 1 b n ) , B = ( c 1 d 1 c n d n ) . A=\begin{pmatrix}a_{1}&\dots&a_{n}\\ b_{1}&\dots&b_{n}\end{pmatrix},\quad B=\begin{pmatrix}c_{1}&d_{1}\\ \vdots&\vdots\\ c_{n}&d_{n}\end{pmatrix}.

Binomial_number.html

  1. x n ± y n \scriptstyle x^{n}\,\pm\,y^{n}
  2. x > y \scriptstyle x\,>\,y
  3. n > 1 \scriptstyle n\,>\,1
  4. x n - y n \scriptstyle x^{n}\,-\,y^{n}
  5. x - y \scriptstyle x\,-\,y
  6. x - y \scriptstyle x\,-\,y
  7. U n ( a + b , a b ) = a n - b n a - b , U_{n}(a+b,ab)=\frac{a^{n}-b^{n}}{a-b},\,
  8. V n ( a + b , a b ) = a n + b n V_{n}(a+b,ab)=a^{n}+b^{n}\,
  9. y = 1 \scriptstyle y\,=\,1
  10. x n ± y n \scriptstyle x^{n}\,\pm\,y^{n}
  11. x m ± y m \scriptstyle x^{m}\,\pm\,y^{m}
  12. m < n \scriptstyle m\,<\,n
  13. k n + 1 \scriptstyle kn\,+\,1
  14. n \scriptstyle n\,
  15. 2 k n + 1 \scriptstyle 2kn\,+\,1

Binomial_proportion_confidence_interval.html

  1. p ^ \hat{p}
  2. p ^ ± z 1 n p ^ ( 1 - p ^ ) \hat{p}\pm z\sqrt{\frac{1}{n}\hat{p}\left(1-\hat{p}\right)}
  3. p ^ \hat{p}
  4. z z
  5. 1 - 1 2 α \scriptstyle 1-\frac{1}{2}\alpha
  6. α \alpha
  7. α \alpha
  8. 1 - 1 2 α \scriptstyle 1-\frac{1}{2}\alpha
  9. z z
  10. θ \theta
  11. { θ | y p ^ - θ 1 n p ^ ( 1 - p ^ ) z } \left\{\theta\bigg|y\leq\frac{\hat{p}-\theta}{\sqrt{\frac{1}{n}\hat{p}\left(1-% \hat{p}\right)}}\leq z\right\}
  12. y y
  13. 1 2 α \scriptstyle\frac{1}{2}\alpha
  14. 1 1 + 1 n z 2 [ p ^ + 1 2 n z 2 ± z 1 n p ^ ( 1 - p ^ ) + 1 4 n 2 z 2 ] \frac{1}{1+\frac{1}{n}z^{2}}\left[\hat{p}+\frac{1}{2n}z^{2}\pm z\sqrt{\frac{1}% {n}\hat{p}\left(1-\hat{p}\right)+\frac{1}{4n^{2}}z^{2}}\right]
  15. P P
  16. 1 n P ( 1 - P ) \scriptstyle\sqrt{\frac{1}{n}P\left(1-P\right)}
  17. p ^ \hat{p}
  18. P P
  19. p ^ \hat{p}
  20. α \alpha
  21. P P
  22. { θ | y p ^ - θ 1 n θ ( 1 - θ ) z } \left\{\theta\bigg|y\leq\frac{\hat{p}-\theta}{\sqrt{\frac{1}{n}\theta\left({1-% \theta}\right)}}\leq z\right\}
  23. θ \theta
  24. p ^ + 1 2 n z 2 1 + 1 n z 2 \frac{\hat{p}+\frac{1}{2n}z^{2}}{1+\frac{1}{n}z^{2}}
  25. p ^ = X n \hat{p}=\scriptstyle\frac{X}{n}
  26. 1 2 \scriptstyle\frac{1}{2}
  27. p ^ \hat{p}
  28. p ~ = X + 2 n + 4 \tilde{p}\,=\,\scriptstyle\frac{X+2}{n+4}
  29. p ^ \hat{p}
  30. ( w - , w + ) (w^{-},w^{+})
  31. w - = max { 0 , 2 n p ^ + z 2 - [ z z 2 - 1 n + 4 n p ^ ( 1 - p ^ ) + ( 4 p ^ - 2 ) + 1 ] 2 ( n + z 2 ) } w^{-}=\operatorname{max}\left\{0,\frac{2n\hat{p}+z^{2}-[z\sqrt{z^{2}-\frac{1}{% n}+4n\hat{p}(1-\hat{p})+(4\hat{p}-2)}+1]}{2(n+z^{2})}\right\}
  32. w + = min { 1 , 2 n p ^ + z 2 + [ z z 2 - 1 n + 4 n p ^ ( 1 - p ^ ) - ( 4 p ^ - 2 ) + 1 ] 2 ( n + z 2 ) } w^{+}=\operatorname{min}\left\{1,\frac{2n\hat{p}+z^{2}+[z\sqrt{z^{2}-\frac{1}{% n}+4n\hat{p}(1-\hat{p})-(4\hat{p}-2)}+1]}{2(n+z^{2})}\right\}
  33. p p
  34. ( 1 / 2 , 1 / 2 ) (1/2, 1/2)
  35. x x
  36. n n
  37. p p
  38. ( x + 1 / 2 , n x + 1 / 2 ) (x+ 1/2,n–x+ 1/2)
  39. x 0 x≠0
  40. x n x≠n
  41. 100 ( 1 α ) % 100(1 –α)\%
  42. α / 2 α/ 2
  43. 1 α / 2 1 –α/ 2
  44. ( x + 1 / 2 , n x + 1 / 2 ) (x+ 1/2,n–x+ 1/2)
  45. p 0 p→ 0
  46. 1 1
  47. x = 0 x=0
  48. x = n x=n
  49. S S or equivalently ( inf S , sup S ) S_{\leq}\cap S_{\geq}\mathrm{~{}~{}or~{}equivalently~{}~{}}(\inf S_{\geq}\,,\,% \sup S_{\leq})
  50. S := { θ | P [ Bin ( n ; θ ) X ] > α 2 } and S := { θ | P [ Bin ( n ; θ ) X ] > α 2 } , S_{\leq}:=\left\{\theta\Big|P\left[\mathrm{Bin}\left(n;\theta\right)\leq X% \right]>\frac{\alpha}{2}\right\}\mathrm{~{}~{}and~{}~{}}S_{\geq}:=\left\{% \theta\Big|P\left[\mathrm{Bin}\left(n;\theta\right)\geq X\right]>\frac{\alpha}% {2}\right\},
  51. B ( α 2 ; x , n - x + 1 ) < θ < B ( 1 - α 2 ; x + 1 , n - x ) B\left(\frac{\alpha}{2};x,n-x+1\right)<\theta<B\left(1-\frac{\alpha}{2};x+1,n-% x\right)
  52. ( 1 + n - x + 1 x F [ 1 - 1 2 α ; 2 x , 2 ( n - x + 1 ) ] ) - 1 < θ < ( 1 + n - x [ x + 1 ] F [ α 2 ; 2 ( x + 1 ) , 2 ( n - x ) ] ) - 1 \left(1+\frac{n-x+1}{x\,\,F\!\left[1-\frac{1}{2}\alpha;2x,2(n-x+1)\right]}% \right)^{-1}<\theta<\left(1+\frac{n-x}{\left[x+1\right]\,F\!\left[\frac{\alpha% }{2};2(x+1),2(n-x)\right]}\right)^{-1}
  53. X X
  54. n n
  55. n ~ = n + z 2 \tilde{n}=n+z^{2}
  56. p ~ = 1 n ~ ( X + 1 2 z 2 ) \tilde{p}=\frac{1}{\tilde{n}}\left(X+\frac{1}{2}z^{2}\right)
  57. p p
  58. p ~ ± z 1 n ~ p ~ ( 1 - p ~ ) \tilde{p}\pm z\sqrt{\frac{1}{\tilde{n}}\tilde{p}\left(1-\tilde{p}\right)}
  59. z z
  60. 1 - 1 2 α 1-\frac{1}{2}\alpha
  61. α = 0.05 \alpha=0.05
  62. z z
  63. z 2 z^{2}
  64. z z
  65. v a r ( p ) = p ( 1 - p ) n var(p)=\frac{p(1-p)}{n}
  66. v a r ( arcsin ( p ) ) v a r ( p ) 4 p ( 1 - p ) = p ( 1 - p ) 4 n p ( 1 - p ) = 1 4 n var(\arcsin(\sqrt{p}))\approx\frac{var(p)}{4p(1-p)}=\frac{p(1-p)}{4np(1-p)}=% \frac{1}{4n}
  67. sin 2 ( arcsin ( p ) - z 2 n ) < θ < sin 2 ( arcsin ( p ) + z 2 n ) \sin^{2}\left(\arcsin(\sqrt{p})-\frac{z}{2\sqrt{n}}\right)<\theta<\sin^{2}% \left(\arcsin(\sqrt{p})+\frac{z}{2\sqrt{n}}\right)
  68. z z
  69. 1 - α / 2 1-\alpha/2
  70. t a = log ( p a ( 1 - p ) 2 - a ) = a log ( p ) - ( 2 - a ) log ( 1 - p ) t_{a}=\log\left(\frac{p^{a}}{(1-p)^{2-a}}\right)=a\log(p)-(2-a)\log(1-p)
  71. p ^ = 0 \hat{p}=0
  72. p ^ = 1 \hat{p}=1

Binomial_regression.html

  1. L ( s y m b o l μ Y ) = i = 1 n ( 1 y i = 1 ( μ i ) + 1 y i = 0 ( 1 - μ i ) ) , L(symbol{\mu}\mid Y)=\prod_{i=1}^{n}\left(1_{y_{i}=1}(\mu_{i})+1_{y_{i}=0}(1-% \mu_{i})\right),\,\!
  2. s y m b o l μ = g ( s y m b o l η ) . symbol{\mu}=g(symbol{\eta})\,.
  3. U n = s y m b o l β 𝐬 𝐧 + ε n U_{n}=symbol\beta\cdot\mathbf{s_{n}}+\varepsilon_{n}
  4. s y m b o l β symbol\beta
  5. 𝐬 𝐧 \mathbf{s_{n}}
  6. ε n \varepsilon_{n}
  7. Y n = { 1 , if U n > 0 , 0 , if U n 0 Y_{n}=\begin{cases}1,&\,\text{if }U_{n}>0,\\ 0,&\,\text{if }U_{n}\leq 0\end{cases}
  8. Y n = { 1 , if U n > 0 , 0 , if U n 0 Y_{n}=\begin{cases}1,&\,\text{if }U_{n}>0,\\ 0,&\,\text{if }U_{n}\leq 0\end{cases}
  9. e e
  10. F e , F_{e},
  11. e e
  12. F e - 1 . F^{-1}_{e}.
  13. Pr ( Y n = 1 ) \displaystyle\Pr(Y_{n}=1)
  14. Y n Y_{n}
  15. 𝔼 [ Y n ] = Pr ( Y n = 1 ) , \mathbb{E}[Y_{n}]=\Pr(Y_{n}=1),
  16. 𝔼 [ Y n ] = F e ( s y m b o l β 𝐬 𝐧 ) \mathbb{E}[Y_{n}]=F_{e}(symbol\beta\cdot\mathbf{s_{n}})
  17. F e - 1 ( 𝔼 [ Y n ] ) = s y m b o l β 𝐬 𝐧 . F^{-1}_{e}(\mathbb{E}[Y_{n}])=symbol\beta\cdot\mathbf{s_{n}}.
  18. e n 𝒩 ( 0 , 1 ) , e_{n}\sim\mathcal{N}(0,1),
  19. Φ - 1 ( 𝔼 [ Y n ] ) = s y m b o l β 𝐬 𝐧 \Phi^{-1}(\mathbb{E}[Y_{n}])=symbol\beta\cdot\mathbf{s_{n}}
  20. e n Logistic ( 0 , 1 ) , e_{n}\sim\operatorname{Logistic}(0,1),
  21. logit ( 𝔼 [ Y n ] ) = s y m b o l β 𝐬 𝐧 \operatorname{logit}(\mathbb{E}[Y_{n}])=symbol\beta\cdot\mathbf{s_{n}}
  22. Y = { 0 , if Y * > 0 1 , if Y * < 0. Y=\begin{cases}0,&\mbox{if }~{}Y^{*}>0\\ 1,&\mbox{if }~{}Y^{*}<0.\end{cases}
  23. Y * = X β + ϵ . Y^{*}=X\beta+\epsilon\ .

Binomial_transform.html

  1. s n = k = 0 n ( - 1 ) k ( n k ) a k . s_{n}=\sum_{k=0}^{n}(-1)^{k}{n\choose k}a_{k}.
  2. s n = ( T a ) n = k = 0 T n k a k . s_{n}=(Ta)_{n}=\sum_{k=0}^{\infty}T_{nk}a_{k}.
  3. T T = 1 TT=1\,
  4. k = 0 T n k T k m = δ n m \sum_{k=0}^{\infty}T_{nk}T_{km}=\delta_{nm}
  5. δ n m \delta_{nm}
  6. a n = k = 0 n ( - 1 ) k ( n k ) s k . a_{n}=\sum_{k=0}^{n}(-1)^{k}{n\choose k}s_{k}.
  7. s 0 = a 0 s_{0}=a_{0}
  8. s 1 = - ( a ) 0 = - a 1 + a 0 s_{1}=-(\triangle a)_{0}=-a_{1}+a_{0}
  9. s 2 = ( 2 a ) 0 = - ( - a 2 + a 1 ) + ( - a 1 + a 0 ) = a 2 - 2 a 1 + a 0 s_{2}=(\triangle^{2}a)_{0}=-(-a_{2}+a_{1})+(-a_{1}+a_{0})=a_{2}-2a_{1}+a_{0}
  10. \vdots\,
  11. s n = ( - 1 ) n ( n a ) 0 s_{n}=(-1)^{n}(\triangle^{n}a)_{0}
  12. t n = k = 0 n ( - 1 ) n - k ( n k ) a k t_{n}=\sum_{k=0}^{n}(-1)^{n-k}{n\choose k}a_{k}
  13. a n = k = 0 n ( n k ) t k . a_{n}=\sum_{k=0}^{n}{n\choose k}t_{k}.
  14. B n + 1 = k = 0 n ( n k ) B k B_{n+1}=\sum_{k=0}^{n}{n\choose k}B_{k}
  15. f ( x ) = n = 0 a n x n f(x)=\sum_{n=0}^{\infty}a_{n}x^{n}
  16. g ( x ) = n = 0 s n x n g(x)=\sum_{n=0}^{\infty}s_{n}x^{n}
  17. g ( x ) = ( T f ) ( x ) = 1 1 - x f ( x x - 1 ) . g(x)=(Tf)(x)=\frac{1}{1-x}f\left(\frac{x}{x-1}\right).
  18. n = 0 ( - 1 ) n a n = n = 0 ( - 1 ) n Δ n a 0 2 n + 1 \sum_{n=0}^{\infty}(-1)^{n}a_{n}=\sum_{n=0}^{\infty}(-1)^{n}\frac{\Delta^{n}a_% {0}}{2^{n+1}}
  19. n = 0 ( - 1 ) n ( n + p n ) a n = n = 0 ( - 1 ) n ( n + p n ) Δ n a 0 2 n + p + 1 \sum_{n=0}^{\infty}(-1)^{n}{n+p\choose n}a_{n}=\sum_{n=0}^{\infty}(-1)^{n}{n+p% \choose n}\frac{\Delta^{n}a_{0}}{2^{n+p+1}}
  20. F 1 2 \,{}_{2}F_{1}
  21. F 1 2 ( a , b ; c ; z ) = ( 1 - z ) 2 - b F 1 ( c - a , b ; c ; z z - 1 ) . \,{}_{2}F_{1}(a,b;c;z)=(1-z)^{-b}\,_{2}F_{1}\left(c-a,b;c;\frac{z}{z-1}\right).
  22. 0 < x < 1 0<x<1
  23. x = [ 0 ; a 1 , a 2 , a 3 , ] x=[0;a_{1},a_{2},a_{3},\cdots]
  24. x 1 - x = [ 0 ; a 1 - 1 , a 2 , a 3 , ] \frac{x}{1-x}=[0;a_{1}-1,a_{2},a_{3},\cdots]
  25. x 1 + x = [ 0 ; a 1 + 1 , a 2 , a 3 , ] . \frac{x}{1+x}=[0;a_{1}+1,a_{2},a_{3},\cdots].
  26. f ¯ ( x ) = n = 0 a n x n n ! \overline{f}(x)=\sum_{n=0}^{\infty}a_{n}\frac{x^{n}}{n!}
  27. g ¯ ( x ) = n = 0 s n x n n ! \overline{g}(x)=\sum_{n=0}^{\infty}s_{n}\frac{x^{n}}{n!}
  28. g ¯ ( x ) = ( T f ¯ ) ( x ) = e x f ¯ ( - x ) . \overline{g}(x)=(T\overline{f})(x)=e^{x}\overline{f}(-x).
  29. u n = k = 0 n ( n k ) a k ( - c ) n - k b k u_{n}=\sum_{k=0}^{n}{n\choose k}a^{k}(-c)^{n-k}b_{k}
  30. U ( x ) = 1 c x + 1 B ( a x c x + 1 ) U(x)=\frac{1}{cx+1}B\left(\frac{ax}{cx+1}\right)
  31. { u n } \{u_{n}\}
  32. { b n } \{b_{n}\}
  33. j = 0 n ( n j ) j k a j . \sum_{j=0}^{n}{n\choose j}j^{k}a_{j}.
  34. j = 0 n ( n j ) j n - k a j \sum_{j=0}^{n}{n\choose j}j^{n-k}a_{j}
  35. i = 0 n ( - 1 ) n - i ( n i ) a i = b n . \sum_{i=0}^{n}(-1)^{n-i}{\left({{n}\atop{i}}\right)}a_{i}=b_{n}.
  36. 𝔍 ( a ) n = b n . \mathfrak{J}(a)_{n}=b_{n}.
  37. { b n } \{b_{n}\}
  38. 𝔍 2 ( a ) n = i = 0 n ( - 2 ) n - i ( n i ) a i . \mathfrak{J}^{2}(a)_{n}=\sum_{i=0}^{n}(-2)^{n-i}{\left({{n}\atop{i}}\right)}a_% {i}.
  39. 𝔍 k ( a ) n = b n = i = 0 n ( - k ) n - i ( n i ) a i . \mathfrak{J}^{k}(a)_{n}=b_{n}=\sum_{i=0}^{n}(-k)^{n-i}{\left({{n}\atop{i}}% \right)}a_{i}.
  40. 𝔍 - k ( b ) n = a n = i = 0 n k n - i ( n i ) b i . \mathfrak{J}^{-k}(b)_{n}=a_{n}=\sum_{i=0}^{n}k^{n-i}{\left({{n}\atop{i}}\right% )}b_{i}.
  41. 𝔍 k ( a ) n = b n = ( 𝐄 - k ) n a 0 \mathfrak{J}^{k}(a)_{n}=b_{n}=(\mathbf{E}-k)^{n}a_{0}
  42. 𝐄 \mathbf{E}
  43. 𝔍 - k ( b ) n = a n = ( 𝐄 + k ) n b 0 . \mathfrak{J}^{-k}(b)_{n}=a_{n}=(\mathbf{E}+k)^{n}b_{0}.

Biological_half-life.html

  1. t 1 2 t_{\frac{1}{2}}
  2. t 1 2 = 0.5 A 0 k 0 t_{\frac{1}{2}}=\frac{0.5A_{0}}{k_{0}}\,
  3. C t = C 0 e - k t C_{t}=C_{0}e^{-kt}\,
  4. k = ln 2 t 1 2 k=\frac{\ln 2}{t_{\frac{1}{2}}}\,
  5. t 1 2 = ln 2 V D C L t_{\frac{1}{2}}=\frac{{\ln 2}\cdot{V_{D}}}{CL}\,

Biorthogonal_wavelet.html

  1. ϕ , ϕ ~ \phi,\tilde{\phi}
  2. ψ , ψ ~ \psi,\tilde{\psi}
  3. a , a ~ a,\tilde{a}
  4. n \Z a n a ~ n + 2 m = 2 δ m , 0 \sum_{n\in\Z}a_{n}\tilde{a}_{n+2m}=2\cdot\delta_{m,0}
  5. b n = ( - 1 ) n a ~ M - 1 - n ( n = 0 , , N - 1 ) b_{n}=(-1)^{n}\tilde{a}_{M-1-n}\quad\quad(n=0,\dots,N-1)
  6. b ~ n = ( - 1 ) n a M - 1 - n ( n = 0 , , N - 1 ) \tilde{b}_{n}=(-1)^{n}a_{M-1-n}\quad\quad(n=0,\dots,N-1)

Bipolar_cylindrical_coordinates.html

  1. z z
  2. F 1 F_{1}
  3. F 2 F_{2}
  4. x = - a x=-a
  5. x = + a x=+a
  6. y = 0 y=0
  7. ( σ , τ , z ) (\sigma,\tau,z)
  8. x = a sinh τ cosh τ - cos σ x=a\ \frac{\sinh\tau}{\cosh\tau-\cos\sigma}
  9. y = a sin σ cosh τ - cos σ y=a\ \frac{\sin\sigma}{\cosh\tau-\cos\sigma}
  10. z = z z=\ z
  11. σ \sigma
  12. P P
  13. F 1 P F 2 F_{1}PF_{2}
  14. τ \tau
  15. d 1 d_{1}
  16. d 2 d_{2}
  17. τ = ln d 1 d 2 \tau=\ln\frac{d_{1}}{d_{2}}
  18. F 1 F_{1}
  19. F 2 F_{2}
  20. x = - a x=-a
  21. x = + a x=+a
  22. σ \sigma
  23. x 2 + ( y - a cot σ ) 2 = a 2 sin 2 σ x^{2}+\left(y-a\cot\sigma\right)^{2}=\frac{a^{2}}{\sin^{2}\sigma}
  24. τ \tau
  25. y 2 + ( x - a coth τ ) 2 = a 2 sinh 2 τ y^{2}+\left(x-a\coth\tau\right)^{2}=\frac{a^{2}}{\sinh^{2}\tau}
  26. z z
  27. z = 0 z=0
  28. σ \sigma
  29. τ \tau
  30. y y
  31. x x
  32. σ \sigma
  33. τ \tau
  34. h σ = h τ = a cosh τ - cos σ h_{\sigma}=h_{\tau}=\frac{a}{\cosh\tau-\cos\sigma}
  35. h z = 1 h_{z}=1
  36. d V = a 2 ( cosh τ - cos σ ) 2 d σ d τ d z dV=\frac{a^{2}}{\left(\cosh\tau-\cos\sigma\right)^{2}}d\sigma d\tau dz
  37. 2 Φ = 1 a 2 ( cosh τ - cos σ ) 2 ( 2 Φ σ 2 + 2 Φ τ 2 ) + 2 Φ z 2 \nabla^{2}\Phi=\frac{1}{a^{2}}\left(\cosh\tau-\cos\sigma\right)^{2}\left(\frac% {\partial^{2}\Phi}{\partial\sigma^{2}}+\frac{\partial^{2}\Phi}{\partial\tau^{2% }}\right)+\frac{\partial^{2}\Phi}{\partial z^{2}}
  38. 𝐅 \nabla\cdot\mathbf{F}
  39. × 𝐅 \nabla\times\mathbf{F}
  40. ( σ , τ ) (\sigma,\tau)

Bispherical_coordinates.html

  1. F 1 F_{1}
  2. F 2 F_{2}
  3. z z
  4. ( σ , τ , ϕ ) (\sigma,\tau,\phi)
  5. x = a sin σ cosh τ - cos σ cos ϕ x=a\ \frac{\sin\sigma}{\cosh\tau-\cos\sigma}\cos\phi
  6. y = a sin σ cosh τ - cos σ sin ϕ y=a\ \frac{\sin\sigma}{\cosh\tau-\cos\sigma}\sin\phi
  7. z = a sinh τ cosh τ - cos σ z=a\ \frac{\sinh\tau}{\cosh\tau-\cos\sigma}
  8. σ \sigma
  9. P P
  10. F 1 P F 2 F_{1}PF_{2}
  11. τ \tau
  12. d 1 d_{1}
  13. d 2 d_{2}
  14. τ = ln d 1 d 2 \tau=\ln\frac{d_{1}}{d_{2}}
  15. σ \sigma
  16. z 2 + ( x 2 + y 2 - a cot σ ) 2 = a 2 sin 2 σ z^{2}+\left(\sqrt{x^{2}+y^{2}}-a\cot\sigma\right)^{2}=\frac{a^{2}}{\sin^{2}\sigma}
  17. τ \tau
  18. ( x 2 + y 2 ) + ( z - a coth τ ) 2 = a 2 sinh 2 τ \left(x^{2}+y^{2}\right)+\left(z-a\coth\tau\right)^{2}=\frac{a^{2}}{\sinh^{2}\tau}
  19. τ \tau
  20. z z
  21. σ \sigma
  22. x y xy
  23. σ = arccos ( ( R 2 - a 2 ) / Q ) \sigma=\arccos((R^{2}-a^{2})/Q)
  24. τ = arsinh ( 2 a z / Q ) \tau=\operatorname{arsinh}(2az/Q)
  25. ϕ = atan ( y / x ) \phi=\operatorname{atan}(y/x)
  26. R = x 2 + y 2 + z 2 R=\sqrt{x^{2}+y^{2}+z^{2}}
  27. Q = ( R 2 + a 2 ) 2 - ( 2 a z ) 2 . Q=\sqrt{(R^{2}+a^{2})^{2}-(2az)^{2}}.
  28. σ \sigma
  29. τ \tau
  30. h σ = h τ = a cosh τ - cos σ h_{\sigma}=h_{\tau}=\frac{a}{\cosh\tau-\cos\sigma}
  31. h ϕ = a sin σ cosh τ - cos σ h_{\phi}=\frac{a\sin\sigma}{\cosh\tau-\cos\sigma}
  32. d V = a 3 sin σ ( cosh τ - cos σ ) 3 d σ d τ d ϕ dV=\frac{a^{3}\sin\sigma}{\left(\cosh\tau-\cos\sigma\right)^{3}}\,d\sigma\,d% \tau\,d\phi
  33. 2 Φ = ( cosh τ - cos σ ) 3 a 2 sin σ [ σ ( sin σ cosh τ - cos σ Φ σ ) + sin σ τ ( 1 cosh τ - cos σ Φ τ ) + 1 sin σ ( cosh τ - cos σ ) 2 Φ ϕ 2 ] \begin{aligned}\displaystyle\nabla^{2}\Phi=\frac{\left(\cosh\tau-\cos\sigma% \right)^{3}}{a^{2}\sin\sigma}&\displaystyle\left[\frac{\partial}{\partial% \sigma}\left(\frac{\sin\sigma}{\cosh\tau-\cos\sigma}\frac{\partial\Phi}{% \partial\sigma}\right)\right.\\ &\displaystyle{}\quad+\left.\sin\sigma\frac{\partial}{\partial\tau}\left(\frac% {1}{\cosh\tau-\cos\sigma}\frac{\partial\Phi}{\partial\tau}\right)+\frac{1}{% \sin\sigma\left(\cosh\tau-\cos\sigma\right)}\frac{\partial^{2}\Phi}{\partial% \phi^{2}}\right]\end{aligned}
  34. 𝐅 \nabla\cdot\mathbf{F}
  35. × 𝐅 \nabla\times\mathbf{F}
  36. ( σ , τ ) (\sigma,\tau)

Bitruncated_cubic_honeycomb.html

  1. 3 ¯ \overline{3}
  2. C ~ 3 {\tilde{C}}_{3}
  3. A ~ 3 {\tilde{A}}_{3}
  4. 3 ¯ \overline{3}
  5. 3 ¯ \overline{3}
  6. 3 ¯ \overline{3}
  7. 4 ¯ \overline{4}
  8. 3 ¯ \overline{3}
  9. C ~ 3 {\tilde{C}}_{3}
  10. C ~ 3 {\tilde{C}}_{3}
  11. B ~ 3 {\tilde{B}}_{3}
  12. A ~ 3 {\tilde{A}}_{3}
  13. A ~ 3 {\tilde{A}}_{3}
  14. A ~ 3 {\tilde{A}}_{3}
  15. A ~ 3 {\tilde{A}}_{3}
  16. C ~ 2 {\tilde{C}}_{2}
  17. 3 ¯ \overline{3}
  18. 3 ¯ \overline{3}
  19. 3 ¯ \overline{3}
  20. 3 ¯ \overline{3}

BK-tree.html

  1. d ( x , y ) d(x,y)
  2. d ( a , b ) = k d(a,b)=k

Blancmange_curve.html

  1. blanc ( x ) = n = 0 s ( 2 n x ) 2 n , {\rm blanc}(x)=\sum_{n=0}^{\infty}{s(2^{n}x)\over 2^{n}},
  2. s ( x ) s(x)
  3. s ( x ) = min n Z | x - n | s(x)=\min_{n\in{Z}}|x-n|
  4. s ( x ) s(x)
  5. T w ( x ) = n = 0 w n s ( 2 n x ) T_{w}(x)=\sum_{n=0}^{\infty}w^{n}s(2^{n}x)
  6. w = 1 / 2 w=1/2
  7. H = - log 2 w H=-\log_{2}w
  8. T w ( x ) T_{w}(x)
  9. 0 s ( x ) 1 / 2 0\leq s(x)\leq 1/2
  10. x x\in\mathbb{R}
  11. n = 0 | w n s ( 2 n x ) | 1 / 2 n = 0 | w | n = 1 2 1 1 - | w | \sum_{n=0}^{\infty}|w^{n}s(2^{n}x)|\leq 1/2\sum_{n=0}^{\infty}|w|^{n}=\frac{1}% {2}\cdot\frac{1}{1-|w|}
  12. | w | < 1 |w|<1
  13. \mathbb{R}
  14. | w | < 1 |w|<1
  15. T w , n T_{w,n}
  16. T w , n ( x ) = k = 0 n w k s ( 2 k x ) T_{w,n}(x)=\sum_{k=0}^{n}w^{k}s(2^{k}x)
  17. T w T_{w}
  18. | T w ( x ) - T w , n ( x ) | = | k = n + 1 w k s ( 2 k x ) | = | w n + 1 k = 0 w k s ( 2 k + n + 1 x ) | | w | n + 1 2 1 1 - | w | \left|T_{w}(x)-T_{w,n}(x)\right|=\left|\sum_{k=n+1}^{\infty}w^{k}s(2^{k}x)% \right|=\left|w^{n+1}\sum_{k=0}^{\infty}w^{k}s(2^{k+n+1}x)\right|\leq\frac{|w|% ^{n+1}}{2}\cdot\frac{1}{1-|w|}
  19. | w | < 1 |w|<1
  20. T w T_{w}
  21. w 1 / 4 \scriptstyle w\neq 1/4
  22. T w ( x ) = m = 0 a m cos ( 2 π m x ) T_{w}(x)=\sum_{m=0}^{\infty}a_{m}\cos(2\pi mx)
  23. a 0 = 1 / 4 ( 1 - w ) \scriptstyle a_{0}=1/4(1-w)
  24. m 1 \scriptstyle m\geq 1
  25. a m := - 2 π 2 m 2 ( 4 w ) ν ( m ) , a_{m}:=-\frac{2}{\pi^{2}m^{2}}(4w)^{\nu(m)}\,,
  26. 2 ν ( m ) \scriptstyle 2^{\nu(m)}
  27. 2 2
  28. m m
  29. s ( x ) s(x)
  30. s ( x ) = 1 4 - 2 π 2 k = 0 1 ( 2 k + 1 ) 2 cos ( 2 π ( 2 k + 1 ) x ) . s(x)=\frac{1}{4}-\frac{2}{\pi^{2}}\sum_{k=0}^{\infty}\frac{1}{(2k+1)^{2}}\cos% \big(2\pi(2k+1)x\big).
  31. T w ( x ) T_{w}(x)
  32. T w ( x ) := n = 0 w n s ( 2 n x ) = 1 4 n = 0 w n - 2 π 2 n = 0 k = 0 w n ( 2 k + 1 ) 2 cos ( 2 π 2 n ( 2 k + 1 ) x ) : T_{w}(x):=\sum_{n=0}^{\infty}w^{n}s(2^{n}x)=\frac{1}{4}\sum_{n=0}^{\infty}w^{n% }-\frac{2}{\pi^{2}}\sum_{n=0}^{\infty}\sum_{k=0}^{\infty}\frac{w^{n}}{(2k+1)^{% 2}}\cos\big(2\pi 2^{n}(2k+1)x\big)\,:
  33. m := 2 n ( 2 k + 1 ) \scriptstyle m:=2^{n}(2k+1)
  34. T w ( x ) \scriptstyle T_{w}(x)
  35. T w ( x ) = s ( x ) + w T w ( 2 x ) T_{w}(x)=s(x)+wT_{w}(2x)
  36. T w ( x ) = { x + w T w ( 2 x ) if 0 x 1 / 2 ( 1 - x ) + w T w ( 2 x - 1 ) if 1 / 2 < x 1. T_{w}(x)=\begin{cases}x+wT_{w}(2x)&\,\text{if }0\leq x\leq 1/2\\ (1-x)+wT_{w}(2x-1)&\,\text{if }1/2<x\leq 1.\end{cases}
  37. T w ( x ) \displaystyle T_{w}(x)
  38. blanc ( x ) {\rm blanc}(x)
  39. blanc ( x ) = blanc ( 2 x ) / 2 + s ( x ) {\rm blanc}(x)={\rm blanc}(2x)/2+s(x)
  40. I ( x ) \displaystyle I(x)
  41. N = ( n t t ) + ( n t - 1 t - 1 ) + + ( n j j ) , n t > n t - 1 > > n j j 1. N={\left({{n_{t}}\atop{t}}\right)}+{\left({{n_{t-1}}\atop{t-1}}\right)}+\ldots% +{\left({{n_{j}}\atop{j}}\right)},\quad n_{t}>n_{t-1}>\ldots>n_{j}\geq j\geq 1.
  42. κ t ( N ) = ( n t t + 1 ) + ( n t - 1 t ) + + ( n j j + 1 ) . \kappa_{t}(N)={n_{t}\choose t+1}+{n_{t-1}\choose t}+\dots+{n_{j}\choose j+1}.
  43. κ t ( N ) - N \kappa_{t}(N)-N

Blasius_boundary_layer.html

  1. U U
  2. u ( η ) / U ( x ) u(\eta)/U(x)
  3. η \eta
  4. U 2 L ν U δ 2 \frac{U^{2}}{L}\approx\nu\frac{U}{\delta^{2}}
  5. δ \delta
  6. ν \nu
  7. L L
  8. u x + v y = 0 \dfrac{\partial u}{\partial x}+\dfrac{\partial v}{\partial y}=0
  9. u u x + v u y = ν 2 u y 2 u\dfrac{\partial u}{\partial x}+v\dfrac{\partial u}{\partial y}={\nu}\dfrac{% \partial^{2}u}{\partial y^{2}}
  10. U U
  11. u u
  12. v v
  13. x x
  14. δ ( x ) ( ν x U ) 1 / 2 . \delta(x)\approx\left(\frac{\nu x}{U}\right)^{1/2}.
  15. η = y δ ( x ) = y ( U ν x ) 1 / 2 \eta=\frac{y}{\delta(x)}=y\left(\frac{U}{\nu x}\right)^{1/2}
  16. u = U f ( η ) . u=Uf^{\prime}(\eta).
  17. ψ \psi
  18. ψ = ( ν U x ) 1 / 2 f ( η ) \psi=(\nu Ux)^{1/2}f(\eta)
  19. f ′′′ + 1 2 f f ′′ = 0 f^{\prime\prime\prime}+\frac{1}{2}ff^{\prime\prime}=0
  20. f = f = 0 f=f^{\prime}=0
  21. η = 0 \eta=0
  22. f 1 f^{\prime}\rightarrow 1
  23. η \eta\rightarrow\infty
  24. τ x y = f ′′ ( 0 ) ρ U 2 ν U x . \tau_{xy}=\frac{f^{\prime\prime}(0)\rho U^{2}\sqrt{\nu}}{\sqrt{Ux}}.
  25. f ′′ ( 0 ) 0.332 f^{\prime\prime}(0)\approx 0.332
  26. β {\beta}
  27. U 0 U_{0}
  28. u e ( x ) = U 0 ( x / L ) m u_{e}(x)=U_{0}\left(x/L\right)^{m}
  29. L L
  30. β = 2 m m + 1 {\beta}=\frac{2m}{m+1}
  31. η {\eta}
  32. η = y U 0 ( m + 1 ) 2 ν L ( x L ) m - 1 2 {\eta}=y\sqrt{\frac{U_{0}(m+1)}{2{\nu}L}}\left(\frac{x}{L}\right)^{\frac{m-1}{% 2}}
  33. ψ = U ( x ) δ ( x ) f ( η ) = y 2 ν U 0 L m + 1 ( x L ) m + 1 2 f ( η ) \psi=U(x)\delta(x)f(\eta)=y\sqrt{\frac{2{\nu}U_{0}L}{m+1}}\left(\frac{x}{L}% \right)^{\frac{m+1}{2}}f(\eta)
  34. u u x + v u y = c 2 m x 2 m - 1 + ν 2 u y 2 . u{\partial u\over\partial x}+v{\partial u\over\partial y}=c^{2}mx^{2m-1}+{\nu}% {\partial^{2}u\over\partial y^{2}}.
  35. 3 f η 3 + f 2 f η 2 + β [ 1 - ( d f d η ) 2 ] = 0 \frac{\partial^{3}f}{\partial\eta^{3}}+f\frac{\partial^{2}f}{\partial\eta^{2}}% +\beta\left[1-\left(\frac{\mathrm{d}f}{\mathrm{d}\eta}\right)^{2}\right]=0
  36. m = 0 m=0
  37. - 0.0905 m 2 -0.0905\leq m\leq 2

BLEU.html

  1. P = m w t = 7 7 = 1 P=\frac{m}{w_{t}}=\frac{7}{7}=1
  2. m ~{}m
  3. w t ~{}w_{t}
  4. m m a x ~{}m_{max}
  5. m m a x = 2 ~{}m_{max}=2
  6. m w m_{w}
  7. m m a x m_{max}
  8. m w = 7 ~{}m_{w}=7
  9. m m a x = 2 ~{}m_{max}=2
  10. m w ~{}m_{w}
  11. m w ~{}m_{w}
  12. P = 2 7 P=\frac{2}{7}
  13. n n
  14. P = 1 2 + 1 2 = 2 2 P=\frac{1}{2}+\frac{1}{2}=\frac{2}{2}
  15. 1 / 1 1/1
  16. 2 / 6 2/6
  17. 2 / 7 2/7
  18. r r
  19. c c
  20. c r c\leq r
  21. e ( 1 - r / c ) e^{(1-r/c)}
  22. r r

Blinn–Phong_shading_model.html

  1. R V R\cdot V
  2. H = L + V | L + V | H=\frac{L+V}{\left|L+V\right|}
  3. R V R\cdot V
  4. N H N\cdot H
  5. N N
  6. L L
  7. V V
  8. H H
  9. V = P H ( - L ) , V=P_{H}(-L),
  10. P H P_{H}
  11. H . H.
  12. ( R V ) α , \left(R\cdot V\right)^{\alpha},
  13. α > α \alpha^{\prime}>\alpha
  14. ( N H ) α \left(N\cdot H\right)^{\alpha^{\prime}}
  15. α = 4 α \alpha^{\prime}=4\,\alpha
  16. R R
  17. H N H\cdot N
  18. 0 , 11 0,11

Block-stacking_problem.html

  1. N N
  2. i = 1 N 1 2 i \sum_{i=1}^{N}\frac{1}{2i}
  3. N N

Block_matrix_pseudoinverse.html

  1. [ 𝐀 , 𝐁 ] , 𝐀 \reals m × n , 𝐁 \reals m × p , m n + p . [\mathbf{A},\mathbf{B}],\qquad\mathbf{A}\in\reals^{m\times n},\qquad\mathbf{B}% \in\reals^{m\times p},\qquad m\geq n+p.
  2. [ 𝐀 , 𝐁 ] + = ( [ 𝐀 , 𝐁 ] T [ 𝐀 , 𝐁 ] ) - 1 [ 𝐀 , 𝐁 ] T , \begin{bmatrix}\mathbf{A},&\mathbf{B}\end{bmatrix}^{+}=([\mathbf{A},\mathbf{B}% ]^{T}[\mathbf{A},\mathbf{B}])^{-1}[\mathbf{A},\mathbf{B}]^{T},
  3. [ 𝐀 T 𝐁 T ] + = [ 𝐀 , 𝐁 ] ( [ 𝐀 , 𝐁 ] T [ 𝐀 , 𝐁 ] ) - 1 . \begin{bmatrix}\mathbf{A}^{T}\\ \mathbf{B}^{T}\end{bmatrix}^{+}=[\mathbf{A},\mathbf{B}]([\mathbf{A},\mathbf{B}% ]^{T}[\mathbf{A},\mathbf{B}])^{-1}.
  4. ( [ 𝐀 , 𝐁 ] T [ 𝐀 , 𝐁 ] ) - 1 \mathbf{(}[\mathbf{A},\mathbf{B}]^{T}[\mathbf{A},\mathbf{B}])^{-1}
  5. [ 𝐀 , 𝐁 ] + = [ 𝐏 B 𝐀 ( 𝐀 T 𝐏 B 𝐀 ) - 1 , 𝐏 A 𝐁 ( 𝐁 T 𝐏 A 𝐁 ) - 1 ] T , \begin{bmatrix}\mathbf{A},&\mathbf{B}\end{bmatrix}^{+}=\left[\mathbf{P}_{B}^{% \perp}\mathbf{A}(\mathbf{A}^{T}\mathbf{P}_{B}^{\perp}\mathbf{A})^{-1},\quad% \mathbf{P}_{A}^{\perp}\mathbf{B}(\mathbf{B}^{T}\mathbf{P}_{A}^{\perp}\mathbf{B% })^{-1}\right]^{T},
  6. [ 𝐀 T 𝐁 T ] + = [ 𝐏 B 𝐀 ( 𝐀 T 𝐏 B 𝐀 ) - 1 , 𝐏 A 𝐁 ( 𝐁 T 𝐏 A 𝐁 ) - 1 ] , \begin{bmatrix}\mathbf{A}^{T}\\ \mathbf{B}^{T}\end{bmatrix}^{+}=\left[\mathbf{P}_{B}^{\perp}\mathbf{A}(\mathbf% {A}^{T}\mathbf{P}_{B}^{\perp}\mathbf{A})^{-1},\quad\mathbf{P}_{A}^{\perp}% \mathbf{B}(\mathbf{B}^{T}\mathbf{P}_{A}^{\perp}\mathbf{B})^{-1}\right],
  7. 𝐏 A \displaystyle\mathbf{P}_{A}^{\perp}
  8. [ 𝐀 , 𝐁 ] + = [ ( 𝐏 B 𝐀 ) + ( 𝐏 A 𝐁 ) + ] , \begin{bmatrix}\mathbf{A},&\mathbf{B}\end{bmatrix}^{+}=\begin{bmatrix}(\mathbf% {P}_{B}^{\perp}\mathbf{A})^{+}\\ (\mathbf{P}_{A}^{\perp}\mathbf{B})^{+}\end{bmatrix},
  9. [ 𝐀 T 𝐁 T ] + = [ ( 𝐀 T 𝐏 B ) + , ( 𝐁 T 𝐏 A ) + ] . \begin{bmatrix}\mathbf{A}^{T}\\ \mathbf{B}^{T}\end{bmatrix}^{+}=[(\mathbf{A}^{T}\mathbf{P}_{B}^{\perp})^{+},% \quad(\mathbf{B}^{T}\mathbf{P}_{A}^{\perp})^{+}].
  10. [ 𝐀 , 𝐁 ] [\mathbf{A},\mathbf{B}]
  11. 𝐀 0 \mathbf{A}\neq 0
  12. [ 𝐀 , 𝐀 ] + = 1 2 [ 𝐀 + 𝐀 + ] [ ( 𝐏 A 𝐀 ) + ( 𝐏 A 𝐀 ) + ] = 0 \begin{bmatrix}\mathbf{A},&\mathbf{A}\end{bmatrix}^{+}=\frac{1}{2}\begin{% bmatrix}\mathbf{A}^{+}\\ \mathbf{A}^{+}\end{bmatrix}\neq\begin{bmatrix}(\mathbf{P}_{A}^{\perp}\mathbf{A% })^{+}\\ (\mathbf{P}_{A}^{\perp}\mathbf{A})^{+}\end{bmatrix}=0
  13. 𝐱 = [ 𝐱 1 𝐱 2 ] \mathbf{x}=\begin{bmatrix}\mathbf{x}_{1}\\ \mathbf{x}_{2}\\ \end{bmatrix}
  14. [ 𝐀 , 𝐁 ] [ 𝐱 1 𝐱 2 ] = 𝐝 , 𝐝 \reals m × 1 . \begin{bmatrix}\mathbf{A},&\mathbf{B}\end{bmatrix}\begin{bmatrix}\mathbf{x}_{1% }\\ \mathbf{x}_{2}\\ \end{bmatrix}=\mathbf{d},\qquad\mathbf{d}\in\reals^{m\times 1}.
  15. 𝐱 = [ 𝐀 , 𝐁 ] + 𝐝 = [ ( 𝐏 B 𝐀 ) + ( 𝐏 A 𝐁 ) + ] 𝐝 . \mathbf{x}=\begin{bmatrix}\mathbf{A},&\mathbf{B}\end{bmatrix}^{+}\,\mathbf{d}=% \begin{bmatrix}(\mathbf{P}_{B}^{\perp}\mathbf{A})^{+}\\ (\mathbf{P}_{A}^{\perp}\mathbf{B})^{+}\end{bmatrix}\mathbf{d}.
  16. 𝐱 1 = ( 𝐏 B 𝐀 ) + 𝐝 , 𝐱 2 = ( 𝐏 A 𝐁 ) + 𝐝 . \mathbf{x}_{1}=(\mathbf{P}_{B}^{\perp}\mathbf{A})^{+}\,\mathbf{d},\qquad% \mathbf{x}_{2}=(\mathbf{P}_{A}^{\perp}\mathbf{B})^{+}\,\mathbf{d}.
  17. 𝐱 \mathbf{x}
  18. [ 𝐀 T 𝐁 T ] 𝐱 = [ 𝐞 𝐟 ] , 𝐞 \reals n × 1 , 𝐟 \reals p × 1 . \begin{bmatrix}\mathbf{A}^{T}\\ \mathbf{B}^{T}\end{bmatrix}\mathbf{x}=\begin{bmatrix}\mathbf{e}\\ \mathbf{f}\end{bmatrix},\qquad\mathbf{e}\in\reals^{n\times 1},\qquad\mathbf{f}% \in\reals^{p\times 1}.
  19. 𝐱 = [ 𝐀 T 𝐁 T ] + [ 𝐞 𝐟 ] . \mathbf{x}=\begin{bmatrix}\mathbf{A}^{T}\\ \mathbf{B}^{T}\end{bmatrix}^{+}\,\begin{bmatrix}\mathbf{e}\\ \mathbf{f}\end{bmatrix}.
  20. 𝐱 = [ ( 𝐀 T 𝐏 B ) + , ( 𝐁 T 𝐏 A ) + ] [ 𝐞 𝐟 ] = ( 𝐀 T 𝐏 B ) + 𝐞 + ( 𝐁 T 𝐏 A ) + 𝐟 . \mathbf{x}=[(\mathbf{A}^{T}\mathbf{P}_{B}^{\perp})^{+},\quad(\mathbf{B}^{T}% \mathbf{P}_{A}^{\perp})^{+}]\begin{bmatrix}\mathbf{e}\\ \mathbf{f}\end{bmatrix}=(\mathbf{A}^{T}\mathbf{P}_{B}^{\perp})^{+}\,\mathbf{e}% +(\mathbf{B}^{T}\mathbf{P}_{A}^{\perp})^{+}\,\mathbf{f}.
  21. ( [ 𝐀 , 𝐁 ] T [ 𝐀 , 𝐁 ] ) - 1 \mathbf{(}[\mathbf{A},\mathbf{B}]^{T}[\mathbf{A},\mathbf{B}])^{-1}
  22. ( 𝐀 T 𝐀 ) - 1 , ( 𝐁 T 𝐁 ) - 1 , ( 𝐀 T 𝐏 B 𝐀 ) - 1 , ( 𝐁 T 𝐏 A 𝐁 ) - 1 . \quad(\mathbf{A}^{T}\mathbf{A})^{-1},\quad(\mathbf{B}^{T}\mathbf{B})^{-1},% \quad(\mathbf{A}^{T}\mathbf{P}_{B}^{\perp}\mathbf{A})^{-1},\quad(\mathbf{B}^{T% }\mathbf{P}_{A}^{\perp}\mathbf{B})^{-1}.
  23. ( 𝐀 T 𝐀 ) - 1 (\mathbf{A}^{T}\mathbf{A})^{-1}
  24. ( 𝐁 T 𝐁 ) - 1 (\mathbf{B}^{T}\mathbf{B})^{-1}
  25. ( 𝐀 T 𝐏 B 𝐀 ) - 1 (\mathbf{A}^{T}\mathbf{P}_{B}^{\perp}\mathbf{A})^{-1}
  26. ( 𝐁 T 𝐏 A 𝐁 ) - 1 (\mathbf{B}^{T}\mathbf{P}_{A}^{\perp}\mathbf{B})^{-1}
  27. [ A B C D ] . \begin{bmatrix}A&B\\ C&D\end{bmatrix}.
  28. [ A B C D ] - 1 = [ ( A - B D - 1 C ) - 1 - A - 1 B ( D - C A - 1 B ) - 1 - D - 1 C ( A - B D - 1 C ) - 1 ( D - C A - 1 B ) - 1 ] = [ S D - 1 - A - 1 B S A - 1 - D - 1 C S D - 1 S A - 1 ] , \begin{bmatrix}A&B\\ C&D\end{bmatrix}^{-1}=\begin{bmatrix}(A-BD^{-1}C)^{-1}&-A^{-1}B(D-CA^{-1}B)^{-% 1}\\ -D^{-1}C(A-BD^{-1}C)^{-1}&(D-CA^{-1}B)^{-1}\end{bmatrix}=\begin{bmatrix}S^{-1}% _{D}&-A^{-1}BS^{-1}_{A}\\ -D^{-1}CS^{-1}_{D}&S^{-1}_{A}\end{bmatrix},
  29. S A S_{A}
  30. S D S_{D}
  31. A A
  32. D D
  33. S A = D - C A - 1 B S_{A}=D-CA^{-1}B
  34. S D = A - B D - 1 C S_{D}=A-BD^{-1}C
  35. [ 𝐀 T 𝐀 𝐀 T 𝐁 𝐁 T 𝐀 𝐁 T 𝐁 ] - 1 = [ ( 𝐀 T 𝐀 - 𝐀 T 𝐁 ( 𝐁 T 𝐁 ) - 1 𝐁 T 𝐀 ) - 1 - ( 𝐀 T 𝐀 ) - 1 𝐀 T 𝐁 ( 𝐁 T 𝐁 - 𝐁 T 𝐀 ( 𝐀 T 𝐀 ) - 1 𝐀 T 𝐁 ) - 1 - ( 𝐁 T 𝐁 ) - 1 𝐁 T 𝐀 ( 𝐀 T 𝐀 - 𝐀 T 𝐁 ( 𝐁 T 𝐁 ) - 1 𝐁 T 𝐀 ) - 1 ( 𝐁 T 𝐁 - 𝐁 T 𝐀 ( 𝐀 T 𝐀 ) - 1 𝐀 T 𝐁 ) - 1 ] \begin{bmatrix}\mathbf{A}^{T}\mathbf{A}&\mathbf{A}^{T}\mathbf{B}\\ \mathbf{B}^{T}\mathbf{A}&\mathbf{B}^{T}\mathbf{B}\end{bmatrix}^{-1}=\begin{% bmatrix}(\mathbf{A}^{T}\mathbf{A}-\mathbf{A}^{T}\mathbf{B}(\mathbf{B}^{T}% \mathbf{B})^{-1}\mathbf{B}^{T}\mathbf{A})^{-1}&-(\mathbf{A}^{T}\mathbf{A})^{-1% }\mathbf{A}^{T}\mathbf{B}(\mathbf{B}^{T}\mathbf{B}-\mathbf{B}^{T}\mathbf{A}(% \mathbf{A}^{T}\mathbf{A})^{-1}\mathbf{A}^{T}\mathbf{B})^{-1}\\ -(\mathbf{B}^{T}\mathbf{B})^{-1}\mathbf{B}^{T}\mathbf{A}(\mathbf{A}^{T}\mathbf% {A}-\mathbf{A}^{T}\mathbf{B}(\mathbf{B}^{T}\mathbf{B})^{-1}\mathbf{B}^{T}% \mathbf{A})^{-1}&(\mathbf{B}^{T}\mathbf{B}-\mathbf{B}^{T}\mathbf{A}(\mathbf{A}% ^{T}\mathbf{A})^{-1}\mathbf{A}^{T}\mathbf{B})^{-1}\end{bmatrix}
  36. = [ ( 𝐀 T 𝐏 B 𝐀 ) - 1 - ( 𝐀 T 𝐀 ) - 1 𝐀 T 𝐁 ( 𝐁 T 𝐏 A 𝐁 ) - 1 - ( 𝐁 T 𝐁 ) - 1 𝐁 T 𝐀 ( 𝐀 T 𝐏 B 𝐀 ) - 1 ( 𝐁 T 𝐏 A 𝐁 ) - 1 ] =\begin{bmatrix}(\mathbf{A}^{T}\mathbf{P}_{B}^{\perp}\mathbf{A})^{-1}&-(% \mathbf{A}^{T}\mathbf{A})^{-1}\mathbf{A}^{T}\mathbf{B}(\mathbf{B}^{T}\mathbf{P% }_{A}^{\perp}\mathbf{B})^{-1}\\ -(\mathbf{B}^{T}\mathbf{B})^{-1}\mathbf{B}^{T}\mathbf{A}(\mathbf{A}^{T}\mathbf% {P}_{B}^{\perp}\mathbf{A})^{-1}&(\mathbf{B}^{T}\mathbf{P}_{A}^{\perp}\mathbf{B% })^{-1}\end{bmatrix}
  37. [ 𝐀 T 𝐀 𝐀 T 𝐁 𝐁 T 𝐀 𝐁 T 𝐁 ] - 1 = [ ( 𝐀 T 𝐏 B 𝐀 ) - 1 - ( 𝐀 T 𝐏 B 𝐀 ) - 1 𝐀 T 𝐁 ( 𝐁 T 𝐁 ) - 1 - ( 𝐁 T 𝐏 A 𝐁 ) - 1 𝐁 T 𝐀 ( 𝐀 T 𝐀 ) - 1 ( 𝐁 T 𝐏 A 𝐁 ) - 1 ] . \begin{bmatrix}\mathbf{A}^{T}\mathbf{A}&\mathbf{A}^{T}\mathbf{B}\\ \mathbf{B}^{T}\mathbf{A}&\mathbf{B}^{T}\mathbf{B}\end{bmatrix}^{-1}=\begin{% bmatrix}(\mathbf{A}^{T}\mathbf{P}_{B}^{\perp}\mathbf{A})^{-1}&-(\mathbf{A}^{T}% \mathbf{P}_{B}^{\perp}\mathbf{A})^{-1}\mathbf{A}^{T}\mathbf{B}(\mathbf{B}^{T}% \mathbf{B})^{-1}\\ -(\mathbf{B}^{T}\mathbf{P}_{A}^{\perp}\mathbf{B})^{-1}\mathbf{B}^{T}\mathbf{A}% (\mathbf{A}^{T}\mathbf{A})^{-1}&(\mathbf{B}^{T}\mathbf{P}_{A}^{\perp}\mathbf{B% })^{-1}\end{bmatrix}.

Block_Truncation_Coding.html

  1. y ( i , j ) = { 1 , x ( i , j ) > x ¯ 0 , x ( i , j ) x ¯ y(i,j)=\begin{cases}1,&x(i,j)>\bar{x}\\ 0,&x(i,j)\leq\bar{x}\end{cases}
  2. x ( i , j ) x(i,j)
  3. y ( i , j ) y(i,j)
  4. a = x ¯ - σ q m - q a=\bar{x}-\sigma\sqrt{\cfrac{q}{m-q}}
  5. b = x ¯ + σ m - q q b=\bar{x}+\sigma\sqrt{\cfrac{m-q}{q}}
  6. σ \sigma
  7. x ¯ \bar{x}
  8. x ( i , j ) = { a , y ( i , j ) = 0 b , y ( i , j ) = 1 x(i,j)=\begin{cases}a,&y(i,j)=0\\ b,&y(i,j)=1\end{cases}
  9. 245 239 249 239 245 245 239 235 245 245 245 245 245 235 235 239 \begin{matrix}245&239&249&239\\ 245&245&239&235\\ 245&245&245&245\\ 245&235&235&239\end{matrix}
  10. 1 0 1 0 1 1 0 0 1 1 1 1 1 0 0 0 \begin{matrix}1&0&1&0\\ 1&1&0&0\\ 1&1&1&1\\ 1&0&0&0\end{matrix}
  11. 245 236 245 236 245 245 236 236 245 245 245 245 245 236 236 236 \begin{matrix}245&236&245&236\\ 245&245&236&236\\ 245&245&245&245\\ 245&236&236&236\end{matrix}

Block_walking.html

  1. ( 0 , 0 ) (0,0)
  2. ( e , n ) (e,n)
  3. e + n = k e+n=k
  4. e e
  5. n n
  6. X = ( x 1 , x 2 ) X=(x_{1},x_{2})
  7. 0 x 1 e 0\leq x_{1}\leq e
  8. 0 x 2 n 0\leq x_{2}\leq n
  9. ( 0 , 0 ) (0,0)
  10. ( 1 , 0 ) (1,0)
  11. k k
  12. E = 1 , 0 E=\langle 1,0\rangle
  13. N = 0 , 1 N=\langle 0,1\rangle
  14. e e
  15. n n
  16. E E E e times N N N n times \overbrace{EE\cdots E}^{e\,\text{ times}}\underbrace{NN\cdots N}_{n\,\text{ % times}}
  17. e e
  18. k k
  19. k k
  20. ( k e ) = ( k n ) = k ! e ! n ! . {\left({{k}\atop{e}}\right)}={\left({{k}\atop{n}}\right)}=\frac{k!}{e!n!}.
  21. k = 0 n ( n k ) 2 = ( 2 n n ) \sum_{k=0}^{n}{n\choose k}^{2}\ ={2n\choose n}

Bochner's_formula.html

  1. ( M , g ) (M,g)
  2. u : M u:M\rightarrow\mathbb{R}
  3. Δ g u = 0 \Delta_{g}u=0
  4. Δ g \Delta_{g}
  5. g g
  6. Δ 1 2 | u | 2 = | 2 u | 2 + Ric ( u , u ) \Delta\frac{1}{2}|\nabla u|^{2}=|\nabla^{2}u|^{2}+\mbox{Ric}~{}(\nabla u,% \nabla u)
  7. u \nabla u
  8. u u
  9. g g

Bochner_integral.html

  1. s ( x ) = i = 1 n χ E i ( x ) b i s(x)=\sum_{i=1}^{n}\chi_{E_{i}}(x)b_{i}
  2. X [ i = 1 n χ E i ( x ) b i ] d μ = i = 1 n μ ( E i ) b i \int_{X}\left[\sum_{i=1}^{n}\chi_{E_{i}}(x)b_{i}\right]\,d\mu=\sum_{i=1}^{n}% \mu(E_{i})b_{i}
  3. lim n X f - s n B d μ = 0 , \lim_{n\to\infty}\int_{X}\|f-s_{n}\|_{B}\,d\mu=0,
  4. X f d μ = lim n X s n d μ . \int_{X}f\,d\mu=\lim_{n\to\infty}\int_{X}s_{n}\,d\mu.
  5. L 1 L^{1}
  6. X f B d μ < . \int_{X}\|f\|_{B}\,d\mu<\infty.
  7. T T
  8. f f
  9. T f Tf
  10. T T
  11. X T f d μ = T X f d μ . \int_{X}Tfd\mu=T\int_{X}fd\mu.
  12. T f Tf
  13. T T
  14. f n ( x ) B g ( x ) \|f_{n}(x)\|_{B}\leq g(x)
  15. X f - f n B d μ 0 \int_{X}\|f-f_{n}\|_{B}\,d\mu\to 0
  16. E f n d μ E f d μ \int_{E}f_{n}\,d\mu\to\int_{E}f\,d\mu
  17. E f d μ B E f B d μ \left\|\int_{E}f\,d\mu\right\|_{B}\leq\int_{E}\|f\|_{B}\,d\mu
  18. E E f d μ E\mapsto\int_{E}f\,d\mu
  19. γ \gamma
  20. γ ( E ) = E g d μ \gamma(E)=\int_{E}g\,d\mu
  21. l 1 l_{1}
  22. c 0 c_{0}
  23. L ( Ω ) L^{\infty}(\Omega)
  24. L 1 ( Ω ) L^{1}(\Omega)
  25. Ω \Omega
  26. n \mathbb{R}^{n}
  27. C ( K ) C(K)

Bol_loop.html

  1. a ( b ( a c ) ) = ( a ( b a ) ) c a(b(ac))=(a(ba))c
  2. ( ( c a ) b ) a = c ( ( a b ) a ) ((ca)b)a=c((ab)a)

Bombieri–Vinogradov_theorem.html

  1. x x
  2. Q Q
  3. x 1 / 2 log - A x Q x 1 / 2 . x^{1/2}\log^{-A}x\leq Q\leq x^{1/2}.
  4. q Q max y < x max 1 a q ( a , q ) = 1 | ψ ( y ; q , a ) - y φ ( q ) | = O ( x 1 / 2 Q ( log x ) 5 ) . \sum_{q\leq Q}\max_{y<x}\max_{1\leq a\leq q\atop(a,q)=1}\left|\psi(y;q,a)-{y% \over\varphi(q)}\right|=O\left(x^{1/2}Q(\log x)^{5}\right).
  5. φ ( q ) \varphi(q)
  6. ψ ( x ; q , a ) = n x n a mod q Λ ( n ) , \psi(x;q,a)=\sum_{n\leq x\atop n\equiv a\bmod q}\Lambda(n),
  7. Λ \Lambda
  8. x \sqrt{x}
  9. x \sqrt{x}

Bond_convexity_closed-form_formula.html

  1. C o n v = - D P { ( m - 1 + a + 1 ) ( m - 1 + a + 2 ) ( 1 / ( 1 + i ) ) ( m - 1 + a + 2 ) i + 2 ( m - 1 + a + 2 ) ( 1 / ( 1 + i ) ) ( m - 1 + a + 2 ) - ( 1 / ( 1 + i ) ) i 2 + 2 ( 1 / ( 1 + i ) ) ( m - 1 + a + 2 ) - ( 1 / ( 1 + i ) i 3 } + B P ( m - 1 + a ) ( m - 1 + a + 1 ) ( 1 + i ) ( m - 1 + a + 2 ) Conv=-\frac{D}{P}\begin{Bmatrix}\frac{(m-1+a+1)(m-1+a+2)(1/(1+i))^{(m-1+a+2)}}% {i}+\\ 2\frac{(m-1+a+2)(1/(1+i))^{(m-1+a+2)}-(1/(1+i))}{i^{2}}+\\ 2\frac{(1/(1+i))^{(m-1+a+2)}-(1/(1+i)}{i^{3}}\end{Bmatrix}+\frac{B}{P}\frac{(m% -1+a)(m-1+a+1)}{(1+i)^{(m-1+a+2)}}

Bond_duration_closed-form_formula.html

  1. D u r = C ( 1 + a i ) ( 1 + i ) m - ( 1 + i ) - ( m - 1 + a ) i i 2 ( 1 + i ) ( m - 1 + a ) + 100 ( m - 1 + a ) ( 1 + i ) ( m - 1 + a ) P Dur=\frac{C\frac{(1+ai)(1+i)^{m}-(1+i)-(m-1+a)i}{i^{2}(1+i)^{(m-1+a)}}+\frac{1% 00(m-1+a)}{(1+i)^{(m-1+a)}}}{P}

Bond_graph.html

  1. engine - - - - - 𝜔 𝜏 wheel {\,\text{engine}}\;\overset{\textstyle\tau}{\underset{\textstyle\omega}{-\!\!% \!-\!\!\!-\!\!\!-\!\!\!-}}\;\,\text{wheel}
  2. engine - - - 𝜔 𝜏 wheel {\,\text{engine}}\;\overset{\textstyle\tau}{\underset{\textstyle\omega}{-\!\!% \!-\!\!\!-\!\!\!\rightharpoondown}}\;\,\text{wheel}
  3. wheel - - - 𝜔 tachometer {\,\text{wheel}}\;\overset{\textstyle}{\underset{\textstyle\omega}{-\!\!\!-\!% \!\!-\!\!\!\rightarrow}}\;\,\text{tachometer}
  4. v 1 i 1 and and and 𝑅 v 2 i 2 = i 1 \frac{v_{1}\qquad}{i_{1}\qquad}\overset{\textstyle R}{\!\!\and\!\!\and\!\!\and% \!}\frac{\qquad v_{2}}{\qquad i_{2}=i_{1}}
  5. - - - i 1 v 1 1 R - - - i 2 v 2 \overset{\textstyle v_{1}}{\underset{\textstyle i_{1}}{-\!\!\!-\!\!\!-\!\!\!% \rightharpoondown}}\stackrel{\textstyle\stackrel{\textstyle R}{\upharpoonright% }}{1}\overset{\textstyle v_{2}}{\underset{\textstyle i_{2}}{-\!\!\!-\!\!\!-\!% \!\!\rightharpoondown}}
  6. motor S E - - - | 𝜔 𝜏 wheel \begin{array}[b]{r}\,\text{motor}\\ SE\end{array}\;\overset{\textstyle\tau}{\underset{\textstyle\omega}{-\!\!\!-\!% \!\!-\!\!\!\rightharpoonup\!\!\!|}}\;\,\text{wheel}
  7. J - - - | I and J | - - - C J\;\overset{\textstyle}{\underset{\textstyle}{-\!\!\!-\!\!\!-\!\!\!% \rightharpoonup\!\!\!|}}\;I\qquad\,\text{and}\qquad J\;\overset{\textstyle}{% \underset{\textstyle}{|\!\!\!-\!\!\!-\!\!\!-\!\!\!\rightharpoonup}}\;C
  8. J - - - | R and J | - - - R J\;\overset{\textstyle}{\underset{\textstyle}{-\!\!\!-\!\!\!-\!\!\!% \rightharpoonup\!\!\!|}}\;R\qquad\,\text{and}\qquad J\;\overset{\textstyle}{% \underset{\textstyle}{|\!\!\!-\!\!\!-\!\!\!-\!\!\!\rightharpoonup}}\;R
  9. - - - - - | T ˙ F ˙ - - - - - | or | - - - - - T ˙ F ˙ | - - - - - \;\overset{\textstyle}{\underset{\textstyle}{-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!|}% }\;\dot{T}\dot{F}\;\overset{\textstyle}{\underset{\textstyle}{-\!\!\!-\!\!\!-% \!\!\!-\!\!\!-\!|}}\;\qquad\,\text{or}\qquad\;\overset{\textstyle}{\underset{% \textstyle}{|\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-}}\;\dot{T}\dot{F}\;\overset{% \textstyle}{\underset{\textstyle}{|\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!-}}\;
  10. | - - - - - G ˙ Y ˙ - - - - - | or - - - - - | G ˙ Y ˙ | - - - - - \;\overset{\textstyle}{\underset{\textstyle}{|\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!% \!-}}\;\dot{G}\dot{Y}\;\overset{\textstyle}{\underset{\textstyle}{-\!\!\!-\!\!% \!-\!\!\!-\!\!\!-\!|}}\;\qquad\,\text{or}\qquad\;\overset{\textstyle}{% \underset{\textstyle}{-\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!|}}\;\dot{G}\dot{Y}\;% \overset{\textstyle}{\underset{\textstyle}{|\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!\!\!% -}}\;
  11. strong bond 0 and strong bond 1 \,\text{strong bond}\rightarrow\;\dashv\!\overset{\textstyle\top}{\underset{% \textstyle\bot}{0}}\!\dashv\qquad\,\text{and}\qquad\,\text{strong bond}% \rightarrow\;\vdash\!\overset{\textstyle\bot}{\underset{\textstyle\top}{1}}\!\vdash
  12. S F - - - - - 1 - - - - - I SF\;\overset{\textstyle}{\underset{\textstyle}{-\!\!\!-\!\!\!-\!\!\!-\!\!\!-}}% 1\overset{\textstyle}{\underset{\textstyle}{-\!\!\!-\!\!\!-\!\!\!-\!\!\!-}}\;I
  13. S F | - - - - - 1 - - - - - I SF\;\overset{\textstyle}{\underset{\textstyle}{|\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!% \!\!-}}1\overset{\textstyle}{\underset{\textstyle}{-\!\!\!-\!\!\!-\!\!\!-\!\!% \!-}}\;I
  14. S F | - - - - - 1 | - - - - - I SF\;\overset{\textstyle}{\underset{\textstyle}{|\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!% \!\!-}}1\overset{\textstyle}{\underset{\textstyle}{|\!\!\!-\!\!\!-\!\!\!-\!\!% \!-\!\!\!-}}\;I
  15. S F | - - - - - 1 | - - - - - | I SF\;\overset{\textstyle}{\underset{\textstyle}{|\!\!\!-\!\!\!-\!\!\!-\!\!\!-\!% \!\!-}}1\overset{\textstyle}{\underset{\textstyle}{|\!\!\!-\!\!\!-\!\!\!-\!\!% \!-\!\!\!-\!|}}\;I
  16. S e , i n S_{e},in
  17. v ˙ 2 = 1 C i 2 \dot{v}_{2}={1\over C}i_{2}
  18. v 2 v_{2}
  19. i 2 i_{2}
  20. i 2 i_{2}
  21. i 3 i_{3}
  22. i 2 = i 3 i_{2}=i_{3}
  23. v ˙ 2 = 1 C i 3 \dot{v}_{2}={1\over C}i_{3}
  24. i 3 i_{3}
  25. v 3 v_{3}
  26. i 3 = v 3 R i_{3}={v_{3}\over R}
  27. v ˙ 2 = 1 C v 3 R \dot{v}_{2}={1\over C}{v_{3}\over R}
  28. v 3 v_{3}
  29. v 3 = v 1 - v 2 v_{3}=v_{1}-v_{2}
  30. v ˙ 2 = 1 C v 1 - v 2 R \dot{v}_{2}={1\over C}{{v_{1}-v_{2}}\over R}
  31. v 1 v_{1}
  32. v 2 v_{2}
  33. v 4 v_{4}
  34. v ˙ 4 = 1 C 4 i 4 = 1 C 4 ( i 5 - i 7 ) = 1 C 4 ( i 6 - i 2 ) \dot{v}_{4}={1\over C_{4}}i_{4}={1\over C_{4}}(i_{5}-i_{7})={1\over C_{4}}(i_{% 6}-i_{2})
  35. v ˙ 4 = 1 C 4 ( v 6 R 6 - v 2 R 2 ) \dot{v}_{4}={1\over C_{4}}({v_{6}\over R_{6}}-{v_{2}\over R_{2}})
  36. v ˙ 4 = 1 C 4 ( v 1 - v 5 R 6 - v 7 - v 3 R 2 ) \dot{v}_{4}={1\over C_{4}}({{v1-v5}\over R_{6}}-{{v7-v3}\over R_{2}})
  37. v ˙ 4 = 1 C 4 ( v 1 - v 4 R 6 - v 4 - v 3 R 2 ) \dot{v}_{4}={1\over C_{4}}({{v1-v4}\over R_{6}}-{{v4-v3}\over R_{2}})
  38. v ˙ 4 \dot{v}_{4}
  39. v ˙ 3 \dot{v}_{3}
  40. v ˙ 3 = 1 C 3 R 2 v 4 - 1 C 3 R 2 v 3 \dot{v}_{3}={1\over{C_{3}R_{2}}}v_{4}-{1\over{C_{3}R_{2}}}v_{3}
  41. v ˙ 4 = - ( 1 C 4 R 6 + 1 C 4 R 2 ) v 4 + 1 C 4 R 2 v 3 + 1 C 4 R 6 v 1 \dot{v}_{4}=-({1\over{C_{4}R_{6}}}+{1\over{C_{4}R_{2}}})v_{4}+{1\over{C_{4}R_{% 2}}}v_{3}+{1\over{C_{4}R_{6}}}v_{1}
  42. [ v ˙ 4 v ˙ 3 ] = [ - ( 1 C 4 R 6 + 1 C 4 R 2 ) 1 C 4 R 2 1 C 3 R 2 - 1 C 3 R 2 ] [ v 4 v 3 ] + [ 1 C 4 R 6 0 ] v 1 \begin{bmatrix}\dot{v}_{4}\\ \dot{v}_{3}\end{bmatrix}=\begin{bmatrix}-({1\over{C_{4}R_{6}}}+{1\over{C_{4}R_% {2}}})&{1\over{C_{4}R_{2}}}\\ {1\over{C_{3}R_{2}}}&-{1\over{C_{3}R_{2}}}\end{bmatrix}\begin{bmatrix}v_{4}\\ v_{3}\end{bmatrix}+\begin{bmatrix}{1\over{C_{4}R_{6}}}\\ 0\end{bmatrix}v_{1}
  43. f 2 = r f 1 f_{2}=rf_{1}\,
  44. e 2 = 1 r e 1 e_{2}=\frac{1}{r}e_{1}\,
  45. e 2 = μ f 1 e_{2}=\mu f_{1}\,
  46. e 1 = μ f 2 e_{1}=\mu f_{2}\,

Bond_plus_option.html

  1. Z C B ( U S D , 5 y , 4.7 % ) = e - 5 * 0.047 0.7906 ZCB(USD,5y,4.7\%)=e^{-5*0.047}\approx 0.7906

Bondy's_theorem.html

  1. [ 1 1 0 1 0 1 0 1 0 0 1 1 0 1 1 0 ] \begin{bmatrix}1&1&0&1\\ 0&1&0&1\\ 0&0&1&1\\ 0&1&1&0\end{bmatrix}
  2. [ 1 0 1 1 0 1 0 1 1 1 1 0 ] \begin{bmatrix}1&0&1\\ 1&0&1\\ 0&1&1\\ 1&1&0\end{bmatrix}
  3. [ 1 1 1 0 1 1 0 0 1 0 1 0 ] \begin{bmatrix}1&1&1\\ 0&1&1\\ 0&0&1\\ 0&1&0\end{bmatrix}

Book_embedding.html

  1. n / 2 \lceil n/2\rceil
  2. K 5 K_{5}
  3. n 4 n\geq 4
  4. n / 2 \lceil n/2\rceil
  5. n n
  6. K n K_{n}
  7. ( 1 / 4 ) n 2 n - 1 2 n - 2 2 n - 3 2 , (1/4)\left\lfloor\frac{n}{2}\right\rfloor\left\lfloor\frac{n-1}{2}\right% \rfloor\left\lfloor\frac{n-2}{2}\right\rfloor\left\lfloor\frac{n-3}{2}\right\rfloor,
  8. K a , b K_{a,b}
  9. min { a , b } \min\{a,b\}
  10. K 4 , 4 K_{4,4}
  11. b > a ( a - 1 ) b>a(a-1)
  12. K a , b K_{a,b}
  13. a a

Boolean_domain.html

  1. { , } \left\{\bot,\top\right\}
  2. 𝔹 . \mathbb{B}.
  3. [ 0 , 1 ] [0,1]
  4. 1 - x , 1-x,
  5. x y xy
  6. 1 - ( 1 - x ) ( 1 - y ) 1-(1-x)(1-y)

Boolean_grammar.html

  1. A α 1 & & α m & ¬ β 1 & & ¬ β n A\to\alpha_{1}\And\ldots\And\alpha_{m}\And\lnot\beta_{1}\And\ldots\And\lnot% \beta_{n}
  2. A A
  3. m + n 1 m+n\geq 1
  4. α 1 \alpha_{1}
  5. α m \alpha_{m}
  6. β 1 \beta_{1}
  7. β n \beta_{n}
  8. Σ \Sigma
  9. N N
  10. w w
  11. Σ \Sigma
  12. α 1 \alpha_{1}
  13. α m \alpha_{m}
  14. β 1 \beta_{1}
  15. β n \beta_{n}
  16. A A

Boost_converter.html

  1. P = V I P=VI
  2. P = V 2 / R P=V^{2}/R
  3. I L I_{L}
  4. V i V_{i}
  5. I L I_{L}
  6. Δ I L Δ t = V i L \frac{\Delta I_{L}}{\Delta t}=\frac{V_{i}}{L}
  7. Δ I L O n = 1 L 0 D T V i d t = D T L V i \Delta I_{L_{On}}=\frac{1}{L}\int_{0}^{DT}V_{i}dt=\frac{DT}{L}V_{i}
  8. V i - V o = L d I L d t V_{i}-V_{o}=L\frac{dI_{L}}{dt}
  9. Δ I L O f f = D T T ( V i - V o ) d t L = ( V i - V o ) ( 1 - D ) T L \Delta I_{L_{Off}}=\int_{DT}^{T}\frac{\left(V_{i}-V_{o}\right)dt}{L}=\frac{% \left(V_{i}-V_{o}\right)\left(1-D\right)T}{L}
  10. E = 1 2 L I L 2 E=\frac{1}{2}LI_{L}^{2}
  11. Δ I L O n + Δ I L O f f = 0 \Delta I_{L_{On}}+\Delta I_{L_{Off}}=0
  12. Δ I L O n \Delta I_{L_{On}}
  13. Δ I L O f f \Delta I_{L_{Off}}
  14. Δ I L O n + Δ I L O f f = V i D T L + ( V i - V o ) ( 1 - D ) T L = 0 \Delta I_{L_{On}}+\Delta I_{L_{Off}}=\frac{V_{i}DT}{L}+\frac{\left(V_{i}-V_{o}% \right)\left(1-D\right)T}{L}=0
  15. V o V i = 1 1 - D \frac{V_{o}}{V_{i}}=\frac{1}{1-D}
  16. D = 1 - V i V o D={1-\frac{V_{i}}{V_{o}}}
  17. I L M a x I_{L_{Max}}
  18. t = D T t=DT
  19. I L M a x = V i D T L I_{L_{Max}}=\frac{V_{i}DT}{L}
  20. δ T \delta T
  21. I L M a x + ( V i - V o ) δ T L = 0 I_{L_{Max}}+\frac{\left(V_{i}-V_{o}\right)\delta T}{L}=0
  22. δ = V i D V o - V i \delta=\frac{V_{i}D}{V_{o}-V_{i}}
  23. I o = I D ¯ = I L m a x 2 δ I_{o}=\bar{I_{D}}=\frac{I_{L_{max}}}{2}\delta
  24. I o = V i D T 2 L V i D V o - V i = V i 2 D 2 T 2 L ( V o - V i ) I_{o}=\frac{V_{i}DT}{2L}\cdot\frac{V_{i}D}{V_{o}-V_{i}}=\frac{V_{i}^{2}D^{2}T}% {2L\left(V_{o}-V_{i}\right)}
  25. V o V i = 1 + V i D 2 T 2 L I o \frac{V_{o}}{V_{i}}=1+\frac{V_{i}D^{2}T}{2LI_{o}}

Born_rule.html

  1. A A
  2. | ψ \scriptstyle|\psi\rangle
  3. λ \lambda
  4. A A
  5. λ i \lambda_{i}
  6. ψ | P i | ψ \scriptstyle\langle\psi|P_{i}|\psi\rangle
  7. P i P_{i}
  8. A A
  9. λ i \lambda_{i}
  10. A A
  11. λ i \lambda_{i}
  12. | λ i \scriptstyle|\lambda_{i}\rangle
  13. P i P_{i}
  14. | λ i λ i | \scriptstyle|\lambda_{i}\rangle\langle\lambda_{i}|
  15. ψ | P i | ψ \scriptstyle\langle\psi|P_{i}|\psi\rangle
  16. ψ | λ i λ i | ψ \scriptstyle\langle\psi|\lambda_{i}\rangle\langle\lambda_{i}|\psi\rangle
  17. λ i | ψ \scriptstyle\langle\lambda_{i}|\psi\rangle
  18. | ψ \scriptstyle|\psi\rangle
  19. | λ i \scriptstyle|\lambda_{i}\rangle
  20. | λ i | ψ | 2 \scriptstyle|\langle\lambda_{i}|\psi\rangle|^{2}
  21. A A
  22. Q Q
  23. A A
  24. M M
  25. ψ | Q ( M ) | ψ \scriptstyle\langle\psi|Q(M)|\psi\rangle
  26. ψ \scriptstyle\psi
  27. p ( x , y , z ) p(x,y,z)
  28. t 0 t_{0}
  29. p ( x , y , z ) = p(x,y,z)=
  30. | ψ ( x , y , z , t 0 ) | 2 . \scriptstyle|\psi(x,y,z,t_{0})|^{2}.

Born–Huang_approximation.html

  1. χ k ( 𝐫 ; 𝐑 ) | T n | χ k ( 𝐫 ; 𝐑 ) ( 𝐫 ) = 𝒯 k ( 𝐑 ) δ k k \langle\chi_{k^{\prime}}(\mathbf{r};\mathbf{R})|T_{\mathrm{n}}|\chi_{k}(% \mathbf{r};\mathbf{R})\rangle_{(\mathbf{r})}=\mathcal{T}_{\mathrm{k}}(\mathbf{% R})\delta_{k^{\prime}k}
  2. [ T n + E k ( 𝐑 ) + 𝒯 k ( 𝐑 ) ] ϕ k ( 𝐑 ) = E ϕ k ( 𝐑 ) for k = 1 , , K , \left[T_{\mathrm{n}}+E_{k}(\mathbf{R})+\mathcal{T}_{\mathrm{k}}(\mathbf{R})% \right]\;\phi_{k}(\mathbf{R})=E\phi_{k}(\mathbf{R})\quad\mathrm{for}\quad k=1,% \ldots,K,
  3. [ E k ( 𝐑 ) + 𝒯 k ( 𝐑 ) ] \left[E_{k}(\mathbf{R})+\mathcal{T}_{\mathrm{k}}(\mathbf{R})\right]

Born–von_Karman_boundary_condition.html

  1. ψ ( r + N i a i ) = ψ ( r ) , \psi({r}+N_{i}{a}_{i})=\psi({r}),\,
  2. ψ ( r + T ) = ψ ( r ) \psi({r}+{T})=\psi({r})
  3. T = i N i a i . {T}=\sum_{i}N_{i}{a}_{i}.

Bose–Hubbard_model.html

  1. H = - t i , j b i b j + U 2 i n ^ i ( n ^ i - 1 ) - μ i n ^ i H=-t\sum_{\left\langle i,j\right\rangle}b^{\dagger}_{i}b_{j}+\frac{U}{2}\sum_{% i}\hat{n}_{i}\left(\hat{n}_{i}-1\right)-\mu\sum_{i}\hat{n}_{i}
  2. i , j \left\langle i,j\right\rangle
  3. b i b^{\dagger}_{i}
  4. b i b_{i}
  5. n ^ i = b i b i \hat{n}_{i}=b^{\dagger}_{i}b_{i}
  6. t t
  7. U U
  8. U > 0 U>0
  9. U < 0 U<0
  10. μ \mu
  11. D b = ( N b + L - 1 ) ! N b ! ( L - 1 ) ! D_{b}=\frac{(N_{b}+L-1)!}{N_{b}!(L-1)!}
  12. D f = L ! N f ! ( L - N f ) ! . D_{f}=\frac{L!}{N_{f}!(L-N_{f})!}.
  13. t / U t/U
  14. t / U t/U
  15. H = d 3 r [ ψ ^ ( r ) ( - 2 2 m 2 + V latt . ( x ) ) ψ ^ ( r ) + g 2 ψ ^ ( r ) ψ ^ ( r ) ψ ^ ( r ) ψ ^ ( r ) - μ ψ ^ ( r ) ψ ^ ( r ) ] H=\int{\rm d}^{3}r\!\left[\hat{\psi}^{\dagger}(\vec{r})\left(-\frac{\hbar^{2}}% {2m}\nabla^{2}+V_{\rm latt.}(x)\right)\hat{\psi}(\vec{r})+\frac{g}{2}\hat{\psi% }^{\dagger}(\vec{r})\hat{\psi}^{\dagger}(\vec{r})\hat{\psi}(\vec{r})\hat{\psi}% (\vec{r})-\mu\hat{\psi}^{\dagger}(\vec{r})\hat{\psi}(\vec{r})\right]
  16. V l a t t V_{latt}
  17. μ \mu
  18. ψ ^ ( r ) = i w i α ( r ) b i α \hat{\psi}(\vec{r})=\sum\limits_{i}w_{i}^{\alpha}(\vec{r})b_{i}^{\alpha}
  19. w i α ( r ) w j β ( r ) w k γ ( r ) w l δ ( r ) d 3 r = 0 \int w_{i}^{\alpha}(\vec{r})w_{j}^{\beta}(\vec{r})w_{k}^{\gamma}(r)w_{l}^{% \delta}(\vec{r})\,{\rm d}^{3}r=0
  20. i = j = k = l , α = β = γ = δ = 0 i=j=k=l,\alpha=\beta=\gamma=\delta=0
  21. w i α ( r ) w_{i}^{\alpha}(\vec{r})
  22. α \alpha
  23. U n U_{n}
  24. U n U_{n}
  25. n 2 U n^{2}U
  26. d < d<\infty
  27. d + 1. d+1.
  28. V n i n j Vn_{i}n_{j}
  29. a i ( n i + n j ) a j a_{i}^{\dagger}(n_{i}+n_{j})a_{j}

Boundary_layer_thickness.html

  1. u 0 u_{0}
  2. u ( y ) = 0.99 u o u(y)=0.99u_{o}
  3. δ 4.91 ν x u 0 \delta\approx 4.91\sqrt{{\nu x}\over u_{0}}
  4. δ 4.91 x / Re x \delta\approx 4.91x/\sqrt{\mathrm{Re}_{x}}
  5. δ 0.382 x / Re x 1 / 5 \delta\approx 0.382x/{\mathrm{Re}_{x}}^{1/5}
  6. Re x = ρ u 0 x / μ \mathrm{Re}_{x}=\rho u_{0}x/\mu
  7. δ \delta
  8. Re x \mathrm{Re}_{x}
  9. ρ \rho
  10. u 0 u_{0}
  11. x x
  12. ν \nu
  13. μ \mu
  14. u 0 u_{0}
  15. δ * = 0 ( 1 - ρ ( y ) u ( y ) ρ 0 u 0 ) d y {\delta^{*}}=\int_{0}^{\infty}{\left(1-{\rho(y)u(y)\over\rho_{0}u_{0}}\right)% \,\mathrm{d}y}
  16. δ * = 0 ( 1 - u ( y ) u 0 ) d y {\delta^{*}}=\int_{0}^{\infty}{\left(1-{u(y)\over u_{0}}\right)\,\mathrm{d}y}
  17. ρ 0 \rho_{0}
  18. u 0 u_{0}
  19. y y
  20. ρ 0 \rho_{0}
  21. u 0 u_{0}
  22. ρ e \rho_{e}
  23. u e u_{e}
  24. u 0 u_{0}
  25. θ = 0 ρ ( y ) u ( y ) ρ 0 u o ( 1 - u ( y ) u o ) d y \theta=\int_{0}^{\infty}{{\rho(y)u(y)\over\rho_{0}u_{o}}{\left(1-{u(y)\over u_% {o}}\right)}}\,\mathrm{d}y
  26. θ = 0 u ( y ) u o ( 1 - u ( y ) u o ) d y \theta=\int_{0}^{\infty}{{u(y)\over u_{o}}{\left(1-{u(y)\over u_{o}}\right)}}% \,\mathrm{d}y
  27. ρ 0 \rho_{0}
  28. u 0 u_{0}
  29. y y
  30. ρ 0 \rho_{0}
  31. u 0 u_{0}
  32. ρ e \rho_{e}
  33. u e u_{e}
  34. θ 0.664 ν x u o \theta\approx 0.664\sqrt{{\nu x}\over u_{o}}
  35. τ w \tau_{w}
  36. ρ 0 u ( y ) ( u o - u ( y ) ) d y \rho\int_{0}^{\infty}{u(y)\left(u_{o}-u(y)\right)}\,\mathrm{d}y
  37. δ 3 \delta_{3}
  38. H = δ * θ H=\frac{\delta^{*}}{\theta}
  39. δ * \delta^{*}

Bounded_mean_oscillation.html

  1. Ω Ω
  2. 1 | Q | Q | u ( y ) - u Q | d y \frac{1}{|Q|}\int_{Q}|u(y)-u_{Q}|\,\mathrm{d}y
  3. u Q = 1 | Q | Q u ( y ) d y u_{Q}=\frac{1}{|Q|}\int_{Q}u(y)\,\mathrm{d}y
  4. Ω Ω
  5. Ω Ω
  6. Ω Ω
  7. | f R - f Q | C || f || B M O |f_{R}-f_{Q}|\leq C||f||_{BMO}
  8. ( f , g ) = n f ( x ) g ( x ) d x (f,g)=\int_{\mathbb{R}^{n}}f(x)g(x)\,\mathrm{d}x
  9. | { x Q : | f - f Q | > λ } | c 1 exp ( - c 2 λ f B M O ) | Q | . \left|\left\{x\in Q:|f-f_{Q}|>\lambda\right\}\right|\leq c_{1}\exp\left(-c_{2}% \frac{\lambda}{\|f\|_{BMO}}\right)|Q|.
  10. sup Q 𝐑 n 1 | Q | Q e | f - f Q | A d x < . \sup_{Q\subseteq\mathbf{R}^{n}}\frac{1}{|Q|}\int_{Q}e^{\frac{|f-f_{Q}|}{A}}% \mathrm{d}x<\infty.
  11. 1 C A ( f ) inf g L || f - g || B M O C A ( f ) . \frac{1}{C}A(f)\leq\inf_{g\in L^{\infty}}||f-g||_{BMO}\leq CA(f).
  12. 1 | I | I | f ( y ) - f I | d y < C < + \frac{1}{|I|}\int_{I}|f(y)-f_{I}|\,\mathrm{d}y<C<+\infty
  13. u ( a ) = 1 2 π 𝐓 1 - | a | 2 | a - e i θ | 2 f ( e i θ ) d θ u(a)=\frac{1}{2\pi}\int_{\mathbf{T}}\frac{1-|a|^{2}}{|a-e^{i\theta}|^{2}}f(e^{% i\theta})\,\mathrm{d}\theta
  14. u B M O H = sup | a | < 1 { 1 2 π 𝐓 1 - | a | 2 | a - e i θ | 2 | f ( e i θ ) - u ( a ) | d θ } \|u\|_{BMOH}=\sup_{|a|<1}\left\{\frac{1}{2\pi}\int_{\mathbf{T}}\frac{1-|a|^{2}% }{|a-e^{i\theta}|^{2}}|f(e^{i\theta})-u(a)|\,\mathrm{d}\theta\right\}
  15. T g ( f ) = lim r 1 - π π g ¯ ( e i θ ) f ( r e i θ ) d θ T_{g}(f)=\lim_{r\rightarrow 1}\int_{-\pi}^{\pi}\bar{g}(e^{i\theta})f(re^{i% \theta})\,\mathrm{d}\theta
  16. f = f 1 + H f 2 + α f=f_{1}+Hf_{2}+\alpha
  17. f 1 + f 2 \|f_{1}\|_{\infty}+\|f_{2}\|_{\infty}

Bôcher's_theorem.html

  1. r ( z ) r^{\prime}(z)
  2. r ( z ) r(z)
  3. r ( z ) r(z)
  4. r ( z ) r(z)
  5. r ( z ) r(z)
  6. r ( z ) r(z)

BPST_instanton.html

  1. = - 1 4 F μ ν a F μ ν a \mathcal{L}=-\frac{1}{4}F_{\mu\nu}^{a}F_{\mu\nu}^{a}
  2. F ~ μ ν F ~ μ ν = F μ ν F μ ν \tilde{F}_{\mu\nu}\tilde{F}^{\mu\nu}=F_{\mu\nu}F^{\mu\nu}
  3. F ~ μ ν = 1 2 ϵ μ ν ρ σ F ρ σ \tilde{F}_{\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu}^{\rho\sigma}F_{\rho\sigma}
  4. S = d x 4 1 4 F 2 = d x 4 1 8 ( F ± F ~ ) 2 d x 4 1 4 F F ~ S=\int dx^{4}\frac{1}{4}F^{2}=\int dx^{4}\frac{1}{8}(F\pm\tilde{F})^{2}\mp\int dx% ^{4}\frac{1}{4}F\tilde{F}
  5. x x\rightarrow\infty
  6. 8 π 2 g 2 \frac{8\pi^{2}}{g^{2}}
  7. A μ a ( x ) = 2 g η μ ν a ( x - z ) ν ( x - z ) 2 + ρ 2 A_{\mu}^{a}(x)=\frac{2}{g}\frac{\eta^{a}_{\mu\nu}(x-z)_{\nu}}{(x-z)^{2}+\rho^{% 2}}
  8. η μ ν a = { ϵ a μ ν μ , ν = 1 , 2 , 3 - δ a ν μ = 4 δ a μ ν = 4 0 μ = ν = 4 . \eta^{a}_{\mu\nu}=\begin{cases}\epsilon^{a\mu\nu}&\mu,\nu=1,2,3\\ -\delta^{a\nu}&\mu=4\\ \delta^{a\mu}&\nu=4\\ 0&\mu=\nu=4\end{cases}.
  9. x 0 + i x σ x 2 \frac{x^{0}+i{x}\cdot{\sigma}}{\sqrt{x^{2}}}
  10. 1 2 ϵ i j k F a j k = F a 0 i = 4 ρ 2 δ a i g ( x 2 + ρ 2 ) 2 \frac{1}{2}\epsilon_{ijk}{F^{a}}_{jk}={F^{a}}_{0i}=\frac{4{\rho}^{2}\delta_{ai% }}{g(x^{2}+\rho^{2})^{2}}
  11. η ¯ μ ν a \bar{\eta}^{a}_{\mu\nu}
  12. S = 8 π 2 g 2 . S=\frac{8\pi^{2}}{g^{2}}.
  13. A μ a ( x ) = 2 g ρ 2 ( x - z ) 2 η ¯ μ ν a ( x - z ) ν ( x - z ) 2 + ρ 2 . A_{\mu}^{a}(x)=\frac{2}{g}\frac{\rho^{2}}{(x-z)^{2}}\frac{\bar{\eta}^{a}_{\mu% \nu}(x-z)_{\nu}}{(x-z)^{2}+\rho^{2}}.

Bracket_polynomial.html

  1. L L
  2. L \langle L\rangle
  3. A A
  4. O = 1 \langle O\rangle=1
  5. O O
  6. O L = ( - A 2 - A - 2 ) L \langle O\cup L\rangle=(-A^{2}-A^{-2})\langle L\rangle
  7. - A 2 - A - 2 -A^{2}-A^{-2}

Bragg_plane.html

  1. 𝐊 \mathbf{K}
  2. e i 𝐤 𝐫 = cos ( 𝐤 𝐫 ) + i sin ( 𝐤 𝐫 ) e^{i\mathbf{k}\cdot\mathbf{r}}=\cos{(\mathbf{k}\cdot\mathbf{r})}+i\sin{(% \mathbf{k}\cdot\mathbf{r})}
  3. 𝐤 \mathbf{k}
  4. 𝐤 = 2 π λ n ^ \mathbf{k}=\frac{2\pi}{\lambda}\hat{n}
  5. λ \lambda
  6. 𝐤 = 2 π λ n ^ \mathbf{k^{\prime}}=\frac{2\pi}{\lambda}\hat{n}^{\prime}
  7. n ^ \hat{n}^{\prime}
  8. | 𝐝 | cos θ + | 𝐝 | cos θ = 𝐝 ( n ^ - n ^ ) = m λ |\mathbf{d}|\cos{\theta}+|\mathbf{d}|\cos{\theta^{\prime}}=\mathbf{d}\cdot(% \hat{n}-\hat{n}^{\prime})=m\lambda
  9. m m\in\mathbb{Z}
  10. 2 π / λ 2\pi/\lambda
  11. 𝐤 \mathbf{k}
  12. 𝐤 \mathbf{k^{\prime}}
  13. 𝐝 ( 𝐤 - 𝐤 ) = 2 π m \mathbf{d}\cdot(\mathbf{k}-\mathbf{k^{\prime}})=2\pi m
  14. 𝐑 \mathbf{R}
  15. 𝐑 \mathbf{R}
  16. 𝐑 ( 𝐤 - 𝐤 ) = 2 π m \mathbf{R}\cdot(\mathbf{k}-\mathbf{k^{\prime}})=2\pi m
  17. e i ( 𝐤 - 𝐤 ) 𝐑 = 1 e^{i(\mathbf{k}-\mathbf{k^{\prime}})\cdot\mathbf{R}}=1
  18. 𝐊 = 𝐤 - 𝐤 \mathbf{K}=\mathbf{k}-\mathbf{k^{\prime}}
  19. 𝐤 \mathbf{k}
  20. 𝐤 \mathbf{k^{\prime}}
  21. 𝐤 \mathbf{k}
  22. 𝐊 \mathbf{K}

Bragg–Gray_cavity_theory.html

  1. E ν = J ν W ρ E_{\nu}=J_{\nu}\cdot W\cdot\rho
  2. E ν E_{\nu}
  3. J ν J_{\nu}
  4. W W
  5. ρ \rho
  6. γ \gamma

Brahmagupta's_problem.html

  1. x 2 - 92 y 2 = 1. x^{2}-92y^{2}=1.
  2. ( x , y ) = ( 1151 , 120 ) . (x,y)=(1151,120).

Brahmagupta_matrix.html

  1. B ( x , y ) = [ x y ± t y ± x ] . B(x,y)=\begin{bmatrix}x&y\\ \pm ty&\pm x\end{bmatrix}.
  2. B ( x 1 , y 1 ) B ( x 2 , y 2 ) = B ( x 1 x 2 ± t y 1 y 2 , x 1 y 2 ± y 1 x 2 ) . B(x_{1},y_{1})B(x_{2},y_{2})=B(x_{1}x_{2}\pm ty_{1}y_{2},x_{1}y_{2}\pm y_{1}x_% {2}).\,
  3. B n = [ x y t y x ] n = [ x n y n t y n x n ] B n . B^{n}=\begin{bmatrix}x&y\\ ty&x\end{bmatrix}^{n}=\begin{bmatrix}x_{n}&y_{n}\\ ty_{n}&x_{n}\end{bmatrix}\equiv B_{n}.
  4. x n \ x_{n}
  5. y n \ y_{n}
  6. B - n = [ x y t y x ] - n = [ x - n y - n t y - n x - n ] B - n . B^{-n}=\begin{bmatrix}x&y\\ ty&x\end{bmatrix}^{-n}=\begin{bmatrix}x_{-n}&y_{-n}\\ ty_{-n}&x_{-n}\end{bmatrix}\equiv B_{-n}.

Branch_and_cut.html

  1. L L
  2. x * = null x^{*}=\,\text{null}
  3. v * = - v^{*}=-\infty
  4. L L
  5. L L
  6. x x
  7. v v
  8. v v * v\leq v^{*}
  9. x x
  10. v * v , x * x v^{*}\leftarrow v,x^{*}\leftarrow x
  11. x x
  12. L L
  13. x * x^{*}
  14. x i x_{i}
  15. x i x_{i}^{\prime}
  16. x i x i x_{i}\leq\lfloor x_{i}^{\prime}\rfloor
  17. x i x i x_{i}\geq\lceil x_{i}^{\prime}\rceil
  18. x i x_{i}

Branching_quantifier.html

  1. Q x 1 Q x n \langle Qx_{1}\dots Qx_{n}\rangle
  2. Q H Q_{H}
  3. ( Q H x 1 , x 2 , y 1 , y 2 ) ϕ ( x 1 , x 2 , y 1 , y 2 ) ( x 1 y 1 x 2 y 2 ) ϕ ( x 1 , x 2 , y 1 , y 2 ) (Q_{H}x_{1},x_{2},y_{1},y_{2})\phi(x_{1},x_{2},y_{1},y_{2})\equiv\begin{% pmatrix}\forall x_{1}\exists y_{1}\\ \forall x_{2}\exists y_{2}\end{pmatrix}\phi(x_{1},x_{2},y_{1},y_{2})
  4. f g x 1 x 2 ϕ ( x 1 , x 2 , f ( x 1 ) , g ( x 2 ) ) \exists f\exists g\forall x_{1}\forall x_{2}\phi(x_{1},x_{2},f(x_{1}),g(x_{2}))
  5. Q Q_{\geq\mathbb{N}}
  6. ( Q x ) ϕ ( x ) a ( Q H x 1 , x 2 , y 1 , y 2 ) [ ϕ a ( x 1 = x 2 y 1 = y 2 ) ( ϕ ( x 1 ) ( ϕ ( y 1 ) y 1 a ) ) ] (Q_{\geq\mathbb{N}}x)\phi(x)\equiv\exists a(Q_{H}x_{1},x_{2},y_{1},y_{2})[\phi a% \land(x_{1}=x_{2}\leftrightarrow y_{1}=y_{2})\land(\phi(x_{1})\rightarrow(\phi% (y_{1})\land y_{1}\neq a))]
  7. Q H Q_{H}
  8. Σ 1 1 \Sigma_{1}^{1}
  9. Q H Q_{H}
  10. ( Q L x ) ( ϕ x , ψ x ) C a r d ( { x : ϕ x } ) C a r d ( { x : ψ x } ) ( Q H x 1 x 2 y 1 y 2 ) [ ( x 1 = x 2 y 1 = y 2 ) ( ϕ x 1 ψ y 1 ) ] (Q_{L}x)(\phi x,\psi x)\equiv Card(\{x\colon\phi x\})\leq Card(\{x\colon\psi x% \})\equiv(Q_{H}x_{1}x_{2}y_{1}y_{2})[(x_{1}=x_{2}\leftrightarrow y_{1}=y_{2})% \land(\phi x_{1}\rightarrow\psi y_{1})]
  11. ( Q I x ) ( ϕ x , ψ x ) ( Q L x ) ( ϕ x , ψ x ) ( Q L x ) ( ψ x , ϕ x ) (Q_{I}x)(\phi x,\psi x)\equiv(Q_{L}x)(\phi x,\psi x)\land(Q_{L}x)(\psi x,\phi x)
  12. ( Q C x ) ( ϕ x ) ( Q L x ) ( x = x , ϕ x ) (Q_{C}x)(\phi x)\equiv(Q_{L}x)(x=x,\phi x)
  13. Q H Q_{H}
  14. ( x 1 y 1 x 2 y 2 ) [ ( V ( x 1 ) T ( x 2 ) ) ( R ( x 1 , y 1 ) R ( x 2 , y 2 ) H ( y 1 , y 2 ) H ( y 2 , y 1 ) ) ] \begin{pmatrix}\forall x_{1}\exists y_{1}\\ \forall x_{2}\exists y_{2}\end{pmatrix}[(V(x_{1})\wedge T(x_{2}))\rightarrow(R% (x_{1},y_{1})\wedge R(x_{2},y_{2})\wedge H(y_{1},y_{2})\wedge H(y_{2},y_{1}))]
  15. Σ 1 1 \Sigma_{1}^{1}
  16. Π 1 1 \Pi_{1}^{1}
  17. [ x 1 y 1 x 2 y 2 ϕ ( x 1 , x 2 , y 1 , y 2 ) ] [ x 2 y 2 x 1 y 1 ϕ ( x 1 , x 2 , y 1 , y 2 ) ] [\forall x_{1}\exists y_{1}\forall x_{2}\exists y_{2}\,\phi(x_{1},x_{2},y_{1},% y_{2})]\wedge[\forall x_{2}\exists y_{2}\forall x_{1}\exists y_{1}\,\phi(x_{1}% ,x_{2},y_{1},y_{2})]
  18. ϕ ( x 1 , x 2 , y 1 , y 2 ) \phi(x_{1},x_{2},y_{1},y_{2})
  19. ( V ( x 1 ) T ( x 2 ) ) ( R ( x 1 , y 1 ) R ( x 2 , y 2 ) H ( y 1 , y 2 ) H ( y 2 , y 1 ) ) (V(x_{1})\wedge T(x_{2}))\rightarrow(R(x_{1},y_{1})\wedge R(x_{2},y_{2})\wedge H% (y_{1},y_{2})\wedge H(y_{2},y_{1}))

Brauer's_theorem_on_forms.html

  1. ( * ) a 1 x 1 r + + a n x n r = 0 , a i K , i = 1 , , n (*)\qquad a_{1}x_{1}^{r}+\cdots+a_{n}x_{n}^{r}=0,\quad a_{i}\in K,\quad i=1,% \ldots,n
  2. f 1 ( x 1 , , x n ) = = f k ( x 1 , , x n ) = 0 , ( x 1 , , x n ) M . f_{1}(x_{1},\ldots,x_{n})=\cdots=f_{k}(x_{1},\ldots,x_{n})=0,\quad\forall(x_{1% },\ldots,x_{n})\in M.
  3. p * / ( p * ) b \mathbb{Q}_{p}^{*}/\left(\mathbb{Q}_{p}^{*}\right)^{b}

Break_junction.html

  1. G Q = 2 e 2 / h G_{Q}=2e^{2}/h
  2. h h

Bregman_divergence.html

  1. F : Ω F:\Omega\to\mathbb{R}
  2. Ω \Omega
  3. p , q Ω p,q\in\Omega
  4. D F ( p , q ) = F ( p ) - F ( q ) - F ( q ) , p - q . D_{F}(p,q)=F(p)-F(q)-\langle\nabla F(q),p-q\rangle.
  5. D F ( p , q ) 0 D_{F}(p,q)\geq 0
  6. D F ( p , q ) D_{F}(p,q)
  7. F 1 , F 2 F_{1},F_{2}
  8. λ 0 \lambda\geq 0
  9. D F 1 + λ F 2 ( p , q ) = D F 1 ( p , q ) + λ D F 2 ( p , q ) D_{F_{1}+\lambda F_{2}}(p,q)=D_{F_{1}}(p,q)+\lambda D_{F_{2}}(p,q)
  10. F * F^{*}
  11. F * F^{*}
  12. D F ( p , q ) D_{F}(p,q)
  13. D F * ( p * , q * ) = D F ( q , p ) D_{F^{*}}(p^{*},q^{*})=D_{F}(q,p)
  14. p * = F ( p ) p^{*}=\nabla F(p)
  15. q * = F ( q ) q^{*}=\nabla F(q)
  16. D F ( x , y ) = x - y 2 D_{F}(x,y)=\|x-y\|^{2}
  17. F ( x ) = x 2 F(x)=\|x\|^{2}
  18. D F ( x , y ) = 1 2 ( x - y ) T Q ( x - y ) D_{F}(x,y)=\tfrac{1}{2}(x-y)^{T}Q(x-y)
  19. F ( x ) = 1 2 x T Q x F(x)=\tfrac{1}{2}x^{T}Qx
  20. D F ( p , q ) = p ( i ) log p ( i ) q ( i ) - p ( i ) + q ( i ) D_{F}(p,q)=\sum p(i)\log\frac{p(i)}{q(i)}-\sum p(i)+\sum q(i)
  21. F ( p ) = i p ( i ) log p ( i ) - p ( i ) F(p)=\sum_{i}p(i)\log p(i)-\sum p(i)
  22. D F ( p , q ) = i ( p ( i ) q ( i ) - log p ( i ) q ( i ) - 1 ) D_{F}(p,q)=\sum_{i}\left(\frac{p(i)}{q(i)}-\log\frac{p(i)}{q(i)}-1\right)
  23. F ( p ) = - log p ( i ) F(p)=-\sum\log p(i)
  24. p = ( p 1 , p d ) p=(p_{1},\ldots p_{d})
  25. x d + 1 = 1 d 2 p i x i x_{d+1}=\sum_{1}^{d}2p_{i}x_{i}
  26. p * = F ( p ) p^{*}=\nabla F(p)
  27. x d + 1 = x i 2 x_{d+1}=\sum x_{i}^{2}

Brewster_(unit).html

  1. 10 - 12 10^{-12}
  2. 10 - 13 10^{-13}

Brillouin_and_Langevin_functions.html

  1. B J ( x ) = 2 J + 1 2 J coth ( 2 J + 1 2 J x ) - 1 2 J coth ( 1 2 J x ) B_{J}(x)=\frac{2J+1}{2J}\coth\left(\frac{2J+1}{2J}x\right)-\frac{1}{2J}\coth% \left(\frac{1}{2J}x\right)
  2. x + x\to+\infty
  3. x - x\to-\infty
  4. M M
  5. B B
  6. M = N g μ B J B J ( x ) M=Ng\mu_{B}J\cdot B_{J}(x)
  7. N N
  8. g g
  9. μ B \mu_{B}
  10. x x
  11. k B T k_{B}T
  12. x = g μ B J B k B T x=\frac{g\mu_{B}JB}{k_{B}T}
  13. k B k_{B}
  14. T T
  15. B B
  16. B = μ 0 H B=\mu_{0}H
  17. H H
  18. μ 0 \mu_{0}
  19. E m = - m g μ B B = - k B T x m / J E_{m}=-mg\mu_{B}B=-k_{B}Txm/J
  20. P ( m ) = e - E m / ( k B T ) / Z = e x m / J / Z P(m)=e^{-E_{m}/(k_{B}T)}/Z=e^{xm/J}/Z
  21. P ( m ) = e x m / J / ( m = - J J e x m / J ) P(m)=e^{xm/J}/\left(\sum_{m^{\prime}=-J}^{J}e^{xm^{\prime}/J}\right)
  22. m = ( - J ) × P ( - J ) + + J × P ( J ) = ( m = - J J m e x m / J ) / ( m = - J J e x m / J ) \langle m\rangle=(-J)\times P(-J)+\cdots+J\times P(J)=\left(\sum_{m=-J}^{J}me^% {xm/J}\right)/\left(\sum_{m=-J}^{J}e^{xm/J}\right)
  23. m = J B J ( x ) \langle m\rangle=JB_{J}(x)
  24. M = N g μ B m = N g J μ B B J ( x ) M=Ng\mu_{B}\langle m\rangle=NgJ\mu_{B}B_{J}(x)
  25. tanh ( x / 3 ) \tanh(x/3)
  26. J J
  27. J J\to\infty
  28. L ( x ) = coth ( x ) - 1 x L(x)=\coth(x)-\frac{1}{x}
  29. x x
  30. L ( x ) = 1 3 x - 1 45 x 3 + 2 945 x 5 - 1 4725 x 7 + L(x)=\tfrac{1}{3}x-\tfrac{1}{45}x^{3}+\tfrac{2}{945}x^{5}-\tfrac{1}{4725}x^{7}+\dots
  31. t a n h ( x ) tanh(x)
  32. L ( x ) = x 3 + x 2 5 + x 2 7 + x 2 9 + L(x)=\frac{x}{3+\tfrac{x^{2}}{5+\tfrac{x^{2}}{7+\tfrac{x^{2}}{9+\ldots}}}}
  33. x x
  34. x x
  35. L - 1 ( x ) = 3 x + 9 5 x 3 + 297 175 x 5 + 1539 875 x 7 + L^{-1}(x)=3x+\tfrac{9}{5}x^{3}+\tfrac{297}{175}x^{5}+\tfrac{1539}{875}x^{7}+\dots
  36. L - 1 ( x ) = 3 x 35 - 12 x 2 35 - 33 x 2 + O ( x 7 ) . L^{-1}(x)=3x\frac{35-12x^{2}}{35-33x^{2}}+O(x^{7}).
  37. x x
  38. L - 1 ( x ) x 3 - x 2 1 - x 2 . L^{-1}(x)\approx x\frac{3-x^{2}}{1-x^{2}}.
  39. x = ± 0.8 x=±0.8
  40. L - 1 ( x ) x 3.0 - 2.6 x + 0.7 x 2 ( 1 - x ) ( 1 + 0.1 x ) , L^{-1}(x)\approx x\frac{3.0-2.6x+0.7x^{2}}{(1-x)(1+0.1x)},
  41. x 0 x≥0
  42. x 1 x\ll 1
  43. μ B B / k B T \mu_{B}B/k_{B}T
  44. M = C B T M=C\cdot\frac{B}{T}
  45. C = N g 2 J ( J + 1 ) μ B 2 3 k B C=\frac{Ng^{2}J(J+1)\mu_{B}^{2}}{3k_{B}}
  46. g J ( J + 1 ) g\sqrt{J(J+1)}
  47. x x\to\infty
  48. M = N g μ B J M=Ng\mu_{B}J

Bring_radical.html

  1. x 5 + x + a . x^{5}+x+a.\,
  2. x 5 + a 4 x 4 + a 3 x 3 + a 2 x 2 + a 1 x + a 0 = 0 x^{5}+a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}=0\,
  3. x 5 + c 2 x 2 + c 1 x + c 0 = 0 x^{5}+c_{2}x^{2}+c_{1}x+c_{0}=0\,
  4. y k = x k 2 + α x k + β y_{k}=x_{k}^{2}+\alpha x_{k}+\beta\,
  5. x 5 + d 1 x + d 0 = 0 x^{5}+d_{1}x+d_{0}=0\,
  6. z k = x k 4 + α x k 3 + β x k 2 + γ x k + δ z_{k}=x^{4}_{k}+\alpha x^{3}_{k}+\beta x^{2}_{k}+\gamma x_{k}+\delta\,
  7. x 5 + d 1 x + d 0 = 0 x^{5}+d_{1}x+d_{0}=0\,
  8. z = x - d 1 5 4 z={x\over\sqrt[4]{-\frac{d_{1}}{5}}}\,
  9. z 5 - 5 z - 4 t = 0 z^{5}-5z-4t=0\,
  10. t = - ( d 0 / 4 ) ( - d 1 / 5 ) - 5 / 4 t=-(d_{0}/4)(-d_{1}/5)^{-5/4}
  11. y 5 - y + a = 0 y^{5}-y+a=0\,
  12. x 5 - 10 C x 3 + 45 C 2 x - C 2 = 0 x^{5}-10Cx^{3}+45C^{2}x-C^{2}=0\,
  13. z k = λ + μ y k y k 2 C - 3 z_{k}=\frac{\lambda+\mu y_{k}}{\frac{y_{k}^{2}}{C}-3}\,
  14. λ \lambda\,
  15. μ \mu\,
  16. x 5 + x + a = 0 x^{5}+x+a=0\,
  17. x 5 + x = - a x^{5}+x=-a\,
  18. f ( x ) = x 5 + x f(x)=x^{5}+x\,
  19. x = f - 1 ( - a ) x=f^{-1}(-a)\,
  20. f - 1 f^{-1}\,
  21. f ( x ) f(x)\,
  22. x + x 5 x+x^{5}\,
  23. f - 1 ( a ) = k = 0 ( 5 k k ) ( - 1 ) k a 4 k + 1 4 k + 1 = a - a 5 + 5 a 9 - 35 a 13 + f^{-1}(a)=\sum_{k=0}^{\infty}{\left({{5k}\atop{k}}\right)}\frac{(-1)^{k}a^{4k+% 1}}{4k+1}=a-a^{5}+5a^{9}-35a^{13}+...
  24. f - 1 ( a ) f^{-1}(a)\,
  25. B R ( a ) = f - 1 ( - a ) = - f - 1 ( a ) = - a + a 5 - 5 a 9 + 35 a 13 + BR(a)=f^{-1}(-a)=-f^{-1}(a)=-a+a^{5}-5a^{9}+35a^{13}+...\,
  26. | a | < 4 / ( 5 5 4 ) 0.53499 |a|<4/(5\cdot\sqrt[4]{5})\approx 0.53499
  27. B R ( a ) = - a 4 F 3 ( 1 5 , 2 5 , 3 5 , 4 5 ; 1 2 , 3 4 , 5 4 ; - 5 ( 5 a 4 ) 4 ) BR(a)=-a\,\,_{4}F_{3}\left(\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5};% \frac{1}{2},\frac{3}{4},\frac{5}{4};-5\left(\frac{5a}{4}\right)^{4}\right)
  28. x 5 + p x + q x^{5}+px+q\,
  29. - p 5 4 BR ( - 1 4 ( - 5 p ) 5 4 q ) \sqrt[4]{-\frac{p}{5}}\operatorname{BR}\left(-\frac{1}{4}\left(-\frac{5}{p}% \right)^{\frac{5}{4}}q\right)
  30. x 5 - x + a = 0 x^{5}-x+a=0\,
  31. K K\,
  32. K K^{\prime}\,
  33. K = 0 π 2 d φ 1 - k 2 sin 2 φ K=\int_{0}^{\frac{\pi}{2}}\frac{d\varphi}{\sqrt{1-k^{2}\sin^{2}\varphi}}
  34. K = 0 π 2 d φ 1 - k 2 sin 2 φ K^{\prime}=\int_{0}^{\frac{\pi}{2}}\frac{d\varphi}{\sqrt{1-k^{\prime 2}\sin^{2% }\varphi}}
  35. q = e - π K K q=e^{-\frac{\pi K^{\prime}}{K}}\,
  36. k 2 + k 2 = 1 k^{2}+k^{\prime 2}=1\,
  37. q = e - π K K = e i π τ q=e^{-\frac{\pi K^{\prime}}{K}}=e^{{\mathrm{i}}\pi\tau}\,
  38. k 4 = φ ( τ ) = ϑ 10 ( 0 ; τ ) ϑ 00 ( 0 ; τ ) \sqrt[4]{k}=\varphi(\tau)=\sqrt{\frac{\vartheta_{10}(0;\tau)}{\vartheta_{00}(0% ;\tau)}}
  39. k 4 = ψ ( τ ) = ϑ 01 ( 0 ; τ ) ϑ 00 ( 0 ; τ ) \sqrt[4]{k^{\prime}}=\psi(\tau)=\sqrt{\frac{\vartheta_{01}(0;\tau)}{\vartheta_% {00}(0;\tau)}}
  40. ϑ 00 ( 0 ; τ ) \vartheta_{00}(0;\tau)
  41. v = φ ( n τ ) v=\varphi(n\tau)\,
  42. u = φ ( τ ) u=\varphi(\tau)\,
  43. u u\,
  44. v v\,
  45. ϵ φ ( n τ ) \epsilon\varphi(n\tau)\,
  46. φ ( τ + 16 m n ) \varphi\left(\frac{\tau+16m}{n}\right)\,
  47. u 6 - v 6 + 5 u 2 v 2 ( u 2 - v 2 ) + 4 u v ( 1 - u 4 v 4 ) = 0 u^{6}-v^{6}+5u^{2}v^{2}(u^{2}-v^{2})+4uv(1-u^{4}v^{4})=0\,
  48. Φ ( τ ) = [ φ ( 5 τ ) + φ ( τ 5 ) ] [ φ ( τ + 16 5 ) - φ ( τ + 64 5 ) ] [ φ ( τ + 32 5 ) - φ ( τ + 48 5 ) ] \Phi(\tau)=\left[\varphi(5\tau)+\varphi\left(\frac{\tau}{5}\right)\right]\left% [\varphi\left(\frac{\tau+16}{5}\right)-\varphi\left(\frac{\tau+64}{5}\right)% \right]\left[\varphi\left(\frac{\tau+32}{5}\right)-\varphi\left(\frac{\tau+48}% {5}\right)\right]\,
  49. Φ ( τ ) \Phi(\tau)\,
  50. Φ ( τ + 16 ) \Phi(\tau+16)\,
  51. Φ ( τ + 32 ) \Phi(\tau+32)\,
  52. Φ ( τ + 48 ) \Phi(\tau+48)\,
  53. Φ ( τ + 64 ) \Phi(\tau+64)\,
  54. φ ( τ ) \varphi(\tau)\,
  55. Φ 5 - 2000 φ 4 ( τ ) ψ 16 ( τ ) Φ - 1600 5 φ 3 ( τ ) ψ 16 ( τ ) [ 1 + φ 8 ( τ ) ] = 0 \Phi^{5}-2000\varphi^{4}(\tau)\psi^{16}(\tau)\Phi-1600\sqrt{5}\varphi^{3}(\tau% )\psi^{16}(\tau)\left[1+\varphi^{8}(\tau)\right]=0\,
  56. Φ = 2 125 4 φ ( τ ) ψ 4 ( τ ) x \Phi=2\sqrt[4]{125}\varphi(\tau)\psi^{4}(\tau)x\,
  57. x 5 - x + a = 0 x^{5}-x+a=0\,
  58. a = 2 [ 1 + φ 8 ( τ ) ] 5 5 4 φ 2 ( τ ) ψ 4 ( τ ) a=\frac{2[1+\varphi^{8}(\tau)]}{\sqrt[4]{5^{5}}\varphi^{2}(\tau)\psi^{4}(\tau)}\,
  59. A = a 5 5 4 2 A=\frac{a\sqrt[4]{5^{5}}}{2}
  60. k k
  61. k 4 + A 2 k 3 + 2 k 2 - A 2 k + 1 = 0 k^{4}+A^{2}k^{3}+2k^{2}-A^{2}k+1=0\,
  62. k = tan α 4 , tan α + 2 π 4 , tan π - α 4 , tan 3 π - α 4 k=\tan\frac{\alpha}{4},\tan\frac{\alpha+2\pi}{4},\tan\frac{\pi-\alpha}{4},\tan% \frac{3\pi-\alpha}{4}\,
  63. sin α = 4 A 2 \sin\alpha=\frac{4}{A^{2}}\,
  64. sin α = 1 4 A 2 \sin\alpha=\frac{1}{4A^{2}}\,
  65. τ \tau
  66. k k\,
  67. x i = Φ ( τ + 16 i ) 2 125 4 φ ( τ ) ψ 4 ( τ ) x_{i}=\frac{\Phi(\tau+16i)}{2\sqrt[4]{125}\varphi(\tau)\psi^{4}(\tau)}
  68. i = 0 , , 4 i=0,\ldots,4
  69. x n = exp ( 1 n ln x ) \sqrt[n]{x}=\exp\left({\frac{1}{n}\ln x}\right)
  70. x n = exp ( 1 n 1 x d t t ) . \sqrt[n]{x}=\exp\left(\frac{1}{n}\int^{x}_{1}\frac{dt}{t}\right).
  71. 1 x d t t \int^{x}_{1}\frac{dt}{t}
  72. x N - x + t = 0 x^{N}-x+t=0\,\!
  73. x = ζ - 1 N - 1 x=\zeta^{-\frac{1}{N-1}}\,
  74. ζ = e 2 π i + t ϕ ( ζ ) \zeta=e^{2\pi i}+t\phi(\zeta)\,\!
  75. ϕ ( ζ ) = ζ N N - 1 \phi(\zeta)=\zeta^{\frac{N}{N-1}}\,\!
  76. f f\,
  77. ζ \zeta\,
  78. f ( ζ ) = f ( e 2 π i ) + n = 1 t n n ! d n - 1 d a n - 1 [ f ( a ) | ϕ ( a ) | n ] a = e 2 π i f(\zeta)=f(e^{2\pi{\mathrm{i}}})+\sum^{\infty}_{n=1}\frac{t^{n}}{n!}\frac{d^{n% -1}}{da^{n-1}}[f^{\prime}(a)|\phi(a)|^{n}]_{a=e^{2\pi{\mathrm{i}}}}
  79. f ( ζ ) = ζ - 1 N - 1 f(\zeta)=\zeta^{-\frac{1}{N-1}}\,
  80. x k = e - 2 k π i N - 1 - t N - 1 n = 0 ( t e 2 k π i N - 1 ) n Γ ( n + 2 ) Γ ( N n N - 1 + 1 ) Γ ( n N - 1 + 1 ) x_{k}=e^{-\frac{2k\pi{\rm{i}}}{N-1}}-\frac{t}{N-1}\sum^{\infty}_{n=0}\frac{(te% ^{\frac{2k\pi{\rm{i}}}{N-1}})^{n}}{\Gamma(n+2)}\cdot\frac{\Gamma\left(\frac{Nn% }{N-1}+1\right)}{\Gamma\left(\frac{n}{N-1}+1\right)}
  81. k = 1 , 2 , 3 , , N - 1 k=1,2,3,\dots,N-1\,
  82. ψ n ( q ) = ( e 2 n π i N - 1 t N - 1 ) q N q N N - 1 k = 0 N - 1 Γ ( q N - 1 + 1 + k N ) Γ ( q N - 1 + 1 ) k = 0 N - 2 Γ ( q + k + 2 N - 1 ) = = ( t e 2 n π i N - 1 N - 1 ) q N q N N - 1 k = = 2 N Γ ( q N - 1 + k - 1 N ) Γ ( q + k N - 1 ) \psi_{n}(q)=\left(\frac{e^{\frac{2n\pi{\rm{i}}}{N-1}}t}{N-1}\right)^{q}N^{% \frac{qN}{N-1}}\frac{\prod_{k=0}^{N-1}\Gamma\left(\frac{q}{N-1}+\frac{1+k}{N}% \right)}{\Gamma\left(\frac{q}{N-1}+1\right)\prod^{N-2}_{k=0}\Gamma\left(\frac{% q+k+2}{N-1}\right)}==\left(\frac{te^{\frac{2n\pi{\rm{i}}}{N-1}}}{N-1}\right)^{% q}N^{\frac{qN}{N-1}}\prod_{k==2}^{N}\frac{\Gamma\left(\frac{q}{N-1}+\frac{k-1}% {N}\right)}{\Gamma\left(\frac{q+k}{N-1}\right)}
  83. x n = e - 2 n π i N - 1 - t ( N - 1 ) 2 N 2 π ( N - 1 ) q = 0 N - 2 ψ n ( q ) ( N + 1 ) F N [ q N + N - 1 N ( N - 1 ) , , q + N - 1 N - 1 , 1 ; q + 2 N - 1 , , q + N N - 1 , q + N - 1 N - 1 ; ( t e 2 n π i N - 1 N - 1 ) N - 1 N N ] , n = 1 , 2 , 3 , , N - 1 x_{n}=e^{-\frac{2n\pi{\rm{i}}}{N-1}}-\frac{t}{(N-1)^{2}}\sqrt{\frac{N}{2\pi(N-% 1)}}\sum^{N-2}_{q=0}\psi_{n}(q)_{(N+1)}F_{N}\begin{bmatrix}\frac{qN+N-1}{N(N-1% )},\ldots,\frac{q+N-1}{N-1},1;\\ \frac{q+2}{N-1},\ldots,\frac{q+N}{N-1},\frac{q+N-1}{N-1};\\ \left(\frac{te^{\frac{2n\pi{\rm{i}}}{N-1}}}{N-1}\right)^{N-1}N^{N}\end{bmatrix% },\quad n=1,2,3,\dots,N-1
  84. x N = m = 1 N - 1 t ( N - 1 ) 2 N 2 π ( N - 1 ) q = 0 N - 2 ψ m ( q ) ( N + 1 ) F N [ q N + N - 1 N ( N - 1 ) , , q + N - 1 N - 1 , 1 ; q + 2 N - 1 , , q + N N - 1 , q + N - 1 N - 1 ; ( t e 2 m π i N - 1 N - 1 ) N - 1 N N ] x_{N}=\sum_{m=1}^{N-1}\frac{t}{(N-1)^{2}}\sqrt{\frac{N}{2\pi(N-1)}}\sum^{N-2}_% {q=0}\psi_{m}(q)_{(N+1)}F_{N}\begin{bmatrix}\frac{qN+N-1}{N(N-1)},\ldots,\frac% {q+N-1}{N-1},1;\\ \frac{q+2}{N-1},\ldots,\frac{q+N}{N-1},\frac{q+N-1}{N-1};\\ \left(\frac{te^{\frac{2m\pi{\rm{i}}}{N-1}}}{N-1}\right)^{N-1}N^{N}\end{bmatrix}
  85. a x N + b x 2 + c = 0 , N 1 ( mod 2 ) {}_{ax^{N}+bx^{2}+c=0,N\equiv 1\;\;(\mathop{{\rm mod}}2)}\,\!
  86. x N = - a 2 b ( c b ) N - 1 F N - 2 N - 1 [ N + 1 2 N , N + 3 2 N , , N - 2 N , N - 1 N , N + 1 N , N + 2 N , , 3 N - 3 2 N , 3 N - 1 2 N ; N + 1 2 N - 4 , N + 3 2 N - 4 , , N - 4 N - 2 , N - 3 N - 2 , N - 1 N - 2 , N N - 2 , , 3 N - 5 2 N - 4 , 3 2 ; - a 2 c N - 2 4 b N ( N - 2 ) N - 2 ] + c b i F N - 2 N - 1 [ 1 2 N , 3 2 N , , N - 4 2 N , N - 2 2 N , N + 2 2 N , N + 4 2 N , , 2 N - 3 2 N , 2 N - 1 2 N ; 3 2 N - 4 , 5 2 N - 4 , , 2 N - 3 2 N - 4 ; - a 2 c N - 2 4 b N ( N - 2 ) N - 2 ] {}_{x_{N}=-\frac{a}{2b}\sqrt{\left(\frac{c}{b}\right)^{N-1}}{}_{N-1}F_{N-2}% \begin{bmatrix}\frac{N+1}{2N},\frac{N+3}{2N},\cdots,\frac{N-2}{N},\frac{N-1}{N% },\frac{N+1}{N},\frac{N+2}{N},\cdots,\frac{3N-3}{2N},\frac{3N-1}{2N};\\ \frac{N+1}{2N-4},\frac{N+3}{2N-4},\cdots,\frac{N-4}{N-2},\frac{N-3}{N-2},\frac% {N-1}{N-2},\frac{N}{N-2},\cdots,\frac{3N-5}{2N-4},\frac{3}{2};\\ -\frac{a^{2}c^{N-2}}{4b^{N}\left(N-2\right)^{N-2}}\end{bmatrix}+\sqrt{\frac{c}% {b}}{\rm{i}}{}_{N-1}F_{N-2}\begin{bmatrix}\frac{1}{2N},\frac{3}{2N},\cdots,% \frac{N-4}{2N},\frac{N-2}{2N},\frac{N+2}{2N},\frac{N+4}{2N},\cdots,\frac{2N-3}% {2N},\frac{2N-1}{2N};\\ \frac{3}{2N-4},\frac{5}{2N-4},\cdots,\frac{2N-3}{2N-4};\\ -\frac{a^{2}c^{N-2}}{4b^{N}\left(N-2\right)^{N-2}}\end{bmatrix}}
  87. x N - 1 = - a 2 b ( c b ) N - 1 F N - 2 N - 1 [ N + 1 2 N , N + 3 2 N , , N - 2 N , N - 1 N , N + 1 N , N + 2 N , , 3 N - 3 2 N , 3 N - 1 2 N ; N + 1 2 N - 4 , N + 3 2 N - 4 , , N - 4 N - 2 , N - 3 N - 2 , N - 1 N - 2 , N N - 2 , , 3 N - 5 2 N - 4 , 3 2 ; - a 2 c N - 2 4 b N ( N - 2 ) N - 2 ] - c b i F N - 2 N - 1 [ 1 2 N , 3 2 N , , N - 4 2 N , N - 2 2 N , N + 2 2 N , N + 4 2 N , , 2 N - 3 2 N , 2 N - 1 2 N ; 3 2 N - 4 , 5 2 N - 4 , , 2 N - 3 2 N - 4 ; - a 2 c N - 2 4 b N ( N - 2 ) N - 2 ] {}_{x_{N-1}=-\frac{a}{2b}\sqrt{\left(\frac{c}{b}\right)^{N-1}}{}_{N-1}F_{N-2}% \begin{bmatrix}\frac{N+1}{2N},\frac{N+3}{2N},\cdots,\frac{N-2}{N},\frac{N-1}{N% },\frac{N+1}{N},\frac{N+2}{N},\cdots,\frac{3N-3}{2N},\frac{3N-1}{2N};\\ \frac{N+1}{2N-4},\frac{N+3}{2N-4},\cdots,\frac{N-4}{N-2},\frac{N-3}{N-2},\frac% {N-1}{N-2},\frac{N}{N-2},\cdots,\frac{3N-5}{2N-4},\frac{3}{2};\\ -\frac{a^{2}c^{N-2}}{4b^{N}\left(N-2\right)^{N-2}}\end{bmatrix}-\sqrt{\frac{c}% {b}}{\rm{i}}{}_{N-1}F_{N-2}\begin{bmatrix}\frac{1}{2N},\frac{3}{2N},\cdots,% \frac{N-4}{2N},\frac{N-2}{2N},\frac{N+2}{2N},\frac{N+4}{2N},\cdots,\frac{2N-3}% {2N},\frac{2N-1}{2N};\\ \frac{3}{2N-4},\frac{5}{2N-4},\cdots,\frac{2N-3}{2N-4};\\ -\frac{a^{2}c^{N-2}}{4b^{N}\left(N-2\right)^{N-2}}\end{bmatrix}}
  88. x n = - e 2 n π i N - 2 b a N - 2 F N - 2 N - 1 [ - 1 N ( N - 2 ) , - 1 N ( N - 2 ) + 1 N , - 1 N ( N - 2 ) + 2 N , , - 1 N ( N - 2 ) + 1 N , N - 5 2 N , - 1 N ( N - 2 ) + N - 3 2 N , - 1 N ( N - 2 ) + N + 1 2 N , - 1 N ( N - 2 ) + N + 3 2 N , , - 1 N ( N - 2 ) + N - 1 N , ; 1 N - 2 , 2 N - 2 , , 2 N - 5 2 N - 4 , ; - a 2 c N - 2 4 b N ( N - 2 ) N - 2 ] + {}_{x_{n}=-e^{\frac{2n\pi{\rm{i}}}{N-2}}\sqrt[N-2]{\frac{b}{a}}{}_{N-1}F_{N-2}% \begin{bmatrix}-\frac{1}{N\left(N-2\right)},-\frac{1}{N\left(N-2\right)}+\frac% {1}{N},-\frac{1}{N\left(N-2\right)}+\frac{2}{N},\cdots,-\frac{1}{N\left(N-2% \right)}+\frac{1}{N},\frac{N-5}{2N},-\frac{1}{N\left(N-2\right)}+\frac{N-3}{2N% },-\frac{1}{N\left(N-2\right)}+\frac{N+1}{2N},-\frac{1}{N\left(N-2\right)}+% \frac{N+3}{2N},\cdots,-\frac{1}{N\left(N-2\right)}+\frac{N-1}{N},;\\ \frac{1}{N-2},\frac{2}{N-2},\cdots,\frac{2N-5}{2N-4},;\\ -\frac{a^{2}c^{N-2}}{4b^{N}\left(N-2\right)^{N-2}}\end{bmatrix}+}
  89. + b a N - 2 q = 1 N - 3 Γ ( 2 q - 1 N - 2 + q ) Γ ( 2 q - 1 N - 2 + 1 ) ( - c b a 2 b 2 N - 2 ) q e 2 n ( 1 - 2 q ) N - 2 π i q ! F N - 2 N - 1 [ N q - 1 N ( N - 2 ) , N q - 1 N ( N - 2 ) + 1 N , N q - 1 N ( N - 2 ) + 2 N , , N q - 1 N ( N - 2 ) + N - 3 2 N , N q - 1 N ( N - 2 ) + N + 1 2 N , , N q - 1 N ( N - 2 ) + N - 1 N ; q + 1 N - 2 , q + 2 N - 2 , , N - 4 N - 2 , N - 3 N - 2 , N - 1 N - 2 , N N - 2 , , q + N - 2 N - 2 , 2 q + 2 N - 5 2 N - 4 ; - a 2 c N - 2 4 b N ( N - 2 ) N - 2 ] , n = 1 , 2 , , N - 2 {}_{+\sqrt[N-2]{\frac{b}{a}}\sum^{N-3}_{q=1}\frac{\Gamma\left(\frac{2q-1}{N-2}% +q\right)}{\Gamma\left(\frac{2q-1}{N-2}+1\right)}\cdot\left(-\frac{c}{b}\sqrt[% N-2]{\frac{a^{2}}{b^{2}}}\right)^{q}\cdot\frac{e^{\frac{2n\left(1-2q\right)}{N% -2}\pi{\rm{i}}}}{q!}{}_{N-1}F_{N-2}\begin{bmatrix}\frac{Nq-1}{N\left(N-2\right% )},\frac{Nq-1}{N\left(N-2\right)}+\frac{1}{N},\frac{Nq-1}{N\left(N-2\right)}+% \frac{2}{N},\cdots,\frac{Nq-1}{N\left(N-2\right)}+\frac{N-3}{2N},\frac{Nq-1}{N% \left(N-2\right)}+\frac{N+1}{2N},\cdots,\frac{Nq-1}{N\left(N-2\right)}+\frac{N% -1}{N};\\ \frac{q+1}{N-2},\frac{q+2}{N-2},\cdots,\frac{N-4}{N-2},\frac{N-3}{N-2},\frac{N% -1}{N-2},\frac{N}{N-2},\cdots,\frac{q+N-2}{N-2},\frac{2q+2N-5}{2N-4};\\ -\frac{a^{2}c^{N-2}}{4b^{N}\left(N-2\right)^{N-2}}\end{bmatrix},n=1,2,\cdots,N% -2}
  90. F 1 ( t ) \displaystyle F_{1}(t)
  91. x 1 = - t F 2 ( t ) x 2 = - F 1 ( t ) + 1 4 t F 2 ( t ) + 5 32 t 2 F 3 ( t ) + 5 32 t 3 F 4 ( t ) x 3 = F 1 ( t ) + 1 4 t F 2 ( t ) + 5 32 t 2 F 3 ( t ) + 5 32 t 3 F 4 ( t ) x 4 = - i F 1 ( t ) + 1 4 t F 2 ( t ) - 5 32 i t 2 F 3 ( t ) - 5 32 t 3 F 4 ( t ) x 5 = i F 1 ( t ) + 1 4 t F 2 ( t ) - 5 32 i t 2 F 3 ( t ) - 5 32 t 3 F 4 ( t ) \begin{array}[]{rcrcccccc}x_{1}&=&{}-tF_{2}(t)\\ x_{2}&=&{}-F_{1}(t)&+&\frac{1}{4}tF_{2}(t)&+&\frac{5}{32}t^{2}F_{3}(t)&+&\frac% {5}{32}t^{3}F_{4}(t)\\ x_{3}&=&F_{1}(t)&+&\frac{1}{4}tF_{2}(t)&+&\frac{5}{32}t^{2}F_{3}(t)&+&\frac{5}% {32}t^{3}F_{4}(t)\\ x_{4}&=&{}-{\mathrm{i}}F_{1}(t)&+&\frac{1}{4}tF_{2}(t)&-&\frac{5}{32}{\mathrm{% i}}t^{2}F_{3}(t)&-&\frac{5}{32}t^{3}F_{4}(t)\\ x_{5}&=&{\mathrm{i}}F_{1}(t)&+&\frac{1}{4}tF_{2}(t)&-&\frac{5}{32}{\mathrm{i}}% t^{2}F_{3}(t)&-&\frac{5}{32}t^{3}F_{4}(t)\end{array}
  92. f ( x ) = x 5 - x + a f(x)=x^{5}-x+a\,
  93. ϕ ( a ) \phi(a)\,
  94. f [ ϕ ( a ) ] = 0 f[\phi(a)]=0\,
  95. ϕ \phi\,
  96. d f [ ϕ ( a ) ] d a = 0 \displaystyle\frac{df[\phi(a)]}{da}=0
  97. ( 256 - 3125 a 4 ) 1155 d 4 ϕ d a 4 - 6250 a 3 231 d 3 ϕ d a 3 - 4875 a 2 77 d 2 ϕ d a 2 - 2125 a 77 d ϕ d a + ϕ = 0 \frac{(256-3125a^{4})}{1155}\frac{d^{4}\phi}{da^{4}}-\frac{6250a^{3}}{231}% \frac{d^{3}\phi}{da^{3}}-\frac{4875a^{2}}{77}\frac{d^{2}\phi}{da^{2}}-\frac{21% 25a}{77}\frac{d\phi}{da}+\phi=0
  98. x 5 - 10 C x 3 + 45 C 2 x - C 2 = 0. x^{5}-10Cx^{3}+45C^{2}x-C^{2}=0.\,
  99. Z = 1 - 1728 C Z=1-1728C\,
  100. T Z ( w ) = w - 12 g ( Z , w ) g ( Z , w ) T_{Z}(w)=w-12\frac{g(Z,w)}{g^{\prime}(Z,w)}\,
  101. g ( Z , w ) g(Z,w)\,
  102. g g^{\prime}\,
  103. g ( Z , w ) g(Z,w)\,
  104. w w\,
  105. T Z [ T Z ( w ) ] T_{Z}[T_{Z}(w)]\,
  106. w 1 w_{1}\,
  107. w 2 = T Z ( w 1 ) w_{2}=T_{Z}(w_{1})\,
  108. μ i = 100 Z ( Z - 1 ) h ( Z , w i ) g ( Z , w i ) \mu_{i}=\frac{100Z(Z-1)h(Z,w_{i})}{g(Z,w_{i})}\,
  109. h ( Z , w ) h(Z,w)\,
  110. w 1 w_{1}\,
  111. w 2 = T Z ( w 1 ) w_{2}=T_{Z}(w_{1})\,
  112. x i = ( 9 + 15 i ) μ i + ( 9 - 15 i ) μ 3 - i 90 x_{i}=\frac{(9+\sqrt{15}{\mathrm{i}})\mu_{i}+(9-\sqrt{15}{\mathrm{i}})\mu_{3-i% }}{90}
  113. g ( Z , w ) g(Z,w)\,
  114. h ( Z , w ) h(Z,w)\,
  115. g ( Z , w ) \displaystyle g(Z,w)

Broken_diagonal.html

  1. 3 + 12 + 14 + 5 = 34 3+12+14+5=34
  2. 10 + 1 + 7 + 16 = 34 10+1+7+16=34
  3. 10 + 13 + 7 + 4 = 34 10+13+7+4=34

Buck_converter.html

  1. I L I_{L}
  2. V L = V i - V o V_{L}=V_{i}-V_{o}
  3. V L = - V o V_{L}=-V_{o}
  4. I L I_{L}
  5. E = 1 2 L I L 2 E=\frac{1}{2}L\cdot I_{L}^{2}
  6. I L I_{L}
  7. I L I_{L}
  8. V L = L d I L d t V_{L}=L\frac{dI_{L}}{dt}
  9. V L V_{L}
  10. V i - V o V_{i}-V_{o}
  11. - V o -V_{o}
  12. Δ I L 𝑜𝑛 = 0 t 𝑜𝑛 V L L d t = ( V i - V o ) L t 𝑜𝑛 , t 𝑜𝑛 = D T \Delta I_{L_{\mathit{on}}}=\int_{0}^{t_{\mathit{on}}}\frac{V_{L}}{L}\,dt=\frac% {\left(V_{i}-V_{o}\right)}{L}t_{\mathit{on}},\;t_{\mathit{on}}=DT
  13. Δ I L 𝑜𝑓𝑓 = t 𝑜𝑛 T = t 𝑜𝑛 + t 𝑜𝑓𝑓 V L L d t = - V o L t 𝑜𝑓𝑓 , t 𝑜𝑓𝑓 = ( 1 - D ) T \Delta I_{L_{\mathit{off}}}=\int_{t_{\mathit{on}}}^{T=t_{\mathit{on}}+t_{% \mathit{off}}}\frac{V_{L}}{L}\,dt=-\frac{V_{o}}{L}t_{\mathit{off}},\;t_{% \mathit{off}}=(1-D)T
  14. I L I_{L}
  15. t = 0 t=0
  16. t = T t=T
  17. V i - V o L t 𝑜𝑛 - V o L t 𝑜𝑓𝑓 = 0 \frac{V_{i}-V_{o}}{L}t_{\mathit{on}}-\frac{V_{o}}{L}t_{\mathit{off}}=0
  18. Δ I L 𝑜𝑛 \Delta I_{L_{\mathit{on}}}
  19. Δ I L 𝑜𝑓𝑓 \Delta I_{L_{\mathit{off}}}
  20. ( V i - V o ) t 𝑜𝑛 \left(V_{i}-V_{o}\right)t_{\mathit{on}}
  21. - V o t 𝑜𝑓𝑓 -V_{o}t_{\mathit{off}}
  22. t o n = D T t_{on}=DT
  23. t o f f = ( 1 - D ) T t_{off}=(1-D)T
  24. D D
  25. ( V i - V o ) D T - V o ( 1 - D ) T = 0 \displaystyle(V_{i}-V_{o})DT-V_{o}(1-D)T=0
  26. D D
  27. t o n t_{o}n
  28. T T
  29. V o V i V_{o}\leq V_{i}
  30. ( V i - V o ) D T - V o δ T = 0 \left(V_{i}-V_{o}\right)DT-V_{o}\delta T=0
  31. δ = V i - V o V o D \delta=\frac{V_{i}-V_{o}}{V_{o}}D
  32. I o I_{o}
  33. I L ¯ = I o \bar{I_{L}}=I_{o}
  34. I L ¯ \bar{I_{L}}
  35. I L ¯ \displaystyle\bar{I_{L}}
  36. I L M a x = V i - V o L D T I_{L_{Max}}=\frac{V_{i}-V_{o}}{L}DT
  37. I o = ( V i - V o ) D T ( D + δ ) 2 L I_{o}=\frac{\left(V_{i}-V_{o}\right)DT\left(D+\delta\right)}{2L}
  38. I o = ( V i - V o ) D T ( D + V i - V o V o D ) 2 L I_{o}=\frac{\left(V_{i}-V_{o}\right)DT\left(D+\frac{V_{i}-V_{o}}{V_{o}}D\right% )}{2L}
  39. V o = V i 1 2 L I o D 2 V i T + 1 V_{o}=V_{i}\frac{1}{\frac{2LI_{o}}{D^{2}V_{i}T}+1}
  40. D T + δ T = T \displaystyle DT+\delta T=T
  41. I o l i m = I L m a x 2 ( D + δ ) = I L m a x 2 I_{o_{lim}}=\frac{I_{L_{max}}}{2}\left(D+\delta\right)=\frac{I_{L_{max}}}{2}
  42. I o l i m = V i - V o 2 L D T I_{o_{lim}}=\frac{V_{i}-V_{o}}{2L}DT
  43. V o = D V i V_{o}=DV_{i}
  44. I o l i m = V i ( 1 - D ) 2 L D T I_{o_{lim}}=\frac{V_{i}\left(1-D\right)}{2L}DT
  45. | V o | = V o V i \left|V_{o}\right|=\frac{V_{o}}{V_{i}}
  46. V o = 0 V_{o}=0
  47. V o = V i V_{o}=V_{i}
  48. | I o | = L T V i I o \left|I_{o}\right|=\frac{L}{TV_{i}}I_{o}
  49. T V i L \frac{TV_{i}}{L}
  50. | I o | \left|I_{o}\right|
  51. | V o | = D \left|V_{o}\right|=D
  52. | V o | = 1 2 L I o D 2 V i T + 1 = 1 2 | I o | D 2 + 1 = D 2 2 | I o | + D 2 \begin{aligned}\displaystyle\left|V_{o}\right|&\displaystyle=\frac{1}{\frac{2% LI_{o}}{D^{2}V_{i}T}+1}\\ &\displaystyle=\frac{1}{\frac{2\left|I_{o}\right|}{D^{2}}+1}\\ &\displaystyle=\frac{D^{2}}{2\left|I_{o}\right|+D^{2}}\end{aligned}
  53. I o l i m \displaystyle I_{o_{lim}}
  54. ( 1 - D ) D 2 | I o | = 1 \frac{\left(1-D\right)D}{2\left|I_{o}\right|}=1
  55. d V o = i d T C dV_{o}=\frac{idT}{C}
  56. d T o n = D T = D f dT_{on}=DT=\frac{D}{f}
  57. d T o f f = ( 1 - D ) T = 1 - D f dT_{off}=(1-D)T=\frac{1-D}{f}
  58. I 2 R I^{2}R
  59. D = V o + ( V 𝑆𝑌𝑁𝐶𝑆𝑊 + V L ) V i - V 𝑆𝑊𝐼𝑇𝐶𝐻 + V 𝑆𝑌𝑁𝐶𝑆𝑊 D=\frac{V_{o}+(V_{\mathit{SYNCSW}}+V_{L})}{V_{i}-V_{\mathit{SWITCH}}+V_{% \mathit{SYNCSW}}}
  60. V 𝑆𝑊𝐼𝑇𝐶𝐻 = I 𝑆𝑊𝐼𝑇𝐶𝐻 R 𝑜𝑛 = D I o R 𝑜𝑛 V_{\mathit{SWITCH}}=I_{\mathit{SWITCH}}R_{\mathit{on}}=DI_{o}R_{\mathit{on}}
  61. V 𝑆𝑌𝑁𝐶𝑆𝑊 = I 𝑆𝑌𝑁𝐶𝑆𝑊 R 𝑜𝑛 = ( 1 - D ) I o R 𝑜𝑛 V_{\mathit{SYNCSW}}=I_{\mathit{SYNCSW}}R_{\mathit{on}}=(1-D)I_{o}R_{\mathit{on}}
  62. V L = I L R 𝐷𝐶𝑅 V_{L}=I_{L}R_{\mathit{DCR}}
  63. P 𝑙𝑒𝑎𝑘𝑎𝑔𝑒 = I 𝑙𝑒𝑎𝑘𝑎𝑔𝑒 V P_{\mathit{leakage}}=I_{\mathit{leakage}}V
  64. P 𝑆𝑊 = V I o ( t 𝑟𝑖𝑠𝑒 + t 𝑓𝑎𝑙𝑙 ) 6 T P_{\mathit{SW}}=\frac{VI_{o}(t_{\mathit{rise}}+t_{\mathit{fall}})}{6T}
  65. P 𝑆𝑊 = V I o ( t 𝑟𝑖𝑠𝑒 + t 𝑓𝑎𝑙𝑙 ) 2 T P_{\mathit{SW}}=\frac{VI_{o}(t_{\mathit{rise}}+t_{\mathit{fall}})}{2T}
  66. P 𝐵𝑂𝐷𝑌𝐷𝐼𝑂𝐷𝐸 = V F I o t 𝑛𝑜 f 𝑆𝑊 P_{\mathit{BODYDIODE}}=V_{F}I_{o}t_{\mathit{no}}f_{\mathit{SW}}
  67. P 𝐺𝐴𝑇𝐸𝐷𝑅𝐼𝑉𝐸 = Q G V 𝐺𝑆 f 𝑆𝑊 P_{\mathit{GATEDRIVE}}=Q_{G}V_{\mathit{GS}}f_{\mathit{SW}}
  68. P D = V D ( 1 - D ) I o P_{D}=V_{D}(1-D)I_{o}
  69. P S 2 = I o 2 R D S O N ( 1 - D ) P_{S2}=I_{o}^{2}R_{DSON}(1-D)
  70. V o I o = η V i I i \displaystyle V_{o}I_{o}=\eta V_{i}I_{i}
  71. I o = V o / Z o \displaystyle I_{o}=V_{o}/Z_{o}
  72. I i = V i / Z i \displaystyle I_{i}=V_{i}/Z_{i}
  73. V o 2 / Z o = η V i 2 / Z i V_{o}^{2}/Z_{o}=\eta V_{i}^{2}/Z_{i}
  74. V o = D V i \displaystyle V_{o}=DV_{i}
  75. ( D V i ) 2 / Z o = η V i 2 / Z i (DV_{i})^{2}/Z_{o}=\eta V_{i}^{2}/Z_{i}
  76. D 2 / Z o = η / Z i \displaystyle D^{2}/Z_{o}=\eta/Z_{i}
  77. D = η Z o / Z i D=\sqrt{\eta Z_{o}/Z_{i}}

Buck–boost_converter.html

  1. - \scriptstyle-\infty
  2. V i \scriptstyle V_{i}
  3. V i \scriptstyle V_{i}
  4. \scriptstyle\infty
  5. t = 0 \scriptstyle t=0
  6. t = D T \scriptstyle t=D\,T
  7. d I L d t = V i L \frac{\operatorname{d}I_{\,\text{L}}}{\operatorname{d}t}=\frac{V_{i}}{L}
  8. Δ I L On = 0 D T d I L = 0 D T V i L d t = V i D T L \Delta I_{\,\text{L}_{\,\text{On}}}=\int_{0}^{D\,T}\operatorname{d}I_{\,\text{% L}}=\int_{0}^{D\,T}\frac{V_{i}}{L}\,\operatorname{d}t=\frac{V_{i}\,D\,T}{L}
  9. d I L d t = V o L \frac{\operatorname{d}I_{\,\text{L}}}{\operatorname{d}t}=\frac{V_{o}}{L}
  10. Δ I L Off = 0 ( 1 - D ) T d I L = 0 ( 1 - D ) T V o d t L = V o ( 1 - D ) T L \Delta I_{\,\text{L}_{\,\text{Off}}}=\int_{0}^{\left(1-D\right)T}\operatorname% {d}I_{\,\text{L}}=\int_{0}^{\left(1-D\right)T}\frac{V_{o}\,\operatorname{d}t}{% L}=\frac{V_{o}\left(1-D\right)T}{L}
  11. E = 1 2 L I L 2 E=\frac{1}{2}L\,I_{\,\text{L}}^{2}
  12. Δ I L On + Δ I L Off = 0 \Delta I_{\,\text{L}_{\,\text{On}}}+\Delta I_{\,\text{L}_{\,\text{Off}}}=0
  13. Δ I L On \Delta I_{\,\text{L}_{\,\text{On}}}
  14. Δ I L Off \Delta I_{\,\text{L}_{\,\text{Off}}}
  15. Δ I L On + Δ I L Off = V i D T L + V o ( 1 - D ) T L = 0 \Delta I_{\,\text{L}_{\,\text{On}}}+\Delta I_{\,\text{L}_{\,\text{Off}}}=\frac% {V_{i}\,D\,T}{L}+\frac{V_{o}\left(1-D\right)T}{L}=0
  16. V o V i = D D - 1 \frac{V_{o}}{V_{i}}=\frac{D}{D-1}
  17. D = V o V o - V i D=\frac{V_{o}}{V_{o}-V_{i}}
  18. I L max I_{L_{\,\text{max}}}
  19. t = D T t=D\,T
  20. I L max = V i D T L I_{L_{\,\text{max}}}=\frac{V_{i}\,D\,T}{L}
  21. I L max + V o δ T L = 0 I_{L_{\,\text{max}}}+\frac{V_{o}\,\delta\,T}{L}=0
  22. δ = - V i D V o \delta=-\frac{V_{i}\,D}{V_{o}}
  23. I o I_{o}
  24. I D I_{D}
  25. I o = I D ¯ = I L max 2 δ I_{o}=\bar{I_{D}}=\frac{I_{L_{\,\text{max}}}}{2}\delta
  26. I L max I_{L_{\,\text{max}}}
  27. I o = - V i D T 2 L V i D V o = - V i 2 D 2 T 2 L V o I_{o}=-\frac{V_{i}\,D\,T}{2L}\frac{V_{i}\,D}{V_{o}}=-\frac{V_{i}^{2}\,D^{2}\,T% }{2L\,V_{o}}
  28. V o V i = - V i D 2 T 2 L I o \frac{V_{o}}{V_{i}}=-\frac{V_{i}\,D^{2}\,T}{2L\,I_{o}}
  29. D T + δ T = T D\,T+\delta\,T=T
  30. D + δ = 1 D+\delta=1\,
  31. I o lim \scriptstyle I_{o_{\,\text{lim}}}
  32. I o lim = I D ¯ = I L max 2 ( 1 - D ) I_{o_{\,\text{lim}}}=\bar{I_{D}}=\frac{I_{L_{\,\text{max}}}}{2}\left(1-D\right)
  33. I L max \scriptstyle I_{L_{\,\text{max}}}
  34. I o lim = V i D T 2 L ( 1 - D ) I_{o_{\,\text{lim}}}=\frac{V_{i}\,D\,T}{2L}\left(1-D\right)
  35. I o lim \scriptstyle I_{o_{\,\text{lim}}}
  36. I o lim = V i D T 2 L V i V o ( - D ) I_{o_{\,\text{lim}}}=\frac{V_{i}\,D\,T}{2L}\frac{V_{i}}{V_{o}}\left(-D\right)
  37. | V o | = V o V i \scriptstyle\left|V_{o}\right|=\frac{V_{o}}{V_{i}}
  38. | I o | = L T V i I o \scriptstyle\left|I_{o}\right|=\frac{L}{T\,V_{i}}I_{o}
  39. T V i L \scriptstyle\frac{T\,V_{i}}{L}
  40. | I o | \scriptstyle\left|I_{o}\right|
  41. | V o | = - D 1 - D \scriptstyle\left|V_{o}\right|=-\frac{D}{1-D}
  42. | V o | = - D 2 2 | I o | \scriptstyle\left|V_{o}\right|=-\frac{D^{2}}{2\left|I_{o}\right|}
  43. I o lim = V i T 2 L D ( 1 - D ) = I o lim 2 | I o | D ( 1 - D ) \scriptstyle I_{o_{\,\text{lim}}}=\frac{V_{i}\,T}{2L}D\left(1-D\right)=\frac{I% _{o_{\,\text{lim}}}}{2\left|I_{o}\right|}D\left(1-D\right)
  44. 1 2 | I o | D ( 1 - D ) = 1 \scriptstyle\frac{1}{2\left|I_{o}\right|}D\left(1-D\right)=1
  45. V i = V ¯ L + V ¯ S V_{i}=\bar{V}_{\,\text{L}}+\bar{V}_{S}
  46. V ¯ L \scriptstyle\bar{V}_{\,\text{L}}
  47. V ¯ S \scriptstyle\bar{V}_{S}
  48. V ¯ L = L d I L ¯ d t + R L I ¯ L = R L I ¯ L \bar{V}_{\,\text{L}}=L\frac{\bar{dI_{\,\text{L}}}}{dt}+R_{\,\text{L}}\bar{I}_{% \,\text{L}}=R_{\,\text{L}}\bar{I}_{\,\text{L}}
  49. V S = 0 \scriptstyle V_{S}=0
  50. V S = V i - V o \scriptstyle V_{S}=V_{i}-V_{o}
  51. V ¯ S = D 0 + ( 1 - D ) ( V i - V o ) = ( 1 - D ) ( V i - V o ) \bar{V}_{S}=D\,0+(1-D)(V_{i}-V_{o})=(1-D)(V_{i}-V_{o})
  52. I ¯ L = - I o 1 - D \bar{I}_{\,\text{L}}=\frac{-I_{o}}{1-D}
  53. I ¯ L = - V o ( 1 - D ) R \bar{I}_{\,\text{L}}=\frac{-V_{o}}{(1-D)R}
  54. V i = R L - V o ( 1 - D ) R + ( 1 - D ) ( V i - V o ) V_{i}=R_{\,\text{L}}\frac{-V_{o}}{(1-D)R}+(1-D)(V_{i}-V_{o})
  55. V o V i = - D R L R ( 1 - D ) + 1 - D \frac{V_{o}}{V_{i}}=\frac{-D}{\frac{R_{\,\text{L}}}{R(1-D)}+1-D}

Burning_Ship_fractal.html

  1. z n + 1 = ( | Re ( z n ) | + i | Im ( z n ) | ) 2 + c , z 0 = 0 z_{n+1}=(|\operatorname{Re}\left(z_{n}\right)|+i|\operatorname{Im}\left(z_{n}% \right)|)^{2}+c,\quad z_{0}=0
  2. \mathbb{C}

Burnside_ring.html

  1. X = i X i X=\cup_{i}X_{i}
  2. i = 1 N a i [ G / G i ] , \sum_{i=1}^{N}a_{i}[G/G_{i}],
  3. m X ( H ) = | X H | m_{X}(H)=\left|X^{H}\right|
  4. X H = { x X h x = x , h H } . X^{H}=\{x\in X\mid h\cdot x=x,\forall h\in H\}.
  5. m ( K , H ) = | [ G / K ] H | = # { g K G / K H g K = g K } . m(K,H)=\left|[G/K]^{H}\right|=\#\left\{gK\in G/K\mid HgK=gK\right\}.
  6. [ G / 𝐙 2 ] [ G / 𝐙 3 ] = [ G / 1 ] , [G/\mathbf{Z}_{2}]\cdot[G/\mathbf{Z}_{3}]=[G/1],
  7. χ ( g ) = m X ( g ) \chi(g)=m_{X}(\langle g\rangle)
  8. β : Ω ( G ) R ( G ) \beta:\Omega(G)\longrightarrow R(G)
  9. β ( 2 [ S 3 / 𝐙 2 ] + [ S 3 / 𝐙 3 ] ) = β ( [ S 3 ] + 2 [ S 3 / S 3 ] ) . \beta(2[S_{3}/\mathbf{Z}_{2}]+[S_{3}/\mathbf{Z}_{3}])=\beta([S_{3}]+2[S_{3}/S_% {3}]).

Bursting.html

  1. x ˙ = f ( x , u ) \dot{x}=f(x,u)
  2. u ˙ = μ g ( x , u ) \dot{u}=\mu g(x,u)
  3. f f
  4. g g
  5. x ˙ \dot{x}
  6. u ˙ \dot{u}
  7. μ 1 \mu\ll 1

Butterfly_graph.html

  1. - ( x - 1 ) ( x + 1 ) 2 ( x 2 - x - 4 ) -(x-1)(x+1)^{2}(x^{2}-x-4)

Buzen's_algorithm.html

  1. ( N + M - 1 M - 1 ) {\textstyle\left({{N+M-1}\atop{M-1}}\right)}
  2. i = 1 M n i = N \scriptstyle\sum_{i=1}^{M}n_{i}=N
  3. ( n 1 , n 2 , , n M ) = 1 G ( N ) i = 1 M ( X i ) n i \mathbb{P}(n_{1},n_{2},\cdots,n_{M})=\frac{1}{\,\text{G}(N)}\prod_{i=1}^{M}% \left(X_{i}\right)^{n_{i}}
  4. μ j X j = i = 1 M μ i X i p i j for j = 1 , , M . \mu_{j}X_{j}=\sum_{i=1}^{M}\mu_{i}X_{i}p_{ij}\quad\,\text{ for }j=1,\ldots,M.
  5. g ( 0 , m ) = 1 for m = 1 , 2 , , M g(0,m)=1\,\text{ for }m=1,2,\cdots,M
  6. g ( n , 1 ) = ( X 1 ) n for n = 0 , 1 , , N . g(n,1)=(X_{1})^{n}\,\text{ for }n=0,1,\cdots,N.
  7. g ( n , m ) = g ( n , m - 1 ) + X m g ( n - 1 , m ) . g(n,m)=g(n,m-1)+X_{m}g(n-1,m).
  8. ( n i = k ) = X i k G ( N ) [ G ( N - k ) - X i G ( N - k - 1 ) ] for k = 0 , 1 , , N - 1 , \mathbb{P}(n_{i}=k)=\frac{X_{i}^{k}}{G(N)}[G(N-k)-X_{i}G(N-k-1)]\quad\,\text{ % for }k=0,1,\ldots,N-1,
  9. ( n i = N ) = X i N G ( N ) [ G ( 0 ) ] . \mathbb{P}(n_{i}=N)=\frac{X_{i}^{N}}{G(N)}[G(0)].
  10. 𝔼 ( n i ) = k = 1 N X i k G ( N - k ) G ( N ) . \mathbb{E}(n_{i})=\sum_{k=1}^{N}X_{i}^{k}\frac{G(N-k)}{G(N)}.

C-normal_subgroup.html

  1. H H
  2. G G
  3. T T
  4. G G
  5. H T = G HT=G
  6. H H
  7. T T
  8. H H
  9. T T

CAIDI.html

  1. CAIDI = U i N i λ i N i \mbox{CAIDI}~{}=\frac{\sum{U_{i}N_{i}}}{\sum{\lambda_{i}N_{i}}}
  2. λ i \lambda_{i}
  3. N i N_{i}
  4. U i U_{i}
  5. i i
  6. CAIDI = sum of all customer interruption durations total number of customer interruptions = SAIDI SAIFI \mbox{CAIDI}~{}=\frac{\mbox{sum of all customer interruption durations}~{}}{% \mbox{total number of customer interruptions}~{}}=\frac{\mbox{SAIDI}~{}}{\mbox% {SAIFI}~{}}

Cake_number.html

  1. ( n k ) = n ! k ! ( n - k ) ! , {n\choose k}=\frac{n!}{k!\,(n-k)!},
  2. C n = ( n 3 ) + ( n 2 ) + ( n 1 ) + ( n 0 ) = 1 6 ( n 3 + 5 n + 6 ) . C_{n}={n\choose 3}+{n\choose 2}+{n\choose 1}+{n\choose 0}=\frac{1}{6}(n^{3}+5n% +6).

Calculating_demand_forecast_accuracy.html

  1. ( | A - F | ) A \sum{(|A-F|)}\over\sum{A}
  2. A A
  3. F F
  4. M A P E = ( w | A - F | ) ( w A ) MAPE=\frac{\sum(w\cdot|A-F|)}{\sum(w\cdot A)}

Callan–Symanzik_equation.html

  1. [ M M + β ( g ) g + n γ ] G ( n ) ( x 1 , x 2 , , x n ; M , g ) = 0 \left[M\frac{\partial}{\partial M}+\beta(g)\frac{\partial}{\partial g}+n\gamma% \right]G^{(n)}(x_{1},x_{2},\ldots,x_{n};M,g)=0
  2. β ( g ) \beta(g)
  3. γ \gamma
  4. [ M M + β ( e ) e + n γ 2 + m γ 3 ] G ( n , m ) ( x 1 , x 2 , , x n ; M , e ) = 0 \left[M\frac{\partial}{\partial M}+\beta(e)\frac{\partial}{\partial e}+n\gamma% _{2}+m\gamma_{3}\right]G^{(n,m)}(x_{1},x_{2},\ldots,x_{n};M,e)=0

Callendar–Van_Dusen_equation.html

  1. R ( t ) = R ( 0 ) [ 1 + A * t + B * t 2 + ( t - 100 ) C * t 3 ] . R(t)=R(0)[1+A*t+B*t^{2}+(t-100)C*t^{3}].
  2. R ( t ) = R ( 0 ) ( 1 + A * t + B * t 2 ) . R(t)=R(0)(1+A*t+B*t^{2}).

Calogero_conjecture.html

  1. h α G 1 / 2 m 3 / 2 [ R ( t ) ] 1 / 2 . h\approx\alpha G^{1/2}m^{3/2}[R(t)]^{1/2}.
  2. h h ( t ) = A R ( t ) = h 0 a ( t ) h\equiv h(t)=A\sqrt{R(t)}=h_{0}\sqrt{a(t)}
  3. G G
  4. m m
  5. R ( t ) R(t)
  6. A A

Caloron.html

  1. A μ a ( x ) = η ¯ μ ν a Π ( x ) ν Π - 1 ( x ) with Π ( x ) = 1 + π ρ 2 T r sinh ( 2 π r T ) cosh ( 2 π r T ) - cos ( 2 π r T ) , A_{\mu}^{a}(x)=\bar{\eta}_{\mu\nu}^{a}\Pi(x)\partial_{\nu}\Pi^{-1}(x)\quad\,% \text{with}\quad\Pi(x)=1+\frac{\pi\rho^{2}T}{r}\frac{\sinh(2\pi rT)}{\cosh(2% \pi rT)-\cos(2\pi rT)}\ ,
  2. η ¯ μ ν a \bar{\eta}_{\mu\nu}^{a}

Camber_(aerodynamics).html

  1. Z upper ( x ) = Z ( x ) + 1 2 T ( x ) Z\text{upper}(x)=Z(x)+\frac{1}{2}T(x)
  2. Z lower ( x ) = Z ( x ) - 1 2 T ( x ) Z\text{lower}(x)=Z(x)-\frac{1}{2}T(x)
  3. Z ¯ ( x ) = a [ ( b - 1 ) x ¯ 3 - b x ¯ 2 + x ¯ ] \overline{Z}(x)=a\left[\left(b-1\right){\overline{x}}^{3}-b{\overline{x}}^{2}+% \overline{x}\right]
  4. T ¯ ( x ) = t 0.2 ( 0.2969 x ¯ - 0.1260 x ¯ - 0.3516 x ¯ 2 + 0.2843 x ¯ 3 - 0.1015 x ¯ 4 ) \overline{T}(x)=\frac{t}{0.2}\left(0.2969\sqrt{\overline{x}}-0.1260\overline{x% }-0.3516{\overline{x}}^{2}+0.2843{\overline{x}}^{3}-0.1015{\overline{x}}^{4}\right)

Canonical_quantum_gravity.html

  1. { q i , p j } = δ i j \{q_{i},p_{j}\}=\delta_{ij}
  2. { f , g } = i = 1 N ( f q i g p i - f p i g q i ) . \{f,g\}=\sum_{i=1}^{N}\left(\frac{\partial f}{\partial q_{i}}\frac{\partial g}% {\partial p_{i}}-\frac{\partial f}{\partial p_{i}}\frac{\partial g}{\partial q% _{i}}\right).
  3. f ( q i , p j ) f(q_{i},p_{j})
  4. g ( q i , p j ) g(q_{i},p_{j})
  5. q ˙ i = { q i , H } \dot{q}_{i}=\{q_{i},H\}
  6. p ˙ i = { p i , H } \dot{p}_{i}=\{p_{i},H\}
  7. H H
  8. F ( q , p ) F(q,p)
  9. d d t F ( q i , p i ) = { F , H } . {d\over dt}F(q_{i},p_{i})=\{F,H\}.
  10. [ q ^ , p ^ ] = i . [\hat{q},\hat{p}]=i\hbar.
  11. q ^ ψ ( q ) = q ψ ( q ) \hat{q}\psi(q)=q\psi(q)
  12. p ^ ψ ( q ) = - i d d q ψ ( q ) \hat{p}\psi(q)=-i\hbar{d\over dq}\psi(q)
  13. i t ψ = H ^ ψ i\hbar{\partial\over\partial t}\psi=\hat{H}\psi
  14. H ^ \hat{H}
  15. H ( q , p ) H(q,p)
  16. q q q\mapsto q
  17. p - i d d q p\mapsto-i\hbar{d\over dq}
  18. C I = 0 C_{I}=0
  19. Ψ \Psi
  20. C ^ I Ψ = 0 \hat{C}_{I}\Psi=0
  21. C ^ I \hat{C}_{I}
  22. g μ ν d x μ d x ν = ( - N 2 + β k β k ) d t 2 + 2 β k d x k d t + γ i j d x i d x j g_{\mu\nu}dx^{\mu}\,dx^{\nu}=(-\,N^{2}+\beta_{k}\beta^{k})dt^{2}+2\beta_{k}\,% dx^{k}\,dt+\gamma_{ij}\,dx^{i}\,dx^{j}
  23. τ = x 0 \tau=x^{0}
  24. N N
  25. β k \beta_{k}
  26. γ i j \gamma_{ij}
  27. γ i j \gamma^{ij}
  28. γ i j γ j k = δ i k \gamma_{ij}\gamma^{jk}=\delta_{i}{}^{k}
  29. β i = γ i j β j \beta^{i}=\gamma^{ij}\beta_{j}
  30. γ = det γ i j \gamma=\det\gamma_{ij}
  31. δ \delta
  32. L = d 3 x N γ 1 / 2 ( K i j K i j - K 2 + R ( 3 ) ) L=\int d^{3}x\,N\gamma^{1/2}(K_{ij}K^{ij}-K^{2}+{}^{(3)}R)
  33. R ( 3 ) {}^{(3)}R
  34. γ i j \gamma_{ij}
  35. K i j K_{ij}
  36. K i j = - 1 2 ( n γ ) i j = 1 2 N - 1 ( j β i + i β j - γ i j t ) , K_{ij}=-\frac{1}{2}(\mathcal{L}_{n}\gamma)_{ij}=\frac{1}{2}N^{-1}\left(\nabla_% {j}\beta_{i}+\nabla_{i}\beta_{j}-\frac{\partial\gamma_{ij}}{\partial t}\right),
  37. \mathcal{L}
  38. n n
  39. t t
  40. i \nabla_{i}
  41. γ i j \gamma_{ij}
  42. γ μ ν = g μ ν + n μ n ν \gamma_{\mu\nu}=g_{\mu\nu}+n_{\mu}n_{\nu}
  43. π \pi
  44. π i \pi^{i}
  45. N = 1 N=1
  46. β i = 0 \beta_{i}=0
  47. H = d 3 x , H=\int d^{3}x\mathcal{H},
  48. = 1 2 γ - 1 / 2 ( γ i k γ j l + γ i l γ j k - γ i j γ k l ) π i j π k l - γ 1 / 2 R ( 3 ) \mathcal{H}=\frac{1}{2}\gamma^{-1/2}(\gamma_{ik}\gamma_{jl}+\gamma_{il}\gamma_% {jk}-\gamma_{ij}\gamma_{kl})\pi^{ij}\pi^{kl}-\gamma^{1/2}{}^{(3)}R
  49. π i j \pi^{ij}
  50. γ i j \gamma_{ij}
  51. = 0 \mathcal{H}=0
  52. j π i j = 0 \nabla_{j}\pi^{ij}=0
  53. C a ( x ) = 0 C_{a}(x)=0
  54. x x
  55. N ( x ) \vec{N}(x)
  56. C ( N ) = d 3 x C a ( x ) N a ( x ) C(\vec{N})=\int d^{3}xC_{a}(x)N^{a}(x)
  57. N a ( x ) N^{a}(x)
  58. H ( x ) = 0 H(x)=0
  59. N ( x ) N(x)
  60. H ( N ) = d 3 x H ( x ) N ( x ) H(N)=\int d^{3}xH(x)N(x)
  61. t t
  62. t t
  63. H ^ Ψ = 0. \hat{H}\Psi=0.

Cantellated_tesseract.html

  1. ( ± 1 , ± 1 , ± ( 1 + 2 ) , ± ( 1 + 2 ) ) \left(\pm 1,\ \pm 1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2})\right)
  2. ( ± 1 , ± ( 1 + 2 ) , ± ( 1 + 2 2 ) , ± ( 1 + 2 2 ) ) \left(\pm 1,\ \pm(1+\sqrt{2}),\ \pm(1+2\sqrt{2}),\ \pm(1+2\sqrt{2})\right)

Capacitor.html

  1. C = Q V C=\frac{Q}{V}
  2. C = d Q d V C=\frac{\mathrm{d}Q}{\mathrm{d}V}
  3. W = 0 Q V ( q ) d q = 0 Q q C d q = 1 2 Q 2 C = 1 2 C V 2 = 1 2 V Q W=\int_{0}^{Q}V(q)\mathrm{d}q=\int_{0}^{Q}\frac{q}{C}\mathrm{d}q={1\over 2}{Q^% {2}\over C}={1\over 2}CV^{2}={1\over 2}VQ
  4. P = d W d t = d d t ( 1 2 C V 2 ) = C V ( t ) d V d t P=\frac{\mathrm{d}W}{\mathrm{d}t}=\frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}% {2}CV^{2}\right)=CV(t)\frac{\mathrm{d}V}{\mathrm{d}t}
  5. V ( t ) = Q ( t ) C = 1 C t 0 t I ( τ ) d τ + V ( t 0 ) V(t)=\frac{Q(t)}{C}=\frac{1}{C}\int_{t_{0}}^{t}I(\tau)\mathrm{d}\tau+V(t_{0})
  6. I ( t ) = d Q ( t ) d t = C d V ( t ) d t I(t)=\frac{\mathrm{d}Q(t)}{\mathrm{d}t}=C\frac{\mathrm{d}V(t)}{\mathrm{d}t}
  7. V 0 = v resistor ( t ) + v capacitor ( t ) = i ( t ) R + 1 C t 0 t i ( τ ) d τ V_{0}=v\text{resistor}(t)+v\text{capacitor}(t)=i(t)R+\frac{1}{C}\int_{t_{0}}^{% t}i(\tau)\mathrm{d}\tau
  8. R C d i ( t ) d t + i ( t ) = 0 RC\frac{\mathrm{d}i(t)}{\mathrm{d}t}+i(t)=0
  9. I ( t ) = V 0 R e - t τ 0 V ( t ) = V 0 ( 1 - e - t τ 0 ) \begin{aligned}\displaystyle I(t)&\displaystyle=\frac{V_{0}}{R}e^{-\frac{t}{% \tau_{0}}}\\ \displaystyle V(t)&\displaystyle=V_{0}\left(1-e^{-\frac{t}{\tau_{0}}}\right)% \end{aligned}
  10. X = - 1 ω C = - 1 2 π f C Z = 1 j ω C = - j ω C = - j 2 π f C \begin{aligned}\displaystyle X&\displaystyle=-\frac{1}{\omega C}=-\frac{1}{2% \pi fC}\\ \displaystyle Z&\displaystyle=\frac{1}{j\omega C}=-\frac{j}{\omega C}=-\frac{j% }{2\pi fC}\end{aligned}
  11. I = C d V d t = - ω C V 0 sin ( ω t ) I=C\frac{dV}{dt}=-\omega{C}{V\text{0}}\sin(\omega t)
  12. X C = V 0 I 0 = V 0 ω C V 0 = 1 ω C X_{C}=\frac{V\text{0}}{I\text{0}}=\frac{V\text{0}}{\omega CV\text{0}}=\frac{1}% {\omega C}
  13. I = - I 0 sin ( ω t ) = I 0 cos ( ω t + 90 ) I=-{I\text{0}}{\sin({\omega t}})={I\text{0}}{\cos({\omega t}+{90^{\circ}})}
  14. Z ( s ) = 1 s C Z(s)=\frac{1}{sC}
  15. V = 0 d E d z = 0 d ρ ε d z = ρ d ε = Q d ε A V=\int_{0}^{d}E\,\mathrm{d}z=\int_{0}^{d}\frac{\rho}{\varepsilon}\,\mathrm{d}z% =\frac{\rho d}{\varepsilon}=\frac{Qd}{\varepsilon A}
  16. C = ε A d C=\frac{\varepsilon A}{d}
  17. E = 1 2 C V 2 = 1 2 ε A d ( U d d ) 2 = 1 2 ε A d U d 2 E=\frac{1}{2}CV^{2}=\frac{1}{2}\frac{\varepsilon A}{d}(U_{d}d)^{2}=\frac{1}{2}% \varepsilon AdU_{d}^{2}
  18. C eq = C 1 + C 2 + + C n C_{\mathrm{eq}}=C_{1}+C_{2}+\cdots+C_{n}
  19. 1 C eq = 1 C 1 + 1 C 2 + + 1 C n \frac{1}{C_{\mathrm{eq}}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\cdots+\frac{1}{C_{n}}
  20. ( A ) \left(A\right)
  21. ( B n ) \left(B\text{n}\right)
  22. ( v o l t s ) A eq \displaystyle(volts)A_{\mathrm{eq}}
  23. P = 1 R 1 n + 1 A volts ( A farads + B farads ) P=\frac{1}{R}\cdot\frac{1}{n+1}A\text{volts}\left(A\text{farads}+B\text{farads% }\right)
  24. V bd = E ds d V_{\,\text{bd}}=E_{\,\text{ds}}d
  25. R C = Z + R ESR = 1 j ω C + R ESR R\text{C}=Z+R\text{ESR}=\frac{1}{j\omega C}+R\text{ESR}
  26. Q = X C R C = 1 ω C R C , Q=\frac{X_{C}}{R_{C}}=\frac{1}{\omega CR_{C}},
  27. ω \omega
  28. C C
  29. X C X_{C}
  30. R C R_{C}
  31. f = 1 2 π L C f=\frac{1}{2\pi\sqrt{LC}}

Capacity_factor.html

  1. 648 , 000 MW·h ( 30 days ) × ( 24 hours/day ) × ( 1000 MW ) = 0.9 = 90 % \frac{648,000\ \mbox{MW·h}~{}}{(30\ \mbox{days}~{})\times(24\ \mbox{hours/day}% ~{})\times(1000\ \mbox{MW}~{})}=0.9={90\%}
  2. 43 , 416 MW·h ( 366 days ) × ( 24 hours/day ) × ( 20 MW ) = 0.2471 25 % \frac{43,416\ \mbox{MW·h}~{}}{(366\ \mbox{days}~{})\times(24\ \mbox{hours/day}% ~{})\times(20\ \mbox{MW}~{})}=0.2471\approx{25\%}
  3. 79 , 470 , 000 MW·h ( 365 days ) × ( 24 hours/day ) × ( 18 , 300 MW ) = 0.4957 50 % \frac{79,470,000\ \mbox{MW·h}~{}}{(365\ \mbox{days}~{})\times(24\ \mbox{hours/% day}~{})\times(18,300\ \mbox{MW}~{})}=0.4957\approx{50\%}
  4. 4 , 200 , 000 MW·h ( 365 days ) × ( 24 hours/day ) × ( 2 , 080 MW ) = 0.23 = 23 % \frac{4,200,000\ \mbox{MW·h}~{}}{(365\ \mbox{days}~{})\times(24\ \mbox{hours/% day}~{})\times(2,080\ \mbox{MW}~{})}=0.23=23\%

Capillary_pressure.html

  1. p c = p non-wetting phase - p wetting phase p_{c}=p_{\,\text{non-wetting phase}}-p_{\,\text{wetting phase}}
  2. γ \gamma
  3. r r
  4. θ \theta
  5. p c = 2 γ cos θ r p_{c}=\frac{2\gamma\cos\theta}{r}
  6. p c = p non-wetting phase - p wetting phase. p_{c}=p_{\,\text{non-wetting phase}}-p_{\,\text{wetting phase.}}
  7. p c p_{c}
  8. p non-wetting phase p_{\,\text{non-wetting phase}}
  9. p wetting phase p_{\,\text{wetting phase}}
  10. p c = c S w - a p_{c}=cS_{w}^{-a}
  11. c c
  12. 1 / a 1/a
  13. S w S_{w}

Capitalization-weighted_index.html

  1. i n d e x new i n d e x old = # s h a r e s p r i c e new # s h a r e s p r i c e old \frac{index_{{}_{new}}}{index_{{}_{old}}}=\frac{\sum{\#shares}\cdot price_{{}_% {new}}}{\sum{\#shares}\cdot price_{\rm{old}}}

Carathéodory's_extension_theorem.html

  1. 𝒫 ( Ω ) \mathcal{P}(\Omega)
  2. A B = i = 1 n K i A\setminus B=\bigcup_{i=1}^{n}K_{i}
  3. 𝒫 ( Ω ) \mathcal{P}(\Omega)
  4. R ( S ) = { A : A = i = 1 n A i , A i S } R(S)=\{A:A=\bigcup_{i=1}^{n}{A_{i}},A_{i}\in S\}
  5. μ ( A ) = i = 1 n μ ( A i ) \mu(A)=\sum_{i=1}^{n}{\mu(A_{i})}
  6. A = i = 1 n A i A=\bigcup_{i=1}^{n}{A_{i}}
  7. 𝒫 ( ) \mathcal{P}(\mathbb{R})
  8. 0 1 n ( x ) d x \int_{0}^{1}n(x)dx

Carathéodory–Jacobi–Lie_theorem.html

  1. d f 1 ( p ) d f r ( p ) 0 , df_{1}(p)\wedge\ldots\wedge df_{r}(p)\neq 0,
  2. ω = i = 1 n d f i d g i . \omega=\sum_{i=1}^{n}df_{i}\wedge dg_{i}.
  3. ( M , ω , H ) (M,\omega,H)
  4. ω \omega
  5. d H 0 dH\neq 0

Carbonate_hardness.html

  1. CT (mEq/L) = [ HCO 3 - ] + 2 * [ CO 3 2 - ] \,\text{CT (mEq/L)}=[\,\text{HCO}_{3}^{-}]+2*[\,\text{CO}_{3}^{2-}]

Cardinal_point_(optics).html

  1. sin θ θ \sin\theta\approx\theta
  2. cos θ 1 \cos\theta\approx 1

Carlson's_theorem.html

  1. f ( z ) f(z)
  2. | f ( z ) | C e τ | z | , z |f(z)|\leq Ce^{\tau|z|},\quad z\in\mathbb{C}
  3. π \pi
  4. | f ( i y ) | C e c | y | , y |f(iy)|\leq Ce^{c|y|},\quad y\in\mathbb{R}
  5. f f
  6. f f
  7. f f
  8. R e z > 0 Rez> 0
  9. R e z 0 Rez≥ 0
  10. | f ( z ) | C e τ | z | , z 0 |f(z)|\leq Ce^{\tau|z|},\quad\Im z\geq 0
  11. f ( z ) f(z)
  12. s i n sin
  13. π \pi
  14. z z
  15. c c
  16. π \pi
  17. f f
  18. f f
  19. lim sup n # ( A { 0 , 1 , , n - 1 } ) n = 1. \limsup_{n\to\infty}\frac{\#\big(A\cap\{0,1,\cdots,n-1\}\big)}{n}=1.
  20. A A
  21. Δ n f ( 0 ) \Delta^{n}f(0)
  22. g ( z ) = n = 0 ( z n ) Δ n f ( 0 ) g(z)=\sum_{n=0}^{\infty}{z\choose n}\Delta^{n}f(0)
  23. ( z n ) {z\choose n}
  24. Δ n f ( 0 ) \Delta^{n}f(0)

Carol_number.html

  1. 4 n - 2 n + 1 - 1 4^{n}-2^{n+1}-1
  2. ( 2 n - 1 ) 2 - 2 (2^{n}-1)^{2}-2
  3. i n + 2 2 n 2 i - 1 . \sum_{i\neq n+2}^{2n}2^{i-1}.
  4. 2 n + 1 2^{n+1}
  5. ( 2 2 n - 1 ) - 2 n + 1 (2^{2n}-1)-2^{n+1}

Carrier-to-noise_ratio.html

  1. CNR = C N \mathrm{CNR}=\frac{C}{N}
  2. CNR = ( V C V N ) 2 \mathrm{CNR}=\left(\frac{V_{C}}{V_{N}}\right)^{2}
  3. V C V_{C}
  4. V N V_{N}
  5. CNR dB = 10 log 10 ( C N ) = C d B m - N d B m \mathrm{CNR_{dB}}=10\log_{10}\left(\frac{C}{N}\right)=C_{dBm}-N_{dBm}
  6. CNR dB = 10 log 10 ( V C V N ) 2 = 20 log 10 ( V C V N ) \mathrm{CNR_{dB}}=10\log_{10}\left(\frac{V_{C}}{V_{N}}\right)^{2}=20\log_{10}% \left(\frac{V_{C}}{V_{N}}\right)

Carry-save_adder.html

  1. p s i = a i b i c i ps_{i}=a_{i}\oplus b_{i}\oplus c_{i}
  2. s c i = ( a i b i ) ( a i c i ) ( b i c i ) sc_{i}=(a_{i}\wedge b_{i})\vee(a_{i}\wedge c_{i})\vee(b_{i}\wedge c_{i})

Case_analysis.html

  1. ( ( ( P Q ) ( R Q ) ) ( P R ) ) Q (((P\rightarrow Q)\land(R\rightarrow Q))\land(P\vee R))\rightarrow Q\,

Castigliano's_method.html

  1. Q i = U q i Q_{i}=\frac{\partial{U}}{\partial q_{i}}
  2. q i = U Q i . q_{i}=\frac{\partial{U}}{\partial Q_{i}}.
  3. δ \delta
  4. δ = U P \delta=\frac{\partial{U}}{\partial P}
  5. δ = P 0 L [ M ( x ) ] 2 2 E I d x = P 0 L [ P x ] 2 2 E I d x \delta=\frac{\partial}{\partial P}\int_{0}^{L}{\frac{[M(x)]^{2}}{2EI}dx}=\frac% {\partial}{\partial P}\int_{0}^{L}{\frac{[Px]^{2}}{2EI}dx}
  6. = 0 L P x 2 E I d x =\int_{0}^{L}{\frac{Px^{2}}{EI}dx}
  7. = P L 3 3 E I . =\frac{PL^{3}}{3EI}.

Catamorphism.html

  1. h = cata f h=\mathrm{cata}\ f
  2. h i n = f F h h\circ in=f\circ Fh
  3. ( | f | ) (\!|f|\!)

Category:Functional_subgroups.html

  1. G G

Category_of_elements.html

  1. F : C 𝐒𝐞𝐭 F:C\to\mathbf{Set}
  2. el ( F ) \mathop{\rm el}(F)
  3. ( A , a ) (A,a)
  4. A Ob ( C ) A\in\mathop{\rm Ob}(C)
  5. a F A a\in FA
  6. ( A , a ) ( B , b ) (A,a)\to(B,b)
  7. f : A B f:A\to B
  8. ( F f ) a = b (Ff)a=b
  9. F \ast\downarrow F
  10. \ast
  11. el ( F ) C \mathop{\rm el}(F)\to C
  12. ( A , a ) ( B , b ) (A,a)\to(B,b)
  13. P C ^ := 𝐒𝐞𝐭 C o p P\in\hat{C}:=\mathbf{Set}^{C^{op}}
  14. el ( P ) \mathop{\rm el}(P)
  15. ( A , a ) (A,a)
  16. A Ob ( C ) A\in\mathop{\rm Ob}(C)
  17. a P ( A ) a\in P(A)
  18. ( A , a ) ( B , b ) (A,a)\to(B,b)
  19. f : A B f:A\to B
  20. ( P f ) b = a (Pf)b=a
  21. ( P ) op (\ast\downarrow P)^{\rm op}
  22. C ^ \hat{C}
  23. 𝐂𝐚𝐭 \mathbf{Cat}
  24. 𝐲 P \cong\mathop{\,\textbf{y}}\downarrow P
  25. 𝐲 : C C ^ \mathop{\,\textbf{y}}:C\to\hat{C}
  26. 𝐲 - : C ^ 𝐂𝐚𝐭 \mathop{\,\textbf{y}}\downarrow-:\hat{C}\to\,\textbf{Cat}

Cauchy's_convergence_test.html

  1. i = 0 a i \sum_{i=0}^{\infty}a_{i}
  2. ε > 0 \varepsilon>0
  3. | a n + 1 + a n + 2 + + a n + p | < ε |a_{n+1}+a_{n+2}+\cdots+a_{n+p}|<\varepsilon
  4. s n := i = 0 n a i s_{n}:=\sum_{i=0}^{n}a_{i}
  5. s n s_{n}
  6. s n s_{n}
  7. ε > 0 \varepsilon>0
  8. | s m - s n | < ε . |s_{m}-s_{n}|<\varepsilon.
  9. | s n + p - s n | = | a n + 1 + a n + 2 + + a n + p | < ε . |s_{n+p}-s_{n}|=|a_{n+1}+a_{n+2}+\cdots+a_{n+p}|<\varepsilon.
  10. a k a_{k}
  11. k = 1 a k \sum_{k=1}^{\infty}a_{k}
  12. ε > 0 \varepsilon>0
  13. | s m - s n | = | k = n m a k | < ε . |s_{m}-s_{n}|=|\sum_{k=n}^{m}a_{k}|<\varepsilon.

Cauchy_condensation_test.html

  1. f ( n ) f(n)
  2. n = 1 f ( n ) \textstyle\sum_{n=1}^{\infty}f(n)
  3. n = 0 2 n f ( 2 n ) \textstyle\sum_{n=0}^{\infty}2^{n}f(2^{n})
  4. 0 n = 1 f ( n ) n = 0 2 n f ( 2 n ) 2 n = 1 f ( n ) + 0\ \leq\ \sum_{n=1}^{\infty}f(n)\ \leq\ \sum_{n=0}^{\infty}2^{n}f(2^{n})\ \leq% \ 2\sum_{n=1}^{\infty}f(n)\ \leq\ +\infty
  5. n = 1 f ( n ) = f ( 1 ) + f ( 2 ) + f ( 3 ) + f ( 4 ) + f ( 5 ) + f ( 6 ) + f ( 7 ) + = f ( 1 ) + ( f ( 2 ) + f ( 3 ) ) + ( f ( 4 ) + f ( 5 ) + f ( 6 ) + f ( 7 ) ) + f ( 1 ) + ( f ( 2 ) + f ( 2 ) ) + ( f ( 4 ) + f ( 4 ) + f ( 4 ) + f ( 4 ) ) + = f ( 1 ) + 2 f ( 2 ) + 4 f ( 4 ) + = n = 0 2 n f ( 2 n ) \begin{array}[]{rcccccccl}\sum_{n=1}^{\infty}f(n)&=&f(1)&+&f(2)+f(3)&+&f(4)+f(% 5)+f(6)+f(7)&+&\cdots\\ &=&f(1)&+&\Big(f(2)+f(3)\Big)&+&\Big(f(4)+f(5)+f(6)+f(7)\Big)&+&\cdots\\ &\leq&f(1)&+&\Big(f(2)+f(2)\Big)&+&\Big(f(4)+f(4)+f(4)+f(4)\Big)&+&\cdots\\ &=&f(1)&+&2f(2)&+&4f(4)&+&\cdots=\sum_{n=0}^{\infty}2^{n}f(2^{n})\end{array}
  6. 2 n = 1 f ( n ) \textstyle 2\sum_{n=1}^{\infty}f(n)
  7. f ( 2 n ) \textstyle f(2^{n})
  8. n = 0 2 n f ( 2 n ) \textstyle\sum_{n=0}^{\infty}2^{n}f(2^{n})
  9. f ( 2 n ) \textstyle f(2^{n})
  10. n = 0 2 n f ( 2 n ) = f ( 1 ) + ( f ( 2 ) + f ( 2 ) ) + ( f ( 4 ) + f ( 4 ) + f ( 4 ) + f ( 4 ) ) + = ( f ( 1 ) + f ( 2 ) ) + ( f ( 2 ) + f ( 4 ) + f ( 4 ) + f ( 4 ) ) + ( f ( 1 ) + f ( 1 ) ) + ( f ( 2 ) + f ( 2 ) + f ( 3 ) + f ( 3 ) ) + = 2 n = 1 f ( n ) \begin{array}[]{rcl}\sum_{n=0}^{\infty}2^{n}f(2^{n})&=&f(1)+\Big(f(2)+f(2)\Big% )+\Big(f(4)+f(4)+f(4)+f(4)\Big)+\cdots\\ &=&\Big(f(1)+f(2)\Big)+\Big(f(2)+f(4)+f(4)+f(4)\Big)+\cdots\\ &\leq&\Big(f(1)+f(1)\Big)+\Big(f(2)+f(2)+f(3)+f(3)\Big)+\cdots=2\sum_{n=1}^{% \infty}f(n)\end{array}
  11. f ( n ) \textstyle\sum f(n)
  12. 2 n f ( 2 n ) \sum 2^{n}f(2^{n})
  13. 2 f ( n ) 2\sum f(n)
  14. f ( n ) 2 n f ( 2 n ) \textstyle f(n)\rightarrow 2^{n}f(2^{n})
  15. x e x \textstyle x\rightarrow e^{x}
  16. f ( x ) d x e x f ( e x ) d x \textstyle f(x)\,\mathrm{d}x\rightarrow e^{x}f(e^{x})\,\mathrm{d}x
  17. n = 1 f ( n ) \textstyle\sum_{n=1}^{\infty}f(n)
  18. 1 f ( x ) d x \textstyle\int_{1}^{\infty}f(x)\,\mathrm{d}x
  19. x 2 x \textstyle x\rightarrow 2^{x}
  20. log 2 0 2 x f ( 2 x ) d x \textstyle\log 2\,\int_{0}^{\infty}2^{x}f(2^{x})\,\mathrm{d}x
  21. n = 0 2 n f ( 2 n ) \textstyle\sum_{n=0}^{\infty}2^{n}f(2^{n})
  22. n = 1 1 / n \textstyle\sum_{n=1}^{\infty}1/n
  23. 1 \textstyle\sum 1
  24. f ( n ) := n - a ( log n ) - b ( log log n ) - c f(n):=n^{-a}(\log n)^{-b}(\log\log n)^{-c}
  25. Δ u ( n ) \Delta u(n)
  26. n = 1 f ( n ) \textstyle\sum_{n=1}^{\infty}f(n)
  27. n = 0 Δ u ( n ) f ( u ( n ) ) = n = 0 ( u ( n + 1 ) - u ( n ) ) f ( u ( n ) ) \sum_{n=0}^{\infty}{\Delta u(n)}f(u(n))=\sum_{n=0}^{\infty}\Big(u(n+1)-u(n)% \Big)f(u(n))
  28. u ( n ) = 2 n \textstyle u(n)=2^{n}
  29. Δ u ( n ) = 2 n \textstyle\Delta u(n)=2^{n}

Cauchy_stress_tensor.html

  1. s y m b o l σ symbol\sigma\,\!
  2. σ i j \sigma_{ij}\,\!
  3. 𝐓 ( 𝐧 ) = 𝐧 \cdotsymbol σ or T j ( n ) = σ i j n i . \mathbf{T}^{(\mathbf{n})}=\mathbf{n}\cdotsymbol{\sigma}\quad\,\text{or}\quad T% _{j}^{(n)}=\sigma_{ij}n_{i}.\,\!
  4. s y m b o l σ = [ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 ] [ σ x x σ x y σ x z σ y x σ y y σ y z σ z x σ z y σ z z ] [ σ x τ x y τ x z τ y x σ y τ y z τ z x τ z y σ z ] symbol{\sigma}=\left[{\begin{matrix}\sigma_{11}&\sigma_{12}&\sigma_{13}\\ \sigma_{21}&\sigma_{22}&\sigma_{23}\\ \sigma_{31}&\sigma_{32}&\sigma_{33}\\ \end{matrix}}\right]\equiv\left[{\begin{matrix}\sigma_{xx}&\sigma_{xy}&\sigma_% {xz}\\ \sigma_{yx}&\sigma_{yy}&\sigma_{yz}\\ \sigma_{zx}&\sigma_{zy}&\sigma_{zz}\\ \end{matrix}}\right]\equiv\left[{\begin{matrix}\sigma_{x}&\tau_{xy}&\tau_{xz}% \\ \tau_{yx}&\sigma_{y}&\tau_{yz}\\ \tau_{zx}&\tau_{zy}&\sigma_{z}\\ \end{matrix}}\right]\,\!
  5. 𝐓 ( 𝐧 ) \mathbf{T}^{(\mathbf{n})}
  6. S S\,\!
  7. 𝐧 \mathbf{n}\,\!
  8. S S\,\!
  9. P P\,\!
  10. S S\,\!
  11. 𝐅 \mathbf{F}
  12. 𝐛 \mathbf{b}
  13. \mathcal{F}
  14. = 𝐛 + 𝐅 \mathcal{F}=\mathbf{b}+\mathbf{F}
  15. 𝐅 \mathbf{F}\,\!
  16. S S\,\!
  17. Δ S \Delta S\,\!
  18. P P\,\!
  19. 𝐧 \mathbf{n}
  20. Δ 𝐅 \Delta\mathbf{F}\,\!
  21. Δ 𝐌 \Delta\mathbf{M}\,\!
  22. Δ 𝐅 = 𝐓 ( 𝐧 ) Δ S \Delta\mathbf{F}=\mathbf{T}^{(\mathbf{n})}\,\Delta S
  23. 𝐓 ( 𝐧 ) \mathbf{T}^{(\mathbf{n})}
  24. Δ S \Delta S\,\!
  25. Δ 𝐅 / Δ S \Delta\mathbf{F}/\Delta S\,\!
  26. d 𝐅 / d S d\mathbf{F}/dS\,\!
  27. Δ 𝐌 \Delta\mathbf{M}\,\!
  28. d 𝐅 / d S d\mathbf{F}/dS\,\!
  29. 𝐓 ( 𝐧 ) = T i ( 𝐧 ) 𝐞 i \mathbf{T}^{(\mathbf{n})}=T_{i}^{(\mathbf{n})}\mathbf{e}_{i}\,\!
  30. P P\,\!
  31. 𝐧 \mathbf{n}\,\!
  32. T i ( 𝐧 ) = lim Δ S 0 Δ F i Δ S = d F i d S . T^{(\mathbf{n})}_{i}=\lim_{\Delta S\to 0}\frac{\Delta F_{i}}{\Delta S}={dF_{i}% \over dS}.
  33. 𝐓 ( 𝐧 , 𝐱 , t ) \mathbf{T}(\mathbf{n},\mathbf{x},t)
  34. t t\,\!
  35. 𝐱 \mathbf{x}
  36. 𝐧 \mathbf{n}
  37. 𝐧 \mathbf{n}\,\!
  38. σ n = lim Δ S 0 Δ F n Δ S = d F n d S , \mathbf{\sigma_{\mathrm{n}}}=\lim_{\Delta S\to 0}\frac{\Delta F_{\mathrm{n}}}{% \Delta S}=\frac{dF_{\mathrm{n}}}{dS},
  39. d F n dF_{\mathrm{n}}\,\!
  40. d 𝐅 d\mathbf{F}\,\!
  41. d S dS\,\!
  42. τ = lim Δ S 0 Δ F s Δ S = d F s d S , \mathbf{\tau}=\lim_{\Delta S\to 0}\frac{\Delta F_{\mathrm{s}}}{\Delta S}=\frac% {dF_{\mathrm{s}}}{dS},
  43. d F s dF_{\mathrm{s}}\,\!
  44. d 𝐅 d\mathbf{F}\,\!
  45. d S dS\,\!
  46. 𝐓 ( 𝐧 ) \mathbf{T}^{(\mathbf{n})}
  47. P P\,\!
  48. 𝐧 \mathbf{n}\,\!
  49. P P\,\!
  50. P P\,\!
  51. 𝐧 \mathbf{n}\,\!
  52. - 𝐓 ( 𝐧 ) = 𝐓 ( - 𝐧 ) . -\mathbf{T}^{(\mathbf{n})}=\mathbf{T}^{(-\mathbf{n})}.\,\!
  53. 𝐓 ( 𝐧 ) = 𝐧 \cdotsymbol σ or T j ( n ) = σ i j n i . \mathbf{T}^{(\mathbf{n})}=\mathbf{n}\cdotsymbol{\sigma}\quad\,\text{or}\quad T% _{j}^{(n)}=\sigma_{ij}n_{i}.\,\!
  54. 𝐓 ( 𝐧 ) d A - 𝐓 ( 𝐞 1 ) d A 1 - 𝐓 ( 𝐞 2 ) d A 2 - 𝐓 ( 𝐞 3 ) d A 3 = ρ ( h 3 d A ) 𝐚 , \mathbf{T}^{(\mathbf{n})}\,dA-\mathbf{T}^{(\mathbf{e}_{1})}\,dA_{1}-\mathbf{T}% ^{(\mathbf{e}_{2})}\,dA_{2}-\mathbf{T}^{(\mathbf{e}_{3})}\,dA_{3}=\rho\left(% \frac{h}{3}dA\right)\mathbf{a},\,\!
  55. d A 1 = ( 𝐧 𝐞 1 ) d A = n 1 d A , dA_{1}=\left(\mathbf{n}\cdot\mathbf{e}_{1}\right)dA=n_{1}\;dA,\,\!
  56. d A 2 = ( 𝐧 𝐞 2 ) d A = n 2 d A , dA_{2}=\left(\mathbf{n}\cdot\mathbf{e}_{2}\right)dA=n_{2}\;dA,\,\!
  57. d A 3 = ( 𝐧 𝐞 3 ) d A = n 3 d A , dA_{3}=\left(\mathbf{n}\cdot\mathbf{e}_{3}\right)dA=n_{3}\;dA,\,\!
  58. 𝐓 ( 𝐧 ) - 𝐓 ( 𝐞 1 ) n 1 - 𝐓 ( 𝐞 2 ) n 2 - 𝐓 ( 𝐞 3 ) n 3 = ρ ( h 3 ) 𝐚 . \mathbf{T}^{(\mathbf{n})}-\mathbf{T}^{(\mathbf{e}_{1})}n_{1}-\mathbf{T}^{(% \mathbf{e}_{2})}n_{2}-\mathbf{T}^{(\mathbf{e}_{3})}n_{3}=\rho\left(\frac{h}{3}% \right)\mathbf{a}.\,\!
  59. 𝐓 ( 𝐧 ) = 𝐓 ( 𝐞 1 ) n 1 + 𝐓 ( 𝐞 2 ) n 2 + 𝐓 ( 𝐞 3 ) n 3 . \mathbf{T}^{(\mathbf{n})}=\mathbf{T}^{(\mathbf{e}_{1})}n_{1}+\mathbf{T}^{(% \mathbf{e}_{2})}n_{2}+\mathbf{T}^{(\mathbf{e}_{3})}n_{3}.\,\!
  60. 𝐓 ( 𝐞 1 ) = T 1 ( 𝐞 1 ) 𝐞 1 + T 2 ( 𝐞 1 ) 𝐞 2 + T 3 ( 𝐞 1 ) 𝐞 3 = σ 11 𝐞 1 + σ 12 𝐞 2 + σ 13 𝐞 3 , \mathbf{T}^{(\mathbf{e}_{1})}=T_{1}^{(\mathbf{e}_{1})}\mathbf{e}_{1}+T_{2}^{(% \mathbf{e}_{1})}\mathbf{e}_{2}+T_{3}^{(\mathbf{e}_{1})}\mathbf{e}_{3}=\sigma_{% 11}\mathbf{e}_{1}+\sigma_{12}\mathbf{e}_{2}+\sigma_{13}\mathbf{e}_{3},
  61. 𝐓 ( 𝐞 2 ) = T 1 ( 𝐞 2 ) 𝐞 1 + T 2 ( 𝐞 2 ) 𝐞 2 + T 3 ( 𝐞 2 ) 𝐞 3 = σ 21 𝐞 1 + σ 22 𝐞 2 + σ 23 𝐞 3 , \mathbf{T}^{(\mathbf{e}_{2})}=T_{1}^{(\mathbf{e}_{2})}\mathbf{e}_{1}+T_{2}^{(% \mathbf{e}_{2})}\mathbf{e}_{2}+T_{3}^{(\mathbf{e}_{2})}\mathbf{e}_{3}=\sigma_{% 21}\mathbf{e}_{1}+\sigma_{22}\mathbf{e}_{2}+\sigma_{23}\mathbf{e}_{3},
  62. 𝐓 ( 𝐞 3 ) = T 1 ( 𝐞 3 ) 𝐞 1 + T 2 ( 𝐞 3 ) 𝐞 2 + T 3 ( 𝐞 3 ) 𝐞 3 = σ 31 𝐞 1 + σ 32 𝐞 2 + σ 33 𝐞 3 , \mathbf{T}^{(\mathbf{e}_{3})}=T_{1}^{(\mathbf{e}_{3})}\mathbf{e}_{1}+T_{2}^{(% \mathbf{e}_{3})}\mathbf{e}_{2}+T_{3}^{(\mathbf{e}_{3})}\mathbf{e}_{3}=\sigma_{% 31}\mathbf{e}_{1}+\sigma_{32}\mathbf{e}_{2}+\sigma_{33}\mathbf{e}_{3},
  63. 𝐓 ( 𝐞 i ) = T j ( 𝐞 i ) 𝐞 j = σ i j 𝐞 j . \mathbf{T}^{(\mathbf{e}_{i})}=T_{j}^{(\mathbf{e}_{i})}\mathbf{e}_{j}=\sigma_{% ij}\mathbf{e}_{j}.
  64. s y m b o l σ = σ i j = [ 𝐓 ( 𝐞 1 ) 𝐓 ( 𝐞 2 ) 𝐓 ( 𝐞 3 ) ] = [ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 ] [ σ x x σ x y σ x z σ y x σ y y σ y z σ z x σ z y σ z z ] [ σ x τ x y τ x z τ y x σ y τ y z τ z x τ z y σ z ] , symbol{\sigma}=\sigma_{ij}=\left[{\begin{matrix}\mathbf{T}^{(\mathbf{e}_{1})}% \\ \mathbf{T}^{(\mathbf{e}_{2})}\\ \mathbf{T}^{(\mathbf{e}_{3})}\\ \end{matrix}}\right]=\left[{\begin{matrix}\sigma_{11}&\sigma_{12}&\sigma_{13}% \\ \sigma_{21}&\sigma_{22}&\sigma_{23}\\ \sigma_{31}&\sigma_{32}&\sigma_{33}\\ \end{matrix}}\right]\equiv\left[{\begin{matrix}\sigma_{xx}&\sigma_{xy}&\sigma_% {xz}\\ \sigma_{yx}&\sigma_{yy}&\sigma_{yz}\\ \sigma_{zx}&\sigma_{zy}&\sigma_{zz}\\ \end{matrix}}\right]\equiv\left[{\begin{matrix}\sigma_{x}&\tau_{xy}&\tau_{xz}% \\ \tau_{yx}&\sigma_{y}&\tau_{yz}\\ \tau_{zx}&\tau_{zy}&\sigma_{z}\\ \end{matrix}}\right],
  65. 𝐓 ( 𝐧 ) = 𝐓 ( 𝐞 1 ) n 1 + 𝐓 ( 𝐞 2 ) n 2 + 𝐓 ( 𝐞 3 ) n 3 = i = 1 3 𝐓 ( 𝐞 i ) n i = ( σ i j 𝐞 j ) n i = σ i j n i 𝐞 j \begin{aligned}\displaystyle\mathbf{T}^{(\mathbf{n})}&\displaystyle=\mathbf{T}% ^{(\mathbf{e}_{1})}n_{1}+\mathbf{T}^{(\mathbf{e}_{2})}n_{2}+\mathbf{T}^{(% \mathbf{e}_{3})}n_{3}\\ &\displaystyle=\sum_{i=1}^{3}\mathbf{T}^{(\mathbf{e}_{i})}n_{i}\\ &\displaystyle=\left(\sigma_{ij}\mathbf{e}_{j}\right)n_{i}\\ &\displaystyle=\sigma_{ij}n_{i}\mathbf{e}_{j}\end{aligned}
  66. T j ( 𝐧 ) = σ i j n i . T_{j}^{(\mathbf{n})}=\sigma_{ij}n_{i}.
  67. [ T 1 ( 𝐧 ) T 2 ( 𝐧 ) T 3 ( 𝐧 ) ] = [ n 1 n 2 n 3 ] [ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 ] . \left[{\begin{matrix}T^{(\mathbf{n})}_{1}&T^{(\mathbf{n})}_{2}&T^{(\mathbf{n})% }_{3}\end{matrix}}\right]=\left[{\begin{matrix}n_{1}&n_{2}&n_{3}\end{matrix}}% \right]\cdot\left[{\begin{matrix}\sigma_{11}&\sigma_{12}&\sigma_{13}\\ \sigma_{21}&\sigma_{22}&\sigma_{23}\\ \sigma_{31}&\sigma_{32}&\sigma_{33}\\ \end{matrix}}\right].
  68. s y m b o l σ = [ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 ] T [ σ 11 σ 22 σ 33 σ 23 σ 13 σ 12 ] T . symbol{\sigma}=\begin{bmatrix}\sigma_{1}&\sigma_{2}&\sigma_{3}&\sigma_{4}&% \sigma_{5}&\sigma_{6}\end{bmatrix}^{T}\equiv\begin{bmatrix}\sigma_{11}&\sigma_% {22}&\sigma_{33}&\sigma_{23}&\sigma_{13}&\sigma_{12}\end{bmatrix}^{T}.\,\!
  69. σ i j = a i m a j n σ m n or s y m b o l σ = 𝐀 s y m b o l σ 𝐀 T , \sigma^{\prime}_{ij}=a_{im}a_{jn}\sigma_{mn}\quad\,\text{or}\quad symbol{% \sigma}^{\prime}=\mathbf{A}symbol{\sigma}\mathbf{A}^{T},
  70. [ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 ] = [ a 11 a 12 a 13 a 21 a 22 a 23 a 31 a 32 a 33 ] [ σ 11 σ 12 σ 13 σ 21 σ 22 σ 23 σ 31 σ 32 σ 33 ] [ a 11 a 21 a 31 a 12 a 22 a 32 a 13 a 23 a 33 ] . \left[{\begin{matrix}\sigma^{\prime}_{11}&\sigma^{\prime}_{12}&\sigma^{\prime}% _{13}\\ \sigma^{\prime}_{21}&\sigma^{\prime}_{22}&\sigma^{\prime}_{23}\\ \sigma^{\prime}_{31}&\sigma^{\prime}_{32}&\sigma^{\prime}_{33}\\ \end{matrix}}\right]=\left[{\begin{matrix}a_{11}&a_{12}&a_{13}\\ a_{21}&a_{22}&a_{23}\\ a_{31}&a_{32}&a_{33}\\ \end{matrix}}\right]\left[{\begin{matrix}\sigma_{11}&\sigma_{12}&\sigma_{13}\\ \sigma_{21}&\sigma_{22}&\sigma_{23}\\ \sigma_{31}&\sigma_{32}&\sigma_{33}\\ \end{matrix}}\right]\left[{\begin{matrix}a_{11}&a_{21}&a_{31}\\ a_{12}&a_{22}&a_{32}\\ a_{13}&a_{23}&a_{33}\\ \end{matrix}}\right].
  71. σ 11 = a 11 2 σ 11 + a 12 2 σ 22 + a 13 2 σ 33 + 2 a 11 a 12 σ 12 + 2 a 11 a 13 σ 13 + 2 a 12 a 13 σ 23 , \sigma_{11}^{\prime}=a_{11}^{2}\sigma_{11}+a_{12}^{2}\sigma_{22}+a_{13}^{2}% \sigma_{33}+2a_{11}a_{12}\sigma_{12}+2a_{11}a_{13}\sigma_{13}+2a_{12}a_{13}% \sigma_{23},
  72. σ 22 = a 21 2 σ 11 + a 22 2 σ 22 + a 23 2 σ 33 + 2 a 21 a 22 σ 12 + 2 a 21 a 23 σ 13 + 2 a 22 a 23 σ 23 , \sigma_{22}^{\prime}=a_{21}^{2}\sigma_{11}+a_{22}^{2}\sigma_{22}+a_{23}^{2}% \sigma_{33}+2a_{21}a_{22}\sigma_{12}+2a_{21}a_{23}\sigma_{13}+2a_{22}a_{23}% \sigma_{23},
  73. σ 33 = a 31 2 σ 11 + a 32 2 σ 22 + a 33 2 σ 33 + 2 a 31 a 32 σ 12 + 2 a 31 a 33 σ 13 + 2 a 32 a 33 σ 23 , \sigma_{33}^{\prime}=a_{31}^{2}\sigma_{11}+a_{32}^{2}\sigma_{22}+a_{33}^{2}% \sigma_{33}+2a_{31}a_{32}\sigma_{12}+2a_{31}a_{33}\sigma_{13}+2a_{32}a_{33}% \sigma_{23},
  74. σ 12 = a 11 a 21 σ 11 + a 12 a 22 σ 22 + a 13 a 23 σ 33 + ( a 11 a 22 + a 12 a 21 ) σ 12 + ( a 12 a 23 + a 13 a 22 ) σ 23 + ( a 11 a 23 + a 13 a 21 ) σ 13 , \begin{aligned}\displaystyle\sigma_{12}^{\prime}=&\displaystyle a_{11}a_{21}% \sigma_{11}+a_{12}a_{22}\sigma_{22}+a_{13}a_{23}\sigma_{33}\\ &\displaystyle+(a_{11}a_{22}+a_{12}a_{21})\sigma_{12}+(a_{12}a_{23}+a_{13}a_{2% 2})\sigma_{23}+(a_{11}a_{23}+a_{13}a_{21})\sigma_{13},\end{aligned}
  75. σ 23 = a 21 a 31 σ 11 + a 22 a 32 σ 22 + a 23 a 33 σ 33 + ( a 21 a 32 + a 22 a 31 ) σ 12 + ( a 22 a 33 + a 23 a 32 ) σ 23 + ( a 21 a 33 + a 23 a 31 ) σ 13 , \begin{aligned}\displaystyle\sigma_{23}^{\prime}=&\displaystyle a_{21}a_{31}% \sigma_{11}+a_{22}a_{32}\sigma_{22}+a_{23}a_{33}\sigma_{33}\\ &\displaystyle+(a_{21}a_{32}+a_{22}a_{31})\sigma_{12}+(a_{22}a_{33}+a_{23}a_{3% 2})\sigma_{23}+(a_{21}a_{33}+a_{23}a_{31})\sigma_{13},\end{aligned}
  76. σ 13 = a 11 a 31 σ 11 + a 12 a 32 σ 22 + a 13 a 33 σ 33 + ( a 11 a 32 + a 12 a 31 ) σ 12 + ( a 12 a 33 + a 13 a 32 ) σ 23 + ( a 11 a 33 + a 13 a 31 ) σ 13 . \begin{aligned}\displaystyle\sigma_{13}^{\prime}=&\displaystyle a_{11}a_{31}% \sigma_{11}+a_{12}a_{32}\sigma_{22}+a_{13}a_{33}\sigma_{33}\\ &\displaystyle+(a_{11}a_{32}+a_{12}a_{31})\sigma_{12}+(a_{12}a_{33}+a_{13}a_{3% 2})\sigma_{23}+(a_{11}a_{33}+a_{13}a_{31})\sigma_{13}.\end{aligned}
  77. σ n \displaystyle\sigma_{\mathrm{n}}
  78. τ n \displaystyle\tau_{\mathrm{n}}
  79. ( T ( 𝐧 ) ) 2 = T i ( 𝐧 ) T i ( 𝐧 ) = ( σ i j n j ) ( σ i k n k ) = σ i j σ i k n j n k . \left(T^{(\mathbf{n})}\right)^{2}=T_{i}^{(\mathbf{n})}T_{i}^{(\mathbf{n})}=% \left(\sigma_{ij}n_{j}\right)\left(\sigma_{ik}n_{k}\right)=\sigma_{ij}\sigma_{% ik}n_{j}n_{k}.
  80. σ j i , j + F i = 0 \sigma_{ji,j}+F_{i}=0\,\!
  81. σ i j = - p δ i j {\sigma_{ij}}=-p{\delta_{ij}}
  82. p p
  83. δ i j {\delta_{ij}}
  84. V V\,\!
  85. S S\,\!
  86. T i ( n ) T_{i}^{(n)}\,\!
  87. F i F_{i}\,\!
  88. V V\,\!
  89. S T i ( n ) d S + V F i d V = 0 \int_{S}T_{i}^{(n)}dS+\int_{V}F_{i}dV=0\,\!
  90. T i ( n ) = σ j i n j T_{i}^{(n)}=\sigma_{ji}n_{j}\,\!
  91. S σ j i n j d S + V F i d V = 0 \int_{S}\sigma_{ji}n_{j}\,dS+\int_{V}F_{i}\,dV=0\,\!
  92. V σ j i , j d V + V F i d V = 0 \int_{V}\sigma_{ji,j}\,dV+\int_{V}F_{i}\,dV=0\,\!
  93. V ( σ j i , j + F i ) d V = 0 \int_{V}(\sigma_{ji,j}+F_{i}\,)dV=0\,\!
  94. σ j i , j + F i = 0 \sigma_{ji,j}+F_{i}=0\,\!
  95. σ i j = σ j i \sigma_{ij}=\sigma_{ji}\,\!
  96. M O = S ( 𝐫 × 𝐓 ) d S + V ( 𝐫 × 𝐅 ) d V = 0 0 = S ε i j k x j T k ( n ) d S + V ε i j k x j F k d V \begin{aligned}\displaystyle M_{O}&\displaystyle=\int_{S}(\mathbf{r}\times% \mathbf{T})dS+\int_{V}(\mathbf{r}\times\mathbf{F})dV=0\\ \displaystyle 0&\displaystyle=\int_{S}\varepsilon_{ijk}x_{j}T_{k}^{(n)}dS+\int% _{V}\varepsilon_{ijk}x_{j}F_{k}dV\\ \end{aligned}\,\!
  97. 𝐫 \mathbf{r}\,\!
  98. 𝐫 = x j 𝐞 j \mathbf{r}=x_{j}\mathbf{e}_{j}\,\!
  99. T k ( n ) = σ m k n m T_{k}^{(n)}=\sigma_{mk}n_{m}\,\!
  100. 0 = S ε i j k x j σ m k n m d S + V ε i j k x j F k d V = V ( ε i j k x j σ m k ) , m d V + V ε i j k x j F k d V = V ( ε i j k x j , m σ m k + ε i j k x j σ m k , m ) d V + V ε i j k x j F k d V = V ( ε i j k x j , m σ m k ) d V + V ε i j k x j ( σ m k , m + F k ) d V \begin{aligned}\displaystyle 0&\displaystyle=\int_{S}\varepsilon_{ijk}x_{j}% \sigma_{mk}n_{m}\,dS+\int_{V}\varepsilon_{ijk}x_{j}F_{k}\,dV\\ &\displaystyle=\int_{V}(\varepsilon_{ijk}x_{j}\sigma_{mk})_{,m}dV+\int_{V}% \varepsilon_{ijk}x_{j}F_{k}\,dV\\ &\displaystyle=\int_{V}(\varepsilon_{ijk}x_{j,m}\sigma_{mk}+\varepsilon_{ijk}x% _{j}\sigma_{mk,m})dV+\int_{V}\varepsilon_{ijk}x_{j}F_{k}\,dV\\ &\displaystyle=\int_{V}(\varepsilon_{ijk}x_{j,m}\sigma_{mk})dV+\int_{V}% \varepsilon_{ijk}x_{j}(\sigma_{mk,m}+F_{k})dV\\ \end{aligned}\,\!
  101. x j , m = δ j m x_{j,m}=\delta_{jm}\,\!
  102. V ( ε i j k σ j k ) d V = 0 \int_{V}(\varepsilon_{ijk}\sigma_{jk})dV=0\,\!
  103. ε i j k σ j k = 0 \varepsilon_{ijk}\sigma_{jk}=0\,\!
  104. σ 12 = σ 21 \sigma_{12}=\sigma_{21}\,\!
  105. σ 23 = σ 32 \sigma_{23}=\sigma_{32}\,\!
  106. σ 13 = σ 31 \sigma_{13}=\sigma_{31}\,\!
  107. σ i j = σ j i \sigma_{ij}=\sigma_{ji}\,\!
  108. K n 1 K_{n}\rightarrow 1\,\!
  109. 𝐧 \mathbf{n}\,\!
  110. 𝐧 \mathbf{n}\,\!
  111. τ n \tau_{\mathrm{n}}\,\!
  112. σ i j \sigma_{ij}\,\!
  113. 𝐧 \mathbf{n}\,\!
  114. 𝐓 ( 𝐧 ) = λ 𝐧 = σ n 𝐧 \mathbf{T}^{(\mathbf{n})}=\lambda\mathbf{n}=\mathbf{\sigma}_{\mathrm{n}}% \mathbf{n}\,\!
  115. λ \lambda\,\!
  116. σ n \sigma_{\mathrm{n}}\,\!
  117. T i ( n ) = σ i j n j T_{i}^{(n)}=\sigma_{ij}n_{j}\,\!
  118. n i = δ i j n j n_{i}=\delta_{ij}n_{j}\,\!
  119. T i ( n ) = λ n i σ i j n j = λ n i σ i j n j - λ n i = 0 ( σ i j - λ δ i j ) n j = 0 \begin{aligned}\displaystyle T_{i}^{(n)}&\displaystyle=\lambda n_{i}\\ \displaystyle\sigma_{ij}n_{j}&\displaystyle=\lambda n_{i}\\ \displaystyle\sigma_{ij}n_{j}-\lambda n_{i}&\displaystyle=0\\ \displaystyle\left(\sigma_{ij}-\lambda\delta_{ij}\right)n_{j}&\displaystyle=0% \\ \end{aligned}\,\!
  120. n j n_{j}\,\!
  121. n j n_{j}\,\!
  122. | σ i j - λ δ i j | = | σ 11 - λ σ 12 σ 13 σ 21 σ 22 - λ σ 23 σ 31 σ 32 σ 33 - λ | = 0 \left|\sigma_{ij}-\lambda\delta_{ij}\right|=\begin{vmatrix}\sigma_{11}-\lambda% &\sigma_{12}&\sigma_{13}\\ \sigma_{21}&\sigma_{22}-\lambda&\sigma_{23}\\ \sigma_{31}&\sigma_{32}&\sigma_{33}-\lambda\\ \end{vmatrix}=0\,\!
  123. | σ i j - λ δ i j | = - λ 3 + I 1 λ 2 - I 2 λ + I 3 = 0 \left|\sigma_{ij}-\lambda\delta_{ij}\right|=-\lambda^{3}+I_{1}\lambda^{2}-I_{2% }\lambda+I_{3}=0\,\!
  124. I 1 = σ 11 + σ 22 + σ 33 = σ k k = tr ( s y m b o l σ ) I 2 = | σ 22 σ 23 σ 32 σ 33 | + | σ 11 σ 13 σ 31 σ 33 | + | σ 11 σ 12 σ 21 σ 22 | = σ 11 σ 22 + σ 22 σ 33 + σ 11 σ 33 - σ 12 2 - σ 23 2 - σ 31 2 = 1 2 ( σ i i σ j j - σ i j σ j i ) = 1 2 [ tr ( s y m b o l σ ) 2 - tr ( s y m b o l σ 2 ) ] I 3 = det ( σ i j ) = det ( s y m b o l σ ) = σ 11 σ 22 σ 33 + 2 σ 12 σ 23 σ 31 - σ 12 2 σ 33 - σ 23 2 σ 11 - σ 31 2 σ 22 \begin{aligned}\displaystyle I_{1}&\displaystyle=\sigma_{11}+\sigma_{22}+% \sigma_{33}\\ &\displaystyle=\sigma_{kk}=\,\text{tr}(symbol{\sigma})\\ \displaystyle I_{2}&\displaystyle=\begin{vmatrix}\sigma_{22}&\sigma_{23}\\ \sigma_{32}&\sigma_{33}\\ \end{vmatrix}+\begin{vmatrix}\sigma_{11}&\sigma_{13}\\ \sigma_{31}&\sigma_{33}\\ \end{vmatrix}+\begin{vmatrix}\sigma_{11}&\sigma_{12}\\ \sigma_{21}&\sigma_{22}\\ \end{vmatrix}\\ &\displaystyle=\sigma_{11}\sigma_{22}+\sigma_{22}\sigma_{33}+\sigma_{11}\sigma% _{33}-\sigma_{12}^{2}-\sigma_{23}^{2}-\sigma_{31}^{2}\\ &\displaystyle=\frac{1}{2}\left(\sigma_{ii}\sigma_{jj}-\sigma_{ij}\sigma_{ji}% \right)=\frac{1}{2}\left[\,\text{tr}(symbol{\sigma})^{2}-\,\text{tr}(symbol{% \sigma}^{2})\right]\\ \displaystyle I_{3}&\displaystyle=\det(\sigma_{ij})=\det(symbol{\sigma})\\ &\displaystyle=\sigma_{11}\sigma_{22}\sigma_{33}+2\sigma_{12}\sigma_{23}\sigma% _{31}-\sigma_{12}^{2}\sigma_{33}-\sigma_{23}^{2}\sigma_{11}-\sigma_{31}^{2}% \sigma_{22}\\ \end{aligned}\,\!
  125. λ i \lambda_{i}\,\!
  126. σ 1 = max ( λ 1 , λ 2 , λ 3 ) \sigma_{1}=\max\left(\lambda_{1},\lambda_{2},\lambda_{3}\right)\,\!
  127. σ 3 = min ( λ 1 , λ 2 , λ 3 ) \sigma_{3}=\min\left(\lambda_{1},\lambda_{2},\lambda_{3}\right)\,\!
  128. σ 2 = I 1 - σ 1 - σ 3 \sigma_{2}=I_{1}-\sigma_{1}-\sigma_{3}\,\!
  129. λ i \lambda_{i}\,\!
  130. I 1 I_{1}\,\!
  131. I 2 I_{2}\,\!
  132. I 3 I_{3}\,\!
  133. n j n_{j}\,\!
  134. ( σ i j - λ δ i j ) n j = 0 \left(\sigma_{ij}-\lambda\delta_{ij}\right)n_{j}=0\,\!
  135. σ i j = [ σ 1 0 0 0 σ 2 0 0 0 σ 3 ] \sigma_{ij}=\begin{bmatrix}\sigma_{1}&0&0\\ 0&\sigma_{2}&0\\ 0&0&\sigma_{3}\end{bmatrix}\,\!
  136. I 1 I_{1}\,\!
  137. I 2 I_{2}\,\!
  138. I 3 I_{3}\,\!
  139. I 1 = σ 1 + σ 2 + σ 3 I 2 = σ 1 σ 2 + σ 2 σ 3 + σ 3 σ 1 I 3 = σ 1 σ 2 σ 3 \begin{aligned}\displaystyle I_{1}&\displaystyle=\sigma_{1}+\sigma_{2}+\sigma_% {3}\\ \displaystyle I_{2}&\displaystyle=\sigma_{1}\sigma_{2}+\sigma_{2}\sigma_{3}+% \sigma_{3}\sigma_{1}\\ \displaystyle I_{3}&\displaystyle=\sigma_{1}\sigma_{2}\sigma_{3}\\ \end{aligned}\,\!
  140. σ 1 , σ 2 = σ x + σ y 2 ± ( σ x - σ y 2 ) 2 + τ x y 2 \sigma_{1},\sigma_{2}=\frac{\sigma_{x}+\sigma_{y}}{2}\pm\sqrt{\left(\frac{% \sigma_{x}-\sigma_{y}}{2}\right)^{2}+\tau_{xy}^{2}}\,\!
  141. τ m a x , τ m i n = ± ( σ x - σ y 2 ) 2 + τ x y 2 \tau_{max},\tau_{min}=\pm\sqrt{\left(\frac{\sigma_{x}-\sigma_{y}}{2}\right)^{2% }+\tau_{xy}^{2}}\,\!
  142. 45 45^{\circ}
  143. τ max = 1 2 | σ max - σ min | \tau_{\max}=\frac{1}{2}\left|\sigma_{\max}-\sigma_{\min}\right|\,\!
  144. σ 1 σ 2 σ 3 \sigma_{1}\geq\sigma_{2}\geq\sigma_{3}\,\!
  145. τ max = 1 2 | σ 1 - σ 3 | \tau_{\max}=\frac{1}{2}\left|\sigma_{1}-\sigma_{3}\right|\,\!
  146. σ n = 1 2 ( σ 1 + σ 3 ) \sigma_{\mathrm{n}}=\frac{1}{2}\left(\sigma_{1}+\sigma_{3}\right)\,\!
  147. ( σ 1 σ 2 σ 3 ) (\sigma_{1}\geq\sigma_{2}\geq\sigma_{3})\,\!
  148. σ n = σ i j n i n j = σ 1 n 1 2 + σ 2 n 2 2 + σ 3 n 3 2 \begin{aligned}\displaystyle\sigma_{\mathrm{n}}&\displaystyle=\sigma_{ij}n_{i}% n_{j}\\ &\displaystyle=\sigma_{1}n_{1}^{2}+\sigma_{2}n_{2}^{2}+\sigma_{3}n_{3}^{2}\\ \end{aligned}\,\!
  149. ( T ( n ) ) 2 = σ i j σ i k n j n k \left(T^{(n)}\right)^{2}=\sigma_{ij}\sigma_{ik}n_{j}n_{k}
  150. τ n 2 = ( T ( n ) ) 2 - σ n 2 = σ 1 2 n 1 2 + σ 2 2 n 2 2 + σ 3 2 n 3 2 - ( σ 1 n 1 2 + σ 2 n 2 2 + σ 3 n 3 2 ) 2 = ( σ 1 2 - σ 2 2 ) n 1 2 + ( σ 2 2 - σ 3 2 ) n 2 2 + σ 3 2 - [ ( σ 1 - σ 3 ) n 1 2 + ( σ 2 - σ 2 ) n 2 2 + σ 3 ] 2 = ( σ 1 - σ 2 ) 2 n 1 2 n 2 2 + ( σ 2 - σ 3 ) 2 n 2 2 n 3 2 + ( σ 1 - σ 3 ) 2 n 1 2 n 3 2 \begin{aligned}\displaystyle\tau_{\mathrm{n}}^{2}&\displaystyle=\left(T^{(n)}% \right)^{2}-\sigma_{\mathrm{n}}^{2}\\ &\displaystyle=\sigma_{1}^{2}n_{1}^{2}+\sigma_{2}^{2}n_{2}^{2}+\sigma_{3}^{2}n% _{3}^{2}-\left(\sigma_{1}n_{1}^{2}+\sigma_{2}n_{2}^{2}+\sigma_{3}n_{3}^{2}% \right)^{2}\\ &\displaystyle=(\sigma_{1}^{2}-\sigma_{2}^{2})n_{1}^{2}+(\sigma_{2}^{2}-\sigma% _{3}^{2})n_{2}^{2}+\sigma_{3}^{2}-\left[\left(\sigma_{1}-\sigma_{3}\right)n_{1% }^{2}+\left(\sigma_{2}-\sigma_{2}\right)n_{2}^{2}+\sigma_{3}\right]^{2}\\ &\displaystyle=(\sigma_{1}-\sigma_{2})^{2}n_{1}^{2}n_{2}^{2}+(\sigma_{2}-% \sigma_{3})^{2}n_{2}^{2}n_{3}^{2}+(\sigma_{1}-\sigma_{3})^{2}n_{1}^{2}n_{3}^{2% }\\ \end{aligned}\,\!
  151. τ n 2 \tau_{\mathrm{n}}^{2}\,\!
  152. n i n i = n 1 2 + n 2 2 + n 3 2 = 1 n_{i}n_{i}=n_{1}^{2}+n_{2}^{2}+n_{3}^{2}=1\,\!
  153. τ n 2 \tau_{\mathrm{n}}^{2}\,\!
  154. τ n 2 \tau_{\mathrm{n}}^{2}\,\!
  155. F F\,\!
  156. F ( n 1 , n 2 , n 3 , λ ) = τ 2 + λ ( g ( n 1 , n 2 , n 3 ) - 1 ) = σ 1 2 n 1 2 + σ 2 2 n 2 2 + σ 3 2 n 3 2 - ( σ 1 n 1 2 + σ 2 n 2 2 + σ 3 n 3 2 ) 2 + λ ( n 1 2 + n 2 2 + n 3 2 - 1 ) \begin{aligned}\displaystyle F\left(n_{1},n_{2},n_{3},\lambda\right)&% \displaystyle=\tau^{2}+\lambda\left(g\left(n_{1},n_{2},n_{3}\right)-1\right)\\ &\displaystyle=\sigma_{1}^{2}n_{1}^{2}+\sigma_{2}^{2}n_{2}^{2}+\sigma_{3}^{2}n% _{3}^{2}-\left(\sigma_{1}n_{1}^{2}+\sigma_{2}n_{2}^{2}+\sigma_{3}n_{3}^{2}% \right)^{2}+\lambda\left(n_{1}^{2}+n_{2}^{2}+n_{3}^{2}-1\right)\\ \end{aligned}\,\!
  157. λ \lambda\,\!
  158. λ \lambda\,\!
  159. F n 1 = 0 F n 2 = 0 F n 3 = 0 \frac{\partial F}{\partial n_{1}}=0\qquad\frac{\partial F}{\partial n_{2}}=0% \qquad\frac{\partial F}{\partial n_{3}}=0\,\!
  160. F n 1 = n 1 σ 1 2 - 2 n 1 σ 1 ( σ 1 n 1 2 + σ 2 n 2 2 + σ 3 n 3 2 ) + λ n 1 = 0 \frac{\partial F}{\partial n_{1}}=n_{1}\sigma_{1}^{2}-2n_{1}\sigma_{1}\left(% \sigma_{1}n_{1}^{2}+\sigma_{2}n_{2}^{2}+\sigma_{3}n_{3}^{2}\right)+\lambda n_{% 1}=0\,\!
  161. F n 2 = n 2 σ 2 2 - 2 n 2 σ 2 ( σ 1 n 1 2 + σ 2 n 2 2 + σ 3 n 3 2 ) + λ n 2 = 0 \frac{\partial F}{\partial n_{2}}=n_{2}\sigma_{2}^{2}-2n_{2}\sigma_{2}\left(% \sigma_{1}n_{1}^{2}+\sigma_{2}n_{2}^{2}+\sigma_{3}n_{3}^{2}\right)+\lambda n_{% 2}=0\,\!
  162. F n 3 = n 3 σ 3 2 - 2 n 3 σ 3 ( σ 1 n 1 2 + σ 2 n 2 2 + σ 3 n 3 2 ) + λ n 3 = 0 \frac{\partial F}{\partial n_{3}}=n_{3}\sigma_{3}^{2}-2n_{3}\sigma_{3}\left(% \sigma_{1}n_{1}^{2}+\sigma_{2}n_{2}^{2}+\sigma_{3}n_{3}^{2}\right)+\lambda n_{% 3}=0\,\!
  163. n i n i = 1 n_{i}n_{i}=1\,\!
  164. λ , n 1 , n 2 , \lambda,n_{1},n_{2},\,\!
  165. n 3 n_{3}\,\!
  166. n 1 , n 2 , n_{1},\,n_{2},\,\!
  167. n 3 n_{3}\,\!
  168. σ n = σ i j n i n j = σ 1 n 1 2 + σ 2 n 2 2 + σ 3 n 3 2 \sigma_{\mathrm{n}}=\sigma_{ij}n_{i}n_{j}=\sigma_{1}n_{1}^{2}+\sigma_{2}n_{2}^% {2}+\sigma_{3}n_{3}^{2}\,\!
  169. n 1 2 σ 1 2 - 2 σ 1 n 1 2 σ n + n 1 2 λ = 0 n_{1}^{2}\sigma_{1}^{2}-2\sigma_{1}n_{1}^{2}\sigma_{\mathrm{n}}+n_{1}^{2}% \lambda=0\,\!
  170. n 2 2 σ 2 2 - 2 σ 2 n 2 2 σ n + n 2 2 λ = 0 n_{2}^{2}\sigma_{2}^{2}-2\sigma_{2}n_{2}^{2}\sigma_{\mathrm{n}}+n_{2}^{2}% \lambda=0\,\!
  171. n 3 2 σ 3 2 - 2 σ 1 n 3 2 σ n + n 3 2 λ = 0 n_{3}^{2}\sigma_{3}^{2}-2\sigma_{1}n_{3}^{2}\sigma_{\mathrm{n}}+n_{3}^{2}% \lambda=0\,\!
  172. [ n 1 2 σ 1 2 + n 2 2 σ 2 2 + n 3 2 σ 3 2 ] - 2 ( σ 1 n 1 2 + σ 2 n 2 2 + σ 3 n 3 2 ) σ n + λ ( n 1 2 + n 2 2 + n 3 2 ) = 0 [ τ n 2 + ( σ 1 n 1 2 + σ 2 n 2 2 + σ 3 n 3 2 ) 2 ] - 2 σ n 2 + λ = 0 [ τ n 2 + σ n 2 ] - 2 σ n 2 + λ = 0 λ = σ n 2 - τ n 2 \begin{aligned}\displaystyle\left[n_{1}^{2}\sigma_{1}^{2}+n_{2}^{2}\sigma_{2}^% {2}+n_{3}^{2}\sigma_{3}^{2}\right]-2\left(\sigma_{1}n_{1}^{2}+\sigma_{2}n_{2}^% {2}+\sigma_{3}n_{3}^{2}\right)\sigma_{\mathrm{n}}+\lambda\left(n_{1}^{2}+n_{2}% ^{2}+n_{3}^{2}\right)&\displaystyle=0\\ \displaystyle\left[\tau_{\mathrm{n}}^{2}+\left(\sigma_{1}n_{1}^{2}+\sigma_{2}n% _{2}^{2}+\sigma_{3}n_{3}^{2}\right)^{2}\right]-2\sigma_{\mathrm{n}}^{2}+% \lambda&\displaystyle=0\\ \displaystyle\left[\tau_{\mathrm{n}}^{2}+\sigma_{\mathrm{n}}^{2}\right]-2% \sigma_{\mathrm{n}}^{2}+\lambda&\displaystyle=0\\ \displaystyle\lambda&\displaystyle=\sigma_{\mathrm{n}}^{2}-\tau_{\mathrm{n}}^{% 2}\end{aligned}\,\!
  173. F n 1 = n 1 σ 1 2 - 2 n 1 σ 1 ( σ 1 n 1 2 + σ 2 n 2 2 + σ 3 n 3 2 ) + ( σ n 2 - τ n 2 ) n 1 = 0 n 1 σ 1 2 - 2 n 1 σ 1 σ n + ( σ n 2 - τ n 2 ) n 1 = 0 ( σ 1 2 - 2 σ 1 σ n + σ n 2 - τ n 2 ) n 1 = 0 \begin{aligned}\displaystyle\frac{\partial F}{\partial n_{1}}=n_{1}\sigma_{1}^% {2}-2n_{1}\sigma_{1}\left(\sigma_{1}n_{1}^{2}+\sigma_{2}n_{2}^{2}+\sigma_{3}n_% {3}^{2}\right)+\left(\sigma_{\mathrm{n}}^{2}-\tau_{\mathrm{n}}^{2}\right)n_{1}% &\displaystyle=0\\ \displaystyle n_{1}\sigma_{1}^{2}-2n_{1}\sigma_{1}\sigma_{\mathrm{n}}+\left(% \sigma_{\mathrm{n}}^{2}-\tau_{\mathrm{n}}^{2}\right)n_{1}&\displaystyle=0\\ \displaystyle\left(\sigma_{1}^{2}-2\sigma_{1}\sigma_{\mathrm{n}}+\sigma_{% \mathrm{n}}^{2}-\tau_{\mathrm{n}}^{2}\right)n_{1}&\displaystyle=0\\ \end{aligned}\,\!
  174. F n 2 = ( σ 2 2 - 2 σ 2 σ n + σ n 2 - τ n 2 ) n 2 = 0 \frac{\partial F}{\partial n_{2}}=\left(\sigma_{2}^{2}-2\sigma_{2}\sigma_{% \mathrm{n}}+\sigma_{\mathrm{n}}^{2}-\tau_{\mathrm{n}}^{2}\right)n_{2}=0\,\!
  175. F n 3 = ( σ 3 2 - 2 σ 3 σ n + σ n 2 - τ n 2 ) n 3 = 0 \frac{\partial F}{\partial n_{3}}=\left(\sigma_{3}^{2}-2\sigma_{3}\sigma_{% \mathrm{n}}+\sigma_{\mathrm{n}}^{2}-\tau_{\mathrm{n}}^{2}\right)n_{3}=0\,\!
  176. n i = 0 n_{i}=0\,\!
  177. n i n i = 1 n_{i}n_{i}=1\,\!
  178. n 1 = n 2 = 0 n_{1}=n_{2}=0\,\!
  179. n 3 0 n_{3}\neq 0\,\!
  180. n i n i = 1 n_{i}n_{i}=1\,\!
  181. n 3 = ± 1 n_{3}=\pm 1\,\!
  182. τ n 2 \tau_{\mathrm{n}}^{2}\,\!
  183. τ n = 0 \tau_{\mathrm{n}}=0\,\!
  184. τ n \tau_{\mathrm{n}}\,\!
  185. n 1 = n 3 = 0 n_{1}=n_{3}=0\,\!
  186. n 2 0 n_{2}\neq 0\,\!
  187. n 2 = n 3 = 0 n_{2}=n_{3}=0\,\!
  188. n 1 0 n_{1}\neq 0\,\!
  189. n 1 = 0 , n 2 = 0 , n 3 = ± 1 , τ n = 0 n 1 = 0 , n 2 = ± 1 , n 3 = 0 , τ n = 0 n 1 = ± 1 , n 2 = 0 , n 3 = 0 , τ n = 0 \begin{aligned}\displaystyle n_{1}&\displaystyle=0,\,\,n_{2}&\displaystyle=0,% \,\,n_{3}&\displaystyle=\pm 1,\,\,\tau_{\mathrm{n}}&\displaystyle=0\\ \displaystyle n_{1}&\displaystyle=0,\,\,n_{2}&\displaystyle=\pm 1,\,\,n_{3}&% \displaystyle=0,\,\,\tau_{\mathrm{n}}&\displaystyle=0\\ \displaystyle n_{1}&\displaystyle=\pm 1,\,\,n_{2}&\displaystyle=0,\,\,n_{3}&% \displaystyle=0,\,\,\tau_{\mathrm{n}}&\displaystyle=0\end{aligned}\,\!
  190. τ n \tau_{\mathrm{n}}\,\!
  191. n 1 = 0 , n 2 0 n_{1}=0,\,n_{2}\neq 0\,\!
  192. n 3 0 n_{3}\neq 0\,\!
  193. F n 2 = σ 2 2 - 2 σ 2 σ n + σ n 2 - τ n 2 = 0 \frac{\partial F}{\partial n_{2}}=\sigma_{2}^{2}-2\sigma_{2}\sigma_{\mathrm{n}% }+\sigma_{\mathrm{n}}^{2}-\tau_{\mathrm{n}}^{2}=0\,\!
  194. F n 3 = σ 3 2 - 2 σ 3 σ n + σ n 2 - τ n 2 = 0 \frac{\partial F}{\partial n_{3}}=\sigma_{3}^{2}-2\sigma_{3}\sigma_{\mathrm{n}% }+\sigma_{\mathrm{n}}^{2}-\tau_{\mathrm{n}}^{2}=0\,\!
  195. n 2 n_{2}\,\!
  196. n 3 n_{3}\,\!
  197. σ 2 2 - σ 3 2 - 2 σ 2 σ n + 2 σ 3 σ n = 0 σ 2 2 - σ 3 2 - 2 σ n ( σ 2 - σ 3 ) = 0 σ 2 + σ 3 = 2 σ n \begin{aligned}\displaystyle\sigma_{2}^{2}-\sigma_{3}^{2}-2\sigma_{2}\sigma_{n% }+2\sigma_{3}\sigma_{n}&\displaystyle=0\\ \displaystyle\sigma_{2}^{2}-\sigma_{3}^{2}-2\sigma_{n}\left(\sigma_{2}-\sigma_% {3}\right)&\displaystyle=0\\ \displaystyle\sigma_{2}+\sigma_{3}&\displaystyle=2\sigma_{n}\end{aligned}\,\!
  198. n 1 = 0 n_{1}=0\,\!
  199. σ n = σ 1 n 1 2 + σ 2 n 2 2 + σ 3 n 3 2 = σ 2 n 2 2 + σ 3 n 3 2 \sigma_{\mathrm{n}}=\sigma_{1}n_{1}^{2}+\sigma_{2}n_{2}^{2}+\sigma_{3}n_{3}^{2% }=\sigma_{2}n_{2}^{2}+\sigma_{3}n_{3}^{2}\,\!
  200. n 1 2 + n 2 2 + n 3 2 = n 2 2 + n 3 2 = 1 n_{1}^{2}+n_{2}^{2}+n_{3}^{2}=n_{2}^{2}+n_{3}^{2}=1\,\!
  201. σ 2 + σ 3 = 2 σ n σ 2 + σ 3 = 2 ( σ 2 n 2 2 + σ 3 n 3 2 ) σ 2 + σ 3 = 2 ( σ 2 n 2 2 + σ 3 ( 1 - n 2 2 ) ) = 0 \begin{aligned}\displaystyle\sigma_{2}+\sigma_{3}&\displaystyle=2\sigma_{n}\\ \displaystyle\sigma_{2}+\sigma_{3}&\displaystyle=2\left(\sigma_{2}n_{2}^{2}+% \sigma_{3}n_{3}^{2}\right)\\ \displaystyle\sigma_{2}+\sigma_{3}&\displaystyle=2\left(\sigma_{2}n_{2}^{2}+% \sigma_{3}\left(1-n_{2}^{2}\right)\right)&\displaystyle=0\end{aligned}\,\!
  202. n 2 n_{2}\,\!
  203. n 2 = ± 1 2 n_{2}=\pm\frac{1}{\sqrt{2}}\,\!
  204. n 3 n_{3}\,\!
  205. n 3 = 1 - n 2 2 = ± 1 2 n_{3}=\sqrt{1-n_{2}^{2}}=\pm\frac{1}{\sqrt{2}}\,\!
  206. τ n 2 = ( σ 2 - σ 3 ) 2 n 2 2 n 3 2 τ n = σ 2 - σ 3 2 \begin{aligned}\displaystyle\tau_{\mathrm{n}}^{2}&\displaystyle=(\sigma_{2}-% \sigma_{3})^{2}n_{2}^{2}n_{3}^{2}\\ \displaystyle\tau_{\mathrm{n}}&\displaystyle=\frac{\sigma_{2}-\sigma_{3}}{2}% \end{aligned}\,\!
  207. τ n \tau_{\mathrm{n}}\,\!
  208. n 2 = 0 , n 1 0 n_{2}=0,\,n_{1}\neq 0\,\!
  209. n 3 0 n_{3}\neq 0\,\!
  210. n 3 = 0 , n 1 0 n_{3}=0,\,n_{1}\neq 0\,\!
  211. n 2 0 n_{2}\neq 0\,\!
  212. F n 1 = 0 \frac{\partial F}{\partial n_{1}}=0\,\!
  213. τ n \tau_{\mathrm{n}}\,\!
  214. n 1 = 0 , n 2 = ± 1 2 , n 3 = ± 1 2 , τ n = ± σ 2 - σ 3 2 n_{1}=0,\,\,n_{2}=\pm\frac{1}{\sqrt{2}},\,\,n_{3}=\pm\frac{1}{\sqrt{2}},\,\,% \tau_{\mathrm{n}}=\pm\frac{\sigma_{2}-\sigma_{3}}{2}\,\!
  215. n 1 = ± 1 2 , n 2 = 0 , n 3 = ± 1 2 , τ n = ± σ 1 - σ 3 2 n_{1}=\pm\frac{1}{\sqrt{2}},\,\,n_{2}=0,\,\,n_{3}=\pm\frac{1}{\sqrt{2}},\,\,% \tau_{\mathrm{n}}=\pm\frac{\sigma_{1}-\sigma_{3}}{2}\,\!
  216. n 1 = ± 1 2 , n 2 = ± 1 2 , n 3 = 0 , τ n = ± σ 2 - σ 3 2 n_{1}=\pm\frac{1}{\sqrt{2}},\,\,n_{2}=\pm\frac{1}{\sqrt{2}},\,\,n_{3}=0,\,\,% \tau_{\mathrm{n}}=\pm\frac{\sigma_{2}-\sigma_{3}}{2}\,\!
  217. σ 1 σ 2 σ 3 \sigma_{1}\geq\sigma_{2}\geq\sigma_{3}\,\!
  218. τ max = 1 2 | σ 1 - σ 3 | = 1 2 | σ max - σ min | \tau_{\mathrm{max}}=\frac{1}{2}\left|\sigma_{1}-\sigma_{3}\right|=\frac{1}{2}% \left|\sigma_{\mathrm{max}}-\sigma_{\mathrm{min}}\right|\,\!
  219. σ i j \sigma_{ij}\,\!
  220. π δ i j \pi\delta_{ij}\,\!
  221. s i j s_{ij}\,\!
  222. σ i j = s i j + π δ i j , \sigma_{ij}=s_{ij}+\pi\delta_{ij},\,
  223. π \pi\,\!
  224. π = σ k k 3 = σ 11 + σ 22 + σ 33 3 = 1 3 I 1 . \pi=\frac{\sigma_{kk}}{3}=\frac{\sigma_{11}+\sigma_{22}+\sigma_{33}}{3}=\tfrac% {1}{3}I_{1}.\,
  225. p p
  226. p = λ u - π = λ u k x k - π = k λ u k x k - π , p=\lambda\,\nabla\cdot\vec{u}-\pi=\lambda\,\frac{\partial u_{k}}{\partial x_{k% }}-\pi=\sum_{k}\lambda\,\frac{\partial u_{k}}{\partial x_{k}}-\pi,
  227. λ \lambda
  228. \nabla
  229. x k x_{k}
  230. u \vec{u}
  231. u k u_{k}
  232. u \vec{u}
  233. s i j \displaystyle\ s_{ij}
  234. s i j s_{ij}\,\!
  235. σ i j \sigma_{ij}\,\!
  236. | s i j - λ δ i j | = λ 3 - J 1 λ 2 + J 2 λ - J 3 = 0 , \left|s_{ij}-\lambda\delta_{ij}\right|=\lambda^{3}-J_{1}\lambda^{2}+J_{2}% \lambda-J_{3}=0,\,
  237. J 1 J_{1}\,\!
  238. J 2 J_{2}\,\!
  239. J 3 J_{3}\,\!
  240. s i j s_{ij}\,\!
  241. s 1 s_{1}\,\!
  242. s 2 s_{2}\,\!
  243. s 3 s_{3}\,\!
  244. σ i j \sigma_{ij}\,\!
  245. σ 1 \sigma_{1}\,\!
  246. σ 2 \sigma_{2}\,\!
  247. σ 3 \sigma_{3}\,\!
  248. J 1 = s k k = 0 , J 2 = 1 2 s i j s j i = 1 2 tr ( s y m b o l s 2 ) = 1 2 ( s 1 2 + s 2 2 + s 3 2 ) = 1 6 [ ( σ 11 - σ 22 ) 2 + ( σ 22 - σ 33 ) 2 + ( σ 33 - σ 11 ) 2 ] + σ 12 2 + σ 23 2 + σ 31 2 = 1 6 [ ( σ 1 - σ 2 ) 2 + ( σ 2 - σ 3 ) 2 + ( σ 3 - σ 1 ) 2 ] = 1 3 I 1 2 - I 2 = 1 2 [ tr ( s y m b o l σ 2 ) - 1 3 tr ( s y m b o l σ ) 2 ] , J 3 = det ( s i j ) = 1 3 s i j s j k s k i = 1 3 tr ( s y m b o l s 3 ) = s 1 s 2 s 3 = 2 27 I 1 3 - 1 3 I 1 I 2 + I 3 = 1 3 [ tr ( s y m b o l σ 3 ) - tr ( s y m b o l σ 2 ) tr ( s y m b o l σ ) + 2 9 tr ( s y m b o l σ ) 3 ] . \begin{aligned}\displaystyle J_{1}&\displaystyle=s_{kk}=0,\\ \displaystyle J_{2}&\displaystyle=\textstyle{\frac{1}{2}}s_{ij}s_{ji}=\tfrac{1% }{2}\,\text{tr}(symbol{s}^{2})\\ &\displaystyle=\tfrac{1}{2}(s_{1}^{2}+s_{2}^{2}+s_{3}^{2})\\ &\displaystyle=\tfrac{1}{6}\left[(\sigma_{11}-\sigma_{22})^{2}+(\sigma_{22}-% \sigma_{33})^{2}+(\sigma_{33}-\sigma_{11})^{2}\right]+\sigma_{12}^{2}+\sigma_{% 23}^{2}+\sigma_{31}^{2}\\ &\displaystyle=\tfrac{1}{6}\left[(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-% \sigma_{3})^{2}+(\sigma_{3}-\sigma_{1})^{2}\right]\\ &\displaystyle=\tfrac{1}{3}I_{1}^{2}-I_{2}=\frac{1}{2}\left[\,\text{tr}(symbol% {\sigma}^{2})-\frac{1}{3}\,\text{tr}(symbol{\sigma})^{2}\right],\\ \displaystyle J_{3}&\displaystyle=\det(s_{ij})\\ &\displaystyle=\tfrac{1}{3}s_{ij}s_{jk}s_{ki}=\tfrac{1}{3}\,\text{tr}(symbol{s% }^{3})\\ &\displaystyle=s_{1}s_{2}s_{3}\\ &\displaystyle=\tfrac{2}{27}I_{1}^{3}-\tfrac{1}{3}I_{1}I_{2}+I_{3}=\tfrac{1}{3% }\left[\,\text{tr}(symbol{\sigma}^{3})-\,\text{tr}(symbol{\sigma}^{2})\,\text{% tr}(symbol{\sigma})+\tfrac{2}{9}\,\text{tr}(symbol{\sigma})^{3}\right].\end{aligned}
  249. s k k = 0 s_{kk}=0\,\!
  250. σ e = 3 J 2 = 1 2 [ ( σ 1 - σ 2 ) 2 + ( σ 2 - σ 3 ) 2 + ( σ 3 - σ 1 ) 2 ] . \sigma_{\mathrm{e}}=\sqrt{3~{}J_{2}}=\sqrt{\tfrac{1}{2}~{}\left[(\sigma_{1}-% \sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{2}+(\sigma_{3}-\sigma_{1})^{2}\right]% }\,.
  251. | 1 / 3 | |1/\sqrt{3}|\,\!
  252. σ oct \sigma_{\mathrm{oct}}\,\!
  253. τ oct \tau_{\mathrm{oct}}\,\!
  254. σ i j = [ σ 1 0 0 0 σ 2 0 0 0 σ 3 ] \sigma_{ij}=\begin{bmatrix}\sigma_{1}&0&0\\ 0&\sigma_{2}&0\\ 0&0&\sigma_{3}\end{bmatrix}\,\!
  255. 𝐓 oct ( 𝐧 ) = σ i j n i 𝐞 j = σ 1 n 1 𝐞 1 + σ 2 n 2 𝐞 2 + σ 3 n 3 𝐞 3 = 1 3 ( σ 1 𝐞 1 + σ 2 𝐞 2 + σ 3 𝐞 3 ) \begin{aligned}\displaystyle\mathbf{T}_{\mathrm{oct}}^{(\mathbf{n})}&% \displaystyle=\sigma_{ij}n_{i}\mathbf{e}_{j}\\ &\displaystyle=\sigma_{1}n_{1}\mathbf{e}_{1}+\sigma_{2}n_{2}\mathbf{e}_{2}+% \sigma_{3}n_{3}\mathbf{e}_{3}\\ &\displaystyle=\tfrac{1}{\sqrt{3}}(\sigma_{1}\mathbf{e}_{1}+\sigma_{2}\mathbf{% e}_{2}+\sigma_{3}\mathbf{e}_{3})\end{aligned}\,\!
  256. σ oct = T i ( n ) n i = σ i j n i n j = σ 1 n 1 n 1 + σ 2 n 2 n 2 + σ 3 n 3 n 3 = 1 3 ( σ 1 + σ 2 + σ 3 ) = 1 3 I 1 \begin{aligned}\displaystyle\sigma_{\mathrm{oct}}&\displaystyle=T^{(n)}_{i}n_{% i}\\ &\displaystyle=\sigma_{ij}n_{i}n_{j}\\ &\displaystyle=\sigma_{1}n_{1}n_{1}+\sigma_{2}n_{2}n_{2}+\sigma_{3}n_{3}n_{3}% \\ &\displaystyle=\tfrac{1}{3}(\sigma_{1}+\sigma_{2}+\sigma_{3})=\tfrac{1}{3}I_{1% }\end{aligned}\,\!
  257. τ oct = T i ( n ) T i ( n ) - σ n 2 = [ 1 3 ( σ 1 2 + σ 2 2 + σ 3 2 ) - 1 9 ( σ 1 + σ 2 + σ 3 ) 2 ] 1 / 2 = 1 3 [ ( σ 1 - σ 2 ) 2 + ( σ 2 - σ 3 ) 2 + ( σ 3 - σ 1 ) 2 ] 1 / 2 = 1 3 2 I 1 2 - 6 I 2 = 2 3 J 2 \begin{aligned}\displaystyle\tau_{\mathrm{oct}}&\displaystyle=\sqrt{T_{i}^{(n)% }T_{i}^{(n)}-\sigma_{\mathrm{n}}^{2}}\\ &\displaystyle=\left[\tfrac{1}{3}(\sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}% )-\tfrac{1}{9}(\sigma_{1}+\sigma_{2}+\sigma_{3})^{2}\right]^{1/2}\\ &\displaystyle=\tfrac{1}{3}\left[(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-% \sigma_{3})^{2}+(\sigma_{3}-\sigma_{1})^{2}\right]^{1/2}=\tfrac{1}{3}\sqrt{2I_% {1}^{2}-6I_{2}}=\sqrt{\tfrac{2}{3}J_{2}}\end{aligned}\,\!

Causal_filter.html

  1. t , t,
  2. s ( x ) s(x)\,
  3. f ( x ) = x - 1 x + 1 s ( τ ) d τ = - 1 + 1 s ( x + τ ) d τ f(x)=\int_{x-1}^{x+1}s(\tau)\,d\tau\ =\int_{-1}^{+1}s(x+\tau)\,d\tau\,
  4. x x\,
  5. ( t ) (t)\,
  6. f ( t ) f(t)\,
  7. s ( t + 1 ) s(t+1)\,
  8. f ( t - 1 ) = - 2 0 s ( t + τ ) d τ = 0 + 2 s ( t - τ ) d τ f(t-1)=\int_{-2}^{0}s(t+\tau)\,d\tau=\int_{0}^{+2}s(t-\tau)\,d\tau\,
  9. f ( t ) = ( h * s ) ( t ) = - h ( τ ) s ( t - τ ) d τ . f(t)=(h*s)(t)=\int_{-\infty}^{\infty}h(\tau)s(t-\tau)\,d\tau.\,
  10. f ( t ) = 0 h ( τ ) s ( t - τ ) d τ f(t)=\int_{0}^{\infty}h(\tau)s(t-\tau)\,d\tau
  11. h ( t ) = 2 Θ ( t ) g ( t ) h(t)=2\,\Theta(t)\cdot g(t)\,
  12. H ( ω ) = ( δ ( ω ) - i π ω ) * G ( ω ) = G ( ω ) - i G ^ ( ω ) H(\omega)=\left(\delta(\omega)-{i\over\pi\omega}\right)*G(\omega)=G(\omega)-i% \cdot\widehat{G}(\omega)\,
  13. G ^ ( ω ) \widehat{G}(\omega)\,
  14. G ^ ( ω ) \widehat{G}(\omega)\,
  15. H ^ ( ω ) = i H ( ω ) \widehat{H}(\omega)=iH(\omega)

Cellular_repeater.html

  1. P dB = 10 log 10 ( P P 0 ) \quad P_{\mathrm{dB}}=10\log_{10}\left(\frac{P}{P_{0}}\right)

Center_manifold.html

  1. d 𝐱 d t = 𝐟 ( 𝐱 ) \frac{d\,\textbf{x}}{dt}=\,\textbf{f}(\,\textbf{x})
  2. 𝐱 * \,\textbf{x}^{*}
  3. d 𝐱 d t = A 𝐱 , where Jacobian A = d 𝐟 d 𝐱 ( 𝐱 * ) . \frac{d\,\textbf{x}}{dt}=A\,\textbf{x},\quad\,\text{where Jacobian }A=\frac{d% \,\textbf{f}}{d\,\textbf{x}}(\,\textbf{x}^{*}).
  4. A A
  5. λ \lambda
  6. Re λ < 0 \operatorname{Re}\lambda<0
  7. λ \lambda
  8. Re λ > 0 \operatorname{Re}\lambda>0
  9. λ \lambda
  10. Re λ = 0 \operatorname{Re}\lambda=0
  11. 𝐟 ( 𝐱 ) \,\textbf{f}(\,\textbf{x})
  12. C r C^{r}
  13. r r
  14. C r C^{r}
  15. C r C^{r}
  16. C r - 1 C^{r-1}
  17. 𝐲 ( t ) \,\textbf{y}(t)
  18. 𝐱 ( t ) = 𝐲 ( t ) + O ( e - β t ) as t , \,\textbf{x}(t)=\,\textbf{y}(t)+O(e^{-\beta^{\prime}t})\quad\,\text{as }t\to% \infty\,,
  19. β \beta^{\prime}
  20. 𝐱 = 𝐗 ( 𝐬 ) \,\textbf{x}=\,\textbf{X}(\,\textbf{s})
  21. O ( | 𝐬 | p ) O(|\,\textbf{s}|^{p})
  22. 𝐬 𝟎 \,\textbf{s}\to\,\textbf{0}
  23. 𝐱 = 𝐗 ( 𝐬 ) \,\textbf{x}=\,\textbf{X}(\,\textbf{s})
  24. O ( | 𝐬 | p ) O(|\,\textbf{s}|^{p})
  25. d 𝐱 d t = 𝐟 ( 𝐱 , t ) \frac{d\,\textbf{x}}{dt}=\,\textbf{f}(\,\textbf{x},t)
  26. | Re λ | α |\operatorname{Re}\lambda|\leq\alpha
  27. Re λ - β < - r α \operatorname{Re}\lambda\leq-\beta<-r\alpha
  28. Re λ β > r α \operatorname{Re}\lambda\geq\beta>r\alpha
  29. x ˙ = x 2 , y ˙ = y . \dot{x}=x^{2},\quad\dot{y}=y.
  30. y = A e - 1 / x y=Ae^{-1/x}
  31. y = A e - 1 / x y=Ae^{-1/x}
  32. a 4 a\approx 4
  33. d x / d t = - a x ( t - 1 ) - 2 x 2 - x 3 {dx}/{dt}=-ax(t-1)-2x^{2}-x^{3}
  34. u 1 ( t ) = x ( t ) u_{1}(t)=x(t)
  35. x ( t - 1 ) u 3 ( t ) x(t-1)\approx u_{3}(t)
  36. d u 2 / d t = 2 ( u 1 - u 2 ) {du_{2}}/{dt}=2(u_{1}-u_{2})
  37. d u 3 / d t = 2 ( u 2 - u 3 ) {du_{3}}/{dt}=2(u_{2}-u_{3})
  38. a = 4 + α a=4+\alpha
  39. d 𝐮 d t = [ 0 0 - 4 2 - 2 0 0 2 - 2 ] 𝐮 + [ - α u 3 - 2 u 1 2 - u 1 3 0 0 ] . \frac{d\,\textbf{u}}{dt}=\left[\begin{array}[]{ccc}0&0&-4\\ 2&-2&0\\ 0&2&-2\end{array}\right]\,\textbf{u}+\left[\begin{array}[]{c}-\alpha u_{3}-2u_% {1}^{2}-u_{1}^{3}\\ 0\\ 0\end{array}\right].
  40. s ( t ) s(t)
  41. s ¯ ( t ) \bar{s}(t)
  42. 𝐮 = [ e i 2 t s + e - i 2 t s ¯ 1 - i 2 e i 2 t s + 1 + i 2 e - i 2 t s ¯ - i 2 e i 2 t s + i 2 e - i 2 t s ¯ ] + O ( α + | s | 2 ) \,\textbf{u}=\left[\begin{array}[]{c}e^{i2t}s+e^{-i2t}\bar{s}\\ \frac{1-i}{2}e^{i2t}s+\frac{1+i}{2}e^{-i2t}\bar{s}\\ -\frac{i}{2}e^{i2t}s+\frac{i}{2}e^{-i2t}\bar{s}\end{array}\right]+{O}(\alpha+|% s|^{2})
  43. d s d t = [ 1 + 2 i 10 α s - 3 + 16 i 15 | s | 2 s ] + O ( α 2 + | s | 4 ) \frac{ds}{dt}=\left[\frac{1+2i}{10}\alpha s-\frac{3+16i}{15}|s|^{2}s\right]+{O% }(\alpha^{2}+|s|^{4})
  44. α > 0 ( a > 4 ) \alpha>0\ (a>4)

Central_subgroup.html

  1. G G
  2. G G
  3. Z ( G ) Z(G)
  4. H H
  5. G G
  6. H Z ( G ) H\leq Z(G)

CEP_subgroup.html

  1. H H
  2. G G
  3. N N
  4. H H
  5. H M H\cap M
  6. M M
  7. G G

Ceramic_engineering.html

  1. σ y = σ 0 + k y d \sigma_{y}=\sigma_{0}+{k_{y}\over\sqrt{d}}

Chakravala_method.html

  1. x 2 = N y 2 + 1 , \,x^{2}=Ny^{2}+1,
  2. x 2 = 61 y 2 + 1 , \,x^{2}=61y^{2}+1,
  3. x = 1766319049 , y = 226153980. \,x=1766319049,y=226153980.
  4. ( x 1 2 - N y 1 2 ) ( x 2 2 - N y 2 2 ) = ( x 1 x 2 + N y 1 y 2 ) 2 - N ( x 1 y 2 + x 2 y 1 ) 2 (x_{1}^{2}-Ny_{1}^{2})(x_{2}^{2}-Ny_{2}^{2})=(x_{1}x_{2}+Ny_{1}y_{2})^{2}-N(x_% {1}y_{2}+x_{2}y_{1})^{2}
  5. x 2 - N y 2 = k x^{2}-Ny^{2}=k
  6. ( x 1 , y 1 , k 1 ) (x_{1},y_{1},k_{1})
  7. ( x 2 , y 2 , k 2 ) (x_{2},y_{2},k_{2})
  8. ( x 1 x 2 + N y 1 y 2 , x 1 y 2 + x 2 y 1 , k 1 k 2 ) . (x_{1}x_{2}+Ny_{1}y_{2}\,,\,x_{1}y_{2}+x_{2}y_{1}\,,\,k_{1}k_{2}).
  9. ( a , b , k ) (a,b,k)
  10. a 2 - N b 2 = k a^{2}-Nb^{2}=k
  11. ( m , 1 , m 2 - N ) (m,1,m^{2}-N)
  12. ( a m + N b , a + b m , k ( m 2 - N ) ) (am+Nb,a+bm,k(m^{2}-N))
  13. gcd ( a , b ) = 1 \gcd(a,b)=1
  14. a 2 - N b 2 = k ( a m + N b k ) 2 - N ( a + b m k ) 2 = m 2 - N k a^{2}-Nb^{2}=k\Rightarrow\left(\frac{am+Nb}{k}\right)^{2}-N\left(\frac{a+bm}{k% }\right)^{2}=\frac{m^{2}-N}{k}
  15. a a m + N b | k | , b a + b m | k | , k m 2 - N k a\leftarrow\frac{am+Nb}{|k|},b\leftarrow\frac{a+bm}{|k|},k\leftarrow\frac{m^{2% }-N}{k}
  16. k = 1 k=1
  17. a 2 - 61 b 2 = 1 a^{2}-61b^{2}=1
  18. a 2 - 61 b 2 = k a^{2}-61b^{2}=k
  19. 8 2 - 61 1 2 = 3 8^{2}-61\cdot 1^{2}=3
  20. ( a , b , k ) = ( 8 , 1 , 3 ) (a,b,k)=(8,1,3)
  21. ( m , 1 , m 2 - 61 ) (m,1,m^{2}-61)
  22. ( 8 m + 61 , 8 + m , 3 ( m 2 - 61 ) ) (8m+61,8+m,3(m^{2}-61))
  23. ( 8 m + 61 3 , 8 + m 3 , m 2 - 61 3 ) . \left(\frac{8m+61}{3},\frac{8+m}{3},\frac{m^{2}-61}{3}\right).
  24. 8 + m 8+m
  25. | m 2 - 61 | |m^{2}-61|
  26. m = 7 m=7
  27. ( 39 , 5 , - 4 ) (39,5,-4)
  28. ( 39 / 2 , 5 / 2 , - 1 ) (39/2,5/2,-1)\,
  29. m = 7 , 11 , 9 m={7,11,9}
  30. ( 1523 / 2 , 195 / 2 , 1 ) (1523/2,195/2,1)\,
  31. ( 1766319049 , 226153980 , 1 ) (1766319049,\,226153980,\,1)
  32. x 2 - 67 y 2 = 1 x^{2}-67y^{2}=1
  33. a 2 - 67 b 2 = k a^{2}-67b^{2}=k
  34. 8 2 - 67 1 2 = - 3 8^{2}-67\cdot 1^{2}=-3
  35. a m + N b | k | , a + b m | k | , and m 2 - N k \frac{am+Nb}{|k|},\frac{a+bm}{|k|},\,\text{ and }\frac{m^{2}-N}{k}
  36. ( a , b , k ) = ( 8 , 1 , - 3 ) (a,b,k)=(8,1,-3)
  37. ( a m + N b | k | , a + b m | k | , m 2 - N k ) \left(\frac{am+Nb}{|k|},\frac{a+bm}{|k|},\frac{m^{2}-N}{k}\right)
  38. a = ( 8 7 + 67 1 ) / 3 = 41 , b = ( 8 + 1 7 ) / 3 = 5 , k = ( 7 2 - 67 ) / ( - 3 ) = 6 a=(8\cdot 7+67\cdot 1)/3=41,b=(8+1\cdot 7)/3=5,k=(7^{2}-67)/(-3)=6
  39. 41 2 - 67 ( 5 ) 2 = 6. 41^{2}-67\cdot(5)^{2}=6.
  40. ( a , b , k ) = ( 41 , 5 , 6 ) (a,b,k)=(41,5,6)
  41. 90 2 - 67 11 2 = - 7. 90^{2}-67\cdot 11^{2}=-7.
  42. 221 2 - 67 27 2 = - 2. 221^{2}-67\cdot 27^{2}=-2.
  43. ( 221 2 + 67 27 2 2 ) 2 - 67 ( 221 27 ) 2 = 1 , \left(\frac{221^{2}+67\cdot 27^{2}}{2}\right)^{2}-67\cdot(221\cdot 27)^{2}=1,
  44. 48842 2 - 67 5967 2 = 1. 48842^{2}-67\cdot 5967^{2}=1.
  45. 67 \sqrt{67}
  46. 48842 5967 \frac{48842}{5967}
  47. 2 × 10 - 9 2\times 10^{-9}
  48. Q n Q_{n}
  49. P n P_{n}

Channel_state_information.html

  1. 𝐲 = 𝐇𝐱 + 𝐧 \mathbf{y}=\mathbf{H}\mathbf{x}+\mathbf{n}
  2. 𝐲 \scriptstyle\mathbf{y}
  3. 𝐱 \scriptstyle\mathbf{x}
  4. 𝐇 \scriptstyle\mathbf{H}
  5. 𝐧 \scriptstyle\mathbf{n}
  6. 𝐧 𝒞 𝒩 ( 𝟎 , 𝐒 ) \mathbf{n}\sim\mathcal{CN}(\mathbf{0},\,\mathbf{S})
  7. 𝐒 \scriptstyle\mathbf{S}
  8. 𝐇 \scriptstyle\mathbf{H}
  9. vec ( 𝐇 estimate ) 𝒞 𝒩 ( vec ( 𝐇 ) , 𝐑 error ) \mbox{vec}~{}(\mathbf{H}_{\textrm{estimate}})\sim\mathcal{CN}(\mbox{vec}~{}(% \mathbf{H}),\,\mathbf{R}_{\textrm{error}})
  10. 𝐇 estimate \scriptstyle\mathbf{H}_{\textrm{estimate}}
  11. 𝐑 error \scriptstyle\mathbf{R}_{\textrm{error}}
  12. vec ( ) \mbox{vec}~{}()
  13. 𝐇 \scriptstyle\mathbf{H}
  14. 𝐇 \scriptstyle\mathbf{H}
  15. vec ( 𝐇 ) 𝒞 𝒩 ( 𝟎 , 𝐑 ) \mbox{vec}~{}(\mathbf{H})\sim\mathcal{CN}(\mathbf{0},\,\mathbf{R})
  16. 𝐑 \scriptstyle\mathbf{R}
  17. 𝐇 \scriptstyle\mathbf{H}
  18. 𝐩 1 , , 𝐩 N \mathbf{p}_{1},\ldots,\mathbf{p}_{N}
  19. 𝐩 i \mathbf{p}_{i}
  20. 𝐲 i = 𝐇𝐩 i + 𝐧 i . \mathbf{y}_{i}=\mathbf{H}\mathbf{p}_{i}+\mathbf{n}_{i}.
  21. 𝐲 i \mathbf{y}_{i}
  22. i = 1 , , N i=1,\ldots,N
  23. 𝐘 = [ 𝐲 1 , , 𝐲 N ] = 𝐇𝐏 + 𝐍 \mathbf{Y}=[\mathbf{y}_{1},\ldots,\mathbf{y}_{N}]=\mathbf{H}\mathbf{P}+\mathbf% {N}
  24. 𝐏 = [ 𝐩 1 , , 𝐩 N ] \scriptstyle\mathbf{P}=[\mathbf{p}_{1},\ldots,\mathbf{p}_{N}]
  25. 𝐍 = [ 𝐧 1 , , 𝐧 N ] \scriptstyle\mathbf{N}=[\mathbf{n}_{1},\ldots,\mathbf{n}_{N}]
  26. 𝐇 \scriptstyle\mathbf{H}
  27. 𝐘 \scriptstyle\mathbf{Y}
  28. 𝐏 \scriptstyle\mathbf{P}
  29. 𝐇 LS-estimate = 𝐘𝐏 H ( 𝐏𝐏 H ) - 1 \mathbf{H}_{\textrm{LS-estimate}}=\mathbf{Y}\mathbf{P}^{H}(\mathbf{P}\mathbf{P% }^{H})^{-1}
  30. ( ) H ()^{H}
  31. tr ( 𝐏𝐏 H ) - 1 \mathrm{tr}(\mathbf{P}\mathbf{P}^{H})^{-1}
  32. tr \mathrm{tr}
  33. 𝐏𝐏 H \mathbf{P}\mathbf{P}^{H}
  34. N N
  35. 𝐏 \scriptstyle\mathbf{P}
  36. vec ( 𝐇 ) 𝒞 𝒩 ( 0 , 𝐑 ) , vec ( 𝐍 ) 𝒞 𝒩 ( 0 , 𝐒 ) . \mbox{vec}~{}(\mathbf{H})\sim\mathcal{CN}(0,\,\mathbf{R}),\quad\mbox{vec}~{}(% \mathbf{N})\sim\mathcal{CN}(0,\,\mathbf{S}).
  37. vec ( 𝐇 MMSE-estimate ) = ( 𝐑 - 1 + ( 𝐏 T 𝐈 ) H 𝐒 - 1 ( 𝐏 T 𝐈 ) ) - 1 ( 𝐏 T 𝐈 ) H 𝐒 - 1 vec ( 𝐘 ) \mbox{vec}~{}(\mathbf{H}_{\textrm{MMSE-estimate}})=\left(\mathbf{R}^{-1}+(% \mathbf{P}^{T}\,\otimes\,\mathbf{I})^{H}\mathbf{S}^{-1}(\mathbf{P}^{T}\,% \otimes\,\mathbf{I})\right)^{-1}(\mathbf{P}^{T}\,\otimes\,\mathbf{I})^{H}% \mathbf{S}^{-1}\mbox{vec}~{}(\mathbf{Y})
  38. \otimes
  39. 𝐈 \scriptstyle\mathbf{I}
  40. tr ( 𝐑 - 1 + ( 𝐏 T 𝐈 ) H 𝐒 - 1 ( 𝐏 T 𝐈 ) ) - 1 \mathrm{tr}\left(\mathbf{R}^{-1}+(\mathbf{P}^{T}\,\otimes\,\mathbf{I})^{H}% \mathbf{S}^{-1}(\mathbf{P}^{T}\,\otimes\,\mathbf{I})\right)^{-1}
  41. 𝐏 \scriptstyle\mathbf{P}
  42. N N
  43. 𝐑 \scriptstyle\mathbf{R}
  44. 𝐒 \scriptstyle\mathbf{S}

Characteristic_function_(probability_theory).html

  1. F X ( x ) = E [ 𝟏 { X x } ] , F_{X}(x)=\operatorname{E}\left[\mathbf{1}_{\{X\leq x\}}\right],
  2. φ X ( t ) = E [ e i t X ] \varphi_{X}(t)=\operatorname{E}\left[e^{itX}\right]
  3. φ X ( - i t ) = M X ( t ) . \varphi_{X}(-it)=M_{X}(t).
  4. { φ X : 𝐑 𝐂 φ X ( t ) = E [ e i t X ] = 𝐑 e i t x d F X ( x ) = 𝐑 e i t x f X ( x ) d x = 0 1 e i t Q X ( p ) d p \begin{cases}\varphi_{X}\!:\mathbf{R}\to\mathbf{C}\\ \varphi_{X}(t)=\operatorname{E}\left[e^{itX}\right]=\int_{\mathbf{R}}e^{itx}\,% dF_{X}(x)=\int_{\mathbf{R}}e^{itx}f_{X}(x)\,dx=\int_{0}^{1}e^{itQ_{X}(p)}\,dp% \end{cases}
  5. p ^ \scriptstyle\hat{p}
  6. f ^ \scriptstyle\hat{f}
  7. φ X ( t ) = E [ exp ( i t T X ) ] , \varphi_{X}(t)=\operatorname{E}\left[\exp({i\,t^{T}\!X})\right],
  8. φ X ( t ) = E [ exp ( i tr ( t T X ) ) ] , \varphi_{X}(t)=\operatorname{E}\left[\exp\left({i\,\operatorname{tr}(t^{T}\!X)% }\right)\right],
  9. φ X ( t ) = E [ exp ( i Re ( t ¯ X ) ) ] , \varphi_{X}(t)=\operatorname{E}\left[\exp({i\,\operatorname{Re}(\overline{t}X)% })\right],
  10. φ X ( t ) = E [ exp ( i Re ( t * X ) ) ] , \varphi_{X}(t)=\operatorname{E}\left[\exp({i\,\operatorname{Re}(t^{*}\!X)})% \right],
  11. φ X ( t ) = E [ exp ( i 𝐑 t ( s ) X ( s ) d s ) ] . \varphi_{X}(t)=\operatorname{E}\left[\exp\left({i\int_{\mathbf{R}}t(s)X(s)ds}% \right)\right].
  12. T {}^{T}
  13. e i t a \!e^{ita}
  14. 1 - p + p e i t \!1-p+pe^{it}
  15. ( 1 - p + p e i t ) n \!(1-p+pe^{it})^{n}
  16. ( 1 - p 1 - p e i t ) r \biggl(\frac{1-p}{1-pe^{i\,t}}\biggr)^{\!r}
  17. e λ ( e i t - 1 ) \!e^{\lambda(e^{it}-1)}
  18. e i t b - e i t a i t ( b - a ) \!\frac{e^{itb}-e^{ita}}{it(b-a)}
  19. e i t μ 1 + b 2 t 2 \!\frac{e^{it\mu}}{1+b^{2}t^{2}}
  20. e i t μ - 1 2 σ 2 t 2 \!e^{it\mu-\frac{1}{2}\sigma^{2}t^{2}}
  21. ( 1 - 2 i t ) - k / 2 \!(1-2it)^{-k/2}
  22. e i t μ - θ | t | \!e^{it\mu-\theta|t|}
  23. ( 1 - i t θ ) - k \!(1-it\theta)^{-k}
  24. ( 1 - i t λ - 1 ) - 1 \!(1-it\lambda^{-1})^{-1}
  25. e i t T μ - 1 2 t T Σ t \!e^{it^{T}\mu-\frac{1}{2}t^{T}\Sigma t}
  26. e i t T μ - t T Σ t \!e^{it^{T}\mu-\sqrt{t^{T}\Sigma t}}
  27. φ X 1 = φ X 2 \varphi_{X_{1}}=\varphi_{X_{2}}
  28. E [ X k ] = ( - i ) k φ X ( k ) ( 0 ) . \operatorname{E}[X^{k}]=(-i)^{k}\varphi_{X}^{(k)}(0).
  29. φ X ( k ) ( 0 ) = i k E [ X k ] \varphi_{X}^{(k)}(0)=i^{k}\operatorname{E}[X^{k}]
  30. φ a 1 X 1 + + a n X n ( t ) = φ X 1 ( a 1 t ) φ X n ( a n t ) . \varphi_{a_{1}X_{1}+\cdots+a_{n}X_{n}}(t)=\varphi_{X_{1}}(a_{1}t)\cdots\varphi% _{X_{n}}(a_{n}t).
  31. φ X 1 + X 2 ( t ) = φ X 1 ( t ) φ X 2 ( t ) . \varphi_{X_{1}+X_{2}}(t)=\varphi_{X_{1}}(t)\cdot\varphi_{X_{2}}(t).
  32. f X ( x ) = F X ( x ) = 1 2 π 𝐑 e - i t x φ X ( t ) d t , f_{X}(x)=F_{X}^{\prime}(x)=\frac{1}{2\pi}\int_{\mathbf{R}}e^{-itx}\varphi_{X}(% t)dt,
  33. f X ( x ) = d μ X d λ ( x ) = 1 ( 2 π ) n 𝐑 n e - i ( t x ) φ X ( t ) λ ( d t ) . f_{X}(x)=\frac{d\mu_{X}}{d\lambda}(x)=\frac{1}{(2\pi)^{n}}\int_{\mathbf{R}^{n}% }e^{-i(t\cdot x)}\varphi_{X}(t)\lambda(dt).
  34. F ( x + h ) - F ( x - h ) 2 h = 1 2 π - sin h t h t e - i t x φ X ( t ) d t . \frac{F(x+h)-F(x-h)}{2h}=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{\sin ht}{% ht}e^{-itx}\varphi_{X}(t)\,dt.
  35. F ( b ) F(b)
  36. a a
  37. F ( a ) = 0. F(a)=0.
  38. a - a\to-\infty
  39. F ( b ) F(b)
  40. μ X ( { a < x < b } ) = 1 ( 2 π ) n lim T 1 lim T n - T 1 t 1 T 1 - T n t n T n k = 1 n ( e - i t k a k - e - i t k b k i t k ) φ X ( t ) λ ( d t 1 × × d t n ) \mu_{X}\big(\{a<x<b\}\big)=\frac{1}{(2\pi)^{n}}\lim_{T_{1}\to\infty}\cdots\lim% _{T_{n}\to\infty}\int\limits_{-T_{1}\leq t_{1}\leq T_{1}}\cdots\int\limits_{-T% _{n}\leq t_{n}\leq T_{n}}\prod_{k=1}^{n}\left(\frac{e^{-it_{k}a_{k}}-e^{-it_{k% }b_{k}}}{it_{k}}\right)\varphi_{X}(t)\lambda(dt_{1}\times\cdots\times dt_{n})
  41. F X ( a ) - F X ( a - 0 ) = lim T 1 2 T - T + T e - i t a φ X ( t ) d t F_{X}(a)-F_{X}(a-0)=\lim_{T\to\infty}\frac{1}{2T}\int_{-T}^{+T}e^{-ita}\varphi% _{X}(t)dt
  42. μ X ( { a } ) = lim T 1 lim T n ( k = 1 n 1 2 T k ) - T T e - i ( t a ) φ X ( t ) λ ( d t ) \mu_{X}(\{a\})=\lim_{T_{1}\to\infty}\cdots\lim_{T_{n}\to\infty}\left(\prod_{k=% 1}^{n}\frac{1}{2T_{k}}\right)\int_{-T}^{T}e^{-i(t\cdot a)}\varphi_{X}(t)% \lambda(dt)
  43. F X ( x ) = 1 2 - 1 π 0 Im [ e - i t x φ X ( t ) ] t d t . F_{X}(x)=\frac{1}{2}-\frac{1}{\pi}\int_{0}^{\infty}\frac{\operatorname{Im}[e^{% -itx}\varphi_{X}(t)]}{t}\,dt.
  44. z z
  45. Im ( z ) = ( z - z * ) / 2 i \mathrm{Im}(z)=(z-z^{*})/2i
  46. n a n φ n ( t ) \scriptstyle\sum_{n}a_{n}\varphi_{n}(t)
  47. a n 0 , n a n = 1 \scriptstyle a_{n}\geq 0,\ \sum_{n}a_{n}=1
  48. φ ¯ \bar{\varphi}
  49. φ ( t ) = 𝐑 g ( t + θ ) g ( θ ) ¯ d θ . \varphi(t)=\int_{\mathbf{R}}g(t+\theta)\overline{g(\theta)}d\theta.
  50. ( - 1 ) n ( 𝐑 φ ( p t ) e - t 2 2 H 2 n ( t ) d t ) 0 (-1)^{n}\left(\int_{\mathbf{R}}\varphi(pt)e^{-\frac{t^{2}}{2}}H_{2n}(t)dt% \right)\geq 0
  51. S n = i = 1 n a i X i , S_{n}=\sum_{i=1}^{n}a_{i}X_{i},\,\!
  52. φ S n ( t ) = φ X 1 ( a 1 t ) φ X 2 ( a 2 t ) φ X n ( a n t ) \varphi_{S_{n}}(t)=\varphi_{X_{1}}(a_{1}t)\varphi_{X_{2}}(a_{2}t)\cdots\varphi% _{X_{n}}(a_{n}t)\,\!
  53. φ X + Y ( t ) = E [ e i t ( X + Y ) ] = E [ e i t X e i t Y ] = E [ e i t X ] E [ e i t Y ] = φ X ( t ) φ Y ( t ) \varphi_{X+Y}(t)=\operatorname{E}\left[e^{it(X+Y)}\right]=\operatorname{E}% \left[e^{itX}e^{itY}\right]=\operatorname{E}\left[e^{itX}\right]E\left[e^{itY}% \right]=\varphi_{X}(t)\varphi_{Y}(t)
  54. φ X ¯ ( t ) = φ X ( t n ) n \varphi_{\overline{X}}(t)=\varphi_{X}\!\left(\tfrac{t}{n}\right)^{n}
  55. E [ X n ] = i - n φ X ( n ) ( 0 ) = i - n [ d n d t n φ X ( t ) ] t = 0 \operatorname{E}\left[X^{n}\right]=i^{-n}\,\varphi_{X}^{(n)}(0)=i^{-n}\,\left[% \frac{d^{n}}{dt^{n}}\varphi_{X}(t)\right]_{t=0}\,\!
  56. ( 1 - θ i t ) - k . (1-\theta\,i\,t)^{-k}.
  57. X Γ ( k 1 , θ ) and Y Γ ( k 2 , θ ) X~{}\sim\Gamma(k_{1},\theta)\mbox{ and }~{}Y\sim\Gamma(k_{2},\theta)\,
  58. φ X ( t ) = ( 1 - θ i t ) - k 1 , φ Y ( t ) = ( 1 - θ i t ) - k 2 \varphi_{X}(t)=(1-\theta\,i\,t)^{-k_{1}},\,\qquad\varphi_{Y}(t)=(1-\theta\,i\,% t)^{-k_{2}}
  59. φ X + Y ( t ) = φ X ( t ) φ Y ( t ) = ( 1 - θ i t ) - k 1 ( 1 - θ i t ) - k 2 = ( 1 - θ i t ) - ( k 1 + k 2 ) . \varphi_{X+Y}(t)=\varphi_{X}(t)\varphi_{Y}(t)=(1-\theta\,i\,t)^{-k_{1}}(1-% \theta\,i\,t)^{-k_{2}}=\left(1-\theta\,i\,t\right)^{-(k_{1}+k_{2})}.
  60. X + Y Γ ( k 1 + k 2 , θ ) X+Y\sim\Gamma(k_{1}+k_{2},\theta)\,
  61. i { 1 , , n } : X i Γ ( k i , θ ) i = 1 n X i Γ ( i = 1 n k i , θ ) . \forall i\in\{1,\ldots,n\}:X_{i}\sim\Gamma(k_{i},\theta)\qquad\Rightarrow% \qquad\sum_{i=1}^{n}X_{i}\sim\Gamma\left(\sum_{i=1}^{n}k_{i},\theta\right).
  62. φ X ( t ) = e i t X = 𝐑 e i t x p ( x ) d x = ( 𝐑 e - i t x p ( x ) d x ) ¯ = P ( t ) ¯ , \varphi_{X}(t)=\langle e^{itX}\rangle=\int_{\mathbf{R}}e^{itx}p(x)\,dx=% \overline{\left(\int_{\mathbf{R}}e^{-itx}p(x)\,dx\right)}=\overline{P(t)},
  63. p ( x ) = 1 2 π 𝐑 e i t x P ( t ) d t = 1 2 π 𝐑 e i t x φ X ( t ) ¯ d t . p(x)=\frac{1}{2\pi}\int_{\mathbf{R}}e^{itx}P(t)\,dt=\frac{1}{2\pi}\int_{% \mathbf{R}}e^{itx}\overline{\varphi_{X}(t)}\,dt.

Charge_amplifier.html

  1. v i n R 1 \frac{v_{in}}{R_{1}}
  2. i 1 = I B + i F i_{\,\text{1}}=I_{\,\text{B}}+i_{\,\text{F}}
  3. I B = 0 I_{\,\text{B}}=0
  4. i 1 = i F i_{\,\text{1}}=i_{\,\text{F}}
  5. I C = C d V c d t I_{\,\text{C}}=C\frac{dV_{\,\text{c}}}{dt}
  6. v in - v 2 R 1 = C F d ( v 2 - v o ) d t \frac{v_{\,\text{in}}-v_{\,\text{2}}}{R_{\,\text{1}}}=C_{\,\text{F}}\frac{d(v_% {\,\text{2}}-v_{\,\text{o}})}{dt}
  7. v 2 = v 1 = 0 v_{2}=v_{1}=0
  8. v in R 1 = - C F d v o d t \frac{v_{\,\text{in}}}{R_{\,\text{1}}}=-C_{\,\text{F}}\frac{dv_{\,\text{o}}}{dt}
  9. 0 t v in R 1 d t = - 0 t C F d v o d t d t \int_{0}^{t}\frac{v_{\,\text{in}}}{R_{\,\text{1}}}\ dt\ =-\int_{0}^{t}C_{\,% \text{F}}\frac{dv_{\,\text{o}}}{dt}\,dt
  10. v o = - 1 R 1 C F 0 t v in d t v_{\,\text{o}}=-\frac{1}{R_{\,\text{1}}C_{\,\text{F}}}\int_{0}^{t}v_{\,\text{% in}}\,dt
  11. I B I_{B}
  12. v in = 0 v_{\,\text{in}}=0
  13. I B I_{B}
  14. v in v_{\,\text{in}}
  15. R on = R 1 | | R f R_{\,\text{on}}=R_{1}||R_{f}
  16. V E = ( R f R 1 + 1 ) V I O S V\text{E}=\left(\frac{R\text{f}}{R_{1}}+1\right)V_{IOS}
  17. R F R_{F}
  18. V E = ( R f R 1 + 1 ) ( V I O S + I B I ( R f R 1 ) ) V\text{E}=\left(\frac{R\text{f}}{R_{1}}+1\right)\left(V_{IOS}+I_{BI}\left(R% \text{f}\parallel R_{1}\right)\right)
  19. V I O S V_{IOS}
  20. I B I I_{BI}
  21. R f R 1 R_{f}\parallel R_{1}

Charge_number.html

  1. Cl - \mathrm{Cl}^{-}
  2. - 1 e -1\cdot e
  3. - 1 -1
  4. z z
  5. Q = z e Q=ze

Charles_Read_(mathematician).html

  1. 1 \ell_{1}

Chebyshev_equation.html

  1. ( 1 - x 2 ) d 2 y d x 2 - x d y d x + p 2 y = 0 (1-x^{2}){d^{2}y\over dx^{2}}-x{dy\over dx}+p^{2}y=0
  2. y = n = 0 a n x n y=\sum_{n=0}^{\infty}a_{n}x^{n}
  3. a n + 2 = ( n - p ) ( n + p ) ( n + 1 ) ( n + 2 ) a n . a_{n+2}={(n-p)(n+p)\over(n+1)(n+2)}a_{n}.
  4. [ - 1 , 1 ] [-1,1]
  5. F ( x ) = 1 - p 2 2 ! x 2 + ( p - 2 ) p 2 ( p + 2 ) 4 ! x 4 - ( p - 4 ) ( p - 2 ) p 2 ( p + 2 ) ( p + 4 ) 6 ! x 6 + F(x)=1-\frac{p^{2}}{2!}x^{2}+\frac{(p-2)p^{2}(p+2)}{4!}x^{4}-\frac{(p-4)(p-2)p% ^{2}(p+2)(p+4)}{6!}x^{6}+\cdots
  6. G ( x ) = x - ( p - 1 ) ( p + 1 ) 3 ! x 3 + ( p - 3 ) ( p - 1 ) ( p + 1 ) ( p + 3 ) 5 ! x 5 - . G(x)=x-\frac{(p-1)(p+1)}{3!}x^{3}+\frac{(p-3)(p-1)(p+1)(p+3)}{5!}x^{5}-\cdots.
  7. T p ( x ) = ( - 1 ) p / 2 F ( x ) T_{p}(x)=(-1)^{p/2}\ F(x)\,
  8. T p ( x ) = ( - 1 ) ( p - 1 ) / 2 p G ( x ) T_{p}(x)=(-1)^{(p-1)/2}\ p\ G(x)\,

Cheeger_bound.html

  1. X X
  2. K ( x , y ) K(x,y)
  3. X X
  4. π \pi
  5. Q ( x , y ) = π ( x ) K ( x , y ) Q(x,y)=\pi(x)K(x,y)
  6. A , B X A,B\subset X
  7. Q ( A × B ) = x A , y B Q ( x , y ) . Q(A\times B)=\sum_{x\in A,y\in B}Q(x,y).
  8. Φ \Phi
  9. Φ = min S X , π ( S ) 1 2 Q ( S × S c ) π ( S ) . \Phi=\min_{S\subset X,\pi(S)\leq\frac{1}{2}}\frac{Q(S\times S^{c})}{\pi(S)}.
  10. K , K,
  11. | X | |X|
  12. | X | |X|
  13. ( K ϕ ) ( x ) = y K ( x , y ) ϕ ( y ) (K\phi)(x)=\sum_{y}K(x,y)\phi(y)\,
  14. λ 1 λ 2 λ n \lambda_{1}\geq\lambda_{2}\geq\cdots\geq\lambda_{n}
  15. λ 1 = 1 \lambda_{1}=1
  16. λ 2 \lambda_{2}
  17. 1 - 2 Φ λ 2 1 - Φ 2 2 . 1-2\Phi\leq\lambda_{2}\leq 1-\frac{\Phi^{2}}{2}.