wpmath0000008_10

Quasi-open_map.html

  1. f : X Y f:X\to Y
  2. X X
  3. Y Y
  4. U X U\subset X
  5. f ( U ) f(U)
  6. Y Y
  7. f : X Y f:X\to Y
  8. f f
  9. f f
  10. f f
  11. f f
  12. f f
  13. f f
  14. f : X Y f:X\to Y
  15. g : Y Z g:Y\to Z
  16. h = g f : X Z h=g\circ f:X\to Z

Quasi-set_theory.html

  1. 𝔔 \mathfrak{Q}
  2. 𝔔 \mathfrak{Q}
  3. 𝔔 \mathfrak{Q}
  4. 𝔔 \mathfrak{Q}
  5. 𝔔 \mathfrak{Q}
  6. 𝔔 \mathfrak{Q}
  7. 𝔔 \mathfrak{Q}
  8. 𝔔 \mathfrak{Q}
  9. 𝔔 \mathfrak{Q}
  10. 𝔔 \mathfrak{Q}
  11. 𝔔 \mathfrak{Q}
  12. 𝔔 \mathfrak{Q}
  13. 𝔔 \mathfrak{Q}
  14. \equiv
  15. 𝔔 \mathfrak{Q}
  16. 𝔔 \mathfrak{Q}
  17. 𝔔 \mathfrak{Q}
  18. 𝔔 \mathfrak{Q}
  19. 𝔔 \mathfrak{Q}
  20. 𝔔 \mathfrak{Q}
  21. 𝔔 \mathfrak{Q}
  22. 𝔔 \mathfrak{Q}
  23. 𝔔 \mathfrak{Q}
  24. 𝔔 \mathfrak{Q}
  25. 𝔔 \mathfrak{Q}
  26. 𝔔 \mathfrak{Q}
  27. 𝔔 \mathfrak{Q}
  28. 𝔔 \mathfrak{Q}
  29. { x : φ ( x ) } \{x:\varphi(x)\}
  30. [ x : φ ( x ) ] [x:\varphi(x)]
  31. 𝔔 \mathfrak{Q}
  32. x y x\not=y
  33. \equiv
  34. x = E y x=_{E}y
  35. z ( z x z y ) \forall z(z\in x\Rightarrow z\in y)
  36. z ( x z y z ) \forall z(x\in z\Rightarrow y\in z)
  37. = E =_{E}
  38. α \alpha
  39. β < α \beta<\alpha
  40. β \beta
  41. α \alpha
  42. q c ( x ) qc(x)
  43. w x w\in x
  44. ( x - { w } ) { z } = x (x-\{w\})\cup\{z\}=x
  45. z = w z=w
  46. 𝔔 \mathfrak{Q}
  47. [ [ z ] ] [[z]]
  48. [ [ w ] ] [[w]]
  49. z x z\in x
  50. w z w\equiv z
  51. w x w\notin x
  52. [ [ w ] ] [[w]]
  53. ( x - [ [ z ] ] ) [ [ w ] ] x (x-[[z]])\cup[[w]]\equiv x
  54. ( x - [ [ z ] ] ) [ [ w ] ] (x-[[z]])\cup[[w]]
  55. ( x - [ [ z ] ] ) [ [ w ] ] (x-[[z]])\cup[[w]]
  56. 𝔔 \mathfrak{Q}

Quasiconvex_function.html

  1. ( - , a ) (-\infty,a)
  2. f : S f:S\to\mathbb{R}
  3. x , y S x,y\in S
  4. λ [ 0 , 1 ] \lambda\in[0,1]
  5. f ( λ x + ( 1 - λ ) y ) max { f ( x ) , f ( y ) } . f(\lambda x+(1-\lambda)y)\leq\max\big\{f(x),f(y)\big\}.
  6. f ( x ) f(x)
  7. S α ( f ) = { x | f ( x ) α } S_{\alpha}(f)=\{x|f(x)\leq\alpha\}
  8. f ( λ x + ( 1 - λ ) y ) < max { f ( x ) , f ( y ) } f(\lambda x+(1-\lambda)y)<\max\big\{f(x),f(y)\big\}
  9. x y x\neq y
  10. λ ( 0 , 1 ) \lambda\in(0,1)
  11. f f
  12. f f
  13. f ( λ x + ( 1 - λ ) y ) min { f ( x ) , f ( y ) } . f(\lambda x+(1-\lambda)y)\geq\min\big\{f(x),f(y)\big\}.
  14. f ( λ x + ( 1 - λ ) y ) > min { f ( x ) , f ( y ) } f(\lambda x+(1-\lambda)y)>\min\big\{f(x),f(y)\big\}
  15. S S\subset\mathbb{R}
  16. f = max { w 1 f 1 , , w n f n } f=\max\left\{w_{1}f_{1},\ldots,w_{n}f_{n}\right\}
  17. w i w_{i}
  18. g : n g:\mathbb{R}^{n}\rightarrow\mathbb{R}
  19. h : h:\mathbb{R}\rightarrow\mathbb{R}
  20. f = h g f=h\circ g
  21. f ( x , y ) f(x,y)
  22. C C
  23. h ( x ) = inf y C f ( x , y ) h(x)=\inf_{y\in C}f(x,y)
  24. f ( x ) , g ( x ) f(x),g(x)
  25. ( f + g ) ( x ) = f ( x ) + g ( x ) (f+g)(x)=f(x)+g(x)
  26. f ( x ) , g ( y ) f(x),g(y)
  27. h ( x , y ) = f ( x ) + g ( y ) h(x,y)=f(x)+g(y)
  28. x log ( x ) x\mapsto\log(x)
  29. x x x\mapsto\lfloor x\rfloor
  30. x f ( x ) x\mapsto f(x)
  31. y g ( y ) y\mapsto g(y)
  32. ( x , y ) f ( x ) g ( y ) (x,y)\mapsto f(x)g(y)

Quasinorm.html

  1. x + y K ( x + y ) \|x+y\|\leq K(\|x\|+\|y\|)
  2. K > 1. K>1.
  3. ( A , ) (A,\|\cdot\|)
  4. x y K x y \|xy\|\leq K\|x\|\cdot\|y\|
  5. x , y A x,y\in A

Quasinormal_operator.html

  1. A ( A * A ) = ( A * A ) A . A(A^{*}A)=(A^{*}A)A.\,
  2. U P P = P U P . UPP=PUP.\,
  3. A * A = ( U P ) * U P = P U ( P U ) * = A A * . A^{*}A=(UP)^{*}UP=PU(PU)^{*}=AA^{*}.\,
  4. A = σ ( A ) λ d E ( λ ) . A=\int_{\sigma(A)}\lambda dE(\lambda).\,

Quasitransitive_relation.html

  1. ( a T b ) ¬ ( b T a ) ( b T c ) ¬ ( c T b ) ( a T c ) ¬ ( c T a ) . (a\operatorname{T}b)\wedge\neg(b\operatorname{T}a)\wedge(b\operatorname{T}c)% \wedge\neg(c\operatorname{T}b)\Rightarrow(a\operatorname{T}c)\wedge\neg(c% \operatorname{T}a).
  2. ( a P b ) ( a T b ) ¬ ( b T a ) . (a\operatorname{P}b)\Leftrightarrow(a\operatorname{T}b)\wedge\neg(b% \operatorname{T}a).

Query_(complexity).html

  1. σ \sigma
  2. τ \tau
  3. STRUC [ σ ] \mbox{STRUC}~{}[\sigma]
  4. STRUC [ τ ] \mbox{STRUC}~{}[\tau]
  5. I : STRUC [ σ ] STRUC [ τ ] I:\mbox{STRUC}~{}[\sigma]\to\mbox{STRUC}~{}[\tau]
  6. I ( 𝔄 ) I ( 𝔅 ) I(\mathfrak{A})\equiv I(\mathfrak{B})
  7. 𝔄 \mathfrak{A}
  8. 𝔅 \mathfrak{B}

Quotition_and_partition.html

  1. 6 ÷ 2 6\div 2
  2. 6 = 2 + 2 + 2 3 parts . 6=\underbrace{2+2+2}_{\,\text{3 parts}}.
  3. 6 ÷ 2 = 3. 6\div 2=3.\,
  4. 6 = 3 + 3 2 parts . 6=\underbrace{3+3}_{\,\text{2 parts}}.
  5. 6 ÷ 2 = 3. 6\div 2=3.

R-factor_(crystallography).html

  1. R = | | F obs | - | F calc | | | F obs | R=\frac{\sum{||F\text{obs}|-|F\text{calc}||}}{\sum{|F\text{obs}|}}
  2. I h k l | F ( h k l ) | 2 I_{hkl}\propto|F(hkl)|^{2}
  3. R R
  4. R R
  5. R F r e e R_{Free}
  6. R F r e e R_{Free}
  7. R F r e e R_{Free}
  8. R R
  9. R F r e e R_{Free}
  10. R sym R\text{sym}
  11. R merge R\text{merge}

R_(disambiguation).html

  1. \mathbb{R}

Rabi_frequency.html

  1. χ i , j = d i , j E 0 \chi_{i,j}={\vec{d}_{i,j}\cdot\vec{E}_{0}\over\hbar}
  2. d i , j \scriptstyle{\vec{d}_{i,j}}
  3. i j \scriptstyle{i\rightarrow j}
  4. E 0 = ϵ ^ E 0 \scriptstyle{\vec{E}_{0}=\hat{\epsilon}E_{0}}
  5. \scriptstyle{\hbar}
  6. d i , j = d j , i * \scriptstyle{\vec{d}_{i,j}=\vec{d}_{j,i}^{*}}
  7. χ i , j = χ j , i * \scriptstyle{\chi_{i,j}=\chi_{j,i}^{*}}
  8. ϵ ^ E 0 \scriptstyle{\hat{\epsilon}E_{0}}
  9. cos θ \cos\theta
  10. θ \theta
  11. Ω i , j \Omega_{i,j}
  12. Ω i , j = | χ i , j | 2 + Δ 2 \Omega_{i,j}=\sqrt{|\chi_{i,j}|^{2}+\Delta^{2}}
  13. Δ = ω light - ω transition \Delta=\omega\text{light}-\omega\text{transition}

Rabi_problem.html

  1. x ¨ a + 2 τ 0 x ˙ a + ω a 2 x a = e m E ( t , 𝐫 a ) \ddot{x}_{a}+\frac{2}{\tau_{0}}\dot{x}_{a}+\omega_{a}^{2}x_{a}=\frac{e}{m}E(t,% \mathbf{r}_{a})
  2. ω a \omega_{a}
  3. τ 0 \tau_{0}
  4. 2 τ 0 = 2 e 2 ω a 2 3 m c 3 \frac{2}{\tau_{0}}=\frac{2e^{2}\omega_{a}^{2}}{3mc^{3}}
  5. E = E 0 [ e i ω t + e - i ω t ] E=E_{0}[e^{i\omega t}+e^{-i\omega t}]
  6. x a = x 0 ( u a cos ω t + v a sin ω t ) x_{a}=x_{0}(u_{a}\cos\omega t+v_{a}\sin\omega t)
  7. ω ω a \omega\approx\omega_{a}
  8. u ˙ a ω u a \dot{u}_{a}\ll\omega u_{a}
  9. v ˙ a ω v a \dot{v}_{a}\ll\omega v_{a}
  10. u ¨ a ω 2 u a \ddot{u}_{a}\ll\omega^{2}u_{a}
  11. v ¨ a ω 2 v a \ddot{v}_{a}\ll\omega^{2}v_{a}
  12. u ˙ = - δ v - u T \dot{u}=-\delta v-\frac{u}{T}
  13. v ˙ = δ u - v T + κ E 0 \dot{v}=\delta u-\frac{v}{T}+\kappa E_{0}
  14. τ 0 \tau_{0}
  15. δ = ω - ω a \delta=\omega-\omega_{a}
  16. κ \kappa
  17. κ = def e m ω x 0 \kappa\ \stackrel{\mathrm{def}}{=}\ \frac{e}{m\omega x_{0}}
  18. u ( t ; δ ) = [ u 0 cos δ t - v 0 sin δ t ] e - t / T + κ E 0 0 t d t sin δ ( t - t ) e - ( t - t ) / T u(t;\delta)=[u_{0}\cos\delta t-v_{0}\sin\delta t]e^{-t/T}+\kappa E_{0}\int_{0}% ^{t}dt^{\prime}\sin\delta(t-t^{\prime})e^{-(t-t^{\prime})/T}
  19. v ( t ; δ ) = [ u 0 cos δ t + v 0 sin δ t ] e - t / T - κ E 0 0 t d t cos δ ( t - t ) e - ( t - t ) / T v(t;\delta)=[u_{0}\cos\delta t+v_{0}\sin\delta t]e^{-t/T}-\kappa E_{0}\int_{0}% ^{t}dt^{\prime}\cos\delta(t-t^{\prime})e^{-(t-t^{\prime})/T}
  20. x a ( t ) = e m E 0 ( e i ω t ω a 2 - ω 2 + 2 i ω / T + c . c . ) x_{a}(t)=\frac{e}{m}E_{0}\left(\frac{e^{i\omega t}}{\omega_{a}^{2}-\omega^{2}+% 2i\omega/T}+\mathrm{c.c.}\right)
  21. u ˙ = - δ v \dot{u}=-\delta v
  22. v ˙ = δ u + κ E w \dot{v}=\delta u+\kappa Ew
  23. w ˙ = - κ E v \dot{w}=-\kappa Ev
  24. ω \omega
  25. d d t [ u v w ] = [ 0 - δ 0 δ 0 κ E 0 - κ E 0 ] [ u v w ] \frac{d}{dt}\begin{bmatrix}u\\ v\\ w\\ \end{bmatrix}=\begin{bmatrix}0&-\delta&0\\ \delta&0&\kappa E\\ 0&-\kappa E&0\end{bmatrix}\begin{bmatrix}u\\ v\\ w\\ \end{bmatrix}
  26. d ρ d t = 𝛀 × ρ \frac{d\mathbf{\rho}}{dt}=\mathbf{\Omega}\times\mathbf{\rho}
  27. ρ = ( u , v , w ) \mathbf{\rho}=(u,v,w)
  28. 𝛀 = ( - κ E , 0 , δ ) \mathbf{\Omega}=(-\kappa E,0,\delta)
  29. E = E 0 ( e i ω t + c . c . ) E=E_{0}(e^{i\omega t}+\mathrm{c.c.})
  30. [ u v w ] = [ cos χ 0 sin χ 0 1 0 - sin χ 0 cos χ ] [ u v w ] \begin{bmatrix}u\\ v\\ w\end{bmatrix}=\begin{bmatrix}\cos\chi&0&\sin\chi\\ 0&1&0\\ -\sin\chi&0&\cos\chi\end{bmatrix}\begin{bmatrix}u^{\prime}\\ v^{\prime}\\ w^{\prime}\end{bmatrix}
  31. [ u v w ] = [ 1 0 0 0 cos Ω t sin Ω t 0 - sin Ω t cos Ω t ] [ u ′′ v ′′ w ′′ ] \begin{bmatrix}u^{\prime}\\ v^{\prime}\\ w^{\prime}\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&\cos\Omega t&\sin\Omega t\\ 0&-\sin\Omega t&\cos\Omega t\end{bmatrix}\begin{bmatrix}u^{\prime\prime}\\ v^{\prime\prime}\\ w^{\prime\prime}\end{bmatrix}
  32. tan χ = δ κ E 0 \tan\chi=\frac{\delta}{\kappa E_{0}}
  33. Ω ( δ ) = δ 2 + ( κ E 0 ) 2 \Omega(\delta)=\sqrt{\delta^{2}+(\kappa E_{0})^{2}}
  34. Ω ( δ ) \Omega(\delta)
  35. δ = 0 \delta=0
  36. κ E 0 \kappa E_{0}
  37. Δ t = π / κ E 0 \Delta t=\pi/\kappa E_{0}
  38. π \pi
  39. π \pi
  40. [ u v w ] = [ ( κ E 0 ) 2 + δ 2 cos Ω t Ω 2 - δ Ω sin Ω t - δ κ E 0 Ω 2 ( 1 - cos Ω t ) δ Ω sin Ω t cos Ω t κ E 0 Ω sin Ω t δ κ E 0 Ω 2 ( 1 - cos Ω t ) - κ E 0 Ω sin Ω t δ 2 + ( κ E 0 ) 2 cos Ω t Ω 2 ] [ u 0 v 0 w 0 ] \begin{bmatrix}u\\ v\\ w\end{bmatrix}=\begin{bmatrix}\frac{(\kappa E_{0})^{2}+\delta^{2}\cos\Omega t}% {\Omega^{2}}&-\frac{\delta}{\Omega}\sin{\Omega t}&-\frac{\delta\kappa E_{0}}{% \Omega^{2}}(1-\cos\Omega t)\\ \frac{\delta}{\Omega}\sin\Omega t&\cos\Omega t&\frac{\kappa E_{0}}{\Omega}\sin% \Omega t\\ \frac{\delta\kappa E_{0}}{\Omega^{2}}(1-\cos\Omega t)&-\frac{\kappa E_{0}}{% \Omega}\sin{\Omega t}&\frac{\delta^{2}+(\kappa E_{0})^{2}\cos\Omega t}{\Omega^% {2}}\end{bmatrix}\begin{bmatrix}u_{0}\\ v_{0}\\ w_{0}\end{bmatrix}
  41. w ( t ; δ ) = - 1 + 2 ( κ E 0 ) 2 ( κ E 0 ) 2 + δ 2 sin 2 ( Ω t 2 ) w(t;\delta)=-1+\frac{2(\kappa E_{0})^{2}}{(\kappa E_{0})^{2}+\delta^{2}}\sin^{% 2}\left(\frac{\Omega t}{2}\right)

Rabin_signature_algorithm.html

  1. H : { 0 , 1 } * { 0 , 1 } k H:\{0,1\}^{*}\rightarrow\{0,1\}^{k}
  2. n = p q n=pq
  3. { 1 , , n } \{1,\ldots,n\}
  4. x ( x + b ) = H ( m U ) mod n x(x+b)=H(mU)\mod n
  5. n = p q n=pq
  6. x 2 = H ( m U ) mod n x^{2}=H(mU)\mod n
  7. ( a p ) = - ( a q ) = 1 (\tfrac{a}{p})=-(\tfrac{a}{q})=1
  8. ( b q ) = - ( b p ) = 1 (\tfrac{b}{q})=-(\tfrac{b}{p})=1
  9. ( ) (\cdot)
  10. r , a r , b r , a b r r,ar,br,abr
  11. / n \mathbb{Z}/n\mathbb{Z}
  12. / n \mathbb{Z}/n\mathbb{Z}
  13. x 2 = y 2 mod n x^{2}=y^{2}\mod n
  14. x ± y mod n x\neq\pm y\mod n
  15. x 2 - y 2 = ( x - y ) ( x + y ) x^{2}-y^{2}=(x-y)(x+y)
  16. g c d ( x ± y , n ) gcd(x\pm y,n)
  17. r 2 = R mod n r^{2}=R\mod n
  18. r r^{\prime}
  19. r ± r mod n r\neq\pm r^{\prime}\mod n

Radical_polynomial.html

  1. k [ x 1 , x 2 , , x n ] k[x_{1},x_{2},\ldots,x_{n}]
  2. i = 1 n x i 2 . \sum_{i=1}^{n}x_{i}^{2}.

Radix_economy.html

  1. \lfloor\rfloor
  2. E ( b , N ) = b log b ( N ) + 1 . E(b,N)=b\lfloor\log_{b}(N)+1\rfloor\,.
  3. y = x ln ( x ) y=\frac{x}{\ln(x)}\,
  4. E ( b , N ) b log b ( N ) = b ln ( N ) ln ( b ) {E(b,N)}\approx{b{\log_{b}(N)}}={b{\ln(N)\over\ln(b)}}
  5. E ( b , N ) {E(b,N)}
  6. E ( b 1 , N ) E ( b 2 , N ) b 1 log b 1 ( N ) b 2 log b 2 ( N ) = ( b 1 ln ( N ) ln ( b 1 ) ) ( b 2 ln ( N ) ln ( b 2 ) ) = b 1 ln ( b 2 ) b 2 ln ( b 1 ) . {{E(b_{1},N)}\over{E(b_{2},N)}}\approx{{b_{1}{\log_{b_{1}}(N)}}\over{b_{2}{% \log_{b_{2}}(N)}}}={\left(\dfrac{b_{1}\ln(N)}{\ln(b_{1})}\right)\over\left(% \dfrac{b_{2}\ln(N)}{\ln(b_{2})}\right)}={{b_{1}\ln(b_{2})}\over{b_{2}\ln(b_{1}% )}}\,.
  7. E ( b ) E ( e ) b ln ( e ) e ln ( b ) = b e ln ( b ) {{E(b)}\over{E(e)}}\approx{{b\ln(e)}\over{e\ln(b)}}={{b}\over{e\ln(b)}}\,
  8. E ( b ) E ( e ) {{E(b)}\over{E(e)}}
  9. \infty

Rado_graph.html

  1. u : v : u v \forall u:\exists v:u\sim v
  2. \sim
  3. S S
  4. G G
  5. S S
  6. G S G\models S
  7. S S
  8. G G
  9. E i , j E_{i,j}
  10. i i
  11. A A
  12. j j
  13. B B
  14. A A
  15. B B
  16. E 1 , 1 E_{1,1}
  17. a : b : a b c : c a c b c a ¬ ( c b ) . \forall a:\forall b:a\neq b\rightarrow\exists c:c\neq a\wedge c\neq b\wedge c% \sim a\wedge\lnot(c\sim b).
  18. S S
  19. S S

Radonifying_function.html

  1. E E
  2. G G
  3. { μ T | T 𝒜 ( E ) } \{\mu_{T}|T\in\mathcal{A}(E)\}
  4. E E
  5. θ Lin ( E ; G ) \theta\in\mathrm{Lin}(E;G)
  6. θ \theta
  7. { ( θ * ( μ ) ) S | S 𝒜 ( G ) } \left\{\left.\left(\theta_{*}(\mu_{\cdot})\right)_{S}\right|S\in\mathcal{A}(G)\right\}
  8. G G
  9. ν \nu
  10. G G
  11. ( θ * ( μ ) ) S = S * ( ν ) \left(\theta_{*}(\mu_{\cdot})\right)_{S}=S_{*}(\nu)
  12. S 𝒜 ( G ) S\in\mathcal{A}(G)
  13. S * ( ν ) S_{*}(\nu)
  14. ν \nu
  15. S : G F S S:G\to F_{S}
  16. G G
  17. 𝒜 ( G ) \mathcal{A}(G)
  18. { ( θ * ( μ ) ) S | S 𝒜 ( G ) } \left\{\left.\left(\theta_{*}(\mu_{\cdot})\right)_{S}\right|S\in\mathcal{A}(G)\right\}
  19. ( θ * ( μ ) ) S = μ S θ \left(\theta_{*}(\mu_{\cdot})\right)_{S}=\mu_{S\circ\theta}
  20. S θ : E F S S\circ\theta:E\to F_{S}
  21. S θ S\circ\theta
  22. F ~ \tilde{F}
  23. S θ S\circ\theta
  24. i : F ~ F S i:\tilde{F}\to F_{S}
  25. ( θ * ( μ ) ) S = i * ( μ Σ ) \left(\theta_{*}(\mu_{\cdot})\right)_{S}=i_{*}\left(\mu_{\Sigma}\right)
  26. Σ : E F ~ \Sigma:E\to\tilde{F}
  27. Σ 𝒜 ( E ) \Sigma\in\mathcal{A}(E)
  28. i Σ = S θ i\circ\Sigma=S\circ\theta

Ragone_chart.html

  1. Specific Energy = V × I × t m , \,\text{Specific Energy}=\frac{V\times I\times t}{m},
  2. Specific Power = V × I m , \,\text{Specific Power}=\frac{V\times I}{m},

Ramanujan's_congruences.html

  1. p ( 5 k + 4 ) 0 ( mod 5 ) p(5k+4)\equiv 0\;\;(\mathop{{\rm mod}}5)
  2. p ( 7 k + 5 ) 0 ( mod 7 ) p(7k+5)\equiv 0\;\;(\mathop{{\rm mod}}7)
  3. p ( 11 k + 6 ) 0 ( mod 11 ) . p(11k+6)\equiv 0\;\;(\mathop{{\rm mod}}11).
  4. k = 0 p ( 5 k + 4 ) q k = 5 ( q 5 ) 5 ( q ) 6 \sum_{k=0}^{\infty}p(5k+4)q^{k}=5\frac{(q^{5})_{\infty}^{5}}{(q)_{\infty}^{6}}
  5. k = 0 p ( 7 k + 5 ) q k = 7 ( q 7 ) 3 ( q ) 4 + 49 q ( q 7 ) 7 ( q ) 8 . \sum_{k=0}^{\infty}p(7k+5)q^{k}=7\frac{(q^{7})_{\infty}^{3}}{(q)_{\infty}^{4}}% +49q\frac{(q^{7})_{\infty}^{7}}{(q)_{\infty}^{8}}.
  6. p ( 4063467631 k + 30064597 ) 0 ( mod 31 ) . p(4063467631k+30064597)\equiv 0\;\;(\mathop{{\rm mod}}31).
  7. P P
  8. P l ( b ; z ) := n = 0 p ( l b n + 1 24 ) q n / 24 . P_{l}(b;z):=\sum_{n=0}^{\infty}p\left(\frac{l^{b}n+1}{24}\right)q^{n/24}.
  9. p ( 28995244292486005245947069 k + 28995221336976431135321047 ) 0 ( mod 29 ) . p(28995244292486005245947069k+28995221336976431135321047)\equiv 0\;\;(\mathop{% {\rm mod}}29).

Ramanujan's_sum.html

  1. c q ( n ) = a = 1 ( a , q ) = 1 q e 2 π i a q n , c_{q}(n)=\sum_{a=1\atop(a,q)=1}^{q}e^{2\pi i\tfrac{a}{q}n},
  2. a b a\mid b
  3. a b a\nmid b
  4. d m f ( d ) \sum_{d\,\mid\,m}f(d)
  5. d 12 f ( d ) = f ( 1 ) + f ( 2 ) + f ( 3 ) + f ( 4 ) + f ( 6 ) + f ( 12 ) . \sum_{d\,\mid\,12}f(d)=f(1)+f(2)+f(3)+f(4)+f(6)+f(12).
  6. ( a , b ) (a,\,b)\;
  7. ϕ ( n ) \phi(n)\;
  8. μ ( n ) \mu(n)\;
  9. ζ ( s ) \zeta(s)\;
  10. e i x = cos x + i sin x , e^{ix}=\cos x+i\sin x,
  11. c 1 ( n ) = 1 c 2 ( n ) = cos n π c 3 ( n ) = 2 cos 2 3 n π c 4 ( n ) = 2 cos 1 2 n π c 5 ( n ) = 2 cos 2 5 n π + 2 cos 4 5 n π c 6 ( n ) = 2 cos 1 3 n π c 7 ( n ) = 2 cos 2 7 n π + 2 cos 4 7 n π + 2 cos 6 7 n π c 8 ( n ) = 2 cos 1 4 n π + 2 cos 3 4 n π c 9 ( n ) = 2 cos 2 9 n π + 2 cos 4 9 n π + 2 cos 8 9 n π c 10 ( n ) = 2 cos 1 5 n π + 2 cos 3 5 n π \begin{aligned}\displaystyle c_{1}(n)&\displaystyle=1\\ \displaystyle c_{2}(n)&\displaystyle=\cos n\pi\\ \displaystyle c_{3}(n)&\displaystyle=2\cos\tfrac{2}{3}n\pi\\ \displaystyle c_{4}(n)&\displaystyle=2\cos\tfrac{1}{2}n\pi\\ \displaystyle c_{5}(n)&\displaystyle=2\cos\tfrac{2}{5}n\pi+2\cos\tfrac{4}{5}n% \pi\\ \displaystyle c_{6}(n)&\displaystyle=2\cos\tfrac{1}{3}n\pi\\ \displaystyle c_{7}(n)&\displaystyle=2\cos\tfrac{2}{7}n\pi+2\cos\tfrac{4}{7}n% \pi+2\cos\tfrac{6}{7}n\pi\\ \displaystyle c_{8}(n)&\displaystyle=2\cos\tfrac{1}{4}n\pi+2\cos\tfrac{3}{4}n% \pi\\ \displaystyle c_{9}(n)&\displaystyle=2\cos\tfrac{2}{9}n\pi+2\cos\tfrac{4}{9}n% \pi+2\cos\tfrac{8}{9}n\pi\\ \displaystyle c_{10}(n)&\displaystyle=2\cos\tfrac{1}{5}n\pi+2\cos\tfrac{3}{5}n% \pi\\ \end{aligned}
  12. ζ q = e 2 π i q . \zeta_{q}=e^{\frac{2\pi i}{q}}.
  13. η q ( n ) = k = 1 q ζ q k n \eta_{q}(n)=\sum_{k=1}^{q}\zeta_{q}^{kn}
  14. η q ( n ) = d q c d ( n ) , \eta_{q}(n)=\sum_{d\,\mid\,q}c_{d}(n),
  15. c q ( n ) = d q μ ( q d ) η d ( n ) . c_{q}(n)=\sum_{d\,\mid\,q}\mu\left(\frac{q}{d}\right)\eta_{d}(n).
  16. η q ( n ) = { 0 if q n q if q n \eta_{q}(n)=\begin{cases}0&\;\mbox{ if }~{}q\nmid n\\ q&\;\mbox{ if }~{}q\mid n\\ \end{cases}
  17. c q ( n ) = d ( q , n ) μ ( q d ) d , c_{q}(n)=\sum_{d\,\mid\,(q,n)}\mu\left(\frac{q}{d}\right)d,
  18. ϕ ( q ) = d q μ ( q d ) d . \phi(q)=\sum_{d\,\mid\,q}\mu\left(\frac{q}{d}\right)d.
  19. If ( q , r ) = 1 then c q ( n ) c r ( n ) = c q r ( n ) . \mbox{If }~{}\;(q,r)=1\;\mbox{ then }~{}\;c_{q}(n)c_{r}(n)=c_{qr}(n).
  20. c p ( n ) = { - 1 if p n ϕ ( p ) if p n , c_{p}(n)=\begin{cases}-1&\mbox{ if }~{}p\nmid n\\ \phi(p)&\mbox{ if }~{}p\mid n\\ \end{cases},
  21. c p k ( n ) = { 0 if p k - 1 n - p k - 1 if p k - 1 n and p k n ϕ ( p k ) if p k n . c_{p^{k}}(n)=\begin{cases}0&\mbox{ if }~{}p^{k-1}\nmid n\\ -p^{k-1}&\mbox{ if }~{}p^{k-1}\mid n\mbox{ and }~{}p^{k}\nmid n\\ \phi(p^{k})&\mbox{ if }~{}p^{k}\mid n\\ \end{cases}.
  22. c q ( n ) = μ ( q ( q , n ) ) ϕ ( q ) ϕ ( q ( q , n ) ) . c_{q}(n)=\mu\left(\frac{q}{(q,n)}\right)\frac{\phi(q)}{\phi\left(\frac{q}{(q,n% )}\right)}.
  23. c 1 ( q ) = 1 , c q ( 1 ) = μ ( q ) , and c q ( q ) = ϕ ( q ) . c_{1}(q)=1,\;\;c_{q}(1)=\mu(q),\;\mbox{ and }~{}\;c_{q}(q)=\phi(q).
  24. If m n ( mod q ) then c q ( m ) = c q ( n ) . \mbox{If }~{}m\equiv n\;\;(\mathop{{\rm mod}}q)\mbox{ then }~{}c_{q}(m)=c_{q}(% n).
  25. n = a a + q - 1 c q ( n ) = 0. \sum_{n=a}^{a+q-1}c_{q}(n)=0.
  26. 1 m k = 1 m c m 1 ( k ) c m 2 ( k ) = { ϕ ( m ) m 1 = m 2 = m , 0 otherwise \frac{1}{m}\sum_{k=1}^{m}c_{m_{1}}(k)c_{m_{2}}(k)=\begin{cases}\phi(m)&m_{1}=m% _{2}=m,\\ 0&\,\text{otherwise}\end{cases}
  27. gcd ( d , k ) = 1 d n d μ ( n d ) ϕ ( d ) = μ ( n ) c n ( k ) ϕ ( n ) , \sum_{\stackrel{d\mid n}{\gcd(d,k)=1}}d\;\frac{\mu(\tfrac{n}{d})}{\phi(d)}=% \frac{\mu(n)c_{n}(k)}{\phi(n)},
  28. gcd ( k , n ) = 1 1 k n c n ( k - a ) = μ ( n ) c n ( a ) , \sum_{\stackrel{1\leq k\leq n}{\gcd(k,n)=1}}c_{n}(k-a)=\mu(n)c_{n}(a),
  29. f ( n ) = q = 1 a q c q ( n ) f(n)=\sum_{q=1}^{\infty}a_{q}c_{q}(n)
  30. f ( q ) = n = 1 a n c q ( n ) f(q)=\sum_{n=1}^{\infty}a_{n}c_{q}(n)
  31. n = 1 μ ( n ) n \sum_{n=1}^{\infty}\frac{\mu(n)}{n}
  32. ζ ( s ) δ q μ ( q δ ) δ 1 - s = n = 1 c q ( n ) n s \zeta(s)\sum_{\delta\,\mid\,q}\mu\left(\frac{q}{\delta}\right)\delta^{1-s}=% \sum_{n=1}^{\infty}\frac{c_{q}(n)}{n^{s}}
  33. σ r - 1 ( n ) n r - 1 ζ ( r ) = q = 1 c q ( n ) q r \frac{\sigma_{r-1}(n)}{n^{r-1}\zeta(r)}=\sum_{q=1}^{\infty}\frac{c_{q}(n)}{q^{% r}}
  34. ζ ( s ) ζ ( r + s - 1 ) ζ ( r ) = q = 1 n = 1 c q ( n ) q r n s . \frac{\zeta(s)\zeta(r+s-1)}{\zeta(r)}=\sum_{q=1}^{\infty}\sum_{n=1}^{\infty}% \frac{c_{q}(n)}{q^{r}n^{s}}.
  35. σ s ( n ) = n s ζ ( s + 1 ) ( c 1 ( n ) 1 s + 1 + c 2 ( n ) 2 s + 1 + c 3 ( n ) 3 s + 1 + ) \sigma_{s}(n)=n^{s}\zeta(s+1)\left(\frac{c_{1}(n)}{1^{s+1}}+\frac{c_{2}(n)}{2^% {s+1}}+\frac{c_{3}(n)}{3^{s+1}}+\dots\right)
  36. σ - s ( n ) = ζ ( s + 1 ) ( c 1 ( n ) 1 s + 1 + c 2 ( n ) 2 s + 1 + c 3 ( n ) 3 s + 1 + ) . \sigma_{-s}(n)=\zeta(s+1)\left(\frac{c_{1}(n)}{1^{s+1}}+\frac{c_{2}(n)}{2^{s+1% }}+\frac{c_{3}(n)}{3^{s+1}}+\dots\right).
  37. σ ( n ) = π 2 6 n ( c 1 ( n ) 1 + c 2 ( n ) 4 + c 3 ( n ) 9 + ) . \sigma(n)=\frac{\pi^{2}}{6}n\left(\frac{c_{1}(n)}{1}+\frac{c_{2}(n)}{4}+\frac{% c_{3}(n)}{9}+\dots\right).
  38. - 1 2 < s < 1 2 , -\tfrac{1}{2}<s<\tfrac{1}{2},
  39. σ s ( n ) = ζ ( 1 - s ) ( c 1 ( n ) 1 1 - s + c 2 ( n ) 2 1 - s + c 3 ( n ) 3 1 - s + ) = n s ζ ( 1 + s ) ( c 1 ( n ) 1 1 + s + c 2 ( n ) 2 1 + s + c 3 ( n ) 3 1 + s + ) . \sigma_{s}(n)=\zeta(1-s)\left(\frac{c_{1}(n)}{1^{1-s}}+\frac{c_{2}(n)}{2^{1-s}% }+\frac{c_{3}(n)}{3^{1-s}}+\dots\right)=n^{s}\zeta(1+s)\left(\frac{c_{1}(n)}{1% ^{1+s}}+\frac{c_{2}(n)}{2^{1+s}}+\frac{c_{3}(n)}{3^{1+s}}+\dots\right).
  40. - d ( n ) = log 1 1 c 1 ( n ) + log 2 2 c 2 ( n ) + log 3 3 c 3 ( n ) + - d ( n ) ( 2 γ + log n ) = log 2 1 1 c 1 ( n ) + log 2 2 2 c 2 ( n ) + log 2 3 3 c 3 ( n ) + \begin{aligned}\displaystyle-d(n)&\displaystyle=\frac{\log 1}{1}c_{1}(n)+\frac% {\log 2}{2}c_{2}(n)+\frac{\log 3}{3}c_{3}(n)+\dots\\ \displaystyle-d(n)(2\gamma+\log n)&\displaystyle=\frac{\log^{2}1}{1}c_{1}(n)+% \frac{\log^{2}2}{2}c_{2}(n)+\frac{\log^{2}3}{3}c_{3}(n)+\cdots\end{aligned}
  41. n = p 1 a 1 p 2 a 2 p 3 a 3 n=p_{1}^{a_{1}}p_{2}^{a_{2}}p_{3}^{a_{3}}\cdots
  42. ϕ s ( n ) = n s ( 1 - p 1 - s ) ( 1 - p 2 - s ) ( 1 - p 3 - s ) , \phi_{s}(n)=n^{s}(1-p_{1}^{-s})(1-p_{2}^{-s})(1-p_{3}^{-s})\cdots,
  43. μ ( n ) n s ϕ s ( n ) ζ ( s ) = ν = 1 μ ( n ν ) ν s \frac{\mu(n)n^{s}}{\phi_{s}(n)\zeta(s)}=\sum_{\nu=1}^{\infty}\frac{\mu(n\nu)}{% \nu^{s}}
  44. ϕ s ( n ) ζ ( s + 1 ) n s = μ ( 1 ) c 1 ( n ) ϕ s + 1 ( 1 ) + μ ( 2 ) c 2 ( n ) ϕ s + 1 ( 2 ) + μ ( 3 ) c 3 ( n ) ϕ s + 1 ( 3 ) + . \frac{\phi_{s}(n)\zeta(s+1)}{n^{s}}=\frac{\mu(1)c_{1}(n)}{\phi_{s+1}(1)}+\frac% {\mu(2)c_{2}(n)}{\phi_{s+1}(2)}+\frac{\mu(3)c_{3}(n)}{\phi_{s+1}(3)}+\dots.
  45. ϕ ( n ) = 6 π 2 n ( c 1 ( n ) - c 2 ( n ) 2 2 - 1 - c 3 ( n ) 3 2 - 1 - c 5 ( n ) 5 2 - 1 + c 6 ( n ) ( 2 2 - 1 ) ( 3 2 - 1 ) - c 7 ( n ) 7 2 - 1 + c 10 ( n ) ( 2 2 - 1 ) ( 5 2 - 1 ) - ) . \phi(n)=\frac{6}{\pi^{2}}n\left(c_{1}(n)-\frac{c_{2}(n)}{2^{2}-1}-\frac{c_{3}(% n)}{3^{2}-1}-\frac{c_{5}(n)}{5^{2}-1}+\frac{c_{6}(n)}{(2^{2}-1)(3^{2}-1)}-% \frac{c_{7}(n)}{7^{2}-1}+\frac{c_{10}(n)}{(2^{2}-1)(5^{2}-1)}-\dots\right).
  46. Λ ( n ) = 0 Λ(n)=0
  47. - Λ ( m ) = c m ( 1 ) + 1 2 c m ( 2 ) + 1 3 c m ( 3 ) + -\Lambda(m)=c_{m}(1)+\frac{1}{2}c_{m}(2)+\frac{1}{3}c_{m}(3)+\dots
  48. 0 = c 1 ( n ) + 1 2 c 2 ( n ) + 1 3 c 3 ( n ) + . 0=c_{1}(n)+\frac{1}{2}c_{2}(n)+\frac{1}{3}c_{3}(n)+\dots.
  49. δ 2 ( n ) = π ( c 1 ( n ) 1 - c 3 ( n ) 3 + c 5 ( n ) 5 - ) . \delta_{2}(n)=\pi\left(\frac{c_{1}(n)}{1}-\frac{c_{3}(n)}{3}+\frac{c_{5}(n)}{5% }-\dots\right).
  50. δ 2 s ( n ) = π s n s - 1 ( s - 1 ) ! ( c 1 ( n ) 1 s + c 4 ( n ) 2 s + c 3 ( n ) 3 s + c 8 ( n ) 4 s + c 5 ( n ) 5 s + c 12 ( n ) 6 s + c 7 ( n ) 7 s + c 16 ( n ) 8 s + ) \delta_{2s}(n)=\frac{\pi^{s}n^{s-1}}{(s-1)!}\left(\frac{c_{1}(n)}{1^{s}}+\frac% {c_{4}(n)}{2^{s}}+\frac{c_{3}(n)}{3^{s}}+\frac{c_{8}(n)}{4^{s}}+\frac{c_{5}(n)% }{5^{s}}+\frac{c_{12}(n)}{6^{s}}+\frac{c_{7}(n)}{7^{s}}+\frac{c_{16}(n)}{8^{s}% }+\dots\right)
  51. δ 2 s ( n ) = π s n s - 1 ( s - 1 ) ! ( c 1 ( n ) 1 s - c 4 ( n ) 2 s + c 3 ( n ) 3 s - c 8 ( n ) 4 s + c 5 ( n ) 5 s - c 12 ( n ) 6 s + c 7 ( n ) 7 s - c 16 ( n ) 8 s + ) \delta_{2s}(n)=\frac{\pi^{s}n^{s-1}}{(s-1)!}\left(\frac{c_{1}(n)}{1^{s}}-\frac% {c_{4}(n)}{2^{s}}+\frac{c_{3}(n)}{3^{s}}-\frac{c_{8}(n)}{4^{s}}+\frac{c_{5}(n)% }{5^{s}}-\frac{c_{12}(n)}{6^{s}}+\frac{c_{7}(n)}{7^{s}}-\frac{c_{16}(n)}{8^{s}% }+\dots\right)
  52. δ 2 s ( n ) = π s n s - 1 ( s - 1 ) ! ( c 1 ( n ) 1 s + c 4 ( n ) 2 s - c 3 ( n ) 3 s + c 8 ( n ) 4 s + c 5 ( n ) 5 s + c 12 ( n ) 6 s - c 7 ( n ) 7 s + c 16 ( n ) 8 s + ) \delta_{2s}(n)=\frac{\pi^{s}n^{s-1}}{(s-1)!}\left(\frac{c_{1}(n)}{1^{s}}+\frac% {c_{4}(n)}{2^{s}}-\frac{c_{3}(n)}{3^{s}}+\frac{c_{8}(n)}{4^{s}}+\frac{c_{5}(n)% }{5^{s}}+\frac{c_{12}(n)}{6^{s}}-\frac{c_{7}(n)}{7^{s}}+\frac{c_{16}(n)}{8^{s}% }+\dots\right)
  53. δ 2 s ( n ) = π s n s - 1 ( s - 1 ) ! ( c 1 ( n ) 1 s - c 4 ( n ) 2 s - c 3 ( n ) 3 s - c 8 ( n ) 4 s + c 5 ( n ) 5 s - c 12 ( n ) 6 s - c 7 ( n ) 7 s - c 16 ( n ) 8 s + ) \delta_{2s}(n)=\frac{\pi^{s}n^{s-1}}{(s-1)!}\left(\frac{c_{1}(n)}{1^{s}}-\frac% {c_{4}(n)}{2^{s}}-\frac{c_{3}(n)}{3^{s}}-\frac{c_{8}(n)}{4^{s}}+\frac{c_{5}(n)% }{5^{s}}-\frac{c_{12}(n)}{6^{s}}-\frac{c_{7}(n)}{7^{s}}-\frac{c_{16}(n)}{8^{s}% }+\dots\right)
  54. r 2 ( n ) = π ( c 1 ( n ) 1 - c 3 ( n ) 3 + c 5 ( n ) 5 - c 7 ( n ) 7 + c 11 ( n ) 11 - c 13 ( n ) 13 + c 15 ( n ) 15 - c 17 ( n ) 17 + ) r 4 ( n ) = π 2 n ( c 1 ( n ) 1 - c 4 ( n ) 4 + c 3 ( n ) 9 - c 8 ( n ) 16 + c 5 ( n ) 25 - c 12 ( n ) 36 + c 7 ( n ) 49 - c 16 ( n ) 64 + ) r 6 ( n ) = π 3 n 2 2 ( c 1 ( n ) 1 - c 4 ( n ) 8 - c 3 ( n ) 27 - c 8 ( n ) 64 + c 5 ( n ) 125 - c 12 ( n ) 216 - c 7 ( n ) 343 - c 16 ( n ) 512 + ) r 8 ( n ) = π 4 n 3 6 ( c 1 ( n ) 1 + c 4 ( n ) 16 + c 3 ( n ) 81 + c 8 ( n ) 256 + c 5 ( n ) 625 + c 12 ( n ) 1296 + c 7 ( n ) 2401 + c 16 ( n ) 4096 + ) \begin{aligned}\displaystyle r_{2}(n)&\displaystyle=\pi\left(\frac{c_{1}(n)}{1% }-\frac{c_{3}(n)}{3}+\frac{c_{5}(n)}{5}-\frac{c_{7}(n)}{7}+\frac{c_{11}(n)}{11% }-\frac{c_{13}(n)}{13}+\frac{c_{15}(n)}{15}-\frac{c_{17}(n)}{17}+\cdots\right)% \\ \displaystyle r_{4}(n)&\displaystyle=\pi^{2}n\left(\frac{c_{1}(n)}{1}-\frac{c_% {4}(n)}{4}+\frac{c_{3}(n)}{9}-\frac{c_{8}(n)}{16}+\frac{c_{5}(n)}{25}-\frac{c_% {12}(n)}{36}+\frac{c_{7}(n)}{49}-\frac{c_{16}(n)}{64}+\cdots\right)\\ \displaystyle r_{6}(n)&\displaystyle=\frac{\pi^{3}n^{2}}{2}\left(\frac{c_{1}(n% )}{1}-\frac{c_{4}(n)}{8}-\frac{c_{3}(n)}{27}-\frac{c_{8}(n)}{64}+\frac{c_{5}(n% )}{125}-\frac{c_{12}(n)}{216}-\frac{c_{7}(n)}{343}-\frac{c_{16}(n)}{512}+% \cdots\right)\\ \displaystyle r_{8}(n)&\displaystyle=\frac{\pi^{4}n^{3}}{6}\left(\frac{c_{1}(n% )}{1}+\frac{c_{4}(n)}{16}+\frac{c_{3}(n)}{81}+\frac{c_{8}(n)}{256}+\frac{c_{5}% (n)}{625}+\frac{c_{12}(n)}{1296}+\frac{c_{7}(n)}{2401}+\frac{c_{16}(n)}{4096}+% \cdots\right)\end{aligned}
  55. δ 2 ( n ) = π 4 ( c 1 ( 4 n + 1 ) 1 - c 3 ( 4 n + 1 ) 3 + c 5 ( 4 n + 1 ) 5 - c 7 ( 4 n + 1 ) 7 + ) . \delta^{\prime}_{2}(n)=\frac{\pi}{4}\left(\frac{c_{1}(4n+1)}{1}-\frac{c_{3}(4n% +1)}{3}+\frac{c_{5}(4n+1)}{5}-\frac{c_{7}(4n+1)}{7}+\dots\right).
  56. δ 2 s ( n ) = ( π 2 ) s ( s - 1 ) ! ( n + s 4 ) s - 1 ( c 1 ( n + s 4 ) 1 s + c 3 ( n + s 4 ) 3 s + c 5 ( n + s 4 ) 5 s + ) . \delta^{\prime}_{2s}(n)=\frac{(\frac{\pi}{2})^{s}}{(s-1)!}\left(n+\frac{s}{4}% \right)^{s-1}\left(\frac{c_{1}(n+\frac{s}{4})}{1^{s}}+\frac{c_{3}(n+\frac{s}{4% })}{3^{s}}+\frac{c_{5}(n+\frac{s}{4})}{5^{s}}+\dots\right).
  57. δ 2 s ( n ) = ( π 2 ) s ( s - 1 ) ! ( n + s 4 ) s - 1 ( c 1 ( 2 n + s 2 ) 1 s + c 3 ( 2 n + s 2 ) 3 s + c 5 ( 2 n + s 2 ) 5 s + ) . \delta^{\prime}_{2s}(n)=\frac{(\frac{\pi}{2})^{s}}{(s-1)!}\left(n+\frac{s}{4}% \right)^{s-1}\left(\frac{c_{1}(2n+\frac{s}{2})}{1^{s}}+\frac{c_{3}(2n+\frac{s}% {2})}{3^{s}}+\frac{c_{5}(2n+\frac{s}{2})}{5^{s}}+\dots\right).
  58. δ 2 s ( n ) = ( π 2 ) s ( s - 1 ) ! ( n + s 4 ) s - 1 ( c 1 ( 4 n + s ) 1 s - c 3 ( 4 n + s ) 3 s + c 5 ( 4 n + s ) 5 s - ) . \delta^{\prime}_{2s}(n)=\frac{(\frac{\pi}{2})^{s}}{(s-1)!}\left(n+\frac{s}{4}% \right)^{s-1}\left(\frac{c_{1}(4n+s)}{1^{s}}-\frac{c_{3}(4n+s)}{3^{s}}+\frac{c% _{5}(4n+s)}{5^{s}}-\dots\right).
  59. r 2 ( n ) = π 4 ( c 1 ( 4 n + 1 ) 1 - c 3 ( 4 n + 1 ) 3 + c 5 ( 4 n + 1 ) 5 - c 7 ( 4 n + 1 ) 7 + ) r 4 ( n ) = ( π 2 ) 2 ( n + 1 2 ) ( c 1 ( 2 n + 1 ) 1 + c 3 ( 2 n + 1 ) 9 + c 5 ( 2 n + 1 ) 25 + ) r 6 ( n ) = ( π 2 ) 3 2 ( n + 3 4 ) 2 ( c 1 ( 4 n + 3 ) 1 - c 3 ( 4 n + 3 ) 27 + c 5 ( 4 n + 3 ) 125 - ) r 8 ( n ) = ( 1 2 π ) 4 6 ( n + 1 ) 3 ( c 1 ( n + 1 ) 1 + c 3 ( n + 1 ) 81 + c 5 ( n + 1 ) 625 + ) \begin{aligned}\displaystyle r^{\prime}_{2}(n)&\displaystyle=\frac{\pi}{4}% \left(\frac{c_{1}(4n+1)}{1}-\frac{c_{3}(4n+1)}{3}+\frac{c_{5}(4n+1)}{5}-\frac{% c_{7}(4n+1)}{7}+\dots\right)\\ \displaystyle r^{\prime}_{4}(n)&\displaystyle=\left(\tfrac{\pi}{2}\right)^{2}% \left(n+\tfrac{1}{2}\right)\left(\frac{c_{1}(2n+1)}{1}+\frac{c_{3}(2n+1)}{9}+% \frac{c_{5}(2n+1)}{25}+\dots\right)\\ \displaystyle r^{\prime}_{6}(n)&\displaystyle=\frac{(\tfrac{\pi}{2})^{3}}{2}% \left(n+\tfrac{3}{4}\right)^{2}\left(\frac{c_{1}(4n+3)}{1}-\frac{c_{3}(4n+3)}{% 27}+\frac{c_{5}(4n+3)}{125}-\dots\right)\\ \displaystyle r^{\prime}_{8}(n)&\displaystyle=\frac{(\frac{1}{2}\pi)^{4}}{6}(n% +1)^{3}\left(\frac{c_{1}(n+1)}{1}+\frac{c_{3}(n+1)}{81}+\frac{c_{5}(n+1)}{625}% +\dots\right)\end{aligned}
  60. T q ( n ) = c q ( 1 ) + c q ( 2 ) + + c q ( n ) U q ( n ) = T q ( n ) + 1 2 ϕ ( q ) \begin{aligned}\displaystyle T_{q}(n)&\displaystyle=c_{q}(1)+c_{q}(2)+\dots+c_% {q}(n)\\ \displaystyle U_{q}(n)&\displaystyle=T_{q}(n)+\tfrac{1}{2}\phi(q)\end{aligned}
  61. s > 1 s>1
  62. σ - s ( 1 ) + + σ - s ( n ) = ζ ( s + 1 ) ( n + T 2 ( n ) 2 s + 1 + T 3 ( n ) 3 s + 1 + T 4 ( n ) 4 s + 1 + ) = ζ ( s + 1 ) ( n + 1 2 + U 2 ( n ) 2 s + 1 + U 3 ( n ) 3 s + 1 + U 4 ( n ) 4 s + 1 + ) - 1 2 ζ ( s ) d ( 1 ) + + d ( n ) = - T 2 ( n ) log 2 2 - T 3 ( n ) log 3 3 - T 4 ( n ) log 4 4 - , d ( 1 ) log 1 + + d ( n ) log n = - T 2 ( n ) ( 2 γ log 2 - log 2 2 ) 2 - T 3 ( n ) ( 2 γ log 3 - log 2 3 ) 3 - T 4 ( n ) ( 2 γ log 4 - log 2 4 ) 4 - , r 2 ( 1 ) + + r 2 ( n ) = π ( n - T 3 ( n ) 3 + T 5 ( n ) 5 - T 7 ( n ) 7 + ) . \begin{aligned}\displaystyle\sigma_{-s}(1)+\cdots+\sigma_{-s}(n)&\displaystyle% =\zeta(s+1)\left(n+\frac{T_{2}(n)}{2^{s+1}}+\frac{T_{3}(n)}{3^{s+1}}+\frac{T_{% 4}(n)}{4^{s+1}}+\dots\right)\\ &\displaystyle=\zeta(s+1)\left(n+\tfrac{1}{2}+\frac{U_{2}(n)}{2^{s+1}}+\frac{U% _{3}(n)}{3^{s+1}}+\frac{U_{4}(n)}{4^{s+1}}+\cdots\right)-\tfrac{1}{2}\zeta(s)% \\ \displaystyle d(1)+\cdots+d(n)&\displaystyle=-\frac{T_{2}(n)\log 2}{2}-\frac{T% _{3}(n)\log 3}{3}-\frac{T_{4}(n)\log 4}{4}-\cdots,\\ \displaystyle d(1)\log 1+\cdots+d(n)\log n&\displaystyle=-\frac{T_{2}(n)(2% \gamma\log 2-\log^{2}2)}{2}-\frac{T_{3}(n)(2\gamma\log 3-\log^{2}3)}{3}-\frac{% T_{4}(n)(2\gamma\log 4-\log^{2}4)}{4}-\cdots,\\ \displaystyle r_{2}(1)+\cdots+r_{2}(n)&\displaystyle=\pi\left(n-\frac{T_{3}(n)% }{3}+\frac{T_{5}(n)}{5}-\frac{T_{7}(n)}{7}+\cdots\right).\end{aligned}
  63. 6 π 2 < σ ( n ) ϕ ( n ) n 2 < 1. \;\frac{6}{\pi^{2}}<\frac{\sigma(n)\phi(n)}{n^{2}}<1.

Ramanujan_prime.html

  1. π ( x ) - π ( x / 2 ) \pi(x)-\pi(x/2)
  2. π ( x ) \pi(x)
  3. π ( x ) - π ( x / 2 ) \pi(x)-\pi(x/2)
  4. π ( x ) - π ( x / 2 ) \pi(x)-\pi(x/2)
  5. π ( x ) \pi(x)
  6. π ( x ) - π ( x / 2 ) \pi(x)-\pi(x/2)
  7. π ( \pi(
  8. ) - π ( )-\pi(
  9. / 2 ) = n /2)=n
  10. R n 41 47 p 3 n R_{n}\leq\frac{41}{47}\ p_{3n}
  11. R n < p t n R_{n}<p_{\lceil t\cdot n\rceil}
  12. \lceil\cdot\rceil
  13. π ( x ) - π ( c x ) n \pi(x)-\pi(cx)\geq n
  14. \lfloor
  15. \rfloor
  16. . \lfloor.\rfloor
  17. 2 p i - n > p i for i > k where k = π ( p k ) = π ( R n ) , 2p_{i-n}>p_{i}\,\text{ for }i>k\,\text{ where }k=\pi(p_{k})=\pi(R_{n})\,,
  18. π ( R n ) \pi(R_{n})\,

Ramberg–Osgood_relationship.html

  1. ε = σ E + K ( σ E ) n \varepsilon=\frac{\sigma}{E}+K\left(\frac{\sigma}{E}\right)^{n}
  2. ε \varepsilon
  3. σ \sigma
  4. E E
  5. K K
  6. n n
  7. σ / E {\sigma}/{E}\,
  8. K ( σ / E ) n \ K({\sigma}/{E})^{n}
  9. K K
  10. n n
  11. σ 0 \sigma_{0}
  12. α \alpha
  13. K K
  14. α = K ( σ 0 / E ) n - 1 \alpha=K({\sigma_{0}}/{E})^{n-1}\,
  15. K ( σ E ) n = α σ E ( σ σ 0 ) n - 1 \ K\left(\frac{\sigma}{E}\right)^{n}=\alpha\frac{\sigma}{E}\left(\frac{\sigma}% {\sigma_{0}}\right)^{n-1}
  16. ε = σ E + α σ E ( σ σ 0 ) n - 1 \varepsilon=\frac{\sigma}{E}+\alpha\frac{\sigma}{E}\left(\frac{\sigma}{\sigma_% {0}}\right)^{n-1}
  17. α \alpha\,
  18. n n\,
  19. α \alpha
  20. n n
  21. σ 0 \sigma_{0}
  22. α σ 0 E \alpha\frac{\sigma_{0}}{E}
  23. ε = ( 1 + α ) σ 0 / E \varepsilon=(1+\alpha){{\sigma_{0}}/{E}}\,
  24. σ = σ 0 \sigma=\sigma_{0}\,
  25. σ 0 / E {{\sigma_{0}}/{E}}\,
  26. α ( σ 0 / E ) \alpha({\sigma_{0}}/E)\,
  27. n n\,
  28. α \alpha\,
  29. α σ 0 E = 0 , 002 \alpha\frac{\sigma_{0}}{E}=0,002

Rami_Grossberg.html

  1. L ω 1 , ω \mathit{L}_{{\omega_{1}},\omega}
  2. ψ \psi
  3. ψ \psi
  4. > ω 1 \;>\beth_{\omega_{1}}
  5. ψ \psi
  6. > ω 1 \;>\beth_{\omega_{1}}
  7. L ( Q ) \mathit{L(Q)}
  8. L ω 1 , ω \mathit{L}_{\omega_{1},\omega}
  9. L ω 1 , ω \mathit{L}_{\omega_{1},\omega}

Ramp_function.html

  1. R ( x ) : R(x):\mathbb{R}\rightarrow\mathbb{R}
  2. R ( x ) := { x , x 0 ; 0 , x < 0 R(x):=\begin{cases}x,&x\geq 0;\\ 0,&x<0\end{cases}
  3. R ( x ) := max ( x , 0 ) R(x):=\operatorname{max}(x,0)
  4. R ( x ) := x + | x | 2 R(x):=\frac{x+|x|}{2}
  5. max ( a , b ) \operatorname{max}(a,b)
  6. max ( a , b ) = a + b + | a - b | 2 \operatorname{max}(a,b)=\frac{a+b+|a-b|}{2}
  7. a = x a=x
  8. b = 0 b=0
  9. R ( x ) := x H ( x ) R\left(x\right):=xH\left(x\right)
  10. R ( x ) := H ( x ) * H ( x ) R\left(x\right):=H\left(x\right)*H\left(x\right)
  11. R ( x ) := - x H ( ξ ) d ξ R(x):=\int_{-\infty}^{x}H(\xi)\,\mathrm{d}\xi
  12. R ( x ) := x R(x):=\langle x\rangle
  13. x : R ( x ) 0 \forall x\in\mathbb{R}:R(x)\geqslant 0
  14. | R ( x ) | = R ( x ) \left|R\left(x\right)\right|=R\left(x\right)
  15. R ( x ) = H ( x ) if x 0 R^{\prime}(x)=H(x)\ \mathrm{if}\ x\neq 0
  16. { R ( x ) } ( f ) \mathcal{F}\left\{R(x)\right\}(f)
  17. = =
  18. - R ( x ) e - 2 π i f x d x \int_{-\infty}^{\infty}R(x)e^{-2\pi ifx}dx
  19. = =
  20. i δ ( f ) 4 π - 1 4 π 2 f 2 \frac{i\delta^{\prime}(f)}{4\pi}-\frac{1}{4\pi^{2}f^{2}}
  21. R ( x ) R(x)
  22. { R ( x ) } ( s ) = 0 e - s x R ( x ) d x = 1 s 2 . \mathcal{L}\left\{R\left(x\right)\right\}(s)=\int_{0}^{\infty}e^{-sx}R(x)dx=% \frac{1}{s^{2}}.
  23. R ( R ( x ) ) = R ( x ) R\left(R\left(x\right)\right)=R\left(x\right)
  24. R ( R ( x ) ) := R ( x ) + | R ( x ) | 2 = R ( x ) + R ( x ) 2 R(R(x)):=\frac{R(x)+|R(x)|}{2}=\frac{R(x)+R(x)}{2}
  25. = =
  26. = =
  27. 2 R ( x ) 2 = R ( x ) \frac{2R(x)}{2}=R(x)

Ramsey_RESET_test.html

  1. y ^ = E { y | x } = β x . \hat{y}=E\{y|x\}=\beta x.
  2. ( β x ) 2 , ( β x ) 3 , ( β x ) k (\beta x)^{2},(\beta x)^{3}...,(\beta x)^{k}
  3. y y
  4. y = α x + γ 1 y ^ 2 + + γ k - 1 y ^ k + ϵ y=\alpha x+\gamma_{1}\hat{y}^{2}+...+\gamma_{k-1}\hat{y}^{k}+\epsilon
  5. γ 1 \gamma_{1}~{}
  6. γ k - 1 ~{}\gamma_{k-1}
  7. γ \gamma~{}

Random_compact_set.html

  1. ( M , d ) (M,d)
  2. 𝒦 \mathcal{K}
  3. M M
  4. h h
  5. 𝒦 \mathcal{K}
  6. h ( K 1 , K 2 ) := max { sup a K 1 inf b K 2 d ( a , b ) , sup b K 2 inf a K 1 d ( a , b ) } . h(K_{1},K_{2}):=\max\left\{\sup_{a\in K_{1}}\inf_{b\in K_{2}}d(a,b),\sup_{b\in K% _{2}}\inf_{a\in K_{1}}d(a,b)\right\}.
  7. ( 𝒦 , h ) (\mathcal{K},h)
  8. 𝒦 \mathcal{K}
  9. ( 𝒦 ) \mathcal{B}(\mathcal{K})
  10. 𝒦 \mathcal{K}
  11. K K
  12. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  13. ( 𝒦 , ( 𝒦 ) ) (\mathcal{K},\mathcal{B}(\mathcal{K}))
  14. K : Ω 2 M K\colon\Omega\to 2^{M}
  15. K ( ω ) K(\omega)
  16. ω inf b K ( ω ) d ( x , b ) \omega\mapsto\inf_{b\in K(\omega)}d(x,b)
  17. x M x\in M
  18. ( X K = ) \mathbb{P}(X\cap K=\emptyset)
  19. K 𝒦 . K\in\mathcal{K}.
  20. ( X K ) . \mathbb{P}(X\subset K).
  21. K = { x } K=\{x\}
  22. ( x X ) \mathbb{P}(x\in X)
  23. ( x X ) = 1 - ( x X ) . \mathbb{P}(x\in X)=1-\mathbb{P}(x\not\in X).
  24. p X p_{X}
  25. p X ( x ) = ( x X ) p_{X}(x)=\mathbb{P}(x\in X)
  26. x M . x\in M.
  27. p X p_{X}
  28. 𝟏 X \mathbf{1}_{X}
  29. p X ( x ) = 𝔼 𝟏 X ( x ) . p_{X}(x)=\mathbb{E}\mathbf{1}_{X}(x).
  30. 0
  31. 1 1
  32. b X b_{X}
  33. x M x\in M
  34. p X ( x ) > 0 p_{X}(x)>0
  35. X X
  36. k X k_{X}
  37. x M x\in M
  38. p X ( x ) = 1 p_{X}(x)=1
  39. e ( X ) e(X)
  40. X 1 , X 2 , X_{1},X_{2},\ldots
  41. i = 1 X i = e ( X ) \bigcap_{i=1}^{\infty}X_{i}=e(X)
  42. i = 1 X i \bigcap_{i=1}^{\infty}X_{i}
  43. e ( X ) . e(X).

Random_dynamical_system.html

  1. X S X\in S
  2. f : d d f:\mathbb{R}^{d}\to\mathbb{R}^{d}
  3. d d
  4. ε > 0 \varepsilon>0
  5. X ( t , ω ; x 0 ) X(t,\omega;x_{0})
  6. { d X = f ( X ) d t + ε d W ( t ) ; X ( 0 ) = x 0 ; \left\{\begin{matrix}\mathrm{d}X=f(X)\,\mathrm{d}t+\varepsilon\,\mathrm{d}W(t)% ;\\ X(0)=x_{0};\end{matrix}\right.
  7. ω Ω \omega\in\Omega
  8. W : × Ω d W:\mathbb{R}\times\Omega\to\mathbb{R}^{d}
  9. d d
  10. ( Ω , , ) := ( C 0 ( ; d ) , ( C 0 ( ; d ) ) , γ ) . (\Omega,\mathcal{F},\mathbb{P}):=\left(C_{0}(\mathbb{R};\mathbb{R}^{d}),% \mathcal{B}(C_{0}(\mathbb{R};\mathbb{R}^{d})),\gamma\right).
  11. φ : × Ω × d d \varphi:\mathbb{R}\times\Omega\times\mathbb{R}^{d}\to\mathbb{R}^{d}
  12. φ ( t , ω , x 0 ) := X ( t , ω ; x 0 ) \varphi(t,\omega,x_{0}):=X(t,\omega;x_{0})
  13. φ \varphi
  14. ( d , φ ) (\mathbb{R}^{d},\varphi)
  15. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  16. ϑ : × Ω Ω \vartheta:\mathbb{R}\times\Omega\to\Omega
  17. s s\in\mathbb{R}
  18. ϑ s : Ω Ω \vartheta_{s}:\Omega\to\Omega
  19. ( E ) = ( ϑ s - 1 ( E ) ) \mathbb{P}(E)=\mathbb{P}(\vartheta_{s}^{-1}(E))
  20. E E\in\mathcal{F}
  21. s s\in\mathbb{R}
  22. ϑ 0 = id Ω : Ω Ω \vartheta_{0}=\mathrm{id}_{\Omega}:\Omega\to\Omega
  23. Ω \Omega
  24. s , t s,t\in\mathbb{R}
  25. ϑ s ϑ t = ϑ s + t \vartheta_{s}\circ\vartheta_{t}=\vartheta_{s+t}
  26. ϑ s \vartheta_{s}
  27. s s\in\mathbb{R}
  28. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  29. s s
  30. s s
  31. ϑ s \vartheta_{s}
  32. ( Ω , , , ϑ ) (\Omega,\mathcal{F},\mathbb{P},\vartheta)
  33. ( X , d ) (X,d)
  34. φ : × Ω × X X \varphi:\mathbb{R}\times\Omega\times X\to X
  35. ( ( ) ( X ) , ( X ) ) (\mathcal{B}(\mathbb{R})\otimes\mathcal{F}\otimes\mathcal{B}(X),\mathcal{B}(X))
  36. ω Ω \omega\in\Omega
  37. φ ( 0 , ω ) = id X : X X \varphi(0,\omega)=\mathrm{id}_{X}:X\to X
  38. X X
  39. ω Ω \omega\in\Omega
  40. ( t , ω , x ) φ ( t , ω , x ) (t,\omega,x)\mapsto\varphi(t,\omega,x)
  41. t t
  42. x x
  43. φ \varphi
  44. ω Ω \omega\in\Omega
  45. φ ( t , ϑ s ( ω ) ) φ ( s , ω ) = φ ( t + s , ω ) . \varphi(t,\vartheta_{s}(\omega))\circ\varphi(s,\omega)=\varphi(t+s,\omega).
  46. W : × Ω X W:\mathbb{R}\times\Omega\to X
  47. ϑ s : Ω Ω \vartheta_{s}:\Omega\to\Omega
  48. W ( t , ϑ s ( ω ) ) = W ( t + s , ω ) - W ( s , ω ) W(t,\vartheta_{s}(\omega))=W(t+s,\omega)-W(s,\omega)
  49. ϑ s \vartheta_{s}
  50. s s
  51. x 0 x_{0}
  52. ω \omega
  53. s s
  54. t t
  55. s s
  56. x 0 x_{0}
  57. ( t + s ) (t+s)
  58. ω \omega

Random_effects_model.html

  1. Y i j = μ + U i + W i j , Y_{ij}=\mu+U_{i}+W_{ij},\,
  2. Y i j = μ + β 1 Sex i j + β 2 Race i j + β 3 ParentsEduc i j + U i + W i j , Y_{ij}=\mu+\beta_{1}\mathrm{Sex}_{ij}+\beta_{2}\mathrm{Race}_{ij}+\beta_{3}% \mathrm{ParentsEduc}_{ij}+U_{i}+W_{ij},\,
  3. Y ¯ i = 1 n j = 1 n Y i j \overline{Y}_{i\bullet}=\frac{1}{n}\sum_{j=1}^{n}Y_{ij}
  4. Y ¯ = 1 m n i = 1 m j = 1 n Y i j \overline{Y}_{\bullet\bullet}=\frac{1}{mn}\sum_{i=1}^{m}\sum_{j=1}^{n}Y_{ij}
  5. S S W = i = 1 m j = 1 n ( Y i j - Y ¯ i ) 2 SSW=\sum_{i=1}^{m}\sum_{j=1}^{n}(Y_{ij}-\overline{Y}_{i\bullet})^{2}\,
  6. S S B = n i = 1 m ( Y ¯ i - Y ¯ ) 2 SSB=n\sum_{i=1}^{m}(\overline{Y}_{i\bullet}-\overline{Y}_{\bullet\bullet})^{2}\,
  7. 1 m ( n - 1 ) E ( S S W ) = σ 2 \frac{1}{m(n-1)}E(SSW)=\sigma^{2}
  8. 1 ( m - 1 ) n E ( S S B ) = σ 2 n + τ 2 . \frac{1}{(m-1)n}E(SSB)=\frac{\sigma^{2}}{n}+\tau^{2}.

Random_element.html

  1. ( Ω , , P ) (\Omega,\mathcal{F},P)
  2. ( E , ) (E,\mathcal{E})
  3. ( , ) (\mathcal{F},\mathcal{E})
  4. B B\in\mathcal{E}
  5. \mathcal{F}
  6. E E
  7. E E
  8. ( E , ) = ( , ( ) ) (E,\mathcal{E})=(\mathbb{R},\mathcal{B}(\mathbb{R}))
  9. \mathbb{R}
  10. ( ) \mathcal{B}(\mathbb{R})
  11. X X
  12. B B
  13. σ \sigma
  14. X : Ω B X:\Omega\rightarrow B
  15. f X f\circ X
  16. X X
  17. X : Ω X\colon\Omega\to\mathbb{R}
  18. Ω \Omega
  19. \mathbb{R}
  20. X X
  21. X X
  22. X X
  23. X X
  24. 𝐗 = ( X 1 , , X n ) T \mathbf{X}=(X_{1},...,X_{n})^{T}
  25. ( Ω , , P ) (\Omega,\mathcal{F},P)
  26. Ω \Omega
  27. \mathcal{F}
  28. P P
  29. ( Ω , , P ) (\Omega,\mathcal{F},P)
  30. { F t : t T } \{F_{t}:t\in T\}
  31. F t F_{t}
  32. P ( X i = x i | X j = x j , i j ) = P ( X i = x i | i ) , P(X_{i}=x_{i}|X_{j}=x_{j},i\neq j)=P(X_{i}=x_{i}|\partial_{i}),\,
  33. i \partial_{i}
  34. P ( X i = x i | i ) = P ( ω ) ω P ( ω ) , P(X_{i}=x_{i}|\partial_{i})=\frac{P(\omega)}{\sum_{\omega^{\prime}}P(\omega^{% \prime})},
  35. 𝔅 ( X ) \mathfrak{B}(X)
  36. 𝔅 ( X ) \mathfrak{B}(X)
  37. μ = μ d + μ a = μ d + n = 1 N κ n δ X n , \mu=\mu_{d}+\mu_{a}=\mu_{d}+\sum_{n=1}^{N}\kappa_{n}\delta_{X_{n}},
  38. μ d \mu_{d}
  39. μ a \mu_{a}
  40. ( M , d ) (M,d)
  41. 𝒦 \mathcal{K}
  42. M M
  43. h h
  44. 𝒦 \mathcal{K}
  45. h ( K 1 , K 2 ) := max { sup a K 1 inf b K 2 d ( a , b ) , sup b K 2 inf a K 1 d ( a , b ) } . h(K_{1},K_{2}):=\max\left\{\sup_{a\in K_{1}}\inf_{b\in K_{2}}d(a,b),\sup_{b\in K% _{2}}\inf_{a\in K_{1}}d(a,b)\right\}.
  46. ( 𝒦 , h ) (\mathcal{K},h)
  47. 𝒦 \mathcal{K}
  48. ( 𝒦 ) \mathcal{B}(\mathcal{K})
  49. 𝒦 \mathcal{K}
  50. K K
  51. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  52. ( 𝒦 , ( 𝒦 ) ) (\mathcal{K},\mathcal{B}(\mathcal{K}))
  53. K : Ω 2 M K\colon\Omega\to 2^{M}
  54. K ( ω ) K(\omega)
  55. ω inf b K ( ω ) d ( x , b ) \omega\mapsto\inf_{b\in K(\omega)}d(x,b)
  56. x M x\in M

Random_energy_model.html

  1. N N
  2. 2 N 2^{N}
  3. r r
  4. r r
  5. r r\to\infty
  6. H ( σ ) = { i 1 , , i r } J i 1 , i r σ i 1 σ i r , H(\sigma)=\sum_{\{i_{1},\ldots,i_{r}\}}J_{i_{1},\ldots i_{r}}\sigma_{i_{1}}% \cdots\sigma_{i_{r}},
  7. ( N r ) {N\choose r}
  8. r r
  9. { i 1 , , i r } \{i_{1},\ldots,i_{r}\}
  10. J i 1 , , i r J_{i_{1},\ldots,i_{r}}
  11. J 2 r ! / ( 2 N r - 1 ) J^{2}r!/(2N^{r-1})
  12. r r\to\infty
  13. P ( E ) = δ ( E - H ( σ ) ) P(E)=\delta(E-H(\sigma))
  14. σ = ( σ i ) \sigma=(\sigma_{i})
  15. H ( σ ) H(\sigma)
  16. P ( E ) P(E)
  17. E E
  18. [ P ( E ) ] = 1 N π J 2 exp ( - E 2 J 2 N ) , [P(E)]=\sqrt{\dfrac{1}{N\pi J^{2}}}\exp\left(-\dfrac{E^{2}}{J^{2}N}\right),
  19. [ ] [\ldots]
  20. σ \sigma
  21. σ \sigma^{\prime}
  22. [ P ( E , E ) ] = [ P ( E ) ] [ P ( E ) ] . [P(E,E^{\prime})]=[P(E)]\,[P(E^{\prime})].
  23. S ( E ) = N [ log 2 - ( E N J ) 2 ] S(E)=N\left[\log 2-\left(\dfrac{E}{NJ}\right)^{2}\right]
  24. | E | < N J log 2 |E|<NJ\sqrt{\log 2}
  25. lim N S ( E ) / N \lim_{N\to\infty}S(E)/N
  26. | E | < - N J log 2 . |E|<-NJ\sqrt{\log 2}.
  27. ( 1 / T ) = S / E (1/T)=\partial S/\partial E
  28. T > T c = 1 / ( 2 log 2 ) T>T_{c}=1/(2\sqrt{\log 2})
  29. T < T c T<T_{c}
  30. E - N J log 2 E\simeq-NJ\sqrt{\log 2}

Random_phase_approximation.html

  1. | 𝐑𝐏𝐀 \left|\mathbf{RPA}\right\rangle
  2. | 𝐌𝐅𝐓 \left|\mathbf{MFT}\right\rangle
  3. 𝐚 i \mathbf{a}_{i}^{\dagger}
  4. | RPA = 𝒩 𝐞 Z i j 𝐚 i 𝐚 j / 2 | MFT \left|\mathrm{RPA}\right\rangle=\mathcal{N}\mathbf{e}^{Z_{ij}\mathbf{a}_{i}^{% \dagger}\mathbf{a}_{j}^{\dagger}/2}\left|\mathrm{MFT}\right\rangle
  5. | Z | 1 |Z|\leq 1
  6. 𝒩 = MFT | RPA MFT | MFT \mathcal{N}=\frac{\left\langle\mathrm{MFT}\right|\left.\mathrm{RPA}\right% \rangle}{\left\langle\mathrm{MFT}\right|\left.\mathrm{MFT}\right\rangle}
  7. RPA | RPA = 𝒩 2 MFT | 𝐞 z i ( 𝐪 ~ i ) 2 / 2 𝐞 z j ( 𝐪 ~ j ) 2 / 2 | MFT = 1 \langle\mathrm{RPA}|\mathrm{RPA}\rangle=\mathcal{N}^{2}\langle\mathrm{MFT}|% \mathbf{e}^{z_{i}(\tilde{\mathbf{q}}_{i})^{2}/2}\mathbf{e}^{z_{j}(\tilde{% \mathbf{q}}^{\dagger}_{j})^{2}/2}|\mathrm{MFT}\rangle=1
  8. Z i j = ( X t ) i k z k X j k Z_{ij}=(X^{\mathrm{t}})_{i}^{k}z_{k}X^{k}_{j}
  9. Z i j Z_{ij}
  10. 𝐪 ~ i = ( X ) j i 𝐚 j \tilde{\mathbf{q}}^{i}=(X^{\dagger})^{i}_{j}\mathbf{a}^{j}
  11. 𝒩 - 2 = m i n j ( z i / 2 ) m i ( z j / 2 ) n j m ! n ! MFT | i j ( 𝐪 ~ i ) 2 m i ( 𝐪 ~ j ) 2 n j | MFT \mathcal{N}^{-2}=\sum_{m_{i}}\sum_{n_{j}}\frac{(z_{i}/2)^{m_{i}}(z_{j}/2)^{n_{% j}}}{m!n!}\langle\mathrm{MFT}|\prod_{i\,j}(\tilde{\mathbf{q}}_{i})^{2m_{i}}(% \tilde{\mathbf{q}}^{\dagger}_{j})^{2n_{j}}|\mathrm{MFT}\rangle
  12. = i m i ( z i / 2 ) 2 m i ( 2 m i ) ! m i ! 2 = =\prod_{i}\sum_{m_{i}}(z_{i}/2)^{2m_{i}}\frac{(2m_{i})!}{m_{i}!^{2}}=
  13. i m i ( z i ) 2 m i ( 1 / 2 m i ) = det ( 1 - | Z | 2 ) \prod_{i}\sum_{m_{i}}(z_{i})^{2m_{i}}{1/2\choose m_{i}}=\sqrt{\det(1-|Z|^{2})}
  14. 𝐚 ~ i = ( 1 1 - Z 2 ) i j 𝐚 j + ( 1 1 - Z 2 Z ) i j 𝐚 j \tilde{\mathbf{a}}_{i}=\left(\frac{1}{\sqrt{1-Z^{2}}}\right)_{ij}\mathbf{a}_{j% }+\left(\frac{1}{\sqrt{1-Z^{2}}}Z\right)_{ij}\mathbf{a}^{\dagger}_{j}

Randomized_response.html

  1. Y A = p × E P + ( 1 - p ) ( 1 - E P ) YA=p\times EP+(1-p)(1-EP)
  2. E P = Y A + p - 1 2 p - 1 EP=\frac{YA+p-1}{2p-1}
  3. p = 1 6 p=\tfrac{1}{6}
  4. Y A = 3 4 YA=\tfrac{3}{4}
  5. E P = ( 3 4 + 1 6 - 1 ) / ( 2 × 1 6 - 1 ) = 1 8 EP=(\tfrac{3}{4}+\tfrac{1}{6}-1)/(2\times\tfrac{1}{6}-1)=\tfrac{1}{8}

Range_criterion.html

  1. H = H 1 H n H=H_{1}\otimes\cdots\otimes H_{n}
  2. M = i v i v i * M=\sum_{i}v_{i}v_{i}^{*}
  3. { v i } \;\{v_{i}\}
  4. v i v_{i}
  5. M = v 1 v 1 * + T M=v_{1}v_{1}^{*}+T
  6. { v 1 } \{v_{1}\}\subset
  7. v 1 v_{1}\in
  8. \;{}^{\perp}\subset
  9. { v 1 } \{v_{1}\}^{\perp}
  10. \perp
  11. \subset
  12. { v 1 } \{v_{1}\}^{\perp}
  13. \cap
  14. { v 1 } \{v_{1}\}
  15. T w = α v 1 \;Tw=\alpha v_{1}
  16. M w = w , v 1 v 1 + T w = ( w , v 1 + α ) v 1 . Mw=\langle w,v_{1}\rangle v_{1}+Tw=(\langle w,v_{1}\rangle+\alpha)v_{1}.
  17. v 1 v_{1}
  18. { v i } \;\{v_{i}\}
  19. ρ = i ψ 1 , i ψ 1 , i * ψ n , i ψ n , i * \rho=\sum_{i}\psi_{1,i}\psi_{1,i}^{*}\otimes\cdots\otimes\psi_{n,i}\psi_{n,i}^% {*}
  20. ψ j , i ψ j , i * \psi_{j,i}\psi_{j,i}^{*}
  21. ρ = i ( ψ 1 , i ψ n , i ) ( ψ 1 , i * ψ n , i * ) . \rho=\sum_{i}(\psi_{1,i}\otimes\cdots\otimes\psi_{n,i})(\psi_{1,i}^{*}\otimes% \cdots\otimes\psi_{n,i}^{*}).
  22. ψ 1 , i ψ n , i \psi_{1,i}\otimes\cdots\otimes\psi_{n,i}
  23. v i v_{i}

Rank-dependent_expected_utility.html

  1. 𝐲 [ ] \mathbf{y}_{[\;]}
  2. 𝐲 \mathbf{y}
  3. y [ 1 ] y [ 2 ] y [ S ] y_{[1]}\leq y_{[2]}\leq...\leq y_{[S]}
  4. W ( 𝐲 ) = s Ω h [ s ] ( π ) u ( y [ s ] ) W(\mathbf{y})=\sum_{s\in\Omega}h_{[s]}(\mathbf{\pi})u(y_{[s]})
  5. π Π , u : , \mathbf{\pi}\in\Pi,u:\mathbb{R}\rightarrow\mathbb{R},
  6. h [ s ] ( π ) h_{[s]}(\mathbf{\pi})
  7. h [ s ] ( π ) = q ( t = 1 s π [ t ] ) - q ( t = 1 s - 1 π [ t ] ) h_{[s]}(\mathbf{\pi})=q\left(\sum\limits_{t=1}^{s}\pi_{[t]}\right)-q\left(\sum% \limits_{t=1}^{s-1}\pi_{[t]}\right)
  8. q : [ 0 , 1 ] [ 0 , 1 ] q:[0,1]\rightarrow[0,1]
  9. q ( 0 ) = 0 q(0)=0
  10. q ( 1 ) = 1 q(1)=1
  11. s Ω h [ s ] ( π ) = q ( t = 1 S π [ t ] ) = q ( 1 ) = 1 \sum_{s\in\Omega}h_{[s]}(\mathbf{\pi})=q\left(\sum\limits_{t=1}^{S}\pi_{[t]}% \right)=q(1)=1

Rank_(differential_topology).html

  1. d p f : T p M T f ( p ) N d_{p}f:T_{p}M\to T_{f(p)}N\,
  2. rank ( f ) p = dim ( im ( d p f ) ) . \operatorname{rank}(f)_{p}=\dim(\operatorname{im}(d_{p}f)).
  3. f ( x 1 , , x m ) = ( x 1 , , x k , 0 , , 0 ) f(x^{1},\ldots,x^{m})=(x^{1},\ldots,x^{k},0,\ldots,0)\,

Rational_representation.html

  1. G G

Rayl.html

  1. Z ¯ ( 𝐫 , ω ) = p ¯ ( 𝐫 , ω ) v ¯ ( 𝐫 , ω ) {\underline{Z}(\mathbf{r},\omega)=\frac{\underline{p}(\mathbf{r},\omega)}{% \underline{v}(\mathbf{r},\omega)}}
  2. Z ¯ , p ¯ \underline{Z},\underline{p}
  3. v ¯ \underline{v}
  4. 𝐫 \mathbf{r}
  5. ω \omega
  6. Z 0 = ρ 0 c 0 {Z_{0}=\rho_{0}c_{0}}
  7. Z 0 {Z_{0}}
  8. ρ 0 {\rho_{0}}
  9. c 0 {c_{0}}
  10. Z ¯ \underline{Z}
  11. Z 0 {Z_{0}}
  12. 1 Rayl MKS = 1 N s m 3 = 1 Pa s m = 1 kg s m 2 {\rm 1~{}Rayl_{MKS}=1~{}\frac{N\cdot s}{m^{3}}=1~{}\frac{Pa\cdot s}{m}=1~{}% \frac{kg}{s\cdot m^{2}}}
  13. 1 Rayl CGS = 1 dyn s cm 3 = 1 ba s cm = 1 g s cm 2 {\rm 1~{}Rayl_{CGS}=1~{}\frac{dyn\cdot s}{cm^{3}}=1~{}\frac{ba\cdot s}{cm}=1~{% }\frac{g}{s\cdot cm^{2}}}

Reaction_calorimeter.html

  1. Q = U A ( T r - T j ) Q\quad=UA(Tr-Tj)
  2. Q Q
  3. U U
  4. A A
  5. T r Tr
  6. T j Tj
  7. Q = m s C p s ( T i - T o ) Q\quad=m_{s}C_{ps}(T_{i}-T_{o})
  8. Q Q
  9. m s m_{s}
  10. C p s C_{ps}
  11. T i T_{i}
  12. T o T_{o}
  13. Q = I V Q\quad=IV
  14. I - I 0 \quad I-I_{0}
  15. I I
  16. V V
  17. I 0 I_{0}

Reactive_inhibition.html

  1. I R = I R x 10 - a t I^{\prime}_{R}=I_{R}x10^{-at}
  2. I ( t ) = I ( 0 ) e - b t I(t)=I(0)e^{-bt}
  3. b = a ln ( 10 ) b=a\ln(10)
  4. I ( 0 ) I(0)
  5. Y ( t ) = Y ( 0 ) - b t Y(t)=Y(0)-bt
  6. Y ( t ) = ln I ( t ) Y(t)=\ln I(t)
  7. Y ( 0 ) = ln I ( 0 ) Y(0)=\ln I(0)

Real_analytic_Eisenstein_series.html

  1. E ( z , s ) = 1 2 ( m , n ) = 1 y s | m z + n | 2 s E(z,s)={1\over 2}\sum_{(m,n)=1}{y^{s}\over|mz+n|^{2s}}
  2. y 2 ( 2 x 2 + 2 y 2 ) E ( z , s ) = s ( s - 1 ) E ( z , s ) , y^{2}\left(\frac{\partial^{2}}{\partial x^{2}}+\frac{\partial^{2}}{\partial y^% {2}}\right)E(z,s)=s(s-1)E(z,s),
  3. z = x + y i . z=x+yi.
  4. E * ( z , s ) = π - s Γ ( s ) ζ ( 2 s ) E ( z , s ) E^{*}(z,s)=\pi^{-s}\Gamma(s)\zeta(2s)E(z,s)
  5. E * ( z , s ) = E * ( z , 1 - s ) E^{*}(z,s)=E^{*}(z,1-s)
  6. E ( z , s ) = y s + ζ ^ ( 2 s - 1 ) ζ ( 2 s ) y 1 - s + 4 ζ ^ ( 2 s ) m = 1 m s - 1 / 2 σ 1 - 2 s ( m ) y K s - 1 / 2 ( 2 π m y ) cos ( 2 π m x ) , E(z,s)=y^{s}+\frac{\hat{\zeta}(2s-1)}{\zeta(2s)}y^{1-s}+\frac{4}{\hat{\zeta}(2% s)}\sum_{m=1}^{\infty}m^{s-1/2}\sigma_{1-2s}(m)\sqrt{y}K_{s-1/2}(2\pi my)\cos(% 2\pi mx)\ ,
  7. ζ ^ ( s ) = π - s / 2 Γ ( s 2 ) ζ ( s ) , \hat{\zeta}(s)=\pi^{-s/2}\Gamma\biggl(\frac{s}{2}\biggr)\zeta(s)\ ,
  8. σ s ( m ) = d | m d s , \sigma_{s}(m)=\sum_{d|m}d^{s}\ ,
  9. K s ( z ) = 1 2 0 exp ( - z 2 ( u + 1 u ) ) u s - 1 d u π 2 z e - z . ( z ) \begin{aligned}\displaystyle K_{s}(z)&\displaystyle=\frac{1}{2}\int^{\infty}_{% 0}\exp\biggl(-\frac{z}{2}(u+\frac{1}{u})\biggr)\cdot u^{s-1}du\\ &\displaystyle\sim\sqrt{\frac{\pi}{2z}}e^{-z}\ .\ \ \ \ \ \ \ \ \ (z% \rightarrow\infty)\end{aligned}
  10. ζ Q ( s ) = ( m , n ) ( 0 , 0 ) 1 Q ( m , n ) s . \zeta_{Q}(s)=\sum_{(m,n)\neq(0,0)}{1\over Q(m,n)^{s}}.
  11. Q ( m , n ) = a | m z + n | 2 Q(m,n)=a|mz+n|^{2}
  12. z = - b 2 a + i - b 2 + 4 a c 2 a . z=\frac{-b}{2a}+\frac{i\sqrt{-b^{2}+4ac}}{2a}.

Real_point.html

  1. ( x , y , z ) (x,y,z)
  2. λ λ
  3. λ x λx
  4. λ y λy
  5. λ z λz
  6. ( u 1 , u 2 , , u n ) (u_{1},u_{2},\ldots,u_{n})
  7. λ λ
  8. ( λ u 1 , λ u 2 , , λ u n ) (\lambda u_{1},\lambda u_{2},\ldots,\lambda u_{n})
  9. λ u λu
  10. λ λ
  11. u u

Realcompact_space.html

  1. n \mathbb{R}^{n}

Receptor–ligand_kinetics.html

  1. K K
  2. R + L RL {\rm R}+{\rm L}\rightleftharpoons{\rm RL}
  3. k on k_{\rm on}
  4. k off k_{\rm off}
  5. R + L RL {\rm R}+{\rm L}\to{\rm RL}
  6. RL R + L {\rm RL}\to{\rm R}+{\rm L}
  7. k on [ R ] [ L ] = k off [ RL ] k_{\rm on}\,[{\rm R}]\,[{\rm L}]=k_{\rm off}\,[{\rm RL}]
  8. [ R ] [{\rm R}]
  9. [ L ] [{\rm L}]
  10. [ RL ] [{\rm RL}]
  11. K a K_{\rm a}
  12. K a = k on k off = [ RL ] [ R ] [ L ] K_{\rm a}={k_{\rm on}\over k_{\rm off}}={[{\rm RL}]\over{[{\rm R}]\,[{\rm L}]}}
  13. R + L C \mathrm{R}+\mathrm{L}\leftrightarrow\mathrm{C}
  14. K d = def k - 1 k 1 = [ R ] e q [ L ] e q [ C ] e q K_{d}\ \stackrel{\mathrm{def}}{=}\ \frac{k_{-1}}{k_{1}}=\frac{[\mathrm{R}]_{eq% }[\mathrm{L}]_{eq}}{[\mathrm{C}]_{eq}}
  15. R t o t = def [ R ] + [ C ] R_{tot}\ \stackrel{\mathrm{def}}{=}\ [\mathrm{R}]+[\mathrm{C}]
  16. L t o t = def [ L ] + [ C ] L_{tot}\ \stackrel{\mathrm{def}}{=}\ [\mathrm{L}]+[\mathrm{C}]
  17. R = def [ R ] R\ \stackrel{\mathrm{def}}{=}\ [\mathrm{R}]
  18. d R d t = - k 1 R L + k - 1 C = - k 1 R ( L t o t - R t o t + R ) + k - 1 ( R t o t - R ) \frac{dR}{dt}=-k_{1}RL+k_{-1}C=-k_{1}R(L_{tot}-R_{tot}+R)+k_{-1}(R_{tot}-R)
  19. 1 k 1 d R d t = - R 2 + 2 E R + K d R t o t = - ( R - R + ) ( R - R - ) \frac{1}{k_{1}}\frac{dR}{dt}=-R^{2}+2ER+K_{d}R_{tot}=-\left(R-R_{+}\right)% \left(R-R_{-}\right)
  20. R ± = def E ± D R_{\pm}\ \stackrel{\mathrm{def}}{=}\ E\pm D
  21. D = def E 2 + R t o t K d D\ \stackrel{\mathrm{def}}{=}\ \sqrt{E^{2}+R_{tot}K_{d}}
  22. R - R_{-}
  23. { 1 R - R + - 1 R - R - } d R = - 2 D k 1 d t \left\{\frac{1}{R-R_{+}}-\frac{1}{R-R_{-}}\right\}dR=-2Dk_{1}dt
  24. log | R - R + | - log | R - R - | = - 2 D k 1 t + ϕ 0 \log\left|R-R_{+}\right|-\log\left|R-R_{-}\right|=-2Dk_{1}t+\phi_{0}
  25. g = e x p ( - 2 D k 1 t + ϕ 0 ) g=exp(-2Dk_{1}t+\phi_{0})
  26. R ( t ) = R + - g R - 1 - g R(t)=\frac{R_{+}-gR_{-}}{1-g}
  27. ϕ 0 = def log | R ( t = 0 ) - R + | - log | R ( t = 0 ) - R - | \phi_{0}\ \stackrel{\mathrm{def}}{=}\ \log\left|R(t=0)-R_{+}\right|-\log\left|% R(t=0)-R_{-}\right|
  28. C ( t ) C(t)
  29. L ( t ) L(t)

Rectangular_potential_barrier.html

  1. V 0 V_{0}
  2. E > V 0 E>V_{0}
  3. ψ ( x ) \psi(x)
  4. H ψ ( x ) = [ - 2 2 m d 2 d x 2 + V ( x ) ] ψ ( x ) = E ψ ( x ) H\psi(x)=\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}+V(x)\right]\psi(x)=E% \psi(x)
  5. H H
  6. \hbar
  7. m m
  8. E E
  9. V ( x ) = V 0 [ Θ ( x ) - Θ ( x - a ) ] V(x)=V_{0}[\Theta(x)-\Theta(x-a)]
  10. V 0 > 0 V_{0}>0
  11. a a
  12. Θ ( x ) = 0 , x < 0 ; Θ ( x ) = 1 , x > 0 \Theta(x)=0,\;x<0;\;\Theta(x)=1,\;x>0
  13. x = 0 x=0
  14. x = a x=a
  15. x x
  16. - 2 2 m d 2 d x 2 ψ -\frac{\hbar^{2}}{2m}\frac{d^{2}}{dx^{2}}\psi
  17. x < 0 , 0 < x < a , x > a x<0,0<x<a,x>a
  18. E > V 0 E>V_{0}
  19. ψ L ( x ) = A r e i k 0 x + A l e - i k 0 x x < 0 \psi_{L}(x)=A_{r}e^{ik_{0}x}+A_{l}e^{-ik_{0}x}\quad x<0
  20. ψ C ( x ) = B r e i k 1 x + B l e - i k 1 x 0 < x < a \psi_{C}(x)=B_{r}e^{ik_{1}x}+B_{l}e^{-ik_{1}x}\quad 0<x<a
  21. ψ R ( x ) = C r e i k 0 x + C l e - i k 0 x x > a \psi_{R}(x)=C_{r}e^{ik_{0}x}+C_{l}e^{-ik_{0}x}\quad x>a
  22. k 0 = 2 m E / 2 x < 0 o r x > a k_{0}=\sqrt{2mE/\hbar^{2}}\quad\quad\quad\quad x<0\quad or\quad x>a
  23. k 1 = 2 m ( E - V 0 ) / 2 0 < x < a k_{1}=\sqrt{2m(E-V_{0})/\hbar^{2}}\quad 0<x<a
  24. k 1 k_{1}
  25. E V 0 E\neq V_{0}
  26. E = V 0 E=V_{0}
  27. A , B , C A,B,C
  28. x = 0 x=0
  29. x = a x=a
  30. ψ L ( 0 ) = ψ C ( 0 ) \psi_{L}(0)=\psi_{C}(0)
  31. d d x ψ L ( 0 ) = d d x ψ C ( 0 ) \frac{d}{dx}\psi_{L}(0)=\frac{d}{dx}\psi_{C}(0)
  32. ψ C ( a ) = ψ R ( a ) \psi_{C}(a)=\psi_{R}(a)
  33. d d x ψ C ( a ) = d d x ψ R ( a ) \frac{d}{dx}\psi_{C}(a)=\frac{d}{dx}\psi_{R}(a)
  34. A r + A l = B r + B l A_{r}+A_{l}=B_{r}+B_{l}
  35. i k 0 ( A r - A l ) = i k 1 ( B r - B l ) ik_{0}(A_{r}-A_{l})=ik_{1}(B_{r}-B_{l})
  36. B r e i a k 1 + B l e - i a k 1 = C r e i a k 0 + C l e - i a k 0 B_{r}e^{iak_{1}}+B_{l}e^{-iak_{1}}=C_{r}e^{iak_{0}}+C_{l}e^{-iak_{0}}
  37. i k 1 ( B r e i a k 1 - B l e - i a k 1 ) = i k 0 ( C r e i a k 0 - C l e - i a k 0 ) ik_{1}(B_{r}e^{iak_{1}}-B_{l}e^{-iak_{1}})=ik_{0}(C_{r}e^{iak_{0}}-C_{l}e^{-% iak_{0}})
  38. ψ C ( x ) = B 1 + B 2 x 0 < x < a . \psi_{C}(x)=B_{1}+B_{2}x\quad 0<x<a.
  39. x = 0 x=0
  40. x = a x=a
  41. A r + A l = B 1 A_{r}+A_{l}=B_{1}\,\!
  42. i k 0 ( A r - A l ) = B 2 ik_{0}(A_{r}-A_{l})=B_{2}\,\!
  43. B 1 + B 2 a = C r e i a k 0 + C l e - i a k 0 B_{1}+B_{2}a=C_{r}e^{iak_{0}}+C_{l}e^{-iak_{0}}
  44. B 2 = i k 0 ( C r e i a k 0 - C l e - i a k 0 ) B_{2}=ik_{0}(C_{r}e^{iak_{0}}-C_{l}e^{-iak_{0}})
  45. E E
  46. V 0 V_{0}
  47. E < V 0 E<V_{0}
  48. A r A_{r}
  49. A l A_{l}
  50. C r C_{r}
  51. A r = 1 A_{r}=1
  52. A l = r A_{l}=r
  53. C l C_{l}
  54. C r = t C_{r}=t
  55. B l , B r B_{l},B_{r}
  56. r r
  57. t t
  58. t = 4 k 0 k 1 e - i a ( k 0 - k 1 ) ( k 0 + k 1 ) 2 - e 2 i a k 1 ( k 0 - k 1 ) 2 t=\frac{4k_{0}k_{1}e^{-ia(k_{0}-k_{1})}}{(k_{0}+k_{1})^{2}-e^{2iak_{1}}(k_{0}-% k_{1})^{2}}
  59. r = ( k 0 2 - k 1 2 ) sin ( a k 1 ) 2 i k 0 k 1 cos ( a k 1 ) + ( k 0 2 + k 1 2 ) sin ( a k 1 ) . r=\frac{(k_{0}^{2}-k_{1}^{2})\sin(ak_{1})}{2ik_{0}k_{1}\cos(ak_{1})+(k_{0}^{2}% +k_{1}^{2})\sin(ak_{1})}.
  60. E > 0 E>0
  61. 2 m V 0 a / = 7 \sqrt{2mV_{0}}a/\hbar=7
  62. E < V 0 E<V_{0}
  63. T = | t | 2 = 1 1 + V 0 2 sinh 2 ( k 1 a ) 4 E ( V 0 - E ) T=|t|^{2}=\frac{1}{1+\frac{V_{0}^{2}\sinh^{2}(k_{1}a)}{4E(V_{0}-E)}}
  64. k 1 = 2 m ( V 0 - E ) / 2 k_{1}=\sqrt{2m(V_{0}-E)/\hbar^{2}}
  65. k 0 k_{0}
  66. 1 / k 1 1/k_{1}
  67. T = | t | 2 = 1 1 + V 0 2 sin 2 ( k 1 a ) 4 E ( E - V 0 ) T=|t|^{2}=\frac{1}{1+\frac{V_{0}^{2}\sin^{2}(k_{1}a)}{4E(E-V_{0})}}
  68. E > V 0 E>V_{0}
  69. R = | r | 2 = 1 - T . \,R=|r|^{2}=1-T.
  70. k 1 a k_{1}a
  71. E V 0 E\gg V_{0}
  72. r = 0 r=0
  73. E = V 0 E=V_{0}
  74. T = 1 1 + m a 2 V 0 / 2 2 T=\frac{1}{1+ma^{2}V_{0}/2\hbar^{2}}
  75. m m
  76. Ψ ( x , y , z ) = ψ ( x ) ϕ ( y , z ) \Psi(x,y,z)=\psi(x)\phi(y,z)
  77. V 0 , a 0 V_{0}\to\infty,\quad a\to 0
  78. V 0 a = λ 2 m 2 V_{0}a=\frac{\lambda^{2}}{m^{2}}

Rectified_24-cell.html

  1. D 3 {D}_{3}
  2. C 3 {C}_{3}
  3. D 3 {D}_{3}
  4. F 3 {F}_{3}
  5. D 3 {D}_{3}
  6. C 3 {C}_{3}
  7. F 3 {F}_{3}
  8. F 3 {F}_{3}
  9. C 3 {C}_{3}
  10. D 3 {D}_{3}

Recurrent_point.html

  1. X X
  2. f : X X f\colon X\to X
  3. x X x\in X
  4. f f
  5. x ω ( x ) x\in\omega(x)
  6. x x
  7. ω \omega
  8. U U
  9. x x
  10. n > 0 n>0
  11. f n ( x ) U f^{n}(x)\in U
  12. f f
  13. R ( f ) R(f)
  14. f f
  15. f f
  16. f f
  17. X X
  18. R ( f ) R(f)
  19. f f

Recursive_ordinal.html

  1. α \alpha
  2. α \alpha
  3. ω \omega
  4. ω 1 C K \omega^{CK}_{1}
  5. ω 1 C K \omega^{CK}_{1}
  6. ω 1 C K \omega^{CK}_{1}
  7. 𝒪 \mathcal{O}

Recursive_tree.html

  1. T n = ( n - 1 ) ! . T_{n}=(n-1)!.\,
  2. T ( z ) = n 1 T n z n n ! = log ( 1 1 - z ) . T(z)=\sum_{n\geq 1}T_{n}\frac{z^{n}}{n!}=\log\left(\frac{1}{1-z}\right).
  3. F = + 1 1 ! × F + 1 2 ! × F * F + 1 3 ! × F * F * F * = × exp ( F ) , F=\circ+\frac{1}{1!}\cdot\circ\times F+\frac{1}{2!}\cdot\circ\times F*F+\frac{% 1}{3!}\cdot\circ\times F*F*F*\cdots=\circ\times\exp(F),
  4. \circ
  5. * *
  6. T ( z ) = exp ( T ( z ) ) , T^{\prime}(z)=\exp(T(z)),

Redmond–Sun_conjecture.html

  1. [ 2 3 , 3 2 ] , [ 5 2 , 3 3 ] , [ 2 5 , 6 2 ] , [ 11 2 , 5 3 ] , [ 3 7 , 13 3 ] , [2^{3},\,3^{2}],\ [5^{2},\,3^{3}],\ [2^{5},\,6^{2}],\ [11^{2},\,5^{3}],\ [3^{7% },\,13^{3}],
  2. [ 5 5 , 56 2 ] , [ 181 2 , 2 15 ] , [ 43 3 , 282 2 ] , [ 46 3 , 312 2 ] , [ 22434 2 , 55 5 ] . [5^{5},\,56^{2}],\ [181^{2},\,2^{15}],\ [43^{3},\,282^{2}],\ [46^{3},\,312^{2}% ],\ [22434^{2},\,55^{5}].

Reduction_criterion.html

  1. H = H 1 H 2 . H=H_{1}\otimes H_{2}.
  2. ( I Φ ) ( ρ ) (I\otimes\Phi)(\rho)
  3. ( I Φ ) ( ρ ) 0. (I\otimes\Phi)(\rho)\geq 0.
  4. Φ ( A ) = Tr A - A . \Phi(A)=\operatorname{Tr}A-A.
  5. ( I Φ ) ( ρ ) 0. (I\otimes\Phi)(\rho)\geq 0.
  6. I ρ 1 - ρ 0 I\otimes\rho_{1}-\rho\geq 0
  7. ρ 2 I - ρ 0 \rho_{2}\otimes I-\rho\geq 0

Refactorable_number.html

  1. τ ( n ) | n \tau(n)|n
  2. n 0 a mod m n_{0}\equiv a\mod m
  3. n > n 0 n>n_{0}
  4. n a mod m n\equiv a\mod m

Regular_dodecahedron.html

  1. r u = a 3 4 ( 1 + 5 ) 1.401258538 a r_{u}=a\frac{\sqrt{3}}{4}\left(1+\sqrt{5}\right)\approx 1.401258538\cdot a
  2. r i = a 1 2 5 2 + 11 10 5 1.113516364 a r_{i}=a\frac{1}{2}\sqrt{\frac{5}{2}+\frac{11}{10}\sqrt{5}}\approx 1.113516364\cdot a
  3. r m = a 1 4 ( 3 + 5 ) 1.309016994 a r_{m}=a\frac{1}{4}\left(3+\sqrt{5}\right)\approx 1.309016994\cdot a
  4. r u = a 3 2 ϕ r_{u}=a\,\frac{\sqrt{3}}{2}\phi
  5. r i = a ϕ 2 2 3 - ϕ r_{i}=a\,\frac{\phi^{2}}{2\sqrt{3-\phi}}
  6. r m = a ϕ 2 2 r_{m}=a\,\frac{\phi^{2}}{2}
  7. A = 3 25 + 10 5 a 2 20.645728807 a 2 A=3\sqrt{25+10\sqrt{5}}a^{2}\approx 20.645728807a^{2}
  8. V = 1 4 ( 15 + 7 5 ) a 3 7.6631189606 a 3 V=\frac{1}{4}(15+7\sqrt{5})a^{3}\approx 7.6631189606a^{3}

Regular_homotopy.html

  1. f , g : M N f,g:M\to N
  2. C ( M , N ) C(M,N)
  3. C ( M , N ) C(M,N)
  4. I m m ( M , N ) Imm(M,N)
  5. f , g : M N f,g:M\to N
  6. I m m ( M , N ) Imm(M,N)
  7. n \mathbb{R}^{n}
  8. 3 \mathbb{R}^{3}

Regular_number.html

  1. 60 max ( i / 2 , j , k ) \scriptstyle 60^{\max(\lceil i\,/2\rceil,j,k)}{}
  2. ( ln 2 ) i + ( ln 3 ) j + ( ln 5 ) k ln N , (\ln 2)i+(\ln 3)j+(\ln 5)k\leq\ln N,
  3. log 2 N log 3 N log 5 N 6 . \frac{\log_{2}N\,\log_{3}N\,\log_{5}N}{6}.
  4. ( ln ( N 30 ) ) 3 6 ln 2 ln 3 ln 5 + O ( log N ) , \frac{\left(\ln(N\sqrt{30})\right)^{3}}{6\ln 2\ln 3\ln 5}+O(\log N),
  5. O ( log log N ) O(\log\log N)
  6. ( p 2 - q 2 , 2 p q , p 2 + q 2 ) (p^{2}-q^{2},\,2pq,\,p^{2}+q^{2})

Regular_part.html

  1. f ( z ) = n = - a n ( z - c ) n , f(z)=\sum_{n=-\infty}^{\infty}a_{n}(z-c)^{n},
  2. n = 0 a n ( z - c ) n . \sum_{n=0}^{\infty}a_{n}(z-c)^{n}.

Regular_semigroup.html

  1. a b a\,\mathcal{L}\,b
  2. a b a\,\mathcal{R}\,b
  3. a 𝒥 b a\,\mathcal{J}\,b
  4. \mathcal{L}
  5. \mathcal{R}
  6. \mathcal{L}
  7. \mathcal{R}
  8. a b a\,\mathcal{L}\,b
  9. a b a\,\mathcal{R}\,b
  10. \mathcal{L}
  11. \mathcal{R}

Regular_solution.html

  1. Δ S m = - n R ( x 1 ln x 1 + x 2 ln x 2 ) \Delta S_{m}=-nR(x_{1}\ln x_{1}+x_{2}\ln x_{2})\,
  2. R R\,
  3. n n\,
  4. x i x_{i}\,
  5. α \alpha
  6. P 1 = x 1 P 1 * f 1 , M \ P_{1}=x_{1}P^{*}_{1}f_{1,M}\,
  7. P 2 = x 2 P 2 * f 2 , M \ P_{2}=x_{2}P^{*}_{2}f_{2,M}\,
  8. f 1 , M = exp ( α x 2 2 ) \ f_{1,M}={\rm exp}(\alpha x_{2}^{2})\,
  9. f 2 , M = exp ( α x 1 2 ) \ f_{2,M}={\rm exp}(\alpha x_{1}^{2})\,
  10. α \alpha

Regularity_theorem_for_Lebesgue_measure.html

  1. A = B N = ( B N ) ( N B ) . A=B\triangle N=\left(B\setminus N\right)\cup\left(N\setminus B\right).

Regulated_integral.html

  1. Π = { a = t 0 < t 1 < < t k = b } \Pi=\{a=t_{0}<t_{1}<\cdots<t_{k}=b\}
  2. a b φ ( t ) d t := i = 0 k - 1 c i | t i + 1 - t i | . \int_{a}^{b}\varphi(t)\,\mathrm{d}t:=\sum_{i=0}^{k-1}c_{i}|t_{i+1}-t_{i}|.
  3. f ( t + ) = lim s t f ( s ) f(t+)=\lim_{s\downarrow t}f(s)
  4. f ( t - ) = lim s t f ( s ) f(t-)=\lim_{s\uparrow t}f(s)
  5. a b f ( t ) d t := lim n a b φ n ( t ) d t , \int_{a}^{b}f(t)\,\mathrm{d}t:=\lim_{n\to\infty}\int_{a}^{b}\varphi_{n}(t)\,% \mathrm{d}t,
  6. a b α f ( t ) + β g ( t ) d t = α a b f ( t ) d t + β a b g ( t ) d t . \int_{a}^{b}\alpha f(t)+\beta g(t)\,\mathrm{d}t=\alpha\int_{a}^{b}f(t)\,% \mathrm{d}t+\beta\int_{a}^{b}g(t)\,\mathrm{d}t.
  7. m | b - a | a b f ( t ) d t M | b - a | . m|b-a|\leq\int_{a}^{b}f(t)\,\mathrm{d}t\leq M|b-a|.
  8. | a b f ( t ) d t | a b | f ( t ) | d t . \left|\int_{a}^{b}f(t)\,\mathrm{d}t\right|\leq\int_{a}^{b}|f(t)|\,\mathrm{d}t.

Reid_index.html

  1. R I = g l a n d w a l l \ RI=\frac{gland}{wall}

Relational_quantum_mechanics.html

  1. O O
  2. S S
  3. O O
  4. O O
  5. | ψ |\psi\rangle
  6. O O^{\prime}
  7. O O
  8. S S
  9. O O^{\prime}
  10. S S
  11. | |\uparrow\rangle
  12. | |\downarrow\rangle
  13. H S H_{S}
  14. O O
  15. t 1 t_{1}
  16. | ψ = α | + β | |\psi\rangle=\alpha|\uparrow\rangle+\beta|\downarrow\rangle
  17. | α | 2 |\alpha|^{2}
  18. | β | 2 |\beta|^{2}
  19. | |\uparrow\rangle
  20. | |\downarrow\rangle
  21. O O
  22. t 1 t 2 α | + β | | . \begin{matrix}t_{1}&\rightarrow&t_{2}\\ \alpha|\uparrow\rangle+\beta|\downarrow\rangle&\rightarrow&|\uparrow\rangle.% \end{matrix}
  23. O O
  24. H S H O H_{S}\otimes H_{O}
  25. H O H_{O}
  26. O O
  27. O O
  28. | i n i t |init\rangle
  29. O O
  30. S S
  31. | O |O_{\uparrow}\rangle
  32. | O |O_{\downarrow}\rangle
  33. O O
  34. S S
  35. O O^{\prime}
  36. S + O S+O
  37. O O^{\prime}
  38. t 1 t 2 ( α | + β | ) | i n i t α | | O + β | | O . \begin{matrix}t_{1}&\rightarrow&t_{2}\\ \left(\alpha|\uparrow\rangle+\beta|\downarrow\rangle\right)\otimes|init\rangle% &\rightarrow&\alpha|\uparrow\rangle\otimes|O_{\uparrow}\rangle+\beta|% \downarrow\rangle\otimes|O_{\downarrow}\rangle.\end{matrix}
  39. O O
  40. O O^{\prime}
  41. t 1 t 2 t_{1}\rightarrow t_{2}
  42. O O
  43. t 2 t_{2}
  44. S S
  45. O O^{\prime}
  46. S S
  47. O O
  48. t 2 t_{2}
  49. O O^{\prime}
  50. O O
  51. O O^{\prime}
  52. O O
  53. S S
  54. S S
  55. O O
  56. S S
  57. O O
  58. O O^{\prime}
  59. S + O S+O
  60. O O^{\prime}
  61. O O
  62. S S
  63. O O
  64. O O
  65. S + O S+O
  66. S S
  67. ( S + O ) + O (S+O)+O^{\prime}
  68. O O^{\prime}
  69. S + O S+O
  70. S S
  71. O O
  72. O O^{\prime}
  73. O O
  74. S S
  75. O O
  76. O O
  77. O O
  78. O O^{\prime}
  79. O O
  80. S S
  81. O O^{\prime}
  82. M M
  83. M ( | | O ) = | | O M\left(|\uparrow\rangle\otimes|O_{\uparrow}\rangle\right)=|\uparrow\rangle% \otimes|O_{\uparrow}\rangle
  84. M ( | | O ) = 0 M\left(|\uparrow\rangle\otimes|O_{\downarrow}\rangle\right)=0
  85. M ( | | O ) = 0 M\left(|\downarrow\rangle\otimes|O_{\uparrow}\rangle\right)=0
  86. M ( | | O ) = | | O M\left(|\downarrow\rangle\otimes|O_{\downarrow}\rangle\right)=|\downarrow% \rangle\otimes|O_{\downarrow}\rangle
  87. O O
  88. S S
  89. O O
  90. S S
  91. | |\uparrow\rangle
  92. | |\downarrow\rangle
  93. t 2 t_{2}
  94. O O^{\prime}
  95. S + O S+O
  96. M M
  97. O O^{\prime}
  98. S + O S+O
  99. O O^{\prime}
  100. S S
  101. O O
  102. t 2 t_{2}
  103. O O
  104. S S
  105. O O
  106. O O^{\prime}
  107. S S
  108. O O
  109. O O
  110. O O^{\prime}
  111. S + O S+O
  112. O O
  113. S S
  114. O ′′ O^{\prime\prime}
  115. t 1 t_{1}
  116. t 2 t_{2}
  117. t 3 t_{3}
  118. M A ( α ) M_{A}(\alpha)
  119. A A
  120. α \alpha
  121. t 2 t_{2}
  122. M A ( α ) M_{A}(\alpha)
  123. M A ( α ) + M A ( β ) = 0 , M_{A}(\alpha)+M_{A}(\beta)=0,
  124. σ \sigma
  125. β \beta
  126. - σ -\sigma
  127. t 3 t_{3}
  128. M A ( B ) = M A ( β ) M_{A}(B)=M_{A}(\beta)
  129. M C ( A ) = M C ( α ) M_{C}(A)=M_{C}(\alpha)
  130. M C ( B ) = M C ( β ) M_{C}(B)=M_{C}(\beta)
  131. M C ( α ) + M C ( β ) = 0. M_{C}(\alpha)+M_{C}(\beta)=0.
  132. W ( S ) W\left(S\right)
  133. Q i Q_{i}
  134. i W i\in W
  135. { , , ¬ , , } \left\{\land,\lor,\neg,\supset,\bot\right\}
  136. Q 1 Q 2 Q 1 ¬ Q 2 Q_{1}\bot Q_{2}\equiv Q_{1}\supset\neg Q_{2}
  137. Q c ( i ) Q_{c}^{(i)}
  138. N N
  139. N N
  140. Q c ( i ) Q_{c}^{(i)}
  141. Q c ( i ) Q_{c}^{(i)}
  142. 2 N = k 2^{N}=k
  143. { , } \left\{\land,\lor\right\}
  144. Q i Q_{i}
  145. W ( S ) W\left(S\right)
  146. Q c ( i ) Q_{c}^{(i)}
  147. O 1 O_{1}
  148. S S
  149. O 1 O_{1}
  150. p ( Q | Q c ( j ) ) p\left(Q|Q_{c}^{(j)}\right)
  151. Q Q
  152. Q c ( j ) Q_{c}^{(j)}
  153. Q Q
  154. Q c ( j ) Q_{c}^{(j)}
  155. p = 0.5 p=0.5
  156. Q c ( j ) Q_{c}^{(j)}
  157. p = 1 p=1
  158. O 1 O_{1}
  159. Q b ( i ) Q_{b}^{(i)}
  160. p i j = p ( Q b ( i ) | Q c ( j ) ) p^{ij}=p\left(Q_{b}^{(i)}|Q_{c}^{(j)}\right)
  161. 0 p i j 1 , 0\leq p^{ij}\leq 1,
  162. i p i j = 1 , \sum_{i}p^{ij}=1,
  163. j p i j = 1. \sum_{j}p^{ij}=1.
  164. p i j = | U i j | 2 p^{ij}=\left|U^{ij}\right|^{2}
  165. U i j U^{ij}
  166. b b
  167. c c
  168. U b c U_{bc}
  169. U c d = U c b U b d U_{cd}=U_{cb}U_{bd}
  170. b , c b,c
  171. d d
  172. | Q c ( i ) |Q^{(i)}_{c}\rangle
  173. | Q b ( j ) |Q^{(j)}_{b}\rangle
  174. | Q b ( j ) = i U b c i j | Q c ( i ) . |Q^{(j)}_{b}\rangle=\sum_{i}U^{ij}_{bc}|Q^{(i)}_{c}\rangle.
  175. p i j p^{ij}
  176. p i j = | Q c ( i ) | Q b ( j ) | 2 = | U b c i j | 2 . p^{ij}=|\langle Q^{(i)}_{c}|Q^{(j)}_{b}\rangle|^{2}=|U_{bc}^{ij}|^{2}.
  177. t Q ( t ) t\rightarrow Q(t)
  178. t 2 t_{2}
  179. t 1 t_{1}
  180. W ( S ) W(S)
  181. U ( t 2 - t 1 ) U\left(t_{2}-t_{1}\right)
  182. Q ( t 2 ) = U ( t 2 - t 1 ) Q ( t 1 ) U - 1 ( t 2 - t 1 ) Q(t_{2})=U\left(t_{2}-t_{1}\right)Q(t_{1})U^{-1}\left(t_{2}-t_{1}\right)
  183. U ( t 2 - t 1 ) = exp ( - i ( t 2 - t 1 ) H ) U\left(t_{2}-t_{1}\right)=\exp({-i\left(t_{2}-t_{1}\right)H})
  184. H H

Relative_survival.html

  1. λ = λ * + ν \lambda=\lambda^{*}+\nu\,\!
  2. λ = Overall Death Rate , λ * = Expected death rate , ν = Disease-specific death rate \lambda=\,\text{Overall Death Rate},~{}\lambda^{*}=\,\text{Expected death rate% },~{}\nu=\,\text{Disease-specific death rate}

Relaxation_(approximation).html

  1. z = min { c ( x ) : x X 𝐑 n } z=\min\{c(x):x\in X\subseteq\mathbf{R}^{n}\}
  2. z R = min { c R ( x ) : x X R 𝐑 n } z_{R}=\min\{c_{R}(x):x\in X_{R}\subseteq\mathbf{R}^{n}\}
  3. X R X X_{R}\supseteq X
  4. c R ( x ) c ( x ) c_{R}(x)\leq c(x)
  5. x X x\in X
  6. x * x^{*}
  7. x * X X R x^{*}\in X\subseteq X_{R}
  8. z = c ( x * ) c R ( x * ) z R z=c(x^{*})\geq c_{R}(x^{*})\geq z_{R}
  9. x * X R x^{*}\in X_{R}
  10. z R z_{R}
  11. c R ( x ) = c ( x ) c_{R}(x)=c(x)
  12. x X \forall x\in X

Rellich–Kondrachov_theorem.html

  1. p * := n p n - p . p^{*}:=\frac{np}{n-p}.
  2. W 1 , p ( Ω ) L p * ( Ω ) W^{1,p}(\Omega)\hookrightarrow L^{p^{*}}(\Omega)
  3. W 1 , p ( Ω ) L q ( Ω ) for 1 q < p * . W^{1,p}(\Omega)\subset\subset L^{q}(\Omega)\mbox{ for }~{}1\leq q<p^{*}.
  4. k > k>ℓ
  5. k n / p > n / q k−n/p>ℓ−n/q
  6. W k , p ( M ) W , q ( M ) W^{k,p}(M)\subset W^{\ell,q}(M)
  7. u - u Ω L p ( Ω ) C u L p ( Ω ) \|u-u_{\Omega}\|_{L^{p}(\Omega)}\leq C\|\nabla u\|_{L^{p}(\Omega)}
  8. u Ω := 1 meas ( Ω ) Ω u ( x ) d x u_{\Omega}:=\frac{1}{\mathrm{meas}(\Omega)}\int_{\Omega}u(x)\,\mathrm{d}x

Rendleman–Bartter_model.html

  1. d r t = θ r t d t + σ r t d W t dr_{t}=\theta r_{t}\,dt+\sigma r_{t}\,dW_{t}
  2. θ \theta
  3. σ \sigma

Repeat-accumulate_code.html

  1. N {N}
  2. q {q}
  3. q N {qN}
  4. 1 / ( 1 + D ) {1/(1+D)}
  5. ( z 1 , , z n ) {(z_{1},\ldots,z_{n})}
  6. ( x 1 , , x n ) {(x_{1},\ldots,x_{n})}
  7. x 1 = z 1 {x_{1}=z_{1}}
  8. x i = x i - 1 + z i x_{i}=x_{i-1}+z_{i}
  9. i > 1 i>1
  10. 1 / q 1/q

Representation_rigid_group.html

  1. n n
  2. n n

Residual_(numerical_analysis).html

  1. f ( x ) = b . f(x)=b.\,
  2. b - f ( x 0 ) b-f(x_{0})\,
  3. x 0 - x . x_{0}-x.\,
  4. f a ~{}f_{\rm a}~{}
  5. f ~{}f~{}
  6. T ( f ) ( x ) = g ( x ) T(f)(x)=g(x)
  7. g ( x ) - T ( f a ) ( x ) ~{}g(x)~{}-~{}T(f_{\rm a})(x)
  8. max x 𝒳 | g ( x ) - T ( f a ) ( x ) | \max_{x\in\mathcal{X}}|g(x)-T(f_{\rm a})(x)|
  9. 𝒳 \mathcal{X}
  10. f a ~{}f_{\rm a}~{}
  11. f ~{}f~{}
  12. 𝒳 | g ( x ) - T ( f a ) ( x ) | 2 d x . ~{}\int_{\mathcal{X}}|g(x)-T(f_{\rm a})(x)|^{2}~{}{\rm d}x.
  13. | f a ( x ) - f ( x ) f ( x ) | 1. ~{}\left|\frac{f_{\rm a}(x)-f(x)}{f(x)}\right|\ll 1.~{}

Residual_dipolar_coupling.html

  1. I I
  2. S , S,
  3. H D = γ I γ S 4 π 2 r I S 3 [ 1 - 3 cos 2 θ ] ( 3 I z S z - I S ) H_{\mathrm{D}}={\frac{\hbar\gamma_{I}\gamma_{S}}{4\pi^{2}r^{3}_{IS}}}[1-3\cos^% {2}\theta](3I_{z}S_{z}-\vec{I}\cdot\vec{S})
  4. \hbar
  5. γ I \gamma_{I}
  6. γ S \gamma_{S}
  7. I I
  8. S S
  9. r I S r_{IS}
  10. θ \theta
  11. I \vec{I}
  12. S \vec{S}
  13. H D = D I S ( θ ) [ 2 I z S z - ( I x S x + I y S y ) ] H_{\mathrm{D}}=D_{IS}(\theta)[2I_{z}S_{z}-(I_{x}S_{x}+I_{y}S_{y})]\!
  14. D I S ( θ ) = γ I γ S 4 π 2 r I S 3 [ 1 - 3 cos 2 θ ] . D_{IS}(\theta)=\frac{\hbar\gamma_{I}\gamma_{S}}{4\pi^{2}r^{3}_{IS}}[1-3\cos^{2% }\theta].\!
  15. D I S D_{IS}
  16. D I S D_{IS}
  17. D I S = - μ 0 γ I γ S h ( 2 π r I S ) 3 B A D_{IS}=-\frac{\mu_{0}\gamma_{I}\gamma_{S}h}{(2\pi r_{IS})^{3}}BA\!
  18. θ \theta

Residually_finite_group.html

  1. h ( g ) 1. h(g)\neq 1.\,
  2. Π C \Pi_{C}\,

Respirometry.html

  1. V O 2 = F R ( F i n O 2 - F e x O 2 ) 1 - F e x O 2 VO_{2}=\frac{FR\cdot(F_{in}O_{2}-F_{ex}O_{2})}{1-F_{ex}O_{2}}
  2. V O 2 = F R ( F i n O 2 - F e x O 2 ) 1 - F i n O 2 VO_{2}=\frac{FR\cdot(F_{in}O_{2}-F_{ex}O_{2})}{1-F_{in}O_{2}}

Restricted_sumset.html

  1. S = { a 1 + + a n : a 1 A 1 , , a n A n and P ( a 1 , , a n ) 0 } , S=\{a_{1}+\cdots+a_{n}:\ a_{1}\in A_{1},\ldots,a_{n}\in A_{n}\ \mathrm{and}\ P% (a_{1},\ldots,a_{n})\not=0\},
  2. A 1 , , A n A_{1},\ldots,A_{n}
  3. P ( x 1 , , x n ) P(x_{1},\ldots,x_{n})
  4. P ( x 1 , , x n ) = 1 P(x_{1},\ldots,x_{n})=1
  5. A 1 + + A n A_{1}+\cdots+A_{n}
  6. A 1 = = A n = A A_{1}=\cdots=A_{n}=A
  7. P ( x 1 , , x n ) = 1 i < j n ( x j - x i ) , P(x_{1},\ldots,x_{n})=\prod_{1\leq i<j\leq n}(x_{j}-x_{i}),
  8. A 1 A n A_{1}\dotplus\cdots\dotplus A_{n}
  9. n A n^{\wedge}A
  10. A 1 = = A n = A A_{1}=\cdots=A_{n}=A
  11. a 1 A 1 , , a n A n a_{1}\in A_{1},\ldots,a_{n}\in A_{n}
  12. P ( a 1 , , a n ) 0 P(a_{1},\ldots,a_{n})\not=0
  13. | A + B | min { p , | A | + | B | - 1 } . |A+B|\geq\min\{p,\ |A|+|B|-1\}.\,
  14. | 2 A | min { p , 2 | A | - 3 } |2^{\wedge}A|\geq\min\{p,2|A|-3\}
  15. | n A | min { p ( F ) , n | A | - n 2 + 1 } , |n^{\wedge}A|\geq\min\{p(F),\ n|A|-n^{2}+1\},
  16. f ( x 1 , , x n ) f(x_{1},\ldots,x_{n})
  17. x 1 k 1 x n k n x_{1}^{k_{1}}\cdots x_{n}^{k_{n}}
  18. f ( x 1 , , x n ) f(x_{1},\ldots,x_{n})
  19. k 1 + + k n k_{1}+\cdots+k_{n}
  20. f ( x 1 , , x n ) f(x_{1},\ldots,x_{n})
  21. A 1 , , A n A_{1},\ldots,A_{n}
  22. | A i | > k i |A_{i}|>k_{i}
  23. i = 1 , , n i=1,\ldots,n
  24. a 1 A 1 , , a n A n a_{1}\in A_{1},\ldots,a_{n}\in A_{n}
  25. f ( a 1 , , a n ) 0 f(a_{1},\ldots,a_{n})\not=0

Retiming.html

  1. G := ( V , E ) G:=(V,E)
  2. e := ( u , v ) e:=(u,v)
  3. w ( e ) w(e)
  4. e e
  5. d ( v ) d(v)
  6. v v
  7. r ( v ) r(v)
  8. w r ( e ) := w ( e ) + r ( v ) - r ( u ) w_{r}(e):=w(e)+r(v)-r(u)
  9. T T
  10. T T
  11. W ( u , v ) W(u,v)
  12. u u
  13. v v
  14. D ( u , v ) D(u,v)
  15. u u
  16. v v
  17. W W
  18. D D
  19. w ( e ) , W ( u , v ) , D ( u , v ) w(e),W(u,v),D(u,v)
  20. T T
  21. r ( v ) : V r(v):V\to\mathbb{Z}
  22. r ( u ) - r ( v ) r(u)-r(v)
  23. w ( e ) \leq w(e)
  24. r ( u ) - r ( v ) r(u)-r(v)
  25. W ( u , v ) - 1 \leq W(u,v)-1
  26. D ( u , v ) > c D(u,v)>c
  27. T T
  28. r ( v ) r(v)
  29. w ( e ) , d ( v ) w(e),d(v)
  30. T T
  31. r ( v ) : V r(v):V\to\mathbb{Z}
  32. R ( v ) : V R(v):V\to\mathcal{R}
  33. r ( v ) - R ( V ) r(v)-R(V)
  34. - d ( v ) / T \leq-d(v)/T
  35. R ( v ) - r ( v ) R(v)-r(v)
  36. 1 \leq 1
  37. r ( u ) - r ( v ) r(u)-r(v)
  38. w ( e ) \leq w(e)
  39. R ( u ) - R ( v ) R(u)-R(v)
  40. w ( e ) - d ( v ) / T \leq w(e)-d(v)/T

Reuleaux_tetrahedron.html

  1. ( 3 - 2 2 ) s 1.0249 s . \left(\sqrt{3}-\frac{\sqrt{2}}{2}\right)\cdot s\approx 1.0249s.
  2. s 3 12 ( 3 2 - 49 π + 162 tan - 1 2 ) 0.422 s 3 \frac{s^{3}}{12}(3\sqrt{2}-49\pi+162\tan^{-1}\sqrt{2})\approx 0.422s^{3}

Reversible-jump_Markov_chain_Monte_Carlo.html

  1. n m N m = { 1 , 2 , , I } n_{m}\in N_{m}=\{1,2,\ldots,I\}\,
  2. M = n m = 1 I \R d m M=\bigcup_{n_{m}=1}^{I}\R^{d_{m}}
  3. d m d_{m}
  4. n m n_{m}
  5. ( M , N m ) (M,N_{m})
  6. ( m , n m ) (m,n_{m})
  7. m m^{\prime}
  8. g 1 m m g_{1mm^{\prime}}
  9. m m
  10. u u
  11. u u
  12. U U
  13. q q
  14. \R d m m \R^{d_{mm^{\prime}}}
  15. ( m , n m ) (m^{\prime},n_{m}^{\prime})
  16. ( m , n m ) = ( g 1 m m ( m , u ) , n m ) (m^{\prime},n_{m}^{\prime})=(g_{1mm^{\prime}}(m,u),n_{m}^{\prime})\,
  17. g m m := ( ( m , u ) ( ( m , u ) = ( g 1 m m ( m , u ) , g 2 m m ( m , u ) ) ) ) g_{mm^{\prime}}:=\Bigg((m,u)\mapsto\bigg((m^{\prime},u^{\prime})=\big(g_{1mm^{% \prime}}(m,u),g_{2mm^{\prime}}(m,u)\big)\bigg)\Bigg)\,
  18. supp ( g m m ) \mathrm{supp}(g_{mm^{\prime}})\neq\varnothing\,
  19. g m m - 1 = g m m g^{-1}_{mm^{\prime}}=g_{m^{\prime}m}\,
  20. ( m , u ) (m,u)
  21. ( m , u ) (m^{\prime},u^{\prime})
  22. d m + d m m = d m + d m m d_{m}+d_{mm^{\prime}}=d_{m^{\prime}}+d_{m^{\prime}m}\,
  23. d m m d_{mm^{\prime}}
  24. u u
  25. \R d m \R d m \R^{d_{m}}\subset\R^{d_{m^{\prime}}}
  26. d m + d m m = d m d_{m}+d_{mm^{\prime}}=d_{m^{\prime}}\,
  27. ( m , u ) = g m m ( m ) . (m,u)=g_{m^{\prime}m}(m).\,
  28. a ( m , m ) = min ( 1 , p m m p m f m ( m ) p m m q m m ( m , u ) p m f m ( m ) | det ( g m m ( m , u ) ( m , u ) ) | ) , a(m,m^{\prime})=\min\left(1,\frac{p_{m^{\prime}m}p_{m^{\prime}}f_{m^{\prime}}(% m^{\prime})}{p_{mm^{\prime}}q_{mm^{\prime}}(m,u)p_{m}f_{m}(m)}\left|\det\left(% \frac{\partial g_{mm^{\prime}}(m,u)}{\partial(m,u)}\right)\right|\right),
  29. | | |\cdot|
  30. p m f m p_{m}f_{m}
  31. p m f m = c - 1 p ( y | m , n m ) p ( m | n m ) p ( n m ) , p_{m}f_{m}=c^{-1}p(y|m,n_{m})p(m|n_{m})p(n_{m}),\,
  32. c c

RF_power_amplifier.html

  1. P o u t ( V b r - V k ) 2 8 Z o P_{out}\leq\frac{(V_{br}-V_{k})^{2}}{8Z_{o}}
  2. V b r V_{br}
  3. V k V_{k}
  4. Z o = 50 Ω Z_{o}=50\Omega\,

Rheobase.html

  1. I = b ( 1 + c d ) , I=b(1+{c\over d}\,),
  2. Q = b ( d + c ) Q=b(d+c)
  3. Q = I d , Q=Id,

Rheonomous.html

  1. x 2 + y 2 - L = 0 \sqrt{x^{2}+y^{2}}-L=0\,\!
  2. ( x , y ) (x,\ y)\,\!
  3. L L\,\!
  4. x t = x 0 cos ω t x_{t}=x_{0}\cos\omega t\,\!
  5. x 0 x_{0}\,\!
  6. ω \omega\,\!
  7. t t\,\!
  8. ( x - x 0 cos ω t ) 2 + y 2 - L = 0 \sqrt{(x-x_{0}\cos\omega t)^{2}+y^{2}}-L=0\,\!

Ribbon_Hopf_algebra.html

  1. ( A , m , Δ , u , ε , S , , ν ) (A,m,\Delta,u,\varepsilon,S,\mathcal{R},\nu)
  2. ν \nu
  3. ν 2 = u S ( u ) , S ( ν ) = ν , ε ( ν ) = 1 \nu^{2}=uS(u),\;S(\nu)=\nu,\;\varepsilon(\nu)=1
  4. Δ ( ν ) = ( 21 12 ) - 1 ( ν ν ) \Delta(\nu)=(\mathcal{R}_{21}\mathcal{R}_{12})^{-1}(\nu\otimes\nu)
  5. u = m ( S id ) ( 21 ) u=m(S\otimes\,\text{id})(\mathcal{R}_{21})
  6. u S ( u ) uS(u)
  7. S ( u S ( u ) ) = u S ( u ) , ε ( u S ( u ) ) = 1 , Δ ( u S ( u ) ) = ( 21 12 ) - 2 ( u S ( u ) u S ( u ) ) S(uS(u))=uS(u),\varepsilon(uS(u))=1,\Delta(uS(u))=(\mathcal{R}_{21}\mathcal{R}% _{12})^{-2}(uS(u)\otimes uS(u))
  8. A A
  9. m m
  10. m : A A A m:A\otimes A\rightarrow A
  11. Δ \Delta
  12. Δ : A A A \Delta:A\rightarrow A\otimes A
  13. u u
  14. u : A u:\mathbb{C}\rightarrow A
  15. ε \varepsilon
  16. ε : A \varepsilon:A\rightarrow\mathbb{C}
  17. S S
  18. S : A A S:A\rightarrow A
  19. \mathcal{R}
  20. K K
  21. \mathbb{C}

Ribonucleoside-triphosphate_reductase.html

  1. \rightleftharpoons

Richards_equation.html

  1. θ t = z [ K ( θ ) ( ψ z + 1 ) ] \frac{\partial\theta}{\partial t}=\frac{\partial}{\partial z}\left[K(\theta)% \left(\frac{\partial\psi}{\partial z}+1\right)\right]
  2. K K
  3. ψ \psi
  4. z z
  5. θ \theta
  6. t t
  7. θ t = ( i = 1 n q i , in - j = 1 m q j , out ) \frac{\partial\theta}{\partial t}=\vec{\nabla}\cdot\left(\sum_{i=1}^{n}{\vec{q% }_{i,\,\,\text{in}}}-\sum_{j=1}^{m}{\vec{q}_{j,\,\,\text{out}}}\right)
  8. k ^ \hat{k}
  9. θ t = - z q \frac{\partial\theta}{\partial t}=-\frac{\partial}{\partial z}q
  10. q = - K h z q=-K\frac{\partial h}{\partial z}
  11. θ t = z [ K h z ] \frac{\partial\theta}{\partial t}=\frac{\partial}{\partial z}\left[K\frac{% \partial h}{\partial z}\right]
  12. θ t = z [ K ( ψ z + z z ) ] = z [ K ( ψ z + 1 ) ] \frac{\partial\theta}{\partial t}=\frac{\partial}{\partial z}\left[K\left(% \frac{\partial\psi}{\partial z}+\frac{\partial z}{\partial z}\right)\right]=% \frac{\partial}{\partial z}\left[K\left(\frac{\partial\psi}{\partial z}+1% \right)\right]
  13. C ( h ) h t = K ( h ) h C(h)\frac{\partial h}{\partial t}=\nabla\cdot K(h)\nabla h
  14. C ( h ) θ h C(h)\equiv\frac{\partial\theta}{\partial h}
  15. θ t = D ( θ ) θ \frac{\partial\theta}{\partial t}=\nabla\cdot D(\theta)\nabla\theta
  16. D ( θ ) K ( θ ) C ( θ ) K ( θ ) h θ D(\theta)\equiv\frac{K(\theta)}{C(\theta)}\equiv K(\theta)\frac{\partial h}{% \partial\theta}

Ricker_model.html

  1. a t + 1 = a t e r ( 1 - a t k ) . a_{t+1}=a_{t}e^{r\left(1-\frac{a_{t}}{k}\right)}.\,
  2. a t + 1 = k 1 a t ( 1 + k 2 a t ) c . a_{t+1}=k_{1}\frac{a_{t}}{\left(1+k_{2}a_{t}\right)^{c}}.

Ridge_detection.html

  1. N N
  2. N - 1 N-1
  3. f ( x , y ) f(x,y)
  4. L L
  5. f ( x , y ) f(x,y)
  6. f ( x , y ) f(x,y)
  7. g ( x , y , t ) = 1 2 π t e - ( x 2 + y 2 ) / 2 t g(x,y,t)=\frac{1}{2\pi t}e^{-(x^{2}+y^{2})/2t}
  8. L p p L_{pp}
  9. L q q L_{qq}
  10. H = [ L x x L x y L x y L y y ] H=\begin{bmatrix}L_{xx}&L_{xy}\\ L_{xy}&L_{yy}\end{bmatrix}
  11. L L
  12. p = sin β x - cos β y , q = cos β x + sin β y \partial_{p}=\sin\beta\partial_{x}-\cos\beta\partial_{y},\partial_{q}=\cos% \beta\partial_{x}+\sin\beta\partial_{y}
  13. L p q L_{pq}
  14. cos β = 1 2 ( 1 + L x x - L y y ( L x x - L y y ) 2 + 4 L x y 2 ) \cos\beta=\sqrt{\frac{1}{2}\left(1+\frac{L_{xx}-L_{yy}}{\sqrt{(L_{xx}-L_{yy})^% {2}+4L_{xy}^{2}}}\right)}
  15. sin β = sgn ( L x y ) 1 2 ( 1 - L x x - L y y ( L x x - L y y ) 2 + 4 L x y 2 ) \sin\beta=\operatorname{sgn}(L_{xy})\sqrt{\frac{1}{2}\left(1-\frac{L_{xx}-L_{% yy}}{\sqrt{(L_{xx}-L_{yy})^{2}+4L_{xy}^{2}}}\right)}
  16. f ( x , y ) f(x,y)
  17. t t
  18. L p = 0 , L p p 0 , | L p p | | L q q | . L_{p}=0,L_{pp}\leq 0,|L_{pp}|\geq|L_{qq}|.
  19. f ( x , y ) f(x,y)
  20. t t
  21. L q = 0 , L q q 0 , | L q q | | L p p | . L_{q}=0,L_{qq}\geq 0,|L_{qq}|\geq|L_{pp}|.
  22. ( u , v ) (u,v)
  23. v v
  24. u = sin α x - cos α y , v = cos α x + sin α y \partial_{u}=\sin\alpha\partial_{x}-\cos\alpha\partial_{y},\partial_{v}=\cos% \alpha\partial_{x}+\sin\alpha\partial_{y}
  25. cos α = L x L x 2 + L y 2 , sin α = L y L x 2 + L y 2 \cos\alpha=\frac{L_{x}}{\sqrt{L_{x}^{2}+L_{y}^{2}}},\sin\alpha=\frac{L_{y}}{% \sqrt{L_{x}^{2}+L_{y}^{2}}}
  26. L u v = 0 , L u u 2 - L v v 2 0 L_{uv}=0,L_{uu}^{2}-L_{vv}^{2}\geq 0
  27. L v 2 L u u = L x 2 L y y - 2 L x L y L x y + L y 2 L x x , L_{v}^{2}L_{uu}=L_{x}^{2}L_{yy}-2L_{x}L_{y}L_{xy}+L_{y}^{2}L_{xx},
  28. L v 2 L u v = L x L y ( L x x - L y y ) - ( L x 2 - L y 2 ) L x y , L_{v}^{2}L_{uv}=L_{x}L_{y}(L_{xx}-L_{yy})-(L_{x}^{2}-L_{y}^{2})L_{xy},
  29. L v 2 L v v = L x 2 L x x + 2 L x L y L x y + L y 2 L y y L_{v}^{2}L_{vv}=L_{x}^{2}L_{xx}+2L_{x}L_{y}L_{xy}+L_{y}^{2}L_{yy}
  30. L u u L_{uu}
  31. L u u < 0 L_{uu}<0
  32. L u u > 0 L_{uu}>0
  33. R ( x , y , t ) R(x,y,t)
  34. L p = 0 , L p p 0 , t ( R ) = 0 , t t ( R ) 0 , L_{p}=0,L_{pp}\leq 0,\partial_{t}(R)=0,\partial_{tt}(R)\leq 0,
  35. t t
  36. L q = 0 , L q q 0 , t ( R ) = 0 , t t ( R ) 0. L_{q}=0,L_{qq}\geq 0,\partial_{t}(R)=0,\partial_{tt}(R)\leq 0.
  37. L p p , γ - n o r m = t γ 2 ( L x x + L y y - ( L x x - L y y ) 2 + 4 L x y 2 ) L_{pp,\gamma-norm}=\frac{t^{\gamma}}{2}\left(L_{xx}+L_{yy}-\sqrt{(L_{xx}-L_{yy% })^{2}+4L_{xy}^{2}}\right)
  38. γ \gamma
  39. ξ = t γ / 2 x , η = t γ / 2 y \partial_{\xi}=t^{\gamma/2}\partial_{x},\partial_{\eta}=t^{\gamma/2}\partial_{y}
  40. γ \gamma
  41. N γ - n o r m = ( L p p , γ - n o r m 2 - L q q , γ - n o r m 2 ) 2 = t 4 γ ( L x x + L y y ) 2 ( ( L x x - L y y ) 2 + 4 L x y 2 ) . N_{\gamma-norm}=\left(L_{pp,\gamma-norm}^{2}-L_{qq,\gamma-norm}^{2}\right)^{2}% =t^{4\gamma}(L_{xx}+L_{yy})^{2}\left((L_{xx}-L_{yy})^{2}+4L_{xy}^{2}\right).
  42. γ \gamma
  43. A γ - n o r m = ( L p p , γ - n o r m - L q q , γ - n o r m ) 2 = t 2 γ ( ( L x x - L y y ) 2 + 4 L x y 2 ) . A_{\gamma-norm}=\left(L_{pp,\gamma-norm}-L_{qq,\gamma-norm}\right)^{2}=t^{2% \gamma}\left((L_{xx}-L_{yy})^{2}+4L_{xy}^{2}\right).
  44. γ \gamma
  45. γ = 3 / 4 \gamma=3/4
  46. L p p , γ - n o r m L_{pp,\gamma-norm}
  47. A γ - n o r m A_{\gamma-norm}
  48. γ = 1 \gamma=1
  49. γ = 1 \gamma=1
  50. γ = 3 / 4 \gamma=3/4
  51. γ \gamma
  52. 𝐱 0 \mathbf{x}_{0}
  53. f : n f:\mathbb{R}^{n}\rightarrow\mathbb{R}
  54. δ > 0 \delta>0
  55. 𝐱 \mathbf{x}
  56. δ \delta
  57. 𝐱 0 \mathbf{x}_{0}
  58. f ( 𝐱 ) < f ( 𝐱 0 ) f(\mathbf{x})<f(\mathbf{x}_{0})
  59. f ( 𝐱 ) < f ( 𝐱 0 ) f(\mathbf{x})<f(\mathbf{x}_{0})
  60. 𝐱 \mathbf{x}
  61. 𝐱 0 \mathbf{x}_{0}
  62. n - 1 n-1
  63. U n U\subset\mathbb{R}^{n}
  64. f : U f:U\rightarrow\mathbb{R}
  65. 𝐱 0 U \mathbf{x}_{0}\in U
  66. 𝐱 0 f \nabla_{\mathbf{x}_{0}}f
  67. f f
  68. 𝐱 0 \mathbf{x}_{0}
  69. H 𝐱 0 ( f ) H_{\mathbf{x}_{0}}(f)
  70. n × n n\times n
  71. f f
  72. 𝐱 0 \mathbf{x}_{0}
  73. λ 1 λ 2 λ n \lambda_{1}\leq\lambda_{2}\leq\cdots\leq\lambda_{n}
  74. n n
  75. H 𝐱 0 ( f ) H_{\mathbf{x}_{0}}(f)
  76. 𝐞 i \mathbf{e}_{i}
  77. λ i \lambda_{i}
  78. 𝐱 0 \mathbf{x}_{0}
  79. f f
  80. λ n - 1 < 0 \lambda_{n-1}<0
  81. 𝐱 0 f 𝐞 i = 0 \nabla_{\mathbf{x}_{0}}f\cdot\mathbf{e}_{i}=0
  82. i = 1 , 2 , , n - 1 i=1,2,\ldots,n-1
  83. f f
  84. n - 1 n-1
  85. 𝐱 0 \mathbf{x}_{0}
  86. 𝐱 0 \mathbf{x}_{0}
  87. f f
  88. λ n - k < 0 \lambda_{n-k}<0
  89. 𝐱 0 f 𝐞 i = 0 \nabla_{\mathbf{x}_{0}}f\cdot\mathbf{e}_{i}=0
  90. i = 1 , 2 , , n - k i=1,2,\ldots,n-k
  91. ( 𝐱 , σ ) (\mathbf{x},\sigma)
  92. f ( 𝐱 , σ ) f(\mathbf{x},\sigma)
  93. U 2 × + U\subset\mathbb{R}^{2}\times\mathbb{R}_{+}
  94. ( 𝐱 , σ ) (\mathbf{x},\sigma)
  95. f σ = 0 \frac{\partial f}{\partial\sigma}=0
  96. 2 f σ 2 < 0 \frac{\partial^{2}f}{\partial\sigma^{2}}<0
  97. f 𝐞 1 = 0 \nabla f\cdot\mathbf{e}_{1}=0
  98. 𝐞 1 t H ( f ) 𝐞 1 < 0 \mathbf{e}_{1}^{t}H(f)\mathbf{e}_{1}<0
  99. ( u , v ) (u,v)
  100. v v
  101. L v L_{v}
  102. v v
  103. v ( L v ) = 0 \partial_{v}(L_{v})=0
  104. v v
  105. L v L_{v}
  106. v v ( L v ) 0 \partial_{vv}(L_{v})\leq 0
  107. L x L_{x}
  108. L y L_{y}
  109. L y y y L_{yyy}
  110. L v 2 L v v = L x 2 L x x + 2 L x L y L x y + L y 2 L y y = 0 , L_{v}^{2}L_{vv}=L_{x}^{2}\,L_{xx}+2\,L_{x}\,L_{y}\,L_{xy}+L_{y}^{2}\,L_{yy}=0,
  111. L v 3 L v v v = L x 3 L x x x + 3 L x 2 L y L x x y + 3 L x L y 2 L x y y + L y 3 L y y y 0 L_{v}^{3}L_{vvv}=L_{x}^{3}\,L_{xxx}+3\,L_{x}^{2}\,L_{y}\,L_{xxy}+3\,L_{x}\,L_{% y}^{2}\,L_{xyy}+L_{y}^{3}\,L_{yyy}\leq 0
  112. n \mathbb{R}^{n}

Riemann_Xi_function.html

  1. ξ ( s ) \xi(s)
  2. s s
  3. ξ ( s ) = 1 2 s ( s - 1 ) π - s / 2 Γ ( 1 2 s ) ζ ( s ) \xi(s)=\tfrac{1}{2}s(s-1)\pi^{-s/2}\Gamma\left(\tfrac{1}{2}s\right)\zeta(s)
  4. s s\in\mathbb{C}
  5. ξ ( 1 - s ) = ξ ( s ) . \xi(1-s)=\xi(s).
  6. Ξ ( z ) = ξ ( 1 2 + z i ) \Xi(z)=\xi(\frac{1}{2}+zi)
  7. Ξ ( - z ) = Ξ ( z ) . \Xi(-z)=\Xi(z).
  8. ξ ( 2 n ) = ( - 1 ) n + 1 1 ( 2 n ) ! B 2 n 2 2 n - 1 π n ( 2 n 2 - n ) ( n - 1 ) ! \xi(2n)=(-1)^{n+1}\frac{1}{(2n)!}B_{2n}2^{2n-1}\pi^{n}(2n^{2}-n)(n-1)!
  9. ξ ( 2 ) = π 6 \xi(2)={\pi\over 6}
  10. ξ \xi
  11. d d z ln ξ ( - z 1 - z ) = n = 0 λ n + 1 z n , \frac{d}{dz}\ln\xi\left(\frac{-z}{1-z}\right)=\sum_{n=0}^{\infty}\lambda_{n+1}% z^{n},
  12. λ n = 1 ( n - 1 ) ! d n d s n [ s n - 1 log ξ ( s ) ] | s = 1 = ρ [ 1 - ( 1 - 1 ρ ) n ] , \lambda_{n}=\frac{1}{(n-1)!}\left.\frac{d^{n}}{ds^{n}}\left[s^{n-1}\log\xi(s)% \right]\right|_{s=1}=\sum_{\rho}\left[1-\left(1-\frac{1}{\rho}\right)^{n}% \right],
  13. | ( ρ ) | |\Im(\rho)|
  14. Ξ ( s ) = Ξ ( 0 ) ρ ( 1 - s ρ ) , \Xi(s)=\Xi(0)\prod_{\rho}\left(1-\frac{s}{\rho}\right),\!

Riemann–von_Mangoldt_formula.html

  1. N ( T ) = T 2 π log T 2 π - T 2 π + O ( log T ) . N(T)=\frac{T}{2\pi}\log{\frac{T}{2\pi}}-\frac{T}{2\pi}+O(\log{T}).
  2. | N ( T ) - ( T 2 π log T 2 π - T 2 π - 7 8 ) | < 0.137 log T + 0.443 log log T + 4.350 . \left|{N(T)-\left({\frac{T}{2\pi}\log{\frac{T}{2\pi}}-\frac{T}{2\pi}}-\frac{7}% {8}\right)}\right|<0.137\log T+0.443\log\log T+4.350\ .

Riesz's_lemma.html

  1. d ( x , Y ) = inf y Y | x - y | . d(x,Y)=\inf_{y\in Y}|x-y|.
  2. d ( x n , Y n - 1 ) > α d(x_{n},Y_{n-1})>\alpha
  3. C = i = 1 n ( c i + 1 2 C ) . C=\bigcup_{i=1}^{n}\;\left(c_{i}+\frac{1}{2}C\right).
  4. C \sub Y + 1 2 m C C\sub Y+\frac{1}{2^{m}}C
  5. | x n - x m | > α |x_{n}-x_{m}|>\alpha

Riesz_mean.html

  1. { s n } \{s_{n}\}
  2. s δ ( λ ) = n λ ( 1 - n λ ) δ s n s^{\delta}(\lambda)=\sum_{n\leq\lambda}\left(1-\frac{n}{\lambda}\right)^{% \delta}s_{n}
  3. R n = 1 λ n k = 0 n ( λ k - λ k - 1 ) δ s k R_{n}=\frac{1}{\lambda_{n}}\sum_{k=0}^{n}(\lambda_{k}-\lambda_{k-1})^{\delta}s% _{k}
  4. λ n \lambda_{n}
  5. λ n \lambda_{n}\to\infty
  6. λ n + 1 / λ n 1 \lambda_{n+1}/\lambda_{n}\to 1
  7. n n\to\infty
  8. λ n \lambda_{n}
  9. s n = k = 0 n a n s_{n}=\sum_{k=0}^{n}a_{n}
  10. { a n } \{a_{n}\}
  11. lim n R n \lim_{n\to\infty}R_{n}
  12. lim δ 1 , λ s δ ( λ ) \lim_{\delta\to 1,\lambda\to\infty}s^{\delta}(\lambda)
  13. a n = 1 a_{n}=1
  14. n n
  15. n λ ( 1 - n λ ) δ = 1 2 π i c - i c + i Γ ( 1 + δ ) Γ ( s ) Γ ( 1 + δ + s ) ζ ( s ) λ s d s = λ 1 + δ + n b n λ - n . \sum_{n\leq\lambda}\left(1-\frac{n}{\lambda}\right)^{\delta}=\frac{1}{2\pi i}% \int_{c-i\infty}^{c+i\infty}\frac{\Gamma(1+\delta)\Gamma(s)}{\Gamma(1+\delta+s% )}\zeta(s)\lambda^{s}\,ds=\frac{\lambda}{1+\delta}+\sum_{n}b_{n}\lambda^{-n}.
  16. c > 1 c>1
  17. Γ ( s ) \Gamma(s)
  18. ζ ( s ) \zeta(s)
  19. n b n λ - n \sum_{n}b_{n}\lambda^{-n}
  20. λ > 1 \lambda>1
  21. a n = Λ ( n ) a_{n}=\Lambda(n)
  22. Λ ( n ) \Lambda(n)
  23. n λ ( 1 - n λ ) δ Λ ( n ) = - 1 2 π i c - i c + i Γ ( 1 + δ ) Γ ( s ) Γ ( 1 + δ + s ) ζ ( s ) ζ ( s ) λ s d s = λ 1 + δ + ρ Γ ( 1 + δ ) Γ ( ρ ) Γ ( 1 + δ + ρ ) + n c n λ - n . \sum_{n\leq\lambda}\left(1-\frac{n}{\lambda}\right)^{\delta}\Lambda(n)=-\frac{% 1}{2\pi i}\int_{c-i\infty}^{c+i\infty}\frac{\Gamma(1+\delta)\Gamma(s)}{\Gamma(% 1+\delta+s)}\frac{\zeta^{\prime}(s)}{\zeta(s)}\lambda^{s}\,ds=\frac{\lambda}{1% +\delta}+\sum_{\rho}\frac{\Gamma(1+\delta)\Gamma(\rho)}{\Gamma(1+\delta+\rho)}% +\sum_{n}c_{n}\lambda^{-n}.
  24. n c n λ - n \sum_{n}c_{n}\lambda^{-n}\,

Riesz_potential.html

  1. ( I α f ) ( x ) = 1 c α n f ( y ) | x - y | n - α d y (I_{\alpha}f)(x)=\frac{1}{c_{\alpha}}\int_{{\mathbb{R}}^{n}}\frac{f(y)}{|x-y|^% {n-\alpha}}\,\mathrm{d}y
  2. c α = π n / 2 2 α Γ ( α / 2 ) Γ ( ( n - α ) / 2 ) . c_{\alpha}=\pi^{n/2}2^{\alpha}\frac{\Gamma(\alpha/2)}{\Gamma((n-\alpha)/2)}.
  3. I α f p * C p f p , p * = n p n - α p . \|I_{\alpha}f\|_{p^{*}}\leq C_{p}\|f\|_{p},\quad p^{*}=\frac{np}{n-\alpha p}.
  4. K α ( x ) = 1 c α 1 | x | n - α . K_{\alpha}(x)=\frac{1}{c_{\alpha}}\frac{1}{|x|^{n-\alpha}}.
  5. K α ^ ( ξ ) = | 2 π ξ | - α \widehat{K_{\alpha}}(\xi)=|2\pi\xi|^{-\alpha}
  6. I α f ^ ( ξ ) = | 2 π ξ | - α f ^ ( ξ ) . \widehat{I_{\alpha}f}(\xi)=|2\pi\xi|^{-\alpha}\hat{f}(\xi).
  7. I α I β = I α + β I_{\alpha}I_{\beta}=I_{\alpha+\beta}
  8. 0 < Re α , Re β < n , 0 < Re ( α + β ) < n . 0<\operatorname{Re\,}\alpha,\operatorname{Re\,}\beta<n,\quad 0<\operatorname{% Re\,}(\alpha+\beta)<n.
  9. lim α 0 + ( I α f ) ( x ) = f ( x ) . \lim_{\alpha\to 0^{+}}(I^{\alpha}f)(x)=f(x).

Rigid_analytic_space.html

  1. ( X , 𝒪 X ) (X,\mathcal{O}_{X})

Ring_laser.html

  1. Ω A λ L \frac{\vec{\Omega}\cdot\vec{A}}{\lambda L}
  2. A \vec{A}
  3. Ω \vec{\Omega}
  4. A = 1 2 r × d l \vec{A}=\frac{1}{2}\oint{\vec{r}\times}d\vec{l}
  5. d l \oint{dl}
  6. S δ f = h f 3 P Q 2 S_{\delta f}=\frac{hf^{3}}{PQ^{2}}
  7. S δ f S_{\delta f}
  8. σ = h f 0 3 2 P Q 2 T \sigma=\sqrt{\frac{hf_{0}^{3}}{2PQ^{2}T}}
  9. S 1 / f = A Q 4 ( f 0 2 / f ) S_{1/f}=\frac{A}{Q^{4}}(f_{0}^{2}/f)
  10. | E | = | E 0 | e - r 2 w 2 \left|E\right|=\left|E_{0}\right|e^{-\frac{r^{2}}{w^{2}}}
  11. E 0 E_{0}
  12. 1 q = 1 R - j λ π w 2 \frac{1}{q}=\frac{1}{R}-j\frac{\lambda}{\pi w^{2}}
  13. ( 1 d 0 1 ) \left(\begin{matrix}1&d\\ 0&1\\ \end{matrix}\right)
  14. ( 1 0 - 1 f 1 ) \left(\begin{matrix}1&0\\ -\frac{1}{f}&1\\ \end{matrix}\right)
  15. f = f x = R M 2 cos θ f=f_{x}=\frac{R_{M}}{2}\cdot\cos\theta
  16. f = f y = R M 2 1 cos θ f=f_{y}=\frac{R_{M}}{2}\cdot\frac{1}{\cos\theta}
  17. | M 1 | = | M 2 | = 1 \left|M_{1}\right|=\left|M_{2}\right|=1
  18. ( r r ) 4 = ( r r ) 1 = ( M 1 M 2 ) 4 ( M 1 M 2 ) 3 ( M 1 M 2 ) 2 ( M 1 M 2 ) 1 ( r r ) 1 \left(\begin{matrix}r\\ r^{{}^{\prime}}\\ \end{matrix}\right)_{4}=\left(\begin{matrix}r\\ r^{{}^{\prime}}\\ \end{matrix}\right)_{1}=\left(M_{1}\cdot M_{2}\right)_{4}\cdot\left(M_{1}\cdot M% _{2}\right)_{3}\cdot\left(M_{1}\cdot M_{2}\right)_{2}\cdot\left(M_{1}\cdot M_{% 2}\right)_{1}\cdot\left(\begin{matrix}r\\ r^{{}^{\prime}}\\ \end{matrix}\right)_{1}
  19. ( A B C D ) ( r r ) 1 \left(\begin{matrix}A&B\\ C&D\\ \end{matrix}\right)\cdot\left(\begin{matrix}r\\ r^{{}^{\prime}}\\ \end{matrix}\right)_{1}
  20. | A B C D | = 1 \left|\begin{matrix}A&B\\ C&D\\ \end{matrix}\right|=1
  21. ( A s B s C s D s ) \left(\begin{matrix}A_{s}&B_{s}\\ C_{s}&D_{s}\\ \end{matrix}\right)
  22. q o u t = A s q i n + B s C s q i n + D s q_{out}=\frac{A_{s}q_{in}+B_{s}}{C_{s}q_{in}+D_{s}}
  23. q = A q + B C q + D q=\frac{Aq+B}{Cq+D}
  24. 1 q = 1 R - j λ π w 2 = D - A 2 B - j 1 - ( A + D 2 ) 2 B \frac{1}{q}=\frac{1}{R}-j\frac{\lambda}{\pi w^{2}}=\frac{D-A}{2B}-j\frac{\sqrt% {1-(\frac{A+D}{2})^{2}}}{B}
  25. | A + D 2 | 1 \left|\frac{A+D}{2}\right|\leq 1
  26. L \sqrt{L}
  27. ( E p E s ) \left(\begin{matrix}E_{p}\\ E_{s}\\ \end{matrix}\right)
  28. M r e f l = ( r p e j χ p 0 0 - r s e j χ s ) M_{refl}=\left(\begin{matrix}r_{p}e^{j\chi_{p}}&0\\ 0&-r_{s}e^{j\chi_{s}}\\ \end{matrix}\right)
  29. M r o t = ( cos θ sin θ - sin θ cos θ ) M_{rot}=\left(\begin{matrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\\ \end{matrix}\right)
  30. ( E p E s ) = ( M r e f l 4 ) ( M r o t 4 ) ( M r e f l 1 ) ( M r o t 1 ) ( E p E s ) \left(\begin{matrix}E_{p}\\ E_{s}\\ \end{matrix}\right)=\left(M_{refl_{4}}\right)\left(M_{rot_{4}}\right).........% ...\left(M_{refl_{1}}\right)\left(M_{rot_{1}}\right)\left(\begin{matrix}E_{p}% \\ E_{s}\\ \end{matrix}\right)
  31. δ = δ p - δ s = ( 1 - r p ) - ( 1 - r s ) \delta=\delta_{p}-\delta_{s}=(1-r_{p})-(1-r_{s})
  32. χ = χ p - χ s \chi=\chi_{p}-\chi_{s}
  33. E p / E s E_{p}/E_{s}
  34. E p E s = ± j 1 - ( γ / θ ) 2 + γ / θ \frac{E_{p}}{E_{s}}=\pm j\sqrt{1-(\gamma/\theta)^{2}}+\gamma/\theta
  35. γ = 1 2 ( δ - j χ ) \gamma=\frac{1}{\sqrt{2}}(\delta-j\chi)
  36. γ / θ 1 \gamma/\theta<<1
  37. E p / E s = ± j E_{p}/E_{s}=\pm j
  38. γ / θ 1 \gamma/\theta>>1
  39. E p E_{p}
  40. f = f t 1 - ( f L f t ) 2 f=f_{t}\sqrt{1-(\frac{f_{L}}{f_{t}})^{2}}
  41. f t - f f t 1 2 ( f L f t ) 2 1 L 4 \frac{f_{t}-f}{f_{t}}\cong\frac{1}{2}(\frac{f_{L}}{f_{t}})^{2}\propto\frac{1}{% L^{4}}
  42. Q = 2 π f 0 W - d W d t Q=2\pi f_{0}\frac{W}{-\frac{dW}{dt}}
  43. f 0 f_{0}
  44. Q = π L 2 λ ( 1 - r ) Q=\frac{\pi L}{2\lambda(1-r)}
  45. W = W 0 e - ω t Q W 0 e - t τ W=W_{0}e^{-\frac{\omega t}{Q}}\equiv W_{0}e^{-\frac{t}{\tau}}
  46. r 2 cos ( π / n ) n \frac{r}{2}\frac{\cos(\pi/n)}{n}
  47. ( 1 j / ( n r - j k r ) ( n r - j k r ) 1 ) \left(\begin{matrix}1&j/(n_{r}-jk_{r})\\ (n_{r}-jk_{r})&1\\ \end{matrix}\right)
  48. S / N L 3 [ 1 - e - ( L c r i t L ) 2 ] 1 2 \,\text{ }S/N\propto L^{3}[1-e^{-(\frac{L_{crit}}{L})^{2}}]^{\frac{1}{2}}

Ring_of_periods.html

  1. n \mathbb{R}^{n}
  2. a b f ( x ) d x = f ( b ) - f ( a ) \int_{a}^{b}f^{\prime}(x)\,dx=f(b)-f(a)

Ring_signature.html

  1. O ( n ) O(n)
  2. n n
  3. O ( n ) O(n)

Road_speed_limit_enforcement_in_Australia.html

  1. s p e e d = d i s t a n c e t i m e speed={distance\over time}

Robbins_algebra.html

  1. \lor
  2. ¬ \neg
  3. a ( b c ) = ( a b ) c a\lor\left(b\lor c\right)=\left(a\lor b\right)\lor c
  4. a b = b a a\lor b=b\lor a
  5. ¬ ( ¬ ( a b ) ¬ ( a ¬ b ) ) = a \neg\left(\neg\left(a\lor b\right)\lor\neg\left(a\lor\neg b\right)\right)=a
  6. ¬ ( ¬ a b ) ¬ ( ¬ a ¬ b ) = a . \neg(\neg a\lor b)\lor\neg(\neg a\lor\neg b)=a.
  7. \lor
  8. ¬ \neg
  9. ¬ ( ¬ ( a b ) ¬ ( a ¬ b ) ) = a \neg\left(\neg\left(a\lor b\right)\lor\neg\left(a\lor\neg b\right)\right)=a
  10. ¬ ( ¬ ( a ¬ b ) ¬ ( a b ) ) = a \neg\left(\neg\left(a\lor\neg b\right)\lor\neg\left(a\lor b\right)\right)=a
  11. ¬ ( ¬ ( b ¬ a ) ¬ ( b a ) ) = b \neg\left(\neg\left(b\lor\neg a\right)\lor\neg\left(b\lor a\right)\right)=b
  12. ( b ¬ a ) \left(b\lor\neg a\right)
  13. ¬ ( ¬ ( ¬ a b ) ¬ ( b a ) ) = b \neg\left(\neg\left(\neg a\lor b\right)\lor\neg\left(b\lor a\right)\right)=b
  14. ¬ \neg
  15. ¬ ( ¬ a b ) ¬ ( b a ) = ¬ b \neg\left(\neg a\lor b\right)\lor\neg\left(b\lor a\right)=\neg b
  16. ¬ a \neg a
  17. ¬ b \neg b
  18. ¬ ( a ¬ b ) ¬ ( ¬ b ¬ a ) = b \neg\left(a\lor\neg b\right)\lor\neg\left(\neg b\lor\neg a\right)=b
  19. ¬ ( b ¬ a ) ¬ ( ¬ a ¬ b ) = a \neg\left(b\lor\neg a\right)\lor\neg\left(\neg a\lor\neg b\right)=a
  20. ( b ¬ a ) \left(b\lor\neg a\right)
  21. ¬ ( ¬ a b ) ¬ ( ¬ a ¬ b ) = a \neg\left(\neg a\lor b\right)\lor\neg\left(\neg a\lor\neg b\right)=a

Robocopy.html

  1. D = B A - B D B A × B D × 512 × 1000 D={B_{A}-B_{D}\over B_{A}\times B_{D}}\times 512\times 1000

Robust_optimization.html

  1. max x , y { 3 x + 2 y } subject to x , y 0 ; c x + d y 10 , ( c , d ) P \max_{x,y}\ \{3x+2y\}\ \ \mathrm{subject\ to}\ \ x,y\geq 0;cx+dy\leq 10,% \forall(c,d)\in P
  2. P P
  3. 2 \mathbb{R}^{2}
  4. ( c , d ) P \forall(c,d)\in P
  5. ( x , y ) (x,y)
  6. c x + d y 10 cx+dy\leq 10
  7. ( c , d ) P (c,d)\in P
  8. ( x , y ) (x,y)
  9. ( c , d ) P (c,d)\in P
  10. c x + d y cx+dy
  11. ( x , y ) (x,y)
  12. P P
  13. ( c , d ) P (c,d)\in P
  14. c x + d y 10 cx+dy\leq 10
  15. P P
  16. ρ ^ ( x , u ^ ) := max ρ 0 { ρ : u S ( x ) , u B ( ρ , u ^ ) } \hat{\rho}(x,\hat{u}):=\max_{\rho\geq 0}\ \{\rho:u\in S(x),\forall u\in B(\rho% ,\hat{u})\}
  17. u ^ \hat{u}
  18. B ( ρ , u ^ ) B(\rho,\hat{u})
  19. ρ \rho
  20. u ^ \hat{u}
  21. S ( x ) S(x)
  22. u u
  23. x x
  24. x x
  25. u ^ \hat{u}
  26. x x
  27. U ( x ) U(x)
  28. u u
  29. x x
  30. max x X { f ( x ) : g ( x , u ) b , u U } \max_{x\in X}\ \{f(x):g(x,u)\leq b,\forall u\in U\}
  31. U U
  32. u u
  33. g ( x , u ) b , u U g(x,u)\leq b,\forall u\in U
  34. u u
  35. x X x\in X
  36. x X x\in X
  37. x X x\in X
  38. f ( x ) f(x)
  39. X X
  40. x X x^{\prime}\in X
  41. f ( x ) f(x^{\prime})
  42. g ( x , u ) b , g(x,u)\leq b,
  43. x X x\in X
  44. u u
  45. U U
  46. x X x\in X
  47. g ( x , u ) b , u U g(x,u)\leq b,\forall u\in U
  48. ρ ( x ) := max Y U { s i z e ( Y ) : g ( x , u ) b , u Y } , x X \rho(x):=\max_{Y\subseteq U}\ \{size(Y):g(x,u)\leq b,\forall u\in Y\}\ ,\ x\in X
  49. s i z e ( Y ) size(Y)
  50. Y Y
  51. U U
  52. s i z e ( Y ) size(Y)
  53. Y Y
  54. U U
  55. g ( x , u ) b g(x,u)\leq b
  56. u u
  57. max x X , Y U { s i z e ( Y ) : g ( x , u ) b , u Y } \max_{x\in X,Y\subseteq U}\ \{size(Y):g(x,u)\leq b,\forall u\in Y\}
  58. z ( U ) := max x X { f ( x ) : g ( x , u ) b , u U } z(U):=\max_{x\in X}\ \{f(x):g(x,u)\leq b,\forall u\in U\}
  59. g g
  60. X × U X\times U
  61. g ( x , u ) b , u U g(x,u)\leq b,\forall u\in U
  62. 𝒩 \mathcal{N}
  63. U U
  64. u u
  65. z ( 𝒩 ) := max x X { f ( x ) : g ( x , u ) b , u 𝒩 } z(\mathcal{N}):=\max_{x\in X}\ \{f(x):g(x,u)\leq b,\forall u\in\mathcal{N}\}
  66. 𝒩 \mathcal{N}
  67. U U
  68. U U
  69. u u
  70. U U
  71. g ( x , u ) b g(x,u)\leq b
  72. u u
  73. 𝒩 \mathcal{N}
  74. u u
  75. 𝒩 \mathcal{N}
  76. g ( x , u ) b + β d i s t ( u , 𝒩 ) , u U g(x,u)\leq b+\beta\cdot dist(u,\mathcal{N})\ ,\ \forall u\in U
  77. β 0 \beta\geq 0
  78. d i s t ( u , 𝒩 ) dist(u,\mathcal{N})
  79. u u
  80. 𝒩 \mathcal{N}
  81. β = 0 \beta=0
  82. z ( 𝒩 , U ) := max x X { f ( x ) : g ( x , u ) b + β d i s t ( u , 𝒩 ) , u U } z(\mathcal{N},U):=\max_{x\in X}\ \{f(x):g(x,u)\leq b+\beta\cdot dist(u,% \mathcal{N})\ ,\ \forall u\in U\}
  83. d i s t dist
  84. d i s t ( u , 𝒩 ) 0 , u U dist(u,\mathcal{N})\geq 0,\forall u\in U
  85. d i s t ( u , 𝒩 ) = 0 , u 𝒩 dist(u,\mathcal{N})=0,\forall u\not\in\mathcal{N}
  86. g ( x , u ) b g(x,u)\leq b
  87. u u
  88. 𝒩 \mathcal{N}
  89. g ( x , u ) b + β d i s t ( u , 𝒩 ) g(x,u)\leq b+\beta\cdot dist(u,\mathcal{N})
  90. 𝒩 \mathcal{N}
  91. max x X min u U ( x ) f ( x , u ) \max_{x\in X}\min_{u\in U(x)}f(x,u)
  92. max \max
  93. min \min
  94. X X
  95. U ( x ) U(x)
  96. u u
  97. x x
  98. max x X , v { v : v f ( x , u ) , u U ( x ) } \max_{x\in X,v\in\mathbb{R}}\ \{v:v\leq f(x,u),\forall u\in U(x)\}
  99. max x X min u U ( x ) { f ( x , u ) : g ( x , u ) b , u U ( x ) } \max_{x\in X}\min_{u\in U(x)}\ \{f(x,u):g(x,u)\leq b,\forall u\in U(x)\}
  100. max x X , v { v : v f ( x , u ) , g ( x , u ) b , u U ( x ) } \max_{x\in X,v\in\mathbb{R}}\ \{v:v\leq f(x,u),g(x,u)\leq b,\forall u\in U(x)\}

Rodrigues_equation.html

  1. H E T P = A + B u + C f ( λ ) u HETP=A+\frac{B}{u}+C\cdot f(\lambda)\cdot u
  2. f ( λ ) = 3 λ [ 1 t a n h ( λ ) - 1 λ ] f(\lambda)=\frac{3}{\lambda}\left[\frac{1}{tanh(\lambda)}-\frac{1}{\lambda}\right]
  3. λ \lambda

Rokhlin's_theorem.html

  1. Q M : H 2 ( M , ) × H 2 ( M , ) Q_{M}:H^{2}(M,\mathbb{Z})\times H^{2}(M,\mathbb{Z})\rightarrow\mathbb{Z}
  2. \mathbb{Z}
  3. M M
  4. s s
  5. M M
  6. μ ( M , s ) \mu(M,s)
  7. / 16 \mathbb{Z}/16\mathbb{Z}
  8. ( M , s ) (M,s)
  9. S 4 S^{4}
  10. ( N , s ) (N,s)

Rook's_graph.html

  1. n 2 n^{2}
  2. n 3 - n 2 n^{3}-n^{2}
  3. \square

Root_datum.html

  1. ( X , Φ , X , Φ ) (X^{\ast},\Phi,X_{\ast},\Phi^{\vee})
  2. X X^{\ast}
  3. X X_{\ast}
  4. \mathbb{Z}
  5. Φ \Phi
  6. X X^{\ast}
  7. Φ \Phi^{\vee}
  8. X X_{\ast}
  9. Φ \Phi
  10. Φ \Phi^{\vee}
  11. α α \alpha\mapsto\alpha^{\vee}
  12. α \alpha
  13. ( α , α ) = 2 (\alpha,\alpha^{\vee})=2
  14. α \alpha
  15. x x - ( x , α ) α x\mapsto x-(x,\alpha^{\vee})\alpha
  16. Φ \Phi
  17. Φ \Phi
  18. X X_{\ast}
  19. Φ \Phi^{\vee}
  20. Φ \Phi^{\vee}
  21. Φ \Phi
  22. Φ \Phi^{\vee}
  23. X X^{\ast}
  24. X X_{\ast}
  25. Φ \Phi
  26. 2 α 2\alpha
  27. α Φ \alpha\in\Phi

Root_mean_square_fluctuation.html

  1. M S F = 1 T t j = 1 T ( x i ( t j ) - x ~ i ) 2 MSF=\sqrt{\frac{1}{T}\sum_{t_{j}=1}^{T}(x_{i}(t_{j})-\tilde{x}_{i})^{2}}
  2. x ~ i \tilde{x}_{i}
  3. x ¯ i \overline{x}_{i}

Roshko_number.html

  1. Ro = f L 2 ν = St Re \mathrm{Ro}={fL^{2}\over\nu}=\mathrm{St}\,\mathrm{Re}

Rotation_formalisms_in_three_dimensions.html

  1. 𝐮 ^ \scriptstyle\hat{\mathbf{u}}
  2. 𝐯 ^ \scriptstyle\hat{\mathbf{v}}
  3. 𝐰 ^ \scriptstyle\hat{\mathbf{w}}
  4. 𝐀 = [ 𝐮 ^ x 𝐯 ^ x 𝐰 ^ x 𝐮 ^ y 𝐯 ^ y 𝐰 ^ y 𝐮 ^ z 𝐯 ^ z 𝐰 ^ z ] \mathbf{A}=\left[{\begin{array}[]{ccc}\hat{\mathbf{u}}_{x}&\hat{\mathbf{v}}_{x% }&\hat{\mathbf{w}}_{x}\\ \hat{\mathbf{u}}_{y}&\hat{\mathbf{v}}_{y}&\hat{\mathbf{w}}_{y}\\ \hat{\mathbf{u}}_{z}&\hat{\mathbf{v}}_{z}&\hat{\mathbf{w}}_{z}\\ \end{array}}\right]
  5. { 1 , e ± i θ } = { 1 , cos ( θ ) + i sin ( θ ) , cos ( θ ) - i sin ( θ ) } \{1,e^{\pm i\theta}\}=\{1,\ \cos(\theta)+i\sin(\theta),\ \cos(\theta)-i\sin(% \theta)\}
  6. 1 + 2 cos ( θ ) \scriptstyle 1\,+\,2\cos(\theta)
  7. θ \scriptstyle\theta
  8. | 𝐮 ^ | = | 𝐯 ^ | \displaystyle|\hat{\mathbf{u}}|=|\hat{\mathbf{v}}|
  9. ( 𝐮 ^ , 𝐯 ^ , 𝐰 ^ ) (\hat{\mathbf{u}},\,\hat{\mathbf{v}},\,\hat{\mathbf{w}})
  10. 𝐀 1 \scriptstyle\mathbf{A}_{1}
  11. 𝐀 2 \scriptstyle\mathbf{A}_{2}
  12. 𝐀 total = 𝐀 2 𝐀 1 \scriptstyle\mathbf{A}\text{total}\;=\;\mathbf{A}_{2}\mathbf{A}_{1}
  13. 𝐞 ^ = [ e x e y e z ] T \scriptstyle\hat{\mathbf{e}}\;=\;[e_{x}\ e_{y}\ e_{z}]^{\mathrm{T}}
  14. θ \scriptstyle\theta
  15. 𝐫 = θ 𝐞 ^ . \mathbf{r}=\theta\hat{\mathbf{e}}~{}.
  16. θ \scriptstyle\theta
  17. 𝐪 ^ = 𝐢 q i + 𝐣 q j + 𝐤 q k + q r = [ q i q j q k q r ] T \hat{\mathbf{q}}=\mathbf{i}q_{i}+\mathbf{j}q_{j}+\mathbf{k}q_{k}+q_{r}=[q_{i}% \ q_{j}\ q_{k}\ q_{r}]^{\mathrm{T}}
  18. 𝐞 ^ = [ e x e y e z ] T \hat{\mathbf{e}}=[e_{x}\ e_{y}\ e_{z}]^{\mathrm{T}}
  19. θ \theta
  20. q i \displaystyle q_{i}
  21. q i 2 + q j 2 + q k 2 + q r 2 = 1 q_{i}^{2}+q_{j}^{2}+q_{k}^{2}+q_{r}^{2}=1
  22. a + b i + c j + d k a+bi+cj+dk
  23. { a , b , c , d } \{a,b,c,d\}\in\mathbb{R}
  24. { i , j , k } \scriptstyle\{i,\,j,\,k\}
  25. i 2 = j 2 = k 2 = - 1 i j = - j i = k j k = - k j = i k i = - i k = j \begin{array}[]{lclrlcl}i^{2}&=&j^{2}&=&k^{2}&=&-1\\ ij&=&-ji&=&k&&\\ jk&=&-kj&=&i&&\\ ki&=&-ik&=&j&&\end{array}
  26. 𝐪 ~ 𝐪 = [ q r q k - q j q i - q k q r q i q j q j - q i q r q k - q i - q j - q k q r ] [ q ~ i q ~ j q ~ k q ~ r ] = [ q ~ r - q ~ k q ~ j q ~ i q ~ k q ~ r - q ~ i q ~ j - q ~ j q ~ i q ~ r q ~ k - q ~ i - q ~ j - q ~ k q ~ r ] [ q i q j q k q r ] \tilde{\mathbf{q}}\otimes\mathbf{q}=\left[{\begin{array}[]{rrrr}q_{r}&q_{k}&-q% _{j}&q_{i}\\ -q_{k}&q_{r}&q_{i}&q_{j}\\ q_{j}&-q_{i}&q_{r}&q_{k}\\ -q_{i}&-q_{j}&-q_{k}&q_{r}\end{array}}\right]\left[{\begin{array}[]{c}\tilde{q% }_{i}\\ \tilde{q}_{j}\\ \tilde{q}_{k}\\ \tilde{q}_{r}\end{array}}\right]=\left[{\begin{array}[]{rrrr}\tilde{q}_{r}&-% \tilde{q}_{k}&\tilde{q}_{j}&\tilde{q}_{i}\\ \tilde{q}_{k}&\tilde{q}_{r}&-\tilde{q}_{i}&\tilde{q}_{j}\\ -\tilde{q}_{j}&\tilde{q}_{i}&\tilde{q}_{r}&\tilde{q}_{k}\\ -\tilde{q}_{i}&-\tilde{q}_{j}&-\tilde{q}_{k}&\tilde{q}_{r}\end{array}}\right]% \left[{\begin{array}[]{c}q_{i}\\ q_{j}\\ q_{k}\\ q_{r}\end{array}}\right]
  27. 𝐀 1 \scriptstyle\mathbf{A}_{1}
  28. 𝐀 2 \scriptstyle\mathbf{A}_{2}
  29. 𝐀 3 = 𝐀 2 𝐀 1 \mathbf{A}_{3}=\mathbf{A}_{2}\mathbf{A}_{1}
  30. 𝐪 3 = 𝐪 2 𝐪 1 \mathbf{q}_{3}=\mathbf{q}_{2}\otimes\mathbf{q}_{1}
  31. 4 \scriptstyle\mathbb{R}^{4}
  32. S 3 \scriptstyle S^{3}
  33. 𝐫 = 𝐞 ^ θ \mathbf{r}=\hat{\mathbf{e}}\theta
  34. π π
  35. 𝐠 = 𝐞 ^ tan ( θ 2 ) \mathbf{g}=\hat{\mathbf{e}}\tan\left(\frac{\theta}{2}\right)
  36. ( 𝐠 , 𝐟 ) = 𝐠 + 𝐟 - 𝐟 × 𝐠 1 - 𝐠 𝐟 . (\mathbf{g},\mathbf{f})=\frac{\mathbf{g}+\mathbf{f}-\mathbf{f}\times\mathbf{g}% }{1-\mathbf{g}\cdot\mathbf{f}}~{}.
  37. ( , , 0 ) \scriptstyle(\infty,\,\infty,\,0)
  38. 𝐩 = 𝐞 ^ tan ( θ 4 ) . \mathbf{p}=\hat{\mathbf{e}}\tan\left(\frac{\theta}{4}\right)~{}.
  39. ( ϕ , θ , ψ ) \scriptstyle(\phi,\,\theta,\,\psi)
  40. 𝐀 \scriptstyle\mathbf{A}
  41. ϕ \scriptstyle\phi
  42. θ \scriptstyle\theta
  43. ψ \scriptstyle\psi
  44. Z \scriptstyle Z
  45. X \scriptstyle X
  46. Z \scriptstyle Z
  47. ϕ \displaystyle\phi
  48. arctan2 ( a , b ) \scriptstyle\operatorname{arctan2}(a,\,b)
  49. arctan ( a / b ) \scriptstyle\arctan(a/b)
  50. ( a , b ) \scriptstyle(a,\,b)
  51. θ \theta
  52. A 33 = 0 \scriptstyle A_{33}\;=\;0
  53. ϕ , ψ \scriptstyle\phi,\,\psi
  54. A 11 , A 12 \scriptstyle A_{11},\,A_{12}
  55. 𝐀 \scriptstyle\mathbf{A}
  56. 𝐀 = 𝐀 3 𝐀 2 𝐀 1 = 𝐀 Z 𝐀 Y 𝐀 X \mathbf{A}=\mathbf{A}_{3}\mathbf{A}_{2}\mathbf{A}_{1}=\mathbf{A}_{Z}\mathbf{A}% _{Y}\mathbf{A}_{X}
  57. X \scriptstyle X
  58. Y \scriptstyle Y
  59. Z \scriptstyle Z
  60. ϕ \scriptstyle\phi
  61. θ \scriptstyle\theta
  62. ψ \scriptstyle\psi
  63. 𝐀 X \displaystyle\mathbf{A}_{X}
  64. 𝐀 = [ cos θ cos ψ cos ϕ sin ψ + sin ϕ sin θ cos ψ sin ϕ sin ψ - cos ϕ sin θ cos ψ - cos θ sin ψ cos ϕ cos ψ - sin ϕ sin θ sin ψ sin ϕ cos ψ + cos ϕ sin θ sin ψ sin θ - sin ϕ cos θ cos ϕ cos θ ] \begin{array}[]{lcl}\mathbf{A}&=&\begin{bmatrix}\cos\theta\cos\psi&\cos\phi% \sin\psi+\sin\phi\sin\theta\cos\psi&\sin\phi\sin\psi-\cos\phi\sin\theta\cos% \psi\\ -\cos\theta\sin\psi&\cos\phi\cos\psi-\sin\phi\sin\theta\sin\psi&\sin\phi\cos% \psi+\cos\phi\sin\theta\sin\psi\\ \sin\theta&-\sin\phi\cos\theta&\cos\phi\cos\theta\\ \end{bmatrix}\end{array}
  65. θ \scriptstyle\theta
  66. π \scriptstyle\pi
  67. 𝐞 ^ = [ e 1 e 2 e 3 ] T \scriptstyle\hat{\mathbf{e}}\;=\;[e_{1}\ e_{2}\ e_{3}]^{\mathrm{T}}
  68. θ \scriptstyle\theta
  69. 𝐀 \scriptstyle\mathbf{A}
  70. θ \displaystyle\theta
  71. cos θ ± i sin θ \scriptstyle\cos\theta\pm i\sin\theta
  72. θ \scriptstyle\theta
  73. I - A \scriptstyle I\,-\,A
  74. 𝐞 ^ = [ e 1 e 2 e 3 ] T \scriptstyle\hat{\mathbf{e}}\;=\;[e_{1}\ e_{2}\ e_{3}]^{\mathrm{T}}
  75. θ \scriptstyle\theta
  76. 𝐀 = 𝐈 3 cos θ + ( 1 - cos θ ) 𝐞 ^ 𝐞 ^ T + [ 𝐞 ^ ] × sin θ \mathbf{A}=\mathbf{I}_{3}\cos\theta+(1-\cos\theta)\hat{\mathbf{e}}\hat{\mathbf% {e}}^{\mathrm{T}}+[\hat{\mathbf{e}}]_{\times}\sin\theta
  77. 𝐈 3 \scriptstyle\mathbf{I}_{3}
  78. [ 𝐞 ^ ] × = [ 0 - e 3 e 2 e 3 0 - e 1 - e 2 e 1 0 ] [\hat{\mathbf{e}}]_{\times}=\left[\begin{array}[]{ccc}0&-e_{3}&e_{2}\\ e_{3}&0&-e_{1}\\ -e_{2}&e_{1}&0\end{array}\right]
  79. A 11 = ( 1 - cos θ ) e 1 2 + cos θ A_{11}=(1-\cos\theta)e_{1}^{2}+\cos\theta
  80. A 12 = ( 1 - cos θ ) e 1 e 2 - e 3 sin θ A_{12}=(1-\cos\theta)e_{1}e_{2}-e_{3}\sin\theta
  81. A 13 = ( 1 - cos θ ) e 1 e 3 + e 2 sin θ A_{13}=(1-\cos\theta)e_{1}e_{3}+e_{2}\sin\theta
  82. A 21 = ( 1 - cos θ ) e 2 e 1 + e 3 sin θ A_{21}=(1-\cos\theta)e_{2}e_{1}+e_{3}\sin\theta
  83. A 22 = ( 1 - cos θ ) e 2 2 + cos θ A_{22}=(1-\cos\theta)e_{2}^{2}+\cos\theta
  84. A 23 = ( 1 - cos θ ) e 2 e 3 - e 1 sin θ A_{23}=(1-\cos\theta)e_{2}e_{3}-e_{1}\sin\theta
  85. A 31 = ( 1 - cos θ ) e 3 e 1 - e 2 sin θ A_{31}=(1-\cos\theta)e_{3}e_{1}-e_{2}\sin\theta
  86. A 32 = ( 1 - cos θ ) e 3 e 2 + e 1 sin θ A_{32}=(1-\cos\theta)e_{3}e_{2}+e_{1}\sin\theta
  87. A 33 = ( 1 - cos θ ) e 3 2 + cos θ A_{33}=(1-\cos\theta)e_{3}^{2}+\cos\theta
  88. 𝐪 \scriptstyle\mathbf{q}
  89. - 𝐪 \scriptstyle-\mathbf{q}
  90. 𝐪 = [ q i q j q k q r ] T = 𝐢 q i + 𝐣 q j + 𝐤 q k + q r \scriptstyle\mathbf{q}\;=\;[q_{i}\ q_{j}\ q_{k}\ q_{r}]^{\mathrm{T}}=\mathbf{i% }q_{i}+\mathbf{j}q_{j}+\mathbf{k}q_{k}+q_{r}
  91. 𝐀 \scriptstyle\mathbf{A}
  92. q r \displaystyle q_{r}
  93. 𝐪 \scriptstyle\mathbf{q}
  94. q i = 1 2 1 + A 11 - A 22 - A 33 q j = 1 4 q i ( A 12 + A 21 ) q k = 1 4 q i ( A 13 + A 31 ) q r = 1 4 q i ( A 32 - A 23 ) \begin{aligned}\displaystyle q_{i}&\displaystyle=\frac{1}{2}\sqrt{1+A_{11}-A_{% 22}-A_{33}}\\ \displaystyle q_{j}&\displaystyle=\frac{1}{4q_{i}}(A_{12}+A_{21})\\ \displaystyle q_{k}&\displaystyle=\frac{1}{4q_{i}}(A_{13}+A_{31})\\ \displaystyle q_{r}&\displaystyle=\frac{1}{4q_{i}}(A_{32}-A_{23})\end{aligned}
  95. 𝐪 = [ q i q j q k q r ] T = 𝐢 q i + 𝐣 q j + 𝐤 q k + q r \scriptstyle\mathbf{q}\;=\;[q_{i}\ q_{j}\ q_{k}\ q_{r}]^{\mathrm{T}}=\mathbf{i% }q_{i}+\mathbf{j}q_{j}+\mathbf{k}q_{k}+q_{r}
  96. 𝐀 = ( q r 2 - 𝐪 ˇ T 𝐪 ˇ ) 𝐈 3 + 2 𝐪 ˇ 𝐪 ˇ T - 2 q r 𝒬 \mathbf{A}=(q_{r}^{2}-\check{\mathbf{q}}^{\mathrm{T}}\check{\mathbf{q}})% \mathbf{I}_{3}+2\check{\mathbf{q}}\check{\mathbf{q}}^{\mathrm{T}}-2q_{r}% \mathbf{\mathcal{Q}}
  97. 𝐈 3 \scriptstyle\mathbf{I}_{3}
  98. 𝐪 ˇ = [ q i q j q k ] , 𝒬 = [ 0 - q k q j q k 0 - q i - q j q i 0 ] \check{\mathbf{q}}=\left[\begin{array}[]{c}q_{i}\\ q_{j}\\ q_{k}\end{array}\right],\ \ \ \mathbf{\mathcal{Q}}=\left[\begin{array}[]{ccc}0% &-q_{k}&q_{j}\\ q_{k}&0&-q_{i}\\ -q_{j}&q_{i}&0\end{array}\right]
  99. 𝐀 = [ 1 - 2 q j 2 - 2 q k 2 2 ( q i q j + q k q r ) 2 ( q i q k - q j q r ) 2 ( q i q j - q k q r ) 1 - 2 q i 2 - 2 q k 2 2 ( q i q r + q j q k ) 2 ( q i q k + q j q r ) 2 ( q j q k - q i q r ) 1 - 2 q i 2 - 2 q j 2 ] \mathbf{A}=\left[\begin{array}[]{ccc}1-2q_{j}^{2}-2q_{k}^{2}&2(q_{i}q_{j}+q_{k% }q_{r})&2(q_{i}q_{k}-q_{j}q_{r})\\ 2(q_{i}q_{j}-q_{k}q_{r})&1-2q_{i}^{2}-2q_{k}^{2}&2(q_{i}q_{r}+q_{j}q_{k})\\ 2(q_{i}q_{k}+q_{j}q_{r})&2(q_{j}q_{k}-q_{i}q_{r})&1-2q_{i}^{2}-2q_{j}^{2}\end{% array}\right]
  100. 𝐀 = [ - 1 + 2 q i 2 + 2 q r 2 2 ( q i q j + q k q r ) 2 ( q i q k - q j q r ) 2 ( q i q j - q k q r - 1 + 2 q j 2 + 2 q r 2 2 ( q i q r + q j q k ) 2 ( q i q k + q j q r ) 2 ( q j q k - q i q r ) - 1 + 2 q k 2 + 2 q r 2 ] \mathbf{A}=\left[\begin{array}[]{ccc}-1+2q_{i}^{2}+2q_{r}^{2}&2(q_{i}q_{j}+q_{% k}q_{r})&2(q_{i}q_{k}-q_{j}q_{r})\\ 2(q_{i}q_{j}-q_{k}q_{r}&-1+2q_{j}^{2}+2q_{r}^{2}&2(q_{i}q_{r}+q_{j}q_{k})\\ 2(q_{i}q_{k}+q_{j}q_{r})&2(q_{j}q_{k}-q_{i}q_{r})&-1+2q_{k}^{2}+2q_{r}^{2}\end% {array}\right]
  101. 𝐪 = [ q i q j q k q r ] T = 𝐢 q i + 𝐣 q j + 𝐤 q k + q r \scriptstyle\mathbf{q}=[q_{i}\ q_{j}\ q_{k}\ q_{r}]^{\mathrm{T}}=\mathbf{i}q_{% i}+\mathbf{j}q_{j}+\mathbf{k}q_{k}+q_{r}
  102. ( ϕ , θ , ψ ) \scriptstyle(\phi,\,\theta,\,\psi)
  103. q i \displaystyle q_{i}
  104. ψ \psi
  105. θ \theta
  106. ϕ \phi
  107. q i = sin ( ϕ / 2 ) cos ( θ / 2 ) cos ( ψ / 2 ) - cos ( ϕ / 2 ) sin ( θ / 2 ) sin ( ψ / 2 ) q j = sin ( ϕ / 2 ) cos ( θ / 2 ) cos ( ψ / 2 ) - cos ( ϕ / 2 ) sin ( θ / 2 ) sin ( ψ / 2 ) q k = cos ( ϕ / 2 ) sin ( θ / 2 ) cos ( ψ / 2 ) + sin ( ϕ / 2 ) cos ( θ / 2 ) sin ( ψ / 2 ) q r = cos ( ϕ / 2 ) cos ( θ / 2 ) cos ( ψ / 2 ) + sin ( ϕ / 2 ) sin ( θ / 2 ) sin ( ψ / 2 ) \begin{aligned}\displaystyle q_{i}&\displaystyle=\sin(\phi/2)\cos(\theta/2)% \cos(\psi/2)-\cos(\phi/2)\sin(\theta/2)\sin(\psi/2)\\ \displaystyle q_{j}&\displaystyle=\sin(\phi/2)\cos(\theta/2)\cos(\psi/2)-\cos(% \phi/2)\sin(\theta/2)\sin(\psi/2)\\ \displaystyle q_{k}&\displaystyle=\cos(\phi/2)\sin(\theta/2)\cos(\psi/2)+\sin(% \phi/2)\cos(\theta/2)\sin(\psi/2)\\ \displaystyle q_{r}&\displaystyle=\cos(\phi/2)\cos(\theta/2)\cos(\psi/2)+\sin(% \phi/2)\sin(\theta/2)\sin(\psi/2)\end{aligned}
  108. 𝐪 = [ q i q j q k q r ] T = 𝐢 q i + 𝐣 q j + 𝐤 q k + q r \scriptstyle\mathbf{q}=[q_{i}\ q_{j}\ q_{k}\ q_{r}]^{\mathrm{T}}=\mathbf{i}q_{% i}+\mathbf{j}q_{j}+\mathbf{k}q_{k}+q_{r}
  109. ( ϕ , θ , ψ ) \scriptstyle(\phi,\,\theta,\,\psi)
  110. ϕ = arctan 2 ( ( q i q k + q j q r ) , - ( q j q k - q i q r ) ) θ = arccos ( - q i 2 - q j 2 + q k 2 + q r 2 ) ψ = arctan 2 ( ( q i q k - q j q r ) , ( q j q k + q i q r ) ) \begin{aligned}\displaystyle\phi&\displaystyle=\arctan 2((q_{i}q_{k}+q_{j}q_{r% }),-(q_{j}q_{k}-q_{i}q_{r}))\\ \displaystyle\theta&\displaystyle=\arccos(-q_{i}^{2}-q_{j}^{2}+q_{k}^{2}+q_{r}% ^{2})\\ \displaystyle\psi&\displaystyle=\arctan 2((q_{i}q_{k}-q_{j}q_{r}),(q_{j}q_{k}+% q_{i}q_{r}))\end{aligned}
  111. 𝐪 = 𝐢 q i + 𝐣 q j + 𝐤 q k + q r \scriptstyle\mathbf{q}=\mathbf{i}q_{i}+\mathbf{j}q_{j}+\mathbf{k}q_{k}+q_{r}
  112. y a w = atan2 ( 2 ( q r q i + q j q k ) , 1 - 2 ( q i 2 + q j 2 ) ) p i t c h = arcsin ( 2 ( q r q j - q k q i ) ) r o l l = atan2 ( 2 ( q r q k + q i q j ) , 1 - 2 ( q j 2 + q k 2 ) ) \begin{aligned}\displaystyle yaw&\displaystyle=\mbox{atan2}~{}(2(q_{r}q_{i}+q_% {j}q_{k}),1-2(q_{i}^{2}+q_{j}^{2}))\\ \displaystyle pitch&\displaystyle=\mbox{arcsin}~{}(2(q_{r}q_{j}-q_{k}q_{i}))\\ \displaystyle roll&\displaystyle=\mbox{atan2}~{}(2(q_{r}q_{k}+q_{i}q_{j}),1-2(% q_{j}^{2}+q_{k}^{2}))\end{aligned}
  113. 𝐞 ^ \scriptstyle\hat{\mathbf{e}}
  114. θ \scriptstyle\theta
  115. 𝐪 = [ q i q j q k q r ] T = 𝐢 q i + 𝐣 q j + 𝐤 q k + q r \mathbf{q}=[q_{i}\ q_{j}\ q_{k}\ q_{r}]^{\mathrm{T}}=\mathbf{i}q_{i}+\mathbf{j% }q_{j}+\mathbf{k}q_{k}+q_{r}
  116. q i = e ^ 1 sin ( θ 2 ) q j = e ^ 2 sin ( θ 2 ) q k = e ^ 3 sin ( θ 2 ) q r = cos ( θ 2 ) \begin{aligned}\displaystyle q_{i}&\displaystyle=\hat{e}_{1}\sin\left(\frac{% \theta}{2}\right)\\ \displaystyle q_{j}&\displaystyle=\hat{e}_{2}\sin\left(\frac{\theta}{2}\right)% \\ \displaystyle q_{k}&\displaystyle=\hat{e}_{3}\sin\left(\frac{\theta}{2}\right)% \\ \displaystyle q_{r}&\displaystyle=\cos\left(\frac{\theta}{2}\right)\end{aligned}
  117. 𝐪 = [ q i q j q k q r ] T = 𝐢 q i + 𝐣 q j + 𝐤 q k + q r \scriptstyle\mathbf{q}\;=\;[q_{i}\ q_{j}\ q_{k}\ q_{r}]^{\mathrm{T}}=\mathbf{i% }q_{i}+\mathbf{j}q_{j}+\mathbf{k}q_{k}+q_{r}
  118. 𝐪 ˇ = [ q i q j q k ] T \scriptstyle\check{\mathbf{q}}\;=\;[q_{i}\ q_{j}\ q_{k}]^{\mathrm{T}}
  119. 𝐞 ^ \scriptstyle\hat{\mathbf{e}}
  120. θ \scriptstyle\theta
  121. 𝐞 ^ = 𝐪 ˇ 𝐪 ˇ θ = 2 arccos ( q r ) \begin{aligned}\displaystyle\hat{\mathbf{e}}&\displaystyle=\frac{\check{% \mathbf{q}}}{\|\check{\mathbf{q}}\|}\\ \displaystyle\theta&\displaystyle=2\arccos(q_{r})\end{aligned}
  122. ω = ( ω x , ω y , ω z ) \scriptstyle\mathbf{\omega}\;=\;(\omega_{x},\,\omega_{y},\,\omega_{z})
  123. d 𝐀 d t \scriptstyle\frac{d\mathbf{A}}{dt}
  124. [ ω ] × = [ 0 - ω z ω y ω z 0 - ω x - ω y ω x 0 ] = d 𝐀 d t 𝐀 T [\mathbf{\omega}]_{\times}=\left[\begin{array}[]{ccc}0&-\omega_{z}&\omega_{y}% \\ \omega_{z}&0&-\omega_{x}\\ -\omega_{y}&\omega_{x}&0\end{array}\right]=\frac{d\mathbf{A}}{dt}\mathbf{A}^{% \mathrm{T}}
  125. r 0 \scriptstyle r_{0}
  126. r ( t ) = 𝐀 ( t ) r 0 \scriptstyle r(t)\;=\;\mathbf{A}(t)r_{0}
  127. d r d t = d 𝐀 d t r 0 = d 𝐀 d t 𝐀 T ( t ) r ( t ) \frac{dr}{dt}=\frac{d\mathbf{A}}{dt}r_{0}=\frac{d\mathbf{A}}{dt}\mathbf{A}^{% \mathrm{T}}(t)r(t)
  128. r ( t ) \scriptstyle r(t)
  129. r 0 \scriptstyle r_{0}
  130. r ( t ) \scriptstyle r(t)
  131. r ( t ) \scriptstyle r(t)
  132. d r d t = ω ( t ) × r ( t ) = [ ω ] × r ( t ) \frac{dr}{dt}=\mathbf{\omega}(t)\times r(t)=[\mathbf{\omega}]_{\times}r(t)
  133. d 𝐀 d t 𝐀 T ( t ) r ( t ) = [ ω ] × r ( t ) \frac{d\mathbf{A}}{dt}\mathbf{A}^{\mathrm{T}}(t)r(t)=[\mathbf{\omega}]_{\times% }r(t)
  134. d 𝐀 d t 𝐀 T ( t ) = [ ω ] × \frac{d\mathbf{A}}{dt}\mathbf{A}^{\mathrm{T}}(t)=[\mathbf{\omega}]_{\times}
  135. ω = ( ω x , ω y , ω z ) \scriptstyle\mathbf{\omega}\;=\;(\omega_{x},\,\omega_{y},\,\omega_{z})
  136. d 𝐪 d t \scriptstyle\frac{d\mathbf{q}}{dt}
  137. [ 0 ω x ω y ω z ] = 2 d 𝐪 d t 𝐪 ~ \left[{\begin{array}[]{c}0\\ \omega_{x}\\ \omega_{y}\\ \omega_{z}\end{array}}\right]=2\frac{d\mathbf{q}}{dt}\otimes\tilde{\mathbf{q}}
  138. 𝐪 ~ \tilde{\mathbf{q}}
  139. 𝐪 \mathbf{q}
  140. d 𝐪 d t = 1 2 [ 0 ω x ω y ω z ] 𝐪 \frac{d\mathbf{q}}{dt}=\frac{1}{2}\left[{\begin{array}[]{c}0\\ \omega_{x}\\ \omega_{y}\\ \omega_{z}\end{array}}\right]\otimes\mathbf{q}
  141. a b = a b + a b ab=a\cdot b+a\wedge b
  142. \scriptstyle\wedge
  143. a , b \scriptstyle a,\,b
  144. x ^ y ^ \scriptstyle\hat{x}\hat{y}
  145. x y \scriptstyle xy
  146. ( x ^ y ^ ) 2 = x ^ y ^ x ^ y ^ \scriptstyle(\hat{x}\hat{y})^{2}\;=\;\hat{x}\hat{y}\hat{x}\hat{y}
  147. x ^ \scriptstyle\hat{x}
  148. y ^ \scriptstyle\hat{y}
  149. - x ^ x ^ y ^ y ^ = - 1 \scriptstyle-\hat{x}\hat{x}\hat{y}\hat{y}\;=\;-1
  150. R = exp ( - B ^ θ 2 ) = cos θ 2 - B ^ sin θ 2 \scriptstyle R\;=\;\exp\left(\frac{-\hat{B}\theta}{2}\right)\;=\;\cos\frac{% \theta}{2}\,-\,\hat{B}\sin\frac{\theta}{2}
  151. B ^ \scriptstyle\hat{B}
  152. B ^ \scriptstyle\hat{B}
  153. R \scriptstyle R
  154. a \scriptstyle a
  155. b \scriptstyle b
  156. b = R a R b=RaR^{\dagger}
  157. R = exp ( 1 2 B ^ θ ) = cos 1 2 θ + B ^ sin 1 2 θ \scriptstyle R^{\dagger}\;=\;\exp\left(\frac{1}{2}\hat{B}\theta\right)\;=\;% \cos\frac{1}{2}\theta\,+\,\hat{B}\sin\frac{1}{2}\theta
  158. R \scriptstyle R
  159. B \scriptstyle B
  160. v ^ = 1 3 ( x ^ + y ^ + z ^ ) \scriptstyle\hat{v}\;=\;\frac{1}{\sqrt{3}}(\hat{x}\,+\,\hat{y}\,+\,\hat{z})
  161. v ^ \scriptstyle\hat{v}
  162. B ^ = x ^ y ^ z ^ v ^ = i v ^ \scriptstyle\hat{B}\;=\;\hat{x}\hat{y}\hat{z}\hat{v}\;=\;i\hat{v}
  163. i = x ^ y ^ z ^ \scriptstyle i\;=\;\hat{x}\hat{y}\hat{z}
  164. B ^ = 1 3 ( y ^ z ^ + z ^ x ^ + x ^ y ^ ) \scriptstyle\hat{B}\;=\;\frac{1}{\sqrt{3}}(\hat{y}\hat{z}\,+\,\hat{z}\hat{x}\,% +\,\hat{x}\hat{y})
  165. B ^ = i v ^ \scriptstyle\hat{B}\;=\;i\hat{v}
  166. i \scriptstyle i
  167. x ^ \scriptstyle\hat{x}
  168. θ \scriptstyle\theta
  169. x ^ = R x ^ R = e - i v ^ θ 2 x ^ e i v ^ θ 2 = x ^ cos 2 θ 2 + i ( x ^ v ^ - v ^ x ^ ) cos θ 2 sin θ 2 + v ^ x ^ v ^ sin 2 θ 2 \hat{x}^{\prime}=R\hat{x}R^{\dagger}=e^{-i\hat{v}\frac{\theta}{2}}\hat{x}e^{i% \hat{v}\frac{\theta}{2}}=\hat{x}\cos^{2}\frac{\theta}{2}+i(\hat{x}\hat{v}-\hat% {v}\hat{x})\cos\frac{\theta}{2}\sin\frac{\theta}{2}+\hat{v}\hat{x}\hat{v}\sin^% {2}\frac{\theta}{2}
  170. i ( x ^ v ^ - v ^ x ^ ) = 2 i ( x ^ v ^ ) \scriptstyle i(\hat{x}\hat{v}\,-\,\hat{v}\hat{x})\;=\;2i(\hat{x}\,\wedge\,\hat% {v})
  171. - v ^ x ^ v ^ \scriptstyle-\hat{v}\hat{x}\hat{v}
  172. x ^ \scriptstyle\hat{x}
  173. v ^ \scriptstyle\hat{v}
  174. v ^ \scriptstyle\hat{v}
  175. v ^ x ^ v ^ \displaystyle\hat{v}\hat{x}\hat{v}
  176. x ^ = x ^ ( cos 2 θ 2 - 1 3 sin 2 θ 2 ) + 2 3 y ^ sin θ 2 ( sin θ 2 + 3 cos θ 2 ) + 2 3 z ^ sin θ 2 ( sin θ 2 - 3 cos θ 2 ) \hat{x}^{\prime}=\hat{x}\left(\cos^{2}\frac{\theta}{2}-\frac{1}{3}\sin^{2}% \frac{\theta}{2}\right)+\frac{2}{3}\hat{y}\sin\frac{\theta}{2}\left(\sin\frac{% \theta}{2}+\sqrt{3}\cos\frac{\theta}{2}\right)+\frac{2}{3}\hat{z}\sin\frac{% \theta}{2}\left(\sin\frac{\theta}{2}-\sqrt{3}\cos\frac{\theta}{2}\right)
  177. θ = 2 3 π \scriptstyle\theta\;=\;\frac{2}{3}\pi
  178. x ^ \scriptstyle\hat{x}
  179. y ^ \scriptstyle\hat{y}
  180. x ^ = x ^ ( 1 4 - 1 3 3 4 ) + 2 3 y ^ 3 2 ( 3 2 + 3 1 2 ) + 2 3 z ^ 3 2 ( 3 2 - 3 1 2 ) = 0 x ^ + y ^ + 0 z ^ = y ^ \begin{aligned}\displaystyle\hat{x}^{\prime}&\displaystyle=\hat{x}\left(\frac{% 1}{4}-\frac{1}{3}\frac{3}{4}\right)+\frac{2}{3}\hat{y}\frac{\sqrt{3}}{2}\left(% \frac{\sqrt{3}}{2}+\sqrt{3}\frac{1}{2}\right)+\frac{2}{3}\hat{z}\frac{\sqrt{3}% }{2}\left(\frac{\sqrt{3}}{2}-\sqrt{3}\frac{1}{2}\right)\\ &\displaystyle=0\hat{x}+\hat{y}+0\hat{z}=\hat{y}\end{aligned}
  181. R = R γ R β R α = exp ( - i z ^ γ 2 ) exp ( - i x ^ β 2 ) exp ( - i z ^ α 2 ) R=R_{\gamma^{\prime}}R_{\beta^{\prime}}R_{\alpha}=\exp\left(\frac{-i\hat{z}^{% \prime}\gamma}{2}\right)\exp\left(\frac{-i\hat{x}^{\prime}\beta}{2}\right)\exp% \left(\frac{-i\hat{z}\alpha}{2}\right)
  182. x ^ = R α x ^ R α \scriptstyle\hat{x}^{\prime}\;=\;R_{\alpha}\hat{x}R_{\alpha}^{\dagger}
  183. z ^ = R β z ^ R β \scriptstyle\hat{z}^{\prime}\;=\;R_{\beta^{\prime}}\hat{z}R_{\beta^{\prime}}^{\dagger}
  184. R β = cos β 2 - i R α x ^ R α sin β 2 = R α R β R α R_{\beta^{\prime}}=\cos\frac{\beta}{2}-iR_{\alpha}\hat{x}R_{\alpha}^{\dagger}% \sin\frac{\beta}{2}=R_{\alpha}R_{\beta}R_{\alpha}^{\dagger}
  185. R β \scriptstyle R_{\beta}
  186. γ \scriptstyle\gamma
  187. R γ = R β R γ R β = R α R β R α R γ R α R β R α \scriptstyle R_{\gamma^{\prime}}\;=\;R_{\beta^{\prime}}R_{\gamma}R_{\beta^{% \prime}}^{\dagger}\;=\;R_{\alpha}R_{\beta}R_{\alpha}^{\dagger}R_{\gamma}R_{% \alpha}R_{\beta}^{\dagger}R_{\alpha}^{\dagger}
  188. R γ \scriptstyle R_{\gamma}
  189. R α \scriptstyle R_{\alpha}
  190. R = R α R β R γ R=R_{\alpha}R_{\beta}R_{\gamma}

Rotation_of_axes.html

  1. θ \theta
  2. θ \theta
  3. θ \theta
  4. θ \theta
  5. ( r , α ) (r,\alpha)
  6. ( r , α - θ ) (r,\alpha-\theta)
  7. ( x y ) = ( cos θ sin θ - sin θ cos θ ) ( x y ) , \begin{pmatrix}x^{\prime}\\ y^{\prime}\end{pmatrix}=\begin{pmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix},
  8. ( x y ) = ( cos θ - sin θ sin θ cos θ ) ( x y ) . \begin{pmatrix}x\\ y\end{pmatrix}=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}x^{\prime}\\ y^{\prime}\end{pmatrix}.
  9. P 1 = ( x , y ) = ( 3 , 1 ) P_{1}=(x,y)=(\sqrt{3},1)
  10. θ 1 = π / 6 \theta_{1}=\pi/6
  11. x = 3 cos ( π / 6 ) + 1 sin ( π / 6 ) = ( 3 ) ( 3 / 2 ) + ( 1 ) ( 1 / 2 ) = 2 x^{\prime}=\sqrt{3}\cos(\pi/6)+1\sin(\pi/6)=(\sqrt{3})({\sqrt{3}}/2)+(1)(1/2)=2
  12. y = 1 cos ( π / 6 ) - 3 sin ( π / 6 ) = ( 1 ) ( 3 / 2 ) - ( 3 ) ( 1 / 2 ) = 0. y^{\prime}=1\cos(\pi/6)-\sqrt{3}\sin(\pi/6)=(1)({\sqrt{3}}/2)-(\sqrt{3})(1/2)=0.
  13. θ 1 = π / 6 \theta_{1}=\pi/6
  14. P 1 = ( x , y ) = ( 2 , 0 ) P_{1}=(x^{\prime},y^{\prime})=(2,0)
  15. π / 6 \pi/6
  16. P 2 = ( x , y ) = ( 7 , 7 ) P_{2}=(x,y)=(7,7)
  17. θ 2 = - π / 2 \theta_{2}=-\pi/2
  18. ( x y ) = ( cos ( - π / 2 ) sin ( - π / 2 ) - sin ( - π / 2 ) cos ( - π / 2 ) ) ( 7 7 ) = ( 0 - 1 1 0 ) ( 7 7 ) = ( - 7 7 ) . \begin{pmatrix}x^{\prime}\\ y^{\prime}\end{pmatrix}=\begin{pmatrix}\cos(-\pi/2)&\sin(-\pi/2)\\ -\sin(-\pi/2)&\cos(-\pi/2)\end{pmatrix}\begin{pmatrix}7\\ 7\end{pmatrix}=\begin{pmatrix}0&-1\\ 1&0\end{pmatrix}\begin{pmatrix}7\\ 7\end{pmatrix}=\begin{pmatrix}-7\\ 7\end{pmatrix}.
  19. θ 2 = - π / 2 \theta_{2}=-\pi/2
  20. P 2 = ( x , y ) = ( - 7 , 7 ) P_{2}=(x^{\prime},y^{\prime})=(-7,7)
  21. π / 2 \pi/2
  22. θ \theta
  23. cot 2 θ = ( A - C ) / B \cot 2\theta=(A-C)/B
  24. B = 0 B^{\prime}=0
  25. B 2 - 4 A C B^{2}\ -\ 4AC
  26. { an ellipse or a circle , if B 2 - 4 A C < 0 ; a parabola , if B 2 - 4 A C = 0 ; a hyperbola , if B 2 - 4 A C > 0. \begin{cases}\mbox{an ellipse or a circle}~{},\ \mbox{if}~{}\ B^{2}\ -\ 4AC\ <% \ 0;\\ \mbox{a parabola}~{},\ \mbox{if}~{}\ B^{2}\ -\ 4AC\ =\ 0;\\ \mbox{a hyperbola}~{},\ \mbox{if}~{}\ B^{2}\ -\ 4AC\ >\ 0.\end{cases}
  27. θ \theta
  28. ( x y z ) = ( cos θ sin θ 0 - sin θ cos θ 0 0 0 1 ) ( x y z ) . \begin{pmatrix}x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{pmatrix}=\begin{pmatrix}\cos\theta&\sin\theta&0\\ -\sin\theta&\cos\theta&0\\ 0&0&1\end{pmatrix}\begin{pmatrix}x\\ y\\ z\end{pmatrix}.
  29. A A
  30. a i i = a j j = cos θ a_{ii}=a_{jj}=\cos\theta
  31. a i j = - a j i = sin θ , a_{ij}=-a_{ji}=\sin\theta,
  32. θ \theta
  33. P 3 = ( w , x , y , z ) = ( 1 , 1 , 1 , 1 ) P_{3}=(w,x,y,z)=(1,1,1,1)
  34. θ 3 = π / 12 \theta_{3}=\pi/12
  35. ( w x y z ) = ( cos ( π / 12 ) 0 0 sin ( π / 12 ) 0 1 0 0 0 0 1 0 - sin ( π / 12 ) 0 0 cos ( π / 12 ) ) ( w x y z ) \begin{pmatrix}w^{\prime}\\ x^{\prime}\\ y^{\prime}\\ z^{\prime}\end{pmatrix}=\begin{pmatrix}\cos(\pi/12)&0&0&\sin(\pi/12)\\ 0&1&0&0\\ 0&0&1&0\\ -\sin(\pi/12)&0&0&\cos(\pi/12)\end{pmatrix}\begin{pmatrix}w\\ x\\ y\\ z\end{pmatrix}
  36. ( 0.96593 0.0 0.0 0.25882 0.0 1.0 0.0 0.0 0.0 0.0 1.0 0.0 - 0.25882 0.0 0.0 0.96593 ) ( 1.0 1.0 1.0 1.0 ) = ( 1.22475 1.00000 1.00000 0.70711 ) . \approx\begin{pmatrix}0.96593&0.0&0.0&0.25882\\ 0.0&1.0&0.0&0.0\\ 0.0&0.0&1.0&0.0\\ -0.25882&0.0&0.0&0.96593\end{pmatrix}\begin{pmatrix}1.0\\ 1.0\\ 1.0\\ 1.0\end{pmatrix}=\begin{pmatrix}1.22475\\ 1.00000\\ 1.00000\\ 0.70711\end{pmatrix}.

Rothe–Hagen_identity.html

  1. x , y , z x,y,z
  2. k = 0 n x x + k z ( x + k z k ) y y + ( n - k ) z ( y + ( n - k ) z n - k ) = x + y x + y + n z ( x + y + n z n ) . \sum_{k=0}^{n}\frac{x}{x+kz}{x+kz\choose k}\frac{y}{y+(n-k)z}{y+(n-k)z\choose n% -k}=\frac{x+y}{x+y+nz}{x+y+nz\choose n}.

Rotordynamics.html

  1. M q ¨ ( t ) + ( C + G ) q ˙ ( t ) + ( K + N ) q ( t ) = f ( t ) \begin{matrix}{M}\ddot{{q}}(t)+({C}+{G})\dot{{q}}(t)+({K}+{N}){{q}}(t)&=&{f}(t% )\\ \end{matrix}

Roundness_(object).html

  1. θ i \theta_{i}
  2. R i R_{i}
  3. R ^ = 1 N i = 1 N R i \hat{R}=\frac{1}{N}\sum\limits_{i=1}^{N}R_{i}
  4. a ^ = 2 N i = 1 N R i cos θ i \hat{a}=\frac{2}{N}\sum\limits_{i=1}^{N}R_{i}\cos{\theta_{i}}
  5. b ^ = 2 N i = 1 N R i sin θ i \hat{b}=\frac{2}{N}\sum\limits_{i=1}^{N}R_{i}\sin{\theta_{i}}
  6. Δ ^ = R i - R ^ - a ^ cos θ i - b ^ sin θ i \hat{\Delta}=R_{i}-\hat{R}-\hat{a}\cos{\theta_{i}}-\hat{b}\sin{\theta_{i}}

Row_equivalence.html

  1. ( 1 0 0 0 1 1 ) and ( 1 0 0 1 1 1 ) \begin{pmatrix}1&0&0\\ 0&1&1\end{pmatrix}\;\;\;\;\,\text{and}\;\;\;\;\begin{pmatrix}1&0&0\\ 1&1&1\end{pmatrix}
  2. ( a b b ) \begin{pmatrix}a&b&b\end{pmatrix}
  3. x = 0 y + z = 0 and x = 0 x + y + z = 0. \begin{matrix}x=0\\ y+z=0\end{matrix}\;\;\;\;\,\text{and}\;\;\;\;\begin{matrix}x=0\\ x+y+z=0.\end{matrix}
  4. a x + b y + b z = 0. ax+by+bz=0.\,

Rubber_ducky_antenna.html

  1. A e = 3 λ 2 8 π A_{e}=\frac{3\lambda^{2}}{8\pi}

Rubber_elasticity.html

  1. b b\,
  2. n n\,
  3. b b\,
  4. r r\,
  5. L c L_{c}\,
  6. n b nb\,
  7. r r\,
  8. r r\,
  9. P ( r , n ) d r = 4 π r 2 ( 2 n b 2 π 3 ) - 3 2 exp ( - 3 r 2 2 n b 2 ) d r P(r,n)dr=4\pi r^{2}\left(\frac{2nb^{2}\pi}{3}\right)^{-\frac{3}{2}}\exp\left(% \frac{-3r^{2}}{2nb^{2}}\right)dr\,
  10. r \langle r\rangle
  11. r = 0 r 2 = n b 2 r 2 1 2 = n b \begin{aligned}\displaystyle\langle r\rangle&\displaystyle=0\\ \displaystyle\langle r^{2}\rangle&\displaystyle=nb^{2}\\ \displaystyle\langle r^{2}\rangle^{\frac{1}{2}}&\displaystyle=\sqrt{n}b\end{aligned}
  12. S = k B ln Ω k B ln ( P ( r , n ) d r ) A - T S = - k B T 3 r 2 2 L c b F - d A d r = 3 k B T L c b r \begin{aligned}\displaystyle S&\displaystyle=k_{B}\ln\Omega\,\approx k_{B}\ln(% P(r,n)dr)\\ \displaystyle A&\displaystyle\approx-TS=-k_{B}T\frac{3r^{2}}{2L_{c}b}\\ \displaystyle F&\displaystyle\approx\frac{-dA}{dr}=\frac{3k_{B}T}{L_{c}b}r\end% {aligned}
  13. 3 k B T L c b \frac{3k_{B}T}{L_{c}b}
  14. L p L_{p}\,
  15. b b\,
  16. F k B T L p ( 1 4 ( 1 - r L c ) 2 - 1 4 + r L c ) F\approx\frac{k_{B}T}{L_{p}}\left(\frac{1}{4\left(1-\frac{r}{L_{c}}\right)^{2}% }-\frac{1}{4}+\frac{r}{L_{c}}\right)\,
  17. r = L c r=L_{c}\,
  18. r r\,
  19. L c L_{c}\,

Rubin_causal_model.html

  1. t 1 t_{1}
  2. t 2 t_{2}
  3. t 2 t_{2}
  4. t 1 t_{1}
  5. t 2 t_{2}
  6. t 1 t_{1}
  7. Y t ( u ) Y_{t}(u)
  8. Y c ( u ) Y_{c}(u)
  9. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  10. Y t ( u ) Y_{t}(u)
  11. Y c ( u ) Y_{c}(u)
  12. Y c ( u ) Y_{c}(u)
  13. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  14. Y t ( u ) Y_{t}(u)
  15. Y c ( u ) Y_{c}(u)
  16. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  17. 10 - 20 10-20
  18. 0 - 5 0-5
  19. 20 - 5 20-5
  20. Y t ( u ) Y_{t}(u)
  21. Y c ( u ) Y_{c}(u)
  22. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  23. Y t ( u ) Y_{t}(u)
  24. Y c ( u ) Y_{c}(u)
  25. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  26. Y t ( u ) Y_{t}(u)
  27. Y c ( u ) Y_{c}(u)
  28. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  29. Y t ( u ) Y_{t}(u)
  30. Y c ( u ) Y_{c}(u)
  31. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  32. Y t ( u ) = T + Y c ( u ) Y_{t}(u)=T+Y_{c}(u)
  33. Y t ( u ) - T = Y c ( u ) . Y_{t}(u)-T=Y_{c}(u).
  34. Y t ( u ) Y_{t}(u)
  35. Y c ( u ) Y_{c}(u)
  36. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  37. Y t 1 ( u ) = Y t 2 ( u ) Y_{t1}(u)=Y_{t2}(u)
  38. Y c 1 ( u ) = Y c 2 ( u ) Y_{c1}(u)=Y_{c2}(u)
  39. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  40. Y t ( u ) Y_{t}(u)
  41. Y c ( u ) Y_{c}(u)
  42. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  43. Y t ( u ) Y_{t}(u)
  44. Y c ( u ) Y_{c}(u)
  45. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  46. Y t ( u ) Y_{t}(u)
  47. Y c ( u ) Y_{c}(u)
  48. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  49. Y t ( u ) Y_{t}(u)
  50. Y c ( u ) Y_{c}(u)
  51. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  52. Y t ( u ) Y_{t}(u)
  53. Y c ( u ) Y_{c}(u)
  54. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  55. Y t ( u ) Y_{t}(u)
  56. Y c ( u ) Y_{c}(u)
  57. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  58. Y t ( u ) Y_{t}(u)
  59. Y c ( u ) Y_{c}(u)
  60. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  61. Y t ( u ) Y_{t}(u)
  62. Y c ( u ) Y_{c}(u)
  63. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  64. Y t ( u ) Y_{t}(u)
  65. Y c ( u ) Y_{c}(u)
  66. Y t ( u ) - Y c ( u ) Y_{t}(u)-Y_{c}(u)
  67. E u [ Y t ( u ) - Y c ( u ) ] E_{u}[Y_{t}(u)-Y_{c}(u)]

Runge–Kutta_method_(SDE).html

  1. X X
  2. d X t = a ( X t ) d t + b ( X t ) d W t , {{d}X_{t}}=a(X_{t})\,{d}t+b(X_{t})\,{d}W_{t},
  3. X 0 = x 0 X_{0}=x_{0}
  4. W t W_{t}
  5. [ 0 , T ] [0,T]
  6. X X
  7. Y Y
  8. [ 0 , T ] [0,T]
  9. N N
  10. δ = T / N > 0 \delta=T/N>0
  11. 0 = τ 0 < τ 1 < < τ N = T ; 0=\tau_{0}<\tau_{1}<\dots<\tau_{N}=T;
  12. Y 0 := x 0 Y_{0}:=x_{0}
  13. Y n Y_{n}
  14. 1 n N 1\leq n\leq N
  15. Y n + 1 := Y n + a ( Y n ) δ + b ( Y n ) Δ W n + 1 2 ( b ( Υ ^ n ) - b ( Y n ) ) ( ( Δ W n ) 2 - δ ) δ - 1 / 2 , Y_{n+1}:=Y_{n}+a(Y_{n})\delta+b(Y_{n})\Delta W_{n}+\frac{1}{2}\left(b(\hat{% \Upsilon}_{n})-b(Y_{n})\right)\left((\Delta W_{n})^{2}-\delta\right)\delta^{-1% /2},
  16. Δ W n = W τ n + 1 - W τ n \Delta W_{n}=W_{\tau_{n+1}}-W_{\tau_{n}}
  17. Υ ^ n = Y n + a ( Y n ) δ + b ( Y n ) δ 1 / 2 . \hat{\Upsilon}_{n}=Y_{n}+a(Y_{n})\delta+b(Y_{n})\delta^{1/2}.
  18. Δ W n \Delta W_{n}
  19. δ \delta
  20. δ \delta
  21. δ \delta
  22. a a
  23. b b
  24. X ( t ) n \vec{X}(t)\in\mathbb{R}^{n}
  25. d X = a ( t , X ) d t + b ( t , X ) d W , d\vec{X}=\vec{a}(t,\vec{X})\,dt+\vec{b}(t,\vec{X})\,dW,
  26. a \vec{a}
  27. b \vec{b}
  28. h h
  29. X ( t k ) = X k \vec{X}(t_{k})=\vec{X}_{k}
  30. X ( t k + 1 ) \vec{X}(t_{k+1})
  31. X k + 1 \vec{X}_{k+1}
  32. t k + 1 = t k + h t_{k+1}=t_{k}+h
  33. K 1 = h a ( t k , X k ) + ( Δ W k - S k h ) b ( t k , X k ) , K 2 = h a ( t k + 1 , X k + K 1 ) + ( Δ W k + S k h ) b ( t k + 1 , X k + K 1 ) , X k + 1 = X k + 1 2 ( K 1 + K 2 ) , \begin{array}[]{rl}&\vec{K}_{1}=h\vec{a}(t_{k},\vec{X}_{k})+(\Delta W_{k}-S_{k% }\sqrt{h})\vec{b}(t_{k},\vec{X}_{k}),\\ &\vec{K}_{2}=h\vec{a}(t_{k+1},\vec{X}_{k}+\vec{K}_{1})+(\Delta W_{k}+S_{k}% \sqrt{h})\vec{b}(t_{k+1},\vec{X}_{k}+\vec{K}_{1}),\\ &\vec{X}_{k+1}=\vec{X}_{k}+\frac{1}{2}(\vec{K}_{1}+\vec{K}_{2}),\end{array}
  34. Δ W k = h Z k \Delta W_{k}=\sqrt{h}Z_{k}
  35. Z k N ( 0 , 1 ) Z_{k}\sim N(0,1)
  36. S k = ± 1 S_{k}=\pm 1
  37. 1 / 2 1/2
  38. ( t m - t 0 ) / h (t_{m}-t_{0})/h
  39. t = t 0 t=t_{0}
  40. t = t m t=t_{m}
  41. O ( h ) O(h)
  42. S k = 0 S_{k}=0
  43. ± 1 \pm 1

Runoff_curve_number.html

  1. Q = { 0 for P I a ( P - I a ) 2 P - I a + S for P > I a Q=\begin{cases}0&\,\text{for }P\leq I_{a}\\ \frac{(P-I_{a})^{2}}{{P-I_{a}}+S}&\,\text{for }P>I_{a}\end{cases}
  2. Q Q
  3. P P
  4. S S
  5. I a I_{a}
  6. I a = 0.2 S I_{a}=0.2S
  7. I a = 0.05 S I_{a}=0.05S
  8. C N CN
  9. S = 1000 C N - 10 S=\frac{1000}{CN}-10
  10. C N CN
  11. 1 / 8 {1}/{8}
  12. 1 / 4 {1}/{4}
  13. 1 / 3 {1}/{3}
  14. 1 / 2 {1}/{2}
  15. C N I I CN_{II}
  16. C N I CN_{I}
  17. C N I I I CN_{III}
  18. C N I I CN_{II}
  19. C N I CN_{I}
  20. C N CN
  21. C N I I I CN_{III}
  22. C N CN
  23. I a = 0.2 S I_{a}=0.2S
  24. I a I_{a}
  25. S S
  26. I a I_{a}
  27. S S
  28. I a / S = 0.20 I_{a}/S=0.20
  29. I a / S I_{a}/S
  30. I a / S I_{a}/S
  31. Q = { 0 for P 0.05 S ( P - 0.05 S 0.05 ) 2 P + 0.95 S 0.05 for P > 0.05 S Q=\begin{cases}0&\,\text{for }P\leq 0.05S\\ \frac{(P-0.05S_{0.05})^{2}}{P+0.95S_{0.05}}&\,\text{for }P>0.05S\end{cases}
  32. S 0.05 S_{0.05}
  33. I a / S I_{a}/S
  34. S 0.05 S_{0.05}
  35. S 0.20 S_{0.20}
  36. S 0.05 = 1.33 S 0.20 1.15 S_{0.05}=1.33{S_{0.20}}^{1.15}
  37. S 0.20 S_{0.20}
  38. S = 1000 C N - 10 S=\frac{1000}{CN}-10
  39. S 0.05 S_{0.05}

Ryll-Nardzewski_fixed-point_theorem.html

  1. E E
  2. K K
  3. E E
  4. K K

S4_Index.html

  1. S 4 S_{4}

SABR_volatility_model.html

  1. F F
  2. F F
  3. σ \sigma
  4. F F
  5. σ \sigma
  6. d F t = σ t F t β d W t , dF_{t}=\sigma_{t}F^{\beta}_{t}\,dW_{t},
  7. d σ t = α σ t d Z t , d\sigma_{t}=\alpha\sigma_{t}\,dZ_{t},
  8. F 0 F_{0}
  9. σ 0 \sigma_{0}
  10. W t W_{t}
  11. Z t Z_{t}
  12. - 1 < ρ < 1 -1<\rho<1
  13. d W t d Z t = ρ d t dW_{t}dZ_{t}=\rho dt
  14. β , α \beta,\;\alpha
  15. 0 β 1 , α 0 0\leq\beta\leq 1,\;\alpha\geq 0
  16. β \beta
  17. α = 0 \alpha=0
  18. α \alpha
  19. σ \sigma
  20. F F
  21. K K
  22. T T
  23. max ( F T - K , 0 ) \max\left(F_{T}-K,\;0\right)
  24. F t F_{t}
  25. β = 0 \beta=0
  26. β = 1 \beta=1
  27. ε = T α 2 \varepsilon=T\alpha^{2}
  28. σ impl = α log ( F 0 / K ) D ( ζ ) { 1 + [ 2 γ 2 - γ 1 2 + 1 / F mid 2 24 ( σ 0 C ( F mid ) α ) 2 + ρ γ 1 4 σ 0 C ( F mid ) α + 2 - 3 ρ 2 24 ] ε } , \sigma_{\,\text{impl}}=\alpha\;\frac{\log\left(F_{0}/K\right)}{D\left(\zeta% \right)}\;\left\{1+\left[\frac{2\gamma_{2}-\gamma_{1}^{2}+1/F_{\,\text{mid}}^{% 2}}{24}\;\left(\frac{\sigma_{0}C\left(F_{\,\text{mid}}\right)}{\alpha}\right)^% {2}+\frac{\rho\gamma_{1}}{4}\;\frac{\sigma_{0}C\left(F_{\,\text{mid}}\right)}{% \alpha}+\frac{2-3\rho^{2}}{24}\right]\varepsilon\right\},
  29. C ( F ) = F β C\left(F\right)=F^{\beta}
  30. F mid F_{\,\text{mid}}
  31. F 0 F_{0}
  32. K K
  33. F 0 K \sqrt{F_{0}K}
  34. ( F 0 + K ) / 2 \left(F_{0}+K\right)/2
  35. ζ = α σ 0 K F 0 d x C ( x ) = α σ 0 ( 1 - β ) ( F 0 1 - β - K 1 - β ) , \zeta=\frac{\alpha}{\sigma_{0}}\;\int_{K}^{F_{0}}\frac{dx}{C\left(x\right)}=% \frac{\alpha}{\sigma_{0}\left(1-\beta\right)}\;\left(F_{0}^{1-\beta}-K^{1-% \beta}\right),
  36. γ 1 = C ( F mid ) C ( F mid ) = β F mid , \gamma_{1}=\frac{C^{\prime}\left(F_{\,\text{mid}}\right)}{C\left(F_{\,\text{% mid}}\right)}=\frac{\beta}{F_{\,\text{mid}}}\;,
  37. γ 2 = C ′′ ( F mid ) C ( F mid ) = - β ( 1 - β ) F mid 2 . \gamma_{2}=\frac{C^{\prime\prime}\left(F_{\,\text{mid}}\right)}{C\left(F_{\,% \text{mid}}\right)}=-\frac{\beta\left(1-\beta\right)}{F_{\,\text{mid}}^{2}}\;.
  38. D ( ζ ) D\left(\zeta\right)
  39. D ( ζ ) = log ( 1 - 2 ρ ζ + ζ 2 + ζ - ρ 1 - ρ ) . D\left(\zeta\right)=\log\left(\frac{\sqrt{1-2\rho\zeta+\zeta^{2}}+\zeta-\rho}{% 1-\rho}\right).
  40. σ impl n = α F 0 - K D ( ζ ) { 1 + [ 2 γ 2 - γ 1 2 24 ( σ 0 C ( F mid ) α ) 2 + ρ γ 1 4 σ 0 C ( F mid ) α + 2 - 3 ρ 2 24 ] ε } . \sigma_{\,\text{impl}}^{\,\text{n}}=\alpha\;\frac{F_{0}-K}{D\left(\zeta\right)% }\;\left\{1+\left[\frac{2\gamma_{2}-\gamma_{1}^{2}}{24}\;\left(\frac{\sigma_{0% }C\left(F_{\,\text{mid}}\right)}{\alpha}\right)^{2}+\frac{\rho\gamma_{1}}{4}\;% \frac{\sigma_{0}C\left(F_{\,\text{mid}}\right)}{\alpha}+\frac{2-3\rho^{2}}{24}% \right]\varepsilon\right\}.
  41. d F t = σ t | F t | β d W t , dF_{t}=\sigma_{t}|F_{t}|^{\beta}\,dW_{t},
  42. d σ t = α σ t d Z t , d\sigma_{t}=\alpha\sigma_{t}\,dZ_{t},
  43. 0 β 1 / 2 0\leq\beta\leq 1/2
  44. F = 0 F=0
  45. d F t = σ t ( F t + s ) β d W t , dF_{t}=\sigma_{t}(F_{t}+s)^{\beta}\,dW_{t},
  46. d σ t = α σ t d Z t , d\sigma_{t}=\alpha\sigma_{t}\,dZ_{t},
  47. s s

Saddle_surface.html

  1. z = x 2 - y 2 z=x^{2}-y^{2}

SAGE_(Soviet–American_Gallium_Experiment).html

  1. ( ν e , e - ) \left(\nu_{e},e^{-}\right)
  2. t 1 / 2 = 11.43 t_{1/2}=11.43

Salat_times.html

  1. T Z T_{Z}
  2. T E T_{E}
  3. L n g Lng
  4. L L
  5. D D
  6. α \alpha
  7. T ( α ) = 1 15 arccos ( - sin ( α ) - sin ( L ) * sin ( D ) cos ( L ) * cos ( D ) ) T(\alpha)={1\over 15}\arccos\left({-\sin(\alpha)-\sin(L)*\sin(D)\over\cos(L)*% \cos(D)}\right)
  8. T D h u h r = 12 + T Z - ( L n g / 15 + T E ) T_{Dhuhr}=12+T_{Z}-(Lng/15+T_{E})
  9. T ( 0.833 ) T(0.833)
  10. α = 0 \alpha=0
  11. T C h o r o k = T D h u h r - T ( 0.833 ) T_{Chorok}=T_{Dhuhr}-T(0.833)
  12. T S u n s e t = T D h u h r + T ( 0.833 ) T_{Sunset}=T_{Dhuhr}+T(0.833)
  13. 0.0347 × h 0.0347\times\sqrt{h}
  14. α \alpha
  15. T T
  16. A ( T ) = 1 15 arccos ( sin ( \arccot ( t + tan ( L - D ) ) ) - sin ( L ) * sin ( D ) cos ( L ) * cos ( D ) ) A(T)={1\over 15}\arccos\left({\sin(\arccot(t+\tan(L-D)))-\sin(L)*\sin(D)\over% \cos(L)*\cos(D)}\right)

Salt_equivalent.html

  1. m NaCl m Na = m Na + m Cl m Na 23 u + 35.5 u 23 u = 58.5 23 2.5 \frac{m_{\mathrm{NaCl}}}{m_{\mathrm{Na}}}=\frac{m_{\mathrm{Na}}+m_{\mathrm{Cl}% }}{m_{\mathrm{Na}}}\approx\frac{23\mathrm{u}+35.5\mathrm{u}}{23\mathrm{u}}=% \frac{58.5}{23}\approx 2.5

Sample-continuous_process.html

  1. { X t Unif ( { X t - 1 - 1 , X t - 1 + 1 } ) , t an integer; X t = X t , t not an integer; \begin{cases}X_{t}\sim\mathrm{Unif}(\{X_{t-1}-1,X_{t-1}+1\}),&t\mbox{ an % integer;}\\ X_{t}=X_{\lfloor t\rfloor},&t\mbox{ not an integer;}\end{cases}

Sankar_Das_Sarma.html

  1. ν = 5 / 2 \nu=5/2

Satisfiability_modulo_theories.html

  1. 3 x + 2 y - z 4 3x+2y-z\geq 4
  2. f ( f ( u , v ) , v ) = f ( u , v ) f(f(u,v),v)=f(u,v)
  3. f f
  4. x - y > c x-y>c
  5. x x
  6. y y
  7. c c
  8. x + y = y + x x+y=y+x
  9. ( sin ( x ) 3 = cos ( log ( y ) x ) b - x 2 2.3 y ) ( ¬ b y < - 34.4 exp ( x ) > y x ) \begin{array}[]{lr}&(\sin(x)^{3}=\cos(\log(y)\cdot x)\vee b\vee-x^{2}\geq 2.3y% )\wedge\left(\neg b\vee y<-34.4\vee\exp(x)>{y\over x}\right)\end{array}
  10. b 𝔹 , x , y b\in{\mathbb{B}},x,y\in{\mathbb{R}}

Savart.html

  1. f 2 f 1 \frac{f_{2}}{f_{1}}
  2. s = 1000 log 10 f 2 f 1 s=1000\log_{10}{\frac{f_{2}}{f_{1}}}
  3. f 2 f 1 = 10 s / 1000 \frac{f_{2}}{f_{1}}=10^{s/1000}
  4. 1 savart = 1.2 log 10 2 cent 3.9863 cent 1\ \mathrm{savart}=\frac{1.2}{\log_{10}{2}}\ \mathrm{cent}\approx 3.9863\ % \mathrm{cent}
  5. 1 savart = 1 log 10 2 millioctave 3.3219 millioctave 1\ \mathrm{savart}=\frac{1}{\log_{10}{2}}\ \mathrm{millioctave}\approx 3.3219% \ \mathrm{millioctave}

Saybolt_universal_second.html

  1. Saybolt Furol viscosity = Saybolt universal viscosity 10 \,\text{Saybolt Furol viscosity}=\frac{\,\text{Saybolt universal viscosity}}{10}

Scalar_field_theory.html

  1. η μ ν \eta^{\mu\nu}
  2. 𝒮 = d D - 1 x d t = d D - 1 x d t [ 1 2 η μ ν μ ϕ ν ϕ - 1 2 m 2 ϕ 2 ] \mathcal{S}=\int\mathrm{d}^{D-1}x\mathrm{d}t\mathcal{L}=\int\mathrm{d}^{D-1}x% \mathrm{d}t\left[\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi-% \frac{1}{2}m^{2}\phi^{2}\right]
  3. = d D - 1 x d t [ 1 2 ( t ϕ ) 2 - 1 2 δ i j i ϕ j ϕ - 1 2 m 2 ϕ 2 ] , =\int\mathrm{d}^{D-1}x\mathrm{d}t\left[\frac{1}{2}(\partial_{t}\phi)^{2}-\frac% {1}{2}\delta^{ij}\partial_{i}\phi\partial_{j}\phi-\frac{1}{2}m^{2}\phi^{2}% \right],
  4. \mathcal{L}
  5. δ i j \delta^{ij}
  6. φ φ
  7. φ φ
  8. η μ ν μ ν ϕ + m 2 ϕ = t 2 ϕ - 2 ϕ + m 2 ϕ = 0 , \eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi+m^{2}\phi=\partial^{2}_{t}\phi-% \nabla^{2}\phi+m^{2}\phi=0~{},
  9. 2 \nabla^{2}
  10. V ( φ ) V(φ)
  11. φ φ
  12. 𝒮 = d D - 1 x d t = d D - 1 x d t [ 1 2 η μ ν μ ϕ ν ϕ - V ( ϕ ) ] \mathcal{S}=\int\mathrm{d}^{D-1}x\,\mathrm{d}t\mathcal{L}=\int\mathrm{d}^{D-1}% x\mathrm{d}t\left[\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi% -V(\phi)\right]
  13. = d D - 1 x d t [ 1 2 ( t ϕ ) 2 - 1 2 δ i j i ϕ j ϕ - 1 2 m 2 ϕ 2 - n = 3 1 n ! g n ϕ n ] =\int\mathrm{d}^{D-1}x\,\mathrm{d}t\left[\frac{1}{2}(\partial_{t}\phi)^{2}-% \frac{1}{2}\delta^{ij}\partial_{i}\phi\partial_{j}\phi-\frac{1}{2}m^{2}\phi^{2% }-\sum_{n=3}^{\infty}\frac{1}{n!}g_{n}\phi^{n}\right]
  14. η μ ν μ ν ϕ + V ( ϕ ) = t 2 ϕ - 2 ϕ + V ( ϕ ) = 0 \eta^{\mu\nu}\partial_{\mu}\partial_{\nu}\phi+V^{\prime}(\phi)=\partial^{2}_{t% }\phi-\nabla^{2}\phi+V^{\prime}(\phi)=0
  15. t t
  16. l = c t l=ct
  17. c c
  18. l l
  19. ħ ħ
  20. ħ ħ
  21. ħ ħ
  22. c c
  23. Δ Δ
  24. φ φ
  25. x λ x x\rightarrow\lambda x
  26. ϕ λ - Δ ϕ . \phi\rightarrow\lambda^{-\Delta}\phi~{}.
  27. ħ ħ
  28. φ φ
  29. Δ = D - 2 2 \Delta=\frac{D-2}{2}
  30. x λ x x\rightarrow\lambda x
  31. ϕ λ - Δ ϕ . \phi\rightarrow\lambda^{-\Delta}\phi~{}.
  32. g n g_{n}
  33. g n g_{n}
  34. n = 2 D D - 2 n=\frac{2D}{D-2}
  35. g 4 g_{4}
  36. φ φ
  37. x x ~ ( x ) x\rightarrow\tilde{x}(x)
  38. x μ ~ x ρ x ν ~ x σ η μ ν = λ 2 ( x ) η ρ σ \frac{\partial\tilde{x^{\mu}}}{\partial x^{\rho}}\frac{\partial\tilde{x^{\nu}}% }{\partial x^{\sigma}}\eta_{\mu\nu}=\lambda^{2}(x)\eta_{\rho\sigma}
  39. λ 2 ( x ) \lambda^{2}(x)
  40. η μ ν \eta_{\mu\nu}
  41. φ φ
  42. ϕ 4 \phi^{4}
  43. = 1 2 ( t ϕ ) 2 - 1 2 δ i j i ϕ j ϕ - 1 2 m 2 ϕ 2 - g 4 ! ϕ 4 . \mathcal{L}=\frac{1}{2}(\partial_{t}\phi)^{2}-\frac{1}{2}\delta^{ij}\partial_{% i}\phi\partial_{j}\phi-\frac{1}{2}m^{2}\phi^{2}-\frac{g}{4!}\phi^{4}.
  44. Z 2 Z_{2}
  45. ϕ - ϕ \phi\rightarrow-\phi
  46. m 2 m^{2}
  47. V ( ϕ ) = 1 2 m 2 ϕ 2 + g 4 ! ϕ 4 V(\phi)=\frac{1}{2}m^{2}\phi^{2}+\frac{g}{4!}\phi^{4}
  48. ϕ = 0 \phi=0
  49. Z 2 Z_{2}
  50. m 2 m^{2}
  51. V ( ϕ ) = 1 2 m 2 ϕ 2 + g 4 ! ϕ 4 \,V(\phi)=\frac{1}{2}m^{2}\phi^{2}+\frac{g}{4!}\phi^{4}\!
  52. Z 2 Z_{2}
  53. Z 2 Z_{2}
  54. ϕ 4 \phi^{4}
  55. m 2 m^{2}
  56. ϕ ( x , t ) = ± m 2 g 4 ! tanh ( m ( x - x 0 ) 2 ) \phi(\vec{x},t)=\pm\frac{m}{2\sqrt{\frac{g}{4!}}}\tanh\left(\frac{m(x-x_{0})}{% \sqrt{2}}\right)
  57. x x
  58. φ φ
  59. t t
  60. 𝒮 = d D - 1 x d t = d D - 1 x d t [ η μ ν μ ϕ * ν ϕ - V ( | ϕ | 2 ) ] \mathcal{S}=\int\mathrm{d}^{D-1}x\,\mathrm{d}t\mathcal{L}=\int\mathrm{d}^{D-1}% x\,\mathrm{d}t\left[\eta^{\mu\nu}\partial_{\mu}\phi^{*}\partial_{\nu}\phi-V(|% \phi|^{2})\right]
  61. ϕ e i α ϕ \phi\rightarrow e^{i\alpha}\phi
  62. α α
  63. ( ϕ ) (\phi)
  64. ϕ 1 = R e ϕ \phi^{1}=Re{\phi}
  65. ϕ 2 = I m ϕ \phi^{2}=Im{\phi}
  66. U ( 1 ) = O ( 2 ) U(1)=O(2)
  67. = 1 2 η μ ν μ ϕ ν ϕ - V ( ϕ ϕ ) \mathcal{L}=\frac{1}{2}\eta^{\mu\nu}\partial_{\mu}\phi\cdot\partial_{\nu}\phi-% V(\phi\cdot\phi)
  68. O ( N ) O(N)
  69. φ φ
  70. π π
  71. x , y \vec{x},\vec{y}
  72. [ ϕ ( x ) , ϕ ( y ) ] = [ π ( x ) , π ( y ) ] = 0 , [\phi(\vec{x}),\phi(\vec{y})]=[\pi(\vec{x}),\pi(\vec{y})]=0,
  73. [ ϕ ( x ) , π ( y ) ] = i δ ( x - y ) , [\phi(\vec{x}),\pi(\vec{y})]=i\delta(\vec{x}-\vec{y}),
  74. H = d 3 x [ 1 2 π 2 + 1 2 ( ϕ ) 2 + m 2 2 ϕ 2 ] . H=\int d^{3}x\left[{1\over 2}\pi^{2}+{1\over 2}(\nabla\phi)^{2}+{m^{2}\over 2}% \phi^{2}\right].
  75. ϕ ~ ( k ) = d 3 x e - i k x ϕ ( x ) , \tilde{\phi}(\vec{k})=\int d^{3}xe^{-i\vec{k}\cdot\vec{x}}\phi(\vec{x}),
  76. π ~ ( k ) = d 3 x e - i k x π ( x ) \tilde{\pi}(\vec{k})=\int d^{3}xe^{-i\vec{k}\cdot\vec{x}}\pi(\vec{x})
  77. a ( k ) = ( E ϕ ~ ( k ) + i π ~ ( k ) ) , a(\vec{k})=\left(E\tilde{\phi}(\vec{k})+i\tilde{\pi}(\vec{k})\right),
  78. a ( k ) = ( E ϕ ~ ( k ) - i π ~ ( k ) ) , a^{\dagger}(\vec{k})=\left(E\tilde{\phi}(\vec{k})-i\tilde{\pi}(\vec{k})\right),
  79. E = k 2 + m 2 E=\sqrt{k^{2}+m^{2}}
  80. [ a ( k 1 ) , a ( k 2 ) ] = [ a ( k 1 ) , a ( k 2 ) ] = 0 , [a(\vec{k}_{1}),a(\vec{k}_{2})]=[a^{\dagger}(\vec{k}_{1}),a^{\dagger}(\vec{k}_% {2})]=0,
  81. [ a ( k 1 ) , a ( k 2 ) ] = ( 2 π ) 3 2 E δ ( k 1 - k 2 ) . [a(\vec{k}_{1}),a^{\dagger}(\vec{k}_{2})]=(2\pi)^{3}2E\delta(\vec{k}_{1}-\vec{% k}_{2}).
  82. | 0 |0\rangle
  83. k \vec{k}
  84. a ( k ) a^{\dagger}(\vec{k})
  85. H = d 3 k ( 2 π ) 3 1 2 a ( k ) a ( k ) , H=\int{d^{3}k\over(2\pi)^{3}}\frac{1}{2}a^{\dagger}(\vec{k})a(\vec{k}),
  86. 0 | 𝒯 { ϕ ( x 1 ) ϕ ( x n ) } | 0 \langle 0|\mathcal{T}\{{\phi}(x_{1})\cdots{\phi}(x_{n})\}|0\rangle
  87. 0 | 𝒯 { ϕ ( x 1 ) ϕ ( x n ) } | 0 = 𝒟 ϕ ϕ ( x 1 ) ϕ ( x n ) e i d 4 x ( 1 2 μ ϕ μ ϕ - m 2 2 ϕ 2 - g 4 ! ϕ 4 ) 𝒟 ϕ e i d 4 x ( 1 2 μ ϕ μ ϕ - m 2 2 ϕ 2 - g 4 ! ϕ 4 ) . \langle 0|\mathcal{T}\{{\phi}(x_{1})\cdots{\phi}(x_{n})\}|0\rangle=\frac{\int% \mathcal{D}\phi\phi(x_{1})\cdots\phi(x_{n})e^{i\int d^{4}x\left({1\over 2}% \partial^{\mu}\phi\partial_{\mu}\phi-{m^{2}\over 2}\phi^{2}-{g\over 4!}\phi^{4% }\right)}}{\int\mathcal{D}\phi e^{i\int d^{4}x\left({1\over 2}\partial^{\mu}% \phi\partial_{\mu}\phi-{m^{2}\over 2}\phi^{2}-{g\over 4!}\phi^{4}\right)}}.
  88. Z [ J ] = 𝒟 ϕ e i d 4 x ( 1 2 μ ϕ μ ϕ - m 2 2 ϕ 2 - g 4 ! ϕ 4 + J ϕ ) = Z [ 0 ] n = 0 i n J ( x 1 ) J ( x n ) n ! 0 | 𝒯 { ϕ ( x 1 ) ϕ ( x n ) } | 0 . Z[J]=\int\mathcal{D}\phi e^{i\int d^{4}x\left({1\over 2}\partial^{\mu}\phi% \partial_{\mu}\phi-{m^{2}\over 2}\phi^{2}-{g\over 4!}\phi^{4}+J\phi\right)}=Z[% 0]\sum_{n=0}^{\infty}\frac{i^{n}J(x_{1})\cdots J(x_{n})}{n!}\langle 0|\mathcal% {T}\{{\phi}(x_{1})\cdots{\phi}(x_{n})\}|0\rangle.
  89. Z [ J ] = 𝒟 ϕ e - d 4 x ( 1 2 ( ϕ ) 2 + m 2 2 ϕ 2 + g 4 ! ϕ 4 + J ϕ ) . Z[J]=\int\mathcal{D}\phi e^{-\int d^{4}x\left({1\over 2}(\nabla\phi)^{2}+{m^{2% }\over 2}\phi^{2}+{g\over 4!}\phi^{4}+J\phi\right)}.
  90. Z ~ [ J ~ ] = 𝒟 ϕ ~ e - d 4 p ( 1 2 ( p 2 + m 2 ) ϕ ~ 2 + λ 4 ! ϕ ~ 4 - J ~ ϕ ~ ) . \tilde{Z}[\tilde{J}]=\int\mathcal{D}\tilde{\phi}e^{-\int d^{4}p\left({1\over 2% }(p^{2}+m^{2})\tilde{\phi}^{2}+{\lambda\over 4!}\tilde{\phi}^{4}-\tilde{J}% \tilde{\phi}\right)}.
  91. Z ~ [ J ~ ] 𝒟 ϕ ~ p [ e - ( p 2 + m 2 ) ϕ ~ 2 / 2 e - g ϕ ~ 4 / 4 ! e J ~ ϕ ~ ] . \tilde{Z}[\tilde{J}]\sim\int\mathcal{D}\tilde{\phi}\prod_{p}\left[e^{-(p^{2}+m% ^{2})\tilde{\phi}^{2}/2}e^{-g\tilde{\phi}^{4}/4!}e^{\tilde{J}\tilde{\phi}}% \right].
  92. ϕ ~ ( p ) \tilde{\phi}(p)
  93. Z ~ [ 0 ] \tilde{Z}[0]
  94. g g
  95. λ λ
  96. β ( g ) β(g)
  97. β ( g ) = λ g λ . \beta(g)=\lambda\,\frac{\partial g}{\partial\lambda}~{}.
  98. ϕ 4 \phi^{4}
  99. β ( g ) = 3 16 π 2 g 2 + O ( g 3 ) \beta(g)=\frac{3}{16\pi^{2}}g^{2}+O(g^{3})
  100. D 5 D\geq 5
  101. D = 4 D=4

Scale-space_axioms.html

  1. L ( x , y , t ) = ( T t f ) ( x , y ) = g ( x , y , t ) * f ( x , y ) L(x,y,t)=(T_{t}f)(x,y)=g(x,y,t)*f(x,y)
  2. f ( x , y ) f(x,y)
  3. g ( x , y , t ) g(x,y,t)
  4. T t ( a f + b h ) = a T t f + b T t h T_{t}(af+bh)=aT_{t}f+bT_{t}h
  5. f f
  6. h h
  7. a a
  8. b b
  9. T t S ( Δ x , Δ y ) f = S ( Δ x , Δ y ) T t f T_{t}S_{(\Delta x,\Delta_{y})}f=S_{(\Delta x,\Delta_{y})}T_{t}f
  10. S ( Δ x , Δ y ) S_{(\Delta x,\Delta_{y})}
  11. ( S ( Δ x , Δ y ) f ) ( x , y ) = f ( x - Δ x , y - Δ y ) (S_{(\Delta x,\Delta_{y})}f)(x,y)=f(x-\Delta x,y-\Delta y)
  12. g ( x , y , t 1 ) * g ( x , y , t 2 ) = g ( x , y , t 1 + t 2 ) g(x,y,t_{1})*g(x,y,t_{2})=g(x,y,t_{1}+t_{2})
  13. L ( x , y , t 2 ) = g ( x , y , t 2 - t 1 ) * L ( x , y , t 1 ) L(x,y,t_{2})=g(x,y,t_{2}-t_{1})*L(x,y,t_{1})
  14. A A
  15. t L ( x , y , t ) = ( A L ) ( x , y , t ) \partial_{t}L(x,y,t)=(AL)(x,y,t)
  16. t L ( x , y , t ) 0 \partial_{t}L(x,y,t)\leq 0
  17. t L ( x , y , t ) 0 \partial_{t}L(x,y,t)\geq 0
  18. g ( x , y , t ) = h ( x 2 + y 2 , t ) g(x,y,t)=h(x^{2}+y^{2},t)
  19. h h
  20. g ^ ( ω x , ω y , t ) = h ^ ( ω x φ ( t ) , ω x φ ( t ) ) \hat{g}(\omega_{x},\omega_{y},t)=\hat{h}(\frac{\omega_{x}}{\varphi(t)},\frac{% \omega_{x}}{\varphi(t)})
  21. φ \varphi
  22. h ^ \hat{h}
  23. g ^ \hat{g}
  24. g g
  25. g ( x , y , t ) 0 g(x,y,t)\geq 0
  26. x = - y = - g ( x , y , t ) d x d y = 1 \int_{x=-\infty}^{\infty}\int_{y=-\infty}^{\infty}g(x,y,t)\,dx\,dy=1
  27. g ( x , y , t ) = g ( x , t ) g ( y , t ) g(x,y,t)=g(x,t)\,g(y,t)

Scale_space_implementation.html

  1. f C ( x 1 , , x N , t ) , f_{C}\left(x_{1},\cdots,x_{N},t\right),
  2. g N ( x 1 , , x N , t ) . g_{N}\left(x_{1},\cdots,x_{N},t\right).
  3. L ( x 1 , , x N , t ) = u 1 = - u N = - f C ( x 1 - u 1 , , x N - u N , t ) g N ( u 1 , , u N , t ) d u 1 d u N . L\left(x_{1},\cdots,x_{N},t\right)=\int_{u_{1}=-\infty}^{\infty}\cdots\int_{u_% {N}=-\infty}^{\infty}f_{C}\left(x_{1}-u_{1},\cdots,x_{N}-u_{N},t\right)\cdot g% _{N}\left(u_{1},\cdots,u_{N},t\right)\,du_{1}\cdots du_{N}.
  4. g N ( x 1 , , x N , t ) = G ( x 1 , t ) G ( x N , t ) g_{N}\left(x_{1},\dots,x_{N},t\right)=G\left(x_{1},t\right)\cdots G\left(x_{N}% ,t\right)
  5. L ( x 1 , , x N , t ) = u 1 = - u N = - f C ( x 1 - u 1 , , x N - u N , t ) G ( u 1 , t ) d u 1 G ( u N , t ) d u N , L(x_{1},\cdots,x_{N},t)=\int_{u_{1}=-\infty}^{\infty}\cdots\int_{u_{N}=-\infty% }^{\infty}f_{C}(x_{1}-u_{1},\cdots,x_{N}-u_{N},t)G(u_{1},t)\,du_{1}\cdots G(u_% {N},t)\,du_{N},
  6. G ( x , t ) = 1 2 π t e - x 2 2 t G(x,t)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{x^{2}}{2t}}
  7. L ( x , t ) = n = - f ( x - n ) G ( n , t ) L(x,t)=\sum_{n=-\infty}^{\infty}f(x-n)\,G(n,t)
  8. G ( n , t ) = 1 2 π t e - n 2 2 t G(n,t)=\frac{1}{\sqrt{2\pi t}}e^{-\frac{n^{2}}{2t}}
  9. L ( x , t ) = n = - M M f ( x - n ) G ( n , t ) L(x,t)=\sum_{n=-M}^{M}f(x-n)\,G(n,t)
  10. 2 M G ( u , t ) d u = 2 M t G ( v , 1 ) d v < ε . 2\int_{M}^{\infty}G(u,t)\,du=2\int_{\frac{M}{\sqrt{t}}}^{\infty}G(v,1)\,dv<\varepsilon.
  11. M = C σ + 1 = C t + 1 M=C\sigma+1=C\sqrt{t}+1
  12. L ( x , t ) = n = - f ( x - n ) T ( n , t ) L(x,t)=\sum_{n=-\infty}^{\infty}f(x-n)\,T(n,t)
  13. T ( n , t ) = e - t I n ( t ) T(n,t)=e^{-t}I_{n}(t)
  14. I n ( t ) I_{n}(t)
  15. L ( x , t ) = n = - M M f ( x - n ) T ( n , t ) L(x,t)=\sum_{n=-M}^{M}f(x-n)\,T(n,t)
  16. T ^ ( θ , t ) = n = - T ( n , t ) e - i θ n = e t ( cos θ - 1 ) . \widehat{T}(\theta,t)=\sum_{n=-\infty}^{\infty}T(n,t)\,e^{-i\theta n}=e^{t(% \cos\theta-1)}.
  17. T ^ ( θ , t ) = 1 e t ( 1 - cos θ ) 1 < m t p l > 1 + t ( 1 - cos θ ) = H 1 ( θ , t ) , \widehat{T}(\theta,t)=\frac{1}{e^{t(1-\cos\theta)}}\approx\frac{1}{<}mtpl>{{1+% t(1-\cos\theta)}}=H_{1}(\theta,t),
  18. t = 2 p ( 1 - p ) 2 . t=\frac{2p}{(1-p)^{2}}.
  19. 1 ( 1 + t N ( 1 - cos θ ) ) N = H N ( θ , t ) , \frac{1}{\left(1+\frac{t}{N}(1-\cos\theta)\right)^{N}}=H_{N}(\theta,t),
  20. t N = 2 p ( 1 - p ) 2 . \frac{t}{N}=\frac{2p}{(1-p)^{2}}.
  21. [ t / 2 , 1 - t , t / 2 ] [t/2,\ 1-t,\ t/2]
  22. t 0.5 t\leq 0.5
  23. t = - 2 z ( 1 - z ) 2 , t=-\frac{2z}{(1-z)^{2}},
  24. t 0.5 t\leq 0.5
  25. ( 1 - t N ( 1 - cos θ ) ) N = F N ( θ , t ) , \left(1-\frac{t}{N}(1-\cos\theta)\right)^{N}=F_{N}(\theta,t),
  26. t N = - 2 z ( 1 - z ) 2 . \frac{t}{N}=-\frac{2z}{(1-z)^{2}}.

Scheil_equation.html

  1. D S = 0 \ D_{S}=0
  2. D L = \ D_{L}=\infty
  3. ( C L - C S ) d f S = ( f L ) d C L (C_{L}-C_{S})\ df_{S}=(f_{L})\ dC_{L}
  4. k = C S C L k=\frac{C_{S}}{C_{L}}
  5. f S + f L = 1 \ f_{S}+f_{L}=1
  6. C L ( 1 - k ) d f S = ( 1 - f S ) d C L C_{L}(1-k)\ df_{S}=(1-f_{S})\ dC_{L}
  7. C L = C o \ C_{L}=C_{o}
  8. f S = 0 \ f_{S}=0
  9. 0 f S d f S 1 - f S = 1 1 - k C o C L d C L C L \displaystyle\int^{f_{S}}_{0}\frac{df_{S}}{1-f_{S}}=\frac{1}{1-k}\displaystyle% \int^{C_{L}}_{C_{o}}\frac{dC_{L}}{C_{L}}
  10. C L = C o ( f L ) k - 1 \ C_{L}=C_{o}(f_{L})^{k-1}
  11. C S = k C o ( 1 - f S ) k - 1 \ C_{S}=kC_{o}(1-f_{S})^{k-1}

Scherk_surface.html

  1. u n : ( - π 2 , + π 2 ) × ( - π 2 , + π 2 ) u_{n}:\left(-\frac{\pi}{2},+\frac{\pi}{2}\right)\times\left(-\frac{\pi}{2},+% \frac{\pi}{2}\right)\to\mathbb{R}
  2. lim y ± π / 2 u n ( x , y ) = + n for - π 2 < x < + π 2 , \lim_{y\to\pm\pi/2}u_{n}\left(x,y\right)=+n\,\text{ for }-\frac{\pi}{2}<x<+% \frac{\pi}{2},
  3. lim x ± π / 2 u n ( x , y ) = - n for - π 2 < y < + π 2 . \lim_{x\to\pm\pi/2}u_{n}\left(x,y\right)=-n\,\text{ for }-\frac{\pi}{2}<y<+% \frac{\pi}{2}.
  4. div ( u n ( x , y ) 1 + | u n ( x , y ) | 2 ) 0 \mathrm{div}\left(\frac{\nabla u_{n}(x,y)}{\sqrt{1+|\nabla u_{n}(x,y)|^{2}}}% \right)\equiv 0
  5. Σ n = { ( x , y , u n ( x , y ) ) 3 | - π 2 < x , y < + π 2 } . \Sigma_{n}=\left\{(x,y,u_{n}(x,y))\in\mathbb{R}^{3}\left|-\frac{\pi}{2}<x,y<+% \frac{\pi}{2}\right.\right\}.
  6. u : ( - π 2 , + π 2 ) × ( - π 2 , + π 2 ) , u:\left(-\frac{\pi}{2},+\frac{\pi}{2}\right)\times\left(-\frac{\pi}{2},+\frac{% \pi}{2}\right)\to\mathbb{R},
  7. u ( x , y ) = log ( cos ( x ) cos ( y ) ) . u(x,y)=\log\left(\frac{\cos(x)}{\cos(y)}\right).
  8. Σ = { ( x , y , log ( cos ( x ) cos ( y ) ) ) 3 | - π 2 < x , y < + π 2 } . \Sigma=\left\{\left.\left(x,y,\log\left(\frac{\cos(x)}{\cos(y)}\right)\right)% \in\mathbb{R}^{3}\right|-\frac{\pi}{2}<x,y<+\frac{\pi}{2}\right\}.
  9. sin ( z ) - sinh ( x ) sinh ( y ) = 0 \sin(z)-\sinh(x)\sinh(y)=0
  10. f ( z ) = 4 1 - z 4 f(z)=\frac{4}{1-z^{4}}
  11. g ( z ) = i z g(z)=iz
  12. x ( r , θ ) = 2 ( ln ( 1 + r e i θ ) - ln ( 1 - r e i θ ) ) = ln ( 1 + r 2 + 2 r cos θ 1 + r 2 - 2 r cos θ ) x(r,\theta)=2\Re(\ln(1+re^{i\theta})-\ln(1-re^{i\theta}))=\ln\left(\frac{1+r^{% 2}+2r\cos\theta}{1+r^{2}-2r\cos\theta}\right)
  13. y ( r , θ ) = ( 4 i tan - 1 ( r e i θ ) ) = 1 + r 2 - 2 r sin θ 1 + r 2 + 2 r sin θ y(r,\theta)=\Re(4i\tan^{-1}(re^{i\theta}))=\frac{1+r^{2}-2r\sin\theta}{1+r^{2}% +2r\sin\theta}
  14. z ( r , θ ) = ( 2 i ( - ln ( 1 - r 2 e 2 i θ ) + ln ( 1 + r 2 e 2 i θ ) ) = 2 tan - 1 ( 2 r 2 sin 2 θ r 4 - 1 ) z(r,\theta)=\Re(2i(-\ln(1-r^{2}e^{2i\theta})+\ln(1+r^{2}e^{2i\theta}))=2\tan^{% -1}\left(\frac{2r^{2}\sin 2\theta}{r^{4}-1}\right)
  15. θ [ 0 , 2 π ) \theta\in[0,2\pi)
  16. r ( 0 , 1 ) r\in(0,1)

Schlegel_diagram.html

  1. R d R^{d}
  2. R d - 1 R^{d-1}
  3. R d - 1 R^{d-1}

Schmidt_decomposition.html

  1. H 1 H_{1}
  2. H 2 H_{2}
  3. n m n\geq m
  4. v v
  5. H 1 H 2 H_{1}\otimes H_{2}
  6. { u 1 , , u m } H 1 \{u_{1},\ldots,u_{m}\}\subset H_{1}
  7. { v 1 , , v m } H 2 \{v_{1},\ldots,v_{m}\}\subset H_{2}
  8. v = i = 1 m α i u i v i v=\sum_{i=1}^{m}\alpha_{i}u_{i}\otimes v_{i}
  9. α i \alpha_{i}
  10. v v
  11. { e 1 , , e n } H 1 \{e_{1},\ldots,e_{n}\}\subset H_{1}
  12. { f 1 , , f m } H 2 \{f_{1},\ldots,f_{m}\}\subset H_{2}
  13. e i f j e_{i}\otimes f_{j}
  14. e i f j T e_{i}f_{j}^{T}
  15. f j T f_{j}^{T}
  16. f j f_{j}
  17. v = 1 i n , 1 j m β i j e i f j v=\sum_{1\leq i\leq n,1\leq j\leq m}\beta_{ij}e_{i}\otimes f_{j}
  18. M v = ( β i j ) i j . \;M_{v}=(\beta_{ij})_{ij}.
  19. M v = U [ Σ 0 ] V . M_{v}=U\begin{bmatrix}\Sigma\\ 0\end{bmatrix}V^{\star}.
  20. U = [ U 1 U 2 ] U=\begin{bmatrix}U_{1}&U_{2}\end{bmatrix}
  21. U 1 U_{1}
  22. M v = U 1 Σ V . \;M_{v}=U_{1}\Sigma V^{\star}.
  23. { u 1 , , u m } \{u_{1},\ldots,u_{m}\}
  24. U 1 U_{1}
  25. { v 1 , , v m } \{v_{1},\ldots,v_{m}\}
  26. α 1 , , α m \alpha_{1},\ldots,\alpha_{m}
  27. M v = k = 1 m α k u k v k , M_{v}=\sum_{k=1}^{m}\alpha_{k}u_{k}v_{k}^{\star},
  28. v = k = 1 m α k u k v k , v=\sum_{k=1}^{m}\alpha_{k}u_{k}\otimes v_{k},
  29. H 1 H 2 H_{1}\otimes H_{2}
  30. w = i = 1 m α i u i v i . w=\sum_{i=1}^{m}\alpha_{i}u_{i}\otimes v_{i}.
  31. α i \alpha_{i}
  32. w w
  33. u v u\otimes v
  34. - i | α i | 2 log | α i | 2 -\sum_{i}|\alpha_{i}|^{2}\log|\alpha_{i}|^{2}
  35. P = μ ν P=\mu\otimes\nu

Schoen–Yau_conjecture.html

  1. \mathbb{C}
  2. \mathbb{H}
  3. := { ( x , y ) 2 | x 2 + y 2 < 1 } \mathbb{H}:=\{(x,y)\in\mathbb{R}^{2}|x^{2}+y^{2}<1\}
  4. d s 2 = 4 d x 2 + d y 2 ( 1 - ( x 2 + y 2 ) ) 2 . \mathrm{d}s^{2}=4\frac{\mathrm{d}x^{2}+\mathrm{d}y^{2}}{(1-(x^{2}+y^{2}))^{2}}.
  5. f : . f:\mathbb{H}\to\mathbb{C}.\,
  6. g : . g:\mathbb{C}\to\mathbb{H}.\,
  7. M N M\sim N\,
  8. M N M\propto N
  9. \sim
  10. \sim
  11. M N N M . M\sim N\iff N\sim M.
  12. , \mathbb{H}\sim\mathbb{C},
  13. \propto
  14. but ∝̸ . \mathbb{C}\propto\mathbb{H}\,\text{ but }\mathbb{H}\not\propto\mathbb{C}.

Schoolmaster_snapper.html

  1. W = c L b W=cL^{b}\!\,

Schouten_tensor.html

  1. P = 1 n - 2 ( R i c - R 2 ( n - 1 ) g ) R i c = ( n - 2 ) P + J g , P=\frac{1}{n-2}\left(Ric-\frac{R}{2(n-1)}g\right)\,\Leftrightarrow Ric=(n-2)P+% Jg\,,
  2. J = 1 2 ( n - 1 ) R J=\frac{1}{2(n-1)}R
  3. R i j k l = W i j k l + g i k P j l - g j k P i l - g i l P j k + g j l P i k . R_{ijkl}=W_{ijkl}+g_{ik}P_{jl}-g_{jk}P_{il}-g_{il}P_{jk}+g_{jl}P_{ik}\,.
  4. g i j Ω 2 g i j P i j P i j - i Υ j + Υ i Υ j - 1 2 Υ k Υ k g i j , g_{ij}\mapsto\Omega^{2}g_{ij}\Rightarrow P_{ij}\mapsto P_{ij}-\nabla_{i}% \Upsilon_{j}+\Upsilon_{i}\Upsilon_{j}-\frac{1}{2}\Upsilon_{k}\Upsilon^{k}g_{ij% }\,,
  5. Υ i := Ω - 1 i Ω . \Upsilon_{i}:=\Omega^{-1}\partial_{i}\Omega\,.