wpmath0000009_10

Quantum_amplifier.html

  1. E phys ~{}E_{\rm phys}~{}
  2. E phys ( x ) = e a ^ M ( x ) exp ( i k z - i ω t ) + Hermitian conjugate \vec{E}_{\rm phys}(\vec{x})~{}=~{}\vec{e}~{}\hat{a}~{}M(\vec{x})~{}\exp(ikz-{% \rm i}\omega t)~{}+~{}{\rm Hermitian~{}conjugate}~{}
  3. x = { x 1 , x 2 , z } ~{}\vec{x}=\{x_{1},x_{2},z\}~{}
  4. e ~{}\vec{e}~{}
  5. k ~{}k~{}
  6. a ^ ~{}\hat{a}~{}
  7. M ( x ) ~{}M(\vec{x})~{}
  8. M ( x ) ~{}M(\vec{x})~{}
  9. a ^ initial ~{}{\left\langle\hat{a}\right\rangle_{\rm initial}}~{}
  10. U ^ \hat{U}
  11. | initial ~{}|{\rm initial}\rangle~{}
  12. | final ~{}|{\rm final}\rangle~{}
  13. | final = U | initial ~{}|{\rm final}\rangle=U|\rm initial\rangle
  14. a ^ ~{}\langle\hat{a}\rangle~{}
  15. a ^ ~{}\hat{a}~{}
  16. a ^ a ^ - a ^ a ^ ~{}\langle\hat{a}^{\dagger}\hat{a}\rangle-\langle\hat{a}^{\dagger}\rangle% \langle\hat{a}\rangle~{}
  17. G ~{}G~{}
  18. G = a ^ final a ^ initial G=\frac{\left\langle\hat{a}\right\rangle_{\rm final}}{\left\langle\hat{a}% \right\rangle_{\rm initial}}
  19. A ^ = U ^ a ^ U ^ ~{}\hat{A}=\hat{U}^{\dagger}\hat{a}\hat{U}~{}
  20. G = A ^ initial a ^ initial ~{}G=\frac{\left\langle\hat{A}\right\rangle_{\rm initial}}{\left\langle\hat{a}% \right\rangle_{\rm initial}}~{}
  21. G ~{}G~{}
  22. α ~{}\alpha~{}
  23. | initial = | α ~{}~{}|{\rm initial}\rangle=|\alpha\rangle~{}
  24. noise = A ^ A ^ - A ^ A ^ - ( a ^ a ^ - a ^ a ^ ) {\rm noise}=\langle\hat{A}^{\dagger}\hat{A}\rangle-\langle\hat{A}^{\dagger}% \rangle\langle\hat{A}\rangle-\left(\langle\hat{a}^{\dagger}\hat{a}\rangle-% \langle\hat{a}^{\dagger}\rangle\langle\hat{a}\rangle\right)
  25. U ^ ~{}\hat{U}~{}
  26. a ^ ~{}\hat{a}~{}
  27. A ^ = U ^ a ^ U ^ ~{}\hat{A}={\hat{U}}^{\dagger}\hat{a}\hat{U}~{}
  28. A ^ = c a ^ + s b ^ , ~{}\hat{A}=c\hat{a}+s\hat{b}^{\dagger},
  29. c ~{}c~{}
  30. s ~{}s~{}
  31. b ^ ~{}\hat{b}^{\dagger}~{}
  32. c ~{}c~{}
  33. s ~{}s~{}
  34. U ^ ~{}\hat{U}~{}
  35. A ^ A ^ - A ^ A ^ = a ^ a ^ - a ^ a ^ = 1. \hat{A}\hat{A}^{\dagger}-\hat{A}^{\dagger}\hat{A}=\hat{a}\hat{a}^{\dagger}-% \hat{a}^{\dagger}\hat{a}=1.
  36. U ^ ~{}\hat{U}~{}
  37. b ^ ~{}\hat{b}~{}
  38. b ^ b ^ - b ^ b ^ = 1 ~{}\hat{b}\hat{b}^{\dagger}-\hat{b}^{\dagger}\hat{b}=1~{}
  39. c 2 - s 2 = 1 . ~{}c^{2}\!-\!s^{2}=1~{}.
  40. G = c ~{}~{}G\!=\!c~{}~{}
  41. noise = c 2 - 1. ~{}~{}{\rm noise}=c^{2}\!-\!1.
  42. g = | G | 2 ~{}~{}g\!=\!|G|^{2}~{}~{}
  43. g - 1 g-1

Quantum_concentration.html

  1. n Q = ( M k T 2 π 2 ) 3 / 2 n_{\rm Q}=\left(\frac{MkT}{2\pi\hbar^{2}}\right)^{3/2}
  2. M M
  3. k k
  4. T T
  5. \hbar
  6. S ( T , V , N ) = N k B [ 5 2 + ln ( n Q n ) ] S(T,V,N)=Nk_{\rm B}\left[\frac{5}{2}+\ln\left(\frac{n_{\rm Q}}{n}\right)\right]

Quantum_graph.html

  1. V V
  2. E E
  3. e = ( v 1 , v 2 ) E e=(v_{1},v_{2})\in E
  4. [ 0 , L e ] [0,L_{e}]
  5. x e x_{e}
  6. v 1 v_{1}
  7. x e = 0 x_{e}=0
  8. v 2 v_{2}
  9. x e = L e x_{e}=L_{e}
  10. x , y x,y
  11. ρ ( x , y ) \rho(x,y)
  12. [ 0 , ) [0,\infty)
  13. x e = 0 x_{e}=0
  14. f f
  15. | E | |E|
  16. f e ( x e ) f_{e}(x_{e})
  17. e E L 2 ( [ 0 , L e ] ) \bigoplus_{e\in E}L^{2}([0,L_{e}])
  18. f , g = e E 0 L e f e * ( x e ) g e ( x e ) d x e , \langle f,g\rangle=\sum_{e\in E}\int_{0}^{L_{e}}f_{e}^{*}(x_{e})g_{e}(x_{e})\,% dx_{e},
  19. L e L_{e}
  20. - d 2 d x e 2 -\frac{\textrm{d}^{2}}{\textrm{d}x_{e}^{2}}
  21. x e x_{e}
  22. H 2 H^{2}
  23. f e ( 0 ) = f e ( L e ) = 0 f_{e}(0)=f_{e}(L_{e})=0
  24. f e ( x e ) = sin ( n π x e L e ) f_{e}(x_{e})=\sin\left(\frac{n\pi x_{e}}{L_{e}}\right)
  25. n n
  26. n 2 π 2 L e 2 \frac{n^{2}\pi^{2}}{L_{e}^{2}}
  27. f f
  28. e v f ( v ) = 0 , \sum_{e\sim v}f^{\prime}(v)=0\ ,
  29. f ( v ) = f ( 0 ) f^{\prime}(v)=f^{\prime}(0)
  30. v v
  31. x = 0 x=0
  32. f ( v ) = - f ( L e ) f^{\prime}(v)=-f^{\prime}(L_{e})
  33. v v
  34. x = L e x=L_{e}
  35. ( i d d x e + A e ( x e ) ) 2 + V e ( x e ) , \left(i\frac{\textrm{d}}{\textrm{d}x_{e}}+A_{e}(x_{e})\right)^{2}+V_{e}(x_{e})\ ,
  36. A e A_{e}
  37. V e V_{e}
  38. v v
  39. d d
  40. v v
  41. x e = 0 x_{e}=0
  42. v v
  43. f f
  44. 𝐟 = ( f e 1 ( 0 ) , f e 2 ( 0 ) , , f e d ( 0 ) ) T , 𝐟 = ( f e 1 ( 0 ) , f e 2 ( 0 ) , , f e d ( 0 ) ) T . \mathbf{f}=(f_{e_{1}}(0),f_{e_{2}}(0),\dots,f_{e_{d}}(0))^{T},\qquad\mathbf{f}% ^{\prime}=(f^{\prime}_{e_{1}}(0),f^{\prime}_{e_{2}}(0),\dots,f^{\prime}_{e_{d}% }(0))^{T}.
  45. v v
  46. A A
  47. B B
  48. A 𝐟 + B 𝐟 = 0. A\mathbf{f}+B\mathbf{f}^{\prime}=\mathbf{0}.
  49. ( A , B ) (A,B)
  50. d d
  51. A B * = B A * . AB^{*}=BA^{*}.
  52. - d 2 d x e 2 f e ( x e ) = k 2 f e ( x e ) . -\frac{d^{2}}{dx_{e}^{2}}f_{e}(x_{e})=k^{2}f_{e}(x_{e}).\,
  53. f e ( x e ) = c e e i k x e + c ^ e e - i k x e . f_{e}(x_{e})=c_{e}\textrm{e}^{ikx_{e}}+\hat{c}_{e}\textrm{e}^{-ikx_{e}}.\,
  54. c c
  55. 0
  56. c ^ \hat{c}
  57. 0
  58. v v
  59. S ( k ) = - ( A + i k B ) - 1 ( A - i k B ) . S(k)=-(A+ikB)^{-1}(A-ikB).\,
  60. v v
  61. 𝐜 = S ( k ) 𝐜 ^ \mathbf{c}=S(k)\hat{\mathbf{c}}
  62. S S
  63. σ ( u v ) ( v w ) \sigma_{(uv)(vw)}
  64. S S
  65. ( u v ) (uv)
  66. ( v w ) (vw)
  67. k k
  68. k k
  69. A = ( 1 - 1 0 0 0 1 - 1 0 0 0 1 - 1 0 0 0 0 ) , B = ( 0 0 0 0 0 0 1 1 1 ) . A=\left(\begin{array}[]{ccccc}1&-1&0&0&\dots\\ 0&1&-1&0&\dots\\ &&\ddots&\ddots&\\ 0&\dots&0&1&-1\\ 0&\dots&0&0&0\\ \end{array}\right),\quad B=\left(\begin{array}[]{cccc}0&0&\dots&0\\ \vdots&\vdots&&\vdots\\ 0&0&\dots&0\\ 1&1&\dots&1\\ \end{array}\right).
  70. S S
  71. k k
  72. σ ( u v ) ( v w ) = 2 d - δ u w . \sigma_{(uv)(vw)}=\frac{2}{d}-\delta_{uw}.\,
  73. δ u w \delta_{uw}
  74. u = w u=w
  75. 2 | E | × 2 | E | 2|E|\times 2|E|
  76. U ( u v ) ( l m ) ( k ) = δ v l σ ( u v ) ( v m ) ( k ) e i k L ( u v ) . U_{(uv)(lm)}(k)=\delta_{vl}\sigma_{(uv)(vm)}(k)\textrm{e}^{ikL_{(uv)}}.\,
  77. U U
  78. 2 | E | 2|E|
  79. c ( u v ) c_{(uv)}
  80. u u
  81. v v
  82. e i k L ( u v ) \textrm{e}^{ikL_{(uv)}}
  83. u u
  84. v v
  85. 2 | E | 2|E|
  86. | U ( k ) - I | = 0. |U(k)-I|=0.\,
  87. k j k_{j}
  88. k k
  89. U ( k ) U(k)
  90. 0 k 0 k 1 0\leqslant k_{0}\leqslant k_{1}\leqslant\dots
  91. k k
  92. d ( k ) := j = 0 δ ( k - k j ) = L π + 1 π p L p r p A p cos ( k L p ) . d(k):=\sum_{j=0}^{\infty}\delta(k-k_{j})=\frac{L}{\pi}+\frac{1}{\pi}\sum_{p}% \frac{L_{p}}{r_{p}}A_{p}\cos(kL_{p}).
  93. d ( k ) d(k)
  94. L π \frac{L}{\pi}
  95. p = ( e 1 , e 2 , , e n ) p=(e_{1},e_{2},\dots,e_{n})
  96. L p = e p L e L_{p}=\sum_{e\in p}L_{e}
  97. L = e E L e L=\sum_{e\in E}L_{e}
  98. r p r_{p}
  99. A p = σ e 1 e 2 σ e 2 e 3 σ e n e 1 A_{p}=\sigma_{e_{1}e_{2}}\sigma_{e_{2}e_{3}}\dots\sigma_{e_{n}e_{1}}
  100. e e
  101. f f
  102. | σ e f | 2 |\sigma_{ef}|^{2}
  103. 2 {\mathbb{R}}^{2}

Quantum_instrument.html

  1. k \mathcal{E}_{k}
  2. ρ A B ρ ~ A A B := k k ( ρ A B ) | k k | A \rho^{AB}\rightarrow\tilde{\rho}^{AA^{\prime}B}:=\sum_{k}\mathcal{E}_{k}\left(% \rho^{AB}\right)\otimes|k\rangle\langle k|^{A^{\prime}}

Quantum_nonlocality.html

  1. 1 2 \scriptstyle\frac{1}{2}
  2. 2 3 \scriptstyle\frac{2}{3}
  3. ( A X , B Y , a x , b y ) \textstyle(AX,BY,ax,by)
  4. P ( a x , b y | A X , B Y ) = { 1 2 , if x y = X Y 0 , otherwise P({ax,by}{|}{AX,BY})=\begin{cases}\frac{1}{2},&\mbox{if }~{}x\oplus y=XY\\ 0,&\mbox{otherwise}\end{cases}
  5. \scriptstyle\oplus
  6. X Y = 1 XY=1
  7. x y = 1 x y x⊕y=1⇒x≠y
  8. X Y = 0 ) XY=0)
  9. x y = 0 x = y x⊕y=0⇒x=y
  10. | ψ A B = 1 2 ( | A | B - | A | B ) \left|\psi_{AB}\right\rangle=\frac{1}{\sqrt{2}}\bigg(\left|\uparrow\right% \rangle_{A}\left|\downarrow\right\rangle_{B}-\left|\downarrow\right\rangle_{A}% \left|\uparrow\right\rangle_{B}\bigg)
  11. | ψ A B = | 𝐧 A | 𝐧 B \left|\psi_{AB}\right\rangle=\left|\uparrow_{\mathbf{n}}\right\rangle_{A}\left% |\downarrow_{\mathbf{n}}\right\rangle_{B}
  12. λ \lambda
  13. P ( a , b | A , B , λ ) = P ( a | A , λ ) P ( b | B , λ ) P\left({a,b}{|}{A,B,\lambda}\right)=P\left({a}{|}{A,\lambda}\right)P\left({b}{% |}{B,\lambda}\right)
  14. P ( a | A , λ ) \scriptstyle P\left({a}{|}{A,\lambda}\right)
  15. λ i \lambda_{i}
  16. λ i \lambda_{i}
  17. λ i \lambda_{i}
  18. P ( a , b | A , B ) = i = 1 k P ( a , b | A , B , λ i ) ρ ( λ i ) P\left({a,b}{|}{A,B}\right)=\sum_{i=1}^{k}P\left({a,b}{|}{A,B,\lambda_{i}}% \right)\rho\left(\lambda_{i}\right)
  19. E ( A , B ) = a , b a b P ( a , b | A , B ) E\left({A,B}\right)=\sum_{a,b}abP\left({a,b}{|}{A,B}\right)
  20. A 0 A_{0}
  21. A 1 A_{1}
  22. B 0 B_{0}
  23. B 1 B_{1}
  24. S C H S H = E ( A 0 , B 0 ) + E ( A 0 , B 1 ) + E ( A 1 , B 0 ) - E ( A 1 , B 1 ) S_{CHSH}=E\left({A_{0},B_{0}}\right)+E\left({A_{0},B_{1}}\right)+E\left({A_{1}% ,B_{0}}\right)-E\left({A_{1},B_{1}}\right)
  25. x y = X Y \scriptstyle x\oplus y=XY
  26. S C H S H S_{CHSH}
  27. A 1 A_{1}
  28. B 1 B_{1}
  29. x = y x=y
  30. X Y = 1 XY=1
  31. x x
  32. y y
  33. X Y = 0 XY=0
  34. - 2 S C H S H 2 -2\leq S_{CHSH}\leq 2
  35. A 0 , A 1 , B 0 , B 1 \scriptstyle A_{0},A_{1},B_{0},B_{1}
  36. S C H S H = 2 2 \scriptstyle S_{CHSH}\;=\;2\sqrt{2}
  37. 2 2 \scriptstyle 2\sqrt{2}
  38. S C H S H = 4 S_{CHSH}=4
  39. { 0 , 1 } \scriptstyle\{0,1\}
  40. P ( a , b | A , B ) \scriptstyle P\left({{a,b}{|}{A,B}}\right)
  41. P ( a , b | A , B ) 0 a , b , A , B P\left({{a,b}{|}{A,B}}\right)\geq 0\quad\forall{a,b,A,B}
  42. a , b P ( a , b | A , B ) = 1 A , B \sum_{a,b}P\left({{a,b}{|}{A,B}}\right)=1\quad\forall{A,B}
  43. P ( a , b | A , B ) = λ p ( λ ) P ( a | A , λ ) P ( b | B , λ ) P\left({{a,b}{|}{A,B}}\right)=\sum_{\lambda}p(\lambda)\;P\left({{a}{|}{A,% \lambda}}\right)\;P\left({{b}{|}{B,\lambda}}\right)
  44. P ( a | A , λ ) \scriptstyle P\left({{a}{|}{A,\lambda}}\right)
  45. P ( b | B , λ ) \scriptstyle P\left({{b}{|}{B,\lambda}}\right)
  46. λ \lambda
  47. p ( λ ) p(\lambda)
  48. λ \lambda
  49. b P ( a , b | A , B ) = b P ( a , b | A , B ) P ( a | A ) a , A , B , B \sum_{b}P\left({a,b}{|}{A,B}\right)=\sum_{b}P\left({a,b}{|}{A,B^{\prime}}% \right)\equiv P\left({a}{|}{A}\right)\quad\forall{a,A,B,B^{\prime}}
  50. a P ( a , b | A , B ) = a P ( a , b | A , B ) P ( b | B ) b , B , A , A \sum_{a}P\left({a,b}{|}{A,B}\right)=\sum_{a}P\left({a,b}{|}{A^{\prime},B}% \right)\equiv P\left({b}{|}{B}\right)\quad\forall{b,B,A,A^{\prime}}
  51. λ \lambda
  52. p ( λ ) p(\lambda)
  53. P ( a , b | A , B ) = { 1 2 , if a b = A B 0 , otherwise P\left({a,b}{|}{A,B}\right)=\begin{cases}\frac{1}{2},&\mbox{if }~{}a\oplus b=% AB\\ 0,&\mbox{otherwise}\end{cases}
  54. \oplus

Quantum_stirring,_ratchets,_and_pumping.html

  1. Q Q
  2. Q Q
  3. Q Q
  4. ( X ) \mathcal{H}(X)
  5. X = ( X 1 , X 2 , X 3 ) X=(X_{1},X_{2},X_{3})
  6. X 3 X_{3}
  7. - X 3 ˙ -\dot{X_{3}}
  8. I = - G 33 X ˙ 3 I=-G^{33}\dot{X}_{3}
  9. G 33 G^{33}
  10. X 1 X_{1}
  11. I = - G 31 X ˙ 1 I=-G^{31}\dot{X}_{1}
  12. X 2 X_{2}
  13. I = - G 32 X ˙ 2 I=-G^{32}\dot{X}_{2}
  14. G 31 G^{31}
  15. G 32 G^{32}
  16. Q = cycle I d t = - ( G 31 d X 1 + G 32 d X 2 ) Q=\oint\limits_{\,\text{cycle}}I\,dt=-\oint(G^{31}\,dX_{1}+G^{32}\,dX_{2})
  17. G 3 j = i 0 [ ( t ) , j ( 0 ) ] t d t G^{3j}=\frac{i}{\hbar}\int_{0}^{\infty}\left\langle\left[\mathcal{I}(t),% \mathcal{F}^{j}(0)\right]\right\rangle\,t\,dt
  18. \mathcal{I}
  19. j = - / X j \mathcal{F}^{j}=-\partial\mathcal{H}/\partial X_{j}
  20. X j X_{j}
  21. G G
  22. B B
  23. B B
  24. B B
  25. B 1 = - G 32 B_{1}=-G^{32}
  26. B 2 = G 31 B_{2}=G^{31}
  27. Q = B d s Q=\oint\vec{B}\cdot\vec{ds}
  28. d s = ( d X 2 , - d X 1 ) \vec{ds}=(dX_{2},-dX_{1})
  29. Q Q
  30. B \vec{B}
  31. X X
  32. B \vec{B}
  33. B j = n ( n 0 ) 2 Im [ n 0 n n n 0 j ] ( E n - E n 0 ) 2 {B}_{j}=\sum_{n(\neq n_{0})}\frac{2\hbar\ {\rm Im}[\mathcal{I}_{n_{0}n}\ % \mathcal{F}^{j}_{nn_{0}}]}{(E_{n}-E_{n_{0}})^{2}}
  34. n n
  35. n 0 n_{0}
  36. B B
  37. A A
  38. B = A \vec{B}=\nabla\wedge\vec{A}
  39. Berry phase = 1 A d X \,\text{Berry phase}=\frac{1}{\hbar}\oint\vec{A}\cdot d\vec{X}
  40. B B
  41. X X
  42. n 0 n_{0}
  43. S S
  44. G 3 j = 1 2 π i trace ( P A S X j S ) G^{3j}=\frac{1}{2\pi i}\mathrm{trace}\left(P_{A}\frac{\partial S}{\partial X_{% j}}S^{\dagger}\right)
  45. P A P_{A}

Quark–lepton_complementarity.html

  1. θ 12 P M N S + θ 12 C K M 45 , \theta_{12}^{PMNS}+\theta_{12}^{CKM}\simeq 45^{\circ}\ ,
  2. θ 23 P M N S + θ 23 C K M 45 . \quad\quad\theta_{23}^{PMNS}+\theta_{23}^{CKM}\simeq 45^{\circ}\ .
  3. V M = U CKM U PMNS , V_{\mathrm{M}}=U_{\mathrm{CKM}}\cdot U_{\mathrm{PMNS}}\ ,
  4. U PMNS = U CKM V M . U_{\mathrm{PMNS}}=U^{\dagger}_{\mathrm{CKM}}V_{\mathrm{M}}\ .
  5. V M V_{M}

Quartile_coefficient_of_dispersion.html

  1. Q 3 - Q 1 Q 3 + Q 1 . {Q_{3}-Q_{1}\over Q_{3}+Q_{1}}.

Quasi-birth–death_process.html

  1. P = ( A 1 A 2 A 0 A 1 A 2 A 0 A 1 A 2 A 0 A 1 A 2 ) P=\begin{pmatrix}A_{1}^{\ast}&A_{2}^{\ast}\\ A_{0}^{\ast}&A_{1}&A_{2}\\ &A_{0}&A_{1}&A_{2}\\ &&A_{0}&A_{1}&A_{2}\\ &&&\ddots&\ddots&\ddots\end{pmatrix}
  2. Q = ( B 00 B 01 B 10 A 1 A 2 A 0 A 1 A 2 A 0 A 1 A 2 A 0 A 1 A 2 ) Q=\begin{pmatrix}B_{00}&B_{01}\\ B_{10}&A_{1}&A_{2}\\ &A_{0}&A_{1}&A_{2}\\ &&A_{0}&A_{1}&A_{2}\\ &&&A_{0}&A_{1}&A_{2}\\ &&&&\ddots&\ddots&\ddots\end{pmatrix}

Quasi-compact_morphism.html

  1. f : X Y f:X\to Y
  2. V i V_{i}
  3. f - 1 ( V i ) f^{-1}(V_{i})
  4. X = Spec A X=\operatorname{Spec}A
  5. f : X Y f:X\to Y
  6. f : X Y f:X\to Y
  7. f ( X ) f(X)

Quasi-polynomial.html

  1. q ( k ) = c d ( k ) k d + c d - 1 ( k ) k d - 1 + + c 0 ( k ) q(k)=c_{d}(k)k^{d}+c_{d-1}(k)k^{d-1}+\cdots+c_{0}(k)
  2. c i ( k ) c_{i}(k)
  3. c d ( k ) c_{d}(k)
  4. f : f\colon\mathbb{N}\to\mathbb{N}
  5. p 0 , , p s - 1 p_{0},\dots,p_{s-1}
  6. f ( n ) = p i ( n ) f(n)=p_{i}(n)
  7. n i mod s n\equiv i\bmod s
  8. p i p_{i}
  9. v 1 , , v n v_{1},\dots,v_{n}
  10. t v 1 , , t v n tv_{1},\dots,tv_{n}
  11. L ( P , t ) = # ( t P d ) L(P,t)=\#(tP\cap\mathbb{Z}^{d})
  12. \mathbb{N}\to\mathbb{N}
  13. ( F * G ) ( k ) = m = 0 k F ( m ) G ( k - m ) (F*G)(k)=\sum_{m=0}^{k}F(m)G(k-m)
  14. deg F + deg G + 1. \leq\deg F+\deg G+1.

Quasiregular_polyhedron.html

  1. { 3 3 } \begin{Bmatrix}3\\ 3\end{Bmatrix}
  2. { 3 4 } \begin{Bmatrix}3\\ 4\end{Bmatrix}
  3. { 3 5 } \begin{Bmatrix}3\\ 5\end{Bmatrix}
  4. { 3 6 } \begin{Bmatrix}3\\ 6\end{Bmatrix}
  5. { 3 7 } \begin{Bmatrix}3\\ 7\end{Bmatrix}
  6. { 3 8 } \begin{Bmatrix}3\\ 8\end{Bmatrix}
  7. { 3 } \begin{Bmatrix}3\\ \infty\end{Bmatrix}
  8. { p q } \begin{Bmatrix}p\\ q\end{Bmatrix}
  9. { 3 3 } \begin{Bmatrix}3\\ 3\end{Bmatrix}
  10. { 4 4 } \begin{Bmatrix}4\\ 4\end{Bmatrix}
  11. { 5 5 } \begin{Bmatrix}5\\ 5\end{Bmatrix}
  12. { 6 6 } \begin{Bmatrix}6\\ 6\end{Bmatrix}
  13. { 7 7 } \begin{Bmatrix}7\\ 7\end{Bmatrix}
  14. { 8 8 } \begin{Bmatrix}8\\ 8\end{Bmatrix}
  15. { } \begin{Bmatrix}\infty\\ \infty\end{Bmatrix}
  16. { p , q } \begin{Bmatrix}p,q\end{Bmatrix}
  17. { q , p } \begin{Bmatrix}q,p\end{Bmatrix}
  18. { p q } \begin{Bmatrix}p\\ q\end{Bmatrix}
  19. { 3 4 } \begin{Bmatrix}3\\ 4\end{Bmatrix}
  20. { 3 5 } \begin{Bmatrix}3\\ 5\end{Bmatrix}
  21. { 3 3 } \begin{Bmatrix}3\\ 3\end{Bmatrix}
  22. { 3 5 / 2 } \begin{Bmatrix}3\\ 5/2\end{Bmatrix}
  23. { 5 5 / 2 } \begin{Bmatrix}5\\ 5/2\end{Bmatrix}

Quaternion-Kähler_symmetric_space.html

  1. H = K Sp ( 1 ) . H=K\cdot\mathrm{Sp}(1).\,
  2. SU ( p + 2 ) \mathrm{SU}(p+2)\,
  3. S ( U ( p ) × U ( 2 ) ) \mathrm{S}(\mathrm{U}(p)\times\mathrm{U}(2))
  4. p + 2 \mathbb{C}^{p+2}
  5. SO ( p + 4 ) \mathrm{SO}(p+4)\,
  6. SO ( p ) SO ( 4 ) \mathrm{SO}(p)\cdot\mathrm{SO}(4)
  7. p + 4 \mathbb{R}^{p+4}
  8. Sp ( p + 1 ) \mathrm{Sp}(p+1)\,
  9. Sp ( p ) Sp ( 1 ) \mathrm{Sp}(p)\cdot\mathrm{Sp}(1)
  10. p + 1 \mathbb{H}^{p+1}
  11. E 6 E_{6}\,
  12. SU ( 6 ) SU ( 2 ) \mathrm{SU}(6)\cdot\mathrm{SU}(2)
  13. ( 𝕆 ) P 2 (\mathbb{C}\otimes\mathbb{O})P^{2}
  14. ( ) P 2 (\mathbb{C}\otimes\mathbb{H})P^{2}
  15. E 7 E_{7}\,
  16. Spin ( 12 ) Sp ( 1 ) \mathrm{Spin}(12)\cdot\mathrm{Sp}(1)
  17. ( 𝕆 ) P 2 (\mathbb{H}\otimes\mathbb{O})P^{2}
  18. 𝕆 \mathbb{H}\otimes\mathbb{O}
  19. E 8 E_{8}\,
  20. E 7 Sp ( 1 ) E_{7}\cdot\mathrm{Sp}(1)
  21. ( 𝕆 𝕆 ) P 2 (\mathbb{O}\otimes\mathbb{O})P^{2}
  22. ( 𝕆 ) P 2 (\mathbb{H}\otimes\mathbb{O})P^{2}
  23. F 4 F_{4}\,
  24. Sp ( 3 ) Sp ( 1 ) \mathrm{Sp}(3)\cdot\mathrm{Sp}(1)
  25. 𝕆 2 \mathbb{OP}^{2}
  26. 2 \mathbb{HP}^{2}
  27. G 2 G_{2}\,
  28. SO ( 4 ) \mathrm{SO}(4)\,
  29. 𝕆 \mathbb{O}
  30. \mathbb{H}

Quillen_adjunction.html

  1. \leftrightarrows

Quinhydrone_electrode.html

  1. E = E 0 + R T n F ln a H + E=E^{0}+\frac{RT}{nF}\ln a_{H^{+}}

Quotient_of_subspace_theorem.html

  1. e = min y e y , e E , \|e\|=\min_{y\in e}\|y\|,\quad e\in E,
  2. Q ( e ) K e K Q ( e ) \frac{\sqrt{Q(e)}}{K}\leq\|e\|\leq K\sqrt{Q(e)}
  3. e E , e\in E,
  4. c ( K ) 1 - const / log log K . c(K)\approx 1-\,\text{const}/\log\log K.

Rabin_fingerprint.html

  1. f ( x ) = m 0 + m 1 x + + m n - 1 x n - 1 f(x)=m_{0}+m_{1}x+\ldots+m_{n-1}x^{n-1}
  2. r ( x ) r(x)
  3. f ( x ) f(x)
  4. p ( x ) p(x)
  5. 2 - 13 2^{-13}

Racah_W-coefficient.html

  1. W ( j 1 j 2 J j 3 ; J 12 J 23 ) [ ( 2 J 12 + 1 ) ( 2 J 23 + 1 ) ] - 1 2 ( j 1 , ( j 2 j 3 ) J 23 ) J | ( ( j 1 j 2 ) J 12 , j 3 ) J . W(j_{1}j_{2}Jj_{3};J_{12}J_{23})\equiv[(2J_{12}+1)(2J_{23}+1)]^{-\frac{1}{2}}% \langle(j_{1},(j_{2}j_{3})J_{23})J|((j_{1}j_{2})J_{12},j_{3})J\rangle.
  2. 𝐣 1 \mathbf{j}_{1}
  3. 𝐣 2 \mathbf{j}_{2}
  4. 𝐉 2 \mathbf{J}^{2}
  5. J z J_{z}
  6. 𝐉 = 𝐣 1 + 𝐣 2 \mathbf{J}=\mathbf{j}_{1}+\mathbf{j}_{2}
  7. | ( j 1 j 2 ) J M = m 1 = - j 1 j 1 m 2 = - j 2 j 2 | j 1 m 1 | j 2 m 2 j 1 m 1 j 2 m 2 | J M , |(j_{1}j_{2})JM\rangle=\sum_{m_{1}=-j_{1}}^{j_{1}}\sum_{m_{2}=-j_{2}}^{j_{2}}|% j_{1}m_{1}\rangle|j_{2}m_{2}\rangle\langle j_{1}m_{1}j_{2}m_{2}|JM\rangle,
  8. J = | j 1 - j 2 | , , j 1 + j 2 J=|j_{1}-j_{2}|,\ldots,j_{1}+j_{2}
  9. M = - J , , J M=-J,\ldots,J
  10. 𝐣 1 \mathbf{j}_{1}
  11. 𝐣 2 \mathbf{j}_{2}
  12. 𝐣 3 \mathbf{j}_{3}
  13. 𝐣 1 \mathbf{j}_{1}
  14. 𝐣 2 \mathbf{j}_{2}
  15. 𝐉 12 \mathbf{J}_{12}
  16. 𝐉 12 \mathbf{J}_{12}
  17. 𝐣 3 \mathbf{j}_{3}
  18. 𝐉 \mathbf{J}
  19. | ( ( j 1 j 2 ) J 12 j 3 ) J M = M 12 = - J 12 J 12 m 3 = - j 3 j 3 | ( j 1 j 2 ) J 12 M 12 | j 3 m 3 J 12 M 12 j 3 m 3 | J M |((j_{1}j_{2})J_{12}j_{3})JM\rangle=\sum_{M_{12}=-J_{12}}^{J_{12}}\sum_{m_{3}=% -j_{3}}^{j_{3}}|(j_{1}j_{2})J_{12}M_{12}\rangle|j_{3}m_{3}\rangle\langle J_{12% }M_{12}j_{3}m_{3}|JM\rangle
  20. 𝐣 2 \mathbf{j}_{2}
  21. 𝐣 3 \mathbf{j}_{3}
  22. 𝐉 23 \mathbf{J}_{23}
  23. 𝐣 1 \mathbf{j}_{1}
  24. 𝐉 23 \mathbf{J}_{23}
  25. 𝐉 \mathbf{J}
  26. | ( j 1 , ( j 2 j 3 ) J 23 ) J M = m 1 = - j 1 j 1 M 23 = - J 23 J 23 | j 1 m 1 | ( j 2 j 3 ) J 23 M 23 j 1 m 1 J 23 M 23 | J M |(j_{1},(j_{2}j_{3})J_{23})JM\rangle=\sum_{m_{1}=-j_{1}}^{j_{1}}\sum_{M_{23}=-% J_{23}}^{J_{23}}|j_{1}m_{1}\rangle|(j_{2}j_{3})J_{23}M_{23}\rangle\langle j_{1% }m_{1}J_{23}M_{23}|JM\rangle
  27. ( 2 j 1 + 1 ) ( 2 j 2 + 1 ) ( 2 j 3 + 1 ) (2j_{1}+1)(2j_{2}+1)(2j_{3}+1)
  28. | j 1 m 1 | j 2 m 2 | j 3 m 3 , m 1 = - j 1 , , j 1 ; m 2 = - j 2 , , j 2 ; m 3 = - j 3 , , j 3 . |j_{1}m_{1}\rangle|j_{2}m_{2}\rangle|j_{3}m_{3}\rangle,\;\;m_{1}=-j_{1},\ldots% ,j_{1};\;\;m_{2}=-j_{2},\ldots,j_{2};\;\;m_{3}=-j_{3},\ldots,j_{3}.
  29. M M
  30. | ( ( j 1 j 2 ) J 12 j 3 ) J M = J 23 | ( j 1 , ( j 2 j 3 ) J 23 ) J M ( j 1 , ( j 2 j 3 ) J 23 ) J | ( ( j 1 j 2 ) J 12 j 3 ) J . |((j_{1}j_{2})J_{12}j_{3})JM\rangle=\sum_{J_{23}}|(j_{1},(j_{2}j_{3})J_{23})JM% \rangle\langle(j_{1},(j_{2}j_{3})J_{23})J|((j_{1}j_{2})J_{12}j_{3})J\rangle.
  31. M M
  32. M = J M=J
  33. J - J_{-}
  34. Δ ( a , b , c ) = [ ( a + b - c ) ! ( a - b + c ) ! ( - a + b + c ) ! / ( a + b + c + 1 ) ! ] 1 / 2 \Delta(a,b,c)=[(a+b-c)!(a-b+c)!(-a+b+c)!/(a+b+c+1)!]^{1/2}
  35. W ( a b c d ; e f ) = Δ ( a , b , e ) Δ ( c , d , e ) Δ ( a , c , f ) Δ ( b , d , f ) w ( a b c d ; e f ) W(abcd;ef)=\Delta(a,b,e)\Delta(c,d,e)\Delta(a,c,f)\Delta(b,d,f)w(abcd;ef)
  36. w ( a b c d ; e f ) z ( - 1 ) z + β 1 ( z + 1 ) ! ( z - α 1 ) ! ( z - α 2 ) ! ( z - α 3 ) ! ( z - α 4 ) ! ( β 1 - z ) ! ( β 2 - z ) ! ( β 3 - z ) ! w(abcd;ef)\equiv\sum_{z}\frac{(-1)^{z+\beta_{1}}(z+1)!}{(z-\alpha_{1})!(z-% \alpha_{2})!(z-\alpha_{3})!(z-\alpha_{4})!(\beta_{1}-z)!(\beta_{2}-z)!(\beta_{% 3}-z)!}
  37. α 1 = a + b + e ; β 1 = a + b + c + d ; \alpha_{1}=a+b+e;\quad\beta_{1}=a+b+c+d;
  38. α 2 = c + d + e ; β 2 = a + d + e + f ; \alpha_{2}=c+d+e;\quad\beta_{2}=a+d+e+f;
  39. α 3 = a + c + f ; β 3 = b + c + e + f ; \alpha_{3}=a+c+f;\quad\beta_{3}=b+c+e+f;
  40. α 4 = b + d + f . \alpha_{4}=b+d+f.
  41. z z
  42. max ( α 1 , α 2 , α 3 , α 4 ) z min ( β 1 , β 2 , β 3 ) . \max(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4})\leq z\leq\min(\beta_{1},% \beta_{2},\beta_{3}).
  43. W ( a b c d ; e f ) ( - 1 ) a + b + c + d = { a b e d c f } . W(abcd;ef)(-1)^{a+b+c+d}=\begin{Bmatrix}a&b&e\\ d&c&f\end{Bmatrix}.
  44. W ( j 1 j 2 J j 3 ; J 12 J 23 ) = ( - 1 ) j 1 + j 2 + j 3 + J { j 1 j 2 J 12 j 3 J J 23 } . W(j_{1}j_{2}Jj_{3};J_{12}J_{23})=(-1)^{j_{1}+j_{2}+j_{3}+J}\begin{Bmatrix}j_{1% }&j_{2}&J_{12}\\ j_{3}&J&J_{23}\end{Bmatrix}.
  45. r d {}^{rd}

Radar_engineering_details.html

  1. Δ τ \Delta\tau
  2. θ \theta
  3. β \beta
  4. Δ τ \Delta\tau
  5. k d cos θ = β ( f ) = 2 π c λ 0 Δ τ k\,d\,\cos{\theta}=\beta\left(f\right)=2\,\pi\,\frac{c}{\lambda_{0}}\,\Delta\tau
  6. θ = arccos ( c d Δ τ ) \theta=\arccos{\left(\frac{c}{d}\,\Delta\tau\right)}
  7. θ \theta
  8. Σ / Δ \Sigma/\Delta

Radar_signal_characteristics.html

  1. τ \,\tau
  2. R a n g e m a x u n a m b i g u o u s = ( c 2 P R F ) Range_{maxunambiguous}=\left(\frac{c}{2\,PRF}\right)
  3. M U R = ( c * 0.5 * T S P ) MUR=\left(c*0.5*TSP\right)
  4. 1 π f τ \frac{1}{\pi\,f\tau}
  5. τ \,\tau
  6. T π f τ \frac{T}{\pi\,f\tau}
  7. ± c P R F 4 f \pm\frac{c\,PRF}{4\,f}

Radial_basis_function_network.html

  1. x x
  2. 𝐱 n \mathbf{x}\in\mathbb{R}^{n}
  3. φ : n \varphi:\mathbb{R}^{n}\to\mathbb{R}
  4. φ ( 𝐱 ) = i = 1 N a i ρ ( || 𝐱 - 𝐜 i || ) \varphi(\mathbf{x})=\sum_{i=1}^{N}a_{i}\rho(||\mathbf{x}-\mathbf{c}_{i}||)
  5. N N
  6. 𝐜 i \mathbf{c}_{i}
  7. i i
  8. a i a_{i}
  9. i i
  10. ρ ( 𝐱 - 𝐜 i ) = exp [ - β 𝐱 - 𝐜 i 2 ] \rho\big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big)=\exp\left[-\beta\left\|% \mathbf{x}-\mathbf{c}_{i}\right\|^{2}\right]
  11. lim || x || ρ ( 𝐱 - 𝐜 i ) = 0 \lim_{||x||\to\infty}\rho(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|)=0
  12. n \mathbb{R}^{n}
  13. a i a_{i}
  14. 𝐜 i \mathbf{c}_{i}
  15. β i \beta_{i}
  16. φ \varphi
  17. c 1 = 0.75 c_{1}=0.75
  18. c 2 = 3.25 c_{2}=3.25
  19. φ ( 𝐱 ) = def i = 1 N a i ρ ( 𝐱 - 𝐜 i ) i = 1 N ρ ( 𝐱 - 𝐜 i ) = i = 1 N a i u ( 𝐱 - 𝐜 i ) \varphi(\mathbf{x})\ \stackrel{\mathrm{def}}{=}\ \frac{\sum_{i=1}^{N}a_{i}\rho% \big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big)}{\sum_{i=1}^{N}\rho\big(% \left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big)}=\sum_{i=1}^{N}a_{i}u\big(\left% \|\mathbf{x}-\mathbf{c}_{i}\right\|\big)
  20. u ( 𝐱 - 𝐜 i ) = def ρ ( 𝐱 - 𝐜 i ) j = 1 N ρ ( 𝐱 - 𝐜 j ) u\big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big)\ \stackrel{\mathrm{def}}{=% }\ \frac{\rho\big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big)}{\sum_{j=1}^{N% }\rho\big(\left\|\mathbf{x}-\mathbf{c}_{j}\right\|\big)}
  21. P ( 𝐱 y ) = 1 N i = 1 N ρ ( 𝐱 - 𝐜 i ) σ ( | y - e i | ) P\left(\mathbf{x}\land y\right)={1\over N}\sum_{i=1}^{N}\,\rho\big(\left\|% \mathbf{x}-\mathbf{c}_{i}\right\|\big)\,\sigma\big(\left|y-e_{i}\right|\big)
  22. 𝐜 i \mathbf{c}_{i}
  23. e i e_{i}
  24. ρ ( 𝐱 - 𝐜 i ) d n 𝐱 = 1 \int\rho\big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big)\,d^{n}\mathbf{x}=1
  25. σ ( | y - e i | ) d y = 1 \int\sigma\big(\left|y-e_{i}\right|\big)\,dy=1
  26. P ( 𝐱 ) = P ( 𝐱 y ) d y = 1 N i = 1 N ρ ( 𝐱 - 𝐜 i ) P\left(\mathbf{x}\right)=\int P\left(\mathbf{x}\land y\right)\,dy={1\over N}% \sum_{i=1}^{N}\,\rho\big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big)
  27. 𝐱 \mathbf{x}
  28. φ ( 𝐱 ) = def E ( y 𝐱 ) = y P ( y 𝐱 ) d y \varphi\left(\mathbf{x}\right)\ \stackrel{\mathrm{def}}{=}\ E\left(y\mid% \mathbf{x}\right)=\int y\,P\left(y\mid\mathbf{x}\right)dy
  29. P ( y 𝐱 ) P\left(y\mid\mathbf{x}\right)
  30. 𝐱 \mathbf{x}
  31. P ( y 𝐱 ) = P ( 𝐱 y ) P ( 𝐱 ) P\left(y\mid\mathbf{x}\right)=\frac{P\left(\mathbf{x}\land y\right)}{P\left(% \mathbf{x}\right)}
  32. φ ( 𝐱 ) = y P ( 𝐱 y ) P ( 𝐱 ) d y \varphi\left(\mathbf{x}\right)=\int y\,\frac{P\left(\mathbf{x}\land y\right)}{% P\left(\mathbf{x}\right)}\,dy
  33. φ ( 𝐱 ) = i = 1 N e i ρ ( 𝐱 - 𝐜 i ) i = 1 N ρ ( 𝐱 - 𝐜 i ) = i = 1 N e i u ( 𝐱 - 𝐜 i ) \varphi\left(\mathbf{x}\right)=\frac{\sum_{i=1}^{N}e_{i}\rho\big(\left\|% \mathbf{x}-\mathbf{c}_{i}\right\|\big)}{\sum_{i=1}^{N}\rho\big(\left\|\mathbf{% x}-\mathbf{c}_{i}\right\|\big)}=\sum_{i=1}^{N}e_{i}u\big(\left\|\mathbf{x}-% \mathbf{c}_{i}\right\|\big)
  34. φ ( 𝐱 ) = i = 1 N ( a i + 𝐛 i ( 𝐱 - 𝐜 i ) ) ρ ( 𝐱 - 𝐜 i ) \varphi\left(\mathbf{x}\right)=\sum_{i=1}^{N}\left(a_{i}+\mathbf{b}_{i}\cdot% \left(\mathbf{x}-\mathbf{c}_{i}\right)\right)\rho\big(\left\|\mathbf{x}-% \mathbf{c}_{i}\right\|\big)
  35. φ ( 𝐱 ) = i = 1 N ( a i + 𝐛 i ( 𝐱 - 𝐜 i ) ) u ( 𝐱 - 𝐜 i ) \varphi\left(\mathbf{x}\right)=\sum_{i=1}^{N}\left(a_{i}+\mathbf{b}_{i}\cdot% \left(\mathbf{x}-\mathbf{c}_{i}\right)\right)u\big(\left\|\mathbf{x}-\mathbf{c% }_{i}\right\|\big)
  36. 𝐛 i \mathbf{b}_{i}
  37. φ ( 𝐱 ) = i = 1 2 N j = 1 n e i j v i j ( 𝐱 - 𝐜 i ) \varphi\left(\mathbf{x}\right)=\sum_{i=1}^{2N}\sum_{j=1}^{n}e_{ij}v_{ij}\big(% \mathbf{x}-\mathbf{c}_{i}\big)
  38. e i j = { a i , if i [ 1 , N ] b i j , if i [ N + 1 , 2 N ] e_{ij}=\begin{cases}a_{i},&\mbox{if }~{}i\in[1,N]\\ b_{ij},&\mbox{if }~{}i\in[N+1,2N]\end{cases}
  39. v i j ( 𝐱 - 𝐜 i ) = def { δ i j ρ ( 𝐱 - 𝐜 i ) , if i [ 1 , N ] ( x i j - c i j ) ρ ( 𝐱 - 𝐜 i ) , if i [ N + 1 , 2 N ] v_{ij}\big(\mathbf{x}-\mathbf{c}_{i}\big)\ \stackrel{\mathrm{def}}{=}\ \begin{% cases}\delta_{ij}\rho\big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big),&\mbox% {if }~{}i\in[1,N]\\ \left(x_{ij}-c_{ij}\right)\rho\big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|% \big),&\mbox{if }~{}i\in[N+1,2N]\end{cases}
  40. v i j ( 𝐱 - 𝐜 i ) = def { δ i j u ( 𝐱 - 𝐜 i ) , if i [ 1 , N ] ( x i j - c i j ) u ( 𝐱 - 𝐜 i ) , if i [ N + 1 , 2 N ] v_{ij}\big(\mathbf{x}-\mathbf{c}_{i}\big)\ \stackrel{\mathrm{def}}{=}\ \begin{% cases}\delta_{ij}u\big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big),&\mbox{if% }~{}i\in[1,N]\\ \left(x_{ij}-c_{ij}\right)u\big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big),% &\mbox{if }~{}i\in[N+1,2N]\end{cases}
  41. δ i j \delta_{ij}
  42. δ i j = { 1 , if i = j 0 , if i j \delta_{ij}=\begin{cases}1,&\mbox{if }~{}i=j\\ 0,&\mbox{if }~{}i\neq j\end{cases}
  43. 𝐜 i \mathbf{c}_{i}
  44. w i w_{i}
  45. K ( 𝐰 ) = def t = 1 K t ( 𝐰 ) K(\mathbf{w})\ \stackrel{\mathrm{def}}{=}\ \sum_{t=1}^{\infty}K_{t}(\mathbf{w})
  46. K t ( 𝐰 ) = def [ y ( t ) - φ ( 𝐱 ( t ) , 𝐰 ) ] 2 K_{t}(\mathbf{w})\ \stackrel{\mathrm{def}}{=}\ \big[y(t)-\varphi\big(\mathbf{x% }(t),\mathbf{w}\big)\big]^{2}
  47. H ( 𝐰 ) = def K ( 𝐰 ) + λ S ( 𝐰 ) = def t = 1 H t ( 𝐰 ) H(\mathbf{w})\ \stackrel{\mathrm{def}}{=}\ K(\mathbf{w})+\lambda S(\mathbf{w})% \ \stackrel{\mathrm{def}}{=}\ \sum_{t=1}^{\infty}H_{t}(\mathbf{w})
  48. S ( 𝐰 ) = def t = 1 S t ( 𝐰 ) S(\mathbf{w})\ \stackrel{\mathrm{def}}{=}\ \sum_{t=1}^{\infty}S_{t}(\mathbf{w})
  49. H t ( 𝐰 ) = def K t ( 𝐰 ) + λ S t ( 𝐰 ) H_{t}(\mathbf{w})\ \stackrel{\mathrm{def}}{=}\ K_{t}(\mathbf{w})+\lambda S_{t}% (\mathbf{w})
  50. λ \lambda
  51. y : n y:\mathbb{R}^{n}\to\mathbb{R}
  52. y ( 𝐱 i ) = b i , i = 1 , , N y(\mathbf{x}_{i})=b_{i},i=1,\ldots,N
  53. 𝐱 i \mathbf{x}_{i}
  54. g i j = ρ ( || 𝐱 j - 𝐱 i || ) g_{ij}=\rho(||\mathbf{x}_{j}-\mathbf{x}_{i}||)
  55. [ g 11 g 12 g 1 N g 21 g 22 g 2 N g N 1 g N 2 g N N ] [ w 1 w 2 w N ] = [ b 1 b 2 b N ] \left[\begin{matrix}g_{11}&g_{12}&\cdots&g_{1N}\\ g_{21}&g_{22}&\cdots&g_{2N}\\ \vdots&&\ddots&\vdots\\ g_{N1}&g_{N2}&\cdots&g_{NN}\end{matrix}\right]\left[\begin{matrix}w_{1}\\ w_{2}\\ \vdots\\ w_{N}\end{matrix}\right]=\left[\begin{matrix}b_{1}\\ b_{2}\\ \vdots\\ b_{N}\end{matrix}\right]
  56. 𝐱 i \mathbf{x}_{i}
  57. w w
  58. 𝐰 = 𝐆 - 1 𝐛 \mathbf{w}=\mathbf{G}^{-1}\mathbf{b}
  59. c i c_{i}
  60. 𝐰 = 𝐆 + 𝐛 \mathbf{w}=\mathbf{G}^{+}\mathbf{b}
  61. x i x_{i}
  62. g j i = ρ ( || x j - c i || ) g_{ji}=\rho(||x_{j}-c_{i}||)
  63. 𝐰 ( t + 1 ) = 𝐰 ( t ) - ν d d 𝐰 H t ( 𝐰 ) \mathbf{w}(t+1)=\mathbf{w}(t)-\nu\frac{d}{d\mathbf{w}}H_{t}(\mathbf{w})
  64. ν \nu
  65. a i a_{i}
  66. a i ( t + 1 ) = a i ( t ) + ν [ y ( t ) - φ ( 𝐱 ( t ) , 𝐰 ) ] ρ ( 𝐱 ( t ) - 𝐜 i ) a_{i}(t+1)=a_{i}(t)+\nu\big[y(t)-\varphi\big(\mathbf{x}(t),\mathbf{w}\big)\big% ]\rho\big(\left\|\mathbf{x}(t)-\mathbf{c}_{i}\right\|\big)
  67. a i ( t + 1 ) = a i ( t ) + ν [ y ( t ) - φ ( 𝐱 ( t ) , 𝐰 ) ] u ( 𝐱 ( t ) - 𝐜 i ) a_{i}(t+1)=a_{i}(t)+\nu\big[y(t)-\varphi\big(\mathbf{x}(t),\mathbf{w}\big)\big% ]u\big(\left\|\mathbf{x}(t)-\mathbf{c}_{i}\right\|\big)
  68. e i j ( t + 1 ) = e i j ( t ) + ν [ y ( t ) - φ ( 𝐱 ( t ) , 𝐰 ) ] v i j ( 𝐱 ( t ) - 𝐜 i ) e_{ij}(t+1)=e_{ij}(t)+\nu\big[y(t)-\varphi\big(\mathbf{x}(t),\mathbf{w}\big)% \big]v_{ij}\big(\mathbf{x}(t)-\mathbf{c}_{i}\big)
  69. a i a_{i}
  70. e i j e_{ij}
  71. a i ( t + 1 ) = a i ( t ) + ν [ y ( t ) - φ ( 𝐱 ( t ) , 𝐰 ) ] ρ ( 𝐱 ( t ) - 𝐜 i ) i = 1 N ρ 2 ( 𝐱 ( t ) - 𝐜 i ) a_{i}(t+1)=a_{i}(t)+\nu\big[y(t)-\varphi\big(\mathbf{x}(t),\mathbf{w}\big)\big% ]\frac{\rho\big(\left\|\mathbf{x}(t)-\mathbf{c}_{i}\right\|\big)}{\sum_{i=1}^{% N}\rho^{2}\big(\left\|\mathbf{x}(t)-\mathbf{c}_{i}\right\|\big)}
  72. a i ( t + 1 ) = a i ( t ) + ν [ y ( t ) - φ ( 𝐱 ( t ) , 𝐰 ) ] u ( 𝐱 ( t ) - 𝐜 i ) i = 1 N u 2 ( 𝐱 ( t ) - 𝐜 i ) a_{i}(t+1)=a_{i}(t)+\nu\big[y(t)-\varphi\big(\mathbf{x}(t),\mathbf{w}\big)\big% ]\frac{u\big(\left\|\mathbf{x}(t)-\mathbf{c}_{i}\right\|\big)}{\sum_{i=1}^{N}u% ^{2}\big(\left\|\mathbf{x}(t)-\mathbf{c}_{i}\right\|\big)}
  73. e i j ( t + 1 ) = e i j ( t ) + ν [ y ( t ) - φ ( 𝐱 ( t ) , 𝐰 ) ] v i j ( 𝐱 ( t ) - 𝐜 i ) i = 1 N j = 1 n v i j 2 ( 𝐱 ( t ) - 𝐜 i ) e_{ij}(t+1)=e_{ij}(t)+\nu\big[y(t)-\varphi\big(\mathbf{x}(t),\mathbf{w}\big)% \big]\frac{v_{ij}\big(\mathbf{x}(t)-\mathbf{c}_{i}\big)}{\sum_{i=1}^{N}\sum_{j% =1}^{n}v_{ij}^{2}\big(\mathbf{x}(t)-\mathbf{c}_{i}\big)}
  74. x ( t + 1 ) = def f [ x ( t ) ] = 4 x ( t ) [ 1 - x ( t ) ] x(t+1)\ \stackrel{\mathrm{def}}{=}\ f\left[x(t)\right]=4x(t)\left[1-x(t)\right]
  75. x ( t + 1 ) = f [ x ( t ) ] φ ( t ) = φ [ x ( t ) ] x(t+1)=f\left[x(t)\right]\approx\varphi(t)=\varphi\left[x(t)\right]
  76. φ ( 𝐱 ) = def i = 1 N a i ρ ( 𝐱 - 𝐜 i ) \varphi(\mathbf{x})\ \stackrel{\mathrm{def}}{=}\ \sum_{i=1}^{N}a_{i}\rho\big(% \left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big)
  77. ρ ( 𝐱 - 𝐜 i ) = exp [ - β 𝐱 - 𝐜 i 2 ] = exp [ - β ( x ( t ) - c i ) 2 ] \rho\big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big)=\exp\left[-\beta\left\|% \mathbf{x}-\mathbf{c}_{i}\right\|^{2}\right]=\exp\left[-\beta\left(x(t)-c_{i}% \right)^{2}\right]
  78. β \beta
  79. c i c_{i}
  80. a i a_{i}
  81. a i ( t + 1 ) = a i ( t ) + ν [ x ( t + 1 ) - φ ( 𝐱 ( t ) , 𝐰 ) ] ρ ( 𝐱 ( t ) - 𝐜 i ) i = 1 N ρ 2 ( 𝐱 ( t ) - 𝐜 i ) a_{i}(t+1)=a_{i}(t)+\nu\big[x(t+1)-\varphi\big(\mathbf{x}(t),\mathbf{w}\big)% \big]\frac{\rho\big(\left\|\mathbf{x}(t)-\mathbf{c}_{i}\right\|\big)}{\sum_{i=% 1}^{N}\rho^{2}\big(\left\|\mathbf{x}(t)-\mathbf{c}_{i}\right\|\big)}
  82. ν \nu
  83. φ ( 𝐱 ) = def i = 1 N a i ρ ( 𝐱 - 𝐜 i ) i = 1 N ρ ( 𝐱 - 𝐜 i ) = i = 1 N a i u ( 𝐱 - 𝐜 i ) \varphi(\mathbf{x})\ \stackrel{\mathrm{def}}{=}\ \frac{\sum_{i=1}^{N}a_{i}\rho% \big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big)}{\sum_{i=1}^{N}\rho\big(% \left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big)}=\sum_{i=1}^{N}a_{i}u\big(\left% \|\mathbf{x}-\mathbf{c}_{i}\right\|\big)
  84. u ( 𝐱 - 𝐜 i ) = def ρ ( 𝐱 - 𝐜 i ) i = 1 N ρ ( 𝐱 - 𝐜 i ) u\big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big)\ \stackrel{\mathrm{def}}{=% }\ \frac{\rho\big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big)}{\sum_{i=1}^{N% }\rho\big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big)}
  85. ρ ( 𝐱 - 𝐜 i ) = exp [ - β 𝐱 - 𝐜 i 2 ] = exp [ - β ( x ( t ) - c i ) 2 ] \rho\big(\left\|\mathbf{x}-\mathbf{c}_{i}\right\|\big)=\exp\left[-\beta\left\|% \mathbf{x}-\mathbf{c}_{i}\right\|^{2}\right]=\exp\left[-\beta\left(x(t)-c_{i}% \right)^{2}\right]
  86. β \beta
  87. c i c_{i}
  88. a i a_{i}
  89. a i ( t + 1 ) = a i ( t ) + ν [ x ( t + 1 ) - φ ( 𝐱 ( t ) , 𝐰 ) ] u ( 𝐱 ( t ) - 𝐜 i ) i = 1 N u 2 ( 𝐱 ( t ) - 𝐜 i ) a_{i}(t+1)=a_{i}(t)+\nu\big[x(t+1)-\varphi\big(\mathbf{x}(t),\mathbf{w}\big)% \big]\frac{u\big(\left\|\mathbf{x}(t)-\mathbf{c}_{i}\right\|\big)}{\sum_{i=1}^% {N}u^{2}\big(\left\|\mathbf{x}(t)-\mathbf{c}_{i}\right\|\big)}
  90. ν \nu
  91. φ ( 0 ) = x ( 1 ) \varphi(0)=x(1)
  92. x ( t ) φ ( t - 1 ) {x}(t)\approx\varphi(t-1)
  93. x ( t + 1 ) φ ( t ) = φ [ φ ( t - 1 ) ] {x}(t+1)\approx\varphi(t)=\varphi[\varphi(t-1)]
  94. c [ x ( t ) , t ] c[x(t),t]
  95. x ( t + 1 ) = 4 x ( t ) [ 1 - x ( t ) ] + c [ x ( t ) , t ] {x}(t+1)=4x(t)[1-x(t)]+c[x(t),t]
  96. d ( t ) d(t)
  97. c [ x ( t ) , t ] = def - φ [ x ( t ) ] + d ( t + 1 ) c[x(t),t]\ \stackrel{\mathrm{def}}{=}\ -\varphi[x(t)]+d(t+1)
  98. y [ x ( t ) ] f [ x ( t ) ] = x ( t + 1 ) - c [ x ( t ) , t ] y[x(t)]\approx f[x(t)]=x(t+1)-c[x(t),t]
  99. a i ( t + 1 ) = a i ( t ) + ν ε u ( 𝐱 ( t ) - 𝐜 i ) i = 1 N u 2 ( 𝐱 ( t ) - 𝐜 i ) a_{i}(t+1)=a_{i}(t)+\nu\varepsilon\frac{u\big(\left\|\mathbf{x}(t)-\mathbf{c}_% {i}\right\|\big)}{\sum_{i=1}^{N}u^{2}\big(\left\|\mathbf{x}(t)-\mathbf{c}_{i}% \right\|\big)}
  100. ε = def f [ x ( t ) ] - φ [ x ( t ) ] = x ( t + 1 ) - c [ x ( t ) , t ] - φ [ x ( t ) ] = x ( t + 1 ) - d ( t + 1 ) \varepsilon\ \stackrel{\mathrm{def}}{=}\ f[x(t)]-\varphi[x(t)]=x(t+1)-c[x(t),t% ]-\varphi[x(t)]=x(t+1)-d(t+1)

Radial_turbine.html

  1. tan β 2 = c r 2 ( c θ 2 - u 2 ) \,\tan{\beta_{2}}=\frac{c_{r2}}{(c_{\theta 2}-u_{2})}
  2. C 0 = 2 C p T 01 ( 1 - ( p 3 p 01 ) γ - 1 γ ) \,C_{0}=\sqrt{2C_{p}\,T_{01}\,(1-(\frac{p_{3}}{p_{01}})^{\frac{\gamma-1}{% \gamma}})}
  3. η t s = ( h 01 - h 03 ) ( h 01 - h 3 s s ) \,\eta_{t}s\,=\,\frac{(h_{01}-h_{03})}{(h_{01}-h_{3ss})}
  4. η t s = ψ u 2 2 [ C p T 01 ( 1 - ( p 3 p 01 ) ( γ - 1 ) γ ] \,\eta_{t}s\,=\,\frac{\psi\,u_{2}^{2}}{[C_{p}\,T_{01}(1-(\frac{p_{3}}{p_{01}})% ^{\frac{(\gamma-1)}{\gamma}}]}
  5. s t a t i c e n t h a p l y d r o p i n r o t o r ′′ s t a g n a t i o n e n t h a l p y d r o p i n s t a g e ′′ \frac{\textrm{'}^{\prime}{static\,\,enthaply\,\,drop\,\,in\,\,rotor}^{\prime% \prime}}{\textrm{'}^{\prime}{stagnation\,\,enthalpy\,\,drop\,\,in\,\,stage}^{% \prime\prime}}
  6. σ s = u 2 c 0 = [ 2 ( 1 + ϕ 2 cot β 2 ] - 1 2 \,\sigma_{s}=\frac{u_{2}}{c_{0}}=[2(1+\phi_{2}\cot{\beta_{2}}]^{-1\over{2}}

Radical_of_a_Lie_algebra.html

  1. 𝔤 \mathfrak{g}
  2. 𝔤 . \mathfrak{g}.
  3. k k
  4. 𝔤 \mathfrak{g}
  5. k k
  6. 𝔞 \mathfrak{a}
  7. 𝔟 \mathfrak{b}
  8. 𝔤 \mathfrak{g}
  9. 𝔞 + 𝔟 \mathfrak{a}+\mathfrak{b}
  10. 𝔤 \mathfrak{g}
  11. ( 𝔞 + 𝔟 ) / 𝔞 𝔟 / ( 𝔞 𝔟 ) (\mathfrak{a}+\mathfrak{b})/\mathfrak{a}\simeq\mathfrak{b}/(\mathfrak{a}\cap% \mathfrak{b})
  12. 𝔞 \mathfrak{a}
  13. 𝔤 \mathfrak{g}
  14. 𝔤 \mathfrak{g}
  15. 𝔤 \mathfrak{g}
  16. { 0 } \{0\}
  17. 𝔤 \mathfrak{g}
  18. 𝔤 \mathfrak{g}
  19. 0

Radical_of_a_module.html

  1. rad ( M ) = { N N is a maximal submodule of M } \mathrm{rad}(M)=\bigcap\{N\mid N\mbox{ is a maximal submodule of M}~{}\}\,
  2. rad ( M ) = { S S is a superfluous submodule of M } \mathrm{rad}(M)=\sum\{S\mid S\mbox{ is a superfluous submodule of M}~{}\}\,

Radio_frequency_microelectromechanical_system.html

  1. R = λ 2 EIRP G R / T σ 64 π 3 k B BW SNR 4 {\mathrm{R=\sqrt[4]{\frac{\displaystyle{\mathrm{\lambda^{2}\,EIRP\,G_{R}/T\,% \sigma}}}{{\mathrm{\displaystyle 64\,\pi^{3}\,k_{B}\,BW\,SNR}}}}}}

Radiosity_(radiometry).html

  1. J e = Φ e A = J e , em + J e , r + J e , tr , J_{\mathrm{e}}=\frac{\partial\Phi_{\mathrm{e}}}{\partial A}=J_{\mathrm{e,em}}+% J_{\mathrm{e,r}}+J_{\mathrm{e,tr}},
  2. J e = M e + J e , r . J_{\mathrm{e}}=M_{\mathrm{e}}+J_{\mathrm{e,r}}.
  3. J e , ν = J e ν , J_{\mathrm{e},\nu}=\frac{\partial J_{\mathrm{e}}}{\partial\nu},
  4. J e , λ = J e λ , J_{\mathrm{e},\lambda}=\frac{\partial J_{\mathrm{e}}}{\partial\lambda},
  5. J e = M e + J e , r = ε σ T 4 + ( 1 - ε ) E e , J_{\mathrm{e}}=M_{\mathrm{e}}+J_{\mathrm{e,r}}=\varepsilon\sigma T^{4}+(1-% \varepsilon)E_{\mathrm{e}},
  6. E e , i = j = 1 N F j i A j J e , j A i . E_{\mathrm{e},i}=\frac{\sum_{j=1}^{N}F_{ji}A_{j}J_{\mathrm{e},j}}{A_{i}}.
  7. E e , i = j = 1 N F i j J e , j , E_{\mathrm{e},i}=\sum_{j=1}^{N}F_{ij}J_{\mathrm{e},j},
  8. J e , i = ε i σ T i 4 + ( 1 - ε i ) j = 1 N F i j J e , j . J_{\mathrm{e},i}=\varepsilon_{i}\sigma T_{i}^{4}+(1-\varepsilon_{i})\sum_{j=1}% ^{N}F_{ij}J_{\mathrm{e},j}.
  9. Q ˙ i = A i ( J e , i - E e , i ) . \dot{Q}_{i}=A_{i}(J_{\mathrm{e},i}-E_{\mathrm{e},i}).
  10. Q ˙ i = A i ε i 1 - ε i ( σ T i 4 - J e , i ) = A i ε i 1 - ε i ( M e , i - J e , i ) , \dot{Q}_{i}=\frac{A_{i}\varepsilon_{i}}{1-\varepsilon_{i}}(\sigma T_{i}^{4}-J_% {\mathrm{e},i})=\frac{A_{i}\varepsilon_{i}}{1-\varepsilon_{i}}(M_{\mathrm{e},i% }^{\circ}-J_{\mathrm{e},i}),
  11. Q i ˙ = M e , i - J e , i R i , \dot{Q_{i}}=\frac{M_{\mathrm{e},i}^{\circ}-J_{\mathrm{e},i}}{R_{i}},
  12. Q ˙ i j = A i F i j ( J e , i - J e , j ) = J e , i - J e , j R i j , \dot{Q}_{ij}=A_{i}F_{ij}(J_{\mathrm{e},i}-J_{\mathrm{e},j})=\frac{J_{\mathrm{e% },i}-J_{\mathrm{e},j}}{R_{ij}},

Ragsdale_conjecture.html

  1. p 3 2 k ( k - 1 ) + 1 and n 3 2 k ( k - 1 ) . p\leq\tfrac{3}{2}k(k-1)+1\quad\,\text{and}\quad n\leq\tfrac{3}{2}k(k-1).
  2. | 2 ( p - n ) - 1 | 3 k 2 - 3 k + 1 , |2(p-n)-1|\leq 3k^{2}-3k+1,

Raising_and_lowering_indices.html

  1. g i j A j = B i , g^{ij}A_{j}=B^{i},
  2. g i j A j = A i . g^{ij}A_{j}=A^{i}.
  3. g i j A j = A i . g_{ij}A^{j}=A_{i}.
  4. g i j g j k = g k j g j i = δ i = k δ k i g^{ij}g_{jk}=g_{kj}g^{ji}=\delta^{i}{}_{k}=\delta_{k}{}^{i}
  5. g i j A j = \cancel g i \cancel j A \cancel j = A i , g^{ij}A_{j}=\cancel{g}^{i\cancel{j}}A_{\cancel}{j}=A^{i}\,,
  6. X μ = ( c t , - x , - y , - z ) X_{\mu}=(ct,-x,-y,-z)
  7. X 0 = c t , X j = - x j X_{0}=ct,\quad X_{j}=-x_{j}
  8. η μ ν = η μ ν = ( 1 0 0 0 0 - 1 0 0 0 0 - 1 0 0 0 0 - 1 ) \eta_{\mu\nu}=\eta^{\mu\nu}=\begin{pmatrix}1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{pmatrix}
  9. η 00 = 1 , η i 0 = η 0 i = 0 , η i j = - δ i j . \eta_{00}=1,\quad\eta_{i0}=\eta_{0i}=0,\quad\eta_{ij}=-\delta_{ij}\,.
  10. X λ = η λ μ X μ = η λ 0 X 0 + η λ i X i X^{\lambda}=\eta^{\lambda\mu}X_{\mu}=\eta^{\lambda 0}X_{0}+\eta^{\lambda i}X_{i}
  11. X 0 = η 00 X 0 + η 0 i X i = X 0 X^{0}=\eta^{00}X_{0}+\eta^{0i}X_{i}=X_{0}
  12. X j = η j 0 X 0 + η j i X i = - δ j i X i = - X j X^{j}=\eta^{j0}X_{0}+\eta^{ji}X_{i}=-\delta^{ji}X_{i}=-X_{j}
  13. X μ = ( c t , x , y , z ) . X^{\mu}=(ct,x,y,z)\,.
  14. A α β = g α γ g β δ A γ δ A^{\alpha\beta}=g^{\alpha\gamma}g^{\beta\delta}A_{\gamma\delta}
  15. A α β = g α γ g β δ A γ δ A_{\alpha\beta}=g_{\alpha\gamma}g_{\beta\delta}A^{\gamma\delta}
  16. F α β = ( 0 - E x / c - E y / c - E z / c E x / c 0 - B z B y E y / c B z 0 - B x E z / c - B y B x 0 ) F^{\alpha\beta}=\begin{pmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\ E_{x}/c&0&-B_{z}&B_{y}\\ E_{y}/c&B_{z}&0&-B_{x}\\ E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}
  17. F 0 i = - F i 0 = - E i / c , F i j = - ε i j k B k F^{0i}=-F^{i0}=-E^{i}/c,\quad F^{ij}=-\varepsilon^{ijk}B_{k}
  18. F α β = η α γ η β δ F γ δ = η α 0 η β 0 F 00 + η α i η β 0 F i 0 + η α 0 η β i F 0 i + η α i η β j F i j \begin{aligned}\displaystyle F_{\alpha\beta}&\displaystyle=\eta_{\alpha\gamma}% \eta_{\beta\delta}F^{\gamma\delta}\\ &\displaystyle=\eta_{\alpha 0}\eta_{\beta 0}F^{00}+\eta_{\alpha i}\eta_{\beta 0% }F^{i0}+\eta_{\alpha 0}\eta_{\beta i}F^{0i}+\eta_{\alpha i}\eta_{\beta j}F^{ij% }\end{aligned}\,
  19. F α β = ( η α i η β 0 - η α 0 η β i ) F i 0 + η α i η β j F i j F_{\alpha\beta}=(\eta_{\alpha i}\eta_{\beta 0}-\eta_{\alpha 0}\eta_{\beta i})F% ^{i0}+\eta_{\alpha i}\eta_{\beta j}F^{ij}\,
  20. F 0 k \displaystyle F_{0k}
  21. F 0 k = - F k 0 F_{0k}=-F_{k0}
  22. F k = ( η k i η 0 - η k 0 η i ) F i 0 + η k i η j F i j = 0 + δ k i δ j F i j = F k \begin{aligned}\displaystyle F_{k\ell}&\displaystyle=(\eta_{ki}\eta_{\ell 0}-% \eta_{k0}\eta_{\ell i})F^{i0}+\eta_{ki}\eta_{\ell j}F^{ij}\\ &\displaystyle=0+\delta_{ki}\delta_{\ell j}F^{ij}\\ &\displaystyle=F^{k\ell}\\ \end{aligned}
  23. F α β = ( 0 E x / c E y / c E z / c - E x / c 0 - B z B y - E y / c B z 0 - B x - E z / c - B y B x 0 ) F_{\alpha\beta}=\begin{pmatrix}0&E_{x}/c&E_{y}/c&E_{z}/c\\ -E_{x}/c&0&-B_{z}&B_{y}\\ -E_{y}/c&B_{z}&0&-B_{x}\\ -E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}
  24. g i 1 j 1 g i 2 j 2 g i n j n A i 1 i 2 i n = A j 1 j 2 j n g^{i_{1}j_{1}}g^{i_{2}j_{2}}\cdots g^{i_{n}j_{n}}A_{i_{1}i_{2}\cdots i_{n}}=A^% {j_{1}j_{2}\cdots j_{n}}
  25. g i 1 j 1 g i 2 j 2 g i n j n A i 1 i 2 i n = A j 1 j 2 j n g_{i_{1}j_{1}}g_{i_{2}j_{2}}\cdots g_{i_{n}j_{n}}A^{i_{1}i_{2}\cdots i_{n}}=A_% {j_{1}j_{2}\cdots j_{n}}
  26. g i 1 p 1 g i 2 p 2 g i n p n g j 1 q 1 g j 2 q 2 g j m q m A i 1 i 2 i n = j 1 j 2 j m A p 1 p 2 p n q 1 q 2 q m g_{i_{1}p_{1}}g_{i_{2}p_{2}}\cdots g_{i_{n}p_{n}}g^{j_{1}q_{1}}g^{j_{2}q_{2}}% \cdots g^{j_{m}q_{m}}A^{i_{1}i_{2}\cdots i_{n}}{}_{j_{1}j_{2}\cdots j_{m}}=A_{% p_{1}p_{2}\cdots p_{n}}{}^{q_{1}q_{2}\cdots q_{m}}

Ramanujan_summation.html

  1. 1 2 f ( 0 ) + f ( 1 ) + + f ( n - 1 ) + 1 2 f ( n ) \displaystyle\frac{1}{2}f\left(0\right)+f\left(1\right)+\cdots+f\left(n-1% \right)+\frac{1}{2}f\left(n\right)
  2. k = 1 x f ( k ) = C + 0 x f ( t ) d t + 1 2 f ( x ) + k = 1 B 2 k ( 2 k ) ! f ( 2 k - 1 ) ( x ) \sum_{k=1}^{x}f(k)=C+\int_{0}^{x}f(t)\,dt+\frac{1}{2}f(x)+\sum_{k=1}^{\infty}% \frac{B_{2k}}{(2k)!}f^{(2k-1)}(x)
  3. C ( a ) = 0 a f ( t ) d t - 1 2 f ( 0 ) - k = 1 B 2 k ( 2 k ) ! f ( 2 k - 1 ) ( 0 ) C(a)=\int_{0}^{a}f(t)\,dt-\frac{1}{2}f(0)-\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k% )!}f^{(2k-1)}(0)
  4. a = 0 \scriptstyle a\,=\,0
  5. a = \scriptstyle a\,=\,\infty
  6. C ( a ) = 1 a f ( t ) d t + 1 2 f ( 1 ) - k = 1 B 2 k ( 2 k ) ! f ( 2 k - 1 ) ( 1 ) C(a)=\int_{1}^{a}f(t)\,dt+\frac{1}{2}f(1)-\sum_{k=1}^{\infty}\frac{B_{2k}}{(2k% )!}f^{(2k-1)}(1)
  7. ( ) \scriptstyle(\Re)
  8. ( ) \scriptstyle(\Re)
  9. 1 - 1 + 1 - = 1 2 ( ) 1-1+1-\cdots=\frac{1}{2}\ (\Re)
  10. ( ) \scriptstyle(\Re)
  11. ( ) \scriptstyle(\Re)
  12. 1 + 2 + 3 + = - 1 12 ( ) 1+2+3+\cdots=-\frac{1}{12}\ (\Re)
  13. 1 + 2 2 k + 3 2 k + = 0 ( ) 1+2^{2k}+3^{2k}+\cdots=0\ (\Re)
  14. 1 + 2 2 k - 1 + 3 2 k - 1 + = - B 2 k 2 k ( ) 1+2^{2k-1}+3^{2k-1}+\cdots=-\frac{B_{2k}}{2k}\ (\Re)
  15. k = 1 f ( k ) \scriptstyle\sum_{k=1}^{\infty}f(k)
  16. R ( x ) - R ( x + 1 ) = f ( x ) \scriptstyle R(x)\,-\,R(x\,+\,1)\,=\,f(x)
  17. 1 2 R ( t ) d t = 0 \scriptstyle\int_{1}^{2}R(t)\,dt\,=\,0
  18. n 1 f ( n ) \scriptstyle\sum_{n\geq 1}^{\Re}f(n)
  19. n 1 f ( n ) \scriptstyle\sum_{n\geq 1}^{\Re}f(n)
  20. n 1 f ( n ) = lim N [ n = 1 N f ( n ) - 1 N f ( t ) d t ] \sum_{n\geq 1}^{\Re}f(n)=\lim_{N\to\infty}\left[\sum_{n=1}^{N}f(n)-\int_{1}^{N% }f(t)\,dt\right]
  21. n 1 1 n = γ \sum_{n\geq 1}^{\Re}\frac{1}{n}=\gamma
  22. a x m - s d x = m - s 2 a x m - 1 - s d x + ζ ( s - m ) - i = 1 a i m - s + a m - s - r = 1 B 2 r Γ ( m - s + 1 ) ( 2 r ) ! Γ ( m - 2 r + 2 - s ) ( m - 2 r + 1 - s ) a x m - 2 r - s d x \begin{array}[]{l}\int\nolimits_{a}^{\infty}x^{m-s}dx=\frac{m-s}{2}\int% \nolimits_{a}^{\infty}x^{m-1-s}dx+\zeta(s-m)-\sum\limits_{i=1}^{a}i^{m-s}+a^{m% -s}\\ -\sum\limits_{r=1}^{\infty}\frac{B_{2r}\Gamma(m-s+1)}{(2r)!\Gamma(m-2r+2-s)}(m% -2r+1-s)\int\nolimits_{a}^{\infty}x^{m-2r-s}dx\end{array}
  23. m - 2 r < - 1 m-2r<-1
  24. a d x x m - 2 r = - a m - 2 r + 1 m - 2 r + 1 \qquad\int_{a}^{\infty}dxx^{m-2r}=-\frac{a^{m-2r+1}}{m-2r+1}
  25. I ( n , Λ ) = 0 Λ d x x n I(n,\,\Lambda)\,=\,\int_{0}^{\Lambda}dxx^{n}
  26. Λ \Lambda\rightarrow\infty

Ramp_travel_index.html

  1. r = d b × 1000 r=\frac{d}{b}\times 1000
  2. sin 20 = h d \sin 20^{\circ}=\frac{h}{d}
  3. d = h sin 20 d=\frac{h}{\sin 20^{\circ}}
  4. r = h b × 1000 sin 20 r=\frac{h}{b}\times\frac{1000}{\sin 20^{\circ}}
  5. R T I 20 = h b × 2924 RTI_{20}=\frac{h}{b}\times 2924

Random_close_pack.html

  1. K = ϵ 3 36 k ( 1 - ϵ ) 2 d 2 K=\frac{\epsilon^{3}}{36k(1-\epsilon)^{2}}d^{2}
  2. ϵ \epsilon
  3. K = ϵ 5.5 5.6 d 2 K=\frac{\epsilon^{5.5}}{5.6}d^{2}

Random_measure.html

  1. 𝔅 ( X ) \mathfrak{B}(X)
  2. 𝔅 ( X ) \mathfrak{B}(X)
  3. 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}},
  4. μ d \mu_{d}
  5. μ a \mu_{a}
  6. μ = n = 1 N δ X n , \mu=\sum_{n=1}^{N}\delta_{X_{n}},
  7. δ \delta
  8. X n X_{n}
  9. X n X_{n}
  10. μ d \mu_{d}
  11. N X N_{X}
  12. N M X N\in M_{X}

Random_regular_graph.html

  1. 𝒢 n , r \mathcal{G}_{n,r}
  2. r 3 r\geq 3
  3. ϵ \epsilon
  4. ( r - 1 ) d - 1 ( 2 + ϵ ) r n ln n (r-1)^{d-1}\geq(2+\epsilon)rn\ln n
  5. λ i = ( r - 1 ) i 2 i \lambda_{i}=\frac{(r-1)^{i}}{2i}

Range_of_a_projectile.html

  1. d = v cos θ g ( v sin θ + v 2 sin 2 θ + 2 g y 0 ) d=\frac{v\cos\theta}{g}\left(v\sin\theta+\sqrt{v^{2}\sin^{2}\theta+2gy_{0}}\right)
  2. d = v 2 g sin ( 2 θ ) d=\frac{v^{2}}{g}\sin(2\theta)
  3. R = v 2 sin 2 θ g R=\frac{v^{2}\sin 2\theta}{g}
  4. sin 2 θ \sin 2\theta
  5. 2 θ 2\theta
  6. θ \theta
  7. x ( t ) = v t cos θ x(t)=vt\cos\theta
  8. y ( t ) = v t sin θ - 1 2 g t 2 y(t)=vt\sin\theta-\frac{1}{2}gt^{2}
  9. 0 = v t sin θ - 1 2 g t 2 0=vt\sin\theta-\frac{1}{2}gt^{2}
  10. t = 0 t=0
  11. t = 2 v sin θ g t=\frac{2v\sin\theta}{g}
  12. T = 2 v sin θ g T=\frac{2v\sin\theta}{g}
  13. x = 2 v 2 cos θ sin θ g x=\frac{2v^{2}\cos\theta\,\sin\theta}{g}
  14. sin ( x + y ) = sin x cos y + sin y cos x \sin(x+y)=\sin x\,\cos y\ +\ \sin y\,\cos x
  15. sin 2 θ = 2 sin θ cos θ \sin 2\theta=2\sin\theta\,\cos\theta
  16. R = v 2 sin 2 θ g R=\frac{v^{2}\sin 2\theta}{g}
  17. R max = v 2 g R_{\max}=\frac{v^{2}}{g}
  18. x ( t ) = v t cos θ x(t)=vt\cos\theta
  19. y ( t ) = y 0 + v t sin θ - 1 2 g t 2 y(t)=y_{0}+vt\sin\theta-\frac{1}{2}gt^{2}
  20. 0 = y 0 + v t sin θ - 1 2 g t 2 0=y_{0}+vt\sin\theta-\frac{1}{2}gt^{2}
  21. t = v sin θ g ± v 2 sin 2 θ + 2 g y 0 g t=\frac{v\sin\theta}{g}\pm\frac{\sqrt{v^{2}\sin^{2}\theta+2gy_{0}}}{g}
  22. t = v sin θ g + v 2 sin 2 θ + 2 g y 0 g t=\frac{v\sin\theta}{g}+\frac{\sqrt{v^{2}\sin^{2}\theta+2gy_{0}}}{g}
  23. R = v cos θ g [ v sin θ + v 2 sin 2 θ + 2 g y 0 ] R=\frac{v\cos\theta}{g}\left[v\sin\theta+\sqrt{v^{2}\sin^{2}\theta+2gy_{0}}\right]
  24. θ = arccos 2 g y 0 + v 2 2 g y 0 + 2 v 2 \theta=\arccos\sqrt{\frac{2gy_{0}+v^{2}}{2gy_{0}+2v^{2}}}
  25. y 0 y_{0}
  26. lim y 0 0 arccos 2 g y 0 + v 2 2 g y 0 + 2 v 2 = π 4 \lim_{y_{0}\to 0}\arccos\sqrt{\frac{2gy_{0}+v^{2}}{2gy_{0}+2v^{2}}}=\frac{\pi}% {4}
  27. tan ψ = - v y ( t d ) v x ( t d ) = v 2 sin 2 θ + 2 g y 0 v cos θ \tan\psi=\frac{-v_{y}(t_{d})}{v_{x}(t_{d})}=\frac{\sqrt{v^{2}\sin^{2}\theta+2% gy_{0}}}{v\cos\theta}
  28. tan 2 ψ = 2 g y 0 + v 2 v 2 = C + 1 \tan^{2}\psi=\frac{2gy_{0}+v^{2}}{v^{2}}=C+1
  29. tan 2 θ = 1 - cos 2 θ cos 2 θ = v 2 2 g y 0 + v 2 = 1 C + 1 \tan^{2}\theta=\frac{1-\cos^{2}\theta}{\cos^{2}\theta}=\frac{v^{2}}{2gy_{0}+v^% {2}}=\frac{1}{C+1}
  30. tan 2 ψ tan 2 θ = 2 g y 0 + v 2 v 2 v 2 2 g y 0 + v 2 = 1 \tan^{2}\psi\,\tan^{2}\theta=\frac{2gy_{0}+v^{2}}{v^{2}}\frac{v^{2}}{2gy_{0}+v% ^{2}}=1
  31. tan ( θ + ψ ) = tan θ + tan ψ 1 - tan θ tan ψ \tan(\theta+\psi)=\frac{\tan\theta+\tan\psi}{1-\tan\theta\tan\psi}

Rank_product.html

  1. e g , i e_{g,i}
  2. r g , i r_{g,i}
  3. R P ( g ) = ( Π i = 1 k r g , i ) 1 / k RP(g)=(\Pi_{i=1}^{k}r_{g,i})^{1/k}
  4. E RP ( g ) = c / p \mathrm{E}_{\mathrm{RP}}(g)=c/p
  5. pfp ( g ) = E R P ( g ) / rank ( g ) \mathrm{pfp}(g)=\mathrm{E}_{RP}(g)/\mathrm{rank}(g)
  6. rank ( g ) \mathrm{rank}(g)
  7. RP \mathrm{RP}

Ratio_distribution.html

  1. Z = X / Y Z=X/Y
  2. C 1 C_{1}
  3. C 2 C_{2}
  4. C 1 = G 1 / G 2 C_{1}=G_{1}/G_{2}
  5. C 2 = G 3 / G 4 C_{2}=G_{3}/G_{4}
  6. C 1 C 2 = G 1 / G 2 G 3 / G 4 = G 1 G 4 G 2 G 3 = G 1 G 2 × G 4 G 3 = C 1 × C 3 , \frac{C_{1}}{C_{2}}=\frac{{G_{1}}/{G_{2}}}{{G_{3}}/{G_{4}}}=\frac{G_{1}G_{4}}{% G_{2}G_{3}}=\frac{G_{1}}{G_{2}}\times\frac{G_{4}}{G_{3}}=C_{1}\times C_{3},
  7. C 3 = G 4 / G 3 C_{3}=G_{4}/G_{3}
  8. p Z ( z ) = - + | y | p X , Y ( z y , y ) d y . p_{Z}(z)=\int^{+\infty}_{-\infty}|y|\,p_{X,Y}(zy,y)\,dy.
  9. p Z ( z ) = b ( z ) d ( z ) a 3 ( z ) 1 2 π σ x σ y [ Φ ( b ( z ) a ( z ) ) - Φ ( - b ( z ) a ( z ) ) ] + 1 a 2 ( z ) π σ x σ y e - c 2 p_{Z}(z)=\frac{b(z)\cdot d(z)}{a^{3}(z)}\frac{1}{\sqrt{2\pi}\sigma_{x}\sigma_{% y}}\left[\Phi\left(\frac{b(z)}{a(z)}\right)-\Phi\left(-\frac{b(z)}{a(z)}\right% )\right]+\frac{1}{a^{2}(z)\cdot\pi\sigma_{x}\sigma_{y}}e^{-\frac{c}{2}}
  10. a ( z ) = 1 σ x 2 z 2 + 1 σ y 2 a(z)=\sqrt{\frac{1}{\sigma_{x}^{2}}z^{2}+\frac{1}{\sigma_{y}^{2}}}
  11. b ( z ) = μ x σ x 2 z + μ y σ y 2 b(z)=\frac{\mu_{x}}{\sigma_{x}^{2}}z+\frac{\mu_{y}}{\sigma_{y}^{2}}
  12. c = μ x 2 σ x 2 + μ y 2 σ y 2 c=\frac{\mu_{x}^{2}}{\sigma_{x}^{2}}+\frac{\mu_{y}^{2}}{\sigma_{y}^{2}}
  13. d ( z ) = e b 2 ( z ) - c a 2 ( z ) 2 a 2 ( z ) d(z)=e^{\frac{b^{2}(z)-ca^{2}(z)}{2a^{2}(z)}}
  14. Φ \Phi
  15. Φ ( t ) = - t 1 2 π e - 1 2 u 2 d u \Phi(t)=\int_{-\infty}^{t}\,\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}u^{2}}\ du
  16. p ( z ) = 1 π 1 1 + z 2 p(z)=\frac{1}{\pi}\frac{1}{1+z^{2}}
  17. σ X 1 \sigma_{X}\neq 1
  18. σ Y 1 \sigma_{Y}\neq 1
  19. ρ 0 \rho\neq 0
  20. p Z ( z ) = 1 π β ( z - α ) 2 + β 2 , p_{Z}(z)=\frac{1}{\pi}\frac{\beta}{(z-\alpha)^{2}+\beta^{2}},
  21. α = ρ σ x σ y , \alpha=\rho\frac{\sigma_{x}}{\sigma_{y}},
  22. β = σ x σ y 1 - ρ 2 . \beta=\frac{\sigma_{x}}{\sigma_{y}}\sqrt{1-\rho^{2}}.
  23. t = μ y z - μ x σ y 2 z 2 - 2 ρ σ x σ y z + σ x 2 t=\frac{\mu_{y}z-\mu_{x}}{\sqrt{\sigma_{y}^{2}z^{2}-2\rho\sigma_{x}\sigma_{y}z% +\sigma_{x}^{2}}}
  24. p X ( x ) = { 1 0 < x < 1 0 otherwise p_{X}(x)=\begin{cases}1\qquad 0<x<1\\ 0\qquad\mbox{otherwise}\end{cases}
  25. p Z ( z ) = { 1 / 2 0 < z < 1 1 2 z 2 z 1 0 otherwise p_{Z}(z)=\begin{cases}1/2&0<z<1\\ \frac{1}{2z^{2}}&z\geq 1\\ 0&\mbox{otherwise}\end{cases}
  26. a a
  27. p X ( x | a ) = a π ( a 2 + x 2 ) p_{X}(x|a)=\frac{a}{\pi(a^{2}+x^{2})}
  28. Z = X / Y Z=X/Y
  29. p Z ( z | a ) = 1 π 2 ( z 2 - 1 ) ln ( z 2 ) . p_{Z}(z|a)=\frac{1}{\pi^{2}(z^{2}-1)}\ln\left(z^{2}\right).
  30. a a
  31. W = X Y W=XY
  32. p W ( w | a ) = a 2 π 2 ( w 2 - a 4 ) ln ( w 2 a 4 ) . p_{W}(w|a)=\frac{a^{2}}{\pi^{2}(w^{2}-a^{4})}\ln\left(\frac{w^{2}}{a^{4}}% \right).
  33. a a
  34. b b
  35. Z = X / Y Z=X/Y
  36. p Z ( z | a , b ) = a b π 2 ( b 2 z 2 - a 2 ) ln ( b 2 z 2 a 2 ) . p_{Z}(z|a,b)=\frac{ab}{\pi^{2}(b^{2}z^{2}-a^{2})}\ln\left(\frac{b^{2}z^{2}}{a^% {2}}\right).
  37. W = X Y W=XY
  38. p W ( w | a , b ) = a b π 2 ( w 2 - a 2 b 2 ) ln ( w 2 a 2 b 2 ) . p_{W}(w|a,b)=\frac{ab}{\pi^{2}(w^{2}-a^{2}b^{2})}\ln\left(\frac{w^{2}}{a^{2}b^% {2}}\right).
  39. b b
  40. 1 b . \frac{1}{b}.
  41. p Z ( z ) = { [ ϕ ( 0 ) - ϕ ( z ) ] / z 2 z 0 ϕ ( 0 ) / 2 z = 0 , p_{Z}(z)=\begin{cases}\left[\phi(0)-\phi(z)\right]/z^{2}&z\neq 0\\ \phi(0)/2&z=0,\\ \end{cases}
  42. X Y / m = t m \frac{X}{\sqrt{Y/m}}=t_{m}
  43. Y / m Z / n = F m , n \frac{Y/m}{Z/n}=F_{m,n}
  44. Y Y + Z = β ( m / 2 , n / 2 ) \frac{Y}{Y+Z}=\beta(m/2,n/2)
  45. F F
  46. β \beta
  47. ϕ = | 𝐗 | / | 𝐘 | \phi=|\mathbf{X}|/|\mathbf{Y}|
  48. Λ = | 𝐗 | / | 𝐗 + 𝐘 | \Lambda={|\mathbf{X}|/|\mathbf{X}+\mathbf{Y}|}

Rational_consequence_relation.html

  1. θ θ \theta\vdash\theta
  2. θ ψ θ ϕ ϕ ψ \frac{\theta\vdash\psi\quad\theta\equiv\phi}{\phi\vdash\psi}
  3. θ ϕ ϕ ψ θ ψ \frac{\theta\vdash\phi\quad\phi\models\psi}{\theta\vdash\psi}
  4. θ ϕ θ ψ θ ψ ϕ \frac{\theta\vdash\phi\quad\theta\vdash\psi}{\theta\wedge\psi\vdash\phi}
  5. θ ψ ϕ ψ θ ϕ ψ \frac{\theta\vdash\psi\quad\phi\vdash\psi}{\theta\vee\phi\vdash\psi}
  6. θ ϕ θ ψ θ ϕ ψ \frac{\theta\vdash\phi\quad\theta\vdash\psi}{\theta\vdash\phi\wedge\psi}
  7. ϕ ⊬ ¬ θ ϕ ψ ϕ θ ψ \frac{\phi\not\vdash\neg\theta\quad\phi\vdash\psi}{\phi\wedge\theta\vdash\psi}
  8. θ ϕ \theta\vdash\phi
  9. θ ϕ θ ( ϕ ψ ) θ ψ \frac{\theta\vdash\phi\quad\theta\vdash\left(\phi\rightarrow\psi\right)}{% \theta\vdash\psi}
  10. θ ϕ ψ θ ( ϕ ψ ) \frac{\theta\wedge\phi\vdash\psi}{\theta\vdash\left(\phi\rightarrow\psi\right)}
  11. θ ϕ θ ϕ ψ θ ψ \frac{\theta\vdash\phi\quad\theta\wedge\phi\vdash\psi}{\theta\vdash\psi}
  12. θ ϕ θ ϕ \frac{\theta\models\phi}{\theta\vdash\phi}
  13. L = { p 1 , , p n } L=\{p_{1},\ldots,p_{n}\}
  14. i = 1 n p i ϵ \bigwedge_{i=1}^{n}p^{\epsilon}_{i}
  15. p 1 = p p^{1}=p
  16. p - 1 = ¬ p p^{-1}=\neg p
  17. A t L At^{L}
  18. θ \theta\in
  19. S θ = { α A t L | α S C θ } S_{\theta}=\{\alpha\in At^{L}|\alpha\models^{SC}\theta\}
  20. s = s 1 , , s m \vec{s}=s_{1},\ldots,s_{m}
  21. A t L At^{L}
  22. θ \theta
  23. ϕ \phi
  24. s \vdash_{\vec{s}}
  25. θ s ϕ \theta\vdash_{\vec{s}}\phi
  26. S θ s i = S_{\theta}\cap s_{i}=\emptyset
  27. 1 i m 1\leq i\leq m
  28. S θ s i S_{\theta}\cap s_{i}\neq\emptyset
  29. 1 i m 1\leq i\leq m
  30. S θ s i S ϕ S_{\theta}\cap s_{i}\subseteq S_{\phi}
  31. s \vdash_{\vec{s}}
  32. s \vdash_{\vec{s}}
  33. s 2 s_{2}
  34. s 2 s 1 s_{2}\setminus s_{1}
  35. s 3 s_{3}
  36. s 3 s 2 s 1 s_{3}\setminus s_{2}\setminus s_{1}
  37. s m s_{m}
  38. s m i = 1 m - 1 s i s_{m}\setminus\bigcup_{i=1}^{m-1}s_{i}
  39. s i s_{i}
  40. s \vdash_{\vec{s}}
  41. s i s_{i}
  42. s i s_{i}
  43. \vdash
  44. s = s 1 , , s m \vec{s}=s_{1},\ldots,s_{m}
  45. A t L At^{L}
  46. s \vdash_{\vec{s}}
  47. s = \vdash_{\vec{s}}=\vdash
  48. s \vdash_{\vec{s}}
  49. \vdash
  50. s i s_{i}

Rayleigh_length.html

  1. w ( z ) w(z)
  2. z z
  3. w 0 w_{0}
  4. b b
  5. z R z_{\mathrm{R}}
  6. Θ \Theta
  7. z ^ \hat{z}
  8. z R = π w 0 2 λ , z_{\mathrm{R}}=\frac{\pi w_{0}^{2}}{\lambda},
  9. λ \lambda
  10. w 0 w_{0}
  11. w 0 2 λ / π w_{0}\geq 2\lambda/\pi
  12. z z
  13. w ( z ) = w 0 1 + ( z z R ) 2 . w(z)=w_{0}\,\sqrt{1+{\left(\frac{z}{z_{\mathrm{R}}}\right)}^{2}}.
  14. w ( z ) w(z)
  15. w ( 0 ) = w 0 w(0)=w_{0}
  16. z R z_{\mathrm{R}}
  17. 2 \sqrt{2}
  18. Θ div 2 w 0 z R . \Theta_{\mathrm{div}}\simeq 2\frac{w_{0}}{z_{R}}.
  19. D = 2 w 0 4 λ π Θ div D=2\,w_{0}\simeq\frac{4\lambda}{\pi\,\Theta_{\mathrm{div}}}

Reaction_engine.html

  1. 1 2 M V e 2 \begin{matrix}\frac{1}{2}\end{matrix}MV_{e}^{2}
  2. V e V_{e}
  3. I s p I_{sp}
  4. I s p I_{sp}
  5. v e v_{e}
  6. v e v_{e}
  7. E = F × d E=F\times d\;
  8. E t = P = F × d t = F × v \frac{E}{t}=P=\frac{F\times d}{t}=F\times v
  9. P = F × v P=F\times v\;

Reaction–diffusion_system.html

  1. t s y m b o l q = s y m b o l D ¯ ¯ 2 s y m b o l q + s y m b o l R ( s y m b o l q ) , \partial_{t}symbol{q}=\underline{\underline{symbol{D}}}\,\nabla^{2}symbol{q}+% symbol{R}(symbol{q}),
  2. 𝐪 ( 𝐱 , t ) \mathbf{q}(\mathbf{x},t)
  3. [ u u u l i n e , u m a t h b f D ] [u^{\prime}uuline^{\prime},u^{\prime}\\ mathbf{D}^{\prime}]
  4. 𝐑 \mathbf{R}
  5. u u
  6. t u = D x 2 u + R ( u ) , \partial_{t}u=D\partial^{2}_{x}u+R(u),
  7. R ( u ) = u ( 1 u ) R(u)=u(1−u)
  8. R ( u ) = u ( 1 u ) ( u α ) R(u)=u(1−u)(u−α)
  9. t u = - δ 𝔏 δ u \partial_{t}u=-\frac{\delta\mathfrak{L}}{\delta u}
  10. 𝔏 \mathfrak{L}
  11. 𝔏 = - [ D 2 ( x u ) 2 - V ( u ) ] d x \mathfrak{L}=\int_{-\infty}^{\infty}\left[\tfrac{D}{2}\left(\partial_{x}u% \right)^{2}-V(u)\right]\,\text{d}x
  12. V ( u ) V(u)
  13. R ( u ) = d V ( . R(u)=d\frac{V}{(}.
  14. u ( x , t ) = û ( ξ ) u(x,t)=û(ξ)
  15. ξ = x c t ξ=x−ct
  16. c c
  17. c = 0 c=0
  18. t u ~ = D x 2 u ~ - U ( x ) u ~ , U ( x ) = - R ( u ) | u = u 0 ( x ) . \partial_{t}\tilde{u}=D\partial_{x}^{2}\tilde{u}-U(x)\tilde{u},\qquad U(x)=-R^% {\prime}(u)|_{u=u_{0}(x)}.
  19. ũ = ψ ( x ) e x p ( λ t ) ũ=ψ(x)exp(−λt)
  20. H ^ ψ = λ ψ , H ^ = - D x 2 + U ( x ) , \hat{H}\psi=\lambda\psi,\qquad\hat{H}=-D\partial_{x}^{2}+U(x),
  21. λ = 0 λ=0
  22. λ = 0 λ=0
  23. c c
  24. D ξ 2 u ^ ( ξ ) + c ξ u ^ ( ξ ) + R ( u ^ ( ξ ) ) = 0. D\partial^{2}_{\xi}\hat{u}(\xi)+c\partial_{\xi}\hat{u}(\xi)+R(\hat{u}(\xi))=0.
  25. D D
  26. û û
  27. ξ ξ
  28. R R
  29. c c
  30. ( t u t v ) = ( D u 0 0 D v ) ( x x u x x v ) + ( F ( u , v ) G ( u , v ) ) \begin{pmatrix}\partial_{t}u&\partial_{t}v\end{pmatrix}=\begin{pmatrix}D_{u}&0% \\ 0&D_{v}\end{pmatrix}\begin{pmatrix}\partial_{xx}u\\ \partial_{xx}v\end{pmatrix}+\begin{pmatrix}F(u,v)\\ G(u,v)\end{pmatrix}
  31. s y m b o l q ~ s y m b o l k ( s y m b o l x , t ) = ( u ~ ( t ) v ~ ( t ) ) e i s y m b o l k s y m b o l x \tilde{symbol{q}}_{symbol{k}}(symbol{x},t)=\begin{pmatrix}\tilde{u}(t)\\ \tilde{v}(t)\end{pmatrix}e^{isymbol{k}\cdot symbol{x}}
  32. ( t u ~ s y m b o l k ( t ) t v ~ s y m b o l k ( t ) ) = - k 2 ( D u u ~ s y m b o l k ( t ) D v v ~ s y m b o l k ( t ) ) + s y m b o l R ( u ~ s y m b o l k ( t ) v ~ s y m b o l k ( t ) ) . \begin{pmatrix}\partial_{t}\tilde{u}_{symbol{k}}(t)\\ \partial_{t}\tilde{v}_{symbol{k}}(t)\end{pmatrix}=-k^{2}\begin{pmatrix}D_{u}% \tilde{u}_{symbol{k}}(t)\\ D_{v}\tilde{v}_{symbol{k}}(t)\end{pmatrix}+symbol{R}^{\prime}\begin{pmatrix}% \tilde{u}_{symbol{k}}(t)\\ \tilde{v}_{symbol{k}}(t)\end{pmatrix}.
  33. 𝐑 \mathbf{R}′
  34. 𝐤 \mathbf{k}
  35. ( + - + - ) , ( + + - - ) , ( - + - + ) , ( - - + + ) . \begin{pmatrix}+&-\\ +&-\end{pmatrix},\quad\begin{pmatrix}+&+\\ -&-\end{pmatrix},\quad\begin{pmatrix}-&+\\ -&+\end{pmatrix},\quad\begin{pmatrix}-&-\\ +&+\end{pmatrix}.
  36. t u = d u 2 2 u + f ( u ) - σ v , τ t v = d v 2 2 v + u - v \begin{aligned}\displaystyle\partial_{t}u&\displaystyle=d_{u}^{2}\,\nabla^{2}u% +f(u)-\sigma v,\\ \displaystyle\tau\partial_{t}v&\displaystyle=d_{v}^{2}\,\nabla^{2}v+u-v\end{aligned}
  37. λ λ
  38. k = 0 k=0
  39. q n H ( k ) : 1 τ + ( d u 2 + 1 τ d v 2 ) k 2 = f ( u h ) , q n T ( k ) : κ 1 + d v 2 k 2 + d u 2 k 2 = f ( u h ) . \begin{aligned}\displaystyle q_{\,\text{n}}^{H}(k):&\displaystyle{}\quad\frac{% 1}{\tau}+\left(d_{u}^{2}+\frac{1}{\tau}d_{v}^{2}\right)k^{2}&\displaystyle=f^{% \prime}(u_{h}),\\ \displaystyle q_{\,\text{n}}^{T}(k):&\displaystyle{}\quad\frac{\kappa}{1+d_{v}% ^{2}k^{2}}+d_{u}^{2}k^{2}&\displaystyle=f^{\prime}(u_{h}).\end{aligned}

Reactive_programming.html

  1. a := b + c a:=b+c
  2. a a
  3. b + c b+c
  4. b b
  5. c c
  6. a a
  7. a a

Read-only_right_moving_Turing_machines.html

  1. M = Q , Γ , b , Σ , δ , q 0 , F M=\langle Q,\Gamma,b,\Sigma,\delta,q_{0},F\rangle
  2. Q Q
  3. Γ \Gamma
  4. b Γ b\in\Gamma
  5. Σ \Sigma
  6. Γ \Gamma
  7. δ : Q × Γ Q × Γ × { R } \delta:Q\times\Gamma\rightarrow Q\times\Gamma\times\{R\}
  8. q 0 Q q_{0}\in Q
  9. F Q F\subseteq Q
  10. F F
  11. \varnothing

Real_net_output_ratio.html

  1. R e a l N e t O u t p u t R a t i o = i n t e r n a l p r o d u c t i o n t o t a l p r o d u c t i o n v a l u e = i n t e r n a l p r o d u c t i o n i n t e r n a l p r o d u c t i o n + e x t e r n a l l y p r o d u c e d g o o d s + e x t e r n a l l y p r o d u c e d s e r v i c e s \textstyle Real\,Net\,Output\,Ratio=\frac{internal\,production}{total\,% production\,value}=\frac{internal\,production}{internal\,production\,+\,% externally\,produced\,goods\,+\,externally\,produced\,services}

Real_structure.html

  1. σ : \sigma:{\mathbb{C}}\to{\mathbb{C}}\,
  2. σ ( z ) = z ¯ \sigma(z)={\bar{z}}
  3. {\mathbb{C}}\,
  4. = i {\mathbb{C}}={\mathbb{R}}\oplus i{\mathbb{R}}\,
  5. σ ( λ z ) = λ ¯ σ ( z ) \sigma(\lambda z)={\bar{\lambda}}\sigma(z)\,
  6. σ ( z 1 + z 2 ) = σ ( z 1 ) + σ ( z 2 ) \sigma(z_{1}+z_{2})=\sigma(z_{1})+\sigma(z_{2})\,
  7. σ : V V \sigma:V\to V
  8. V V V_{\mathbb{R}}\subset V
  9. V V V_{\mathbb{R}}\otimes_{\mathbb{R}}{\mathbb{C}}\to V
  10. t V t\in V\,
  11. t 0 t\neq 0
  12. t t\,
  13. i t it\,
  14. dim V = 2 dim V \dim_{\mathbb{R}}V=2\dim_{\mathbb{C}}V
  15. σ : V V \sigma:V\to V\,
  16. σ σ = i d V \sigma\circ\sigma=id_{V}\,
  17. t V t\in V\,
  18. t = t + + t - {t=t^{+}+t^{-}}\,
  19. t + = 1 2 ( t + σ t ) t^{+}={1\over{2}}(t+\sigma t)
  20. t - = 1 2 ( t - σ t ) t^{-}={1\over{2}}(t-\sigma t)\,
  21. V = V + V - V=V^{+}\oplus V^{-}\,
  22. V + = { t V | σ t = t } V^{+}=\{t\in V|\sigma t=t\}
  23. V - = { t V | σ t = - t } V^{-}=\{t\in V|\sigma t=-t\}\,
  24. V + V^{+}\,
  25. V - V^{-}\,
  26. K : V + V - K:V^{+}\to V^{-}\,
  27. K ( t ) = i t K(t)=it\,
  28. dim V + = dim V - = dim V \dim_{\mathbb{R}}V^{+}=\dim_{\mathbb{R}}V^{-}=\dim_{\mathbb{C}}V\,
  29. V + V^{+}\,
  30. V V_{\mathbb{R}}\,
  31. σ \sigma\,
  32. σ ( V ) V \sigma(V_{\mathbb{R}})\subset V_{\mathbb{R}}\,
  33. V - V^{-}\,
  34. i V iV_{\mathbb{R}}\,
  35. V = V + V - V=V^{+}\oplus V^{-}\,
  36. V = V i V V=V_{\mathbb{R}}\oplus iV_{\mathbb{R}}\,
  37. V V_{\mathbb{R}}\,
  38. i V iV_{\mathbb{R}}\,
  39. V V_{\mathbb{R}}\,
  40. V = V V^{\mathbb{C}}=V_{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}\,
  41. V V_{\mathbb{R}}\,
  42. V = V i V V_{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}=V_{\mathbb{R}}\oplus iV_{\mathbb{% R}}\,
  43. V V V_{\mathbb{R}}\otimes_{\mathbb{R}}\mathbb{C}\to V\,
  44. σ : V V \sigma:V\to V\,
  45. σ ^ : V V ¯ \hat{\sigma}:V\to\bar{V}\,
  46. V V\,
  47. V ¯ \bar{V}\,
  48. v σ ^ ( v ) := σ ( v ) ¯ v\mapsto\hat{\sigma}(v):=\overline{\sigma(v)}\,

Reality_structure.html

  1. V = V i V . V=V_{\mathbb{R}}\oplus iV_{\mathbb{R}}.
  2. n \mathbb{C}^{n}
  3. n = n i n . \mathbb{C}^{n}=\mathbb{R}^{n}\oplus i\,\mathbb{R}^{n}.
  4. v = Re { v } + i Im { v } v=\operatorname{Re}\{v\}+i\,\operatorname{Im}\{v\}
  5. v ¯ = Re { v } - i Im { v } \overline{v}=\operatorname{Re}\{v\}-i\,\operatorname{Im}\{v\}
  6. v v ¯ v\mapsto\overline{v}
  7. v ¯ ¯ = v , v + w ¯ = v ¯ + w ¯ , and α v ¯ = α ¯ v ¯ . \overline{\overline{v}}=v,\quad\overline{v+w}=\overline{v}+\overline{w},\quad% \,\text{and}\quad\overline{\alpha v}=\overline{\alpha}\,\overline{v}.
  8. v c ( v ) v\mapsto c(v)
  9. Re { v } = 1 2 ( v + c ( v ) ) , \operatorname{Re}\{v\}=\frac{1}{2}\left(v+c(v)\right),
  10. V = { Re { v } v V } . V_{\mathbb{R}}=\left\{\operatorname{Re}\{v\}\mid v\in V\right\}.
  11. V = V i V . V=V_{\mathbb{R}}\oplus iV_{\mathbb{R}}.
  12. i i

Realizability.html

  1. ( A x P ( x ) ) x ( A P ( x ) ) (A\rightarrow\exists x\;P(x))\rightarrow\exists x\;(A\rightarrow P(x))

Realization_(probability).html

  1. x = X ( ω ) x=X(\omega)

Realization_(systems).html

  1. [ A ( t ) , B ( t ) , C ( t ) , D ( t ) ] [A(t),B(t),C(t),D(t)]
  2. 𝐱 ˙ ( t ) = A ( t ) 𝐱 ( t ) + B ( t ) 𝐮 ( t ) \dot{\mathbf{x}}(t)=A(t)\mathbf{x}(t)+B(t)\mathbf{u}(t)
  3. 𝐲 ( t ) = C ( t ) 𝐱 ( t ) + D ( t ) 𝐮 ( t ) \mathbf{y}(t)=C(t)\mathbf{x}(t)+D(t)\mathbf{u}(t)
  4. ( u ( t ) , y ( t ) ) (u(t),y(t))
  5. t t
  6. H ( s ) H(s)
  7. ( A , B , C , D ) (A,B,C,D)
  8. H ( s ) = C ( s I - A ) - 1 B + D H(s)=C(sI-A)^{-1}B+D
  9. H ( s ) = n 1 s 3 + n 2 s 2 + n 3 s + n 4 s 4 + d 1 s 3 + d 2 s 2 + d 3 s + d 4 H(s)=\frac{n_{1}s^{3}+n_{2}s^{2}+n_{3}s+n_{4}}{s^{4}+d_{1}s^{3}+d_{2}s^{2}+d_{% 3}s+d_{4}}
  10. 𝐱 ˙ ( t ) = [ - d 1 - d 2 - d 3 - d 4 1 0 0 0 0 1 0 0 0 0 1 0 ] 𝐱 ( t ) + [ 1 0 0 0 ] 𝐮 ( t ) \dot{\,\textbf{x}}(t)=\begin{bmatrix}-d_{1}&-d_{2}&-d_{3}&-d_{4}\\ 1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\end{bmatrix}\,\textbf{x}(t)+\begin{bmatrix}1\\ 0\\ 0\\ 0\\ \end{bmatrix}\,\textbf{u}(t)
  11. 𝐲 ( t ) = [ n 1 n 2 n 3 n 4 ] 𝐱 ( t ) \,\textbf{y}(t)=\begin{bmatrix}n_{1}&n_{2}&n_{3}&n_{4}\end{bmatrix}\,\textbf{x% }(t)
  12. 𝐱 ˙ ( t ) = [ - d 1 1 0 0 - d 2 0 1 0 - d 3 0 0 1 - d 4 0 0 0 ] 𝐱 ( t ) + [ n 1 n 2 n 3 n 4 ] 𝐮 ( t ) \dot{\,\textbf{x}}(t)=\begin{bmatrix}-d_{1}&1&0&0\\ -d_{2}&0&1&0\\ -d_{3}&0&0&1\\ -d_{4}&0&0&0\end{bmatrix}\,\textbf{x}(t)+\begin{bmatrix}n_{1}\\ n_{2}\\ n_{3}\\ n_{4}\end{bmatrix}\,\textbf{u}(t)
  13. 𝐲 ( t ) = [ 1 0 0 0 ] 𝐱 ( t ) \,\textbf{y}(t)=\begin{bmatrix}1&0&0&0\end{bmatrix}\,\textbf{x}(t)
  14. D = 0 D=0
  15. u ( t ) u(t)
  16. y ( t ) y(t)
  17. T ( t , σ ) T(t,\sigma)
  18. [ A ( t ) , B ( t ) , C ( t ) ] [A(t),B(t),C(t)]
  19. T ( t , σ ) = C ( t ) ϕ ( t , σ ) B ( σ ) T(t,\sigma)=C(t)\phi(t,\sigma)B(\sigma)
  20. ϕ \phi

Reassignment_method.html

  1. x ( t ) x(t)
  2. ϵ ( t , ω ) \epsilon(t,\omega)
  3. h ω ( t ) h_{\omega}(t)
  4. h ω ( t ) = h ( t ) e j ω t h_{\omega}(t)=h(t)e^{j\omega t}
  5. h ( t ) h(t)
  6. ϵ ( t , ω ) = x ( τ ) h ( t - τ ) e - j ω [ τ - t ] d τ = e j ω t x ( τ ) h ( t - τ ) e - j ω τ d τ = e j ω t X ( t , ω ) = X t ( ω ) = M t ( ω ) e j ϕ τ ( ω ) \begin{aligned}\displaystyle\epsilon(t,\omega)&\displaystyle=\int x(\tau)h(t-% \tau)e^{-j\omega\left[\tau-t\right]}d\tau\\ &\displaystyle=e^{j\omega t}\int x(\tau)h(t-\tau)e^{-j\omega\tau}d\tau\\ &\displaystyle=e^{j\omega t}X(t,\omega)\\ &\displaystyle=X_{t}(\omega)=M_{t}(\omega)e^{j\phi_{\tau}(\omega)}\end{aligned}
  7. M t ( ω ) M_{t}(\omega)
  8. ϕ τ ( ω ) \phi_{\tau}(\omega)
  9. X t ( ω ) X_{t}(\omega)
  10. x ( t ) x(t)
  11. t t
  12. h ( t ) h(t)
  13. x ( t ) x(t)
  14. x ( t ) = X τ ( ω ) h ω * ( τ - t ) d ω d τ = X τ ( ω ) h ( τ - t ) e - j ω [ τ - t ] d ω d τ = M τ ( ω ) e j ϕ τ ( ω ) h ( τ - t ) e - j ω [ τ - t ] d ω d τ = M τ ( ω ) h ( τ - t ) e j [ ϕ τ ( ω ) - ω τ + ω t ] d ω d τ \begin{aligned}\displaystyle x(t)&\displaystyle=\iint X_{\tau}(\omega)h^{*}_{% \omega}(\tau-t)d\omega d\tau\\ &\displaystyle=\iint X_{\tau}(\omega)h(\tau-t)e^{-j\omega\left[\tau-t\right]}d% \omega d\tau\\ &\displaystyle=\iint M_{\tau}(\omega)e^{j\phi_{\tau}(\omega)}h(\tau-t)e^{-j% \omega\left[\tau-t\right]}d\omega d\tau\\ &\displaystyle=\iint M_{\tau}(\omega)h(\tau-t)e^{j\left[\phi_{\tau}(\omega)-% \omega\tau+\omega t\right]}d\omega d\tau\end{aligned}
  15. M ( t , ω ) M(t,\omega)
  16. t , ω t,\omega
  17. ω [ ϕ τ ( ω ) - ω τ + ω t ] = 0 τ [ ϕ τ ( ω ) - ω τ + ω t ] = 0 \begin{matrix}\frac{\partial}{\partial\omega}\left[\phi_{\tau}(\omega)-\omega% \tau+\omega t\right]&=0\\ \frac{\partial}{\partial\tau}\left[\phi_{\tau}(\omega)-\omega\tau+\omega t% \right]&=0\end{matrix}
  18. t ^ , ω ^ \hat{t},\hat{\omega}
  19. t ^ ( τ , ω ) = τ - ϕ τ ( ω ) ω = - ϕ ( τ , ω ) ω ω ^ ( τ , ω ) = ϕ τ ( ω ) τ = ω + ϕ ( τ , ω ) τ . \begin{aligned}\displaystyle\hat{t}(\tau,\omega)&\displaystyle=\tau-\frac{% \partial\phi_{\tau}(\omega)}{\partial\omega}=-\frac{\partial\phi(\tau,\omega)}% {\partial\omega}\\ \displaystyle\hat{\omega}(\tau,\omega)&\displaystyle=\frac{\partial\phi_{\tau}% (\omega)}{\partial\tau}=\omega+\frac{\partial\phi(\tau,\omega)}{\partial\tau}.% \end{aligned}
  20. t ^ g ( t , ω ) \hat{t}_{g}(t,\omega)
  21. ω ^ i ( t , ω ) \hat{\omega}_{i}(t,\omega)
  22. ϵ ( t , ω ) \epsilon(t,\omega)
  23. t ^ ( t , ω ) , ω ^ ( t , ω ) \hat{t}(t,\omega),\hat{\omega}(t,\omega)
  24. t , ω t,\omega
  25. X ( k ) X(k)
  26. x ( n ) x(n)
  27. ϕ ( t , ω ) t 1 Δ t [ ϕ ( t + Δ t 2 , ω ) - ϕ ( t - Δ t 2 , ω ) ] ϕ ( t , ω ) ω 1 Δ ω [ ϕ ( t , ω + Δ ω 2 ) - ϕ ( t , ω - Δ ω 2 ) ] \begin{matrix}\frac{\partial\phi(t,\omega)}{\partial t}&\approx\frac{1}{\Delta t% }\left[\phi(t+\frac{\Delta t}{2},\omega)-\phi(t-\frac{\Delta t}{2},\omega)% \right]\\ \frac{\partial\phi(t,\omega)}{\partial\omega}&\approx\frac{1}{\Delta\omega}% \left[\phi(t,\omega+\frac{\Delta\omega}{2})-\phi(t,\omega-\frac{\Delta\omega}{% 2})\right]\end{matrix}
  28. Δ t \Delta t
  29. Δ ω \Delta\omega
  30. t ^ ( t , ω ) = t - τ W x ( t - τ , ω - ν ) Φ ( τ , ν ) d τ d ν W x ( t - τ , ω - ν ) Φ ( τ , ν ) d τ d ν ω ^ ( t , ω ) = ω - ν W x ( t - τ , ω - ν ) Φ ( τ , ν ) d τ d ν W x ( t - τ , ω - ν ) Φ ( τ , ν ) d τ d ν \begin{matrix}\hat{t}(t,\omega)&=t-\frac{\iint\tau\cdot W_{x}(t-\tau,\omega-% \nu)\cdot\Phi(\tau,\nu)d\tau d\nu}{\iint W_{x}(t-\tau,\omega-\nu)\cdot\Phi(% \tau,\nu)d\tau d\nu}\\ \hat{\omega}(t,\omega)&=\omega-\frac{\iint\nu\cdot W_{x}(t-\tau,\omega-\nu)% \cdot\Phi(\tau,\nu)d\tau d\nu}{\iint W_{x}(t-\tau,\omega-\nu)\cdot\Phi(\tau,% \nu)d\tau d\nu}\end{matrix}
  31. W x ( t , ω ) W_{x}(t,\omega)
  32. x ( t ) x(t)
  33. Φ ( t , ω ) \Phi(t,\omega)
  34. t ^ ( t , ω ) = t - { X 𝒯 h ( t , ω ) X * ( t , ω ) | X ( t , ω ) | 2 } ω ^ ( t , ω ) = ω + { X 𝒟 h ( t , ω ) X * ( t , ω ) | X ( t , ω ) | 2 } \begin{matrix}\hat{t}(t,\omega)&=t-\Re\Bigg\{\frac{X_{\mathcal{T}h}(t,\omega)% \cdot X^{*}(t,\omega)}{|X(t,\omega)|^{2}}\Bigg\}\\ \hat{\omega}(t,\omega)&=\omega+\Im\Bigg\{\frac{X_{\mathcal{D}h}(t,\omega)\cdot X% ^{*}(t,\omega)}{|X(t,\omega)|^{2}}\Bigg\}\end{matrix}
  35. X ( t , ω ) X(t,\omega)
  36. h ( t ) h(t)
  37. X 𝒯 h ( t , ω ) X_{\mathcal{T}h}(t,\omega)
  38. h 𝒯 ( t ) = t h ( t ) h_{\mathcal{T}}(t)=t\cdot h(t)
  39. X 𝒟 h ( t , ω ) X_{\mathcal{D}h}(t,\omega)
  40. h 𝒟 ( t ) = d d t h ( t ) h_{\mathcal{D}}(t)=\frac{d}{dt}h(t)
  41. h 𝒯 ( t ) h_{\mathcal{T}}(t)
  42. h 𝒟 ( t ) h_{\mathcal{D}}(t)
  43. t , ω t,\omega
  44. t , ω t,\omega
  45. | X ( t , ω ) | 2 |X(t,\omega)|^{2}
  46. x ( t ) = n A n ( t ) e j θ n ( t ) x(t)=\sum_{n}A_{n}(t)e^{j\theta_{n}(t)}
  47. ω n ( t ) = d θ n ( t ) d t , \omega_{n}(t)=\frac{d\theta_{n}(t)}{dt},
  48. X ( t , ω 0 ) X(t,\omega_{0})
  49. ω 0 \omega_{0}
  50. x n ( t ) x_{n}(t)
  51. ω n ( t ) \omega_{n}(t)
  52. ω 0 \omega_{0}
  53. ω 0 \omega_{0}
  54. ω n ( t ) = t arg { x n ( t ) } = t arg { X ( t , ω 0 ) } \begin{matrix}\omega_{n}(t)&=\frac{\partial}{\partial t}\arg\{x_{n}(t)\}\\ &=\frac{\partial}{\partial t}\arg\{X(t,\omega_{0})\}\end{matrix}
  55. h ( t ) h(t)

Rebound_rate.html

  1. Rebound Rate = 100 × Rebounds × Team Minutes Played 5 Minutes Played × ( Team Total Rebounds + Opposing Team Total Rebounds ) \,\text{Rebound Rate}=\dfrac{100\times\,\text{Rebounds}\times\dfrac{\,\text{% Team Minutes Played}}{5}}{\,\text{Minutes Played}\times\left(\,\text{Team % Total Rebounds}+\,\text{Opposing Team Total Rebounds}\right)}
  2. Offensive Rebound Rate = 100 × Offensive Rebounds × Team Minutes Played 5 Minutes Played × ( Team Offensive Rebounds + Opposing Team Defensive Rebounds ) \,\text{Offensive Rebound Rate}=\dfrac{100\times\,\text{Offensive Rebounds}% \times\dfrac{\,\text{Team Minutes Played}}{5}}{\,\text{Minutes Played}\times% \left(\,\text{Team Offensive Rebounds}+\,\text{Opposing Team Defensive % Rebounds}\right)}
  3. Defensive Rebound Rate = 100 × Defensive Rebounds × Team Minutes Played 5 Minutes Played × ( Team Defensive Rebounds + Opposing Team Offensive Rebounds ) \,\text{Defensive Rebound Rate}=\dfrac{100\times\,\text{Defensive Rebounds}% \times\dfrac{\,\text{Team Minutes Played}}{5}}{\,\text{Minutes Played}\times% \left(\,\text{Team Defensive Rebounds}+\,\text{Opposing Team Offensive % Rebounds}\right)}

Reciprocal_Fibonacci_constant.html

  1. ψ = k = 1 1 F k = 1 1 + 1 1 + 1 2 + 1 3 + 1 5 + 1 8 + 1 13 + 1 21 + . \psi=\sum_{k=1}^{\infty}\frac{1}{F_{k}}=\frac{1}{1}+\frac{1}{1}+\frac{1}{2}+% \frac{1}{3}+\frac{1}{5}+\frac{1}{8}+\frac{1}{13}+\frac{1}{21}+\cdots.
  2. ψ = 3.359885666243177553172011302918927179688905133731 . \psi=3.359885666243177553172011302918927179688905133731\dots.
  3. ψ = [ 3 ; 2 , 1 , 3 , 1 , 1 , 13 , 2 , 3 , 3 , 2 , 1 , 1 , 6 , 3 , 2 , 4 , 362 , 2 , 4 , 8 , 6 , 30 , 50 , 1 , 6 , 3 , 3 , 2 , 7 , 2 , 3 , 1 , 3 , 2 , ] . \psi=[3;2,1,3,1,1,13,2,3,3,2,1,1,6,3,2,4,362,2,4,8,6,30,50,1,6,3,3,2,7,2,3,1,3% ,2,\dots]\!\,.

Reciprocity_(network_science).html

  1. L < - > L^{<->}
  2. r = L < - > L r=\frac{L^{<->}}{L}
  3. r = 1 r=1
  4. r = 0 r=0
  5. a i j = 1 a_{ij}=1
  6. a i j = 0 a_{ij}=0
  7. ρ i j ( a i j - a ¯ ) ( a j i - a ¯ ) i j ( a i j - a ¯ ) 2 \rho\equiv\frac{\sum_{i\neq j}(a_{ij}-\bar{a})(a_{ji}-\bar{a})}{\sum_{i\neq j}% (a_{ij}-\bar{a})^{2}}
  8. a ¯ i j a i j N ( N - 1 ) = L N ( N - 1 ) \bar{a}\equiv\frac{\sum_{i\neq j}a_{ij}}{N(N-1)}=\frac{L}{N(N-1)}
  9. a ¯ \bar{a}
  10. ρ = r - a ¯ 1 - a ¯ \rho=\frac{r-\bar{a}}{1-\bar{a}}
  11. ρ > 0 \rho>0
  12. ρ < 0 \rho<0
  13. ρ = 1 \rho=1
  14. ρ = ρ m i n \rho=\rho_{min}
  15. ρ m i n - a ¯ 1 - a ¯ \rho_{min}\equiv\frac{-\bar{a}}{1-\bar{a}}
  16. ρ \rho

Recurrence_period_density_entropy.html

  1. H norm \scriptstyle H_{\mathrm{norm}}
  2. H norm = 0 \scriptstyle H_{\mathrm{norm}}=0
  3. H norm 1 \scriptstyle H_{\mathrm{norm}}\approx 1
  4. 𝐗 n = [ x n , x n + τ , x n + 2 τ , , x n + ( M - 1 ) τ ] \mathbf{X}_{n}=[x_{n},x_{n+\tau},x_{n+2\tau},\ldots,x_{n+(M-1)\tau}]
  5. 𝐗 n \scriptstyle\mathbf{X}_{n}
  6. ε \varepsilon
  7. H norm = - ( ln T max ) - 1 t = 1 T max P ( t ) ln P ( t ) H_{\mathrm{norm}}=-(\ln{T_{\max})}^{-1}\sum_{t=1}^{T_{\max}}P(t)\ln{P(t)}
  8. T max \scriptstyle T_{\max}
  9. ε \scriptstyle\varepsilon
  10. H norm \scriptstyle H_{\mathrm{norm}}
  11. H n o r m H_{norm}

Reduced_residue_system.html

  1. φ \varphi
  2. φ ( 12 ) = 4 \varphi(12)=4
  3. r i 0 ( mod n ) \sum r_{i}\equiv 0\;\;(\mathop{{\rm mod}}n)

Reduction_(recursion_theory).html

  1. B ( n ) B^{(n)}
  2. Δ 1 1 \Delta^{1}_{1}
  3. B ( α ) B^{(\alpha)}

Reeb_vector_field.html

  1. α \alpha
  2. R ker d α , α ( R ) = 1 R\in\mathrm{ker}\ d\alpha,\ \alpha(R)=1

Reed–Muller_expansion.html

  1. f ( x 1 , , x n ) f(x_{1},\ldots,x_{n})
  2. x i x_{i}
  3. f < m t p l > x i ( x ) \displaystyle f_{<}mtpl>{{x_{i}}}(x)
  4. f f
  5. f f
  6. f = f x ¯ i x i f x i . f=f_{\overline{x}_{i}}\oplus x_{i}\frac{\partial f}{\partial x_{i}}.

Refinement_monoid.html

  1. [ X ] [X]
  2. [ X ] [X]
  3. [ X ] + [ Y ] = [ X × Y ] [X]+[Y]=[X\times Y]
  4. [ R ] [R]
  5. + = { 0 , 1 , 2 , } \mathbb{Z}^{+}=\{0,1,2,\dots\}
  6. F n × n F^{n\times n}
  7. n \mathbb{Z}^{n}

Regular_category.html

  1. R X Y R\rightrightarrows X\to Y
  2. R 𝑠 𝑟 X Y R\overset{r}{\underset{s}{\rightrightarrows}}X\to Y
  3. 0 R r - s X Y 0 0\to R\xrightarrow{r-s}X\to Y\to 0
  4. x ( ϕ ( x ) ψ ( x ) ) \forall x(\phi(x)\to\psi(x))
  5. ϕ \phi
  6. ψ \psi
  7. x ( ϕ ( x ) ψ ( x ) ) \forall x(\phi(x)\to\psi(x))
  8. ϕ \phi
  9. ψ \psi
  10. 𝐌𝐨𝐝 ( T , C ) 𝐑𝐞𝐠𝐂𝐚𝐭 ( R ( T ) , C ) \mathbf{Mod}(T,C)\cong\mathbf{RegCat}(R(T),C)
  11. X X
  12. X × X X\times X
  13. p 0 , p 1 : R X p_{0},p_{1}:R\rightarrow X
  14. R X × X R\rightarrow X\times X

Regulated_function.html

  1. f - φ δ = sup t [ 0 , T ] f ( t ) - φ δ ( t ) X < δ ; \|f-\varphi_{\delta}\|_{\infty}=\sup_{t\in[0,T]}\|f(t)-\varphi_{\delta}(t)\|_{% X}<\delta;
  2. Reg ( [ 0 , T ] ; X ) = BV ( [ 0 , T ] ; X ) ¯ w.r.t. . \mathrm{Reg}([0,T];X)=\overline{\mathrm{BV}([0,T];X)}\mbox{ w.r.t. }~{}\|\cdot% \|_{\infty}.
  3. Reg ( [ 0 , T ] ; X ) = φ BV φ ( [ 0 , T ] ; X ) . \mathrm{Reg}([0,T];X)=\bigcup_{\varphi}\mathrm{BV}_{\varphi}([0,T];X).
  4. ϵ > 0 \epsilon>0
  5. ϵ \epsilon

Relative_change_and_difference.html

  1. Relative change ( x , x r e f e r e n c e ) = Actual change x r e f e r e n c e = Δ x r e f e r e n c e = x - x r e f e r e n c e x r e f e r e n c e . \,\text{Relative change}(x,x_{reference})=\frac{\,\text{Actual change}}{x_{% reference}}=\frac{\Delta}{x_{reference}}=\frac{x-x_{reference}}{x_{reference}}.
  2. Relative change ( x , x r e f e r e n c e ) = Actual change | x r e f e r e n c e | = Δ | x r e f e r e n c e | = x - x r e f e r e n c e | x r e f e r e n c e | . \,\text{Relative change}(x,x_{reference})=\frac{\,\text{Actual change}}{|x_{% reference}|}=\frac{\Delta}{|x_{reference}|}=\frac{x-x_{reference}}{|x_{% reference}|}.
  3. Relative difference ( x , y ) = Absolute difference | f ( x , y ) | = | Δ | | f ( x , y ) | = | x - y f ( x , y ) | . \,\text{Relative difference}(x,y)=\frac{\,\text{Absolute difference}}{|f(x,y)|% }=\frac{|\Delta|}{|f(x,y)|}=\left|\frac{x-y}{f(x,y)}\right|.
  4. d r = | x - y | max ( | x | , | y | ) d_{r}=\frac{|x-y|}{\max(|x|,|y|)}\,
  5. d r = | x - y | ( | x + y | 2 ) . d_{r}=\frac{|x-y|}{\left(\frac{|x+y|}{2}\right)}\,.
  6. | x + y | 2 = 0 , \frac{|x+y|}{2}=0,
  7. d r = | x - y | ( | x | + | y | 2 ) . d_{r}=\frac{|x-y|}{\left(\frac{|x|+|y|}{2}\right)}\,.
  8. % Error = | Experimental - Theoretical | | Theoretical | × 100 \%\,\text{ Error}=\frac{|\,\text{Experimental}-\,\text{Theoretical}|}{|\,\text% {Theoretical}|}\times 100
  9. % Error = Experimental - Theoretical | Theoretical | × 100. \%\,\text{ Error}=\frac{\,\text{Experimental}-\,\text{Theoretical}}{|\,\text{% Theoretical}|}\times 100.
  10. 110000 - 100000 100000 = 0.1 = 10 % . \frac{110000-100000}{100000}=0.1=10\%.
  11. Percentage change = Δ V V 1 = V 2 - V 1 V 1 × 100. \,\text{Percentage change}=\frac{\Delta V}{V_{1}}=\frac{V_{2}-V_{1}}{V_{1}}% \times 100.
  12. 4 % - 3 % 3 % = 0.333 = 33 1 3 % . \frac{4\%-3\%}{3\%}=0.333\ldots=33\frac{1}{3}\%.
  13. 33 1 3 % . 33\frac{1}{3}\%.
  14. D c N p = 100 ln V 2 V 1 100 V 2 - V 1 V 1 = Percentage change when | V 2 - V 1 V 1 | 1 D_{cNp}=100\cdot\ln\frac{V_{2}}{V_{1}}\approx 100\cdot\frac{V_{2}-V_{1}}{V_{1}% }=\,\text{Percentage change}\,\text{ when }\left|\frac{V_{2}-V_{1}}{V_{1}}% \right|<<1\,
  15. $ 10 , 000 $ 40 , 000 = 0.25 = 25 % , \frac{\$10,000}{\$40,000}=0.25=25\%,
  16. $ 50 , 000 $ 40 , 000 = 1.25 = 125 % , \frac{\$50,000}{\$40,000}=1.25=125\%,
  17. - $ 10 , 000 $ 50 , 000 = - 0.20 = - 20 % \frac{-\$10,000}{\$50,000}=-0.20=-20\%
  18. $ 40 , 000 $ 50 , 000 = 0.8 = 80 % \frac{\$40,000}{\$50,000}=0.8=80\%

Relative_interior.html

  1. relint ( S ) \operatorname{relint}(S)
  2. relint ( S ) := { x S : ϵ > 0 , N ϵ ( x ) aff ( S ) S } , \operatorname{relint}(S):=\{x\in S:\exists\epsilon>0,N_{\epsilon}(x)\cap% \operatorname{aff}(S)\subseteq S\},
  3. aff ( S ) \operatorname{aff}(S)
  4. N ϵ ( x ) N_{\epsilon}(x)
  5. ϵ \epsilon
  6. x x
  7. C n C\subseteq\mathbb{R}^{n}
  8. relint ( C ) := { x C : y C λ > 1 : λ x + ( 1 - λ ) y C } . \operatorname{relint}(C):=\{x\in C:\forall{y\in C}\;\exists{\lambda>1}:\lambda x% +(1-\lambda)y\in C\}.

Relative_permeability.html

  1. q i = - κ i μ i P i for i = 1 , 2 q_{i}=-\frac{\kappa_{i}}{\mu_{i}}\nabla P_{i}\qquad\,\text{for}\quad i=1,2
  2. q i q_{i}
  3. P i \nabla P_{i}
  4. μ i \mu_{i}
  5. i i
  6. i i
  7. κ i \kappa_{i}
  8. i i
  9. κ 𝑟𝑖 \kappa_{\mathit{ri}}
  10. i i
  11. κ i = κ 𝑟𝑖 κ \kappa_{i}=\kappa_{\mathit{ri}}\kappa
  12. κ 𝑟𝑖 = κ i / κ \kappa_{\mathit{ri}}=\kappa_{i}/\kappa
  13. κ \kappa
  14. S w S_{w}
  15. S 𝑤𝑖 S_{\mathit{wi}}
  16. S 𝑤𝑖𝑟 S_{\mathit{wir}}
  17. S 𝑤𝑟 S_{\mathit{wr}}
  18. S 𝑤𝑐 S_{\mathit{wc}}
  19. S 𝑜𝑟𝑤 S_{\mathit{orw}}
  20. S 𝑤𝑛 = S 𝑤𝑛 ( S w ) = S w - S 𝑤𝑖 1 - S 𝑤𝑖 - S 𝑜𝑟𝑤 S_{\mathit{wn}}=S_{\mathit{wn}}(S_{w})=\frac{S_{w}-S_{\mathit{wi}}}{1-S_{% \mathit{wi}}-S_{\mathit{orw}}}
  21. K 𝑟𝑜𝑤 ( S w ) = ( 1 - S 𝑤𝑛 ) N o K_{\mathit{row}}(S_{w})=(1-S_{\mathit{wn}})^{N_{\mathit{o}}}
  22. K 𝑟𝑤 ( S w ) = K S 𝑤𝑛 N w 𝑟𝑤 o K_{\mathit{rw}}(S_{w})=K{{}_{\mathit{rw}}^{o}}S_{\mathit{wn}}^{N_{\mathit{w}}}
  23. K 𝑟𝑜𝑤 ( S 𝑤𝑖 ) = 1 K 𝑟𝑜𝑤 ( 1 - S 𝑜𝑟𝑤 ) = 0 K 𝑟𝑤 ( S 𝑤𝑖 ) = 0 K 𝑟𝑤 ( 1 - S 𝑜𝑟𝑤 ) = K 𝑟𝑤 o \begin{aligned}\displaystyle K_{\mathit{row}}(S_{\mathit{wi}})&\displaystyle=1% &\displaystyle K_{\mathit{row}}(1-S_{\mathit{orw}})&\displaystyle=0\\ \displaystyle K_{\mathit{rw}}(S_{\mathit{wi}})&\displaystyle=0&\displaystyle K% _{\mathit{rw}}(1-S_{\mathit{orw}})&\displaystyle=K_{\mathit{rw}}^{o}\end{aligned}
  24. N o N_{\mathit{o}}
  25. N w N_{\mathit{w}}
  26. N o N_{\mathit{o}}
  27. N w = 2 N_{\mathit{w}}=2
  28. K 𝑟𝑤 o K_{\mathit{rw}}^{o}
  29. N o N_{\mathit{o}}
  30. N w N_{\mathit{w}}
  31. S w n S_{wn}
  32. K 𝑟𝑤 = K 𝑟𝑤 o S 𝑤𝑛 L w S 𝑤𝑛 L w + E w ( 1 - S 𝑤𝑛 ) T w K_{\mathit{rw}}=\frac{{K_{\mathit{rw}}^{o}}S_{\mathit{wn}}^{L_{\mathit{w}}}}{{% S_{\mathit{wn}}}^{L_{\mathit{w}}}+{E_{\mathit{w}}}{(1-S_{\mathit{wn}})}^{T_{% \mathit{w}}}}
  33. K 𝑟𝑜𝑤 = ( 1 - S 𝑤𝑛 ) L o ( 1 - S 𝑤𝑛 ) L o + E o S 𝑤𝑛 T o K_{\mathit{row}}=\frac{(1-S_{\mathit{wn}})^{L_{o}}}{{(1-S_{\mathit{wn}})^{L_{o% }}}+{E_{\mathit{o}}}S_{\mathit{wn}}^{T_{\mathit{o}}}}
  34. S w S_{w}
  35. S 𝑤𝑖 S_{\mathit{wi}}
  36. S 𝑜𝑟𝑤 S_{\mathit{orw}}
  37. K 𝑟𝑤 o K_{\mathit{rw}}^{o}

Relaxation_(iterative_method).html

  1. d 2 φ ( x ) d x 2 = φ ( x - h ) - 2 φ ( x ) + φ ( x + h ) h 2 + 𝒪 ( h 2 ) . \frac{d^{2}\varphi(x)}{{dx}^{2}}=\frac{\varphi(x{-}h)-2\varphi(x)+\varphi(x{+}% h)}{h^{2}}\,+\,\mathcal{O}(h^{2})\,.
  2. φ ( x , y ) = 1 4 ( φ ( x + h , y ) + φ ( x , y + h ) + φ ( x - h , y ) + φ ( x , y - h ) - h 2 2 φ ( x , y ) ) + 𝒪 ( h 4 ) . \varphi(x,y)=\tfrac{1}{4}\left(\varphi(x{+}h,y)+\varphi(x,y{+}h)+\varphi(x{-}h% ,y)+\varphi(x,y{-}h)\,-\,h^{2}{\nabla}^{2}\varphi(x,y)\right)\,+\,\mathcal{O}(% h^{4})\,.
  3. 2 φ = f {\nabla}^{2}\varphi=f\,
  4. φ * ( x , y ) = 1 4 ( φ ( x + h , y ) + φ ( x , y + h ) + φ ( x - h , y ) + φ ( x , y - h ) - h 2 f ( x , y ) ) , \varphi^{*}(x,y)=\tfrac{1}{4}\left(\varphi(x{+}h,y)+\varphi(x,y{+}h)+\varphi(x% {-}h,y)+\varphi(x,y{-}h)\,-\,h^{2}f(x,y)\right)\,,

Remez_inequality.html

  1. | p ( x ) | 1 |p(x)|\leq 1
  2. sup p π n ( σ ) p = T n \sup_{p\in\pi_{n}(\sigma)}\|p\|_{\infty}=\|T_{n}\|_{\infty}
  3. [ 1 , + ] [1,+\infty]
  4. T n = T n ( 1 + σ ) . \|T_{n}\|_{\infty}=T_{n}(1+\sigma).
  5. max x J | p ( x ) | ( 4 mes J mes E ) n sup x E | p ( x ) | ( * ) \max_{x\in J}|p(x)|\leq\left(\frac{4\,\,\textrm{mes }J}{\textrm{mes }E}\right)% ^{n}\sup_{x\in E}|p(x)|\qquad\qquad(*)
  6. p ( x ) = k = 1 n a k e λ k x p(x)=\sum_{k=1}^{n}a_{k}e^{\lambda_{k}x}
  7. max x J | p ( x ) | e max k | λ k | mes J ( C mes J mes E ) n - 1 sup x E | p ( x ) | , \max_{x\in J}|p(x)|\leq e^{\max_{k}|\Re\lambda_{k}|\,\mathrm{mes}J}\left(\frac% {C\,\,\textrm{mes}J}{\textrm{mes}E}\right)^{n-1}\sup_{x\in E}|p(x)|~{},
  8. L p ( 𝕋 ) , 0 p 2 L^{p}(\mathbb{T}),\ 0\leq p\leq 2
  9. p L p ( 𝕋 ) e A ( n - 1 ) mes ( 𝕋 E ) p L p ( E ) \|p\|_{L^{p}(\mathbb{T})}\leq e^{A(n-1)\textrm{mes }(\mathbb{T}\setminus E)}\|% p\|_{L^{p}(E)}
  10. mes E < 1 - log n n \mathrm{mes}E<1-\frac{\log n}{n}
  11. E = E λ = { x : | p ( x ) | λ } , λ > 0 E=E_{\lambda}=\{x\,:\ |p(x)|\leq\lambda\},\ \lambda>0
  12. max x J | p ( x ) | e max k | λ k | mes J ( C mes J mes E λ ) n - 1 sup x E λ | p ( x ) | , e max k | λ k | mes J ( C mes J mes E λ ) n - 1 λ \max_{x\in J}|p(x)|\leq e^{\max_{k}|\Re\lambda_{k}|\,\mathrm{mes}J}\left(\frac% {C\,\,\textrm{mes}J}{\textrm{mes}E_{\lambda}}\right)^{n-1}\sup_{x\in E_{% \lambda}}|p(x)|~{},\leq e^{\max_{k}|\Re\lambda_{k}|\,\mathrm{mes}J}\left(\frac% {C\,\,\textrm{mes}J}{\textrm{mes}E_{\lambda}}\right)^{n-1}\lambda
  13. mes E λ C mes J ( λ e max k | λ k | mes J max x J | p ( x ) | ) 1 / ( n - 1 ) \textrm{mes}E_{\lambda}\leq C\,\,\textrm{mes}J\left(\frac{\lambda e^{\max_{k}|% \Re\lambda_{k}|\,\mathrm{mes}J}}{\max_{x\in J}|p(x)|}\right)^{1/(n-1)}
  14. E E
  15. λ \lambda
  16. mes E λ mes E 2 \textrm{mes}E_{\lambda}\leq\frac{\textrm{mes}E}{2}
  17. λ = ( mes E 2 C mes J ) n - 1 e - max k | λ k | mes J max x J | p ( x ) | \lambda=\left(\frac{\textrm{mes}E}{2C\mathrm{mes}J}\right)^{n-1}e^{-\max_{k}|% \Re\lambda_{k}|\,\mathrm{mes}J}\max_{x\in J}|p(x)|
  18. mes E ( J E λ ) mes E - mes E λ mes E 2 \textrm{mes}E\cap(J\setminus E_{\lambda})\geq\textrm{mes}E-\textrm{mes}E_{% \lambda}\geq\frac{\textrm{mes}E}{2}
  19. x E | p ( x ) | p d x x E ( J E λ ) | p ( x ) | p d x λ p mes E ( J E λ ) mes E 2 ( mes E 2 C mes J ) p ( n - 1 ) e - p max k | λ k | mes J max x J | p ( x ) | p mes E 2 mes J ( mes E 2 C mes J ) p ( n - 1 ) e - p max k | λ k | mes J x J | p ( x ) | p d x \begin{aligned}\displaystyle\int_{x\in E}|p(x)|^{p}\,\mbox{d}~{}x&% \displaystyle\geq&\displaystyle\int_{x\in E\cap(J\setminus E_{\lambda})}|p(x)|% ^{p}\,\mbox{d}~{}x\\ &\displaystyle\geq&\displaystyle\lambda^{p}\mathrm{mes}E\cap(J\setminus E_{% \lambda})\\ &\displaystyle\geq&\displaystyle\frac{\textrm{mes}E}{2}\left(\frac{\textrm{mes% }E}{2C\mathrm{mes}J}\right)^{p(n-1)}e^{-p\max_{k}|\Re\lambda_{k}|\,\mathrm{mes% }J}\max_{x\in J}|p(x)|^{p}\\ &\displaystyle\geq&\displaystyle\frac{\textrm{mes}E}{2\textrm{mes}J}\left(% \frac{\textrm{mes}E}{2C\mathrm{mes}J}\right)^{p(n-1)}e^{-p\max_{k}|\Re\lambda_% {k}|\,\mathrm{mes}J}\int_{x\in J}|p(x)|^{p}\,\mbox{d}~{}x\end{aligned}
  20. mes { x | P ( x ) | a } 4 ( a 2 L C ( p ) ) 1 / n , a > 0 . \textrm{mes}\left\{x\in\mathbb{R}\,\mid\,|P(x)|\leq a\right\}\leq 4\left(\frac% {a}{2\mathrm{LC}(p)}\right)^{1/n}~{},\quad a>0~{}.

Rename_(relational_algebra).html

  1. ρ a / b ( R ) \rho_{a/b}(R)
  2. R R
  3. a a
  4. b b
  5. b b
  6. R R
  7. R R
  8. b b
  9. a a
  10. ρ \rho
  11. E m p l o y e e Employee
  12. E m p l o y e e Employee
  13. ρ E m p l o y e e N a m e / N a m e ( E m p l o y e e ) \rho_{EmployeeName/Name}(Employee)
  14. ρ a / b ( R ) = { t [ a / b ] : t R } \rho_{a/b}(R)=\{\ t[a/b]:t\in R\ \}
  15. t [ a / b ] t[a/b]
  16. t t
  17. b b
  18. a a
  19. t [ a / b ] = { ( c , v ) | ( c , v ) t , c b } { ( a , t ( b ) ) } t[a/b]=\{\ (c,v)\ |\ (c,v)\in t,\ c\neq b\ \}\cup\{\ (a,\ t(b))\ \}

Renormalon.html

  1. ( Λ / Q ) p \left(\Lambda/Q\right)^{p}
  2. Q Q
  3. Λ \Lambda
  4. ln ( Λ / Q ) \ln\left(\Lambda/Q\right)
  5. N N
  6. N N
  7. N ! N!
  8. N N
  9. N ! N!
  10. ϕ 4 \phi^{4}
  11. β ( g ) \beta(g)
  12. β ( g ) \beta(g)
  13. ϕ 4 \phi^{4}

Representation_theory_of_SL2(R).html

  1. [ H , X ] = 2 X , [ H , Y ] = - 2 Y , [ X , Y ] = H . [H,X]=2X,\quad[H,Y]=-2Y,\quad[X,Y]=H.
  2. H = ( 0 - i i 0 ) H=\begin{pmatrix}0&-i\\ i&0\end{pmatrix}
  3. ( cos ( θ ) - sin ( θ ) sin ( θ ) cos ( θ ) ) \begin{pmatrix}\cos(\theta)&-\sin(\theta)\\ \sin(\theta)&\cos(\theta)\end{pmatrix}
  4. X = 1 2 ( 1 i i - 1 ) X={1\over 2}\begin{pmatrix}1&i\\ i&-1\end{pmatrix}
  5. Y = 1 2 ( 1 - i - i - 1 ) Y={1\over 2}\begin{pmatrix}1&-i\\ -i&-1\end{pmatrix}
  6. Ω = H 2 + 1 + 2 X Y + 2 Y X . \Omega=H^{2}+1+2XY+2YX.
  7. H ( w j ) = j w j H(w_{j})=jw_{j}
  8. X ( w j ) = μ + j + 1 2 w j + 2 X(w_{j})={\mu+j+1\over 2}w_{j+2}
  9. Y ( w j ) = μ - j + 1 2 w j - 2 Y(w_{j})={\mu-j+1\over 2}w_{j-2}

Rescaled_range.html

  1. X = X 1 , X 2 , , X n X=X_{1},X_{2},\dots,X_{n}\,
  2. m = 1 n i = 1 n X i m=\frac{1}{n}\sum_{i=1}^{n}X_{i}\,
  3. Y t = X t - m for t = 1 , 2 , , n Y_{t}=X_{t}-m\,\text{ for }t=1,2,\dots,n\,
  4. Z t = i = 1 t Y i for t = 1 , 2 , , n Z_{t}=\sum_{i=1}^{t}Y_{i}\,\text{ for }t=1,2,\dots,n\,
  5. R t = max ( Z 1 , Z 2 , , Z t ) - min ( Z 1 , Z 2 , , Z t ) for t = 1 , 2 , , n R_{t}=\max\left(Z_{1},Z_{2},\dots,Z_{t}\right)-\min\left(Z_{1},Z_{2},\dots,Z_{% t}\right)\,\text{ for }t=1,2,\dots,n\,
  6. S t = 1 t i = 1 t ( X i - m ( t ) ) 2 for t = 1 , 2 , , n S_{t}=\sqrt{\frac{1}{t}\sum_{i=1}^{t}\left(X_{i}-m(t)\right)^{2}}\,\text{ for % }t=1,2,\dots,n\,
  7. t t
  8. X 1 , X 2 , , X t X_{1},X_{2},\dots,X_{t}\,
  9. ( R / S ) t = R t S t for t = 1 , 2 , , n \left(R/S\right)_{t}=\frac{R_{t}}{S_{t}}\,\text{ for }t=1,2,\dots,n\,
  10. S S
  11. R R
  12. S S
  13. S ^ \hat{S}
  14. S ^ 2 = S 2 + 2 j = 1 q ( 1 - j q + 1 ) C ( j ) , \hat{S}^{2}=S^{2}+2\sum_{j=1}^{q}\left(1-\frac{j}{q+1}\right)C(j),
  15. q q
  16. C ( j ) C(j)
  17. j j

Reserve_design.html

  1. S = c A z S=cA^{z}

Residuated_Boolean_algebra.html

  1. ˘ \breve{}
  2. ˘ ˘ \breve{\ }\breve{\ }
  3. ˘ \breve{}
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  7. ˘ ˘ \breve{\ }\breve{\ }
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  10. ˘ ˘ \breve{\ }\breve{\ }
  11. ˘ \breve{\ }
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Residue_field.html

  1. k [ t ] ( t - a ) / ( t - a ) k [ t ] ( t - a ) k k[t]_{(t-a)}/(t-a)k[t]_{(t-a)}\cong k
  2. k [ t ] ( 0 ) k ( t ) k[t]_{(0)}\cong k(t)

Resolvent_set.html

  1. L : D ( L ) X L\colon D(L)\rightarrow X
  2. D ( L ) X D(L)\subseteq X
  3. λ \lambda\in\mathbb{C}
  4. L λ = L - λ id . L_{\lambda}=L-\lambda\mathrm{id}.
  5. λ \lambda
  6. R ( λ , L ) R(\lambda,L)
  7. L λ L_{\lambda}
  8. ρ ( L ) = { λ | λ is a regular value of L } . \rho(L)=\{\lambda\in\mathbb{C}|\lambda\mbox{ is a regular value of }~{}L\}.
  9. σ ( L ) = ρ ( L ) . \sigma(L)=\mathbb{C}\setminus\rho(L).
  10. ρ ( L ) \rho(L)\subseteq\mathbb{C}

Respiratory_minute_volume.html

  1. V ˙ \dot{V}
  2. V ˙ = V T × f \dot{V}=V_{T}\times f
  3. V ˙ = V ˙ A + V ˙ D \dot{V}=\dot{V}_{A}+\dot{V}_{D}
  4. V ˙ A \dot{V}_{A}
  5. V ˙ D \dot{V}_{D}

Restricted_partial_quotients.html

  1. x = [ a 0 ; a 1 , a 2 , ] = a 0 + 1 a 1 + 1 a 2 + 1 a 3 + 1 a 4 + = a 0 + K i = 1 1 a i , x=[a_{0};a_{1},a_{2},\dots]=a_{0}+\cfrac{1}{a_{1}+\cfrac{1}{a_{2}+\cfrac{1}{a_% {3}+\cfrac{1}{a_{4}+\ddots}}}}=a_{0}+\underset{i=1}{\overset{\infty}{K}}\frac{% 1}{a_{i}},\,
  2. ζ = [ a 0 ; a 1 , a 2 , , a k , a k + 1 , a k + 2 , , a k + m ¯ ] , \zeta=[a_{0};a_{1},a_{2},\dots,a_{k},\overline{a_{k+1},a_{k+2},\dots,a_{k+m}}],\,
  3. CF ( M ) = { [ 0 ; a 1 , a 2 , a 3 , ] : 1 a i M } . \mathrm{CF}(M)=\{[0;a_{1},a_{2},a_{3},\dots]:1\leq a_{i}\leq M\}.\,
  4. ( 2 × [ 0 ; M , 1 ¯ ] , 2 × [ 0 ; 1 , M ¯ ] ) = ( 1 M [ M 2 + 4 M - M ] , M 2 + 4 M - M ) . (2\times[0;\overline{M,1}],2\times[0;\overline{1,M}])=\left(\frac{1}{M}\left[% \sqrt{M^{2}+4M}-M\right],\sqrt{M^{2}+4M}-M\right).
  5. [ 0 ; 1 , M ¯ ] - [ 0 ; M , 1 ¯ ] 1 2 {\scriptstyle[0;\overline{1,M}]-[0;\overline{M,1}]\geq\frac{1}{2}}

Retarded_potential.html

  1. φ = - ρ ϵ 0 , 𝐀 = - μ 0 𝐉 \Box\varphi=-\dfrac{\rho}{\epsilon_{0}}\,,\quad\Box\mathbf{A}=-\mu_{0}\mathbf{J}
  2. \Box
  3. φ ( 𝐫 , t ) = 1 4 π ϵ 0 ρ ( 𝐫 , t r ) | 𝐫 - 𝐫 | d 3 𝐫 \mathrm{\varphi}(\mathbf{r},t)=\frac{1}{4\pi\epsilon_{0}}\int\frac{\rho(% \mathbf{r}^{\prime},t_{r})}{|\mathbf{r}-\mathbf{r}^{\prime}|}\,\mathrm{d}^{3}% \mathbf{r}^{\prime}
  4. 𝐀 ( 𝐫 , t ) = μ 0 4 π 𝐉 ( 𝐫 , t r ) | 𝐫 - 𝐫 | d 3 𝐫 . \mathbf{A}(\mathbf{r},t)=\frac{\mu_{0}}{4\pi}\int\frac{\mathbf{J}(\mathbf{r}^{% \prime},t_{r})}{|\mathbf{r}-\mathbf{r}^{\prime}|}\,\mathrm{d}^{3}\mathbf{r}^{% \prime}\,.
  5. t r = t - | 𝐫 - 𝐫 | c t_{r}=t-\frac{|\mathbf{r}-\mathbf{r}^{\prime}|}{c}
  6. - 𝐄 = φ + 𝐀 t , 𝐁 = × 𝐀 . -\mathbf{E}=\nabla\varphi+\frac{\partial\mathbf{A}}{\partial t}\,,\quad\mathbf% {B}=\nabla\times\mathbf{A}\,.
  7. t a = t + | 𝐫 - 𝐫 | c t_{a}=t+\frac{|\mathbf{r}-\mathbf{r}^{\prime}|}{c}
  8. \Box
  9. 2 φ = - ρ ϵ 0 , 2 𝐀 = - μ 0 𝐉 , \nabla^{2}\varphi=-\dfrac{\rho}{\epsilon_{0}}\,,\quad\nabla^{2}\mathbf{A}=-\mu% _{0}\mathbf{J}\,,
  10. φ ( 𝐫 ) = 1 4 π ϵ 0 ρ ( 𝐫 ) | 𝐫 - 𝐫 | d 3 𝐫 \mathrm{\varphi}(\mathbf{r})=\frac{1}{4\pi\epsilon_{0}}\int\frac{\rho(\mathbf{% r}^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}\,\mathrm{d}^{3}\mathbf{r}^{\prime}
  11. 𝐀 ( 𝐫 ) = μ 0 4 π 𝐉 ( 𝐫 ) | 𝐫 - 𝐫 | d 3 𝐫 . \mathbf{A}(\mathbf{r})=\frac{\mu_{0}}{4\pi}\int\frac{\mathbf{J}(\mathbf{r}^{% \prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|}\,\mathrm{d}^{3}\mathbf{r}^{\prime}\,.
  12. 2 φ = - ρ ϵ 0 \nabla^{2}\varphi=-\dfrac{\rho}{\epsilon_{0}}
  13. 2 𝐀 - 1 c 2 2 𝐀 t 2 = - μ 0 𝐉 + 1 c 2 ( φ t ) , \nabla^{2}\mathbf{A}-\dfrac{1}{c^{2}}\dfrac{\partial^{2}\mathbf{A}}{\partial t% ^{2}}=-\mu_{0}\mathbf{J}+\dfrac{1}{c^{2}}\nabla\left(\dfrac{\partial\varphi}{% \partial t}\right)\,,
  14. φ ( 𝐫 , t ) = 1 4 π ϵ 0 ρ ( 𝐫 , t ) | 𝐫 - 𝐫 | d 3 𝐫 \varphi(\mathbf{r},t)=\dfrac{1}{4\pi\epsilon_{0}}\int\dfrac{\rho(\mathbf{r}^{% \prime},t)}{|\mathbf{r}-\mathbf{r}^{\prime}|}\mathrm{d}^{3}\mathbf{r}^{\prime}
  15. 𝐀 ( 𝐫 , t ) = 1 4 π ε 0 × d 3 𝐫 0 | 𝐫 - 𝐫 | / c d t r t r 𝐉 ( 𝐫 , t - t r ) | 𝐫 - 𝐫 | 3 × ( 𝐫 - 𝐫 ) . \mathbf{A}(\mathbf{r},t)=\dfrac{1}{4\pi\varepsilon_{0}}\nabla\times\int\mathrm% {d}^{3}\mathbf{r^{\prime}}\int_{0}^{|\mathbf{r}-\mathbf{r}^{\prime}|/c}\mathrm% {d}t_{r}\dfrac{t_{r}\mathbf{J}(\mathbf{r^{\prime}},t-t_{r})}{|\mathbf{r}-% \mathbf{r}^{\prime}|^{3}}\times(\mathbf{r}-\mathbf{r}^{\prime})\,.
  16. φ ( 𝐫 , t ) = 1 4 π 𝐄 ( r , t ) | 𝐫 - 𝐫 | d 3 𝐫 \varphi(\mathbf{r},t)=\dfrac{1}{4\pi}\int\dfrac{\nabla\cdot\mathbf{E}({r}^{% \prime},t)}{|\mathbf{r}-\mathbf{r}^{\prime}|}\mathrm{d}^{3}\mathbf{r}^{\prime}
  17. 𝐀 ( 𝐫 , t ) = 1 4 π × 𝐁 ( r , t ) | 𝐫 - 𝐫 | d 3 𝐫 \mathbf{A}(\mathbf{r},t)=\dfrac{1}{4\pi}\int\dfrac{\nabla\times\mathbf{B}({r}^% {\prime},t)}{|\mathbf{r}-\mathbf{r}^{\prime}|}\mathrm{d}^{3}\mathbf{r}^{\prime}

Retention_distance.html

  1. R D = [ ( n + 1 ) ( n + 1 ) i = 0 n ( R F ( i + 1 ) - R F i ) ] 1 n R_{D}=\Bigg[(n+1)^{(n+1)}\prod^{n}_{i=0}{(R_{F(i+1)}-R_{Fi})\Bigg]^{\frac{1}{n% }}}

Retention_uniformity.html

  1. R U = 1 - 6 ( n + 1 ) n ( 2 n + 1 ) i = 1 n ( R F i - i n + 1 ) 2 R_{U}=1-\sqrt{\frac{6(n+1)}{n(2n+1)}\sum_{i=1}^{n}{\left(R_{Fi}-\frac{i}{n+1}% \right)^{2}}}

Revenue_equivalence.html

  1. { ( x , t ) + I × I | x i { 0 , 1 } , i = 1 I x i = 1 } . \{(x,t)\in\mathbb{R}^{I}_{+}\times\mathbb{R}^{I}|x_{i}\in\{0,1\},\sum_{i=1}^{I% }x_{i}=1\}.
  2. θ i \theta_{i}
  3. u i ( x , t , θ i ) = θ i x i + t i . u_{i}(x,t,\theta_{i})=\theta_{i}x_{i}+t_{i}.
  4. b i ( θ ) = I - 1 I ( θ I - 1 ) b_{i}(\theta)=\frac{I-1}{I}(\theta^{I-1})
  5. i i
  6. b i = v i b_{i}=v_{i}
  7. i i
  8. max j i b j \max_{j\neq i}b_{j}
  9. max j i b j \max_{j\neq i}b_{j}
  10. b ( v ) = E ( max j i v j | v j v j ) b(v)=E(\max_{j\neq i}v_{j}~{}|~{}v_{j}\leq v~{}\forall~{}j)
  11. max j i b j \max_{j\neq i}b_{j}
  12. i i
  13. b ( z ) < v b(z)<v
  14. z z
  15. v v
  16. z z
  17. P r ( max i > 1 v i < z ) Pr(\max_{i>1}v_{i}<z)
  18. E ( max i > 1 v i | v i < z i ) E(\max_{i>1}v_{i}~{}|~{}v_{i}<z~{}\forall~{}i)
  19. P r ( max i > 1 v i < z ) ( v - E ( max i > 1 v i | v i < z i ) ) Pr(\max_{i>1}v_{i}<z)(v-E(\max_{i>1}v_{i}~{}|~{}v_{i}<z~{}\forall~{}i))
  20. X = max i > 1 v i X=\max_{i>1}v_{i}
  21. P r ( X < z ) ( v - E ( X | X z ) ) Pr(X<z)(v-E(X~{}|X\leq z))
  22. E ( X | X z ) P r ( X < z ) = 0 P r ( X < z ) - P r ( X < y ) d y E(X~{}|~{}X\leq z)\cdot Pr(X<z)=\int_{0}^{\infty}Pr(X<z)-Pr(X<y)dy
  23. P r ( X < z ) v - P r ( X < z ) z + 0 z P r ( X < y ) d y Pr(X<z)\cdot v-Pr(X<z)\cdot z+\int_{0}^{z}Pr(X<y)dy
  24. z z
  25. P r ( X < z ) ( v - z ) = 0 v = z Pr(X<z)^{\prime}(v-z)=0\Rightarrow v=z
  26. v v
  27. P r ( X < z ) Pr(X<z)
  28. [ 0 , 1 ] [0,1]
  29. E ( Payment | Player 1 wins ) P ( Player 1 wins ) + E ( Payment | Player 1 loses ) P ( Player 1 loses ) E(\,\text{Payment}~{}|~{}\,\text{Player 1 wins})P(\,\text{Player 1 wins})+E(\,% \text{Payment}~{}|~{}\,\text{Player 1 loses})P(\,\text{Player 1 loses})
  30. p 2 = v 2 p_{2}=v_{2}
  31. p 1 = v 1 p_{1}=v_{1}
  32. E ( v 2 | v 2 < v 1 ) P ( v 2 < v 1 ) + 0 E(v_{2}~{}|~{}v_{2}<v_{1})P(v_{2}<v_{1})+0
  33. v 1 , v 2 v_{1},v_{2}
  34. v 1 2 v 1 = v 1 2 2 \frac{v_{1}}{2}\cdot v_{1}=\frac{v_{1}^{2}}{2}
  35. b ( v ) b(v)
  36. E ( Payment | Player 1 wins ) P ( Player 1 wins ) + E ( Payment | Player 1 loses ) P ( Player 1 loses ) E(\,\text{Payment}~{}|~{}\,\text{Player 1 wins})P(\,\text{Player 1 wins})+E(\,% \text{Payment}~{}|~{}\,\text{Player 1 loses})P(\,\text{Player 1 loses})
  37. b ( v 1 ) v 1 + 0 b(v_{1})\cdot v_{1}+0
  38. b ( v 1 ) v 1 = v 1 2 2 b ( v 1 ) = v 1 2 b(v_{1})\cdot v_{1}=\frac{v_{1}^{2}}{2}\Rightarrow b(v_{1})=\frac{v_{1}}{2}
  39. v 1 2 2 \frac{v_{1}^{2}}{2}
  40. v 1 2 2 = b ( v 1 ) \frac{v_{1}^{2}}{2}=b(v_{1})
  41. v 2 2 \frac{v^{2}}{2}

Reversible_reference_system_propagation_algorithm.html

  1. Γ ( t ) = e i L t Γ ( t = 0 ) \Gamma(t)=e^{iLt}\Gamma(t=0)\,

RevPAR.html

  1. R e v P A R = R o o m s R e v e n u e / R o o m s A v a i l a b l e RevPAR=RoomsRevenue/RoomsAvailable\,

Reynolds_analogy.html

  1. f 2 = h C p × G = k c V a v \frac{f}{2}=\frac{h}{C_{p}\times G}=\frac{k^{\prime}_{c}}{V_{av}}

Rhind_Mathematical_Papyrus.html

  1. 2 / 15 = 1 / 10 + 1 / 30 2/15=1/10+1/30
  2. 2 / 101 = 1 / 101 + 1 / 202 + 1 / 303 + 1 / 606 2/101=1/101+1/202+1/303+1/606
  3. V = [ ( 1 - 1 / 9 ) d ] 2 h V=[(1-1/9)d]^{2}h
  4. V = ( 8 / 9 ) 2 d 2 h = ( 256 / 81 ) r 2 h V=(8/9)^{2}d^{2}h=(256/81)r^{2}h
  5. ( ( 1 + 1 / 3 ) d ) 2 ( ( 2 / 3 ) h ) ((1+1/3)d)^{2}((2/3)h)
  6. ( 32 / 27 ) d 2 h = ( 128 / 27 ) r 2 h (32/27)d^{2}h=(128/27)r^{2}h
  7. ( 256 / 81 ) r 2 h (256/81)r^{2}h
  8. 9 2 - 4 × [ 1 2 ( 3 ) ( 3 ) ] = 63 9^{2}-4\times\left[\frac{1}{2}(3)(3)\right]=63
  9. 64 = 8 2 64=8^{2}
  10. π ( 9 2 ) 2 8 2 \pi\left(\frac{9}{2}\right)^{2}\approx 8^{2}
  11. π 256 81 3.1605 \pi\approx\frac{256}{81}\approx 3.1605
  12. 2 3 n = 1 2 n + 1 6 n \frac{2}{3n}=\frac{1}{2n}+\frac{1}{6n}

Rhodopsin_kinase.html

  1. \rightleftharpoons

Ribbon_knot.html

  1. D 2 D^{2}
  2. D 4 D^{4}
  3. M D 4 = M S 3 M\cap\partial D^{4}=\partial M\subset S^{3}
  4. f : D 4 f:D^{4}\to\mathbb{R}
  5. f ( x , y , z , w ) = x 2 + y 2 + z 2 + w 2 f(x,y,z,w)=x^{2}+y^{2}+z^{2}+w^{2}
  6. M D 4 = S 3 \partial M\subset\partial D^{4}=S^{3}
  7. f | M : M f_{|M}:M\to\mathbb{R}

Richard_J._Lipton.html

  1. 1 + ϵ \vartriangle^{1+\epsilon}
  2. \vartriangle
  3. 1 + ϵ \vartriangle^{1+\epsilon}
  4. f ( x 1 , , x n ) f(x_{1},\dots,x_{n})
  5. P 0 P_{0}
  6. P 1 , , P k - 1 P_{1},\dots,P_{k-1}
  7. a 0 a_{0}
  8. a 1 a_{1}
  9. a k - 1 a_{k-1}
  10. P i P_{i}
  11. a i a_{i}
  12. a i a_{i}
  13. n 1.1 n^{1.1}
  14. n 0.1 n^{0.1}

Riemann_problem.html

  1. [ ρ u ] = [ ρ L u L ] for x 0 and [ ρ u ] = [ ρ R u R ] for x > 0 \begin{bmatrix}\rho\\ u\end{bmatrix}=\begin{bmatrix}\rho_{L}\\ u_{L}\end{bmatrix}\,\text{ for }x\leq 0\qquad\,\text{and}\qquad\begin{bmatrix}% \rho\\ u\end{bmatrix}=\begin{bmatrix}\rho_{R}\\ u_{R}\end{bmatrix}\,\text{ for }x>0
  2. ρ t + ρ 0 u x \displaystyle\frac{\partial\rho}{\partial t}+\rho_{0}\frac{\partial u}{% \partial x}
  3. a 0 a\geq 0
  4. U t + A U x = 0 U_{t}+A\cdot U_{x}=0
  5. U = [ ρ u ] , A = [ 0 ρ 0 a 2 ρ 0 0 ] U=\begin{bmatrix}\rho\\ u\end{bmatrix},\quad A=\begin{bmatrix}0&\rho_{0}\\ \frac{a^{2}}{\rho_{0}}&0\end{bmatrix}
  6. λ 1 = - a , λ 2 = a \lambda_{1}=-a,\lambda_{2}=a
  7. 𝐞 ( 1 ) = [ ρ 0 - a ] , 𝐞 ( 2 ) = [ ρ 0 a ] . \mathbf{e}^{(1)}=\begin{bmatrix}\rho_{0}\\ -a\end{bmatrix},\quad\mathbf{e}^{(2)}=\begin{bmatrix}\rho_{0}\\ a\end{bmatrix}.
  8. u L u_{L}
  9. α 1 , α 2 \alpha_{1},\alpha_{2}
  10. U L = [ ρ L u L ] = α 1 𝐞 ( 1 ) + α 2 𝐞 ( 2 ) . U_{L}=\begin{bmatrix}\rho_{L}\\ u_{L}\end{bmatrix}=\alpha_{1}\mathbf{e}^{(1)}+\alpha_{2}\mathbf{e}^{(2)}.
  11. α 1 \alpha_{1}
  12. α 2 \alpha_{2}
  13. α 1 \displaystyle\alpha_{1}
  14. U R = [ ρ R u R ] = β 1 𝐞 ( 1 ) + β 2 𝐞 ( 2 ) U_{R}=\begin{bmatrix}\rho_{R}\\ u_{R}\end{bmatrix}=\beta_{1}\mathbf{e}^{(1)}+\beta_{2}\mathbf{e}^{(2)}
  15. β 1 \displaystyle\beta_{1}
  16. t = a | x | t=a|x|
  17. U * = [ ρ * u * ] = β 1 𝐞 ( 1 ) + α 2 𝐞 ( 2 ) = β 1 [ ρ 0 - a ] + α 2 [ ρ 0 a ] U_{*}=\begin{bmatrix}\rho_{*}\\ u_{*}\end{bmatrix}=\beta_{1}\mathbf{e}^{(1)}+\alpha_{2}\mathbf{e}^{(2)}=\beta_% {1}\begin{bmatrix}\rho_{0}\\ -a\end{bmatrix}+\alpha_{2}\begin{bmatrix}\rho_{0}\\ a\end{bmatrix}
  18. t > 0 t>0
  19. U ( t , x ) = [ ρ ( t , x ) u ( t , x ) ] = { U L , 0 < t - a x U * , 0 a | x | < t U R , 0 < t a x U(t,x)=\begin{bmatrix}\rho(t,x)\\ u(t,x)\end{bmatrix}=\begin{cases}U_{L},&0<t\leq-ax\\ U_{*},&0\leq a|x|<t\\ U_{R},&0<t\leq ax\end{cases}

Riemann–Hilbert_correspondence.html

  1. d f d z = a z f \frac{df}{dz}=\frac{a}{z}f
  2. d f d z = f \frac{df}{dz}=f
  3. d f d w = - 1 w 2 f . \frac{df}{dw}=-\frac{1}{w^{2}}f.

Riemann–Hilbert_problem.html

  1. M + ( z ) = u ( z ) + i v ( z ) M_{+}(z)=u(z)+iv(z)\!
  2. a ( z ) u ( z ) - b ( z ) v ( z ) = c ( z ) a(z)u(z)-b(z)v(z)=c(z)\!
  3. M - ( z ) = M + ( z ¯ - 1 ) ¯ . M_{-}(z)=\overline{M_{+}\left(\bar{z}^{-1}\right)}.
  4. z = 1 / z ¯ z=1/\bar{z}
  5. M - ( z ) = M + ( z ) ¯ , z Σ . M_{-}(z)=\overline{M_{+}(z)},\quad z\in\Sigma.
  6. a ( z ) + i b ( z ) 2 M + ( z ) + a ( z ) - i b ( z ) 2 M - ( z ) = c ( z ) , \frac{a(z)+ib(z)}{2}M_{+}(z)+\frac{a(z)-ib(z)}{2}M_{-}(z)=c(z),
  7. lim z M - ( z ) = M ¯ + ( 0 ) . \lim_{z\to\infty}M_{-}(z)=\bar{M}_{+}(0).
  8. α ( z ) M + ( z ) + β ( z ) M - ( z ) = c ( z ) \alpha(z)M_{+}(z)+\beta(z)M_{-}(z)=c(z)\,
  9. α ( z ) M + ( z ) + β ( z ) M - ( z ) = c ( z ) \alpha(z)M_{+}(z)+\beta(z)M_{-}(z)=c(z)\,
  10. M + = M - V M_{+}=M_{-}V
  11. log M + ( z ) = log M - ( z ) + log 2. \log M_{+}(z)=\log M_{-}(z)+\log 2.
  12. C + - C - = I C_{+}-C_{-}=I
  13. C + , C - C_{+},C_{-}
  14. 1 2 π i Σ + log 2 ζ - z d ζ - 1 2 π i Σ - log 2 ζ - z d ζ = log 2 when z Σ . \frac{1}{2\pi i}\int_{\Sigma_{+}}\frac{\log{2}}{\zeta-z}d\zeta-\frac{1}{2\pi i% }\int_{\Sigma_{-}}\frac{\log{2}}{\zeta-z}d\zeta=\log{2}\,\,\,\mathrm{when}\,\,% z\in\Sigma.
  15. log M = 1 2 π i Σ log 2 ζ - z d ζ = log 2 2 π i - 1 - z 1 - z 1 ζ d ζ = log 2 2 π i log z - 1 z + 1 , \log M=\frac{1}{2\pi i}\int_{\Sigma}\frac{\log{2}}{\zeta-z}d\zeta=\frac{\log 2% }{2\pi i}\int^{1-z}_{-1-z}\frac{1}{\zeta}d\zeta=\frac{\log 2}{2\pi i}\log{% \frac{z-1}{z+1}},
  16. M ( z ) = ( z - 1 z + 1 ) log 2 2 π i M(z)=\left(\frac{z-1}{z+1}\right)^{\frac{\log{2}}{2\pi i}}
  17. Σ \Sigma
  18. M + ( 0 ) = ( e i π ) log 2 2 π i = e log 2 2 , M - ( 0 ) = ( e - i π ) log 2 2 π i = e - log 2 2 ; M_{+}(0)=(e^{i\pi})^{\frac{\log 2}{2\pi i}}=e^{\frac{\log 2}{2}},M_{-}(0)=(e^{% -i\pi})^{\frac{\log 2}{2\pi i}}=e^{-\frac{\log 2}{2}};
  19. M + ( 0 ) = M - ( 0 ) e log 2 = M - ( 0 ) 2 M_{+}(0)=M_{-}(0)e^{\log{2}}=M_{-}(0)2

Risk_dominance.html

  1. E [ π H ] = p A + ( 1 - p ) C E[\pi_{H}]=pA+(1-p)C
  2. E [ π G ] = p B + ( 1 - p ) D E[\pi_{G}]=pB+(1-p)D
  3. 1 - p 1-p

Risk_measure.html

  1. X X
  2. ρ ( X ) \rho(X)
  3. ρ : { + } \rho:\mathcal{L}\to\mathbb{R}\cup\{+\infty\}
  4. ρ ( 0 ) = 0 \rho(0)=0
  5. If a and Z , then ρ ( Z + a ) = ρ ( Z ) - a \mathrm{If}\;a\in\mathbb{R}\;\mathrm{and}\;Z\in\mathcal{L},\;\mathrm{then}\;% \rho(Z+a)=\rho(Z)-a
  6. If Z 1 , Z 2 and Z 1 Z 2 , then ρ ( Z 2 ) ρ ( Z 1 ) \mathrm{If}\;Z_{1},Z_{2}\in\mathcal{L}\;\mathrm{and}\;Z_{1}\leq Z_{2},\;% \mathrm{then}\;\rho(Z_{2})\leq\rho(Z_{1})
  7. d \mathbb{R}^{d}
  8. m d m\leq d
  9. R : L d p 𝔽 M R:L_{d}^{p}\rightarrow\mathbb{F}_{M}
  10. L d p L_{d}^{p}
  11. d d
  12. 𝔽 M = { D M : D = c l ( D + K M ) } \mathbb{F}_{M}=\{D\subseteq M:D=cl(D+K_{M})\}
  13. K M = K M K_{M}=K\cap M
  14. K K
  15. M M
  16. m m
  17. R R
  18. K M R ( 0 ) and R ( 0 ) - int K M = K_{M}\subseteq R(0)\;\mathrm{and}\;R(0)\cap-\mathrm{int}K_{M}=\emptyset
  19. X L d p , u M : R ( X + u 1 ) = R ( X ) - u \forall X\in L_{d}^{p},\forall u\in M:R(X+u1)=R(X)-u
  20. X 2 - X 1 L d p ( K ) R ( X 2 ) R ( X 1 ) \forall X_{2}-X_{1}\in L_{d}^{p}(K)\Rightarrow R(X_{2})\supseteq R(X_{1})
  21. V a r ( X + a ) = V a r ( X ) V a r ( X ) - a Var(X+a)=Var(X)\neq Var(X)-a
  22. a a\in\mathbb{R}
  23. R A R ( X ) = R ( X ) R_{A_{R}}(X)=R(X)
  24. A R A = A A_{R_{A}}=A
  25. ρ \rho
  26. A ρ = { X L p : ρ ( X ) 0 } A_{\rho}=\{X\in L^{p}:\rho(X)\leq 0\}
  27. R R
  28. A R = { X L d p : 0 R ( X ) } A_{R}=\{X\in L^{p}_{d}:0\in R(X)\}
  29. A A
  30. ρ A ( X ) = inf { u : X + u 1 A } \rho_{A}(X)=\inf\{u\in\mathbb{R}:X+u1\in A\}
  31. A A
  32. R A ( X ) = { u M : X + u 1 A } R_{A}(X)=\{u\in M:X+u1\in A\}
  33. ρ \rho
  34. X 2 X\in\mathcal{L}^{2}
  35. D ( X ) = ρ ( X - 𝔼 [ X ] ) D(X)=\rho(X-\mathbb{E}[X])
  36. ρ ( X ) = D ( X ) - 𝔼 [ X ] \rho(X)=D(X)-\mathbb{E}[X]
  37. ρ \rho
  38. ρ ( X ) > 𝔼 [ - X ] \rho(X)>\mathbb{E}[-X]
  39. ρ ( X ) = 𝔼 [ - X ] \rho(X)=\mathbb{E}[-X]

RiskMetrics.html

  1. Σ \Sigma

River_regime.html

  1. Q = u ¯ b h Q=\bar{u}bh
  2. Q Q
  3. u ¯ \bar{u}
  4. b b
  5. h h
  6. b Q 0.5 b\propto Q^{0.5}
  7. h Q 0.4 h\propto Q^{0.4}
  8. u Q 0.1 u\propto Q^{0.1}
  9. Q Q

RNA_helicase.html

  1. \rightleftharpoons

Robbins_constant.html

  1. 4 + 17 2 - 6 3 - 7 π 105 + ln ( 1 + 2 ) 5 + 2 ln ( 2 + 3 ) 5 . \frac{4+17\sqrt{2}-6\sqrt{3}-7\pi}{105}+\frac{\ln(1+\sqrt{2})}{5}+\frac{2\ln(2% +\sqrt{3})}{5}.
  2. 0.66170718226717623515583. 0.66170718226717623515583.

Robin_boundary_condition.html

  1. Ω \partial\Omega
  2. a u + b u n = g on Ω au+b\frac{\partial u}{\partial n}=g\qquad\,\text{on}~{}\partial\Omega\,
  3. Ω \partial\Omega
  4. Ω \Omega
  5. u / n {\partial u}/{\partial n}
  6. Ω = [ 0 , 1 ] \Omega=[0,1]
  7. a u ( 0 ) - b u ( 0 ) = g ( 0 ) au(0)-bu^{\prime}(0)=g(0)\,
  8. a u ( 1 ) + b u ( 1 ) = g ( 1 ) . au(1)+bu^{\prime}(1)=g(1).\,
  9. [ 0 , 1 ] [0,1]
  10. u x ( 0 ) c ( 0 ) - D c ( 0 ) x = 0 u_{x}(0)\,c(0)-D\frac{\partial c(0)}{\partial x}=0\,

Robotics_conventions.html

  1. L ( p , d ) L(p,d)
  2. p p
  3. L L
  4. d d
  5. x x
  6. t t
  7. x = p + t d x=p+td
  8. p p
  9. d d
  10. L ( p , d ) L(p,d)
  11. d d
  12. p p
  13. p p
  14. d d
  15. L p l ( d , m ) L_{pl}(d,m)
  16. d d
  17. m m
  18. d d
  19. m m
  20. d d
  21. m = p × d m=p\times d
  22. m m
  23. p p
  24. z z
  25. x x
  26. x n = z n × z n - 1 x_{n}=z_{n}\times z_{n-1}
  27. z z
  28. d d
  29. y y
  30. x x
  31. z z
  32. θ \theta\,
  33. z z
  34. x x
  35. x x
  36. d d\,
  37. z z
  38. r r\,
  39. a a
  40. α \alpha
  41. z z
  42. α \alpha\,
  43. z z
  44. z z
  45. L h r ( e x , e y , l x , l y ) L_{hr}(e_{x},e_{y},l_{x},l_{y})
  46. e x e_{x}
  47. e y e_{y}
  48. X X
  49. Y Y
  50. e e
  51. Z Z
  52. e z = ( 1 - e x 2 - e y 2 ) 1 2 e_{z}=(1-e_{x}^{2}-e_{y}^{2})^{\frac{1}{2}}
  53. l x l_{x}
  54. l y l_{y}
  55. X X
  56. Y Y
  57. X X
  58. Y Y
  59. X X
  60. Y Y

Rod_calculus.html

  1. 1 7 \frac{1}{7}
  2. 309 7 \frac{309}{7}
  3. 1 7 \frac{1}{7}
  4. 1 3 \frac{1}{3}
  5. 2 5 \frac{2}{5}
  6. 1 3 \frac{1}{3}
  7. 2 5 \frac{2}{5}
  8. 32 , 450 , 625 59 , 056 , 400 \frac{32,450,625}{59,056,400}
  9. 1 , 298 , 025 2 , 362 , 256 \frac{1,298,025}{2,362,256}
  10. 99 36 = 2 3 4 \frac{99}{36}=2\frac{3}{4}
  11. 1 4 \frac{1}{4}
  12. 1 4 \frac{1}{4}
  13. 234567 484 311 968 \sqrt{234567}\approx 484\tfrac{311}{968}
  14. 234567 484 311 968 \sqrt{234567}\approx 484\tfrac{311}{968}
  15. ( 3 1860867 ) = 123 \sqrt[3]{(}1860867)=123
  16. x 4 = a x^{4}=a
  17. - x 4 + 15245 x 2 - 6262506.25 = 0 -x^{4}+15245x^{2}-6262506.25=0
  18. 32450625 59056400 \frac{32450625}{59056400}
  19. x = 20 1298205 2362256 x=20\frac{1298205}{2362256}
  20. - x 2 - 680 x + 96000 = 0 -x^{2}-680x+96000=0
  21. - y - z - y 2 * x - x + x y z = 0 -y-z-y^{2}*x-x+xyz=0
  22. - y - z + x - x 2 + x z = 0 -y-z+x-x^{2}+xz=0
  23. y 2 - z 2 + x 2 = 0 ; y^{2}-z^{2}+x^{2}=0;
  24. x 4 - 6 x 3 + 4 x 2 + 6 x - 5 = 0 x^{4}-6x^{3}+4x^{2}+6x-5=0

Root-mean-square_deviation.html

  1. θ ^ \hat{\theta}
  2. θ \theta
  3. RMSD ( θ ^ ) = MSE ( θ ^ ) = E ( ( θ ^ - θ ) 2 ) . \operatorname{RMSD}(\hat{\theta})=\sqrt{\operatorname{MSE}(\hat{\theta})}=% \sqrt{\operatorname{E}((\hat{\theta}-\theta)^{2})}.
  4. y ^ t \hat{y}_{t}
  5. y y
  6. RMSD = t = 1 n ( y ^ t - y ) 2 n . \operatorname{RMSD}=\sqrt{\frac{\sum_{t=1}^{n}(\hat{y}_{t}-y)^{2}}{n}}.
  7. x 1 , t x_{1,t}
  8. x 2 , t x_{2,t}
  9. RMSD = t = 1 n ( x 1 , t - x 2 , t ) 2 n . \operatorname{RMSD}=\sqrt{\frac{\sum_{t=1}^{n}(x_{1,t}-x_{2,t})^{2}}{n}}.
  10. NRMSD = RMSD y max - y min \mathrm{NRMSD}=\frac{\mathrm{RMSD}}{y_{\max}-y_{\min}}
  11. CV ( RMSE ) = RMSE y ¯ . \mathrm{CV(RMSE)}=\frac{\mathrm{RMSE}}{\bar{y}}.

Ropelength.html

  1. L ( C ) = Len ( C ) / τ ( C ) {\scriptstyle L(C)\,=\,\operatorname{Len}(C)/\tau(C)}
  2. L ( C ) = Ω ( Cr ( C ) 3 / 4 ) L(C)=\Omega(\operatorname{Cr}(C)^{3/4})\,
  3. L ( C ) = O ( Cr ( C ) log 5 ( Cr ( C ) ) ) L(C)=O(\operatorname{Cr}(C)\log^{5}(\operatorname{Cr}(C)))\,

Rotational_diffusion.html

  1. t t
  2. θ 2 = 2 D r t \langle\theta^{2}\rangle=2D_{r}t\!
  3. D r D_{r}
  4. Ω d = ( d θ / d t ) d r i f t \Omega_{d}=(d\theta/dt)_{drift}
  5. Γ θ \Gamma_{\theta}
  6. Ω d = Γ θ f r \Omega_{d}=\frac{\Gamma_{\theta}}{f_{r}}
  7. f r f_{r}
  8. D r = k B T f r D_{r}=\frac{k_{B}T}{f_{r}}
  9. k B k_{B}
  10. T T
  11. R R
  12. f r , sphere = 8 π η R 3 f_{r,\textrm{sphere}}=8\pi\eta R^{3}\!
  13. η \eta
  14. 1 D rot f t = 2 f = 1 sin θ θ ( sin θ f θ ) + 1 sin 2 θ 2 f ϕ 2 \frac{1}{D_{\mathrm{rot}}}\frac{\partial f}{\partial t}=\nabla^{2}f=\frac{1}{% \sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{\partial f}{% \partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{\partial^{2}f}{\partial% \phi^{2}}
  15. 1 sin θ θ ( sin θ Y l m θ ) + 1 sin 2 θ 2 Y l m ϕ 2 = - l ( l + 1 ) Y l m \frac{1}{\sin\theta}\frac{\partial}{\partial\theta}\left(\sin\theta\frac{% \partial Y^{m}_{l}}{\partial\theta}\right)+\frac{1}{\sin^{2}\theta}\frac{% \partial^{2}Y^{m}_{l}}{\partial\phi^{2}}=-l(l+1)Y^{m}_{l}
  16. f ( θ , ϕ , t ) = l = 0 m = - l l C l m Y l m ( θ , ϕ ) e - t / τ l f(\theta,\phi,t)=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}C_{lm}Y^{m}_{l}(\theta,\phi% )e^{-t/\tau_{l}}
  17. τ l = 1 D rot l ( l + 1 ) \tau_{l}=\frac{1}{D_{\mathrm{rot}}l(l+1)}

Rotor_(electric).html

  1. E = B L ( V s y n - V m ) E=BL(V_{syn}-V_{m})
  2. E E
  3. B B
  4. L L
  5. V s y n V_{syn}
  6. V m V_{m}
  7. F = ( B x I ) L F=(BxI)L
  8. T = F x r T=Fxr
  9. F F
  10. T T
  11. r r
  12. I I
  13. n s n_{s}
  14. n s = 120 f p n_{s}=\frac{120f}{p}
  15. f f
  16. p p
  17. n m n_{m}
  18. s = n s - n m n s × 100 % s=\frac{n_{s}-n_{m}}{n_{s}}\times 100\%
  19. n m = ( 1 - s ) n s n_{m}=(1-s)n_{s}
  20. ω m = ( 1 - s ) ω s \omega_{m}=(1-s)\omega_{s}
  21. n s l i p = n s - n m n_{slip}=n_{s}-n_{m}
  22. f r = s f e f_{r}=sf_{e}

Rotor_(mathematics).html

  1. - v M v - 1 -vMv^{-1}
  2. R M R - 1 . RMR^{-1}.
  3. - v M v , v 2 = 1 , -vMv,\quad v^{2}=1,
  4. R R ~ = R ~ R = 1. R\tilde{R}=\tilde{R}R=1.
  5. R M R ~ RM\tilde{R}

Round-trip_gain.html

  1. G ( x , y , z ) ~{}G(x,y,z)~{}
  2. x ~{}x~{}
  3. y ~{}y~{}
  4. z ~{}z~{}
  5. g ~{}g~{}
  6. g = G ( x ( a ) , y ( a ) , z ( a ) ) d a ~{}g=\int G(x(a),y(a),z(a))~{}{\rm d}a~{}
  7. a ~{}a~{}
  8. x ( a ) ~{}x(a)~{}
  9. y ( a ) ~{}y(a)~{}
  10. z ( a ) ~{}z(a)~{}
  11. G ~{}G~{}
  12. g = 2 G h ~{}g=2Gh~{}
  13. h ~{}h~{}
  14. θ ~{}\theta~{}
  15. β ~{}\beta~{}
  16. g ~{}g~{}
  17. exp ( g ) ( 1 - β - θ ) = 1 ~{}\exp(g)~{}(1-\beta-\theta)=1~{}
  18. g 1 ~{}g~{}\ll 1~{}
  19. g = β + θ ~{}g=\beta+\theta~{}
  20. β ~{}\beta~{}
  21. G ~{}G~{}
  22. g ~{}g~{}
  23. β ~{}\beta~{}
  24. β + θ ~{}\beta+\theta~{}

Rowland_ring.html

  1. 𝐇 \mathbf{H}
  2. 𝐁 \mathbf{B}

Roy's_safety-first_criterion.html

  1. min 𝑖 Pr ( R i < R ¯ ) \underset{i}{\min}\Pr(R_{i}<\underline{R})
  2. Pr ( R i < R ¯ ) \Pr(R_{i}<\underline{R})
  3. R i R_{i}
  4. R ¯ \underline{R}
  5. SFRatio i = E ( R i ) - R ¯ Var ( R i ) \,\text{SFRatio}_{i}=\frac{\,\text{E}(R_{i})-\underline{R}}{\sqrt{\,\text{Var}% (R_{i})}}
  6. E ( R i ) \,\text{E}(R_{i})
  7. Var ( R i ) \sqrt{\,\text{Var}(R_{i})}
  8. R ¯ \underline{R}
  9. SFRatio = Expected Return - Minimum Return standard deviation of Return . \,\text{SFRatio}=\frac{\,\text{ Expected Return - Minimum Return}}{\,\text{% standard deviation of Return}}.
  10. Sharpe ratio = Expected Return - Risk-Free Return standard deviation of (Return - Risk-Free Return) \,\text{Sharpe ratio}=\frac{\,\text{ Expected Return - Risk-Free Return}}{\,% \text{standard deviation of (Return - Risk-Free Return)}}

Rugosity.html

  1. f r = A r / A g f_{r}=A_{r}/A_{g}

Rule_184.html

  1. 1 - ρ ρ \tfrac{1-\rho}{\rho}

Run_average.html

  1. RA = 9 R IP \mathrm{RA}=9\cdot\frac{\mathrm{R}}{\mathrm{IP}}
  2. RA + = 100 lgRA RA \mathrm{RA+}=100\cdot\frac{\mathrm{lgRA}}{\mathrm{RA}}

Ruppeiner_geometry.html

  1. d s 2 = g i j R d x i d x j , ds^{2}=g^{R}_{ij}dx^{i}dx^{j},\,
  2. g i j R = - i j S ( U , N a ) g^{R}_{ij}=-\partial_{i}\partial_{j}S(U,N^{a})
  3. d s R 2 = 1 T d s W 2 ds^{2}_{R}=\frac{1}{T}ds^{2}_{W}\,
  4. g i j W = i j U ( S , N a ) g^{W}_{ij}=\partial_{i}\partial_{j}U(S,N^{a})
  5. S = k B c 3 A 4 G S=\frac{k_{B}c^{3}A}{4G\hbar}
  6. k B k_{B}
  7. c c
  8. G G
  9. A A
  10. S = S ( M , N a ) S=S(M,N^{a})
  11. M M
  12. N a N^{a}
  13. a a
  14. S = S ( M ) S=S(M)

Rupture_field.html

  1. P ( X ) P(X)
  2. K K
  3. P ( X ) K [ X ] P(X)\in K[X]
  4. K K
  5. a a
  6. P ( X ) P(X)
  7. K = K=\mathbb{Q}
  8. P ( X ) = X 3 - 2 P(X)=X^{3}-2
  9. [ 2 3 ] \mathbb{Q}[\sqrt[3]{2}]
  10. P ( X ) P(X)
  11. P ( X ) P(X)
  12. K K
  13. P ( X ) P(X)
  14. K K
  15. K P = K [ X ] / ( P ( X ) ) K_{P}=K[X]/(P(X))
  16. L = K [ a ] L=K[a]
  17. a a
  18. P ( X ) P(X)
  19. f f
  20. f ( k ) = k f(k)=k
  21. k K k\in K
  22. f ( X mod P ) = a f(X\mod P)=a
  23. P P
  24. [ 2 3 ] \mathbb{Q}[\sqrt[3]{2}]
  25. P ( X ) P(X)
  26. ω 2 3 \omega\sqrt[3]{2}
  27. ω 2 2 3 \omega^{2}\sqrt[3]{2}
  28. ω \omega
  29. X 2 + 1 X^{2}+1
  30. \mathbb{R}
  31. \mathbb{C}
  32. X 2 + 1 X^{2}+1
  33. 𝔽 3 \mathbb{F}_{3}
  34. 𝔽 9 \mathbb{F}_{9}
  35. 𝔽 3 \mathbb{F}_{3}
  36. - 1 -1
  37. 𝔽 3 \mathbb{F}_{3}
  38. 𝔽 9 \mathbb{F}_{9}

Russo–Vallois_integral.html

  1. f d g = f g d s \int f\,dg=\int fg^{\prime}\,ds
  2. f f
  3. g g
  4. g g^{\prime}
  5. g ( s + ε ) - g ( s ) ε g(s+\varepsilon)-g(s)\over\varepsilon
  6. H n H_{n}
  7. H , H,
  8. H = ucp- lim n H n , H=\,\text{ucp-}\lim_{n\rightarrow\infty}H_{n},
  9. ε > 0 \varepsilon>0
  10. T > 0 , T>0,
  11. lim n ( sup 0 t T | H n ( t ) - H ( t ) | > ε ) = 0. \lim_{n\rightarrow\infty}\mathbb{P}(\sup_{0\leq t\leq T}|H_{n}(t)-H(t)|>% \varepsilon)=0.
  12. I - ( ε , t , f , d g ) = 1 ε 0 t f ( s ) ( g ( s + ε ) - g ( s ) ) d s I^{-}(\varepsilon,t,f,dg)={1\over\varepsilon}\int_{0}^{t}f(s)(g(s+\varepsilon)% -g(s))\,ds
  13. I + ( ε , t , f , d g ) = 1 ε 0 t f ( s ) ( g ( s ) - g ( s - ε ) ) d s I^{+}(\varepsilon,t,f,dg)={1\over\varepsilon}\int_{0}^{t}f(s)(g(s)-g(s-% \varepsilon))\,ds
  14. [ f , g ] ε ( t ) = 1 ε 0 t ( f ( s + ε ) - f ( s ) ) ( g ( s + ε ) - g ( s ) ) d s . [f,g]_{\varepsilon}(t)={1\over\varepsilon}\int_{0}^{t}(f(s+\varepsilon)-f(s))(% g(s+\varepsilon)-g(s))\,ds.
  15. I - I^{-}
  16. 0 t f d - g = ucp- lim ε I - ( ε , t , f , d g ) . \int_{0}^{t}fd^{-}g=\,\text{ucp-}\lim_{\varepsilon\rightarrow\infty}I^{-}(% \varepsilon,t,f,dg).
  17. I + I^{+}
  18. 0 t f d + g = ucp- lim ε I + ( ε , t , f , d g ) . \int_{0}^{t}f\,d^{+}g=\,\text{ucp-}\lim_{\varepsilon\rightarrow\infty}I^{+}(% \varepsilon,t,f,dg).
  19. [ f , g ] ε [f,g]_{\varepsilon}
  20. [ f , g ] ε = ucp- lim ε [ f , g ] ε ( t ) . [f,g]_{\varepsilon}=\,\text{ucp-}\lim_{\varepsilon\rightarrow\infty}[f,g]_{% \varepsilon}(t).
  21. X , Y X,Y
  22. 0 t H s d X s = 0 t H d - X . \int_{0}^{t}H_{s}\,dX_{s}=\int_{0}^{t}H\,d^{-}X.
  23. [ X ] := [ X , X ] [X]:=[X,X]\,
  24. X X
  25. f C 2 ( ) , f\in C_{2}(\mathbb{R}),
  26. f ( X t ) = f ( X 0 ) + 0 t f ( X s ) d X s + 1 2 0 t f ′′ ( X s ) d [ X ] s . f(X_{t})=f(X_{0})+\int_{0}^{t}f^{\prime}(X_{s})\,dX_{s}+{1\over 2}\int_{0}^{t}% f^{\prime\prime}(X_{s})\,d[X]_{s}.
  27. B p , q λ ( N ) B_{p,q}^{\lambda}(\mathbb{R}^{N})
  28. || f || p , q λ = || f || L p + ( 0 1 | h | 1 + λ q ( || f ( x + h ) - f ( x ) || L p ) q d h ) 1 / q ||f||_{p,q}^{\lambda}=||f||_{L_{p}}+\left(\int_{0}^{\infty}{1\over|h|^{1+% \lambda q}}(||f(x+h)-f(x)||_{L_{p}})^{q}\,dh\right)^{1/q}
  29. q = q=\infty
  30. f B p , q λ , f\in B_{p,q}^{\lambda},
  31. g B p , q 1 - λ , g\in B_{p^{\prime},q^{\prime}}^{1-\lambda},
  32. 1 / p + 1 / p = 1 and 1 / q + 1 / q = 1. 1/p+1/p^{\prime}=1\,\text{ and }1/q+1/q^{\prime}=1.
  33. f d g \int f\,dg
  34. c c
  35. | f d g | c || f || p , q α || g || p , q 1 - α . \left|\int f\,dg\right|\leq c||f||_{p,q}^{\alpha}||g||_{p^{\prime},q^{\prime}}% ^{1-\alpha}.

Ryszard_Engelking.html

  1. 2 μ 2^{\mu}
  2. μ \mu

Saint-Venant's_compatibility_condition.html

  1. ε \varepsilon
  2. u \ u
  3. ε i j = 1 2 ( u i x j + u j x i ) \varepsilon_{ij}=\frac{1}{2}\left(\frac{\partial u_{i}}{\partial x_{j}}+\frac{% \partial u_{j}}{\partial x_{i}}\right)
  4. 1 i , j 3 1\leq i,j\leq 3
  5. n 2 n\geq 2
  6. F F
  7. W ( F ) W(F)
  8. W i j k l = 2 F i j x k x l + 2 F k l x i x j - 2 F i l x j x k - 2 F j k x i x l W_{ijkl}=\frac{\partial^{2}F_{ij}}{\partial x_{k}\partial x_{l}}+\frac{% \partial^{2}F_{kl}}{\partial x_{i}\partial x_{j}}-\frac{\partial^{2}F_{il}}{% \partial x_{j}\partial x_{k}}-\frac{\partial^{2}F_{jk}}{\partial x_{i}\partial x% _{l}}
  9. W W
  10. R i j k l R_{ijkl}
  11. R R
  12. F F
  13. W W
  14. W W
  15. R R
  16. n 2 ( n 2 - 1 ) 12 \frac{n^{2}(n^{2}-1)}{12}
  17. n = 2 n=2
  18. W W
  19. n = 3 n=3
  20. F F
  21. W W
  22. K i 1 i n - 2 j 1 j n - 2 = ϵ i 1 i n - 2 k l ϵ j 1 j n - 2 m p F l m , k p K_{i_{1}...i_{n-2}j_{1}...j_{n-2}}=\epsilon_{i_{1}...i_{n-2}kl}\epsilon_{j_{1}% ...j_{n-2}mp}F_{lm,kp}
  23. ϵ \epsilon
  24. n = 3 n=3
  25. K K
  26. K K
  27. W W
  28. T i j = ( U g ) i j = U i ; j + U j ; i T_{ij}=(\mathcal{L}_{U}g)_{ij}=U_{i;j}+U_{j;i}
  29. W ( T ) W(T)
  30. U U
  31. ( d F ) i 1 i k i k + 1 = F ( i 1 i k , i k + 1 ) (dF)_{i_{1}...i_{k}i_{k+1}}=F_{(i_{1}...i_{k},i_{k+1})}
  32. W W
  33. T T
  34. W i 1 . . i k j 1 j k = V ( i 1 . . i k ) ( j 1 j k ) W_{i_{1}..i_{k}j_{1}...j_{k}}=V_{(i_{1}..i_{k})(j_{1}...j_{k})}
  35. V i 1 . . i k j 1 j k = p = 0 k ( - 1 ) p ( k p ) T i 1 . . i k - p j 1 j p , j p + 1 j k i k - p + 1 i k V_{i_{1}..i_{k}j_{1}...j_{k}}=\sum\limits_{p=0}^{k}(-1)^{p}{k\choose p}T_{i_{1% }..i_{k-p}j_{1}...j_{p},j_{p+1}...j_{k}i_{k-p+1}...i_{k}}
  36. W = 0 W=0
  37. T = d F T=dF
  38. F F

Saint-Venant's_Principle.html

  1. 1 / r i + 2 1/r^{i+2}

Sales_(accounting).html

  1. Net Sales = Gross Sales - ( Customer Discounts , Returns , Allowances ) \mathrm{Net\ Sales}=\mathrm{Gross\ Sales}-\mathrm{(Customer\ Discounts,\ % Returns,\ Allowances)}

Sample_mean_and_sample_covariance.html

  1. x i j x_{ij}
  2. 𝐱 i \mathbf{x}_{i}
  3. 𝐱 ¯ \mathbf{\bar{x}}
  4. x ¯ j \bar{x}_{j}
  5. x ¯ j = 1 N i = 1 N x i j , j = 1 , , K . \bar{x}_{j}=\frac{1}{N}\sum_{i=1}^{N}x_{ij},\quad j=1,\ldots,K.
  6. 𝐱 ¯ = 1 N i = 1 N 𝐱 i . \mathbf{\bar{x}}=\frac{1}{N}\sum_{i=1}^{N}\mathbf{x}_{i}.
  7. 𝐐 = [ q j k ] \textstyle\mathbf{Q}=\left[q_{jk}\right]
  8. q j k = 1 N - 1 i = 1 N ( x i j - x ¯ j ) ( x i k - x ¯ k ) , q_{jk}=\frac{1}{N-1}\sum_{i=1}^{N}\left(x_{ij}-\bar{x}_{j}\right)\left(x_{ik}-% \bar{x}_{k}\right),
  9. q j k q_{jk}
  10. j j
  11. k k
  12. 𝐐 = 1 N - 1 i = 1 N ( 𝐱 i - 𝐱 ¯ ) ( 𝐱 i - 𝐱 ¯ ) T , \mathbf{Q}={1\over{N-1}}\sum_{i=1}^{N}(\mathbf{x}_{i}-\mathbf{\bar{x}})(% \mathbf{x}_{i}-\mathbf{\bar{x}})^{\mathrm{T}},
  13. 𝐅 = [ 𝐱 1 𝐱 2 𝐱 N ] \mathbf{F}=\begin{bmatrix}\mathbf{x}_{1}&\mathbf{x}_{2}&\dots&\mathbf{x}_{N}% \end{bmatrix}
  14. 𝐐 = 1 N - 1 ( 𝐅 - 𝐱 ¯ 1 N T ) ( 𝐅 - 𝐱 ¯ 1 N T ) T \mathbf{Q}=\frac{1}{N-1}(\mathbf{F}-\mathbf{\bar{x}}\,\mathbf{1}_{N}^{\mathrm{% T}})(\mathbf{F}-\mathbf{\bar{x}}\,\mathbf{1}_{N}^{\mathrm{T}})^{\mathrm{T}}
  15. 𝟏 N \mathbf{1}_{N}
  16. 1 1
  17. 𝐱 ¯ \mathbf{\bar{x}}
  18. 𝐌 = 𝐅 T \mathbf{M}=\mathbf{F}^{\mathrm{T}}
  19. 𝐐 = 1 N - 1 ( 𝐌 - 𝟏 N 𝐱 ¯ ) T ( 𝐌 - 𝟏 N 𝐱 ¯ ) . \mathbf{Q}=\frac{1}{N-1}(\mathbf{M}-\mathbf{1}_{N}\mathbf{\bar{x}})^{\mathrm{T% }}(\mathbf{M}-\mathbf{1}_{N}\mathbf{\bar{x}}).
  20. 𝐗 \textstyle\mathbf{X}
  21. N - 1 \textstyle N-1
  22. N \textstyle N
  23. E ( 𝐗 ) \operatorname{E}(\mathbf{X})
  24. q j k = 1 N i = 1 N ( x i j - E ( X j ) ) ( x i k - E ( X k ) ) , q_{jk}=\frac{1}{N}\sum_{i=1}^{N}\left(x_{ij}-\operatorname{E}(X_{j})\right)% \left(x_{ik}-\operatorname{E}(X_{k})\right),
  25. N \textstyle N
  26. q j k = 1 N i = 1 N ( x i j - x ¯ j ) ( x i k - x ¯ k ) q_{jk}=\frac{1}{N}\sum_{i=1}^{N}\left(x_{ij}-\bar{x}_{j}\right)\left(x_{ik}-% \bar{x}_{k}\right)
  27. E ( X j ) E(X_{j})
  28. σ j 2 N , \frac{\sigma^{2}_{j}}{N},
  29. σ j 2 \sigma^{2}_{j}
  30. 𝐱 i \textstyle\,\textbf{x}_{i}
  31. w i 0 \textstyle w_{i}\geq 0
  32. i = 1 N w i = 1. \sum_{i=1}^{N}w_{i}=1.
  33. 𝐱 ¯ \textstyle\mathbf{\bar{x}}
  34. 𝐱 ¯ = i = 1 N w i 𝐱 i . \mathbf{\bar{x}}=\sum_{i=1}^{N}w_{i}\mathbf{x}_{i}.
  35. q j k q_{jk}
  36. 𝐐 \textstyle\mathbf{Q}
  37. q j k = i = 1 N w i ( i = 1 N w i ) 2 - i = 1 N w i 2 i = 1 N w i ( x i j - x ¯ j ) ( x i k - x ¯ k ) . q_{jk}=\frac{\sum_{i=1}^{N}w_{i}}{\left(\sum_{i=1}^{N}w_{i}\right)^{2}-\sum_{i% =1}^{N}w_{i}^{2}}\sum_{i=1}^{N}w_{i}\left(x_{ij}-\bar{x}_{j}\right)\left(x_{ik% }-\bar{x}_{k}\right).
  38. w i = 1 / N \textstyle w_{i}=1/N

Satellite_knot.html

  1. K K
  2. K K^{\prime}
  3. V V
  4. K K^{\prime}
  5. V V
  6. K K^{\prime}
  7. f : V S 3 f\colon V\to S^{3}
  8. K = f ( K ) K=f(K^{\prime})
  9. V V
  10. H H
  11. K K
  12. f ( V ) f(\partial V)
  13. K K
  14. V V
  15. S 3 V S^{3}\setminus V
  16. J J
  17. K J K^{\prime}\cup J
  18. f f
  19. f : V S 3 f\colon V\to S^{3}
  20. f f
  21. V V
  22. f ( V ) f(V)
  23. c 1 , c 2 V c_{1},c_{2}\subset V
  24. f f
  25. l k ( f ( c 1 ) , f ( c 2 ) ) = l k ( c 1 , c 2 ) lk(f(c_{1}),f(c_{2}))=lk(c_{1},c_{2})
  26. K V K^{\prime}\subset\partial V
  27. K K
  28. K K^{\prime}
  29. S 3 S^{3}
  30. V V
  31. K K^{\prime}
  32. K K
  33. K J K^{\prime}\cup J
  34. K K^{\prime}
  35. K J K^{\prime}\cup J
  36. K K
  37. f f
  38. K K
  39. S 3 S^{3}
  40. S 3 S^{3}

Sazonov's_theorem.html

  1. i I T ( e i ) H 2 < + . \sum_{i\in I}\|T(e_{i})\|_{H}^{2}<+\infty.

Scatter_matrix.html

  1. X = [ 𝐱 1 , 𝐱 2 , , 𝐱 n ] X=[\mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{n}]
  2. 𝐱 ¯ = 1 n j = 1 n 𝐱 j \overline{\mathbf{x}}=\frac{1}{n}\sum_{j=1}^{n}\mathbf{x}_{j}
  3. 𝐱 j \mathbf{x}_{j}
  4. X X\,
  5. S = j = 1 n ( 𝐱 j - 𝐱 ¯ ) ( 𝐱 j - 𝐱 ¯ ) T = j = 1 n ( 𝐱 j - 𝐱 ¯ ) ( 𝐱 j - 𝐱 ¯ ) = ( j = 1 n 𝐱 j 𝐱 j T ) - n 𝐱 ¯ 𝐱 ¯ T S=\sum_{j=1}^{n}(\mathbf{x}_{j}-\overline{\mathbf{x}})(\mathbf{x}_{j}-% \overline{\mathbf{x}})^{T}=\sum_{j=1}^{n}(\mathbf{x}_{j}-\overline{\mathbf{x}}% )\otimes(\mathbf{x}_{j}-\overline{\mathbf{x}})=\left(\sum_{j=1}^{n}\mathbf{x}_% {j}\mathbf{x}_{j}^{T}\right)-n\overline{\mathbf{x}}\overline{\mathbf{x}}^{T}
  6. T T
  7. S = X C n X T S=X\,C_{n}\,X^{T}
  8. C n \,C_{n}
  9. C M L = 1 n S . C_{ML}=\frac{1}{n}S.
  10. X X\,
  11. S S\,
  12. X X XX^{\top}

Scattering_from_rough_surfaces.html

  1. Δ z ( 𝐫 ) \Delta_{z}(\mathbf{r})
  2. Δ \Delta
  3. Λ \Lambda
  4. Δ z ( 𝐫 ) Δ z ( 𝐫 ) = Δ 2 exp ( - | 𝐫 - 𝐫 | 2 Λ 2 ) \langle\Delta_{z}(\mathbf{r})\Delta_{z}(\mathbf{r^{\prime}})\rangle=\Delta^{2}% \exp\left(-\frac{|\mathbf{r}-\mathbf{r^{\prime}}|^{2}}{\Lambda^{2}}\right)

Scattering_length.html

  1. lim k 0 k cot δ ( k ) = - 1 a , \lim_{k\to 0}k\cot\delta(k)=-\frac{1}{a}\;,
  2. a a
  3. k k
  4. δ ( k ) \delta(k)
  5. σ e \sigma_{e}
  6. lim k 0 σ e = 4 π a 2 . \lim_{k\to 0}\sigma_{e}=4\pi a^{2}\;.
  7. V ( r ) V(r)
  8. l = 0 l=0
  9. l = 1 , 2 l=1,2
  10. l = 0 l=0
  11. r 0 r_{0}
  12. l = 0 l=0
  13. - 2 2 m u ′′ ( r ) = E u ( r ) , -\frac{\hbar^{2}}{2m}u^{\prime\prime}(r)=Eu(r),
  14. u ( r ) u(r)
  15. r = r 0 r=r_{0}
  16. u ( r 0 ) = 0 u(r_{0})=0
  17. u ( r ) = A sin ( k r + δ s ) u(r)=A\sin(kr+\delta_{s})
  18. k = 2 m E / k=\sqrt{2mE}/\hbar
  19. δ s = - k r 0 \delta_{s}=-k\cdot r_{0}
  20. u ( r 0 ) = 0 u(r_{0})=0
  21. A A
  22. δ s ( k ) - k a s + O ( k 2 ) \delta_{s}(k)\approx-k\cdot a_{s}+O(k^{2})
  23. k k
  24. a s a_{s}
  25. a = r 0 a=r_{0}
  26. a s a_{s}
  27. a s a_{s}
  28. σ \sigma
  29. z z
  30. ψ ( r , θ ) = e i k z + f ( θ ) e i k r r \psi(r,\theta)=e^{ikz}+f(\theta)\frac{e^{ikr}}{r}
  31. f f
  32. d σ / d Ω = | f ( θ ) | 2 d\sigma/d\Omega=|f(\theta)|^{2}
  33. 𝐤 \mathbf{k}
  34. θ \theta
  35. σ = 4 π | f | 2 \sigma=4\pi|f|^{2}
  36. ψ ( r , θ ) \psi(r,\theta)
  37. P l ( cos θ ) P_{l}(\cos\theta)
  38. e i k z 1 2 i k r l = 0 ( 2 l + 1 ) P l ( cos θ ) [ ( - 1 ) l + 1 e - i k r + e i k r ] e^{ikz}\approx\frac{1}{2ikr}\sum_{l=0}^{\infty}(2l+1)P_{l}(\cos\theta)\left[(-% 1)^{l+1}e^{-ikr}+e^{ikr}\right]
  39. l = 0 l=0
  40. ψ ( r , θ ) \psi(r,\theta)
  41. ψ ( r ) = A sin ( k r + δ s ) / r \psi(r)=A\sin(kr+\delta_{s})/r
  42. A A
  43. e i k z e^{ikz}
  44. f = 1 2 i k ( e 2 i δ s - 1 ) δ s / k - a s f=\frac{1}{2ik}(e^{2i\delta_{s}}-1)\approx\delta_{s}/k\approx-a_{s}
  45. σ = 4 π k 2 sin 2 δ s = 4 π a s 2 \sigma=\frac{4\pi}{k^{2}}\sin^{2}\delta_{s}=4\pi a_{s}^{2}

Schatten_class_operator.html

  1. S T S 1 S S p T S q if S S p , T S q and 1 / p + 1 / q = 1. \|ST\|_{S_{1}}\leq\|S\|_{S_{p}}\|T\|_{S_{q}}\ \mbox{if}~{}\ S\in S_{p},\ T\in S% _{q}\mbox{ and }~{}1/p+1/q=1.
  2. S S_{\infty}
  3. p [ 1 , ] p\in[1,\infty]
  4. ϕ : S p S q \phi:S_{p}\rightarrow S_{q}^{\prime}
  5. T tr ( T ) T\mapsto\mathrm{tr}(T\cdot)

Schatten_norm.html

  1. H 1 H_{1}
  2. H 2 H_{2}
  3. T T
  4. H 1 H_{1}
  5. H 2 H_{2}
  6. p [ 1 , ) p\in[1,\infty)
  7. T T
  8. T p := ( n 1 s n p ( T ) ) 1 / p \|T\|_{p}:=\bigg(\sum_{n\geq 1}s^{p}_{n}(T)\bigg)^{1/p}
  9. s 1 ( T ) s 2 ( T ) s n ( T ) 0 s_{1}(T)\geq s_{2}(T)\geq\cdots s_{n}(T)\geq\cdots\geq 0
  10. T T
  11. | T | := ( T * T ) |T|:=\sqrt{(T^{*}T)}
  12. T p p = tr ( | T | p ) . \|T\|_{p}^{p}=\mathrm{tr}(|T|^{p}).
  13. p p
  14. [ 1 , ] [1,\infty]
  15. p = p=\infty
  16. q = 1 q=1
  17. U U
  18. V V
  19. p [ 1 , ] p\in[1,\infty]
  20. U T V p = T p . \|UTV\|_{p}=\|T\|_{p}.
  21. p [ 1 , ] p\in[1,\infty]
  22. q q
  23. 1 p + 1 q = 1 \frac{1}{p}+\frac{1}{q}=1
  24. S ( H 2 , H 3 ) , T ( H 1 , H 2 ) S\in\mathcal{L}(H_{2},H_{3}),T\in\mathcal{L}(H_{1},H_{2})
  25. H 1 , H 2 , H_{1},H_{2},
  26. H 3 H_{3}
  27. S T 1 S p T q . \|ST\|_{1}\leq\|S\|_{p}\|T\|_{q}.
  28. p [ 1 , ] p\in[1,\infty]
  29. S ( H 2 , H 3 ) , T ( H 1 , H 2 ) S\in\mathcal{L}(H_{2},H_{3}),T\in\mathcal{L}(H_{1},H_{2})
  30. H 1 , H 2 , H_{1},H_{2},
  31. H 3 H_{3}
  32. S T p S p T p . \|ST\|_{p}\leq\|S\|_{p}\|T\|_{p}.
  33. 1 p p 1\leq p\leq p^{\prime}\leq\infty
  34. T 1 T p T p T . \|T\|_{1}\geq\|T\|_{p}\geq\|T\|_{p^{\prime}}\geq\|T\|_{\infty}.
  35. H 1 , H 2 H_{1},H_{2}
  36. p [ 1 , ] p\in[1,\infty]
  37. q q
  38. 1 p + 1 q = 1 \frac{1}{p}+\frac{1}{q}=1
  39. S p = sup { | S , T | T q = 1 } , \|S\|_{p}=\sup\{|\langle S,T\rangle|\mid\|T\|_{q}=1\},
  40. S , T = tr ( S * T ) \langle S,T\rangle=\mathrm{tr}(S^{*}T)
  41. . 2 \|.\|_{2}
  42. . 1 \|.\|_{1}
  43. p ( 0 , 1 ) p\in(0,1)
  44. . p \|.\|_{p}
  45. S p ( H 1 , H 2 ) S_{p}(H_{1},H_{2})
  46. S p ( H 1 , H 2 ) S_{p}(H_{1},H_{2})
  47. S p ( H 1 , H 2 ) 𝒦 ( H 1 , H 2 ) S_{p}(H_{1},H_{2})\subseteq\mathcal{K}(H_{1},H_{2})

Schild_regression.html

  1. d r = 1 + [ B ] / K B dr=1+[B]/K_{B}
  2. K d = k - 1 k 1 K_{d}=\frac{k_{-1}}{k_{1}}
  3. [ A R ] = [ R ] t [ A ] [ A ] + K d [AR]=\frac{[R]\,t\,[A]}{[A]+K_{d}}
  4. K d = K d 1 + [ B ] K b K_{d}^{\prime}=K_{d}\frac{1+[B]}{K_{b}}
  5. K d = K d K B + [ B ] K B + [ B ] α K_{d}^{\prime}=K_{d}\frac{K_{B}+[B]}{K_{B}+\frac{[B]}{\alpha}}
  6. [ R ] t = [ R ] t 1 + [ B ] K b [R]\,t^{\prime}=\frac{[R]\,t}{1+\frac{[B]}{K_{b}}}

Schlick's_approximation.html

  1. R ( θ ) = R 0 + ( 1 - R 0 ) ( 1 - cos θ ) 5 R(\theta)=R_{0}+(1-R_{0})(1-\cos\theta)^{5}
  2. R 0 = ( n 1 - n 2 n 1 + n 2 ) 2 R_{0}=\left(\frac{n_{1}-n_{2}}{n_{1}+n_{2}}\right)^{2}
  3. θ \theta
  4. cos θ = ( N V ) \cos\theta=(N\cdot V)
  5. n 1 , n 2 n_{1},\,n_{2}
  6. R 0 R_{0}
  7. θ = 0 \theta=0
  8. n 1 n_{1}

Schouten–Nijenhuis_bracket.html

  1. ω ( a 1 a 2 a p ) = { ω ( a 1 , , a p ) ( ω Ω p M ) 0 ( ω Ω p M ) \omega(a_{1}a_{2}\dots a_{p})=\left\{\begin{matrix}\omega(a_{1},\dots,a_{p})&(% \omega\in\Omega^{p}M)\\ 0&(\omega\not\in\Omega^{p}M)\end{matrix}\right.
  2. [ a 1 a m , b 1 b n ] = i , j ( - 1 ) i + j [ a i , b j ] a 1 a i - 1 a i + 1 a m b 1 b j - 1 b j + 1 b n [a_{1}\cdots a_{m},b_{1}\cdots b_{n}]=\sum_{i,j}(-1)^{i+j}[a_{i},b_{j}]a_{1}% \cdots a_{i-1}a_{i+1}\cdots a_{m}b_{1}\cdots b_{j-1}b_{j+1}\cdots b_{n}
  3. [ f , a 1 a m ] = - i d f ( a 1 a m ) [f,a_{1}\cdots a_{m}]=-i_{df}(a_{1}\cdots a_{m})
  4. ( - 1 ) ( | a | - 1 ) ( | c | - 1 ) [ a , [ b , c ] ] + ( - 1 ) ( | b | - 1 ) ( | a | - 1 ) [ b , [ c , a ] ] + ( - 1 ) ( | c | - 1 ) ( | b | - 1 ) [ c , [ a , b ] ] = 0. (-1)^{(|a|-1)(|c|-1)}[a,[b,c]]+(-1)^{(|b|-1)(|a|-1)}[b,[c,a]]+(-1)^{(|c|-1)(|b% |-1)}[c,[a,b]]=0.\,

Schröder–Bernstein_theorems_for_operator_algebras.html

  1. M = M 0 N 0 M=M_{0}\supset N_{0}
  2. M = M 0 N 0 M 1 . M=M_{0}\supset N_{0}\supset M_{1}.
  3. M = M 0 N 0 M 1 N 1 M 2 N 2 . M=M_{0}\supset N_{0}\supset M_{1}\supset N_{1}\supset M_{2}\supset N_{2}% \supset\cdots.
  4. R = i 0 M i = i 0 N i . R=\cap_{i\geq 0}M_{i}=\cap_{i\geq 0}N_{i}.
  5. M N = def M ( N ) . M\ominus N\stackrel{\mathrm{def}}{=}M\cap(N)^{\perp}.
  6. M = i 0 ( M i N i ) j 0 ( N j M j + 1 ) R M=\oplus_{i\geq 0}(M_{i}\ominus N_{i})\quad\oplus\quad\oplus_{j\geq 0}(N_{j}% \ominus M_{j+1})\quad\oplus R
  7. N 0 = i 1 ( M i N i ) j 0 ( N j M j + 1 ) R . N_{0}=\oplus_{i\geq 1}(M_{i}\ominus N_{i})\quad\oplus\quad\oplus_{j\geq 0}(N_{% j}\ominus M_{j+1})\quad\oplus R.
  8. M i N i M N for all i . M_{i}\ominus N_{i}\sim M\ominus N\quad\mbox{for all}~{}\quad i.
  9. ρ = ρ 1 ρ 1 σ 1 where σ 1 σ . \rho=\rho_{1}\simeq\rho_{1}^{\prime}\oplus\sigma_{1}\quad\mbox{where}~{}\quad% \sigma_{1}\simeq\sigma.
  10. ρ 1 ρ 1 ( σ 1 ρ 2 ) where ρ 2 ρ . \rho_{1}\simeq\rho_{1}^{\prime}\oplus(\sigma_{1}^{\prime}\oplus\rho_{2})\quad% \mbox{where}~{}\quad\rho_{2}\simeq\rho.
  11. ρ 1 ρ 1 σ 1 ρ 2 σ 2 ( i 1 ρ i ) ( i 1 σ i ) , \rho_{1}\simeq\rho_{1}^{\prime}\oplus\sigma_{1}^{\prime}\oplus\rho_{2}^{\prime% }\oplus\sigma_{2}^{\prime}\cdots\simeq(\oplus_{i\geq 1}\rho_{i}^{\prime})% \oplus(\oplus_{i\geq 1}\sigma_{i}^{\prime}),
  12. σ 1 σ 1 ρ 2 σ 2 ( i 2 ρ i ) ( i 1 σ i ) . \sigma_{1}\simeq\sigma_{1}^{\prime}\oplus\rho_{2}^{\prime}\oplus\sigma_{2}^{% \prime}\cdots\simeq(\oplus_{i\geq 2}\rho_{i}^{\prime})\oplus(\oplus_{i\geq 1}% \sigma_{i}^{\prime}).
  13. ρ i ρ j and σ i σ j for all i , j . \rho_{i}^{\prime}\simeq\rho_{j}^{\prime}\quad\mbox{and}~{}\quad\sigma_{i}^{% \prime}\simeq\sigma_{j}^{\prime}\quad\mbox{for all}~{}\quad i,j\;.

Schubert_variety.html

  1. dim ( V W j ) j \dim(V\cap W_{j})\geq j
  2. W 1 W 2 W k , dim W j = a j W_{1}\subset W_{2}\subset\cdots\subset W_{k},\quad\dim W_{j}=a_{j}

Schur_functor.html

  1. φ : E × n M \varphi:E^{\times n}\to M
  2. φ \varphi
  3. φ \varphi
  4. φ \varphi
  5. I { 1 , 2 , , n } I\subset\{1,2,\dots,n\}
  6. φ ( x ) = x φ ( x ) \varphi(x)=\sum_{x^{\prime}}\varphi(x^{\prime})
  7. | I | |I|
  8. i - 1 i-1
  9. 𝕊 λ E \mathbb{S}^{\lambda}E
  10. φ \varphi
  11. φ ~ : 𝕊 λ E M \tilde{\varphi}:\mathbb{S}^{\lambda}E\to M
  12. φ \varphi
  13. ( 2 , 2 , 1 ) (2,2,1)
  14. I = { 4 , 5 } I=\{4,5\}
  15. φ ( x 1 , x 2 , x 3 , x 4 , x 5 ) = φ ( x 4 , x 5 , x 3 , x 1 , x 2 ) + φ ( x 4 , x 2 , x 5 , x 1 , x 3 ) + φ ( x 1 , x 4 , x 5 , x 2 , x 3 ) , \varphi(x_{1},x_{2},x_{3},x_{4},x_{5})=\varphi(x_{4},x_{5},x_{3},x_{1},x_{2})+% \varphi(x_{4},x_{2},x_{5},x_{1},x_{3})+\varphi(x_{1},x_{4},x_{5},x_{2},x_{3}),
  16. I = { 5 } I=\{5\}
  17. φ ( x 1 , x 2 , x 3 , x 4 , x 5 ) = φ ( x 5 , x 2 , x 3 , x 4 , x 1 ) + φ ( x 1 , x 5 , x 3 , x 4 , x 2 ) + φ ( x 1 , x 2 , x 5 , x 4 , x 3 ) . \varphi(x_{1},x_{2},x_{3},x_{4},x_{5})=\varphi(x_{5},x_{2},x_{3},x_{4},x_{1})+% \varphi(x_{1},x_{5},x_{3},x_{4},x_{2})+\varphi(x_{1},x_{2},x_{5},x_{4},x_{3}).
  18. 𝕊 λ V \mathbb{S}^{\lambda}V
  19. G L ( V ) GL(V)
  20. G L ( V ) GL(V)
  21. V n = λ n : ( λ ) k ( 𝕊 λ V ) f λ V^{\otimes n}=\bigoplus_{\lambda\vdash n:\ell(\lambda)\leq k}(\mathbb{S}^{% \lambda}V)^{\oplus f^{\lambda}}
  22. f λ f^{\lambda}
  23. G L ( V ) × 𝔖 n GL(V)\times\mathfrak{S}_{n}
  24. V n = λ n : ( λ ) k ( 𝕊 λ V ) Specht ( λ ) V^{\otimes n}=\bigoplus_{\lambda\vdash n:\ell(\lambda)\leq k}(\mathbb{S}^{% \lambda}V)\otimes\operatorname{Specht}(\lambda)
  25. Specht ( λ ) \operatorname{Specht}(\lambda)

Schwarz_integral_formula.html

  1. f ( z ) = 1 2 π i | ζ | = 1 ζ + z ζ - z Re ( f ( ζ ) ) d ζ ζ + i Im ( f ( 0 ) ) f(z)=\frac{1}{2\pi i}\oint_{|\zeta|=1}\frac{\zeta+z}{\zeta-z}\,\text{Re}(f(% \zeta))\,\frac{d\zeta}{\zeta}+i\,\text{Im}(f(0))
  2. f ( z ) = 1 π i - u ( ζ , 0 ) ζ - z d ζ = 1 π i - R e ( f ) ( ζ + 0 i ) ζ - z d ζ f(z)=\frac{1}{\pi i}\int_{-\infty}^{\infty}\frac{u(\zeta,0)}{\zeta-z}\,d\zeta=% \frac{1}{\pi i}\int_{-\infty}^{\infty}\frac{Re(f)(\zeta+0i)}{\zeta-z}\,d\zeta
  3. u ( z ) = 1 2 π 0 2 π u ( e i ψ ) Re e i ψ + z e i ψ - z d ψ for | z | < 1. u(z)=\frac{1}{2\pi}\int_{0}^{2\pi}u(e^{i\psi})\operatorname{Re}{e^{i\psi}+z% \over e^{i\psi}-z}\,d\psi\,\text{ for }|z|<1.

Schwinger_variational_principle.html

  1. ϕ | T ( E ) | ϕ = T [ ψ , ψ ] ψ | V | ϕ + ϕ | V | ψ - ψ | V - V G 0 ( + ) ( E ) V | ψ , \langle\phi^{\prime}|T(E)|\phi\rangle=T[\psi^{\prime},\psi]\equiv\langle\psi^{% \prime}|V|\phi\rangle+\langle\phi^{\prime}|V|\psi\rangle-\langle\psi^{\prime}|% V-VG_{0}^{(+)}(E)V|\psi\rangle,
  2. ϕ \phi
  3. ϕ \phi^{\prime}
  4. V V
  5. G 0 ( + ) ( E ) G_{0}^{(+)}(E)
  6. E E
  7. ψ \psi
  8. ψ \psi^{\prime}
  9. | ψ = | ϕ + G 0 ( + ) ( E ) V | ψ |\psi\rangle=|\phi\rangle+G_{0}^{(+)}(E)V|\psi\rangle
  10. | ψ = | ϕ + G 0 ( - ) ( E ) V | ψ . |\psi^{\prime}\rangle=|\phi^{\prime}\rangle+G_{0}^{(-)}(E)V|\psi^{\prime}\rangle.
  11. ϕ | T ( E ) | ϕ = T [ ψ , ψ ] ψ | V | ϕ ϕ | V | ψ ψ | V - V G 0 ( + ) ( E ) V | ψ . \langle\phi^{\prime}|T(E)|\phi\rangle=T[\psi^{\prime},\psi]\equiv\frac{\langle% \psi^{\prime}|V|\phi\rangle\langle\phi^{\prime}|V|\psi\rangle}{\langle\psi^{% \prime}|V-VG_{0}^{(+)}(E)V|\psi\rangle}.
  12. ψ \psi
  13. ψ \psi^{\prime}
  14. ψ , ψ \psi,\psi^{\prime}

Scoring_algorithm.html

  1. Y 1 , , Y n Y_{1},\ldots,Y_{n}
  2. f ( y ; θ ) f(y;\theta)
  3. θ * \theta^{*}
  4. θ \theta
  5. θ 0 \theta_{0}
  6. V ( θ ) V(\theta)
  7. θ 0 \theta_{0}
  8. V ( θ ) V ( θ 0 ) - 𝒥 ( θ 0 ) ( θ - θ 0 ) , V(\theta)\approx V(\theta_{0})-\mathcal{J}(\theta_{0})(\theta-\theta_{0}),\,
  9. 𝒥 ( θ 0 ) = - i = 1 n | θ = θ 0 log f ( Y i ; θ ) \mathcal{J}(\theta_{0})=-\sum_{i=1}^{n}\left.\nabla\nabla^{\top}\right|_{% \theta=\theta_{0}}\log f(Y_{i};\theta)
  10. θ 0 \theta_{0}
  11. θ = θ * \theta=\theta^{*}
  12. V ( θ * ) = 0 V(\theta^{*})=0
  13. θ * θ 0 + 𝒥 - 1 ( θ 0 ) V ( θ 0 ) . \theta^{*}\approx\theta_{0}+\mathcal{J}^{-1}(\theta_{0})V(\theta_{0}).\,
  14. θ m + 1 = θ m + 𝒥 - 1 ( θ m ) V ( θ m ) , \theta_{m+1}=\theta_{m}+\mathcal{J}^{-1}(\theta_{m})V(\theta_{m}),\,
  15. θ m θ * \theta_{m}\rightarrow\theta^{*}
  16. 𝒥 ( θ ) \mathcal{J}(\theta)
  17. ( θ ) = E [ 𝒥 ( θ ) ] \mathcal{I}(\theta)=\mathrm{E}[\mathcal{J}(\theta)]
  18. θ m + 1 = θ m + - 1 ( θ m ) V ( θ m ) \theta_{m+1}=\theta_{m}+\mathcal{I}^{-1}(\theta_{m})V(\theta_{m})

Scott's_Pi.html

  1. π = Pr ( a ) - Pr ( e ) 1 - Pr ( e ) , \pi=\frac{\Pr(a)-\Pr(e)}{1-\Pr(e)},
  2. Pr ( a ) = 1 + 5 + 9 45 = 0.333. \Pr(a)=\frac{1+5+9}{45}=0.333.
  3. π = 0.333 - 0.369 1 - 0.369 = - 0.059. \pi=\frac{0.333-0.369}{1-0.369}=-0.059.

Scott_information_system.html

  1. ( T , C o n , ) (T,Con,\vdash)
  2. T is a set of tokens (the basic units of information) T\mbox{ is a set of tokens (the basic units of information)}~{}
  3. C o n 𝒫 f ( T ) the finite subsets of T Con\subseteq\mathcal{P}_{f}(T)\mbox{ the finite subsets of T}~{}
  4. ( C o n { } ) × T \vdash\subseteq(Con\setminus\{\emptyset\})\times T
  5. If a X C o n then X a \mbox{If }~{}a\in X\in Con\mbox{ then }~{}X\vdash a
  6. If X Y and Y a , then X a \mbox{If }~{}X\vdash Y\mbox{ and }~{}Y\vdash a\mbox{, then }~{}X\vdash a
  7. If X a then X { a } C o n \mbox{If }~{}X\vdash a\mbox{ then }~{}X\cup\{a\}\in Con
  8. a T : { a } C o n \forall a\in T:\{a\}\in Con
  9. If X C o n and X X then X C o n . \mbox{If }~{}X\in Con\mbox{ and }~{}X^{\prime}\,\subseteq X\mbox{ then }~{}X^{% \prime}\in Con.
  10. X Y X\vdash Y
  11. a Y , X a . \forall a\in Y,X\vdash a.
  12. T := T:=\mathbb{N}
  13. C o n := { } { { n } n } Con:=\{\}\cup\{\{n\}\mid n\in\mathbb{N}\}
  14. X a iff a X . X\vdash a\mbox{ iff }~{}a\in X.
  15. { n } \{n\}
  16. $\empty$
  17. \mathbb{N}
  18. T := { ϕ ϕ is satisfiable } T:=\{\phi\mid\phi\mbox{ is satisfiable}~{}\}
  19. C o n := { X 𝒫 f ( T ) X is consistent } Con:=\{X\in\mathcal{P}_{f}(T)\mid X\mbox{ is consistent}~{}\}
  20. X a iff X a in the propositional calculus . X\vdash a\mbox{ iff }~{}X\vdash a\mbox{ in the propositional calculus}~{}.
  21. T := D 0 T:=D^{0}
  22. C o n := { X 𝒫 f ( T ) X has an upper bound } Con:=\{X\in\mathcal{P}_{f}(T)\mid X\mbox{ has an upper bound}~{}\}
  23. X d iff d X . X\vdash d\mbox{ iff }~{}d\sqsubseteq\bigsqcup X.
  24. \mathcal{I}
  25. A = ( T , C o n , ) A=(T,Con,\vdash)
  26. x T x\subseteq T
  27. If X f x then X C o n \mbox{If }~{}X\subseteq_{f}x\mbox{ then }~{}X\in Con
  28. If X a and X f x then a x . \mbox{If }~{}X\vdash a\mbox{ and }~{}X\subseteq_{f}x\mbox{ then }~{}a\in x.
  29. 𝒟 ( A ) \mathcal{D}(A)
  30. 𝒟 ( A ) \mathcal{D}(A)
  31. 𝒟 ( ( D ) ) D \mathcal{D}(\mathcal{I}(D))\cong D
  32. ( 𝒟 ( A ) ) A \mathcal{I}(\mathcal{D}(A))\cong A

ScRGB.html

  1. 8192 x + 4096 8192x+4096
  2. - 0.5..7.4999 -0.5..7.4999
  3. 1280 x + 1024 1280x+1024
  4. 5 x + 1024 5x+1024
  5. - 0.6038..7.5913 -0.6038..7.5913
  6. 1280 Y + 1024 1280Y^{\prime}+1024
  7. 1280 C x + 2048 1280Cx+2048
  8. 2 n - 1 2^{n}-1
  9. n n

Sears–Haack_body.html

  1. D wave [ S ′′ ( x ) ] 2 D\text{wave}\sim[S^{\prime\prime}(x)]^{2}
  2. S ( x ) S(x)
  3. S ( x ) = 16 V 3 L π [ 4 x ( 1 - x ) ] 3 / 2 = π R m a x 2 [ 4 x ( 1 - x ) ] 3 / 2 S(x)=\frac{16V}{3L\pi}[4x(1-x)]^{3/2}=\pi R_{max}^{2}[4x(1-x)]^{3/2}
  4. V = 3 π 2 16 R m a x 2 L V=\frac{3\pi^{2}}{16}R_{max}^{2}L
  5. r ( x ) = R m a x [ 4 x ( 1 - x ) ] 3 / 4 r(x)=R_{max}[4x(1-x)]^{3/4}
  6. r ( x ) = 3 R m a x [ 4 x ( 1 - x ) ] - 1 / 4 ( 1 - 2 x ) r^{\prime}(x)=3R_{max}[4x(1-x)]^{-1/4}(1-2x)
  7. r ′′ ( x ) = - 3 R m a x { [ 4 x ( 1 - x ) ] - 5 / 4 ( 1 - 2 x ) 2 + 2 [ 4 x ( 1 - x ) ] - 1 / 4 } r^{\prime\prime}(x)=-3R_{max}\{[4x(1-x)]^{-5/4}(1-2x)^{2}+2[4x(1-x)]^{-1/4}\}
  8. R m a x R_{max}
  9. ρ \rho
  10. D wave = - 1 4 π ρ U 2 0 0 S ′′ ( x 1 ) S ′′ ( x 2 ) ln | x 1 - x 2 | d x 1 d x 2 D\text{wave}=-\frac{1}{4\pi}\rho U^{2}\int_{0}^{\ell}\int_{0}^{\ell}S^{\prime% \prime}(x_{1})S^{\prime\prime}(x_{2})\ln|x_{1}-x_{2}|\mathrm{d}x_{1}\mathrm{d}% x_{2}
  11. D wave = - 1 2 π ρ U 2 0 S ′′ ( x ) d x 0 x S ′′ ( x 1 ) ln ( x - x 1 ) d x 1 D\text{wave}=-\frac{1}{2\pi}\rho U^{2}\int_{0}^{\ell}S^{\prime\prime}(x)% \mathrm{d}x\int_{0}^{x}S^{\prime\prime}(x_{1})\ln(x-x_{1})\mathrm{d}x_{1}
  12. D wave = 64 V 2 π L 4 ρ U 2 = 9 π 3 R m a x 4 4 L 2 ρ U 2 D\text{wave}=\frac{64V^{2}}{\pi L^{4}}\rho U^{2}=\frac{9\pi^{3}R_{max}^{4}}{4L% ^{2}}\rho U^{2}
  13. C D wave = 24 V L 3 = 9 π 2 R m a x 2 2 L 2 C_{D\text{wave}}=\frac{24V}{L^{3}}=\frac{9\pi^{2}R_{max}^{2}}{2L^{2}}
  14. S ( x ) S(x)
  15. x x
  16. x = constant x=\,\text{constant}

Seawanhaka_Corinthian_Yacht_Club.html

  1. R a t i n g = L o a d W a t e r l i n e L e n g t h + S a i l A r e a 2 Rating=\frac{Load\ Waterline\ Length+\sqrt{Sail\ Area}}{2}

Second-harmonic_imaging_microscopy.html

  1. I p a r - I p e r p I p a r + 2 I p e r p = r \frac{I_{par}-I_{perp}}{I_{par}+2I_{perp}}=r
  2. r r
  3. r r
  4. r = 0.7 r=0.7

Secondary_measure.html

  1. x [ 0 , 1 ] , μ ( x ) = ρ ( x ) φ 2 ( x ) 4 + π 2 ρ 2 ( x ) \forall x\in[0,1],\qquad\mu(x)=\frac{\rho(x)}{\frac{\varphi^{2}(x)}{4}+\pi^{2}% \rho^{2}(x)}
  2. φ ( x ) = lim ε 0 + 2 0 1 ( x - t ) ρ ( t ) ( x - t ) 2 + ε 2 d t \varphi(x)=\lim_{\varepsilon\to 0^{+}}2\int_{0}^{1}\frac{(x-t)\rho(t)}{(x-t)^{% 2}+\varepsilon^{2}}\,dt
  3. φ ( x ) = 2 ρ ( x ) ln ( x 1 - x ) - 2 0 1 ρ ( t ) - ρ ( x ) t - x d t \varphi(x)=2\rho(x)\,\text{ln}\left(\frac{x}{1-x}\right)-2\int_{0}^{1}\frac{% \rho(t)-\rho(x)}{t-x}\,dt
  4. S μ ( z ) = z - c 1 - 1 S ρ ( z ) S_{\mu}(z)=z-c_{1}-\frac{1}{S_{\rho}(z)}
  5. f ( x ) = 0 1 g ( t ) - g ( x ) t - x ρ ( t ) d t f(x)=\int_{0}^{1}\frac{g(t)-g(x)}{t-x}\rho(t)\,dt
  6. S μ ( z ) = a ( z - c 1 - 1 S ρ ( z ) ) , S_{\mu}(z)=a\left(z-c_{1}-\frac{1}{S_{\rho}(z)}\right),
  7. f ( x ) I f ( t ) - f ( x ) t - x ρ ( t ) d t f(x)\mapsto\int_{I}\frac{f(t)-f(x)}{t-x}\rho(t)dt
  8. f / g ρ - f / 1 ρ × g / 1 ρ = T ρ ( f ) / T ρ ( g ) μ . \langle f/g\rangle_{\rho}-\langle f/1\rangle_{\rho}\times\langle g/1\rangle_{% \rho}=\langle T_{\rho}(f)/T_{\rho}(g)\rangle_{\mu}.
  9. f / φ ρ = T ρ ( f ) / 1 ρ \langle f/\varphi\rangle_{\rho}=\langle T_{\rho}(f)/1\rangle_{\rho}
  10. f φ × f - T ρ ( f ) f\mapsto\varphi\times f-T_{\rho}(f)
  11. T ρ S ρ ( f ) = ρ μ × ( f ) . T_{\rho}\circ S_{\rho}\left(f\right)=\frac{\rho}{\mu}\times(f).
  12. P n ( x ) = d n d x n ( x n ( 1 - x ) n ) . P_{n}(x)=\frac{d^{n}}{dx^{n}}\left(x^{n}(1-x)^{n}\right).
  13. n ! 2 n + 1 . \frac{n!}{\sqrt{2n+1}}.
  14. 2 ( 2 n + 1 ) X P n ( X ) = - P n + 1 ( X ) + ( 2 n + 1 ) P n ( X ) - n 2 P n - 1 ( X ) . 2(2n+1)XP_{n}(X)=-P_{n+1}(X)+(2n+1)P_{n}(X)-n^{2}P_{n-1}(X).
  15. φ ( x ) = 2 ln ( x 1 - x ) . \varphi(x)=2\ln\left(\frac{x}{1-x}\right).
  16. μ ( x ) = 1 ln 2 ( x 1 - x ) + π 2 \mu(x)=\frac{1}{\ln^{2}\left(\frac{x}{1-x}\right)+\pi^{2}}
  17. C n ( φ ) = - 4 2 n + 1 n ( n + 1 ) C_{n}(\varphi)=-\frac{4\sqrt{2n+1}}{n(n+1)}
  18. L n ( x ) = e x n ! d n d x n ( x n e - x ) = k = 0 n ( n k ) ( - 1 ) k x k k ! L_{n}(x)=\frac{e^{x}}{n!}\frac{d^{n}}{dx^{n}}(x^{n}e^{-x})=\sum_{k=0}^{n}{% \left({{n}\atop{k}}\right)}(-1)^{k}\frac{x^{k}}{k!}
  19. φ ( x ) = 2 ( ln ( x ) - 0 e - t ln | x - t | d t ) . \varphi(x)=2\left(\ln(x)-\int_{0}^{\infty}e^{-t}\ln|x-t|dt\right).
  20. C n ( φ ) = - 1 n k = 0 n - 1 1 ( n - 1 k ) . C_{n}(\varphi)=-\frac{1}{n}\sum_{k=0}^{n-1}\frac{1}{{\left({{n-1}\atop{k}}% \right)}}.
  21. ρ ( x ) = e - x 2 2 2 π \rho(x)=\frac{e^{-\frac{x^{2}}{2}}}{\sqrt{2\pi}}
  22. H n ( x ) = 1 n ! e x 2 2 d n d x n ( e - x 2 2 ) H_{n}(x)=\frac{1}{\sqrt{n!}}e^{\frac{x^{2}}{2}}\frac{d^{n}}{dx^{n}}\left(e^{-% \frac{x^{2}}{2}}\right)
  23. φ ( x ) = - 2 2 π - t e - t 2 2 ln | x - t | d t . \varphi(x)=-\frac{2}{\sqrt{2\pi}}\int_{-\infty}^{\infty}te^{-\frac{t^{2}}{2}}% \ln|x-t|\,dt.
  24. C n ( φ ) = ( - 1 ) n + 1 2 ( n - 1 2 ) ! n ! C_{n}(\varphi)=(-1)^{\frac{n+1}{2}}\frac{\left(\frac{n-1}{2}\right)!}{\sqrt{n!}}
  25. ρ ( x ) = 8 π x ( 1 - x ) \rho(x)=\frac{8}{\pi}\sqrt{x(1-x)}
  26. ρ ( x ) = 2 π 1 - x x . \rho(x)=\frac{2}{\pi}\sqrt{\frac{1-x}{x}}.
  27. ρ ( x ) = 1 π 1 - x 2 . \rho(x)=\frac{1}{\pi\sqrt{1-x^{2}}}.
  28. d 0 = c 2 - c 1 2 , d_{0}=c_{2}-c_{1}^{2},
  29. ρ n ( x ) = 1 d 0 n - 1 ρ ( x ) ( P n - 1 ( x ) φ ( x ) 2 - Q n - 1 ( x ) ) 2 + π 2 ρ 2 ( x ) P n - 1 2 ( x ) . \rho_{n}(x)=\frac{1}{d_{0}^{n-1}}\frac{\rho(x)}{\left(P_{n-1}(x)\frac{\varphi(% x)}{2}-Q_{n-1}(x)\right)^{2}+\pi^{2}\rho^{2}(x)P_{n-1}^{2}(x)}.
  30. d 0 n - 1 d_{0}^{n-1}
  31. x P n ( x ) = t n P n + 1 ( x ) + s n P n ( x ) + t n - 1 P n - 1 ( x ) xP_{n}(x)=t_{n}P_{n+1}(x)+s_{n}P_{n}(x)+t_{n-1}P_{n-1}(x)
  32. lim n t n = 1 4 , lim n s n = 1 2 , \lim_{n\mapsto\infty}t_{n}=\tfrac{1}{4},\quad\lim_{n\mapsto\infty}s_{n}=\tfrac% {1}{2},
  33. ρ t c h ( x ) = 8 π x ( 1 - x ) \rho_{tch}(x)=\frac{8}{\pi}\sqrt{x(1-x)}
  34. ρ t ( x ) = t ρ ( x ) ( 1 2 ( t - 1 ) ( x - c 1 ) φ ( x ) - t ) 2 + π 2 ρ 2 ( x ) ( t - 1 ) 2 ( x - c 1 ) 2 , \rho_{t}(x)=\frac{t\rho(x)}{\left(\tfrac{1}{2}(t-1)(x-c_{1})\varphi(x)-t\right% )^{2}+\pi^{2}\rho^{2}(x)(t-1)^{2}(x-c_{1})^{2}},
  35. φ t ( x ) = 2 ( x - c 1 ) - t G ( x ) ( ( x - c 1 ) - t 1 2 G ( x ) ) 2 + t 2 π 2 μ 2 ( x ) \varphi_{t}(x)=\frac{2(x-c_{1})-tG(x)}{\left((x-c_{1})-t\tfrac{1}{2}G(x)\right% )^{2}+t^{2}\pi^{2}\mu^{2}(x)}
  36. P n t ( x ) = t P n ( x ) + ( 1 - t ) ( x - c 1 ) Q n ( x ) t P_{n}^{t}(x)=\frac{tP_{n}(x)+(1-t)(x-c_{1})Q_{n}(x)}{\sqrt{t}}
  37. ρ t ( x ) = 2 t 1 - x 2 π [ t 2 + 4 ( 1 - t ) x 2 ] , \rho_{t}(x)=\frac{2t\sqrt{1-x^{2}}}{\pi\left[t^{2}+4(1-t)x^{2}\right]},
  38. 1 ln ( p ) = 1 p - 1 + 0 1 ( x + p ) ( ln 2 ( x ) + π 2 ) d x p > 1 \frac{1}{\ln(p)}=\frac{1}{p-1}+\int_{0}^{\infty}\frac{1}{(x+p)(\ln^{2}(x)+\pi^% {2})}dx\qquad\qquad\forall p>1
  39. γ = 0 ln ( 1 + 1 x ) ln 2 ( x ) + π 2 d x \gamma=\int_{0}^{\infty}\frac{\ln(1+\frac{1}{x})}{\ln^{2}(x)+\pi^{2}}dx
  40. γ = 1 2 + 0 ( x + 1 ) cos ( π x ) ¯ x + 1 d x \gamma=\tfrac{1}{2}+\int_{0}^{\infty}\frac{\overline{(x+1)\cos(\pi x)}}{x+1}dx
  41. x ( x + 1 ) cos ( π x ) ¯ x\mapsto\overline{(x+1)\cos(\pi x)}
  42. x ( x + 1 ) cos ( π x ) x\mapsto(x+1)\cos(\pi x)
  43. γ = 1 2 + k = 1 n β 2 k 2 k - β 2 n ζ ( 2 n ) 1 t cos ( 2 π t ) t - 2 n - 1 d t \gamma=\tfrac{1}{2}+\sum_{k=1}^{n}\frac{\beta_{2k}}{2k}-\frac{\beta_{2n}}{% \zeta(2n)}\int_{1}^{\infty}\lfloor t\rfloor\cos(2\pi t)t^{-2n-1}dt
  44. β k = ( - 1 ) k k ! π Im ( - e x ( 1 + e x ) ( x - i π ) k d x ) \beta_{k}=\frac{(-1)^{k}k!}{\pi}\,\text{Im}\left(\int_{-\infty}^{\infty}\frac{% e^{x}}{(1+e^{x})(x-i\pi)^{k}}dx\right)
  45. 0 1 ln 2 n ( x 1 - x ) d x = ( - 1 ) n + 1 ( 2 2 n - 2 ) β 2 n π 2 n \int_{0}^{1}\ln^{2n}\left(\frac{x}{1-x}\right)\,dx=(-1)^{n+1}(2^{2n}-2)\beta_{% 2n}\pi^{2n}
  46. 0 1 0 1 ( k = 1 2 n ln ( t k ) i k ( t k - t i ) ) d t 1 d t 2 n = 1 2 ( - 1 ) n + 1 ( 2 π ) 2 n β 2 n \int_{0}^{1}\cdots\int_{0}^{1}\left(\sum_{k=1}^{2n}\frac{\ln(t_{k})}{\prod_{i% \neq k}(t_{k}-t_{i})}\right)\,dt_{1}\cdots dt_{2n}=\tfrac{1}{2}(-1)^{n+1}(2\pi% )^{2n}\beta_{2n}
  47. 0 e - α x Γ ( x + 1 ) d x = e e - α - 1 + 0 1 - e - x ( ln ( x ) + α ) 2 + π 2 d x x α 𝐑 \int_{0}^{\infty}\frac{e^{-\alpha x}}{\Gamma(x+1)}dx=e^{e^{-\alpha}}-1+\int_{0% }^{\infty}\frac{1-e^{-x}}{(\ln(x)+\alpha)^{2}+\pi^{2}}\frac{dx}{x}\qquad\qquad% \forall\alpha\in\mathbf{R}
  48. n = 1 ( 1 n k = 0 n - 1 1 ( n - 1 k ) ) 2 = 4 9 π 2 = 0 4 ( Ei ( 1 , - x ) + i π ) 2 e - 3 x d x . \sum_{n=1}^{\infty}\left(\frac{1}{n}\sum_{k=0}^{n-1}\frac{1}{{\left({{n-1}% \atop{k}}\right)}}\right)^{2}=\tfrac{4}{9}\pi^{2}=\int_{0}^{\infty}4\left(% \mathrm{Ei}(1,-x)+i\pi\right)^{2}e^{-3x}\,dx.
  49. 23 15 - ln ( 2 ) = n = 0 1575 2 ( n + 1 ) ( 2 n + 1 ) ( 4 n - 3 ) ( 4 n - 1 ) ( 4 n + 1 ) ( 4 n + 5 ) ( 4 n + 7 ) ( 4 n + 9 ) \tfrac{23}{15}-\ln(2)=\sum_{n=0}^{\infty}\frac{1575}{2(n+1)(2n+1)(4n-3)(4n-1)(% 4n+1)(4n+5)(4n+7)(4n+9)}
  50. G = k = 0 ( - 1 ) k 4 k + 1 ( 1 ( 4 k + 3 ) 2 + 2 ( 4 k + 2 ) 2 + 2 ( 4 k + 1 ) 2 ) + π 8 ln ( 2 ) G=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{4^{k+1}}\left(\frac{1}{(4k+3)^{2}}+\frac{% 2}{(4k+2)^{2}}+\frac{2}{(4k+1)^{2}}\right)+\tfrac{\pi}{8}\ln(2)
  51. G = π 8 ln ( 2 ) + n = 0 ( - 1 ) n H 2 n + 1 2 n + 1 . G=\tfrac{\pi}{8}\ln(2)+\sum_{n=0}^{\infty}(-1)^{n}\frac{H_{2n+1}}{2n+1}.
  52. I φ 2 ( x ) ρ ( x ) d x = 4 π 2 3 I ρ 3 ( x ) d x . \int_{I}\varphi^{2}(x)\rho(x)\,dx=\frac{4\pi^{2}}{3}\int_{I}\rho^{3}(x)\,dx.
  53. f ( x ) = I g ( t ) - g ( x ) t - x ρ ( t ) d t g ( x ) = ( x - c 1 ) f ( x ) - T μ ( f ( x ) ) = φ ( x ) μ ( x ) ρ ( x ) f ( x ) - T ρ ( μ ( x ) ρ ( x ) f ( x ) ) f(x)=\int_{I}\frac{g(t)-g(x)}{t-x}\rho(t)dt\Leftrightarrow g(x)=(x-c_{1})f(x)-% T_{\mu}(f(x))=\frac{\varphi(x)\mu(x)}{\rho(x)}f(x)-T_{\rho}\left(\frac{\mu(x)}% {\rho(x)}f(x)\right)
  54. g ( x ) I g ( t ) - g ( x ) t - x ρ ( t ) d t . g(x)\mapsto\int_{I}\frac{g(t)-g(x)}{t-x}\rho(t)\,dt.

Secondary_polynomials.html

  1. { q n ( x ) } \{q_{n}(x)\}
  2. { p n ( x ) } \{p_{n}(x)\}
  3. ρ ( x ) \rho(x)
  4. q n ( x ) = p n ( t ) - p n ( x ) t - x ρ ( t ) d t . q_{n}(x)=\int_{\mathbb{R}}\!\frac{p_{n}(t)-p_{n}(x)}{t-x}\rho(t)\,dt.
  5. q n ( x ) q_{n}(x)
  6. p 0 ( x ) = x 3 . p_{0}(x)=x^{3}.
  7. q 0 ( x ) = t 3 - x 3 t - x ρ ( t ) d t = ( t - x ) ( t 2 + t x + x 2 ) t - x ρ ( t ) d t = ( t 2 + t x + x 2 ) ρ ( t ) d t = t 2 ρ ( t ) d t + x t ρ ( t ) d t + x 2 ρ ( t ) d t \begin{aligned}\displaystyle q_{0}(x)&\displaystyle{}=\int_{\mathbb{R}}\!\frac% {t^{3}-x^{3}}{t-x}\rho(t)\,dt\\ &\displaystyle{}=\int_{\mathbb{R}}\!\frac{(t-x)(t^{2}+tx+x^{2})}{t-x}\rho(t)\,% dt\\ &\displaystyle{}=\int_{\mathbb{R}}\!(t^{2}+tx+x^{2})\rho(t)\,dt\\ &\displaystyle{}=\int_{\mathbb{R}}\!t^{2}\rho(t)\,dt+x\int_{\mathbb{R}}\!t\rho% (t)\,dt+x^{2}\int_{\mathbb{R}}\!\rho(t)\,dt\end{aligned}
  8. x x
  9. t t
  10. ρ \rho

Section_(category_theory).html

  1. X ¯ \bar{X}
  2. π : X X ¯ \pi\colon X\to\bar{X}
  3. π \pi

Segmented_regression.html

  1. R 1 2 = 1 - ( y - Y r ) 2 ( y - Y a 1 ) 2 R_{1}^{2}=1-\frac{\sum(y-Y_{r})^{2}}{\sum(y-Y_{a1})^{2}}
  2. R 2 2 = 1 - ( y - Y r ) 2 ( y - Y a 2 ) 2 R_{2}^{2}=1-\frac{\sum(y-Y_{r})^{2}}{\sum(y-Y_{a2})^{2}}
  3. ( y - Y r ) 2 \sum(y-Y_{r})^{2}
  4. C d = 1 - ( y - Y r ) 2 ( y - Y a ) 2 C_{d}=1-{\sum(y-Y_{r})^{2}\over\sum(y-Y_{a})^{2}}

Seismic_migration.html

  1. d = v t 2 , d=\frac{vt}{2},
  2. tan ξ a = sin ξ , \tan\xi_{a}=\sin\xi,
  3. ξ < s u b > a ξ<sub>a

Self-avoiding_walk.html

  1. d = 2 d=2
  2. 4 3 \frac{4}{3}
  3. d = 3 d=3
  4. 5 3 \frac{5}{3}
  5. d 4 d≥4
  6. 2 2
  7. n n
  8. ( m + n m , n ) {m+n\choose m,n}
  9. m × n m×n
  10. 2 2
  11. n n
  12. ( n + m ) (n+m)
  13. n n
  14. m m
  15. μ = lim n c n 1 n . \mu=\lim_{n\to\infty}c_{n}^{\frac{1}{n}}.
  16. μ μ
  17. μ μ
  18. μ μ
  19. 2 + 2 . \sqrt{2+\sqrt{2}}.
  20. μ μ
  21. c n μ n n 11 32 c_{n}\approx\mu^{n}n^{\frac{11}{32}}
  22. n n→∞
  23. μ μ
  24. n 11 32 n^{\frac{11}{32}}
  25. n n
  26. n n→∞
  27. κ = 8 3 . κ=\frac{8}{3}.

Self-diffusion.html

  1. D i * D_{i}^{*}
  2. i i
  3. D i D_{i}
  4. D i * = D i ln c i ln a i . D_{i}^{*}=D_{i}\frac{\partial\ln c_{i}}{\partial\ln a_{i}}.
  5. a i a_{i}
  6. i i
  7. c i c_{i}
  8. i i

Self-focusing.html

  1. n n
  2. n = n 0 + n 2 I n=n_{0}+n_{2}I
  3. P c r = α λ 2 4 π n 0 n 2 P_{cr}=\alpha\frac{\lambda^{2}}{4\pi n_{0}n_{2}}
  4. n r e l = 1 - ω p 2 ω 2 n_{rel}=\sqrt{1-\frac{\omega_{p}^{2}}{\omega^{2}}}
  5. ω p = n e 2 γ m ϵ 0 \omega_{p}=\sqrt{\frac{ne^{2}}{\gamma m\epsilon_{0}}}
  6. P c r = m e 2 c 5 ω 2 e 2 ω p 2 17 ( ω ω p ) 2 GW P_{cr}=\frac{m_{e}^{2}c^{5}\omega^{2}}{e^{2}\omega_{p}^{2}}\simeq 17\bigg(% \frac{\omega}{\omega_{p}}\bigg)^{2}\ \textrm{GW}

Self-pulsation.html

  1. X X
  2. Y ~{}Y~{}
  3. d X / d t = K X Y - U X d Y / d t = - K X Y - V Y + W ~{}\begin{aligned}\displaystyle{{\rm d}X}/{{\rm d}t}&\displaystyle=KXY-UX\\ \displaystyle{{\rm d}Y}/{{\rm d}t}&\displaystyle=-KXY-VY+W\end{aligned}
  4. K = σ / ( s t r ) ~{}K=\sigma/(st_{\rm r})~{}
  5. U = θ L ~{}U=\theta L~{}
  6. V = 1 / τ ~{}V=1/\tau~{}
  7. W = P p / ( ω p ) ~{}W=P_{\rm p}/({\hbar\omega_{\rm p}})~{}
  8. t r ~{}t_{\rm r}~{}
  9. s ~{}s~{}
  10. σ ~{}\sigma~{}
  11. ω s ~{}\omega_{\rm s}~{}
  12. θ ~{}\theta~{}
  13. τ ~{}\tau~{}
  14. P p P_{\rm p}
  15. X 0 \displaystyle X_{0}
  16. Γ \displaystyle\Gamma
  17. w = K W - U V w=\sqrt{KW-UV}
  18. U / V 1 U/V\ll 1
  19. u u
  20. v ~{}v~{}
  21. X \displaystyle X

Self-similar_process.html

  1. P ( a ) = ( μ a a ! ) e - μ , P(a)=\left(\frac{\mu^{a}}{a!}\right)e^{-\mu},
  2. μ \mu
  3. P ( d ) = ( λ d d ! ) e - λ , P(d)=\left(\frac{\lambda^{d}}{d!}\right)e^{-\lambda},
  4. λ \lambda
  5. P [ T t ] = e - t / h , P[T\geq\ t]=e^{-t/h},\,
  6. lim x e λ x Pr [ X > x ] = for all λ > 0. \lim_{x\to\infty}e^{\lambda x}\Pr[X>x]=\infty\quad\mbox{for all }~{}\lambda>0.\,
  7. Y = ( Y i : i = 0 , 1 , 2 , , N ) Y=(Y_{i}:i=0,1,2,...,N)
  8. μ ^ = E ( Y i ) \hat{\mu}=\,\text{E}(Y_{i})
  9. y i = Y i - μ ^ y_{i}=Y_{i}-\hat{\mu}
  10. σ ^ 2 = E ( y i 2 ) \hat{\sigma}^{2}=\,\text{E}(y_{i}^{2})
  11. r ( k ) = E ( y i , y i + k ) / E ( y i 2 ) r(k)=\,\text{E}(y_{i},y_{i+k})/\,\text{E}(y_{i}^{2})
  12. r ( k ) k - d L ( k ) r(k)\sim k^{-d}L(k)
  13. Y i ( m ) = ( Y i m - m + 1 + + Y i m ) / m Y_{i}^{(m)}=(Y_{im-m+1}+...+Y_{im})/m
  14. var [ Y ( m ) ] = σ ^ 2 m - d \,\text{var}[Y^{(m)}]=\hat{\sigma}^{2}m^{-d}
  15. lim k r ( k ) / k - d = ( 2 - d ) ( 1 - d ) / 2 \lim_{k\to\infty}r(k)/k^{-d}=(2-d)(1-d)/2
  16. Z i ( m ) = m Y i ( m ) Z_{i}^{(m)}=mY_{i}^{(m)}
  17. Z i ( m ) = ( Y i m - m + 1 + + Y i m ) Z_{i}^{(m)}=(Y_{im-m+1}+...+Y_{im})
  18. var [ Z i ( m ) ] = m 2 var [ Y ( m ) ] = ( σ ^ 2 / μ ^ 2 - d ) E [ Z i ( m ) ] 2 - d \,\text{var}[Z_{i}^{(m)}]=m^{2}\,\text{var}[Y^{(m)}]=(\hat{\sigma}^{2}/\hat{% \mu}^{2-d})\,\text{E}[Z_{i}^{(m)}]^{2-d}
  19. μ ^ \hat{\mu}
  20. σ ^ 2 \hat{\sigma}^{2}
  21. var ( Y ) = a [ E ( Y ) ] p , \,\text{var}\,(Y)=a[\,\text{E}\,(Y)]^{p},
  22. K p * ( s ; θ , λ ) = λ κ p ( θ ) [ ( 1 + s / θ ) α - 1 ] K^{*}_{p}(s;\theta,\lambda)=\lambda\kappa_{p}(\theta)[(1+s/\theta)^{\alpha}-1]
  23. κ p ( θ ) = α - 1 α ( θ α - 1 ) α \kappa_{p}(\theta)=\dfrac{\alpha-1}{\alpha}\left(\dfrac{\theta}{\alpha-1}% \right)^{\alpha}
  24. α = p - 2 p - 1 \alpha=\dfrac{p-2}{p-1}
  25. var ( Z ) E ( Z ) p \mathrm{var}(Z)\propto\mathrm{E}(Z)^{p}

Semi-elliptic_operator.html

  1. P f ( x ) = i , j = 1 n a i j ( x ) 2 f x i x j ( x ) + i = 1 n b i ( x ) f x i ( x ) + c ( x ) f ( x ) , Pf(x)=\sum_{i,j=1}^{n}a_{ij}(x)\frac{\partial^{2}f}{\partial x_{i}\,\partial x% _{j}}(x)+\sum_{i=1}^{n}b_{i}(x)\frac{\partial f}{\partial x_{i}}(x)+c(x)f(x),

Semi-implicit_Euler_method.html

  1. d x d t = f ( t , v ) {dx\over dt}=f(t,v)
  2. d v d t = g ( t , x ) , {dv\over dt}=g(t,x),
  3. H = T ( t , v ) + V ( t , x ) . H=T(t,v)+V(t,x).\,
  4. x ( t 0 ) = x 0 , v ( t 0 ) = v 0 . x(t_{0})=x_{0},\qquad v(t_{0})=v_{0}.
  5. v n + 1 = v n + g ( t n , x n ) Δ t x n + 1 = x n + f ( t n , v n + 1 ) Δ t \begin{aligned}\displaystyle v_{n+1}&\displaystyle=v_{n}+g(t_{n},x_{n})\,% \Delta t\\ \displaystyle x_{n+1}&\displaystyle=x_{n}+f(t_{n},v_{n+1})\,\Delta t\end{aligned}
  6. ( x n , v n ) (x_{n},v_{n})
  7. ( x n + 1 , v n + 1 ) (x_{n+1},v_{n+1})
  8. x n + 1 = x n + f ( t n , v n ) Δ t v n + 1 = v n + g ( t n , x n + 1 ) Δ t \begin{aligned}\displaystyle x_{n+1}&\displaystyle=x_{n}+f(t_{n},v_{n})\,% \Delta t\\ \displaystyle v_{n+1}&\displaystyle=v_{n}+g(t_{n},x_{n+1})\,\Delta t\end{aligned}
  9. s > - 2 / Δ t s>-2/\Delta t
  10. d x d t = v ( t ) d v d t = - k m x = - ω 2 x . \begin{aligned}\displaystyle\frac{dx}{dt}&\displaystyle=v(t)\\ \displaystyle\frac{dv}{dt}&\displaystyle=-\frac{k}{m}\,x=-\omega^{2}\,x.\end{aligned}
  11. v n + 1 = v n - ω 2 x n Δ t x n + 1 = x n + v n + 1 Δ t . \begin{aligned}\displaystyle v_{n+1}&\displaystyle=v_{n}-\omega^{2}\,x_{n}\,% \Delta t\\ \displaystyle x_{n+1}&\displaystyle=x_{n}+v_{n+1}\,\Delta t.\end{aligned}
  12. E h ( x , v ) = 1 2 ( v 2 + ω 2 x 2 - ω 2 Δ t v x ) E_{h}(x,v)=\tfrac{1}{2}\left(v^{2}+\omega^{2}\,x^{2}-\omega^{2}\Delta t\,vx\right)
  13. O ( Δ t ) O(\Delta t)
  14. ω \omega
  15. 1 + 1 24 ω 2 Δ t 2 + O ( Δ t 4 ) 1+\tfrac{1}{24}\omega^{2}\Delta t^{2}+O(\Delta t^{4})

Semiautomaton.html

  1. ( M , Q ) (M,Q)
  2. m : Q Q m\colon Q\to Q
  3. ( s t ) ( q ) = ( s t ) ( q ) = s ( t ( q ) ) (st)(q)=(s\circ t)(q)=s(t(q))
  4. μ : Q × M Q \mu\colon Q\times M\to Q
  5. ( q , m ) q m = μ ( q , m ) (q,m)\mapsto qm=\mu(q,m)
  6. q 1 = q q1=q
  7. q ( s t ) = ( q s ) t q(st)=(qs)t
  8. q Q q\in Q
  9. s , t M s,t\in M
  10. ( Q , M , μ ) (Q,M,\mu)
  11. μ \mu
  12. Q M Q_{M}
  13. μ : M × Q Q \mu\colon M\times Q\to Q
  14. ( m , q ) m q = μ ( m , q ) (m,q)\mapsto mq=\mu(m,q)
  15. Q M \,{}_{M}Q
  16. μ \mu
  17. μ ( q , s t ) = μ ( μ ( q , s ) , t ) \mu(q,st)=\mu(\mu(q,s),t)
  18. μ \mu
  19. Q M Q_{M}
  20. B M B_{M}
  21. M M
  22. f : Q M B M f\colon Q_{M}\to B_{M}
  23. f : Q B f\colon Q\to B
  24. f ( q m ) = f ( q ) m f(qm)=f(q)m
  25. q Q q\in Q
  26. m M m\in M
  27. Hom ( Q M , B M ) \mathrm{Hom}(Q_{M},B_{M})
  28. Hom M ( Q , B ) \mathrm{Hom}_{M}(Q,B)
  29. ( Q , Σ , T ) (Q,\Sigma,T)
  30. Σ \Sigma
  31. T : Q × Σ Q . T\colon Q\times\Sigma\to Q.
  32. ( Q , Σ , T , q 0 , A ) (Q,\Sigma,T,q_{0},A)
  33. q 0 q_{0}
  34. Σ * \Sigma^{*}
  35. Σ \Sigma
  36. Σ \Sigma
  37. Σ * \Sigma^{*}
  38. T w : Q Q T_{w}\colon Q\to Q
  39. w = ε w=\varepsilon
  40. T ε ( q ) = q T_{\varepsilon}(q)=q
  41. ε \varepsilon
  42. w = σ w=\sigma
  43. Σ \Sigma
  44. T σ ( q ) = T ( q , σ ) T_{\sigma}(q)=T(q,\sigma)
  45. w = σ v w=\sigma v
  46. σ Σ \sigma\in\Sigma
  47. v Σ * v\in\Sigma^{*}
  48. T w ( q ) = T v ( T σ ( q ) ) T_{w}(q)=T_{v}(T_{\sigma}(q))
  49. M ( Q , Σ , T ) M(Q,\Sigma,T)
  50. M ( Q , Σ , T ) = { T w | w Σ * } . M(Q,\Sigma,T)=\{T_{w}|w\in\Sigma^{*}\}.
  51. M ( Q , Σ , T ) M(Q,\Sigma,T)
  52. v , w Σ * v,w\in\Sigma^{*}
  53. T w T v = T v w T_{w}\circ T_{v}=T_{vw}
  54. T ε T_{\varepsilon}
  55. M ( Q , Σ , T ) M(Q,\Sigma,T)
  56. ( Q , Σ , T ) (Q,\Sigma,T)
  57. P n \mathbb{C}P^{n}
  58. Σ \Sigma
  59. ( P n , Σ , { U σ 1 , U σ 2 , , U σ p , } ) (\mathbb{C}P^{n},\Sigma,\{U_{\sigma_{1}},U_{\sigma_{2}},\ldots,U_{\sigma_{p}},\})
  60. Σ \Sigma
  61. U σ U_{\sigma}
  62. σ Σ \sigma\in\Sigma
  63. P n \mathbb{C}P^{n}

Semimartingale.html

  1. X t = M t + A t X_{t}=M_{t}+A_{t}
  2. H X t 1 { t > T } A ( X t - X T ) . H\cdot X_{t}\equiv 1_{\{t>T\}}A(X_{t}-X_{T}).
  3. { H X t : H is simple predictable and | H | 1 } \left\{H\cdot X_{t}:H{\rm\ is\ simple\ predictable\ and\ }|H|\leq 1\right\}
  4. M t = X 0 + 0 t σ s d W s , A t = 0 t b s d s . M_{t}=X_{0}+\int_{0}^{t}\sigma_{s}\,dW_{s},\ A_{t}=\int_{0}^{t}b_{s}\,ds.
  5. [ X ] t = s t Δ X s 2 [X]_{t}=\sum_{s\leq t}\Delta X_{s}^{2}

Separation_principle.html

  1. x ˙ ( t ) \displaystyle\dot{x}(t)
  2. u ( t ) u(t)
  3. y ( t ) y(t)
  4. x ( t ) x(t)
  5. x ^ ˙ = ( A - L C ) x ^ + B u + L y \dot{\hat{x}}=(A-LC)\hat{x}+Bu+Ly\,
  6. u ( t ) = - K x ^ . u(t)=-K\hat{x}\,.
  7. e = x - x ^ . e=x-\hat{x}\,.
  8. e ˙ = ( A - L C ) e \dot{e}=(A-LC)e\,
  9. u ( t ) = - K ( x - e ) . u(t)=-K(x-e)\,.
  10. [ x ˙ e ˙ ] = [ A - B K B K 0 A - L C ] [ x e ] . \begin{bmatrix}\dot{x}\\ \dot{e}\\ \end{bmatrix}=\begin{bmatrix}A-BK&BK\\ 0&A-LC\\ \end{bmatrix}\begin{bmatrix}x\\ e\\ \end{bmatrix}.

Separation_property_(finance).html

  1. 4 t h 4^{th}

Sequentially_compact_space.html

  1. 2 0 = 𝔠 2^{\aleph_{0}}=\mathfrak{c}

SERF.html

  1. R s e R_{se}
  2. R s e = 1 2 π T s e ( 2 I ( 2 I - 1 ) 3 ( 2 I + 1 ) 2 ) R_{se}=\frac{1}{2\pi T_{se}}\left(\frac{2I(2I-1)}{3(2I+1)^{2}}\right)
  3. T s e T_{se}
  4. I I
  5. ν \nu
  6. γ e \gamma_{e}
  7. R s e = γ e 2 B 2 T s e 2 π 1 2 ( 1 - ( 2 I + 1 ) 2 Q 2 ) R_{se}=\frac{\gamma_{e}^{2}B^{2}T_{se}}{2\pi}\frac{1}{2}\left(1-\frac{(2I+1)^{% 2}}{Q^{2}}\right)
  8. Q Q
  9. Q ( I = 3 / 2 ) = 4 ( 2 - 4 3 + P 2 ) - 1 Q(I=3/2)=4\left(2-\frac{4}{3+P^{2}}\right)^{-1}
  10. Q ( I = 5 / 2 ) = 6 ( 3 - 48 ( 1 + P 2 ) 19 + 26 P 2 + 3 P 4 ) - 1 Q(I=5/2)=6\left(3-\frac{48(1+P^{2})}{19+26P^{2}+3P^{4}}\right)^{-1}
  11. Q ( I = 7 / 2 ) = 8 ( 4 ( 1 + 7 P 2 + 7 P 4 + P 6 ) 11 + 35 P 2 + 17 P 4 + P 6 ) - 1 Q(I=7/2)=8\left(\frac{4(1+7P^{2}+7P^{4}+P^{6})}{11+35P^{2}+17P^{4}+P^{6}}% \right)^{-1}
  12. P P
  13. R t o t = Q Δ ν R_{tot}=Q\Delta\nu
  14. δ B \delta B
  15. N N
  16. T 2 T_{2}
  17. δ B = 1 γ 2 R t o t Q F z N \delta B=\frac{1}{\gamma}\sqrt{\frac{2R_{tot}Q}{F_{z}N}}
  18. γ \gamma
  19. F z F_{z}
  20. F = I + S F=I+S
  21. R t o t = R D + R s d , s e l f + R s d , He + R s d , N 2 R_{tot}=R_{D}+R_{sd,self}+R_{sd,\mathrm{He}}+R_{sd,\mathrm{N_{2}}}
  22. R D R_{D}
  23. R s d , X R_{sd,X}
  24. R t o t R_{tot}
  25. D 1 D_{1}

Series_acceleration.html

  1. S = { s n } n 𝒩 S=\{s_{n}\}_{n\in\mathcal{N}}
  2. lim n s n = , \lim_{n\to\infty}s_{n}=\ell,
  3. S = { s n } n 𝒩 S^{\prime}=\{s^{\prime}_{n}\}_{n\in\mathcal{N}}
  4. \ell
  5. lim n s n - s n - = 0. \lim_{n\to\infty}\frac{s^{\prime}_{n}-\ell}{s_{n}-\ell}=0.
  6. \ell
  7. 5.828 - n 5.828^{-n}
  8. 17.93 - n 17.93^{-n}
  9. n n
  10. n = 0 ( - 1 ) n a n = n = 0 ( - 1 ) n Δ n a 0 2 n + 1 \sum_{n=0}^{\infty}(-1)^{n}a_{n}=\sum_{n=0}^{\infty}(-1)^{n}\frac{\Delta^{n}a_% {0}}{2^{n+1}}
  11. Δ \Delta
  12. Δ n a 0 = k = 0 n ( - 1 ) k ( n k ) a n - k . \Delta^{n}a_{0}=\sum_{k=0}^{n}(-1)^{k}{n\choose k}a_{n-k}.
  13. n = 0 a n \sum_{n=0}^{\infty}a_{n}
  14. f ( z ) = n = 0 a n z n f(z)=\sum_{n=0}^{\infty}a_{n}z^{n}
  15. z = Φ ( w ) z=\Phi(w)
  16. Φ ( 0 ) = 0 \Phi(0)=0
  17. Φ ( 1 ) = 1 \Phi(1)=1
  18. Φ \Phi
  19. g ( w ) = f ( Φ ( w ) ) g(w)=f\left(\Phi(w)\right)
  20. Φ ( 1 ) = 1 \Phi(1)=1
  21. z = Φ ( w ) z=\Phi(w)
  22. Φ ( 0 ) = 0 \Phi(0)=0
  23. Φ ( 0 ) 0 \Phi^{\prime}(0)\neq 0
  24. 𝔸 : S S = 𝔸 ( S ) = ( s n ) n 𝒩 \mathbb{A}:S\to S^{\prime}=\mathbb{A}(S)={(s^{\prime}_{n})}_{n\in\mathcal{N}}
  25. s n = s n + 2 - ( s n + 2 - s n + 1 ) 2 s n + 2 - 2 s n + 1 + s n . s^{\prime}_{n}=s_{n+2}-\frac{(s_{n+2}-s_{n+1})^{2}}{s_{n+2}-2s_{n+1}+s_{n}}.

Shallow_water_equations.html

  1. η t + ( η u ) x + ( η v ) y = 0 ( η u ) t + x ( η u 2 + 1 2 g η 2 ) + ( η u v ) y = 0 ( η v ) t + ( η u v ) x + y ( η v 2 + 1 2 g η 2 ) = 0. \begin{aligned}\displaystyle\frac{\partial\eta}{\partial t}+\frac{\partial(% \eta u)}{\partial x}+\frac{\partial(\eta v)}{\partial y}&\displaystyle=0\\ \displaystyle\frac{\partial(\eta u)}{\partial t}+\frac{\partial}{\partial x}% \left(\eta u^{2}+\frac{1}{2}g\eta^{2}\right)+\frac{\partial(\eta uv)}{\partial y% }&\displaystyle=0\\ \displaystyle\frac{\partial(\eta v)}{\partial t}+\frac{\partial(\eta uv)}{% \partial x}+\frac{\partial}{\partial y}\left(\eta v^{2}+\frac{1}{2}g\eta^{2}% \right)&\displaystyle=0.\end{aligned}
  2. u t + u u x + v u y - f v = - g h x - b u , v t + u v x + v v y + f u = - g h y - b v , h t = - x ( u ( H + h ) ) - y ( v ( H + h ) ) , \begin{aligned}\displaystyle\frac{\partial u}{\partial t}+u\frac{\partial u}{% \partial x}+v\frac{\partial u}{\partial y}-fv&\displaystyle=-g\frac{\partial h% }{\partial x}-bu,\\ \displaystyle\frac{\partial v}{\partial t}+u\frac{\partial v}{\partial x}+v% \frac{\partial v}{\partial y}+fu&\displaystyle=-g\frac{\partial h}{\partial y}% -bv,\\ \displaystyle\frac{\partial h}{\partial t}&\displaystyle=-\frac{\partial}{% \partial x}\Bigl(u\left(H+h\right)\Bigr)-\frac{\partial}{\partial y}\Bigl(v% \left(H+h\right)\Bigr),\end{aligned}
  3. u t - f v = - g h x - b u , v t + f u = - g h y - b v , h t = - H ( u x + v y ) \begin{aligned}\displaystyle\frac{\partial u}{\partial t}-fv&\displaystyle=-g% \frac{\partial h}{\partial x}-bu,\\ \displaystyle\frac{\partial v}{\partial t}+fu&\displaystyle=-g\frac{\partial h% }{\partial y}-bv,\\ \displaystyle\frac{\partial h}{\partial t}&\displaystyle=-H\Bigl(\frac{% \partial u}{\partial x}+\frac{\partial v}{\partial y}\Bigr)\end{aligned}

Shamir's_Secret_Sharing.html

  1. k k
  2. S S
  3. n n
  4. S 1 , , S n S_{1},\ldots,S_{n}
  5. k k
  6. S i S_{i}
  7. S S
  8. k - 1 k-1
  9. S i S_{i}
  10. S S
  11. ( k , n ) \left(k,n\right)
  12. k = n k=n
  13. k k\,\!
  14. k - 1 k-1\,\!
  15. ( k , n ) \left(k,n\right)\,\!
  16. S S\,\!
  17. F F
  18. P P
  19. 0 < k n < P ; S < P 0<k\leq n<P;S<P
  20. P P
  21. k - 1 k-1\,\!
  22. a 1 , , a k - 1 a_{1},\cdots,a_{k-1}\,\!
  23. a i < P a_{i}<P
  24. a 0 = S a_{0}=S\,\!
  25. f ( x ) = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + + a k - 1 x k - 1 f\left(x\right)=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots+a_{k-1}x^{k-1}\,\!
  26. n n\,\!
  27. i = 1 , , n i=1,\cdots,n\,\!
  28. ( i , f ( i ) ) \left(i,f\left(i\right)\right)\,\!
  29. k k\,\!
  30. a 0 a_{0}\,\!
  31. ( S = 1234 ) (S=1234)\,\!
  32. ( n = 6 ) (n=6)\,\!
  33. ( k = 3 ) (k=3)\,\!
  34. k - 1 k-1
  35. ( a 1 = 166 ; a 2 = 94 ) (a_{1}=166;a_{2}=94)\,\!
  36. f ( x ) = 1234 + 166 x + 94 x 2 f\left(x\right)=1234+166x+94x^{2}\,\!
  37. D x - 1 = ( x , f ( x ) ) D_{x-1}=(x,f(x))
  38. D 0 = ( 1 , 1494 ) ; D 1 = ( 2 , 1942 ) ; D 2 = ( 3 , 2578 ) ; D 3 = ( 4 , 3402 ) ; D 4 = ( 5 , 4414 ) ; D 5 = ( 6 , 5614 ) D_{0}=\left(1,1494\right);D_{1}=\left(2,1942\right);D_{2}=\left(3,2578\right);% D_{3}=\left(4,3402\right);D_{4}=\left(5,4414\right);D_{5}=\left(6,5614\right)\,\!
  39. x x\,\!
  40. f ( x ) f\left(x\right)\,\!
  41. D x - 1 D_{x-1}
  42. D x D_{x}
  43. ( 1 , f ( 1 ) ) (1,f(1))
  44. ( 0 , f ( 0 ) ) (0,f(0))
  45. ( 0 , f ( 0 ) ) (0,f(0))
  46. S = f ( 0 ) S=f(0)
  47. ( x 0 , y 0 ) = ( 2 , 1942 ) ; ( x 1 , y 1 ) = ( 4 , 3402 ) ; ( x 2 , y 2 ) = ( 5 , 4414 ) \left(x_{0},y_{0}\right)=\left(2,1942\right);\left(x_{1},y_{1}\right)=\left(4,% 3402\right);\left(x_{2},y_{2}\right)=\left(5,4414\right)\,\!
  48. 0 = x - x 1 x 0 - x 1 x - x 2 x 0 - x 2 = x - 4 2 - 4 x - 5 2 - 5 = 1 6 x 2 - 3 2 x + 10 3 \ell_{0}=\frac{x-x_{1}}{x_{0}-x_{1}}\cdot\frac{x-x_{2}}{x_{0}-x_{2}}=\frac{x-4% }{2-4}\cdot\frac{x-5}{2-5}=\frac{1}{6}x^{2}-\frac{3}{2}x+\frac{10}{3}\,\!
  49. 1 = x - x 0 x 1 - x 0 x - x 2 x 1 - x 2 = x - 2 4 - 2 x - 5 4 - 5 = - 1 2 x 2 + 7 2 x - 5 \ell_{1}=\frac{x-x_{0}}{x_{1}-x_{0}}\cdot\frac{x-x_{2}}{x_{1}-x_{2}}=\frac{x-2% }{4-2}\cdot\frac{x-5}{4-5}=-\frac{1}{2}x^{2}+\frac{7}{2}x-5\,\!
  50. 2 = x - x 0 x 2 - x 0 x - x 1 x 2 - x 1 = x - 2 5 - 2 x - 4 5 - 4 = 1 3 x 2 - 2 x + 8 3 \ell_{2}=\frac{x-x_{0}}{x_{2}-x_{0}}\cdot\frac{x-x_{1}}{x_{2}-x_{1}}=\frac{x-2% }{5-2}\cdot\frac{x-4}{5-4}=\frac{1}{3}x^{2}-2x+\frac{8}{3}\,\!
  51. f ( x ) = j = 0 2 y j j ( x ) f(x)=\sum_{j=0}^{2}y_{j}\cdot\ell_{j}(x)\,\!
  52. = 1234 + 166 x + 94 x 2 =1234+166x+94x^{2}\,\!
  53. S = 1234 S=1234\,\!
  54. S S
  55. D i D_{i}
  56. D 0 = ( 1 , 1494 ) D_{0}=(1,1494)
  57. D 1 = ( 2 , 1942 ) D_{1}=(2,1942)
  58. k = 3 k=3
  59. S S
  60. n = 6 , k = 3 , f ( x ) = a 0 + a 1 x + + a k - 1 x k - 1 , a 0 = S , a i n=6,k=3,f(x)=a_{0}+a_{1}x+\dots+a_{k-1}x^{k-1},a_{0}=S,a_{i}\in\mathbb{N}
  61. S [ 1046 , 1048 , , 1342 , 1344 ] S\in[1046,1048,\dots,1342,1344]
  62. p : p > S , p > n p\in\mathbb{P}:p>S,p>n
  63. p p
  64. a i a_{i}
  65. a 0 = S a_{0}=S
  66. ( x , f ( x ) ( mod p ) ) (x,f(x)\;\;(\mathop{{\rm mod}}p))
  67. ( x , f ( x ) ) (x,f(x))
  68. p p
  69. p p
  70. p p
  71. p > S S [ 0 , 1 , , p - 2 , p - 1 ] p>S\Rightarrow{}S\in{[0,1,\dots,p-2,p-1]}
  72. p p
  73. S S
  74. p p
  75. f ( x ) ( mod p ) = f ( x ) f(x)\;\;(\mathop{{\rm mod}}p)=f(x)
  76. p p
  77. S S
  78. p = 1613 p=1613
  79. f ( x ) = 1234 + 166 x + 94 x 2 mod 1613 f\left(x\right)=1234+166x+94x^{2}\mod{1613}
  80. ( 1 , 1494 ) ; ( 2 , 329 ) ; ( 3 , 965 ) ; ( 4 , 176 ) ; ( 5 , 1188 ) ; ( 6 , 775 ) \left(1,1494\right);\left(2,329\right);\left(3,965\right);\left(4,176\right);% \left(5,1188\right);\left(6,775\right)
  81. D x D_{x}
  82. k k
  83. D 0 = ( 1 , 1494 ) D_{0}=\left(1,1494\right)
  84. D 1 = ( 2 , 329 ) D_{1}=\left(2,329\right)
  85. n = 6 , k = 3 , p = 1613 , f ( x ) = a 0 + a 1 x + + a k - 1 x k - 1 mod p , a 0 = S , a i n=6,k=3,p=1613,f(x)=a_{0}+a_{1}x+\dots+a_{k-1}x^{k-1}\mod{p},a_{0}=S,a_{i}\in% \mathbb{N}
  86. ( m 1 - m 2 ) (m_{1}-m_{2})
  87. m 2 > m 1 m_{2}>m_{1}
  88. a 1 a_{1}
  89. [ 448 , 445 , 442 , ] [448,445,442,...]
  90. 1613 1613
  91. 3 3
  92. a 1 [ 1 , 4 , 7 , ] a_{1}\in[1,4,7,\dots]
  93. ( k , n ) \left(k,n\right)\,\!
  94. k k\,\!
  95. D i D_{i}\,\!