wpmath0000016_10

Multisymplectic_integrator.html

  1. K z t + L z x = S ( z ) , Kz_{t}+Lz_{x}=\nabla S(z),
  2. z ( t , x ) z(t,x)
  3. K K
  4. L L
  5. S \nabla S
  6. S S
  7. J z t = H ( z ) Jz_{t}=\nabla H(z)
  8. u t t - u x x = V ( u ) u_{tt}-u_{xx}=V^{\prime}(u)
  9. u t t = x σ ( u x ) - f ( u ) u_{tt}=\partial_{x}\sigma^{\prime}(u_{x})-f^{\prime}(u)
  10. u t + u u x + u x x x = 0 u_{t}+uu_{x}+u_{xxx}=0
  11. ω \omega
  12. κ \kappa
  13. ω ( u , v ) = K u , v and κ ( u , v ) = L u , v \omega(u,v)=\langle Ku,v\rangle\quad\,\text{and}\quad\kappa(u,v)=\langle Lu,v\rangle
  14. , \langle\,\cdot\,,\,\cdot\,\rangle
  15. t ω + x κ = 0. \partial_{t}\omega+\partial_{x}\kappa=0.
  16. u t u_{t}
  17. t E ( u ) + x F ( u ) = 0 where E ( u ) = S ( u ) - 1 2 κ ( u x , u ) , F ( u ) = 1 2 κ ( u t , u ) . \partial_{t}E(u)+\partial_{x}F(u)=0\quad\,\text{where}\quad E(u)=S(u)-\tfrac{1% }{2}\kappa(u_{x},u),\,F(u)=\tfrac{1}{2}\kappa(u_{t},u).
  18. t I ( u ) + x G ( u ) = 0 where I ( u ) = 1 2 ω ( u x , u ) , G ( u ) = S ( u ) - 1 2 ω ( u t , u ) . \partial_{t}I(u)+\partial_{x}G(u)=0\quad\,\text{where}\quad I(u)=\tfrac{1}{2}% \omega(u_{x},u),\,G(u)=S(u)-\tfrac{1}{2}\omega(u_{t},u).
  19. K K
  20. L L
  21. K \displaystyle K
  22. K + K_{+}
  23. L + L_{+}
  24. K K
  25. L L
  26. u n , i u_{n,i}
  27. u ( n Δ t , i Δ x ) u(n\Delta{t},i\Delta{x})
  28. Δ t \Delta{t}
  29. Δ x \Delta{x}
  30. K + t + u n , i + K - t - u n , i + L + x + u n , i + L - x - u n , i = S ( u n , i ) K_{+}\partial_{t}^{+}u_{n,i}+K_{-}\partial_{t}^{-}u_{n,i}+L_{+}\partial_{x}^{+% }u_{n,i}+L_{-}\partial_{x}^{-}u_{n,i}=\nabla{S}(u_{n,i})
  31. t + u n , i = u n + 1 , i - u n , i Δ t , x + u n , i = u n , i + 1 - u n , i Δ x , t - u n , i = u n , i - u n - 1 , i Δ t , x - u n , i = u n , i - u n , i - 1 Δ x . \begin{aligned}\displaystyle\partial_{t}^{+}u_{n,i}&\displaystyle=\frac{u_{n+1% ,i}-u_{n,i}}{\Delta{t}},&\displaystyle\partial_{x}^{+}u_{n,i}&\displaystyle=% \frac{u_{n,i+1}-u_{n,i}}{\Delta{x}},\\ \displaystyle\partial_{t}^{-}u_{n,i}&\displaystyle=\frac{u_{n,i}-u_{n-1,i}}{% \Delta{t}},&\displaystyle\partial_{x}^{-}u_{n,i}&\displaystyle=\frac{u_{n,i}-u% _{n,i-1}}{\Delta{x}}.\end{aligned}
  32. t + ω n , i + x + κ n , i = 0 where ω n , i = d u n , i - 1 K + d u n , i and κ n , i = d u n - 1 , i L + d u n , i . \partial_{t}^{+}\omega_{n,i}+\partial_{x}^{+}\kappa_{n,i}=0\quad\,\text{where}% \quad\omega_{n,i}=\mathrm{d}u_{n,i-1}\wedge K_{+}\,\mathrm{d}u_{n,i}\quad\,% \text{and}\quad\kappa_{n,i}=\mathrm{d}u_{n-1,i}\wedge L_{+}\,\mathrm{d}u_{n,i}.
  33. K t + u n , i + 1 / 2 + L x + u n + 1 / 2 , i = S ( u n + 1 / 2 , i + 1 / 2 ) , K\partial_{t}^{+}u_{n,i+1/2}+L\partial_{x}^{+}u_{n+1/2,i}=\nabla{S}(u_{n+1/2,i% +1/2}),
  34. t + \partial_{t}^{+}
  35. x + \partial_{x}^{+}
  36. u n , i + 1 / 2 = u n , i + u n , i + 1 2 , u n + 1 / 2 , i = u n , i + u n + 1 , i 2 , u n + 1 / 2 , i + 1 / 2 = u n , i + u n , i + 1 + u n + 1 , i + u n + 1 , i + 1 4 . u_{n,i+1/2}=\frac{u_{n,i}+u_{n,i+1}}{2},\quad u_{n+1/2,i}=\frac{u_{n,i}+u_{n+1% ,i}}{2},u_{n+1/2,i+1/2}=\frac{u_{n,i}+u_{n,i+1}+u_{n+1,i}+u_{n+1,i+1}}{4}.
  37. t + ω n , i + x + κ n , i = 0 where ω n , i = d u n , i + 1 / 2 K d u n , i + 1 / 2 and κ n , i = d u n + 1 / 2 , i L d u n + 1 / 2 , i . \partial_{t}^{+}\omega_{n,i}+\partial_{x}^{+}\kappa_{n,i}=0\quad\,\text{where}% \quad\omega_{n,i}=\mathrm{d}u_{n,i+1/2}\wedge K\,\mathrm{d}u_{n,i+1/2}\quad\,% \text{and}\quad\kappa_{n,i}=\mathrm{d}u_{n+1/2,i}\wedge L\,\mathrm{d}u_{n+1/2,% i}.

Munchausen_number.html

  1. n = a k a k - 1 a 0 , n=a_{k}a_{k-1}\cdots a_{0},
  2. n = i = 0 k a i a i . n=\sum_{i=0}^{k}a_{i}^{a_{i}}.
  3. 3435 = 3 3 + 4 4 + 3 3 + 5 5 = 27 + 256 + 27 + 3125. 3435=3^{3}+4^{4}+3^{3}+5^{5}=27+256+27+3125.

Muon_tomography.html

  1. d N d θ = 1 2 π σ 2 e - θ 2 2 θ 0 2 \frac{dN}{d\theta}={\frac{1}{2\pi\sigma^{2}}}e^{-\frac{\theta^{2}}{2\theta_{0}% ^{2}}}
  2. θ 0 \theta_{0}
  3. θ 0 = 14.1 M e V ρ β l X 0 \theta_{0}={\frac{14.1MeV}{\rho\beta}}\sqrt{\frac{l}{X_{0}}}
  4. X \vec{X}
  5. X \vec{X}

Murnaghan–Nakayama_rule.html

  1. χ ρ λ = T ( - 1 ) h t ( T ) \chi^{\lambda}_{\rho}=\sum_{T}(-1)^{ht(T)}

Musselman's_theorem.html

  1. T T
  2. A A
  3. B B
  4. C C
  5. A * A^{*}
  6. B * B^{*}
  7. C * C^{*}
  8. T * T^{*}
  9. T T
  10. O O
  11. T T
  12. S A S_{A}
  13. S B S_{B}
  14. S C S_{C}
  15. A O A * A\,O\,A^{*}
  16. B O B * B\,O\,B^{*}
  17. C O C * C\,O\,C^{*}
  18. M M
  19. T T
  20. T T
  21. M M
  22. T T
  23. X 1157 X_{1157}
  24. A A
  25. B B
  26. C C
  27. T T
  28. O O
  29. H H
  30. T T
  31. A A^{\prime}
  32. B B^{\prime}
  33. C C^{\prime}
  34. O A OA
  35. O B OB
  36. O C OC
  37. O A / O A = O B / O B = O C / O C = t OA^{\prime}/OA=OB^{\prime}/OB=OC^{\prime}/OC=t
  38. L A L_{A}
  39. L B L_{B}
  40. L C L_{C}
  41. O A OA
  42. O B OB
  43. O C OC
  44. A A^{\prime}
  45. B B^{\prime}
  46. C C^{\prime}
  47. P A P_{A}
  48. P B P_{B}
  49. P C P_{C}
  50. B C BC
  51. C A CA
  52. A B AB
  53. P A P_{A}
  54. P B P_{B}
  55. P C P_{C}
  56. R R
  57. N N
  58. O O
  59. R R
  60. N N^{\prime}
  61. O N ON
  62. O N / O N = t ON^{\prime}/ON=t
  63. N N^{\prime}
  64. T T
  65. Q Q
  66. O H OH
  67. Q H / Q O = 2 t QH/QO=2t

Mutation_(algebra).html

  1. A ( a ) A(a)
  2. x * y = ( x a ) y . x*y=(xa)y.\,
  3. A ( a , b ) A(a,b)
  4. x * y = ( x a ) y - ( y b ) x . x*y=(xa)y-(yb)x.\,
  5. ( x y ) ( x x ) = x ( y ( x x ) ) (xy)(xx)=x(y(xx))
  6. { a , b , c } = ( a b ) c + ( c b ) a - ( a c ) b . \{a,b,c\}=(ab)c+(cb)a-(ac)b.\,
  7. a b = { a , y , b } . a\circ b=\{a,y,b\}.\,

N-acetylmuramic_acid_6-phosphate_etherase.html

  1. \rightleftharpoons

N-acetylneuraminate_epimerase.html

  1. \rightleftharpoons

N-transform.html

  1. R ( u , s ) = 𝒩 { f ( t ) } = 0 f ( u t ) e - s t d t . ( 1 ) R(u,s)=\mathcal{N}\{f(t)\}=\int_{0}^{\infty}f(ut)e^{-st}\,dt.\qquad(1)

N2-citryl-N6-acetyl-N6-hydroxylysine_synthase.html

  1. \rightleftharpoons

N_conjecture.html

  1. n 3 {n\geq 3}
  2. a 1 , a 2 , , a n {a_{1},a_{2},...,a_{n}\in\mathbb{Z}}
  3. gcd ( a 1 , a 2 , , a n ) = 1 \gcd(a_{1},a_{2},...,a_{n})=1
  4. a 1 + a 2 + + a n = 0 {a_{1}+a_{2}+...+a_{n}=0}
  5. a 1 , a 2 , , a n {a_{1},a_{2},...,a_{n}}
  6. 0 {0}
  7. ε > 0 {\varepsilon>0}
  8. C C
  9. n {n}
  10. ε {\varepsilon}
  11. max ( | a 1 | , | a 2 | , , | a n | ) < C n , ε rad ( | a 1 | | a 2 | | a n | ) 2 n - 5 + ε \operatorname{max}(|a_{1}|,|a_{2}|,...,|a_{n}|)<C_{n,\varepsilon}\operatorname% {rad}(|a_{1}|\cdot|a_{2}|\cdot...\cdot|a_{n}|)^{2n-5+\varepsilon}
  12. rad ( m ) \operatorname{rad}(m)
  13. m {m}
  14. m {m}
  15. a 1 , a 2 , , a n {a_{1},a_{2},...,a_{n}}
  16. q ( a 1 , a 2 , , a n ) = log ( max ( | a 1 | , | a 2 | , , | a n | ) ) log ( rad ( | a 1 | | a 2 | | a n | ) ) q(a_{1},a_{2},...,a_{n})=\frac{\log(\operatorname{max}(|a_{1}|,|a_{2}|,...,|a_% {n}|))}{\log(\operatorname{rad}(|a_{1}|\cdot|a_{2}|\cdot...\cdot|a_{n}|))}
  17. lim sup q ( a 1 , a 2 , , a n ) = 2 n - 5 \limsup q(a_{1},a_{2},...,a_{n})=2n-5
  18. a 1 , a 2 , , a n {a_{1},a_{2},...,a_{n}}
  19. a 1 , a 2 , , a n {a_{1},a_{2},...,a_{n}}
  20. n 3 {n\geq 3}
  21. a 1 , a 2 , , a n {a_{1},a_{2},...,a_{n}\in\mathbb{Z}}
  22. a 1 , a 2 , , a n {a_{1},a_{2},...,a_{n}}
  23. a 1 + a 2 + + a n = 0 {a_{1}+a_{2}+...+a_{n}=0}
  24. a 1 , a 2 , , a n {a_{1},a_{2},...,a_{n}}
  25. 0 {0}
  26. ε > 0 {\varepsilon>0}
  27. C C
  28. n {n}
  29. ε {\varepsilon}
  30. max ( | a 1 | , | a 2 | , , | a n | ) < C n , ε rad ( | a 1 | | a 2 | | a n | ) 1 + ε \operatorname{max}(|a_{1}|,|a_{2}|,...,|a_{n}|)<C_{n,\varepsilon}\operatorname% {rad}(|a_{1}|\cdot|a_{2}|\cdot...\cdot|a_{n}|)^{1+\varepsilon}
  31. a 1 , a 2 , , a n {a_{1},a_{2},...,a_{n}}
  32. q ( a 1 , a 2 , , a n ) = log ( max ( | a 1 | , | a 2 | , , | a n | ) ) log ( rad ( | a 1 | | a 2 | | a n | ) ) q(a_{1},a_{2},...,a_{n})=\frac{\log(\operatorname{max}(|a_{1}|,|a_{2}|,...,|a_% {n}|))}{\log(\operatorname{rad}(|a_{1}|\cdot|a_{2}|\cdot...\cdot|a_{n}|))}
  33. lim sup q ( a 1 , a 2 , , a n ) = 1 \limsup q(a_{1},a_{2},...,a_{n})=1

NAD(P)H-hydrate_epimerase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Narrowing_of_algebraic_value_sets.html

  1. x 2 = 4 x^{2}=4
  2. x = 2 x = - 2 x=2x=-2
  3. x = ± 2 x=\pm 2
  4. x + x { 4 , 0 , - 4 } x+x\in\{4,0,-4\}
  5. V ( { 2 : : x 1 , - 2 : : x 2 } ) V(\{2::x_{1},-2::x_{2}\})
  6. 2 : : x 1 2::x_{1}
  7. - 2 : : x 2 -2::x_{2}
  8. x 1 x_{1}
  9. x 2 x_{2}
  10. x 1 x_{1}
  11. x 2 x_{2}
  12. x 2 = 4 x^{2}=4
  13. x = V ( { 2 : : x 1 , - 2 : : x 2 } ) x=V(\{2::x_{1},-2::x_{2}\})
  14. { x 1 , x 2 } \{x_{1},x_{2}\}
  15. x 1 x_{1}
  16. x 2 x_{2}
  17. x 1 x_{1}
  18. x 2 x_{2}
  19. ( x = 2 and x = - 2 ) = false (x=2\and x=-2)=\,\text{false}
  20. V ( M ) V ( N ) = V ( { ( m v n v , m l n l ) : m v : : m l M and n v : : n l N } ) V(M)\ V(N)=V(\{(m_{v}\ n_{v},m_{l}\cap n_{l}):m_{v}::m_{l}\in M\and n_{v}::n_{% l}\in N\})
  21. x + x = V ( { 2 : : x 1 , - 2 : : x 2 } ) + V ( { 2 : : x 1 , - 2 : : x 2 } ) x+x=V(\{2::x_{1},-2::x_{2}\})+V(\{2::x_{1},-2::x_{2}\})
  22. = V ( { - 2 + - 2 : : x 1 x 1 , - 2 + 2 : : x 1 x 2 , 2 + - 2 : : x 1 x 2 , 2 + 2 : : x 2 } ) =V(\{-2+-2::x_{1}\cap x_{1},-2+2::x_{1}\cap x_{2},2+-2::x_{1}\cap x_{2},2+2::x% _{2}\})
  23. = V ( { - 4 : : x 1 x 1 , 0 : : x 1 x 2 , 0 : : x 2 and x 1 , 2 + 2 : : x 2 x 1 } ) =V(\{-4::x_{1}\cap x_{1},0::x_{1}\cap x_{2},0::x_{2}\and x_{1},2+2::x_{2}\cap x% _{1}\})
  24. x 1 x 1 = x 1 x_{1}\cap x_{1}=x_{1}
  25. x 2 x 2 = x 2 x_{2}\cap x_{2}=x_{2}
  26. x 1 x 2 = { } x_{1}\cap x_{2}=\{\}
  27. x 2 x 1 = { } x_{2}\cap x_{1}=\{\}
  28. = V ( { - 4 : : x 1 , 0 : : { } , 0 : : { } , 4 : : x 2 } ) =V(\{-4::x_{1},0::\{\},0::\{\},4::x_{2}\})
  29. V ( K ) = V ( { ( v , l ) : ( v , l ) K and l { } } V(K)=V(\{(v,l):(v,l)\in K\and l\neq\{\}\}
  30. = V ( { - 4 : : x 1 , 4 : : x 2 } ) =V(\{-4::x_{1},4::x_{2}\})
  31. x + x x+x
  32. a and b a\and b
  33. a b ab
  34. a = V ( { false : : a 1 , true : : a 2 } ) a=V(\{\operatorname{false}::a_{1},\operatorname{true}::a_{2}\})
  35. b = V ( { true : : a 1 , false : : a 2 , true : : a 2 } ) b=V(\{\operatorname{true}::a_{1},\operatorname{false}::a_{2},\operatorname{% true}::a_{2}\})
  36. x = 2 x = - 2 x=2x=-2
  37. x = V ( { _ : : a 1 , 2 : : a 2 } ) x=V(\{\_::a_{1},2::a_{2}\})
  38. x = V ( { - 2 : : a 1 , _ : : a 2 , - 2 : : a 2 } ) x=V(\{-2::a_{1},\_::a_{2},-2::a_{2}\})
  39. x = V ( { - 2 : : a 1 , 2 : : a 2 } ) x=V(\{-2::a_{1},2::a_{2}\})
  40. - 2 : : a 2 -2::a_{2}
  41. m v m l n v n l ( ( m v , m l ) M and ( n v , n l ) N ) ( m v n v m l n l = { } ) \forall m_{v}\forall m_{l}\forall n_{v}\forall n_{l}((m_{v},m_{l})\in M\and(n_% {v},n_{l})\in N)\to(m_{v}\neq n_{v}\to m_{l}\cap n_{l}=\{\})
  42. - 2 : : a 2 -2::a_{2}
  43. 2 : : a 2 2::a_{2}
  44. X = V ( { 1 : : x 1 , 3 : : x 2 , 4 : : x 3 } ) X=V(\{1::x_{1},3::x_{2},4::x_{3}\})
  45. Y = V ( { 8 : : y 1 , 9 : : y 2 } ) Y=V(\{8::y_{1},9::y_{2}\})
  46. X * Y < 25 X*Y<25
  47. X + Y > 10 X+Y>10
  48. X * Y < 25 X*Y<25
  49. V ( { 8 : : x 1 y 1 , 24 : : x 2 y 1 , 32 : : x 3 y 1 , 9 : : x 1 y 2 , 27 : : x 2 y 2 , 36 : : x 3 y 2 } ) < 25 V(\{8::x_{1}\cap y_{1},24::x_{2}\cap y_{1},32::x_{3}\cap y_{1},9::x_{1}\cap y_% {2},27::x_{2}\cap y_{2},36::x_{3}\cap y_{2}\})<25
  50. V ( { 8 < 25 : : x 1 y 1 , 24 < 25 : : x 2 y 1 , 32 < 25 : : x 3 y 1 , 9 < 25 : : x 1 y 2 , 27 < 25 : : x 2 y 2 , 36 < 25 : : x 3 y 2 } ) < 25 V(\{8<25::x_{1}\cap y_{1},24<25::x_{2}\cap y_{1},32<25::x_{3}\cap y_{1},9<25::% x_{1}\cap y_{2},27<25::x_{2}\cap y_{2},36<25::x_{3}\cap y_{2}\})<25
  51. V ( { true : : x 1 y 1 , true : : x 2 y 1 , false : : x 3 y 1 , true : : x 1 y 2 , false : : x 2 y 2 , false : : x 3 y 2 } ) V(\{\,\text{true}::x_{1}\cap y_{1},\,\text{true}::x_{2}\cap y_{1},\,\text{% false}::x_{3}\cap y_{1},\,\text{true}::x_{1}\cap y_{2},\,\text{false}::x_{2}% \cap y_{2},\,\text{false}::x_{3}\cap y_{2}\})
  52. V ( { true : : x 1 y 1 , true : : x 2 y 2 , true : : x 1 y 2 } ) V(\{\,\text{true}::x_{1}\cap y_{1},\,\text{true}::x_{2}\cap y_{2},\,\text{true% }::x_{1}\cap y_{2}\})
  53. x 3 y 1 x_{3}\cap y_{1}
  54. x 2 y 2 x_{2}\cap y_{2}
  55. x 3 y 2 x_{3}\cap y_{2}
  56. X 3 X_{3}
  57. X X
  58. X = V ( { 1 : : x 1 , 3 : : x 2 } ) X=V(\{1::x_{1},3::x_{2}\})
  59. X + Y > 10 X+Y>10
  60. X = V ( { 1 : : x 1 , 3 : : x 2 } ) X=V(\{1::x_{1},3::x_{2}\})
  61. Y = V ( { 8 : : y 1 , 9 : : y 2 } ) Y=V(\{8::y_{1},9::y_{2}\})
  62. V ( { 1 + 8 : : x 1 y 1 , 3 + 8 : : x 2 y 1 , 2 + 9 : : x 1 y 2 , 1 + 9 : : x 2 y 2 } ) > 10 V(\{1+8::x_{1}\cap y_{1},3+8::x_{2}\cap y_{1},2+9::x_{1}\cap y_{2},1+9::x_{2}% \cap y_{2}\})>10
  63. x 2 y 2 x_{2}\cap y_{2}
  64. V ( { 9 > 10 : : x 1 y 1 , 11 > 10 : : x 2 y 1 , 10 > 10 : : x 1 y 2 } ) V(\{9>10::x_{1}\cap y_{1},11>10::x_{2}\cap y_{1},10>10::x_{1}\cap y_{2}\})
  65. x 1 y 1 x_{1}\cap y_{1}
  66. x 1 y 2 x_{1}\cap y_{2}
  67. V ( { 11 > 10 : : x 2 y 1 } ) V(\{11>10::x_{2}\cap y_{1}\})
  68. X 1 X_{1}
  69. Y 2 Y_{2}
  70. X = V ( { 3 : : x 2 } ) = 3 X=V(\{3::x_{2}\})=3
  71. Y = V ( { 8 : : y 1 } ) = 8 Y=V(\{8::y_{1}\})=8
  72. PizzaShop ( V ( { Carlton : : p 1 , Richmond : : p 2 , South Melbourne : : p 3 , Footscray : : p 4 , St Kilda : : p 5 , Toorak : : p 6 } ) ) \,\text{PizzaShop}(V(\{\,\text{Carlton}::p_{1},\,\text{Richmond}::p_{2},\,% \text{South Melbourne}::p_{3},\,\text{Footscray}::p_{4},\,\text{St Kilda}::p_{% 5},\,\text{Toorak}::p_{6}\}))
  73. BottleshopWithBeer ( V ( { South Melbourne : : b 1 , St Kilda : : b 2 , Carlton : : b 3 , Docklands : : b 4 } ) ) \,\text{BottleshopWithBeer}(V(\{\,\text{South Melbourne}::b_{1},\,\text{St % Kilda}::b_{2},\,\text{Carlton}::b_{3},\,\text{Docklands}::b_{4}\}))
  74. BottleshopWithWhiskey ( V ( { Essendon : : w 1 , South Melbourne : : w 2 } ) ) \,\text{BottleshopWithWhiskey}(V(\{\,\text{Essendon}::w_{1},\,\text{South % Melbourne}::w_{2}\}))
  75. WhereToGo ( x ) = PizzaShop ( x ) and BottleshopWithBeer ( x ) and BottleshopWithWhiskey ( x ) \,\text{WhereToGo}(x)=\,\text{PizzaShop}(x)\and\,\text{BottleshopWithBeer}(x)% \and\,\text{BottleshopWithWhiskey}(x)
  76. x = V ( { Carlton : : p 1 , Richmond : : p 2 , South Melbourne : : p 3 , Footscray : : p 4 , St Kilda : : p 5 , Toorak : : p 6 } ) x=V(\{\,\text{Carlton}::p_{1},\,\text{Richmond}::p_{2},\,\text{South Melbourne% }::p_{3},\,\text{Footscray}::p_{4},\,\text{St Kilda}::p_{5},\,\text{Toorak}::p% _{6}\})
  77. x = V ( { South Melbourne : : b 1 , St Kilda : : b 2 , Carlton : : b 3 , Docklands : : b 4 } ) x=V(\{\,\text{South Melbourne}::b_{1},\,\text{St Kilda}::b_{2},\,\text{Carlton% }::b_{3},\,\text{Docklands}::b_{4}\})
  78. x = V ( { South Melbourne : : b 1 p 3 , St Kilda : : b 2 p 5 , Carlton : : b 3 p 1 } ) x=V(\{\,\text{South Melbourne}::b_{1}\cap p_{3},\,\text{St Kilda}::b_{2}\cap p% _{5},\,\text{Carlton}::b_{3}\cap p_{1}\})
  79. x = V ( { Essendon : : w 1 , South Melbourne : : w 2 } ) x=V(\{\,\text{Essendon}::w_{1},\,\text{South Melbourne}::w_{2}\})
  80. x = V ( { South Melbourne : : b 1 p 3 w 2 } ) x=V(\{\,\text{South Melbourne}::b_{1}\cap p_{3}\cap w_{2}\})
  81. x = V ( { Essendon : : w 1 , South Melbourne : : w 2 } ) x=V(\{\,\text{Essendon}::w_{1},\,\text{South Melbourne}::w_{2}\})
  82. x = V ( { South Melbourne : : b 1 , St Kilda : : b 2 , Carlton : : b 3 , Docklands : : b 4 } ) x=V(\{\,\text{South Melbourne}::b_{1},\,\text{St Kilda}::b_{2},\,\text{Carlton% }::b_{3},\,\text{Docklands}::b_{4}\})
  83. x = V ( { South Melbourne : : b 1 w 2 } ) x=V(\{\,\text{South Melbourne}::b_{1}\cap w_{2}\})
  84. x = { Carlton : : p 1 , Richmond : : p 2 , South Melbourne : : p 3 , Footscray : : p 4 , St Kilda : : p 5 , Toorak : : p 6 } x=\{\,\text{Carlton}::p_{1},\,\text{Richmond}::p_{2},\,\text{South Melbourne}:% :p_{3},\,\text{Footscray}::p_{4},\,\text{St Kilda}::p_{5},\,\text{Toorak}::p_{% 6}\}
  85. x = V ( { South Melbourne : : b 1 w 2 p 3 } ) x=V(\{\,\text{South Melbourne}::b_{1}\cap w_{2}\ \cap p_{3}\})
  86. let x X in x \operatorname{let}x\in X\operatorname{in}x
  87. x X x\in X
  88. ( let x X in x ) ( let y Y in y ) (\operatorname{let}x\in X\operatorname{in}x)\ (\operatorname{let}y\in Y% \operatorname{in}y)
  89. = let x X and y Y in x y =\operatorname{let}x\in X\and y\in Y\operatorname{in}x\ y
  90. = let ( x , y ) X × Y in x y =\operatorname{let}(x,y)\in X\times Y\operatorname{in}x\ y
  91. ( let x X in x ) ( let x X in x ) (\operatorname{let}x\in X\operatorname{in}x)\ (\operatorname{let}x\in X% \operatorname{in}x)
  92. = let x X in x x =\operatorname{let}x\in X\operatorname{in}x\ x
  93. ( v , l ) (v,l)
  94. V ( K ) V(K)
  95. C ( l ) C(l)
  96. C ( l ) = ( ( r , z , u ) l z = u ) ) = ( r z u ( r , z , u ) l z = u ) C(l)=({\bigwedge_{(r,z,u)\in l}z=u}))=(\forall r\forall z\forall u(r,z,u)\in l% \to z=u)
  97. v l ( ( v , l ) K and C ( l ) v = V ( K ) \forall v\forall l((v,l)\in K\and C(l)\to v=V(K)
  98. v l ( v , l ) K and C ( l ) \exists v\exists l(v,l)\in K\and C(l)
  99. v 1 l 1 v 2 l 2 ( ( v 1 , l 1 ) K and ( v 2 , l 2 ) K and ( v 1 , l 1 ) ( v 2 , l 2 ) ) ¬ ( C ( l 1 ) and C ( l 2 ) ) \forall v_{1}\forall l_{1}\forall v_{2}\forall l_{2}((v_{1},l_{1})\in K\and(v_% {2},l_{2})\in K\and(v_{1},l_{1})\neq(v_{2},l_{2}))\to\neg(C(l_{1})\and C(l_{2}))
  100. v l ( v , l ) K and C ( l ) and v = V ( K ) \exists v\exists l(v,l)\in K\and C(l)\and v=V(K)
  101. V ( K ) = let ( v , l ) K and C ( l ) in v V(K)=\operatorname{let}(v,l)\in K\and C(l)\operatorname{in}v
  102. k = V ( { ( k , { } ) } ) k=V(\{(k,\{\})\})
  103. V ( { ( k , { } ) } ) V(\{(k,\{\})\})
  104. = let ( v , l ) { ( k , { } ) } and C ( l ) in v =\operatorname{let}(v,l)\in\{(k,\{\})\}\and C(l)\operatorname{in}v
  105. = let v = k and C ( { } ) in v =\operatorname{let}v=k\and C(\{\})\operatorname{in}v
  106. = let v = k in v =\operatorname{let}v=k\operatorname{in}v
  107. = k =k
  108. x X ( let x X in x ) = let R = V ( { ( w , { ( R , x , w ) } ) : w X } ) in R \forall x\forall X(\operatorname{let}x\in X\operatorname{in}x)=\operatorname{% let}R=V(\{(w,\{(R,x,w)\}):w\in X\})\operatorname{in}R
  109. x = V ( R ) x=V(R)
  110. x X ( let x X in x ) = V ( { ( w , { ( _ , x , w ) } ) : w X } ) \forall x\forall X(\operatorname{let}x\in X\operatorname{in}x)=V(\{(w,\{(\_,x,% w)\}):w\in X\})
  111. V ( { ( w , { ( r , x , w ) : w X } ) V(\{(w,\{(r,x,w):w\in X\})
  112. = let ( v , l ) { ( w , { ( r , x , w ) } ) : w X } and C ( l ) in v =\operatorname{let}(v,l)\in\{(w,\{(r,x,w)\}):w\in X\}\and C(l)\operatorname{in}v
  113. = let v X and ( ( r , z , u ) { ( _ , x , v ) } z = u ) in v =\operatorname{let}v\in X\and({\bigwedge_{(r,z,u)\in\{(\_,x,v)\}}z=u})% \operatorname{in}v
  114. = let v X and ( r z u ( r , z , u ) { ( _ , x , v ) } z = u ) in v =\operatorname{let}v\in X\and(\forall r\forall z\forall u(r,z,u)\in\{(\_,x,v)% \}\to z=u)\operatorname{in}v
  115. = let v X and x = v in v =\operatorname{let}v\in X\and x=v\operatorname{in}v
  116. = let x X in x =\operatorname{let}x\in X\operatorname{in}x
  117. V ( M ) V ( N ) = V ( { ( m v n v , m l n l ) : ( m v , m l ) M and ( n v , n l ) N } ) V(M)\ V(N)=V(\{(m_{v}\ n_{v},m_{l}\cup n_{l}):(m_{v},m_{l})\in M\and(n_{v},n_{% l})\in N\})
  118. V ( M ) V ( N ) V(M)\ V(N)
  119. = let ( m v , m l ) M and C ( m l ) in m v ) ( let ( n v , n l ) N and C ( n l ) in n v ) =\operatorname{let}(m_{v},m_{l})\in M\and C(m_{l})\operatorname{in}m_{v})\ (% \operatorname{let}(n_{v},n_{l})\in N\and C(n_{l})\operatorname{in}n_{v})
  120. = let ( m v , m l ) M and ( n v , n l ) N and C ( n l ) and C ( n l ) in m v n v ) =\operatorname{let}(m_{v},m_{l})\in M\and(n_{v},n_{l})\in N\and C(n_{l})\and C% (n_{l})\operatorname{in}m_{v}\ n_{v})
  121. C ( m l ) and C ( n l ) C(m_{l})\and C(n_{l})
  122. = ( ( z , u ) m l z = u ) and ( ( z , u ) n l z = u ) =({\bigwedge_{(z,u)\in m_{l}}z=u})\and({\bigwedge_{(z,u)\in n_{l}}z=u})
  123. = ( ( z , u ) m l n l z = u ) =({\bigwedge_{(z,u)\in m_{l}\cup n_{l}}z=u})
  124. = C ( m l n l ) =C(m_{l}\cup n_{l})
  125. = let ( m v , m l ) M and ( n v , n l ) N and C ( n l n l ) in m v n v ) =\operatorname{let}(m_{v},m_{l})\in M\and(n_{v},n_{l})\in N\and C(n_{l}\cup n_% {l})\operatorname{in}m_{v}\ n_{v})
  126. = let ( v , l ) { ( m v n v , m l n l ) : ( m v , m l ) M and ( n v , n l ) N } and C ( l ) in v =\operatorname{let}(v,l)\in\{(m_{v}\ n_{v},m_{l}\cup n_{l}):(m_{v},m_{l})\in M% \and(n_{v},n_{l})\in N\}\and C(l)\operatorname{in}v
  127. = V ( { ( m v n v , m l n l ) : ( m v , m l ) M and ( n v , n l ) N } ) =V(\{(m_{v}\ n_{v},m_{l}\cup n_{l}):(m_{v},m_{l})\in M\and(n_{v},n_{l})\in N\})
  128. V ( M ) s ( v l ( ( v , l ) M and v s ) ¬ C ( l ) ) V(M)\in s\iff(\forall v\forall l((v,l)\in M\and v\not\in s)\to\neg C(l))
  129. V ( M ) s V(M)\in s
  130. v l ( ( v , l ) M and C ( l ) ) ( v = V [ M ] and V ( M ) s ) \to\forall v\forall l((v,l)\in M\and C(l))\to(v=V[M]\and V(M)\in s)
  131. v l ( ( v , l ) M and C ( l ) ) v S \to\forall v\forall l((v,l)\in M\and C(l))\to v\in S
  132. v l ( ( v , l ) M and v S ) ¬ C ( l ) \to\forall v\forall l((v,l)\in M\and v\not\in S)\to\neg C(l)
  133. V ( K ) = V ( { ( v , l ) : ( v , l ) K and C ( l ) } V(K)=V(\{(v,l):(v,l)\in K\and C(l)\}
  134. V ( { ( v , l ) : ( v , l ) K and C ( l ) } ) V(\{(v,l):(v,l)\in K\and C(l)\})
  135. = let ( v , l ) { ( v , l ) : ( v , l ) K and C ( l ) } and C ( l ) in v =\operatorname{let}(v,l)\in\{(v,l):(v,l)\in K\and C(l)\}\and C(l)\operatorname% {in}v
  136. = let ( v , l ) K and C ( l ) and C ( l ) in v =\operatorname{let}(v,l)\in K\and C(l)\and C(l)\operatorname{in}v
  137. = let ( v , l ) K and C ( l ) in v =\operatorname{let}(v,l)\in K\and C(l)\operatorname{in}v
  138. = V ( K ) =V(K)
  139. V ( K ) = let ( v , l ) K and C ( l ) in v V(K)=\operatorname{let}(v,l)\in K\and C(l)\operatorname{in}v
  140. k = V ( { ( k , { } ) } ) k=V(\{(k,\{\})\})
  141. x X ( let x X in x ) = let R = V ( { ( w , { ( R , x , w ) } ) : w X } ) in R \forall x\forall X(\operatorname{let}x\in X\operatorname{in}x)=\operatorname{% let}R=V(\{(w,\{(R,x,w)\}):w\in X\})\operatorname{in}R
  142. V ( M ) V ( N ) = V ( { ( m v n v , m l n l ) : ( m v , m l ) M and ( n v , n l ) N } ) V(M)\ V(N)=V(\{(m_{v}\ n_{v},m_{l}\cup n_{l}):(m_{v},m_{l})\in M\and(n_{v},n_{% l})\in N\})
  143. V ( M ) s ( v l ( ( v , l ) M and v s ) ¬ C ( l ) ) V(M)\in s\iff(\forall v\forall l((v,l)\in M\and v\not\in s)\to\neg C(l))
  144. V ( K ) = V ( { ( v , l ) : ( v , l ) K and C ( l ) } ) V(K)=V(\{(v,l):(v,l)\in K\and C(l)\})
  145. V ( M ) = V ( N ) ( v l ( ( v , l ) N and v S ( M ) ) ¬ C ( l ) ) V(M)=V(N)\to(\forall v\forall l((v,l)\in N\and v\not\in S(M))\to\neg C(l))
  146. x # u \forall x\#u
  147. x # u X ( let x X in x ) = let R = V ( ( u , { ( w , { ( R , x , w ) } ) : w X } ) ) in R \forall x\#u\forall X(\operatorname{let}x\in X\operatorname{in}x)=% \operatorname{let}R=V((u,\{(w,\{(R,x,w)\}):w\in X\}))\operatorname{in}R
  148. m v m l n v n l ( ( m v , m l ) M and ( n v , n l ) N ) ( m v n v ¬ ( C ( m l ) and C ( n l ) ) ) \forall m_{v}\forall m_{l}\forall n_{v}\forall n_{l}((m_{v},m_{l})\in M\and(n_% {v},n_{l})\in N)\to(m_{v}\neq n_{v}\to\neg(C(m_{l})\and C(n_{l})))
  149. V ( M ) = V ( N ) V(M)=V(N)
  150. ( m v m l ( ( m v , m l ) M and C ( m l ) v = V ( M ) ) and ( n v n l ( ( n v , k ) N and C ( n l ) ) n v = V ( N ) ) and V ( M ) = V ( N ) (\forall m_{v}\forall m_{l}((m_{v},m_{l})\in M\and C(m_{l})\to v=V(M))\and(% \forall n_{v}\forall n_{l}((n_{v},k)\in N\and C(n_{l}))\to n_{v}=V(N))\and V(M% )=V(N)
  151. m v m l n v n l ( ( ( m v , m l ) M and C ( m l ) ) m v = V ( M ) ) and ( ( ( n v , n l ) N and C ( n l ) ) n v = V ( N ) ) and V ( M ) = V ( N ) \forall m_{v}\forall m_{l}\forall n_{v}\forall n_{l}(((m_{v},m_{l})\in M\and C% (m_{l}))\to m_{v}=V(M))\and(((n_{v},n_{l})\in N\and C(n_{l}))\to n_{v}=V(N))% \and V(M)=V(N)
  152. m v m l n v n l ( ( m v , m l ) M and ( n v , n l ) N ) ( C ( j ) and C ( k ) v = u ) \forall m_{v}\forall m_{l}\forall n_{v}\forall n_{l}((m_{v},m_{l})\in M\and(n_% {v},n_{l})\in N)\to(C(j)\and C(k)\to v=u)
  153. m v m l n v n l ( ( m v , m l ) M and ( n v , n l ) N ) ( m v n v ¬ ( C ( m l ) and C ( n l ) ) ) \forall m_{v}\forall m_{l}\forall n_{v}\forall n_{l}((m_{v},m_{l})\in M\and(n_% {v},n_{l})\in N)\to(m_{v}\neq n_{v}\to\neg(C(m_{l})\and C(n_{l})))
  154. C ( l ) = ( ( r , z , u ) l z = u ) ) C(l)=({\bigwedge_{(r,z,u)\in l}z=u}))
  155. C ( { ( r , z , u ) } = ( z = u ) C(\{(r,z,u)\}=(z=u)
  156. C ( l ) C(l)
  157. v 1 l 1 v 2 l 2 ( ( v 1 , l 1 ) K and ( v 2 , l 2 ) K and ( v 1 , l 1 ) ( v 2 , l 2 ) ) ¬ ( C ( l 1 ) and C ( l 2 ) ) \forall v_{1}\forall l_{1}\forall v_{2}\forall l_{2}((v_{1},l_{1})\in K\and(v_% {2},l_{2})\in K\and(v_{1},l_{1})\neq(v_{2},l_{2}))\implies\neg(C(l_{1})\and C(% l_{2}))
  158. ( ( K , x , v 1 ) l and ( K , x , v 2 ) l and v 1 v 2 ((K,x,v_{1})\in l\ \and(K,x,v_{2})\in l\and v_{1}\neq v_{2}
  159. ( v 1 , { ( K , x , v 1 ) } ) K and ( v 2 , { ( K , x , v 2 ) } ) K and ( v 1 , l 1 ) ( v 2 , l 2 ) ) \implies(v_{1},\{(K,x,v_{1})\})\in K\and(v_{2},\{(K,x,v_{2})\})\in K\and(v_{1}% ,l_{1})\neq(v_{2},l_{2}))
  160. ¬ ( C ( { ( K , x , v 1 ) } ) and C ( { ( K , x , v 2 ) } ) ) \implies\neg(C(\{(K,x,v_{1})\})\and C(\{(K,x,v_{2})\}))
  161. ¬ C ( l ) \implies\neg C(l)
  162. v l ( v , l ) K C ( l ) \exists v\exists\ l(v,l)\in K\to C(l)
  163. C ( l ) = ( ( r , z , u ) l z = u ) ) C(l)=({\bigwedge_{(r,z,u)\in l}z=u}))
  164. L C ( l ) ( L , z , u ) l z = u \forall L\ C(l)\to(L,z,u)\in l\to z=u
  165. L v l ( v , l ) K C ( l ) ( L , z , u ) l z = u \forall L\exists v\exists l\ (v,l)\in K\to C(l)\to(L,z,u)\in l\to z=u
  166. z = V ( L ) z=V(L)
  167. V ( L ) V(L)
  168. E ( K , L ) = { w : ( v , l ) K and C ( l ) and ( L , z , w ) l } E(K,L)=\{w:(v,l)\in K\and C(l)\and(L,z,w)\in l\}
  169. V ( L ) E ( K , L ) V(L)\in E(K,L)
  170. V ( M ) s ( v l ( ( v , l ) M and v s ) ¬ C ( l ) ) V(M)\in s\iff(\forall v\forall l((v,l)\in M\and v\not\in s)\to\neg C(l))
  171. ( K L v l ( ( v , l ) L and v E ( K , L ) ) ¬ C ( l ) ) (\forall K\forall L\forall v\forall l((v,l)\in L\and v\not\in E(K,L))\to\neg C% (l))
  172. E ( K , L ) E(K,L)
  173. C ( l ) = ( ( r , z , u ) l z = u ) ) C(l)=({\bigwedge_{(r,z,u)\in l}z=u}))
  174. P ( l ) = ( ( r , z , u ) l P ( z = u ) ) ) P(l)=({\prod_{(r,z,u)\in l}P(z=u)}))
  175. ( v , l ) (v,l)
  176. V ( K ) V(K)
  177. P ( l ) P(l)
  178. P ( l ) = ( ( r , z , u ) l P ( z = u ) ) ) P(l)=({\prod_{(r,z,u)\in l}P(z=u)}))
  179. v P ( v = V ( K ) ) = ( v , l ) K P ( l ) \forall vP(v=V(K))=\sum_{(v,l)\in K}P(l)
  180. ( v , l ) K P ( l ) = 1 \sum_{(v,l)\in K}P(l)=1
  181. ( v , l ) K and v gset ( V ( K ) ) P ( l ) = 1 \sum_{(v,l)\in K\and v\in\operatorname{gset}(V(K))}P(l)=1
  182. v 1 l 1 v 2 l 2 ( ( v 1 , l 1 ) K and ( v 2 , l 2 ) K and ( v 1 , l 1 ) ( v 2 , l 2 ) ) P ( C ( l 1 ) and C ( l 2 ) ) = 0 \forall v_{1}\forall l_{1}\forall v_{2}\forall l_{2}((v_{1},l_{1})\in K\and(v_% {2},l_{2})\in K\and(v_{1},l_{1})\neq(v_{2},l_{2}))\to P(C(l_{1})\and C(l_{2}))=0
  183. k = V ( { ( k , { } ) } ) k=V(\{(k,\{\})\})
  184. ( v , l ) K P ( l ) \sum_{(v,l)\in K}P(l)
  185. = ( v , l ) { ( k , { } ) } P ( l ) =\sum_{(v,l)\in\{(k,\{\})\}}P(l)
  186. = P ( { } ) =P(\{\})
  187. = 1 =1
  188. x X ( let x X in x ) = V ( { ( w , { ( _ , x , w ) } ) : w X } ) \forall x\forall X(\operatorname{let}x\in X\operatorname{in}x)=V(\{(w,\{(\_,x,% w)\}):w\in X\})
  189. ( v , l ) K and v gset ( V ( K ) ) P ( l ) = 1 \sum_{(v,l)\in K\and v\in\operatorname{gset}(V(K))}P(l)=1
  190. v gset ( V ( K ) ) P ( x = v ) = 1 \sum_{v\in\operatorname{gset}(V(K))}P(x=v)=1
  191. v gset ( V ( K ) ) , P ( x = v ) = w * P i ( x = v ) \forall v\in\operatorname{gset}(V(K)),P(x=v)=w*P_{i}(x=v)
  192. v gset ( V ( K ) ) , P ( x = v ) = 0 \forall v\not\in\operatorname{gset}(V(K)),P(x=v)=0
  193. v gset ( V ( K ) ) w * P i ( x = v ) = 1 \sum_{v\in\operatorname{gset}(V(K))}w*P_{i}(x=v)=1
  194. w = 1 v gset ( V ( K ) ) P i ( x = v ) w=\frac{1}{\sum_{v\in\operatorname{gset}(V(K))}P_{i}(x=v)}
  195. v gset ( V ( K ) ) , P ( x = v ) = P i ( x = v ) v gset ( V ( K ) ) P i ( x = v ) \forall v\in\operatorname{gset}(V(K)),P(x=v)=\frac{P_{i}(x=v)}{\sum_{v\in% \operatorname{gset}(V(K))}P_{i}(x=v)}
  196. v gset ( V ( K ) ) , P ( x = v ) = 0 \forall v\not\in\operatorname{gset}(V(K)),P(x=v)=0
  197. v gset ( V ( K ) ) , P ( x = v ) = 1 | gset ( V ( K ) ) | \forall v\in\operatorname{gset}(V(K)),P(x=v)=\frac{1}{|\operatorname{gset}(V(K% ))|}
  198. v P ( v = V ( K ) ) = ( v , l ) K and v gset ( V ( K ) ) ( ( r , z , u ) l P ( z = u ) ) ) \forall vP(v=V(K))=\sum_{(v,l)\in K\and v\in\operatorname{gset}(V(K))}({\prod_% {(r,z,u)\in l}P(z=u)}))
  199. E ( x ) E(x)
  200. { z : E ( z ) } \{z:E(z)\}
  201. gset ( x ) = { z : E ( z ) } \operatorname{gset}(x)=\{z:E(z)\}
  202. xmath [ E ] = M , let T [ M ] = math [ E ] in T [ _ ] \operatorname{xmath}[E]=\forall M,\operatorname{let}T[M]=\operatorname{math}[E% ]\operatorname{in}T[\_]
  203. x # u , N , gset ( x ) = let M [ u ] = x in { z : z = N [ u ] and T [ N ] } \forall x\#u,\exists N,\operatorname{gset}(x)=\operatorname{let}M[u]=x% \operatorname{in}\{z:z=N[u]\and T[N]\}
  204. x # u \forall x\#u
  205. x 2 = 4 x^{2}=4
  206. x 2 = 4 x^{2}=4
  207. and s = gset ( x ) \and s=\operatorname{gset}(x)
  208. and ( x # u , N gset ( x ) = let M [ u ] = x in { z : z = N [ u ] and T [ N ] } ) \and(\forall x\#u,\exists N\operatorname{gset}(x)=\operatorname{let}M[u]=x% \operatorname{in}\{z:z=N[u]\and T[N]\})
  209. M , let T [ M ] = \forall M,\operatorname{let}T[M]=
  210. x , s , ( x 2 = 4 \exists x,\exists s,(x^{2}=4
  211. and s = gset ( x ) \and s=\operatorname{gset}(x)
  212. and ( x # u , N , gset ( x ) = let M [ u ] = x in { z : z = N [ u ] and T [ N ] } ) \and(\forall x\#u,\exists N,\operatorname{gset}(x)=\operatorname{let}M[u]=x% \operatorname{in}\{z:z=N[u]\and T[N]\})
  213. in T [ _ ] \operatorname{in}T[\_]
  214. M [ 1 ] = x and s = { z : z = N [ 1 ] and T [ N ] } M[1]=x\and s=\{z:z=N[1]\and T[N]\}
  215. T [ N ] T[N]
  216. s = { z : z = N [ 1 ] and x 2 = 4 and x = N [ 1 ] } s=\{z:z=N[1]\and x^{2}=4\and x=N[1]\}
  217. s = { z : z 2 = 4 } s=\{z:z^{2}=4\}

Natural_Earth_projection.html

  1. x = 0.8707 × l ( φ ) × λ x=0.8707\times l(\varphi)\times\lambda
  2. y = 0.8707 × 0.52 × d ( φ ) × π y=0.8707\times 0.52\times d(\varphi)\times\pi
  3. x x
  4. y y
  5. λ \lambda
  6. φ \varphi
  7. l ( φ ) l(\varphi)
  8. φ \varphi
  9. d ( φ ) d(\varphi)
  10. φ \varphi
  11. l ( φ ) l(\varphi)
  12. d ( φ ) d(\varphi)
  13. φ \varphi
  14. l ( φ ) l(\varphi)
  15. d ( φ ) d(\varphi)

Near_polygon.html

  1. P , L , I P,L,I
  2. P P
  3. L L
  4. I P × L I\subseteq P\times L
  5. x x
  6. L L
  7. L L
  8. x x
  9. 2 d 2d
  10. ( s , t ) (s,t)
  11. t i , i { 0 , 1 , , d } t_{i},i\in\{0,1,\ldots,d\}
  12. x x
  13. y y
  14. i i
  15. t i + 1 t_{i}+1
  16. y y
  17. i - 1 i-1
  18. x x
  19. 2 d 2d
  20. 2 d 2d
  21. 2 d 2d
  22. 2 d 2d
  23. ( s , t ) (s,t)
  24. t 0 = - 1 , t 1 = 0 , t 2 = 0 , , t d = t t_{0}=-1,t_{1}=0,t_{2}=0,\ldots,t_{d}=t

Nearest_neighbour_distribution.html

  1. 𝐑 d \textstyle\,\textbf{R}^{d}
  2. x \textstyle x
  3. N \textstyle{N}
  4. x N , \textstyle x\in{N},
  5. N \textstyle{N}
  6. B \textstyle B
  7. N ( B ) , \textstyle{N}(B),
  8. N \textstyle{N}
  9. 𝐑 d \textstyle\,\textbf{R}^{d}
  10. o \textstyle o
  11. d \textstyle d
  12. b ( o , r ) \textstyle b(o,r)
  13. r \textstyle r
  14. N \textstyle{N}
  15. o \textstyle o
  16. D o ( r ) = 1 - P ( N ( b ( o , r ) ) = 1 o ) . D_{o}(r)=1-P({N}(b(o,r))=1\mid o).
  17. P ( N ( b ( o , r ) ) = 1 o ) \textstyle P({N}(b(o,r))=1\mid o)
  18. N \textstyle{N}
  19. b ( o , r ) \textstyle b(o,r)
  20. N \textstyle{N}
  21. o \textstyle o
  22. x 𝐑 d \textstyle x\in\,\textbf{R}^{d}
  23. N \textstyle{N}
  24. o \textstyle o
  25. D x ( r ) = 1 - P ( N ( b ( x , r ) ) = 1 x ) . D_{x}(r)=1-P({N}(b(x,r))=1\mid x).
  26. N \textstyle{N}
  27. 𝐑 d \textstyle\,\textbf{R}^{d}
  28. Λ \textstyle\Lambda
  29. D x ( r ) = 1 - e - Λ ( b ( x , r ) ) , D_{x}(r)=1-e^{-\Lambda(b(x,r))},
  30. D x ( r ) = 1 - e - λ | b ( x , r ) | , D_{x}(r)=1-e^{-\lambda|b(x,r)|},
  31. | b ( x , r ) | \textstyle|b(x,r)|
  32. r \textstyle r
  33. 𝐑 2 \textstyle\,\textbf{R}^{2}
  34. D x ( r ) = 1 - e - λ π r 2 . D_{x}(r)=1-e^{-\lambda\pi r^{2}}.
  35. J J
  36. J J
  37. r r
  38. J ( r ) = 1 - D o ( r ) 1 - H s ( r ) J(r)=\frac{1-D_{o}(r)}{1-H_{s}(r)}
  39. J J
  40. J ( r ) J(r)
  41. J ( r ) J(r)
  42. J J

Negation_introduction.html

  1. ( P Q ) and ( P ¬ Q ) ¬ P (P\rightarrow Q)\and(P\rightarrow\neg Q)\leftrightarrow\neg P

Negative_hypergeometric_distribution.html

  1. N N
  2. K K
  3. R R
  4. k k
  5. N H G N , K , R ( k ) NHG_{N,K,R}(k)
  6. k k
  7. n n
  8. H G N , K , n ( k ) HG_{N,K,n}(k)
  9. N H G N , K , R ( k ) = 1 - H G N , N - K , k ( R - 1 ) NHG_{N,K,R}(k)=1-HG_{N,N-K,k}(R-1)

Neoabietadiene_synthase.html

  1. \rightleftharpoons

NER_model.html

  1. N E R v a l u e = N - E - R N * 100 NERvalue=\frac{N-E-R}{N}*100

Network_robustness.html

  1. κ = k 2 k > 2 \kappa={{{\left\langle k^{2}\right\rangle}\over{\left\langle k\right\rangle}}>2}
  2. k 2 {\left\langle k^{2}\right\rangle}
  3. k {\left\langle k\right\rangle}
  4. f c = 1 - 1 k 2 k - 1 f_{c}=1-{1\over{{{\left\langle k^{2}\right\rangle}\over{\left\langle k\right% \rangle}}}-1}
  5. κ E R = k \kappa^{ER}={\left\langle k\right\rangle}
  6. f c = 1 - 1 k - 1 f_{c}=1-{1\over{{\left\langle k\right\rangle}-1}}
  7. f ( c ) = { 1 - 1 γ - 2 3 - γ k m i n γ - 2 k m a x 3 - γ - 1 2 < γ < 3 1 - 1 γ - 2 3 - γ k m i n - 1 γ > 3 f(c)=\begin{cases}1-{1\over{{{\gamma-2}\over{3-\gamma}}k_{min}^{\gamma-2}k_{% max}^{3-\gamma}-1}}&2<\gamma<3\\ 1-{1\over{{{\gamma-2}\over{3-\gamma}}k_{min}-1}}&\gamma>3\end{cases}

Network_Science_Based_Basketball_Analytics.html

  1. F = i j p i j ( x i - x j ) F=\sum_{i\neq j}p_{ij}\ (x_{i}-x_{j})
  2. C D = v V d e g ( v * ) - d e g ( v ) V - 1 C_{D}=\sum_{v\in V}\frac{deg(v^{*})-deg(v)}{\mid V\mid-1}
  3. f ( x 1 ) = f ( x 2 ) = f ( x i ) f(x_{1})=f(x_{2})...=f(x_{i})
  4. d F d x 1 = d F d x 2 = d F d x i \frac{dF}{dx_{1}}=\frac{dF}{dx_{2}}...=\frac{dF}{dx_{i}}
  5. F = x 1 f ( x 1 ) + x 2 f ( x 2 ) + x 3 f ( x 3 ) F=x_{1}f(x_{1})+x_{2}f(x_{2})...+x_{3}f(x_{3})
  6. p i = k = 1 J ( I | π i * k π i 0 ) J p_{i}=\frac{\sum_{k=1}^{J}(I|\pi_{i^{*}k}\geq\pi_{i}^{0})}{J}

Network_simplex_algorithm.html

  1. O ( V 2 E log ( V C ) ) O(V^{2}E\log(VC))
  2. C C
  3. O ( V E log V log ( V C ) ) O(VE\log V\log(VC))

Neural_control_of_limb_stiffness.html

  1. K vert = F max Δ y K\text{vert}=\frac{F\text{max}}{\Delta y}
  2. K vert = m ( 2 π P ) 2 K\text{vert}=m(\frac{2\pi}{P})^{2}
  3. K vert = m ω 0 2 K\text{vert}=m\omega\text{0}^{2}
  4. K limb = F max Δ L K\text{limb}=\frac{F\text{max}}{\Delta L}
  5. K joint = Δ M Δ θ K\text{joint}=\frac{\Delta M}{\Delta\theta}
  6. K joint = 2 W Δ θ K\text{joint}=\frac{2W}{\Delta\theta}

Neville_theta_functions.html

  1. θ c ( z , m ) = 2 π q ( m ) 4 k = 0 ( q ( m ) ) k ( k + 1 ) cos ( 1 2 ( 2 k + 1 ) π z K ( m ) ) 1 K ( m ) 1 m 4 \theta_{c}(z,m)=\sqrt{2}\sqrt{\pi}\sqrt[4]{q(m)}\sum_{k=0}^{\infty}(q(m))^{k(k% +1)}\cos\left(\frac{1}{2}\cdot\frac{(2k+1)\pi z}{K(m)}\right)\frac{1}{\sqrt{K(% m)}}\frac{1}{\sqrt[4]{m}}
  2. θ d ( z , m ) = 1 / 2 2 π ( 1 + 2 k = 1 ( q ( m ) ) k 2 cos ( π z k K ( m ) ) ) 1 K ( m ) \theta_{d}(z,m)=1/2\,\sqrt{2}\sqrt{\pi}\left(1+2\,\sum_{k=1}^{\infty}(q(m))^{k% ^{2}}\cos\left(\frac{\pi zk}{K(m)}\right)\right)\frac{1}{\sqrt{K(m)}}
  3. θ n ( z , m ) = 1 / 2 π 2 ( 1 + 2 k = 1 ( - 1 ) k ( q ( m ) ) k 2 cos ( π z k K ( m ) ) ) 1 1 - m 4 1 K ( m ) \theta_{n}(z,m)=1/2\,\sqrt{\pi}\sqrt{2}\left(1+2\sum_{k=1}^{\infty}(-1)^{k}(q(% m))^{k^{2}}\cos\left(\frac{\pi zk}{K(m)}\right)\right)\frac{1}{\sqrt[4]{1-m}}% \frac{1}{\sqrt{K(m)}}
  4. θ s ( z , m ) = π 2 q ( m ) 4 k = 0 ( - 1 ) k ( q ( m ) ) k ( k + 1 ) sin ( 1 / 2 ( 2 k + 1 ) π z K ( m ) ) 1 1 - m 4 1 m 4 1 K ( m ) \theta_{s}(z,m)=\sqrt{\pi}\sqrt{2}\sqrt[4]{q(m)}\sum_{k=0}^{\infty}(-1)^{k}(q(% m))^{k(k+1)}\sin\left(1/2\,\frac{(2k+1)\pi z}{K(m)}\right)\frac{1}{\sqrt[4]{1-% m}}\frac{1}{\sqrt[4]{m}}\frac{1}{\sqrt{K(m)}}
  5. K ( m ) = EllipticK ( m ) K(m)=\operatorname{EllipticK}(\sqrt{m})
  6. K ( m ) = EllipticK ( 1 - m ) K^{\prime}(m)=\operatorname{EllipticK}(\sqrt{1-m})
  7. q ( m ) = e - π K ( m ) / K ( m ) q(m)=e^{-\pi K(m)/K^{\prime}(m)}
  8. θ c ( 2.5 , 0.3 ) = - 0.65900466676738154967 \theta_{c}(2.5,0.3)=-0.65900466676738154967
  9. θ d ( 2.5 , 0.3 ) = 0.95182196661267561994 \theta_{d}(2.5,0.3)=0.95182196661267561994
  10. θ n ( 2.5 , 0.3 ) = 1.0526693354651613637 \theta_{n}(2.5,0.3)=1.0526693354651613637
  11. θ s ( 2.5 , 0.3 ) = 0.82086879524530400536 \theta_{s}(2.5,0.3)=0.82086879524530400536
  12. θ c ( z , m ) = θ c ( - z , m ) \theta_{c}(z,m)=\theta_{c}(-z,m)
  13. θ d ( z , m ) = θ d ( - z , m ) \theta_{d}(z,m)=\theta_{d}(-z,m)
  14. θ n ( z , m ) = θ n ( - z , m ) \theta_{n}(z,m)=\theta_{n}(-z,m)
  15. θ s ( z , m ) = - θ s ( - z , m ) \theta_{s}(z,m)=-\theta_{s}(-z,m)

Newton's_theorem_(quadrilateral).html

  1. A ( P A B ) + A ( P C D ) = 1 2 r a + 1 2 r c = 1 2 r ( a + c ) = 1 2 r ( b + d ) = 1 2 r c + 1 2 r d = A ( P B C ) + A ( P A D ) \begin{aligned}&\displaystyle A(\triangle PAB)+A(\triangle PCD)\\ \displaystyle=&\displaystyle\tfrac{1}{2}ra+\tfrac{1}{2}rc=\tfrac{1}{2}r(a+c)\\ \displaystyle=&\displaystyle\tfrac{1}{2}r(b+d)=\tfrac{1}{2}rc+\tfrac{1}{2}rd\\ \displaystyle=&\displaystyle A(\triangle PBC)+A(\triangle PAD)\end{aligned}

Newton-X.html

  1. s y m b o l τ L M 𝐯 1 4 Δ t ( 3 S L M ( t ) - 3 S M L ( t ) - S L M ( t - Δ t ) + S M L ( t - Δ t ) ) symbol{\tau}_{LM}\cdot\mathbf{v}\approx\frac{1}{4\Delta t}\left(3S_{LM}(t)-3S_% {ML}(t)-S_{LM}(t-\Delta t)+S_{ML}(t-\Delta t)\right)
  2. S L M ( t ) Ψ L ( t - Δ t ) Ψ M ( t ) S_{LM}(t)\equiv\left\langle\Psi_{L}(t-\Delta t)\mid\Psi_{M}(t)\right\rangle
  3. σ ( E ) = π e 2 2 m c ϵ 0 n r E n N f s 1 N p l N p Δ E 0 , n ( 𝐑 l ) f 0 , n ( 𝐑 l ) g ( E - Δ E 0 , n ( 𝐑 l ) , δ ) \sigma(E)=\frac{\pi e^{2}\hbar}{2mc\epsilon_{0}n_{r}E}\sum_{n}^{N_{fs}}\frac{1% }{N_{p}}\sum_{l}^{N_{p}}\Delta E_{0,n}(\mathbf{R}_{l})f_{0,n}(\mathbf{R}_{l})g% \left(E-\Delta E_{0,n}(\mathbf{R}_{l}),\delta\right)
  4. g ( E - Δ E 0 , n , δ ) = 1 ( 2 π ( δ / 2 ) 2 ) 1 / 2 e x p ( - ( E - Δ E 0 , n ) 2 2 ( δ / 2 ) 2 ) g\left(E-\Delta E_{0,n},\delta\right)=\frac{1}{\left(2\pi(\delta/2)^{2}\right)% ^{1/2}}exp\left(\frac{-(E-\Delta E_{0,n})^{2}}{2(\delta/2)^{2}}\right)
  5. Γ ( E ) = e 2 n r 3 2 π m c 3 ϵ 0 1 N p l N p Δ E 1 , 0 ( 𝐑 l ) 2 | f 1 , 0 ( 𝐑 l ) | g ( E - Δ E 1 , 0 ( 𝐑 l ) , δ ) \Gamma(E)=\frac{e^{2}n_{r}^{3}}{2\pi\hbar mc^{3}\epsilon_{0}}\frac{1}{N_{p}}% \sum_{l}^{N_{p}}\Delta E_{1,0}(\mathbf{R}_{l})^{2}\left|f_{1,0}(\mathbf{R}_{l}% )\right|g\left(E-\Delta E_{1,0}(\mathbf{R}_{l}),\delta\right)

NEWUOA.html

  1. Q k Q_{k}
  2. Q k Q_{k}
  3. f f
  4. n n
  5. Q k ( x ) = f ( x 0 ) + f ( x 0 ) T ( x - x 0 ) + 1 2 ( x - x 0 ) T H ( x - x 0 ) Q_{k}(x)=f(x_{0})+\nabla f(x_{0})^{T}(x-x_{0})+\frac{1}{2}(x-x_{0})^{T}H(x-x_{% 0})
  6. f f
  7. f f
  8. ( n + 1 ) ( n + 2 ) / 2 (n+1)(n+2)/2
  9. f ( x 0 ) f(x_{0})
  10. n n
  11. x 0 x_{0}
  12. n ( n + 1 ) / 2 n(n+1)/2
  13. x 0 x_{0}
  14. n n
  15. Q k Q_{k}
  16. m m
  17. n + 2 n+2
  18. ( n + 1 ) ( n + 2 ) / 2 (n+1)(n+2)/2
  19. n n
  20. f f
  21. 2 Q k - 2 Q k - 1 \nabla^{2}Q_{k}-\nabla^{2}Q_{k-1}
  22. ( n + 1 ) ( n + 2 ) / 2 (n+1)(n+2)/2
  23. m m
  24. 2 n + 1 2n+1

Next-generation_matrix.html

  1. n n
  2. m < n m<n
  3. x i , i = 1 , 2 , 3 , , m x_{i},i=1,2,3,\ldots,m
  4. i t h i^{th}
  5. d x i d t = F i ( x ) - V i ( x ) \frac{\mathrm{d}x_{i}}{\mathrm{d}t}=F_{i}(x)-V_{i}(x)
  6. V i ( x ) = [ V i - ( x ) - V i + ( x ) ] V_{i}(x)=[V^{-}_{i}(x)-V^{+}_{i}(x)]
  7. F i ( x ) F_{i}(x)
  8. i i
  9. V i + V^{+}_{i}
  10. i i
  11. V i - ( x ) V^{-}_{i}(x)
  12. i i
  13. d x i d t = F ( x ) - V ( x ) \frac{\mathrm{d}x_{i}}{\mathrm{d}t}=F(x)-V(x)
  14. F ( x ) = ( F 1 ( x ) , F 2 ( x ) , , F n ( x ) ) T F(x)=\begin{pmatrix}F_{1}(x),&F_{2}(x),&\ldots,&F_{n}(x)\end{pmatrix}^{T}
  15. V ( x ) = ( V 1 ( x ) , V 2 ( x ) , , V n ( x ) ) T . V(x)=\begin{pmatrix}V_{1}(x),&V_{2}(x),&\ldots,&V_{n}(x)\end{pmatrix}^{T}.
  16. x 0 x_{0}
  17. F ( x ) F(x)
  18. V ( x ) V(x)
  19. D F ( x 0 ) = ( F 0 0 0 ) DF(x_{0})=\begin{pmatrix}F&0\\ 0&0\end{pmatrix}
  20. D V ( x 0 ) = ( V 0 J 3 J 4 ) DV(x_{0})=\begin{pmatrix}V&0\\ J_{3}&J_{4}\end{pmatrix}
  21. F F
  22. V V
  23. F = F i x j ( x 0 ) F=\frac{\partial F_{i}}{\partial x_{j}}(x_{0})
  24. V = V i x j ( x 0 ) V=\frac{\partial V_{i}}{\partial x_{j}}(x_{0})
  25. F V - 1 FV^{-1}
  26. F V - 1 FV^{-1}

Nickel_superoxide_dismutase.html

  1. 2 O 2 - + 2 H + Nickel Superoxide Dismutase H 2 O 2 + O 2 \rm 2O_{2}^{-}+2H^{+}\xrightarrow{Nickel\ Superoxide\ Dismutase}H_{2}O_{2}+O_{2}
  2. 2 O 2 - + Ni ( III ) SOD Ni ( II ) SOD + O 2 \rm 2O_{2}^{-}+Ni(III)SOD\xrightarrow{}Ni(II)SOD+O_{2}
  3. 2 O 2 - + 2 H + + Ni ( II ) SOD H 2 O 2 + Ni ( III ) SOD \rm 2O_{2}^{-}+2H^{+}+Ni(II)SOD\xrightarrow{}H_{2}O_{2}+Ni(III)SOD

Nilpotent_algebra_(ring_theory).html

  1. A A
  2. R R
  3. n n
  4. 0 = y 1 y 2 y n 0=y_{1}\ y_{2}\ \cdots\ y_{n}
  5. y 1 , y 2 , , y n y_{1},\ y_{2},\ \ldots,\ y_{n}
  6. A A
  7. n n
  8. A A
  9. n n

Nilpotent_space.html

  1. π = π 1 X \pi=\pi_{1}X
  2. π \pi
  3. π i X , i 2 \pi_{i}X,i\geq 2

Niobium_capacitor.html

  1. C = ε A d C=\varepsilon\cdot\frac{A}{d}

Node_deletion.html

  1. r < 1 r<1
  2. γ = 3 + 2 r 1 - r \gamma=3+\frac{2r}{1-r}
  3. r = 1 r=1
  4. r > 1 r>1

Node_influence_metric.html

  1. h h
  2. h h
  3. p j ( h ) p_{j}^{(h)}
  4. i i
  5. h h
  6. κ i ( h ) = exp ( - j p j ( h ) log p j ( h ) ) \kappa_{i}^{(h)}=\exp\left(-\sum_{j}p_{j}^{(h)}\log p_{j}^{(h)}\right)
  7. i i
  8. κ i = - j = 1 J d j log ( d j ) \kappa_{i}=-\sum_{j=1}^{J}d_{j}\log(d_{j})
  9. J J
  10. i i
  11. d j d_{j}
  12. j J j\in J
  13. J J
  14. d j d_{j}
  15. J J

Non-commutative_conditional_expectation.html

  1. σ \sigma
  2. ( X , μ ) (X,\mu)
  3. Φ \Phi
  4. 𝒮 \mathcal{S}
  5. \mathcal{R}
  6. 𝒮 \mathcal{S}
  7. \mathcal{R}
  8. 𝒮 \mathcal{S}
  9. \mathcal{R}
  10. Φ ( I ) = I \Phi(I)=I
  11. Φ ( R 1 S R 2 ) = R 1 Φ ( S ) R 2 \Phi(R_{1}SR_{2})=R_{1}\Phi(S)R_{2}
  12. R 1 , R 2 R_{1},R_{2}\in\mathcal{R}
  13. S 𝒮 S\in\mathcal{S}
  14. \mathcal{B}
  15. 𝔄 , φ 0 \mathfrak{A},\varphi_{0}
  16. 𝔄 \mathfrak{A}
  17. \mathcal{B}
  18. φ 0 = 1 , 𝔄 \|\varphi_{0}\|=1,\mathfrak{A}
  19. \mathcal{H}
  20. 𝔄 \mathfrak{A}
  21. φ 0 \varphi_{0}
  22. φ \varphi
  23. 𝔄 - \mathfrak{A}^{-}
  24. 𝔄 \mathfrak{A}
  25. - \mathcal{B}^{-}
  26. \mathcal{B}
  27. 𝔄 , , φ , φ 0 \mathfrak{A},\mathcal{B},\varphi,\varphi_{0}
  28. φ \varphi
  29. 𝔄 - \mathfrak{A}^{-}
  30. - \mathcal{B}^{-}
  31. φ 0 \varphi_{0}
  32. 𝔄 \mathfrak{A}
  33. \mathcal{B}

Non-commutative_cryptography.html

  1. \oplus
  2. \oplus
  3. \oplus
  4. \oplus
  5. B n = x 1 , x 2 , , x n - 1 | x i x j = x j x i if | i - j | > 1 and x i x j x i = x j x i x j if | i - j | = 1 B_{n}=\left\langle x_{1},x_{2},\ldots,x_{n-1}\big|x_{i}x_{j}=x_{j}x_{i}\,\text% { if }|i-j|>1\,\text{ and }x_{i}x_{j}x_{i}=x_{j}x_{i}x_{j}\,\text{ if }|i-j|=1\right\rangle
  6. F = x 0 , x 1 , x 2 , | x k - 1 x n x k = x n + 1 for k < n F=\left\langle x_{0},x_{1},x_{2},\ldots\big|x_{k}^{-1}x_{n}x_{k}=x_{n+1}\,% \text{ for }k<n\right\rangle
  7. a ( b 1 , b 2 , , b n ) = ( 1 - b 1 , b 2 , , b n ) a(b_{1},b_{2},\ldots,b_{n})=(1-b_{1},b_{2},\ldots,b_{n})
  8. b ( b 1 , b 2 , , b n ) = { ( b 1 , 1 - b 2 , , b n ) if b 1 = 0 ( b 1 , c ( b 2 , , b n ) ) if b 1 = 1 b(b_{1},b_{2},\ldots,b_{n})=\begin{cases}(b_{1},1-b_{2},\ldots,b_{n})&\,\text{% if }b_{1}=0\\ (b_{1},c(b_{2},\ldots,b_{n}))&\,\text{ if }b_{1}=1\end{cases}
  9. c ( b 1 , b 2 , , b n ) = { ( b 1 , 1 - b 2 , , b n ) if b 1 = 0 ( b 1 , d ( b 2 , , b n ) ) if b 1 = 1 c(b_{1},b_{2},\ldots,b_{n})=\begin{cases}(b_{1},1-b_{2},\ldots,b_{n})&\,\text{% if }b_{1}=0\\ (b_{1},d(b_{2},\ldots,b_{n}))&\,\text{ if }b_{1}=1\end{cases}
  10. d ( b 1 , b 2 , , b n ) = { ( b 1 , 1 - b 2 , , b n ) if b 1 = 0 ( b 1 , b ( b 2 , , b n ) ) if b 1 = 1 d(b_{1},b_{2},\ldots,b_{n})=\begin{cases}(b_{1},1-b_{2},\ldots,b_{n})&\,\text{% if }b_{1}=0\\ (b_{1},b(b_{2},\ldots,b_{n}))&\,\text{ if }b_{1}=1\end{cases}
  11. A ( Γ ) = a 1 , a 2 , , a n | μ i j = μ j i for 1 i < j n A(\Gamma)=\left\langle a_{1},a_{2},\ldots,a_{n}|\mu_{ij}=\mu_{ji}\,\text{ for % }1\leq i<j\leq n\right\rangle
  12. μ i j = a i a j a i \mu_{ij}=a_{i}a_{j}a_{i}\ldots
  13. m i j m_{ij}
  14. m i j = m j i m_{ij}=m_{ji}

Non-contact_atomic_force_microscopy.html

  1. Δ F \Delta F
  2. F t s F_{ts}
  3. Δ f = f 0 k A 2 F t s q \Delta f=\frac{f_{0}}{kA^{2}}\langle F_{ts}q^{\prime}\rangle\,
  4. q q^{\prime}
  5. k k
  6. f 0 f_{0}
  7. A A

Non-negative_least_squares.html

  1. 𝐀 \mathbf{A}
  2. 𝐲 \mathbf{y}
  3. arg min 𝐱 𝐀𝐱 - 𝐲 2 \operatorname*{arg\,min}_{\mathbf{x}}\|\mathbf{Ax}-\mathbf{y}\|_{2}
  4. 𝐱 0 \mathbf{x}≥0
  5. 𝐱 0 \mathbf{x}≥0
  6. 𝐱 \mathbf{x}
  7. · ‖·‖₂
  8. α 𝐱 β αᵢ≤\mathbf{x}ᵢ≤βᵢ
  9. arg min 𝐱 𝟎 1 2 𝐱 𝖳 𝐐𝐱 + 𝐜 𝖳 𝐱 \operatorname*{arg\,min}_{\mathbf{x\geq 0}}\frac{1}{2}\mathbf{x}^{\mathsf{T}}% \mathbf{Q}\mathbf{x}+\mathbf{c}^{\mathsf{T}}\mathbf{x}
  10. 𝐐 \mathbf{Q}
  11. 𝐀 𝐀 \mathbf{A}ᵀ\mathbf{A}
  12. 𝐜 \mathbf{c}
  13. 𝐀 𝐲 −\mathbf{A}ᵀ\mathbf{y}
  14. 𝐐 \mathbf{Q}
  15. ( ( 𝐀 ) 𝐀 ) ¹ ((\mathbf{A}ᴾ)ᵀ\mathbf{A}ᴾ)⁻¹

Noncommutative_projective_geometry.html

  1. k x , y / ( y x - q x y ) k\langle x,y\rangle/(yx-qxy)
  2. k x 1 , , x n / ( x i x j - q i j x j x i ) k\langle x_{1},\dots,x_{n}\rangle/(x_{i}x_{j}-q_{ij}x_{j}x_{i})

Nonlinear_system_identification.html

  1. y ( k ) \displaystyle y(k)
  2. h ( m 1 , , m ) h_{\ell}(m_{1},\ldots,m_{\ell})
  3. y ( k ) \displaystyle y(k)
  4. n y n_{y}
  5. n u n_{u}
  6. n e n_{e}

Normal_form_(bifurcation_theory).html

  1. d x d t = μ + x 2 \frac{\mathrm{d}x}{\mathrm{d}t}=\mu+x^{2}
  2. μ \mu

Norman_F._Carnahan.html

  1. P V / N k T = ( 1 + y + y 2 - y 3 ) / ( 1 - y ) 3 PV/NkT=(1+y+y^{2}-y^{3})/(1-y)^{3}

Norton's_dome.html

  1. h = 2 3 g r 3 2 h={\frac{2}{3g}}r^{\frac{3}{2}}

Nowcast_(Air_Quality_Index).html

  1. w * = c m i n c m a x w^{*}=\frac{c_{min}}{c_{max}}
  2. w = { w * if w * > 1 2 , 1 2 if w * 1 2 . w=\begin{cases}w^{*}&\mbox{if }~{}w^{*}>\frac{1}{2},\\ \frac{1}{2}&\mbox{if }~{}w^{*}\leq\frac{1}{2}.\\ \end{cases}
  3. N o w C a s t = i = 1 12 w i - 1 c i i = 1 12 w i - 1 . NowCast=\frac{\sum_{i=1}^{12}w^{i-1}c_{i}}{\sum_{i=1}^{12}w^{i-1}}.
  4. N o w C a s t = i = 1 12 c i 12 . NowCast=\frac{\sum_{i=1}^{12}c_{i}}{12}.
  5. N o w C a s t = c 1 + ( 1 2 ) c 2 + + ( 1 2 ) 11 c 12 1 + 1 2 + ( 1 2 ) 2 + + ( 1 2 ) 11 NowCast=\frac{c_{1}+({\frac{1}{2}})c_{2}+...+{(\frac{1}{2}})^{11}{c_{12}}}{1+% \frac{1}{2}+({\frac{1}{2}})^{2}+...+{(\frac{1}{2}})^{11}}
  6. I = I h i g h - I l o w C h i g h - C l o w ( C - C l o w ) + I l o w I=\frac{I_{high}-I_{low}}{C_{high}-C_{low}}(C-C_{low})+I_{low}
  7. I I
  8. C C
  9. C l o w C_{low}
  10. C C
  11. C h i g h C_{high}
  12. C C
  13. I l o w I_{low}
  14. C l o w C_{low}
  15. I h i g h I_{high}
  16. C h i g h C_{high}
  17. C l o w C_{low}
  18. C h i g h C_{high}
  19. I l o w I_{low}
  20. I h i g h I_{high}

Nucleus_(order_theory).html

  1. F F
  2. 𝔄 \mathfrak{A}
  3. p p
  4. 𝔄 \mathfrak{A}
  5. p F ( p ) p\leq F(p)
  6. F ( F ( p ) ) = F ( p ) F(F(p))=F(p)
  7. F ( p q ) = F ( p ) F ( q ) F(p\wedge q)=F(p)\wedge F(q)
  8. 𝔄 \mathfrak{A}
  9. F F
  10. 𝔄 \mathfrak{A}
  11. Fix ( F ) \operatorname{Fix}(F)
  12. F F
  13. 𝔄 \mathfrak{A}

Number_theoretic_Hilbert_transform.html

  1. p p
  2. m \mathbb{Z}_{m}
  3. m m
  4. n × n n\times n
  5. n = 2 m n=2m
  6. N H T = [ 0 a m 0 a 1 a 1 0 a m 0 a 1 0 0 a m a m 0 a 1 0 ] . NHT=\begin{bmatrix}0&a_{m}&\dots&0&a_{1}\\ a_{1}&0&a_{m}&&0\\ \vdots&a_{1}&0&\ddots&\vdots\\ 0&&\ddots&\ddots&a_{m}\\ a_{m}&0&\dots&a_{1}&0\\ \end{bmatrix}.
  7. N H T T N H T = N H T N H T T = I mod p , NHT^{\mathrm{T}}NHT=NHTNHT^{\mathrm{T}}=I\bmod\ p,\,

Numerical_methods_in_fluid_mechanics.html

  1. lim Δ x 0 f ( n ) = f ( x + Δ x ) - f ( x ) Δ x \lim_{\Delta x\to 0}f^{\prime}(n)=\frac{f(x+\Delta x)-f(x)}{\Delta x}
  2. f ( x ) = f ( x + Δ x ) - f ( x ) Δ x + O ( Δ x ) f^{\prime}(x)=\frac{f(x+\Delta x)-f(x)}{\Delta x}+O(\Delta x)
  3. u n + 1 - u n Δ t κ u n \frac{u^{n+1}-u^{n}}{\Delta t}\approx\kappa u^{n}

Numerical_solution_of_the_convection–diffusion_equation.html

  1. c ρ [ T ( x , t ) t + ϵ u T ( x , t ) x ] = λ 2 T ( x , t ) x 2 + Q ( x , t ) c\rho\left[\frac{\partial T(x,t)}{\partial t}+\epsilon u\frac{\partial T(x,t)}% {\partial x}\right]=\lambda\frac{\partial^{2}T(x,t)}{\partial x^{2}}+Q(x,t)
  2. ϵ \epsilon
  3. ρ \rho
  4. λ \lambda
  5. T t = a 2 T x 2 - ϵ u T x + Q c ρ \frac{\partial T}{\partial t}=a\frac{\partial^{2}T}{\partial x^{2}}-\epsilon u% \frac{\partial T}{\partial x}+\frac{Q}{c\rho}
  6. a = λ c ρ a=\frac{\lambda}{c\rho}
  7. θ = 0 \theta=0
  8. θ \theta
  9. T i f - T i f - 1 Δ t = a T i - 1 f - 1 - 2 T i f - 1 + T i + 1 f - 1 h 2 - ϵ u T i + 1 f - 1 - T i - 1 f - 1 2 h + Q i f - 1 c ρ \frac{T_{i}^{f}-T_{i}^{f-1}}{\Delta t}=a\frac{T_{i-1}^{f-1}-2T_{i}^{f-1}+T_{i+% 1}^{f-1}}{h^{2}}-\epsilon u\frac{T_{i+1}^{f-1}-T_{i-1}^{f-1}}{2h}+\frac{Q_{i}^% {f-1}}{c\rho}
  10. Δ t = t f - t f - 1 \Delta t=t^{f}-t^{f-1}
  11. T i f = ( 1 - 2 a Δ t h 2 ) T i f - 1 + ( a Δ t h 2 + ϵ u Δ t 2 h ) T i - 1 f - 1 + ( a Δ t h 2 - ϵ u Δ t 2 h ) T i + 1 f - 1 + Q i f - 1 c ρ Δ t T_{i}^{f}=\left(1-\frac{2a\Delta t}{h^{2}}\right)T_{i}^{f-1}+\left(\frac{a% \Delta t}{h^{2}}+\frac{\epsilon u\Delta t}{2h}\right)T_{i-1}^{f-1}+\left(\frac% {a\Delta t}{h^{2}}-\frac{\epsilon u\Delta t}{2h}\right)T_{i+1}^{f-1}+\frac{Q_{% i}^{f-1}}{c\rho}\Delta t
  12. h < 2 a ϵ u h<\frac{2a}{\epsilon u}
  13. Δ t < h 2 2 a \Delta t<\frac{h^{2}}{2a}
  14. t + Δ t t+\Delta t
  15. Δ t \Delta t
  16. t + Δ t t+\Delta t
  17. Δ t < h 2 a \Delta t<\frac{h^{2}}{a}

O-phospho-L-serine—tRNA_ligase.html

  1. \rightleftharpoons

Objective_stress_rates.html

  1. s y m b o l Q symbol{Q}
  2. s y m b o l σ symbol{\sigma}
  3. s y m b o l σ r = s y m b o l Q \cdotsymbol σ \cdotsymbol Q T ; s y m b o l Q \cdotsymbol Q T = s y m b o l 1 symbol{\sigma}_{r}=symbol{Q}\cdotsymbol{\sigma}\cdotsymbol{Q}^{T}~{};~{}~{}% symbol{Q}\cdotsymbol{Q}^{T}=symbol{\mathit{1}}
  4. s y m b o l σ symbol{\sigma}
  5. s y m b o l σ symbol{\sigma}
  6. d d t ( s y m b o l σ r ) = s y m b o l σ ˙ r = s y m b o l Q ˙ \cdotsymbol σ \cdotsymbol Q T + s y m b o l Q s y m b o l σ ˙ \cdotsymbol Q T + s y m b o l Q \cdotsymbol σ s y m b o l Q ˙ T s y m b o l Q s y m b o l σ ˙ \cdotsymbol Q T . \cfrac{d}{dt}(symbol{\sigma}_{r})=\dot{symbol{\sigma}}_{r}=\dot{symbol{Q}}% \cdotsymbol{\sigma}\cdotsymbol{Q}^{T}+symbol{Q}\cdot\dot{symbol{\sigma}}% \cdotsymbol{Q}^{T}+symbol{Q}\cdotsymbol{\sigma}\cdot\dot{symbol{Q}}^{T}\neq symbol% {Q}\cdot\dot{symbol{\sigma}}\cdotsymbol{Q}^{T}\,.
  7. s y m b o l Q symbol{Q}
  8. S i j S_{ij}
  9. x i x_{i}
  10. S i j \overset{\circ}{S}_{ij}
  11. Δ S i j = S i j Δ t \Delta S_{ij}=\overset{\circ}{S}_{ij}\Delta t
  12. S i j \overset{\circ}{S}_{ij}
  13. S i j \overset{\circ}{S}_{ij}
  14. s y m b o l σ symbol{\sigma}
  15. s y m b o l S symbol{S}
  16. s y m b o l S = J ϕ * [ s y m b o l σ ] ; s y m b o l σ = J - 1 ϕ * [ s y m b o l S ] symbol{S}=J~{}\phi^{*}[symbol{\sigma}]~{};~{}~{}symbol{\sigma}=J^{-1}~{}\phi_{% *}[symbol{S}]
  17. s y m b o l σ = J - 1 ϕ * [ s y m b o l S ˙ ] \overset{\circ}{symbol{\sigma}}=J^{-1}~{}\phi_{*}[\dot{symbol{S}}]
  18. s y m b o l σ = J - 1 s y m b o l F s y m b o l S ˙ \cdotsymbol F T = J - 1 s y m b o l F [ d d t ( J s y m b o l F - 1 \cdotsymbol σ \cdotsymbol F - T ) ] \cdotsymbol F T = J - 1 φ [ s y m b o l τ ] \overset{\circ}{symbol{\sigma}}=J^{-1}~{}symbol{F}\cdot\dot{symbol{S}}% \cdotsymbol{F}^{T}=J^{-1}~{}symbol{F}\cdot\left[\cfrac{d}{dt}\left(J~{}symbol{% F}^{-1}\cdotsymbol{\sigma}\cdotsymbol{F}^{-T}\right)\right]\cdotsymbol{F}^{T}=% J^{-1}~{}\mathcal{L}_{\varphi}[symbol{\tau}]
  19. s y m b o l τ = J s y m b o l σ symbol{\tau}=J~{}symbol{\sigma}
  20. φ [ s y m b o l τ ] = s y m b o l F [ d d t ( s y m b o l F - 1 \cdotsymbol τ \cdotsymbol F - T ) ] \cdotsymbol F T . \mathcal{L}_{\varphi}[symbol{\tau}]=symbol{F}\cdot\left[\cfrac{d}{dt}\left(% symbol{F}^{-1}\cdotsymbol{\tau}\cdotsymbol{F}^{-T}\right)\right]\cdotsymbol{F}% ^{T}~{}.
  21. s y m b o l σ = s y m b o l σ ˙ - s y m b o l l \cdotsymbol σ - s y m b o l σ \cdotsymbol l T + tr ( s y m b o l l ) s y m b o l σ \overset{\circ}{symbol{\sigma}}=\dot{symbol{\sigma}}-symbol{l}\cdotsymbol{% \sigma}-symbol{\sigma}\cdotsymbol{l}^{T}+\,\text{tr}(symbol{l})~{}symbol{\sigma}
  22. s y m b o l σ = J - 1 s y m b o l F [ d d t ( J s y m b o l F - 1 \cdotsymbol σ \cdotsymbol F - T ) ] \cdotsymbol F T . \overset{\circ}{symbol{\sigma}}=J^{-1}~{}symbol{F}\cdot\left[\cfrac{d}{dt}% \left(J~{}symbol{F}^{-1}\cdotsymbol{\sigma}\cdotsymbol{F}^{-T}\right)\right]% \cdotsymbol{F}^{T}~{}.
  23. s y m b o l σ = J - 1 s y m b o l F ( J ˙ s y m b o l F - 1 \cdotsymbol σ \cdotsymbol F - T ) \cdotsymbol F T + J - 1 s y m b o l F ( J s y m b o l F - 1 ˙ \cdotsymbol σ \cdotsymbol F - T ) \cdotsymbol F T + J - 1 s y m b o l F ( J s y m b o l F - 1 s y m b o l σ ˙ \cdotsymbol F - T ) \cdotsymbol F T + J - 1 s y m b o l F ( J s y m b o l F - 1 \cdotsymbol σ s y m b o l F - T ˙ ) \cdotsymbol F T \begin{aligned}\displaystyle\overset{\circ}{symbol{\sigma}}&\displaystyle=J^{-% 1}~{}symbol{F}\cdot(\dot{J}~{}symbol{F}^{-1}\cdotsymbol{\sigma}\cdotsymbol{F}^% {-T})\cdotsymbol{F}^{T}+J^{-1}~{}symbol{F}\cdot(J~{}\dot{symbol{F}^{-1}}% \cdotsymbol{\sigma}\cdotsymbol{F}^{-T})\cdotsymbol{F}^{T}\\ &\displaystyle+J^{-1}~{}symbol{F}\cdot(J~{}symbol{F}^{-1}\cdot\dot{symbol{% \sigma}}\cdotsymbol{F}^{-T})\cdotsymbol{F}^{T}+J^{-1}~{}symbol{F}\cdot(J~{}% symbol{F}^{-1}\cdotsymbol{\sigma}\cdot\dot{symbol{F}^{-T}})\cdotsymbol{F}^{T}% \end{aligned}
  24. s y m b o l σ = J - 1 J ˙ s y m b o l σ + s y m b o l F s y m b o l F - 1 ˙ \cdotsymbol σ + s y m b o l σ ˙ + s y m b o l σ s y m b o l F - T ˙ \cdotsymbol F T \overset{\circ}{symbol{\sigma}}=J^{-1}~{}\dot{J}~{}symbol{\sigma}+symbol{F}% \cdot\dot{symbol{F}^{-1}}\cdotsymbol{\sigma}+\dot{symbol{\sigma}}+symbol{% \sigma}\cdot\dot{symbol{F}^{-T}}\cdotsymbol{F}^{T}
  25. s y m b o l F \cdotsymbol F - 1 = s y m b o l 1 symbol{F}\cdotsymbol{F}^{-1}=symbol{\mathit{1}}
  26. d d t ( s y m b o l F \cdotsymbol F - 1 ) = 0 s y m b o l F ˙ \cdotsymbol F - 1 + s y m b o l F s y m b o l F - 1 ˙ = 0 \cfrac{d}{dt}\left(symbol{F}\cdotsymbol{F}^{-1}\right)=0\quad\implies\quad\dot% {symbol{F}}\cdotsymbol{F}^{-1}+symbol{F}\cdot\dot{symbol{F}^{-1}}=0
  27. s y m b o l F - 1 ˙ = - s y m b o l F - 1 \cdotsymbol l s y m b o l F - T ˙ = - s y m b o l l T \cdotsymbol F - T \dot{symbol{F}^{-1}}=-symbol{F}^{-1}\cdotsymbol{l}\quad\implies\quad\dot{% symbol{F}^{-T}}=-symbol{l}^{T}\cdotsymbol{F}^{-T}
  28. s y m b o l l = s y m b o l F ˙ \cdotsymbol F - 1 symbol{l}=\dot{symbol{F}}\cdotsymbol{F}^{-1}
  29. J ˙ = J tr ( s y m b o l d ) = J tr ( s y m b o l l ) \dot{J}=J~{}\,\text{tr}(symbol{d})=J~{}\,\text{tr}(symbol{l})
  30. s y m b o l d symbol{d}
  31. s y m b o l σ = J - 1 J tr ( s y m b o l l ) s y m b o l σ - s y m b o l F \cdotsymbol F - 1 \cdotsymbol l \cdotsymbol σ + s y m b o l σ ˙ - s y m b o l σ \cdotsymbol l T \cdotsymbol F - T \cdotsymbol F T \overset{\circ}{symbol{\sigma}}=J^{-1}~{}J~{}\,\text{tr}(symbol{l})~{}symbol{% \sigma}-symbol{F}\cdotsymbol{F}^{-1}\cdotsymbol{l}\cdotsymbol{\sigma}+\dot{% symbol{\sigma}}-symbol{\sigma}\cdotsymbol{l}^{T}\cdotsymbol{F}^{-T}\cdotsymbol% {F}^{T}
  32. s y m b o l σ = s y m b o l σ ˙ - s y m b o l l \cdotsymbol σ - s y m b o l σ \cdotsymbol l T + tr ( s y m b o l l ) s y m b o l σ \overset{\circ}{symbol{\sigma}}=\dot{symbol{\sigma}}-symbol{l}\cdotsymbol{% \sigma}-symbol{\sigma}\cdotsymbol{l}^{T}+\,\text{tr}(symbol{l})~{}symbol{\sigma}
  33. s y m b o l S = ϕ * [ s y m b o l τ ] ; s y m b o l τ = ϕ * [ s y m b o l S ] symbol{S}=\phi^{*}[symbol{\tau}]~{};~{}~{}symbol{\tau}=\phi_{*}[symbol{S}]
  34. s y m b o l τ = ϕ * [ s y m b o l S ˙ ] \overset{\circ}{symbol{\tau}}=\phi_{*}[\dot{symbol{S}}]
  35. s y m b o l τ = s y m b o l F s y m b o l S ˙ \cdotsymbol F T = s y m b o l F [ d d t ( s y m b o l F - 1 \cdotsymbol τ \cdotsymbol F - T ) ] \cdotsymbol F T = φ [ s y m b o l τ ] \overset{\circ}{symbol{\tau}}=symbol{F}\cdot\dot{symbol{S}}\cdotsymbol{F}^{T}=% symbol{F}\cdot\left[\cfrac{d}{dt}\left(symbol{F}^{-1}\cdotsymbol{\tau}% \cdotsymbol{F}^{-T}\right)\right]\cdotsymbol{F}^{T}=\mathcal{L}_{\varphi}[% symbol{\tau}]
  36. s y m b o l τ symbol{\tau}
  37. s y m b o l τ = s y m b o l τ ˙ - s y m b o l l \cdotsymbol τ - s y m b o l τ \cdotsymbol l T \overset{\circ}{symbol{\tau}}=\dot{symbol{\tau}}-symbol{l}\cdotsymbol{\tau}-% symbol{\tau}\cdotsymbol{l}^{T}
  38. s y m b o l σ = J - 1 s y m b o l F [ d d t ( J s y m b o l F - 1 \cdotsymbol σ \cdotsymbol F - T ) ] \cdotsymbol F T . \overset{\circ}{symbol{\sigma}}=J^{-1}~{}symbol{F}\cdot\left[\cfrac{d}{dt}% \left(J~{}symbol{F}^{-1}\cdotsymbol{\sigma}\cdotsymbol{F}^{-T}\right)\right]% \cdotsymbol{F}^{T}~{}.
  39. s y m b o l F = s y m b o l R \cdotsymbol U symbol{F}=symbol{R}\cdotsymbol{U}
  40. s y m b o l R symbol{R}
  41. s y m b o l R - 1 = s y m b o l R T symbol{R}^{-1}=symbol{R}^{T}
  42. s y m b o l U symbol{U}
  43. s y m b o l U = s y m b o l 1 symbol{U}=symbol{\mathit{1}}
  44. s y m b o l F = s y m b o l R symbol{F}=symbol{R}
  45. J = 1 J=1
  46. s y m b o l τ = s y m b o l σ symbol{\tau}=symbol{\sigma}
  47. s y m b o l σ = s y m b o l R [ d d t ( s y m b o l R - 1 \cdotsymbol σ \cdotsymbol R - T ) ] \cdotsymbol R T = s y m b o l R [ d d t ( s y m b o l R T \cdotsymbol σ \cdotsymbol R ) ] \cdotsymbol R T \overset{\circ}{symbol{\sigma}}=symbol{R}\cdot\left[\cfrac{d}{dt}\left(symbol{% R}^{-1}\cdotsymbol{\sigma}\cdotsymbol{R}^{-T}\right)\right]\cdotsymbol{R}^{T}=% symbol{R}\cdot\left[\cfrac{d}{dt}\left(symbol{R}^{T}\cdotsymbol{\sigma}% \cdotsymbol{R}\right)\right]\cdotsymbol{R}^{T}
  48. s y m b o l σ = s y m b o l σ ˙ + s y m b o l σ \cdotsymbol Ω - s y m b o l Ω \cdotsymbol σ \overset{\square}{symbol{\sigma}}=\dot{symbol{\sigma}}+symbol{\sigma}% \cdotsymbol{\Omega}-symbol{\Omega}\cdotsymbol{\sigma}
  49. s y m b o l Ω = s y m b o l R ˙ \cdotsymbol R T symbol{\Omega}=\dot{symbol{R}}\cdotsymbol{R}^{T}
  50. s y m b o l σ = s y m b o l R s y m b o l R T ˙ \cdotsymbol σ \cdotsymbol R \cdotsymbol R T + s y m b o l R \cdotsymbol R T s y m b o l σ ˙ \cdotsymbol R \cdotsymbol R T + s y m b o l R \cdotsymbol R T \cdotsymbol σ s y m b o l R ˙ \cdotsymbol R T \overset{\circ}{symbol{\sigma}}=symbol{R}\cdot\dot{symbol{R}^{T}}\cdotsymbol{% \sigma}\cdotsymbol{R}\cdotsymbol{R}^{T}+symbol{R}\cdotsymbol{R}^{T}\cdot\dot{% symbol{\sigma}}\cdotsymbol{R}\cdotsymbol{R}^{T}+symbol{R}\cdotsymbol{R}^{T}% \cdotsymbol{\sigma}\cdot\dot{symbol{R}}\cdotsymbol{R}^{T}
  51. s y m b o l σ = s y m b o l R s y m b o l R T ˙ \cdotsymbol σ + s y m b o l σ ˙ + s y m b o l σ s y m b o l R ˙ \cdotsymbol R T \overset{\circ}{symbol{\sigma}}=symbol{R}\cdot\dot{symbol{R}^{T}}\cdotsymbol{% \sigma}+\dot{symbol{\sigma}}+symbol{\sigma}\cdot\dot{symbol{R}}\cdotsymbol{R}^% {T}
  52. s y m b o l R \cdotsymbol R T = s y m b o l 1 s y m b o l R ˙ \cdotsymbol R T = - s y m b o l R s y m b o l R T ˙ symbol{R}\cdotsymbol{R}^{T}=symbol{\mathit{1}}\quad\implies\quad\dot{symbol{R}% }\cdotsymbol{R}^{T}=-symbol{R}\cdot\dot{symbol{R}^{T}}
  53. s y m b o l σ = s y m b o l σ ˙ + s y m b o l σ s y m b o l R ˙ \cdotsymbol R T - s y m b o l R ˙ \cdotsymbol R T \cdotsymbol σ \overset{\circ}{symbol{\sigma}}=\dot{symbol{\sigma}}+symbol{\sigma}\cdot\dot{% symbol{R}}\cdotsymbol{R}^{T}-\dot{symbol{R}}\cdotsymbol{R}^{T}\cdotsymbol{\sigma}
  54. s y m b o l Ω = s y m b o l R ˙ \cdotsymbol R T symbol{\Omega}=\dot{symbol{R}}\cdotsymbol{R}^{T}
  55. s y m b o l σ = s y m b o l σ ˙ + s y m b o l σ \cdotsymbol Ω - s y m b o l Ω \cdotsymbol σ \overset{\square}{symbol{\sigma}}=\dot{symbol{\sigma}}+symbol{\sigma}% \cdotsymbol{\Omega}-symbol{\Omega}\cdotsymbol{\sigma}
  56. s y m b o l τ = s y m b o l τ ˙ + s y m b o l τ \cdotsymbol Ω - s y m b o l Ω \cdotsymbol τ \overset{\square}{symbol{\tau}}=\dot{symbol{\tau}}+symbol{\tau}\cdotsymbol{% \Omega}-symbol{\Omega}\cdotsymbol{\tau}
  57. s y m b o l σ = s y m b o l σ ˙ + s y m b o l σ \cdotsymbol w - s y m b o l w \cdotsymbol σ \overset{\triangle}{symbol{\sigma}}=\dot{symbol{\sigma}}+symbol{\sigma}% \cdotsymbol{w}-symbol{w}\cdotsymbol{\sigma}
  58. s y m b o l w symbol{w}
  59. s y m b o l w symbol{w}
  60. s y m b o l w = s y m b o l R ˙ \cdotsymbol R T + 1 2 s y m b o l R ( s y m b o l U ˙ \cdotsymbol U - 1 - s y m b o l U - 1 s y m b o l U ˙ ) \cdotsymbol R T symbol{w}=\dot{symbol{R}}\cdotsymbol{R}^{T}+\frac{1}{2}~{}symbol{R}\cdot(\dot{% symbol{U}}\cdotsymbol{U}^{-1}-symbol{U}^{-1}\cdot\dot{symbol{U}})\cdotsymbol{R% }^{T}
  61. s y m b o l w = s y m b o l R ˙ \cdotsymbol R T = s y m b o l Ω symbol{w}=\dot{symbol{R}}\cdotsymbol{R}^{T}=symbol{\Omega}
  62. s y m b o l U = [ λ X λ Y λ Z ] symbol{U}=\left[\begin{array}[]{ccc}\lambda_{X}\\ &\lambda_{Y}\\ &&\lambda_{Z}\end{array}\right]
  63. s y m b o l U ˙ = [ λ ˙ X λ ˙ Y λ ˙ Z ] \dot{symbol{U}}=\left[\begin{array}[]{ccc}\dot{\lambda}_{X}\\ &\dot{\lambda}_{Y}\\ &&\dot{\lambda}_{Z}\end{array}\right]
  64. s y m b o l U - 1 = [ 1 / λ X 1 / λ Y 1 / λ Z ] symbol{U}^{-1}=\left[\begin{array}[]{ccc}1/\lambda_{X}\\ &1/\lambda_{Y}\\ &&1/\lambda_{Z}\end{array}\right]
  65. s y m b o l U ˙ \cdotsymbol U - 1 = [ λ ˙ X / λ X λ ˙ Y / λ Y λ ˙ Z / λ Z ] = U - 1 U ˙ \dot{symbol{U}}\cdotsymbol{U}^{-1}=\left[\begin{array}[]{ccc}\dot{\lambda}_{X}% /\lambda_{X}\\ &\dot{\lambda}_{Y}/\lambda_{Y}\\ &&\dot{\lambda}_{Z}/\lambda_{Z}\end{array}\right]=U^{-1}\dot{U}
  66. s y m b o l w = s y m b o l R ˙ \cdotsymbol R T = s y m b o l Ω symbol{w}=\dot{symbol{R}}\cdotsymbol{R}^{T}=symbol{\Omega}
  67. s y m b o l w s y m b o l R ˙ \cdotsymbol R T symbol{w}\approx\dot{symbol{R}}\cdotsymbol{R}^{T}
  68. s y m b o l σ = s y m b o l σ ˙ + s y m b o l σ \cdotsymbol w - s y m b o l w \cdotsymbol σ \overset{\triangle}{symbol{\sigma}}=\dot{symbol{\sigma}}+symbol{\sigma}% \cdotsymbol{w}-symbol{w}\cdotsymbol{\sigma}
  69. s y m b o l σ = φ [ s y m b o l σ ] = s y m b o l F [ d d t ( s y m b o l F - 1 \cdotsymbol σ \cdotsymbol F - T ) ] \cdotsymbol F T \overset{\triangledown}{symbol{\sigma}}=\mathcal{L}_{\varphi}[symbol{\sigma}]=% symbol{F}\cdot\left[\cfrac{d}{dt}\left(symbol{F}^{-1}\cdotsymbol{\sigma}% \cdotsymbol{F}^{-T}\right)\right]\cdotsymbol{F}^{T}
  70. s y m b o l σ = s y m b o l σ ˙ - s y m b o l l \cdotsymbol σ - s y m b o l σ \cdotsymbol l T \overset{\triangledown}{symbol{\sigma}}=\dot{symbol{\sigma}}-symbol{l}% \cdotsymbol{\sigma}-symbol{\sigma}\cdotsymbol{l}^{T}
  71. s y m b o l F T symbol{F}^{T}
  72. s y m b o l F - T symbol{F}^{-T}
  73. s y m b o l σ = s y m b o l F - T [ d d t ( s y m b o l F T \cdotsymbol σ \cdotsymbol F ) ] \cdotsymbol F - 1 \overset{\diamond}{symbol{\sigma}}=symbol{F}^{-T}\cdot\left[\cfrac{d}{dt}\left% (symbol{F}^{T}\cdotsymbol{\sigma}\cdotsymbol{F}\right)\right]\cdotsymbol{F}^{-1}
  74. s y m b o l σ = s y m b o l σ ˙ + s y m b o l l \cdotsymbol σ + s y m b o l σ \cdotsymbol l T \overset{\diamond}{symbol{\sigma}}=\dot{symbol{\sigma}}+symbol{l}\cdotsymbol{% \sigma}+symbol{\sigma}\cdotsymbol{l}^{T}
  75. s y m b o l e = 1 2 [ s y m b o l 𝐮 + ( s y m b o l 𝐮 ) T ] e i j = 1 2 ( u i , j + u j , i ) symbol{e}=\tfrac{1}{2}\left[symbol{\nabla}\mathbf{u}+(symbol{\nabla}\mathbf{u}% )^{T}\right]\quad\equiv\quad e_{ij}=\tfrac{1}{2}(u_{i,j}+u_{j,i})
  76. 𝐮 \mathbf{u}
  77. \partialsymbol e t = s y m b o l e ˙ = 1 2 [ s y m b o l 𝐯 + ( s y m b o l 𝐯 ) T ] e ˙ i j = 1 2 ( v i , j + v j , i ) \frac{\partialsymbol{e}}{\partial t}=\dot{symbol{e}}=\tfrac{1}{2}\left[symbol{% \nabla}\mathbf{v}+(symbol{\nabla}\mathbf{v})^{T}\right]\quad\equiv\quad\dot{e}% _{ij}=\tfrac{1}{2}(v_{i,j}+v_{j,i})
  78. 𝐯 = 𝐮 ˙ \mathbf{v}=\dot{\mathbf{u}}
  79. 𝐄 ( m ) = 1 2 m ( 𝐔 2 m - 𝐈 ) \mathbf{E}_{(m)}=\frac{1}{2m}(\mathbf{U}^{2m}-\mathbf{I})
  80. 𝐔 \mathbf{U}
  81. 𝐄 ( m ) s y m b o l e + 1 2 ( 𝐮 ) T 𝐮 - ( 1 - m ) s y m b o l e \cdotsymbol e \mathbf{E}_{(m)}\approx symbol{e}+{\tfrac{1}{2}}(\nabla\mathbf{u})^{T}\cdot% \nabla\mathbf{u}-(1-m)symbol{e}\cdotsymbol{e}
  82. s y m b o l σ 0 symbol{\sigma}_{0}
  83. s y m b o l σ symbol{\sigma}
  84. W W
  85. δ W \delta W
  86. δ 𝐮 \delta\mathbf{u}
  87. s y m b o l S ( m ) symbol{S}_{(m)}
  88. s y m b o l S = s y m b o l S ( m ) - s y m b o l σ 0 symbol{S}=symbol{S}_{(m)}-symbol{\sigma}_{0}
  89. s y m b o l P symbol{P}
  90. s y m b o l T = s y m b o l P - s y m b o l σ 0 symbol{T}=symbol{P}-symbol{\sigma}_{0}
  91. δ W = s y m b o l S ( m ) : \deltasymbol E ( m ) = s y m b o l P : δ 𝐮 \delta W=symbol{S}_{(m)}:\deltasymbol{E}_{(m)}=symbol{P}:\delta\nabla\mathbf{u}
  92. s y m b o l E ( m ) symbol{E}_{(m)}
  93. s y m b o l σ ( m ) symbol{\sigma}^{(m)}
  94. δ W = ( s y m b o l S + s y m b o l σ 0 ) : \deltasymbol E ( m ) = ( s y m b o l T + s y m b o l σ 0 ) : δ 𝐮 . \delta W=\left(symbol{S}+symbol{\sigma}_{0}\right):\deltasymbol{E}_{(m)}=\left% (symbol{T}+symbol{\sigma}_{0}\right):\delta\nabla\mathbf{u}\,.
  95. s y m b o l S ( m ) symbol{S}_{(m)}
  96. s y m b o l S ( m ) : \deltasymbol E ( m ) symbol{S}_{(m)}:\deltasymbol{E}_{(m)}
  97. s y m b o l σ 0 : δ 𝐮 = s y m b o l σ 0 : \deltasymbol e . symbol{\sigma}_{0}:\delta\nabla\mathbf{u}=symbol{\sigma}_{0}:\deltasymbol{e}\,.
  98. s y m b o l S : \deltasymbol E ( m ) s y m b o l S : δ 𝐮 symbol{S}:\deltasymbol{E}_{(m)}\approx symbol{S}:\delta\nabla\mathbf{u}
  99. s y m b o l σ 0 : \deltasymbol E ( m ) = s y m b o l σ 0 : [ s y m b o l E ( m ) 𝐮 : δ 𝐮 ] , s y m b o l σ 0 : \deltasymbol e = s y m b o l σ 0 : [ s y m b o l e 𝐮 : δ 𝐮 ] symbol{\sigma}_{0}:\deltasymbol{E}_{(m)}=symbol{\sigma}_{0}:\left[\frac{% \partial symbol{E}_{(m)}}{\partial\nabla\mathbf{u}}:\delta\nabla\mathbf{u}% \right]~{},~{}~{}symbol{\sigma}_{0}:\deltasymbol{e}=symbol{\sigma}_{0}:\left[% \frac{\partial symbol{e}}{\partial\nabla\mathbf{u}}:\delta\nabla\mathbf{u}\right]
  100. s y m b o l σ 0 : [ s y m b o l E ( m ) 𝐮 : δ 𝐮 ] + s y m b o l S : δ 𝐮 = s y m b o l σ 0 : [ s y m b o l e 𝐮 : δ 𝐮 ] + s y m b o l T : δ 𝐮 . symbol{\sigma}_{0}:\left[\frac{\partial symbol{E}_{(m)}}{\partial\nabla\mathbf% {u}}:\delta\nabla\mathbf{u}\right]+symbol{S}:\delta\nabla\mathbf{u}=symbol{% \sigma}_{0}:\left[\frac{\partial symbol{e}}{\partial\nabla\mathbf{u}}:\delta% \nabla\mathbf{u}\right]+symbol{T}:\delta\nabla\mathbf{u}\,.
  101. δ 𝐮 \delta\nabla\mathbf{u}
  102. s y m b o l S = s y m b o l T - s y m b o l σ 0 : [ s y m b o l E ( m ) 𝐮 - s y m b o l e 𝐮 ] symbol{S}=symbol{T}-symbol{\sigma}_{0}:\left[\frac{\partial symbol{E}_{(m)}}{% \partial\nabla\mathbf{u}}-\frac{\partial symbol{e}}{\partial\nabla\mathbf{u}}\right]
  103. m m
  104. ( 3 ) S ^ i j ( m ) = T ˙ i j - S p q t ( ϵ p q ( m ) - e p q ) u i , j (3)~{}~{}~{}{\hat{S}_{ij}^{(m)}=\dot{T}_{ij}-S_{pq}\ \frac{\partial}{\partial t% }\ \frac{\partial(\epsilon_{pq}^{(m)}-e_{pq})}{\partial u_{i,j}}}
  105. ( 4 ) T ˙ i j = S ˙ i j - S i k v j , k + S i j v k , k (4)~{}~{}~{}\dot{T}_{ij}=\dot{S}_{ij}-S_{ik}v_{j,k}+S_{ij}v_{k,k}
  106. S ˙ i j = S i j / t \dot{S}_{ij}=\partial S_{ij}/\partial t
  107. T i j = S i j 0 + τ i j T_{ij}=S_{ij}^{0}+\tau_{ij}
  108. T ˙ i j = T i j / t = τ i j / t \dot{T}_{ij}=\partial T_{ij}/\partial t=\partial\tau_{ij}/\partial t
  109. ϵ i j \epsilon_{ij}
  110. m m
  111. m = 2 m=2
  112. ( 5 ) S ^ i j ( m ) = S ^ i j + 1 2 ( 2 - m ) ( S i k e ˙ k j + S j k e ˙ k i ) (5)~{}~{}~{}\hat{S}_{ij}^{(m)}=\hat{S}_{ij}+\tfrac{1}{2}(2-m)(S_{ik}\dot{e}_{% kj}+S_{jk}\dot{e}_{ki})
  113. S ^ i j = S ^ i j ( 2 ) \hat{S}_{ij}=\hat{S}_{ij}^{(2)}
  114. m = 2 m=2
  115. m = 2 m=2
  116. S ^ i j ( 2 ) = S ˙ i j - S k j v i , k - S k i v j , k + S i j v k , k \hat{S}_{ij}^{(2)}=\dot{S}_{ij}-S_{kj}v_{i,k}-S_{ki}v_{j,k}+S_{ij}v_{k,k}
  117. m = 0 m=0
  118. S ^ i j ( 0 ) = S ˙ i j - S k j ω ˙ i k + S i k ω ˙ k j + S i j v k , k \hat{S}_{ij}^{(0)}=\dot{S}_{ij}-S_{kj}\dot{\omega}_{ik}+S_{ik}\dot{\omega}_{kj% }+S_{ij}v_{k,k}
  119. ω ˙ j k = 1 2 ( v i , j - v j , i ) \dot{\omega}_{jk}=\tfrac{1}{2}(v_{i,j}-v_{j,i})
  120. m = 1 m=1
  121. S ^ i j J = S ˙ i j - S k j ω ˙ i k + S i k ω ˙ k j \hat{S}_{ij}^{J}=\dot{S}_{ij}-S_{kj}\dot{\omega}_{ik}+S_{ik}\dot{\omega}_{kj}
  122. S i j v k , k S_{ij}v_{k,k}
  123. v k , k = e ˙ k k v_{k,k}=\dot{e}_{kk}
  124. S i j v k , k S_{ij}v_{k,k}
  125. m = - 2 m=-2
  126. S i j v k , k S_{ij}v_{k,k}
  127. S ^ i j G N = S ˙ i j - S k j Ω ˙ i k + S i k Ω ˙ k j \hat{S}^{GN}_{ij}=\dot{S}_{ij}-S_{kj}\dot{\Omega}_{ik}+S_{ik}\dot{\Omega}_{kj}
  128. Ω i j \Omega_{ij}
  129. Ω \Omega
  130. ω \omega
  131. ( 6 ) S i j ( m ) = C i j k l ( m ) e ˙ k l (6)~{}~{}~{}S_{ij}^{(m)}=C_{ijkl}^{(m)}\dot{e}_{kl}
  132. C i j k l ( m ) C_{ijkl}^{(m)}
  133. ϵ i j ( m ) \epsilon_{ij}^{(m)}
  134. m m
  135. ( 7 ) [ C i j k l ( m ) - C i j k l ( 2 ) - 1 4 ( 2 - m ) ( S i k δ j l + S j k δ i l + S i l δ j k + S j l δ i k ) ] v k , l = 0 (7)~{}~{}~{}\left[C_{ijkl}^{(m)}-C^{(2)}_{ijkl}-{\tfrac{1}{4}}(2-m)(S_{ik}% \delta_{jl}+S_{jk}\delta_{il}+S_{il}\delta_{jk}+S_{jl}\delta_{ik})\right]v_{k,% l}=0
  136. v k , l v_{k,l}
  137. ( 8 ) C i j k l ( m ) = C i j k l ( 2 ) + ( 2 - m ) [ S i k δ j l ] s y m , [ S i k δ j l ] s y m = 1 4 ( S i k δ j l + S j k δ i l + S i l δ j k + S j l δ i k ) (8)~{}~{}~{}C_{ijkl}^{(m)}=C^{(2)}_{ijkl}+(2-m)[S_{ik}\delta_{jl}]_{sym},~{}~{% }[S_{ik}\delta_{jl}]_{sym}={\tfrac{1}{4}}(S_{ik}\delta_{jl}+S_{jk}\delta_{il}+% S_{il}\delta_{jk}+S_{jl}\delta_{ik})
  138. C i j k l ( 2 ) C_{ijkl}^{(2)}
  139. m = 2 m=2
  140. S i j S_{ij}
  141. δ i j \delta_{ij}
  142. S i j e ˙ k k = ( S i j δ k l ) δ e k l S_{ij}\dot{e}_{kk}=(S_{ij}\delta_{kl})\delta e_{kl}
  143. ( 9 ) C i j k l conj = C i j k l nonconj + S i j δ k l (9)~{}~{}~{}C_{ijkl}^{\mathrm{conj}}=C_{ijkl}^{\mathrm{nonconj}}+S_{ij}\delta_% {kl}
  144. S i j v k , k S_{ij}v_{k,k}
  145. S i j δ k m S_{ij}\delta_{km}
  146. i j ij
  147. k l kl
  148. C i j k l nonconj C_{ijkl}^{\mathrm{nonconj}}
  149. C i j k l C_{ijkl}
  150. m m

Oblate_spheroidal_wave_function.html

  1. Δ Φ + k 2 Φ = 0 \Delta\Phi+k^{2}\Phi=0
  2. ( ξ , η , ϕ ) (\xi,\eta,\phi)
  3. x = ( d / 2 ) ξ η , \ x=(d/2)\xi\eta,
  4. y = ( d / 2 ) ( ξ 2 + 1 ) ( 1 - η 2 ) cos ϕ , \ y=(d/2)\sqrt{(\xi^{2}+1)(1-\eta^{2})}\cos\phi,
  5. z = ( d / 2 ) ( ξ 2 + 1 ) ( 1 - η 2 ) sin ϕ , \ z=(d/2)\sqrt{(\xi^{2}+1)(1-\eta^{2})}\sin\phi,
  6. ξ 0 and | η | 1. \ \xi\geq 0\,\text{ and }|\eta|\leq 1.
  7. Φ ( ξ , η , ϕ ) \Phi(\xi,\eta,\phi)
  8. R m n ( - i c , i ξ ) R_{mn}(-ic,i\xi)
  9. S m n ( - i c , η ) S_{mn}(-ic,\eta)
  10. e i m ϕ e^{im\phi}
  11. c = k d / 2 c=kd/2
  12. d d
  13. R m n ( - i c , i ξ ) R_{mn}(-ic,i\xi)
  14. ( ξ 2 + 1 ) d 2 R m n ( - i c , i ξ ) d ξ 2 + 2 ξ d R m n ( - i c , i ξ ) d ξ - ( λ m n ( c ) - c 2 ξ 2 - m 2 ξ 2 + 1 ) R m n ( - i c , i ξ ) = 0 \ (\xi^{2}+1)\frac{d^{2}R_{mn}(-ic,i\xi)}{d\xi^{2}}+2\xi\frac{dR_{mn}(-ic,i\xi% )}{d\xi}-\left(\lambda_{mn}(c)-c^{2}\xi^{2}-\frac{m^{2}}{\xi^{2}+1}\right){R_{% mn}(-ic,i\xi)}=0
  15. ( 1 - η 2 ) d 2 S m n ( - i c , η ) d η 2 - 2 η d S m n ( - i c , η ) d η + ( λ m n ( c ) + c 2 η 2 - m 2 1 - η 2 ) S m n ( - i c , η ) = 0 \ (1-\eta^{2})\frac{d^{2}S_{mn}(-ic,\eta)}{d\eta^{2}}-2\eta\frac{dS_{mn}(-ic,% \eta)}{d\eta}+\left(\lambda_{mn}(c)+c^{2}\eta^{2}-\frac{m^{2}}{1-\eta^{2}}% \right){S_{mn}(-ic,\eta)}=0
  16. ξ \xi
  17. η \eta
  18. λ m n ( - i c ) \lambda_{mn}(-ic)
  19. S m n ( - i c , η ) {S_{mn}(-ic,\eta)}
  20. | η | = 1 |\eta|=1
  21. c = 0 c=0
  22. c 0 c\neq 0
  23. - i c -ic
  24. c c
  25. i ξ i\xi
  26. ξ \xi

Occam_learning.html

  1. c c
  2. C C
  3. s i z e ( c ) size(c)
  4. c c
  5. C C
  6. C C
  7. H H
  8. S S
  9. m m
  10. n n
  11. α 0 \alpha\geq 0
  12. 0 β < 1 0\leq\beta<1
  13. L L
  14. C C
  15. H H
  16. S S
  17. c C c\in C
  18. L L
  19. h H h\in H
  20. h h
  21. c c
  22. S S
  23. h ( x ) = c ( x ) , x S h(x)=c(x),\forall x\in S
  24. s i z e ( h ) ( n s i z e ( c ) ) α m β size(h)\leq(n\cdot size(c))^{\alpha}m^{\beta}
  25. L L
  26. n n
  27. m m
  28. s i z e ( c ) size(c)
  29. L L
  30. C C
  31. H H
  32. m m
  33. n n
  34. m α ( 1 ϵ log 1 δ + ( ( n s i z e ( c ) ) α ) ϵ ) 1 1 - β ) m\geq\alpha\left(\frac{1}{\epsilon}\log\frac{1}{\delta}+\left(\frac{(n\cdot size% (c))^{\alpha})}{\epsilon}\right)^{\frac{1}{1-\beta}}\right)
  35. h H h\in H
  36. e r r o r ( h ) ϵ error(h)\leq\epsilon
  37. 1 - δ 1-\delta
  38. b > 0 b>0
  39. m 1 b ϵ ( log | H | + log 1 δ ) m\geq\frac{1}{b\epsilon}\left(\log|H|+\log\frac{1}{\delta}\right)
  40. L L
  41. h H h\in H
  42. e r r o r ( h ) ϵ error(h)\leq\epsilon
  43. 1 - δ 1-\delta

Ohlson_o-score.html

  1. T = \displaystyle T=

Olivetolic_acid_cyclase.html

  1. \rightleftharpoons

Omnitruncated_tesseractic_honeycomb.html

  1. C ~ 4 {\tilde{C}}_{4}

Operation_Reduction_for_Low_Power.html

  1. a a + ( b × c ) \ a\leftarrow a+(b\times c)

Optic_equation.html

  1. 1 a + 1 b = 1 c . \frac{1}{a}+\frac{1}{b}=\frac{1}{c}.
  2. a = m ( m + n ) , b = n ( m + n ) , c = m n . a=m(m+n),\quad b=n(m+n),\quad c=mn.
  3. 1 h = 1 A + 1 B . \tfrac{1}{h}=\tfrac{1}{A}+\tfrac{1}{B}.
  4. 1 ( R - x ) 2 + 1 ( R + x ) 2 = 1 r 2 , \frac{1}{(R-x)^{2}}+\frac{1}{(R+x)^{2}}=\frac{1}{r^{2}},
  5. 1 I A 2 + 1 I C 2 = 1 I B 2 + 1 I D 2 = 1 r 2 . \frac{1}{IA^{2}}+\frac{1}{IC^{2}}=\frac{1}{IB^{2}}+\frac{1}{ID^{2}}=\frac{1}{r% ^{2}}.
  6. 1 h = 1 A + 1 B . \tfrac{1}{h}=\tfrac{1}{A}+\tfrac{1}{B}.
  7. 1 a + 1 b = 1 c . \tfrac{1}{a}+\tfrac{1}{b}=\tfrac{1}{c}.
  8. 1 x n + 1 y n = 1 z n , \tfrac{1}{x^{n}}+\tfrac{1}{y^{n}}=\tfrac{1}{z^{n}},
  9. ( x y z ) n (xyz)^{n}
  10. ( y z ) n + ( x z ) n = ( x y ) n , (yz)^{n}+(xz)^{n}=(xy)^{n},

Optimal_binary_search_tree.html

  1. n n
  2. 2 n + 1 2n+1
  3. a 1 a_{1}
  4. a n a_{n}
  5. A 1 A_{1}
  6. A n A_{n}
  7. B 0 B_{0}
  8. B n B_{n}
  9. A i A_{i}
  10. a i a_{i}
  11. 1 i < n 1\leq i<n
  12. B i B_{i}
  13. a i a_{i}
  14. a i + 1 a_{i+1}
  15. B 0 B_{0}
  16. a 0 a_{0}
  17. B n B_{n}
  18. a n a_{n}
  19. 2 n + 1 2n+1
  20. 2 n + 1 2n+1
  21. n n
  22. ( 2 n n ) 1 n + 1 {2n\choose n}\frac{1}{n+1}
  23. n n
  24. P = P L + P R + W P=P_{L}+P_{R}+W
  25. P i j P_{ij}
  26. W i j W_{ij}
  27. R i j R_{ij}
  28. P i i = W i i \displaystyle P_{ii}=W_{ii}
  29. 2 + ( 1 - log ( 5 - 1 ) ) - 1 H 2+(1-\log(\sqrt{5}-1))^{-1}H
  30. ( 1 / log 3 ) H (1/\log 3)H
  31. O ( log log n OPT ( X ) ) O(\log\log n\operatorname{OPT}(X))
  32. log log n \log\log n

Optimistic_knowledge_gradient.html

  1. i = { 1 , 2 , , k } i=\{1,2,\ldots,k\}
  2. Z i Z_{i}
  3. H * H^{*}
  4. θ i \theta_{i}
  5. θ i \theta_{i}
  6. θ i \theta_{i}
  7. θ i \theta_{i}
  8. θ i \theta_{i}
  9. Y i Y_{i}
  10. θ i \theta_{i}
  11. θ i \theta_{i}
  12. θ i \theta_{i}
  13. i = 1 k ( 𝟏 ( i H ) 𝟏 ( i H ) + 𝟏 ( i H ) 𝟏 ( i H ) ) \sum_{i=1}^{k}(\,\textbf{1}_{(i\in H)}\,\textbf{1}_{(i\in H^{\star})}+\,% \textbf{1}_{(i\notin H)}\,\textbf{1}_{(i\notin H^{\star})})
  14. | H H | + | H c H c | |H\cap H^{\star}|+|H^{c}\cap H^{\star c}|
  15. θ i \theta_{i}
  16. θ i \theta_{i}
  17. θ i Beta ( a i o , b i o ) \theta_{i}\sim\mathrm{Beta}(a_{i}^{o},b_{i}^{o})
  18. s o = ( a i o , b i o ) i = 1 k 𝐑 k × 2 s^{o}=\left\langle(a_{i}^{o},b_{i}^{o})\right\rangle_{i=1}^{k}\in\,\textbf{R}^% {k\times 2}
  19. θ i Beta ( a i t , b i t ) \theta_{i}\sim\mathrm{Beta}(a_{i}^{t},b_{i}^{t})
  20. y i θ i Bernoulli ( θ i ) y_{i}\mid\theta_{i}\sim\mathrm{Bernoulli}(\theta_{i})
  21. θ i y i = 1 Beta ( a i t + 1 , b i t ) \theta_{i}\mid y_{i}=1\sim\mathrm{Beta}(a_{i}^{t}+1,b_{i}^{t})
  22. θ i y i = - 1 Beta ( a i t + 1 , b i t ) \theta_{i}\mid y_{i}=-1\sim\mathrm{Beta}(a_{i}^{t}+1,b_{i}^{t})
  23. 0 t T - 1 0\leq t\leq T-1
  24. θ i \theta_{i}
  25. s t = ( a i t , b i t ) i = 1 k 𝐑 k × 2 s^{t}=\left\langle(a_{i}^{t},b_{i}^{t})\right\rangle_{i=1}^{k}\in\,\textbf{R}^% {k\times 2}
  26. i t i_{t}
  27. i t { 1 , 2 , , k } i_{t}\in\{1,2,\ldots,k\}
  28. θ i Beta ( a i t , b i t ) \theta_{i}\sim\mathrm{Beta}(a_{i}^{t},b_{i}^{t})
  29. y i θ i Bernoulli ( θ i ) y_{i}\mid\theta_{i}\sim\mathrm{Bernoulli}(\theta_{i})
  30. θ i y i = 1 Beta ( a i t + 1 , b i t ) \theta_{i}\mid y_{i}=1\sim\mathrm{Beta}(a_{i}^{t}+1,b_{i}^{t})
  31. θ i y i = - 1 Beta ( a i t + 1 , b i t ) \theta_{i}\mid y_{i}=-1\sim\mathrm{Beta}(a_{i}^{t}+1,b_{i}^{t})
  32. H t \displaystyle H_{t}
  33. S t S_{t}
  34. H t H_{t}
  35. H t H_{t}
  36. S t S_{t}
  37. t t
  38. h ( x ) = max ( x , 1 - x ) h(x)=\max(x,1-x)
  39. R ( s t , i t , y i t ) \displaystyle R(s^{t},i_{t},y_{i_{t}})
  40. h ( Pr ( a i t t + 1 , b i t t + 1 ) ) - h ( Pr ( a i t t , b i t t ) ) h(\Pr(a_{i_{t}}^{t+1},b_{i_{t}}^{t+1}))-h(\Pr(a_{i_{t}}^{t},b_{i_{t}}^{t}))
  41. i t \displaystyle i_{t}
  42. H t H_{t}
  43. H * H^{*}
  44. i t = argmax i { 1 , , k } ( R + ( S t , i ) ) = max ( R ( S t , i , 1 ) , R ( S t , i , - 1 ) ) i_{t}=\operatorname{argmax}\limits_{i\in\{1,\ldots,k\}}(R^{+}(S^{t},i))=\max(R% (S^{t},i,1),R(S^{t},i,-1))
  45. 1 i k 1\leq i\leq k
  46. θ i ( 0 , 1 ) Bet a ( a i o , b i o ) \theta_{i}\in(0,1)\sim\mathrm{Bet}a(a_{i}^{o},b_{i}^{o})
  47. i i
  48. 1 j M 1\leq j\leq M
  49. ρ j ( 0 , 1 ) Beta ( c j o , d j o ) \rho_{j}\in(0,1)\sim\mathrm{Beta}(c_{j}^{o},d_{j}^{o})
  50. j j
  51. Z i j Z_{ij}
  52. j j
  53. i i
  54. Pr ( Z i j = 1 θ i , ρ j ) = Pr ( Z i j = 1 Y i = 1 ) Pr ( Y i = 1 ) + Pr ( Z i j = 1 Y i = - 1 ) Pr ( Y i = - 1 ) = ρ j θ i t ( 1 - ρ j ) ( 1 - θ i ) \Pr(Z_{ij}=1\mid\theta_{i},\rho_{j})=\Pr(Z_{ij}=1\mid Y_{i}=1)\Pr(Y_{i}=1)+\Pr% (Z_{ij}=1\mid Y_{i}=-1)\Pr(Y_{i}=-1)=\rho_{j}\theta_{i}t(1-\rho_{j})(1-\theta_% {i})
  55. Pr ( Z i j = 1 θ i , ρ j ) = P r ( Z i j = 1 Y i = 1 ) Pr ( Y i = 1 ) + Pr ( Z i j = 1 Y i = - 1 ) Pr ( Y i = - 1 ) = ρ j θ i t ( 1 - ρ j ) ( 1 - θ i ) = ρ j θ i t ( 1 - ρ j ) ( 1 - θ i ) , \Pr(Z_{ij}=1\mid\theta_{i},\rho_{j})=Pr(Z_{ij}=1\mid Y_{i}=1)\Pr(Y_{i}=1)+\Pr(% Z_{ij}=1\mid Y_{i}=-1)\Pr(Y_{i}=-1)=\rho_{j}\theta_{i}t(1-\rho_{j})(1-\theta_{% i})=\rho_{j}\theta_{i}t(1-\rho_{j})(1-\theta_{i}),
  56. ( i , j ) { 1 , 2 , , k } × { 1 , 2 , , M } \qquad\qquad(i,j)\in\{1,2,\ldots,k\}\times\{1,2,\ldots,M\}
  57. Z i j { - 1 , 1 } Z_{ij}\in\{-1,1\}
  58. Pr ( i H S t ) \Pr(i\in H^{\star}\mid S^{t})

Optimization_mechanism.html

  1. C i = m i n j [ δ * d i j + h j ] C_{i}=min_{j}[\delta*d_{ij}+h_{j}]
  2. C i C_{i}
  3. d i j d_{ij}
  4. h j h_{j}
  5. δ \delta
  6. d i j d_{ij}
  7. h j h_{j}
  8. δ \delta
  9. δ < ( 1 / 2 ) 1 / 2 \delta<(1/2)^{1/2}
  10. δ > N 1 / 2 \delta>N^{1/2}
  11. δ \delta
  12. 4 < δ < N 1 / 2 4<\delta<N^{1/2}
  13. δ \delta

OptiSLang.html

  1. C o P = 1 - S S E pred S S T CoP=1-\frac{SS_{E}^{\,\text{pred}}}{SS_{T}}
  2. S S E pred SS_{E}^{\,\text{pred}}
  3. q q
  4. i i
  5. y i y_{i}

Order-4_hexagonal_tiling_honeycomb.html

  1. B V ¯ \overline{BV}
  2. D V ¯ \overline{DV}
  3. B V ¯ \overline{BV}
  4. D V ¯ \overline{DV}
  5. B V ¯ \overline{BV}
  6. D V ¯ \overline{DV}
  7. B V ¯ \overline{BV}
  8. D V ¯ \overline{DV}
  9. B V ¯ \overline{BV}
  10. D V ¯ \overline{DV}
  11. B V ¯ \overline{BV}
  12. D V ¯ \overline{DV}
  13. B V ¯ \overline{BV}
  14. B V ¯ \overline{BV}
  15. D P ¯ 3 {\bar{DP}}_{3}

Order-5_hexagonal_tiling_honeycomb.html

  1. H V ¯ \overline{HV}
  2. V H ¯ 3 {\bar{VH}}_{3}
  3. V H ¯ 3 {\bar{VH}}_{3}
  4. V H ¯ 3 {\bar{VH}}_{3}

Order-6_dodecahedral_honeycomb.html

  1. H V ¯ \overline{HV}
  2. H P ¯ \overline{HP}
  3. V H ¯ 3 {\bar{VH}}_{3}
  4. V H ¯ 3 {\bar{VH}}_{3}

Order-6_hexagonal_tiling_honeycomb.html

  1. Z ¯ \overline{Z}
  2. V P ¯ \overline{VP}
  3. Z ¯ \overline{Z}
  4. D V ¯ \overline{DV}
  5. Z ¯ \overline{Z}
  6. D V ¯ \overline{DV}
  7. Z ¯ \overline{Z}
  8. D V ¯ \overline{DV}
  9. Z ¯ \overline{Z}
  10. D V ¯ \overline{DV}
  11. Z ¯ \overline{Z}
  12. Z ¯ \overline{Z}
  13. Z ¯ \overline{Z}
  14. V ¯ 3 {\bar{V}}_{3}
  15. V ¯ 3 {\bar{V}}_{3}

Order-6_tetrahedral_honeycomb.html

  1. V ¯ 3 {\bar{V}}_{3}
  2. P ¯ 3 {\bar{P}}_{3}
  3. V ¯ 3 {\bar{V}}_{3}
  4. P ¯ 3 {\bar{P}}_{3}
  5. V ¯ 3 {\bar{V}}_{3}
  6. P ¯ 3 {\bar{P}}_{3}
  7. V ¯ 3 {\bar{V}}_{3}
  8. P ¯ 3 {\bar{P}}_{3}
  9. V ¯ 3 {\bar{V}}_{3}
  10. P ¯ 3 {\bar{P}}_{3}

Order_of_a_polynomial.html

  1. P ( x ) = i = 0 n p i x i P(x)=\sum^{n}_{i=0}p_{i}x^{i}
  2. ( p 0 , , p n ) \left(p_{0},\ldots,p_{n}\right)
  3. { 1 , x , , x n } \left\{1,x,\ldots,x^{n}\right\}
  4. ( 1 , 2 , 0 , , 0 ) \left(1,2,0,\ldots,0\right)
  5. P ( x ) P(x)
  6. P ( x ) = 1 + 2 x P(x)=1+2x
  7. n n
  8. { x - 0.5 0.5 - 0 x - 1 0 - 1 ; x - 0 0.5 - 0 x - 1 0.5 - 1 ; x - 0 1 - 0 x - 0.5 1 - 0.5 } \left\{\frac{x-0.5}{0.5-0}\cdot\frac{x-1}{0-1};\frac{x-0}{0.5-0}\cdot\frac{x-1% }{0.5-1};\frac{x-0}{1-0}\cdot\frac{x-0.5}{1-0.5}\right\}
  9. p = { 1 , 1.5 , 2 } p=\left\{1,1.5,2\right\}
  10. 2 2

Ore_algebra.html

  1. K K
  2. A = K [ x 1 , , x s ] A=K[x_{1},\ldots,x_{s}]
  3. A = K A=K
  4. s = 0 s=0
  5. A [ 1 ; σ 1 , δ 1 ] [ r ; σ r , δ r ] A[\partial_{1};\sigma_{1},\delta_{1}]\cdots[\partial_{r};\sigma_{r},\delta_{r}]
  6. σ i \sigma_{i}
  7. δ j \delta_{j}
  8. i j i\neq j
  9. σ i ( j ) = j \sigma_{i}(\partial_{j})=\partial_{j}
  10. δ i ( j ) = 0 \delta_{i}(\partial_{j})=0
  11. i > j i>j

Orientation_of_a_vector_bundle.html

  1. ϕ U : π - 1 ( U ) U × 𝐑 n \phi_{U}:\pi^{-1}(U)\to U\times\mathbf{R}^{n}
  2. u H n ( T ( E ) ; Λ ) u\in H^{n}(T(E);\Lambda)
  3. H ~ * ( T ( E ) ; Λ ) \tilde{H}^{*}(T(E);\Lambda)
  4. H * ( E ; Λ ) H^{*}(E;\Lambda)
  5. H * ( E ; Λ ) H ~ * ( T ( E ) ; Λ ) , x x u H^{*}(E;\Lambda)\to\tilde{H}^{*}(T(E);\Lambda),x\mapsto x\cup u
  6. H * ( π - 1 ( U ) ; Λ ) H ~ * ( T ( E | U ) ; Λ ) H^{*}(\pi^{-1}(U);\Lambda)\to\tilde{H}^{*}(T(E|_{U});\Lambda)
  7. π - 1 ( U ) U × 𝐑 n \pi^{-1}(U)\simeq U\times\mathbf{R}^{n}

Ortelius_oval_projection.html

  1. y = R φ y=R\varphi
  2. x = ± R ( | λ - λ 0 | - F + F 2 - y 2 R 2 ) x=\pm R\left(|\lambda-\lambda_{0}|-F+\sqrt{F^{2}-\frac{y^{2}}{R^{2}}}\right)
  3. F = 1 2 ( π 2 4 | λ - λ 0 | + | λ - λ 0 | ) F=\frac{1}{2}\left(\frac{\pi^{2}}{4|\lambda-\lambda_{0}|}+|\lambda-\lambda_{0}% |\right)
  4. x = ± R ( π 2 4 - φ 2 + | λ - λ 0 | - π 2 ) x=\pm R\left(\sqrt{\frac{\pi^{2}}{4}-\varphi^{2}}+|\lambda-\lambda_{0}|-\frac{% \pi}{2}\right)

Orthocentroidal_circle.html

  1. D 2 - 4 9 ( a 2 + b 2 + c 2 ) , D^{2}-\tfrac{4}{9}(a^{2}+b^{2}+c^{2}),

Ostrogradsky_instability.html

  1. L ( q , q ˙ , q ¨ ) L(q,{\dot{q}},{\ddot{q}})
  2. d L d q - d d t d L d q ˙ + d 2 d t 2 d L d q ¨ = 0. \frac{dL}{dq}-\frac{d}{dt}\frac{dL}{d{\dot{q}}}+\frac{d^{2}}{dt^{2}}\frac{dL}{% d{\ddot{q}}}=0.
  3. L L
  4. q {q}
  5. d L / d q ¨ dL/d{\ddot{q}}
  6. q ¨ {\ddot{q}}
  7. det [ d 2 L / ( d q ¨ i d q ¨ j ) ] \det[d^{2}L/(d{\ddot{q}_{i}}\,d{\ddot{q}}_{j})]
  8. L L
  9. q ( 4 ) = F ( q , q ˙ , q ¨ , q ( 3 ) ) q^{(4)}=F(q,{\dot{q}},{\ddot{q}},q^{(3)})
  10. q = G ( t , q 0 , q ˙ 0 , q ¨ 0 , q 0 ( 3 ) ) q=G(t,q_{0},{\dot{q}}_{0},{\ddot{q}}_{0},q^{(3)}_{0})
  11. q q
  12. Q 1 := q Q_{1}:=q
  13. Q 2 := q ˙ Q_{2}:={\dot{q}}
  14. P 1 := d L d q ˙ - d d t d L d q ¨ P_{1}:=\frac{dL}{d{\dot{q}}}-\frac{d}{dt}\frac{dL}{d{\ddot{q}}}
  15. P 2 := d L d q ¨ P_{2}:=\frac{dL}{d{\ddot{q}}}
  16. q ¨ {\ddot{q}}
  17. q ¨ = a ( Q 1 , Q 2 , P 2 ) {\ddot{q}}=a(Q_{1},Q_{2},P_{2})
  18. H = P 1 Q 2 - P 2 a ( Q 1 , Q 2 , P 2 ) H=P_{1}Q_{2}-P_{2}a(Q_{1},Q_{2},P_{2})
  19. P 1 P_{1}

Outermorphism.html

  1. 𝖿 ¯ ( x ) = f ( x ) \underline{\mathsf{f}}(x)=f(x)
  2. 𝖿 ¯ ( A B ) = 𝖿 ¯ ( A ) 𝖿 ¯ ( B ) \underline{\mathsf{f}}(A\wedge B)=\underline{\mathsf{f}}(A)\wedge\underline{% \mathsf{f}}(B)
  3. 𝖿 ¯ ( A + B ) = 𝖿 ¯ ( A ) + 𝖿 ¯ ( B ) \underline{\mathsf{f}}(A+B)=\underline{\mathsf{f}}(A)+\underline{\mathsf{f}}(B)
  4. 𝖿 ¯ ( 1 ) = 1 \underline{\mathsf{f}}(1)=1
  5. 𝖿 ¯ ( α x z + β y z ) \displaystyle\underline{\mathsf{f}}(\alpha x\wedge z+\beta y\wedge z)
  6. 𝖿 ¯ \underline{\mathsf{f}}
  7. 𝖿 ¯ \overline{\mathsf{f}}
  8. b 𝖿 ¯ ( a ) = a 𝖿 ¯ ( b ) . b\cdot\overline{\mathsf{f}}(a)=a\cdot\underline{\mathsf{f}}(b).
  9. 𝖿 ¯ ( a ) = b a 𝖿 ¯ ( b ) \overline{\mathsf{f}}(a)=\nabla_{b}\left\langle a\underline{\mathsf{f}}(b)\right\rangle
  10. f ¯ ( A r ) = f ¯ ( A ) r \underline{f}(\left\langle A\right\rangle_{r})=\left\langle\underline{f}(A)% \right\rangle_{r}
  11. \langle
  12. \rangle
  13. x = 1 x x=1\wedge x
  14. 𝖿 ( 1 ) = 1 \mathsf{f}(1)=1
  15. 𝖿 ( I ) I \mathsf{f}(I)\propto I
  16. det 𝖿 = 𝖿 ¯ ( I ) I - 1 \det\mathsf{f}=\underline{\mathsf{f}}(I)I^{-1}
  17. det ( 𝖿 𝗀 ) = det 𝖿 det 𝗀 \det(\mathsf{f}\circ\mathsf{g})=\det\mathsf{f}\det\mathsf{g}
  18. 𝖿 ¯ - 1 ( X ) = 𝖿 ¯ ( X I ) I - 1 det 𝖿 = 𝖿 ¯ ( X I ) [ 𝖿 ¯ ( I ) ] - 1 , \underline{\mathsf{f}}^{-1}(X)=\frac{\overline{\mathsf{f}}(XI)I^{-1}}{\det% \mathsf{f}}=\overline{\mathsf{f}}(XI)[\overline{\mathsf{f}}(I)]^{-1},
  19. 𝖿 ¯ - 1 ( X ) = I - 1 𝖿 ¯ ( I X ) det 𝖿 = [ 𝖿 ¯ ( I ) ] - 1 𝖿 ¯ ( I X ) . \overline{\mathsf{f}}^{-1}(X)=\frac{I^{-1}\underline{\mathsf{f}}(IX)}{\det% \mathsf{f}}=[\underline{\mathsf{f}}(I)]^{-1}\underline{\mathsf{f}}(IX).
  20. 𝖿 ¯ ( B ) = λ B , \underline{\mathsf{f}}(B)=\lambda B,
  21. f ( x ) = R x R f(x)=RxR^{\dagger}\,
  22. 𝖿 ¯ ( X ) = R X R . \underline{\mathsf{f}}(X)=RXR^{\dagger}.
  23. x y = ( R x R ) ( R y R ) x\cdot y=(RxR^{\dagger})\cdot(RyR^{\dagger})
  24. 𝖿 ¯ ( x y ) \displaystyle\underline{\mathsf{f}}(x\wedge y)
  25. ( x y ) 1 = 0 \langle(x\wedge y)\rangle_{1}=0
  26. x 1 y 1 = x y \langle x\rangle_{1}\wedge\langle y\rangle_{1}=x\wedge y

Oviaivo's_polyhedra_:_annoviaivo.html

  1. \color B l u e R {\color{Blue}R}
  2. 4 sin 4 ( π n ) \color B l u e R 4 - 4 sin 3 ( π n ) \color B l u e R 3 + ( 4 cos ( π n ) - 7 ) sin 2 ( π n ) \color B l u e R 2 + 2 ( 1 + cos ( π n ) ) sin ( π n ) \color B l u e R - 3 cos 2 ( π n ) + 2 = 0 {4}\sin^{4}\!\left(\frac{\pi}{n}\right)\!{\color{Blue}R}^{4}-{4}\sin^{3}\!% \left(\frac{\pi}{n}\right)\!{\color{Blue}R}^{3}+({4}\cos\!\left(\frac{\pi}{n}% \right)-7)\sin^{2}\!\left(\frac{\pi}{n}\right)\!{\color{Blue}R}^{2}+{2}({1}+% \cos\!\left(\frac{\pi}{n}\right))\sin\!\left(\frac{\pi}{n}\right)\!{\color{% Blue}R}-{3}\cos^{2}\!\left(\frac{\pi}{n}\right)+2=0
  3. h h
  4. h = 3 4 - ( \color B l u e R - 1 2 cot ( π n ) ) 2 + 1 - ( \color B l u e R - 1 2 csc ( π n ) ) 2 h=\sqrt{\frac{3}{4}-\left(\!{\color{Blue}R}-\frac{1}{2}\cot\!\left(\frac{\pi}{% n}\right)\!\right)^{2}}+\sqrt{{1}-\left(\!{\color{Blue}R}-\frac{1}{2}\csc\!% \left(\frac{\pi}{n}\right)\!\right)^{2}}
  5. n n
  6. 3 n 9 3\leq n\leq 9
  7. h = 3 4 - ( \color B l u e R - 1 2 cot ( π n ) ) 2 - 1 - ( \color B l u e R - 1 2 csc ( π n ) ) 2 h=\sqrt{\frac{3}{4}-\left(\!{\color{Blue}R}-\frac{1}{2}\cot\!\left(\frac{\pi}{% n}\right)\!\right)^{2}}-\sqrt{{1}-\left(\!{\color{Blue}R}-\frac{1}{2}\csc\!% \left(\frac{\pi}{n}\right)\!\right)^{2}}
  8. n n
  9. 9 n 9\leq n
  10. h h
  11. H H
  12. h H = \color B l u e R tan ( π 2 n ) hH={\color{Blue}R}\tan\!\left(\frac{\pi}{2n}\right)\!
  13. χ \chi
  14. χ = V - E + F \chi=V-E+F\,\!
  15. χ = V - E + F = 0. \chi=V-E+F=0.\,\!
  16. V = 4 n V=4n
  17. E = 12 n E=12n
  18. F = 8 n F=8n

P-FEM.html

  1. p min p_{\min}
  2. h max h_{\max}

P-group_generation_algorithm.html

  1. p n p^{n}
  2. p p
  3. n 0 n\geq 0
  4. G G
  5. G G
  6. ( P j ( G ) ) j 0 (P_{j}(G))_{j\geq 0}
  7. G G
  8. P 0 ( G ) := G P_{0}(G):=G
  9. P j ( G ) := [ P j - 1 ( G ) , G ] P j - 1 ( G ) p P_{j}(G):=[P_{j-1}(G),G]\cdot P_{j-1}(G)^{p}
  10. j 1 j\geq 1
  11. G > 1 G>1
  12. c 1 c\geq 1
  13. P c - 1 ( G ) > P c ( G ) = 1 P_{c-1}(G)>P_{c}(G)=1
  14. cl p ( G ) := c \mathrm{cl}_{p}(G):=c
  15. G G
  16. 1 1
  17. cl p ( 1 ) = 0 \mathrm{cl}_{p}(1)=0
  18. G G
  19. cl p ( G ) := min { c 0 P c ( G ) = 1 } \mathrm{cl}_{p}(G):=\min\{c\geq 0\mid P_{c}(G)=1\}
  20. G = P 0 ( G ) > Φ ( G ) = P 1 ( G ) > P 2 ( G ) > > P c - 1 ( G ) > P c ( G ) = 1 G=P_{0}(G)>\Phi(G)=P_{1}(G)>P_{2}(G)>\cdots>P_{c-1}(G)>P_{c}(G)=1
  21. P 1 ( G ) = [ P 0 ( G ) , G ] P 0 ( G ) p = [ G , G ] G p = Φ ( G ) P_{1}(G)=[P_{0}(G),G]\cdot P_{0}(G)^{p}=[G,G]\cdot G^{p}=\Phi(G)
  22. G G
  23. G G
  24. ( γ j ( G ) ) j 1 (\gamma_{j}(G))_{j\geq 1}
  25. G G
  26. γ 1 ( G ) := G \gamma_{1}(G):=G
  27. γ j ( G ) := [ γ j - 1 ( G ) , G ] \gamma_{j}(G):=[\gamma_{j-1}(G),G]
  28. j 2 j\geq 2
  29. G > 1 G>1
  30. c 1 c\geq 1
  31. γ c ( G ) > γ c + 1 ( G ) = 1 \gamma_{c}(G)>\gamma_{c+1}(G)=1
  32. cl p ( G ) := c \mathrm{cl}_{p}(G):=c
  33. G G
  34. c + 1 c+1
  35. G G
  36. 1 1
  37. cl ( 1 ) = 0 \mathrm{cl}(1)=0
  38. G = γ 1 ( G ) > G = γ 2 ( G ) > γ 3 ( G ) > > γ c ( G ) > γ c + 1 ( G ) = 1 G=\gamma_{1}(G)>G^{\prime}=\gamma_{2}(G)>\gamma_{3}(G)>\cdots>\gamma_{c}(G)>% \gamma_{c+1}(G)=1
  39. γ 2 ( G ) = [ γ 1 ( G ) , G ] = [ G , G ] = G \gamma_{2}(G)=[\gamma_{1}(G),G]=[G,G]=G^{\prime}
  40. G G
  41. G G
  42. cl ( G ) cl p ( G ) \mathrm{cl}(G)\leq\mathrm{cl}_{p}(G)
  43. γ j ( G ) \gamma_{j}(G)
  44. P j ( G ) P_{j}(G)
  45. ϑ Hom ( G , G ~ ) \vartheta\in\mathrm{Hom}(G,\tilde{G})
  46. G ~ \tilde{G}
  47. \Rightarrow
  48. ϑ ( P j ( G ) ) = P j ( ϑ ( G ) ) \vartheta(P_{j}(G))=P_{j}(\vartheta(G))
  49. j 0 j\geq 0
  50. c 0 c\geq 0
  51. N G N\triangleleft G
  52. cl p ( G / N ) = c \mathrm{cl}_{p}(G/N)=c
  53. P c ( G ) N P_{c}(G)\leq N
  54. c 0 c\geq 0
  55. cl p ( G ) = c \mathrm{cl}_{p}(G)=c
  56. \Rightarrow
  57. cl p ( G / P k ( G ) ) = min ( k , c ) \mathrm{cl}_{p}(G/P_{k}(G))=\min(k,c)
  58. k 0 k\geq 0
  59. cl p ( G / P k ( G ) ) = k \mathrm{cl}_{p}(G/P_{k}(G))=k
  60. 0 k c 0\leq k\leq c
  61. π ( G ) \pi(G)
  62. G > 1 G>1
  63. cl p ( G ) = c 1 \mathrm{cl}_{p}(G)=c\geq 1
  64. π ( G ) := G / P c - 1 ( G ) \pi(G):=G/P_{c-1}(G)
  65. G G
  66. P c - 1 ( G ) > 1 P_{c-1}(G)>1
  67. G G
  68. G G
  69. π ( G ) \pi(G)
  70. cl p ( G ) = cl p ( π ( G ) ) + 1 \mathrm{cl}_{p}(G)=\mathrm{cl}_{p}(\pi(G))+1
  71. π ( G ) \pi(G)
  72. G G
  73. G π ( G ) G\to\pi(G)
  74. π : G π ( G ) \pi:G\to\pi(G)
  75. π ( G ) = G / P c - 1 ( G ) \pi(G)=G/P_{c-1}(G)
  76. R R
  77. P P
  78. P P
  79. R R
  80. R R
  81. P P
  82. R = Q 0 Q 1 Q m - 1 Q m = P R=Q_{0}\to Q_{1}\to\cdots\to Q_{m-1}\to Q_{m}=P
  83. m 1 m\geq 1
  84. R R
  85. P P
  86. Q j = π j ( R ) Q_{j}=\pi^{j}(R)
  87. R R
  88. 0 j m 0\leq j\leq m
  89. Q j = R / P c - j ( R ) Q_{j}=R/P_{c-j}(R)
  90. c - j c-j
  91. R R
  92. R R
  93. c m c\geq m
  94. G > 1 G>1
  95. c c
  96. G = G / 1 = G / P c ( G ) π ( G ) = G / P c - 1 ( G ) π 2 ( G ) = G / P c - 2 ( G ) G=G/1=G/P_{c}(G)\to\pi(G)=G/P_{c-1}(G)\to\pi^{2}(G)=G/P_{c-2}(G)\to\cdots
  97. π c - 1 ( G ) = G / P 1 ( G ) π c ( G ) = G / P 0 ( G ) = G / G = 1 \to\pi^{c-1}(G)=G/P_{1}(G)\to\pi^{c}(G)=G/P_{0}(G)=G/G=1
  98. π c ( G ) = 1 \pi^{c}(G)=1
  99. G G
  100. π c - 1 ( G ) = G / P 1 ( G ) C p d \pi^{c-1}(G)=G/P_{1}(G)\simeq C_{p}^{d}
  101. d = d ( G ) d=d(G)
  102. d ( G ) = dim 𝔽 p ( H 1 ( G , 𝔽 p ) ) d(G)=\dim_{\mathbb{F}_{p}}(H^{1}(G,\mathbb{F}_{p}))
  103. G G
  104. 𝒯 ( G ) \mathcal{T}(G)
  105. G G
  106. G G
  107. G G
  108. 𝒯 ( 1 ) \mathcal{T}(1)
  109. 1 1
  110. 1 1
  111. d 1 d\geq 1
  112. p p
  113. G G
  114. d d
  115. G G
  116. G G^{\ast}
  117. G G
  118. G G
  119. G G
  120. 1 R F G 1 1\longrightarrow R\longrightarrow F\longrightarrow G\longrightarrow 1
  121. F F
  122. d d
  123. ϑ : F G \vartheta:\ F\longrightarrow G
  124. R := ker ( ϑ ) R:=\ker(\vartheta)
  125. R F R\triangleleft F
  126. F F
  127. G = F / R G=F/R
  128. r R r\in R
  129. f F f\in F
  130. f - 1 r f R f^{-1}rf\in R
  131. [ r , f ] = r - 1 f - 1 r f R [r,f]=r^{-1}f^{-1}rf\in R
  132. R R
  133. R := [ R , F ] R p R^{\ast}:=[R,F]\cdot R^{p}
  134. R R
  135. R / R R/R^{\ast}
  136. G G
  137. [ R , R ] R p [ R , F ] R p = R [R,R]\cdot R^{p}\leq[R,F]\cdot R^{p}=R^{\ast}
  138. G G
  139. G := F / R G^{\ast}:=F/R^{\ast}
  140. 1 R / R F / R F / R 1 1\longrightarrow R/R^{\ast}\longrightarrow F/R^{\ast}\longrightarrow F/R\longrightarrow 1
  141. G G^{\ast}
  142. G G
  143. μ ( G ) := dim 𝔽 p ( R / R ) \mu(G):=\dim_{\mathbb{F}_{p}}(R/R^{\ast})
  144. G G
  145. G = F / R G=F/R
  146. cl p ( G ) = c \mathrm{cl}_{p}(G)=c
  147. R F R\triangleleft F
  148. cl p ( F / R ) = c \mathrm{cl}_{p}(F/R)=c
  149. P c ( F ) R P_{c}(F)\leq R
  150. G G
  151. P c ( G ) = P c ( F ) R / R R / R P_{c}(G^{\ast})=P_{c}(F)\cdot R^{\ast}/R^{\ast}\leq R/R^{\ast}
  152. ν ( G ) := dim 𝔽 p ( P c ( G ) ) μ ( G ) \nu(G):=\dim_{\mathbb{F}_{p}}(P_{c}(G^{\ast}))\leq\mu(G)
  153. G G
  154. G G
  155. d d
  156. 1 Z H G 1 1\to Z\to H\to G\to 1
  157. G G
  158. Z ζ 1 ( H ) Z\leq\zeta_{1}(H)
  159. d ( H ) = d ( G ) = d d(H)=d(G)=d
  160. G G^{\ast}
  161. G G
  162. ψ : F H \psi:\ F\to H
  163. ϑ = ω ψ \vartheta=\omega\circ\psi
  164. ω : H G = H / Z \omega:\ H\to G=H/Z
  165. R = ker ( ϑ ) = ker ( ω ψ ) = ψ - 1 ( Z ) R=\ker(\vartheta)=\ker(\omega\circ\psi)=\psi^{-1}(Z)
  166. ψ ( R ) = ψ ( ψ - 1 ( Z ) ) = Z \psi(R)=\psi(\psi^{-1}(Z))=Z
  167. ψ ( R p ) Z p = 1 \psi(R^{p})\leq Z^{p}=1
  168. Z Z
  169. ψ ( [ R , F ] ) [ Z , Z ] = 1 \psi([R,F])\leq[Z,Z]=1
  170. Z Z
  171. ψ ( R ) = ψ ( [ R , F ] R p ) = 1 \psi(R^{\ast})=\psi([R,F]\cdot R^{p})=1
  172. ψ \psi
  173. ψ : G H \psi^{\ast}:\ G^{\ast}\to H
  174. H G / ker ( ψ ) H\simeq G^{\ast}/\ker(\psi^{\ast})
  175. H H
  176. G G
  177. 1 P c - 1 ( H ) H G 1 1\to P_{c-1}(H)\to H\to G\to 1
  178. G G
  179. 1 = P c ( H ) = [ P c - 1 ( H ) , H ] P c - 1 ( H ) p 1=P_{c}(H)=[P_{c-1}(H),H]\cdot P_{c-1}(H)^{p}
  180. P c - 1 ( H ) p = 1 P_{c-1}(H)^{p}=1
  181. P c - 1 ( H ) ζ 1 ( H ) P_{c-1}(H)\leq\zeta_{1}(H)
  182. c = cl p ( H ) c=\mathrm{cl}_{p}(H)
  183. M / R R / R M/R^{\ast}\leq R/R^{\ast}
  184. G G
  185. M / R = ker ( ψ ) M/R^{\ast}=\ker(\psi^{\ast})
  186. ψ : G H \psi^{\ast}:\ G^{\ast}\to H
  187. H H
  188. G G
  189. 1 < M / R < R / R 1<M/R^{\ast}<R/R^{\ast}
  190. ( M / R ) ( P c ( F ) R / R ) = R / R (M/R^{\ast})\cdot(P_{c}(F)\cdot R^{\ast}/R^{\ast})=R/R^{\ast}
  191. G G
  192. R / R R/R^{\ast}
  193. P c ( G ) = P c ( F ) R / R P_{c}(G^{\ast})=P_{c}(F)\cdot R^{\ast}/R^{\ast}
  194. c = cl p ( G ) c=\mathrm{cl}_{p}(G)
  195. { G / ( M / R ) = ( F / R ) / ( M / R ) F / M M / R is allowable } \{G^{\ast}/(M/R^{\ast})=(F/R^{\ast})/(M/R^{\ast})\simeq F/M\mid M/R^{\ast}\,% \text{ is allowable }\}
  196. F / M 1 F / M 2 F/M_{1}\simeq F/M_{2}
  197. M 1 / R M_{1}/R^{\ast}
  198. M 2 / R M_{2}/R^{\ast}
  199. F / M 1 F / M 2 F/M_{1}\simeq F/M_{2}
  200. G G
  201. φ : F / M 1 F / M 2 \varphi:\ F/M_{1}\to F/M_{2}
  202. G = F / R G=F/R
  203. c = cl p ( G ) c=\mathrm{cl}_{p}(G)
  204. φ ( R / M 1 ) = φ ( P c ( F / M 1 ) ) = P c ( φ ( F / M 1 ) ) = P c ( F / M 2 ) = R / M 2 \varphi(R/M_{1})=\varphi(P_{c}(F/M_{1}))=P_{c}(\varphi(F/M_{1}))=P_{c}(F/M_{2}% )=R/M_{2}
  205. α Aut ( G ) \alpha\in\mathrm{Aut}(G)
  206. G G
  207. α Aut ( G ) \alpha^{\ast}\in\mathrm{Aut}(G^{\ast})
  208. G = F / R G^{\ast}=F/R^{\ast}
  209. G G
  210. α \alpha^{\ast}
  211. R / R R/R^{\ast}
  212. G G
  213. α \alpha
  214. α ( M / R ) P c ( F / R ) = α [ M / R P c ( F / R ) ] = α ( R / R ) = R / R \alpha^{\ast}(M/R^{\ast})\cdot P_{c}(F/R^{\ast})=\alpha^{\ast}[M/R^{\ast}\cdot P% _{c}(F/R^{\ast})]=\alpha^{\ast}(R/R^{\ast})=R/R^{\ast}
  215. α Aut ( G ) \alpha^{\ast}\in\mathrm{Aut}(G^{\ast})
  216. α \alpha^{\prime}
  217. M / R R / R M/R^{\ast}\leq R/R^{\ast}
  218. P := α α Aut ( G ) P:=\langle\alpha^{\prime}\mid\alpha\in\mathrm{Aut}(G)\rangle
  219. G G
  220. Aut ( G ) P \mathrm{Aut}(G)\to P
  221. α α \alpha\mapsto\alpha^{\prime}
  222. M / R R / R M/R^{\ast}\leq R/R^{\ast}
  223. P P
  224. { F / M i 1 i N } \{F/M_{i}\mid 1\leq i\leq N\}
  225. G G
  226. M i / R M_{i}/R^{\ast}
  227. N N
  228. R / R R/R^{\ast}
  229. P P
  230. G G
  231. ν ( G ) \nu(G)
  232. G G
  233. G G
  234. G G
  235. ν ( G ) = 0 \nu(G)=0
  236. G G
  237. ν ( G ) 1 \nu(G)\geq 1
  238. G = F / R G=F/R
  239. ν = ν ( G ) \nu=\nu(G)
  240. 1 s ν 1\leq s\leq\nu
  241. ( R / R : M / R ) = p s (R/R^{\ast}:M/R^{\ast})=p^{s}
  242. M / R M/R^{\ast}
  243. R / R R/R^{\ast}
  244. G G
  245. | G | = p n |G|=p^{n}
  246. s s
  247. # ( F / M ) = ( F / R : M / R ) = ( F / R : R / R ) ( R / R : M / R ) \#(F/M)=(F/R^{\ast}:M/R^{\ast})=(F/R^{\ast}:R/R^{\ast})\cdot(R/R^{\ast}:M/R^{% \ast})
  248. = # ( F / R ) p s = | G | p s = p n p s = p n + s =\#(F/R)\cdot p^{s}=|G|\cdot p^{s}=p^{n}\cdot p^{s}=p^{n+s}
  249. G G
  250. ν ( G ) 2 \nu(G)\geq 2
  251. 1 s ν 1\leq s\leq\nu
  252. s s
  253. G G
  254. N N
  255. N s N_{s}
  256. N = s = 1 ν N s N=\sum_{s=1}^{\nu}\,N_{s}
  257. 0 C s N s 0\leq C_{s}\leq N_{s}
  258. ( N 1 / C 1 ; ; N ν / C ν ) (N_{1}/C_{1};\ldots;N_{\nu}/C_{\nu})
  259. p = 3 p=3
  260. ( 3 , 3 ) (3,3)
  261. 27 , 3 \langle 27,3\rangle
  262. 1 1
  263. ν = 2 \nu=2
  264. μ = 4 \mu=4
  265. ( 4 / 1 ; 7 / 5 ) (4/1;7/5)
  266. N = 11 N=11
  267. 243 , 3 = 27 , 3 - # 2 ; 1 \langle 243,3\rangle=\langle 27,3\rangle-\#2;1
  268. 2 2
  269. ν = 2 \nu=2
  270. μ = 4 \mu=4
  271. ( 10 / 6 ; 15 / 15 ) (10/6;15/15)
  272. N = 25 N=25
  273. 729 , 40 = 243 , 3 - # 1 ; 7 \langle 729,40\rangle=\langle 243,3\rangle-\#1;7
  274. ν = 2 \nu=2
  275. μ = 5 \mu=5
  276. ( 16 / 2 ; 27 / 4 ) (16/2;27/4)
  277. N = 43 N=43
  278. ( 3 , 3 , 3 ) (3,3,3)
  279. 81 , 12 \langle 81,12\rangle
  280. 2 2
  281. ν = 2 \nu=2
  282. μ = 7 \mu=7
  283. ( 10 / 2 ; 100 / 50 ) (10/2;100/50)
  284. N = 110 N=110
  285. 243 , 37 \langle 243,37\rangle
  286. 3 3
  287. ν = 5 \nu=5
  288. μ = 9 \mu=9
  289. ( 35 / 3 ; 2783 / 186 ; 81711 / 10202 ; 350652 / 202266 ; ) (35/3;2783/186;81711/10202;350652/202266;\ldots)
  290. N > 4 10 5 N>4\cdot 10^{5}
  291. 729 , 122 \langle 729,122\rangle
  292. 4 4
  293. ν = 8 \nu=8
  294. μ = 11 \mu=11
  295. ( 45 / 3 ; 117919 / 1377 ; ) (45/3;117919/1377;\ldots)
  296. N > 10 5 N>10^{5}
  297. p = 5 p=5
  298. ( 5 , 5 ) (5,5)
  299. p = 3 p=3
  300. 125 , 3 \langle 125,3\rangle
  301. 1 1
  302. ν = 2 \nu=2
  303. μ = 4 \mu=4
  304. ( 4 / 1 ; 12 / 6 ) (4/1;12/6)
  305. N = 16 N=16
  306. 3125 , 3 = 125 , 3 - # 2 ; 1 \langle 3125,3\rangle=\langle 125,3\rangle-\#2;1
  307. 2 2
  308. ν = 3 \nu=3
  309. μ = 5 \mu=5
  310. ( 8 / 3 ; 61 / 61 ; 47 / 47 ) (8/3;61/61;47/47)
  311. N = 116 N=116
  312. / μ \mathbb{Q}/\mathbb{Z}\to\mu_{\infty}
  313. n d exp ( n d 2 π i ) \frac{n}{d}\mapsto\exp(\frac{n}{d}\cdot 2\pi i)
  314. / = { n d d 1 , 0 n d - 1 } \mathbb{Q}/\mathbb{Z}=\{\frac{n}{d}\cdot\mathbb{Z}\mid d\geq 1,\ 0\leq n\leq d% -1\}
  315. μ = { z z d = 1 for some integer d 1 } \mu_{\infty}=\{z\in\mathbb{C}\mid z^{d}=1\,\text{ for some integer }d\geq 1\}
  316. p p
  317. G G
  318. G = F / R G=F/R
  319. M ( G ) := H 2 ( G , / ) M(G):=H^{2}(G,\mathbb{Q}/\mathbb{Z})
  320. G G
  321. / \mathbb{Q}/\mathbb{Z}
  322. G G
  323. M ( G ) = ( R [ F , F ] ) / [ F , R ] M(G)=(R\cap[F,F])/[F,R]
  324. r ( G ) = dim 𝔽 p ( H 2 ( G , 𝔽 p ) ) r(G)=\dim_{\mathbb{F}_{p}}(H^{2}(G,\mathbb{F}_{p}))
  325. G G
  326. d ( G ) = dim 𝔽 p ( H 1 ( G , 𝔽 p ) ) d(G)=\dim_{\mathbb{F}_{p}}(H^{1}(G,\mathbb{F}_{p}))
  327. G G
  328. G G
  329. r ( G ) - d ( G ) = d ( M ( G ) ) r(G)-d(G)=d(M(G))
  330. μ ( G j ) - ν ( G j ) r ( G ) \mu(G_{j})-\nu(G_{j})\leq r(G)
  331. G j := G / P j ( G ) G_{j}:=G/P_{j}(G)
  332. cl p ( G j ) = j \mathrm{cl}_{p}(G_{j})=j
  333. j 0 j\geq 0
  334. G G
  335. G / G G/G^{\prime}
  336. G G
  337. M ( G ) M(G)
  338. 𝒯 ( 1 ) \mathcal{T}(1)
  339. 1 1
  340. M ( G ) = 1 M(G)=1
  341. \Rightarrow
  342. ν ( G ) = 0 \nu(G)=0
  343. G G
  344. r ( G ) = d ( G ) r(G)=d(G)
  345. r ( G ) - d ( G ) = 0 = d ( M ( G ) ) r(G)-d(G)=0=d(M(G))
  346. M ( G ) = 1 M(G)=1
  347. 𝒯 ( 1 ) \mathcal{T}(1)
  348. G G
  349. r ( G ) = d ( G ) + 1 r(G)=d(G)+1
  350. r ( G ) - d ( G ) = 1 = d ( M ( G ) ) r(G)-d(G)=1=d(M(G))
  351. M ( G ) M(G)

Packing_density.html

  1. X X
  2. X X
  3. η = i = 1 n μ ( K i ) μ ( X ) \eta=\frac{\sum_{i=1}^{n}\mu(K_{i})}{\mu(X)}
  4. t t
  5. η = lim t i = 1 μ ( K i B t ) μ ( B t ) \eta=\lim_{t\to\infty}\frac{\sum_{i=1}^{\infty}\mu(K_{i}\cap B_{t})}{\mu(B_{t})}
  6. μ ( K < s u b > i B t ) μ(K<sub>i∩B_{t})

Painlevé_conjecture.html

  1. ( 𝐪 , 𝐩 ) (\mathbf{q},\mathbf{p})
  2. 𝐪 ˙ = M - 1 𝐩 , 𝐩 ˙ = U ( 𝐪 ) \dot{\mathbf{q}}=M^{-1}\mathbf{p},\dot{\mathbf{p}}=\nabla U(\mathbf{q})
  3. t n t_{n}
  4. t * t^{*}
  5. U ( 𝐪 ( t n ) ) \nabla U(\mathbf{q}(t_{n}))\rightarrow\infty
  6. 𝐪 ( t ) \mathbf{q}(t)
  7. t t * , t < t * t\rightarrow t^{*},t<t^{*}
  8. J ( 𝐪 ( t ) ) J(\mathbf{q}(t))
  9. t t * t\rightarrow t^{*}
  10. J ( 𝐪 ) = i m i | 𝐪 i | 2 J(\mathbf{q})=\sum_{i}m_{i}|\mathbf{q}_{i}|^{2}
  11. 𝐪 \mathbf{q}

Palatini_identity.html

  1. δ R μ ν = ( δ Γ λ ) μ ν ; λ - ( δ Γ λ ) μ λ ; ν \delta R_{\mu\nu}{}=(\delta\Gamma^{\lambda}{}_{\mu\nu})_{;\lambda}-(\delta% \Gamma^{\lambda}{}_{\mu\lambda})_{;\nu}
  2. δ Γ λ μ ν \delta\Gamma^{\lambda}{}_{\mu\nu}

Pappus'_area_theorem.html

  1. A A B D E + A A C F G = A B C L M A_{ABDE}+A_{ACFG}=A_{BCLM}
  2. A A B D E + A A C F G \displaystyle A_{ABDE}+A_{ACFG}

Paracompact_uniform_honeycombs.html

  1. R ¯ 3 {\bar{R}}_{3}
  2. N ¯ 3 {\bar{N}}_{3}
  3. V ¯ 3 {\bar{V}}_{3}
  4. B V ¯ 3 {\bar{BV}}_{3}
  5. H V ¯ 3 {\bar{HV}}_{3}
  6. Y ¯ 3 {\bar{Y}}_{3}
  7. Z ¯ 3 {\bar{Z}}_{3}
  8. D V ¯ 3 {\bar{DV}}_{3}
  9. O ¯ 3 {\bar{O}}_{3}
  10. M ¯ 3 {\bar{M}}_{3}
  11. C R ^ 3 {\widehat{CR}}_{3}
  12. R R ^ 3 {\widehat{RR}}_{3}
  13. A V ^ 3 {\widehat{AV}}_{3}
  14. B V ^ 3 {\widehat{BV}}_{3}
  15. H V ^ 3 {\widehat{HV}}_{3}
  16. V V ^ 3 {\widehat{VV}}_{3}
  17. B R ^ 3 {\widehat{BR}}_{3}
  18. D P ¯ 3 {\bar{DP}}_{3}
  19. P P ¯ 3 {\bar{PP}}_{3}
  20. P ¯ 3 {\bar{P}}_{3}
  21. B P ¯ 3 {\bar{BP}}_{3}
  22. H P ¯ 3 {\bar{HP}}_{3}
  23. V P ¯ 3 {\bar{VP}}_{3}

Paradox_of_a_charge_in_a_gravitational_field.html

  1. 1 / r 1/r
  2. 1 / r 2 1/r^{2}
  3. c 2 d τ 2 = u 2 ( z ) c 2 d t 2 - ( c 2 g d u d z ) 2 d z 2 - d x 2 - d y 2 c^{2}d\tau^{2}=u^{2}(z)c^{2}dt^{2}-\left({c^{2}\over g}{du\over dz}\right)^{2}% dz^{2}-dx^{2}-dy^{2}
  4. c c
  5. τ \tau
  6. x , y , z , t x,y,z,t
  7. g g
  8. u ( z ) u(z)
  9. 1 + g z / c 2 1+gz/c^{2}
  10. c 2 d τ 2 = c 2 d t 2 - d z 2 - d y 2 - d z 2 c^{2}d\tau^{2}=c^{2}dt^{\prime 2}-dz^{\prime 2}-dy^{\prime 2}-dz^{\prime 2}
  11. x = x \displaystyle x^{\prime}=x
  12. e e
  13. g g
  14. R = 2 3 e 2 c 3 g 2 R={2\over 3}{e^{2}\over c^{3}}g^{2}
  15. 1 / r 2 1/r^{2}

Parametric_programming.html

  1. J * ( θ ) = \displaystyle J^{*}(\theta)=
  2. x x
  3. θ \theta
  4. f ( x , θ ) f(x,\theta)
  5. g ( x , θ ) g(x,\theta)
  6. Θ \Theta
  7. f ( x , θ ) f(x,\theta)
  8. g ( x , θ ) g(x,\theta)
  9. m > 1 m>1

Paramodular_group.html

  1. ( 0 F - F 0 ) \begin{pmatrix}0&F\\ -F&0\end{pmatrix}
  2. ( A B C D ) \begin{pmatrix}A&B\\ C&D\end{pmatrix}
  3. ( A B C D ) t ( 0 F - F 0 ) ( A B C D ) = ( 0 F - F 0 ) . \begin{pmatrix}A&B\\ C&D\end{pmatrix}^{t}\begin{pmatrix}0&F\\ -F&0\end{pmatrix}\begin{pmatrix}A&B\\ C&D\end{pmatrix}=\begin{pmatrix}0&F\\ -F&0\end{pmatrix}.
  4. ( I 0 0 F ) \begin{pmatrix}I&0\\ 0&F\end{pmatrix}
  5. ( I 0 0 F ) t ( 0 I - I 0 ) ( I 0 0 F ) = ( 0 F - F 0 ) \begin{pmatrix}I&0\\ 0&F\end{pmatrix}^{t}\begin{pmatrix}0&I\\ -I&0\end{pmatrix}\begin{pmatrix}I&0\\ 0&F\end{pmatrix}=\begin{pmatrix}0&F\\ -F&0\end{pmatrix}
  6. F = ( 1 0 0 N ) F=\begin{pmatrix}1&0\\ 0&N\end{pmatrix}
  7. ( * * * * / N * * * * / N * * * * / N N * N * N * * ) \begin{pmatrix}*&*&*&*/N\\ {}*&*&*&*/N\\ {}*&*&*&*/N\\ N*&N*&N*&*\end{pmatrix}
  8. ( * N * * * * * * * / N * N * * * N * N * N * * ) \begin{pmatrix}*&N*&*&*\\ {}*&*&*&*/N\\ {}*&N*&*&*\\ N*&N*&N*&*\end{pmatrix}
  9. ( 1 0 0 0 0 1 0 0 x N y 1 0 N y N z 0 1 ) \begin{pmatrix}1&0&0&0\\ {}0&1&0&0\\ {}x&Ny&1&0\\ Ny&Nz&0&1\end{pmatrix}
  10. ( 1 0 x y 0 1 y z / N 0 0 1 0 0 0 0 1 ) \begin{pmatrix}1&0&x&y\\ {}0&1&y&z/N\\ {}0&0&1&0\\ 0&0&0&1\end{pmatrix}
  11. F = ( N 0 0 1 ) F=\begin{pmatrix}N&0\\ 0&1\end{pmatrix}
  12. ( 1 0 0 N ) \begin{pmatrix}1&0\\ 0&N\end{pmatrix}
  13. ( * * * / N * N * * * * N * N * * N * N * * * * ) \begin{pmatrix}*&*&*/N&*\\ {}N*&*&*&*\\ {}N*&N*&*&N*\\ N*&*&*&*\end{pmatrix}

Parkeol_synthase.html

  1. \rightleftharpoons

Parshin's_conjecture.html

  1. K i ( X ) 𝐐 = 0 i > 0. K_{i}(X)\otimes\mathbf{Q}=0\ \,i>0.

Partial-matching_Meet-in-the-Middle_attack.html

  1. i i
  2. j j
  3. i i
  4. j j
  5. i i
  6. j j
  7. i i
  8. j j
  9. 2 - 3 = 1 / 8 2^{-3}=1/8
  10. 2 - | i | 2^{-|i|}
  11. | i | |i|

PatchMatch.html

  1. f : 2 2 f:\mathbb{R}^{2}\to\mathbb{R}^{2}
  2. D D
  3. a a
  4. A A
  5. b b
  6. B B
  7. f ( a ) f(a)
  8. b - a b-a
  9. B B
  10. B B
  11. N N F NNF
  12. f ( x , y ) f(x,y)
  13. f ( x - 1 , y ) f(x-1,y)
  14. f ( x , y - 1 ) f(x,y-1)
  15. f ( x , y ) f(x,y)
  16. arg min ( x , y ) D ( f ( x , y ) ) , D ( f ( x - 1 , y ) ) , D ( f ( x , y - 1 ) ) \arg\min\limits_{(x,y)}{D(f(x,y)),D(f(x-1,y)),D(f(x,y-1))}
  17. f ( x , y ) f(x,y)
  18. R R
  19. R R
  20. f ( x , y ) f(x,y)
  21. arg min ( x , y ) { D ( f ( x , y ) ) , D ( f ( x + 1 , y ) ) , D ( f ( x , y + 1 ) ) } \arg\min\limits_{(x,y)}\{D(f(x,y)),D(f(x+1,y)),D(f(x,y+1))\}
  22. v 0 = f ( x , y ) v_{0}=f(x,y)
  23. f ( x , y ) f(x,y)
  24. v 0 v_{0}
  25. u i = v 0 + w α i R i u_{i}=v_{0}+w\alpha^{i}R_{i}
  26. R i R_{i}
  27. [ - 1 , 1 ] × [ - 1 , 1 ] [-1,1]\times[-1,1]
  28. w w
  29. α \alpha
  30. f ( x , y ) f(x,y)

Patchoulol_synthase.html

  1. \rightleftharpoons

Path_integrals_in_polymer_science.html

  1. G [ f ( x ) ] 𝒟 f ( x ) \int G[f(x)]\mathcal{D}f(x)
  2. G [ f ( x ) ] G[f(x)]
  3. 𝒟 f ( x ) \mathcal{D}f(x)
  4. ψ ( r ) = 1 4 π l 2 δ ( | r | - l ) \psi(\vec{r})=\frac{1}{4\pi l^{2}}\delta(\left|\vec{r}\right|-l)
  5. δ ( ) \delta()
  6. l l
  7. r n \vec{r}_{n}
  8. Ψ ( { r n } ) = n = 1 N ψ ( r n ) \Psi(\left\{\vec{r}_{n}\right\})=\prod_{n=1}^{N}\psi(\vec{r}_{n})
  9. N \textstyle N
  10. n \textstyle n
  11. { } \left\{\right\}
  12. Ψ \Psi
  13. r n \vec{r}_{n}
  14. R 2 = N l 2 \left\langle\vec{R}^{2}\right\rangle=Nl^{2}
  15. R n = 1 N r n \textstyle\vec{R}\equiv\sum_{n=1}^{N}\vec{r}_{n}
  16. R 0 R 2 = N l R_{0}\equiv\sqrt{\left\langle\vec{R}^{2}\right\rangle}=\sqrt{N}l
  17. Φ ( R , N ) = ( 3 2 π N l 2 ) 3 2 exp ( - 3 R 2 2 N l 2 ) \Phi(\vec{R},N)=\left(\frac{3}{2\pi Nl^{2}}\right)^{\frac{3}{2}}\exp\left(-% \frac{3\vec{R}^{2}}{2Nl^{2}}\right)
  18. N l Nl
  19. N N\rightarrow\infty
  20. N N\rightarrow\infty
  21. l 0 , l\rightarrow 0,
  22. N l = c o n s t Nl=const
  23. Φ N = l 2 6 2 Φ \frac{\partial\Phi}{\partial N}=\frac{l^{2}}{6}\nabla^{2}\Phi
  24. 2 \textstyle\nabla^{2}
  25. Φ ( R , N \Phi(\vec{R},N
  26. Φ ( R , N + Δ N ) . \Phi(\vec{R},N+\Delta N).
  27. Φ ( R , N ) = 0 , 0 R , N exp { - 0 N L 0 d ν } 𝒟 R ( ν ) \Phi(\vec{R},N)=\int_{0,0}^{\vec{R},N}\exp\left\{-\int_{0}^{N}L_{0}d\nu\right% \}\mathcal{D}\vec{R}(\nu)
  28. L 0 = 3 2 l 2 ( d R d ν ) 2 . \textstyle L_{0}=\frac{3}{2l^{2}}\left(\frac{d\vec{R}}{d\nu}\right)^{2}.
  29. ν \nu
  30. f ( R ) f(\vec{R})
  31. 0 f ( R ) 1 0\leq f(\vec{R})\leq 1
  32. Φ ( R , N + Δ N ) . \Phi(\vec{R},N+\Delta N).
  33. Φ N = l 2 6 2 - f Φ \frac{\partial\Phi}{\partial N}=\frac{l^{2}}{6}\nabla^{2}-f\Phi
  34. Φ ( R , N ) = 0 , 0 R , N exp { - 0 N [ L 0 + f ( R ) ] d ν } 𝒟 R ( ν ) \Phi(\vec{R},N)=\int_{0,0}^{\vec{R},N}\exp\left\{-\int_{0}^{N}[L_{0}+f(\vec{R}% )]d\nu\right\}\mathcal{D}\vec{R}(\nu)
  35. f ( R ) l 2 + \textstyle\frac{f(\vec{R})}{l^{2}}\rightarrow+\infty
  36. V ( R ) \textstyle V(\vec{R})
  37. exp { - β j = 0 N V ( R j ) } exp { - β 0 N V ( R ( ν ) ) } \exp\left\{-\beta\sum_{j=0}^{N}V(\vec{R}_{j})\right\}\cong\exp\left\{-\beta% \int_{0}^{N}V(\vec{R}(\nu))\right\}
  38. β = ( k b T ) - 1 \beta=(k_{b}T)^{-}1
  39. T T
  40. k b k_{b}
  41. N & L 0 N\rightarrow\infty\quad\&\quad L\rightarrow 0
  42. Q V ( R N , N | R 0 , 0 ) = R 0 , 0 R N , N exp { - 0 N [ L 0 ] d ν } 𝒟 R ( ν ) Q_{V}(\vec{R}_{N},N|\vec{R}_{0},0)=\int_{\vec{R}_{0},0}^{\vec{R}_{N},N}\exp% \left\{-\int_{0}^{N}[L_{0}]d\nu\right\}\mathcal{D}\vec{R}(\nu)
  43. f N = l 2 6 2 f - β V ( R ) f \frac{\partial f}{\partial N}=\frac{l^{2}}{6}\nabla^{2}f-\beta V(\vec{R})f
  44. Q V ( R N , N | R 0 , 0 ) = n f n ( R N ) f n * ( R 0 ) exp ( - E N N ) Q_{V}(\vec{R}_{N},N|\vec{R}_{0},0)=\sum_{n}f_{n}(\vec{R}_{N})f_{n}^{*}(\vec{R}% _{0})\exp(-E_{N}N)
  45. [ l 2 6 2 f - β V ( R ) ] f n ( R n ) = E + n f n ( R n ) \left[\frac{l^{2}}{6}\nabla^{2}f-\beta V(\vec{R})\right]f_{n}(\vec{R}_{n})=E+% nf_{n}(\vec{R}_{n})
  46. T c T_{c}
  47. l , V ( R ) l,V(\vec{R})
  48. T > T c T>T_{c}
  49. < ( x ) <(x\rightarrow\infty)
  50. f n A n sin ( 6 λ n / l 2 x ) + B m cos ( 6 λ m / l 2 x ) f_{n}\cong A_{n}\sin(\sqrt{6\lambda_{n}/l^{2}}x)+B_{m}\cos(\sqrt{6\lambda_{m}/% l^{2}}x)
  51. λ n \lambda_{n}
  52. x = 0 x=0
  53. T < T c T<T_{c}
  54. ( x ) (x\rightarrow\infty)
  55. f ( x 0 ) A 0 exp ( - 6 | λ 0 | / l 2 x ) f(x_{0})\cong A_{0}\exp(-\sqrt{6|\lambda_{0}|/l^{2}}x)
  56. l 6 | λ 0 | \textstyle\frac{l}{\sqrt{6|\lambda_{0}|}}
  57. r < l / 2 r<l/2
  58. f ( R ) f(\vec{R})
  59. Q V ( R N , N | R 0 .0 ) \textstyle Q_{V}(\vec{R}_{N},N|\vec{R}_{0}.0)
  60. R * ( ν ) \vec{R}^{*}(\nu)
  61. S [ R ( ν ) ] 0 N { 3 2 l 2 ( d R d ν ) 2 + f ( R ) } d ν S[\vec{R}(\nu)]\equiv\int_{0}^{N}\left\{\frac{3}{2l^{2}}\left(\frac{d\vec{R}}{% d\nu}\right)^{2}+f(\vec{R})\right\}d\nu
  62. 3 l 2 d 2 R * d ν 2 = f ( R * ) \frac{3}{l^{2}}\frac{d^{2}\vec{R}^{*}}{d\nu^{2}}=\nabla f(\vec{R}^{*})
  63. R R * R\equiv R^{*}
  64. f ( R ) f(\vec{R})
  65. R R
  66. d R dR
  67. 4 π R 2 4\pi R^{2}
  68. 4 π R 2 ( 4 / 3 ) π r 3 f ( R ) d R \textstyle\frac{4\pi R^{2}}{(4/3)\pi r^{3}}f(R)dR
  69. d ν = ( d R d ν ) - 1 \textstyle d\nu=\left(\frac{dR}{d\nu}\right)^{-1}
  70. ν \nu
  71. 0 ν N 0\leq\nu\leq N
  72. f ( R ) = ( 4 / 3 ) π r 3 4 π R 2 ( d R d ν ) - 1 f(\vec{R})=\frac{(4/3)\pi r^{3}}{4\pi}R^{2}\left(\frac{dR}{d\nu}\right)^{-1}
  73. S [ R ( ν ) ] S[\vec{R}(\nu)]
  74. S [ R ( ν ) ] = 0 N { 3 2 l 2 ( d R d ν ) 2 + ( 4 / 3 ) π r 3 4 π R 2 ( d R d ν ) - 1 } d ν S[\vec{R}(\nu)]=\int_{0}^{N}\left\{\frac{3}{2l^{2}}\left(\frac{dR}{d\nu}\right% )^{2}+\frac{(4/3)\pi r^{3}}{4\pi}R^{2}\left(\frac{dR}{d\nu}\right)^{-1}\right% \}d\nu
  75. { 3 l 2 + 2 ( 4 / 3 ) π r 3 4 π R 2 ( d R d ν ) - 3 } d 2 R d ν 2 + 4 ( 4 / 3 ) π r 3 4 π R 2 ( d R d ν ) - 1 = 0 \left\{\frac{3}{l^{2}}+\frac{2(4/3)\pi r^{3}}{4\pi}R^{2}\left(\frac{dR}{d\nu}% \right)^{-3}\right\}\frac{d^{2}R}{d\nu^{2}}+4\frac{(4/3)\pi r^{3}}{4\pi}R^{2}% \left(\frac{dR}{d\nu}\right)^{-1}=0
  76. R ( ν ) R(\nu)
  77. f ( R * ) f(\vec{R}^{*})
  78. R ( ν ) = ( 3 π ( 4 / 3 ) π r 3 l 2 ) - 1 / 5 ( 3 5 ) - 3 / 5 ν 3 / 5 R(\nu)=\left(\frac{3\pi}{(4/3)\pi r^{3}l^{2}}\right)^{-1/5}\left(\frac{3}{5}% \right)^{-3/5}\nu^{3/5}
  79. R ( 3 π ( 4 / 3 ) π r 3 l 2 ) - 1 / 5 N 3 / 5 R\cong\left(\frac{3\pi}{(4/3)\pi r^{3}l^{2}}\right)^{-1/5}N^{3/5}
  80. R N R\sim\sqrt{N}
  81. N N
  82. l l
  83. ψ ( R ) = ( 3 2 π l 2 ) 3 / 2 exp ( - 3 R 2 2 l 2 ) \psi(\vec{R})=\left(\frac{3}{2\pi l^{2}}\right)^{3/2}\exp\left(-\frac{3\vec{R}% ^{2}}{2l^{2}}\right)
  84. R 2 = l 2 \langle\vec{R}^{2}\rangle=l^{2}
  85. ψ ( R ) \psi(\vec{R})
  86. l l
  87. Ψ ( { r n } ) \displaystyle\Psi(\left\{\vec{r}_{n}\right\})
  88. r n \vec{r}_{n}
  89. ( R n - R n - 1 ) (\vec{R}_{n}-\vec{R}_{n-1})
  90. ψ ( r ) \psi(\vec{r})
  91. U 0 ( { R n } ) = 3 2 l 2 k b T n = 1 N ( R n - R n - 1 ) U_{0}(\{\vec{R}_{n}\})=\frac{3}{2l^{2}}k_{b}T\sum_{n=1}^{N}(\vec{R}_{n}-\vec{R% }_{n-1})
  92. Ψ ( { r n } ) \Psi(\left\{\vec{r}_{n}\right\})
  93. R n - R m \vec{R}_{n}-\vec{R}_{m}
  94. l l
  95. ( n - m ) (n-m)
  96. ϕ ( R n - R m | n - m ) = ( 3 2 π l 2 | n - m | ) 3 / 2 exp [ - 3 ( R n - R m ) 2 2 | n - m | l 2 ] \phi(\vec{R}_{n}-\vec{R}_{m}|n-m)=\left(\frac{3}{2\pi l^{2}|n-m|}\right)^{3/2}% \exp\left[-\frac{3(\vec{R}_{n}-\vec{R}_{m})^{2}}{2|n-m|l^{2}}\right]
  97. < ( R n - R m ) 2 | n - m | l 2 <(\vec{R}_{n}-\vec{R}_{m})^{2}>=|n-m|l^{2}
  98. n n
  99. R n - R m \vec{R}_{n}-\vec{R}_{m}
  100. R n / n \partial\vec{R}_{n}/\partial n
  101. Ψ ( { r n } ) = ( 3 2 π l 2 ) 3 N / 2 exp [ - 3 2 l 2 0 N d n ( R n n ) 2 ] . \Psi(\left\{\vec{r}_{n}\right\})=\left(\frac{3}{2\pi l^{2}}\right)^{3N/2}\exp% \left[-\frac{3}{2l^{2}}\int_{0}^{N}dn\left(\frac{\partial\vec{R}_{n}}{\partial n% }\right)^{2}\right].
  102. Ψ [ R ( n ) ] \Psi[\vec{R}(n)]
  103. U e ( R ) U_{e}(\vec{R})
  104. Ψ ( { r n } ) = ( 3 2 π l 2 ) 3 N / 2 exp [ - 3 2 l 2 0 N d n ( R n n ) 2 - β 0 N d n U e [ R ( n ) ] ] . \Psi(\left\{\vec{r}_{n}\right\})=\left(\frac{3}{2\pi l^{2}}\right)^{3N/2}\exp% \left[-\frac{3}{2l^{2}}\int_{0}^{N}dn\left(\frac{\partial\vec{R}_{n}}{\partial n% }\right)^{2}-\beta\int_{0}^{N}dnU_{e}[\vec{R}(n)]\right].
  105. G ( R , R ; N ) R 0 = R R N = R 𝒟 R ( n ) exp [ - 3 2 l 2 0 N d n ( R n n ) 2 - β 0 N d u U e [ R ( n ) ] ] d R d R R 0 = R R N = R 𝒟 R n exp [ - 3 2 l 2 0 N d n ( R n n ) 2 ] G(\vec{R},\vec{R}^{\prime};N)\equiv\frac{\displaystyle\int_{\vec{R}_{0}=\vec{R% }^{\prime}}^{\vec{R}_{N}=\vec{R}}\mathcal{D}\vec{R}(n)\exp\left[-\frac{3}{2l^{% 2}}\displaystyle\int_{0}^{N}dn\left(\frac{\partial\vec{R}_{n}}{\partial n}% \right)^{2}-\beta\displaystyle\int_{0}^{N}duU_{e}[\vec{R}(n)]\right]}{% \displaystyle\int d\vec{R}^{\prime}\displaystyle\int d\vec{R}\displaystyle\int% _{\vec{R}_{0}=\vec{R}^{\prime}}^{\vec{R}_{N}=\vec{R}}\mathcal{D}\vec{R}_{n}% \exp\left[-\frac{3}{2l^{2}}\displaystyle\int_{0}^{N}dn\left(\frac{\partial\vec% {R}_{n}}{\partial n}\right)^{2}\right]}
  106. R ( n ) \vec{R}(n)
  107. R 0 = R \vec{R}_{0}=\vec{R}^{\prime}
  108. R N = R \vec{R}_{N}=\vec{R}
  109. U e = 0 U_{e}=0
  110. G ( R - R ; N ) = ( 3 2 π l 2 N ) 3 / 2 exp [ - 3 ( R - R ) 2 2 N l 2 ] G(\vec{R}-\vec{R}^{\prime};N)=\left(\frac{3}{2\pi l^{2}N}\right)^{3/2}\exp% \left[-\frac{3(\vec{R}-\vec{R}^{\prime})^{2}}{2Nl^{2}}\right]
  111. G ( R - R ; N ) G(\vec{R}-\vec{R}^{\prime};N)
  112. Z = d R d R G ( R - R ; N ) . Z=\int d\vec{R}~{}d\vec{R}^{\prime}~{}G(\vec{R}-\vec{R}^{\prime};N).
  113. G ( R , R ; N ) = d R ′′ G ( R , R ′′ ; N - n ) G ( R ′′ , R ; N ) , ( 0 < n < N ) . G(\vec{R},\vec{R}^{\prime};N)=\int d\vec{R}^{\prime\prime}G(\vec{R},\vec{R}^{% \prime\prime};N-n)G(\vec{R}^{\prime\prime},\vec{R}^{\prime};N),\quad(0<n<N).
  114. G ( R , R ′′ ; N - n ) G ( R ′′ , R ; N ) ) \textstyle G(\vec{R},\vec{R}^{\prime\prime};N-n)G(\vec{R}^{\prime\prime},\vec{% R}^{\prime};N))
  115. R R^{\prime}
  116. R ′′ R^{\prime\prime}
  117. n n
  118. R R
  119. N N
  120. R ′′ R^{\prime\prime}
  121. R R^{\prime}
  122. R R
  123. G ( R , R ; N ) G(\vec{R},\vec{R}^{\prime};N)
  124. A A
  125. A \textstyle A
  126. n n
  127. A ( R n ) = d R N d R n d R 0 G ( R N , R n ; N - n ) G ( R n , R 0 ; n ) A ( R n ) d R N d R 0 G ( R N , R 0 ; N ) \left\langle A(\vec{R}_{n})\right\rangle=\frac{\displaystyle\int d\vec{R}_{N}~% {}d\vec{R}_{n}~{}d\vec{R}_{0}~{}G(\vec{R}_{N},\vec{R}_{n};N-n)G(\vec{R}_{n},% \vec{R}_{0};n)A(\vec{R}_{n})}{\displaystyle\int d\vec{R}_{N}~{}\vec{d}R_{0}~{}% G(\vec{R}_{N},\vec{R}_{0};N)}
  128. R m \vec{R}_{m}
  129. R n \vec{R}_{n}
  130. A ( R n , R m ) = d R N d R n d R m d R 0 G ( R N , R n ; N - n ) G ( R n , R m ; n - m ) A ( R n , R m ) d R N d R 0 G ( R N , R 0 ; N ) \left\langle A(\vec{R}_{n},\vec{R}_{m})\right\rangle=\frac{\displaystyle\int d% \vec{R}_{N}~{}d\vec{R}_{n}~{}d\vec{R}_{m}~{}d\vec{R}_{0}~{}G(\vec{R}_{N},\vec{% R}_{n};N-n)G(\vec{R}_{n},\vec{R}_{m};n-m)A(\vec{R}_{n},\vec{R}_{m})}{% \displaystyle\int d\vec{R}_{N}~{}\vec{d}R_{0}~{}G(\vec{R}_{N},\vec{R}_{0};N)}
  131. G ( R , R ; N < 0 ) = 0 \displaystyle G(\vec{R},\vec{R}^{\prime};N<0)=0
  132. G ( R , R ; N + Δ N ) G(\vec{R},\vec{R}^{\prime};N+\Delta N)
  133. G G
  134. ( N - l 2 6 2 R 2 + β U e ( R ) ) ) G ( R , R ; N ) = δ 3 ( R - R ) δ ( N ) . \left(\frac{\partial}{\partial N}-\frac{l^{2}}{6}\frac{\partial^{2}}{\partial% \vec{R}^{2}}+\beta U_{e}(\vec{R}))\right)G(\vec{R},\vec{R}^{\prime};N)=\delta^% {3}(\vec{R}-\vec{R}^{\prime})\delta(N).
  135. G ( R , R ; N ) G(\vec{R},\vec{R}^{\prime};N)
  136. R 2 N α \left\langle\vec{R}^{2}\right\rangle\propto N^{\alpha}
  137. Φ ( R , N ) = 0 , 0 R , N e - 𝒜 [ η ] P η ( N , l ) 𝒟 η \Phi(\vec{R},N)=\int_{0,0}^{\vec{R},N}e^{-\mathcal{A}[\eta]}P^{\eta}(N,l)% \mathcal{D}\eta
  138. η ( R ) \eta(\vec{R})
  139. 𝒜 [ η ] = - 1 2 d R d R η ( R ) V - 1 ( R , R ) η ( R ) \mathcal{A}[\eta]=-\frac{1}{2}\int d\vec{R}~{}d\vec{R}^{\prime}~{}\eta(\vec{R}% )V^{-1}(\vec{R},\vec{R}^{\prime})\eta(\vec{R}^{\prime})
  140. V ( R , R ) V(\vec{R},\vec{R}^{\prime})
  141. P η ( N , L ) = exp { - 0 N d ν [ M 2 R ˙ + η ( R ( ν ) ) ] } 𝒟 R P^{\eta}(N,L)=\int\exp\left\{-\int_{0}^{N}d\nu\left[\frac{M}{2}\dot{\vec{R}}+% \eta(\vec{R}(\nu))\right]\right\}\mathcal{D}\vec{R}
  142. [ N - 1 2 M 2 + η ( R ) ] P η ( N , L ) = δ ( 3 ) ( R - R ) δ ( N ) \left[\frac{\partial}{\partial N}-\frac{1}{2M}\nabla^{2}+\eta(\vec{R})\right]P% ^{\eta}(N,L)=\delta^{(3)}(\vec{R}-\vec{R}^{\prime})\delta(N)
  143. M M
  144. R ( ν ) \vec{R}(\nu)
  145. η ( R ) \eta(\vec{R})
  146. 𝒜 \mathcal{A}
  147. η ( R ) \eta(\vec{R})
  148. R ( ν ) \vec{R}(\nu)
  149. η ( R ) \eta(\vec{R})
  150. e - 𝒜 [ η ] e^{-\mathcal{A}[\eta]}
  151. η ( R ) \eta(\vec{R})
  152. Φ ( R , N ) \Phi(\vec{R},N)
  153. A ( R n , R m ) \left\langle A(\vec{R}_{n},\vec{R}_{m})\right\rangle
  154. R 0 < ξ R_{0}<\xi
  155. R 0 > ξ R_{0}>\xi
  156. ξ \xi
  157. C * = N / R 0 3 N / N 3 σ = N 1 - 3 σ C^{*}=N/R_{0}^{3}\sim N/N^{3\sigma}=N^{1-3\sigma}
  158. R 0 N σ R_{0}\sim N^{\sigma}
  159. Z = α = 1 n p 𝒟 R α ( ν ) exp { - β ( [ R α ( ν ) ] ) } Z=\int\prod_{\alpha=1}^{n_{p}}\mathcal{D}\vec{R}_{\alpha}(\nu)\exp\{-\beta% \mathcal{H}([\vec{R}_{\alpha}(\nu)])\}
  160. β ( [ R α ( ν ) ] ) = 3 2 l 2 α = 1 n p 0 N α ( R α ν ) 2 d ν + 1 2 σ α , β = 1 n p 0 N α d ν 0 N β d ν δ ( R α ( ν ) - R β ( ν ) ) \displaystyle\beta\mathcal{H}([\vec{R}_{\alpha}(\nu)])=\frac{3}{2l^{2}}\sum_{% \alpha=1}^{n_{p}}\int_{0}^{N_{\alpha}}\left(\frac{\partial\vec{R}_{\alpha}}{% \partial\nu}\right)^{2}d\nu+\frac{1}{2}\sigma\sum_{\alpha,\beta=1}^{n_{p}}\int% _{0}^{N_{\alpha}}d\nu\int_{0}^{N_{\beta}}d\nu^{\prime}\delta(\vec{R}_{\alpha}(% \nu)-\vec{R}_{\beta}(\nu^{\prime}))
  161. n p n_{p}
  162. N α = N β α , β N_{\alpha}=N_{\beta}\quad\forall\ \alpha,\beta
  163. n p n_{p}
  164. ρ ( x ) = 1 V α = 1 n p 0 N d ν δ ( x - R α ( ν ) ) . \rho(\vec{x})=\frac{1}{V}\sum_{\alpha=1}^{n_{p}}\int_{0}^{N}d\nu\delta(\vec{x}% -\vec{R}_{\alpha}(\nu)).
  165. V V
  166. ρ ( x ) \rho(\vec{x})
  167. x \vec{x}
  168. ( [ R α ( ν ) ] ) ( [ ρ ( x ) ] ) \mathcal{H}([\vec{R}_{\alpha}(\nu)])\rightarrow\mathcal{H}([\rho(\vec{x})])

Paul_C._Yang.html

  1. S 2 S^{2}
  2. S 2 S^{2}

Paul_Gruner.html

  1. S S
  2. S S^{\prime}
  3. t t
  4. t t
  5. t t
  6. τ \tau
  7. S 0 S_{0}
  8. S S
  9. S S^{\prime}
  10. t t
  11. τ \tau

Pearcey_integral.html

  1. Pe ( x , y ) = - exp ( i ( t 4 + x t 2 + y t ) ) d t . \operatorname{Pe}(x,y)=\int_{-\infty}^{\infty}\exp(i(t^{4}+xt^{2}+yt))\,dt.

Peeler_centrifuge.html

  1. a = V θ 2 r \,\text{a}=\frac{V_{\theta}^{2}}{r}
  2. V θ 2 \textstyle V_{\theta}^{2}
  3. r \textstyle r
  4. V θ = Ω 2 r V_{\theta}=\Omega^{2}r
  5. Ω \textstyle\Omega
  6. r \textstyle r
  7. V θ \textstyle V_{\theta}
  8. δ = μ / ρ Ω \delta=\sqrt{\frac{\mu/\rho}{\Omega}}
  9. μ / ρ \textstyle\mu/\rho
  10. δ \textstyle\delta
  11. ϵ = 1 - ω s V s ρ c \epsilon=1-\frac{\omega_{s}}{V_{s}\rho_{c}}
  12. ω s \textstyle\omega_{s}
  13. V s \textstyle V_{s}
  14. ρ c \textstyle\rho_{c}
  15. S = ( 1 - W W ) ( 1 - ϵ ϵ ) ( ρ s ρ L ) S=(\frac{1-W}{W})(\frac{1-\epsilon}{\epsilon})(\frac{\rho_{s}}{\rho_{L}})
  16. W \textstyle W
  17. ρ L \textstyle\rho_{L}
  18. ϵ = ( 1 + ρ L ρ s W 1 - W ) - 1 \epsilon=(1+\frac{\rho_{L}}{\rho_{s}}\frac{W}{1-W})^{-1}
  19. r e c = m c W c m f W f rec=\frac{m_{c}W_{c}}{m_{f}W_{f}}
  20. m \textstyle m

Peirce's_law.html

  1. ( p q ) p p q ¯ p p ¯ q ¯ p ( p and q ¯ ) p ( p and q ¯ ) ( p and 1 ) p and ( q ¯ 1 ) p and 1 p . (p\rightarrow q)\rightarrow p\Rightarrow\overline{p\rightarrow q}p\Rightarrow% \overline{\overline{p}q}p\Rightarrow(p\and\overline{q})p\Rightarrow(p\and% \overline{q})(p\and 1)\Rightarrow p\and(\overline{q}1)\Rightarrow p\and 1% \Rightarrow p.

Pentaborane(11).html

  1. 𝟧 𝖡 𝟤 𝖧 𝟨 𝟣𝟣𝟧 𝗈 𝖢 2 𝖡 𝟧 𝖧 𝟣𝟣 + 𝟦 𝖧 𝟤 \mathsf{5B_{2}H_{6}\ \xrightarrow{115^{o}C}\ 2B_{5}H_{11}+4H_{2}}

Perchlorate_reductase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Perovskite_solar_cell.html

  1. J ( V ) = J t ( V ) + J e ( V ) - J p J(V)={J_{t}}(V)+{J_{e}}(V)-{J_{p}}
  2. V V
  3. J p J_{p}
  4. J e J_{e}
  5. J t J_{t}
  6. J p = q 0 a ( λ , W ) < m t p l > Γ ( λ ) λ h c 0 d λ {J_{p}}=q\int_{0}^{\infty}{a(\lambda,W)}\frac{<}{m}tpl>{{\Gamma(\lambda)% \lambda}}{{h{c_{0}}}}d\lambda
  7. c 0 c_{0}
  8. Γ \Gamma
  9. λ \lambda
  10. q q
  11. a a
  12. J e ( V ) = J 0 [ exp ( < m t p l > q V k B T ) - 1 ] {J_{e}}(V)={J_{0}}\left[{\exp\left({\frac{<}{m}tpl>{{qV}}{{{k_{B}}T}}}\right)-% 1}\right]
  13. k B k_{B}
  14. T T
  15. J 0 J_{0}
  16. J 0 = q 0 a ( λ , W ) Γ 0 ( λ ) λ h c 0 d λ {J_{0}}=q\int_{0}^{\infty}{a(\lambda,W)}\frac{{{\Gamma_{0}}(\lambda)\lambda}}{% {h{c_{0}}}}d\lambda
  17. Γ \Gamma
  18. Γ 0 \Gamma_{0}
  19. J t ( V ) = q γ n i W exp ( < m t p l > 0.5 q V k B T ) {J_{t}}(V)=q\gamma{n_{i}}W\exp\left({\frac{<}{m}tpl>{{0.5qV}}{{{k_{B}}T}}}\right)
  20. γ \gamma
  21. n i n_{i}
  22. W W

Personal_Composite_Instrument.html

  1. P C I = i = 1 M N i P B i j = 1 N N j P Q j = B ( U S D ) Q ( U S D ) PCI=\frac{\sum_{i=1}^{M}N_{i}\cdot P_{B}^{i}}{\sum_{j=1}^{N}N_{j}\cdot P_{Q}^{% j}}=\frac{B(USD)}{Q(USD)}

Pest_insect_population_dynamics.html

  1. x x
  2. a x a_{x}
  3. l x l_{x}
  4. d x d_{x}
  5. q x q_{x}
  6. x x
  7. L x L_{x}
  8. x x
  9. L x = l x + l x + 1 2 L_{x}=\frac{l_{x}+l_{x+1}}{2}
  10. T x T_{x}
  11. x x
  12. T x = L x + L x + 1 + L x + 2 + T_{x}=L_{x}+L_{x+1}+L_{x+2}+...
  13. e x e_{x}
  14. x x
  15. e x = T x l x e_{x}=\frac{T_{x}}{l_{x}}
  16. F x F_{x}
  17. x x
  18. m x m_{x}
  19. m x = F x a x m_{x}=\frac{F_{x}}{a_{x}}
  20. a x a_{x}
  21. R 0 R_{0}
  22. R 0 = x l x m x R_{0}=\sum_{x}l_{x}m_{x}
  23. l x m x l_{x}m_{x}
  24. x x
  25. x x
  26. x x
  27. N 0 N_{0}
  28. N T N_{T}
  29. R 0 = N T N 0 R_{0}=\frac{N_{T}}{N_{0}}
  30. T c T_{c}
  31. x x
  32. T c = x x l x m x x l x m x T_{c}=\frac{\sum_{x}xl_{x}m_{x}}{\sum_{x}l_{x}m_{x}}
  33. R 0 R_{0}
  34. r r
  35. r ln R 0 T c r\approx\frac{\ln R_{0}}{T_{c}}
  36. ln R 0 = ln N T N 0 = ln N 0 + Δ N N 0 = ln ( 1 + Δ N N 0 ) Δ N N 0 \ln R_{0}=\ln\frac{N_{T}}{N_{0}}=\ln\frac{N_{0}+\Delta N}{N_{0}}=\ln\left(1+% \frac{\Delta N}{N_{0}}\right)\approx\frac{\Delta N}{N_{0}}
  37. T c T_{c}
  38. Δ t \Delta t
  39. r Δ N Δ t 1 N 0 r\approx\frac{\Delta N}{\Delta t}\frac{1}{N_{0}}
  40. r r
  41. r = { T - T 0 K , T T 0 0 , T < T 0 r=\begin{cases}\frac{T-T_{0}}{K},&T\geq T_{0}\\ 0,&T<T_{0}\end{cases}
  42. T T
  43. T 0 T_{0}
  44. K K
  45. r r
  46. Δ T \Delta T
  47. Δ T \Delta T
  48. Δ N \Delta N
  49. Δ N Δ T K ( 206.7 + 12.46 ( m - T 0 ) ) \Delta N\approx\frac{\Delta T}{K}\left(206.7+12.46(m-T_{0})\right)
  50. m m
  51. X X
  52. α \alpha
  53. σ \sigma
  54. d X = α X d t + σ X d z dX=\alpha Xdt+\sigma Xdz
  55. d t dt
  56. d z = ξ t d t dz=\xi_{t}\sqrt{dt}
  57. ξ t \xi_{t}
  58. σ X d z \sigma Xdz
  59. α X d t \alpha Xdt
  60. t t
  61. X t X_{t}
  62. X t Log - 𝒩 ( X 0 e α t , X 0 2 e 2 α t ( e σ 2 t - 1 ) ) X_{t}\sim\operatorname{Log-\mathcal{N}}\left(X_{0}e^{\alpha t},X_{0}^{2}e^{2% \alpha t}(e^{\sigma^{2}t}-1)\right)
  63. X 0 X_{0}
  64. α = 0.1199 \alpha=0.1199
  65. σ = 0.1152 \sigma=0.1152
  66. e α t e^{\alpha t}
  67. ln 2 / 0.1199 = 5.8 \ln 2/0.1199=5.8

Peter_Duren.html

  1. H p H^{p}

Peter_Keevash.html

  1. R ( 3 , k ) R(3,k)
  2. R ( 3 , k ) ( 1 4 - o ( 1 ) ) k 2 / log k . R(3,k)\geq(\frac{1}{4}-o(1))k^{2}/\log k.

Peter_Minkowski.html

  1. S p Sp
  2. p ¯ \overline{p}
  3. S S
  4. ν \nu
  5. ν ¯ \overline{\nu}

Petroleomics.html

  1. Kendrick mass defect = nominal Kendrick mass - Kendrick mass \textrm{Kendrick~{}mass~{}defect}=\textrm{nominal~{}Kendrick~{}mass}-\textrm{% Kendrick~{}mass}
  2. R i n g s + π B o n d s = C - H 2 - X 2 + N 2 + 1 Rings+\pi Bonds=C-\frac{H}{2}-\frac{X}{2}+\frac{N}{2}+1\,

Pfister's_sixteen-square_identity.html

  1. ( x 1 2 + x 2 2 + x 3 2 + x 4 2 + + x 16 2 ) ( y 1 2 + y 2 2 + y 3 2 + y 4 2 + + y 16 2 ) = z 1 2 + z 2 2 + z 3 2 + z 4 2 + + z 16 2 \begin{aligned}&\displaystyle{}\quad(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+% \cdots+x_{16}^{2})(y_{1}^{2}+y_{2}^{2}+y_{3}^{2}+y_{4}^{2}+\cdots+y_{16}^{2})% \\ &\displaystyle=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+z_{4}^{2}+\cdots+z_{16}^{2}\end{aligned}
  2. z 1 = \color b l u e x 1 y 1 - x 2 y 2 - x 3 y 3 - x 4 y 4 - x 5 y 5 - x 6 y 6 - x 7 y 7 - x 8 y 8 + u 1 y 9 - u 2 y 10 - u 3 y 11 - u 4 y 12 - u 5 y 13 - u 6 y 14 - u 7 y 15 - u 8 y 16 \,{}^{z_{1}={\color{blue}{x_{1}y_{1}-x_{2}y_{2}-x_{3}y_{3}-x_{4}y_{4}-x_{5}y_{% 5}-x_{6}y_{6}-x_{7}y_{7}-x_{8}y_{8}}}+u_{1}y_{9}-u_{2}y_{10}-u_{3}y_{11}-u_{4}% y_{12}-u_{5}y_{13}-u_{6}y_{14}-u_{7}y_{15}-u_{8}y_{16}}
  3. z 2 = \color b l u e x 2 y 1 + x 1 y 2 + x 4 y 3 - x 3 y 4 + x 6 y 5 - x 5 y 6 - x 8 y 7 + x 7 y 8 + u 2 y 9 + u 1 y 10 + u 4 y 11 - u 3 y 12 + u 6 y 13 - u 5 y 14 - u 8 y 15 + u 7 y 16 \,{}^{z_{2}={\color{blue}{x_{2}y_{1}+x_{1}y_{2}+x_{4}y_{3}-x_{3}y_{4}+x_{6}y_{% 5}-x_{5}y_{6}-x_{8}y_{7}+x_{7}y_{8}}}+u_{2}y_{9}+u_{1}y_{10}+u_{4}y_{11}-u_{3}% y_{12}+u_{6}y_{13}-u_{5}y_{14}-u_{8}y_{15}+u_{7}y_{16}}
  4. z 3 = \color b l u e x 3 y 1 - x 4 y 2 + x 1 y 3 + x 2 y 4 + x 7 y 5 + x 8 y 6 - x 5 y 7 - x 6 y 8 + u 3 y 9 - u 4 y 10 + u 1 y 11 + u 2 y 12 + u 7 y 13 + u 8 y 14 - u 5 y 15 - u 6 y 16 \,{}^{z_{3}={\color{blue}{x_{3}y_{1}-x_{4}y_{2}+x_{1}y_{3}+x_{2}y_{4}+x_{7}y_{% 5}+x_{8}y_{6}-x_{5}y_{7}-x_{6}y_{8}}}+u_{3}y_{9}-u_{4}y_{10}+u_{1}y_{11}+u_{2}% y_{12}+u_{7}y_{13}+u_{8}y_{14}-u_{5}y_{15}-u_{6}y_{16}}
  5. z 4 = \color b l u e x 4 y 1 + x 3 y 2 - x 2 y 3 + x 1 y 4 + x 8 y 5 - x 7 y 6 + x 6 y 7 - x 5 y 8 + u 4 y 9 + u 3 y 10 - u 2 y 11 + u 1 y 12 + u 8 y 13 - u 7 y 14 + u 6 y 15 - u 5 y 16 \,{}^{z_{4}={\color{blue}{x_{4}y_{1}+x_{3}y_{2}-x_{2}y_{3}+x_{1}y_{4}+x_{8}y_{% 5}-x_{7}y_{6}+x_{6}y_{7}-x_{5}y_{8}}}+u_{4}y_{9}+u_{3}y_{10}-u_{2}y_{11}+u_{1}% y_{12}+u_{8}y_{13}-u_{7}y_{14}+u_{6}y_{15}-u_{5}y_{16}}
  6. z 5 = \color b l u e x 5 y 1 - x 6 y 2 - x 7 y 3 - x 8 y 4 + x 1 y 5 + x 2 y 6 + x 3 y 7 + x 4 y 8 + u 5 y 9 - u 6 y 10 - u 7 y 11 - u 8 y 12 + u 1 y 13 + u 2 y 14 + u 3 y 15 + u 4 y 16 \,{}^{z_{5}={\color{blue}{x_{5}y_{1}-x_{6}y_{2}-x_{7}y_{3}-x_{8}y_{4}+x_{1}y_{% 5}+x_{2}y_{6}+x_{3}y_{7}+x_{4}y_{8}}}+u_{5}y_{9}-u_{6}y_{10}-u_{7}y_{11}-u_{8}% y_{12}+u_{1}y_{13}+u_{2}y_{14}+u_{3}y_{15}+u_{4}y_{16}}
  7. z 6 = \color b l u e x 6 y 1 + x 5 y 2 - x 8 y 3 + x 7 y 4 - x 2 y 5 + x 1 y 6 - x 4 y 7 + x 3 y 8 + u 6 y 9 + u 5 y 10 - u 8 y 11 + u 7 y 12 - u 2 y 13 + u 1 y 14 - u 4 y 15 + u 3 y 16 \,{}^{z_{6}={\color{blue}{x_{6}y_{1}+x_{5}y_{2}-x_{8}y_{3}+x_{7}y_{4}-x_{2}y_{% 5}+x_{1}y_{6}-x_{4}y_{7}+x_{3}y_{8}}}+u_{6}y_{9}+u_{5}y_{10}-u_{8}y_{11}+u_{7}% y_{12}-u_{2}y_{13}+u_{1}y_{14}-u_{4}y_{15}+u_{3}y_{16}}
  8. z 7 = \color b l u e x 7 y 1 + x 8 y 2 + x 5 y 3 - x 6 y 4 - x 3 y 5 + x 4 y 6 + x 1 y 7 - x 2 y 8 + u 7 y 9 + u 8 y 10 + u 5 y 11 - u 6 y 12 - u 3 y 13 + u 4 y 14 + u 1 y 15 - u 2 y 16 \,{}^{z_{7}={\color{blue}{x_{7}y_{1}+x_{8}y_{2}+x_{5}y_{3}-x_{6}y_{4}-x_{3}y_{% 5}+x_{4}y_{6}+x_{1}y_{7}-x_{2}y_{8}}}+u_{7}y_{9}+u_{8}y_{10}+u_{5}y_{11}-u_{6}% y_{12}-u_{3}y_{13}+u_{4}y_{14}+u_{1}y_{15}-u_{2}y_{16}}
  9. z 8 = \color b l u e x 8 y 1 - x 7 y 2 + x 6 y 3 + x 5 y 4 - x 4 y 5 - x 3 y 6 + x 2 y 7 + x 1 y 8 + u 8 y 9 - u 7 y 10 + u 6 y 11 + u 5 y 12 - u 4 y 13 - u 3 y 14 + u 2 y 15 + u 1 y 16 \,{}^{z_{8}={\color{blue}{x_{8}y_{1}-x_{7}y_{2}+x_{6}y_{3}+x_{5}y_{4}-x_{4}y_{% 5}-x_{3}y_{6}+x_{2}y_{7}+x_{1}y_{8}}}+u_{8}y_{9}-u_{7}y_{10}+u_{6}y_{11}+u_{5}% y_{12}-u_{4}y_{13}-u_{3}y_{14}+u_{2}y_{15}+u_{1}y_{16}}
  10. z 9 = x 9 y 1 - x 10 y 2 - x 11 y 3 - x 12 y 4 - x 13 y 5 - x 14 y 6 - x 15 y 7 - x 16 y 8 + x 1 y 9 - x 2 y 10 - x 3 y 11 - x 4 y 12 - x 5 y 13 - x 6 y 14 - x 7 y 15 - x 8 y 16 \,{}^{z_{9}=x_{9}y_{1}-x_{10}y_{2}-x_{11}y_{3}-x_{12}y_{4}-x_{13}y_{5}-x_{14}y% _{6}-x_{15}y_{7}-x_{16}y_{8}+x_{1}y_{9}-x_{2}y_{10}-x_{3}y_{11}-x_{4}y_{12}-x_% {5}y_{13}-x_{6}y_{14}-x_{7}y_{15}-x_{8}y_{16}}
  11. z 10 = x 10 y 1 + x 9 y 2 + x 12 y 3 - x 11 y 4 + x 14 y 5 - x 13 y 6 - x 16 y 7 + x 15 y 8 + x 2 y 9 + x 1 y 10 + x 4 y 11 - x 3 y 12 + x 6 y 13 - x 5 y 14 - x 8 y 15 + x 7 y 16 \,{}^{z_{10}=x_{10}y_{1}+x_{9}y_{2}+x_{12}y_{3}-x_{11}y_{4}+x_{14}y_{5}-x_{13}% y_{6}-x_{16}y_{7}+x_{15}y_{8}+x_{2}y_{9}+x_{1}y_{10}+x_{4}y_{11}-x_{3}y_{12}+x% _{6}y_{13}-x_{5}y_{14}-x_{8}y_{15}+x_{7}y_{16}}
  12. z 11 = x 11 y 1 - x 12 y 2 + x 9 y 3 + x 10 y 4 + x 15 y 5 + x 16 y 6 - x 13 y 7 - x 14 y 8 + x 3 y 9 - x 4 y 10 + x 1 y 11 + x 2 y 12 + x 7 y 13 + x 8 y 14 - x 5 y 15 - x 6 y 16 \,{}^{z_{11}=x_{11}y_{1}-x_{12}y_{2}+x_{9}y_{3}+x_{10}y_{4}+x_{15}y_{5}+x_{16}% y_{6}-x_{13}y_{7}-x_{14}y_{8}+x_{3}y_{9}-x_{4}y_{10}+x_{1}y_{11}+x_{2}y_{12}+x% _{7}y_{13}+x_{8}y_{14}-x_{5}y_{15}-x_{6}y_{16}}
  13. z 12 = x 12 y 1 + x 11 y 2 - x 10 y 3 + x 9 y 4 + x 16 y 5 - x 15 y 6 + x 14 y 7 - x 13 y 8 + x 4 y 9 + x 3 y 10 - x 2 y 11 + x 1 y 12 + x 8 y 13 - x 7 y 14 + x 6 y 15 - x 5 y 16 \,{}^{z_{12}=x_{12}y_{1}+x_{11}y_{2}-x_{10}y_{3}+x_{9}y_{4}+x_{16}y_{5}-x_{15}% y_{6}+x_{14}y_{7}-x_{13}y_{8}+x_{4}y_{9}+x_{3}y_{10}-x_{2}y_{11}+x_{1}y_{12}+x% _{8}y_{13}-x_{7}y_{14}+x_{6}y_{15}-x_{5}y_{16}}
  14. z 13 = x 13 y 1 - x 14 y 2 - x 15 y 3 - x 16 y 4 + x 9 y 5 + x 10 y 6 + x 11 y 7 + x 12 y 8 + x 5 y 9 - x 6 y 10 - x 7 y 11 - x 8 y 12 + x 1 y 13 + x 2 y 14 + x 3 y 15 + x 4 y 16 \,{}^{z_{13}=x_{13}y_{1}-x_{14}y_{2}-x_{15}y_{3}-x_{16}y_{4}+x_{9}y_{5}+x_{10}% y_{6}+x_{11}y_{7}+x_{12}y_{8}+x_{5}y_{9}-x_{6}y_{10}-x_{7}y_{11}-x_{8}y_{12}+x% _{1}y_{13}+x_{2}y_{14}+x_{3}y_{15}+x_{4}y_{16}}
  15. z 14 = x 14 y 1 + x 13 y 2 - x 16 y 3 + x 15 y 4 - x 10 y 5 + x 9 y 6 - x 12 y 7 + x 11 y 8 + x 6 y 9 + x 5 y 10 - x 8 y 11 + x 7 y 12 - x 2 y 13 + x 1 y 14 - x 4 y 15 + x 3 y 16 \,{}^{z_{14}=x_{14}y_{1}+x_{13}y_{2}-x_{16}y_{3}+x_{15}y_{4}-x_{10}y_{5}+x_{9}% y_{6}-x_{12}y_{7}+x_{11}y_{8}+x_{6}y_{9}+x_{5}y_{10}-x_{8}y_{11}+x_{7}y_{12}-x% _{2}y_{13}+x_{1}y_{14}-x_{4}y_{15}+x_{3}y_{16}}
  16. z 15 = x 15 y 1 + x 16 y 2 + x 13 y 3 - x 14 y 4 - x 11 y 5 + x 12 y 6 + x 9 y 7 - x 10 y 8 + x 7 y 9 + x 8 y 10 + x 5 y 11 - x 6 y 12 - x 3 y 13 + x 4 y 14 + x 1 y 15 - x 2 y 16 \,{}^{z_{15}=x_{15}y_{1}+x_{16}y_{2}+x_{13}y_{3}-x_{14}y_{4}-x_{11}y_{5}+x_{12% }y_{6}+x_{9}y_{7}-x_{10}y_{8}+x_{7}y_{9}+x_{8}y_{10}+x_{5}y_{11}-x_{6}y_{12}-x% _{3}y_{13}+x_{4}y_{14}+x_{1}y_{15}-x_{2}y_{16}}
  17. z 16 = x 16 y 1 - x 15 y 2 + x 14 y 3 + x 13 y 4 - x 12 y 5 - x 11 y 6 + x 10 y 7 + x 9 y 8 + x 8 y 9 - x 7 y 10 + x 6 y 11 + x 5 y 12 - x 4 y 13 - x 3 y 14 + x 2 y 15 + x 1 y 16 \,{}^{z_{16}=x_{16}y_{1}-x_{15}y_{2}+x_{14}y_{3}+x_{13}y_{4}-x_{12}y_{5}-x_{11% }y_{6}+x_{10}y_{7}+x_{9}y_{8}+x_{8}y_{9}-x_{7}y_{10}+x_{6}y_{11}+x_{5}y_{12}-x% _{4}y_{13}-x_{3}y_{14}+x_{2}y_{15}+x_{1}y_{16}}
  18. x i , y i x_{i},y_{i}
  19. i > 8 i>8
  20. u i u_{i}
  21. u 1 = ( a x 1 2 + x 2 2 + x 3 2 + x 4 2 + x 5 2 + x 6 2 + x 7 2 + x 8 2 ) x 9 - 2 x 1 ( b x 1 x 9 + x 2 x 10 + x 3 x 11 + x 4 x 12 + x 5 x 13 + x 6 x 14 + x 7 x 15 + x 8 x 16 ) c u_{1}=\tfrac{(ax_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{% 7}^{2}+x_{8}^{2})x_{9}-2x_{1}(bx_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+% x_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16})}{c}
  22. u 2 = ( x 1 2 + a x 2 2 + x 3 2 + x 4 2 + x 5 2 + x 6 2 + x 7 2 + x 8 2 ) x 10 - 2 x 2 ( x 1 x 9 + b x 2 x 10 + x 3 x 11 + x 4 x 12 + x 5 x 13 + x 6 x 14 + x 7 x 15 + x 8 x 16 ) c u_{2}=\tfrac{(x_{1}^{2}+ax_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{% 7}^{2}+x_{8}^{2})x_{10}-2x_{2}(x_{1}x_{9}+bx_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}% +x_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16})}{c}
  23. u 3 = ( x 1 2 + x 2 2 + a x 3 2 + x 4 2 + x 5 2 + x 6 2 + x 7 2 + x 8 2 ) x 11 - 2 x 3 ( x 1 x 9 + x 2 x 10 + b x 3 x 11 + x 4 x 12 + x 5 x 13 + x 6 x 14 + x 7 x 15 + x 8 x 16 ) c u_{3}=\tfrac{(x_{1}^{2}+x_{2}^{2}+ax_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{% 7}^{2}+x_{8}^{2})x_{11}-2x_{3}(x_{1}x_{9}+x_{2}x_{10}+bx_{3}x_{11}+x_{4}x_{12}% +x_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16})}{c}
  24. u 4 = ( x 1 2 + x 2 2 + x 3 2 + a x 4 2 + x 5 2 + x 6 2 + x 7 2 + x 8 2 ) x 12 - 2 x 4 ( x 1 x 9 + x 2 x 10 + x 3 x 11 + b x 4 x 12 + x 5 x 13 + x 6 x 14 + x 7 x 15 + x 8 x 16 ) c u_{4}=\tfrac{(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+ax_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{% 7}^{2}+x_{8}^{2})x_{12}-2x_{4}(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+bx_{4}x_{12}% +x_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16})}{c}
  25. u 5 = ( x 1 2 + x 2 2 + x 3 2 + x 4 2 + a x 5 2 + x 6 2 + x 7 2 + x 8 2 ) x 13 - 2 x 5 ( x 1 x 9 + x 2 x 10 + x 3 x 11 + x 4 x 12 + b x 5 x 13 + x 6 x 14 + x 7 x 15 + x 8 x 16 ) c u_{5}=\tfrac{(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+ax_{5}^{2}+x_{6}^{2}+x_{% 7}^{2}+x_{8}^{2})x_{13}-2x_{5}(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+% bx_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16})}{c}
  26. u 6 = ( x 1 2 + x 2 2 + x 3 2 + x 4 2 + x 5 2 + a x 6 2 + x 7 2 + x 8 2 ) x 14 - 2 x 6 ( x 1 x 9 + x 2 x 10 + x 3 x 11 + x 4 x 12 + x 5 x 13 + b x 6 x 14 + x 7 x 15 + x 8 x 16 ) c u_{6}=\tfrac{(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+ax_{6}^{2}+x_{% 7}^{2}+x_{8}^{2})x_{14}-2x_{6}(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+% x_{5}x_{13}+bx_{6}x_{14}+x_{7}x_{15}+x_{8}x_{16})}{c}
  27. u 7 = ( x 1 2 + x 2 2 + x 3 2 + x 4 2 + x 5 2 + x 6 2 + a x 7 2 + x 8 2 ) x 15 - 2 x 7 ( x 1 x 9 + x 2 x 10 + x 3 x 11 + x 4 x 12 + x 5 x 13 + x 6 x 14 + b x 7 x 15 + x 8 x 16 ) c u_{7}=\tfrac{(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+ax_{% 7}^{2}+x_{8}^{2})x_{15}-2x_{7}(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+% x_{5}x_{13}+x_{6}x_{14}+bx_{7}x_{15}+x_{8}x_{16})}{c}
  28. u 8 = ( x 1 2 + x 2 2 + x 3 2 + x 4 2 + x 5 2 + x 6 2 + x 7 2 + a x 8 2 ) x 16 - 2 x 8 ( x 1 x 9 + x 2 x 10 + x 3 x 11 + x 4 x 12 + x 5 x 13 + x 6 x 14 + x 7 x 15 + b x 8 x 16 ) c u_{8}=\tfrac{(x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}+x_{7% }^{2}+ax_{8}^{2})x_{16}-2x_{8}(x_{1}x_{9}+x_{2}x_{10}+x_{3}x_{11}+x_{4}x_{12}+% x_{5}x_{13}+x_{6}x_{14}+x_{7}x_{15}+bx_{8}x_{16})}{c}
  29. a = - 1 , b = 0 , c = x 1 2 + x 2 2 + x 3 2 + x 4 2 + x 5 2 + x 6 2 + x 7 2 + x 8 2 . a=-1,\;\;b=0,\;\;c=x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+x_{4}^{2}+x_{5}^{2}+x_{6}^{2}% +x_{7}^{2}+x_{8}^{2}\,.
  30. u i u_{i}
  31. u 1 2 + u 2 2 + u 3 2 + u 4 2 + u 5 2 + u 6 2 + u 7 2 + u 8 2 = x 9 2 + x 10 2 + x 11 2 + x 12 2 + x 13 2 + x 14 2 + x 15 2 + x 16 2 u_{1}^{2}+u_{2}^{2}+u_{3}^{2}+u_{4}^{2}+u_{5}^{2}+u_{6}^{2}+u_{7}^{2}+u_{8}^{2% }=x_{9}^{2}+x_{10}^{2}+x_{11}^{2}+x_{12}^{2}+x_{13}^{2}+x_{14}^{2}+x_{15}^{2}+% x_{16}^{2}\,
  32. ( x 1 2 + x 2 2 + x 3 2 + + x n 2 ) ( y 1 2 + y 2 2 + y 3 2 + + y n 2 ) = z 1 2 + z 2 2 + z 3 2 + + z n 2 (x_{1}^{2}+x_{2}^{2}+x_{3}^{2}+\cdots+x_{n}^{2})(y_{1}^{2}+y_{2}^{2}+y_{3}^{2}% +\cdots+y_{n}^{2})=z_{1}^{2}+z_{2}^{2}+z_{3}^{2}+\cdots+z_{n}^{2}\,
  33. z i z_{i}
  34. x i x_{i}
  35. y i y_{i}
  36. z i z_{i}
  37. n = 2 m n=2^{m}

Phase_resetting_in_neurons.html

  1. F ( ) = ( T 1 - T 0 ) T 0 F(\varnothing)=\frac{(T_{1}-T_{0})}{T_{0}}

Phenotypic_disease_network_(PDN).html

  1. R R i j = C i j N P i P j RR_{ij}=\frac{C_{ij}N}{P_{i}P_{j}}
  2. C i j C_{ij}
  3. P i P_{i}
  4. P j P_{j}
  5. ϕ \phi
  6. ϕ i j = C i j N - P i P j P i P j ( N - P i ) ( N - P j ) . \phi_{ij}=\frac{C_{ij}N-P_{i}P_{j}}{\sqrt{P_{i}P_{j}(N-P_{i})(N-P_{j})}}.
  7. ϕ \phi
  8. ϕ \phi

Phenylalanine::tyrosine_ammonia-lyase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Photoredox_catalysis.html

  1. ( I 0 I ) = 1 + k q * τ 0 × [ Q ] \left(\frac{I_{0}}{I}\right)=1+{k_{q}}*{\tau_{0}}\times[Q]
  2. E * 1 / 2 r e d = E 1 / 2 r e d + E 0 , 0 + w r {{E^{*}}_{1/2}}^{red}={E_{1/2}}^{red}+E_{0,0}+w_{r}
  3. E * 1 / 2 o x = E 1 / 2 o x - E 0 , 0 + w r {{E^{*}}_{1/2}}^{ox}={E_{1/2}}^{ox}-E_{0,0}+w_{r}

Phyllocladan-16alpha-ol_synthase.html

  1. \rightleftharpoons

Physics_applications_of_asymptotically_safe_gravity.html

  1. A λ A_{\lambda}
  2. A λ < 0 A_{\lambda}<0
  3. m H m\text{H}
  4. 126 GeV < m H < 174 GeV 126\,\,\text{GeV}<m\text{H}<174\,\,\text{GeV}
  5. A λ > 0 A_{\lambda}>0
  6. m H m\text{H}
  7. m H = 126 GeV , m\text{H}=126\,\,\text{GeV},
  8. m H m\text{H}
  9. α \alpha
  10. α \alpha
  11. α \alpha
  12. α IR \alpha\text{IR}
  13. α \alpha

Pillai's_arithmetical_function.html

  1. n n
  2. P ( n ) = k = 1 n gcd ( k , n ) P(n)=\sum_{k=1}^{n}\gcd(k,n)
  3. P ( n ) = d n d φ ( n / d ) P(n)=\sum_{d\mid n}d\varphi(n/d)
  4. d d
  5. n n
  6. φ \varphi
  7. P ( n ) = d n d σ ( d ) μ ( n / d ) P(n)=\sum_{d\mid n}d\sigma(d)\mu(n/d)
  8. σ \sigma
  9. μ \mu

Pisier–Ringrose_inequality.html

  1. γ \gamma
  2. 𝔄 \mathfrak{A}
  3. 𝔅 \mathfrak{B}
  4. j = 1 n γ ( A j ) * γ ( A j ) + γ ( A j ) γ ( A j ) * 4 γ 2 j = 1 n A j * A j + A j A j * \left\|\sum_{j=1}^{n}\gamma(A_{j})^{*}\gamma(A_{j})+\gamma(A_{j})\gamma(A_{j})% ^{*}\right\|\leq 4\|\gamma\|^{2}\left\|\sum_{j=1}^{n}A_{j}^{*}A_{j}+A_{j}A_{j}% ^{*}\right\|
  5. { A 1 , A 2 , , A n } \{A_{1},A_{2},\ldots,A_{n}\}
  6. A j A_{j}
  7. 𝔄 \mathfrak{A}

PISO_algorithm.html

  1. p * p^{*}
  2. u * u^{*}
  3. v * v^{*}
  4. p * * p^{**}
  5. u * * u^{**}
  6. v * * v^{**}
  7. p = p * * - p * p^{\prime}=p^{**}-p^{*}
  8. v = v * * - v * v^{\prime}=v^{**}-v^{*}
  9. u = u * * - u * u^{\prime}=u^{**}-u^{*}
  10. p * * , u * * , v * * p^{**},u^{**},v^{**}
  11. p , u , v p^{\prime},u^{\prime},v^{\prime}
  12. p * , u * , v * p^{*},u^{*},v^{*}
  13. p , u , v p^{\prime},u^{\prime},v^{\prime}
  14. p * * p^{**}
  15. v * * v^{**}
  16. u * * u^{**}
  17. p p^{\prime}
  18. u u^{\prime}
  19. v v^{\prime}
  20. p * * * = p * * + p ′′ p^{***}=p^{**}+p^{\prime\prime}
  21. p ′′ = p * + p p^{\prime\prime}=p^{*}+p^{\prime}
  22. u * * * = u * * + u ′′ u^{***}=u^{**}+u^{\prime\prime}
  23. u ′′ = u * + u u^{\prime\prime}=u^{*}+u^{\prime}
  24. v * * * = v * * + v ′′ v^{***}=v^{**}+v^{\prime\prime}
  25. v ′′ = v * + v v^{\prime\prime}=v^{*}+v^{\prime}
  26. p * * * , v * * * , u * * * p^{***},v^{***},u^{***}
  27. p ′′ , v ′′ , u ′′ p^{\prime\prime},v^{\prime\prime},u^{\prime\prime}
  28. p = p * * * , v = v * * * , u = u * * * p=p^{***},v=v^{***},u=u^{***}
  29. p , v , u p,v,u

Piston-cylinder_apparatus.html

  1. P = F A P=\frac{F}{A}
  2. 1 2 \tfrac{1}{2}
  3. 3 4 \tfrac{3}{4}

Place-permutation_action.html

  1. σ \sigma
  2. 𝔖 n \mathfrak{S}_{n}
  3. n n
  4. 𝔖 n \mathfrak{S}_{n}
  5. n n
  6. n n
  7. n n
  8. n n
  9. n n
  10. x 1 , , x n x_{1},\dots,x_{n}
  11. x = x 1 x n x=x_{1}\cdots x_{n}
  12. n n
  13. x x
  14. σ 𝔖 n \sigma\in\mathfrak{S}_{n}
  15. j j
  16. σ ( j ) \sigma(j)
  17. σ ( j ) \sigma(j)
  18. j j
  19. σ \sigma
  20. j j
  21. j j
  22. σ ( j ) \sigma(j)
  23. j j
  24. σ - 1 ( j ) \sigma^{-1}(j)
  25. j j
  26. j j
  27. j j
  28. σ - 1 ( j ) \sigma^{-1}(j)
  29. x 1 x n 𝜎 x σ - 1 ( 1 ) x σ - 1 ( n ) x_{1}\cdots x_{n}\overset{\sigma}{\longrightarrow}x_{\sigma^{-1}(1)}\cdots x_{% \sigma^{-1}(n)}
  30. τ \tau
  31. y 1 y n = x σ - 1 ( 1 ) x σ - 1 ( n ) y_{1}\cdots y_{n}=x_{\sigma^{-1}(1)}\cdots x_{\sigma^{-1}(n)}
  32. τ \tau
  33. y τ - 1 ( 1 ) y τ - 1 ( n ) = x σ - 1 τ - 1 ( 1 ) x σ - 1 τ - 1 ( n ) = x ( τ σ ) - 1 ( 1 ) x ( τ σ ) - 1 ( n ) . y_{\tau^{-1}(1)}\cdots y_{\tau^{-1}(n)}=x_{\sigma^{-1}\tau^{-1}(1)}\cdots x_{% \sigma^{-1}\tau^{-1}(n)}=x_{(\tau\sigma)^{-1}(1)}\cdots x_{(\tau\sigma)^{-1}(n% )}.
  34. τ ( σ x ) = ( τ σ ) x \tau\cdot(\sigma\cdot x)=(\tau\sigma)\cdot x
  35. σ \sigma
  36. j j
  37. j j
  38. σ - 1 ( j ) \sigma^{-1}(j)
  39. j j
  40. σ ( j ) \sigma(j)
  41. j j
  42. j j
  43. j j
  44. σ ( j ) \sigma(j)
  45. x 1 x n 𝜎 x σ ( 1 ) x σ ( n ) x_{1}\cdots x_{n}\overset{\sigma}{\longrightarrow}x_{\sigma(1)}\cdots x_{% \sigma(n)}
  46. τ \tau
  47. y 1 y n = x σ ( 1 ) x σ ( n ) y_{1}\cdots y_{n}=x_{\sigma(1)}\cdots x_{\sigma(n)}
  48. τ \tau
  49. y τ ( 1 ) y τ ( n ) = x σ τ ( 1 ) x σ τ ( n ) = x ( σ τ ) ( 1 ) x ( σ τ ) ( n ) . y_{\tau(1)}\cdots y_{\tau(n)}=x_{\sigma\tau(1)}\cdots x_{\sigma\tau(n)}=x_{(% \sigma\tau)(1)}\cdots x_{(\sigma\tau)(n)}.
  50. ( x σ ) τ = x ( σ τ ) (x\cdot\sigma)\cdot\tau=x\cdot(\sigma\tau)
  51. σ = ( 1 , 2 , 3 ) \sigma=(1,2,3)
  52. 1 2 3 1 1\to 2\to 3\to 1
  53. τ = ( 1 , 3 ) \tau=(1,3)
  54. 1 3 1 1\to 3\to 1
  55. σ τ = ( 1 , 2 , 3 ) ( 1 , 3 ) = ( 2 , 3 ) , τ σ = ( 1 , 3 ) ( 1 , 2 , 3 ) = ( 1 , 2 ) . \sigma\tau=(1,2,3)(1,3)=(2,3),\quad\tau\sigma=(1,3)(1,2,3)=(1,2).
  56. x = x 1 x 2 x 3 𝜎 x 3 x 1 x 2 𝜏 x 2 x 1 x 3 x=x_{1}x_{2}x_{3}\overset{\sigma}{\longrightarrow}x_{3}x_{1}x_{2}\overset{\tau% }{\longrightarrow}x_{2}x_{1}x_{3}
  57. τ σ = ( 1 , 2 ) \tau\sigma=(1,2)
  58. x = x 1 x 2 x 3 x=x_{1}x_{2}x_{3}
  59. τ ( σ x ) = ( τ σ ) x \tau\cdot(\sigma\cdot x)=(\tau\sigma)\cdot x
  60. x = x 1 x 2 x 3 𝜎 x 2 x 3 x 1 𝜏 x 1 x 3 x 2 x=x_{1}x_{2}x_{3}\overset{\sigma}{\longrightarrow}x_{2}x_{3}x_{1}\overset{\tau% }{\longrightarrow}x_{1}x_{3}x_{2}
  61. σ τ = ( 2 , 3 ) \sigma\tau=(2,3)
  62. x = x 1 x 2 x 3 x=x_{1}x_{2}x_{3}
  63. ( x σ ) τ = x ( σ τ ) (x\cdot\sigma)\cdot\tau=x\cdot(\sigma\tau)
  64. 𝔖 n \mathfrak{S}_{n}
  65. 𝔖 n op \mathfrak{S}_{n}^{\,\text{op}}
  66. x 1 x n 𝜎 x 1 σ - 1 x n σ - 1 x_{1}\cdots x_{n}\overset{\sigma}{\longrightarrow}x_{1\sigma^{-1}}\cdots x_{n% \sigma^{-1}}
  67. x 1 x n 𝜎 x 1 σ x n σ x_{1}\cdots x_{n}\overset{\sigma}{\longrightarrow}x_{1\sigma}\cdots x_{n\sigma}
  68. σ = ( 1 , 2 , 3 ) \sigma=(1,2,3)
  69. 1 2 3 1 1\to 2\to 3\to 1
  70. τ = ( 1 , 3 ) \tau=(1,3)
  71. 1 3 1 1\to 3\to 1
  72. σ τ = ( 1 , 2 , 3 ) ( 1 , 3 ) = ( 1 , 2 ) , τ σ = ( 1 , 3 ) ( 1 , 2 , 3 ) = ( 2 , 3 ) . \sigma\tau=(1,2,3)(1,3)=(1,2),\quad\tau\sigma=(1,3)(1,2,3)=(2,3).
  73. x = x 1 x 2 x 3 𝜎 x 3 x 1 x 2 𝜏 x 2 x 1 x 3 x=x_{1}x_{2}x_{3}\overset{\sigma}{\longrightarrow}x_{3}x_{1}x_{2}\overset{\tau% }{\longrightarrow}x_{2}x_{1}x_{3}
  74. σ τ = ( 1 , 2 ) \sigma\tau=(1,2)
  75. x = x 1 x 2 x 3 x=x_{1}x_{2}x_{3}
  76. ( x σ ) τ = x ( σ τ ) (x\cdot\sigma)\cdot\tau=x\cdot(\sigma\tau)
  77. x = x 1 x 2 x 3 𝜎 x 2 x 3 x 1 𝜏 x 1 x 3 x 2 x=x_{1}x_{2}x_{3}\overset{\sigma}{\longrightarrow}x_{2}x_{3}x_{1}\overset{\tau% }{\longrightarrow}x_{1}x_{3}x_{2}
  78. τ σ = ( 2 , 3 ) \tau\sigma=(2,3)
  79. x = x 1 x 2 x 3 x=x_{1}x_{2}x_{3}
  80. τ ( σ x ) = ( τ σ ) x \tau\cdot(\sigma\cdot x)=(\tau\sigma)\cdot x
  81. x 1 x n 𝜎 x σ - 1 ( 1 ) x σ - 1 ( n ) x_{1}\cdots x_{n}\overset{\sigma}{\longrightarrow}x_{\sigma^{-1}(1)}\cdots x_{% \sigma^{-1}(n)}
  82. x 1 x n 𝜎 x σ ( 1 ) x σ ( n ) x_{1}\cdots x_{n}\overset{\sigma}{\longrightarrow}x_{\sigma(1)}\cdots x_{% \sigma(n)}
  83. x 1 x n 𝜎 x 1 σ - 1 x n σ - 1 x_{1}\cdots x_{n}\overset{\sigma}{\longrightarrow}x_{1\sigma^{-1}}\cdots x_{n% \sigma^{-1}}
  84. x 1 x n 𝜎 x 1 σ x n σ x_{1}\cdots x_{n}\overset{\sigma}{\longrightarrow}x_{1\sigma}\cdots x_{n\sigma}

Planck–Einstein_relation.html

  1. E = h ν E=h\nu
  2. ν \nu
  3. λ \lambda
  4. ν ~ \tilde{\nu}
  5. ω \omega
  6. y y
  7. k k
  8. ν = c λ = c ν ~ = c ω 2 π = 2 π c y = c k 2 π , \nu=\frac{c}{\lambda}=c\tilde{\nu}=\frac{c\omega}{2\pi}=\frac{2\pi c}{y}=\frac% {ck}{2\pi},
  9. E = h ν = h c λ = h c ν ~ , E=h\nu=\frac{hc}{\lambda}=hc\tilde{\nu},
  10. E = ω = c y = c k . E=\hbar\omega=\frac{\hbar c}{y}=\hbar ck.
  11. = h 2 π \hbar=\frac{h}{2\pi}
  12. c c
  13. E = h ν E=h\nu
  14. λ = h p \lambda=\frac{h}{p}
  15. p = h ν ~ p=h\tilde{\nu}
  16. p = k . p=\hbar k.
  17. 𝐩 = 𝐤 , \mathbf{p}=\hbar\mathbf{k},
  18. 𝐩 \mathbf{p}
  19. 𝐤 \mathbf{k}
  20. Δ E \Delta E
  21. Δ E = h ν . \Delta E=h\nu.

Planted_motif_search.html

  1. ( l d ) 3 d {\textstyle\left({{l}\atop{d}}\right)}3^{d}
  2. 2 d - 4 d 2 3 l 2d-\tfrac{4d^{2}}{3l}
  3. O ( N 2 d + 1 ) O(N^{2d+1})
  4. O ( m 2 n l ( l d ) 3 d ) O(m^{2}nl{\textstyle\left({{l}\atop{d}}\right)}3^{d})
  5. i = 1 n L i . \bigcap_{i=1}^{n}L_{i}.
  6. M l d ( S ) M_{l}^{d}(S)
  7. L s 1 M l d ( L , S ) \bigcup_{L\in s_{1}}M_{l}^{d}(L,S^{^{\prime}})
  8. B d ( y ) B d ( z ) B_{d}(y)\bigcap B_{d}(z)
  9. B d ( y ) B d ( z ) B_{d}(y)\bigcap B_{d}(z)
  10. B d ( y ) B d ( z ) B_{d}(y)\bigcap B_{d}(z)
  11. x s i {x\in s_{i}}
  12. x s i x\in s_{i}
  13. y s j y\in s_{j}
  14. B d ( x ) B d ( y ) B_{d}(x)\bigcap B_{d}(y)
  15. B d ( x ) B d ( y ) B_{d}(x)\bigcap B_{d}(y)
  16. B d ( x ) B d ( y ) B_{d}(x)\bigcap B_{d}(y)

Plasma_fusion_preface.html

  1. n i n n = 2.4 * 10 21 T 3 / 2 n i exp - ( U i k T ) \frac{n_{i}}{n_{n}}=2.4*10^{21}\frac{T^{3/2}}{n_{i}}\exp-(\frac{U_{i}}{kT})
  2. n n = 3 * 10 25 m - 3 n_{n}=3*10^{25}m^{-3}
  3. T = 300 K T=300K
  4. U i = 14 , 5 e V ( n i t r o g e n ) U_{i}=14,5eV(nitrogen)
  5. n i n n = 10 - 122 \frac{n_{i}}{n_{n}}=10^{-122}
  6. m d v d t = q v X B m\frac{dv}{dt}=qvXB
  7. v = v 0 e j w t v=v_{0}e^{jwt}
  8. m j w v 0 e j w t = q v X B = m j w v mjwv_{0}e^{jwt}=qvXB=mjwv
  9. w c = | q | B m w_{c}=\frac{|q|B}{m}
  10. r L = m v | q | B r_{L}=\frac{mv}{|q|B}
  11. E A V = 1 4 m v 2 = 1 2 k T E_{AV}=\frac{1}{4}mv^{2}=\frac{1}{2}kT
  12. v = 2 k T m v=\sqrt{\frac{2kT}{m}}
  13. f ( v ) = A exp ( - m v 2 / 2 k T ) f(v)=A\exp{(-\frac{mv^{2}/2}{kT})}
  14. n = - f ( v ) d v n=\int_{-\infty}^{\infty}f(v)dv
  15. A = n m 2 π k T A=n\sqrt{\frac{m}{2\pi kT}}
  16. m v 2 2 = k T \frac{mv^{2}}{2}=kT
  17. k T = e V kT=eV
  18. k = 1 , 38 * 10 - 23 [ J / K ] k=1,38*10^{-23}[J/K]
  19. e = 1 , 6 * 10 - 19 [ A s ] e=1,6*10^{-19}[As]
  20. m p = 1 , 67 * 10 - 27 [ k g ] m_{p}=1,67*10^{-27}[kg]
  21. μ 0 = 4 π * 10 - 7 [ V s / A m ] \mu_{0}=4\pi*10^{-7}[Vs/Am]
  22. k T = 10 k e V kT=10keV
  23. B = 5 T B=5T
  24. R = 2 m R=2m
  25. I D C = M A I_{DC}=MA
  26. E p r o t o n s = 10 M e V E_{protons}=10MeV
  27. T = 10 4 * 1 , 6 * 10 - 19 1 , 38 * 10 - 23 T=\frac{10^{4}*1,6*10^{-19}}{1,38*10^{-23}}
  28. E A V = 1 4 m v 2 = 1 2 k T E_{AV}=\frac{1}{4}mv^{2}=\frac{1}{2}kT
  29. m = 2 m p m=2m_{p}
  30. v = 2 k T 2 m p = k T m p v=\sqrt{\frac{2kT}{2m_{p}}}=\sqrt{\frac{kT}{m_{p}}}
  31. r L = 2 m p v | 2 q | B = m p v | q | B r_{L}=\frac{2m_{p}v}{|2q|B}=\frac{m_{p}v}{|q|B}
  32. w c = | 2 q | B 2 m p = | q | B m p w_{c}=\frac{|2q|B}{2m_{p}}=\frac{|q|B}{m_{p}}
  33. v = 50 , 000 k m / s v=50,000km/s
  34. r L = 0 , 1 m r_{L}=0,1m
  35. B = μ 0 N I A 2 π r 3 = μ 0 N I R 2 2 r 3 = μ 0 N I 2 R B=\mu_{0}\frac{NIA}{2\pi r^{3}}=\mu_{0}\frac{NIR^{2}}{2r^{3}}=\mu_{0}\frac{NI}% {2R}
  36. I = 2 R B μ 0 N = 16 M A / N I=\frac{2RB}{\mu_{0}N}=16MA/N
  37. B = 0 \nabla\cdot B=0
  38. B = X A B=\nabla XA
  39. A = μ 0 4 π v J R d v A=\frac{\mu_{0}}{4\pi}\int_{v}{\frac{J}{R}dv}
  40. J d v = J S d l = I d l Jdv=JSdl=Idl
  41. B = μ 0 I 4 π c d l X a R R 2 B=\frac{\mu_{0}I}{4\pi}\oint_{c}\frac{dlXa_{R}}{R^{2}}
  42. d l = b d ϕ a ϕ dl=bd\phi a_{\phi}
  43. R = a z z - a r b R=a_{z}z-a_{r}b
  44. d l X R = a ϕ b d ϕ X ( a z z - a r b ) = a r b z d ϕ + a z b 2 d ϕ dlXR=a_{\phi}bd\phi X(a_{z}z-a_{r}b)=a_{r}bzd\phi+a_{z}b^{2}d\phi
  45. B = μ 0 I 4 π 0 2 π a z b 2 d ϕ ( z 2 + b 2 ) 3 / 2 B=\frac{\mu_{0}I}{4\pi}\int_{0}^{2\pi}a_{z}\frac{b^{2}d\phi}{(z^{2}+b^{2})^{3/% 2}}
  46. B = μ 0 I 2 b 2 ( z 2 + b 2 ) 3 / 2 = μ 0 I 2 b 2 R 3 B=\frac{\mu_{0}I}{2}\frac{b^{2}}{(z^{2}+b^{2})^{3/2}}=\frac{\mu_{0}I}{2}\frac{% b^{2}}{R^{3}}
  47. m d v d t = q ( E + v X B ) m\frac{dv}{dt}=q(E+vXB)
  48. E X B = B X ( v X B ) = v B 2 - B ( v B ) EXB=BX(vXB)=vB^{2}-B(v\cdot B)
  49. v g c = E X B B 2 v_{gc}=\frac{EXB}{B^{2}}
  50. v g c = E B v_{gc}=\frac{E}{B}
  51. F = q E F=qE
  52. v f o r c e = 1 q F X B B 2 v_{force}=\frac{1}{q}\frac{FXB}{B^{2}}
  53. F E = q E F_{E}=qE
  54. F g = m g F_{g}=mg
  55. F c f = a r m v / / 2 R c F_{cf}=a_{r}\frac{mv_{//}^{2}}{R_{c}}
  56. v E = E X B B 2 v_{E}=\frac{EXB}{B^{2}}
  57. v g = m q g X B B 2 v_{g}=\frac{m}{q}\frac{gXB}{B^{2}}
  58. v R = 1 q F c f X B B 2 = m v / / 2 q B 2 R c X B R c 2 v_{R}=\frac{1}{q}\frac{F_{cf}XB}{B^{2}}=\frac{mv_{//}^{2}}{qB^{2}}\frac{R_{c}% XB}{R_{c}^{2}}
  59. | v E | = | E B | |v_{E}|=|\frac{E}{B}|
  60. F y = - / + q v p r L 2 d B d y a y F_{y}=-/+\frac{qv_{p}r_{L}}{2}\frac{dB}{dy}a_{y}
  61. v g c = 1 q F X B B 2 = 1 q F y | B | a x = - / + v p r L 2 B d B d y a x v_{gc}=\frac{1}{q}\frac{FXB}{B^{2}}=\frac{1}{q}\frac{F_{y}}{|B|}a_{x}=-/+\frac% {v_{p}r_{L}}{2B}\frac{dB}{dy}a_{x}
  62. v B = - / + v p r L 2 B X B B 2 v_{\nabla B}=-/+\frac{v_{p}r_{L}}{2}\frac{\nabla BXB}{B^{2}}
  63. v c v = v R + v B = m q B 2 R c X B R c 2 ( v / / 2 + 1 2 v p 2 ) v_{cv}=v_{R}+v_{\nabla B}=\frac{m}{qB^{2}}\frac{RcXB}{Rc^{2}}(v_{//}^{2}+\frac% {1}{2}v_{p}^{2})
  64. m n [ d v d t + ( v ) v ] = q n ( E + v X B ) - p mn[\frac{dv}{dt}+(v\cdot\nabla)v]=qn(E+vXB)-\nabla p
  65. 0 = q n [ E X B + ( v p X B ) X B ] - p X B 0=qn[EXB+(v_{p}XB)XB]-\nabla pXB
  66. 0 = q n [ E X B - v p B 2 ] - p X B 0=qn[EXB-v_{p}B^{2}]-\nabla pXB
  67. v p = E X B B 2 - p X B q n B 2 = v E + v D v_{p}=\frac{EXB}{B^{2}}-\frac{\nabla pXB}{qnB^{2}}=v_{E}+v_{D}
  68. v D = - p X B q n B 2 v_{D}=-\frac{\nabla pXB}{qnB^{2}}
  69. F D = - p n F_{D}=-\frac{\nabla p}{n}
  70. p = n k T p=nkT
  71. p = k T n \nabla p=kT\nabla n
  72. N B d S = L I N\int BdS=LI
  73. B = μ 0 N I l m B=\mu_{0}\frac{NI}{l_{m}}
  74. B = μ 0 N I 2 R = μ 0 N I D B=\mu_{0}\frac{NI}{2R}=\mu_{0}\frac{NI}{D}
  75. L = N B A I = μ 0 N 2 A 2 R L=\frac{NBA}{I}=\mu_{0}\frac{N^{2}A}{2R}
  76. L = μ 0 N 2 π R 2 = μ 0 N 2 π D 4 L=\mu_{0}\frac{N^{2}\pi R}{2}=\mu_{0}\frac{N^{2}\pi D}{4}
  77. L = 0 , 4 m H L=0,4mH
  78. L = 5 m H L=5mH
  79. e = - L d i d t e=-L\frac{di}{dt}
  80. B = μ 0 N I R 2 2 r 3 B=\mu_{0}\frac{NIR^{2}}{2r^{3}}
  81. 1 / r 3 1/r^{3}
  82. z - a x i s z-axis
  83. ϕ - a x i s \phi-axis
  84. q / C = R c u i + L d i d t q/C=R_{cu}i+L\frac{di}{dt}
  85. i = - d q d t i=-\frac{dq}{dt}
  86. L i ′′ + R c u i + i / C = 0 Li^{\prime\prime}+R_{cu}i^{\prime}+i/C=0
  87. i ′′ + R c u L i + 1 L C i = 0 i^{\prime\prime}+\frac{R_{cu}}{L}i^{\prime}+\frac{1}{LC}i=0
  88. b = R c u / 2 L b=R_{cu}/2L
  89. w = 1 L C w=\frac{1}{\sqrt{LC}}
  90. ( r 2 + 2 b r + w 2 ) i = 0 (r^{2}+2br+w^{2})i=0
  91. r 1 , 2 = - b + / - b 2 - w 2 r_{1,2}=-b+/-\sqrt{b^{2}-w^{2}}
  92. b > w b>w
  93. i ( t ) = C 1 e r 1 t + C 2 e r 2 t i(t)=C_{1}e^{r_{1}t}+C_{2}e^{r_{2}t}
  94. i ( 0 ) = 0 i(0)=0
  95. i ( 0 ) = E / L i^{\prime}(0)=E/L
  96. E = - L d i d t E=-L\frac{di}{dt}
  97. i ( 0 ) = C 1 + C 2 = = 0 i(0)=C_{1}+C_{2}==0
  98. C 2 = - C 1 C_{2}=-C_{1}
  99. i ( t ) = C 1 ( e r 1 t - e r 2 t ) i(t)=C_{1}(e^{r_{1}t}-e^{r_{2}t})
  100. i ( 0 ) = C 1 ( r 1 - r 2 ) = = E / L i^{\prime}(0)=C_{1}(r_{1}-r_{2})==E/L
  101. C 1 = E / L r 1 - r 2 C_{1}=\frac{E/L}{r_{1}-r_{2}}
  102. i ( t ) = E / L r 1 - r 2 ( e r 1 t - e r 2 t ) i(t)=\frac{E/L}{r_{1}-r_{2}}(e^{r_{1}t}-e^{r_{2}t})
  103. r 1 e r 1 t = r 2 e r 2 t r_{1}e^{r_{1}t}=r_{2}e^{r_{2}t}
  104. l n ( r 1 r 2 ) = t ( r 2 - r 1 ) ln(\frac{r_{1}}{r_{2}})=t(r_{2}-r_{1})
  105. t m a x = l n ( r 1 / r 2 ) r 2 - r 1 t_{max}=\frac{ln(r_{1}/r_{2})}{r_{2}-r_{1}}
  106. i ( t ) = E / L b 2 - w 2 e - b t e b 2 - w 2 t - e - b 2 - w 2 t 2 i(t)=\frac{E/L}{\sqrt{b^{2}-w^{2}}}e^{-bt}\frac{e^{\sqrt{b^{2}-w^{2}}t}-e^{-% \sqrt{b^{2}-w^{2}}t}}{2}
  107. b = w b=w
  108. i ( t ) = ( C 1 t + C 2 ) e - b t i(t)=(C_{1}t+C_{2})e^{-bt}
  109. b < w b<w
  110. γ = w 2 - b 2 > 0 \gamma=\sqrt{w^{2}-b^{2}}>0
  111. r 1 , 2 = - b + / - j γ r_{1,2}=-b+/-j\gamma
  112. i ( t ) = C e - b t s i n ( γ t + α ) i(t)=Ce^{-bt}sin(\gamma t+\alpha)
  113. γ \gamma
  114. C e - b t Ce^{-bt}
  115. b = R c u / 2 L 1 L C = w b=R_{cu}/2L\frac{1}{\sqrt{LC}}=w
  116. i m a x = E / L r 1 - r 2 ( e r 1 r 2 - r 1 l n ( r 1 / r 2 ) - e r 2 r 2 - r 1 l n ( r 1 / r 2 ) ) i_{max}=\frac{E/L}{r_{1}-r_{2}}(e^{\frac{r_{1}}{r_{2}-r_{1}}ln(r1/r2)}-e^{% \frac{r_{2}}{r_{2}-r_{1}}ln(r1/r2)})
  117. e 4 l n 5 = 5 4 e^{4ln5}=5^{4}
  118. i m a x = E / L r 1 - r 2 ( ( r 1 r 2 ) r 1 r 2 - r 1 - ( r 1 r 2 ) r 2 r 2 - r 1 ) i_{max}=\frac{E/L}{r_{1}-r_{2}}((\frac{r_{1}}{r_{2}})^{\frac{r_{1}}{r_{2}-r_{1% }}}-(\frac{r_{1}}{r_{2}})^{\frac{r_{2}}{r_{2}-r_{1}}})
  119. u c ( t ) = E ( 1 - e - t / R C ) u_{c}(t)=E(1-e^{-t/RC})
  120. i = C d u d t i=C\frac{du}{dt}
  121. E = R i + u c = R C d u d t + u E=Ri+u_{c}=RC\frac{du}{dt}+u
  122. u c ( t ) = A e - k t + B u_{c}(t)=Ae^{-kt}+B
  123. u = - k A e - k t u^{\prime}=-kAe^{-kt}
  124. u ( 0 ) = 0 u(0)=0
  125. u ( 0 ) = E / R C u^{\prime}(0)=E/RC
  126. i = C d u d t i=C\frac{du}{dt}
  127. i ( 0 ) C = E / R C \frac{i(0)}{C}=\frac{E/R}{C}
  128. u ( ) = E u(\infty)=E
  129. u ( 0 ) = A + B = = 0 = > B = - A u(0)=A+B==0=>B=-A
  130. u ( ) = B = E u(\infty)=B=E
  131. u ( t ) = E ( 1 - e - k t ) u(t)=E(1-e^{-kt})
  132. u = E k e - k t u^{\prime}=Eke^{-kt}
  133. u ( 0 ) = E k = = E / R C u^{\prime}(0)=Ek==E/RC
  134. k = 1 / R C k=1/RC
  135. u c ( t ) = E ( 1 - e - t / R C ) u_{c}(t)=E(1-e^{-t/RC})
  136. i ( t ) = C d u d t = E R e - t / R C i(t)=C\frac{du}{dt}=\frac{E}{R}e^{-t/RC}
  137. E t o t = - k e 2 2 r E_{tot}=-\frac{ke^{2}}{2r}
  138. r n 2 r\propto n^{2}
  139. P s T 4 P_{s}\propto T^{4}
  140. B = 2 h f 3 c 2 1 e h f k T - 1 B=\frac{2hf^{3}}{c^{2}}\frac{1}{e^{\frac{hf}{kT}}-1}
  141. λ m a x T = b \lambda_{max}T=b
  142. U = - L d I d t = - j w L I U=-L\frac{dI}{dt}=-jwLI
  143. I = I 0 e j w t I=I_{0}e^{jwt}
  144. ϕ = B d s = B S = μ 0 H S = μ 0 N I S / l m [ W b ] \phi=\int Bds=BS=\mu_{0}HS=\mu_{0}NIS/l_{m}[Wb]
  145. e = E d l = - d ϕ d t = - j w ϕ [ V ] e=\oint Edl=-\frac{d\phi}{dt}=-jw\phi[V]
  146. N ϕ = L I N\phi=LI
  147. X E = - d B d t \nabla XE=-\frac{dB}{dt}
  148. m v r B = mvr_{B}=\hbar
  149. n λ = 2 π r n\lambda=2\pi r
  150. λ = h / p \lambda=h/p
  151. = h / 2 π \hbar=h/2\pi
  152. p r = m v r = n pr=mvr=n\hbar
  153. m v 2 r = k e 2 r 2 \frac{mv^{2}}{r}=\frac{ke^{2}}{r^{2}}
  154. k = 1 4 π ϵ 0 k=\frac{1}{4\pi\epsilon_{0}}
  155. v = k e 2 m r v=\sqrt{\frac{ke^{2}}{mr}}
  156. E p = - k e 2 r E_{p}=-\frac{ke^{2}}{r}
  157. E k = m v 2 2 E_{k}=\frac{mv^{2}}{2}
  158. E t o t = E p / 2 = - k e 2 2 r E_{tot}=E_{p}/2=-\frac{ke^{2}}{2r}
  159. m r v = m r k e 2 m r = k e 2 m r = = n mrv=mr\sqrt{\frac{ke^{2}}{mr}}=\sqrt{ke^{2}mr}==n\hbar
  160. r = n 2 2 k e 2 m r=\frac{n^{2}\hbar^{2}}{ke^{2}m}
  161. v = k e 2 n v=\frac{ke^{2}}{n\hbar}
  162. p p = ρ g h [ N / m 2 ] p_{p}=\rho gh[N/m^{2}]
  163. p = n k T [ J / m 3 ] p=nkT[J/m^{3}]
  164. ρ \rho
  165. 1 a t m = 10 5 P a = 10 5 N / m 2 = 10 4 k g / m 2 = 1 k g / c m 2 1atm=10^{5}Pa=10^{5}N/m^{2}=10^{4}kg/m^{2}=1kg/cm^{2}
  166. p k = 1 / 2 ρ v 2 p_{k}=1/2\rho v^{2}
  167. p = p 0 - ρ g h p=p_{0}-\rho gh
  168. ρ \rho
  169. p = p 0 e - m g h k T p=p_{0}e^{-\frac{mgh}{kT}}
  170. k T i t e r = 10 k e V = 1 , 6 * 10 - 15 10 - 15 [ J ] kT_{iter}=10keV=1,6*10^{-15}\propto 10^{-15}[J]
  171. p n * 10 - 15 [ J / m 3 ] p\propto n*10^{-15}[J/m^{3}]
  172. n n i t r o g e n = 3 * 10 25 10 25 n_{nitrogen}=3*10^{25}\propto 10^{25}
  173. p 10 10 [ J / m 3 ] p\propto 10^{10}[J/m^{3}]
  174. p 10 5 [ J / m 3 ] p\propto 10^{5}[J/m^{3}]
  175. p 10 5 [ a t m ] p\propto 10^{5}[atm]
  176. p = N m o l V R T = N V k T = n k T p=\frac{N_{mol}}{V}RT=\frac{N}{V}kT=nkT
  177. ( P V ) N = k T = E k \frac{(PV)}{N}=kT=E_{k}
  178. Q = Δ U + W Q=\Delta U+W
  179. U = K E + P E U=KE+PE
  180. W = F d r W=\int Fdr

Platt_scaling.html

  1. x x
  2. + 1 +1
  3. 1 −1
  4. f f
  5. y = s i g n ( f ( x ) ) y=sign(f(x))
  6. P ( y = 1 | x ) P(y=1|x)
  7. P ( y = 1 | x ) = 1 1 + exp ( A f ( x ) + B ) \mathrm{P}(y=1|x)=\frac{1}{1+\exp(Af(x)+B)}
  8. f ( x ) f(x)
  9. A A
  10. B B
  11. y = 1 y=1
  12. P ( y = 1 | x ) > ½ P(y=1|x)>½
  13. B 0 B≠0
  14. y = s i g n ( f ( x ) ) y=sign(f(x))
  15. A A
  16. B B
  17. f f
  18. y y
  19. t + = N + + 1 N + + 2 t_{+}=\frac{N_{+}+1}{N_{+}+2}
  20. y = 1 y=1
  21. t - = 1 N - + 2 t_{-}=\frac{1}{N_{-}+2}
  22. y = - 1 y=-1
  23. N N₊
  24. N N₋

Plethystic_substitution.html

  1. Λ R ( x 1 , x 2 , ) \Lambda_{R}(x_{1},x_{2},\ldots)
  2. p k = x 1 k + x 2 k + x 3 k + . p_{k}=x_{1}^{k}+x_{2}^{k}+x_{3}^{k}+\cdots.
  3. f f
  4. A = a 1 + a 2 + A=a_{1}+a_{2}+\cdots
  5. p k a 1 k + a 2 k + a 3 k + p_{k}\longrightarrow a_{1}^{k}+a_{2}^{k}+a_{3}^{k}+\cdots
  6. f f
  7. X X
  8. X = x 1 + x 2 + X=x_{1}+x_{2}+\cdots
  9. f [ X ] = f ( x 1 , x 2 , ) f[X]=f(x_{1},x_{2},\ldots)
  10. 1 / ( 1 - t ) 1/(1-t)
  11. 1 + t + t 2 + t 3 + 1+t+t^{2}+t^{3}+\cdots
  12. f [ 1 / ( 1 - t ) ] f[1/(1-t)]
  13. x i = t i - 1 x_{i}=t^{i-1}
  14. f [ 1 1 - t ] = f ( 1 , t , t 2 , t 3 , ) f\left[\frac{1}{1-t}\right]=f(1,t,t^{2},t^{3},\ldots)
  15. X = x 1 + x 2 + , x n X=x_{1}+x_{2}+\cdots,x_{n}
  16. f [ X ] = f ( x 1 , , x n ) f[X]=f(x_{1},\ldots,x_{n})
  17. Λ R ( x 1 , , x n ) \Lambda_{R}(x_{1},\ldots,x_{n})
  18. X = x 1 + x 2 + X=x_{1}+x_{2}+\cdots
  19. Y = y 1 + y 2 + Y=y_{1}+y_{2}+\cdots
  20. f f
  21. d d
  22. f [ t X ] = t d f ( x 1 , x 2 , ) f[tX]=t^{d}f(x_{1},x_{2},\ldots)
  23. f f
  24. d d
  25. f [ - X ] = ( - 1 ) d ω f ( x 1 , x 2 , ) f[-X]=(-1)^{d}\omega f(x_{1},x_{2},\ldots)
  26. ω \omega
  27. s λ s_{\lambda}
  28. s λ s_{\lambda^{\ast}}
  29. S : f f [ - X ] S:f\mapsto f[-X]
  30. p n [ X + Y ] = p n [ X ] + p n [ Y ] p_{n}[X+Y]=p_{n}[X]+p_{n}[Y]
  31. Δ : f f [ X + Y ] \Delta:f\mapsto f[X+Y]
  32. h n [ X ( 1 - t ) ] h_{n}\left[X(1-t)\right]
  33. h n h_{n}
  34. n n
  35. h n [ X / ( 1 - t ) ] h_{n}\left[X/(1-t)\right]

Poincaré_separation_theorem.html

  1. λ i \lambda_{i}
  2. μ i \mu_{i}
  3. λ i μ i λ n - r + i , \lambda_{i}\geq\mu_{i}\geq\lambda_{n-r+i},

Poincaré–Miranda_theorem.html

  1. x i x_{i}
  2. f i f_{i}
  3. x i = 0 x_{i}=0
  4. x i = 1 x_{i}=1
  5. n = 2 n=2
  6. x i x_{i}
  7. a i a_{i}
  8. [ sup x i = 0 f i , inf x i = 1 f i ] [\sup_{x_{i}=0}{f_{i}},\inf_{x_{i}=1}{f_{i}}]
  9. f i = a i f_{i}=a_{i}

Point_process_notation.html

  1. N {N}
  2. x N , x\in{N},
  3. x x
  4. N {N}
  5. { x 1 , x 2 , } = { x } i , \{x_{1},x_{2},\dots\}=\{x\}_{i},
  6. N {N}
  7. B B
  8. Φ ( B ) = # ( B N ) , \Phi(B)=\#(B\cap{N}),
  9. Φ ( B ) \Phi(B)
  10. # \#
  11. N {N}
  12. Φ \Phi
  13. N {N}
  14. B B
  15. Φ ( B ) = n \Phi(B)=n
  16. B B
  17. n n
  18. N {N}
  19. N {N}
  20. N ( B ) {N}(B)
  21. N {N}
  22. B B
  23. # \#
  24. N ( B ) = # ( B N ) . {N}(B)=\#(B\cap{N}).
  25. f f
  26. f ( x ) f(x)
  27. x x
  28. N {N}
  29. f ( x 1 ) + f ( x 2 ) + f(x_{1})+f(x_{2})+\cdots
  30. x N f ( x ) \sum_{x\in{N}}f(x)
  31. 𝐍 f ( x ) N ( d x ) \int_{\,\textbf{N}}f(x){N}(dx)
  32. 𝐍 \,\textbf{N}
  33. N {N}
  34. 𝐍 f d N \int_{\,\textbf{N}}f\,d{N}
  35. N {N}
  36. B B
  37. N ( B ) = x N 1 B ( x ) {N}(B)=\sum_{x\in{N}}1_{B}(x)
  38. 1 B ( x ) = 1 1_{B}(x)=1
  39. x x
  40. B B
  41. E [ x N f ( x ) ] or 𝐍 x N f ( x ) P ( d N ) , E\left[\sum_{x\in{N}}f(x)\right]\qquad\,\text{or}\qquad\int_{\,\textbf{N}}\sum% _{x\in{N}}f(x)P(d{N}),
  42. P P
  43. 𝐍 \,\textbf{N}
  44. N ( B ) {N}(B)
  45. E [ N ( B ) ] = E ( x N 1 B ( x ) ) or 𝐍 x N 1 B ( x ) P ( d N ) . E[{N}(B)]=E\left(\sum_{x\in{N}}1_{B}(x)\right)\qquad\,\text{or}\qquad\int_{\,% \textbf{N}}\sum_{x\in{N}}1_{B}(x)P(d{N}).
  46. N {N}