wpmath0000007_9

Nef_line_bundle.html

  1. D C 0 D\cdot C\geq 0\,

Nernst–Planck_equation.html

  1. c \nabla c
  2. E = - ϕ - 𝐀 t E=-\nabla\phi-\frac{\partial\mathbf{A}}{\partial t}
  3. c t = [ D c - u c + D z e k B T c ( ϕ + 𝐀 t ) ] \frac{\partial c}{\partial t}=\nabla\cdot\left[D\nabla c-uc+\frac{Dze}{k_{B}T}% c(\nabla\phi+\frac{\partial\mathbf{A}}{\partial t})\right]
  4. k B k_{B}
  5. J = - u c J=-uc
  6. J J
  7. [ D c + J + D z e k B T c ( ϕ ) ] = 0 \nabla\cdot\left[D\nabla c+J+\frac{Dze}{k_{B}T}c(\nabla\phi)\right]=0
  8. J = - [ D c + D z e k B T c ( ϕ ) ] J=-\left[D\nabla c+\frac{Dze}{k_{B}T}c(\nabla\phi)\right]
  9. C o u l o m b s M 2 \frac{Coulombs}{M^{2}}
  10. J = - z F D [ c + F z c R T ( ϕ ) ] J=-zFD\left[\nabla c+\frac{Fzc}{RT}(\nabla\phi)\right]

Net_acid_excretion.html

  1. N A E = V ( U N H 4 + U T A - U H C O 3 ) NAE=V(U_{NH_{4}}+U_{TA}-U_{HCO_{3}})

Net_run_rate.html

  1. run rate = total runs scored total overs faced \,\text{run rate }=\frac{\,\text{total runs scored}}{\,\text{total overs faced}}
  2. 250 50 = 5 \frac{250}{50}=5
  3. 250 47 5 6 5.226 \frac{250}{47\frac{5}{6}}\approx 5.226
  4. net run rate = total runs scored total overs faced - total runs conceded total overs bowled \,\text{net run rate }=\frac{\,\text{total runs scored}}{\,\text{total overs % faced}}-\frac{\,\text{total runs conceded }}{\,\text{total overs bowled}}
  5. tournament net run rate = total runs scored in all matches total overs faced in all matches - total runs conceded in all matches total overs bowled in all matches \,\text{tournament net run rate }=\frac{\,\text{total runs scored in all % matches}}{\,\text{total overs faced in all matches}}-\frac{\,\text{total runs % conceded in all matches}}{\,\text{total overs bowled in all matches}}
  6. 287 50 = 5.74 \frac{287}{50}=5.74
  7. 243 50 = 4.86 \frac{243}{50}=4.86
  8. 265 50 = 5.300 \frac{265}{50}=5.300
  9. 267 47 2 6 5.64 \frac{267}{47\frac{2}{6}}\approx 5.64
  10. 287 + 265 50 + 50 - 243 + 267 50 + 47.33 = 552 100 - 510 97.33 0.28 \frac{287+265}{50+50}-\frac{243+267}{50+47.33}=\frac{552}{100}-\frac{510}{97.3% 3}\approx 0.28
  11. = 149 44 3.39 =\frac{149}{44}\approx 3.39
  12. 150 26.33 5.70 \frac{150}{26.33}\approx 5.70
  13. 254 47.333 - 253 50 . \frac{\mbox{254}~{}}{\mbox{47.333}~{}}-\frac{\mbox{253}~{}}{\mbox{50}~{}}.
  14. 254 + 199 47.333 + 50 - 253 + 110 50 + 50 , \frac{\mbox{254 + 199}~{}}{\mbox{47.333 + 50}~{}}-\frac{\mbox{253 + 110}~{}}{% \mbox{50 + 50}~{}},
  15. 254 97.333 + 199 97.333 - 253 100 - 110 100 . \frac{\mbox{254}~{}}{\mbox{97.333}~{}}+\frac{\mbox{199}~{}}{\mbox{97.333}~{}}-% \frac{\mbox{253}~{}}{\mbox{100}~{}}-\frac{\mbox{110}~{}}{\mbox{100}~{}}.
  16. 254 + 199 + 225 47.333 + 50 + 50 - 253 + 110 + 103 50 + 50 + 50 , \frac{\mbox{254 + 199 + 225}~{}}{\mbox{47.333 + 50 + 50}~{}}-\frac{\mbox{253 +% 110 + 103}~{}}{\mbox{50 + 50 + 50}~{}},
  17. 254 147.333 + 199 147.333 + 225 147.333 - 253 150 - 110 150 - 103 150 . \frac{\mbox{254}~{}}{\mbox{147.333}~{}}+\frac{\mbox{199}~{}}{\mbox{147.333}~{}% }+\frac{\mbox{225}~{}}{\mbox{147.333}~{}}-\frac{\mbox{253}~{}}{\mbox{150}~{}}-% \frac{\mbox{110}~{}}{\mbox{150}~{}}-\frac{\mbox{103}~{}}{\mbox{150}~{}}.
  18. ( 5.37 × 47.333 97.333 ) + ( 3.98 × 50 97.333 ) - ( 5.06 × 50 100 ) - ( 2.20 × 50 100 ) \left(5.37\times\frac{\mbox{47.333}~{}}{\mbox{97.333}~{}}\right)+\left(3.98% \times\frac{\mbox{50}~{}}{\mbox{97.333}~{}}\right)-\left(5.06\times\frac{\mbox% {50}~{}}{\mbox{100}~{}}\right)-\left(2.20\times\frac{\mbox{50}~{}}{\mbox{100}~% {}}\right)
  19. = ( 5.37 × 48.6 % ) + ( 3.98 × 51.4 % ) - ( 5.06 × 50 % ) - ( 2.20 × 50 % ) . =\left(5.37\times 48.6\%\right)+\left(3.98\times 51.4\%\right)-\left(5.06% \times 50\%\right)-\left(2.20\times 50\%\right).
  20. ( 5.37 × 47.333 147.333 ) + ( 3.98 × 50 147.333 ) + ( 4.50 × 50 147.333 ) - ( 5.06 × 50 150 ) - ( 2.20 × 50 150 ) - ( 2.06 × 50 150 ) \left(5.37\times\frac{\mbox{47.333}~{}}{\mbox{147.333}~{}}\right)+\left(3.98% \times\frac{\mbox{50}~{}}{\mbox{147.333}~{}}\right)+\left(4.50\times\frac{% \mbox{50}~{}}{\mbox{147.333}~{}}\right)-\left(5.06\times\frac{\mbox{50}~{}}{% \mbox{150}~{}}\right)-\left(2.20\times\frac{\mbox{50}~{}}{\mbox{150}~{}}\right% )-\left(2.06\times\frac{\mbox{50}~{}}{\mbox{150}~{}}\right)
  21. = ( 5.37 × 32.1 % ) + ( 3.98 × 33.9 % ) + ( 4.50 × 33.9 % ) - ( 5.06 × 33.3 % ) - ( 2.20 × 33.3 % ) - ( 2.06 × 33.3 % ) . =\left(5.37\times 32.1\%\right)+\left(3.98\times 33.9\%\right)+\left(4.50% \times 33.9\%\right)-\left(5.06\times 33.3\%\right)-\left(2.20\times 33.3\%% \right)-\left(2.06\times 33.3\%\right).

Network_analyzer_(electrical).html

  1. S 11 S_{11}\,
  2. S 21 S_{21}\,
  3. S 12 S_{12}\,
  4. S 22 S_{22}\,
  5. S 11 S_{11}\,
  6. S 21 S_{21}\,
  7. S 22 S_{22}\,
  8. S 12 S_{12}\,

Network_calculus.html

  1. A A
  2. A ( t ) A(t)
  3. 0 , t ) ) 0,t))
  4. A : + + A:\mathbb{R}^{+}\rightarrow\mathbb{R}^{+}
  5. u , t + : u < t A ( u ) A ( t ) \forall u,t\in\mathbb{R}^{+}:u<t\implies A(u)\leq A(t)
  6. A A
  7. D D
  8. A D A≥D
  9. A A
  10. D D
  11. t t
  12. b ( A , D , t ) b(A,D,t)
  13. A A
  14. D D
  15. t t
  16. d ( A , D , t ) d(A,D,t)
  17. b ( A , D , t ) := A ( t ) - D ( t ) b(A,D,t):=A(t)-D(t)
  18. d ( A , D , t ) := inf { d + s . t . D ( t + d ) A ( t ) } d(A,D,t):=\inf\left\{d\in\mathbb{R}^{+}~{}s.t.~{}D(t+d)\geq A(t)\right\}
  19. b ( A , D ) := sup t 0 { A ( t ) - D ( t ) } b(A,D):=\sup_{t\geq 0}\left\{A(t)-D(t)\right\}
  20. d ( A , D ) := sup t 0 { inf { d + s . t . D ( t + d ) A ( t ) } } d(A,D):=\sup_{t\geq 0}\left\{\inf\left\{d\in\mathbb{R}^{+}~{}s.t.~{}D(t+d)\geq A% (t)\right\}\right\}
  21. f f
  22. g g
  23. ( f g ) ( t ) := 0 τ f ( τ ) g ( t - τ ) d τ (f\ast g)(t):=\int_{0}^{\tau}f(\tau)\cdot g(t-\tau)d\tau
  24. f f
  25. g g
  26. ( f g ) ( t ) := inf 0 τ t { f ( τ ) + g ( t - τ ) } (f\otimes g)(t):=\inf_{0\leq\tau\leq t}\left\{f(\tau)+g(t-\tau)\right\}
  27. ( f g ) ( t ) := sup τ 0 { f ( t + τ ) - g ( τ ) } (f\oslash g)(t):=\sup_{\tau\geq 0}\left\{f(t+\tau)-g(\tau)\right\}
  28. b ( f , g ) = ( f g ) ( 0 ) b(f,g)=(f\oslash g)(0)
  29. d ( f , g ) = inf { w : ( f g ) ( - w ) 0 } d(f,g)=\inf\{w:(f\oslash g)(-w)\leq 0\}
  30. A A
  31. E E
  32. t t
  33. E ( t ) sup τ 0 { A ( t + τ ) - A ( τ ) } = ( A A ) ( t ) . E(t)\geq\sup_{\tau\geq 0}\{A(t+\tau)-A(\tau)\}=(A\oslash A)(t).
  34. A A E A\leq A\otimes E
  35. E E
  36. A A
  37. E E
  38. t t
  39. τ τ
  40. A A
  41. D D
  42. S S
  43. ( A , B ) (A,B)
  44. t t
  45. D ( t ) ( A S ) ( t ) . D(t)\geq(A\otimes S)(t).
  46. A A
  47. D D
  48. I I
  49. t I t∈I
  50. A ( t ) > D ( t ) A(t)>D(t)
  51. S S
  52. ( A , B ) (A,B)
  53. s , t + \forall s,t\in\mathbb{R}^{+}
  54. s t s\leq t
  55. ( s , t ] (s,t]
  56. D ( t ) - D ( s ) S ( t - s ) D(t)-D(s)\geq S(t-s)
  57. S S
  58. S S
  59. A A
  60. D D
  61. E E
  62. S S
  63. b ( A , D ) b ( E , S ) b(A,D)\leq b(E,S)
  64. d ( A , D ) d ( E , S ) d(A,D)\leq d(E,S)
  65. E = E S E^{\prime}=E\oslash S
  66. E E
  67. S S
  68. b ( A , D ) b(A,D)
  69. b ( E , S ) b(E,S)
  70. v ( A , D ) v(A,D)
  71. v ( E , S ) v(E,S)
  72. S 1 S_{1}
  73. S 2 S_{2}
  74. S e 2 e = S 1 S 2 S_{e2e}=S_{1}\otimes S_{2}
  75. X A S 1 X\geq A\otimes S_{1}
  76. D X S 2 D\geq X\otimes S_{2}
  77. D ( X S 2 ) S 1 D\geq(X\otimes S_{2})\otimes S_{1}
  78. D X ( S 2 S 1 ) D\geq X\otimes(S_{2}\otimes S_{1})
  79. d ( E , S 2 S 1 ) d ( E , S 1 ) + d ( E S 1 , S 2 ) d(E,S_{2}\otimes S_{1})\leq d(E,S_{1})+d(E\oslash S_{1},S_{2})

Neugebauer_equations.html

  1. R ( λ ) = i = 1 16 w i R i ( λ ) R(\lambda)=\sum_{i=1}^{16}w_{i}R_{i}(\lambda)
  2. R i ( λ ) R_{i}(\lambda)
  3. w i w_{i}
  4. R ( λ ) = [ i = 1 16 w i R i ( λ ) 1 / n ] n R(\lambda)=\left[\sum_{i=1}^{16}w_{i}R_{i}(\lambda)^{1/n}\right]^{n}

Neutral_current.html

  1. 𝔐 NC J μ ( NC ) ( ν e ) J ( NC ) μ ( e - ) \mathfrak{M}^{\mathrm{NC}}\propto J_{\mu}^{\mathrm{(NC)}}(\nu_{\mathrm{e}})\;J% ^{\mathrm{(NC)}\mu}(\mathrm{e^{-}})
  2. J ( NC ) μ ( f ) = u ¯ f γ μ 1 2 ( g V f - g A f γ 5 ) u f , J^{\mathrm{(NC)}\mu}(f)=\bar{u}_{f}\gamma^{\mu}\frac{1}{2}\left(g^{f}_{V}-g^{f% }_{A}\gamma^{5}\right)u_{f},
  3. g V f g^{f}_{V}
  4. g A f g^{f}_{A}
  5. f f
  6. ( E / M Z ) 2 ~{}(E/M_{Z})^{2}

New_Zealand_mud_snail.html

  1. 1 5 \begin{matrix}\frac{1}{5}\end{matrix}
  2. 1 2 \begin{matrix}\frac{1}{2}\end{matrix}

Newsvendor_model.html

  1. q q
  2. q q
  3. π = E [ p min ( q , D ) ] - c q \pi=E\left[p\min(q,D)\right]-cq
  4. D D
  5. D D
  6. p p
  7. c c
  8. q q
  9. E E
  10. F - 1 F^{-1}
  11. D D
  12. ( p - c ) (p-c)
  13. c c
  14. p p
  15. p = 7 p=7
  16. c = 5 c=5
  17. D D
  18. D min = 50 D_{\min}=50
  19. D max = 80 D_{\max}=80
  20. q opt = F - 1 ( 7 - 5 7 ) = F - 1 ( 0.285 ) = D min + ( D max - D min ) 0.285 = 58.55 59. q\text{opt}=F^{-1}\left(\frac{7-5}{7}\right)=F^{-1}\left(0.285\right)=D_{\min}% +(D_{\max}-D_{\min})\cdot 0.285=58.55\approx 59.
  21. p = 7 p=7
  22. c = 5 c=5
  23. D D
  24. μ \mu
  25. σ \sigma
  26. q opt = F - 1 ( 7 - 5 7 ) = μ + σ Z - 1 ( 0.285 ) = 50 + 20 - 0.56595 = 38.68 39. q\text{opt}=F^{-1}\left(\frac{7-5}{7}\right)=\mu+\sigma Z^{-1}\left(0.285% \right)=50+20\cdot-0.56595=38.68\approx 39.
  27. p = 7 p=7
  28. c = 5 c=5
  29. D D
  30. μ \mu
  31. σ \sigma
  32. q opt = F - 1 ( 7 - 5 7 ) = μ e Z - 1 ( 0.285 ) σ = 50 e ( 0.2 - 0.56595 ) = 44.64 45. q\text{opt}=F^{-1}\left(\frac{7-5}{7}\right)=\mu e^{Z^{-1}\left(0.285\right)% \sigma}=50e^{\left(0.2\cdot-0.56595\right)}=44.64\approx 45.
  33. p < c p<c
  34. K ( q ) = c f + c v ( q - x ) + p E [ max ( D - q , 0 ) ] + h E [ max ( q - D , 0 ) ] K(q)=c_{f}+c_{v}(q-x)+pE\left[\max(D-q,0)\right]+hE\left[\max(q-D,0)\right]
  35. c f c_{f}
  36. c v c_{v}
  37. q q
  38. x x
  39. x x
  40. p p
  41. E [ D ] E[D]
  42. D D
  43. D D
  44. h h
  45. q opt = F - 1 ( p - c v p + h ) q\text{opt}=F^{-1}\left(\frac{p-c_{v}}{p+h}\right)

Newton_fractal.html

  1. p ( Z ) [ Z ] p(Z)\in\mathbb{C}[Z]
  2. z z - p ( z ) p ( z ) z\mapsto z-\tfrac{p(z)}{p^{\prime}(z)}
  3. G k G_{k}
  4. ζ k \zeta_{k}
  5. k = 1 , . . , deg ( p ) k=1,..,\operatorname{deg}(p)
  6. deg ( p ) \operatorname{deg}(p)
  7. z 0 z_{0}
  8. z n + 1 := z n - p ( z n ) p ( z n ) z_{n+1}:=z_{n}-\frac{p(z_{n})}{p^{\prime}(z_{n})}
  9. z 1 z_{1}
  10. z 2 z_{2}
  11. ζ k \zeta_{k}
  12. z 0 z_{0}
  13. G k G_{k}
  14. z 3 - 2 z + 2 z^{3}-2z+2
  15. d d
  16. ( ζ 1 , , ζ d ) (\zeta_{1},...,\zeta_{d})
  17. ( p 1 , , p d ) (p_{1},...,p_{d})
  18. p ( z ) = z d + p 1 z d - 1 + + p d - 1 z + p d := ( z - ζ 1 ) ( z - ζ d ) p(z)=z^{d}+p_{1}z^{d-1}+\cdots+p_{d-1}z+p_{d}:=(z-\zeta_{1})\cdot\cdots\cdot(z% -\zeta_{d})
  19. z m n = z 00 + m Δ x + i n Δ y z_{mn}=z_{00}+m\Delta x+in\Delta y
  20. m = 0 m=0
  21. M - 1 M-1
  22. n = 0 n=0
  23. N - 1 N-1
  24. \mathbb{C}
  25. k ( m , n ) k(m,n)
  26. ζ k ( m , n ) \zeta_{k(m,n)}
  27. M M
  28. N N
  29. ( m , n ) (m,n)
  30. f k ( m , n ) f_{k(m,n)}
  31. D ( m , n ) D(m,n)
  32. D D
  33. | z D - ζ k ( m , n ) | < ϵ |z_{D}-\zeta_{k(m,n)}|<\epsilon
  34. ϵ > 0 \epsilon>0
  35. z n + 1 = z n - a p ( z n ) p ( z n ) z_{n+1}=z_{n}-a\frac{p(z_{n})}{p^{\prime}(z_{n})}
  36. a a
  37. a = 1 a=1
  38. a a
  39. a a
  40. p p
  41. n n
  42. z n z_{n}
  43. a a
  44. n n
  45. n n
  46. p ( z ) = z 3 - 1 p(z)=z^{3}-1
  47. p ( z ) = z 3 - 1 p(z)=z^{3}-1
  48. p ( z ) = z 3 - 2 z + 2 p(z)=z^{3}-2z+2
  49. x 8 + 15 x 4 - 16 x^{8}+15x^{4}-16
  50. p ( z ) = z 5 - 3 i z 3 - ( 5 + 2 i ) z 2 + 3 z + 1 p(z)=z^{5}-3iz^{3}-(5+2i)z^{2}+3z+1
  51. p ( z ) = sin ( z ) p(z)=\sin(z)
  52. sin ( x ) \sin(x)
  53. p ( z ) = z 3 - 1 p(z)=z^{3}-1
  54. a = - 0.5. a=-0.5.
  55. p ( z ) = z 2 - 1 p(z)=z^{2}-1
  56. a = 1 + i . a=1+i.
  57. p ( z ) = z 3 - 1 p(z)=z^{3}-1
  58. a = 2. a=2.
  59. p ( z ) = z 4 + 3 i - 1 p(z)=z^{4+3i}-1
  60. a = 2.1. a=2.1.
  61. p ( z ) = z 6 + z 3 - 1 p(z)=z^{6}+z^{3}-1
  62. p ( z ) = sin ( z ) - 1 p(z)=\sin(z)-1
  63. p ( z ) = cosh ( z ) - 1 p(z)=\cosh(z)-1

Newtonian_gauge.html

  1. d s 2 = - ( 1 + 2 Ψ ) d t 2 + a 2 ( t ) ( 1 - 2 Φ ) δ a b d x a d x b , ds^{2}=-(1+2\Psi)dt^{2}+a^{2}(t)(1-2\Phi)\delta_{ab}dx^{a}dx^{b},
  2. δ a b \delta_{ab}
  3. d s 2 = a 2 ( t ) [ - ( 1 + 2 Ψ ) d τ 2 + ( 1 - 2 Φ ) δ a b d x a d x b ] ds^{2}=a^{2}(t)[-(1+2\Psi)d\tau^{2}+(1-2\Phi)\delta_{ab}dx^{a}dx^{b}]
  4. d t = a ( t ) d τ dt=a(t)d\tau
  5. Ψ \Psi
  6. 2 Ψ = 4 π G ρ \nabla^{2}\Psi=4\pi G\rho
  7. Φ = Ψ \Phi=\Psi

Newtonian_motivations_for_general_relativity.html

  1. r r
  2. v v
  3. v 2 r = G M r 2 {v^{2}\over r}={GM\over r^{2}}
  4. M M
  5. 𝐡 \mathbf{h}
  6. d 2 𝐡 d τ 2 + R 𝐡 = 0 {d^{2}\mathbf{h}\over d\tau^{2}}+R\mathbf{h}=0
  7. R = 1 r 2 v 2 c 2 R={1\over r^{2}}{v^{2}\over c^{2}}
  8. τ = c t \tau=ct
  9. M M
  10. R = G M r 3 R={GM\over{r^{3}}}
  11. ρ ( r ) \rho(r)
  12. r r
  13. M = 4 π ρ ( r ) r 3 3 M={4\pi\rho(r)r^{3}\over 3}
  14. R = 4 π G 3 ρ ( r ) R={4\pi G\over{3}}\rho(r)
  15. z z
  16. 𝐫 \mathbf{r}
  17. x x
  18. 𝐯 \mathbf{v}
  19. y y
  20. 𝐡 \mathbf{h}
  21. c c
  22. d 2 𝐫 d t 2 = - G M r 3 𝐫 {d^{2}\mathbf{r}\over dt^{2}}=-{GM\over r^{3}}\mathbf{r}
  23. d 2 𝐡 d τ 2 + R 𝐡 = 0 {d^{2}\mathbf{h}\over d\tau^{2}}+R\mathbf{h}=0
  24. R = R = G M c 2 r 3 = 4 π G 3 c 2 ρ ( r ) R=R_{\perp}={GM\over{c^{2}r^{3}}}={4\pi G\over{3c^{2}}}\rho(r)
  25. 𝐫 \mathbf{r}
  26. R = R = - 2 G M c 2 r 3 = - 8 π G 3 c 2 ρ ( r ) R=R_{\|}=-{2GM\over{c^{2}r^{3}}}=-{8\pi G\over{3c^{2}}}\rho(r)
  27. 𝐫 \mathbf{r}
  28. R R_{\|}
  29. 𝐡 \mathbf{h}
  30. z z
  31. 𝐫 \mathbf{r}
  32. x x
  33. 𝐫 \mathbf{r}
  34. y y
  35. 𝐫 ^ \mathbf{\hat{r}}
  36. d 2 h i d s 2 + R j i h j = 0 {d^{2}h^{i}\over ds^{2}}+R^{i}_{j}h^{j}=0
  37. R j i R^{i}_{j}
  38. R 1 1 R 1 2 R 1 3 R 2 1 R 2 2 R 2 3 R 3 1 R 3 2 R 3 3 = R 0 0 0 R 0 0 0 R \begin{Vmatrix}R^{1}_{1}&R^{2}_{1}&R^{3}_{1}\\ R^{1}_{2}&R^{2}_{2}&R^{3}_{2}\\ R^{1}_{3}&R^{2}_{3}&R^{3}_{3}\end{Vmatrix}=\begin{Vmatrix}R_{\perp}&0&0\\ 0&R_{\perp}&0\\ 0&0&R_{\|}\end{Vmatrix}
  39. h 1 h 2 h 3 = 𝐡 𝐱 ^ 𝐡 𝐲 ^ 𝐡 𝐳 ^ \begin{Vmatrix}h^{1}&h^{2}&h^{3}\end{Vmatrix}=\begin{Vmatrix}\mathbf{h}\cdot% \mathbf{\hat{x}}&\mathbf{h}\cdot\mathbf{\hat{y}}&\mathbf{h}\cdot\mathbf{\hat{z% }}\end{Vmatrix}
  40. 𝐡 𝐱 ^ \mathbf{h}\cdot\mathbf{\hat{x}}
  41. 𝐡 \mathbf{h}
  42. 𝐱 ^ \mathbf{\hat{x}}
  43. 𝐡 𝐲 ^ \mathbf{h}\cdot\mathbf{\hat{y}}
  44. 𝐲 ^ \mathbf{\hat{y}}
  45. 𝐡 𝐳 ^ \mathbf{h}\cdot\mathbf{\hat{z}}
  46. 𝐳 ^ \mathbf{\hat{z}}
  47. 𝐫 ^ \mathbf{\hat{r}}
  48. 𝐫 ^ \mathbf{\hat{r}}
  49. 𝒞 \mathcal{C}
  50. 𝐡 ¯ \mathbf{\bar{h}}
  51. 𝐡 \mathbf{h}
  52. h ¯ i = j i h j \bar{h}^{i}=\mathcal{M}^{i}_{j}h^{j}
  53. ¯ \bar{\mathcal{M}}
  54. \mathcal{M}
  55. ¯ j i k j = δ k i \bar{\mathcal{M}}^{i}_{j}\mathcal{M}^{j}_{k}=\delta^{i}_{k}
  56. h i = ¯ j i h ¯ j h^{i}=\bar{\mathcal{M}}^{i}_{j}\bar{h}^{j}
  57. δ k i \delta^{i}_{k}
  58. θ \theta
  59. x x
  60. 1 1 1 2 1 3 2 1 2 2 2 3 3 1 3 2 3 3 = 1 0 0 0 cos ( θ ) sin ( θ ) 0 - sin ( θ ) cos ( θ ) \begin{Vmatrix}\mathcal{M}^{1}_{1}&\mathcal{M}^{2}_{1}&\mathcal{M}^{3}_{1}\\ \mathcal{M}^{1}_{2}&\mathcal{M}^{2}_{2}&\mathcal{M}^{3}_{2}\\ \mathcal{M}^{1}_{3}&\mathcal{M}^{2}_{3}&\mathcal{M}^{3}_{3}\end{Vmatrix}=% \begin{Vmatrix}1&0&0\\ 0&\cos(\theta)&\sin(\theta)\\ 0&-\sin(\theta)&\cos(\theta)\end{Vmatrix}
  61. θ \theta
  62. d 2 h ¯ i d s 2 + R ¯ j i h ¯ j = 0 {d^{2}\bar{h}^{i}\over ds^{2}}+\bar{R}^{i}_{j}\bar{h}^{j}=0
  63. R ¯ j i = k i R l k ¯ j l \bar{R}^{i}_{j}=\mathcal{M}^{i}_{k}R^{k}_{l}\bar{\mathcal{M}}^{l}_{j}
  64. i k d h i d s = i k d ¯ j i h ¯ j d s \mathcal{M}^{k}_{i}{dh^{i}\over ds}=\mathcal{M}^{k}_{i}{d\bar{\mathcal{M}}^{i}% _{j}\bar{h}^{j}\over ds}
  65. d h ¯ k d s + Γ j k h ¯ j = def D h ¯ k D s {d\bar{h}^{k}\over ds}+\Gamma^{k}_{j}\bar{h}^{j}\ \stackrel{\mathrm{def}}{=}\ % {D\bar{h}^{k}\over Ds}
  66. Γ j k = def i k d ¯ j i d s \Gamma^{k}_{j}\ \stackrel{\mathrm{def}}{=}\ \mathcal{M}^{k}_{i}{d\bar{\mathcal% {M}}^{i}_{j}\over ds}
  67. D 2 h ¯ i D s 2 + R ¯ j i h ¯ j = 0 {D^{2}\bar{h}^{i}\over Ds^{2}}+\bar{R}^{i}_{j}\bar{h}^{j}=0
  68. u ¯ i = def D h ¯ i D s \bar{u}^{i}\ \stackrel{\mathrm{def}}{=}\ {D\bar{h}^{i}\over Ds}
  69. d u ¯ i d s + Γ j i u ¯ j + R ¯ j i h ¯ j = 0 {d\bar{u}^{i}\over ds}+\Gamma^{i}_{j}\bar{u}^{j}+\bar{R}^{i}_{j}\bar{h}^{j}=0
  70. d 2 h ¯ i d s 2 + 2 Γ j i d h ¯ i d s + d Γ j i d s h ¯ j + Γ j i Γ k j h ¯ k + R ¯ j i h ¯ j = 0 {d^{2}\bar{h}^{i}\over ds^{2}}+2\Gamma^{i}_{j}{d\bar{h}^{i}\over ds}+{d\Gamma^% {i}_{j}\over ds}\bar{h}^{j}+\Gamma^{i}_{j}\Gamma^{j}_{k}\bar{h}^{k}+\bar{R}^{i% }_{j}\bar{h}^{j}=0
  71. Γ j i = 0 \Gamma^{i}_{j}=0
  72. D 2 h ¯ i D s 2 + R ¯ j i h ¯ j = 0 {D^{2}\bar{h}^{i}\over Ds^{2}}+\bar{R}^{i}_{j}\bar{h}^{j}=0
  73. D D s = def d d s + Γ {D\over Ds}\ \stackrel{\mathrm{def}}{=}\ {d\over ds}+\Gamma
  74. Γ \Gamma
  75. R ¯ j i = k i R l k ¯ j l \bar{R}^{i}_{j}=\mathcal{M}^{i}_{k}R^{k}_{l}\bar{\mathcal{M}}^{l}_{j}
  76. j i \mathcal{M}^{i}_{j}
  77. R 1 1 R 1 2 R 1 3 R 2 1 R 2 2 R 2 3 R 3 1 R 3 2 R 3 3 = R 0 0 0 R 0 0 0 R \begin{Vmatrix}R^{1}_{1}&R^{2}_{1}&R^{3}_{1}\\ R^{1}_{2}&R^{2}_{2}&R^{3}_{2}\\ R^{1}_{3}&R^{2}_{3}&R^{3}_{3}\end{Vmatrix}=\begin{Vmatrix}R_{\perp}&0&0\\ 0&R_{\perp}&0\\ 0&0&R_{\|}\end{Vmatrix}
  78. R = 4 π G 3 c 2 ρ ( r ) R_{\perp}={4\pi G\over{3c^{2}}}\rho(r)
  79. R = - 8 π G 3 c 2 ρ ( r ) R_{\|}=-{8\pi G\over{3c^{2}}}\rho(r)

Next-to-Minimal_Supersymmetric_Standard_Model.html

  1. μ \mu
  2. μ \mu
  3. μ \mu
  4. μ H u H d \mu H_{u}H_{d}
  5. μ \mu
  6. S S
  7. S S
  8. S ^ \hat{S}
  9. S ~ \tilde{S}
  10. W NMSSM = W Yuk + λ S H u H d + κ 3 S 3 W_{\,\text{NMSSM}}=W_{\,\text{Yuk}}+\lambda SH_{u}H_{d}+\frac{\kappa}{3}S^{3}
  11. W Yuk W_{\,\text{Yuk}}
  12. λ \lambda
  13. κ \kappa
  14. μ \mu
  15. λ \lambda
  16. μ \mu
  17. S ^ \hat{S}
  18. S ^ \langle\hat{S}\rangle
  19. μ eff = λ S ^ \mu_{\,\text{eff}}=\lambda\langle\hat{S}\rangle
  20. κ \kappa
  21. κ \kappa
  22. κ \kappa
  23. κ \kappa
  24. 3 \mathbb{Z}_{3}
  25. 3 \mathbb{Z}_{3}
  26. μ \mu
  27. κ \kappa
  28. Z Z^{\prime}
  29. S S
  30. S ^ \hat{S}
  31. H 1 , H 2 , , H 7 H_{1},H_{2},...,H_{7}
  32. H 1 H_{1}
  33. H 1 , H 2 , H 3 H_{1},H_{2},H_{3}
  34. A 1 , A 2 A_{1},A_{2}
  35. H + , H - H^{+},H^{-}
  36. S ~ \tilde{S}

Nielsen_theory.html

  1. 𝑀𝐹 [ f ] = min { # Fix ( g ) | g f } , \mathit{MF}[f]=\min\{\#\mathrm{Fix}(g)\,|\,g\sim f\},
  2. N ( f ) 𝑀𝐹 [ f ] , N(f)\leq\mathit{MF}[f],

Nijenhuis–Richardson_bracket.html

  1. A l t p ( V ) = ( p + 1 V * ) V Alt^{p}(V)=(\wedge^{p+1}V^{*})\otimes V
  2. [ P , Q ] and = i P Q - ( - 1 ) p q i Q P . [P,Q]^{\and}=i_{P}Q-(-1)^{pq}i_{Q}P.\,
  3. ( i P Q ) ( X 0 , X 1 , , X p + q ) = σ S h p , q sgn ( σ ) P ( Q ( X σ ( 0 ) , X σ ( 1 ) , , X σ ( q ) ) , X < m t p l > σ ( q + 1 ) , , X σ ( p + q ) ) (i_{P}Q)(X_{0},X_{1},\ldots,X_{p+q})=\sum_{\sigma\in Sh_{p,q}}\mathrm{sgn}(% \sigma)P(Q(X_{\sigma(0)},X_{\sigma(1)},\ldots,X_{\sigma(q)}),X_{<}mtpl>{{% \sigma(q+1)}},\ldots,X_{{\sigma(p+q)}})

Nilmanifold.html

  1. N / H N/H
  2. Γ \Gamma
  3. Γ \Gamma
  4. N / Γ N/\Gamma
  5. Γ \Gamma
  6. Γ \Gamma
  7. Γ \Gamma
  8. Γ \ N \Gamma\backslash N
  9. N / Γ N/\Gamma
  10. Z = [ N , N ] Z=[N,N]
  11. Z Γ Z\cap\Gamma
  12. G = Z / ( Z Γ ) G=Z/(Z\cap\Gamma)
  13. P = N / Γ P=N/\Gamma
  14. M = P / G M=P/G
  15. I : g g I:\;g\rightarrow g
  16. ± - 1 \pm\sqrt{-1}
  17. g g\otimes{\mathbb{C}}
  18. Γ \Gamma
  19. ( 1 x z 1 y 1 ) Γ \begin{pmatrix}1&x&z\\ &1&y\\ &&1\end{pmatrix}\Gamma
  20. ( 1 { x } { z - x y } 1 { y } 1 ) \begin{pmatrix}1&\{x\}&\{z-x\lfloor y\rfloor\}\\ &1&\{y\}\\ &&1\end{pmatrix}
  21. x \lfloor x\rfloor
  22. { x } \{x\}
  23. / \mathbb{R}/\mathbb{Z}

Nilpotent_Lie_algebra.html

  1. 𝐠 \mathbf{g}
  2. 𝐠 \mathbf{g}
  3. n n∈ℕ
  4. [ X , [ X , [ [ X , Y ] ] = ad X n - 1 Y 𝔤 n = 0 X , Y 𝔤 . [X,[X,[\cdots[X,Y]\cdots]={\mathrm{ad}_{X}}^{n-1}Y\in\mathfrak{g}_{n}=0\quad% \forall X,Y\in\mathfrak{g}.
  5. X 𝐠 X∈\mathbf{g}
  6. [ X 1 , [ X 2 , [ [ X n - 1 , Y ] ] = ad X 1 ad X 2 ad X n - 1 Y 𝔤 n = 0 X 1 , X 2 , , X n - 1 , Y 𝔤 , [X_{1},[X_{2},[\cdots[X_{n-1},Y]\cdots]=\mathrm{ad}_{X_{1}}\mathrm{ad}_{X_{2}}% \mathrm{ad}_{X_{n-1}}Y\in\mathfrak{g}_{n}=0\quad\forall X_{1},X_{2},\ldots,X_{% n-1},Y\in\mathfrak{g},
  7. ( n 1 ) (n−1)
  8. a d ad
  9. [ [ [ [ X n + 1 , X n ] , X n - 1 ] , , X 1 ] = ad [ [ X n + 1 , X n ] , , X 2 ] ( X 1 ) , [[\cdots[[X_{n+1},X_{n}],X_{n-1}],\cdots,X_{1}]=\mathrm{ad}[\cdots[X_{n+1},X_{% n}],\cdots,X_{2}](X_{1}),
  10. ad [ [ X n + 1 , X n ] , , X 2 ] = [ ad [ [ X n + 1 , X n ] , X 3 ] , ad X 2 ] = = [ [ ad X n + 1 , ad X n ] , ad X 2 ] . \begin{aligned}\displaystyle\mathrm{ad}[\cdots[X_{n+1},X_{n}],\cdots,X_{2}]&% \displaystyle=[\mathrm{ad}[\cdots[X_{n+1},X_{n}],\cdots X_{3}],\mathrm{ad}_{X_% {2}}]\\ &\displaystyle=\ldots=[\cdots[\mathrm{ad}_{X_{n+1}},\mathrm{ad}_{X_{n}}],% \cdots\mathrm{ad}_{X_{2}}].\end{aligned}
  11. a d 𝐠 ad\mathbf{g}
  12. 𝐠 \mathbf{g}
  13. a d 𝐠 ad\mathbf{g}
  14. 𝐠 \mathbf{g}
  15. 𝐠 \mathbf{g}
  16. n n
  17. 𝐠 \mathbf{g}
  18. ad x 1 ad x 2 ad x n ( y ) = 0 x i , y 𝔤 . \mathrm{ad}x_{1}\mathrm{ad}x_{2}\dots\mathrm{ad}x_{n}(y)=0\quad\forall x_{i},y% \in\mathfrak{g}.
  19. a d 𝐠 ad\mathbf{g}
  20. a d < s u b > x n = 0 ad<sub>x^{n}=0

Nine-point_center.html

  1. N O = N H = 3 N G . NO=NH=3NG.
  2. I N < 1 2 I O , IN<\tfrac{1}{2}IO,
  3. I N = 1 2 ( R - 2 r ) < R 2 , IN=\tfrac{1}{2}(R-2r)<\frac{R}{2},
  4. 2 R I N = O I 2 , 2R\cdot IN=OI^{2},
  5. cos ( B - C ) : cos ( C - A ) : cos ( A - B ) \cos(B-C):\cos(C-A):\cos(A-B)
  6. = cos A + 2 cos B cos C : cos B + 2 cos C cos A : cos C + 2 cos A cos B =\cos A+2\cos B\cos C:\cos B+2\cos C\cos A:\cos C+2\cos A\cos B
  7. = cos A - 2 sin B sin C : cos B - 2 sin C sin A : cos C - 2 sin A sin B =\cos A-2\sin B\sin C:\cos B-2\sin C\sin A:\cos C-2\sin A\sin B
  8. = b c [ a 2 ( b 2 + c 2 ) - ( b 2 - c 2 ) 2 ] : c a [ b 2 ( c 2 + a 2 ) - ( c 2 - a 2 ) 2 ] : a b [ c 2 ( a 2 + b 2 ) - ( a 2 - b 2 ) 2 ] . =bc[a^{2}(b^{2}+c^{2})-(b^{2}-c^{2})^{2}]:ca[b^{2}(c^{2}+a^{2})-(c^{2}-a^{2})^% {2}]:ab[c^{2}(a^{2}+b^{2})-(a^{2}-b^{2})^{2}].
  9. a cos ( B - C ) : b cos ( C - A ) : c cos ( A - B ) a\cos(B-C):b\cos(C-A):c\cos(A-B)
  10. = a 2 ( b 2 + c 2 ) - ( b 2 - c 2 ) 2 : b 2 ( c 2 + a 2 ) - ( c 2 - a 2 ) 2 : c 2 ( a 2 + b 2 ) - ( a 2 - b 2 ) 2 . =a^{2}(b^{2}+c^{2})-(b^{2}-c^{2})^{2}:b^{2}(c^{2}+a^{2})-(c^{2}-a^{2})^{2}:c^{% 2}(a^{2}+b^{2})-(a^{2}-b^{2})^{2}.

Nitroso.html

  1. \overrightarrow{\leftarrow}
  2. \overrightarrow{\leftarrow}

No-slip_condition.html

  1. u - u W a l l = β u n u-u_{Wall}=\beta\frac{\partial u}{\partial n}
  2. n n
  3. β \beta
  4. β 1.15 \beta\approx 1.15\ell
  5. \ell

Nodal_admittance_matrix.html

  1. Y = [ Y 11 Y 12 Y 1 n Y 21 Y 22 Y 2 n Y n 1 Y n 2 Y n n ] Y=\begin{bmatrix}Y_{11}&Y_{12}&\cdots&Y_{1n}\\ Y_{21}&Y_{22}&\cdots&Y_{2n}\\ \cdots&\cdots&\cdots&\cdots\\ Y_{n1}&Y_{n2}&\cdots&Y_{nn}\end{bmatrix}
  2. Y i j = { y i i + k i y i k , if j = i - y i j , if j i Y_{ij}=\begin{cases}y_{ii}+\sum_{k\neq i}{y_{ik}},&\mbox{if}~{}\quad j=i\\ -y_{ij},&\mbox{if}~{}\quad j\neq i\end{cases}
  3. y i k = g i k + - 1 b i k y_{ik}=g_{ik}+\sqrt{-1}b_{ik}
  4. i i
  5. k k
  6. y i i y_{ii}
  7. Y 11 , Y 22 , , Y n n Y_{11},Y_{22},...,Y_{nn}

Noise-equivalent_flux_density.html

  1. ν \nu
  2. N E F D = η N E P A ν NEFD=\eta\frac{NEP}{A\nu}
  3. η \eta

Noise_(electronics).html

  1. v n v_{n}
  2. v n = 4 k B T R Δ f v_{n}=\sqrt{4k_{B}TR\Delta f}
  3. i n = 2 I q Δ B i_{n}=\sqrt{2Iq\Delta B}
  4. V / Hz \scriptstyle\mathrm{V}/\sqrt{\mathrm{Hz}}

Noisy-channel_coding_theorem.html

  1. R < C R<C
  2. R > C R>C
  3. C = sup p X I ( X ; Y ) \ C=\sup_{p_{X}}I(X;Y)\,
  4. R ( p b ) = C 1 - H 2 ( p b ) . R(p_{b})=\frac{C}{1-H_{2}(p_{b})}.
  5. H 2 ( p b ) H_{2}(p_{b})
  6. H 2 ( p b ) = - [ p b log 2 p b + ( 1 - p b ) log 2 ( 1 - p b ) ] H_{2}(p_{b})=-\left[p_{b}\log_{2}{p_{b}}+(1-p_{b})\log_{2}({1-p_{b}})\right]
  7. n n
  8. X 1 n X_{1}^{n}
  9. n n
  10. Y 1 n Y_{1}^{n}
  11. A ε ( n ) = { ( x n , y n ) 𝒳 n × 𝒴 n A_{\varepsilon}^{(n)}=\{(x^{n},y^{n})\in\mathcal{X}^{n}\times\mathcal{Y}^{n}
  12. 2 - n ( H ( X ) + ε ) p ( X 1 n ) 2 - n ( H ( X ) - ε ) 2^{-n(H(X)+\varepsilon)}\leq p(X_{1}^{n})\leq 2^{-n(H(X)-\varepsilon)}
  13. 2 - n ( H ( Y ) + ε ) p ( Y 1 n ) 2 - n ( H ( Y ) - ε ) 2^{-n(H(Y)+\varepsilon)}\leq p(Y_{1}^{n})\leq 2^{-n(H(Y)-\varepsilon)}
  14. 2 - n ( H ( X , Y ) + ε ) p ( X 1 n , Y 1 n ) 2 - n ( H ( X , Y ) - ε ) } {2^{-n(H(X,Y)+\varepsilon)}}\leq p(X_{1}^{n},Y_{1}^{n})\leq 2^{-n(H(X,Y)-% \varepsilon)}\}
  15. X 1 n {X_{1}^{n}}
  16. Y 1 n Y_{1}^{n}
  17. 2 n R 2^{nR}
  18. p ( y | x ) p(y|x)
  19. P r ( W = w ) = 2 - n R , w = 1 , 2 , , 2 n R Pr(W=w)=2^{-nR},w=1,2,\dots,2^{nR}
  20. P ( y n | x n ( w ) ) = i = 1 n p ( y i | x i ( w ) ) P(y^{n}|x^{n}(w))=\prod_{i=1}^{n}p(y_{i}|x_{i}(w))
  21. Y 1 n Y_{1}^{n}
  22. ε \varepsilon
  23. X 1 n ( i ) X_{1}^{n}(i)
  24. Y 1 n Y_{1}^{n}
  25. 2 - n ( I ( X ; Y ) - 3 ε ) \leq 2^{-n(I(X;Y)-3\varepsilon)}
  26. E i = { ( X 1 n ( i ) , Y 1 n ) A ε ( n ) } , i = 1 , 2 , , 2 n R E_{i}=\{(X_{1}^{n}(i),Y_{1}^{n})\in A_{\varepsilon}^{(n)}\},i=1,2,\dots,2^{nR}
  27. P ( error ) \displaystyle P(\,\text{error})
  28. n n
  29. R < I ( X ; Y ) R<I(X;Y)
  30. 2 n R 2^{nR}
  31. X n X^{n}
  32. Y n Y^{n}
  33. n R = H ( W ) = H ( W | Y n ) + I ( W ; Y n ) nR=H(W)=H(W|Y^{n})+I(W;Y^{n})\;
  34. H ( W | Y n ) + I ( X n ( W ) ; Y n ) \leq H(W|Y^{n})+I(X^{n}(W);Y^{n})
  35. 1 + P e ( n ) n R + I ( X n ( W ) ; Y n ) \leq 1+P_{e}^{(n)}nR+I(X^{n}(W);Y^{n})
  36. 1 + P e ( n ) n R + n C \leq 1+P_{e}^{(n)}nR+nC
  37. P e ( n ) 1 - 1 n R - C R P_{e}^{(n)}\geq 1-\frac{1}{nR}-\frac{C}{R}
  38. n n
  39. P e ( n ) P_{e}^{(n)}
  40. P e 1 - 4 A n ( R - C ) 2 - e - n ( R - C ) 2 P_{e}\geq 1-\frac{4A}{n(R-C)^{2}}-e^{-\frac{n(R-C)}{2}}
  41. A A
  42. n n
  43. C C
  44. C = lim inf max p ( X 1 ) , p ( X 2 ) , 1 n i = 1 n I ( X i ; Y i ) . C=\lim\inf\max_{p^{(}X_{1}),p^{(}X_{2}),...}\frac{1}{n}\sum_{i=1}^{n}I(X_{i};Y% _{i}).
  45. C = lim inf 1 n i = 1 n C i C=\lim\inf\frac{1}{n}\sum_{i=1}^{n}C_{i}
  46. C i C_{i}
  47. 1 n i = 1 n C i \frac{1}{n}\sum_{i=1}^{n}C_{i}

Non-classical_logic.html

  1. \Box

Non-negative_matrix_factorization.html

  1. 𝐕 = 𝐖𝐇 . \mathbf{V}=\mathbf{W}\mathbf{H}\,.
  2. 𝐯 i = 𝐖𝐡 i , \mathbf{v}_{i}=\mathbf{W}\mathbf{h}_{i}\,,
  3. W + m × k W\in\Re^{m\times k}_{+}
  4. W W
  5. ( v 1 , , v n ) (v_{1},\cdots,v_{n})
  6. 𝐕 \mathbf{V}
  7. F ( 𝐖 , 𝐇 ) = 𝐕 - 𝐖𝐇 F 2 F(\mathbf{W},\mathbf{H})=\|\mathbf{V}-\mathbf{WH}\|^{2}_{F}
  8. 𝐖𝐇 = 𝐖𝐁𝐁 - 1 𝐇 \mathbf{WH}=\mathbf{WBB}^{-1}\mathbf{H}
  9. 𝐖 ~ = 𝐖𝐁 \mathbf{\tilde{W}=WB}
  10. 𝐇 ~ = 𝐁 - 1 𝐇 \mathbf{\tilde{H}}=\mathbf{B}^{-1}\mathbf{H}
  11. 𝐖 ~ \mathbf{\tilde{W}}
  12. 𝐇 ~ \mathbf{\tilde{H}}
  13. 𝐕 = ( v 1 , , v n ) \mathbf{V}=(v_{1},\cdots,v_{n})
  14. 𝐕 \mathbf{V}
  15. 𝐕 𝐖𝐇 \mathbf{V}\simeq\mathbf{W}\mathbf{H}
  16. min W , H || V - W H || F , \min_{W,H}||V-WH||_{F},
  17. W 0 , H 0. W\geq 0,H\geq 0.
  18. H H
  19. H H T = I HH^{T}=I
  20. H H
  21. 𝐇 k j > 0 \mathbf{H}_{kj}>0
  22. v j v_{j}
  23. k t h k^{th}
  24. W W
  25. k t h k^{th}
  26. k t h k^{th}
  27. H H T = I HH^{T}=I
  28. N N
  29. N 2 N^{2}
  30. O ( N ) O(N)

Non-standard_positional_numeral_systems.html

  1. p q r s pqrs
  2. p × b 3 + q × b 2 + r × b + s p\times b^{3}+q\times b^{2}+r\times b+s
  3. 7 × 16 3 + 10 × 16 2 + 3 × 16 + 15 7\times 16^{3}+10\times 16^{2}+3\times 16+15
  4. d 3 d 2 d 1 d 0 d_{3}d_{2}d_{1}d_{0}
  5. d 3 + d 2 + d 1 + d 0 d_{3}+d_{2}+d_{1}+d_{0}
  6. b n = 1 b^{n}=1

Noncentral_t-distribution.html

  1. T = Z + μ V / ν T=\frac{Z+\mu}{\sqrt{V/\nu}}
  2. F ν , μ ( x ) = { F ~ ν , μ ( x ) , if x 0 ; 1 - F ~ ν , - μ ( x ) , if x < 0 , F_{\nu,\mu}(x)=\begin{cases}\tilde{F}_{\nu,\mu}(x),&\mbox{if }~{}x\geq 0;\\ 1-\tilde{F}_{\nu,-\mu}(x),&\mbox{if }~{}x<0,\end{cases}
  3. F ~ ν , μ ( x ) = Φ ( - μ ) + 1 2 j = 0 [ p j I y ( j + 1 2 , ν 2 ) + q j I y ( j + 1 , ν 2 ) ] , \tilde{F}_{\nu,\mu}(x)=\Phi(-\mu)+\frac{1}{2}\sum_{j=0}^{\infty}\left[p_{j}I_{% y}\left(j+\frac{1}{2},\frac{\nu}{2}\right)+q_{j}I_{y}\left(j+1,\frac{\nu}{2}% \right)\right],
  4. I y ( a , b ) I_{y}\,\!(a,b)
  5. y = x 2 x 2 + ν , y=\frac{x^{2}}{x^{2}+\nu},
  6. p j = 1 j ! exp { - μ 2 2 } ( μ 2 2 ) j , p_{j}=\frac{1}{j!}\exp\left\{-\frac{\mu^{2}}{2}\right\}\left(\frac{\mu^{2}}{2}% \right)^{j},
  7. q j = μ 2 Γ ( j + 3 / 2 ) exp { - μ 2 2 } ( μ 2 2 ) j , q_{j}=\frac{\mu}{\sqrt{2}\Gamma(j+3/2)}\exp\left\{-\frac{\mu^{2}}{2}\right\}% \left(\frac{\mu^{2}}{2}\right)^{j},
  8. F v , μ ( x ) = { 1 2 j = 0 1 j ! ( - μ 2 ) j e - μ 2 2 Γ ( j + 1 2 ) Γ ( 1 / 2 ) I ( v v + x 2 ; v 2 , j + 1 2 ) , x 0 1 - 1 2 j = 0 1 j ! ( - μ 2 ) j e - μ 2 2 Γ ( j + 1 2 ) Γ ( 1 / 2 ) I ( v v + x 2 ; v 2 , j + 1 2 ) , x < 0 F_{v,\mu}(x)=\begin{cases}\frac{1}{2}\sum_{j=0}^{\infty}\frac{1}{j!}(-\mu\sqrt% {2})^{j}e^{\frac{-\mu^{2}}{2}}\frac{\Gamma(\frac{j+1}{2})}{\Gamma(1/2)}I\left(% \frac{v}{v+x^{2}};\frac{v}{2},\frac{j+1}{2}\right),&x\geq 0\\ 1-\frac{1}{2}\sum_{j=0}^{\infty}\frac{1}{j!}(-\mu\sqrt{2})^{j}e^{\frac{-\mu^{2% }}{2}}\frac{\Gamma(\frac{j+1}{2})}{\Gamma(1/2)}I\left(\frac{v}{v+x^{2}};\frac{% v}{2},\frac{j+1}{2}\right),&x<0\end{cases}
  9. f ( x ) = ν ν 2 Γ ( ν + 1 ) exp ( - μ 2 2 ) 2 ν ( ν + x 2 ) ν 2 Γ ( ν 2 ) { 2 μ x F 1 1 ( ν 2 + 1 ; 3 2 ; μ 2 x 2 2 ( ν + x 2 ) ) ( ν + x 2 ) Γ ( ν + 1 2 ) + F 1 1 ( ν + 1 2 ; 1 2 ; μ 2 x 2 2 ( ν + x 2 ) ) ν + x 2 Γ ( ν 2 + 1 ) } f(x)=\frac{\nu^{\frac{\nu}{2}}\Gamma(\nu+1)\exp\left(-\frac{\mu^{2}}{2}\right)% }{2^{\nu}(\nu+x^{2})^{\frac{\nu}{2}}\Gamma(\frac{\nu}{2})}\left\{\sqrt{2}\mu x% \frac{{}_{1}F_{1}\left(\frac{\nu}{2}+1;\,\frac{3}{2};\,\frac{\mu^{2}x^{2}}{2(% \nu+x^{2})}\right)}{(\nu+x^{2})\Gamma(\frac{\nu+1}{2})}+\frac{{}_{1}F_{1}\left% (\frac{\nu+1}{2};\,\frac{1}{2};\,\frac{\mu^{2}x^{2}}{2(\nu+x^{2})}\right)}{% \sqrt{\nu+x^{2}}\Gamma(\frac{\nu}{2}+1)}\right\}
  10. f ( x ) = ν ν 2 exp ( - ν μ 2 2 ( x 2 + ν ) ) π Γ ( ν 2 ) 2 ν - 1 2 ( x 2 + ν ) ν + 1 2 0 y ν exp ( - 1 2 ( y - μ x x 2 + ν ) 2 ) d y . f(x)=\frac{\nu^{\frac{\nu}{2}}\exp\left(-\frac{\nu\mu^{2}}{2(x^{2}+\nu)}\right% )}{\sqrt{\pi}\Gamma(\frac{\nu}{2})2^{\frac{\nu-1}{2}}(x^{2}+\nu)^{\frac{\nu+1}% {2}}}\int_{0}^{\infty}y^{\nu}\exp\left(-\frac{1}{2}\left(y-\frac{\mu x}{\sqrt{% x^{2}+\nu}}\right)^{2}\right)dy.
  11. f ( x ) = { ν x { F ν + 2 , μ ( x 1 + 2 ν ) - F ν , μ ( x ) } , if x 0 ; Γ ( ν + 1 2 ) π ν Γ ( ν 2 ) exp ( - μ 2 2 ) , if x = 0. f(x)=\begin{cases}\frac{\nu}{x}\left\{F_{\nu+2,\mu}\left(x\sqrt{1+\frac{2}{\nu% }}\right)-F_{\nu,\mu}(x)\right\},&\mbox{if }~{}x\neq 0;\\ \frac{\Gamma(\frac{\nu+1}{2})}{\sqrt{\pi\nu}\Gamma(\frac{\nu}{2})}\exp\left(-% \frac{\mu^{2}}{2}\right),&\mbox{if }~{}x=0.\end{cases}
  12. { ( ν + x 2 ) 2 f ′′ ( x ) + x f ( x ) ( ν ( - μ 2 + 2 ν + 5 ) + ( 2 ν + 5 ) x 2 ) + ( ν + 1 ) f ( x ) ( - μ 2 ν + ν + ( ν + 3 ) x 2 ) = 0 , f ( 0 ) = exp ( - μ 2 2 ) Γ ( ν + 1 2 ) π ν Γ ( ν 2 ) , f ( 0 ) = - exp ( - μ 2 2 ) μ 2 π } \left\{\begin{array}[]{l}\left(\nu+x^{2}\right)^{2}f^{\prime\prime}(x)+xf^{% \prime}(x)\left(\nu\left(-\mu^{2}+2\nu+5\right)+(2\nu+5)x^{2}\right)+(\nu+1)f(% x)\left(-\mu^{2}\nu+\nu+(\nu+3)x^{2}\right)=0,\\ f(0)=\frac{\exp\left(-\frac{\mu^{2}}{2}\right)\Gamma\left(\frac{\nu+1}{2}% \right)}{\sqrt{\pi}\sqrt{\nu}\Gamma\left(\frac{\nu}{2}\right)},\\ f^{\prime}(0)=-\frac{\exp\left(-\frac{\mu^{2}}{2}\right)\mu}{\sqrt{2\pi}}\end{% array}\right\}
  13. E [ T k ] = { ( ν 2 ) k 2 Γ ( ν - k 2 ) Γ ( ν 2 ) exp ( - μ 2 2 ) d k d μ k exp ( μ 2 2 ) , if ν > k ; Does not exist , if ν k . \mbox{E}~{}\left[T^{k}\right]=\begin{cases}\left(\frac{\nu}{2}\right)^{\frac{k% }{2}}\frac{\Gamma\left(\frac{\nu-k}{2}\right)}{\Gamma\left(\frac{\nu}{2}\right% )}\mbox{exp}~{}\left(-\frac{\mu^{2}}{2}\right)\frac{d^{k}}{d\mu^{k}}\mbox{exp}% ~{}\left(\frac{\mu^{2}}{2}\right),&\mbox{if }~{}\nu>k;\\ \mbox{Does not exist}~{},&\mbox{if }~{}\nu\leq k.\\ \end{cases}
  14. E [ T ] = { μ ν 2 Γ ( ( ν - 1 ) / 2 ) Γ ( ν / 2 ) , if ν > 1 ; Does not exist , if ν 1 , Var [ T ] = { ν ( 1 + μ 2 ) ν - 2 - μ 2 ν 2 ( Γ ( ( ν - 1 ) / 2 ) Γ ( ν / 2 ) ) 2 , if ν > 2 ; Does not exist , if ν 2. \begin{aligned}\displaystyle\mbox{E}~{}\left[T\right]&\displaystyle=\begin{% cases}\mu\sqrt{\frac{\nu}{2}}\frac{\Gamma((\nu-1)/2)}{\Gamma(\nu/2)},&\mbox{if% }~{}\nu>1;\\ \mbox{Does not exist}~{},&\mbox{if }~{}\nu\leq 1,\\ \end{cases}\\ \displaystyle\mbox{Var}~{}\left[T\right]&\displaystyle=\begin{cases}\frac{\nu(% 1+\mu^{2})}{\nu-2}-\frac{\mu^{2}\nu}{2}\left(\frac{\Gamma((\nu-1)/2)}{\Gamma(% \nu/2)}\right)^{2},&\mbox{if }~{}\nu>2;\\ \mbox{Does not exist}~{},&\mbox{if }~{}\nu\leq 2.\\ \end{cases}\end{aligned}
  15. ν 2 Γ ( ( ν - 1 ) / 2 ) Γ ( ν / 2 ) \sqrt{\frac{\nu}{2}}\frac{\Gamma((\nu-1)/2)}{\Gamma(\nu/2)}
  16. ( 1 - 3 4 ν - 1 ) - 1 \left(1-\frac{3}{4\nu-1}\right)^{-1}
  17. ( 2 ν 2 ν + 5 μ , ν ν + 1 μ ) \left(\sqrt{\frac{2\nu}{2\nu+5}}\mu,\,\sqrt{\frac{\nu}{\nu+1}}\mu\right)
  18. ( ν ν + 1 μ , 2 ν 2 ν + 5 μ ) \left(\sqrt{\frac{\nu}{\nu+1}}\mu,\,\sqrt{\frac{2\nu}{2\nu+5}}\mu\right)
  19. ν ν + 1 μ . \sqrt{\frac{\nu}{\nu+1}}\mu.
  20. T = n X ¯ σ ^ = n X ¯ - θ σ + n θ σ ( n - 1 ) σ ^ 2 σ 2 1 n - 1 T=\frac{\sqrt{n}\bar{X}}{\hat{\sigma}}=\frac{\sqrt{n}\frac{\bar{X}-\theta}{% \sigma}+\frac{\sqrt{n}\theta}{\sigma}}{\sqrt{\frac{(n-1)\hat{\sigma}^{2}}{% \sigma^{2}}\frac{1}{n-1}}}
  21. X ¯ \bar{X}
  22. σ ^ 2 \hat{\sigma}^{2}\,\!
  23. n θ / σ \sqrt{n}\theta/\sigma\,\!
  24. | T | > t 1 - α / 2 |T|>t_{1-\alpha/2}\,\!
  25. t 1 - α / 2 t_{1-\alpha/2}\,\!
  26. 1 - F n - 1 , n θ / σ ( t 1 - α / 2 ) + F n - 1 , n θ / σ ( - t 1 - α / 2 ) . 1-F_{n-1,\sqrt{n}\theta/\sigma}(t_{1-\alpha/2})+F_{n-1,\sqrt{n}\theta/\sigma}(% -t_{1-\alpha/2}).
  27. Z = lim ν T Z=\lim_{\nu\rightarrow\infty}T

Nonlinear_control.html

  1. Φ ( y ) y [ a , b ] , a < b y \frac{\Phi(y)}{y}\in[a,b],\quad a<b\quad\forall y
  2. x ˙ = A x + b u ξ ˙ = u y = c x + d ξ ( 1 ) \begin{matrix}\dot{x}&=&Ax+bu\\ \dot{\xi}&=&u\\ y&=&cx+d\xi\quad(1)\end{matrix}
  3. u = - ϕ ( y ) ( 2 ) \begin{matrix}u=-\phi(y)\quad(2)\end{matrix}
  4. H ( s ) = d s + c ( s I - A ) - 1 b H(s)=\frac{d}{s}+c(sI-A)^{-1}b\quad\quad
  5. x ˙ = i = 1 k f i ( x ) u i ( t ) \dot{x}=\sum_{i=1}^{k}f_{i}(x)u_{i}(t)\,
  6. x R n x\in R^{n}
  7. f 1 , , f k f_{1},\dots,f_{k}
  8. Δ \Delta
  9. u i ( t ) u_{i}(t)
  10. x x
  11. m m
  12. Δ ) = m \Delta)=m
  13. Δ \Delta

Nonlocal_Lagrangian.html

  1. [ ϕ ( x ) ] \mathcal{L}[\phi(x)]
  2. ϕ ( x ) \ \phi(x)
  3. = 1 2 ( x ϕ ( x ) ) 2 - 1 2 m 2 ϕ ( x ) 2 + ϕ ( x ) ϕ ( y ) ( x - y ) 2 d n y \mathcal{L}=\frac{1}{2}(\partial_{x}\phi(x))^{2}-\frac{1}{2}m^{2}\phi(x)^{2}+% \phi(x)\int{\frac{\phi(y)}{(x-y)^{2}}\,d^{n}y}
  4. = - 1 4 μ ν ( 1 + m 2 2 ) μ ν \mathcal{L}=-\frac{1}{4}\mathcal{F}_{\mu\nu}(1+\frac{m^{2}}{\partial^{2}})% \mathcal{F}^{\mu\nu}
  5. S = d t d d x [ ψ * ( i t + μ ) ψ - 2 2 m ψ * ψ ] - 1 2 d t d d x d d y V ( y - x ) ψ * ( x ) ψ ( x ) ψ * ( y ) ψ ( y ) S=\int dt\,d^{d}x\left[\psi^{*}(i\hbar\frac{\partial}{\partial t}+\mu)\psi-% \frac{\hbar^{2}}{2m}\nabla\psi^{*}\cdot\nabla\psi\right]-\frac{1}{2}\int dt\,d% ^{d}x\,d^{d}y\,V(\vec{y}-\vec{x})\psi^{*}(\vec{x})\psi(\vec{x})\psi^{*}(\vec{y% })\psi(\vec{y})

Nonnegative_matrix.html

  1. 𝐗 0 , i , j x i j 0. \mathbf{X}\geq 0,\qquad\forall{i,j}\quad x_{ij}\geq 0.
  2. n > 1. n>1.

Nontransitive_dice.html

  1. P ( A > B ) = P ( B > C ) = P ( C > D ) = P ( D > A ) = 2 3 P(A>B)=P(B>C)=P(C>D)=P(D>A)={2\over 3}
  2. ( 1 3 × 1 ) + ( 2 3 × 1 2 ) = 2 3 \left({1\over 3}\times 1\right)+\left({2\over 3}\times{1\over 2}\right)={2% \over 3}
  3. ( 1 2 × 1 ) + ( 1 2 × 1 3 ) = 2 3 \left({1\over 2}\times 1\right)+\left({1\over 2}\times{1\over 3}\right)={2% \over 3}
  4. 1 3 × ( 2 3 + 1 3 + 4 9 ) = 13 27 {1\over 3}\times\left({2\over 3}+{1\over 3}+{4\over 9}\right)={13\over 27}
  5. 1 3 × ( 2 3 + 1 3 + 1 2 ) = 1 2 {1\over 3}\times\left({2\over 3}+{1\over 3}+{1\over 2}\right)={1\over 2}
  6. 1 3 × ( 2 3 + 1 3 + 5 9 ) = 14 27 {1\over 3}\times\left({2\over 3}+{1\over 3}+{5\over 9}\right)={14\over 27}
  7. 1 3 × ( 2 3 + 1 3 + 1 2 ) = 1 2 {1\over 3}\times\left({2\over 3}+{1\over 3}+{1\over 2}\right)={1\over 2}

Normalization_(image_processing).html

  1. I : { 𝕏 n } { Min , . . , Max } I:\{\mathbb{X}\subseteq\mathbb{R}^{n}\}\rightarrow\{\,\text{Min},..,\,\text{% Max}\}
  2. I N : { 𝕏 n } { newMin , . . , newMax } I_{N}:\{\mathbb{X}\subseteq\mathbb{R}^{n}\}\rightarrow\{\,\text{newMin},..,\,% \text{newMax}\}
  3. I N = ( I - Min ) newMax - newMin Max - Min + newMin I_{N}=(I-\,\text{Min})\frac{\,\text{newMax}-\,\text{newMin}}{\,\text{Max}-\,% \text{Min}}+\,\text{newMin}
  4. I I
  5. I N I_{N}
  6. I N = ( newMax - newMin ) 1 1 + e - I - β α + newMin I_{N}=(\,\text{newMax}-\,\text{newMin})\frac{1}{1+e^{-\frac{I-\beta}{\alpha}}}% +\,\text{newMin}
  7. α \alpha
  8. β \beta

Normalized_Difference_Vegetation_Index.html

  1. NDVI = ( NIR - VIS ) ( NIR + VIS ) \mbox{NDVI}~{}=\frac{(\mbox{NIR}~{}-\mbox{VIS}~{})}{(\mbox{NIR}~{}+\mbox{VIS}~% {})}

Norsk_Tipping.html

  1. 7 ! 27 ! 34 ! \frac{7!27!}{34!}

Nörlund–Rice_integral.html

  1. Δ n [ f ] ( x ) = k = 0 n ( n k ) ( - 1 ) n - k f ( x + k ) \Delta^{n}[f](x)=\sum_{k=0}^{n}{n\choose k}(-1)^{n-k}f(x+k)
  2. ( n k ) {n\choose k}
  3. k = α n ( n k ) ( - 1 ) n - k f ( k ) = n ! 2 π i γ f ( z ) z ( z - 1 ) ( z - 2 ) ( z - n ) d z \sum_{k=\alpha}^{n}{n\choose k}(-1)^{n-k}f(k)=\frac{n!}{2\pi i}\oint_{\gamma}% \frac{f(z)}{z(z-1)(z-2)\cdots(z-n)}\,\mathrm{d}z
  4. 0 α n 0\leq\alpha\leq n
  5. k = α n ( n k ) ( - 1 ) k f ( k ) = - 1 2 π i γ B ( n + 1 , - z ) f ( z ) d z \sum_{k=\alpha}^{n}{n\choose k}(-1)^{k}f(k)=-\frac{1}{2\pi i}\oint_{\gamma}B(n% +1,-z)f(z)\,\mathrm{d}z
  6. f ( z ) f(z)
  7. k = α n ( n k ) ( - 1 ) n - k f ( k ) = - n ! 2 π i c - i c + i f ( z ) z ( z - 1 ) ( z - 2 ) ( z - n ) d z \sum_{k=\alpha}^{n}{n\choose k}(-1)^{n-k}f(k)=\frac{-n!}{2\pi i}\int_{c-i% \infty}^{c+i\infty}\frac{f(z)}{z(z-1)(z-2)\cdots(z-n)}\,\mathrm{d}z
  8. { f n } \{f_{n}\}
  9. g ( t ) = e - t n = 0 f n t n . g(t)=e^{-t}\sum_{n=0}^{\infty}f_{n}t^{n}.
  10. ϕ ( s ) = 0 g ( t ) t s - 1 d t , \phi(s)=\int_{0}^{\infty}g(t)t^{s-1}\,\mathrm{d}t,
  11. f n = ( - 1 ) n 2 π i γ ϕ ( s ) Γ ( - s ) n ! s ( s - 1 ) ( s - n ) d s f_{n}=\frac{(-1)^{n}}{2\pi i}\int_{\gamma}\frac{\phi(s)}{\Gamma(-s)}\frac{n!}{% s(s-1)\cdots(s-n)}\,\mathrm{d}s

Nullcline.html

  1. x 1 = f 1 ( x 1 , , x n ) x_{1}^{\prime}=f_{1}(x_{1},\ldots,x_{n})
  2. x 2 = f 2 ( x 1 , , x n ) x_{2}^{\prime}=f_{2}(x_{1},\ldots,x_{n})
  3. \vdots
  4. x n = f n ( x 1 , , x n ) x_{n}^{\prime}=f_{n}(x_{1},\ldots,x_{n})
  5. x x^{\prime}
  6. x x
  7. t t
  8. j j
  9. x j = 0 x_{j}^{\prime}=0
  10. 𝐰 = sign ( P ) 𝐢 + sign ( Q ) 𝐣 \mathbf{w}=\mathrm{sign}(P)\mathbf{i}+\mathrm{sign}(Q)\mathbf{j}

Nyquist_ISI_criterion.html

  1. h ( t ) h(t)
  2. h ( n T s ) = { 1 ; n = 0 0 ; n 0 h(nT_{s})=\begin{cases}1;&n=0\\ 0;&n\neq 0\end{cases}
  3. n n
  4. T s T_{s}
  5. 1 T s k = - + H ( f - k T s ) = 1 f \frac{1}{T_{s}}\sum_{k=-\infty}^{+\infty}H\left(f-\frac{k}{T_{s}}\right)=1% \quad\forall f
  6. H ( f ) H(f)
  7. h ( t ) h(t)
  8. y ( t ) = n = - x [ n ] h ( t - n T s ) y(t)=\sum_{n=-\infty}^{\infty}x[n]\cdot h(t-nT_{s})
  9. y [ k ] = y ( k T s ) = n = - x [ n ] h [ k - n ] y[k]=y(kT_{s})=\sum_{n=-\infty}^{\infty}x[n]\cdot h[k-n]
  10. y [ k ] = x [ k ] h [ 0 ] + n k x [ n ] h [ k - n ] y[k]=x[k]\cdot h[0]+\sum_{n\neq k}x[n]\cdot h[k-n]
  11. h [ n ] = { 1 ; n = 0 0 ; n 0 h[n]=\begin{cases}1;&n=0\\ 0;&n\neq 0\end{cases}
  12. h ( n T s ) = { 1 ; n = 0 0 ; n 0 h(nT_{s})=\begin{cases}1;&n=0\\ 0;&n\neq 0\end{cases}
  13. n n
  14. δ ( t ) \delta(t)
  15. h ( t ) k = - + δ ( t - k T s ) = δ ( t ) h(t)\cdot\sum_{k=-\infty}^{+\infty}\delta(t-kT_{s})=\delta(t)
  16. H ( f ) * 1 T s k = - + δ ( f - k T s ) = 1 H\left(f\right)*\frac{1}{T_{s}}\sum_{k=-\infty}^{+\infty}\delta\left(f-\frac{k% }{T_{s}}\right)=1
  17. 1 T s k = - + H ( f - k T s ) = 1 \frac{1}{T_{s}}\sum_{k=-\infty}^{+\infty}H\left(f-\frac{k}{T_{s}}\right)=1

OBD-II_PIDs.html

  1. ( A × 256 + B ) / 10 - 40 (A\times 256+B)/10-40

Oblate_spheroidal_coordinates.html

  1. a a
  2. x = a cosh μ cos ν cos ϕ x=a\ \cosh\mu\ \cos\nu\ \cos\phi
  3. y = a cosh μ cos ν sin ϕ y=a\ \cosh\mu\ \cos\nu\ \sin\phi
  4. z = a sinh μ sin ν z=a\ \sinh\mu\ \sin\nu
  5. x 2 + y 2 a 2 cosh 2 μ + z 2 a 2 sinh 2 μ = cos 2 ν + sin 2 ν = 1 \frac{x^{2}+y^{2}}{a^{2}\cosh^{2}\mu}+\frac{z^{2}}{a^{2}\sinh^{2}\mu}=\cos^{2}% \nu+\sin^{2}\nu=1
  6. x 2 + y 2 a 2 cos 2 ν - z 2 a 2 sin 2 ν = cosh 2 μ - sinh 2 μ = 1 \frac{x^{2}+y^{2}}{a^{2}\cos^{2}\nu}-\frac{z^{2}}{a^{2}\sin^{2}\nu}=\cosh^{2}% \mu-\sinh^{2}\mu=1
  7. tan ϕ = y x \tan\phi=\frac{y}{x}
  8. ρ 2 = x 2 + y 2 \rho^{2}=x^{2}+y^{2}
  9. d 1 2 = ( ρ + a ) 2 + z 2 d_{1}^{2}=(\rho+a)^{2}+z^{2}
  10. d 2 2 = ( ρ - a ) 2 + z 2 d_{2}^{2}=(\rho-a)^{2}+z^{2}
  11. cosh μ = d 1 + d 2 2 a \cosh\mu=\frac{d_{1}+d_{2}}{2a}
  12. cos ν = d 1 - d 2 2 a \cos\nu=\frac{d_{1}-d_{2}}{2a}
  13. μ = Re arccosh ρ + z i a \mu=\operatorname{Re}\operatorname{arccosh}\frac{\rho+zi}{a}
  14. ν = Im arccosh ρ + z i a \nu=\operatorname{Im}\operatorname{arccosh}\frac{\rho+zi}{a}
  15. ϕ = arctan y x \phi=\arctan\frac{y}{x}
  16. ρ = x 2 + y 2 \rho=\sqrt{x^{2}+y^{2}}
  17. h μ = h ν = a sinh 2 μ + sin 2 ν h_{\mu}=h_{\nu}=a\sqrt{\sinh^{2}\mu+\sin^{2}\nu}
  18. h ϕ = a cosh μ cos ν h_{\phi}=a\cosh\mu\ \cos\nu
  19. d V = a 3 cosh μ cos ν ( sinh 2 μ + sin 2 ν ) d μ d ν d ϕ dV=a^{3}\cosh\mu\ \cos\nu\ \left(\sinh^{2}\mu+\sin^{2}\nu\right)d\mu d\nu d\phi
  20. 2 Φ = 1 a 2 ( sinh 2 μ + sin 2 ν ) [ 1 cosh μ μ ( cosh μ Φ μ ) + 1 cos ν ν ( cos ν Φ ν ) ] + 1 a 2 cosh 2 μ cos 2 ν 2 Φ ϕ 2 \nabla^{2}\Phi=\frac{1}{a^{2}\left(\sinh^{2}\mu+\sin^{2}\nu\right)}\left[\frac% {1}{\cosh\mu}\frac{\partial}{\partial\mu}\left(\cosh\mu\frac{\partial\Phi}{% \partial\mu}\right)+\frac{1}{\cos\nu}\frac{\partial}{\partial\nu}\left(\cos\nu% \frac{\partial\Phi}{\partial\nu}\right)\right]+\frac{1}{a^{2}\cosh^{2}\mu\cos^% {2}\nu}\frac{\partial^{2}\Phi}{\partial\phi^{2}}
  21. 𝐅 \nabla\cdot\mathbf{F}
  22. × 𝐅 \nabla\times\mathbf{F}
  23. μ , ν , ϕ \mu,\nu,\phi
  24. e ^ μ = 1 sinh 2 μ + sin 2 ν ( sinh μ cos ν cos ϕ s y m b o l i ^ + sinh μ cos ν sin ϕ s y m b o l j ^ + cosh μ sin ν s y m b o l k ^ ) \hat{e}_{\mu}=\frac{1}{\sqrt{\sinh^{2}\mu+\sin^{2}\nu}}\left(\sinh\mu\cos\nu% \cos\phi symbol{\hat{i}}+\sinh\mu\cos\nu\sin\phi symbol{\hat{j}}+\cosh\mu\sin% \nu symbol{\hat{k}}\right)
  25. e ^ ν = 1 sinh 2 μ + sin 2 ν ( - cosh μ sin ν cos ϕ s y m b o l i ^ - cosh μ sin ν sin ϕ s y m b o l j ^ + sinh μ cos ν s y m b o l k ^ ) \hat{e}_{\nu}=\frac{1}{\sqrt{\sinh^{2}\mu+\sin^{2}\nu}}\left(-\cosh\mu\sin\nu% \cos\phi symbol{\hat{i}}-\cosh\mu\sin\nu\sin\phi symbol{\hat{j}}+\sinh\mu\cos% \nu symbol{\hat{k}}\right)
  26. e ^ ϕ = - sin ϕ s y m b o l i ^ + cos ϕ s y m b o l j ^ \hat{e}_{\phi}=-\sin\phi symbol{\hat{i}}+\cos\phi symbol{\hat{j}}
  27. s y m b o l i ^ , s y m b o l j ^ , s y m b o l k ^ symbol{\hat{i}},symbol{\hat{j}},symbol{\hat{k}}
  28. e ^ μ \hat{e}_{\mu}
  29. μ \mu
  30. e ^ ϕ \hat{e}_{\phi}
  31. e ^ ν \hat{e}_{\nu}
  32. ( ζ , ξ , ϕ ) (\zeta,\xi,\phi)
  33. ζ = sinh μ \zeta=\sinh\mu
  34. ξ = sin ν \xi=\sin\nu
  35. ζ \zeta
  36. ξ \xi
  37. ζ \zeta
  38. 0 ζ < 0\leq\zeta<\infty
  39. ξ \xi
  40. - 1 ξ < 1 -1\leq\xi<1
  41. x = a ( 1 + ζ 2 ) ( 1 - ξ 2 ) cos ϕ x=a\sqrt{(1+\zeta^{2})(1-\xi^{2})}\,\cos\phi\,
  42. y = a ( 1 + ζ 2 ) ( 1 - ξ 2 ) sin ϕ y=a\sqrt{(1+\zeta^{2})(1-\xi^{2})}\,\sin\phi\,
  43. z = a ζ ξ z=a\zeta\xi\,
  44. ( ζ , ξ , ϕ ) (\zeta,\xi,\phi)
  45. h ζ = a ζ 2 + ξ 2 1 + ζ 2 h_{\zeta}=a\sqrt{\frac{\zeta^{2}+\xi^{2}}{1+\zeta^{2}}}
  46. h ξ = a ζ 2 + ξ 2 1 - ξ 2 h_{\xi}=a\sqrt{\frac{\zeta^{2}+\xi^{2}}{1-\xi^{2}}}
  47. h ϕ = a ( 1 + ζ 2 ) ( 1 - ξ 2 ) h_{\phi}=a\sqrt{(1+\zeta^{2})(1-\xi^{2})}
  48. d V = a 3 ( ζ 2 + ξ 2 ) d ζ d ξ d ϕ dV=a^{3}(\zeta^{2}+\xi^{2})\,d\zeta\,d\xi\,d\phi
  49. V = 1 h ζ V ζ ζ ^ + 1 h ξ V ξ ξ ^ + 1 h ϕ V ϕ ϕ ^ \nabla V=\frac{1}{h_{\zeta}}\frac{\partial V}{\partial\zeta}\,\hat{\zeta}+% \frac{1}{h_{\xi}}\frac{\partial V}{\partial\xi}\,\hat{\xi}+\frac{1}{h_{\phi}}% \frac{\partial V}{\partial\phi}\,\hat{\phi}
  50. 𝐅 = 1 a ( ζ 2 + ξ 2 ) { ζ ( 1 + ζ 2 ζ 2 + ξ 2 F ζ ) + ξ ( 1 - ξ 2 ζ 2 + ξ 2 F ξ ) } + 1 1 + ζ 2 1 - ξ 2 F ϕ ϕ \nabla\mathbf{F}=\frac{1}{a(\zeta^{2}+\xi^{2})}\left\{\frac{\partial}{\partial% \zeta}\left(\sqrt{1+\zeta^{2}}\sqrt{\zeta^{2}+\xi^{2}}F_{\zeta}\right)+\frac{% \partial}{\partial\xi}\left(\sqrt{1-\xi^{2}}\sqrt{\zeta^{2}+\xi^{2}}F_{\xi}% \right)\right\}+\frac{1}{\sqrt{1+\zeta^{2}}\sqrt{1-\xi^{2}}}\frac{\partial F_{% \phi}}{\partial\phi}
  51. 2 V = 1 a 2 ( ζ 2 + ξ 2 ) { ζ [ ( 1 + ζ 2 ) V ζ ] + ξ [ ( 1 - ξ 2 ) V ξ ] } + 1 a 2 ( 1 + ζ 2 ) ( 1 - ξ 2 ) 2 V ϕ 2 \nabla^{2}V=\frac{1}{a^{2}\left(\zeta^{2}+\xi^{2}\right)}\left\{\frac{\partial% }{\partial\zeta}\left[\left(1+\zeta^{2}\right)\frac{\partial V}{\partial\zeta}% \right]+\frac{\partial}{\partial\xi}\left[\left(1-\xi^{2}\right)\frac{\partial V% }{\partial\xi}\right]\right\}+\frac{1}{a^{2}\left(1+\zeta^{2}\right)\left(1-% \xi^{2}\right)}\frac{\partial^{2}V}{\partial\phi^{2}}
  52. V = Z ( ζ ) Ξ ( ξ ) Φ ( ϕ ) V=Z(\zeta)\,\Xi(\xi)\,\Phi(\phi)\,
  53. d d ζ [ ( 1 + ζ 2 ) d Z d ζ ] + m 2 Z 1 + ζ 2 - n ( n + 1 ) Z = 0 \frac{d}{d\zeta}\left[(1+\zeta^{2})\frac{dZ}{d\zeta}\right]+\frac{m^{2}Z}{1+% \zeta^{2}}-n(n+1)Z=0
  54. d d ξ [ ( 1 - ξ 2 ) d Ξ d ξ ] - m 2 Ξ 1 - ξ 2 + n ( n + 1 ) Ξ = 0 \frac{d}{d\xi}\left[(1-\xi^{2})\frac{d\Xi}{d\xi}\right]-\frac{m^{2}\Xi}{1-\xi^% {2}}+n(n+1)\Xi=0
  55. d 2 Φ d ϕ 2 = - m 2 Φ \frac{d^{2}\Phi}{d\phi^{2}}=-m^{2}\Phi
  56. Z m n = A 1 P n m ( i ζ ) + A 2 Q n m ( i ζ ) Z_{mn}=A_{1}P_{n}^{m}(i\zeta)+A_{2}Q_{n}^{m}(i\zeta)
  57. Ξ m n = A 3 P n m ( ξ ) + A 4 Q n m ( ξ ) \Xi_{mn}=A_{3}P_{n}^{m}(\xi)+A_{4}Q_{n}^{m}(\xi)
  58. Φ m = A 5 e i m ϕ + A 6 e - i m ϕ \Phi_{m}=A_{5}e^{im\phi}+A_{6}e^{-im\phi}\,
  59. A i A_{i}
  60. P n m ( z ) P_{n}^{m}(z)
  61. Q n m ( z ) Q_{n}^{m}(z)
  62. V = n = 0 m = 0 Z m n ( ζ ) Ξ m n ( ξ ) Φ m ( ϕ ) V=\sum_{n=0}^{\infty}\sum_{m=0}^{\infty}\,Z_{mn}(\zeta)\,\Xi_{mn}(\xi)\,\Phi_{% m}(\phi)
  63. x = a σ τ cos ϕ x=a\sigma\tau\cos\phi\,
  64. y = a σ τ sin ϕ y=a\sigma\tau\sin\phi\,
  65. z 2 = a 2 ( σ 2 - 1 ) ( 1 - τ 2 ) z^{2}=a^{2}\left(\sigma^{2}-1\right)\left(1-\tau^{2}\right)
  66. σ \sigma
  67. τ \tau
  68. d 1 + d 2 d_{1}+d_{2}
  69. 2 a σ 2a\sigma
  70. d 1 - d 2 d_{1}-d_{2}
  71. 2 a τ 2a\tau
  72. a ( σ + τ ) a(\sigma+\tau)
  73. a ( σ - τ ) a(\sigma-\tau)
  74. x 2 + y 2 a 2 σ 2 + z 2 a 2 ( σ 2 - 1 ) = 1 \frac{x^{2}+y^{2}}{a^{2}\sigma^{2}}+\frac{z^{2}}{a^{2}\left(\sigma^{2}-1\right% )}=1
  75. x 2 + y 2 a 2 τ 2 - z 2 a 2 ( 1 - τ 2 ) = 1 \frac{x^{2}+y^{2}}{a^{2}\tau^{2}}-\frac{z^{2}}{a^{2}\left(1-\tau^{2}\right)}=1
  76. ( σ , τ , ϕ ) (\sigma,\tau,\phi)
  77. h σ = a σ 2 - τ 2 σ 2 - 1 h_{\sigma}=a\sqrt{\frac{\sigma^{2}-\tau^{2}}{\sigma^{2}-1}}
  78. h τ = a σ 2 - τ 2 1 - τ 2 h_{\tau}=a\sqrt{\frac{\sigma^{2}-\tau^{2}}{1-\tau^{2}}}
  79. h ϕ = a σ τ h_{\phi}=a\sigma\tau
  80. d V = a 3 σ τ σ 2 - τ 2 ( σ 2 - 1 ) ( 1 - τ 2 ) d σ d τ d ϕ dV=a^{3}\sigma\tau\frac{\sigma^{2}-\tau^{2}}{\sqrt{\left(\sigma^{2}-1\right)% \left(1-\tau^{2}\right)}}\,d\sigma\,d\tau\,d\phi
  81. 2 Φ = 1 a 2 ( σ 2 - τ 2 ) { σ 2 - 1 σ σ [ ( σ σ 2 - 1 ) Φ σ ] + 1 - τ 2 τ τ [ ( τ 1 - τ 2 ) Φ τ ] } + 1 a 2 σ 2 τ 2 2 Φ ϕ 2 \nabla^{2}\Phi=\frac{1}{a^{2}\left(\sigma^{2}-\tau^{2}\right)}\left\{\frac{% \sqrt{\sigma^{2}-1}}{\sigma}\frac{\partial}{\partial\sigma}\left[\left(\sigma% \sqrt{\sigma^{2}-1}\right)\frac{\partial\Phi}{\partial\sigma}\right]+\frac{% \sqrt{1-\tau^{2}}}{\tau}\frac{\partial}{\partial\tau}\left[\left(\tau\sqrt{1-% \tau^{2}}\right)\frac{\partial\Phi}{\partial\tau}\right]\right\}+\frac{1}{a^{2% }\sigma^{2}\tau^{2}}\frac{\partial^{2}\Phi}{\partial\phi^{2}}
  82. 𝐅 \nabla\cdot\mathbf{F}
  83. × 𝐅 \nabla\times\mathbf{F}
  84. ( σ , τ ) (\sigma,\tau)

Ohm.html

  1. Ω = V A = 1 S = W A 2 = V 2 W = s F = J s C 2 = kg m 2 s C 2 = J s A 2 = kg m 2 s 3 A 2 \Omega=\dfrac{\mbox{V}~{}}{\mbox{A}~{}}=\dfrac{\mbox{1}~{}}{\mbox{S}~{}}=% \dfrac{\mbox{W}~{}}{\mbox{A}~{}^{2}}=\dfrac{\mbox{V}~{}^{2}}{\mbox{W}~{}}=% \dfrac{\mbox{s}~{}}{\mbox{F}~{}}=\dfrac{\mbox{J}~{}\cdot\mbox{s}~{}}{\mbox{C}~% {}^{2}}=\dfrac{\mbox{kg}~{}\cdot\mbox{m}~{}^{2}}{\mbox{s}~{}\cdot\mbox{C}~{}^{% 2}}=\dfrac{\mbox{J}~{}}{\mbox{s}~{}\cdot\mbox{A}~{}^{2}}=\dfrac{\mbox{kg}~{}% \cdot\mbox{m}~{}^{2}}{\mbox{s}~{}^{3}\cdot\mbox{A}~{}^{2}}
  2. P = V I = V 2 R = I 2 R P=V\cdot I=\frac{V^{2}}{R}=I^{2}\cdot R

Oleg_D._Jefimenko.html

  1. 1 4 \tfrac{1}{4}
  2. π \pi
  3. π \pi
  4. 1 4 \tfrac{1}{4}
  5. π \pi
  6. π \pi

Omega_language.html

  1. \mathbb{N}
  2. \cdot
  3. \mathbb{N}
  4. L \vec{L}

On_Denoting.html

  1. C ( E ) x C ( x ) C(E)\leftrightarrow\forall xC(x)
  2. C ( N ) x ¬ C ( x ) C(N)\leftrightarrow\forall x\lnot C(x)
  3. C ( S ) ¬ x ¬ C ( x ) C(S)\leftrightarrow\lnot\forall x\lnot C(x)
  4. x ( ( F ( x ) y ( F ( y ) x = y ) ) E ( x ) ) \exists x\left(\left(F\left(x\right)\land\forall y\left(F\left(y\right)% \rightarrow x=y\right)\right)\land E\left(x\right)\right)
  5. x ( F ( x ) y ( F ( y ) x = y ) ) \exists x\left(F\left(x\right)\land\forall y\left(F\left(y\right)\rightarrow x% =y\right)\right)
  6. x ( K ( x ) y ( K ( y ) y = x ) ) ¬ B ( x ) ) \exists x(K(x)\land\forall y(K(y)\rightarrow y=x))\land\lnot B(x))
  7. ¬ x ( K ( x ) y ( K ( y ) y = x ) ) B ( x ) ) \lnot\exists x(K(x)\land\forall y(K(y)\rightarrow y=x))\land B(x))

On_Sizes_and_Distances.html

  1. s = 1880 3 12 + 1 / 3 s=\sqrt[3]{1880}\approx 12+1/3
  2. = 1 / 27 3 = 1 / 3 \ell=\sqrt[3]{1/27}=1/3
  3. S = s L 12 + 1 / 3 1 / 3 ( 67 + 1 / 3 ) = 2491 + 1 / 3 2490 S=\frac{s}{\ell}L\approx\frac{12+1/3}{1/3}(67+1/3)=2491+1/3\approx 2490
  4. = L tan θ L sin θ \ell=L\tan\theta\approx L\sin\theta
  5. h = tan φ tan θ φ θ φ θ L sin θ h=\frac{\tan\varphi}{\tan\theta}\ell\approx\frac{\varphi}{\theta}\ell\approx% \frac{\varphi}{\theta}L\sin\theta
  6. + x = t + ( t - h ) = 2 t - h x 2 - ( φ θ + 1 ) L sin θ \ell+x=t+(t-h)=2t-h\Rightarrow x\approx 2-\left(\frac{\varphi}{\theta}+1\right% )L\sin\theta
  7. t x = S S - L S = L 1 - x \frac{t}{x}=\frac{S}{S-L}\Rightarrow S=\frac{L}{1-x}
  8. S L ( φ θ + 1 ) L sin θ - 1 = 1 / ( ( φ θ + 1 ) sin θ - 1 L ) S\approx\frac{L}{\left(\frac{\varphi}{\theta}+1\right)L\sin\theta-1}=1\left/% \left(\left(\frac{\varphi}{\theta}+1\right)\sin\theta-\frac{1}{L}\right)\right.
  9. θ \theta
  10. 360 / 650 = 0.554 360^{\circ}/650=0.554^{\circ}
  11. φ \varphi
  12. 2.5 θ = 1.385 2.5\theta=1.385^{\circ}
  13. L L
  14. L sin θ 0 ; 19 , 30 \ell\approx L\sin\theta\approx 0;19,30
  15. h φ θ 0 ; 48 , 45 h\approx\frac{\varphi}{\theta}\ell\approx 0;48,45
  16. + h φ θ L sin θ + L sin θ = ( φ θ + 1 ) L sin θ \ell+h\approx\frac{\varphi}{\theta}L\sin\theta+L\sin\theta=\left(\frac{\varphi% }{\theta}+1\right)L\sin\theta
  17. S L / ( + h - 1 ) 67 ; 20 / 0 ; 8 , 15 489.70 490 S\approx L/(\ell+h-1)\approx 67;20/0;8,15\approx 489.70\approx 490
  18. L S ( φ θ + 1 ) S sin θ - 1 = 1 / ( ( φ θ + 1 ) sin θ - 1 S ) L\approx\frac{S}{\left(\frac{\varphi}{\theta}+1\right)S\sin\theta-1}=1\left/% \left(\left(\frac{\varphi}{\theta}+1\right)\sin\theta-\frac{1}{S}\right)\right.
  19. L 1 / ( ( 2.5 + 1 ) sin 0.277 - 1 490 ) 67.203 67 1 3 L\approx 1\left/\left((2.5+1)\sin 0.277^{\circ}-\frac{1}{490}\right)\right.% \approx 67.203\approx 67\tfrac{1}{3}
  20. L 1 / ( ( 2.5 + 1 ) sin 0.277 ) 59.10 L\approx 1\left/\left((2.5+1)\sin 0.277^{\circ}\right)\right.\approx 59.10
  21. φ A \varphi_{A}
  22. φ H \varphi_{H}
  23. δ \delta
  24. μ \mu
  25. A H = t Crd A O H = t Crd ( φ H - φ A ) AH=t\ \,\text{Crd}\ \angle AOH=t\ \,\text{Crd}(\varphi_{H}-\varphi_{A})
  26. ζ ζ \zeta^{\prime}\approx\zeta
  27. ζ = φ H - δ \zeta=\varphi_{H}-\delta
  28. Z H A = 180 - O H A \angle ZHA=180^{\circ}-\angle OHA
  29. O H A = 180 - A O H 2 = 180 - ( φ H - φ A ) 2 \angle OHA=\frac{180^{\circ}-\angle AOH}{2}=\frac{180^{\circ}-(\varphi_{H}-% \varphi_{A})}{2}
  30. Z H A = 90 + 1 2 ( φ H - φ A ) \angle ZHA=90^{\circ}+\frac{1}{2}(\varphi_{H}-\varphi_{A})
  31. M H A = θ = Z H A - ζ Z H A - ζ = 90 - 1 2 ( φ H + φ A ) + δ \angle MHA=\theta=\angle ZHA-\zeta^{\prime}\approx\angle ZHA-\zeta=90^{\circ}-% \frac{1}{2}(\varphi_{H}+\varphi_{A})+\delta
  32. A H AH
  33. θ \theta
  34. μ \mu
  35. D D^{\prime}
  36. μ \mu
  37. D D + t D\approx D^{\prime}+t
  38. δ = - 3 \delta=-3^{\circ}
  39. θ = 54 + δ \theta=54^{\circ}+\delta
  40. A H = t Crd 10 t 600 3438 AH=t\ \,\text{Crd}\ 10^{\circ}\approx t\ \frac{600}{3438}
  41. μ = 360 60 5 650 \mu=\frac{360\cdot 60}{5\cdot 650}
  42. D = Crd 2 θ A H Crd μ 2 R = Crd ( 108 + 2 δ ) 600 5 650 21600 2 3438 t D^{\prime}=\frac{\,\text{Crd}\ 2\theta\cdot AH}{\,\text{Crd}\ \mu\cdot 2R}=% \frac{\,\text{Crd}\ (108^{\circ}+2\delta)\cdot 600\cdot 5\cdot 650}{21600\cdot 2% \cdot 3438}t
  43. Crd ( 108 + 2 ( - 3 ) ) = Crd 102 2 3438 sin 56 5340 \,\text{Crd}\ (108^{\circ}+2(-3^{\circ}))=\,\text{Crd}\ 102^{\circ}\approx 2% \cdot 3438\sin 56^{\circ}\approx 5340
  44. D = 5340 600 5 650 21600 2 3438 t 70.1 t D D + t 71.1 t D^{\prime}=\frac{5340\cdot 600\cdot 5\cdot 650}{21600\cdot 2\cdot 3438}t% \approx 70.1t\Rightarrow D\approx D^{\prime}+t\approx 71.1t

On_the_Sizes_and_Distances_(Aristarchus).html

  1. S L = 1 cos φ = sec φ . \frac{S}{L}=\frac{1}{\cos\varphi}=\sec\varphi.
  2. 18 < S L < 20. 18<\frac{S}{L}<20.
  3. D L = t t - d \frac{D}{L}=\frac{t}{t-d}\quad
  4. D S = t s - t . \quad\frac{D}{S}=\frac{t}{s-t}.
  5. L S = s \frac{L}{S}=\frac{\ell}{s}
  6. s = t - d s - t s - t s = t - d 1 - t s = t - d t + t s = 1 + d . \frac{\ell}{s}=\frac{t-d}{s-t}\ \ \Rightarrow\ \ \frac{s-t}{s}=\frac{t-d}{\ell% }\ \ \Rightarrow\ \ 1-\frac{t}{s}=\frac{t}{\ell}-\frac{d}{\ell}\ \ \Rightarrow% \ \ \frac{t}{\ell}+\frac{t}{s}=1+\frac{d}{\ell}.
  7. t ( 1 + s ) = 1 + d t = 1 + s 1 + d . \frac{t}{\ell}(1+\frac{\ell}{s})=1+\frac{d}{\ell}\ \ \Rightarrow\ \ \frac{\ell% }{t}=\frac{1+\frac{\ell}{s}}{1+\frac{d}{\ell}}.
  8. t s ( 1 + s ) = 1 + d s t = 1 + s 1 + d . \frac{t}{s}(1+\frac{s}{\ell})=1+\frac{d}{\ell}\ \ \Rightarrow\ \ \frac{s}{t}=% \frac{1+\frac{s}{\ell}}{1+\frac{d}{\ell}}.
  9. t = 1 + x x ( 1 + n ) \frac{\ell}{t}=\frac{1+x}{x(1+n)}
  10. s t = 1 + x 1 + n \frac{s}{t}=\frac{1+x}{1+n}
  11. L t = ( t ) ( 180 π θ ) \frac{L}{t}=\left(\frac{\ell}{t}\right)\left(\frac{180}{\pi\theta}\right)
  12. S t = ( s t ) ( 180 π θ ) \frac{S}{t}=\left(\frac{s}{t}\right)\left(\frac{180}{\pi\theta}\right)

One-way_compression_function.html

  1. R h = | m i | s n R_{h}=\frac{\left|m_{i}\right|}{s\cdot n}
  2. \oplus
  3. H i = E m i ( H i - 1 ) H i - 1 . H_{i}=E_{m_{i}}{(H_{i-1})}\oplus{H_{i-1}}.
  4. R D M = k 1 n = k n . R_{DM}=\frac{k}{1\cdot n}=\frac{k}{n}.
  5. m m
  6. h h
  7. E m ( h ) h = h E_{m}(h)\oplus h=h
  8. h = E m - 1 ( 0 ) h=E_{m}^{-1}(0)
  9. \oplus
  10. H i = E g ( H i - 1 ) ( m i ) m i . H_{i}=E_{g(H_{i-1})}(m_{i})\oplus m_{i}.
  11. R M M O = n 1 n = 1. R_{MMO}=\frac{n}{1\cdot n}=1.
  12. \oplus
  13. H i = E g ( H i - 1 ) ( m i ) H i - 1 m i . H_{i}=E_{g(H_{i-1})}(m_{i})\oplus H_{i-1}\oplus m_{i}.
  14. R M P = n 1 n = 1. R_{MP}=\frac{n}{1\cdot n}=1.

Ontology_alignment.html

  1. i = C i , R i , I i , A i i=\langle C_{i},R_{i},I_{i},A_{i}\rangle
  2. j = C j , R j , I j , A j j=\langle C_{j},R_{j},I_{j},A_{j}\rangle
  3. s [ 0 , 1 ] s\in[0,1]
  4. i i
  5. j j
  6. m = i d , t i , t j , s m=\langle id,t_{i},t_{j},s\rangle
  7. t i t_{i}
  8. t j t_{j}
  9. s s
  10. m m
  11. μ = t i , t j \mu=\langle t_{i},t_{j}\rangle
  12. t i t_{i}
  13. t j t_{j}

Opacity_(optics).html

  1. κ ν \kappa_{\nu}
  2. ν \nu
  3. ν \nu
  4. κ ν \kappa_{\nu}
  5. ρ \rho
  6. I ( x ) = I 0 e - κ ν ρ x I(x)=I_{0}e^{-\kappa_{\nu}\rho x}
  7. I ( x ) I(x)
  8. I 0 I_{0}
  9. x = 0 x=0
  10. κ ν \kappa_{\nu}
  11. u ( ν , T ) = B ν ( T ) / T u(\nu,T)=\partial B_{\nu}(T)/\partial T
  12. κ ν - 1 \kappa_{\nu}^{-1}
  13. 1 κ = 0 κ ν - 1 u ( ν , T ) d ν 0 u ( ν , T ) d ν \frac{1}{\kappa}=\frac{\int_{0}^{\infty}\kappa_{\nu}^{-1}u(\nu,T)d\nu}{\int_{0% }^{\infty}u(\nu,T)d\nu}
  14. λ ν = ( κ ν ρ ) - 1 \lambda_{\nu}=(\kappa_{\nu}\rho)^{-1}
  15. κ es = 0.20 ( 1 + X ) cm 2 g - 1 \kappa_{\rm es}=0.20(1+X){\rm\,cm}^{2}{\rm\,g}^{-1}
  16. X X
  17. κ ff ( ρ , T ) = 0.64 × 10 23 ( ρ [ g cm - 3 ] ) ( T [ K ] ) - 7 / 2 cm 2 g - 1 \kappa_{\rm ff}(\rho,T)=0.64\times 10^{23}(\rho[{\rm g}~{}{\rm\,cm}^{-3}])(T[{% \rm K}])^{-7/2}{\rm\,cm}^{2}{\rm\,g}^{-1}
  18. 1 κ = 0 ( κ ν , es + κ ν , ff ) - 1 u ( ν , T ) d ν 0 u ( ν , T ) d ν \frac{1}{\kappa}=\frac{\int_{0}^{\infty}(\kappa_{\nu,{\rm es}}+\kappa_{\nu,{% \rm ff}})^{-1}u(\nu,T)d\nu}{\int_{0}^{\infty}u(\nu,T)d\nu}

Opaque_context.html

  1. [ t ] [t]
  2. t t
  3. L L
  4. [ x ] [x]
  5. L ( [ C i c e r o ] ) L([Cicero])
  6. L ( [ T u l l y ] ) L([Tully])
  7. [ x ] [x]

Operation_(mathematics).html

  1. + +
  2. - -
  3. × \times
  4. ÷ \div

OPLS.html

  1. E ( r N ) = E bonds + E angles + E dihedrals + E nonbonded E\left(r^{N}\right)=E_{\mathrm{bonds}}+E_{\mathrm{angles}}+E_{\mathrm{% dihedrals}}+E_{\mathrm{nonbonded}}
  2. E bonds = bonds K r ( r - r 0 ) 2 E_{\mathrm{bonds}}=\sum_{\mathrm{bonds}}K_{r}(r-r_{0})^{2}\,
  3. E angles = angles k θ ( θ - θ 0 ) 2 E_{\mathrm{angles}}=\sum_{\mathrm{angles}}k_{\theta}(\theta-\theta_{0})^{2}\,
  4. E dihedrals = V 1 2 [ 1 + cos ( ϕ - ϕ 0 ) ] + V 2 2 [ 1 - cos 2 ( ϕ - ϕ 0 ) ] + V 3 2 [ 1 + cos 3 ( ϕ - ϕ 0 ) ] + V 4 2 [ 1 - cos 4 ( ϕ - ϕ 0 ) ] E_{\mathrm{dihedrals}}=\frac{V_{1}}{2}\left[1+\cos(\phi-\phi_{0})\right]+\frac% {V_{2}}{2}\left[1-\cos 2(\phi-\phi_{0})\right]+\frac{V_{3}}{2}\left[1+\cos 3(% \phi-\phi_{0})\right]+\frac{V_{4}}{2}\left[1-\cos 4(\phi-\phi_{0})\right]
  5. E nonbonded = i > j f i j ( A i j r i j 12 - C i j r i j 6 + q i q j e 2 4 π ϵ 0 r i j ) E_{\mathrm{nonbonded}}=\sum_{i>j}f_{ij}\left(\frac{A_{ij}}{r_{ij}^{12}}-\frac{% C_{ij}}{r_{ij}^{6}}+\frac{q_{i}q_{j}e^{2}}{4\pi\epsilon_{0}r_{ij}}\right)
  6. A i j = A i i A j j A_{ij}=\sqrt{A_{ii}A_{jj}}
  7. C i j = C i i C j j C_{ij}=\sqrt{C_{ii}C_{jj}}
  8. E nonbonded E_{\mathrm{nonbonded}}
  9. f i j = 0.5 f_{ij}=0.5
  10. f i j = 1.0 f_{ij}=1.0

Optical_buffer.html

  1. D D
  2. 2 D , 3 D , 2\cdot D,3\cdot D,\ldots

Optical_cross_section.html

  1. m 2 s r \frac{m^{2}}{sr}
  2. r ( λ ) r(\lambda)
  3. OCS = r ( λ ) D 4 1.4876 λ 2 \mbox{OCS}~{}=r(\lambda)\frac{D^{4}}{1.4876\lambda^{2}}
  4. D D

Optical_flat.html

  1. λ \lambda
  2. λ \lambda

Optical_proximity_correction.html

  1. 0.61 λ / N A , 0.61\lambda/NA,
  2. N A NA
  3. λ \lambda
  4. k 1 , k_{1},
  5. k 1 λ / N A . k_{1}\lambda/NA.
  6. k 1 < 1 k_{1}<1
  7. σ \sigma
  8. 0.5 λ / ( σ N A ) . 0.5\lambda/(\sigma NA).
  9. σ \sigma
  10. k 1 k_{1}
  11. k 1 k_{1}

Optical_pumping.html

  1. m F m_{F}\!
  2. m F m_{F}\!
  3. m F m_{F}\!

Optical_transfer_function.html

  1. OTF ( ν ) = MTF ( ν ) e i PhTF ( ν ) \mathrm{OTF}(\nu)=\mathrm{MTF}(\nu)e^{i\,\mathrm{PhTF}(\nu)}
  2. MTF ( ν ) = | OTF ( ν ) | \mathrm{MTF}(\nu)=\left|\mathrm{OTF}(\nu)\right|
  3. PhTF ( ν ) = arg ( OTF ( ν ) ) , \mathrm{PhTF}(\nu)=\mathrm{arg}(\mathrm{OTF}(\nu)),
  4. arg ( ) \mathrm{arg}(\cdot)
  5. ν \nu
  6. ν \nu
  7. OTF ( 0 ) = MTF ( 0 ) \mathrm{OTF}(0)=\mathrm{MTF}(0)
  8. MTF ( 0 ) 1 \mathrm{MTF}(0)\equiv 1
  9. t h {}^{th}
  10. 2 ν 2\nu
  11. ν \nu
  12. ν = 0 \nu=0
  13. π \pi
  14. θ / 2 - sin ( θ ) / 2 \theta/2-\sin(\theta)/2
  15. θ \theta
  16. | ν | = cos ( θ / 2 ) |\nu|=\cos(\theta/2)
  17. sin ( θ ) / 2 = sin ( θ / 2 ) cos ( θ / 2 ) \sin(\theta)/2=\sin(\theta/2)\cos(\theta/2)
  18. 1 = ν 2 + sin ( arccos ( | ν | ) ) 2 1=\nu^{2}+\sin(\arccos(|\nu|))^{2}
  19. arccos ( | ν | ) - | ν | 1 - ν 2 \arccos(|\nu|)-|\nu|\sqrt{1-\nu^{2}}
  20. 𝑂𝑇𝐹 ( ν ) = 2 π ( arccos ( | ν | ) - | ν | 1 - ν 2 ) \mathit{OTF}(\nu)=\frac{2}{\pi}\left(\arccos(|\nu|)-|\nu|\sqrt{1-\nu^{2}}\right)
  21. MTF = [ LSF ] MTF = f ( x ) e - i 2 π x s d x \,\text{MTF}=\mathcal{F}\left[\,\text{LSF}\right]\qquad\qquad\,\text{MTF}=\int f% (x)e^{-i2\pi\,xs}\,dx
  22. 𝒟 𝒯 \mathcal{DFT}
  23. MTF = 𝒟 𝒯 [ LSF ] = Y k = n = 0 N - 1 y n e - i k 2 π N n k [ 0 , N - 1 ] \,\text{MTF}=\mathcal{DFT}[\,\text{LSF}]=Y_{k}=\sum_{n=0}^{N-1}y_{n}e^{-ik% \frac{2\pi}{N}n}\qquad k\in[0,N-1]
  24. Y k Y_{k}\,
  25. k t h k^{th}
  26. N N\,
  27. n n\,
  28. k k\,
  29. k t h k^{th}
  30. y n y_{n}\,
  31. n t h n^{th}\,
  32. i = - 1 i=\sqrt{-1}
  33. e ± i a = cos ( a ) ± i sin ( a ) e^{\pm ia}=\cos(a)\,\pm\,i\sin(a)
  34. MTF = 𝒟 𝒯 [ LSF ] = Y k = n = 0 N - 1 y n [ cos ( k 2 π N n ) - i sin ( k 2 π N n ) ] k [ 0 , N - 1 ] \,\text{MTF}=\mathcal{DFT}[\,\text{LSF}]=Y_{k}=\sum_{n=0}^{N-1}y_{n}\left[\cos% \left(k\frac{2\pi}{N}n\right)-i\sin\left(k\frac{2\pi}{N}n\right)\right]\qquad k% \in[0,N-1]
  35. ESF = X - μ σ σ = i = 0 n - 1 ( x i - μ ) 2 n μ = i = 0 n - 1 x i n \,\text{ESF}=\frac{X-\mu}{\sigma}\qquad\qquad\sigma\,=\sqrt{\frac{\sum_{i=0}^{% n-1}(x_{i}-\mu\,)^{2}}{n}}\qquad\qquad\mu\,=\frac{\sum_{i=0}^{n-1}x_{i}}{n}
  36. X X\,
  37. x i x_{i}\,
  38. X X\,
  39. μ \mu\,
  40. σ \sigma\,
  41. n n\,
  42. LSF = d d x ESF ( x ) \,\text{LSF}=\frac{d}{dx}\,\text{ESF}(x)
  43. LSF = d d x ESF ( x ) Δ ESF Δ x \,\text{LSF}=\frac{d}{dx}\,\text{ESF}(x)\approx\frac{\Delta\,\text{ESF}}{% \Delta x}
  44. LSF ESF i + 1 - ESF i - 1 2 ( x i + 1 - x i ) \,\text{LSF}\approx\frac{\,\text{ESF}_{i+1}-\,\text{ESF}_{i-1}}{2(x_{i+1}-x_{i% })}
  45. i i\,
  46. i = 1 , 2 , , n - 1 i=1,2,\dots,n-1
  47. x i x_{i}\,
  48. i t h i^{th}\,
  49. i t h i^{th}\,
  50. ESF i \,\text{ESF}_{i}\,
  51. i t h i^{th}\,

Orbit_method.html

  1. G G
  2. j j
  3. 𝔤 * \mathfrak{g}^{*}

Order_type.html

  1. n 2 n n\mapsto 2n
  2. y = 2 x - 1 1 - | 2 x - 1 | y=\frac{2x-1}{1-|{2x-1}|}
  3. η \eta
  4. σ \sigma
  5. σ * \sigma^{*}

Ordered_graph.html

  1. n n
  2. m m
  3. N , E , < \langle N,E,<\rangle
  4. ( n , m ) E (n,m)\in E
  5. n < m n<m
  6. n n
  7. m m
  8. l l
  9. ( m , l ) (m,l)
  10. a a
  11. b b
  12. a a
  13. b b
  14. c c
  15. a a
  16. a a
  17. b b
  18. d d
  19. c c
  20. c c
  21. b b
  22. b b
  23. c c
  24. d d
  25. d d
  26. b b
  27. c c
  28. b b
  29. c c
  30. a a
  31. b b
  32. c c
  33. a a
  34. c c
  35. b b
  36. b b
  37. d d
  38. b b
  39. c c
  40. d d

Ordered_logit.html

  1. poor , log p 1 p 2 + p 3 + p 4 + p 5 , 0 poor or fair , log p 1 + p 2 p 3 + p 4 + p 5 , 1 poor, fair, or good , log p 1 + p 2 + p 3 p 4 + p 5 , 2 poor, fair, good, or very good , log p 1 + p 2 + p 3 + p 4 p 5 , 3 \begin{array}[]{rll}\,\text{poor},&\log\frac{p_{1}}{p_{2}+p_{3}+p_{4}+p_{5}},&% 0\\ \,\text{poor or fair},&\log\frac{p_{1}+p_{2}}{p_{3}+p_{4}+p_{5}},&1\\ \,\text{poor, fair, or good},&\log\frac{p_{1}+p_{2}+p_{3}}{p_{4}+p_{5}},&2\\ \,\text{poor, fair, good, or very good},&\log\frac{p_{1}+p_{2}+p_{3}+p_{4}}{p_% {5}},&3\end{array}
  2. y * = x β + ε , y^{*}=x^{\prime}\beta+\varepsilon,\,
  3. β \beta
  4. y = { 0 if y * μ 1 , 1 if μ 1 < y * μ 2 , 2 if μ 2 < y * μ 3 , N if μ N < y * . y=\begin{cases}0&\,\text{if }y^{*}\leq\mu_{1},\\ 1&\,\text{if }\mu_{1}<y^{*}\leq\mu_{2},\\ 2&\,\text{if }\mu_{2}<y^{*}\leq\mu_{3},\\ \vdots\\ N&\,\text{if }\mu_{N}<y^{*}.\end{cases}
  5. β \beta

Ordered_probit.html

  1. y * = y^{*}=
  2. β + ϵ \beta+\epsilon
  3. β \beta
  4. y = { 0 if y * 0 , 1 if 0 < y * μ 1 , 2 if μ 1 < y * μ 2 N if μ N - 1 < y * . y=\begin{cases}0~{}~{}\,\text{if}~{}~{}y^{*}\leq 0,\\ 1~{}~{}\,\text{if}~{}~{}0<y^{*}\leq\mu_{1},\\ 2~{}~{}\,\text{if}~{}~{}\mu_{1}<y^{*}\leq\mu_{2}\\ \vdots\\ N~{}~{}\,\text{if}~{}~{}\mu_{N-1}<y^{*}.\end{cases}
  5. β \beta

Orthogonal_coordinates.html

  1. d s 2 = k = 1 d ( h k d q k ) 2 ds^{2}=\sum_{k=1}^{d}\left(h_{k}\,dq^{k}\right)^{2}
  2. h k ( 𝐪 ) = def g k k ( 𝐪 ) = | 𝐞 k | h_{k}(\mathbf{q})\ \stackrel{\mathrm{def}}{=}\ \sqrt{g_{kk}(\mathbf{q})}=|% \mathbf{e}_{k}|
  3. 𝐞 k \mathbf{e}_{k}
  4. 𝐞 i 𝐞 j = 0 if i j \mathbf{e}_{i}\cdot\mathbf{e}_{j}=0\quad\,\text{if}\quad i\neq j
  5. 𝐞 i = 𝐫 q i \mathbf{e}_{i}=\frac{\partial\mathbf{r}}{\partial q^{i}}
  6. h i h_{i}
  7. 𝐞 ^ i \hat{\mathbf{e}}_{i}
  8. 𝐞 ^ i = 𝐞 i h i = 𝐞 i | 𝐞 i | \hat{\mathbf{e}}_{i}=\frac{{\mathbf{e}}_{i}}{h_{i}}=\frac{{\mathbf{e}}_{i}}{% \left|{\mathbf{e}}_{i}\right|}
  9. 𝐞 i = 𝐞 ^ i h i = 𝐞 i h i 2 \mathbf{e}^{i}=\frac{\hat{\mathbf{e}}_{i}}{h_{i}}=\frac{\mathbf{e}_{i}}{h_{i}^% {2}}
  10. 𝐞 i 𝐞 j = δ i j \mathbf{e}_{i}\cdot\mathbf{e}^{j}=\delta^{j}_{i}
  11. 𝐞 ^ i = 𝐞 i h i = h i 𝐞 i = 𝐞 ^ i \hat{\mathbf{e}}_{i}=\frac{\mathbf{e}_{i}}{h_{i}}=h_{i}\mathbf{e}^{i}=\hat{% \mathbf{e}}^{i}
  12. 𝐱 = i x i 𝐞 i = i x i 𝐞 i \mathbf{x}=\sum_{i}x^{i}\mathbf{e}_{i}=\sum_{i}x_{i}\mathbf{e}^{i}
  13. h i 2 x i = x i h_{i}^{2}x^{i}=x_{i}\,
  14. 𝐱 𝐲 = i x i 𝐞 ^ i j y j 𝐞 ^ j = i x i y i \mathbf{x}\cdot\mathbf{y}=\sum_{i}x_{i}\hat{\mathbf{e}}_{i}\cdot\sum_{j}y_{j}% \hat{\mathbf{e}}_{j}=\sum_{i}x_{i}y_{i}
  15. 𝐱 𝐲 = i h i 2 x i y i = i x i y i h i 2 = i x i y i = i x i y i \mathbf{x}\cdot\mathbf{y}=\sum_{i}h_{i}^{2}x^{i}y^{i}=\sum_{i}\frac{x_{i}y_{i}% }{h_{i}^{2}}=\sum_{i}x^{i}y_{i}=\sum_{i}x_{i}y^{i}
  16. 𝐱 𝐲 \displaystyle\mathbf{x}\cdot\mathbf{y}
  17. 𝐱 × 𝐲 = ( x 2 y 3 - x 3 y 2 ) 𝐞 ^ 1 + ( x 3 y 1 - x 1 y 3 ) 𝐞 ^ 2 + ( x 1 y 2 - x 2 y 1 ) 𝐞 ^ 3 \mathbf{x}\times\mathbf{y}=(x_{2}y_{3}-x_{3}y_{2})\hat{\mathbf{e}}_{1}+(x_{3}y% _{1}-x_{1}y_{3})\hat{\mathbf{e}}_{2}+(x_{1}y_{2}-x_{2}y_{1})\hat{\mathbf{e}}_{3}
  18. 𝐱 × 𝐲 = i x i 𝐞 i × j y j 𝐞 j = i x i h i 𝐞 ^ i × j y j h j 𝐞 ^ j \mathbf{x}\times\mathbf{y}=\sum_{i}x^{i}\mathbf{e}_{i}\times\sum_{j}y^{j}% \mathbf{e}_{j}=\sum_{i}x^{i}h_{i}\hat{\mathbf{e}}_{i}\times\sum_{j}y^{j}h_{j}% \hat{\mathbf{e}}_{j}
  19. 𝐱 × 𝐲 = ( x 2 y 3 - x 3 y 2 ) h 2 h 3 h 1 𝐞 1 + ( x 3 y 1 - x 1 y 3 ) h 1 h 3 h 2 𝐞 2 + ( x 1 y 2 - x 2 y 1 ) h 1 h 2 h 3 𝐞 3 \mathbf{x}\times\mathbf{y}=(x^{2}y^{3}-x^{3}y^{2})\frac{h_{2}h_{3}}{h_{1}}% \mathbf{e}_{1}+(x^{3}y^{1}-x^{1}y^{3})\frac{h_{1}h_{3}}{h_{2}}\mathbf{e}_{2}+(% x^{1}y^{2}-x^{2}y^{1})\frac{h_{1}h_{2}}{h_{3}}\mathbf{e}_{3}
  20. d 𝐫 = i 𝐫 q i d q i = i 𝐞 i d q i d\mathbf{r}=\sum_{i}\frac{\partial\mathbf{r}}{\partial q^{i}}\,dq^{i}=\sum_{i}% \mathbf{e}_{i}\,dq^{i}
  21. d f = f d 𝐫 d f = f i 𝐞 i d q i df=\nabla f\cdot d\mathbf{r}\quad\Rightarrow\quad df=\nabla f\cdot\sum_{i}% \mathbf{e}_{i}\,dq^{i}
  22. = i 𝐞 i q i \nabla=\sum_{i}\mathbf{e}^{i}\frac{\partial}{\partial q^{i}}
  23. d s y m b o l = h i d q i 𝐞 ^ i = 𝐫 q i d q i dsymbol{\ell}=h_{i}dq_{i}\hat{\mathbf{e}}_{i}=\frac{\partial\mathbf{r}}{% \partial q_{i}}dq_{i}
  24. d = d 𝐫 d 𝐫 = h 1 2 d q 1 2 + h 2 2 d q 2 2 + h 3 2 d q 3 2 d\ell=\sqrt{d\mathbf{r}\cdot d\mathbf{r}}=\sqrt{h_{1}^{2}\,dq_{1}^{2}+h_{2}^{2% }\,dq_{2}^{2}+h_{3}^{2}\,dq_{3}^{2}}
  25. d 𝐒 \displaystyle d\mathbf{S}
  26. d S k = h i h j d q i d q j dS_{k}=h_{i}h_{j}\,dq^{i}\,dq^{j}
  27. d V = | ( h 1 d q 1 𝐞 ^ 1 ) ( h 2 d q 2 𝐞 ^ 2 ) × ( h 3 d q 3 𝐞 ^ 3 ) | = | 𝐞 ^ 1 𝐞 ^ 2 × 𝐞 ^ 3 | h 1 h 2 h 3 d q 1 d q 2 d q 3 = J d q 1 d q 2 d q 3 = h 1 h 2 h 3 d q 1 d q 2 d q 3 \begin{aligned}\displaystyle dV&\displaystyle=|(h_{1}\,dq_{1}\hat{\mathbf{e}}_% {1})\cdot(h_{2}\,dq_{2}\hat{\mathbf{e}}_{2})\times(h_{3}\,dq_{3}\hat{\mathbf{e% }}_{3})|\\ &\displaystyle=|\hat{\mathbf{e}}_{1}\cdot\hat{\mathbf{e}}_{2}\times\hat{% \mathbf{e}}_{3}|h_{1}h_{2}h_{3}\,dq_{1}\,dq_{2}\,dq_{3}\\ &\displaystyle=J\,dq_{1}\,dq_{2}\,dq_{3}\\ &\displaystyle=h_{1}h_{2}h_{3}\,dq_{1}\,dq_{2}\,dq_{3}\end{aligned}
  28. J = | 𝐫 q 1 ( 𝐫 q 2 × 𝐫 q 3 ) | = | ( x , y , z ) ( q 1 , q 2 , q 3 ) | = h 1 h 2 h 3 J=\left|\frac{\partial\mathbf{r}}{\partial q_{1}}\cdot\left(\frac{\partial% \mathbf{r}}{\partial q_{2}}\times\frac{\partial\mathbf{r}}{\partial q_{3}}% \right)\right|=\left|\frac{\partial(x,y,z)}{\partial(q_{1},q_{2},q_{3})}\right% |=h_{1}h_{2}h_{3}
  29. 𝒫 \scriptstyle\mathcal{P}
  30. 𝒫 𝐅 d 𝐫 = 𝒫 i F i 𝐞 i j 𝐞 j d q j = i 𝒫 F i d q i \int_{\mathcal{P}}\mathbf{F}\cdot d\mathbf{r}=\int_{\mathcal{P}}\sum_{i}F_{i}% \mathbf{e}^{i}\cdot\sum_{j}\mathbf{e}_{j}\,dq^{j}=\sum_{i}\int_{\mathcal{P}}F_% {i}\,dq^{i}
  31. d A = i k d s i = i k h i d q i dA=\prod_{i\neq k}ds_{i}=\prod_{i\neq k}h_{i}\,dq^{i}\,
  32. d V = i d s i = i h i d q i dV=\prod_{i}ds_{i}=\prod_{i}h_{i}\,dq^{i}
  33. 𝒮 \scriptstyle\mathcal{S}
  34. 𝒮 𝐅 d 𝐀 = 𝒮 𝐅 𝐧 ^ d A = 𝒮 𝐅 𝐞 ^ 1 d A = 𝒮 F 1 h 2 h 3 h 1 d q 2 d q 3 \int_{\mathcal{S}}\mathbf{F}\cdot d\mathbf{A}=\int_{\mathcal{S}}\mathbf{F}% \cdot\hat{\mathbf{n}}\ dA=\int_{\mathcal{S}}\mathbf{F}\cdot\hat{\mathbf{e}}_{1% }\ dA=\int_{\mathcal{S}}F^{1}\frac{h_{2}h_{3}}{h_{1}}\,dq^{2}\,dq^{3}
  35. F i = 𝐅 𝐞 ^ i F_{i}=\mathbf{F}\cdot\hat{\mathbf{e}}_{i}
  36. ϕ = 𝐞 ^ 1 h 1 ϕ q 1 + 𝐞 ^ 2 h 2 ϕ q 2 + 𝐞 ^ 3 h 3 ϕ q 3 \nabla\phi=\frac{\hat{\mathbf{e}}_{1}}{h_{1}}\frac{\partial\phi}{\partial q^{1% }}+\frac{\hat{\mathbf{e}}_{2}}{h_{2}}\frac{\partial\phi}{\partial q^{2}}+\frac% {\hat{\mathbf{e}}_{3}}{h_{3}}\frac{\partial\phi}{\partial q^{3}}
  37. 𝐅 = 1 h 1 h 2 h 3 [ q 1 ( F 1 h 2 h 3 ) + q 2 ( F 2 h 3 h 1 ) + q 3 ( F 3 h 1 h 2 ) ] \nabla\cdot\mathbf{F}=\frac{1}{h_{1}h_{2}h_{3}}\left[\frac{\partial}{\partial q% ^{1}}\left(F_{1}h_{2}h_{3}\right)+\frac{\partial}{\partial q^{2}}\left(F_{2}h_% {3}h_{1}\right)+\frac{\partial}{\partial q^{3}}\left(F_{3}h_{1}h_{2}\right)\right]
  38. × 𝐅 = 𝐞 ^ 1 h 2 h 3 [ q 2 ( h 3 F 3 ) - q 3 ( h 2 F 2 ) ] + 𝐞 ^ 2 h 3 h 1 [ q 3 ( h 1 F 1 ) - q 1 ( h 3 F 3 ) ] + 𝐞 ^ 3 h 1 h 2 [ q 1 ( h 2 F 2 ) - q 2 ( h 1 F 1 ) ] = 1 h 1 h 2 h 3 | h 1 𝐞 ^ 1 h 2 𝐞 ^ 2 h 3 𝐞 ^ 3 q 1 q 2 q 3 h 1 F 1 h 2 F 2 h 3 F 3 | \begin{aligned}\displaystyle\nabla\times\mathbf{F}&\displaystyle=\frac{\hat{% \mathbf{e}}_{1}}{h_{2}h_{3}}\left[\frac{\partial}{\partial q^{2}}\left(h_{3}F_% {3}\right)-\frac{\partial}{\partial q^{3}}\left(h_{2}F_{2}\right)\right]+\frac% {\hat{\mathbf{e}}_{2}}{h_{3}h_{1}}\left[\frac{\partial}{\partial q^{3}}\left(h% _{1}F_{1}\right)-\frac{\partial}{\partial q^{1}}\left(h_{3}F_{3}\right)\right]% \\ &\displaystyle+\frac{\hat{\mathbf{e}}_{3}}{h_{1}h_{2}}\left[\frac{\partial}{% \partial q^{1}}\left(h_{2}F_{2}\right)-\frac{\partial}{\partial q^{2}}\left(h_% {1}F_{1}\right)\right]=\frac{1}{h_{1}h_{2}h_{3}}\begin{vmatrix}h_{1}\hat{% \mathbf{e}}_{1}&h_{2}\hat{\mathbf{e}}_{2}&h_{3}\hat{\mathbf{e}}_{3}\\ \dfrac{\partial}{\partial q^{1}}&\dfrac{\partial}{\partial q^{2}}&\dfrac{% \partial}{\partial q^{3}}\\ h_{1}F_{1}&h_{2}F_{2}&h_{3}F_{3}\end{vmatrix}\end{aligned}
  39. 2 ϕ = 1 h 1 h 2 h 3 [ q 1 ( h 2 h 3 h 1 ϕ q 1 ) + q 2 ( h 3 h 1 h 2 ϕ q 2 ) + q 3 ( h 1 h 2 h 3 ϕ q 3 ) ] \nabla^{2}\phi=\frac{1}{h_{1}h_{2}h_{3}}\left[\frac{\partial}{\partial q^{1}}% \left(\frac{h_{2}h_{3}}{h_{1}}\frac{\partial\phi}{\partial q^{1}}\right)+\frac% {\partial}{\partial q^{2}}\left(\frac{h_{3}h_{1}}{h_{2}}\frac{\partial\phi}{% \partial q^{2}}\right)+\frac{\partial}{\partial q^{3}}\left(\frac{h_{1}h_{2}}{% h_{3}}\frac{\partial\phi}{\partial q^{3}}\right)\right]
  40. H = h 1 h 2 h 3 H=h_{1}h_{2}h_{3}
  41. ( ϕ ) k = 𝐞 ^ k h k ϕ q k (\nabla\phi)_{k}=\frac{\hat{\mathbf{e}}_{k}}{h_{k}}\frac{\partial\phi}{% \partial q^{k}}
  42. 𝐅 = 1 H q k ( H h k F k ) \nabla\cdot\mathbf{F}=\frac{1}{H}\frac{\partial}{\partial q^{k}}\left(\frac{H}% {h_{k}}F_{k}\right)
  43. ( × 𝐅 ) k = h k 𝐞 ^ k H ϵ i j k q i ( h j F j ) \left(\nabla\times\mathbf{F}\right)_{k}=\frac{h_{k}\hat{\mathbf{e}}_{k}}{H}% \epsilon_{ijk}\frac{\partial}{\partial q^{i}}\left(h_{j}F_{j}\right)
  44. 2 ϕ = 1 H q k ( H h k 2 ϕ q k ) \nabla^{2}\phi=\frac{1}{H}\frac{\partial}{\partial q^{k}}\left(\frac{H}{h_{k}^% {2}}\frac{\partial\phi}{\partial q^{k}}\right)
  45. ( r , θ , ϕ ) [ 0 , ) × [ 0 , π ] × [ 0 , 2 π ) (r,\theta,\phi)\in[0,\infty)\times[0,\pi]\times[0,2\pi)
  46. x = r sin θ cos ϕ y = r sin θ sin ϕ z = r cos θ \begin{aligned}\displaystyle x&\displaystyle=r\sin\theta\cos\phi\\ \displaystyle y&\displaystyle=r\sin\theta\sin\phi\\ \displaystyle z&\displaystyle=r\cos\theta\end{aligned}
  47. h 1 = 1 h 2 = r h 3 = r sin θ \begin{aligned}\displaystyle h_{1}&\displaystyle=1\\ \displaystyle h_{2}&\displaystyle=r\\ \displaystyle h_{3}&\displaystyle=r\sin\theta\end{aligned}
  48. ( r , ϕ , z ) [ 0 , ) × [ 0 , 2 π ) × ( - , ) (r,\phi,z)\in[0,\infty)\times[0,2\pi)\times(-\infty,\infty)
  49. x = r cos ϕ y = r sin ϕ z = z \begin{aligned}\displaystyle x&\displaystyle=r\cos\phi\\ \displaystyle y&\displaystyle=r\sin\phi\\ \displaystyle z&\displaystyle=z\end{aligned}
  50. h 1 = h 3 = 1 h 2 = r \begin{aligned}\displaystyle h_{1}&\displaystyle=h_{3}=1\\ \displaystyle h_{2}&\displaystyle=r\end{aligned}
  51. ( u , v , z ) ( - , ) × [ 0 , ) × ( - , ) (u,v,z)\in(-\infty,\infty)\times[0,\infty)\times(-\infty,\infty)
  52. x = 1 2 ( u 2 - v 2 ) y = u v z = z \begin{aligned}\displaystyle x&\displaystyle=\frac{1}{2}(u^{2}-v^{2})\\ \displaystyle y&\displaystyle=uv\\ \displaystyle z&\displaystyle=z\end{aligned}
  53. h 1 = h 2 = u 2 + v 2 h 3 = 1 \begin{aligned}\displaystyle h_{1}&\displaystyle=h_{2}=\sqrt{u^{2}+v^{2}}\\ \displaystyle h_{3}&\displaystyle=1\end{aligned}
  54. ( u , v , ϕ ) [ 0 , ) × [ 0 , ) × [ 0 , 2 π ) (u,v,\phi)\in[0,\infty)\times[0,\infty)\times[0,2\pi)
  55. x = u v cos ϕ y = u v sin ϕ z = 1 2 ( u 2 - v 2 ) \begin{aligned}\displaystyle x&\displaystyle=uv\cos\phi\\ \displaystyle y&\displaystyle=uv\sin\phi\\ \displaystyle z&\displaystyle=\frac{1}{2}(u^{2}-v^{2})\end{aligned}
  56. h 1 = h 2 = u 2 + v 2 h 3 = u v \begin{aligned}\displaystyle h_{1}&\displaystyle=h_{2}=\sqrt{u^{2}+v^{2}}\\ \displaystyle h_{3}&\displaystyle=uv\end{aligned}
  57. ( u , v , z ) [ 0 , ) × [ 0 , 2 π ) × ( - , ) (u,v,z)\in[0,\infty)\times[0,2\pi)\times(-\infty,\infty)
  58. x = a cosh u cos v y = a sinh u sin v z = z \begin{aligned}\displaystyle x&\displaystyle=a\cosh u\cos v\\ \displaystyle y&\displaystyle=a\sinh u\sin v\\ \displaystyle z&\displaystyle=z\end{aligned}
  59. h 1 = h 2 = a sinh 2 u + sin 2 v h 3 = 1 \begin{aligned}\displaystyle h_{1}&\displaystyle=h_{2}=a\sqrt{\sinh^{2}u+\sin^% {2}v}\\ \displaystyle h_{3}&\displaystyle=1\end{aligned}
  60. ( ξ , η , ϕ ) [ 0 , ) × [ 0 , π ] × [ 0 , 2 π ) (\xi,\eta,\phi)\in[0,\infty)\times[0,\pi]\times[0,2\pi)
  61. x = a sinh ξ sin η cos ϕ y = a sinh ξ sin η sin ϕ z = a cosh ξ cos η \begin{aligned}\displaystyle x&\displaystyle=a\sinh\xi\sin\eta\cos\phi\\ \displaystyle y&\displaystyle=a\sinh\xi\sin\eta\sin\phi\\ \displaystyle z&\displaystyle=a\cosh\xi\cos\eta\end{aligned}
  62. h 1 = h 2 = a sinh 2 ξ + sin 2 η h 3 = a sinh ξ sin η \begin{aligned}\displaystyle h_{1}&\displaystyle=h_{2}=a\sqrt{\sinh^{2}\xi+% \sin^{2}\eta}\\ \displaystyle h_{3}&\displaystyle=a\sinh\xi\sin\eta\end{aligned}
  63. ( ξ , η , ϕ ) [ 0 , ) × [ - π 2 , π 2 ] × [ 0 , 2 π ) (\xi,\eta,\phi)\in[0,\infty)\times\left[-\frac{\pi}{2},\frac{\pi}{2}\right]% \times[0,2\pi)
  64. x = a cosh ξ cos η cos ϕ y = a cosh ξ cos η sin ϕ z = a sinh ξ sin η \begin{aligned}\displaystyle x&\displaystyle=a\cosh\xi\cos\eta\cos\phi\\ \displaystyle y&\displaystyle=a\cosh\xi\cos\eta\sin\phi\\ \displaystyle z&\displaystyle=a\sinh\xi\sin\eta\end{aligned}
  65. h 1 = h 2 = a sinh 2 ξ + sin 2 η h 3 = a cosh ξ cos η \begin{aligned}\displaystyle h_{1}&\displaystyle=h_{2}=a\sqrt{\sinh^{2}\xi+% \sin^{2}\eta}\\ \displaystyle h_{3}&\displaystyle=a\cosh\xi\cos\eta\end{aligned}
  66. ( λ , μ , ν ) λ < c 2 < b 2 < a 2 , c 2 < μ < b 2 < a 2 , c 2 < b 2 < ν < a 2 , \begin{aligned}&\displaystyle(\lambda,\mu,\nu)\\ &\displaystyle\lambda<c^{2}<b^{2}<a^{2},\\ &\displaystyle c^{2}<\mu<b^{2}<a^{2},\\ &\displaystyle c^{2}<b^{2}<\nu<a^{2},\end{aligned}
  67. x 2 a 2 - q i + y 2 b 2 - q i + z 2 c 2 - q i = 1 \frac{x^{2}}{a^{2}-q_{i}}+\frac{y^{2}}{b^{2}-q_{i}}+\frac{z^{2}}{c^{2}-q_{i}}=1
  68. ( q 1 , q 2 , q 3 ) = ( λ , μ , ν ) (q_{1},q_{2},q_{3})=(\lambda,\mu,\nu)
  69. h i = 1 2 ( q j - q i ) ( q k - q i ) ( a 2 - q i ) ( b 2 - q i ) ( c 2 - q i ) h_{i}=\frac{1}{2}\sqrt{\frac{(q_{j}-q_{i})(q_{k}-q_{i})}{(a^{2}-q_{i})(b^{2}-q% _{i})(c^{2}-q_{i})}}
  70. ( u , v , z ) [ 0 , 2 π ) × ( - , ) × ( - , ) (u,v,z)\in[0,2\pi)\times(-\infty,\infty)\times(-\infty,\infty)
  71. x = a sinh v cosh v - cos u y = a sin u cosh v - cos u z = z \begin{aligned}\displaystyle x&\displaystyle=\frac{a\sinh v}{\cosh v-\cos u}\\ \displaystyle y&\displaystyle=\frac{a\sin u}{\cosh v-\cos u}\\ \displaystyle z&\displaystyle=z\end{aligned}
  72. h 1 = h 2 = a cosh v - cos u h 3 = 1 \begin{aligned}\displaystyle h_{1}&\displaystyle=h_{2}=\frac{a}{\cosh v-\cos u% }\\ \displaystyle h_{3}&\displaystyle=1\end{aligned}
  73. ( u , v , ϕ ) ( - π , π ] × [ 0 , ) × [ 0 , 2 π ) (u,v,\phi)\in(-\pi,\pi]\times[0,\infty)\times[0,2\pi)
  74. x = a sinh v cos ϕ cosh v - cos u y = a sinh v sin ϕ cosh v - cos u z = a sin u cosh v - cos u \begin{aligned}\displaystyle x&\displaystyle=\frac{a\sinh v\cos\phi}{\cosh v-% \cos u}\\ \displaystyle y&\displaystyle=\frac{a\sinh v\sin\phi}{\cosh v-\cos u}\\ \displaystyle z&\displaystyle=\frac{a\sin u}{\cosh v-\cos u}\end{aligned}
  75. h 1 = h 2 = a cosh v - cos u h 3 = a sinh v cosh v - cos u \begin{aligned}\displaystyle h_{1}&\displaystyle=h_{2}=\frac{a}{\cosh v-\cos u% }\\ \displaystyle h_{3}&\displaystyle=\frac{a\sinh v}{\cosh v-\cos u}\end{aligned}
  76. ( λ , μ , ν ) ν 2 < b 2 < μ 2 < a 2 λ [ 0 , ) \begin{aligned}&\displaystyle(\lambda,\mu,\nu)\\ &\displaystyle\nu^{2}<b^{2}<\mu^{2}<a^{2}\\ &\displaystyle\lambda\in[0,\infty)\end{aligned}
  77. x = λ μ ν a b y = λ a ( μ 2 - a 2 ) ( ν 2 - a 2 ) a 2 - b 2 z = λ b ( μ 2 - b 2 ) ( ν 2 - b 2 ) a 2 - b 2 \begin{aligned}\displaystyle x&\displaystyle=\frac{\lambda\mu\nu}{ab}\\ \displaystyle y&\displaystyle=\frac{\lambda}{a}\sqrt{\frac{(\mu^{2}-a^{2})(\nu% ^{2}-a^{2})}{a^{2}-b^{2}}}\\ \displaystyle z&\displaystyle=\frac{\lambda}{b}\sqrt{\frac{(\mu^{2}-b^{2})(\nu% ^{2}-b^{2})}{a^{2}-b^{2}}}\end{aligned}
  78. h 1 = 1 h 2 2 = λ 2 ( μ 2 - ν 2 ) ( μ 2 - a 2 ) ( b 2 - μ 2 ) h 3 2 = λ 2 ( μ 2 - ν 2 ) ( ν 2 - a 2 ) ( ν 2 - b 2 ) \begin{aligned}\displaystyle h_{1}&\displaystyle=1\\ \displaystyle h_{2}^{2}&\displaystyle=\frac{\lambda^{2}(\mu^{2}-\nu^{2})}{(\mu% ^{2}-a^{2})(b^{2}-\mu^{2})}\\ \displaystyle h_{3}^{2}&\displaystyle=\frac{\lambda^{2}(\mu^{2}-\nu^{2})}{(\nu% ^{2}-a^{2})(\nu^{2}-b^{2})}\end{aligned}

Orthogonal_wavelet.html

  1. ϕ ( x ) = k = 0 N - 1 a k ϕ ( 2 x - k ) \phi(x)=\sum_{k=0}^{N-1}a_{k}\phi(2x-k)
  2. ( a 0 , , a N - 1 ) (a_{0},\dots,a_{N-1})
  3. ψ ( x ) = k = 0 M - 1 b k ϕ ( 2 x - k ) \psi(x)=\sum_{k=0}^{M-1}b_{k}\phi(2x-k)
  4. ( b 0 , , b M - 1 ) (b_{0},\dots,b_{M-1})
  5. n \Z a n a n + 2 m = 2 δ m , 0 \sum_{n\in\Z}a_{n}a_{n+2m}=2\delta_{m,0}
  6. b n = ( - 1 ) n a N - 1 - n b_{n}=(-1)^{n}a_{N-1-n}
  7. a ( Z ) := a 0 + a 1 Z + + a N - 1 Z N - 1 a(Z):=a_{0}+a_{1}Z+\dots+a_{N-1}Z^{N-1}
  8. ϕ \phi
  9. ψ ~ \tilde{\psi}
  10. ψ ~ \tilde{\psi}
  11. ϕ ~ \tilde{\phi}
  12. ψ \psi
  13. a ( Z ) = 2 1 - A ( 1 + Z ) A p ( Z ) a(Z)=2^{1-A}(1+Z)^{A}p(Z)
  14. 1 sup t [ 0 , 2 π ] | p ( e i t ) | < 2 A - 1 - n 1\leq\sup_{t\in[0,2\pi]}|p(e^{it})|<2^{A-1-n}
  15. n 𝒩 n\in\mathcal{N}
  16. a ( Z ) = 2 1 - A ( 1 + Z ) A a(Z)=2^{1-A}(1+Z)^{A}
  17. a ( Z ) = 1 4 ( 1 + Z ) 2 ( ( 1 + Z ) + c ( 1 - Z ) ) a(Z)=\frac{1}{4}(1+Z)^{2}\,((1+Z)+c(1-Z))

Orthotropic_material.html

  1. 𝐟 = s y m b o l K 𝐝 \mathbf{f}=symbol{K}\cdot\mathbf{d}
  2. 𝐝 , 𝐟 \mathbf{d},\mathbf{f}
  3. s y m b o l K symbol{K}
  4. f i = K i j d j . f_{i}=K_{ij}~{}d_{j}~{}.
  5. 𝐟 ¯ = s y m b o l K ¯ ¯ 𝐝 ¯ [ f 1 f 2 f 3 ] = [ K 11 K 12 K 13 K 21 K 22 K 23 K 31 K 32 K 33 ] [ d 1 d 2 d 3 ] \underline{\mathbf{f}}=\underline{\underline{symbol{K}}}~{}\underline{\mathbf{% d}}\implies\begin{bmatrix}f_{1}\\ f_{2}\\ f_{3}\end{bmatrix}=\begin{bmatrix}K_{11}&K_{12}&K_{13}\\ K_{21}&K_{22}&K_{23}\\ K_{31}&K_{32}&K_{33}\end{bmatrix}\begin{bmatrix}d_{1}\\ d_{2}\\ d_{3}\end{bmatrix}
  6. 𝐟 \mathbf{f}
  7. 𝐝 \mathbf{d}
  8. s y m b o l K symbol{K}
  9. 𝐉 \mathbf{J}
  10. 𝐄 \mathbf{E}
  11. s y m b o l σ symbol{\sigma}
  12. 𝐃 \mathbf{D}
  13. 𝐄 \mathbf{E}
  14. s y m b o l ε symbol{\varepsilon}
  15. 𝐁 \mathbf{B}
  16. 𝐇 \mathbf{H}
  17. s y m b o l μ symbol{\mu}
  18. 𝐪 \mathbf{q}
  19. - s y m b o l T -symbol{\nabla}T
  20. s y m b o l κ symbol{\kappa}
  21. 𝐉 \mathbf{J}
  22. - s y m b o l c -symbol{\nabla}c
  23. s y m b o l D symbol{D}
  24. η μ 𝐯 \eta_{\mu}\mathbf{v}
  25. s y m b o l P symbol{\nabla}P
  26. s y m b o l κ symbol{\kappa}
  27. s y m b o l K ¯ ¯ \underline{\underline{symbol{K}}}
  28. s y m b o l A symbol{A}
  29. s y m b o l A 𝐟 = s y m b o l K ( s y m b o l A \cdotsymbol d ) 𝐟 = ( s y m b o l A - 1 \cdotsymbol K \cdotsymbol A ) \cdotsymbol d symbol{A}\cdot\mathbf{f}=symbol{K}\cdot(symbol{A}\cdotsymbol{d})\implies% \mathbf{f}=(symbol{A}^{-1}\cdotsymbol{K}\cdotsymbol{A})\cdotsymbol{d}
  30. s y m b o l K = s y m b o l A - 1 \cdotsymbol K \cdotsymbol A = s y m b o l A T \cdotsymbol K \cdotsymbol A symbol{K}=symbol{A}^{-1}\cdotsymbol{K}\cdotsymbol{A}=symbol{A}^{T}\cdotsymbol{% K}\cdotsymbol{A}
  31. 3 × 3 3\times 3
  32. s y m b o l A ¯ ¯ \underline{\underline{symbol{A}}}
  33. s y m b o l A ¯ ¯ = [ A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 ] . \underline{\underline{symbol{A}}}=\begin{bmatrix}A_{11}&A_{12}&A_{13}\\ A_{21}&A_{22}&A_{23}\\ A_{31}&A_{32}&A_{33}\end{bmatrix}~{}.
  34. s y m b o l K ¯ ¯ = s y m b o l A T ¯ ¯ s y m b o l K ¯ ¯ s y m b o l A ¯ ¯ \underline{\underline{symbol{K}}}=\underline{\underline{symbol{A}^{T}}}~{}% \underline{\underline{symbol{K}}}~{}\underline{\underline{symbol{A}}}
  35. s y m b o l A 1 ¯ ¯ = [ - 1 0 0 0 1 0 0 0 1 ] ; s y m b o l A 2 ¯ ¯ = [ 1 0 0 0 - 1 0 0 0 1 ] ; s y m b o l A 3 ¯ ¯ = [ 1 0 0 0 1 0 0 0 - 1 ] \underline{\underline{symbol{A}_{1}}}=\begin{bmatrix}-1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}~{};~{}~{}\underline{\underline{symbol{A}_{2}}}=\begin{% bmatrix}1&0&0\\ 0&-1&0\\ 0&0&1\end{bmatrix}~{};~{}~{}\underline{\underline{symbol{A}_{3}}}=\begin{% bmatrix}1&0&0\\ 0&1&0\\ 0&0&-1\end{bmatrix}
  36. s y m b o l K ¯ ¯ \underline{\underline{symbol{K}}}
  37. s y m b o l A 3 ¯ ¯ \underline{\underline{symbol{A}_{3}}}
  38. 1 - 2 1-2\,
  39. s y m b o l K ¯ ¯ = s y m b o l A 3 T ¯ ¯ s y m b o l K ¯ ¯ s y m b o l A 3 ¯ ¯ = [ K 11 K 12 - K 13 K 21 K 22 - K 23 - K 31 - K 32 K 33 ] \underline{\underline{symbol{K}}}=\underline{\underline{symbol{A}^{T}_{3}}}~{}% \underline{\underline{symbol{K}}}~{}\underline{\underline{symbol{A}_{3}}}=% \begin{bmatrix}K_{11}&K_{12}&-K_{13}\\ K_{21}&K_{22}&-K_{23}\\ -K_{31}&-K_{32}&K_{33}\end{bmatrix}
  40. K 13 = K 23 = K 31 = K 32 = 0 K_{13}=K_{23}=K_{31}=K_{32}=0
  41. s y m b o l A 2 ¯ ¯ \underline{\underline{symbol{A}_{2}}}
  42. 1 - 3 1-3\,
  43. s y m b o l K ¯ ¯ = s y m b o l A 2 T ¯ ¯ s y m b o l K ¯ ¯ s y m b o l A 2 ¯ ¯ = [ K 11 - K 12 0 - K 21 K 22 0 0 0 K 33 ] \underline{\underline{symbol{K}}}=\underline{\underline{symbol{A}^{T}_{2}}}~{}% \underline{\underline{symbol{K}}}~{}\underline{\underline{symbol{A}_{2}}}=% \begin{bmatrix}K_{11}&-K_{12}&0\\ -K_{21}&K_{22}&0\\ 0&0&K_{33}\end{bmatrix}
  44. K 12 = K 21 = 0 K_{12}=K_{21}=0
  45. s y m b o l K ¯ ¯ = [ K 11 0 0 0 K 22 0 0 0 K 33 ] \underline{\underline{symbol{K}}}=\begin{bmatrix}K_{11}&0&0\\ 0&K_{22}&0\\ 0&0&K_{33}\end{bmatrix}
  46. s y m b o l σ = 𝖼 \cdotsymbol ε symbol{\sigma}=\mathsf{c}\cdotsymbol{\varepsilon}
  47. s y m b o l σ symbol{\sigma}
  48. s y m b o l ε symbol{\varepsilon}
  49. 𝖼 \mathsf{c}
  50. σ i j = c i j k ε k \sigma_{ij}=c_{ijk\ell}~{}\varepsilon_{k\ell}
  51. c i j k = c j i k , c i j k = c i j k , c i j k = c k i j . c_{ijk\ell}=c_{jik\ell}~{},~{}~{}c_{ijk\ell}=c_{ij\ell k}~{},~{}~{}c_{ijk\ell}% =c_{k\ell ij}~{}.
  52. [ σ 11 σ 22 σ 33 σ 23 σ 31 σ 12 ] = [ c 1111 c 1122 c 1133 c 1123 c 1131 c 1112 c 2211 c 2222 c 2233 c 2223 c 2231 c 2212 c 3311 c 3322 c 3333 c 3323 c 3331 c 3312 c 2311 c 2322 c 2333 c 2323 c 2331 c 2312 c 3111 c 3122 c 3133 c 3123 c 3131 c 3112 c 1211 c 1222 c 1233 c 1223 c 1231 c 1212 ] [ ε 11 ε 22 ε 33 2 ε 23 2 ε 31 2 ε 12 ] \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{33}\\ \sigma_{23}\\ \sigma_{31}\\ \sigma_{12}\end{bmatrix}=\begin{bmatrix}c_{1111}&c_{1122}&c_{1133}&c_{1123}&c_% {1131}&c_{1112}\\ c_{2211}&c_{2222}&c_{2233}&c_{2223}&c_{2231}&c_{2212}\\ c_{3311}&c_{3322}&c_{3333}&c_{3323}&c_{3331}&c_{3312}\\ c_{2311}&c_{2322}&c_{2333}&c_{2323}&c_{2331}&c_{2312}\\ c_{3111}&c_{3122}&c_{3133}&c_{3123}&c_{3131}&c_{3112}\\ c_{1211}&c_{1222}&c_{1233}&c_{1223}&c_{1231}&c_{1212}\end{bmatrix}\begin{% bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{33}\\ 2\varepsilon_{23}\\ 2\varepsilon_{31}\\ 2\varepsilon_{12}\end{bmatrix}
  53. [ σ 1 σ 2 σ 3 σ 4 σ 5 σ 6 ] = [ C 11 C 12 C 13 C 14 C 15 C 16 C 12 C 22 C 23 C 24 C 25 C 26 C 13 C 23 C 33 C 34 C 35 C 36 C 14 C 24 C 34 C 44 C 45 C 46 C 15 C 25 C 35 C 45 C 55 C 56 C 16 C 26 C 36 C 46 C 56 C 66 ] [ ε 1 ε 2 ε 3 ε 4 ε 5 ε 6 ] \begin{bmatrix}\sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\\ \sigma_{4}\\ \sigma_{5}\\ \sigma_{6}\end{bmatrix}=\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{1% 6}\\ C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\ C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\ C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\ C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\ C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}\begin{bmatrix}% \varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{3}\\ \varepsilon_{4}\\ \varepsilon_{5}\\ \varepsilon_{6}\end{bmatrix}
  54. s y m b o l σ ¯ ¯ = 𝖢 ¯ ¯ s y m b o l ε ¯ ¯ \underline{\underline{symbol{\sigma}}}=\underline{\underline{\mathsf{C}}}~{}% \underline{\underline{symbol{\varepsilon}}}
  55. 𝖢 ¯ ¯ \underline{\underline{\mathsf{C}}}
  56. 𝖢 ¯ ¯ \underline{\underline{\mathsf{C}}}
  57. 3 × 3 3\times 3
  58. 𝐀 ¯ ¯ \underline{\underline{\mathbf{A}}}
  59. 𝐀 ¯ ¯ = [ A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 ] . \underline{\underline{\mathbf{A}}}=\begin{bmatrix}A_{11}&A_{12}&A_{13}\\ A_{21}&A_{22}&A_{23}\\ A_{31}&A_{32}&A_{33}\end{bmatrix}~{}.
  60. 6 × 6 6\times 6
  61. 𝖠 σ ¯ ¯ \underline{\underline{\mathsf{A}_{\sigma}}}
  62. 𝖠 σ ¯ ¯ = [ A 11 2 A 12 2 A 13 2 2 A 12 A 13 2 A 11 A 13 2 A 11 A 12 A 21 2 A 22 2 A 23 2 2 A 22 A 23 2 A 21 A 23 2 A 21 A 22 A 31 2 A 32 2 A 33 2 2 A 32 A 33 2 A 31 A 33 2 A 31 A 32 A 21 A 31 A 22 A 32 A 23 A 33 A 22 A 33 + A 23 A 32 A 21 A 33 + A 23 A 31 A 21 A 32 + A 22 A 31 A 11 A 31 A 12 A 32 A 13 A 33 A 12 A 33 + A 13 A 32 A 11 A 33 + A 13 A 31 A 11 A 32 + A 12 A 31 A 11 A 21 A 12 A 22 A 13 A 23 A 12 A 23 + A 13 A 22 A 11 A 23 + A 13 A 21 A 11 A 22 + A 12 A 21 ] \underline{\underline{\mathsf{A}_{\sigma}}}=\begin{bmatrix}A_{11}^{2}&A_{12}^{% 2}&A_{13}^{2}&2A_{12}A_{13}&2A_{11}A_{13}&2A_{11}A_{12}\\ A_{21}^{2}&A_{22}^{2}&A_{23}^{2}&2A_{22}A_{23}&2A_{21}A_{23}&2A_{21}A_{22}\\ A_{31}^{2}&A_{32}^{2}&A_{33}^{2}&2A_{32}A_{33}&2A_{31}A_{33}&2A_{31}A_{32}\\ A_{21}A_{31}&A_{22}A_{32}&A_{23}A_{33}&A_{22}A_{33}+A_{23}A_{32}&A_{21}A_{33}+% A_{23}A_{31}&A_{21}A_{32}+A_{22}A_{31}\\ A_{11}A_{31}&A_{12}A_{32}&A_{13}A_{33}&A_{12}A_{33}+A_{13}A_{32}&A_{11}A_{33}+% A_{13}A_{31}&A_{11}A_{32}+A_{12}A_{31}\\ A_{11}A_{21}&A_{12}A_{22}&A_{13}A_{23}&A_{12}A_{23}+A_{13}A_{22}&A_{11}A_{23}+% A_{13}A_{21}&A_{11}A_{22}+A_{12}A_{21}\end{bmatrix}
  63. 𝖠 ε ¯ ¯ = [ A 11 2 A 12 2 A 13 2 A 12 A 13 A 11 A 13 A 11 A 12 A 21 2 A 22 2 A 23 2 A 22 A 23 A 21 A 23 A 21 A 22 A 31 2 A 32 2 A 33 2 A 32 A 33 A 31 A 33 A 31 A 32 2 A 21 A 31 2 A 22 A 32 2 A 23 A 33 A 22 A 33 + A 23 A 32 A 21 A 33 + A 23 A 31 A 21 A 32 + A 22 A 31 2 A 11 A 31 2 A 12 A 32 2 A 13 A 33 A 12 A 33 + A 13 A 32 A 11 A 33 + A 13 A 31 A 11 A 32 + A 12 A 31 2 A 11 A 21 2 A 12 A 22 2 A 13 A 23 A 12 A 23 + A 13 A 22 A 11 A 23 + A 13 A 21 A 11 A 22 + A 12 A 21 ] \underline{\underline{\mathsf{A}_{\varepsilon}}}=\begin{bmatrix}A_{11}^{2}&A_{% 12}^{2}&A_{13}^{2}&A_{12}A_{13}&A_{11}A_{13}&A_{11}A_{12}\\ A_{21}^{2}&A_{22}^{2}&A_{23}^{2}&A_{22}A_{23}&A_{21}A_{23}&A_{21}A_{22}\\ A_{31}^{2}&A_{32}^{2}&A_{33}^{2}&A_{32}A_{33}&A_{31}A_{33}&A_{31}A_{32}\\ 2A_{21}A_{31}&2A_{22}A_{32}&2A_{23}A_{33}&A_{22}A_{33}+A_{23}A_{32}&A_{21}A_{3% 3}+A_{23}A_{31}&A_{21}A_{32}+A_{22}A_{31}\\ 2A_{11}A_{31}&2A_{12}A_{32}&2A_{13}A_{33}&A_{12}A_{33}+A_{13}A_{32}&A_{11}A_{3% 3}+A_{13}A_{31}&A_{11}A_{32}+A_{12}A_{31}\\ 2A_{11}A_{21}&2A_{12}A_{22}&2A_{13}A_{23}&A_{12}A_{23}+A_{13}A_{22}&A_{11}A_{2% 3}+A_{13}A_{21}&A_{11}A_{22}+A_{12}A_{21}\end{bmatrix}
  64. 𝖠 ε ¯ ¯ T = 𝖠 σ ¯ ¯ - 1 \underline{\underline{\mathsf{A}_{\varepsilon}}}^{T}=\underline{\underline{% \mathsf{A}_{\sigma}}}^{-1}
  65. 𝐀 ¯ ¯ \underline{\underline{\mathbf{A}}}
  66. 𝖢 ¯ ¯ = 𝖠 ε ¯ ¯ T 𝖢 ¯ ¯ 𝖠 ε ¯ ¯ \underline{\underline{\mathsf{C}}}=\underline{\underline{\mathsf{A}_{% \varepsilon}}}^{T}~{}\underline{\underline{\mathsf{C}}}~{}\underline{% \underline{\mathsf{A}_{\varepsilon}}}
  67. 𝐀 1 ¯ ¯ = [ - 1 0 0 0 1 0 0 0 1 ] ; 𝐀 2 ¯ ¯ = [ 1 0 0 0 - 1 0 0 0 1 ] ; 𝐀 3 ¯ ¯ = [ 1 0 0 0 1 0 0 0 - 1 ] \underline{\underline{\mathbf{A}_{1}}}=\begin{bmatrix}-1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}~{};~{}~{}\underline{\underline{\mathbf{A}_{2}}}=\begin{% bmatrix}1&0&0\\ 0&-1&0\\ 0&0&1\end{bmatrix}~{};~{}~{}\underline{\underline{\mathbf{A}_{3}}}=\begin{% bmatrix}1&0&0\\ 0&1&0\\ 0&0&-1\end{bmatrix}
  68. 𝖢 ¯ ¯ \underline{\underline{\mathsf{C}}}
  69. 𝐀 3 ¯ ¯ \underline{\underline{\mathbf{A}_{3}}}
  70. 1 - 2 1-2\,
  71. 𝖠 ε ¯ ¯ = [ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 - 1 0 0 0 0 0 0 - 1 0 0 0 0 0 0 1 ] \underline{\underline{\mathsf{A}_{\varepsilon}}}=\begin{bmatrix}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&-1&0&0\\ 0&0&0&0&-1&0\\ 0&0&0&0&0&1\end{bmatrix}
  72. 𝖢 ¯ ¯ = 𝖠 ε ¯ ¯ T 𝖢 ¯ ¯ 𝖠 ε ¯ ¯ \underline{\underline{\mathsf{C}}}=\underline{\underline{\mathsf{A}_{% \varepsilon}}}^{T}~{}\underline{\underline{\mathsf{C}}}~{}\underline{% \underline{\mathsf{A}_{\varepsilon}}}
  73. [ C 11 C 12 C 13 C 14 C 15 C 16 C 12 C 22 C 23 C 24 C 25 C 26 C 13 C 23 C 33 C 34 C 35 C 36 C 14 C 24 C 34 C 44 C 45 C 46 C 15 C 25 C 35 C 45 C 55 C 56 C 16 C 26 C 36 C 46 C 56 C 66 ] = [ C 11 C 12 C 13 - C 14 - C 15 C 16 C 12 C 22 C 23 - C 24 - C 25 C 26 C 13 C 23 C 33 - C 34 - C 35 C 36 - C 14 - C 24 - C 34 C 44 C 45 - C 46 - C 15 - C 25 - C 35 C 45 C 55 - C 56 C 16 C 26 C 36 - C 46 - C 56 C 66 ] \begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{16}\\ C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\ C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\ C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\ C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\ C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}=\begin{bmatrix}C_{11}&C% _{12}&C_{13}&-C_{14}&-C_{15}&C_{16}\\ C_{12}&C_{22}&C_{23}&-C_{24}&-C_{25}&C_{26}\\ C_{13}&C_{23}&C_{33}&-C_{34}&-C_{35}&C_{36}\\ -C_{14}&-C_{24}&-C_{34}&C_{44}&C_{45}&-C_{46}\\ -C_{15}&-C_{25}&-C_{35}&C_{45}&C_{55}&-C_{56}\\ C_{16}&C_{26}&C_{36}&-C_{46}&-C_{56}&C_{66}\end{bmatrix}
  74. C 14 = C 15 = C 24 = C 25 = C 34 = C 35 = C 46 = C 56 = 0 . C_{14}=C_{15}=C_{24}=C_{25}=C_{34}=C_{35}=C_{46}=C_{56}=0~{}.
  75. 𝐀 2 ¯ ¯ \underline{\underline{\mathbf{A}_{2}}}
  76. 1 - 3 1-3\,
  77. 𝖠 ε ¯ ¯ = [ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 - 1 0 0 0 0 0 0 1 0 0 0 0 0 0 - 1 ] \underline{\underline{\mathsf{A}_{\varepsilon}}}=\begin{bmatrix}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&-1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&-1\end{bmatrix}
  78. C 16 = C 26 = C 36 = C 45 = 0 . C_{16}=C_{26}=C_{36}=C_{45}=0~{}.
  79. 𝖢 ¯ ¯ = [ C 11 C 12 C 13 0 0 0 C 12 C 22 C 23 0 0 0 C 13 C 23 C 33 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 55 0 0 0 0 0 0 C 66 ] \underline{\underline{\mathsf{C}}}=\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\ C_{12}&C_{22}&C_{23}&0&0&0\\ C_{13}&C_{23}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{55}&0\\ 0&0&0&0&0&C_{66}\end{bmatrix}
  80. 𝖲 ¯ ¯ = [ 1 E 1 - ν 21 E 2 - ν 31 E 3 0 0 0 - ν 12 E 1 1 E 2 - ν 32 E 3 0 0 0 - ν 13 E 1 - ν 23 E 2 1 E 3 0 0 0 0 0 0 1 G 23 0 0 0 0 0 0 1 G 31 0 0 0 0 0 0 1 G 12 ] \underline{\underline{\mathsf{S}}}=\begin{bmatrix}\tfrac{1}{E_{\rm 1}}&-\tfrac% {\nu_{\rm 21}}{E_{\rm 2}}&-\tfrac{\nu_{\rm 31}}{E_{\rm 3}}&0&0&0\\ -\tfrac{\nu_{\rm 12}}{E_{\rm 1}}&\tfrac{1}{E_{\rm 2}}&-\tfrac{\nu_{\rm 32}}{E_% {\rm 3}}&0&0&0\\ -\tfrac{\nu_{\rm 13}}{E_{\rm 1}}&-\tfrac{\nu_{\rm 23}}{E_{\rm 2}}&\tfrac{1}{E_% {\rm 3}}&0&0&0\\ 0&0&0&\tfrac{1}{G_{\rm 23}}&0&0\\ 0&0&0&0&\tfrac{1}{G_{\rm 31}}&0\\ 0&0&0&0&0&\tfrac{1}{G_{\rm 12}}\\ \end{bmatrix}
  81. E i {E}_{\rm i}\,
  82. i i
  83. G ij G_{\rm ij}\,
  84. j j
  85. i i
  86. ν ij \nu_{\rm ij}\,
  87. j j
  88. i i
  89. s y m b o l ε ¯ ¯ = 𝖲 ¯ ¯ s y m b o l σ ¯ ¯ \underline{\underline{symbol{\varepsilon}}}=\underline{\underline{\mathsf{S}}}% ~{}\underline{\underline{symbol{\sigma}}}
  90. 𝖲 ¯ ¯ \underline{\underline{\mathsf{S}}}
  91. 𝖲 ¯ ¯ = [ S 11 S 12 S 13 0 0 0 S 12 S 22 S 23 0 0 0 S 13 S 23 S 33 0 0 0 0 0 0 S 44 0 0 0 0 0 0 S 55 0 0 0 0 0 0 S 66 ] \underline{\underline{\mathsf{S}}}=\begin{bmatrix}S_{11}&S_{12}&S_{13}&0&0&0\\ S_{12}&S_{22}&S_{23}&0&0&0\\ S_{13}&S_{23}&S_{33}&0&0&0\\ 0&0&0&S_{44}&0&0\\ 0&0&0&0&S_{55}&0\\ 0&0&0&0&0&S_{66}\end{bmatrix}
  92. Δ k := det ( 𝖲 k ¯ ¯ ) > 0 \Delta_{k}:=\det(\underline{\underline{\mathsf{S}_{k}}})>0
  93. 𝖲 k ¯ ¯ \underline{\underline{\mathsf{S}_{k}}}
  94. k × k k\times k
  95. 𝖲 ¯ ¯ \underline{\underline{\mathsf{S}}}
  96. Δ 1 > 0 S 11 > 0 Δ 2 > 0 S 11 S 22 - S 12 2 > 0 Δ 3 > 0 ( S 11 S 22 - S 12 2 ) S 33 - S 11 S 23 2 + 2 S 12 S 23 S 13 - S 22 S 13 2 > 0 Δ 4 > 0 S 44 Δ 3 > 0 S 44 > 0 Δ 5 > 0 S 44 S 55 Δ 3 > 0 S 55 > 0 Δ 6 > 0 S 44 S 55 S 66 Δ 3 > 0 S 66 > 0 \begin{aligned}\displaystyle\Delta_{1}>0&\displaystyle\implies\quad S_{11}>0\\ \displaystyle\Delta_{2}>0&\displaystyle\implies\quad S_{11}S_{22}-S_{12}^{2}>0% \\ \displaystyle\Delta_{3}>0&\displaystyle\implies\quad(S_{11}S_{22}-S_{12}^{2})S% _{33}-S_{11}S_{23}^{2}+2S_{12}S_{23}S_{13}-S_{22}S_{13}^{2}>0\\ \displaystyle\Delta_{4}>0&\displaystyle\implies\quad S_{44}\Delta_{3}>0% \implies S_{44}>0\\ \displaystyle\Delta_{5}>0&\displaystyle\implies\quad S_{44}S_{55}\Delta_{3}>0% \implies S_{55}>0\\ \displaystyle\Delta_{6}>0&\displaystyle\implies\quad S_{44}S_{55}S_{66}\Delta_% {3}>0\implies S_{66}>0\end{aligned}
  97. S 11 > 0 , S 22 > 0 , S 33 > 0 , S 44 > 0 , S 55 > 0 , S 66 > 0 S_{11}>0~{},~{}~{}S_{22}>0~{},~{}~{}S_{33}>0~{},~{}~{}S_{44}>0~{},~{}~{}S_{55}% >0~{},~{}~{}S_{66}>0
  98. E 1 > 0 , E 2 > 0 , E 3 > 0 , G 12 > 0 , G 23 > 0 , G 13 > 0 E_{1}>0,E_{2}>0,E_{3}>0,G_{12}>0,G_{23}>0,G_{13}>0
  99. ν i j \nu_{ij}

Oscillatory_integral.html

  1. f ( x ) f(x)
  2. f ( x ) = e i ϕ ( x , ξ ) a ( x , ξ ) d ξ f(x)=\int e^{i\phi(x,\xi)}\,a(x,\xi)\,\mathrm{d}\xi
  3. ϕ ( x , ξ ) \phi(x,\xi)
  4. a ( x , ξ ) a(x,\xi)
  5. x n × R ξ N \mathbb{R}_{x}^{n}\times\mathrm{R}^{N}_{\xi}
  6. ϕ \phi
  7. { ξ = 0 } \{\xi=0\}
  8. ϕ \phi
  9. a a
  10. ϕ \phi
  11. a a
  12. S 1 , 0 m ( x n × R ξ N ) S^{m}_{1,0}(\mathbb{R}_{x}^{n}\times\mathrm{R}^{N}_{\xi})
  13. m m\in\mathbb{R}
  14. m m
  15. ϕ \phi
  16. a a
  17. m < - n + 1 m<-n+1
  18. f ( x ) f(x)
  19. x x
  20. f ( x ) f(x)
  21. m - n + 1 m\geq-n+1
  22. n \mathbb{R}^{n}
  23. f ( x ) f(x)
  24. a ( x , ξ ) S 1 , 0 m ( x n × R ξ N ) a(x,\xi)\in S^{m}_{1,0}(\mathbb{R}_{x}^{n}\times\mathrm{R}^{N}_{\xi})
  25. ξ \xi
  26. f ( x ) = lim ϵ 0 + e i ϕ ( x , ξ ) a ( x , ξ ) e - ϵ | ξ | 2 / 2 d ξ f(x)=\lim\limits_{\epsilon\rightarrow 0^{+}}\int e^{i\phi(x,\xi)}\,a(x,\xi)e^{% -\epsilon|\xi|^{2}/2}\,\mathrm{d}\xi
  27. L L
  28. f ( x ) f(x)
  29. ψ \psi
  30. f , ψ = e i ϕ ( x , ξ ) L ( a ( x , ξ ) ψ ( x ) ) d x d ξ \langle f,\psi\rangle=\int e^{i\phi(x,\xi)}L\left(a(x,\xi)\,\psi(x)\right)\,% \mathrm{d}x\,\mathrm{d}\xi
  31. L L
  32. ϕ \phi
  33. m m
  34. a a
  35. N N
  36. M M
  37. L L
  38. C ( 1 + | ξ | ) - M C(1+|\xi|)^{-M}
  39. | ξ | |\xi|
  40. δ ( x ) \delta(x)
  41. 1 ( 2 π ) n n e i x ξ d ξ . \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{ix\cdot\xi}\,\mathrm{d}\xi.
  42. δ ( x ) = lim ε 0 + 1 ( 2 π ) n n e i x ξ e - ε | ξ | 2 / 2 d ξ = lim ε 0 + 1 ( 2 π ε ) n e - | ξ | 2 / ( 2 ε ) . \delta(x)=\lim_{\varepsilon\rightarrow 0^{+}}\frac{1}{(2\pi)^{n}}\int_{\mathbb% {R}^{n}}e^{ix\cdot\xi}e^{-\varepsilon|\xi|^{2}/2}\mathrm{d}\xi=\lim_{% \varepsilon\rightarrow 0^{+}}\frac{1}{(\sqrt{2\pi\varepsilon})^{n}}e^{-|\xi|^{% 2}/(2\varepsilon)}.
  43. L L
  44. L = ( 1 - Δ x ) k ( 1 + | ξ | 2 ) k L=\frac{(1-\Delta_{x})^{k}}{(1+|\xi|^{2})^{k}}
  45. Δ x \Delta_{x}
  46. x x
  47. k k
  48. ( n - 1 ) / 2 (n-1)/2
  49. L L
  50. δ , ψ = ψ ( 0 ) = 1 ( 2 π ) n n e i x ξ L ( ψ ) ( x , ξ ) d ξ d x , \langle\delta,\psi\rangle=\psi(0)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{% ix\cdot\xi}L(\psi)(x,\xi)\,\mathrm{d}\xi\,\mathrm{d}x,
  51. L = | α | m p α ( x ) D α L=\sum\limits_{|\alpha|\leq m}p_{\alpha}(x)D^{\alpha}
  52. D α = x α / i | α | D^{\alpha}=\partial^{\alpha}_{x}/i^{|\alpha|}
  53. L L
  54. 1 ( 2 π ) n n e i ξ ( x - y ) | α | m p α ( x ) ξ α d ξ . \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}e^{i\xi\cdot(x-y)}\sum\limits_{|% \alpha|\leq m}p_{\alpha}(x)\,\xi^{\alpha}\,\mathrm{d}\xi.

Ostrowski's_theorem.html

  1. p p
  2. | | |\cdot|
  3. | | |\cdot|_{\ast}
  4. c > 0 c>0
  5. | x | = | x | c for all x 𝐊 . |x|_{\ast}=|x|^{c}\,\text{ for all }x\in\mathbf{K}.
  6. | x | 0 := { 0 , if x = 0 1 , if x 0. |x|_{0}:=\begin{cases}0,&\,\text{if }x=0\\ 1,&\,\text{if }x\neq 0.\end{cases}
  7. | x | := { x , if x 0 - x , if x < 0. |x|_{\infty}:=\begin{cases}x,&\,\text{if }x\geq 0\\ -x,&\,\text{if }x<0.\end{cases}
  8. p p
  9. p p
  10. x x
  11. x = p n a b x=p^{n}\dfrac{a}{b}
  12. a a
  13. b b
  14. p p
  15. n 𝐙 n\in\mathbf{Z}
  16. | x | p := { 0 , if x = 0 p - n , if x 0. |x|_{p}:=\begin{cases}0,&\,\text{if }x=0\\ p^{-n},&\,\text{if }x\neq 0.\end{cases}
  17. ( 𝐐 , | | ) (\mathbf{Q},|\cdot|_{\ast})
  18. n 𝐍 , | n | > 1 \exists{n\in\mathbf{N}},|n|_{\ast}>1
  19. n 𝐍 , | n | 1 \forall{n\in\mathbf{N}},|n|_{\ast}\leq 1
  20. c c
  21. | n | = | n | c |n|_{\ast}=|n|^{c}_{\ast\ast}
  22. | m n | = | m | | n | = | m | c | n | c = ( | m | | n | ) c = | m n | c , \left|\frac{m}{n}\right|_{\ast}=\frac{|m|_{\ast}}{|n|_{\ast}}=\frac{|m|^{c}_{% \ast\ast}}{|n|^{c}_{\ast\ast}}=\left(\frac{|m|_{\ast\ast}}{|n|_{\ast\ast}}% \right)^{c}=\left|\frac{m}{n}\right|^{c}_{\ast\ast},
  23. | - x | = | x | = | x | c = | - x | c . |-x|_{\ast}=|x|_{\ast}=|x|^{c}_{\ast\ast}=|-x|^{c}_{\ast\ast}.
  24. a , b a,b
  25. n n
  26. a , b > 1 a,b>1
  27. a a
  28. b n = i < m c i a i , c i { 0 , 1 , , a - 1 } , m n log b log a + 1. b^{n}=\sum_{i<m}c_{i}a^{i},\qquad c_{i}\in\{0,1,\ldots,a-1\},\quad m\leq n% \tfrac{\log b}{\log a}+1.
  29. | b | n \displaystyle|b|_{\ast}^{n}
  30. | b | ( a ( n log a b + 1 ) ) 1 n max { | a | log a b , 1 } |b|_{\ast}\leq\left(a(n\log_{a}b+1)\right)^{\frac{1}{n}}\max\left\{|a|_{\ast}^% {\log_{a}b},1\right\}
  31. ( a ( n log a b + 1 ) ) 1 n 1 , as n \left(a(n\log_{a}b+1)\right)^{\frac{1}{n}}\to 1,\quad\,\text{as}\quad n\to\infty
  32. | b | max { | a | log a b , 1 } . |b|_{\ast}\leq\max\left\{|a|_{\ast}^{\log_{a}b},1\right\}.
  33. a a
  34. | a | log a b 1 |a|_{\ast}^{\log_{a}b}\leq 1
  35. | b | 1 |b|_{\ast}\leq 1
  36. a , b > 1 a,b>1
  37. | b | | a | log b log a , |b|_{\ast}\leq|a|_{\ast}^{\frac{\log b}{\log a}},
  38. log | b | log b log | a | log a . \frac{\log|b|_{\ast}}{\log b}\leq\frac{\log|a|_{\ast}}{\log a}.
  39. a , b a,b
  40. λ 𝐑 + \lambda\in\mathbf{R}^{+}
  41. log | n | = λ log n \log|n|_{\ast}=\lambda\log n
  42. | n | = n λ = | n | λ |n|_{\ast}=n^{\lambda}=|n|_{\infty}^{\lambda}
  43. n > 1 n>1
  44. | x | = | x | λ |x|_{\ast}=|x|_{\infty}^{\lambda}
  45. n = i < r p i e i n=\prod_{i<r}p_{i}^{e_{i}}
  46. p , q p,q
  47. e 𝐍 + e\in\mathbf{N}^{+}
  48. | p | e , | q | e < 1 2 |p|_{\ast}^{e},|q|_{\ast}^{e}<\tfrac{1}{2}
  49. m , n 𝐙 m,n∈\mathbf{Z}
  50. m p e + n q e = 1 mp^{e}+nq^{e}=1
  51. 1 = | 1 | | m | | p | e + | n | | q | e < | m | + | n | 2 1 , 1=|1|_{\ast}\leq|m|_{\ast}|p|_{\ast}^{e}+|n|_{\ast}|q|_{\ast}^{e}<\frac{|m|_{% \ast}+|n|_{\ast}}{2}\leq 1,
  52. j j
  53. i j i≠j
  54. c = - log α log p , c=-\tfrac{\log\alpha}{\log p},
  55. | n | = | i < r p i e i | = i < r | p i | e i = | p j | e j = ( p - e j ) c = | n | p c |n|_{\ast}=\left|\prod_{i<r}p_{i}^{e_{i}}\right|_{\ast}=\prod_{i<r}\left|p_{i}% \right|_{\ast}^{e_{i}}=\left|p_{j}\right|_{\ast}^{e_{j}}=(p^{-e_{j}})^{c}=|n|_% {p}^{c}
  56. | x | = | x | p c |x|_{\ast}=|x|_{p}^{c}
  57. p p
  58. | | : 𝐐 𝐑 |\cdot|_{\ast}:\mathbf{Q}\to\mathbf{R}
  59. | | = | | c |\cdot|_{\ast}=|\cdot|_{\infty}^{c}
  60. c ( 0 , 1 ] c\in(0,1]
  61. | | = | | p c |\cdot|_{\ast}=|\cdot|_{p}^{c}
  62. c ( 0 , ) , p 𝐏 c\in(0,\infty),p\in\mathbf{P}

Ostwald_ripening.html

  1. R 3 - R 0 3 = 8 γ c v 2 D 9 R g T t \langle R\rangle^{3}-\langle R\rangle_{0}^{3}=\frac{8\gamma c_{\infty}v^{2}D}{% 9R_{g}T}t
  2. R \langle R\rangle
  3. γ \gamma
  4. c c_{\infty}
  5. v v
  6. D D
  7. R g R_{g}
  8. T T
  9. t t
  10. t t
  11. i i
  12. f ( R , t ) f(R,t)
  13. f ( R , t ) = 4 9 ρ 2 ( 3 3 + ρ ) 7 3 ( 1.5 1.5 - ρ ) 11 3 exp ( - 1.5 1.5 - ρ ) , ρ < 1.5 f(R,t)=\frac{4}{9}\rho^{2}\left(\frac{3}{3+\rho}\right)^{\frac{7}{3}}\left(% \frac{1.5}{1.5-\rho}\right)^{\frac{11}{3}}\exp\left(-\frac{1.5}{1.5-\rho}% \right),\rho<1.5
  14. R 2 = 64 γ c v 2 k s 81 R g T t \langle R\rangle^{2}=\frac{64\gamma c_{\infty}v^{2}k_{s}}{81R_{g}T}t
  15. k < s u b > s k<sub>s

Output_gap.html

  1. ( G D P a c t u a l - G D P p o t e n t i a l ) G D P p o t e n t i a l {(GDP_{actual}-GDP_{potential})}\over{GDP_{potential}}
  2. ( Y - Y * ) Y * = - β ( u - u ¯ ) {{(Y-Y^{*})}\over{Y^{*}}}=-\beta{}(u-\bar{u})

Oval_(projective_plane).html

  1. { ( t , t 2 , 1 ) t G F ( q ) } { ( 0 , 1 , 0 ) } . \{(t,t^{2},1)\mid t\in GF(q)\}\cup\{(0,1,0)\}.
  2. { ( t , t 2 , 1 ) t G F ( q ) } { ( 0 , 1 , 0 ) } { ( 1 , 0 , 0 ) } . \{(t,t^{2},1)\mid t\in GF(q)\}\cup\{(0,1,0)\}\cup\{(1,0,0)\}.
  3. f ( t ) = t 2 i f(t)=t^{2^{i}}
  4. γ 4 σ 2 2 mod ( 2 h - 1 ) \gamma^{4}\equiv\sigma^{2}\equiv 2(\bmod(2^{h}-1))
  5. f ( t ) = d 2 t 4 + d 2 ( 1 + d + d 2 ) t 3 + d 2 ( 1 + d + d 2 ) t 2 + d 2 t t 4 + d 2 t 2 + 1 + t 1 / 2 , f(t)={{d^{2}t^{4}+d^{2}(1+d+d^{2})t^{3}+d^{2}(1+d+d^{2})t^{2}+d^{2}t}\over{t^{% 4}+d^{2}t^{2}+1}}+t^{1/2},
  6. t r ( 1 / d ) = 1 and d G F ( 4 ) if h 2 mod 4 tr(1/d)=1\hbox{ and }d\not\in GF(4)\hbox{ if }h\equiv 2(\bmod 4)
  7. h 2 mod 4 h\not\equiv 2(\bmod 4)
  8. h 2 mod 4 , h > 2 h\equiv 2(\bmod 4),h>2
  9. b K b\in K
  10. b 1 b\neq 1
  11. t F t\in F
  12. n n
  13. ( F , 𝔊 ) (F,{\mathfrak{G}})
  14. F F
  15. n + 1 n+1
  16. 𝔊 {\mathfrak{G}}
  17. F F
  18. ( a 1 , a 2 ) , ( b 1 , b 2 ) F (a_{1},a_{2}),(b_{1},b_{2})\in F
  19. a i b j a_{i}\neq b_{j}
  20. i , j { 1 , 2 } i,j\in\{1,2\}
  21. σ 𝔊 \sigma\in{\mathfrak{G}}
  22. σ ( a 1 ) = a 2 \sigma(a_{1})=a_{2}
  23. σ ( b 1 ) = b 2 \sigma(b_{1})=b_{2}
  24. q q
  25. n 8 n\geq 8
  26. n = 8 n=8
  27. n n
  28. n n
  29. ( F , 𝔊 ) (F,{\mathfrak{G}})
  30. F F
  31. n + 2 n+2
  32. 𝔊 {\mathfrak{G}}
  33. F F
  34. a , b , c , d F a,b,c,d\in F
  35. σ 𝔊 \sigma\in{\mathfrak{G}}
  36. σ ( a ) = b , σ ( c ) = d \sigma(a)=b,\sigma(c)=d

Overconstrained_mechanism.html

  1. M = 6 ( N - 1 - j ) + i = 1 j f i , M=6(N-1-j)+\sum_{i=1}^{j}f_{i},
  2. M = 3 ( N - 1 - j ) + i = 1 j f i . M=3(N-1-j)+\sum_{i=1}^{j}f_{i}.
  3. M = 6 ( N - 1 - j ) + i = 1 j f i = 6 ( 6 - 1 - 6 ) + 6 = 0 , M=6(N-1-j)+\sum_{i=1}^{j}f_{i}=6(6-1-6)+6=0,
  4. M = 6 ( N - 1 - j ) + i = 1 j f i = 6 ( 4 - 1 - 4 ) + 4 = - 2 , M=6(N-1-j)+\sum_{i=1}^{j}f_{i}=6(4-1-4)+4=-2,
  5. d 1 = d 3 , a 1 = a 3 \displaystyle d_{1}=d_{3},\quad a_{1}=a_{3}

Overlapping_interval_topology.html

  1. [ - 1 , 1 ] [-1,1]
  2. [ - 1 , b ) [-1,b)
  3. ( a , 1 ] (a,1]
  4. a < 0 < b a<0<b
  5. [ - 1 , b ) [-1,b)
  6. ( a , b ) (a,b)
  7. ( a , 1 ] (a,1]
  8. a < 0 < b a<0<b
  9. [ - 1 , 1 ] [-1,1]
  10. [ - 1 , 1 ] [-1,1]
  11. [ - 1 , 1 ] [-1,1]
  12. [ - 1 , 1 ] [-1,1]
  13. [ - 1 , s ) [-1,s)
  14. ( r , s ) (r,s)
  15. ( r , 1 ] (r,1]
  16. r < 0 < s r<0<s

Overline.html

  1. AB ¯ \overline{\rm AB}
  2. 1 / 7 {1}/{7}
  3. 142857 ¯ \overline{142857}
  4. 3 ¯ \overline{3}
  5. x ¯ \overline{x}
  6. x i x_{i}
  7. A B ¯ A ¯ B ¯ \overline{A\cup B}\equiv\overline{A}\cap\overline{B}
  8. A B ¯ A ¯ B ¯ \overline{A\cap B}\equiv\overline{A}\cup\overline{B}
  9. A B ¯ A ¯ + B ¯ \overline{A\cdot B}\equiv\overline{A}+\overline{B}
  10. A + B ¯ A ¯ B ¯ \overline{A+B}\equiv\overline{A}\cdot\overline{B}
  11. x = a + i b x=a+ib
  12. x ¯ = a - i b . \overline{x}=a-ib.
  13. x ¯ = | x | x ^ \overline{x}=|x|\hat{x}
  14. 3 ¯ \overline{3}
  15. [ 1 ¯ 1 2 ¯ ] [\overline{1}1\overline{2}]
  16. h = - 1 h=-1
  17. k = 1 k=1
  18. l = - 2 l=-2
  19. X ¯ \overline{X}
  20. t e x t ¯ \overline{text}
  21. x ¯ \overline{x}
  22. x y z ¯ \overline{xyz}
  23. 123 ¯ \overline{123}

Ovoid_(projective_geometry).html

  1. q 2 + 1 q^{2}+1
  2. q = 2 q=2
  3. q 2 + 1 q^{2}+1
  4. q = 4 q=4
  5. q = 2 2 h + 1 q=2^{2h+1}
  6. q + 1 q+1
  7. q = 4 q=4

Oxygen-18.html

  1. T = A + B ( ( δ O 18 ) calcite - ( δ O 18 ) water ) T=A+B\cdot\left(\left(\delta{}^{18}\,\text{O}\right)\,\text{calcite}-\left(% \delta{}^{18}\,\text{O}\right)\,\text{water}\right)

Oxygen–hemoglobin_dissociation_curve.html

  1. S ( t ) = 1 1 + e - t . S(t)=\frac{1}{1+e^{-t}}.

P-Coumaric_acid.html

  1. C 4 H \xrightarrow{C4H}
  2. T A L \xrightarrow{TAL}

P-y_method.html

  1. p = k y p=ky

Pacific_halibut.html

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

Pairing_heap.html

  1. O ( l o g n ) O(logn)
  2. O ( 1 ) O(1)
  3. O ( 1 ) O(1)
  4. Ω ( log log n ) \Omega(\log\log n)
  5. O ( 2 2 log log n ) O(2^{2\sqrt{\log\log n}})
  6. o ( log n ) o(\log n)
  7. Θ ( log log n ) \Theta(\log\log n)
  8. O ( 1 ) O(1)

Pairwise_comparison.html

  1. Pr { X j i = 1 } = e δ j - δ i 1 + e δ j - δ i = σ ( δ j - δ i ) , \Pr\{X_{ji}=1\}=\frac{e^{{\delta_{j}}-{\delta_{i}}}}{1+e^{{\delta_{j}}-{\delta% _{i}}}}=\sigma(\delta_{j}-\delta_{i}),
  2. δ i \delta_{i}
  3. i i
  4. σ \sigma
  5. a > b > c a>b>c
  6. a > c > b a>c>b
  7. b > a > c b>a>c
  8. b > c > a b>c>a
  9. c > a > b c>a>b
  10. c > b > a c>b>a
  11. a > b = c a>b=c
  12. b = c > a b=c>a
  13. b > a = c b>a=c
  14. a = c > b a=c>b
  15. c > a = b c>a=b
  16. a = b > c a=b>c
  17. a = b = c a=b=c
  18. k = 1 n k ! S 2 ( n , k ) , \sum_{k=1}^{n}k!S_{2}(n,k),

Palais–Smale_compactness_condition.html

  1. I C 1 ( H , ) I\in C^{1}(H,\mathbb{R})
  2. { u k } k = 1 H \{u_{k}\}_{k=1}^{\infty}\subset H
  3. { I [ u k ] } k = 1 \{I[u_{k}]\}_{k=1}^{\infty}
  4. I [ u k ] 0 I^{\prime}[u_{k}]\rightarrow 0
  5. Φ : X 𝐑 \Phi\colon X\to\mathbf{R}
  6. Φ \Phi
  7. { x n } X \{x_{n}\}\subset X
  8. sup | Φ ( x n ) | < \sup|\Phi(x_{n})|<\infty
  9. lim Φ ( x n ) = 0 \lim\Phi^{\prime}(x_{n})=0
  10. X * X^{*}
  11. Φ ( x n ) 0 \Phi(x_{n})\neq 0
  12. n 𝐍 n\in\mathbf{N}
  13. x ¯ X \overline{x}\in X
  14. Φ \Phi
  15. lim inf Φ ( x n ) Φ ( x ¯ ) lim sup Φ ( x n ) . \liminf\Phi(x_{n})\leq\Phi(\overline{x})\leq\limsup\Phi(x_{n}).

Parabolic_cylindrical_coordinates.html

  1. z z
  2. ( σ , τ , z ) (\sigma,\tau,z)
  3. x = σ τ x=\sigma\tau\,
  4. y = 1 2 ( τ 2 - σ 2 ) y=\frac{1}{2}\left(\tau^{2}-\sigma^{2}\right)
  5. z = z z=z\,
  6. σ \sigma
  7. 2 y = x 2 σ 2 - σ 2 2y=\frac{x^{2}}{\sigma^{2}}-\sigma^{2}
  8. + y +y
  9. τ \tau
  10. 2 y = - x 2 τ 2 + τ 2 2y=-\frac{x^{2}}{\tau^{2}}+\tau^{2}
  11. - y -y
  12. x = y = 0 x=y=0
  13. r = x 2 + y 2 = 1 2 ( σ 2 + τ 2 ) r=\sqrt{x^{2}+y^{2}}=\frac{1}{2}\left(\sigma^{2}+\tau^{2}\right)
  14. σ \sigma
  15. τ \tau
  16. h σ = h τ = σ 2 + τ 2 h_{\sigma}=h_{\tau}=\sqrt{\sigma^{2}+\tau^{2}}
  17. h z = 1 h_{z}=1\,
  18. d V = h σ h τ h z = ( σ 2 + τ 2 ) d σ d τ d z dV=h_{\sigma}h_{\tau}h_{z}=\left(\sigma^{2}+\tau^{2}\right)d\sigma d\tau dz
  19. 2 Φ = 1 σ 2 + τ 2 ( 2 Φ σ 2 + 2 Φ τ 2 ) + 2 Φ z 2 \nabla^{2}\Phi=\frac{1}{\sigma^{2}+\tau^{2}}\left(\frac{\partial^{2}\Phi}{% \partial\sigma^{2}}+\frac{\partial^{2}\Phi}{\partial\tau^{2}}\right)+\frac{% \partial^{2}\Phi}{\partial z^{2}}
  20. 𝐅 \nabla\cdot\mathbf{F}
  21. × 𝐅 \nabla\times\mathbf{F}
  22. ( σ , τ ) (\sigma,\tau)
  23. V = S ( σ ) T ( τ ) Z ( z ) V=S(\sigma)\,T(\tau)\,Z(z)
  24. 1 σ 2 + τ 2 [ S ¨ S + T ¨ T ] + Z ¨ Z = 0 \frac{1}{\sigma^{2}+\tau^{2}}\left[\frac{\ddot{S}}{S}+\frac{\ddot{T}}{T}\right% ]+\frac{\ddot{Z}}{Z}=0
  25. Z ¨ Z = - m 2 \frac{\ddot{Z}}{Z}=-m^{2}
  26. Z m ( z ) = A 1 e i m z + A 2 e - i m z Z_{m}(z)=A_{1}\,e^{imz}+A_{2}\,e^{-imz}\,
  27. - m 2 -m^{2}
  28. Z ¨ / Z \ddot{Z}/Z
  29. [ S ¨ S + T ¨ T ] = m 2 ( σ 2 + τ 2 ) \left[\frac{\ddot{S}}{S}+\frac{\ddot{T}}{T}\right]=m^{2}(\sigma^{2}+\tau^{2})
  30. n 2 n^{2}
  31. S ¨ - ( m 2 σ 2 + n 2 ) S = 0 \ddot{S}-(m^{2}\sigma^{2}+n^{2})S=0
  32. T ¨ - ( m 2 τ 2 - n 2 ) T = 0 \ddot{T}-(m^{2}\tau^{2}-n^{2})T=0
  33. S m n ( σ ) = A 3 y 1 ( n 2 / 2 m , σ 2 m ) + A 4 y 2 ( n 2 / 2 m , σ 2 m ) S_{mn}(\sigma)=A_{3}\,y_{1}(n^{2}/2m,\sigma\sqrt{2m})+A_{4}\,y_{2}(n^{2}/2m,% \sigma\sqrt{2m})
  34. T m n ( τ ) = A 5 y 1 ( n 2 / 2 m , i τ 2 m ) + A 6 y 2 ( n 2 / 2 m , i τ 2 m ) T_{mn}(\tau)=A_{5}\,y_{1}(n^{2}/2m,i\tau\sqrt{2m})+A_{6}\,y_{2}(n^{2}/2m,i\tau% \sqrt{2m})
  35. V ( σ , τ , z ) = m , n A m n S m n T m n Z m V(\sigma,\tau,z)=\sum_{m,n}A_{mn}S_{mn}T_{mn}Z_{m}\,

Paraboloidal_coordinates.html

  1. ( λ , μ , ν ) (\lambda,\mu,\nu)
  2. ( x , y , z ) (x,y,z)
  3. ( λ , μ , ν ) (\lambda,\mu,\nu)
  4. x 2 = ( A - λ ) ( A - μ ) ( A - ν ) B - A x^{2}=\frac{\left(A-\lambda\right)\left(A-\mu\right)\left(A-\nu\right)}{B-A}
  5. y 2 = ( B - λ ) ( B - μ ) ( B - ν ) A - B y^{2}=\frac{\left(B-\lambda\right)\left(B-\mu\right)\left(B-\nu\right)}{A-B}
  6. z = 1 2 ( A + B - λ - μ - ν ) z=\frac{1}{2}\left(A+B-\lambda-\mu-\nu\right)
  7. λ < B < μ < A < ν \lambda<B<\mu<A<\nu
  8. λ \lambda
  9. x 2 λ - A + y 2 λ - B = 2 z + λ \frac{x^{2}}{\lambda-A}+\frac{y^{2}}{\lambda-B}=2z+\lambda
  10. ν \nu
  11. x 2 ν - A + y 2 ν - B = 2 z + ν \frac{x^{2}}{\nu-A}+\frac{y^{2}}{\nu-B}=2z+\nu
  12. μ \mu
  13. x 2 μ - A + y 2 μ - B = 2 z + μ \frac{x^{2}}{\mu-A}+\frac{y^{2}}{\mu-B}=2z+\mu
  14. ( λ , μ , ν ) (\lambda,\mu,\nu)
  15. h λ = 1 2 ( μ - λ ) ( ν - λ ) ( A - λ ) ( B - λ ) h_{\lambda}=\frac{1}{2}\sqrt{\frac{\left(\mu-\lambda\right)\left(\nu-\lambda% \right)}{\left(A-\lambda\right)\left(B-\lambda\right)}}
  16. h μ = 1 2 ( ν - μ ) ( λ - μ ) ( A - μ ) ( B - μ ) h_{\mu}=\frac{1}{2}\sqrt{\frac{\left(\nu-\mu\right)\left(\lambda-\mu\right)}{% \left(A-\mu\right)\left(B-\mu\right)}}
  17. h ν = 1 2 ( λ - ν ) ( μ - ν ) ( A - ν ) ( B - ν ) h_{\nu}=\frac{1}{2}\sqrt{\frac{\left(\lambda-\nu\right)\left(\mu-\nu\right)}{% \left(A-\nu\right)\left(B-\nu\right)}}
  18. d V = ( μ - λ ) ( ν - λ ) ( ν - μ ) 8 ( A - λ ) ( B - λ ) ( A - μ ) ( μ - B ) ( ν - A ) ( ν - B ) d λ d μ d ν dV=\frac{\left(\mu-\lambda\right)\left(\nu-\lambda\right)\left(\nu-\mu\right)}% {8\sqrt{\left(A-\lambda\right)\left(B-\lambda\right)\left(A-\mu\right)\left(% \mu-B\right)\left(\nu-A\right)\left(\nu-B\right)}}\ d\lambda d\mu d\nu
  19. 𝐅 \nabla\cdot\mathbf{F}
  20. × 𝐅 \nabla\times\mathbf{F}
  21. ( λ , μ , ν ) (\lambda,\mu,\nu)

Paradox_of_enrichment.html

  1. d x d t = f ( x , y ) = x ( 1 - x K ) - y x 1 + x \frac{dx}{dt}=f(x,y)=x\left(1-\frac{x}{K}\right)-y\frac{x}{1+x}
  2. d y d t = g ( x , y ) = - y ( γ - δ x 1 + x ) \frac{dy}{dt}=g(x,y)=-y\left(\gamma-\delta\frac{x}{1+x}\right)
  3. x ( 1 - x K ) x\left(1-\frac{x}{K}\right)
  4. x 1 + x \frac{x}{1+x}
  5. x = 0 \ x=0
  6. y = ( 1 + x ) ( 1 - x / K ) y=(1+x)\left(1-x/K\right)
  7. y = 0 \ y=0
  8. x = α 1 - α x=\frac{\alpha}{1-\alpha}
  9. α = γ δ \alpha=\frac{\gamma}{\delta}
  10. x 1 = 0 , y 1 = 0 x_{1}=0,\;y_{1}=0
  11. x 2 = K , y 2 = 0 x_{2}=K,\;y_{2}=0
  12. x 3 = α 1 - α , y 3 = ( 1 + x 3 ) ( 1 - x 3 K ) x_{3}=\frac{\alpha}{1-\alpha},\;y_{3}=(1+x_{3})\left(1-\frac{x_{3}}{K}\right)
  13. f f
  14. g g
  15. x x
  16. y y
  17. ( x 3 , y 3 ) (x_{3},y_{3})
  18. d d t ( x - x 3 y - y 3 ) ( α ( 1 - ( 1 + 2 x 3 ) / K ) - α δ ( 1 - α ) 2 y 3 0 ) ( x - x 3 y - y 3 ) \frac{d}{dt}\begin{pmatrix}x-x_{3}\\ y-y_{3}\\ \end{pmatrix}\approx\begin{pmatrix}\alpha\left(1-(1+2x_{3})/K\right)&-\alpha\\ \delta(1-\alpha)^{2}y_{3}&0\\ \end{pmatrix}\begin{pmatrix}x-x_{3}\\ y-y_{3}\\ \end{pmatrix}
  19. α ( 1 - 1 + 2 x 3 K ) < 0 , or K < 1 + 2 α 1 - α \alpha\left(1-\frac{1+2x_{3}}{K}\right)<0,\,\text{ or }K<1+2\frac{\alpha}{1-\alpha}

Parallel_computation_thesis.html

  1. t ( n ) t(n)
  2. t ( n ) k t(n)^{k}
  3. s ( n ) s(n)
  4. s ( n ) k s(n)^{k}
  5. 2 2 O ( T ( n ) ) 2^{2^{O(T(n))}}
  6. T ( n ) T(n)
  7. 2 O ( T ( n ) ) 2^{O(T(n))}
  8. 2 T ( n ) O ( 1 ) 2^{T(n)^{O(1)}}

Parameter_identification_problem.html

  1. Q = a S + b S P + c X Q=a_{S}+b_{S}P+cX\,
  2. Q = a D + b D P + d Z Q=a_{D}+b_{D}P+dZ\,
  3. M - 1 M-1
  4. M - 1 M-1

Parametric_oscillator.html

  1. ω \omega
  2. β \beta
  3. ω s , ω i \omega_{s},\omega_{i}
  4. d 2 x d t 2 + β ( t ) d x d t + ω 2 ( t ) x = 0 \frac{d^{2}x}{dt^{2}}+\beta(t)\frac{dx}{dt}+\omega^{2}(t)x=0
  5. x ( t ) x(t)
  6. ω 2 \omega^{2}
  7. β \beta
  8. β ( t ) \beta(t)
  9. ω 2 ( t ) \omega^{2}(t)
  10. T T
  11. β \beta
  12. q ( t ) = def e D ( t ) x ( t ) q(t)\ \stackrel{\mathrm{def}}{=}\ e^{D(t)}x(t)
  13. D ( t ) D(t)
  14. D ( t ) = def 1 2 t d τ β ( τ ) . D(t)\ \stackrel{\mathrm{def}}{=}\ \frac{1}{2}\int^{t}d\tau\ \beta(\tau).
  15. d 2 q d t 2 + Ω 2 ( t ) q = 0 \frac{d^{2}q}{dt^{2}}+\Omega^{2}(t)q=0
  16. Ω 2 ( t ) = ω 2 ( t ) - 1 2 ( d β d t ) - 1 4 β 2 . \Omega^{2}(t)=\omega^{2}(t)-\frac{1}{2}\left(\frac{d\beta}{dt}\right)-\frac{1}% {4}\beta^{2}.
  17. β ( t ) = ω 0 [ b + g ( t ) ] \beta(t)=\omega_{0}\left[b+g(t)\right]
  18. ω 2 ( t ) = ω 0 2 [ 1 + h ( t ) ] \omega^{2}(t)=\omega_{0}^{2}\left[1+h(t)\right]
  19. ω 0 \omega_{0}
  20. b ω 0 b\omega_{0}
  21. Ω 2 ( t ) = ω n 2 [ 1 + f ( t ) ] , \Omega^{2}(t)=\omega_{n}^{2}\left[1+f(t)\right],
  22. ω n \omega_{n}
  23. ω n 2 = def ω 0 2 ( 1 - b 2 4 ) \omega_{n}^{2}\ \stackrel{\mathrm{def}}{=}\ \omega_{0}^{2}\left(1-\frac{b^{2}}% {4}\right)
  24. ω n 2 f ( t ) = def ω 0 2 { h ( t ) - 1 2 ω 0 ( d g d t ) - b 2 g ( t ) - 1 4 g 2 ( t ) } . \omega_{n}^{2}f(t)\ \stackrel{\mathrm{def}}{=}\ \omega_{0}^{2}\left\{h(t)-% \frac{1}{2\omega_{0}}\left(\frac{dg}{dt}\right)-\frac{b}{2}g(t)-\frac{1}{4}g^{% 2}(t)\right\}.
  25. d 2 q d t 2 + ω n 2 [ 1 + f ( t ) ] q = 0. \frac{d^{2}q}{dt^{2}}+\omega_{n}^{2}\left[1+f(t)\right]q=0.
  26. g ( t ) g(t)
  27. h ( t ) h(t)
  28. f ( t ) f(t)
  29. f ( t ) f(t)
  30. f ( t ) = f 0 sin 2 ω p t f(t)=f_{0}\sin 2\omega_{p}t
  31. ω p 2 ω n \omega_{p}\approx 2\omega_{n}
  32. 2 ω n 2\omega_{n}
  33. q ( t ) q(t)
  34. q ( t ) = A ( t ) cos ω p t + B ( t ) sin ω p t q(t)=A(t)\cos\omega_{p}t+B(t)\sin\omega_{p}t
  35. cos ω p t \cos\omega_{p}t
  36. sin ω p t \sin\omega_{p}t
  37. A ( t ) A(t)
  38. B ( t ) B(t)
  39. f 0 1 f_{0}\ll 1
  40. 2 ω p d A d t = ( f 0 2 ) ω n 2 A - ( ω p 2 - ω n 2 ) B 2\omega_{p}\frac{dA}{dt}=\left(\frac{f_{0}}{2}\right)\omega_{n}^{2}A-\left(% \omega_{p}^{2}-\omega_{n}^{2}\right)B
  41. 2 ω p d B d t = - ( f 0 2 ) ω n 2 B + ( ω p 2 - ω n 2 ) A 2\omega_{p}\frac{dB}{dt}=-\left(\frac{f_{0}}{2}\right)\omega_{n}^{2}B+\left(% \omega_{p}^{2}-\omega_{n}^{2}\right)A
  42. A ( t ) = def r ( t ) cos θ ( t ) A(t)\ \stackrel{\mathrm{def}}{=}\ r(t)\cos\theta(t)
  43. B ( t ) = def r ( t ) sin θ ( t ) B(t)\ \stackrel{\mathrm{def}}{=}\ r(t)\sin\theta(t)
  44. d r d t = ( α max cos 2 θ ) r \frac{dr}{dt}=\left(\alpha_{\mathrm{max}}\cos 2\theta\right)r
  45. d θ d t = - α max [ sin 2 θ - sin 2 θ eq ] \frac{d\theta}{dt}=-\alpha_{\mathrm{max}}\left[\sin 2\theta-\sin 2\theta_{% \mathrm{eq}}\right]
  46. α max = def f 0 ω n 2 4 ω p \alpha_{\mathrm{max}}\ \stackrel{\mathrm{def}}{=}\ \frac{f_{0}\omega_{n}^{2}}{% 4\omega_{p}}
  47. sin 2 θ eq = def ( 2 f 0 ) ϵ \sin 2\theta_{\mathrm{eq}}\ \stackrel{\mathrm{def}}{=}\ \left(\frac{2}{f_{0}}% \right)\epsilon
  48. ϵ = def ω p 2 - ω n 2 ω n 2 \epsilon\ \stackrel{\mathrm{def}}{=}\ \frac{\omega_{p}^{2}-\omega_{n}^{2}}{% \omega_{n}^{2}}
  49. θ \theta
  50. r r
  51. θ eq \theta_{\mathrm{eq}}
  52. θ \theta
  53. θ ( t ) = θ eq + ( θ 0 - θ eq ) e - 2 α t \theta(t)=\theta_{\mathrm{eq}}+\left(\theta_{0}-\theta_{\mathrm{eq}}\right)e^{% -2\alpha t}
  54. α = def α max cos 2 θ eq \alpha\ \stackrel{\mathrm{def}}{=}\ \alpha_{\mathrm{max}}\cos 2\theta_{\mathrm% {eq}}
  55. f ( t ) f(t)
  56. θ ( t ) = θ eq \theta(t)=\theta_{\mathrm{eq}}
  57. r r
  58. d r d t = α r \frac{dr}{dt}=\alpha r
  59. r ( t ) = r 0 e α t r(t)=r_{0}e^{\alpha t}
  60. q ( t ) q(t)
  61. R ( t ) R(t)
  62. x = def q e - D ( t ) x\ \stackrel{\mathrm{def}}{=}\ qe^{-D(t)}
  63. R ( t ) = r ( t ) e - D ( t ) = r 0 e α t - D ( t ) R(t)=r(t)e^{-D(t)}=r_{0}e^{\alpha t-D(t)}
  64. R ( t ) R(t)
  65. α t \alpha t
  66. D ( t ) D(t)
  67. ω p = ω n \omega_{p}=\omega_{n}
  68. θ eq \theta_{\mathrm{eq}}
  69. cos 2 θ eq = 1 \cos 2\theta_{\mathrm{eq}}=1
  70. α = α max \alpha=\alpha_{\mathrm{max}}
  71. ω p \omega_{p}
  72. ω n \omega_{n}
  73. θ eq \theta_{\mathrm{eq}}
  74. α < α max \alpha<\alpha_{\mathrm{max}}
  75. ω p \omega_{p}
  76. α \alpha
  77. α = α max 1 - ( 2 f 0 ) 2 ϵ 2 \alpha=\alpha_{\mathrm{max}}\sqrt{1-\left(\frac{2}{f_{0}}\right)^{2}\epsilon^{% 2}}
  78. ϵ \epsilon
  79. f 0 / 2 f_{0}/2
  80. α \alpha
  81. q ( t ) q(t)
  82. ϵ \epsilon
  83. 2 ω p 2\omega_{p}
  84. 2 ω n 1 - f 0 2 2\omega_{n}\sqrt{1-\frac{f_{0}}{2}}
  85. 2 ω n 1 + f 0 2 2\omega_{n}\sqrt{1+\frac{f_{0}}{2}}
  86. q q
  87. q q
  88. ω n f 0 \omega_{n}f_{0}
  89. q q
  90. d 2 q d t 2 + ω n 2 q = - ω n 2 f ( t ) q \frac{d^{2}q}{dt^{2}}+\omega_{n}^{2}q=-\omega_{n}^{2}f(t)q
  91. - ω n 2 f ( t ) q -\omega_{n}^{2}f(t)q
  92. q q
  93. q ( t ) = A cos ω p t q(t)=A\cos\omega_{p}t
  94. ω p \omega_{p}
  95. f ( t ) = f 0 sin 2 ω p t f(t)=f_{0}\sin 2\omega_{p}t
  96. f 0 1 f_{0}\ll 1
  97. q ( t ) f ( t ) q(t)f(t)
  98. ω p \omega_{p}
  99. 3 ω p 3\omega_{p}
  100. f ( t ) q ( t ) = f 0 2 A ( sin ω p t + sin 3 ω p t ) f(t)q(t)=\frac{f_{0}}{2}A\left(\sin\omega_{p}t+\sin 3\omega_{p}t\right)
  101. 3 ω p 3\omega_{p}
  102. ω p \omega_{p}
  103. q q
  104. A A
  105. q q
  106. f ( t ) q ( t ) f(t)q(t)
  107. F ~ ( ω ) \tilde{F}(\omega)
  108. Q ~ ( ω ) \tilde{Q}(\omega)
  109. + 2 ω p +2\omega_{p}
  110. f ( t ) f(t)
  111. - ω p -\omega_{p}
  112. q ( t ) q(t)
  113. + ω p +\omega_{p}
  114. 2 ω n 2\omega_{n}
  115. - ω p -\omega_{p}
  116. + ω p +\omega_{p}
  117. q ( t ) q(t)
  118. u ¨ + ( a + B cos t ) u = 0 \ddot{u}+(a+B\cos t)u=0
  119. u u
  120. B cos ( t ) B\ \cos(t)
  121. E ( t ) E(t)
  122. d 2 x d t 2 + β ( t ) d x d t + ω 2 ( t ) x = E ( t ) . \frac{d^{2}x}{dt^{2}}+\beta(t)\frac{dx}{dt}+\omega^{2}(t)x=E(t).
  123. D D
  124. E E
  125. α t < D \alpha t<D
  126. β ( t ) = ω 0 b \beta(t)=\omega_{0}b
  127. ω 0 \omega_{0}
  128. E ( t ) = E 0 sin ω 0 t E(t)=E_{0}\sin\omega_{0}t
  129. d 2 x d t 2 + b ω 0 d x d t + ω 0 2 [ 1 + h 0 sin 2 ω 0 t ] x = E 0 sin ω 0 t \frac{d^{2}x}{dt^{2}}+b\omega_{0}\frac{dx}{dt}+\omega_{0}^{2}\left[1+h_{0}\sin 2% \omega_{0}t\right]x=E_{0}\sin\omega_{0}t
  130. x ( t ) = 2 E 0 ω 0 2 ( 2 b - h 0 ) cos ω 0 t . x(t)=\frac{2E_{0}}{\omega_{0}^{2}\left(2b-h_{0}\right)}\cos\omega_{0}t.
  131. h 0 h_{0}
  132. 2 b 2b
  133. h 2 b h\geq 2b
  134. E ( t ) E(t)
  135. q q
  136. f ( t ) f(t)
  137. f ( t ) f(t)

Parametric_polymorphism.html

  1. τ \tau
  2. α \alpha
  3. α \alpha
  4. τ \tau
  5. τ \tau
  6. T = X . X X T=\forall X.X\to X
  7. E q α α [ α ] B o o l {\scriptstyle Eq\,\alpha\,\Rightarrow\alpha\,\rightarrow\left[\alpha\right]% \rightarrow Bool}

Parametric_surface.html

  1. r : 2 3 . \quad\vec{r}:\mathbb{R}^{2}\rightarrow\mathbb{R}^{3}.
  2. z = f ( x , y ) , r ( x , y ) = ( x , y , f ( x , y ) ) . z=f(x,y),\quad\vec{r}(x,y)=(x,y,f(x,y)).
  3. r ( u , ϕ ) = ( u cos ϕ , u sin ϕ , f ( u ) ) , a u b , 0 ϕ < 2 π . \vec{r}(u,\phi)=(u\cos\phi,u\sin\phi,f(u)),\quad a\leq u\leq b,0\leq\phi<2\pi.
  4. r ( u , v ) = ( u 1 - v 2 1 + v 2 , u 2 v 1 + v 2 , f ( u ) ) , a u b , \vec{r}(u,v)=(u\frac{1-v^{2}}{1+v^{2}},u\frac{2v}{1+v^{2}},f(u)),\quad a\leq u% \leq b,
  5. f f
  6. r ( x , ϕ ) = ( x , R cos ϕ , R sin ϕ ) . \vec{r}(x,\phi)=(x,R\cos\phi,R\sin\phi).
  7. r ( θ , ϕ ) = ( cos θ sin ϕ , sin θ sin ϕ , cos ϕ ) , 0 θ < 2 π , 0 ϕ π . \vec{r}(\theta,\phi)=(\cos\theta\sin\phi,\sin\theta\sin\phi,\cos\phi),\quad 0% \leq\theta<2\pi,0\leq\phi\leq\pi.
  8. r ( u , v ) = ( a u + b v , c u + d v , 0 ) \vec{r}(u,v)=(au+bv,cu+dv,0)
  9. [ a b c d ] \begin{bmatrix}a&b\\ c&d\end{bmatrix}
  10. r = r ( u , v ) , \vec{r}=\vec{r}(u,v),
  11. r \vec{r}
  12. r u := r u \vec{r}_{u}:=\frac{\partial\vec{r}}{\partial u}
  13. r v , \vec{r}_{v},
  14. r u u , r u v , r v v . \vec{r}_{uu},\vec{r}_{uv},\vec{r}_{vv}.
  15. r s , r t , 2 r s 2 , 2 r s t , 2 r t 2 . \frac{\partial\vec{r}}{\partial s},\frac{\partial\vec{r}}{\partial t},\frac{% \partial^{2}\vec{r}}{\partial s^{2}},\frac{\partial^{2}\vec{r}}{\partial s% \partial t},\frac{\partial^{2}\vec{r}}{\partial t^{2}}.
  16. r u , r v \vec{r}_{u},\vec{r}_{v}
  17. r u \vec{r}_{u}
  18. r v . \vec{r}_{v}.
  19. n = r u × r v | r u × r v | . \vec{n}=\frac{\vec{r}_{u}\times\vec{r}_{v}}{\left|\vec{r}_{u}\times\vec{r}_{v}% \right|}.
  20. r u × r v \vec{r}_{u}\times\vec{r}_{v}
  21. A ( D ) = D | r u × r v | d u d v . A(D)=\iint_{D}\left|\vec{r}_{u}\times\vec{r}_{v}\right|dudv.
  22. S 1 d S . \int_{S}1\,dS.
  23. I = E d u 2 + 2 F d u d v + G d v 2 I=Edu^{2}+2Fdudv+Gdv^{2}
  24. r = r ( u , v ) , \vec{r}=\vec{r}(u,v),
  25. E = r u r u , F = r u r v , G = r v r v . E=\vec{r}_{u}\cdot\vec{r}_{u},\quad F=\vec{r}_{u}\cdot\vec{r}_{v},\quad G=\vec% {r}_{v}\cdot\vec{r}_{v}.
  26. a b E u ( t ) 2 + 2 F u ( t ) v ( t ) + G v ( t ) 2 d t . \int_{a}^{b}\sqrt{E\,u^{\prime}(t)^{2}+2F\,u^{\prime}(t)v^{\prime}(t)+G\,v^{% \prime}(t)^{2}}\,dt.
  27. cos θ = a b | a | | b | \cos\theta=\frac{\vec{a}\cdot\vec{b}}{\left|\vec{a}\right||\vec{b}|}
  28. A ( D ) = D E G - F 2 d u d v . A(D)=\iint_{D}\sqrt{EG-F^{2}}\,dudv.
  29. | r u × r v | 2 \left|\vec{r}_{u}\times\vec{r}_{v}\right|^{2}
  30. II = L d u 2 + 2 M d u d v + N d v 2 \mathrm{II}=L\,\,\text{d}u^{2}+2M\,\,\text{d}u\,\,\text{d}v+N\,\,\text{d}v^{2}
  31. r \vec{r}
  32. n \vec{n}
  33. L = r u u n , M = r u v n , N = r v v n . L=\vec{r}_{uu}\cdot\vec{n},\quad M=\vec{r}_{uv}\cdot\vec{n},\quad N=\vec{r}_{% vv}\cdot\vec{n}.\quad
  34. det ( II - κ I ) = 0 , det | L - κ E M - κ F M - κ F N - κ G | = 0. \det(\mathrm{II}-\kappa\mathrm{I})=0,\quad\det\left|\begin{matrix}L-\kappa E&M% -\kappa F\\ M-\kappa F&N-\kappa G\end{matrix}\right|=0.
  35. K = L N - M 2 E G - F 2 , H = E N - 2 F M + G L 2 ( E G - F 2 ) . K={LN-M^{2}\over EG-F^{2}},\quad H={EN-2FM+GL\over 2(EG-F^{2})}.
  36. F 1 = [ E F F G ] . F_{1}=\begin{bmatrix}E&F\\ F&G\end{bmatrix}.
  37. F 2 = [ L M M N ] . F_{2}=\begin{bmatrix}L&M\\ M&N\end{bmatrix}.
  38. A = F 1 - 1 F 2 A=F_{1}^{-1}F_{2}
  39. t 1 = v 11 r u + v 12 r v \vec{t}_{1}=v_{11}\vec{r}_{u}+v_{12}\vec{r}_{v}
  40. t 2 = v 21 r u + v 22 r v \vec{t}_{2}=v_{21}\vec{r}_{u}+v_{22}\vec{r}_{v}

Paranormal_subgroup.html

  1. H H
  2. G G
  3. g g
  4. G G
  5. K K
  6. H H
  7. H g H^{g}
  8. H K H^{K}

Parasitic_capacitance.html

  1. i = C d v d t i=C\frac{dv}{dt}\,
  2. v o = - A v i v\text{o}=-Av\text{i}\,
  3. i i = C d d t ( v i - v o ) i\text{i}=C{d\over dt}(v\text{i}-v\text{o})\,
  4. i i = C d d t ( v i + A v i ) i\text{i}=C{d\over dt}(v\text{i}+Av\text{i})\,
  5. i i = C ( 1 + A ) d v i d t i\text{i}=C(1+A){dv\text{i}\over dt}\,
  6. C M = C ( 1 + A ) C\text{M}=C(1+A)\,
  7. V o = A 1 + j ω R i C M V i V\text{o}=\frac{A}{1+j\omega R\text{i}C\text{M}}V\text{i}\,
  8. f = 1 2 π R i C M = 1 2 π R i C ( 1 + A ) f={1\over 2\pi R\text{i}C\text{M}}={1\over 2\pi R\text{i}C(1+A)}\,

Paris'_law.html

  1. d a d N = C Δ K m \frac{{\rm d}a}{{\rm d}N}=C\Delta K^{m}
  2. Δ K \Delta K
  3. Δ K = K m a x - K m i n \Delta K=K_{max}-K_{min}
  4. K m a x K_{max}
  5. K m i n K_{min}
  6. K = σ Y π a K=\sigma Y\sqrt{\pi a}
  7. σ \sigma
  8. Δ K = Δ σ Y π a \Delta K=\Delta\sigma Y\sqrt{\pi a}
  9. Δ σ \Delta\sigma
  10. d a d N = C Δ K m = C ( Δ σ Y π a ) m \frac{{\rm d}a}{{\rm d}N}=C\Delta K^{m}=C(\Delta\sigma Y\sqrt{\pi a})^{m}
  11. 0 N f d N = a i a c d a C ( Δ σ Y π a ) m = 1 C ( Δ σ Y π ) m a i a c a - m 2 d a \int^{N_{f}}_{0}{\rm d}N=\int^{a_{c}}_{a_{i}}\frac{{\rm d}a}{C(\Delta\sigma Y% \sqrt{\pi a})^{m}}=\frac{1}{C(\Delta\sigma Y\sqrt{\pi})^{m}}\int^{a_{c}}_{a_{i% }}a^{-\frac{m}{2}}\;{\rm d}a
  12. N f = 2 ( a c 2 - m 2 - a i 2 - m 2 ) ( 2 - m ) C ( Δ σ Y π ) m N_{f}=\frac{2\;(a_{c}^{\frac{2-m}{2}}-a_{i}^{\frac{2-m}{2}})}{(2-m)\;C(\Delta% \sigma Y\sqrt{\pi})^{m}}
  13. N f N_{f}
  14. a c a_{c}
  15. a i a_{i}
  16. Δ σ \Delta\sigma

Paris–Harrington_theorem.html

  1. K 𝒫 n ( S ) K_{\mathcal{P}_{n}(S)}
  2. ( N n ) = | 𝒫 n ( S ) | R ( m , m , , m k ) . {\left({{N}\atop{n}}\right)}=|\mathcal{P}_{n}(S)|\geq R(\underbrace{m,m,\ldots% ,m}_{k}).

Parrondo's_paradox.html

  1. C t C_{t}
  2. C t + 1 = C t + 1 C_{t+1}=C_{t}+1
  3. C t + 1 = C t - 1 C_{t+1}=C_{t}-1
  4. P 1 = ( 1 / 2 ) - ϵ P_{1}=(1/2)-\epsilon
  5. ϵ > 0 \epsilon>0
  6. M M
  7. P 2 = ( 1 / 10 ) - ϵ P_{2}=(1/10)-\epsilon
  8. P 3 = ( 3 / 4 ) - ϵ P_{3}=(3/4)-\epsilon
  9. M M
  10. M = 3 M=3
  11. ϵ = 0.005 , \epsilon=0.005,
  12. C t C_{t}
  13. M M
  14. C t C_{t}
  15. M M
  16. C t C_{t}
  17. M M
  18. M M

Partial_equivalence_relation.html

  1. R R
  2. X X
  3. a , b , c X a,b,c\in X
  4. a R b aRb
  5. b R a bRa
  6. a R b aRb
  7. b R c bRc
  8. a R c aRc
  9. R R
  10. R R
  11. R R
  12. X X
  13. Y = { x X | x R x } X Y=\{x\in X|x\,R\,x\}\subseteq X
  14. Y Y
  15. X X
  16. Y Y
  17. X Y X\setminus Y
  18. R R
  19. R R
  20. Y Y
  21. Y Y
  22. R R
  23. Y Y
  24. x R y xRy
  25. y R x yRx
  26. x R x xRx
  27. y R y yRy
  28. R = R=\emptyset
  29. X = X=\emptyset
  30. X X
  31. A A
  32. f f
  33. A A
  34. \approx
  35. x y x\approx y
  36. f f
  37. x x
  38. f f
  39. y y
  40. f ( x ) = f ( y ) f(x)=f(y)
  41. f ( x ) f(x)
  42. x x x\not\approx x
  43. x x
  44. y A y\in A
  45. x y x\approx y
  46. A A
  47. \approx
  48. f f
  49. X , Y \approx_{X},\approx_{Y}
  50. f , g : X Y f,g:X\to Y
  51. f g f\approx g
  52. x 0 x 1 , x 0 X x 1 f ( x 0 ) Y g ( x 1 ) \forall x_{0}\;x_{1},\quad x_{0}\approx_{X}x_{1}\Rightarrow f(x_{0})\approx_{Y% }g(x_{1})
  53. f f f\approx f
  54. X / X Y / Y X/\approx_{X}\;\to\;Y/\approx_{Y}
  55. \approx

Particle_aggregation.html

  1. M R g d M\propto R_{g}^{d}

Particle_number_operator.html

  1. | Ψ ν = | ϕ 1 , ϕ 2 , , ϕ n ν |\Psi\rangle_{\nu}=|\phi_{1},\phi_{2},\cdots,\phi_{n}\rangle_{\nu}
  2. | ϕ i |\phi_{i}\rangle
  3. a ( ϕ i ) a^{\dagger}(\phi_{i})
  4. a ( ϕ i ) a(\phi_{i})\,
  5. N i ^ = def a ( ϕ i ) a ( ϕ i ) \hat{N_{i}}\ \stackrel{\mathrm{def}}{=}\ a^{\dagger}(\phi_{i})a(\phi_{i})
  6. N i ^ | Ψ ν = N i | Ψ ν \hat{N_{i}}|\Psi\rangle_{\nu}=N_{i}|\Psi\rangle_{\nu}
  7. N i N_{i}
  8. | ϕ i |\phi_{i}\rangle
  9. a ( ϕ i ) | ϕ 1 , ϕ 2 , , ϕ i - 1 , ϕ i , ϕ i + 1 , , ϕ n ν = N i | ϕ 1 , ϕ 2 , , ϕ i - 1 , ϕ i + 1 , , ϕ n ν a ( ϕ i ) | ϕ 1 , ϕ 2 , , ϕ i - 1 , ϕ i + 1 , , ϕ n ν = N i | ϕ 1 , ϕ 2 , , ϕ i - 1 , ϕ i , ϕ i + 1 , , ϕ n ν \begin{matrix}a(\phi_{i})|\phi_{1},\phi_{2},\cdots,\phi_{i-1},\phi_{i},\phi_{i% +1},\cdots,\phi_{n}\rangle_{\nu}&=&\sqrt{N_{i}}|\phi_{1},\phi_{2},\cdots,\phi_% {i-1},\phi_{i+1},\cdots,\phi_{n}\rangle_{\nu}\\ a^{\dagger}(\phi_{i})|\phi_{1},\phi_{2},\cdots,\phi_{i-1},\phi_{i+1},\cdots,% \phi_{n}\rangle_{\nu}&=&\sqrt{N_{i}}|\phi_{1},\phi_{2},\cdots,\phi_{i-1},\phi_% {i},\phi_{i+1},\cdots,\phi_{n}\rangle_{\nu}\end{matrix}
  10. N i ^ | Ψ ν = a ( ϕ i ) a ( ϕ i ) | ϕ 1 , ϕ 2 , , ϕ i - 1 , ϕ i , ϕ i + 1 , , ϕ n ν = N i a ( ϕ i ) | ϕ 1 , ϕ 2 , , ϕ i - 1 , ϕ i + 1 , , ϕ n ν = N i N i | ϕ 1 , ϕ 2 , , ϕ i - 1 , ϕ i , ϕ i + 1 , , ϕ n ν = N i | Ψ ν \begin{matrix}\hat{N_{i}}|\Psi\rangle_{\nu}=a^{\dagger}(\phi_{i})a(\phi_{i})|% \phi_{1},\phi_{2},\cdots,\phi_{i-1},\phi_{i},\phi_{i+1},\cdots,\phi_{n}\rangle% _{\nu}&=&\sqrt{N_{i}}a^{\dagger}(\phi_{i})|\phi_{1},\phi_{2},\cdots,\phi_{i-1}% ,\phi_{i+1},\cdots,\phi_{n}\rangle_{\nu}\\ &=&\sqrt{N_{i}}\sqrt{N_{i}}|\phi_{1},\phi_{2},\cdots,\phi_{i-1},\phi_{i},\phi_% {i+1},\cdots,\phi_{n}\rangle_{\nu}\\ &=&N_{i}|\Psi\rangle_{\nu}\\ \end{matrix}

Particle_physics_and_representation_theory.html

  1. | p 0 |p_{0}\rangle
  2. | p g = g | p 0 |p_{g}\rangle=g|p_{0}\rangle
  3. up quark ( 1 0 ) , down quark ( 0 1 ) \,\text{up quark}\rightarrow\begin{pmatrix}1\\ 0\end{pmatrix},\qquad\,\text{down quark}\rightarrow\begin{pmatrix}0\\ 1\end{pmatrix}
  4. ( x y ) A ( x y ) , where A is in S U ( 2 ) \begin{pmatrix}x\\ y\end{pmatrix}\mapsto A\begin{pmatrix}x\\ y\end{pmatrix},\quad\,\text{where }A\,\text{ is in }SU(2)
  5. A = ( 0 1 - 1 0 ) A=\begin{pmatrix}0&1\\ -1&0\end{pmatrix}

Particular_values_of_the_Gamma_function.html

  1. Γ ( n ) = ( n - 1 ) ! n 0 , \Gamma(n)=(n-1)!\qquad n\in\mathbb{N}_{0},
  2. Γ ( 1 ) = 1 , Γ ( 2 ) = 1 , Γ ( 3 ) = 2 , Γ ( 4 ) = 6 , Γ ( 5 ) = 24. \begin{aligned}\displaystyle\Gamma(1)&\displaystyle=1,\\ \displaystyle\Gamma(2)&\displaystyle=1,\\ \displaystyle\Gamma(3)&\displaystyle=2,\\ \displaystyle\Gamma(4)&\displaystyle=6,\\ \displaystyle\Gamma(5)&\displaystyle=24.\end{aligned}
  3. Γ ( n 2 ) = π ( n - 2 ) ! ! 2 ( n - 1 ) / 2 , \Gamma\left(\tfrac{n}{2}\right)=\sqrt{\pi}\frac{(n-2)!!}{2^{(n-1)/2}}\,,
  4. Γ ( 1 2 + n ) \displaystyle\Gamma\left(\tfrac{1}{2}+n\right)
  5. Γ ( 1 2 ) \Gamma(\tfrac{1}{2})\,
  6. = π =\sqrt{\pi}\,
  7. 1.7724538509055160273 , \approx 1.7724538509055160273\,,
  8. Γ ( 3 2 ) \Gamma(\tfrac{3}{2})\,
  9. = 1 2 π =\frac{1}{2}\sqrt{\pi}\,
  10. 0.8862269254527580137 , \approx 0.8862269254527580137\,,
  11. Γ ( 5 2 ) \Gamma(\tfrac{5}{2})\,
  12. = 3 4 π =\frac{3}{4}\sqrt{\pi}\,
  13. 1.3293403881791370205 , \approx 1.3293403881791370205\,,
  14. Γ ( 7 2 ) \Gamma(\tfrac{7}{2})\,
  15. = 15 8 π =\frac{15}{8}\sqrt{\pi}\,
  16. 3.3233509704478425512 , \approx 3.3233509704478425512\,,
  17. Γ ( - 1 2 ) \Gamma(-\tfrac{1}{2})\,
  18. = - 2 π =-2\sqrt{\pi}\,
  19. - 3.5449077018110320546 , \approx-3.5449077018110320546\,,
  20. Γ ( - 3 2 ) \Gamma(-\tfrac{3}{2})\,
  21. = 4 3 π =\frac{4}{3}\sqrt{\pi}\,
  22. 2.3632718012073547031 , \approx 2.3632718012073547031\,,
  23. Γ ( - 5 2 ) \Gamma(-\tfrac{5}{2})\,
  24. = - 8 15 π =-\frac{8}{15}\sqrt{\pi}\,
  25. - 0.9453087204829418812 , \approx-0.9453087204829418812\,,
  26. Γ ( n + 1 3 ) = Γ ( 1 3 ) ( 3 n - 2 ) ! ( 3 ) 3 n Γ ( n + 1 4 ) = Γ ( 1 4 ) ( 4 n - 3 ) ! ( 4 ) 4 n Γ ( n + 1 p ) = Γ ( 1 p ) ( p n - ( p - 1 ) ) ! ( p ) p n \begin{aligned}\displaystyle\Gamma\left(n+\tfrac{1}{3}\right)&\displaystyle=% \Gamma\left(\tfrac{1}{3}\right)\frac{(3n-2)!^{(3)}}{3^{n}}\\ \displaystyle\Gamma\left(n+\tfrac{1}{4}\right)&\displaystyle=\Gamma\left(% \tfrac{1}{4}\right)\frac{(4n-3)!^{(4)}}{4^{n}}\\ \displaystyle\Gamma\left(n+\tfrac{1}{p}\right)&\displaystyle=\Gamma\left(% \tfrac{1}{p}\right)\frac{(pn-(p-1))!^{(p)}}{p^{n}}\end{aligned}
  27. n ! ( k ) n!^{(k)}
  28. Γ ( 1 3 ) 2.6789385347077476337 \Gamma(\tfrac{1}{3})\approx 2.6789385347077476337
  29. Γ ( 1 4 ) 3.6256099082219083119 \Gamma(\tfrac{1}{4})\approx 3.6256099082219083119
  30. Γ ( 1 5 ) 4.5908437119988030532 \Gamma(\tfrac{1}{5})\approx 4.5908437119988030532
  31. Γ ( 1 6 ) 5.5663160017802352043 \Gamma(\tfrac{1}{6})\approx 5.5663160017802352043
  32. Γ ( 1 7 ) 6.5480629402478244377 \Gamma(\tfrac{1}{7})\approx 6.5480629402478244377
  33. Γ ( 1 8 ) 7.5339415987976119047 \Gamma(\tfrac{1}{8})\approx 7.5339415987976119047
  34. Γ ( 1 3 ) Γ(\frac{1}{3})
  35. Γ ( 1 4 ) Γ(\frac{1}{4})
  36. Γ ( 1 4 ) / π 1 / 4 \Gamma(\tfrac{1}{4})/\pi^{1/4}
  37. Γ ( 1 4 ) , π Γ(\frac{1}{4}),π
  38. Γ ( 1 4 ) Γ(\frac{1}{4})
  39. S S
  40. Γ ( 1 4 ) = 2 π S , \Gamma(\tfrac{1}{4})=\sqrt{\sqrt{2\pi}S},
  41. Γ ( 1 4 ) = ( 4 π 3 e 2 γ - δ + 1 ) 1 4 \Gamma\left(\tfrac{1}{4}\right)=\left(4\pi^{3}e^{2\gamma-\mathrm{\delta}+1}% \right)^{\frac{1}{4}}
  42. Γ ( n 24 ) Γ(\frac{n}{24})
  43. π π
  44. K ( k ( 1 ) ) , K ( k ( 2 ) ) , K ( k ( 3 ) ) K(k(1)),K(k(2)),K(k(3))
  45. K ( k ( 6 ) ) K(k(6))
  46. K ( k ( N ) ) K(k(N))
  47. Γ ( 1 5 ) Γ(\frac{1}{5})
  48. Γ ( 1 4 ) Γ(\frac{1}{4})
  49. Γ ( 1 4 ) = ( 2 π ) 3 2 A G M ( 2 , 1 ) . \Gamma(\tfrac{1}{4})=\sqrt{\frac{(2\pi)^{\frac{3}{2}}}{AGM(\sqrt{2},1)}}.
  50. Γ ( 1 4 ) = ( 2 π ) 3 4 k = 1 tanh ( π k 2 ) \Gamma(\tfrac{1}{4})=(2\pi)^{\frac{3}{4}}\prod_{k=1}^{\infty}\tanh\left(\frac{% \pi k}{2}\right)
  51. Γ ( 1 4 ) = A 3 e - G π π 2 1 6 k = 1 ( 1 - 1 2 k ) k ( - 1 ) k \Gamma(\tfrac{1}{4})=A^{3}e^{-\frac{G}{\pi}}\sqrt{\pi}2^{\frac{1}{6}}\prod_{k=% 1}^{\infty}\left(1-\frac{1}{2k}\right)^{k(-1)^{k}}
  52. ( Γ ( 1 3 ) ) 6 12 π 4 = 1 10 k = 0 ( 6 k ) ! ( - 1 ) k ( k ! ) 3 ( 3 k ) ! 3 k 160 3 k ( Γ ( 1 4 ) ) 4 128 π 3 = 1 u k = 0 ( 6 k ) ! ( 2 w ) k ( k ! ) 3 ( 3 k ) ! 6486 3 k \begin{aligned}\displaystyle\frac{\left(\Gamma(\tfrac{1}{3})\right)^{6}}{12\pi% ^{4}}&\displaystyle=\frac{1}{\sqrt{10}}\sum_{k=0}^{\infty}\frac{(6k)!(-1)^{k}}% {(k!)^{3}(3k)!3^{k}160^{3k}}\\ \displaystyle\frac{\left(\Gamma(\tfrac{1}{4})\right)^{4}}{128\pi^{3}}&% \displaystyle=\frac{1}{\sqrt{u}}\,\sum_{k=0}^{\infty}\frac{(6k)!(2w)^{k}}{(k!)% ^{3}(3k)!6486^{3k}}\end{aligned}
  53. u = 273 + 180 2 v = 1 + 2 w = - 761354780 + 538359129 2 = 6486 3 2 ( u v 2 2 ) 3 \begin{aligned}\displaystyle u&\displaystyle=273+180\sqrt{2}\\ \displaystyle v&\displaystyle=1+\sqrt{2}\\ \displaystyle w&\displaystyle=-761354780+538359129\sqrt{2}=\frac{6486^{3}}{2% \bigl(uv^{2}\sqrt{2}\bigr)^{3}}\end{aligned}
  54. ( Γ ( 1 4 ) ) 4 128 π 3 = 1 u k = 0 ( 6 k ) ! ( k ! ) 3 ( 3 k ) ! 1 ( u v 2 2 ) 3 k . \frac{\left(\Gamma(\tfrac{1}{4})\right)^{4}}{128\pi^{3}}=\frac{1}{\sqrt{u}}% \sum_{k=0}^{\infty}\frac{(6k)!}{(k!)^{3}(3k)!}\frac{1}{(uv^{2}\sqrt{2})^{3k}}.
  55. r = 1 2 Γ ( r 3 ) = 2 π 3 3.6275987284684357012 \prod_{r=1}^{2}\Gamma(\tfrac{r}{3})=\frac{2\pi}{\sqrt{3}}\approx 3.62759872846% 84357012
  56. r = 1 3 Γ ( r 4 ) = 2 π 3 7.8748049728612098721 \prod_{r=1}^{3}\Gamma(\tfrac{r}{4})=\sqrt{2\pi^{3}}\approx 7.8748049728612098721
  57. r = 1 4 Γ ( r 5 ) = 4 π 2 5 17.6552850814935242483 \prod_{r=1}^{4}\Gamma(\tfrac{r}{5})=\frac{4\pi^{2}}{\sqrt{5}}\approx 17.655285% 0814935242483
  58. r = 1 5 Γ ( r 6 ) = 4 π 5 3 40.3993191220037900785 \prod_{r=1}^{5}\Gamma(\tfrac{r}{6})=4\sqrt{\frac{\pi^{5}}{3}}\approx 40.399319% 1220037900785
  59. r = 1 6 Γ ( r 7 ) = 8 π 3 7 93.7541682035825037970 \prod_{r=1}^{6}\Gamma(\tfrac{r}{7})=\frac{8\pi^{3}}{\sqrt{7}}\approx 93.754168% 2035825037970
  60. r = 1 7 Γ ( r 8 ) = 4 π 7 219.8287780169572636207 \prod_{r=1}^{7}\Gamma(\tfrac{r}{8})=4\sqrt{\pi^{7}}\approx 219.828778016957263% 6207
  61. r = 1 n Γ ( r n + 1 ) = ( 2 π ) n n + 1 \prod_{r=1}^{n}\Gamma(\tfrac{r}{n+1})=\sqrt{\frac{(2\pi)^{n}}{n+1}}
  62. Γ ( 1 5 ) Γ ( 4 15 ) Γ ( 1 3 ) Γ ( 2 15 ) = 2 3 20 5 6 5 - 7 5 + 6 - 6 5 4 \frac{\Gamma(\tfrac{1}{5})\Gamma(\tfrac{4}{15})}{\Gamma(\tfrac{1}{3})\Gamma(% \tfrac{2}{15})}=\frac{\sqrt{2}\sqrt[20]{3}}{\sqrt[6]{5}\sqrt[4]{5-\frac{7}{% \sqrt{5}}+\sqrt{6-\frac{6}{\sqrt{5}}}}}
  63. Γ ( 1 20 ) Γ ( 9 20 ) Γ ( 3 20 ) Γ ( 7 20 ) = 5 4 ( 1 + 5 ) 2 \frac{\Gamma(\tfrac{1}{20})\Gamma(\tfrac{9}{20})}{\Gamma(\tfrac{3}{20})\Gamma(% \tfrac{7}{20})}=\frac{\sqrt[4]{5}\left(1+\sqrt{5}\right)}{2}
  64. i = - 1 i=\sqrt{-1}
  65. Γ ( i ) = ( - 1 + i ) ! - 0.1549 - 0.4980 i . \Gamma(i)=(-1+i)!\approx-0.1549-0.4980i.
  66. Γ ( i ) = G ( 1 + i ) G ( i ) = e - log G ( i ) + log G ( 1 + i ) . \Gamma(i)=\frac{G(1+i)}{G(i)}=e^{-\log G(i)+\log G(1+i)}.
  67. i = - 1 i=\sqrt{-1}
  68. Γ ( 1 + i ) = i Γ ( i ) 0.498 - 0.155 i \Gamma(1+i)=i\Gamma(i)\approx 0.498-0.155i
  69. Γ ( 1 - i ) = - i Γ ( - i ) 0.498 + 0.155 i \Gamma(1-i)=-i\Gamma(-i)\approx 0.498+0.155i
  70. Γ ( 0.5 + 0.5 i ) 0.8181639995 - 0.7633138287 i \Gamma(0.5+0.5i)\approx 0.8181639995-0.7633138287i
  71. Γ ( 0.5 - 0.5 i ) 0.8181639995 + 0.7633138287 i \Gamma(0.5-0.5i)\approx 0.8181639995+0.7633138287i
  72. Γ ( 5 + 3 i ) 0.0160418827 - 9.4332932898 i \Gamma(5+3i)\approx 0.0160418827-9.4332932898i
  73. Γ ( 5 - 3 i ) 0.0160418827 + 9.4332932897 i . \Gamma(5-3i)\approx 0.0160418827+9.4332932897i.
  74. x min = 1.461632144968362341262 x_{\mathrm{min}}=1.461632144968362341262\ldots\,
  75. Γ ( x min ) = 0.885603194410888 \Gamma(x_{\mathrm{min}})=0.885603194410888\ldots\,

Partition_of_sums_of_squares.html

  1. y i - y ¯ y_{i}-\overline{y}
  2. y i y_{i}
  3. y ¯ \overline{y}
  4. i = 1 n ( y i - y ¯ ) 2 \sum_{i=1}^{n}\left(y_{i}-\overline{y}\,\right)^{2}
  5. y i = β 0 + β 1 x i 1 + + β p x i p + ε i y_{i}=\beta_{0}+\beta_{1}x_{i1}+\cdots+\beta_{p}x_{ip}+\varepsilon_{i}
  6. ( y i , x i 1 , , x i p ) , i = 1 , , n (y_{i},x_{i1},\ldots,x_{ip}),\,i=1,\ldots,n
  7. T S S = i = 1 n ( y i - y ¯ ) 2 TSS=\sum_{i=1}^{n}(y_{i}-\bar{y})^{2}
  8. TSS = ESS + RSS , \mathrm{TSS}=\mathrm{ESS}+\mathrm{RSS},
  9. y - y ¯ 𝟏 2 \displaystyle\left\|y-\bar{y}\mathbf{1}\right\|^{2}
  10. i = 1 n ( y i - y ¯ ) 2 = i = 1 n ( y i - y ¯ + y ^ i - y ^ i ) 2 = i = 1 n ( ( y ^ i - y ¯ ) + ( y i - y ^ i ) ε ^ i ) 2 = i = 1 n ( ( y ^ i - y ¯ ) 2 + 2 ε ^ i ( y ^ i - y ¯ ) + ε ^ i 2 ) = i = 1 n ( y ^ i - y ¯ ) 2 + i = 1 n ε ^ i 2 + 2 i = 1 n ε ^ i ( y ^ i - y ¯ ) = i = 1 n ( y ^ i - y ¯ ) 2 + i = 1 n ε ^ i 2 + 2 i = 1 n ε ^ i ( β ^ 0 + β ^ 1 x i 1 + + β ^ p x i p - y ¯ ) = i = 1 n ( y ^ i - y ¯ ) 2 + i = 1 n ε ^ i 2 + 2 ( β ^ 0 - y ¯ ) i = 1 n ε ^ i 0 + 2 β ^ 1 i = 1 n ε ^ i x i 1 0 + + 2 β ^ p i = 1 n ε ^ i x i p 0 = i = 1 n ( y ^ i - y ¯ ) 2 + i = 1 n ε ^ i 2 = ESS + RSS \begin{aligned}\displaystyle\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}&% \displaystyle=\sum_{i=1}^{n}(y_{i}-\overline{y}+\hat{y}_{i}-\hat{y}_{i})^{2}=% \sum_{i=1}^{n}((\hat{y}_{i}-\bar{y})+\underbrace{(y_{i}-\hat{y}_{i})}_{\hat{% \varepsilon}_{i}})^{2}\\ &\displaystyle=\sum_{i=1}^{n}((\hat{y}_{i}-\bar{y})^{2}+2\hat{\varepsilon}_{i}% (\hat{y}_{i}-\bar{y})+\hat{\varepsilon}_{i}^{2})\\ &\displaystyle=\sum_{i=1}^{n}(\hat{y}_{i}-\bar{y})^{2}+\sum_{i=1}^{n}\hat{% \varepsilon}_{i}^{2}+2\sum_{i=1}^{n}\hat{\varepsilon}_{i}(\hat{y}_{i}-\bar{y})% \\ &\displaystyle=\sum_{i=1}^{n}(\hat{y}_{i}-\bar{y})^{2}+\sum_{i=1}^{n}\hat{% \varepsilon}_{i}^{2}+2\sum_{i=1}^{n}\hat{\varepsilon}_{i}(\hat{\beta}_{0}+\hat% {\beta}_{1}x_{i1}+\cdots+\hat{\beta}_{p}x_{ip}-\overline{y})\\ &\displaystyle=\sum_{i=1}^{n}(\hat{y}_{i}-\bar{y})^{2}+\sum_{i=1}^{n}\hat{% \varepsilon}_{i}^{2}+2(\hat{\beta}_{0}-\overline{y})\underbrace{\sum_{i=1}^{n}% \hat{\varepsilon}_{i}}_{0}+2\hat{\beta}_{1}\underbrace{\sum_{i=1}^{n}\hat{% \varepsilon}_{i}x_{i1}}_{0}+\cdots+2\hat{\beta}_{p}\underbrace{\sum_{i=1}^{n}% \hat{\varepsilon}_{i}x_{ip}}_{0}\\ &\displaystyle=\sum_{i=1}^{n}(\hat{y}_{i}-\bar{y})^{2}+\sum_{i=1}^{n}\hat{% \varepsilon}_{i}^{2}=\mathrm{ESS}+\mathrm{RSS}\\ \end{aligned}
  11. i = 1 n ε ^ i = 0 \sum_{i=1}^{n}\hat{\varepsilon}_{i}=0
  12. S S total = 𝐲 - 𝐲 ¯ 2 \displaystyle SS_{{\,\text{total}}}=\|{{\mathbf{y}}-\bar{\mathbf{y}}}\|^{2}
  13. ε ^ T X = ( 𝐲 - 𝐲 ^ ) T X = 𝐲 T ( I - X ( X T X ) - 1 X T ) X = 𝐲 T ( X - X ) = 0. \hat{\varepsilon}^{T}X=\left({\mathbf{y}}-{\mathbf{\hat{y}}}\right)^{T}X={% \mathbf{y}}^{T}\left({I-X\left({X^{T}X}\right)^{-1}X^{T}}\right)X={\mathbf{y}}% ^{T}\left(X-X\right)={\mathbf{0}}.

Partition_problem.html

  1. N / 2 \lfloor N/2\rfloor
  2. N / 2 \lfloor N/2\rfloor
  3. N / 2 \lceil N/2\rceil
  4. N / 2 \lfloor N/2\rfloor
  5. N / 2 \lfloor N/2\rfloor
  6. N / 2 \lfloor N/2\rfloor
  7. N / 2 \lfloor N/2\rfloor
  8. N / 2 \lfloor N/2\rfloor
  9. N / 2 \lfloor N/2\rfloor
  10. N / 2 \lfloor N/2\rfloor
  11. N / 2 \lfloor N/2\rfloor
  12. N / 2 \lfloor N/2\rfloor
  13. m / n < 1 m/n<1
  14. m / n > 1 m/n>1

Partition_regularity.html

  1. X X
  2. 𝕊 𝒫 ( X ) \mathbb{S}\subset\mathcal{P}(X)
  3. A 𝕊 A\in\mathbb{S}
  4. A = C 1 C 2 C n A=C_{1}\cup C_{2}\cup\cdots\cup C_{n}
  5. C i C_{i}
  6. 𝕊 \mathbb{S}
  7. 𝕊 \mathbb{S}
  8. \mathbb{N}
  9. d ¯ ( A ) \overline{d}(A)
  10. A A\subset\mathbb{N}
  11. d ¯ ( A ) = lim sup n | { 1 , 2 , , n } A | n . \overline{d}(A)=\limsup_{n\rightarrow\infty}\frac{|\{1,2,\ldots,n\}\cap A|}{n}.
  12. 𝕌 \mathbb{U}
  13. X X
  14. 𝕌 \mathbb{U}
  15. 𝕌 A = 1 n C i \mathbb{U}\ni A=\bigcup_{1}^{n}C_{i}
  16. i i
  17. C i 𝕌 C_{i}\in\mathbb{U}
  18. T T
  19. A β A\in\ \beta
  20. n R n\in R
  21. μ ( A T n A ) > 0 \mu(A\cap T^{n}A)>0
  22. [ A ] n [A]^{n}
  23. A A\subset\mathbb{N}
  24. 𝕊 n = A [ A ] n \mathbb{S}^{n}=\bigcup_{A\subset\mathbb{N}}[A]^{n}
  25. 𝕊 n \mathbb{S}^{n}
  26. κ \kappa
  27. κ \kappa
  28. S S
  29. S = α < λ S α S=\bigcup_{\alpha<\lambda}S_{\alpha}
  30. λ < κ \lambda<\kappa
  31. S α S_{\alpha}
  32. Δ \Delta
  33. A A\subset\mathbb{N}
  34. Δ \Delta
  35. A A
  36. { s m - s n : m , n , n < m } \{s_{m}-s_{n}:m,n\in\mathbb{N},n<m\}
  37. s n n = 1 ω \langle s_{n}\rangle^{\omega}_{n=1}
  38. \mathbb{N}
  39. 𝔹 \mathbb{B}
  40. \mathbb{N}
  41. X , Y 𝔹 , X Y \forall X,Y\in\mathbb{B},X\not\subset Y
  42. I 𝔹 I\subset\cup\mathbb{B}
  43. X 𝔹 X\in\mathbb{B}
  44. X I X\subset I
  45. i I X , x X , x < i \forall i\in I\setminus X,\forall x\in X,x<i
  46. [ A ] n [A]^{n}
  47. β \beta\mathbb{N}

Passive_integrator_circuit.html

  1. Y = X Z C Z C + Z R = X 1 j ω C 1 j ω C + R = X 1 1 + j ω R C , Y=X\frac{Z_{C}}{Z_{C}+Z_{R}}=X\frac{\frac{1}{j\omega C}}{\frac{1}{j\omega C}+R% }=X\frac{1}{1+j\omega RC},
  2. X X
  3. Y Y
  4. Z R Z_{R}
  5. Z C Z_{C}
  6. K ( j ω ) = 1 1 + j ω R C = 1 1 + j ω ω 0 , K(j\omega)=\frac{1}{1+j\omega RC}=\frac{1}{1+\frac{j\omega}{\omega_{0}}},
  7. ω 0 = 1 R C . \omega_{0}=\frac{1}{RC}.
  8. H ( ω ) = | K ( j ω ) | = 1 1 + ( ω ω 0 ) 2 . H(\omega)=|K(j\omega)|=\frac{1}{\sqrt{1+\left(\frac{\omega}{\omega_{0}}\right)% ^{2}}}.
  9. ϕ ( ω ) = arg K ( j ω ) = - arctan ω ω 0 . \phi(\omega)=\arg K(j\omega)=-\arctan\frac{\omega}{\omega_{0}}.
  10. ω 0 = R L \omega_{0}=\frac{R}{L}
  11. h ( t ) = - 1 { K ( p ) } = β - j β + j K ( p ) e p t d p = ω 0 e - ω 0 t = 1 τ e - t τ , h(t)=\mathcal{L}^{-1}\left\{K(p)\right\}=\int_{\beta-j\infty}^{\beta+j\infty}K% (p)e^{pt}\,dp=\omega_{0}e^{-\omega_{0}t}=\frac{1}{\tau}e^{-\frac{t}{\tau}},
  12. τ = 1 ω 0 \tau=\frac{1}{\omega_{0}}

Password_strength.html

  1. H = log 2 N L = L log 2 N = L log N log 2 H=\log_{2}N^{L}=L\log_{2}N=L{\log N\over\log 2}
  2. L = H log 2 N L={H\over\log_{2}N}

Path_coloring.html

  1. R R
  2. G G
  3. R R
  4. G G
  5. R R
  6. G G
  7. R R
  8. R R
  9. R R
  10. R R
  11. G G
  12. R R
  13. G G
  14. R R
  15. I I
  16. I I
  17. G G
  18. G G
  19. L L
  20. L L
  21. P P
  22. P v P_{v}
  23. P P
  24. v v
  25. P v P_{v}
  26. v v
  27. P v P_{v}

Patlak_plot.html

  1. R ( t ) = K 0 t C p ( τ ) d τ + V 0 C p ( t ) R(t)=K\int_{0}^{t}C_{p}(\tau)\,d\tau+V_{0}C_{p}(t)
  2. t t
  3. R ( t ) R(t)
  4. C p ( t ) C_{p}(t)
  5. K K
  6. V 0 V_{0}
  7. C p ( t ) C_{p}(t)
  8. R ( t ) C p ( t ) = K 0 t C p ( τ ) d τ C p ( t ) + V 0 {R(t)\over C_{p}(t)}=K{\int_{0}^{t}C_{p}(\tau)\,d\tau\over C_{p}(t)}+V_{0}
  9. K K
  10. V 0 V_{0}
  11. R ( t ) C p ( t ) {R(t)\over C_{p}(t)}
  12. 0 t C p ( τ ) d τ / C p ( t ) \int_{0}^{t}C_{p}(\tau)\,d\tau/C_{p}(t)

Pattern_formation.html

  1. u t = F ( u , t ) \frac{\partial u}{\partial t}=F(u,t)
  2. u ( 𝐱 , t ) = j z j ( t ) e i 𝐤 j 𝐱 + z j ( t ) * e - i 𝐤 j 𝐱 u(\mathbf{x},t)=\sum_{j}z_{j}(t)e^{i\mathbf{k}_{j}\cdot\mathbf{x}}+z_{j}(t)^{*% }e^{-i\mathbf{k}_{j}\cdot\mathbf{x}}
  3. z j z_{j}
  4. 𝐤 j \mathbf{k}_{j}

Pattern_theory.html

  1. g G g{\in}G
  2. b B b{\in}B
  3. g g
  4. s S s{\in}S
  5. ρ \rho
  6. ρ \rho
  7. ρ ( c ) = neighboring bonds b , b ′′ c ρ ( b , b ′′ ) . \rho(c)=\prod_{\,\text{neighboring bonds }b^{\prime},b^{\prime\prime}\in c}% \rho(b^{\prime},b^{\prime\prime}).
  8. ρ \rho
  9. p ( c ) = neighboring bonds b , b ′′ c A ( b , b ′′ ) p(c)=\prod_{\,\text{neighboring bonds }b^{\prime},b^{\prime\prime}\in c}A(b^{% \prime},b^{\prime\prime})
  10. = n 0 ( v 0 | j = 1 l v j ) j = 1 l n ( v j | e j ) v j ! j = 1 m t ( f j | e a j ) j : a j 0 m d ( j | a j , l , m ) . =n_{0}(v_{0}|\sum_{j=1}^{l}{v_{j}})\cdot\prod_{j=1}^{l}n(v_{j}|e_{j})v_{j}!% \cdot\prod_{j=1}^{m}t(f_{j}|e_{a_{j}})\cdot\prod_{j:a_{j}\not=0}^{m}d(j|a_{j},% l,m).\,
  11. 1 / 2 {1}/{2}
  12. , x k , x k + 1 , Pr ( x , s ) = k p 1 ( x k | x k - 1 ) p 2 ( s k | x k ) \dots,x_{k},x_{k+1},\dots\Pr(x,s)=\prod_{k}p_{1}(x_{k}|x_{k-1})p_{2}(s_{k}|x_{% k})

Paul_Leyland.html

  1. x y + y x x^{y}+y^{x}

Paul_Thagard.html

  1. ( p , q ) C + (p,q)\in C^{+}
  2. p p
  3. q q
  4. ( p , q ) C - (p,q)\in C^{-}
  5. ( p , q ) C + C - (p,q)\in C^{+}\cup C^{-}
  6. w ( p , q ) w(p,q)
  7. A A
  8. R R
  9. ( p , q ) (p,q)
  10. p p
  11. q q
  12. p , q A p,q\in A
  13. p p
  14. q q
  15. p , q R p,q\in R
  16. ( p , q ) (p,q)
  17. p A p\in A
  18. q R q\in R

Pauli_group.html

  1. G 1 G_{1}
  2. G 1 G_{1}
  3. I I
  4. X = σ 1 = ( 0 1 1 0 ) , Y = σ 2 = ( 0 - i i 0 ) , Z = σ 3 = ( 1 0 0 - 1 ) X=\sigma_{1}=\begin{pmatrix}0&1\\ 1&0\end{pmatrix},\quad Y=\sigma_{2}=\begin{pmatrix}0&-i\\ i&0\end{pmatrix},\quad Z=\sigma_{3}=\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}
  5. - 1 -1
  6. ± i \pm i
  7. G 1 = def { ± I , ± i I , ± X , ± i X , ± Y , ± i Y , ± Z , ± i Z } X , Y , Z G_{1}\ \stackrel{\mathrm{def}}{=}\ \{\pm I,\pm iI,\pm X,\pm iX,\pm Y,\pm iY,% \pm Z,\pm iZ\}\equiv\langle X,Y,Z\rangle
  8. G n G_{n}
  9. n n
  10. ( 2 ) n (\mathbb{C}^{2})^{\otimes n}

Pauling's_rules.html

  1. r + / r - r_{+}/r_{-}
  2. r c / r a r_{c}/r_{a}
  3. 2 r - = 2 ( r - + r + ) 2r_{-}=\sqrt{2}(r_{-}+r_{+})
  4. 2 r - = r - + r + \sqrt{2}r_{-}=r_{-}+r_{+}
  5. r + = ( 2 - 1 ) r - = 0.414 r - r_{+}=(\sqrt{2}-1)r_{-}=0.414r_{-}
  6. s = z ν s=\frac{z}{\nu}
  7. ξ = i s i \xi=\sum_{i}s_{i}
  8. ξ \xi
  9. s i s_{i}

PDE_surface.html

  1. ( 2 u 2 + a 2 2 v 2 ) 2 X ( u , v ) = 0. \left(\frac{\partial^{2}}{\partial u^{2}}+a^{2}\frac{\partial^{2}}{\partial v^% {2}}\right)^{2}X(u,v)=0.
  2. X ( u , v ) X(u,v)
  3. u u
  4. v v
  5. X ( u , v ) = ( x ( u , v ) , y ( u , v ) , z ( u , v ) ) X(u,v)=(x(u,v),y(u,v),z(u,v))
  6. x x
  7. y y
  8. z z
  9. X ( u , v ) X(u,v)
  10. X / < m t p l > n \partial{X}/\partial<mtpl>{{n}}
  11. a a
  12. u u
  13. v v

Pearson–Anson_effect.html

  1. v ( t ) = V S ( 1 - e - t / R C ) v(t)=V_{S}(1-e^{-t/RC})\,
  2. t = R C ln [ V S V S - v ] t=RC\ln\Bigl[{V_{S}\over{V_{S}-v}}\Bigr]\,
  3. T = t ( V b ) - t ( V e ) T=t(V_{b})-t(V_{e})\,
  4. T = R C ln [ V S V S - V b ] - R C ln [ V S V S - V e ] T=RC\ln\Bigl[{V_{S}\over{V_{S}-V_{b}}}\Bigr]-RC\ln\Bigl[{V_{S}\over{V_{S}-V_{e% }}}\Bigr]\,
  5. T = R C ln [ V S - V e V S - V b ] T=RC\ln\Bigl[{{V_{S}-V_{e}}\over{V_{S}-V_{b}}}\Bigr]\,

Peirce's_criterion.html

  1. m m
  2. n n

Penrose_interpretation.html

  1. \scriptstyle\hbar
  2. G \scriptstyle G

Penrose_method.html

  1. M M
  2. 1 / 2 + 1 / π M 1/2+1/\sqrt{\pi M}

Perfect_power.html

  1. 2 2 = 4 , 2 3 = 8 , 3 2 = 9 , 2 4 = 16 , 4 2 = 16 , 5 2 = 25 , 3 3 = 27 , 2^{2}=4,\ 2^{3}=8,\ 3^{2}=9,\ 2^{4}=16,\ 4^{2}=16,\ 5^{2}=25,\ 3^{3}=27,
  2. 2 5 = 32 , 6 2 = 36 , 7 2 = 49 , 2 6 = 64 , 4 3 = 64 , 8 2 = 64 , 2^{5}=32,\ 6^{2}=36,\ 7^{2}=49,\ 2^{6}=64,\ 4^{3}=64,\ 8^{2}=64,\dots
  3. m = 2 k = 2 1 m k = 1. \sum_{m=2}^{\infty}\sum_{k=2}^{\infty}\frac{1}{m^{k}}=1.
  4. m = 2 k = 2 1 m k = m = 2 1 m 2 k = 0 1 m k = m = 2 1 m 2 ( m m - 1 ) = m = 2 1 m ( m - 1 ) = m = 2 ( 1 m - 1 - 1 m ) = 1 . \sum_{m=2}^{\infty}\sum_{k=2}^{\infty}\frac{1}{m^{k}}=\sum_{m=2}^{\infty}\frac% {1}{m^{2}}\sum_{k=0}^{\infty}\frac{1}{m^{k}}=\sum_{m=2}^{\infty}\frac{1}{m^{2}% }\left(\frac{m}{m-1}\right)=\sum_{m=2}^{\infty}\frac{1}{m(m-1)}=\sum_{m=2}^{% \infty}\left(\frac{1}{m-1}-\frac{1}{m}\right)=1\,.
  5. p 1 p = k = 2 μ ( k ) ( 1 - ζ ( k ) ) 0.874464368 \sum_{p}\frac{1}{p}=\sum_{k=2}^{\infty}\mu(k)(1-\zeta(k))\approx 0.874464368\dots
  6. p 1 p - 1 = 1 3 + 1 7 + 1 8 + 1 15 + 1 24 + 1 26 + 1 31 + = 1. \sum_{p}\frac{1}{p-1}={\frac{1}{3}+\frac{1}{7}+\frac{1}{8}+\frac{1}{15}+\frac{% 1}{24}+\frac{1}{26}+\frac{1}{31}}+\cdots=1.
  7. k log 2 n k\leq\log_{2}n
  8. n n
  9. n 1 , n 2 , , n j n_{1},n_{2},\dots,n_{j}
  10. n 1 2 , n 2 2 , , n j 2 , n 1 3 , n 2 3 , n_{1}^{2},n_{2}^{2},\dots,n_{j}^{2},n_{1}^{3},n_{2}^{3},\dots
  11. n = m k n=m^{k}
  12. k = a p k=ap
  13. n = m k = m a p = ( m a ) p n=m^{k}=m^{ap}=(m^{a})^{p}
  14. n = p 1 α 1 p 2 α 2 p r α r n=p_{1}^{\alpha_{1}}p_{2}^{\alpha_{2}}\cdots p_{r}^{\alpha_{r}}
  15. p i p_{i}
  16. gcd ( α 1 , α 2 , , α r ) > 1 \gcd(\alpha_{1},\alpha_{2},\ldots,\alpha_{r})>1
  17. K ( a , b ) = 1 K(a,b)=1
  18. K ( a , b ) = 0 K(a,b)=0
  19. K ( a , b ) = b K ( a - b , b ) + ( a - b + 1 ) K ( a - b + 1 , b - 1 ) K(a,b)=bK(a-b,b)+(a-b+1)K(a-b+1,b-1)
  20. P ( a , b ) = 1 P(a,b)=1
  21. P ( a , b ) = P ( a - 1 , b ) + P ( a , b - 1 ) P(a,b)=P(a-1,b)+P(a,b-1)
  22. a b = x = 1 b P ( a - x + 1 , b + 1 ) K ( b , x ) a^{b}=\sum_{x=1}^{b}\,\!P(a-x+1,b+1)K(b,x)
  23. 7 3 = x = 1 b P ( 8 - x , 4 ) K ( 3 , x ) = P ( 7 , 4 ) K ( 3 , 1 ) + P ( 6 , 4 ) K ( 3 , 2 ) + P ( 5 , 4 ) K ( 3 , 3 ) 7^{3}=\sum_{x=1}^{b}P(8-x,4)K(3,x)=P(7,4)K(3,1)+P(6,4)K(3,2)+P(5,4)K(3,3)
  24. P ( 7 , 4 ) = P ( 7 , 3 ) + P ( 6 , 4 ) P ( 7 , 2 ) + 2 P ( 6 , 3 ) + P ( 5 , 4 ) P(7,4)=P(7,3)+P(6,4)P(7,2)+2P(6,3)+P(5,4)
  25. = P ( 7 , 1 ) + 3 P ( 6 , 2 ) + 3 P ( 5 , 3 ) + P ( 4 , 4 ) =P(7,1)+3P(6,2)+3P(5,3)+P(4,4)
  26. = 1 + 3 P ( 6 , 1 ) + 6 P ( 5 , 2 ) + 4 P ( 4 , 3 ) + P ( 3 , 4 ) =1+3P(6,1)+6P(5,2)+4P(4,3)+P(3,4)
  27. = 4 + 6 P ( 5 , 1 ) + 10 P ( 4 , 2 ) + 5 P ( 3 , 3 ) + P ( 2 , 4 ) =4+6P(5,1)+10P(4,2)+5P(3,3)+P(2,4)
  28. = 10 + 10 P ( 4 , 1 ) + 15 P ( 3 , 2 ) + 6 P ( 2 , 3 ) + P ( 1 , 4 ) =10+10P(4,1)+15P(3,2)+6P(2,3)+P(1,4)
  29. = 21 + 15 P ( 3 , 1 ) + 21 P ( 2 , 2 ) + 6 P ( 1 , 3 ) =21+15P(3,1)+21P(2,2)+6P(1,3)
  30. = 42 + 21 P ( 2 , 1 ) + 21 P ( 1 , 2 ) = 84 =42+21P(2,1)+21P(1,2)=84
  31. K ( 3 , 2 ) = 2 K ( 1 , 2 ) + 2 K ( 2 , 1 ) = 4 K(3,2)=2K(1,2)+2K(2,1)=4
  32. 7 3 = P ( 7 , 4 ) K ( 3 , 1 ) + P ( 6 , 4 ) K ( 3 , 2 ) + P ( 5 , 4 ) K ( 3 , 3 ) = 84 * 1 + 56 * 4 + 35 * 1 = 343 7^{3}=P(7,4)K(3,1)+P(6,4)K(3,2)+P(5,4)K(3,3)=84*1+56*4+35*1=343
  33. 7 4 = P ( 7 , 5 ) K ( 4 , 1 ) + P ( 6 , 5 ) K ( 4 , 2 ) + P ( 5 , 5 ) K ( 4 , 3 ) + P ( 4 , 5 ) K ( 4 , 4 ) = 210 * 1 + 126 * 11 + 70 * 11 + 35 * 1 = 2401 7^{4}=P(7,5)K(4,1)+P(6,5)K(4,2)+P(5,5)K(4,3)+P(4,5)K(4,4)=210*1+126*11+70*11+3% 5*1=2401
  34. 7 5 = 462 * 1 + 252 * 26 + 126 * 66 + 56 * 26 + 21 * 1 = 16807 7^{5}=462*1+252*26+126*66+56*26+21*1=16807

Perfectly_matched_layer.html

  1. / x \partial/\partial x
  2. x 1 1 + i σ ( x ) ω x \frac{\partial}{\partial x}\to\frac{1}{1+\frac{i\sigma(x)}{\omega}}\frac{% \partial}{\partial x}
  3. ω \omega
  4. σ \sigma
  5. σ \sigma
  6. e i ( k x - ω t ) e i ( k x - ω t ) - k ω x σ ( x ) d x , e^{i(kx-\omega t)}\to e^{i(kx-\omega t)-\frac{k}{\omega}\int^{x}\sigma(x^{% \prime})dx^{\prime}},
  7. k > 0 k>0
  8. x x + i ω x σ ( x ) d x x\to x+\frac{i}{\omega}\int^{x}\sigma(x^{\prime})dx^{\prime}
  9. d x d x ( 1 + i σ / ω ) dx\to dx(1+i\sigma/\omega)
  10. e i k x e^{ikx}

Period-doubling_bifurcation.html

  1. π t = f ( u t ) + a π t e \pi_{t}=f(u_{t})+a\pi_{t}^{e}
  2. π t + 1 = π t e + c ( π t - π t e ) \pi_{t+1}=\pi_{t}^{e}+c(\pi_{t}-\pi_{t}^{e})
  3. f ( u ) = β 1 + β 2 e - u f(u)=\beta_{1}+\beta_{2}e^{-u}\,
  4. b > 0 , 0 c 1 , d f d u < 0 b>0,0\leq c\leq 1,\frac{df}{du}<0
  5. π \pi
  6. π e \pi^{e}
  7. m - π m-\pi
  8. β 1 = - 2.5 , β 2 = 20 , c = 0.75 \beta_{1}=-2.5,\ \beta_{2}=20,\ c=0.75
  9. b b

Periodic_point.html

  1. f : X X f:X\to X
  2. f n ( x ) = x \ f_{n}(x)=x
  3. f n f_{n}
  4. f n ( x ) = f m ( x ) f_{n}(x)=f_{m}(x)
  5. f n f_{n}^{\prime}
  6. | f n | 1 , |f_{n}^{\prime}|\neq 1,
  7. | f n | < 1 , |f_{n}^{\prime}|<1,
  8. | f n | > 1. |f_{n}^{\prime}|>1.
  9. Φ : × X X \Phi:\mathbb{R}\times X\to X
  10. Φ ( t , x ) = x \Phi(t,x)=x\,
  11. Φ ( t , x ) = Φ ( t + p , x ) \Phi(t,x)=\Phi(t+p,x)\,
  12. γ x \gamma_{x}
  13. x t + 1 = r x t ( 1 - x t ) , 0 x t 1 , 0 r 4 x_{t+1}=rx_{t}(1-x_{t}),\qquad 0\leq x_{t}\leq 1,\qquad 0\leq r\leq 4

Permanent_income_hypothesis.html

  1. c c
  2. t t
  3. T T
  4. u ( ) u(\cdot)
  5. t t
  6. y t y_{t}
  7. c t c_{t}
  8. A t A_{t}
  9. r r
  10. β ( 0 , 1 ) \beta\in(0,1)
  11. 𝔼 t [ ] \mathbb{E}_{t}[\cdot]
  12. t t
  13. maximize { c k } k = t T 𝔼 t k = 0 T - t β k u ( c t + k ) \operatorname{maximize}\limits_{\{c_{k}\}_{k=t}^{T}}\mathbb{E}_{t}\sum_{k=0}^{% T-t}\beta^{k}u(c_{t+k})
  14. A t + 1 = ( 1 + r ) ( A t + y t - c t ) . A_{t+1}=(1+r)(A_{t}+y_{t}-c_{t}).
  15. ( 1 + r ) β = 1 (1+r)\beta=1
  16. c t = 𝔼 t [ c t + 1 ] . c_{t}=\mathbb{E}_{t}[c_{t+1}].
  17. T - t T-t
  18. A T + 1 = 0 A_{T+1}=0
  19. c t = r ( 1 + r ) - ( 1 + r ) - ( T - t ) [ A t + k = 0 T - t ( 1 1 + r ) k 𝔼 t [ y t + k ] ] . c_{t}=\frac{r}{(1+r)-(1+r)^{-(T-t)}}\left[A_{t}+\sum_{k=0}^{T-t}\left(\frac{1}% {1+r}\right)^{k}\mathbb{E}_{t}[y_{t+k}]\right].
  20. lim t ( 1 1 + r ) t A t = 0. \lim_{t\to\infty}\left(\frac{1}{1+r}\right)^{t}A_{t}=0.
  21. c t = r 1 + r [ A t + k = 0 ( 1 1 + r ) k 𝔼 t [ y t + k ] ] . c_{t}=\frac{r}{1+r}\left[A_{t}+\sum_{k=0}^{\infty}\left(\frac{1}{1+r}\right)^{% k}\mathbb{E}_{t}[y_{t+k}]\right].
  22. A t A_{t}
  23. y t y_{t}
  24. r r

Permeation.html

  1. . J = - D ϕ x . \bigg.J=-D\frac{\partial\phi}{\partial x}\bigg.
  2. . J = - D ( C 2 - C 1 ) δ . \bigg.J=-D\frac{(C_{2}-C_{1})}{\delta}\bigg.
  3. J J
  4. D \,D
  5. C C
  6. δ \,\delta
  7. S S
  8. p p
  9. C C
  10. C = S p C=Sp
  11. . J = - D S ( p 2 - p 1 ) δ . \bigg.J=-D\frac{S(p_{2}-p_{1})}{\delta}\bigg.
  12. P P
  13. P = S D P=SD
  14. . J = - P ( p 2 - p 1 ) δ . \bigg.J=-\frac{P(p_{2}-p_{1})}{\delta}\bigg.
  15. . J = - D ( S 1 - S 2 ) δ . \bigg.J=-D\frac{(S_{1}-S_{2})}{\delta}\bigg.
  16. K K
  17. S = K p N S={K\sqrt{p_{N}}}
  18. . J = - D K ( p 1 - p 2 ) δ . \bigg.J=-D\frac{K(\sqrt{p_{1}}-\sqrt{p_{2}})}{\delta}\bigg.
  19. P P
  20. P = K D P=KD
  21. . J = - P ( p 1 - p 2 ) δ . \bigg.J=-\frac{P(\sqrt{p_{1}}-\sqrt{p_{2}})}{\delta}\bigg.

Perplexity.html

  1. 2 H ( p ) = 2 - x p ( x ) log 2 p ( x ) 2^{H(p)}=2^{-\sum_{x}p(x)\log_{2}p(x)}
  2. b - 1 N i = 1 N log b q ( x i ) b^{-\frac{1}{N}\sum_{i=1}^{N}\log_{b}q(x_{i})}
  3. b b
  4. H ( p ~ , q ) = - x p ~ ( x ) log 2 q ( x ) H(\tilde{p},q)=-\sum_{x}\tilde{p}(x)\log_{2}q(x)
  5. p ~ \tilde{p}
  6. p ~ ( x ) = n / N \tilde{p}(x)=n/N

Perron's_formula.html

  1. { a ( n ) } \{a(n)\}
  2. g ( s ) = n = 1 a ( n ) n s g(s)=\sum_{n=1}^{\infty}\frac{a(n)}{n^{s}}
  3. ( s ) > σ \Re(s)>\sigma
  4. A ( x ) = n x a ( n ) = 1 2 π i c - i c + i g ( z ) x z z d z . A(x)={\sum_{n\leq x}}^{\prime}a(n)=\frac{1}{2\pi i}\int_{c-i\infty}^{c+i\infty% }g(z)\frac{x^{z}}{z}dz.\;
  5. g ( s ) = n = 1 a ( n ) n s = s 0 A ( x ) x - ( s + 1 ) d x . g(s)=\sum_{n=1}^{\infty}\frac{a(n)}{n^{s}}=s\int_{0}^{\infty}A(x)x^{-(s+1)}dx.
  6. x = e t . x=e^{t}.
  7. ζ ( s ) = s 1 x x s + 1 d x \zeta(s)=s\int_{1}^{\infty}\frac{\lfloor x\rfloor}{x^{s+1}}\,dx
  8. L ( s , χ ) = s 1 A ( x ) x s + 1 d x L(s,\chi)=s\int_{1}^{\infty}\frac{A(x)}{x^{s+1}}\,dx
  9. A ( x ) = n x χ ( n ) A(x)=\sum_{n\leq x}\chi(n)
  10. χ ( n ) \chi(n)

Persistence_length.html

  1. cos θ = e - ( L / P ) \langle\cos{\theta}\rangle=e^{-(L/P)}\,
  2. B s B_{s}
  3. P = B s k B T P=\frac{B_{s}}{k_{B}T}\,
  4. B s = E I B_{s}=EI\,
  5. I = π a 4 4 I=\frac{\pi a^{4}}{4}\,
  6. 10 18 10^{18}
  7. 10 18 10^{18}

Petersson_inner_product.html

  1. 𝕄 k \mathbb{M}_{k}
  2. k k
  3. 𝕊 k \mathbb{S}_{k}
  4. , : 𝕄 k × 𝕊 k \langle\cdot,\cdot\rangle:\mathbb{M}_{k}\times\mathbb{S}_{k}\rightarrow\mathbb% {C}
  5. f , g := F f ( τ ) g ( τ ) ¯ ( Im τ ) k d ν ( τ ) \langle f,g\rangle:=\int_{\mathrm{F}}f(\tau)\overline{g(\tau)}(\operatorname{% Im}\tau)^{k}d\nu(\tau)
  6. F = { τ H : | Re τ | 1 2 , | τ | 1 } \mathrm{F}=\left\{\tau\in\mathrm{H}:\left|\operatorname{Re}\tau\right|\leq% \frac{1}{2},\left|\tau\right|\geq 1\right\}
  7. Γ \Gamma
  8. τ = x + i y \tau=x+iy
  9. d ν ( τ ) = y - 2 d x d y d\nu(\tau)=y^{-2}dxdy
  10. T n T_{n}
  11. f , g f,g
  12. Γ 0 \Gamma_{0}
  13. T n f , g = f , T n g \langle T_{n}f,g\rangle=\langle f,T_{n}g\rangle
  14. Γ 0 \Gamma_{0}

Petite_size.html

  1. z - 0.25 z<=-0.25
  2. z - 0.75 z<=-0.75

Petrick's_method.html

  1. P 1 P_{1}
  2. P 2 P_{2}
  3. P 3 P_{3}
  4. P 4 P_{4}
  5. P P
  6. ( P i 0 + P i 1 + (P_{i0}+P_{i1}+
  7. \cdots
  8. + P i N ) +P_{iN})
  9. P i j P_{ij}
  10. i i
  11. P P
  12. X + X Y = X X+XY=X
  13. f ( A , B , C ) = m ( 0 , 1 , 2 , 5 , 6 , 7 ) f(A,B,C)=\sum m(0,1,2,5,6,7)\,

Pépin's_test.html

  1. F n = 2 2 n + 1 F_{n}=2^{2^{n}}+1
  2. F n F_{n}
  3. 3 ( F n - 1 ) / 2 - 1 ( mod F n ) . 3^{(F_{n}-1)/2}\equiv-1\;\;(\mathop{{\rm mod}}F_{n}).
  4. 3 ( F n - 1 ) / 2 3^{(F_{n}-1)/2}
  5. F n F_{n}
  6. 3 ( F n - 1 ) / 2 - 1 ( mod F n ) 3^{(F_{n}-1)/2}\equiv-1\;\;(\mathop{{\rm mod}}F_{n})
  7. 3 F n - 1 1 ( mod F n ) 3^{F_{n}-1}\equiv 1\;\;(\mathop{{\rm mod}}F_{n})
  8. F n F_{n}
  9. F n - 1 = 2 2 n F_{n}-1=2^{2^{n}}
  10. ( F n - 1 ) / 2 (F_{n}-1)/2
  11. F n - 1 F_{n}-1
  12. F n - 1 F_{n}-1
  13. F n F_{n}
  14. F n F_{n}
  15. F n F_{n}
  16. F n F_{n}
  17. 3 ( F n - 1 ) / 2 ( 3 F n ) ( mod F n ) 3^{(F_{n}-1)/2}\equiv\left(\frac{3}{F_{n}}\right)\;\;(\mathop{{\rm mod}}F_{n})
  18. ( 3 F n ) \left(\frac{3}{F_{n}}\right)
  19. 2 2 n 1 ( mod 3 ) 2^{2^{n}}\equiv 1\;\;(\mathop{{\rm mod}}3)
  20. F n 2 ( mod 3 ) F_{n}\equiv 2\;\;(\mathop{{\rm mod}}3)
  21. ( F n 3 ) = - 1 \left(\frac{F_{n}}{3}\right)=-1
  22. F n 1 ( mod 4 ) F_{n}\equiv 1\;\;(\mathop{{\rm mod}}4)
  23. ( 3 F n ) = - 1 \left(\frac{3}{F_{n}}\right)=-1
  24. F 33 F_{33}
  25. F 7 F_{7}
  26. F 8 F_{8}
  27. F 10 F_{10}
  28. F 13 F_{13}
  29. F 14 F_{14}
  30. F 20 F_{20}
  31. F 22 F_{22}
  32. F 24 F_{24}
  33. 2 2 n + 1 2^{2^{n}}+1

Pfund_telescope.html

  1. 2 = 1.4142 \sqrt{2}=1.4142

Phase-shift_oscillator.html

  1. f oscillation = 1 2 π R C 6 f_{\mathrm{oscillation}}=\frac{1}{2\pi RC\sqrt{6}}
  2. R fb = 29 R R_{\mathrm{fb}}=29\cdot R
  3. f oscillation = 1 2 π R 2 R 3 ( C 1 C 2 + C 1 C 3 + C 2 C 3 ) + R 1 R 3 ( C 1 C 2 + C 1 C 3 ) + R 1 R 2 C 1 C 2 f_{\mathrm{oscillation}}=\frac{1}{2\pi\sqrt{R_{2}R_{3}(C_{1}C_{2}+C_{1}C_{3}+C% _{2}C_{3})+R_{1}R_{3}(C_{1}C_{2}+C_{1}C_{3})+R_{1}R_{2}C_{1}C_{2}}}
  4. R fb = 2 ( R 1 + R 2 + R 3 ) + 2 R 1 R 3 R 2 + C 2 R 2 + C 2 R 3 + C 3 R 3 C 1 R_{\mathrm{fb}}=2(R_{1}+R_{2}+R_{3})+\frac{2R_{1}R_{3}}{R_{2}}+\frac{C_{2}R_{2% }+C_{2}R_{3}+C_{3}R_{3}}{C_{1}}
  5. + 2 C 1 R 1 + C 1 R 2 + C 3 R 3 C 2 + 2 C 1 R 1 + 2 C 2 R 1 + C 1 R 2 + C 2 R 2 + C 2 R 3 C 3 +\frac{2C_{1}R_{1}+C_{1}R_{2}+C_{3}R_{3}}{C_{2}}+\frac{2C_{1}R_{1}+2C_{2}R_{1}% +C_{1}R_{2}+C_{2}R_{2}+C_{2}R_{3}}{C_{3}}
  6. + C 1 R 1 2 + C 3 R 1 R 3 C 2 R 2 + C 2 R 1 R 3 + C 1 R 1 2 C 3 R 2 + C 1 R 1 2 + C 1 R 1 R 2 + C 2 R 1 R 2 C 3 R 3 +\frac{C_{1}R_{1}^{2}+C_{3}R_{1}R_{3}}{C_{2}R_{2}}+\frac{C_{2}R_{1}R_{3}+C_{1}% R_{1}^{2}}{C_{3}R_{2}}+\frac{C_{1}R_{1}^{2}+C_{1}R_{1}R_{2}+C_{2}R_{1}R_{2}}{C% _{3}R_{3}}

Phase-type_distribution.html

  1. Q = [ 0 𝟎 𝐒 0 S ] , {Q}=\left[\begin{matrix}0&\mathbf{0}\\ \mathbf{S}^{0}&{S}\\ \end{matrix}\right],
  2. F ( x ) = 1 - s y m b o l α exp ( S x ) 𝟏 , F(x)=1-symbol{\alpha}\exp({S}x)\mathbf{1},
  3. f ( x ) = s y m b o l α exp ( S x ) 𝐒 𝟎 , f(x)=symbol{\alpha}\exp({S}x)\mathbf{S^{0}},
  4. E [ X n ] = ( - 1 ) n n ! s y m b o l α S - n 1. E[X^{n}]=(-1)^{n}n!symbol{\alpha}{S}^{-n}\mathbf{1}.
  5. s y m b o l α = ( α 1 , α 2 , α 3 , α 4 , , α n ) symbol{\alpha}=(\alpha_{1},\alpha_{2},\alpha_{3},\alpha_{4},...,\alpha_{n})
  6. i = 1 n α i = 1 \sum_{i=1}^{n}\alpha_{i}=1
  7. S = [ - λ 1 0 0 0 0 0 - λ 2 0 0 0 0 0 - λ 3 0 0 0 0 0 - λ 4 0 0 0 0 0 - λ 5 ] . {S}=\left[\begin{matrix}-\lambda_{1}&0&0&0&0\\ 0&-\lambda_{2}&0&0&0\\ 0&0&-\lambda_{3}&0&0\\ 0&0&0&-\lambda_{4}&0\\ 0&0&0&0&-\lambda_{5}\\ \end{matrix}\right].
  8. f ( x ) = i = 1 n α i λ i e - λ i x = i = 1 n α i f X i ( x ) , f(x)=\sum_{i=1}^{n}\alpha_{i}\lambda_{i}e^{-\lambda_{i}x}=\sum_{i=1}^{n}\alpha% _{i}f_{X_{i}}(x),
  9. F ( x ) = 1 - i = 1 n α i e - λ i x = i = 1 n α i F X i ( x ) . F(x)=1-\sum_{i=1}^{n}\alpha_{i}e^{-\lambda_{i}x}=\sum_{i=1}^{n}\alpha_{i}F_{X_% {i}}(x).
  10. X i E x p ( λ i ) X_{i}\sim Exp(\lambda_{i})
  11. s y m b o l α = ( 1 , 0 , 0 , 0 , 0 ) , symbol{\alpha}=(1,0,0,0,0),
  12. S = [ - λ λ 0 0 0 0 - λ λ 0 0 0 0 - λ λ 0 0 0 0 - λ λ 0 0 0 0 - λ ] . {S}=\left[\begin{matrix}-\lambda&\lambda&0&0&0\\ 0&-\lambda&\lambda&0&0\\ 0&0&-\lambda&\lambda&0\\ 0&0&0&-\lambda&\lambda\\ 0&0&0&0&-\lambda\\ \end{matrix}\right].
  13. s y m b o l α = ( α 1 , 0 , 0 , α 2 , 0 , 0 ) , symbol{\alpha}=(\alpha_{1},0,0,\alpha_{2},0,0),
  14. S = [ - β 1 β 1 0 0 0 0 0 - β 1 β 1 0 0 0 0 0 - β 1 0 0 0 0 0 0 - β 2 β 2 0 0 0 0 0 - β 2 β 2 0 0 0 0 0 - β 2 ] . {S}=\left[\begin{matrix}-\beta_{1}&\beta_{1}&0&0&0&0\\ 0&-\beta_{1}&\beta_{1}&0&0&0\\ 0&0&-\beta_{1}&0&0&0\\ 0&0&0&-\beta_{2}&\beta_{2}&0\\ 0&0&0&0&-\beta_{2}&\beta_{2}\\ 0&0&0&0&0&-\beta_{2}\\ \end{matrix}\right].
  15. S = [ - λ 1 p 1 λ 1 0 0 0 0 - λ 2 p 2 λ 2 0 0 0 0 - λ k - 2 p k - 2 λ k - 2 0 0 0 0 - λ k - 1 p k - 1 λ k - 1 0 0 0 0 - λ k ] S=\left[\begin{matrix}-\lambda_{1}&p_{1}\lambda_{1}&0&\dots&0&0\\ 0&-\lambda_{2}&p_{2}\lambda_{2}&\ddots&0&0\\ \vdots&\ddots&\ddots&\ddots&\ddots&\vdots\\ 0&0&\ddots&-\lambda_{k-2}&p_{k-2}\lambda_{k-2}&0\\ 0&0&\dots&0&-\lambda_{k-1}&p_{k-1}\lambda_{k-1}\\ 0&0&\dots&0&0&-\lambda_{k}\end{matrix}\right]
  16. s y m b o l α = ( 1 , 0 , , 0 ) , symbol{\alpha}=(1,0,\dots,0),

Phase_boundary.html

  1. d 2 d\geq 2

Phosphoinositide_phospholipase_C.html

  1. \rightleftharpoons

Photochemical_logic_gate.html

  1. υ c \upsilon_{c}
  2. \searrow
  3. \swarrow
  4. \searrow
  5. \swarrow
  6. \rightarrow
  7. υ a \upsilon_{a}
  8. υ c 2 \upsilon_{c2}
  9. \uparrow
  10. υ c 2 \upsilon_{c2}
  11. \uparrow
  12. υ c \upsilon_{c}
  13. \rightarrow
  14. \rightarrow
  15. υ \upsilon
  16. υ \upsilon
  17. \rightarrow
  18. υ c \upsilon_{c}
  19. υ c \upsilon_{c}
  20. υ c 2 \upsilon_{c2}
  21. υ c 3 \upsilon_{c3}
  22. c 2 \rightarrow_{c2}
  23. \rightarrow
  24. υ c 2 \upsilon_{c2}

Photodissociation.html

  1. h ν h\nu
  2. ν \nu
  3. H 2 O + h ν H + O H H_{2}O+h\nu\rightarrow H+OH
  4. C H 4 + h ν C H 3 + H CH_{4}+h\nu\rightarrow CH_{3}+H

Photon_entanglement.html

  1. | 45 \left|45\right\rangle
  2. | 45 \left|45\right\rangle
  3. | 1 , V \left|1,V\right\rangle
  4. | 2 , H \left|2,H\right\rangle
  5. | Ψ = | 1 , V | 2 , H \left|\Psi\right\rangle=\left|1,V\right\rangle\left|2,H\right\rangle
  6. | Ψ = | 1 , V | 2 , H \left|\Psi\right\rangle=\left|1,V\right\rangle\left|2,H\right\rangle
  7. | 1 , V \left|1,V\right\rangle
  8. | 1 , V = 1 2 | 1 , 45 + 1 2 | 1 , 135 \left|1,V\right\rangle={1\over\sqrt{2}}\left|1,45\right\rangle+{1\over\sqrt{2}% }\left|1,135\right\rangle
  9. | Ψ \left|\Psi\right\rangle
  10. | Ψ = | 1 , V | 2 , H \left|\Psi\right\rangle=\left|1,V\right\rangle\left|2,H\right\rangle
  11. = 1 2 | 1 , 45 | 2 , H + 1 2 | 1 , 135 | 2 , H ={1\over\sqrt{2}}\left|1,45\right\rangle\left|2,H\right\rangle+{1\over\sqrt{2}% }\left|1,135\right\rangle\left|2,H\right\rangle
  12. | Ψ = | 1 , V | 2 , H \left|\Psi\right\rangle=\left|1,V\right\rangle\left|2,H\right\rangle
  13. = 1 2 | 1 , 45 | 2 , H + 1 2 | 1 , 135 | 2 , H ={1\over\sqrt{2}}\left|1,45\right\rangle\left|2,H\right\rangle+{1\over\sqrt{2}% }\left|1,135\right\rangle\left|2,H\right\rangle
  14. | 1 , 45 | 2 , H \Rightarrow\left|1,45\right\rangle\left|2,H\right\rangle
  15. | Θ = | 1 , V | 2 , V \left|\Theta\right\rangle=\left|1,V\right\rangle\left|2,V\right\rangle
  16. | Λ = | 1 , H | 2 , H \left|\Lambda\right\rangle=\left|1,H\right\rangle\left|2,H\right\rangle
  17. | Θ \left|\Theta\right\rangle
  18. | Λ \left|\Lambda\right\rangle
  19. | Ψ = 1 2 [ | Θ + | Λ ] \left|\Psi\right\rangle={1\over\sqrt{2}}[\left|\Theta\right\rangle+\left|% \Lambda\right\rangle]
  20. = 1 2 | 1 , V | 2 , V + 1 2 | 1 , H | 2 , H ={1\over\sqrt{2}}\left|1,V\right\rangle\left|2,V\right\rangle+{1\over\sqrt{2}}% \left|1,H\right\rangle\left|2,H\right\rangle
  21. | Ψ = 1 2 1 2 [ | 1 , 45 + | 1 , 135 1 2 [ | 2 , 45 + | 2 , 135 ] + 1 2 [ | 1 , 45 - | 1 , 135 ] 1 2 [ | 2 , 45 - | 2 , 135 ] \left|\Psi\right\rangle={1\over\sqrt{2}}{1\over\sqrt{2}}[\left|1,45\right% \rangle+\left|1,135\right\rangle{1\over\sqrt{2}}[\left|2,45\right\rangle+\left% |2,135\right\rangle]+{1\over\sqrt{2}}[\left|1,45\right\rangle-\left|1,135% \right\rangle]{1\over\sqrt{2}}[\left|2,45\right\rangle-\left|2,135\right\rangle]
  22. = 1 2 ( 2 ) [ | 1 , 45 | 2 , 45 + | 1 , 45 | 2 , 135 + | 1 , 135 | 2 , 45 + | 1 , 135 | 2 , 135 + | 1 , 45 | 2 , 45 - | 1 , 45 | 2 , 135 - | 1 , 135 | 2 , 45 + | 1 , 135 | 2 , 135 ={1\over 2}(\sqrt{2}){[\left|1,45\right\rangle\left|2,45\right\rangle+\left|1,% 45\right\rangle\left|2,135\right\rangle+\left|1,135\right\rangle\left|2,45% \right\rangle+\left|1,135\right\rangle\left|2,135\right\rangle+\left|1,45% \right\rangle\left|2,45\right\rangle-\left|1,45\right\rangle\left|2,135\right% \rangle-\left|1,135\right\rangle\left|2,45\right\rangle+\left|1,135\right% \rangle\left|2,135\right\rangle}
  23. = 1 2 | 1 , 45 | 2 , 45 + | 1 , 135 | 2 , 135 ={1\over\sqrt{2}}{\left|1,45\right\rangle\left|2,45\right\rangle+\left|1,135% \right\rangle\left|2,135\right\rangle}
  24. | Ψ = 1 2 | 1 , n | 2 , n + 1 2 | 1 , n | 2 , n \left|\Psi\right\rangle={1\over\sqrt{2}}\left|1,n\right\rangle\left|2,n\right% \rangle+{1\over\sqrt{2}}\left|1,n\bot\right\rangle\left|2,n\bot\right\rangle
  25. | Ψ \left|\Psi\right\rangle
  26. n n\bot
  27. | Ψ = 1 2 | 1 , n | 2 , n + 1 2 | 1 , n | 2 , n \left|\Psi\right\rangle={1\over\sqrt{2}}\left|1,n\right\rangle\left|2,n\right% \rangle+{1\over\sqrt{2}}\left|1,n\bot\right\rangle\left|2,n\bot\right\rangle
  28. | 1 , n | 2 , n \Rightarrow\left|1,n\right\rangle\left|2,n\right\rangle

Photosynthetic_reaction_centre.html

  1. l i g h t 2 Q + 2 H 2 O O 2 + 2 Q H 2 \begin{matrix}&light&\\ 2Q+2H_{2}O&\Longrightarrow&O_{2}+2QH_{2}\end{matrix}
  2. l i g h t P c ( C u + ) + F d o x P c ( C u 2 + ) + F d r e d \begin{matrix}&light&\\ Pc(Cu^{+})+Fd_{ox}&\Longrightarrow&Pc(Cu^{2+})+Fd_{red}\end{matrix}

Pi_system.html

  1. σ ( E 1 , E 2 , ) \sigma(E_{1},E_{2},\ldots)
  2. n σ ( E 1 , , E n ) \bigcup_{n}\sigma(E_{1},\ldots,E_{n})
  3. \mathbb{R}
  4. ( - , a ] (-\infty,a]
  5. ( a , b ] (a,b]
  6. Σ \mathcal{I}_{\Sigma}
  7. f : Ω f\colon\Omega\rightarrow\mathbb{R}
  8. f = { f - 1 ( ( - , x ] ) : x } \mathcal{I}_{f}=\left\{f^{-1}\left(\left(-\infty,x\right]\right)\colon x\in% \mathbb{R}\right\}
  9. { f - 1 ( ( a , b ] ) : a , b , a < b } { } \left\{f^{-1}\left(\left(a,b\right]\right)\colon a,b\in\mathbb{R},a<b\right\}% \cup\{\emptyset\}
  10. f f
  11. { A 1 × A 2 : A 1 P 1 , A 2 P 2 } \{A_{1}\times A_{2}:A_{1}\in P_{1},A_{2}\in P_{2}\}
  12. Ω D \Omega\in D
  13. A D A\in D
  14. A c D A^{c}\in D
  15. A 1 , A 2 , A 3 , A_{1},A_{2},A_{3},\dots
  16. D D
  17. n = 1 A n D \cup_{n=1}^{\infty}A_{n}\in D
  18. D D
  19. D \mathcal{I}\subseteq D
  20. D D
  21. σ ( ) \sigma(\mathcal{I})
  22. \mathcal{I}
  23. D D
  24. σ ( ) D \sigma(\mathcal{I})\subset D
  25. D = { A σ ( I ) : μ 1 ( A ) = μ 2 ( A ) } . D=\left\{A\in\sigma(I)\colon\mu_{1}(A)=\mu_{2}(A)\right\}.
  26. X : ( Ω , , ) X\colon(\Omega,\mathcal{F},\mathbb{P})\rightarrow\mathbb{R}
  27. F X ( a ) = [ X a ] , a F_{X}(a)=\mathbb{P}\left[X\leq a\right],\qquad a\in\mathbb{R}
  28. X ( B ) = [ X - 1 ( B ) ] , B ( ) \mathcal{L}_{X}(B)=\mathbb{P}\left[X^{-1}(B)\right],\qquad B\in\mathcal{B}(% \mathbb{R})
  29. ( ) \mathcal{B}(\mathbb{R})
  30. X : ( Ω , , ) X\colon(\Omega,\mathcal{F},\mathbb{P})
  31. Y : ( Ω ~ , ~ , ~ ) Y\colon(\tilde{\Omega},\tilde{\mathcal{F}},\tilde{\mathbb{P}})\rightarrow% \mathbb{R}
  32. X = 𝒟 Y X\stackrel{\mathcal{D}}{=}Y
  33. X \mathcal{L}_{X}
  34. Y \mathcal{L}_{Y}
  35. { ( - , a ] : a } \left\{(-\infty,a]\colon a\in\mathbb{R}\right\}
  36. ( ) \mathcal{B}(\mathbb{R})
  37. X = Y \mathcal{L}_{X}=\mathcal{L}_{Y}
  38. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  39. X \mathcal{I}_{X}
  40. Y \mathcal{I}_{Y}
  41. F X , Y ( a , b ) = [ X a , Y b ] = [ X - 1 ( ( - , a ] ) Y - 1 ( ( - , b ] ) ] , a , b F_{X,Y}(a,b)=\mathbb{P}\left[X\leq a,Y\leq b\right]=\mathbb{P}\left[X^{-1}((-% \infty,a])\cap Y^{-1}((-\infty,b])\right],\qquad a,b\in\mathbb{R}
  42. A = X - 1 ( ( - , a ] ) X A=X^{-1}((-\infty,a])\in\mathcal{I}_{X}
  43. B = Y - 1 ( ( - , b ] ) Y B=Y^{-1}((-\infty,b])\in\mathcal{I}_{Y}
  44. X , Y = { A B : A X , B Y } \mathcal{I}_{X,Y}=\{A\cap B:A\in\mathcal{I}_{X},\,B\in\mathcal{I}_{Y}\}
  45. ( X t ) t T , ( Y t ) t T (X_{t})_{t\in T},(Y_{t})_{t\in T}
  46. t 1 , , t n T , n t_{1},\ldots,t_{n}\in T,\,n\in\mathbb{N}
  47. ( X t 1 , , X t n ) = 𝒟 ( Y t 1 , , Y t n ) (X_{t_{1}},\ldots,X_{t_{n}})\stackrel{\mathcal{D}}{=}(Y_{t_{1}},\ldots,Y_{t_{n% }})
  48. ( Ω , , ) (\Omega,\mathcal{F},\mathbb{P})
  49. X , Y \mathcal{I}_{X},\mathcal{I}_{Y}
  50. [ A B ] = [ A ] [ B ] , A X , B Y , \mathbb{P}\left[A\cap B\right]=\mathbb{P}\left[A\right]\mathbb{P}\left[B\right% ],\qquad\forall A\in\mathcal{I}_{X},\,B\in\mathcal{I}_{Y},
  51. X , Y \mathcal{I}_{X},\mathcal{I}_{Y}
  52. Z = ( Z 1 , Z 2 ) Z=(Z_{1},Z_{2})
  53. Z 1 , Z 2 𝒩 ( 0 , 1 ) Z_{1},Z_{2}\sim\mathcal{N}(0,1)
  54. R = Z 1 2 + Z 2 2 , Θ = tan - 1 ( Z 2 / Z 1 ) R=\sqrt{Z_{1}^{2}+Z_{2}^{2}},\qquad\Theta=\tan^{-1}(Z_{2}/Z_{1})
  55. R R
  56. Θ \Theta
  57. R , Θ \mathcal{I}_{R},\mathcal{I}_{\Theta}
  58. [ R ρ , Θ θ ] = [ R ρ ] [ Θ θ ] ρ [ 0 , ) , θ [ 0 , 2 π ] . \mathbb{P}[R\leq\rho,\Theta\leq\theta]=\mathbb{P}[R\leq\rho]\mathbb{P}[\Theta% \leq\theta]\quad\forall\rho\in[0,\infty),\,\theta\in[0,2\pi].
  59. ρ [ 0 , ) , θ [ 0 , 2 π ] \rho\in[0,\infty),\,\theta\in[0,2\pi]
  60. Z Z
  61. [ R ρ , Θ θ ] \displaystyle\mathbb{P}[R\leq\rho,\Theta\leq\theta]

Piecewise_syndetic_set.html

  1. S \sub S\sub\mathbb{N}
  2. \mathbb{N}
  3. \mathbb{N}
  4. x x\in\mathbb{N}
  5. x + F n G ( S - n ) x+F\subset\bigcup_{n\in G}(S-n)
  6. S - n = { m : m + n S } S-n=\{m\in\mathbb{N}:m+n\in S\}
  7. \mathbb{N}
  8. β \beta\mathbb{N}
  9. S S
  10. S = C 1 C 2 C n S=C_{1}\cup C_{2}\cup...\cup C_{n}
  11. i n i\leq n
  12. C i C_{i}
  13. \mathbb{N}
  14. A + B = { a + b : a A , b B } A+B=\{a+b:a\in A,b\in B\}

Piezoelectric_sensor.html

  1. C x C_{x}
  2. a , b , c a,b,c
  3. C x = d x y F y b / a C_{x}=d_{xy}F_{y}b/a~{}
  4. a a
  5. b b
  6. d d
  7. C x = d x x F x n C_{x}=d_{xx}F_{x}n~{}
  8. d x x d_{xx}
  9. F x F_{x}
  10. n n
  11. n n
  12. C x = 2 d x x F x n C_{x}=2d_{xx}F_{x}n
  13. F = m a F=ma

Piezoresistive_effect.html

  1. ρ σ = ( ρ ρ ) ε \rho_{\sigma}=\frac{\left(\frac{\partial\rho}{\rho}\right)}{\varepsilon}
  2. R = ρ A R=\rho\frac{\ell}{A}\,
  3. \ell
  4. V r = R 0 I [ 1 + π L σ x x + π T ( σ y y + σ z z ) ] \ V_{r}=R_{0}I[1+\pi_{L}\sigma_{xx}+\pi_{T}(\sigma_{yy}+\sigma_{zz})]
  5. R 0 R_{0}
  6. π T \pi_{T}
  7. π L \pi_{L}
  8. σ i j \sigma_{ij}

Pion_decay_constant.html

  1. f π ± = 130.41 ± 0.03 ± 0.20 MeV f_{\pi^{\pm}}=130.41\pm 0.03\pm 0.20~{}\mbox{MeV}~{}
  2. f π 0 = 130 ± 5 MeV f_{\pi^{0}}=130\pm 5~{}\mbox{MeV}~{}
  3. 2 \scriptstyle\sqrt{2}

Pitch_angle_(particle_motion).html

  1. P \displaystyle P
  2. Ω \Omega
  3. α 0 \alpha_{0}
  4. B 0 B_{0}
  5. B m B_{m}
  6. μ \mu
  7. μ = 1 2 m v 0 2 B 0 = 1 2 m v 2 sin 2 ( α 0 ) B 0 = 1 2 m v 2 B m \mu=\frac{1}{2}\frac{mv^{2}_{\perp 0}}{B_{0}}=\frac{1}{2}\frac{mv^{2}\sin^{2}% \left(\alpha_{0}\right)}{B_{0}}=\frac{1}{2}\frac{mv^{2}}{B_{m}}
  8. sin 2 ( α 0 ) = B 0 B m \sin^{2}\left(\alpha_{0}\right)=\frac{B_{0}}{B_{m}}