wpmath0000001_11

History_of_mathematics.html

  1. 10 / 71 {10}/{71}
  2. π \pi

Holographic_principle.html

  1. d S = d M T . {\rm d}S=\frac{{\rm d}M}{T}.

Holography.html

  1. | U O + U R | 2 = U O U R * + | U R | 2 + | U O | 2 + U O * U R |U_{O}+U_{R}|^{2}=U_{O}U_{R}^{*}+|U_{R}|^{2}+|U_{O}|^{2}+U_{O}^{*}U_{R}
  2. T = k U O U R * + k | U R | 2 + k | U O | 2 + k U O * U R T=kU_{O}U_{R}^{*}+k|U_{R}|^{2}+k|U_{O}|^{2}+kU_{O}^{*}U_{R}
  3. U H = T U R = k U O | U R | 2 + k | U R | 2 U R + k | U O | 2 U R + k U O * U R 2 U_{H}=TU_{R}=kU_{O}|U_{R}|^{2}+k|U_{R}|^{2}U_{R}+k|U_{O}|^{2}U_{R}+kU_{O}^{*}U% _{R}^{2}

Holomorphic_function.html

  1. f ( z 0 ) = lim z z 0 f ( z ) - f ( z 0 ) z - z 0 . f^{\prime}(z_{0})=\lim_{z\to z_{0}}{f(z)-f(z_{0})\over z-z_{0}}.
  2. u x = v y and u y = - v x \frac{\partial u}{\partial x}=\frac{\partial v}{\partial y}\qquad\mbox{and}~{}% \qquad\frac{\partial u}{\partial y}=-\frac{\partial v}{\partial x}\,
  3. f z ¯ = 0 , \frac{\partial f}{\partial\overline{z}}=0,
  4. γ f ( z ) d z = 0. \oint_{\gamma}f(z)\,dz=0.
  5. f ( a ) = 1 2 π i γ f ( z ) z - a d z f(a)=\frac{1}{2\pi i}\oint_{\gamma}\frac{f(z)}{z-a}\,dz
  6. f ( a ) = 1 2 π i γ f ( z ) ( z - a ) 2 d z , f^{\prime}(a)={1\over 2\pi i}\oint_{\gamma}{f(z)\over(z-a)^{2}}\,dz,
  7. f ( a ) = lim γ a i 2 𝒜 ( γ ) γ f ( z ) d z ¯ , f^{\prime}(a)=\lim\limits_{\gamma\to a}\frac{i}{2\mathcal{A}(\gamma)}\oint_{% \gamma}f(z)d\bar{z},
  8. 0 = d 2 f = d ( f d z ) = d f d z \textstyle 0=d^{2}f=d(f^{\prime}dz)=df^{\prime}\wedge dz
  9. F γ ( z ) = F 0 + γ f d z \textstyle F_{\gamma}(z)=F_{0}+\int_{\gamma}fdz
  10. z = e 1 2 log z \sqrt{z}=e^{\frac{1}{2}\log z}
  11. z ¯ \overline{z}

Homeomorphism.html

  1. t a n tan
  2. x x - 1 x\mapsto x^{-1}
  3. x G x\in G
  4. y x y y\mapsto xy
  5. y y x y\mapsto yx
  6. y x y x - 1 y\mapsto xyx^{-1}
  7. 2 \mathbb{R}^{2}
  8. S 1 S^{1}
  9. D 2 D^{2}

Homomorphism.html

  1. f ( r ) = ( r 0 0 r ) f(r)=\begin{pmatrix}r&0\\ 0&r\end{pmatrix}
  2. f ( r + s ) = ( r + s 0 0 r + s ) = ( r 0 0 r ) + ( s 0 0 s ) = f ( r ) + f ( s ) f(r+s)=\begin{pmatrix}r+s&0\\ 0&r+s\end{pmatrix}=\begin{pmatrix}r&0\\ 0&r\end{pmatrix}+\begin{pmatrix}s&0\\ 0&s\end{pmatrix}=f(r)+f(s)
  3. f ( r s ) = ( r s 0 0 r s ) = ( r 0 0 r ) ( s 0 0 s ) = f ( r ) f ( s ) . f(rs)=\begin{pmatrix}rs&0\\ 0&rs\end{pmatrix}=\begin{pmatrix}r&0\\ 0&r\end{pmatrix}\begin{pmatrix}s&0\\ 0&s\end{pmatrix}=f(r)\,f(s).

Horizon.html

  1. d 3.57 h , d\approx 3.57\sqrt{h}\,,
  2. d 1.22 h . d\approx 1.22\sqrt{h}\,.
  3. OC 2 = OA × OB . \mathrm{OC}^{2}=\mathrm{OA}\times\mathrm{OB}\,.
  4. d 2 = h ( D + h ) d^{2}=h(D+h)\,\!
  5. d = h ( D + h ) = h ( 2 R + h ) , d=\sqrt{h(D+h)}=\sqrt{h(2R+h)}\,,
  6. ( R + h ) 2 = R 2 + d 2 (R+h)^{2}=R^{2}+d^{2}\,\!
  7. R 2 + 2 R h + h 2 = R 2 + d 2 R^{2}+2Rh+h^{2}=R^{2}+d^{2}\,\!
  8. d = h ( 2 R + h ) . d=\sqrt{h(2R+h)}\,.
  9. s = R γ ; s=R\gamma\,;
  10. cos γ = cos s R = R R + h . \cos\gamma=\cos\frac{s}{R}=\frac{R}{R+h}\,.
  11. s = R cos - 1 R R + h . s=R\cos^{-1}\frac{R}{R+h}\,.
  12. tan γ = d R ; \tan\gamma=\frac{d}{R}\,;
  13. s = R tan - 1 d R . s=R\tan^{-1}\frac{d}{R}\,.
  14. d = 2 R h . d=\sqrt{2Rh}\,.
  15. d 2 6371 h / 1000 3.570 h , d\approx\sqrt{2\cdot 6371\cdot{h/1000}}\approx 3.570\sqrt{h}\,,
  16. d 2 3963 h / 5280 1.22 h d\approx\sqrt{2\cdot 3963\cdot{h/5280}}\approx 1.22\sqrt{h}
  17. d h d\approx\sqrt{h}
  18. d = 2 R h + h 2 , d=\sqrt{2Rh+h^{2}}\,,
  19. D BL < 3.57 ( h B + h L ) , D_{\mathrm{BL}}<3.57\,(\sqrt{h_{\mathrm{B}}}+\sqrt{h_{\mathrm{L}}})\,,
  20. D 3.57 ( 2 + 70 ) D\approx 3.57(\sqrt{2}+\sqrt{70})
  21. 3.57 10 3.57\sqrt{10}
  22. h ( 8.7 3.57 ) 2 h\approx\left(\frac{8.7}{3.57}\right)^{2}
  23. d = R E ( ψ + δ ) , d={{R}_{\,\text{E}}}\left(\psi+\delta\right)\,,
  24. cos ψ = R E μ 0 ( R E + h ) μ , \cos\psi=\frac{{R}_{\,\text{E}}{\mu}_{0}}{\left({{R}_{\,\text{E}}}+h\right)\mu% }\,,
  25. δ = - 0 h tan ϕ d μ μ , \delta=-\int_{0}^{h}{\tan\phi\frac{\,\text{d}\mu}{\mu}}\,,
  26. ϕ \phi\,\!
  27. ϕ \phi\,\!
  28. ϕ = 90 - ψ . \phi=90{}^{\circ}-\psi\,.
  29. d = 2 R h . d=\sqrt{2R^{\prime}h}\,.
  30. d 3.86 h ; d\approx 3.86\sqrt{h}\,;
  31. d 1.32 h . d\approx 1.32\sqrt{h}\,.
  32. κ \kappa
  33. κ = ( 1 + h R ) 2 - 1 . \kappa=\sqrt{\left(1+\frac{h}{R}\right)^{2}-1}\ .

Horizontal_line_test.html

  1. f : f\colon\mathbb{R}\to\mathbb{R}
  2. y = c y=c
  3. y = c y=c
  4. f : X Y f\colon X\to Y
  5. X × Y X\times Y
  6. X × Y X\times Y
  7. { ( x , y 0 ) X × Y : y 0 is constant } = X × { y 0 } \{(x,y_{0})\in X\times Y:y_{0}\,\text{ is constant}\}=X\times\{y_{0}\}

Horner's_method.html

  1. p ( x ) = i = 0 n a i x i = a 0 + a 1 x + a 2 x 2 + a 3 x 3 + + a n x n , p(x)=\sum_{i=0}^{n}a_{i}x^{i}=a_{0}+a_{1}x+a_{2}x^{2}+a_{3}x^{3}+\cdots+a_{n}x% ^{n},
  2. a 0 , , a n a_{0},\ldots,a_{n}
  3. x x
  4. x 0 x_{0}
  5. b n \displaystyle b_{n}
  6. b 0 b_{0}
  7. p ( x 0 ) p(x_{0})
  8. p ( x ) = a 0 + x ( a 1 + x ( a 2 + + x ( a n - 1 + a n x ) ) ) . p(x)=a_{0}+x(a_{1}+x(a_{2}+\cdots+x(a_{n-1}+a_{n}x)\cdots)).\,
  9. b i b_{i}
  10. p ( x 0 ) \displaystyle p(x_{0})
  11. f ( x ) = 2 x 3 - 6 x 2 + 2 x - 1 f(x)=2x^{3}-6x^{2}+2x-1\,
  12. x = 3. x=3.\;
  13. f ( x ) f(x)
  14. x - 3 x-3
  15. f ( 3 ) f(3)
  16. f ( 3 ) = 5 f(3)=5
  17. a 3 = 2 , a 2 = - 6 , a 1 = 2 , a 0 = - 1 a_{3}=2,a_{2}=-6,a_{1}=2,a_{0}=-1
  18. b 3 = 2 , b 2 = 0 , b 1 = 2 , b 0 = 5 b_{3}=2,b_{2}=0,b_{1}=2,b_{0}=5
  19. f ( x ) f(x)
  20. x - 3 x-3
  21. x 3 - 6 x 2 + 11 x - 6 x^{3}-6x^{2}+11x-6\,
  22. x - 2 x-2\,
  23. x 2 - 4 x + 3 x^{2}-4x+3\,
  24. f 1 ( x ) = 4 x 4 - 6 x 3 + 3 x - 5 f_{1}(x)=4x^{4}-6x^{3}+3x-5\,
  25. f 2 ( x ) = 2 x - 1 f_{2}(x)=2x-1\,
  26. f 1 ( x ) f_{1}(x)\,
  27. f 2 ( x ) f_{2}\,(x)
  28. f 1 ( x ) f 2 ( x ) = 2 x 3 - 2 x 2 - x + 1 - 4 2 x - 1 . \frac{f_{1}(x)}{f_{2}(x)}=2x^{3}-2x^{2}-x+1-\frac{4}{2x-1}.
  29. ( 0.15625 ) m \displaystyle(0.15625)m
  30. d 3 d 2 d 1 d 0 d_{3}d_{2}d_{1}d_{0}
  31. ( d 3 2 3 + d 2 2 2 + d 1 2 1 + d 0 2 0 ) m = d 3 2 3 m + d 2 2 2 m + d 1 2 1 m + d 0 2 0 m (d_{3}2^{3}+d_{2}2^{2}+d_{1}2^{1}+d_{0}2^{0})m=d_{3}2^{3}m+d_{2}2^{2}m+d_{1}2^% {1}m+d_{0}2^{0}m
  32. = d 0 ( m + 2 d 1 d 0 ( m + 2 d 2 d 1 ( m + 2 d 3 d 2 ( m ) ) ) ) . =d_{0}(m+2\frac{d_{1}}{d_{0}}(m+2\frac{d_{2}}{d_{1}}(m+2\frac{d_{3}}{d_{2}}(m)% ))).
  33. = d 0 ( m + 2 d 1 ( m + 2 d 2 ( m + 2 d 3 ( m ) ) ) ) , =d_{0}(m+2{d_{1}}(m+2{d_{2}}(m+2{d_{3}}(m)))),
  34. = d 3 ( m + 2 - 1 d 2 ( m + 2 - 1 d 1 ( m + d 0 ( m ) ) ) ) . =d_{3}(m+2^{-1}{d_{2}}(m+2^{-1}{d_{1}}(m+{d_{0}}(m)))).
  35. p n ( x ) p_{n}(x)
  36. n n
  37. z n < z n - 1 < < z 1 z_{n}<z_{n-1}<\cdots<z_{1}
  38. x 0 x_{0}
  39. x 0 > z 1 x_{0}>z_{1}
  40. z 1 z_{1}
  41. p n ( x ) p_{n}(x)
  42. x 0 x_{0}
  43. ( x - z 1 ) (x-z_{1})
  44. p n - 1 p_{n-1}
  45. p n - 1 p_{n-1}
  46. z 1 z_{1}
  47. p 6 ( x ) = ( x - 3 ) ( x + 3 ) ( x + 5 ) ( x + 8 ) ( x - 2 ) ( x - 7 ) p_{6}(x)=(x-3)(x+3)(x+5)(x+8)(x-2)(x-7)
  48. p 6 ( x ) = x 6 + 4 x 5 - 72 x 4 - 214 x 3 + 1127 x 2 + 1602 x - 5040. p_{6}(x)=x^{6}+4x^{5}-72x^{4}-214x^{3}+1127x^{2}+1602x-5040.
  49. p ( x ) p(x)
  50. ( x - 7 ) (x-7)
  51. p 5 ( x ) = x 5 + 11 x 4 + 5 x 3 - 179 x 2 - 126 x + 720 p_{5}(x)=x^{5}+11x^{4}+5x^{3}-179x^{2}-126x+720\,
  52. ( x - 3 ) (x-3)
  53. p 4 ( x ) = x 4 + 14 x 3 + 47 x 2 - 38 x - 240 p_{4}(x)=x^{4}+14x^{3}+47x^{2}-38x-240\,
  54. p 3 ( x ) = x 3 + 16 x 2 + 79 x + 120 p_{3}(x)=x^{3}+16x^{2}+79x+120\,
  55. p 2 ( x ) = x 2 + 13 x + 40 p_{2}(x)=x^{2}+13x+40\,
  56. p 2 ( x ) p_{2}(x)
  57. n \sqrt{n}
  58. n / 2 + 2 {\scriptstyle{\left\lfloor n/2\right\rfloor+2}}

Horsepower.html

  1. P = W t = F d t = 180 lbf 2.4 2 π 12 ft 1 min = 32 , 572 ft lbf min . P=\frac{W}{t}=\frac{F\cdot d}{t}=\frac{180\,\mathrm{lbf}\cdot 2.4\cdot 2\,\pi% \cdot 12\,\mathrm{ft}}{1\,\mathrm{min}}=32,572\cdot\frac{\mathrm{ft}\cdot% \mathrm{lbf}}{\mathrm{min}}.
  2. T T
  3. ( N ) (N)
  4. P ( hp ) = T ft lbf × N rpm 5252 P\mathrm{(hp)}=\frac{T\,\mathrm{ft}\cdot\mathrm{lbf}\times N\,\mathrm{rpm}}{52% 52}
  5. T T
  6. P ( hp ) = T in lbf × N rpm 63 , 025 P\mathrm{(hp)}=\frac{T\,\mathrm{in}\cdot\mathrm{lbf}\times N\,\mathrm{rpm}}{63% {,}025}
  7. 33 , 000 ft lbf min 12 in ft 2 π rad 63 , 025 33{,}000\,\frac{\mathrm{ft}\cdot\mathrm{lbf}}{\mathrm{min}}\cdot\frac{12\,% \frac{\mathrm{in}}{\mathrm{ft}}}{2\,\pi\mathrm{rad}}\approx 63{,}025
  8. P = τ ω P=\tau\cdot\omega
  9. P P
  10. τ \tau
  11. ω \omega
  12. ( P 40 ) 1.6 + U 45 \scriptstyle\left(\tfrac{P}{40}\right)^{1.6}+\tfrac{U}{45}
  13. F F
  14. v v
  15. P P
  16. P / hp = ( F / lbf ) ( v / mph ) 375 P/{\rm hp}={(F/{\rm lbf})(v/{\rm mph})\over 375}
  17. P / hp = 2025 × 5 375 = 27 P/{\rm hp}={{2025\times 5}\over 375}=27
  18. P = F v P=Fv
  19. P / hp = 4000 × 400 375 = 4266.7 P/{\rm hp}={{4000\times 400}\over 375}=4266.7
  20. R A C h . p . = ( D 2 * n ) / 2.5 RACh.p.=(D^{2}*n)/2.5\,

Hour_angle.html

  1. LHA object = LST - α object \,\text{LHA}_{\,\text{object}}={\,\text{LST}}-\alpha_{\,\text{object}}
  2. LHA object = GST - λ observer - α object \,\text{LHA}_{\,\text{object}}={\,\text{GST}}-\lambda_{\,\text{observer}}-% \alpha_{\,\text{object}}
  3. α object \alpha_{\,\text{object}}
  4. λ observer \lambda_{\,\text{observer}}

Hubble's_law.html

  1. v = H 0 D v=H_{0}\,D
  2. v v
  3. H H
  4. D D
  5. r H S = c H 0 . r_{HS}=\frac{c}{H_{0}}\ .
  6. v r s c z , v_{rs}\equiv cz\ ,
  7. z = λ o λ e - 1 = 1 + v / c 1 - v / c - 1 v c . z=\frac{\lambda_{o}}{\lambda_{e}}-1=\sqrt{\frac{1+v/c}{1-v/c}}-1\approx\frac{v% }{c}\ .
  8. D ( t ) D ( t 0 ) = R ( t ) R ( t 0 ) , \frac{D(t)}{D(t_{0})}=\frac{R(t)}{R(t_{0})}\ ,
  9. z = R ( t 0 ) R ( t e ) - 1 . z=\frac{R(t_{0})}{R(t_{e})}-1\ .
  10. v r = d t D = d t R R D . v_{r}=d_{t}D=\frac{d_{t}R}{R}D\ .
  11. H d t R R , H\equiv\frac{d_{t}R}{R}\ ,
  12. v r = H D . v_{r}=HD\ .
  13. z = R ( t 0 ) R ( t e ) - 1 R ( t 0 ) R ( t 0 ) ( 1 + ( t e - t 0 ) H ( t 0 ) ) - 1 ( t 0 - t e ) H ( t 0 ) , z=\frac{R(t_{0})}{R(t_{e})}-1\approx\frac{R(t_{0})}{R(t_{0})\left(1+(t_{e}-t_{% 0})H(t_{0})\right)}-1\approx(t_{0}-t_{e})H(t_{0})\ ,
  14. z ( t 0 - t e ) H ( t 0 ) D c H ( t 0 ) , z\approx(t_{0}-t_{e})H(t_{0})\approx\frac{D}{c}H(t_{0})\ ,
  15. c z D H ( t 0 ) = v r . cz\approx DH(t_{0})=v_{r}\ .
  16. v = z c v=zc
  17. q q
  18. q = - ( 1 + H ˙ H 2 ) . q=-\left(1+\frac{\dot{H}}{H^{2}}\right).
  19. q q
  20. H 0 H_{0}
  21. q q
  22. H 2 ( a ˙ a ) 2 = 8 π G 3 ρ - k c 2 a 2 + Λ c 2 3 , H^{2}\equiv\left(\frac{\dot{a}}{a}\right)^{2}=\frac{8\pi G}{3}\rho-\frac{kc^{2% }}{a^{2}}+\frac{\Lambda c^{2}}{3},
  23. H H
  24. a a
  25. k k
  26. Λ \Lambda
  27. ρ \rho
  28. ρ = ρ m ( a ) = ρ m 0 a 3 , \rho=\rho_{m}(a)=\frac{\rho_{m_{0}}}{a^{3}},
  29. ρ m 0 \rho_{m_{0}}
  30. Ω m \Omega_{m}
  31. ρ c = 3 H 2 8 π G ; \rho_{c}=\frac{3H^{2}}{8\pi G};
  32. Ω m ρ m 0 ρ c = 8 π G 3 H 0 2 ρ m 0 ; \Omega_{m}\equiv\frac{\rho_{m_{0}}}{\rho_{c}}=\frac{8\pi G}{3H_{0}^{2}}\rho_{m% _{0}};
  33. ρ = ρ c Ω m / a 3 . \rho=\rho_{c}\Omega_{m}/a^{3}.
  34. Ω k - k c 2 ( a 0 H 0 ) 2 \Omega_{k}\equiv\frac{-kc^{2}}{(a_{0}H_{0})^{2}}
  35. Ω Λ Λ c 2 3 H 0 2 , \Omega_{\Lambda}\equiv\frac{\Lambda c^{2}}{3H_{0}^{2}},
  36. a 0 = 1 a_{0}=1
  37. a a
  38. a = 1 / ( 1 + z ) a=1/(1+z)
  39. H 2 ( z ) = H 0 2 ( Ω M ( 1 + z ) 3 + Ω k ( 1 + z ) 2 + Ω Λ ) . H^{2}(z)=H_{0}^{2}\left(\Omega_{M}(1+z)^{3}+\Omega_{k}(1+z)^{2}+\Omega_{% \Lambda}\right).
  40. ρ = ρ m ( a ) + ρ d e ( a ) , \rho=\rho_{m}(a)+\rho_{de}(a),
  41. ρ d e \rho_{de}
  42. P = w ρ c 2 P=w\rho c^{2}
  43. ρ ˙ + 3 a ˙ a ( ρ + P c 2 ) = 0 ; \dot{\rho}+3\frac{\dot{a}}{a}\left(\rho+\frac{P}{c^{2}}\right)=0;
  44. d ρ ρ = - 3 d a a ( 1 + w ) . \frac{d\rho}{\rho}=-3\frac{da}{a}\left(1+w\right).
  45. ln ρ = - 3 ( 1 + w ) ln a ; \ln{\rho}=-3\left(1+w\right)\ln{a};
  46. ρ = a - 3 ( 1 + w ) . \rho=a^{-3\left(1+w\right)}.
  47. ρ d e ( a ) = ρ d e 0 a - 3 ( 1 + w ) \rho_{de}(a)=\rho_{de0}a^{-3\left(1+w\right)}
  48. k = 0 k=0
  49. H 2 ( z ) = H 0 2 ( Ω M ( 1 + z ) 3 + Ω d e ( 1 + z ) 3 ( 1 + w ) ) . H^{2}(z)=H_{0}^{2}\left(\Omega_{M}(1+z)^{3}+\Omega_{de}(1+z)^{3\left(1+w\right% )}\right).
  50. ρ d e ( a ) = ρ d e 0 e - 3 d a a ( 1 + w ( a ) ) , \rho_{de}(a)=\rho_{de0}e^{-3\int\frac{da}{a}\left(1+w(a)\right)},
  51. w ( a ) w(a)
  52. w ( a ) = w 0 + w a ( 1 - a ) w(a)=w_{0}+w_{a}(1-a)
  53. H 2 ( z ) = H 0 2 ( Ω M a - 3 + Ω d e a - 3 ( 1 + w 0 + w a ) e - 3 w a ( 1 - a ) ) . H^{2}(z)=H_{0}^{2}\left(\Omega_{M}a^{-3}+\Omega_{de}a^{-3\left(1+w_{0}+w_{a}% \right)}e^{-3w_{a}(1-a)}\right).
  54. H 0 H_{0}
  55. t H 1 H 0 = 1 67.8 km/(s Mpc) = 4.55 10 17 s t_{H}\equiv{1\over H_{0}}={1\over 67.8\textrm{km/(s Mpc)}}=4.55\cdot 10^{17}% \textrm{s}
  56. t 0 t_{0}\approx
  57. H a ˙ a = const. a e H t = e t / t H H\equiv\frac{\dot{a}}{a}=\textrm{const.}\Rightarrow a\propto\textrm{e}^{Ht}=% \textrm{e}^{t/t_{H}}

Hubble_sequence.html

  1. n n
  2. n n
  3. e = 1 - b a \begin{matrix}e=1-\frac{b}{a}\end{matrix}
  4. a a
  5. b b

Hue.html

  1. h a b = atan2 ( b * , a * ) h_{ab}=\mathrm{atan2}(b^{*},a^{*})\;
  2. h u v = atan2 ( v * , u * ) = atan2 ( v , u ) h_{uv}=\mathrm{atan2}(v^{*},u^{*})=\mathrm{atan2}(v^{\prime},u^{\prime})\;
  3. h r g b = atan2 ( 3 ( G - B ) , 2 R - G - B ) h_{rgb}=\mathrm{atan2}\left(\sqrt{3}\cdot(G-B),2\cdot R-G-B\right)
  4. tan ( h r g b ) = 3 ( G - B ) 2 R - G - B \tan(h_{rgb})=\frac{\sqrt{3}\cdot(G-B)}{2\cdot R-G-B}
  5. R G B R\geq G\geq B
  6. h P r e u c i l c i r c l e = 60 G - B R - B h_{Preucil\ circle}=60^{\circ}\cdot\frac{G-B}{R-B}
  7. G > R B G>R\geq B
  8. h P r e u c i l c i r c l e = 60 ( 2 - R - B G - B ) h_{Preucil\ circle}=60^{\circ}\cdot\left(2-\frac{R-B}{G-B}\right)
  9. G B > R G\geq B>R
  10. h P r e u c i l c i r c l e = 60 ( 2 + B - R G - R ) h_{Preucil\ circle}=60^{\circ}\cdot\left(2+\frac{B-R}{G-R}\right)
  11. B > G > R \ B>G>R
  12. h P r e u c i l c i r c l e = 60 ( 4 - G - R B - R ) h_{Preucil\ circle}=60^{\circ}\cdot\left(4-\frac{G-R}{B-R}\right)
  13. B > R G B>R\geq G
  14. h P r e u c i l c i r c l e = 60 ( 4 + R - G B - G ) h_{Preucil\ circle}=60^{\circ}\cdot\left(4+\frac{R-G}{B-G}\right)
  15. R B > G R\geq B>G
  16. h P r e u c i l c i r c l e = 60 ( 6 - B - G R - G ) h_{Preucil\ circle}=60^{\circ}\cdot\left(6-\frac{B-G}{R-G}\right)
  17. M - L H - L \frac{M-L}{H-L}
  18. Δ h \Delta h
  19. Δ H * \Delta H^{*}
  20. Δ h a b \Delta h_{ab}
  21. Δ h u v \Delta h_{uv}
  22. Δ H a b * \Delta H^{*}_{ab}
  23. Δ H u v * \Delta H^{*}_{uv}

Huffman_coding.html

  1. A = { a 1 , a 2 , , a n } A=\left\{a_{1},a_{2},\cdots,a_{n}\right\}
  2. n n
  3. W = { w 1 , w 2 , , w n } W=\left\{w_{1},w_{2},\cdots,w_{n}\right\}
  4. w i = weight ( a i ) , 1 i n w_{i}=\mathrm{weight}\left(a_{i}\right),1\leq i\leq n
  5. C ( A , W ) = ( c 1 , c 2 , , c n ) C\left(A,W\right)=(c_{1},c_{2},\cdots,c_{n})
  6. c i c_{i}
  7. a i , 1 i n a_{i},1\leq i\leq n
  8. L ( C ) = i = 1 n w i × length ( c i ) L\left(C\right)=\sum_{i=1}^{n}{w_{i}\times\mathrm{length}\left(c_{i}\right)}
  9. C C
  10. L ( C ) L ( T ) L\left(C\right)\leq L\left(T\right)
  11. T ( A , W ) T\left(A,W\right)
  12. h ( a i ) = log 2 1 w i . h(a_{i})=\log_{2}{1\over w_{i}}.
  13. H ( A ) = w i > 0 w i h ( a i ) = w i > 0 w i log 2 1 w i = - w i > 0 w i log 2 w i . H(A)=\sum_{w_{i}>0}w_{i}h(a_{i})=\sum_{w_{i}>0}w_{i}\log_{2}{1\over w_{i}}=-% \sum_{w_{i}>0}w_{i}\log_{2}{w_{i}}.
  14. lim w 0 + w log 2 w = 0 \lim_{w\to 0^{+}}w\log_{2}w=0
  15. L ( C ) L(C)
  16. { a 1 , a 2 , a 3 , a 4 } \{a_{1},a_{2},a_{3},a_{4}\}
  17. { 0.4 ; 0.35 ; 0.2 ; 0.05 } \{0.4;0.35;0.2;0.05\}
  18. n n
  19. n n
  20. n - 1 n-1
  21. B 2 B B2^{B}
  22. B B
  23. max i [ w i + length ( c i ) ] \max_{i}\left[w_{i}+\mathrm{length}\left(c_{i}\right)\right]
  24. O ( n L ) O(nL)
  25. L L
  26. A = { a , b , c } A=\left\{a,b,c\right\}
  27. H ( A , C ) = { 00 , 1 , 01 } H\left(A,C\right)=\left\{00,1,01\right\}
  28. H ( A , C ) = { 00 , 01 , 1 } H\left(A,C\right)=\left\{00,01,1\right\}
  29. H ( A , C ) = { 0 , 10 , 11 } H\left(A,C\right)=\left\{0,10,11\right\}
  30. { 000 , 001 , 01 , 10 , 11 } \{000,001,01,10,11\}
  31. { 110 , 111 , 00 , 01 , 10 } \{110,111,00,01,10\}

Hull_(watercraft).html

  1. C b = V L W L B T C_{b}=\frac{V}{L_{WL}\cdot B\cdot T}
  2. C m = A m B T C_{m}=\frac{A_{m}}{B\cdot T}
  3. C p = V L p p A m C_{p}=\frac{V}{L_{pp}\cdot A_{m}}
  4. C w = A w L p p B C_{w}=\frac{A_{w}}{L_{pp}\cdot B}
  5. C b = C p C m C_{b}={C_{p}\cdot C_{m}}

Humidity.html

  1. m w m_{w}
  2. p n e t p_{net}
  3. A H = m w p n e t . AH={m_{w}\over p_{net}}.
  4. ( ϕ ) \left(\phi\right)
  5. ( e w ) \left({e_{w}}\right)
  6. ( e * w ) \left({{e^{*}}_{w}}\right)
  7. ϕ = e w e * w × 100 % \phi={{e_{w}}\over{{e^{*}}_{w}}}\times 100\%
  8. m v m_{v}
  9. m a m_{a}
  10. S H = m v m a . SH={m_{v}\over m_{a}}.
  11. S H = 0.622 × p ( H 2 O ) p ( d r y a i r ) SH={0.622\times{p_{(H_{2}O)}}\over{p_{(dry\,air)}}}
  12. 0.622 = M M H 2 O M M d r y a i r 0.622={{MM_{H_{2}O}}\over{MM_{dry\,air}}}
  13. S H = 0.622 × p ( H 2 O ) p - 0.378 × p ( H 2 O ) . SH={{0.622\times p_{(H_{2}O)}}\over{p-0.378\times p_{(H_{2}O)}}}.
  14. ϕ = S H × p ( 0.622 + 0.378 × S H ) p ( H 2 O ) * × 100 \phi={{SH\times p}\over{(0.622+0.378\times SH)p^{*}_{(H_{2}O)}}}\times 100

Hurwitz_polynomial.html

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

Huygens–Fresnel_principle.html

  1. U ( r 0 ) = U 0 e i k r 0 r 0 U(r_{0})=\frac{U_{0}e^{ikr_{0}}}{r_{0}}
  2. U ( P ) = - i λ U ( r 0 ) S e i k s s K ( χ ) d S U(P)=-\frac{i}{\lambda}U(r_{0})\int_{S}\frac{e^{iks}}{s}K(\chi)\,dS
  3. K ( χ ) = 1 2 ( 1 + cos χ ) ~{}K(\chi)=\frac{1}{2}(1+\cos\chi)

Hybrid_(biology).html

  1. ( 1 2 ) ( 2 × 2 ) = 1 16 \left(\frac{1}{2}\right)^{(2\times 2)}=\frac{1}{16}

Hybrid_rocket.html

  1. I s p I_{sp}
  2. I s p I_{sp}
  3. I s p I_{sp}
  4. I s p I_{sp}
  5. I s p I_{sp}

Hydrogen_atom.html

  1. E n = - m e c 2 α 2 2 n 2 E_{n}=-\frac{m_{e}c^{2}\alpha^{2}}{2n^{2}}
  2. E j n = - m e c 2 [ 1 - ( 1 + [ α n - j - 1 2 + ( j + 1 2 ) 2 - α 2 ] 2 ) - 1 / 2 ] - m e c 2 α 2 2 n 2 [ 1 + α 2 n 2 ( n j + 1 2 - 3 4 ) ] , \begin{array}[]{rl}E_{j\,n}&=-m\text{e}c^{2}\left[1-\left(1+\left[\dfrac{% \alpha}{n-j-\frac{1}{2}+\sqrt{\left(j+\frac{1}{2}\right)^{2}-\alpha^{2}}}% \right]^{2}\right)^{-1/2}\right]\\ &\approx-\dfrac{m\text{e}c^{2}\alpha^{2}}{2n^{2}}\left[1+\dfrac{\alpha^{2}}{n^% {2}}\left(\dfrac{n}{j+\frac{1}{2}}-\dfrac{3}{4}\right)\right],\end{array}
  3. 1 / 2 {1}/{2}
  4. m e c 2 α 2 2 = 0.51 MeV 2 137 2 = 13.6 eV \frac{m_{\,\text{e}}c^{2}\alpha^{2}}{2}=\frac{0.51\,\,\text{MeV}}{2\cdot 137^{% 2}}=13.6\,\,\text{eV}
  5. - 13.6 eV = - m e e 4 8 h 2 ε 0 2 , -13.6\,\,\text{eV}=-\frac{m_{\,\text{e}}e^{4}}{8h^{2}\varepsilon_{0}^{2}},
  6. 1 Ry h c R = 13.605 692 53 ( 30 ) eV . 1\,\,\text{Ry}\equiv hcR_{\infty}=13.605\;692\;53(30)\,\,\text{eV}.
  7. R M = R 1 + m e / M , R_{M}=\frac{R_{\infty}}{1+m_{\,\text{e}}/M},
  8. m e / M , m_{\,\text{e}}/M,
  9. ψ n m ( r , ϑ , φ ) = ( 2 n a 0 ) 3 ( n - - 1 ) ! 2 n ( n + ) ! e - ρ / 2 ρ L n - - 1 2 + 1 ( ρ ) Y m ( ϑ , φ ) \psi_{n\ell m}(r,\vartheta,\varphi)=\sqrt{{\left(\frac{2}{na_{0}}\right)}^{3}% \frac{(n-\ell-1)!}{2n(n+\ell)!}}e^{-\rho/2}\rho^{\ell}L_{n-\ell-1}^{2\ell+1}(% \rho)Y_{\ell}^{m}(\vartheta,\varphi)
  10. ρ = 2 r n a 0 \rho={2r\over{na_{0}}}
  11. a 0 a_{0}
  12. L n - - 1 2 + 1 ( ρ ) L_{n-\ell-1}^{2\ell+1}(\rho)
  13. Y m ( ϑ , φ ) Y_{\ell}^{m}(\vartheta,\varphi)\,
  14. ( n + ) ! (n+\ell)!
  15. L n + 2 + 1 ( ρ ) L_{n+\ell}^{2\ell+1}(\rho)
  16. n = 1 , 2 , 3 , n=1,2,3,\ldots
  17. = 0 , 1 , 2 , , n - 1 \ell=0,1,2,\ldots,n-1
  18. m = - , , . m=-\ell,\ldots,\ell.
  19. 0 r 2 d r 0 π sin ϑ d ϑ 0 2 π d φ ψ n m * ( r , ϑ , φ ) ψ n m ( r , ϑ , φ ) = n , , m | n , , m = δ n n δ δ m m , \int_{0}^{\infty}r^{2}dr\int_{0}^{\pi}\sin\vartheta d\vartheta\int_{0}^{2\pi}d% \varphi\;\psi^{*}_{n\ell m}(r,\vartheta,\varphi)\psi_{n^{\prime}\ell^{\prime}m% ^{\prime}}(r,\vartheta,\varphi)=\langle n,\ell,m|n^{\prime},\ell^{\prime},m^{% \prime}\rangle=\delta_{nn^{\prime}}\delta_{\ell\ell^{\prime}}\delta_{mm^{% \prime}},
  20. | n , , m |n,\ell,m\rangle
  21. ψ n m \psi_{n\ell m}
  22. δ \delta
  23. ϕ ( p , ϑ p , φ p ) = ( 2 π ) - 3 / 2 e - i p r / ψ ( r , ϑ , φ ) d V , \phi(p,\vartheta_{p},\varphi_{p})=(2\pi\hbar)^{-3/2}\int e^{-i\vec{p}\cdot\vec% {r}/\hbar}\psi(r,\vartheta,\varphi)dV,
  24. ϕ ( p , ϑ p , φ p ) = 2 π ( n - l - 1 ) ! ( n + l ) ! n 2 2 2 l + 2 l ! n l p l ( n 2 p 2 + 1 ) l + 2 C n - l - 1 l + 1 ( n 2 p 2 - 1 n 2 p 2 + 1 ) Y l m ( ϑ p , φ p ) , \phi(p,\vartheta_{p},\varphi_{p})=\sqrt{\frac{2}{\pi}\frac{(n-l-1)!}{(n+l)!}}n% ^{2}2^{2l+2}l!\frac{n^{l}p^{l}}{(n^{2}p^{2}+1)^{l+2}}C_{n-l-1}^{l+1}\left(% \frac{n^{2}p^{2}-1}{n^{2}p^{2}+1}\right)Y_{l}^{m}({\vartheta_{p},\varphi_{p}}),
  25. C N α ( x ) C_{N}^{\alpha}(x)
  26. p p
  27. / a 0 \hbar/a_{0}
  28. L 2 | n , , m = 2 ( + 1 ) | n , , m L^{2}\,|n,\ell,m\rangle={\hbar}^{2}\ell(\ell+1)\,|n,\ell,m\rangle
  29. L z | n , , m = m | n , , m . L_{z}\,|n,\ell,m\rangle=\hbar m\,|n,\ell,m\rangle.
  30. n - 1 n-1
  31. l l
  32. m m
  33. ϕ \phi
  34. l - m l-m
  35. θ \theta
  36. n - l - 1 n-l-1
  37. 1 / 2 {1}/{2}
  38. 1 / 2 {1}/{2}

Hydrogen_peroxide.html

  1. D 4 4 P 4 1 2 1 D_{4}^{4}P4_{1}2_{1}

Hydrophobe.html

  1. γ SG = γ SL + γ LG cos θ \gamma\text{SG}\ =\gamma\text{SL}+\gamma\text{LG}\cos{\theta}\,
  2. γ SG \gamma\text{SG}
  3. γ SL \gamma\text{SL}
  4. γ LG \gamma\text{LG}
  5. cos θ W * = r cos θ \cos{\theta}_{W}*=r\cos{\theta}\,
  6. cos θ CB * = φ ( cos θ + 1 ) - 1 \cos{\theta}\text{CB}*=\varphi(\cos\theta+1)-1\,
  7. cos θ > φ - 1 r - φ \cos\theta>\frac{\varphi-1}{r-\varphi}

Hydropower.html

  1. P = η ρ Q g h P=\eta\rho\,Qgh\!
  2. Power (W) = 0.85 × 1000 × 80 × 9.81 × 145 \,\text{Power (W)}=0.85\times 1000\times 80\times 9.81\times 145
  3. Power (W) = 0.85 × 62.5 × 2800 × 480 × 1.356 \,\text{Power (W)}=0.85\times 62.5\times 2800\times 480\times 1.356

Hydrostatic_equilibrium.html

  1. F t o p = - P t o p A . F_{top}=-P_{top}\cdot A.
  2. F b o t t o m = P b o t t o m A . F_{bottom}=P_{bottom}\cdot A.
  3. F w e i g h t = - ρ g V . F_{weight}=-\rho\cdot g\cdot V.
  4. F w e i g h t = - ρ g A h F_{weight}=-\rho\cdot g\cdot A\cdot h
  5. F = F b o t t o m + F t o p + F w e i g h t = P b o t t o m A - P t o p A - ρ g A h . \sum F=F_{bottom}+F_{top}+F_{weight}=P_{bottom}\cdot A-P_{top}\cdot A-\rho% \cdot g\cdot A\cdot h.
  6. 0 = P b o t t o m - P t o p - ρ g h . 0=P_{bottom}-P_{top}-\rho\cdot g\cdot h.
  7. P t o p - P b o t t o m = - ρ g h . P_{top}-P_{bottom}=-\rho\cdot g\cdot h.
  8. d P = - ρ g d h . dP=-\rho\cdot g\cdot dh.
  9. d P = - ρ ( P ) g ( h ) d h . dP=-\rho(P)\cdot g(h)\cdot dh.
  10. u = v = p x = p y = 0. u=v=\frac{\partial p}{\partial x}=\frac{\partial p}{\partial y}=0.
  11. z z
  12. p z + ρ g = 0. \frac{\partial p}{\partial z}+\rho g=0.
  13. T μ ν = ( ρ c - 2 + P ) U μ U ν + P g μ ν T^{\mu\nu}=(\rho c^{-2}+P)U^{\mu}U^{\nu}+Pg^{\mu\nu}
  14. R μ ν = 8 π G c 4 ( T μ ν - 1 2 g μ ν T ) R_{\mu\nu}=\frac{8\pi G}{c^{4}}(T_{\mu\nu}-\frac{1}{2}g_{\mu\nu}T)
  15. D μ T μ ν = 0 D_{\mu}T^{\mu\nu}=0
  16. d P d r = - G M ( r ) ρ ( r ) r 2 ( 1 + P ( r ) ρ ( r ) c 2 ) ( 1 + 4 π r 3 P ( r ) M ( r ) c 2 ) ( 1 - 2 G M ( r ) r c 2 ) - 1 \frac{dP}{dr}=-\frac{GM(r)\rho(r)}{r^{2}}\left(1+\frac{P(r)}{\rho(r)c^{2}}% \right)\left(1+\frac{4\pi r^{3}P(r)}{M(r)c^{2}}\right)\left(1-\frac{2GM(r)}{rc% ^{2}}\right)^{-1}
  17. M ( r ) = 4 π 0 r d r r 2 ρ ( r ) . M(r)=4\pi\int_{0}^{r}dr^{\prime}r^{\prime 2}\rho(r^{\prime}).
  18. ( 1 + P ( r ) ρ ( r ) c 2 ) ( 1 + 4 π r 3 P ( r ) M ( r ) c 2 ) ( 1 - 2 G M ( r ) r c 2 ) - 1 1 \left(1+\frac{P(r)}{\rho(r)c^{2}}\right)\left(1+\frac{4\pi r^{3}P(r)}{M(r)c^{2% }}\right)\left(1-\frac{2GM(r)}{rc^{2}}\right)^{-1}\rightarrow 1
  19. d P d r = - G M ( r ) ρ ( r ) r 2 = - g ( r ) ρ ( r ) d P = - ρ ( P ) g ( h ) d h \frac{dP}{dr}=-\frac{GM(r)\rho(r)}{r^{2}}=-g(r)\,\rho(r)\longrightarrow dP=-% \rho(P)\,g(h)\,dh
  20. X = Λ ( T B ) ρ B 2 \mathcal{L}_{X}=\Lambda(T_{B})\rho_{B}^{2}
  21. T B T_{B}
  22. ρ B \rho_{B}
  23. Λ ( T ) \Lambda(T)
  24. d P = - ρ g d r dP=-\rho gdr
  25. p B ( r + d r ) - p B ( r ) = - d r ρ B ( r ) G r 2 0 r 4 π r 2 ρ M ( r ) d r . p_{B}(r+dr)-p_{B}(r)=-dr\frac{\rho_{B}(r)G}{r^{2}}\int_{0}^{r}4\pi r^{2}\,\rho% _{M}(r)\,dr.
  26. r r
  27. p B = k T B ρ B / m B p_{B}=kT_{B}\rho_{B}/m_{B}
  28. k k
  29. m B m_{B}
  30. d d r ( k T B ( r ) ρ B ( r ) m B ) = - ρ B ( r ) G r 2 0 r 4 π r 2 ρ M ( r ) d r . \frac{d}{dr}\left(\frac{kT_{B}(r)\rho_{B}(r)}{m_{B}}\right)=-\frac{\rho_{B}(r)% G}{r^{2}}\int_{0}^{r}4\pi r^{2}\,\rho_{M}(r)\,dr.
  31. r 2 / ρ B ( r ) r^{2}/\rho_{B}(r)
  32. r r
  33. d d r [ r 2 ρ B ( r ) d d r ( k T B ( r ) ρ B ( r ) m B ) ] = - 4 π G r 2 ρ M ( r ) . \frac{d}{dr}\left[\frac{r^{2}}{\rho_{B}(r)}\frac{d}{dr}\left(\frac{kT_{B}(r)% \rho_{B}(r)}{m_{B}}\right)\right]=-4\pi Gr^{2}\rho_{M}(r).
  34. ρ D = ρ M - ρ B \rho_{D}=\rho_{M}-\rho_{B}
  35. d d r [ r 2 ρ D ( r ) d d r ( k T D ( r ) ρ D ( r ) m D ) ] = - 4 π G r 2 ρ M ( r ) . \frac{d}{dr}\left[\frac{r^{2}}{\rho_{D}(r)}\frac{d}{dr}\left(\frac{kT_{D}(r)% \rho_{D}(r)}{m_{D}}\right)\right]=-4\pi Gr^{2}\rho_{M}(r).
  36. σ D 2 \sigma^{2}_{D}
  37. σ D 2 = k T D m D . \sigma^{2}_{D}=\frac{kT_{D}}{m_{D}}.
  38. ρ B ( 0 ) / ρ M ( 0 ) \rho_{B}(0)/\rho_{M}(0)
  39. z z
  40. ρ B ( 0 ) / ρ M ( 0 ) ( 1 + z ) 2 ( θ s ) 3 / 2 \rho_{B}(0)/\rho_{M}(0)\propto(1+z)^{2}\left(\frac{\theta}{s}\right)^{3/2}
  41. θ \theta
  42. s s

Hyperbola.html

  1. f ( x ) = 1 / x f(x)=1/x
  2. f ( x ) = 1 / x f(x)=1/x
  3. b / a {b}/{a}
  4. = =
  5. x 2 a 2 - y 2 b 2 = 1. \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1.
  6. y 2 a 2 - x 2 b 2 = 1. \frac{y^{2}}{a^{2}}-\frac{x^{2}}{b^{2}}=1.
  7. c c
  8. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}
  9. ε = c a = a 2 + b 2 a = 1 + b 2 a 2 = sec θ . \varepsilon=\frac{c}{a}=\frac{\sqrt{a^{2}+b^{2}}}{a}=\sqrt{1+\frac{b^{2}}{a^{2% }}}=\sec\theta.
  10. 2 \sqrt{2}
  11. x 2 a 2 - y 2 b 2 = 1 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
  12. x 2 a 2 - y 2 b 2 = - 1 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1
  13. 2 b 2 a \frac{2b^{2}}{a}
  14. l = b 2 a l=\frac{b^{2}}{a}
  15. p = b 2 c p=\frac{b^{2}}{c}
  16. ϵ \epsilon
  17. ϵ = B C ¯ r \epsilon=\frac{\overline{BC}}{r}
  18. A x x x 2 + 2 A x y x y + A y y y 2 + 2 B x x + 2 B y y + C = 0 A_{xx}x^{2}+2A_{xy}xy+A_{yy}y^{2}+2B_{x}x+2B_{y}y+C=0
  19. D = | A x x A x y A x y A y y | < 0 D=\begin{vmatrix}A_{xx}&A_{xy}\\ A_{xy}&A_{yy}\end{vmatrix}<0\,
  20. Δ := | A x x A x y B x A x y A y y B y B x B y C | = 0 \Delta:=\begin{vmatrix}A_{xx}&A_{xy}&B_{x}\\ A_{xy}&A_{yy}&B_{y}\\ B_{x}&B_{y}&C\end{vmatrix}=0
  21. x c = - 1 D | B x A x y B y A y y | x_{c}=-\frac{1}{D}\begin{vmatrix}B_{x}&A_{xy}\\ B_{y}&A_{yy}\end{vmatrix}
  22. y c = - 1 D | A x x B x A x y B y | y_{c}=-\frac{1}{D}\begin{vmatrix}A_{xx}&B_{x}\\ A_{xy}&B_{y}\end{vmatrix}
  23. A x x ξ 2 + 2 A x y ξ η + A y y η 2 + Δ D = 0 A_{xx}\xi^{2}+2A_{xy}\xi\eta+A_{yy}\eta^{2}+\frac{\Delta}{D}=0
  24. tan 2 Φ = 2 A x y A x x - A y y \tan 2\Phi=\frac{2A_{xy}}{A_{xx}-A_{yy}}
  25. x 2 a 2 - y 2 b 2 = 1 \frac{{x}^{2}}{a^{2}}-\frac{{y}^{2}}{b^{2}}=1
  26. a 2 = - Δ λ 1 D = - Δ λ 1 2 λ 2 a^{2}=-\frac{\Delta}{\lambda_{1}D}=-\frac{\Delta}{\lambda_{1}^{2}\lambda_{2}}
  27. b 2 = - Δ λ 2 D = - Δ λ 1 λ 2 2 b^{2}=-\frac{\Delta}{\lambda_{2}D}=-\frac{\Delta}{\lambda_{1}\lambda_{2}^{2}}
  28. λ 2 - ( A x x + A y y ) λ + D = 0 \lambda^{2}-\left(A_{xx}+A_{yy}\right)\lambda+D=0
  29. x 2 a 2 - y 2 b 2 = 0 \frac{{x}^{2}}{a^{2}}-\frac{{y}^{2}}{b^{2}}=0
  30. E x + F y + G = 0 Ex+Fy+G=0
  31. E = A x x x 0 + A x y y 0 + B x E=A_{xx}x_{0}+A_{xy}y_{0}+B_{x}
  32. F = A x y x 0 + A y y y 0 + B y F=A_{xy}x_{0}+A_{yy}y_{0}+B_{y}
  33. G = B x x 0 + B y y 0 + C G=B_{x}x_{0}+B_{y}y_{0}+C
  34. F ( x - x 0 ) - E ( y - y 0 ) = 0 F\left(x-x_{0}\right)-E\left(y-y_{0}\right)=0
  35. x 2 a 2 - y 2 b 2 = 1 0 < b a \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\qquad 0<b\leq a
  36. r 1 r_{1}\,\!
  37. r 2 r_{2}\,\!
  38. ( x , y ) (x,y)\,\!
  39. ( - a e , 0 ) (-ae,0)\,\!
  40. ( a e , 0 ) (ae,0)\,\!
  41. r 1 - r 2 = 2 a r_{1}-r_{2}=2a\,\!
  42. r 2 - r 1 = 2 a r_{2}-r_{1}=2a\,\!
  43. r 1 2 = ( x + a e ) 2 + y 2 = x 2 + 2 x a e + a 2 e 2 + ( x 2 - a 2 ) ( e 2 - 1 ) = ( e x + a ) 2 r_{1}^{2}=(x+ae)^{2}+y^{2}=x^{2}+2xae+a^{2}e^{2}+(x^{2}-a^{2})(e^{2}-1)=(ex+a)% ^{2}
  44. r 2 2 = ( x - a e ) 2 + y 2 = x 2 - 2 x a e + a 2 e 2 + ( x 2 - a 2 ) ( e 2 - 1 ) = ( e x - a ) 2 r_{2}^{2}=(x-ae)^{2}+y^{2}=x^{2}-2xae+a^{2}e^{2}+(x^{2}-a^{2})(e^{2}-1)=(ex-a)% ^{2}
  45. e x > a ex>a\,\!
  46. r 1 = e x + a r_{1}=ex+a\,\!
  47. r 2 = e x - a r_{2}=ex-a\,\!
  48. r 1 - r 2 = 2 a r_{1}-r_{2}=2a\,\!
  49. e x < - a ex<-a\,\!
  50. r 1 = - e x - a r_{1}=-ex-a\,\!
  51. r 2 = - e x + a r_{2}=-ex+a\,\!
  52. r 2 - r 1 = 2 a r_{2}-r_{1}=2a\,\!
  53. x 2 a 2 - y 2 b 2 = 1 \frac{{x}^{2}}{a^{2}}-\frac{{y}^{2}}{b^{2}}=1
  54. r r
  55. ( x , y ) (x\ ,\ y)
  56. ( - e a , 0 ) (-ea\ ,\ 0)
  57. r = - e x - a r=-ex-a\,\!
  58. ( r , θ ) (r\ ,\ \theta)
  59. x = - a e + r cos θ x\ =\ -ae+r\cos\theta
  60. y = r sin θ y\ =r\sin\theta
  61. r = - e ( - a e + r cos θ ) - a r=-e(-ae+r\cos\theta)-a\,\!
  62. r = a ( e 2 - 1 ) 1 + e cos θ r=\frac{a(e^{2}-1)}{1+e\cos\theta}
  63. θ \theta
  64. L 1 - L 2 = ( S - L B ) - ( L - L B ) = S - L = 2 a L_{1}-L_{2}=\left(S-L_{B}\right)-\left(L-L_{B}\right)=S-L=2a
  65. cosh 2 μ - sinh 2 μ = 1 \cosh^{2}\mu-\sinh^{2}\mu=1
  66. μ \mu
  67. x = a cosh μ x=a\ \cosh\ \mu
  68. y = b sinh μ y=b\ \sinh\ \mu
  69. x 2 a 2 - y 2 b 2 = 1 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
  70. - < μ < -\infty<\mu<\infty
  71. ( x , y ) (x\ ,\ y)
  72. x < 0 x<0
  73. x = - a cosh μ x=-a\ \cosh\ \mu
  74. y = b sinh μ y=b\ \sinh\ \mu
  75. ( x k , y k ) (x_{k}\ ,\ y_{k})
  76. x k = - a cosh μ k x_{k}=-a\ \cosh\mu_{k}
  77. y k = b sinh μ k y_{k}=b\ \sinh\mu_{k}
  78. μ k = 0.3 k k = - 5 , - 4 , , 5 \mu_{k}\ =\ 0.3\ k\quad k=-5,-4,\cdots,5
  79. x 2 a 2 - y 2 b 2 = 1. \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1.
  80. x 2 a 2 + y 2 b 2 = 1 \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1
  81. cosh μ = cos i μ \cosh\mu=\cos i\mu
  82. ( x - h ) 2 a 2 - ( y - k ) 2 b 2 = 1. \frac{\left(x-h\right)^{2}}{a^{2}}-\frac{\left(y-k\right)^{2}}{b^{2}}=1.
  83. ε = 1 + b 2 a 2 = sec ( arctan ( b a ) ) = cosh ( arcsinh ( b a ) ) \varepsilon=\sqrt{1+\frac{b^{2}}{a^{2}}}=\sec\left(\arctan\left(\frac{b}{a}% \right)\right)=\cosh\left(\operatorname{arcsinh}\left(\frac{b}{a}\right)\right)
  84. ε = c a \varepsilon=\frac{c}{a}
  85. c = a 2 + b 2 c=\sqrt{a^{2}+b^{2}}
  86. ( h ± c , k ) \left(h\pm c,k\right)
  87. ( h , k ± c ) \left(h,k\pm c\right)
  88. x = h ± a cos ( arctan ( b a ) ) x=h\pm a\;\cos\left(\arctan\left(\frac{b}{a}\right)\right)
  89. y = k ± a cos ( arctan ( b a ) ) y=k\pm a\;\cos\left(\arctan\left(\frac{b}{a}\right)\right)
  90. r = a ( e 2 - 1 ) 1 + e cos θ r=\frac{a(e^{2}-1)}{1+e\cos\theta}
  91. θ \theta
  92. - arccos ( - 1 e ) < θ < arccos ( - 1 e ) -\arccos{\left(-\frac{1}{e}\right)}<\theta<\arccos{\left(-\frac{1}{e}\right)}
  93. x = R cos t x=R\,\cos t
  94. y = R sin t y=R\,\sin t
  95. R 2 = b 2 e 2 cos 2 t - 1 R^{2}=\frac{b^{2}}{e^{2}\cos^{2}t-1}\,
  96. t t
  97. - arccos ( 1 e ) < t < arccos ( 1 e ) -\arccos{\left(\frac{1}{e}\right)}<t<\arccos{\left(\frac{1}{e}\right)}
  98. x = a sec t + h y = b tan t + k or x = ± a cosh t + h y = b sinh t + k \begin{matrix}x=a\sec t+h\\ y=b\tan t+k\\ \end{matrix}\qquad\mathrm{or}\qquad\begin{matrix}x=\pm a\cosh t+h\\ y=b\sinh t+k\\ \end{matrix}
  99. x = b tan t + h y = a sec t + k or x = b sinh t + h y = ± a cosh t + k \begin{matrix}x=b\tan t+h\\ y=a\sec t+k\\ \end{matrix}\qquad\mathrm{or}\qquad\begin{matrix}x=b\sinh t+h\\ y=\pm a\cosh t+k\\ \end{matrix}
  100. ( x c cos θ ) 2 - ( y c sin θ ) 2 = 1 \left(\frac{x}{c\cos\theta}\right)^{2}-\left(\frac{y}{c\sin\theta}\right)^{2}=1
  101. y = 1 x y=\tfrac{1}{x}
  102. ( x - h ) ( y - k ) = m (x-h)(y-k)=m\,\,\,
  103. ε = 2 \varepsilon=\sqrt{2}
  104. a = b = 2 m a=b=\sqrt{2m}
  105. y = m x y=\frac{m}{x}\,
  106. ( x , y ) (x^{\prime},y^{\prime})
  107. ( x ) 2 ( 2 m ) 2 - ( y ) 2 ( 2 m ) 2 = 1 \frac{(x^{\prime})^{2}}{(\sqrt{2m})^{2}}-\frac{(y^{\prime})^{2}}{(\sqrt{2m})^{% 2}}=1
  108. x 2 a 2 - y 2 b 2 = 1 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
  109. a 2 + b 2 4 \frac{a^{2}+b^{2}}{4}
  110. x 2 a 2 - y 2 b 2 = 1 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
  111. x 2 a 2 - y 2 b 2 = - 1 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=-1
  112. a 2 b 2 a 2 + b 2 \frac{a^{2}b^{2}}{a^{2}+b^{2}}
  113. ( b / e ) 2 (b/e)^{2}
  114. x 2 a 2 - y 2 b 2 = 1 \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1
  115. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}
  116. sech x \operatorname{sech}\,x
  117. \ell
  118. \ell
  119. \ell

Hyperbolic_function.html

  1. sinh x = e x - e - x 2 = e 2 x - 1 2 e x = 1 - e - 2 x 2 e - x \sinh x=\frac{e^{x}-e^{-x}}{2}=\frac{e^{2x}-1}{2e^{x}}=\frac{1-e^{-2x}}{2e^{-x}}
  2. cosh x = e x + e - x 2 = e 2 x + 1 2 e x = 1 + e - 2 x 2 e - x \cosh x=\frac{e^{x}+e^{-x}}{2}=\frac{e^{2x}+1}{2e^{x}}=\frac{1+e^{-2x}}{2e^{-x}}
  3. tanh x = sinh x cosh x = e x - e - x e x + e - x = \tanh x=\frac{\sinh x}{\cosh x}=\frac{e^{x}-e^{-x}}{e^{x}+e^{-x}}=
  4. = e 2 x - 1 e 2 x + 1 = 1 - e - 2 x 1 + e - 2 x =\frac{e^{2x}-1}{e^{2x}+1}=\frac{1-e^{-2x}}{1+e^{-2x}}
  5. coth x = cosh x sinh x = e x + e - x e x - e - x = \coth x=\frac{\cosh x}{\sinh x}=\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=
  6. = e 2 x + 1 e 2 x - 1 = 1 + e - 2 x 1 - e - 2 x =\frac{e^{2x}+1}{e^{2x}-1}=\frac{1+e^{-2x}}{1-e^{-2x}}
  7. sech x = ( cosh x ) - 1 = 2 e x + e - x = \operatorname{sech}\,x=\left(\cosh x\right)^{-1}=\frac{2}{e^{x}+e^{-x}}=
  8. = 2 e x e 2 x + 1 = 2 e - x 1 + e - 2 x =\frac{2e^{x}}{e^{2x}+1}=\frac{2e^{-x}}{1+e^{-2x}}
  9. csch x = ( sinh x ) - 1 = 2 e x - e - x = \operatorname{csch}\,x=\left(\sinh x\right)^{-1}=\frac{2}{e^{x}-e^{-x}}=
  10. = 2 e x e 2 x - 1 = 2 e - x 1 - e - 2 x =\frac{2e^{x}}{e^{2x}-1}=\frac{2e^{-x}}{1-e^{-2x}}
  11. sinh x = - i sin ( i x ) \sinh x=-i\sin(ix)
  12. cosh x = cos ( i x ) \cosh x=\cos(ix)
  13. tanh x = - i tan ( i x ) \tanh x=-i\tan(ix)
  14. coth x = i cot ( i x ) \coth x=i\cot(ix)
  15. sech x = sec ( i x ) \operatorname{sech}x=\sec(ix)
  16. csch x = i csc ( i x ) \operatorname{csch}x=i\csc(ix)
  17. sinh ( - x ) \displaystyle\sinh(-x)
  18. tanh ( - x ) \displaystyle\tanh(-x)
  19. arsech x = arcosh 1 x arcsch x = arsinh 1 x arcoth x = artanh 1 x \begin{aligned}\displaystyle\operatorname{arsech}x&\displaystyle=\operatorname% {arcosh}\frac{1}{x}\\ \displaystyle\operatorname{arcsch}x&\displaystyle=\operatorname{arsinh}\frac{1% }{x}\\ \displaystyle\operatorname{arcoth}x&\displaystyle=\operatorname{artanh}\frac{1% }{x}\end{aligned}
  20. cosh 2 x - sinh 2 x = 1 \cosh^{2}x-\sinh^{2}x=1
  21. sech 2 x = 1 - tanh 2 x csch 2 x = coth 2 x - 1 \begin{aligned}\displaystyle\operatorname{sech}^{2}x&\displaystyle=1-\tanh^{2}% x\\ \displaystyle\operatorname{csch}^{2}x&\displaystyle=\coth^{2}x-1\end{aligned}
  22. f = 1 - f 2 f^{\prime}=1-f^{2}
  23. 1 2 f ′′ = f 3 - f ; f ( 0 ) = f ( ) = 0 \frac{1}{2}f^{\prime\prime}=f^{3}-f;\quad f(0)=f^{\prime}(\infty)=0
  24. area = a b cosh ( x ) d x = a b 1 + ( d d x cosh ( x ) ) 2 d x = arc length \,\text{area}=\int_{a}^{b}{\cosh{(x)}}\ dx=\int_{a}^{b}\sqrt{1+\left(\frac{d}{% dx}\cosh{(x)}\right)^{2}}\ dx=\,\text{arc length}
  25. sinh ( x + y ) \displaystyle\sinh(x+y)
  26. cosh ( 2 x ) = sinh 2 x + cosh 2 x = 2 sinh 2 x + 1 = 2 cosh 2 x - 1 sinh ( 2 x ) = 2 sinh x cosh x \begin{aligned}\displaystyle\cosh(2x)&\displaystyle=\sinh^{2}{x}+\cosh^{2}{x}=% 2\sinh^{2}x+1=2\cosh^{2}x-1\\ \displaystyle\sinh(2x)&\displaystyle=2\sinh x\cosh x\end{aligned}
  27. cosh x + sinh x \displaystyle\cosh x+\sinh x
  28. arsinh ( x ) = ln ( x + x 2 + 1 ) arcosh ( x ) = ln ( x + x 2 - 1 ) ; x 1 artanh ( x ) = 1 2 ln ( 1 + x 1 - x ) ; | x | < 1 arcoth ( x ) = 1 2 ln ( x + 1 x - 1 ) ; | x | > 1 arsech ( x ) = ln ( 1 x + 1 - x 2 x ) ; 0 < x 1 arcsch ( x ) = ln ( 1 x + 1 + x 2 | x | ) ; x 0 \begin{aligned}\displaystyle\operatorname{arsinh}(x)&\displaystyle=\ln\left(x+% \sqrt{x^{2}+1}\right)\\ \displaystyle\operatorname{arcosh}(x)&\displaystyle=\ln\left(x+\sqrt{x^{2}-1}% \right);x\geq 1\\ \displaystyle\operatorname{artanh}(x)&\displaystyle=\frac{1}{2}\ln\left(\frac{% 1+x}{1-x}\right);\left|x\right|<1\\ \displaystyle\operatorname{arcoth}(x)&\displaystyle=\frac{1}{2}\ln\left(\frac{% x+1}{x-1}\right);\left|x\right|>1\\ \displaystyle\operatorname{arsech}(x)&\displaystyle=\ln\left(\frac{1}{x}+\frac% {\sqrt{1-x^{2}}}{x}\right);0<x\leq 1\\ \displaystyle\operatorname{arcsch}(x)&\displaystyle=\ln\left(\frac{1}{x}+\frac% {\sqrt{1+x^{2}}}{\left|x\right|}\right);x\neq 0\end{aligned}
  29. d d x sinh x = cosh x \frac{d}{dx}\sinh x=\cosh x\,
  30. d d x cosh x = sinh x \frac{d}{dx}\cosh x=\sinh x\,
  31. d d x tanh x = 1 - tanh 2 x = sech 2 x = 1 / cosh 2 x \frac{d}{dx}\tanh x=1-\tanh^{2}x=\operatorname{sech}^{2}x=1/\cosh^{2}x\,
  32. d d x coth x = 1 - coth 2 x = - csch 2 x = - 1 / sinh 2 x \frac{d}{dx}\coth x=1-\coth^{2}x=-\operatorname{csch}^{2}x=-1/\sinh^{2}x\,
  33. d d x sech x = - tanh x sech x \frac{d}{dx}\ \operatorname{sech}\,x=-\tanh x\ \operatorname{sech}\,x\,
  34. d d x csch x = - coth x csch x \frac{d}{dx}\ \operatorname{csch}\,x=-\coth x\ \operatorname{csch}\,x\,
  35. d d x arsinh x = 1 x 2 + 1 \frac{d}{dx}\,\operatorname{arsinh}\,x=\frac{1}{\sqrt{x^{2}+1}}
  36. d d x arcosh x = 1 x 2 - 1 \frac{d}{dx}\,\operatorname{arcosh}\,x=\frac{1}{\sqrt{x^{2}-1}}
  37. d d x artanh x = 1 1 - x 2 \frac{d}{dx}\,\operatorname{artanh}\,x=\frac{1}{1-x^{2}}
  38. d d x arcoth x = 1 1 - x 2 \frac{d}{dx}\,\operatorname{arcoth}\,x=\frac{1}{1-x^{2}}
  39. d d x arsech x = - 1 x 1 - x 2 \frac{d}{dx}\,\operatorname{arsech}\,x=-\frac{1}{x\sqrt{1-x^{2}}}
  40. d d x arcsch x = - 1 | x | 1 + x 2 \frac{d}{dx}\,\operatorname{arcsch}\,x=-\frac{1}{\left|x\right|\sqrt{1+x^{2}}}
  41. d 2 d x 2 sinh x = sinh x \frac{d^{2}}{dx^{2}}\sinh x=\sinh x\,
  42. d 2 d x 2 cosh x = cosh x . \frac{d^{2}}{dx^{2}}\cosh x=\cosh x\,.
  43. e x e^{x}
  44. e - x e^{-x}
  45. x = 0 x=0
  46. sinh ( a x ) d x = a - 1 cosh ( a x ) + C cosh ( a x ) d x = a - 1 sinh ( a x ) + C tanh ( a x ) d x = a - 1 ln ( cosh ( a x ) ) + C coth ( a x ) d x = a - 1 ln ( sinh ( a x ) ) + C sech ( a x ) d x = a - 1 arctan ( sinh ( a x ) ) + C csch ( a x ) d x = a - 1 ln ( tanh ( a x 2 ) ) + C = a - 1 ln | csch ( a x ) - coth ( a x ) | + C \begin{aligned}\displaystyle\int\sinh(ax)\,dx&\displaystyle=a^{-1}\cosh(ax)+C% \\ \displaystyle\int\cosh(ax)\,dx&\displaystyle=a^{-1}\sinh(ax)+C\\ \displaystyle\int\tanh(ax)\,dx&\displaystyle=a^{-1}\ln(\cosh(ax))+C\\ \displaystyle\int\coth(ax)\,dx&\displaystyle=a^{-1}\ln(\sinh(ax))+C\\ \displaystyle\int\operatorname{sech}(ax)\,dx&\displaystyle=a^{-1}\arctan(\sinh% (ax))+C\\ \displaystyle\int\operatorname{csch}(ax)\,dx&\displaystyle=a^{-1}\ln\left(% \tanh\left(\frac{ax}{2}\right)\right)+C&\displaystyle=a^{-1}\ln\left|% \operatorname{csch}(ax)-\coth(ax)\right|+C\end{aligned}
  47. d u a 2 + u 2 = arsinh ( u a ) + C d u u 2 - a 2 = arcosh ( u a ) + C d u a 2 - u 2 = a - 1 artanh ( u a ) + C ; u 2 < a 2 d u a 2 - u 2 = a - 1 arcoth ( u a ) + C ; u 2 > a 2 d u u a 2 - u 2 = - a - 1 arsech ( u a ) + C d u u a 2 + u 2 = - a - 1 arcsch | u a | + C \begin{aligned}\displaystyle\int{\frac{du}{\sqrt{a^{2}+u^{2}}}}&\displaystyle=% \operatorname{arsinh}\left(\frac{u}{a}\right)+C\\ \displaystyle\int{\frac{du}{\sqrt{u^{2}-a^{2}}}}&\displaystyle=\operatorname{% arcosh}\left(\frac{u}{a}\right)+C\\ \displaystyle\int{\frac{du}{a^{2}-u^{2}}}&\displaystyle=a^{-1}\operatorname{% artanh}\left(\frac{u}{a}\right)+C;u^{2}<a^{2}\\ \displaystyle\int{\frac{du}{a^{2}-u^{2}}}&\displaystyle=a^{-1}\operatorname{% arcoth}\left(\frac{u}{a}\right)+C;u^{2}>a^{2}\\ \displaystyle\int{\frac{du}{u\sqrt{a^{2}-u^{2}}}}&\displaystyle=-a^{-1}% \operatorname{arsech}\left(\frac{u}{a}\right)+C\\ \displaystyle\int{\frac{du}{u\sqrt{a^{2}+u^{2}}}}&\displaystyle=-a^{-1}% \operatorname{arcsch}\left|\frac{u}{a}\right|+C\end{aligned}
  48. sinh x = x + x 3 3 ! + x 5 5 ! + x 7 7 ! + = n = 0 x 2 n + 1 ( 2 n + 1 ) ! \sinh x=x+\frac{x^{3}}{3!}+\frac{x^{5}}{5!}+\frac{x^{7}}{7!}+\cdots=\sum_{n=0}% ^{\infty}\frac{x^{2n+1}}{(2n+1)!}
  49. cosh x = 1 + x 2 2 ! + x 4 4 ! + x 6 6 ! + = n = 0 x 2 n ( 2 n ) ! \cosh x=1+\frac{x^{2}}{2!}+\frac{x^{4}}{4!}+\frac{x^{6}}{6!}+\cdots=\sum_{n=0}% ^{\infty}\frac{x^{2n}}{(2n)!}
  50. tanh x = x - x 3 3 + 2 x 5 15 - 17 x 7 315 + = n = 1 2 2 n ( 2 2 n - 1 ) B 2 n x 2 n - 1 ( 2 n ) ! , | x | < π 2 coth x = x - 1 + x 3 - x 3 45 + 2 x 5 945 + = x - 1 + n = 1 2 2 n B 2 n x 2 n - 1 ( 2 n ) ! , 0 < | x | < π sech x = 1 - x 2 2 + 5 x 4 24 - 61 x 6 720 + = n = 0 E 2 n x 2 n ( 2 n ) ! , | x | < π 2 csch x = x - 1 - x 6 + 7 x 3 360 - 31 x 5 15120 + = x - 1 + n = 1 2 ( 1 - 2 2 n - 1 ) B 2 n x 2 n - 1 ( 2 n ) ! , 0 < | x | < π \begin{aligned}\displaystyle\tanh x&\displaystyle=x-\frac{x^{3}}{3}+\frac{2x^{% 5}}{15}-\frac{17x^{7}}{315}+\cdots=\sum_{n=1}^{\infty}\frac{2^{2n}(2^{2n}-1)B_% {2n}x^{2n-1}}{(2n)!},\left|x\right|<\frac{\pi}{2}\\ \displaystyle\coth x&\displaystyle=x^{-1}+\frac{x}{3}-\frac{x^{3}}{45}+\frac{2% x^{5}}{945}+\cdots=x^{-1}+\sum_{n=1}^{\infty}\frac{2^{2n}B_{2n}x^{2n-1}}{(2n)!% },0<\left|x\right|<\pi\\ \displaystyle\operatorname{sech}\,x&\displaystyle=1-\frac{x^{2}}{2}+\frac{5x^{% 4}}{24}-\frac{61x^{6}}{720}+\cdots=\sum_{n=0}^{\infty}\frac{E_{2n}x^{2n}}{(2n)% !},\left|x\right|<\frac{\pi}{2}\\ \displaystyle\operatorname{csch}\,x&\displaystyle=x^{-1}-\frac{x}{6}+\frac{7x^% {3}}{360}-\frac{31x^{5}}{15120}+\cdots=x^{-1}+\sum_{n=1}^{\infty}\frac{2(1-2^{% 2n-1})B_{2n}x^{2n-1}}{(2n)!},0<\left|x\right|<\pi\end{aligned}
  51. B n B_{n}\,
  52. E n E_{n}\,
  53. r 2 u 2 , \frac{r^{2}u}{2},
  54. sinh ( x + y ) = sinh ( x ) cosh ( y ) + cosh ( x ) sinh ( y ) cosh ( x + y ) = cosh ( x ) cosh ( y ) + sinh ( x ) sinh ( y ) tanh ( x + y ) = tanh ( x ) + tanh ( y ) 1 + tanh ( x ) tanh ( y ) \begin{aligned}\displaystyle\sinh(x+y)&\displaystyle=\sinh(x)\cosh(y)+\cosh(x)% \sinh(y)\\ \displaystyle\cosh(x+y)&\displaystyle=\cosh(x)\cosh(y)+\sinh(x)\sinh(y)\\ \displaystyle\tanh(x+y)&\displaystyle=\frac{\tanh(x)+\tanh(y)}{1+\tanh(x)\tanh% (y)}\end{aligned}
  55. sinh 2 x = 2 sinh x cosh x cosh 2 x = cosh 2 x + sinh 2 x = 2 cosh 2 x - 1 = 2 sinh 2 x + 1 tanh 2 x = 2 tanh x 1 + tanh 2 x sinh 2 x = 2 tanh x 1 - tanh 2 x cosh 2 x = 1 + tanh 2 x 1 - tanh 2 x \begin{aligned}\displaystyle\sinh 2x&\displaystyle=2\sinh x\cosh x\\ \displaystyle\cosh 2x&\displaystyle=\cosh^{2}x+\sinh^{2}x=2\cosh^{2}x-1=2\sinh% ^{2}x+1\\ \displaystyle\tanh 2x&\displaystyle=\frac{2\tanh x}{1+\tanh^{2}x}\\ \displaystyle\sinh 2x&\displaystyle=\frac{2\tanh x}{1-\tanh^{2}x}\\ \displaystyle\cosh 2x&\displaystyle=\frac{1+\tanh^{2}x}{1-\tanh^{2}x}\end{aligned}
  56. sinh x 2 = 1 2 ( cosh x - 1 ) \sinh\frac{x}{2}=\sqrt{\frac{1}{2}(\cosh x-1)}\,
  57. cosh x 2 = 1 2 ( cosh x + 1 ) \cosh\frac{x}{2}=\sqrt{\frac{1}{2}(\cosh x+1)}\,
  58. tanh x 2 = cosh x - 1 cosh x + 1 = sinh x cosh x + 1 = cosh x - 1 sinh x = coth x - csch x . \tanh\frac{x}{2}=\sqrt{\frac{\cosh x-1}{\cosh x+1}}=\frac{\sinh x}{\cosh x+1}=% \frac{\cosh x-1}{\sinh x}=\coth x-\operatorname{csch}x.
  59. coth x 2 = coth x + csch x . \coth\frac{x}{2}=\coth x+\operatorname{csch}x.
  60. e x = cosh x + sinh x e^{x}=\cosh x+\sinh x
  61. e - x = cosh x - sinh x e^{-x}=\cosh x-\sinh x
  62. e i x \displaystyle e^{ix}
  63. cosh i x \displaystyle\cosh ix
  64. 2 π i 2\pi i
  65. π i \pi i
  66. sinh ( z ) \operatorname{sinh}(z)
  67. cosh ( z ) \operatorname{cosh}(z)
  68. tanh ( z ) \operatorname{tanh}(z)
  69. coth ( z ) \operatorname{coth}(z)
  70. sech ( z ) \operatorname{sech}(z)
  71. csch ( z ) \operatorname{csch}(z)

Hypercomplex_number.html

  1. ( a 0 , , a n ) (a_{0},\dots,a_{n})
  2. { 1 , i 1 , , i n } \{1,i_{1},\dots,i_{n}\}
  3. i k 2 { - 1 , 0 , + 1 } i_{k}^{2}\in\{-1,0,+1\}
  4. u 2 = a 0 + a 1 u u^{2}=a_{0}+a_{1}u
  5. u 2 - a 1 u + a 1 2 4 = a 0 + a 1 2 4 . u^{2}-a_{1}u+\frac{a_{1}^{2}}{4}=a_{0}+\frac{a_{1}^{2}}{4}.
  6. u 2 - a 1 u + a 1 2 4 = ( u - a 1 2 ) 2 = u ~ 2 u^{2}-a_{1}u+\frac{a_{1}^{2}}{4}=\left(u-\frac{a_{1}}{2}\right)^{2}=\tilde{u}^% {2}
  7. u ~ 2 = a 0 + a 1 2 4 . \tilde{u}^{2}~{}=a_{0}+\frac{a_{1}^{2}}{4}.
  8. ϵ \epsilon
  9. { 1 , ϵ } \{1,~{}\epsilon\}
  10. { 1 , j } \{1,~{}j\}
  11. j 2 = + 1 j^{2}=+1
  12. a := a 0 + a 1 2 4 a:=\sqrt{a_{0}+\frac{a_{1}^{2}}{4}}
  13. { 1 , i } \{1,~{}i\}
  14. i 2 = - 1 i^{2}=-1
  15. a := a 1 2 4 - a 0 a:=\sqrt{\frac{a_{1}^{2}}{4}-a_{0}}
  16. 1 2 ( 1 ± j ) \tfrac{1}{2}(1\pm j)
  17. ( 1 + j ) ( 1 - j ) = 0 (1+j)(1-j)=0
  18. 1 2 ( e i e j + e j e i ) = { - 1 , 0 , + 1 i = j , 0 i j . \tfrac{1}{2}(e_{i}e_{j}+e_{j}e_{i})=\Bigg\{\begin{matrix}-1,0,+1&i=j,\\ 0&i\not=j.\end{matrix}
  19. e 0 2 = e 1 2 = - 1 e_{0}^{2}=e_{1}^{2}=-1
  20. e 0 2 = e 1 2 = 1 e_{0}^{2}=e_{1}^{2}=1
  21. e 0 2 = e 1 2 = e 2 2 = - 1 e_{0}^{2}=e_{1}^{2}=e_{2}^{2}=-1
  22. e 0 2 = e 1 2 = e 2 2 = 1 e_{0}^{2}=e_{1}^{2}=e_{2}^{2}=1
  23. { 1 , i 1 , , i 2 n - 1 } \{1,i_{1},\dots,i_{2^{n}-1}\}
  24. i m 2 = - 1 i_{m}^{2}=-1
  25. { 1 , i 1 , i 2 , i 3 } \{1,i_{1},i_{2},i_{3}\}
  26. i 1 2 = - 1 , i 2 2 = i 3 2 = + 1 \ i_{1}^{2}=-1,i_{2}^{2}=i_{3}^{2}=+1
  27. { 1 , i 1 , , i 7 } \{1,i_{1},\dots,i_{7}\}
  28. i 1 2 = i 2 2 = i 3 2 = - 1 \ i_{1}^{2}=i_{2}^{2}=i_{3}^{2}=-1
  29. i 4 2 = = i 7 2 = + 1. \ i_{4}^{2}=\cdots=i_{7}^{2}=+1.
  30. \mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}
  31. \mathbb{C}\otimes_{\mathbb{R}}\mathbb{H}
  32. 𝕆 \mathbb{C}\otimes_{\mathbb{R}}\mathbb{O}

Hypercube.html

  1. n \sqrt{n}
  2. ( ± 1 2 , ± 1 2 , , ± 1 2 ) \left(\pm\frac{1}{2},\pm\frac{1}{2},\cdots,\pm\frac{1}{2}\right)
  3. ( ± 1 , ± 1 , , ± 1 ) (\pm 1,\pm 1,\cdots,\pm 1)
  4. 2 n 2^{n}
  5. 2 3 2^{3}
  6. 2 n 2 - 2 n 2n^{2}-2n
  7. E m , n = 2 n - m ( n m ) E_{m,n}=2^{n-m}{n\choose m}
  8. ( n m ) = n ! m ! ( n - m ) ! {n\choose m}=\frac{n!}{m!\,(n-m)!}
  9. 2 n 2^{n}
  10. m m
  11. ( n m ) {n\choose m}
  12. 2 m 2^{m}
  13. 2 n s n - 1 2ns^{n-1}
  14. E m , n = 2 E m , n - 1 + E m - 1 , n - 1 E_{m,n}=2E_{m,n-1}+E_{m-1,n-1}\!
  15. E 0 , 0 = 1 E_{0,0}=1\!
  16. m < n m<n
  17. m < 0 m<0
  18. n < 0 n<0
  19. E 1 , 3 E_{1,3}\!
  20. E m , n E_{m,n}\!

Hyperinflation.html

  1. × 10 2 9 \times 10^{2}9
  2. 10 1 0 10^{1}0
  3. 10 1 3 10^{1}3
  4. 10 1 2 10^{1}2
  5. 10 2 2 10^{2}2
  6. 10 2 5 10^{2}5
  7. 10 1 0 10^{1}0
  8. 10 1 2 10^{1}2
  9. 10 2 5 10^{2}5
  10. New price y years later = old price × ( 1 + inflation 100 ) y \hbox{New price }y\hbox{ years later}=\hbox{old price }\times\left(1+\frac{% \hbox{inflation}}{100}\right)^{y}
  11. Monthly inflation = 100 × ( ( 1 + inflation 100 ) 1 12 - 1 ) \hbox{Monthly inflation }=100\times\left(\left(1+\frac{\hbox{inflation}}{100}% \right)^{\frac{1}{12}}-1\right)
  12. Price doubling time = log e 2 log e ( 1 + inflation 100 ) \hbox{Price doubling time}=\frac{\log_{e}2}{\log_{e}\left(1+\frac{\hbox{% inflation}}{100}\right)}
  13. Years per added zero of the price = 1 log 10 ( 1 + inflation 100 ) \hbox{Years per added zero of the price }=\frac{1}{\log_{10}\left(1+\frac{% \hbox{inflation}}{100}\right)}

Hyperreal_number.html

  1. 1 + 1 + + 1. 1+1+\cdots+1.\,
  2. sin π n = 0 \sin{\pi n}=0
  3. sin π H = 0 \sin{\pi H}=0
  4. f ( x ) = st ( f ( x + Δ x ) - f ( x ) Δ x ) f^{\prime}(x)={\rm st}\left(\frac{f(x+\Delta x)-f(x)}{\Delta x}\right)
  5. Δ x \Delta x
  6. 1 < ω , 1 + 1 < ω , 1 + 1 + 1 < ω , 1 + 1 + 1 + 1 < ω , . 1<\omega,\quad 1+1<\omega,\quad 1+1+1<\omega,\quad 1+1+1+1<\omega,\ldots.
  7. f ( x ) f^{\prime}(x)\,
  8. = st ( f ( x + d x ) - f ( x ) d x ) =\operatorname{st}\left(\frac{f(x+dx)-f(x)}{dx}\right)
  9. = st ( x 2 + 2 x d x + d x 2 - x 2 d x ) =\operatorname{st}\left(\frac{x^{2}+2x\cdot dx+dx^{2}-x^{2}}{dx}\right)
  10. = st ( 2 x d x + d x 2 d x ) =\operatorname{st}\left(\frac{2x\cdot dx+dx^{2}}{dx}\right)
  11. = st ( 2 x d x d x + d x 2 d x ) =\operatorname{st}\left(\frac{2x\cdot dx}{dx}+\frac{dx^{2}}{dx}\right)
  12. = st ( 2 x + d x ) =\operatorname{st}\left(2x+dx\right)
  13. = 2 x =2x\,
  14. ( a 0 , a 1 , a 2 , ) + ( b 0 , b 1 , b 2 , ) = ( a 0 + b 0 , a 1 + b 1 , a 2 + b 2 , ) (a_{0},a_{1},a_{2},\ldots)+(b_{0},b_{1},b_{2},\ldots)=(a_{0}+b_{0},a_{1}+b_{1}% ,a_{2}+b_{2},\ldots)
  15. 7 + ϵ 7+\epsilon
  16. ϵ \epsilon
  17. ( a 0 , a 1 , a 2 , ) ( b 0 , b 1 , b 2 , ) a 0 b 0 a 1 b 1 a 2 b 2 (a_{0},a_{1},a_{2},\ldots)\leq(b_{0},b_{1},b_{2},\ldots)\iff a_{0}\leq b_{0}% \wedge a_{1}\leq b_{1}\wedge a_{2}\leq b_{2}\ldots
  18. ( 2 0 ) 0 = 2 0 2 = 2 0 , (2^{\aleph_{0}})^{\aleph_{0}}=2^{\aleph_{0}^{2}}=2^{\aleph_{0}},\,
  19. 2 0 2^{\aleph_{0}}
  20. a , b a,b\quad
  21. z ( a ) = { i : a i = 0 } z(a)=\{i:a_{i}=0\}\quad
  22. z ( a ) z(a)\quad
  23. i i\quad
  24. a i = 0 a_{i}=0\quad
  25. a b = 0 ab=0\quad
  26. z ( a ) z(a)\quad
  27. z ( b ) z(b)\quad
  28. a a\quad
  29. a b ab\quad
  30. b b\quad
  31. a a\quad
  32. b b\quad
  33. a 2 + b 2 a^{2}+b^{2}\quad
  34. a = 0 a=0\quad
  35. z ( a ) z(a)\quad
  36. f f\quad
  37. x x\quad
  38. f f\quad
  39. f ( { x n } ) = { f ( x n ) } f(\{x_{n}\})=\{f(x_{n})\}\,
  40. { } \{\dots\}
  41. \dots
  42. | x | < a |x|<a\quad
  43. a a\quad
  44. x x\quad
  45. y + d y+d\quad
  46. y y\quad
  47. d d\quad
  48. f f\quad
  49. f ( a ) - f ( x ) f(a)-f(x)\quad
  50. x - a x-a\quad
  51. f f\quad
  52. ( f ( x ) - f ( a ) ) / ( x - a ) - f ( a ) (f(x)-f(a))/(x-a)-f^{\prime}(a)\quad
  53. x - a x-a\quad
  54. a a\quad
  55. f f\quad
  56. x x\quad
  57. f ( x ) - ( f ( x ) - f ( a ) ) / ( x - a ) = f ( x ) - ( f ( a ) - f ( x ) ) / ( a - x ) f^{\prime}(x)-(f(x)-f(a))/(x-a)=f^{\prime}(x)-(f(a)-f(x))/(a-x)\quad
  58. x - a x-a\quad
  59. f ( x ) - f ( a ) f^{\prime}(x)-f^{\prime}(a)\quad
  60. x - a x-a\quad
  61. x y x\leq y
  62. st ( x ) st ( y ) \operatorname{st}(x)\leq\operatorname{st}(y)
  63. x < y x<y
  64. st ( x ) < st ( y ) \operatorname{st}(x)<\operatorname{st}(y)
  65. st ( x + y ) = st ( x ) + st ( y ) \operatorname{st}(x+y)=\operatorname{st}(x)+\operatorname{st}(y)
  66. st ( x y ) = st ( x ) st ( y ) \operatorname{st}(xy)=\operatorname{st}(x)\operatorname{st}(y)
  67. st ( 1 / x ) = 1 / st ( x ) \operatorname{st}(1/x)=1/\operatorname{st}(x)
  68. st ( x ) = x \operatorname{st}(x)=x
  69. κ \mathbb{R}^{\kappa}

Hypoxia_(medical).html

  1. H b C O ( % ) = C O - 2.34 5.09 Hb_{CO}(\%)=\frac{CO-2.34}{5.09}
  2. = 5 - 2.34 5.09 = .5 % =\frac{5-2.34}{5.09}=.5\%

Icosidodecahedron.html

  1. A = ( 5 3 + 3 25 + 10 5 ) a 2 29.3059828 a 2 A=\left(5\sqrt{3}+3\sqrt{25+10\sqrt{5}}\right)a^{2}\approx 29.3059828a^{2}
  2. V = 1 6 ( 45 + 17 5 ) a 3 13.8355259 a 3 . V=\frac{1}{6}\left(45+17\sqrt{5}\right)a^{3}\approx 13.8355259a^{3}.

Ideal_(ring_theory).html

  1. ( R , + , ) (R,+,\cdot)
  2. ( R , + ) (R,+)
  3. I I
  4. R R
  5. I I
  6. ( I , + ) (I,+)
  7. ( R , + ) (R,+)
  8. x I , r R : x r I \forall x\in I,\forall r\in R:\quad x\cdot r\in I
  9. x I , r R : r x I . \forall x\in I,\forall r\in R:\quad r\cdot x\in I.
  10. I I
  11. R R
  12. R R
  13. ( I , + ) (I,+)
  14. ( R , + ) (R,+)
  15. x I , r R : x r I . \forall x\in I,\forall r\in R:\quad x\cdot r\in I.
  16. R R
  17. R R
  18. R R
  19. I I
  20. R R
  21. R R
  22. ( I , + ) (I,+)
  23. ( R , + ) (R,+)
  24. x I , r R : r x I . \forall x\in I,\forall r\in R:\quad r\cdot x\in I.
  25. R R
  26. R R
  27. R R
  28. I I
  29. x , y I : x - y I \forall x,y\in I:x-y\in I
  30. z 1 = 0 z_{1}=0
  31. z 2 = 0 z_{2}=0
  32. z 1 + z 2 z_{1}+z_{2}
  33. r z 1 rz_{1}
  34. z 1 r z_{1}r
  35. r r
  36. \mathbb{Z}
  37. 2 2\mathbb{Z}
  38. n n\mathbb{Z}
  39. C ( ) C(\mathbb{R})
  40. \mathbb{R}
  41. \mathbb{R}
  42. C ( ) C(\mathbb{R})
  43. { r 1 x 1 + + r n x n n , r i R , x i X } . \{r_{1}x_{1}+\dots+r_{n}x_{n}\mid n\in\mathbb{N},r_{i}\in R,x_{i}\in X\}.\,
  44. { x 1 r 1 + + x n r n n , r i R , x i X } \{x_{1}r_{1}+\dots+x_{n}r_{n}\mid n\in\mathbb{N},r_{i}\in R,x_{i}\in X\}\,
  45. { r 1 x 1 s 1 + + r n x n s n n , r i R , s i R , x i X } . \{r_{1}x_{1}s_{1}+\dots+r_{n}x_{n}s_{n}\mid n\in\mathbb{N},r_{i}\in R,s_{i}\in R% ,x_{i}\in X\}.\,
  46. R a = { r a r R } Ra=\{ra\mid r\in R\}\,
  47. a R = { a r r R } aR=\{ar\mid r\in R\}\,
  48. R a R = { r 1 a s 1 + + r n a s n n , r i R , s i R } . RaR=\{r_{1}as_{1}+\dots+r_{n}as_{n}\mid n\in\mathbb{N},r_{i}\in R,s_{i}\in R\}.\,
  49. 𝔦 , 𝔧 \mathfrak{i},\mathfrak{j}
  50. x + y = 1 x+y=1
  51. x 𝔦 x\in\mathfrak{i}
  52. y 𝔧 y\in\mathfrak{j}
  53. 𝔞 \mathfrak{a}
  54. 𝔟 \mathfrak{b}
  55. 𝔞 + 𝔟 := { a + b a 𝔞 and b 𝔟 } \mathfrak{a}+\mathfrak{b}:=\{a+b\mid a\in\mathfrak{a}\mbox{ and }~{}b\in% \mathfrak{b}\}
  56. 𝔞 𝔟 := { a 1 b 1 + + a n b n a i 𝔞 and b i 𝔟 , i = 1 , 2 , , n ; for n = 1 , 2 , } , \mathfrak{a}\mathfrak{b}:=\{a_{1}b_{1}+\dots+a_{n}b_{n}\mid a_{i}\in\mathfrak{% a}\mbox{ and }~{}b_{i}\in\mathfrak{b},i=1,2,\dots,n;\mbox{ for }~{}n=1,2,\dots\},
  57. 𝔞 \mathfrak{a}
  58. 𝔟 \mathfrak{b}
  59. 𝔞 𝔟 \mathfrak{a}\mathfrak{b}
  60. 𝔞 \mathfrak{a}
  61. 𝔟 \mathfrak{b}
  62. 𝔞 𝔟 \mathfrak{a}\mathfrak{b}
  63. 𝔞 \mathfrak{a}
  64. 𝔟 \mathfrak{b}

Ideal_gas.html

  1. P V = n R T PV=nRT\,
  2. V = k / P V=k/P
  3. V = b T V=bT
  4. V = a n V=an
  5. 3 V = k b a ( T n P ) 3V=kba\left(\frac{Tn}{P}\right)
  6. V = ( k b a 3 ) ( T n P ) V=\left(\frac{kba}{3}\right)\left(\frac{Tn}{P}\right)
  7. V = R ( T n P ) V=R\left(\frac{Tn}{P}\right)
  8. P V = n R T PV=nRT
  9. U = c ^ V n R T U=\hat{c}_{V}nRT
  10. P P
  11. V V
  12. n n
  13. R R
  14. T T
  15. k k
  16. b b
  17. V / T V/T
  18. a a
  19. V / n V/n
  20. U U
  21. c ^ V \hat{c}_{V}
  22. n R = N k B nR=Nk_{B}
  23. N N
  24. k B k_{B}
  25. c ^ V = 1 n R T ( S T ) V = 1 n R ( U T ) V \hat{c}_{V}=\frac{1}{nR}T\left(\frac{\partial S}{\partial T}\right)_{V}=\frac{% 1}{nR}\left(\frac{\partial U}{\partial T}\right)_{V}
  26. c ^ V = 3 / 2 \hat{c}_{V}=3/2
  27. c ^ V = 5 / 2 \hat{c}_{V}=5/2
  28. c ^ p = 1 n R T ( S T ) p = 1 n R ( H T ) p = c ^ V + 1 \hat{c}_{p}=\frac{1}{nR}T\left(\frac{\partial S}{\partial T}\right)_{p}=\frac{% 1}{nR}\left(\frac{\partial H}{\partial T}\right)_{p}=\hat{c}_{V}+1
  29. H = U + p V H=U+pV
  30. c ^ V \hat{c}_{V}
  31. c ^ p \hat{c}_{p}
  32. γ = c P c V \gamma=\frac{c_{P}}{c_{V}}
  33. Δ S \Delta S
  34. Δ S = S 0 S d S = T 0 T ( S T ) V d T + V 0 V ( S V ) T d V \Delta S=\int_{S_{0}}^{S}dS=\int_{T_{0}}^{T}\left(\frac{\partial S}{\partial T% }\right)_{V}\!dT+\int_{V_{0}}^{V}\left(\frac{\partial S}{\partial V}\right)_{T% }\!dV
  35. Δ S = T 0 T C v T d T + V 0 V ( P T ) V d V . \Delta S=\int_{T_{0}}^{T}\frac{C_{v}}{T}\,dT+\int_{V_{0}}^{V}\left(\frac{% \partial P}{\partial T}\right)_{V}dV.
  36. C V C_{V}
  37. c ^ V \hat{c}_{V}
  38. Δ S = c ^ V N k ln ( T T 0 ) + N k ln ( V V 0 ) \Delta S=\hat{c}_{V}Nk\ln\left(\frac{T}{T_{0}}\right)+Nk\ln\left(\frac{V}{V_{0% }}\right)
  39. S = N k ln ( V T c ^ v f ( N ) ) S=Nk\ln\left(\frac{VT^{\hat{c}_{v}}}{f(N)}\right)
  40. V T c ^ v VT^{\hat{c}_{v}}
  41. S ( T , a V , a N ) = a S ( T , V , N ) . S(T,aV,aN)=aS(T,V,N).\,
  42. a f ( N ) = f ( a N ) . af(N)=f(aN).\,
  43. f ( N ) = Φ N f(N)=\Phi N\,
  44. Φ \Phi
  45. V T c ^ v / N VT^{\hat{c}_{v}}/N
  46. S N k = ln ( V T c ^ v N Φ ) . \frac{S}{Nk}=\ln\left(\frac{VT^{\hat{c}_{v}}}{N\Phi}\right).\,
  47. S N k = ln [ V N ( U c ^ v k N ) c ^ v 1 Φ ] \frac{S}{Nk}=\ln\left[\frac{V}{N}\,\left(\frac{U}{\hat{c}_{v}kN}\right)^{\hat{% c}_{v}}\,\frac{1}{\Phi}\right]
  48. ( c ^ v = 3 / 2 ) (\hat{c}_{v}=3/2)
  49. S k N = ln ( V T c ^ V N Φ ) \frac{S}{kN}=\ln\left(\frac{VT^{\hat{c}_{V}}}{N\Phi}\right)
  50. μ = ( G N ) T , P \mu=\left(\frac{\partial G}{\partial N}\right)_{T,P}
  51. U + P V - T S U+PV-TS
  52. μ ( T , V , N ) = k T ( c ^ P - ln ( V T c ^ V N Φ ) ) \mu(T,V,N)=kT\left(\hat{c}_{P}-\ln\left(\frac{VT^{\hat{c}_{V}}}{N\Phi}\right)\right)
  53. U U\,
  54. = c ^ V N k T =\hat{c}_{V}NkT\,
  55. A = A=\,
  56. U - T S U-TS\,
  57. = μ N - N k T =\mu N-NkT\,
  58. H = H=\,
  59. U + P V U+PV\,
  60. = c ^ P N k T =\hat{c}_{P}NkT\,
  61. G = G=\,
  62. U + P V - T S U+PV-TS\,
  63. = μ N =\mu N\,
  64. c ^ P = c ^ V + 1 \hat{c}_{P}=\hat{c}_{V}+1
  65. U ( S , V , N ) = c ^ V N k ( N Φ V e S / N k ) 1 / c ^ V U(S,V,N)=\hat{c}_{V}Nk\left(\frac{N\Phi}{V}\,e^{S/Nk}\right)^{1/\hat{c}_{V}}
  66. A ( T , V , N ) = N k T ( c ^ V - ln ( V T c ^ V N Φ ) ) A(T,V,N)=NkT\left(\hat{c}_{V}-\ln\left(\frac{VT^{\hat{c}_{V}}}{N\Phi}\right)\right)
  67. H ( S , P , N ) = c ^ P N k ( P Φ k e S / N k ) 1 / c ^ P H(S,P,N)=\hat{c}_{P}Nk\left(\frac{P\Phi}{k}\,e^{S/Nk}\right)^{1/\hat{c}_{P}}
  68. G ( T , P , N ) = N k T ( c ^ P - ln ( k T c ^ P P Φ ) ) G(T,P,N)=NkT\left(\hat{c}_{P}-\ln\left(\frac{kT^{\hat{c}_{P}}}{P\Phi}\right)\right)
  69. c s o u n d = ( P ρ ) s = γ P ρ = γ R T M c_{sound}=\sqrt{\left(\frac{\partial P}{\partial\rho}\right)_{s}}=\sqrt{\frac{% \gamma P}{\rho}}=\sqrt{\frac{\gamma RT}{M}}
  70. γ \gamma\,
  71. ( c ^ P / c ^ V ) (\hat{c}_{P}/\hat{c}_{V})
  72. s s\,
  73. ρ \rho\,
  74. P P\,
  75. R R\,
  76. T T\,
  77. M M\,
  78. Φ = T 3 / 2 Λ 3 g \Phi=\frac{T^{3/2}\Lambda^{3}}{g}

Ideal_gas_law.html

  1. P V = n R T PV=nRT\,
  2. P V = n R T PV=nRT\,
  3. n = m M n={\frac{m}{M}}
  4. P V = m M R T \ PV=\frac{m}{M}RT
  5. P = ρ R M T \ P=\rho\frac{R}{M}T
  6. P = ρ R specific T \ P=\rho R_{\rm specific}T
  7. P v = R specific T \ Pv=R_{\rm specific}T
  8. P V = N k B T \ PV=Nk_{B}T
  9. Y = m μ m u Y=\frac{m}{\mu m_{\mathrm{u}}}
  10. P = 1 V m μ m u k T = k μ m u ρ T . P=\frac{1}{V}\frac{m}{\mu m_{\mathrm{u}}}kT=\frac{k}{\mu m_{\mathrm{u}}}\rho T.
  11. P V T = C \frac{PV}{T}=C
  12. P V = n R T PV=nRT\,
  13. 𝐪 𝐅 \displaystyle\langle\mathbf{q}\cdot\mathbf{F}\rangle
  14. 3 N k B T = - k = 1 N 𝐪 k 𝐅 k . 3Nk_{B}T=-\biggl\langle\sum_{k=1}^{N}\mathbf{q}_{k}\cdot\mathbf{F}_{k}\biggr\rangle.
  15. - k = 1 N 𝐪 k 𝐅 k = P surface 𝐪 d 𝐒 , -\biggl\langle\sum_{k=1}^{N}\mathbf{q}_{k}\cdot\mathbf{F}_{k}\biggr\rangle=P% \oint_{\mathrm{surface}}\mathbf{q}\cdot d\mathbf{S},
  16. 𝐪 = q x q x + q y q y + q z q z = 3 , \nabla\cdot\mathbf{q}=\frac{\partial q_{x}}{\partial q_{x}}+\frac{\partial q_{% y}}{\partial q_{y}}+\frac{\partial q_{z}}{\partial q_{z}}=3,
  17. P surface 𝐪 d 𝐒 = P volume ( 𝐪 ) d V = 3 P V , P\oint_{\mathrm{surface}}\mathbf{q}\cdot d\mathbf{S}=P\int_{\mathrm{volume}}% \left(\nabla\cdot\mathbf{q}\right)dV=3PV,
  18. 3 N k B T = - k = 1 N 𝐪 k 𝐅 k = 3 P V , 3Nk_{B}T=-\biggl\langle\sum_{k=1}^{N}\mathbf{q}_{k}\cdot\mathbf{F}_{k}\biggr% \rangle=3PV,
  19. P V = N k B T = n R T , PV=Nk_{B}T=nRT,\,

Idempotence.html

  1. f f
  2. S S
  3. x x
  4. S S
  5. f ( f ( x ) ) = f ( x ) f\!\left(f\!\left(x\right)\right)=f\!\left(x\right)
  6. id S \,\text{id}_{S}
  7. id S ( x ) = x \,\text{id}_{S}\left(x\right)=x
  8. K c K_{c}
  9. c c
  10. S S
  11. K c ( x ) = c K_{c}\left(x\right)=c
  12. π x y \pi_{xy}
  13. π x y ( x , y , z ) = ( x , y , 0 ) \pi_{xy}\left(x,y,z\right)=\left(x,y,0\right)
  14. x y xy
  15. z z
  16. f : S S f\colon S\to S
  17. S S
  18. f f
  19. n n
  20. k k
  21. n - k n-k
  22. k n - k k^{n-k}
  23. k = 0 n ( n k ) k n - k \sum_{k=0}^{n}{n\choose k}k^{n-k}
  24. n = { 0 , 1 , 2 , } n=\left\{0,1,2,\dots\right\}
  25. 1 , 1 , 3 , 10 , 41 , 196 , 1057 , 6322 , 41393 , 1,1,3,10,41,196,1057,6322,41393,\dots
  26. \bigstar
  27. S S
  28. x x
  29. \bigstar
  30. x x = x x\,\bigstar\,x=x
  31. \bigstar
  32. \bigstar
  33. S S
  34. x S x\in S
  35. \in
  36. x x = x x\,\bigstar\,x=x
  37. U U
  38. X X
  39. U U
  40. 𝒫 ( X ) \mathcal{P}\left(X\right)
  41. X X
  42. 3 , 6 , 8 , 8 , and 10 3,6,8,8,\,\text{and }10
  43. 1 n x n n x n \frac{\sum_{1}^{n}x_{n}}{n}\,\forall x_{n}
  44. 3 + 6 + 8 + 8 + 10 5 = 35 5 = 7 \frac{3+6+8+8+10}{5}=\frac{35}{5}=7
  45. ( - 4 ) , ( - 1 ) , 1 , 1 , 3 \left(-4\right),\left(-1\right),1,1,3
  46. 1 n x n n x n \frac{\sum_{1}^{n}x_{n}}{n}\,\forall x_{n}
  47. ( - 4 ) + ( - 1 ) + 1 + 1 + 3 5 = 0 5 = 0 \frac{\left(-4\right)+\left(-1\right)+1+1+3}{5}=\frac{0}{5}=0
  48. x x = x x\wedge x=x
  49. x x = x x\vee x=x
  50. x x

Identical_particles.html

  1. | n 1 | n 2 |n_{1}\rangle|n_{2}\rangle
  2. H H H\otimes H
  3. | n 1 | n 2 ± | n 2 | n 1 |n_{1}\rangle|n_{2}\rangle\pm|n_{2}\rangle|n_{1}\rangle
  4. n 1 n_{1}
  5. n 2 n_{2}
  6. n 1 n_{1}
  7. n 2 n_{2}
  8. | n 1 , n 2 ; S constant × ( | n 1 | n 2 + | n 2 | n 1 ) |n_{1},n_{2};S\rangle\equiv\mbox{constant}~{}\times\bigg(|n_{1}\rangle|n_{2}% \rangle+|n_{2}\rangle|n_{1}\rangle\bigg)
  9. | n 1 , n 2 ; A constant × ( | n 1 | n 2 - | n 2 | n 1 ) |n_{1},n_{2};A\rangle\equiv\mbox{constant}~{}\times\bigg(|n_{1}\rangle|n_{2}% \rangle-|n_{2}\rangle|n_{1}\rangle\bigg)
  10. | n 1 , n 2 ; ? = constant × ( | n 1 | n 2 + i | n 2 | n 1 ) |n_{1},n_{2};?\rangle=\mbox{constant}~{}\times\bigg(|n_{1}\rangle|n_{2}\rangle% +i|n_{2}\rangle|n_{1}\rangle\bigg)
  11. P ( | ψ | ϕ ) | ϕ | ψ P\bigg(|\psi\rangle|\phi\rangle\bigg)\equiv|\phi\rangle|\psi\rangle
  12. P 2 = 1 P^{2}=1
  13. P | n 1 , n 2 ; S = + | n 1 , n 2 ; S P|n_{1},n_{2};S\rangle=+|n_{1},n_{2};S\rangle
  14. P | n 1 , n 2 ; A = - | n 1 , n 2 ; A P|n_{1},n_{2};A\rangle=-|n_{1},n_{2};A\rangle
  15. H = p 1 2 2 m + p 2 2 2 m + U ( | x 1 - x 2 | ) + V ( x 1 ) + V ( x 2 ) H=\frac{p_{1}^{2}}{2m}+\frac{p_{2}^{2}}{2m}+U(|x_{1}-x_{2}|)+V(x_{1})+V(x_{2})
  16. [ P , H ] = 0 \left[P,H\right]=0
  17. | n 1 n 2 n N ; S = 1 N ! n m n ! p | n p ( 1 ) | n p ( 2 ) | n p ( N ) |n_{1}n_{2}\cdots n_{N};S\rangle=\frac{1}{\sqrt{N!\prod_{n}m_{n}!}}\sum_{p}|n_% {p(1)}\rangle|n_{p(2)}\rangle\cdots|n_{p(N)}\rangle
  18. | n 1 n 2 n N ; A = 1 N ! p sgn ( p ) | n p ( 1 ) | n p ( 2 ) | n p ( N ) |n_{1}n_{2}\cdots n_{N};A\rangle=\frac{1}{\sqrt{N!}}\sum_{p}\mathrm{sgn}(p)|n_% {p(1)}\rangle|n_{p(2)}\rangle\cdots|n_{p(N)}\rangle
  19. s g n ( p ) sgn(p)
  20. + 1 +1
  21. p p
  22. - 1 -1
  23. Π n m n \Pi_{n}m_{n}
  24. n 1 n 2 n N ; S | n 1 n 2 n N ; S = 1 , n 1 n 2 n N ; A | n 1 n 2 n N ; A = 1. \langle n_{1}n_{2}\cdots n_{N};S|n_{1}n_{2}\cdots n_{N};S\rangle=1,\qquad% \langle n_{1}n_{2}\cdots n_{N};A|n_{1}n_{2}\cdots n_{N};A\rangle=1.
  25. | n 1 n 2 n N ; S / A |n_{1}n_{2}\cdots n_{N};S/A\rangle
  26. | m 1 m 2 m N ; S / A |m_{1}m_{2}\cdots m_{N};S/A\rangle
  27. P S / A ( n 1 , n N m 1 , m N ) | m 1 m N ; S / A | n 1 n N ; S / A | 2 P_{S/A}(n_{1},\cdots n_{N}\rightarrow m_{1},\cdots m_{N})\equiv\bigg|\langle m% _{1}\cdots m_{N};S/A\,|\,n_{1}\cdots n_{N};S/A\rangle\bigg|^{2}
  28. m 1 m 2 m N P S / A ( n 1 , n N m 1 , m N ) = 1 \sum_{m_{1}\leq m_{2}\leq\dots\leq m_{N}}P_{S/A}(n_{1},\cdots n_{N}\rightarrow m% _{1},\cdots m_{N})=1
  29. | x | ψ | 2 d 3 x |\langle x|\psi\rangle|^{2}\;d^{3}x
  30. x | x = δ 3 ( x - x ) \langle x|x^{\prime}\rangle=\delta^{3}(x-x^{\prime})
  31. | x 1 x 2 x N ; S = j n j ! N ! p | x p ( 1 ) | x p ( 2 ) | x p ( N ) |x_{1}x_{2}\cdots x_{N};S\rangle=\frac{\prod_{j}n_{j}!}{N!}\sum_{p}|x_{p(1)}% \rangle|x_{p(2)}\rangle\cdots|x_{p(N)}\rangle
  32. | x 1 x 2 x N ; A = 1 N ! p sgn ( p ) | x p ( 1 ) | x p ( 2 ) | x p ( N ) |x_{1}x_{2}\cdots x_{N};A\rangle=\frac{1}{N!}\sum_{p}\mathrm{sgn}(p)|x_{p(1)}% \rangle|x_{p(2)}\rangle\cdots|x_{p(N)}\rangle
  33. Ψ n 1 n 2 n N ( S ) ( x 1 , x 2 , x N ) \displaystyle\Psi^{(S)}_{n_{1}n_{2}\cdots n_{N}}(x_{1},x_{2},\cdots x_{N})
  34. Ψ n 1 n 2 n N ( A ) ( x 1 , x 2 , x N ) \displaystyle\Psi^{(A)}_{n_{1}n_{2}\cdots n_{N}}(x_{1},x_{2},\cdots x_{N})
  35. ψ n ( x ) x | n \psi_{n}(x)\equiv\langle x|n\rangle
  36. Ψ n 1 n N ( S ) ( x i x j ) = Ψ n 1 n N ( S ) ( x j x i ) \Psi^{(S)}_{n_{1}\cdots n_{N}}(\cdots x_{i}\cdots x_{j}\cdots)=\Psi^{(S)}_{n_{% 1}\cdots n_{N}}(\cdots x_{j}\cdots x_{i}\cdots)
  37. Ψ n 1 n N ( A ) ( x i x j ) = - Ψ n 1 n N ( A ) ( x j x i ) \Psi^{(A)}_{n_{1}\cdots n_{N}}(\cdots x_{i}\cdots x_{j}\cdots)=-\Psi^{(A)}_{n_% {1}\cdots n_{N}}(\cdots x_{j}\cdots x_{i}\cdots)
  38. N ! | Ψ n 1 n 2 n N ( S / A ) ( x 1 , x 2 , x N ) | 2 d 3 N x N!\;\left|\Psi^{(S/A)}_{n_{1}n_{2}\cdots n_{N}}(x_{1},x_{2},\cdots x_{N})% \right|^{2}\;d^{3N}\!x
  39. | Ψ n 1 n 2 n N ( S / A ) ( x 1 , x 2 , x N ) | 2 d 3 x 1 d 3 x 2 d 3 x N = 1 \int\!\int\!\cdots\!\int\;\left|\Psi^{(S/A)}_{n_{1}n_{2}\cdots n_{N}}(x_{1},x_% {2},\cdots x_{N})\right|^{2}d^{3}\!x_{1}d^{3}\!x_{2}\cdots d^{3}\!x_{N}=1
  40. Ψ n 1 n N ( A ) ( x 1 , x N ) = 1 N ! | ψ n 1 ( x 1 ) ψ n 1 ( x 2 ) ψ n 1 ( x N ) ψ n 2 ( x 1 ) ψ n 2 ( x 2 ) ψ n 2 ( x N ) ψ n N ( x 1 ) ψ n N ( x 2 ) ψ n N ( x N ) | \Psi^{(A)}_{n_{1}\cdots n_{N}}(x_{1},\cdots x_{N})=\frac{1}{\sqrt{N!}}\left|% \begin{matrix}\psi_{n_{1}}(x_{1})&\psi_{n_{1}}(x_{2})&\cdots&\psi_{n_{1}}(x_{N% })\\ \psi_{n_{2}}(x_{1})&\psi_{n_{2}}(x_{2})&\cdots&\psi_{n_{2}}(x_{N})\\ \cdots&\cdots&\cdots&\cdots\\ \psi_{n_{N}}(x_{1})&\psi_{n_{N}}(x_{2})&\cdots&\psi_{n_{N}}(x_{N})\\ \end{matrix}\right|
  41. Z = n 1 , n 2 , n N exp { - 1 k T [ ε ( n 1 ) + ε ( n 2 ) + + ε ( n N ) ] } Z=\sum_{n_{1},n_{2},\cdots n_{N}}\exp\left\{-\frac{1}{kT}\left[\varepsilon(n_{% 1})+\varepsilon(n_{2})+\cdots+\varepsilon(n_{N})\right]\right\}
  42. Z = ξ N Z=\xi^{N}
  43. ξ = n exp [ - ε ( n ) k T ] . \xi=\sum_{n}\exp\left[-\frac{\varepsilon(n)}{kT}\right].
  44. Z = ξ N N ! . Z=\frac{\xi^{N}}{N!}.
  45. S = N k ln ( V ) + N f ( T ) S=Nk\ln\left(V\right)+Nf(T)
  46. S = N k ln ( V N ) + N f ( T ) S=Nk\ln\left(\frac{V}{N}\right)+Nf(T)
  47. | 0 |0\rangle
  48. | 1 |1\rangle
  49. | 0 |0\rangle
  50. | 1 |1\rangle
  51. | 0 | 0 |0\rangle|0\rangle
  52. | 1 | 1 |1\rangle|1\rangle
  53. | 0 | 1 |0\rangle|1\rangle
  54. | 1 | 0 |1\rangle|0\rangle
  55. | 0 |0\rangle
  56. | 1 |1\rangle
  57. | 0 |0\rangle
  58. | 1 |1\rangle
  59. | 0 | 0 |0\rangle|0\rangle
  60. | 1 | 1 |1\rangle|1\rangle
  61. 1 2 ( | 0 | 1 + | 1 | 0 ) \frac{1}{\sqrt{2}}(|0\rangle|1\rangle+|1\rangle|0\rangle)
  62. | 0 |0\rangle
  63. | 1 |1\rangle
  64. | 0 |0\rangle
  65. | 1 |1\rangle
  66. 1 2 ( | 0 | 1 - | 1 | 0 ) \frac{1}{\sqrt{2}}(|0\rangle|1\rangle-|1\rangle|0\rangle)
  67. | 0 |0\rangle
  68. | 1 |1\rangle

Identity_matrix.html

  1. I 1 = [ 1 ] , I 2 = [ 1 0 0 1 ] , I 3 = [ 1 0 0 0 1 0 0 0 1 ] , , I n = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] I_{1}=\begin{bmatrix}1\end{bmatrix},\ I_{2}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix},\ I_{3}=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix},\ \cdots,\ I_{n}=\begin{bmatrix}1&0&0&\cdots&0\\ 0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&1\end{bmatrix}
  2. I m A = A I n = A . I_{m}A=AI_{n}=A.\,
  3. I n = diag ( 1 , 1 , , 1 ) . I_{n}=\mathrm{diag}(1,1,...,1).\,
  4. ( I n ) i j = δ i j . (I_{n})_{ij}=\delta_{ij}.\,

IEEE_754-1985.html

  1. × 10 - 38 \times 10^{-}38
  2. × 10 3 8 \times 10^{3}8
  3. × 10 - 308 \times 10^{-}308
  4. × 10 3 08 \times 10^{3}08
  5. 0.00101 2 = 1.01 2 × 2 - 3 0.00101_{2}=1.01_{2}\times 2^{-3}
  6. × 10 45 \times 10^{−}45
  7. × 10 38 \times 10^{−}38
  8. × 10 3 8 \times 10^{3}8
  9. × 10 324 \times 10^{−}324
  10. × 10 308 \times 10^{−}308
  11. × 10 3 08 \times 10^{3}08
  12. × 10 - 45 \times 10^{-}45
  13. × 10 - 39 \times 10^{-}39
  14. × 10 - 38 \times 10^{-}38
  15. × 10 - 38 \times 10^{-}38
  16. × 10 3 8 \times 10^{3}8
  17. x ( r o u n d ( x / y ) · y ) x–(round(x/y)·y)

Imaginary_number.html

  1. ...
  2. i < s u p > 3 = i i<sup>−3=i
  3. i i
  4. b i bi
  5. 5 i 5i
  6. 25 −25
  7. b i bi
  8. a a
  9. a + b i a+bi
  10. a a
  11. b b
  12. 0
  13. b i bi
  14. + 1 +1
  15. x x
  16. y y
  17. i iℝ
  18. 𝕀 \scriptstyle\mathbb{I}
  19. 1 –1
  20. i i
  21. i −i
  22. - 1 = i 2 = - 1 - 1 = ( - 1 ) ( - 1 ) = 1 = 1 -1=i^{2}=\sqrt{-1}\sqrt{-1}=\sqrt{(-1)(-1)}=\sqrt{1}=1
  23. x y = x y \sqrt{x}\sqrt{y}=\sqrt{xy}
  24. x x
  25. y y
  26. 1 \sqrt{−1}
  27. i i
  28. i −i
  29. ( π / 2 , π / 2 ] (−π/2,π/2]
  30. A r g Arg
  31. ( π , π ] (−π,π]
  32. π < A r g ( x ) + A r g ( y ) π −π<Arg(x)+Arg(y)≤π

Imaginary_unit.html

  1. i i
  2. P ( x ) P(x)
  3. 1 −1
  4. i i
  5. i −i
  6. i i
  7. j j
  8. ι ι
  9. j j
  10. i i
  11. i i
  12. i i
  13. ...
  14. i < s u p > 3 = i i<sup>−3=i
  15. i i
  16. i 2 = - 1 . i^{2}=-1\ .
  17. i i
  18. i i
  19. i −i
  20. i i
  21. i i
  22. i −i
  23. i i
  24. i 3 = i 2 i = ( - 1 ) i = - i i^{3}=i^{2}i=(-1)i=-i\,
  25. i 4 = i 3 i = ( - i ) i = - ( i 2 ) = - ( - 1 ) = 1 i^{4}=i^{3}i=(-i)i=-(i^{2})=-(-1)=1\,
  26. i 5 = i 4 i = ( 1 ) i = i i^{5}=i^{4}i=(1)i=i\,
  27. i 0 = i 1 - 1 = i 1 i - 1 = i 1 1 i = i 1 i = i i = 1 i^{0}=i^{1-1}=i^{1}i^{-1}=i^{1}\frac{1}{i}=i\frac{1}{i}=\frac{i}{i}=1\,
  28. i i
  29. 0 + i 0 +i
  30. i i
  31. i i
  32. i i
  33. i −i
  34. i i
  35. i −i
  36. i i
  37. i i
  38. i i
  39. i −i
  40. i −i
  41. i i
  42. i i
  43. i −i
  44. i −i
  45. + i +i
  46. i −i
  47. ( i ) = + i −(−i)=+i
  48. x x
  49. x x
  50. x −x
  51. X = ( 0 - 1 1 0 ) X=\begin{pmatrix}0&-1\\ 1&\;\;0\end{pmatrix}
  52. X = ( 0 1 - 1 0 ) X=\begin{pmatrix}\;\;0&1\\ -1&0\end{pmatrix}
  53. X 2 = - I = - ( 1 0 0 1 ) = ( - 1 0 0 - 1 ) . X^{2}=-I=-\begin{pmatrix}1&0\\ 0&1\end{pmatrix}=\begin{pmatrix}-1&\;\;0\\ \;\;0&-1\end{pmatrix}.
  54. \mathbf{ℝ}
  55. 1 \sqrt{−1}
  56. x 0 x≥0
  57. - 1 = i i = - 1 - 1 = ( - 1 ) ( - 1 ) = 1 = 1 -1=i\cdot i=\sqrt{-1}\cdot\sqrt{-1}=\sqrt{(-1)\cdot(-1)}=\sqrt{1}=1
  58. - 1 = i i = ± - 1 ± - 1 = ± ( - 1 ) ( - 1 ) = ± 1 = ± 1 -1=i\cdot i=\pm\sqrt{-1}\cdot\pm\sqrt{-1}=\pm\sqrt{(-1)\cdot(-1)}=\pm\sqrt{1}=\pm 1
  59. 1 i = 1 - 1 = 1 - 1 = - 1 1 = - 1 = i \frac{1}{i}=\frac{\sqrt{1}}{\sqrt{-1}}=\sqrt{\frac{1}{-1}}=\sqrt{\frac{-1}{1}}% =\sqrt{-1}=i
  60. a b = a b \sqrt{a}\cdot\sqrt{b}=\sqrt{a\cdot b}
  61. a b = a b \frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}
  62. a a
  63. b b
  64. i 7 i\sqrt{7}
  65. 7 \sqrt{−7}
  66. i i
  67. ( x + i y ) < s u p > 2 = i (x+iy)<sup>2=i
  68. i = ± ( 2 2 + 2 2 i ) = ± 2 2 ( 1 + i ) . \sqrt{i}=\pm\left(\frac{\sqrt{2}}{2}+\frac{\sqrt{2}}{2}i\right)=\pm\frac{\sqrt% {2}}{2}(1+i).
  69. ( ± 2 2 ( 1 + i ) ) 2 = ( ± 2 2 ) 2 ( 1 + i ) 2 = 1 2 ( 1 + 2 i + i 2 ) = 1 2 ( 1 + 2 i - 1 ) = i . \begin{aligned}\displaystyle\left(\pm\frac{\sqrt{2}}{2}(1+i)\right)^{2}&% \displaystyle=\left(\pm\frac{\sqrt{2}}{2}\right)^{2}(1+i)^{2}\\ &\displaystyle=\frac{1}{2}(1+2i+i^{2})\\ &\displaystyle=\frac{1}{2}(1+2i-1)\\ &\displaystyle=i.\\ \end{aligned}
  70. e i x = cos ( x ) + i sin ( x ) e^{ix}=\cos(x)+i\sin(x)\,
  71. x = π / 2 x=π/2
  72. e i ( π / 2 ) = cos ( π / 2 ) + i sin ( π / 2 ) = 0 + i 1 = i . e^{i(\pi/2)}=\cos(\pi/2)+i\sin(\pi/2)=0+i1=i\,\!.
  73. i = ± e i ( π / 4 ) , \sqrt{i}=\pm e^{i(\pi/4)}\,\!,
  74. x = π / 4 x=π/4
  75. i = ± ( cos ( π / 4 ) + i sin ( π / 4 ) ) = 1 ± 2 + i ± 2 = 1 + i ± 2 = ± 2 2 ( 1 + i ) . \begin{aligned}\displaystyle\sqrt{i}&\displaystyle=\pm(\cos(\pi/4)+i\sin(\pi/4% ))\\ &\displaystyle=\frac{1}{\pm\sqrt{2}}+\frac{i}{\pm\sqrt{2}}\\ &\displaystyle=\frac{1+i}{\pm\sqrt{2}}\\ &\displaystyle=\pm\frac{\sqrt{2}}{2}(1+i).\\ \end{aligned}
  76. i −i
  77. e i x = cos ( x ) + i sin ( x ) e^{ix}=\cos(x)+i\sin(x)\,
  78. x = 3 π / 2 x=3π/2
  79. e i ( 3 π / 2 ) = cos ( 3 π / 2 ) + i sin ( 3 π / 2 ) = 0 - i 1 = - i . e^{i(3\pi/2)}=\cos(3\pi/2)+i\sin(3\pi/2)=0-i1=-i\,\!.
  80. - i = ± e i ( 3 π / 4 ) , \sqrt{-i}=\pm e^{i(3\pi/4)}\,\!,
  81. x = 3 π / 4 x=3π/4
  82. - i = ± ( cos ( 3 π / 4 ) + i sin ( 3 π / 4 ) ) = - 1 ± 2 + i 1 ± 2 = - 1 + i ± 2 = ± 2 2 ( i - 1 ) . \begin{aligned}\displaystyle\sqrt{-i}&\displaystyle=\pm(\cos(3\pi/4)+i\sin(3% \pi/4))\\ &\displaystyle=-\frac{1}{\pm\sqrt{2}}+i\frac{1}{\pm\sqrt{2}}\\ &\displaystyle=\frac{-1+i}{\pm\sqrt{2}}\\ &\displaystyle=\pm\frac{\sqrt{2}}{2}(i-1).\\ \end{aligned}
  83. i i
  84. i i
  85. - i = ( i ) ( ± 1 2 ( 1 + i ) ) \displaystyle\sqrt{-i}=(i)\cdot(\pm\frac{1}{\sqrt{2}}(1+i))
  86. i i
  87. i ( a + b i ) = a i + b i 2 = - b + a i . i\,(a+bi)=ai+bi^{2}=-b+ai.
  88. i i
  89. i i
  90. 1 i = 1 i i i = i i 2 = i - 1 = - i . \frac{1}{i}=\frac{1}{i}\cdot\frac{i}{i}=\frac{i}{i^{2}}=\frac{i}{-1}=-i.
  91. i i
  92. a + b i i = - i ( a + b i ) = - a i - b i 2 = b - a i . \frac{a+bi}{i}=-i\,(a+bi)=-ai-bi^{2}=b-ai.
  93. i i
  94. n n
  95. i 4 n = 1 i^{4n}=1\,
  96. i 4 n + 1 = i i^{4n+1}=i\,
  97. i 4 n + 2 = - 1 i^{4n+2}=-1\,
  98. i 4 n + 3 = - i . i^{4n+3}=-i.\,
  99. i n = i n mod 4 i^{n}=i^{n\bmod 4}\,
  100. i n = cos ( n π / 2 ) + i sin ( n π / 2 ) i^{n}=\cos(n\pi/2)+i\sin(n\pi/2)
  101. i i
  102. i i
  103. i i = ( e i ( π / 2 + 2 k π ) ) i = e i 2 ( π / 2 + 2 k π ) = e - ( π / 2 + 2 k π ) i^{i}=\left(e^{i(\pi/2+2k\pi)}\right)^{i}=e^{i^{2}(\pi/2+2k\pi)}=e^{-(\pi/2+2k% \pi)}
  104. k k\in\mathbb{Z}
  105. k = 0 k=0
  106. i i
  107. 1 + i 1+i
  108. i ! = Γ ( 1 + i ) 0.4980 - 0.1549 i . i!=\Gamma(1+i)\approx 0.4980-0.1549i.
  109. | i ! | = π sinh π |i!|=\sqrt{\pi\over\sinh\pi}
  110. i i
  111. n i ni
  112. x n i = cos ( ln x n ) + i sin ( ln x n ) . \!\ x^{ni}=\cos(\ln x^{n})+i\sin(\ln x^{n}).
  113. x n i = cos ( ln x n ) - i sin ( ln x n ) . \!\ \sqrt[ni]{x}=\cos(\ln\sqrt[n]{x})-i\sin(\ln\sqrt[n]{x}).
  114. log i ( x ) = 2 ln x i π . \log_{i}(x)={{2\ln x}\over i\pi}.
  115. i i
  116. i i
  117. cos ( i ) = cosh ( 1 ) = e + 1 / e 2 = e 2 + 1 2 e 1.54308064.... \cos(i)=\cosh(1)={{e+1/e}\over 2}={{e^{2}+1}\over 2e}\approx 1.54308064....
  118. i i
  119. sin ( i ) = i sinh ( 1 ) = e - 1 / e 2 i = e 2 - 1 2 e i 1.17520119 i . \sin(i)=i\sinh(1)\,={{e-1/e}\over 2}\,i={{e^{2}-1}\over 2e}\,i\approx 1.175201% 19\,i....
  120. j j
  121. i ( t ) i(t)
  122. i i
  123. j j
  124. i i
  125. j j
  126. 1 i 1i
  127. 1 j 1j
  128. ι ι
  129. i i
  130. j j
  131. k k
  132. h h

Inclined_plane.html

  1. θ = tan - 1 ( Rise Run ) \theta=\tan^{-1}\bigg(\frac{\,\text{Rise}}{\,\text{Run}}\bigg)\,
  2. MA = F w F i . \mathrm{MA}=\frac{F_{w}}{F_{i}}.\,
  3. W o u t = W i n W_{out}=W_{in}\,
  4. W o u t = F w Rise W_{out}=F_{w}\cdot\,\text{Rise}\,
  5. W i n = F i Length W_{in}=F_{i}\cdot\,\text{Length}\,
  6. MA = F w F i = Length Rise \,\text{MA}=\frac{F_{w}}{F_{i}}=\frac{\,\text{Length}}{\,\text{Rise}}\,
  7. sin θ = Rise Length \sin\theta=\frac{\,\text{Rise}}{\,\text{Length}}\,
  8. MA = F w F i = 1 sin θ \,\text{MA}=\frac{F_{w}}{F_{i}}=\frac{1}{\sin\theta}\,
  9. W in = W fric + W out W\text{in}=W\text{fric}+W\text{out}\,
  10. F f = μ F n F_{f}=\mu F_{n}\,
  11. ϕ = tan - 1 μ \phi=\tan^{-1}\mu\,
  12. MA = F w F i = cos ϕ sin ( θ + ϕ ) \mathrm{MA}=\frac{F_{w}}{F_{i}}=\frac{\cos\phi}{\sin(\theta+\phi)}\,
  13. ϕ = tan - 1 μ \phi=\tan^{-1}\mu\,
  14. MA = F w F i = cos ϕ sin ( θ - ϕ ) \mathrm{MA}=\frac{F_{w}}{F_{i}}=\frac{\cos\phi}{\sin(\theta-\phi)}\,
  15. θ < ϕ \theta<\phi\,
  16. θ = ϕ \theta=\phi\,
  17. θ > ϕ \theta>\phi\,
  18. 𝐫 = R ( cos θ , sin θ ) , \mathbf{r}=R(\cos\theta,\sin\theta),
  19. 𝐯 = V ( cos θ , sin θ ) . \mathbf{v}=V(\cos\theta,\sin\theta).
  20. P in = F V , P_{\mathrm{in}}=FV,\!
  21. P out = 𝐖 𝐯 = ( 0 , W ) V ( cos θ , sin θ ) = W V sin θ . P_{\mathrm{out}}=\mathbf{W}\cdot\mathbf{v}=(0,W)\cdot V(\cos\theta,\sin\theta)% =WV\sin\theta.
  22. MA = W F = 1 sin θ . \mathrm{MA}=\frac{W}{F}=\frac{1}{\sin\theta}.
  23. sin θ = H L , \sin\theta=\frac{H}{L},
  24. MA = W F = L H . \mathrm{MA}=\frac{W}{F}=\frac{L}{H}.
  25. MA = W F = 5 , \mathrm{MA}=\frac{W}{F}=5,
  26. MA = W F = 1804 / 37.50 = 48.1 , \mathrm{MA}=\frac{W}{F}=1804/37.50=48.1,

Income.html

  1. Y Y
  2. x x
  3. y y
  4. P x P_{x}
  5. P y P_{y}
  6. Y = P x x + P y y Y=P_{x}\cdot x+P_{y}\cdot y
  7. P x P y \frac{P_{x}}{P_{y}}
  8. P x P y \frac{P_{x}}{P_{y}}
  9. Y Y

Independence_(probability_theory).html

  1. A B A\perp B
  2. A B A\perp\!\!\!\perp B
  3. P ( A B ) = P ( A ) P ( B ) \mathrm{P}(A\cap B)=\mathrm{P}(A)\mathrm{P}(B)
  4. P ( A B ) = P ( A ) P ( B ) P ( A ) = P ( A ) P ( B ) P ( B ) = P ( A B ) P ( B ) = P ( A B ) \mathrm{P}(A\cap B)=\mathrm{P}(A)\mathrm{P}(B)\Leftrightarrow\mathrm{P}(A)=% \frac{\mathrm{P}(A)\mathrm{P}(B)}{\mathrm{P}(B)}=\frac{\mathrm{P}(A\cap B)}{% \mathrm{P}(B)}=\mathrm{P}(A\mid B)
  5. P ( A B ) = P ( A ) P ( B ) P ( B ) = P ( B A ) \mathrm{P}(A\cap B)=\mathrm{P}(A)\mathrm{P}(B)\Leftrightarrow\mathrm{P}(B)=% \mathrm{P}(B\mid A)
  6. P ( A m A k ) = P ( A m ) P ( A k ) . \mathrm{P}(A_{m}\cap A_{k})=\mathrm{P}(A_{m})\mathrm{P}(A_{k}).
  7. P ( i = 1 n A i ) = i = 1 n P ( A i ) . \mathrm{P}\left(\bigcap_{i=1}^{n}A_{i}\right)=\prod_{i=1}^{n}\mathrm{P}(A_{i}).
  8. F X ( x ) F_{X}(x)
  9. F Y ( y ) F_{Y}(y)
  10. f X ( x ) f_{X}(x)
  11. f Y ( y ) f_{Y}(y)
  12. F X , Y ( x , y ) = F X ( x ) F Y ( y ) , F_{X,Y}(x,y)=F_{X}(x)F_{Y}(y),
  13. f X , Y ( x , y ) = f X ( x ) f Y ( y ) . f_{X,Y}(x,y)=f_{X}(x)f_{Y}(y).
  14. X 1 , , X n X_{1},\ldots,X_{n}
  15. a 1 , , a n a_{1},\ldots,a_{n}
  16. { X 1 a 1 } , , { X n a n } \{X_{1}\leq a_{1}\},\ldots,\{X_{n}\leq a_{n}\}
  17. P ( X x , Y y | Z = z ) = P ( X x | Z = z ) P ( Y y | Z = z ) \mathrm{P}(X\leq x,Y\leq y\;|\;Z=z)=\mathrm{P}(X\leq x\;|\;Z=z)\cdot\mathrm{P}% (Y\leq y\;|\;Z=z)
  18. p X Y | Z ( x , y | z ) = p X | Z ( x | z ) p Y | Z ( y | z ) p_{XY|Z}(x,y|z)=p_{X|Z}(x|z)\cdot p_{Y|Z}(y|z)
  19. P ( X = x | Y = y , Z = z ) = P ( X = x | Z = z ) \mathrm{P}(X=x|Y=y,Z=z)=\mathrm{P}(X=x|Z=z)
  20. P ( A B ) = P ( A ) P ( B ) . \mathrm{P}(A\cap B)=\mathrm{P}(A)\mathrm{P}(B).
  21. ( τ i ) i I (\tau_{i})_{i\in I}
  22. ( A i ) i I i I τ i : P ( i I A i ) = i I P ( A i ) \forall\left(A_{i}\right)_{i\in I}\in\prod\nolimits_{i\in I}\tau_{i}\ :\ % \mathrm{P}\left(\bigcap\nolimits_{i\in I}A_{i}\right)=\prod\nolimits_{i\in I}% \mathrm{P}\left(A_{i}\right)
  23. σ ( { E } ) = { , E , Ω E , Ω } . \sigma(\{E\})=\{\emptyset,E,\Omega\setminus E,\Omega\}.
  24. P ( A ) = P ( A A ) = P ( A ) P ( A ) P ( A ) = 0 or 1 \mathrm{P}(A)=\mathrm{P}(A\cap A)=\mathrm{P}(A)\cdot\mathrm{P}(A)% \Leftrightarrow\mathrm{P}(A)=0\,\text{ or }1
  25. E [ X Y ] = E [ X ] E [ Y ] , E[XY]=E[X]E[Y],
  26. cov [ X , Y ] = E [ X Y ] - E [ X ] E [ Y ] . \,\text{cov}[X,Y]=E[XY]-E[X]E[Y].
  27. φ ( X , Y ) ( t , s ) = φ X ( t ) φ Y ( s ) . \varphi_{(X,Y)}(t,s)=\varphi_{X}(t)\cdot\varphi_{Y}(s).
  28. φ X + Y ( t ) = φ X ( t ) φ Y ( t ) , \varphi_{X+Y}(t)=\varphi_{X}(t)\cdot\varphi_{Y}(t),
  29. P ( A | B C ) = 4 40 4 40 + 1 40 = 4 5 P ( A ) \mathrm{P}(A|BC)=\frac{\frac{4}{40}}{\frac{4}{40}+\frac{1}{40}}=\tfrac{4}{5}% \neq\mathrm{P}(A)
  30. P ( B | A C ) = 4 40 4 40 + 1 40 = 4 5 P ( B ) \mathrm{P}(B|AC)=\frac{\frac{4}{40}}{\frac{4}{40}+\frac{1}{40}}=\tfrac{4}{5}% \neq\mathrm{P}(B)
  31. P ( C | A B ) = 4 40 4 40 + 6 40 = 2 5 P ( C ) \mathrm{P}(C|AB)=\frac{\frac{4}{40}}{\frac{4}{40}+\frac{6}{40}}=\tfrac{2}{5}% \neq\mathrm{P}(C)
  32. P ( A | B C ) = 1 16 1 16 + 1 16 = 1 2 = P ( A ) \mathrm{P}(A|BC)=\frac{\frac{1}{16}}{\frac{1}{16}+\frac{1}{16}}=\tfrac{1}{2}=% \mathrm{P}(A)
  33. P ( B | A C ) = 1 16 1 16 + 1 16 = 1 2 = P ( B ) \mathrm{P}(B|AC)=\frac{\frac{1}{16}}{\frac{1}{16}+\frac{1}{16}}=\tfrac{1}{2}=% \mathrm{P}(B)
  34. P ( C | A B ) = 1 16 1 16 + 3 16 = 1 4 = P ( C ) \mathrm{P}(C|AB)=\frac{\frac{1}{16}}{\frac{1}{16}+\frac{3}{16}}=\tfrac{1}{4}=% \mathrm{P}(C)
  35. P ( A B C ) = P ( A ) P ( B ) P ( C ) , \mathrm{P}(A\cap B\cap C)=\mathrm{P}(A)\mathrm{P}(B)\mathrm{P}(C),

Index.html

  1. x a v a ( x ) b ( v b ( x ) ) 2 x^{a}\mapsto\frac{v^{a}(x)}{\sqrt{\sum_{b}(v^{b}(x))^{2}}}

Indifference_curve.html

  1. U ( x , y ) = x α y 1 - α , 0 α 1 \scriptstyle U\left(x,y\right)=x^{\alpha}y^{1-\alpha},0\leq\alpha\leq 1
  2. U ( x , y ) = α x + β y \scriptstyle U\left(x,y\right)=\alpha x+\beta y
  3. U ( x , y ) = min { α x , β y } \scriptstyle U\left(x,y\right)=\min\{\alpha x,\beta y\}
  4. A A\;
  5. a a\;
  6. b b\;
  7. A A\;
  8. A A\;
  9. a a\;
  10. b b\;
  11. \succeq
  12. A A\;
  13. a b a\succeq b\;
  14. a a\;
  15. b b\;
  16. a a\;
  17. b b\;
  18. a b a\sim b\;
  19. a a\;
  20. b b\;
  21. b b\;
  22. a a\;
  23. a a\;
  24. b b\;
  25. a b a\succ b\;
  26. a a\;
  27. b b\;
  28. b b\;
  29. a a\;
  30. a a\;
  31. b b\;
  32. \succeq
  33. a , b a,b\;
  34. a b a\succeq b\;
  35. b c , b\succeq c,\;
  36. a c a\succeq c\;
  37. a A a\in A\;
  38. 𝒞 a \mathcal{C}_{a}
  39. A A\;
  40. a a
  41. 𝒞 a = { b A : b a } \mathcal{C}_{a}=\{b\in A:b\sim a\}
  42. a a\;
  43. A A\;
  44. x , x,\;
  45. y . y.\;
  46. ( x , y ) \left(x,y\right)
  47. U ( x , y ) U\left(x,y\right)
  48. ( x , y ) \left(x,y\right)
  49. ( x , y ) U ( x , y ) \left(x,y\right)\to U\left(x,y\right)
  50. U ( x , y ) U ( x , y ) U(x,y)\geq U(x^{\prime},y^{\prime})
  51. ( x , y ) \left(x,y\right)
  52. ( x , y ) \left(x^{\prime},y^{\prime}\right)
  53. U ( x , y ) > U ( x , y ) U\left(x,y\right)>U\left(x^{\prime},y^{\prime}\right)
  54. ( x , y ) \left(x,y\right)
  55. ( x , y ) \left(x^{\prime},y^{\prime}\right)
  56. ( x 0 , y 0 ) \left(x_{0},y_{0}\right)
  57. U ( x , y ) U\left(x,y\right)
  58. d U ( x 0 , y 0 ) = U 1 ( x 0 , y 0 ) d x + U 2 ( x 0 , y 0 ) d y dU\left(x_{0},y_{0}\right)=U_{1}\left(x_{0},y_{0}\right)dx+U_{2}\left(x_{0},y_% {0}\right)dy
  59. d U ( x 0 , y 0 ) d x = U 1 ( x 0 , y 0 ) .1 + U 2 ( x 0 , y 0 ) d y d x \frac{dU\left(x_{0},y_{0}\right)}{dx}=U_{1}(x_{0},y_{0}).1+U_{2}(x_{0},y_{0})% \frac{dy}{dx}
  60. U 1 ( x , y ) U_{1}\left(x,y\right)
  61. U ( x , y ) U\left(x,y\right)
  62. ( x , y ) \left(x,y\right)
  63. U 2 ( x , y ) . U_{2}\left(x,y\right).
  64. ( x 0 , y 0 ) \left(x_{0},y_{0}\right)
  65. ( x 0 , y 0 ) \left(x_{0},y_{0}\right)
  66. x x\,
  67. d x dx\,
  68. y y\,
  69. d y dy\,
  70. d U ( x 0 , y 0 ) d x = 0 \frac{dU\left(x_{0},y_{0}\right)}{dx}=0
  71. d U ( x 0 , y 0 ) d x = 0 d y d x = - U 1 ( x 0 , y 0 ) U 2 ( x 0 , y 0 ) \frac{dU\left(x_{0},y_{0}\right)}{dx}=0\Leftrightarrow\frac{dy}{dx}=-\frac{U_{% 1}(x_{0},y_{0})}{U_{2}(x_{0},y_{0})}
  72. ( x 0 , y 0 ) \left(x_{0},y_{0}\right)
  73. x x\,
  74. y y\,
  75. U ( x , y ) = α x + β y U\left(x,y\right)=\alpha x+\beta y
  76. x x\,
  77. U 1 ( x , y ) = α U_{1}\left(x,y\right)=\alpha
  78. y y\,
  79. U 2 ( x , y ) = β U_{2}\left(x,y\right)=\beta
  80. d x d y = - β α . \frac{dx}{dy}=-\frac{\beta}{\alpha}.
  81. x x\,
  82. y y\,
  83. U ( x , y ) = x α y 1 - α U\left(x,y\right)=x^{\alpha}y^{1-\alpha}
  84. x x\,
  85. U 1 ( x , y ) = α ( x / y ) α - 1 U_{1}\left(x,y\right)=\alpha\left(x/y\right)^{\alpha-1}
  86. y y\,
  87. U 2 ( x , y ) = ( 1 - α ) ( x / y ) α U_{2}\left(x,y\right)=(1-\alpha)\left(x/y\right)^{\alpha}
  88. α < 1 \alpha<1
  89. d x d y = - 1 - α α ( x y ) . \frac{dx}{dy}=-\frac{1-\alpha}{\alpha}\left(\frac{x}{y}\right).
  90. U ( x , y ) = ( α x ρ + ( 1 - α ) y ρ ) 1 / ρ U(x,y)=\left(\alpha x^{\rho}+(1-\alpha)y^{\rho}\right)^{1/\rho}
  91. α ( 0 , 1 ) \alpha\in(0,1)
  92. ρ 1 \rho\leq 1
  93. ρ = 1 \rho=1\,
  94. U 1 ( x , y ) = α ( α x ρ + ( 1 - α ) y ρ ) ( 1 / ρ ) - 1 x ρ - 1 U_{1}(x,y)=\alpha\left(\alpha x^{\rho}+(1-\alpha)y^{\rho}\right)^{\left(1/\rho% \right)-1}x^{\rho-1}
  95. U 2 ( x , y ) = ( 1 - α ) ( α x ρ + ( 1 - α ) y ρ ) ( 1 / ρ ) - 1 y ρ - 1 . U_{2}(x,y)=(1-\alpha)\left(\alpha x^{\rho}+(1-\alpha)y^{\rho}\right)^{\left(1/% \rho\right)-1}y^{\rho-1}.
  96. d x d y = - 1 - α α ( x y ) ρ - 1 . \frac{dx}{dy}=-\frac{1-\alpha}{\alpha}\left(\frac{x}{y}\right)^{\rho-1}.

Inductive_logic_programming.html

  1. B B
  2. E + E^{+}
  3. E - E^{-}
  4. h h
  5. B B
  6. ⊧̸ \not\models
  7. E + E^{+}
  8. B h B\land h
  9. \models
  10. E + E^{+}
  11. B h B\land h
  12. ⊧̸ \not\models
  13. 𝑓𝑎𝑙𝑠𝑒 \,\textit{false}
  14. B h E - B\land h\land E^{-}
  15. ⊧̸ \not\models
  16. 𝑓𝑎𝑙𝑠𝑒 \,\textit{false}
  17. h h
  18. h h
  19. E + E^{+}
  20. h h
  21. B B
  22. h h
  23. E - E^{-}
  24. B B
  25. 𝑝𝑎𝑟 : 𝑝𝑎𝑟𝑒𝑛𝑡 \,\textit{par}:\,\textit{parent}
  26. 𝑓𝑒𝑚 : 𝑓𝑒𝑚𝑎𝑙𝑒 \,\textit{fem}:\,\textit{female}
  27. 𝑑𝑎𝑢 : 𝑑𝑎𝑢𝑔ℎ𝑡𝑒𝑟 \,\textit{dau}:\,\textit{daughter}
  28. g : 𝐺𝑒𝑜𝑟𝑔𝑒 g:\,\textit{George}
  29. h : 𝐻𝑒𝑙𝑒𝑛 h:\,\textit{Helen}
  30. m : 𝑀𝑎𝑟𝑦 m:\,\textit{Mary}
  31. t : 𝑇𝑜𝑚 t:\,\textit{Tom}
  32. n : 𝑁𝑎𝑛𝑐𝑦 n:\,\textit{Nancy}
  33. e : 𝐸𝑣𝑒 e:\,\textit{Eve}
  34. 𝑝𝑎𝑟 ( h , m ) 𝑝𝑎𝑟 ( h , t ) 𝑝𝑎𝑟 ( g , m ) 𝑝𝑎𝑟 ( t , e ) 𝑝𝑎𝑟 ( n , e ) 𝑓𝑒𝑚 ( h ) 𝑓𝑒𝑚 ( m ) 𝑓𝑒𝑚 ( n ) 𝑓𝑒𝑚 ( e ) \,\textit{par}(h,m)\land\,\textit{par}(h,t)\land\,\textit{par}(g,m)\land\,% \textit{par}(t,e)\land\,\textit{par}(n,e)\land\,\textit{fem}(h)\land\,\textit{% fem}(m)\land\,\textit{fem}(n)\land\,\textit{fem}(e)
  35. 𝑑𝑎𝑢 ( m , h ) 𝑑𝑎𝑢 ( e , t ) \,\textit{dau}(m,h)\land\,\textit{dau}(e,t)
  36. 𝑡𝑟𝑢𝑒 \,\textit{true}
  37. 𝑑𝑎𝑢 \,\textit{dau}
  38. 𝑑𝑎𝑢 ( m , h ) 𝑝𝑎𝑟 ( h , m ) 𝑝𝑎𝑟 ( h , t ) 𝑝𝑎𝑟 ( g , m ) 𝑝𝑎𝑟 ( t , e ) 𝑝𝑎𝑟 ( n , e ) 𝑓𝑒𝑚 ( h ) 𝑓𝑒𝑚 ( m ) 𝑓𝑒𝑚 ( n ) 𝑓𝑒𝑚 ( e ) \,\textit{dau}(m,h)\leftarrow\,\textit{par}(h,m)\land\,\textit{par}(h,t)\land% \,\textit{par}(g,m)\land\,\textit{par}(t,e)\land\,\textit{par}(n,e)\land\,% \textit{fem}(h)\land\,\textit{fem}(m)\land\,\textit{fem}(n)\land\,\textit{fem}% (e)
  39. 𝑑𝑎𝑢 ( e , t ) 𝑝𝑎𝑟 ( h , m ) 𝑝𝑎𝑟 ( h , t ) 𝑝𝑎𝑟 ( g , m ) 𝑝𝑎𝑟 ( t , e ) 𝑝𝑎𝑟 ( n , e ) 𝑓𝑒𝑚 ( h ) 𝑓𝑒𝑚 ( m ) 𝑓𝑒𝑚 ( n ) 𝑓𝑒𝑚 ( e ) \,\textit{dau}(e,t)\leftarrow\,\textit{par}(h,m)\land\,\textit{par}(h,t)\land% \,\textit{par}(g,m)\land\,\textit{par}(t,e)\land\,\textit{par}(n,e)\land\,% \textit{fem}(h)\land\,\textit{fem}(m)\land\,\textit{fem}(n)\land\,\textit{fem}% (e)
  40. 𝑑𝑎𝑢 ( m , h ) ¬ 𝑝𝑎𝑟 ( h , m ) ¬ 𝑝𝑎𝑟 ( h , t ) ¬ 𝑝𝑎𝑟 ( g , m ) ¬ 𝑝𝑎𝑟 ( t , e ) ¬ 𝑝𝑎𝑟 ( n , e ) ¬ 𝑓𝑒𝑚 ( h ) ¬ 𝑓𝑒𝑚 ( m ) ¬ 𝑓𝑒𝑚 ( n ) ¬ 𝑓𝑒𝑚 ( e ) \,\textit{dau}(m,h)\lor\lnot\,\textit{par}(h,m)\lor\lnot\,\textit{par}(h,t)% \lor\lnot\,\textit{par}(g,m)\lor\lnot\,\textit{par}(t,e)\lor\lnot\,\textit{par% }(n,e)\lor\lnot\,\textit{fem}(h)\lor\lnot\,\textit{fem}(m)\lor\lnot\,\textit{% fem}(n)\lor\lnot\,\textit{fem}(e)
  41. 𝑑𝑎𝑢 ( e , t ) ¬ 𝑝𝑎𝑟 ( h , m ) ¬ 𝑝𝑎𝑟 ( h , t ) ¬ 𝑝𝑎𝑟 ( g , m ) ¬ 𝑝𝑎𝑟 ( t , e ) ¬ 𝑝𝑎𝑟 ( n , e ) ¬ 𝑓𝑒𝑚 ( h ) ¬ 𝑓𝑒𝑚 ( m ) ¬ 𝑓𝑒𝑚 ( n ) ¬ 𝑓𝑒𝑚 ( e ) \,\textit{dau}(e,t)\lor\lnot\,\textit{par}(h,m)\lor\lnot\,\textit{par}(h,t)% \lor\lnot\,\textit{par}(g,m)\lor\lnot\,\textit{par}(t,e)\lor\lnot\,\textit{par% }(n,e)\lor\lnot\,\textit{fem}(h)\lor\lnot\,\textit{fem}(m)\lor\lnot\,\textit{% fem}(n)\lor\lnot\,\textit{fem}(e)
  42. 𝑑𝑎𝑢 ( x m e , x h t ) \,\textit{dau}(x_{me},x_{ht})
  43. 𝑑𝑎𝑢 ( m , h ) \,\textit{dau}(m,h)
  44. 𝑑𝑎𝑢 ( e , t ) \,\textit{dau}(e,t)
  45. ¬ 𝑝𝑎𝑟 ( x h t , x m e ) \lnot\,\textit{par}(x_{ht},x_{me})
  46. ¬ 𝑝𝑎𝑟 ( h , m ) \lnot\,\textit{par}(h,m)
  47. ¬ 𝑝𝑎𝑟 ( t , e ) \lnot\,\textit{par}(t,e)
  48. ¬ 𝑓𝑒𝑚 ( x m e ) \lnot\,\textit{fem}(x_{me})
  49. ¬ 𝑓𝑒𝑚 ( m ) \lnot\,\textit{fem}(m)
  50. ¬ 𝑓𝑒𝑚 ( e ) \lnot\,\textit{fem}(e)
  51. ¬ 𝑝𝑎𝑟 ( g , m ) \lnot\,\textit{par}(g,m)
  52. ¬ 𝑝𝑎𝑟 ( g , m ) \lnot\,\textit{par}(g,m)
  53. ¬ 𝑝𝑎𝑟 ( g , m ) \lnot\,\textit{par}(g,m)
  54. ¬ 𝑝𝑎𝑟 ( x g t , x m e ) \lnot\,\textit{par}(x_{gt},x_{me})
  55. ¬ 𝑝𝑎𝑟 ( g , m ) \lnot\,\textit{par}(g,m)
  56. ¬ 𝑝𝑎𝑟 ( t , e ) \lnot\,\textit{par}(t,e)
  57. x m e , x h t x_{me},x_{ht}
  58. 𝑑𝑎𝑢 ( x m e , x h t ) ¬ 𝑝𝑎𝑟 ( x h t , x m e ) ¬ 𝑓𝑒𝑚 ( x m e ) \,\textit{dau}(x_{me},x_{ht})\lor\lnot\,\textit{par}(x_{ht},x_{me})\lor\lnot\,% \textit{fem}(x_{me})
  59. 𝑑𝑎𝑢 ( x m e , x h t ) 𝑝𝑎𝑟 ( x h t , x m e ) 𝑓𝑒𝑚 ( x m e ) ( all background knowledge facts ) \,\textit{dau}(x_{me},x_{ht})\leftarrow\,\textit{par}(x_{ht},x_{me})\land\,% \textit{fem}(x_{me})\land(\,\text{all background knowledge facts})
  60. h h
  61. x m e x_{me}
  62. x h t x_{ht}
  63. x h t x_{ht}
  64. x m e x_{me}
  65. x m e x_{me}
  66. 𝑑𝑎𝑢 \,\textit{dau}
  67. h h
  68. 𝑝𝑎𝑟 ( h , m ) 𝑓𝑒𝑚 ( m ) \,\textit{par}(h,m)\land\,\textit{fem}(m)
  69. 𝑑𝑎𝑢 ( m , h ) \,\textit{dau}(m,h)
  70. h h
  71. 𝑝𝑎𝑟 ( t , e ) 𝑓𝑒𝑚 ( e ) \,\textit{par}(t,e)\land\,\textit{fem}(e)
  72. 𝑑𝑎𝑢 ( e , t ) \,\textit{dau}(e,t)
  73. h h
  74. h h
  75. 𝑔𝑟𝑎 ( x , z ) 𝑓𝑒𝑚 ( x ) 𝑝𝑎𝑟 ( x , y ) 𝑝𝑎𝑟 ( y , z ) \,\textit{gra}(x,z)\leftarrow\,\textit{fem}(x)\land\,\textit{par}(x,y)\land\,% \textit{par}(y,z)
  76. y y
  77. B , E + , E - B,E^{+},E^{-}
  78. H H
  79. B , E + , E - B,E^{+},E^{-}
  80. B , E + , E - B,E^{+},E^{-}
  81. H H
  82. H H
  83. B B
  84. E E
  85. H H
  86. B H E B ¬ E ¬ H B\land H\models E\iff B\land\neg E\models\neg H
  87. F F
  88. B ¬ E F B\land\neg E\models F
  89. F ¬ H F\models\neg H
  90. H ¬ F H\models\neg F
  91. F F
  92. n n
  93. n n

Inductor.html

  1. L = ϕ i L={\phi\over i}
  2. L = d ϕ d i L={d\phi\over di}\,
  3. v = d ϕ d t v={d\phi\over dt}\,
  4. v = d d t ( L i ) = L d i d t v={d\over dt}(Li)=L{di\over dt}\,
  5. v ( t ) = L d i ( t ) d t v(t)=L\frac{di(t)}{dt}
  6. i ( t ) = I P sin ( 2 π f t ) d i ( t ) d t = 2 π f I P cos ( 2 π f t ) v ( t ) = 2 π f L I P cos ( 2 π f t ) \begin{aligned}\displaystyle i(t)&\displaystyle=I_{\mathrm{P}}\sin(2\pi ft)\\ \displaystyle\frac{di(t)}{dt}&\displaystyle=2\pi fI_{\mathrm{P}}\cos(2\pi ft)% \\ \displaystyle v(t)&\displaystyle=2\pi fLI_{\mathrm{P}}\cos(2\pi ft)\end{aligned}
  7. i ( t ) = I e - R L t i(t)=Ie^{-\frac{R}{L}t}
  8. X L = V P I P = 2 π f L I P I P X_{\mathrm{L}}=\frac{V_{\mathrm{P}}}{I_{\mathrm{P}}}=\frac{2\pi fLI_{\mathrm{P% }}}{I_{\mathrm{P}}}
  9. X L = 2 π f L X_{\mathrm{L}}=2\pi fL
  10. Z ( s ) = L s Z(s)=Ls\,
  11. L L
  12. s s
  13. L I 0 LI_{0}\,
  14. L L
  15. I 0 I_{0}
  16. I 0 s \frac{I_{0}}{s}
  17. I 0 I_{0}
  18. s s
  19. 1 L eq = 1 L 1 + 1 L 2 + + 1 L n \frac{1}{L_{\mathrm{eq}}}=\frac{1}{L_{1}}+\frac{1}{L_{2}}+\cdots+\frac{1}{L_{n}}
  20. L eq = L 1 + L 2 + + L n L_{\mathrm{eq}}=L_{1}+L_{2}+\cdots+L_{n}\,\!
  21. E stored = 1 2 L I 2 E_{\mathrm{stored}}={1\over 2}LI^{2}
  22. t 0 t_{0}
  23. t 1 t_{1}
  24. E = t 0 t 1 P ( t ) d t = 1 2 L I ( t 1 ) 2 - 1 2 L I ( t 0 ) 2 E=\int_{t_{0}}^{t_{1}}\!P(t)\,dt=\frac{1}{2}LI(t_{1})^{2}-\frac{1}{2}LI(t_{0})% ^{2}
  25. Q = ω L R Q=\frac{\omega L}{R}
  26. L = 1 l μ 0 K N 2 A L=\frac{1}{l}\mu_{0}KN^{2}A
  27. π \pi
  28. L = μ 0 2 π ( l ln [ 1 c ( l + l 2 + c 2 ) ] - l 2 + c 2 + c + l 4 + c 2 ρ ω μ ) L=\frac{\mu_{0}}{2\pi}\left(l\ln\left[\frac{1}{c}\left(l+\sqrt{l^{2}+c^{2}}% \right)\right]-\sqrt{l^{2}+c^{2}}+c+\frac{l}{4+c\sqrt{\frac{2}{\rho}\omega\mu}% }\right)
  29. π \pi
  30. L = 1 5 l [ ln ( 4 l d ) - 1 ] L=\frac{1}{5}l\left[\ln\left(\frac{4l}{d}\right)-1\right]
  31. L = 1 5 l [ ln ( 4 l d ) - 3 4 ] L=\frac{1}{5}l\left[\ln\left(\frac{4l}{d}\right)-\frac{3}{4}\right]
  32. L = r 2 N 2 9 r + 10 l L=\frac{r^{2}N^{2}}{9r+10l}
  33. L = 4 5 r 2 N 2 6 r + 9 l + 10 d L=\frac{4}{5}\cdot\frac{r^{2}N^{2}}{6r+9l+10d}
  34. L = r 2 N 2 20 r + 28 d L=\frac{r^{2}N^{2}}{20r+28d}
  35. L = r 2 N 2 8 r + 11 d L=\frac{r^{2}N^{2}}{8r+11d}
  36. L = 0.01595 N 2 ( D - D 2 - d 2 ) L=0.01595N^{2}\left(D-\sqrt{D^{2}-d^{2}}\right)
  37. L 0.007975 d 2 N 2 D L\approx 0.007975{d^{2}N^{2}\over D}
  38. L = 0.00508 N 2 h ln ( d 2 d 1 ) L=0.00508N^{2}h\ln\left({\frac{d_{2}}{d_{1}}}\right)

Inelastic_collision.html

  1. v a = C R m b ( u b - u a ) + m a u a + m b u b m a + m b v_{a}=\frac{C_{R}m_{b}(u_{b}-u_{a})+m_{a}u_{a}+m_{b}u_{b}}{m_{a}+m_{b}}
  2. v b = C R m a ( u a - u b ) + m a u a + m b u b m a + m b v_{b}=\frac{C_{R}m_{a}(u_{a}-u_{b})+m_{a}u_{a}+m_{b}u_{b}}{m_{a}+m_{b}}
  3. v a = - C R u a v_{a}=-C_{R}u_{a}
  4. v b = - C R u b v_{b}=-C_{R}u_{b}
  5. m a u a + m b u b = ( m a + m b ) v m_{a}u_{a}+m_{b}u_{b}=\left(m_{a}+m_{b}\right)v\,
  6. v = m a u a + m b u b m a + m b v=\frac{m_{a}u_{a}+m_{b}u_{b}}{m_{a}+m_{b}}

Inertia.html

  1. p p
  2. v v
  3. p = m v p=mv
  4. F = m a F=ma
  5. m m

Inertial_frame_of_reference.html

  1. 𝐅 = m 𝐚 , \mathbf{F}=m\mathbf{a}\ ,
  2. 𝐅 = m 𝐚 , \mathbf{F}^{\prime}=m\mathbf{a}\ ,
  3. 𝐅 = 𝐅 - 2 m 𝛀 × 𝐯 B - m 𝛀 × ( 𝛀 × 𝐱 B ) - m d 𝛀 d t × 𝐱 B , \mathbf{F}^{\prime}=\mathbf{F}-2m\mathbf{\Omega}\times\mathbf{v}_{B}-m\mathbf{% \Omega}\times(\mathbf{\Omega}\times\mathbf{x}_{B})-m\frac{d\mathbf{\Omega}}{dt% }\times\mathbf{x}_{B}\ ,
  4. 𝐫 = 𝐫 - 𝐫 0 - 𝐯 t \mathbf{r}^{\prime}=\mathbf{r}-\mathbf{r}_{0}-\mathbf{v}t
  5. t = t - t 0 t^{\prime}=t-t_{0}
  6. x = γ ( x - v t ) x^{\prime}=\gamma\left(x-vt\right)
  7. y = y y^{\prime}=y
  8. z = z z^{\prime}=z
  9. t = γ ( t - v x c 0 2 ) t^{\prime}=\gamma\left(t-\frac{vx}{c_{0}^{2}}\right)
  10. x x
  11. γ = def 1 1 - ( v / c 0 ) 2 1. \gamma\ \stackrel{\mathrm{def}}{=}\ \frac{1}{\sqrt{1-(v/c_{0})^{2}}}\ \geq 1.
  12. s 2 = ( x 2 - x 1 ) 2 + ( y 2 - y 1 ) 2 + ( z 2 - z 1 ) 2 - c 0 2 ( t 2 - t 1 ) 2 s^{2}=\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_% {1}\right)^{2}-c_{0}^{2}\left(t_{2}-t_{1}\right)^{2}

Infimum_and_supremum.html

  1. inf { 1 , 2 , 3 } = 1. \inf\,\{1,2,3\}=1.
  2. inf { x : 0 < x < 1 } = 0. \inf\,\{x\in\mathbb{R}:0<x<1\}=0.
  3. inf { x : x 3 > 2 } = 2 3 . \inf\,\{x\in\mathbb{Q}:x^{3}>2\}=\sqrt[3]{2}.
  4. inf { ( - 1 ) n + 1 / n : n = 1 , 2 , 3 , } = - 1. \inf\,\{(-1)^{n}+1/n:n=1,2,3,\dots\}=-1.
  5. inf ( S ) = - sup ( - S ) \inf(S)=-\sup(-S)
  6. - S = { - s | s S } . -S=\{-s|s\in S\}.
  7. S S
  8. S S
  9. M M\in\mathbb{R}
  10. M M
  11. S S
  12. ε > 0 \varepsilon>0
  13. s S s\in S
  14. s > M - ε s>M-\varepsilon
  15. M M
  16. S S
  17. M = sup S M=\sup\ S
  18. \bigvee
  19. \bigwedge
  20. S S
  21. S S
  22. m m\in\mathbb{R}
  23. m m
  24. S S
  25. ε > 0 \varepsilon>0
  26. s S s\in S
  27. s < m + ε s<m+\varepsilon
  28. m m
  29. S S
  30. m = inf S m=\inf\ S
  31. 2 ¯ \overline{2}
  32. 2 ¯ \overline{2}

Infinite_monkey_theorem.html

  1. X n = ( 1 - 1 50 6 ) n . X_{n}=\left(1-\frac{1}{50^{6}}\right)^{n}.
  2. k = 1 P ( E k ) = k = 1 p = , \sum_{k=1}^{\infty}P(E_{k})=\sum_{k=1}^{\infty}p=\infty,

Inflation.html

  1. ( 211.080 - 202.416 202.416 ) × 100 % = 4.28 % \left(\frac{211.080-202.416}{202.416}\right)\times 100\%=4.28\%
  2. GDP Deflator = Nominal GDP Real GDP \mbox{GDP Deflator}~{}=\frac{\mbox{Nominal GDP}~{}}{\mbox{Real GDP}~{}}
  3. M V = P Q MV=PQ
  4. M M
  5. V V
  6. P P
  7. Q Q
  8. P Q PQ

Inflation_(cosmology).html

  1. d s 2 = - ( 1 - Λ r 2 ) d t 2 + 1 1 - Λ r 2 d r 2 + r 2 d Ω 2 . ds^{2}=-(1-\Lambda r^{2})\,dt^{2}+{1\over 1-\Lambda r^{2}}\,dr^{2}+r^{2}\,d% \Omega^{2}.
  2. d t dt
  3. r = 0 r=0
  4. Λ \Lambda
  5. p = - ρ \!p=-\rho
  6. e H t e^{Ht}
  7. Λ \Lambda
  8. ϵ = ( 1 / 2 ) ( V / V ) 2 \epsilon=(1/2)(V^{\prime}/V)^{2}
  9. η = V ′′ / V \eta=V^{\prime\prime}/V
  10. V V

Information_retrieval.html

  1. X Y X\cap Y
  2. | X | |X|
  3. \int
  4. \sum
  5. Δ \Delta
  6. precision = | { relevant documents } { retrieved documents } | | { retrieved documents } | \mbox{precision}~{}=\frac{|\{\mbox{relevant documents}~{}\}\cap\{\mbox{% retrieved documents}~{}\}|}{|\{\mbox{retrieved documents}~{}\}|}
  7. recall = | { relevant documents } { retrieved documents } | | { relevant documents } | \mbox{recall}~{}=\frac{|\{\mbox{relevant documents}~{}\}\cap\{\mbox{retrieved % documents}~{}\}|}{|\{\mbox{relevant documents}~{}\}|}
  8. fall-out = | { non-relevant documents } { retrieved documents } | | { non-relevant documents } | \mbox{fall-out}~{}=\frac{|\{\mbox{non-relevant documents}~{}\}\cap\{\mbox{% retrieved documents}~{}\}|}{|\{\mbox{non-relevant documents}~{}\}|}
  9. ( 1 - specificity ) (1-\mbox{specificity}~{})
  10. F = 2 precision recall ( precision + recall ) F=\frac{2\cdot\mathrm{precision}\cdot\mathrm{recall}}{(\mathrm{precision}+% \mathrm{recall})}
  11. F 1 F_{1}
  12. β \beta
  13. F β = ( 1 + β 2 ) ( precision recall ) ( β 2 precision + recall ) F_{\beta}=\frac{(1+\beta^{2})\cdot(\mathrm{precision}\cdot\mathrm{recall})}{(% \beta^{2}\cdot\mathrm{precision}+\mathrm{recall})}\,
  14. F 2 F_{2}
  15. F 0.5 F_{0.5}
  16. F β F_{\beta}
  17. β \beta
  18. E = 1 - 1 α P + 1 - α R E=1-\frac{1}{\frac{\alpha}{P}+\frac{1-\alpha}{R}}
  19. F β = 1 - E F_{\beta}=1-E
  20. α = 1 1 + β 2 \alpha=\frac{1}{1+\beta^{2}}
  21. p ( r ) p(r)
  22. r r
  23. p ( r ) p(r)
  24. r = 0 r=0
  25. r = 1 r=1
  26. AveP = 0 1 p ( r ) d r \operatorname{AveP}=\int_{0}^{1}p(r)dr
  27. AveP = k = 1 n P ( k ) Δ r ( k ) \operatorname{AveP}=\sum_{k=1}^{n}P(k)\Delta r(k)
  28. k k
  29. n n
  30. P ( k ) P(k)
  31. k k
  32. Δ r ( k ) \Delta r(k)
  33. k - 1 k-1
  34. k k
  35. AveP = k = 1 n ( P ( k ) × rel ( k ) ) number of relevant documents \operatorname{AveP}=\frac{\sum_{k=1}^{n}(P(k)\times\operatorname{rel}(k))}{% \mbox{number of relevant documents}~{}}\!
  36. rel ( k ) \operatorname{rel}(k)
  37. k k
  38. p ( r ) p(r)
  39. AveP = 1 11 r { 0 , 0.1 , , 1.0 } p interp ( r ) \operatorname{AveP}=\frac{1}{11}\sum_{r\in\{0,0.1,\ldots,1.0\}}p_{% \operatorname{interp}}(r)
  40. p interp ( r ) p_{\operatorname{interp}}(r)
  41. r r
  42. p interp ( r ) = max r ~ : r ~ r p ( r ~ ) p_{\operatorname{interp}}(r)=\operatorname{max}_{\tilde{r}:\tilde{r}\geq r}p(% \tilde{r})
  43. p ( r ) p(r)
  44. R R
  45. r r
  46. r / R = r / 15 r/R=r/15
  47. MAP = q = 1 Q AveP ( q ) Q \operatorname{MAP}=\frac{\sum_{q=1}^{Q}\operatorname{AveP(q)}}{Q}\!
  48. p p
  49. DCG p = r e l 1 + i = 2 p r e l i log 2 i . \mathrm{DCG_{p}}=rel_{1}+\sum_{i=2}^{p}\frac{rel_{i}}{\log_{2}i}.
  50. I D C G p IDCG_{p}
  51. nDCG p = D C G p I D C G p . \mathrm{nDCG_{p}}=\frac{DCG_{p}}{IDCG{p}}.
  52. D C G p DCG_{p}
  53. I D C G p IDCG_{p}

Information_theory.html

  1. W = K log m W=K\log m
  2. H = log S n = n log S H=\log S^{n}=n\log S
  3. p log p p\log p\,
  4. p = 0. p=0.
  5. lim p 0 + p log p = 0 \lim_{p\rightarrow 0+}p\log p=0
  6. H H
  7. X X
  8. X X
  9. 1 1
  10. 0
  11. 𝕏 \mathbb{X}
  12. { x 1 , , x n } \{x_{1},...,x_{n}\}
  13. X X
  14. p ( x ) p(x)
  15. x 𝕏 x\in\mathbb{X}
  16. H H
  17. X X
  18. H ( X ) = 𝔼 X [ I ( x ) ] = - x 𝕏 p ( x ) log p ( x ) . H(X)=\mathbb{E}_{X}[I(x)]=-\sum_{x\in\mathbb{X}}p(x)\log p(x).
  19. I ( x ) I(x)
  20. 𝔼 X \mathbb{E}_{X}
  21. p ( x ) = 1 / n p(x)=1/n
  22. H ( X ) = log n H(X)=\log n
  23. H b ( p ) = - p log 2 p - ( 1 - p ) log 2 ( 1 - p ) . H_{\mathrm{b}}(p)=-p\log_{2}p-(1-p)\log_{2}(1-p).
  24. X X
  25. Y Y
  26. ( X , Y ) (X,Y)
  27. X X
  28. Y Y
  29. ( X , Y ) (X,Y)
  30. X X
  31. Y Y
  32. H ( X , Y ) = 𝔼 X , Y [ - log p ( x , y ) ] = - x , y p ( x , y ) log p ( x , y ) H(X,Y)=\mathbb{E}_{X,Y}[-\log p(x,y)]=-\sum_{x,y}p(x,y)\log p(x,y)\,
  33. X X
  34. Y Y
  35. X X
  36. Y Y
  37. Y Y
  38. H ( X | Y ) = 𝔼 Y [ H ( X | y ) ] = - y Y p ( y ) x X p ( x | y ) log p ( x | y ) = - x , y p ( x , y ) log p ( x , y ) p ( y ) . H(X|Y)=\mathbb{E}_{Y}[H(X|y)]=-\sum_{y\in Y}p(y)\sum_{x\in X}p(x|y)\log p(x|y)% =-\sum_{x,y}p(x,y)\log\frac{p(x,y)}{p(y)}.
  39. H ( X | Y ) = H ( X , Y ) - H ( Y ) . H(X|Y)=H(X,Y)-H(Y).\,
  40. X X
  41. Y Y
  42. I ( X ; Y ) = 𝔼 X , Y [ S I ( x , y ) ] = x , y p ( x , y ) log p ( x , y ) p ( x ) p ( y ) I(X;Y)=\mathbb{E}_{X,Y}[SI(x,y)]=\sum_{x,y}p(x,y)\log\frac{p(x,y)}{p(x)\,p(y)}
  43. S I SI
  44. I ( X ; Y ) = H ( X ) - H ( X | Y ) . I(X;Y)=H(X)-H(X|Y).\,
  45. I ( X ; Y ) I(X;Y)
  46. I ( X ; Y ) = I ( Y ; X ) = H ( X ) + H ( Y ) - H ( X , Y ) . I(X;Y)=I(Y;X)=H(X)+H(Y)-H(X,Y).\,
  47. I ( X ; Y ) = 𝔼 p ( y ) [ D KL ( p ( X | Y = y ) p ( X ) ) ] . I(X;Y)=\mathbb{E}_{p(y)}[D_{\mathrm{KL}}(p(X|Y=y)\|p(X))].
  48. I ( X ; Y ) = D KL ( p ( X , Y ) p ( X ) p ( Y ) ) . I(X;Y)=D_{\mathrm{KL}}(p(X,Y)\|p(X)p(Y)).
  49. D KL ( p ( X ) q ( X ) ) = x X - p ( x ) log q ( x ) - x X - p ( x ) log p ( x ) = x X p ( x ) log p ( x ) q ( x ) . D_{\mathrm{KL}}(p(X)\|q(X))=\sum_{x\in X}-p(x)\log{q(x)}\,-\,\sum_{x\in X}-p(x% )\log{p(x)}=\sum_{x\in X}p(x)\log\frac{p(x)}{q(x)}.
  50. r = lim n H ( X n | X n - 1 , X n - 2 , X n - 3 , ) ; r=\lim_{n\to\infty}H(X_{n}|X_{n-1},X_{n-2},X_{n-3},\ldots);
  51. r = lim n 1 n H ( X 1 , X 2 , X n ) ; r=\lim_{n\to\infty}\frac{1}{n}H(X_{1},X_{2},\dots X_{n});
  52. p ( y | x ) p(y|x)
  53. p ( y | x ) p(y|x)
  54. f ( x ) f(x)
  55. C = max f I ( X ; Y ) . C=\max_{f}I(X;Y).\!
  56. 1 - H b ( p ) 1-H_{\mbox{b}}~{}(p)
  57. H b H_{\mbox{b}}~{}

Infrared.html

  1. E = J ( T , n ) E=J(T,n)

Infrared_spectroscopy.html

  1. ν = 1 2 π c k μ \nu=\frac{1}{2\pi c}\sqrt{\frac{k}{\mu}}
  2. μ = m A m B m A + m B \mu=\frac{m_{A}m_{B}}{m_{A}+m_{B}}
  3. m i m_{i}
  4. i i
  5. ν ( 16 O ) ν ( 18 O ) = 9 8 832 788 . \frac{\nu(^{16}O)}{\nu(^{18}O)}=\sqrt{\frac{9}{8}}\approx\frac{832}{788}.
  6. ν \nu
  7. τ 1 \tau_{1}
  8. τ 2 \tau_{2}
  9. τ 1 \tau_{1}

Injective_function.html

  1. a , b A , f ( a ) = f ( b ) a = b \forall a,b\in A,\;\;f(a)=f(b)\Rightarrow a=b
  2. a , b A , a b f ( a ) f ( b ) \forall a,b\in A,\;\;a\neq b\Rightarrow f(a)\neq f(b)
  3. Y Y
  4. X X
  5. X X
  6. Y Y

Inner_automorphism.html

  1. G {}_{G}
  2. 𝔤 \mathfrak{g}
  3. 𝔤 \mathfrak{g}

Inner_product_space.html

  1. F F
  2. 𝐑 \mathbf{R}
  3. 𝐂 \mathbf{C}
  4. V V
  5. F F
  6. , : V × V F \langle\cdot,\cdot\rangle:V\times V\to F
  7. x , y , z V x,y,z\in V
  8. a F a\in F
  9. x , y = y , x ¯ \langle x,y\rangle=\overline{\langle y,x\rangle}
  10. a x , y = a x , y \langle ax,y\rangle=a\langle x,y\rangle
  11. x + y , z = x , z + y , z \langle x+y,z\rangle=\langle x,z\rangle+\langle y,z\rangle
  12. x , x 0 \langle x,x\rangle\geq 0
  13. x , x = 0 x = 0 \langle x,x\rangle=0\Rightarrow x=0
  14. x , y \langle x,y\rangle
  15. y | x \langle y|x\rangle
  16. y x y^{\dagger}x
  17. A B AB
  18. A A
  19. B B
  20. V V
  21. x , y \langle x,y\rangle
  22. x x
  23. y y
  24. , \langle\cdot,\cdot\rangle
  25. | \langle\cdot|\cdot\rangle
  26. 𝐑 \mathbf{R}
  27. 𝐂 \mathbf{C}
  28. 𝐑 \mathbf{R}
  29. 𝐂 \mathbf{C}
  30. 𝐑 \mathbf{R}
  31. 𝐂 \mathbf{C}
  32. 𝐑 \mathbf{R}
  33. 𝐂 \mathbf{C}
  34. x , x \langle x,x\rangle
  35. F = 𝐑 F=\mathbf{R}
  36. x , y = y , x \langle x,y\rangle=\langle y,x\rangle
  37. F = 𝐑 F=\mathbf{R}
  38. F = 𝐂 F=\mathbf{C}
  39. x , y \langle x,y\rangle
  40. x , x \langle x,x\rangle
  41. x x
  42. x , x = x , x ¯ . \langle x,x\rangle=\overline{\langle x,x\rangle}.
  43. - x , x = - 1 x , x = - 1 ¯ x , x = x , - x . \langle-x,x\rangle=-1\langle x,x\rangle=\overline{-1}\langle x,x\rangle=% \langle x,-x\rangle.
  44. x , a y = a y , x ¯ = a ¯ y , x ¯ = a ¯ x , y \langle x,ay\rangle=\overline{\langle ay,x\rangle}=\overline{a}\overline{% \langle y,x\rangle}=\overline{a}\langle x,y\rangle
  45. x , y + z = y + z , x ¯ = y , x ¯ + z , x ¯ = x , y + x , z , \langle x,y+z\rangle=\overline{\langle y+z,x\rangle}=\overline{\langle y,x% \rangle}+\overline{\langle z,x\rangle}=\langle x,y\rangle+\langle x,z\rangle,
  46. F = 𝐑 F=\mathbf{R}
  47. x = 0 x=0
  48. x , x = 0 , \langle x,x\rangle=0,
  49. x , x = 0 \langle x,x\rangle=0
  50. x = 0 x=0
  51. x , x = 0 \langle x,x\rangle=0
  52. x = 0 x=0
  53. x + y , x + y = x , x + x , y + y , x + y , y . \langle x+y,x+y\rangle=\langle x,x\rangle+\langle x,y\rangle+\langle y,x% \rangle+\langle y,y\rangle.
  54. 𝐑 \mathbf{R}
  55. x ± y , x ± y = x , x ± 2 x , y + y , y , \langle x\pm y,x\pm y\rangle=\langle x,x\rangle\pm 2\langle x,y\rangle+\langle y% ,y\rangle,
  56. V V
  57. x + y , z = x , z + y , z , \langle x+y,z\rangle=\langle x,z\rangle+\langle y,z\rangle,
  58. x , y + z = x , y + x , z \langle x,y+z\rangle=\langle x,y\rangle+\langle x,z\rangle
  59. x , y := x y . \langle x,y\rangle:=xy.
  60. n n
  61. n n
  62. [ x 1 x n ] , [ y 1 y n ] := x T y = i = 1 n x i y i = x 1 y 1 + + x n y n , \left\langle\begin{bmatrix}x_{1}\\ \vdots\\ x_{n}\end{bmatrix},\begin{bmatrix}y_{1}\\ \vdots\\ y_{n}\end{bmatrix}\right\rangle:=x\text{T}y=\sum_{i=1}^{n}x_{i}y_{i}=x_{1}y_{1% }+\cdots+x_{n}y_{n},
  63. x < s u p > T x<sup>T
  64. 𝐱 , 𝐲 := 𝐲 𝐌𝐱 = 𝐱 𝐌𝐲 ¯ , \langle\mathbf{x},\mathbf{y}\rangle:=\mathbf{y}^{\dagger}\mathbf{M}\mathbf{x}=% \overline{\mathbf{x}^{\dagger}\mathbf{M}\mathbf{y}},
  65. 𝐌 \mathbf{M}
  66. 𝐲 \mathbf{y}
  67. a a , b aa,b
  68. f , g := a b f ( t ) g ( t ) ¯ d t . \langle f,g\rangle:=\int_{a}^{b}f(t)\overline{g(t)}\,dt.
  69. [ 1 , 1 ] [−1,1]
  70. f k ( t ) = { 0 t [ - 1 , 0 ] 1 t [ 1 k , 1 ] k t t ( 0 , 1 k ) f_{k}(t)=\begin{cases}0&t\in[-1,0]\\ 1&t\in\left[\tfrac{1}{k},1\right]\\ kt&t\in\left(0,\tfrac{1}{k}\right)\end{cases}
  71. X X
  72. Y Y
  73. X , Y := E ( X Y ) \langle X,Y\rangle:=\operatorname{E}(XY)
  74. A , B := tr ( A B T ) \langle A,B\rangle:=\mathrm{tr}(AB\text{T})
  75. ( A , B = B T , A T ) \left(\langle A,B\rangle=\langle B\text{T},A\text{T}\rangle\right)
  76. x p = ( i = 1 | ξ i | p ) 1 p x = { ξ i } p , p 2 , \|x\|_{p}=\left(\sum_{i=1}^{\infty}\left|\xi_{i}\right|^{p}\right)^{\frac{1}{p% }}\qquad x=\left\{\xi_{i}\right\}\in\ell^{p},p\neq 2,
  77. x = x , x . \|x\|=\sqrt{\langle x,x\rangle}.
  78. x x
  79. x x
  80. y y
  81. V V
  82. | x , y | x y |\langle x,y\rangle|\leq\|x\|\cdot\|y\|
  83. x x
  84. y y
  85. x x
  86. y y
  87. F = 𝐑 F=\mathbf{R}
  88. angle ( x , y ) = arccos x , y x y . \operatorname{angle}(x,y)=\arccos\frac{\langle x,y\rangle}{\|x\|\cdot\|y\|}.
  89. [ 0 , π ] [0, π]
  90. F = 𝐂 F=\mathbf{C}
  91. [ 0 , π / 2 ] [0, π/2]
  92. angle ( x , y ) = arccos | x , y | x y . \operatorname{angle}(x,y)=\arccos\frac{|\langle x,y\rangle|}{\|x\|\cdot\|y\|}.
  93. x x
  94. y y
  95. V V
  96. x x
  97. V V
  98. r r
  99. r x = | r | x . \|r\cdot x\|=|r|\cdot\|x\|.
  100. x x
  101. y y
  102. V V
  103. x + y x + y . \|x+y\|\leq\|x\|+\|y\|.
  104. V V
  105. V V
  106. V V
  107. V V
  108. x x
  109. y y
  110. x 2 + y 2 = x + y 2 . \|x\|^{2}+\|y\|^{2}=\|x+y\|^{2}.
  111. x j , x k = 0 \langle x_{j},x_{k}\rangle=0
  112. j j
  113. k k
  114. i = 1 n x i 2 = i = 1 n x i 2 . \sum_{i=1}^{n}\|x_{i}\|^{2}=\left\|\sum_{i=1}^{n}x_{i}\right\|^{2}.
  115. , \langle\cdot,\cdot\rangle
  116. V × V V×V
  117. F F
  118. V V
  119. V V
  120. i = 1 x i 2 = i = 1 x i 2 , \sum_{i=1}^{\infty}\|x_{i}\|^{2}=\left\|\sum_{i=1}^{\infty}x_{i}\right\|^{2},
  121. S k = i = 1 k x i , S_{k}=\sum_{i=1}^{k}x_{i},
  122. x x
  123. y y
  124. V V
  125. x + y 2 + x - y 2 = 2 x 2 + 2 y 2 . \|x+y\|^{2}+\|x-y\|^{2}=2\|x\|^{2}+2\|y\|^{2}.
  126. x + y 2 = x 2 + y 2 + 2 \real x , y . \|x+y\|^{2}=\|x\|^{2}+\|y\|^{2}+2\real\langle x,y\rangle.
  127. V V
  128. n n
  129. V V
  130. n n
  131. { e 1 , , e n } \textstyle{\{e_{1},\ldots,e_{n}\}}
  132. e i , e j = 0 \textstyle\langle e_{i},e_{j}\rangle=0
  133. i j \textstyle i\neq j
  134. e i , e i = e i = 1 \textstyle\langle e_{i},e_{i}\rangle=\|e_{i}\|=1
  135. V V
  136. E = { e α } α A E=\left\{e_{\alpha}\right\}_{\alpha\in A}
  137. V V
  138. V V
  139. E E
  140. V V
  141. E E
  142. V V
  143. e α , e β = 0 \left\langle e_{\alpha},e_{\beta}\right\rangle=0
  144. α β \textstyle\alpha\neq\beta
  145. e α , e α = e α = 1 \textstyle\langle e_{\alpha},e_{\alpha}\rangle=\|e_{\alpha}\|=1
  146. α , β A \textstyle\alpha,\beta\in A
  147. V V
  148. V V
  149. G G
  150. H H
  151. G G
  152. H H
  153. H H
  154. G G
  155. H H
  156. K K
  157. 0 \aleph_{0}
  158. E E
  159. K K
  160. | E | = 0 |E|=\aleph_{0}
  161. E E
  162. E F E\cup F
  163. K K
  164. E F = E\cap F=\emptyset
  165. K K
  166. c c
  167. | F | = c |F|=c
  168. L L
  169. c c
  170. B B
  171. L L
  172. ϕ : F B \phi:F\to B
  173. T : K L T:K\to L
  174. T f = ϕ ( f ) Tf=\phi(f)
  175. f F f\in F
  176. T e = 0 Te=0
  177. e E e\in E
  178. H = K L H=K\oplus L
  179. G = { ( k , T k ) : k K ) } G=\{(k,Tk):k\in K)\}
  180. T T
  181. G ¯ \bar{G}
  182. G G
  183. H H
  184. G ¯ = H \bar{G}=H
  185. e E e\in E
  186. ( e , 0 ) G (e,0)\in G
  187. K 0 G ¯ K\oplus 0\subset\bar{G}
  188. b B b\in B
  189. b = T f b=Tf
  190. f F K f\in F\subset K
  191. ( f , b ) G G ¯ (f,b)\in G\subset\bar{G}
  192. ( f , 0 ) G ¯ (f,0)\in\bar{G}
  193. ( 0 , b ) G ¯ (0,b)\in\bar{G}
  194. 0 L G ¯ 0\oplus L\subset\bar{G}
  195. G ¯ = H \bar{G}=H
  196. G G
  197. H H
  198. { ( e , 0 ) : e E } \{(e,0):e\in E\}
  199. G G
  200. 0 = ( e , 0 ) , ( k , T k ) = e , k + 0 , T k = e , k 0=\langle(e,0),(k,Tk)\rangle=\langle e,k\rangle+\langle 0,Tk\rangle=\langle e,k\rangle
  201. e E e\in E
  202. k = 0 k=0
  203. ( k , T k ) = ( 0 , 0 ) (k,Tk)=(0,0)
  204. G G
  205. G G
  206. | E | = 0 |E|=\aleph_{0}
  207. H H
  208. c c
  209. V V
  210. V V
  211. x { e k , x } k x\mapsto\{\langle e_{k},x\rangle\}_{k\in\mathbb{N}}
  212. V V
  213. C [ - π , π ] C[-\pi,\pi]
  214. e k ( t ) = e i k t 2 π e_{k}(t)=\frac{e^{ikt}}{\sqrt{2\pi}}
  215. C [ - π , π ] C[-\pi,\pi]
  216. f 1 2 π { - π π f ( t ) e - i k t d t } k f\mapsto\frac{1}{\sqrt{2\pi}}\left\{\int_{-\pi}^{\pi}f(t)e^{-ikt}\,dt\right\}_% {k\in\mathbb{Z}}
  217. - π π e - i ( j - k ) t d t = 0. \int_{-\pi}^{\pi}e^{-i(j-k)t}\,dt=0.
  218. [ - π , π ] [-\pi,\pi]
  219. A A
  220. V V
  221. W W
  222. A A
  223. A A
  224. x x
  225. V V
  226. A A
  227. A x , y = x , A y \langle Ax,y\rangle=\langle x,Ay\rangle
  228. x x
  229. y y
  230. V V
  231. A A
  232. A x , A y = x , y \langle Ax,Ay\rangle=\langle x,y\rangle
  233. x x
  234. y y
  235. V V
  236. A A
  237. A x = x ‖Ax‖=‖x‖
  238. x x
  239. V V
  240. A A
  241. V V
  242. , \langle\cdot,\cdot\rangle
  243. x = x , x \|x\|=\sqrt{\langle x,x\rangle}
  244. x = 0 x=0
  245. , \langle\cdot,\cdots\rangle
  246. W W
  247. x x
  248. y y
  249. x , y 0 , \langle x,y\rangle\neq 0,
  250. y y
  251. x x
  252. W × V * Hom ( V , W ) W\times V^{*}\to\operatorname{Hom}(V,W)
  253. V * × V F V^{*}\times V\to F

Insertion_loss.html

  1. 10 log 10 P T P R 10\log_{10}{P_{\mathrm{T}}\over P_{\mathrm{R}}}
  2. Insertion loss (dB) = 10 log 10 | V 1 | 2 | V 2 | 2 = 20 log 10 | V 1 | | V 2 | \mbox{Insertion loss (dB)}~{}=10\log_{10}{\left|V_{1}\right|^{2}\over\left|V_{% 2}\right|^{2}}=20\log_{10}{\left|V_{1}\right|\over\left|V_{2}\right|}
  3. I L IL
  4. I L = - 20 log 10 | S 21 | dB IL=-20\log_{10}\left|S_{21}\right|\,\,\text{dB}
  5. I L = 10 log 10 | S 21 | 2 1 - | S 11 | 2 IL=10\log_{10}\frac{\left|S_{21}\right|^{2}}{1-\left|S_{11}\right|^{2}}\,
  6. S 11 S_{11}
  7. S 21 S_{21}

Integer.html

  1. 2 \sqrt{2}
  2. \mathbb{Z}
  3. ( a , b ) ( c , d ) (a,b)\sim(c,d)\,\!
  4. a + d = b + c . a+d=b+c.\,\!
  5. [ ( a , b ) ] + [ ( c , d ) ] := [ ( a + c , b + d ) ] . [(a,b)]+[(c,d)]:=[(a+c,b+d)].\,
  6. [ ( a , b ) ] [ ( c , d ) ] := [ ( a c + b d , a d + b c ) ] . [(a,b)]\cdot[(c,d)]:=[(ac+bd,ad+bc)].\,
  7. - [ ( a , b ) ] := [ ( b , a ) ] . -[(a,b)]:=[(b,a)].\,
  8. [ ( a , b ) ] - [ ( c , d ) ] := [ ( a + d , b + c ) ] . [(a,b)]-[(c,d)]:=[(a+d,b+c)].\,
  9. [ ( a , b ) ] < [ ( c , d ) ] [(a,b)]<[(c,d)]\,
  10. a + d < b + c . a+d<b+c.\,
  11. { a - b , if a b - ( b - a ) , if a < b . \begin{cases}a-b,&\mbox{if }~{}a\geq b\\ -(b-a),&\mbox{if }~{}a<b.\end{cases}
  12. 0 \displaystyle 0
  13. 0 \aleph_{0}
  14. f ( x ) = { 2 | x | , if x 0 2 x - 1 , if x > 0. f(x)=\begin{cases}2|x|,&\mbox{if }~{}x\leq 0\\ 2x-1,&\mbox{if }~{}x>0.\end{cases}
  15. g ( x ) = { 2 | x | , if x < 0 2 x + 1 , if x 0. g(x)=\begin{cases}2|x|,&\mbox{if }~{}x<0\\ 2x+1,&\mbox{if }~{}x\geq 0.\end{cases}

Integer_factorization.html

  1. O ( exp ( ( 64 9 b ) 1 3 ( log b ) 2 3 ) ) . O\left(\exp\left(\left(\begin{matrix}\frac{64}{9}\end{matrix}b\right)^{1\over 3% }(\log b)^{2\over 3}\right)\right).
  2. M = N M=\sqrt{N}
  3. log N \log{N}
  4. L n [ 1 / 2 , 1 + o ( 1 ) ] = e ( 1 + o ( 1 ) ) ( log n ) 1 2 ( log log n ) 1 2 L_{n}\left[1/2,1+o(1)\right]=e^{(1+o(1))(\log n)^{\frac{1}{2}}(\log\log n)^{% \frac{1}{2}}}
  5. L n [ 1 / 2 , 1 + o ( 1 ) ] L_{n}\left[1/2,1+o(1)\right]
  6. ( Δ q ) = 1 \left(\tfrac{\Delta}{q}\right)=1
  7. c 0 ( log | Δ | ) 2 c_{0}(\log|\Delta|)^{2}
  8. c 0 c_{0}
  9. p 1 = 2 , p 2 = 3 , p 3 = 5 , , p t p_{1}=2,p_{2}=3,p_{3}=5,\dots,p_{t}
  10. t t\in{\mathbb{N}}
  11. f q f_{q}
  12. ( Δ q ) = 1 \left(\tfrac{\Delta}{q}\right)=1
  13. ( x X x r ( x ) ) . ( q P Δ f q t ( q ) ) = 1 \left(\prod_{x\in X}x^{r(x)}\right).\left(\prod_{q\in P_{\Delta}}f^{t(q)}_{q}% \right)=1
  14. L n [ 1 / 2 , 1 + o ( 1 ) ] L_{n}\left[1/2,1+o(1)\right]

Integral.html

  1. f f
  2. x x
  3. a a , b aa,b
  4. a b f ( x ) d x \int_{a}^{b}\!f(x)\,dx
  5. x y xy
  6. f f
  7. x x
  8. x = a x=a
  9. x = b x=b
  10. x x
  11. x x
  12. F F
  13. f f
  14. F ( x ) = f ( x ) d x . F(x)=\int f(x)\,dx.
  15. f f
  16. a a , b aa,b
  17. F F
  18. f f
  19. f f
  20. a b f ( x ) d x = F ( b ) - F ( a ) . \int_{a}^{b}\!f(x)\,dx=F(b)-F(a).
  21. a a , b aa,b
  22. n = 9 n=9
  23. x x
  24. [ u o v e r s e t , u m a t h b f . , u x ] [u^{\prime}overset^{\prime},u^{\prime}\\ mathbf{.}^{\prime},u^{\prime}x^{\prime}]
  25. x x′
  26. x x
  27. f ( x ) f(x)
  28. f ( x ) d x . \int f(x)\,dx.
  29. d x dx
  30. x x
  31. f ( x ) f(x)
  32. d x dx
  33. d d
  34. d x dx
  35. d x dx
  36. d x dx
  37. f ( x ) f(x)
  38. D D
  39. D f ( x ) d x \int_{D}f(x)\,dx
  40. a a , b aa,b
  41. a b f ( x ) d x \displaystyle\int_{a}^{b}f(x)\,dx
  42. D D
  43. a a , b aa,b
  44. d x dx
  45. d x dx
  46. x x
  47. d x dx
  48. x x
  49. d x dx
  50. x x
  51. d x dx
  52. x x
  53. μ μ
  54. d μ ( dμ(
  55. y = f ( x ) y=f(x)
  56. x = 0 x=0
  57. x = 1 x=1
  58. f ( x ) = x f(x)=\sqrt{x}
  59. f f
  60. f f
  61. 0 1 x d x . \int_{0}^{1}\sqrt{x}\,dx\,\!.
  62. x = 0 x=0
  63. x = 1 x=1
  64. y = f ( 0 ) = 0 y=f(0)=0
  65. y = f ( 1 ) = 1 y=f(1)=1
  66. 1 / 5 \sqrt{1/5}
  67. 2 / 5 \sqrt{2/5}
  68. 1 = 1 \sqrt{1}=1
  69. 1 5 ( 1 5 - 0 ) + 2 5 ( 2 5 - 1 5 ) + + 5 5 ( 5 5 - 4 5 ) 0.7497. \textstyle\sqrt{\frac{1}{5}}\left(\frac{1}{5}-0\right)+\sqrt{\frac{2}{5}}\left% (\frac{2}{5}-\frac{1}{5}\right)+\cdots+\sqrt{\frac{5}{5}}\left(\frac{5}{5}-% \frac{4}{5}\right)\approx 0.7497.\,\!
  70. f f
  71. F ( 1 ) F ( 0 ) F(1)−F(0)
  72. 0 1 x d x = 0 1 x 1 / 2 d x = F ( 1 ) - F ( 0 ) = 2 3 . \int_{0}^{1}\sqrt{x}\,dx=\int_{0}^{1}x^{1/2}\,dx=F(1)-F(0)=\frac{2}{3}.
  73. q 1 q≠−1
  74. f ( x ) d x \int f(x)\,dx\,\!
  75. s s
  76. f ( x ) f(x)
  77. d x dx
  78. d x dx
  79. A f ( x ) d μ \int_{A}f(x)\,d\mu\,\!
  80. μ μ
  81. A A
  82. f ( x ) f(x)
  83. d x dx
  84. ω = f ( x ) d x ω=f(x)dx
  85. d d
  86. A d ω = A ω , \int_{A}d\omega=\int_{\partial A}\omega,\,\!
  87. a a , b aa,b
  88. a a , b aa,b
  89. a = x 0 t 1 x 1 t 2 x 2 x n - 1 t n x n = b . a=x_{0}\leq t_{1}\leq x_{1}\leq t_{2}\leq x_{2}\leq\cdots\leq x_{n-1}\leq t_{n% }\leq x_{n}=b.\,\!
  90. a a , b aa,b
  91. n n
  92. i i
  93. f f
  94. i = 1 n f ( t i ) Δ i ; \sum_{i=1}^{n}f(t_{i})\Delta_{i};
  95. i i
  96. f f
  97. a a , b aa,b
  98. S S
  99. ε > 0 ε>0
  100. δ > 0 δ>0
  101. a a , b aa,b
  102. δ δ
  103. | S - i = 1 n f ( t i ) Δ i | < ε . \left|S-\sum_{i=1}^{n}f(t_{i})\Delta_{i}\right|<\varepsilon.
  104. f f
  105. a a , b aa,b
  106. f f
  107. μ ( A ) μ(A)
  108. A = a a , b A=aa,b
  109. b a b−a
  110. f f
  111. f : 𝐑 𝐑 f:\mathbf{R}→\mathbf{R}
  112. t t
  113. y = t y=t
  114. y = t + d t y=t+dt
  115. f f
  116. f = 0 f * ( t ) d t \int f=\int_{0}^{\infty}f^{*}(t)\,dt
  117. f f
  118. f f
  119. x x
  120. E | f | d μ < + . \int_{E}|f|\,d\mu<+\infty.
  121. x x
  122. x x
  123. E f d μ = E f + d μ - E f - d μ \int_{E}f\,d\mu=\int_{E}f^{+}\,d\mu-\int_{E}f^{-}\,d\mu
  124. f + ( x ) = max ( { f ( x ) , 0 } ) = { f ( x ) , if f ( x ) > 0 , 0 , otherwise, f - ( x ) = max ( { - f ( x ) , 0 } ) = { - f ( x ) , if f ( x ) < 0 , 0 , otherwise. \begin{aligned}\displaystyle f^{+}(x)&\displaystyle=\max(\{f(x),0\})&% \displaystyle=&\displaystyle\begin{cases}f(x),&\,\text{if }f(x)>0,\\ 0,&\,\text{otherwise,}\end{cases}\\ \displaystyle f^{-}(x)&\displaystyle=\max(\{-f(x),0\})&\displaystyle=&% \displaystyle\begin{cases}-f(x),&\,\text{if }f(x)<0,\\ 0,&\,\text{otherwise.}\end{cases}\end{aligned}
  125. a a , b aa,b
  126. f a b f ( x ) d x f\mapsto\int_{a}^{b}f(x)\;dx
  127. a b ( α f + β g ) ( x ) d x = α a b f ( x ) d x + β a b g ( x ) d x . \int_{a}^{b}(\alpha f+\beta g)(x)\,dx=\alpha\int_{a}^{b}f(x)\,dx+\beta\int_{a}% ^{b}g(x)\,dx.\,
  128. E E
  129. μ μ
  130. f E f d μ f\mapsto\int_{E}f\,d\mu
  131. E ( α f + β g ) d μ = α E f d μ + β E g d μ . \int_{E}(\alpha f+\beta g)\,d\mu=\alpha\int_{E}f\,d\mu+\beta\int_{E}g\,d\mu.
  132. ( E , μ ) (E,μ)
  133. V V
  134. K , f : E V K,f:E→V
  135. f f
  136. V V
  137. f E f d μ , f\mapsto\int_{E}f\,d\mu,\,
  138. V V
  139. K K
  140. 𝐑 \mathbf{R}
  141. 𝐂 \mathbf{C}
  142. V V
  143. K K
  144. K = 𝐂 K=\mathbf{C}
  145. V V
  146. X X
  147. a a , b aa,b
  148. f f
  149. a a , b aa,b
  150. m m
  151. M M
  152. m f ( x ) M m≤f(x)≤M
  153. x x
  154. a a , b aa,b
  155. f f
  156. a a , b aa,b
  157. m ( b a ) m(b−a)
  158. M ( b a ) M(b−a)
  159. m ( b - a ) a b f ( x ) d x M ( b - a ) . m(b-a)\leq\int_{a}^{b}f(x)\,dx\leq M(b-a).
  160. f ( x ) g ( x ) f(x)≤g(x)
  161. x x
  162. a a , b aa,b
  163. f f
  164. g g
  165. a b f ( x ) d x a b g ( x ) d x . \int_{a}^{b}f(x)\,dx\leq\int_{a}^{b}g(x)\,dx.
  166. M ( b a ) M(b−a)
  167. M M
  168. a a , b aa,b
  169. x x
  170. a a , b aa,b
  171. a b f ( x ) d x < a b g ( x ) d x . \int_{a}^{b}f(x)\,dx<\int_{a}^{b}g(x)\,dx.
  172. c c , d cc,d
  173. a a , b aa,b
  174. f ( x ) f(x)
  175. x x
  176. c d f ( x ) d x a b f ( x ) d x . \int_{c}^{d}f(x)\,dx\leq\int_{a}^{b}f(x)\,dx.
  177. f f
  178. g g
  179. ( f g ) ( x ) = f ( x ) g ( x ) , f 2 ( x ) = ( f ( x ) ) 2 , | f | ( x ) = | f ( x ) | . (fg)(x)=f(x)g(x),\;f^{2}(x)=(f(x))^{2},\;|f|(x)=|f(x)|.\,
  180. f f
  181. a a , b aa,b
  182. | f | |f|
  183. | a b f ( x ) d x | a b | f ( x ) | d x . \left|\int_{a}^{b}f(x)\,dx\right|\leq\int_{a}^{b}|f(x)|\,dx.
  184. f f
  185. g g
  186. f g fg
  187. ( a b ( f g ) ( x ) d x ) 2 ( a b f ( x ) 2 d x ) ( a b g ( x ) 2 d x ) . \left(\int_{a}^{b}(fg)(x)\,dx\right)^{2}\leq\left(\int_{a}^{b}f(x)^{2}\,dx% \right)\left(\int_{a}^{b}g(x)^{2}\,dx\right).
  188. f f
  189. g g
  190. a a , b aa,b
  191. p p
  192. q q
  193. 1 p , q 1≤p,q≤∞
  194. 1 p + 1 q = 1 \frac{1}{p}+\frac{1}{q}=1
  195. f f
  196. g g
  197. | f ( x ) g ( x ) d x | ( | f ( x ) | p d x ) 1 / p ( | g ( x ) | q d x ) 1 / q . \left|\int f(x)g(x)\,dx\right|\leq\left(\int\left|f(x)\right|^{p}\,dx\right)^{% 1/p}\left(\int\left|g(x)\right|^{q}\,dx\right)^{1/q}.
  198. p p
  199. q q
  200. p 1 p≥1
  201. f f
  202. g g
  203. ( | f ( x ) + g ( x ) | p d x ) 1 / p ( | f ( x ) | p d x ) 1 / p + ( | g ( x ) | p d x ) 1 / p . \left(\int\left|f(x)+g(x)\right|^{p}\,dx\right)^{1/p}\leq\left(\int\left|f(x)% \right|^{p}\,dx\right)^{1/p}+\left(\int\left|g(x)\right|^{p}\,dx\right)^{1/p}.
  204. f f
  205. a b f ( x ) d x \int_{a}^{b}f(x)\,dx
  206. a a , b aa,b
  207. a Align l t ; b a&lt;b
  208. f f
  209. f f
  210. a a
  211. b b
  212. f f
  213. a > b a>b
  214. a > b a>b
  215. a b f ( x ) d x = - b a f ( x ) d x . \int_{a}^{b}f(x)\,dx=-\int_{b}^{a}f(x)\,dx.
  216. a = b a=b
  217. a a
  218. a a f ( x ) d x = 0. \int_{a}^{a}f(x)\,dx=0.
  219. a a , b aa,b
  220. f f
  221. a a , b aa,b
  222. f f
  223. c c , d cc,d
  224. c c
  225. a a , b aa,b
  226. a b f ( x ) d x = a c f ( x ) d x + c b f ( x ) d x . \int_{a}^{b}f(x)\,dx=\int_{a}^{c}f(x)\,dx+\int_{c}^{b}f(x)\,dx.
  227. a c f ( x ) d x \displaystyle\int_{a}^{c}f(x)\,dx
  228. a a
  229. b b
  230. c c
  231. M M
  232. m m
  233. M M′
  234. ω ω
  235. m m
  236. M ω = - M ω . \int_{M}\omega=-\int_{M^{\prime}}\omega\,.
  237. f f
  238. μ μ
  239. A A
  240. A f d μ = [ a , b ] f d μ \textstyle{\int_{A}f\,d\mu=\int_{[a,b]}f\,d\mu}
  241. A A
  242. f f
  243. a a , b aa,b
  244. F F
  245. x x
  246. a a , b aa,b
  247. F ( x ) = a x f ( t ) d t . F(x)=\int_{a}^{x}f(t)\,dt.
  248. F F
  249. a a , b aa,b
  250. ( a , b ) (a,b)
  251. F ( x ) = f ( x ) F^{\prime}(x)=f(x)
  252. x x
  253. ( a , b ) (a,b)
  254. f f
  255. a a
  256. F F
  257. a a , b aa,b
  258. f f
  259. F F
  260. x x
  261. a a , b aa,b
  262. f ( x ) = F ( x ) . f(x)=F^{\prime}(x).
  263. f f
  264. a a , b aa,b
  265. a b f ( x ) d x = F ( b ) - F ( a ) . \int_{a}^{b}f(x)\,dx=F(b)-F(a).
  266. a f ( x ) d x = lim b a b f ( x ) d x \int_{a}^{\infty}f(x)\,dx=\lim_{b\to\infty}\int_{a}^{b}f(x)\,dx
  267. a b f ( x ) d x = lim ϵ 0 a + ϵ b f ( x ) d x \int_{a}^{b}f(x)\,dx=\lim_{\epsilon\to 0}\int_{a+\epsilon}^{b}f(x)\,dx
  268. −∞
  269. 1 / ( ( x + 1 ) x ) 1/((x+1)\sqrt{x})
  270. x x
  271. π \pi
  272. t t
  273. t > 1 t>1
  274. 2 a r c t a n ( t ) π / 2 2arctan(\sqrt{t})−\pi/2
  275. t t
  276. π \pi
  277. π \pi
  278. s s
  279. s Align l t ; 1 s&lt;1
  280. π / 2 2 a r c t a n ( s ) \pi/2−2arctan(\sqrt{s})
  281. s s
  282. π \pi
  283. 0 d x ( x + 1 ) x = lim s 0 s 1 d x ( x + 1 ) x + lim t 1 t d x ( x + 1 ) x = lim s 0 ( π 2 - 2 arctan s ) + lim t ( 2 arctan t - π 2 ) = π 2 + ( π - π 2 ) = π 2 + π 2 = π . \begin{aligned}\displaystyle\int_{0}^{\infty}\frac{dx}{(x+1)\sqrt{x}}&% \displaystyle{}=\lim_{s\to 0}\int_{s}^{1}\frac{dx}{(x+1)\sqrt{x}}+\lim_{t\to% \infty}\int_{1}^{t}\frac{dx}{(x+1)\sqrt{x}}\\ &\displaystyle{}=\lim_{s\to 0}\left(\frac{\pi}{2}-2\arctan{\sqrt{s}}\right)+% \lim_{t\to\infty}\left(2\arctan{\sqrt{t}}-\frac{\pi}{2}\right)\\ &\displaystyle{}=\frac{\pi}{2}+\left(\pi-\frac{\pi}{2}\right)\\ &\displaystyle{}=\frac{\pi}{2}+\frac{\pi}{2}\\ &\displaystyle{}=\pi.\end{aligned}
  284. x x
  285. x \sqrt{x}
  286. - 1 1 d x x 2 3 \displaystyle\int_{-1}^{1}\frac{dx}{\sqrt[3]{x^{2}}}
  287. - 1 1 d x x \int_{-1}^{1}\frac{dx}{x}\,\!
  288. E E
  289. f f
  290. E f ( x ) d x . \int_{E}f(x)\,dx.
  291. x x
  292. f ( x , y , z ) = 1 f(x,y,z)=1
  293. D 5 d x d y \iint_{D}5\ dx\,dy
  294. f ( x , y ) = 5 f(x,y)=5
  295. D D
  296. x y xy
  297. 3 x 7 , 4 y 10 3≤x≤7,4≤y≤10
  298. 4 10 [ 3 7 5 d x ] d y . \int_{4}^{10}\left[\int_{3}^{7}\ 5\ dx\right]dy.
  299. x x
  300. y y
  301. x x
  302. x x
  303. F ( b ) F ( a ) F(b)−F(a)
  304. cuboid 1 d x d y d z \iiint\text{cuboid}1\,dx\,dy\,dz
  305. 1 1
  306. 𝐅 \mathbf{F}
  307. 𝐬 \mathbf{s}
  308. W = 𝐅 𝐬 . W=\mathbf{F}\cdot\mathbf{s}.
  309. C C
  310. 𝐅 \mathbf{F}
  311. 𝐬 \mathbf{s}
  312. 𝐬 + d 𝐬 \mathbf{s}+d\mathbf{s}
  313. W = C 𝐅 d 𝐬 . W=\int_{C}\mathbf{F}\cdot d\mathbf{s}.
  314. 𝐯 \mathbf{v}
  315. S S
  316. x x
  317. S S
  318. 𝐯 ( x ) \mathbf{v}(x)
  319. S S
  320. 𝐯 ( x ) \mathbf{v}(x)
  321. x x
  322. S S
  323. 𝐯 \mathbf{v}
  324. S S
  325. S 𝐯 d 𝐒 . \int_{S}{\mathbf{v}}\cdot\,d{\mathbf{S}}.
  326. f f
  327. f f
  328. m m
  329. S S
  330. S f d x 1 d x m . \int_{S}f\,dx^{1}\ldots dx^{m}.
  331. d x a d x a = 0 dx^{a}\wedge dx^{a}=0\,\!
  332. a a
  333. f f
  334. n n
  335. k > n k>n
  336. d d
  337. d d
  338. d ω = i = 1 n f x i d x i d x a . d\omega=\sum_{i=1}^{n}\frac{\partial f}{\partial x_{i}}dx^{i}\wedge dx^{a}.
  339. Ω d ω = Ω ω \int_{\Omega}d\omega=\int_{\partial\Omega}\omega\,\!
  340. ω ω
  341. Ω ∂Ω
  342. Ω Ω
  343. ω ω
  344. Ω Ω
  345. ω ω
  346. Ω Ω
  347. f ( x ) f(x)
  348. x x
  349. a a , b aa,b
  350. f f
  351. F F
  352. F = f F′=f
  353. a b f ( x ) d x = F ( b ) - F ( a ) . \textstyle\int_{a}^{b}f(x)\,dx=F(b)-F(a).
  354. f f
  355. ( s i n x ) / x (sinx)/x
  356. - 2 2 1 5 ( 1 100 ( 322 + 3 x ( 98 + x ( 37 + x ) ) ) - 24 x 1 + x 2 ) d x \int_{-2}^{2}\tfrac{1}{5}\left(\tfrac{1}{100}(322+3x(98+x(37+x)))-24\frac{x}{1% +x^{2}}\right)dx
  357. h h
  358. T ( 0 ) T(0)
  359. h = 0 h=0
  360. x x
  361. ± 2 3 ±2⁄\sqrt{3}
  362. n n
  363. 2 n 1 2n−1
  364. - 2.25 1.75 f ( x ) d x = 4.1639019006585897075 \textstyle\int_{-2.25}^{1.75}f(x)\,dx=4.1639019006585897075\ldots
  365. a b f ( x ) d x b - a 6 [ f ( a ) + 4 f ( a + b 2 ) + f ( b ) ] , \int_{a}^{b}f(x)\,dx\approx\frac{b-a}{6}\left[f(a)+4f\left(\frac{a+b}{2}\right% )+f(b)\right],
  366. | - ( b - a ) 5 2880 f ( 4 ) ( ξ ) | . \left|-\frac{(b-a)^{5}}{2880}f^{(4)}(\xi)\right|.
  367. γ = 1 ( 1 x - 1 x ) d x , \gamma=\int_{1}^{\infty}\left({1\over\lfloor x\rfloor}-{1\over x}\right)\,dx\,,
  368. Γ ( z ) = 0 e - t t z - 1 d t , \Gamma(z)=\int_{0}^{\infty}e^{-t}t^{z-1}dt,
  369. F ( ξ ) = - f ( x ) e - 2 π i x ξ d x , F(\xi)=\int_{-\infty}^{\infty}f(x)e^{-2\pi ix\xi}\,dx,
  370. F ( s ) = 0 f ( t ) e - s t d t , F(s)=\int_{0}^{\infty}f(t)e^{-st}\,dt,
  371. - e - x 2 d x = π . \int_{-\infty}^{\infty}e^{-x^{2}}\,dx=\sqrt{\pi}.

Integral_domain.html

  1. 𝐙 2 𝐙 2 n 𝐙 2 n + 1 𝐙 \mathbf{Z}\;\supset\;2\mathbf{Z}\;\supset\;\cdots\;\supset\;2^{n}\mathbf{Z}\;% \supset\;2^{n+1}\mathbf{Z}\;\supset\;\cdots
  2. \mathbb{C}\otimes_{\mathbb{R}}\mathbb{C}
  3. ( i 1 - 1 i ) ( i 1 + 1 i ) = 0 (i\otimes 1-1\otimes i)\,(i\otimes 1+1\otimes i)=0
  4. k [ x , y ] / ( x y ) k[x,y]/(xy)
  5. k k
  6. ( x y ) (xy)
  7. [ - 5 ] \mathbb{Z}\left[\sqrt{-5}\right]
  8. a 2 + 5 b 2 = 3 a^{2}+5b^{2}=3
  9. ( 2 + - 5 ) ( 2 - - 5 ) \left(2+\sqrt{-5}\right)\left(2-\sqrt{-5}\right)
  10. [ - 5 ] \mathbb{Z}\left[\sqrt{-5}\right]
  11. \mapsto

Intensity_(physics).html

  1. P = I d A P=\int I\,\cdot\mathrm{d}A
  2. P = | I | A surf = | I | 4 π r 2 P=|I|\cdot A_{\mathrm{surf}}=|I|\cdot 4\pi r^{2}\,
  3. A surf = 4 π r 2 A_{\mathrm{surf}}=4\pi r^{2}
  4. | I | = P A surf = P 4 π r 2 |I|=\frac{P}{A_{\mathrm{surf}}}=\frac{P}{4\pi r^{2}}
  5. U = n 2 ϵ 0 2 | E | 2 \left\langle U\right\rangle=\frac{n^{2}\epsilon_{0}}{2}|E|^{2}
  6. I = c n ϵ 0 2 | E | 2 I=\frac{\mathrm{c}n\epsilon_{0}}{2}|E|^{2}
  7. ϵ 0 \epsilon_{0}

Interference_(wave_propagation).html

  1. Δ φ = 2 π d λ = 2 π x sin θ λ \Delta\varphi=\frac{2\pi d}{\lambda}=\frac{2\pi x\sin\theta}{\lambda}
  2. x sin θ λ = 0 , ± 1 , ± 2 , \frac{x\sin\theta}{\lambda}=0,\pm 1,\pm 2,...
  3. x sin θ λ = ± 1 2 , ± 3 2 , \frac{x\sin\theta}{\lambda}=\pm\frac{1}{2},\pm\frac{3}{2},...
  4. d f = λ sin θ d_{f}=\frac{\lambda}{\sin\theta}
  5. θ θ
  6. A e i φ n Ae^{i\varphi_{n}}
  7. N N
  8. n = 0 n=0
  9. n = N - 1 n=N-1
  10. φ n - φ n - 1 = 2 π N \varphi_{n}-\varphi_{n-1}=\frac{2\pi}{N}
  11. n = 0 N - 1 A e i φ n = 0 \sum_{n=0}^{N-1}Ae^{i\varphi_{n}}=0
  12. e i 2 π N e^{i\frac{2\pi}{N}}
  13. 𝐫 \mathbf{r}
  14. U 1 ( 𝐫 , t ) = A 1 ( 𝐫 ) e i [ φ 1 ( 𝐫 ) - ω t ] U_{1}(\mathbf{r},t)=A_{1}(\mathbf{r})e^{i[\varphi_{1}(\mathbf{r})-\omega t]}
  15. U 2 ( 𝐫 , t ) = A 2 ( 𝐫 ) e i [ φ 2 ( 𝐫 ) - ω t ] U_{2}(\mathbf{r},t)=A_{2}(\mathbf{r})e^{i[\varphi_{2}(\mathbf{r})-\omega t]}
  16. A A
  17. φ φ
  18. ω ω
  19. U ( 𝐫 , t ) = A 1 ( 𝐫 ) e i [ φ 1 ( 𝐫 ) - ω t ] + A 2 ( 𝐫 ) e i [ φ 2 ( 𝐫 ) - ω t ] U(\mathbf{r},t)=A_{1}(\mathbf{r})e^{i[\varphi_{1}(\mathbf{r})-\omega t]}+A_{2}% (\mathbf{r})e^{i[\varphi_{2}(\mathbf{r})-\omega t]}
  20. 𝐫 \mathbf{r}
  21. I ( 𝐫 ) = U ( 𝐫 , t ) U * ( 𝐫 , t ) d t A 1 2 ( 𝐫 ) + A 2 2 ( 𝐫 ) + 2 A 1 ( 𝐫 ) A 2 ( 𝐫 ) cos [ φ 1 ( 𝐫 ) - φ 2 ( 𝐫 ) ] I(\mathbf{r})=\int U(\mathbf{r},t)U^{*}(\mathbf{r},t)dt\propto A_{1}^{2}(% \mathbf{r})+A_{2}^{2}(\mathbf{r})+2A_{1}(\mathbf{r})A_{2}(\mathbf{r})\cos{[% \varphi_{1}(\mathbf{r})-\varphi_{2}(\mathbf{r})]}
  22. I ( 𝐫 ) = I 1 ( 𝐫 ) + I 2 ( 𝐫 ) + 2 I 1 ( 𝐫 ) I 2 ( 𝐫 ) cos [ φ 1 ( 𝐫 ) - φ 2 ( 𝐫 ) ] I(\mathbf{r})=I_{1}(\mathbf{r})+I_{2}(\mathbf{r})+2\sqrt{I_{1}(\mathbf{r})I_{2% }(\mathbf{r})}\cos{[\varphi_{1}(\mathbf{r})-\varphi_{2}(\mathbf{r})]}
  23. ψ \psi
  24. | ψ = i | i ψ i |\psi\rangle=\sum_{i}|i\rangle\psi_{i}
  25. | i |i\rangle
  26. ψ i \psi_{i}
  27. ψ \psi
  28. φ \varphi
  29. prob ( ψ φ ) = | ψ | φ | 2 = | i ψ i * φ i | 2 \operatorname{prob}(\psi\Rightarrow\varphi)=|\langle\psi|\varphi\rangle|^{2}=|% \sum_{i}\psi^{*}_{i}\varphi_{i}|^{2}
  30. = i j ψ i * ψ j φ j * φ i = i | ψ i | 2 | φ i | 2 + i j ; i j ψ i * ψ j φ j * φ i =\sum_{ij}\psi^{*}_{i}\psi_{j}\varphi^{*}_{j}\varphi_{i}=\sum_{i}|\psi_{i}|^{2% }|\varphi_{i}|^{2}+\sum_{ij;i\neq j}\psi^{*}_{i}\psi_{j}\varphi^{*}_{j}\varphi% _{i}
  31. ψ i = i | ψ \psi_{i}=\langle i|\psi\rangle
  32. φ i = i | φ \varphi_{i}=\langle i|\varphi\rangle
  33. ψ i * = ψ | i \psi_{i}^{*}=\langle\psi|i\rangle
  34. | ψ |\psi\rangle
  35. | φ |\varphi\rangle
  36. | i |i\rangle
  37. prob ( ψ φ ) = i prob ( ψ i φ ) \operatorname{prob}(\psi\Rightarrow\varphi)=\sum_{i}\operatorname{prob}(\psi% \Rightarrow i\Rightarrow\varphi)
  38. = i | ψ | i | 2 | i | φ | 2 = i | ψ i | 2 | φ i | 2 =\sum_{i}|\langle\psi|i\rangle|^{2}|\langle i|\varphi\rangle|^{2}=\sum_{i}|% \psi_{i}|^{2}|\varphi_{i}|^{2}
  39. i j ; i j ψ i * ψ j φ j * φ i \sum_{ij;i\neq j}\psi^{*}_{i}\psi_{j}\varphi^{*}_{j}\varphi_{i}
  40. i j i\neq j
  41. | i |i\rangle

Interference_filter.html

  1. λ 0 = λ c 1 - sin 2 θ n * 2 \lambda_{0}=\lambda_{c}\sqrt{1-\frac{\sin^{2}\theta}{{n^{*}}^{2}}}

Interior_(topology).html

  1. A X A\subset X
  2. \mathbb{R}
  3. \mathbb{R}
  4. \mathbb{Q}
  5. = 2 \mathbb{C}=\mathbb{R}^{2}
  6. ( { z : | z | 1 } ) = { z : | z | < 1 } . (\{z\in\mathbb{C}:|z|\leq 1\})=\{z\in\mathbb{C}:|z|<1\}.
  7. X = X=\mathbb{R}
  8. \mathbb{R}
  9. \mathbb{R}
  10. \mathbb{R}
  11. \mathbb{R}

Intermolecular_force.html

  1. - 2 m 1 2 m 2 2 48 π 2 ε o 2 ε r 2 k b T r 6 = V \frac{-2m_{1}^{2}m_{2}^{2}}{48\pi^{2}\varepsilon_{o}^{2}\varepsilon_{r}^{2}k_{% b}Tr^{6}}=V
  2. ε o \varepsilon_{o}
  3. ε r \varepsilon_{r}
  4. k b k_{b}
  5. - m 1 2 α 2 16 π 2 ε o 2 ε r 2 r 6 = V \frac{-m_{1}^{2}\alpha_{2}}{16\pi^{2}\varepsilon_{o}^{2}\varepsilon_{r}^{2}r^{% 6}}=V
  6. α \alpha

Internal_rate_of_return.html

  1. n n
  2. C n C_{n}
  3. n n
  4. N N
  5. NPV \mathrm{NPV}
  6. r r
  7. NPV = n = 0 N C n ( 1 + r ) n = 0 \mathrm{NPV}=\sum_{n=0}^{N}\frac{C_{n}}{(1+r)^{n}}=0
  8. r r
  9. r r
  10. n n
  11. C n C_{n}
  12. r r
  13. NPV = - 123400 + 36200 ( 1 + r ) 1 + 54800 ( 1 + r ) 2 + 48100 ( 1 + r ) 3 = 0. \mathrm{NPV}=-123400+\frac{36200}{(1+r)^{1}}+\frac{54800}{(1+r)^{2}}+\frac{481% 00}{(1+r)^{3}}=0.
  14. NPV ( r ) = 0 \mathrm{NPV}(r)=0
  15. r r
  16. r r
  17. r n + 1 = r n - NPV n ( r n - r n - 1 NPV n - NPV n - 1 ) . r_{n+1}=r_{n}-\mathrm{NPV}_{n}\left(\frac{r_{n}-r_{n-1}}{\mathrm{NPV}_{n}-% \mathrm{NPV}_{n-1}}\right).
  18. r n r_{n}
  19. n n
  20. r r
  21. NPV ( i ) \mathrm{NPV}(i)
  22. r r
  23. r r
  24. NPV ( i ) \mathrm{NPV}(i)
  25. n n
  26. r 1 , r 2 , , r n \scriptstyle r_{1},r_{2},\dots,r_{n}
  27. NPV ( i ) \mathrm{NPV}(i)
  28. r 1 > r 0 \scriptstyle{r_{1}>r_{0}}
  29. NPV 0 > 0 \mathrm{NPV}_{0}>0
  30. r 1 < r 0 \scriptstyle{r_{1}<r_{0}}
  31. NPV 0 < 0 \mathrm{NPV}_{0}<0
  32. r n r_{n}
  33. r r
  34. C 0 < 0 , C n 0 for n 1. C_{0}<0,\quad C_{n}\geq 0\,\text{ for }n\geq 1.\,
  35. r 1 r_{1}
  36. r 2 r_{2}
  37. n = 2 n=2
  38. r 3 r_{3}
  39. r n + 1 = r n - NPV n ( r n - r n - 1 NPV n - NPV n - 1 ) ( 1 - 1.4 NPV n - 1 NPV n - 1 - 3 N P V n + 2 C 0 ) r_{n+1}=r_{n}-\mathrm{NPV}_{n}\left(\frac{r_{n}-r_{n-1}}{\mathrm{NPV}_{n}-% \mathrm{NPV}_{n-1}}\right)\left(1-1.4\frac{\mathrm{NPV}_{n-1}}{\mathrm{NPV}_{n% -1}-3\mathrm{NPV}_{n}+2C_{0}}\right)
  40. 0 > NPV n > NPV n - 1 0>\mathrm{NPV}_{n}>\mathrm{NPV}_{n-1}
  41. r 1 = 0.25 r_{1}=0.25
  42. r 2 = 0.2 r_{2}=0.2
  43. r 1 = ( A / | C 0 | ) 2 / ( N + 1 ) - 1 r_{1}=\left(A/|C_{0}|\right)^{2/(N+1)}-1\,
  44. r 2 = ( 1 + r 1 ) p - 1 r_{2}=(1+r_{1})^{p}-1\,
  45. A = sum of inflows = C 1 + + C N A=\,\text{ sum of inflows }=C_{1}+\cdots+C_{N}\,
  46. p = log ( A / | C 0 | ) log ( A / NPV 1 , i n ) . p=\frac{\log(\mathrm{A}/|C_{0}|)}{\log(\mathrm{A}/\mathrm{NPV}_{1,in})}.
  47. NPV 1 , i n \mathrm{NPV}_{1,in}
  48. C 0 = 0 \mathrm{C_{0}=0}

International_Standard_Book_Number.html

  1. s \displaystyle s
  2. ( 10 x 1 + 9 x 2 + 8 x 3 + 7 x 4 + 6 x 5 + 5 x 6 + 4 x 7 + 3 x 8 + 2 x 9 + x 10 ) 0 ( mod 11 ) . (10x_{1}+9x_{2}+8x_{3}+7x_{4}+6x_{5}+5x_{6}+4x_{7}+3x_{8}+2x_{9}+x_{10})\equiv 0% \;\;(\mathop{{\rm mod}}11).
  3. s \displaystyle s
  4. ( x 1 + 2 x 2 + 3 x 3 + 4 x 4 + 5 x 5 + 6 x 6 + 7 x 7 + 8 x 8 + 9 x 9 + 10 x 10 ) 0 ( mod 11 ) . (x_{1}+2x_{2}+3x_{3}+4x_{4}+5x_{5}+6x_{6}+7x_{7}+8x_{8}+9x_{9}+10x_{10})\equiv 0% \;\;(\mathop{{\rm mod}}11).
  5. s \displaystyle s
  6. x 10 x_{10}
  7. ( x 1 + 3 x 2 + x 3 + 3 x 4 + x 5 + 3 x 6 + x 7 + 3 x 8 + x 9 + 3 x 10 + x 11 + 3 x 12 + x 13 ) 0 ( mod 10 ) . (x_{1}+3x_{2}+x_{3}+3x_{4}+x_{5}+3x_{6}+x_{7}+3x_{8}+x_{9}+3x_{10}+x_{11}+3x_{% 12}+x_{13})\equiv 0\;\;(\mathop{{\rm mod}}10).
  8. x 13 = ( 10 - ( x 1 + 3 x 2 + x 3 + 3 x 4 + + x 11 + 3 x 12 ) mod 10 ) mod 10. x_{13}=\big(10-\big(x_{1}+3x_{2}+x_{3}+3x_{4}+\cdots+x_{11}+3x_{12}\big)\,% \bmod\,10\big)\,\bmod\,10.

Interpolation.html

  1. y = y a + ( y b - y a ) x - x a x b - x a at the point ( x , y ) y=y_{a}+\left(y_{b}-y_{a}\right)\frac{x-x_{a}}{x_{b}-x_{a}}\,\text{ at the % point }\left(x,y\right)
  2. y - y a y b - y a = x - x a x b - x a \frac{y-y_{a}}{y_{b}-y_{a}}=\frac{x-x_{a}}{x_{b}-x_{a}}
  3. y - y a x - x a = y b - y a x b - x a \frac{y-y_{a}}{x-x_{a}}=\frac{y_{b}-y_{a}}{x_{b}-x_{a}}
  4. ( x a , y a ) (x_{a},y_{a})
  5. ( x , y ) (x,y)
  6. ( x a , y a ) (x_{a},y_{a})
  7. ( x b , y b ) (x_{b},y_{b})
  8. | f ( x ) - g ( x ) | C ( x b - x a ) 2 where C = 1 8 max r [ x a , x b ] | g ′′ ( r ) | . |f(x)-g(x)|\leq C(x_{b}-x_{a})^{2}\quad\,\text{where}\quad C=\frac{1}{8}\max_{% r\in[x_{a},x_{b}]}|g^{\prime\prime}(r)|.
  9. f ( x ) = - 0.0001521 x 6 - 0.003130 x 5 + 0.07321 x 4 - 0.3577 x 3 + 0.2255 x 2 + 0.9038 x . f(x)=-0.0001521x^{6}-0.003130x^{5}+0.07321x^{4}-0.3577x^{3}+0.2255x^{2}+0.9038x.
  10. f ( x ) = { - 0.1522 x 3 + 0.9937 x , if x [ 0 , 1 ] , - 0.01258 x 3 - 0.4189 x 2 + 1.4126 x - 0.1396 , if x [ 1 , 2 ] , 0.1403 x 3 - 1.3359 x 2 + 3.2467 x - 1.3623 , if x [ 2 , 3 ] , 0.1579 x 3 - 1.4945 x 2 + 3.7225 x - 1.8381 , if x [ 3 , 4 ] , 0.05375 x 3 - 0.2450 x 2 - 1.2756 x + 4.8259 , if x [ 4 , 5 ] , - 0.1871 x 3 + 3.3673 x 2 - 19.3370 x + 34.9282 , if x [ 5 , 6 ] . f(x)=\begin{cases}-0.1522x^{3}+0.9937x,&\,\text{if }x\in[0,1],\\ -0.01258x^{3}-0.4189x^{2}+1.4126x-0.1396,&\,\text{if }x\in[1,2],\\ 0.1403x^{3}-1.3359x^{2}+3.2467x-1.3623,&\,\text{if }x\in[2,3],\\ 0.1579x^{3}-1.4945x^{2}+3.7225x-1.8381,&\,\text{if }x\in[3,4],\\ 0.05375x^{3}-0.2450x^{2}-1.2756x+4.8259,&\,\text{if }x\in[4,5],\\ -0.1871x^{3}+3.3673x^{2}-19.3370x+34.9282,&\,\text{if }x\in[5,6].\end{cases}

Interpretations_of_quantum_mechanics.html

  1. ψ | A | ψ . \langle\psi|A|\psi\rangle.
  2. | ϕ |\phi\rangle
  3. Π = | ϕ ϕ | . \Pi=|\phi\rangle\langle\phi|.
  4. P = ψ | Π | ψ = | ϕ | ψ | 2 . P=\langle\psi|\Pi|\psi\rangle=|\langle\phi|\psi\rangle|^{2}.