wpmath0000001_22

Tractatus_Logico-Philosophicus.html

  1. [ p ¯ , ξ ¯ , N ( ξ ¯ ) ] [\bar{p},\bar{\xi},N(\bar{\xi})]
  2. [ p ¯ , ξ ¯ , N ( ξ ¯ ) ] [\bar{p},\bar{\xi},N(\bar{\xi})]
  3. p ¯ \bar{p}
  4. ξ ¯ \bar{\xi}
  5. N ( ξ ¯ ) N(\bar{\xi})
  6. ξ ¯ \bar{\xi}

Traffic_intensity.html

  1. a L R \frac{aL}{R}

Transcendental_number.html

  1. k = 1 10 - k ! = 0.1100010000000000000000010000 \sum_{k=1}^{\infty}10^{-k!}=0.1100010000000000000000010000\ldots
  2. e a e^{a}
  3. 2 ¯ \overline{2}
  4. π ¯ \overline{π}
  5. 3 ¯ \overline{3}
  6. π \pi
  7. π \pi
  8. π \pi
  9. 2 2 , 2^{\sqrt{2}},
  10. 1 + 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + {1+\cfrac{1}{2+\cfrac{1}{3+\cfrac{1}{4+\cfrac{1}{5+\cfrac{1}{6+\ddots}}}}}}
  11. n = 0 2 - 2 n \sum_{n=0}^{\infty}2^{-2^{n}}
  12. n = 0 β 2 n \sum_{n=0}^{\infty}\beta^{2^{n}}
  13. n = 1 10 - n ! ; \sum_{n=1}^{\infty}10^{-n!};
  14. n = 1 β n ! \sum_{n=1}^{\infty}\beta^{n!}
  15. k = 0 10 - β k ; \sum_{k=0}^{\infty}10^{-\left\lfloor\beta^{k}\right\rfloor};
  16. β β \beta\mapsto\lfloor\beta\rfloor
  17. 2 ¯ \overline{2}
  18. n ¯ \overline{n}
  19. c 0 + c 1 e + c 2 e 2 + + c n e n = 0 , c 0 , c n 0. c_{0}+c_{1}e+c_{2}e^{2}+\cdots+c_{n}e^{n}=0,\qquad c_{0},c_{n}\neq 0.
  20. f k ( x ) = x k [ ( x - 1 ) ( x - n ) ] k + 1 , f_{k}(x)=x^{k}\left[(x-1)\cdots(x-n)\right]^{k+1},
  21. 0 f k e - x d x , \int^{\infty}_{0}f_{k}e^{-x}\,dx,
  22. c 0 ( 0 f k e - x d x ) + c 1 e ( 0 f k e - x d x ) + + c n e n ( 0 f k e - x d x ) = 0. c_{0}\left(\int^{\infty}_{0}f_{k}e^{-x}\,dx\right)+c_{1}e\left(\int^{\infty}_{% 0}f_{k}e^{-x}\,dx\right)+\cdots+c_{n}e^{n}\left(\int^{\infty}_{0}f_{k}e^{-x}\,% dx\right)=0.
  23. P + Q = 0 P+Q=0
  24. P = c 0 ( 0 f k e - x d x ) + c 1 e ( 1 f k e - x d x ) + c 2 e 2 ( 2 f k e - x d x ) + + c n e n ( n f k e - x d x ) P=c_{0}\left(\int^{\infty}_{0}f_{k}e^{-x}\,dx\right)+c_{1}e\left(\int^{\infty}% _{1}f_{k}e^{-x}\,dx\right)+c_{2}e^{2}\left(\int^{\infty}_{2}f_{k}e^{-x}\,dx% \right)+\cdots+c_{n}e^{n}\left(\int^{\infty}_{n}f_{k}e^{-x}\,dx\right)
  25. Q = c 1 e ( 0 1 f k e - x d x ) + c 2 e 2 ( 0 2 f k e - x d x ) + + c n e n ( 0 n f k e - x d x ) Q=c_{1}e\left(\int^{1}_{0}f_{k}e^{-x}\,dx\right)+c_{2}e^{2}\left(\int^{2}_{0}f% _{k}e^{-x}\,dx\right)+\cdots+c_{n}e^{n}\left(\int^{n}_{0}f_{k}e^{-x}\,dx\right)
  26. P k ! \tfrac{P}{k!}
  27. 0 x j e - x d x = j ! \int^{\infty}_{0}x^{j}e^{-x}\,dx=j!
  28. 0 x j e - x d x \int^{\infty}_{0}x^{j}e^{-x}\,dx
  29. 0 f k e - x d x = 0 ( [ ( - 1 ) n ( n ! ) ] k + 1 e - x x k + ) d x \int^{\infty}_{0}f_{k}e^{-x}\,dx=\int^{\infty}_{0}\left([(-1)^{n}(n!)]^{k+1}e^% {-x}x^{k}+\cdots\right)dx
  30. 1 k ! c 0 0 f k e - x d x = c 0 [ ( - 1 ) n ( n ! ) ] k + 1 mod ( k + 1 ) . {\frac{1}{k!}}c_{0}\int^{\infty}_{0}f_{k}e^{-x}\,dx=c_{0}[(-1)^{n}(n!)]^{k+1}% \qquad\mod(k+1).
  31. P k ! \tfrac{P}{k!}
  32. | Q k ! | < 1 \left|\tfrac{Q}{k!}\right|<1
  33. f k e - x = x k [ ( x - 1 ) ( x - 2 ) ( x - n ) ] k + 1 e - x = ( [ x ( x - 1 ) ( x - n ) ] k ) ( ( x - 1 ) ( x - n ) e - x ) f_{k}e^{-x}=x^{k}[(x-1)(x-2)\cdots(x-n)]^{k+1}e^{-x}=\left([x(x-1)\cdots(x-n)]% ^{k}\right)\left((x-1)\cdots(x-n)e^{-x}\right)
  34. | x ( x - 1 ) ( x - n ) | |x(x-1)\cdots(x-n)|
  35. | ( x - 1 ) ( x - n ) e - x | |(x-1)\cdots(x-n)e^{-x}|
  36. | Q | < G k H ( | c 1 | e + 2 | c 2 | e 2 + + n | c n | e n ) |Q|<G^{k}H(|c_{1}|e+2|c_{2}|e^{2}+\cdots+n|c_{n}|e^{n})
  37. lim k G k k ! = 0 \lim_{k\to\infty}\frac{G^{k}}{k!}=0
  38. lim k Q k ! = 0 \lim_{k\to\infty}\frac{Q}{k!}=0
  39. ω ( x , 1 , H ) = - log m ( x , 1 , H ) log H \omega(x,1,H)=-\frac{\log m(x,1,H)}{\log H}
  40. ω ( x , 1 ) = lim sup H ω ( x , 1 , H ) . \omega(x,1)=\limsup_{H\to\infty}\omega(x,1,H).
  41. ω ( x , n , H ) = - log m ( x , n , H ) n log H \omega(x,n,H)=-\frac{\log m(x,n,H)}{n\log H}
  42. ω ( x , n ) = lim sup H ω ( x , n , H ) . \omega(x,n)=\limsup_{H\to\infty}\omega(x,n,H).
  43. ω ( x ) = lim sup n ω ( x , n ) . \omega(x)=\limsup_{n\to\infty}\omega(x,n).
  44. | x - α | = H - n ω * ( x , H , n ) - 1 . |x-\alpha|=H^{-n\omega^{*}(x,H,n)-1}.
  45. ω * ( x , n ) = lim sup H ω * ( x , n , H ) . \omega^{*}(x,n)=\limsup_{H\to\infty}\omega^{*}(x,n,H).
  46. λ = 1 3 + k = 1 10 - k ! \lambda=\tfrac{1}{3}+\sum_{k=1}^{\infty}10^{-k!}
  47. π \pi
  48. π \pi

Transfer_function.html

  1. x ( t ) x(t)
  2. y ( t ) y(t)
  3. H ( s ) H(s)
  4. X ( s ) = { x ( t ) } X(s)=\mathcal{L}\left\{x(t)\right\}
  5. Y ( s ) = { y ( t ) } Y(s)=\mathcal{L}\left\{y(t)\right\}
  6. Y ( s ) = H ( s ) X ( s ) Y(s)=H(s)\;X(s)
  7. H ( s ) = Y ( s ) X ( s ) = { y ( t ) } { x ( t ) } H(s)=\frac{Y(s)}{X(s)}=\frac{\mathcal{L}\left\{y(t)\right\}}{\mathcal{L}\left% \{x(t)\right\}}
  8. x ( t ) x(t)
  9. y ( t ) y(t)
  10. H ( z ) = Y ( z ) X ( z ) H(z)=\frac{Y(z)}{X(z)}
  11. L [ u ] = d n u d t n + a 1 d n - 1 u d t n - 1 + + a n - 1 d u d t + a n u = r ( t ) L[u]=\frac{d^{n}u}{dt^{n}}+a_{1}\frac{d^{n-1}u}{dt^{n-1}}+\cdots+a_{n-1}\frac{% du}{dt}+a_{n}u=r(t)
  12. F [ r ] = u F[r]=u
  13. L [ F [ r ] ] = r L[F[r]]=r
  14. L [ u ] = 0 L[u]=0
  15. u = e λ t u=e^{\lambda t}
  16. p L ( λ ) = λ n + a 1 λ n - 1 + + a n - 1 λ + a n p_{L}(\lambda)=\lambda^{n}+a_{1}\lambda^{n-1}+\cdots+a_{n-1}\lambda+a_{n}\,
  17. r ( t ) = e s t r(t)=e^{st}
  18. u = H ( s ) e s t u=H(s)e^{st}
  19. L [ H ( s ) e s t ] = e s t L[H(s)e^{st}]=e^{st}
  20. H ( s ) = 1 p L ( s ) , p L ( s ) 0. H(s)=\frac{1}{p_{L}(s)},\qquad p_{L}(s)\neq 0.
  21. 1 / p L ( i k ) . 1/p_{L}(ik).
  22. x ( t ) x(t)
  23. y ( t ) y(t)
  24. x ( t ) x(t)
  25. y ( t ) y(t)
  26. X ( s ) \displaystyle X(s)
  27. H ( s ) H(s)
  28. Y ( s ) = H ( s ) X ( s ) Y(s)=H(s)X(s)\,
  29. H ( s ) = Y ( s ) X ( s ) . H(s)=\frac{Y(s)}{X(s)}.
  30. | X | |X|
  31. ω \omega
  32. arg ( X ) \arg(X)
  33. x ( t ) = X e j ω t = | X | e j ( ω t + arg ( X ) ) x(t)=Xe^{j\omega t}=|X|e^{j(\omega t+\arg(X))}
  34. X = | X | e j arg ( X ) X=|X|e^{j\arg(X)}
  35. y ( t ) \displaystyle y(t)
  36. ω \omega
  37. H ( j ω ) H(j\omega)
  38. ω \omega
  39. G ( ω ) = | Y | | X | = | H ( j ω ) | G(\omega)=\frac{|Y|}{|X|}=|H(j\omega)|
  40. ϕ ( ω ) = arg ( Y ) - arg ( X ) = arg ( H ( j ω ) ) . \phi(\omega)=\arg(Y)-\arg(X)=\arg(H(j\omega)).
  41. τ ϕ ( ω ) = - ϕ ( ω ) ω . \tau_{\phi}(\omega)=-\frac{\phi(\omega)}{\omega}.
  42. ω \omega
  43. τ g ( ω ) = - d ϕ ( ω ) d ω . \tau_{g}(\omega)=-\frac{d\phi(\omega)}{d\omega}.
  44. s = j ω s=j\omega
  45. MTF ( f ) = M ( image ) M ( source ) \mathrm{MTF}(f)=\frac{M(\mathrm{image})}{M(\mathrm{source})}
  46. M = L max - L min L max + L min . M=\frac{L_{\max}-L_{\min}}{L_{\max}+L_{\min}}.

Transfinite_induction.html

  1. v 0 v_{0}
  2. { v β | β < α } \{v_{\beta}|\beta<\alpha\}
  3. \upharpoonright
  4. \upharpoonright
  5. \upharpoonright
  6. r α | α < β \langle r_{\alpha}|\alpha<\beta\rangle

Transformational_grammar.html

  1. \Rightarrow

Transformer.html

  1. V S = - N S d Φ d t V\text{S}=-N\text{S}\frac{\mathrm{d}\Phi}{\mathrm{d}t}
  2. V P = - N P d Φ d t V\text{P}=-N\text{P}\frac{\mathrm{d}\Phi}{\mathrm{d}t}
  3. = V P V S = - N P - N S = a =\frac{V\text{P}}{V\text{S}}=\frac{-N\text{P}}{-N\text{S}}=a
  4. S = I P V P = I S V S S=I\text{P}V\text{P}=I\text{S}V\text{S}
  5. V P V S = I S I P = N P N S = L P L S = a \frac{V\text{P}}{V\text{S}}=\frac{I\text{S}}{I\text{P}}=\frac{N\text{P}}{N% \text{S}}=\sqrt{\frac{L\text{P}}{L\text{S}}}=a
  6. Z L = V S I S Z\text{L}=\frac{V\text{S}}{I\text{S}}
  7. E rms = 2 π f N a B peak 2 4.44 f N a B peak E\text{rms}={\frac{2\pi fNaB\text{peak}}{\sqrt{2}}}\approx 4.44fNaB\text{peak}
  8. E avg = 4 f N a B peak E\text{avg}=4fNaB\text{peak}\!
  9. W h η β max 1.6 W\text{h}\approx\eta\beta^{1.6}_{\,\text{max}}
  10. P h W h f η f β max 1.6 P\text{h}\approx{W}\text{h}f\approx\eta{f}\beta^{1.6}_{\,\text{max}}
  11. | | = | d Φ B d t | |\mathcal{E}|=\left|{{d\Phi\text{B}}\over dt}\right|
  12. | | |\mathcal{E}|

Transistor.html

  1. f T f_{\mathrm{T}}
  2. I d s ( V g s - V T ) 2 I_{ds}\propto(V_{gs}-V_{T})^{2}

Transmission_line.html

  1. R R
  2. L L
  3. C C
  4. G G
  5. R R
  6. L L
  7. C C
  8. G G
  9. R R^{\prime}
  10. L L^{\prime}
  11. C C^{\prime}
  12. G G^{\prime}
  13. V ( x ) V(x)
  14. I ( x ) I(x)
  15. V ( x ) x = - ( R + j ω L ) I ( x ) \frac{\partial V(x)}{\partial x}=-(R+j\omega L)I(x)
  16. I ( x ) x = - ( G + j ω C ) V ( x ) . \frac{\partial I(x)}{\partial x}=-(G+j\omega C)V(x).
  17. R R
  18. G G
  19. L L
  20. C C
  21. 2 V ( x ) x 2 + ω 2 L C V ( x ) = 0 \frac{\partial^{2}V(x)}{\partial x^{2}}+\omega^{2}LC\cdot V(x)=0
  22. 2 I ( x ) x 2 + ω 2 L C I ( x ) = 0. \frac{\partial^{2}I(x)}{\partial x^{2}}+\omega^{2}LC\cdot I(x)=0.
  23. R R
  24. G G
  25. 2 V ( x ) x 2 = γ 2 V ( x ) \frac{\partial^{2}V(x)}{\partial x^{2}}=\gamma^{2}V(x)
  26. 2 I ( x ) x 2 = γ 2 I ( x ) \frac{\partial^{2}I(x)}{\partial x^{2}}=\gamma^{2}I(x)
  27. γ γ
  28. γ = ( R + j ω L ) ( G + j ω C ) \gamma=\sqrt{(R+j\omega L)(G+j\omega C)}
  29. Z 0 = R + j ω L G + j ω C . Z_{0}=\sqrt{\frac{R+j\omega L}{G+j\omega C}}.
  30. V ( x ) V(x)
  31. I ( x ) I(x)
  32. V ( x ) = V + e - γ x + V - e γ x V(x)=V^{+}e^{-\gamma x}+V^{-}e^{\gamma x}\,
  33. I ( x ) = 1 Z 0 ( V + e - γ x - V - e γ x ) . I(x)=\frac{1}{Z_{0}}(V^{+}e^{-\gamma x}-V^{-}e^{\gamma x}).\,
  34. V ± V^{\pm}
  35. I ± I^{\pm}
  36. V in ( t ) V_{\mathrm{in}}(t)\,
  37. x = 0 x=0
  38. x x
  39. V out ( x , t ) V_{\mathrm{out}}(x,t)\,
  40. x x
  41. V ~ ( ω ) \tilde{V}(\omega)
  42. V in ( t ) V_{\mathrm{in}}(t)\,
  43. e - Re ( γ ) x e^{\mathrm{-Re}(\gamma)x}\,
  44. - Im ( γ ) x \mathrm{-Im}(\gamma)x\,
  45. γ \gamma
  46. Re ( γ ) = ( a 2 + b 2 ) 1 / 4 cos ( atan2 ( b , a ) / 2 ) \mathrm{Re}(\gamma)=(a^{2}+b^{2})^{1/4}\cos(\mathrm{atan2}(b,a)/2)\,
  47. Im ( γ ) = ( a 2 + b 2 ) 1 / 4 sin ( atan2 ( b , a ) / 2 ) \mathrm{Im}(\gamma)=(a^{2}+b^{2})^{1/4}\sin(\mathrm{atan2}(b,a)/2)\,
  48. a ω 2 L C [ ( R ω L ) ( G ω C ) - 1 ] a\equiv\omega^{2}LC\left[\left(\frac{R}{\omega L}\right)\left(\frac{G}{\omega C% }\right)-1\right]
  49. b ω 2 L C ( R ω L + G ω C ) . b\equiv\omega^{2}LC\left(\frac{R}{\omega L}+\frac{G}{\omega C}\right).
  50. R / ω L R/\omega L
  51. G / ω C G/\omega C
  52. Re ( γ ) L C 2 ( R L + G C ) \mathrm{Re}(\gamma)\approx\frac{\sqrt{LC}}{2}\left(\frac{R}{L}+\frac{G}{C}% \right)\,
  53. Im ( γ ) ω L C . \mathrm{Im}(\gamma)\approx\omega\sqrt{LC}.\,
  54. - ω δ -\omega\delta
  55. δ \delta
  56. V o u t ( t ) V_{out}(t)
  57. V out ( x , t ) V in ( t - L C x ) e - L C 2 ( R L + G C ) x . V_{\mathrm{out}}(x,t)\approx V_{\mathrm{in}}(t-\sqrt{LC}x)e^{-\frac{\sqrt{LC}}% {2}\left(\frac{R}{L}+\frac{G}{C}\right)x}.\,
  58. l l
  59. l l
  60. l l
  61. Z i n ( l ) = V ( l ) I ( l ) = Z 0 1 + Γ L e - 2 γ l 1 - Γ L e - 2 γ l Z_{in}\left(l\right)=\frac{V(l)}{I(l)}=Z_{0}\frac{1+\Gamma_{L}e^{-2\gamma l}}{% 1-\Gamma_{L}e^{-2\gamma l}}
  62. γ γ
  63. Γ L = ( Z L - Z 0 ) / ( Z L + Z 0 ) \Gamma_{L}=\left(Z_{L}-Z_{0}\right)/\left(Z_{L}+Z_{0}\right)
  64. Z i n ( l ) = Z 0 Z L + Z 0 tanh ( γ l ) Z 0 + Z L tanh ( γ l ) Z_{in}\left(l\right)=Z_{0}\frac{Z_{L}+Z_{0}\tanh\left(\gamma l\right)}{Z_{0}+Z% _{L}\tanh\left(\gamma l\right)}
  65. γ = j β γ=jβ
  66. Z in ( l ) = Z 0 Z L + j Z 0 tan ( β l ) Z 0 + j Z L tan ( β l ) Z_{\mathrm{in}}(l)=Z_{0}\frac{Z_{L}+jZ_{0}\tan(\beta l)}{Z_{0}+jZ_{L}\tan(% \beta l)}
  67. β = 2 π λ \beta=\frac{2\pi}{\lambda}
  68. β β
  69. β l = n π \beta l=n\pi
  70. Z in = Z L Z_{\mathrm{in}}=Z_{L}\,
  71. n n
  72. n = 0 n=0
  73. Z in = Z 0 2 Z L . Z_{\mathrm{in}}=\frac{{Z_{0}}^{2}}{Z_{L}}.\,
  74. Z in = Z L = Z 0 Z_{\mathrm{in}}=Z_{L}=Z_{0}\,
  75. l l
  76. λ \lambda
  77. Z L = 0 Z_{L}=0
  78. Z in ( l ) = j Z 0 tan ( β l ) . Z_{\mathrm{in}}(l)=jZ_{0}\tan(\beta l).\,
  79. Z L = Z_{L}=\infty
  80. Z in ( l ) = - j Z 0 cot ( β l ) . Z_{\mathrm{in}}(l)=-jZ_{0}\cot(\beta l).\,
  81. Z i + 1 = Z 0 , i Z i + j Z 0 , i tan ( β i l i ) Z 0 , i + j Z i tan ( β i l i ) Z_{\mathrm{i+1}}=Z_{\mathrm{0,i}}\frac{Z_{i}+jZ_{\mathrm{0,i}}\tan(\beta_{i}l_% {i})}{Z_{\mathrm{0,i}}+jZ_{i}\tan(\beta_{i}l_{i})}
  82. β i \beta_{i}

Travelling_salesman_problem.html

  1. x i j = { 1 the path goes from city i to city j 0 otherwise x_{ij}=\begin{cases}1&\,\text{the path goes from city }i\,\text{ to city }j\\ 0&\,\text{otherwise}\end{cases}
  2. u i u_{i}
  3. c i j c_{ij}
  4. min \displaystyle\min
  5. u i u_{i}
  6. x i j = 1 x_{ij}=1
  7. n k ( n - 1 ) k , nk\leq(n-1)k,
  8. u i u_{i}
  9. u i = t u_{i}=t
  10. u i - u j n - 1 , u_{i}-u_{j}\leq n-1,
  11. u i u_{i}
  12. u j u_{j}
  13. x i j = 0. x_{ij}=0.
  14. x i j = 1 x_{ij}=1
  15. u i - u j + n x i j = ( t ) - ( t + 1 ) + n = n - 1 , u_{i}-u_{j}+nx_{ij}=(t)-(t+1)+n=n-1,
  16. O ( n ! ) O(n!)
  17. O ( n 2 2 n ) O(n^{2}2^{n})
  18. O ( 1.9999 n ) O(1.9999^{n})
  19. Θ ( log | V | ) \Theta(\log|V|)
  20. d A B d A C + d C B d_{AB}\leq d_{AC}+d_{CB}
  21. d A B d_{AB}
  22. O ( n ( log n ) ( O ( c d ) ) d - 1 ) , O\left(n(\log n)^{(O(c\sqrt{d}))^{d-1}}\right),
  23. X 1 , , X n X_{1},\ldots,X_{n}
  24. n n
  25. [ 0 , 1 ] 2 [0,1]^{2}
  26. L n L^{\ast}_{n}
  27. L n * n β when n , \frac{L^{*}_{n}}{\sqrt{n}}\rightarrow\beta\qquad\,\text{when }n\to\infty,
  28. β \beta
  29. L n * 2 n + 2 L^{*}_{n}\leq 2\sqrt{n}+2
  30. β = lim n 𝔼 [ L n * ] / n \beta=\lim_{n\to\infty}\mathbb{E}[L^{*}_{n}]/\sqrt{n}
  31. β \beta
  32. 𝔼 [ L n * ] \mathbb{E}[L^{*}_{n}]
  33. L * 2 n + 2 L^{*}\leq 2\sqrt{n}+2
  34. β 2 \beta\leq 2
  35. n \sqrt{n}
  36. 1 / n 1/\sqrt{n}
  37. L n * 2 n + 1.75 L^{*}_{n}\leq\sqrt{2n}+1.75
  38. β 2 \beta\leq\sqrt{2}
  39. β 0.984 2 \beta\leq 0.984\sqrt{2}
  40. β 0.92 \beta\leq 0.92\dots
  41. 𝔼 [ L n * ] \mathbb{E}[L^{*}_{n}]
  42. n n
  43. X 0 X_{0}
  44. X i X 0 X_{i}\neq X_{0}
  45. 𝔼 [ L n * ] 1 2 n . \mathbb{E}[L^{*}_{n}]\geq\tfrac{1}{2}\sqrt{n}.
  46. 𝔼 [ L n * ] \mathbb{E}[L^{*}_{n}]
  47. 1 2 n \tfrac{1}{2}n
  48. X 0 X_{0}
  49. X i , X j X 0 X_{i},X_{j}\neq X_{0}
  50. 𝔼 [ L n * ] ( 1 4 + 3 8 ) n = 5 8 n , \mathbb{E}[L^{*}_{n}]\geq\left(\tfrac{1}{4}+\tfrac{3}{8}\right)\sqrt{n}=\tfrac% {5}{8}\sqrt{n},
  51. 𝔼 [ L n * ] ( 5 8 + 19 5184 ) n , \mathbb{E}[L^{*}_{n}]\geq(\tfrac{5}{8}+\tfrac{19}{5184})\sqrt{n},
  52. L n * L^{*}_{n}
  53. β ( L n * / n ) \beta(\simeq L^{*}_{n}/{\sqrt{n}})
  54. L n * 0.7080 n + 0.522 , L^{*}_{n}\gtrsim 0.7080\sqrt{n}+0.522,
  55. L n * 0.7078 n + 0.551 L^{*}_{n}\gtrsim 0.7078\sqrt{n}+0.551
  56. 1 25 ( 33 + ε ) \tfrac{1}{25}(33+\varepsilon)

Tree_(graph_theory).html

  1. K 3 K_{3}
  2. ( n - 2 d 1 - 1 , d 2 - 1 , , d n - 1 ) . {n-2\choose d_{1}-1,d_{2}-1,\ldots,d_{n}-1}.
  3. t ( n ) C α n n - 5 / 2 as n , {t(n)\sim C\alpha^{n}n^{-5/2}\quad\,\text{as }n\to\infty,}
  4. f g f\sim g
  5. lim n f / g = 1 \lim_{n\to\infty}f/g=1
  6. r ( n ) r(n)
  7. r ( n ) D α n n - 3 / 2 as n , r(n)\sim D\alpha^{n}n^{-3/2}\quad\,\text{as }n\to\infty,

Triangle.html

  1. A B C \triangle ABC
  2. \underbrace{\qquad\qquad\qquad\qquad\qquad\qquad}
  3. a 2 + b 2 = c 2 . a^{2}+b^{2}=c^{2}.\,
  4. a + b + 90 = 180 a + b = 90 a = 90 - b a+b+90^{\circ}=180^{\circ}\Rightarrow a+b=90^{\circ}\Rightarrow a=90^{\circ}-b
  5. c = 2 a c=2a\,
  6. b = a × 3 . b=a\times\sqrt{3}.
  7. tan α 2 tan β 2 + tan β 2 tan γ 2 + tan γ 2 tan α 2 = 1 , \tan{\frac{\alpha}{2}}\tan{\frac{\beta}{2}}+\tan{\frac{\beta}{2}}\tan{\frac{% \gamma}{2}}+\tan{\frac{\gamma}{2}}\tan{\frac{\alpha}{2}}=1,
  8. sin 2 α 2 + sin 2 β 2 + sin 2 γ 2 + 2 sin α 2 sin β 2 sin γ 2 = 1 , \sin^{2}{\frac{\alpha}{2}}+\sin^{2}{\frac{\beta}{2}}+\sin^{2}{\frac{\gamma}{2}% }+2\sin{\frac{\alpha}{2}}\sin{\frac{\beta}{2}}\sin{\frac{\gamma}{2}}=1,
  9. sin ( 2 α ) + sin ( 2 β ) + sin ( 2 γ ) = 4 sin ( α ) sin ( β ) sin ( γ ) , \sin(2\alpha)+\sin(2\beta)+\sin(2\gamma)=4\sin(\alpha)\sin(\beta)\sin(\gamma),
  10. cos 2 α + cos 2 β + cos 2 γ + 2 cos ( α ) cos ( β ) cos ( γ ) = 1 , \cos^{2}\alpha+\cos^{2}\beta+\cos^{2}\gamma+2\cos(\alpha)\cos(\beta)\cos(% \gamma)=1,
  11. tan ( α ) + tan ( β ) + tan ( γ ) = tan ( α ) tan ( β ) tan ( γ ) , \tan(\alpha)+\tan(\beta)+\tan(\gamma)=\tan(\alpha)\tan(\beta)\tan(\gamma),
  12. sin A = opposite side hypotenuse = a h . \sin A=\frac{\,\text{opposite side}}{\,\text{hypotenuse}}=\frac{a}{h}\,.
  13. cos A = adjacent side hypotenuse = b h . \cos A=\frac{\,\text{adjacent side}}{\,\text{hypotenuse}}=\frac{b}{h}\,.
  14. tan A = opposite side adjacent side = a b = sin A cos A . \tan A=\frac{\,\text{opposite side}}{\,\text{adjacent side}}=\frac{a}{b}=\frac% {\sin A}{\cos A}\,.
  15. θ = arcsin ( opposite side hypotenuse ) \theta=\arcsin\left(\frac{\,\text{opposite side}}{\,\text{hypotenuse}}\right)
  16. θ = arccos ( adjacent side hypotenuse ) \theta=\arccos\left(\frac{\,\text{adjacent side}}{\,\text{hypotenuse}}\right)
  17. θ = arctan ( opposite side adjacent side ) \theta=\arctan\left(\frac{\,\text{opposite side}}{\,\text{adjacent side}}\right)
  18. a sin α = b sin β = c sin γ . \frac{a}{\sin\alpha}=\frac{b}{\sin\beta}=\frac{c}{\sin\gamma}.
  19. c 2 = a 2 + b 2 - 2 a b cos ( γ ) c^{2}\ =a^{2}+b^{2}-2ab\cos(\gamma)
  20. b 2 = a 2 + c 2 - 2 a c cos ( β ) b^{2}\ =a^{2}+c^{2}-2ac\cos(\beta)
  21. a 2 = b 2 + c 2 - 2 b c cos ( α ) a^{2}\ =b^{2}+c^{2}-2bc\cos(\alpha)
  22. α = arccos ( b 2 + c 2 - a 2 2 b c ) \alpha=\arccos\left(\frac{b^{2}+c^{2}-a^{2}}{2bc}\right)
  23. β = arccos ( a 2 + c 2 - b 2 2 a c ) \beta=\arccos\left(\frac{a^{2}+c^{2}-b^{2}}{2ac}\right)
  24. γ = arccos ( a 2 + b 2 - c 2 2 a b ) \gamma=\arccos\left(\frac{a^{2}+b^{2}-c^{2}}{2ab}\right)
  25. a - b a + b = tan [ 1 2 ( α - β ) ] tan [ 1 2 ( α + β ) ] . \frac{a-b}{a+b}=\frac{\tan[\frac{1}{2}(\alpha-\beta)]}{\tan[\frac{1}{2}(\alpha% +\beta)]}.
  26. T = 1 2 b h T=\frac{1}{2}bh
  27. T = 1 2 b h T=\frac{1}{2}bh
  28. T = 1 2 a b sin γ = 1 2 b c sin α = 1 2 c a sin β T=\frac{1}{2}ab\sin\gamma=\frac{1}{2}bc\sin\alpha=\frac{1}{2}ca\sin\beta
  29. γ \gamma
  30. γ \gamma
  31. T = 1 2 a b sin ( α + β ) = 1 2 b c sin ( β + γ ) = 1 2 c a sin ( γ + α ) . T=\frac{1}{2}ab\sin(\alpha+\beta)=\frac{1}{2}bc\sin(\beta+\gamma)=\frac{1}{2}% ca\sin(\gamma+\alpha).
  32. T = b 2 ( sin α ) ( sin ( α + β ) ) 2 sin β , T=\frac{b^{2}(\sin\alpha)(\sin(\alpha+\beta))}{2\sin\beta},
  33. T = a 2 2 ( cot β + cot γ ) = a 2 ( sin β ) ( sin γ ) 2 sin ( β + γ ) , T=\frac{a^{2}}{2(\cot\beta+\cot\gamma)}=\frac{a^{2}(\sin\beta)(\sin\gamma)}{2% \sin(\beta+\gamma)},
  34. T = s ( s - a ) ( s - b ) ( s - c ) T=\sqrt{s(s-a)(s-b)(s-c)}
  35. s = a + b + c 2 s=\tfrac{a+b+c}{2}
  36. T = 1 4 ( a 2 + b 2 + c 2 ) 2 - 2 ( a 4 + b 4 + c 4 ) T=\frac{1}{4}\sqrt{(a^{2}+b^{2}+c^{2})^{2}-2(a^{4}+b^{4}+c^{4})}
  37. T = 1 4 2 ( a 2 b 2 + a 2 c 2 + b 2 c 2 ) - ( a 4 + b 4 + c 4 ) T=\frac{1}{4}\sqrt{2(a^{2}b^{2}+a^{2}c^{2}+b^{2}c^{2})-(a^{4}+b^{4}+c^{4})}
  38. T = 1 4 ( a + b - c ) ( a - b + c ) ( - a + b + c ) ( a + b + c ) . T=\frac{1}{4}\sqrt{(a+b-c)(a-b+c)(-a+b+c)(a+b+c)}.
  39. | 𝐀𝐁 × 𝐀𝐂 | , |\mathbf{AB}\times\mathbf{AC}|,
  40. 1 2 | 𝐀𝐁 × 𝐀𝐂 | . \frac{1}{2}|\mathbf{AB}\times\mathbf{AC}|.
  41. 1 2 ( 𝐀𝐁 𝐀𝐁 ) ( 𝐀𝐂 𝐀𝐂 ) - ( 𝐀𝐁 𝐀𝐂 ) 2 = 1 2 | 𝐀𝐁 | 2 | 𝐀𝐂 | 2 - ( 𝐀𝐁 𝐀𝐂 ) 2 . \frac{1}{2}\sqrt{(\mathbf{AB}\cdot\mathbf{AB})(\mathbf{AC}\cdot\mathbf{AC})-(% \mathbf{AB}\cdot\mathbf{AC})^{2}}=\frac{1}{2}\sqrt{|\mathbf{AB}|^{2}|\mathbf{% AC}|^{2}-(\mathbf{AB}\cdot\mathbf{AC})^{2}}.\,
  42. 1 2 | x 1 y 2 - x 2 y 1 | . \frac{1}{2}\,|x_{1}y_{2}-x_{2}y_{1}|.\,
  43. 1 / 2 {1}/{2}
  44. T = 1 2 | det ( x B x C y B y C ) | = 1 2 | x B y C - x C y B | . T=\frac{1}{2}\left|\det\begin{pmatrix}x_{B}&x_{C}\\ y_{B}&y_{C}\end{pmatrix}\right|=\frac{1}{2}|x_{B}y_{C}-x_{C}y_{B}|.
  45. T = 1 2 | det ( x A x B x C y A y B y C 1 1 1 ) | = 1 2 | x A y B - x A y C + x B y C - x B y A + x C y A - x C y B | , T=\frac{1}{2}\left|\det\begin{pmatrix}x_{A}&x_{B}&x_{C}\\ y_{A}&y_{B}&y_{C}\\ 1&1&1\end{pmatrix}\right|=\frac{1}{2}\big|x_{A}y_{B}-x_{A}y_{C}+x_{B}y_{C}-x_{% B}y_{A}+x_{C}y_{A}-x_{C}y_{B}\big|,
  46. T = 1 2 | ( x A - x C ) ( y B - y A ) - ( x A - x B ) ( y C - y A ) | . T=\frac{1}{2}\big|(x_{A}-x_{C})(y_{B}-y_{A})-(x_{A}-x_{B})(y_{C}-y_{A})\big|.
  47. a ¯ \bar{a}
  48. b ¯ \bar{b}
  49. c ¯ \bar{c}
  50. T = i 4 | a a ¯ 1 b b ¯ 1 c c ¯ 1 | T=\frac{i}{4}\begin{vmatrix}a&\bar{a}&1\\ b&\bar{b}&1\\ c&\bar{c}&1\end{vmatrix}
  51. T = 1 2 | x A x B x C y A y B y C 1 1 1 | 2 + | y A y B y C z A z B z C 1 1 1 | 2 + | z A z B z C x A x B x C 1 1 1 | 2 . T=\frac{1}{2}\sqrt{\begin{vmatrix}x_{A}&x_{B}&x_{C}\\ y_{A}&y_{B}&y_{C}\\ 1&1&1\end{vmatrix}^{2}+\begin{vmatrix}y_{A}&y_{B}&y_{C}\\ z_{A}&z_{B}&z_{C}\\ 1&1&1\end{vmatrix}^{2}+\begin{vmatrix}z_{A}&z_{B}&z_{C}\\ x_{A}&x_{B}&x_{C}\\ 1&1&1\end{vmatrix}^{2}}.
  52. T = 4 3 σ ( σ - m a ) ( σ - m b ) ( σ - m c ) . T=\frac{4}{3}\sqrt{\sigma(\sigma-m_{a})(\sigma-m_{b})(\sigma-m_{c})}.
  53. H = ( h a - 1 + h b - 1 + h c - 1 ) / 2 H=(h_{a}^{-1}+h_{b}^{-1}+h_{c}^{-1})/2
  54. T - 1 = 4 H ( H - h a - 1 ) ( H - h b - 1 ) ( H - h c - 1 ) . T^{-1}=4\sqrt{H(H-h_{a}^{-1})(H-h_{b}^{-1})(H-h_{c}^{-1})}.
  55. T = D 2 S ( S - sin α ) ( S - sin β ) ( S - sin γ ) T=D^{2}\sqrt{S(S-\sin\alpha)(S-\sin\beta)(S-\sin\gamma)}
  56. D = a sin α = b sin β = c sin γ . D=\tfrac{a}{\sin\alpha}=\tfrac{b}{\sin\beta}=\tfrac{c}{\sin\gamma}.
  57. T = I + 1 2 B - 1 T=I+\frac{1}{2}B-1
  58. I I
  59. T = r s , T=r\cdot s,
  60. T = 1 2 D 2 ( sin α ) ( sin β ) ( sin γ ) T=\frac{1}{2}D^{2}(\sin\alpha)(\sin\beta)(\sin\gamma)
  61. T = a b c 2 D = a b c 4 R T=\frac{abc}{2D}=\frac{abc}{4R}
  62. T = tan α 4 ( b 2 + c 2 - a 2 ) T=\frac{\tan\alpha}{4}(b^{2}+c^{2}-a^{2})
  63. T = r r 1 r 2 r 3 . T=\sqrt{rr_{1}r_{2}r_{3}}.
  64. T = 1 2 [ a b c h a h b h c ] 1 / 3 , T=\frac{1}{2}[abch_{a}h_{b}h_{c}]^{1/3},
  65. T = 1 2 a b h a h b , T=\frac{1}{2}\sqrt{abh_{a}h_{b}},
  66. T = a + b 2 ( h a - 1 + h b - 1 ) , T=\frac{a+b}{2(h_{a}^{-1}+h_{b}^{-1})},
  67. T = R h b h c a T=\frac{Rh_{b}h_{c}}{a}
  68. T = h a h b 2 sin γ . T=\frac{h_{a}h_{b}}{2\sin\gamma}.
  69. p 2 12 3 , \tfrac{p^{2}}{12\sqrt{3}},
  70. 4 3 T a 2 + b 2 + c 2 4\sqrt{3}T\leq a^{2}+b^{2}+c^{2}
  71. 4 3 T 9 a b c a + b + c , 4\sqrt{3}T\leq\frac{9abc}{a+b+c},
  72. 3 4 ( a 2 + b 2 + c 2 ) = m a 2 + m b 2 + m c 2 \frac{3}{4}(a^{2}+b^{2}+c^{2})=m_{a}^{2}+m_{b}^{2}+m_{c}^{2}
  73. m a = 1 2 2 b 2 + 2 c 2 - a 2 = 1 2 ( a 2 + b 2 + c 2 ) - 3 4 a 2 m_{a}=\frac{1}{2}\sqrt{2b^{2}+2c^{2}-a^{2}}=\sqrt{\frac{1}{2}(a^{2}+b^{2}+c^{2% })-\frac{3}{4}a^{2}}
  74. w a = 2 b c s ( s - a ) b + c = b c [ 1 - a 2 ( b + c ) 2 ] w_{a}=\frac{2\sqrt{bcs(s-a)}}{b+c}=\sqrt{bc\left[1-\frac{a^{2}}{(b+c)^{2}}% \right]}
  75. p a = 2 a T a 2 + b 2 - c 2 , p_{a}=\frac{2aT}{a^{2}+b^{2}-c^{2}},
  76. p b = 2 b T a 2 + b 2 - c 2 , p_{b}=\frac{2bT}{a^{2}+b^{2}-c^{2}},
  77. p c = 2 c T a 2 - b 2 + c 2 , p_{c}=\frac{2cT}{a^{2}-b^{2}+c^{2}},
  78. a b c a\geq b\geq c
  79. T . T.
  80. h a = 2 T a . h_{a}=\frac{2T}{a}.
  81. R = a 2 b 2 c 2 ( a + b + c ) ( - a + b + c ) ( a - b + c ) ( a + b - c ) ; R=\sqrt{\frac{a^{2}b^{2}c^{2}}{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}};
  82. r = ( - a + b + c ) ( a - b + c ) ( a + b - c ) 4 ( a + b + c ) ; r=\sqrt{\frac{(-a+b+c)(a-b+c)(a+b-c)}{4(a+b+c)}};
  83. 1 r = 1 h a + 1 h b + 1 h c \frac{1}{r}=\frac{1}{h_{a}}+\frac{1}{h_{b}}+\frac{1}{h_{c}}
  84. r R = 4 T 2 s a b c = cos α + cos β + cos γ - 1 ; \frac{r}{R}=\frac{4T^{2}}{sabc}=\cos\alpha+\cos\beta+\cos\gamma-1;
  85. 2 R r = a b c a + b + c 2Rr=\frac{abc}{a+b+c}
  86. f 2 = ( a c + b d ) ( a d + b c ) ( a b + c d ) . f^{2}=\frac{(ac+bd)(ad+bc)}{(ab+cd)}.\,
  87. ( P A ) 2 + ( P B ) 2 + ( P C ) 2 = ( M A ) 2 + ( M B ) 2 + ( M C ) 2 + 3 ( P M ) 2 . (PA)^{2}+(PB)^{2}+(PC)^{2}=(MA)^{2}+(MB)^{2}+(MC)^{2}+3(PM)^{2}.\,
  88. p a p b = b a , p b p c = c b , p a p c = c a \frac{p_{a}}{p_{b}}=\frac{b}{a},\ \ \ \ \frac{p_{b}}{p_{c}}=\frac{c}{b},\ \ \ % \ \frac{p_{a}}{p_{c}}=\frac{c}{a}\,
  89. p a a = p b b = p c c = 2 3 T . p_{a}\cdot a=p_{b}\cdot b=p_{c}\cdot c=\frac{2}{3}T.\,
  90. d 2 = R ( R - 2 r ) \displaystyle d^{2}=R(R-2r)
  91. 1 R - d + 1 R + d = 1 r , \frac{1}{R-d}+\frac{1}{R+d}=\frac{1}{r},
  92. a = b cos C + c cos B , b = c cos A + a cos C , c = a cos B + b cos A . a=b\cos C+c\cos B,\quad b=c\cos A+a\cos C,\quad c=a\cos B+b\cos A.
  93. P A ¯ Q A ¯ C A ¯ A B ¯ + P B ¯ Q B ¯ A B ¯ B C ¯ + P C ¯ Q C ¯ B C ¯ C A ¯ = 1. \frac{\overline{PA}\cdot\overline{QA}}{\overline{CA}\cdot\overline{AB}}+\frac{% \overline{PB}\cdot\overline{QB}}{\overline{AB}\cdot\overline{BC}}+\frac{% \overline{PC}\cdot\overline{QC}}{\overline{BC}\cdot\overline{CA}}=1.
  94. q = 2 T a a 2 + 2 T . q=\frac{2Ta}{a^{2}+2T}.
  95. 2 2 / 3 = 0.94.... 2\sqrt{2}/3=0.94....
  96. x : y : z x:y:z
  97. x : y x:y
  98. α : β : γ \alpha:\beta:\gamma

Triangle_inequality.html

  1. x x
  2. y y
  3. z z
  4. z z
  5. x + y x+y
  6. z z
  7. x + y x+y
  8. x x
  9. y y
  10. z z
  11. z z
  12. z x + y , z\leq x+y,
  13. 𝐱 + 𝐲 𝐱 + 𝐲 , \|\mathbf{x}+\mathbf{y}\|\leq\|\mathbf{x}\|+\|\mathbf{y}\|,
  14. z z
  15. 𝐱 + 𝐲 \mathbf{x}+\mathbf{y}
  16. 𝐱 \mathbf{x}
  17. 𝐲 \mathbf{y}
  18. 180 ° 180°
  19. 0 °
  20. 0 , π 0,π
  21. p 1 p≥1
  22. A B C ABC
  23. B C BC
  24. B D BD
  25. A B AB
  26. β > α β>α
  27. A D ¯ > A C ¯ \overline{AD}>\overline{AC}
  28. A D ¯ = A B ¯ + B D ¯ = A B ¯ + B C ¯ \overline{AD}=\overline{AB}+\overline{BD}=\overline{AB}+\overline{BC}
  29. A B ¯ + B C ¯ > A C ¯ \overline{AB}+\overline{BC}>\overline{AC}
  30. a a
  31. b b
  32. c c
  33. a + b > c , b + c > a , c + a > b . a+b>c,\quad b+c>a,\quad c+a>b.
  34. | a - b | < c < a + b . |a-b|<c<a+b.
  35. max ( a , b , c ) < a + b + c - max ( a , b , c ) \max(a,b,c)<a+b+c-\max(a,b,c)
  36. 2 max ( a , b , c ) < a + b + c 2\max(a,b,c)<a+b+c
  37. 4 Area = ( a + b + c ) ( - a + b + c ) ( a - b + c ) ( a + b - c ) 4\cdot\,\text{Area}=\sqrt{(a+b+c)(-a+b+c)(a-b+c)(a+b-c)}
  38. = - a 4 - b 4 - c 4 + 2 a 2 b 2 + 2 a 2 c 2 + 2 b 2 c 2 . =\sqrt{-a^{4}-b^{4}-c^{4}+2a^{2}b^{2}+2a^{2}c^{2}+2b^{2}c^{2}}.
  39. A B ¯ = A C ¯ \overline{AB}=\overline{AC}
  40. A D C ADC
  41. A B C ABC
  42. A B ¯ = A C ¯ \overline{AB}=\overline{AC}
  43. A D C ADC
  44. α + γ = π / 2 . \alpha+\gamma=\pi/2\ .
  45. A B C ABC
  46. 2 β + γ = π . 2\beta+\gamma=\pi\ .
  47. α = π / 2 - γ , while β = π / 2 - γ / 2 , \alpha=\pi/2-\gamma,\ \mathrm{while}\ \beta=\pi/2-\gamma/2\ ,
  48. α < β . \alpha<\beta\ .
  49. A D AD
  50. α α
  51. A B AB
  52. β β
  53. A B ¯ = A C ¯ \overline{AB}=\overline{AC}
  54. AC ¯ > AD ¯ . \overline{\mathrm{AC}}>\overline{\mathrm{AD}}\ .
  55. A C ¯ > D C ¯ \overline{AC}>\overline{DC}
  56. B B
  57. B B
  58. D D
  59. γ γ
  60. B B
  61. A A
  62. D D
  63. A B C ABC
  64. B D C BDC
  65. π / 2 π/2
  66. π / 2 π/2
  67. π π
  68. a a
  69. a + d a+d
  70. a + 2 d a+2d
  71. 0 < a < 2 a + 3 d 0<a<2a+3d\,
  72. 0 < a + d < 2 a + 2 d 0<a+d<2a+2d\,
  73. 0 < a + 2 d < 2 a + d . 0<a+2d<2a+d.\,
  74. a > 0 and - a 3 < d < a . a>0\,\text{ and }-\frac{a}{3}<d<a.
  75. d d
  76. d = a / 3 d=a/3
  77. 3 3
  78. 4 4
  79. 5 5
  80. a a
  81. a r ar
  82. 0 < a < a r + a r 2 0<a<ar+ar^{2}\,
  83. 0 < a r < a + a r 2 0<ar<a+ar^{2}\,
  84. 0 < a r 2 < a + a r . 0<ar^{2}<a+ar.\,
  85. a > 0 a>0
  86. a > 0 a>0
  87. r > 0 r>0
  88. r 2 + r - 1 > 0 r 2 - r - 1 < 0. \begin{aligned}\displaystyle r^{2}+r-1&\displaystyle{}>0\\ \displaystyle r^{2}-r-1&\displaystyle{}<0.\end{aligned}\,
  89. r r
  90. r > φ 1 r>φ−1
  91. φ φ
  92. r r
  93. φ - 1 < r < φ and a > 0. \varphi-1<r<\varphi\,\,\text{ and }a>0.\,
  94. r r
  95. r = φ r=\sqrt{φ}
  96. a a
  97. a r ar
  98. 0 < a < a r + a r 2 + a r 3 0<a<ar+ar^{2}+ar^{3}\,
  99. 0 < a r < a + a r 2 + a r 3 0<ar<a+ar^{2}+ar^{3}\,
  100. 0 < a r 2 < a + a r + a r 3 0<ar^{2}<a+ar+ar^{3}\,
  101. 0 < a r 3 < a + a r + a r 2 . 0<ar^{3}<a+ar+ar^{2}.\,
  102. a > 0 a>0
  103. r 3 + r 2 + r - 1 > 0 r^{3}+r^{2}+r-1>0\,
  104. r 3 - r 2 - r - 1 < 0. r^{3}-r^{2}-r-1<0.\,
  105. r r
  106. ( a 2 - b 2 + c 2 2 a , c 2 - ( a 2 - b 2 + c 2 2 a ) 2 ) . \left(\tfrac{a^{2}-b^{2}+c^{2}}{2a},\sqrt{c^{2}-\left(\tfrac{a^{2}-b^{2}+c^{2}% }{2a}\right)^{2}}\right).
  107. ( n 1 ) (n−1)
  108. n n
  109. n n
  110. V V
  111. x + y x + y x , y V \displaystyle\|x+y\|\leq\|x\|+\|y\|\quad\forall\,x,y\in V
  112. x + y = x + y \|x+y\|=\|x\|+\|y\|
  113. x x
  114. y y
  115. x + y x+y
  116. x x
  117. y y
  118. x = 0 x=0
  119. y = 0 y=0
  120. x = α y x=αy
  121. α > 0 α>0
  122. x = ( 1 , 0 ) x=(1,0)
  123. y = ( 0 , 1 ) y=(0,1)
  124. x x
  125. y y
  126. x + y x+y
  127. x + y = ( 1 , 1 ) = | 1 | + | 1 | = 2 = x + y . \|x+y\|=\|(1,1)\|=|1|+|1|=2=\|x\|+\|y\|.
  128. x x
  129. y y
  130. | x + y | | x | + | y | , |x+y|\leq|x|+|y|,
  131. x x
  132. y y
  133. | x - y | | | x | - | y | | . |x-y|\geq\bigg||x|-|y|\bigg|.
  134. x x
  135. y y
  136. x , y ⟨x,y⟩
  137. x + y 2 \|x+y\|^{2}
  138. = x + y , x + y =\langle x+y,x+y\rangle
  139. = x 2 + x , y + y , x + y 2 =\|x\|^{2}+\langle x,y\rangle+\langle y,x\rangle+\|y\|^{2}
  140. x 2 + 2 | x , y | + y 2 \leq\|x\|^{2}+2|\langle x,y\rangle|+\|y\|^{2}
  141. x 2 + 2 x y + y 2 \leq\|x\|^{2}+2\|x\|\|y\|+\|y\|^{2}
  142. = ( x + y ) 2 =\left(\|x\|+\|y\|\right)^{2}
  143. x 2 + 2 x y + y 2 = ( x + y ) 2 . \|x\|^{2}+2\|x\|\|y\|+\|y\|^{2}=\left(\|x\|+\|y\|\right)^{2}\ .
  144. x x
  145. y y
  146. x , y + y , x 2 | x , y | \langle x,y\rangle+\langle y,x\rangle\leq 2|\langle x,y\rangle|
  147. x x
  148. y y
  149. x x
  150. y y
  151. p p
  152. x p = ( i = 1 n | x i | p ) 1 / p , \|x\|_{p}=\left(\sum_{i=1}^{n}|x_{i}|^{p}\right)^{1/p}\ ,
  153. x x
  154. p = 2 p=2
  155. p p
  156. x 2 = ( i = 1 n | x i | 2 ) 1 / 2 = ( i = 1 n x i 2 ) 1 / 2 , \|x\|_{2}=\left(\sum_{i=1}^{n}|x_{i}|^{2}\right)^{1/2}=\left(\sum_{i=1}^{n}x_{% i}^{2}\right)^{1/2}\ ,
  157. n n
  158. p = 2 p=2
  159. p p
  160. p p
  161. x + y p x p + y p . \|x+y\|_{p}\leq\|x\|_{p}+\|y\|_{p}\ .
  162. M M
  163. d d
  164. d ( x , z ) d ( x , y ) + d ( y , z ) , d(x,\ z)\leq d(x,\ y)+d(y,\ z)\ ,
  165. x x
  166. y y
  167. z z
  168. M M
  169. x x
  170. z z
  171. x x
  172. y y
  173. y y
  174. z z
  175. ε > 0 ε>0
  176. d ( x , y ) x y d(x,y)≔‖x−y‖
  177. x y x−y
  178. y y
  179. x x
  180. | x - y | x - y , \bigg|\|x\|-\|y\|\bigg|\leq\|x-y\|,
  181. | d ( y , x ) d ( x , z ) | d ( y , z ) |d(y,x)−d(x,z)|≤d(y,z)
  182. ‖–‖
  183. d ( x , ) d(x,–)
  184. 1 1
  185. y - x = - 1 ( x - y ) = | - 1 | x - y = x - y \|y-x\|=\|-1(x-y)\|=|-1|\|x-y\|=\|x-y\|
  186. x = ( x - y ) + y x - y + y x - y x - y , \|x\|=\|(x-y)+y\|\leq\|x-y\|+\|y\|\Rightarrow\|x\|-\|y\|\leq\|x-y\|,
  187. y = ( y - x ) + x y - x + x x - y - x - y , \|y\|=\|(y-x)+x\|\leq\|y-x\|+\|x\|\Rightarrow\|x\|-\|y\|\geq-\|x-y\|,
  188. - x - y x - y x - y | x - y | x - y . -\|x-y\|\leq\|x\|-\|y\|\leq\|x-y\|\Rightarrow\bigg|\|x\|-\|y\|\bigg|\leq\|x-y\|.
  189. x + y x + y x , y V such that x , y 0 and t x , t y 0. \|x+y\|\geq\|x\|+\|y\|\;\forall x,y\in V\,\text{ such that }\|x\|,\|y\|\geq 0% \,\text{ and }t_{x},t_{y}\geq 0.

Triangle_wave.html

  1. f f
  2. t t
  3. x triangle ( t ) \displaystyle x_{\mathrm{triangle}}(t)
  4. x ( t ) = 2 a ( t - a t a + 1 2 ) ( - 1 ) t a + 1 2 x(t)=\frac{2}{a}\left(t-a\left\lfloor\frac{t}{a}+\frac{1}{2}\right\rfloor% \right)(-1)^{\left\lfloor}\frac{t}{a}+\frac{1}{2}\right\rfloor
  5. n \scriptstyle\lfloor n\rfloor
  6. x ( t ) = | 2 ( t a - t a + 1 2 ) | x(t)=\left|2\left({t\over a}-\left\lfloor{t\over a}+{1\over 2}\right\rfloor% \right)\right|
  7. x ( t ) = 2 | 2 ( t a - t a + 1 2 ) | - 1 x(t)=2\left|2\left({t\over a}-\left\lfloor{t\over a}+{1\over 2}\right\rfloor% \right)\right|-1
  8. sgn ( sin ( x ) ) d x \int\operatorname{sgn}(\sin(x))\,dx\,
  9. y ( 0 ) = 1 y(0)=1
  10. y ( x ) = | x mod 4 - 2 | - 1 y(x)=|x\,\bmod\,4-2|-1
  11. y ( 0 ) = 0 y(0)=0
  12. y ( x ) = 4 a p { | [ ( x - p 4 ) mod p ] - p 2 | - p 4 } y(x)=\frac{4a}{p}\Biggl\{\biggl|\left[\left(x-\frac{p}{4}\right)\,\bmod\,p% \right]-\frac{p}{2}\biggr|-\frac{p}{4}\Biggr\}
  13. y ( x ) = 2 a π arcsin ( sin ( 2 π p x ) ) y(x)=\frac{2a}{\pi}\arcsin\left(\sin\left(\frac{2\pi}{p}x\right)\right)
  14. s = ( 4 a ) 2 + p 2 s=\sqrt{(4a)^{2}+p^{2}}

Trigonometric_functions.html

  1. sin θ = cos ( π 2 - θ ) = 1 csc θ \sin\theta=\cos\left(\frac{\pi}{2}-\theta\right)=\frac{1}{\csc\theta}
  2. cos θ = sin ( π 2 - θ ) = 1 sec θ \cos\theta=\sin\left(\frac{\pi}{2}-\theta\right)=\frac{1}{\sec\theta}\,
  3. tan θ = sin θ cos θ = cot ( π 2 - θ ) = 1 cot θ \tan\theta=\frac{\sin\theta}{\cos\theta}=\cot\left(\frac{\pi}{2}-\theta\right)% =\frac{1}{\cot\theta}
  4. cot θ = cos θ sin θ = tan ( π 2 - θ ) = 1 tan θ \cot\theta=\frac{\cos\theta}{\sin\theta}=\tan\left(\frac{\pi}{2}-\theta\right)% =\frac{1}{\tan\theta}
  5. sec θ = csc ( π 2 - θ ) = 1 cos θ \sec\theta=\csc\left(\frac{\pi}{2}-\theta\right)=\frac{1}{\cos\theta}
  6. csc θ = sec ( π 2 - θ ) = 1 sin θ \csc\theta=\sec\left(\frac{\pi}{2}-\theta\right)=\frac{1}{\sin\theta}
  7. sin A = opposite hypotenuse = a h . \sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{h}.
  8. cos A = adjacent hypotenuse = b h . \cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{h}.
  9. tan A = opposite adjacent = a b . \tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{b}.
  10. csc A = 1 sin A = hypotenuse opposite = h a . \csc A=\frac{1}{\sin A}=\frac{\textrm{hypotenuse}}{\textrm{opposite}}=\frac{h}% {a}.
  11. sec A = 1 cos A = hypotenuse adjacent = h b . \sec A=\frac{1}{\cos A}=\frac{\textrm{hypotenuse}}{\textrm{adjacent}}=\frac{h}% {b}.
  12. cot A = 1 tan A = adjacent opposite = b a . \cot A=\frac{1}{\tan A}=\frac{\textrm{adjacent}}{\textrm{opposite}}=\frac{b}{a}.
  13. r i s e / r u n {rise}/{run}
  14. x 2 + y 2 = 1. x^{2}+y^{2}=1.\,
  15. 1 2 0 , 1 2 1 , 1 2 2 , 1 2 3 , 1 2 4 , \frac{1}{2}\sqrt{0},\quad\frac{1}{2}\sqrt{1},\quad\frac{1}{2}\sqrt{2},\quad% \frac{1}{2}\sqrt{3},\quad\frac{1}{2}\sqrt{4},
  16. sin 15 = cos 75 = 6 - 2 4 \sin 15^{\circ}=\cos 75^{\circ}=\frac{\sqrt{6}-\sqrt{2}}{4}\,\!
  17. sin 18 = cos 72 = 5 - 1 4 \sin 18^{\circ}=\cos 72^{\circ}=\frac{\sqrt{5}-1}{4}
  18. sin 36 = cos 54 = 10 - 2 5 4 \sin 36^{\circ}=\cos 54^{\circ}=\frac{\sqrt{10-2\sqrt{5}}}{4}
  19. sin 54 = cos 36 = 5 + 1 4 \sin 54^{\circ}=\cos 36^{\circ}=\frac{\sqrt{5}+1}{4}\,\!
  20. sin 72 = cos 18 = 10 + 2 5 4 \sin 72^{\circ}=\cos 18^{\circ}=\frac{\sqrt{10+2\sqrt{5}}}{4}
  21. sin 75 = cos 15 = 6 + 2 4 . \sin 75^{\circ}=\cos 15^{\circ}=\frac{\sqrt{6}+\sqrt{2}}{4}.\,
  22. sin 3 = cos 87 = 2 ( 1 - 3 ) 5 + 5 + ( 10 - 2 ) ( 3 + 1 ) 16 \sin 3^{\circ}=\cos 87^{\circ}=\frac{2(1-\sqrt{3})\sqrt{5+\sqrt{5}}+(\sqrt{10}% -\sqrt{2})(\sqrt{3}+1)}{16}\,\!
  23. sin 6 = cos 84 = 30 - 6 5 - 5 - 1 8 \sin 6^{\circ}=\cos 84^{\circ}=\frac{\sqrt{30-6\sqrt{5}}-\sqrt{5}-1}{8}\,\!
  24. sin 9 = cos 81 = 10 + 2 - 2 5 - 5 8 \sin 9^{\circ}=\cos 81^{\circ}=\frac{\sqrt{10}+\sqrt{2}-2\sqrt{5-\sqrt{5}}}{8}\,\!
  25. sin 84 = cos 6 = 10 - 2 5 + 3 + 15 8 \sin 84^{\circ}=\cos 6^{\circ}=\frac{\sqrt{10-2\sqrt{5}}+\sqrt{3}+\sqrt{15}}{8}
  26. sin 87 = cos 3 = 2 ( 1 + 3 ) 5 + 5 + 30 + 10 - 6 - 2 16 \sin 87^{\circ}=\cos 3^{\circ}=\frac{2(1+\sqrt{3})\sqrt{5+\sqrt{5}}+\sqrt{30}+% \sqrt{10}-\sqrt{6}-\sqrt{2}}{16}\,\!
  27. sin 3 = 3 sin 1 - 4 sin 3 1 \sin 3^{\circ}=3\sin 1^{\circ}-4\sin^{3}1^{\circ}
  28. sin θ = sin ( θ + 2 π k ) , \sin\theta=\sin\left(\theta+2\pi k\right),\,
  29. cos θ = cos ( θ + 2 π k ) , \cos\theta=\cos\left(\theta+2\pi k\right),\,
  30. tan θ \displaystyle\tan\theta
  31. sin x = x - x 3 3 ! + x 5 5 ! - x 7 7 ! + = n = 0 ( - 1 ) n x 2 n + 1 ( 2 n + 1 ) ! , cos x = 1 - x 2 2 ! + x 4 4 ! - x 6 6 ! + = n = 0 ( - 1 ) n x 2 n ( 2 n ) ! . \begin{aligned}\displaystyle\sin x&\displaystyle=x-\frac{x^{3}}{3!}+\frac{x^{5% }}{5!}-\frac{x^{7}}{7!}+\cdots\\ &\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n+1}}{(2n+1)!},\\ \displaystyle\cos x&\displaystyle=1-\frac{x^{2}}{2!}+\frac{x^{4}}{4!}-\frac{x^% {6}}{6!}+\cdots\\ &\displaystyle=\sum_{n=0}^{\infty}\frac{(-1)^{n}x^{2n}}{(2n)!}.\end{aligned}
  32. π \pi
  33. tan x \displaystyle\tan x
  34. csc x \displaystyle\csc x
  35. sec x \displaystyle\sec x
  36. cot x \displaystyle\cot x
  37. π cot ( π x ) = lim N n = - N N 1 x + n . \pi\cdot\cot(\pi x)=\lim_{N\to\infty}\sum_{n=-N}^{N}\frac{1}{x+n}.
  38. ( n ) (–n)
  39. n n
  40. π cot ( π x ) = 1 x + n = 1 2 x x 2 - n 2 , π sin ( π x ) = 1 x + n = 1 ( - 1 ) n 2 x x 2 - n 2 . \pi\cdot\cot(\pi x)=\frac{1}{x}+\sum_{n=1}^{\infty}\frac{2x}{x^{2}-n^{2}}\ ,% \quad\frac{\pi}{\sin(\pi x)}=\frac{1}{x}+\sum_{n=1}^{\infty}\frac{(-1)^{n}\,2x% }{x^{2}-n^{2}}.
  41. e i θ = cos θ + i sin θ . e^{i\theta}=\cos\theta+i\sin\theta.\,
  42. sin θ = e i θ - e - i θ 2 i \sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i}\;
  43. cos θ = e i θ + e - i θ 2 \cos\theta=\frac{e^{i\theta}+e^{-i\theta}}{2}\;
  44. sin z = n = 0 ( - 1 ) n ( 2 n + 1 ) ! z 2 n + 1 = e i z - e - i z 2 i = sinh ( i z ) i \sin z=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)!}z^{2n+1}=\frac{e^{iz}-e^{-iz% }}{2i}\,=\frac{\sinh\left(iz\right)}{i}
  45. cos z = n = 0 ( - 1 ) n ( 2 n ) ! z 2 n = e i z + e - i z 2 = cosh ( i z ) \cos z=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n)!}z^{2n}=\frac{e^{iz}+e^{-iz}}{2% }\,=\cosh\left(iz\right)
  46. cos x = Re ( e i x ) \cos x=\operatorname{Re}(e^{ix})\,
  47. sin x = Im ( e i x ) \sin x=\operatorname{Im}(e^{ix})\,
  48. sin ( x + i y ) = sin x cosh y + i cos x sinh y , \sin(x+iy)=\sin x\cosh y+i\cos x\sinh y,\,
  49. cos ( x + i y ) = cos x cosh y - i sin x sinh y . \cos(x+iy)=\cos x\cosh y-i\sin x\sinh y.\,
  50. sin z \sin z\,
  51. cos z \cos z\,
  52. tan z \tan z\,
  53. cot z \cot z\,
  54. sec z \sec z\,
  55. csc z \csc z\,
  56. y ′′ = - y . y^{\prime\prime}=-y.\,
  57. ( y ( 0 ) , y ( 0 ) ) = ( 1 , 0 ) \scriptstyle\left(y^{\prime}(0),y(0)\right)=(1,0)\,
  58. ( y ( 0 ) , y ( 0 ) ) = ( 0 , 1 ) \scriptstyle\left(y^{\prime}(0),y(0)\right)=(0,1)\,
  59. y = 1 + y 2 y^{\prime}=1+y^{2}\,
  60. f ( x ) = sin k x , f(x)=\sin kx,\,
  61. f ( x ) = k cos k x . f^{\prime}(x)=k\cos kx.\,
  62. k = π 180 . k=\frac{\pi}{\textstyle 180^{\circ}}.
  63. y ′′ = - y y^{\prime\prime}=-y\,
  64. y ′′ = - k 2 y . y^{\prime\prime}=-k^{2}y.\,
  65. sin 2 x + cos 2 x = 1 \sin^{2}x+\cos^{2}x=1
  66. sin 2 x + cos 2 x = 1 \textstyle\sin^{2}x+\cos^{2}x=1
  67. ( sin x ) 2 + ( cos x ) 2 = 1 \textstyle(\sin x)^{2}+(\cos x)^{2}=1
  68. sin ( x + y ) = sin x cos y + cos x sin y , \sin\left(x+y\right)=\sin x\cos y+\cos x\sin y,\,
  69. cos ( x + y ) = cos x cos y - sin x sin y , \cos\left(x+y\right)=\cos x\cos y-\sin x\sin y,\,
  70. sin ( x - y ) = sin x cos y - cos x sin y , \sin\left(x-y\right)=\sin x\cos y-\cos x\sin y,\,
  71. cos ( x - y ) = cos x cos y + sin x sin y . \cos\left(x-y\right)=\cos x\cos y+\sin x\sin y.\,
  72. sin ( x + y + z ) = sin x cos y cos z + sin y cos z cos x + sin z cos y cos x - sin x sin y sin z , \sin\left(x+y+z\right)=\sin x\cos y\cos z+\sin y\cos z\cos x+\sin z\cos y\cos x% -\sin x\sin y\sin z,\,
  73. cos ( x + y + z ) = cos x cos y cos z - cos x sin y sin z - cos y sin x sin z - cos z sin x sin y , \cos\left(x+y+z\right)=\cos x\cos y\cos z-\cos x\sin y\sin z-\cos y\sin x\sin z% -\cos z\sin x\sin y,\,
  74. sin 2 x = 2 sin x cos x , \sin 2x=2\sin x\cos x,\,
  75. cos 2 x = cos 2 x - sin 2 x = 2 cos 2 x - 1 = 1 - 2 sin 2 x . \cos 2x=\cos^{2}x-\sin^{2}x=2\cos^{2}x-1=1-2\sin^{2}x.\,
  76. sin 3 x = 3 sin x - 4 sin 3 x . \sin 3x=3\sin x-4\sin^{3}x.\,
  77. cos 3 x = 4 cos 3 x - 3 cos x . \cos 3x=4\cos^{3}x-3\cos x.\,
  78. f ( x ) \ \ \ \ f(x)
  79. f ( x ) \ \ \ \ f^{\prime}(x)
  80. f ( x ) d x \int f(x)\,dx
  81. sin x \,\ \sin x
  82. cos x \,\ \cos x
  83. - cos x + C \,\ -\cos x+C
  84. cos x \,\ \cos x
  85. - sin x \,\ -\sin x
  86. sin x + C \,\ \sin x+C
  87. tan x \,\ \tan x
  88. sec 2 x = 1 + tan 2 x \,\ \sec^{2}x=1+\tan^{2}x
  89. - ln | cos x | + C -\ln\left|\cos x\right|+C
  90. cot x \,\ \cot x
  91. - csc 2 x = - ( 1 + cot 2 x ) \,\ -\csc^{2}x=-(1+\cot^{2}x)
  92. ln | sin x | + C \ln\left|\sin x\right|+C
  93. sec x \,\ \sec x
  94. sec x tan x \,\ \sec x\tan x
  95. ln | sec x + tan x | + C \ln\left|\sec x+\tan x\right|+C
  96. csc x \,\ \csc x
  97. - csc x cot x \,\ -\csc x\cot x
  98. - ln | csc x + cot x | + C \ -\ln\left|\csc x+\cot x\right|+C
  99. sin \scriptstyle\sin\,
  100. cos \scriptstyle\cos\,
  101. x \scriptstyle x\,
  102. y \scriptstyle y\,
  103. cos ( x - y ) = cos x cos y + sin x sin y \cos(x-y)=\cos x\cos y+\sin x\sin y\,
  104. 0 < x cos x < sin x < x for 0 < x < 1. 0<x\cos x<\sin x<x\hbox{ for }0<x<1.\,
  105. π / 60 \pi/60
  106. π / 4 \pi/4
  107. a = b = 1 a=b=1
  108. π / 4 \pi/4
  109. c = a 2 + b 2 = 2 . c=\sqrt{a^{2}+b^{2}}=\sqrt{2}\,.
  110. sin π 4 = sin 45 = cos π 4 = cos 45 = 1 2 = 2 2 , \sin\frac{\pi}{4}=\sin 45^{\circ}=\cos\frac{\pi}{4}=\cos 45^{\circ}={1\over% \sqrt{2}}={\sqrt{2}\over 2},\,
  111. tan π 4 = tan 45 = sin π 4 cos π 4 = 1 2 2 1 = 2 2 = 1. \tan\frac{\pi}{4}=\tan 45^{\circ}={{\sin\frac{\pi}{4}}\over{\cos\frac{\pi}{4}}% }={1\over\sqrt{2}}\cdot{\sqrt{2}\over 1}={\sqrt{2}\over\sqrt{2}}=1.\,
  112. sin π 6 = sin 30 = cos π 3 = cos 60 = 1 2 , \sin\frac{\pi}{6}=\sin 30^{\circ}=\cos\frac{\pi}{3}=\cos 60^{\circ}={1\over 2}\,,
  113. cos π 6 = cos 30 = sin π 3 = sin 60 = 3 2 , \cos\frac{\pi}{6}=\cos 30^{\circ}=\sin\frac{\pi}{3}=\sin 60^{\circ}={\sqrt{3}% \over 2}\,,
  114. tan π 6 = tan 30 = cot π 3 = cot 60 = 1 3 = 3 3 . \tan\frac{\pi}{6}=\tan 30^{\circ}=\cot\frac{\pi}{3}=\cot 60^{\circ}={1\over% \sqrt{3}}={\sqrt{3}\over 3}\,.
  115. 0 ( 0 ) 0\ (0^{\circ})
  116. π 12 ( 15 ) \frac{\pi}{12}\ (15^{\circ})
  117. π 8 ( 22.5 ) \frac{\pi}{8}\ (22.5^{\circ})
  118. π 6 ( 30 ) \frac{\pi}{6}\ (30^{\circ})
  119. π 4 ( 45 ) \frac{\pi}{4}\ (45^{\circ})
  120. π 3 ( 60 ) \frac{\pi}{3}\ (60^{\circ})
  121. 5 π 12 ( 75 ) \frac{5\pi}{12}\ (75^{\circ})
  122. π 2 ( 90 ) \frac{\pi}{2}\ (90^{\circ})
  123. 0
  124. 6 - 2 4 \frac{\sqrt{6}-\sqrt{2}}{4}
  125. 2 - 2 2 \frac{\sqrt{2-\sqrt{2}}}{2}
  126. 1 2 \frac{1}{2}
  127. 2 2 \frac{\sqrt{2}}{2}
  128. 3 2 \frac{\sqrt{3}}{2}
  129. 6 + 2 4 \frac{\sqrt{6}+\sqrt{2}}{4}
  130. 1 1
  131. 1 1
  132. 6 + 2 4 \frac{\sqrt{6}+\sqrt{2}}{4}
  133. 2 + 2 2 \frac{\sqrt{2+\sqrt{2}}}{2}
  134. 3 2 \frac{\sqrt{3}}{2}
  135. 2 2 \frac{\sqrt{2}}{2}
  136. 1 2 \frac{1}{2}
  137. 6 - 2 4 \frac{\sqrt{6}-\sqrt{2}}{4}
  138. 0
  139. 0
  140. 2 - 3 2-\sqrt{3}
  141. 2 - 1 \sqrt{2}-1
  142. 3 3 \frac{\sqrt{3}}{3}
  143. 1 1
  144. 3 \sqrt{3}
  145. 2 + 3 2+\sqrt{3}
  146. \infty
  147. \infty
  148. 2 + 3 2+\sqrt{3}
  149. 2 + 1 \sqrt{2}+1
  150. 3 \sqrt{3}
  151. 1 1
  152. 3 3 \frac{\sqrt{3}}{3}
  153. 2 - 3 2-\sqrt{3}
  154. 0
  155. 1 1
  156. 6 - 2 \sqrt{6}-\sqrt{2}
  157. 2 2 - 2 \sqrt{2}\sqrt{2-\sqrt{2}}
  158. 2 3 3 \frac{2\sqrt{3}}{3}
  159. 2 \sqrt{2}
  160. 2 2
  161. 6 + 2 \sqrt{6}+\sqrt{2}
  162. \infty
  163. \infty
  164. 6 + 2 \sqrt{6}+\sqrt{2}
  165. 2 2 + 2 \sqrt{2}\sqrt{2+\sqrt{2}}
  166. 2 2
  167. 2 \sqrt{2}
  168. 2 3 3 \frac{2\sqrt{3}}{3}
  169. 6 - 2 \sqrt{6}-\sqrt{2}
  170. 1 1
  171. \infty
  172. + +\infty
  173. - -\infty
  174. arcsin x = y \arcsin x=y\,
  175. sin y = x \sin y=x\,
  176. - π 2 y π 2 -\frac{\pi}{2}\leq y\leq\frac{\pi}{2}\,
  177. arccos x = y \arccos x=y\,
  178. cos y = x \cos y=x\,
  179. 0 y π 0\leq y\leq\pi\,
  180. arctan x = y \arctan x=y\,
  181. tan y = x \tan y=x\,
  182. - π 2 < y < π 2 -\frac{\pi}{2}<y<\frac{\pi}{2}\,
  183. \arccot x = y \arccot x=y\,
  184. cot y = x \cot y=x\,
  185. 0 < y < π 0<y<\pi\,
  186. \arcsec x = y \arcsec x=y\,
  187. sec y = x \sec y=x\,
  188. 0 y π , y π 2 0\leq y\leq\pi,y\neq\frac{\pi}{2}\,
  189. \arccsc x = y \arccsc x=y\,
  190. csc y = x \csc y=x\,
  191. - π 2 y π 2 , y 0 -\frac{\pi}{2}\leq y\leq\frac{\pi}{2},y\neq 0\,
  192. arcsin z = z + ( 1 2 ) z 3 3 + ( 1 3 2 4 ) z 5 5 + ( 1 3 5 2 4 6 ) z 7 7 + . \arcsin z=z+\left(\frac{1}{2}\right)\frac{z^{3}}{3}+\left(\frac{1\cdot 3}{2% \cdot 4}\right)\frac{z^{5}}{5}+\left(\frac{1\cdot 3\cdot 5}{2\cdot 4\cdot 6}% \right)\frac{z^{7}}{7}+\cdots\,.
  193. arcsin z = 0 z ( 1 - x 2 ) - 1 / 2 d x , | z | < 1. \arcsin z=\int_{0}^{z}(1-x^{2})^{-1/2}\,dx,\quad|z|<1.
  194. arcsin z = - i log ( i z + 1 - z 2 ) , \arcsin z=-i\log\left(iz+\sqrt{1-z^{2}}\right),\,
  195. arccos z = - i log ( z + z 2 - 1 ) , \arccos z=-i\log\left(z+\sqrt{z^{2}-1}\right),\,
  196. arctan z = 1 2 i log ( 1 - i z 1 + i z ) . \arctan z=\frac{1}{2}i\log\left(\frac{1-iz}{1+iz}\right).
  197. angle ( x , y ) = arccos x , y x y . \operatorname{angle}(x,y)=\arccos\frac{\langle x,y\rangle}{\|x\|\cdot\|y\|}.
  198. sin A a = sin B b = sin C c = 2 Δ a b c , \frac{\sin A}{a}=\frac{\sin B}{b}=\frac{\sin C}{c}=\frac{2\Delta}{abc},
  199. Δ \Delta
  200. a sin A = b sin B = c sin C = 2 R , \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R,
  201. c 2 = a 2 + b 2 - 2 a b cos C , c^{2}=a^{2}+b^{2}-2ab\cos C,\,
  202. cos C = a 2 + b 2 - c 2 2 a b . \cos C=\frac{a^{2}+b^{2}-c^{2}}{2ab}.
  203. tan A - B 2 tan A + B 2 \displaystyle\frac{\tan\frac{A-B}{2}}{\tan\frac{A+B}{2}}
  204. ζ = 1 s ( s - a ) ( s - b ) ( s - c ) \zeta=\sqrt{\frac{1}{s}(s-a)(s-b)(s-c)}
  205. s = a + b + c 2 s=\frac{a+b+c}{2}
  206. cot A 2 = s - a ζ \cot{\frac{A}{2}}=\frac{s-a}{\zeta}
  207. cot B 2 = s - b ζ \cot{\frac{B}{2}}=\frac{s-b}{\zeta}
  208. cot C 2 = s - c ζ \cot{\frac{C}{2}}=\frac{s-c}{\zeta}
  209. cot ( A / 2 ) s - a = cot ( B / 2 ) s - b = cot ( C / 2 ) s - c . \frac{\cot(A/2)}{s-a}=\frac{\cot(B/2)}{s-b}=\frac{\cot(C/2)}{s-c}.
  210. f ( t ) = k = 1 c k φ k ( t ) . f(t)=\sum_{k=1}^{\infty}c_{k}\varphi_{k}(t).
  211. f square ( t ) = 4 π k = 1 sin ( ( 2 k - 1 ) t ) 2 k - 1 . f\text{square}(t)=\frac{4}{\pi}\sum_{k=1}^{\infty}{\sin\left((2k-1)t\right)% \over 2k-1}.

Tropic_of_Cancer.html

  1. = π 2 r \ell=\pi\cdot 2r

Troposphere.html

  1. d p d z = - ρ g n = - m p g R T \frac{dp}{dz}=-\rho g_{n}=-\frac{mpg}{RT}
  2. - d T / d z -dT/dz
  3. d Q dQ
  4. d S dS
  5. d Q = T d S dQ=TdS
  6. d S d z = 0 \frac{dS}{dz}=0
  7. p ( z ) T ( z ) - γ γ - 1 = c o n s t a n t p(z)T(z)^{-\frac{\gamma}{\gamma-1}}=constant
  8. γ \gamma
  9. γ \gamma
  10. d T d z = - m g R γ - 1 γ = - 9.8 C / km \frac{dT}{dz}=-\frac{mg}{R}\frac{\gamma-1}{\gamma}=-9.8^{\circ}\mathrm{C}/% \mathrm{km}
  11. d T / d z dT/dz
  12. d S / d z 0 dS/dz\neq 0
  13. d S / d z > 0 dS/dz>0

Truncated_icosahedron.html

  1. r u = a 2 1 + 9 φ 2 = a 4 58 + 18 5 2.47801866 a r_{u}=\frac{a}{2}\sqrt{1+9\varphi^{2}}=\frac{a}{4}\sqrt{58+18\sqrt{5}}\approx 2% .47801866\cdot a
  2. A \displaystyle A

Tsunami.html

  1. I = 1 2 + log 2 H a v \,\mathit{I}=\frac{1}{2}+\log_{2}\mathit{H}_{av}
  2. H a v \mathit{H}_{av}
  3. M t \mathit{M}_{t}
  4. M t = a log h + b log R = D \,\mathit{M}_{t}={a}\log h+{b}\log R=\mathit{D}

Tuning_fork.html

  1. f = 1.875 2 2 π l 2 E I ρ A f=\frac{1.875^{2}}{2\pi l^{2}}\sqrt{\frac{EI}{\rho A}}
  2. I A \frac{I}{A}
  3. r 2 / 4 r^{2}/4
  4. a 2 / 12 a^{2}/12

Turbine.html

  1. Δ h = u Δ v w \Delta\;h=u\cdot\Delta\;v_{w}
  2. ( Δ h T ) = ( u T ) ( Δ v w T ) \left(\frac{\Delta\;h}{T}\right)=\left(\frac{u}{\sqrt{T}}\right)\cdot\left(% \frac{\Delta\;v_{w}}{\sqrt{T}}\right)
  3. Δ h = \Delta\;h=\,
  4. T = T=\,
  5. u = u=\,
  6. Δ v w = \Delta\;v_{w}=\,
  7. ( Δ H T ) \left(\frac{\Delta\;H}{T}\right)

Turing_machine.html

  1. M = Q , Γ , b , Σ , δ , q 0 , F M=\langle Q,\Gamma,b,\Sigma,\delta,q_{0},F\rangle
  2. Q Q
  3. Γ \Gamma
  4. b Γ b\in\Gamma
  5. Σ Γ { b } \Sigma\subseteq\Gamma\setminus\{b\}
  6. δ : ( Q F ) × Γ Q × Γ × { L , R } \delta:(Q\setminus F)\times\Gamma\rightarrow Q\times\Gamma\times\{L,R\}
  7. δ \delta
  8. q 0 Q q_{0}\in Q
  9. F Q F\subseteq Q
  10. M M
  11. F F
  12. Q = { A , B , C , HALT } Q=\{\mbox{A}~{},\mbox{B}~{},\mbox{C}~{},\mbox{HALT}~{}\}
  13. Γ = { 0 , 1 } \Gamma=\{0,1\}
  14. b = 0 b=0
  15. Σ = { 1 } \Sigma=\{1\}
  16. q 0 = A q_{0}=\mbox{A}~{}
  17. F = { HALT } F=\{\mbox{HALT}~{}\}
  18. δ = \delta=
  19. { L , R } \{L,R\}
  20. { L , R , N } \{L,R,N\}
  21. δ \delta
  22. F F

Twin_paradox.html

  1. ϵ = 1 - v 2 / c 2 \scriptstyle{\epsilon=\sqrt{1-v^{2}/c^{2}}}
  2. 1 / 3 {1}/{3}
  3. 1 / 3 {1}/{3}
  4. f obs = f rest ( 1 - v / c ) / ( 1 + v / c ) f_{\mathrm{obs}}=f_{\mathrm{rest}}\sqrt{\left({1-v/c}\right)/\left({1+v/c}% \right)}
  5. 1 / 3 {1}/{3}
  6. 1 / 3 {1}/{3}
  7. 1 / 3 {1}/{3}
  8. 1 / 3 {1}/{3}
  9. f obs = f rest ( 1 + v / c ) / ( 1 - v / c ) f_{\mathrm{obs}}=f_{\mathrm{rest}}\sqrt{\left({1+v/c}\right)/\left({1-v/c}% \right)}
  10. f rest ( 1 + v / c ) / ( 1 - v / c ) \scriptstyle{f_{\mathrm{rest}}\sqrt{\left({1+v/c}\right)/\left({1-v/c}\right)}}
  11. f rest ( 1 + v / c ) / ( 1 - v / c ) × ( 1 - v / c ) = f rest 1 - v 2 / c 2 ϵ f rest f_{\mathrm{rest}}\sqrt{\left({1+v/c}\right)/\left({1-v/c}\right)}\times\left(1% -v/c\right)=f_{\mathrm{rest}}\sqrt{1-v^{2}/c^{2}}\equiv\epsilon f_{\mathrm{% rest}}
  12. Δ τ = 1 - ( v ( t ) / c ) 2 d t \Delta\tau=\int\sqrt{1-(v(t)/c)^{2}}\ dt
  13. v ( t ) = a t 1 + ( a t c ) 2 . v(t)=\frac{at}{\sqrt{1+\left(\frac{at}{c}\right)^{2}}}.
  14. : c / a arsinh ( a T a / c ) :\quad c/a\ \,\text{arsinh}(a\ T_{a}/c)\,
  15. : T c 1 - V 2 / c 2 :\quad T_{c}\ \sqrt{1-V^{2}/c^{2}}
  16. : c / a arsinh ( a T a / c ) :\quad c/a\ \,\text{arsinh}(a\ T_{a}/c)\,
  17. : c / a arsinh ( a T a / c ) :\quad c/a\ \,\text{arsinh}(a\ T_{a}/c)\,
  18. : T c 1 - V 2 / c 2 :\quad T_{c}\ \sqrt{1-V^{2}/c^{2}}
  19. : c / a arsinh ( a T a / c ) :\quad c/a\ \,\text{arsinh}(a\ T_{a}/c)\,
  20. V = a T a / 1 + ( a T a / c ) 2 V=a\ T_{a}/\sqrt{1+(a\ T_{a}/c)^{2}}
  21. a T a = V / 1 - V 2 / c 2 a\ T_{a}=V/\sqrt{1-V^{2}/c^{2}}
  22. Δ τ = 2 T c 1 - V 2 / c 2 + 4 c / a arsinh ( a T a / c ) \Delta\tau=2T_{c}\sqrt{1-V^{2}/c^{2}}+4c/a\ \,\text{arsinh}(a\ T_{a}/c)
  23. Δ τ = 2 T c / 1 + ( a T a / c ) 2 + 4 c / a arsinh ( a T a / c ) \Delta\tau=2T_{c}/\sqrt{1+(a\ T_{a}/c)^{2}}+4c/a\ \,\text{arsinh}(a\ T_{a}/c)
  24. Δ t = 2 T c + 4 T a \Delta t=2T_{c}+4T_{a}\,
  25. Δ t > Δ τ \Delta t>\Delta\tau\,
  26. Δ τ = 0 Δ t 1 - ( v ( t ) c ) 2 d t , \Delta\tau=\int_{0}^{\Delta t}\sqrt{1-\left(\frac{v(t)}{c}\right)^{2}}\ dt,
  27. Δ t 2 = [ 0 Δ τ e 0 τ ¯ a ( τ ) d τ d τ ¯ ] [ 0 Δ τ e - 0 τ ¯ a ( τ ) d τ d τ ¯ ] , \Delta t^{2}=\left[\int^{\Delta\tau}_{0}e^{\int^{\bar{\tau}}_{0}a(\tau^{\prime% })d\tau^{\prime}}\,d\bar{\tau}\right]\,\left[\int^{\Delta\tau}_{0}e^{-\int^{% \bar{\tau}}_{0}a(\tau^{\prime})d\tau^{\prime}}\,d\bar{\tau}\right],
  28. Δ t 2 \displaystyle\Delta t^{2}
  29. Δ t = 1 1 - v 2 c 2 Δ τ . \Delta t=\frac{1}{\sqrt{1-\tfrac{v^{2}}{c^{2}}}}\Delta\tau.
  30. Δ t = 0 Δ τ e ± 0 τ ¯ a ( τ ) d τ d τ ¯ , \Delta t=\int^{\Delta\tau}_{0}e^{\pm\int^{\bar{\tau}}_{0}a(\tau^{\prime})d\tau% ^{\prime}}\,d\bar{\tau},
  31. Δ t = 4 a sinh ( a 4 Δ τ ) \Delta t=\tfrac{4}{a}\sinh(\tfrac{a}{4}\Delta\tau)

Twin_prime.html

  1. C N ( log N ) 2 \frac{CN}{(\log N)^{2}}
  2. lim inf n p n + 1 - p n log p n = 0. \liminf_{n\to\infty}\frac{p_{n+1}-p_{n}}{\log p_{n}}=0.
  3. lim inf n ( p n + 1 - p n ) < N with N = 7 × 10 7 , \liminf_{n\to\infty}(p_{n+1}-p_{n})<N\;\,\text{ with }\;N=7\times 10^{7},
  4. 4 ( ( m - 1 ) ! + 1 ) - m ( mod m ( m + 2 ) ) . 4((m-1)!+1)\equiv-m\;\;(\mathop{{\rm mod}}m(m+2)).
  5. 2 p prime p 3 ( 1 - 1 ( p - 1 ) 2 ) = 1.3203236 ; 2\prod_{\textstyle{p\;{\rm prime}\atop p\geq 3}}\left(1-\frac{1}{(p-1)^{2}}% \right)=1.3203236\ldots;
  6. π 2 ( x ) 2 C 2 li 2 ( x ) = 2 C 2 2 x d t ( log e t ) 2 . \pi_{2}(x)\approx 2C_{2}\;\operatorname{li}_{2}(x)=2C_{2}\int_{2}^{x}\frac{dt}% {\left(\log_{e}t\right)^{2}}.
  7. C 2 = p 3 p ( p - 2 ) ( p - 1 ) 2 0.660161815846869573927812110014 C_{2}=\prod_{p\geq 3}\frac{p(p-2)}{(p-1)^{2}}\approx 0.66016181584686957392781% 2110014\dots
  8. π 2 ( n ) 2 C 2 n ( ln n ) 2 2 C 2 2 n d t ( ln t ) 2 \pi_{2}(n)\sim 2C_{2}\frac{n}{(\ln n)^{2}}\sim 2C_{2}\int_{2}^{n}{dt\over(\ln t% )^{2}}

Tychonoff_space.html

  1. f : ( X , τ ) Y f:(X,\tau)\to Y

Type_theory.html

  1. M : A M:A
  2. M M
  3. A A
  4. nat \mathrm{nat}
  5. 0 , 1 , 2 , 0,1,2,...
  6. 2 2
  7. nat \mathrm{nat}
  8. 2 : nat 2:\mathrm{nat}
  9. \to
  10. addOne \mathrm{addOne}
  11. addOne : nat nat \mathrm{addOne}:\mathrm{nat}\to\mathrm{nat}
  12. addOne 2 \mathrm{addOne}\ 2
  13. addOne ( 2 ) \mathrm{addOne}(2)
  14. 2 + 1 2+1
  15. 3 3
  16. 2 + 1 3 2+1\twoheadrightarrow 3
  17. 2 + 1 2+1
  18. 3 3
  19. 3 3
  20. x + 1 x+1
  21. x x
  22. 2 + 1 2+1
  23. 3 + 0 3+0
  24. 3 3
  25. 2 + 1 2+1
  26. 1 + 2 1+2
  27. x + ( 1 + 1 ) x+(1+1)
  28. x + 2 x+2
  29. x + 1 x+1
  30. 1 + x 1+x
  31. x x
  32. nat \mathrm{nat}
  33. nat \mathrm{nat}
  34. I I
  35. I A a b I\ A\ a\ b
  36. A A
  37. a a
  38. b b
  39. A A
  40. I A a b I\ A\ a\ b
  41. a a
  42. b b
  43. I nat 3 4 I\ \mathrm{nat}\ 3\ 4
  44. 3 3
  45. nat \mathrm{nat}
  46. I nat 3 3 I\ \mathrm{nat}\ 3\ 3
  47. 2 + 1 2+1
  48. nat \mathrm{nat}
  49. I nat ( 2 + 1 ) ( 2 + 1 ) I\ \mathrm{nat}\ (2+1)\ (2+1)
  50. I nat ( 2 + 1 ) 3 I\ \mathrm{nat}\ (2+1)\ 3
  51. ¬ ( a = b ) \neg(a=b)
  52. ( a = b ) (a=b)\to\bot
  53. \bot
  54. ( I nat 3 4 ) (I\ \mathrm{nat}\ 3\ 4)\to\bot
  55. ( I nat 3 3 ) (I\ \mathrm{nat}\ 3\ 3)\to\bot
  56. e e
  57. t t
  58. a a
  59. b b
  60. a , b \langle a,b\rangle
  61. a , b \langle a,b\rangle
  62. a a
  63. b b
  64. e , t \langle e,t\rangle
  65. e , t , t \langle\langle e,t\rangle,t\rangle

Ulam_spiral.html

  1. f ( n ) = 4 n 2 + b n + c f(n)=4n^{2}+bn+c
  2. P ( n ) A 1 a n log n P(n)\sim A\frac{1}{\sqrt{a}}\frac{\sqrt{n}}{\log n}
  3. A = ε p ( p p - 1 ) ϖ ( 1 - 1 ϖ - 1 ( Δ ϖ ) ) . A=\varepsilon\prod_{p}\left(\frac{p}{p-1}\right)\prod_{\varpi}\left(1-\frac{1}% {\varpi-1}\left(\frac{\Delta}{\varpi}\right)\right).
  4. A = p 1 - ω ( p ) p 1 - 1 p A=\prod\limits_{p}\frac{1-\frac{\omega(p)}{p}}{1-\frac{1}{p}}~{}
  5. p p
  6. ω ( p ) \omega(p)
  7. ϖ \varpi
  8. ( Δ ϖ ) \left(\frac{\Delta}{\varpi}\right)

Ultrafilter.html

  1. U F U\subseteq F
  2. A B U A\cup B\in U
  3. A U A\in U
  4. B U B\in U
  5. A X : A U \forall A\subseteq X\colon A\in U
  6. X A U X\setminus A\in U
  7. 0 \aleph_{0}
  8. 0 \aleph_{0}
  9. σ \sigma
  10. a , b 𝐁 a,b\in\mathbf{B}
  11. a b F a\vee b\in F
  12. a F a\in F
  13. b F b\in F
  14. \vee
  15. V R K U V\leq_{RK}U
  16. C V f - 1 [ C ] U C\in V\iff f^{-1}[C]\in U
  17. U R K V U\equiv_{RK}V
  18. A U A\in U
  19. B V B\in V
  20. R K \equiv_{RK}
  21. R K \leq_{RK}
  22. U R K V U\equiv_{RK}V
  23. U R K V U\leq_{RK}V
  24. V R K U V\leq_{RK}U
  25. { C n n < ω } \{C_{n}\mid n<\omega\}
  26. C n U , n < ω C_{n}\not\in U,\forall n<\omega
  27. A U A\in U
  28. | A C n | < ω , n < ω |A\cap C_{n}|<\omega,\forall n<\omega
  29. { C n n < ω } \{C_{n}\mid n<\omega\}
  30. C n U , n < ω C_{n}\not\in U,\forall n<\omega
  31. A U A\in U
  32. | A C n | = 1 , n < ω |A\cap C_{n}|=1,\forall n<\omega
  33. [ ω ] 2 [\omega]^{2}
  34. 0 F 0\notin F

Unbinilium.html

  1. Pu 94 244 + 26 58 Fe 120 302 Ubn * 𝑓𝑖𝑠𝑠𝑖𝑜𝑛 𝑜𝑛𝑙𝑦 \,{}^{244}_{94}\mathrm{Pu}+\,^{58}_{26}\mathrm{Fe}\to\,^{302}_{120}\mathrm{Ubn% }^{*}\to\ \mathit{fission\ only}
  2. U 92 238 + 28 64 Ni 120 302 Ubn * 𝑓𝑖𝑠𝑠𝑖𝑜𝑛 𝑜𝑛𝑙𝑦 \,{}^{238}_{92}\mathrm{U}+\,^{64}_{28}\mathrm{Ni}\to\,^{302}_{120}\mathrm{Ubn}% ^{*}\to\ \mathit{fission\ only}
  3. Cm 96 248 + 24 54 Cr 120 302 Ubn * \,{}^{248}_{96}\mathrm{Cm}+\,^{54}_{24}\mathrm{Cr}\to\,^{302}_{120}\mathrm{Ubn% }^{*}
  4. Cf 98 249 + 22 50 Ti 120 299 Ubn * \,{}^{249}_{98}\mathrm{Cf}+\,^{50}_{22}\mathrm{Ti}\to\,^{299}_{120}\mathrm{Ubn% }^{*}
  5. U 92 238 + 28 n a t Ni 296 , 298 , 299 , 300 , 302 Ubn * 𝑓𝑖𝑠𝑠𝑖𝑜𝑛 . \,{}^{238}_{92}\mathrm{U}+\,^{nat}_{28}\mathrm{Ni}\to\,^{296,298,299,300,302}% \mathrm{Ubn}^{*}\to\ \mathit{fission}.

Uncertainty_principle.html

  1. ħ ħ
  2. h h
  3. 2 π
  4. p = ħ k p=ħk
  5. k k
  6. A A
  7. Ψ Ψ
  8. A A
  9. B B
  10. Ψ ( x , t ) \Psi(x,t)
  11. ψ ( x ) e i k 0 x = e i p 0 x / . \psi(x)\propto e^{ik_{0}x}=e^{ip_{0}x/\hbar}~{}.
  12. P [ a X b ] = a b | ψ ( x ) | 2 d x . \operatorname{P}[a\leq X\leq b]=\int_{a}^{b}|\psi(x)|^{2}\,\mathrm{d}x~{}.
  13. | ψ ( x ) | 2 |\psi(x)|^{2}
  14. ψ ( x ) n A n e i p n x / , \psi(x)\propto\sum_{n}A_{n}e^{ip_{n}x/\hbar}~{},
  15. ψ ( x ) = 1 2 π - ϕ ( p ) e i p x / d p , \psi(x)=\frac{1}{\sqrt{2\pi\hbar}}\int_{-\infty}^{\infty}\phi(p)\cdot e^{ipx/% \hbar}\,dp~{},
  16. ϕ ( p ) \phi(p)
  17. ϕ ( p ) \phi(p)
  18. ψ ( x ) \psi(x)
  19. | ψ ( x ) | 2 |\psi(x)|^{2}
  20. σ x 2 = - x 2 | ψ ( x ) | 2 d x - ( - x | ψ ( x ) | 2 d x ) 2 \sigma_{x}^{2}=\int_{-\infty}^{\infty}x^{2}\cdot|\psi(x)|^{2}\,dx-\left(\int_{% -\infty}^{\infty}x\cdot|\psi(x)|^{2}\,dx\right)^{2}
  21. σ p 2 = - p 2 | ϕ ( p ) | 2 d p - ( - p | ϕ ( p ) | 2 d p ) 2 . \sigma_{p}^{2}=\int_{-\infty}^{\infty}p^{2}\cdot|\phi(p)|^{2}\,dp-\left(\int_{% -\infty}^{\infty}p\cdot|\phi(p)|^{2}\,dp\right)^{2}~{}.
  22. σ x 2 = - x 2 | ψ ( x ) | 2 d x \sigma_{x}^{2}=\int_{-\infty}^{\infty}x^{2}\cdot|\psi(x)|^{2}\,dx
  23. σ p 2 = - p 2 | ϕ ( p ) | 2 d p . \sigma_{p}^{2}=\int_{-\infty}^{\infty}p^{2}\cdot|\phi(p)|^{2}\,dp~{}.
  24. f ( x ) = x ψ ( x ) f(x)=x\cdot\psi(x)
  25. u | v = - u * ( x ) v ( x ) d x , \langle u|v\rangle=\int_{-\infty}^{\infty}u^{*}(x)\cdot v(x)\,dx,
  26. σ x 2 = - | f ( x ) | 2 d x = f | f . \sigma_{x}^{2}=\int_{-\infty}^{\infty}|f(x)|^{2}\,dx=\langle f|f\rangle~{}.
  27. g ~ ( p ) = p ϕ ( p ) \tilde{g}(p)=p\cdot\phi(p)
  28. ψ ( x ) \psi(x)
  29. ϕ ( p ) \phi(p)
  30. g ( x ) \displaystyle g(x)
  31. - i d d x -i\hbar\frac{d}{dx}
  32. σ p 2 = - | g ~ ( p ) | 2 d p = - | g ( x ) | 2 d x = g | g . \sigma_{p}^{2}=\int_{-\infty}^{\infty}|\tilde{g}(p)|^{2}\,dp=\int_{-\infty}^{% \infty}|g(x)|^{2}\,dx=\langle g|g\rangle.
  33. σ x 2 σ p 2 = f | f g | g | f | g | 2 . \sigma_{x}^{2}\sigma_{p}^{2}=\langle f|f\rangle\cdot\langle g|g\rangle\geq|% \langle f|g\rangle|^{2}~{}.
  34. | z | 2 = ( Re ( z ) ) 2 + ( Im ( z ) ) 2 ( Im ( z ) ) 2 = ( z - z 2 i ) 2 . |z|^{2}=\Big(\,\text{Re}(z)\Big)^{2}+\Big(\,\text{Im}(z)\Big)^{2}\geq\Big(\,% \text{Im}(z)\Big)^{2}=\Big(\frac{z-z^{\ast}}{2i}\Big)^{2}.
  35. z = f | g z=\langle f|g\rangle
  36. z * = g | f z^{*}=\langle g|f\rangle
  37. | f | g | 2 ( f | g - g | f 2 i ) 2 . |\langle f|g\rangle|^{2}\geq\bigg(\frac{\langle f|g\rangle-\langle g|f\rangle}% {2i}\bigg)^{2}~{}.
  38. f | g - g | f = - ψ * ( x ) x ( - i d d x ) ψ ( x ) d x - - ψ * ( x ) ( - i d d x ) x ψ ( x ) d x = i - ψ * ( x ) [ ( - x d ψ ( x ) d x ) + d ( x ψ ( x ) ) d x ] d x = i - ψ * ( x ) [ ( - x d ψ ( x ) d x ) + ψ ( x ) + ( x d ψ ( x ) d x ) ] d x = i - ψ * ( x ) ψ ( x ) d x = i - | ψ ( x ) | 2 d x = i \begin{aligned}\displaystyle\langle f|g\rangle-\langle g|f\rangle&% \displaystyle=\int_{-\infty}^{\infty}\psi^{*}(x)\,x\cdot\left(-i\hbar\frac{d}{% dx}\right)\,\psi(x)\,dx\\ &\displaystyle{}\,\,\,\,\,-\int_{-\infty}^{\infty}\psi^{*}(x)\,\left(-i\hbar% \frac{d}{dx}\right)\cdot x\,\psi(x)dx\\ &\displaystyle=i\hbar\cdot\int_{-\infty}^{\infty}\psi^{*}(x)\left[\left(-x% \cdot\frac{d\psi(x)}{dx}\right)+\frac{d(x\psi(x))}{dx}\right]\,dx\\ &\displaystyle=i\hbar\cdot\int_{-\infty}^{\infty}\psi^{*}(x)\left[\left(-x% \cdot\frac{d\psi(x)}{dx}\right)+\psi(x)+\left(x\cdot\frac{d\psi(x)}{dx}\right)% \right]\,dx\\ &\displaystyle=i\hbar\cdot\int_{-\infty}^{\infty}\psi^{*}(x)\psi(x)\,dx\\ &\displaystyle=i\hbar\cdot\int_{-\infty}^{\infty}|\psi(x)|^{2}\,dx\\ &\displaystyle=i\hbar\end{aligned}
  39. σ x 2 σ p 2 | f | g | 2 ( f | g - g | f 2 i ) 2 = ( i 2 i ) 2 = 2 4 \sigma_{x}^{2}\sigma_{p}^{2}\geq|\langle f|g\rangle|^{2}\geq\left(\frac{% \langle f|g\rangle-\langle g|f\rangle}{2i}\right)^{2}=\left(\frac{i\hbar}{2i}% \right)^{2}=\frac{\hbar^{2}}{4}
  40. σ x σ p 2 . \sigma_{x}\sigma_{p}\geq\frac{\hbar}{2}~{}.
  41. ψ ( x ) \psi(x)
  42. ϕ ( p ) \phi(p)
  43. Â Â
  44. [ A ^ , B ^ ] = A ^ B ^ - B ^ A ^ . [\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}.
  45. [ x ^ , p ^ ] = i . [\hat{x},\hat{p}]=i\hbar.
  46. | ψ |\psi\rangle
  47. x ^ | ψ = x 0 | ψ . \hat{x}|\psi\rangle=x_{0}|\psi\rangle.
  48. | ψ |\psi\rangle
  49. [ x ^ , p ^ ] | ψ = ( x ^ p ^ - p ^ x ^ ) | ψ = ( x ^ - x 0 I ^ ) p ^ | ψ = i | ψ , [\hat{x},\hat{p}]|\psi\rangle=(\hat{x}\hat{p}-\hat{p}\hat{x})|\psi\rangle=(% \hat{x}-x_{0}\hat{I})\cdot\hat{p}\,|\psi\rangle=i\hbar|\psi\rangle,
  50. Î Î
  51. | ψ |\psi\rangle
  52. ( x ^ - x 0 I ^ ) p ^ | ψ = ( x ^ - x 0 I ^ ) p 0 | ψ = ( x 0 I ^ - x 0 I ^ ) p 0 | ψ = 0. (\hat{x}-x_{0}\hat{I})\cdot\hat{p}\,|\psi\rangle=(\hat{x}-x_{0}\hat{I})\cdot p% _{0}\,|\psi\rangle=(x_{0}\hat{I}-x_{0}\hat{I})\cdot p_{0}\,|\psi\rangle=0.
  53. [ x ^ , p ^ ] | ψ = i | ψ 0. [\hat{x},\hat{p}]|\psi\rangle=i\hbar|\psi\rangle\neq 0.
  54. σ x = x ^ 2 - x ^ 2 \sigma_{x}=\sqrt{\langle\hat{x}^{2}\rangle-\langle\hat{x}\rangle^{2}}
  55. σ p = p ^ 2 - p ^ 2 . \sigma_{p}=\sqrt{\langle\hat{p}^{2}\rangle-\langle\hat{p}\rangle^{2}}.
  56. 𝒪 ^ \hat{\mathcal{O}}
  57. σ 𝒪 = 𝒪 ^ 2 - 𝒪 ^ 2 \sigma_{\mathcal{O}}=\sqrt{\langle\hat{\mathcal{O}}^{2}\rangle-\langle\hat{% \mathcal{O}}\rangle^{2}}
  58. 𝒪 \langle\mathcal{O}\rangle
  59. Â Â
  60. [ A ^ , B ^ ] = A ^ B ^ - B ^ A ^ [\hat{A},\hat{B}]=\hat{A}\hat{B}-\hat{B}\hat{A}
  61. σ A σ B | 1 2 i [ A ^ , B ^ ] | = 1 2 | [ A ^ , B ^ ] | \sigma_{A}\sigma_{B}\geq\left|\frac{1}{2i}\langle[\hat{A},\hat{B}]\rangle% \right|=\frac{1}{2}\left|\langle[\hat{A},\hat{B}]\rangle\right|
  62. { A ^ , B ^ } = A ^ B ^ + B ^ A ^ \{\hat{A},\hat{B}\}=\hat{A}\hat{B}+\hat{B}\hat{A}
  63. A ^ \hat{A}
  64. σ A 2 = ( A ^ - A ^ ) Ψ | ( A ^ - A ^ ) Ψ . \sigma_{A}^{2}=\langle(\hat{A}-\langle\hat{A}\rangle)\Psi|(\hat{A}-\langle\hat% {A}\rangle)\Psi\rangle.
  65. | f = | ( A ^ - A ^ ) Ψ |f\rangle=|(\hat{A}-\langle\hat{A}\rangle)\Psi\rangle
  66. σ A 2 = f | f . \sigma_{A}^{2}=\langle f|f\rangle\,.
  67. B ^ \hat{B}
  68. σ B 2 = ( B ^ - B ^ ) Ψ | ( B ^ - B ^ ) Ψ = g | g \sigma_{B}^{2}=\langle(\hat{B}-\langle\hat{B}\rangle)\Psi|(\hat{B}-\langle\hat% {B}\rangle)\Psi\rangle=\langle g|g\rangle
  69. | g = | ( B ^ - B ^ ) Ψ |g\rangle=|(\hat{B}-\langle\hat{B}\rangle)\Psi\rangle
  70. | f |f\rangle
  71. | g |g\rangle
  72. f | f g | g | f | g | 2 , \langle f|f\rangle\langle g|g\rangle\geq|\langle f|g\rangle|^{2},\,
  73. f | g \langle f|g\rangle
  74. z z
  75. | z | 2 = z z * |z|^{2}=zz^{*}
  76. z * z^{*}
  77. z z
  78. | z | 2 = ( Re ( z ) ) 2 + ( Im ( z ) ) 2 = ( z + z 2 ) 2 + ( z - z 2 i ) 2 . |z|^{2}=\Big(\,\text{Re}(z)\Big)^{2}+\Big(\,\text{Im}(z)\Big)^{2}=\Big(\frac{z% +z^{\ast}}{2}\Big)^{2}+\Big(\frac{z-z^{\ast}}{2i}\Big)^{2}.
  79. z = f | g z=\langle f|g\rangle
  80. z * = g | f z^{*}=\langle g|f\rangle
  81. | f | g | 2 = ( f | g + g | f 2 ) 2 + ( f | g - g | f 2 i ) 2 |\langle f|g\rangle|^{2}=\bigg(\frac{\langle f|g\rangle+\langle g|f\rangle}{2}% \bigg)^{2}+\bigg(\frac{\langle f|g\rangle-\langle g|f\rangle}{2i}\bigg)^{2}
  82. f | g \langle f|g\rangle
  83. f | g = ( A ^ - A ^ ) Ψ | ( B ^ - B ^ ) Ψ , \langle f|g\rangle=\langle(\hat{A}-\langle\hat{A}\rangle)\Psi|(\hat{B}-\langle% \hat{B}\rangle)\Psi\rangle,
  84. A ^ \hat{A}
  85. B ^ \hat{B}
  86. f | g = Ψ | ( A ^ - A ^ ) ( B ^ - B ^ ) Ψ \langle f|g\rangle=\langle\Psi|(\hat{A}-\langle\hat{A}\rangle)(\hat{B}-\langle% \hat{B}\rangle)\Psi\rangle
  87. = Ψ | ( A ^ B ^ - A ^ B ^ - B ^ A ^ + A ^ B ^ ) Ψ =\langle\Psi|(\hat{A}\hat{B}-\hat{A}\langle\hat{B}\rangle-\hat{B}\langle\hat{A% }\rangle+\langle\hat{A}\rangle\langle\hat{B}\rangle)\Psi\rangle
  88. = Ψ | A ^ B ^ Ψ - Ψ | A ^ B ^ Ψ - Ψ | B ^ A ^ Ψ + Ψ | A ^ B ^ Ψ =\langle\Psi|\hat{A}\hat{B}\Psi\rangle-\langle\Psi|\hat{A}\langle\hat{B}% \rangle\Psi\rangle-\langle\Psi|\hat{B}\langle\hat{A}\rangle\Psi\rangle+\langle% \Psi|\langle\hat{A}\rangle\langle\hat{B}\rangle\Psi\rangle
  89. = A ^ B ^ - A ^ B ^ - A ^ B ^ + A ^ B ^ =\langle\hat{A}\hat{B}\rangle-\langle\hat{A}\rangle\langle\hat{B}\rangle-% \langle\hat{A}\rangle\langle\hat{B}\rangle+\langle\hat{A}\rangle\langle\hat{B}\rangle
  90. = A ^ B ^ - A ^ B ^ =\langle\hat{A}\hat{B}\rangle-\langle\hat{A}\rangle\langle\hat{B}\rangle\,
  91. g | f = B ^ A ^ - A ^ B ^ . \langle g|f\rangle=\langle\hat{B}\hat{A}\rangle-\langle\hat{A}\rangle\langle% \hat{B}\rangle.
  92. f | g - g | f = A ^ B ^ - A ^ B ^ - B ^ A ^ + A ^ B ^ = [ A ^ , B ^ ] \langle f|g\rangle-\langle g|f\rangle=\langle\hat{A}\hat{B}\rangle-\langle\hat% {A}\rangle\langle\hat{B}\rangle-\langle\hat{B}\hat{A}\rangle+\langle\hat{A}% \rangle\langle\hat{B}\rangle=\langle[\hat{A},\hat{B}]\rangle
  93. f | g + g | f = A ^ B ^ - A ^ B ^ + B ^ A ^ - A ^ B ^ = { A ^ , B ^ } - 2 A ^ B ^ \langle f|g\rangle+\langle g|f\rangle=\langle\hat{A}\hat{B}\rangle-\langle\hat% {A}\rangle\langle\hat{B}\rangle+\langle\hat{B}\hat{A}\rangle-\langle\hat{A}% \rangle\langle\hat{B}\rangle=\langle\{\hat{A},\hat{B}\}\rangle-2\langle\hat{A}% \rangle\langle\hat{B}\rangle
  94. < t d > < p > f | g | 2 = ( 1 2 { A ^ , B ^ } - A ^ B ^ ) 2 + ( 1 2 i [ A ^ , B ^ ] ) 2 . < / p > S u b s t i t u t i n g t h e a b o v e i n t o E q . ( E q u a t i o n N o t e | 2 ) w e g e t t h e S c h r ö d i n g e r u n c e r t a i n t y r e l a t i o n ; : ; ; ; : < m a t h > σ A σ B ( 1 2 { A ^ , B ^ } - A ^ B ^ ) 2 + ( 1 2 i [ A ^ , B ^ ] ) 2 <td><p>\langle f|g\rangle|^{2}=\Big(\frac{1}{2}\langle\{\hat{A},\hat{B}\}% \rangle-\langle\hat{A}\rangle\langle\hat{B}\rangle\Big)^{2}+\Big(\frac{1}{2i}% \langle[\hat{A},\hat{B}]\rangle\Big)^{2}\,.</p>\par \par SubstitutingtheaboveintoEq% .({{EquationNote|2}})wegettheSchrödingeruncertaintyrelation\par ;:;;;:<math>% \sigma_{A}\sigma_{B}\geq\sqrt{\Big(\frac{1}{2}\langle\{\hat{A},\hat{B}\}% \rangle-\langle\hat{A}\rangle\langle\hat{B}\rangle\Big)^{2}+\Big(\frac{1}{2i}% \langle[\hat{A},\hat{B}]\rangle\Big)^{2}}\,
  95. B ^ | Ψ \hat{B}|\Psi\rangle
  96. A ^ \hat{A}
  97. A ^ \hat{A}
  98. B ^ \hat{B}
  99. A ^ \hat{A}
  100. B ^ \hat{B}
  101. [ x ^ , p ^ ] = i [\hat{x},\hat{p}]=i\hbar
  102. σ x σ p 2 \sigma_{x}\sigma_{p}\geq\frac{\hbar}{2}
  103. σ J i σ J j 2 | J k | \sigma_{J_{i}}\sigma_{J_{j}}\geq\tfrac{\hbar}{2}\left|\left\langle J_{k}\right% \rangle\right|~{}
  104. [ J x , J y ] = i ϵ x y z J z [{J_{x}},{J_{y}}]=i\hbar\epsilon_{xyz}{J_{z}}
  105. A ^ = J x , B ^ = J y \hat{A}=J_{x},~{}~{}\hat{B}=J_{y}
  106. J x 2 + J y 2 + J z 2 \langle J_{x}^{2}+J_{y}^{2}+J_{z}^{2}\rangle
  107. ψ ψ
  108. B ^ \hat{B}
  109. σ E σ B | d B ^ d t | 2 \sigma_{E}~{}\frac{\sigma_{B}}{\left|\frac{\mathrm{d}\langle\hat{B}\rangle}{% \mathrm{d}t}\right|}\geq\frac{\hbar}{2}
  110. ψ ψ
  111. ψ ψ
  112. B ^ \langle\hat{B}\rangle
  113. Δ N Δ ϕ 1 \Delta N\Delta\phi\geq 1~{}
  114. x ^ = 2 m ω ( a + a ) \hat{x}=\sqrt{\frac{\hbar}{2m\omega}}(a+a^{\dagger})
  115. p ^ = i m ω 2 ( a - a ) \hat{p}=i\sqrt{\frac{m\omega\hbar}{2}}(a^{\dagger}-a)
  116. a | n = n + 1 | n + 1 a^{\dagger}|n\rangle=\sqrt{n+1}|n+1\rangle
  117. a | n = n | n - 1 a|n\rangle=\sqrt{n}|n-1\rangle~{}
  118. σ x 2 = m ω ( n + 1 2 ) \sigma_{x}^{2}=\frac{\hbar}{m\omega}\left(n+\frac{1}{2}\right)
  119. σ p 2 = m ω ( n + 1 2 ) . \sigma_{p}^{2}=\hbar m\omega\left(n+\frac{1}{2}\right)\,.
  120. σ x σ p = ( n + 1 2 ) 2 \sigma_{x}\sigma_{p}=\hbar\left(n+\frac{1}{2}\right)\geq\frac{\hbar}{2}~{}
  121. n = 0 n=0
  122. ψ ( x ) = ( m Ω π ) 1 / 4 exp ( - m Ω ( x - x 0 ) 2 2 ) \psi(x)=\left(\frac{m\Omega}{\pi\hbar}\right)^{1/4}\exp{\left(-\frac{m\Omega(x% -x_{0})^{2}}{2\hbar}\right)}
  123. | Ψ ( x , t ) | 2 𝒩 ( x 0 cos ( ω t ) , 2 m Ω ( cos 2 ( ω t ) + Ω 2 ω 2 sin 2 ( ω t ) ) ) |\Psi(x,t)|^{2}\sim\mathcal{N}\left(x_{0}\cos{(\omega t)},\frac{\hbar}{2m% \Omega}\left(\cos^{2}{(\omega t)}+\frac{\Omega^{2}}{\omega^{2}}\sin^{2}{(% \omega t)}\right)\right)
  124. | Φ ( p , t ) | 2 𝒩 ( - m x 0 ω sin ( ω t ) , m Ω 2 ( cos 2 ( ω t ) + ω 2 Ω 2 sin 2 ( ω t ) ) ) |\Phi(p,t)|^{2}\sim\mathcal{N}\left(-mx_{0}\omega\sin{(\omega t)},\frac{\hbar m% \Omega}{2}\left(\cos^{2}{(\omega t)}+\frac{\omega^{2}}{\Omega^{2}}\sin^{2}{(% \omega t)}\right)\right)
  125. 𝒩 ( μ , σ 2 ) \mathcal{N}(\mu,\sigma^{2})
  126. σ x σ p = 2 ( cos 2 ( ω t ) + Ω 2 ω 2 sin 2 ( ω t ) ) ( cos 2 ( ω t ) + ω 2 Ω 2 sin 2 ( ω t ) ) = 4 3 + 1 2 ( Ω 2 ω 2 + ω 2 Ω 2 ) - ( 1 2 ( Ω 2 ω 2 + ω 2 Ω 2 ) - 1 ) cos ( 4 ω t ) \begin{aligned}\displaystyle\sigma_{x}\sigma_{p}&\displaystyle=\frac{\hbar}{2}% \sqrt{\left(\cos^{2}{(\omega t)}+\frac{\Omega^{2}}{\omega^{2}}\sin^{2}{(\omega t% )}\right)\left(\cos^{2}{(\omega t)}+\frac{\omega^{2}}{\Omega^{2}}\sin^{2}{(% \omega t)}\right)}\\ &\displaystyle=\frac{\hbar}{4}\sqrt{3+\frac{1}{2}\left(\frac{\Omega^{2}}{% \omega^{2}}+\frac{\omega^{2}}{\Omega^{2}}\right)-\left(\frac{1}{2}\left(\frac{% \Omega^{2}}{\omega^{2}}+\frac{\omega^{2}}{\Omega^{2}}\right)-1\right)\cos{(4% \omega t)}}\end{aligned}
  127. Ω 2 ω 2 + ω 2 Ω 2 2 , | cos ( 4 ω t ) | 1 \frac{\Omega^{2}}{\omega^{2}}+\frac{\omega^{2}}{\Omega^{2}}\geq 2,\,\,\,|\cos{% (4\omega t)}|\leq 1
  128. σ x σ p 4 3 + 1 2 ( Ω 2 ω 2 + ω 2 Ω 2 ) - ( 1 2 ( Ω 2 ω 2 + ω 2 Ω 2 ) - 1 ) = 2 \sigma_{x}\sigma_{p}\geq\frac{\hbar}{4}\sqrt{3+\frac{1}{2}\left(\frac{\Omega^{% 2}}{\omega^{2}}+\frac{\omega^{2}}{\Omega^{2}}\right)-\left(\frac{1}{2}\left(% \frac{\Omega^{2}}{\omega^{2}}+\frac{\omega^{2}}{\Omega^{2}}\right)-1\right)}=% \frac{\hbar}{2}
  129. a ^ | α = α | α \hat{a}|\alpha\rangle=\alpha|\alpha\rangle
  130. | α = e - | α | 2 2 n = 0 α n n ! | n |\alpha\rangle=e^{-{|\alpha|^{2}\over 2}}\sum_{n=0}^{\infty}{\alpha^{n}\over% \sqrt{n!}}|n\rangle
  131. σ x 2 = 2 m ω \sigma_{x}^{2}=\frac{\hbar}{2m\omega}
  132. σ p 2 = m ω 2 \sigma_{p}^{2}=\frac{\hbar m\omega}{2}
  133. σ x σ p = 2 m ω m ω 2 = 2 \sigma_{x}\sigma_{p}=\sqrt{\frac{\hbar}{2m\omega}}\,\sqrt{\frac{\hbar m\omega}% {2}}=\frac{\hbar}{2}
  134. / 2 \sqrt{\hbar/2}
  135. L L
  136. ψ n ( x , t ) = { A sin ( k n x ) e - i ω n t , 0 < x < L , 0 , otherwise, \psi_{n}(x,t)=\begin{cases}A\sin(k_{n}x)\mathrm{e}^{-\mathrm{i}\omega_{n}t},&0% <x<L,\\ 0,&\,\text{otherwise,}\end{cases}
  137. ϕ n ( p , t ) = π L n ( 1 - ( - 1 ) n e - i k L ) e - i ω n t π 2 n 2 - k 2 L 2 \phi_{n}(p,t)=\sqrt{\frac{\pi L}{\hbar}}\,\,\frac{n\left(1-(-1)^{n}e^{-ikL}% \right)e^{-i\omega_{n}t}}{\pi^{2}n^{2}-k^{2}L^{2}}
  138. ω n = π 2 n 2 8 L 2 m \omega_{n}=\frac{\pi^{2}\hbar n^{2}}{8L^{2}m}
  139. p = k p=\hbar k
  140. x x
  141. p p
  142. σ x 2 = L 2 12 ( 1 - 6 n 2 π 2 ) \sigma_{x}^{2}=\frac{L^{2}}{12}\left(1-\frac{6}{n^{2}\pi^{2}}\right)
  143. σ p 2 = ( n π L ) 2 \sigma_{p}^{2}=\left(\frac{\hbar n\pi}{L}\right)^{2}
  144. σ x σ p = 2 n 2 π 2 3 - 2 . \sigma_{x}\sigma_{p}=\frac{\hbar}{2}\sqrt{\frac{n^{2}\pi^{2}}{3}-2}.
  145. n = 1 , 2 , 3 ... n=1,\,2,\,3\,...
  146. n 2 π 2 3 - 2 \sqrt{\frac{n^{2}\pi^{2}}{3}-2}
  147. n = 1 n=1
  148. σ x σ p = 2 π 2 3 - 2 0.568 > 2 \sigma_{x}\sigma_{p}=\frac{\hbar}{2}\sqrt{\frac{\pi^{2}}{3}-2}\approx 0.568% \hbar>\frac{\hbar}{2}
  149. ϕ ( p ) = ( x 0 π ) 1 / 2 exp ( - x 0 2 ( p - p 0 ) 2 2 2 ) , \phi(p)=\left(\frac{x_{0}}{\hbar\sqrt{\pi}}\right)^{1/2}\cdot\exp{\left(\frac{% -x_{0}^{2}(p-p_{0})^{2}}{2\hbar^{2}}\right)},
  150. x 0 = / m ω 0 x_{0}=\sqrt{\hbar/m\omega_{0}}
  151. ω 0 > 0 \omega_{0}>0
  152. Φ ( p , t ) = ( x 0 π ) 1 / 2 exp ( - x 0 2 ( p - p 0 ) 2 2 2 - i p 2 t 2 m ) , \Phi(p,t)=\left(\frac{x_{0}}{\hbar\sqrt{\pi}}\right)^{1/2}\cdot\exp{\left(% \frac{-x_{0}^{2}(p-p_{0})^{2}}{2\hbar^{2}}-\frac{ip^{2}t}{2m\hbar}\right)},
  153. Ψ ( x , t ) = ( 1 x 0 π ) 1 / 2 e - x 0 2 p 0 2 / 2 2 1 + i ω 0 t exp ( - ( x - i x 0 2 p 0 / ) 2 2 x 0 2 ( 1 + i ω 0 t ) ) . \Psi(x,t)=\left(\frac{1}{x_{0}\sqrt{\pi}}\right)^{1/2}\cdot\frac{e^{-x_{0}^{2}% p_{0}^{2}/2\hbar^{2}}}{\sqrt{1+i\omega_{0}t}}\cdot\exp{\left(-\frac{(x-ix_{0}^% {2}p_{0}/\hbar)^{2}}{2x_{0}^{2}(1+i\omega_{0}t)}\right)}.
  154. p ( t ) = p 0 \langle p(t)\rangle=p_{0}
  155. σ p ( t ) = / x 0 2 , \sigma_{p}(t)=\hbar/x_{0}\sqrt{2},
  156. σ x = x 0 2 1 + ω 0 2 t 2 \sigma_{x}=\frac{x_{0}}{\sqrt{2}}\sqrt{1+\omega_{0}^{2}t^{2}}
  157. σ x ( t ) σ p ( t ) = 2 1 + ω 0 2 t 2 \sigma_{x}(t)\sigma_{p}(t)=\frac{\hbar}{2}\sqrt{1+\omega_{0}^{2}t^{2}}
  158. σ A 2 σ B 2 ( 1 2 tr ( ρ { A , B } ) - tr ( ρ A ) tr ( ρ B ) ) 2 + ( 1 2 i tr ( ρ [ A , B ] ) ) 2 \sigma_{A}^{2}\sigma_{B}^{2}\geq\left(\frac{1}{2}\mathrm{tr}(\rho\{A,B\})-% \operatorname{tr}(\rho A)\mathrm{tr}(\rho B)\right)^{2}+\left(\frac{1}{2i}% \mathrm{tr}(\rho[A,B])\right)^{2}
  159. W ( x , p ) W(x,p)
  160. f * f = ( f * f ) W ( x , p ) d x d p 0. \langle f^{*}\star f\rangle=\int(f^{*}\star f)\,W(x,p)\,dxdp\geq 0.
  161. f = a + b x + c p f=a+bx+cp
  162. f * f = [ a * b * c * ] [ 1 x p x x x x p p p x p p ] [ a b c ] 0. \langle f^{*}\star f\rangle=\begin{bmatrix}a^{*}&b^{*}&c^{*}\end{bmatrix}% \begin{bmatrix}1&\langle x\rangle&\langle p\rangle\\ \langle x\rangle&\langle x\star x\rangle&\langle x\star p\rangle\\ \langle p\rangle&\langle p\star x\rangle&\langle p\star p\rangle\end{bmatrix}% \begin{bmatrix}a\\ b\\ c\end{bmatrix}\geq 0.
  163. det [ 1 x p x x x x p p p x p p ] = det [ 1 x p x x 2 x p + i 2 p x p - i 2 p 2 ] 0 , \det\begin{bmatrix}1&\langle x\rangle&\langle p\rangle\\ \langle x\rangle&\langle x\star x\rangle&\langle x\star p\rangle\\ \langle p\rangle&\langle p\star x\rangle&\langle p\star p\rangle\end{bmatrix}=% \det\begin{bmatrix}1&\langle x\rangle&\langle p\rangle\\ \langle x\rangle&\langle x^{2}\rangle&\left\langle xp+\frac{i\hbar}{2}\right% \rangle\\ \langle p\rangle&\left\langle xp-\frac{i\hbar}{2}\right\rangle&\langle p^{2}% \rangle\end{bmatrix}\geq 0,
  164. σ x 2 σ p 2 = ( x 2 - x 2 ) ( p 2 - p 2 ) ( x p - x p ) 2 + 2 4 . \sigma_{x}^{2}\sigma_{p}^{2}=\left(\langle x^{2}\rangle-\langle x\rangle^{2}% \right)\left(\langle p^{2}\rangle-\langle p\rangle^{2}\right)\geq\left(\langle xp% \rangle-\langle x\rangle\langle p\rangle\right)^{2}+\frac{\hbar^{2}}{4}~{}.
  165. ϵ 𝒪 \epsilon_{\mathcal{O}}
  166. 𝒪 \mathcal{O}
  167. η 𝒪 \eta_{\mathcal{O}}
  168. \cancel ϵ x η p 2 \cancel{\epsilon_{x}\eta_{p}\sim\frac{\hbar}{2}}\,\,
  169. H x = - | ψ ( x ) | 2 ln ( | ψ ( x ) | 2 ) d x = - ln ( | ψ ( x ) | 2 ) H_{x}=-\int|\psi(x)|^{2}\ln(|\psi(x)|^{2}\cdot\ell)\,dx=-\left\langle\ln(|\psi% (x)|^{2}\cdot\ell)\right\rangle
  170. H p = - | ϕ ( p ) | 2 ln ( | ϕ ( p ) | 2 / ) d p = - ln ( | ϕ ( p ) | 2 / ) H_{p}=-\int|\phi(p)|^{2}\ln(|\phi(p)|^{2}\cdot\hbar/\ell)\,dp=-\left\langle\ln% (|\phi(p)|^{2}\cdot\hbar/\ell)\right\rangle
  171. \ell
  172. H x 1 2 ln ( 2 e π σ x 2 / 2 ) , H_{x}\leq\frac{1}{2}\ln(2e\pi\sigma_{x}^{2}/\ell^{2})~{},
  173. H p 1 2 ln ( 2 e π σ p 2 2 / 2 ) , H_{p}\leq\frac{1}{2}\ln(2e\pi\sigma_{p}^{2}\ell^{2}/\hbar^{2})~{},
  174. σ x σ p 2 exp ( H x + H p - ln ( e π ) ) 2 . \sigma_{x}\sigma_{p}\geq\frac{\hbar}{2}\cdot\exp\left(H_{x}+H_{p}-\ln(e\pi)% \right)\geq\frac{\hbar}{2}~{}.
  175. = 2 m ω \ell=\sqrt{\frac{\hbar}{2m\omega}}
  176. ψ ( x ) = ( m ω π ) 1 / 4 exp ( - m ω x 2 2 ) = ( 1 2 π 2 ) 1 / 4 exp ( - x 2 4 2 ) \begin{aligned}\displaystyle\psi(x)&\displaystyle=\left(\frac{m\omega}{\pi% \hbar}\right)^{1/4}\exp{\left(-\frac{m\omega x^{2}}{2\hbar}\right)}\\ &\displaystyle=\left(\frac{1}{2\pi\ell^{2}}\right)^{1/4}\exp{\left(-\frac{x^{2% }}{4\ell^{2}}\right)}\end{aligned}
  177. | ψ ( x ) | 2 = 1 2 π exp ( - x 2 2 2 ) |\psi(x)|^{2}=\frac{1}{\ell\sqrt{2\pi}}\exp{\left(-\frac{x^{2}}{2\ell^{2}}% \right)}
  178. H x = - | ψ ( x ) | 2 ln ( | ψ ( x ) | 2 ) d x = - 1 2 π - exp ( - x 2 2 2 ) ln [ 1 2 π exp ( - x 2 2 2 ) ] d x = 1 2 π - exp ( - u 2 2 ) [ ln ( 2 π ) + u 2 2 ] d u = ln ( 2 π ) + 1 2 . \begin{aligned}\displaystyle H_{x}&\displaystyle=-\int|\psi(x)|^{2}\ln(|\psi(x% )|^{2}\cdot\ell)\,dx\\ &\displaystyle=-\frac{1}{\ell\sqrt{2\pi}}\int_{-\infty}^{\infty}\exp{\left(-% \frac{x^{2}}{2\ell^{2}}\right)}\ln\left[\frac{1}{\sqrt{2\pi}}\exp{\left(-\frac% {x^{2}}{2\ell^{2}}\right)}\right]\,dx\\ &\displaystyle=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\exp{\left(-\frac{u% ^{2}}{2}\right)}\left[\ln(\sqrt{2\pi})+\frac{u^{2}}{2}\right]\,du\\ &\displaystyle=\ln(\sqrt{2\pi})+\frac{1}{2}.\end{aligned}
  179. ϕ ( p ) = ( 2 2 π 2 ) 1 / 4 exp ( - 2 p 2 2 ) \phi(p)=\left(\frac{2\ell^{2}}{\pi\hbar^{2}}\right)^{1/4}\exp{\left(-\frac{% \ell^{2}p^{2}}{\hbar^{2}}\right)}
  180. | ϕ ( p ) | 2 = 2 2 π 2 exp ( - 2 2 p 2 2 ) |\phi(p)|^{2}=\sqrt{\frac{2\ell^{2}}{\pi\hbar^{2}}}\exp{\left(-\frac{2\ell^{2}% p^{2}}{\hbar^{2}}\right)}
  181. H p = - | ϕ ( p ) | 2 ln ( | ϕ ( p ) | 2 / ) d p = - 2 2 π 2 - exp ( - 2 2 p 2 2 ) ln [ 2 π exp ( - 2 2 p 2 2 ) ] d p = 2 π - exp ( - 2 v 2 ) [ ln ( π 2 ) + 2 v 2 ] d v = ln ( π 2 ) + 1 2 . \begin{aligned}\displaystyle H_{p}&\displaystyle=-\int|\phi(p)|^{2}\ln(|\phi(p% )|^{2}\cdot\hbar/\ell)\,dp\\ &\displaystyle=-\sqrt{\frac{2\ell^{2}}{\pi\hbar^{2}}}\int_{-\infty}^{\infty}% \exp{\left(-\frac{2\ell^{2}p^{2}}{\hbar^{2}}\right)}\ln\left[\sqrt{\frac{2}{% \pi}}\exp{\left(-\frac{2\ell^{2}p^{2}}{\hbar^{2}}\right)}\right]\,dp\\ &\displaystyle=\sqrt{\frac{2}{\pi}}\int_{-\infty}^{\infty}\exp{\left(-2v^{2}% \right)}\left[\ln\left(\sqrt{\frac{\pi}{2}}\right)+2v^{2}\right]\,dv\\ &\displaystyle=\ln\left(\sqrt{\frac{\pi}{2}}\right)+\frac{1}{2}.\end{aligned}
  182. H x + H p = ln ( 2 π ) + 1 2 + ln ( π 2 ) + 1 2 = 1 + ln π = ln ( e π ) . \begin{aligned}\displaystyle H_{x}+H_{p}&\displaystyle=\ln(\sqrt{2\pi})+\frac{% 1}{2}+\ln\left(\sqrt{\frac{\pi}{2}}\right)+\frac{1}{2}\\ &\displaystyle=1+\ln\pi=\ln(e\pi).\end{aligned}
  183. P [ x j ] = ( j - 1 / 2 ) δ x - c ( j + 1 / 2 ) δ x - c | ψ ( x ) | 2 d x \operatorname{P}[x_{j}]=\int_{(j-1/2)\delta x-c}^{(j+1/2)\delta x-c}|\psi(x)|^% {2}\,dx
  184. H x = - j = - P [ x j ] ln P [ x j ] . H_{x}=-\sum_{j=-\infty}^{\infty}\operatorname{P}[x_{j}]\ln\operatorname{P}[x_{% j}].
  185. H x + H p > ln ( e 2 ) - ln ( δ x δ p h ) . H_{x}+H_{p}>\ln\left(\frac{e}{2}\right)-\ln\left(\frac{\delta x\delta p}{h}% \right).
  186. ψ ( x ) = ( m ω π ) 1 / 4 exp ( - m ω x 2 2 ) \psi(x)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}\exp{\left(-\frac{m\omega x% ^{2}}{2\hbar}\right)}
  187. P [ x j ] = m ω π ( j - 1 / 2 ) δ x ( j + 1 / 2 ) δ x exp ( - m ω x 2 ) d x = 1 π ( j - 1 / 2 ) δ x m ω / ( j + 1 / 2 ) δ x m ω / e u 2 d u = 1 2 [ erf ( ( j + 1 2 ) δ x m ω ) - erf ( ( j - 1 2 ) δ x m ω ) ] \begin{aligned}\displaystyle\operatorname{P}[x_{j}]&\displaystyle=\sqrt{\frac{% m\omega}{\pi\hbar}}\int_{(j-1/2)\delta x}^{(j+1/2)\delta x}\exp{\left(-\frac{m% \omega x^{2}}{\hbar}\right)}\,dx\\ &\displaystyle=\sqrt{\frac{1}{\pi}}\int_{(j-1/2)\delta x\sqrt{m\omega/\hbar}}^% {(j+1/2)\delta x\sqrt{m\omega/\hbar}}e^{u^{2}}\,du\\ &\displaystyle=\frac{1}{2}\left[\operatorname{erf}\left(\left(j+\frac{1}{2}% \right)\delta x\cdot\sqrt{\frac{m\omega}{\hbar}}\right)-\operatorname{erf}% \left(\left(j-\frac{1}{2}\right)\delta x\cdot\sqrt{\frac{m\omega}{\hbar}}% \right)\right]\end{aligned}
  188. P [ p j ] = 1 2 [ erf ( ( j + 1 2 ) δ p 1 m ω ) - erf ( ( j - 1 2 ) δ x 1 m ω ) ] \operatorname{P}[p_{j}]=\frac{1}{2}\left[\operatorname{erf}\left(\left(j+\frac% {1}{2}\right)\delta p\cdot\frac{1}{\sqrt{\hbar m\omega}}\right)-\operatorname{% erf}\left(\left(j-\frac{1}{2}\right)\delta x\cdot\frac{1}{\sqrt{\hbar m\omega}% }\right)\right]
  189. δ x = h m ω \delta x=\sqrt{\frac{h}{m\omega}}
  190. δ p = h m ω \delta p=\sqrt{hm\omega}
  191. P [ x j ] = P [ p j ] = 1 2 [ erf ( ( j + 1 2 ) 2 π ) - erf ( ( j - 1 2 ) 2 π ) ] \operatorname{P}[x_{j}]=\operatorname{P}[p_{j}]=\frac{1}{2}\left[\operatorname% {erf}\left(\left(j+\frac{1}{2}\right)\sqrt{2\pi}\right)-\operatorname{erf}% \left(\left(j-\frac{1}{2}\right)\sqrt{2\pi}\right)\right]
  192. H x = H p = - j = - P [ x j ] ln P [ x j ] = - j = - 1 2 [ erf ( ( j + 1 2 ) 2 π ) - erf ( ( j - 1 2 ) 2 π ) ] ln 1 2 [ erf ( ( j + 1 2 ) 2 π ) - erf ( ( j - 1 2 ) 2 π ) ] 0.3226 \begin{aligned}\displaystyle H_{x}=H_{p}&\displaystyle=-\sum_{j=-\infty}^{% \infty}\operatorname{P}[x_{j}]\ln\operatorname{P}[x_{j}]\\ &\displaystyle=-\sum_{j=-\infty}^{\infty}\frac{1}{2}\left[\operatorname{erf}% \left(\left(j+\frac{1}{2}\right)\sqrt{2\pi}\right)-\operatorname{erf}\left(% \left(j-\frac{1}{2}\right)\sqrt{2\pi}\right)\right]\ln\frac{1}{2}\left[% \operatorname{erf}\left(\left(j+\frac{1}{2}\right)\sqrt{2\pi}\right)-% \operatorname{erf}\left(\left(j-\frac{1}{2}\right)\sqrt{2\pi}\right)\right]&% \displaystyle\approx 0.3226\end{aligned}
  193. H x + H p 0.3226 + 0.3226 = 0.6452 > ln ( e 2 ) - ln 1 0.3069 H_{x}+H_{p}\approx 0.3226+0.3226=0.6452>\ln\left(\frac{e}{2}\right)-\ln 1% \approx 0.3069
  194. ψ ( x ) = { 1 2 a for | x | a , 0 for | x | > a \psi(x)=\begin{cases}\frac{1}{\sqrt{2a}}&\mathrm{for}\ |x|\leq a,\\ 0&\mathrm{for}\ |x|>a\end{cases}
  195. ϕ ( p ) = a π sinc ( a p ) \phi(p)=\sqrt{\frac{a}{\pi\hbar}}\cdot\operatorname{sinc}\left(\frac{ap}{\hbar% }\right)
  196. P [ x 0 ] = - a 0 1 2 a d x = 1 2 \operatorname{P}[x_{0}]=\int_{-a}^{0}\frac{1}{2a}\,dx=\frac{1}{2}
  197. P [ x 1 ] = 0 a 1 2 a d x = 1 2 \operatorname{P}[x_{1}]=\int_{0}^{a}\frac{1}{2a}\,dx=\frac{1}{2}
  198. H x = - j = 0 1 P [ x j ] ln P [ x j ] = - 1 2 ln 1 2 - 1 2 ln 1 2 = ln 2 H_{x}=-\sum_{j=0}^{1}\operatorname{P}[x_{j}]\ln\operatorname{P}[x_{j}]=-\frac{% 1}{2}\ln\frac{1}{2}-\frac{1}{2}\ln\frac{1}{2}=\ln 2
  199. P [ p j ] = a π ( j - 1 / 2 ) δ p ( j + 1 / 2 ) δ p sinc 2 ( a p ) d p = 1 π 2 π ( j - 1 / 2 ) 2 π ( j + 1 / 2 ) sinc 2 ( u ) d u = 1 π [ Si ( ( 4 j + 2 ) π ) - Si ( ( 4 j - 2 ) π ) ] \begin{aligned}\displaystyle\operatorname{P}[p_{j}]&\displaystyle=\frac{a}{\pi% \hbar}\int_{(j-1/2)\delta p}^{(j+1/2)\delta p}\mathrm{sinc}^{2}\left(\frac{ap}% {\hbar}\right)\,dp\\ &\displaystyle=\frac{1}{\pi}\int_{2\pi(j-1/2)}^{2\pi(j+1/2)}\mathrm{sinc}^{2}(% u)\,du\\ &\displaystyle=\frac{1}{\pi}\left[\operatorname{Si}((4j+2)\pi)-\operatorname{% Si}((4j-2)\pi)\right]\end{aligned}
  200. H p = - j = - P [ p j ] ln P [ p j ] = - P [ p 0 ] ln P [ p 0 ] - 2 j = 1 P [ p j ] ln P [ p j ] 0.53 H_{p}=-\sum_{j=-\infty}^{\infty}\operatorname{P}[p_{j}]\ln\operatorname{P}[p_{% j}]=-\operatorname{P}[p_{0}]\ln\operatorname{P}[p_{0}]-2\cdot\sum_{j=1}^{% \infty}\operatorname{P}[p_{j}]\ln\operatorname{P}[p_{j}]\approx 0.53
  201. H x + H p 0.69 + 0.53 = 1.22 > ln ( e 2 ) - ln 1 0.31 H_{x}+H_{p}\approx 0.69+0.53=1.22>\ln\left(\frac{e}{2}\right)-\ln 1\approx 0.31
  202. ( - x 2 | f ( x ) | 2 d x ) ( - ξ 2 | f ^ ( ξ ) | 2 d ξ ) f 2 4 16 π 2 . \left(\int_{-\infty}^{\infty}x^{2}|f(x)|^{2}\,dx\right)\left(\int_{-\infty}^{% \infty}\xi^{2}|\hat{f}(\xi)|^{2}\,d\xi\right)\geq\frac{\|f\|_{2}^{4}}{16\pi^{2% }}.
  203. f f
  204. ƒ̂ ƒ̂
  205. f f
  206. ƒ̂ ƒ̂
  207. f f
  208. ƒ̂ ƒ̂
  209. f f
  210. ƒ̂ ƒ̂
  211. f L 2 ( 𝐑 d ) C e C | S | | Σ | ( f L 2 ( S c ) + f ^ L 2 ( Σ c ) ) . \|f\|_{L^{2}(\mathbf{R}^{d})}\leq Ce^{C|S||\Sigma|}\bigl(\|f\|_{L^{2}(S^{c})}+% \|\hat{f}\|_{L^{2}(\Sigma^{c})}\bigr)~{}.
  212. C e C | S | | Σ | Ce^{C|S||\Sigma|}
  213. C e C ( | S | | Σ | ) 1 / d Ce^{C(|S||\Sigma|)^{1/d}}
  214. S S
  215. Σ Σ
  216. f f
  217. ƒ̂ ƒ̂
  218. f f
  219. | f ( x ) | C ( 1 + | x | ) N e - a π x 2 |f(x)|\leq C(1+|x|)^{N}e^{-a\pi x^{2}}
  220. | f ^ ( ξ ) | C ( 1 + | ξ | ) N e - b π ξ 2 |\hat{f}(\xi)|\leq C(1+|\xi|)^{N}e^{-b\pi\xi^{2}}
  221. C > 0 , N C>0,N
  222. a b > 1 , f = 0 ab>1,f=0
  223. a b = 1 ab=1
  224. P P
  225. N ≤N
  226. f ( x ) = P ( x ) e - a π x 2 . f(x)=P(x)e^{-a\pi x^{2}}.\,
  227. 𝐑 d 𝐑 d | f ( x ) | | f ^ ( ξ ) | e π | x , ξ | ( 1 + | x | + | ξ | ) N d x d ξ < + , \int_{\mathbf{R}^{d}}\int_{\mathbf{R}^{d}}|f(x)||\hat{f}(\xi)|\frac{e^{\pi|% \langle x,\xi\rangle|}}{(1+|x|+|\xi|)^{N}}\,dx\,d\xi<+\infty~{},
  228. f ( x ) = P ( x ) e - π A x , x , f(x)=P(x)e^{-\pi\langle Ax,x\rangle}~{},
  229. P P
  230. ( N d ) / 2 (N−d)/2
  231. A A
  232. d × d d×d
  233. d = 1 , N = 0 d=1,N=0
  234. a b > 1 ab>1
  235. f 𝒮 ( \R d ) f\in\mathcal{S}^{\prime}(\R^{d})
  236. e π | x | 2 f 𝒮 ( \R d ) e^{\pi|x|^{2}}f\in\mathcal{S}^{\prime}(\R^{d})
  237. e π | ξ | 2 f ^ 𝒮 ( \R d ) , e^{\pi|\xi|^{2}}\hat{f}\in\mathcal{S}^{\prime}(\R^{d})~{},
  238. f ( x ) = P ( x ) e - π A x , x , f(x)=P(x)e^{-\pi\langle Ax,x\rangle}~{},
  239. P P
  240. A A
  241. d × d d×d
  242. ħ = h 2 π ħ=h\frac{2}{π}
  243. σ < s u b > x σ<sub>x

Uncountable_set.html

  1. 0 \aleph_{0}
  2. 0 \aleph_{0}
  3. 2 0 2^{\aleph_{0}}
  4. 1 \beth_{1}
  5. 2 \beth_{2}
  6. 1 \beth_{1}
  7. 1 \aleph_{1}
  8. 1 \aleph_{1}
  9. 1 \beth_{1}
  10. 1 \aleph_{1}
  11. 1 \beth_{1}
  12. 1 \aleph_{1}
  13. 1 = 1 \aleph_{1}=\beth_{1}
  14. 0 \aleph_{0}
  15. κ \kappa\!
  16. κ 0 ; \kappa\nleq\aleph_{0};
  17. κ > 0 ; \kappa>\aleph_{0};
  18. κ 1 \kappa\geq\aleph_{1}
  19. 1 = | ω 1 | \aleph_{1}=|\omega_{1}|
  20. ω 1 \omega_{1}\,
  21. ω . \omega.\!

Unemployment.html

  1. Unemployment rate = Unemployed workers Total labor force * 100 % \,\text{Unemployment rate}=\frac{\,\text{Unemployed workers}}{\,\text{Total % labor force}}*100\%

Uniform_continuity.html

  1. ε > 0 δ > 0 x I y I ( | y - x | < δ | f ( y ) - f ( x ) | < ε ) \forall\varepsilon>0\,\exists\delta>0\,\forall x\in I\,\forall y\in I\,(\,|y-x% |<\delta\,\Rightarrow\,|f(y)-f(x)|<\varepsilon\,)
  2. x δ , \cdots\forall x\,\exists\delta\cdots,
  3. δ x y . \cdots\exists\delta\,\forall x\,\forall y\cdots.
  4. \scriptstyle\mapsto\,
  5. f : , x x 2 f\colon\mathbb{R}\rightarrow\mathbb{R},x\mapsto x^{2}
  6. ϵ \epsilon
  7. δ \delta
  8. x 1 , x 2 x_{1},x_{2}
  9. | x 1 - x 2 | < δ |x_{1}-x_{2}|<\delta
  10. | f ( x 1 ) - f ( x 2 ) | < ϵ |f(x_{1})-f(x_{2})|<\epsilon
  11. f ( x + δ ) - f ( x ) = 2 x δ + δ 2 = δ ( 2 x + δ ) , f(x+\delta)-f(x)=2x\delta+\delta^{2}=\delta(2x+\delta)\ ,
  12. ϵ \epsilon
  13. f f
  14. [ 0 , ) [0,\infty)
  15. lim x f ( x ) \lim_{x\to\infty}f(x)
  16. f f
  17. C 0 ( ) C_{0}(\mathbb{R})
  18. \mathbb{R}
  19. C c ( ) C 0 ( ) C_{c}(\mathbb{R})\subset C_{0}(\mathbb{R})
  20. lim n | x n - y n | = 0 \lim_{n\to\infty}|x_{n}-y_{n}|=0\,
  21. lim n | f ( x n ) - f ( y n ) | = 0. \lim_{n\to\infty}|f(x_{n})-f(y_{n})|=0.\,
  22. f : S R f:S\rightarrow R
  23. f : S R f:S\rightarrow R
  24. f : R R , x x 2 f:R\rightarrow R,x\mapsto x^{2}
  25. f : x a x f:x\mapsto a^{x}
  26. f ( x + δ ) - f ( x ) = a x ( a δ - 1 ) f(x+\delta)-f(x)=a^{x}(a^{\delta}-1)\,
  27. Q I Q\cap I
  28. f : S R f:S\rightarrow R
  29. V V
  30. W W
  31. f : V W f:V\to W
  32. B B
  33. W W
  34. A A
  35. V V
  36. v 1 - v 2 A v_{1}-v_{2}\in A
  37. f ( v 1 ) - f ( v 2 ) B . f(v_{1})-f(v_{2})\in B.
  38. f : V W f:V\to W

Uniform_convergence.html

  1. n = 1 f n ( x , ϕ , ψ ) \textstyle{\sum_{n=1}^{\infty}f_{n}(x,\phi,\psi)}
  2. ϕ \phi
  3. ψ . \psi.
  4. S S
  5. f n : S f_{n}:S\to\mathbb{R}
  6. n n
  7. ( f n ) n (f_{n})_{n\in\mathbb{N}}
  8. f : S f:S\to\mathbb{R}
  9. ϵ > 0 \epsilon>0
  10. N N
  11. x S x\in S
  12. n N n\geq N
  13. | f n ( x ) - f ( x ) | < ϵ |f_{n}(x)-f(x)|<\epsilon
  14. a n = sup x | f n ( x ) - f ( x ) | a_{n}=\sup_{x}|f_{n}(x)-f(x)|
  15. x S x\in S
  16. f n f_{n}
  17. f f
  18. a n a_{n}
  19. ( f n ) n (f_{n})_{n\in\mathbb{N}}
  20. f f
  21. x x
  22. S S
  23. r > 0 r>0
  24. ( f n ) (f_{n})
  25. B ( x , r ) S B(x,r)\cap S
  26. N N
  27. x x
  28. f : S 𝐑 f:S→\mathbf{R}
  29. x S x∈S
  30. ε > 0 ε>0
  31. n N n≥N
  32. x x
  33. ε ε
  34. N N
  35. N N
  36. ε ε
  37. N N
  38. ε ε
  39. x x
  40. S S
  41. n n
  42. f f
  43. f ( x ) = 0 f(x)=0
  44. x Align l t ; 1 x&lt;1
  45. f ( 1 ) = 1 f(1)=1
  46. ε = 1 / 4 ε=1/4
  47. N N
  48. n n
  49. n > l o g ε / l o g x n>logε/logx
  50. x x
  51. ε ε
  52. n n
  53. x x
  54. ε ε
  55. l o g ε / l o g x logε/logx
  56. x x
  57. f n f_{n}
  58. f n * ( x ) f_{n}^{*}(x)
  59. f * ( x ) f^{*}(x)
  60. f n : [ 0 , 1 ] [ 0 , 1 ] f_{n}:[0,1]\rightarrow[0,1]
  61. f n ( x ) := x n f_{n}(x):=x^{n}
  62. lim n f n ( x ) = { 0 , x [ 0 , 1 ) 1 , x = 1. \lim_{n\rightarrow\infty}f_{n}(x)=\begin{cases}0,&x\in[0,1)\\ 1,&x=1.\end{cases}
  63. f n C ( [ 0 , 1 ] ) f_{n}\in C^{\infty}([0,1])
  64. lim n f n \lim_{n\rightarrow\infty}f_{n}
  65. \mathbb{C}
  66. n = 0 z n n ! . \sum_{n=0}^{\infty}\frac{z^{n}}{n!}.
  67. D R D_{R}
  68. M n M_{n}
  69. M n M_{n}
  70. | z n n ! | M n , z D R . \left|\frac{z^{n}}{n!}\right|\leq M_{n},\forall z\in D_{R}.
  71. | z n n ! | | z | n n ! R n n ! \left|\frac{z^{n}}{n!}\right|\leq\frac{\left|z\right|^{n}}{n!}\leq\frac{R^{n}}% {n!}
  72. M n = R n n ! . \Rightarrow M_{n}=\frac{R^{n}}{n!}.
  73. n = 0 M n \sum_{n=0}^{\infty}M_{n}
  74. lim n M n + 1 M n = lim n R n + 1 R n n ! ( n + 1 ) ! = lim n R n + 1 = 0 \lim_{n\to\infty}\frac{M_{n+1}}{M_{n}}=\lim_{n\to\infty}\frac{R^{n+1}}{R^{n}}% \frac{n!}{(n+1)!}=\lim_{n\to\infty}\frac{R}{n+1}=0
  75. M n M_{n}
  76. z D R z\in D_{R}
  77. S D R S\subset D_{R}
  78. S S
  79. ( f n ) (f_{n})
  80. f n ( x ) f n + 1 ( x ) f_{n}(x)\leq f_{n+1}(x)
  81. f f
  82. S S
  83. ( f n ) (f_{n})
  84. I I
  85. f n f_{n}
  86. f f
  87. ( f n ) n (f_{n})_{n}
  88. I I
  89. f f
  90. I I
  91. f f
  92. S S
  93. ϵ / 3 \epsilon/3
  94. < ϵ <\epsilon
  95. < ϵ / 3 <\epsilon/3
  96. f ( x ) f(x)
  97. I I
  98. x 0 I x_{0}\in I
  99. ε > 0 \varepsilon>0
  100. δ ( ε ) > 0 \delta(\varepsilon)>0
  101. x x
  102. | x - x 0 | < δ ( ε ) |x-x_{0}|<\delta(\varepsilon)
  103. | f ( x ) - f ( x 0 ) | < ε . |f(x)-f(x_{0})|<\varepsilon.
  104. | f ( x ) - f ( x 0 ) | = | f ( x ) - f n ( x ) + f n ( x ) - f n ( x 0 ) + f n ( x 0 ) - f ( x 0 ) | | f ( x ) - f n ( x ) | + | f n ( x ) - f n ( x 0 ) | + | f n ( x 0 ) - f ( x 0 ) | \Big|f(x)-f(x_{0})\Big|=\Big|f(x)-f_{n}(x)+f_{n}(x)-f_{n}(x_{0})+f_{n}(x_{0})-% f(x_{0})\Big|\leq\Big|f(x)-f_{n}(x)\Big|+\Big|f_{n}(x)-f_{n}(x_{0})\Big|+\Big|% f_{n}(x_{0})-f(x_{0})\Big|
  105. f n ( x ) f_{n}(x)
  106. f ( x ) f(x)
  107. ε > 0 \varepsilon>0
  108. N ( ε ) N(\varepsilon)
  109. n > N ( ε ) n>N(\varepsilon)
  110. x I x\in I
  111. | f ( x ) - f n ( x ) | < ε 3 , \Big|f(x)-f_{n}(x)\Big|<\frac{\varepsilon}{3},
  112. | f n ( x 0 ) - f ( x 0 ) | < ε 3 . \Big|f_{n}(x_{0})-f(x_{0})\Big|<\frac{\varepsilon}{3}.
  113. f n ( x ) f_{n}(x)
  114. x 0 x_{0}
  115. ε > 0 \varepsilon>0
  116. δ ( ε ) > 0 \delta(\varepsilon)>0
  117. x x
  118. | x - x 0 | < δ ( ε ) |x-x_{0}|<\delta(\varepsilon)
  119. | f n ( x ) - f n ( x 0 ) | < ε 3 . \Big|f_{n}(x)-f_{n}(x_{0})\Big|<\frac{\varepsilon}{3}.
  120. ε > 0 \varepsilon>0
  121. δ ( ε ) > 0 \delta(\varepsilon)>0
  122. n > N ( ε ) n>N(\varepsilon)
  123. x I x\in I
  124. | x - x 0 | < δ ( ε ) |x-x_{0}|<\delta(\varepsilon)
  125. | f ( x ) - f ( x 0 ) | | f ( x ) - f n ( x ) | + | f n ( x ) - f n ( x 0 ) | + | f n ( x 0 ) - f ( x 0 ) | < ε 3 + ε 3 + ε 3 = ε . \Big|f(x)-f(x_{0})\Big|\leq\Big|f(x)-f_{n}(x)\Big|+\Big|f_{n}(x)-f_{n}(x_{0})% \Big|+\Big|f_{n}(x_{0})-f(x_{0})\Big|<\frac{\varepsilon}{3}+\frac{\varepsilon}% {3}+\frac{\varepsilon}{3}=\varepsilon.
  126. ε > 0 \varepsilon>0
  127. δ ( ε ) > 0 \delta(\varepsilon)>0
  128. x I x\in I
  129. | x - x 0 | < δ ( ε ) |x-x_{0}|<\delta(\varepsilon)
  130. | f ( x ) - f ( x 0 ) | < ε . \Big|f(x)-f(x_{0})\Big|<\varepsilon.
  131. f f
  132. f f
  133. x 0 I x_{0}\in I
  134. f f
  135. I I
  136. S \scriptstyle S
  137. f n \scriptstyle f_{n}
  138. f \scriptstyle f
  139. f \scriptstyle f
  140. f n \scriptstyle f_{n}
  141. f n ( x ) = 1 n sin ( n x ) \scriptstyle f_{n}(x)=\frac{1}{n}\sin(nx)
  142. f n \scriptstyle{f_{n}}
  143. [ a , b ] \scriptstyle[a,b]
  144. f n ( x 0 ) \scriptstyle{f_{n}(x_{0})}
  145. x 0 \scriptstyle x_{0}
  146. [ a , b ] \scriptstyle[a,b]
  147. f n \scriptstyle f^{\prime}_{n}
  148. [ a , b ] \scriptstyle[a,b]
  149. f n \scriptstyle{f_{n}}
  150. f \scriptstyle f
  151. f ( x ) = lim n f n ( x ) \scriptstyle f^{\prime}(x)=\lim_{n\to\infty}f^{\prime}_{n}(x)
  152. x [ a , b ] \scriptstyle x\in[a,b]
  153. ( f n ) n = 1 (f_{n})_{n=1}^{\infty}
  154. I I
  155. f f
  156. f f
  157. f n f_{n}
  158. I f = lim n I f n . \int_{I}f=\lim_{n\to\infty}\int_{I}f_{n}.
  159. f n f_{n}
  160. ε ε
  161. f n f_{n}
  162. ε | I | \varepsilon|I|
  163. f f
  164. n = 1 f n \textstyle\sum_{n=1}^{\infty}f_{n}
  165. n = 1 | f n | \textstyle\sum_{n=1}^{\infty}|f_{n}|
  166. n = 1 f n \textstyle\sum_{n=1}^{\infty}f_{n}
  167. [ a , b ] [a,b]
  168. [ a , b ] [a,b]
  169. n = 1 f n \textstyle\sum_{n=1}^{\infty}f_{n}
  170. [ a , b ] [a,b]
  171. [ a , b ] [a,b]
  172. ( f n ) (f_{n})
  173. δ > 0 \delta>0
  174. E δ E_{\delta}
  175. δ \delta
  176. ( f n ) (f_{n})
  177. E E δ E\setminus E_{\delta}

Uniform_space.html

  1. U X × X U\subseteq X\times X
  2. U Φ U\in\Phi
  3. Δ U \Delta\subseteq U
  4. Δ = { ( x , x ) : x X } \Delta=\{(x,x):x\in X\}
  5. X × X X\times X
  6. U Φ U\in\Phi
  7. U V U\subseteq V
  8. V X × X V\subseteq X\times X
  9. V Φ V\in\Phi
  10. U Φ U\in\Phi
  11. V Φ V\in\Phi
  12. U V Φ U\cap V\in\Phi
  13. U Φ U\in\Phi
  14. V Φ V\in\Phi
  15. V V U V\circ V\subseteq U
  16. V V V\circ V
  17. V V
  18. V V
  19. U U
  20. X × X X\times X
  21. V U = { ( x , z ) : y X : ( x , y ) U ( y , z ) V } V\circ U=\{(x,z):\exists y\in X:(x,y)\in U\wedge(y,z)\in V\}
  22. U Φ U\in\Phi
  23. U - 1 Φ U^{-1}\in\Phi
  24. U - 1 = { ( y , x ) : ( x , y ) U } U^{-1}=\{(y,x):(x,y)\in U\}
  25. U a = { ( x , y ) X × X : d ( x , y ) a } where a > 0 U_{a}=\{(x,y)\in X\times X:d(x,y)\leq a\}\quad\,\text{where}\quad a>0
  26. U a d - 1 ( [ 0 , a ] ) = { ( m , n ) M × M : d ( m , n ) a } . \qquad U_{a}\triangleq d^{-1}([0,a])=\{(m,n)\in M\times M:d(m,n)\leq a\}.
  27. U ~ = { ( s , t ) G / H × G / H : t U s } \tilde{U}=\{(s,t)\in G/H\times G/H:\ \ t\in U\cdot s\}

Union_(set_theory).html

  1. A B = { x : x A or x B } A\cup B=\{x:x\in A\,\text{ or }x\in B\}
  2. A B = { 2 , 3 , 4 , 5 , 6 , } A\cup B=\{2,3,4,5,6,\dots\}
  3. x 𝐌 A 𝐌 , x A . x\in\bigcup\mathbf{M}\iff\exists A\in\mathbf{M},\ x\in A.
  4. S 1 , S 2 , S 3 , , S n S_{1},S_{2},S_{3},\dots,S_{n}\,\!
  5. S 1 S 2 S 3 S n S_{1}\cup S_{2}\cup S_{3}\cup\dots\cup S_{n}
  6. 𝐌 \bigcup\mathbf{M}
  7. A 𝐌 A \bigcup_{A\in\mathbf{M}}A
  8. i I A i \bigcup_{i\in I}A_{i}
  9. { A i : i I } \left\{A_{i}:i\in I\right\}
  10. A i A_{i}
  11. i I i\in I
  12. i = 1 A i \bigcup_{i=1}^{\infty}A_{i}
  13. A ( B C ) = ( A B ) ( A C ) A\cap(B\cup C)=(A\cap B)\cup(A\cap C)
  14. A ( B C ) = ( A B ) ( A C ) A\cup(B\cap C)=(A\cup B)\cap(A\cup C)
  15. A B = ( A C B C ) C A\cup B=\left(A^{C}\cap B^{C}\right)^{C}
  16. i I ( j J A i , j ) j J ( i I A i , j ) \bigcup_{i\in I}\bigg(\bigcap_{j\in J}A_{i,j}\bigg)\subseteq\bigcap_{j\in J}% \bigg(\bigcup_{i\in I}A_{i,j}\bigg)

Unique_factorization_domain.html

  1. R R
  2. R [ X , Y , Z , W ] / ( X Y - Z W ) R[X,Y,Z,W]/(XY-ZW)
  3. X Y XY
  4. Z W ZW
  5. X Y XY
  6. Z W ZW
  7. sin π z = π z n = 1 ( 1 - z 2 n 2 ) . \sin\pi z=\pi z\prod_{n=1}^{\infty}\left(1-{{z^{2}}\over{n^{2}}}\right).
  8. [ - 5 ] \mathbb{Z}[\sqrt{-5}]
  9. a + b - 5 a+b\sqrt{-5}
  10. ( 1 + - 5 ) ( 1 - - 5 ) \left(1+\sqrt{-5}\right)\left(1-\sqrt{-5}\right)
  11. 1 + - 5 1+\sqrt{-5}
  12. 1 - - 5 1-\sqrt{-5}
  13. z K [ x , y , z ] / ( z 2 - x y ) z\in K[x,y,z]/(z^{2}-xy)
  14. S - 1 A S^{-1}A

Unit_interval.html

  1. [ 0 , 1 ] [0,1]
  2. I I
  3. ( 0 , 1 ] (0,1]
  4. [ 0 , 1 ) [0,1)
  5. ( 0 , 1 ) (0,1)
  6. I I
  7. [ 0 , 1 ] [0,1]
  8. \mathbb{R}
  9. n \mathbb{R}^{n}
  10. 1 - x 1-x
  11. x y xy
  12. 1 - ( 1 - x ) ( 1 - y ) 1-(1-x)(1-y)

Unit_of_alcohol.html

  1. 568 ml × 4 1000 < m t p l 2.3 units \frac{568\mbox{ ml}~{}\times 4}{1000}<mtpl>{{=}}2.3\mbox{ units}~{}
  2. < v a r > m l ÷ 1000 <var>ml÷1000

United_States_customary_units.html

  1. 33 / 50 {33}/{50}
  2. 1200 / 3937 {1200}/{3937}
  3. 1200 / 3937 {1200}/{3937}
  4. 1 / 8 {1}/{8}
  5. 1 / 2 {1}/{2}
  6. 2 / 3 {2}/{3}
  7. 1 / 16 {1}/{16}
  8. 1 / 32 {1}/{32}
  9. 1 / 128 {1}/{128}
  10. 1 / 6 {1}/{6}
  11. 1 / 2 {1}/{2}
  12. 1 / 128 {1}/{128}
  13. 1 / 7000 {1}/{7000}
  14. 2711 / 32 27{11}/{32}
  15. 1 / 7000 {1}/{7000}
  16. 1 / 5760 {1}/{5760}
  17. F = 9 5 C + 32 F=\frac{9}{5}C+32
  18. C = 5 9 ( F - 32 ) . C=\frac{5}{9}(F-32).

Universal_algebra.html

  1. α J x α \textstyle\bigwedge_{\alpha\in J}x_{\alpha}
  2. Ω \Omega
  3. Ω \Omega
  4. \forall
  5. \exists
  6. Ω \Omega

Universal_property.html

  1. ( X U ) (X\downarrow U)
  2. ( U X ) (U\downarrow X)
  3. X i A i X_{i}\mapsto A_{i}
  4. h g h\mapsto g

Universal_Time.html

  1. U T 2 = U T 1 + 0.022 sin ( 2 π t ) - 0.012 cos ( 2 π t ) - 0.006 sin ( 4 π t ) + 0.007 cos ( 4 π t ) seconds UT2=UT1+0.022\cdot\sin(2\pi t)-0.012\cdot\cos(2\pi t)-0.006\cdot\sin(4\pi t)+0% .007\cdot\cos(4\pi t)\;\mbox{seconds}~{}

Universe.html

  1. ( x , y , z , t ) (x,y,z,t)
  2. × 10 2 3 \times 10^{2}3
  3. 𝐕 = 1 2 π 2 R 4 \mathbf{V}=\begin{matrix}\frac{1}{2}\end{matrix}\pi^{2}R^{4}
  4. d s 2 = - c 2 d t 2 + R ( t ) 2 ( d r 2 1 - k r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 ) ds^{2}=-c^{2}dt^{2}+R(t)^{2}\left(\frac{dr^{2}}{1-kr^{2}}+r^{2}d\theta^{2}+r^{% 2}\sin^{2}\theta\,d\phi^{2}\right)

Unknot.html

  1. Δ ( t ) = 1 , ( z ) = 1 , V ( q ) = 1. \Delta(t)=1,\quad\nabla(z)=1,\quad V(q)=1.

Ununennium.html

  1. Es 99 254 + 20 48 Ca 119 302 Uue * \,{}^{254}_{99}\mathrm{Es}+\,^{48}_{20}\mathrm{Ca}\to\,^{302}_{119}\mathrm{Uue% }^{*}
  2. Bk 97 249 + 22 50 Ti 119 296 Uue + 3 0 1 n \,{}^{249}_{97}\mathrm{Bk}+\,^{50}_{22}\mathrm{Ti}\to\,^{296}_{119}\mathrm{Uue% }\,+3\,^{1}_{0}\mathrm{n}
  3. Bk 97 249 + 22 50 Ti 119 295 Uue + 4 0 1 n \,{}^{249}_{97}\mathrm{Bk}+\,^{50}_{22}\mathrm{Ti}\to\,^{295}_{119}\mathrm{Uue% }\,+4\,^{1}_{0}\mathrm{n}

Up_to.html

  1. f ( x ) f(x)
  2. f ( x ) f(x)
  3. 3709440 8 ! \tfrac{3709440}{8!}

Uranium.html

  1. × 10 1 3 \times 10^{1}3
  2. U 92 238 + n U 92 239 + γ β - Np 93 239 β - Pu 94 239 {}_{\ 92}^{238}\mathrm{U}+\mathrm{n}\rightarrow{}_{\ 92}^{239}\mathrm{U}+% \gamma\xrightarrow{\beta^{-}}{}_{\ 93}^{239}\mathrm{Np}\xrightarrow{\beta^{-}}% {}_{\ 94}^{239}\mathrm{Pu}
  3. × 10 9 \times 10^{9}
  4. × 10 1 7 \times 10^{1}7
  5. × 10 1 3 \times 10^{1}3
  6. × 10 9 \times 10^{9}
  7. × 10 8 \times 10^{8}
  8. × 10 5 \times 10^{5}

Usability_testing.html

  1. U = 1 - ( 1 - p ) n U=1-(1-p)^{n}

Utility.html

  1. u ( x ) u(x)
  2. u ( x ) 3 u(x)^{3}
  3. u ( x ) u(x)
  4. u ( x ) 3 u(x)^{3}
  5. u : X \R u\colon X\rightarrow\R
  6. u ( x ) u ( y ) u(x)\geq u(y)
  7. \R + L \R^{L}_{+}
  8. x \R + L x\in\R^{L}_{+}
  9. X = \R + 2 X=\R^{2}_{+}
  10. u : X \R u\colon X\rightarrow\R
  11. \preceq
  12. x , y X x,y\in X
  13. u ( x ) u ( y ) u(x)\leq u(y)
  14. x y x\preceq y
  15. \preceq
  16. \preceq
  17. L = p A + ( 1 - p ) B L=pA+(1-p)B
  18. L = i p i A i , L=\sum_{i}p_{i}A_{i},
  19. i p i = 1 \textstyle\sum_{i}p_{i}=1
  20. B A B\preceq A
  21. L L
  22. M M
  23. L M L\preceq M
  24. M L M\preceq L
  25. L , M , N L,M,N
  26. L M L\preceq M
  27. M N M\preceq N
  28. L N L\preceq N
  29. L M N L\preceq M\preceq N
  30. p p
  31. p L + ( 1 - p ) N pL+(1-p)N
  32. M M
  33. L , M , N L,M,N
  34. L M L\preceq M
  35. p L + ( 1 - p ) N p M + ( 1 - p ) N pL+(1-p)N\preceq pM+(1-p)N
  36. L L
  37. N N
  38. M M
  39. N , N,
  40. L M L\preceq M
  41. u : X \R u\colon X\rightarrow\R
  42. L 2 L_{2}
  43. L 1 L_{1}
  44. L 2 L_{2}
  45. L 1 L_{1}
  46. L 1 L 2 iff u ( L 1 ) u ( L 2 ) L_{1}\preceq L_{2}\,\text{ iff }u(L_{1})\leq u(L_{2})
  47. u u

Vacuous_truth.html

  1. S S
  2. P Q P\Rightarrow Q
  3. P P
  4. x : P ( x ) Q ( x ) \forall x:P(x)\Rightarrow Q(x)
  5. x : ¬ P ( x ) \forall x:\neg P(x)
  6. x A : Q ( x ) \forall x\in A:Q(x)
  7. A A
  8. ξ : Q ( ξ ) \forall\xi:Q(\xi)
  9. ξ \xi

Vacuum.html

  1. s y m b o l D ( s y m b o l r , t ) = ε 0 s y m b o l E ( s y m b o l r , t ) symbolD(symbolr,\ t)=\varepsilon_{0}symbolE(symbolr,\ t)\,
  2. s y m b o l H ( s y m b o l r , t ) = 1 μ 0 s y m b o l B ( s y m b o l r , t ) symbolH(symbolr,\ t)=\frac{1}{\mu_{0}}symbolB(symbolr,\ t)\,

Vacuum_tube.html

  1. g m = μ R p gm={\mu\over R_{p}}

Vapor_pressure.html

  1. log P = A - B C + T \log P=A-\frac{B}{C+T}
  2. T = B A - log P - C T=\frac{B}{A-\log P}-C
  3. P P
  4. T T
  5. A A
  6. B B
  7. C C
  8. log \log
  9. log 10 \log_{10}
  10. log e \log_{e}
  11. log P = A - B T \log P=A-\frac{B}{T}
  12. T = B A - log P T=\frac{B}{A-\log P}
  13. p tot = i p i χ i p\text{tot}=\sum_{i}p_{i}\chi_{i}\,
  14. ln P s o l i d S = ln P l i q u i d S - Δ H m R ( 1 T - 1 T m ) \ln\,P^{S}_{solid}=\ln\,P^{S}_{liquid}-\frac{\Delta H_{m}}{R}\left(\frac{1}{T}% -\frac{1}{T_{m}}\right)
  15. P s o l i d S P^{S}_{solid}
  16. T < T m T<T_{m}
  17. P l i q u i d S P^{S}_{liquid}
  18. T < T m T<T_{m}
  19. Δ H m \Delta H_{m}
  20. R R
  21. T T
  22. T m T_{m}
  23. log 10 P = 8.07131 - 1730.63 233.426 + T b \log_{10}P=8.07131-\frac{1730.63}{233.426+T_{b}}
  24. T b = 1730.63 8.07131 - log 10 P - 233.426 T_{b}=\frac{1730.63}{8.07131-\log_{10}P}-233.426
  25. T b T_{b}
  26. P P

Variance.html

  1. Var ( X ) = E [ ( X - μ ) 2 ] . \operatorname{Var}(X)=\operatorname{E}\left[(X-\mu)^{2}\right].
  2. Var ( X ) = Cov ( X , X ) . \operatorname{Var}(X)=\operatorname{Cov}(X,X).
  3. σ X 2 \scriptstyle\sigma_{X}^{2}
  4. Var ( X ) \displaystyle\operatorname{Var}(X)
  5. Var ( X ) = σ 2 = ( x - μ ) 2 f ( x ) d x = x 2 f ( x ) d x - μ 2 \operatorname{Var}(X)=\sigma^{2}=\int(x-\mu)^{2}\,f(x)\,dx\,=\int x^{2}\,f(x)% \,dx\,-\mu^{2}
  6. μ \mu
  7. μ = x f ( x ) d x \mu=\int x\,f(x)\,dx\,
  8. Var ( X ) = i = 1 n p i ( x i - μ ) 2 , \operatorname{Var}(X)=\sum_{i=1}^{n}p_{i}\cdot(x_{i}-\mu)^{2},
  9. Var ( X ) = i = 1 n p i x i 2 - μ 2 , \operatorname{Var}(X)=\sum_{i=1}^{n}p_{i}x_{i}^{2}-\mu^{2},
  10. μ \mu
  11. μ = i = 1 n p i x i . \mu=\sum_{i=1}^{n}p_{i}\cdot x_{i}.
  12. Var ( X ) = 1 n i = 1 n ( x i - μ ) 2 . \operatorname{Var}(X)=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\mu)^{2}.
  13. μ \mu
  14. μ = 1 n i = 1 n x i \mu=\frac{1}{n}\sum_{i=1}^{n}x_{i}
  15. Var ( X ) = 1 n 2 i = 1 n j = 1 n 1 2 ( x i - x j ) 2 = 1 n 2 i j > i ( x i - x j ) 2 . \operatorname{Var}(X)=\frac{1}{n^{2}}\sum_{i=1}^{n}\sum_{j=1}^{n}\frac{1}{2}(x% _{i}-x_{j})^{2}=\frac{1}{n^{2}}\sum_{i}\sum_{j>i}(x_{i}-x_{j})^{2}.
  16. f ( x ) = 1 2 π σ 2 e - ( x - μ ) 2 2 σ 2 . f(x)=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}.
  17. Var ( X ) = - ( x - μ ) 2 2 π σ 2 e - ( x - μ ) 2 2 σ 2 d x = σ 2 . \operatorname{Var}(X)=\int_{-\infty}^{\infty}\frac{(x-\mu)^{2}}{\sqrt{2\pi% \sigma^{2}}}e^{-\frac{(x-\mu)^{2}}{2\sigma^{2}}}\,dx=\sigma^{2}.
  18. f ( x ) = λ e - λ x , f(x)=\lambda e^{-\lambda x},\,
  19. Var ( X ) = 0 ( x - λ - 1 ) 2 λ e - λ x d x = λ - 2 . \operatorname{Var}(X)=\int_{0}^{\infty}(x-\lambda^{-1})^{2}\,\lambda e^{-% \lambda x}dx=\lambda^{-2}.\,
  20. p ( k ) = λ k k ! e - λ , p(k)=\frac{\lambda^{k}}{k!}e^{-\lambda},
  21. Var ( X ) = k = 0 λ k k ! e - λ ( k - λ ) 2 = λ , \operatorname{Var}(X)=\sum_{k=0}^{\infty}\frac{\lambda^{k}}{k!}e^{-\lambda}(k-% \lambda)^{2}=\lambda,
  22. p ( k ) = ( n k ) p k ( 1 - p ) n - k , p(k)={n\choose k}p^{k}(1-p)^{n-k},
  23. Var ( X ) = k = 0 n ( n k ) p k ( 1 - p ) n - k ( k - n p ) 2 = n p ( 1 - p ) , \operatorname{Var}(X)=\sum_{k=0}^{n}{n\choose k}p^{k}(1-p)^{n-k}(k-np)^{2}=np(% 1-p),
  24. p = 0.5 p=0.5
  25. k k
  26. n n
  27. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  28. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  29. 1 6 \textstyle\frac{1}{6}
  30. i = 1 6 1 6 ( i - 3.5 ) 2 = 1 6 i = 1 6 ( i - 3.5 ) 2 \displaystyle\sum_{i=1}^{6}\tfrac{1}{6}(i-3.5)^{2}=\tfrac{1}{6}\sum_{i=1}^{6}(% i-3.5)^{2}
  31. σ 2 = E ( X 2 ) - ( E ( X ) ) 2 \displaystyle\sigma^{2}=E(X^{2})-(E(X))^{2}
  32. Var ( X ) 0. \operatorname{Var}(X)\geq 0.
  33. P ( X = a ) = 1 Var ( X ) = 0. P(X=a)=1\Leftrightarrow\operatorname{Var}(X)=0.
  34. Var ( X + a ) = Var ( X ) . \operatorname{Var}(X+a)=\operatorname{Var}(X).
  35. Var ( a X ) = a 2 Var ( X ) . \operatorname{Var}(aX)=a^{2}\operatorname{Var}(X).
  36. Var ( a X + b Y ) = a 2 Var ( X ) + b 2 Var ( Y ) + 2 a b Cov ( X , Y ) , \operatorname{Var}(aX+bY)=a^{2}\operatorname{Var}(X)+b^{2}\operatorname{Var}(Y% )+2ab\,\operatorname{Cov}(X,Y),
  37. Var ( a X - b Y ) = a 2 Var ( X ) + b 2 Var ( Y ) - 2 a b Cov ( X , Y ) , \operatorname{Var}(aX-bY)=a^{2}\operatorname{Var}(X)+b^{2}\operatorname{Var}(Y% )-2ab\,\operatorname{Cov}(X,Y),
  38. C o v ( , ) Cov(⋅,⋅)
  39. N N
  40. { X 1 , , X N } \{X_{1},\dots,X_{N}\}
  41. Var ( i = 1 N X i ) = i , j = 1 N Cov ( X i , X j ) = i = 1 N Var ( X i ) + i j Cov ( X i , X j ) . \operatorname{Var}\left(\sum_{i=1}^{N}X_{i}\right)=\sum_{i,j=1}^{N}% \operatorname{Cov}(X_{i},X_{j})=\sum_{i=1}^{N}\operatorname{Var}(X_{i})+\sum_{% i\neq j}\operatorname{Cov}(X_{i},X_{j}).
  42. Var ( i = 1 N a i X i ) \displaystyle\operatorname{Var}\left(\sum_{i=1}^{N}a_{i}X_{i}\right)
  43. X 1 , , X N X_{1},\dots,X_{N}
  44. Cov ( X i , X j ) = 0 , ( i j ) , \operatorname{Cov}(X_{i},X_{j})=0\ ,\ \forall\ (i\neq j),
  45. X 1 , , X N X_{1},\dots,X_{N}
  46. Var ( i = 1 N X i ) = i = 1 N Var ( X i ) . \operatorname{Var}\left(\sum_{i=1}^{N}X_{i}\right)=\sum_{i=1}^{N}\operatorname% {Var}(X_{i}).
  47. X 1 , , X n X_{1},\dots,X_{n}
  48. Var ( i = 1 n X i ) = i = 1 n Var ( X i ) . \operatorname{Var}\Big(\sum_{i=1}^{n}X_{i}\Big)=\sum_{i=1}^{n}\operatorname{% Var}(X_{i}).
  49. Var ( X ¯ ) = Var ( 1 n i = 1 n X i ) = 1 n 2 i = 1 n Var ( X i ) = σ 2 n . \operatorname{Var}\left(\overline{X}\right)=\operatorname{Var}\left(\frac{1}{n% }\sum_{i=1}^{n}X_{i}\right)=\frac{1}{n^{2}}\sum_{i=1}^{n}\operatorname{Var}% \left(X_{i}\right)=\frac{\sigma^{2}}{n}.
  50. Var ( X Y ) = [ E ( X ) ] 2 Var ( Y ) + [ E ( Y ) ] 2 Var ( X ) + Var ( X ) Var ( Y ) = E ( X 2 ) E ( Y 2 ) - [ E ( X ) ] 2 [ E ( Y ) ] 2 . \begin{aligned}\displaystyle\operatorname{Var}(XY)&\displaystyle=[E(X)]^{2}% \operatorname{Var}(Y)+[E(Y)]^{2}\operatorname{Var}(X)+\operatorname{Var}(X)% \operatorname{Var}(Y)\\ &\displaystyle=E(X^{2})E(Y^{2})-[E(X)]^{2}[E(Y)]^{2}.\end{aligned}
  51. Var ( i = 1 n X i ) = i = 1 n j = 1 n Cov ( X i , X j ) = i = 1 n Var ( X i ) + 2 1 i < j n Cov ( X i , X j ) . \operatorname{Var}\left(\sum_{i=1}^{n}X_{i}\right)=\sum_{i=1}^{n}\sum_{j=1}^{n% }\operatorname{Cov}(X_{i},X_{j})=\sum_{i=1}^{n}\operatorname{Var}(X_{i})+2\sum% _{1\leq i}\sum_{<j\leq n}\operatorname{Cov}(X_{i},X_{j}).
  52. C o v ( , ) Cov(⋅,⋅)
  53. Var ( X ¯ ) = σ 2 n + n - 1 n ρ σ 2 . \operatorname{Var}(\overline{X})=\frac{\sigma^{2}}{n}+\frac{n-1}{n}\rho\sigma^% {2}.
  54. Var ( X ¯ ) = 1 n + n - 1 n ρ . \operatorname{Var}(\overline{X})=\frac{1}{n}+\frac{n-1}{n}\rho.
  55. lim n Var ( X ¯ ) = ρ . \lim_{n\to\infty}\operatorname{Var}(\overline{X})=\rho.
  56. C o v ( a X , b Y ) = a b C o v ( X , Y ) Cov(aX,bY)=abCov(X,Y)
  57. Var ( a X + b Y ) = a 2 Var ( X ) + b 2 Var ( Y ) + 2 a b Cov ( X , Y ) . \operatorname{Var}(aX+bY)=a^{2}\operatorname{Var}(X)+b^{2}\operatorname{Var}(Y% )+2ab\,\operatorname{Cov}(X,Y).
  58. Var ( i n a i X i ) = i = 1 n a i 2 Var ( X i ) + 2 1 i < j n a i a j Cov ( X i , X j ) \operatorname{Var}\left(\sum_{i}^{n}a_{i}X_{i}\right)=\sum_{i=1}^{n}a_{i}^{2}% \operatorname{Var}(X_{i})+2\sum_{1\leq i}\sum_{<j\leq n}a_{i}a_{j}% \operatorname{Cov}(X_{i},X_{j})
  59. X X
  60. Y Y
  61. X X
  62. Var ( X ) = Var ( E ( X | Y ) ) + E ( Var ( X | Y ) ) . \operatorname{Var}(X)=\operatorname{Var}(\operatorname{E}(X|Y))+\operatorname{% E}(\operatorname{Var}(X|Y)).
  63. E ( X | Y ) \operatorname{E}(X|Y)
  64. X X
  65. Y Y
  66. Var ( X | Y ) \operatorname{Var}(X|Y)
  67. X X
  68. Y Y
  69. Y Y
  70. X X
  71. E ( X | Y ) \operatorname{E}(X|Y)
  72. Var ( X | Y ) \operatorname{Var}(X|Y)
  73. Var ( X ) \operatorname{Var}(X)
  74. Y Y
  75. 𝑀𝑆 total = 𝑀𝑆 between + 𝑀𝑆 within ; \mathit{MS}\text{total}=\mathit{MS}\text{between}+\mathit{MS}\text{within};
  76. 𝑀𝑆 \mathit{MS}
  77. 𝑀𝑆 total = 𝑀𝑆 regression + 𝑀𝑆 residual . \mathit{MS}\text{total}=\mathit{MS}\text{regression}+\mathit{MS}\text{residual}.
  78. 𝑆𝑆 \mathit{SS}
  79. 𝑆𝑆 total = 𝑆𝑆 between + 𝑆𝑆 within , \mathit{SS}\text{total}=\mathit{SS}\text{between}+\mathit{SS}\text{within},
  80. 𝑆𝑆 total = 𝑆𝑆 regression + 𝑆𝑆 residual . \mathit{SS}\text{total}=\mathit{SS}\text{regression}+\mathit{SS}\text{residual}.
  81. Var ( X ) = E ( X 2 ) - ( E ( X ) ) 2 . \operatorname{Var}(X)=\operatorname{E}(X^{2})-(\operatorname{E}(X))^{2}.
  82. 2 0 u ( 1 - F ( u ) ) d u - ( 0 1 - F ( u ) d u ) 2 . 2\int_{0}^{\infty}u(1-F(u))\,du-\Big(\int_{0}^{\infty}1-F(u)\,du\Big)^{2}.
  83. argmin m E ( ( X - m ) 2 ) = E ( X ) \mathrm{argmin}_{m}\,\mathrm{E}((X-m)^{2})=\mathrm{E}(X)\,
  84. φ \varphi
  85. argmin m E ( φ ( X - m ) ) = E ( X ) \mathrm{argmin}_{m}\,\mathrm{E}(\varphi(X-m))=\mathrm{E}(X)\,
  86. φ ( x ) = a x 2 + b \varphi(x)=ax^{2}+b
  87. X X
  88. n n
  89. X 1 , , X n X_{1},\ldots,X_{n}
  90. c c
  91. n n
  92. c 1 , , c n c_{1},\ldots,c_{n}
  93. c T X c^{T}X
  94. c T c^{T}
  95. c c
  96. Σ \Sigma
  97. X X
  98. c T X c^{T}X
  99. Var ( c T X ) = c T Σ c . \operatorname{Var}(c^{T}X)=c^{T}\Sigma c.
  100. Var [ f ( X ) ] ( f ( E [ X ] ) ) 2 Var [ X ] \operatorname{Var}\left[f(X)\right]\approx\left(f^{\prime}(\operatorname{E}% \left[X\right])\right)^{2}\operatorname{Var}\left[X\right]
  101. σ 2 = 1 N i = 1 N ( x i - μ ) 2 = ( 1 N i = 1 N x i 2 ) - μ 2 \sigma^{2}=\frac{1}{N}\sum_{i=1}^{N}\left(x_{i}-\mu\right)^{2}=\left(\frac{1}{% N}\sum_{i=1}^{N}x_{i}^{2}\right)-\mu^{2}
  102. μ = 1 N i = 1 N x i \mu=\frac{1}{N}\sum_{i=1}^{N}x_{i}
  103. σ y 2 = 1 n i = 1 n ( y i - y ¯ ) 2 = ( 1 n i = 1 n y i 2 ) - y ¯ 2 \sigma_{y}^{2}=\frac{1}{n}\sum_{i=1}^{n}\left(y_{i}-\overline{y}\right)^{2}=% \left(\frac{1}{n}\sum_{i=1}^{n}y_{i}^{2}\right)-\overline{y}^{2}
  104. y ¯ \overline{y}
  105. y ¯ = 1 n i = 1 n y i . \overline{y}=\frac{1}{n}\sum_{i=1}^{n}y_{i}.
  106. y ¯ \scriptstyle\overline{y}
  107. σ y 2 \scriptstyle\sigma_{y}^{2}
  108. σ y 2 \scriptstyle\sigma_{y}^{2}
  109. E [ σ y 2 ] \displaystyle E[\sigma_{y}^{2}]
  110. σ y 2 \scriptstyle\sigma_{y}^{2}
  111. n - 1 n \frac{n-1}{n}
  112. σ y 2 \scriptstyle\sigma_{y}^{2}
  113. s 2 = n n - 1 σ y 2 = n n - 1 ( 1 n i = 1 n ( y i - y ¯ ) 2 ) = 1 n - 1 i = 1 n ( y i - y ¯ ) 2 s^{2}=\frac{n}{n-1}\sigma_{y}^{2}=\frac{n}{n-1}\left(\frac{1}{n}\sum_{i=1}^{n}% \left(y_{i}-\overline{y}\right)^{2}\right)=\frac{1}{n-1}\sum_{i=1}^{n}\left(y_% {i}-\overline{y}\right)^{2}
  114. ( n - 1 ) s 2 σ 2 χ n - 1 2 . (n-1)\frac{s^{2}}{\sigma^{2}}\sim\chi^{2}_{n-1}.
  115. E ( s 2 ) = E ( σ 2 n - 1 χ n - 1 2 ) = σ 2 , \operatorname{E}(s^{2})=\operatorname{E}\left(\frac{\sigma^{2}}{n-1}\chi^{2}_{% n-1}\right)=\sigma^{2},
  116. Var [ s 2 ] = Var ( σ 2 n - 1 χ n - 1 2 ) = σ 4 ( n - 1 ) 2 Var ( χ n - 1 2 ) = 2 σ 4 n - 1 . \operatorname{Var}[s^{2}]=\operatorname{Var}\left(\frac{\sigma^{2}}{n-1}\chi^{% 2}_{n-1}\right)=\frac{\sigma^{4}}{(n-1)^{2}}\operatorname{Var}\left(\chi^{2}_{% n-1}\right)=\frac{2\sigma^{4}}{n-1}.
  117. E [ s 2 ] = σ 2 , Var [ s 2 ] = σ 4 ( 2 n - 1 + κ n ) = 1 n ( μ 4 - n - 3 n - 1 σ 4 ) , \operatorname{E}[s^{2}]=\sigma^{2},\quad\operatorname{Var}[s^{2}]=\sigma^{4}% \left(\frac{2}{n-1}+\frac{\kappa}{n}\right)=\frac{1}{n}\left(\mu_{4}-\frac{n-3% }{n-1}\sigma^{4}\right),
  118. y ¯ ± σ y ( n - 1 ) 1 / 2 . \bar{y}\pm\sigma_{y}(n-1)^{1/2}.
  119. σ y 2 2 y max ( A - H ) , \sigma_{y}^{2}\leq 2y_{\max}(A-H),
  120. σ y 2 \sigma_{y}^{2}
  121. σ y 2 y max ( A - H ) ( y max - A ) y max - H , \sigma_{y}^{2}\leq\frac{y_{\max}(A-H)(y_{\max}-A)}{y_{\max}-H},
  122. σ y 2 y min ( A - H ) ( A - y min ) H - y min , \sigma_{y}^{2}\geq\frac{y_{\min}(A-H)(A-y_{\min})}{H-y_{\min}},
  123. x x
  124. \mathbb{C}
  125. E ( ( x - μ ) ( x - μ ) * ) \operatorname{E}((x-\mu)(x-\mu)^{*})
  126. x * x^{*}
  127. x x
  128. X X
  129. n \mathbb{R}^{n}
  130. E ( ( X - μ ) ( X - μ ) T ) \operatorname{E}((X-\mu)(X-\mu)^{\operatorname{T}})
  131. μ = E ( X ) \mu=\operatorname{E}(X)
  132. X T X^{\operatorname{T}}
  133. X X
  134. X X
  135. n \mathbb{C}^{n}
  136. E ( ( X - μ ) ( X - μ ) ) \operatorname{E}((X-\mu)(X-\mu)^{\dagger})
  137. X X^{\dagger}
  138. X X
  139. σ 1 \sigma_{1}
  140. σ 2 \sigma_{2}
  141. σ 1 2 + σ 2 2 \sqrt{\sigma_{1}^{2}+\sigma_{2}^{2}}
  142. Σ \Sigma
  143. I = n ( 𝟏 3 × 3 tr ( Σ ) - Σ ) . I=n(\mathbf{1}_{3\times 3}\operatorname{tr}(\Sigma)-\Sigma).
  144. Σ = [ 10 0 0 0 0.1 0 0 0 0.1 ] . \Sigma=\begin{bmatrix}10&0&0\\ 0&0.1&0\\ 0&0&0.1\end{bmatrix}.
  145. I = n [ 0.2 0 0 0 10.1 0 0 0 10.1 ] . I=n\begin{bmatrix}0.2&0&0\\ 0&10.1&0\\ 0&0&10.1\end{bmatrix}.

Vector_calculus.html

  1. 3 . \mathbb{R}^{3}.
  2. a v a{v}
  3. v 1 + v 2 {v}_{1}+{v}_{2}
  4. v 1 v 2 {v}_{1}\cdot{v}_{2}
  5. v 1 × v 2 {v}_{1}\times{v}_{2}
  6. v 1 ( v 2 × v 3 ) {v}_{1}\cdot\left({v}_{2}\times{v}_{3}\right)
  7. v 1 × ( v 2 × v 3 ) {v}_{1}\times\left({v}_{2}\times{v}_{3}\right)
  8. ( v 3 × v 2 ) × v 1 \left({v}_{3}\times{v}_{2}\right)\times{v}_{1}
  9. \nabla
  10. grad ( f ) = f \operatorname{grad}(f)=\nabla f
  11. curl ( 𝐅 ) = × 𝐅 \operatorname{curl}(\mathbf{F})=\nabla\times\mathbf{F}
  12. div ( 𝐅 ) = 𝐅 \operatorname{div}(\mathbf{F})=\nabla\cdot\mathbf{F}
  13. 2 𝐅 = ( 𝐅 ) - × ( × 𝐅 ) \nabla^{2}\mathbf{F}=\nabla(\nabla\cdot\mathbf{F})-\nabla\times(\nabla\times% \mathbf{F})
  14. Δ f = 2 f = f \Delta f=\nabla^{2}f=\nabla\cdot\nabla f
  15. f f
  16. F F
  17. L [ 𝐩 𝐪 ] n φ d 𝐫 = φ ( 𝐪 ) - φ ( 𝐩 ) \int_{L[\mathbf{p}\to\mathbf{q}]\subset\mathbb{R}^{n}}\nabla\varphi\cdot d% \mathbf{r}=\varphi\left(\mathbf{q}\right)-\varphi\left(\mathbf{p}\right)
  18. A 2 ( M x - L y ) d 𝐀 = A ( L d x + M d y ) \int\!\!\!\!\int_{A\,\subset\mathbb{R}^{2}}\left(\frac{\partial M}{\partial x}% -\frac{\partial L}{\partial y}\right)\,d\mathbf{A}=\oint_{\partial A}\left(L\,% dx+M\,dy\right)
  19. Σ 3 × 𝐅 d 𝚺 = Σ 𝐅 d 𝐫 \int\!\!\!\!\int_{\Sigma\,\subset\mathbb{R}^{3}}\nabla\times\mathbf{F}\cdot d% \mathbf{\Sigma}=\oint_{\partial\Sigma}\mathbf{F}\cdot d\mathbf{r}
  20. 3 \mathbb{R}^{3}
  21. f ( x , y ) f(x,y)
  22. f ( x , y ) f(x,y)
  23. ( x , y ) (x,y)
  24. ( a , b ) (a,b)
  25. f ( x , y ) f ( a , b ) + f x ( a , b ) ( x - a ) + f y ( a , b ) ( y - b ) . f\left(x,y\right)\approx f\left(a,b\right)+\frac{\partial f}{\partial x}\left(% a,b\right)\left(x-a\right)+\frac{\partial f}{\partial y}\left(a,b\right)\left(% y-b\right).
  26. z = f ( x , y ) z=f(x,y)
  27. ( a , b ) . (a,b).
  28. 3 , \mathbb{R}^{3},
  29. n - 1 n-1
  30. ( n 2 ) = 1 2 n ( n - 1 ) \textstyle{{\left({{n}\atop{2}}\right)}=\frac{1}{2}n(n-1)}
  31. v 1 v 2 = | v 1 × v 2 | = | v 1 | | v 2 | sin θ {v}_{1}\bot\cdot{v}_{2}=\left|{v}_{1}\times{v}_{2}\right|=\left|{v}_{1}\right|% \left|{v}_{2}\right|\sin\theta

Vector_field.html

  1. ( f V ) ( p ) := f ( p ) V ( p ) (fV)(p):=f(p)V(p)\,
  2. ( V + W ) ( p ) := V ( p ) + W ( p ) (V+W)(p):=V(p)+W(p)\,
  3. V x = ( V 1 , x , , V n , x ) V_{x}=(V_{1,x},\dots,V_{n,x})
  4. V i , y = j = 1 n y i x j V j , x . V_{i,y}=\sum_{j=1}^{n}\frac{\partial y_{i}}{\partial x_{j}}V_{j,x}.
  5. p F p\circ F
  6. 𝔛 ( M ) \scriptstyle\mathfrak{X}(M)
  7. V = f = ( f x 1 , f x 2 , f x 3 , , f x n ) . V=\nabla f=\bigg(\frac{\partial f}{\partial x_{1}},\frac{\partial f}{\partial x% _{2}},\frac{\partial f}{\partial x_{3}},\dots,\frac{\partial f}{\partial x_{n}% }\bigg).
  8. γ V ( x ) , d x = γ f ( x ) , d x = f ( γ ( 1 ) ) - f ( γ ( 0 ) ) . \oint_{\gamma}\langle V(x),\mathrm{d}x\rangle=\oint_{\gamma}\langle\nabla f(x)% ,\mathrm{d}x\rangle=f(\gamma(1))-f(\gamma(0)).
  9. \langle
  10. \rangle
  11. V ( T ( p ) ) = T ( V ( p ) ) ( T O ( n , 𝐑 ) ) V(T(p))=T(V(p))\qquad(T\in\mathrm{O}(n,\mathbf{R}))
  12. γ V ( x ) , d x = a b V ( γ ( t ) ) , γ ( t ) d t . \int_{\gamma}\langle V(x),\mathrm{d}x\rangle=\int_{a}^{b}\langle V(\gamma(t)),% \gamma^{\prime}(t)\;\mathrm{d}t\rangle.
  13. div 𝐅 = 𝐅 = F 1 x + F 2 y + F 3 z , \operatorname{div}\mathbf{F}=\nabla\cdot\mathbf{F}=\frac{\partial F_{1}}{% \partial x}+\frac{\partial F_{2}}{\partial y}+\frac{\partial F_{3}}{\partial z},
  14. curl 𝐅 = × 𝐅 = ( F 3 y - F 2 z ) 𝐞 1 - ( F 3 x - F 1 z ) 𝐞 2 + ( F 2 x - F 1 y ) 𝐞 3 . \operatorname{curl}\,\mathbf{F}=\nabla\times\mathbf{F}=\left(\frac{\partial F_% {3}}{\partial y}-\frac{\partial F_{2}}{\partial z}\right)\mathbf{e}_{1}-\left(% \frac{\partial F_{3}}{\partial x}-\frac{\partial F_{1}}{\partial z}\right)% \mathbf{e}_{2}+\left(\frac{\partial F_{2}}{\partial x}-\frac{\partial F_{1}}{% \partial y}\right)\mathbf{e}_{3}.
  15. γ ( t ) = V ( γ ( t ) ) . \gamma^{\prime}(t)=V(\gamma(t))\,.
  16. γ x ( 0 ) = x \gamma_{x}(0)=x\,
  17. γ x ( t ) = V ( γ x ( t ) ) ( t ( - ε , + ε ) 𝐑 ) . \gamma^{\prime}_{x}(t)=V(\gamma_{x}(t))\qquad(t\in(-\varepsilon,+\varepsilon)% \subset\mathbf{R}).
  18. s polar : ( r , θ ) 1 , v polar : ( r , θ ) ( 1 , 0 ) . s_{\mathrm{polar}}:(r,\theta)\mapsto 1,\quad v_{\mathrm{polar}}:(r,\theta)% \mapsto(1,0).
  19. s Euclidean : ( x , y ) 1 , s_{\mathrm{Euclidean}}:(x,y)\mapsto 1,
  20. v Euclidean : ( x , y ) ( cos θ , sin θ ) = ( x < m t p l > x 2 + y 2 , y x 2 + y 2 ) . v_{\mathrm{Euclidean}}:(x,y)\mapsto(\cos\theta,\sin\theta)=\left(\frac{x}{% \sqrt{<}mtpl>{{x^{2}+y^{2}}}},\frac{y}{\sqrt{x^{2}+y^{2}}}\right).
  21. s Euclidean : x 1 , v Euclidean : x 1. s_{\mathrm{Euclidean}}:x\mapsto 1,\quad v_{\mathrm{Euclidean}}:x\mapsto 1.
  22. s unusual : ξ 1 , v unusual : ξ 2 s_{\mathrm{unusual}}:\xi\mapsto 1,\quad v_{\mathrm{unusual}}:\xi\mapsto 2

Vector_processor.html

  1. r / [ ( 1 - f ) * r + f ] r/[(1-f)*r+f]
  2. r = i n f i n i t y r=infinity
  3. 1 / ( 1 - f ) 1/(1-f)

Vector_quantization.html

  1. [ x 1 , x 2 , , x k ] [x_{1},x_{2},...,x_{k}]
  2. [ y 1 , y 2 , , y n ] [y_{1},y_{2},...,y_{n}]

Vector_space.html

  1. 𝐯 \mathbf{v}
  2. 𝐰 \mathbf{w}
  3. 𝐯 + 𝐰 \mathbf{v}+\mathbf{w}
  4. a a
  5. 𝐯 \mathbf{v}
  6. a a
  7. 𝐯 \mathbf{v}
  8. a a
  9. a 𝐯 a\mathbf{v}
  10. a a
  11. a 𝐯 a\mathbf{v}
  12. a = 2 a=2
  13. a 𝐰 a\mathbf{w}
  14. 𝐰 \mathbf{w}
  15. 𝐰 \mathbf{w}
  16. 2 𝐰 2\mathbf{w}
  17. 𝐰 + 𝐰 \mathbf{w}+\mathbf{w}
  18. ( 1 ) 𝐯 = 𝐯 (−1)\mathbf{v}=−\mathbf{v}
  19. 𝐯 \mathbf{v}
  20. 𝐯 + 𝐰 \mathbf{v}+\mathbf{w}
  21. 𝐯 \mathbf{v}
  22. 𝐰 \mathbf{w}
  23. 𝐯 −\mathbf{v}
  24. 2 𝐰 2\mathbf{w}
  25. x x
  26. y y
  27. x x
  28. y y
  29. ( x , y ) (x,y)
  30. a ( x , y ) = ( a x , a y ) a(x,y)=(ax,ay)
  31. F F
  32. V V
  33. V V
  34. F F
  35. 𝐯 \mathbf{v}
  36. 𝐰 \mathbf{w}
  37. 𝐯 + 𝐰 \mathbf{v}+\mathbf{w}
  38. a a
  39. 𝐯 \mathbf{v}
  40. a 𝐯 a\mathbf{v}
  41. V V
  42. 𝐮 \mathbf{u}
  43. 𝐯 \mathbf{v}
  44. 𝐰 \mathbf{w}
  45. V V
  46. a a
  47. b b
  48. F F
  49. 𝐮 + ( 𝐯 + 𝐰 ) = ( 𝐮 + 𝐯 ) + 𝐰 \mathbf{u}+(\mathbf{v}+\mathbf{w})=(\mathbf{u}+\mathbf{v})+\mathbf{w}
  50. 𝐮 + 𝐯 = 𝐯 + 𝐮 \mathbf{u}+\mathbf{v}=\mathbf{v}+\mathbf{u}
  51. 𝟎 V \mathbf{0}∈V
  52. 𝐯 + 𝟎 = 𝐯 \mathbf{v}+\mathbf{0}=\mathbf{v}
  53. 𝐯 V \mathbf{v}∈V
  54. 𝐯 V \mathbf{v}∈V
  55. 𝐯 V −\mathbf{v}∈V
  56. 𝐯 \mathbf{v}
  57. 𝐯 + ( 𝐯 ) = 𝟎 \mathbf{v}+(−\mathbf{v})=\mathbf{0}
  58. a ( b 𝐯 ) = ( a b ) 𝐯 a(b\mathbf{v})=(ab)\mathbf{v}
  59. 1 𝐯 = 𝐯 1\mathbf{v}=\mathbf{v}
  60. 1 1
  61. F F
  62. a ( 𝐮 + 𝐯 ) = a 𝐮 + a 𝐯 a(\mathbf{u}+\mathbf{v})=a\mathbf{u}+a\mathbf{v}
  63. ( a + b ) 𝐯 = a 𝐯 + b 𝐯 (a+b)\mathbf{v}=a\mathbf{v}+b\mathbf{v}
  64. 𝐯 + 𝐰 = 𝐰 + 𝐯 \mathbf{v}+\mathbf{w}=\mathbf{w}+\mathbf{v}
  65. 𝐯 𝐰 = 𝐯 + ( 𝐰 ) \mathbf{v}−\mathbf{w}=\mathbf{v}+(−\mathbf{w})
  66. 𝐯 / a = ( 1 / a ) 𝐯 \mathbf{v}/a=(1/a)\mathbf{v}
  67. F F
  68. 𝐑 \mathbf{R}
  69. F F
  70. F F
  71. F F
  72. 𝐮 + 𝐯 \mathbf{u}+\mathbf{v}
  73. a 𝐯 a\mathbf{v}
  74. V V
  75. a a
  76. F F
  77. 𝐮 \mathbf{u}
  78. 𝐯 \mathbf{v}
  79. V V
  80. F F
  81. f f
  82. F F
  83. a 𝐯 a\mathbf{v}
  84. ( f ( a ) ) ( 𝐯 ) (f(a))(\mathbf{v})
  85. 𝟎 \mathbf{0}
  86. V V
  87. 𝐯 −\mathbf{v}
  88. 𝐯 \mathbf{v}
  89. a 𝐯 a\mathbf{v}
  90. 𝟎 \mathbf{0}
  91. a a
  92. 0
  93. 𝐯 \mathbf{v}
  94. 𝟎 \mathbf{0}
  95. F F
  96. n n
  97. F F
  98. F F
  99. n n
  100. F F
  101. n = 1 n=1
  102. F F
  103. F = 𝐑 F=\mathbf{R}
  104. n = 2 n=2
  105. 𝐂 \mathbf{C}
  106. x + i y x+iy
  107. x x
  108. y y
  109. i i
  110. ( x + i y ) + ( a + i b ) = ( x + a ) + i ( y + b ) (x+iy)+(a+ib)=(x+a)+i(y+b)
  111. c ( x + i y ) = ( c x ) + i ( c y ) c⋅(x+iy)=(c⋅x)+i(c⋅y)
  112. x x
  113. y y
  114. a a
  115. b b
  116. c c
  117. x + i y x+iy
  118. ( x , y ) (x,y)
  119. F F
  120. E E
  121. E E
  122. F F
  123. 𝐑 \mathbf{R}
  124. 𝐐 ( i 5 ) \mathbf{Q}(i\sqrt{5})
  125. 𝐐 \mathbf{Q}
  126. Ω Ω
  127. F F
  128. f f
  129. g g
  130. ( f + g ) (f+g)
  131. ( f + g ) ( w ) = f ( w ) + g ( w ) (f+g)(w)=f(w)+g(w)
  132. Ω Ω
  133. 𝐑 \mathbf{R}
  134. F x x Fxx
  135. F F
  136. a a
  137. + +
  138. 3 b 3b
  139. + +
  140. c c
  141. = 0 =0
  142. 4 a 4a
  143. + +
  144. 2 b 2b
  145. + +
  146. 2 c 2c
  147. = 0 =0
  148. a a
  149. b = a / 2 b=a/2
  150. c = 5 a / 2 c=−5a/2
  151. A 𝐱 = 𝟎 A\mathbf{x}=\mathbf{0}
  152. A = A=
  153. [ 1 3 1 4 2 2 ] \begin{bmatrix}1&3&1\\ 4&2&2\end{bmatrix}
  154. 𝐱 \mathbf{x}
  155. ( a , b , c ) (a,b,c)
  156. A 𝐱 A\mathbf{x}
  157. 𝟎 = ( 0 , 0 ) \mathbf{0}=(0,0)
  158. f ( x ) + 2 f ( x ) + f ( x ) = 0 f′′(x)+2f′(x)+f(x)=0
  159. a a
  160. b b
  161. I I
  162. 𝐯 \mathbf{v}
  163. 𝐯 \mathbf{v}
  164. B B
  165. B B
  166. ...
  167. n n
  168. 1 1
  169. x x
  170. ...
  171. 𝐑 \mathbf{R}
  172. 𝐐 \mathbf{Q}
  173. 𝐐 \mathbf{Q}
  174. 𝐐 \mathbf{Q}
  175. 𝐐 ( α ) \mathbf{Q}(α)
  176. 𝐐 \mathbf{Q}
  177. α α
  178. α α
  179. f : V W f:V→W
  180. g : W V g:W→V
  181. f g : W W f∘g:W→W
  182. g f : V V g∘f:V→V
  183. V V
  184. f : V W f:V→W
  185. d i m V = d i m W dimV=dimW
  186. V V
  187. W W
  188. V V
  189. W W
  190. F F
  191. V V
  192. V V
  193. V V
  194. φ φ
  195. 𝐱 = ( x 1 , x 2 , , x n ) ( j = 1 n a 1 j x j , j = 1 n a 2 j x j , , j = 1 n a m j x j ) \mathbf{x}=(x_{1},x_{2},\cdots,x_{n})\mapsto\left(\sum_{j=1}^{n}a_{1j}x_{j},% \sum_{j=1}^{n}a_{2j}x_{j},\cdots,\sum_{j=1}^{n}a_{mj}x_{j}\right)
  196. \sum
  197. A A
  198. 𝐱 \mathbf{x}
  199. 𝐱 A 𝐱 \mathbf{x}↦A\mathbf{x}
  200. V V
  201. W W
  202. f : V W f:V→W
  203. d e t ( A ) det(A)
  204. A A
  205. f : V V f:V→V
  206. 𝐯 \mathbf{v}
  207. f f
  208. f ( 𝐯 ) f(\mathbf{v})
  209. 𝐯 \mathbf{v}
  210. λ 𝐯 = f ( 𝐯 ) λ\mathbf{v}=f(\mathbf{v})
  211. λ λ
  212. f f
  213. λ λ
  214. 𝐯 \mathbf{v}
  215. f λ · I d f−λ·Id
  216. V V ) V→V)
  217. V V
  218. f f
  219. λ λ
  220. d e t ( f λ · I d ) = 0 det(f−λ·Id)=0
  221. λ λ
  222. f f
  223. F F
  224. F F
  225. F = 𝐂 F=\mathbf{C}
  226. V V
  227. f f
  228. f f
  229. X X
  230. X X
  231. W V W⊂V
  232. a · ( 𝐯 + W ) = ( a · 𝐯 ) + W a·(\mathbf{v}+W)=(a·\mathbf{v})+W
  233. f : V W f:V→W
  234. 𝐱 A 𝐱 \mathbf{x}↦A\mathbf{x}
  235. A 𝐱 = 0 A\mathbf{x}=0
  236. a 0 f + a 1 d f d x + a 2 d 2 f d x 2 + + a n d n f d x n = 0 a_{0}f+a_{1}\frac{df}{dx}+a_{2}\frac{d^{2}f}{dx^{2}}+\cdots+a_{n}\frac{d^{n}f}% {dx^{n}}=0
  237. f D ( f ) = i = 0 n a i d i f d x i f\mapsto D(f)=\sum_{i=0}^{n}a_{i}\frac{d^{i}f}{dx^{i}}
  238. ( f + g ) = f + g (f+g)′=f′+g′
  239. ( c · f ) = c · f (c·f)′=c·f′
  240. c c
  241. D ( f ) = 0 D(f)=0
  242. 𝐑 \mathbf{R}
  243. 𝐂 \mathbf{C}
  244. i I V i \textstyle{\prod_{i\in I}V_{i}}
  245. i I V i \oplus_{i\in I}V_{i}
  246. i I V i \textstyle{\coprod_{i\in I}V_{i}}
  247. V W V⊗W
  248. g : V × W X g:V×W→X
  249. 𝐯 g ( 𝐯 , 𝐰 ) \mathbf{v}↦g(\mathbf{v},\mathbf{w})
  250. V × W V×W
  251. V W V⊗W
  252. ( 𝐯 , 𝐰 ) (\mathbf{v},\mathbf{w})
  253. 𝐯 𝐰 \mathbf{v}⊗\mathbf{w}
  254. g : V × W X g:V×W→X
  255. u ( 𝐯 𝐰 ) = g ( 𝐯 , 𝐰 ) u(\mathbf{v}⊗\mathbf{w})=g(\mathbf{v},\mathbf{w})
  256. | 𝐯 | |\mathbf{v}|
  257. 𝐯 , 𝐰 \langle\mathbf{v},\mathbf{w}\rangle
  258. | 𝐯 | := 𝐯 , 𝐯 |\mathbf{v}|:=\sqrt{\langle\mathbf{v},\mathbf{v}\rangle}
  259. 𝐱 , 𝐲 = 𝐱 𝐲 = x 1 y 1 + + x n y n . \langle\mathbf{x},\mathbf{y}\rangle=\mathbf{x}\cdot\mathbf{y}=x_{1}y_{1}+% \cdots+x_{n}y_{n}.
  260. 𝐱 𝐲 = cos ( ( 𝐱 , 𝐲 ) ) | 𝐱 | | 𝐲 | . \mathbf{x}\cdot\mathbf{y}=\cos\left(\angle(\mathbf{x},\mathbf{y})\right)\cdot|% \mathbf{x}|\cdot|\mathbf{y}|.
  261. 𝐱 , 𝐲 = 0 \langle\mathbf{x},\mathbf{y}\rangle=0
  262. 𝐱 | 𝐲 = x 1 y 1 + x 2 y 2 + x 3 y 3 - x 4 y 4 . \langle\mathbf{x}|\mathbf{y}\rangle=x_{1}y_{1}+x_{2}y_{2}+x_{3}y_{3}-x_{4}y_{4}.
  263. 𝐱 | 𝐱 \langle\mathbf{x}|\mathbf{x}\rangle
  264. 𝐱 = ( 0 , 0 , 0 , 1 ) \mathbf{x}=(0,0,0,1)
  265. 𝐱 + 𝐲 \mathbf{x}+\mathbf{y}
  266. a 𝐱 a\mathbf{x}
  267. i = 0 f i \sum_{i=0}^{\infty}f_{i}
  268. lim n | 𝐯 n - 𝐯 | = 0. \,\text{lim}_{n\rightarrow\infty}|\mathbf{v}_{n}-\mathbf{v}|=0.
  269. V W V→W
  270. V < s u p > V<sup>∗
  271. V 𝐑 V→\mathbf{R}
  272. 𝐂 \mathbf{C}
  273. ( 1 p ) (1≤p≤∞)
  274. | 𝐱 | p := ( i | x i | p ) 1 / p |\mathbf{x}|_{p}:=\left(\sum_{i}|x_{i}|^{p}\right)^{1/p}
  275. p = p=∞
  276. p = 1 p=1
  277. | x n | = sup ( 2 - n , 0 ) = 2 - n 0 |x_{n}|_{\infty}=\sup(2^{-n},0)=2^{-n}\rightarrow 0
  278. | x n | 1 = i = 1 2 n 2 - n = 2 n 2 - n = 1. |x_{n}|_{1}=\sum_{i=1}^{2^{n}}2^{-n}=2^{n}\cdot 2^{-n}=1.
  279. f : Ω 𝐑 f:Ω→\mathbf{R}
  280. | f | p := ( Ω | f ( x ) | p d x ) 1 / p . |f|_{p}:=\left(\int_{\Omega}|f(x)|^{p}\,dx\right)^{1/p}.
  281. x x , y xx,y
  282. x x
  283. y y
  284. x x , y = y y , x xx,y=−−yy,x
  285. x x , y y , z xx,yy,z
  286. x x , y = x y y x xx,y=xy−yx
  287. n n
  288. F F
  289. F F
  290. I ( f ) = Ω f ( x ) d x . I(f)=\int_{\Omega}f(x)\,dx.
  291. p : δ ( f ) = f ( p ) p:δ(f)=f(p)
  292. a 0 2 + m = 1 [ a m cos ( m x ) + b m sin ( m x ) ] . \frac{a_{0}}{2}+\sum_{m=1}^{\infty}\left[a_{m}\cos\left(mx\right)+b_{m}\sin% \left(mx\right)\right].
  293. a m = 1 π 0 2 π f ( t ) cos ( m t ) d t a_{m}=\frac{1}{\pi}\int_{0}^{2\pi}f(t)\cos(mt)\,dt
  294. b m = 1 π 0 2 π f ( t ) sin ( m t ) d t . b_{m}=\frac{1}{\pi}\int_{0}^{2\pi}f(t)\sin(mt)\,dt.
  295. V = 1 V=1
  296. X × V X X×V→X
  297. X × V X×V
  298. U × V U U×V→U
  299. X × V X×V
  300. S < s u p > 1 × 𝐑 S<sup>1×\mathbf{R}
  301. v \vec{v}
  302. b 𝐯 b\mathbf{v}
  303. a b ab
  304. 𝐑 \mathbf{R}
  305. 𝐂 \mathbf{C}
  306. f ( a ) f(a)
  307. V × U V×U

Vega.html

  1. × 10 - 3 \times 10^{-}3
  2. 10 - 0.5 = 0.316 \begin{smallmatrix}10^{-0.5}\ =\ 0.316\end{smallmatrix}
  3. v sp = 10.7 2 + 8.0 2 + 9.7 2 = 16.5 km/s . \begin{smallmatrix}v_{\,\text{sp}}=\sqrt{10.7^{2}+8.0^{2}+9.7^{2}}=16.5\,\text% {km/s}.\end{smallmatrix}
  4. m = M v - 5 ( log 10 π + 1 ) = 4.3. \begin{smallmatrix}m\ =\ M_{v}-5(\log_{10}\pi+1)\ =\ 4.3.\end{smallmatrix}

Venn_diagram.html

  1. A = { 1 , 2 , 5 } A=\{1,\,2,\,5\}
  2. B = { 1 , 6 } B=\{1,\,6\}
  3. C = { 4 , 7 } C=\{4,\,7\}
  4. y i = sin ( 2 i x ) 2 i where 0 i n - 2 and i . y_{i}=\frac{\sin(2^{i}x)}{2i}\,\text{ where }0\leq i\leq n-2\,\text{ and }i\in% \mathbb{N}.
  5. x A x\in A
  6. x B x\in B

Very_high_frequency.html

  1. 1.23 × A f 1.23\times\sqrt{A_{f}}
  2. A f A_{f}
  3. 12.746 × A m \sqrt{12.746\times A_{m}}
  4. A m A_{m}

Vigenère_cipher.html

  1. E E
  2. K K
  3. C i = E K ( M i ) = ( M i + K i ) mod 26 C_{i}=E_{K}(M_{i})=(M_{i}+K_{i})\mod{26}
  4. D D
  5. K K
  6. M i = D K ( C i ) = ( C i - K i ) mod 26 M_{i}=D_{K}(C_{i})=(C_{i}-K_{i})\mod{26}
  7. M = M 1 M n M=M_{1}\dots M_{n}
  8. C = C 1 C n C=C_{1}\dots C_{n}
  9. K = K 1 K n K=K_{1}\dots K_{n}
  10. n / m \lceil n/m\rceil
  11. m m
  12. A = ^ 0 A\widehat{=}0
  13. L = ^ 11 L\widehat{=}11
  14. 11 = ^ L 11\widehat{=}L
  15. 11 = ( 0 + 11 ) mod 26 11=(0+11)\mod{26}
  16. R = ^ 17 R\widehat{=}17
  17. E = ^ 4 E\widehat{=}4
  18. 13 = ^ N 13\widehat{=}N
  19. 13 = ( 17 - 4 ) mod 26 13=(17-4)\mod{26}
  20. κ p \kappa_{p}
  21. κ r \kappa_{r}
  22. κ p - κ r κ o - κ r {\kappa_{p}-\kappa_{r}}\over{\kappa_{o}-\kappa_{r}}
  23. κ o = i = 1 c n i ( n i - 1 ) N ( N - 1 ) \kappa_{o}=\frac{\sum_{i=1}^{c}n_{i}(n_{i}-1)}{N(N-1)}

Vilfredo_Pareto.html

  1. log N = log A + m log x \log N=\log A+m\log x

Virial_theorem.html

  1. T \left\langle T\right\rangle
  2. V TOT \left\langle V\text{TOT}\right\rangle
  3. T = - 1 2 k = 1 N 𝐅 k 𝐫 k \left\langle T\right\rangle=-\frac{1}{2}\,\sum_{k=1}^{N}\left\langle\mathbf{F}% _{k}\cdot\mathbf{r}_{k}\right\rangle
  4. 2 T = n V TOT . 2\langle T\rangle=n\langle V\text{TOT}\rangle.
  5. T \left\langle T\right\rangle
  6. V TOT \left\langle V\text{TOT}\right\rangle
  7. I = k = 1 N m k | 𝐫 k | 2 = k = 1 N m k r k 2 I=\sum_{k=1}^{N}m_{k}|\mathbf{r}_{k}|^{2}=\sum_{k=1}^{N}m_{k}r_{k}^{2}
  8. G = k = 1 N 𝐩 k 𝐫 k G=\sum_{k=1}^{N}\mathbf{p}_{k}\cdot\mathbf{r}_{k}
  9. 1 2 d I d t = 1 2 d d t k = 1 N m k 𝐫 k 𝐫 k = k = 1 N m k d 𝐫 k d t 𝐫 k = k = 1 N 𝐩 k 𝐫 k = G . \frac{1}{2}\frac{dI}{dt}=\frac{1}{2}\frac{d}{dt}\sum_{k=1}^{N}m_{k}\,\mathbf{r% }_{k}\cdot\mathbf{r}_{k}=\sum_{k=1}^{N}m_{k}\,\frac{d\mathbf{r}_{k}}{dt}\cdot% \mathbf{r}_{k}=\sum_{k=1}^{N}\mathbf{p}_{k}\cdot\mathbf{r}_{k}=G\,.
  10. d G d t = k = 1 N 𝐩 k d 𝐫 k d t + k = 1 N d 𝐩 k d t 𝐫 k = k = 1 N m k d 𝐫 k d t d 𝐫 k d t + k = 1 N 𝐅 k 𝐫 k = 2 T + k = 1 N 𝐅 k 𝐫 k \begin{aligned}\displaystyle\frac{dG}{dt}&\displaystyle=\sum_{k=1}^{N}\mathbf{% p}_{k}\cdot\frac{d\mathbf{r}_{k}}{dt}+\sum_{k=1}^{N}\frac{d\mathbf{p}_{k}}{dt}% \cdot\mathbf{r}_{k}\\ &\displaystyle=\sum_{k=1}^{N}m_{k}\frac{d\mathbf{r}_{k}}{dt}\cdot\frac{d% \mathbf{r}_{k}}{dt}+\sum_{k=1}^{N}\mathbf{F}_{k}\cdot\mathbf{r}_{k}\\ &\displaystyle=2T+\sum_{k=1}^{N}\mathbf{F}_{k}\cdot\mathbf{r}_{k}\end{aligned}
  11. 𝐅 k = d 𝐩 k d t \mathbf{F}_{k}=\frac{d\mathbf{p}_{k}}{dt}
  12. T = 1 2 k = 1 N m k v k 2 = 1 2 k = 1 N m k d 𝐫 k d t d 𝐫 k d t . T=\frac{1}{2}\sum_{k=1}^{N}m_{k}v_{k}^{2}=\frac{1}{2}\sum_{k=1}^{N}m_{k}\frac{% d\mathbf{r}_{k}}{dt}\cdot\frac{d\mathbf{r}_{k}}{dt}.
  13. 𝐅 k \mathbf{F}_{k}
  14. 𝐅 k = j = 1 N 𝐅 j k \mathbf{F}_{k}=\sum_{j=1}^{N}\mathbf{F}_{jk}
  15. 𝐅 j k \mathbf{F}_{jk}
  16. - 1 2 k = 1 N 𝐅 k 𝐫 k = - 1 2 k = 1 N j = 1 N 𝐅 j k 𝐫 k . -\frac{1}{2}\,\sum_{k=1}^{N}\mathbf{F}_{k}\cdot\mathbf{r}_{k}=-\frac{1}{2}\,% \sum_{k=1}^{N}\sum_{j=1}^{N}\mathbf{F}_{jk}\cdot\mathbf{r}_{k}.
  17. 𝐅 j k = 0 \mathbf{F}_{jk}=0
  18. j = k j=k
  19. k = 1 N 𝐅 k 𝐫 k = k = 1 N j < k 𝐅 j k 𝐫 k + k = 1 N j > k 𝐅 j k 𝐫 k = k = 1 N j < k 𝐅 j k ( 𝐫 k - 𝐫 j ) . \sum_{k=1}^{N}\mathbf{F}_{k}\cdot\mathbf{r}_{k}=\sum_{k=1}^{N}\sum_{j<k}% \mathbf{F}_{jk}\cdot\mathbf{r}_{k}+\sum_{k=1}^{N}\sum_{j>k}\mathbf{F}_{jk}% \cdot\mathbf{r}_{k}=\sum_{k=1}^{N}\sum_{j<k}\mathbf{F}_{jk}\cdot\left(\mathbf{% r}_{k}-\mathbf{r}_{j}\right).
  20. 𝐅 j k = - 𝐅 k j \mathbf{F}_{jk}=-\mathbf{F}_{kj}
  21. 𝐅 j k = - 𝐫 k V = - d V d r ( 𝐫 k - 𝐫 j r j k ) , \mathbf{F}_{jk}=-\nabla_{\mathbf{r}_{k}}V=-\frac{dV}{dr}\left(\frac{\mathbf{r}% _{k}-\mathbf{r}_{j}}{r_{jk}}\right),
  22. 𝐅 k j = - 𝐫 j V \mathbf{F}_{kj}=-\nabla_{\mathbf{r}_{j}}V
  23. k k
  24. k = 1 N 𝐅 k 𝐫 k = k = 1 N j < k 𝐅 j k ( 𝐫 k - 𝐫 j ) = - k = 1 N j < k d V d r | 𝐫 k - 𝐫 j | 2 r j k = - k = 1 N j < k d V d r r j k . \sum_{k=1}^{N}\mathbf{F}_{k}\cdot\mathbf{r}_{k}=\sum_{k=1}^{N}\sum_{j<k}% \mathbf{F}_{jk}\cdot\left(\mathbf{r}_{k}-\mathbf{r}_{j}\right)=-\sum_{k=1}^{N}% \sum_{j<k}\frac{dV}{dr}\frac{|\mathbf{r}_{k}-\mathbf{r}_{j}|^{2}}{r_{jk}}=-% \sum_{k=1}^{N}\sum_{j<k}\frac{dV}{dr}r_{jk}.
  25. d G d t = 2 T + k = 1 N 𝐅 k 𝐫 k = 2 T - k = 1 N j < k d V d r r j k . \frac{dG}{dt}=2T+\sum_{k=1}^{N}\mathbf{F}_{k}\cdot\mathbf{r}_{k}=2T-\sum_{k=1}% ^{N}\sum_{j<k}\frac{dV}{dr}r_{jk}.
  26. V ( r j k ) = α r j k n , V(r_{jk})=\alpha r_{jk}^{n},
  27. - 1 2 k = 1 N 𝐅 k 𝐫 k = 1 2 k = 1 N j < k d V d r r j k = 1 2 k = 1 N j < k n V ( r j k ) = n 2 V TOT -\frac{1}{2}\,\sum_{k=1}^{N}\mathbf{F}_{k}\cdot\mathbf{r}_{k}=\frac{1}{2}\,% \sum_{k=1}^{N}\sum_{j<k}\frac{dV}{dr}r_{jk}=\frac{1}{2}\,\sum_{k=1}^{N}\sum_{j% <k}nV(r_{jk})=\frac{n}{2}\,V\text{TOT}
  28. V TOT = k = 1 N j < k V ( r j k ) . V\text{TOT}=\sum_{k=1}^{N}\sum_{j<k}V(r_{jk}).
  29. d G d t = 2 T + k = 1 N 𝐅 k 𝐫 k = 2 T - n V TOT . \frac{dG}{dt}=2T+\sum_{k=1}^{N}\mathbf{F}_{k}\cdot\mathbf{r}_{k}=2T-nV\text{% TOT}.
  30. d G d t = 1 2 d 2 I d t 2 = 2 T + V TOT \frac{dG}{dt}=\frac{1}{2}\frac{d^{2}I}{dt^{2}}=2T+V\text{TOT}
  31. d G d t τ = 1 τ 0 τ d G d t d t = 1 τ G ( 0 ) G ( τ ) d G = G ( τ ) - G ( 0 ) τ , \left\langle\frac{dG}{dt}\right\rangle_{\tau}=\frac{1}{\tau}\int_{0}^{\tau}% \frac{dG}{dt}\,dt=\frac{1}{\tau}\int_{G(0)}^{G(\tau)}\,dG=\frac{G(\tau)-G(0)}{% \tau},
  32. d G d t τ = 2 T τ + k = 1 N 𝐅 k 𝐫 k τ . \left\langle\frac{dG}{dt}\right\rangle_{\tau}=2\left\langle T\right\rangle_{% \tau}+\sum_{k=1}^{N}\left\langle\mathbf{F}_{k}\cdot\mathbf{r}_{k}\right\rangle% _{\tau}.
  33. d G / d t τ = 0 \left\langle{dG}/{dt}\right\rangle_{\tau}=0
  34. 2 T τ = - k = 1 N 𝐅 k 𝐫 k τ . 2\left\langle T\right\rangle_{\tau}=-\sum_{k=1}^{N}\left\langle\mathbf{F}_{k}% \cdot\mathbf{r}_{k}\right\rangle_{\tau}.
  35. d G / d t τ = 0 \left\langle{dG}/{dt}\right\rangle_{\tau}=0
  36. lim τ | d G bound d t τ | = lim τ | G ( τ ) - G ( 0 ) τ | lim τ G max - G min τ = 0. \lim_{\tau\rightarrow\infty}\left|\left\langle\frac{dG^{\mathrm{bound}}}{dt}% \right\rangle_{\tau}\right|=\lim_{\tau\rightarrow\infty}\left|\frac{G(\tau)-G(% 0)}{\tau}\right|\leq\lim_{\tau\rightarrow\infty}\frac{G_{\max}-G_{\min}}{\tau}% =0.
  37. T τ = - 1 2 k = 1 N 𝐅 k 𝐫 k τ = n 2 V TOT τ . \langle T\rangle_{\tau}=-\frac{1}{2}\sum_{k=1}^{N}\langle\mathbf{F}_{k}\cdot% \mathbf{r}_{k}\rangle_{\tau}=\frac{n}{2}\langle V\text{TOT}\rangle_{\tau}.
  38. T τ = - 1 2 V TOT τ . \langle T\rangle_{\tau}=-\frac{1}{2}\langle V\text{TOT}\rangle_{\tau}.
  39. H = V ( { X i } ) + n P n 2 / 2 m H=V(\{X_{i}\})+\sum_{n}P_{n}^{2}/2m
  40. X n P n X_{n}P_{n}
  41. X n X_{n}
  42. P n = - i d / d X n P_{n}=-i\hbar d/dX_{n}
  43. n n
  44. [ H , X n P n ] = X n [ H , P n ] + [ H , X n ] P n = i X n d V d X n - i P n 2 m , [H,X_{n}P_{n}]=X_{n}[H,P_{n}]+[H,X_{n}]P_{n}=i\hbar X_{n}\frac{dV}{dX_{n}}-i% \hbar\frac{P_{n}^{2}}{m},
  45. Q = n X n P n Q=\sum_{n}X_{n}P_{n}
  46. i [ H , Q ] = 2 T - n X n d V d X n \frac{i}{\hbar}[H,Q]=2T-\sum_{n}X_{n}\frac{dV}{dX_{n}}
  47. T = n P n 2 / 2 m T=\sum_{n}P_{n}^{2}/2m
  48. - d Q / d t -dQ/dt
  49. d Q / d t \langle dQ/dt\rangle
  50. 2 T = n X n d V d X n . 2\langle T\rangle=\sum_{n}\langle X_{n}\frac{dV}{dX_{n}}\rangle.
  51. T = 1 2 𝐩 𝐯 T=\frac{1}{2}\mathbf{p}\cdot\mathbf{v}
  52. T = ( γ - 1 ) m c 2 T=(\gamma-1)mc^{2}\,
  53. 1 2 𝐩 𝐯 = 1 2 β γ m c β c = 1 2 γ β 2 m c 2 = ( γ β 2 2 ( γ - 1 ) ) T . \begin{aligned}\displaystyle\frac{1}{2}\mathbf{p}\cdot\mathbf{v}&\displaystyle% =\frac{1}{2}\vec{\beta}\gamma mc\cdot\vec{\beta}c=\frac{1}{2}\gamma\beta^{2}mc% ^{2}=\left(\frac{\gamma\beta^{2}}{2(\gamma-1)}\right)T\,.\end{aligned}
  54. ( 1 + 1 - β 2 2 ) T \left(\frac{1+\sqrt{1-\beta^{2}}}{2}\right)T
  55. ( γ + 1 2 γ ) T \left(\frac{\gamma+1}{2\gamma}\right)T
  56. 𝐅 j k = - 𝐅 k j \mathbf{F}_{jk}=-\mathbf{F}_{kj}
  57. N N
  58. n 2 V TOT τ = k = 1 N ( 1 + 1 - β k 2 2 ) T k τ = k = 1 N ( γ k + 1 2 γ k ) T k τ . \frac{n}{2}\langle V_{\mathrm{TOT}}\rangle_{\tau}=\left\langle\sum_{k=1}^{N}% \left(\frac{1+\sqrt{1-\beta_{k}^{2}}}{2}\right)T_{k}\right\rangle_{\tau}=\left% \langle\sum_{k=1}^{N}\left(\frac{\gamma_{k}+1}{2\gamma_{k}}\right)T_{k}\right% \rangle_{\tau}\,.
  59. 2 T TOT n V TOT [ 1 , 2 ] , \frac{2\langle T_{\mathrm{TOT}}\rangle}{n\langle V_{\mathrm{TOT}}\rangle}\in% \left[1,2\right]\,,
  60. 2 lim τ + T τ = lim τ + U τ 2\lim\limits_{\tau\rightarrow+\infty}\langle T\rangle_{\tau}=\lim\limits_{\tau% \rightarrow+\infty}\langle U\rangle_{\tau}
  61. lim τ + τ - 2 I ( τ ) = 0. \lim\limits_{\tau\rightarrow+\infty}{\tau}^{-2}I(\tau)=0.
  62. 1 2 d 2 I d t 2 + V x k G k t d 3 r = 2 ( T + U ) + W E + W M - x k ( p i k + T i k ) d S i , \frac{1}{2}\frac{d^{2}I}{dt^{2}}+\int_{V}x_{k}\frac{\partial G_{k}}{\partial t% }\,d^{3}r=2(T+U)+W^{E}+W^{M}-\int x_{k}(p_{ik}+T_{ik})\,dS_{i},
  63. p i k = Σ n σ m σ v i v k σ - V i V k Σ m σ n σ , p_{ik}=\Sigma n^{\sigma}m^{\sigma}\langle v_{i}v_{k}\rangle^{\sigma}-V_{i}V_{k% }\Sigma m^{\sigma}n^{\sigma},
  64. T i k = ( ε 0 E 2 2 + B 2 2 μ 0 ) δ i k - ( ε 0 E i E k + B i B k μ 0 ) . T_{ik}=\left(\frac{\varepsilon_{0}E^{2}}{2}+\frac{B^{2}}{2\mu_{0}}\right)% \delta_{ik}-\left(\varepsilon_{0}E_{i}E_{k}+\frac{B_{i}B_{k}}{\mu_{0}}\right).
  65. τ R / c s , \tau\,\sim R/c_{s},
  66. 3 5 G M R = 3 2 k B T m p = 1 2 v 2 , \frac{3}{5}\frac{GM}{R}=\frac{3}{2}\frac{k_{B}T}{m_{p}}=\frac{1}{2}v^{2},
  67. M M
  68. R R
  69. v v
  70. T T
  71. G G
  72. k B k_{B}
  73. m p m_{p}
  74. σ \sigma
  75. T = ( 1 / 2 ) v 2 ( 3 / 2 ) σ 2 T=(1/2)v^{2}\sim(3/2)\sigma^{2}
  76. G M R σ 2 . \frac{GM}{R}\approx\sigma^{2}.
  77. R R
  78. M M
  79. G M vir R vir σ max 2 . \frac{GM\text{vir}}{R\text{vir}}\approx\sigma_{\max}^{2}.
  80. ρ crit = 3 H 2 8 π G \rho\text{crit}=\frac{3H^{2}}{8\pi G}
  81. H H
  82. G G
  83. r vir r 200 = r ( ρ = 200 ρ crit ) r\text{vir}\approx r_{200}=r(\rho=200\cdot\rho\text{crit})
  84. M vir M 200 = ( 4 / 3 ) π r 200 3 200 ρ crit M\text{vir}\approx M_{200}=(4/3)\pi r_{200}^{3}\cdot 200\rho\text{crit}

Viscometer.html

  1. F = 6 π r η v F=6\pi r\eta v\,
  2. F F
  3. r r
  4. η \eta
  5. v v
  6. V s = 2 9 r 2 g ( ρ p - ρ f ) μ V_{s}=\frac{2}{9}\frac{r^{2}g(\rho_{p}-\rho_{f})}{\mu}
  7. ρ p > ρ f \rho_{p}>\rho_{f}
  8. ρ p < ρ f \rho_{p}<\rho_{f}
  9. r r
  10. μ \mu
  11. ν = η ρ \nu=\frac{\eta}{\rho}
  12. ν {\nu}
  13. η {\eta}
  14. ρ {\rho}

Visual_binary.html

  1. d d
  2. d = 1 A U tan ( p ) d=\frac{1AU}{\tan(p)}
  3. p p
  4. b = L 4 π d 2 b=\frac{L}{4\pi d^{2}}
  5. b b
  6. L L
  7. L L = ( d 2 b ) ( d 2 b ) \frac{L}{L_{\odot}}=(\frac{d^{2}_{\odot}}{b})(\frac{d^{2}}{b_{\odot}})
  8. \odot
  9. d 2 d^{2}
  10. d 2 = ( L L ) ( b b ) d^{2}=(\frac{L}{L_{\odot}})(\frac{b_{\odot}}{b})
  11. T 2 a 3 T^{2}\propto a^{3}
  12. T T
  13. a a
  14. m 1 m_{1}
  15. m 2 m_{2}
  16. m 1 m_{1}
  17. r 1 r_{1}
  18. v 1 v_{1}
  19. m 2 m_{2}
  20. r 2 r_{2}
  21. v 2 v_{2}
  22. r r
  23. F n e t = F i = m a F_{net}=\sum\,F_{i}=ma
  24. F n e t F_{net}
  25. m m
  26. a a
  27. F = m v 2 r F=\frac{mv^{2}}{r}
  28. v = 2 π r T v=\frac{2\pi r}{T}
  29. F 1 = 4 π 2 m 1 r 1 T 2 F_{1}=\frac{4\pi^{2}m_{1}r_{1}}{T^{2}}
  30. F 2 = 4 π 2 m 2 r 2 T 2 F_{2}=\frac{4\pi^{2}m_{2}r_{2}}{T^{2}}
  31. F 12 = - F 21 F_{12}=-F_{21}
  32. 4 π 2 m 1 r 1 T 2 = 4 π 2 m 2 r 2 T 2 \frac{4\pi^{2}m_{1}r_{1}}{T^{2}}=\frac{4\pi^{2}m_{2}r_{2}}{T^{2}}
  33. r 1 m 1 = r 2 m 2 r_{1}m_{1}=r_{2}m_{2}
  34. r r
  35. r = r 1 + r 2 r=r_{1}+r_{2}
  36. r 1 r_{1}
  37. r 2 r_{2}
  38. r 1 r_{1}
  39. r 2 r_{2}
  40. r 1 = m 2 a ( m 1 + m 2 ) r_{1}=\frac{m_{2}a}{(m_{1}+m_{2})}
  41. F = G m 1 m 2 / a 2 F=Gm_{1}m_{2}/a^{2}
  42. T 2 = 4 π 2 a 3 G ( m 1 + m 2 ) T^{2}=\frac{4\pi^{2}a^{3}}{G(m_{1}+m_{2})}
  43. G G
  44. a = a ′′ p ′′ a=\frac{a^{\prime\prime}}{p^{\prime\prime}}
  45. a ′′ a^{\prime\prime}
  46. p ′′ p^{\prime\prime}
  47. ( m 1 + m 2 ) T 2 = 4 π 2 ( a ′′ / p ′′ ) 3 G (m_{1}+m_{2})T^{2}=\frac{4\pi^{2}(a^{\prime\prime}/p^{\prime\prime})^{3}}{G}
  48. r 1 m 1 = r 2 m 2 r_{1}m_{1}=r_{2}m_{2}
  49. r 1 + r 2 = r r_{1}+r_{2}=r
  50. a 1 ′′ a 2 ′′ = a 1 a 2 \frac{a_{1}^{\prime\prime}}{a_{2}^{\prime\prime}}=\frac{a_{1}}{a_{2}}
  51. a 1 a 2 = m 1 m 2 \frac{a_{1}}{a_{2}}=\frac{m_{1}}{m_{2}}
  52. L L = ( M M ) α \frac{L}{L_{\odot}}=\left(\frac{M}{M_{\odot}}\right)^{\alpha}
  53. α \alpha
  54. L L .23 ( M M ) 2.3 ( M < .43 M ) \frac{L}{L_{\odot}}\approx.23\left(\frac{M}{M_{\odot}}\right)^{2.3}\qquad(M<.4% 3M_{\odot})
  55. L L = ( M M ) 4 ( .43 M < M < 2 M ) \frac{L}{L_{\odot}}=\left(\frac{M}{M_{\odot}}\right)^{4}\qquad\qquad(.43M_{% \odot}<M<2M_{\odot})
  56. L L 1.5 ( M M ) 3.5 ( 2 M < M < 20 M ) \frac{L}{L_{\odot}}\approx 1.5\left(\frac{M}{M_{\odot}}\right)^{3.5}\qquad(2M_% {\odot}<M<20M_{\odot})
  57. L L M M ( M > 20 M ) \frac{L}{L_{\odot}}\varpropto\frac{M}{M_{\odot}}\qquad(M>20M_{\odot})

Volt-ampere_reactive.html

  1. ϕ \phi
  2. Q Q
  3. Q = V rms I rms sin ( ϕ ) Q=V_{\mathrm{rms}}I_{\mathrm{rms}}\sin\left(\phi\right)\,
  4. ϕ \phi

Volt.html

  1. 𝐕 \mathbf{V}
  2. V = Potential Energy charge = N m coulomb = kg m m s 2 A s = kg m 2 A s 3 \mathrm{V}=\frac{\mathrm{Potential\,Energy}}{\mathrm{charge}}=\frac{\mathrm{N}% \cdot\mathrm{m}}{\mathrm{coulomb}}=\frac{\mathrm{kg}\cdot\mathrm{m}\cdot% \mathrm{m}}{\mathrm{s}^{2}\cdot\mathrm{A}\cdot\mathrm{s}}=\frac{\mathrm{kg}% \cdot\mathrm{m}^{2}}{\mathrm{A}\cdot\mathrm{s}^{3}}
  3. V = A Ω = W A = J C \mathrm{V}=\mathrm{A}\cdot\Omega=\frac{\mathrm{W}}{\mathrm{A}}=\frac{\mathrm{J% }}{\mathrm{C}}
  4. K J - 90 = 0.4835979 GHz μ V K_{\mathrm{J-90}}=0.4835979\,\frac{\mathrm{GHz}}{\mathrm{\mu V}}

Voltage.html

  1. V ∆V
  2. U ∆U
  3. Δ V B A = V B - V A = - r 0 B E d l - ( - r 0 A E d l ) \Delta V_{BA}=V_{B}-V_{A}=-\int_{r_{0}}^{B}\vec{E}\cdot d\vec{l}-\left(-\int_{% r_{0}}^{A}\vec{E}\cdot d\vec{l}\right)
  4. = B r 0 E d l + r 0 A E d l = B A E d l =\int_{B}^{r_{0}}\vec{E}\cdot d\vec{l}+\int_{r_{0}}^{A}\vec{E}\cdot d\vec{l}=% \int_{B}^{A}\vec{E}\cdot d\vec{l}

Volume.html

  1. f ( x , y , z ) = 1 f(x,y,z)=1
  2. D 1 d x d y d z . \iiint\limits_{D}1\,dx\,dy\,dz.
  3. D r d r d θ d z , \iiint\limits_{D}r\,dr\,d\theta\,dz,
  4. θ \theta
  5. ϕ \phi
  6. D ρ 2 sin ϕ d ρ d θ d ϕ . \iiint\limits_{D}\rho^{2}\sin\phi\,d\rho\,d\theta\,d\phi.
  7. a 3 a^{3}\;
  8. π r 2 h \pi r^{2}h\;
  9. B h B\cdot h
  10. l w h l\cdot w\cdot h
  11. 1 2 b h l \frac{1}{2}bhl
  12. 4 3 π r 3 \frac{4}{3}\pi r^{3}
  13. 4 3 π a b c \frac{4}{3}\pi abc
  14. ( π r 2 ) ( 2 π R ) = 2 π 2 R r 2 (\pi r^{2})(2\pi R)=2\pi^{2}Rr^{2}
  15. 1 3 B h \frac{1}{3}Bh
  16. 1 3 s 2 h \frac{1}{3}s^{2}h\;
  17. 1 3 l w h \frac{1}{3}lwh
  18. 1 3 π r 2 h \frac{1}{3}\pi r^{2}h
  19. 2 12 a 3 {\sqrt{2}\over 12}a^{3}\,
  20. a a
  21. a b c K abc\sqrt{K}
  22. K = 1 + 2 cos ( α ) cos ( β ) cos ( γ ) - cos 2 ( α ) - cos 2 ( β ) - cos 2 ( γ ) \begin{aligned}\displaystyle K=&\displaystyle 1+2\cos(\alpha)\cos(\beta)\cos(% \gamma)\\ &\displaystyle-\cos^{2}(\alpha)-\cos^{2}(\beta)-\cos^{2}(\gamma)\end{aligned}
  23. a b A ( h ) d h \int_{a}^{b}A(h)\,\mathrm{d}h
  24. π a b ( [ R O ( x ) ] 2 - [ R I ( x ) ] 2 ) d x \pi\int_{a}^{b}\left({\left[R_{O}(x)\right]}^{2}-{\left[R_{I}(x)\right]}^{2}% \right)\mathrm{d}x
  25. R O R_{O}
  26. R I R_{I}
  27. 1 3 π r 2 h = 1 3 π r 2 ( 2 r ) = ( 2 3 π r 3 ) × 1 , \tfrac{1}{3}\pi r^{2}h=\tfrac{1}{3}\pi r^{2}(2r)=(\tfrac{2}{3}\pi r^{3})\times 1,
  28. 4 3 π r 3 = ( 2 3 π r 3 ) × 2 , \tfrac{4}{3}\pi r^{3}=(\tfrac{2}{3}\pi r^{3})\times 2,
  29. π r 2 h = π r 2 ( 2 r ) = ( 2 3 π r 3 ) × 3. \pi r^{2}h=\pi r^{2}(2r)=(\tfrac{2}{3}\pi r^{3})\times 3.
  30. π r 2 \pi r^{2}
  31. y = r 2 - x 2 y=\sqrt{r^{2}-x^{2}}
  32. z = r 2 - x 2 z=\sqrt{r^{2}-x^{2}}
  33. - r r π y 2 d x = - r r π ( r 2 - x 2 ) d x . \int_{-r}^{r}\pi y^{2}\,dx=\int_{-r}^{r}\pi(r^{2}-x^{2})\,dx.
  34. - r r π r 2 d x - - r r π x 2 d x = π ( r 3 + r 3 ) - π 3 ( r 3 + r 3 ) = 2 π r 3 - 2 π r 3 3 . \int_{-r}^{r}\pi r^{2}\,dx-\int_{-r}^{r}\pi x^{2}\,dx=\pi(r^{3}+r^{3})-\frac{% \pi}{3}(r^{3}+r^{3})=2\pi r^{3}-\frac{2\pi r^{3}}{3}.
  35. V = 4 3 π r 3 . V=\frac{4}{3}\pi r^{3}.
  36. 4 π r 2 4\pi r^{2}
  37. 0 r 4 π u 2 d u \int_{0}^{r}4\pi u^{2}\,du
  38. 4 3 π r 3 . \frac{4}{3}\pi r^{3}.
  39. r ( h - x ) h . r\frac{(h-x)}{h}.
  40. π ( r ( h - x ) h ) 2 = π r 2 ( h - x ) 2 h 2 . \pi\left(r\frac{(h-x)}{h}\right)^{2}=\pi r^{2}\frac{(h-x)^{2}}{h^{2}}.
  41. 0 h π r 2 ( h - x ) 2 h 2 d x , \int_{0}^{h}\pi r^{2}\frac{(h-x)^{2}}{h^{2}}dx,
  42. π r 2 h 2 0 h ( h - x ) 2 d x \frac{\pi r^{2}}{h^{2}}\int_{0}^{h}(h-x)^{2}dx
  43. π r 2 h 2 ( h 3 3 ) = 1 3 π r 2 h . \frac{\pi r^{2}}{h^{2}}\left(\frac{h^{3}}{3}\right)=\frac{1}{3}\pi r^{2}h.
  44. ω = | g | d x 1 d x n \omega=\sqrt{|g|}dx^{1}\wedge\dots\wedge dx^{n}
  45. d x i dx^{i}
  46. | g | |g|

Volumetric_heat_capacity.html

  1. I = k ρ c I=\sqrt{k\rho c}
  2. k k
  3. ρ \rho
  4. c c
  5. I I

W._T._Tutte.html

  1. ψ \psi
  2. ψ \psi
  3. ψ \psi
  4. ψ \psi
  5. ψ \psi
  6. μ \mu
  7. μ \mu
  8. χ \chi
  9. χ \chi
  10. χ \chi
  11. χ \chi
  12. χ \chi
  13. χ \chi
  14. ψ \psi
  15. χ \chi
  16. χ \chi
  17. χ \chi
  18. χ \chi
  19. χ \chi
  20. ψ \psi
  21. ψ \psi
  22. ψ \psi
  23. ψ \psi
  24. ψ \psi
  25. χ \chi
  26. ψ \psi
  27. ψ \psi
  28. ψ \psi
  29. ψ \psi

Wafer_(electronics).html

  1. D P W = π d 2 4 S DPW=\left\lfloor\frac{\pi d^{2}}{4S}\right\rfloor
  2. d d
  3. S S
  4. D P W = π d 2 4 S - π d 2 S DPW=\frac{\displaystyle\pi d^{2}}{4S}-\frac{\displaystyle\pi d}{\sqrt{2S}}
  5. D P W = ( π d 2 4 S ) exp ( - 2 S / d ) DPW=\left(\frac{\displaystyle\pi d^{2}}{4S}\right)\exp(-2\sqrt{S}/d)
  6. D P W = π d 2 4 S ( 1 - 2 S d ) 2 DPW=\frac{\displaystyle\pi d^{2}}{4S}\left(1-\frac{\displaystyle 2\sqrt{S}}{d}% \right)^{2}
  7. S \sqrt{S}
  8. ( H + W ) / 2 (H+W)/2
  9. D P W = π d 2 4 S - 0.58 * π d S DPW=\frac{\displaystyle\pi d^{2}}{4S}-0.58^{*}\frac{\displaystyle\pi d}{\sqrt{% S}}
  10. D P W = ( π d 2 4 S ) exp ( - 2.32 * S / d ) DPW=\left(\frac{\displaystyle\pi d^{2}}{4S}\right)\exp(-2.32^{*}\sqrt{S}/d)
  11. D P W = π d 2 4 S ( 1 - 1.16 * S d ) 2 DPW=\frac{\displaystyle\pi d^{2}}{4S}\left(1-\frac{\displaystyle 1.16^{*}\sqrt% {S}}{d}\right)^{2}

Walleye.html

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

Waring's_problem.html

  1. × 10 9 \times 10^{9}
  2. × 10 9 \times 10^{9}
  3. × 10 9 \times 10^{9}
  4. G ( k ) k ( 3 log k + 11 ) G(k)\leq k(3\log k+11)
  5. G ( k ) k ( 2 log k + 2 log log k + C log log log k ) G(k)\leq k(2\log k+2\log\log k+C\log\log\log k)
  6. G ( k ) G(k)
  7. k 400 k\geq 400
  8. G ( k ) < 2 k log k + 2 k log log k + 12 k . \!G(k)<2k\log k+2k\log\log k+12k.
  9. G ( k ) k log k + k log log k + C k . G(k)\leq k\log k+k\log\log k+Ck.

Water_turbine.html

  1. P = η ρ g h q ˙ P=\eta\cdot\rho\cdot g\cdot h\cdot\dot{q}
  2. P = P=
  3. η = \eta=
  4. ρ = \rho=
  5. g = g=
  6. h = h=
  7. q ˙ \dot{q}
  8. n s n_{s}

Wave.html

  1. x x
  2. x x
  3. x x
  4. u u
  5. v v
  6. v v
  7. u ( x , t ) = F ( x - v t ) u(x,t)=F(x-v\ t)
  8. F F
  9. u ( x , t ) = G ( x + v t ) u(x,t)=G(x+v\ t)
  10. G G
  11. u ( x , t ) = F ( x - v t ) + G ( x + v t ) . u(x,t)=F(x-vt)+G(x+vt).\,
  12. F F
  13. G G
  14. 1 v 2 2 u t 2 = 2 u x 2 . \frac{1}{v^{2}}\frac{\partial^{2}u}{\partial t^{2}}=\frac{\partial^{2}u}{% \partial x^{2}}.\,
  15. u ( x , t ) = A ( x , t ) sin ( k x - ω t + ϕ ) , u(x,t)=A(x,t)\sin(kx-\omega t+\phi)\ ,
  16. A ( x , t ) A(x,\ t)
  17. k k
  18. ϕ \phi
  19. v g v_{g}
  20. u ( x , t ) = A ( x - v g t ) sin ( k x - ω t + ϕ ) , u(x,t)=A(x-v_{g}\ t)\sin(kx-\omega t+\phi)\ ,
  21. ψ ( x , t ) = A e i ( k x - ω t ) , \psi(x,t)=Ae^{i\left(kx-\omega t\right)}\ ,
  22. k x - ω t = ( 2 π λ ) ( x - v t ) kx-\omega t=\left(\frac{2\pi}{\lambda}\right)(x-vt)
  23. λ = 2 π k \lambda=\frac{2\pi}{k}
  24. v p = ω k v_{p}=\frac{\omega}{k}\,
  25. ψ ( x , t ) = - d k 1 A ( k 1 ) e i ( k 1 x - ω t ) , \psi(x,t)=\int_{-\infty}^{\infty}\ dk_{1}\ A(k_{1})\ e^{i\left(k_{1}x-\omega t% \right)}\ ,
  26. A = A o ( k 1 ) e i α ( k 1 ) , A=A_{o}(k_{1})e^{i\alpha(k_{1})}\ ,
  27. A o ( k 1 ) = N e - σ 2 ( k 1 - k ) 2 / 2 , A_{o}(k_{1})=N\ e^{-\sigma^{2}(k_{1}-k)^{2}/2}\ ,
  28. d φ d k 1 | k 1 = k = x - t d ω d k 1 | k 1 = k + d α d k 1 | k 1 = k , \left.\frac{d\varphi}{dk_{1}}\right|_{k_{1}=k}=x-t\left.\frac{d\omega}{dk_{1}}% \right|_{k_{1}=k}+\left.\frac{d\alpha}{dk_{1}}\right|_{k_{1}=k}\ ,
  29. v g = d ω d k . v_{g}=\frac{d\omega}{dk}\ .
  30. v g = k m , v_{g}=\frac{\hbar k}{m}\ ,
  31. u u
  32. u ( x , t ) = A sin ( k x - ω t + ϕ ) , u(x,t)=A\sin(kx-\omega t+\phi)\ ,
  33. A A
  34. x x
  35. t t
  36. k k
  37. ω \omega
  38. ϕ \phi
  39. λ \lambda
  40. k k
  41. k = 2 π λ . k=\frac{2\pi}{\lambda}.\,
  42. T T
  43. f f
  44. f = 1 T . f=\frac{1}{T}.\,
  45. ω \omega
  46. ω = 2 π f = 2 π T . \omega=2\pi f=\frac{2\pi}{T}.\,
  47. λ \lambda
  48. v v
  49. λ = v f , \lambda=\frac{v}{f},
  50. v v
  51. f f
  52. v = T μ , v=\sqrt{\frac{T}{\mu}},\,
  53. v = B ρ 0 , v=\sqrt{\frac{B}{\rho_{0}}},\,
  54. c = e m c=\sqrt{\frac{e}{m}}
  55. λ = h p , \lambda=\frac{h}{p},
  56. ψ ( 𝐫 , t = 0 ) = A e i 𝐤 𝐫 , \psi(\mathbf{r},\ t=0)=A\ e^{i\mathbf{k\cdot r}}\ ,
  57. λ = 2 π k , \lambda=\frac{2\pi}{k}\ ,
  58. 𝐩 = 𝐤 . \mathbf{p}=\hbar\mathbf{k}\ .
  59. ψ ( x , t = 0 ) = A exp ( - x 2 2 σ 2 + i k 0 x ) , \psi(x,\ t=0)=A\ \exp\left(-\frac{x^{2}}{2\sigma^{2}}+ik_{0}x\right)\ ,
  60. f ( x ) = e - x 2 / ( 2 σ 2 ) , f(x)=e^{-x^{2}/(2\sigma^{2})}\ ,
  61. f ~ ( k ) = σ e - σ 2 k 2 / 2 . \tilde{f}(k)=\sigma e^{-\sigma^{2}k^{2}/2}\ .
  62. f ( x ) = 1 2 π - f ~ ( k ) e i k x d k ; f(x)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}\ \tilde{f}(k)e^{ikx}\ dk\ ;