wpmath0000001_15

Monetarism.html

  1. G + B - T ( Y + B ) = M ˙ + B ˙ r G+B-T(Y+B)=\dot{M}+\frac{\dot{B}}{r}
  2. M ˙ = B ˙ = 0 \dot{M}=\dot{B}=0
  3. G + B = T ( Y + B ) . G+B=T(Y+B).
  4. d Y d G = 1 T \frac{dY}{dG}=\frac{1}{T^{\prime}}
  5. d B d G = 0 \frac{dB}{dG}=0
  6. d Y d G = 1 + ( 1 - T ) d B d G T , d B d G > 0 \frac{dY}{dG}=\frac{1+(1-T^{\prime})\frac{dB}{dG}}{T^{\prime}}\;,\quad\frac{dB% }{dG}>0

Monoid.html

  1. ( a b ) c = a ( b c ) (ab)c=a(bc)
  2. e a = a e = a ea=ae=a
  3. f \langle f\rangle
  4. f = { f 0 , f 1 , , f n - 1 } \langle f\rangle=\{f^{0},f^{1},\dots,f^{n-1}\}
  5. f n = f k f^{n}=f^{k}
  6. 0 k < n 0\leq k<n
  7. { 0 , 1 , 2 , , n - 1 } \{0,1,2,\dots,n-1\}
  8. [ 0 1 2 n - 2 n - 1 1 2 3 n - 1 k ] \begin{bmatrix}0&1&2&\dots&n-2&n-1\\ 1&2&3&\dots&n-1&k\end{bmatrix}
  9. f ( i ) := { i + 1 , if 0 i < n - 1 k , if i = n - 1. f(i):=\begin{cases}i+1,&\mbox{if }~{}0\leq i<n-1\\ k,&\mbox{if }~{}i=n-1.\end{cases}
  10. f \langle f\rangle
  11. k = 0 k=0
  12. { 0 , 1 , 2 , , n - 1 } \{0,1,2,\dots,n-1\}
  13. R = { u 1 = v 1 , , u n = v n } R=\{u_{1}=v_{1},\cdots,u_{n}=v_{n}\}
  14. p , q | p q = 1 \langle p,q\,|\;pq=1\rangle
  15. a , b | a b a = b a a , b b a = b a b \langle a,b\,|\;aba=baa,bba=bab\rangle
  16. a i b j ( b a ) k a^{i}b^{j}(ba)^{k}
  17. ε \varepsilon
  18. * *
  19. fold : M * M = l { ε if l = nil m * fold l if l = cons m l \mathrm{fold}:M^{*}\rightarrow M=l\mapsto\begin{cases}\varepsilon&\mbox{if }~{% }l=\mathrm{nil}\\ m*\mathrm{fold}\,l^{\prime}&\mbox{if }~{}l=\mathrm{cons}\,m\,l^{\prime}\end{cases}
  20. Σ I \Sigma_{I}
  21. i m i = 0 ; i { j } m i = m j ; i { j , k } m i = m j + m k for j k \sum_{i\in\emptyset}{m_{i}}=0;\quad\sum_{i\in\{j\}}{m_{i}}=m_{j};\quad\sum_{i% \in\{j,k\}}{m_{i}}=m_{j}+m_{k}\quad\textrm{for}\;j\neq k
  22. j J i I j m i = i I ( m i ) if j J I j = I and I j I j = for j j \sum_{j\in J}{\sum_{i\in I_{j}}{m_{i}}}=\sum_{i\in I}(m_{i})\;\textrm{if}% \bigcup_{j\in J}I_{j}=I\;\textrm{and}\;I_{j}\cap I_{j^{\prime}}=\emptyset\;% \textrm{for}\;j\neq j^{\prime}
  23. a + sup S = sup ( a + S ) . a+\sup S=\sup(a+S)\ .
  24. I a i = sup E a i \sum_{I}a_{i}=\sup\sum_{E}a_{i}

Monoid_ring.html

  1. g G r g g \sum_{g\in G}r_{g}g
  2. r g R r_{g}\in R
  3. g G g\in G
  4. ( ϕ ψ ) ( g ) = k = g ϕ ( k ) ψ ( ) (\phi\psi)(g)=\sum_{k\ell=g}\phi(k)\psi(\ell)
  5. η ( g G r g g ) = g G r g . \eta(\sum_{g\in G}r_{g}g)=\sum_{g\in G}r_{g}.

Monomorphism.html

  1. X Y X\hookrightarrow Y
  2. f g 1 = f g 2 g 1 = g 2 . f\circ g_{1}=f\circ g_{2}\Rightarrow g_{1}=g_{2}.
  3. l f = id X l\circ f=\operatorname{id}_{X}
  4. f g 1 = f g 2 l f g 1 = l f g 2 g 1 = g 2 . f\circ g_{1}=f\circ g_{2}\Rightarrow lfg_{1}=lfg_{2}\Rightarrow g_{1}=g_{2}.
  5. 0 h ( x ) h ( x ) + 1 = h ( y ) < 1 0\leq\frac{h(x)}{h(x)+1}=h(y)<1

Monopoly.html

  1. 50 - 4 Q = 0 50-4Q=0
  2. - 4 Q = - 50 -4Q=-50
  3. Q = 12.5 Q=12.5

Monotonic_function.html

  1. f f
  2. x x
  3. y y
  4. x y x\leq y
  5. f ( x ) f ( y ) f\!\left(x\right)\leq f\!\left(y\right)
  6. f f
  7. x y x\leq y
  8. f ( x ) f ( y ) f\!\left(x\right)\geq f\!\left(y\right)
  9. \leq
  10. < <
  11. x x
  12. y y
  13. x < y x<y
  14. x > y x>y
  15. f ( x ) < f ( y ) f\!\left(x\right)<f\!\left(y\right)
  16. f ( x ) > f ( y ) f\!\left(x\right)>f\!\left(y\right)
  17. f ( x ) f\!\left(x\right)
  18. f ( y ) f\!\left(y\right)
  19. f ( x ) f\!\left(x\right)
  20. ( a , b ) \left(a,b\right)
  21. f f
  22. f : f\colon\mathbb{R}\to\mathbb{R}
  23. f f
  24. f f
  25. ± \pm\infty
  26. \infty
  27. ( - ) \left(-\infty\right)
  28. f f
  29. f f
  30. f f
  31. I I
  32. f f
  33. I I
  34. { x : x I } \left\{x:x\in I\right\}
  35. x x
  36. I I
  37. f f
  38. x x
  39. f f
  40. [ a , b ] \left[a,b\right]
  41. f f
  42. X X
  43. F X ( x ) = Prob ( X x ) F_{X}\!\left(x\right)=\,\text{Prob}\!\left(X\leq x\right)
  44. f f
  45. f f
  46. T T
  47. f f
  48. T T
  49. f f
  50. ( T u - T v , u - v ) 0 u , v X . (Tu-Tv,u-v)\geq 0\quad\forall u,v\in X.
  51. ( w 1 - w 2 , u 1 - u 2 ) 0. (w_{1}-w_{2},u_{1}-u_{2})\geq 0.
  52. h ( n ) c ( n , a , n ) + h ( n ) . h(n)\leq c(n,a,n^{\prime})+h(n^{\prime}).

Monster_group.html

  1. × 10 5 3 \times 10^{5}3
  2. E ~ 8 , \tilde{E}_{8},
  3. E ~ 6 , E ~ 7 , E ~ 8 \tilde{E}_{6},\tilde{E}_{7},\tilde{E}_{8}

Monte_Carlo_method.html

  1. π \pi
  2. π \pi
  3. π \pi
  4. π \pi
  5. π \pi
  6. π \pi
  7. 1 / N \scriptstyle 1/\sqrt{N}

Moon.html

  1. 1.14 2 1.30 \scriptstyle 1.14^{2}\approx 1.30
  2. perceived reduction % = 100 × actual reduction % 100 \,\text{perceived reduction}\%=100\times\sqrt{\,\text{actual reduction}\%\over 1% 00}

MOSFET.html

  1. N A N_{A}
  2. V G B V_{GB}
  3. V G B V_{GB}
  4. V G S V_{GS}
  5. V t h V_{th}
  6. V G S V_{GS}
  7. I D I D 0 e V G S - V t h n V T , I_{D}\approx I_{D0}e^{\begin{matrix}\frac{V_{GS}-V_{th}}{nV_{T}}\end{matrix}},
  8. I D 0 I_{D0}
  9. V G S = V t h V_{GS}=V_{th}
  10. V T = k T / q V_{T}=kT/q
  11. n = 1 + C D / C O X , n=1+C_{D}/C_{OX},\,
  12. C D C_{D}
  13. C O X C_{OX}
  14. I D I D 0 e κ ( V G - V t h ) - V S V T , I_{D}\approx I_{D0}e^{\begin{matrix}\frac{\kappa(V_{G}-V_{th})-V_{S}}{V_{T}}% \end{matrix}},
  15. κ \kappa
  16. κ = C O X C O X + C D , \kappa=\frac{C_{OX}}{{C_{OX}+C_{D}}},
  17. C D C_{D}
  18. C O X C_{OX}
  19. V D S V T V_{DS}\gg V_{T}
  20. g m / I D = 1 / ( n V T ) g_{m}/I_{D}=1/(nV_{T})
  21. V G S - V t h V_{GS}-V_{th}
  22. I D = μ n C o x W L ( ( V G S - V t h ) V D S - V D S 2 2 ) I_{D}=\mu_{n}C_{ox}\frac{W}{L}\left((V_{GS}-V_{th})V_{DS}-\frac{V_{DS}^{2}}{2}\right)
  23. μ n \mu_{n}
  24. W W
  25. L L
  26. C o x C_{ox}
  27. I D = μ n C o x 2 W L ( V G S - V t h ) 2 [ 1 + λ ( V D S - V D S s a t ) ] . I_{D}=\frac{\mu_{n}C_{ox}}{2}\frac{W}{L}(V_{GS}-V_{th})^{2}\left[1+\lambda(V_{% DS}-V_{DSsat})\right].
  28. g m = I D V G S = 2 I D V G S - V t h = 2 I D V o v , g_{m}=\frac{\partial I_{D}}{\partial V_{GS}}=\frac{2I_{D}}{V_{GS}-V_{th}}=% \frac{2I_{D}}{V_{ov}},
  29. I D I_{D}
  30. r o u t = 1 λ I D r_{out}=\frac{1}{\lambda I_{D}}
  31. g D S = I D S V D S g_{DS}=\frac{\partial I_{DS}}{\partial V_{DS}}
  32. V T B = V T 0 + γ ( V S B + 2 φ B - 2 φ B ) , V_{TB}=V_{T0}+\gamma\left(\sqrt{V_{SB}+2\varphi_{B}}-\sqrt{2\varphi_{B}}\right),
  33. γ \gamma
  34. φ B = ( k B T / q ) ln ( N A / n i ) , \varphi_{B}=(k_{B}T/q)\ln(N_{A}/n_{i})\ ,
  35. Q = κ ϵ 0 E , Q=\kappa\epsilon_{0}\ E,
  36. V G = V c h + E t i n s = V c h + Q t i n s κ ϵ 0 , V_{G}=V_{ch}+E\ t_{ins}=V_{ch}+\frac{Qt_{ins}}{\kappa\epsilon_{0}},

Möbius_function.html

  1. n n
  2. μ ( a b ) = μ ( a ) μ ( b ) μ(ab)=μ(a)μ(b)
  3. a a
  4. b b
  5. n n
  6. n n
  7. n = 1 n=1
  8. d | n μ ( d ) = { 1 if n = 1 0 if n > 1. \sum_{d|n}\mu(d)=\begin{cases}1&\mbox{ if }~{}n=1\\ 0&\mbox{ if }~{}n>1.\end{cases}
  9. n n
  10. n n
  11. d d
  12. d d
  13. n n
  14. μ μ
  15. μ ( n ) μ(n)
  16. M ( n ) = k = 1 n μ ( k ) M(n)=\sum_{k=1}^{n}\mu(k)
  17. n n
  18. M ( n ) M(n)
  19. μ ( n ) = gcd ( k , n ) = 1 1 k n e 2 π i k n , \mu(n)=\sum_{\stackrel{1\leq k\leq n}{\gcd(k,\,n)=1}}e^{2\pi i\tfrac{k}{n}},
  20. μ ( n ) μ(n)
  21. n n
  22. M ( n ) = a n e 2 π i a , M(n)=\sum_{a\in\mathcal{F}_{n}}e^{2\pi ia},
  23. n \mathcal{F}_{n}
  24. n n
  25. d | n μ ( d ) \sum_{d|n}\mu(d)
  26. d | n μ ( d ) = { 1 if n = 1 0 if n > 1. \sum_{d|n}\mu(d)=\begin{cases}1&\mbox{ if }~{}n=1\\ 0&\mbox{ if }~{}n>1.\end{cases}
  27. n = 1 n=1
  28. n > 1 n>1
  29. d d
  30. n n
  31. μ ( d ) 0 μ(d)≠0
  32. n n
  33. | S | |S|
  34. S S
  35. | S | = 1 |S|=1
  36. S S
  37. S S
  38. | S | > 1 |S|>1
  39. S S
  40. x x
  41. S S
  42. S S
  43. n = 1 μ ( n ) n s = 1 ζ ( s ) . \sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}=\frac{1}{\zeta(s)}.
  44. 1 ζ ( s ) = p ( 1 - 1 p s ) = ( 1 - 1 2 s ) ( 1 - 1 3 s ) ( 1 - 1 5 s ) . \frac{1}{\zeta(s)}=\prod_{p\in\mathbb{P}}{\left(1-\frac{1}{p^{s}}\right)}=% \left(1-\frac{1}{2^{s}}\right)\left(1-\frac{1}{3^{s}}\right)\left(1-\frac{1}{5% ^{s}}\right)\cdots.
  45. n = 1 μ ( n ) n s = 1 - a = 2 1 a s + a = 2 b = 2 1 ( a b ) s - a = 2 b = 2 c = 2 1 ( a b c ) s + a = 2 b = 2 c = 2 d = 2 1 ( a b c d ) s - . \sum_{n=1}^{\infty}\frac{\mu(n)}{n^{s}}=1-\sum_{a=2}^{\infty}\frac{1}{a^{s}}+% \sum_{a=2}^{\infty}\sum_{b=2}^{\infty}\frac{1}{(a\cdot b)^{s}}-\sum_{a=2}^{% \infty}\sum_{b=2}^{\infty}\sum_{c=2}^{\infty}\frac{1}{(a\cdot b\cdot c)^{s}}+% \sum_{a=2}^{\infty}\sum_{b=2}^{\infty}\sum_{c=2}^{\infty}\sum_{d=2}^{\infty}% \frac{1}{(a\cdot b\cdot c\cdot d)^{s}}-\cdots.
  46. n = 1 μ ( n ) q n 1 - q n = q \sum_{n=1}^{\infty}\frac{\mu(n)q^{n}}{1-q^{n}}=q
  47. ( I + X ) - 1 (I+X)^{-1}
  48. n = 1 μ ( n ) x n = x - a = 2 x a + a = 2 b = 2 x a b - a = 2 b = 2 c = 2 x a b c + a = 2 b = 2 c = 2 d = 2 x a b c d - \sum_{n=1}^{\infty}\mu(n)x^{n}=x-\sum_{a=2}^{\infty}x^{a}+\sum_{a=2}^{\infty}% \sum_{b=2}^{\infty}x^{ab}-\sum_{a=2}^{\infty}\sum_{b=2}^{\infty}\sum_{c=2}^{% \infty}x^{abc}+\sum_{a=2}^{\infty}\sum_{b=2}^{\infty}\sum_{c=2}^{\infty}\sum_{% d=2}^{\infty}x^{abcd}-\cdots
  49. p p
  50. μ ( p 1 ) ( m o d p ) μ(p−1)(modp)
  51. q q
  52. q q
  53. N N
  54. n n
  55. N ( q , n ) = 1 n d | n μ ( d ) q n d . N(q,n)=\frac{1}{n}\sum_{d|n}\mu(d)q^{\frac{n}{d}}.
  56. M M
  57. M ( 1 , 1 ) = 1 M(1,1)=1
  58. n = k : M ( n , k ) = - i = 1 i = k - 1 M ( n , k - i ) n=k:M(n,k)=-\sum\limits_{i=1}^{i=k-1}M(n,k-i)
  59. n > k : M ( n , k ) = M ( n - k , k ) n>k:M(n,k)=M(n-k,k)
  60. n < k : M ( n , k ) = 0 n<k:M(n,k)=0
  61. M = ( 1 0 0 0 0 0 0 1 - 1 0 0 0 0 0 1 0 - 1 0 0 0 0 1 - 1 0 0 0 0 0 1 0 0 0 - 1 0 0 1 - 1 - 1 0 0 1 0 1 0 0 0 0 0 - 1 ) M=\left(\begin{array}[]{ccccccc}1&0&0&0&0&0&0\\ 1&-1&0&0&0&0&0\\ 1&0&-1&0&0&0&0\\ 1&-1&0&0&0&0&0\\ 1&0&0&0&-1&0&0\\ 1&-1&-1&0&0&1&0\\ 1&0&0&0&0&0&-1\end{array}\right)
  62. M M
  63. M ( n , k ) M(n,k)
  64. 1 1
  65. k k
  66. n n
  67. 0
  68. M = ( 1 0 0 0 0 0 0 1 1 0 0 0 0 0 1 0 1 0 0 0 0 1 1 0 1 0 0 0 1 0 0 0 1 0 0 1 1 1 0 0 1 0 1 0 0 0 0 0 1 ) M=\left(\begin{array}[]{ccccccc}1&0&0&0&0&0&0\\ 1&1&0&0&0&0&0\\ 1&0&1&0&0&0&0\\ 1&1&0&1&0&0&0\\ 1&0&0&0&1&0&0\\ 1&1&1&0&0&1&0\\ 1&0&0&0&0&0&1\end{array}\right)
  69. μ ( n / k ) \mu(n/k)
  70. k k
  71. n n
  72. 0
  73. M - 1 = ( 1 0 0 0 0 0 0 - 1 1 0 0 0 0 0 - 1 0 1 0 0 0 0 0 - 1 0 1 0 0 0 - 1 0 0 0 1 0 0 1 - 1 - 1 0 0 1 0 - 1 0 0 0 0 0 1 ) M^{-1}=\left(\begin{array}[]{ccccccc}1&0&0&0&0&0&0\\ -1&1&0&0&0&0&0\\ -1&0&1&0&0&0&0\\ 0&-1&0&1&0&0&0\\ -1&0&0&0&1&0&0\\ 1&-1&-1&0&0&1&0\\ -1&0&0&0&0&0&1\end{array}\right)
  74. μ k = μ μ \mu_{k}=\mu\star\cdots\star\mu
  75. μ k ( p a ) = ( - 1 ) a ( k a ) \mu_{k}(p^{a})=(-1)^{a}{\left({{k}\atop{a}}\right)}

Möbius_inversion_formula.html

  1. g ( n ) = d n f ( d ) for every integer n 1 g(n)=\sum_{d\,\mid\,n}f(d)\quad\,\text{for every integer }n\geq 1
  2. f ( n ) = d n μ ( d ) g ( n / d ) for every integer n 1 f(n)=\sum_{d\,\mid\,n}\mu(d)g(n/d)\quad\,\text{for every integer }n\geq 1
  3. \mathbb{Z}
  4. g = f * 1 g=f*1
  5. 1 ( n ) = 1 1(n)=1
  6. f = μ * g . f=\mu*g.
  7. * *
  8. 1 * μ = ϵ 1*\mu=\epsilon
  9. ϵ \epsilon
  10. ϵ ( 1 ) = 1 , ϵ ( n ) = 0 \epsilon(1)=1,\epsilon(n)=0
  11. n > 1 n>1
  12. μ * g = μ * ( 1 * f ) = ( μ * 1 ) * f = ϵ * f = f \mu*g=\mu*(1*f)=(\mu*1)*f=\epsilon*f=f
  13. a n = d n b d a_{n}=\sum_{d\mid n}b_{d}
  14. b n = d n μ ( n d ) a d b_{n}=\sum_{d\mid n}\mu\left(\frac{n}{d}\right)a_{d}
  15. n = 1 a n x n = n = 1 b n x n 1 - x n \sum_{n=1}^{\infty}a_{n}x^{n}=\sum_{n=1}^{\infty}b_{n}\frac{x^{n}}{1-x^{n}}
  16. n = 1 a n n s = ζ ( s ) n = 1 b n n s \sum_{n=1}^{\infty}\frac{a_{n}}{n^{s}}=\zeta(s)\sum_{n=1}^{\infty}\frac{b_{n}}% {n^{s}}
  17. ζ ( s ) \zeta(s)
  18. φ \varphi
  19. φ \varphi
  20. φ * 1 = Id \varphi*1=\operatorname{Id}
  21. Id ( n ) = n \operatorname{Id}(n)=n
  22. Id * 1 = σ 1 = σ \operatorname{Id}*1=\sigma_{1}=\sigma
  23. μ \mu
  24. μ * 1 = ε \mu*1=\varepsilon
  25. ε ( n ) = { 1 , if n = 1 0 , if n > 1 \varepsilon(n)=\begin{cases}1,&\mbox{if }~{}n=1\\ 0,&\mbox{if }~{}n>1\end{cases}
  26. ε * 1 = 1 \varepsilon*1=1
  27. 1 * 1 = σ 0 = d = τ 1*1=\sigma_{0}=\operatorname{d}=\tau
  28. d = τ \operatorname{d}=\tau
  29. φ \varphi
  30. f n = { μ * * μ - n factors * φ if n < 0 φ if n = 0 φ * 1 * * 1 n factors if n > 0 f_{n}=\begin{cases}\underbrace{\mu*\ldots*\mu}_{-n\,\text{ factors}}*\varphi&% \,\text{if }n<0\\ \varphi&\,\text{if }n=0\\ \varphi*\underbrace{1*\ldots*1}_{n\,\text{ factors}}&\,\text{if }n>0\end{cases}
  31. G ( x ) = 1 n x F ( x / n ) for all x 1 G(x)=\sum_{1\leq n\leq x}F(x/n)\quad\mbox{ for all }~{}x\geq 1
  32. F ( x ) = 1 n x μ ( n ) G ( x / n ) for all x 1. F(x)=\sum_{1\leq n\leq x}\mu(n)G(x/n)\quad\mbox{ for all }~{}x\geq 1.
  33. α ( n ) \alpha(n)
  34. α - 1 ( n ) \alpha^{-1}(n)
  35. G ( x ) = 1 n x α ( n ) F ( x / n ) for all x 1 G(x)=\sum_{1\leq n\leq x}\alpha(n)F(x/n)\quad\mbox{ for all }~{}x\geq 1
  36. F ( x ) = 1 n x α - 1 ( n ) G ( x / n ) for all x 1. F(x)=\sum_{1\leq n\leq x}\alpha^{-1}(n)G(x/n)\quad\mbox{ for all }~{}x\geq 1.
  37. α ( n ) = 1 \alpha(n)=1
  38. α - 1 ( n ) = μ ( n ) \alpha^{-1}(n)=\mu(n)
  39. g ( n ) = 1 m n f ( n m ) for all n 1. g(n)=\sum_{1\leq m\leq n}f\left(\left\lfloor\frac{n}{m}\right\rfloor\right)% \quad\mbox{ for all }~{}n\geq 1.
  40. F ( x ) = f ( x ) F(x)=f(\lfloor x\rfloor)
  41. G ( x ) = g ( x ) G(x)=g(\lfloor x\rfloor)
  42. f ( n ) = 1 m n μ ( m ) g ( n m ) for all n 1. f(n)=\sum_{1\leq m\leq n}\mu(m)g\left(\left\lfloor\frac{n}{m}\right\rfloor% \right)\quad\mbox{ for all }~{}n\geq 1.
  43. α ( n ) \alpha(n)
  44. α - 1 ( n ) \alpha^{-1}(n)
  45. g ( x ) = m = 1 α ( m ) f ( m x ) m s for all x 1 f ( x ) = m = 1 α - 1 ( m ) g ( m x ) m s for all x 1. g(x)=\sum_{m=1}^{\infty}\alpha(m)\frac{f(mx)}{m^{s}}\quad\mbox{ for all }~{}x% \geq 1\quad\Longleftrightarrow\quad f(x)=\sum_{m=1}^{\infty}\alpha^{-1}(m)% \frac{g(mx)}{m^{s}}\quad\mbox{ for all }~{}x\geq 1.
  46. If F ( n ) = d | n f ( d ) , then f ( n ) = d | n F ( n / d ) μ ( d ) . \mbox{If }~{}F(n)=\prod_{d|n}f(d),\mbox{ then }~{}f(n)=\prod_{d|n}F(n/d)^{\mu(% d)}.\,
  47. d | n μ ( d ) = i ( n ) \sum_{d|n}\mu(d)=i(n)
  48. 1 n x μ ( n ) g ( x n ) \displaystyle\sum_{1\leq n\leq x}\mu(n)g\left(\frac{x}{n}\right)

Möbius_strip.html

  1. x ( u , v ) = ( 1 + v 2 cos u 2 ) cos u x(u,v)=\left(1+\frac{v}{2}\cos\frac{u}{2}\right)\cos u
  2. y ( u , v ) = ( 1 + v 2 cos u 2 ) sin u y(u,v)=\left(1+\frac{v}{2}\cos\frac{u}{2}\right)\sin u
  3. z ( u , v ) = v 2 sin u 2 z(u,v)=\frac{v}{2}\sin\frac{u}{2}
  4. log ( r ) sin ( 1 2 θ ) = z cos ( 1 2 θ ) . \log(r)\sin\left(\frac{1}{2}\theta\right)=z\cos\left(\frac{1}{2}\theta\right).
  5. 3 \sqrt{3}
  6. z 1 = sin η e i φ z_{1}=\sin\eta\,e^{i\varphi}
  7. z 2 = cos η e i φ / 2 . z_{2}=\cos\eta\,e^{i\varphi/2}.
  8. { 1 / 2 , i / 2 } \left\{1/\sqrt{2},i/\sqrt{2}\right\}

Mössbauer_effect.html

  1. | P R | = | P γ | |P_{\mathrm{R}}|=|P_{\mathrm{\gamma}}|\,
  2. P γ P_{γ}
  3. E R = E γ 2 2 M c 2 E_{\mathrm{R}}=\frac{E_{\mathrm{\gamma}}^{2}}{2Mc^{2}}
  4. E < s m a l l > R E<small>_{R}

MPEG-1.html

  1. [ - 415 - 30 - 61 27 56 - 20 - 2 0 4 - 22 - 61 10 13 - 7 - 9 5 - 47 7 77 - 25 - 29 10 5 - 6 - 49 12 34 - 15 - 10 6 2 2 12 - 7 - 13 - 4 - 2 2 - 3 3 - 8 3 2 - 6 - 2 1 4 2 - 1 0 0 - 2 - 1 - 3 4 - 1 0 0 - 1 - 4 - 1 0 1 2 ] \begin{bmatrix}-415&-30&-61&27&56&-20&-2&0\\ 4&-22&-61&10&13&-7&-9&5\\ -47&7&77&-25&-29&10&5&-6\\ -49&12&34&-15&-10&6&2&2\\ 12&-7&-13&-4&-2&2&-3&3\\ -8&3&2&-6&-2&1&4&2\\ -1&0&0&-2&-1&-3&4&-1\\ 0&0&-1&-4&-1&0&1&2\end{bmatrix}
  2. [ 16 11 10 16 24 40 51 61 12 12 14 19 26 58 60 55 14 13 16 24 40 57 69 56 14 17 22 29 51 87 80 62 18 22 37 56 68 109 103 77 24 35 55 64 81 104 113 92 49 64 78 87 103 121 120 101 72 92 95 98 112 100 103 99 ] \begin{bmatrix}16&11&10&16&24&40&51&61\\ 12&12&14&19&26&58&60&55\\ 14&13&16&24&40&57&69&56\\ 14&17&22&29&51&87&80&62\\ 18&22&37&56&68&109&103&77\\ 24&35&55&64&81&104&113&92\\ 49&64&78&87&103&121&120&101\\ 72&92&95&98&112&100&103&99\end{bmatrix}
  3. [ - 26 - 3 - 6 2 2 - 1 0 0 0 - 2 - 4 1 1 0 0 0 - 3 1 5 - 1 - 1 0 0 0 - 4 1 2 - 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] \begin{bmatrix}-26&-3&-6&2&2&-1&0&0\\ 0&-2&-4&1&1&0&0&0\\ -3&1&5&-1&-1&0&0&0\\ -4&1&2&-1&0&0&0&0\\ 1&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\end{bmatrix}

Multipath_propagation.html

  1. x ( t ) = δ ( t ) x(t)=\delta(t)
  2. y ( t ) = h ( t ) = n = 0 N - 1 ρ n e j ϕ n δ ( t - τ n ) y(t)=h(t)=\sum_{n=0}^{N-1}{\rho_{n}e^{j\phi_{n}}\delta(t-\tau_{n})}
  3. N N
  4. τ n \tau_{n}
  5. n t h n^{th}
  6. ρ n e j ϕ n \rho_{n}e^{j\phi_{n}}
  7. y ( t ) y(t)
  8. h ( t ) h(t)
  9. τ n = τ n ( t ) \tau_{n}=\tau_{n}(t)
  10. ρ n = ρ n ( t ) \rho_{n}=\rho_{n}(t)
  11. ϕ n = ϕ n ( t ) \phi_{n}=\phi_{n}(t)
  12. T M T_{M}
  13. T M = τ N - 1 - τ 0 T_{M}=\tau_{N-1}-\tau_{0}
  14. H ( f ) H(f)
  15. h ( t ) h(t)
  16. H ( f ) = 𝔉 ( h ( t ) ) = - + h ( t ) e - j 2 π f t d t = n = 0 N - 1 ρ n e j ϕ n e - j 2 π f τ n H(f)=\mathfrak{F}(h(t))=\int_{-\infty}^{+\infty}{h(t)e^{-j2\pi ft}dt}=\sum_{n=% 0}^{N-1}{\rho_{n}e^{j\phi_{n}}e^{-j2\pi f\tau_{n}}}
  17. B C 1 T M B_{C}\approx\frac{1}{T_{M}}

Multiplexer.html

  1. s e l sel
  2. I 0 \scriptstyle I_{0}
  3. I 1 \scriptstyle I_{1}
  4. log 2 ( n ) \scriptstyle\left\lceil\log_{2}(n)\right\rceil
  5. n \scriptstyle n
  6. A \scriptstyle A
  7. B \scriptstyle B
  8. S \scriptstyle S
  9. Z \scriptstyle Z
  10. Z = ( A S ¯ ) + ( B S ) Z=(A\cdot\overline{S})+(B\cdot S)
  11. S \scriptstyle S
  12. A \scriptstyle A
  13. B \scriptstyle B
  14. Z \scriptstyle Z
  15. S \scriptstyle S
  16. Z \scriptstyle Z
  17. S = 0 \scriptstyle S=0
  18. Z = A \scriptstyle Z=A
  19. S = 1 \scriptstyle S=1
  20. Z = B \scriptstyle Z=B
  21. log 2 ( n ) \scriptstyle\left\lceil\log_{2}(n)\right\rceil
  22. n n
  23. Z = ( A S 0 ¯ S 1 ¯ ) + ( B S 0 S 1 ¯ ) + ( C S 0 ¯ S 1 ) + ( D S 0 S 1 ) Z=(A\cdot\overline{S_{0}}\cdot\overline{S_{1}})+(B\cdot S_{0}\cdot\overline{S_% {1}})+(C\cdot\overline{S_{0}}\cdot S_{1})+(D\cdot S_{0}\cdot S_{1})
  24. I n \scriptstyle I_{n}
  25. A = ( X S ¯ ) A=(X\cdot\overline{S})
  26. B = ( X S ) B=(X\cdot S)

Multiplication.html

  1. a × b = b + + b a a\times b=\underbrace{b+\cdots+b}_{a}
  2. 3 × 4 = 4 + 4 + 4 = 12 3\times 4=4+4+4=12
  3. 3 × 4 = 3 + 3 + 3 + 3 = 12 3\times 4=3+3+3+3=12
  4. 2 × 3 = 6 2\times 3=6
  5. 3 × 4 = 12 3\times 4=12
  6. 2 × 3 × 5 = 6 × 5 = 30 2\times 3\times 5=6\times 5=30
  7. 2 × 2 × 2 × 2 × 2 = 32 2\times 2\times 2\times 2\times 2=32
  8. 5 2 or 5 . 2 5\cdot 2\quad\,\text{or}\quad 5\,.\,2
  9. 2.5 meters × 4.5 meters = 11.25 square meters 2.5\mbox{ meters}~{}\times 4.5\mbox{ meters}~{}=11.25\mbox{ square meters}~{}
  10. 11 meters/second × 9 seconds = 99 meters 11\mbox{ meters/second}~{}\times 9\mbox{ seconds}~{}=99\mbox{ meters}~{}
  11. i = 1 4 i = 1 2 3 4 , \prod_{i=1}^{4}i=1\cdot 2\cdot 3\cdot 4,
  12. i = 1 4 i = 24. \prod_{i=1}^{4}i=24.
  13. i = 1 6 i = 1 2 3 4 5 6 = 720 \prod_{i=1}^{6}i=1\cdot 2\cdot 3\cdot 4\cdot 5\cdot 6=720
  14. i = m n x i = x m x m + 1 x m + 2 x n - 1 x n , \prod_{i=m}^{n}x_{i}=x_{m}\cdot x_{m+1}\cdot x_{m+2}\cdot\,\,\cdots\,\,\cdot x% _{n-1}\cdot x_{n},
  15. i = m x i = lim n i = m n x i . \prod_{i=m}^{\infty}x_{i}=\lim_{n\to\infty}\prod_{i=m}^{n}x_{i}.
  16. i = - x i = ( lim m - i = m 0 x i ) ( lim n i = 1 n x i ) , \prod_{i=-\infty}^{\infty}x_{i}=\left(\lim_{m\to-\infty}\prod_{i=m}^{0}x_{i}% \right)\cdot\left(\lim_{n\to\infty}\prod_{i=1}^{n}x_{i}\right),
  17. x y = y x x\cdot y=y\cdot x
  18. ( x y ) z = x ( y z ) (x\cdot y)\cdot z=x\cdot(y\cdot z)
  19. x ( y + z ) = x y + x z x\cdot(y+z)=x\cdot y+x\cdot z
  20. x 1 = x x\cdot 1=x
  21. x 0 = 0 x\cdot 0=0
  22. ( - 1 ) x = ( - x ) (-1)\cdot x=(-x)
  23. ( - 1 ) ( - 1 ) = 1 (-1)\cdot(-1)=1
  24. 1 x \frac{1}{x}
  25. x ( 1 x ) = 1 x\cdot\left(\frac{1}{x}\right)=1
  26. x × 0 = 0 x\times 0=0
  27. x × S ( y ) = ( x × y ) + x x\times S(y)=(x\times y)+x
  28. x × 1 = x × S ( 0 ) = ( x × 0 ) + x = 0 + x = x x\times 1=x\times S(0)=(x\times 0)+x=0+x=x
  29. ( x p , x m ) × ( y p , y m ) = ( x p × y p + x m × y m , x p × y m + x m × y p ) (x_{p},\,x_{m})\times(y_{p},\,y_{m})=(x_{p}\times y_{p}+x_{m}\times y_{m},\;x_% {p}\times y_{m}+x_{m}\times y_{p})
  30. ( 0 , 1 ) × ( 0 , 1 ) = ( 0 × 0 + 1 × 1 , 0 × 1 + 1 × 0 ) = ( 1 , 0 ) (0,1)\times(0,1)=(0\times 0+1\times 1,\,0\times 1+1\times 0)=(1,0)
  31. \cdot
  32. ( { 0 } , ) \left(\mathbb{Q}\smallsetminus\{0\},\cdot\right)
  33. N × M N\times M
  34. N × ( - M ) = ( - N ) × M = - ( N × M ) N\times(-M)=(-N)\times M=-(N\times M)
  35. ( - N ) × ( - M ) = N × M (-N)\times(-M)=N\times M
  36. A B × C D \frac{A}{B}\times\frac{C}{D}
  37. A B × C D = ( A × C ) ( B × D ) \frac{A}{B}\times\frac{C}{D}=\frac{(A\times C)}{(B\times D)}
  38. A B \frac{A}{B}
  39. C D \frac{C}{D}
  40. z 1 z_{1}
  41. z 2 z_{2}
  42. ( a 1 , b 1 ) (a_{1},b_{1})
  43. ( a 2 , b 2 ) (a_{2},b_{2})
  44. z 1 × z 2 z_{1}\times z_{2}
  45. ( a 1 × a 2 - b 1 × b 2 , a 1 × b 2 + a 2 × b 1 ) (a_{1}\times a_{2}-b_{1}\times b_{2},a_{1}\times b_{2}+a_{2}\times b_{1})
  46. a 1 × a 2 a_{1}\times a_{2}
  47. b 1 b_{1}
  48. b 2 b_{2}
  49. - 1 \sqrt{-1}
  50. z 1 × z 2 = ( a 1 + b 1 i ) ( a 2 + b 2 i ) = ( a 1 × a 2 ) + ( a 1 × b 2 i ) + ( b 1 × a 2 i ) + ( b 1 × b 2 i 2 ) = ( a 1 a 2 - b 1 b 2 ) + ( a 1 b 2 + b 1 a 2 ) i . z_{1}\times z_{2}=(a_{1}+b_{1}i)(a_{2}+b_{2}i)=(a_{1}\times a_{2})+(a_{1}% \times b_{2}i)+(b_{1}\times a_{2}i)+(b_{1}\times b_{2}i^{2})=(a_{1}a_{2}-b_{1}% b_{2})+(a_{1}b_{2}+b_{1}a_{2})i.
  51. x y \frac{x}{y}
  52. x ( 1 y ) x\left(\frac{1}{y}\right)
  53. 1 x \frac{1}{x}
  54. x y \frac{x}{y}
  55. x y \frac{x}{y}
  56. x ( 1 y ) x\left(\frac{1}{y}\right)
  57. ( 1 y ) x \left(\frac{1}{y}\right)x
  58. a n = a × a × × a n a^{n}=\underbrace{a\times a\times\cdots\times a}_{n}

Multiplication_algorithm.html

  1. r i = ( c i - 1 + j + k = i a j b k ) mod D r_{i}=(c_{i-1}+\sum_{j+k=i}a_{j}b_{k})\mod D
  2. c i = ( c i - 1 + j + k = i a j b k ) / D c_{i}=\lfloor(c_{i-1}+\sum_{j+k=i}a_{j}b_{k})/D\rfloor
  3. c 0 = 0 c_{0}=0
  4. 3 × 11 \displaystyle 3\times 11
  5. 2 n ± 1 2^{n}\pm 1
  6. ( x + y ) 2 4 - ( x - y ) 2 4 = 1 4 ( ( x 2 + 2 x y + y 2 ) - ( x 2 - 2 x y + y 2 ) ) = 1 4 ( 4 x y ) = x y . \left\lfloor\frac{\left(x+y\right)^{2}}{4}\right\rfloor-\left\lfloor\frac{% \left(x-y\right)^{2}}{4}\right\rfloor=\frac{1}{4}\left(\left(x^{2}+2xy+y^{2}% \right)-\left(x^{2}-2xy+y^{2}\right)\right)=\frac{1}{4}\left(4xy\right)=xy.
  7. x + y x+y
  8. x - y x-y
  9. 9 × 9 9×9
  10. n n
  11. n < s u p > 2 / 4 ⌊n<sup>2/4⌋
  12. x x
  13. y y
  14. N N
  15. x = N + x x=N+x^{\prime}
  16. y = N + y y=N+y^{\prime}
  17. x y = ( N + x ) ( N + y ) = N 2 + N ( x + y ) + x y = N ( N + x + y ) + x y = N ( x + y ) + x y xy=\left(N+x^{\prime}\right)\left(N+y^{\prime}\right)=N^{2}+N\left(x^{\prime}+% y^{\prime}\right)+x^{\prime}y^{\prime}=N\left(N+x^{\prime}+y^{\prime}\right)+x% ^{\prime}y^{\prime}=N\left(x+y^{\prime}\right)+x^{\prime}y^{\prime}
  18. N = 100 N=100
  19. 92 8 87 13 \begin{array}[]{c|c}92&8\\ 87&13\end{array}
  20. - x -x^{\prime}
  21. - y -y^{\prime}
  22. N = 10 N=10
  23. 92 8 2 87 13 - 3 \begin{array}[]{c|c|c}92&8&2\\ 87&13&-3\end{array}
  24. ( a + b i ) ( c + d i ) = ( a c - b d ) + ( b c + a d ) i . (a+bi)(c+di)=(ac-bd)+(bc+ad)i.
  25. × a b i c a c b c i d i a d i - b d \begin{array}[]{c|c|c}\times&a&bi\\ \hline c&ac&bci\\ \hline di&adi&-bd\end{array}
  26. a = i = 0 m - 1 a i 2 w i and b = j = 0 m - 1 b j 2 w j . a=\sum_{i=0}^{m-1}{a_{i}2^{wi}}\,\text{ and }b=\sum_{j=0}^{m-1}{b_{j}2^{wj}}.
  27. a = i = 0 m - 1 a i x i and b = j = 0 m - 1 b j x j . a=\sum_{i=0}^{m-1}{a_{i}x^{i}}\,\text{ and }b=\sum_{j=0}^{m-1}{b_{j}x^{j}}.
  28. a b = i = 0 m - 1 j = 0 m - 1 a i b j x ( i + j ) = k = 0 2 m - 2 c k x k ab=\sum_{i=0}^{m-1}\sum_{j=0}^{m-1}a_{i}b_{j}x^{(i+j)}=\sum_{k=0}^{2m-2}c_{k}x% ^{k}
  29. N = 2 3 w + 1 N=2^{3w}+1

Multiplicative_function.html

  1. 1 C ( n ) 1_{C}(n)
  2. C C\subset\mathbb{Z}
  3. ϵ \epsilon
  4. ϵ \epsilon
  5. μ \mu
  6. φ \varphi
  7. φ \varphi
  8. μ \mu
  9. σ \sigma
  10. σ \sigma
  11. σ \sigma
  12. σ \sigma
  13. a ( n ) a(n)
  14. λ \lambda
  15. γ \gamma
  16. γ \gamma
  17. ω \omega
  18. ω \omega
  19. σ \sigma
  20. σ \sigma
  21. σ \sigma
  22. σ \sigma
  23. σ \sigma
  24. σ \sigma
  25. σ \sigma
  26. σ \sigma
  27. σ \sigma
  28. σ \sigma
  29. φ \varphi
  30. φ \varphi
  31. φ \varphi
  32. ( f * g ) ( n ) = d | n f ( d ) g ( n d ) (f\,*\,g)(n)=\sum_{d|n}f(d)\,g\left(\frac{n}{d}\right)
  33. ϵ \epsilon
  34. μ \mu
  35. ϵ \epsilon
  36. μ \mu
  37. ϵ \epsilon
  38. φ \varphi
  39. σ \sigma
  40. φ \varphi
  41. σ \sigma
  42. φ \varphi
  43. σ \sigma
  44. μ \mu
  45. σ \sigma
  46. μ \mu
  47. n 1 μ ( n ) n s = 1 ζ ( s ) \sum_{n\geq 1}\frac{\mu(n)}{n^{s}}=\frac{1}{\zeta(s)}
  48. n 1 φ ( n ) n s = ζ ( s - 1 ) ζ ( s ) \sum_{n\geq 1}\frac{\varphi(n)}{n^{s}}=\frac{\zeta(s-1)}{\zeta(s)}
  49. n 1 d ( n ) 2 n s = ζ ( s ) 4 ζ ( 2 s ) \sum_{n\geq 1}\frac{d(n)^{2}}{n^{s}}=\frac{\zeta(s)^{4}}{\zeta(2s)}
  50. n 1 2 ω ( n ) n s = ζ ( s ) 2 ζ ( 2 s ) \sum_{n\geq 1}\frac{2^{\omega(n)}}{n^{s}}=\frac{\zeta(s)^{2}}{\zeta(2s)}
  51. λ \lambda
  52. λ ( f g ) = λ ( f ) λ ( g ) \lambda(fg)=\lambda(f)\lambda(g)
  53. D h ( s ) = f monic h ( f ) | f | - s D_{h}(s)=\sum_{f\,\text{ monic}}h(f)|f|^{-s}
  54. g A g\in A
  55. | g | = q d e g ( g ) |g|=q^{deg(g)}
  56. g 0 g\neq 0
  57. | g | = 0 |g|=0
  58. ζ A ( s ) = f monic | f | - s \zeta_{A}(s)=\sum_{f\,\text{ monic}}|f|^{-s}
  59. 𝐍 \mathbf{N}
  60. D h ( s ) = P ( n = 0 h ( P n ) | P | - s n ) D_{h}(s)=\prod_{P}(\sum_{n\mathop{=}0}^{\infty}h(P^{n})|P|^{-sn})
  61. ζ A ( s ) = P ( 1 - | P | - s ) - 1 \zeta_{A}(s)=\prod_{P}(1-|P|^{-s})^{-1}
  62. ζ A ( s ) \zeta_{A}(s)
  63. ζ A ( s ) = f ( | f | - s ) = n deg(f)=n q - s n = n ( q n - s n ) = ( 1 - q 1 - s ) - 1 \zeta_{A}(s)=\sum_{f}(|f|^{-s})=\sum_{n}\sum_{\,\text{deg(f)=n}}q^{-sn}=\sum_{% n}(q^{n-sn})=(1-q^{1-s})^{-1}
  64. ( f * g ) ( m ) = d m f ( m ) g ( m d ) = a b = f f ( a ) g ( b ) \begin{aligned}\displaystyle(f*g)(m)&\displaystyle=\sum_{d\,\mid\,m}f(m)g\left% (\frac{m}{d}\right)\\ &\displaystyle=\sum_{ab\,=\,f}f(a)g(b)\end{aligned}
  65. D h D g = D h * g D_{h}D_{g}=D_{h*g}

Multivariate_normal_distribution.html

  1. k 2 ( 1 + ln ( 2 π ) ) + 1 2 ln | s y m b o l Σ | \frac{k}{2}(1+\ln(2\pi))+\frac{1}{2}\ln|symbol\Sigma|
  2. exp ( s y m b o l μ 𝐭 + 1 2 𝐭 s y m b o l Σ 𝐭 ) \exp\!\Big(symbol\mu^{\prime}\mathbf{t}+\tfrac{1}{2}\mathbf{t}^{\prime}symbol% \Sigma\mathbf{t}\Big)
  3. exp ( i s y m b o l μ 𝐭 - 1 2 𝐭 s y m b o l Σ 𝐭 ) \exp\!\Big(isymbol\mu^{\prime}\mathbf{t}-\tfrac{1}{2}\mathbf{t}^{\prime}symbol% \Sigma\mathbf{t}\Big)
  4. 𝐱 𝒩 ( s y m b o l μ , s y m b o l Σ ) , \mathbf{x}\ \sim\ \mathcal{N}(symbol\mu,\,symbol\Sigma),
  5. 𝐱 𝒩 k ( s y m b o l μ , s y m b o l Σ ) . \mathbf{x}\ \sim\ \mathcal{N}_{k}(symbol\mu,\,symbol\Sigma).
  6. s y m b o l μ = [ E [ X 1 ] , E [ X 2 ] , , E [ X k ] ] symbol\mu=[\operatorname{E}[X_{1}],\operatorname{E}[X_{2}],\ldots,% \operatorname{E}[X_{k}]]
  7. k × k k\times k
  8. s y m b o l Σ = [ Cov [ X i , X j ] ] , i = 1 , 2 , , k ; j = 1 , 2 , , k symbol\Sigma=[\operatorname{Cov}[X_{i},X_{j}]],i=1,2,\ldots,k;j=1,2,\ldots,k
  9. φ 𝐱 ( 𝐮 ) = exp ( i 𝐮 s y m b o l μ - 1 2 𝐮 s y m b o l Σ 𝐮 ) . \varphi_{\mathbf{x}}(\mathbf{u})=\exp\Big(i\mathbf{u}^{\prime}symbol\mu-\tfrac% {1}{2}\mathbf{u}^{\prime}symbol\Sigma\mathbf{u}\Big).
  10. s y m b o l Σ symbol\Sigma
  11. f 𝐱 ( x 1 , , x k ) = 1 ( 2 π ) k | s y m b o l Σ | exp ( - 1 2 ( 𝐱 - s y m b o l μ ) T s y m b o l Σ - 1 ( 𝐱 - s y m b o l μ ) ) , f_{\mathbf{x}}(x_{1},\ldots,x_{k})=\frac{1}{\sqrt{(2\pi)^{k}|symbol\Sigma|}}% \exp\left(-\frac{1}{2}({\mathbf{x}}-{symbol\mu})^{\mathrm{T}}{symbol\Sigma}^{-% 1}({\mathbf{x}}-{symbol\mu})\right),
  12. 𝐱 {\mathbf{x}}
  13. | s y m b o l Σ | |symbol\Sigma|
  14. s y m b o l Σ symbol\Sigma
  15. s y m b o l Σ symbol\Sigma
  16. 1 × 1 1\times 1
  17. ( 𝐱 - s y m b o l μ ) T s y m b o l Σ - 1 ( 𝐱 - s y m b o l μ ) ({\mathbf{x}}-{symbol\mu})^{\mathrm{T}}{symbol\Sigma}^{-1}({\mathbf{x}}-{% symbol\mu})
  18. 𝐱 {\mathbf{x}}
  19. s y m b o l μ {symbol\mu}
  20. k = 1 k=1
  21. f ( x , y ) \displaystyle f(x,y)
  22. σ X > 0 \sigma_{X}>0
  23. σ Y > 0 \sigma_{Y}>0
  24. s y m b o l μ = ( μ X μ Y ) , s y m b o l Σ = ( σ X 2 ρ σ X σ Y ρ σ X σ Y σ Y 2 ) . symbol\mu=\begin{pmatrix}\mu_{X}\\ \mu_{Y}\end{pmatrix},\quad symbol\Sigma=\begin{pmatrix}\sigma_{X}^{2}&\rho% \sigma_{X}\sigma_{Y}\\ \rho\sigma_{X}\sigma_{Y}&\sigma_{Y}^{2}\end{pmatrix}.
  25. y ( x ) = sgn ( ρ ) σ Y σ X ( x - μ X ) + μ Y . y(x)=\mathop{\rm sgn}(\rho)\frac{\sigma_{Y}}{\sigma_{X}}(x-\mu_{X})+\mu_{Y}.
  26. s y m b o l Σ symbol\Sigma
  27. rank ( s y m b o l Σ ) \operatorname{rank}(symbol\Sigma)
  28. 𝐱 \mathbf{x}
  29. rank ( s y m b o l Σ ) \operatorname{rank}(symbol\Sigma)
  30. k \mathbb{R}^{k}
  31. { s y m b o l μ + s y m b o l Σ 1 / 2 𝐯 : 𝐯 k } \{symbol\mu+symbol{\Sigma^{1/2}}\mathbf{v}:\mathbf{v}\in\mathbb{R}^{k}\}
  32. f ( 𝐱 ) = ( det * ( 2 \pisymbol Σ ) ) - 1 2 e - 1 2 ( 𝐱 - s y m b o l μ ) s y m b o l Σ + ( 𝐱 - s y m b o l μ ) f(\mathbf{x})=\left(\det^{*}(2\pisymbol\Sigma)\right)^{-\frac{1}{2}}\,e^{-% \frac{1}{2}(\mathbf{x}-symbol\mu)^{\prime}symbol\Sigma^{+}(\mathbf{x}-symbol% \mu)}
  33. s y m b o l Σ + symbol\Sigma^{+}
  34. μ 1 , , N ( 𝐱 ) = def μ r 1 , , r N ( 𝐱 ) = def E [ j = 1 N x j r j ] \mu_{1,\dots,N}(\mathbf{x})\ \stackrel{\mathrm{def}}{=}\ \mu_{r_{1},\dots,r_{N% }}(\mathbf{x})\ \stackrel{\mathrm{def}}{=}\ E\left[\prod\limits_{j=1}^{N}x_{j}% ^{r_{j}}\right]
  35. μ 1 , , 2 λ ( 𝐱 - s y m b o l μ ) = ( Σ i j Σ k Σ X Z ) \mu_{1,\dots,2\lambda}(\mathbf{x}-symbol\mu)=\sum\left(\Sigma_{ij}\Sigma_{k% \ell}\cdots\Sigma_{XZ}\right)
  36. { 1 , , 2 λ } \left\{1,\dots,2\lambda\right\}
  37. E [ x 1 x 2 x 3 x 4 x 5 x 6 ] \displaystyle{}E[x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}]
  38. ( 2 λ - 1 ) ! / ( 2 λ - 1 ( λ - 1 ) ! ) (2\lambda-1)!/(2^{\lambda-1}(\lambda-1)!)
  39. [ 1 , , 2 λ ] \left[1,\dots,2\lambda\right]
  40. E [ x i 4 ] = 3 Σ i i 2 E\left[x_{i}^{4}\right]=3\Sigma_{ii}^{2}
  41. E [ x i 3 x j ] = 3 Σ i i Σ i j E\left[x_{i}^{3}x_{j}\right]=3\Sigma_{ii}\Sigma_{ij}
  42. E [ x i 2 x j 2 ] = Σ i i Σ j j + 2 ( Σ i j ) 2 E\left[x_{i}^{2}x_{j}^{2}\right]=\Sigma_{ii}\Sigma_{jj}+2\left(\Sigma_{ij}% \right)^{2}
  43. E [ x i 2 x j x k ] = Σ i i Σ j k + 2 Σ i j Σ i k E\left[x_{i}^{2}x_{j}x_{k}\right]=\Sigma_{ii}\Sigma_{jk}+2\Sigma_{ij}\Sigma_{ik}
  44. E [ x i x j x k x n ] = Σ i j Σ k n + Σ i k Σ j n + Σ i n Σ j k . E\left[x_{i}x_{j}x_{k}x_{n}\right]=\Sigma_{ij}\Sigma_{kn}+\Sigma_{ik}\Sigma_{% jn}+\Sigma_{in}\Sigma_{jk}.
  45. Σ i j \Sigma_{ij}
  46. E [ x i x j x k x n ] E\left[x_{i}x_{j}x_{k}x_{n}\right]
  47. E [ x i 2 x k x n ] E\left[x_{i}^{2}x_{k}x_{n}\right]
  48. ln ( L ) = - 1 2 ln ( | s y m b o l Σ | ) - 1 2 ( 𝐱 - s y m b o l μ ) T s y m b o l Σ - 1 ( 𝐱 - s y m b o l μ ) - k 2 ln ( 2 π ) \ln(L)=-\frac{1}{2}\ln(|symbol\Sigma|\,)-\frac{1}{2}(\mathbf{x}-symbol\mu)^{% \rm T}symbol\Sigma^{-1}(\mathbf{x}-symbol\mu)-\frac{k}{2}\ln(2\pi)
  49. ln ( L ) = - ln ( | s y m b o l Σ | ) - ( 𝐳 - s y m b o l μ ) \daggersymbol Σ - 1 ( 𝐳 - s y m b o l μ ) - k ln ( π ) \ln(L)=-\ln(|symbol\Sigma|\,)-(\mathbf{z}-symbol\mu)^{\daggersymbol}\Sigma^{-1% }(\mathbf{z}-symbol\mu)-k\ln(\pi)
  50. \dagger
  51. T {}^{\rm T}
  52. h ( f ) = - - - - f ( 𝐱 ) ln f ( 𝐱 ) d 𝐱 , = 1 2 ln ( ( 2 π e ) n | s y m b o l Σ | ) , \begin{aligned}\displaystyle h\left(f\right)&\displaystyle=-\int_{-\infty}^{% \infty}\int_{-\infty}^{\infty}\cdots\int_{-\infty}^{\infty}f(\mathbf{x})\ln f(% \mathbf{x})\,d\mathbf{x},\\ &\displaystyle=\frac{1}{2}\ln\left((2\pi e)^{n}\cdot\left|symbol\Sigma\right|% \right),\\ \end{aligned}
  53. 𝒩 0 ( s y m b o l μ 0 , s y m b o l Σ 0 ) \mathcal{N}_{0}(symbol\mu_{0},symbol\Sigma_{0})
  54. 𝒩 1 ( s y m b o l μ 1 , s y m b o l Σ 1 ) \mathcal{N}_{1}(symbol\mu_{1},symbol\Sigma_{1})
  55. D KL ( 𝒩 0 𝒩 1 ) = 1 2 { tr ( s y m b o l Σ 1 - 1 s y m b o l Σ 0 ) + ( s y m b o l μ 1 - s y m b o l μ 0 ) T s y m b o l Σ 1 - 1 ( s y m b o l μ 1 - s y m b o l μ 0 ) - K + ln | s y m b o l Σ 1 | | s y m b o l Σ 0 | } , D\text{KL}(\mathcal{N}_{0}\|\mathcal{N}_{1})={1\over 2}\left\{\mathrm{tr}\left% (symbol\Sigma_{1}^{-1}symbol\Sigma_{0}\right)+\left(symbol\mu_{1}-symbol\mu_{0% }\right)^{\rm T}symbol\Sigma_{1}^{-1}(symbol\mu_{1}-symbol\mu_{0})-K+\ln{|% symbol\Sigma_{1}|\over|symbol\Sigma_{0}|}\right\},
  56. K K
  57. F ( r ) F(r)
  58. r r
  59. ( 𝐱 - s y m b o l μ ) T s y m b o l Σ - 1 ( 𝐱 - s y m b o l μ ) χ k 2 ( p ) . ({\mathbf{x}}-{symbol\mu})^{T}{symbol\Sigma}^{-1}({\mathbf{x}}-{symbol\mu})% \leq\chi^{2}_{k}(p).
  60. 𝐱 {\mathbf{x}}
  61. k k
  62. s y m b o l μ {symbol\mu}
  63. k k
  64. s y m b o l Σ symbol\Sigma
  65. χ k 2 ( p ) \chi^{2}_{k}(p)
  66. p p
  67. k k
  68. k = 2 , k=2,
  69. ρ = 0 \rho=0
  70. 𝐱 = [ 𝐱 1 𝐱 2 ] with sizes [ q × 1 ( N - q ) × 1 ] \mathbf{x}=\begin{bmatrix}\mathbf{x}_{1}\\ \mathbf{x}_{2}\end{bmatrix}\,\text{ with sizes }\begin{bmatrix}q\times 1\\ (N-q)\times 1\end{bmatrix}
  71. s y m b o l μ = [ s y m b o l μ 1 s y m b o l μ 2 ] with sizes [ q × 1 ( N - q ) × 1 ] symbol\mu=\begin{bmatrix}symbol\mu_{1}\\ symbol\mu_{2}\end{bmatrix}\,\text{ with sizes }\begin{bmatrix}q\times 1\\ (N-q)\times 1\end{bmatrix}
  72. s y m b o l Σ = [ s y m b o l Σ 11 s y m b o l Σ 12 s y m b o l Σ 21 s y m b o l Σ 22 ] with sizes [ q × q q × ( N - q ) ( N - q ) × q ( N - q ) × ( N - q ) ] symbol\Sigma=\begin{bmatrix}symbol\Sigma_{11}&symbol\Sigma_{12}\\ symbol\Sigma_{21}&symbol\Sigma_{22}\end{bmatrix}\,\text{ with sizes }\begin{% bmatrix}q\times q&q\times(N-q)\\ (N-q)\times q&(N-q)\times(N-q)\end{bmatrix}
  73. s y m b o l μ ¯ = s y m b o l μ 1 + s y m b o l Σ 12 s y m b o l Σ 22 - 1 ( 𝐚 - s y m b o l μ 2 ) \bar{symbol\mu}=symbol\mu_{1}+symbol\Sigma_{12}symbol\Sigma_{22}^{-1}\left(% \mathbf{a}-symbol\mu_{2}\right)
  74. s y m b o l Σ ¯ = s y m b o l Σ 11 - s y m b o l Σ 12 s y m b o l Σ 22 - 1 s y m b o l Σ 21 . \overline{symbol\Sigma}=symbol\Sigma_{11}-symbol\Sigma_{12}symbol\Sigma_{22}^{% -1}symbol\Sigma_{21}.
  75. s y m b o l Σ 22 - 1 symbol\Sigma_{22}^{-1}
  76. s y m b o l Σ 22 symbol\Sigma_{22}
  77. s y m b o l Σ 12 s y m b o l Σ 22 - 1 ( 𝐚 - s y m b o l μ 2 ) symbol\Sigma_{12}symbol\Sigma_{22}^{-1}\left(\mathbf{a}-symbol\mu_{2}\right)
  78. 𝒩 q ( s y m b o l μ 1 , s y m b o l Σ 11 ) \mathcal{N}_{q}\left(symbol\mu_{1},symbol\Sigma_{11}\right)
  79. 𝐱 2 \mathbf{x}_{2}
  80. 𝐲 1 = 𝐱 1 - s y m b o l Σ 12 s y m b o l Σ 22 - 1 𝐱 2 \mathbf{y}_{1}=\mathbf{x}_{1}-symbol\Sigma_{12}symbol\Sigma_{22}^{-1}\mathbf{x% }_{2}
  81. X 1 | X 2 = x 2 𝒩 ( μ 1 + σ 1 σ 2 ρ ( x 2 - μ 2 ) , ( 1 - ρ 2 ) σ 1 2 ) . X_{1}|X_{2}=x_{2}\ \sim\ \mathcal{N}\left(\mu_{1}+\frac{\sigma_{1}}{\sigma_{2}% }\rho(x_{2}-\mu_{2}),\,(1-\rho^{2})\sigma_{1}^{2}\right).
  82. ρ \rho
  83. ( X 1 X 2 ) 𝒩 ( ( μ 1 μ 2 ) , ( σ 1 2 ρ σ 1 σ 2 ρ σ 1 σ 2 σ 2 2 ) ) \begin{pmatrix}X_{1}\\ X_{2}\end{pmatrix}\sim\mathcal{N}\left(\begin{pmatrix}\mu_{1}\\ \mu_{2}\end{pmatrix},\begin{pmatrix}\sigma^{2}_{1}&\rho\sigma_{1}\sigma_{2}\\ \rho\sigma_{1}\sigma_{2}&\sigma^{2}_{2}\end{pmatrix}\right)
  84. E ( X 1 | X 2 = x 2 ) = μ 1 + ρ σ 1 σ 2 ( x 2 - μ 2 ) \operatorname{E}(X_{1}|X_{2}=x_{2})=\mu_{1}+\rho\frac{\sigma_{1}}{\sigma_{2}}(% x_{2}-\mu_{2})
  85. X 1 | X 2 X_{1}|X_{2}
  86. ( X 1 X 2 ) 𝒩 ( ( 0 0 ) , ( 1 ρ ρ 1 ) ) \begin{pmatrix}X_{1}\\ X_{2}\end{pmatrix}\sim\mathcal{N}\left(\begin{pmatrix}0\\ 0\end{pmatrix},\begin{pmatrix}1&\rho\\ \rho&1\end{pmatrix}\right)
  87. E ( X 1 X 2 = x 2 ) = ρ x 2 \operatorname{E}(X_{1}\mid X_{2}=x_{2})=\rho x_{2}
  88. E ( X 1 X 2 < z ) = - ρ ϕ ( z ) Φ ( z ) , \operatorname{E}(X_{1}\mid X_{2}<z)=-\rho{\phi(z)\over\Phi(z)},
  89. E ( X 1 X 2 > z ) = ρ ϕ ( z ) ( 1 - Φ ( z ) ) , \operatorname{E}(X_{1}\mid X_{2}>z)=\rho{\phi(z)\over(1-\Phi(z))},
  90. E ( X 1 X 2 = x 2 ) = ρ x 2 \operatorname{E}(X_{1}\mid X_{2}=x_{2})=\rho x_{2}
  91. E ( X 1 X 2 < z ) = ρ E ( X 2 X 2 < z ) \operatorname{E}(X_{1}\mid X_{2}<z)=\rho E(X_{2}\mid X_{2}<z)
  92. s y m b o l Σ = [ s y m b o l Σ 11 s y m b o l Σ 13 s y m b o l Σ 31 s y m b o l Σ 33 ] symbol\Sigma^{\prime}=\begin{bmatrix}symbol\Sigma_{11}&symbol\Sigma_{13}\\ symbol\Sigma_{31}&symbol\Sigma_{33}\end{bmatrix}
  93. 𝐱 𝒩 ( s y m b o l μ , s y m b o l Σ ) , \mathbf{x}\ \sim\mathcal{N}(symbol\mu,symbol\Sigma),
  94. M × 1 M\times 1
  95. M × N M\times N
  96. 𝐲 𝒩 ( 𝐜 + 𝐁 s y m b o l μ , 𝐁 s y m b o l Σ 𝐁 T ) \mathbf{y}\sim\mathcal{N}\left(\mathbf{c}+\mathbf{B}symbol\mu,\mathbf{B}symbol% \Sigma\mathbf{B}^{\rm T}\right)
  97. 𝐁 = [ 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 0 ] \mathbf{B}=\begin{bmatrix}1&0&0&0&0&\ldots&0\\ 0&1&0&0&0&\ldots&0\\ 0&0&0&1&0&\ldots&0\end{bmatrix}
  98. Z 𝒩 ( 𝐛 \cdotsymbol μ , 𝐛 T s y m b o l Σ 𝐛 ) Z\sim\mathcal{N}\left(\mathbf{b}\cdotsymbol\mu,\mathbf{b}^{\rm T}symbol\Sigma% \mathbf{b}\right)
  99. 𝐁 = [ b 1 b 2 b n ] = 𝐛 T . \mathbf{B}=\begin{bmatrix}b_{1}&b_{2}&\ldots&b_{n}\end{bmatrix}=\mathbf{b}^{% \rm T}.
  100. 𝐱 𝒩 ( s y m b o l μ , s y m b o l Σ ) 𝐱 s y m b o l μ + 𝐔 s y m b o l Λ 1 / 2 𝒩 ( 0 , 𝐈 ) 𝐱 s y m b o l μ + 𝐔 𝒩 ( 0 , s y m b o l Λ ) . \mathbf{x}\ \sim\mathcal{N}(symbol\mu,symbol\Sigma)\iff\mathbf{x}\ \sim symbol% \mu+\mathbf{U}symbol\Lambda^{1/2}\mathcal{N}(0,\mathbf{I})\iff\mathbf{x}\ \sim symbol% \mu+\mathbf{U}\mathcal{N}(0,symbol\Lambda).
  101. f ( 𝐱 ) = 1 ( 2 π ) k | s y m b o l Σ | exp ( - 1 2 ( 𝐱 - s y m b o l μ ) T s y m b o l Σ - 1 ( 𝐱 - s y m b o l μ ) ) f(\mathbf{x})=\frac{1}{\sqrt{(2\pi)^{k}|symbol\Sigma|}}\exp\left(-{1\over 2}(% \mathbf{x}-symbol\mu)^{\rm T}symbol\Sigma^{-1}({\mathbf{x}}-symbol\mu)\right)
  102. s y m b o l Σ ^ = 1 n i = 1 n ( 𝐱 i - 𝐱 ¯ ) ( 𝐱 i - 𝐱 ¯ ) T \widehat{symbol\Sigma}={1\over n}\sum_{i=1}^{n}({\mathbf{x}}_{i}-\overline{% \mathbf{x}})({\mathbf{x}}_{i}-\overline{\mathbf{x}})^{T}
  103. E [ s y m b o l Σ ^ ] = n - 1 n s y m b o l Σ . E[\widehat{symbol\Sigma}]=\frac{n-1}{n}symbol\Sigma.
  104. s y m b o l Σ ^ = 1 n - 1 i = 1 n ( 𝐱 i - 𝐱 ¯ ) ( 𝐱 i - 𝐱 ¯ ) T . \widehat{symbol\Sigma}={1\over n-1}\sum_{i=1}^{n}(\mathbf{x}_{i}-\overline{% \mathbf{x}})(\mathbf{x}_{i}-\overline{\mathbf{x}})^{\rm T}.
  105. 𝒲 - 1 \mathcal{W}^{-1}
  106. 𝐗 = { 𝐱 1 , , 𝐱 n } 𝒩 ( s y m b o l μ , s y m b o l Σ ) \mathbf{X}=\{\mathbf{x}_{1},\dots,\mathbf{x}_{n}\}\sim\mathcal{N}(symbol\mu,% symbol\Sigma)
  107. p ( s y m b o l μ , s y m b o l Σ ) = p ( s y m b o l μ \midsymbol Σ ) p ( s y m b o l Σ ) , p(symbol\mu,symbol\Sigma)=p(symbol\mu\midsymbol\Sigma)\ p(symbol\Sigma),
  108. p ( s y m b o l μ \midsymbol Σ ) 𝒩 ( s y m b o l μ 0 , m - 1 s y m b o l Σ ) , p(symbol\mu\midsymbol\Sigma)\sim\mathcal{N}(symbol\mu_{0},m^{-1}symbol\Sigma),
  109. p ( s y m b o l Σ ) 𝒲 - 1 ( s y m b o l Ψ , n 0 ) . p(symbol\Sigma)\sim\mathcal{W}^{-1}(symbol\Psi,n_{0}).
  110. p ( s y m b o l μ \midsymbol Σ , 𝐗 ) 𝒩 ( n 𝐱 ¯ + m s y m b o l μ 0 n + m , 1 n + m s y m b o l Σ ) , p ( s y m b o l Σ 𝐗 ) 𝒲 - 1 ( s y m b o l Ψ + n 𝐒 + n m n + m ( 𝐱 ¯ - s y m b o l μ 0 ) ( 𝐱 ¯ - s y m b o l μ 0 ) , n + n 0 ) , \begin{array}[]{rcl}p(symbol\mu\midsymbol\Sigma,\mathbf{X})&\sim&\mathcal{N}% \left(\frac{n\bar{\mathbf{x}}+msymbol\mu_{0}}{n+m},\frac{1}{n+m}symbol\Sigma% \right),\\ p(symbol\Sigma\mid\mathbf{X})&\sim&\mathcal{W}^{-1}\left(symbol\Psi+n\mathbf{S% }+\frac{nm}{n+m}(\bar{\mathbf{x}}-symbol\mu_{0})(\bar{\mathbf{x}}-symbol\mu_{0% })^{\prime},n+n_{0}\right),\end{array}
  111. 𝐱 ¯ = n - 1 i = 1 n 𝐱 i , 𝐒 = n - 1 i = 1 n ( 𝐱 i - 𝐱 ¯ ) ( 𝐱 i - 𝐱 ¯ ) . \begin{array}[]{rcl}\bar{\mathbf{x}}&=&n^{-1}\sum_{i=1}^{n}\mathbf{x}_{i},\\ \mathbf{S}&=&n^{-1}\sum_{i=1}^{n}(\mathbf{x}_{i}-\bar{\mathbf{x}})(\mathbf{x}_% {i}-\bar{\mathbf{x}})^{\prime}.\end{array}
  112. s y m b o l Σ ^ = 1 n j = 1 n ( 𝐱 j - 𝐱 ¯ ) ( 𝐱 j - 𝐱 ¯ ) T \displaystyle\widehat{symbol\Sigma}={1\over n}\sum_{j=1}^{n}\left(\mathbf{x}_{% j}-\bar{\mathbf{x}}\right)\left(\mathbf{x}_{j}-\bar{\mathbf{x}}\right)^{T}
  113. ( 50 n < 400 ) (50\leq n<400)
  114. n < 50 n<50
  115. μ β ( 𝐭 ) = ( 2 π β 2 ) - k / 2 e - | 𝐭 | 2 / ( 2 β 2 ) \scriptstyle\mu_{\beta}(\mathbf{t})=(2\pi\beta^{2})^{-k/2}e^{-|\mathbf{t}|^{2}% /(2\beta^{2})}
  116. T β \displaystyle T_{\beta}

Multivariate_random_variable.html

  1. 𝐗 = ( X 1 , , X n ) T \mathbf{X}=(X_{1},...,X_{n})^{T}
  2. ( Ω , , P ) (\Omega,\mathcal{F},P)
  3. Ω \Omega
  4. \mathcal{F}
  5. P P
  6. n \mathbb{R}^{n}
  7. X i X_{i}
  8. X i X_{i}
  9. X j X_{j}
  10. X i X_{i}
  11. X j X_{j}
  12. 𝐘 \mathbf{Y}
  13. g : n n g\colon\mathbb{R}^{n}\to\mathbb{R}^{n}
  14. 𝐗 \mathbf{X}
  15. 𝐘 = 𝒜 𝐗 + b \mathbf{Y}=\mathcal{A}\mathbf{X}+b
  16. 𝒜 \mathcal{A}
  17. n × n n\times n
  18. b b
  19. n × 1 n\times 1
  20. 𝒜 \mathcal{A}
  21. 𝐗 \textstyle\mathbf{X}
  22. f 𝐗 f_{\mathbf{X}}
  23. 𝐘 \mathbf{Y}
  24. f 𝐘 ( y ) = f 𝐗 ( 𝒜 - 1 ( y - b ) ) | det 𝒜 | f_{\mathbf{Y}}(y)=\frac{f_{\mathbf{X}}(\mathcal{A}^{-1}(y-b))}{|\det\mathcal{A% }|}
  25. 𝐗 \mathbf{X}
  26. E [ 𝐗 ] \operatorname{E}[\mathbf{X}]
  27. n × 1 n\times 1
  28. n × n n\times n
  29. i , j t h i,j^{th}
  30. i t h i^{th}
  31. j t h j^{th}
  32. n × n n\times n
  33. [ 𝐗 - E [ 𝐗 ] ] [ 𝐗 - E [ 𝐗 ] ] T [\mathbf{X}-\operatorname{E}[\mathbf{X}]][\mathbf{X}-\operatorname{E}[\mathbf{% X}]]^{T}
  34. Var [ 𝐗 ] = E [ ( 𝐗 - E [ 𝐗 ] ) ( 𝐗 - E [ 𝐗 ] ) T ] . \operatorname{Var}[\mathbf{X}]=\operatorname{E}[(\mathbf{X}-\operatorname{E}[% \mathbf{X}])(\mathbf{X}-\operatorname{E}[\mathbf{X}])^{T}].
  35. 𝐗 \mathbf{X}
  36. 𝐘 \mathbf{Y}
  37. 𝐗 \mathbf{X}
  38. n n
  39. 𝐘 \mathbf{Y}
  40. p p
  41. n × p n\times p
  42. Cov [ 𝐗 , 𝐘 ] = E [ ( 𝐗 - E [ 𝐗 ] ) ( 𝐘 - E [ 𝐘 ] ) T ] , \operatorname{Cov}[\mathbf{X},\mathbf{Y}]=\operatorname{E}[(\mathbf{X}-% \operatorname{E}[\mathbf{X}])(\mathbf{Y}-\operatorname{E}[\mathbf{Y}])^{T}],
  43. Cov [ 𝐘 , 𝐗 ] \operatorname{Cov}[\mathbf{Y},\mathbf{X}]
  44. Cov [ 𝐗 , 𝐘 ] \operatorname{Cov}[\mathbf{X},\mathbf{Y}]
  45. E ( X T A X ) = [ E ( X ) ] T A [ E ( X ) ] + tr ( A C ) , \operatorname{E}(X^{T}AX)=[\operatorname{E}(X)]^{T}A[\operatorname{E}(X)]+% \operatorname{tr}(AC),
  46. 𝐳 \mathbf{z}
  47. m × 1 m\times 1
  48. E [ 𝐳 ] = μ \operatorname{E}[\mathbf{z}]=\mu
  49. Cov [ 𝐳 ] = V \operatorname{Cov}[\mathbf{z}]=V
  50. A A
  51. m × m m\times m
  52. 𝐳 = 𝐗 \mathbf{z}^{\prime}=\mathbf{X}
  53. 𝐳 A = 𝐘 \mathbf{z}^{\prime}A^{\prime}=\mathbf{Y}
  54. Cov [ 𝐗 , 𝐘 ] = E [ 𝐗𝐘 ] - E [ 𝐗 ] E [ 𝐘 ] \operatorname{Cov}[\mathbf{X},\mathbf{Y}]=\operatorname{E}[\mathbf{X}\mathbf{Y% }^{\prime}]-\operatorname{E}[\mathbf{X}]\operatorname{E}[\mathbf{Y}]^{\prime}
  55. E ( X Y ) = Cov ( X , Y ) + E ( X ) E ( Y ) E ( z A z ) = Cov ( z , z A ) + E ( z ) E ( z A ) = Cov ( z , z A ) + μ ( μ A ) = Cov ( z , z A ) + μ A μ , \begin{aligned}\displaystyle E(XY^{\prime})&\displaystyle=\operatorname{Cov}(X% ,Y)+E(X)E(Y)^{\prime}\\ \displaystyle E(z^{\prime}Az)&\displaystyle=\operatorname{Cov}(z^{\prime},z^{% \prime}A^{\prime})+E(z^{\prime})E(z^{\prime}A^{\prime})^{\prime}\\ &\displaystyle=\operatorname{Cov}(z^{\prime},z^{\prime}A^{\prime})+\mu^{\prime% }(\mu^{\prime}A^{\prime})^{\prime}\\ &\displaystyle=\operatorname{Cov}(z^{\prime},z^{\prime}A^{\prime})+\mu^{\prime% }A\mu,\end{aligned}
  56. Cov ( z , z A ) = t ( A V ) . \operatorname{Cov}(z^{\prime},z^{\prime}A^{\prime})=\operatorname{t}(AV).
  57. Cov ( z , z A ) = E [ ( z - E ( z ) ) ( z A - E ( z A ) ) ] = E [ ( z - μ ) ( z A - μ A ) ] = E [ ( z - μ ) ( A z - A μ ) ] . \begin{aligned}\displaystyle\operatorname{Cov}(z^{\prime},z^{\prime}A^{\prime}% )&\displaystyle=E\left[\left(z^{\prime}-E(z^{\prime})\right)\left(z^{\prime}A^% {\prime}-E\left(z^{\prime}A^{\prime}\right)\right)^{\prime}\right]\\ &\displaystyle=E\left[(z^{\prime}-\mu^{\prime})(z^{\prime}A^{\prime}-\mu^{% \prime}A^{\prime})^{\prime}\right]\\ &\displaystyle=E\left[(z-\mu)^{\prime}(Az-A\mu)\right].\end{aligned}
  58. ( z - μ ) ( A z - A μ ) \left({z-\mu}\right)^{\prime}\left({Az-A\mu}\right)
  59. ( z - μ ) ( A z - A μ ) = trace [ ( z - μ ) ( A z - A μ ) ] = trace [ ( z - μ ) A ( z - μ ) ] (z-\mu)^{\prime}(Az-A\mu)=\operatorname{trace}\left[{(z-\mu)^{\prime}(Az-A\mu)% }\right]=\operatorname{trace}\left[(z-\mu)^{\prime}A(z-\mu)\right]
  60. trace [ ( z - μ ) A ( z - μ ) ] = trace [ A ( z - μ ) ( z - μ ) ] , \operatorname{trace}\left[{(z-\mu)^{\prime}A(z-\mu)}\right]=\operatorname{% trace}\left[{A(z-\mu)(z-\mu)^{\prime}}\right],
  61. Cov ( z , z A ) \displaystyle\operatorname{Cov}\left({z^{\prime},z^{\prime}A^{\prime}}\right)
  62. E [ X T A X ] [ X T B X ] = 2 trace ( A C B C ) + trace ( A C ) trace ( B C ) \operatorname{E}[X^{T}AX][X^{T}BX]=2\operatorname{trace}(ACBC)+\operatorname{% trace}(AC)\operatorname{trace}(BC)
  63. y = X β + e , y=X\beta+e,
  64. β ^ \hat{\beta}
  65. e ^ \hat{e}
  66. e ^ = y - X β ^ . \hat{e}=y-X\hat{\beta}.
  67. β ^ \hat{\beta}
  68. e ^ \hat{e}

Multivibrator.html

  1. V cap ( t ) = [ ( V capinit - V charging ) × e - t R C ] + V charging V\text{cap}(t)=\left[\left(V\text{capinit}-V\text{charging}\right)\times e^{-% \frac{t}{RC}}\right]+V\text{charging}
  2. V cap ( t ) V\text{cap}(t)
  3. V capinit V\text{capinit}
  4. V charging V\text{charging}
  5. V BE _ Q1 = ( [ ( V BE _ Q1 - V CC ) - V CC ] × e - t R C ) + V CC V_{\,\text{BE}\_\,\text{Q1}}=\left(\left[\left(V_{\,\text{BE}\_\,\text{Q1}}-V% \text{CC}\right)-V\text{CC}\right]\times e^{-\frac{t}{RC}}\right)+V\text{CC}
  6. t = - R C × ln ( V BE _ Q1 - V CC V BE _ Q1 - 2 V CC ) t=-RC\times\ln\left(\frac{V_{\,\text{BE}\_\,\text{Q1}}-V\text{CC}}{V_{\,\text{% BE}\_\,\text{Q1}}-2V\text{CC}}\right)
  7. t = - R C × ln ( - V CC - 2 V CC ) t=-RC\times\ln\left(\frac{-V\text{CC}}{-2V\text{CC}}\right)
  8. t = - R C × ln ( 1 2 ) t=-RC\times\ln\left(\frac{1}{2}\right)
  9. t = R C × ln ( 2 ) t=RC\times\ln(2)
  10. f = 1 T = 1 ln ( 2 ) ( R 2 C 1 + R 3 C 2 ) 1 0.693 ( R 2 C 1 + R 3 C 2 ) f=\frac{1}{T}=\frac{1}{\ln(2)\cdot(R_{2}C_{1}+R_{3}C_{2})}\approx\frac{1}{0.69% 3\cdot(R_{2}C_{1}+R_{3}C_{2})}
  11. f = 1 T = 1 ln ( 2 ) 2 R C 0.72 R C f=\frac{1}{T}=\frac{1}{\ln(2)\cdot 2RC}\approx\frac{0.72}{RC}

Muon.html

  1. 1 / 2 {1}/{2}
  2. τ τ
  3. Γ Γ
  4. Γ = G F 2 m μ 5 192 π 3 I ( m e 2 m μ 2 ) , \Gamma=\frac{G_{F}^{2}m_{\mu}^{5}}{192\pi^{3}}I\left(\frac{m_{e}^{2}}{m_{\mu}^% {2}}\right),
  5. I ( x ) = 1 - 8 x - 12 x 2 ln x + 8 x 3 - x 4 I(x)=1-8x-12x^{2}\ln x+8x^{3}-x^{4}
  6. G F G_{F}
  7. x = 2 E e / m μ c 2 x=2E_{e}/{m_{\mu}}c^{2}
  8. d 2 Γ d x d cos θ x 2 [ ( 3 - 2 x ) + P μ cos θ ( 1 - 2 x ) ] \frac{d^{2}\Gamma}{dx\,d\cos\theta}\sim x^{2}[(3-2x)+P_{\mu}\cos\theta(1-2x)]
  9. θ \theta
  10. 𝐏 μ \mathbf{P}_{\mu}
  11. P μ = | 𝐏 μ | P_{\mu}=|\mathbf{P}_{\mu}|
  12. d Γ d cos θ 1 - 1 3 P μ cos θ . \frac{d\Gamma}{d\cos\theta}\sim 1-\frac{1}{3}P_{\mu}\cos\theta.
  13. x < 1 x<1
  14. d Γ d x ( 3 x 2 - 2 x 3 ) . \frac{d\Gamma}{dx}\sim(3x^{2}-2x^{3}).
  15. cos θ \cos\theta
  16. cos ω t \cos\omega t
  17. ω ω
  18. ω = e g B 2 m \omega=\frac{egB}{2m}
  19. m m
  20. e e
  21. g g
  22. B B
  23. a = g - 2 2 = 0.00116592080 ( 54 ) ( 33 ) a=\frac{g-2}{2}=0.00116592080(54)(33)

Musical_note.html

  1. 2 12 \sqrt[12]{2}
  2. f = 2 n / 12 × 440 Hz f=2^{n/12}\times 440\,\,\text{Hz}\,
  3. f = 2 3 / 12 × 440 Hz 523.2 Hz f=2^{3/12}\times 440\,\,\text{Hz}\approx 523.2\,\,\text{Hz}
  4. f = 2 - 4 / 12 × 440 Hz 349.2 Hz f=2^{-4/12}\times 440\,\,\text{Hz}\approx 349.2\,\,\text{Hz}
  5. f = 2 12 k / 12 × 440 Hz = 2 k × 440 Hz f=2^{12k/12}\times 440\,\,\text{Hz}=2^{k}\times 440\,\,\text{Hz}
  6. p = 69 + 12 × log 2 f 440 Hz p=69+12\times\log_{2}{f\over 440\,\,\text{Hz}}
  7. f = 2 ( p - 69 ) / 12 × 440 Hz f=2^{(p-69)/12}\times 440\,\,\text{Hz}

Muskellunge.html

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

Mutual_recursion.html

  1. F = f 1 ( 0 ) , f 2 ( 0 ) , f 1 ( 1 ) , f 2 ( 1 ) , , F=f_{1}(0),f_{2}(0),f_{1}(1),f_{2}(1),\dots,

Mutualism_(biology).html

  1. d N d t \displaystyle\frac{dN}{dt}
  2. a x 1 + a x T H \cfrac{ax}{1+axT_{H}}
  3. d N d t = N [ r ( 1 - c N ) + β M ( X + M ) ] \frac{dN}{dt}=N[r(1-cN)+\beta M(X+M)]
  4. d N d t = N [ r ( 1 - c N ) + b a M 1 + a T H M ] \frac{dN}{dt}=N\left[r(1-cN)+\cfrac{baM}{1+aT_{H}M}\right]

Myriad.html

  1. M ¯ \overline{M}
  2. \Mu δ ϕ π β \stackrel{\delta\phi\pi\beta}{\Mu}
  3. M M ¯ \overline{MM}

N-sphere.html

  1. S n = { x n + 1 : x = r } . S^{n}=\left\{x\in\mathbb{R}^{n+1}:\left\|x\right\|=r\right\}.
  2. r 2 = i = 1 n + 1 ( x i - c i ) 2 . r^{2}=\sum_{i=1}^{n+1}(x_{i}-c_{i})^{2}.\,
  3. ω = 1 r j = 1 n + 1 ( - 1 ) j - 1 x j d x 1 d x j - 1 d x j + 1 d x n + 1 = * d r \omega={1\over r}\sum_{j=1}^{n+1}(-1)^{j-1}x_{j}\,dx_{1}\wedge\cdots\wedge dx_% {j-1}\wedge dx_{j+1}\wedge\cdots\wedge dx_{n+1}=*dr
  4. d r ω = d x 1 d x n + 1 . \scriptstyle{dr\wedge\omega=dx_{1}\wedge\cdots\wedge dx_{n+1}}.
  5. 𝕊 n = n { } \mathbb{S}^{n}=\mathbb{R}^{n}\cup\{\infty\}
  6. n \mathbb{R}^{n}
  7. V n ( R ) V_{n}(R)
  8. S n ( R ) S_{n}(R)
  9. R R
  10. V n V_{n}
  11. S n S_{n}
  12. V 0 = 1 V n + 1 = S n / ( n + 1 ) V_{0}=1\qquad V_{n+1}=S_{n}/(n+1)
  13. S 0 = 2 S n + 1 = 2 π V n S_{0}=2\qquad S_{n+1}=2\pi V_{n}
  14. S n - 1 ( R ) = n π n / 2 Γ ( n 2 + 1 ) R n - 1 V n ( R ) = π n / 2 Γ ( n 2 + 1 ) R n \begin{array}[]{ll}S_{n-1}(R)&=\displaystyle{\frac{n\pi^{n/2}}{\Gamma(\frac{n}% {2}+1)}R^{n-1}}\\ V_{n}(R)&=\displaystyle{\frac{\pi^{n/2}}{\Gamma(\frac{n}{2}+1)}}R^{n}\end{array}
  15. Γ \Gamma\,
  16. V n ( R ) = V n R n V_{n}(R)=V_{n}R^{n}
  17. S n ( R ) = S n R n S_{n}(R)=S_{n}R^{n}
  18. R R
  19. V 0 = 1 V_{0}=1
  20. [ - 1 , 1 ] [-1,1]
  21. V 1 = 2. V_{1}=2.
  22. { - 1 , 1 } \{-1,1\}
  23. S 0 = 2 S_{0}=2
  24. S 1 = 2 π . S_{1}=2\pi.\,
  25. V 2 = π . V_{2}=\pi.\,
  26. S 2 = 4 π S_{2}=4\pi\,
  27. V 3 = 4 3 π . V_{3}=\frac{4}{3}\pi.\,
  28. R R
  29. S n R n = d V n + 1 R n + 1 d R = ( n + 1 ) V n + 1 R n S_{n}R^{n}=\frac{dV_{n+1}R^{n+1}}{dR}={(n+1)V_{n+1}R^{n}}
  30. V n + 1 = 0 1 S n r n d r V_{n+1}=\int_{0}^{1}S_{n}r^{n}\,dr
  31. V n + 1 = S n n + 1 V_{n+1}=\frac{S_{n}}{n+1}
  32. r = cos θ r=\cos\theta
  33. r 2 + R 2 = 1 r^{2}+R^{2}=1
  34. R = sin θ R=\sin\theta
  35. d R = cos θ d θ dR=\cos\theta\,d\theta
  36. S n + 2 = 0 π / 2 S 1 r . S n R n d θ = 0 π / 2 S 1 . S n R n cos θ d θ = 0 1 S 1 . S n R n d R = S 1 0 1 S n R n d R = 2 π V n + 1 \begin{aligned}\displaystyle S_{n+2}&\displaystyle=\int_{0}^{\pi/2}S_{1}r.S_{n% }R^{n}\,d\theta=\int_{0}^{\pi/2}S_{1}.S_{n}R^{n}\cos\theta\,d\theta\\ &\displaystyle=\int_{0}^{1}S_{1}.S_{n}R^{n}\,dR=S_{1}\int_{0}^{1}S_{n}R^{n}\,% dR\\ &\displaystyle=2\pi V_{n+1}\end{aligned}
  37. S 1 = 2 π V 0 S_{1}=2\pi V_{0}
  38. S n + 1 = 2 π V n S_{n+1}=2\pi V_{n}
  39. V 0 = 1 V n + 1 = S n / ( n + 1 ) V_{0}=1\qquad V_{n+1}=S_{n}/(n+1)
  40. S 0 = 2 S n + 1 = 2 π V n S_{0}=2\qquad S_{n+1}=2\pi V_{n}
  41. V n + 2 = 2 π V n / ( n + 2 ) V_{n+2}=2\pi V_{n}/(n+2)
  42. V 2 k = π k k ! V_{2k}=\frac{\pi^{k}}{k!}
  43. V 2 k + 1 = 2 ( 2 π ) k ( 2 k + 1 ) ! ! = 2 k ! ( 4 π ) k ( 2 k + 1 ) ! V_{2k+1}=\frac{2(2\pi)^{k}}{(2k+1)!!}=\frac{2k!(4\pi)^{k}}{(2k+1)!}
  44. ! ! !!
  45. V n = π n 2 Γ ( n 2 + 1 ) V_{n}=\frac{\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2}+1)}
  46. Γ \Gamma\,
  47. Γ ( 1 / 2 ) = π ; Γ ( 1 ) = 1 ; Γ ( x + 1 ) = x Γ ( x ) \Gamma(1/2)=\sqrt{\pi};\Gamma(1)=1;\Gamma(x+1)=x\Gamma(x)
  48. V n V_{n}
  49. R n R^{n}
  50. R R
  51. R = 1 R=1
  52. S n - 1 = 2 π n 2 Γ ( n 2 ) S_{n-1}=\frac{2\pi^{\frac{n}{2}}}{\Gamma(\frac{n}{2})}
  53. S n - 1 = n 2 π S n - 1 + 2 S_{n-1}=\frac{n}{2\pi}S_{n-1+2}
  54. V n = 2 π n V n - 2 V_{n}=\frac{2\pi}{n}V_{n-2}
  55. S n - 1 = 2 π n - 2 S n - 1 - 2 S_{n-1}=\frac{2\pi}{n-2}S_{n-1-2}
  56. π \pi
  57. π \pi
  58. V n V_{n}
  59. V n \displaystyle V_{n}
  60. r , r\,,
  61. ϕ 1 , ϕ 2 , , ϕ n - 1 \phi_{1},\phi_{2},\dots,\phi_{n-1}\,
  62. ϕ n - 1 \phi_{n-1}\,
  63. [ 0 , 2 π ) [0,2\pi)\,
  64. [ 0 , π ] [0,\pi]\,
  65. x i \ x_{i}
  66. x 1 , , x n x_{1},\ldots,x_{n}
  67. r , ϕ 1 , , ϕ n - 1 r,\phi_{1},\ldots,\phi_{n-1}
  68. x 1 \displaystyle x_{1}
  69. r \displaystyle r
  70. x k 0 x_{k}\neq 0
  71. k k
  72. x k + 1 , , x n x_{k+1},\ldots,x_{n}
  73. ϕ k = 0 \phi_{k}=0
  74. x k > 0 x_{k}>0
  75. ϕ k = π \phi_{k}=\pi
  76. x k < 0 x_{k}<0
  77. ϕ k \phi_{k}
  78. k k
  79. x k , x k + 1 , , x n x_{k},x_{k+1},\ldots,x_{n}
  80. ϕ k \phi_{k}
  81. d n V \displaystyle d^{n}V
  82. V n = ϕ n - 1 = 0 2 π ϕ n - 2 = 0 π ϕ 1 = 0 π r = 0 R d n V . V_{n}=\int_{\phi_{n-1}=0}^{2\pi}\int_{\phi_{n-2}=0}^{\pi}\cdots\int_{\phi_{1}=% 0}^{\pi}\int_{r=0}^{R}d^{n}V.\,
  83. d S n - 1 V = sin n - 2 ( ϕ 1 ) sin n - 3 ( ϕ 2 ) sin ( ϕ n - 2 ) d ϕ 1 d ϕ 2 d ϕ n - 1 . d_{S^{n-1}}V=\sin^{n-2}(\phi_{1})\sin^{n-3}(\phi_{2})\cdots\sin(\phi_{n-2})\,d% \phi_{1}\,d\phi_{2}\cdots d\phi_{n-1}.
  84. 0 π sin n - j - 1 ( ϕ j ) C s ( ( n - j - 1 ) / 2 ) ( cos ϕ j ) C s ( ( n - j - 1 ) / 2 ) ( cos ϕ j ) d ϕ j \displaystyle{}\quad\int_{0}^{\pi}\sin^{n-j-1}(\phi_{j})C_{s}^{((n-j-1)/2)}(% \cos\phi_{j})C_{s^{\prime}}^{((n-j-1)/2)}(\cos\phi_{j})\,d\phi_{j}
  85. [ x , y , z ] \ [x,y,z]
  86. [ x 1 - z , y 1 - z ] \left[\frac{x}{1-z},\frac{y}{1-z}\right]
  87. x y \ xy
  88. [ x , y , z ] [ x 1 - z , y 1 - z ] . \ [x,y,z]\mapsto\left[\frac{x}{1-z},\frac{y}{1-z}\right].
  89. 𝐒 n - 1 \mathbf{S}^{n-1}
  90. n - 1 n-1
  91. 𝐑 n - 1 \mathbf{R}^{n-1}
  92. x n \ x_{n}
  93. [ x 1 , x 2 , , x n ] [ x 1 1 - x n , x 2 1 - x n , , x n - 1 1 - x n ] . [x_{1},x_{2},\ldots,x_{n}]\mapsto\left[\frac{x_{1}}{1-x_{n}},\frac{x_{2}}{1-x_% {n}},\ldots,\frac{x_{n-1}}{1-x_{n}}\right].
  94. 𝐱 = ( x 1 , x 2 , , x n ) \mathbf{x}=(x_{1},x_{2},\ldots,x_{n})
  95. r = x 1 2 + x 2 2 + + x n 2 . r=\sqrt{x_{1}^{2}+x_{2}^{2}+\cdots+x_{n}^{2}}.
  96. 1 r 𝐱 \frac{1}{r}\mathbf{x}
  97. Sp ( 1 ) SO ( 4 ) / SO ( 3 ) SU ( 2 ) Spin ( 3 ) \mathrm{Sp}(1)\cong\mathrm{SO}(4)/\mathrm{SO}(3)\cong\mathrm{SU}(2)\cong% \mathrm{Spin}(3)

N.html

  1. \mathbb{N}

Naive_set_theory.html

  1. S = { x : P ( x ) } S=\{x:P(x)\}
  2. P ( x ) P(x)
  3. x x
  4. P P
  5. P ( x ) " x P(x)"x
  6. { } \{\}
  7. 1 1
  8. 2 2
  9. P P
  10. x x
  11. A A
  12. P P
  13. x x
  14. 𝐙 \mathbf{Z}
  15. A A
  16. F F
  17. 2 A 2^{A}
  18. P ( A ) P(A)
  19. P ( A ) P(A)
  20. 2 n 2^{n}
  21. P ( A ) P(A)
  22. \mathbb{N}
  23. \mathbb{Z}
  24. \mathbb{Q}
  25. 𝔸 \mathbb{A}
  26. ¯ \overline{\mathbb{Q}}
  27. \mathbb{R}
  28. \mathbb{R}
  29. \mathbb{C}
  30. P P
  31. P P
  32. x x
  33. x x
  34. Y Y
  35. P P
  36. X X
  37. U U
  38. X = U X=U
  39. P ( x ) P(x)
  40. x x x∉x
  41. U U
  42. Y Y
  43. Y Y Y∈Y
  44. x x x∈x
  45. P ( x ) P(x)
  46. Y Y Y∈Y
  47. x x x∈x
  48. X X
  49. X X X∈X

Naked_singularity.html

  1. r r
  2. r ± = μ ± ( μ 2 - a 2 ) 1 / 2 r_{\pm}=\mu\pm(\mu^{2}-a^{2})^{1/2}
  3. μ = G M / c 2 \mu=GM/c^{2}
  4. a = J / M c a=J/Mc
  5. r ± r_{\pm}
  6. μ 2 < a 2 \mu^{2}<a^{2}
  7. r ± = μ ± ( μ 2 - q 2 ) 1 / 2 r_{\pm}=\mu\pm(\mu^{2}-q^{2})^{1/2}
  8. μ = G M / c 2 \mu=GM/c^{2}
  9. q = G Q 2 / ( 4 π ϵ 0 c 4 ) q=GQ^{2}/(4\pi\epsilon_{0}c^{4})
  10. μ \mu
  11. q q
  12. μ 2 < q 2 \mu^{2}<q^{2}
  13. r ± r_{\pm}
  14. r r

Nanowire.html

  1. 2 e 2 h 77.41 μ S \frac{2e^{2}}{h}\simeq 77.41\;\mu S

Nash_embedding_theorem.html

  1. ϵ \epsilon
  2. u , v = d f p ( u ) d f p ( v ) \langle u,v\rangle=df_{p}(u)\cdot df_{p}(v)

Nash_equilibrium.html

  1. ( S , f ) (S,f)
  2. n n
  3. S i S_{i}
  4. i i
  5. S = S 1 × S 2 × × S n S=S_{1}\times S_{2}\times\cdots\times S_{n}
  6. f = ( f 1 ( x ) , , f n ( x ) ) f=(f_{1}(x),\ldots,f_{n}(x))
  7. x S x\in S
  8. x i x_{i}
  9. i i
  10. x - i x_{-i}
  11. i i
  12. i { 1 , , n } i\in\{1,\ldots,n\}
  13. x i x_{i}
  14. x = ( x 1 , , x n ) x=(x_{1},\ldots,x_{n})
  15. i i
  16. f i ( x ) f_{i}(x)
  17. i i
  18. x * S x^{*}\in S
  19. i , x i S i : f i ( x i * , x - i * ) f i ( x i , x - i * ) . \forall i,x_{i}\in S_{i}:f_{i}(x^{*}_{i},x^{*}_{-i})\geq f_{i}(x_{i},x^{*}_{-i% }).
  20. x i * x^{*}_{i}
  21. S S
  22. x x
  23. A B D ABD
  24. A B C D ABCD
  25. A C D ACD
  26. A B D ABD
  27. ( 1 + x / 100 ) + 2 (1+x/100)+2
  28. x x
  29. A B AB
  30. A B D ABD
  31. A B C D ABCD
  32. A C D ACD
  33. A B AB
  34. C D CD
  35. A B D ABD
  36. A C D ACD
  37. r i ( σ - i ) r_{i}(\sigma_{-i})
  38. r i ( σ - i ) = arg max σ i u i ( σ i , σ - i ) r_{i}(\sigma_{-i})=\arg\max_{\sigma_{i}}u_{i}(\sigma_{i},\sigma_{-i})
  39. σ Σ \sigma\in\Sigma
  40. Σ = Σ i × Σ - i \Sigma=\Sigma_{i}\times\Sigma_{-i}
  41. u i u_{i}
  42. r : Σ 2 Σ r\colon\Sigma\rightarrow 2^{\Sigma}
  43. r = ( r i ( σ - i ) , r - i ( σ i ) ) r=(r_{i}(\sigma_{-i}),r_{-i}(\sigma_{i}))
  44. r r
  45. Σ \Sigma
  46. r ( σ ) r(\sigma)
  47. r ( σ ) r(\sigma)
  48. r ( σ ) r(\sigma)
  49. Σ \Sigma
  50. Σ \Sigma
  51. σ i , σ i r ( σ - i ) \sigma_{i},\sigma^{\prime}_{i}\in r(\sigma_{-i})
  52. λ σ i + ( 1 - λ ) σ i r ( σ - i ) \lambda\sigma_{i}+(1-\lambda)\sigma^{\prime}_{i}\in r(\sigma_{-i})
  53. u i u_{i}
  54. r ( σ i ) r(\sigma_{i})
  55. r r
  56. G = ( N , A , u ) G=(N,A,u)
  57. N N
  58. A = A 1 × × A N A=A_{1}\times\cdots\times A_{N}
  59. A i A_{i}
  60. Δ = Δ 1 × × Δ N \Delta=\Delta_{1}\times\cdots\times\Delta_{N}
  61. A i A_{i}
  62. Δ \Delta
  63. σ Δ \sigma\in\Delta
  64. i i
  65. a A i a\in A_{i}
  66. Gain i ( σ , a ) = max { 0 , u i ( a , σ - i ) - u i ( σ i , σ - i ) } . \,\text{Gain}_{i}(\sigma,a)=\max\{0,u_{i}(a,\sigma_{-i})-u_{i}(\sigma_{i},% \sigma_{-i})\}.
  67. g = ( g 1 , , g N ) g=(g_{1},\ldots,g_{N})
  68. g i ( σ ) ( a ) = σ i ( a ) + Gain i ( σ , a ) g_{i}(\sigma)(a)=\sigma_{i}(a)+\,\text{Gain}_{i}(\sigma,a)
  69. σ Δ , a A i \sigma\in\Delta,a\in A_{i}
  70. a A i g i ( σ ) ( a ) = a A i σ i ( a ) + Gain i ( σ , a ) = 1 + a A i Gain i ( σ , a ) > 0. \sum_{a\in A_{i}}g_{i}(\sigma)(a)=\sum_{a\in A_{i}}\sigma_{i}(a)+\,\text{Gain}% _{i}(\sigma,a)=1+\sum_{a\in A_{i}}\,\text{Gain}_{i}(\sigma,a)>0.
  71. g g
  72. f : Δ Δ f\colon\Delta\rightarrow\Delta
  73. f i ( σ ) ( a ) = g i ( σ ) ( a ) b A i g i ( σ ) ( b ) f_{i}(\sigma)(a)=\frac{g_{i}(\sigma)(a)}{\sum_{b\in A_{i}}g_{i}(\sigma)(b)}
  74. a A i a\in A_{i}
  75. f i f_{i}
  76. Δ i \Delta_{i}
  77. f i f_{i}
  78. σ \sigma
  79. f f
  80. Δ \Delta
  81. Δ \Delta
  82. f f
  83. f f
  84. Δ \Delta
  85. σ * \sigma^{*}
  86. σ * \sigma^{*}
  87. G G
  88. 1 i N , a A i , Gain i ( σ * , a ) = 0 . \forall 1\leq i\leq N,~{}\forall a\in A_{i},~{}\,\text{Gain}_{i}(\sigma^{*},a)% =0\,\text{.}
  89. i \exists i
  90. 1 i N 1\leq i\leq N
  91. a A i a\in A_{i}
  92. Gain i ( σ * , a ) > 0 \,\text{Gain}_{i}(\sigma^{*},a)>0
  93. a A i g i ( σ * , a ) = 1 + a A i Gain i ( σ * , a ) > 1. \sum_{a\in A_{i}}g_{i}(\sigma^{*},a)=1+\sum_{a\in A_{i}}\,\text{Gain}_{i}(% \sigma^{*},a)>1.
  94. C = a A i g i ( σ * , a ) C=\sum_{a\in A_{i}}g_{i}(\sigma^{*},a)
  95. Gain ( i , ) \,\text{Gain}(i,\cdot)
  96. A i A_{i}
  97. f ( σ * ) = σ * f(\sigma^{*})=\sigma^{*}
  98. f i ( σ * ) = σ i * f_{i}(\sigma^{*})=\sigma^{*}_{i}
  99. σ i * = g i ( σ * ) a A i g i ( σ * ) ( a ) σ i * = σ i * + Gain i ( σ * , ) C C σ i * = σ i * + Gain i ( σ * , ) \sigma^{*}_{i}=\frac{g_{i}(\sigma^{*})}{\sum_{a\in A_{i}}g_{i}(\sigma^{*})(a)}% \Rightarrow\sigma^{*}_{i}=\frac{\sigma^{*}_{i}+\,\text{Gain}_{i}(\sigma^{*},% \cdot)}{C}\Rightarrow C\sigma^{*}_{i}=\sigma^{*}_{i}+\,\text{Gain}_{i}(\sigma^% {*},\cdot)
  100. ( C - 1 ) σ i * = Gain i ( σ * , ) σ i * = ( 1 C - 1 ) Gain i ( σ * , ) . \left(C-1\right)\sigma^{*}_{i}=\,\text{Gain}_{i}(\sigma^{*},\cdot)\Rightarrow% \sigma^{*}_{i}=\left(\frac{1}{C-1}\right)\,\text{Gain}_{i}(\sigma^{*},\cdot).
  101. C > 1 C>1
  102. σ i * \sigma^{*}_{i}
  103. Gain i ( σ * , ) \,\text{Gain}_{i}(\sigma^{*},\cdot)
  104. σ i * ( a ) ( u i ( a i , σ - i * ) - u i ( σ i * , σ - i * ) ) = σ i * ( a ) Gain i ( σ * , a ) \sigma^{*}_{i}(a)(u_{i}(a_{i},\sigma^{*}_{-i})-u_{i}(\sigma^{*}_{i},\sigma^{*}% _{-i}))=\sigma^{*}_{i}(a)\,\text{Gain}_{i}(\sigma^{*},a)
  105. a A i \forall a\in A_{i}
  106. Gain i ( σ * , a ) > 0 \,\text{Gain}_{i}(\sigma^{*},a)>0
  107. Gain i ( σ * , a ) = 0 \,\text{Gain}_{i}(\sigma^{*},a)=0
  108. σ i * ( a ) = ( 1 C - 1 ) Gain i ( σ * , a ) = 0 \sigma^{*}_{i}(a)=\left(\frac{1}{C-1}\right)\,\text{Gain}_{i}(\sigma^{*},a)=0
  109. 0
  110. 0 = u i ( σ i * , σ - i * ) - u i ( σ i * , σ - i * ) 0=u_{i}(\sigma^{*}_{i},\sigma^{*}_{-i})-u_{i}(\sigma^{*}_{i},\sigma^{*}_{-i})
  111. = ( a A i σ i * ( a ) u i ( a i , σ - i * ) ) - u i ( σ i * , σ - i * ) =\left(\sum_{a\in A_{i}}\sigma^{*}_{i}(a)u_{i}(a_{i},\sigma^{*}_{-i})\right)-u% _{i}(\sigma^{*}_{i},\sigma^{*}_{-i})
  112. = a A i σ i * ( a ) ( u i ( a i , σ - i * ) - u i ( σ i * , σ - i * ) ) =\sum_{a\in A_{i}}\sigma^{*}_{i}(a)(u_{i}(a_{i},\sigma^{*}_{-i})-u_{i}(\sigma^% {*}_{i},\sigma^{*}_{-i}))
  113. = a A i σ i * ( a ) Gain i ( σ * , a ) by the previous statements =\sum_{a\in A_{i}}\sigma^{*}_{i}(a)\,\text{Gain}_{i}(\sigma^{*},a)\quad\,\text% { by the previous statements }
  114. = a A i ( C - 1 ) σ i * ( a ) 2 > 0 =\sum_{a\in A_{i}}\left(C-1\right)\sigma^{*}_{i}(a)^{2}>0
  115. σ i * \sigma^{*}_{i}
  116. σ * \sigma^{*}
  117. G G
  118. s A s_{A}
  119. s A s_{A}
  120. s A s_{A}
  121. s A s_{A}
  122. s A s_{A}
  123. s A s_{A}

Natural_deduction.html

  1. A B A\wedge B
  2. A prop B prop ( A B ) prop F \frac{A\hbox{ prop}\qquad B\hbox{ prop}}{(A\wedge B)\hbox{ prop}}\ \wedge_{F}
  3. A prop B prop A B prop F \frac{A\hbox{ prop}\qquad B\hbox{ prop}}{A\wedge B\hbox{ prop}}\ \wedge_{F}
  4. J 1 J 2 J n J name \frac{J_{1}\qquad J_{2}\qquad\cdots\qquad J_{n}}{J}\ \hbox{name}
  5. J i J_{i}
  6. A B A\vee B
  7. ¬ A \neg A
  8. A B A\supset B
  9. \top
  10. \bot
  11. A prop B prop A B prop F A prop B prop A B prop F prop F prop F \frac{A\hbox{ prop}\qquad B\hbox{ prop}}{A\vee B\hbox{ prop}}\ \vee_{F}\qquad% \frac{A\hbox{ prop}\qquad B\hbox{ prop}}{A\supset B\hbox{ prop}}\ \supset_{F}% \qquad\frac{\hbox{ }}{\top\hbox{ prop}}\ \top_{F}\qquad\frac{\hbox{ }}{\bot% \hbox{ prop}}\ \bot_{F}
  12. A prop ¬ A prop ¬ F \qquad\frac{A\hbox{ prop}}{\neg A\hbox{ prop}}\ \neg_{F}
  13. A true B true ( A B ) true I \frac{A\hbox{ true}\qquad B\hbox{ true}}{(A\wedge B)\hbox{ true}}\ \wedge_{I}
  14. A prop B prop A true B true ( A B ) true I \frac{A\hbox{ prop}\qquad B\hbox{ prop}\qquad A\hbox{ true}\qquad B\hbox{ true% }}{(A\wedge B)\hbox{ true}}\ \wedge_{I}
  15. A B prop A true B true ( A B ) true I \frac{A\wedge B\hbox{ prop}\qquad A\hbox{ true}\qquad B\hbox{ true}}{(A\wedge B% )\hbox{ true}}\ \wedge_{I}
  16. F \wedge_{F}
  17. true I \frac{\ }{\top\hbox{ true}}\ \top_{I}
  18. A true A B true I 1 B true A B true I 2 \frac{A\hbox{ true}}{A\vee B\hbox{ true}}\ \vee_{I1}\qquad\frac{B\hbox{ true}}% {A\vee B\hbox{ true}}\ \vee_{I2}
  19. A B true A true E 1 A B true B true E 2 \frac{A\wedge B\hbox{ true}}{A\hbox{ true}}\ \wedge_{E1}\qquad\frac{A\wedge B% \hbox{ true}}{B\hbox{ true}}\ \wedge_{E2}
  20. A B true B true E 2 A B true A true E 1 B A true I \cfrac{\cfrac{A\wedge B\hbox{ true}}{B\hbox{ true}}\ \wedge_{E2}\qquad\cfrac{A% \wedge B\hbox{ true}}{A\hbox{ true}}\ \wedge_{E1}}{B\wedge A\hbox{ true}}\ % \wedge_{I}
  21. A ( B C ) t r u e B C t r u e B t r u e E 1 E 2 \cfrac{A\wedge\left(B\wedge C\right)\ true}{\cfrac{B\wedge C\ true}{B\ true}% \wedge_{E_{1}}}\wedge_{E_{2}}
  22. A ( B C ) t r u e B t r u e \begin{matrix}A\wedge\left(B\wedge C\right)\ true\\ \vdots\\ B\ true\end{matrix}
  23. D 1 D 2 D n J \begin{matrix}D_{1}\quad D_{2}\cdots D_{n}\\ \vdots\\ J\end{matrix}
  24. A t r u e u B t r u e A B t r u e I u A B t r u e A t r u e B t r u e E \cfrac{\begin{matrix}\cfrac{}{A\ true}u\\ \vdots\\ B\ true\end{matrix}}{A\supset B\ true}\supset_{I^{u}}\qquad\cfrac{A\supset B\ % true\quad A\ true}{B\ true}\supset_{E}
  25. < m t p l > A t r u e u B t r u e w A B t r u e I B ( A B ) t r u e A ( B ( A B ) ) t r u e I u I w \cfrac{\cfrac{\cfrac{<}{m}tpl>{{}}{A\ true}u\quad\cfrac{{}}{B\ true}w}{A\wedge B% \ true}\wedge_{I}}{\cfrac{B\supset\left(A\wedge B\right)\ true}{A\supset\left(% B\supset\left(A\wedge B\right)\right)\ true}\supset_{I^{u}}}\supset_{I^{w}}
  26. A B true A t r u e u C t r u e B t r u e w C t r u e C t r u e E u , w \cfrac{A\vee B\hbox{ true}\quad\begin{matrix}\cfrac{}{A\ true}u\\ \vdots\\ C\ true\end{matrix}\quad\begin{matrix}\cfrac{}{B\ true}w\\ \vdots\\ C\ true\end{matrix}}{C\ true}\vee_{E^{u,w}}
  27. t r u e C t r u e E \frac{\perp true}{C\ true}\perp_{E}
  28. A t r u e u p t r u e ¬ A t r u e ¬ I u , p ¬ A t r u e A t r u e C t r u e ¬ E \cfrac{\begin{matrix}\cfrac{}{A\ true}u\\ \vdots\\ p\ true\end{matrix}}{\lnot A\ true}\lnot_{I^{u,p}}\qquad\cfrac{\lnot A\ true% \quad A\ true}{C\ true}\lnot_{E}

Natural_logarithm.html

  1. ln ( e x ) = x . \ln(e^{x})=x.
  2. ln ( x y ) = ln ( x ) + ln ( y ) . \ln(xy)=\ln(x)+\ln(y).
  3. ln : + . \ln\colon\mathbb{R}^{+}\to\mathbb{R}.
  4. ln x \ln x
  5. e x e^{x}
  6. 1 x \frac{1}{x}
  7. x ln x - x + C x\ln x-x+C
  8. d d x log b ( x ) = d d x ( 1 ln ( b ) ln x ) = 1 ln ( b ) d d x ln x = 1 x ln ( b ) . \frac{d}{dx}\log_{b}(x)=\frac{d}{dx}\left(\frac{1}{\ln(b)}\ln{x}\right)=\frac{% 1}{\ln(b)}\frac{d}{dx}\ln{x}=\frac{1}{x\ln(b)}.
  9. ln ( a ) = 1 a 1 x d x . \ln(a)=\int_{1}^{a}\frac{1}{x}\,dx.
  10. ln ( a b ) = ln ( a ) + ln ( b ) . \ln(ab)=\ln(a)+\ln(b).\,\!
  11. ln ( a b ) = 1 a b 1 x d x \displaystyle\ln(ab)=\int_{1}^{ab}\frac{1}{x}\;dx
  12. ln ( 1 ) = 0 \ln(1)=0
  13. ln ( e ) = 1 \ln(e)=1
  14. ln ( x y ) = ln ( x ) + ln ( y ) , for x > 0 , y > 0 \ln(xy)=\ln(x)+\ln(y),\quad\,\text{for}\quad x>0,y>0
  15. ln ( x ) < ln ( y ) for 0 < x < y \ln(x)<\ln(y)\quad{\rm for}\quad 0<x<y
  16. lim x 0 ln ( 1 + x ) x = 1 \lim_{x\to 0}\frac{\ln(1+x)}{x}=1
  17. ln ( x y ) = y ln ( x ) for x > 0 \ln(x^{y})=y\,\ln(x)\quad\,\text{for}\quad x>0
  18. x - 1 x ln ( x ) x - 1 for x > 0 \frac{x-1}{x}\leq\ln(x)\leq x-1\quad{\rm for}\quad x>0
  19. ln ( 1 + x α ) α x for x 0 , α 1 \ln{(1+x^{\alpha})}\leq\alpha x\quad{\rm for}\quad x\geq 0,\alpha\geq 1
  20. x = 0 x=0
  21. d d x ln ( 1 + x α ) d d x ( α x ) \frac{d}{dx}\ln{(1+x^{\alpha})}\leq\frac{d}{dx}(\alpha x)
  22. x x
  23. d d x ln ( 1 + x α ) = α x α - 1 1 + x α α = d d x ( α x ) \frac{d}{dx}\ln{(1+x^{\alpha})}=\frac{\alpha x^{\alpha-1}}{1+x^{\alpha}}\leq% \alpha=\frac{d}{dx}(\alpha x)
  24. ( 1 + x α ) / α (1+x^{\alpha})/\alpha
  25. x α x^{\alpha}
  26. x α - 1 x α + 1 x^{\alpha-1}\leq x^{\alpha}+1
  27. x α - 1 ( 1 - x ) 1 x^{\alpha-1}(1-x)\leq 1
  28. x 1 x\geq 1
  29. 0 x < 1 0\leq x<1
  30. α 1 \alpha\geq 1
  31. d d x ln ( 1 + x α ) d d x ( α x ) \frac{d}{dx}\ln{(1+x^{\alpha})}\leq\frac{d}{dx}(\alpha x)
  32. x x
  33. d d x ln ( x ) = 1 x . \frac{d}{dx}\ln(x)=\frac{1}{x}.\,
  34. d d x ln ( x ) = lim h 0 ln ( x + h ) - ln ( x ) h \frac{d}{dx}\ln(x)=\lim_{h\to 0}\frac{\ln(x+h)-\ln(x)}{h}
  35. = lim h 0 ln ( x + h x ) h =\lim_{h\to 0}\frac{\ln(\frac{x+h}{x})}{h}
  36. = lim h 0 [ 1 h ln ( 1 + h x ) ] =\lim_{h\to 0}\left[\frac{1}{h}\ln\left(1+\frac{h}{x}\right)\right]\quad
  37. = lim h 0 ln ( 1 + h x ) 1 h =\lim_{h\to 0}\ln\left(1+\frac{h}{x}\right)^{\frac{1}{h}}
  38. Let u = h x u x = h \,\text{Let }u=\frac{h}{x}\Rightarrow ux=h
  39. 1 h = 1 u x \frac{1}{h}=\frac{1}{ux}
  40. d d x ln ( x ) = lim u 0 ln ( 1 + u ) 1 u x \frac{d}{dx}\ln(x)=\lim_{u\to 0}\ln(1+u)^{\frac{1}{ux}}
  41. = lim u 0 ln [ ( 1 + u ) 1 u ] 1 x =\lim_{u\to 0}\ln\left[(1+u)^{\frac{1}{u}}\right]^{\frac{1}{x}}
  42. = 1 x lim u 0 ln ( 1 + u ) 1 u =\frac{1}{x}\lim_{u\to 0}\ln(1+u)^{\frac{1}{u}}
  43. Let n = 1 u u = 1 n \text{Let }n=\frac{1}{u}\Rightarrow u=\frac{1}{n}
  44. d d x ln ( x ) = 1 x lim n ln ( 1 + 1 n ) n \frac{d}{dx}\ln(x)=\frac{1}{x}\lim_{n\to\infty}\ln\left(1+\frac{1}{n}\right)^{n}
  45. = 1 x ln [ lim n ( 1 + 1 n ) n ] =\frac{1}{x}\ln\left[\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}\right]
  46. = 1 x ln e =\frac{1}{x}\ln e
  47. = 1 x =\frac{1}{x}
  48. ln ( 1 + x ) = n = 1 ( - 1 ) n + 1 n x n = x - x 2 2 + x 3 3 - for | x | 1 , \ln(1+x)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}x^{n}=x-\frac{x^{2}}{2}+\frac{% x^{3}}{3}-\cdots\quad{\rm for}\quad\left|x\right|\leq 1,\quad
  49. unless x = - 1. {\rm unless}\quad x=-1.
  50. ln ( x ) = n = 1 ( - 1 ) n + 1 n ( x - 1 ) n \ln(x)=\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}(x-1)^{n}
  51. ln ( x ) = ( x - 1 ) - ( x - 1 ) 2 2 + ( x - 1 ) 3 3 - ( x - 1 ) 4 4 \ln(x)=(x-1)-\frac{(x-1)^{2}}{2}+\frac{(x-1)^{3}}{3}-\frac{(x-1)^{4}}{4}\cdots
  52. for | x - 1 | 1 unless x = 0 . {\rm for}\quad\left|x-1\right|\leq 1\quad{\rm unless}\quad x=0\,.
  53. ln x x - 1 = n = 1 1 n x n = 1 x + 1 2 x 2 + 1 3 x 3 + . \ln{x\over{x-1}}=\sum_{n=1}^{\infty}{1\over{nx^{n}}}={1\over x}+{1\over{2x^{2}% }}+{1\over{3x^{3}}}+\cdots\,.
  54. x x - 1 x\over{x-1}
  55. y y - 1 y\over{y-1}
  56. ln x = n = 1 1 n ( x - 1 x ) n = ( x - 1 x ) + 1 2 ( x - 1 x ) 2 + 1 3 ( x - 1 x ) 3 + \ln{x}=\sum_{n=1}^{\infty}{1\over{n}}\left({x-1\over x}\right)^{n}=\left({x-1% \over x}\right)+{1\over 2}\left({x-1\over x}\right)^{2}+{1\over 3}\left({x-1% \over x}\right)^{3}+\cdots\,
  57. for Re ( x ) 1 2 . {\rm for}\quad\operatorname{Re}(x)\geq\frac{1}{2}\,.
  58. d d x ln | x | = 1 x . \ {d\over dx}\ln\left|x\right|={1\over x}.
  59. 1 x d x = ln | x | + C \int{1\over x}dx=\ln|x|+C
  60. f ( x ) f ( x ) d x = ln | f ( x ) | + C . \int{\frac{f^{\prime}(x)}{f(x)}\,dx}=\ln|f(x)|+C.
  61. tan ( x ) d x = sin ( x ) cos ( x ) d x \int\tan(x)\,dx=\int{\sin(x)\over\cos(x)}\,dx
  62. tan ( x ) d x = - d d x cos ( x ) cos ( x ) d x . \int\tan(x)\,dx=\int{-{d\over dx}\cos(x)\over{\cos(x)}}\,dx.
  63. tan ( x ) d x = - ln | cos ( x ) | + C \int\tan(x)\,dx=-\ln{\left|\cos(x)\right|}+C
  64. tan ( x ) d x = ln | sec ( x ) | + C \int\tan(x)\,dx=\ln{\left|\sec(x)\right|}+C
  65. ln ( x ) d x = x ln ( x ) - x + C . \int\ln(x)\,dx=x\ln(x)-x+C.
  66. u = ln ( x ) d u = d x x u=\ln(x)\Rightarrow du=\frac{dx}{x}
  67. d v = d x v = x dv=dx\Rightarrow v=x\,
  68. ln ( x ) d x \displaystyle\int\ln(x)\,dx
  69. ln ( 1 + x ) = x ( 1 1 - x ( 1 2 - x ( 1 3 - x ( 1 4 - x ( 1 5 - ) ) ) ) ) for | x | < 1. \ln(1+x)=x\,\left(\frac{1}{1}-x\,\left(\frac{1}{2}-x\,\left(\frac{1}{3}-x\,% \left(\frac{1}{4}-x\,\left(\frac{1}{5}-\cdots\right)\right)\right)\right)% \right)\quad{\rm for}\quad\left|x\right|<1.\,\!
  70. ln ( x ) = ln ( 1 + y 1 - y ) = 2 y ( 1 1 + 1 3 y 2 + 1 5 y 4 + 1 7 y 6 + 1 9 y 8 + ) = 2 y ( 1 1 + y 2 ( 1 3 + y 2 ( 1 5 + y 2 ( 1 7 + y 2 ( 1 9 + ) ) ) ) ) \begin{aligned}\displaystyle\ln(x)=\ln\left(\frac{1+y}{1-y}\right)&% \displaystyle=2\,y\,\left(\frac{1}{1}+\frac{1}{3}y^{2}+\frac{1}{5}y^{4}+\frac{% 1}{7}y^{6}+\frac{1}{9}y^{8}+\cdots\right)\\ &\displaystyle=2\,y\,\left(\frac{1}{1}+y^{2}\,\left(\frac{1}{3}+y^{2}\,\left(% \frac{1}{5}+y^{2}\,\left(\frac{1}{7}+y^{2}\,\left(\frac{1}{9}+\cdots\right)% \right)\right)\right)\right)\end{aligned}
  71. ln ( 123.456 ) \displaystyle\ln(123.456)
  72. ln ( a × 10 n ) = ln a + n ln 10. \ln(a\times 10^{n})=\ln a+n\ln 10.
  73. [ 1 , 10 ) [1,10)
  74. y n + 1 = y n + 2 x - exp ( y n ) x + exp ( y n ) y_{n+1}=y_{n}+2\cdot\frac{x-\exp(y_{n})}{x+\exp(y_{n})}\,
  75. ln x π 2 M ( 1 , 4 / s ) - m ln 2 , \ln x\approx\frac{\pi}{2M(1,4/s)}-m\ln 2,
  76. s = x 2 m > 2 p / 2 , s=x\,2^{m}>2^{p/2},
  77. ln ( 1 + x ) = x 1 1 - x 2 2 + x 3 3 - x 4 4 + x 5 5 - = x 1 - 0 x + 1 2 x 2 - 1 x + 2 2 x 3 - 2 x + 3 2 x 4 - 3 x + 4 2 x 5 - 4 x + \ln(1+x)=\frac{x^{1}}{1}-\frac{x^{2}}{2}+\frac{x^{3}}{3}-\frac{x^{4}}{4}+\frac% {x^{5}}{5}-\cdots=\cfrac{x}{1-0x+\cfrac{1^{2}x}{2-1x+\cfrac{2^{2}x}{3-2x+% \cfrac{3^{2}x}{4-3x+\cfrac{4^{2}x}{5-4x+\ddots}}}}}
  78. ln ( 1 + x y ) = x y + 1 x 2 + 1 x 3 y + 2 x 2 + 2 x 5 y + 3 x 2 + = 2 x 2 y + x - ( 1 x ) 2 3 ( 2 y + x ) - ( 2 x ) 2 5 ( 2 y + x ) - ( 3 x ) 2 7 ( 2 y + x ) - \ln\left(1+\frac{x}{y}\right)=\cfrac{x}{y+\cfrac{1x}{2+\cfrac{1x}{3y+\cfrac{2x% }{2+\cfrac{2x}{5y+\cfrac{3x}{2+\ddots}}}}}}=\cfrac{2x}{2y+x-\cfrac{(1x)^{2}}{3% (2y+x)-\cfrac{(2x)^{2}}{5(2y+x)-\cfrac{(3x)^{2}}{7(2y+x)-\ddots}}}}
  79. ln 2 = 3 ln ( 1 + 1 4 ) + ln ( 1 + 3 125 ) = 6 9 - 1 2 27 - 2 2 45 - 3 2 63 - + 6 253 - 3 2 759 - 6 2 1265 - 9 2 1771 - . \ln 2=3\ln\left(1+\frac{1}{4}\right)+\ln\left(1+\frac{3}{125}\right)=\cfrac{6}% {9-\cfrac{1^{2}}{27-\cfrac{2^{2}}{45-\cfrac{3^{2}}{63-\ddots}}}}+\cfrac{6}{253% -\cfrac{3^{2}}{759-\cfrac{6^{2}}{1265-\cfrac{9^{2}}{1771-\ddots}}}}.
  80. ln 10 = 10 ln ( 1 + 1 4 ) + 3 ln ( 1 + 3 125 ) = 20 9 - 1 2 27 - 2 2 45 - 3 2 63 - + 18 253 - 3 2 759 - 6 2 1265 - 9 2 1771 - . \ln 10=10\ln\left(1+\frac{1}{4}\right)+3\ln\left(1+\frac{3}{125}\right)=\cfrac% {20}{9-\cfrac{1^{2}}{27-\cfrac{2^{2}}{45-\cfrac{3^{2}}{63-\ddots}}}}+\cfrac{18% }{253-\cfrac{3^{2}}{759-\cfrac{6^{2}}{1265-\cfrac{9^{2}}{1771-\ddots}}}}.

Natural_number.html

  1. \mathbb{N}
  2. ( 0 ) (\aleph_{0})
  3. * *
  4. 1 1
  5. 0 = 0 = { 0 , 1 , 2 , } \mathbb{N}^{0}=\mathbb{N}_{0}=\{0,1,2,\ldots\}
  6. * = + = 1 = > 0 = { 1 , 2 , } . \mathbb{N}^{*}=\mathbb{N}^{+}=\mathbb{N}_{1}=\mathbb{N}_{>0}=\{1,2,\ldots\}.
  7. 0 \aleph_{0}
  8. ω \omega
  9. 0 \aleph_{0}
  10. ω \omega
  11. 0 \aleph_{0}
  12. ω \omega
  13. \mathbb{N}
  14. 0 ω 0\in\omega
  15. 0 = 0=\varnothing
  16. ω \omega

Natural_selection.html

  1. d N d t = r N ( 1 - N K ) \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)\qquad\!

Natural_transformation.html

  1. η Y F ( f ) = G ( f ) η X \eta_{Y}\circ F(f)=G(f)\circ\eta_{X}
  2. η : X G ( X ) , \eta\colon X\to G(X),
  3. A : X X A\colon X\to X
  4. η A G ( A ) η \eta\circ A\neq G(A)\circ\eta
  5. π n ( ( X , x 0 ) × ( Y , y 0 ) ) π n ( ( X , x 0 ) ) × π n ( ( Y , y 0 ) ) , \pi_{n}((X,x_{0})\times(Y,y_{0}))\cong\pi_{n}((X,x_{0}))\times\pi_{n}((Y,y_{0}% )),
  6. π 1 ( T , t 0 ) 𝐙 × 𝐙 \pi_{1}(T,t_{0})\approx\mathbf{Z}\times\mathbf{Z}
  7. \approx
  8. \cong
  9. = =
  10. π 1 ( T , t 0 ) π 1 ( S 1 , x 0 ) × π 1 ( S 1 , y 0 ) 𝐙 × 𝐙 = 𝐙 2 . \pi_{1}(T,t_{0})\approx\pi_{1}(S^{1},x_{0})\times\pi_{1}(S^{1},y_{0})\cong% \mathbf{Z}\times\mathbf{Z}=\mathbf{Z}^{2}.
  11. ( 1 1 0 1 ) \left(\begin{smallmatrix}1&1\\ 0&1\end{smallmatrix}\right)
  12. ( T , t 0 ) = ( S 1 , x 0 ) × ( S 1 , y 0 ) (T,t_{0})=(S^{1},x_{0})\times(S^{1},y_{0})
  13. η V : V V * . \eta_{V}\colon V\to V^{*}.
  14. b V : V × V K . b_{V}\colon V\times V\to K.
  15. T * ( η T ( V ) ( T ( v ) ) ) = η V ( v ) T^{*}(\eta_{T(V)}(T(v)))=\eta_{V}(v)
  16. b T ( V ) ( T ( v ) , T ( w ) ) = b V ( v , w ) . b_{T(V)}(T(v),T(w))=b_{V}(v,w).
  17. ( H η ) X = H η X . (H\eta)_{X}=H\eta_{X}.
  18. ( η K ) X = η K ( X ) . (\eta K)_{X}=\eta_{K(X)}.\,
  19. C C
  20. D D
  21. C C
  22. D D
  23. F : C D F:C\to D
  24. G : C D G:C\to D
  25. F F
  26. G G
  27. C I C^{I}

Navier–Stokes_equations.html

  1. 𝐲 t + 𝐀 ( 𝐲 ) 𝐲 x = 0 \mathbf{y}_{t}+\mathbf{A}(\mathbf{y})\mathbf{y}_{x}=0
  2. ρ \rho
  3. 𝐮 \mathbf{u}
  4. \nabla
  5. p p
  6. 𝐈 \mathbf{I}
  7. s y m b o l τ symbol\tau
  8. 𝐠 \mathbf{g}
  9. 𝐮 \nabla\mathbf{u}
  10. 𝐕 \mathbf{V}
  11. s y m b o l ε ( 𝐮 ) 1 2 𝐮 + 1 2 ( 𝐮 ) T symbol\varepsilon(\nabla\mathbf{u})\equiv\tfrac{1}{2}\nabla\mathbf{u}+\tfrac{1% }{2}\left(\nabla\mathbf{u}\right)^{\mathrm{T}}
  12. μ μ
  13. s y m b o l τ = μ ( 2 ε ) = μ ( 𝐮 + ( 𝐮 ) T ) = μ 2 𝐮 \nabla\cdot symbol\tau=\mu\nabla\cdot(2\varepsilon)=\mu\nabla\cdot(\nabla% \mathbf{u}+(\nabla\mathbf{u})^{\mathrm{T}})=\mu\nabla^{2}\mathbf{u}
  14. 1 ρ 0 p = ( p ρ 0 ) w \frac{1}{\rho_{0}}\nabla p=\nabla\left(\frac{p}{\rho_{0}}\right)\equiv\nabla w
  15. w w
  16. u ( x ) = 0 u(x)=0
  17. u ( y ) = u u(y)=u
  18. u ( z ) = 0 u(z)=0
  19. y y
  20. 0 = - d P d x + μ ( d u 2 d y 2 ) 0=-\frac{\mbox{d}~{}P}{\mbox{d}~{}x}+\mu\left(\frac{\mbox{d}~{}^{2}u}{\mbox{d}% ~{}y^{2}}\right)
  21. y = h y=h
  22. u = 0 u=0
  23. y = - h y=-h
  24. u = 0 u=0
  25. u = 1 2 μ ( d P d x ) y 2 + A y + B u=\frac{1}{2\mu}\left(\frac{\mbox{d}~{}P}{\mbox{d}~{}x}\right)y^{2}+Ay+B
  26. 0 = 1 2 μ d P d x h 2 + A h + B 0=\frac{1}{2\mu}\frac{\mathrm{d}P}{\mathrm{d}x}h^{2}+Ah+B
  27. 0 = 1 2 μ d P d x h 2 - A h + B 0=\frac{1}{2\mu}\frac{\mathrm{d}P}{\mathrm{d}x}h^{2}-Ah+B
  28. B = - 1 2 μ d P d x h 2 B=-\frac{1}{2\mu}\frac{\mathrm{d}P}{\mathrm{d}x}h^{2}
  29. A = 0 A=0
  30. u = 1 2 μ d P d x ( y 2 - h 2 ) u=\frac{1}{2\mu}\frac{\mathrm{d}P}{\mathrm{d}x}(y^{2}-h^{2})
  31. 𝐮 t Variation + ( 𝐮 ) 𝐮 Convection Inertia (per volume) - ν 2 𝐮 Diffusion = - w Internal source Divergence of stress + 𝐠 External source . \overbrace{\underbrace{\frac{\partial\mathbf{u}}{\partial t}}_{\begin{% smallmatrix}\,\text{Variation}\end{smallmatrix}}+\underbrace{(\mathbf{u}\cdot% \nabla)\mathbf{u}}_{\begin{smallmatrix}\,\text{Convection}\end{smallmatrix}}}^% {\,\text{Inertia (per volume)}}\overbrace{-\underbrace{\nu\nabla^{2}\mathbf{u}% }_{\,\text{Diffusion}}=\underbrace{-\nabla w}_{\begin{smallmatrix}\,\text{% Internal}\\ \,\text{source}\end{smallmatrix}}}^{\,\text{Divergence of stress}}+\underbrace% {\mathbf{g}}_{\begin{smallmatrix}\,\text{External}\\ \,\text{source}\end{smallmatrix}}.
  32. τ ∇⋅τ
  33. 𝐠 = - ϕ \mathbf{g}=-\nabla\phi
  34. h w + ϕ h\equiv w+\phi
  35. 𝐮 t + ( 𝐮 ) 𝐮 - ν 2 𝐮 = - h . \frac{\partial\mathbf{u}}{\partial t}+(\mathbf{u}\cdot\nabla)\mathbf{u}-\nu% \nabla^{2}\mathbf{u}=-\nabla h.
  36. 𝐮 t = Π S ( - ( 𝐮 ) 𝐮 + ν 2 𝐮 ) + 𝐟 S ρ - 1 p = Π I ( - ( 𝐮 ) 𝐮 + ν 2 𝐮 ) + 𝐟 I \begin{aligned}\displaystyle\frac{\partial\mathbf{u}}{\partial t}&% \displaystyle=\Pi^{S}\left(-(\mathbf{u}\cdot\nabla)\mathbf{u}+\nu\nabla^{2}% \mathbf{u}\right)+\mathbf{f}^{S}\\ \displaystyle\rho^{-1}\nabla p&\displaystyle=\Pi^{I}\left(-(\mathbf{u}\cdot% \nabla)\mathbf{u}+\nu\nabla^{2}\mathbf{u}\right)+\mathbf{f}^{I}\end{aligned}
  37. 𝐟 S \mathbf{f}^{S}
  38. Π S 𝐅 ( 𝐫 ) = 1 4 π × × 𝐅 ( 𝐫 ) | 𝐫 - 𝐫 | d V , Π I = 1 - Π S \Pi^{S}\,\mathbf{F}(\mathbf{r})=\frac{1}{4\pi}\nabla\times\int\frac{\nabla^{% \prime}\times\mathbf{F}(\mathbf{r}^{\prime})}{|\mathbf{r}-\mathbf{r}^{\prime}|% }dV^{\prime},\quad\Pi^{I}=1-\Pi^{S}
  39. ( 𝐰 , 𝐮 t ) = - ( 𝐰 , ( 𝐮 ) 𝐮 ) - ν ( 𝐰 : 𝐮 ) + ( 𝐰 , 𝐟 S ) \left(\mathbf{w},\frac{\partial\mathbf{u}}{\partial t}\right)=-(\mathbf{w},(% \mathbf{u}\cdot\nabla)\mathbf{u})-\nu(\nabla\mathbf{w}:\nabla\mathbf{u})+(% \mathbf{w},\mathbf{f}^{S})
  40. 𝐰 \mathbf{w}
  41. ( 𝐰 i , 𝐮 j t ) = - ( 𝐰 i , ( 𝐮 ) 𝐮 j ) - ν ( 𝐰 i : 𝐮 j ) + ( 𝐰 i , 𝐟 S ) . \left(\mathbf{w}_{i},\frac{\partial\mathbf{u}_{j}}{\partial t}\right)=-(% \mathbf{w}_{i},(\mathbf{u}\cdot\nabla)\mathbf{u}_{j})-\nu(\nabla\mathbf{w}_{i}% :\nabla\mathbf{u}_{j})+\left(\mathbf{w}_{i},\mathbf{f}^{S}\right).
  42. ϕ = [ ϕ x , ϕ y ] T , × ϕ = [ ϕ y , - ϕ x ] T . \nabla\phi=\left[\frac{\partial\phi}{\partial x},\,\frac{\partial\phi}{% \partial y}\right]^{\mathrm{T}},\quad\nabla\times\phi=\left[\frac{\partial\phi% }{\partial y},\,-\frac{\partial\phi}{\partial x}\right]^{\mathrm{T}}.
  43. ( 𝐠 i , p ) = - ( 𝐠 i , ( 𝐮 ) 𝐮 j ) - ν ( 𝐠 i : 𝐮 j ) + ( 𝐠 i , 𝐟 I ) (\mathbf{g}_{i},\nabla p)=-(\mathbf{g}_{i},(\mathbf{u}\cdot\nabla)\mathbf{u}_{% j})-\nu(\nabla\mathbf{g}_{i}:\nabla\mathbf{u}_{j})+(\mathbf{g}_{i},\mathbf{f}^% {I})
  44. 𝐮 . ∇\mathbf{u}.
  45. σ ( 𝐮 ) = 𝐂 : ( 𝐮 ) , \mathbf{σ}(∇\mathbf{u})=\mathbf{C}:(∇\mathbf{u}),
  46. 𝐂 \mathbf{C}
  47. 𝐕 \mathbf{V}
  48. λ λ
  49. μ , μ,
  50. 𝐈 \mathbf{I}
  51. 𝐮 ∇⋅\mathbf{u}
  52. s y m b o l σ = λ ( 𝐮 ) 𝐈 + μ ( 𝐮 + ( 𝐮 ) T ) . symbol\sigma=\lambda(\nabla\cdot\mathbf{u})\mathbf{I}+\mu(\nabla\mathbf{u}+% \left(\nabla\mathbf{u}\right)^{\mathrm{T}}).
  53. tr ( s y m b o l ε ) = 𝐮 . \mathrm{tr}(symbol\varepsilon)=\nabla\cdot\mathbf{u}.
  54. tr ( s y m b o l σ ) = ( 3 λ + 2 μ ) 𝐮 . \mathrm{tr}(symbol\sigma)=(3\lambda+2\mu)\nabla\cdot\mathbf{u}.
  55. s y m b o l σ = ( λ + 2 3 μ ) ( 𝐮 ) 𝐈 + μ ( 𝐮 + ( 𝐮 ) T - 2 3 ( 𝐮 ) 𝐈 ) symbol\sigma=(\lambda+\tfrac{2}{3}\mu)(\nabla\cdot\mathbf{u})\mathbf{I}+\mu(% \nabla\mathbf{u}+\left(\nabla\mathbf{u}\right)^{\mathrm{T}}-\tfrac{2}{3}(% \nabla\cdot\mathbf{u})\mathbf{I})
  56. ζ ζ
  57. ζ λ + 2 3 μ , \zeta\equiv\lambda+\tfrac{2}{3}\mu,
  58. λ λ
  59. μ μ
  60. 𝐮 ∇\mathbf{u}
  61. ( 𝐮 ) . ∇(∇⋅\mathbf{u}).
  62. ζ ζ
  63. p ¯ p - ζ 𝐮 , \bar{p}\equiv p-\zeta\nabla\cdot\mathbf{u},
  64. ζ ζ
  65. 𝐮 = 0. ∇⋅\mathbf{u}=0.
  66. ρ t + ( ρ 𝐮 ) = 0 \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\mathbf{u})=0
  67. D ρ D t + ρ ( 𝐮 ) = 0. \frac{D\rho}{Dt}+\rho(\nabla\cdot\mathbf{u})=0.
  68. ρ \rho
  69. μ \mu
  70. Δ x Δ y Δ z P t \displaystyle\Delta x\Delta y\Delta z\frac{\partial P}{\partial t}
  71. P t = 0 \frac{\partial P}{\partial t}=0
  72. 0 = - ρ [ U x x + U y y + U z z ] 0=-\rho\left[{\partial U_{x}\over\partial x}+{\partial U_{y}\over\partial y}+{% \partial U_{z}\over\partial z}\right]
  73. ρ \rho
  74. u z = 0 u_{z}=0
  75. ρ ( u x t + u x u x x + u y u x y ) \displaystyle\rho\left(\frac{\partial u_{x}}{\partial t}+u_{x}\frac{\partial u% _{x}}{\partial x}+u_{y}\frac{\partial u_{x}}{\partial y}\right)
  76. ψ \psi
  77. u x = ψ y ; u y = - ψ x u_{x}=\frac{\partial\psi}{\partial y};\quad u_{y}=-\frac{\partial\psi}{% \partial x}
  78. t ( 2 ψ ) + ψ y x ( 2 ψ ) - ψ x y ( 2 ψ ) = ν 4 ψ \frac{\partial}{\partial t}\left(\nabla^{2}\psi\right)+\frac{\partial\psi}{% \partial y}\frac{\partial}{\partial x}\left(\nabla^{2}\psi\right)-\frac{% \partial\psi}{\partial x}\frac{\partial}{\partial y}\left(\nabla^{2}\psi\right% )=\nu\nabla^{4}\psi
  79. 4 \nabla^{4}
  80. ν \nu
  81. ν = μ ρ \nu=\frac{\mu}{\rho}
  82. t ( 2 ψ ) + ( ψ , 2 ψ ) ( y , x ) = ν 4 ψ . \frac{\partial}{\partial t}\left(\nabla^{2}\psi\right)+\frac{\partial\left(% \psi,\nabla^{2}\psi\right)}{\partial\left(y,x\right)}=\nu\nabla^{4}\psi.
  83. d 2 u d y 2 = - 1 ; u ( 0 ) = u ( 1 ) = 0. \frac{d^{2}u}{dy^{2}}=-1;\quad u(0)=u(1)=0.
  84. u ( y ) = y - y 2 2 . u(y)=\frac{y-y^{2}}{2}.
  85. f ( z ) f(z)
  86. d 2 f d z 2 + R f 2 = - 1 ; f ( - 1 ) = f ( 1 ) = 0. \frac{d^{2}f}{dz^{2}}+Rf^{2}=-1;\quad f(-1)=f(1)=0.
  87. A < - 2 ν A<-2\nu
  88. ν = 0 \nu=0
  89. 𝐯 ( x , y ) = 1 x 2 + y 2 ( A x + B y A y - B x ) , p ( x , y ) = - A 2 + B 2 2 ( x 2 + y 2 ) \mathbf{v}(x,y)=\frac{1}{x^{2}+y^{2}}\begin{pmatrix}Ax+By\\ Ay-Bx\end{pmatrix},\qquad p(x,y)=-\frac{A^{2}+B^{2}}{2(x^{2}+y^{2})}
  90. ρ ( x , y , z ) = 3 B r 2 + x 2 + y 2 + z 2 p ( x , y , z ) = - A 2 B ( r 2 + x 2 + y 2 + z 2 ) 3 𝐮 ( x , y , z ) = A ( r 2 + x 2 + y 2 + z 2 ) 2 ( 2 ( - r y + x z ) 2 ( r x + y z ) r 2 - x 2 - y 2 + z 2 ) g = 0 μ = 0 \begin{aligned}\displaystyle\rho(x,y,z)&\displaystyle=\frac{3B}{r^{2}+x^{2}+y^% {2}+z^{2}}\\ \displaystyle p(x,y,z)&\displaystyle=\frac{-A^{2}B}{(r^{2}+x^{2}+y^{2}+z^{2})^% {3}}\\ \displaystyle\mathbf{u}(x,y,z)&\displaystyle=\frac{A}{(r^{2}+x^{2}+y^{2}+z^{2}% )^{2}}\begin{pmatrix}2(-ry+xz)\\ 2(rx+yz)\\ r^{2}-x^{2}-y^{2}+z^{2}\end{pmatrix}\\ \displaystyle g&\displaystyle=0\\ \displaystyle\mu&\displaystyle=0\end{aligned}
  91. ρ \rho
  92. ρ t + 1 r r ( ρ r u r ) + 1 r ( ρ u ϕ ) ϕ + ( ρ u z ) z = 0. \frac{\partial\rho}{\partial t}+\frac{1}{r}\frac{\partial}{\partial r}\left(% \rho ru_{r}\right)+\frac{1}{r}\frac{\partial(\rho u_{\phi})}{\partial\phi}+% \frac{\partial(\rho u_{z})}{\partial z}=0.
  93. u ϕ = 0 u_{\phi}=0
  94. ϕ \phi
  95. ρ ( u r t + u r u r r + u z u r z ) \displaystyle\rho\left(\frac{\partial u_{r}}{\partial t}+u_{r}\frac{\partial u% _{r}}{\partial r}+u_{z}\frac{\partial u_{r}}{\partial z}\right)
  96. ρ t + 1 r 2 r ( ρ r 2 u r ) + 1 r sin ( θ ) ρ u ϕ ϕ + 1 r sin ( θ ) θ ( sin ( θ ) ρ u θ ) = 0. \frac{\partial\rho}{\partial t}+\frac{1}{r^{2}}\frac{\partial}{\partial r}% \left(\rho r^{2}u_{r}\right)+\frac{1}{r\sin(\theta)}\frac{\partial\rho u_{\phi% }}{\partial\phi}+\frac{1}{r\sin(\theta)}\frac{\partial}{\partial\theta}\left(% \sin(\theta)\rho u_{\theta}\right)=0.
  97. 1 / r 2 1/r^{2}

NC_(complexity).html

  1. 𝐍𝐂 1 𝐍𝐂 2 𝐍𝐂 i 𝐍𝐂 \mathbf{NC}^{1}\subseteq\mathbf{NC}^{2}\subseteq\cdots\subseteq\mathbf{NC}^{i}% \subseteq\cdots\mathbf{NC}
  2. 𝐍𝐂 1 𝐋 𝐍𝐋 𝐀𝐂 1 𝐍𝐂 2 𝐏 . \mathbf{NC}^{1}\subseteq\mathbf{L}\subseteq\mathbf{NL}\subseteq\mathbf{AC}^{1}% \subseteq\mathbf{NC}^{2}\subseteq\mathbf{P}.
  3. 𝐍𝐂 i 𝐀𝐂 i 𝐍𝐂 i + 1 . \mathbf{NC}^{i}\subseteq\mathbf{AC}^{i}\subseteq\mathbf{NC}^{i+1}.
  4. ( log n ) O ( 1 ) (\log n)^{O(1)}
  5. 𝐍𝐂 1 𝐍𝐂 2 \,\textbf{NC}^{1}\subseteq\,\textbf{NC}^{2}\subseteq\cdots
  6. 𝐍𝐂 1 𝐍𝐂 i 𝐍𝐂 i + j 𝐍𝐂 \,\textbf{NC}^{1}\subset\cdots\subset\,\textbf{NC}^{i}\subset...\subset\,% \textbf{NC}^{i+j}\subset\cdots\,\textbf{NC}
  7. 𝐍𝐂 1 𝐍𝐂 i = = 𝐍𝐂 i + j = 𝐍𝐂 \,\textbf{NC}^{1}\subset\cdots\subset\,\textbf{NC}^{i}=...=\,\textbf{NC}^{i+j}% =\cdots\,\textbf{NC}
  8. γ δ γ - 1 δ - 1 = ϵ \gamma\delta\gamma^{-1}\delta^{-1}=\epsilon
  9. ¬ C \neg C
  10. α - 1 \alpha^{-1}
  11. i d id
  12. ¬ C \neg C
  13. C D C\wedge D
  14. δ - 1 \delta^{-1}
  15. γ - 1 \gamma^{-1}
  16. C D C\wedge D
  17. C D C\wedge D
  18. 4 d 4^{d}

Near-Earth_object.html

  1. f B = 0.03 E - 0.8 f_{B}=0.03E^{-0.8}\;
  2. f B = 0.00737 E - 0.9 f_{B}=0.00737E^{-0.9}\;

Negative_binomial_distribution.html

  1. X NB ( r ; p ) X\ \sim\ \,\text{NB}(r;p)
  2. f ( k ; r , p ) Pr ( X = k ) = ( k + r - 1 k ) p k ( 1 - p ) r for k = 0 , 1 , 2 , f(k;r,p)\equiv\Pr(X=k)={k+r-1\choose k}p^{k}(1-p)^{r}\quad\,\text{for }k=0,1,2,\dots
  3. ( k + r - 1 k ) = ( k + r - 1 ) ! k ! ( r - 1 ) ! = ( k + r - 1 ) ( k + r - 2 ) ( r ) k ! . {k+r-1\choose k}=\frac{(k+r-1)!}{k!\,(r-1)!}=\frac{(k+r-1)(k+r-2)\cdots(r)}{k!}.
  4. ( k + r - 1 ) ( r ) k ! = ( - 1 ) k ( - r ) ( - r - 1 ) ( - r - 2 ) ( - r - k + 1 ) k ! = ( - 1 ) k ( - r k ) . ( * ) \frac{(k+r-1)\cdots(r)}{k!}=(-1)^{k}\frac{(-r)(-r-1)(-r-2)\cdots(-r-k+1)}{k!}=% (-1)^{k}{-r\choose k}.\qquad(*)
  5. { ( k + 1 ) Pr ( k + 1 ) - p Pr ( k ) ( k + r ) = 0 , Pr ( 0 ) = ( 1 - p ) r } \left\{(k+1)\Pr(k+1)-p\Pr(k)(k+r)=0,\Pr(0)=(1-p)^{r}\right\}
  6. f ( k ; r , p ) Pr ( X = k ) = ( k + r - 1 k ) p k ( 1 - p ) r for k = 0 , 1 , 2 , f(k;r,p)\equiv\Pr(X=k)={k+r-1\choose k}p^{k}(1-p)^{r}\quad\,\text{for }k=0,1,2,\dots
  7. ( k + r - 1 k ) = ( k + r - 1 ) ( k + r - 2 ) ( r ) k ! = Γ ( k + r ) k ! Γ ( r ) . {k+r-1\choose k}=\frac{(k+r-1)(k+r-2)\cdots(r)}{k!}=\frac{\Gamma(k+r)}{k!\,% \Gamma(r)}.
  8. ( 1 - p ) - r = k = 0 ( - r k ) ( - p ) k = k = 0 ( k + r - 1 k ) p k , (1-p)^{-r}=\sum_{k=0}^{\infty}{-r\choose k}(-p)^{k}=\sum_{k=0}^{\infty}{k+r-1% \choose k}p^{k},
  9. f ( k ; r , p ) = ( 1 - p ) p k f(k;r,p)=(1-p)\cdot p^{k}\!
  10. Geom ( p ) = NB ( 1 , 1 - p ) . \,\text{Geom}(p)=\,\text{NB}(1,\,1-p).\,
  11. λ = r p 1 - p p = λ r + λ . \lambda=r\,\frac{p}{1-p}\quad\Rightarrow\quad p=\frac{\lambda}{r+\lambda}.
  12. f ( k ; r , p ) = Γ ( k + r ) k ! Γ ( r ) p k ( 1 - p ) r = λ k k ! Γ ( r + k ) Γ ( r ) ( r + λ ) k 1 ( 1 + λ r ) r f(k;r,p)=\frac{\Gamma(k+r)}{k!\cdot\Gamma(r)}p^{k}(1-p)^{r}=\frac{\lambda^{k}}% {k!}\cdot\frac{\Gamma(r+k)}{\Gamma(r)\;(r+\lambda)^{k}}\cdot\frac{1}{\left(1+% \frac{\lambda}{r}\right)^{r}}
  13. lim r f ( k ; r , p ) = λ k k ! 1 1 e λ , \lim_{r\to\infty}f(k;r,p)=\frac{\lambda^{k}}{k!}\cdot 1\cdot\frac{1}{e^{% \lambda}},
  14. Poisson ( λ ) = lim r NB ( r , λ λ + r ) . \,\text{Poisson}(\lambda)=\lim_{r\to\infty}\,\text{NB}\Big(r,\ \frac{\lambda}{% \lambda+r}\Big).
  15. f ( k ; r , p ) \displaystyle f(k;r,p)
  16. Pr ( Y r s ) \displaystyle\Pr(Y_{r}\leq s)
  17. f ( k ; r , p ) = - p k k ln ( 1 - p ) , k . f(k;r,p)=\frac{-p^{k}}{k\ln(1-p)},\qquad k\in{\mathbb{N}}.
  18. X = n = 1 N Y n X=\sum_{n=1}^{N}Y_{n}
  19. G N ( z ) = exp ( λ ( z - 1 ) ) , z , G_{N}(z)=\exp(\lambda(z-1)),\qquad z\in\mathbb{R},
  20. G Y 1 ( z ) = ln ( 1 - p z ) ln ( 1 - p ) , | z | < 1 p , G_{Y_{1}}(z)=\frac{\ln(1-pz)}{\ln(1-p)},\qquad|z|<\frac{1}{p},
  21. G X ( z ) \displaystyle G_{X}(z)
  22. f ( k ; r , p ) Pr ( X k ) = 1 - I p ( k + 1 , r ) = I 1 - p ( r , k + 1 ) . f(k;r,p)\equiv\Pr(X\leq k)=1-I_{p}(k+1,r)=I_{1-p}(r,k+1).
  23. p ^ = r - 1 r + k - 1 . \hat{p}=\frac{r-1}{r+k-1}.
  24. p ~ = r r + k , \tilde{p}=\frac{r}{r+k},
  25. 1 = 1 n = ( p + q ) n = k = 0 n ( n k ) p k q n - k . 1=1^{n}=(p+q)^{n}=\sum_{k=0}^{n}{n\choose k}p^{k}q^{n-k}.
  26. ( p + q ) n = k = 0 ( n k ) p k q n - k , (p+q)^{n}=\sum_{k=0}^{\infty}{n\choose k}p^{k}q^{n-k},
  27. ( n k ) = n ( n - 1 ) ( n - 2 ) ( n - k + 1 ) k ! . {n\choose k}={n(n-1)(n-2)\cdots(n-k+1)\over k!}.
  28. ( p + q ) 8.3 = k = 0 ( 8.3 k ) p k q 8.3 - k . (p+q)^{8.3}=\sum_{k=0}^{\infty}{8.3\choose k}p^{k}q^{8.3-k}.
  29. 1 = p r p - r = p r ( 1 - q ) - r = p r k = 0 ( - r k ) ( - q ) k . 1=p^{r}\cdot p^{-r}=p^{r}(1-q)^{-r}=p^{r}\sum_{k=0}^{\infty}{-r\choose k}(-q)^% {k}.
  30. p r ( - r k ) ( - q ) k p^{r}{-r\choose k}(-q)^{k}
  31. L ( r , p ) = i = 1 N f ( k i ; r , p ) L(r,p)=\prod_{i=1}^{N}f(k_{i};r,p)\,\!
  32. ( r , p ) = i = 1 N ln ( Γ ( k i + r ) ) - i = 1 N ln ( k i ! ) - N ln ( Γ ( r ) ) + i = 1 N k i ln ( p ) + N r ln ( 1 - p ) . \ell(r,p)=\sum_{i=1}^{N}\ln{(\Gamma(k_{i}+r))}-\sum_{i=1}^{N}\ln(k_{i}!)-N\ln{% (\Gamma(r))}+\sum_{i=1}^{N}k_{i}\ln{(p)}+Nr\ln(1-p).
  33. ( r , p ) p = i = 1 N k i 1 p - N r 1 1 - p = 0 \frac{\partial\ell(r,p)}{\partial p}=\sum_{i=1}^{N}k_{i}\frac{1}{p}-Nr\frac{1}% {1-p}=0
  34. ( r , p ) r = i = 1 N ψ ( k i + r ) - N ψ ( r ) + N ln ( 1 - p ) = 0 \frac{\partial\ell(r,p)}{\partial r}=\sum_{i=1}^{N}\psi(k_{i}+r)-N\psi(r)+N\ln% {(1-p)}=0
  35. ψ ( k ) = Γ ( k ) Γ ( k ) \psi(k)=\frac{\Gamma^{\prime}(k)}{\Gamma(k)}\!
  36. p = i = 1 N k i N r + i = 1 N k i p=\frac{\sum_{i=1}^{N}k_{i}}{Nr+\sum_{i=1}^{N}k_{i}}
  37. ( r , p ) r = i = 1 N ψ ( k i + r ) - N ψ ( r ) + N ln ( r r + i = 1 N k i / N ) = 0 \frac{\partial\ell(r,p)}{\partial r}=\sum_{i=1}^{N}\psi(k_{i}+r)-N\psi(r)+N\ln% {\left(\frac{r}{r+\sum_{i=1}^{N}k_{i}/N}\right)}=0
  38. f ( n ) = ( ( n - 5 ) + 5 - 1 n - 5 ) 0.4 5 0.6 n - 5 = ( n - 1 n - 5 ) 2 5 3 n - 5 5 n . f(n)={(n-5)+5-1\choose n-5}\;0.4^{5}\;0.6^{n-5}={n-1\choose n-5}\;2^{5}\;\frac% {3^{n-5}}{5^{n}}.
  39. f ( 10 ) = 0.1003290624. f(10)=0.1003290624.\,
  40. f ( 5 ) = 0.01024 f(5)=0.01024\,
  41. f ( 6 ) = 0.03072 f(6)=0.03072\,
  42. f ( 7 ) = 0.055296 f(7)=0.055296\,
  43. f ( 8 ) = 0.0774144 f(8)=0.0774144\,
  44. j = 5 8 f ( j ) = 0.17367. \sum_{j=5}^{8}f(j)=0.17367.
  45. 1 - j = 5 30 f ( j ) = 1 - I 0.4 ( 5 , 30 - 5 + 1 ) 1 - 0.99849 = 0.00151. 1-\sum_{j=5}^{30}f(j)=1-I_{0.4}(5,30-5+1)\approx 1-0.99849=0.00151.

Neper.html

  1. L Np = ln x 1 x 2 = ln x 1 - ln x 2 . L_{\rm Np}=\ln\frac{x_{1}}{x_{2}}=\ln x_{1}-\ln x_{2}.\,
  2. x 1 x_{1}
  3. x 2 x_{2}
  4. 1 Np = 20 / ln 10 dB 8.685889638 dB 1\ {\rm Np}=20/\ln 10\ {\rm dB}\approx 8{.}685889638\ {\rm dB}\,
  5. 1 dB = 1 20 log 10 e Np 0.115129254 Np . 1\ {\rm dB}=\frac{1}{20\log_{10}e}\ {\rm Np}\approx 0{.}115129254\ {\rm Np}.\,
  6. L = 10 log 10 x 1 2 x 2 2 dB = 10 log 10 ( x 1 x 2 ) 2 dB = 20 log 10 x 1 x 2 dB = ln x 1 x 2 Np . \begin{aligned}\displaystyle L&\displaystyle=10\log_{10}\frac{x_{1}^{2}}{x_{2}% ^{2}}&\displaystyle\mathrm{dB}\\ &\displaystyle=10\log_{10}{\left(\frac{x_{1}}{x_{2}}\right)}^{2}&\displaystyle% \mathrm{dB}\\ &\displaystyle=20\log_{10}\frac{x_{1}}{x_{2}}&\displaystyle\mathrm{dB}\\ &\displaystyle=\ln\frac{x_{1}}{x_{2}}&\displaystyle\mathrm{Np}.\\ \end{aligned}
  7. ( 1 + δ ) ( 1 + ϵ ) = 1 + δ + ϵ + δ ϵ 1 + δ + ϵ , (1+\delta)(1+\epsilon)=1+\delta+\epsilon+\delta\epsilon\approx 1+\delta+\epsilon,
  8. δ \delta
  9. D δ = 100 ln [ ( 1 + δ ) - 1 1 ] = 100 ln ( δ ) D_{\delta}=100\cdot\ln\left[\frac{(1+\delta)-1}{1}\right]=100\cdot\ln(\delta)

Neptunium.html

  1. U 92 238 + 0 1 n 92 239 U β - 23 min 93 239 Np β - 2.355 days 94 239 Pu \mathrm{{}^{238}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{239}_{\ 92}U\ % \xrightarrow[23\ min]{\beta^{-}}\ ^{239}_{\ 93}Np\ \xrightarrow[2.355\ days]{% \beta^{-}}\ ^{239}_{\ 94}Pu}
  2. U 92 235 + 0 1 n 92 236 U m 120 ns 92 236 U + γ \mathrm{{}^{235}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{236}_{\ 92}U_{m}\ % \xrightarrow[120\ ns]{}\ ^{236}_{\ 92}U\ +\ \gamma}
  3. U 92 236 + 0 1 n 92 237 U β - 6.75 d 93 237 Np \mathrm{{}^{236}_{\ 92}U\ +\ ^{1}_{0}n\ \longrightarrow\ ^{237}_{\ 92}U\ % \xrightarrow[6.75\ d]{\beta^{-}}\ ^{237}_{\ 93}Np}
  4. 1 / 2 {1}/{2}
  5. 1 / 2 {1}/{2}
  6. 1 / 2 {1}/{2}
  7. Np 93 237 + 0 1 n 93 238 Np β - 2.117 d 94 238 Pu \mathrm{{}^{237}_{\ 93}Np\ +\ ^{1}_{0}n\ \longrightarrow\ ^{238}_{\ 93}Np\ % \xrightarrow[2.117\ d]{\beta^{-}}\ ^{238}_{\ 94}Pu}

Net_(mathematics).html

  1. x α α A \langle x_{\alpha}\rangle_{\alpha\in A}
  2. y X y\in X
  3. x α α A \langle x_{\alpha}\rangle_{\alpha\in A}
  4. x α α A \langle x_{\alpha}\rangle_{\alpha\in A}
  5. x α α A \langle x_{\alpha}\rangle_{\alpha\in A}
  6. ( U , α ) (U,\alpha)
  7. α A \alpha\in A
  8. x α U x_{\alpha}\in U
  9. h : B A h:B\rightarrow A
  10. ( U , α ) (U,\alpha)
  11. α \alpha
  12. y β β B \langle y_{\beta}\rangle_{\beta\in B}
  13. y β = x h ( β ) y_{\beta}=x_{h(\beta)}
  14. { C i } i I \{C_{i}\}_{i\in I}
  15. i J C i \bigcap_{i\in J}C_{i}\neq\emptyset
  16. J I J\subseteq I
  17. i I C i \bigcap_{i\in I}C_{i}\neq\emptyset
  18. { C i c } i I \{C_{i}^{c}\}_{i\in I}
  19. x α α A \langle x_{\alpha}\rangle_{\alpha\in A}
  20. α A \alpha\in A
  21. E α { x β : β α } . E_{\alpha}\triangleq\{x_{\beta}:\beta\geq\alpha\}.
  22. { cl ( E α ) : α A } \{\operatorname{cl}(E_{\alpha}):\alpha\in A\}
  23. α A cl ( E α ) \bigcap_{\alpha\in A}\operatorname{cl}(E_{\alpha})\neq\emptyset
  24. x α α A \langle x_{\alpha}\rangle_{\alpha\in A}
  25. x α α A \langle x_{\alpha}\rangle_{\alpha\in A}
  26. x α α A \langle x_{\alpha}\rangle_{\alpha\in A}
  27. { U i : i I } \{U_{i}:i\in I\}
  28. D { J I : | J | < } D\triangleq\{J\subset I:|J|<\infty\}
  29. C D C\in D
  30. x C X x_{C}\in X
  31. x C U a x_{C}\notin U_{a}
  32. a C a\in C
  33. x C C D \langle x_{C}\rangle_{C\in D}
  34. x X x\in X
  35. c I c\in I
  36. U c U_{c}
  37. B { c } B\supseteq\{c\}
  38. x B U c x_{B}\notin U_{c}
  39. π i ( x α ) π i ( x ) \pi_{i}(x_{\alpha})\to\pi_{i}(x)
  40. ( x α ) α I (x_{\alpha})_{\alpha\in I}
  41. lim sup x α = lim α I sup β α x β = inf α I sup β α x β . \limsup x_{\alpha}=\lim_{\alpha\in I}\sup_{\beta\succeq\alpha}x_{\beta}=\inf_{% \alpha\in I}\sup_{\beta\succeq\alpha}x_{\beta}.
  42. lim sup ( x α + y α ) lim sup x α + lim sup y α , \limsup(x_{\alpha}+y_{\alpha})\leq\limsup x_{\alpha}+\limsup y_{\alpha},

Net_present_value.html

  1. R t ( 1 + i ) t \frac{R_{t}}{(1+i)^{t}}
  2. t t
  3. i i
  4. R t R_{t}
  5. R 0 R_{0}
  6. t t
  7. R t R_{t}
  8. N N
  9. NPV \mathrm{NPV}
  10. NPV ( i , N ) = t = 0 N R t ( 1 + i ) t \mathrm{NPV}(i,N)=\sum_{t=0}^{N}\frac{R_{t}}{(1+i)^{t}}
  11. R t R_{t}
  12. R t R_{t}
  13. NPV ( i ) = t = 0 N R t ( 1 + i ) t \mathrm{NPV}(i)=\sum_{t=0}^{N}\frac{R_{t}}{(1+i)^{t}}
  14. NPV ( i ) = t = 0 ( 1 + i ) - t r ( t ) d t \mathrm{NPV}(i)=\int_{t=0}^{\infty}(1+i)^{-t}\cdot r(t)\,dt
  15. F ( s ) = { f } ( s ) = 0 e - s t f ( t ) d t F(s)=\left\{\mathcal{L}f\right\}(s)=\int_{0}^{\infty}e^{-st}f(t)\,dt
  16. - 100 , 000 ( 1 + 0.10 ) 0 \frac{-100,000}{(1+0.10)^{0}}
  17. 10 , 000 ( 1 + 0.10 ) 1 \frac{10,000}{(1+0.10)^{1}}
  18. 10 , 000 ( 1 + 0.10 ) 2 \frac{10,000}{(1+0.10)^{2}}
  19. 10 , 000 ( 1 + 0.10 ) 3 \frac{10,000}{(1+0.10)^{3}}
  20. 10 , 000 ( 1 + 0.10 ) 4 \frac{10,000}{(1+0.10)^{4}}
  21. 10 , 000 ( 1 + 0.10 ) 5 \frac{10,000}{(1+0.10)^{5}}
  22. 10 , 000 ( 1 + 0.10 ) 6 \frac{10,000}{(1+0.10)^{6}}
  23. 10 , 000 ( 1 + 0.10 ) 7 \frac{10,000}{(1+0.10)^{7}}
  24. 10 , 000 ( 1 + 0.10 ) 8 \frac{10,000}{(1+0.10)^{8}}
  25. 10 , 000 ( 1 + 0.10 ) 9 \frac{10,000}{(1+0.10)^{9}}
  26. 10 , 000 ( 1 + 0.10 ) 10 \frac{10,000}{(1+0.10)^{10}}
  27. 10 , 000 ( 1 + 0.10 ) 11 \frac{10,000}{(1+0.10)^{11}}
  28. 10 , 000 ( 1 + 0.10 ) 12 \frac{10,000}{(1+0.10)^{12}}
  29. NPV = P V ( B e n e f i t s ) - P V ( C o s t s ) \mathrm{NPV}=PV(Benefits)-PV(Costs)
  30. NPV = - P V ( C o s t s ) + P V ( B e n e f i t s ) \mathrm{NPV}=-PV(Costs)+PV(Benefits)
  31. NPV = - 100 , 000 + 68 , 136.92 \mathrm{NPV}=-100,000+68,136.92
  32. NPV = - 31 , 863.08 \mathrm{NPV}=-31,863.08

Network_topology.html

  1. c = n ( n - 1 ) 2 . c=\frac{n(n-1)}{2}.\,

Neutrino.html

  1. ħ / 2 {ħ}/{2}
  2. δ t = - 15 ± 31 \delta t=-15\pm 31

Neutron.html

  1. B d B_{d}
  2. E r d E_{rd}
  3. m n = m d - m p + B d - E r d m_{n}=m_{d}-m_{p}+B_{d}-E_{rd}

Neutron_star.html

  1. × 10 1 1 \times 10^{1}1
  2. 3 G M / c 2 3GM/c^{2}
  3. B E = 0.60 β 1 - β 2 BE=\frac{0.60\,\beta}{1-\frac{\beta}{2}}
  4. β = G M / R c 2 \beta\ =G\,M/R\,{c}^{2}
  5. G = 6.6742 × 10 - 11 m 3 kg - 1 s - 2 G=6.6742\times 10^{-11}\,\,\text{m}^{3}\,\text{kg}^{-1}\,\text{s}^{-2}
  6. c 2 = 8.98755 × 10 16 m 2 s - 2 c^{2}=8.98755\times 10^{16}\,\,\text{m}^{2}\,\text{s}^{-2}
  7. M s o l a r = 1.98844 × 10 30 kg M_{solar}=1.98844\times 10^{30}\,\,\text{kg}
  8. M x = M M M_{x}=\frac{M}{M_{\odot}}
  9. B E = 885.975 M x R - 738.313 M x BE=\frac{885.975\,M_{x}}{R-738.313\,M_{x}}
  10. 2.01 ± 0.04 M 2.01\pm 0.04M_{\odot}

New_moon.html

  1. d = 5.597661 + 29.5305888610 × N + ( 102.026 × 10 - 12 ) × N 2 d=5.597661+29.5305888610\times N+(102.026\times 10^{-12})\times N^{2}
  2. - 0.000739 - ( 235 × 10 - 12 ) × N 2 -0.000739-(235\times 10^{-12})\times N^{2}
  3. 2449128.59 + 29.53058867 * ( BLN - 871 ) ± 0.25 2449128.59+29.53058867*(\mathrm{BLN}-871)\pm 0.25

Newton's_laws_of_motion.html

  1. 𝐅 \mathbf{F}
  2. m m
  3. 𝐚 \mathbf{a}
  4. 𝐅 = m 𝐚 \mathbf{F}=m\mathbf{a}
  5. 𝐅 = 0 d 𝐯 d t = 0. \sum\mathbf{F}=0\;\Leftrightarrow\;\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}=0.
  6. 𝐅 = d 𝐩 d t = d ( m 𝐯 ) d t . \mathbf{F}=\frac{\mathrm{d}\mathbf{p}}{\mathrm{d}t}=\frac{\mathrm{d}(m\mathbf{% v})}{\mathrm{d}t}.
  7. 𝐅 = m d 𝐯 d t = m 𝐚 , \mathbf{F}=m\,\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}=m\mathbf{a},
  8. 𝐉 = Δ t 𝐅 d t . \mathbf{J}=\int_{\Delta t}\mathbf{F}\,\mathrm{d}t.
  9. 𝐉 = Δ 𝐩 = m Δ 𝐯 . \mathbf{J}=\Delta\mathbf{p}=m\Delta\mathbf{v}.
  10. 𝐅 net = d d t [ m ( t ) 𝐯 ( t ) ] = m ( t ) d 𝐯 d t + 𝐯 ( t ) d m d t . ( wrong ) \mathbf{F}_{\mathrm{net}}=\frac{\mathrm{d}}{\mathrm{d}t}\big[m(t)\mathbf{v}(t)% \big]=m(t)\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}+\mathbf{v}(t)\frac{\mathrm{% d}m}{\mathrm{d}t}.\qquad\mathrm{(wrong)}
  11. 𝐅 + 𝐮 d m d t = m d 𝐯 d t \mathbf{F}+\mathbf{u}\frac{\mathrm{d}m}{\mathrm{d}t}=m{\mathrm{d}\mathbf{v}% \over\mathrm{d}t}

Newton's_method.html

  1. x : f ( x ) = 0 . x:f(x)=0\,.
  2. x 1 = x 0 - f ( x 0 ) f ( x 0 ) . x_{1}=x_{0}-\frac{f(x_{0})}{f^{\prime}(x_{0})}\,.
  3. x n + 1 = x n - f ( x n ) f ( x n ) x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}\,
  4. y = f ( x n ) ( x - x n ) + f ( x n ) , y=f^{\prime}(x_{n})\,(x-x_{n})+f(x_{n}),
  5. 0 = f ( x n ) ( x n + 1 - x n ) + f ( x n ) . 0=f^{\prime}(x_{n})\,(x_{n+1}-x_{n})+f(x_{n}).
  6. x n + 1 = x n - f ( x n ) f ( x n ) . x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}.\,\!
  7. x n x_{n}
  8. x P - N = 0 x^{P}-N=0
  9. f ( x ) = | x | a , 0 < a < 1 2 f(x)=|x|^{a},\quad 0<a<\tfrac{1}{2}
  10. x x
  11. a = 1 / 2 a=1/2
  12. m m
  13. x n + 1 = x n - m f ( x n ) f ( x n ) . x_{n+1}=x_{n}-m\frac{f(x_{n})}{f^{\prime}(x_{n})}.\,\!
  14. m m
  15. m m
  16. Δ x i + 1 = f ′′ ( α ) 2 f ( α ) ( Δ x i ) 2 + O [ Δ x i ] 3 , \Delta x_{i+1}=\frac{f^{\prime\prime}(\alpha)}{2f^{\prime}(\alpha)}(\Delta x_{% i})^{2}+O[\Delta x_{i}]^{3}\,,
  17. Δ x i x i - α . \Delta x_{i}\triangleq x_{i}-\alpha\,.
  18. f 0 f^{\prime}\neq 0\!
  19. f f ′′ > 0 f\cdot f^{\prime\prime}>0\!
  20. α . \alpha\,.
  21. R 1 = 1 2 ! f ′′ ( ξ n ) ( α - x n ) 2 , R_{1}=\frac{1}{2!}f^{\prime\prime}(\xi_{n})(\alpha-x_{n})^{2}\,,
  22. α . \alpha\,.
  23. α \alpha\,
  24. 0 = f ( α ) = f ( x n ) + f ( x n ) ( α - x n ) + 1 2 f ′′ ( ξ n ) ( α - x n ) 2 0=f(\alpha)=f(x_{n})+f^{\prime}(x_{n})(\alpha-x_{n})+\frac{1}{2}f^{\prime% \prime}(\xi_{n})(\alpha-x_{n})^{2}\,
  25. f ( x n ) f^{\prime}(x_{n})\,
  26. f ( x n ) f ( x n ) + ( α - x n ) = - f ′′ ( ξ n ) 2 f ( x n ) ( α - x n ) 2 \frac{f(x_{n})}{f^{\prime}(x_{n})}+\left(\alpha-x_{n}\right)=\frac{-f^{\prime% \prime}(\xi_{n})}{2f^{\prime}(x_{n})}\left(\alpha-x_{n}\right)^{2}
  27. x n + 1 = x n - f ( x n ) f ( x n ) , x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}\,,
  28. α - x n + 1 ϵ n + 1 = - f ′′ ( ξ n ) 2 f ( x n ) ( α - x n ϵ n ) 2 . \underbrace{\alpha-x_{n+1}}_{\epsilon_{n+1}}=\frac{-f^{\prime\prime}(\xi_{n})}% {2f^{\prime}(x_{n})}(\underbrace{\alpha-x_{n}}_{\epsilon_{n}})^{2}\,.
  29. ϵ n + 1 = - f ′′ ( ξ n ) 2 f ( x n ) ϵ n 2 . \epsilon_{n+1}=\frac{-f^{\prime\prime}(\xi_{n})}{2f^{\prime}(x_{n})}\,{% \epsilon_{n}}^{2}\,.
  30. f ( x ) 0 ; x I , where I is the interval [ α - r , α + r ] for some r | ( α - x 0 ) | ; f^{\prime}(x)\neq 0;\forall x\in I\,\text{, where }I\,\text{ is the interval }% [\alpha-r,\alpha+r]\,\text{ for some }r\geq\left|(\alpha-x_{0})\right|;\,
  31. f ′′ ( x ) is finite , x I ; f^{\prime\prime}(x)\,\text{ is finite },\forall x\in I;\,
  32. x 0 x_{0}\,
  33. α \alpha\,
  34. 1 2 | f ′′ ( x n ) f ( x n ) | < C | f ′′ ( α ) f ( α ) | , for some C < , \frac{1}{2}\left|{\frac{f^{\prime\prime}(x_{n})}{f^{\prime}(x_{n})}}\right|<C% \left|{\frac{f^{\prime\prime}(\alpha)}{f^{\prime}(\alpha)}}\right|,\,\text{ % for some }C<\infty,\,
  35. C | f ′′ ( α ) f ( α ) | ϵ n < 1 , for n \Zeta + { 0 } and C satisfying condition (b) . C\left|{\frac{f^{\prime\prime}(\alpha)}{f^{\prime}(\alpha)}}\right|\epsilon_{n% }<1,\,\text{ for }n\in\Zeta^{+}\cup\{0\}\,\text{ and }C\,\text{ satisfying % condition (b) }.\,
  36. | ϵ n + 1 | M ϵ n 2 \left|{\epsilon_{n+1}}\right|\leq M{{\epsilon}_{n}}^{2}\,
  37. ϵ n 2 {\epsilon_{n}}^{2}\,
  38. I I\,
  39. M = sup x I 1 2 | f ′′ ( x ) f ( x ) | . M=\sup_{x\in I}\frac{1}{2}\left|{\frac{f^{\prime\prime}(x)}{f^{\prime}(x)}}% \right|.\,
  40. x 0 x_{0}\,
  41. M | ϵ 0 | < 1. M\left|\epsilon_{0}\right|<1.\,
  42. f ( x ) = x 3 - 2 x 2 - 11 x + 12 f(x)=x^{3}-2x^{2}-11x+12
  43. \infty
  44. - -\infty
  45. f ( x ) = 1 - x 2 . f(x)=1-x^{2}.\!
  46. x 1 = x 0 - f ( x 0 ) f ( x 0 ) = 0 - 1 0 . x_{1}=x_{0}-\frac{f(x_{0})}{f^{\prime}(x_{0})}=0-\frac{1}{0}.
  47. f ( x ) = x 3 - 2 x + 2 f(x)=x^{3}-2x+2\!
  48. f ( x ) = x 3 . f(x)=\sqrt[3]{x}.
  49. x n + 1 = x n - f ( x n ) f ( x n ) = x n - x n 1 3 1 3 x n 1 3 - 1 = x n - 3 x n = - 2 x n . x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}=x_{n}-\frac{{x_{n}}^{\frac{1}% {3}}}{\frac{1}{3}\,{x_{n}}^{\frac{1}{3}-1}}=x_{n}-3\,x_{n}=-2\,x_{n}.
  50. f ( x ) = | x | α f(x)=|x|^{\alpha}
  51. 0 < α < 1 2 0<\alpha<\tfrac{1}{2}
  52. α = 1 2 \alpha=\tfrac{1}{2}
  53. f ( x ) = { 0 if x = 0 , x + x 2 sin ( 2 x ) if x 0. f(x)=\begin{cases}0&\,\text{if }x=0,\\ x+x^{2}\sin\left(\frac{2}{x}\right)&\,\text{if }x\neq 0.\end{cases}
  54. f ( x ) = { 1 if x = 0 , 1 + 2 x sin ( 2 x ) - 2 cos ( 2 x ) if x 0. f^{\prime}(x)=\begin{cases}1&\,\text{if }x=0,\\ 1+2\,x\,\sin\left(\frac{2}{x}\right)-2\,\cos\left(\frac{2}{x}\right)&\,\text{% if }x\neq 0.\end{cases}
  55. f ( x ) = x 2 f(x)=x^{2}\!
  56. f ( x ) = 2 x f^{\prime}(x)=2x\!
  57. x - f ( x ) / f ( x ) = x / 2 x-f(x)/f^{\prime}(x)=x/2\!
  58. f ( x ) = x 2 ( x - 1000 ) + 1. f(x)=x^{2}(x-1000)+1.\!
  59. f ( x ) = x + x 4 3 . f(x)=x+x^{\frac{4}{3}}.\!
  60. f ( x ) = 1 + 4 3 x 1 3 . f^{\prime}(x)=1+\frac{4}{3}x^{\frac{1}{3}}.\!
  61. f ′′ ( x ) = 4 9 x - 2 3 f^{\prime\prime}(x)=\frac{4}{9}x^{-\frac{2}{3}}\!
  62. x = 0 x=0\!
  63. x n x_{n}\!
  64. x n + 1 = x n - f ( x n ) f ( x n ) = 1 3 x n 4 3 ( 1 + 4 3 x n 1 3 ) x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}=\frac{\frac{1}{3}{x_{n}}^{% \frac{4}{3}}}{(1+\frac{4}{3}{x_{n}}^{\frac{1}{3}})}\!
  65. x n x_{n}\!
  66. f f\!
  67. x 2 + 1 x^{2}+1
  68. J F ( x n ) ( x n + 1 - x n ) = - F ( x n ) J_{F}(x_{n})(x_{n+1}-x_{n})=-F(x_{n})\,\!
  69. X n + 1 = X n - [ F ( X n ) ] - 1 F ( X n ) , X_{n+1}=X_{n}-[F^{\prime}(X_{n})]^{-1}F(X_{n}),\,
  70. F ( X n ) F^{\prime}(X_{n})
  71. X n X_{n}
  72. X n X_{n}
  73. f ( x ) f(x)
  74. [ a , b ] [a,b]
  75. f f
  76. f ( x ) f ′′ ( x ) 0 f^{\prime}(x)f^{\prime\prime}(x)\neq 0
  77. f ( a ) < 0 f(a)<0
  78. f ( b ) > 0 f(b)>0
  79. f ( x ) > 0 f^{\prime}(x)>0
  80. f ′′ ( x ) > 0 f^{\prime\prime}(x)>0
  81. α \alpha
  82. f ( x ) f(x)
  83. - f ( x ) -f(x)
  84. x 0 = b x_{0}=b
  85. z 0 = a z_{0}=a
  86. x n x_{n}
  87. x n + 1 = x n - f ( x n ) f ( x n ) x_{n+1}=x_{n}-\frac{f(x_{n})}{f^{\prime}(x_{n})}
  88. z n + 1 = z n - f ( z n ) f ( x n ) z_{n+1}=z_{n}-\frac{f(z_{n})}{f^{\prime}(x_{n})}
  89. f ( x n ) f^{\prime}(x_{n})
  90. f ( z n ) f^{\prime}(z_{n})
  91. x n x_{n}
  92. z n z_{n}
  93. lim n x n + 1 - z n + 1 ( x n - z n ) 2 = f ′′ ( α ) 2 f ( α ) \lim_{n\to\infty}\frac{x_{n+1}-z_{n+1}}{(x_{n}-z_{n})^{2}}=\frac{f^{\prime% \prime}(\alpha)}{2f^{\prime}(\alpha)}
  94. x n x_{n}
  95. z n z_{n}
  96. x n + 1 = x n - f ( x n ) f ′′ ( x n ) . x_{n+1}=x_{n}-\frac{f^{\prime}(x_{n})}{f^{\prime\prime}(x_{n})}.\,\!
  97. f ( x ) = a - 1 x f(x)=a-\frac{1}{x}
  98. x n + 1 = x n - f ( x n ) f ( x n ) = x n - a - 1 x n 1 x n 2 = x n ( 2 - a x n ) \begin{aligned}\displaystyle x_{n+1}&\displaystyle=x_{n}-\frac{f(x_{n})}{f^{% \prime}(x_{n})}\\ &\displaystyle=x_{n}-\frac{a-\frac{1}{x_{n}}}{\frac{1}{x_{n}^{2}}}\\ &\displaystyle=x_{n}\,(2-ax_{n})\end{aligned}
  99. g ( x ) = h ( x ) , g(x)=h(x),\,\!
  100. f ( x ) = g ( x ) - h ( x ) . f(x)=g(x)-h(x).\,\!
  101. x 2 = 612 \,x^{2}=612
  102. f ( x ) = x 2 - 612 \,f(x)=x^{2}-612
  103. f ( x ) = 2 x . f^{\prime}(x)=2x.\,
  104. x 1 = x 0 - f ( x 0 ) f ( x 0 ) = 10 - 10 2 - 612 2 10 = 35.6 x 2 = x 1 - f ( x 1 ) f ( x 1 ) = 35.6 - 35.6 2 - 612 2 35.6 = 2 ¯ 6.395505617978 x 3 = = = 24.7 ¯ 90635492455 x 4 = = = 24.7386 ¯ 88294075 x 5 = = = 24.7386337537 ¯ 67 \begin{matrix}x_{1}&=&x_{0}-\dfrac{f(x_{0})}{f^{\prime}(x_{0})}&=&10-\dfrac{10% ^{2}-612}{2\cdot 10}&=&35.6\\ x_{2}&=&x_{1}-\dfrac{f(x_{1})}{f^{\prime}(x_{1})}&=&35.6-\dfrac{35.6^{2}-612}{% 2\cdot 35.6}&=&\underline{2}6.395505617978\dots\\ x_{3}&=&\vdots&=&\vdots&=&\underline{24.7}90635492455\dots\\ x_{4}&=&\vdots&=&\vdots&=&\underline{24.7386}88294075\dots\\ x_{5}&=&\vdots&=&\vdots&=&\underline{24.7386337537}67\dots\end{matrix}
  105. x 1 = x 0 - f ( x 0 ) f ( x 0 ) = 0.5 - cos ( 0.5 ) - ( 0.5 ) 3 - sin ( 0.5 ) - 3 ( 0.5 ) 2 = 1.112141637097 x 2 = x 1 - f ( x 1 ) f ( x 1 ) = = 0. ¯ 909672693736 x 3 = = = 0.86 ¯ 7263818209 x 4 = = = 0.86547 ¯ 7135298 x 5 = = = 0.8654740331 ¯ 11 x 6 = = = 0.865474033102 ¯ \begin{matrix}x_{1}&=&x_{0}-\dfrac{f(x_{0})}{f^{\prime}(x_{0})}&=&0.5-\dfrac{% \cos(0.5)-(0.5)^{3}}{-\sin(0.5)-3(0.5)^{2}}&=&1.112141637097\\ x_{2}&=&x_{1}-\dfrac{f(x_{1})}{f^{\prime}(x_{1})}&=&\vdots&=&\underline{0.}909% 672693736\\ x_{3}&=&\vdots&=&\vdots&=&\underline{0.86}7263818209\\ x_{4}&=&\vdots&=&\vdots&=&\underline{0.86547}7135298\\ x_{5}&=&\vdots&=&\vdots&=&\underline{0.8654740331}11\\ x_{6}&=&\vdots&=&\vdots&=&\underline{0.865474033102}\end{matrix}
  106. x 0 = 1 x_{0}=1
  107. f ( x ) = x 2 - 2 f(x)=x^{2}-2
  108. f ( x ) = 2 x f^{\prime}(x)=2x
  109. f ( x n ) 0 f^{\prime}(x_{n})\approx 0

Niccolò_Fontana_Tartaglia.html

  1. V 2 = 1 288 det [ 0 d 12 2 d 13 2 d 14 2 1 d 21 2 0 d 23 2 d 24 2 1 d 31 2 d 32 2 0 d 34 2 1 d 41 2 d 42 2 d 43 2 0 1 1 1 1 1 0 ] V^{2}=\frac{1}{288}\det\begin{bmatrix}0&d_{12}^{2}&d_{13}^{2}&d_{14}^{2}&1\\ d_{21}^{2}&0&d_{23}^{2}&d_{24}^{2}&1\\ d_{31}^{2}&d_{32}^{2}&0&d_{34}^{2}&1\\ d_{41}^{2}&d_{42}^{2}&d_{43}^{2}&0&1\\ 1&1&1&1&0\end{bmatrix}

Nicolas_Léonard_Sadi_Carnot.html

  1. Δ S ln V V 0 . \Delta S\propto\ln\frac{V}{V_{0}}.

Niels_Bohr.html

  1. 1 λ = R H ( 1 2 2 - 1 n 2 ) for n = 3 , 4 , 5 , \frac{1}{\lambda}=R_{\mathrm{H}}\left(\frac{1}{2^{2}}-\frac{1}{n^{2}}\right)% \quad\mathrm{for~{}}n=3,4,5,...
  2. R Z = 2 π 2 m e Z 2 e 4 h 3 R_{Z}={2\pi^{2}m_{e}Z^{2}e^{4}\over h^{3}}
  3. h ν = ϵ 2 - ϵ 1 h\nu=\epsilon_{2}-\epsilon_{1}\,

Nim.html

  1. n = 2 a 1 3 a 2 5 a 3 7 a 4 p k a k n=2^{a_{1}}3^{a_{2}}5^{a_{3}}7^{a_{4}}\cdots p_{k}^{a_{k}}
  2. p k p_{k}
  3. a r a_{r}
  4. 0.123 k 0123 k 0123 = 0 ˙ .123 k ˙ , 0.123\ldots k0123\ldots k0123\dots=\dot{0}.123\ldots\dot{k},\,
  5. k {}_{k}

Nitrox.html

  1. f O 2 , max = p O 2 , max p = Maximum acceptable partial pressure of oxygen Maximum ambient pressure of the dive f_{\,\text{O}_{2},\,\text{max}}=\frac{p_{\,\text{O}_{2},\,\text{max}}}{p}=% \frac{\,\text{Maximum acceptable partial pressure of oxygen}}{\,\text{Maximum % ambient pressure of the dive}}

No-cloning_theorem.html

  1. | ϕ A |\phi\rangle_{A}
  2. | e B |e\rangle_{B}
  3. | ϕ A |\phi\rangle_{A}
  4. | ϕ A | e B |\phi\rangle_{A}\otimes|e\rangle_{B}\,
  5. \otimes
  6. U ( t ) = e - i H t / U(t)=e^{-iHt/\hbar}
  7. - H / -H/\hbar
  8. U | ϕ A | e B = | ϕ A | ϕ B U|\phi\rangle_{A}|e\rangle_{B}=|\phi\rangle_{A}|\phi\rangle_{B}\,
  9. | ϕ |\phi\rangle
  10. | ϕ A |\phi\rangle_{A}
  11. | ψ A |\psi\rangle_{A}
  12. e | B ϕ | A | ψ A | e B = e | B ϕ | A U U | ψ A | e B = ϕ | B ϕ | A | ψ A | ψ B , \langle e|_{B}\langle\phi|_{A}|\psi\rangle_{A}|e\rangle_{B}=\langle e|_{B}% \langle\phi|_{A}U^{\dagger}U|\psi\rangle_{A}|e\rangle_{B}=\langle\phi|_{B}% \langle\phi|_{A}|\psi\rangle_{A}|\psi\rangle_{B},
  13. ϕ | ψ = ϕ | ψ 2 . \langle\phi|\psi\rangle=\langle\phi|\psi\rangle^{2}.\,
  14. ϕ | ψ = 1 \langle\phi|\psi\rangle=1
  15. ϕ | ψ = 0 \langle\phi|\psi\rangle=0
  16. ϕ = ψ \phi=\psi
  17. ϕ \phi
  18. ψ \psi
  19. | ϕ = 1 2 ( | 0 + | 1 ) , | ψ = 1 2 ( | 0 - | 1 ) |\phi\rangle={1\over\sqrt{2}}\bigg(|0\rangle+|1\rangle\bigg),\qquad|\psi% \rangle={1\over\sqrt{2}}\bigg(|0\rangle-|1\rangle\bigg)
  20. ϕ | ψ = 0 = ϕ | ψ 2 \langle\phi|\psi\rangle=0=\langle\phi|\psi\rangle^{2}
  21. | z + B |z+\rangle_{B}
  22. | z - B |z-\rangle_{B}
  23. | z + B |z+\rangle_{B}
  24. | z - B |z-\rangle_{B}

Noetherian_ring.html

  1. I 1 I k - 1 I k I k + 1 I_{1}\subseteq\cdots\subseteq I_{k-1}\subseteq I_{k}\subseteq I_{k+1}\subseteq\cdots
  2. I n = I n + 1 = . I_{n}=I_{n+1}=\cdots.
  3. 𝔤 \mathfrak{g}
  4. Sym ( 𝔤 ) \operatorname{Sym}(\mathfrak{g})
  5. R = { [ a β 0 γ ] | a , β , γ } . R=\left\{\left.\begin{bmatrix}a&\beta\\ 0&\gamma\end{bmatrix}\,\right|\,a\in\mathbb{Z},\beta\in\mathbb{Q},\gamma\in% \mathbb{Q}\right\}.
  6. n = i p i e i n=\prod_{i}{p_{i}}^{e_{i}}
  7. ( n ) = i ( p i e i ) (n)=\cap_{i}({p_{i}}^{e_{i}})
  8. n = i p i e i n=\prod_{i}{p_{i}}^{e_{i}}
  9. I = i = 1 t Q i I=\bigcap_{i=1}^{t}Q_{i}
  10. I = i = 1 k P i I=\bigcap_{i=1}^{k}P_{i}

Noise-equivalent_power.html

  1. A / Hz \mathrm{A}/\sqrt{\mathrm{Hz}}
  2. V / Hz \mathrm{V}/\sqrt{\mathrm{Hz}}
  3. A / W \mathrm{A}/\mathrm{W}
  4. V / W \mathrm{V}/\mathrm{W}
  5. 10 - 12 W / Hz 10^{-12}\mathrm{W}/\sqrt{\mathrm{Hz}}

Noise_figure.html

  1. F = SNR in SNR out F=\frac{\mathrm{SNR}_{\mathrm{in}}}{\mathrm{SNR}_{\mathrm{out}}}
  2. NF = 10 log 10 ( F ) = 10 log 10 ( SNR in SNR out ) = SNR in , dB - SNR out , dB \mathrm{NF}=10\log_{10}(F)=10\log_{10}\left(\frac{\mathrm{SNR}_{\mathrm{in}}}{% \mathrm{SNR}_{\mathrm{out}}}\right)=\mathrm{SNR}_{\mathrm{in,dB}}-\mathrm{SNR}% _{\mathrm{out,dB}}
  3. F = 1 + T e T 0 F=1+\frac{T_{e}}{T_{0}}
  4. T e = ( L - 1 ) T T_{\mathrm{e}}=(L-1)T
  5. F = 1 + ( L - 1 ) T T 0 F=1+\frac{(L-1)T}{T_{0}}
  6. F = F 1 + F 2 - 1 G 1 + F 3 - 1 G 1 G 2 + F 4 - 1 G 1 G 2 G 3 + + F n - 1 G 1 G 2 G 3 G n - 1 , F=F_{1}+\frac{F_{2}-1}{G_{1}}+\frac{F_{3}-1}{G_{1}G_{2}}+\frac{F_{4}-1}{G_{1}G% _{2}G_{3}}+\cdots+\frac{F_{n}-1}{G_{1}G_{2}G_{3}\cdots G_{n-1}},

Noise_temperature.html

  1. P B = k B T {{P}\over{B}}\,=\,k_{B}\,T
  2. P P
  3. B B
  4. k B k_{B}
  5. T T
  6. P / B P/B
  7. P B {{P}\over{B}}
  8. v n 2 ¯ B = 4 k B R T {{\bar{v_{n}^{2}}}\over{B}}=4k_{B}RT
  9. i n 2 ¯ B = 4 k B G T {{\bar{i_{n}^{2}}}\over{B}}=4k_{B}GT
  10. P / B P/B
  11. T T
  12. T = P / B k B T=\frac{P/B}{k_{B}}
  13. T e q T_{eq}
  14. T e q = T a n t + T s y s T_{eq}=T_{ant}+T_{sys}
  15. T a n t T_{ant}
  16. T s y s T_{sys}
  17. T e q T_{eq}
  18. T a n t T_{ant}
  19. T e q T_{eq}
  20. B k B T s y s B{k_{B}}T_{sys}
  21. T s y s T_{sys}
  22. 1 / T a n t - 1 / ( T a n t + T s y s ) 1/T_{ant}-1/(T_{ant}+T_{sys})
  23. T 0 T_{0}
  24. F = T 0 + T s y s T 0 F=\frac{T_{0}+T_{sys}}{T_{0}}
  25. T 0 T_{0}
  26. N F = 10 log 10 ( F ) NF=10\ \log_{10}(F)
  27. T e q = T 1 + T 2 G 1 + T 3 G 1 G 2 + T_{eq}=T_{1}+\frac{T_{2}}{G_{1}}+\frac{T_{3}}{G_{1}G_{2}}+\cdots
  28. T e q T_{eq}
  29. T 1 T_{1}
  30. T 2 T_{2}
  31. T 3 T_{3}
  32. G 1 G_{1}
  33. G 2 G_{2}
  34. G 1 G 2 G 3 G_{1}\cdot G_{2}\cdot G_{3}\cdots
  35. N F = 10 log 10 ( 1 + T e q / 290 ) NF=10\ \log_{10}(1+T_{eq}/290)
  36. G 1 = 1 4 {G_{1}}=\frac{1}{4}
  37. T e q = T 1 + 4 T 2 + T_{eq}=T_{1}+4T_{2}+\cdots
  38. T 2 T_{2}
  39. T 1 T_{1}
  40. G 1 {G_{1}}
  41. T a n t T_{ant}

Non-deterministic_Turing_machine.html

  1. M = ( Q , Σ , ι , , A , δ ) M=(Q,\Sigma,\iota,\sqcup,A,\delta)
  2. Q Q
  3. Σ \Sigma
  4. ι Q \iota\in Q
  5. Σ \sqcup\in\Sigma
  6. A Q A\subseteq Q
  7. δ ( Q \ A × Σ ) × ( Q × Σ × { L , R } ) \delta\subseteq\left(Q\backslash A\times\Sigma\right)\times\left(Q\times\Sigma% \times\{L,R\}\right)
  8. L L
  9. R R
  10. A A

Non-Euclidean_geometry.html

  1. z z = ( x + y ϵ ) ( x - y ϵ ) = x 2 + y 2 zz^{\ast}=(x+y\epsilon)(x-y\epsilon)=x^{2}+y^{2}
  2. ϵ 2 = + 1 \epsilon^{2}=+1
  3. z z = ( x + y j ) ( x - y j ) = x 2 - y 2 zz^{\ast}=(x+yj)(x-yj)=x^{2}-y^{2}\!
  4. ϵ 2 = 0 \epsilon^{2}=0
  5. z = x + y ϵ , ϵ 2 = 0 , z=x+y\epsilon,\quad\epsilon^{2}=0,
  6. x = x + v t , t = t x^{\prime}=x+vt,\quad t^{\prime}=t
  7. ( x t ) = ( 1 v 0 1 ) ( x t ) . \begin{pmatrix}x^{\prime}\\ t^{\prime}\end{pmatrix}=\begin{pmatrix}1&v\\ 0&1\end{pmatrix}\begin{pmatrix}x\\ t\end{pmatrix}.
  8. t + x ϵ = ( 1 + v ϵ ) ( t + x ϵ ) = t + ( x + v t ) ϵ . t^{\prime}+x^{\prime}\epsilon=(1+v\epsilon)(t+x\epsilon)=t+(x+vt)\epsilon.

Non-standard_analysis.html

  1. 𝔽 \mathbb{F}
  2. 𝔽 \mathbb{F}
  3. 1 n \frac{1}{n}
  4. n n
  5. * {}^{*}\mathbb{R}
  6. \mathbb{R}
  7. \mathbb{N}
  8. \mathbb{R}^{\mathbb{N}}
  9. * {}^{*}\mathbb{R}
  10. \mathbb{R}^{\mathbb{N}}
  11. F P ( ) F\subset P(\mathbb{N})
  12. F F
  13. u = ( u n ) , v = ( v n ) u=(u_{n}),v=(v_{n})\in\mathbb{R}^{\mathbb{N}}
  14. u u
  15. v v
  16. { n : u n = v n } F \{n\in\mathbb{N}:u_{n}=v_{n}\}\in F
  17. \mathbb{R}^{\mathbb{N}}
  18. * {}^{*}\mathbb{R}
  19. * = / F {}^{*}\mathbb{R}={\mathbb{R}^{\mathbb{N}}}/{F}
  20. V ( S ) V(S)
  21. S S
  22. V ( S ) V(S)
  23. * V ( S ) *V(S)
  24. V ( S ) * V ( S ) V(S)→*V(S)
  25. V ( S ) V(S)
  26. * V ( S ) *V(S)
  27. T T
  28. H H
  29. v v
  30. H H
  31. T T
  32. H H
  33. H H
  34. T T
  35. T T
  36. w w
  37. * H *H
  38. y y
  39. T ( y ) T(y)
  40. P w T P_{w}\circ T
  41. q q
  42. q ( T ) q(T)
  43. k k
  44. 1 1
  45. w w
  46. k k
  47. T T
  48. x x
  49. H H
  50. q ( T ) ( x ) q(T)(x)
  51. | q ( T ) ( x ) | > 1 |q(T)(x)|>1
  52. q ( T ) q(T)
  53. q ( T ) ( x ) q(T)(x)
  54. j j
  55. | q ( T w ) ( Π j ( x ) ) | < 1 2 |q(T_{w})\left(\Pi_{j}(x)\right)|<\tfrac{1}{2}
  56. r r
  57. r r
  58. s t ( r ) st(r)
  59. r r
  60. S S
  61. S S
  62. V ( S ) V(S)
  63. V 0 ( S ) = S , V_{0}(S)=S,
  64. V n + 1 ( S ) = V n ( S ) 2 V n ( S ) , V_{n+1}(S)=V_{n}(S)\cup 2^{V_{n}(S)},
  65. V ( S ) = n 𝐍 V n ( S ) . V(S)=\bigcup_{n\in\mathbf{N}}V_{n}(S).
  66. S S
  67. S S
  68. S S
  69. V ( 𝐑 ) V(\mathbf{R})
  70. * 𝐑 *\mathbf{R}
  71. * : V ( 𝐑 ) V ( * 𝐑 ) *:V(\mathbf{R})→V(*\mathbf{R})
  72. x A , Φ ( x , α 1 , , α n ) \forall x\in A,\Phi(x,\alpha_{1},\ldots,\alpha_{n})
  73. x A , Φ ( x , α 1 , , α n ) \exists x\in A,\Phi(x,\alpha_{1},\ldots,\alpha_{n})
  74. x A , y 2 B , x y \forall x\in A,\ \exists y\in 2^{B},\quad x\in y
  75. x x
  76. A A
  77. y y
  78. B B
  79. x A , y , x y \forall x\in A,\ \exists y,\quad x\in y
  80. V ( 𝐑 ) V(\mathbf{R})
  81. V ( 𝐑 ) V(\mathbf{R})
  82. 𝐑 \mathbf{R}
  83. V ( 𝐑 ) V(\mathbf{R})
  84. P ( A 1 , , A n ) P ( * A 1 , , * A n ) P(A_{1},\ldots,A_{n})\iff P(*A_{1},\ldots,*A_{n})
  85. k A k \bigcap_{k}A_{k}\neq\emptyset
  86. V ( 𝐑 ) V(\mathbf{R})
  87. * 𝐑 *\mathbf{R}
  88. * 𝐍 *\mathbf{N}
  89. 𝐍 \mathbf{N}
  90. * 𝐍 𝐍 *\mathbf{N}−\mathbf{N}
  91. A n = { k 𝐍 * : k n } A_{n}=\{k\in{{}^{*}\mathbf{N}}:k\geq n\}
  92. r s θ 𝐑 + , | r - s | θ r\cong s\iff\forall\theta\in\mathbf{R}^{+},\ |r-s|\leq\theta
  93. r r
  94. n n
  95. * 𝐍 𝐍 *\mathbf{N}−\mathbf{N}
  96. 1 / n 1/n
  97. r r
  98. * 𝐑 *\mathbf{R}
  99. V ( * 𝐑 ) V(*\mathbf{R})
  100. ( x , y ) (x,y)
  101. x x
  102. y y
  103. * 𝐑 *\mathbf{R}
  104. x x
  105. y y
  106. ( 0 , 1 ) (0,1)
  107. y y
  108. x x
  109. r r
  110. s t ( r ) st(r)
  111. r r
  112. s t st
  113. 𝐑 \mathbf{R}
  114. s s
  115. ( L , U ) (L,U)
  116. U U
  117. a a
  118. s s
  119. U U
  120. b b
  121. s s
  122. ( L , U ) (L,U)
  123. s s
  124. f f
  125. a a , b aa,b
  126. x x
  127. * a a , b *aa,b
  128. * f ( x ) * f ( s t ( x ) ) *f(x)≅*f(st(x))
  129. f f
  130. x x
  131. h h
  132. f ( x ) = st ( f * ( x + h ) - f * ( x ) h ) f^{\prime}(x)=\operatorname{st}\left(\frac{{{}^{*}f}(x+h)-{{}^{*}f}(x)}{h}\right)
  133. h h
  134. f ( x ) f′(x)
  135. f f
  136. x x
  137. κ κ
  138. κ κ
  139. { A i } i I \{A_{i}\}_{i\in I}
  140. | I | κ |I|\leq\kappa
  141. i I A i \bigcap_{i\in I}A_{i}\neq\emptyset
  142. X X
  143. | 2 < s u p > X | |2<sup>X|

Nonlinear_optics.html

  1. χ ( 3 ) \chi^{(3)}
  2. χ ( 3 ) \chi^{(3)}
  3. 𝐏 ( t ) = ε 0 ( χ ( 1 ) 𝐄 ( t ) + χ ( 2 ) 𝐄 2 ( t ) + χ ( 3 ) 𝐄 3 ( t ) + ) . \mathbf{P}(t)=\varepsilon_{0}(\chi^{(1)}\mathbf{E}(t)+\chi^{(2)}\mathbf{E}^{2}% (t)+\chi^{(3)}\mathbf{E}^{3}(t)+\ldots)\ .
  4. × × 𝐄 + n 2 c 2 2 t 2 𝐄 = - 1 ε 0 c 2 2 t 2 𝐏 N L , \nabla\times\nabla\times\mathbf{E}+\frac{n^{2}}{c^{2}}\frac{\partial^{2}}{% \partial t^{2}}\mathbf{E}=-\frac{1}{\varepsilon_{0}c^{2}}\frac{\partial^{2}}{% \partial t^{2}}\mathbf{P}^{NL},
  5. × ( × 𝐕 ) = ( 𝐕 ) - 2 𝐕 \nabla\times\left(\nabla\times\mathbf{V}\right)=\nabla\left(\nabla\cdot\mathbf% {V}\right)-\nabla^{2}\mathbf{V}
  6. ρ free = 0 \rho\text{free}=0
  7. 𝐃 = 0 , \nabla\cdot\mathbf{D}=0,
  8. 2 𝐄 - n 2 c 2 2 t 2 𝐄 = 0. \nabla^{2}\mathbf{E}-\frac{n^{2}}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}% \mathbf{E}=0.
  9. 𝐄 = 0 \nabla\cdot\mathbf{E}=0
  10. 2 𝐄 - n 2 c 2 2 t 2 𝐄 = 1 ε 0 c 2 2 t 2 𝐏 N L . \nabla^{2}\mathbf{E}-\frac{n^{2}}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}% \mathbf{E}=\frac{1}{\varepsilon_{0}c^{2}}\frac{\partial^{2}}{\partial t^{2}}% \mathbf{P}^{NL}.
  11. 2 𝐄 - n 2 c 2 2 t 2 𝐄 = 0. \nabla^{2}\mathbf{E}-\frac{n^{2}}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}% \mathbf{E}=0.
  12. 1 ε 0 c 2 2 t 2 𝐏 N L , \frac{1}{\varepsilon_{0}c^{2}}\frac{\partial^{2}}{\partial t^{2}}\mathbf{P}^{% NL},
  13. 𝐏 N L = ε 0 χ ( 2 ) 𝐄 2 ( t ) . \mathbf{P}^{NL}=\varepsilon_{0}\chi^{(2)}\mathbf{E}^{2}(t).
  14. 𝐄 ( t ) = E 1 cos ( ω 1 t ) + E 2 cos ( ω 2 t ) \mathbf{E}(t)=E_{1}\cos(\omega_{1}t)+E_{2}\cos(\omega_{2}t)
  15. 𝐄 ( t ) = 1 2 E 1 e - i ω 1 t + 1 2 E 2 e - i ω 2 t + c . c . \mathbf{E}(t)=\frac{1}{2}E_{1}e^{-i\omega_{1}t}+\frac{1}{2}E_{2}e^{-i\omega_{2% }t}+c.c.
  16. 𝐏 N L = ε 0 χ ( 2 ) 𝐄 2 ( t ) = ε 0 4 χ ( 2 ) [ | E 1 | 2 e - i 2 ω 1 t + | E 2 | 2 e - i 2 ω 2 t + 2 E 1 E 2 e - i ( ω 1 + ω 2 ) t + 2 E 1 E 2 * e - i ( ω 1 - ω 2 ) t + ( | E 1 | 2 + | E 2 | 2 ) e 0 + c . c . ] , \begin{aligned}\displaystyle\mathbf{P}^{NL}=\varepsilon_{0}\chi^{(2)}\mathbf{E% }^{2}(t)&\displaystyle=\frac{\varepsilon_{0}}{4}\chi^{(2)}[|E_{1}|^{2}e^{-i2% \omega_{1}t}+|E_{2}|^{2}e^{-i2\omega_{2}t}\\ &\displaystyle\qquad+2E_{1}E_{2}e^{-i(\omega_{1}+\omega_{2})t}\\ &\displaystyle\qquad+2E_{1}E_{2}^{*}e^{-i(\omega_{1}-\omega_{2})t}\\ &\displaystyle\qquad+\left(|E_{1}|^{2}+|E_{2}|^{2}\right)e^{0}+c.c.],\end{aligned}
  17. E j ( 𝐱 , t ) = e i ( 𝐤 j 𝐱 - ω j t ) + c . c . , E_{j}(\mathbf{x},t)=e^{i(\mathbf{k}_{j}\cdot\mathbf{x}-\omega_{j}t)}+c.c.,
  18. 𝐱 \mathbf{x}
  19. 𝐤 j = n ( ω j ) ω j / c \left\|\mathbf{k}_{j}\right\|=n(\omega_{j})\omega_{j}/c
  20. c c
  21. n ( ω j ) n(\omega_{j})
  22. ω j \omega_{j}
  23. ω 3 = ω 1 + ω 2 \omega_{3}=\omega_{1}+\omega_{2}
  24. P ( 2 ) ( 𝐱 , t ) E 1 n 1 E 2 n 2 e i ( ( 𝐤 1 + 𝐤 2 ) 𝐱 - ω 3 t ) + c . c . P^{(2)}(\mathbf{x},t)\propto E_{1}^{n_{1}}E_{2}^{n_{2}}e^{i((\mathbf{k}_{1}+% \mathbf{k}_{2})\cdot\mathbf{x}-\omega_{3}t)}+c.c.
  25. 𝐱 \mathbf{x}
  26. ω 3 \omega_{3}
  27. 𝐤 3 = n ( ω 3 ) ω 3 / c \left\|\mathbf{k}_{3}\right\|=n(\omega_{3})\omega_{3}/c
  28. ω 3 \omega_{3}
  29. 𝐤 3 = 𝐤 1 + 𝐤 2 . \vec{\mathbf{k}}_{3}=\vec{\mathbf{k}}_{1}+\vec{\mathbf{k}}_{2}.
  30. λ p λ s λ i \lambda_{p}\leq\lambda_{s}\leq\lambda_{i}
  31. χ ( 2 ) \chi^{(2)}
  32. χ ( 3 ) \chi^{(3)}
  33. χ ( 3 ) \chi^{(3)}
  34. χ ( 3 ) \chi^{(3)}
  35. χ ( 2 ) \chi^{(2)}
  36. χ ( 3 ) \chi^{(3)}
  37. Ξ j ( 𝐱 , t ) = 1 2 E j ( 𝐱 ) e i ( ω j t - 𝐤 𝐱 ) + c.c. \Xi_{j}(\mathbf{x},t)=\frac{1}{2}E_{j}(\mathbf{x})e^{i(\omega_{j}t-\mathbf{k}% \cdot\mathbf{x})}+\mbox{c.c.}~{}
  38. P NL = ϵ 0 χ ( 3 ) ( Ξ 1 + Ξ 2 + Ξ 3 ) 3 P_{\mbox{NL}~{}}=\epsilon_{0}\chi^{(3)}(\Xi_{1}+\Xi_{2}+\Xi_{3})^{3}
  39. P ω = 1 2 χ ( 3 ) ϵ 0 E 1 E 2 E 3 * e i ( ω t - 𝐤 𝐱 ) + c.c. . P_{\omega}=\frac{1}{2}\chi^{(3)}\epsilon_{0}E_{1}E_{2}E_{3}^{*}e^{i(\omega t-% \mathbf{k}\cdot\mathbf{x})}+\mbox{c.c.}~{}.
  40. E 4 = i ω l 2 n c χ ( 3 ) E 1 E 2 E 3 * E_{4}=\frac{i\omega l}{2nc}\chi^{(3)}E_{1}E_{2}E_{3}^{*}
  41. E 4 ( 𝐱 ) E 3 * ( 𝐱 ) ; E_{4}(\mathbf{x})\propto E_{3}^{*}(\mathbf{x});

Normal_distribution.html

  1. 1 2 [ 1 + erf ( x - μ σ 2 ) ] \frac{1}{2}\left[1+\operatorname{erf}\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right]
  2. μ + σ 2 erf - 1 ( 2 F - 1 ) \mu+\sigma\sqrt{2}\,\operatorname{erf}^{-1}(2F-1)
  3. μ μ
  4. μ μ
  5. μ μ
  6. σ 2 \sigma^{2}\,
  7. 1 2 ln ( 2 π e σ 2 ) \frac{1}{2}\ln(2\pi e\,\sigma^{2})
  8. exp { μ t + 1 2 σ 2 t 2 } \exp\{\mu t+\frac{1}{2}\sigma^{2}t^{2}\}
  9. exp { i μ t - 1 2 σ 2 t 2 } \exp\{i\mu t-\frac{1}{2}\sigma^{2}t^{2}\}
  10. ( 1 / σ 2 0 0 1 / ( 2 σ 4 ) ) \begin{pmatrix}1/\sigma^{2}&0\\ 0&1/(2\sigma^{4})\end{pmatrix}
  11. f ( x | μ , σ ) = 1 σ 2 π e - ( x - μ ) 2 2 σ 2 f(x\;|\;\mu,\sigma)=\frac{1}{\sigma\sqrt{2\pi}}\;e^{-\frac{(x-\mu)^{2}}{2% \sigma^{2}}}
  12. μ \mu
  13. σ \sigma
  14. σ 2 \sigma^{2}
  15. μ = 0 \mu=0
  16. σ = 1 \sigma=1
  17. N ( 0 , 1 ) N(0,1)
  18. ϕ ( x ) = e - 1 2 x 2 2 π \phi(x)=\frac{e^{-\frac{\scriptscriptstyle 1}{\scriptscriptstyle 2}x^{2}}}{% \sqrt{2\pi}}\,
  19. 1 / 2 π \scriptstyle\ 1/\sqrt{2\pi}
  20. 1 / 2 π 1/\sqrt{2\pi}
  21. ϕ ( x ) = e - x 2 π \phi(x)=\frac{e^{-x^{2}}}{\sqrt{\pi}}\,
  22. ϕ ( x ) = e - π x 2 \phi(x)=e^{-\pi x^{2}}
  23. f ( x , μ , σ ) = 1 σ ϕ ( x - μ σ ) . f(x,\mu,\sigma)=\frac{1}{\sigma}\phi\left(\frac{x-\mu}{\sigma}\right).
  24. 1 / σ 1/\sigma
  25. f ( x ) = e a x 2 + b x + c f(x)=e^{ax^{2}+bx+c}
  26. b 2 / ( 4 a ) + ln ( - a / π ) / 2 b^{2}/(4a)+\ln(-a/\pi)/2
  27. - ln ( 2 π ) / 2 -\ln(2\pi)/2
  28. X 𝒩 ( μ , σ 2 ) . X\ \sim\ \mathcal{N}(\mu,\,\sigma^{2}).
  29. τ \tau
  30. f ( x ) = τ 2 π e - τ ( x - μ ) 2 2 . f(x)=\sqrt{\frac{\tau}{2\pi}}\,e^{\frac{-\tau(x-\mu)^{2}}{2}}.
  31. τ = 1 / σ \tau^{\prime}=1/\sigma
  32. f ( x ) = τ 2 π e - ( τ ) 2 ( x - μ ) 2 2 . f(x)=\frac{\tau^{\prime}}{\sqrt{2\pi}}\,e^{\frac{-(\tau^{\prime})^{2}(x-\mu)^{% 2}}{2}}.
  33. σ 2 f ( x ) + f ( x ) ( x - μ ) = 0 , f ( 0 ) = e - μ 2 / ( 2 σ 2 ) 2 π σ \sigma^{2}f^{\prime}(x)+f(x)(x-\mu)=0,\qquad f(0)=\frac{e^{-\mu^{2}/(2\sigma^{% 2})}}{\sqrt{2\pi}\sigma}
  34. f ( x ) + τ f ( x ) ( x - μ ) = 0 , f ( 0 ) = τ e - μ 2 τ / 2 2 π . f^{\prime}(x)+\tau f(x)(x-\mu)=0,\qquad f(0)=\frac{\sqrt{\tau}e^{-\mu^{2}\tau/% 2}}{\sqrt{2\pi}}.
  35. E [ X p ] = { 0 if p is odd, σ p ( p - 1 ) ! ! if p is even. \mathrm{E}\left[X^{p}\right]=\begin{cases}0&\,\text{if }p\,\text{ is odd,}\\ \sigma^{p}\,(p-1)!!&\,\text{if }p\,\text{ is even.}\end{cases}
  36. E [ | X | p ] = σ p ( p - 1 ) ! ! { 2 π if p is odd 1 if p is even } = σ p 2 p 2 Γ ( p + 1 2 ) π \operatorname{E}\left[|X|^{p}\right]=\sigma^{p}\,(p-1)!!\cdot\left.\begin{% cases}\sqrt{\frac{2}{\pi}}&\,\text{if }p\,\text{ is odd}\\ 1&\,\text{if }p\,\text{ is even}\end{cases}\right\}=\sigma^{p}\cdot\frac{2^{% \frac{p}{2}}\Gamma\left(\frac{p+1}{2}\right)}{\sqrt{\pi}}
  37. E [ X p ] = σ p ( - i 2 ) p U ( - 1 2 p , 1 2 , - 1 2 ( μ / σ ) 2 ) , \operatorname{E}\left[X^{p}\right]=\sigma^{p}\cdot(-i\sqrt{2})^{p}\;U\left({-% \frac{1}{2}p},\,\frac{1}{2},\,-\frac{1}{2}(\mu/\sigma)^{2}\right),
  38. E [ | X | p ] = σ p 2 p 2 Γ ( 1 + p 2 ) π 1 F 1 ( - 1 2 p , 1 2 , - 1 2 ( μ / σ ) 2 ) . \operatorname{E}\left[|X|^{p}\right]=\sigma^{p}\cdot 2^{\frac{p}{2}}\frac{% \Gamma\left(\frac{1+p}{2}\right)}{\sqrt{\pi}}\;_{1}F_{1}\left({-\frac{1}{2}p},% \,\frac{1}{2},\,-\frac{1}{2}(\mu/\sigma)^{2}\right).
  39. ϕ ^ ( t ) = - f ( x ) e i t x d x = e i μ t e - 1 2 ( σ t ) 2 \hat{\phi}(t)=\int_{-\infty}^{\infty}\!f(x)e^{itx}dx=e^{i\mu t}e^{-\frac{1}{2}% (\sigma t)^{2}}
  40. M ( t ) = ϕ ^ ( - i t ) = e μ t e 1 2 σ 2 t 2 M(t)=\hat{\phi}(-it)=e^{\mu t}e^{\frac{1}{2}\sigma^{2}t^{2}}
  41. g ( t ) = ln M ( t ) = μ t + 1 2 σ 2 t 2 g(t)=\ln M(t)=\mu t+\frac{1}{2}\sigma^{2}t^{2}
  42. Φ \Phi
  43. Φ ( x ) = 1 2 π - x e - t 2 / 2 d t \Phi(x)\;=\;\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{x}e^{-t^{2}/2}\,dt
  44. [ - x , x ] [-x,x]
  45. erf ( x ) = 1 π - x x e - t 2 d t \operatorname{erf}(x)\;=\;\frac{1}{\sqrt{\pi}}\int_{-x}^{x}e^{-t^{2}}\,dt
  46. Φ ( x ) = 1 2 [ 1 + erf ( x 2 ) ] \Phi(x)\;=\;\frac{1}{2}\left[1+\operatorname{erf}\left(\frac{x}{\sqrt{2}}% \right)\right]
  47. F ( x ) = Φ ( x - μ σ ) = 1 2 [ 1 + erf ( x - μ σ 2 ) ] F(x)\;=\;\Phi\left(\frac{x-\mu}{\sigma}\right)\;=\;\frac{1}{2}\left[1+% \operatorname{erf}\left(\frac{x-\mu}{\sigma\sqrt{2}}\right)\right]
  48. Q ( x ) = 1 - Φ ( x ) Q(x)=1-\Phi(x)
  49. Φ \Phi
  50. Φ \Phi
  51. Φ ( - x ) = 1 - Φ ( x ) \Phi(-x)=1-\Phi(x)
  52. Φ ( x ) d x \int\Phi(x)\,dx
  53. Φ ( x ) d x = x Φ ( x ) + ϕ ( x ) \int\Phi(x)\,dx=x\Phi(x)+\phi(x)
  54. Φ ( x ) = 0.5 + 1 2 π e - x 2 / 2 [ x + x 3 3 + x 5 3 5 + + x 2 n + 1 ( 2 n + 1 ) ! ! + ] \Phi(x)\;=\;0.5+\frac{1}{\sqrt{2\pi}}\cdot e^{-x^{2}/2}\left[x+\frac{x^{3}}{3}% +\frac{x^{5}}{3\cdot 5}+\cdots+\frac{x^{2n+1}}{(2n+1)!!}+\cdots\right]
  55. ! ! !!
  56. F ( μ + n σ ) - F ( μ - n σ ) = Φ ( n ) - Φ ( - n ) = erf ( n 2 ) , F(\mu+n\sigma)-F(\mu-n\sigma)=\Phi(n)-\Phi(-n)=\mathrm{erf}\left(\frac{n}{% \sqrt{2}}\right),
  57. Φ - 1 ( p ) = 2 erf - 1 ( 2 p - 1 ) , p ( 0 , 1 ) . \Phi^{-1}(p)\;=\;\sqrt{2}\;\operatorname{erf}^{-1}(2p-1),\quad p\in(0,1).
  58. F - 1 ( p ) = μ + σ Φ - 1 ( p ) = μ + σ 2 erf - 1 ( 2 p - 1 ) , p ( 0 , 1 ) . F^{-1}(p)=\mu+\sigma\Phi^{-1}(p)=\mu+\sigma\sqrt{2}\,\operatorname{erf}^{-1}(2% p-1),\quad p\in(0,1).
  59. Φ - 1 ( p ) \Phi^{-1}(p)
  60. F ( x ) = { 0 if x < μ 1 if x μ F(x)=\begin{cases}0&\,\text{if }x<\mu\\ 1&\,\text{if }x\geq\mu\end{cases}
  61. n \sqrt{n}
  62. Z = n ( 1 n i = 1 n X i ) Z=\sqrt{n}\left(\frac{1}{n}\sum_{i=1}^{n}X_{i}\right)
  63. H ( X ) = - - f ( x ) log f ( x ) d x H(X)=-\int_{-\infty}^{\infty}f(x)\log f(x)dx
  64. L = - f ( x ) ln ( f ( x ) ) d x - λ 0 ( 1 - - f ( x ) d x ) - λ ( σ 2 - - f ( x ) ( x - μ ) 2 d x ) L=\int_{-\infty}^{\infty}f(x)\ln(f(x))\,dx-\lambda_{0}\left(1-\int_{-\infty}^{% \infty}f(x)\,dx\right)-\lambda\left(\sigma^{2}-\int_{-\infty}^{\infty}f(x)(x-% \mu)^{2}\,dx\right)
  65. σ \sigma
  66. 0 = δ L = - δ f ( x ) ( ln ( f ( x ) ) + 1 + λ 0 + λ ( x - μ ) 2 ) d x 0=\delta L=\int_{-\infty}^{\infty}\delta f(x)\left(\ln(f(x))+1+\lambda_{0}+% \lambda(x-\mu)^{2}\right)\,dx
  67. f ( x ) = e - λ 0 - 1 - λ ( x - μ ) 2 f(x)=e^{-\lambda_{0}-1-\lambda(x-\mu)^{2}}
  68. f ( x , μ , σ ) = 1 2 π σ 2 e - ( x - μ ) 2 2 σ 2 f(x,\mu,\sigma)=\frac{1}{\sqrt{2\pi\sigma^{2}}}e^{-\frac{(x-\mu)^{2}}{2\sigma^% {2}}}
  69. σ 1 2 + σ 2 2 \sigma_{1}^{2}+\sigma_{2}^{2}
  70. X 3 = a X 1 + b X 2 - ( a + b ) μ a 2 + b 2 + μ X_{3}=\frac{aX_{1}+bX_{2}-(a+b)\mu}{\sqrt{a^{2}+b^{2}}}+\mu
  71. σ k 2 {σ}_{k}^{2}
  72. D KL ( X 1 X 2 ) = ( μ 1 - μ 2 ) 2 2 σ 2 2 + 1 2 ( σ 1 2 σ 2 2 - 1 - ln σ 1 2 σ 2 2 ) . D_{\mathrm{KL}}(X_{1}\,\|\,X_{2})=\frac{(\mu_{1}-\mu_{2})^{2}}{2\sigma_{2}^{2}% }\,+\,\frac{1}{2}\left(\,\frac{\sigma_{1}^{2}}{\sigma_{2}^{2}}-1-\ln\frac{% \sigma_{1}^{2}}{\sigma_{2}^{2}}\,\right)\ .
  73. H 2 ( X 1 , X 2 ) = 1 - 2 σ 1 σ 2 σ 1 2 + σ 2 2 e - 1 4 ( μ 1 - μ 2 ) 2 σ 1 2 + σ 2 2 . H^{2}(X_{1},X_{2})=1\,-\,\sqrt{\frac{2\sigma_{1}\sigma_{2}}{\sigma_{1}^{2}+% \sigma_{2}^{2}}}\;e^{-\frac{1}{4}\frac{(\mu_{1}-\mu_{2})^{2}}{\sigma_{1}^{2}+% \sigma_{2}^{2}}}\ .
  74. = ( 1 σ 2 0 0 1 2 σ 4 ) \mathcal{I}=\begin{pmatrix}\frac{1}{\sigma^{2}}&0\\ 0&\frac{1}{2\sigma^{4}}\end{pmatrix}
  75. θ 1 = μ σ 2 \scriptstyle\theta_{1}=\frac{\mu}{\sigma^{2}}
  76. θ 2 = - 1 2 σ 2 \scriptstyle\theta_{2}=\frac{-1}{2\sigma^{2}}
  77. μ | x 1 , , x n 𝒩 ( σ 2 n μ 0 + σ 0 2 x ¯ σ 2 n + σ 0 2 , ( n σ 2 + 1 σ 0 2 ) - 1 ) \mu|x_{1},\ldots,x_{n}\ \sim\ \mathcal{N}\left(\frac{\frac{\sigma^{2}}{n}\mu_{% 0}+\sigma_{0}^{2}\bar{x}}{\frac{\sigma^{2}}{n}+\sigma_{0}^{2}},\ \left(\frac{n% }{\sigma^{2}}+\frac{1}{\sigma_{0}^{2}}\right)^{\!-1}\right)
  78. X 1 2 + X 2 2 \scriptstyle\sqrt{X_{1}^{2}\,+\,X_{2}^{2}}
  79. X 1 2 + + X n 2 χ n 2 . X_{1}^{2}+\cdots+X_{n}^{2}\ \sim\ \chi_{n}^{2}.
  80. t = X ¯ - μ S / n = 1 n ( X 1 + + X n ) - μ 1 n ( n - 1 ) [ ( X 1 - X ¯ ) 2 + + ( X n - X ¯ ) 2 ] t n - 1 . t=\frac{\overline{X}-\mu}{S/\sqrt{n}}=\frac{\frac{1}{n}(X_{1}+\cdots+X_{n})-% \mu}{\sqrt{\frac{1}{n(n-1)}\left[(X_{1}-\overline{X})^{2}+\cdots+(X_{n}-% \overline{X})^{2}\right]}}\ \sim\ t_{n-1}.
  81. F = ( X 1 2 + X 2 2 + + X n 2 ) / n ( Y 1 2 + Y 2 2 + + Y m 2 ) / m F n , m . F=\frac{\left(X_{1}^{2}+X_{2}^{2}+\cdots+X_{n}^{2}\right)/n}{\left(Y_{1}^{2}+Y% _{2}^{2}+\cdots+Y_{m}^{2}\right)/m}\ \sim\ F_{n,\,m}.
  82. z ( k ) = ( x ( k ) - μ ^ ) / σ ^ \scriptstyle z_{(k)}=(x_{(k)}-\hat{\mu})/\hat{\sigma}
  83. ln ( μ , σ 2 ) = i = 1 n ln f ( x i ; μ , σ 2 ) = - n 2 ln ( 2 π ) - n 2 ln σ 2 - 1 2 σ 2 i = 1 n ( x i - μ ) 2 . \ln\mathcal{L}(\mu,\sigma^{2})=\sum_{i=1}^{n}\ln f(x_{i};\,\mu,\sigma^{2})=-% \frac{n}{2}\ln(2\pi)-\frac{n}{2}\ln\sigma^{2}-\frac{1}{2\sigma^{2}}\sum_{i=1}^% {n}(x_{i}-\mu)^{2}.
  84. μ ^ = x ¯ 1 n i = 1 n x i , σ ^ 2 = 1 n i = 1 n ( x i - x ¯ ) 2 . \hat{\mu}=\overline{x}\equiv\frac{1}{n}\sum_{i=1}^{n}x_{i},\qquad\hat{\sigma}^% {2}=\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}.
  85. μ ^ \scriptstyle\hat{\mu}
  86. x ¯ \scriptstyle\overline{x}
  87. μ ^ \scriptstyle\hat{\mu}
  88. μ ^ 𝒩 ( μ , σ 2 / n ) . \hat{\mu}\ \sim\ \mathcal{N}(\mu,\,\,\sigma^{2}\!\!\;/n).
  89. - 1 \scriptstyle\mathcal{I}^{-1}
  90. μ ^ \scriptstyle\hat{\mu}
  91. 1 / n \scriptstyle 1/\sqrt{n}
  92. μ ^ \scriptstyle\hat{\mu}
  93. n ( μ ^ - μ ) 𝑑 𝒩 ( 0 , σ 2 ) . \sqrt{n}(\hat{\mu}-\mu)\ \xrightarrow{d}\ \mathcal{N}(0,\,\sigma^{2}).
  94. σ ^ 2 \scriptstyle\hat{\sigma}^{2}
  95. σ ^ 2 \scriptstyle\hat{\sigma}^{2}
  96. σ ^ 2 \scriptstyle\hat{\sigma}^{2}
  97. s 2 = n n - 1 σ ^ 2 = 1 n - 1 i = 1 n ( x i - x ¯ ) 2 . s^{2}=\frac{n}{n-1}\,\hat{\sigma}^{2}=\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-% \overline{x})^{2}.
  98. σ ^ 2 \scriptstyle\hat{\sigma}^{2}
  99. σ ^ 2 \scriptstyle\hat{\sigma}^{2}
  100. σ ^ 2 \scriptstyle\hat{\sigma}^{2}
  101. σ ^ 2 \scriptstyle\hat{\sigma}^{2}
  102. s 2 σ 2 n - 1 χ n - 1 2 , σ ^ 2 σ 2 n χ n - 1 2 . s^{2}\ \sim\ \frac{\sigma^{2}}{n-1}\cdot\chi^{2}_{n-1},\qquad\hat{\sigma}^{2}% \ \sim\ \frac{\sigma^{2}}{n}\cdot\chi^{2}_{n-1}\ .
  103. - 1 \scriptstyle\mathcal{I}^{-1}
  104. σ ^ 2 \scriptstyle\hat{\sigma}^{2}
  105. n ( σ ^ 2 - σ 2 ) n ( s 2 - σ 2 ) 𝑑 𝒩 ( 0 , 2 σ 4 ) . \sqrt{n}(\hat{\sigma}^{2}-\sigma^{2})\simeq\sqrt{n}(s^{2}-\sigma^{2})\ % \xrightarrow{d}\ \mathcal{N}(0,\,2\sigma^{4}).
  106. μ ^ \scriptstyle\hat{\mu}
  107. μ ^ \scriptstyle\hat{\mu}
  108. t = μ ^ - μ s / n = x ¯ - μ 1 n ( n - 1 ) ( x i - x ¯ ) 2 t n - 1 t=\frac{\hat{\mu}-\mu}{s/\sqrt{n}}=\frac{\overline{x}-\mu}{\sqrt{\frac{1}{n(n-% 1)}\sum(x_{i}-\overline{x})^{2}}}\ \sim\ t_{n-1}
  109. μ [ μ ^ - t n - 1 , 1 - α / 2 1 n s , μ ^ + t n - 1 , 1 - α / 2 1 n s ] [ μ ^ - | z α / 2 | 1 n s , μ ^ + | z α / 2 | 1 n s ] , \displaystyle\mu\in\left[\,\hat{\mu}-t_{n-1,1-\alpha/2}\,\frac{1}{\sqrt{n}}s,% \ \ \hat{\mu}+t_{n-1,1-\alpha/2}\,\frac{1}{\sqrt{n}}s\,\right]\approx\left[\,% \hat{\mu}-|z_{\alpha/2}|\frac{1}{\sqrt{n}}s,\ \ \hat{\mu}+|z_{\alpha/2}|\frac{% 1}{\sqrt{n}}s\,\right],
  110. χ k , p 2 {χ}_{k,p}^{2}
  111. μ ^ \scriptstyle\hat{\mu}
  112. a ( x - y ) 2 + b ( x - z ) 2 = ( a + b ) ( x - a y + b z a + b ) 2 + a b a + b ( y - z ) 2 a(x-y)^{2}+b(x-z)^{2}=(a+b)\left(x-\frac{ay+bz}{a+b}\right)^{2}+\frac{ab}{a+b}% (y-z)^{2}
  113. a y + b z a + b \frac{ay+bz}{a+b}
  114. a b a + b = 1 1 a + 1 b = ( a - 1 + b - 1 ) - 1 . \frac{ab}{a+b}=\frac{1}{\frac{1}{a}+\frac{1}{b}}=(a^{-1}+b^{-1})^{-1}.
  115. a b a + b \frac{ab}{a+b}
  116. k × k k\times k
  117. ( 𝐲 - 𝐱 ) 𝐀 ( 𝐲 - 𝐱 ) + ( 𝐱 - 𝐳 ) 𝐁 ( 𝐱 - 𝐳 ) = ( 𝐱 - 𝐜 ) ( 𝐀 + 𝐁 ) ( 𝐱 - 𝐜 ) + ( 𝐲 - 𝐳 ) ( 𝐀 - 1 + 𝐁 - 1 ) - 1 ( 𝐲 - 𝐳 ) (\mathbf{y}-\mathbf{x})^{\prime}\mathbf{A}(\mathbf{y}-\mathbf{x})+(\mathbf{x}-% \mathbf{z})^{\prime}\mathbf{B}(\mathbf{x}-\mathbf{z})=(\mathbf{x}-\mathbf{c})^% {\prime}(\mathbf{A}+\mathbf{B})(\mathbf{x}-\mathbf{c})+(\mathbf{y}-\mathbf{z})% ^{\prime}(\mathbf{A}^{-1}+\mathbf{B}^{-1})^{-1}(\mathbf{y}-\mathbf{z})
  118. 𝐜 = ( 𝐀 + 𝐁 ) - 1 ( 𝐀𝐲 + 𝐁𝐳 ) \mathbf{c}=(\mathbf{A}+\mathbf{B})^{-1}(\mathbf{A}\mathbf{y}+\mathbf{B}\mathbf% {z})
  119. 𝐱 𝐀𝐱 = i , j a i j x i x j \mathbf{x}^{\prime}\mathbf{A}\mathbf{x}=\sum_{i,j}a_{ij}x_{i}x_{j}
  120. x i x j = x j x i x_{i}x_{j}=x_{j}x_{i}
  121. a i j + a j i a_{ij}+a_{ji}
  122. 𝐱 𝐀𝐲 = 𝐲 𝐀𝐱 \mathbf{x}^{\prime}\mathbf{A}\mathbf{y}=\mathbf{y}^{\prime}\mathbf{A}\mathbf{x}
  123. i = 1 n ( x i - μ ) 2 = i = 1 n ( x i - x ¯ ) 2 + n ( x ¯ - μ ) 2 \sum_{i=1}^{n}(x_{i}-\mu)^{2}=\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}+n(\bar{x}-\mu)% ^{2}
  124. x ¯ = 1 n i = 1 n x i . \bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}.
  125. x 𝒩 ( μ , σ 2 ) x\sim\mathcal{N}(\mu,\sigma^{2})
  126. x 𝒩 ( μ , 1 / τ ) x\sim\mathcal{N}(\mu,1/\tau)
  127. μ 𝒩 ( μ 0 , 1 / τ 0 ) , \mu\sim\mathcal{N}(\mu_{0},1/\tau_{0}),
  128. p ( 𝐗 | μ , τ ) \displaystyle p(\mathbf{X}|\mu,\tau)
  129. p ( μ | 𝐗 ) \displaystyle p(\mu|\mathbf{X})
  130. n τ x ¯ + τ 0 μ 0 n τ + τ 0 \frac{n\tau\bar{x}+\tau_{0}\mu_{0}}{n\tau+\tau_{0}}
  131. n τ + τ 0 n\tau+\tau_{0}
  132. p ( μ | 𝐗 ) 𝒩 ( n τ x ¯ + τ 0 μ 0 n τ + τ 0 , 1 n τ + τ 0 ) p(\mu|\mathbf{X})\sim\mathcal{N}\left(\frac{n\tau\bar{x}+\tau_{0}\mu_{0}}{n% \tau+\tau_{0}},\frac{1}{n\tau+\tau_{0}}\right)
  133. τ 0 \displaystyle\tau_{0}^{\prime}
  134. x ¯ \bar{x}
  135. σ 0 2 = 1 n σ 2 + 1 σ 0 2 μ 0 = n x ¯ σ 2 + μ 0 σ 0 2 n σ 2 + 1 σ 0 2 x ¯ = 1 n i = 1 n x i \begin{aligned}\displaystyle{\sigma^{2}_{0}}^{\prime}&\displaystyle=\frac{1}{% \frac{n}{\sigma^{2}}+\frac{1}{\sigma_{0}^{2}}}\\ \displaystyle\mu_{0}^{\prime}&\displaystyle=\frac{\frac{n\bar{x}}{\sigma^{2}}+% \frac{\mu_{0}}{\sigma_{0}^{2}}}{\frac{n}{\sigma^{2}}+\frac{1}{\sigma_{0}^{2}}}% \\ \displaystyle\bar{x}&\displaystyle=\frac{1}{n}\sum_{i=1}^{n}x_{i}\end{aligned}
  136. x 𝒩 ( μ , σ 2 ) x\sim\mathcal{N}(\mu,\sigma^{2})
  137. p ( σ 2 | ν 0 , σ 0 2 ) = ( σ 0 2 ν 0 2 ) ν 0 2 Γ ( ν 0 2 ) exp [ - ν 0 σ 0 2 2 σ 2 ] ( σ 2 ) 1 + ν 0 2 exp [ - ν 0 σ 0 2 2 σ 2 ] ( σ 2 ) 1 + ν 0 2 p(\sigma^{2}|\nu_{0},\sigma_{0}^{2})=\frac{(\sigma_{0}^{2}\frac{\nu_{0}}{2})^{% \frac{\nu_{0}}{2}}}{\Gamma\left(\frac{\nu_{0}}{2}\right)}~{}\frac{\exp\left[% \frac{-\nu_{0}\sigma_{0}^{2}}{2\sigma^{2}}\right]}{(\sigma^{2})^{1+\frac{\nu_{% 0}}{2}}}\propto\frac{\exp\left[\frac{-\nu_{0}\sigma_{0}^{2}}{2\sigma^{2}}% \right]}{(\sigma^{2})^{1+\frac{\nu_{0}}{2}}}
  138. p ( 𝐗 | μ , σ 2 ) \displaystyle p(\mathbf{X}|\mu,\sigma^{2})
  139. S = i = 1 n ( x i - μ ) 2 . S=\sum_{i=1}^{n}(x_{i}-\mu)^{2}.
  140. p ( σ 2 | 𝐗 ) \displaystyle p(\sigma^{2}|\mathbf{X})
  141. ν 0 = ν 0 + n ν 0 σ 0 2 = ν 0 σ 0 2 + i = 1 n ( x i - μ ) 2 \begin{aligned}\displaystyle\nu_{0}^{\prime}&\displaystyle=\nu_{0}+n\\ \displaystyle\nu_{0}^{\prime}{\sigma_{0}^{2}}^{\prime}&\displaystyle=\nu_{0}% \sigma_{0}^{2}+\sum_{i=1}^{n}(x_{i}-\mu)^{2}\end{aligned}
  142. ν 0 = ν 0 + n σ 0 2 = ν 0 σ 0 2 + i = 1 n ( x i - μ ) 2 ν 0 + n \begin{aligned}\displaystyle\nu_{0}^{\prime}&\displaystyle=\nu_{0}+n\\ \displaystyle{\sigma_{0}^{2}}^{\prime}&\displaystyle=\frac{\nu_{0}\sigma_{0}^{% 2}+\sum_{i=1}^{n}(x_{i}-\mu)^{2}}{\nu_{0}+n}\end{aligned}
  143. α \displaystyle\alpha^{\prime}
  144. x 𝒩 ( μ , σ 2 ) x\sim\mathcal{N}(\mu,\sigma^{2})
  145. p ( μ | σ 2 ; μ 0 , n 0 ) \displaystyle p(\mu|\sigma^{2};\mu_{0},n_{0})
  146. x ¯ \displaystyle\bar{x}
  147. ν 0 σ 0 2 \nu_{0}^{\prime}{\sigma_{0}^{2}}^{\prime}
  148. p ( μ | σ 2 ; μ 0 , n 0 ) 𝒩 ( μ 0 , σ 2 / n 0 ) = 1 2 π σ 2 n 0 exp ( - n 0 2 σ 2 ( μ - μ 0 ) 2 ) ( σ 2 ) - 1 / 2 exp ( - n 0 2 σ 2 ( μ - μ 0 ) 2 ) p ( σ 2 ; ν 0 , σ 0 2 ) I χ 2 ( ν 0 , σ 0 2 ) = I G ( ν 0 / 2 , ν 0 σ 0 2 / 2 ) = ( σ 0 2 ν 0 / 2 ) ν 0 / 2 Γ ( ν 0 / 2 ) exp [ - ν 0 σ 0 2 2 σ 2 ] ( σ 2 ) 1 + ν 0 / 2 ( σ 2 ) - ( 1 + ν 0 / 2 ) exp [ - ν 0 σ 0 2 2 σ 2 ] \begin{aligned}\displaystyle p(\mu|\sigma^{2};\mu_{0},n_{0})&\displaystyle\sim% \mathcal{N}(\mu_{0},\sigma^{2}/n_{0})=\frac{1}{\sqrt{2\pi\frac{\sigma^{2}}{n_{% 0}}}}\exp\left(-\frac{n_{0}}{2\sigma^{2}}(\mu-\mu_{0})^{2}\right)\\ &\displaystyle\propto(\sigma^{2})^{-1/2}\exp\left(-\frac{n_{0}}{2\sigma^{2}}(% \mu-\mu_{0})^{2}\right)\\ \displaystyle p(\sigma^{2};\nu_{0},\sigma_{0}^{2})&\displaystyle\sim I\chi^{2}% (\nu_{0},\sigma_{0}^{2})=IG(\nu_{0}/2,\nu_{0}\sigma_{0}^{2}/2)\\ &\displaystyle=\frac{(\sigma_{0}^{2}\nu_{0}/2)^{\nu_{0}/2}}{\Gamma(\nu_{0}/2)}% ~{}\frac{\exp\left[\frac{-\nu_{0}\sigma_{0}^{2}}{2\sigma^{2}}\right]}{(\sigma^% {2})^{1+\nu_{0}/2}}\\ &\displaystyle\propto{(\sigma^{2})^{-(1+\nu_{0}/2)}}\exp\left[\frac{-\nu_{0}% \sigma_{0}^{2}}{2\sigma^{2}}\right]\end{aligned}
  149. p ( μ , σ 2 ; μ 0 , n 0 , ν 0 , σ 0 2 ) = p ( μ | σ 2 ; μ 0 , n 0 ) p ( σ 2 ; ν 0 , σ 0 2 ) ( σ 2 ) - ( ν 0 + 3 ) / 2 exp [ - 1 2 σ 2 ( ν 0 σ 0 2 + n 0 ( μ - μ 0 ) 2 ) ] \begin{aligned}\displaystyle p(\mu,\sigma^{2};\mu_{0},n_{0},\nu_{0},\sigma_{0}% ^{2})&\displaystyle=p(\mu|\sigma^{2};\mu_{0},n_{0})\,p(\sigma^{2};\nu_{0},% \sigma_{0}^{2})\\ &\displaystyle\propto(\sigma^{2})^{-(\nu_{0}+3)/2}\exp\left[-\frac{1}{2\sigma^% {2}}\left(\nu_{0}\sigma_{0}^{2}+n_{0}(\mu-\mu_{0})^{2}\right)\right]\end{aligned}
  150. p ( 𝐗 | μ , σ 2 ) \displaystyle p(\mathbf{X}|\mu,\sigma^{2})
  151. p ( 𝐗 | μ , σ 2 ) \displaystyle p(\mathbf{X}|\mu,\sigma^{2})
  152. S = i = 1 n ( x i - x ¯ ) 2 . S=\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}.
  153. p ( μ , σ 2 | 𝐗 ) \displaystyle p(\mu,\sigma^{2}|\mathbf{X})
  154. X = - 2 ln U cos ( 2 π V ) , Y = - 2 ln U sin ( 2 π V ) . X=\sqrt{-2\ln U}\,\cos(2\pi V),\qquad Y=\sqrt{-2\ln U}\,\sin(2\pi V).
  155. X = U - 2 ln S S , Y = V - 2 ln S S X=U\sqrt{\frac{-2\ln S}{S}},\qquad Y=V\sqrt{\frac{-2\ln S}{S}}
  156. 8 / e \sqrt{8/}{e}
  157. 2 / 2 π n 2/\sqrt{2\pi n}
  158. - 2 n -\frac{2\ell\ell}{n}
  159. φ Δ = h π e - hh Δ Δ , \varphi\mathit{\Delta}=\frac{h}{\surd\pi}\,e^{-\mathrm{hh}\Delta\Delta},
  160. N 1 α π e - x 2 α 2 d x \mathrm{N}\;\frac{1}{\alpha\;\sqrt{\pi}}\;e^{-\frac{x^{2}}{\alpha^{2}}}dx
  161. d f = 1 σ 2 π e - ( x - m ) 2 2 σ 2 d x df=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{(x-m)^{2}}{2\sigma^{2}}}dx

Normal_subgroup.html

  1. N G n N , g G , g n g - 1 N . N\triangleleft G\,\,\Leftrightarrow\,\forall\,n\in{N},\forall\,g\in{G},\,gng^{% -1}\in{N}.
  2. N M := N M N\wedge M:=N\cap M
  3. N M := N M = { n m | n N , and m M } . N\vee M:=NM=\{nm\,|\,n\in N\,\text{, and }m\in M\}.

Normed_vector_space.html

  1. x > 0 \|x\|>0
  2. x 0 x\neq 0
  3. α x = | α | x \|\alpha x\|=|\alpha|\|x\|
  4. α . \alpha.
  5. x + y x + y \|x+y\|\leq\|x\|+\|y\|
  6. x = 0 \|x\|=0
  7. x = 0 x=0
  8. x - y | x - y | \|x-y\|\geq|\|x\|-\|y\||
  9. | α | |\alpha|
  10. \mathbb{R}
  11. \mathbb{C}
  12. \mathbb{Q}
  13. | α | |\alpha|
  14. 𝒩 ( 0 ) \mathcal{N}(0)
  15. 𝒩 ( x ) = x + 𝒩 ( 0 ) := { x + N N 𝒩 ( 0 ) } \mathcal{N}(x)=x+\mathcal{N}(0):=\{x+N\mid N\in\mathcal{N}(0)\}
  16. x + N := { x + n n N } x+N:=\{x+n\mid n\in N\}
  17. f p = ( | f ( x ) | p d x ) 1 / p \|f\|_{p}=\left(\int|f(x)|^{p}\;dx\right)^{1/p}
  18. X := i = 1 n X i X:=\prod_{i=1}^{n}X_{i}
  19. ( x 1 , , x n ) + ( y 1 , , y n ) := ( x 1 + y 1 , , x n + y n ) (x_{1},\ldots,x_{n})+(y_{1},\ldots,y_{n}):=(x_{1}+y_{1},\ldots,x_{n}+y_{n})
  20. α ( x 1 , , x n ) := ( α x 1 , , α x n ) \alpha(x_{1},\ldots,x_{n}):=(\alpha x_{1},\ldots,\alpha x_{n})
  21. q : X q:X\mapsto\mathbb{R}
  22. q : ( x 1 , , x n ) i = 1 n q i ( x i ) q:(x_{1},\ldots,x_{n})\to\sum_{i=1}^{n}q_{i}(x_{i})
  23. q : ( x 1 , , x n ) ( i = 1 n q i ( x i ) p ) 1 p . q:(x_{1},\ldots,x_{n})\to\left(\sum_{i=1}^{n}q_{i}(x_{i})^{p}\right)^{\frac{1}% {p}}.

Nowhere_dense_set.html

  1. \mathbb{Z}
  2. \mathbb{R}
  3. S = { 1 n : n } S=\left\{\frac{1}{n}:n\in\mathbb{N}\right\}
  4. \mathbb{R}
  5. S { 0 } S\cup\{0\}
  6. [ ( 0 , 1 ) ] \mathbb{Z}\cup\left[(0,1)\cap\mathbb{Q}\right]
  7. \mathbb{R}
  8. [ 0 , 1 ] [0,1]
  9. ( 0 , 1 ) (0,1)
  10. { 1 n n } \left\{\frac{1}{n}\mid n\in\mathbb{N}\right\}

NP_(complexity).html

  1. NP = k NTIME ( n k ) . \mbox{NP}~{}=\bigcup_{k\in\mathbb{N}}\mbox{NTIME}~{}(n^{k}).
  2. \subsetneq
  3. \subsetneq

Nuclear_chain_reaction.html

  1. E c 2 = m original - m final \frac{E}{c^{2}}=m_{\mbox{original}}~{}-m_{\mbox{final}}~{}
  2. U 235 + neutron fission fragments + 2.4 neutrons + 192.9 MeV {}^{235}\mbox{U}~{}+\mbox{neutron}~{}\rightarrow\mbox{fission fragments}~{}+2.% 4\mbox{ neutrons}~{}+192.9\mbox{ MeV}~{}
  3. Pu 239 + neutron fission fragments + 2.9 neutrons + 198.5 MeV {}^{239}\mbox{Pu}~{}+\mbox{neutron}~{}\rightarrow\mbox{fission fragments}~{}+2% .9\mbox{ neutrons}~{}+198.5\mbox{ MeV}~{}
  4. Λ = l k \Lambda=\frac{l}{k}
  5. e ( k - 1 ) t / Λ e^{(k-1)t/\Lambda}

Nuclear_fusion.html

  1. f = n 1 n 2 σ v . f=n_{1}n_{2}\langle\sigma v\rangle.
  2. n 1 n 2 n_{1}n_{2}
  3. ( 1 / 2 ) n 2 (1/2)n^{2}
  4. σ v \langle\sigma v\rangle
  5. σ v \langle\sigma v\rangle
  6. × 10 - 24 \times 10^{-}24

Nuclear_photonic_rocket.html

  1. 1 / c 1/c
  2. 2 / v 2/v
  3. v c v\ll c

Nuclear_shell_model.html

  1. 1 / 2 {1}/{2}
  2. 1 / 2 {1}/{2}
  3. 1 / 2 {1}/{2}
  4. 1 / 2 {1}/{2}
  5. 1 / 2 {1}/{2}
  6. 1 / 2 {1}/{2}
  7. 1 / 2 {1}/{2}
  8. 1 / 2 {1}/{2}
  9. ( n + 1 ) ( n + 2 ) 2 {(n+1)(n+2)\over 2}
  10. ( n + 1 ) ( n + 2 ) (n+1)(n+2)
  11. n = 0 k ( n + 1 ) ( n + 2 ) = ( k + 1 ) ( k + 2 ) ( k + 3 ) 3 {\sum_{n=0}^{k}}(n+1)(n+2)={(k+1)(k+2)(k+3)\over 3}
  12. 1 / 2 {1}/{2}
  13. 1 / 2 {1}/{2}
  14. 3 / 2 {3}/{2}
  15. 1 / 2 {1}/{2}
  16. 3 / 2 {3}/{2}
  17. 5 / 2 {5}/{2}
  18. 1 / 2 {1}/{2}
  19. 3 / 2 {3}/{2}
  20. 5 / 2 {5}/{2}
  21. 7 / 2 {7}/{2}
  22. 1 / 2 {1}/{2}
  23. 3 / 2 {3}/{2}
  24. 5 / 2 {5}/{2}
  25. 7 / 2 {7}/{2}
  26. 9 / 2 {9}/{2}
  27. 1 / 2 {1}/{2}
  28. 3 / 2 {3}/{2}
  29. 5 / 2 {5}/{2}
  30. 7 / 2 {7}/{2}
  31. 9 / 2 {9}/{2}
  32. 11 / 2 {11}/{2}
  33. s \scriptstyle\vec{s}
  34. l \scriptstyle\vec{l}
  35. 1 / 2 {1}/{2}
  36. s \scriptstyle\vec{s}
  37. l \scriptstyle\vec{l}
  38. 9 / 2 {9}/{2}
  39. 7 / 2 {7}/{2}
  40. 5 / 2 {5}/{2}
  41. 9 / 2 {9}/{2}
  42. 3 / 2 {3}/{2}
  43. 7 / 2 {7}/{2}
  44. 1 / 2 {1}/{2}
  45. V ( r ) = μ ω 2 r 2 / 2 V(r)=\mu\omega^{2}r^{2}/2
  46. 2 l ( l + 1 ) / 2 m r 2 \scriptstyle\hbar^{2}l(l+1)/2mr^{2}
  47. 1 / 2 {1}/{2}
  48. 1 / 2 {1}/{2}
  49. 3 / 2 {3}/{2}
  50. 1 / 2 {1}/{2}
  51. 3 / 2 {3}/{2}
  52. 5 / 2 {5}/{2}
  53. 7 / 2 {7}/{2}
  54. 1 / 2 {1}/{2}
  55. 3 / 2 {3}/{2}
  56. 5 / 2 {5}/{2}
  57. 9 / 2 {9}/{2}
  58. 1 / 2 {1}/{2}
  59. 3 / 2 {3}/{2}
  60. 5 / 2 {5}/{2}
  61. 7 / 2 {7}/{2}
  62. 11 / 2 {11}/{2}
  63. 1 / 2 {1}/{2}
  64. 3 / 2 {3}/{2}
  65. 5 / 2 {5}/{2}
  66. 7 / 2 {7}/{2}
  67. 9 / 2 {9}/{2}
  68. 13 / 2 {13}/{2}
  69. 1 / 2 {1}/{2}
  70. 3 / 2 {3}/{2}
  71. 5 / 2 {5}/{2}
  72. 7 / 2 {7}/{2}
  73. 9 / 2 {9}/{2}
  74. 11 / 2 {11}/{2}
  75. 15 / 2 {15}/{2}
  76. p = ( - 1 ) l p=(-1)^{l}
  77. 5 / 2 {5}/{2}
  78. 5 / 2 {5}/{2}
  79. 3 / 2 {3}/{2}
  80. 1 / 2 {1}/{2}
  81. 5 / 2 {5}/{2}
  82. 3 / 2 {3}/{2}
  83. ψ 2 \psi^{2}
  84. ψ \psi

Nuclear_thermal_rocket.html

  1. P = T * V e / 2 P=T*V_{e}/2
  2. V e V_{e}
  3. V e V_{e}
  4. V e = I s p * g n V_{e}=I_{sp}*g_{n}
  5. m = T / V e m=T/V_{e}
  6. Δ v = V e ln m 0 m 1 \Delta v\ =V_{e}\ln\frac{m_{0}}{m_{1}}
  7. m 0 {m_{0}}
  8. m 1 {m_{1}}
  9. m 1 {m_{1}}
  10. m 0 {m_{0}}
  11. m 0 {m_{0}}
  12. m 1 {m_{1}}
  13. m 0 {m_{0}}

Nucleon.html

  1. 2 / 3 {2}/{3}
  2. 1 / 3 {1}/{3}
  3. 1 / 2 {1}/{2}
  4. 1 / 2 {1}/{2}
  5. | m p - m p ¯ | m p \frac{|m_{p}-m_{\bar{p}}|}{m_{p}}
  6. | q p ¯ m p ¯ | ( q p m p ) \frac{|\frac{q_{\bar{p}}}{m_{\bar{p}}}|}{(\frac{q_{p}}{m_{p}})}
  7. | q p ¯ m p ¯ | - q p m p q p m p \frac{|\frac{q_{\bar{p}}}{m_{\bar{p}}}|-\frac{q_{p}}{m_{p}}}{\frac{q_{p}}{m_{p% }}}
  8. | q p + q p ¯ | e \frac{|q_{p}+q_{\bar{p}}|}{e}
  9. | q p + q e | e \frac{|q_{p}+q_{e}|}{e}
  10. | μ p + μ p ¯ | μ p \frac{|\mu_{p}+\mu_{\bar{p}}|}{\mu_{p}}
  11. 1 / 2 {1}/{2}
  12. 1 / 2 {1}/{2}
  13. 3 / 2 {3}/{2}
  14. 1 / 2 {1}/{2}
  15. 1 / 2 {1}/{2}

Nucleophile.html

  1. log 10 ( k k 0 ) = s n \log_{10}\left(\frac{k}{k_{0}}\right)=sn
  2. log 10 ( k k 0 ) = N + \log_{10}\left(\frac{k}{k_{0}}\right)=N^{+}
  3. log ( k ) = s ( N + E ) \log(k)=s(N+E)
  4. log ( k ) = s E s N ( N + E ) \log(k)=s_{E}s_{N}(N+E)
  5. log ( k ) = 0.6 s E N + 0.6 s E E \log(k)=0.6s_{E}N+0.6s_{E}E
  6. log ( k ) = log ( k 0 ) + s E N \log(k)=\log(k_{0})+s_{E}N
  7. log ( k ) = 0.6 N + 0.6 E \log(k)=0.6N+0.6E
  8. log ( k ) - log ( k 0 ) = N + \log(k)-\log(k_{0})=N^{+}

Null.html

  1. A ¯ \overline{A}

Number.html

  1. 1 2 \frac{1}{2}
  2. - 2 3 -\frac{2}{3}
  3. 2 \sqrt{2}
  4. π \pi
  5. - 1 \sqrt{-1}
  6. \mathbb{N}
  7. \mathbb{Z}
  8. \mathbb{Q}
  9. a b \frac{a}{b}
  10. \mathbb{R}
  11. \mathbb{C}
  12. \mathbb{N}
  13. 0 \mathbb{N}_{0}
  14. 1 \mathbb{N}_{1}
  15. \mathbb{Z}
  16. m n \frac{m}{n}
  17. 1 2 \frac{1}{2}
  18. 2 4 \frac{2}{4}
  19. 1 2 = 2 4 . {1\over 2}={2\over 4}.\,
  20. 7 1 \frac{−7}{1}
  21. \mathbb{Q}
  22. \mathbb{R}
  23. 123456 1000 \frac{123456}{1000}
  24. 1234555 10000 \frac{1234555}{10000}
  25. 1234565 10000 \frac{1234565}{10000}
  26. 123456 1000 \frac{123456}{1000}
  27. 123457 1000 \frac{123457}{1000}
  28. 3 ¯ \overline{3}
  29. π \pi
  30. π = 3.14159265358979 \pi=3.14159265358979\dots
  31. 2 = 1.41421356237 \sqrt{2}=1.41421356237\dots\,
  32. a + b i \,a+bi
  33. \mathbb{C}
  34. \mathbb{N}\subset\mathbb{Z}\subset\mathbb{Q}\subset\mathbb{R}\subset\mathbb{C}
  35. ( - 1 ) 2 = - 1 - 1 = - 1 \left(\sqrt{-1}\right)^{2}=\sqrt{-1}\sqrt{-1}=-1
  36. a b = a b , \sqrt{a}\sqrt{b}=\sqrt{ab},
  37. 1 a = 1 a \frac{1}{\sqrt{a}}=\sqrt{\frac{1}{a}}
  38. - 1 \sqrt{-1}
  39. ( cos θ + i sin θ ) n = cos n θ + i sin n θ (\cos\theta+i\sin\theta)^{n}=\cos n\theta+i\sin n\theta\,
  40. cos θ + i sin θ = e i θ . \cos\theta+i\sin\theta=e^{i\theta}.\,

Number_sign.html

  1. S = { s 1 , s 2 , s 3 , , s n } S=\{s_{1},s_{2},s_{3},\dots,s_{n}\}
  2. # S = n . \#S=n.

Number_theory.html

  1. ( a , b , c ) (a,b,c)
  2. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}
  3. ( 1 2 ( x - 1 x ) ) 2 + 1 = ( 1 2 ( x + 1 x ) ) 2 , \left(\frac{1}{2}\left(x-\frac{1}{x}\right)\right)^{2}+1=\left(\frac{1}{2}% \left(x+\frac{1}{x}\right)\right)^{2},
  4. c / a c/a
  5. 2 \sqrt{2}
  6. 2 \sqrt{2}
  7. 3 , 5 , , 17 \sqrt{3},\sqrt{5},\dots,\sqrt{17}
  8. f ( x , y ) = z 2 f(x,y)=z^{2}
  9. f ( x , y , z ) = w 2 f(x,y,z)=w^{2}
  10. f ( x 1 , x 2 , x 3 ) = 0 f(x_{1},x_{2},x_{3})=0
  11. g 1 , g 2 , g 3 g_{1},g_{2},g_{3}
  12. r r
  13. s s
  14. x i = g i ( r , s ) x_{i}=g_{i}(r,s)
  15. i = 1 , 2 , 3 i=1,2,3
  16. f ( x 1 , x 2 , x 3 ) = 0. f(x_{1},x_{2},x_{3})=0.
  17. n a 1 ( mod m ) 1 n\equiv a_{1}\;\;(\mathop{{\rm mod}}m)_{1}
  18. n a 2 ( mod m ) 2 n\equiv a_{2}\;\;(\mathop{{\rm mod}}m)_{2}
  19. a p - 1 1 ( mod p ) . a^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p).
  20. a 2 + b 2 a^{2}+b^{2}
  21. a 2 + b 2 a^{2}+b^{2}
  22. x 2 - N y 2 = 1 x^{2}-Ny^{2}=1
  23. x 4 + y 4 = z 4 x^{4}+y^{4}=z^{4}
  24. x 3 + y 3 = z 3 x^{3}+y^{3}=z^{3}
  25. x n + y n = z n x^{n}+y^{n}=z^{n}
  26. n 3 n\geq 3
  27. p = x 2 + y 2 p=x^{2}+y^{2}
  28. p 1 m o d 4 p\equiv 1\;mod\;4
  29. x 4 + y 4 = z 2 x^{4}+y^{4}=z^{2}
  30. x 2 + N y 2 x^{2}+Ny^{2}
  31. m X 2 + n Y 2 mX^{2}+nY^{2}
  32. a x 2 + b y 2 + c z 2 = 0 ax^{2}+by^{2}+cz^{2}=0
  33. n = 5 n=5
  34. f ( x ) = 0 f(x)=0
  35. x x
  36. x 5 + ( 11 / 2 ) x 3 - 7 x 2 + 9 = 0 x^{5}+(11/2)x^{3}-7x^{2}+9=0
  37. a + b d a+b\sqrt{d}
  38. a a
  39. b b
  40. d d
  41. - 5 \sqrt{-5}
  42. 6 6
  43. 6 = 2 3 6=2\cdot 3
  44. 6 = ( 1 + - 5 ) ( 1 - - 5 ) 6=(1+\sqrt{-5})(1-\sqrt{-5})
  45. 2 2
  46. 3 3
  47. 1 + - 5 1+\sqrt{-5}
  48. 1 - - 5 1-\sqrt{-5}
  49. x 2 + y 2 = 1 x^{2}+y^{2}=1
  50. ( x , y ) (x,y)
  51. a 2 + b 2 = c 2 a^{2}+b^{2}=c^{2}
  52. x = a / c x=a/c
  53. y = b / c y=b/c
  54. x 2 + y 2 = 1 x^{2}+y^{2}=1
  55. f ( x , y ) = 0 f(x,y)=0
  56. f f
  57. f ( x , y ) = 0 f(x,y)=0
  58. f ( x , y ) = 0 f(x,y)=0
  59. f ( x , y ) = 0 f(x,y)=0
  60. x x
  61. a / q a/q
  62. gcd ( a , q ) = 1 \gcd(a,q)=1
  63. x x
  64. | x - a / q | < 1 q c |x-a/q|<\frac{1}{q^{c}}
  65. c c
  66. x x
  67. x x
  68. π \pi
  69. 0
  70. A A
  71. a a
  72. a + b a+b
  73. a + 2 b a+2b
  74. a + 3 b a+3b
  75. \ldots
  76. a + 10 b a+10b
  77. A A
  78. A A
  79. A A
  80. A + A A+A
  81. A A
  82. A A
  83. a x + b y = c ax+by=c
  84. ( p 2 - q 2 , 2 p q , p 2 + q 2 ) (p^{2}-q^{2},2pq,p^{2}+q^{2})
  85. a b ( mod m ) a\equiv b\;\;(\mathop{{\rm mod}}m)
  86. x a 1 ( mod p ) xa\equiv 1\;\;(\mathop{{\rm mod}}p)
  87. y 2 = x 3 + 7 y^{2}=x^{3}+7
  88. ( a + b i ) 2 = ( c + d i ) 3 + 7 (a+bi)^{2}=(c+di)^{3}+7

Numeral_system.html

  1. ( a n a n - 1 a 1 a 0 . c 1 c 2 c 3 ) b = k = 0 n a k b k + k = 1 c k b - k . (a_{n}a_{n-1}\cdots a_{1}a_{0}.c_{1}c_{2}c_{3}\cdots)_{b}=\sum_{k=0}^{n}a_{k}b% ^{k}+\sum_{k=1}^{\infty}c_{k}b^{-k}.
  2. k = log b w = log b b k k=\log_{b}w=\log_{b}b^{k}
  3. k + 1 = k+1=
  4. log b w \log_{b}w
  5. + 1 +1
  6. log 10 1000 + 1 = 3 + 1 \log_{10}1000+1=3+1
  7. log b k + 1 = log b log b w + 1 \log_{b}k+1=\log_{b}\log_{b}w+1
  8. b 3 b^{3}
  9. b 2 b^{2}
  10. b 1 b^{1}
  11. b 0 b^{0}
  12. b - 1 b^{-1}
  13. b - 2 b^{-2}
  14. \dots
  15. a 3 a_{3}
  16. a 2 a_{2}
  17. a 1 a_{1}
  18. a 0 a_{0}
  19. c 1 c_{1}
  20. c 2 c_{2}
  21. \dots
  22. n ¯ \overline{n}
  23. 27 ¯ \overline{27}
  24. a 0 a 1 a 2 a_{0}a_{1}a_{2}
  25. a 0 + a 1 b 1 + a 2 b 1 b 2 a_{0}+a_{1}b_{1}+a_{2}b_{1}b_{2}

Numerical_analysis.html

  1. 2 \sqrt{2}
  2. 2 \sqrt{2}
  3. 3 ′′ x 3 + 4 = 28 ′′ {}^{\prime\prime}3x^{3}+4=28^{\prime\prime}
  4. 2 \sqrt{2}
  5. f ( x ) = x ( x + 1 - x ) and g ( x ) = x x + 1 + x . f(x)=x\left(\sqrt{x+1}-\sqrt{x}\right)\,\text{ and }g(x)=\frac{x}{\sqrt{x+1}+% \sqrt{x}}.
  6. f ( 500 ) = 500 ( 501 - 500 ) = 500 ( 22.3830 - 22.3607 ) = 500 ( 0.0223 ) = 11.1500 f(500)=500\left(\sqrt{501}-\sqrt{500}\right)=500\left(22.3830-22.3607\right)=5% 00(0.0223)=11.1500
  7. g ( 500 ) = 500 501 + 500 = 500 22.3830 + 22.3607 = 500 44.7437 = 11.1748 \begin{aligned}\displaystyle g(500)&\displaystyle=\frac{500}{\sqrt{501}+\sqrt{% 500}}\\ &\displaystyle=\frac{500}{22.3830+22.3607}\\ &\displaystyle=\frac{500}{44.7437}=11.1748\end{aligned}
  8. f ( x ) \displaystyle f(x)
  9. 2 x + 5 = 3 2x+5=3
  10. 2 x 2 + 5 = 3 2x^{2}+5=3
  11. x = ( x 2 - 2 ) 2 + x = f ( x ) x=(x^{2}-2)^{2}+x=f(x)
  12. 2 \sqrt{2}
  13. f ( x ) x f(x)\geq x
  14. x 1 = 1.4 < 2 x_{1}=1.4<\sqrt{2}
  15. x 1 = 1.42 > 2 x_{1}=1.42>\sqrt{2}

Numerical_aperture.html

  1. NA = n sin θ \mathrm{NA}=n\sin\theta\;
  2. n sin θ n\sin\theta
  3. N N
  4. f f
  5. D D
  6. N = f / D \ N=f/D
  7. NA i = n sin θ = n sin [ arctan ( D 2 f ) ] n D 2 f \mathrm{NA_{i}}=n\sin\theta=n\sin\left[\arctan\left(\frac{D}{2f}\right)\right]% \approx n\frac{D}{2f}
  8. N 1 2 NA i N\approx\frac{1}{2\;\mathrm{NA_{i}}}
  9. n = 1 n=1
  10. N N
  11. 1 / ( 2 N A i ) 1/(2\mathrm{NA_{i}})
  12. D / 2 f D/2f
  13. tan θ \tan\theta
  14. sin θ \sin\theta
  15. 1 2 N A i = N w = ( 1 - m ) N , \frac{1}{2\mathrm{NA_{i}}}=N_{\mathrm{w}}=(1-m)\,N,
  16. N w N_{\mathrm{w}}
  17. m m
  18. 1 2 N A o = m - 1 m N . \frac{1}{2\mathrm{NA_{o}}}=\frac{m-1}{m}\,N.
  19. NA = n sin θ , \mathrm{NA}=n\sin\theta,\;
  20. NA λ 0 π w 0 , \mathrm{NA}\simeq\frac{\lambda_{0}}{\pi w_{0}},
  21. n sin θ max = n core 2 - n clad 2 , n\sin\theta_{\max}=\sqrt{n\text{core}^{2}-n\text{clad}^{2}},
  22. n sin θ max = n core sin θ r . n\sin\theta_{\mathrm{max}}=n\text{core}\sin\theta_{r}.
  23. sin θ r = sin ( 90 - θ c ) = cos θ c \sin\theta_{r}=\sin\left({90^{\circ}}-\theta_{c}\right)=\cos\theta_{c}
  24. θ c = sin - 1 n clad n core \theta_{c}=\sin^{-1}\frac{n\text{clad}}{n\text{core}}
  25. n n core sin θ max = cos θ c . \frac{n}{n\text{core}}\sin\theta_{\mathrm{max}}=\cos\theta_{c}.
  26. n 2 n core 2 sin 2 θ max = cos 2 θ c = 1 - sin 2 θ c = 1 - n clad 2 n core 2 . \frac{n^{2}}{n\text{core}^{2}}\sin^{2}\theta_{\mathrm{max}}=\cos^{2}\theta_{c}% =1-\sin^{2}\theta_{c}=1-\frac{n\text{clad}^{2}}{n\text{core}^{2}}.
  27. n sin θ max = n core 2 - n clad 2 , n\sin\theta_{\mathrm{max}}=\sqrt{n\text{core}^{2}-n\text{clad}^{2}},
  28. NA = n core 2 - n clad 2 , \mathrm{NA}=\sqrt{n\text{core}^{2}-n\text{clad}^{2}},

Nusselt_number.html

  1. Nu L = Convective heat transfer Conductive heat transfer = h L k \mathrm{Nu}_{L}=\frac{\mbox{Convective heat transfer }~{}}{\mbox{Conductive % heat transfer }~{}}=\frac{hL}{k}
  2. N u x = h x x k Nu_{x}=\frac{h_{x}x}{k}
  3. N u ¯ = 1 H 0 H N u ( y ) d y \overline{Nu}=\frac{1}{H}\int_{0}^{H}{Nu(y)\cdot dy}
  4. Q y = h A ( T s - T ) {{Q}_{y}}=hA\left({{T}_{s}}-{{T}_{\infty}}\right)
  5. Q y = - k A y ( T - T s ) | y = 0 {{Q}_{y}}=-kA\frac{\partial}{\partial y}{{\left.\left(T-{{T}_{s}}\right)\right% |}_{y=0}}
  6. - k A y ( T - T s ) | y = 0 = h A ( T s - T ) -kA\frac{\partial}{\partial y}{{\left.\left(T-{{T}_{s}}\right)\right|}_{y=0}}=% hA\left({{T}_{s}}-{{T}_{\infty}}\right)
  7. h k = ( T s - T ) y | y = 0 ( T s - T ) \frac{h}{k}=\frac{{{\left.\frac{\partial\left({{T}_{s}}-T\right)}{\partial y}% \right|}_{y=0}}}{{\left({{T}_{s}}-{{T}_{\infty}}\right)}}
  8. h L k = ( T s - T ) y | y = 0 ( T s - T ) L \frac{hL}{k}=\frac{{{\left.\frac{\partial\left({{T}_{s}}-T\right)}{\partial y}% \right|}_{y=0}}}{\frac{\left({{T}_{s}}-{{T}_{\infty}}\right)}{L}}
  9. Nu = h L k \mathrm{Nu}=\frac{hL}{k}
  10. q = - k A T q=-kA\nabla T
  11. = - L \nabla^{\prime}=-L\nabla
  12. T = T h - T 0 T h - T c T^{\prime}=\frac{T_{h}-T_{0}}{T_{h}-T_{c}}
  13. - T = - L k A ( T h - T c ) q = h L k -\nabla^{\prime}T^{\prime}=-\frac{L}{kA(T_{h}-T_{c})}q=\frac{hL}{k}
  14. Nu L = h L k \mathrm{Nu}_{L}=\frac{hL}{k}
  15. Nu L = - T \mathrm{Nu}_{L}=-\nabla^{\prime}T^{\prime}
  16. Nu ¯ = - 1 S S Nu d S \overline{\mathrm{Nu}}=-{{1}\over{S^{\prime}}}\int_{S^{\prime}}\mathrm{Nu}\,% \mathrm{d}S^{\prime}\!
  17. S = S L 2 S^{\prime}=\frac{S}{L^{2}}
  18. N u = f ( R a , P r ) Nu=f(Ra,Pr)
  19. N u = f ( R e , P r ) Nu=f(Re,Pr)
  20. Nu ¯ L = 0.68 + 0.67 Ra L 1 / 4 [ 1 + ( 0.492 / Pr ) 9 / 16 ] 4 / 9 Ra L 10 9 \overline{\mathrm{Nu}}_{L}\ =0.68+\frac{0.67\,\mathrm{Ra}_{L}^{1/4}}{\left[1+(% 0.492/\mathrm{Pr})^{9/16}\,\right]^{4/9}\,}\quad\mathrm{Ra}_{L}\leq 10^{9}
  21. L = A s P L\ =\frac{A_{s}}{P}
  22. A s \mathrm{A}_{s}
  23. P P
  24. Nu ¯ L = 0.54 Ra L 1 / 4 10 4 Ra L 10 7 \overline{\mathrm{Nu}}_{L}\ =0.54\,\mathrm{Ra}_{L}^{1/4}\,\quad 10^{4}\leq% \mathrm{Ra}_{L}\leq 10^{7}
  25. Nu ¯ L = 0.15 Ra L 1 / 3 10 7 Ra L 10 11 \overline{\mathrm{Nu}}_{L}\ =0.15\,\mathrm{Ra}_{L}^{1/3}\,\quad 10^{7}\leq% \mathrm{Ra}_{L}\leq 10^{11}
  26. Nu ¯ L = 0.27 Ra L 1 / 4 10 5 Ra L 10 10 \overline{\mathrm{Nu}}_{L}\ =0.27\,\mathrm{Ra}_{L}^{1/4}\,\quad 10^{5}\leq% \mathrm{Ra}_{L}\leq 10^{10}
  27. Nu x = 0.332 Re x 1 / 2 Pr 1 / 3 , ( Pr > 0.6 ) \mathrm{Nu}_{x}\ =0.332\,\mathrm{Re}_{x}^{1/2}\,\mathrm{Pr}^{1/3},(\mathrm{Pr}% >0.6)
  28. Nu ¯ x = 0.664 Re x 1 / 2 Pr 1 / 3 , ( Pr > 0.6 ) \mathrm{\bar{Nu}_{x}}\ =0.664\,\mathrm{Re}_{x}^{1/2}\,\mathrm{Pr}^{1/3},(% \mathrm{Pr}>0.6)
  29. Nu D = ( f / 8 ) ( Re D - 1000 ) Pr 1 + 12.7 ( f / 8 ) 1 / 2 ( Pr 2 / 3 - 1 ) \mathrm{Nu}_{D}=\frac{\left(f/8\right)\left(\mathrm{Re}_{D}-1000\right)\mathrm% {Pr}}{1+12.7(f/8)^{1/2}\left(\mathrm{Pr}^{2/3}-1\right)}
  30. f = ( 0.79 ln ( Re D ) - 1.64 ) - 2 f=\left(0.79\ln\left(\mathrm{Re}_{D}\right)-1.64\right)^{-2}
  31. 0.5 Pr 2000 0.5\leq\mathrm{Pr}\leq 2000
  32. 3000 Re D 5 × 10 6 3000\leq\mathrm{Re}_{D}\leq 5\times 10^{6}
  33. Nu D = 0.023 Re D 4 / 5 Pr n \mathrm{Nu}_{D}=0.023\,\mathrm{Re}_{D}^{4/5}\,\mathrm{Pr}^{n}
  34. D D
  35. Pr \mathrm{Pr}
  36. n = 0.4 n=0.4
  37. n = 0.3 n=0.3
  38. 0.6 Pr 160 0.6\leq\mathrm{Pr}\leq 160
  39. Re D 10 000 \mathrm{Re}_{D}\gtrsim 10\,000
  40. L D 10 \frac{L}{D}\gtrsim 10
  41. ( μ / μ s ) ({\mu}/{\mu_{s}})
  42. μ \mu
  43. μ s \mu_{s}
  44. Nu D = 0.027 Re D 4 / 5 Pr 1 / 3 ( μ μ s ) 0.14 \mathrm{Nu}_{D}=0.027\,\mathrm{Re}_{D}^{4/5}\,\mathrm{Pr}^{1/3}\left(\frac{\mu% }{\mu_{s}}\right)^{0.14}
  45. μ \mu
  46. μ s \mu_{s}
  47. 0.7 Pr 16 700 0.7\leq\mathrm{Pr}\leq 16\,700
  48. Re D 10 000 \mathrm{Re}_{D}\gtrsim 10\,000
  49. L D 10 \frac{L}{D}\gtrsim 10
  50. Nu = h D h k f \mathrm{Nu}=\frac{hD_{h}}{k_{f}}
  51. Nu D = 48 11 4.36 \mathrm{Nu}_{D}=\frac{48}{11}\simeq 4.36
  52. Nu D = 3.66 \mathrm{Nu}_{D}=3.66

Nutation.html

  1. τ \mathbf{τ}
  2. 𝛀 \mathbf{Ω}
  3. τ = 𝛀 × 𝐋 , \mathbf{\tau}=\mathbf{\Omega}\times\mathbf{L},
  4. 𝐋 \mathbf{L}
  5. θ θ
  6. φ φ
  7. ψ ψ
  8. φ φ
  9. θ θ
  10. M M
  11. l l
  12. V = M g l cos θ . V=Mgl\cos\theta.
  13. z z
  14. x , y x,y
  15. z z
  16. T = 1 2 I 1 ( ω 1 2 + ω 2 2 ) + 1 2 I 3 ω 3 2 . T=\frac{1}{2}I_{1}\left(\omega_{1}^{2}+\omega_{2}^{2}\right)+\frac{1}{2}I_{3}% \omega_{3}^{2}.
  17. T = 1 2 I 1 ( θ ˙ 2 + ϕ ˙ 2 sin 2 θ ) + 1 2 I 3 ( ψ ˙ + ϕ ˙ cos θ ) 2 . T=\frac{1}{2}I_{1}\left(\dot{\theta}^{2}+\dot{\phi}^{2}\sin^{2}\theta\right)+% \frac{1}{2}I_{3}\left(\dot{\psi}+\dot{\phi}\cos\theta\right)^{2}.
  18. a a
  19. b b
  20. ϕ ˙ = b - a cos θ sin 2 θ . \dot{\phi}=\frac{b-a\cos\theta}{\sin^{2}\theta}.
  21. u = c o s θ u=cosθ
  22. u ˙ 2 = f ( u ) \dot{u}^{2}=f(u)
  23. f f
  24. a a
  25. b b
  26. f f
  27. θ θ

Nyquist–Shannon_sampling_theorem.html

  1. X ( f ) = def - x ( t ) e - i 2 π f t d t , X(f)\ \stackrel{\mathrm{def}}{=}\ \int_{-\infty}^{\infty}x(t)\ e^{-i2\pi ft}\ % {\rm d}t,
  2. X s ( f ) = def k = - X ( f - k f s ) = n = - T x ( n T ) e - i 2 π n T f , X_{s}(f)\ \stackrel{\mathrm{def}}{=}\sum_{k=-\infty}^{\infty}X\left(f-kf_{s}% \right)=\sum_{n=-\infty}^{\infty}T\cdot x(nT)\ e^{-i2\pi nTf},
  3. X ( f ) = H ( f ) X s ( f ) , X(f)=H(f)\cdot X_{s}(f),\,
  4. H ( f ) = def { 1 | f | < B 0 | f | > f s - B . H(f)\ \stackrel{\mathrm{def}}{=}\ \begin{cases}1&|f|<B\\ 0&|f|>f_{s}-B.\end{cases}
  5. H ( f ) = rect ( f f s ) = { 1 | f | < f s 2 0 | f | > f s 2 , H(f)=\mathrm{rect}\left(\frac{f}{f_{s}}\right)=\begin{cases}1&|f|<\frac{f_{s}}% {2}\\ 0&|f|>\frac{f_{s}}{2},\end{cases}
  6. X ( f ) = rect ( f f s ) X s ( f ) X(f)=\mathrm{rect}\left(\frac{f}{f_{s}}\right)\cdot X_{s}(f)
  7. = rect ( T f ) n = - T x ( n T ) e - i 2 π n T f =\mathrm{rect}(Tf)\cdot\sum_{n=-\infty}^{\infty}T\cdot x(nT)\ e^{-i2\pi nTf}
  8. = n = - x ( n T ) T rect ( T f ) e - i 2 π n T f { sinc ( t - n T T ) } . =\sum_{n=-\infty}^{\infty}x(nT)\cdot\underbrace{T\cdot\mathrm{rect}(Tf)\cdot e% ^{-i2\pi nTf}}_{\mathcal{F}\left\{\mathrm{sinc}\left(\frac{t-nT}{T}\right)% \right\}}.
  9. x ( t ) = n = - x ( n T ) sinc ( t - n T T ) , x(t)=\sum_{n=-\infty}^{\infty}x(nT)\cdot\mathrm{sinc}\left(\frac{t-nT}{T}% \right),
  10. n = - x ( n T ) δ ( t - n T ) , \textstyle\sum_{n=-\infty}^{\infty}x(nT)\cdot\delta(t-nT),
  11. X ( ω ) \scriptstyle X(\omega)
  12. x ( t ) . \scriptstyle x(t).
  13. x ( t ) x(t)\,
  14. = 1 2 π - X ( ω ) e i ω t d ω ={1\over 2\pi}\int_{-\infty}^{\infty}X(\omega)e^{i\omega t}\;{\rm d}\omega
  15. = 1 2 π - 2 π B 2 π B X ( ω ) e i ω t d ω ={1\over 2\pi}\int_{-2\pi B}^{2\pi B}X(\omega)e^{i\omega t}\;{\rm d}\omega
  16. X ( ω ) \scriptstyle X(\omega)
  17. | ω 2 π | < B \scriptstyle|\frac{\omega}{2\pi}|<B
  18. t = n 2 B t={n\over{2B}}\,
  19. x ( n 2 B ) = 1 2 π - 2 π B 2 π B X ( ω ) e i ω n 2 B d ω . x\left(\tfrac{n}{2B}\right)={1\over 2\pi}\int_{-2\pi B}^{2\pi B}X(\omega)e^{i% \omega{n\over{2B}}}\;{\rm d}\omega.
  20. x ( t ) \scriptstyle x(t)
  21. X ( ω ) , \scriptstyle X(\omega),
  22. x ( n / 2 B ) \scriptstyle x(n/2B)
  23. X ( ω ) . \scriptstyle X(\omega).
  24. X ( ω ) , \scriptstyle X(\omega),
  25. X ( ω ) \scriptstyle X(\omega)
  26. X ( ω ) \scriptstyle X(\omega)
  27. X ( ω ) \scriptstyle X(\omega)
  28. x ( t ) \scriptstyle x(t)
  29. x ( t ) \scriptstyle x(t)
  30. x n \scriptstyle x_{n}
  31. x ( t ) \scriptstyle x(t)
  32. x ( t ) = n = - x n sin π ( 2 B t - n ) π ( 2 B t - n ) . x(t)=\sum_{n=-\infty}^{\infty}x_{n}{\sin\pi(2Bt-n)\over\pi(2Bt-n)}.\,
  33. x ( t ) = cos ( 2 π B t + θ ) cos ( θ ) = cos ( 2 π B t ) - sin ( 2 π B t ) tan ( θ ) , - π / 2 < θ < π / 2. x(t)=\frac{\cos(2\pi Bt+\theta)}{\cos(\theta)}\ =\ \cos(2\pi Bt)-\sin(2\pi Bt)% \tan(\theta),\quad-\pi/2<\theta<\pi/2.
  34. x ( n T ) = cos ( π n ) - sin ( π n ) 0 tan ( θ ) = ( - 1 ) n x(nT)=\cos(\pi n)-\underbrace{\sin(\pi n)}_{0}\tan(\theta)=(-1)^{n}
  35. ( N 2 f s , N + 1 2 f s ) , \left(\frac{N}{2}f_{\mathrm{s}},\frac{N+1}{2}f_{\mathrm{s}}\right),
  36. ( N + 1 ) sinc ( ( N + 1 ) t T ) - N sinc ( N t T ) . (N+1)\,\operatorname{sinc}\left(\frac{(N+1)t}{T}\right)-N\,\operatorname{sinc}% \left(\frac{Nt}{T}\right).
  37. sin ( x - x i ) x - x i . \frac{\sin(x-x_{i})}{x-x_{i}}.
  38. 1 / 2 B = T . \scriptstyle 1/2B=T.
  39. T x ( n T ) , \scriptstyle T\cdot x(nT),

Obesity.html

  1. BMI = m h 2 \mathrm{BMI}=\frac{m}{h^{2}}

Ocean_thermal_energy_conversion.html

  1. - d I ( y ) d y = μ I -\frac{dI(y)}{dy}=\mu I
  2. I ( y ) = I 0 exp ( - μ y ) I(y)=I_{0}\exp(-\mu y)\,
  3. H 1 = H f H_{1}=H_{f}\,
  4. f {}_{f}
  5. 1 {}_{1}
  6. 2 {}_{2}
  7. H 2 = H 1 = H f + x 2 H f g H_{2}=H_{1}=H_{f}+x_{2}H_{fg}\,
  8. 2 {}_{2}
  9. 2 {}_{2}
  10. H 3 = H g H_{3}=H_{g}\,
  11. g {}_{g}
  12. 2 {}_{2}
  13. s 5 , s = s 3 = s f + x 5 , s s f g s_{5,s}=s_{3}=s_{f}+x_{5,s}s_{fg}\,
  14. 5 {}_{5}
  15. 5 , s {}_{5,s}
  16. 5 {}_{5}
  17. H 5 , s = H f + x 5 , s H f g H_{5,s}=H_{f}+x_{5,s}H_{fg}\,
  18. 3 {}_{3}
  19. 5 , s {}_{5,s}
  20. H 5 = H 3 - actual work H_{5}=H_{3}-\ \mathrm{actual}\ \mathrm{work}
  21. 6 {}_{6}
  22. f {}_{f}
  23. 5 {}_{5}
  24. 7 {}_{7}
  25. H 7 H f a t T 7 H_{7}\approx H_{f}\,\ at\ T_{7}\,
  26. m c = H 5 - H 6 H 6 - H 7 ˙ \dot{m_{c}=\frac{H_{5}-\ H_{6}}{H_{6}-\ H_{7}}}\,
  27. M T ˙ = turbine work required W T \dot{M_{T}}=\frac{\mathrm{turbine}\ \mathrm{work}\ \mathrm{required}}{W_{T}}
  28. M w ˙ = M T m w ˙ ˙ \dot{M_{w}}=\dot{M_{T}\dot{m_{w}}}\,
  29. M c ˙ = M T m C ˙ ˙ \dot{\dot{M_{c}}=\dot{M_{T}m_{C}}}\,
  30. H {}_{H}
  31. T {}_{T}
  32. C {}_{C}
  33. C {}_{C}
  34. C {}_{C}
  35. C T {}_{CT}
  36. H T {}_{HT}
  37. A {}_{A}
  38. N P {}_{NP}
  39. W N P = W T - W C - W C T - W H T - W A W_{NP}=W_{T}-W_{C}-W_{CT}-W_{HT}-W_{A}\,
  40. W N = Q H - Q C W_{N}=Q_{H}-Q_{C}\,
  41. W N = W T + W C W_{N}=W_{T}+W_{C}
  42. Q H = H T H d s Q_{H}=\int_{H}T_{H}ds\,
  43. Q C = C T C d s Q_{C}=\int_{C}T_{C}ds\,
  44. W N = H T H d s - C T C d s W_{N}=\int_{H}T_{H}ds-\int_{C}T_{C}ds\,