wpmath0000006_8

Metric_tensor_(general_relativity).html

  1. g x ( u , v ) = g x ( v , u ) . g_{x}(u,v)=g_{x}(v,u)\in\mathbb{R}.
  2. x μ x^{\mu}
  3. μ \mu
  4. g = g μ ν d x μ d x ν . g=g_{\mu\nu}dx^{\mu}\otimes dx^{\nu}.
  5. d x μ dx^{\mu}
  6. x μ x^{\mu}
  7. g μ ν g_{\mu\nu}
  8. g μ ν = g ν μ g_{\mu\nu}=g_{\nu\mu}\,
  9. d x μ d x ν = d x ν d x μ dx^{\mu}dx^{\nu}=dx^{\nu}dx^{\mu}
  10. g = g μ ν d x μ d x ν . g=g_{\mu\nu}dx^{\mu}dx^{\nu}.\,
  11. g μ ν g_{\mu\nu}
  12. g μ ν g_{\mu\nu}
  13. g μ ν g_{\mu\nu}
  14. d x μ dx^{\mu}
  15. d s 2 ds^{2}
  16. d s 2 = g μ ν d x μ d x ν . ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}.\,
  17. d s 2 ds^{2}
  18. d s 2 < 0 ds^{2}<0
  19. d s 2 = 0 ds^{2}=0
  20. d s 2 > 0 ds^{2}>0
  21. x μ x μ ¯ x^{\mu}\to x^{\bar{\mu}}
  22. g μ ¯ ν ¯ = x ρ x μ ¯ x σ x ν ¯ g ρ σ = Λ ρ Λ σ μ ¯ g ρ σ ν ¯ . g_{\bar{\mu}\bar{\nu}}=\frac{\partial x^{\rho}}{\partial x^{\bar{\mu}}}\frac{% \partial x^{\sigma}}{\partial x^{\bar{\nu}}}g_{\rho\sigma}=\Lambda^{\rho}{}_{% \bar{\mu}}\,\Lambda^{\sigma}{}_{\bar{\nu}}\,g_{\rho\sigma}.
  23. ( t , x , y , z ) (t,x,y,z)
  24. d s 2 = - c 2 d t 2 + d x 2 + d y 2 + d z 2 = η μ ν d x μ d x ν . ds^{2}=-c^{2}dt^{2}+dx^{2}+dy^{2}+dz^{2}=\eta_{\mu\nu}dx^{\mu}dx^{\nu}.\,
  25. η = ( - c 2 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) \eta=\begin{pmatrix}-c^{2}&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}
  26. ( t , r , θ , ϕ ) (t,r,\theta,\phi)
  27. d s 2 = - c 2 d t 2 + d r 2 + r 2 d Ω 2 ds^{2}=-c^{2}dt^{2}+dr^{2}+r^{2}d\Omega^{2}\,
  28. d Ω 2 = d θ 2 + sin 2 θ d ϕ 2 d\Omega^{2}=d\theta^{2}+\sin^{2}\theta\,d\phi^{2}
  29. d s 2 = - ( 1 - 2 G M r c 2 ) c 2 d t 2 + ( 1 - 2 G M r c 2 ) - 1 d r 2 + r 2 d Ω 2 ds^{2}=-\left(1-\frac{2GM}{rc^{2}}\right)c^{2}dt^{2}+\left(1-\frac{2GM}{rc^{2}% }\right)^{-1}dr^{2}+r^{2}d\Omega^{2}
  30. d Ω 2 d\Omega^{2}
  31. x μ x^{\mu}
  32. vol g = | det g | d x 0 d x 1 d x 2 d x 3 \mathrm{vol}_{g}=\sqrt{|\det g|}\,dx^{0}\wedge dx^{1}\wedge dx^{2}\wedge dx^{3}
  33. x μ x^{\mu}
  34. Γ λ = μ ν 1 2 g λ ρ ( g ρ μ x ν + g ρ ν x μ - g μ ν x ρ ) \Gamma^{\lambda}{}_{\mu\nu}={1\over 2}g^{\lambda\rho}\left({\partial g_{\rho% \mu}\over\partial x^{\nu}}+{\partial g_{\rho\nu}\over\partial x^{\mu}}-{% \partial g_{\mu\nu}\over\partial x^{\rho}}\right)
  35. R ρ σ μ ν = μ Γ ρ - ν σ ν Γ ρ + μ σ Γ ρ Γ λ μ λ - ν σ Γ ρ Γ λ ν λ . μ σ {R^{\rho}}_{\sigma\mu\nu}=\partial_{\mu}\Gamma^{\rho}{}_{\nu\sigma}-\partial_{% \nu}\Gamma^{\rho}{}_{\mu\sigma}+\Gamma^{\rho}{}_{\mu\lambda}\Gamma^{\lambda}{}% _{\nu\sigma}-\Gamma^{\rho}{}_{\nu\lambda}\Gamma^{\lambda}{}_{\mu\sigma}.
  36. g g
  37. R μ ν - 1 2 R g μ ν = 8 π G T μ ν R_{\mu\nu}-{1\over 2}Rg_{\mu\nu}=8\pi G\,T_{\mu\nu}
  38. R ν ρ = def R μ ν μ ρ R_{\nu\rho}\ \stackrel{\mathrm{def}}{=}\ {R^{\mu}}_{\nu\mu\rho}
  39. T μ ν T_{\mu\nu}

Mex_(mathematics).html

  1. mex ( { 1 , 2 , 3 } ) = 0 \mbox{mex}~{}(\left\{1,2,3\right\})=0
  2. mex ( { 0 , 1 , 4 , 7 , 12 } ) = 2 \mbox{mex}~{}(\left\{0,1,4,7,12\right\})=2
  3. mex ( { 0 , 1 , 2 , 3 , } ) = ω \mbox{mex}~{}(\left\{0,1,2,3,\ldots\right\})=\omega
  4. mex ( { 0 , 2 , 4 , 6 , } ) = 1 \mbox{mex}~{}(\left\{0,2,4,6,\ldots\right\})=1
  5. mex ( { 0 , 1 , 2 , 3 , , ω } ) = ω + 1 \mbox{mex}~{}(\left\{0,1,2,3,\ldots,\omega\right\})=\omega+1

Michał_Kalecki.html

  1. P + W = C W + C P + I \textstyle P+W=C_{W}+C_{P}+I\,
  2. P P\,
  3. W W\,
  4. C P C_{P}\,
  5. C W C_{W}\,
  6. I I\,
  7. W = C W W=C_{W}\,
  8. P = C P + I \textstyle P=C_{P}+I
  9. S = I \textstyle S=I\,
  10. P N = C P + I + D g + E e - S w \textstyle P_{N}=C_{P}+I+D_{g}+E_{e}-S_{w}\,
  11. C P = A + q P \textstyle C_{P}=A+q\cdot P\,
  12. A A\,
  13. q q\,
  14. P = A + q P + I \textstyle P=A+q\cdot P+I\,
  15. P P\,
  16. P = A + I 1 - q \textstyle P=\frac{A+I}{1-q}\,
  17. P + B = k ( W + M ) \textstyle P+B=k\cdot(W+M)\,
  18. P P\,
  19. W W\,
  20. k k\,
  21. M M\,
  22. B B\,
  23. W W\,
  24. Y = k ( W + M ) + W \textstyle Y=k\cdot(W+M)+W\,
  25. W W\,
  26. α = W W + k ( W + M ) \alpha=\frac{W}{W+k\cdot(W+M)}\,
  27. α = 1 1 + k ( 1 + j ) \alpha=\frac{1}{1+k\cdot(1+j)}\,
  28. α \alpha\,
  29. j j\,
  30. k k\,
  31. j j\,
  32. j j\,
  33. 1 - α = P + B Y 1-\alpha=\frac{P+B}{Y}\,
  34. Y Y\,
  35. Y = P + B 1 - α Y=\frac{P+B}{1-\alpha}\,
  36. P P\,
  37. Y = I + A 1 - q + B 1 - α Y=\frac{\frac{I+A}{1-q}+B}{1-\alpha}\,
  38. A A\,
  39. B B\,
  40. Δ Y = Δ I ( 1 - q ) ( 1 - α ) \Delta Y=\frac{\Delta I}{(1-q)\cdot(1-\alpha)}\,
  41. D = a S + b Δ P Δ t - c Δ K Δ t + d D=a\cdot S+b\cdot\frac{\Delta P}{\Delta t}-c\cdot\frac{\Delta K}{\Delta t}+d\,
  42. D D\,
  43. a a\,
  44. b b\,
  45. c c\,
  46. d d\,
  47. P P\,
  48. S S\,
  49. K K\,

Michelson–Gale–Pearson_experiment.html

  1. Δ = 4 A ω sin ϕ λ c \Delta=\frac{4A\omega\sin\phi}{\lambda c}
  2. Δ \Delta
  3. A A
  4. ϕ \phi
  5. c c
  6. ω \omega
  7. λ \lambda

Microstate_(statistical_mechanics).html

  1. N N
  2. i i
  3. p i p_{i}
  4. E i E_{i}
  5. E i E_{i}
  6. U = E = i = 1 N p i E i . U=\langle E\rangle=\sum_{i=1}^{N}p_{i}\,E_{i}\ .
  7. S = - k B i p i ln p i , S=-k_{B}\,\sum_{i}p_{i}\ln\,p_{i},
  8. k B k_{B}
  9. S = k B ln W S=k_{B}\,\ln W
  10. W W
  11. δ W = i = 1 N p i d E i \delta W=\sum_{i=1}^{N}p_{i}\,dE_{i}
  12. δ Q = i = 1 N E i d p i \delta Q=\sum_{i=1}^{N}E_{i}\,dp_{i}
  13. d U = δ W + δ Q . ~{}dU=\delta W+\delta Q.

Mid-range.html

  1. M = max x + min x 2 . M=\frac{\max x+\min x}{2}.
  2. x / 2 , x/2,
  3. x / n . x/n.
  4. var ( M ) = π 2 24 ln ( n ) . \operatorname{var}(M)=\frac{\pi^{2}}{24\ln(n)}.
  5. var ( M ) = π 2 12 \operatorname{var}(M)=\frac{\pi^{2}}{12}
  6. max | x i - m | \max\left|x_{i}-m\right|

Milling_cutter.html

  1. S = 1000 V c π D S=\frac{1000V_{c}}{\pi D}\,
  2. F = z S F z F=zSF_{z}\,

Millman's_theorem.html

  1. v = ± e k R k + ± a m 1 R k + 1 R i v=\frac{\sum\frac{\pm e_{k}}{R_{k}}+\sum\pm a_{m}}{\sum\frac{1}{R_{k}}+\sum% \frac{1}{R_{i}}}

Milnor_conjecture.html

  1. K n M ( F ) / 2 H e ´ t n ( F , / 2 ) K_{n}^{M}(F)/2\cong H_{\acute{e}t}^{n}(F,\mathbb{Z}/2\mathbb{Z})

Milnor_K-theory.html

  1. K * M ( F ) := T * F × / ( a ( 1 - a ) ) , K^{M}_{*}(F):=T^{*}F^{\times}/(a\otimes(1-a)),\,
  2. a ( 1 - a ) a\otimes(1-a)\,
  3. { a 1 , , a n } \{a_{1},\ldots,a_{n}\}
  4. a 1 a n a_{1}\otimes\cdots\otimes a_{n}
  5. K m × K n K m + n K_{m}\times K_{n}\rightarrow K_{m+n}
  6. K * M ( F ) K^{M}_{*}(F)
  7. K n M ( 𝔽 q ) = 0 K^{M}_{n}(\mathbb{F}_{q})=0
  8. K 2 M ( ) K^{M}_{2}(\mathbb{C})
  9. K 2 M ( ) K^{M}_{2}(\mathbb{R})
  10. K 2 M ( p ) K^{M}_{2}(\mathbb{Q}_{p})
  11. 𝔽 p \mathbb{F}_{p}
  12. K 2 M ( ) K^{M}_{2}(\mathbb{Q})
  13. p - 1 p-1
  14. p p
  15. K 1 M K^{M}_{1}
  16. { a 1 , , a n } a 1 , a 2 , , a n = 1 , a 1 1 , a 2 1 , a n , \{a_{1},\ldots,a_{n}\}\mapsto\langle\langle a_{1},a_{2},...,a_{n}\rangle% \rangle=\langle 1,a_{1}\rangle\otimes\langle 1,a_{2}\rangle\otimes...\otimes% \langle 1,a_{n}\rangle\ ,

Milnor_map.html

  1. f ( z 0 , , z n ) f(z_{0},\dots,z_{n})
  2. n + 1 n+1
  3. z 0 , , z n z_{0},\dots,z_{n}
  4. f ( 0 , , 0 ) = 0 f(0,\dots,0)=0
  5. V f V_{f}
  6. ( n + 1 ) (n+1)
  7. ( z 0 , , z n ) (z_{0},\dots,z_{n})
  8. f ( z 0 , , z n ) = 0 f(z_{0},\dots,z_{n})=0
  9. n n
  10. ( n + 1 ) (n+1)
  11. n = 1 n=1
  12. V f V_{f}
  13. ( 0 , 0 ) (0,0)
  14. f f
  15. f / | f | f/|f|
  16. V f V_{f}
  17. ( n + 1 ) (n+1)
  18. S 1 S^{1}
  19. r > 0 r>0
  20. f f
  21. V f V_{f}
  22. ( 2 n + 1 ) (2n+1)
  23. r r
  24. f f
  25. r r
  26. f f
  27. V f V_{f}
  28. f f
  29. ϵ \epsilon
  30. f | f | : ( S ε 2 n + 1 - V f ) S 1 \dfrac{f}{|f|}:\left(S^{2n+1}_{\varepsilon}-V_{f}\right)\rightarrow S^{1}
  31. 2 n 2n
  32. V f V_{f}
  33. ( 2 n + 1 ) (2n+1)
  34. ( 2 n - 1 ) (2n-1)
  35. V f V_{f}
  36. ( 2 n + 2 ) (2n+2)
  37. ( 2 n + 1 ) (2n+1)
  38. V g V_{g}
  39. g = f - e g=f-e
  40. e e
  41. f ( z , w ) = z 2 + w 3 f(z,w)=z^{2}+w^{3}

Mimic_function.html

  1. A A
  2. B B
  3. p ( t , A ) p(t,A)
  4. t t
  5. A A
  6. f f
  7. A A
  8. p ( t , f ( A ) ) p(t,f(A))
  9. p ( t , B ) p(t,B)
  10. t t
  11. n n
  12. x x
  13. p ( x , A ) p(x,A)
  14. y y
  15. p ( y , A ) p(y,A)
  16. x x
  17. y y
  18. p ( x , A ) p(x,A)
  19. p ( y , A ) p(y,A)
  20. B B

Minimal_models.html

  1. c = 1 - 6 ( p - q ) 2 p q c=1-6{(p-q)^{2}\over pq}
  2. h = h r , s ( c ) = ( p r - q s ) 2 - ( p - q ) 2 4 p q h=h_{r,s}(c)={{(pr-qs)^{2}-(p-q)^{2}}\over 4pq}
  3. c = 1 - 6 m ( m + 1 ) = 0 , 1 / 2 , 7 / 10 , 4 / 5 , 6 / 7 , 25 / 28 , c=1-{6\over m(m+1)}=0,\quad 1/2,\quad 7/10,\quad 4/5,\quad 6/7,\quad 25/28,\ldots
  4. h = h r , s ( c ) = ( ( m + 1 ) r - m s ) 2 - 1 4 m ( m + 1 ) h=h_{r,s}(c)={((m+1)r-ms)^{2}-1\over 4m(m+1)}

Minimax_Condorcet.html

  1. s c o r e ( X , Y ) score(X,Y)
  2. X X
  3. Y Y
  4. W W
  5. W = arg min X ( max Y s c o r e ( Y , X ) ) W=\arg\min_{X}(\max_{Y}score(Y,X))
  6. d ( X , Y ) d(X,Y)
  7. s c o r e ( X , Y ) score(X,Y)
  8. s c o r e ( X , Y ) := { d ( X , Y ) , d ( X , Y ) > d ( Y , X ) 0 , e l s e score(X,Y):=\begin{cases}d(X,Y)&,d(X,Y)>d(Y,X)\\ 0&,else\end{cases}
  9. s c o r e ( X , Y ) := d ( X , Y ) - d ( Y , X ) score(X,Y):=d(X,Y)-d(Y,X)
  10. s c o r e ( X , Y ) := d ( X , Y ) score(X,Y):=d(X,Y)

Minimum-shift_keying.html

  1. δ \delta
  2. s ( t ) = a I ( t ) cos ( π t 2 T ) cos ( 2 π f c t ) - a Q ( t ) sin ( π t 2 T ) sin ( 2 π f c t ) s(t)=a_{I}(t)\cos{\left(\frac{{\pi}t}{2T}\right)}\cos{(2{\pi}f_{c}t)}-a_{Q}(t)% \sin{\left(\frac{{\pi}t}{2T}\right)}\sin{\left(2{\pi}f_{c}t\right)}
  3. a I ( t ) a_{I}(t)
  4. a Q ( t ) a_{Q}(t)
  5. a I ( t ) a_{I}(t)
  6. t = [ - T , T , 3 T , ] t=[-T,T,3T,...]
  7. a Q ( t ) a_{Q}(t)
  8. t = [ 0 , 2 T , 4 T , ] t=[0,2T,4T,...]
  9. f c f_{c}
  10. s ( t ) = cos [ 2 π f c t + b k ( t ) π t 2 T + ϕ k ] s(t)=\cos\left[2\pi f_{c}t+b_{k}(t)\frac{\pi t}{2T}+\phi_{k}\right]
  11. a I ( t ) = a Q ( t ) a_{I}(t)=a_{Q}(t)
  12. ϕ k \phi_{k}
  13. a I ( t ) a_{I}(t)
  14. π \pi

Minimum_degree_spanning_tree.html

  1. G G
  2. T T
  3. G G
  4. G G
  5. T T
  6. T T
  7. | V | - 1 |V|-1
  8. V V
  9. G G
  10. T T^{\prime}
  11. T T^{\prime}
  12. G G

Minkowski–Hlawka_theorem.html

  1. Δ ζ ( n ) 2 n - 1 , \Delta\geq\frac{\zeta(n)}{2^{n-1}},

MINOS.html

  1. δ t = - 15 ± 31 \delta t=-15\pm 31

Mired.html

  1. M = 1000000 T M=\frac{1000000}{T}
  2. 10 6 5700 - 10 6 3200 - 137 MK - 1 \frac{10^{6}}{5700}-\frac{10^{6}}{3200}\approx-137\ \mbox{MK}~{}^{-1}

Mirimanoff's_congruence.html

  1. ϕ n ( t ) = 1 n - 1 t + 2 n - 1 t 2 + + ( p - 1 ) n - 1 t p - 1 . \phi_{n}(t)=1^{n-1}t+2^{n-1}t^{2}+...+(p-1)^{n-1}t^{p-1}.
  2. 3 p - 1 ( - 2 3 { 1 + 1 2 + 1 3 + 1 4 + + p / 3 - 1 } ) p + 1 ( mod p 2 ) 3^{p-1}\equiv\left(-\frac{2}{3}\cdot\left\{1+\frac{1}{2}+\frac{1}{3}+\frac{1}{% 4}+\ldots+\left\lfloor p/3\right\rfloor^{-1}\right\}\right)p+1\;\;(\mathop{{% \rm mod}}p^{2})

Mittag-Leffler's_theorem.html

  1. D D
  2. \mathbb{C}
  3. E D E\subset D
  4. a a
  5. E E
  6. p a ( z ) p_{a}(z)
  7. 1 / ( z - a ) 1/(z-a)
  8. f f
  9. D D
  10. a E a\in E
  11. f ( z ) - p a ( z ) f(z)-p_{a}(z)
  12. a a
  13. f f
  14. a a
  15. p a ( z ) p_{a}(z)
  16. E E
  17. f ( z ) = a E p a ( z ) f(z)=\sum_{a\in E}p_{a}(z)
  18. E E
  19. S F ( z ) = a F p a ( z ) S_{F}(z)=\sum_{a\in F}p_{a}(z)
  20. F F
  21. E E
  22. S F ( z ) S_{F}(z)
  23. S F ( z ) S_{F}(z)
  24. p k = 1 z - k p_{k}=\frac{1}{z-k}
  25. E = + E=\mathbb{Z}^{+}
  26. f f
  27. p k ( z ) p_{k}(z)
  28. z = k z=k
  29. k k
  30. f f
  31. f ( z ) = z k = 1 1 k ( z - k ) f(z)=z\sum_{k=1}^{\infty}\frac{1}{k(z-k)}
  32. \mathbb{C}
  33. 1 sin ( z ) = n ( - 1 ) n z - n π = 1 z + n = 1 ( - 1 ) n 2 z z 2 - n 2 π 2 \frac{1}{\sin(z)}=\sum_{n\in\mathbb{Z}}\frac{(-1)^{n}}{z-n\pi}=\frac{1}{z}+% \sum_{n=1}^{\infty}(-1)^{n}\frac{2z}{z^{2}-n^{2}\pi^{2}}
  34. cot ( z ) cos ( z ) sin ( z ) = n 1 z - n π = 1 z + k = 1 2 z z 2 - k 2 π 2 \cot(z)\equiv\frac{\cos(z)}{\sin(z)}=\sum_{n\in\mathbb{Z}}\frac{1}{z-n\pi}=% \frac{1}{z}+\sum_{k=1}^{\infty}\frac{2z}{z^{2}-k^{2}\pi^{2}}
  35. 1 sin 2 ( z ) = n 1 ( z - n π ) 2 \frac{1}{\sin^{2}(z)}=\sum_{n\in\mathbb{Z}}\frac{1}{(z-n\pi)^{2}}
  36. 1 z sin ( z ) = 1 z 2 + n 0 ( - 1 ) n π n ( z - π n ) = 1 z 2 + n = 1 ( - 1 ) n n π 2 z z 2 - π 2 n 2 \frac{1}{z\sin(z)}=\frac{1}{z^{2}}+\sum_{n\neq 0}\frac{(-1)^{n}}{\pi n(z-\pi n% )}=\frac{1}{z^{2}}+\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n\pi}\frac{2z}{z^{2}-\pi% ^{2}n^{2}}

Mixed_boundary_condition.html

  1. u u
  2. Ω Ω
  3. Ω ∂Ω
  4. Ω ∂Ω
  5. Γ [ u s u , u b = 1 ] Γ[u^{\prime}su^{\prime},u^{\prime}b=1^{\prime}]
  6. Γ [ u s u , u b = 2 ] Γ[u^{\prime}su^{\prime},u^{\prime}b=2^{\prime}]
  7. u u
  8. u | Γ 1 = u 0 \left.u\right|_{\Gamma_{1}}=u_{0}
  9. u n | Γ 2 = g , \left.\frac{\partial u}{\partial n}\right|_{\Gamma_{2}}=g,
  10. u [ u s u , u b = 0 ] u[u^{\prime}su^{\prime},u^{\prime}b=0^{\prime}]
  11. g g
  12. u [ u s u , u b = 0 ] u[u^{\prime}su^{\prime},u^{\prime}b=0^{\prime}]
  13. g g

Mixing_ratio.html

  1. r i r_{i}
  2. n i n_{i}
  3. r i = n i n t o t - n i r_{i}=\frac{n_{i}}{n_{tot}-n_{i}}
  4. n i n_{i}
  5. n t o t n_{tot}
  6. ζ i \zeta_{i}
  7. m i m_{i}
  8. ζ i = m i m t o t - m i \zeta_{i}=\frac{m_{i}}{m_{tot}-m_{i}}

Model-based_reasoning.html

  1. \forall
  2. \rightarrow
  3. \land
  4. \forall
  5. \rightarrow
  6. \forall
  7. \rightarrow

Model_category.html

  1. ( α , β ) (\alpha,\beta)
  2. ( γ , δ ) (\gamma,\delta)
  3. C 2 C^{2}
  4. p i p\circ i
  5. p i p\circ i
  6. 𝒞 o p \mathcal{C}^{op}
  7. F : C D : G F:C\leftrightarrows D:G
  8. L F : H o ( C ) H o ( D ) : R G LF:Ho(C)\leftrightarrows Ho(D):RG
  9. | - | : s 𝐒𝐞𝐭 𝐓𝐨𝐩 : S i n g |-|:s\mathbf{Set}\leftrightarrows\mathbf{Top}:Sing

Modern_valence_bond_theory.html

  1. Φ C = \Phi_{C}=
  2. Φ I = \Phi_{I}=
  3. Φ C F = \Phi_{CF}=

Moduli_of_algebraic_curves.html

  1. g \mathcal{M}_{g}
  2. ¯ g \overline{\mathcal{M}}_{g}
  3. 3 g - 3 3g-3
  4. 0 \mathcal{M}_{0}
  5. 1 \mathcal{M}_{1}
  6. g , n \mathcal{M}_{g,n}
  7. ¯ g , n \overline{\mathcal{M}}_{g,n}
  8. ¯ 1 , 1 \overline{\mathcal{M}}_{1,1}
  9. ¯ g , n \overline{\mathcal{M}}_{g,n}
  10. ¯ g , n \overline{\mathcal{M}}_{g,n}
  11. ¯ g , n \overline{\mathcal{M}}_{g,n}
  12. v ¯ g v , n v \prod_{v}\overline{\mathcal{M}}_{g_{v},n_{v}}

Moishezon_manifold.html

  1. M M
  2. M M
  3. dim 𝐂 M = a ( M ) = tr . deg . 𝐂 𝐂 ( M ) . \dim_{\mathbf{C}}M=a(M)=\operatorname{tr.deg.}_{\mathbf{C}}\mathbf{C}(M).
  4. S p e c ( 𝐂 ) Spec(\mathbf{C})

Momentum_diffusion.html

  1. \Tau = - μ * ( d u / d y ) \Tau=-\mu*(du/dy)
  2. \Tau \Tau
  3. d u / d y du/dy
  4. μ \mu
  5. N / m 2 N/m^{2}
  6. K g / m . s 2 Kg/m.s^{2}

Momentum_operator.html

  1. ψ ( 𝐫 , t ) ψ(\mathbf{r},t)
  2. ψ = e i ( k x - ω t ) \psi=e^{i(kx-\omega t)}
  3. ψ x = i k e i ( k x - ω t ) = i k ψ \frac{\partial\psi}{\partial x}=ike^{i(kx-\omega t)}=ik\psi
  4. k k
  5. p = k p=\hbar k
  6. ψ ψ
  7. ψ x = i p ψ \frac{\partial\psi}{\partial x}=i\frac{p}{\hbar}\psi
  8. p ^ = - i x \hat{p}=-i\hbar\frac{\partial}{\partial x}
  9. p p
  10. ψ = e i ( k r - ω t ) \psi=e^{i({k}\cdot{r}-\omega t)}
  11. ψ = e x ψ x + e y ψ y + e z ψ z = i k x ψ e x + i k y ψ e y + i k z ψ e z = i ( p x e x + p y e y + p z e z ) ψ = i p ^ ψ \begin{aligned}\displaystyle\nabla\psi&\displaystyle={e}_{x}\frac{\partial\psi% }{\partial x}+{e}_{y}\frac{\partial\psi}{\partial y}+{e}_{z}\frac{\partial\psi% }{\partial z}\\ &\displaystyle=ik_{x}\psi{e}_{x}+ik_{y}\psi{e}_{y}+ik_{z}\psi{e}_{z}\\ &\displaystyle=\frac{i}{\hbar}\left(p_{x}{e}_{x}+p_{y}{e}_{y}+p_{z}{e}_{z}% \right)\psi\\ &\displaystyle=\frac{i}{\hbar}{\hat{p}}\psi\end{aligned}
  12. p ^ = - i {\hat{p}}=-i\hbar\nabla
  13. p ^ = - i {\hat{p}}=-i\hbar\nabla
  14. ħ ħ
  15. i i
  16. p ^ = p ^ x = - i x . \hat{p}=\hat{p}_{x}=-i\hbar{\partial\over\partial x}.
  17. q q
  18. φ φ
  19. 𝐀 \mathbf{A}
  20. p ^ = - i - q A {\hat{p}}=-i\hbar\nabla-q{A}
  21. P ^ = - i {\hat{P}}=-i\hbar\nabla
  22. q = 0 q=0
  23. [ x ^ , p ^ ] = x ^ p ^ - p ^ x ^ = i . \left[\hat{x},\hat{p}\right]=\hat{x}\hat{p}-\hat{p}\hat{x}=i\hbar.
  24. x | p ^ | ψ = - i d d x ψ ( x ) \langle x|\hat{p}|\psi\rangle=-i\hbar\frac{d}{dx}\psi(x)
  25. p | x ^ | ψ = i d d p ψ ( p ) \langle p|\hat{x}|\psi\rangle=i\hbar\frac{d}{dp}\psi(p)
  26. p | x ^ | p = i d d p δ ( p - p ) \langle p|\hat{x}|p^{\prime}\rangle=i\hbar\frac{d}{dp}\delta(p-p^{\prime})
  27. x | p ^ | x = - i d d x δ ( x - x ) \langle x|\hat{p}|x^{\prime}\rangle=-i\hbar\frac{d}{dx}\delta(x-x^{\prime})
  28. δ δ
  29. T ( ε ) T(ε)
  30. ε ε
  31. T ( ε ) | ψ = d x T ( ε ) | x x | ψ T(\varepsilon)|\psi\rangle=\int dxT(\varepsilon)|x\rangle\langle x|\psi\rangle
  32. d x | x + ε x | ψ = d x | x x - ε | ψ = d x | x ψ ( x - ε ) \int dx|x+\varepsilon\rangle\langle x|\psi\rangle=\int dx|x\rangle\langle x-% \varepsilon|\psi\rangle=\int dx|x\rangle\psi(x-\varepsilon)
  33. ψ ψ
  34. x x
  35. ψ ( x - ε ) = ψ ( x ) - ε d ψ d x \psi(x-\varepsilon)=\psi(x)-\varepsilon\frac{d\psi}{dx}
  36. ε ε
  37. T ( ε ) = 1 - ε d d x = 1 - i ε ( - i d d x ) T(\varepsilon)=1-\varepsilon{d\over dx}=1-{i\over\hbar}\varepsilon\left(-i% \hbar{d\over dx}\right)
  38. T ( ε ) = 1 - i ε p ^ T(\varepsilon)=1-{i\over\hbar}\varepsilon\hat{p}
  39. p ^ = - i d d x . \hat{p}=-i\hbar{d\over dx}.
  40. ( + ) (+ − − −)
  41. P μ = ( E c , - p ) P_{\mu}=\left(\frac{E}{c},-{p}\right)
  42. P ^ μ = ( 1 c E ^ , - p ^ ) = i ( 1 c t , ) = i μ \hat{P}_{\mu}=\left(\frac{1}{c}\hat{E},-{\hat{p}}\right)=i\hbar\left(\frac{1}{% c}\frac{\partial}{\partial t},\nabla\right)=i\hbar\partial_{\mu}
  43. i ħ −iħ
  44. + i ħ +iħ
  45. γ μ P ^ μ = i γ μ μ = P ^ = i \gamma^{\mu}\hat{P}_{\mu}=i\hbar\gamma^{\mu}\partial_{\mu}=\hat{P}=i\hbar\partial
  46. ( + + + ) (− + + +)
  47. P ^ μ = ( - 1 c E ^ , p ^ ) = - i ( 1 c t , ) = - i μ \hat{P}_{\mu}=\left(-\frac{1}{c}\hat{E},{\hat{p}}\right)=-i\hbar\left(\frac{1}% {c}\frac{\partial}{\partial t},\nabla\right)=-i\hbar\partial_{\mu}

Momentum_transfer.html

  1. p i 1 , p i 2 \vec{p}_{i1},\vec{p}_{i2}
  2. p f 1 , p f 2 \vec{p}_{f1},\vec{p}_{f2}
  3. q = p i 1 - p f 1 = p f 2 - p i 2 \vec{q}=\vec{p}_{i1}-\vec{p}_{f1}=\vec{p}_{f2}-\vec{p}_{i2}
  4. Δ x = / | q | \Delta x=\hbar/|q|
  5. p = k p=\hbar k
  6. k = q / k=q/\hbar
  7. k = 2 π / λ k=2\pi/\lambda
  8. Q = k f - k i Q=k_{f}-k_{i}
  9. k f k_{f}
  10. k i k_{i}
  11. G = Q = k f - k i G=Q=k_{f}-k_{i}
  12. G = 2 π / d G=2\pi/d
  13. Q Q
  14. 2 θ 2\theta
  15. α \alpha
  16. Q Q
  17. Q = 4 π sin ( θ ) λ Q=\frac{4\pi\sin\left(\theta\right)}{\lambda}
  18. 2 θ 2\theta
  19. Q Q

Monochromatic_electromagnetic_plane_wave.html

  1. q q
  2. ω ω
  3. d s 2 = - 2 d u d v + C 2 ( q 2 ω 2 , 2 q 2 ω 2 , ω u ) ( d x 2 + d y 2 ) , - < v , x , y < , - u 0 < u < u 0 . ds^{2}=-2\,du\,dv+C^{2}\left(\frac{q^{2}}{\omega^{2}},2\frac{q^{2}}{\omega^{2}% },\omega u\right)\,\left(dx^{2}+dy^{2}\right),\qquad-\infty<v,x,y<\infty,\quad% -u_{0}<u<u_{0}.
  4. ξ = u 0 ω \xi=\frac{u_{0}}{\omega}
  5. C ( a , 2 a , ξ ) = 0 C(a,2a,ξ)=0
  6. a = q 2 ω 2 a=\frac{q^{2}}{\omega^{2}}
  7. C ( a , b , ξ ) C(a,b,ξ)
  8. C ( a , b , 0 ) = 1 C(a,b,0)=1
  9. A = 2 a C ( u 2 ω 2 , q 2 2 ω 2 , ω u ) sin ( ω u ) d u C ( q 2 ω 2 , q 2 2 ω 2 , ω u ) x \vec{A}=\frac{\sqrt{2}a\int C\left(\frac{u^{2}}{\omega^{2}},\frac{q^{2}}{2% \omega^{2}},\omega u\right)\,\sin(\omega u)\,du}{C\left(\frac{q^{2}}{\omega^{2% }},\frac{q^{2}}{2\omega^{2}},\omega u\right)}\;\partial_{x}
  10. ξ 1 = v \vec{\xi}_{1}=\partial_{v}
  11. ξ 2 = x , ξ 3 = y , ξ 4 = - y x + x y \vec{\xi}_{2}=\partial_{x},\;\vec{\xi}_{3}=\partial_{y},\;\vec{\xi}_{4}=-y\,% \partial_{x}+x\,\partial_{y}
  12. ξ 5 = x v + d u C ( q 2 ω 2 , q 2 2 ω 2 , ω u ) x ξ 6 = y v + d u C ( q 2 ω 2 , q 2 2 ω 2 , ω u ) y \begin{aligned}\displaystyle\vec{\xi}_{5}&\displaystyle=x\,\partial_{v}+\int% \frac{du}{C\left(\frac{q^{2}}{\omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u% \right)}\,\partial_{x}\\ \displaystyle\vec{\xi}_{6}&\displaystyle=y\,\partial_{v}+\int\frac{du}{C\left(% \frac{q^{2}}{\omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u\right)}\,\partial_% {y}\end{aligned}
  13. ξ 2 , ξ 3 , ξ 4 \vec{\xi}_{2},\,\vec{\xi}_{3},\,\vec{\xi}_{4}
  14. ξ 5 , ξ 6 \vec{\xi}_{5},\,\vec{\xi}_{6}
  15. e 0 = 1 2 ( u + v ) \vec{e}_{0}=\frac{1}{\sqrt{2}}\left(\partial_{u}+\partial_{v}\right)
  16. e 1 = 1 2 ( - u + v ) \vec{e}_{1}=\frac{1}{\sqrt{2}}\left(-\partial_{u}+\partial_{v}\right)
  17. e 2 = 1 C ( q 2 ω 2 , 2 q 2 ω 2 , ω u ) x \vec{e}_{2}=\frac{1}{C\left(\frac{q^{2}}{\omega^{2}},\,\frac{2q^{2}}{\omega^{2% }},\,\omega u\right)}\partial_{x}
  18. e 3 = 1 C ( q 2 ω 2 , 2 q 2 ω 2 , ω u ) y \vec{e}_{3}=\frac{1}{C\left(\frac{q^{2}}{\omega^{2}},\,\frac{2q^{2}}{\omega^{2% }},\,\omega u\right)}\partial_{y}
  19. e 0 e 0 = 0 \nabla_{\vec{e}_{0}}\vec{e}_{0}=0
  20. e 0 e 1 = e 0 e 2 = e 0 e 3 = 0 \nabla_{\vec{e}_{0}}\vec{e}_{1}=\nabla_{\vec{e}_{0}}\vec{e}_{2}=\nabla_{\vec{e% }_{0}}\vec{e}_{3}=0
  21. e 0 \vec{e}_{0}
  22. e 1 , e 2 , e 3 \vec{e}_{1},\vec{e}_{2},\vec{e}_{3}
  23. E = q sin ( ω u ) e 2 \vec{E}=q\,\sin(\omega u)\,\vec{e}_{2}
  24. B = - q sin ( ω u ) e 3 \vec{B}=-q\,\sin(\omega u)\,\vec{e}_{3}
  25. q q
  26. ω ω
  27. T j ^ k ^ = q 2 sin 2 ( ω u ) 4 π [ 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ] T^{\hat{j}\hat{k}}=\frac{q^{2}\sin^{2}(\omega u)}{4\pi}\begin{bmatrix}1&1&0&0% \\ 1&1&0&0\\ 0&0&0&0\\ 0&0&0&0\end{bmatrix}
  28. v u , x , y v−u,x,y
  29. X = e 0 \vec{X}=\vec{e}_{0}
  30. θ [ X ] i ^ j ^ = ω 2 C ( q 2 ω 2 , q 2 2 ω 2 , ω u ) C ( q 2 ω 2 , q 2 2 ω 2 , ω u ) diag ( 0 , 1 , 1 ) \theta[\vec{X}]_{\hat{i}\hat{j}}=\frac{\omega}{\sqrt{2}}\,\frac{C^{\prime}(% \frac{q^{2}}{\omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u)}{C(\frac{q^{2}}{% \omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u)}\,\operatorname{diag}(0,1,1)
  31. C ( a , q , ξ ) = C ( a , q , ξ ) ξ . C^{\prime}(a,q,\xi)=\frac{\partial C(a,q,\xi)}{\partial\xi}.
  32. - u 0 < u < u 0 -u_{0}<u<u_{0}
  33. u = 0 u=0
  34. u = 0 u=0
  35. q ω 1 \frac{q}{\omega}\ll 1
  36. g x x cos ( q u ) 2 g_{xx}\approx\cos(qu)^{2}
  37. θ [ X ] 22 - q tan ( q u ) \theta[\vec{X}]_{22}\approx-q\,\tan(qu)
  38. q = 1 / 2 , ω = 5 q=1/2,\omega=5
  39. R 1212 3 = R 1313 3 = q 2 sin ( ω u ) 2 {{}^{3}R}_{1212}={{}^{3}R}_{1313}=q^{2}\,\sin(\omega u)^{2}
  40. R 2323 3 = - ω 2 2 C ( q 2 ω 2 , q 2 2 ω 2 , ω u ) 2 C ( q 2 ω 2 , q 2 2 ω 2 , ω u ) 2 {{}^{3}R}_{2323}=\frac{-\omega^{2}}{2}\,\frac{C^{\prime}\left(\frac{q^{2}}{% \omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u\right)^{2}}{C\left(\frac{q^{2}}% {\omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u\right)^{2}}
  41. X = e 0 \vec{X}=\vec{e}_{0}
  42. E [ X ] m ^ n ^ = q 2 sin ( ω u ) 2 diag ( 0 , 1 , 1 ) E[\vec{X}]_{\hat{m}\hat{n}}=q^{2}\,\sin(\omega u)^{2}\,\operatorname{diag}(0,1% ,1)
  43. B [ X ] m ^ n ^ = q 2 sin ( ω u ) 2 [ 0 0 0 0 0 - 1 0 1 0 ] B[\vec{X}]_{\hat{m}\hat{n}}=q^{2}\,\sin(\omega u)^{2}\begin{bmatrix}0&0&0\\ 0&0&-1\\ 0&1&0\end{bmatrix}
  44. ω ω
  45. k = e 0 + e 1 \vec{k}=\vec{e}_{0}+\vec{e}_{1}
  46. = e 0 - e 1 \vec{\ell}=\vec{e}_{0}-\vec{e}_{1}
  47. θ = 2 ω C ( q 2 ω 2 , q 2 2 ω 2 , ω u ) C ( q 2 ω 2 , q 2 2 ω 2 , ω u ) \theta=\sqrt{2}\omega\,\frac{C^{\prime}\left(\frac{q^{2}}{\omega^{2}},\frac{q^% {2}}{2\omega^{2}},\omega u\right)}{C\left(\frac{q^{2}}{\omega^{2}},\frac{q^{2}% }{2\omega^{2}},\omega u\right)}
  48. u u u\to u
  49. v v - r ˙ 2 r ( x 2 + y 2 ) v\to v-\frac{\dot{r}}{2r}\left(x^{2}+y^{2}\right)
  50. x x / r x\to x/r
  51. y y / r y\to y/r
  52. - r ¨ ( u ) r ( u ) = q sin ( ω u ) 2 -\frac{\ddot{r}(u)}{r(u)}=q\sin(\omega u)^{2}
  53. d s 2 = - q sin ( ω u ) 2 d u 2 - 2 d u d v + d x 2 + d y 2 , - < u , v , x , y < ds^{2}=-q\,\sin(\omega u)^{2}\,du^{2}-2\,du\,dv+dx^{2}+dy^{2},\qquad-\infty<u,% v,x,y<\infty
  54. e 0 = u + v 2 + { x 2 + y 2 2 ( - q 2 sin ( ω u ) 2 + ω 2 2 C ( q 2 ω 2 , q 2 2 ω 2 , ω u ) 2 C ( q 2 ω 2 , q 2 2 ω 2 , ω u ) 2 ) v } + { ω 2 C ( q 2 ω 2 , q 2 2 ω 2 , ω u ) C ( q 2 ω 2 , q 2 2 ω 2 , ω u ) ( x x + y y ) } \vec{e}_{0}=\frac{\partial_{u}+\partial_{v}}{\sqrt{2}}+\left\{\frac{x^{2}+y^{2% }}{\sqrt{2}}\left(-q^{2}\sin(\omega u)^{2}+\frac{\omega^{2}}{2}\,\frac{C^{% \prime}\left(\frac{q^{2}}{\omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u\right% )^{2}}{C\left(\frac{q^{2}}{\omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u% \right)^{2}}\right)\partial_{v}\right\}+\left\{\frac{\omega}{2}\frac{C^{\prime% }\left(\frac{q^{2}}{\omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u\right)}{C% \left(\frac{q^{2}}{\omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u\right)}\left% (x\partial_{x}+y\partial_{y}\right)\right\}
  55. X = e 0 \vec{X}=\vec{e}_{0}
  56. ω ω
  57. X u + v 2 - q tan ( q u ) ( x x + y y ) + x 2 + y 2 2 ( - q 2 sin ( ω u ) 2 + q 2 tan ( q u ) 2 ) v \vec{X}\approx\frac{\partial_{u}+\partial_{v}}{\sqrt{2}}-q\tan(qu)\left(x% \partial_{x}+y\partial_{y}\right)+\frac{x^{2}+y^{2}}{\sqrt{2}}\,\left(-q^{2}% \sin(\omega u)^{2}+q^{2}\tan(qu)^{2}\right)\partial_{v}
  58. X t - q tan ( q u 2 ) ( x x + y y ) \vec{X}\approx\partial_{t}-q\tan\left(\frac{qu}{\sqrt{2}}\right)\left(x% \partial_{x}+y\partial_{y}\right)
  59. 2 π / q 2\pi/q
  60. 2 π / ω 2\pi/\omega

Monod-Wyman-Changeux_model.html

  1. Y ¯ = L c α ( 1 + c α ) n - 1 + α ( 1 + α ) n - 1 ( 1 + α ) n + L ( 1 + c α ) n \bar{Y}=\frac{Lc\alpha(1+c\alpha)^{n-1}+\alpha(1+\alpha)^{n-1}}{(1+\alpha)^{n}% +L(1+c\alpha)^{n}}
  2. R ¯ = ( 1 + α ) n ( 1 + α ) n + L ( 1 + c α ) n \bar{R}=\frac{(1+\alpha)^{n}}{(1+\alpha)^{n}+L(1+c\alpha)^{n}}
  3. L = [ T ] 0 / [ R ] 0 L=[T]_{0}/[R]_{0}
  4. c = K R / K T c=K_{R}/K_{T}
  5. α = [ X ] / K R \alpha=[X]/K_{R}

Monopole_antenna.html

  1. i i
  2. r r
  3. P = i 2 r P=i^{2}r

Montonen–Olive_duality.html

  1. τ = θ 2 π + 4 π i g 2 . \tau=\frac{\theta}{2\pi}+\frac{4\pi i}{g^{2}}.
  2. τ τ + 1. \tau\mapsto\tau+1.
  3. τ - 1 n G τ \tau\mapsto\frac{-1}{n_{G}\tau}
  4. n G n_{G}

Mooney–Rivlin_solid.html

  1. W W\,
  2. s y m b o l B symbol{B}
  3. W = C 1 ( I ¯ 1 - 3 ) + C 2 ( I ¯ 2 - 3 ) , W=C_{1}(\overline{I}_{1}-3)+C_{2}(\overline{I}_{2}-3),\,
  4. C 1 C_{1}
  5. C 2 C_{2}
  6. I 1 I_{1}
  7. I 2 I_{2}
  8. p B ( λ ) = λ 3 - a 1 λ 2 + a 2 λ - a 3 p_{B}(\lambda)=\lambda^{3}-a_{1}\,\lambda^{2}+a_{2}\,\lambda-a_{3}\,
  9. a 1 a_{1}
  10. I 1 I_{1}
  11. a 2 a_{2}
  12. I 2 I_{2}
  13. a 3 a_{3}
  14. I 3 I_{3}
  15. I ¯ 1 = J - 2 / 3 I 1 ; I 1 = λ 1 2 + λ 2 2 + λ 3 2 ; J = det ( s y m b o l F ) I ¯ 2 = J - 4 / 3 I 2 ; I 2 = λ 1 2 λ 2 2 + λ 2 2 λ 3 2 + λ 3 2 λ 1 2 \begin{aligned}\displaystyle\bar{I}_{1}&\displaystyle=J^{-2/3}~{}I_{1}~{};~{}~% {}I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}~{};~{}~{}J=\det(symbol% {F})\\ \displaystyle\bar{I}_{2}&\displaystyle=J^{-4/3}~{}I_{2}~{};~{}~{}I_{2}=\lambda% _{1}^{2}\lambda_{2}^{2}+\lambda_{2}^{2}\lambda_{3}^{2}+\lambda_{3}^{2}\lambda_% {1}^{2}\end{aligned}
  16. s y m b o l F symbol{F}
  17. J = 1 J=1
  18. W = p , q = 0 N C p q ( I ¯ 1 - 3 ) p ( I ¯ 2 - 3 ) q + m = 1 M D m ( J - 1 ) 2 m W=\sum_{p,q=0}^{N}C_{pq}(\bar{I}_{1}-3)^{p}~{}(\bar{I}_{2}-3)^{q}+\sum_{m=1}^{% M}D_{m}~{}(J-1)^{2m}
  19. C 00 = 0 C_{00}=0
  20. C p q C_{pq}
  21. D m D_{m}
  22. N = 1 , C 01 = C 2 , C 11 = 0 , C 10 = C 1 , M = 1 N=1,C_{01}=C_{2},C_{11}=0,C_{10}=C_{1},M=1
  23. W = C 01 ( I ¯ 2 - 3 ) + C 10 ( I ¯ 1 - 3 ) + D 1 ( J - 1 ) 2 W=C_{01}~{}(\bar{I}_{2}-3)+C_{10}~{}(\bar{I}_{1}-3)+D_{1}~{}(J-1)^{2}
  24. C 01 = 0 C_{01}=0
  25. κ = 2 D 1 ; μ = 2 ( C 01 + C 10 ) \kappa=2\cdot D_{1}~{};~{}~{}\mu=2~{}(C_{01}+C_{10})
  26. κ \kappa
  27. μ \mu
  28. s y m b o l σ = 2 J [ 1 J 2 / 3 ( W I ¯ 1 + I ¯ 1 W I ¯ 2 ) s y m b o l B - 1 J 4 / 3 W I ¯ 2 s y m b o l B \cdotsymbol B ] + [ W J - 2 3 J ( I ¯ 1 W I ¯ 1 + 2 I ¯ 2 W I ¯ 2 ) ] s y m b o l 1 symbol{\sigma}=\cfrac{2}{J}\left[\cfrac{1}{J^{2/3}}\left(\cfrac{\partial{W}}{% \partial\bar{I}_{1}}+\bar{I}_{1}~{}\cfrac{\partial{W}}{\partial\bar{I}_{2}}% \right)symbol{B}-\cfrac{1}{J^{4/3}}~{}\cfrac{\partial{W}}{\partial\bar{I}_{2}}% ~{}symbol{B}\cdotsymbol{B}\right]+\left[\cfrac{\partial{W}}{\partial J}-\cfrac% {2}{3J}\left(\bar{I}_{1}~{}\cfrac{\partial{W}}{\partial\bar{I}_{1}}+2~{}\bar{I% }_{2}~{}\cfrac{\partial{W}}{\partial\bar{I}_{2}}\right)\right]~{}symbol{% \mathit{1}}
  29. W I ¯ 1 = C 1 ; W I ¯ 2 = C 2 ; W J = 2 D 1 ( J - 1 ) \cfrac{\partial{W}}{\partial\bar{I}_{1}}=C_{1}~{};~{}~{}\cfrac{\partial{W}}{% \partial\bar{I}_{2}}=C_{2}~{};~{}~{}\cfrac{\partial{W}}{\partial J}=2D_{1}(J-1)
  30. s y m b o l σ = 2 J [ 1 J 2 / 3 ( C 1 + I ¯ 1 C 2 ) s y m b o l B - 1 J 4 / 3 C 2 s y m b o l B \cdotsymbol B ] + [ 2 D 1 ( J - 1 ) - 2 3 J ( C 1 I ¯ 1 + 2 C 2 I ¯ 2 ) ] s y m b o l 1 symbol{\sigma}=\cfrac{2}{J}\left[\cfrac{1}{J^{2/3}}\left(C_{1}+\bar{I}_{1}~{}C% _{2}\right)symbol{B}-\cfrac{1}{J^{4/3}}~{}C_{2}~{}symbol{B}\cdotsymbol{B}% \right]+\left[2D_{1}(J-1)-\cfrac{2}{3J}\left(C_{1}\bar{I}_{1}+2C_{2}\bar{I}_{2% }~{}\right)\right]symbol{\mathit{1}}
  31. p := - 1 3 tr ( s y m b o l σ ) = - W J = - 2 D 1 ( J - 1 ) . p:=-\tfrac{1}{3}\,\,\text{tr}(symbol{\sigma})=-\frac{\partial W}{\partial J}=-% 2D_{1}(J-1)\,.
  32. s y m b o l σ = 1 J [ - p s y m b o l 1 + 2 J 2 / 3 ( C 1 + I ¯ 1 C 2 ) s y m b o l B - 2 J 4 / 3 C 2 s y m b o l B \cdotsymbol B - 2 3 ( C 1 I ¯ 1 + 2 C 2 I ¯ 2 ) s y m b o l 1 ] . symbol{\sigma}=\cfrac{1}{J}\left[-p~{}symbol{\mathit{1}}+\cfrac{2}{J^{2/3}}% \left(C_{1}+\bar{I}_{1}~{}C_{2}\right)symbol{B}-\cfrac{2}{J^{4/3}}~{}C_{2}~{}% symbol{B}\cdotsymbol{B}-\cfrac{2}{3}\left(C_{1}\,\bar{I}_{1}+2C_{2}\,\bar{I}_{% 2}\right)symbol{\mathit{1}}\right]\,.
  33. s y m b o l σ = 1 J [ - p s y m b o l 1 + 2 ( C 1 + I ¯ 1 C 2 ) s y m b o l B ¯ - 2 C 2 s y m b o l B ¯ s y m b o l B ¯ - 2 3 ( C 1 I ¯ 1 + 2 C 2 I ¯ 2 ) s y m b o l 1 ] where s y m b o l B ¯ = J - 2 / 3 s y m b o l B . symbol{\sigma}=\cfrac{1}{J}\left[-p~{}symbol{\mathit{1}}+2\left(C_{1}+\bar{I}_% {1}~{}C_{2}\right)\bar{symbol{B}}-2~{}C_{2}~{}\bar{symbol{B}}\cdot\bar{symbol{% B}}-\cfrac{2}{3}\left(C_{1}\,\bar{I}_{1}+2C_{2}\,\bar{I}_{2}\right)symbol{% \mathit{1}}\right]\quad\,\text{where}\quad\bar{symbol{B}}=J^{-2/3}\,symbol{B}\,.
  34. J = 1 J=1
  35. s y m b o l σ = 2 ( C 1 + I ¯ 1 C 2 ) s y m b o l B - 2 C 2 s y m b o l B \cdotsymbol B - 2 3 ( C 1 I ¯ 1 + 2 C 2 I ¯ 2 ) s y m b o l 1 . symbol{\sigma}=2\left(C_{1}+\bar{I}_{1}~{}C_{2}\right)symbol{B}-2C_{2}~{}% symbol{B}\cdotsymbol{B}-\cfrac{2}{3}\left(C_{1}\,\bar{I}_{1}+2C_{2}\,\bar{I}_{% 2}\right)symbol{\mathit{1}}\,.
  36. J = 1 J=1
  37. det ( s y m b o l B ) = det ( s y m b o l F ) det ( s y m b o l F T ) = 1 \det(symbol{B})=\det(symbol{F})\det(symbol{F}^{T})=1
  38. s y m b o l B - 1 = s y m b o l B \cdotsymbol B - I 1 s y m b o l B + I 2 s y m b o l 1 symbol{B}^{-1}=symbol{B}\cdotsymbol{B}-I_{1}~{}symbol{B}+I_{2}~{}symbol{% \mathit{1}}
  39. s y m b o l σ = - p * s y m b o l 1 + 2 C 1 s y m b o l B - 2 C 2 s y m b o l B - 1 symbol{\sigma}=-p^{*}~{}symbol{\mathit{1}}+2C_{1}~{}symbol{B}-2C_{2}~{}symbol{% B}^{-1}
  40. p * := 2 3 ( C 1 I 1 - C 2 I 2 ) . p^{*}:=\tfrac{2}{3}(C_{1}~{}I_{1}-C_{2}~{}I_{2}).\,
  41. σ 11 - σ 33 = λ 1 W λ 1 - λ 3 W λ 3 ; σ 22 - σ 33 = λ 2 W λ 2 - λ 3 W λ 3 \sigma_{11}-\sigma_{33}=\lambda_{1}~{}\cfrac{\partial{W}}{\partial\lambda_{1}}% -\lambda_{3}~{}\cfrac{\partial{W}}{\partial\lambda_{3}}~{};~{}~{}\sigma_{22}-% \sigma_{33}=\lambda_{2}~{}\cfrac{\partial{W}}{\partial\lambda_{2}}-\lambda_{3}% ~{}\cfrac{\partial{W}}{\partial\lambda_{3}}
  42. W = C 1 ( λ 1 2 + λ 2 2 + λ 3 2 - 3 ) + C 2 ( λ 1 2 λ 2 2 + λ 2 2 λ 3 2 + λ 3 2 λ 1 2 - 3 ) ; λ 1 λ 2 λ 3 = 1 W=C_{1}(\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-3)+C_{2}(\lambda_{1}^{% 2}\lambda_{2}^{2}+\lambda_{2}^{2}\lambda_{3}^{2}+\lambda_{3}^{2}\lambda_{1}^{2% }-3)~{};~{}~{}\lambda_{1}\lambda_{2}\lambda_{3}=1
  43. λ 1 W λ 1 = 2 C 1 λ 1 2 + 2 C 2 λ 1 2 ( λ 2 2 + λ 3 2 ) ; λ 2 W λ 2 = 2 C 1 λ 2 2 + 2 C 2 λ 2 2 ( λ 1 2 + λ 3 2 ) ; λ 3 W λ 3 = 2 C 1 λ 3 2 + 2 C 2 λ 3 2 ( λ 1 2 + λ 2 2 ) \lambda_{1}\cfrac{\partial{W}}{\partial\lambda_{1}}=2C_{1}\lambda_{1}^{2}+2C_{% 2}\lambda_{1}^{2}(\lambda_{2}^{2}+\lambda_{3}^{2})~{};~{}~{}\lambda_{2}\cfrac{% \partial{W}}{\partial\lambda_{2}}=2C_{1}\lambda_{2}^{2}+2C_{2}\lambda_{2}^{2}(% \lambda_{1}^{2}+\lambda_{3}^{2})~{};~{}~{}\lambda_{3}\cfrac{\partial{W}}{% \partial\lambda_{3}}=2C_{1}\lambda_{3}^{2}+2C_{2}\lambda_{3}^{2}(\lambda_{1}^{% 2}+\lambda_{2}^{2})
  44. λ 1 λ 2 λ 3 = 1 \lambda_{1}\lambda_{2}\lambda_{3}=1
  45. λ 1 W λ 1 = 2 C 1 λ 1 2 + 2 C 2 ( 1 λ 3 2 + 1 λ 2 2 ) ; λ 2 W λ 2 = 2 C 1 λ 2 2 + 2 C 2 ( 1 λ 3 2 + 1 λ 1 2 ) λ 3 W λ 3 = 2 C 1 λ 3 2 + 2 C 2 ( 1 λ 2 2 + 1 λ 1 2 ) \begin{aligned}\displaystyle\lambda_{1}\cfrac{\partial{W}}{\partial\lambda_{1}% }&\displaystyle=2C_{1}\lambda_{1}^{2}+2C_{2}\left(\cfrac{1}{\lambda_{3}^{2}}+% \cfrac{1}{\lambda_{2}^{2}}\right)~{};~{}~{}\lambda_{2}\cfrac{\partial{W}}{% \partial\lambda_{2}}=2C_{1}\lambda_{2}^{2}+2C_{2}\left(\cfrac{1}{\lambda_{3}^{% 2}}+\cfrac{1}{\lambda_{1}^{2}}\right)\\ \displaystyle\lambda_{3}\cfrac{\partial{W}}{\partial\lambda_{3}}&\displaystyle% =2C_{1}\lambda_{3}^{2}+2C_{2}\left(\cfrac{1}{\lambda_{2}^{2}}+\cfrac{1}{% \lambda_{1}^{2}}\right)\end{aligned}
  46. σ 11 - σ 33 = 2 C 1 ( λ 1 2 - λ 3 2 ) - 2 C 2 ( 1 λ 1 2 - 1 λ 3 2 ) ; σ 22 - σ 33 = 2 C 1 ( λ 2 2 - λ 3 2 ) - 2 C 2 ( 1 λ 2 2 - 1 λ 3 2 ) \sigma_{11}-\sigma_{33}=2C_{1}(\lambda_{1}^{2}-\lambda_{3}^{2})-2C_{2}\left(% \cfrac{1}{\lambda_{1}^{2}}-\cfrac{1}{\lambda_{3}^{2}}\right)~{};~{}~{}\sigma_{% 22}-\sigma_{33}=2C_{1}(\lambda_{2}^{2}-\lambda_{3}^{2})-2C_{2}\left(\cfrac{1}{% \lambda_{2}^{2}}-\cfrac{1}{\lambda_{3}^{2}}\right)
  47. λ 1 = λ \lambda_{1}=\lambda\,
  48. λ 2 = λ 3 = 1 / λ \lambda_{2}=\lambda_{3}=1/\sqrt{\lambda}
  49. σ 11 - σ 33 \displaystyle\sigma_{11}-\sigma_{33}
  50. σ 22 = σ 33 = 0 \sigma_{22}=\sigma_{33}=0
  51. σ 11 = ( 2 C 1 + 2 C 2 λ ) ( λ 2 - 1 λ ) \sigma_{11}=\left(2C_{1}+\cfrac{2C_{2}}{\lambda}\right)\left(\lambda^{2}-% \cfrac{1}{\lambda}\right)
  52. s y m b o l T symbol{T}
  53. α \alpha
  54. T 11 = ( 2 C 1 + 2 C 2 α ) ( α 2 - α - 1 ) T_{11}=\left(2C_{1}+\frac{2C_{2}}{\alpha}\right)\left(\alpha^{2}-\alpha^{-1}\right)
  55. T 11 eng = T 11 α 2 α 3 = T 11 α T_{11}^{\mathrm{eng}}=T_{11}\alpha_{2}\alpha_{3}=\cfrac{T_{11}}{\alpha}
  56. T 11 eng = ( 2 C 1 + 2 C 2 α ) ( α - α - 2 ) T_{11}^{\mathrm{eng}}=\left(2C_{1}+\frac{2C_{2}}{\alpha}\right)\left(\alpha-% \alpha^{-2}\right)
  57. T 11 * := T 11 eng α - α - 2 ; β := 1 α T^{*}_{11}:=\cfrac{T_{11}^{\mathrm{eng}}}{\alpha-\alpha^{-2}}~{};~{}~{}\beta:=% \cfrac{1}{\alpha}
  58. T 11 * = 2 C 1 + 2 C 2 β . T^{*}_{11}=2C_{1}+2C_{2}\beta~{}.
  59. T 11 * T^{*}_{11}
  60. β \beta
  61. C 2 C_{2}
  62. T 11 * T^{*}_{11}
  63. C 1 C_{1}
  64. λ 1 = λ 2 = λ \lambda_{1}=\lambda_{2}=\lambda
  65. λ 3 = 1 / λ 2 \lambda_{3}=1/\lambda^{2}
  66. σ 11 - σ 33 = σ 22 - σ 33 = 2 C 1 ( λ 2 - 1 λ 4 ) - 2 C 2 ( 1 λ 2 - λ 4 ) \sigma_{11}-\sigma_{33}=\sigma_{22}-\sigma_{33}=2C_{1}\left(\lambda^{2}-\cfrac% {1}{\lambda^{4}}\right)-2C_{2}\left(\cfrac{1}{\lambda^{2}}-\lambda^{4}\right)
  67. λ 1 = λ ; λ 2 = 1 λ ; λ 3 = 1 \lambda_{1}=\lambda~{};~{}~{}\lambda_{2}=\cfrac{1}{\lambda}~{};~{}~{}\lambda_{% 3}=1
  68. σ 11 - σ 33 = 2 C 1 ( λ 2 - 1 ) - 2 C 2 ( 1 λ 2 - 1 ) ; σ 22 - σ 33 = 2 C 1 ( 1 λ 2 - 1 ) - 2 C 2 ( λ 2 - 1 ) \sigma_{11}-\sigma_{33}=2C_{1}(\lambda^{2}-1)-2C_{2}\left(\cfrac{1}{\lambda^{2% }}-1\right)~{};~{}~{}\sigma_{22}-\sigma_{33}=2C_{1}\left(\cfrac{1}{\lambda^{2}% }-1\right)-2C_{2}(\lambda^{2}-1)
  69. σ 11 - σ 22 = 2 ( C 1 + C 2 ) ( λ 2 - 1 λ 2 ) \sigma_{11}-\sigma_{22}=2(C_{1}+C_{2})\left(\lambda^{2}-\cfrac{1}{\lambda^{2}}\right)
  70. I 1 = λ 1 2 + λ 2 2 + λ 3 2 = λ 2 + 1 λ 2 + 1 ; I 2 = 1 λ 1 2 + 1 λ 2 2 + 1 λ 3 2 = 1 λ 2 + λ 2 + 1 I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}=\lambda^{2}+\cfrac{1}{% \lambda^{2}}+1~{};~{}~{}I_{2}=\cfrac{1}{\lambda_{1}^{2}}+\cfrac{1}{\lambda_{2}% ^{2}}+\cfrac{1}{\lambda_{3}^{2}}=\cfrac{1}{\lambda^{2}}+\lambda^{2}+1
  71. I 1 = I 2 I_{1}=I_{2}
  72. s y m b o l F = s y m b o l 1 + γ 𝐞 1 𝐞 2 symbol{F}=symbol{1}+\gamma~{}\mathbf{e}_{1}\otimes\mathbf{e}_{2}
  73. 𝐞 1 , 𝐞 2 \mathbf{e}_{1},\mathbf{e}_{2}
  74. γ = λ - 1 λ ; λ 1 = λ ; λ 2 = 1 λ ; λ 3 = 1 \gamma=\lambda-\cfrac{1}{\lambda}~{};~{}~{}\lambda_{1}=\lambda~{};~{}~{}% \lambda_{2}=\cfrac{1}{\lambda}~{};~{}~{}\lambda_{3}=1
  75. s y m b o l F = [ 1 γ 0 0 1 0 0 0 1 ] ; s y m b o l B = s y m b o l F \cdotsymbol F T = [ 1 + γ 2 γ 0 γ 1 0 0 0 1 ] symbol{F}=\begin{bmatrix}1&\gamma&0\\ 0&1&0\\ 0&0&1\end{bmatrix}~{};~{}~{}symbol{B}=symbol{F}\cdotsymbol{F}^{T}=\begin{% bmatrix}1+\gamma^{2}&\gamma&0\\ \gamma&1&0\\ 0&0&1\end{bmatrix}
  76. s y m b o l B - 1 = [ 1 - γ 0 - γ 1 + γ 2 0 0 0 1 ] symbol{B}^{-1}=\begin{bmatrix}1&-\gamma&0\\ -\gamma&1+\gamma^{2}&0\\ 0&0&1\end{bmatrix}
  77. s y m b o l σ = [ - p * + 2 ( C 1 - C 2 ) + 2 C 1 γ 2 2 ( C 1 + C 2 ) γ 0 2 ( C 1 + C 2 ) γ - p * + 2 ( C 1 - C 2 ) - 2 C 2 γ 2 0 0 0 - p * + 2 ( C 1 - C 2 ) ] symbol{\sigma}=\begin{bmatrix}-p^{*}+2(C_{1}-C_{2})+2C_{1}\gamma^{2}&2(C_{1}+C% _{2})\gamma&0\\ 2(C_{1}+C_{2})\gamma&-p^{*}+2(C_{1}-C_{2})-2C_{2}\gamma^{2}&0\\ 0&0&-p^{*}+2(C_{1}-C_{2})\end{bmatrix}
  78. μ = 2 ( C 1 + C 2 ) \mu=2(C_{1}+C_{2})
  79. μ \mu
  80. C 1 , C 2 C_{1},C_{2}

Moore_space_(algebraic_topology).html

  1. H n ( X ) G H_{n}(X)\cong G
  2. H ~ i ( X ) 0 \tilde{H}_{i}(X)\cong 0
  3. H ~ i ( X ) \tilde{H}_{i}(X)
  4. S n S^{n}
  5. \mathbb{Z}
  6. n 1 n\geq 1
  7. 2 \mathbb{RP}^{2}
  8. / 2 \mathbb{Z}/2\mathbb{Z}

Mordell_curve.html

  1. m 2 + 2 = n 3 m^{2}+2=n^{3}
  2. n = 3 , m = ± 5 n=3,m=\pm 5

Morin_surface.html

  1. ( A 1 , B , C , D , A 3 ) = ( A 1 , D ′′ , C ′′ , B ′′ , A 3 ) (A_{1},B,C,D,A_{3})=(A_{1},D^{\prime\prime},C^{\prime\prime},B^{\prime\prime},% A_{3})
  2. ( A 2 , E , F , G , A 3 ) = ( A 2 , H , I , J , A 3 ) (A_{2},E,F,G,A_{3})=(A_{2},H^{\prime},I^{\prime},J^{\prime},A_{3})
  3. ( A 1 , H , I , J , A 2 ) = ( A 1 , E ′′′ , F ′′′ , G ′′′ , A 2 ) (A_{1},H,I,J,A_{2})=(A_{1},E^{\prime\prime\prime},F^{\prime\prime\prime},G^{% \prime\prime\prime},A_{2})
  4. ( A 2 , B , C , D , A 0 ) = ( A 2 , D ′′′ , C ′′′ , B ′′′ , A 0 ) (A_{2},B^{\prime},C^{\prime},D^{\prime},A_{0})=(A_{2},D^{\prime\prime\prime},C% ^{\prime\prime\prime},B^{\prime\prime\prime},A_{0})
  5. ( A 3 , E , F , G , A 0 ) = ( A 3 , H ′′ , I ′′ , J ′′ , A 0 ) (A_{3},E^{\prime},F^{\prime},G^{\prime},A_{0})=(A_{3},H^{\prime\prime},I^{% \prime\prime},J^{\prime\prime},A_{0})
  6. ( A 0 , E ′′ , F ′′ , G ′′ , A 1 ) = ( A 0 , H ′′′ , I ′′′ , J ′′′ , A 1 ) . (A_{0},E^{\prime\prime},F^{\prime\prime},G^{\prime\prime},A_{1})=(A_{0},H^{% \prime\prime\prime},I^{\prime\prime\prime},J^{\prime\prime\prime},A_{1}).

Morse_homology.html

  1. C * ( M , ( f , g ) ) C_{*}(M,(f,g))
  2. C * ( M , ( f 0 , g 0 ) ) C_{*}(M,(f_{0},g_{0}))
  3. C * ( M , ( f 1 , g 1 ) ) C_{*}(M,(f_{1},g_{1}))

Morse–Kelley_set_theory.html

  1. M x , \ Mx,
  2. W ( x W ) ; \exists W(x\in W);
  3. \varnothing
  4. x ( x ) ; \forall x(x\not\in\varnothing);
  5. x ( M x x V ) . \forall x(Mx\to x\in V).
  6. X Y ( z ( z X z Y ) X = Y ) . \forall X\,\forall Y\,(\forall z\,(z\in X\leftrightarrow z\in Y)\rightarrow X=% Y).
  7. A [ A b ( b A and c ( c b c A ) ) ] . \forall A[A\not=\varnothing\rightarrow\exists b(b\in A\and\forall c(c\in b% \rightarrow c\not\in A))].
  8. Y = { x ϕ ( x ) } Y=\{x\mid\phi(x)\}
  9. ϕ ( x ) \phi(x)
  10. W 1 W n Y x [ x Y ( ϕ ( x , W 1 , W n ) and M x ) ] . \forall W_{1}...W_{n}\exists Y\forall x[x\in Y\leftrightarrow(\phi(x,W_{1},...% W_{n})\and Mx)].
  11. z = { x , y } z=\{x,y\}
  12. x y [ ( M x and M y ) z ( M z and s [ s z ( s = x s = y ) ] ) ] . \forall x\,\forall y\,[(Mx\and My)\rightarrow\exists z\,(Mz\and\forall s\,[s% \in z\leftrightarrow(s=x\,\,s=y)])].
  13. x , y \langle x,y\rangle
  14. { { x } , { x , y } } \ \{\{x\},\{x,y\}\}
  15. C [ ¬ M C F ( x [ M x ! s ( s C and x , s F ) ] and \forall C[\lnot MC\leftrightarrow\exists F(\forall x[Mx\rightarrow\exists!s(s% \in C\and\langle x,s\rangle\in F)]\and
  16. x y s [ ( x , s F and y , s F ) x = y ] ) ] . \forall x\forall y\forall s[(\langle x,s\rangle\in F\and\langle y,s\rangle\in F% )\rightarrow x=y])].
  17. a p [ ( M a and x [ x p y ( y x y a ) ] ) M p ] . \forall a\,\forall p\,[(Ma\and\forall x\,[x\in p\leftrightarrow\forall y\,(y% \in x\rightarrow y\in a)])\rightarrow Mp].
  18. s = a s=\bigcup a
  19. a s [ ( M a and x [ x s y ( x y and y a ) ] ) M s ] . \forall a\,\forall s\,[(Ma\and\forall x\,[x\in s\leftrightarrow\exists y\,(x% \in y\and y\in a)])\rightarrow Ms].
  20. x { x } . x\cup\{x\}.
  21. y [ M y and y and z ( z y x [ x y and w ( w x [ w = z w z ] ) ] ) ] . \exists y[My\and\varnothing\in y\and\forall z(z\in y\rightarrow\exists x[x\in y% \and\forall w(w\in x\leftrightarrow[w=zw\in z])])].
  22. ω , \omega,
  23. ω . \omega.
  24. ω \omega
  25. { x : A ( x ) } , \ \{x:A(x)\},
  26. x y x\in y
  27. z x z\in x
  28. z y . z\in y.
  29. β \beta
  30. β { α : A } \beta\in\{\alpha:A\}
  31. β \beta
  32. B , B,
  33. z x z\subseteq x
  34. z y . z\in y.
  35. x y x\cup y
  36. { x } \{x\}
  37. { x , y } \{x,y\}
  38. x \bigcup x
  39. x x\neq\varnothing
  40. x y = . x\cap y=\varnothing.
  41. y \varnothing\in y
  42. x { x } y x\cup\{x\}\in y
  43. x y . x\in y.
  44. . \varnothing.
  45. \varnothing
  46. c ( x ) x c(x)\in x
  47. V - { } . V-\{\varnothing\}.

Morton's_theorem.html

  1. 𝔼 [ Charles | folding ] = 0 \mathbb{E}\left[\mbox{ Charles }~{}|\mbox{ folding }~{}\right]=0
  2. 𝔼 [ Charles | calling ] = 4 42 ( P + 2 ) - 38 42 1 \mathbb{E}\left[\mbox{ Charles }~{}|\mbox{ calling }~{}\right]=\frac{4}{42}% \cdot(P+2)-\frac{38}{42}\cdot 1
  3. 𝔼 [ Charles | folding ] = 𝔼 [ Charles | calling ] \mathbb{E}\left[\mbox{ Charles }~{}|\mbox{ folding }~{}\right]=\mathbb{E}\left% [\mbox{ Charles }~{}|\mbox{ calling }~{}\right]
  4. P = 7.5 big bets \Rightarrow P=7.5\mbox{ big bets }~{}
  5. 𝔼 [ Arnold | Charles folds ] = 42 - 9 42 ( P + 2 ) = 33 42 ( P + 2 ) \mathbb{E}\left[\mbox{ Arnold }~{}|\mbox{ Charles folds }~{}\right]=\frac{42-9% }{42}\cdot(P+2)=\frac{33}{42}\cdot(P+2)
  6. 𝔼 [ Arnold | Charles calls ] = 42 - 9 - 4 42 ( P + 3 ) = 29 42 ( P + 3 ) \mathbb{E}\left[\mbox{ Arnold }~{}|\mbox{ Charles calls }~{}\right]=\frac{42-9% -4}{42}\cdot(P+3)=\frac{29}{42}\cdot(P+3)
  7. 𝔼 [ Arnold | Charles calls ] = 𝔼 [ Arnold | Charles folds ] \mathbb{E}\left[\mbox{ Arnold }~{}|\mbox{ Charles calls }~{}\right]=\mathbb{E}% \left[\mbox{ Arnold }~{}|\mbox{ Charles folds }~{}\right]
  8. P = 5.25 big bets \Rightarrow P=5.25\mbox{ big bets }~{}

Motor_variable.html

  1. f ( z ) = u ( z ) + j v ( z ) , z = x + j y , x , y R , j 2 = + 1 , u ( z ) , v ( z ) R . f(z)=u(z)+j\ v(z),\ z=x+jy,\ x,y\in R,\quad j^{2}=+1,\quad u(z),v(z)\in R.
  2. { z = x + j y : x , y R } \{z=x+jy:x,y\in R\}
  3. u = exp ( a j ) = cosh a + j sinh a u=\exp(aj)=\cosh a+j\sinh a
  4. f ( z ) = u z + c f(z)=uz+c
  5. f ( z ) = z 2 f(z)=z^{2}
  6. f ( - 1 ) = f ( j ) = f ( - j ) = 1. f(-1)=f(j)=f(-j)=1.
  7. U 1 = { z D : y < x } U_{1}=\{z\in D:\mid y\mid<x\}
  8. z z * = 1 zz^{*}=1
  9. x 2 - y 2 = 1 x^{2}-y^{2}=1
  10. f ( z ) = 1 / z = z * / z 2 where z 2 = z z * f(z)=1/z=z^{*}/\mid z\mid^{2}\,\text{where}\mid z\mid^{2}=zz^{*}
  11. f ( z ) = a z + b c z + d . f(z)=\frac{az+b}{cz+d}.
  12. f ( z ) = 1 z + 1 / 2 , f : U 1 T f(z)=\frac{1}{z+1/2},\quad f:U_{1}\to T
  13. e x = n = 0 x n n ! = n = 0 x 2 n ( 2 n ) ! + n = 0 x 2 n + 1 ( 2 n + 1 ) ! = cosh x + sinh x e^{x}=\sum_{n=0}^{\infty}{x^{n}\over n!}=\sum_{n=0}^{\infty}\frac{x^{2n}}{(2n)% !}+\sum_{n=0}^{\infty}\frac{x^{2n+1}}{(2n+1)!}=\cosh x+\sinh x
  14. e z = e a ( cosh b + j sinh b ) e^{z}=e^{a}(\cosh b+j\ \sinh b)
  15. 0 = ( x - j y ) ( u + j v ) 0\ =\ ({\partial\over\partial x}-j{\partial\over\partial y})(u+jv)
  16. = u x - j 2 v y + j ( v x - u y ) . =\ u_{x}-j^{2}v_{y}+j(v_{x}-u_{y}).
  17. u x = v y , v x = u y . u_{x}=v_{y},\quad v_{x}=u_{y}.
  18. w = α z + β γ z + δ where α , β , γ , δ w=\frac{\alpha z+\beta}{\gamma z+\delta}\quad\,\text{where}\quad\alpha,\beta,% \gamma,\delta
  19. ( z - 1 ) ( z + 1 ) = z 2 - 1 = ( z - j ) ( z + j ) (z-1)(z+1)=z^{2}-1=(z-j)(z+j)
  20. H = { ( x , y , z ) : z 2 + x 2 - y 2 = 1 } . H=\{(x,y,z):z^{2}+x^{2}-y^{2}=1\}.
  21. L = { ( s x , s y , 1 - s ) : s R } L=\{(sx,sy,1-s):s\in R\}
  22. ( 1 - s ) 2 + ( s x ) 2 - ( s y ) 2 = 1 , so that s = 2 1 + x 2 - y 2 . (1-s)^{2}+(sx)^{2}-(sy)^{2}=1,\,\text{ so that}\quad s=\frac{2}{1+x^{2}-y^{2}}.
  23. y 2 > 1 + x 2 , y^{2}>1+x^{2},
  24. { ( w , x , y , z ) P 3 R : z 2 + x 2 = y 2 + w 2 } , \{(w,x,y,z)\in P^{3}R:z^{2}+x^{2}=y^{2}+w^{2}\},

Mott_insulator.html

  1. n = 0 n=0
  2. n = 2 n=2
  3. n = 1 n=1
  4. 2 e 2e
  5. e < 0 e<0
  6. n = 1 n=1

Moving_parts.html

  1. a a
  2. n n
  3. a 2 d m \int a^{2}dm
  4. k = 0 n m k × a k 2 \sum_{k=0}^{n}m_{k}\times a_{k}^{2}
  5. I I
  6. 1 2 I ω 2 \frac{1}{2}I\omega^{2}
  7. ω \omega
  8. 1 2 m v 2 \frac{1}{2}mv^{2}
  9. m m
  10. v v
  11. 1 2 I ω 2 + 1 2 m v 2 \frac{1}{2}I\omega^{2}+\frac{1}{2}mv^{2}

Mueller_calculus.html

  1. S i \vec{S}_{i}
  2. S o \vec{S}_{o}
  3. S o = M S i . \vec{S}_{o}=\mathrm{M}\vec{S}_{i}\ .
  4. S o = M 3 ( M 2 ( M 1 S i ) ) \vec{S}_{o}=\mathrm{M}_{3}\big(\mathrm{M}_{2}(\mathrm{M}_{1}\vec{S}_{i})\big)
  5. S o = M 3 M 2 M 1 S i . \vec{S}_{o}=\mathrm{M}_{3}\mathrm{M}_{2}\mathrm{M}_{1}\vec{S}_{i}\ .
  6. M 3 M 2 M 1 S i M 1 M 2 M 3 S i . \mathrm{M}_{3}\mathrm{M}_{2}\mathrm{M}_{1}\vec{S}_{i}\neq\mathrm{M}_{1}\mathrm% {M}_{2}\mathrm{M}_{3}\vec{S}_{i}\ .
  7. M = A ( J J * ) A - 1 \mathrm{M=A(J\otimes J^{*})A^{-1}}
  8. A = ( 1 0 0 1 1 0 0 - 1 0 1 1 0 0 i - i 0 ) \mathrm{A}=\begin{pmatrix}1&0&0&1\\ 1&0&0&-1\\ 0&1&1&0\\ 0&i&-i&0\\ \end{pmatrix}
  9. 1 2 ( 1 1 0 0 1 1 0 0 0 0 0 0 0 0 0 0 ) {1\over 2}\begin{pmatrix}1&1&0&0\\ 1&1&0&0\\ 0&0&0&0\\ 0&0&0&0\end{pmatrix}\quad
  10. 1 2 ( 1 - 1 0 0 - 1 1 0 0 0 0 0 0 0 0 0 0 ) {1\over 2}\begin{pmatrix}1&-1&0&0\\ -1&1&0&0\\ 0&0&0&0\\ 0&0&0&0\end{pmatrix}\quad
  11. 1 2 ( 1 0 1 0 0 0 0 0 1 0 1 0 0 0 0 0 ) {1\over 2}\begin{pmatrix}1&0&1&0\\ 0&0&0&0\\ 1&0&1&0\\ 0&0&0&0\end{pmatrix}\quad
  12. 1 2 ( 1 0 - 1 0 0 0 0 0 - 1 0 1 0 0 0 0 0 ) {1\over 2}\begin{pmatrix}1&0&-1&0\\ 0&0&0&0\\ -1&0&1&0\\ 0&0&0&0\end{pmatrix}\quad
  13. ( 1 0 0 0 0 1 0 0 0 0 0 - 1 0 0 1 0 ) \begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&0&-1\\ 0&0&1&0\end{pmatrix}\quad
  14. ( 1 0 0 0 0 1 0 0 0 0 0 1 0 0 - 1 0 ) \begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\ 0&0&-1&0\end{pmatrix}\quad
  15. ( 1 0 0 0 0 1 0 0 0 0 - 1 0 0 0 0 - 1 ) \begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&-1&0\\ 0&0&0&-1\end{pmatrix}\quad
  16. 1 4 ( 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) {1\over 4}\begin{pmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}\quad

Mulliken_population_analysis.html

  1. 𝐃 μ ν = 𝟐 i 𝐂 μ 𝐢 𝐂 ν 𝐢 * \mathbf{D_{\mu\nu}}=\mathbf{2}\sum_{i}\mathbf{C_{\mu i}}\mathbf{C_{\nu i}^{*}}
  2. 𝐏 \mathbf{P}
  3. 𝐏 μ ν = 𝐃 μ ν 𝐒 μ ν \mathbf{P_{\mu\nu}}=\mathbf{D_{\mu\nu}}\mathbf{S_{\mu\nu}}
  4. 𝐒 \mathbf{S}
  5. 𝐏 ν μ \mathbf{P_{\nu\mu}}
  6. μ \mathbf{\mu}
  7. ν \mathbf{\nu}
  8. 𝐆𝐎𝐏 ν \mathbf{GOP_{\nu}}
  9. 𝐆𝐀𝐏 𝐀 \mathbf{GAP_{A}}
  10. 𝐆𝐎𝐏 ν \mathbf{GOP_{\nu}}
  11. ν \mathbf{\nu}
  12. 𝐐 𝐀 \mathbf{Q_{A}}
  13. 𝐙 𝐀 \mathbf{Z_{A}}
  14. 𝐐 𝐀 = 𝐙 𝐀 - 𝐆𝐀𝐏 𝐀 \mathbf{Q_{A}}=\mathbf{Z_{A}}-\mathbf{GAP_{A}}
  15. 𝐏 μ ν \mathbf{P_{\mu\nu}}
  16. 𝐏 μ μ \mathbf{P_{\mu\mu}}
  17. 𝐏 ν ν \mathbf{P_{\nu\nu}}

Multi-armed_bandit.html

  1. B = { R 1 , , R K } B=\{R_{1},\dots,R_{K}\}
  2. K K\in\mathbb{Z}
  3. μ 1 , , μ K \mu_{1},\dots,\mu_{K}
  4. H H
  5. ρ \rho
  6. T T
  7. ρ = T μ * - t = 1 T r ^ t \rho=T\mu^{*}-\sum_{t=1}^{T}\widehat{r}_{t}
  8. μ * \mu^{*}
  9. μ * = max k { μ k } \mu^{*}=\max_{k}\{\mu_{k}\}
  10. r ^ t \widehat{r}_{t}
  11. ρ / T \rho/T
  12. p p
  13. 1 - ϵ 1-\epsilon
  14. ϵ \epsilon
  15. ϵ = 0.1 \epsilon=0.1
  16. N N
  17. ϵ N \epsilon N
  18. ( 1 - ϵ ) N (1-\epsilon)N
  19. ϵ \epsilon
  20. ϵ \epsilon
  21. K K
  22. K K

Multi-configurational_self-consistent_field.html

  1. φ i = N i ( χ i A ± χ i B ) , \varphi_{i}=N_{i}(\chi_{iA}\pm\chi_{iB}),\,
  2. Φ 1 = φ 1 ( 𝐫 1 ) φ 1 ( 𝐫 2 ) Θ 2 , 0 , \Phi_{1}=\varphi_{1}(\mathbf{r}_{1})\varphi_{1}(\mathbf{r}_{2})\Theta_{2,0},
  3. Φ 1 = N 1 2 [ 1 s A ( 𝐫 1 ) 1 s A ( 𝐫 2 ) + 1 s A ( 𝐫 1 ) 1 s B ( 𝐫 2 ) + 1 s B ( 𝐫 1 ) 1 s A ( 𝐫 2 ) + 1 s B ( 𝐫 1 ) 1 s B ( 𝐫 2 ) ] Θ 2 , 0 . \Phi_{1}=N_{1}^{2}\left[1s_{A}(\mathbf{r}_{1})1s_{A}(\mathbf{r}_{2})+1s_{A}(% \mathbf{r}_{1})1s_{B}(\mathbf{r}_{2})+1s_{B}(\mathbf{r}_{1})1s_{A}(\mathbf{r}_% {2})+1s_{B}(\mathbf{r}_{1})1s_{B}(\mathbf{r}_{2})\right]\Theta_{2,0}.
  4. Ψ 1 = C Ion Φ Ion + C Cov Φ Cov , \Psi_{1}=C\text{Ion}\Phi\text{Ion}+C\text{Cov}\Phi\text{Cov},
  5. ϕ 2 = N 2 ( 1 s A - 1 s B ) , \phi_{2}=N_{2}(1s_{A}-1s_{B}),\,
  6. Ψ M C = C 1 Φ 1 + C 2 Φ 2 , \Psi_{MC}=C_{1}\Phi_{1}+C_{2}\Phi_{2},

Multicategory.html

  1. ( X i ) i n (X_{i})_{i\in n}
  2. n n\in\mathbb{N}
  3. ( X i ) i n (X_{i})_{i\in n}
  4. ( ( X i j ) i n j ) j m ((X_{ij})_{i\in n_{j}})_{j\in m}
  5. ( Y i ) i m (Y_{i})_{i\in m}
  6. j m j\in m
  7. ( X i j ) i n j (X_{ij})_{i\in n_{j}}
  8. ( Y i ) i m (Y_{i})_{i\in m}
  9. g ( f i ) i m g(f_{i})_{i\in m}
  10. ( X i j ) i n j , j m (X_{ij})_{i\in n_{j},j\in m}
  11. i m i\in m
  12. X 0 i = Y i X_{0i}=Y_{i}
  13. g ( f i ) i m = g g(f_{i})_{i\in m}=g
  14. k m k\in m
  15. j n k j\in n_{k}
  16. e j k e_{jk}
  17. ( W i j k ) i o j k (W_{ijk})_{i\in o_{jk}}
  18. X j k X_{jk}
  19. g ( f j ( e i j ) i n j ) j m = g ( f i ) i m ( e i j ) i n j , j m g\left(f_{j}(e_{ij})_{i\in n_{j}}\right)_{j\in m}=g(f_{i})_{i\in m}(e_{ij})_{i% \in n_{j},j\in m}
  20. ( W i j k ) i o j k , j n k , k m (W_{ijk})_{i\in o_{jk},j\in n_{k},k\in m}

Multiferroics.html

  1. P i = α i j H j + β i j k H j H k + P_{i}=\sum\alpha_{ij}H_{j}+\sum\beta_{ijk}H_{j}H_{k}+\ldots
  2. M i = α i j E j + β i j k E j E k + M_{i}=\sum\alpha_{ij}E_{j}+\sum\beta_{ijk}E_{j}E_{k}+\ldots

Multilayer_perceptron.html

  1. y ( v i ) = tanh ( v i ) and y ( v i ) = ( 1 + e - v i ) - 1 y(v_{i})=\tanh(v_{i})~{}~{}\textrm{and}~{}~{}y(v_{i})=(1+e^{-v_{i}})^{-1}
  2. y i y_{i}
  3. i i
  4. v i v_{i}
  5. w i j w_{ij}
  6. w i j w_{ij}
  7. j j
  8. n n
  9. e j ( n ) = d j ( n ) - y j ( n ) e_{j}(n)=d_{j}(n)-y_{j}(n)
  10. d d
  11. y y
  12. ( n ) = 1 2 j e j 2 ( n ) \mathcal{E}(n)=\frac{1}{2}\sum_{j}e_{j}^{2}(n)
  13. Δ w j i ( n ) = - η ( n ) v j ( n ) y i ( n ) \Delta w_{ji}(n)=-\eta\frac{\partial\mathcal{E}(n)}{\partial v_{j}(n)}y_{i}(n)
  14. y i y_{i}
  15. η \eta
  16. v j v_{j}
  17. - ( n ) v j ( n ) = e j ( n ) ϕ ( v j ( n ) ) -\frac{\partial\mathcal{E}(n)}{\partial v_{j}(n)}=e_{j}(n)\phi^{\prime}(v_{j}(% n))
  18. ϕ \phi^{\prime}
  19. - ( n ) v j ( n ) = ϕ ( v j ( n ) ) k - ( n ) v k ( n ) w k j ( n ) -\frac{\partial\mathcal{E}(n)}{\partial v_{j}(n)}=\phi^{\prime}(v_{j}(n))\sum_% {k}-\frac{\partial\mathcal{E}(n)}{\partial v_{k}(n)}w_{kj}(n)
  20. k k

Multiple_integral.html

  1. 𝐃 f ( x 1 , x 2 , , x n ) d x 1 d x n \int\cdots\int_{\mathbf{D}}\;f(x_{1},x_{2},\ldots,x_{n})\;dx_{1}\!\cdots dx_{n}
  2. T = [ a 1 , b 1 ) × [ a 2 , b 2 ) × × [ a n , b n ) 𝐑 n . T=\left[a_{1},b_{1}\right)\times\left[a_{2},b_{2}\right)\times\cdots\times% \left[a_{n},b_{n}\right)\subseteq\mathbf{R}^{n}.
  3. C = I 1 × I 2 × × I n C=I_{1}\times I_{2}\times\cdots\times I_{n}
  4. T = C 1 C 2 C m T=C_{1}\cup C_{2}\cup\cdots\cup C_{m}
  5. k = 1 m f ( P k ) m ( C k ) \sum_{k=1}^{m}f(P_{k})\,\operatorname{m}(C_{k})
  6. S = lim δ 0 k = 1 m f ( P k ) m ( C k ) S=\lim_{\delta\to 0}\sum_{k=1}^{m}f(P_{k})\,\operatorname{m}\,(C_{k})
  7. T f ( x 1 , x 2 , , x n ) d x 1 d x n \int\cdots\int_{T}\;f(x_{1},x_{2},\ldots,x_{n})\;dx_{1}\!\cdots dx_{n}
  8. T f ( 𝐱 ) d n 𝐱 . \int_{T}\!f(\mathbf{x})\,d^{n}\mathbf{x}.
  9. = T f ( x , y ) d x d y \ell=\iint_{T}f(x,y)\,dx\,dy
  10. = T f ( x , y , z ) d x d y d z \ell=\iiint_{T}f(x,y,z)\,dx\,dy\,dz
  11. D = { ( x , y ) 𝐑 2 : 2 x 4 ; 3 y 6 } D=\{(x,y)\in\mathbf{R}^{2}\ :\ 2\leq x\leq 4\ ;\ 3\leq y\leq 6\}
  12. 3 6 2 4 2 d x d y = 2 3 6 2 4 1 d x d y = 2 area ( D ) = ( 2 3 ) 2 = 12 \int_{3}^{6}\int_{2}^{4}\ 2\ dx\,dy=2\int_{3}^{6}\int_{2}^{4}\ 1\ dx\,dy=2% \cdot\mbox{area}~{}(D)=(2\cdot 3)\cdot 2=12
  13. 3 6 2 4 1 d x d y = area ( D ) . \int_{3}^{6}\int_{2}^{4}\ 1\ dx\,dy=\mbox{area}~{}(D).
  14. f ( x , y ) = 2 sin ( x ) - 3 y 3 + 5 f(x,y)=2\sin(x)-3y^{3}+5
  15. T = { ( x , y ) 𝐑 2 : x 2 + y 2 1 } , T=\left\{(x,y)\in\mathbf{R}^{2}\ :\ x^{2}+y^{2}\leq 1\right\},
  16. T ( 2 sin x - 3 y 3 + 5 ) d x d y = T 2 sin x d x d y - T 3 y 3 d x d y + T 5 d x d y \iint_{T}(2\sin x-3y^{3}+5)\,dx\,dy=\iint_{T}2\sin x\,dx\,dy-\iint_{T}3y^{3}\,% dx\,dy+\iint_{T}5\,dx\,dy
  17. T = { ( x , y , z ) 𝐑 3 : x 2 + y 2 + z 2 4 } . T=\left\{(x,y,z)\in\mathbf{R}^{3}\ :\ x^{2}+y^{2}+z^{2}\leq 4\right\}.
  18. D f ( x , y ) d x d y = a b d x α ( x ) β ( x ) f ( x , y ) d y . \iint_{D}f(x,y)\ dx\,dy=\int_{a}^{b}dx\int_{\alpha(x)}^{\beta(x)}f(x,y)\,dy.
  19. D f ( x , y ) d x d y = a b d y α ( y ) β ( y ) f ( x , y ) d x . \iint_{D}f(x,y)\ dx\,dy=\int_{a}^{b}dy\int_{\alpha(y)}^{\beta(y)}f(x,y)\,dx.
  20. D = { ( x , y ) 𝐑 2 : x 0 , y 1 , y x 2 } D=\{(x,y)\in\mathbf{R}^{2}\ :\ x\geq 0,y\leq 1,y\geq x^{2}\}
  21. D ( x + y ) d x d y . \iint_{D}(x+y)\,dx\,dy.
  22. α ( x ) = x 2 and β ( x ) = 1 \alpha(x)=x^{2}\,\text{ and }\beta(x)=1
  23. D ( x + y ) d x d y = 0 1 d x x 2 1 ( x + y ) d y = 0 1 d x [ x y + y 2 2 ] x 2 1 \iint_{D}(x+y)\,dx\,dy=\int_{0}^{1}dx\int_{x^{2}}^{1}(x+y)\,dy=\int_{0}^{1}dx% \ \left[xy+\frac{y^{2}}{2}\right]^{1}_{x^{2}}
  24. 0 1 [ x y + y 2 2 ] x 2 1 d x = 0 1 ( x + 1 2 - x 3 - x 4 2 ) d x = = 13 20 . \int_{0}^{1}\left[xy+\frac{y^{2}}{2}\right]^{1}_{x^{2}}\,dx=\int_{0}^{1}\left(% x+\frac{1}{2}-x^{3}-\frac{x^{4}}{2}\right)dx=\cdots=\frac{13}{20}.
  25. 0 1 d y 0 y ( x + y ) d x . \int_{0}^{1}dy\int_{0}^{\sqrt{y}}(x+y)\,dx.
  26. T f ( x , y , z ) d x d y d z = D α ( x , y ) β ( x , y ) f ( x , y , z ) d z d x d y \iiint_{T}f(x,y,z)\ dx\,dy\,dz=\iint_{D}\int_{\alpha(x,y)}^{\beta(x,y)}f(x,y,z% )\,dzdxdy
  27. f ( x , y ) = ( x - 1 ) 2 + y f(x,y)=(x-1)^{2}+\sqrt{y}
  28. x = x - 1 , y = y x^{\prime}=x-1,\ y^{\prime}=y
  29. x = x + 1 , y = y x=x^{\prime}+1,\ y=y^{\prime}
  30. f 2 ( x , y ) = ( x ) 2 + y f_{2}(x,y)=(x^{\prime})^{2}+\sqrt{y}
  31. f ( x , y ) f ( ρ cos ϕ , ρ sin ϕ ) . f(x,y)\rightarrow f(\rho\cos\phi,\rho\sin\phi).
  32. f ( x , y ) = x + y f(x,y)=x+y
  33. f ( ρ , ϕ ) = ρ cos ϕ + ρ sin ϕ = ρ ( cos ϕ + sin ϕ ) . f(\rho,\phi)=\rho\cos\phi+\rho\sin\phi=\rho(\cos\phi+\sin\phi).
  34. f ( x , y ) = x 2 + y 2 f(x,y)=x^{2}+y^{2}
  35. f ( ρ , ϕ ) = ρ 2 ( cos 2 ϕ + sin 2 ϕ ) = ρ 2 f(\rho,\phi)=\rho^{2}(\cos^{2}\phi+\sin^{2}\phi)=\rho^{2}
  36. D = { x 2 + y 2 4 } D=\{x^{2}+y^{2}\leq 4\}
  37. D = { x 2 + y 2 9 , x 2 + y 2 4 , y 0 } D=\{x^{2}+y^{2}\leq 9,\ x^{2}+y^{2}\geq 4,\ y\geq 0\}
  38. T = { 2 ρ 3 , 0 ϕ π } . T=\{2\leq\rho\leq 3,\ 0\leq\phi\leq\pi\}.
  39. ( x , y ) ( ρ , ϕ ) = | cos ϕ - ρ sin ϕ sin ϕ ρ cos ϕ | = ρ \frac{\partial(x,y)}{\partial(\rho,\phi)}=\begin{vmatrix}\cos\phi&-\rho\sin% \phi\\ \sin\phi&\rho\cos\phi\end{vmatrix}=\rho
  40. D f ( x , y ) d x d y = T f ( ρ cos ϕ , ρ sin ϕ ) ρ d ρ d ϕ . \iint_{D}f(x,y)\ dx\,dy=\iint_{T}f(\rho\cos\phi,\rho\sin\phi)\rho\,d\rho\,d\phi.
  41. f ( x , y ) = x f ( ρ , ϕ ) = ρ cos ϕ . f(x,y)=x\longrightarrow f(\rho,\phi)=\rho\cos\phi.
  42. D x d x d y = T ρ cos ϕ ρ d ρ d ϕ . \iint_{D}x\,dx\,dy=\iint_{T}\rho\cos\phi\rho\,d\rho\,d\phi.
  43. 0 π 2 3 ρ 2 cos ϕ d ρ d ϕ = 0 π cos ϕ d ϕ [ ρ 3 3 ] 2 3 = [ sin ϕ ] 0 π ( 9 - 8 3 ) = 0. \int_{0}^{\pi}\int_{2}^{3}\rho^{2}\cos\phi\,d\rho\,d\phi=\int_{0}^{\pi}\cos% \phi\ d\phi\left[\frac{\rho^{3}}{3}\right]_{2}^{3}=\left[\sin\phi\right]_{0}^{% \pi}\ \left(9-\frac{8}{3}\right)=0.
  44. f ( x , y , z ) f ( ρ cos ϕ , ρ sin ϕ , z ) f(x,y,z)\rightarrow f(\rho\cos\phi,\rho\sin\phi,z)
  45. D = { x 2 + y 2 9 , x 2 + y 2 4 , 0 z 5 } D=\{x^{2}+y^{2}\leq 9,\ x^{2}+y^{2}\geq 4,\ 0\leq z\leq 5\}
  46. T = { 2 ρ 3 , 0 ϕ 2 π , 0 z 5 } T=\{2\leq\rho\leq 3,\ 0\leq\phi\leq 2\pi,\ 0\leq z\leq 5\}
  47. D f ( x , y , z ) d x d y d z = T f ( ρ cos ϕ , ρ sin ϕ , z ) ρ d ρ d ϕ d z . \iiint_{D}f(x,y,z)\,dx\,dy\,dz=\iiint_{T}f(\rho\cos\phi,\rho\sin\phi,z)\rho\,d% \rho\,d\phi\,dz.
  48. f ( x , y , z ) = x 2 + y 2 + z f(x,y,z)=x^{2}+y^{2}+z
  49. D = { x 2 + y 2 9 , - 5 z 5 } D=\{x^{2}+y^{2}\leq 9,\ -5\leq z\leq 5\}
  50. T = { 0 ρ 3 , 0 ϕ 2 π , - 5 z 5 } . T=\{0\leq\rho\leq 3,\ 0\leq\phi\leq 2\pi,\ -5\leq z\leq 5\}.
  51. f ( ρ cos ϕ , ρ sin ϕ , z ) = ρ 2 + z f(\rho\cos\phi,\rho\sin\phi,z)=\rho^{2}+z
  52. D ( x 2 + y 2 + z ) d x d y d z = T ( ρ 2 + z ) ρ d ρ d ϕ d z ; \iiint_{D}(x^{2}+y^{2}+z)\,dx\,dy\,dz=\iiint_{T}(\rho^{2}+z)\rho\,d\rho\,d\phi% \,dz;
  53. - 5 5 d z 0 2 π d ϕ 0 3 ( ρ 3 + ρ z ) d ρ = 2 π - 5 5 [ ρ 4 4 + ρ 2 z 2 ] 0 3 d z = 2 π - 5 5 ( 81 4 + 9 2 z ) d z = = 405 π . \int_{-5}^{5}dz\int_{0}^{2\pi}d\phi\int_{0}^{3}(\rho^{3}+\rho z)\,d\rho=2\pi% \int_{-5}^{5}\left[\frac{\rho^{4}}{4}+\frac{\rho^{2}z}{2}\right]_{0}^{3}\,dz=2% \pi\int_{-5}^{5}\left(\frac{81}{4}+\frac{9}{2}z\right)\,dz=\cdots=405\pi.
  54. f ( x , y , z ) f ( ρ cos θ sin ϕ , ρ sin θ sin ϕ , ρ cos ϕ ) f(x,y,z)\longrightarrow f(\rho\cos\theta\sin\phi,\rho\sin\theta\sin\phi,\rho% \cos\phi)
  55. D = x 2 + y 2 + z 2 16 D=x^{2}+y^{2}+z^{2}\leq 16
  56. T = { 0 ρ 4 , 0 ϕ π , 0 θ 2 π } . T=\{0\leq\rho\leq 4,\ 0\leq\phi\leq\pi,\ 0\leq\theta\leq 2\pi\}.
  57. ( x , y , z ) ( ρ , θ , ϕ ) = | cos θ sin ϕ - ρ sin θ sin ϕ ρ cos θ cos ϕ sin θ sin ϕ ρ cos θ sin ϕ ρ sin θ cos ϕ cos ϕ 0 - ρ sin ϕ | = ρ 2 sin ϕ \frac{\partial(x,y,z)}{\partial(\rho,\theta,\phi)}=\begin{vmatrix}\cos\theta% \sin\phi&-\rho\sin\theta\sin\phi&\rho\cos\theta\cos\phi\\ \sin\theta\sin\phi&\rho\cos\theta\sin\phi&\rho\sin\theta\cos\phi\\ \cos\phi&0&-\rho\sin\phi\end{vmatrix}=\rho^{2}\sin\phi
  58. D f ( x , y , z ) d x d y d z = T f ( ρ sin ϕ cos θ , ρ sin ϕ sin θ , ρ cos ϕ ) ρ 2 sin ϕ d ρ d θ d ϕ . \iiint_{D}f(x,y,z)\,dx\,dy\,dz=\iiint_{T}f(\rho\sin\phi\cos\theta,\rho\sin\phi% \sin\theta,\rho\cos\phi)\rho^{2}\sin\phi\,d\rho\,d\theta\,d\phi.
  59. T f ( a , b , c ) ρ 2 sin ϕ d ρ d θ d ϕ . \iiint_{T}f(a,b,c)\rho^{2}\sin\phi\,d\rho\,d\theta\,d\phi.
  60. ρ 2 \rho^{2}
  61. sin ϕ \sin\phi
  62. f ( x , y , z ) = x 2 + y 2 + z 2 f(x,y,z)=x^{2}+y^{2}+z^{2}
  63. f ( ρ sin ϕ cos θ , ρ sin ϕ sin θ , ρ cos ϕ ) = ρ 2 , f(\rho\sin\phi\cos\theta,\rho\sin\phi\sin\theta,\rho\cos\phi)=\rho^{2},
  64. ( 0 ρ 4 , 0 ϕ π , 0 θ 2 π ) . (0\leq\rho\leq 4,\ 0\leq\phi\leq\pi,\ 0\leq\theta\leq 2\pi).
  65. D ( x 2 + y 2 + z 2 ) d x d y d z = T ρ 2 ρ 2 sin θ d ρ d θ d ϕ , \iiint_{D}(x^{2}+y^{2}+z^{2})\,dx\,dy\,dz=\iiint_{T}\rho^{2}\ \rho^{2}\sin% \theta\,d\rho\,d\theta\,d\phi,
  66. T ρ 4 sin θ d ρ d θ d ϕ = 0 π sin ϕ d ϕ 0 4 ρ 4 d ρ 0 2 π d θ = 2 π 0 π sin ϕ [ ρ 5 5 ] 0 4 d ϕ = 2 π [ ρ 5 5 ] 0 4 [ - cos ϕ ] 0 π = 4096 π 5 . \iiint_{T}\rho^{4}\sin\theta\,d\rho\,d\theta\,d\phi=\int_{0}^{\pi}\sin\phi\,d% \phi\int_{0}^{4}\rho^{4}d\rho\int_{0}^{2\pi}d\theta=2\pi\int_{0}^{\pi}\sin\phi% \left[\frac{\rho^{5}}{5}\right]_{0}^{4}\,d\phi=2\pi\left[\frac{\rho^{5}}{5}% \right]_{0}^{4}\left[-\cos\phi\right]_{0}^{\pi}=\frac{4096\pi}{5}.
  67. D = { x 2 + y 2 + z 2 9 a 2 } D=\left\{x^{2}+y^{2}+z^{2}\leq 9a^{2}\right\}
  68. f ( x , y , z ) = x 2 + y 2 f(x,y,z)=x^{2}+y^{2}
  69. 0 ρ 3 a , 0 ϕ 2 π , 0 θ π . 0\leq\rho\leq 3a,\ 0\leq\phi\leq 2\pi,\ 0\leq\theta\leq\pi.
  70. f ( x , y , z ) = x 2 + y 2 ρ 2 sin 2 θ cos 2 ϕ + ρ 2 sin 2 θ sin 2 ϕ = ρ 2 sin 2 θ f(x,y,z)=x^{2}+y^{2}\longrightarrow\rho^{2}\sin^{2}\theta\cos^{2}\phi+\rho^{2}% \sin^{2}\theta\sin^{2}\phi=\rho^{2}\sin^{2}\theta
  71. T ρ 2 sin 2 θ ρ 2 sin θ d ρ d θ d ϕ = T ρ 4 sin 3 θ d ρ d θ d ϕ \iiint_{T}\rho^{2}\sin^{2}\theta\rho^{2}\sin\theta\,d\rho\,d\theta\,d\phi=% \iiint_{T}\rho^{4}\sin^{3}\theta\,d\rho\,d\theta\,d\phi
  72. 0 ρ 3 a , 0 ϕ 2 π , - 9 a 2 - ρ 2 z 9 a 2 - ρ 2 ; 0\leq\rho\leq 3a,\ 0\leq\phi\leq 2\pi,\ -\sqrt{9a^{2}-\rho^{2}}\leq z\leq\sqrt% {9a^{2}-\rho^{2}};
  73. T ρ 2 ρ d ρ d ϕ d z \iiint_{T}\rho^{2}\rho\ d\rho d\phi dz
  74. 0 2 π d ϕ 0 3 a ρ 3 d ρ - 9 a 2 - ρ 2 9 a 2 - ρ 2 d z = 2 π 0 3 a 2 ρ 3 9 a 2 - ρ 2 d ρ = - 2 π 9 a 2 0 ( 9 a 2 - t ) t d t t = 9 a 2 - ρ 2 = 2 π 0 9 a 2 ( 9 a 2 t - t t ) d t = 2 π [ 0 9 a 2 9 a 2 t d t - 0 9 a 2 t t d t ] = 2 π [ 9 a 2 2 3 t 3 2 - 2 5 t 5 2 ] 0 9 a 2 = 2 27 π a 5 ( 6 - 18 5 ) = 648 π 5 a 5 . \begin{aligned}\displaystyle\int_{0}^{2\pi}d\phi\int_{0}^{3a}\rho^{3}d\rho\int% _{-\sqrt{9a^{2}-\rho^{2}}}^{\sqrt{9a^{2}-\rho^{2}}}\,dz&\displaystyle=2\pi\int% _{0}^{3a}2\rho^{3}\sqrt{9a^{2}-\rho^{2}}\,d\rho\\ &\displaystyle=-2\pi\int_{9a^{2}}^{0}(9a^{2}-t)\sqrt{t}\,dt&&\displaystyle t=9% a^{2}-\rho^{2}\\ &\displaystyle=2\pi\int_{0}^{9a^{2}}\left(9a^{2}\sqrt{t}-t\sqrt{t}\right)\,dt% \\ &\displaystyle=2\pi\left[\int_{0}^{9a^{2}}9a^{2}\sqrt{t}\,dt-\int_{0}^{9a^{2}}% t\sqrt{t}\,dt\right]\\ &\displaystyle=2\pi\left[9a^{2}\frac{2}{3}t^{\frac{3}{2}}-\frac{2}{5}t^{\frac{% 5}{2}}\right]_{0}^{9a^{2}}\\ &\displaystyle=2\cdot 27\pi a^{5}\left(6-\frac{18}{5}\right)\\ &\displaystyle=\frac{648\pi}{5}a^{5}.\end{aligned}
  75. A = { ( x , y ) 𝐑 2 : 11 x 14 ; 7 y 10 } and f ( x , y ) = x 2 + 4 y A=\left\{(x,y)\in\mathbf{R}^{2}\ :\ 11\leq x\leq 14\ ;\ 7\leq y\leq 10\right\}% \,\text{ and }f(x,y)=x^{2}+4y\,
  76. 7 10 11 14 ( x 2 + 4 y ) d x d y \int_{7}^{10}\int_{11}^{14}(x^{2}+4y)\ dx\,dy
  77. 11 14 ( x 2 + 4 y ) d x = ( 1 3 x 3 + 4 y x ) | x = 11 x = 14 = 1 3 ( 14 ) 3 + 4 y ( 14 ) - 1 3 ( 11 ) 3 - 4 y ( 11 ) = 471 + 12 y \begin{aligned}\displaystyle\int_{11}^{14}(x^{2}+4y)\ dx&\displaystyle=\left(% \frac{1}{3}x^{3}+4yx\right)\Big|_{x=11}^{x=14}\\ &\displaystyle=\frac{1}{3}(14)^{3}+4y(14)-\frac{1}{3}(11)^{3}-4y(11)\\ &\displaystyle=471+12y\end{aligned}
  78. 7 10 ( 471 + 12 y ) d y = ( 471 y + 6 y 2 ) | y = 7 y = 10 = 471 ( 10 ) + 6 ( 10 ) 2 - 471 ( 7 ) - 6 ( 7 ) 2 = 1719 \begin{aligned}\displaystyle\int_{7}^{10}(471+12y)\ dy&\displaystyle=(471y+6y^% {2})\big|_{y=7}^{y=10}\\ &\displaystyle=471(10)+6(10)^{2}-471(7)-6(7)^{2}\\ &\displaystyle=1719\end{aligned}
  79. 11 14 7 10 ( x 2 + 4 y ) d y d x \displaystyle\int_{11}^{14}\int_{7}^{10}\ (x^{2}+4y)\ dy\,dx
  80. Volume = 0 2 π d ϕ 0 R h ρ d ρ = h 2 π [ ρ 2 2 ] 0 R = π R 2 h \mathrm{Volume}=\int_{0}^{2\pi}d\phi\int_{0}^{R}h\rho\ d\rho=h2\pi\left[\frac{% \rho^{2}}{2}\right]_{0}^{R}=\pi R^{2}h
  81. Volume = base area × h e i g h t \mathrm{Volume}=\,\text{base area}\times height
  82. Volume \displaystyle\,\text{Volume}
  83. Volume \displaystyle\,\text{Volume}
  84. Volume = 1 3 × base area × height = 1 3 × 2 2 × = 3 6 . \mathrm{Volume}=\frac{1}{3}\times\,\text{base area}\times\,\text{height}=\frac% {1}{3}\times\frac{\ell^{2}}{2}\times\ell=\frac{\ell^{3}}{6}.
  85. A × B | f ( x , y ) | d ( x , y ) < , \iint_{A\times B}|f(x,y)|\,d(x,y)<\infty,
  86. A × B f ( x , y ) d ( x , y ) = A ( B f ( x , y ) d y ) d x = B ( A f ( x , y ) d x ) d y . \iint_{A\times B}f(x,y)\,d(x,y)=\int_{A}\left(\int_{B}f(x,y)\,dy\right)\,dx=% \int_{B}\left(\int_{A}f(x,y)\,dx\right)\,dy.
  87. 0 1 0 1 f ( x , y ) d y d x \int_{0}^{1}\int_{0}^{1}f(x,y)\,dy\,dx
  88. 0 1 d x \int_{0}^{1}\cdots\,dx
  89. g ( x ) = 0 1 f ( x , y ) d y . g(x)=\int_{0}^{1}f(x,y)\,dy.
  90. 0 1 0 1 f ( x , y ) d y d x 0 1 0 1 f ( x , y ) d x d y . \int_{0}^{1}\int_{0}^{1}f(x,y)\,dy\,dx\neq\int_{0}^{1}\int_{0}^{1}f(x,y)\,dx\,dy.
  91. [ 0 , 1 ] × [ 0 , 1 ] f ( x , y ) d x d y \int_{[0,1]\times[0,1]}f(x,y)\,dx\,dy
  92. f ¯ = 1 m ( D ) D f ( x ) d x , \bar{f}=\frac{1}{m(D)}\int_{D}f(x)\,dx,
  93. I z = V ρ r 2 d V . I_{z}=\iiint_{V}\rho r^{2}\,dV.
  94. V ( 𝐱 ) = - 𝐑 3 G | 𝐱 - 𝐲 | d m ( 𝐲 ) . V(\mathbf{x})=-\iiint_{\mathbf{R}^{3}}\frac{G}{|\mathbf{x}-\mathbf{y}|}\,dm(% \mathbf{y}).
  95. V ( 𝐱 ) = - 𝐑 3 G | 𝐱 - 𝐲 | ρ ( 𝐲 ) d 3 𝐲 . V(\mathbf{x})=-\iiint_{\mathbf{R}^{3}}\frac{G}{|\mathbf{x}-\mathbf{y}|}\,\rho(% \mathbf{y})\,d^{3}\mathbf{y}.
  96. ρ ( r ) \rho(\vec{r})
  97. E = 1 4 π ϵ 0 r - r r - r 3 ρ ( r ) d 3 r . \vec{E}=\frac{1}{4\pi\epsilon_{0}}\iiint\frac{\vec{r}-\vec{r}^{\prime}}{\left% \|\vec{r}-\vec{r}^{\prime}\right\|^{3}}\rho(\vec{r}^{\prime})\,\operatorname{d% }^{3}r^{\prime}.

Multiplication_(music).html

  1. M x M_{x}
  2. x x
  3. M x ( y ) x y ( mod 12 ) M_{x}(y)\equiv xy\;\;(\mathop{{\rm mod}}12)
  4. X × Y = { ( x + y ) mod 12 | x X , y Y } X\times Y=\{(x+y)\bmod 12|x\in X,y\in Y\}
  5. { 0 , 4 , 7 } \{0,4,7\}
  6. { 0 , 2 } \{0,2\}
  7. { 0 , 4 , 7 } × { 0 , 2 } = { 0 , 2 , 4 , 6 , 7 , 9 } \{0,4,7\}\times\{0,2\}=\{0,2,4,6,7,9\}

Multireference_configuration_interaction.html

  1. Φ 1 , Φ 2 , Φ 5 , \Phi_{1},\Phi_{2},\Phi_{5},...

Multivector.html

  1. 𝐮 ( α 𝐯 + β 𝐰 ) = α 𝐮 𝐯 + β 𝐮 𝐰 ; \mathbf{u}\wedge(\alpha\mathbf{v}+\beta\mathbf{w})=\alpha\mathbf{u}\wedge% \mathbf{v}+\beta\mathbf{u}\wedge\mathbf{w};
  2. ( 𝐮 𝐯 ) 𝐰 = 𝐮 ( 𝐯 𝐰 ) = 𝐮 𝐯 𝐰 ; (\mathbf{u}\wedge\mathbf{v})\wedge\mathbf{w}=\mathbf{u}\wedge(\mathbf{v}\wedge% \mathbf{w})=\mathbf{u}\wedge\mathbf{v}\wedge\mathbf{w};
  3. 𝐮 𝐯 = - 𝐯 𝐮 , 𝐮 𝐮 = 0. \mathbf{u}\wedge\mathbf{v}=-\mathbf{v}\wedge\mathbf{u},\quad\mathbf{u}\wedge% \mathbf{u}=0.
  4. 𝐮 = u 1 𝐞 1 + u 2 𝐞 2 , 𝐯 = v 1 𝐞 1 + v 2 𝐞 2 , \mathbf{u}=u_{1}\mathbf{e}_{1}+u_{2}\mathbf{e}_{2},\quad\mathbf{v}=v_{1}% \mathbf{e}_{1}+v_{2}\mathbf{e}_{2},
  5. 𝐮 𝐯 = | u 1 v 1 u 2 v 2 | 𝐞 1 𝐞 2 . \mathbf{u}\wedge\mathbf{v}=\begin{vmatrix}u_{1}&v_{1}\\ u_{2}&v_{2}\end{vmatrix}\mathbf{e}_{1}\wedge\mathbf{e}_{2}.
  6. 𝐮 = u 1 𝐞 1 + u 2 𝐞 2 + u 3 𝐞 3 , 𝐯 = v 1 𝐞 1 + v 2 𝐞 2 + v 3 𝐞 3 , 𝐰 = w 1 𝐞 1 + w 2 𝐞 2 + w 3 𝐞 3 , \mathbf{u}=u_{1}\mathbf{e}_{1}+u_{2}\mathbf{e}_{2}+u_{3}\mathbf{e}_{3},\quad% \mathbf{v}=v_{1}\mathbf{e}_{1}+v_{2}\mathbf{e}_{2}+v_{3}\mathbf{e}_{3},\quad% \mathbf{w}=w_{1}\mathbf{e}_{1}+w_{2}\mathbf{e}_{2}+w_{3}\mathbf{e}_{3},
  7. 𝐮 𝐯 = | u 2 v 2 u 3 v 3 | 𝐞 2 𝐞 3 + | u 1 v 1 u 3 v 3 | 𝐞 1 𝐞 3 + | u 1 v 1 u 2 v 2 | 𝐞 1 𝐞 2 . \mathbf{u}\wedge\mathbf{v}=\begin{vmatrix}u_{2}&v_{2}\\ u_{3}&v_{3}\end{vmatrix}\mathbf{e}_{2}\wedge\mathbf{e}_{3}+\begin{vmatrix}u_{1% }&v_{1}\\ u_{3}&v_{3}\end{vmatrix}\mathbf{e}_{1}\wedge\mathbf{e}_{3}+\begin{vmatrix}u_{1% }&v_{1}\\ u_{2}&v_{2}\end{vmatrix}\mathbf{e}_{1}\wedge\mathbf{e}_{2}.
  8. 𝐮 𝐯 𝐰 = | u 1 v 1 w 1 u 2 v 2 w 2 u 3 v 3 w 3 | 𝐞 1 𝐞 2 𝐞 3 . \mathbf{u}\wedge\mathbf{v}\wedge\mathbf{w}=\begin{vmatrix}u_{1}&v_{1}&w_{1}\\ u_{2}&v_{2}&w_{2}\\ u_{3}&v_{3}&w_{3}\end{vmatrix}\mathbf{e}_{1}\wedge\mathbf{e}_{2}\wedge\mathbf{% e}_{3}.
  9. 𝐩 𝐪 = ( p 2 - q 2 ) 𝐞 2 𝐞 3 + ( p 1 - q 1 ) 𝐞 1 𝐞 3 + ( p 1 q 2 - q 1 p 2 ) 𝐞 1 𝐞 2 . \mathbf{p}\wedge\mathbf{q}=(p_{2}-q_{2})\mathbf{e}_{2}\wedge\mathbf{e}_{3}+(p_% {1}-q_{1})\mathbf{e}_{1}\wedge\mathbf{e}_{3}+(p_{1}q_{2}-q_{1}p_{2})\mathbf{e}% _{1}\wedge\mathbf{e}_{2}.
  10. 𝐱 𝐩 𝐪 = ( x 𝐞 1 + y 𝐞 2 + 𝐞 3 ) ( ( p 2 - q 2 ) 𝐞 2 𝐞 3 + ( p 1 - q 1 ) 𝐞 1 𝐞 3 + ( p 1 q 2 - q 1 p 2 ) 𝐞 1 𝐞 2 ) = 0 , \mathbf{x}\wedge\mathbf{p}\wedge\mathbf{q}=(x\mathbf{e}_{1}+y\mathbf{e}_{2}+% \mathbf{e}_{3})\wedge\big((p_{2}-q_{2})\mathbf{e}_{2}\wedge\mathbf{e}_{3}+(p_{% 1}-q_{1})\mathbf{e}_{1}\wedge\mathbf{e}_{3}+(p_{1}q_{2}-q_{1}p_{2})\mathbf{e}_% {1}\wedge\mathbf{e}_{2}\big)=0,
  11. λ : x ( p 2 - q 2 ) + y ( p 1 - q 1 ) + ( p 1 q 2 - q 1 p 2 ) = 0. \lambda:x(p_{2}-q_{2})+y(p_{1}-q_{1})+(p_{1}q_{2}-q_{1}p_{2})=0.
  12. 𝐩 𝐪 𝐫 = | p 2 q 2 r 2 p 3 q 3 r 3 1 1 1 | 𝐞 2 𝐞 3 𝐞 4 + | p 1 q 1 r 1 p 3 q 3 r 3 1 1 1 | 𝐞 1 𝐞 3 𝐞 4 + | p 1 q 1 r 1 p 2 q 2 r 2 1 1 1 | 𝐞 1 𝐞 2 𝐞 4 + | p 1 q 1 r 1 p 2 q 2 r 2 p 3 q 3 r 3 | 𝐞 1 𝐞 2 𝐞 3 . \mathbf{p}\wedge\mathbf{q}\wedge\mathbf{r}=\begin{vmatrix}p_{2}&q_{2}&r_{2}\\ p_{3}&q_{3}&r_{3}\\ 1&1&1\end{vmatrix}\mathbf{e}_{2}\wedge\mathbf{e}_{3}\wedge\mathbf{e}_{4}+% \begin{vmatrix}p_{1}&q_{1}&r_{1}\\ p_{3}&q_{3}&r_{3}\\ 1&1&1\end{vmatrix}\mathbf{e}_{1}\wedge\mathbf{e}_{3}\wedge\mathbf{e}_{4}+% \begin{vmatrix}p_{1}&q_{1}&r_{1}\\ p_{2}&q_{2}&r_{2}\\ 1&1&1\end{vmatrix}\mathbf{e}_{1}\wedge\mathbf{e}_{2}\wedge\mathbf{e}_{4}+% \begin{vmatrix}p_{1}&q_{1}&r_{1}\\ p_{2}&q_{2}&r_{2}\\ p_{3}&q_{3}&r_{3}\end{vmatrix}\mathbf{e}_{1}\wedge\mathbf{e}_{2}\wedge\mathbf{% e}_{3}.
  13. 𝐱 𝐩 𝐪 𝐫 = ( x 𝐞 1 + y 𝐞 2 + z 𝐞 3 + 𝐞 4 ) 𝐩 𝐪 𝐫 = 0 , \mathbf{x}\wedge\mathbf{p}\wedge\mathbf{q}\wedge\mathbf{r}=(x\mathbf{e}_{1}+y% \mathbf{e}_{2}+z\mathbf{e}_{3}+\mathbf{e}_{4})\wedge\mathbf{p}\wedge\mathbf{q}% \wedge\mathbf{r}=0,
  14. λ : x | p 2 q 2 r 2 p 3 q 3 r 3 1 1 1 | + y | p 1 q 1 r 1 p 3 q 3 r 3 1 1 1 | + z | p 1 q 1 r 1 p 2 q 2 r 2 1 1 1 | + | p 1 q 1 r 1 p 2 q 2 r 2 p 3 q 3 r 3 | = 0. \lambda:x\begin{vmatrix}p_{2}&q_{2}&r_{2}\\ p_{3}&q_{3}&r_{3}\\ 1&1&1\end{vmatrix}+y\begin{vmatrix}p_{1}&q_{1}&r_{1}\\ p_{3}&q_{3}&r_{3}\\ 1&1&1\end{vmatrix}+z\begin{vmatrix}p_{1}&q_{1}&r_{1}\\ p_{2}&q_{2}&r_{2}\\ 1&1&1\end{vmatrix}+\begin{vmatrix}p_{1}&q_{1}&r_{1}\\ p_{2}&q_{2}&r_{2}\\ p_{3}&q_{3}&r_{3}\end{vmatrix}=0.
  15. λ : 𝐩 𝐪 = ( p 1 𝐞 1 + p 2 𝐞 2 + p 3 𝐞 3 + 𝐞 4 ) ( q 1 𝐞 1 + q 2 𝐞 2 + q 3 𝐞 3 + 𝐞 4 ) , \lambda:\mathbf{p}\wedge\mathbf{q}=(p_{1}\mathbf{e}_{1}+p_{2}\mathbf{e}_{2}+p_% {3}\mathbf{e}_{3}+\mathbf{e}_{4})\wedge(q_{1}\mathbf{e}_{1}+q_{2}\mathbf{e}_{2% }+q_{3}\mathbf{e}_{3}+\mathbf{e}_{4}),
  16. = | p 1 q 1 1 1 | 𝐞 1 𝐞 4 + | p 2 q 2 1 1 | 𝐞 2 𝐞 4 + | p 3 q 3 1 1 | 𝐞 3 𝐞 4 + | p 2 q 2 p 3 q 3 | 𝐞 2 𝐞 3 + | p 3 q 3 p 1 q 1 | 𝐞 3 𝐞 1 + | p 1 q 1 p 2 q 2 | 𝐞 1 𝐞 2 . \qquad=\begin{vmatrix}p_{1}&q_{1}\\ 1&1\end{vmatrix}\mathbf{e}_{1}\wedge\mathbf{e}_{4}+\begin{vmatrix}p_{2}&q_{2}% \\ 1&1\end{vmatrix}\mathbf{e}_{2}\wedge\mathbf{e}_{4}+\begin{vmatrix}p_{3}&q_{3}% \\ 1&1\end{vmatrix}\mathbf{e}_{3}\wedge\mathbf{e}_{4}+\begin{vmatrix}p_{2}&q_{2}% \\ p_{3}&q_{3}\end{vmatrix}\mathbf{e}_{2}\wedge\mathbf{e}_{3}+\begin{vmatrix}p_{3% }&q_{3}\\ p_{1}&q_{1}\end{vmatrix}\mathbf{e}_{3}\wedge\mathbf{e}_{1}+\begin{vmatrix}p_{1% }&q_{1}\\ p_{2}&q_{2}\end{vmatrix}\mathbf{e}_{1}\wedge\mathbf{e}_{2}.
  17. μ : 𝐱 π = 0 , 𝐱 ρ = 0. \mu:\mathbf{x}\wedge\pi=0,\mathbf{x}\wedge\rho=0.
  18. 𝐞 1 = * 𝐞 2 𝐞 3 𝐞 4 , - 𝐞 2 = * 𝐞 1 𝐞 3 𝐞 4 , 𝐞 3 = * 𝐞 1 𝐞 2 𝐞 4 , - 𝐞 4 = * 𝐞 1 𝐞 2 𝐞 3 , \mathbf{e}_{1}=*\mathbf{e}_{2}\wedge\mathbf{e}_{3}\wedge\mathbf{e}_{4},-% \mathbf{e}_{2}=*\mathbf{e}_{1}\wedge\mathbf{e}_{3}\wedge\mathbf{e}_{4},\mathbf% {e}_{3}=*\mathbf{e}_{1}\wedge\mathbf{e}_{2}\wedge\mathbf{e}_{4},-\mathbf{e}_{4% }=*\mathbf{e}_{1}\wedge\mathbf{e}_{2}\wedge\mathbf{e}_{3},
  19. * π = π 1 𝐞 1 + π 2 𝐞 2 + π 3 𝐞 3 + π 4 𝐞 4 , * ρ = ρ 1 𝐞 1 + ρ 2 𝐞 2 + ρ 3 𝐞 3 + ρ 4 𝐞 4 . *\pi=\pi_{1}\mathbf{e}_{1}+\pi_{2}\mathbf{e}_{2}+\pi_{3}\mathbf{e}_{3}+\pi_{4}% \mathbf{e}_{4},\quad*\rho=\rho_{1}\mathbf{e}_{1}+\rho_{2}\mathbf{e}_{2}+\rho_{% 3}\mathbf{e}_{3}+\rho_{4}\mathbf{e}_{4}.
  20. μ : ( * π ) ( * ρ ) = | π 1 ρ 1 π 4 ρ 4 | 𝐞 1 𝐞 4 + | π 2 ρ 2 π 4 ρ 4 | 𝐞 2 𝐞 4 + | π 3 ρ 3 π 4 ρ 4 | 𝐞 3 𝐞 4 + | π 2 ρ 2 π 3 ρ 3 | 𝐞 2 𝐞 3 + | π 3 ρ 3 π 1 ρ 1 | 𝐞 3 𝐞 1 + | π 1 ρ 1 π 2 ρ 2 | 𝐞 1 𝐞 2 . \mu:(*\pi)\wedge(*\rho)=\begin{vmatrix}\pi_{1}&\rho_{1}\\ \pi_{4}&\rho_{4}\end{vmatrix}\mathbf{e}_{1}\wedge\mathbf{e}_{4}+\begin{vmatrix% }\pi_{2}&\rho_{2}\\ \pi_{4}&\rho_{4}\end{vmatrix}\mathbf{e}_{2}\wedge\mathbf{e}_{4}+\begin{vmatrix% }\pi_{3}&\rho_{3}\\ \pi_{4}&\rho_{4}\end{vmatrix}\mathbf{e}_{3}\wedge\mathbf{e}_{4}+\begin{vmatrix% }\pi_{2}&\rho_{2}\\ \pi_{3}&\rho_{3}\end{vmatrix}\mathbf{e}_{2}\wedge\mathbf{e}_{3}+\begin{vmatrix% }\pi_{3}&\rho_{3}\\ \pi_{1}&\rho_{1}\end{vmatrix}\mathbf{e}_{3}\wedge\mathbf{e}_{1}+\begin{vmatrix% }\pi_{1}&\rho_{1}\\ \pi_{2}&\rho_{2}\end{vmatrix}\mathbf{e}_{1}\wedge\mathbf{e}_{2}.
  21. 𝐮𝐯 + 𝐯𝐮 = - 2 𝐮 𝐯 . \mathbf{u}\mathbf{v}+\mathbf{v}\mathbf{u}=-2\mathbf{u}\cdot\mathbf{v}.
  22. 𝐞 i 𝐞 j + 𝐞 j 𝐞 i = - 2 𝐞 i 𝐞 j = 0 , \mathbf{e}_{i}\mathbf{e}_{j}+\mathbf{e}_{j}\mathbf{e}_{i}=-2\mathbf{e}_{i}% \cdot\mathbf{e}_{j}=0,
  23. 𝐞 i 𝐞 j = - 𝐞 j 𝐞 i , i j = 1 , , n . \mathbf{e}_{i}\mathbf{e}_{j}=-\mathbf{e}_{j}\mathbf{e}_{i},\quad i\neq j=1,% \ldots,n.
  24. 𝐞 i 𝐞 i + 𝐞 i 𝐞 i = - 2 𝐞 i 𝐞 i = - 2 , \mathbf{e}_{i}\mathbf{e}_{i}+\mathbf{e}_{i}\mathbf{e}_{i}=-2\mathbf{e}_{i}% \cdot\mathbf{e}_{i}=-2,
  25. 𝐞 i 𝐞 i = - 1 , i = 1 , , n . \mathbf{e}_{i}\mathbf{e}_{i}=-1,\quad i=1,\ldots,n.
  26. 𝐚 𝐛 = 𝐚 𝐛 sin ( ϕ a , b ) \|\mathbf{a}\wedge\mathbf{b}\|=\|\mathbf{a}\|\,\|\mathbf{b}\|\,\sin(\phi_{a,b})
  27. F = F a b e a e b ( 1 a < b n ) F=F^{ab}e_{a}\wedge e_{b}\quad(1\leq a<b\leq n)
  28. G ( F , H ) := G a b c d F a b H c d G(F,H):=\,G_{abcd}F^{ab}H^{cd}
  29. G a b c d G_{abcd}

Mundell–Fleming_model.html

  1. Y = C + I + G + N X Y=C+I+G+NX\,
  2. M P = L ( i , Y ) \frac{M}{P}=L(i,Y)
  3. B o P = C A + K A BoP=CA+KA\,
  4. C = C ( Y - T ( Y ) , i - E ( π ) ) C=C(Y-T(Y),i-E(\pi))\,
  5. I = I ( i - E ( π ) , Y - 1 ) I=I(i-E(\pi),Y_{-1})\,
  6. N X = N X ( e , Y , Y * ) NX=NX(e,Y,Y*)
  7. C A = N X CA=NX
  8. K A = z ( i - i * ) + k KA=z(i-i*)+k
  9. i * i*
  10. i i^{\star}
  11. i = i + e e - 1 i=i^{\star}+\frac{e^{\prime}}{e}-1
  12. σ \sigma
  13. d i d e = σ - 1 < 0 . \frac{di}{de}=\sigma-1<0\quad.
  14. y = E ( i , y ) + T ( e , y ) y=E(i,y)+T(e,y)
  15. d y d e = E i d i d e + E y d y d e + T e + T y d y d e \frac{dy}{de}=\frac{\partial E}{\partial i}\frac{di}{de}+\frac{\partial E}{% \partial y}\frac{dy}{de}+\frac{\partial T}{\partial e}+\frac{\partial T}{% \partial y}\frac{dy}{de}
  16. d y d e = 1 1 - E y - T y ( E i d i d e + T e ) . \frac{dy}{de}=\frac{1}{1-E_{y}-T_{y}}\left(E_{i}\frac{di}{de}+T_{e}\right)\;.
  17. E i < 0 , E y = 1 - s > 0 E_{i}<0\;,\quad E_{y}=1-s>0
  18. T e > 0 , T y = - m < 0 . T_{e}>0\;,\quad T_{y}=-m<0\;.
  19. d y d e = 1 s + m ( E i d i d e + T e ) \frac{dy}{de}=\frac{1}{s+m}\left(E_{i}\frac{di}{de}+T_{e}\right)
  20. d y d e = 1 s + m ( E i ( σ - 1 ) + T e ) . \frac{dy}{de}=\frac{1}{s+m}\left(E_{i}(\sigma-1)+T_{e}\right)\;.
  21. d T = T e d e + T y d y = T e d e + T y d y dT=\frac{\partial T}{\partial e}de+\frac{\partial T}{\partial y}dy=T_{e}de+T_{% y}dy
  22. d L = L i d i + L y d y = L i d i + L y d y dL=\frac{\partial L}{\partial i}di+\frac{\partial L}{\partial y}dy=L_{i}di+L_{% y}dy
  23. L i < 0 , L y > 0 L_{i}<0\;,\quad L_{y}>0
  24. d T d L = T e ( s + m ) + T y ( E i ( σ - 1 ) + T e ) L i ( σ - 1 ) ( s + m ) + L y ( E i ( σ - 1 ) + T e ) \frac{dT}{dL}=\frac{T_{e}(s+m)+T_{y}(E_{i}(\sigma-1)+T_{e})}{L_{i}(\sigma-1)(s% +m)+L_{y}(E_{i}(\sigma-1)+T_{e})}
  25. d T d L = T e s + T y E i ( σ - 1 ) L i ( σ - 1 ) ( s + m ) + L y ( E i ( σ - 1 ) + T e ) . \frac{dT}{dL}=\frac{T_{e}s+T_{y}E_{i}(\sigma-1)}{L_{i}(\sigma-1)(s+m)+L_{y}(E_% {i}(\sigma-1)+T_{e})}\;.

Muon_spin_spectroscopy.html

  1. 10 4 - 10 7 10^{4}-10^{7}
  2. π + \pi^{+}
  3. p + p p + n + π + p + n n + n + π + \begin{array}[]{lll}p+p&\rightarrow&p+n+\pi^{+}\\ p+n&\rightarrow&n+n+\pi^{+}\\ \end{array}
  4. τ π + \tau_{\pi^{+}}
  5. μ + \mu^{+}
  6. π + μ + + ν μ . \pi^{+}\rightarrow\mu^{+}+\nu_{\mu}.
  7. μ + \mu^{+}
  8. μ + \mu^{+}
  9. μ + e + + ν e + ν ¯ μ . \mu^{+}\rightarrow e^{+}+\nu_{e}+\bar{\nu}_{\mu}~{}.
  10. W ( θ ) d θ ( 1 + a cos θ ) d θ , W(\theta)d\theta\propto(1+a\cos\theta)d\theta~{},
  11. θ \theta
  12. a a
  13. A A
  14. P μ P_{\mu}
  15. A A
  16. A A
  17. N α ( t ) = N 0 exp ( - t / τ μ ) ( 1 + α A ) N_{\alpha}(t)=N_{0}\exp(-t/\tau_{\mu})(1+\alpha A)
  18. α = ± 1 \alpha=\pm 1
  19. B B
  20. ω \omega
  21. N α ( t ) = N 0 exp ( - t / τ μ ) ( 1 + α A cos ω t ) N_{\alpha}(t)=N_{0}\exp(-t/\tau_{\mu})(1+\alpha A\cos\omega t)
  22. ω = γ μ B \omega=\gamma_{\mu}B
  23. γ μ = 851.616 \gamma_{\mu}=851.616

MV-algebra.html

  1. \oplus
  2. ¬ \neg
  3. 0
  4. A , , ¬ , 0 , \langle A,\oplus,\lnot,0\rangle,
  5. A , A,
  6. \oplus
  7. A , A,
  8. ¬ \lnot
  9. A , A,
  10. 0
  11. A , A,
  12. ( x y ) z = x ( y z ) , (x\oplus y)\oplus z=x\oplus(y\oplus z),
  13. x 0 = x , x\oplus 0=x,
  14. x y = y x , x\oplus y=y\oplus x,
  15. ¬ ¬ x = x , \lnot\lnot x=x,
  16. x ¬ 0 = ¬ 0 , x\oplus\lnot 0=\lnot 0,
  17. ¬ ( ¬ x y ) y = ¬ ( ¬ y x ) x . \lnot(\lnot x\oplus y)\oplus y=\lnot(\lnot y\oplus x)\oplus x.
  18. A , , 0 \langle A,\oplus,0\rangle
  19. L , , , , , 0 , 1 \langle L,\wedge,\vee,\otimes,\rightarrow,0,1\rangle
  20. x y = ( x y ) y . x\vee y=(x\rightarrow y)\rightarrow y.
  21. A = [ 0 , 1 ] , A=[0,1],
  22. x y = min ( x + y , 1 ) x\oplus y=\min(x+y,1)
  23. ¬ x = 1 - x . \lnot x=1-x.
  24. 0 0 = 0 0\oplus 0=0
  25. ¬ 0 = 0. \lnot 0=0.
  26. { 0 , 1 } , \{0,1\},
  27. \oplus
  28. ¬ \lnot
  29. x x = x x\oplus x=x
  30. x x x = x x x\oplus x\oplus x=x\oplus x
  31. n n
  32. { 0 , 1 / ( n - 1 ) , 2 / ( n - 1 ) , , 1 } , \{0,1/(n-1),2/(n-1),\dots,1\},
  33. \oplus
  34. ¬ \lnot
  35. , ¬ , \oplus,\lnot,
  36. ¬ \lnot

Myhill_isomorphism_theorem.html

  1. f ( A ) B f(A)\subseteq B
  2. f ( A ) B f(\mathbb{N}\setminus A)\subseteq\mathbb{N}\setminus B

N-skeleton.html

  1. K * K_{*}
  2. K i , i 0 K_{i},\ i\geq 0
  3. s k n ( K * ) sk_{n}(K_{*})
  4. K i K_{i}
  5. i > n i>n
  6. K i K_{i}
  7. i n i\leq n
  8. i > n i>n
  9. i * : Δ o p S e t s Δ n o p S e t s i_{*}:\Delta^{op}Sets\rightarrow\Delta^{op}_{\leq n}Sets
  10. i * i^{*}
  11. i * , i * i^{*},i_{*}
  12. K * K_{*}
  13. s k n ( K ) := i * i * K . sk_{n}(K):=i^{*}i_{*}K.
  14. i * i_{*}
  15. i ! i^{!}
  16. c o s k n ( K ) := i ! i * K . cosk_{n}(K):=i^{!}i_{*}K.
  17. K 0 K_{0}
  18. K 0 × K 0 K 0 . \dots\rightarrow K_{0}\times K_{0}\rightarrow K_{0}.

Nagata–Biran_conjecture.html

  1. ε ( p 1 , , p r ; X , L ) = d r . \varepsilon(p_{1},\ldots,p_{r};X,L)={d\over\sqrt{r}}.

Nambu–Jona-Lasinio_model.html

  1. = i ψ ¯ / ψ + λ 4 [ ( ψ ¯ ψ ) ( ψ ¯ ψ ) - ( ψ ¯ γ 5 ψ ) ( ψ ¯ γ 5 ψ ) ] = i ψ ¯ L / ψ L + i ψ ¯ R / ψ R + λ ( ψ ¯ L ψ R ) ( ψ ¯ R ψ L ) . \mathcal{L}=\,i\,\bar{\psi}\partial\!\!\!/\psi+\frac{\lambda}{4}\,\left[\left(% \bar{\psi}\psi\right)\left(\bar{\psi}\psi\right)-\left(\bar{\psi}\gamma^{5}% \psi\right)\left(\bar{\psi}\gamma^{5}\psi\right)\right]=\,i\,\bar{\psi}_{L}% \partial\!\!\!/\psi_{L}+\,i\,\bar{\psi}_{R}\partial\!\!\!/\psi_{R}+\lambda\,% \left(\bar{\psi}_{L}\psi_{R}\right)\left(\bar{\psi}_{R}\psi_{L}\right).
  2. = i ψ ¯ a / ψ a + λ 4 N [ ( ψ ¯ a ψ b ) ( ψ ¯ b ψ a ) - ( ψ ¯ a γ 5 ψ b ) ( ψ ¯ b γ 5 ψ a ) ] = i ψ ¯ L a / ψ L a + i ψ ¯ R a / ψ R a + λ N ( ψ ¯ L a ψ R b ) ( ψ ¯ R b ψ L a ) . \mathcal{L}=\,i\,\bar{\psi}_{a}\partial\!\!\!/\psi^{a}+\frac{\lambda}{4N}\,% \left[\left(\bar{\psi}_{a}\psi^{b}\right)\left(\bar{\psi}_{b}\psi^{a}\right)-% \left(\bar{\psi}_{a}\gamma^{5}\psi^{b}\right)\left(\bar{\psi}_{b}\gamma^{5}% \psi^{a}\right)\right]=\,i\,\bar{\psi}_{La}\partial\!\!\!/\psi_{L}^{a}+\,i\,% \bar{\psi}_{Ra}\partial\!\!\!/\psi_{R}^{a}+\frac{\lambda}{N}\,\left(\bar{\psi}% _{La}\psi_{R}^{b}\right)\left(\bar{\psi}_{Rb}\psi_{L}^{a}\right).

Napoleon's_problem.html

  1. A H A B = A B A A \frac{AH}{AB}=\frac{AB}{AA^{\prime}}
  2. A H = b 2 2 a AH=\frac{b^{2}}{2a}
  3. A C = R 2 r AC=\frac{R^{2}}{r}
  4. a 2 = R 2 r a_{2}=\frac{R^{2}}{r}
  5. A O = R 2 R 2 / r = r AO=\frac{R^{2}}{R^{2}/r}=r
  6. 3 \sqrt{3}
  7. 2 \sqrt{2}

Napoleon's_theorem.html

  1. Δ \Delta
  2. Area(inner) = - Δ 2 + 3 24 ( a 2 + b 2 + c 2 ) 0 , \text{Area(inner)}=-\frac{\Delta}{2}+\frac{\sqrt{3}}{24}(a^{2}+b^{2}+c^{2})% \geq 0,
  3. Area(outer) = Δ 2 + 3 24 ( a 2 + b 2 + c 2 ) . \text{Area(outer)}=\frac{\Delta}{2}+\frac{\sqrt{3}}{24}(a^{2}+b^{2}+c^{2}).
  4. Side(outer) = a 2 + b 2 + c 2 6 + ( a + b + c ) ( a + b - c ) ( a - b + c ) ( - a + b + c ) 2 3 . \,\text{Side(outer)}=\sqrt{{a^{2}+b^{2}+c^{2}\over 6}+{\sqrt{(a+b+c)(a+b-c)(a-% b+c)(-a+b+c)}\over{2\sqrt{3}}}}.
  5. 3 / 4. \sqrt{3}/4.

Narrow_class_group.html

  1. C K = I K / P K , C_{K}=I_{K}/P_{K},\,\!
  2. C K + = I K / P K + , C_{K}^{+}=I_{K}/P_{K}^{+},
  3. σ : K 𝐑 . \sigma:K\to\mathbf{R}.
  4. K = 𝐐 ( d ) , K=\mathbf{Q}(\sqrt{d}),
  5. { ω 1 , ω 2 } \{\omega_{1},\omega_{2}\}\,\!
  6. q K ( x , y ) = N K / 𝐐 ( ω 1 x + ω 2 y ) q_{K}(x,y)=N_{K/\mathbf{Q}}(\omega_{1}x+\omega_{2}y)
  7. p = q K ( x , y ) p=q_{K}(x,y)\,\!
  8. p d K , p\mid d_{K}\,\!,
  9. p = 2 and d K 1 ( mod 8 ) , p=2\quad\mbox{and}~{}\quad d_{K}\equiv 1\;\;(\mathop{{\rm mod}}8),
  10. p > 2 and ( d K p ) = 1 , p>2\quad\mbox{and}~{}\quad\left(\frac{d_{K}}{p}\right)=1,
  11. ( a b ) \left(\frac{a}{b}\right)
  12. p = 2 or p 1 ( mod 4 ) . p=2\quad\mbox{or}~{}\quad p\equiv 1\;\;(\mathop{{\rm mod}}4).
  13. p = 2 or p 1 , 7 ( mod 8 ) . p=2\quad\mbox{or}~{}\quad p\equiv 1,7\;\;(\mathop{{\rm mod}}8).
  14. p = 3 or p 1 ( mod 3 ) . p=3\quad\mbox{or}~{}\quad p\equiv 1\;\;(\mathop{{\rm mod}}3).

Nash–Moser_theorem.html

  1. F F
  2. G G
  3. P : U F G P:U\subseteq F\rightarrow G
  4. D P ( f ) h = k DP(f)h=k
  5. h = V P ( f ) k h=VP(f)k
  6. f U f\in U
  7. k k
  8. V P : U × G F VP:U\times G\rightarrow F
  9. P - 1 P^{-1}

Nat_(unit).html

  1. H = - i p i ln p i H=-\sum_{i}p_{i}\ln p_{i}\!\,
  2. ( 2 x = e 1 x = 1 ln 2 ) \left(2^{x}=e^{1}\Rightarrow x=\tfrac{1}{\ln 2}\right)
  3. ( 10 x = e 1 x = 1 ln 10 ) \left(10^{x}=e^{1}\Rightarrow x=\tfrac{1}{\ln 10}\right)

Nearest_integer_function.html

  1. [ x ] [x]
  2. x \lfloor x\rceil
  3. x \|x\|
  4. [ 1.25 ] = 1 [1.25]=1
  5. [ 1.50 ] = 2 [1.50]=2
  6. [ 1.75 ] = 2 [1.75]=2
  7. [ 2.25 ] = 2 [2.25]=2
  8. [ 2.50 ] = 2 [2.50]=2
  9. [ 2.75 ] = 3 [2.75]=3
  10. [ 3.25 ] = 3 [3.25]=3
  11. [ 3.50 ] = 4 [3.50]=4
  12. [ 3.75 ] = 4 [3.75]=4
  13. [ 4.50 ] = 4 [4.50]=4

Negation_as_failure.html

  1. not p \mathrm{not}~{}p
  2. p ~{}p
  3. p ~{}p
  4. not p \mathrm{not}~{}p
  5. ¬ p \neg p
  6. p ~{}p
  7. n o t p not~{}p
  8. p q and not r p\leftarrow q\and\mathrm{not}~{}r
  9. q s q\leftarrow s
  10. q t q\leftarrow t
  11. t t\leftarrow
  12. not s \mathrm{not}~{}s
  13. not r \mathrm{not}~{}r
  14. p ~{}p
  15. \leftarrow
  16. \equiv
  17. p q and not r p\equiv q\and\mathrm{not}~{}r
  18. q s t q\equiv st
  19. t true t\equiv\mathrm{true}
  20. r false r\equiv\mathrm{false}
  21. s false s\equiv\mathrm{false}
  22. not p \mathrm{not}~{}p
  23. not p not q r \mathrm{not}~{}p\equiv\mathrm{not}~{}qr
  24. not q not s and not t \mathrm{not}~{}q\equiv\mathrm{not}~{}s\and\mathrm{not}~{}t
  25. not t false \mathrm{not}~{}t\equiv\mathrm{false}
  26. not r true \mathrm{not}~{}r\equiv\mathrm{true}
  27. not s true \mathrm{not}~{}s\equiv\mathrm{true}
  28. p ( a ) p(a)\leftarrow
  29. p ( b ) p(b)\leftarrow
  30. not p ( c ) \mathrm{not}~{}p(c)
  31. p ( X ) X = a X = b p(X)\equiv X=aX=b
  32. not p \mathrm{not}~{}p
  33. ¬ p \neg p
  34. p p
  35. not p \mathrm{not}~{}p
  36. p p
  37. p p
  38. p p
  39. not p \mathrm{not}~{}p
  40. p p
  41. p not p p\leftarrow\mathrm{not}~{}p
  42. p not q p\leftarrow\mathrm{not}~{}q
  43. not q \mathrm{not}~{}q
  44. p not q p\leftarrow\mathrm{not}~{}q
  45. q not p q\leftarrow\mathrm{not}~{}p
  46. not p \mathrm{not}~{}p
  47. not q \mathrm{not}~{}q
  48. ¬ p not p \neg p\leftarrow\mathrm{not}~{}p
  49. p not ¬ p p\leftarrow\mathrm{not}~{}\neg p
  50. p p

Negative_impedance_converter.html

  1. V opamp = V s ( 1 + R 2 R 1 ) V_{\,\text{opamp}}=V_{s}\left(1+\frac{R_{2}}{R_{1}}\right)\,
  2. R 3 R_{3}
  3. V s V_{s}
  4. - I s -I_{s}
  5. - I s = V opamp - V s R 3 = V s R 2 R 1 R 3 . -I_{s}=\frac{V_{\,\text{opamp}}-V_{s}}{R_{3}}=V_{s}\frac{\frac{R_{2}}{R_{1}}}{% R_{3}}.
  6. V s V_{s}
  7. - I s -I_{s}
  8. V s V_{s}
  9. R in V s I s = - R 3 R 1 R 2 . R_{\,\text{in}}\triangleq\frac{V_{s}}{I_{s}}=-R_{3}\frac{R_{1}}{R_{2}}.
  10. R 1 R_{1}
  11. R 2 R_{2}
  12. R 3 R_{3}
  13. R s R_{s}
  14. R s R_{s}
  15. - R s -R_{s}
  16. lim R NIC R s + R s ( - R INIC ) lim R INIC R s + - R s R INIC R s + - R INIC = . \lim\limits_{R_{\,\text{NIC}}\to R_{s}+}R_{s}\|(-R_{\,\text{INIC}})\triangleq% \lim\limits_{R_{\,\text{INIC}}\to R_{s}+}\frac{-R_{s}R_{\,\text{INIC}}}{R_{s}+% -R_{\,\text{INIC}}}=\infty.
  17. Z L Z_{L}
  18. R s R_{s}
  19. R s R_{s}
  20. R NIC R_{\,\text{NIC}}
  21. R INIC > R s R_{\,\text{INIC}}>R_{s}
  22. R s R_{s}
  23. R INIC R s R_{\,\text{INIC}}\gg R_{s}
  24. 1 R INIC > 1 R s + 1 R L , (i.e., when R INIC < R s R L ) \frac{1}{R_{\,\text{INIC}}}>\frac{1}{R_{s}}+\frac{1}{R_{L}},\quad\,\text{(i.e.% , when}\,R_{\,\text{INIC}}<R_{s}\|R_{L}\,\text{)}\,
  25. R INIC < R s R_{\,\text{INIC}}<R_{s}
  26. Z in = v i = - Z Z\text{in}={v\over i}=-Z
  27. R in = v i = - R R\text{in}={v\over i}=-R
  28. Z in = v i = j ω C Z\text{in}={v\over i}={j\over{\omega C}}
  29. Z in = v i = - j ω C R 1 2 Z\text{in}={v\over i}=-j\omega CR_{1}^{2}

Negative_refraction.html

  1. P k < 0 \scriptstyle\vec{P}\cdot\vec{k}<0
  2. ϵ r | μ | + μ r | ϵ | < 0 , \epsilon_{r}|\mu|+\mu_{r}|\epsilon|<0,
  3. ϵ r , μ r \epsilon_{r},\mu_{r}
  4. n = ± ϵ μ \scriptstyle n=\pm\sqrt{\epsilon\mu}

Negative_thermal_expansion.html

  1. Π ′′′ ( a ) > 0 , \Pi^{\prime\prime\prime}(a)>0,
  2. Π \Pi
  3. a a
  4. Π ′′′ ( a ) a > - ( d - 1 ) Π ′′ ( a ) , \Pi^{\prime\prime\prime}(a)a>-(d-1)\Pi^{\prime\prime}(a),
  5. d d
  6. Π ′′ \Pi^{\prime\prime}
  7. Π ′′′ \Pi^{\prime\prime\prime}

Nelder–Mead_method.html

  1. f ( x ) f(x)
  2. x n x\in\mathbb{R}^{n}
  3. x 1 , , x n + 1 x_{1},\ldots,x_{n+1}
  4. f ( 𝐱 1 ) f ( 𝐱 2 ) f ( 𝐱 n + 1 ) f(\,\textbf{x}_{1})\leq f(\,\textbf{x}_{2})\leq\cdots\leq f(\,\textbf{x}_{n+1})
  5. 𝐱 o \,\textbf{x}_{o}
  6. 𝐱 n + 1 \,\textbf{x}_{n+1}
  7. 𝐱 r = 𝐱 o + α ( 𝐱 o - 𝐱 n + 1 ) \,\textbf{x}_{r}=\,\textbf{x}_{o}+\alpha(\,\textbf{x}_{o}-\,\textbf{x}_{n+1})
  8. f ( 𝐱 1 ) f ( 𝐱 r ) < f ( 𝐱 n ) f(\,\textbf{x}_{1})\leq f(\,\textbf{x}_{r})<f(\,\textbf{x}_{n})
  9. 𝐱 n + 1 \,\textbf{x}_{n+1}
  10. 𝐱 r \,\textbf{x}_{r}
  11. f ( 𝐱 r ) < f ( 𝐱 1 ) , f(\,\textbf{x}_{r})<f(\,\textbf{x}_{1}),
  12. 𝐱 e = 𝐱 o + γ ( 𝐱 o - 𝐱 n + 1 ) \,\textbf{x}_{e}=\,\textbf{x}_{o}+\gamma(\,\textbf{x}_{o}-\,\textbf{x}_{n+1})
  13. f ( 𝐱 e ) < f ( 𝐱 r ) f(\,\textbf{x}_{e})<f(\,\textbf{x}_{r})
  14. 𝐱 n + 1 \,\textbf{x}_{n+1}
  15. 𝐱 e \,\textbf{x}_{e}
  16. 𝐱 n + 1 \,\textbf{x}_{n+1}
  17. 𝐱 r \,\textbf{x}_{r}
  18. f ( 𝐱 r ) f ( 𝐱 n ) f(\,\textbf{x}_{r})\geq f(\,\textbf{x}_{n})
  19. 𝐱 c = 𝐱 o + ρ ( 𝐱 o - 𝐱 n + 1 ) \,\textbf{x}_{c}=\,\textbf{x}_{o}+\rho(\,\textbf{x}_{o}-\,\textbf{x}_{n+1})
  20. f ( 𝐱 c ) < f ( 𝐱 n + 1 ) f(\,\textbf{x}_{c})<f(\,\textbf{x}_{n+1})
  21. 𝐱 n + 1 \,\textbf{x}_{n+1}
  22. 𝐱 c \,\textbf{x}_{c}
  23. 𝐱 i = 𝐱 1 + σ ( 𝐱 i - 𝐱 1 ) for all i { 2 , , n + 1 } \,\textbf{x}_{i}=\,\textbf{x}_{1}+\sigma(\,\textbf{x}_{i}-\,\textbf{x}_{1})\,% \text{ for all i }\in\{2,\dots,n+1\}
  24. α \alpha
  25. γ \gamma
  26. ρ \rho
  27. σ \sigma
  28. α = 1 \alpha=1
  29. γ = 2 \gamma=2
  30. ρ = - 1 / 2 \rho=-1/2
  31. σ = 1 / 2 \sigma=1/2
  32. 𝐱 n + 1 \,\textbf{x}_{n+1}
  33. 𝐱 n + 1 \,\textbf{x}_{n+1}
  34. 𝐱 i \,\textbf{x}_{i}
  35. 𝐱 n + 1 \,\textbf{x}_{n+1}
  36. 𝐱 r \,\textbf{x}_{r}
  37. 𝐱 o \,\textbf{x}_{o}
  38. 𝐱 r \,\textbf{x}_{r}
  39. f ( 𝐱 r ) > f ( 𝐱 n ) f(\,\textbf{x}_{r})>f(\,\textbf{x}_{n})
  40. 𝐱 i \,\textbf{x}_{i}
  41. f f

Neo-Hookean_solid.html

  1. W = C 1 ( I 1 - 3 ) W=C_{1}(I_{1}-3)\,
  2. C 1 C_{1}
  3. I 1 I_{1}
  4. I 1 = λ 1 2 + λ 2 2 + λ 3 2 I_{1}=\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}~{}
  5. λ i \lambda_{i}
  6. W = C 1 ( I ¯ 1 - 3 ) + D 1 ( J - 1 ) 2 ; J = det ( s y m b o l F ) = λ 1 λ 2 λ 3 W=C_{1}~{}(\bar{I}_{1}-3)+D_{1}~{}(J-1)^{2}~{};~{}~{}J=\det(symbol{F})=\lambda% _{1}\lambda_{2}\lambda_{3}
  7. D 1 D_{1}
  8. I ¯ 1 = J - 2 / 3 I 1 \bar{I}_{1}=J^{-2/3}I_{1}
  9. s y m b o l F symbol{F}
  10. W = C 1 ( I ¯ 1 - 2 ) + D 1 ( J - 1 ) 2 ; W=C_{1}~{}(\bar{I}_{1}-2)+D_{1}~{}(J-1)^{2}~{};
  11. I ¯ 1 = J - 1 I 1 \bar{I}_{1}=J^{-1}I_{1}
  12. W = C 1 ( I 1 - 3 - 2 ln J ) + D 1 ( ln J ) 2 W=C_{1}~{}(I_{1}-3-2\ln J)+D_{1}~{}(\ln J)^{2}
  13. C 1 = μ 2 ; D 1 = κ 2 C_{1}=\cfrac{\mu}{2}~{};~{}~{}D_{1}=\cfrac{\kappa}{2}
  14. μ \mu
  15. κ \kappa
  16. J s y m b o l σ = - p s y m b o l 1 + 2 C 1 dev ( s y m b o l B ¯ ) = - p s y m b o l 1 + 2 C 1 J 2 / 3 dev ( s y m b o l B ) J~{}symbol{\sigma}=-p~{}symbol{\mathit{1}}+2C_{1}~{}\mathrm{dev}(\bar{symbol{B% }})=-p~{}symbol{\mathit{1}}+\frac{2C_{1}}{J^{2/3}}~{}\mathrm{dev}(symbol{B})
  17. s y m b o l B symbol{B}
  18. p := - 2 D 1 J ( J - 1 ) ; dev ( s y m b o l B ¯ ) = s y m b o l B ¯ - 1 3 I ¯ 1 s y m b o l 1 ; s y m b o l B ¯ = J - 2 / 3 s y m b o l B . p:=-2D_{1}~{}J(J-1)~{};~{}~{}\mathrm{dev}(\bar{symbol{B}})=\bar{symbol{B}}-% \tfrac{1}{3}\bar{I}_{1}symbol{\mathit{1}}~{};~{}~{}\bar{symbol{B}}=J^{-2/3}% symbol{B}~{}.
  19. s y m b o l ε symbol{\varepsilon}
  20. J 1 + tr ( s y m b o l ε ) ; s y m b o l B s y m b o l 1 + 2 s y m b o l ε J\approx 1+\mathrm{tr}(symbol{\varepsilon})~{};~{}~{}symbol{B}\approx symbol{% \mathit{1}}+2symbol{\varepsilon}
  21. s y m b o l σ 4 C 1 ( s y m b o l ε - 1 3 tr ( s y m b o l ε ) s y m b o l 1 ) + 2 D 1 tr ( s y m b o l ε ) s y m b o l 1 symbol{\sigma}\approx 4C_{1}\left(symbol{\varepsilon}-\tfrac{1}{3}\mathrm{tr}(% symbol{\varepsilon})symbol{\mathit{1}}\right)+2D_{1}\mathrm{tr}(symbol{% \varepsilon})symbol{\mathit{1}}
  22. μ = 2 C 1 \mu=2C_{1}
  23. κ = 2 D 1 \kappa=2D_{1}
  24. s y m b o l σ = 2 J [ 1 J 2 / 3 ( W I ¯ 1 + I ¯ 1 W I ¯ 2 ) s y m b o l B - 1 J 4 / 3 W I ¯ 2 s y m b o l B \cdotsymbol B ] + [ W J - 2 3 J ( I ¯ 1 W I ¯ 1 + 2 I ¯ 2 W I ¯ 2 ) ] s y m b o l 1 symbol{\sigma}=\cfrac{2}{J}\left[\cfrac{1}{J^{2/3}}\left(\cfrac{\partial{W}}{% \partial\bar{I}_{1}}+\bar{I}_{1}~{}\cfrac{\partial{W}}{\partial\bar{I}_{2}}% \right)symbol{B}-\cfrac{1}{J^{4/3}}~{}\cfrac{\partial{W}}{\partial\bar{I}_{2}}% ~{}symbol{B}\cdotsymbol{B}\right]+\left[\cfrac{\partial{W}}{\partial J}-\cfrac% {2}{3J}\left(\bar{I}_{1}~{}\cfrac{\partial{W}}{\partial\bar{I}_{1}}+2~{}\bar{I% }_{2}~{}\cfrac{\partial{W}}{\partial\bar{I}_{2}}\right)\right]~{}symbol{% \mathit{1}}
  25. W I ¯ 1 = C 1 ; W I ¯ 2 = 0 ; W J = 2 D 1 ( J - 1 ) \cfrac{\partial{W}}{\partial\bar{I}_{1}}=C_{1}~{};~{}~{}\cfrac{\partial{W}}{% \partial\bar{I}_{2}}=0~{};~{}~{}\cfrac{\partial{W}}{\partial J}=2D_{1}(J-1)
  26. W I ¯ 1 = C 1 ; W I ¯ 2 = 0 ; W J = 2 D 1 ( J - 1 ) - 2 C 1 J \cfrac{\partial{W}}{\partial\bar{I}_{1}}=C_{1}~{};~{}~{}\cfrac{\partial{W}}{% \partial\bar{I}_{2}}=0~{};~{}~{}\cfrac{\partial{W}}{\partial J}=2D_{1}(J-1)-% \cfrac{2C_{1}}{J}
  27. s y m b o l σ = 2 J [ 1 J 2 / 3 C 1 s y m b o l B ] + [ 2 D 1 ( J - 1 ) - 2 3 J C 1 I ¯ 1 ] s y m b o l 1 symbol{\sigma}=\cfrac{2}{J}\left[\cfrac{1}{J^{2/3}}~{}C_{1}~{}symbol{B}\right]% +\left[2D_{1}(J-1)-\cfrac{2}{3J}~{}C_{1}\bar{I}_{1}\right]symbol{\mathit{1}}
  28. s y m b o l σ = 2 J [ 1 J 2 / 3 C 1 s y m b o l B ] + [ 2 D 1 ( J - 1 ) - 2 C 1 J - 2 3 J C 1 I ¯ 1 ] s y m b o l 1 symbol{\sigma}=\cfrac{2}{J}\left[\cfrac{1}{J^{2/3}}~{}C_{1}~{}symbol{B}\right]% +\left[2D_{1}(J-1)-\cfrac{2C_{1}}{J}-\cfrac{2}{3J}~{}C_{1}\bar{I}_{1}\right]% symbol{\mathit{1}}
  29. s y m b o l B ¯ = J - 2 / 3 s y m b o l B \bar{symbol{B}}=J^{-2/3}symbol{B}
  30. s y m b o l σ = 2 C 1 J [ s y m b o l B ¯ - 1 3 I ¯ 1 s y m b o l 1 ] + 2 D 1 ( J - 1 ) s y m b o l 1 = 2 C 1 J dev ( s y m b o l B ¯ ) + 2 D 1 ( J - 1 ) s y m b o l 1 symbol{\sigma}=\cfrac{2C_{1}}{J}\left[\bar{symbol{B}}-\tfrac{1}{3}\bar{I}_{1}% symbol{\mathit{1}}\right]+2D_{1}(J-1)symbol{\mathit{1}}=\cfrac{2C_{1}}{J}% \mathrm{dev}(\bar{symbol{B}})+2D_{1}(J-1)symbol{\mathit{1}}
  31. s y m b o l σ = 2 C 1 J [ s y m b o l B ¯ - 1 3 I ¯ 1 s y m b o l 1 - s y m b o l 1 ] + 2 D 1 ( J - 1 ) s y m b o l 1 = 2 C 1 J [ dev ( s y m b o l B ¯ ) - s y m b o l 1 ] + 2 D 1 ( J - 1 ) s y m b o l 1 symbol{\sigma}=\cfrac{2C_{1}}{J}\left[\bar{symbol{B}}-\tfrac{1}{3}\bar{I}_{1}% symbol{\mathit{1}}-symbol{\mathit{1}}\right]+2D_{1}(J-1)symbol{\mathit{1}}=% \cfrac{2C_{1}}{J}\left[\mathrm{dev}(\bar{symbol{B}})-symbol{\mathit{1}}\right]% +2D_{1}(J-1)symbol{\mathit{1}}
  32. p := - 2 D 1 J ( J - 1 ) ; p * = - 2 D 1 J ( J - 1 ) + 2 C 1 p:=-2D_{1}~{}J(J-1)~{};~{}~{}p^{*}=-2D_{1}~{}J(J-1)+2C_{1}
  33. s y m b o l τ = J s y m b o l σ = - p s y m b o l 1 + 2 C 1 dev ( s y m b o l B ¯ ) symbol{\tau}=J~{}symbol{\sigma}=-psymbol{\mathit{1}}+2C_{1}~{}\mathrm{dev}(% \bar{symbol{B}})
  34. s y m b o l τ = - p * s y m b o l 1 + 2 C 1 dev ( s y m b o l B ¯ ) symbol{\tau}=-p^{*}symbol{\mathit{1}}+2C_{1}~{}\mathrm{dev}(\bar{symbol{B}})
  35. J = 1 J=1
  36. s y m b o l σ = - p s y m b o l 1 + 2 C 1 s y m b o l B symbol{\sigma}=-p~{}symbol{\mathit{1}}+2C_{1}symbol{B}
  37. p p
  38. σ i = 2 C 1 J - 5 / 3 [ λ i 2 - I 1 3 ] + 2 D 1 ( J - 1 ) ; i = 1 , 2 , 3 \sigma_{i}=2C_{1}J^{-5/3}\left[\lambda_{i}^{2}-\cfrac{I_{1}}{3}\right]+2D_{1}(% J-1)~{};~{}~{}i=1,2,3
  39. σ 11 - σ 33 = 2 C 1 J 5 / 3 ( λ 1 2 - λ 3 2 ) ; σ 22 - σ 33 = 2 C 1 J 5 / 3 ( λ 2 2 - λ 3 2 ) \sigma_{11}-\sigma_{33}=\cfrac{2C_{1}}{J^{5/3}}(\lambda_{1}^{2}-\lambda_{3}^{2% })~{};~{}~{}\sigma_{22}-\sigma_{33}=\cfrac{2C_{1}}{J^{5/3}}(\lambda_{2}^{2}-% \lambda_{3}^{2})
  40. σ i = λ i λ 1 λ 2 λ 3 W λ i ; i = 1 , 2 , 3 \sigma_{i}=\cfrac{\lambda_{i}}{\lambda_{1}\lambda_{2}\lambda_{3}}~{}\frac{% \partial W}{\partial\lambda_{i}}~{};~{}~{}i=1,2,3
  41. W = C 1 ( I ¯ 1 - 3 ) + D 1 ( J - 1 ) 2 = C 1 [ J - 2 / 3 ( λ 1 2 + λ 2 2 + λ 3 2 ) - 3 ] + D 1 ( J - 1 ) 2 W=C_{1}(\bar{I}_{1}-3)+D_{1}(J-1)^{2}=C_{1}\left[J^{-2/3}(\lambda_{1}^{2}+% \lambda_{2}^{2}+\lambda_{3}^{2})-3\right]+D_{1}(J-1)^{2}
  42. λ i W λ i = C 1 [ - 2 3 J - 5 / 3 λ i J λ i ( λ 1 2 + λ 2 2 + λ 3 2 ) + 2 J - 2 / 3 λ i 2 ] + 2 D 1 ( J - 1 ) λ i J λ i \lambda_{i}\frac{\partial W}{\partial\lambda_{i}}=C_{1}\left[-\frac{2}{3}J^{-5% /3}\lambda_{i}\frac{\partial J}{\partial\lambda_{i}}(\lambda_{1}^{2}+\lambda_{% 2}^{2}+\lambda_{3}^{2})+2J^{-2/3}\lambda_{i}^{2}\right]+2D_{1}(J-1)\lambda_{i}% \frac{\partial J}{\partial\lambda_{i}}
  43. J = λ 1 λ 2 λ 3 J=\lambda_{1}\lambda_{2}\lambda_{3}
  44. λ i J λ i = λ 1 λ 2 λ 3 = J \lambda_{i}\frac{\partial J}{\partial\lambda_{i}}=\lambda_{1}\lambda_{2}% \lambda_{3}=J
  45. λ i W λ i = C 1 [ - 2 3 J - 2 / 3 ( λ 1 2 + λ 2 2 + λ 3 2 ) + 2 J - 2 / 3 λ i 2 ] + 2 D 1 J ( J - 1 ) = 2 C 1 J - 2 / 3 [ - 1 3 ( λ 1 2 + λ 2 2 + λ 3 2 ) + λ i 2 ] + 2 D 1 J ( J - 1 ) \begin{aligned}\displaystyle\lambda_{i}\frac{\partial W}{\partial\lambda_{i}}&% \displaystyle=C_{1}\left[-\frac{2}{3}J^{-2/3}(\lambda_{1}^{2}+\lambda_{2}^{2}+% \lambda_{3}^{2})+2J^{-2/3}\lambda_{i}^{2}\right]+2D_{1}J(J-1)\\ &\displaystyle=2C_{1}J^{-2/3}\left[-\frac{1}{3}(\lambda_{1}^{2}+\lambda_{2}^{2% }+\lambda_{3}^{2})+\lambda_{i}^{2}\right]+2D_{1}J(J-1)\end{aligned}
  46. σ i = 2 C 1 J - 5 / 3 [ λ i 2 - I 1 3 ] + 2 D 1 ( J - 1 ) \sigma_{i}=2C_{1}J^{-5/3}\left[\lambda_{i}^{2}-\cfrac{I_{1}}{3}\right]+2D_{1}(% J-1)
  47. σ 11 - σ 33 = λ 1 W λ 1 - λ 3 W λ 3 ; σ 22 - σ 33 = λ 2 W λ 2 - λ 3 W λ 3 \sigma_{11}-\sigma_{33}=\lambda_{1}~{}\cfrac{\partial{W}}{\partial\lambda_{1}}% -\lambda_{3}~{}\cfrac{\partial{W}}{\partial\lambda_{3}}~{};~{}~{}\sigma_{22}-% \sigma_{33}=\lambda_{2}~{}\cfrac{\partial{W}}{\partial\lambda_{2}}-\lambda_{3}% ~{}\cfrac{\partial{W}}{\partial\lambda_{3}}
  48. W = C 1 ( λ 1 2 + λ 2 2 + λ 3 2 - 3 ) ; λ 1 λ 2 λ 3 = 1 W=C_{1}(\lambda_{1}^{2}+\lambda_{2}^{2}+\lambda_{3}^{2}-3)~{};~{}~{}\lambda_{1% }\lambda_{2}\lambda_{3}=1
  49. W λ 1 = 2 C 1 λ 1 ; W λ 2 = 2 C 1 λ 2 ; W λ 3 = 2 C 1 λ 3 \cfrac{\partial{W}}{\partial\lambda_{1}}=2C_{1}\lambda_{1}~{};~{}~{}\cfrac{% \partial{W}}{\partial\lambda_{2}}=2C_{1}\lambda_{2}~{};~{}~{}\cfrac{\partial{W% }}{\partial\lambda_{3}}=2C_{1}\lambda_{3}
  50. σ 11 - σ 33 = 2 ( λ 1 2 - λ 3 2 ) C 1 ; σ 22 - σ 33 = 2 ( λ 2 2 - λ 3 2 ) C 1 \sigma_{11}-\sigma_{33}=2(\lambda_{1}^{2}-\lambda_{3}^{2})C_{1}~{};~{}~{}% \sigma_{22}-\sigma_{33}=2(\lambda_{2}^{2}-\lambda_{3}^{2})C_{1}
  51. λ 1 = λ ; λ 2 = λ 3 = J λ ; I 1 = λ 2 + 2 J λ \lambda_{1}=\lambda~{};~{}~{}\lambda_{2}=\lambda_{3}=\sqrt{\tfrac{J}{\lambda}}% ~{};~{}~{}I_{1}=\lambda^{2}+\tfrac{2J}{\lambda}
  52. σ 11 = 4 C 1 3 J 5 / 3 ( λ 2 - J λ ) + 2 D 1 ( J - 1 ) σ 22 = σ 33 = 2 C 1 3 J 5 / 3 ( J λ - λ 2 ) + 2 D 1 ( J - 1 ) \begin{aligned}\displaystyle\sigma_{11}&\displaystyle=\cfrac{4C_{1}}{3J^{5/3}}% \left(\lambda^{2}-\tfrac{J}{\lambda}\right)+2D_{1}(J-1)\\ \displaystyle\sigma_{22}&\displaystyle=\sigma_{33}=\cfrac{2C_{1}}{3J^{5/3}}% \left(\tfrac{J}{\lambda}-\lambda^{2}\right)+2D_{1}(J-1)\end{aligned}
  53. σ 11 - σ 33 = 2 C 1 J 5 / 3 ( λ 2 - J λ ) ; σ 22 - σ 33 = 0 \sigma_{11}-\sigma_{33}=\cfrac{2C_{1}}{J^{5/3}}\left(\lambda^{2}-\tfrac{J}{% \lambda}\right)~{};~{}~{}\sigma_{22}-\sigma_{33}=0
  54. σ 22 = σ 33 = 0 \sigma_{22}=\sigma_{33}=0
  55. σ 11 = 2 C 1 J 5 / 3 ( λ 2 - J λ ) \sigma_{11}=\cfrac{2C_{1}}{J^{5/3}}\left(\lambda^{2}-\tfrac{J}{\lambda}\right)
  56. σ 11 \sigma_{11}
  57. J J
  58. λ \lambda
  59. 4 C 1 3 J 5 / 3 ( λ 2 - J λ ) + 2 D 1 ( J - 1 ) = 2 C 1 J 5 / 3 ( λ 2 - J λ ) \cfrac{4C_{1}}{3J^{5/3}}\left(\lambda^{2}-\tfrac{J}{\lambda}\right)+2D_{1}(J-1% )=\cfrac{2C_{1}}{J^{5/3}}\left(\lambda^{2}-\tfrac{J}{\lambda}\right)
  60. D 1 J 8 / 3 - D 1 J 5 / 3 + C 1 3 λ J - C 1 λ 2 3 = 0 D_{1}J^{8/3}-D_{1}J^{5/3}+\tfrac{C_{1}}{3\lambda}J-\tfrac{C_{1}\lambda^{2}}{3}=0
  61. λ 1 = λ \lambda_{1}=\lambda\,
  62. λ 2 = λ 3 = 1 / λ \lambda_{2}=\lambda_{3}=1/\sqrt{\lambda}
  63. σ 11 - σ 33 = 2 C 1 ( λ 2 - 1 λ ) ; σ 22 - σ 33 = 0 \sigma_{11}-\sigma_{33}=2C_{1}\left(\lambda^{2}-\cfrac{1}{\lambda}\right)~{};~% {}~{}\sigma_{22}-\sigma_{33}=0
  64. σ 22 = σ 33 = 0 \sigma_{22}=\sigma_{33}=0
  65. σ 11 = 2 C 1 ( λ 2 - 1 λ ) = 2 C 1 ( 3 ε 11 + 3 ε 11 2 + ε 11 3 1 + ε 11 ) \sigma_{11}=2C_{1}\left(\lambda^{2}-\cfrac{1}{\lambda}\right)=2C_{1}\left(% \frac{3\varepsilon_{11}+3\varepsilon_{11}^{2}+\varepsilon_{11}^{3}}{1+% \varepsilon_{11}}\right)
  66. ε 11 = λ - 1 \varepsilon_{11}=\lambda-1
  67. T 11 = 2 C 1 ( α 2 - 1 α ) T_{11}=2C_{1}\left(\alpha^{2}-\cfrac{1}{\alpha}\right)
  68. σ 11 eng = 2 C 1 ( λ - 1 λ 2 ) \sigma_{11}^{\mathrm{eng}}=2C_{1}\left(\lambda-\cfrac{1}{\lambda^{2}}\right)
  69. ε 1 \varepsilon\ll 1
  70. σ 11 = 6 C 1 ε = 3 μ ε \sigma_{11}=6C_{1}\varepsilon=3\mu\varepsilon
  71. 3 μ 3\mu
  72. E = 2 μ ( 1 + ν ) E=2\mu(1+\nu)
  73. ν = 0.5 \nu=0.5
  74. λ 1 = λ 2 = λ ; λ 3 = J λ 2 ; I 1 = 2 λ 2 + J 2 λ 4 \lambda_{1}=\lambda_{2}=\lambda~{};~{}~{}\lambda_{3}=\tfrac{J}{\lambda^{2}}~{}% ;~{}~{}I_{1}=2\lambda^{2}+\tfrac{J^{2}}{\lambda^{4}}
  75. σ 11 = 2 C 1 [ λ 2 J 5 / 3 - 1 3 J ( 2 λ 2 + J 2 λ 4 ) ] + 2 D 1 ( J - 1 ) = σ 22 σ 33 = 2 C 1 [ J 1 / 3 λ 4 - 1 3 J ( 2 λ 2 + J 2 λ 4 ) ] + 2 D 1 ( J - 1 ) \begin{aligned}\displaystyle\sigma_{11}&\displaystyle=2C_{1}\left[\cfrac{% \lambda^{2}}{J^{5/3}}-\cfrac{1}{3J}\left(2\lambda^{2}+\cfrac{J^{2}}{\lambda^{4% }}\right)\right]+2D_{1}(J-1)\\ &\displaystyle=\sigma_{22}\\ \displaystyle\sigma_{33}&\displaystyle=2C_{1}\left[\cfrac{J^{1/3}}{\lambda^{4}% }-\cfrac{1}{3J}\left(2\lambda^{2}+\cfrac{J^{2}}{\lambda^{4}}\right)\right]+2D_% {1}(J-1)\end{aligned}
  76. σ 11 - σ 22 = 0 ; σ 11 - σ 33 = 2 C 1 J 5 / 3 ( λ 2 - J 2 λ 4 ) \sigma_{11}-\sigma_{22}=0~{};~{}~{}\sigma_{11}-\sigma_{33}=\cfrac{2C_{1}}{J^{5% /3}}\left(\lambda^{2}-\cfrac{J^{2}}{\lambda^{4}}\right)
  77. σ 33 = 0 \sigma_{33}=0
  78. σ 11 = σ 22 = 2 C 1 J 5 / 3 ( λ 2 - J 2 λ 4 ) \sigma_{11}=\sigma_{22}=\cfrac{2C_{1}}{J^{5/3}}\left(\lambda^{2}-\cfrac{J^{2}}% {\lambda^{4}}\right)
  79. J J
  80. λ \lambda
  81. 2 C 1 [ λ 2 J 5 / 3 - 1 3 J ( 2 λ 2 + J 2 λ 4 ) ] + 2 D 1 ( J - 1 ) = 2 C 1 J 5 / 3 ( λ 2 - J 2 λ 4 ) 2C_{1}\left[\cfrac{\lambda^{2}}{J^{5/3}}-\cfrac{1}{3J}\left(2\lambda^{2}+% \cfrac{J^{2}}{\lambda^{4}}\right)\right]+2D_{1}(J-1)=\cfrac{2C_{1}}{J^{5/3}}% \left(\lambda^{2}-\cfrac{J^{2}}{\lambda^{4}}\right)
  82. ( 2 D 1 - C 1 λ 4 ) J 2 + 3 C 1 λ 4 J 4 / 3 - 3 D 1 J - 2 C 1 λ 2 = 0 \left(2D_{1}-\cfrac{C_{1}}{\lambda^{4}}\right)J^{2}+\cfrac{3C_{1}}{\lambda^{4}% }J^{4/3}-3D_{1}J-2C_{1}\lambda^{2}=0
  83. J J
  84. J = 1 J=1
  85. σ 11 - σ 22 = 0 ; σ 11 - σ 33 = 2 C 1 ( λ 2 - 1 λ 4 ) \sigma_{11}-\sigma_{22}=0~{};~{}~{}\sigma_{11}-\sigma_{33}=2C_{1}\left(\lambda% ^{2}-\cfrac{1}{\lambda^{4}}\right)
  86. σ 11 = 2 C 1 ( λ 2 - 1 λ 4 ) \sigma_{11}=2C_{1}\left(\lambda^{2}-\cfrac{1}{\lambda^{4}}\right)
  87. λ 1 = λ 2 = λ 3 = λ : J = λ 3 ; I 1 = 3 λ 2 \lambda_{1}=\lambda_{2}=\lambda_{3}=\lambda~{}:~{}~{}J=\lambda^{3}~{};~{}~{}I_% {1}=3\lambda^{2}
  88. σ i = 2 C 1 ( 1 λ 3 - 1 λ ) + 2 D 1 ( λ 3 - 1 ) \sigma_{i}=2C_{1}\left(\cfrac{1}{\lambda^{3}}-\cfrac{1}{\lambda}\right)+2D_{1}% (\lambda^{3}-1)
  89. λ 3 = 1 \lambda^{3}=1
  90. C 1 , D 1 C_{1},D_{1}
  91. C 1 , D 1 C_{1},D_{1}
  92. s y m b o l F = [ 1 γ 0 0 1 0 0 0 1 ] symbol{F}=\begin{bmatrix}1&\gamma&0\\ 0&1&0\\ 0&0&1\end{bmatrix}
  93. γ \gamma
  94. s y m b o l B = s y m b o l F \cdotsymbol F T = [ 1 + γ 2 γ 0 γ 1 0 0 0 1 ] symbol{B}=symbol{F}\cdotsymbol{F}^{T}=\begin{bmatrix}1+\gamma^{2}&\gamma&0\\ \gamma&1&0\\ 0&0&1\end{bmatrix}
  95. J = det ( s y m b o l F ) = 1 J=\det(symbol{F})=1
  96. s y m b o l σ = 2 C 1 dev ( s y m b o l B ) symbol{\sigma}=2C_{1}\mathrm{dev}(symbol{B})
  97. dev ( s y m b o l B ) = s y m b o l B - 1 3 tr ( s y m b o l B ) s y m b o l 1 = s y m b o l B - 1 3 ( 3 + γ 2 ) s y m b o l 1 = [ 2 3 γ 2 γ 0 γ - 1 3 γ 2 0 0 0 - 1 3 γ 2 ] \mathrm{dev}(symbol{B})=symbol{B}-\tfrac{1}{3}\mathrm{tr}(symbol{B})symbol{% \mathit{1}}=symbol{B}-\tfrac{1}{3}(3+\gamma^{2})symbol{\mathit{1}}=\begin{% bmatrix}\tfrac{2}{3}\gamma^{2}&\gamma&0\\ \gamma&-\tfrac{1}{3}\gamma^{2}&0\\ 0&0&-\tfrac{1}{3}\gamma^{2}\end{bmatrix}
  98. s y m b o l σ = [ 4 C 1 3 γ 2 2 C 1 γ 0 2 C 1 γ - 2 C 1 3 γ 2 0 0 - 2 C 1 3 γ 2 ] symbol{\sigma}=\begin{bmatrix}\tfrac{4C_{1}}{3}\gamma^{2}&2C_{1}\gamma&0\\ 2C_{1}\gamma&-\tfrac{2C_{1}}{3}\gamma^{2}&\\ 0&0&-\tfrac{2C_{1}}{3}\gamma^{2}\end{bmatrix}
  99. s y m b o l σ = - p s y m b o l 1 + 2 C 1 s y m b o l B = [ 2 C 1 ( 1 + γ 2 ) - p 2 C 1 γ 0 2 C 1 γ 2 C 1 - p 0 0 0 2 C 1 - p ] symbol{\sigma}=-psymbol{\mathit{1}}+2C_{1}symbol{B}=\begin{bmatrix}2C_{1}(1+% \gamma^{2})-p&2C_{1}\gamma&0\\ 2C_{1}\gamma&2C_{1}-p&0\\ 0&0&2C_{1}-p\end{bmatrix}
  100. p p

Nested_radical.html

  1. 5 - 2 5 \sqrt{5-2\sqrt{5}\ }
  2. 5 + 2 6 , \sqrt{5+2\sqrt{6}\ },
  3. 2 + 3 + 4 3 3 . \sqrt[3]{2+\sqrt{3}+\sqrt[3]{4}\ }.
  4. 3 + 2 2 = 1 + 2 , \sqrt{3+2\sqrt{2}}=1+\sqrt{2}\,,
  5. 5 + 2 6 = 2 + 3 , \sqrt{5+2\sqrt{6}}=\sqrt{2}+\sqrt{3},
  6. 2 3 - 1 3 = 1 - 2 3 + 4 3 9 3 . \sqrt[3]{\sqrt[3]{2}-1}=\frac{1-\sqrt[3]{2}+\sqrt[3]{4}}{\sqrt[3]{9}}\,.
  7. a ± b c = d ± e . \sqrt{a\pm b\sqrt{c}\ }=\sqrt{d}\pm\sqrt{e}.
  8. a ± b c = d + e ± 2 d e . a\pm b\sqrt{c}=d+e\pm 2\sqrt{de}.
  9. a = d + e , a=d+e,
  10. d = a - e , d=a-e,
  11. e = a - d . e=a-d.
  12. b c = 2 d e , b\sqrt{c}=2\sqrt{de},
  13. b 2 c = 4 d e . b^{2}c=4de.
  14. b 2 c = 4 ( a - d ) d = 4 a d - 4 d 2 . b^{2}c=4(a-d)d=4ad-4d^{2}.
  15. 4 d 2 - 4 a d + b 2 c = 0 , 4d^{2}-4ad+b^{2}c=0,
  16. d = a ± a 2 - b 2 c 2 . d=\frac{a\pm\sqrt{a^{2}-b^{2}c}}{2}.
  17. d = a + a 2 - b 2 c 2 , d=\frac{a+\sqrt{a^{2}-b^{2}c}}{2},
  18. e = a - a 2 - b 2 c 2 . e=\frac{a-\sqrt{a^{2}-b^{2}c}}{2}.
  19. a ± b c \sqrt{a\pm b\sqrt{c}\ }
  20. a 2 - b 2 c \sqrt{a^{2}-b^{2}c}
  21. 3 + 2 5 4 3 - 2 5 4 4 = 5 4 + 1 5 4 - 1 = 1 2 ( 3 + 5 4 + 5 + 125 4 ) , \sqrt[4]{\frac{3+2\sqrt[4]{5}}{3-2\sqrt[4]{5}}}=\frac{\sqrt[4]{5}+1}{\sqrt[4]{% 5}-1}=\tfrac{1}{2}\left(3+\sqrt[4]{5}+\sqrt{5}+\sqrt[4]{125}\right),
  22. 28 3 - 27 3 = 1 3 ( 98 3 - 28 3 - 1 ) , \sqrt{\sqrt[3]{28}-\sqrt[3]{27}}=\tfrac{1}{3}\left(\sqrt[3]{98}-\sqrt[3]{28}-1% \right),
  23. 32 5 5 - 27 5 5 3 = 1 25 5 + 3 25 5 - 9 25 5 , \sqrt[3]{\sqrt[5]{\frac{32}{5}}-\sqrt[5]{\frac{27}{5}}}=\sqrt[5]{\frac{1}{25}}% +\sqrt[5]{\frac{3}{25}}-\sqrt[5]{\frac{9}{25}},
  24. 2 3 - 1 3 = 1 9 3 - 2 9 3 + 4 9 3 . \sqrt[3]{\ \sqrt[3]{2}\ -1}=\sqrt[3]{\frac{1}{9}}-\sqrt[3]{\frac{2}{9}}+\sqrt[% 3]{\frac{4}{9}}.
  25. 49 + 20 6 4 + 49 - 20 6 4 = 2 3 , \sqrt[4]{49+20\sqrt{6}}+\sqrt[4]{49-20\sqrt{6}}=2\sqrt{3},
  26. ( 2 + 3 ) ( 5 - 6 ) + 3 ( 2 3 + 3 2 ) 3 = 10 - 13 - 5 6 5 + 6 . \sqrt[3]{\left(\sqrt{2}+\sqrt{3}\right)\left(5-\sqrt{6}\right)+3\left(2\sqrt{3% }+3\sqrt{2}\right)}=\sqrt{10-\frac{13-5\sqrt{6}}{5+\sqrt{6}}}.
  27. sin π 60 = sin 3 = 1 16 [ 2 ( 1 - 3 ) 5 + 5 + 2 ( 5 - 1 ) ( 3 + 1 ) ] \sin\frac{\pi}{60}=\sin 3^{\circ}=\tfrac{1}{16}\left[2(1-\sqrt{3})\sqrt{5+% \sqrt{5}}+\sqrt{2}(\sqrt{5}-1)(\sqrt{3}+1)\right]\,
  28. sin π 24 = sin 7.5 = 1 2 2 - 2 + 3 = 1 2 2 - 1 + 3 2 . \sin\frac{\pi}{24}=\sin 7.5^{\circ}=\tfrac{1}{2}\sqrt{2-\sqrt{2+\sqrt{3}}}=% \tfrac{1}{2}\sqrt{2-\tfrac{1+\sqrt{3}}{\sqrt{2}}}.
  29. x 3 + p x + q = 0 , x^{3}+px+q=0,
  30. x = - q 2 + q 2 4 + p 3 27 3 + - q 2 - q 2 4 + p 3 27 3 ; x=\sqrt[3]{-{q\over 2}+\sqrt{{q^{2}\over 4}+{p^{3}\over 27}}}+\sqrt[3]{-{q% \over 2}-\sqrt{{q^{2}\over 4}+{p^{3}\over 27}}};
  31. x 3 - 7 x + 6 = 0 , x^{3}-7x+6=0,
  32. x = - 3 + 10 3 i 9 3 + - 3 - 10 3 i 9 3 . x=\sqrt[3]{-3+\frac{10\sqrt{3}i}{9}}+\sqrt[3]{-3-\frac{10\sqrt{3}i}{9}}.
  33. x = 2 + 2 + 2 + 2 + x=\sqrt{2+\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}}
  34. x = 2 + x . x=\sqrt{2+x}.
  35. n + n + n + n + = 1 2 ( 1 + 1 + 4 n ) \sqrt{n+\sqrt{n+\sqrt{n+\sqrt{n+\cdots}}}}=\tfrac{1}{2}\left(1+\sqrt{1+4n}\right)
  36. n - n - n - n - = 1 2 ( - 1 + 1 + 4 n ) . \sqrt{n-\sqrt{n-\sqrt{n-\sqrt{n-\cdots}}}}=\tfrac{1}{2}\left(-1+\sqrt{1+4n}% \right).
  37. n = x 2 + x . n=x^{2}+x.\,
  38. ? = 1 + 2 1 + 3 1 + . ?=\sqrt{1+2\sqrt{1+3\sqrt{1+\cdots}}}.\,
  39. ? = a x + ( n + a ) 2 + x a ( x + n ) + ( n + a ) 2 + ( x + n ) ?=\sqrt{ax+(n+a)^{2}+x\sqrt{a(x+n)+(n+a)^{2}+(x+n)\sqrt{\mathrm{\cdots}}}}\,
  40. F ( x ) 2 = a x + ( n + a ) 2 + x a ( x + n ) + ( n + a ) 2 + ( x + n ) F(x)^{2}=ax+(n+a)^{2}+x\sqrt{a(x+n)+(n+a)^{2}+(x+n)\sqrt{\mathrm{\cdots}}}\,
  41. F ( x ) 2 = a x + ( n + a ) 2 + x F ( x + n ) F(x)^{2}=ax+(n+a)^{2}+xF(x+n)\,
  42. F ( x ) = x + n + a F(x)={x+n+a}\,
  43. 3 = 1 + 2 1 + 3 1 + . 3=\sqrt{1+2\sqrt{1+3\sqrt{1+\cdots}}}.\,
  44. 5 + 5 + 5 - 5 + 5 + 5 + 5 - = 2 + 5 + 15 - 6 5 2 \sqrt{5+\sqrt{5+\sqrt{5-\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5-\cdots}}}}}}}=\frac{2+% \sqrt{5}+\sqrt{15-6\sqrt{5}}}{2}
  45. ( + , + , - , + ) (+,+,-,+)
  46. 2 π = 2 2 2 + 2 2 2 + 2 + 2 2 . \frac{2}{\pi}=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{% \sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdots.
  47. x = 6 + 6 + 6 + 6 + 3 3 3 3 x=\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+\sqrt[3]{6+\cdots}}}}
  48. x = 6 + x 3 . x=\sqrt[3]{6+x}.
  49. n + n + n + n + 3 3 3 3 \sqrt[3]{n+\sqrt[3]{n+\sqrt[3]{n+\sqrt[3]{n+\cdots}}}}
  50. n - n - n - n - 3 3 3 3 \sqrt[3]{n-\sqrt[3]{n-\sqrt[3]{n-\sqrt[3]{n-\cdots}}}}

Neumann_series.html

  1. k = 0 T k \sum_{k=0}^{\infty}T^{k}
  2. ( Id - T ) - 1 = k = 0 T k (\mathrm{Id}-T)^{-1}=\sum_{k=0}^{\infty}T^{k}
  3. Id \mathrm{Id}
  4. S n := k = 0 n T k S_{n}:=\sum_{k=0}^{n}T^{k}
  5. lim n ( Id - T ) S n = lim n ( k = 0 n T k - k = 0 n T k + 1 ) = lim n ( Id - T n + 1 ) = Id . \lim_{n\rightarrow\infty}(\mathrm{Id}-T)S_{n}=\lim_{n\rightarrow\infty}\left(% \sum_{k=0}^{n}T^{k}-\sum_{k=0}^{n}T^{k+1}\right)=\lim_{n\rightarrow\infty}% \left(\mathrm{Id}-T^{n+1}\right)=\mathrm{Id}.
  6. \mathbb{R}
  7. 1 1 - x = 1 + x + x 2 + \frac{1}{1-x}=1+x+x^{2}+\cdots
  8. T = S ( Id - ( Id - S - 1 T ) ) T=S(\mathrm{Id}-(\mathrm{Id}-S^{-1}T))\,
  9. | T - 1 | 1 1 - q | S - 1 | where q = | S - T | | S - 1 | . |T^{-1}|\leq\tfrac{1}{1-q}|S^{-1}|\quad\,\text{where}\quad q=|S-T|\,|S^{-1}|.

Neural_gas.html

  1. w i k t + 1 = w i k t + ε e - k / λ ( x - w i k t ) w_{i_{k}}^{t+1}=w_{i_{k}}^{t}+\varepsilon\cdot e^{-k/\lambda}\cdot(x-w_{i_{k}}% ^{t})

Neutral_density_filter.html

  1. d = - log 10 I I 0 d=-\log_{10}\frac{I}{I_{0}}

Neutral_particle_oscillation.html

  1. ν e + C l 37 A r 38 + e - {{\nu}_{e}}+C{{l}^{37}}\to A{{r}^{38}}+{{e}^{-}}
  2. ν e + n p + e - {{\nu}_{e}}+n\to p+e^{-}
  3. H 0 {H_{0}}
  4. | 1 \left|1\right\rangle
  5. | 2 \left|2\right\rangle
  6. E 1 {E_{1}}
  7. E 2 {E_{2}}
  8. | Ψ ( t ) \left|{\Psi\left(t\right)}\right\rangle
  9. t t
  10. H 0 {H_{0}}
  11. | Ψ ( 0 ) = | 1 \left|{\Psi\left(0\right)}\right\rangle=\left|1\right\rangle
  12. | Ψ ( t ) = | 1 e - i E 1 t \left|\Psi\left(t\right)\right\rangle=\left|1\right\rangle{{e}^{-i\frac{{{E}_{% 1}}t}{\hbar}}}
  13. | 1 \left|1\right\rangle
  14. { | 1 , | 2 } \left\{{\left|1\right\rangle,\left|2\right\rangle}\right\}
  15. H 0 {H_{0}}
  16. H 0 = ( E 1 0 0 E 2 ) {{H}_{0}}=\left(\begin{matrix}{{E}_{1}}&0\\ 0&{{E}_{2}}\\ \end{matrix}\right)
  17. W W
  18. H 0 {{H}_{0}}
  19. H H
  20. W = ( W 11 W 12 W 12 * W 22 ) W=\left(\begin{matrix}{{W}_{11}}&{{W}_{12}}\\ {{W}_{12}}^{*}&{{W}_{22}}\\ \end{matrix}\right)
  21. W 11 , W 22 {{W}_{11}},{{W}_{22}}\in\mathbb{R}
  22. W 12 {{W}_{12}}\in\mathbb{C}
  23. H H
  24. H H
  25. H = j = 0 3 a j σ j = a 0 σ 0 + H H=\sum\limits_{j=0}^{3}{{{a}_{j}}{{\sigma}_{j}}}={{a}_{0}}{{\sigma}_{0}}+H^{\prime}
  26. H = a . σ = | a | n ^ . σ H^{\prime}=\vec{a}.\vec{\sigma}=\left|a\right|\hat{n}.\vec{\sigma}
  27. a = ( a 1 , a 2 , a 3 ) \vec{a}=\left({{a}_{1}},{{a}_{2}},{{a}_{3}}\right)
  28. n ^ {\hat{n}}
  29. a {\vec{a}}
  30. σ 0 = I = ( 1 0 0 1 ) {{\sigma}_{0}}=I=\left(\begin{matrix}1&0\\ 0&1\\ \end{matrix}\right)
  31. σ 1 = σ x = ( 0 1 1 0 ) {{\sigma}_{1}}={{\sigma}_{x}}=\left(\begin{matrix}0&1\\ 1&0\\ \end{matrix}\right)
  32. σ 2 = σ y = ( 0 - i i 0 ) {{\sigma}_{2}}={{\sigma}_{y}}=\left(\begin{matrix}0&-i\\ i&0\\ \end{matrix}\right)
  33. σ 3 = σ z = ( 1 0 0 - 1 ) {{\sigma}_{3}}={{\sigma}_{z}}=\left(\begin{matrix}1&0\\ 0&-1\\ \end{matrix}\right)
  34. [ H , H ] = 0 \left[H,H^{\prime}\right]=0
  35. H H = a 0 σ 0 H + H H = a 0 σ 0 + H 2 H H = a 0 H σ 0 + H H = a 0 σ 0 + H 2 [ H , H ] = H H - H H = 0 \begin{aligned}&\displaystyle HH^{\prime}={{a}_{0}}{{\sigma}_{0}}H^{\prime}+H^% {\prime}H^{\prime}={{a}_{0}}{{\sigma}_{0}}+H{{{}^{\prime}}^{2}}\\ &\displaystyle H^{\prime}H={{a}_{0}}H^{\prime}{{\sigma}_{0}}+H^{\prime}H^{% \prime}={{a}_{0}}{{\sigma}_{0}}+H{{{}^{\prime}}^{2}}\\ &\displaystyle\therefore\left[H,H^{\prime}\right]=HH^{\prime}-H^{\prime}H=0\\ \end{aligned}
  36. H = 2 I H{{{}^{\prime}}^{2}}=I
  37. H 2 = j = 1 3 n j σ j k = 1 3 n k σ k = j , k = 1 3 n j n k σ j σ k = j , k = 1 3 n j n k ( δ j k I + i l = 1 3 ε j k l σ l ) = ( j = 1 3 n j 2 ) I + i l = 1 3 σ l j , k = 1 3 ε j k l = I \begin{aligned}\displaystyle H{{{}^{\prime}}^{2}}&\displaystyle=\sum\limits_{j% =1}^{3}{{{n}_{j}}{{\sigma}_{j}}}\sum\limits_{k=1}^{3}{{{n}_{k}}{{\sigma}_{k}}}% =\sum\limits_{j,k=1}^{3}{{{n}_{j}}{{n}_{k}}{{\sigma}_{j}}{{\sigma}_{k}}}\\ &\displaystyle=\sum\limits_{j,k=1}^{3}{{{n}_{j}}{{n}_{k}}\left({{\delta}_{jk}}% I+i\sum\limits_{l=1}^{3}{{{\varepsilon}_{jkl}}{{\sigma}_{l}}}\right)}\\ &\displaystyle=\left(\sum\limits_{j=1}^{3}{{{n}_{j}}^{2}}\right)I+i\sum\limits% _{l=1}^{3}{{{\sigma}_{l}}\sum\limits_{j,k=1}^{3}{{{\varepsilon}_{jkl}}}}\\ &\displaystyle=I\\ \end{aligned}
  38. σ j σ k = δ j k I + i l = 1 3 ε j k l σ l {{\sigma}_{j}}{{\sigma}_{k}}={{\delta}_{jk}}I+i\sum\limits_{l=1}^{3}{{{% \varepsilon}_{jkl}}{{\sigma}_{l}}}
  39. n ^ {\hat{n}}
  40. j = 1 3 n j 2 = | n ^ | 2 = 1 \sum\limits_{j=1}^{3}{{{n}_{j}}^{2}}={{\left|{\hat{n}}\right|}^{2}}=1
  41. ε j k l {{\varepsilon}_{jkl}}
  42. j j
  43. k k
  44. j , k = 1 3 ε j k l = 0 \sum\limits_{j,k=1}^{3}{{{\varepsilon}_{jkl}}}=0
  45. ϕ \phi
  46. n ^ = ( sin θ cos ϕ , sin θ sin ϕ , cos θ ) \hat{n}=\left(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta\right)
  47. H H^{\prime}
  48. H H
  49. tan θ = 2 | W 12 | E 1 + W 11 - E 2 - W 22 \tan\theta=\frac{2\left|{{W}_{12}}\right|}{{{E}_{1}}+{{W}_{11}}-{{E}_{2}}-{{W}% _{22}}}
  50. W 12 = | W 12 | e i ϕ {{W}_{12}}=\left|{{W}_{12}}\right|{{e}^{i\phi}}
  51. H 0 {H_{0}}
  52. H H
  53. H 0 {H_{0}}
  54. | 1 \left|1\right\rangle
  55. | Ψ ( 0 ) = | 1 \left|\Psi\left(0\right)\right\rangle=\left|1\right\rangle
  56. | Ψ ( t ) = e i ϕ 2 ( cos θ 2 | + e - i E + t - sin θ 2 | - e - i E - t ) \left|\Psi\left(t\right)\right\rangle={{e}^{i\frac{\phi}{2}}}\left(\cos\frac{% \theta}{2}\left|+\right\rangle{{e}^{-i\frac{{{E}_{+}}t}{\hbar}}}-\sin\frac{% \theta}{2}\left|-\right\rangle{{e}^{-i\frac{{{E}_{-}}t}{\hbar}}}\right)
  57. | 1 \left|1\right\rangle
  58. | 2 \left|2\right\rangle
  59. t t
  60. H 0 {H_{0}}
  61. H 0 {H_{0}}
  62. P 21 ( t ) {{P}_{21}}(t)
  63. | W 12 | 2 0 {{\left|{{W}_{12}}\right|}^{2}}\neq 0
  64. W 12 {{W}_{12}}
  65. H 0 {H_{0}}
  66. H H
  67. E + = E - {{E}_{+}}={{E}_{-}}
  68. H H
  69. H 0 {H_{0}}
  70. | W 12 | 2 0 {{\left|{{W}_{12}}\right|}^{2}}\neq 0
  71. H H
  72. E + E - {{E}_{+}}\neq{{E}_{-}}
  73. H H
  74. H = M - i 2 Γ = ( M 11 M 12 M 12 * M 11 ) - i 2 ( Γ 11 Γ 12 Γ 12 * Γ 11 ) H=M-\frac{i}{2}\Gamma=\left(\begin{matrix}{{M}_{11}}&{{M}_{12}}\\ {{M}_{12}}^{*}&{{M}_{11}}\\ \end{matrix}\right)-\frac{i}{2}\left(\begin{matrix}{{\Gamma}_{11}}&{{\Gamma}_{% 12}}\\ {{\Gamma}_{12}}^{*}&{{\Gamma}_{11}}\\ \end{matrix}\right)
  75. M = ( M 11 M 12 M 21 M 22 ) M=\left(\begin{matrix}{{M}_{11}}&{{M}_{12}}\\ {{M}_{21}}&{{M}_{22}}\\ \end{matrix}\right)
  76. Γ = ( Γ 11 Γ 12 Γ 12 * Γ 11 ) \Gamma=\left(\begin{matrix}{{\Gamma}_{11}}&{{\Gamma}_{12}}\\ {{\Gamma}_{12}}^{*}&{{\Gamma}_{11}}\\ \end{matrix}\right)
  77. M M
  78. Γ \Gamma
  79. M 21 = M 12 * {{M}_{21}}={{M}_{12}}^{*}
  80. Γ 21 = Γ 12 * {{\Gamma}_{21}}={{\Gamma}_{12}}^{*}
  81. M 22 = M 11 {{M}_{22}}={{M}_{11}}
  82. Γ 22 = Γ 11 {{\Gamma}_{22}}={{\Gamma}_{11}}
  83. Θ = C P T \Theta=CPT
  84. Θ \Theta
  85. Θ | 1 = | 2 \Theta\left|1\right\rangle=\left|2\right\rangle
  86. Θ | 2 = | 1 \Theta\left|2\right\rangle=\left|1\right\rangle
  87. H H
  88. M M
  89. Γ \Gamma
  90. Θ - 1 M Θ = M {{\Theta}^{-1}}M\Theta=M
  91. Θ - 1 Γ Θ = Γ {{\Theta}^{-1}}\Gamma\Theta=\Gamma
  92. Θ \Theta
  93. Θ Θ = I {{\Theta}^{\dagger}}\Theta=I
  94. M 22 = 2 | M | 2 = 2 | Θ - 1 M Θ | 2 = 2 | Θ M Θ | 2 = 1 | M | 1 = M 11 {{M}_{22}}=\left\langle 2\right|M\left|2\right\rangle=\left\langle 2\right|{{% \Theta}^{-1}}M\Theta\left|2\right\rangle=\left\langle 2\right|{{\Theta}^{% \dagger}}M\Theta\left|2\right\rangle=\left\langle 1\right|M\left|1\right% \rangle={{M}_{11}}
  95. Γ \Gamma
  96. M M
  97. Γ \Gamma
  98. H H
  99. μ L = M 11 - i 2 Γ 11 - 1 2 ( Δ m - i 2 Δ Γ ) {{\mu}_{L}}={{M}_{11}}-\frac{i}{2}{{\Gamma}_{11}}-\frac{1}{2}\left(\Delta m-% \frac{i}{2}\Delta\Gamma\right)
  100. Δ m \Delta m
  101. Δ Γ \Delta\Gamma
  102. ( Δ m ) 2 - ( Δ Γ 2 ) 2 = 4 | M 12 | 2 - | Γ 12 | 2 {{\left(\Delta m\right)}^{2}}-{{\left(\frac{\Delta\Gamma}{2}\right)}^{2}}=4{{% \left|{{M}_{12}}\right|}^{2}}-{{\left|{{\Gamma}_{12}}\right|}^{2}}
  103. Δ m Δ Γ = 4 Re ( M 12 Γ 12 * ) \Delta m\Delta\Gamma=4\operatorname{Re}\left({{M}_{12}}{{\Gamma}_{12}}^{*}\right)
  104. Δ m \Delta m
  105. μ L {{\mu}_{L}}
  106. μ H {{\mu}_{H}}
  107. { | P , | P ¯ } { ( 1 , 0 ) , ( 0 , 1 ) } \left\{\left|P\right\rangle,\left|{\bar{P}}\right\rangle\right\}\equiv\left\{% \left(1,0\right),\left(0,1\right)\right\}
  108. | p | 2 + | q | 2 = 1 {{\left|p\right|}^{2}}+{{\left|q\right|}^{2}}=1
  109. ( p q ) 2 = M 12 * - i 2 Γ 12 * M 12 - i 2 Γ 12 {{\left(\frac{p}{q}\right)}^{2}}=\frac{{{M}_{12}}^{*}-\frac{i}{2}{{\Gamma}_{12% }}^{*}}{{{M}_{12}}-\frac{i}{2}{{\Gamma}_{12}}}
  110. p p
  111. q q
  112. | P \left|P\right\rangle
  113. | P ( 0 ) = | P = 1 2 p ( | P L + | P H ) \left|P\left(0\right)\right\rangle=\left|P\right\rangle=\frac{1}{2p}\left(% \left|{{P}_{L}}\right\rangle+\left|{{P}_{H}}\right\rangle\right)
  114. | P ( t ) = 1 2 p ( | P L e - i ( m L - i 2 γ L t ) + | P H e - i ( m H - i 2 γ H t ) ) = g + ( t ) | P - q p g - ( t ) | P ¯ \left|P\left(t\right)\right\rangle=\frac{1}{2p}\left(\left|{{P}_{L}}\right% \rangle{{e}^{-i\left({{m}_{L}}-\frac{i}{2}{{\gamma}_{L}}t\right)}}+\left|{{P}_% {H}}\right\rangle{{e}^{-i\left({{m}_{H}}-\frac{i}{2}{{\gamma}_{H}}t\right)}}% \right)={{g}_{+}}\left(t\right)\left|P\right\rangle-\frac{q}{p}{{g}_{-}}\left(% t\right)\left|{\bar{P}}\right\rangle
  115. g ± ( t ) = 1 2 ( e - i ( m H - i 2 γ H ) t ± e - i ( m L - i 2 γ L ) t ) {{g}_{\pm}}\left(t\right)=\frac{1}{2}\left({{e}^{-\frac{i}{\hbar}\left({{m}_{H% }}-\frac{i}{2}{{\gamma}_{H}}\right)t}}\pm{{e}^{-\frac{i}{\hbar}\left({{m}_{L}}% -\frac{i}{2}{{\gamma}_{L}}\right)t}}\right)
  116. | P ¯ \left|{\bar{P}}\right\rangle
  117. | P ¯ ( t ) = 1 2 q ( | P L e - i ( m L - i 2 γ L ) t - | P H e - i ( m H - i 2 γ H ) t ) = - p q g - ( t ) | P + g + ( t ) | P ¯ \left|\bar{P}(t)\right\rangle=\frac{1}{2q}\left(\left|{{P}_{L}}\right\rangle{{% e}^{-\frac{i}{\hbar}\left({{m}_{L}}-\frac{i}{2}{{\gamma}_{L}}\right)t}}-\left|% {{P}_{H}}\right\rangle{{e}^{-\frac{i}{\hbar}\left({{m}_{H}}-\frac{i}{2}{{% \gamma}_{H}}\right)t}}\right)=-\frac{p}{q}{{g}_{-}}\left(t\right)\left|P\right% \rangle+{{g}_{+}}\left(t\right)\left|{\bar{P}}\right\rangle
  118. | P \left|P\right\rangle
  119. | P ¯ \left|{\bar{P}}\right\rangle
  120. C P | P = e i δ | P ¯ CP\left|P\right\rangle={{e}^{i\delta}}\left|{\bar{P}}\right\rangle
  121. C P | P ¯ = e - i δ | P CP\left|{\bar{P}}\right\rangle={{e}^{-i\delta}}\left|P\right\rangle
  122. { | P , | P ¯ } \left\{\left|P\right\rangle,\left|\bar{P}\right\rangle\right\}
  123. { | f , | f ¯ } \left\{\left|f\right\rangle,\left|\bar{f}\right\rangle\right\}
  124. | P \left|P\right\rangle
  125. | f \left|f\right\rangle
  126. P f ( t ) = | f | P ( t ) | 2 = | g + ( t ) A f - q p g - ( t ) A ¯ f | 2 {{\wp}_{P\to f}}\left(t\right)={{\left|\left\langle f|P\left(t\right)\right% \rangle\right|}^{2}}={{\left|{{g}_{+}}\left(t\right){{A}_{f}}-\frac{q}{p}{{g}_% {-}}\left(t\right){{{\bar{A}}}_{f}}\right|}^{2}}
  127. P ¯ f ¯ ( t ) = | f ¯ | P ¯ ( t ) | 2 = | g + ( t ) A ¯ f ¯ - p q g - ( t ) A f ¯ | 2 {{\wp}_{\bar{P}\to\bar{f}}}\left(t\right)={{\left|\left\langle{\bar{f}}|\bar{P% }\left(t\right)\right\rangle\right|}^{2}}={{\left|{{g}_{+}}\left(t\right){{{% \bar{A}}}_{{\bar{f}}}}-\frac{p}{q}{{g}_{-}}\left(t\right){{A}_{{\bar{f}}}}% \right|}^{2}}
  128. A f = f | P {{A}_{f}}=\left\langle f|P\right\rangle
  129. A ¯ f = f | P ¯ {{{\bar{A}}}_{f}}=\left\langle f|{\bar{P}}\right\rangle
  130. A f ¯ = f ¯ | P {{A}_{{\bar{f}}}}=\left\langle{\bar{f}}|P\right\rangle
  131. A ¯ f ¯ = f ¯ | P ¯ {{{\bar{A}}}_{{\bar{f}}}}=\left\langle{\bar{f}}|{\bar{P}}\right\rangle
  132. | q p | = 1 \left|\frac{q}{p}\right|=1
  133. | A f ¯ A ¯ f | 1 \left|\frac{{{A}_{{\bar{f}}}}}{{{{\bar{A}}}_{f}}}\right|\neq 1
  134. | P ¯ \left|{\bar{P}}\right\rangle
  135. | P \left|P\right\rangle
  136. P P ¯ ( t ) = | P ¯ | P ( t ) | 2 = | q p g - ( t ) | 2 {{\wp}_{P\to\bar{P}}}\left(t\right)={{\left|\left\langle{\bar{P}}|P\left(t% \right)\right\rangle\right|}^{2}}={{\left|\frac{q}{p}{{g}_{-}}\left(t\right)% \right|}^{2}}
  137. P ¯ P ( t ) = | P | P ¯ ( t ) | 2 = | p q g - ( t ) | 2 {{\wp}_{\bar{P}\to P}}\left(t\right)={{\left|\left\langle P|\bar{P}\left(t% \right)\right\rangle\right|}^{2}}={{\left|\frac{p}{q}{{g}_{-}}\left(t\right)% \right|}^{2}}
  138. | P \left|P\right\rangle
  139. | P ¯ \left|{\bar{P}}\right\rangle
  140. | f \left|f\right\rangle
  141. | P \left|P\right\rangle
  142. | P ¯ \left|{\bar{P}}\right\rangle
  143. P f ( t ) = | f | P ( t ) | 2 = | A f | 2 e - γ t 2 [ ( 1 + | λ f | 2 ) cosh ( Δ γ 2 t ) + 2 Re ( λ f ) sinh ( Δ γ 2 t ) + ( 1 - | λ f | 2 ) cos ( Δ m t ) + 2 Im ( λ f ) sin ( Δ m t ) ] \begin{aligned}\displaystyle{{\wp}_{P\to f}}\left(t\right)&\displaystyle={{% \left|\left\langle f|P\left(t\right)\right\rangle\right|}^{2}}\\ &\displaystyle={{\left|{{A}_{f}}\right|}^{2}}\frac{{{e}^{-\gamma t}}}{2}\left[% \left(1+{{\left|{{\lambda}_{f}}\right|}^{2}}\right)\cosh\left(\frac{\Delta% \gamma}{2}t\right)+2\operatorname{Re}\left({{\lambda}_{f}}\right)\sinh\left(% \frac{\Delta\gamma}{2}t\right)+\left(1-{{\left|{{\lambda}_{f}}\right|}^{2}}% \right)\cos\left(\Delta mt\right)+2\operatorname{Im}\left({{\lambda}_{f}}% \right)\sin\left(\Delta mt\right)\right]\\ \end{aligned}
  144. P ¯ f ( t ) = | f | P ¯ ( t ) | 2 = | A f | 2 | p q | 2 e - γ t 2 [ ( 1 + | λ f | 2 ) cosh ( Δ γ 2 t ) + 2 Re ( λ f ) sinh ( Δ γ 2 t ) - ( 1 - | λ f | 2 ) cos ( Δ m t ) - 2 Im ( λ f ) sin ( Δ m t ) ] \begin{aligned}\displaystyle{{\wp}_{\bar{P}\to f}}\left(t\right)&\displaystyle% ={{\left|\left\langle f|\bar{P}\left(t\right)\right\rangle\right|}^{2}}\\ &\displaystyle={{\left|{{A}_{f}}\right|}^{2}}{{\left|\frac{p}{q}\right|}^{2}}% \frac{{{e}^{-\gamma t}}}{2}\left[\left(1+{{\left|{{\lambda}_{f}}\right|}^{2}}% \right)\cosh\left(\frac{\Delta\gamma}{2}t\right)+2\operatorname{Re}\left({{% \lambda}_{f}}\right)\sinh\left(\frac{\Delta\gamma}{2}t\right)-\left(1-{{\left|% {{\lambda}_{f}}\right|}^{2}}\right)\cos\left(\Delta mt\right)-2\operatorname{% Im}\left({{\lambda}_{f}}\right)\sin\left(\Delta mt\right)\right]\\ \end{aligned}
  145. γ = γ H + γ L 2 \gamma=\frac{{{\gamma}_{H}}+{{\gamma}_{L}}}{2}
  146. Δ γ = γ H - γ L \Delta\gamma={{\gamma}_{H}}-{{\gamma}_{L}}
  147. Δ m = m H - m L \Delta m={{m}_{H}}-{{m}_{L}}
  148. λ f = q p A ¯ f A f {{\lambda}_{f}}=\frac{q}{p}\frac{{{{\bar{A}}}_{f}}}{{{A}_{f}}}
  149. A f = f | P {{A}_{f}}=\left\langle f|P\right\rangle
  150. A ¯ f = f | P ¯ {{{\bar{A}}}_{f}}=\left\langle f|{\bar{P}}\right\rangle
  151. | q / p | = 1 \left|q/p\right|=1
  152. | A ¯ f / A f | = 1 \left|{{{\bar{A}}}_{f}}/{{A}_{f}}\right|=1
  153. | λ f | = 1 \left|{{\lambda}_{f}}\right|=1
  154. | A ¯ f / A f | 1 \left|{{{\bar{A}}}_{f}}/{{A}_{f}}\right|\neq 1
  155. α \alpha
  156. β \beta
  157. P β α ( t ) = sin 2 θ sin 2 ( E + - E - 2 t ) {{P}_{\beta\alpha}}\left(t\right)={{\sin}^{2}}\theta{{\sin}^{2}}\left(\frac{{{% E}_{+}}-{{E}_{-}}}{2\hbar}t\right)
  158. E + {{E}_{+}}
  159. E - {{E}_{-}}
  160. Δ m 2 = m + 2 - m - 2 \Delta{{m}^{2}}={{m}_{+}}^{2}-{{m}_{-}}^{2}
  161. c c
  162. x x
  163. E E
  164. λ o s c {{\lambda}_{osc}}
  165. E ± = p 2 c 2 + m ± 2 c 4 p c ( 1 + m ± 2 c 2 2 p 2 ) [ m ± c p 1 ] {{E}_{\pm}}=\sqrt{{{p}^{2}}{{c}^{2}}+{{m}_{\pm}}^{2}{{c}^{4}}}\simeq pc\left(1% +\frac{{{m}_{\pm}}^{2}{{c}^{2}}}{2{{p}^{2}}}\right)\left[\because\frac{{{m}_{% \pm}}c}{p}\ll 1\right]
  166. p p
  167. E p c E\simeq pc
  168. t x / c t\simeq x/c
  169. E + - E - 2 t ( m + 2 - m - 2 ) c 3 2 p t Δ m 2 c 3 4 E x = 2 π λ o s c x \frac{{{E}_{+}}-{{E}_{-}}}{2\hbar}t\simeq\frac{\left({{m}_{+}}^{2}-{{m}_{-}}^{% 2}\right){{c}^{3}}}{2p\hbar}t\simeq\frac{\Delta{{m}^{2}}{{c}^{3}}}{4E\hbar}x=% \frac{2\pi}{{{\lambda}_{osc}}}x
  170. λ o s c = 8 π E Δ m 2 c 3 {{\lambda}_{osc}}=\frac{8\pi E\hbar}{\Delta{{m}^{2}}{{c}^{3}}}
  171. ( Δ m 2 ) 12 = m 1 2 - m 2 2 {{\left(\Delta{{m}^{2}}\right)}_{12}}={{m}_{1}}^{2}-{{m}_{2}}^{2}
  172. ( Δ m 2 ) 23 = m 2 2 - m 3 2 {{\left(\Delta{{m}^{2}}\right)}_{23}}={{m}_{2}}^{2}-{{m}_{3}}^{2}
  173. ( Δ m 2 ) 31 = m 3 2 - m 1 2 {{\left(\Delta{{m}^{2}}\right)}_{31}}={{m}_{3}}^{2}-{{m}_{1}}^{2}
  174. ( Δ m 2 ) 12 + ( Δ m 2 ) 23 + ( Δ m 2 ) 31 = 0 {{\left(\Delta{{m}^{2}}\right)}_{12}}+{{\left(\Delta{{m}^{2}}\right)}_{23}}+{{% \left(\Delta{{m}^{2}}\right)}_{31}}=0
  175. ( Δ m 2 ) s o l 8 × 10 - 5 ( e V / c 2 ) 2 {{\left(\Delta{{m}^{2}}\right)}_{sol}}\simeq 8\times{{10}^{-5}}{{\left(eV/{{c}% ^{2}}\right)}^{2}}
  176. ( Δ m 2 ) a t m 3 × 10 - 3 ( e V / c 2 ) 2 {{\left(\Delta{{m}^{2}}\right)}_{atm}}\simeq 3\times{{10}^{-3}}{{\left(eV/{{c}% ^{2}}\right)}^{2}}
  177. Δ m 2 \Delta{{m}^{2}}
  178. Δ m 2 \Delta{{m}^{2}}
  179. λ o s c {{\lambda}_{osc}}
  180. x / λ o s c 1 x/{{\lambda}_{osc}}\ll 1
  181. P β α 0 {{P}_{\beta\alpha}}\simeq 0
  182. x / λ o s c n x/{{\lambda}_{osc}}\simeq n
  183. n n
  184. P β α 0 {{P}_{\beta\alpha}}\simeq 0
  185. x / λ o s c 1 x/{{\lambda}_{osc}}\gg 1
  186. x λ o s c x\sim{{\lambda}_{osc}}
  187. | K 0 \left|{{K}^{0}}\right\rangle
  188. | K ¯ 0 \left|{{{\bar{K}}}^{0}}\right\rangle
  189. | K 1 0 = 1 2 ( | K 0 + | K ¯ 0 ) \left|K_{{}^{1}}^{0}\right\rangle=\frac{1}{\sqrt{2}}\left(\left|{{K}^{0}}% \right\rangle+\left|{{{\bar{K}}}^{0}}\right\rangle\right)
  190. | K 2 0 = 1 2 ( | K 0 - | K ¯ 0 ) \left|K_{2}^{0}\right\rangle=\frac{1}{\sqrt{2}}\left(\left|{{K}^{0}}\right% \rangle-\left|{{{\bar{K}}}^{0}}\right\rangle\right)
  191. | K 1 0 \left|K_{{}^{1}}^{0}\right\rangle
  192. | K 2 0 \left|K_{2}^{0}\right\rangle
  193. | K 1 0 \left|K_{{}^{1}}^{0}\right\rangle
  194. | K S 0 \left|K_{S}^{0}\right\rangle
  195. | K 2 0 \left|K_{2}^{0}\right\rangle
  196. | K L 0 \left|K_{L}^{0}\right\rangle
  197. | K L 0 \left|K_{L}^{0}\right\rangle
  198. | K 2 0 \left|K_{2}^{0}\right\rangle
  199. | K 1 0 \left|K_{{}^{1}}^{0}\right\rangle
  200. | K 2 0 \left|K_{2}^{0}\right\rangle
  201. | K L 0 = 1 1 + | ε | 2 ( | K 2 0 + ε | K 1 0 ) \left|K_{L}^{0}\right\rangle=\frac{1}{\sqrt{1+{{\left|\varepsilon\right|}^{2}}% }}\left(\left|K_{2}^{0}\right\rangle+\varepsilon\left|K_{1}^{0}\right\rangle\right)
  202. | K S 0 = 1 1 + | ε | 2 ( | K 1 0 + ε | K 2 0 ) \left|K_{S}^{0}\right\rangle=\frac{1}{\sqrt{1+{{\left|\varepsilon\right|}^{2}}% }}\left(\left|K_{1}^{0}\right\rangle+\varepsilon\left|K_{2}^{0}\right\rangle\right)
  203. ε \varepsilon
  204. | ε | = ( 2.228 ± 0.011 ) × 10 - 3 \left|\varepsilon\right|=\left(2.228\pm 0.011\right)\times{{10}^{-3}}
  205. | K 1 0 \left|K_{{}^{1}}^{0}\right\rangle
  206. | K 2 0 \left|K_{2}^{0}\right\rangle
  207. | K 0 \left|{{K}^{0}}\right\rangle
  208. | K ¯ 0 \left|{{{\bar{K}}}^{0}}\right\rangle
  209. m K L 0 > m K S 0 {{m}_{K_{{}_{L}}^{0}}}>{{m}_{K_{S}^{0}}}
  210. | K L 0 = ( p | K 0 - q | K ¯ 0 ) \left|K_{L}^{0}\right\rangle=\left(p\left|{{K}^{0}}\right\rangle-q\left|{{{% \bar{K}}}^{0}}\right\rangle\right)
  211. | K S 0 = ( p | K 0 + q | K ¯ 0 ) \left|K_{S}^{0}\right\rangle=\left(p\left|{{K}^{0}}\right\rangle+q\left|{{{% \bar{K}}}^{0}}\right\rangle\right)
  212. q p = 1 - ε 1 + ε \frac{q}{p}=\frac{1-\varepsilon}{1+\varepsilon}
  213. | ε | 0 \left|\varepsilon\right|\neq 0
  214. | K 0 \left|{{K}^{0}}\right\rangle
  215. | K ¯ 0 \left|{{{\bar{K}}}^{0}}\right\rangle
  216. η + - = π + π - | K L 0 π + π - | K S 0 = p A π + π - - q A ¯ π + π - p A π + π - + q A ¯ π + π - = 1 - λ π + π - 1 + λ π + π - {{\eta}_{+-}}=\frac{\left\langle{{\pi}^{+}}{{\pi}^{-}}|K_{L}^{0}\right\rangle}% {\left\langle{{\pi}^{+}}{{\pi}^{-}}|K_{S}^{0}\right\rangle}=\frac{p{{A}_{{{\pi% }^{+}}{{\pi}^{-}}}}-q{{{\bar{A}}}_{{{\pi}^{+}}{{\pi}^{-}}}}}{p{{A}_{{{\pi}^{+}% }{{\pi}^{-}}}}+q{{{\bar{A}}}_{{{\pi}^{+}}{{\pi}^{-}}}}}=\frac{1-{{\lambda}_{{{% \pi}^{+}}{{\pi}^{-}}}}}{1+{{\lambda}_{{{\pi}^{+}}{{\pi}^{-}}}}}
  217. η 00 = π 0 π 0 | K L 0 π 0 π 0 | K S 0 = p A π 0 π 0 - q A ¯ π 0 π 0 p A π 0 π 0 + q A ¯ π 0 π 0 = 1 - λ π 0 π 0 1 + λ π 0 π 0 {{\eta}_{00}}=\frac{\left\langle{{\pi}^{0}}{{\pi}^{0}}|K_{L}^{0}\right\rangle}% {\left\langle{{\pi}^{0}}{{\pi}^{0}}|K_{S}^{0}\right\rangle}=\frac{p{{A}_{{{\pi% }^{0}}{{\pi}^{0}}}}-q{{{\bar{A}}}_{{{\pi}^{0}}{{\pi}^{0}}}}}{p{{A}_{{{\pi}^{0}% }{{\pi}^{0}}}}+q{{{\bar{A}}}_{{{\pi}^{0}}{{\pi}^{0}}}}}=\frac{1-{{\lambda}_{{{% \pi}^{0}}{{\pi}^{0}}}}}{1+{{\lambda}_{{{\pi}^{0}}{{\pi}^{0}}}}}
  218. η + - = ( 2.232 ± 0.011 ) × 10 - 3 {{\eta}_{+-}}=\left(2.232\pm 0.011\right)\times{{10}^{-3}}
  219. η 00 = ( 2.220 ± 0.011 ) × 10 - 3 {{\eta}_{00}}=\left(2.220\pm 0.011\right)\times{{10}^{-3}}
  220. η + - η 00 {{\eta}_{+-}}\neq{{\eta}_{00}}
  221. | A π + π - / A ¯ π + π - | 1 \left|{{A}_{{{\pi}^{+}}{{\pi}^{-}}}}/{{{\bar{A}}}_{{{\pi}^{+}}{{\pi}^{-}}}}% \right|\neq 1
  222. | A π 0 π 0 / A ¯ π 0 π 0 | 1 \left|{{A}_{{{\pi}^{0}}{{\pi}^{0}}}}/{{{\bar{A}}}_{{{\pi}^{0}}{{\pi}^{0}}}}% \right|\neq 1
  223. f C P {{f}_{CP}}
  224. K 0 f C P {{K}^{0}}\to{{f}_{CP}}
  225. K 0 K ¯ 0 f C P {{K}^{0}}\to{{{\bar{K}}}^{0}}\to{{f}_{CP}}
  226. | φ α \left|{{\varphi}_{\alpha}}\right\rangle
  227. | φ β \left|{{\varphi}_{\beta}}\right\rangle
  228. | φ γ \left|{{\varphi}_{\gamma}}\right\rangle
  229. | ψ 1 \left|{{\psi}_{1}}\right\rangle
  230. | ψ 2 \left|{{\psi}_{2}}\right\rangle
  231. | ψ 3 \left|{{\psi}_{3}}\right\rangle
  232. ( | φ α | φ β | φ γ ) = ( Ω α 1 Ω α 2 Ω α 3 Ω β 1 Ω β 2 Ω β 3 Ω γ 1 Ω γ 2 Ω γ 3 ) ( | ψ 1 | ψ 2 | ψ 3 ) \left(\begin{matrix}\left|{{\varphi}_{\alpha}}\right\rangle\\ \left|{{\varphi}_{\beta}}\right\rangle\\ \left|{{\varphi}_{\gamma}}\right\rangle\\ \end{matrix}\right)=\left(\begin{matrix}{{\Omega}_{\alpha 1}}&{{\Omega}_{% \alpha 2}}&{{\Omega}_{\alpha 3}}\\ {{\Omega}_{\beta 1}}&{{\Omega}_{\beta 2}}&{{\Omega}_{\beta 3}}\\ {{\Omega}_{\gamma 1}}&{{\Omega}_{\gamma 2}}&{{\Omega}_{\gamma 3}}\\ \end{matrix}\right)\left(\begin{matrix}\left|{{\psi}_{1}}\right\rangle\\ \left|{{\psi}_{2}}\right\rangle\\ \left|{{\psi}_{3}}\right\rangle\\ \end{matrix}\right)

Neutron_cross_section.html

  1. σ T = σ S + σ A \sigma_{T}=\sigma_{S}+\sigma_{A}
  2. λ ( E ) = h 2 m E \lambda(E)=\frac{h}{\sqrt{2mE}}
  3. λ \lambda
  4. σ \sigma
  5. R R
  6. σ ( E ) π ( R + λ ( E ) ) 2 \sigma(E)\propto\pi(R+\lambda(E))^{2}
  7. R R
  8. σ ( E ) \sigma(E)
  9. λ \lambda
  10. σ = σ 0 ( T 0 T ) 1 2 \sigma=\sigma_{0}\,\left(\frac{T_{0}}{T}\right)^{\frac{1}{2}}
  11. V = σ v d t V=\sigma\,v\,dt
  12. r d t = n V = n σ v d t r\,dt=n\,V=n\,\sigma\,v\,dt
  13. r = σ Φ r=\sigma\,\Phi
  14. R = N r = N Φ σ R=N\,r=N\,\Phi\,\sigma
  15. R = E N Φ ( E ) σ ( E ) d E R=\int_{E}N\,\Phi(E)\,\sigma(E)\,dE
  16. σ = E Φ ( E ) σ ( E ) d E E Φ ( E ) d E = E Φ ( E ) σ ( E ) d E Φ \sigma=\frac{\int_{E}\Phi(E)\,\sigma(E)\,dE}{\int_{E}\Phi(E)\,dE}=\frac{\int_{% E}\Phi(E)\,\sigma(E)\,dE}{\Phi}
  17. \int
  18. R = N Φ σ R=N\,\Phi\,\sigma
  19. Σ = N σ \Sigma=N\,\sigma
  20. R = Σ Φ R=\Sigma\,\Phi
  21. L = l N L=l\,N
  22. l = v d t l=v\,dt
  23. N = n d V N=n\,dV
  24. L = v d t n d V L=v\,dt\,n\,dV
  25. Φ = n v \Phi=n\,v
  26. L = Φ d t d V L=\Phi\,dt\,dV
  27. λ = L R = Φ d t d V R \lambda=\frac{L}{R}=\frac{\Phi\,dt\,dV}{R}
  28. R = Φ Σ d t d V R=\Phi\,\Sigma\,dt\,dV
  29. λ = 1 Σ \lambda=\frac{1}{\Sigma}

Newton's_identities.html

  1. p k ( x 1 , , x n ) = i = 1 n x i k = x 1 k + + x n k , p_{k}(x_{1},\ldots,x_{n})=\sum\nolimits_{i=1}^{n}x_{i}^{k}=x_{1}^{k}+\cdots+x_% {n}^{k},
  2. e 0 ( x 1 , , x n ) \displaystyle e_{0}(x_{1},\ldots,x_{n})
  3. k e k ( x 1 , , x n ) = i = 1 k ( - 1 ) i - 1 e k - i ( x 1 , , x n ) p i ( x 1 , , x n ) , ke_{k}(x_{1},\ldots,x_{n})=\sum_{i=1}^{k}(-1)^{i-1}e_{k-i}(x_{1},\ldots,x_{n})% p_{i}(x_{1},\ldots,x_{n}),
  4. 0 = i = k - n k ( - 1 ) i - 1 e k - i ( x 1 , , x n ) p i ( x 1 , , x n ) , 0=\sum_{i=k-n}^{k}(-1)^{i-1}e_{k-i}(x_{1},\ldots,x_{n})p_{i}(x_{1},\ldots,x_{n% }),
  5. e 1 ( x 1 , , x n ) \displaystyle e_{1}(x_{1},\ldots,x_{n})
  6. e 1 = p 1 , 2 e 2 = e 1 p 1 - p 2 , 3 e 3 = e 2 p 1 - e 1 p 2 + p 3 , 4 e 4 = e 3 p 1 - e 2 p 2 + e 1 p 3 - p 4 , \begin{aligned}\displaystyle e_{1}&\displaystyle=p_{1},\\ \displaystyle 2e_{2}&\displaystyle=e_{1}p_{1}-p_{2},\\ \displaystyle 3e_{3}&\displaystyle=e_{2}p_{1}-e_{1}p_{2}+p_{3},\\ \displaystyle 4e_{4}&\displaystyle=e_{3}p_{1}-e_{2}p_{2}+e_{1}p_{3}-p_{4},\\ \end{aligned}
  7. p 1 = e 1 , p 2 = e 1 p 1 - 2 e 2 , p 3 = e 1 p 2 - e 2 p 1 + 3 e 3 , p 4 = e 1 p 3 - e 2 p 2 + e 3 p 1 - 4 e 4 , \begin{aligned}\displaystyle p_{1}&\displaystyle=e_{1},\\ \displaystyle p_{2}&\displaystyle=e_{1}p_{1}-2e_{2},\\ \displaystyle p_{3}&\displaystyle=e_{1}p_{2}-e_{2}p_{1}+3e_{3},\\ \displaystyle p_{4}&\displaystyle=e_{1}p_{3}-e_{2}p_{2}+e_{3}p_{1}-4e_{4},\\ &\displaystyle{}\ \ \vdots\end{aligned}
  8. p k ( x 1 , , x n ) = ( - 1 ) k - 1 k e k ( x 1 , , x n ) + i = 1 k - 1 ( - 1 ) k - 1 + i e k - i ( x 1 , , x n ) p i ( x 1 , , x n ) , p_{k}(x_{1},\ldots,x_{n})=(-1)^{k-1}ke_{k}(x_{1},\ldots,x_{n})+\sum_{i=1}^{k-1% }(-1)^{k-1+i}e_{k-i}(x_{1},\ldots,x_{n})p_{i}(x_{1},\ldots,x_{n}),
  9. p k ( x 1 , , x n ) = i = k - n k - 1 ( - 1 ) k - 1 + i e k - i ( x 1 , , x n ) p i ( x 1 , , x n ) , p_{k}(x_{1},\ldots,x_{n})=\sum_{i=k-n}^{k-1}(-1)^{k-1+i}e_{k-i}(x_{1},\ldots,x% _{n})p_{i}(x_{1},\ldots,x_{n}),
  10. i = 1 n ( x - x i ) = k = 0 n ( - 1 ) n + k e n - k x k , \prod_{i=1}^{n}\left(x-x_{i}\right)=\sum_{k=0}^{n}(-1)^{n+k}e_{n-k}x^{k},
  11. e k ( x 1 , , x n ) e_{k}(x_{1},\ldots,x_{n})
  12. p k ( x 1 , , x n ) = i = 1 n x i k , p_{k}(x_{1},\ldots,x_{n})=\sum_{i=1}^{n}x_{i}^{k},
  13. x 1 , , x n x_{1},\ldots,x_{n}
  14. e 0 = 1 , e 1 = p 1 , e 2 = 1 2 ( e 1 p 1 - p 2 ) , e 3 = 1 3 ( e 2 p 1 - e 1 p 2 + p 3 ) , e 4 = 1 4 ( e 3 p 1 - e 2 p 2 + e 1 p 3 - p 4 ) , \begin{aligned}\displaystyle e_{0}&\displaystyle=1,\\ \displaystyle e_{1}&\displaystyle=p_{1},\\ \displaystyle e_{2}&\displaystyle=\frac{1}{2}(e_{1}p_{1}-p_{2}),\\ \displaystyle e_{3}&\displaystyle=\frac{1}{3}(e_{2}p_{1}-e_{1}p_{2}+p_{3}),\\ \displaystyle e_{4}&\displaystyle=\frac{1}{4}(e_{3}p_{1}-e_{2}p_{2}+e_{1}p_{3}% -p_{4}),\\ &\displaystyle{}\ \ \vdots\end{aligned}
  15. x i x_{i}
  16. x i x_{i}
  17. s k = tr ( A k ) . s_{k}=\operatorname{tr}(A^{k})\,.
  18. t n + k = 1 n ( - 1 ) k a k t n - k \textstyle t^{n}+\sum_{k=1}^{n}(-1)^{k}a_{k}t^{n-k}
  19. k h k = i = 1 k h k - i p i , kh_{k}=\sum_{i=1}^{k}h_{k-i}p_{i},
  20. h 1 = p 1 , 2 h 2 = h 1 p 1 + p 2 , 3 h 3 = h 2 p 1 + h 1 p 2 + p 3 . \begin{aligned}\displaystyle h_{1}&\displaystyle=p_{1},\\ \displaystyle 2h_{2}&\displaystyle=h_{1}p_{1}+p_{2},\\ \displaystyle 3h_{3}&\displaystyle=h_{2}p_{1}+h_{1}p_{2}+p_{3}.\\ \end{aligned}
  21. k = 0 h k ( X 1 , , X n ) t k = i = 1 n 1 1 - X i t . \sum_{k=0}^{\infty}h_{k}(X_{1},\ldots,X_{n})t^{k}=\prod_{i=1}^{n}\frac{1}{1-X_% {i}t}.
  22. e 1 \displaystyle e_{1}
  23. h 1 = p 1 , h 2 = 1 2 p 1 2 + 1 2 p 2 = 1 2 ( p 1 2 + p 2 ) , h 3 = 1 6 p 1 3 + 1 2 p 1 p 2 + 1 3 p 3 = 1 6 ( p 1 3 + 3 p 1 p 2 + 2 p 3 ) , h 4 = 1 24 p 1 4 + 1 4 p 1 2 p 2 + 1 8 p 2 2 + 1 3 p 1 p 3 + 1 4 p 4 = 1 24 ( p 1 4 + 6 p 1 2 p 2 + 3 p 2 2 + 8 p 1 p 3 + 6 p 4 ) , h m = m 1 + 2 m 2 + + n m n = m m 1 0 , , m n 0 i = 1 n p i m i m i ! i m i \begin{aligned}\displaystyle h_{1}&\displaystyle=p_{1},\\ \displaystyle h_{2}&\displaystyle=\textstyle\frac{1}{2}p_{1}^{2}+\frac{1}{2}p_% {2}&&\displaystyle=\textstyle\frac{1}{2}(p_{1}^{2}+p_{2}),\\ \displaystyle h_{3}&\displaystyle=\textstyle\frac{1}{6}p_{1}^{3}+\frac{1}{2}p_% {1}p_{2}+\frac{1}{3}p_{3}&&\displaystyle=\textstyle\frac{1}{6}(p_{1}^{3}+3p_{1% }p_{2}+2p_{3}),\\ \displaystyle h_{4}&\displaystyle=\textstyle\frac{1}{24}p_{1}^{4}+\frac{1}{4}p% _{1}^{2}p_{2}+\frac{1}{8}p_{2}^{2}+\frac{1}{3}p_{1}p_{3}+\frac{1}{4}p_{4}&&% \displaystyle=\textstyle\frac{1}{24}(p_{1}^{4}+6p_{1}^{2}p_{2}+3p_{2}^{2}+8p_{% 1}p_{3}+6p_{4}),\\ \displaystyle h_{m}&\displaystyle=\sum_{m_{1}+2m_{2}+\cdots+nm_{n}=m\atop m_{1% }\geq 0,\ldots,m_{n}\geq 0}\prod_{i=1}^{n}\frac{p_{i}^{m_{i}}}{m_{i}!i^{m_{i}}% }\\ \end{aligned}
  24. 1 / N 1/N
  25. N = Π i = 1 l ( m i ! i m i ) N=\Pi_{i=1}^{l}(m_{i}!\,i^{m_{i}})
  26. m f ( m ; m 1 , , m n ) \displaystyle mf(m;m_{1},...,m_{n})
  27. p 1 \displaystyle p_{1}
  28. p m = r 1 + 2 r 2 + + n r n = m r 1 0 , , r n 0 ( - 1 ) m m ( r 1 + r 2 + + r n - 1 ) ! r 1 ! r 2 ! r n ! i = 1 n ( - e i ) r i p_{m}=\sum_{r_{1}+2r_{2}+\cdots+nr_{n}=m\atop r_{1}\geq 0,...,r_{n}\geq 0}(-1)% ^{m}\frac{m(r_{1}+r_{2}+\cdots+r_{n}-1)!}{r_{1}!r_{2}!\cdots r_{n}!}\prod_{i=1% }^{n}(-e_{i})^{r_{i}}
  29. f ( m ; r 1 , , r n ) \displaystyle f(m;\;r_{1},\cdots,r_{n})
  30. p 1 \displaystyle p_{1}
  31. Π i = 1 l h i m i \Pi_{i=1}^{l}h_{i}^{m_{i}}
  32. p m = - m 1 + 2 m 2 + + n m n = m m 1 0 , , m n 0 m ( r 1 + r 2 + + r n - 1 ) ! r 1 ! r 2 ! r n ! i = 1 n ( - h i ) r i p_{m}=-\sum_{m_{1}+2m_{2}+\cdots+nm_{n}=m\atop m_{1}\geq 0,...,m_{n}\geq 0}% \frac{m(r_{1}+r_{2}+\cdots+r_{n}-1)!}{r_{1}!r_{2}!\cdots r_{n}!}\prod_{i=1}^{n% }(-h_{i})^{r_{i}}
  33. e 1 = 1 p 1 , 2 e 2 = e 1 p 1 - 1 p 2 , 3 e 3 = e 2 p 1 - e 1 p 2 + 1 p 3 , n e n = e n - 1 p 1 - e n - 2 p 2 + + ( - 1 ) n e 1 p n - 1 + ( - 1 ) n - 1 p n \begin{aligned}\displaystyle e_{1}&\displaystyle=1p_{1},\\ \displaystyle 2e_{2}&\displaystyle=e_{1}p_{1}-1p_{2},\\ \displaystyle 3e_{3}&\displaystyle=e_{2}p_{1}-e_{1}p_{2}+1p_{3},\\ &\displaystyle\vdots\\ \displaystyle ne_{n}&\displaystyle=e_{n-1}p_{1}-e_{n-2}p_{2}+\cdots+(-1)^{n}e_% {1}p_{n-1}+(-1)^{n-1}p_{n}\\ \end{aligned}
  34. p 1 p_{1}
  35. - p 2 {-p_{2}}
  36. p 3 p_{3}
  37. ( - 1 ) n p n - 1 (-1)^{n}p_{n-1}
  38. p n p_{n}
  39. p n = | 1 0 e 1 e 1 1 0 2 e 2 e 2 e 1 1 3 e 3 e n - 1 e 2 e 1 n e n | | 1 0 e 1 1 0 e 2 e 1 1 e n - 1 e 2 e 1 ( - 1 ) n - 1 | - 1 = 1 ( - 1 ) n - 1 | 1 0 e 1 e 1 1 0 2 e 2 e 2 e 1 1 3 e 3 e n - 1 e 2 e 1 n e n | = | e 1 1 0 2 e 2 e 1 1 0 3 e 3 e 2 e 1 1 n e n e n - 1 e 1 | . \begin{aligned}\displaystyle p_{n}=&\displaystyle\begin{vmatrix}1&0&\cdots&&e_% {1}\\ e_{1}&1&0&\cdots&2e_{2}\\ e_{2}&e_{1}&1&&3e_{3}\\ \vdots&&\ddots&\ddots&\vdots\\ e_{n-1}&\cdots&e_{2}&e_{1}&ne_{n}\end{vmatrix}\begin{vmatrix}1&0&\cdots&\\ e_{1}&1&0&\cdots\\ e_{2}&e_{1}&1&\\ \vdots&&\ddots&\ddots\\ e_{n-1}&\cdots&e_{2}&e_{1}&(-1)^{n-1}\end{vmatrix}^{-1}\\ \displaystyle=\frac{1}{(-1)^{n-1}}&\displaystyle\begin{vmatrix}1&0&\cdots&&e_{% 1}\\ e_{1}&1&0&\cdots&2e_{2}\\ e_{2}&e_{1}&1&&3e_{3}\\ \vdots&&\ddots&\ddots&\vdots\\ e_{n-1}&\cdots&e_{2}&e_{1}&ne_{n}\end{vmatrix}\\ \displaystyle=&\displaystyle\begin{vmatrix}e_{1}&1&0&\cdots\\ 2e_{2}&e_{1}&1&0&\cdots\\ 3e_{3}&e_{2}&e_{1}&1\\ \vdots&&&\ddots&\ddots\\ ne_{n}&e_{n-1}&\cdots&&e_{1}\end{vmatrix}.\end{aligned}
  40. e n e_{n}
  41. p n p_{n}
  42. e n = 1 n ! \displaystyle e_{n}=\frac{1}{n!}
  43. h n h_{n}
  44. e n e_{n}
  45. h n h_{n}
  46. e n e_{n}
  47. i = 1 k ( t - x i ) = i = 0 k ( - 1 ) k - i e k - i ( x 1 , , x k ) t i \prod_{i=1}^{k}(t-x_{i})=\sum_{i=0}^{k}(-1)^{k-i}e_{k-i}(x_{1},\ldots,x_{k})t^% {i}
  48. 0 = i = 0 k ( - 1 ) k - i e k - i ( x 1 , , x k ) x j i for 1 j k 0=\sum_{i=0}^{k}(-1)^{k-i}e_{k-i}(x_{1},\ldots,x_{k}){x_{j}}^{i}\quad\,\text{% for }1\leq j\leq k
  49. 0 = ( - 1 ) k k e k ( x 1 , , x k ) + i = 1 k ( - 1 ) k - i e k - i ( x 1 , , x k ) p i ( x 1 , , x k ) , 0=(-1)^{k}ke_{k}(x_{1},\ldots,x_{k})+\sum_{i=1}^{k}(-1)^{k-i}e_{k-i}(x_{1},% \ldots,x_{k})p_{i}(x_{1},\ldots,x_{k}),
  50. i = 1 n ( t - x i ) = k = 0 n ( - 1 ) k a k t n - k \prod_{i=1}^{n}(t-x_{i})=\sum_{k=0}^{n}(-1)^{k}a_{k}t^{n-k}
  51. i = 1 n ( 1 - x i t ) = k = 0 n ( - 1 ) k a k t k . \prod_{i=1}^{n}(1-x_{i}t)=\sum_{k=0}^{n}(-1)^{k}a_{k}t^{k}.
  52. k = 0 n ( - 1 ) k e k ( x 1 , , x n ) t k = i = 1 n ( 1 - x i t ) . \sum_{k=0}^{n}(-1)^{k}e_{k}(x_{1},\ldots,x_{n})t^{k}=\prod_{i=1}^{n}(1-x_{i}t).
  53. k = 0 n ( - 1 ) k k e k ( x 1 , , x n ) t k = t i = 1 n [ ( - x i ) j i ( 1 - x j t ) ] = - ( i = 1 n x i t 1 - x i t ) j = 1 n ( 1 - x j t ) = - [ i = 1 n j = 1 ( x i t ) j ] [ = 0 n ( - 1 ) e ( x 1 , , x n ) t ] = [ j = 1 p j ( x 1 , , x n ) t j ] [ = 0 n ( - 1 ) - 1 e ( x 1 , , x n ) t ] , \begin{aligned}\displaystyle\sum_{k=0}^{n}(-1)^{k}ke_{k}(x_{1},\ldots,x_{n})t^% {k}&\displaystyle=t\sum_{i=1}^{n}\left[(-x_{i})\prod\nolimits_{j\neq i}(1-x_{j% }t)\right]\\ &\displaystyle=-\left(\sum_{i=1}^{n}\frac{x_{i}t}{1-x_{i}t}\right)\prod% \nolimits_{j=1}^{n}(1-x_{j}t)\\ &\displaystyle=-\left[\sum_{i=1}^{n}\sum_{j=1}^{\infty}(x_{i}t)^{j}\right]% \left[\sum_{\ell=0}^{n}(-1)^{\ell}e_{\ell}(x_{1},\ldots,x_{n})t^{\ell}\right]% \\ &\displaystyle=\left[\sum_{j=1}^{\infty}p_{j}(x_{1},\ldots,x_{n})t^{j}\right]% \left[\sum_{\ell=0}^{n}(-1)^{\ell-1}e_{\ell}(x_{1},\ldots,x_{n})t^{\ell}\right% ],\\ \end{aligned}
  54. X 1 - X = X + X 2 + X 3 + X 4 + X 5 + \frac{X}{1-X}=X+X^{2}+X^{3}+X^{4}+X^{5}+\cdots
  55. ( - 1 ) k k e k ( x 1 , , x n ) = j = 1 k ( - 1 ) k - j - 1 p j ( x 1 , , x n ) e k - j ( x 1 , , x n ) , (-1)^{k}ke_{k}(x_{1},\ldots,x_{n})=\sum_{j=1}^{k}(-1)^{k-j-1}p_{j}(x_{1},% \ldots,x_{n})e_{k-j}(x_{1},\ldots,x_{n}),
  56. p i e k - i = r ( i ) + r ( i + 1 ) for 1 < i < k p_{i}e_{k-i}=r(i)+r(i+1)\quad\,\text{for }1<i<k
  57. p k e 0 = p k = r ( k ) p_{k}e_{0}=p_{k}=r(k)\,
  58. p 1 e k - 1 = k e k + r ( 2 ) p_{1}e_{k-1}=ke_{k}+r(2)\,

Newton_second.html

  1. F t = Δ m v \vec{F}\cdot t=\Delta m\vec{v}

Newton–Euler_equations.html

  1. ( F s y m b o l τ ) = ( m I 3 0 0 I cm ) ( a cm s y m b o l α ) + ( 0 s y m b o l ω × I cm s y m b o l ω ) , \left(\begin{matrix}{F}\\ {symbol\tau}\end{matrix}\right)=\left(\begin{matrix}m{I_{3}}&0\\ 0&{I}_{\rm cm}\end{matrix}\right)\left(\begin{matrix}a_{\rm cm}\\ {symbol\alpha}\end{matrix}\right)+\left(\begin{matrix}0\\ {symbol\omega}\times{I}_{\rm cm}\,{symbol\omega}\end{matrix}\right),
  2. ( F s y m b o l τ p ) = ( m I 3 - m [ c ] × m [ c ] × I cm - m [ c ] × [ c ] × ) ( a p s y m b o l α ) + ( m [ s y m b o l ω ] × [ s y m b o l ω ] × c [ s y m b o l ω ] × ( I cm - m [ c ] × [ c ] × ) s y m b o l ω ) , \left(\begin{matrix}{F}\\ {symbol\tau}_{\rm p}\end{matrix}\right)=\left(\begin{matrix}m{I_{3}}&-m[{c}]^{% \times}\\ m[{c}]^{\times}&{I}_{\rm cm}-m[{c}]^{\times}[{c}]^{\times}\end{matrix}\right)% \left(\begin{matrix}a_{\rm p}\\ {symbol\alpha}\end{matrix}\right)+\left(\begin{matrix}m[{symbol\omega}]^{% \times}[{symbol\omega}]^{\times}{c}\\ {[symbol\omega]}^{\times}({I}_{\rm cm}-m[{c}]^{\times}[{c}]^{\times})\,{symbol% \omega}\end{matrix}\right),
  3. [ 𝐜 ] × ( 0 - c z c y c z 0 - c x - c y c x 0 ) [ 𝐬𝐲𝐦𝐛𝐨𝐥 ω ] × ( 0 - ω z ω y ω z 0 - ω x - ω y ω x 0 ) [\mathbf{c}]^{\times}\equiv\left(\begin{matrix}0&-c_{z}&c_{y}\\ c_{z}&0&-c_{x}\\ -c_{y}&c_{x}&0\end{matrix}\right)\qquad\qquad[\mathbf{symbol{\omega}}]^{\times% }\equiv\left(\begin{matrix}0&-\omega_{z}&\omega_{y}\\ \omega_{z}&0&-\omega_{x}\\ -\omega_{y}&\omega_{x}&0\end{matrix}\right)
  4. ( m I 3 - m [ c ] × m [ c ] × I cm - m [ c ] × [ c ] × ) , \left(\begin{matrix}m{I_{3}}&-m[{c}]^{\times}\\ m[{c}]^{\times}&{I}_{\rm cm}-m[{c}]^{\times}[{c}]^{\times}\end{matrix}\right),
  5. ( m [ s y m b o l ω ] × [ s y m b o l ω ] × c [ s y m b o l ω ] × ( I cm - m [ c ] × [ c ] × ) s y m b o l ω ) . \left(\begin{matrix}m{[symbol\omega]}^{\times}{[symbol\omega]}^{\times}{c}\\ {[symbol\omega]}^{\times}({I}_{\rm cm}-m[{c}]^{\times}[{c}]^{\times})\,{symbol% \omega}\end{matrix}\right).

Nilpotent_cone.html

  1. 𝒩 \mathcal{N}
  2. 𝔤 \mathfrak{g}
  3. 𝔤 . \mathfrak{g}.
  4. 𝒩 = { a 𝔤 : ρ ( a ) is nilpotent for all representations ρ : 𝔤 End ( V ) } . \mathcal{N}=\{a\in\mathfrak{g}:\rho(a)\mbox{ is nilpotent for all % representations }~{}\rho:\mathfrak{g}\to\operatorname{End}(V)\}.
  5. 𝔤 \mathfrak{g}
  6. k k
  7. sl 2 \operatorname{sl}_{2}
  8. 1. 1.

Nitrogen_laser.html

  1. t [ ns ] = 36 1 + 12.8 * p [ bar ] . t[\mathrm{ns}]=\cfrac{36}{1+12.8*p[\mathrm{bar}]}.

No-three-in-line_problem.html

  1. ( 1 - ϵ ) n (1-\epsilon)n
  2. ( 3 2 - ϵ ) n (\frac{3}{2}-\epsilon)n
  3. c = 2 π 2 3 3 1.874. c=\sqrt[3]{\frac{2\pi^{2}}{3}}\approx 1.874.
  4. c = π 3 1.814. c=\frac{\pi}{\sqrt{3}}\approx 1.814.
  5. 1 - ϵ 2 n 2 . \frac{1-\epsilon}{2n^{2}}.
  6. Θ ( n 2 ) \Theta(n^{2})

Nodal_analysis.html

  1. V 1 - V S R 1 + V 1 R 2 - I S = 0 \frac{V_{1}-V_{S}}{R_{1}}+\frac{V_{1}}{R_{2}}-I_{S}=0
  2. V 1 = ( V S R 1 + I S ) ( 1 R 1 + 1 R 2 ) V_{1}=\frac{\left(\frac{V_{S}}{R1}+I_{S}\right)}{\left(\frac{1}{R_{1}}+\frac{1% }{R_{2}}\right)}
  3. V 1 = ( 5 V 100 Ω + 20 mA ) ( 1 100 Ω + 1 200 Ω ) 4.667 V V_{1}=\frac{\left(\frac{5\,\text{ V}}{100\,\Omega}+20\,\text{ mA}\right)}{% \left(\frac{1}{100\,\Omega}+\frac{1}{200\,\Omega}\right)}\approx 4.667\,\text{% V}
  4. { V 1 - V B R 1 + V 2 - V B R 2 + V 2 R 3 = 0 V 1 = V 2 + V A \begin{cases}\frac{V_{1}-V\text{B}}{R_{1}}+\frac{V_{2}-V\text{B}}{R_{2}}+\frac% {V_{2}}{R_{3}}=0\\ V_{1}=V_{2}+V\text{A}\\ \end{cases}
  5. V 2 = ( R 1 + R 2 ) R 3 V B - R 2 R 3 V A ( R 1 + R 2 ) R 3 + R 1 R 2 V_{2}=\frac{(R_{1}+R_{2})R_{3}V\text{B}-R_{2}R_{3}V\text{A}}{(R_{1}+R_{2})R_{3% }+R_{1}R_{2}}

Noether_normalization_lemma.html

  1. 𝔸 k d \mathbb{A}^{d}_{k}
  2. X 𝔸 k d X\to\mathbb{A}^{d}_{k}
  3. y 1 , , y m y_{1},...,y_{m}
  4. y 1 , , y d y_{1},...,y_{d}
  5. k [ y 1 , , y d ] k[y_{1},...,y_{d}]
  6. m = d m=d
  7. m > d m>d
  8. m - 1 m-1
  9. x 1 , , x d x_{1},...,x_{d}
  10. k [ x 1 , , x d ] k[x_{1},...,x_{d}]
  11. m > d m>d
  12. f ( y 1 , , y m ) = 0 f(y_{1},...,y_{m})=0
  13. z i = y i - y 1 r i - 1 , 2 i m . z_{i}=y_{i}-y_{1}^{r^{i-1}},\quad 2\leq i\leq m.
  14. f ( y 1 , z 2 + y 1 r , z 3 + y 1 r 2 , , z m + y 1 r m - 1 ) = 0 f(y_{1},z_{2}+y_{1}^{r},z_{3}+y_{1}^{r^{2}},...,z_{m}+y_{1}^{r^{m-1}})=0
  15. y 1 y_{1}
  16. a y 1 α 1 2 m ( z i + y 1 r i - 1 ) α i , a k , ay_{1}^{\alpha_{1}}\prod_{2}^{m}(z_{i}+y_{1}^{r^{i-1}})^{\alpha_{i}},a\in k,
  17. a y 1 α 1 + r α 2 + + α m r m - 1 . ay_{1}^{\alpha_{1}+r\alpha_{2}+...+\alpha_{m}r^{m-1}}.
  18. α i \alpha_{i}
  19. f ( y 1 , z 2 + y 1 r , z 3 + y 1 r 2 , , z m + y 1 r m - 1 ) f(y_{1},z_{2}+y_{1}^{r},z_{3}+y_{1}^{r^{2}},...,z_{m}+y_{1}^{r^{m-1}})
  20. y 1 y_{1}
  21. y 1 y_{1}
  22. S = k [ z 2 , , z m ] S=k[z_{2},...,z_{m}]
  23. y i = z i + y 1 r i - 1 y_{i}=z_{i}+y_{1}^{r^{i-1}}
  24. S = k [ y 1 , , y d ] S=k[y_{1},...,y_{d}]
  25. d = 0 d=0
  26. 0 ( y 1 ) ( y 1 , y 2 ) ( y 1 , , y d ) 0\subsetneq(y_{1})\subsetneq(y_{1},y_{2})\subsetneq\cdots\subsetneq(y_{1},% \dots,y_{d})
  27. 0 𝔭 1 𝔭 m 0\subsetneq\mathfrak{p}_{1}\subsetneq\cdots\subsetneq\mathfrak{p}_{m}
  28. 0 u 𝔭 1 0\neq u\in\mathfrak{p}_{1}
  29. T = k [ u , z 2 , , z d ] T=k[u,z_{2},\dots,z_{d}]
  30. T / ( u ) T/(u)
  31. 𝔭 i T \mathfrak{p}_{i}\cap T
  32. m m
  33. T / ( 𝔭 1 T ) T/(\mathfrak{p}_{1}\cap T)
  34. m - 1 m-1
  35. dim T / ( 𝔭 1 T ) dim T / ( u ) \operatorname{dim}T/(\mathfrak{p}_{1}\cap T)\leq\operatorname{dim}T/(u)
  36. m - 1 d - 1 m-1\leq d-1
  37. dim S d \dim S\leq d
  38. X = Spec A 𝐀 m X=\operatorname{Spec}A\subset\mathbf{A}^{m}
  39. 𝐀 m 𝐀 d \mathbf{A}^{m}\to\mathbf{A}^{d}
  40. X 𝐀 d X\to\mathbf{A}^{d}

Noetherian_scheme.html

  1. Spec A i \operatorname{Spec}A_{i}
  2. A i A_{i}
  3. Spec A \operatorname{Spec}A
  4. Spec A \operatorname{Spec}A
  5. 𝒪 X , x \mathcal{O}_{X,x}

Noetherian_topological_space.html

  1. X X
  2. Y 1 Y 2 Y_{1}\supseteq Y_{2}\supseteq\cdots
  3. Y i Y_{i}
  4. X X
  5. m m
  6. Y m = Y m + 1 = . Y_{m}=Y_{m+1}=\cdots.
  7. X X
  8. X X
  9. X X
  10. 𝔸 k n \mathbb{A}^{n}_{k}
  11. n n
  12. k k
  13. 𝔸 k n \mathbb{A}^{n}_{k}
  14. Y 1 Y 2 Y 3 Y_{1}\supseteq Y_{2}\supseteq Y_{3}\supseteq\cdots
  15. I ( Y 1 ) I ( Y 2 ) I ( Y 3 ) I(Y_{1})\subseteq I(Y_{2})\subseteq I(Y_{3})\subseteq\cdots
  16. k [ x 1 , , x n ] . k[x_{1},\ldots,x_{n}].
  17. k [ x 1 , , x n ] k[x_{1},\ldots,x_{n}]
  18. m m
  19. I ( Y m ) = I ( Y m + 1 ) = I ( Y m + 2 ) = . I(Y_{m})=I(Y_{m+1})=I(Y_{m+2})=\cdots.
  20. V ( I ( Y ) ) V(I(Y))
  21. V ( I ( Y i ) ) = Y i V(I(Y_{i}))=Y_{i}
  22. i . i.
  23. Y m = Y m + 1 = Y m + 2 = Y_{m}=Y_{m+1}=Y_{m+2}=\cdots

Non-competitive_inhibition.html

  1. V m a x a p p = V m a x 1 + [ I ] K I V_{max}^{app}=\frac{V_{max}}{1+\frac{[I]}{K_{I}}}
  2. a p p a r e n t [ E ] 0 = [ E ] 0 1 + [ I ] K I {apparent\ [E]_{0}}=\frac{[E]_{0}}{1+\frac{[I]}{K_{I}}}

Non-deliverable_forward.html

  1. π \pi
  2. π = N S - N F S = N ( 1 - F S ) \pi=\frac{NS-NF}{S}=N\left(1-\frac{F}{S}\right)

Non-line-of-sight_propagation.html

  1. tan δ \tan\delta
  2. tan δ = σ ω ϵ 0 ϵ r \tan\delta=\frac{\sigma}{\omega\epsilon_{0}\epsilon_{r}}
  3. σ \sigma
  4. ω = 2 π f \omega=2\pi f
  5. f f
  6. ϵ 0 \epsilon_{0}
  7. ϵ r \epsilon_{r}
  8. σ ω ϵ 0 ϵ r \sigma\gg\omega\epsilon_{0}\epsilon_{r}
  9. σ ω ϵ 0 ϵ r \sigma\ll\omega\epsilon_{0}\epsilon_{r}
  10. ϵ r \epsilon_{r}
  11. η \eta
  12. η = μ 0 μ r ϵ 0 ϵ r \eta=\sqrt{\frac{\mu_{0}\mu_{r}}{\epsilon_{0}\epsilon_{r}}}
  13. μ 0 \mu_{0}
  14. 4 π x 10 - 7 4\pi x10^{-7}
  15. μ r \mu_{r}
  16. ϵ 0 \epsilon_{0}
  17. 8.85 x 10 - 12 8.85x10^{-12}
  18. ϵ r \epsilon_{r}
  19. μ r = 1 \mu_{r}=1
  20. ϵ r = 1 \epsilon_{r}=1
  21. η 0 \eta_{0}
  22. η 0 = μ 0 ϵ 0 \eta_{0}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}
  23. Ω \Omega
  24. Γ \Gamma
  25. η 1 \eta_{1}
  26. η 2 \eta_{2}
  27. Γ 21 \Gamma_{21}
  28. Γ 21 = η 2 - η 1 η 2 + η 1 \Gamma_{21}=\frac{\eta_{2}-\eta_{1}}{\eta_{2}+\eta_{1}}
  29. T r T_{r}
  30. T r e f = 10 log 10 ( 1 - | Γ 21 | 2 ) d B T_{ref}=10\log_{10}(1-\left|\Gamma_{21}\right|^{2})dB

Noncentral_chi-squared_distribution.html

  1. Q M ( a , b ) Q_{M}(a,b)
  2. k + λ k+\lambda\,
  3. 2 ( k + 2 λ ) 2(k+2\lambda)\,
  4. 2 3 / 2 ( k + 3 λ ) ( k + 2 λ ) 3 / 2 \frac{2^{3/2}(k+3\lambda)}{(k+2\lambda)^{3/2}}
  5. 12 ( k + 4 λ ) ( k + 2 λ ) 2 \frac{12(k+4\lambda)}{(k+2\lambda)^{2}}
  6. exp ( λ t 1 - 2 t ) ( 1 - 2 t ) k / 2 for 2 t < 1 \frac{\exp\left(\frac{\lambda t}{1-2t}\right)}{(1-2t)^{k/2}}\,\text{ for }2t<1
  7. exp ( i λ t 1 - 2 i t ) ( 1 - 2 i t ) k / 2 \frac{\exp\left(\frac{i\lambda t}{1-2it}\right)}{(1-2it)^{k/2}}
  8. χ 2 \chi^{2}
  9. X 1 X_{1}
  10. X 2 , , X_{2},\ldots,
  11. X i , , X_{i},\ldots,
  12. X k X_{k}
  13. μ i \mu_{i}
  14. i = 1 k X i 2 \sum_{i=1}^{k}X_{i}^{2}
  15. k k
  16. X i X_{i}
  17. λ \lambda
  18. X i X_{i}
  19. λ = i = 1 k μ i 2 . \lambda=\sum_{i=1}^{k}\mu_{i}^{2}.
  20. λ \lambda
  21. λ \lambda
  22. N ( 0 k , I k ) N(0_{k},I_{k})
  23. χ 2 \chi^{2}
  24. N ( μ , I k ) N(\mu,I_{k})
  25. 0 k 0_{k}
  26. μ = ( μ 1 , , μ k ) \mu=(\mu_{1},\ldots,\mu_{k})
  27. I k I_{k}
  28. f X ( x ; k , λ ) = i = 0 e - λ / 2 ( λ / 2 ) i i ! f Y k + 2 i ( x ) , f_{X}(x;k,\lambda)=\sum_{i=0}^{\infty}\frac{e^{-\lambda/2}(\lambda/2)^{i}}{i!}% f_{Y_{k+2i}}(x),
  29. Y q Y_{q}
  30. q q
  31. λ / 2 \lambda/2
  32. J = i J=i
  33. λ \lambda
  34. f X ( x ; k , λ ) = 1 2 e - ( x + λ ) / 2 ( x λ ) k / 4 - 1 / 2 I k / 2 - 1 ( λ x ) f_{X}(x;k,\lambda)=\frac{1}{2}e^{-(x+\lambda)/2}\left(\frac{x}{\lambda}\right)% ^{k/4-1/2}I_{k/2-1}(\sqrt{\lambda x})
  35. I ν ( y ) I_{\nu}(y)
  36. I ν ( y ) = ( y / 2 ) ν j = 0 ( y 2 / 4 ) j j ! Γ ( ν + j + 1 ) . I_{\nu}(y)=(y/2)^{\nu}\sum_{j=0}^{\infty}\frac{(y^{2}/4)^{j}}{j!\Gamma(\nu+j+1% )}.
  37. f X ( x ; k , λ ) = e - λ / 2 0 F 1 ( ; k / 2 ; λ x / 4 ) 1 2 k / 2 Γ ( k / 2 ) e - x / 2 x k / 2 - 1 . f_{X}(x;k,\lambda)={{\rm e}^{-\lambda/2}}_{0}F_{1}(;k/2;\lambda x/4)\frac{1}{2% ^{k/2}\Gamma(k/2)}{\rm e}^{-x/2}x^{k/2-1}.
  38. M ( t ; k , λ ) = exp ( λ t 1 - 2 t ) ( 1 - 2 t ) k / 2 . M(t;k,\lambda)=\frac{\exp\left(\frac{\lambda t}{1-2t}\right)}{(1-2t)^{k/2}}.
  39. μ 1 = k + λ \mu^{\prime}_{1}=k+\lambda
  40. μ 2 = ( k + λ ) 2 + 2 ( k + 2 λ ) \mu^{\prime}_{2}=(k+\lambda)^{2}+2(k+2\lambda)
  41. μ 3 = ( k + λ ) 3 + 6 ( k + λ ) ( k + 2 λ ) + 8 ( k + 3 λ ) \mu^{\prime}_{3}=(k+\lambda)^{3}+6(k+\lambda)(k+2\lambda)+8(k+3\lambda)
  42. μ 4 = ( k + λ ) 4 + 12 ( k + λ ) 2 ( k + 2 λ ) + 4 ( 11 k 2 + 44 k λ + 36 λ 2 ) + 48 ( k + 4 λ ) \mu^{\prime}_{4}=(k+\lambda)^{4}+12(k+\lambda)^{2}(k+2\lambda)+4(11k^{2}+44k% \lambda+36\lambda^{2})+48(k+4\lambda)
  43. μ 2 = 2 ( k + 2 λ ) \mu_{2}=2(k+2\lambda)\,
  44. μ 3 = 8 ( k + 3 λ ) \mu_{3}=8(k+3\lambda)\,
  45. μ 4 = 12 ( k + 2 λ ) 2 + 48 ( k + 4 λ ) \mu_{4}=12(k+2\lambda)^{2}+48(k+4\lambda)\,
  46. K n = 2 n - 1 ( n - 1 ) ! ( k + n λ ) . K_{n}=2^{n-1}(n-1)!(k+n\lambda).\,
  47. μ n = 2 n - 1 ( n - 1 ) ! ( k + n λ ) + j = 1 n - 1 ( n - 1 ) ! 2 j - 1 ( n - j ) ! ( k + j λ ) μ n - j . \mu^{\prime}_{n}=2^{n-1}(n-1)!(k+n\lambda)+\sum_{j=1}^{n-1}\frac{(n-1)!2^{j-1}% }{(n-j)!}(k+j\lambda)\mu^{\prime}_{n-j}.
  48. P ( x ; k , λ ) = e - λ / 2 j = 0 ( λ / 2 ) j j ! Q ( x ; k + 2 j ) P(x;k,\lambda)=e^{-\lambda/2}\;\sum_{j=0}^{\infty}\frac{(\lambda/2)^{j}}{j!}Q(% x;k+2j)
  49. Q ( x ; k ) Q(x;k)\,
  50. Q ( x ; k ) = γ ( k / 2 , x / 2 ) Γ ( k / 2 ) Q(x;k)=\frac{\gamma(k/2,x/2)}{\Gamma(k/2)}\,
  51. γ ( k , z ) \gamma(k,z)\,
  52. Q M ( a , b ) Q_{M}(a,b)
  53. P ( x ; k , λ ) = 1 - Q k 2 ( λ , x ) P(x;k,\lambda)=1-Q_{\frac{k}{2}}\left(\sqrt{\lambda},\sqrt{x}\right)
  54. P ( x ; k , λ ) Φ { ( x k + λ ) h - ( 1 + h p ( h - 1 - 0.5 ( 2 - h ) m p ) ) h 2 p ( 1 + 0.5 m p ) } P(x;k,\lambda)\approx\Phi\left\{\frac{(\frac{x}{k+\lambda})^{h}-(1+hp(h-1-0.5(% 2-h)mp))}{h\sqrt{2p}(1+0.5mp)}\right\}
  55. Φ { } \Phi\{\cdot\}\,
  56. h = 1 - 2 3 ( k + λ ) ( k + 3 λ ) ( k + 2 λ ) 2 ; h=1-\frac{2}{3}\frac{(k+\lambda)(k+3\lambda)}{(k+2\lambda)^{2}}\,;
  57. p = k + 2 λ ( k + λ ) 2 ; p=\frac{k+2\lambda}{(k+\lambda)^{2}};
  58. m = ( h - 1 ) ( 1 - 3 h ) . m=(h-1)(1-3h)\,.
  59. λ \lambda\,
  60. P ( x ; k , λ ) Φ { ( x k ) 1 / 3 - ( 1 - 2 9 k ) 2 9 k } , P(x;k,\lambda)\approx\Phi\left\{\frac{\left(\frac{x}{k}\right)^{1/3}-\left(1-% \frac{2}{9k}\right)}{\sqrt{\frac{2}{9k}}}\right\},
  61. x x
  62. { 4 x f ′′ ( x ) + ( - 2 k + 4 x + 8 ) f ( x ) + f ( x ) ( - k - λ + x + 4 ) = 0 f ( 1 ) 2 k / 2 e λ + 1 2 = 0 F ~ 1 ( ; k 2 ; λ 4 ) λ 0 F ~ 1 ( ; k 2 + 1 ; λ 4 ) + 2 ( k - 3 ) 0 F ~ 1 ( ; k 2 ; λ 4 ) = 2 k 2 + 2 e λ + 1 2 f ( 1 ) } \left\{\begin{array}[]{l}4xf^{\prime\prime}(x)+(-2k+4x+8)f^{\prime}(x)+f(x)(-k% -\lambda+x+4)=0\\ f(1)2^{k/2}e^{\frac{\lambda+1}{2}}=\,_{0}\tilde{F}_{1}\left(;\frac{k}{2};\frac% {\lambda}{4}\right)\\ \lambda\,_{0}\tilde{F}_{1}\left(;\frac{k}{2}+1;\frac{\lambda}{4}\right)+2(k-3)% \,_{0}\tilde{F}_{1}\left(;\frac{k}{2};\frac{\lambda}{4}\right)=2^{\frac{k}{2}+% 2}e^{\frac{\lambda+1}{2}}f^{\prime}(1)\end{array}\right\}
  63. σ 1 = = σ k = 1 \sigma_{1}=\cdots=\sigma_{k}=1
  64. X 1 , , X k X_{1},\ldots,X_{k}
  65. X = X 1 2 + + X k 2 X=X_{1}^{2}+\cdots+X_{k}^{2}
  66. λ = μ 1 2 + + μ k 2 \lambda=\mu_{1}^{2}+\cdots+\mu_{k}^{2}
  67. μ 1 = λ \mu_{1}=\sqrt{\lambda}
  68. μ 2 = = μ k = 0 \mu_{2}=\cdots=\mu_{k}=0
  69. X = X 1 2 X=X_{1}^{2}
  70. f X ( x , 1 , λ ) \displaystyle f_{X}(x,1,\lambda)
  71. ϕ ( ) \phi(\cdot)
  72. X 2 , , X k X_{2},\ldots,X_{k}
  73. X 2 2 + + X k 2 X_{2}^{2}+\cdots+X_{k}^{2}
  74. X 1 2 X_{1}^{2}
  75. X 1 2 X_{1}^{2}
  76. V V
  77. V χ k 2 V\sim\chi_{k}^{2}
  78. V V
  79. V χ k 2 ( 0 ) V\sim{\chi^{\prime}}^{2}_{k}(0)
  80. V 1 χ k 1 2 ( λ ) V_{1}\sim{\chi^{\prime}}_{k_{1}}^{2}(\lambda)
  81. V 2 χ k 2 2 ( 0 ) V_{2}\sim{\chi^{\prime}}_{k_{2}}^{2}(0)
  82. V 1 V_{1}
  83. V 2 V_{2}
  84. V 1 / k 1 V 2 / k 2 F k 1 , k 2 ( λ ) \frac{V_{1}/k_{1}}{V_{2}/k_{2}}\sim F^{\prime}_{k_{1},k_{2}}(\lambda)
  85. J Poisson ( λ 2 ) J\sim\mathrm{Poisson}\left(\frac{\lambda}{2}\right)
  86. χ k + 2 J 2 χ k 2 ( λ ) \chi_{k+2J}^{2}\sim{\chi^{\prime}}_{k}^{2}(\lambda)
  87. V χ 2 2 ( λ ) V\sim{\chi^{\prime}}^{2}_{2}(\lambda)
  88. V \sqrt{V}
  89. λ \sqrt{\lambda}
  90. V χ k 2 ( λ ) V\sim{\chi^{\prime}}^{2}_{k}(\lambda)
  91. V - ( k + λ ) 2 ( k + 2 λ ) N ( 0 , 1 ) \frac{V-(k+\lambda)}{\sqrt{2(k+2\lambda)}}\to N(0,1)
  92. k k\to\infty
  93. λ \lambda\to\infty
  94. z = [ ( X - b ) / ( k + λ ) ] 1 / 2 z=[(X-b)/(k+\lambda)]^{1/2}
  95. z z
  96. O ( ( k + λ ) - 4 ) O((k+\lambda)^{-4})
  97. b b
  98. b = ( k - 1 ) / 2 b=(k-1)/2
  99. z z
  100. λ \lambda
  101. b = ( k - 1 ) / 3 b=(k-1)/3
  102. z z
  103. λ \lambda
  104. b = ( k - 1 ) / 4 b=(k-1)/4
  105. z z
  106. λ \lambda
  107. z 1 = ( X - ( k - 1 ) / 2 ) 1 / 2 z_{1}=(X-(k-1)/2)^{1/2}
  108. ( λ + ( k - 1 ) / 2 ) 1 / 2 (\lambda+(k-1)/2)^{1/2}
  109. O ( ( k + λ ) - 2 ) O((k+\lambda)^{-2})
  110. 1 k ( X i - μ i σ i ) 2 \sum_{1}^{k}\left(\frac{X_{i}-\mu_{i}}{\sigma_{i}}\right)^{2}
  111. 1 k ( X i σ i ) 2 \sum_{1}^{k}\left(\frac{X_{i}}{\sigma_{i}}\right)^{2}
  112. 1 k ( X i - μ i σ i ) 2 \sqrt{\sum_{1}^{k}\left(\frac{X_{i}-\mu_{i}}{\sigma_{i}}\right)^{2}}
  113. 1 k ( X i σ i ) 2 \sqrt{\sum_{1}^{k}\left(\frac{X_{i}}{\sigma_{i}}\right)^{2}}

Noncentral_chi_distribution.html

  1. π 2 L 1 / 2 ( k / 2 - 1 ) ( - λ 2 2 ) \sqrt{\frac{\pi}{2}}L_{1/2}^{(k/2-1)}\left(\frac{-\lambda^{2}}{2}\right)\,
  2. k + λ 2 - μ 2 k+\lambda^{2}-\mu^{2}\,
  3. X i X_{i}
  4. μ i \mu_{i}
  5. σ i 2 \sigma_{i}^{2}
  6. Z = i = 1 k ( X i σ i ) 2 Z=\sqrt{\sum_{i=1}^{k}\left(\frac{X_{i}}{\sigma_{i}}\right)^{2}}
  7. k k
  8. X i X_{i}
  9. λ \lambda
  10. X i X_{i}
  11. λ = i = 1 k ( μ i σ i ) 2 \lambda=\sqrt{\sum_{i=1}^{k}\left(\frac{\mu_{i}}{\sigma_{i}}\right)^{2}}
  12. f ( x ; k , λ ) = e - ( x 2 + λ 2 ) / 2 x k λ ( λ x ) k / 2 I k / 2 - 1 ( λ x ) f(x;k,\lambda)=\frac{e^{-(x^{2}+\lambda^{2})/2}x^{k}\lambda}{(\lambda x)^{k/2}% }I_{k/2-1}(\lambda x)
  13. I ν ( z ) I_{\nu}(z)
  14. μ 1 = π 2 L 1 / 2 ( k / 2 - 1 ) ( - λ 2 2 ) \mu^{^{\prime}}_{1}=\sqrt{\frac{\pi}{2}}L_{1/2}^{(k/2-1)}\left(\frac{-\lambda^% {2}}{2}\right)
  15. μ 2 = k + λ 2 \mu^{^{\prime}}_{2}=k+\lambda^{2}
  16. μ 3 = 3 π 2 L 3 / 2 ( k / 2 - 1 ) ( - λ 2 2 ) \mu^{^{\prime}}_{3}=3\sqrt{\frac{\pi}{2}}L_{3/2}^{(k/2-1)}\left(\frac{-\lambda% ^{2}}{2}\right)
  17. μ 4 = ( k + λ 2 ) 2 + 2 ( k + 2 λ 2 ) \mu^{^{\prime}}_{4}=(k+\lambda^{2})^{2}+2(k+2\lambda^{2})
  18. L n ( a ) ( z ) L_{n}^{(a)}(z)
  19. n n
  20. n n
  21. λ \lambda
  22. λ 2 \lambda^{2}
  23. { x 2 f ′′ ( x ) + ( - k x + 2 x 3 + x ) f ( x ) + f ( x ) ( - x 2 ( λ 2 + k - 2 ) + k + x 4 - 1 ) = 0 , f ( 1 ) = e - λ 2 2 - 1 2 λ 1 - k 2 I k - 2 2 ( λ ) , f ( 1 ) = e - λ 2 2 - 1 2 λ 2 - k 2 I k - 4 2 ( λ ) } \left\{\begin{array}[]{l}x^{2}f^{\prime\prime}(x)+\left(-kx+2x^{3}+x\right)f^{% \prime}(x)+f(x)\left(-x^{2}\left(\lambda^{2}+k-2\right)+k+x^{4}-1\right)=0,\\ f(1)=e^{-\frac{\lambda^{2}}{2}-\frac{1}{2}}\lambda^{1-\frac{k}{2}}I_{\frac{k-2% }{2}}(\lambda),f^{\prime}(1)=e^{-\frac{\lambda^{2}}{2}-\frac{1}{2}}\lambda^{2-% \frac{k}{2}}I_{\frac{k-4}{2}}(\lambda)\end{array}\right\}
  24. X j = ( X 1 j , X 2 j ) , j = 1 , 2 , n X_{j}=(X_{1j},X_{2j}),j=1,2,\dots n
  25. N ( μ i , σ i 2 ) , i = 1 , 2 N(\mu_{i},\sigma_{i}^{2}),i=1,2
  26. ρ \rho
  27. E ( X j ) = μ = ( μ 1 , μ 2 ) T , Σ = [ σ 11 σ 12 σ 21 σ 22 ] = [ σ 1 2 ρ σ 1 σ 2 ρ σ 1 σ 2 σ 2 2 ] , E(X_{j})=\mu=(\mu_{1},\mu_{2})^{T},\qquad\Sigma=\begin{bmatrix}\sigma_{11}&% \sigma_{12}\\ \sigma_{21}&\sigma_{22}\end{bmatrix}=\begin{bmatrix}\sigma_{1}^{2}&\rho\sigma_% {1}\sigma_{2}\\ \rho\sigma_{1}\sigma_{2}&\sigma_{2}^{2}\end{bmatrix},
  28. Σ \Sigma
  29. U = [ j = 1 n X 1 j 2 σ 1 2 ] 1 / 2 , V = [ j = 1 n X 2 j 2 σ 2 2 ] 1 / 2 . U=\left[\sum_{j=1}^{n}\frac{X_{1j}^{2}}{\sigma_{1}^{2}}\right]^{1/2},\qquad V=% \left[\sum_{j=1}^{n}\frac{X_{2j}^{2}}{\sigma_{2}^{2}}\right]^{1/2}.
  30. μ 1 0 \mu_{1}\neq 0
  31. μ 2 0 \mu_{2}\neq 0
  32. X X
  33. X 2 X^{2}
  34. X X
  35. X χ k X\sim\chi_{k}
  36. X X
  37. X N C χ k ( 0 ) X\sim NC\chi_{k}(0)
  38. σ = 1 \sigma=1

Nonimaging_optics.html

  1. S = i n i d i S=\textstyle\sum_{i}n_{i}d_{i}
  2. S ( τ B ) - S ( τ A ) = A B d S = τ A τ B d S d τ d τ = τ A τ B S ( τ + d τ ) - S ( τ ) ( τ + d τ ) - τ d τ S(\tau_{B})-S(\tau_{A})=\int_{A}^{B}dS=\int_{\tau_{A}}^{\tau_{B}}\frac{dS}{d% \tau}d\tau=\int_{\tau_{A}}^{\tau_{B}}\frac{S(\tau+d\tau)-S(\tau)}{(\tau+d\tau)% -\tau}d\tau
  3. C = 1 sin 2 θ , C=\frac{1}{\sin^{2}\theta},
  4. θ \theta

Nonlinear_element.html

  1. I = I 0 e V / V T I=I_{0}e^{V/V_{T}}

Nordström's_theory_of_gravitation.html

  1. Δ ϕ = 4 π ρ \Delta\phi=4\pi\rho
  2. ϕ \phi
  3. ρ \rho
  4. d u d t = - ϕ \frac{d\vec{u}}{dt}=-\nabla\phi
  5. ϕ = 4 π ρ \Box\phi=4\pi\,\rho
  6. u ˙ a = - ϕ , a \dot{u}_{a}=-\phi_{,a}
  7. u a u^{a}
  8. u ˙ a = - ϕ , a - ϕ ˙ u a \dot{u}_{a}=-\phi_{,a}-\dot{\phi}\,u_{a}
  9. F T matter = ρ FT_{\rm matter}=\rho
  10. ϕ \phi
  11. ϕ ϕ = - 4 π T matter \phi\,\Box\phi=-4\pi\,T_{\rm matter}
  12. d ( ϕ u a ) d s = - ϕ , a \frac{d\left(\phi\,u_{a}\right)}{ds}=-\phi_{,a}
  13. ϕ u ˙ a = - ϕ , a - ϕ ˙ u a \phi\,\dot{u}_{a}=-\phi_{,a}-\dot{\phi}\,u_{a}
  14. ( T matter ) a b = ϕ ρ u a u b \left(T_{\rm matter}\right)_{ab}=\phi\,\rho\,u_{a}\,u_{b}
  15. 4 π ( T grav ) a b = ϕ , a ϕ , b - 1 / 2 η a b ϕ , m ϕ , m 4\pi\,\left(T_{\rm grav}\right)_{ab}=\phi_{,a}\,\phi_{,b}-1/2\,\eta_{ab}\,\phi% _{,m}\,\phi^{,m}
  16. L = 1 8 π η a b ϕ , a ϕ , b - ρ ϕ L=\frac{1}{8\pi}\,\eta^{ab}\,\phi_{,a}\,\phi_{,b}-\rho\,\phi
  17. L = ϕ 2 η a b u ˙ a u ˙ b L=\phi^{2}\,\eta_{ab}\,\dot{u}^{a}\,\dot{u}^{b}
  18. g a b = ϕ 2 η a b g_{ab}=\phi^{2}\,\eta_{ab}
  19. d σ 2 = η a b d x a d x b d\sigma^{2}=\eta_{ab}\,dx^{a}\,dx^{b}
  20. \Box
  21. d s 2 = ϕ 2 η a b d x a d x b ds^{2}=\phi^{2}\,\eta_{ab}\,dx^{a}\,dx^{b}
  22. R = - 6 ϕ ϕ 3 R=-\frac{6\,\Box\phi}{\phi^{3}}
  23. R = 24 π T R=24\pi\,T
  24. g a b g_{ab}
  25. C a b c d C_{abcd}
  26. R = 24 π T , C a b c d = 0 R=24\pi\,T,\;\;\;C_{abcd}=0
  27. R = 0 , C a b c d = 0 R=0,\;\;\;C_{abcd}=0
  28. d s 2 = exp ( 2 ψ ) η a b d x a d x b , φ = 0 ds^{2}=\exp(2\psi)\,\eta_{ab}\,dx^{a}\,dx^{b},\;\;\;\Box\varphi=0
  29. ϕ = exp ( ψ ) \phi=\exp(\psi)
  30. d σ 2 = η a b d x a d x b d\sigma^{2}=\eta_{ab}\,dx^{a}\,dx^{b}
  31. \Box
  32. ψ \psi
  33. ψ \psi
  34. d s 2 = exp ( 2 ψ ) η a b d x a d x b ( 1 + 2 ψ ) η a b d x a d x b ds^{2}=\exp(2\,\psi)\,\eta_{ab}\,dx^{a}\,dx^{b}\approx(1+2\psi)\,\eta_{ab}\,dx% ^{a}\,dx^{b}
  35. ψ \psi
  36. S a b = R a b - 1 4 R g a b S_{ab}=R_{ab}-\frac{1}{4}\,R\,g_{ab}
  37. S a b ; b = 6 π T ; a {{S_{a}}^{b}}_{;b}=6\,\pi\,T_{;a}
  38. S a b S_{ab}
  39. R R
  40. S a b S_{ab}
  41. C a b c d C_{abcd}
  42. d s 2 = exp ( 2 ψ ) η a b d x a d x b , Δ ψ = 0 ds^{2}=\exp(2\psi)\,\eta_{ab}\,dx^{a}\,dx^{b},\;\;\Delta\psi=0
  43. ψ \psi
  44. d s 2 = ( 1 + 2 ψ ) η a b d x a d x b ds^{2}=(1+2\,\psi)\,\eta_{ab}\,dx^{a}\,dx^{b}
  45. η a b d x a d x b \eta_{ab}\,dx^{a}\,dx^{b}
  46. d s 2 = ( 1 - m / ρ ) ( - d t 2 + d ρ 2 + ρ 2 ( d θ 2 + sin ( θ ) 2 d ϕ 2 ) ) ds^{2}=(1-m/\rho)\,\left(-dt^{2}+d\rho^{2}+\rho^{2}\,(d\theta^{2}+\sin(\theta)% ^{2}\,d\phi^{2})\right)
  47. r = ρ ( 1 - m / ρ ) r=\rho\,(1-m/\rho)
  48. d s 2 = ( 1 + m / r ) - 2 ( - d t 2 + d r 2 ) + r 2 ( d θ 2 + sin ( θ ) 2 d ϕ 2 ) ds^{2}=(1+m/r)^{-2}\,(-dt^{2}+dr^{2})+r^{2}\,(d\theta^{2}+\sin(\theta)^{2}\,d% \phi^{2})
  49. - < t < , 0 < r < , 0 < θ < π , - π < ϕ < π -\infty<t<\infty,\;0<r<\infty,\;0<\theta<\pi,\;-\pi<\phi<\pi
  50. r = r 0 r=r_{0}
  51. 4 π r 0 2 4\pi\,r_{0}^{2}
  52. t \partial_{t}
  53. ϕ \partial_{\phi}
  54. - cos ( ϕ ) θ + cot ( θ ) sin ( ϕ ) ϕ -\cos(\phi)\,\partial_{\theta}+\cot(\theta)\,\sin(\phi)\,\partial_{\phi}
  55. sin ( ϕ ) θ + cot ( θ ) cos ( ϕ ) ϕ \sin(\phi)\,\partial_{\theta}+\cot(\theta)\,\cos(\phi)\,\partial_{\phi}
  56. θ = π / 2 \theta=\pi/2
  57. t ˙ = E ( 1 + m / r ) 2 E ( 1 + 2 m / r ) \dot{t}=E\,\left(1+m/r\right)^{2}\approx E\,\left(1+2m/r\right)
  58. ϕ ˙ = L / r 2 \dot{\phi}=L/r^{2}
  59. ϵ = - 1 , 0 , 1 \epsilon=-1,0,1
  60. r ˙ 2 ( 1 + m / r ) 4 = E 2 - V \frac{\dot{r}^{2}}{\left(1+m/r\right)^{4}}=E^{2}-V
  61. V = L 2 / r 2 - ϵ ( 1 + m / r ) 2 V=\frac{L^{2}/r^{2}-\epsilon}{\left(1+m/r\right)^{2}}
  62. r c = L 2 / m r_{c}=L^{2}/m
  63. r c L 2 / m - 3 m r_{c}\approx L^{2}/m-3m
  64. e 0 = ( 1 + m / r ) t \vec{e}_{0}=\left(1+m/r\right)\,\partial_{t}
  65. e 1 = ( 1 + m / r ) r \vec{e}_{1}=\left(1+m/r\right)\,\partial_{r}
  66. e 2 = 1 r θ \vec{e}_{2}=\frac{1}{r}\,\partial_{\theta}
  67. e 3 = 1 r sin ( θ ) ϕ \vec{e}_{3}=\frac{1}{r\,\sin(\theta)}\,\partial_{\phi}
  68. e 0 e 0 = m r 2 e 2 \nabla_{\vec{e}_{0}}\vec{e}_{0}=\frac{m}{r^{2}}\,\vec{e}_{2}
  69. E [ X ] a b = m r 3 diag ( - 2 , 1 , 1 ) + m 2 r 4 diag ( - 1 , 1 , 1 ) E[\vec{X}]_{ab}=\frac{m}{r^{3}}\,{\rm diag}(-2,1,1)+\frac{m^{2}}{r^{4}}\,{\rm diag% }(-1,1,1)
  70. X = e 0 \vec{X}=\vec{e}_{0}
  71. θ = π / 2 \theta=\pi/2
  72. r ˙ 2 = ( E 2 - V ) ( 1 + m / r ) 4 \dot{r}^{2}=(E^{2}-V)\;(1+m/r)^{4}
  73. V = ( 1 + L 2 / r 2 ) / ( 1 + m / r ) 2 V=(1+L^{2}/r^{2})/(1+m/r)^{2}
  74. 2 r ˙ r ¨ = d d r ( ( E 2 - V ) ( 1 + m / r ) 4 ) r ˙ 2\dot{r}\ddot{r}=\frac{d}{dr}\left((E^{2}-V)\,(1+m/r)^{4}\right)\;\dot{r}
  75. r ˙ \dot{r}
  76. r ¨ = 1 2 d d r ( ( E 2 - V ) ( 1 + m / r ) 4 ) \ddot{r}=\frac{1}{2}\,\frac{d}{dr}\left((E^{2}-V)\,(1+m/r)^{4}\right)
  77. r c = L 2 / m r_{c}=L^{2}/m
  78. E c = L 2 / ( L 2 + m 2 ) E_{c}=L^{2}/(L^{2}+m^{2})
  79. ε = r - L 2 / m 2 \varepsilon=r-L^{2}/m^{2}
  80. ε ¨ = - m 4 L 8 ( m 2 + L 2 ) ε + O ( ε 2 ) \ddot{\varepsilon}=-\frac{m^{4}}{L^{8}}\,(m^{2}+L^{2})\,\varepsilon+O(% \varepsilon^{2})
  81. ω shm m 2 L 4 m 2 + L 2 = 1 r 2 m 2 + m r \omega_{\rm shm}\approx\frac{m^{2}}{L^{4}}\,\sqrt{m^{2}+L^{2}}=\frac{1}{r^{2}}% \,\sqrt{m^{2}+mr}
  82. L = m r L=\sqrt{mr}
  83. r c r_{c}
  84. ω orb = L r 2 = m / r 3 \omega_{\rm orb}=\frac{L}{r^{2}}=\sqrt{m/r^{3}}
  85. Δ ω = ω orb - ω shm = m r 3 - m 2 r 4 + m r 3 - 1 2 m 3 r 5 \Delta\omega=\omega_{\rm orb}-\omega_{\rm shm}=\sqrt{\frac{m}{r^{3}}}-\sqrt{% \frac{m^{2}}{r^{4}}+\frac{m}{r^{3}}}\approx-\frac{1}{2}\sqrt{\frac{m^{3}}{r^{5% }}}
  86. Δ ϕ = 2 π Δ ω - π m 3 r 5 \Delta\phi=2\pi\,\Delta\omega\approx-\pi\,\sqrt{\frac{m^{3}}{r^{5}}}
  87. Δ ϕ ω orb - π m r \frac{\Delta\phi}{\omega_{\rm orb}}\approx-\frac{\pi m}{r}
  88. Δ ϕ ω orb 6 π m r \frac{\Delta\phi}{\omega_{\rm orb}}\approx\frac{6\pi m}{r}
  89. 0 = - d t 2 + d r 2 ( 1 + m / r ) 2 + r 2 d ϕ 2 0=\frac{-dt^{2}+dr^{2}}{(1+m/r)^{2}}+r^{2}\,d\phi^{2}
  90. R 1 , R , R 2 R_{1},\,R,\,R_{2}
  91. R 1 , R 2 R R_{1},\,R_{2}\gg R
  92. ϕ \phi
  93. R = r cos ϕ R=r\,\cos\phi
  94. 0 = - r sin ϕ d ϕ + cos ϕ d r 0=-r\sin\phi\,d\phi+\cos\phi\,dr
  95. r 2 d ϕ 2 = cot ( ϕ ) 2 d r 2 = R 2 r 2 - R 2 d r 2 r^{2}\,d\phi^{2}=\cot(\phi)^{2}\,dr^{2}=\frac{R^{2}}{r^{2}-R^{2}}\,dr^{2}
  96. d t 1 r 2 - R 2 ( r + m R 2 r 2 ) d r dt\approx\frac{1}{\sqrt{r^{2}-R^{2}}}\;\left(r+m\,\frac{R^{2}}{r^{2}}\right)\;dr
  97. ( Δ t ) 1 = R R 1 d t m + R 1 R 1 R 1 2 - R 2 = R 1 2 - R 2 + m 1 - ( R / R 1 ) 2 (\Delta t)_{1}=\int_{R}^{R_{1}}dt\approx\frac{m+R_{1}}{R_{1}}\,\sqrt{R_{1}^{2}% -R^{2}}=\sqrt{R_{1}^{2}-R^{2}}+m\,\sqrt{1-(R/R_{1})^{2}}
  98. ( Δ t ) 2 = R R 2 d t m + R 2 R 2 R 2 2 - R 2 = R 2 2 - R 2 + m 1 - ( R / R 2 ) 2 (\Delta t)_{2}=\int_{R}^{R_{2}}dt\approx\frac{m+R_{2}}{R_{2}}\,\sqrt{R_{2}^{2}% -R^{2}}=\sqrt{R_{2}^{2}-R^{2}}+m\,\sqrt{1-(R/R_{2})^{2}}
  99. R 1 2 - R 2 + R 2 2 - R 2 \sqrt{R_{1}^{2}-R^{2}}+\sqrt{R_{2}^{2}-R^{2}}
  100. Δ t = m ( 1 - ( R / R 1 ) 2 + 1 - ( R / R 2 ) 2 ) \Delta t=m\,\left(\sqrt{1-(R/R_{1})^{2}}+\sqrt{1-(R/R_{2})^{2}}\right)
  101. R / R 1 , R / R 2 R/R_{1},\;R/R_{2}
  102. Δ t = 2 m \Delta t=2m
  103. Δ t = 2 m + 2 m log ( 4 R 1 R 2 R 2 ) \Delta t=2m+2m\,\log\left(\frac{4\,R_{1}\,R_{2}}{R^{2}}\right)
  104. R / R 1 , R / R 2 R/R_{1},\;R/R_{2}
  105. δ ϕ = 2 m R \delta\phi=\frac{2\,m}{R}
  106. δ ϕ = 4 m R \delta\phi=\frac{4\,m}{R}
  107. Δ ϕ ω orb = - π m R \frac{\Delta\phi}{\omega_{\rm orb}}=-\frac{\pi\,m}{R}
  108. Δ ϕ ω orb = 6 π m R \frac{\Delta\phi}{\omega_{\rm orb}}=\frac{6\,\pi\,m}{R}
  109. 2 m 2\,m
  110. 2 m + 2 m log ( 4 R 1 R 2 R 2 ) 2\,m+2\,m\;\log\left(\frac{4\,R_{1}\,R_{2}}{R^{2}}\right)
  111. d s 2 = 2 d u d v + d x 2 + d y 2 , - < u , v , x , y < ds^{2}=2\,du\,dv+dx^{2}+dy^{2},\;\;\;-\infty<u,\,v,\,x,\,y<\infty
  112. 2 ψ u v + ψ x x + ψ y y = 0 2\,\psi_{uv}+\psi_{xx}+\psi_{yy}=0
  113. ψ = f ( u ) \psi=f(u)
  114. d s 2 = exp ( 2 f ( u ) ) ( 2 d u d v + d x 2 + d y 2 ) , - < u , v , x , y < ds^{2}=\exp(2f(u))\;\left(2\,du\,dv+dx^{2}+dy^{2}\right),\;\;\;-\infty<u,\,v,% \,x,\,y<\infty
  115. v \partial_{v}
  116. u \partial_{u}
  117. x , y \partial_{x},\;\;\partial_{y}
  118. - y x + x y -y\,\partial_{x}+x\,\partial_{y}
  119. x v + u x , y v + u y x\,\partial_{v}+u\,\partial_{x},\;\;y\,\partial_{v}+u\,\partial_{y}
  120. x v + u x x\,\partial_{v}+u\,\partial_{x}
  121. ( u , v , x , y ) ( u , v + x λ + u 2 λ 2 , x + u λ , y ) (u,v,x,y)\longrightarrow(u,\;v+x\,\lambda+\frac{u}{2}\,\lambda^{2},\;x+u\,% \lambda,\;y)
  122. z \partial_{z}
  123. u = u 0 u=u_{0}
  124. e 0 = 1 2 ( v + exp ( - 2 f ) u ) \vec{e}_{0}=\frac{1}{\sqrt{2}}\,\left(\partial_{v}+\exp(-2f)\,\partial_{u}\right)
  125. e 1 = 1 2 ( v - exp ( - 2 f ) u ) \vec{e}_{1}=\frac{1}{\sqrt{2}}\,\left(\partial_{v}-\exp(-2f)\,\partial_{u}\right)
  126. e 2 = x \vec{e}_{2}=\partial_{x}
  127. e 3 = y \vec{e}_{3}=\partial_{y}
  128. e 0 e 0 = 0 \nabla_{\vec{e}_{0}}\vec{e}_{0}=0
  129. X = e 0 \vec{X}=\vec{e}_{0}
  130. θ [ X ] p ^ q ^ = 1 2 f ( u ) exp ( - 2 f ( u ) ) diag ( 0 , 1 , 1 ) \theta[\vec{X}]_{\hat{p}\hat{q}}=\frac{1}{\sqrt{2}}\,f^{\prime}(u)\,\exp(-2\,f% (u))\,{\rm diag}(0,1,1)
  131. E [ X ] p ^ q ^ = 1 2 exp ( - 4 f ( u ) ) ( f ( u ) 2 - f ′′ ( u ) ) diag ( 0 , 1 , 1 ) E[\vec{X}]_{\hat{p}\hat{q}}=\frac{1}{2}\,\exp(-4\,f(u))\;\left(f^{\prime}(u)^{% 2}-f^{\prime\prime}(u)\right)\,{\rm diag}(0,1,1)
  132. e 0 \vec{e}_{0}

Normal_bundle.html

  1. ( M , g ) (M,g)
  2. S M S\subset M
  3. p S p\in S
  4. n T p M n\in\mathrm{T}_{p}M
  5. S S
  6. g ( n , v ) = 0 g(n,v)=0
  7. v T p S v\in\mathrm{T}_{p}S
  8. n n
  9. T p S \mathrm{T}_{p}S
  10. N p S \mathrm{N}_{p}S
  11. n n
  12. S S
  13. p p
  14. N S \mathrm{N}S
  15. S S
  16. N S := p S N p S \mathrm{N}S:=\coprod_{p\in S}\mathrm{N}_{p}S
  17. i : N M i\colon N\to M
  18. V V / W V\to V/W
  19. 0 T N T M | i ( N ) T M / N := T M | i ( N ) / T N 0 0\to TN\to TM|_{i(N)}\to T_{M/N}:=TM|_{i(N)}/TN\to 0
  20. T M | i ( N ) TM|_{i(N)}
  21. i * T M i^{*}TM
  22. i i
  23. 𝐑 2 N \mathbf{R}^{2N}
  24. 𝐑 N \mathbf{R}^{N}
  25. 𝐑 N \mathbf{R}^{N}
  26. [ T N ] + [ T M / N ] = [ T M ] [TN]+[T_{M/N}]=[TM]
  27. 𝐑 N \mathbf{R}^{N}
  28. 𝐑 N \mathbf{R}^{N}
  29. [ T N ] + [ T M / N ] = 0 [TN]+[T_{M/N}]=0
  30. [ T M / N ] = - [ T N ] [T_{M/N}]=-[TN]
  31. X X
  32. ( M , ω ) (M,\omega)
  33. X X
  34. ( T i ( x ) X ) ω / ( T i ( x ) X ( T i ( x ) X ) ω ) , x X , (T_{i(x)}X)^{\omega}/(T_{i(x)}X\cap(T_{i(x)}X)^{\omega}),\quad x\in X,
  35. i : X M i:X\rightarrow M
  36. i * ( T M ) i*(TM)
  37. i * ( T M ) T X / ν ( T X ) ω / ν ( ν ν * ) , ν = T X ( T X ) ω , i^{*}(TM)\cong TX/\nu\oplus(TX)^{\omega}/\nu\oplus(\nu\oplus\nu^{*}),\quad\nu=% TX\cap(TX)^{\omega},
  38. X X
  39. S p e c n 0 I n / I n + 1 Spec\oplus_{n\geq 0}I^{n}/I^{n+1}

Normal_height.html

  1. H * H*
  2. H * H*

Normalization_(statistics).html

  1. X - μ σ \frac{X-\mu}{\sigma}
  2. X - X ¯ s \frac{X-\overline{X}}{s}
  3. ϵ ^ i σ ^ i = X i - μ ^ i σ ^ i \frac{\hat{\epsilon}_{i}}{\hat{\sigma}_{i}}=\frac{X_{i}-\hat{\mu}_{i}}{\hat{% \sigma}_{i}}
  4. μ k σ k \frac{\mu_{k}}{\sigma^{k}}
  5. σ \sigma
  6. σ μ \frac{\sigma}{\mu}
  7. μ \mu
  8. X = X - X m i n X m a x - X m i n X^{\prime}=\frac{X-X_{min}}{X_{max}-X_{min}}
  9. a a
  10. b b
  11. X = a + ( X - X m i n ) ( b - a ) X m a x - X m i n X^{\prime}=a+\frac{\left(X-X_{min}\right)\left(b-a\right)}{X_{max}-X_{min}}
  12. ( σ 2 μ ) \left(\frac{\sigma^{2}}{\mu}\right)
  13. | ψ | < s u p > 2 |ψ|<sup>2

Notation_in_probability_and_statistics.html

  1. P ( X > x ) P(X>x)
  2. \mathbb{P}
  3. ( A ) \mathbb{P}(A)
  4. P ( { ω : X ( ω ) A } ) P(\{\omega:X(\omega)\in A\})
  5. ω \omega
  6. X ( ω ) X(\omega)
  7. ( A B ) \mathbb{P}(A\cap B)
  8. [ A B ] \mathbb{P}[A\cap B]
  9. ( A B ) \mathbb{P}(A\cup B)
  10. [ A B ] \mathbb{P}[A\cup B]
  11. \mathcal{F}
  12. F ¯ ( x ) = 1 - F ( x ) \overline{F}(x)=1-F(x)
  13. X Y X\perp Y
  14. X Y X\perp\!\!\!\perp Y
  15. X Y | W X\perp\!\!\!\perp Y\,|\,W
  16. X Y | W X\perp Y\,|\,W
  17. P ( A B ) \textstyle P(A\mid B)
  18. A \textstyle A
  19. B \textstyle B
  20. B \textstyle B
  21. θ ^ \widehat{\theta}
  22. θ \theta
  23. x ¯ \bar{x}
  24. x ¯ \bar{x}
  25. χ α , ν 2 \chi^{2}_{\alpha,\nu}
  26. χ 2 ( α , ν ) \chi^{2}(\alpha,\nu)
  27. F α , ν 1 , ν 2 F_{\alpha,\nu_{1},\nu_{2}}
  28. ν \nu

Novikov_conjecture.html

  1. f : M B G f:M\rightarrow BG
  2. x H n - 4 i ( B G ; ) . x\in H^{n-4i}(BG;\mathbb{Q}).
  3. f * ( x ) L i ( M ) , [ M ] \left\langle f^{*}(x)\cup L_{i}(M),[M]\right\rangle\in\mathbb{Q}
  4. h : M M h:M^{\prime}\rightarrow M
  5. f h f\circ h

NPSH.html

  1. N P S H A = ( p i ρ g + V i 2 2 g ) - p v ρ g NPSH_{A}=\left(\frac{p_{i}}{\rho g}+\frac{V_{i}^{2}}{2g}\right)-\frac{p_{v}}{% \rho g}
  2. p 0 ρ g + z 0 = p i ρ g + V i 2 2 g + z i + h f \frac{p_{0}}{\rho g}+z_{0}=\frac{p_{i}}{\rho g}+\frac{V_{i}^{2}}{2g}+z_{i}+h_{f}
  3. N P S H A = p 0 ρ g - p v ρ g - ( z i - z 0 ) - h f NPSH_{A}=\frac{p_{0}}{\rho g}-\frac{p_{v}}{\rho g}-(z_{i}-z_{0})-h_{f}
  4. N P S H A = ( p e ρ g + V e 2 2 g ) - p v ρ g NPSH_{A}=\left(\frac{p_{e}}{\rho g}+\frac{V_{e}^{2}}{2g}\right)-\frac{p_{v}}{% \rho g}
  5. p e ρ g + V e 2 2 g + z e = p 0 ρ g + z 0 + h f \frac{p_{e}}{\rho g}+\frac{V_{e}^{2}}{2g}+z_{e}=\frac{p_{0}}{\rho g}+z_{0}+h_{f}
  6. N P S H A = p 0 ρ g - p v ρ g - ( z e - z 0 ) + h f NPSH_{A}=\frac{p_{0}}{\rho g}-\frac{p_{v}}{\rho g}-(z_{e}-z_{0})+h_{f}
  7. h f h_{f}
  8. C N P S H = g N P S H n 2 D 2 C_{NPSH}=\frac{gNPSH}{n^{2}D^{2}}
  9. n n
  10. D D
  11. σ = N P S H H \sigma=\frac{NPSH}{H}
  12. H H

NTRU_Cryptosystems,_Inc..html

  1. R [ x ] / ( x N - 1 ) R[x]/(x^{N}-1)
  2. / p \mathbb{Z}/p\mathbb{Z}

Nuclear_binding_energy.html

  1. BE A MeV = a - b A 1 / 3 - c Z 2 A 4 / 3 - d ( N - Z ) 2 A 2 ± e A 7 / 4 \frac{\,\text{BE}}{A\cdot\,\text{MeV}}=a-\frac{b}{A^{1/3}}-\frac{cZ^{2}}{A^{4/% 3}}-\frac{d\left(N-Z\right)^{2}}{A^{2}}\pm\frac{e}{A^{7/4}}
  2. a = 14.0 a=14.0
  3. b = 13.0 b=13.0
  4. c = 0.585 c=0.585
  5. d = 19.3 d=19.3
  6. e = 33 e=33
  7. a a
  8. - b / A 1 / 3 -b/A^{1/3}
  9. - c Z 2 / A 4 / 3 -cZ^{2}/A^{4/3}
  10. Z Z
  11. - d ( N - Z ) 2 / A 2 -d(N-Z)^{2}/A^{2}
  12. ± e / A 7 / 4 \pm e/A^{7/4}

Nuclear_cross_section.html

  1. γ \gamma
  2. r x = Φ σ x ρ A = Φ Σ x r_{x}=\Phi\ \sigma_{x}\ \rho_{A}=\Phi\Sigma_{x}
  3. r x r_{x}
  4. Φ \Phi
  5. σ x \sigma_{x}
  6. x x
  7. ρ A \rho_{A}
  8. Σ x σ x ρ A \Sigma_{x}\equiv\sigma_{x}\ \rho_{A}
  9. γ \gamma
  10. σ s \sigma_{s}
  11. σ γ \sigma_{\gamma}
  12. σ a \sigma_{a}
  13. σ t \sigma_{t}
  14. σ t = σ s + σ γ + σ f + \sigma_{t}=\sigma_{s}+\sigma_{\gamma}+\sigma_{f}+...

Nuclear_fuel.html

  1. \rightarrow
  2. \rightarrow
  3. \rightarrow
  4. \rightarrow
  5. \rightarrow

Nuclear_operator.html

  1. \mathcal{L}
  2. \mathcal{H}
  3. : \mathcal{L}:\mathcal{H}\to\mathcal{H}
  4. = n = 1 N ρ n f n , g n \mathcal{L}=\sum_{n=1}^{N}\rho_{n}\langle f_{n},\cdot\rangle g_{n}
  5. 1 N 1\leq N\leq\infty
  6. f 1 , , f N f_{1},\ldots,f_{N}
  7. g 1 , , g N g_{1},\ldots,g_{N}
  8. ρ 1 , , ρ N \rho_{1},\ldots,\rho_{N}
  9. ρ n 0 \rho_{n}\to 0
  10. N = N=\infty
  11. , \langle\cdot,\cdot\rangle
  12. n = 1 | ρ n | < \sum_{n=1}^{\infty}|\rho_{n}|<\infty
  13. { ψ n } \{\psi_{n}\}
  14. Tr = n ψ n , ψ n \mbox{Tr}~{}\mathcal{L}=\sum_{n}\langle\psi_{n},\mathcal{L}\psi_{n}\rangle
  15. \mathcal{L}
  16. : A B \mathcal{L}:A\to B
  17. { g n } B \{g_{n}\}\in B
  18. g n 1 \|g_{n}\|\leq 1
  19. { f n * } A \{f^{*}_{n}\}\in A^{\prime}
  20. f n * 1 \|f^{*}_{n}\|\leq 1
  21. { ρ n } \{\rho_{n}\}
  22. inf { p 1 : n | ρ n | p < } = q , \inf\left\{p\geq 1:\sum_{n}|\rho_{n}|^{p}<\infty\right\}=q,
  23. = n ρ n f n * ( ) g n \mathcal{L}=\sum_{n}\rho_{n}f^{*}_{n}(\cdot)g_{n}

Nuclear_reactor_physics.html

  1. τ \tau
  2. d N / d t = α N / τ dN/dt=\alpha N/\tau
  3. α \alpha
  4. d N / d t dN/dt
  5. α \alpha
  6. α = P i m p a c t P f i s s i o n n a v g - P a b s o r b - P e s c a p e \alpha=P_{impact}P_{fission}n_{avg}-P_{absorb}-P_{escape}
  7. P i m p a c t P_{impact}
  8. P f i s s i o n P_{fission}
  9. P a b s o r b P_{absorb}
  10. P e s c a p e P_{escape}
  11. n a v g n_{avg}
  12. α \alpha
  13. α \alpha
  14. α \alpha
  15. d N / d t = 0 dN/dt=0
  16. P e s c a p e P_{escape}
  17. P a b s o r b P_{absorb}
  18. P e s c a p e P_{escape}
  19. P f i s s i o n P_{fission}
  20. P a b s o r b P_{absorb}
  21. α \alpha
  22. P a b s o r b P_{absorb}
  23. d N / d t = α N / τ + R e x t dN/dt=\alpha N/\tau+R_{ext}
  24. R e x t R_{ext}
  25. N = τ R e x t / ( - α ) N=\tau R_{ext}/(-\alpha)
  26. α \alpha
  27. α \alpha
  28. P f i s s i o n P_{fission}
  29. P f i s s i o n P_{fission}
  30. P e s c a p e P_{escape}
  31. τ \tau
  32. α \alpha
  33. α \alpha

Nuclear_space.html

  1. 𝐑 n \mathbf{R}^{n}
  2. z j z_{j}
  3. x j A x_{j}\in A
  4. j = 1 n k = 1 n z j z ¯ k C ( x j - x k ) 0. \sum_{j=1}^{n}\sum_{k=1}^{n}z_{j}\bar{z}_{k}C(x_{j}-x_{k})\geq 0.
  5. μ \mu
  6. A A^{\prime}
  7. C ( y ) = A e i x , y d μ ( x ) . C(y)=\int_{A^{\prime}}e^{i\langle x,y\rangle}d\mu(x).
  8. A = k = 0 H k A=\bigcap_{k=0}^{\infty}H_{k}
  9. H k H_{k}
  10. e - 1 2 y H 0 2 e^{-\frac{1}{2}\|y\|_{H_{0}}^{2}}

Nuclear_weapon_yield.html

  1. A = 4 π r 2 A=4\pi r^{2}
  2. V = 4 3 π r 3 \!V=\frac{4}{3}\pi r^{3}
  3. R = S ( E t 2 ρ ) 1 5 R=S\left({\frac{{E{t}}^{2}}{\rho}}\right)^{\frac{1}{5}}
  4. E = [ M . L 2 . T - 2 ] E=[{M}.{L^{2}}.{T^{-2}}]
  5. E = m v 2 2 E=\frac{mv^{2}}{2}
  6. ρ = [ M L - 3 ] \rho=[{M}\cdot{L^{-3}}]
  7. t = [ T ] t=[T]
  8. r = [ L ] r=[L]
  9. α \alpha
  10. β \beta
  11. γ \gamma
  12. E = ρ α t β r γ E={\rho^{\alpha}}\cdot{t^{\beta}}\cdot{r^{\gamma}}

Number_density.html

  1. n = N V , n=\frac{N}{V},
  2. N = V n ( x , y , z ) d V , N=\iiint_{V}n(x,\,y,\,z)\,\mathrm{d}V,
  3. m = V m 0 n ( x , y , z ) d V . m=\iiint_{V}m_{0}n(x,\,y,\,z)\,\mathrm{d}V.
  4. n = N A c , n=\mathrm{N_{A}}c,
  5. n = N A M ρ m . n=\frac{\mathrm{N_{A}}}{M}\rho_{\mathrm{m}}.

Number_needed_to_harm.html

  1. 6054 86 , 318 = 0.0701 \frac{6054}{86,318}=0.0701
  2. 32 516 = 0.0620 \frac{32}{516}=0.0620
  3. 0.0701 0.0620 = 1.13 = \frac{0.0701}{0.0620}=1.13={}
  4. 1 0.0083 = 121 \frac{1}{0.0083}=121

Numbering_(computability_theory).html

  1. S S\!
  2. \mathbb{N}
  3. ν ( i ) \nu(i)\!
  4. \mathbb{N}
  5. γ ( ) = 0 \gamma(\emptyset)=0
  6. γ ( { a 0 , , a k } ) = i k 2 a i \gamma(\{a_{0},\ldots,a_{k}\})=\sum_{i\leq k}2^{a_{i}}
  7. φ i \varphi_{i}
  8. { ( x , y ) : η ( x ) = η ( y ) } \{(x,y):\eta(x)=\eta(y)\}
  9. ν 1 : S 1 \nu_{1}:\subseteq\mathbb{N}\to S_{1}
  10. ν 2 : S 2 \nu_{2}:\subseteq\mathbb{N}\to S_{2}
  11. ν 1 \nu_{1}
  12. ν 2 \nu_{2}
  13. ν 1 ν 2 \nu_{1}\leq\nu_{2}
  14. f 𝐏 ( 1 ) i Domain ( ν 1 ) : ν 1 ( i ) = ν 2 f ( i ) . \exists f\in\mathbf{P}^{(1)}\,\forall i\in\mathrm{Domain}(\nu_{1}):\nu_{1}(i)=% \nu_{2}\circ f(i).
  15. ν 1 ν 2 \nu_{1}\leq\nu_{2}
  16. ν 1 ν 2 \nu_{1}\geq\nu_{2}
  17. ν 1 \nu_{1}
  18. ν 2 \nu_{2}
  19. ν 1 ν 2 \nu_{1}\equiv\nu_{2}
  20. \mathbb{N}

Oblique_reflection.html

  1. ( x - 2 z a c , y - 2 z b c , - z ) . \left(x-\frac{2za}{c},y-\frac{2zb}{c},-z\right).

Oblique_shock.html

  1. tan θ = 2 cot β M 1 2 sin 2 β - 1 M 1 2 ( γ + cos 2 β ) + 2 \tan\theta=2\cot\beta\frac{M_{1}^{2}\sin^{2}\beta-1}{M_{1}^{2}(\gamma+\cos 2% \beta)+2}
  2. p 2 p 1 = 1 + 2 γ γ + 1 ( M 1 2 sin 2 β - 1 ) \frac{p_{2}}{p_{1}}=1+\frac{2\gamma}{\gamma+1}(M_{1}^{2}\sin^{2}\beta-1)
  3. ρ 2 ρ 1 = ( γ + 1 ) M 1 2 sin 2 β ( γ - 1 ) M 1 2 sin 2 β + 2 \frac{\rho_{2}}{\rho_{1}}=\frac{(\gamma+1)M_{1}^{2}\sin^{2}\beta}{(\gamma-1)M_% {1}^{2}\sin^{2}\beta+2}
  4. T 2 T 1 = p 2 p 1 ρ 1 ρ 2 . \frac{T_{2}}{T_{1}}=\frac{p_{2}}{p_{1}}\frac{\rho_{1}}{\rho_{2}}.
  5. M 2 = 1 sin ( β - θ ) 1 + γ - 1 2 M 1 2 sin 2 β γ M 1 2 sin 2 β - γ - 1 2 . M_{2}=\frac{1}{\sin(\beta-\theta)}\sqrt{\frac{1+\frac{\gamma-1}{2}M_{1}^{2}% \sin^{2}\beta}{\gamma M_{1}^{2}\sin^{2}\beta-\frac{\gamma-1}{2}}}.
  6. p 2 p 1 2 γ γ + 1 M 1 2 sin 2 β \frac{p_{2}}{p_{1}}\approx\frac{2\gamma}{\gamma+1}M_{1}^{2}\sin^{2}\beta
  7. ρ 2 ρ 1 γ + 1 γ - 1 . \frac{\rho_{2}}{\rho_{1}}\approx\frac{\gamma+1}{\gamma-1}.
  8. T 2 T 1 2 γ ( γ - 1 ) ( γ + 1 ) 2 M 1 2 sin 2 β . \frac{T_{2}}{T_{1}}\approx\frac{2\gamma(\gamma-1)}{(\gamma+1)^{2}}M_{1}^{2}% \sin^{2}\beta.

Obsidian_hydration_dating.html

  1. c = e a + b x + c x 2 + d x 3 c=e^{a+bx+cx^{2}+dx^{3}}
  2. T = ( C 1 - C 2 ) 2 ( 1.128 1 - 0.177 k C 1 C 2 ) 2 4 D s e f f ( d C d x | x = 0 ) 2 T=\frac{(C_{1}-C_{2})^{2}\left(\frac{1.128}{1-\frac{0.177kC_{1}}{C_{2}}}\right% )^{2}}{4Dse\!f\!\!f\left(\left.\frac{\mathrm{d}C}{\mathrm{d}x}\right|_{x=0}% \right)^{2}}

Olami–Feder–Christensen_model.html

  1. K max = max ( i , j ) S K i j K_{\max}=\underset{(i,j)\in S}{\max}K_{ij}\,
  2. K i j K i j + ( 1 - K max ) K_{ij}\leftarrow K_{ij}+(1-K_{\max})\,
  3. K C i 0 , i = 1 , , m K_{C_{i}}\leftarrow 0,\quad i=1,\ldots,m\,
  4. K j K j + α K C i j Γ C i , i = 1 , , m K_{j}\leftarrow K_{j}+\alpha K^{\prime}_{C_{i}}\,\forall\,j\in\Gamma_{C_{i}},% \quad i=1,\ldots,m
  5. t t + ( 1 - K max ) t\leftarrow t+(1-K_{\max})\,
  6. d K i d t = 1 i S \frac{dK_{i}}{dt}=1\,\forall\,i\in S

Olivia_MFSK.html

  1. + 1.0000000000 + 1.1913785723 cos ( x ) - 0.0793018558 cos ( 2 x ) - 0.2171442026 cos ( 3 x ) - 0.0014526076 cos ( 4 x ) +1.0000000000+1.1913785723\cos(x)-0.0793018558\cos(2x)-0.2171442026\cos(3x)-0.% 0014526076\cos(4x)

On-balance_volume.html

  1. O B V = O B V p r e v + { v o l u m e if c l o s e > c l o s e p r e v 0 if c l o s e = c l o s e p r e v - v o l u m e if c l o s e < c l o s e p r e v OBV=OBV_{prev}+\left\{\begin{matrix}volume&\mathrm{if}\ close>close_{prev}\\ 0&\mathrm{if}\ close=close_{prev}\\ -volume&\mathrm{if}\ close<close_{prev}\end{matrix}\right.

One-dimensional_symmetry_group.html

  1. ( a , 0 ) : x x + a (a,0):x\mapsto x+a
  2. ( a , 1 ) : x a - x (a,1):x\mapsto a-x
  3. ( a , 0 ) ( b , 0 ) = ( a + b , 0 ) (a,0)\circ(b,0)=(a+b,0)
  4. ( a , 0 ) ( b , 1 ) = ( a + b , 1 ) (a,0)\circ(b,1)=(a+b,1)
  5. ( a , 1 ) ( b , 0 ) = ( a - b , 1 ) (a,1)\circ(b,0)=(a-b,1)
  6. ( a , 1 ) ( b , 1 ) = ( a - b , 0 ) (a,1)\circ(b,1)=(a-b,0)
  7. G x = { g x g G } . Gx=\left\{g\cdot x\mid g\in G\right\}.
  8. \in
  9. \in
  10. G x = { g G g x = x } . G_{x}=\{g\in G\mid g\cdot x=x\}.

One-repetition_maximum.html

  1. r r
  2. w w
  3. w w
  4. 1 R M = w ( 1 + r 30 ) 1RM=w\left(1+\frac{r}{30}\right)
  5. 1 R M = w 36 ( 37 - r ) = w [ 37 36 - ( 1 36 r ) ] w [ 1.0278 - ( 0.0278 r ) ] 1RM=w\cdot\frac{36}{\left(37-r\right)}=\frac{w}{\left[\frac{37}{36}-\left(% \frac{1}{36}\cdot r\right)\right]}\approx\frac{w}{\left[1.0278-\left(0.0278% \cdot r\right)\right]}
  6. 1 R M = 100 w 101.3 - 2.67123 r 1RM=\frac{100\cdot w}{101.3-2.67123\cdot r}
  7. 1 R M = w r 0.10 1RM=w\cdot r^{0.10}
  8. 1 R M = 100 w 52.2 + 41.9 e - 0.055 r 1RM=\frac{100\cdot w}{52.2+41.9\cdot e^{-0.055\cdot r}}
  9. 1 R M = w ( 1 + 0.025 r ) 1RM=w\cdot(1+0.025\cdot r)
  10. 1 R M = 100 w 48.8 + 53.8 e - 0.075 r 1RM=\frac{100\cdot w}{48.8+53.8\cdot e^{-0.075\cdot r}}

Oops-Leon.html

  1. Υ \Upsilon\,

Operad_theory.html

  1. Σ \Sigma
  2. ( P ( n ) ) n (P(n))_{n\in\mathbb{N}}
  3. n n
  4. 1 1
  5. P ( 1 ) P(1)
  6. n n
  7. k 1 , , k n k_{1},\ldots,k_{n}
  8. : P ( n ) × P ( k 1 ) × × P ( k n ) P ( k 1 + + k n ) ( θ , θ 1 , , θ n ) θ ( θ 1 , , θ n ) , \begin{matrix}\circ:P(n)\times P(k_{1})\times\cdots\times P(k_{n})&\to&P(k_{1}% +\cdots+k_{n})\\ (\theta,\theta_{1},\ldots,\theta_{n})&\mapsto&\theta\circ(\theta_{1},\ldots,% \theta_{n}),\end{matrix}
  9. θ ( 1 , , 1 ) = θ = 1 θ \theta\circ(1,\ldots,1)=\theta=1\circ\theta
  10. θ ( θ 1 ( θ 1 , 1 , , θ 1 , k 1 ) , , θ n ( θ n , 1 , , θ n , k n ) ) \displaystyle\theta\circ(\theta_{1}\circ(\theta_{1,1},\ldots,\theta_{1,k_{1}})% ,\ldots,\theta_{n}\circ(\theta_{n,1},\ldots,\theta_{n,k_{n}}))
  11. P ( n ) , n P(n),{n\in\mathbb{N}}
  12. Σ n \Sigma_{n}
  13. P ( n ) P(n)
  14. P ( 1 ) P(1)
  15. \circ
  16. s i Σ k i , t Σ n s_{i}\in\Sigma_{k_{i}},t\in\Sigma_{n}
  17. ( θ * t ) ( θ t 1 , , θ t n ) = ( θ ( θ 1 , , θ n ) ) * t ; (\theta*t)\circ(\theta_{t1},\ldots,\theta_{tn})=(\theta\circ(\theta_{1},\ldots% ,\theta_{n}))*t;
  18. θ ( θ 1 * s 1 , , θ n * s n ) = ( θ ( θ 1 , , θ n ) ) * ( s 1 , , s n ) \theta\circ(\theta_{1}*s_{1},\ldots,\theta_{n}*s_{n})=(\theta\circ(\theta_{1},% \ldots,\theta_{n}))*(s_{1},\ldots,s_{n})
  19. f : P Q f:P\to Q
  20. ( f n : P ( n ) Q ( n ) ) n (f_{n}:P(n)\to Q(n))_{n\in\mathbb{N}}
  21. f ( 1 ) = 1 f(1)=1
  22. θ \theta
  23. θ 1 , , θ n \theta_{1},\ldots,\theta_{n}
  24. f ( θ ( θ 1 , , θ n ) ) = f ( θ ) ( f ( θ 1 ) , , f ( θ n ) ) f(\theta\circ(\theta_{1},\ldots,\theta_{n}))=f(\theta)\circ(f(\theta_{1}),% \ldots,f(\theta_{n}))
  25. f ( x * s ) = f ( x ) * s f(x*s)=f(x)*s
  26. \circ
  27. f ( g h ) = ( f g ) h f\circ(g\circ h)=(f\circ g)\circ h
  28. θ \theta
  29. θ ( a , b ) \theta(a,b)
  30. ( a b ) (ab)
  31. θ \theta
  32. ( ( a b ) c ) ((ab)c)
  33. θ ( θ , 1 ) \theta\circ(\theta,1)
  34. ( a , b , c ) (a,b,c)
  35. ( a b , c ) (ab,c)
  36. θ \theta
  37. θ \theta
  38. a b ab
  39. c c
  40. ( ( ( a b ) c ) d ) (((ab)c)d)
  41. θ ( ( θ , 1 ) ( ( θ , 1 ) , 1 ) ) \theta\circ((\theta,1)\circ((\theta,1),1))
  42. ( θ ( θ , 1 ) ) ( ( θ , 1 ) , 1 ) (\theta\circ(\theta,1))\circ((\theta,1),1)
  43. x = θ , y = ( θ , 1 ) , z = ( ( θ , 1 ) , 1 ) x=\theta,y=(\theta,1),z=((\theta,1),1)
  44. x ( y z ) x\circ(y\circ z)
  45. ( x y ) z (x\circ y)\circ z
  46. ( a b ) c d (ab)c\ \ d
  47. θ ( a b ) c d ( ( θ a b c , 1 d ) ( ( θ a b , 1 c ) , 1 d ) ) \theta_{(ab)c\cdot d}\circ((\theta_{ab\cdot c},1_{d})\circ((\theta_{a\cdot b},% 1_{c}),1_{d}))
  48. a b c d ab\quad c\ \ d
  49. ( ( ( a b ) c ) d ) (((ab)c)d)
  50. 1 1 = 1 1\circ 1=1
  51. ψ \psi
  52. ψ ( ψ , 1 ) = ψ ( 1 , ψ ) . \psi\circ(\psi,1)=\psi\circ(1,\psi).
  53. ψ \psi
  54. ψ ( a , b ) \psi(a,b)
  55. ( a b ) c = a ( b c ) (ab)c=a(bc)
  56. x 1 x n x_{1}\cdots x_{n}
  57. S n S_{n}
  58. x 1 x n = x σ ( 1 ) x σ ( n ) x_{1}\cdots x_{n}=x_{\sigma(1)}\cdots x_{\sigma(n)}
  59. σ S n \sigma\in S_{n}
  60. P ( n ) P(n)
  61. P ( n ) P(n)
  62. P ( n ) P(n)
  63. S n S_{n}
  64. σ ( τ 1 , , τ n ) \sigma\circ(\tau_{1},\dots,\tau_{n})
  65. σ \sigma
  66. τ i \tau_{i}
  67. P ( n ) P(n)
  68. B n B_{n}
  69. 𝐑 \mathbf{R}^{\infty}
  70. ( 2 , 3 , - 5 , 0 , ) (2,3,-5,0,\dots)
  71. 2 v 1 + 3 v 2 - 5 v 3 + 0 v 4 + . 2v_{1}+3v_{2}-5v_{3}+0v_{4}+\cdots.
  72. 𝐑 n \mathbf{R}^{n}

Operational_amplifier_applications.html

  1. V out = ( R f + R 1 ) R g ( R g + R 2 ) R 1 V 2 - R f R 1 V 1 = ( R 1 + R f R 1 ) ( R g R g + R 2 ) V 2 - R f R 1 V 1 . V_{\,\text{out}}=\frac{\left(R_{\,\text{f}}+R_{1}\right)R_{\,\text{g}}}{\left(% R_{\,\text{g}}+R_{2}\right)R_{1}}V_{2}-\frac{R_{\,\text{f}}}{R_{1}}V_{1}=\left% (\frac{R_{1}+R_{\,\text{f}}}{R_{1}}\right)\cdot\left(\frac{R_{\,\text{g}}}{R_{% \,\text{g}}+R_{2}}\right)V_{2}-\frac{R_{\,\text{f}}}{R_{1}}V_{1}.
  2. V com = ( V 1 + V 2 ) / 2 ; V dif = V 2 - V 1 , V_{\,\text{com}}=(V_{1}+V_{2})/2;V_{\,\text{dif}}=V_{2}-V_{1}\,,
  3. V out R 1 R f = V com R 1 / R f - R 2 / R g 1 + R 2 / R g + V dif 1 + ( R 2 / R g + R 1 / R f ) / 2 1 + R 2 / R g . V_{\,\text{out}}\frac{R_{1}}{R_{\,\text{f}}}=V_{\,\text{com}}\frac{R_{1}/R_{\,% \text{f}}-R_{2}/R_{\,\text{g}}}{1+R_{2}/R_{\,\text{g}}}+V_{\,\text{dif}}\frac{% 1+(R_{2}/R_{\,\text{g}}+R_{1}/R_{\,\text{f}})/2}{1+R_{2}/R_{\,\text{g}}}.
  4. R 1 / R f = R 2 / R g R_{1}/R_{\,\text{f}}=R_{2}/R_{\,\text{g}}
  5. V out = R f R 1 V dif = R f R 1 ( V 2 - V 1 ) . V_{\,\text{out}}=\frac{R_{\,\text{f}}}{R_{1}}V_{\,\text{dif}}=\frac{R_{\,\text% {f}}}{R_{1}}\left(V_{2}-V_{1}\right).
  6. V out = V 2 - V 1 . V_{\,\text{out}}=V_{2}-V_{1}.\,
  7. V out = - R f R in V in V_{\,\text{out}}=-\frac{R_{\,\text{f}}}{R_{\,\text{in}}}V_{\,\text{in}}\!\,
  8. i in = V in R in i_{\,\text{in}}=\frac{V_{\,\text{in}}}{R_{\,\text{in}}}
  9. V out = - i in R f = - V in R f R in V_{\,\text{out}}=-i_{\,\text{in}}R_{\,\text{f}}=-V_{\,\text{in}}\frac{R_{\,% \text{f}}}{R_{\,\text{in}}}
  10. V out = ( 1 + R 2 R 1 ) V in V_{\,\text{out}}=\left(1+\frac{R_{\,\text{2}}}{R_{\,\text{1}}}\right)V_{\,% \text{in}}\!\,
  11. i 1 = V in R 1 , i_{1}=\frac{V_{\,\text{in}}}{R_{1}}\,,
  12. V out = V in + i in R 2 = V in ( 1 + R 2 R 1 ) V_{\,\text{out}}=V_{\,\text{in}}+i_{\,\text{in}}R_{2}=V_{\,\text{in}}\left(1+% \frac{R_{2}}{R_{1}}\right)
  13. V out = V in V_{\,\text{out}}=V_{\,\text{in}}\!
  14. Z in = Z_{\,\text{in}}=\infty
  15. V out = - R f ( V 1 R 1 + V 2 R 2 + + V n R n ) V_{\,\text{out}}=-R_{\,\text{f}}\left(\frac{V_{1}}{R_{1}}+\frac{V_{2}}{R_{2}}+% \cdots+\frac{V_{n}}{R_{n}}\right)
  16. R 1 = R 2 = = R n R_{1}=R_{2}=\cdots=R_{n}
  17. R f R_{\,\text{f}}
  18. V out = - R f R 1 ( V 1 + V 2 + + V n ) V_{\,\text{out}}=-\frac{R_{\,\text{f}}}{R_{1}}(V_{1}+V_{2}+\cdots+V_{n})\!
  19. R 1 = R 2 = = R n = R f R_{1}=R_{2}=\cdots=R_{n}=R_{\,\text{f}}
  20. V out = - ( V 1 + V 2 + + V n ) V_{\,\text{out}}=-(V_{1}+V_{2}+\cdots+V_{n})\!
  21. Z n = R n Z_{n}=R_{n}
  22. V - V_{-}
  23. V out ( t 1 ) = V out ( t 0 ) - 1 R C t 0 t 1 V in ( t ) d t V_{\,\text{out}}(t_{1})=V_{\,\text{out}}(t_{0})-\frac{1}{RC}\int_{t_{0}}^{t_{1% }}V_{\,\text{in}}(t)\,\operatorname{d}t
  24. - 1 R C t 0 t 1 V in ( t ) d t -\frac{1}{RC}\int_{t_{0}}^{t_{1}}V_{\,\text{in}}(t)\,\operatorname{d}t
  25. ω = 0 \omega=0
  26. R n = 1 1 R i + 1 R f = R i | | R f R_{\,\text{n}}=\frac{1}{\frac{1}{R_{\,\text{i}}}+\frac{1}{R_{\,\text{f}}}}=R_{% \,\text{i}}||R_{\,\text{f}}
  27. V out ( t 1 ) = V out ( t 0 ) - 1 R i C f t 0 t 1 V in ( t ) d t V_{\,\text{out}}(t_{1})=V_{\,\text{out}}(t_{0})-\frac{1}{R_{i}C_{f}}\int_{t_{0% }}^{t_{1}}V_{\,\text{in}}(t)\,\operatorname{d}t
  28. V out = - R C d V in d t where V in and V out are functions of time. V_{\,\text{out}}=-RC\,\frac{\operatorname{d}V_{\,\text{in}}}{\operatorname{d}t% }\,\qquad\,\text{where }V_{\,\text{in}}\,\text{ and }V_{\,\text{out}}\,\text{ % are functions of time.}
  29. ω = 0 \omega=0
  30. R in = - R 3 R 1 R 2 R_{\,\text{in}}=-R_{3}\frac{R_{1}}{R_{2}}
  31. R 1 R_{1}
  32. R 2 R_{2}
  33. R 3 R_{3}
  34. v in v_{\,\text{in}}
  35. v out v_{\,\text{out}}
  36. v out = - V T ln ( v in I S R ) v_{\,\text{out}}=-V_{\,\text{T}}\ln\left(\frac{v_{\,\text{in}}}{I_{\,\text{S}}% \,R}\right)
  37. I S I_{\,\text{S}}
  38. V T V_{\,\text{T}}
  39. v in R = I R = I D \frac{v_{\,\text{in}}}{R}=I_{\,\text{R}}=I_{\,\text{D}}
  40. I D I_{\,\text{D}}
  41. I D = I S ( e V D V T - 1 ) . I_{\,\text{D}}=I_{\,\text{S}}\left(e^{\frac{V_{\,\text{D}}}{V_{\,\text{T}}}}-1% \right).
  42. I D I S e V D V T . I_{\,\text{D}}\simeq I_{\,\text{S}}e^{\frac{V_{\,\text{D}}}{V_{\,\text{T}}}}.
  43. V out = - V D V_{\,\text{out}}=-V_{\,\text{D}}
  44. v in v_{\,\text{in}}
  45. v out v_{\,\text{out}}
  46. v out = - R I S e v in V T v_{\,\text{out}}=-RI_{\,\text{S}}e^{\frac{v_{\,\text{in}}}{V_{\,\text{T}}}}
  47. I S I_{\,\text{S}}
  48. V T V_{\,\text{T}}
  49. I D = I S ( e V D V T - 1 ) I_{\,\text{D}}=I_{\,\text{S}}\left(e^{\frac{V_{\,\text{D}}}{V_{\,\text{T}}}}-1\right)
  50. I D I S e V D V T . I_{\,\text{D}}\simeq I_{\,\text{S}}e^{\frac{V_{\,\text{D}}}{V_{\,\text{T}}}}.
  51. v out = - R I D . v_{\,\text{out}}=-RI_{\,\text{D}}.\,