wpmath0000004_16

Well-quasi-ordering.html

  1. x 0 x_{0}
  2. x 1 x_{1}
  3. x 2 x_{2}
  4. X X
  5. x i x j x_{i}\leq x_{j}
  6. i < j i<j
  7. \leq
  8. X X
  9. + \leq^{+}
  10. X X
  11. P ( X ) P(X)
  12. A + B A\leq^{+}B
  13. A A
  14. B B
  15. \leq
  16. P ( X ) P(X)
  17. X X
  18. x 0 x_{0}
  19. x 1 x_{1}
  20. x 2 x_{2}
  21. X X
  22. x i x_{i}
  23. x j x_{j}
  24. i i
  25. j j
  26. X X
  27. x 0 x_{0}
  28. x 1 x_{1}
  29. x 2 x_{2}
  30. X X
  31. ( , ) (\mathbb{N},\leq)
  32. ( , ) (\mathbb{Z},\leq)
  33. ( , ) (\mathbb{N},\mid)
  34. ( k , ) (\mathbb{N}^{k},\leq)
  35. k k
  36. ( X , ) (X,\leq)
  37. ( X k , k ) (X^{k},\leq^{k})
  38. k k
  39. X X
  40. X * X^{*}
  41. X X
  42. b , a b , a a b , a a a b , b,ab,aab,aaab,\dots
  43. X * X^{*}
  44. X * X^{*}
  45. X X
  46. ( X * , ) (X^{*},\leq)
  47. X X
  48. ( X , ) (X,\leq)
  49. u u
  50. v v
  51. v v
  52. u u
  53. ( X , = ) (X,=)
  54. u v u\leq v
  55. u u
  56. v v
  57. ( X ω , ) (X^{\omega},\leq)
  58. ( X , ) (X,\leq)
  59. ( X , ) (X,\leq)
  60. ( X , ) (X,\leq)
  61. \mathbb{Z}
  62. n m n\equiv m
  63. n = ± m n=\pm m
  64. ( , ) ( , ) (\mathbb{Z},\mid)\;\;\approx\;\;(\mathbb{N},\mid)
  65. X X
  66. x 0 x_{0}
  67. x 1 x_{1}
  68. x 2 x_{2}
  69. x n 0 x_{n0}
  70. x n 1 x_{n1}
  71. x n 2 x_{n2}
  72. n 0 {n0}
  73. n 1 {n1}
  74. n 2 {n2}
  75. ( x i ) i (x_{i})_{i}
  76. I I
  77. i i
  78. x i x_{i}
  79. x j x_{j}
  80. i < j i<j
  81. I I
  82. I I
  83. X X
  84. I I
  85. x n x_{n}
  86. n n
  87. I I
  88. ( X , ) (X,\leq)
  89. ( P ( X ) , + ) (P(X),\leq^{+})
  90. A + B a A b B ( a b ) A\leq^{+}B\iff\forall a\in A\exists b\in B(a\leq b)
  91. ( X , ) (X,\leq)
  92. x y x y y x x\sim y\iff x\leq y\land y\leq x
  93. ( X , ) (X,\leq)
  94. S 0 S 1 X S_{0}\subseteq S_{1}\subseteq...\subseteq X
  95. i , x , y X , x y x S i y S i \forall i\in\mathbb{N},\forall x,y\in X,x\leq y\wedge x\in S_{i}\Rightarrow y% \in S_{i}
  96. n n\in\mathbb{N}
  97. S n = S n + 1 = S_{n}=S_{n+1}=...
  98. S X S\subseteq X
  99. x , y X , x y x S y S \forall x,y\in X,x\leq y\wedge x\in S\Rightarrow y\in S
  100. i j , j > i , x S j S i \forall i\in\mathbb{N}\exists j\in\mathbb{N},j>i,\exists x\in S_{j}\setminus S% _{i}
  101. ( X , ) (X,\leq)
  102. S X S\subseteq X
  103. \leq
  104. \leq
  105. S S

Well_test.html

  1. s = B Q + C Q 2 s=BQ+CQ^{2}
  2. Q Q
  3. B B
  4. C C
  5. B Q BQ
  6. C Q 2 CQ^{2}
  7. Q Q
  8. s / Q s/Q
  9. Δ t \Delta t
  10. s Q = B + C Q \frac{s}{Q}=B+CQ
  11. C C
  12. Q = 0 Q=0
  13. B B
  14. S c = Q h 0 - h S_{c}=\frac{Q}{h_{0}-h}
  15. S c S_{c}
  16. Q Q
  17. h 0 - h h_{0}-h

Weyl_tensor.html

  1. C = R - 1 n - 2 ( Ric - s n g ) g - s 2 n ( n - 1 ) g g C=R-\frac{1}{n-2}\left(\mathrm{Ric}-\frac{s}{n}g\right)\wedge\!\!\!\!\!\!% \bigcirc g-\frac{s}{2n(n-1)}g\wedge\!\!\!\!\!\!\bigcirc g
  2. h k h\wedge\!\!\!\!\!\!\bigcirc k
  3. ( h k ) ( v 1 , v 2 , v 3 , v 4 ) = (h\wedge\!\!\!\!\!\!\bigcirc k)(v_{1},v_{2},v_{3},v_{4})=
  4. h ( v 1 , v 3 ) k ( v 2 , v 4 ) + h ( v 2 , v 4 ) k ( v 1 , v 3 ) h(v_{1},v_{3})k(v_{2},v_{4})+h(v_{2},v_{4})k(v_{1},v_{3})\,
  5. - h ( v 1 , v 4 ) k ( v 2 , v 3 ) - h ( v 2 , v 3 ) k ( v 1 , v 4 ) {}-h(v_{1},v_{4})k(v_{2},v_{3})-h(v_{2},v_{3})k(v_{1},v_{4})\,
  6. | R | 2 = | C | 2 + | 1 n - 2 ( Ric - s n g ) g | 2 + | s 2 n ( n - 1 ) g g | 2 . |R|^{2}=|C|^{2}+\left|\frac{1}{n-2}\left(\mathrm{Ric}-\frac{s}{n}g\right)% \wedge\!\!\!\!\!\!\bigcirc g\right|^{2}+\left|\frac{s}{2n(n-1)}g\wedge\!\!\!\!% \!\!\bigcirc g\right|^{2}.
  7. P = 1 n - 2 ( Ric - s 2 ( n - 1 ) g ) . P=\frac{1}{n-2}\left(\mathrm{Ric}-\frac{s}{2(n-1)}g\right).
  8. C = R - P g . C=R-P\wedge\!\!\!\!\!\!\bigcirc g.
  9. C a b c d = R a b c d - 2 n - 2 ( g a [ c R d ] b - g b [ c R d ] a ) + 2 ( n - 1 ) ( n - 2 ) R g a [ c g d ] b C_{abcd}=R_{abcd}-\frac{2}{n-2}(g_{a[c}R_{d]b}-g_{b[c}R_{d]a})+\frac{2}{(n-1)(% n-2)}R~{}g_{a[c}g_{d]b}
  10. R a b c d R_{abcd}
  11. R a b R_{ab}
  12. R R
  13. C a b c d = R a b c d - 4 S [ a [ c δ b ] d ] {C_{ab}}^{cd}={R_{ab}}^{cd}-4S_{[a}^{[c}\delta_{b]}^{d]}
  14. D d f - d f d f + ( | d f | 2 + Δ f n - 2 ) g = Ric . Ddf-df\otimes df+\left(|df|^{2}+\frac{\Delta f}{n-2}\right)g=\operatorname{Ric}.
  15. C ( u , v ) = - C ( v , u ) C(u,v)=-C(v,u)
  16. C ( u , v ) w , z = - C ( u , v ) z , w \langle C(u,v)w,z\rangle=-\langle C(u,v)z,w\rangle
  17. C ( u , v ) w + C ( v , w ) u + C ( w , u ) v = 0. C(u,v)w+C(v,w)u+C(w,u)v=0.
  18. tr C ( u , ) v = 0 \operatorname{tr}C(u,\cdot)v=0
  19. C a b c d = - C b a c d = - C a b d c C_{abcd}=-C_{bacd}=-C_{abdc}
  20. C a b c d + C a c d b + C a d b c = 0 C_{abcd}+C_{acdb}+C_{adbc}=0
  21. C a b a c = 0. {C^{a}}_{bac}=0.
  22. a C a b c d = 2 ( n - 3 ) [ c S d ] b \nabla_{a}{C^{a}}_{bcd}=2(n-3)\nabla_{[c}S_{d]b}

Whale_vocalization.html

  1. μ \mu

Wheel_sizing.html

  1. sin ( half top angle ) = center distance 2 × radius \sin\left(\,\text{half top angle}\right)=\frac{\,\text{center distance}}{2% \times\,\text{radius}}
  2. radius = center distance 2 × sin ( half top angle ) \,\text{radius}=\frac{\,\text{center distance}}{2\times\sin\left(\,\text{half % top angle}\right)}
  3. 360 5 × 1 2 = 36 \frac{360^{\circ}}{5}\times\frac{1}{2}=36^{\circ}
  4. B C D = 2 × 1 2 × 0.58778 = 1.701 {BCD}=2\times\frac{1}{2\times 0.58778}=1.701
  5. b = d sin ( 1 n 180 ) b=\frac{d}{\sin\left(\frac{1}{n}180^{\circ}\right)}

Wheelbase.html

  1. F f = d r L m g F_{f}={d_{r}\over L}mg
  2. F r = d f L m g F_{r}={d_{f}\over L}mg
  3. F f F_{f}
  4. F r F_{r}
  5. L L
  6. d r d_{r}
  7. d f d_{f}
  8. d f d_{f}
  9. d r d_{r}
  10. L L
  11. m m
  12. g g
  13. F f = d r L m g - h c m L m a F_{f}={d_{r}\over L}mg-{h_{cm}\over L}ma
  14. F r = d f L m g + h c m L m a F_{r}={d_{f}\over L}mg+{h_{cm}\over L}ma
  15. F f F_{f}
  16. F r F_{r}
  17. d r d_{r}
  18. d f d_{f}
  19. L L
  20. m m
  21. g g
  22. h c m h_{cm}
  23. a a

Whitehead_theorem.html

  1. f : X Y f\colon X\to Y
  2. f * : π n ( X , x ) π n ( Y , y ) , f_{*}\colon\pi_{n}(X,x)\to\pi_{n}(Y,y),

Whitehead_torsion.html

  1. M W M\hookrightarrow W
  2. K ~ 1 ( 𝐙 [ G ] ) {\tilde{K}}_{1}(\mathbf{Z}[G])
  3. τ ( h * ) K ~ 1 ( R ) \tau(h_{*})\in{\tilde{K}}_{1}(R)
  4. h * : D * E * h_{*}:D_{*}\to E_{*}
  5. γ * : C * C * + 1 \gamma_{*}:C_{*}\to C_{*+1}
  6. c n + 1 γ n + γ n - 1 c n = id C n c_{n+1}\circ\gamma_{n}+\gamma_{n-1}\circ c_{n}=\operatorname{id}_{C_{n}}
  7. ( c * + γ * ) odd : C odd C even (c_{*}+\gamma_{*})_{\mathrm{odd}}:C_{\mathrm{odd}}\to C_{\mathrm{even}}
  8. C odd := n odd C n C_{\mathrm{odd}}:=\oplus_{n\,\text{ odd}}\,C_{n}
  9. C even := n even C n C_{\mathrm{even}}:=\oplus_{n\,\text{ even}}\,C_{n}
  10. τ ( h * ) := [ A ] K ~ 1 ( R ) \tau(h_{*}):=[A]\in{\tilde{K}}_{1}(R)
  11. f ~ : X ~ Y ~ {\tilde{f}}:{\tilde{X}}\to{\tilde{Y}}
  12. C * ( f ~ ) : C * ( X ~ ) C * ( Y ~ ) C_{*}({\tilde{f}}):C_{*}({\tilde{X}})\to C_{*}({\tilde{Y}})
  13. K ~ 1 ( 𝐙 [ π 1 ( Y ) ] ) {\tilde{K}}_{1}(\mathbf{Z}[\pi_{1}(Y)])
  14. τ ( g f ) = g * τ ( f ) + τ ( g ) \tau(g\circ f)=g_{*}\tau(f)+\tau(g)
  15. \hookrightarrow

Whitney_immersion_theorem.html

  1. m > 1 m>1
  2. m m
  3. 2 m 2m
  4. ( 2 m - 1 ) (2m-1)
  5. m m
  6. 2 m - 1 2m-1
  7. m > 1 m>1
  8. 2 m + 1 2m+1
  9. 𝐑 2 m \mathbf{R}^{2m}
  10. S 2 n - a ( n ) S^{2n-a(n)}
  11. a ( n ) a(n)
  12. n n
  13. S 2 n - 1 - a ( n ) S^{2n-1-a(n)}
  14. S 2 n - a ( n ) S^{2n-a(n)}

Wigner_semicircle_distribution.html

  1. - R x R -R\leq x\leq R
  2. 0 0\,
  3. 0 0\,
  4. 0 0\,
  5. R 2 4 \frac{R^{2}}{4}\!
  6. 0 0\,
  7. - 1 -1\,
  8. ln ( π R ) - 1 2 \ln(\pi R)-\frac{1}{2}\,
  9. 2 I 1 ( R t ) R t 2\,\frac{I_{1}(R\,t)}{R\,t}
  10. 2 J 1 ( R t ) R t 2\,\frac{J_{1}(R\,t)}{R\,t}
  11. f ( x ) = 2 π R 2 R 2 - x 2 f(x)={2\over\pi R^{2}}\sqrt{R^{2}-x^{2}\,}\,
  12. E ( X 2 n ) = ( R 2 ) 2 n C n E(X^{2n})=\left({R\over 2}\right)^{2n}C_{n}\,
  13. C n = 1 n + 1 ( 2 n n ) , C_{n}={1\over n+1}{2n\choose n},\,
  14. x = R cos ( θ ) x=R\cos(\theta)
  15. M ( t ) = 2 π 0 π e R t cos ( θ ) sin 2 ( θ ) d θ M(t)=\frac{2}{\pi}\int_{0}^{\pi}e^{Rt\cos(\theta)}\sin^{2}(\theta)\,d\theta
  16. M ( t ) = 2 I 1 ( R t ) R t M(t)=2\,\frac{I_{1}(Rt)}{Rt}
  17. I 1 ( z ) I_{1}(z)
  18. φ ( t ) = 2 J 1 ( R t ) R t \varphi(t)=2\,\frac{J_{1}(Rt)}{Rt}
  19. J 1 ( z ) J_{1}(z)
  20. sin ( R t cos ( θ ) ) \sin(Rt\cos(\theta))
  21. R R
  22. { ( r 2 - x 2 ) f ( x ) + x f ( x ) = 0 , f ( 1 ) = 2 r 2 - 1 π r 2 } \left\{\left(r^{2}-x^{2}\right)f^{\prime}(x)+xf(x)=0,f(1)=\frac{2\sqrt{r^{2}-1% }}{\pi r^{2}}\right\}

Wigner–Eckart_theorem.html

  1. j m | T q ( k ) | j m = j m k q | j m j T ( k ) j \langle j\,m|T^{(k)}_{q}|j^{\prime}\,m^{\prime}\rangle=\langle j^{\prime}\,m^{% \prime}\,k\,q|j\,m\rangle\langle j\|T^{(k)}\|j^{\prime}\rangle
  2. q q
  3. k k
  4. | j m |jm⟩
  5. j m k q | j m ⟨j′m′kq|jm⟩
  6. j j′
  7. k k
  8. j j
  9. m m
  10. q q
  11. k k
  12. 2 p , m 1 | r i | 4 d , m 2 \langle 2p,m_{1}|r_{i}|4d,m_{2}\rangle
  13. 2 p , m 1 | r i | 4 d , m 2 = K \langle 2p,m_{1}|r_{i}|4d,m_{2}\rangle=K
  14. 2 p , m 1 | \langle 2p,m_{1}|
  15. r i r_{i}
  16. | 4 d , m 2 |4d,m_{2}\rangle
  17. 2 p , m 1 | r i | 4 d , m 2 \langle 2p,m_{1}|r_{i}|4d,m_{2}\rangle
  18. 2 p , m 1 | r i | 4 d , m 2 \langle 2p,m_{1}|r_{i}|4d,m_{2}\rangle
  19. [ J ± , T q ( k ) ] = ( k q ) ( k ± q + 1 ) T q ± 1 ( k ) [J_{\pm},T^{(k)}_{q}]=\hbar\sqrt{(k\mp q)(k\pm q+1)}T_{q\pm 1}^{(k)}
  20. j m | [ J ± , T q ( k ) ] | j m = ( k q ) ( k ± q + 1 ) j m | T q ± 1 ( k ) | j m \langle j\,m|[J_{\pm},T^{(k)}_{q}]|j^{\prime}\,m^{\prime}\rangle=\hbar\sqrt{(k% \mp q)(k\pm q+1)}\langle j\,m|T^{(k)}_{q\pm 1}|j^{\prime}\,m^{\prime}\rangle
  21. j m | [ J ± , T q ( k ) ] | j m = ( j ± m ) ( j m + 1 ) j ( m 1 ) | T q ( k ) | j m - ( j m ) ( j ± m + 1 ) j m | T q ( k ) | j ( m ± 1 ) \begin{aligned}\displaystyle\langle j\,m|[J_{\pm},T^{(k)}_{q}]|j^{\prime}\,m^{% \prime}\rangle&\displaystyle=\sqrt{(j\pm m)(j\mp m+1)}\langle j\,(m\mp 1)|T^{(% k)}_{q}|j^{\prime}\,m^{\prime}\rangle\\ &\displaystyle\qquad{}-\sqrt{(j^{\prime}\mp m^{\prime})(j^{\prime}\pm m^{% \prime}+1)}\langle j\,m|T^{(k)}_{q}|j^{\prime}\,(m^{\prime}\pm 1)\rangle\end{aligned}
  22. ( j ± m ) ( j m + 1 ) j ( m 1 ) | T q ( k ) | j m = ( j m ) ( j ± m + 1 ) j m | T q ( k ) | j ( m ± 1 ) + ( k q ) ( k ± q + 1 ) j m | T q ± 1 ( k ) | j m \begin{aligned}\displaystyle\sqrt{(j\pm m)(j\mp m+1)}\langle j\,(m\mp 1)|T^{(k% )}_{q}|j^{\prime}\,m^{\prime}\rangle&\displaystyle=\sqrt{(j^{\prime}\mp m^{% \prime})(j^{\prime}\pm m^{\prime}+1)}\langle j\,m|T^{(k)}_{q}|j^{\prime}\,(m^{% \prime}\pm 1)\rangle\\ &\displaystyle\qquad{}+\sqrt{(k\mp q)(k\pm q+1)}\langle j\,m|T^{(k)}_{q\pm 1}|% j^{\prime}\,m^{\prime}\rangle\end{aligned}
  23. c a b , c x c = 0 c a b , c y c = 0 \begin{aligned}\displaystyle\sum_{c}a_{b,c}x_{c}&\displaystyle=0&\displaystyle% \sum_{c}a_{b,c}y_{c}&\displaystyle=0\end{aligned}
  24. x c x d = y c y d \frac{x_{c}}{x_{d}}=\frac{y_{c}}{y_{d}}
  25. j m | T q ± 1 ( k ) | j m j m k ( q ± 1 ) | j m \langle j^{\prime}\,m^{\prime}|T^{(k)}_{q\pm 1}|j\,m\rangle\propto\langle j\,m% \,k\,(q\pm 1)|j^{\prime}\,m^{\prime}\rangle
  26. j m | T q ( k ) | j m = j m k q | j m j T ( k ) j * 2 j + 1 \langle j\,m|T^{(k)}_{q}|j^{\prime}\,m^{\prime}\rangle=\frac{\langle j^{\prime% }\,m^{\prime}\,k\,q|j\,m\rangle\langle j\|T^{(k)}\|j^{\prime}\rangle_{*}}{% \sqrt{2j^{\prime}+1}}
  27. n j m | x | n j m ⟨njm|x|njm⟩
  28. x x
  29. 𝐫 \mathbf{r}
  30. x x
  31. x = T - 1 ( 1 ) - T 1 ( 1 ) 2 x=\frac{T^{(1)}_{-1}-T^{(1)}_{1}}{\sqrt{2}}
  32. T q ( 1 ) = 4 π 3 r Y 1 q T^{(1)}_{q}=\sqrt{\frac{4\pi}{3}}rY_{1}^{q}
  33. l l
  34. T ± 1 ( 1 ) = x ± i y 2 T^{(1)}_{\pm 1}=\mp\frac{x\pm iy}{\sqrt{2}}
  35. n j m | x | n j m = n j m | T - 1 ( 1 ) - T 1 ( 1 ) 2 | n j m = 1 2 n j T ( 1 ) n j ( j m 1 ( - 1 ) | j m - j m 1 1 | j m ) \langle n\,j\,m|x|n^{\prime}\,j^{\prime}\,m^{\prime}\rangle=\langle n\,j\,m|% \frac{T^{(1)}_{-1}-T^{(1)}_{1}}{\sqrt{2}}|n^{\prime}\,j^{\prime}\,m^{\prime}% \rangle=\frac{1}{\sqrt{2}}\langle n\,j\|T^{(1)}\|n^{\prime}\,j^{\prime}\rangle% \Bigl(\langle j^{\prime}\,m^{\prime}\,1\,(-1)|j\,m\rangle-\langle j^{\prime}\,% m^{\prime}\,1\,1|j\,m\rangle\Bigr)
  36. x x
  37. | n j m |njm⟩
  38. n = n n′=n
  39. j = j j′=j
  40. m = m m′=m
  41. m m′
  42. m m
  43. m ± 1 = m m±1=m′
  44. T < s u p > ( 1 ) ± 1 T<sup>(1)_{±1}
  45. ( k ) (k)
  46. q q

Wilkinson's_polynomial.html

  1. w ( x ) = i = 1 20 ( x - i ) = ( x - 1 ) ( x - 2 ) ( x - 20 ) . w(x)=\prod_{i=1}^{20}(x-i)=(x-1)(x-2)\ldots(x-20).
  2. p ( x ) = i = 0 n c i x i . p(x)=\sum_{i=0}^{n}c_{i}x^{i}.
  3. w ( x ) = i = 1 20 ( x - i ) = ( x - 1 ) ( x - 2 ) ( x - 20 ) w(x)=\prod_{i=1}^{20}(x-i)=(x-1)(x-2)\ldots(x-20)
  4. w ( x ) = w(x)=\,\!
  5. x 20 - 210 x 19 + 20615 x 18 - 1256850 x 17 + 53327946 x 16 x^{20}-210x^{19}+20615x^{18}-1256850x^{17}+53327946x^{16}\,\!
  6. - 1672280820 x 15 + 40171771630 x 14 - 756111184500 x 13 {}-1672280820x^{15}+40171771630x^{14}-756111184500x^{13}\,\!
  7. + 11310276995381 x 12 - 135585182899530 x 11 {}+11310276995381x^{12}-135585182899530x^{11}\,\!
  8. + 1307535010540395 x 10 - 10142299865511450 x 9 {}+1307535010540395x^{10}-10142299865511450x^{9}\,\!
  9. + 63030812099294896 x 8 - 311333643161390640 x 7 {}+63030812099294896x^{8}-311333643161390640x^{7}\,\!
  10. + 1206647803780373360 x 6 - 3599979517947607200 x 5 {}+1206647803780373360x^{6}-3599979517947607200x^{5}\,\!
  11. + 8037811822645051776 x 4 - 12870931245150988800 x 3 {}+8037811822645051776x^{4}-12870931245150988800x^{3}\,\!
  12. + 13803759753640704000 x 2 - 8752948036761600000 x {}+13803759753640704000x^{2}-8752948036761600000x\,\!
  13. + 2432902008176640000. {}+2432902008176640000.\,\!
  14. d α j d t = - c ( α j ) p ( α j ) . {d\alpha_{j}\over dt}=-{c(\alpha_{j})\over p^{\prime}(\alpha_{j})}.
  15. α j + d α j d t t + d 2 α j d t 2 t 2 2 ! + \alpha_{j}+{d\alpha_{j}\over dt}t+{d^{2}\alpha_{j}\over dt^{2}}{t^{2}\over 2!}+\cdots
  16. d α j d t = - α j 19 k j ( α j - α k ) = - k j α j α j - α k . {d\alpha_{j}\over dt}=-{\alpha_{j}^{19}\over\prod_{k\neq j}(\alpha_{j}-\alpha_% {k})}=-\prod_{k\neq j}{\alpha_{j}\over\alpha_{j}-\alpha_{k}}.\,\!
  17. w 2 ( x ) = i = 1 20 ( x - 2 - i ) = ( x - 2 - 1 ) ( x - 2 - 2 ) ( x - 2 - 20 ) . w_{2}(x)=\prod_{i=1}^{20}(x-2^{-i})=(x-2^{-1})(x-2^{-2})\ldots(x-2^{-20}).
  18. α j α j - α k \alpha_{j}\over\alpha_{j}-\alpha_{k}
  19. p ( x ) = i = 0 n c i x i p(x)=\sum_{i=0}^{n}c_{i}x^{i}
  20. k ( x ) = i { 0 , , 20 } { k } ( x - i ) ( k - i ) , for k = 0 , , 20. \ell_{k}(x)=\prod_{i\in\{0,\ldots,20\}\setminus\{k\}}\frac{(x-i)}{(k-i)},% \qquad\mbox{for}~{}\quad k=0,\ldots,20.
  21. p ( x ) = i = 0 20 d i i ( x ) . p(x)=\sum_{i=0}^{20}d_{i}\ell_{i}(x).
  22. w ( x ) = ( 20 ! ) 0 ( x ) = i = 0 20 d i i ( x ) with d 0 = ( 20 ! ) , d 1 = d 2 = = d 20 = 0. w(x)=(20!)\ell_{0}(x)=\sum_{i=0}^{20}d_{i}\ell_{i}(x)\quad\mbox{with}~{}\quad d% _{0}=(20!),\,d_{1}=d_{2}=\ldots=d_{20}=0.

Wilshire_5000.html

  1. α \alpha
  2. α i = 1 M N i P i \alpha\sum_{i=1}^{M}N_{i}P_{i}

Wind_wave.html

  1. C = L / T C={L}/{T}
  2. C = g L / 2 π C=\sqrt{{gL}/{2\pi}}
  3. C = 1.251 L C=1.251\sqrt{L}
  4. C = g d = 3.1 d C=\sqrt{gd}=3.1\sqrt{d}
  5. c = g λ 2 π tanh ( 2 π d λ ) c=\sqrt{\frac{g\lambda}{2\pi}\tanh\left(\frac{2\pi d}{\lambda}\right)}
  6. d 1 2 λ d\geq\frac{1}{2}\lambda
  7. 2 π d λ π \frac{2\pi d}{\lambda}\geq\pi
  8. 1 1
  9. c c
  10. c deep = g λ 2 π . c\text{deep}=\sqrt{\frac{g\lambda}{2\pi}}.
  11. c deep c\text{deep}
  12. c deep 1.25 λ c\text{deep}\approx 1.25\sqrt{\lambda}
  13. λ \lambda
  14. c c
  15. c shallow = lim λ c = g d . c\text{shallow}=\lim_{\lambda\rightarrow\infty}c=\sqrt{gd}.
  16. c gravity-capillary = g λ 2 π + 2 π S ρ λ c\text{gravity-capillary}=\sqrt{\frac{g\lambda}{2\pi}+\frac{2\pi S}{\rho% \lambda}}
  17. ρ \rho
  18. d d
  19. ρ \rho
  20. g g
  21. H H
  22. a a
  23. E = 1 8 ρ g H 2 = 1 2 ρ g a 2 . E=\frac{1}{8}\rho gH^{2}=\frac{1}{2}\rho ga^{2}.

Wingtip_vortices.html

  1. T c T_{c}
  2. T c T_{c}
  3. T c T_{c}
  4. T c T_{c}
  5. T c T_{c}
  6. T c T_{c}
  7. T c T_{c}
  8. T f T i = ( p f p i ) γ - 1 γ . \frac{T_{\,\text{f}}}{T_{\,\text{i}}}=\left(\frac{p_{\,\text{f}}}{p_{\,\text{i% }}}\right)^{\frac{\gamma-1}{\gamma}}.
  9. T i T_{\,\text{i}}
  10. p i p_{\,\text{i}}
  11. T f T_{\,\text{f}}
  12. p f p_{\,\text{f}}
  13. γ \gamma
  14. T i T_{\,\text{i}}
  15. p i p_{\,\text{i}}
  16. p i p_{\,\text{i}}
  17. \,
  18. T i T_{\,\text{i}}
  19. p f p_{\,\text{f}}
  20. T f = ( 80 000 101 325 ) 0.4 / 1.4 T i = 0.935 × 293.15 = 274 K , T_{\,\text{f}}=\left(\frac{\scriptstyle 80\,000}{\scriptstyle 101\,325}\right)% ^{\scriptscriptstyle 0.4/1.4}\,T_{\,\text{i}}=0.935\,\times\,293.15=274\;\,% \text{K},

Wittgenstein_on_Rules_and_Private_Language.html

  1. x quus y = { x + y if x , y < 57 5 otherwise \,\text{x quus y}=\begin{cases}\,\text{x + y}&\,\text{if }x,y<57\\ 5&\,\text{otherwise}\end{cases}

WKB_approximation.html

  1. ε ε
  2. ϵ d n y d x n + a ( x ) d n - 1 y d x n - 1 + + k ( x ) d y d x + m ( x ) y = 0 , \epsilon\frac{\mathrm{d}^{n}y}{\mathrm{d}x^{n}}+a(x)\frac{\mathrm{d}^{n-1}y}{% \mathrm{d}x^{n-1}}+\cdots+k(x)\frac{\mathrm{d}y}{\mathrm{d}x}+m(x)y=0~{},
  3. y ( x ) exp [ 1 δ n = 0 δ n S n ( x ) ] y(x)\sim\exp\left[\frac{1}{\delta}\sum_{n=0}^{\infty}\delta^{n}S_{n}(x)\right]
  4. δ 0 δ→0
  5. δ δ
  6. ε ε
  7. ϵ 2 d 2 y d x 2 = Q ( x ) y , \epsilon^{2}\frac{d^{2}y}{dx^{2}}=Q(x)y,
  8. Q ( x ) 0 Q(x)\neq 0
  9. y ( x ) = exp [ 1 δ n = 0 δ n S n ( x ) ] y(x)=\exp\left[\frac{1}{\delta}\sum_{n=0}^{\infty}\delta^{n}S_{n}(x)\right]
  10. ϵ 2 [ 1 δ 2 ( n = 0 δ n S n ) 2 + 1 δ n = 0 δ n S n ′′ ] = Q ( x ) . \epsilon^{2}\left[\frac{1}{\delta^{2}}\left(\sum_{n=0}^{\infty}\delta^{n}S_{n}% ^{\prime}\right)^{2}+\frac{1}{\delta}\sum_{n=0}^{\infty}\delta^{n}S_{n}^{% \prime\prime}\right]=Q(x).
  11. ϵ 2 δ 2 S 0 2 + 2 ϵ 2 δ S 0 S 1 + ϵ 2 δ S 0 ′′ = Q ( x ) . \frac{\epsilon^{2}}{\delta^{2}}S_{0}^{\prime 2}+\frac{2\epsilon^{2}}{\delta}S_% {0}^{\prime}S_{1}^{\prime}+\frac{\epsilon^{2}}{\delta}S_{0}^{\prime\prime}=Q(x).
  12. δ 0 δ→0
  13. ϵ 2 δ 2 S 0 2 Q ( x ) . \frac{\epsilon^{2}}{\delta^{2}}S_{0}^{\prime 2}\sim Q(x).
  14. δ δ
  15. ϵ 0 : S 0 2 = Q ( x ) , \epsilon^{0}:\quad S_{0}^{\prime 2}=Q(x),
  16. S 0 ( x ) = ± x 0 x Q ( t ) d t . S_{0}(x)=\pm\int_{x_{0}}^{x}\sqrt{Q(t)}\,dt.
  17. ε ε
  18. ϵ 1 : 2 S 0 S 1 + S 0 ′′ = 0. \epsilon^{1}:\quad 2S_{0}^{\prime}S_{1}^{\prime}+S_{0}^{\prime\prime}=0.
  19. S 1 ( x ) = - 1 4 ln Q ( x ) + k 1 , S_{1}(x)=-\frac{1}{4}\ln Q(x)+k_{1},
  20. y ( x ) c 1 Q - 1 4 ( x ) exp [ 1 ϵ x 0 x Q ( t ) d t ] + c 2 Q - 1 4 ( x ) exp [ - 1 ϵ x 0 x Q ( t ) d t ] . y(x)\approx c_{1}Q^{-\frac{1}{4}}(x)\exp\left[\frac{1}{\epsilon}\int_{x_{0}}^{% x}\sqrt{Q(t)}dt\right]+c_{2}Q^{-\frac{1}{4}}(x)\exp\left[-\frac{1}{\epsilon}% \int_{x_{0}}^{x}\sqrt{Q(t)}dt\right].
  21. δ δ
  22. 2 S 0 S n + S n - 1 ′′ + j = 1 n - 1 S j S n - j = 0 2S_{0}^{\prime}S_{n}^{\prime}+S^{\prime\prime}_{n-1}+\sum_{j=1}^{n-1}S^{\prime% }_{j}S^{\prime}_{n-j}=0
  23. n n
  24. y ( x ) y(x)
  25. ϵ 2 d 2 y d x 2 = Q ( x ) y , \epsilon^{2}\frac{d^{2}y}{dx^{2}}=Q(x)y,
  26. Q ( x ) Q(x)
  27. n max 2 ϵ - 1 | x 0 x d z - Q ( z ) | , n_{\max}\approx 2\epsilon^{-1}\left|\int_{x_{0}}^{x_{\ast}}dz\sqrt{-Q(z)}% \right|,
  28. δ n max S n max ( x 0 ) 2 π n max exp [ - n max ] , \delta^{n_{\max}}S_{n_{\max}}(x_{0})\approx\sqrt{\frac{2\pi}{n_{\max}}}\exp[-n% _{\max}],
  29. x 0 x_{0}
  30. y ( x 0 ) y(x_{0})
  31. x x_{\ast}
  32. Q ( x ) = 0 Q(x_{\ast})=0
  33. x = x 0 x=x_{0}
  34. x 0 x_{0}
  35. ϵ - 1 Q ( x ) \epsilon^{-1}Q(x)
  36. ϵ | d Q d x | Q 2 , \epsilon\left|\frac{dQ}{dx}\right|\ll Q^{2},
  37. - 2 2 m d 2 d x 2 Ψ ( x ) + V ( x ) Ψ ( x ) = E Ψ ( x ) , -\frac{\hbar^{2}}{2m}\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\Psi(x)+V(x)\Psi(x)% =E\Psi(x),
  38. d 2 d x 2 Ψ ( x ) = 2 m 2 ( V ( x ) - E ) Ψ ( x ) . \frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\Psi(x)=\frac{2m}{\hbar^{2}}\left(V(x)-E% \right)\Psi(x).
  39. Φ Φ
  40. Ψ ( x ) = e Φ ( x ) , \Psi(x)=e^{\Phi(x)}~{},
  41. Φ ′′ ( x ) + [ Φ ( x ) ] 2 = 2 m 2 ( V ( x ) - E ) , \Phi^{\prime\prime}(x)+\left[\Phi^{\prime}(x)\right]^{2}=\frac{2m}{\hbar^{2}}% \left(V(x)-E\right),
  42. Φ Φ
  43. Φ Φ
  44. Φ Φ
  45. Φ ( x ) = A ( x ) + i B ( x ) . \Phi^{\prime}(x)=A(x)+iB(x)~{}.
  46. exp [ x A ( x ) d x ] , \exp\left[\int^{x}A(x^{\prime})dx^{\prime}\right]\,\!,
  47. x B ( x ) d x . \int^{x}B(x^{\prime})dx^{\prime}\,\!.
  48. A ( x ) + A ( x ) 2 - B ( x ) 2 = 2 m 2 ( V ( x ) - E ) , A^{\prime}(x)+A(x)^{2}-B(x)^{2}=\frac{2m}{\hbar^{2}}\left(V(x)-E\right),
  49. B ( x ) + 2 A ( x ) B ( x ) = 0 . B^{\prime}(x)+2A(x)B(x)=0~{}.
  50. ħ ħ
  51. ħ ħ
  52. ħ ħ
  53. A ( x ) = 1 n = 0 n A n ( x ) , A(x)=\frac{1}{\hbar}\sum_{n=0}^{\infty}\hbar^{n}A_{n}(x),
  54. B ( x ) = 1 n = 0 n B n ( x ) . B(x)=\frac{1}{\hbar}\sum_{n=0}^{\infty}\hbar^{n}B_{n}(x)~{}.
  55. A 0 ( x ) 2 - B 0 ( x ) 2 = 2 m ( V ( x ) - E ) , A_{0}(x)^{2}-B_{0}(x)^{2}=2m\left(V(x)-E\right),
  56. A 0 ( x ) B 0 ( x ) = 0 . A_{0}(x)B_{0}(x)=0\;.
  57. A ( x ) ) A(x))
  58. B ( x ) ) B(x))
  59. ħ ħ
  60. ħ ħ
  61. A 0 ( x ) = 0 A_{0}(x)=0
  62. B 0 ( x ) = ± 2 m ( E - V ( x ) ) , B_{0}(x)=\pm\sqrt{2m\left(E-V(x)\right)},
  63. Ψ ( x ) C 0 e i d x 2 m 2 ( E - V ( x ) ) + θ 2 m 2 ( E - V ( x ) ) 4 . \Psi(x)\approx C_{0}\frac{e^{i\int\mathrm{d}x\sqrt{\frac{2m}{\hbar^{2}}\left(E% -V(x)\right)}+\theta}}{\sqrt[4]{\frac{2m}{\hbar^{2}}\left(E-V(x)\right)}}.
  64. B 0 ( x ) = 0 B_{0}(x)=0
  65. A 0 ( x ) = ± 2 m ( V ( x ) - E ) , A_{0}(x)=\pm\sqrt{2m\left(V(x)-E\right)},
  66. E = V ( x ) E=V(x)
  67. E = V ( x ) E=V(x)
  68. x x
  69. 2 m 2 ( V ( x ) - E ) \frac{2m}{\hbar^{2}}\left(V(x)-E\right)
  70. 2 m 2 ( V ( x ) - E ) = U 1 ( x - x 1 ) + U 2 ( x - x 1 ) 2 + . \frac{2m}{\hbar^{2}}\left(V(x)-E\right)=U_{1}\cdot(x-x_{1})+U_{2}\cdot(x-x_{1}% )^{2}+\cdots\;.
  71. d 2 d x 2 Ψ ( x ) = U 1 ( x - x 1 ) Ψ ( x ) . \frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}\Psi(x)=U_{1}\cdot(x-x_{1})\cdot\Psi(x).
  72. Ψ ( x ) = C A Ai ( U 1 3 ( x - x 1 ) ) + C B Bi ( U 1 3 ( x - x 1 ) ) . \Psi(x)=C_{A}\textrm{Ai}\left(\sqrt[3]{U_{1}}\cdot(x-x_{1})\right)+C_{B}% \textrm{Bi}\left(\sqrt[3]{U_{1}}\cdot(x-x_{1})\right).
  73. C 0 , θ C_{0},\theta
  74. C + , C - C_{+},C_{-}
  75. C + = + 1 2 C 0 cos ( θ - π 4 ) , C_{+}=+\frac{1}{2}C_{0}\cos{\left(\theta-\frac{\pi}{4}\right)},
  76. C - = - 1 2 C 0 sin ( θ - π 4 ) . C_{-}=-\frac{1}{2}C_{0}\sin{\left(\theta-\frac{\pi}{4}\right)}.

Wolf_number.html

  1. R R
  2. R = k ( 10 g + s ) R=k(10g+s)\,
  3. s s
  4. g g
  5. k k
  6. K K

Wolstenholme's_theorem.html

  1. ( 2 p - 1 p - 1 ) 1 ( mod p 3 ) {2p-1\choose p-1}\equiv 1\;\;(\mathop{{\rm mod}}p^{3})
  2. ( a p b p ) ( a b ) ( mod p 3 ) . {ap\choose bp}\equiv{a\choose b}\;\;(\mathop{{\rm mod}}p^{3}).
  3. 1 + 1 2 + 1 3 + + 1 p - 1 0 ( mod p 2 ) , and 1+{1\over 2}+{1\over 3}+...+{1\over p-1}\equiv 0\;\;(\mathop{{\rm mod}}p^{2})% \mbox{, and}~{}
  4. 1 + 1 2 2 + 1 3 2 + + 1 ( p - 1 ) 2 0 ( mod p ) . 1+{1\over 2^{2}}+{1\over 3^{2}}+...+{1\over(p-1)^{2}}\equiv 0\;\;(\mathop{{\rm mod% }}p).
  5. ( 2 p - 1 p - 1 ) 1 ( mod p 4 ) . {{2p-1}\choose{p-1}}\equiv 1\;\;(\mathop{{\rm mod}}p^{4}).
  6. ( a b ) \textstyle{a\choose b}
  7. ( a p b p ) ( a b ) ( mod p 2 ) . {ap\choose bp}\equiv{a\choose b}\;\;(\mathop{{\rm mod}}p^{2}).
  8. ( a p b p ) ( a b ) + ( a 2 ) ( ( 2 p p ) - 2 ) ( a - 2 b - 1 ) ( mod p 3 ) . {ap\choose bp}\equiv{a\choose b}+{a\choose 2}\left({2p\choose p}-2\right){a-2% \choose b-1}\;\;(\mathop{{\rm mod}}p^{3}).
  9. ( - p p ) ( - 1 1 ) + ( - 1 2 ) ( ( 2 p p ) - 2 ) ( mod p 3 ) . {-p\choose p}\equiv{-1\choose 1}+{-1\choose 2}\left({2p\choose p}-2\right)\;\;% (\mathop{{\rm mod}}p^{3}).
  10. ( 2 p p ) \textstyle{2p\choose p}
  11. ( - p p ) = ( - 1 ) p 2 ( 2 p p ) . {-p\choose p}=\frac{(-1)^{p}}{2}{2p\choose p}.
  12. 3 ( 2 p p ) 6 ( mod p 3 ) . 3{2p\choose p}\equiv 6\;\;(\mathop{{\rm mod}}p^{3}).
  13. ( a p b p ) ( a b ) ( mod p 4 ) {ap\choose bp}\equiv{a\choose b}\;\;(\mathop{{\rm mod}}p^{4})
  14. ( 2 n - 1 n - 1 ) 1 ( mod n k ) {2n-1\choose n-1}\equiv 1\;\;(\mathop{{\rm mod}}n^{k})
  15. ( a n b n ) ( a b ) ( mod n k ) . {an\choose bn}\equiv{a\choose b}\;\;(\mathop{{\rm mod}}n^{k}).
  16. i = 1 ( i , n ) = 1 n - 1 1 i 0 ( mod n 2 ) . \sum_{i=1\atop(i,n)=1}^{n-1}\frac{1}{i}\equiv 0\;\;(\mathop{{\rm mod}}n^{2}).

Working_fluid.html

  1. W = - 1 2 𝐅 d 𝐬 W=-\int_{1}^{2}\mathbf{F}\cdot\mathrm{d}\mathbf{s}
  2. W = - 1 2 𝑝𝐴 d 𝐬 W=-\int_{1}^{2}\mathit{pA}\cdot\mathrm{d}\mathbf{s}
  3. W = - 1 2 p d V W=-\int_{1}^{2}\mathit{p}\cdot\mathrm{d}\mathit{V}
  4. W = - p 1 2 d V W=-\mathit{p}\int_{1}^{2}\mathrm{d}\mathit{V}
  5. W = - p ( V 2 - V 1 ) W=-\mathit{p}\cdot\mathrm{(V_{2}-V_{1})}

Writhe.html

  1. C C
  2. 𝐫 1 \mathbf{r}_{1}
  3. 𝐫 2 \mathbf{r}_{2}
  4. C C
  5. W r = 1 4 π C C d 𝐫 1 × d 𝐫 2 𝐫 1 - 𝐫 2 | 𝐫 1 - 𝐫 2 | 3 Wr=\frac{1}{4\pi}\int_{C}\int_{C}d\mathbf{r}_{1}\times d\mathbf{r}_{2}\cdot% \frac{\mathbf{r}_{1}-\mathbf{r}_{2}}{\left|\mathbf{r}_{1}-\mathbf{r}_{2}\right% |^{3}}
  6. N N
  7. W r = i = 1 N j = 1 N Ω i j 4 π = 2 i = 2 N j < i Ω i j 4 π Wr=\sum_{i=1}^{N}\sum_{j=1}^{N}\frac{\Omega_{ij}}{4\pi}=2\sum_{i=2}^{N}\sum_{j% <i}\frac{\Omega_{ij}}{4\pi}
  8. Ω i j / 4 π \Omega_{ij}/{4\pi}
  9. i i
  10. j j
  11. Ω i j = Ω j i \Omega_{ij}=\Omega_{ji}
  12. Ω i , i + 1 = Ω i i = 0 \Omega_{i,i+1}=\Omega_{ii}=0
  13. Ω i j / 4 π \Omega_{ij}/{4\pi}
  14. i i
  15. j j
  16. r p q r_{pq}
  17. p p
  18. q q
  19. n 1 = r 13 × r 14 | r 13 × r 14 | , n 2 = r 14 × r 24 | r 14 × r 24 | , n 3 = r 24 × r 23 | r 24 × r 23 | , n 4 = r 23 × r 13 | r 23 × r 13 | n_{1}=\frac{r_{13}\times r_{14}}{\left|r_{13}\times r_{14}\right|},\;n_{2}=% \frac{r_{14}\times r_{24}}{\left|r_{14}\times r_{24}\right|},\;n_{3}=\frac{r_{% 24}\times r_{23}}{\left|r_{24}\times r_{23}\right|},\;n_{4}=\frac{r_{23}\times r% _{13}}{\left|r_{23}\times r_{13}\right|}
  20. Ω * = arcsin ( n 1 n 2 ) + arcsin ( n 2 n 3 ) + arcsin ( n 3 n 4 ) + arcsin ( n 4 n 1 ) . \Omega^{*}=\arcsin\left(n_{1}\cdot n_{2}\right)+\arcsin\left(n_{2}\cdot n_{3}% \right)+\arcsin\left(n_{3}\cdot n_{4}\right)+\arcsin\left(n_{4}\cdot n_{1}% \right).
  21. 4 π 4\pi
  22. Ω 4 π = Ω * 4 π sign ( ( r 34 × r 12 ) r 13 ) . \frac{\Omega}{4\pi}=\frac{\Omega^{*}}{4\pi}\,\text{sign}\left(\left(r_{34}% \times r_{12}\right)\cdot r_{13}\right).

X-ray_tube.html

  1. E heat = V eff I eff t E_{\mathrm{heat}}=V_{\mathrm{eff}}I_{\mathrm{eff}}\;t
  2. kV p \mathrm{kV_{p}}
  3. E heat = w × kV p × mA × t E_{\mathrm{heat}}=w\times\mathrm{kV_{p}}\times\mathrm{mA}\times t
  4. 1 2 0.707 \frac{1}{\sqrt{2}}\approx 0.707

XOR_cipher.html

  1. \oplus
  2. \oplus
  3. \oplus
  4. \oplus
  5. \oplus
  6. \oplus
  7. \oplus
  8. \oplus
  9. \oplus
  10. \oplus
  11. \oplus
  12. \oplus
  13. \oplus

Yang–Mills_theory.html

  1. gf = - 1 2 Tr ( F 2 ) = - 1 4 F a μ ν F μ ν a \mathcal{L}_{\mathrm{gf}}=-\frac{1}{2}\operatorname{Tr}(F^{2})=-\frac{1}{4}F^{% a\mu\nu}F_{\mu\nu}^{a}
  2. Tr ( T a T b ) = 1 2 δ a b , [ T a , T b ] = i f a b c T c \operatorname{Tr}(T^{a}T^{b})=\frac{1}{2}\delta^{ab},\quad[T^{a},T^{b}]=if^{% abc}T^{c}
  3. D μ = I μ - i g T a A μ a D_{\mu}=I\partial_{\mu}-igT^{a}A^{a}_{\mu}
  4. A μ a A^{a}_{\mu}
  5. a , b , c = 1 N 2 - 1. a,b,c=1\ldots N^{2}-1.
  6. F μ ν a = μ A ν a - ν A μ a + g f a b c A μ b A ν c F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+gf^{abc}A_{% \mu}^{b}A_{\nu}^{c}
  7. [ D μ , D ν ] = - i g T a F μ ν a . [D_{\mu},D_{\nu}]=-igT^{a}F_{\mu\nu}^{a}.
  8. f a b c = f a b c f^{abc}=f_{abc}
  9. η μ ν = diag ( + - - - ) \eta_{\mu\nu}={\rm diag}(+---)
  10. μ F μ ν a + g f a b c A μ b F μ ν c = 0. \partial^{\mu}F_{\mu\nu}^{a}+gf^{abc}A^{\mu b}F_{\mu\nu}^{c}=0.
  11. F μ ν = T a F μ ν a F_{\mu\nu}=T^{a}F^{a}_{\mu\nu}
  12. ( D μ F μ ν ) a = 0. (D^{\mu}F_{\mu\nu})^{a}=0.
  13. ( D μ F ν κ ) a + ( D κ F μ ν ) a + ( D ν F κ μ ) a = 0 (D_{\mu}F_{\nu\kappa})^{a}+(D_{\kappa}F_{\mu\nu})^{a}+(D_{\nu}F_{\kappa\mu})^{% a}=0
  14. [ D μ , [ D ν , D κ ] ] + [ D κ , [ D μ , D ν ] ] + [ D ν , [ D κ , D μ ] ] = 0 [D_{\mu},[D_{\nu},D_{\kappa}]]+[D_{\kappa},[D_{\mu},D_{\nu}]]+[D_{\nu},[D_{% \kappa},D_{\mu}]]=0
  15. [ D μ , F ν κ a ] = D μ F ν κ a [D_{\mu},F^{a}_{\nu\kappa}]=D_{\mu}F^{a}_{\nu\kappa}
  16. F ~ μ ν = 1 2 ε μ ν ρ σ F ρ σ \tilde{F}^{\mu\nu}=\frac{1}{2}\varepsilon^{\mu\nu\rho\sigma}F_{\rho\sigma}
  17. D μ F ~ μ ν = 0. D_{\mu}\tilde{F}^{\mu\nu}=0.
  18. J μ a J_{\mu}^{a}
  19. μ F μ ν a + g f a b c A b μ F μ ν c = - J ν a . \partial^{\mu}F_{\mu\nu}^{a}+gf^{abc}A^{b\mu}F_{\mu\nu}^{c}=-J_{\nu}^{a}.
  20. [ A ] = [ L 2 - D 2 ] [A]=[L^{\frac{2-D}{2}}]
  21. [ g 2 ] = [ L D - 4 ] [g^{2}]=[L^{D-4}]
  22. Z [ j ] = [ d A ] exp [ - i 2 d 4 x Tr ( F μ ν F μ ν ) + i d 4 x j μ a ( x ) A a μ ( x ) ] , Z[j]=\int[dA]\exp\left[-\frac{i}{2}\int d^{4}x\operatorname{Tr}(F^{\mu\nu}F_{% \mu\nu})+i\int d^{4}x\,j^{a}_{\mu}(x)A^{a\mu}(x)\right],
  23. Z [ j , ε ¯ , ε ] = [ d A ] [ d c ¯ ] [ d c ] exp { i S F [ A , A ] + i S g f [ A ] + i S g [ c , c ¯ , c , c ¯ , A ] } exp { i d 4 x j μ a ( x ) A a μ ( x ) + i d 4 x [ c ¯ a ( x ) ε a ( x ) + ε ¯ a ( x ) c a ( x ) ] } \begin{aligned}\displaystyle Z[j,\bar{\varepsilon},\varepsilon]&\displaystyle=% \int[dA][d\bar{c}][dc]\exp\left\{iS_{F}[\partial A,A]+iS_{gf}[\partial A]+iS_{% g}[\partial c,\partial\bar{c},c,\bar{c},A]\right\}\\ &\displaystyle\exp\left\{i\int d^{4}xj^{a}_{\mu}(x)A^{a\mu}(x)+i\int d^{4}x[% \bar{c}^{a}(x)\varepsilon^{a}(x)+\bar{\varepsilon}^{a}(x)c^{a}(x)]\right\}\end% {aligned}
  24. S F = - 1 2 Tr ( F μ ν F μ ν ) S_{F}=-\frac{1}{2}\operatorname{Tr}(F^{\mu\nu}F_{\mu\nu})
  25. S g f = - 1 2 ξ ( A ) 2 S_{gf}=-\frac{1}{2\xi}(\partial\cdot A)^{2}
  26. S g = - ( c ¯ a μ μ c a + g c ¯ a f a b c μ A b μ c c ) S_{g}=-(\bar{c}^{a}\partial_{\mu}\partial^{\mu}c^{a}+g\bar{c}^{a}f^{abc}% \partial_{\mu}A^{b\mu}c^{c})
  27. Z [ j , ε ¯ , ε ] = exp ( - i g d 4 x δ i δ ε ¯ a ( x ) f a b c μ i δ δ j μ b ( x ) i δ δ ε c ( x ) ) × exp ( - i g d 4 x f a b c μ i δ δ j ν a ( x ) i δ δ j μ b ( x ) i δ δ j c ν ( x ) ) × exp ( - i g 2 4 d 4 x f a b c f a r s i δ δ j μ b ( x ) i δ δ j ν c ( x ) i δ δ j r μ ( x ) i δ δ j s ν ( x ) ) × Z 0 [ j , ε ¯ , ε ] \begin{aligned}\displaystyle Z[j,\bar{\varepsilon},\varepsilon]&\displaystyle=% \exp\left(-ig\int d^{4}x\,\frac{\delta}{i\delta\bar{\varepsilon}^{a}(x)}f^{abc% }\partial_{\mu}\frac{i\delta}{\delta j^{b}_{\mu}(x)}\frac{i\delta}{\delta% \varepsilon^{c}(x)}\right)\\ &\displaystyle\qquad\times\exp\left(-ig\int d^{4}xf^{abc}\partial_{\mu}\frac{i% \delta}{\delta j^{a}_{\nu}(x)}\frac{i\delta}{\delta j^{b}_{\mu}(x)}\frac{i% \delta}{\delta j^{c\nu}(x)}\right)\\ &\displaystyle\qquad\qquad\times\exp\left(-i\frac{g^{2}}{4}\int d^{4}xf^{abc}f% ^{ars}\frac{i\delta}{\delta j^{b}_{\mu}(x)}\frac{i\delta}{\delta j^{c}_{\nu}(x% )}\frac{i\delta}{\delta j^{r\mu}(x)}\frac{i\delta}{\delta j^{s\nu}(x)}\right)% \\ &\displaystyle\qquad\qquad\qquad\times Z_{0}[j,\bar{\varepsilon},\varepsilon]% \end{aligned}
  28. Z 0 [ j , ε ¯ , ε ] = exp ( - d 4 x d 4 y ε ¯ a ( x ) C a b ( x - y ) ε b ( y ) ) exp ( 1 2 d 4 x d 4 y j μ a ( x ) D a b μ ν ( x - y ) j ν b ( y ) ) Z_{0}[j,\bar{\varepsilon},\varepsilon]=\exp\left(-\int d^{4}xd^{4}y\bar{% \varepsilon}^{a}(x)C^{ab}(x-y)\varepsilon^{b}(y)\right)\exp\left(\tfrac{1}{2}% \int d^{4}xd^{4}yj^{a}_{\mu}(x)D^{ab\mu\nu}(x-y)j^{b}_{\nu}(y)\right)
  29. c ¯ a f a b c μ A b μ c c \bar{c}^{a}f^{abc}\partial_{\mu}A^{b\mu}c^{c}
  30. f a b c f^{abc}
  31. D μ ν a b ( p ) = δ a b ( η μ ν - p μ p ν p 2 ) p 2 p 4 + M 4 . D_{\mu\nu}^{ab}(p)=\delta^{ab}\left(\eta_{\mu\nu}-\frac{p_{\mu}p_{\nu}}{p^{2}}% \right)\frac{p^{2}}{p^{4}+M^{4}}.
  32. V ( r ) r V(r)\propto r
  33. 128 4 10 8 128^{4}\sim 10^{8}
  34. D μ ν a b ( p ) = p 0 δ a b ( η μ ν - p μ p ν p 2 ) Z p 2 - M 2 + i 0 , D_{\mu\nu}^{ab}(p)\stackrel{p\rightarrow 0}{=}\delta^{ab}\left(\eta_{\mu\nu}-% \frac{p_{\mu}p_{\nu}}{p^{2}}\right)\frac{Z}{p^{2}-M^{2}+i0},
  35. Z Z
  36. M M
  37. μ 2 d α s d μ 2 = β ( α s ) . \mu^{2}\frac{d\alpha_{s}}{d\mu^{2}}=\beta(\alpha_{s}).
  38. α s = g 2 / 4 π \alpha_{s}=g^{2}/4\pi
  39. β ( α s ) = - 11 N 12 π α s 2 - 17 N 2 24 π 2 α s 3 + O ( α s 4 ) . \beta(\alpha_{s})=-\frac{11N}{12\pi}\alpha_{s}^{2}-\frac{17N^{2}}{24\pi^{2}}% \alpha_{s}^{3}+O\left(\alpha_{s}^{4}\right).
  40. β ( α s ) = - α s 2 4 π 3 N 1 - N α s 2 π . \beta(\alpha_{s})=-\frac{\alpha_{s}^{2}}{4\pi}\frac{3N}{1-\frac{N\alpha_{s}}{2% \pi}}.
  41. β ( α s ) = - α s 2 11 N 12 π 1 1 - 17 N 11 α s 2 π . \beta(\alpha_{s})=-\alpha_{s}^{2}\frac{11N}{12\pi}\frac{1}{1-\frac{17N}{11}% \frac{\alpha_{s}}{2\pi}}.
  42. β ( α s ) 3 2 α s . \beta(\alpha_{s})\approx\frac{3}{2}\alpha_{s}.

YDbDr.html

  1. Y Y
  2. D B D_{B}
  3. D R D_{R}
  4. Y Y
  5. D B D_{B}
  6. D R D_{R}
  7. R R
  8. G G
  9. B B
  10. Y Y
  11. D B D_{B}
  12. Y Y
  13. D R D_{R}
  14. Y Y
  15. R , G , B , Y [ 0 , 1 ] D B , D R [ - 1.333 , 1.333 ] \begin{aligned}\displaystyle R,G,B,Y&\displaystyle\in\left[0,1\right]\\ \displaystyle D_{B},D_{R}&\displaystyle\in\left[-1.333,1.333\right]\end{aligned}
  16. Y \displaystyle Y
  17. R \displaystyle R
  18. Y Y
  19. Y Y
  20. D B D_{B}
  21. D R D_{R}
  22. U U
  23. V V
  24. D B \displaystyle D_{B}

Yield_management.html

  1. R 2 R_{2}
  2. R R
  3. R \geq R
  4. * P r o b ( D *Prob(D
  5. > x ) >x)
  6. R 2 R_{2}
  7. R 1 R_{1}
  8. D 1 D_{1}
  9. x x
  10. y y
  11. = P r o b =Prob
  12. ( R (R
  13. / R /R
  14. ) )
  15. y y
  16. y y

Young_tableau.html

  1. λ λ
  2. n n
  3. λ λ
  4. λ λ
  5. n n
  6. n n
  7. n n
  8. λ λ
  9. μ μ
  10. λ λ
  11. μ μ
  12. λ λ
  13. μ μ
  14. λ λ
  15. μ μ
  16. i i
  17. λ λ
  18. μ μ
  19. λ λ
  20. μ μ
  21. λ λ
  22. μ μ
  23. λ λ
  24. μ μ
  25. λ λ
  26. μ μ
  27. μ μ
  28. T T
  29. λ λ
  30. μ μ
  31. μ μ
  32. i i
  33. μ μ
  34. i i
  35. T T
  36. λ λ
  37. T T
  38. λ λ
  39. μ μ
  40. k k
  41. k k
  42. n n
  43. n n
  44. n 1 n−1
  45. n 1 n−1
  46. n n
  47. λ λ
  48. n n
  49. h o o k ( x ) hook(x)
  50. x x
  51. Y ( λ ) Y(λ)
  52. λ λ
  53. n ! n!
  54. dim π λ = n ! x Y ( λ ) hook ( x ) . \dim\pi_{\lambda}=\frac{n!}{\prod_{x\in Y(\lambda)}\mathrm{hook}(x)}.
  55. dim π λ = 10 ! 7 5 4 3 1 5 3 2 1 1 = 288. \dim\pi_{\lambda}=\frac{10!}{7\cdot 5\cdot 4\cdot 3\cdot 1\cdot 5\cdot 3\cdot 2% \cdot 1\cdot 1}=288.
  56. W ( λ ) W(λ)
  57. dim W ( λ ) = ( i , j ) Y ( λ ) r + j - i hook ( i , j ) , \dim W(\lambda)=\prod_{(i,j)\in Y(\lambda)}\frac{r+j-i}{\mathrm{hook}(i,j)},
  58. dim W ( λ ) = 7 8 9 10 11 6 7 8 9 5 7 5 4 3 1 5 3 2 1 1 = 66528. \dim W(\lambda)=\frac{7\cdot 8\cdot 9\cdot 10\cdot 11\cdot 6\cdot 7\cdot 8% \cdot 9\cdot 5}{7\cdot 5\cdot 4\cdot 3\cdot 1\cdot 5\cdot 3\cdot 2\cdot 1\cdot 1% }=66528.
  59. n n
  60. S < s u b > n S<sub>n
  61. μ μ
  62. = ( 5 , 3 , 2 , 1 ) =(5,3,2,1)
  63. λ = ( 5 , 4 , 2 , 1 ) λ=(5,4,2,1)
  64. A A
  65. B B
  66. A B AB
  67. B A BA
  68. A A
  69. B B

Yuan-Cheng_Fung.html

  1. w = 1 2 [ q + c ( e Q - 1 ) ] w=\frac{1}{2}\left[q+c\left(e^{Q}-1\right)\right]
  2. q = a i j k l E i j E k l Q = b i j k l E i j E k l q=a_{ijkl}E_{ij}E_{kl}\qquad Q=b_{ijkl}E_{ij}E_{kl}
  3. E i j E_{ij}
  4. a i j k l a_{ijkl}
  5. b i j k l b_{ijkl}
  6. c c
  7. w w

Yule–Simon_distribution.html

  1. ρ > 3 \rho>3\,
  2. ρ + 3 + 11 ρ 3 - 49 ρ - 22 ( ρ - 4 ) ( ρ - 3 ) ρ \rho+3+\frac{11\rho^{3}-49\rho-22}{(\rho-4)\;(\rho-3)\;\rho}\,
  3. ρ > 4 \rho>4\,
  4. ρ ρ + 1 F 1 2 ( 1 , 1 ; ρ + 2 ; e t ) e t \frac{\rho}{\rho+1}\;{}_{2}F_{1}(1,1;\rho+2;e^{t})\,e^{t}\,
  5. ρ ρ + 1 F 1 2 ( 1 , 1 ; ρ + 2 ; e i t ) e i t \frac{\rho}{\rho+1}\;{}_{2}F_{1}(1,1;\rho+2;e^{i\,t})\,e^{i\,t}\,
  6. f ( k ; ρ ) = ρ B ( k , ρ + 1 ) , f(k;\rho)=\rho\,\mathrm{B}(k,\rho+1),\,
  7. k 1 k\geq 1
  8. ρ > 0 \rho>0
  9. B \mathrm{B}
  10. f ( k ; ρ ) = ρ Γ ( ρ + 1 ) ( k + ρ ) ρ + 1 ¯ , f(k;\rho)=\frac{\rho\,\Gamma(\rho+1)}{(k+\rho)^{\underline{\rho+1}}},\,
  11. Γ \Gamma
  12. ρ \rho
  13. f ( k ; ρ ) = ρ ρ ! ( k - 1 ) ! ( k + ρ ) ! . f(k;\rho)=\frac{\rho\,\rho!\,(k-1)!}{(k+\rho)!}.\,
  14. ρ \rho
  15. f ( k ; ρ ) ρ Γ ( ρ + 1 ) k ρ + 1 1 k ρ + 1 . f(k;\rho)\approx\frac{\rho\,\Gamma(\rho+1)}{k^{\rho+1}}\propto\frac{1}{k^{\rho% +1}}.\,
  16. f ( k ; ρ ) f(k;\rho)
  17. k k
  18. k k
  19. W W
  20. 1 / ρ 1/\rho
  21. ρ \rho
  22. W Exponential ( ρ ) , W\sim\mathrm{Exponential}(\rho)\,,
  23. h ( w ; ρ ) = ρ exp ( - ρ w ) . h(w;\rho)=\rho\,\exp(-\rho\,w)\,.
  24. K Geometric ( exp ( - W ) ) . K\sim\mathrm{Geometric}(\exp(-W))\,.
  25. g ( k ; p ) = p ( 1 - p ) k - 1 g(k;p)=p\,(1-p)^{k-1}\,
  26. k { 1 , 2 , } k\in\{1,2,\dots\}
  27. f ( k ; ρ ) = 0 g ( k ; exp ( - w ) ) h ( w ; ρ ) d w . f(k;\rho)=\int_{0}^{\infty}\,\,\,g(k;\exp(-w))\,h(w;\rho)\,dw\,.
  28. { k P ( k ) = ( α + k + 1 ) P ( k + 1 ) , P ( 1 ) = α B ( α + 1 , 1 ) } \{kP(k)=(\alpha+k+1)P(k+1),P(1)=\alpha B(\alpha+1,1)\}
  29. f ( k ; ρ , α ) = ρ 1 - α ρ B 1 - α ( k , ρ + 1 ) , f(k;\rho,\alpha)=\frac{\rho}{1-\alpha^{\rho}}\;\mathrm{B}_{1-\alpha}(k,\rho+1),\,
  30. 0 α < 1 0\leq\alpha<1
  31. α = 0 \alpha=0

Zassenhaus_lemma.html

  1. ( G , Ω ) (G,\Omega)
  2. A A
  3. C C
  4. B A B\triangleleft A
  5. D C D\triangleleft C
  6. ( A C ) B / ( A D ) B (A\cap C)B/(A\cap D)B
  7. ( A C ) D / ( B C ) D . (A\cap C)D/(B\cap C)D.

Zech's_logarithm.html

  1. α \alpha
  2. α \alpha
  3. α \alpha
  4. Z α ( n ) = log α ( 1 + α n ) , Z_{\alpha}(n)=\log_{\alpha}(1+\alpha^{n}),
  5. α Z α ( n ) = 1 + α n . \alpha^{Z_{\alpha}(n)}=1+\alpha^{n}.
  6. α \alpha
  7. Z α Z_{\alpha}
  8. α \alpha
  9. - -\infty
  10. α - = 0 \alpha^{-\infty}=0
  11. n + ( - ) = - n+(-\infty)=-\infty
  12. Z α ( - ) = 0 Z_{\alpha}(-\infty)=0
  13. Z α ( e ) = - Z_{\alpha}(e)=-\infty
  14. e e
  15. α e = - 1 \alpha^{e}=-1
  16. e = 0 e=0
  17. e = q - 1 2 e=\frac{q-1}{2}
  18. q q
  19. α m + α n = α m ( 1 + α n - m ) = α m α Z ( n - m ) = α m + Z ( n - m ) \alpha^{m}+\alpha^{n}=\alpha^{m}\cdot(1+\alpha^{n-m})=\alpha^{m}\cdot\alpha^{Z% (n-m)}=\alpha^{m+Z(n-m)}
  20. - α n = ( - 1 ) α n = α e α n = α e + n -\alpha^{n}=(-1)\cdot\alpha^{n}=\alpha^{e}\cdot\alpha^{n}=\alpha^{e+n}
  21. α m - α n = α m + ( - α n ) = α m + Z ( e + n - m ) \alpha^{m}-\alpha^{n}=\alpha^{m}+(-\alpha^{n})=\alpha^{m+Z(e+n-m)}
  22. α m α n = α m + n \alpha^{m}\cdot\alpha^{n}=\alpha^{m+n}
  23. ( α m ) - 1 = α - m \left(\alpha^{m}\right)^{-1}=\alpha^{-m}
  24. α m / α n = α m ( α n ) - 1 = α m - n \alpha^{m}/\alpha^{n}=\alpha^{m}\cdot\left(\alpha^{n}\right)^{-1}=\alpha^{m-n}
  25. - -\infty
  26. - -\infty
  27. m = - m=-\infty
  28. + +\infty
  29. α + = \alpha^{+\infty}=\infty
  30. Z α ( n ) = m Z_{\alpha}(n)=m
  31. Z α ( m ) = n Z_{\alpha}(m)=n
  32. α 3 = α 2 + 1 \alpha^{3}=\alpha^{2}+1
  33. α 4 = α 3 α = ( α 2 + 1 ) α = α 3 + α = α 2 + α + 1 \alpha^{4}=\alpha^{3}\alpha=(\alpha^{2}+1)\alpha=\alpha^{3}+\alpha=\alpha^{2}+% \alpha+1
  34. α 5 = α 4 α = ( α 2 + α + 1 ) α = α 3 + α 2 + α = α 2 + 1 + α 2 + α = α + 1 \alpha^{5}=\alpha^{4}\alpha=(\alpha^{2}+\alpha+1)\alpha=\alpha^{3}+\alpha^{2}+% \alpha=\alpha^{2}+1+\alpha^{2}+\alpha=\alpha+1
  35. α 6 = α 5 α = ( α + 1 ) α = α 2 + α \alpha^{6}=\alpha^{5}\alpha=(\alpha+1)\alpha=\alpha^{2}+\alpha
  36. α 6 + α 3 = α 6 + Z ( - 3 ) = α 6 + Z ( 4 ) = α 6 + 6 = α 12 = α 5 \alpha^{6}+\alpha^{3}=\alpha^{6+Z(-3)}=\alpha^{6+Z(4)}=\alpha^{6+6}=\alpha^{12% }=\alpha^{5}
  37. α 6 + α 3 = α 3 + Z ( 3 ) = α 3 + 2 = α 5 \alpha^{6}+\alpha^{3}=\alpha^{3+Z(3)}=\alpha^{3+2}=\alpha^{5}
  38. α 6 + α 3 = ( α 2 + α ) + ( α 2 + 1 ) = α + 1 = α 5 \alpha^{6}+\alpha^{3}=(\alpha^{2}+\alpha)+(\alpha^{2}+1)=\alpha+1=\alpha^{5}

Zeisel_number.html

  1. p x = a p x - 1 + b p_{x}=ap_{x-1}+b
  2. p 0 = 1 p_{0}=1
  3. p 1 = 7 , p 1 = 1 p 0 + 6 p 2 = 13 , p 2 = 1 p 1 + 6 p 3 = 19 , p 3 = 1 p 2 + 6 \begin{aligned}\displaystyle p_{1}=7,&\displaystyle{}\quad p_{1}=1p_{0}+6\\ \displaystyle p_{2}=13,&\displaystyle{}\quad p_{2}=1p_{1}+6\\ \displaystyle p_{3}=19,&\displaystyle{}\quad p_{3}=1p_{2}+6\end{aligned}
  4. ( 6 n + 1 ) ( 12 n + 1 ) ( 18 n + 1 ) (6n+1)(12n+1)(18n+1)
  5. p x = a p x - 1 + b p_{x}=ap_{x-1}+b
  6. 2 k - 1 + k 2^{k-1}+k

Zero_matrix.html

  1. 0 1 , 1 = [ 0 ] , 0 2 , 2 = [ 0 0 0 0 ] , 0 2 , 3 = [ 0 0 0 0 0 0 ] . 0_{1,1}=\begin{bmatrix}0\end{bmatrix},\ 0_{2,2}=\begin{bmatrix}0&0\\ 0&0\end{bmatrix},\ 0_{2,3}=\begin{bmatrix}0&0&0\\ 0&0&0\end{bmatrix}.
  2. K m , n K_{m,n}\,
  3. 0 K m , n 0_{K_{m,n}}\,
  4. K m , n K_{m,n}\,
  5. 0 K 0_{K}\,
  6. 0 K 0_{K}\,
  7. 0 K m , n = [ 0 K 0 K 0 K 0 K 0 K 0 K 0 K 0 K 0 K ] m × n 0_{K_{m,n}}=\begin{bmatrix}0_{K}&0_{K}&\cdots&0_{K}\\ 0_{K}&0_{K}&\cdots&0_{K}\\ \vdots&\vdots&\ddots&\vdots\\ 0_{K}&0_{K}&\cdots&0_{K}\end{bmatrix}_{m\times n}
  8. K m , n K_{m,n}\,
  9. A K m , n A\in K_{m,n}\,
  10. 0 K m , n + A = A + 0 K m , n = A . 0_{K_{m,n}}+A=A+0_{K_{m,n}}=A.

Zeta_potential.html

  1. κ a 1 {\kappa}\cdot a\gg 1
  2. D u 1 Du\ll 1

Zone_plate.html

  1. r n = n λ f + n 2 λ 2 4 r_{n}=\sqrt{n\lambda f+\frac{n^{2}\lambda^{2}}{4}}
  2. r n n f λ r_{n}\simeq\sqrt{nf\lambda}
  3. f = 2 r N Δ r N λ f=\frac{2r_{N}\Delta r_{N}}{\lambda}
  4. Δ l Δ r N = 1.22 \frac{\Delta l}{\Delta r_{N}}=1.22
  5. 1 ± cos ( k r 2 ) 2 \frac{1\pm\cos(kr^{2})}{2}\,
  6. 1 ± sgn ( cos ( k r 2 ) ) 2 \frac{1\pm\operatorname{sgn}(\cos(kr^{2}))}{2}\,
  7. r r
  8. k = 2 π / λ k=2\pi/\lambda
  9. r n = ( n + α ) λ f + ( n + α ) 2 λ 2 4 r_{n}=\sqrt{(n+\alpha)\lambda f+\frac{(n+\alpha)^{2}\lambda^{2}}{4}}

Μ_operator.html

  1. μ y y < z R ( y ) . The least y < z such that R ( y ) , if ( y ) y < z R ( y ) ; otherwise , z . \mu y_{y<z}R(y).\ \ \mbox{The least}~{}\ y<z\ \mbox{such that}~{}\ R(y),\ % \mbox{if}~{}\ (\exists y)_{y<z}R(y);\ \mbox{otherwise}~{},\ z.
  2. ( y ) μ y R ( y ) = { the least (natural number) y such that R ( y ) } (\exists y)\mu yR(y)=\{\mbox{the least (natural number)}~{}\ y\ \mbox{such % that}~{}\ R(y)\}
  3. ( y ) (\exists y)
  4. μ y R ( y , x 1 , , x k ) \mu yR(y,x_{1},\ldots,x_{k})
  5. μ y R ( y , x 1 , , x k ) \mu yR(y,x_{1},\ldots,x_{k})
  6. μ y R ( y , x 1 , , x k ) \mu yR(y,x_{1},\ldots,x_{k})