wpmath0000014_6

Identity_channel.html

  1. I I
  2. 𝗂𝖽 \mathsf{id}
  3. 𝕀 \mathbb{I}

Ignorance_space.html

  1. d 0 d_{0}
  2. d 0 d_{0}
  3. d 0 d_{0}
  4. d 0 d_{0}
  5. d 0 d_{0}
  6. 𝔇 0 \mathfrak{D}_{0}
  7. d 0 d_{0}
  8. 𝔇 0 \mathfrak{D}_{0}
  9. 𝔇 0 \mathfrak{D}_{0}
  10. 𝔇 0 \mathfrak{D}_{0}
  11. d 0 d_{0}
  12. 𝔇 0 \mathfrak{D}_{0}
  13. 𝔇 \mathfrak{D}
  14. 𝔇 \mathfrak{D}
  15. d 0 d_{0}

Image_texture.html

  1. F e d g e n e s s = | { p | M a g ( p ) > T } | N F_{edgeness}=\frac{|\{p|Mag(p)>T\}|}{N}
  2. F m a g d i r = ( H m a g ( R ) , H d i r ( R ) ) F_{magdir}=(H_{mag}(R),H_{dir}(R))
  3. A n g u l a r 2 n d M o m e n t \displaystyle Angular\,\text{ }2nd\,\text{ }Moment
  4. p [ i , j ] p[i,j]
  5. [ i , j ] [i,j]

Impairment_cost.html

  1. Impairment Cost = Recoverable Amount - Carrying Value \mbox{Impairment Cost}~{}={\mbox{Recoverable Amount}~{}-\mbox{Carrying Value}~% {}}
  2. $ 37500 - $ 33000 = $ 4500 \$37500-\$33000=\$4500
  3. $ 33000 * 0.2 = $ 6600 \$33000*0.2=\$6600

Impedance_analogy.html

  1. v = i R v=iR
  2. F = u R m F=uR_{\mathrm{m}}
  3. v = L d i d t v=L\frac{di}{dt}
  4. F = M d u d t F=M\frac{du}{dt}
  5. Z = j ω L Z=j\omega L
  6. Z m = j ω M Z_{\mathrm{m}}=j\omega M
  7. v = D i d t v=D\int idt
  8. F = S u d t F=S\int udt
  9. Z = D j ω Z=\frac{D}{j\omega}
  10. Z m = S j ω Z_{\mathrm{m}}=\frac{S}{j\omega}
  11. Z m = 1 j ω C m Z_{\mathrm{m}}=\frac{1}{j\omega C_{\mathrm{m}}}
  12. [ v F ] = [ z 11 z 12 z 21 z 22 ] [ i u ] \begin{bmatrix}v\\ F\end{bmatrix}=\begin{bmatrix}z_{11}&z_{12}\\ z_{21}&z_{22}\end{bmatrix}\begin{bmatrix}i\\ u\end{bmatrix}
  13. z 22 z_{22}\,
  14. z 11 z_{11}\,
  15. z 21 z_{21}\,
  16. z 12 z_{12}\,
  17. E = v i d t E=\int vi\ dt
  18. E = F u d t E=\int Fu\ dt
  19. P = v i P=vi
  20. P = F u P=Fu
  21. P = i 2 R = v 2 R P=i^{2}R={v^{2}\over R}
  22. P = u 2 R m = F 2 R m P=u^{2}R_{\mathrm{m}}={F^{2}\over R_{\mathrm{m}}}
  23. E = 1 2 L i 2 E=\tfrac{1}{2}Li^{2}
  24. E = 1 2 M u 2 E=\tfrac{1}{2}Mu^{2}
  25. E = 1 2 C v 2 E=\tfrac{1}{2}Cv^{2}
  26. E = 1 2 C m F 2 E=\tfrac{1}{2}C_{\mathrm{m}}F^{2}
  27. F = B I l F=BIl

Imperialist_competitive_algorithm.html

  1. f ( 𝐱 ) , 𝐱 = ( x 1 , x 2 , , x d ) ; f(\mathbf{x}),\quad\mathbf{x}=(x_{1},x_{2},\dots,x_{d});\,

In-crowd_algorithm.html

  1. min x 1 2 y - A x 2 2 + λ x 1 . \min_{x}\frac{1}{2}\|y-Ax\|^{2}_{2}+\lambda\|x\|_{1}.
  2. y y
  3. x x
  4. A x Ax
  5. x x
  6. λ \lambda
  7. x x
  8. r = y r=y
  9. I I
  10. u j = | r A j | u_{j}=|\langle rA_{j}\rangle|
  11. I c I^{c}
  12. I c I^{c}
  13. u j > λ u_{j}>\lambda
  14. L 25 L\approx 25
  15. I I
  16. I I
  17. I I
  18. r = y - A x r=y-Ax
  19. I I
  20. L L
  21. L L

Ince_equation.html

  1. w ′′ + ξ sin ( 2 z ) w + ( η - p ξ cos ( 2 z ) ) w = 0. w^{\prime\prime}+\xi\sin(2z)w^{\prime}+(\eta-p\xi\cos(2z))w=0.\,

Indian_and_ISO_standards_for_dimensioning.html

  1. 𝖠 - - | {\displaystyle\Box}\!\!\!\!{\scriptstyle\mathsf{A}}\!-\!\!\!-\!\!\!% \blacktriangleleft\!\!\!|

Indifference_price.html

  1. u u
  2. C T C_{T}
  3. T T
  4. V : × V:\mathbb{R}\times\mathbb{R}\to\mathbb{R}
  5. V ( x , k ) = sup X T 𝒜 ( x ) 𝔼 [ u ( X T + k C T ) ] V(x,k)=\sup_{X_{T}\in\mathcal{A}(x)}\mathbb{E}\left[u\left(X_{T}+kC_{T}\right)\right]
  6. x x
  7. 𝒜 ( x ) \mathcal{A}(x)
  8. T T
  9. x x
  10. k k
  11. v b ( k ) v^{b}(k)
  12. k k
  13. C T C_{T}
  14. V ( x - v b ( k ) , k ) = V ( x , 0 ) V(x-v^{b}(k),k)=V(x,0)
  15. v a ( k ) v^{a}(k)
  16. V ( x + v a ( k ) , - k ) = V ( x , 0 ) V(x+v^{a}(k),-k)=V(x,0)
  17. [ v b ( k ) , v a ( k ) ] \left[v^{b}(k),v^{a}(k)\right]
  18. B B
  19. B 0 = 100 B_{0}=100
  20. B T = 110 B_{T}=110
  21. S S
  22. S 0 = 100 S_{0}=100
  23. S T { 90 , 110 , 130 } S_{T}\in\{90,110,130\}
  24. 1 / 3 1/3
  25. u ( x ) = 1 - exp ( - x / 10 ) u(x)=1-\exp(-x/10)
  26. V ( x , 0 ) V(x,0)
  27. V ( x , 0 ) = max α B 0 + β S 0 = x 𝔼 [ 1 - exp ( - .1 × ( α B T + β S T ) ) ] V(x,0)=\max_{\alpha B_{0}+\beta S_{0}=x}\mathbb{E}[1-\exp(-.1\times(\alpha B_{% T}+\beta S_{T}))]
  28. = max β [ 1 - 1 3 [ exp ( - 1.10 x - 20 β 10 ) + exp ( - 1.10 x 10 ) + exp ( - 1.10 x + 20 β 10 ) ] ] =\max_{\beta}\left[1-\frac{1}{3}\left[\exp\left(-\frac{1.10x-20\beta}{10}% \right)+\exp\left(-\frac{1.10x}{10}\right)+\exp\left(-\frac{1.10x+20\beta}{10}% \right)\right]\right]
  29. β = 0 \beta=0
  30. V ( x , 0 ) = 1 - exp ( - 1.10 x 10 ) V(x,0)=1-\exp\left(-\frac{1.10x}{10}\right)
  31. V ( x - v b ( 1 ) , 1 ) V(x-v^{b}(1),1)
  32. V ( x - v b ( 1 ) , 1 ) = max α B 0 + β S 0 = x - v b ( 1 ) 𝔼 [ 1 - exp ( - .1 × ( α B T + β S T + C T ) ) ] V(x-v^{b}(1),1)=\max_{\alpha B_{0}+\beta S_{0}=x-v^{b}(1)}\mathbb{E}[1-\exp(-.% 1\times(\alpha B_{T}+\beta S_{T}+C_{T}))]
  33. = max β [ 1 - 1 3 [ exp ( - 1.10 ( x - v b ( 1 ) ) - 20 β 10 ) + exp ( - 1.10 ( x - v b ( 1 ) ) 10 ) + exp ( - 1.10 ( x - v b ( 1 ) ) + 20 β + 20 10 ) ] ] =\max_{\beta}\left[1-\frac{1}{3}\left[\exp\left(-\frac{1.10(x-v^{b}(1))-20% \beta}{10}\right)+\exp\left(-\frac{1.10(x-v^{b}(1))}{10}\right)+\exp\left(-% \frac{1.10(x-v^{b}(1))+20\beta+20}{10}\right)\right]\right]
  34. β = - 1 2 \beta=-\frac{1}{2}
  35. V ( x - v b ( 1 ) , 1 ) = 1 - 1 3 exp ( - 1.10 x / 10 ) exp ( 1.10 v b ( 1 ) / 10 ) [ 1 + 2 exp ( - 1 ) ] V(x-v^{b}(1),1)=1-\frac{1}{3}\exp(-1.10x/10)\exp(1.10v^{b}(1)/10)\left[1+2\exp% (-1)\right]
  36. V ( x , 0 ) = V ( x - v b ( 1 ) , 1 ) V(x,0)=V(x-v^{b}(1),1)
  37. v b ( 1 ) = 10 1.1 log ( 3 1 + 2 exp ( - 1 ) ) 4.97 v^{b}(1)=\frac{10}{1.1}\log\left(\frac{3}{1+2\exp(-1)}\right)\approx 4.97
  38. v a ( 1 ) v^{a}(1)
  39. [ v b ( k ) , v a ( k ) ] \left[v^{b}(k),v^{a}(k)\right]
  40. v b ( k ) = - v a ( - k ) v^{b}(k)=-v^{a}(-k)
  41. v ( k ) v(k)
  42. v s u p ( k ) , v s u b ( k ) v^{sup}(k),v^{sub}(k)
  43. v s u b ( k ) v ( k ) v s u p ( k ) v^{sub}(k)\leq v(k)\leq v^{sup}(k)

Induced_character.html

  1. Ind ( f ) \operatorname{Ind}(f)
  2. Ind ( f ) ( s ) = 1 | H | t G , t - 1 s t H f ( t - 1 s t ) . \operatorname{Ind}(f)(s)=\frac{1}{|H|}\sum_{t\in G,\ t^{-1}st\in H}f(t^{-1}st).
  3. Ind ( f ) \operatorname{Ind}(f)

Induction_regulator.html

  1. U s t a t o r U r o t o r = ξ s t a t o r N s t a t o r ξ r o t o r N r o t o r \frac{U_{stator}}{U_{rotor}}=\frac{\xi_{stator}N_{stator}}{\xi_{rotor}N_{rotor}}

Inerter_(mechanical_networks).html

  1. F = b ( v ˙ 2 - v ˙ 1 ) F=b(\dot{v}_{2}-\dot{v}_{1})

Inertial_number.html

  1. I I
  2. I = γ ˙ d P / ρ , I=\frac{\dot{\gamma}d}{\sqrt{P/\rho}},
  3. γ ˙ \dot{\gamma}
  4. d d
  5. P P
  6. ρ \rho
  7. I < 10 - 3 I<10^{-3}
  8. 10 - 3 < I < 10 - 1 10^{-3}<I<10^{-1}
  9. I > 10 - 1 I>10^{-1}

Infineta_Systems.html

  1. Throughput RWIN RTT \mathrm{Throughput}\leq\frac{\mathrm{RWIN}}{\mathrm{RTT}}\,\!

Infinite-dimensional_vector_function.html

  1. f k ( t ) = t / k 2 f_{k}(t)=t/k^{2}
  2. f ( t ) = ( f 1 ( t ) , f 2 ( t ) , f 3 ( t ) , ) f(t)=(f_{1}(t),f_{2}(t),f_{3}(t),\ldots)\,
  3. 𝐑 𝐍 \mathbf{R}^{\mathbf{N}}
  4. f ( 2 ) = ( 2 , 2 4 , 2 9 , 2 16 , 2 25 , ) . f(2)=\left(2,\frac{2}{4},\frac{2}{9},\frac{2}{16},\frac{2}{25},\ldots\right).
  5. A K A\rightarrow K
  6. f : [ 0 , 1 ] X f:[0,1]\rightarrow X
  7. f ( t ) := lim h 0 f ( t + h ) - f ( t ) h f^{\prime}(t):=\lim_{h\rightarrow 0}\frac{f(t+h)-f(t)}{h}
  8. L p L^{p}
  9. f ( t ) = lim h 0 f ( t + h ) - f ( t ) h . f^{\prime}(t)=\lim_{h\rightarrow 0}\frac{f(t+h)-f(t)}{h}.
  10. t R n t\in R^{n}
  11. t Y t\in Y
  12. f = ( f 1 , f 2 , f 3 , ) f=(f_{1},f_{2},f_{3},\ldots)
  13. f = f 1 e 1 + f 2 e 2 + f 3 e 3 + f=f_{1}e_{1}+f_{2}e_{2}+f_{3}e_{3}+\cdots
  14. e 1 , e 2 , e 3 , e_{1},e_{2},e_{3},\ldots
  15. f ( t ) f^{\prime}(t)
  16. f ( t ) = ( f 1 ( t ) , f 2 ( t ) , f 3 ( t ) , ) f^{\prime}(t)=(f_{1}^{\prime}(t),f_{2}^{\prime}(t),f_{3}^{\prime}(t),\ldots)

Infinite_compositions_of_analytic_functions.html

  1. lim n F 1 , n ( z ) , lim n G 1 , n ( z ) . \lim_{n\to\infty}F_{1,n}(z),\qquad\lim_{n\to\infty}G_{1,n}(z).
  2. S ¯ \overline{S}
  3. S ¯ \overline{S}
  4. S ¯ \overline{S}
  5. F n ( z ) = ( f f f ) ( z ) α , F_{n}(z)=(f\circ f\circ\cdots\circ f)(z)\to\alpha,
  6. f n ( z ) \displaystyle f_{n}(z)
  7. n = 1 ρ n < \sum_{n=1}^{\infty}\rho_{n}<\infty
  8. n = 1 ε n \displaystyle\sum_{n=1}^{\infty}\varepsilon_{n}
  9. C n = 1 β n < M , C\sum_{n=1}^{\infty}\beta_{n}<M,
  10. | f n ( z ) - z | < C β n , z S . \left|f_{n}(z)-z\right|<C\beta_{n},\qquad z\in S.
  11. R = M - C n = 1 β n > 0 R=M-C\sum_{n=1}^{\infty}\beta_{n}>0
  12. n = 1 β n < . \sum_{n=1}^{\infty}\beta_{n}<\infty.
  13. R = R ( r ) = R 0 - r n = 1 ( 1 + C β n ) . R=R(r)=\frac{R_{0}-r}{\prod_{n=1}^{\infty}\left(1+C\beta_{n}\right)}.
  14. | F ( z ) | n = 1 ( 1 + R 0 r C β n ) \left|F^{\prime}(z)\right|\leq\prod_{n=1}^{\infty}{\left(1+\tfrac{R_{0}}{r}C% \beta_{n}\right)}
  15. z β = lim n β n z\neq\beta=\lim_{n\to\infty}\beta_{n}
  16. n = 1 | γ n - β n | \displaystyle\sum_{n=1}^{\infty}\left|\gamma_{n}-\beta_{n}\right|
  17. a 1 b 1 + a 2 b 2 + \frac{a_{1}}{b_{1}+\frac{a_{2}}{b_{2}+\ldots}}
  18. f n ( z ) = a n b n + z . f_{n}(z)=\frac{a_{n}}{b_{n}+z}.
  19. a 1 ζ 1 + a 2 ζ 1 + \frac{a_{1}\zeta}{1+\frac{a_{2}\zeta}{1+\ldots}}
  20. f n ( z ) = a n ζ 1 + z . f_{n}(z)=\frac{a_{n}\zeta}{1+z}.
  21. φ ( t z ) = t ( φ ( z ) + φ ( z ) 2 ) \varphi(tz)=t\left(\varphi(z)+\varphi(z)^{2}\right)
  22. f n ( z ) = z + z 2 t n F n ( z ) φ ( z ) f_{n}(z)=z+\frac{z^{2}}{t^{n}}\Rightarrow F_{n}(z)\to\varphi(z)
  23. f n ( z ) = z + z 2 2 n F n ( z ) 1 2 ( e 2 z - 1 ) f_{n}(z)=z+\frac{z^{2}}{2^{n}}\Rightarrow F_{n}(z)\to\tfrac{1}{2}\left(e^{2z}-% 1\right)
  24. f n ( z ) = z / ( 1 - 1 4 n z 2 ) F n ( z ) tan ( z ) {{f}_{n}}(z)=z/\left(1-\tfrac{1}{{{4}^{n}}}{{z}^{2}}\right)\,\text{ }% \Rightarrow\,\text{ }{{F}_{n}}(z)\to\tan(z)
  25. g n ( z ) = 2 4 n z ( 1 + 1 4 n z 2 - 1 ) G n ( z ) arctan ( z ) {{g}_{n}}(z)=\frac{2\cdot{{4}^{n}}}{z}\left(\sqrt{1+\tfrac{1}{{{4}^{n}}}{{z}^{% 2}}}-1\right)\,\text{ }\Rightarrow\,\text{ }{{G}_{n}}(z)\to\arctan(z)
  26. G ( ζ ) = e ζ 4 3 + ζ + e ζ 8 3 + ζ + e ζ 12 3 + ζ + G(\zeta)=\frac{\tfrac{e^{\zeta}}{4}}{3+\zeta+\frac{\tfrac{e^{\zeta}}{8}}{3+% \zeta+\frac{\tfrac{e^{\zeta}}{12}}{3+\zeta+\ldots}}}
  27. t n ( z ) \displaystyle t_{n}(z)
  28. G n ( ζ ) = f n f 1 ( ζ ) G_{n}(\zeta)=f_{n}\circ\cdots\circ f_{1}(\zeta)
  29. lim n G n ( ζ ) = α \lim_{n\to\infty}G_{n}(\zeta)=\alpha
  30. g k , n ( z ) = z + φ k , n ( z ) g_{k,n}(z)=z+\varphi_{k,n}(z)
  31. lim n φ k , n ( z ) = 0 \lim_{n\to\infty}\varphi_{k,n}(z)=0
  32. g k , n ( z ) S g_{k,n}(z)\in S
  33. g k , n ( z ) = z + k n 2 f ( z ) . g_{k,n}(z)=z+\frac{k}{n^{2}}f(z).
  34. T 1 , n ( z ) = g 1 , n ( z ) T_{1,n}(z)=g_{1,n}(z)
  35. T k , n ( z ) = g k , n ( T k - 1 , n ( z ) ) T_{k,n}(z)=g_{k,n}\left(T_{k-1,n}(z)\right)
  36. lim n T n , n ( z ) = T ( z ) \lim_{n\to\infty}T_{n,n}(z)=T(z)
  37. g k , n ( z ) z g_{k,n}(z)\approx z
  38. T n , n ( z ) e α 2 z + b β T_{n,n}(z)\to e^{\frac{\alpha}{2}}z+b\beta
  39. lim n k = 1 n ( 1 + 2 k n 2 z ) = e z , z 𝐂 . \lim_{n\to\infty}\prod_{k=1}^{n}\left(1+\frac{2k}{n^{2}}z\right)=e^{z},\qquad z% \in\mathbf{C}.
  40. T ( z ) = z + c 0 1 t d t . T(z)=z+c\int_{0}^{1}tdt.
  41. g n ( z ) = z + c n n φ ( z ) , g_{n}(z)=z+\frac{c_{n}}{n}\varphi(z),
  42. T 1 , n ( z ) = g n ( z ) , T k , n ( z ) = g n ( T k - 1 , n ( z ) ) T_{1,n}(z)=g_{n}(z),T_{k,n}(z)=g_{n}\left(T_{k-1,n}(z)\right)
  43. T ( z ) = lim n T n ( z ) T(z)=\lim_{n\to\infty}T_{n}(z)
  44. c n = n c_{n}=\sqrt{n}
  45. γ φ ( ζ ) d ζ = lim n c n k = 1 n φ 2 ( T k - 1 , n ( z ) ) \oint_{\gamma}\varphi(\zeta)d\zeta=\lim_{n\to\infty}\frac{c}{n}\sum_{k=1}^{n}% \varphi^{2}\left(T_{k-1,n}(z)\right)
  46. L ( γ ( z ) ) = lim n c n k = 1 n | φ ( T k - 1 , n ( z ) ) | , L(\gamma(z))=\lim_{n\to\infty}\frac{c}{n}\sum_{k=1}^{n}\left|\varphi\left(T_{k% -1,n}(z)\right)\right|,
  47. g n ( G n - 1 ( z ) ) g_{n}(G_{n-1}(z))
  48. | z | < R = M - C k = 1 β k > 0 |z|<R=M-C\sum_{k=1}^{\infty}\beta_{k}>0
  49. f n ( z ) = z + 1 ρ n 2 z , ρ > π 6 f_{n}(z)=z+\frac{1}{\rho n^{2}}\sqrt{z},\qquad\rho>\sqrt{\frac{\pi}{6}}
  50. S = { z : | z | < R , Re ( z ) > 0 } S=\left\{z:|z|<R,\operatorname{Re}(z)>0\right\}
  51. f n ( z ) = z ( 1 + g n ( z ) ) f_{n}(z)=z\left(1+g_{n}(z)\right)
  52. G n ( z ) = z k = 1 n ( 1 + g k ( G k - 1 ( z ) ) ) . G_{n}(z)=z\prod_{k=1}^{n}\left(1+g_{k}\left(G_{k-1}(z)\right)\right).
  53. | z g n ( z ) | C β n \left|z\cdot g_{n}(z)\right|\leq C\beta_{n}
  54. k = 1 β k < . \sum_{k=1}^{\infty}\beta_{k}<\infty.
  55. | G n - 1 ( z ) g n ( G n - 1 ( z ) ) | C β n . \left|G_{n-1}(z)\cdot g_{n}(G_{n-1}(z))\right|\leq C\beta_{n}.
  56. n = 1 ( 1 + C β n ) = P \prod_{n=1}^{\infty}\left(1+C\beta_{n}\right)=P
  57. f n ( z ) = z ( 1 + g n ( z ) ) f_{n}(z)=z(1+g_{n}(z))
  58. g n ( z ) = z 2 n 3 g_{n}(z)=\frac{z^{2}}{n^{3}}
  59. G n ( z ) = z k = 1 n - 1 ( 1 + G k ( z ) 2 n 3 ) G_{n}(z)=z\cdot\prod_{k=1}^{n-1}\left(1+\frac{G_{k}(z)^{2}}{n^{3}}\right)
  60. F < s u b > n ( z ) = f n f 1 ( z ) F<sub>n(z)=f_{n}∘...∘f_{1}(z)

Infinite_expression_(mathematics).html

  1. n = 0 a n = a 0 + a 1 + a 2 + ; \sum_{n=0}^{\infty}a_{n}=a_{0}+a_{1}+a_{2}+\cdots\,;
  2. n = 0 b n = b 0 × b 1 × b 2 × \prod_{n=0}^{\infty}b_{n}=b_{0}\times b_{1}\times b_{2}\times\cdots\,
  3. 1 + 2 1 + 3 1 + \sqrt{1+2\sqrt{1+3\sqrt{1+\cdots}}}\,
  4. 2 2 2 \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}}\,
  5. c 0 + K n = 1 1 c n = c 0 + 1 c 1 + 1 c 2 + 1 c 3 + 1 c 4 + c_{0}+\underset{n=1}{\overset{\infty}{\mathrm{K}}}\,\frac{1}{c_{n}}=c_{0}+% \cfrac{1}{c_{1}+\cfrac{1}{c_{2}+\cfrac{1}{c_{3}+\cfrac{1}{c_{4}+\ddots}}}}\,
  6. E E_{\infty}
  7. E n : n \langle E_{n}:n\in\mathbb{N}\rangle
  8. n = H n=H
  9. E = st ( E H ) E_{\infty}=\,\text{st}(E_{H})

Infinity_Laplacian.html

  1. L L^{\infty}
  2. Δ \Delta_{\infty}
  3. Δ u ( x ) = D u , D 2 u D u = i , j 2 u x i x j u x i u x j \Delta_{\infty}u(x)=\langle Du,D^{2}u\,Du\rangle=\sum_{i,j}\frac{\partial^{2}u% }{\partial x_{i}\,\partial x_{j}}\frac{\partial u}{\partial x_{i}}\frac{% \partial u}{\partial x_{j}}
  4. Δ u ( x ) = D u , D 2 u D u | D u | 2 = 1 | D u | 2 i , j 2 u x i x j u x i u x j . \Delta_{\infty}u(x)=\frac{\langle Du,D^{2}u\,Du\rangle}{|Du|^{2}}=\frac{1}{|Du% |^{2}}\sum_{i,j}\frac{\partial^{2}u}{\partial x_{i}\,\partial x_{j}}\frac{% \partial u}{\partial x_{i}}\frac{\partial u}{\partial x_{j}}.
  5. Δ u = 0 \Delta_{\infty}u=0
  6. u ( x , y ) = | x | 4 / 3 - | y | 4 / 3 u(x,y)=|x|^{4/3}-|y|^{4/3}
  7. Δ u = 0 \Delta_{\infty}u=0
  8. D u L \|Du\|_{L^{\infty}}
  9. p p\rightarrow\infty
  10. L 2 L^{2}
  11. f f
  12. G ( V , E ) G(V,E)
  13. U V U\subseteq V
  14. f ( x ) = y x f ( y ) f(x)=\sum_{y\sim x}f(y)
  15. x U x\in U
  16. f ( x ) = sup { f ( y ) : y x } + inf { f ( y ) : y x } 2 f(x)=\frac{\sup\{f(y):y\sim x\}+\inf\{f(y):y\sim x\}}{2}
  17. G ( V , E ) G(V,E)
  18. V ( G ) V(G)
  19. \R d \R^{d}
  20. ( x , y ) E ( G ) (x,y)\in E(G)
  21. ϵ \epsilon
  22. D \R d D\subseteq\R^{d}
  23. D \partial D
  24. f : D \R f:\partial D\longrightarrow\R
  25. L 2 L^{2}
  26. \Z ϵ d \Z_{\epsilon}^{d}
  27. ϵ \epsilon
  28. G ϵ ( V , E ) = D \Z ϵ d G_{\epsilon}(V,E)=D\cap\Z_{\epsilon}^{d}
  29. V V \partial V\subset V
  30. f ϵ : V \R f_{\epsilon}:\partial V\longrightarrow\R
  31. f ϵ f_{\epsilon}
  32. L L^{\infty}
  33. ϵ \epsilon
  34. L 2 L^{2}
  35. f ϵ f_{\epsilon}
  36. V \partial V
  37. V V
  38. f ϵ ( x ) = 𝔼 [ f ϵ ( X τ ) | X 0 = x ] f_{\epsilon}(x)=\mathbb{E}\big[f_{\epsilon}(X_{\tau})\,|\,X_{0}=x\big]
  39. X 0 , X 1 , X_{0},X_{1},\dots
  40. G ϵ ( V , E ) G_{\epsilon}(V,E)
  41. X 0 = x X_{0}=x
  42. τ \tau
  43. V \partial V
  44. L L^{\infty}
  45. x V ( G ) x\in V(G)
  46. f : V \R f:\partial V\longrightarrow\R
  47. V \partial V
  48. τ \tau
  49. X τ X_{\tau}
  50. f ( X τ ) f(X_{\tau})
  51. f ( X τ ) f(X_{\tau})
  52. 𝔼 [ f ( X τ ) | X 0 = x ] \mathbb{E}[f(X_{\tau})\,|\,X_{0}=x]

Inflation-restriction_exact_sequence.html

  1. \in
  2. \in

Information_projection.html

  1. p * = arg min p P D KL ( p | | q ) p^{*}=\underset{p\in P}{\arg\min}\operatorname{D}_{\mathrm{KL}}(p||q)
  2. D KL D_{\mathrm{KL}}
  3. p * p^{*}

Inscribed_square_problem.html

  1. 1 + 2 1+\sqrt{2}

Integer_broom_topology.html

  1. ( n , θ ) { n \Z : n 0 } × { θ = 1 / k : k \Z , k 1 } . (n,\theta)\in\{n\in\Z:n\geq 0\}\times\{\theta=1/k:k\in\Z,\ k\geq 1\}\,.

Integer_circuit.html

  1. n n
  2. n n
  3. {}_{\mathbb{Z}}
  4. {}_{\mathbb{Z}}
  5. \land\oplus
  6. \land\oplus

Integer_sequence_prime.html

  1. a n = n 2 n + 1 . a_{n}=n2^{n}+1\,.
  2. a n = n ! - 1 a_{n}=n!-1
  3. b n = n ! + 1 . b_{n}=n!+1\,.
  4. a n = 2 2 n + 1 . a_{n}=2^{2^{n}}+1\,.
  5. a n = 2 n - 1 . a_{n}=2^{n}-1\,.
  6. a n = n # - 1 a_{n}=n\#-1
  7. b n = n # + 1 . b_{n}=n\#+1\,.
  8. a n = 4 n + 1 . a_{n}=4n+1\,.
  9. a n = n 2 n - 1 . a_{n}=n2^{n}-1\,.

Integer_triangle.html

  1. ( c / + 1 ) ({c}/{+ 1)}
  2. ( c / + 1 ) ({c}/{+ 1)}
  3. c / 2 {c}/{2}
  4. c / 2 {c}/{2}
  5. 4 T = ( a + b + c ) ( a + b - c ) ( a - b + c ) ( - a + b + c ) . 4T=\sqrt{(a+b+c)(a+b-c)(a-b+c)(-a+b+c)}.
  6. π p / q {πp}/{q}
  7. 1 / 3 {1}/{3}
  8. 2 b c s ( s - a ) b + c \tfrac{2\sqrt{bcs(s-a)}}{b+c}
  9. ( 2 b 2 + 2 c 2 - a 2 ) 4 \tfrac{(2b^{2}+2c^{2}-a^{2})}{4}
  10. 4 T 2 s a b c \tfrac{4T^{2}}{sabc}
  11. a b c 2 ( a + b + c ) . \tfrac{abc}{2(a+b+c)}.
  12. a = n ( m 2 + k 2 ) a=n(m^{2}+k^{2})\,
  13. b = m ( n 2 + k 2 ) b=m(n^{2}+k^{2})\,
  14. c = ( m + n ) ( m n - k 2 ) c=(m+n)(mn-k^{2})\,
  15. Semiperimeter = m n ( m + n ) \,\text{Semiperimeter}=mn(m+n)\,
  16. Area = m n k ( m + n ) ( m n - k 2 ) \,\text{Area}=mnk(m+n)(mn-k^{2})\,
  17. gcd ( m , n , k ) = 1 \gcd{(m,n,k)}=1\,
  18. m n > k 2 m 2 n / ( 2 m + n ) mn>k^{2}\geq m^{2}n/(2m+n)\,
  19. m n 1 m\geq n\geq 1\,
  20. p q \frac{p}{q}
  21. q = gcd ( a , b , c ) q=\gcd{(a,b,c)}
  22. p p
  23. ( a , b , c ) (a,b,c)
  24. c c
  25. a = m 2 - n 2 , a=m^{2}-n^{2},\,
  26. b = 2 m n , b=2mn,\,
  27. c = m 2 + n 2 , c=m^{2}+n^{2},\,
  28. Semiperimeter = m ( m + n ) \,\text{Semiperimeter}=m(m+n)\,
  29. Area = m n ( m 2 - n 2 ) \,\text{Area}=mn(m^{2}-n^{2})\,
  30. ( a , b , c ) = ( 3 , 4 , 5 ) (a,b,c)=(3,4,5)
  31. ( a , b , c , d ) = ( x z , y z , z 2 , x y ) . (a,b,c,d)=(xz,yz,z^{2},xy).\,
  32. 1 a 2 + 1 b 2 = 1 d 2 \tfrac{1}{a^{2}}+\tfrac{1}{b^{2}}=\tfrac{1}{d^{2}}
  33. a = ( m 2 - n 2 ) ( m 2 + n 2 ) , a=(m^{2}-n^{2})(m^{2}+n^{2}),\,
  34. b = 2 m n ( m 2 + n 2 ) , b=2mn(m^{2}+n^{2}),\,
  35. c = ( m 2 + n 2 ) 2 , c=(m^{2}+n^{2})^{2},\,
  36. d = 2 m n ( m 2 - n 2 ) , d=2mn(m^{2}-n^{2}),\,
  37. Semiperimeter = m ( m + n ) ( m 2 + n 2 ) \,\text{Semiperimeter}=m(m+n)(m^{2}+n^{2})\,
  38. Area = m n ( m 2 - n 2 ) ( m 2 + n 2 ) 2 \,\text{Area}=mn(m^{2}-n^{2})(m^{2}+n^{2})^{2}\,
  39. b = 2 ( m 2 + 3 n 2 ) / g , b=2(m^{2}+3n^{2})/g,\,
  40. d = ( m 2 - 3 n 2 ) / g , d=(m^{2}-3n^{2})/g,\,
  41. m 2 - 3 n 2 , m^{2}-3n^{2},
  42. 2 m n 2mn
  43. m 2 + 3 n 2 . m^{2}+3n^{2}.
  44. a = k 2 ( s 2 + r 2 ) 2 4 , a=\tfrac{k^{2}(s^{2}+r^{2})^{2}}{4},\,
  45. b = k 2 ( s 4 - r 4 ) 2 , b=\tfrac{k^{2}(s^{4}-r^{4})}{2},\,
  46. c = k 2 ( 3 s 4 - 10 s 2 r 2 + 3 r 4 ) 4 c=\tfrac{k^{2}(3s^{4}-10s^{2}r^{2}+3r^{4})}{4}\,
  47. Area = k 2 c s r ( s 2 - r 2 ) 2 \,\text{Area}=\tfrac{k^{2}csr(s^{2}-r^{2})}{2}\,
  48. a = q 2 ( u 2 + v 2 ) 2 4 a=\tfrac{q^{2}(u^{2}+v^{2})^{2}}{4}\,
  49. b = q 2 u v ( u 2 + v 2 ) b=q^{2}uv(u^{2}+v^{2})\,
  50. c = q 2 ( 14 u 2 v 2 - u 4 - v 4 ) 4 c=\tfrac{q^{2}(14u^{2}v^{2}-u^{4}-v^{4})}{4}\,
  51. Area = q 2 c u v ( v 2 - u 2 ) 2 \,\text{Area}=\tfrac{q^{2}cuv(v^{2}-u^{2})}{2}\,
  52. a = 2 ( u 2 - v 2 ) , a=2(u^{2}-v^{2}),
  53. b = u 2 + v 2 , b=u^{2}+v^{2},
  54. c = u 2 + v 2 , c=u^{2}+v^{2},
  55. p a = 2 a T a 2 + b 2 - c 2 , p_{a}=\tfrac{2aT}{a^{2}+b^{2}-c^{2}},
  56. p b = 2 b T a 2 + b 2 - c 2 , p_{b}=\tfrac{2bT}{a^{2}+b^{2}-c^{2}},
  57. p c = 2 c T a 2 - b 2 + c 2 , p_{c}=\tfrac{2cT}{a^{2}-b^{2}+c^{2}},
  58. 2 T a a 2 + 2 T \tfrac{2Ta}{a^{2}+2T}
  59. a , b , c a,b,c
  60. d d
  61. a = 2 ( k 2 - m 2 ) , a=2(k^{2}-m^{2}),\,
  62. b = ( k - m ) 2 , b=(k-m)^{2},\,
  63. c = ( k + m ) 2 , c=(k+m)^{2},\,
  64. d = 2 k m ( k 2 - m 2 ) k 2 + m 2 , d=\tfrac{2km(k^{2}-m^{2})}{k^{2}+m^{2}},\,
  65. k > m > 0 k>m>0
  66. a = p 2 - 2 p q h + q 2 k 2 , a=p^{2}-2pqh+q^{2}k^{2},
  67. b = p 2 - q 2 k 2 , b=p^{2}-q^{2}k^{2},
  68. c = 2 q k ( p - q h ) , c=2qk(p-qh),
  69. a = 4 m n a=4mn\,
  70. b = 3 m 2 + n 2 b=3m^{2}+n^{2}\,
  71. c = 2 m n + | 3 m 2 - n 2 | c=2mn+|3m^{2}-n^{2}|\,
  72. a = m 2 - m n + n 2 a=m^{2}-mn+n^{2}\,
  73. b = 2 m n - n 2 b=2mn-n^{2}\,
  74. c = m 2 - n 2 c=m^{2}-n^{2}\,
  75. a = m 2 + m n + n 2 , a=m^{2}+mn+n^{2},\,
  76. b = 2 m n + n 2 , b=2mn+n^{2},\,
  77. c = m 2 - n 2 c=m^{2}-n^{2}\,
  78. h α h\alpha
  79. k α k\alpha
  80. π - ( h + k ) α \pi-(h+k)\alpha
  81. a = q h + k - 1 sin h α sin α = q k 0 i h - 1 2 ( - 1 ) i ( h 2 i + 1 ) p h - 2 i - 1 ( q 2 - p 2 ) i , a=q^{h+k-1}\frac{\sin h\alpha}{\sin\alpha}=q^{k}\cdot\sum_{0\leq i\leq\frac{h-% 1}{2}}(-1)^{i}{\left({{h}\atop{2i+1}}\right)}p^{h-2i-1}(q^{2}-p^{2})^{i},
  82. b = q h + k - 1 sin k α sin α = q h 0 i k - 1 2 ( - 1 ) i ( k 2 i + 1 ) p k - 2 i - 1 ( q 2 - p 2 ) i , b=q^{h+k-1}\frac{\sin k\alpha}{\sin\alpha}=q^{h}\cdot\sum_{0\leq i\leq\frac{k-% 1}{2}}(-1)^{i}{\left({{k}\atop{2i+1}}\right)}p^{k-2i-1}(q^{2}-p^{2})^{i},
  83. c = q h + k - 1 sin ( h + k ) α sin α = 0 i h + k - 1 2 ( - 1 ) i ( h + k 2 i + 1 ) p h + k - 2 i - 1 ( q 2 - p 2 ) i , c=q^{h+k-1}\frac{\sin(h+k)\alpha}{\sin\alpha}=\sum_{0\leq i\leq\frac{h+k-1}{2}% }(-1)^{i}{\left({{h+k}\atop{2i+1}}\right)}p^{h+k-2i-1}(q^{2}-p^{2})^{i},
  84. α = cos - 1 p q \alpha=\cos^{-1}\frac{p}{q}
  85. cos π h + k < p q < 1 \cos\frac{\pi}{h+k}<\frac{p}{q}<1
  86. a a
  87. b b
  88. a = n 2 , a=n^{2},\,
  89. b = m n b=mn\,
  90. c = m 2 - n 2 , c=m^{2}-n^{2},\,
  91. a ( a + c ) = b 2 a(a+c)=b^{2}
  92. B = 3 2 A \ B=\tfrac{3}{2}A
  93. a = m n 3 , a=mn^{3},\,
  94. b = n 2 ( m 2 - n 2 ) , b=n^{2}(m^{2}-n^{2}),\,
  95. c = ( m 2 - n 2 ) 2 - m 2 n 2 , c=(m^{2}-n^{2})^{2}-m^{2}n^{2},\,
  96. m , n \ m,n
  97. 0 < φ n < m < 2 n \ 0<\varphi n<m<2n
  98. φ \ \varphi
  99. φ = 1 + 5 2 1.61803 \varphi=\frac{1+\sqrt{5}}{2}\approx 1.61803
  100. B = 3 2 A \ B=\tfrac{3}{2}A
  101. ( b 2 - a 2 ) ( b 2 - a 2 + b c ) = a 2 c 2 \ (b^{2}-a^{2})(b^{2}-a^{2}+bc)=a^{2}c^{2}
  102. a = n 3 , a=n^{3},\,
  103. b = n ( m 2 - n 2 ) , b=n(m^{2}-n^{2}),\,
  104. c = m ( m 2 - 2 n 2 ) , c=m(m^{2}-2n^{2}),\,
  105. m m
  106. n n
  107. 2 n < m < 2 n \sqrt{2}n<m<2n
  108. a c 2 = ( b - a ) 2 ( b + a ) ac^{2}=(b-a)^{2}(b+a)

Integral_of_the_secant_function.html

  1. 0 θ sec ζ d ζ = ln | tan ( θ 2 + π 4 ) | . \int_{0}^{\theta}\sec\zeta\,d\zeta=\ln\left|\tan\left(\frac{\theta}{2}+\frac{% \pi}{4}\right)\right|.
  2. sec θ d θ = d θ cos θ = cos θ d θ cos 2 θ = cos θ d θ 1 - sin 2 θ \int\sec\theta\,d\theta=\int\frac{d\theta}{\cos\theta}=\int\frac{\cos\theta\,d% \theta}{\cos^{2}\theta}=\int\frac{\cos\theta\,d\theta}{1-\sin^{2}\theta}
  3. u u
  4. sin θ \sin\theta
  5. d u 1 - u 2 \displaystyle\int\frac{du}{1-u^{2}}
  6. sec θ d θ = { 1 2 ln | 1 + sin θ 1 - sin θ | + C ln | sec θ + tan θ | + C ln | tan ( θ 2 + π 4 ) | + C } (equivalent forms) \int\sec\theta\,d\theta=\left\{\begin{array}[]{l}\dfrac{1}{2}\ln\left|\dfrac{1% +\sin\theta}{1-\sin\theta}\right|+C\\ \ln\left|\sec\theta+\tan\theta\right|+C\\ \ln\left|\tan\left(\dfrac{\theta}{2}+\dfrac{\pi}{4}\right)\right|+C\end{array}% \right\}\,\text{ (equivalent forms)}
  7. ( 1 + sin θ ) (1+\sin\theta)
  8. cos 2 θ \cos^{2}\theta
  9. sin θ \sin\theta
  10. - cos ( θ + π / 2 ) -\cos(\theta+\pi/2)
  11. cos 2 x \cos 2x
  12. sec θ = 1 sin ( θ + π 2 ) = 1 2 sin ( θ 2 + π 4 ) cos ( θ 2 + π 4 ) = sec 2 ( θ 2 + π 4 ) 2 tan ( θ 2 + π 4 ) . \displaystyle\sec\theta=\frac{1}{\sin\left(\theta+\dfrac{\pi}{2}\right)}=\frac% {1}{2\sin\left(\dfrac{\theta}{2}+\dfrac{\pi}{4}\right)\cos\left(\dfrac{\theta}% {2}+\dfrac{\pi}{4}\right)}=\frac{\sec^{2}\left(\dfrac{\theta}{2}+\dfrac{\pi}{4% }\right)}{2\tan\left(\dfrac{\theta}{2}+\dfrac{\pi}{4}\right)}.
  13. y = ln tan ( ϕ 2 + π 4 ) . y=\ln\tan\!\left(\dfrac{\phi}{2}+\dfrac{\pi}{4}\right).
  14. ψ = ln ( sec θ + tan θ ) , e ψ = sec θ + tan θ , sinh ψ = 1 2 ( e ψ - e - ψ ) = tan θ , cosh ψ = 1 + sinh 2 ψ = sec θ , tanh ψ = sin θ . \begin{aligned}\displaystyle\psi&\displaystyle=\ln(\sec\theta+\tan\theta),\\ \displaystyle{\rm e}^{\psi}&\displaystyle=\sec\theta+\tan\theta,\\ \displaystyle\sinh\psi&\displaystyle=\frac{1}{2}({\rm e}^{\psi}-{\rm e}^{-\psi% })=\tan\theta,\\ \displaystyle\cosh\psi&\displaystyle=\sqrt{1+\sinh^{2}\psi}=\sec\theta,\\ \displaystyle\tanh\psi&\displaystyle=\sin\theta.\end{aligned}
  15. sec θ d θ = ψ = tanh - 1 ( sin θ ) = sinh - 1 ( tan θ ) = cosh - 1 ( sec θ ) . \begin{aligned}\displaystyle\int\sec\theta\,d\theta&\displaystyle=\psi=\tanh^{% -1}\!\left(\sin\theta\right)=\sinh^{-1}\!\left(\tan\theta\right)=\cosh^{-1}\!% \left(\sec\theta\right).\end{aligned}
  16. sec θ d θ \displaystyle\int\sec\theta\,d\theta
  17. y = lam ( ϕ ) . y=\mbox{lam}~{}(\phi).

Integrated_information_theory.html

  1. e i ( X ( m e c h , x 1 ) ) = H [ p ( X 0 ( m e c h , x 1 ) ) p ( X 0 ( m a x H ) ) ] ei(X(mech,x_{1}))=H[p(X_{0}(mech,x_{1}))\parallel p(X_{0}(maxH))]
  2. x 1 x_{1}
  3. p ( X 0 ( m a x H ) ) p(X_{0}(maxH))
  4. Φ ( X ( m e c h , x 1 ) ) = H [ p ( X 0 ( m e c h , x 1 ) ) Π p ( k M 0 ( m e c h , μ 1 ) ) ] \Phi(X(mech,x_{1}))=H[p(X_{0}(mech,x_{1}))\parallel\Pi p(^{k}M_{0}(mech,\mu_{1% }))]
  5. M 0 k M I P {}^{k}M_{0}\in MIP
  6. x 1 x_{1}
  7. Π ( p ( k M 0 ( m e c h , μ 1 ) ) ) \Pi(p(^{k}M_{0}(mech,\mu_{1})))
  8. Φ c = H [ p ( X ( m e c h , x ) ) p ( Y ( m e c h , y ) ) ] \Phi_{c}=H[p(X(mech,x))\parallel p(Y(mech,y))]
  9. Φ c \Phi_{c}

Interlocking_interval_topology.html

  1. X n := ( 0 , 1 n ) ( n , n + 1 ) = { x R + : 0 < x < 1 n or n < x < n + 1 } . X_{n}:=\left(0,\frac{1}{n}\right)\cup(n,n+1)=\left\{x\in{R}^{+}:0<x<\frac{1}{n% }\ \,\text{ or }\ n<x<n+1\right\}.

Internal_category.html

  1. C C
  2. C C
  3. C C
  4. C 0 , C 1 C_{0},C_{1}
  5. C C
  6. d 0 , d 1 : C 1 C 0 , e : C 0 C 1 , m : C 1 × C 0 C 1 C 1 d_{0},d_{1}:C_{1}\rightarrow C_{0},e:C_{0}\rightarrow C_{1},m:C_{1}\times_{C_{% 0}}C_{1}\rightarrow C_{1}

Intravoxel_incoherent_motion.html

  1. S S 0 = f I V I M F p e r f + ( 1 - f I V I M ) F d i f f \frac{S}{S_{0}}=f_{IVIM}F_{perf}+(1-f_{IVIM})F_{diff}\,
  2. f I V I M f_{IVIM}
  3. F p e r f F_{perf}
  4. F d i f f F_{diff}
  5. F p e r f F_{perf}
  6. F p e r f = e x p ( - b . D * ) F_{perf}=exp(-b.D^{*})\,
  7. b b
  8. D * D^{*}
  9. D b l o o d D_{blood}
  10. D * = L . v b l o o d / 6 + D b l o o d D^{*}=L.v_{blood}/6+D_{blood}\,
  11. L L
  12. v b l o o d vblood
  13. F p e r f F_{perf}
  14. F p e r f = s i n c ( v b l o o d c / π ) ( 1 - v b l o o d c / 6 ) F_{perf}=sinc(v_{blood}c/\pi)\approx(1-v_{blood}c/6)\,
  15. c c
  16. A D C D + f I V I M / b ADC\approx D+f_{IVIM}/b\,
  17. D D
  18. D * D*
  19. F d i f f = f s l o w e x p ( - b D s l o w ) + f f a s t e x p ( - b D f a s t ) F_{diff}=f_{slow}exp(-bD_{slow})+f_{fast}exp(-bD_{fast})\,
  20. f f a s t , s l o w f_{fast,slow}
  21. D f a s t , s l o w D_{fast,slow}
  22. F d i f f = e x p ( - b D i n t + K ( b D i n t ) 2 / 6 ) F_{diff}=exp(-bD_{int}+K(bD_{int})^{2}/6)\,
  23. D i n t Dint
  24. K K

Invariant_extended_Kalman_filter.html

  1. x ˙ = f ( x , u ) + M ( x ) w \dot{x}{=}f(x,u)+M(x)w
  2. y = h ( x , u ) + N ( x ) v y{=}h(x,u)+N(x)v
  3. w , v w,v
  4. G G
  5. e e
  6. φ g , ψ g , ρ g \varphi_{g},\psi_{g},\rho_{g}
  7. g G g\in G
  8. ( X , U , Y ) = ( φ g ( x ) , ψ g ( u ) , ρ g ( y ) ) (X,U,Y)=(\varphi_{g}(x),\psi_{g}(u),\rho_{g}(y))
  9. φ g , ψ g , ρ g \varphi_{g},\psi_{g},\rho_{g}
  10. X ˙ = f ( X , U ) + M ( X ) w \dot{X}{=}f(X,U)+M(X)w
  11. Y = h ( X , U ) + N ( X ) v Y{=}h(X,U)+N(X)v
  12. x ^ ˙ = f ( x ^ , u ) + W ( x ^ ) L ( I ( x ^ , u ) , E ( x ^ , u , y ) ) E ( x ^ , u , y ) \dot{\hat{x}}=f(\hat{x},u)+W(\hat{x})L\Bigl(I(\hat{x},u),E(\hat{x},u,y)\Bigr)E% (\hat{x},u,y)
  13. E ( x ^ , u , y ) E(\hat{x},u,y)
  14. y ^ - y \hat{y}-y
  15. W ( x ^ ) = ( w 1 ( x ^ ) , . . , w n ( x ^ ) ) W(\hat{x})=\bigl(w_{1}(\hat{x}),..,w_{n}(\hat{x})\bigr)
  16. I ( x ^ , u ) I(\hat{x},u)
  17. L ( I , E ) L(I,E)
  18. η ( x ^ , x ) \eta(\hat{x},x)
  19. x ^ - x \hat{x}-x
  20. E ( x ^ , u , y ) , W ( x ^ ) , I ( x ^ , u ) , η ( x ^ , x ) E(\hat{x},u,y),W(\hat{x}),I(\hat{x},u),\eta(\hat{x},x)
  21. L ( I , E ) L(I,E)
  22. L = P C T ( N ( e ) N T ( e ) ) - 1 L{=}PC^{T}\bigl(N(e)N^{T}(e)\bigr)^{-1}
  23. P ˙ = A P + P A T + M ( e ) M T ( e ) - P C T ( N ( e ) N T ( e ) ) - 1 C P \dot{P}{=}AP+PA^{T}+M(e)M^{T}(e)-PC^{T}\bigl(N(e)N^{T}(e)\bigr)^{-1}CP
  24. A , C A,C
  25. I ( x ^ , u ) I(\hat{x},u)
  26. ( x ^ , u ) (\hat{x},u)
  27. A , C A,C
  28. q ^ q - 1 \hat{q}q^{-1}

Invariant_manifold.html

  1. d x / d t = f ( x ) , x n , dx/dt=f(x),\ x\in\mathbb{R}^{n},
  2. x ( t ) = ϕ t ( x 0 ) x(t)=\phi_{t}(x_{0})
  3. x ( 0 ) = x 0 x(0)=x_{0}
  4. S n S\subset\mathbb{R}^{n}
  5. x 0 S x_{0}\in S
  6. t ϕ t ( x 0 ) t\mapsto\phi_{t}(x_{0})
  7. S S
  8. x 0 S x_{0}\in S
  9. S S
  10. S S
  11. S S
  12. a a
  13. x ( t ) , y ( t ) x(t),y(t)
  14. d x / d t = a x - x y and d y / d t = - y + x 2 - 2 y 2 . dx/dt=ax-xy\quad\,\text{and}\quad dy/dt=-y+x^{2}-2y^{2}.
  15. x = 0 x=0
  16. x = 0 x=0
  17. x x
  18. d x / d t = 0 dx/dt=0
  19. x x
  20. x = 0 x=0
  21. a 0 a\geq 0
  22. x ( 0 ) = 0 , y ( 0 ) > - 1 / 2 x(0)=0,\ y(0)>-1/2
  23. y = x 2 / ( 1 + 2 a ) y=x^{2}/(1+2a)
  24. a a
  25. d / d t [ y - x 2 / ( 1 + 2 a ) ] d/dt[y-x^{2}/(1+2a)]
  26. y = x 2 / ( 1 + 2 a ) y=x^{2}/(1+2a)
  27. a > 0 a>0
  28. a = 0 a=0
  29. a < 0 a<0
  30. ( x , y ) , y > - 1 / 2 (x,y),\ y>-1/2

Invariant_of_a_binary_form.html

  1. H ( f ) = [ 2 f x 2 2 f x y 2 f y x 2 f y 2 ] . H(f)=\begin{bmatrix}\frac{\partial^{2}f}{\partial x^{2}}&\frac{\partial^{2}f}{% \partial x\,\partial y}\\ \frac{\partial^{2}f}{\partial y\,\partial x}&\frac{\partial^{2}f}{\partial y^{% 2}}\end{bmatrix}.
  2. det [ f x f y g x g y ] \det\begin{bmatrix}\frac{\partial f}{\partial x}&\frac{\partial f}{\partial y}% \\ \frac{\partial g}{\partial x}&\frac{\partial g}{\partial y}\end{bmatrix}

Inventor's_paradox.html

  1. 1 + 2 + 3 + + 97 + 98 + 99 1+2+3+...+97+98+99\,
  2. ( 1 + 99 ) + ( 2 + 98 ) + ( 3 + 97 ) + + ( 48 + 52 ) + ( 49 + 51 ) + ( 50 ) (1+99)+(2+98)+(3+97)+...+(48+52)+(49+51)+(50)\,
  3. 1 + 3 = 4 1+3=4\,
  4. 1 + 3 + 5 = 9 1+3+5=9\,
  5. 1 + 3 + 5 + 7 + 9 = 25 1+3+5+7+9=25\,
  6. k = 1 n ( 2 k - 1 ) = n 2 . \sum_{k=1}^{n}\mathbf{(}2k-1)=n^{2}.

Inverse-variance_weighting.html

  1. y ^ = i y i / σ i 2 i 1 / σ i 2 . \hat{y}=\frac{\sum_{i}y_{i}/\sigma_{i}^{2}}{\sum_{i}1/\sigma_{i}^{2}}.
  2. D 2 ( y ^ ) = 1 i 1 / σ i 2 . D^{2}(\hat{y})=\frac{1}{\sum_{i}1/\sigma_{i}^{2}}.

Irrational_winding_of_a_torus.html

  1. T 2 = 2 / 2 T^{2}=\mathbb{R}^{2}/\mathbb{Z}^{2}
  2. π : 2 T 2 \pi:\mathbb{R}^{2}\to T^{2}
  3. 2 \mathbb{Z}^{2}
  4. 2 \mathbb{R}^{2}
  5. π \pi
  6. [ 0 , 1 ) 2 [0,1)^{2}
  7. 2 \mathbb{R}^{2}
  8. 2 \mathbb{Z}^{2}
  9. π \pi
  10. U ( 1 ) × U ( 1 ) U(1)\times U(1)
  11. \mathbb{R}
  12. x ( e i x , e i k x ) x\mapsto(e^{ix},e^{ikx})

Ising_critical_exponents.html

  1. d = 2 d=2
  2. d = 3 d=3
  3. d = 4 d=4
  4. 2 - d / ( d - Δ ϵ ) 2-d/(d-\Delta_{\epsilon})
  5. Δ σ / ( d - Δ ϵ ) \Delta_{\sigma}/(d-\Delta_{\epsilon})
  6. ( d - 2 Δ σ ) / ( d - Δ ϵ ) (d-2\Delta_{\sigma})/(d-\Delta_{\epsilon})
  7. ( d - Δ σ ) / Δ σ (d-\Delta_{\sigma})/\Delta_{\sigma}
  8. 2 Δ σ - d + 2 2\Delta_{\sigma}-d+2
  9. 1 / ( d - Δ ϵ ) 1/(d-\Delta_{\epsilon})
  10. Δ ϵ - d \Delta_{\epsilon^{\prime}}-d
  11. σ , ϵ , ϵ \sigma,\epsilon,\epsilon^{\prime}
  12. ϕ , ϕ 2 , ϕ 4 \phi,\phi^{2},\phi^{4}
  13. Δ σ \Delta_{\sigma}
  14. Δ ϵ \Delta_{\epsilon}
  15. Δ ϵ \Delta_{\epsilon^{\prime}}
  16. M 3 , 4 M_{3,4}

Isocitrate—homoisocitrate_dehydrogenase.html

  1. \rightleftharpoons
  2. \rightleftharpoons

Isoelastic_function.html

  1. r r
  2. f ( x ) = k x r , f(x)={kx^{r}},
  3. k k
  4. r r
  5. elasticity = f ( x ) x x f ( x ) = ln f ( x ) ln x , \text{elasticity}=\frac{\partial f(x)}{\partial x}\frac{x}{f(x)}=\frac{% \partial\,\text{ln}f(x)}{\partial\,\text{ln}x},
  6. r r
  7. D ( p ) = k p - r , D(p)={kp^{-r}},
  8. r > 0 r>0
  9. U ( x ) = 1 1 - γ x 1 - γ U(x)=\frac{1}{1-\gamma}x^{1-\gamma}
  10. 1 - γ 1-\gamma
  11. γ > 0 \gamma>0
  12. γ \gamma
  13. γ \gamma

Isoline_retrieval.html

  1. j = { 1 ; q < q 0 2 ; q q 0 j=\begin{cases}1;&q<q_{0}\\ 2;&q\geq q_{0}\end{cases}
  2. y \vec{y}
  3. P ( y | j ) P(\vec{y}|j)
  4. a = 1 A A P [ c ( r ) | y ( r ) ] d r a=\frac{1}{A}\int_{A}P\left[c(\vec{r})|\vec{y}(\vec{r})\right]\,d\vec{r}
  5. r \vec{r}
  6. max ( a ) = 1 A A { max j P [ j | y ( r ) ] } d r \max(a)=\frac{1}{A}\int_{A}\left\{\max_{j}P\left[j|\vec{y}(\vec{r})\right]% \right\}\,d\vec{r}
  7. y = f ( x ) \vec{y}=\vec{f}(\vec{x})\,
  8. x \vec{x}
  9. P ( y | j ) P(\vec{y}|j)
  10. C = n c P ( c | y ) - 1 n c - 1 C=\frac{n_{c}P(c|\vec{y})-1}{n_{c}-1}
  11. δ ( C ) = 1 l 0 l h ( C - C ( r ) ) d s \delta(C)=\frac{1}{l}\int_{0}^{l}h(C-C^{\prime}(\vec{r}))\,ds
  12. C C^{\prime}
  13. C C^{\prime}
  14. δ ( C ) \delta(C)
  15. P ( 1 | y ) = - q 0 P ( q | y ) d q P(1|\vec{y})=\int_{-\infty}^{q_{0}}P(q|\vec{y})\,dq
  16. P ( 2 | y ) = q 0 P ( q | y ) d q P(2|\vec{y})=\int^{\infty}_{q_{0}}P(q|\vec{y})\,dq
  17. P ( q | y ) P(q|\vec{y})
  18. P ( q | y ) = 1 2 π σ q exp { - [ q - q ¯ ( y ) ] 2 2 σ q } P(q|\vec{y})=\frac{1}{\sqrt{2\pi}\sigma_{q}}\exp\left\{-\frac{\left[q-\bar{q}(% \vec{y})\right]^{2}}{2\sigma_{q}}\right\}
  19. q ¯ \bar{q}
  20. σ q \sigma_{q}
  21. R = P ( 2 | y ) - P ( 1 | y ) = erf [ q 0 - q ¯ ( y ) 2 σ q ] R=P(2|\vec{y})-P(1|\vec{y})=\mathrm{erf}\left[\frac{q_{0}-\bar{q}(\vec{y})}{% \sqrt{2}\sigma_{q}}\right]
  22. - q 0 P ( q | y ) d q = q 0 P ( q | y ) d q \int_{-\infty}^{q_{0}}P(q|\vec{y})\,dq=\int^{\infty}_{q_{0}}P(q|\vec{y})\,dq

Isotypic_component.html

  1. λ \lambda
  2. λ \lambda
  3. V V
  4. 𝔤 \mathfrak{g}
  5. V = i = 1 N V i V=\oplus_{i=1}^{N}V_{i}
  6. 𝔤 \mathfrak{g}
  7. i { 1 , , N } λ P ( 𝔤 ) : V i M λ \forall i\in\{1,\ldots,N\}\exists\lambda\in P(\mathfrak{g}):V_{i}\simeq M_{\lambda}
  8. M λ M_{\lambda}
  9. λ \lambda
  10. V V
  11. V λ P ( 𝔤 ) ( i = 1 d λ M λ ) V\simeq\oplus_{\lambda\in P(\mathfrak{g})}(\oplus_{i=1}^{d_{\lambda}}M_{% \lambda})
  12. λ \lambda
  13. λ ( V ) := i = 1 d λ V i d λ M λ \lambda(V):=\oplus_{i=1}^{d_{\lambda}}V_{i}\simeq\mathbb{C}^{d_{\lambda}}% \otimes M_{\lambda}
  14. d λ d_{\lambda}

Iterated_filtering.html

  1. y 1 , , y N y_{1},\dots,y_{N}
  2. t 1 < t 2 < < t N t_{1}<t_{2}<\dots<t_{N}
  3. X ( t ) X(t)
  4. f ( x , s , t , θ , W ) f(x,s,t,\theta,W)
  5. X ( t n ) = f ( X ( t n - 1 ) , t n - 1 , t n , θ , W ) X(t_{n})=f(X(t_{n-1}),t_{n-1},t_{n},\theta,W)\,
  6. θ \theta
  7. W W
  8. f ( . ) f(.)
  9. X ( t 0 ) X(t_{0})
  10. t 0 < t 1 t_{0}<t_{1}
  11. X ( t 0 ) = h ( θ ) X(t_{0})=h(\theta)
  12. g ( y n | X n , t n , θ ) g(y_{n}|X_{n},t_{n},\theta)
  13. J J
  14. M M
  15. 0 < a < 1 0<a<1
  16. b b
  17. Φ \Phi
  18. θ ( 1 ) \theta^{(1)}
  19. ${}$
  20. m = 1 m=1
  21. M M
  22. Θ F ( t 0 , j ) Normal ( θ ( m ) , b a m - 1 Φ ) \Theta_{F}(t_{0},j)\sim\mathrm{Normal}(\theta^{(m)},ba^{m-1}\Phi)
  23. j = 1 , , J j=1,\dots,J
  24. X F ( t 0 , j ) = h ( Θ F ( t 0 , j ) ) X_{F}(t_{0},j)=h\big(\Theta_{F}(t_{0},j)\big)
  25. j = 1 , , J j=1,\dots,J
  26. θ ¯ ( t 0 ) = θ ( m ) \bar{\theta}(t_{0})=\theta^{(m)}
  27. n = 1 n=1
  28. N N
  29. Θ P ( t n , j ) Normal ( Θ F ( t n - 1 , j ) , a m - 1 Φ ) \Theta_{P}(t_{n},j)\sim\mathrm{Normal}(\Theta_{F}(t_{n-1},j),a^{m-1}\Phi)
  30. j = 1 , , J j=1,\dots,J
  31. X P ( t n , j ) = f ( X F ( t n - 1 , j ) , t n - 1 , t n , Θ P ( t n , j ) , W ) X_{P}(t_{n},j)=f(X_{F}(t_{n-1},j),t_{n-1},t_{n},\Theta_{P}(t_{n},j),W)
  32. j = 1 , , J j=1,\dots,J
  33. w ( n , j ) = g ( y n | X P ( t n , j ) , t n , Θ P ( t n , j ) ) w(n,j)=g(y_{n}|X_{P}(t_{n},j),t_{n},\Theta_{P}(t_{n},j))
  34. j = 1 , , J j=1,\dots,J
  35. k 1 , , k J k_{1},\dots,k_{J}
  36. P ( k j = i ) = w ( n , i ) / w ( n , ) P(k_{j}=i)=w(n,i)\big/{\sum}_{\ell}w(n,\ell)
  37. X F ( t n , j ) = X P ( t n , k j ) X_{F}(t_{n},j)=X_{P}(t_{n},k_{j})
  38. Θ F ( t n , j ) = Θ P ( t n , k j ) \Theta_{F}(t_{n},j)=\Theta_{P}(t_{n},k_{j})
  39. j = 1 , , J j=1,\dots,J
  40. θ ¯ i ( t n ) \bar{\theta}_{i}(t_{n})
  41. { Θ F , i ( t n , j ) , j = 1 , , J } \{\Theta_{F,i}(t_{n},j),j=1,\dots,J\}
  42. Θ F \Theta_{F}
  43. { Θ F , i } \{\Theta_{F,i}\}
  44. V i ( t n ) V_{i}(t_{n})
  45. { Θ P , i ( t n , j ) , j = 1 , , J } \{\Theta_{P,i}(t_{n},j),j=1,\dots,J\}
  46. θ i ( m + 1 ) = θ i ( m ) + V i ( t 1 ) n = 1 N V i - 1 ( t n ) ( θ ¯ i ( t n ) - θ ¯ i ( t n - 1 ) ) \theta_{i}^{(m+1)}=\theta_{i}^{(m)}+V_{i}(t_{1})\sum_{n=1}^{N}V_{i}^{-1}(t_{n}% )(\bar{\theta}_{i}(t_{n})-\bar{\theta}_{i}(t_{n-1}))
  47. ${}$
  48. θ ^ = θ ( M + 1 ) \hat{\theta}=\theta^{(M+1)}
  49. X ( t 0 ) X(t_{0})
  50. J J
  51. M M
  52. 0 < a < 1 0<a<1
  53. Φ \Phi
  54. { Θ j , j = 1 , , J } \{\Theta_{j},j=1,\dots,J\}
  55. ${}$
  56. m = 1 m=1
  57. M M
  58. Θ F ( t 0 , j ) Normal ( Θ j , a m - 1 Φ ) \Theta_{F}(t_{0},j)\sim\mathrm{Normal}(\Theta_{j},a^{m-1}\Phi)
  59. j = 1 , , J j=1,\dots,J
  60. X F ( t 0 , j ) = h ( Θ F ( t 0 , j ) ) X_{F}(t_{0},j)=h\big(\Theta_{F}(t_{0},j)\big)
  61. j = 1 , , J j=1,\dots,J
  62. n = 1 n=1
  63. N N
  64. Θ P ( t n , j ) Normal ( Θ F ( t n - 1 , k j ) , a m - 1 Φ ) \Theta_{P}(t_{n},j)\sim\mathrm{Normal}(\Theta_{F}(t_{n-1},k_{j}),a^{m-1}\Phi)
  65. j = 1 , , J j=1,\dots,J
  66. X P ( t n , j ) = f ( X F ( t n - 1 , j ) , t n - 1 , t n , Θ P ( t n , j ) , W ) X_{P}(t_{n},j)=f(X_{F}(t_{n-1},j),t_{n-1},t_{n},\Theta_{P}(t_{n},j),W)
  67. j = 1 , , J j=1,\dots,J
  68. w ( n , j ) = g ( y n | X P ( t n , j ) , t n , Θ P ( t n , j ) ) w(n,j)=g(y_{n}|X_{P}(t_{n},j),t_{n},\Theta_{P}(t_{n},j))
  69. j = 1 , , J j=1,\dots,J
  70. k 1 , , k J k_{1},\dots,k_{J}
  71. P ( k j = i ) = w ( n , i ) / w ( n , ) P(k_{j}=i)=w(n,i)\big/{\sum}_{\ell}w(n,\ell)
  72. X F ( t n , j ) = X P ( t n , k j ) X_{F}(t_{n},j)=X_{P}(t_{n},k_{j})
  73. Θ F ( t n , j ) = Θ P ( t n , k j ) \Theta_{F}(t_{n},j)=\Theta_{P}(t_{n},k_{j})
  74. j = 1 , , J j=1,\dots,J
  75. Θ j = Θ F ( t N , j ) \Theta_{j}=\Theta_{F}(t_{N},j)
  76. j = 1 , , J j=1,\dots,J
  77. ${}$
  78. { Θ j , j = 1 , , J } \{\Theta_{j},j=1,\dots,J\}

Ivar_Ekeland.html

  1. f ( x ) = f ( x 1 , , x N ) = n f n ( x n ) . f(x)=f(x_{1},\dots,x_{N})=\sum_{n}f_{n}(x_{n}).
  2. x min = ( x 1 , , x N ) min x_{\min}=(x_{1},\dots,x_{N})_{\min}
  3. ( x j , f ( x j ) ) Conv ( Graph ( f n ) ) . (x_{j},f(x_{j}))\in\mathrm{Conv}(\mathrm{Graph}(f_{n})).\,

Iwasawa_algebra.html

  1. i 𝐙 p [ [ T ] ] / ( p μ i ) j 𝐙 p [ [ T ] ] / ( f j m j ) \bigoplus_{i}\mathbf{Z}_{p}[\![T]\!]/(p^{\mu_{i}})\oplus\bigoplus_{j}\mathbf{Z% }_{p}[\![T]\!]/(f_{j}^{m_{j}})
  2. μ = i μ i \mu=\sum_{i}\mu_{i}
  3. λ = j m j deg ( f j ) . \lambda=\sum_{j}m_{j}\deg(f_{j}).
  4. e n = μ p n + λ n + c e_{n}=\mu p^{n}+\lambda n+c

J-line.html

  1. M ( [ Γ ( 1 ) ] ) = Spec ( R [ j ] ) M([\Gamma(1)])=\mathrm{Spec}(R[j])

Jackiw–Teitelboim_gravity.html

  1. S = 1 κ d 2 x - g [ - R Φ - 1 2 g μ ν μ Φ ν Φ - Λ + κ mat ] S=\frac{1}{\kappa}\int d^{2}x\,\sqrt{-g}\left[-R\Phi-\frac{1}{2}g^{\mu\nu}% \nabla_{\mu}\Phi\nabla_{\nu}\Phi-\Lambda+\kappa\mathcal{L}_{\,\text{mat}}\right]
  2. μ \nabla_{\mu}
  3. R - Λ = κ T R-\Lambda=\kappa T

Jackson_q-Bessel_function.html

  1. J ν ( 1 ) ( x ; q ) = ( q ν + 1 ; q ) ( q ; q ) ( x / 2 ) ν ϕ 1 2 ( 0 , 0 ; q ν + 1 ; q , - x 2 / 4 ) J_{\nu}^{(1)}(x;q)=\frac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}}(x/2)^{\nu}{}_% {2}\phi_{1}(0,0;q^{\nu+1};q,-x^{2}/4)
  2. J ν ( 2 ) ( x ; q ) = ( q ν + 1 ; q ) ( q ; q ) ( x / 2 ) ν ϕ 1 0 ( ; q ν + 1 ; q , - x 2 q ν + 1 / 4 ) J_{\nu}^{(2)}(x;q)=\frac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}}(x/2)^{\nu}{}_% {0}\phi_{1}(;q^{\nu+1};q,-x^{2}q^{\nu+1}/4)
  3. J ν ( 3 ) ( x ; q ) = ( q ν + 1 ; q ) ( q ; q ) ( x / 2 ) ν ϕ 1 1 ( 0 ; q ν + 1 ; q , q x 2 / 4 ) J_{\nu}^{(3)}(x;q)=\frac{(q^{\nu+1};q)_{\infty}}{(q;q)_{\infty}}(x/2)^{\nu}{}_% {1}\phi_{1}(0;q^{\nu+1};q,qx^{2}/4)

Jacobi_method_for_complex_Hermitian_matrices.html

  1. ( R p q ) m , n \displaystyle(R_{pq})_{m,n}
  2. ( R p q M ) m , n \displaystyle(R_{pq}M)_{m,n}
  3. H = H H i , j = H j , i * H^{\dagger}=H\ \Leftrightarrow\ H_{i,j}=H^{*}_{j,i}
  4. R p q \displaystyle R^{\dagger}_{pq}
  5. T T
  6. T \displaystyle T
  7. T p , p = H p , p + H q , q 2 - Re { H p , q e - 2 i θ } , T p , q = H p , p - H q , q 2 + i Im { H p , q e - 2 i θ } , T q , p = H p , p - H q , q 2 - i Im { H p , q e - 2 i θ } , T q , q = H p , p + H q , q 2 + Re { H p , q e - 2 i θ } . \begin{array}[]{clrcl}T_{p,p}&=&&\frac{H_{p,p}+H_{q,q}}{2}&-\ \ \ \mathrm{Re}% \{H_{p,q}e^{-2i\theta}\},\\ T_{p,q}&=&&\frac{H_{p,p}-H_{q,q}}{2}&+\ i\ \mathrm{Im}\{H_{p,q}e^{-2i\theta}\}% ,\\ T_{q,p}&=&&\frac{H_{p,p}-H_{q,q}}{2}&-\ i\ \mathrm{Im}\{H_{p,q}e^{-2i\theta}\}% ,\\ T_{q,q}&=&&\frac{H_{p,p}+H_{q,q}}{2}&+\ \ \ \mathrm{Re}\{H_{p,q}e^{-2i\theta}% \}.\end{array}
  8. R p q J R p q ( θ 2 ) R p q ( θ 1 ) , with θ 1 2 ϕ 1 - π 4 and θ 2 ϕ 2 2 , \begin{aligned}\displaystyle R^{J}_{pq}&\displaystyle\equiv R_{pq}(\theta_{2})% \,R_{pq}(\theta_{1}),\,\text{ with}\\ \displaystyle\theta_{1}&\displaystyle\equiv\frac{2\phi_{1}-\pi}{4}\,\text{ and% }\theta_{2}\equiv\frac{\phi_{2}}{2},\end{aligned}
  9. ϕ 1 \phi_{1}
  10. ϕ 2 \phi_{2}
  11. tan ϕ 1 \displaystyle\tan\phi_{1}
  12. [ R p q ( θ 2 ) R p q ( θ 1 ) ] m , n = { δ m , n m , n p , q , - i e - i θ 1 sin θ 2 m = p and n = p , - e + i θ 1 cos θ 2 m = p and n = q , e - i θ 1 cos θ 2 m = q and n = p , + i e + i θ 1 sin θ 2 m = q and n = q . \displaystyle\left[R_{pq}(\theta_{2})\,R_{pq}(\theta_{1})\right]_{m,n}=\begin{% cases}\ \ \ \ \delta_{m,n}&m,n\neq p,q,\\ -ie^{-i\theta_{1}}\,\sin{\theta_{2}}&m=p\,\text{ and }n=p,\\ -e^{+i\theta_{1}}\,\cos{\theta_{2}}&m=p\,\text{ and }n=q,\\ \ \ \ \ e^{-i\theta_{1}}\,\cos{\theta_{2}}&m=q\,\text{ and }n=p,\\ +ie^{+i\theta_{1}}\,\sin{\theta_{2}}&m=q\,\text{ and }n=q.\end{cases}

Jacobi_operator.html

  1. 2 ( ) \ell^{2}(\mathbb{N})
  2. J f ( 1 ) = a ( 1 ) f ( 2 ) + b ( 1 ) f ( 1 ) , J f ( n ) = a ( n ) f ( n + 1 ) + a ( n - 1 ) f ( n - 1 ) + b ( n ) f ( n ) , n > 1 , J\,f(1)=a(1)f(2)+b(1)f(1),\quad J\,f(n)=a(n)f(n+1)+a(n-1)f(n-1)+b(n)f(n),\quad n% >1,
  3. a ( n ) > 0 , b ( n ) . a(n)>0,\quad b(n)\in\mathbb{R}.
  4. J p ( z , n ) = z p ( z , n ) , p ( z , 1 ) = 1 and p ( z , 0 ) = 0 , J\,p(z,n)=z\,p(z,n),\qquad p(z,1)=1\,\text{ and }p(z,0)=0,
  5. δ 1 ( n ) = δ 1 , n \delta_{1}(n)=\delta_{1,n}

Jacobi_polynomials.html

  1. P [ u s u , u b = , u n , u p = ( , u 3 b 1 , u , , u 3 b 2 , u ) ] ( x ) P[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}n^{\prime},u^{\prime}p% =(^{\prime},u^{\prime}\u{0}3b1^{\prime},u^{\prime},^{\prime},u^{\prime}\u{0}3b% 2^{\prime},u^{\prime})^{\prime}](x)
  2. 1 , 11 −1,11
  3. P n ( α , β ) ( z ) = ( α + 1 ) n n ! F 1 2 ( - n , 1 + α + β + n ; α + 1 ; 1 2 ( 1 - z ) ) , P_{n}^{(\alpha,\beta)}(z)=\frac{(\alpha+1)_{n}}{n!}\,{}_{2}F_{1}\left(-n,1+% \alpha+\beta+n;\alpha+1;\tfrac{1}{2}(1-z)\right),
  4. ( α + 1 ) n (\alpha+1)_{n}
  5. P n ( α , β ) ( z ) = Γ ( α + n + 1 ) n ! Γ ( α + β + n + 1 ) m = 0 n ( n m ) Γ ( α + β + n + m + 1 ) Γ ( α + m + 1 ) ( z - 1 2 ) m . P_{n}^{(\alpha,\beta)}(z)=\frac{\Gamma(\alpha+n+1)}{n!\,\Gamma(\alpha+\beta+n+% 1)}\sum_{m=0}^{n}{n\choose m}\frac{\Gamma(\alpha+\beta+n+m+1)}{\Gamma(\alpha+m% +1)}\left(\frac{z-1}{2}\right)^{m}.
  6. P n ( α , β ) ( z ) = ( - 1 ) n 2 n n ! ( 1 - z ) - α ( 1 + z ) - β d n d z n { ( 1 - z ) α ( 1 + z ) β ( 1 - z 2 ) n } . P_{n}^{(\alpha,\beta)}(z)=\frac{(-1)^{n}}{2^{n}n!}(1-z)^{-\alpha}(1+z)^{-\beta% }\frac{d^{n}}{dz^{n}}\left\{(1-z)^{\alpha}(1+z)^{\beta}\left(1-z^{2}\right)^{n% }\right\}.
  7. α = β = 0 \alpha=\beta=0
  8. P n = 1 2 n n ! d n d z n ( z 2 - 1 ) n . P_{n}=\frac{1}{2^{n}n!}\frac{d^{n}}{dz^{n}}(z^{2}-1)^{n}\;.
  9. P n ( α , β ) ( x ) = s ( n + α s ) ( n + β n - s ) ( x - 1 2 ) n - s ( x + 1 2 ) s , n s 0. P_{n}^{(\alpha,\beta)}(x)=\sum_{s}{n+\alpha\choose s}{n+\beta\choose n-s}\left% (\frac{x-1}{2}\right)^{n-s}\left(\frac{x+1}{2}\right)^{s},\qquad n\geq s\geq 0.
  10. ( z n ) = { Γ ( z + 1 ) Γ ( n + 1 ) Γ ( z - n + 1 ) n 0 0 n < 0 {z\choose n}=\begin{cases}\frac{\Gamma(z+1)}{\Gamma(n+1)\Gamma(z-n+1)}&n\geq 0% \\ 0&n<0\end{cases}
  11. Γ ( z ) Γ(z)
  12. n , n + α , n + β n,n+α,n+β
  13. n + α + β n+α+β
  14. P n ( α , β ) ( x ) = ( n + α ) ! ( n + β ) ! s 1 s ! ( n + α - s ) ! ( β + s ) ! ( n - s ) ! ( x - 1 2 ) n - s ( x + 1 2 ) s . P_{n}^{(\alpha,\beta)}(x)=(n+\alpha)!(n+\beta)!\sum_{s}\frac{1}{s!(n+\alpha-s)% !(\beta+s)!(n-s)!}\left(\frac{x-1}{2}\right)^{n-s}\left(\frac{x+1}{2}\right)^{% s}.
  15. - 1 1 ( 1 - x ) α ( 1 + x ) β P m ( α , β ) ( x ) P n ( α , β ) ( x ) d x = 2 α + β + 1 2 n + α + β + 1 Γ ( n + α + 1 ) Γ ( n + β + 1 ) Γ ( n + α + β + 1 ) n ! δ n m , α , β , α + β > - 1. \int_{-1}^{1}(1-x)^{\alpha}(1+x)^{\beta}P_{m}^{(\alpha,\beta)}(x)P_{n}^{(% \alpha,\beta)}(x)\;dx=\frac{2^{\alpha+\beta+1}}{2n+\alpha+\beta+1}\frac{\Gamma% (n+\alpha+1)\Gamma(n+\beta+1)}{\Gamma(n+\alpha+\beta+1)n!}\delta_{nm},\qquad% \alpha,\beta,\alpha+\beta>-1.
  16. P n ( α , β ) ( 1 ) = ( n + α n ) . P_{n}^{(\alpha,\beta)}(1)={n+\alpha\choose n}.
  17. P n ( α , β ) ( - z ) = ( - 1 ) n P n ( β , α ) ( z ) ; P_{n}^{(\alpha,\beta)}(-z)=(-1)^{n}P_{n}^{(\beta,\alpha)}(z);
  18. P n ( α , β ) ( - 1 ) = ( - 1 ) n ( n + β n ) . P_{n}^{(\alpha,\beta)}(-1)=(-1)^{n}{n+\beta\choose n}.
  19. d k d z k P n ( α , β ) ( z ) = Γ ( α + β + n + 1 + k ) 2 k Γ ( α + β + n + 1 ) P n - k ( α + k , β + k ) ( z ) . \frac{\mathrm{d}^{k}}{\mathrm{d}z^{k}}P_{n}^{(\alpha,\beta)}(z)=\frac{\Gamma(% \alpha+\beta+n+1+k)}{2^{k}\Gamma(\alpha+\beta+n+1)}P_{n-k}^{(\alpha+k,\beta+k)% }(z).
  20. P [ u s u , u b = , u n , u p = ( , u 3 b 1 , u , , u 3 b 2 , u ) ] P[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}n^{\prime},u^{\prime}p% =(^{\prime},u^{\prime}\u{0}3b1^{\prime},u^{\prime},^{\prime},u^{\prime}\u{0}3b% 2^{\prime},u^{\prime})^{\prime}]
  21. ( 1 - x 2 ) y ′′ + ( β - α - ( α + β + 2 ) x ) y + n ( n + α + β + 1 ) y = 0. \left(1-x^{2}\right)y^{\prime\prime}+(\beta-\alpha-(\alpha+\beta+2)x)y^{\prime% }+n(n+\alpha+\beta+1)y=0.
  22. 2 n ( n + α + β ) ( 2 n + α + β - 2 ) P n ( α , β ) ( z ) = = ( 2 n + α + β - 1 ) { ( 2 n + α + β ) ( 2 n + α + β - 2 ) z + α 2 - β 2 } P n - 1 ( α , β ) ( z ) - 2 ( n + α - 1 ) ( n + β - 1 ) ( 2 n + α + β ) P n - 2 ( α , β ) ( z ) , \begin{aligned}&\displaystyle 2n(n+\alpha+\beta)(2n+\alpha+\beta-2)P_{n}^{(% \alpha,\beta)}(z)=\\ &\displaystyle\qquad=(2n+\alpha+\beta-1)\Big\{(2n+\alpha+\beta)(2n+\alpha+% \beta-2)z+\alpha^{2}-\beta^{2}\Big\}P_{n-1}^{(\alpha,\beta)}(z)-2(n+\alpha-1)(% n+\beta-1)(2n+\alpha+\beta)P_{n-2}^{(\alpha,\beta)}(z),\end{aligned}
  23. n = 0 P n ( α , β ) ( z ) t n = 2 α + β R - 1 ( 1 - t + R ) - α ( 1 + t + R ) - β , \sum_{n=0}^{\infty}P_{n}^{(\alpha,\beta)}(z)t^{n}=2^{\alpha+\beta}R^{-1}(1-t+R% )^{-\alpha}(1+t+R)^{-\beta},
  24. R = R ( z , t ) = ( 1 - 2 z t + t 2 ) 1 2 , R=R(z,t)=\left(1-2zt+t^{2}\right)^{\frac{1}{2}}~{},
  25. 1 , 11 −1,11
  26. P [ u s u , u b = , u n , u p = ( , u 3 b 1 , u , , u 3 b 2 , u ) ] P[u^{\prime}su^{\prime},u^{\prime}b=^{\prime},u^{\prime}n^{\prime},u^{\prime}p% =(^{\prime},u^{\prime}\u{0}3b1^{\prime},u^{\prime},^{\prime},u^{\prime}\u{0}3b% 2^{\prime},u^{\prime})^{\prime}]
  27. P n ( α , β ) ( cos θ ) = n - 1 2 k ( θ ) cos ( N θ + γ ) + O ( n - 3 2 ) , P_{n}^{(\alpha,\beta)}(\cos\theta)=n^{-\frac{1}{2}}k(\theta)\cos(N\theta+% \gamma)+O\left(n^{-\frac{3}{2}}\right),
  28. k ( θ ) = π - 1 2 sin - α - 1 2 θ 2 cos - β - 1 2 θ 2 , N = n + 1 2 ( α + β + 1 ) , γ = - π 2 ( α + 1 2 ) , \begin{aligned}\displaystyle k(\theta)&\displaystyle=\pi^{-\frac{1}{2}}\sin^{-% \alpha-\frac{1}{2}}\tfrac{\theta}{2}\cos^{-\beta-\frac{1}{2}}\tfrac{\theta}{2}% ,\\ \displaystyle N&\displaystyle=n+\tfrac{1}{2}(\alpha+\beta+1),\\ \displaystyle\gamma&\displaystyle=-\tfrac{\pi}{2}\left(\alpha+\tfrac{1}{2}% \right),\end{aligned}
  29. π \pi
  30. lim n n - α P n ( α , β ) ( cos ( z n ) ) = ( z 2 ) - α J α ( z ) lim n n - β P n ( α , β ) ( cos ( π - z n ) ) = ( z 2 ) - β J β ( z ) \begin{aligned}\displaystyle\lim_{n\to\infty}n^{-\alpha}P_{n}^{(\alpha,\beta)}% \left(\cos\left(\tfrac{z}{n}\right)\right)&\displaystyle=\left(\tfrac{z}{2}% \right)^{-\alpha}J_{\alpha}(z)\\ \displaystyle\lim_{n\to\infty}n^{-\beta}P_{n}^{(\alpha,\beta)}\left(\cos\left(% \pi-\tfrac{z}{n}\right)\right)&\displaystyle=\left(\tfrac{z}{2}\right)^{-\beta% }J_{\beta}(z)\end{aligned}
  31. 1 , 11 −1,11
  32. π \pi
  33. d m m j ( ϕ ) = [ ( j + m ) ! ( j - m ) ! ( j + m ) ! ( j - m ) ! ] 1 2 ( sin ϕ 2 ) m - m ( cos ϕ 2 ) m + m P j - m ( m - m , m + m ) ( cos ϕ ) . d^{j}_{m^{\prime}m}(\phi)=\left[\frac{(j+m)!(j-m)!}{(j+m^{\prime})!(j-m^{% \prime})!}\right]^{\frac{1}{2}}\left(\sin\tfrac{\phi}{2}\right)^{m-m^{\prime}}% \left(\cos\tfrac{\phi}{2}\right)^{m+m^{\prime}}P_{j-m}^{(m-m^{\prime},m+m^{% \prime})}(\cos\phi).

Jacobi_set.html

  1. f , g : M \R f,g:M\to\R
  2. d d
  3. f f
  4. g - 1 ( t ) g^{-1}(t)
  5. t \R t\in\R
  6. f t : g - 1 ( t ) \R f_{t}:g^{-1}(t)\to\R
  7. J J
  8. f f
  9. g g
  10. J = c l { x M x is critical point of f t } J=cl{\{x\in M\mid x\mbox{ is critical point of }~{}f_{t}\}}
  11. λ \R \lambda\in\R
  12. J = { x M f ( x ) + λ g ( x ) = 0 or λ f ( x ) + g ( x ) = 0 } . J=\{x\in M\mid\nabla{f(x)}+\lambda\nabla{g(x)}=0\mbox{ or }~{}\lambda\nabla{f(% x)}+\nabla{g(x)}=0\}.
  13. f + λ g f+\lambda g
  14. λ \R \lambda\in\R
  15. J = { x M x is a critical point of f + λ g or λ f + g } . J=\{x\in M\mid x\mbox{ is a critical point of }~{}f+\lambda g\mbox{ or }~{}% \lambda f+g\}.

Jacobian_ideal.html

  1. 𝒪 ( x 1 , , x n ) \mathcal{O}(x_{1},\ldots,x_{n})
  2. J f := f x 1 , , f x n . J_{f}:=\left\langle\frac{\partial f}{\partial x_{1}},\ldots,\frac{\partial f}{% \partial x_{n}}\right\rangle.

Jacques_Feldbau.html

  1. S n S^{n}
  2. S n - 1 S^{n-1}

Jamshidian's_trick.html

  1. f i f_{i}
  2. [ 0 , ) [0,\infty)
  3. W W
  4. K 0 K\geq 0
  5. i f i \sum_{i}f_{i}
  6. [ 0 , ) [0,\infty)
  7. w w\in\mathbb{R}
  8. i f i ( w ) = K . \sum_{i}f_{i}(w)=K.
  9. f i f_{i}
  10. ( i f i ( W ) - K ) + = ( i ( f i ( W ) - f i ( w ) ) ) + = i ( f i ( W ) - f i ( w ) ) 1 { W w } = i ( f i ( W ) - f i ( w ) ) + . \left(\sum_{i}f_{i}(W)-K\right)_{+}=\left(\sum_{i}(f_{i}(W)-f_{i}(w))\right)_{% +}=\sum_{i}(f_{i}(W)-f_{i}(w))1_{\{W\geq w\}}=\sum_{i}(f_{i}(W)-f_{i}(w))_{+}.
  11. f i ( W ) f_{i}(W)
  12. K K
  13. f i ( W ) f_{i}(W)
  14. f i ( w ) f_{i}(w)

Jantzen_filtration.html

  1. M ( λ ) = M ( λ ) 0 M ( λ ) 1 M ( λ ) 2 . M(\lambda)=M(\lambda)^{0}\supseteq M(\lambda)^{1}\supseteq M(\lambda)^{2}% \supseteq\cdots.
  2. i > 0 Ch ( M ( λ ) i ) = α > 0 , s α ( λ ) < λ Ch ( M ( s α ( λ ) ) ) \sum_{i>0}\,\text{Ch}(M(\lambda)^{i})=\sum_{\alpha>0,s_{\alpha}(\lambda)<% \lambda}\,\text{Ch}(M(s_{\alpha}(\lambda)))

Janzen–Connell_hypothesis.html

  1. ( R 0 ) (R_{0})
  2. R < s u b > 0 = β L S R<sub>0=βLS

Jarque–Bera_test.html

  1. 𝐽𝐵 = n - k + 1 6 ( S 2 + 1 4 ( C - 3 ) 2 ) \mathit{JB}=\frac{n-k+1}{6}\left(S^{2}+\frac{1}{4}(C-3)^{2}\right)
  2. S = μ ^ 3 σ ^ 3 = 1 n i = 1 n ( x i - x ¯ ) 3 ( 1 n i = 1 n ( x i - x ¯ ) 2 ) 3 / 2 , S=\frac{\hat{\mu}_{3}}{\hat{\sigma}^{3}}=\frac{\frac{1}{n}\sum_{i=1}^{n}(x_{i}% -\bar{x})^{3}}{\left(\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}\right)^{3/2}},
  3. C = μ ^ 4 σ ^ 4 = 1 n i = 1 n ( x i - x ¯ ) 4 ( 1 n i = 1 n ( x i - x ¯ ) 2 ) 2 , C=\frac{\hat{\mu}_{4}}{\hat{\sigma}^{4}}=\frac{\frac{1}{n}\sum_{i=1}^{n}(x_{i}% -\bar{x})^{4}}{\left(\frac{1}{n}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}\right)^{2}},
  4. μ ^ 3 \hat{\mu}_{3}
  5. μ ^ 4 \hat{\mu}_{4}
  6. x ¯ \bar{x}
  7. σ ^ 2 \hat{\sigma}^{2}
  8. 𝐽𝐵 = n - k 6 ( S 2 + 1 4 ( K - 3 ) 2 ) \mathit{JB}=\frac{n-k}{6}\left(S^{2}+\frac{1}{4}(K-3)^{2}\right)

Jean-François_Mertens.html

  1. v n v_{n}
  2. n n
  3. n n
  4. v λ v_{\lambda}
  5. λ {\lambda}
  6. λ 1 {\lambda}\to 1
  7. p p
  8. q q
  9. i i
  10. i i
  11. p p
  12. q q
  13. p p
  14. q q
  15. p p
  16. q q
  17. lim v n \lim v_{n}

Jeans_equations.html

  1. n t + i ( n v i ) x i = 0 , \frac{\partial n}{\partial t}+\sum_{i}\frac{\partial(n\langle{v_{i}}\rangle)}{% \partial x_{i}}=0,
  2. ( n v j ) t + n Φ x j + i ( n v i v j ) x i = 0 ( j = 1 , 2 , 3. ) \frac{\partial(n\langle{v_{j}}\rangle)}{\partial t}+n\frac{\partial\Phi}{% \partial x_{j}}+\sum_{i}\frac{\partial(n\langle{v_{i}v_{j}}\rangle)}{\partial x% _{i}}=0\qquad(j=1,2,3.)
  3. v 1 \langle{v_{1}}\rangle
  4. n v j t + i n v i v j x i = - n Φ x j - i ( n σ i j 2 ) x i ( j = 1 , 2 , 3. ) n\frac{\partial\langle{v_{j}}\rangle}{\partial t}+\sum_{i}n\langle{v_{i}}% \rangle\frac{\partial{\langle{v_{j}}\rangle}}{\partial x_{i}}=-n\frac{\partial% \Phi}{\partial x_{j}}-\sum_{i}\frac{\partial(n\sigma_{ij}^{2})}{\partial x_{i}% }\qquad(j=1,2,3.)
  5. σ i j 2 = v i v j - v i v j \sigma_{ij}^{2}=\langle{v_{i}v_{j}}\rangle-\langle{v_{i}}\rangle\langle{v_{j}}\rangle

Jigu_Suanjing.html

  1. x 3 = N x^{3}=N
  2. x 3 + p x 2 + q x = N , x^{3}+px^{2}+qx=N,\,
  3. x 4 + p x 2 + q = 0 x^{4}+px^{2}+q=0
  4. x 3 + 3 c d b - c x 2 + 3 ( a + c ) h d 2 ( H - h ) ( b - c ) x = 6 V d 2 ( H - h ) ( b - c ) x^{3}+\frac{3cd}{b-c}x^{2}+\frac{3(a+c)hd^{2}}{(H-h)(b-c)}x=\frac{6Vd^{2}}{(H-% h)(b-c)}
  5. x 3 + 5004 x 2 + 1169953 1 3 x = 41107188 1 3 x^{3}+5004x^{2}+1169953\frac{1}{3}x=41107188\frac{1}{3}
  6. x 3 + 62 x 2 + 696 x = 38448 , x = 18 ; x^{3}+62x^{2}+696x=38448,\quad x=18;
  7. x 3 + 594 x 2 = 682803 , x = 33 ; x^{3}+594x^{2}=682803,\quad x=33;
  8. x 3 + 15 x 2 + 66 x - 360 , x = 3 x^{3}+15x^{2}+66x-360,\quad x=3
  9. x 3 + ( D + G ) x 2 + ( D G + D 2 3 ) x = P - D 2 G 3 x^{3}+(D+G)x^{2}+\left(DG+\frac{D^{2}}{3}\right)x=P-\frac{D^{2}G}{3}
  10. X + 3 h s D x 2 + 3 ( h s D ) 2 x = P 3 h 2 D 2 X^{+}3\frac{hs}{D}x^{2}+3\left(\frac{hs}{D}\right)^{2}x=\frac{P^{\prime}}{3}% \frac{h^{2}}{D^{2}}
  11. x 3 + 90 x 2 - 839808 , x = 72 x^{3}+90x^{2}-839808,\quad x=72
  12. x 3 + S 2 x 2 - P 2 2 S = 0 x^{3}+\frac{S}{2}x^{2}-\frac{P^{2}}{2S}=0
  13. x 3 + 5 2 D x 2 + 2 D 2 x = P 2 2 D - D 2 2 x^{3}+\frac{5}{2}Dx^{2}+2D^{2}x=\frac{P^{2}}{2D}-\frac{D^{2}}{2}
  14. x 4 + ( 16 1 2 ) 2 x 2 = ( 164 14 15 ) 2 x^{4}+\left(16\frac{1}{2}\right)^{2}x^{2}=\left(164\frac{14}{15}\right)^{2}

Jitter_(optics).html

  1. M T F j i t t e r ( k ) = e - 1 2 k 2 σ 2 MTF_{jitter}(k)=e^{-\frac{1}{2}k^{2}\sigma^{2}}
  2. σ \sigma
  3. M T F j i t t e r ( u ) = e - 2 π 2 u 2 σ 2 MTF_{jitter}(u)=e^{-2\pi^{2}u^{2}\sigma^{2}}
  4. σ \sigma

Jodie_Williams.html

  1. \infty

Joint_(audio_engineering).html

  1. M = L + R M=L+R
  2. S = L - R S=L-R
  3. L = M + S 2 L=\frac{M+S}{2}
  4. R = M - S 2 R=\frac{M-S}{2}
  5. L + R L+R
  6. L - R L-R

Jorge_Luis_Borges_and_mathematics.html

  1. \aleph

July_1962.html

  1. R ˙ n ¯ \bar{\dot{R}_{n}}
  2. R ˙ n {\dot{R}_{n}}

Jurin's_law.html

  1. h = 2 γ cos θ r ρ g \qquad h=\frac{2\gamma\cos\theta}{r\rho g}

Jørgensen's_inequality.html

  1. | Tr ( A ) 2 - 4 | + | Tr ( A B A - 1 B - 1 ) - 2 | 1. |\,\text{Tr}(A)^{2}-4|+|\,\text{Tr}(ABA^{-1}B^{-1})-2|\geq 1.\,

K-independent_hashing.html

  1. k k
  2. k k
  3. k k
  4. k k
  5. U U
  6. m m
  7. [ m ] = { 0 , , m - 1 } [m]=\{0,\dots,m-1\}
  8. [ m ] [m]
  9. k k
  10. k k
  11. k {}_{k}
  12. H = { h : U [ m ] } H=\{h:U\to[m]\}
  13. k k
  14. k k
  15. ( x 1 , , x k ) U k (x_{1},\dots,x_{k})\in U^{k}
  16. k k
  17. ( y 1 , , y k ) [ m ] k (y_{1},\dots,y_{k})\in[m]^{k}
  18. Pr h H [ h ( x 1 ) = y 1 h ( x k ) = y k ] = m - k \Pr_{h\in H}\left[h(x_{1})=y_{1}\land\cdots\land h(x_{k})=y_{k}\right]=m^{-k}
  19. x U x\in U
  20. h h
  21. H H
  22. h ( x ) h(x)
  23. [ m ] [m]
  24. x 1 , , x k U x_{1},\dots,x_{k}\in U
  25. h h
  26. H H
  27. h ( x 1 ) , , h ( x k ) h(x_{1}),\dots,h(x_{k})
  28. m - k m^{-k}
  29. ( μ , k ) (\mu,k)
  30. \forall
  31. ( x 1 , , x k ) U k (x_{1},\dots,x_{k})\in U^{k}
  32. ( y 1 , , y k ) [ m ] k \forall(y_{1},\dots,y_{k})\in[m]^{k}
  33. Pr h H [ h ( x 1 ) = y 1 h ( x k ) = y k ] μ / m k ~{}~{}\Pr_{h\in H}\left[h(x_{1})=y_{1}\land\cdots\land h(x_{k})=y_{k}\right]% \leq\mu/m^{k}
  34. μ \mu
  35. h ( x i ) h(x_{i})
  36. k k

K_q-flats.html

  1. k k
  2. q q
  3. m m
  4. k k
  5. q q
  6. q q
  7. k k
  8. k k
  9. 0
  10. k k
  11. q q
  12. k k
  13. A A
  14. m m
  15. ( a 1 , a 2 , , a m ) (a_{1},a_{2},\dots,a_{m})
  16. a i a_{i}
  17. k k
  18. q q
  19. m m
  20. k k
  21. q q
  22. q q
  23. R n R^{n}
  24. R q R^{q}
  25. 0
  26. 1 1
  27. n - 1 n-1
  28. q q
  29. F = { x | x R n , W x = γ } F=\{x|x\in R^{n},W^{\prime}x=\gamma\}
  30. W R n × ( n - q ) W\in R^{n\times(n-q)}
  31. γ R 1 × ( n - q ) \gamma\in R^{1\times(n-q)}
  32. { 1 , 2 , , n } \{1,2,\dots,n\}
  33. S = ( S 1 , S 2 , , S k ) S=(S_{1},S_{2},\dots,S_{k})
  34. ( P 1 ) min F l , l = 1 , , k are q-flats min S l = 1 k a j S i a j - P F i ( a j ) 2 , (P1)\min_{F_{l},l=1,\dots,k\,\text{ are q-flats}}\min_{S}\sum_{l=1}^{k}\sum_{a% _{j}\in S_{i}}\|a_{j}-P_{F_{i}}(a_{j})\|^{2},
  35. P F i ( a j ) P_{F_{i}}(a_{j})
  36. a j a_{j}
  37. F i F_{i}
  38. a j - P F i ( a j ) = d i s t ( a j , F l ) \|a_{j}-P_{F_{i}}(a_{j})\|=dist(a_{j},F_{l})
  39. a j a_{j}
  40. F l F_{l}
  41. q q
  42. F l ( 0 ) = { x R n | ( W l ( 0 ) ) x = γ l ( 0 ) } , l = 1 , , k F_{l}^{(0)}=\{x\in R^{n}|(W_{l}^{(0)})^{\prime}x=\gamma_{l}^{(0)}\},l=1,\dots,k
  43. q q
  44. q q
  45. S i ( t ) = { a j | ( W i ( t ) ) a j - γ i ( t ) F = min l = 1 , , k ( W l ( t ) ) a j - γ l ( t ) F } . S_{i}^{(t)}=\{a_{j}|\|(W_{i}^{(t)})^{\prime}a_{j}-\gamma_{i}^{(t)}\|_{F}=\min_% {l=1,\dots,k}\|(W_{l}^{(t)})^{\prime}a_{j}-\gamma_{l}^{(t)}\|_{F}\}.
  46. q q
  47. l = 1 , , k l=1,\dots,k
  48. A ( l ) R m ( l ) × n A(l)\in R^{m(l)\times n}
  49. a i a_{i}
  50. l l
  51. W l ( t + 1 ) W_{l}^{(t+1)}
  52. ( n - q ) (n-q)
  53. A ( l ) ( I - e e m ) A ( l ) A(l)^{\prime}(I-\frac{ee^{\prime}}{m})A(l)
  54. γ l ( t + 1 ) = e A ( l ) W l ( t + 1 ) m \gamma_{l}^{(t+1)}=\frac{e^{\prime}A(l)W_{l}^{(t+1)}}{m}
  55. F l = { x | W x = γ } F_{l}=\{x|W^{\prime}x=\gamma\}
  56. a a
  57. W W = I W^{\prime}W=I
  58. a a
  59. F l F_{l}
  60. d i s t ( a , F l ) = m i n x : W x = γ x - a F 2 = W ( W W ) - 1 ( W x - γ ) F 2 = W x - γ F 2 . dist(a,F_{l})=min_{x:W^{\prime}x=\gamma}\|x-a\|_{F}^{2}=\|W(W^{\prime}W)^{-1}(% W^{\prime}x-\gamma)\|_{F}^{2}=\|W^{\prime}x-\gamma\|_{F}^{2}.
  61. m m
  62. q q
  63. q q
  64. A R m × n , A\in R^{m\times n},
  65. ( P 2 ) min W R n × ( n - q ) , γ R 1 × ( n - q ) A W - e γ F 2 , (P2)\min_{W\in R^{n\times(n-q)},\gamma\in R^{1\times(n-q)}}\|AW-e\gamma\|_{F}^% {2},
  66. W W = I , W^{\prime}W=I,
  67. A R m × n A\in R^{m\times n}
  68. e = ( 1 , , 1 ) R m × 1 e=(1,\dots,1)^{\prime}\in R^{m\times 1}
  69. k m k^{m}
  70. k k
  71. k k
  72. k k
  73. q q
  74. k k
  75. k k
  76. k k
  77. k k
  78. q q
  79. k k
  80. k k
  81. q q
  82. q = 1 q=1
  83. k k
  84. min B , R X - B R F 2 \min_{B,R}\|X-BR\|_{F}^{2}
  85. R i 0 q \|R_{i}\|_{0}\leq q
  86. R i R_{i}
  87. v 0 \|v\|_{0}
  88. V F \|V\|_{F}
  89. k k
  90. q - q-
  91. k k
  92. q - q-
  93. k k
  94. q - q-
  95. B k B_{k}
  96. d × q d\times q
  97. B k B_{k}
  98. k t h k^{th}
  99. k t h k^{th}
  100. B k r k B_{k}r_{k}
  101. r k r_{k}
  102. B = [ B 1 , , B K ] B=[B_{1},\cdots,B_{K}]
  103. min B , R X - B R F 2 \min_{B,R}\|X-BR\|_{F}^{2}
  104. R i 0 q \|R_{i}\|_{0}\leq q
  105. l = K × q l=K\times q
  106. k k
  107. q q
  108. x - P F ( x ) 2 \|x-P_{F}(x)\|^{2}
  109. P F ( x ) P_{F}(x)
  110. k k
  111. x - x c 2 \|x-x_{c}\|^{2}
  112. x c x_{c}
  113. x - y A 2 = ( x - y ) T A ( x - y ) \|x-y\|_{A}^{2}=(x-y)^{T}A(x-y)
  114. x - y A 2 \|x-y\|_{A}^{2}

Kallman–Rota_inequality.html

  1. A f 2 4 f A 2 f . \|Af\|^{2}\leq 4\|f\|\|A^{2}f\|.\,

Kamen_Rider_OOO.html

  1. \infty

Kampé_de_Fériet_function.html

  1. f r + s p + q ( a 1 , , a p : b 1 , b 1 ; ; b q , b q ; c 1 , , c r : d 1 , d 1 ; ; d s , d s ; x , y ) = m = 0 n = 0 ( a 1 ) m + n ( a p ) m + n ( c 1 ) m + n ( c r ) m + n ( b 1 ) m ( b 1 ) n ( b q ) m ( b q ) n ( d 1 ) m ( d 1 ) n ( d s ) m ( d s ) n x m y n m ! n ! . {}^{p+q}f_{r+s}\left(\begin{matrix}a_{1},\cdots,a_{p}\colon b_{1},b_{1}{}^{% \prime};\cdots;b_{q},b_{q}{}^{\prime};\\ c_{1},\cdots,c_{r}\colon d_{1},d_{1}{}^{\prime};\cdots;d_{s},d_{s}{}^{\prime};% \end{matrix}x,y\right)=\sum_{m=0}^{\infty}\sum_{n=0}^{\infty}\frac{(a_{1})_{m+% n}\cdots(a_{p})_{m+n}}{(c_{1})_{m+n}\cdots(c_{r})_{m+n}}\frac{(b_{1})_{m}(b_{1% }{}^{\prime})_{n}\cdots(b_{q})_{m}(b_{q}{}^{\prime})_{n}}{(d_{1})_{m}(d_{1}{}^% {\prime})_{n}\cdots(d_{s})_{m}(d_{s}{}^{\prime})_{n}}\cdot\frac{x^{m}y^{n}}{m!% n!}.

Kamrun_Nahar.html

  1. α f u e l \alpha_{fuel}
  2. P t o t a l P_{total}
  3. X v i X_{vi}
  4. n v i n_{vi}
  5. M v i M_{vi}
  6. L U r e q , f u e l - 1 = 1 C y . α f u e l P t o t a l . i = 1 n [ ( X v i ) × n v i M v i ] LU_{req,fuel-1}={1\over C_{y}}.{\alpha_{fuel}\over P_{total}}.\sum_{i=1}^{n}% \left[\frac{\left(X_{vi}\right)\times n_{vi}}{M_{vi}}\right]
  7. P t h e o , c = A c , t × C y × h i P_{theo,c}=A_{c,t}\times C_{y}\times h_{i}
  8. P m a x , c = C M P , C × h i P_{max,c}=C_{MP,C}\times h_{i}
  9. A c , t A_{c,t}
  10. C y C_{y}
  11. C M P , C C_{MP,C}
  12. h i h_{i}
  13. C y = β x y × ρ x y × ϵ f C_{y}=\beta_{xy}\times\rho_{xy}\times\epsilon_{f}
  14. β x y \beta_{xy}
  15. ρ x y \rho_{xy}
  16. ϵ f \epsilon_{f}

Kapitza's_pendulum.html

  1. y y
  2. x x
  3. x x
  4. y y
  5. ν \nu
  6. a a
  7. ω 0 = g / l \omega_{0}=\sqrt{g/l}
  8. g g
  9. l l
  10. m m
  11. φ \varphi
  12. { x = l sin φ y = - l cos φ - a cos ν t \begin{cases}x&=l\sin\varphi\\ y&=-l\cos\varphi-a\cos\nu t\end{cases}
  13. E POT = - m g ( l cos φ + a cos ν t ) . E_{\mathrm{POT}}=-mg(l\cos\varphi+a\cos\nu t).\,
  14. E KIN = m l 2 φ ˙ 2 / 2 E_{\mathrm{KIN}}=ml^{2}\dot{\varphi}^{2}/2
  15. E KIN = m l 2 2 φ ˙ 2 + m a l ν sin ( ν t ) sin ( φ ) φ ˙ + m a 2 ν 2 2 sin 2 ( ν t ) . E_{\mathrm{KIN}}=\frac{ml^{2}}{2}\dot{\varphi}^{2}+mal\nu~{}\sin(\nu t)\sin(% \varphi)~{}\dot{\varphi}+\frac{ma^{2}\nu^{2}}{2}\sin^{2}(\nu t)\;.
  16. E = E KIN + E POT E=E_{\mathrm{KIN}}+E_{\mathrm{POT}}
  17. L = E KIN - E POT L=E_{\mathrm{KIN}}-E_{\mathrm{POT}}
  18. t t
  19. E POT E_{\mathrm{POT}}
  20. E KIN E_{\mathrm{KIN}}
  21. E POT = m g y E_{\mathrm{POT}}=mgy
  22. - m g ( l + a ) < E POT < m g ( l + a ) -mg(l+a)<E_{\mathrm{POT}}<mg(l+a)
  23. E KIN 0 E_{\mathrm{KIN}}\geq 0
  24. ν \nu
  25. φ \varphi
  26. d d t L φ ˙ = L φ , \frac{d}{dt}\frac{\partial L}{\partial\dot{\varphi}}=\frac{\partial L}{% \partial\varphi},
  27. L L
  28. L = m l 2 2 φ ˙ 2 + m l ( g + a ν 2 cos ν t ) cos φ , L=\frac{ml^{2}}{2}\dot{\varphi}^{2}+ml(g+a~{}\nu^{2}\cos\nu t)\cos\varphi,
  29. φ ¨ = - ( g + a ν 2 cos ν t ) sin φ l , \ddot{\varphi}=-(g+a~{}\nu^{2}\cos\nu t)\frac{\sin\varphi}{l},
  30. sin φ \sin\varphi
  31. a = 0 a=0
  32. x 2 + y 2 = l 2 = constant x^{2}+y^{2}=l^{2}=\,\text{constant}
  33. E > m g l E>mgl
  34. E < m g l E<mgl
  35. ( x , y ) = ( 0 , - l ) (x,y)=(0,-l)
  36. a l a\ll l
  37. ν ω 0 \nu\gg\omega_{0}
  38. ω 0 \omega_{0}
  39. φ \varphi
  40. φ = φ 0 + ξ \varphi=\varphi_{0}+\xi
  41. φ 0 \varphi_{0}
  42. ξ \xi
  43. ( a / l ) , ( ω 0 / ν ) 1 (a/l),(\omega_{0}/\nu)\ll 1
  44. ( a / l ) ( ν / ω 0 ) (a/l)(\nu/\omega_{0})
  45. a 0 , ν a\to 0,\nu\to\infty
  46. ξ \xi
  47. ξ = a l sin φ 0 cos ν t . \xi=\frac{a}{l}\sin\varphi_{0}~{}\cos\nu t.\,
  48. φ 0 \varphi_{0}
  49. φ ¨ 0 = φ ¨ - ξ ¨ \displaystyle\ddot{\varphi}_{0}=\ddot{\varphi}-\ddot{\xi}
  50. ν \nu
  51. φ ¨ 0 = - g l sin φ 0 - 1 2 ( a ν l ) 2 sin φ 0 cos φ 0 . \ddot{\varphi}_{0}=-\frac{g}{l}\sin\varphi_{0}-\frac{1}{2}\left(\frac{a\nu}{l}% \right)^{2}\sin\varphi_{0}\cos\varphi_{0}.
  52. m l 2 φ ¨ 0 = - V eff φ 0 , ml^{2}\ddot{\varphi}_{0}=-\frac{\partial V_{\mathrm{eff}}}{\partial\varphi_{0}% }\;,
  53. V eff = - m g l cos φ 0 + m ( a ν 2 sin φ 0 ) 2 . V_{\mathrm{eff}}=-mgl\cos\varphi_{0}+m\left(\frac{a\nu}{2}\sin\varphi_{0}% \right)^{2}.
  54. V eff V_{\mathrm{eff}}
  55. ( a ν ) 2 > 2 g l (a\nu)^{2}>2gl
  56. ( a / l ) ( ν / ω 0 ) > 2 (a/l)(\nu/\omega_{0})>\sqrt{2}
  57. ( x , y ) = ( 0 , - l ) (x,y)=(0,-l)
  58. ( x , y ) = ( 0 , l ) (x,y)=(0,l)
  59. φ φ ± ν t \varphi\rightarrow\varphi^{\prime}\pm\nu t
  60. φ \varphi
  61. φ ¨ = - 1 l [ 1 2 a ν 2 sin ( φ ) + g sin ( φ ± ν t ) + 1 2 a ν 2 sin ( φ ± 2 ν t ) ] . \ddot{\varphi}^{\prime}=-\frac{1}{l}\left[\frac{1}{2}a\nu^{2}\sin(\varphi^{% \prime})+g\sin(\varphi^{\prime}\pm\nu t)+\frac{1}{2}a\nu^{2}\sin(\varphi^{% \prime}\pm 2\nu t)\right]\;.
  62. ν \nu
  63. ω 0 \omega_{0}
  64. ν \nu
  65. φ 0 \varphi_{0}^{\prime}
  66. φ ¨ 0 = - 1 2 l a ν 2 sin φ 0 . \ddot{\varphi}_{0}^{\prime}=-\frac{1}{2l}a\nu^{2}\sin\varphi_{0}^{\prime}\;.
  67. φ 0 = 0 \varphi_{0}^{\prime}=0
  68. φ 0 = π \varphi_{0}^{\prime}=\pi
  69. a l a\approx l
  70. a = l / 2 a=l/2
  71. a l a\approx l
  72. a a

Kappa_calculus.html

  1. τ = 1 τ × τ \tau=1\mid\tau\times\tau\mid\ldots
  2. e = x i d τ ! τ lift τ ( e ) e e κ x : 1 τ . e e=x\mid id_{\tau}\mid!_{\tau}\mid\operatorname{lift}_{\tau}(e)\mid e\circ e% \mid\kappa x:1{\to}\tau.e
  3. τ 1 \tau_{1}
  4. τ 2 \tau_{2}
  5. τ 1 × τ 2 \tau_{1}\times\tau_{2}
  6. τ \tau
  7. i d τ id_{\tau}
  8. τ \tau
  9. ! τ !_{\tau}
  10. τ \tau
  11. lift τ ( e ) \operatorname{lift}_{\tau}(e)
  12. e 1 e_{1}
  13. e 2 e_{2}
  14. e 1 e 2 e_{1}\circ e_{2}
  15. τ \tau
  16. κ x : 1 τ . e \kappa x{:}1{\to}\tau\;.\;e
  17. : 1 τ :1{\to}\tau
  18. lift \operatorname{lift}
  19. lift \operatorname{lift}
  20. e 1 e 2 = d e f e 1 lift ( e 2 ) e_{1}e_{2}\overset{def}{=}e_{1}\circ\operatorname{lift}(e_{2})
  21. Γ e : τ \Gamma\vdash e:\tau
  22. e : τ 1 τ 2 e:\tau_{1}{\to}\tau_{2}
  23. τ 1 {\tau_{1}}
  24. τ 2 {\tau_{2}}
  25. x : 1 τ Γ Γ x : 1 τ {x{:}1{\to}\tau\;\in\;\Gamma}\over{\Gamma\vdash x:1{\to}\tau}
  26. i d τ : τ τ {}\over{\vdash id_{\tau}\;:\;\tau\to\tau}
  27. ! τ : τ 1 {}\over{\vdash!_{\tau}\;:\;\tau\to 1}
  28. Γ e 1 : τ 1 τ 2 Γ e 2 : τ 2 τ 3 Γ e 2 e 1 : τ 1 τ 3 {\Gamma\vdash e_{1}{:}\tau_{1}{\to}\tau_{2}\;\;\;\;\;\;\Gamma\vdash e_{2}{:}% \tau_{2}{\to}\tau_{3}}\over{\Gamma\vdash e_{2}\circ e_{1}:\tau_{1}{\to}\tau_{3}}
  29. Γ e : 1 τ 1 Γ lift τ 2 ( e ) : τ 2 ( τ 1 × τ 2 ) {\Gamma\vdash e{:}1{\to}\tau_{1}}\over{\Gamma\vdash\operatorname{lift}_{\tau_{% 2}}(e)\;:\;\tau_{2}\to(\tau_{1}\times\tau_{2})}
  30. Γ , x : 1 τ 1 e : τ 2 τ 3 Γ κ x : 1 τ 1 . e : τ 1 × τ 2 τ 3 {\Gamma,\;x{:}1{\to}\tau_{1}\;\vdash\;e:\tau_{2}{\to}\tau_{3}}\over{\Gamma% \vdash\kappa x{:}1{\to}\tau_{1}\,.\,e\;:\;\tau_{1}\times\tau_{2}\to\tau_{3}}
  31. x : 1 τ x:1{\to}\tau
  32. x : 1 τ x:1{\to}\tau
  33. τ \tau
  34. i d τ : τ τ id_{\tau}:\tau{\to}\tau
  35. τ \tau
  36. ! τ : τ 1 !_{\tau}:\tau{\to}1
  37. e 1 e_{1}
  38. e 2 e_{2}
  39. e 2 e 1 e_{2}\circ e_{1}
  40. e 1 e_{1}
  41. e 2 e_{2}
  42. e : 1 τ 1 e:1{\to}\tau_{1}
  43. lift τ 2 ( e ) : τ 2 ( τ 1 × τ 2 ) \operatorname{lift}_{\tau_{2}}(e):\tau_{2}{\to}(\tau_{1}\times\tau_{2})
  44. e : τ 2 τ 3 e:\tau_{2}\to\tau_{3}
  45. x : 1 τ 1 x:1{\to}\tau_{1}
  46. κ x : 1 τ 1 . e : τ 1 × τ 2 τ 3 \kappa x{:}1{\to}\tau_{1}\,.\,e\;:\;\tau_{1}\times\tau_{2}\to\tau_{3}
  47. f : τ 1 τ 2 f:\tau_{1}{\to}\tau_{2}
  48. f i d τ 1 = f f{\circ}id_{\tau_{1}}=f
  49. f = i d τ 2 f f=id_{\tau_{2}}{\circ}f
  50. f : τ 1 τ 2 f:\tau_{1}{\to}\tau_{2}
  51. g : τ 2 τ 3 g:\tau_{2}{\to}\tau_{3}
  52. h : τ 3 τ 4 h:\tau_{3}{\to}\tau_{4}
  53. ( h g ) f = h ( g f ) (h{\circ}g){\circ}f=h{\circ}(g{\circ}f)
  54. f : τ 1 f{:}\tau{\to}1
  55. g : τ 1 g{:}\tau{\to}1
  56. f = g f=g
  57. ( κ x . f ) lift τ ( c ) = f [ c / x ] (\kappa x.f)\circ\operatorname{lift}_{\tau}(c)=f[c/x]
  58. κ x . ( h lift τ ( x ) ) = h \kappa x.(h\circ\operatorname{lift}_{\tau}(x))=h
  59. 1 τ 1{\to}\tau
  60. 1 τ 1{\to}\tau
  61. τ \tau
  62. f : A × ( B × ( C × 1 ) ) D f:A\times(B\times(C\times 1))\to D
  63. ( f lift ( c ) ) (f\circ\operatorname{lift}(c))
  64. a : 1 A a:1{\to}A
  65. b : 1 B b:1{\to}B
  66. c : 1 C c:1{\to}C
  67. f a b c : 1 D f\;a\;b\;c\;:\;1\to D
  68. f a b c fabc
  69. g : ( D × E ) F g:(D\times E){\to}F
  70. g ( f a b c ) : E F g\;(f\;a\;b\;c)\;:\;E\to F
  71. A ( B ( C D ) ) A{\to}(B{\to}(C{\to}D))
  72. f a : B × ( C × 1 ) D f\;a\;:\;B\times(C\times 1)\to D
  73. ( τ τ ) τ (\tau{\to}\tau){\to}\tau
  74. f a fa
  75. h ( f a ) h\;(f\;a)
  76. c : 1 C c:1{\to}C
  77. j c j\;c
  78. ( C × α ) β (C\times\alpha){\to}\beta
  79. α \alpha
  80. β \beta
  81. ! τ !_{\tau}
  82. × \times
  83. \otimes
  84. κ \kappa

Kardar–Parisi–Zhang_equation.html

  1. h ( x , t ) h(\vec{x},t)
  2. x \vec{x}
  3. t t
  4. h ( x , t ) t = ν 2 h + λ 2 ( h ) 2 + η ( x , t ) , \frac{\partial h(\vec{x},t)}{\partial t}=\nu\nabla^{2}h+\frac{\lambda}{2}\left% (\nabla h\right)^{2}+\eta(\vec{x},t)\;,
  5. η ( x , t ) \eta(\vec{x},t)
  6. η ( x , t ) = 0 \langle\eta(\vec{x},t)\rangle=0
  7. η ( x , t ) η ( x , t ) = 2 D δ d ( x - x ) δ ( t - t ) . \langle\eta(\vec{x},t)\eta(\vec{x}^{\prime},t^{\prime})\rangle=2D\delta^{d}(% \vec{x}-\vec{x}^{\prime})\delta(t-t^{\prime}).
  8. ν \nu
  9. λ \lambda
  10. D D
  11. d d
  12. u ( x , t ) u(x,t)
  13. u = - λ h / x u=-\lambda\,\partial h/\partial x
  14. W ( L , t ) W(L,t)
  15. W ( L , t ) = 1 L 0 L ( h ( x , t ) - h ¯ ( t ) ) 2 d x 1 / 2 , W(L,t)=\left\langle\frac{1}{L}\int_{0}^{L}\big(h(x,t)-\bar{h}(t)\big)^{2}\,dx% \right\rangle^{1/2},
  16. h ¯ ( t ) \bar{h}(t)
  17. h ( x , t ) h(x,t)
  18. W ( L , t ) L α f ( t / L z ) , W(L,t)\approx L^{\alpha}f(t/L^{z}),
  19. f ( u ) f(u)
  20. f ( u ) { u β u 1 1 u 1 f(u)\propto\begin{cases}u^{\beta}&\ u\ll 1\\ 1&\ u\gg 1\end{cases}
  21. h ( x , t ) t = ν 2 h + P ( h ) + η ( x , t ) , \frac{\partial h(\vec{x},t)}{\partial t}=\nu\nabla^{2}h+P\left(\nabla h\right)% +\eta(\vec{x},t)\;,
  22. P P

Kasch_ring.html

  1. r . ann ( x ) \mathrm{r.ann}(x)\,
  2. . ann ( T ) { 0 } \mathrm{\ell.ann}(T)\neq\{0\}
  3. . ann ( T ) { 0 } \mathrm{\ell.ann}(T)\neq\{0\}
  4. r . ann ( . ann ( T ) ) = T \mathrm{r.ann}(\mathrm{\ell.ann}(T))=T
  5. [ a 0 b c 0 a 0 d 0 0 a 0 0 0 0 e ] \begin{bmatrix}a&0&b&c\\ 0&a&0&d\\ 0&0&a&0\\ 0&0&0&e\end{bmatrix}
  6. soc ( R R ) = { 0 } \mathrm{soc}(R_{R})=\{0\}

Katz_centrality.html

  1. α \alpha
  2. α \alpha
  3. α d \alpha^{d}
  4. α = 0.5 \alpha=0.5
  5. ( 0.5 ) 1 = 0.5 (0.5)^{1}=0.5
  6. ( 0.5 ) 2 = 0.25 (0.5)^{2}=0.25
  7. ( 0.5 ) 3 = 0.125 (0.5)^{3}=0.125
  8. ( a i j ) (a_{ij})
  9. A 3 A^{3}
  10. ( a 2 , 12 ) = 1 (a_{2,12})=1
  11. C Katz ( i ) C_{\mathrm{Katz}}(i)
  12. C Katz ( i ) = k = 1 j = 1 n α k ( A k ) j i C_{\mathrm{Katz}}(i)=\sum_{k=1}^{\infty}\sum_{j=1}^{n}\alpha^{k}(A^{k})_{ji}
  13. ( i , j ) (i,j)
  14. A A
  15. k k
  16. A k A^{k}
  17. k k
  18. i i
  19. j j
  20. C Katz = ( ( I - α A T ) - 1 - I ) I \overrightarrow{C}_{\mathrm{Katz}}=((I-\alpha A^{T})^{-1}-I)\overrightarrow{I}
  21. I \overrightarrow{I}
  22. A T A^{T}
  23. ( I - α A T ) - 1 (I-\alpha A^{T})^{-1}
  24. ( I - α A T ) (I-\alpha A^{T})

Kazamaki's_condition.html

  1. M = ( M t ) t 0 M=(M_{t})_{t\geq 0}
  2. ( t ) t 0 (\mathcal{F}_{t})_{t\geq 0}
  3. ( exp ( M t / 2 ) ) t 0 (\exp(M_{t}/2))_{t\geq 0}

Källén–Lehmann_spectral_representation.html

  1. Δ ( p ) = 0 d μ 2 ρ ( μ 2 ) 1 p 2 - μ 2 + i ϵ \Delta(p)=\int_{0}^{\infty}d\mu^{2}\rho(\mu^{2})\frac{1}{p^{2}-\mu^{2}+i\epsilon}
  2. ρ ( μ 2 ) \rho(\mu^{2})
  3. Φ ( x ) \Phi(x)
  4. { | n } \{|n\rangle\}
  5. 0 | Φ ( x ) Φ ( y ) | 0 = n 0 | Φ ( x ) | n n | Φ ( y ) | 0 . \langle 0|\Phi(x)\Phi^{\dagger}(y)|0\rangle=\sum_{n}\langle 0|\Phi(x)|n\rangle% \langle n|\Phi^{\dagger}(y)|0\rangle.
  6. 0 | Φ ( x ) Φ ( y ) | 0 = n e - i p n ( x - y ) | 0 | Φ ( 0 ) | n | 2 . \langle 0|\Phi(x)\Phi^{\dagger}(y)|0\rangle=\sum_{n}e^{-ip_{n}\cdot(x-y)}|% \langle 0|\Phi(0)|n\rangle|^{2}.
  7. ρ ( p 2 ) θ ( p 0 ) ( 2 π ) - 3 = n δ 4 ( p - p n ) | 0 | Φ ( 0 ) | n | 2 \rho(p^{2})\theta(p_{0})(2\pi)^{-3}=\sum_{n}\delta^{4}(p-p_{n})|\langle 0|\Phi% (0)|n\rangle|^{2}
  8. p μ p_{\mu}
  9. p 2 p^{2}
  10. p 2 0 p^{2}\geq 0
  11. p 0 > 0 p_{0}>0
  12. 0 | Φ ( x ) Φ ( y ) | 0 = d 4 p ( 2 π ) 3 0 d μ 2 e - i p ( x - y ) ρ ( μ 2 ) θ ( p 0 ) δ ( p 2 - μ 2 ) \langle 0|\Phi(x)\Phi^{\dagger}(y)|0\rangle=\int\frac{d^{4}p}{(2\pi)^{3}}\int_% {0}^{\infty}d\mu^{2}e^{-ip\cdot(x-y)}\rho(\mu^{2})\theta(p_{0})\delta(p^{2}-% \mu^{2})
  13. 0 | Φ ( x ) Φ ( y ) | 0 = 0 d μ 2 ρ ( μ 2 ) Δ ( x - y ; μ 2 ) \langle 0|\Phi(x)\Phi^{\dagger}(y)|0\rangle=\int_{0}^{\infty}d\mu^{2}\rho(\mu^% {2})\Delta^{\prime}(x-y;\mu^{2})
  14. Δ ( x - y ; μ 2 ) = d 4 p ( 2 π ) 3 e - i p ( x - y ) θ ( p 0 ) δ ( p 2 - μ 2 ) \Delta^{\prime}(x-y;\mu^{2})=\int\frac{d^{4}p}{(2\pi)^{3}}e^{-ip\cdot(x-y)}% \theta(p_{0})\delta(p^{2}-\mu^{2})
  15. 0 | Φ ( x ) Φ ( y ) | 0 \langle 0|\Phi^{\dagger}(x)\Phi(y)|0\rangle
  16. 0 | T Φ ( x ) Φ ( y ) | 0 = 0 d μ 2 ρ ( μ 2 ) Δ ( x - y ; μ 2 ) \langle 0|T\Phi(x)\Phi^{\dagger}(y)|0\rangle=\int_{0}^{\infty}d\mu^{2}\rho(\mu% ^{2})\Delta(x-y;\mu^{2})
  17. Δ ( p ; μ 2 ) = 1 p 2 - μ 2 + i ϵ \Delta(p;\mu^{2})=\frac{1}{p^{2}-\mu^{2}+i\epsilon}

Keeper_of_the_Archives.html

  1. \infty

Kennicutt–Schmidt_law.html

  1. n n
  2. Σ S F R ( Σ g a s ) n \Sigma_{SFR}\propto(\Sigma_{gas})^{n}
  3. ( Σ S F R ) (\Sigma_{SFR})
  4. ( M yr - 1 pc - 2 ) (M_{\odot}~{}\textrm{ yr}^{-1}\textrm{ pc}^{-2})
  5. ( g pc - 2 ) (\textrm{g}~{}\textrm{pc}^{-2})
  6. n 2 n\approx 2
  7. n = 1.4 ± 0.15 n=1.4\pm 0.15

Kepler's_equation.html

  1. E = i H E=iH
  2. M = i ( E - ε sin E ) M=i\left(E-\varepsilon\sin E\right)
  3. E = 2 sin - 1 ( x ) E=2\sin^{-1}(\sqrt{x})
  4. t ( x ) = 1 2 [ E - sin ( E ) ] . t(x)=\frac{1}{2}\left[E-\sin(E)\right].
  5. E = { n = 1 M n 3 n ! lim θ 0 + ( d n - 1 d θ n - 1 ( ( θ θ - sin ( θ ) 3 ) n ) ) , ε = 1 n = 1 M n n ! lim θ 0 + ( d n - 1 d θ n - 1 ( ( θ θ - ε sin ( θ ) ) n ) ) , ε 1 E=\begin{cases}\displaystyle\sum_{n=1}^{\infty}{\frac{M^{\frac{n}{3}}}{n!}}% \lim_{\theta\to 0^{+}}\!\Bigg(\frac{\mathrm{d}^{\,n-1}}{\mathrm{d}\theta^{\,n-% 1}}\bigg(\bigg(\frac{\theta}{\sqrt[3]{\theta-\sin(\theta)}}\bigg)^{\!\!\!n}% \bigg)\Bigg),&\varepsilon=1\\ \displaystyle\sum_{n=1}^{\infty}{\frac{M^{n}}{n!}}\lim_{\theta\to 0^{+}}\!% \Bigg(\frac{\mathrm{d}^{\,n-1}}{\mathrm{d}\theta^{\,n-1}}\bigg(\Big(\frac{% \theta}{\theta-\varepsilon\sin(\theta)}\Big)^{\!n}\bigg)\Bigg),&\varepsilon% \neq 1\end{cases}
  6. E = { x + 1 60 x 3 + 1 1400 x 5 + 1 25200 x 7 + 43 17248000 x 9 + 1213 7207200000 x 11 + 151439 12713500800000 x 13 + | x = ( 6 M ) 1 3 , ε = 1 1 1 - ε M - ε ( 1 - ε ) 4 M 3 3 ! + ( 9 ε 2 + ε ) ( 1 - ε ) 7 M 5 5 ! - ( 225 ε 3 + 54 ε 2 + ε ) ( 1 - ε ) 10 M 7 7 ! + ( 11025 ε 4 + 4131 ε 3 + 243 ε 2 + ε ) ( 1 - ε ) 13 M 9 9 ! + , ε 1 E=\begin{cases}\displaystyle x+\frac{1}{60}x^{3}+\frac{1}{1400}x^{5}+\frac{1}{% 25200}x^{7}+\frac{43}{17248000}x^{9}+\frac{1213}{7207200000}x^{11}+\frac{15143% 9}{12713500800000}x^{13}+\cdots\ |\ x=(6M)^{\frac{1}{3}},&\varepsilon=1\\ \\ \displaystyle\frac{1}{1-\varepsilon}M-\frac{\varepsilon}{(1-\varepsilon)^{4}}% \frac{M^{3}}{3!}+\frac{(9\varepsilon^{2}+\varepsilon)}{(1-\varepsilon)^{7}}% \frac{M^{5}}{5!}-\frac{(225\varepsilon^{3}+54\varepsilon^{2}+\varepsilon)}{(1-% \varepsilon)^{10}}\frac{M^{7}}{7!}+\frac{(11025\varepsilon^{4}+4131\varepsilon% ^{3}+243\varepsilon^{2}+\varepsilon)}{(1-\varepsilon)^{13}}\frac{M^{9}}{9!}+% \cdots,&\varepsilon\neq 1\end{cases}
  7. x ( t ) = n = 1 [ lim r 0 + ( t 2 3 n n ! d n - 1 d r n - 1 ( r n ( 3 2 ( sin - 1 ( r ) - r - r 2 ) ) - 2 3 n ) ) ] x(t)=\sum_{n=1}^{\infty}\left[\lim_{r\to 0^{+}}\left({\frac{t^{\frac{2}{3}n}}{% n!}}\frac{\mathrm{d}^{\,n-1}}{\mathrm{d}r^{\,n-1}}\!\left(r^{n}\left(\frac{3}{% 2}\Big(\sin^{-1}(\sqrt{r})-\sqrt{r-r^{2}}\Big)\right)^{\!-\frac{2}{3}n}\right)% \right)\right]
  8. x ( t ) = p - 1 5 p 2 - 3 175 p 3 - 23 7875 p 4 - 1894 3931875 p 5 - 3293 21896875 p 6 - 2418092 62077640625 p 7 - | p = ( 3 2 t ) 2 / 3 x(t)=p-\frac{1}{5}p^{2}-\frac{3}{175}p^{3}-\frac{23}{7875}p^{4}-\frac{1894}{39% 31875}p^{5}-\frac{3293}{21896875}p^{6}-\frac{2418092}{62077640625}p^{7}-\ % \cdots\ \bigg|{p=\left(\tfrac{3}{2}t\right)^{2/3}}
  9. f ( E ) = E - ε sin ( E ) - M ( t ) f(E)=E-\varepsilon\sin(E)-M(t)
  10. E n + 1 = E n - f ( E n ) f ( E n ) = E n - E n - ε sin ( E n ) - M ( t ) 1 - ε cos ( E n ) E_{n+1}=E_{n}-\frac{f(E_{n})}{f^{\prime}(E_{n})}=E_{n}-\frac{E_{n}-\varepsilon% \sin(E_{n})-M(t)}{1-\varepsilon\cos(E_{n})}

Kernel_Fisher_discriminant_analysis.html

  1. 𝐂 1 \mathbf{C}_{1}
  2. 𝐂 2 \mathbf{C}_{2}
  3. 𝐦 1 \mathbf{m}_{1}
  4. 𝐦 2 \mathbf{m}_{2}
  5. 𝐦 i = 1 l i n = 1 l i 𝐱 n i , \mathbf{m}_{i}=\frac{1}{l_{i}}\sum_{n=1}^{l_{i}}\mathbf{x}_{n}^{i},
  6. l i l_{i}
  7. 𝐂 i \mathbf{C}_{i}
  8. J ( 𝐰 ) = 𝐰 T 𝐒 B 𝐰 𝐰 T 𝐒 W 𝐰 , J(\mathbf{w})=\frac{\mathbf{w}^{\,\text{T}}\mathbf{S}_{B}\mathbf{w}}{\mathbf{w% }^{\,\text{T}}\mathbf{S}_{W}\mathbf{w}},
  9. 𝐒 B \mathbf{S}_{B}
  10. 𝐒 W \mathbf{S}_{W}
  11. 𝐒 B \displaystyle\mathbf{S}_{B}
  12. J ( 𝐰 ) J(\mathbf{w})
  13. 𝐰 \mathbf{w}
  14. ( 𝐰 T 𝐒 B 𝐰 ) 𝐒 W 𝐰 = ( 𝐰 T 𝐒 W 𝐰 ) 𝐒 B 𝐰 . (\mathbf{w}^{\,\text{T}}\mathbf{S}_{B}\mathbf{w})\mathbf{S}_{W}\mathbf{w}=(% \mathbf{w}^{\,\text{T}}\mathbf{S}_{W}\mathbf{w})\mathbf{S}_{B}\mathbf{w}.
  15. 𝐰 \mathbf{w}
  16. 𝐒 B 𝐰 \mathbf{S}_{B}\mathbf{w}
  17. ( 𝐦 2 - 𝐦 1 ) (\mathbf{m}_{2}-\mathbf{m}_{1})
  18. 𝐒 B 𝐰 \mathbf{S}_{B}\mathbf{w}
  19. ( 𝐦 2 - 𝐦 1 ) (\mathbf{m}_{2}-\mathbf{m}_{1})
  20. ( 𝐰 T 𝐒 B 𝐰 ) (\mathbf{w}^{\,\text{T}}\mathbf{S}_{B}\mathbf{w})
  21. ( 𝐰 T 𝐒 W 𝐰 ) (\mathbf{w}^{\,\text{T}}\mathbf{S}_{W}\mathbf{w})
  22. 𝐰 𝐒 W - 1 ( 𝐦 2 - 𝐦 1 ) . \mathbf{w}\propto\mathbf{S}^{-1}_{W}(\mathbf{m}_{2}-\mathbf{m}_{1}).
  23. F F
  24. ϕ \phi
  25. J ( 𝐰 ) = 𝐰 T 𝐒 B ϕ 𝐰 𝐰 T 𝐒 W ϕ 𝐰 , J(\mathbf{w})=\frac{\mathbf{w}^{\,\text{T}}\mathbf{S}_{B}^{\phi}\mathbf{w}}{% \mathbf{w}^{\,\text{T}}\mathbf{S}_{W}^{\phi}\mathbf{w}},
  26. 𝐒 B ϕ \displaystyle\mathbf{S}_{B}^{\phi}
  27. 𝐦 i ϕ = 1 l i j = 1 l i ϕ ( 𝐱 j i ) . \mathbf{m}_{i}^{\phi}=\frac{1}{l_{i}}\sum_{j=1}^{l_{i}}\phi(\mathbf{x}_{j}^{i}).
  28. 𝐰 F \mathbf{w}\in F
  29. ϕ ( 𝐱 i ) \phi(\mathbf{x}_{i})
  30. F F
  31. F F
  32. k ( 𝐱 , 𝐲 ) = ϕ ( 𝐱 ) ϕ ( 𝐲 ) k(\mathbf{x},\mathbf{y})=\phi(\mathbf{x})\cdot\phi(\mathbf{y})
  33. 𝐰 \mathbf{w}
  34. 𝐰 = i = 1 l α i ϕ ( 𝐱 i ) . \mathbf{w}=\sum_{i=1}^{l}\alpha_{i}\phi(\mathbf{x}_{i}).
  35. 𝐰 T 𝐦 i ϕ = 1 l i j = 1 l k = 1 l i α j k ( 𝐱 j , 𝐱 k i ) = α T 𝐌 i , \mathbf{w}^{\,\text{T}}\mathbf{m}_{i}^{\phi}=\frac{1}{l_{i}}\sum_{j=1}^{l}\sum% _{k=1}^{l_{i}}\alpha_{j}k(\mathbf{x}_{j},\mathbf{x}_{k}^{i})=\mathbf{\alpha}^{% \,\text{T}}\mathbf{M}_{i},
  36. ( 𝐌 i ) j = 1 l i k = 1 l i k ( 𝐱 j , 𝐱 k i ) . (\mathbf{M}_{i})_{j}=\frac{1}{l_{i}}\sum_{k=1}^{l_{i}}k(\mathbf{x}_{j},\mathbf% {x}_{k}^{i}).
  37. J ( 𝐰 ) J(\mathbf{w})
  38. 𝐰 T 𝐒 B ϕ 𝐰 = 𝐰 T ( 𝐦 2 ϕ - 𝐦 1 ϕ ) ( 𝐦 2 ϕ - 𝐦 1 ϕ ) T 𝐰 = α T 𝐌 α , \begin{aligned}\displaystyle\mathbf{w}^{\,\text{T}}\mathbf{S}_{B}^{\phi}% \mathbf{w}&\displaystyle=\mathbf{w}^{\,\text{T}}(\mathbf{m}_{2}^{\phi}-\mathbf% {m}_{1}^{\phi})(\mathbf{m}_{2}^{\phi}-\mathbf{m}_{1}^{\phi})^{\,\text{T}}% \mathbf{w}\\ &\displaystyle=\mathbf{\alpha}^{\,\text{T}}\mathbf{M}\mathbf{\alpha},\end{aligned}
  39. 𝐌 = ( 𝐌 2 - 𝐌 1 ) ( 𝐌 2 - 𝐌 1 ) T \mathbf{M}=(\mathbf{M}_{2}-\mathbf{M}_{1})(\mathbf{M}_{2}-\mathbf{M}_{1})^{\,% \text{T}}
  40. 𝐰 T 𝐒 W ϕ 𝐰 = α T 𝐍 α , \mathbf{w}^{\,\text{T}}\mathbf{S}_{W}^{\phi}\mathbf{w}=\mathbf{\alpha}^{\,% \text{T}}\mathbf{N}\mathbf{\alpha},
  41. 𝐍 = j = 1 , 2 𝐊 j ( 𝐈 - 𝟏 l j ) 𝐊 j T , \mathbf{N}=\sum_{j=1,2}\mathbf{K}_{j}(\mathbf{I}-\mathbf{1}_{l_{j}})\mathbf{K}% _{j}^{\,\text{T}},
  42. n th , m th n^{\,\text{th}},m^{\,\text{th}}
  43. 𝐊 j \mathbf{K}_{j}
  44. k ( 𝐱 n , 𝐱 m j ) k(\mathbf{x}_{n},\mathbf{x}_{m}^{j})
  45. 𝐈 \mathbf{I}
  46. 𝟏 l j \mathbf{1}_{l_{j}}
  47. 1 / l j 1/l_{j}
  48. 𝐰 T 𝐒 W ϕ 𝐰 \mathbf{w}^{\,\text{T}}\mathbf{S}_{W}^{\phi}\mathbf{w}
  49. 𝐰 \mathbf{w}
  50. 𝐒 W ϕ \mathbf{S}_{W}^{\phi}
  51. 𝐦 i ϕ \mathbf{m}_{i}^{\phi}
  52. 𝐰 T 𝐒 W ϕ 𝐰 \displaystyle\mathbf{w}^{\,\text{T}}\mathbf{S}_{W}^{\phi}\mathbf{w}
  53. J ( 𝐰 ) J(\mathbf{w})
  54. J J
  55. J ( α ) = α T 𝐌 α α T 𝐍 α . J(\mathbf{\alpha})=\frac{\mathbf{\alpha}^{\,\text{T}}\mathbf{M}\mathbf{\alpha}% }{\mathbf{\alpha}^{\,\text{T}}\mathbf{N}\mathbf{\alpha}}.
  56. ( α T 𝐌 α ) 𝐍 α = ( α T 𝐍 α ) 𝐌 α . (\mathbf{\alpha}^{\,\text{T}}\mathbf{M}\mathbf{\alpha})\mathbf{N}\mathbf{% \alpha}=(\mathbf{\alpha}^{\,\text{T}}\mathbf{N}\mathbf{\alpha})\mathbf{M}% \mathbf{\alpha}.
  57. 𝐰 \mathbf{w}
  58. α \mathbf{\alpha}
  59. α \mathbf{\alpha}
  60. α = 𝐍 - 1 ( 𝐌 2 - 𝐌 1 ) . \mathbf{\alpha}=\mathbf{N}^{-1}(\mathbf{M}_{2}-\mathbf{M}_{1}).
  61. 𝐍 \mathbf{N}
  62. 𝐍 ϵ = 𝐍 + ϵ 𝐈 . \mathbf{N}_{\epsilon}=\mathbf{N}+\epsilon\mathbf{I}.
  63. α \mathbf{\alpha}
  64. y ( 𝐱 ) = ( 𝐰 ϕ ( 𝐱 ) ) = i = 1 l α i k ( 𝐱 i , 𝐱 ) . y(\mathbf{x})=(\mathbf{w}\cdot\phi(\mathbf{x}))=\sum_{i=1}^{l}\alpha_{i}k(% \mathbf{x}_{i},\mathbf{x}).
  65. c c
  66. ( c - 1 ) (c-1)
  67. ( c - 1 ) (c-1)
  68. y i = 𝐰 i T ϕ ( 𝐱 ) i = 1 , , c - 1. y_{i}=\mathbf{w}_{i}^{\,\text{T}}\phi(\mathbf{x})\qquad i=1,\ldots,c-1.
  69. 𝐲 = 𝐖 T ϕ ( 𝐱 ) , \mathbf{y}=\mathbf{W}^{\,\text{T}}\phi(\mathbf{x}),
  70. 𝐰 i \mathbf{w}_{i}
  71. 𝐖 \mathbf{W}
  72. 𝐒 B ϕ = i = 1 c l i ( 𝐦 i ϕ - 𝐦 ϕ ) ( 𝐦 i ϕ - 𝐦 ϕ ) T , \mathbf{S}_{B}^{\phi}=\sum_{i=1}^{c}l_{i}(\mathbf{m}_{i}^{\phi}-\mathbf{m}^{% \phi})(\mathbf{m}_{i}^{\phi}-\mathbf{m}^{\phi})^{\,\text{T}},
  73. 𝐦 ϕ \mathbf{m}^{\phi}
  74. 𝐒 W ϕ = i = 1 c n = 1 l i ( ϕ ( 𝐱 n i ) - 𝐦 i ϕ ) ( ϕ ( 𝐱 n i ) - 𝐦 i ϕ ) T , \mathbf{S}_{W}^{\phi}=\sum_{i=1}^{c}\sum_{n=1}^{l_{i}}(\phi(\mathbf{x}_{n}^{i}% )-\mathbf{m}_{i}^{\phi})(\phi(\mathbf{x}_{n}^{i})-\mathbf{m}_{i}^{\phi})^{\,% \text{T}},
  75. J ( 𝐖 ) = | 𝐖 T 𝐒 B ϕ 𝐖 | | 𝐖 T 𝐒 W ϕ 𝐖 | . J(\mathbf{W})=\frac{\left|\mathbf{W}^{\,\text{T}}\mathbf{S}_{B}^{\phi}\mathbf{% W}\right|}{\left|\mathbf{W}^{\,\text{T}}\mathbf{S}_{W}^{\phi}\mathbf{W}\right|}.
  76. 𝐀 * = argmax 𝐀 = | 𝐀 T 𝐌𝐀 | | 𝐀 T 𝐍𝐀 | , \mathbf{A}^{*}=\underset{\mathbf{A}}{\operatorname{argmax}}=\frac{\left|% \mathbf{A}^{\,\text{T}}\mathbf{M}\mathbf{A}\right|}{\left|\mathbf{A}^{\,\text{% T}}\mathbf{N}\mathbf{A}\right|},
  77. A = [ α 1 , , α c - 1 ] A=[\mathbf{\alpha}_{1},\ldots,\mathbf{\alpha}_{c-1}]
  78. M \displaystyle M
  79. 𝐌 i \mathbf{M}_{i}
  80. 𝐌 * \mathbf{M}_{*}
  81. ( 𝐌 * ) j = 1 l k = 1 l k ( 𝐱 j , 𝐱 k ) . (\mathbf{M}_{*})_{j}=\frac{1}{l}\sum_{k=1}^{l}k(\mathbf{x}_{j},\mathbf{x}_{k}).
  82. 𝐀 * \mathbf{A}^{*}
  83. ( c - 1 ) (c-1)
  84. 𝐍 - 1 𝐌 \mathbf{N}^{-1}\mathbf{M}
  85. 𝐱 t \mathbf{x}_{t}
  86. 𝐲 ( 𝐱 t ) = ( 𝐀 * ) T 𝐊 t , \mathbf{y}(\mathbf{x}_{t})=\left(\mathbf{A}^{*}\right)^{\,\text{T}}\mathbf{K}_% {t},
  87. i t h i^{th}
  88. 𝐊 t \mathbf{K}_{t}
  89. k ( 𝐱 i , 𝐱 t ) k(\mathbf{x}_{i},\mathbf{x}_{t})
  90. f ( 𝐱 ) = a r g min j D ( 𝐲 ( 𝐱 ) , 𝐲 ¯ j ) , f(\mathbf{x})=arg\min_{j}D(\mathbf{y}(\mathbf{x}),\bar{\mathbf{y}}_{j}),
  91. 𝐲 ¯ j \bar{\mathbf{y}}_{j}
  92. j j
  93. D ( , ) D(\cdot,\cdot)

Key-recovery_attack.html

  1. 2 128 2^{128}

Kicked_rotator.html

  1. ( p , x , t ) = 1 2 p 2 + K cos ( x ) n = - δ ( t - n ) \mathcal{H}(p,x,t)=\frac{1}{2}p^{2}+K\cos(x)\sum_{n=-\infty}^{\infty}\delta(t-n)
  2. δ \textstyle\delta
  3. x \textstyle x
  4. 2 π \textstyle 2\pi
  5. p \textstyle p
  6. K \textstyle K
  7. p n + 1 = p n + K sin ( x n ) , x n + 1 = x n + p n + 1 p_{n+1}=p_{n}+K\sin(x_{n}),\;\;x_{n+1}=x_{n}+p_{n+1}
  8. p \textstyle p
  9. K > K c 0.971635 K>K_{c}\approx 0.971635\dots
  10. p ( n ) = p ( 0 ) + K i = 0 n - 1 sin ( x ( i ) ) p(n)=p(0)+K\sum_{i=0}^{n-1}\sin(x(i))
  11. n t h {n}^{th}
  12. ( Δ p ) 2 = ( p ( n ) - p ( 0 ) ) 2 = K 2 i = 0 n - 1 sin 2 ( x ( i ) ) + K 2 i j sin ( x ( i ) ) sin ( x ( j ) ) \left\langle{(\Delta p)}^{2}\right\rangle=\left\langle{(p(n)-p(0))}^{2}\right% \rangle=K^{2}\sum_{i=0}^{n-1}\left\langle{\sin}^{2}(x(i))\right\rangle+K^{2}% \sum_{i\neq j}\left\langle\sin(x(i))\sin(x(j))\right\rangle
  13. i j \textstyle i\neq j
  14. n \textstyle n
  15. 1 2 \textstyle\frac{1}{2}
  16. ( Δ p ) 2 = 1 2 K 2 n \textstyle\left\langle{(\Delta p)}^{2}\right\rangle=\frac{1}{2}K^{2}n
  17. 1 2 \textstyle\frac{1}{2}
  18. n 2 \textstyle n^{2}
  19. sin 2 x = 1 2 \textstyle\sim\left\langle{\sin}^{2}x\right\rangle=\frac{1}{2}
  20. ( Δ p ) 2 = 1 2 K 2 n 2 \textstyle\left\langle{(\Delta p)}^{2}\right\rangle=\frac{1}{2}K^{2}n^{2}
  21. 1 2 K 2 n ( Δ p ) 2 1 2 K 2 n 2 \frac{1}{2}K^{2}n\leq\left\langle{(\Delta p)}^{2}\right\rangle\leq\frac{1}{2}K% ^{2}n^{2}
  22. ( Δ p ) 2 \textstyle\left\langle{(\Delta p)}^{2}\right\rangle
  23. p = i x \textstyle p=i\hbar\frac{\partial}{\partial x}
  24. H ( x , t ) = - 1 2 2 x 2 + K cos ( x ) n = - δ ( t - n ) H(x,t)=-\frac{1}{2}\frac{\partial^{2}}{\partial x^{2}}+K\cos(x)\sum_{n=-\infty% }^{\infty}\delta(t-n)
  25. i t ψ ( t ) = H ψ ( t ) i\hbar\frac{\partial}{\partial t}\psi(t)=H\psi(t)
  26. \textstyle\hbar
  27. T \textstyle T
  28. k 0 \textstyle k_{0}
  29. k 0 2 T / m \hbar\rightarrow\hbar{{k}_{0}}^{2}T/m
  30. n t h \textstyle n^{th}
  31. | l \textstyle|l\rangle
  32. ψ ( t n ) = l = - a l ( t n ) | l \psi(t_{n})=\sum_{l=-\infty}^{\infty}a_{l}(t_{n})|l\rangle
  33. a m ( t n + 1 ) = exp [ - i m 2 / 2 ] l = - ( - i ) l - m J l - m ( K / ) a l ( t n ) a_{m}({t}_{n+1})=\exp[-im^{2}\hbar/2]\sum_{l=-\infty}^{\infty}{(-i)}^{l-m}{J}_% {l-m}(K/\hbar)a_{l}(t_{n})
  34. J α \textstyle{J}_{\alpha}
  35. α \textstyle\alpha
  36. U ^ = exp [ - i 1 2 p ^ 2 ] exp [ - i 1 K cos x ^ ] \hat{U}=\exp\left[-i\frac{1}{2\hbar}\hat{p}^{2}\right]\exp\left[-i\frac{1}{% \hbar}K\cos\hat{x}\right]
  37. t * D c l / 2 t^{*}\ \approx\ D_{cl}/\hbar^{2}
  38. D c l D_{cl}
  39. D c l t * \textstyle\sqrt{D_{cl}t^{*}}
  40. D c l K 2 / 2 D_{cl}\approx K^{2}/2
  41. ( p ( t ) - p ( 0 ) ) (p(t)-p(0))
  42. K sin ( x ( n ) ) K\sin(x(n))
  43. D c l D_{cl}
  44. C ( n ) = sin ( x ( n ) ) sin ( x ( 0 ) ) C(n)=\langle\sin(x(n))\sin(x(0))\rangle
  45. D = K 2 C ( n ) D=K^{2}\sum C(n)
  46. C ( 0 ) = 1 / 2 C(0)=1/2
  47. C ( n ) C(n)
  48. C ( n ) C ( n ) e - t / t c C(n)\mapsto C(n)e^{-t/t_{c}}
  49. t c t_{c}
  50. D D c l t * / t c [ assuming t c t * ] D\approx D_{cl}t^{*}/t_{c}\quad[\,\text{assuming }t_{c}\gg t^{*}]
  51. x x

Kig_(software).html

  1. x 2 - y 2 x^{2}-y^{2}
  2. ( x + i y ) 2 = x 2 - y 2 + i ( 2 x y ) (x+iy)^{2}=x^{2}-y^{2}+i(2xy)

Kinematic_wave.html

  1. h h
  2. t t
  3. x x
  4. h t + C h x = D 2 h x 2 , \frac{\partial h}{\partial t}+C\frac{\partial h}{\partial x}=D\frac{\partial^{% 2}h}{\partial x^{2}},
  5. h h
  6. t t
  7. x x
  8. C C
  9. D D
  10. h t + F x = 0 , \frac{\partial h}{\partial t}+\frac{\partial F}{\partial x}=0,
  11. F F
  12. F = h 2 / 2 F=h^{2}/2

Kinetic_scheme.html

  1. A j i A_{ji}
  2. A j i P i ( t ) = A i j P j ( t ) A_{ji}P_{i}(t\rightarrow\infty)=A_{ij}P_{j}(t\rightarrow\infty)
  3. d P d t = 𝐀 P \frac{d\vec{P}}{dt}=\mathbf{A}\vec{P}
  4. P \vec{P}
  5. 𝐀 \mathbf{A}
  6. 𝐀 \mathbf{A}
  7. 𝐀 \mathbf{A}
  8. 𝐀 𝐀 ( t ) \mathbf{A}\rightarrow\mathbf{A}(t)
  9. d P d t = 𝐀 ( t ) P \frac{d\vec{P}}{dt}=\mathbf{A}(t)\vec{P}
  10. d P d t = 0 t 𝐀 ( t - τ ) P ( τ ) d τ \frac{d\vec{P}}{dt}=\int^{t}_{0}\mathbf{A}(t-\tau)\vec{P}(\tau)d\tau

King–Plosser–Rebelo_preferences.html

  1. σ c \sigma_{c}
  2. σ c > 0 \sigma_{c}>0
  3. 0 < σ c < 1 0<\sigma_{c}<1
  4. σ c > 1 \sigma_{c}>1
  5. u ( C , L ) = 1 1 - σ c C 1 - σ c v ( L ) u\left({C,L}\right)=\frac{1}{{1-{\sigma_{c}}}}{C^{1-{\sigma_{c}}}}v\left(L\right)
  6. v ( L ) v\left(L\right)
  7. 0 < σ c < 1 0<\sigma_{c}<1
  8. σ c > 1 \sigma_{c}>1
  9. σ c = 1 \sigma_{c}=1
  10. u ( C , L ) = ln C t + v ( L ) u\left({C,L}\right)=\ln{C_{t}}+v\left(L\right)
  11. v ( L ) v\left(L\right)
  12. u ( C , L ) = 1 1 - σ c C 1 - σ c - z 1 - σ c ( 1 - L ) 1 + κ 1 + κ u\left(C,L\right)=\frac{1}{1-\sigma_{c}}C^{1-\sigma_{c}}-z^{1-\sigma_{c}}\frac% {\left(1-L\right)^{1+\kappa}}{1+\kappa}
  13. κ \kappa

Kirchhoff's_diffraction_formula.html

  1. U ( P ) = - 1 4 π S [ U n ( e i k s s ) - e i k s s U n ] d S U(P)=-\frac{1}{4\pi}\int_{S}\left[U\frac{\partial}{\partial n}\left(\frac{e^{% iks}}{s}\right)-\frac{e^{iks}}{s}\frac{\partial U}{\partial n}\right]dS
  2. U ( r ) = a e i k r r U(r)=\frac{ae^{ikr}}{r}
  3. U ( P ) = 1 4 π [ A 1 + A 2 + A 3 ( U n ( e i k s s ) - e i k s s U n ) ] d S U(P)=\frac{1}{4\pi}\left[\int_{A_{1}}+\int_{A_{2}}+\int_{A_{3}}\left(U\frac{% \partial}{\partial n}\left(\frac{e^{iks}}{s}\right)-\frac{e^{iks}}{s}\frac{% \partial U}{\partial n}\right)\right]dS
  4. U A 1 = a e i k r r U_{A_{1}}=\frac{ae^{ikr}}{r}
  5. U A 1 n = a e i k r r [ i k - 1 r ] cos ( n , r ) \frac{\partial U_{A_{1}}}{\partial n}=\frac{ae^{ikr}}{r}\left[ik-\frac{1}{r}% \right]\cos{(n,r)}
  6. n ( e i k s s ) = e i k s s [ i k - 1 s ] cos ( n , s ) \frac{\partial}{\partial n}\left(\frac{e^{iks}}{s}\right)=\frac{e^{iks}}{s}% \left[ik-\frac{1}{s}\right]\cos{(n,s)}
  7. U ( P ) = - i a 2 λ S e i k ( r + s ) r s [ cos ( n , r ) - cos ( n , s ) ] d S U(P)=-\frac{ia}{2\lambda}\int_{S}{\frac{e^{ik(r+s)}}{rs}[\cos(n,r)-\cos(n,s)]}dS
  8. χ = π - ( r 0 , s ) \chi=\pi-(r_{0},s)
  9. U ( r 0 ) = a e i k r 0 r 0 U(r_{0})=\frac{ae^{ikr_{0}}}{r_{0}}
  10. U ( P ) = - i 2 λ a e i k r 0 r 0 S e i k s s ( 1 + cos χ ) d S U(P)=-\frac{i}{2\lambda}\frac{ae^{ikr_{0}}}{r_{0}}\int_{S}\frac{e^{iks}}{s}(1+% \cos\chi)\,dS
  11. U 0 ( r ) a ( r ) e - i k r U_{0}(r)\approx a(r)e^{-ikr}
  12. U 0 ( r ) n = - i k a ( r ) cos ( n , r ) \frac{\partial{U_{0}(r)}}{\partial n}=-ika(r)\cos{(n,r)}
  13. U ( P ) = - i 2 λ S a ( r ) e i k s s [ cos ( n , r ) - cos ( n , s ) ] d S U(P)=-\frac{i}{2\lambda}\int_{S}{a(r)\frac{e^{iks}}{s}[\cos(n,r)-\cos(n,s)]}dS
  14. U ( P ) = - i a cos β λ r s S e i k ( r + s ) d s U(P)=-\frac{ia\cos\beta}{\lambda r^{\prime}s^{\prime}}\int_{S}e^{ik(r+s)}ds
  15. r 2 = ( x 0 - x ) 2 + ( y 0 - y ) 2 + z 0 2 ~{}r^{2}={(x_{0}-x^{\prime})^{2}+(y_{0}-y^{\prime})^{2}+z_{0}^{2}}
  16. s 2 = ( x - x ) 2 + ( y - y ) 2 + z 2 ~{}s^{2}={(x-x^{\prime})^{2}+(y-y^{\prime})^{2}+z^{2}}
  17. r 2 = x 0 2 + y 0 2 + z 0 2 ~{}r^{\prime 2}=x_{0}^{2}+y_{0}^{2}+z_{0}^{2}
  18. s 2 = x 2 + y 2 + z 2 ~{}s^{\prime 2}=x^{2}+y^{2}+z^{2}
  19. r = r [ 1 - 2 ( x 0 x + y 0 y ) r 2 + x 2 + y 2 r 2 ] 1 / 2 r=r^{\prime}\left[{1-\frac{2(x_{0}x^{\prime}+y_{0}y^{\prime})}{r^{\prime 2}}+% \frac{x^{\prime 2}+y^{\prime 2}}{r^{\prime 2}}}\right]^{1/2}
  20. s = s [ 1 - 2 ( x x + y y ) s 2 + x 2 + y 2 s 2 ] 1 / 2 s=s^{\prime}\left[{1-\frac{2(xx^{\prime}+yy^{\prime})}{s^{\prime 2}}+\frac{x^{% \prime 2}+y^{\prime 2}}{s^{\prime 2}}}\right]^{1/2}
  21. r = r [ 1 - 1 2 r 2 [ 2 ( x 0 x + y 0 y ) + ( x 2 + y 2 ) ] + 1 2 r 2 [ 2 ( x 0 x + y 0 y ) + ( x 2 + y 2 ) ] 2 + ] r=r^{\prime}\left[{1-\frac{1}{2r^{\prime 2}}[2(x_{0}x^{\prime}+y_{0}y^{\prime}% )+(x^{\prime 2}+y^{\prime 2})]+\frac{1}{2r^{\prime 2}}[2(x_{0}x^{\prime}+y_{0}% y^{\prime})+(x^{\prime 2}+y^{\prime 2})]^{2}+\cdots}\right]
  22. s = s [ 1 - 1 2 s 2 [ 2 ( x x + y y ) + ( x 2 + y 2 ) ] + 1 2 s 2 [ 2 ( x x + y y ) + ( x 2 + y 2 ) ] 2 + ] s=s^{\prime}\left[{1-\frac{1}{2s^{\prime 2}}[2(xx^{\prime}+yy^{\prime})+(x^{% \prime 2}+y^{\prime 2})]+\frac{1}{2s^{\prime 2}}[2(xx^{\prime}+yy^{\prime})+(x% ^{\prime 2}+y^{\prime 2})]^{2}+\cdots}\right]
  23. U ( P ) = - i cos β λ a e i k ( r + s ) r s S e i k f ( x , y ) d x d y U(P)=-\frac{i\cos\beta}{\lambda}\frac{ae^{ik(r^{\prime}+s^{\prime})}}{r^{% \prime}s^{\prime}}\int_{S}e^{ikf(x^{\prime},y^{\prime})}dx^{\prime}dy^{\prime}
  24. f ( x , y ) = c 1 x + c 2 y + c 3 x 2 + c 4 y 2 + c 5 x y ~{}f(x^{\prime},y^{\prime})=c_{1}x^{\prime}+c_{2}y^{\prime}+c_{3}x^{\prime 2}+% c_{4}y^{\prime 2}+c_{5}x^{\prime}y^{\prime}\cdots
  25. l 0 = - x 0 / r m 0 = - y 0 / r l = x / s m = y / s \begin{array}[]{rcl}l_{0}&=&-{x_{0}}/{r^{\prime}}\\ m_{0}&=&-{y_{0}}/{r^{\prime}}\\ l&=&{x}/{s^{\prime}}\\ m&=&{y}/{s^{\prime}}\end{array}
  26. U ( P ) = C S e i k [ ( l 0 - l ) x + ( m - m 0 ) y ] d x d y U(P)=C\int_{S}e^{ik[(l_{0}-l)x^{\prime}+(m-m_{0})y^{\prime}]}dx^{\prime}dy^{\prime}
  27. U ( P ) = C S e i ( 𝐤 𝟎 - 𝐤 ) 𝐫 d r U(P)=C\int_{S}e^{i(\mathbf{k_{0}}-\mathbf{k})\cdot\mathbf{r^{\prime}}}dr^{\prime}
  28. U ( P ) S a 0 ( 𝐫 ) e i ( 𝐤 𝟎 - 𝐤 ) 𝐫 d r U(P)\propto{\int_{S}a_{0}(\mathbf{r^{\prime}})e^{i\mathbf{(k_{0}-k)}\cdot% \mathbf{r^{\prime}}}dr^{\prime}}

Kirchhoff–Love_plate_theory.html

  1. 𝐱 \mathbf{x}
  2. 𝐱 = x 1 s y m b o l e 1 + x 2 s y m b o l e 2 + x 3 s y m b o l e 3 x i s y m b o l e i . \mathbf{x}=x_{1}symbol{e}_{1}+x_{2}symbol{e}_{2}+x_{3}symbol{e}_{3}\equiv x_{i% }symbol{e}_{i}\,.
  3. s y m b o l e i symbol{e}_{i}
  4. x 1 x_{1}
  5. x 2 x_{2}
  6. x 3 x_{3}
  7. 𝐮 ( 𝐱 ) \mathbf{u}(\mathbf{x})
  8. 𝐮 = u 1 s y m b o l e 1 + u 2 s y m b o l e 2 + u 3 s y m b o l e 3 u i s y m b o l e i \mathbf{u}=u_{1}symbol{e}_{1}+u_{2}symbol{e}_{2}+u_{3}symbol{e}_{3}\equiv u_{i% }symbol{e}_{i}
  9. w 0 w^{0}
  10. x 3 x_{3}
  11. 𝐮 0 = u 1 0 s y m b o l e 1 + u 2 0 s y m b o l e 2 u \alphasymbol 0 e α \mathbf{u}^{0}=u^{0}_{1}symbol{e}_{1}+u^{0}_{2}symbol{e}_{2}\equiv u^{0}_{% \alphasymbol}{e}_{\alpha}
  12. α \alpha
  13. φ α \varphi_{\alpha}
  14. φ α = w , α 0 \varphi_{\alpha}=w^{0}_{,\alpha}
  15. u α u_{\alpha}
  16. ε α β = 1 2 ( u α x β + u β x α ) 1 2 ( u α , β + u β , α ) ε α 3 = 1 2 ( u α x 3 + u 3 x α ) 1 2 ( u α , 3 + u 3 , α ) ε 33 = u 3 x 3 u 3 , 3 \begin{aligned}\displaystyle\varepsilon_{\alpha\beta}&\displaystyle=\frac{1}{2% }\left(\frac{\partial u_{\alpha}}{\partial x_{\beta}}+\frac{\partial u_{\beta}% }{\partial x_{\alpha}}\right)\equiv\frac{1}{2}(u_{\alpha,\beta}+u_{\beta,% \alpha})\\ \displaystyle\varepsilon_{\alpha 3}&\displaystyle=\frac{1}{2}\left(\frac{% \partial u_{\alpha}}{\partial x_{3}}+\frac{\partial u_{3}}{\partial x_{\alpha}% }\right)\equiv\frac{1}{2}(u_{\alpha,3}+u_{3,\alpha})\\ \displaystyle\varepsilon_{33}&\displaystyle=\frac{\partial u_{3}}{\partial x_{% 3}}\equiv u_{3,3}\end{aligned}
  17. q ( x ) q(x)
  18. N 11 x 1 + N 21 x 2 = 0 N 12 x 1 + N 22 x 2 = 0 2 M 11 x 1 2 + 2 2 M 12 x 1 x 2 + 2 M 22 x 2 2 = q \begin{aligned}&\displaystyle\cfrac{\partial N_{11}}{\partial x_{1}}+\cfrac{% \partial N_{21}}{\partial x_{2}}=0\\ &\displaystyle\cfrac{\partial N_{12}}{\partial x_{1}}+\cfrac{\partial N_{22}}{% \partial x_{2}}=0\\ &\displaystyle\cfrac{\partial^{2}M_{11}}{\partial x_{1}^{2}}+2\cfrac{\partial^% {2}M_{12}}{\partial x_{1}\partial x_{2}}+\cfrac{\partial^{2}M_{22}}{\partial x% _{2}^{2}}=q\end{aligned}
  19. 2 h 2h
  20. σ α β \sigma_{\alpha\beta}
  21. δ U = Ω 0 - h h s y m b o l σ : \deltasymbol ϵ d x 3 d Ω = Ω 0 - h h σ α β δ ε α β d x 3 d Ω = Ω 0 - h h [ 1 2 σ α β ( δ u α , β 0 + δ u β , α 0 ) - x 3 σ α β δ w , α β 0 ] d x 3 d Ω = Ω 0 [ 1 2 N α β ( δ u α , β 0 + δ u β , α 0 ) - M α β δ w , α β 0 ] d Ω \begin{aligned}\displaystyle\delta U&\displaystyle=\int_{\Omega^{0}}\int_{-h}^% {h}symbol{\sigma}:\deltasymbol{\epsilon}~{}dx_{3}~{}d\Omega=\int_{\Omega^{0}}% \int_{-h}^{h}\sigma_{\alpha\beta}~{}\delta\varepsilon_{\alpha\beta}~{}dx_{3}~{% }d\Omega\\ &\displaystyle=\int_{\Omega^{0}}\int_{-h}^{h}\left[\frac{1}{2}~{}\sigma_{% \alpha\beta}~{}(\delta u^{0}_{\alpha,\beta}+\delta u^{0}_{\beta,\alpha})-x_{3}% ~{}\sigma_{\alpha\beta}~{}\delta w^{0}_{,\alpha\beta}\right]~{}dx_{3}~{}d% \Omega\\ &\displaystyle=\int_{\Omega^{0}}\left[\frac{1}{2}~{}N_{\alpha\beta}~{}(\delta u% ^{0}_{\alpha,\beta}+\delta u^{0}_{\beta,\alpha})-M_{\alpha\beta}~{}\delta w^{0% }_{,\alpha\beta}\right]~{}d\Omega\end{aligned}
  22. 2 h 2h
  23. N α β := - h h σ α β d x 3 ; M α β := - h h x 3 σ α β d x 3 N_{\alpha\beta}:=\int_{-h}^{h}\sigma_{\alpha\beta}~{}dx_{3}~{};~{}~{}M_{\alpha% \beta}:=\int_{-h}^{h}x_{3}~{}\sigma_{\alpha\beta}~{}dx_{3}
  24. δ U = Ω 0 [ - 1 2 ( N α β , β δ u α 0 + N α β , α δ u β 0 ) + M α β , β δ w , α 0 ] d Ω + Γ 0 [ 1 2 ( n β N α β δ u α 0 + n α N α β δ u β 0 ) - n β M α β δ w , α 0 ] d Γ \begin{aligned}\displaystyle\delta U&\displaystyle=\int_{\Omega^{0}}\left[-% \frac{1}{2}~{}(N_{\alpha\beta,\beta}~{}\delta u^{0}_{\alpha}+N_{\alpha\beta,% \alpha}~{}\delta u^{0}_{\beta})+M_{\alpha\beta,\beta}~{}\delta w^{0}_{,\alpha}% \right]~{}d\Omega\\ &\displaystyle+\int_{\Gamma^{0}}\left[\frac{1}{2}~{}(n_{\beta}~{}N_{\alpha% \beta}~{}\delta u^{0}_{\alpha}+n_{\alpha}~{}N_{\alpha\beta}~{}\delta u^{0}_{% \beta})-n_{\beta}~{}M_{\alpha\beta}~{}\delta w^{0}_{,\alpha}\right]~{}d\Gamma% \end{aligned}
  25. N α β = N β α N_{\alpha\beta}=N_{\beta\alpha}
  26. δ U = Ω 0 [ - N α β , α δ u β 0 + M α β , β δ w , α 0 ] d Ω + Γ 0 [ n α N α β δ u β 0 - n β M α β δ w , α 0 ] d Γ \delta U=\int_{\Omega^{0}}\left[-N_{\alpha\beta,\alpha}~{}\delta u^{0}_{\beta}% +M_{\alpha\beta,\beta}~{}\delta w^{0}_{,\alpha}\right]~{}d\Omega+\int_{\Gamma^% {0}}\left[n_{\alpha}~{}N_{\alpha\beta}~{}\delta u^{0}_{\beta}-n_{\beta}~{}M_{% \alpha\beta}~{}\delta w^{0}_{,\alpha}\right]~{}d\Gamma
  27. δ U = Ω 0 [ - N α β , α δ u β 0 - M α β , β α δ w 0 ] d Ω + Γ 0 [ n α N α β δ u β 0 + n α M α β , β δ w 0 - n β M α β δ w , α 0 ] d Γ \delta U=\int_{\Omega^{0}}\left[-N_{\alpha\beta,\alpha}~{}\delta u^{0}_{\beta}% -M_{\alpha\beta,\beta\alpha}~{}\delta w^{0}\right]~{}d\Omega+\int_{\Gamma^{0}}% \left[n_{\alpha}~{}N_{\alpha\beta}~{}\delta u^{0}_{\beta}+n_{\alpha}~{}M_{% \alpha\beta,\beta}~{}\delta w^{0}-n_{\beta}~{}M_{\alpha\beta}~{}\delta w^{0}_{% ,\alpha}\right]~{}d\Gamma
  28. δ U = 0 \delta U=0
  29. N α β , α = 0 M α β , α β = 0 \begin{aligned}\displaystyle N_{\alpha\beta,\alpha}&\displaystyle=0\\ \displaystyle M_{\alpha\beta,\alpha\beta}&\displaystyle=0\end{aligned}
  30. q ( x ) q(x)
  31. x 3 x_{3}
  32. δ V ext = Ω 0 q δ w 0 d Ω \delta V_{\mathrm{ext}}=\int_{\Omega^{0}}q~{}\delta w^{0}~{}d\Omega
  33. N α β , α = 0 M α β , α β - q = 0 \begin{aligned}\displaystyle N_{\alpha\beta,\alpha}&\displaystyle=0\\ \displaystyle M_{\alpha\beta,\alpha\beta}-q&\displaystyle=0\end{aligned}
  34. n α N α β or u β 0 n α M α β , β or w 0 n β M α β or w , α 0 \begin{aligned}\displaystyle n_{\alpha}~{}N_{\alpha\beta}&\displaystyle\quad% \mathrm{or}\quad u^{0}_{\beta}\\ \displaystyle n_{\alpha}~{}M_{\alpha\beta,\beta}&\displaystyle\quad\mathrm{or}% \quad w^{0}\\ \displaystyle n_{\beta}~{}M_{\alpha\beta}&\displaystyle\quad\mathrm{or}\quad w% ^{0}_{,\alpha}\end{aligned}
  35. n α M α β , β n_{\alpha}~{}M_{\alpha\beta,\beta}
  36. σ α β = C α β γ θ ε γ θ σ α 3 = C α 3 γ θ ε γ θ σ 33 = C 33 γ θ ε γ θ \begin{aligned}\displaystyle\sigma_{\alpha\beta}&\displaystyle=C_{\alpha\beta% \gamma\theta}~{}\varepsilon_{\gamma\theta}\\ \displaystyle\sigma_{\alpha 3}&\displaystyle=C_{\alpha 3\gamma\theta}~{}% \varepsilon_{\gamma\theta}\\ \displaystyle\sigma_{33}&\displaystyle=C_{33\gamma\theta}~{}\varepsilon_{% \gamma\theta}\end{aligned}
  37. σ α 3 \sigma_{\alpha 3}
  38. σ 33 \sigma_{33}
  39. [ σ 11 σ 22 σ 12 ] = [ C 11 C 12 C 13 C 12 C 22 C 23 C 13 C 23 C 33 ] [ ε 11 ε 22 ε 12 ] \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}=\begin{bmatrix}C_{11}&C_{12}&C_{13}\\ C_{12}&C_{22}&C_{23}\\ C_{13}&C_{23}&C_{33}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}
  40. [ N 11 N 22 N 12 ] = - h h [ C 11 C 12 C 13 C 12 C 22 C 23 C 13 C 23 C 33 ] [ ε 11 ε 22 ε 12 ] d x 3 = { - h h [ C 11 C 12 C 13 C 12 C 22 C 23 C 13 C 23 C 33 ] d x 3 } [ u 1 , 1 0 u 2 , 2 0 1 2 ( u 1 , 2 0 + u 2 , 1 0 ) ] \begin{bmatrix}N_{11}\\ N_{22}\\ N_{12}\end{bmatrix}=\int_{-h}^{h}\begin{bmatrix}C_{11}&C_{12}&C_{13}\\ C_{12}&C_{22}&C_{23}\\ C_{13}&C_{23}&C_{33}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}dx_{3}=\left\{\int_{-h}^{h}\begin{bmatrix}C_{11}&% C_{12}&C_{13}\\ C_{12}&C_{22}&C_{23}\\ C_{13}&C_{23}&C_{33}\end{bmatrix}~{}dx_{3}\right\}\begin{bmatrix}u^{0}_{1,1}\\ u^{0}_{2,2}\\ \frac{1}{2}~{}(u^{0}_{1,2}+u^{0}_{2,1})\end{bmatrix}
  41. [ M 11 M 22 M 12 ] = - h h x 3 [ C 11 C 12 C 13 C 12 C 22 C 23 C 13 C 23 C 33 ] [ ε 11 ε 22 ε 12 ] d x 3 = - { - h h x 3 2 [ C 11 C 12 C 13 C 12 C 22 C 23 C 13 C 23 C 33 ] d x 3 } [ w , 11 0 w , 22 0 w , 12 0 ] \begin{bmatrix}M_{11}\\ M_{22}\\ M_{12}\end{bmatrix}=\int_{-h}^{h}x_{3}~{}\begin{bmatrix}C_{11}&C_{12}&C_{13}\\ C_{12}&C_{22}&C_{23}\\ C_{13}&C_{23}&C_{33}\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}dx_{3}=-\left\{\int_{-h}^{h}x_{3}^{2}~{}\begin{% bmatrix}C_{11}&C_{12}&C_{13}\\ C_{12}&C_{22}&C_{23}\\ C_{13}&C_{23}&C_{33}\end{bmatrix}~{}dx_{3}\right\}\begin{bmatrix}w^{0}_{,11}\\ w^{0}_{,22}\\ w^{0}_{,12}\end{bmatrix}
  42. A α β := - h h C α β d x 3 A_{\alpha\beta}:=\int_{-h}^{h}C_{\alpha\beta}~{}dx_{3}
  43. D α β := - h h x 3 2 C α β d x 3 D_{\alpha\beta}:=\int_{-h}^{h}x_{3}^{2}~{}C_{\alpha\beta}~{}dx_{3}
  44. Q α = - D x α ( 2 w 0 ) . Q_{\alpha}=-D\frac{\partial}{\partial x_{\alpha}}(\nabla^{2}w^{0})\,.
  45. Q α = , α Q_{\alpha}=\mathcal{M}_{,\alpha}
  46. := - D 2 w 0 . \mathcal{M}:=-D\nabla^{2}w^{0}\,.
  47. {}^{\circ}
  48. {}^{\circ}
  49. ε α β = 1 2 ( u α , β + u β , α + u 3 , α u 3 , β ) ε α 3 = 1 2 ( u α , 3 + u 3 , α ) ε 33 = u 3 , 3 \begin{aligned}\displaystyle\varepsilon_{\alpha\beta}&\displaystyle=\tfrac{1}{% 2}(u_{\alpha,\beta}+u_{\beta,\alpha}+u_{3,\alpha}~{}u_{3,\beta})\\ \displaystyle\varepsilon_{\alpha 3}&\displaystyle=\tfrac{1}{2}(u_{\alpha,3}+u_% {3,\alpha})\\ \displaystyle\varepsilon_{33}&\displaystyle=u_{3,3}\end{aligned}
  50. ε α β = 1 2 ( u α , β 0 + u β , α 0 + w , α 0 w , β 0 ) - x 3 w , α β 0 ε α 3 = - w , α 0 + w , α 0 = 0 ε 33 = 0 \begin{aligned}\displaystyle\varepsilon_{\alpha\beta}&\displaystyle=\frac{1}{2% }(u^{0}_{\alpha,\beta}+u^{0}_{\beta,\alpha}+w^{0}_{,\alpha}~{}w^{0}_{,\beta})-% x_{3}~{}w^{0}_{,\alpha\beta}\\ \displaystyle\varepsilon_{\alpha 3}&\displaystyle=-w^{0}_{,\alpha}+w^{0}_{,% \alpha}=0\\ \displaystyle\varepsilon_{33}&\displaystyle=0\end{aligned}
  51. N α β , α = 0 M α β , α β + [ N α β w , β 0 ] , α - q = 0 \begin{aligned}\displaystyle N_{\alpha\beta,\alpha}&\displaystyle=0\\ \displaystyle M_{\alpha\beta,\alpha\beta}+[N_{\alpha\beta}~{}w^{0}_{,\beta}]_{% ,\alpha}-q&\displaystyle=0\end{aligned}
  52. [ σ 11 σ 22 σ 12 ] = E 1 - ν 2 [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ ε 11 ε 22 ε 12 ] . \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}=\cfrac{E}{1-\nu^{2}}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}\,.
  53. [ M 11 M 22 M 12 ] = - 2 h 3 E 3 ( 1 - ν 2 ) [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ w , 11 0 w , 22 0 w , 12 0 ] \begin{bmatrix}M_{11}\\ M_{22}\\ M_{12}\end{bmatrix}=-\cfrac{2h^{3}E}{3(1-\nu^{2})}~{}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}w^{0}_{,11}\\ w^{0}_{,22}\\ w^{0}_{,12}\end{bmatrix}
  54. M 11 = - D ( 2 w 0 x 1 2 + ν 2 w 0 x 2 2 ) M 22 = - D ( 2 w 0 x 2 2 + ν 2 w 0 x 1 2 ) M 12 = - D ( 1 - ν ) 2 w 0 x 1 x 2 \begin{aligned}\displaystyle M_{11}&\displaystyle=-D\left(\frac{\partial^{2}w^% {0}}{\partial x_{1}^{2}}+\nu\frac{\partial^{2}w^{0}}{\partial x_{2}^{2}}\right% )\\ \displaystyle M_{22}&\displaystyle=-D\left(\frac{\partial^{2}w^{0}}{\partial x% _{2}^{2}}+\nu\frac{\partial^{2}w^{0}}{\partial x_{1}^{2}}\right)\\ \displaystyle M_{12}&\displaystyle=-D(1-\nu)\frac{\partial^{2}w^{0}}{\partial x% _{1}\partial x_{2}}\end{aligned}
  55. D = 2 h 3 E / [ 3 ( 1 - ν 2 ) ] = H 3 E / [ 12 ( 1 - ν 2 ) ] D=2h^{3}E/[3(1-\nu^{2})]=H^{3}E/[12(1-\nu^{2})]
  56. H = 2 h H=2h
  57. σ 11 = 3 x 3 2 h 3 M 11 = 12 x 3 H 3 M 11 and σ 22 = 3 x 3 2 h 3 M 22 = 12 x 3 H 3 M 22 . \sigma_{11}=\frac{3x_{3}}{2h^{3}}\,M_{11}=\frac{12x_{3}}{H^{3}}\,M_{11}\quad\,% \text{and}\quad\sigma_{22}=\frac{3x_{3}}{2h^{3}}\,M_{22}=\frac{12x_{3}}{H^{3}}% \,M_{22}\,.
  58. x 3 = h = H / 2 x_{3}=h=H/2
  59. σ 11 = 3 2 h 2 M 11 = 6 H 2 M 11 and σ 22 = 3 2 h 2 M 22 = 6 H 2 M 22 . \sigma_{11}=\frac{3}{2h^{2}}\,M_{11}=\frac{6}{H^{2}}\,M_{11}\quad\,\text{and}% \quad\sigma_{22}=\frac{3}{2h^{2}}\,M_{22}=\frac{6}{H^{2}}\,M_{22}\,.
  60. 4 w 0 x 1 4 + 2 4 w 0 x 1 2 x 2 2 + 4 w 0 x 2 4 = 0 . \frac{\partial^{4}w^{0}}{\partial x_{1}^{4}}+2\frac{\partial^{4}w^{0}}{% \partial x_{1}^{2}\partial x_{2}^{2}}+\frac{\partial^{4}w^{0}}{\partial x_{2}^% {4}}=0\,.
  61. x 1 x_{1}
  62. x 2 x_{2}
  63. w , 1111 0 + 2 w , 1212 0 + w , 2222 0 = 0 w^{0}_{,1111}+2~{}w^{0}_{,1212}+w^{0}_{,2222}=0
  64. [ M 11 M 22 M 12 ] = - 2 h 3 E 3 ( 1 - ν 2 ) [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ w , 11 0 w , 22 0 w , 12 0 ] \begin{bmatrix}M_{11}\\ M_{22}\\ M_{12}\end{bmatrix}=-\cfrac{2h^{3}E}{3(1-\nu^{2})}~{}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}w^{0}_{,11}\\ w^{0}_{,22}\\ w^{0}_{,12}\end{bmatrix}
  65. N α β , α = 0 N 11 , 1 + N 21 , 2 = 0 , N 12 , 1 + N 22 , 2 = 0 M α β , α β = 0 M 11 , 11 + 2 M 12 , 12 + M 22 , 22 = 0 \begin{aligned}\displaystyle N_{\alpha\beta,\alpha}&\displaystyle=0\implies N_% {11,1}+N_{21,2}=0~{},~{}~{}N_{12,1}+N_{22,2}=0\\ \displaystyle M_{\alpha\beta,\alpha\beta}&\displaystyle=0\implies M_{11,11}+2M% _{12,12}+M_{22,22}=0\end{aligned}
  66. [ σ 11 σ 22 σ 12 ] = E 1 - ν 2 [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ ε 11 ε 22 ε 12 ] \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}=\cfrac{E}{1-\nu^{2}}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}
  67. [ N 11 N 22 N 12 ] = 2 h E ( 1 - ν 2 ) [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ u 1 , 1 0 u 2 , 2 0 1 2 ( u 1 , 2 0 + u 2 , 1 0 ) ] \begin{bmatrix}N_{11}\\ N_{22}\\ N_{12}\end{bmatrix}=\cfrac{2hE}{(1-\nu^{2})}~{}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}u^{0}_{1,1}\\ u^{0}_{2,2}\\ \frac{1}{2}~{}(u^{0}_{1,2}+u^{0}_{2,1})\end{bmatrix}
  68. [ M 11 M 22 M 12 ] = - 2 h 3 E 3 ( 1 - ν 2 ) [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ w , 11 0 w , 22 0 w , 12 0 ] \begin{bmatrix}M_{11}\\ M_{22}\\ M_{12}\end{bmatrix}=-\cfrac{2h^{3}E}{3(1-\nu^{2})}~{}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}w^{0}_{,11}\\ w^{0}_{,22}\\ w^{0}_{,12}\end{bmatrix}
  69. N 11 , 1 = 2 h E ( 1 - ν 2 ) ( u 1 , 11 0 + ν u 2 , 21 0 ) ; N 22 , 2 = 2 h E ( 1 - ν 2 ) ( ν u 1 , 12 0 + u 2 , 22 0 ) N 12 , 1 = h E ( 1 - ν ) ( 1 - ν 2 ) ( u 1 , 21 0 + u 2 , 11 0 ) ; N 12 , 2 = h E ( 1 - ν ) ( 1 - ν 2 ) ( u 1 , 22 0 + u 2 , 12 0 ) \begin{aligned}\displaystyle N_{11,1}&\displaystyle=\cfrac{2hE}{(1-\nu^{2})}% \left(u^{0}_{1,11}+\nu~{}u^{0}_{2,21}\right)~{};~{}~{}N_{22,2}=\cfrac{2hE}{(1-% \nu^{2})}\left(\nu~{}u^{0}_{1,12}+u^{0}_{2,22}\right)\\ \displaystyle N_{12,1}&\displaystyle=\cfrac{hE(1-\nu)}{(1-\nu^{2})}\left(u^{0}% _{1,21}+u^{0}_{2,11}\right)~{};~{}~{}N_{12,2}=\cfrac{hE(1-\nu)}{(1-\nu^{2})}% \left(u^{0}_{1,22}+u^{0}_{2,12}\right)\end{aligned}
  70. M 11 , 11 = - 2 h 3 E 3 ( 1 - ν 2 ) ( w , 1111 0 + ν w , 2211 0 ) M 22 , 22 = - 2 h 3 E 3 ( 1 - ν 2 ) ( ν w , 1122 0 + w , 2222 0 ) M 12 , 12 = - 2 h 3 E 3 ( 1 - ν 2 ) ( 1 - ν ) w , 1212 0 \begin{aligned}\displaystyle M_{11,11}&\displaystyle=-\cfrac{2h^{3}E}{3(1-\nu^% {2})}\left(w^{0}_{,1111}+\nu~{}w^{0}_{,2211}\right)\\ \displaystyle M_{22,22}&\displaystyle=-\cfrac{2h^{3}E}{3(1-\nu^{2})}\left(\nu~% {}w^{0}_{,1122}+w^{0}_{,2222}\right)\\ \displaystyle M_{12,12}&\displaystyle=-\cfrac{2h^{3}E}{3(1-\nu^{2})}(1-\nu)~{}% w^{0}_{,1212}\end{aligned}
  71. u 1 , 11 0 + ν u 2 , 21 0 + 1 2 ( 1 - ν ) ( u 1 , 22 0 + u 2 , 12 0 ) = 0 ν u 1 , 12 0 + u 2 , 22 0 + 1 2 ( 1 - ν ) ( u 1 , 21 0 + u 2 , 11 0 ) = 0 w , 1111 0 + ν w , 2211 0 + 2 ( 1 - ν ) w , 1212 0 + ν w , 1122 0 + w , 2222 0 = 0 \begin{aligned}&\displaystyle u^{0}_{1,11}+\nu~{}u^{0}_{2,21}+\tfrac{1}{2}(1-% \nu)\left(u^{0}_{1,22}+u^{0}_{2,12}\right)=0\\ &\displaystyle\nu~{}u^{0}_{1,12}+u^{0}_{2,22}+\tfrac{1}{2}(1-\nu)\left(u^{0}_{% 1,21}+u^{0}_{2,11}\right)=0\\ &\displaystyle w^{0}_{,1111}+\nu~{}w^{0}_{,2211}+2(1-\nu)~{}w^{0}_{,1212}+\nu~% {}w^{0}_{,1122}+w^{0}_{,2222}=0\end{aligned}
  72. u 1 , 12 0 = u 1 , 21 0 u^{0}_{1,12}=u^{0}_{1,21}
  73. u 2 , 21 0 = u 2 , 12 0 u^{0}_{2,21}=u^{0}_{2,12}
  74. w , 2211 0 = w , 1212 0 = w , 1122 0 w^{0}_{,2211}=w^{0}_{,1212}=w^{0}_{,1122}
  75. u 1 , 11 0 + 1 2 ( 1 - ν ) u 1 , 22 0 + 1 2 ( 1 + ν ) u 2 , 12 0 = 0 u 2 , 22 0 + 1 2 ( 1 - ν ) u 2 , 11 0 + 1 2 ( 1 + ν ) u 1 , 12 0 = 0 w , 1111 0 + 2 w , 1212 0 + w , 2222 0 = 0 \begin{aligned}&\displaystyle u^{0}_{1,11}+\tfrac{1}{2}(1-\nu)~{}u^{0}_{1,22}+% \tfrac{1}{2}(1+\nu)~{}u^{0}_{2,12}=0\\ &\displaystyle u^{0}_{2,22}+\tfrac{1}{2}(1-\nu)~{}u^{0}_{2,11}+\tfrac{1}{2}(1+% \nu)~{}u^{0}_{1,12}=0\\ &\displaystyle w^{0}_{,1111}+2~{}w^{0}_{,1212}+w^{0}_{,2222}=0\end{aligned}
  76. 2 2 w = 0 \nabla^{2}\nabla^{2}w=0
  77. u 1 0 , u 2 0 u^{0}_{1},u^{0}_{2}
  78. - q ( x ) -q(x)
  79. M α β , α β = - q M_{\alpha\beta,\alpha\beta}=-q
  80. w , 1111 0 + 2 w , 1212 0 + w , 2222 0 = - q D w^{0}_{,1111}+2\,w^{0}_{,1212}+w^{0}_{,2222}=-\cfrac{q}{D}
  81. 1 r d d r [ r d d r { 1 r d d r ( r d w d r ) } ] = - q D . \frac{1}{r}\cfrac{d}{dr}\left[r\cfrac{d}{dr}\left\{\frac{1}{r}\cfrac{d}{dr}% \left(r\cfrac{dw}{dr}\right)\right\}\right]=-\frac{q}{D}\,.
  82. M α β , α β = q M 11 , 11 + 2 M 12 , 12 + M 22 , 22 = q M_{\alpha\beta,\alpha\beta}=q\implies M_{11,11}+2M_{12,12}+M_{22,22}=q
  83. q q
  84. M α β M_{\alpha\beta}
  85. - 2 h 3 E 3 ( 1 - ν 2 ) [ w , 1111 0 + 2 w , 1212 0 + w , 2222 0 ] = q . -\cfrac{2h^{3}E}{3(1-\nu^{2})}\left[w^{0}_{,1111}+2\,w^{0}_{,1212}+w^{0}_{,222% 2}\right]=q\,.
  86. D := 2 h 3 E 3 ( 1 - ν 2 ) D:=\cfrac{2h^{3}E}{3(1-\nu^{2})}
  87. ( r , θ , z ) (r,\theta,z)
  88. 2 w 1 r r ( r w r ) + 1 r 2 2 w θ 2 + 2 w z 2 . \nabla^{2}w\equiv\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial w% }{\partial r}\right)+\frac{1}{r^{2}}\frac{\partial^{2}w}{\partial\theta^{2}}+% \frac{\partial^{2}w}{\partial z^{2}}\,.
  89. w = w ( r ) w=w(r)
  90. 2 w 1 r d d r ( r d w d r ) . \nabla^{2}w\equiv\frac{1}{r}\cfrac{d}{dr}\left(r\cfrac{dw}{dr}\right)\,.
  91. u 1 = u 1 ( x 1 ) , u 2 = 0 , w = w ( x 1 ) u_{1}=u_{1}(x_{1}),u_{2}=0,w=w(x_{1})
  92. [ N 11 N 22 N 12 ] = 2 h E ( 1 - ν 2 ) [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ u 1 , 1 0 0 0 ] \begin{bmatrix}N_{11}\\ N_{22}\\ N_{12}\end{bmatrix}=\cfrac{2hE}{(1-\nu^{2})}~{}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}u^{0}_{1,1}\\ 0\\ 0\end{bmatrix}
  93. [ M 11 M 22 M 12 ] = - 2 h 3 E 3 ( 1 - ν 2 ) [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ w , 11 0 0 0 ] \begin{bmatrix}M_{11}\\ M_{22}\\ M_{12}\end{bmatrix}=-\cfrac{2h^{3}E}{3(1-\nu^{2})}~{}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}w^{0}_{,11}\\ 0\\ 0\end{bmatrix}
  94. N 11 = A d u d x 1 d 2 u d x 1 2 = 0 M 11 = - D d 2 w d x 1 2 d 4 w d x 1 4 = q D \begin{aligned}\displaystyle N_{11}&\displaystyle=A~{}\cfrac{\mathrm{d}u}{% \mathrm{d}x_{1}}\quad\implies\quad\cfrac{\mathrm{d}^{2}u}{\mathrm{d}x_{1}^{2}}% =0\\ \displaystyle M_{11}&\displaystyle=-D~{}\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x_{1% }^{2}}\quad\implies\quad\cfrac{\mathrm{d}^{4}w}{\mathrm{d}x_{1}^{4}}=\cfrac{q}% {D}\\ \end{aligned}
  95. ρ = ρ ( x ) \rho=\rho(x)
  96. J 1 := - h h ρ d x 3 = 2 ρ h ; J 3 := - h h x 3 2 ρ d x 3 = 2 3 ρ h 3 J_{1}:=\int_{-h}^{h}\rho~{}dx_{3}=2~{}\rho~{}h~{};~{}~{}J_{3}:=\int_{-h}^{h}x_% {3}^{2}~{}\rho~{}dx_{3}=\frac{2}{3}~{}\rho~{}h^{3}
  97. u ˙ i = u i t ; u ¨ i = 2 u i t 2 ; u i , α = u i x α ; u i , α β = 2 u i x α x β \dot{u}_{i}=\frac{\partial u_{i}}{\partial t}~{};~{}~{}\ddot{u}_{i}=\frac{% \partial^{2}u_{i}}{\partial t^{2}}~{};~{}~{}u_{i,\alpha}=\frac{\partial u_{i}}% {\partial x_{\alpha}}~{};~{}~{}u_{i,\alpha\beta}=\frac{\partial^{2}u_{i}}{% \partial x_{\alpha}\partial x_{\beta}}
  98. K = 0 T Ω 0 - h h ρ 2 [ ( u 1 t ) 2 + ( u 2 t ) 2 + ( u 3 t ) 2 ] d x 3 d A d t K=\int_{0}^{T}\int_{\Omega^{0}}\int_{-h}^{h}\cfrac{\rho}{2}\left[\left(\frac{% \partial u_{1}}{\partial t}\right)^{2}+\left(\frac{\partial u_{2}}{\partial t}% \right)^{2}+\left(\frac{\partial u_{3}}{\partial t}\right)^{2}\right]~{}% \mathrm{d}x_{3}~{}\mathrm{d}A~{}\mathrm{d}t
  99. δ K = 0 T Ω 0 - h h ρ 2 [ 2 ( u 1 t ) ( δ u 1 t ) + 2 ( u 2 t ) ( δ u 2 t ) + 2 ( u 3 t ) ( δ u 3 t ) ] d x 3 d A d t \delta K=\int_{0}^{T}\int_{\Omega^{0}}\int_{-h}^{h}\cfrac{\rho}{2}\left[2\left% (\frac{\partial u_{1}}{\partial t}\right)\left(\frac{\partial\delta u_{1}}{% \partial t}\right)+2\left(\frac{\partial u_{2}}{\partial t}\right)\left(\frac{% \partial\delta u_{2}}{\partial t}\right)+2\left(\frac{\partial u_{3}}{\partial t% }\right)\left(\frac{\partial\delta u_{3}}{\partial t}\right)\right]~{}\mathrm{% d}x_{3}~{}\mathrm{d}A~{}\mathrm{d}t
  100. u ˙ i = u i t ; u ¨ i = 2 u i t 2 ; u i , α = u i x α ; u i , α β = 2 u i x α x β \dot{u}_{i}=\frac{\partial u_{i}}{\partial t}~{};~{}~{}\ddot{u}_{i}=\frac{% \partial^{2}u_{i}}{\partial t^{2}}~{};~{}~{}u_{i,\alpha}=\frac{\partial u_{i}}% {\partial x_{\alpha}}~{};~{}~{}u_{i,\alpha\beta}=\frac{\partial^{2}u_{i}}{% \partial x_{\alpha}\partial x_{\beta}}
  101. δ K = 0 T Ω 0 - h h ρ ( u ˙ α δ u ˙ α + u ˙ 3 δ u ˙ 3 ) d x 3 d A d t \delta K=\int_{0}^{T}\int_{\Omega^{0}}\int_{-h}^{h}\rho\left(\dot{u}_{\alpha}~% {}\delta\dot{u}_{\alpha}+\dot{u}_{3}~{}\delta\dot{u}_{3}\right)~{}\mathrm{d}x_% {3}~{}\mathrm{d}A~{}\mathrm{d}t
  102. u α = u α 0 - x 3 w , α 0 ; u 3 = w 0 u_{\alpha}=u^{0}_{\alpha}-x_{3}~{}w^{0}_{,\alpha}~{};~{}~{}u_{3}=w^{0}
  103. δ K = 0 T Ω 0 - h h ρ [ ( u ˙ α 0 - x 3 w ˙ , α 0 ) ( δ u ˙ α 0 - x 3 δ w ˙ , α 0 ) + w ˙ 0 δ w ˙ 0 ] d x 3 d A d t = 0 T Ω 0 - h h ρ ( u ˙ α 0 δ u ˙ α 0 - x 3 w ˙ , α 0 δ u ˙ α 0 - x 3 u ˙ α 0 δ w ˙ , α 0 + x 3 2 w ˙ , α 0 δ w ˙ , α 0 + w ˙ 0 δ w ˙ 0 ) d x 3 d A d t \begin{aligned}\displaystyle\delta K&\displaystyle=\int_{0}^{T}\int_{\Omega^{0% }}\int_{-h}^{h}\rho\left[\left(\dot{u}^{0}_{\alpha}-x_{3}~{}\dot{w}^{0}_{,% \alpha}\right)~{}\left(\delta\dot{u}^{0}_{\alpha}-x_{3}~{}\delta\dot{w}^{0}_{,% \alpha}\right)+\dot{w}^{0}~{}\delta\dot{w}^{0}\right]~{}\mathrm{d}x_{3}~{}% \mathrm{d}A~{}\mathrm{d}t\\ &\displaystyle=\int_{0}^{T}\int_{\Omega^{0}}\int_{-h}^{h}\rho\left(\dot{u}^{0}% _{\alpha}~{}\delta\dot{u}^{0}_{\alpha}-x_{3}~{}\dot{w}^{0}_{,\alpha}~{}\delta% \dot{u}^{0}_{\alpha}-x_{3}~{}\dot{u}^{0}_{\alpha}~{}\delta\dot{w}^{0}_{,\alpha% }+x_{3}^{2}~{}\dot{w}^{0}_{,\alpha}~{}\delta\dot{w}^{0}_{,\alpha}+\dot{w}^{0}~% {}\delta\dot{w}^{0}\right)~{}\mathrm{d}x_{3}~{}\mathrm{d}A~{}\mathrm{d}t\end{aligned}
  104. ρ \rho
  105. J 1 := - h h ρ d x 3 = 2 ρ h ; J 2 := - h h x 3 ρ d x 3 = 0 ; J 3 := - h h x 3 2 ρ d x 3 = 2 3 ρ h 3 J_{1}:=\int_{-h}^{h}\rho~{}dx_{3}=2~{}\rho~{}h~{};~{}~{}J_{2}:=\int_{-h}^{h}x_% {3}~{}\rho~{}dx_{3}=0~{};~{}~{}J_{3}:=\int_{-h}^{h}x_{3}^{2}~{}\rho~{}dx_{3}=% \frac{2}{3}~{}\rho~{}h^{3}
  106. δ K = 0 T Ω 0 [ J 1 ( u ˙ α 0 δ u ˙ α 0 + w ˙ 0 δ w ˙ 0 ) + J 3 w ˙ , α 0 δ w ˙ , α 0 ] d A d t \delta K=\int_{0}^{T}\int_{\Omega^{0}}\left[J_{1}\left(\dot{u}^{0}_{\alpha}~{}% \delta\dot{u}^{0}_{\alpha}+\dot{w}^{0}~{}\delta\dot{w}^{0}\right)+J_{3}~{}\dot% {w}^{0}_{,\alpha}~{}\delta\dot{w}^{0}_{,\alpha}\right]~{}\mathrm{d}A~{}\mathrm% {d}t
  107. δ K = Ω 0 [ 0 T { - J 1 ( u ¨ α 0 δ u α 0 + w ¨ 0 δ w 0 ) - J 3 w ¨ , α 0 δ w , α 0 } d t + | J 1 ( u ˙ α 0 δ u α 0 + w ˙ 0 δ w 0 ) + J 3 w ˙ , α 0 δ w , α 0 | 0 T ] d A \delta K=\int_{\Omega^{0}}\left[\int_{0}^{T}\left\{-J_{1}\left(\ddot{u}^{0}_{% \alpha}~{}\delta u^{0}_{\alpha}+\ddot{w}^{0}~{}\delta w^{0}\right)-J_{3}~{}% \ddot{w}^{0}_{,\alpha}~{}\delta w^{0}_{,\alpha}\right\}~{}\mathrm{d}t+\left|J_% {1}\left(\dot{u}^{0}_{\alpha}~{}\delta u^{0}_{\alpha}+\dot{w}^{0}~{}\delta w^{% 0}\right)+J_{3}~{}\dot{w}^{0}_{,\alpha}~{}\delta w^{0}_{,\alpha}\right|_{0}^{T% }\right]~{}\mathrm{d}A
  108. δ u α 0 \delta u^{0}_{\alpha}
  109. δ w 0 \delta w^{0}
  110. t = 0 t=0
  111. t = T t=T
  112. δ K = - 0 T { Ω 0 [ J 1 ( u ¨ α 0 δ u α 0 + w ¨ 0 δ w 0 ) + J 3 w ¨ , α 0 δ w , α 0 ] d A } d t + | Ω 0 J 3 w ˙ , α 0 δ w , α 0 d A | 0 T \delta K=-\int_{0}^{T}\left\{\int_{\Omega^{0}}\left[J_{1}\left(\ddot{u}^{0}_{% \alpha}~{}\delta u^{0}_{\alpha}+\ddot{w}^{0}~{}\delta w^{0}\right)+J_{3}~{}% \ddot{w}^{0}_{,\alpha}~{}\delta w^{0}_{,\alpha}\right]~{}\mathrm{d}A\right\}~{% }\mathrm{d}t+\left|\int_{\Omega^{0}}J_{3}~{}\dot{w}^{0}_{,\alpha}~{}\delta w^{% 0}_{,\alpha}\mathrm{d}A\right|_{0}^{T}
  113. δ K = - 0 T { Ω 0 [ J 1 ( u ¨ α 0 δ u α 0 + w ¨ 0 δ w 0 ) - J 3 w ¨ , α α 0 δ w 0 ] d A + Γ 0 J 3 n α w ¨ , α 0 δ w 0 d s } d t - | Ω 0 J 3 w ˙ , α α 0 δ w 0 d A - Γ 0 J 3 w ˙ , α 0 δ w 0 d s | 0 T \begin{aligned}\displaystyle\delta K&\displaystyle=-\int_{0}^{T}\left\{\int_{% \Omega^{0}}\left[J_{1}\left(\ddot{u}^{0}_{\alpha}~{}\delta u^{0}_{\alpha}+% \ddot{w}^{0}~{}\delta w^{0}\right)-J_{3}~{}\ddot{w}^{0}_{,\alpha\alpha}~{}% \delta w^{0}\right]~{}\mathrm{d}A+\int_{\Gamma^{0}}J_{3}~{}n_{\alpha}~{}\ddot{% w}^{0}_{,\alpha}~{}\delta w^{0}~{}\mathrm{d}s\right\}~{}\mathrm{d}t\\ &\displaystyle\qquad-\left|\int_{\Omega^{0}}J_{3}~{}\dot{w}^{0}_{,\alpha\alpha% }~{}\delta w^{0}~{}\mathrm{d}A-\int_{\Gamma^{0}}J_{3}~{}\dot{w}^{0}_{,\alpha}~% {}\delta w^{0}~{}\mathrm{d}s\right|_{0}^{T}\end{aligned}
  114. δ K = - 0 T { Ω 0 [ J 1 ( u ¨ α 0 δ u α 0 + w ¨ 0 δ w 0 ) - J 3 w ¨ , α α 0 δ w 0 ] d A + Γ 0 J 3 n α w ¨ , α 0 δ w 0 d s } d t \delta K=-\int_{0}^{T}\left\{\int_{\Omega^{0}}\left[J_{1}\left(\ddot{u}^{0}_{% \alpha}~{}\delta u^{0}_{\alpha}+\ddot{w}^{0}~{}\delta w^{0}\right)-J_{3}~{}% \ddot{w}^{0}_{,\alpha\alpha}~{}\delta w^{0}\right]~{}\mathrm{d}A+\int_{\Gamma^% {0}}J_{3}~{}n_{\alpha}~{}\ddot{w}^{0}_{,\alpha}~{}\delta w^{0}~{}\mathrm{d}s% \right\}~{}\mathrm{d}t
  115. δ U = - 0 T { Ω 0 [ N α β , α δ u β 0 + M α β , β α δ w 0 ] d A - Γ 0 [ n α N α β δ u β 0 + n α M α β , β δ w 0 - n β M α β δ w , α 0 ] d s } d t \delta U=-\int_{0}^{T}\left\{\int_{\Omega^{0}}\left[N_{\alpha\beta,\alpha}~{}% \delta u^{0}_{\beta}+M_{\alpha\beta,\beta\alpha}~{}\delta w^{0}\right]~{}% \mathrm{d}A-\int_{\Gamma^{0}}\left[n_{\alpha}~{}N_{\alpha\beta}~{}\delta u^{0}% _{\beta}+n_{\alpha}~{}M_{\alpha\beta,\beta}~{}\delta w^{0}-n_{\beta}~{}M_{% \alpha\beta}~{}\delta w^{0}_{,\alpha}\right]~{}\mathrm{d}s\right\}\mathrm{d}t
  116. δ U = - 0 T { Ω 0 [ N α β , α δ u β 0 + M α β , β α δ w 0 ] d A - Γ 0 [ n α N α β δ u β 0 + n α M α β , β δ w 0 + n β M α β , α δ w 0 ] d s } d t \delta U=-\int_{0}^{T}\left\{\int_{\Omega^{0}}\left[N_{\alpha\beta,\alpha}~{}% \delta u^{0}_{\beta}+M_{\alpha\beta,\beta\alpha}~{}\delta w^{0}\right]~{}% \mathrm{d}A-\int_{\Gamma^{0}}\left[n_{\alpha}~{}N_{\alpha\beta}~{}\delta u^{0}% _{\beta}+n_{\alpha}~{}M_{\alpha\beta,\beta}~{}\delta w^{0}+n_{\beta}~{}M_{% \alpha\beta,\alpha}~{}\delta w^{0}\right]~{}\mathrm{d}s\right\}\mathrm{d}t
  117. q ( x , t ) q(x,t)
  118. δ V ext = 0 T [ Ω 0 q ( x , t ) δ w 0 d A ] d t \delta V_{\mathrm{ext}}=\int_{0}^{T}\left[\int_{\Omega^{0}}q(x,t)~{}\delta w^{% 0}~{}\mathrm{d}A\right]\mathrm{d}t
  119. δ U + δ V ext = δ K \delta U+\delta V_{\mathrm{ext}}=\delta K
  120. N α β , β = J 1 u ¨ α 0 M α β , α β - q ( x , t ) = J 1 w ¨ 0 - J 3 w ¨ , α α 0 \begin{aligned}\displaystyle N_{\alpha\beta,\beta}&\displaystyle=J_{1}~{}\ddot% {u}^{0}_{\alpha}\\ \displaystyle M_{\alpha\beta,\alpha\beta}-q(x,t)&\displaystyle=J_{1}~{}\ddot{w% }^{0}-J_{3}~{}\ddot{w}^{0}_{,\alpha\alpha}\end{aligned}
  121. D ( 4 w x 4 + 2 4 w x 2 y 2 + 4 w y 4 ) = - q ( x , y , t ) - 2 ρ h 2 w t 2 . D\,\left(\frac{\partial^{4}w}{\partial x^{4}}+2\frac{\partial^{4}w}{\partial x% ^{2}\partial y^{2}}+\frac{\partial^{4}w}{\partial y^{4}}\right)=-q(x,y,t)-2% \rho h\,\frac{\partial^{2}w}{\partial t^{2}}\,.
  122. D D
  123. 2 h 2h
  124. D := 2 h 3 E 3 ( 1 - ν 2 ) . D:=\cfrac{2h^{3}E}{3(1-\nu^{2})}\,.
  125. D 2 2 w = - q ( x , y , t ) - 2 ρ h w ¨ . D\,\nabla^{2}\nabla^{2}w=-q(x,y,t)-2\rho h\,\ddot{w}\,.
  126. D 2 2 w = - 2 ρ h w ¨ . D\,\nabla^{2}\nabla^{2}w=-2\rho h\,\ddot{w}\,.
  127. [ σ 11 σ 22 σ 12 ] = E 1 - ν 2 [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ ε 11 ε 22 ε 12 ] . \begin{bmatrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{bmatrix}=\cfrac{E}{1-\nu^{2}}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}\varepsilon_{11}\\ \varepsilon_{22}\\ \varepsilon_{12}\end{bmatrix}\,.
  128. ε α β \varepsilon_{\alpha\beta}
  129. ε α β = 1 2 ( u α , β + u β , α ) - x 3 w , α β . \varepsilon_{\alpha\beta}=\frac{1}{2}(u_{\alpha,\beta}+u_{\beta,\alpha})-x_{3}% \,w_{,\alpha\beta}\,.
  130. [ M 11 M 22 M 12 ] = - 2 h 3 E 3 ( 1 - ν 2 ) [ 1 ν 0 ν 1 0 0 0 1 - ν ] [ w , 11 w , 22 w , 12 ] \begin{bmatrix}M_{11}\\ M_{22}\\ M_{12}\end{bmatrix}=-\cfrac{2h^{3}E}{3(1-\nu^{2})}~{}\begin{bmatrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{bmatrix}\begin{bmatrix}w_{,11}\\ w_{,22}\\ w_{,12}\end{bmatrix}
  131. 2 h 2h
  132. M 11 , 11 + 2 M 12 , 12 + M 22 , 22 - q ( x , t ) = 2 ρ h w ¨ - 2 3 ρ h 3 ( w ¨ , 11 + w ¨ , 22 + w ¨ , 33 ) . M_{11,11}+2M_{12,12}+M_{22,22}-q(x,t)=2\rho h\ddot{w}-\frac{2}{3}\rho h^{3}% \left(\ddot{w}_{,11}+\ddot{w}_{,22}+\ddot{w}_{,33}\right)\,.
  133. M 11 , 11 = - 2 h 3 E 3 ( 1 - ν 2 ) ( w , 1111 + ν w , 2211 ) M 22 , 22 = - 2 h 3 E 3 ( 1 - ν 2 ) ( ν w , 1122 + w , 2222 ) M 12 , 12 = - 2 h 3 E 3 ( 1 - ν 2 ) ( 1 - ν ) w , 1212 \begin{aligned}\displaystyle M_{11,11}&\displaystyle=-\cfrac{2h^{3}E}{3(1-\nu^% {2})}\left(w_{,1111}+\nu~{}w_{,2211}\right)\\ \displaystyle M_{22,22}&\displaystyle=-\cfrac{2h^{3}E}{3(1-\nu^{2})}\left(\nu~% {}w_{,1122}+w_{,2222}\right)\\ \displaystyle M_{12,12}&\displaystyle=-\cfrac{2h^{3}E}{3(1-\nu^{2})}(1-\nu)~{}% w_{,1212}\end{aligned}
  134. - 2 h 3 E 3 ( 1 - ν 2 ) ( w , 1111 + ν w , 2211 + 2 ( 1 - ν ) w , 1212 + ν w , 1122 + w , 2222 ) = q ( x , t ) + 2 ρ h w ¨ - 2 3 ρ h 3 ( w ¨ , 11 + w ¨ , 22 + w ¨ , 33 ) . \begin{aligned}\displaystyle-\cfrac{2h^{3}E}{3(1-\nu^{2})}&\displaystyle\left(% w_{,1111}+\nu~{}w_{,2211}+2(1-\nu)~{}w_{,1212}+\nu~{}w_{,1122}+w_{,2222}\right% )=\\ &\displaystyle q(x,t)+2\rho h\ddot{w}-\frac{2}{3}\rho h^{3}\left(\ddot{w}_{,11% }+\ddot{w}_{,22}+\ddot{w}_{,33}\right)\,.\end{aligned}
  135. w , 2211 = w , 1212 = w , 1122 w_{,2211}=w_{,1212}=w_{,1122}
  136. - 2 h 3 E 3 ( 1 - ν 2 ) ( w , 1111 + 2 w , 1212 + w , 2222 ) = q ( x , t ) + 2 ρ h w ¨ - 2 3 ρ h 3 ( w ¨ , 11 + w ¨ , 22 + w ¨ , 33 ) . \begin{aligned}\displaystyle-\cfrac{2h^{3}E}{3(1-\nu^{2})}&\displaystyle\left(% w_{,1111}+2w_{,1212}+w_{,2222}\right)=\\ &\displaystyle q(x,t)+2\rho h\ddot{w}-\frac{2}{3}\rho h^{3}\left(\ddot{w}_{,11% }+\ddot{w}_{,22}+\ddot{w}_{,33}\right)\,.\end{aligned}
  137. D := 2 h 3 E 3 ( 1 - ν 2 ) D:=\cfrac{2h^{3}E}{3(1-\nu^{2})}
  138. D ( w , 1111 + 2 w , 1212 + w , 2222 ) = - q ( x , t ) - 2 ρ h w ¨ + 2 3 ρ h 3 ( w ¨ , 11 + w ¨ , 22 + w ¨ , 33 ) . D\left(w_{,1111}+2w_{,1212}+w_{,2222}\right)=-q(x,t)-2\rho h\ddot{w}+\frac{2}{% 3}\rho h^{3}\left(\ddot{w}_{,11}+\ddot{w}_{,22}+\ddot{w}_{,33}\right)\,.
  139. D ( w , 1111 + 2 w , 1212 + w , 2222 ) = - q ( x , t ) - 2 ρ h w ¨ . D\left(w_{,1111}+2w_{,1212}+w_{,2222}\right)=-q(x,t)-2\rho h\ddot{w}\,.
  140. D 2 2 w = - q ( x , t ) - 2 ρ h w ¨ . D\nabla^{2}\nabla^{2}w=-q(x,t)-2\rho h\ddot{w}\,.

Klecka's_tau.html

  1. τ = n c o r r - i = 1 T p i n i N - i = 1 T p i n i \tau=\frac{n_{corr}-\sum_{i=1}^{T}p_{i}n_{i}}{N-\sum_{i=1}^{T}p_{i}n_{i}}

Klein_polyhedron.html

  1. C \textstyle C
  2. n \textstyle\mathbb{R}^{n}
  3. C \textstyle C
  4. C n \textstyle C\cap\mathbb{Z}^{n}
  5. α > 0 \textstyle\alpha>0
  6. 2 \textstyle\mathbb{R}^{2}
  7. { ( 1 , α ) , ( 1 , 0 ) } \textstyle\{(1,\alpha),(1,0)\}
  8. { ( 1 , α ) , ( 0 , 1 ) } \textstyle\{(1,\alpha),(0,1)\}
  9. n \textstyle\mathbb{Z}^{n}
  10. α \textstyle\alpha
  11. C \textstyle C
  12. ( a i ) \textstyle(a_{i})
  13. n \textstyle\mathbb{R}^{n}
  14. C = { i λ i a i : ( i ) λ i 0 } \textstyle C=\{\sum_{i}\lambda_{i}a_{i}:(\forall i)\;\lambda_{i}\geq 0\}
  15. ( w i ) \textstyle(w_{i})
  16. C = { x : ( i ) w i , x 0 } \textstyle C=\{x:(\forall i)\;\langle w_{i},x\rangle\geq 0\}
  17. D ( x ) \textstyle D(x)
  18. x \textstyle x
  19. H ( x ) \textstyle H(x)
  20. x \textstyle x
  21. x n \textstyle x\in\mathbb{R}^{n}
  22. H ( x ) n = { 0 } \textstyle H(x)\cap\mathbb{Q}^{n}=\{0\}
  23. C \textstyle C
  24. a i \textstyle a_{i}
  25. w i \textstyle w_{i}
  26. V \textstyle V
  27. V \textstyle V
  28. Γ e ( V ) \textstyle\Gamma_{\mathrm{e}}(V)
  29. V \textstyle V
  30. V \textstyle V
  31. Γ f ( V ) \textstyle\Gamma_{\mathrm{f}}(V)
  32. ( n - 1 ) \textstyle(n-1)
  33. V \textstyle V
  34. ( n - 2 ) \textstyle(n-2)
  35. Υ n \textstyle\Upsilon_{n}
  36. GL n ( ) \textstyle\mathrm{GL}_{n}(\mathbb{Q})
  37. A \textstyle A
  38. B \textstyle B
  39. A - 1 B \textstyle A^{-1}B
  40. U W \textstyle UW
  41. U = ( 1 0 c 1 0 1 c n - 1 0 0 c n ) U=\left(\begin{array}[]{cccc}1&\cdots&0&c_{1}\\ \vdots&\ddots&\vdots&\vdots\\ 0&\cdots&1&c_{n-1}\\ 0&\cdots&0&c_{n}\end{array}\right)
  42. c i \textstyle c_{i}\in\mathbb{Q}
  43. c n 0 \textstyle c_{n}\neq 0
  44. W \textstyle W
  45. V \textstyle V
  46. Γ e ( V ) \textstyle\Gamma_{\mathrm{e}}(V)
  47. Γ f ( V ) \textstyle\Gamma_{\mathrm{f}}(V)
  48. Υ n \textstyle\Upsilon_{n}
  49. ( x 0 , x 1 , ) \textstyle(x_{0},x_{1},\ldots)
  50. Γ e ( V ) \textstyle\Gamma_{\mathrm{e}}(V)
  51. ( A 0 , A 1 , ) \textstyle(A_{0},A_{1},\ldots)
  52. Υ n \textstyle\Upsilon_{n}
  53. x k = A k ( e ) \textstyle x_{k}=A_{k}(e)
  54. e \textstyle e
  55. ( 1 , , 1 ) n \textstyle(1,\ldots,1)\in\mathbb{R}^{n}
  56. ( σ 0 , σ 1 , ) \textstyle(\sigma_{0},\sigma_{1},\ldots)
  57. Γ f ( V ) \textstyle\Gamma_{\mathrm{f}}(V)
  58. ( A 0 , A 1 , ) \textstyle(A_{0},A_{1},\ldots)
  59. Υ n \textstyle\Upsilon_{n}
  60. σ k = A k ( Δ ) \textstyle\sigma_{k}=A_{k}(\Delta)
  61. Δ \textstyle\Delta
  62. ( n - 1 ) \textstyle(n-1)
  63. n \textstyle\mathbb{R}^{n}
  64. α \textstyle\alpha
  65. α \textstyle\alpha
  66. α \textstyle\alpha
  67. K \textstyle K\subseteq\mathbb{R}
  68. n \textstyle n
  69. α i : K \textstyle\alpha_{i}:K\to\mathbb{R}
  70. n \textstyle n
  71. K \textstyle K
  72. C \textstyle C
  73. K \textstyle K
  74. C = { x n : ( i ) α i ( ω 1 ) x 1 + + α i ( ω n ) x n 0 } \textstyle C=\{x\in\mathbb{R}^{n}:(\forall i)\;\alpha_{i}(\omega_{1})x_{1}+% \ldots+\alpha_{i}(\omega_{n})x_{n}\geq 0\}
  75. ω 1 , , ω n \textstyle\omega_{1},\ldots,\omega_{n}
  76. K \textstyle K
  77. \textstyle\mathbb{Q}
  78. ( A 0 , A 1 , ) \textstyle(A_{0},A_{1},\ldots)
  79. Υ n \textstyle\Upsilon_{n}
  80. R k = A k + 1 A k - 1 \textstyle R_{k}=A_{k+1}A_{k}^{-1}
  81. m \textstyle m
  82. R k + q m = R k \textstyle R_{k+qm}=R_{k}
  83. k , q 0 \textstyle k,q\geq 0
  84. A m A 0 - 1 \textstyle A_{m}A_{0}^{-1}
  85. Γ e ( V ) \textstyle\Gamma_{\mathrm{e}}(V)
  86. Γ f ( V ) \textstyle\Gamma_{\mathrm{f}}(V)
  87. C n \textstyle C\subseteq\mathbb{R}^{n}
  88. ( a i ) \textstyle(a_{i})
  89. ( w i ) \textstyle(w_{i})
  90. V \textstyle V
  91. C \textstyle C
  92. n \textstyle n
  93. a i \textstyle a_{i}
  94. x 0 , x 1 , \textstyle x_{0},x_{1},\ldots
  95. Γ e ( V ) \textstyle\Gamma_{\mathrm{e}}(V)
  96. x k \textstyle x_{k}
  97. D ( a i ) \textstyle D(a_{i})
  98. w i \textstyle w_{i}
  99. σ 0 , σ 1 , \textstyle\sigma_{0},\sigma_{1},\ldots
  100. Γ f ( V ) \textstyle\Gamma_{\mathrm{f}}(V)
  101. σ k \textstyle\sigma_{k}
  102. H ( w i ) \textstyle H(w_{i})
  103. n = 2 \textstyle n=2
  104. K = ( 2 ) \textstyle K=\mathbb{Q}(\sqrt{2})
  105. { ( x , y ) : x 0 , | y | x / 2 } \textstyle\{(x,y):x\geq 0,|y|\leq x/\sqrt{2}\}
  106. K \textstyle K
  107. ( p k , ± q k ) \textstyle(p_{k},\pm q_{k})
  108. p k / q k \textstyle p_{k}/q_{k}
  109. 2 \textstyle\sqrt{2}
  110. ( x k ) \textstyle(x_{k})
  111. ( 1 , 0 ) \textstyle(1,0)
  112. ( ( 1 , 0 ) , ( 3 , 2 ) , ( 17 , 12 ) , ( 99 , 70 ) , ) \textstyle((1,0),(3,2),(17,12),(99,70),\ldots)
  113. σ k \textstyle\sigma_{k}
  114. x k \textstyle x_{k}
  115. x k + 1 \textstyle x_{k+1}
  116. x ¯ k \textstyle\bar{x}_{k}
  117. σ ¯ k \textstyle\bar{\sigma}_{k}
  118. x k \textstyle x_{k}
  119. σ k \textstyle\sigma_{k}
  120. x \textstyle x
  121. T = ( 3 4 2 3 ) \textstyle T=\left(\begin{array}[]{cc}3&4\\ 2&3\end{array}\right)
  122. x k + 1 = T x k \textstyle x_{k+1}=Tx_{k}
  123. R = ( 6 1 - 1 0 ) = ( 1 6 0 - 1 ) ( 0 1 1 0 ) \textstyle R=\left(\begin{array}[]{cc}6&1\\ -1&0\end{array}\right)=\left(\begin{array}[]{cc}1&6\\ 0&-1\end{array}\right)\left(\begin{array}[]{cc}0&1\\ 1&0\end{array}\right)
  124. M e = ( 1 2 1 2 1 4 - 1 4 ) \textstyle M_{\mathrm{e}}=\left(\begin{array}[]{cc}\frac{1}{2}&\frac{1}{2}\\ \frac{1}{4}&-\frac{1}{4}\end{array}\right)
  125. M ¯ e = ( 1 2 1 2 - 1 4 1 4 ) \textstyle\bar{M}_{\mathrm{e}}=\left(\begin{array}[]{cc}\frac{1}{2}&\frac{1}{2% }\\ -\frac{1}{4}&\frac{1}{4}\end{array}\right)
  126. M f = ( 3 1 2 0 ) \textstyle M_{\mathrm{f}}=\left(\begin{array}[]{cc}3&1\\ 2&0\end{array}\right)
  127. M ¯ f = ( 3 1 - 2 0 ) \textstyle\bar{M}_{\mathrm{f}}=\left(\begin{array}[]{cc}3&1\\ -2&0\end{array}\right)
  128. ( M e R k ) \textstyle(M_{\mathrm{e}}R^{k})
  129. ( M ¯ e R k ) \textstyle(\bar{M}_{\mathrm{e}}R^{k})
  130. Υ 2 \textstyle\Upsilon_{2}
  131. M e R M e - 1 = T \textstyle M_{\mathrm{e}}RM_{\mathrm{e}}^{-1}=T
  132. M ¯ e R M ¯ e - 1 = T - 1 \textstyle\bar{M}_{\mathrm{e}}R\bar{M}_{\mathrm{e}}^{-1}=T^{-1}
  133. x k = M e R k ( e ) \textstyle x_{k}=M_{\mathrm{e}}R^{k}(e)
  134. x ¯ k = M ¯ e R k ( e ) \textstyle\bar{x}_{k}=\bar{M}_{\mathrm{e}}R^{k}(e)
  135. ( M f R k ) \textstyle(M_{\mathrm{f}}R^{k})
  136. ( M ¯ f R k ) \textstyle(\bar{M}_{\mathrm{f}}R^{k})
  137. Υ 2 \textstyle\Upsilon_{2}
  138. M f R M f - 1 = T \textstyle M_{\mathrm{f}}RM_{\mathrm{f}}^{-1}=T
  139. M ¯ f R M ¯ f - 1 = T - 1 \textstyle\bar{M}_{\mathrm{f}}R\bar{M}_{\mathrm{f}}^{-1}=T^{-1}
  140. σ k = M f R k ( Δ ) \textstyle\sigma_{k}=M_{\mathrm{f}}R^{k}(\Delta)
  141. σ ¯ k = M ¯ f R k ( Δ ) \textstyle\bar{\sigma}_{k}=\bar{M}_{\mathrm{f}}R^{k}(\Delta)
  142. α > 0 \textstyle\alpha>0
  143. { q ( p α - q ) : p , q , q > 0 } \textstyle\{q(p\alpha-q):p,q\in\mathbb{Z},q>0\}
  144. C = { x : ( i ) w i , x 0 } \textstyle C=\{x:(\forall i)\;\langle w_{i},x\rangle\geq 0\}
  145. n \textstyle\mathbb{R}^{n}
  146. w i , w i = 1 \textstyle\langle w_{i},w_{i}\rangle=1
  147. C \textstyle C
  148. N ( C ) = inf { i w i , x : x n C { 0 } } \textstyle N(C)=\inf\{\prod_{i}\langle w_{i},x\rangle:x\in\mathbb{Z}^{n}\cap C% \setminus\{0\}\}
  149. 𝐯 1 , , 𝐯 m n \textstyle\mathbf{v}_{1},\ldots,\mathbf{v}_{m}\in\mathbb{Z}^{n}
  150. [ 𝐯 1 , , 𝐯 m ] = i 1 < < i n | det ( 𝐯 i 1 𝐯 i n ) | \textstyle[\mathbf{v}_{1},\ldots,\mathbf{v}_{m}]=\sum_{i_{1}<\cdots<i_{n}}|% \det(\mathbf{v}_{i_{1}}\cdots\mathbf{v}_{i_{n}})|
  151. { i λ i 𝐯 i : ( i ) 0 λ i 1 } \textstyle\{\sum_{i}\lambda_{i}\mathbf{v}_{i}:(\forall i)\;0\leq\lambda_{i}% \leq 1\}
  152. V \textstyle V
  153. C \textstyle C
  154. x \textstyle x
  155. Γ e ( V ) \textstyle\Gamma_{\mathrm{e}}(V)
  156. [ x ] = [ 𝐯 1 , , 𝐯 m ] \textstyle[x]=[\mathbf{v}_{1},\ldots,\mathbf{v}_{m}]
  157. 𝐯 1 , , 𝐯 m \textstyle\mathbf{v}_{1},\ldots,\mathbf{v}_{m}
  158. n \textstyle\mathbb{Z}^{n}
  159. x \textstyle x
  160. σ \textstyle\sigma
  161. Γ f ( V ) \textstyle\Gamma_{\mathrm{f}}(V)
  162. [ σ ] = [ 𝐯 1 , , 𝐯 m ] \textstyle[\sigma]=[\mathbf{v}_{1},\ldots,\mathbf{v}_{m}]
  163. 𝐯 1 , , 𝐯 m \textstyle\mathbf{v}_{1},\ldots,\mathbf{v}_{m}
  164. σ \textstyle\sigma
  165. N ( C ) > 0 \textstyle N(C)>0
  166. { [ x ] : x Γ e ( V ) } \textstyle\{[x]:x\in\Gamma_{\mathrm{e}}(V)\}
  167. { [ σ ] : σ Γ f ( V ) } \textstyle\{[\sigma]:\sigma\in\Gamma_{\mathrm{f}}(V)\}
  168. [ x ] \textstyle[x]
  169. [ σ ] \textstyle[\sigma]
  170. { ( 1 , α ) , ( 1 , 0 ) } \textstyle\{(1,\alpha),(1,0)\}
  171. α \textstyle\alpha

Kmetija.html

  1. 1 2 \tfrac{1}{2}

Knaster's_condition.html

  1. ω 1 \omega_{1}

Knudsen_equation.html

  1. q = 1 6 2 π Δ P d 3 l ρ 1 , q=\frac{1}{6}\sqrt{2\pi}\Delta P\frac{d^{3}}{l\sqrt{\rho_{1}}},
  2. C C
  3. C L / s 12 d 3 / cm 3 l / cm \frac{C}{\mathrm{L}/\mathrm{s}}\approx 12\,\frac{d^{3}/\mathrm{cm}^{3}}{{l/% \mathrm{cm}}}

Kobayashi_metric.html

  1. d ( f ( x ) , f ( y ) ) ρ ( x , y ) d(f(x),f(y))\leq\rho(x,y)
  2. ρ ( x , y ) \rho(x,y)

Koenigs_function.html

  1. f f
  2. f f
  3. f f
  4. f f
  5. λ λ
  6. h h
  7. h h
  8. h ( f ( z ) ) = f ( 0 ) h ( z ) . h(f(z))=f^{\prime}(0)h(z)~{}.
  9. g n ( z ) = λ - n f n ( z ) g_{n}(z)=\lambda^{-n}f^{n}(z)
  10. f f
  11. h h
  12. f f
  13. h h
  14. D D
  15. U = h ( D ) U=h(D)
  16. f f
  17. λ λ
  18. U U
  19. k k
  20. H = k h - 1 ( z ) H=k\circ h^{-1}(z)
  21. λ H ( z ) = λ h ( k - 1 ( z ) ) = h ( f ( k - 1 ( z ) ) = h ( k - 1 ( λ z ) = H ( λ z ) . \lambda H(z)=\lambda h(k^{-1}(z))=h(f(k^{-1}(z))=h(k^{-1}(\lambda z)=H(\lambda z% )~{}.
  22. H H
  23. H ( z ) = z H(z)=z
  24. h = k h=k
  25. F ( z ) = f ( z ) / λ z , F(z)=f(z)/\lambda z,
  26. | F ( z ) - 1 | ( 1 + | λ | - 1 ) | z | . |F(z)-1|\leq(1+|\lambda|^{-1})|z|~{}.
  27. g n ( z ) = z j = 0 n - 1 F ( f j ( z ) ) . g_{n}(z)=z\prod_{j=0}^{n-1}F(f^{j}(z))~{}.
  28. sup | z | r | 1 - F f j ( z ) | ( 1 + | λ | - 1 ) M ( r ) j < . \sum\sup_{|z|\leq r}|1-F\circ f^{j}(z)|\leq(1+|\lambda|^{-1})\sum M(r)^{j}<\infty.
  29. h h
  30. D D
  31. t 0 , ) ) t∈0,∞))
  32. f s f_{s}
  33. s s
  34. f s ( f t ( z ) ) = f t + s ( z ) f_{s}(f_{t}(z))=f_{t+s}(z)
  35. f 0 ( z ) = z f_{0}(z)=z
  36. f t ( z ) f_{t}(z)
  37. t t
  38. z z
  39. s s
  40. h ( f s ( z ) ) = f s ( 0 ) h ( z ) . h(f_{s}(z))=f_{s}^{\prime}(0)h(z).
  41. h h
  42. U = h ( D ) U=h(D)
  43. λ ( s ) = f s ( 0 ) \lambda(s)=f_{s}^{\prime}(0)
  44. λ ( s ) = e μ s \lambda(s)=e^{\mu s}
  45. μ μ
  46. μ μ
  47. D D
  48. ( 𝟎 ) \mathbf{(0)}
  49. μ μ
  50. t ( f t ( z ) ) h ( f t ( z ) ) = μ e μ t h ( z ) = μ h ( f t ( z ) ) , \partial_{t}(f_{t}(z))h^{\prime}(f_{t}(z))=\mu e^{\mu t}h(z)=\mu h(f_{t}(z)),
  51. v = v ( 0 ) h h v=v^{\prime}(0){h\over h^{\prime}}
  52. t f t ( z ) = v ( f t ( z ) ) , f t ( z ) = 0 , \partial_{t}f_{t}(z)=v(f_{t}(z)),\,\,\,f_{t}(z)=0~{},
  53. v ( z ) z 0 , \Re{v(z)\over z}\leq 0~{},
  54. v ( z ) v(z)
  55. v ( z ) = z p ( z ) , p ( z ) 0 , p ( 0 ) < 0. v(z)=zp(z),\,\,\,\Re p(z)\leq 0,\,\,\,p^{\prime}(0)<0.
  56. v ( z ) v(z)
  57. h ( z ) = z exp 0 z v ( 0 ) v ( w ) - 1 w d w . h(z)=z\exp\int_{0}^{z}{v^{\prime}(0)\over v(w)}-{1\over w}\,dw.

Kolmogorov_equations_(Markov_jump_process).html

  1. P ( x , s ; y , t ) P(x,s;y,t)
  2. x , y Ω x,y\in\Omega
  3. t > s t>s
  4. i , j i,j
  5. x , y x,y
  6. P i j t ( s ; t ) = k P i k ( s ; t ) A k j ( t ) \frac{\partial P_{ij}}{\partial t}(s;t)=\sum_{k}P_{ik}(s;t)A_{kj}(t)
  7. P i j s ( s ; t ) = - k A i k ( s ) P k j ( s ; t ) \frac{\partial P_{ij}}{\partial s}(s;t)=-\sum_{k}A_{ik}(s)P_{kj}(s;t)
  8. P i j ( s ; t ) P_{ij}(s;t)
  9. i i
  10. s s
  11. j j
  12. t > s t>s
  13. A i j ( t ) A_{ij}(t)
  14. A i j ( t ) = [ P i j u ( t ; u ) ] u = t , A j k ( t ) 0 , j k , k A j k ( t ) = 0. A_{ij}(t)=\left[\frac{\partial P_{ij}}{\partial u}(t;u)\right]_{u=t},\quad A_{% jk}(t)\geq 0,\ j\neq k,\quad\sum_{k}A_{jk}(t)=0.
  15. P ( i , s ; j , t ) P(i,s;j,t)
  16. t > s t>s
  17. s = 0 s=0
  18. i i
  19. A j k ( t ) A_{jk}(t)
  20. P i k ( 0 ; t ) = P k ( t ) P_{ik}(0;t)=P_{k}(t)
  21. d P k d t ( t ) = j A j k ( t ) P j ( t ) ; P k ( 0 ) = δ i k , k = 0 , 1 , . \frac{dP_{k}}{dt}(t)=\sum_{j}A_{jk}(t)P_{j}(t);\quad P_{k}(0)=\delta_{ik},% \qquad k=0,1,\dots.
  22. A j , j - 1 = μ , j 1 A_{j,j-1}=\mu,\ j\geq 1
  23. Ψ ( x , t ) = k x k P k ( t ) , \Psi(x,t)=\sum_{k}x^{k}P_{k}(t),\quad
  24. Ψ ( x , t ) {\Psi}(x,t)
  25. Ψ ( x , 0 ) = x i \Psi(x,0)=x^{i}
  26. Ψ t ( x , t ) = μ ( 1 - x ) Ψ x ( x , t ) ; Ψ ( x , 0 ) = x i , Ψ ( 1 , t ) = 1. \frac{\partial\Psi}{\partial t}(x,t)=\mu(1-x)\frac{\partial{\Psi}}{\partial x}% (x,t);\qquad\Psi(x,0)=x^{i},\quad\Psi(1,t)=1.

Kolmogorov–Zurbenko_filter.html

  1. X ( t ) , t = 0 , ± 1 , ± 2 , {X(t)},t=0,\pm 1,\pm 2,\dots
  2. m m
  3. k k
  4. K Z m , k [ X ( t ) ] = s = - k ( m - 1 ) / 2 k ( m - 1 ) / 2 X ( t + s ) × a s m , k KZ_{m,k}[X(t)]=\sum\limits_{s=-k(m-1)/2}^{k(m-1)/2}{X(t+s)\times{a_{s}^{m,k}}}
  5. a s m , k = c s k , m m k , s = - k ( m - 1 ) 2 , , k ( m - 1 ) 2 a_{s}^{m,k}=\frac{c_{s}^{k,m}}{m^{k}},s=\frac{-k(m-1)}{2},\dots,\frac{k(m-1)}{2}
  6. r = 0 k ( m - 1 ) z r c r - k ( m - 1 ) / 2 k , m = ( 1 + z + + z m - 1 ) k \sum\limits_{r=0}^{k(m-1)}{z^{r}c_{r-k(m-1)/2}^{k,m}=(1+z+\dots+z^{m-1})^{k}}
  7. m m
  8. k k
  9. k k
  10. m m
  11. X ( t ) X(t)
  12. K Z m , k = 1 [ X ( t ) ] = s = - ( m - 1 ) / 2 ( m - 1 ) / 2 X ( t + s ) × 1 m KZ_{m,k=1}[X(t)]=\sum\limits_{s=-(m-1)/2}^{(m-1)/2}{X(t+s)}\times\frac{1}{m}
  13. K Z m , k = 2 [ X ( t ) ] = s = - ( m - 1 ) / 2 ( m - 1 ) / 2 K Z m , k = 1 [ X ( t + s ) ] × 1 m = s = - 2 ( m - 1 ) / 2 2 ( m - 1 ) / 2 X ( t + s ) × a s m , k = 2 \begin{array}[]{l}KZ_{m,k=2}[X(t)]=\sum\limits_{s=-(m-1)/2}^{(m-1)/2}{KZ_{m,k=% 1}[X(t+s)]\times\frac{1}{m}}\\ =\sum\limits_{s=-2(m-1)/2}^{2(m-1)/2}{X(t+s)\times{a_{s}^{m,k=2}}}\end{array}
  14. - ( m - 1 ) / 2 , ( m - 1 ) / 2 -(m-1)/2,(m-1)/2
  15. ( m 1 ) k + 11 (m−1)k+11
  16. k 2 k−2
  17. | B m , k ( ω ) | 2 = { 1 m sin ( π m ω ) sin ( π ω ) } 2 k |B_{m,k}(\omega)|^{2}=\left\{\frac{1}{m}\frac{\sin(\pi m\omega)}{\sin(\pi% \omega)}\right\}^{2k}
  18. ω 0 6 π 1 - ( 1 / 2 ) 1 / 2 k m 2 - ( 1 / 2 ) 1 / 2 k \omega_{0}\approx\frac{\sqrt{6}}{\pi}\sqrt{\frac{1-(1/2)^{1/2k}}{m^{2}-(1/2)^{% 1/2k}}}
  19. K Z F T m , k , ν 0 [ X ( t ) ] = s = - k ( m - 1 ) / 2 k ( m - 1 ) / 2 X ( t + s ) × a s m , k × e - i ( 2 m ν 0 ) s KZFT_{m,k,\nu_{0}}[X(t)]=\sum\limits_{s=-k(m-1)/2}^{k(m-1)/2}{X(t+s)\times{a_{% s}^{m,k}\times{e^{-i(2m\nu_{0})s}}}}
  20. K Z P ( t , m , k , ν 0 ) = 2 | 1 2 S ρ 0 τ = - S ρ 0 S ρ 0 2 R e [ K Z F T m , k , ν + 0 [ X ( τ + t ) ] ] 2 | KZP(t,m,k,\nu_{0})=2|\frac{1}{2S\rho_{0}}\sum\limits_{\tau=-S\rho_{0}}^{S\rho_% {0}}2Re[KZFT_{m,k,\nu+{0}}[X(\tau+t)]]^{2}|
  21. c / ( m k ) c/(m\sqrt{k})
  22. X ( t ) = e < s u p > i ( 2 m ν 0 ) t X(t)=e<sup>i(2mν_{0})t

Komlós–Major–Tusnády_approximation.html

  1. U 1 , U 2 , U_{1},U_{2},\ldots
  2. F U , n ( t ) = 1 n i = 1 n 𝟏 U i t , t [ 0 , 1 ] . F_{U,n}(t)=\frac{1}{n}\sum_{i=1}^{n}\mathbf{1}_{U_{i}\leq t},\quad t\in[0,1].
  3. α U , n ( t ) = n ( F U , n ( t ) - t ) , t [ 0 , 1 ] . \alpha_{U,n}(t)=\sqrt{n}(F_{U,n}(t)-t),\quad t\in[0,1].
  4. α U , n ( t ) \alpha_{U,n}(t)
  5. B ( t ) . B(t).
  6. U 1 , U 2 U_{1},U_{2}\ldots
  7. { α U , n ( t ) , 0 t 1 } \{\alpha_{U,n}(t),0\leq t\leq 1\}
  8. { B n ( t ) , 0 t 0 } \{B_{n}(t),0\leq t\leq 0\}
  9. P { sup 0 t 1 | α U , n ( t ) - B n ( t ) | > 1 n ( a log n + x ) } b e - c x P\left\{\sup_{0\leq t\leq 1}|\alpha_{U,n}(t)-B_{n}(t)|>\frac{1}{\sqrt{n}}(a% \log n+x)\right\}\leq be^{-cx}
  10. x > 0 x>0
  11. X 1 , X 2 , , X_{1},X_{2},\ldots,
  12. F ( t ) , F(t),
  13. α X , n ( t ) = n ( F X , n ( t ) - F ( t ) ) \alpha_{X,n}(t)=\sqrt{n}(F_{X,n}(t)-F(t))
  14. G F , n ( t ) = B n ( F ( t ) ) G_{F,n}(t)=B_{n}(F(t))
  15. lim sup n n ln n α X , n - G F , n < , \limsup_{n\to\infty}\frac{\sqrt{n}}{\ln n}\big\|\alpha_{X,n}-G_{F,n}\big\|_{% \infty}<\infty,

Kondo_insulator.html

  1. ϵ g \epsilon_{\mathrm{g}}

Kondo_model.html

  1. H = k σ ϵ 𝐤 c 𝐤 σ c 𝐤 σ - J 𝐒 ( 0 ) 𝐒 H=\sum_{k\sigma}\epsilon_{\mathbf{k}}c^{\dagger}_{\mathbf{k}\sigma}c_{\mathbf{% k}\sigma}-J\mathbf{S}(0)\cdot\mathbf{S}
  2. 𝐒 \mathbf{S}
  3. 𝐒 ( 0 ) = k , k , σ , σ c 𝐤 σ σ σ , σ c 𝐤 σ \mathbf{S}(0)=\sum_{k,k^{\prime},\sigma,\sigma^{\prime}}c^{\dagger}_{\mathbf{k% }\sigma}\mathbf{\sigma}_{\sigma,\sigma^{\prime}}c_{\mathbf{k^{\prime}}\sigma^{% \prime}}
  4. σ \mathbf{\sigma}
  5. J eff J_{\mathrm{eff}}

Korn–Kreer–Lenssen_model.html

  1. Δ t \Delta t
  2. Δ t \Delta t
  3. Δ t 0 \Delta t\rightarrow 0

Kosmann_lift.html

  1. X X\,
  2. ( M , g ) (M,g)\,
  3. X K X_{K}\,
  4. X ^ \hat{X}\,
  5. Q E Q\subset E\,
  6. π E : E M \pi_{E}\colon E\to M\,
  7. M M
  8. Z Z\,
  9. E E
  10. Z | Q Z|_{Q}\,
  11. Q Q
  12. Q Q
  13. Q Q
  14. i Q : Q E i_{Q}\colon Q\hookrightarrow E
  15. Z | Q Z|_{Q}\,
  16. i Q ( T E ) Q i^{\ast}_{Q}(TE)\to Q\,
  17. i Q ( T E ) = { ( q , v ) Q × T E i ( q ) = τ E ( v ) } Q × T E , i^{\ast}_{Q}(TE)=\{(q,v)\in Q\times TE\mid i(q)=\tau_{E}(v)\}\subset Q\times TE,\,
  18. τ E : T E E \tau_{E}\colon TE\to E\,
  19. E E
  20. i Q ( T E ) Q i^{\ast}_{Q}(TE)\to Q\,
  21. i Q ( T E ) = T Q ( Q ) , i^{\ast}_{Q}(TE)=TQ\oplus\mathcal{M}(Q),\,
  22. q Q q\in Q
  23. T q E = T q Q u , T_{q}E=T_{q}Q\oplus\mathcal{M}_{u}\,,
  24. u \mathcal{M}_{u}
  25. T q E T_{q}E\,
  26. ( Q ) Q \mathcal{M}(Q)\to Q\,
  27. Q Q
  28. Z | Q Z|_{Q}\,
  29. Q Q
  30. Z K Z_{K}\,
  31. Q Q
  32. Z G , Z_{G},\,
  33. ( Q ) Q . \mathcal{M}(Q)\to Q.\,
  34. F S O ( M ) M \mathrm{F}_{SO}(M)\to M
  35. n n
  36. M M
  37. g g\,
  38. SO ( n ) {\mathrm{S}\mathrm{O}}(n)\,
  39. F M \mathrm{F}M\,
  40. M M
  41. GL ( n , ) {\mathrm{G}\mathrm{L}}(n,\mathbb{R})\,
  42. SO ( n ) {\mathrm{S}\mathrm{O}}(n)\,
  43. SO ( n ) {\mathrm{S}\mathrm{O}}(n)\,
  44. GL ( n , ) {\mathrm{G}\mathrm{L}}(n,\mathbb{R})\,
  45. 𝔤 𝔩 ( n ) = 𝔰 𝔬 ( n ) 𝔪 \mathfrak{gl}(n)=\mathfrak{so}(n)\oplus\mathfrak{m}\,
  46. 𝔤 𝔩 ( n ) \mathfrak{gl}(n)\,
  47. GL ( n , ) {\mathrm{G}\mathrm{L}}(n,\mathbb{R})\,
  48. 𝔰 𝔬 ( n ) \mathfrak{so}(n)\,
  49. SO ( n ) {\mathrm{S}\mathrm{O}}(n)\,
  50. 𝔪 \mathfrak{m}\,
  51. Ad SO \mathrm{Ad}_{\mathrm{S}\mathrm{O}}\,
  52. Ad a 𝔪 𝔪 \mathrm{Ad}_{a}\mathfrak{m}\subset\mathfrak{m}\,
  53. a SO ( n ) . a\in{\mathrm{S}\mathrm{O}}(n)\,.
  54. i F S O ( M ) : F S O ( M ) F M i_{\mathrm{F}_{SO}(M)}\colon\mathrm{F}_{SO}(M)\hookrightarrow\mathrm{F}M
  55. i F S O ( M ) ( T F M ) F S O ( M ) i^{\ast}_{\mathrm{F}_{SO}(M)}(T\mathrm{F}M)\to\mathrm{F}_{SO}(M)
  56. i F S O ( M ) ( T F M ) = T F S O ( M ) ( F S O ( M ) ) , i^{\ast}_{\mathrm{F}_{SO}(M)}(T\mathrm{F}M)=T\mathrm{F}_{SO}(M)\oplus\mathcal{% M}(\mathrm{F}_{SO}(M))\,,
  57. u F S O ( M ) u\in\mathrm{F}_{SO}(M)
  58. T u F M = T u F S O ( M ) u , T_{u}\mathrm{F}M=T_{u}\mathrm{F}_{SO}(M)\oplus\mathcal{M}_{u}\,,
  59. u \mathcal{M}_{u}
  60. u u
  61. ( F S O ( M ) ) F S O ( M ) \mathcal{M}(\mathrm{F}_{SO}(M))\to\mathrm{F}_{SO}(M)
  62. i F S O ( M ) ( V F M ) F S O ( M ) i^{\ast}_{\mathrm{F}_{SO}(M)}(V\mathrm{F}M)\to\mathrm{F}_{SO}(M)
  63. V F M V\mathrm{F}M\,
  64. T F M T\mathrm{F}M\,
  65. u F S O ( M ) u\in\mathrm{F}_{SO}(M)
  66. u \mathcal{M}_{u}
  67. 𝔪 \mathfrak{m}
  68. Z | F S O ( M ) Z|_{\mathrm{F}_{SO}(M)}
  69. GL ( n , ) {\mathrm{G}\mathrm{L}}(n,\mathbb{R})
  70. Z Z\,
  71. F M \mathrm{F}M
  72. F S O ( M ) \mathrm{F}_{SO}(M)
  73. SO ( n ) {\mathrm{S}\mathrm{O}}(n)
  74. Z K Z_{K}\,
  75. F S O ( M ) \mathrm{F}_{SO}(M)
  76. Z Z\,
  77. Z G Z_{G}\,
  78. X X\,
  79. ( M , g ) (M,g)\,
  80. X ^ | F S O ( M ) \hat{X}|_{\mathrm{F}_{SO}(M)}\,
  81. F S O ( M ) M \mathrm{F}_{SO}(M)\to M
  82. X ^ \hat{X}\,
  83. F M M \mathrm{F}M\to M
  84. SO ( n ) {\mathrm{S}\mathrm{O}}(n)
  85. X K X_{K}\,
  86. F S O ( M ) \mathrm{F}_{SO}(M)
  87. X X\,
  88. X G X_{G}\,

KR-theory.html

  1. K R p , q ( X , Y ) = K R ( X × B p , q , X × S p , q Y × B p , q ) KR^{p,q}(X,Y)=KR(X\times B^{p,q},X\times S^{p,q}\cup Y\times B^{p,q})

Kramers'_opacity_law.html

  1. κ ¯ ρ T - 7 / 2 , \bar{\kappa}\propto\rho T^{-7/2},
  2. κ ¯ \bar{\kappa}
  3. ρ \rho
  4. T T
  5. κ ¯ b f = 4.34 × 10 25 g b f t Z ( 1 + X ) ρ g / cm 3 ( T K ) - 7 / 2 cm 2 g - 1 , \bar{\kappa}_{bf}=4.34\times 10^{25}\frac{g_{bf}}{t}Z(1+X)\frac{\rho}{\rm g/cm% ^{3}}\left(\frac{T}{\rm K}\right)^{-7/2}{\rm\,cm^{2}\,g^{-1}},
  6. κ ¯ f f = 3.68 × 10 22 g f f ( 1 - Z ) ( 1 + X ) ρ g / cm 3 ( T K ) - 7 / 2 cm 2 g - 1 . \bar{\kappa}_{ff}=3.68\times 10^{22}g_{ff}(1-Z)(1+X)\frac{\rho}{\rm g/cm^{3}}% \left(\frac{T}{\rm K}\right)^{-7/2}{\rm\,cm^{2}\,g^{-1}}.
  7. κ ¯ e s = 0.2 ( 1 + X ) cm 2 g - 1 \bar{\kappa}_{es}=0.2(1+X){\rm\,cm^{2}\,g^{-1}}
  8. g b f g_{bf}
  9. g f f g_{ff}
  10. t t
  11. Z Z
  12. X X

Krein's_condition.html

  1. { k = 1 n a k exp ( i λ k x ) , a k , λ k 0 } , \left\{\sum_{k=1}^{n}a_{k}\exp(i\lambda_{k}x),\quad a_{k}\in\mathbb{C},\,% \lambda_{k}\geq 0\right\},\,
  2. k = 1 n a k exp ( i λ k x ) , a k , λ k 0 \sum_{k=1}^{n}a_{k}\exp(i\lambda_{k}x),\quad a_{k}\in\mathbb{C},\,\lambda_{k}\geq 0
  3. - - ln f ( x ) 1 + x 2 d x = . \int_{-\infty}^{\infty}\frac{-\ln f(x)}{1+x^{2}}\,dx=\infty.
  4. m n = - x n d μ ( x ) , n = 0 , 1 , 2 , m_{n}=\int_{-\infty}^{\infty}x^{n}d\mu(x),\quad n=0,1,2,\ldots
  5. - - ln f ( x ) 1 + x 2 d x < \int_{-\infty}^{\infty}\frac{-\ln f(x)}{1+x^{2}}\,dx<\infty
  6. m n = - x n d ν ( x ) , n = 0 , 1 , 2 , m_{n}=\int_{-\infty}^{\infty}x^{n}\,d\nu(x),\quad n=0,1,2,\ldots
  7. f ( x ) = 1 π exp { - ln 2 x } ; f(x)=\frac{1}{\sqrt{\pi}}\exp\left\{-\ln^{2}x\right\};
  8. - - ln f ( x ) 1 + x 2 d x = - ln 2 x + ln π 1 + x 2 d x < , \int_{-\infty}^{\infty}\frac{-\ln f(x)}{1+x^{2}}dx=\int_{-\infty}^{\infty}% \frac{\ln^{2}x+\ln\sqrt{\pi}}{1+x^{2}}\,dx<\infty,

Kronecker_coefficient.html

  1. V μ V ν = λ g μ ν λ V λ . V_{\mu}\otimes V_{\nu}=\bigoplus_{\lambda}g_{\mu\nu}^{\lambda}V_{\lambda}.
  2. s μ s ν = λ g μ ν λ s λ . s_{\mu}\star s_{\nu}=\sum_{\lambda}g_{\mu\nu}^{\lambda}s_{\lambda}.
  3. S | μ | × S | ν | S | λ | ( V μ V ν ) = λ c μ ν λ V λ , \uparrow_{S_{|\mu|}\times S_{|\nu|}}^{S_{|\lambda|}}\left(V_{\mu}\otimes V_{% \nu}\right)=\bigoplus_{\lambda}c_{\mu\nu}^{\lambda}V_{\lambda},
  4. W μ W ν = λ c μ ν λ W λ . W_{\mu}\otimes W_{\nu}=\bigoplus_{\lambda}c_{\mu\nu}^{\lambda}W_{\lambda}.
  5. g ( λ , μ , ν ) = 1 n ! σ S n χ λ ( σ ) χ μ ( σ ) χ ν ( σ ) g(\lambda,\mu,\nu)=\frac{1}{n!}\sum_{\sigma\in S_{n}}\chi^{\lambda}(\sigma)% \chi^{\mu}(\sigma)\chi^{\nu}(\sigma)
  6. λ , μ , ν g ( λ , μ , ν ) s λ ( x ) s μ ( y ) s ν ( z ) = i , j , k 1 1 - x i y j z k . \sum_{\lambda,\mu,\nu}g(\lambda,\mu,\nu)s_{\lambda}(x)s_{\mu}(y)s_{\nu}(z)=% \prod_{i,j,k}\frac{1}{1-x_{i}y_{j}z_{k}}.

KST_oscillator.html

  1. R O C 1 = ( P r i c e / P r i c e ( X 1 ) - 1 ) * 100 ; ROC1=(Price/Price(X1)-1)*100;
  2. R O C 2 = ( P r i c e / P r i c e ( X 2 ) - 1 ) * 100 ; ROC2=(Price/Price(X2)-1)*100;
  3. R O C 3 = ( P r i c e / P r i c e ( X 3 ) - 1 ) * 100 ; ROC3=(Price/Price(X3)-1)*100;
  4. R O C 4 = ( P r i c e / P r i c e ( X 4 ) - 1 ) * 100 ; ROC4=(Price/Price(X4)-1)*100;
  5. K S T = M O V ( R O C 1 , A V G 1 ) * W 1 + M O V ( R O C 2 , A V G 2 ) * W 2 + M O V ( R O C 3 , A V G 3 ) * W 3 + M O V ( R O C 4 , A V G 4 ) * W 4 KST=MOV(ROC1,AVG1)*W1+MOV(ROC2,AVG2)*W2+MOV(ROC3,AVG3)*W3+MOV(ROC4,AVG4)*W4

Kummer's_congruence.html

  1. B h h B k k ( mod p ) whenever h k ( mod p - 1 ) \frac{B_{h}}{h}\equiv\frac{B_{k}}{k}\;\;(\mathop{{\rm mod}}p)\,\text{ whenever% }h\equiv k\;\;(\mathop{{\rm mod}}p-1)
  2. ( 1 - p h - 1 ) B h h ( 1 - p k - 1 ) B k k ( mod p a + 1 ) (1-p^{h-1})\frac{B_{h}}{h}\equiv(1-p^{k-1})\frac{B_{k}}{k}\;\;(\mathop{{\rm mod% }}p^{a+1})
  3. h k ( mod φ ( p a + 1 ) ) h\equiv k\;\;(\mathop{{\rm mod}}\varphi(p^{a+1}))

Kummer's_theorem.html

  1. ( n m ) {\textstyle\left({{n}\atop{m}}\right)}
  2. ( n m ) {\textstyle\left({{n}\atop{m}}\right)}
  3. n ! m ! ( n - m ) ! \tfrac{n!}{m!(n-m)!}

Kunita–Watanabe_theorem.html

  1. 0 t | H s | | K s | d M , N s 0 t H s 2 d M s 0 t K s 2 d N s \int_{0}^{t}|H_{s}||K_{s}|d\langle M,N\rangle_{s}\leq\sqrt{\int_{0}^{t}H_{s}^{% 2}d\langle M\rangle_{s}}\sqrt{\int_{0}^{t}K_{s}^{2}d\langle N\rangle_{s}}

Kv_(flow_factor).html

  1. K v = F S G Δ P K_{v}=F\sqrt{\dfrac{SG}{\Delta P}}
  2. K v = ( 0.865 ) ( C v ) {K_{v}}=(0.865)({C_{v}})

L-balance_theorem.html

  1. L 2 ( C X ( T ) ) L 2 ( X ) L_{2^{\prime}}(C_{X}(T))\leq L_{2^{\prime}}(X)
  2. L 2 ( L 2 ( C a ) C b ) = L 2 ( L 2 ( C b ) C a ) L_{2^{\prime}}(L_{2^{\prime}}(C_{a})\cap C_{b})=L_{2^{\prime}}(L_{2^{\prime}}(% C_{b})\cap C_{a})

L-stability.html

  1. ϕ ( z ) 0 \phi(z)\to 0
  2. z z\to\infty
  3. ϕ \phi
  4. z + z\to+\infty
  5. z - z\to-\infty

L10a140_link.html

  1. Δ ( u , v , w ) = ( u - 1 ) ( v - 1 ) ( w - 1 ) ( v w + 1 ) 2 v w u v w , \Delta(u,v,w)=\frac{(u-1)(v-1)(w-1)(vw+1)^{2}}{vw\sqrt{uvw}},\,
  2. ( z ) = 4 z 4 + 4 z 6 + z 8 , \nabla(z)=4z^{4}+4z^{6}+z^{8},\,
  3. V ( t ) \displaystyle V(t)
  4. w ( t ) = t 5 - 2 t 4 + t 3 - 2 t 2 + t - 1. w(t)=t^{5}-2t^{4}+t^{3}-2t^{2}+t-1.
  5. w ( t ) w(t)
  6. P ( α , z ) = z - 2 α - 2 - 4 z 2 α - 2 - 4 z 4 α - 2 - z 6 α - 2 - 2 z - 2 + 8 z 2 + 12 z 4 + 6 z 6 + z 8 + z - 2 α 2 - 4 z 2 α 2 - 4 z 4 α 2 - z 6 α 2 , P(\alpha,z)=z^{-2}\alpha^{-2}-4z^{2}\alpha^{-2}-4z^{4}\alpha^{-2}-z^{6}\alpha^% {-2}-2z^{-2}+8z^{2}+12z^{4}+6z^{6}+z^{8}+z^{-2}\alpha^{2}-4z^{2}\alpha^{2}-4z^% {4}\alpha^{2}-z^{6}\alpha^{2},\,
  7. F ( a , z ) \displaystyle F(a,z)

Labelled_enumeration_theorem.html

  1. f n ( z ) f_{n}(z)
  2. f n ( z ) = g ( z ) n | G | . f_{n}(z)=\frac{g(z)^{n}}{|G|}.
  3. F ( z , t ) = n = 0 f n ( z ) t n = n = 0 g ( z ) n n ! t n = e g ( z ) t F(z,t)=\sum_{n=0}^{\infty}f_{n}(z)t^{n}=\sum_{n=0}^{\infty}\frac{g(z)^{n}}{n!}% t^{n}=e^{g(z)t}
  4. G = S 3 G=S_{3}
  5. ω \omega
  6. | ω | |\omega|
  7. z | ω | / | ω | ! z^{|\omega|}/|\omega|!
  8. | ω | = m |\omega|=m\,
  9. | G | |G|
  10. | G | |G|
  11. r 1 r_{1}
  12. r 2 r_{2}
  13. r 1 + r 2 + + r n = k . r_{1}+r_{2}+\cdots+r_{n}=k.\,
  14. ( k r 1 , r 2 , r n ) {k\choose r_{1},r_{2},\ldots r_{n}}
  15. r 1 , r 2 , r n . r_{1},r_{2},\ldots r_{n}.
  16. [ k ] [k]
  17. 1 | G | r 1 + r 2 + + r n = k ( k r 1 , r 2 , r n ) r 1 ! [ z r 1 ] g ( z ) r 2 ! [ z r 2 ] g ( z ) r n ! [ z r n ] g ( z ) \frac{1}{|G|}\sum_{r_{1}+r_{2}+\ldots+r_{n}=k}{k\choose r_{1},r_{2},\ldots r_{% n}}\;r_{1}![z^{r_{1}}]g(z)\;r_{2}![z^{r_{2}}]g(z)\;\cdots\;r_{n}![z^{r_{n}}]g(z)
  18. k ! | G | r 1 + r 2 + + r n = k [ z r 1 ] g ( z ) [ z r 2 ] g ( z ) [ z r n ] g ( z ) = k ! | G | [ z k ] g ( z ) n \frac{k!}{|G|}\sum_{r_{1}+r_{2}+\ldots+r_{n}=k}[z^{r_{1}}]g(z)[z^{r_{2}}]g(z)% \cdots[z^{r_{n}}]g(z)\;=\;\frac{k!}{|G|}[z^{k}]g(z)^{n}
  19. [ z k ] g ( z ) n [z^{k}]g(z)^{n}
  20. n ! | k ! n!|k!
  21. | G | |G|
  22. S n S_{n}
  23. n ! n!
  24. f n ( z ) = k 0 ( k ! | G | [ z k ] g ( z ) n ) z k k ! = 1 | G | k 0 z k [ z k ] g ( z ) n = g ( z ) n | G | . f_{n}(z)=\sum_{k\geq 0}\left(\frac{k!}{|G|}[z^{k}]g(z)^{n}\right)\frac{z^{k}}{% k!}=\frac{1}{|G|}\sum_{k\geq 0}z^{k}[z^{k}]g(z)^{n}=\frac{g(z)^{n}}{|G|}.
  25. 1 / | G | 1/|G|
  26. g ( z ) n g(z)^{n}
  27. | G | |G|
  28. g ( z ) n / | G | . g(z)^{n}/|G|.

Labor_demand.html

  1. Maximize p Q - w L - r K with respect to Q , L , and K \,\text{Maximize}\,\,pQ-wL-rK\,\,\,\text{with respect to}\,\,Q,\,L,\,\,\text{% and}\,K
  2. subject to \,\text{subject to}
  3. Q = f ( L , K ) , Q=f\,(L,K),
  4. L ( p , w , r ) , L(p,w,r),
  5. K ( p , w , r ) , K(p,w,r),
  6. Q ( p , w , r ) . Q(p,w,r).
  7. Maximize p Q ( p ) - w L - r K with respect to L , K , and p \,\text{Maximize}\,\,pQ(p)-wL-rK\,\,\,\text{with respect to}\,L,\,K,\,\,\text{% and}\,p
  8. subject to \,\text{subject to}
  9. Q ( p ) = f ( L , K ) , Q(p)=f\,(L,K),
  10. L ( w , r ) , L(w,r),
  11. K ( w , r ) , K(w,r),
  12. p ( w , r ) . p(w,r).
  13. Maximize p Q - w L ( w ) - r K with respect to Q , w , and K \,\text{Maximize}\,\,pQ-wL(w)-rK\,\,\,\text{with respect to}\,\,Q,\,w,\,\,% \text{and}\,K
  14. subject to \,\text{subject to}
  15. Q = f ( L , K ) , Q=f\,(L,K),

Lady_Windermere's_Fan_(mathematics).html

  1. E ( τ , t 0 , y ( t 0 ) ) E(\ \tau,t_{0},y(t_{0})\ )
  2. y ( t 0 + τ ) = E ( τ , t 0 , y ( t 0 ) ) y ( t 0 ) y(t_{0}+\tau)=E(\tau,t_{0},y(t_{0}))\ y(t_{0})
  3. t 0 t_{0}
  4. y ( t ) y(t)
  5. y ( t 0 ) y(t_{0})
  6. y n y_{n}
  7. n 𝒩 , n N n\in\mathcal{N},\ n\leq N
  8. t n t_{n}
  9. t 0 < t n T = t N t_{0}<t_{n}\leq T=t_{N}
  10. y n y_{n}
  11. Φ ( h n , t n , y ( t n ) ) \Phi(\ h_{n},t_{n},y(t_{n})\ )
  12. y n = Φ ( h n - 1 , t n - 1 , y ( t n - 1 ) ) y n - 1 y_{n}=\Phi(\ h_{n-1},t_{n-1},y(t_{n-1})\ )\ y_{n-1}\quad
  13. h n = t n + 1 - t n h_{n}=t_{n+1}-t_{n}
  14. h h
  15. Φ Euler ( h , t n - 1 , y ( t n - 1 ) ) y ( t n - 1 ) = ( 1 + h d d t ) y ( t n - 1 ) \Phi_{\,\text{Euler}}(\ h,t_{n-1},y(t_{n-1})\ )\ y(t_{n-1})=(1+h\frac{d}{dt})% \ y(t_{n-1})
  16. d n d_{n}
  17. d n := D ( h n - 1 , t n - 1 , y ( t n - 1 ) y n - 1 := [ Φ ( h n - 1 , t n - 1 , y ( t n - 1 ) ) - E ( h n - 1 , t n - 1 , y ( t n - 1 ) ) ] y n - 1 d_{n}:=D(\ h_{n-1},t_{n-1},y(t_{n-1}\ )\ y_{n-1}:=\left[\Phi(\ h_{n-1},t_{n-1}% ,y(t_{n-1})\ )-E(\ h_{n-1},t_{n-1},y(t_{n-1})\ )\right]\ y_{n-1}
  18. Φ ( h n ) := Φ ( h n , t n , y ( t n ) ) \Phi(h_{n}):=\Phi(\ h_{n},t_{n},y(t_{n})\ )
  19. E ( h n ) := E ( h n , t n , y ( t n ) ) E(h_{n}):=E(\ h_{n},t_{n},y(t_{n})\ )
  20. D ( h n ) := D ( h n , t n , y ( t n ) ) D(h_{n}):=D(\ h_{n},t_{n},y(t_{n})\ )
  21. t t
  22. y N - y ( t N ) = j = 0 N - 1 Φ ( h j ) ( y 0 - y ( t 0 ) ) + n = 1 N j = n N - 1 Φ ( h j ) d n y_{N}-y(t_{N})=\prod_{j=0}^{N-1}\Phi(h_{j})\ (y_{0}-y(t_{0}))+\sum_{n=1}^{N}\ % \prod_{j=n}^{N-1}\Phi(h_{j})\ d_{n}
  23. y N - y ( t N ) y_{N}-y(t_{N})
  24. y N - y ( t N ) = y N - j = 0 N - 1 Φ ( h j ) y ( t 0 ) + j = 0 N - 1 Φ ( h j ) y ( t 0 ) = 0 - y ( t N ) = y N - j = 0 N - 1 Φ ( h j ) y ( t 0 ) + n = 0 N - 1 j = n N - 1 Φ ( h j ) y ( t n ) - n = 1 N j = n N - 1 Φ ( h j ) y ( t n ) = n = 0 N - 1 Φ ( h n ) y ( t n ) - n = N N [ j = n N - 1 Φ ( h j ) ] y ( t n ) = j = 0 N - 1 Φ ( h j ) y ( t 0 ) - y ( t N ) = j = 0 N - 1 Φ ( h j ) y 0 - j = 0 N - 1 Φ ( h j ) y ( t 0 ) + n = 1 N j = n - 1 N - 1 Φ ( h j ) y ( t n - 1 ) - n = 1 N j = n N - 1 Φ ( h j ) y ( t n ) = j = 0 N - 1 Φ ( h j ) ( y 0 - y ( t 0 ) ) + n = 1 N j = n N - 1 Φ ( h j ) [ Φ ( h n - 1 ) - E ( h n - 1 ) ] y ( t n - 1 ) = j = 0 N - 1 Φ ( h j ) ( y 0 - y ( t 0 ) ) + n = 1 N j = n N - 1 Φ ( h j ) d n \begin{aligned}\displaystyle y_{N}-y(t_{N})&\displaystyle{}=y_{N}-\underbrace{% \prod_{j=0}^{N-1}\Phi(h_{j})\ y(t_{0})+\prod_{j=0}^{N-1}\Phi(h_{j})\ y(t_{0})}% _{=0}-y(t_{N})\\ &\displaystyle{}=y_{N}-\prod_{j=0}^{N-1}\Phi(h_{j})\ y(t_{0})+\underbrace{\sum% _{n=0}^{N-1}\ \prod_{j=n}^{N-1}\Phi(h_{j})\ y(t_{n})-\sum_{n=1}^{N}\ \prod_{j=% n}^{N-1}\Phi(h_{j})\ y(t_{n})}_{=\prod_{n=0}^{N-1}\Phi(h_{n})\ y(t_{n})-\sum_{% n=N}^{N}\left[\prod_{j=n}^{N-1}\Phi(h_{j})\right]\ y(t_{n})=\prod_{j=0}^{N-1}% \Phi(h_{j})\ y(t_{0})-y(t_{N})}\\ &\displaystyle{}=\prod_{j=0}^{N-1}\Phi(h_{j})\ y_{0}-\prod_{j=0}^{N-1}\Phi(h_{% j})\ y(t_{0})+\sum_{n=1}^{N}\ \prod_{j=n-1}^{N-1}\Phi(h_{j})\ y(t_{n-1})-\sum_% {n=1}^{N}\ \prod_{j=n}^{N-1}\Phi(h_{j})\ y(t_{n})\\ &\displaystyle{}=\prod_{j=0}^{N-1}\Phi(h_{j})\ (y_{0}-y(t_{0}))+\sum_{n=1}^{N}% \ \prod_{j=n}^{N-1}\Phi(h_{j})\left[\Phi(h_{n-1})-E(h_{n-1})\right]\ y(t_{n-1}% )\\ &\displaystyle{}=\prod_{j=0}^{N-1}\Phi(h_{j})\ (y_{0}-y(t_{0}))+\sum_{n=1}^{N}% \ \prod_{j=n}^{N-1}\Phi(h_{j})\ d_{n}\end{aligned}

Lagrange,_Euler_and_Kovalevskaya_tops.html

  1. I 1 = I 2 = 2 I 3 I_{1}=I_{2}=2I_{3}
  2. 𝐞 ^ 1 \hat{\mathbf{e}}^{1}
  3. 𝐞 ^ 2 \hat{\mathbf{e}}^{2}
  4. 𝐞 ^ 3 \hat{\mathbf{e}}^{3}
  5. I 1 I_{1}
  6. I 2 I_{2}
  7. I 3 I_{3}
  8. L \vec{L}
  9. ( l 1 , l 2 , l 3 ) = ( L 𝐞 ^ 1 , L 𝐞 ^ 2 , L 𝐞 ^ 3 ) (l_{1},l_{2},l_{3})=(\vec{L}\cdot\hat{\mathbf{e}}^{1},\vec{L}\cdot\hat{\mathbf% {e}}^{2},\vec{L}\cdot\hat{\mathbf{e}}^{3})
  10. ( n 1 , n 2 , n 3 ) = ( 𝐳 ^ 𝐞 ^ 1 , 𝐳 ^ 𝐞 ^ 2 , 𝐳 ^ 𝐞 ^ 3 ) (n_{1},n_{2},n_{3})=(\mathbf{\hat{z}}\cdot\hat{\mathbf{e}}^{1},\mathbf{\hat{z}% }\cdot\hat{\mathbf{e}}^{2},\mathbf{\hat{z}}\cdot\hat{\mathbf{e}}^{3})
  11. { l a , l b } = ϵ a b c l c , { l a , n b } = ϵ a b c n c , { n a , n b } = 0 \{l_{a},l_{b}\}=\epsilon_{abc}l_{c},\ \{l_{a},n_{b}\}=\epsilon_{abc}n_{c},\ \{% n_{a},n_{b}\}=0
  12. R c m = ( a 𝐞 ^ 1 + b 𝐞 ^ 2 + c 𝐞 ^ 3 ) \vec{R}_{cm}=(a\mathbf{\hat{e}}^{1}+b\mathbf{\hat{e}}^{2}+c\mathbf{\hat{e}}^{3})
  13. H = ( l 1 ) 2 2 I 1 + ( l 2 ) 2 2 I 2 + ( l 3 ) 2 2 I 3 + m g ( a n 1 + b n 2 + c n 3 ) , H=\frac{(l_{1})^{2}}{2I_{1}}+\frac{(l_{2})^{2}}{2I_{2}}+\frac{(l_{3})^{2}}{2I_% {3}}+mg(an_{1}+bn_{2}+cn_{3}),
  14. l ˙ a = { H , l a } , n ˙ a = { H , n a } \dot{l}_{a}=\{H,l_{a}\},\dot{n}_{a}=\{H,n_{a}\}
  15. H E = ( l 1 ) 2 2 I 1 + ( l 2 ) 2 2 I 2 + ( l 3 ) 2 2 I 3 , H_{E}=\frac{(l_{1})^{2}}{2I_{1}}+\frac{(l_{2})^{2}}{2I_{2}}+\frac{(l_{3})^{2}}% {2I_{3}},
  16. H E H_{E}
  17. ( L 1 , L 2 , L 3 ) = l 1 𝐞 ^ 1 + l 2 𝐞 ^ 2 + l 3 𝐞 ^ 3 . (L_{1},L_{2},L_{3})=l_{1}\mathbf{\hat{e}}^{1}+l_{2}\mathbf{\hat{e}}^{2}+l_{3}% \mathbf{\hat{e}}^{3}.
  18. R c m = h 𝐞 ^ 3 \vec{R}_{cm}=h\mathbf{\hat{e}}^{3}
  19. H L = ( l 1 ) 2 + ( l 2 ) 2 2 I + ( l 3 ) 2 2 I 3 + m g h n 3 . H_{L}=\frac{(l_{1})^{2}+(l_{2})^{2}}{2I}+\frac{(l_{3})^{2}}{2I_{3}}+mghn_{3}.
  20. H L H_{L}
  21. l 3 l_{3}
  22. L z = l 1 n 1 + l 2 n 2 + l 3 n 3 , L_{z}=l_{1}n_{1}+l_{2}n_{2}+l_{3}n_{3},
  23. n 2 = n 1 2 + n 2 2 + n 3 2 n^{2}=n_{1}^{2}+n_{2}^{2}+n_{3}^{2}
  24. I 1 = I 2 = 2 I 3 = I I_{1}=I_{2}=2I_{3}=I
  25. R c m = h 𝐞 ^ 1 \vec{R}_{cm}=h\mathbf{\hat{e}}^{1}
  26. H K = ( l 1 ) 2 + ( l 2 ) 2 + 2 ( l 3 ) 2 2 I + m g h n 1 . H_{K}=\frac{(l_{1})^{2}+(l_{2})^{2}+2(l_{3})^{2}}{2I}+mghn_{1}.
  27. H K H_{K}
  28. K = ξ + ξ - K=\xi_{+}\xi_{-}
  29. ξ ± \xi_{\pm}
  30. ξ ± = ( l 1 ± i l 2 ) 2 - 2 m g h I ( n 1 ± i n 2 ) , \xi_{\pm}=(l_{1}\pm il_{2})^{2}-2mghI(n_{1}\pm in_{2}),
  31. L z = l 1 n 1 + l 2 n 2 + l 3 n 3 , L_{z}=l_{1}n_{1}+l_{2}n_{2}+l_{3}n_{3},
  32. n 2 = n 1 2 + n 2 2 + n 3 2 . n^{2}=n_{1}^{2}+n_{2}^{2}+n_{3}^{2}.

Lahun_Mathematical_Papyri.html

  1. V = ( ( 1 + 1 / 3 ) d ) 2 ( ( 2 / 3 ) h ) V=((1+1/3)d)^{2}\ ((2/3)h)
  2. V = 32 27 d 2 h = 128 27 r 2 h V=\frac{32}{27}d^{2}\ h=\frac{128}{27}r^{2}\ h
  3. V = 256 81 r 2 h V=\frac{256}{81}r^{2}\ h

Lamb_Dicke_regime.html

  1. η 2 ( 2 n + 1 ) 1 , \eta^{2}(2n+1)\ll 1,
  2. η \eta
  3. n n
  4. z z
  5. | n |n\rangle
  6. z ^ \hat{z}
  7. z ^ = z 0 ( a ^ + a ^ ) . \hat{z}=z_{0}(\hat{a}+\hat{a}^{\dagger}).
  8. z 0 = ( 0 | z 2 | 0 ) 1 2 = ( / 2 m ω z ) 1 2 z_{0}=(\langle 0|z^{2}|0\rangle)^{\frac{1}{2}}=(\hbar/2m\omega_{z})^{\frac{1}{% 2}}
  9. ω z \omega_{z}
  10. z z
  11. a ^ , a ^ \hat{a},\hat{a}^{\dagger}
  12. Ψ m o t i o n | k z 2 z 2 | Ψ m o t i o n 1 / 2 1 \langle\Psi_{motion}|{k_{z}}^{2}z^{2}|\Psi_{motion}\rangle^{1/2}\ll 1
  13. Ψ m o t i o n | \langle\Psi_{motion}|
  14. k z = 𝐤 z ^ = | 𝐤 | cos θ = cos θ ( 2 π λ ) k_{z}=\mathbf{k}\cdot\hat{z}=|\mathbf{k}|\cos\theta=\cos\theta(\frac{2\pi}{% \lambda})
  15. z z
  16. η = k z z 0 . \eta=k_{z}z_{0}.
  17. k z \hbar k_{z}
  18. E R = ω R E_{R}=\hbar\omega_{R}
  19. ω R = k z 2 2 m . \omega_{R}=\frac{\hbar k_{z}^{2}}{2m}.
  20. η 2 = ω R ω z = change in kinetic energy quantized energy spacing of HO . \eta^{2}=\frac{\omega_{R}}{\omega_{z}}=\frac{\mathrm{change\,in\,kinetic\,% energy}}{\mathrm{quantized\,energy\,spacing\,of\,HO}}.
  21. η \eta
  22. exp ( ± i k z z ) \exp(\pm ik_{z}z)
  23. ± k z \pm\hbar k_{z}
  24. { | n } n 0 \{|n\rangle\}_{n\in\mathbb{N}_{0}}
  25. | n | n |n\rangle\rightarrow|n^{\prime}\rangle
  26. F n n = n | exp ( i k z z ) | n = n | exp ( i η ( a ^ + a ^ ) ) | n . F_{n\rightarrow n^{\prime}}=\langle n^{\prime}|\exp(ik_{z}z)|n\rangle=\langle n% ^{\prime}|\exp(i\eta(\hat{a}+\hat{a}^{\dagger}))|n\rangle.
  27. exp ( i η ( a ^ + a ^ ) ) = 1 + i η ( a ^ + a ^ ) + O ( η 2 ) \exp(i\eta(\hat{a}+\hat{a}^{\dagger}))=1+i\eta(\hat{a}+\hat{a}^{\dagger})+O(% \eta^{2})
  28. n n

Lambda-connectedness.html

  1. λ \lambda
  2. λ \lambda
  3. λ \lambda
  4. λ \lambda
  5. λ \lambda
  6. λ \lambda
  7. G , ρ \langle G,\rho\rangle
  8. ρ \rho
  9. G G
  10. G , ρ \langle G,\rho\rangle
  11. G G
  12. ρ \rho
  13. f = ( red , green , blue ) f=(\,\text{red},\,\text{green},\,\text{blue})
  14. ρ \rho
  15. α ρ ( x , y ) \alpha_{\rho}(x,y)
  16. x 1 , x 2 , , x n x_{1},x_{2},\ldots,x_{n}
  17. ( x i , x i + 1 ) E (x_{i},x_{i+1})\in E
  18. β \beta
  19. π = π ( x 1 , x n ) = { x 1 , x 2 , , x n } \pi=\pi(x_{1},x_{n})=\{x_{1},x_{2},...,x_{n}\}
  20. β ρ ( π ( x 1 , x n ) ) = min { α ρ ( x i , x i + 1 ) | i = 1 , , n - 1 } \beta_{\rho}(\pi(x_{1},x_{n}))=\min\{\alpha_{\rho}(x_{i},x_{i+1})|i=1,\ldots,n% -1\}
  21. ρ \rho
  22. C ρ ( x , y ) = max { β ( π ( x , y ) ) | π is a (simple) path . } C_{\rho}(x,y)=\max\{\beta(\pi(x,y))|\pi\text{ is a (simple) path}.\}
  23. λ [ 0 , 1 ] \lambda\in[0,1]
  24. p = ( x , ρ ( x ) ) p=(x,\rho(x))
  25. q = ( y , ρ ( y ) ) q=(y,\rho(y))
  26. λ \lambda
  27. C ρ ( x , y ) λ C_{\rho}(x,y)\geq\lambda
  28. λ \lambda

Laminar_flame_speed.html

  1. s L = α ω ˙ ( T b - T i T i - T u ) s_{\mathrm{L}}^{\circ}=\sqrt{\alpha\dot{\omega}\left(\dfrac{T_{\mathrm{b}}-T_{% \mathrm{i}}}{T_{\mathrm{i}}-T_{\mathrm{u}}}\right)}
  2. α \alpha
  3. ω ˙ \dot{\omega}

Landweber_exact_functor_theorem.html

  1. M U * ( * ) = M U * [ x 1 , x 2 , ] MU_{*}(*)=MU_{*}\cong\mathbb{Z}[x_{1},x_{2},\dots]
  2. x i x_{i}
  3. L * \mathcal{}L_{*}
  4. R * \mathcal{}R_{*}
  5. L * R * L_{*}\to R_{*}
  6. [ n + 1 ] F x = F ( x , [ n ] F x ) [n+1]^{F}x=F(x,[n]^{F}x)
  7. [ 1 ] F x = x . [1]^{F}x=x.
  8. R * \mathcal{}R_{*}
  9. E * ( X ) = M U * ( X ) M U * R * E_{*}(X)=MU_{*}(X)\otimes_{MU_{*}}R_{*}
  10. R * \mathcal{}R_{*}
  11. M U * \mathcal{}MU_{*}
  12. R * \mathcal{}R_{*}
  13. M U * \mathcal{}MU_{*}
  14. v 1 , v 2 , M U * v_{1},v_{2},\cdots\in MU_{*}
  15. M * \mathcal{}M_{*}
  16. M U * \mathcal{}MU_{*}
  17. ( p , v 1 , v 2 , , v n ) (p,v_{1},v_{2},\dots,v_{n})
  18. E * ( X ) = M U * ( X ) M U * M * E_{*}(X)=MU_{*}(X)\otimes_{MU_{*}}M_{*}
  19. M U * \mathcal{}MU_{*}
  20. M U * R MU_{*}\to R
  21. M U ( p ) MU_{(p)}
  22. ( p ) [ v 1 , v 2 , ] \mathbb{Z}_{(p)}[v_{1},v_{2},\dots]
  23. B P * \mathcal{}BP_{*}
  24. B P * B P \mathcal{}BP_{*}BP
  25. I n = ( p , v 1 , , v n ) I_{n}=(p,v_{1},\dots,v_{n})
  26. B P * / I n \mathcal{}BP_{*}/I_{n}
  27. * \mathcal{E}_{*}
  28. M U * \mathcal{}MU_{*}
  29. \mathcal{E}
  30. π * M \mathcal{}\pi_{*}M
  31. π * : * \pi_{*}:\mathcal{E}\to\mathcal{E}_{*}
  32. M U * \mathcal{}MU_{*}
  33. x + y + x y x+y+xy
  34. M U * K * MU_{*}\to K_{*}
  35. K * ( X ) = M U * ( X ) M U * K * , K_{*}(X)=MU_{*}(X)\otimes_{MU_{*}}K_{*},
  36. E ( n ) E(n)
  37. E n E_{n}
  38. H H\mathbb{Q}
  39. H H\mathbb{Z}
  40. M U * \mathcal{}MU_{*}
  41. \mathcal{F}
  42. Spec L \,\text{Spec }L
  43. M = M U * ( X ) M=\mathcal{}MU_{*}(X)
  44. M U * M U \mathcal{}MU_{*}MU
  45. \mathcal{F}
  46. G \Z [ b 1 , b 2 , ] G\cong\Z[b_{1},b_{2},\dots]
  47. g ( t ) = t + b 1 t 2 + b 2 t 3 + R [ [ t ] ] g(t)=t+b_{1}t^{2}+b_{2}t^{3}+\cdots\in R[[t]]
  48. Spec L ( R ) \,\text{Spec }L(R)
  49. F ( x , y ) g F ( g - 1 x , g - 1 y ) F(x,y)\mapsto gF(g^{-1}x,g^{-1}y)
  50. Spec L / / G \,\text{Spec }L//G
  51. f g \mathcal{M}_{fg}
  52. M = M U * ( X ) \mathcal{}M=MU_{*}(X)
  53. \mathcal{F}
  54. f g \mathcal{M}_{fg}
  55. M U * ( X ) M U * M MU_{*}(X)\otimes_{MU_{*}}M
  56. f g \mathcal{M}_{fg}
  57. E E_{\infty}
  58. M U * \mathcal{}MU_{*}
  59. E E_{\infty}
  60. X f g X\to\mathcal{M}_{fg}
  61. M p ( n ) M_{p}(n)
  62. X M p ( n ) X\to M_{p}(n)
  63. E E_{\infty}
  64. M U * ( M U ) MU_{*}(MU)
  65. B P * ( B P ) BP_{*}(BP)

Langlands–Shahidi_method.html

  1. π = π v \pi=\otimes^{\prime}\pi_{v}
  2. L S ( s , π , r i ) = v S γ i ( s , π v , ψ v ) L S ( 1 - s , π ~ , r i ) . L^{S}(s,\pi,r_{i})=\prod_{v\in S}\gamma_{i}(s,\pi_{v},\psi_{v})L^{S}(1-s,% \tilde{\pi},r_{i}).
  3. π v \pi_{v}
  4. r = r i r=\oplus r_{i}
  5. π v \pi_{v}
  6. ( s , π v , r i , v , ψ v ) (s,\pi_{v},r_{i,v},\psi_{v})
  7. v S v\in S
  8. L ( s , π , r i ) = ϵ ( s , π , r i ) L ( 1 - s , π ~ , r i ) , L(s,\pi,r_{i})=\epsilon(s,\pi,r_{i})L(1-s,\tilde{\pi},r_{i}),
  9. L ( s , π , r i ) L(s,\pi,r_{i})
  10. ϵ ( s , π , r i ) \epsilon(s,\pi,r_{i})
  11. L ( s , π 1 × π 2 ) L(s,\pi_{1}\times\pi_{2})
  12. π 1 \pi_{1}
  13. π 2 \pi_{2}
  14. L ( s , τ × π ) L(s,\tau\times\pi)
  15. L ( s , τ , r ) L(s,\tau,r)
  16. r = r i r=\oplus r_{i}
  17. L ( s , π , r ) , L ( s , π ~ , r ) L(s,\pi,r),\ L(s,\tilde{\pi},r)
  18. L ( s , π , r ) , L ( s , π ~ , r ) L(s,\pi,r),\ L(s,\tilde{\pi},r)
  19. L ( s , π , r ) = ϵ ( s , π , r ) L ( 1 - s , π ~ , r ) L(s,\pi,r)=\epsilon(s,\pi,r)L(1-s,\tilde{\pi},r)
  20. L ( 1 + i t , π 1 × π 2 ) L(1+it,\pi_{1}\times\pi_{2})
  21. I ( π ) = I ( 0 , π ) I(\pi)=I(0,\pi)
  22. I ( s , π ) I(s,\pi)
  23. I ( 1 , π ) I(1,\pi)
  24. I ( s , π ) I(s,\pi)
  25. I ( s , π ) I(s,\pi)
  26. π | det | s \pi\otimes|\det|^{s}
  27. π | det | s / 2 \pi\otimes|\det|^{s/2}

Laplace_functional.html

  1. E ( X , d , μ ) : [ 0 , + ) [ 0 , + ] E_{(X,d,\mu)}\colon[0,+\infty)\to[0,+\infty]
  2. E ( X , d , μ ) ( λ ) := sup { X e λ f ( x ) d μ ( x ) | f : X is bounded, 1-Lipschitz and has X f ( x ) d μ ( x ) = 0 } . E_{(X,d,\mu)}(\lambda):=\sup\left\{\left.\int_{X}e^{\lambda f(x)}\,\mathrm{d}% \mu(x)\right|f\colon X\to\mathbb{R}\,\text{ is bounded, 1-Lipschitz and has }% \int_{X}f(x)\,\mathrm{d}\mu(x)=0\right\}.
  3. α ( X , d , μ ) ( r ) := sup { 1 - μ ( A r ) A X and μ ( A ) 1 2 } , \alpha_{(X,d,\mu)}(r):=\sup\{1-\mu(A_{r})\mid A\subseteq X\,\text{ and }\mu(A)% \geq\tfrac{1}{2}\},
  4. A r := { x X d ( x , A ) r } . A_{r}:=\{x\in X\mid d(x,A)\leq r\}.
  5. α ( X , d , μ ) ( r ) inf λ 0 e - λ r / 2 E ( X , d , μ ) ( λ ) . \alpha_{(X,d,\mu)}(r)\leq\inf_{\lambda\geq 0}e^{-\lambda r/2}E_{(X,d,\mu)}(% \lambda).

Large_margin_nearest_neighbor.html

  1. D = { ( x 1 , y 1 ) , , ( x n , y n ) } R d × C D=\{(\vec{x}_{1},y_{1}),\dots,(\vec{x}_{n},y_{n})\}\subset R^{d}\times C
  2. C = { 1 , , c } C=\{1,\dots,c\}
  3. d ( x i , x j ) = ( x i - x j ) 𝐌 ( x i - x j ) d(\vec{x}_{i},\vec{x}_{j})=(\vec{x}_{i}-\vec{x}_{j})^{\top}\mathbf{M}(\vec{x}_% {i}-\vec{x}_{j})
  4. d ( , ) d(\cdot,\cdot)
  5. 𝐌 \mathbf{M}
  6. 𝐌 \mathbf{M}
  7. 𝐌 \mathbf{M}
  8. x i \vec{x}_{i}
  9. x i \vec{x}_{i}
  10. k k
  11. D D
  12. y i y_{i}
  13. x i \vec{x}_{i}
  14. N i N_{i}
  15. x i \vec{x}_{i}
  16. x j \vec{x}_{j}
  17. y i y j y_{i}\neq y_{j}
  18. x i \vec{x}_{i}
  19. 𝐌 \mathbf{M}
  20. x i \vec{x}_{i}
  21. x i \vec{x}_{i}
  22. k = 3 k=3
  23. i , j N i d ( x i , x j ) \sum_{i,j\in N_{i}}d(\vec{x}_{i},\vec{x}_{j})
  24. x l \vec{x}_{l}
  25. x j \vec{x}_{j}
  26. x i \vec{x}_{i}
  27. i , j N i , l , y l y i d ( x i , x j ) + 1 d ( x i , x l ) \forall_{i,j\in N_{i},l,y_{l}\neq y_{i}}d(\vec{x}_{i},\vec{x}_{j})+1\leq d(% \vec{x}_{i},\vec{x}_{l})
  28. M M
  29. c > 0 c>0
  30. M M
  31. 1 / c 1/c
  32. min 𝐌 i , j N i d ( x i , x j ) + i , j , l ξ i j l \min_{\mathbf{M}}\sum_{i,j\in N_{i}}d(\vec{x}_{i},\vec{x}_{j})+\sum_{i,j,l}\xi% _{ijl}
  33. i , j N i , l , y l y i \forall_{i,j\in N_{i},l,y_{l}\neq y_{i}}
  34. d ( x i , x j ) + 1 d ( x i , x l ) + ξ i j l d(\vec{x}_{i},\vec{x}_{j})+1\leq d(\vec{x}_{i},\vec{x}_{l})+\xi_{ijl}
  35. ξ i j l 0 \xi_{ijl}\geq 0
  36. 𝐌 0 \mathbf{M}\succeq 0
  37. ξ i j l \xi_{ijl}
  38. 𝐌 \mathbf{M}

Lars_Svenonius.html

  1. 0 \aleph_{0}

Laser_beam_quality.html

  1. Θ 00 = 4 λ π D 00 \Theta_{00}={4\lambda\over\pi D_{00}}
  2. Θ 0 = M 2 4 λ π D 0 \Theta_{0}=M^{2}{4\lambda\over\pi D_{0}}
  3. d 00 = 4 λ f π D 00 d_{00}={4\lambda f\over\pi D_{00}}
  4. d 0 = M 2 4 λ f π D 0 d_{0}=M^{2}{4\lambda f\over\pi D_{0}}

Laser_flash_analysis.html

  1. a a
  2. a = 0.1388 d 2 t 1 / 2 a=0.1388\cdot\frac{d^{2}}{t_{1/2}}
  3. a a
  4. d d
  5. t 1 / 2 t_{1/2}

Laser_linewidth.html

  1. Δ λ Δ θ ( Θ λ ) - 1 \Delta\lambda\approx\Delta\theta\left({\partial\Theta\over\partial\lambda}% \right)^{-1}
  2. Δ θ \Delta\theta
  3. Δ ν \Delta\nu
  4. Δ λ \Delta\lambda
  5. Δ ν \Delta\nu
  6. Δ λ \Delta\lambda
  7. Δ ν c Δ x \Delta\nu\approx{c\over\Delta x}
  8. Δ x \Delta x
  9. Δ ν \Delta\nu
  10. Δ λ λ 2 Δ x \Delta\lambda\approx{\lambda^{2}\over\Delta x}
  11. Δ x \Delta x
  12. Δ ν \Delta\nu
  13. Δ λ \Delta\lambda

Laser_surface_velocimeter.html

  1. f = f ( c - v ) = f ( 1 - v c ) f^{\prime}=f(c-v)=f\left(1-\frac{v}{c}\right)
  2. f P1,2 = f 1,2 ( 1 - v e 1,2 c ) f\text{P1,2}=f\text{1,2}\left(1-\vec{v}\ast\frac{\vec{e}\text{1,2}}{c}\right)
  3. e 1,2,e \vec{e}\text{1,2,e}
  4. e e \vec{e}\text{e}
  5. f e1,e2 \displaystyle f\text{e1,e2}
  6. f D \displaystyle f\text{D}
  7. v ( e 1 - e 2 ) = 2 v sin φ \vec{v}\ast(\vec{e}\text{1}-\vec{e}\text{2})=2v\sin\varphi
  8. v e e = 0 \vec{v}\ast\vec{e}\text{e}=0
  9. f D = v Δ s f\text{D}=\frac{v}{\Delta s}
  10. Δ s = λ 2 sin φ \Delta s=\frac{\lambda}{2\sin\varphi}
  11. f D = v p Δ s = 2 v λ sin φ f\text{D}=\frac{v\text{p}}{\Delta s}=\frac{2v}{\lambda}\sin\varphi
  12. f mod = f b + v p Δ s = f b + 2 v λ sin φ f\text{mod}=f\text{b}+\frac{v\text{p}}{\Delta s}=f\text{b}+\frac{2v}{\lambda}\sin\varphi

Last_geometric_statement_of_Jacobi.html

  1. p p

Law_of_cotangents.html

  1. a , b , c a,b,c
  2. A , B , C A,B,C
  3. α \alpha\,
  4. β \beta\,
  5. γ \gamma\,
  6. s s
  7. s = ( a + b + c ) / 2 s=(a+b+c)/2
  8. r r
  9. cot ( α / 2 ) s - a = cot ( β / 2 ) s - b = cot ( γ / 2 ) s - c = 1 r \frac{\cot(\alpha/2)}{s-a}=\frac{\cot(\beta/2)}{s-b}=\frac{\cot(\gamma/2)}{s-c% }=\frac{1}{r}\,
  10. r = ( s - a ) ( s - b ) ( s - c ) s . r=\sqrt{\frac{(s-a)(s-b)(s-c)}{s}}\,.
  11. cot ( α / 2 ) = s - a r \cot(\alpha/2)=\frac{s-a}{r}\,
  12. cot ( u + v + w ) = cot u + cot v + cot w - cot u cot v cot w 1 - cot u cot v - cot v cot w - cot w cot u \cot(u+v+w)=\frac{\cot u+\cot v+\cot w-\cot u\cot v\cot w}{1-\cot u\cot v-\cot v% \cot w-\cot w\cot u}
  13. cot ( α / 2 + β / 2 + γ / 2 ) = cot ( π / 2 ) = 0 \cot(\alpha/2+\beta/2+\gamma/2)=\cot(\pi/2)=0
  14. cot ( α / 2 ) cot ( β / 2 ) cot ( γ / 2 ) = cot ( α / 2 ) + cot ( β / 2 ) + cot ( γ / 2 ) \cot(\alpha/2)\cot(\beta/2)\cot(\gamma/2)=\cot(\alpha/2)+\cot(\beta/2)+\cot(% \gamma/2)
  15. ( s - a ) r ( s - b ) r ( s - c ) r = s - a r + s - b r + s - c r \frac{(s-a)}{r}\frac{(s-b)}{r}\frac{(s-c)}{r}=\frac{s-a}{r}+\frac{s-b}{r}+% \frac{s-c}{r}
  16. s / r s/r
  17. r 3 / s r^{3}/s
  18. r 2 r^{2}
  19. A B C ABC
  20. 1 2 r ( s - a ) \tfrac{1}{2}r(s-a)
  21. r ( s - a ) r(s-a)\,
  22. S S
  23. S = r ( s - a ) + r ( s - b ) + r ( s - c ) = r ( s - a + s - b + s - c ) = r ( 3 s - ( a + b + c ) ) = r ( 3 s - 2 s ) = r s \begin{aligned}\displaystyle S&\displaystyle=r(s-a)+r(s-b)+r(s-c)=r(s-a+s-b+s-% c)\\ &\displaystyle=r(3s-(a+b+c))=r(3s-2s)=rs\\ \end{aligned}
  24. S = s ( s - a ) ( s - b ) ( s - c ) S=\sqrt{s(s-a)(s-b)(s-c)}\,
  25. sin ( α / 2 - β / 2 ) sin ( α / 2 + β / 2 ) = cot ( β / 2 ) - cot ( α / 2 ) cot ( β / 2 ) + cot ( α / 2 ) = a - b 2 s - a - b . \frac{\sin(\alpha/2-\beta/2)}{\sin(\alpha/2+\beta/2)}=\frac{\cot(\beta/2)-\cot% (\alpha/2)}{\cot(\beta/2)+\cot(\alpha/2)}=\frac{a-b}{2s-a-b}.
  26. a - b c = sin ( α / 2 - β / 2 ) cos ( γ / 2 ) \dfrac{a-b}{c}=\dfrac{\sin(\alpha/2-\beta/2)}{\cos(\gamma/2)}
  27. cos ( α / 2 - β / 2 ) cos ( α / 2 + β / 2 ) = cot ( α / 2 ) cot ( β / 2 ) + 1 cot ( α / 2 ) cot ( β / 2 ) - 1 \displaystyle\frac{\cos(\alpha/2-\beta/2)}{\cos(\alpha/2+\beta/2)}=\frac{\cot(% \alpha/2)\cot(\beta/2)+1}{\cot(\alpha/2)\cot(\beta/2)-1}
  28. b + a c = cos ( α / 2 - β / 2 ) sin ( γ / 2 ) \dfrac{b+a}{c}=\dfrac{\cos(\alpha/2-\beta/2)}{\sin(\gamma/2)}

Lawler's_algorithm.html

  1. m i n m a x 0 i n g i ( F i ) min\,max_{0\leq i\leq n}\,g_{i}(F_{i})
  2. g i g_{i}
  3. F i F_{i}
  4. g i ( F i ) = F i - d i = L i g_{i}(F_{i})=F_{i}-d_{i}=L_{i}
  5. d i d_{i}
  6. i i
  7. L i L_{i}
  8. i i
  9. g i ( F i ) = m a x ( F i - d i , 0 ) g_{i}(F_{i})=max{(F_{i}-d_{i},0)}
  10. t = p j t=\sum p_{j}
  11. S S
  12. \emptyset
  13. J J
  14. t t
  15. t = p j t=\sum p_{j}
  16. J J\neq\emptyset
  17. j J j\in J
  18. f j ( t ) = m i n k J f k ( t ) f_{j}(t)=min_{k\in J}f_{k}(t)
  19. j j
  20. t t
  21. j j
  22. S S
  23. j j
  24. J J
  25. J J
  26. t = t - p j t=t-p_{j}

Leaky_wave_antenna.html

  1. β = k 0 2 - ( π a ) 2 < k 0 \beta=\sqrt{k_{0}^{2}-\left(\frac{\pi}{a}\right)^{2}}<k_{0}
  2. β k 0 = c v p h = λ 0 λ g s i n θ m \frac{\beta}{k_{0}}=\frac{c}{v_{ph}}=\frac{\lambda_{0}}{\lambda_{g}}\simeq sin% \theta_{m}
  3. - 3 d B \simeq-3dB
  4. Δ θ 1 L λ 0 c o s θ m \Delta\theta\simeq\frac{1}{\frac{L}{\lambda_{0}}cos\theta_{m}}
  5. L λ 0 0.18 α k 0 Δ θ α k 0 \frac{L}{\lambda_{0}}\simeq\frac{0.18}{\frac{\alpha}{k_{0}}}\Rightarrow\Delta% \theta\propto\frac{\alpha}{k_{0}}
  6. c o s 2 ϑ m 1 - ( β k 0 ) 2 cos^{2}\vartheta_{m}\simeq 1-\left(\frac{\beta}{k_{0}}\right)^{2}
  7. k 0 2 = k t 2 + β 2 ( β k 0 ) 2 = 1 - ( k t k 0 2 ) k_{0}^{2}=k_{t}^{2}+\beta^{2}\Rightarrow\left(\frac{\beta}{k_{0}}\right)^{2}=1% -\left(\frac{k_{t}}{k_{0}}^{2}\right)
  8. c o s ϑ m Δ θ 2 π k t L = λ c L \Rightarrow cos\vartheta_{m}\simeq\Delta\theta\simeq\frac{2\pi}{k_{t}L}=\frac{% \lambda_{c}}{L}

Least_squares_(function_approximation).html

  1. f ( x ) f n ( x ) = a 1 ϕ 1 ( x ) + a 2 ϕ 2 ( x ) + + a n ϕ n ( x ) , f(x)\approx f_{n}(x)=a_{1}\phi_{1}(x)+a_{2}\phi_{2}(x)+\cdots+a_{n}\phi_{n}(x),
  2. ϕ j ( x ) \ \phi_{j}(x)
  3. a j \ a_{j}
  4. g = ( a b g * ( x ) g ( x ) d x ) 1 / 2 \|g\|=\left(\int_{a}^{b}g^{*}(x)g(x)\,dx\right)^{1/2}
  5. a b ϕ i * ( x ) ϕ j ( x ) d x = δ i j , \int_{a}^{b}\phi_{i}^{*}(x)\phi_{j}(x)\,dx=\delta_{ij},
  6. f n 2 = | a 1 | 2 + | a 2 | 2 + + | a n | 2 . \|f_{n}\|^{2}=|a_{1}|^{2}+|a_{2}|^{2}+\cdots+|a_{n}|^{2}.\,
  7. a j = a b ϕ j * ( x ) f ( x ) d x . a_{j}=\int_{a}^{b}\phi_{j}^{*}(x)f(x)\,dx.

Leave-one-out_error.html

  1. i { 1 , , m } , S { sup z Z | V ( f S , z i ) - V ( f S | i , z i ) | β C V } 1 - δ C V \forall i\in\{1,...,m\},\mathbb{P}_{S}\{\sup_{z\in Z}|V(f_{S},z_{i})-V(f_{S^{|% i}},z_{i})|\leq\beta_{CV}\}\geq 1-\delta_{CV}
  2. E l o o e r r Eloo_{err}
  3. E l o o e r r Eloo_{err}
  4. β E L m \beta_{EL}^{m}
  5. δ E L m \delta_{EL}^{m}
  6. i { 1 , , m } , S { | I [ f S ] - 1 m i = 1 m V ( f S | i , z i ) | β E L m } 1 - δ E L m \forall i\in\{1,...,m\},\mathbb{P}_{S}\{|I[f_{S}]-\frac{1}{m}\sum_{i=1}^{m}V(f% _{S^{|i}},z_{i})|\leq\beta_{EL}^{m}\}\geq 1-\delta_{EL}^{m}
  7. β E L m \beta_{EL}^{m}
  8. δ E L m \delta_{EL}^{m}
  9. n inf n\rightarrow\inf
  10. S = { z 1 = ( x 1 , y 1 ) , . . , z m = ( x m , y m ) } S=\{z_{1}=(x_{1},\ y_{1})\ ,..,\ z_{m}=(x_{m},\ y_{m})\}
  11. Z = X × Y Z=X\times Y
  12. f f
  13. Z m Z_{m}
  14. F Y X F\subset YX
  15. f S f_{S}
  16. f f
  17. z = ( x , y ) z=(x,y)
  18. V ( f , z ) = V ( f ( x ) , y ) V(f,z)=V(f(x),y)
  19. I S [ f ] = 1 n V ( f , z i ) I_{S}[f]=\frac{1}{n}\sum V(f,z_{i})
  20. I [ f ] = 𝔼 z V ( f , z ) I[f]=\mathbb{E}_{z}V(f,z)
  21. S | i = { z 1 , , z i - 1 , z i + 1 , , z m } S^{|i}=\{z_{1},...,\ z_{i-1},\ z_{i+1},...,\ z_{m}\}
  22. S i = { z 1 , , z i - 1 , z i , z i + 1 , , z m } S^{i}=\{z_{1},...,\ z_{i-1},\ z_{i}^{\prime},\ z_{i+1},...,\ z_{m}\}

LEDA_074886.html

  1. α \alpha
  2. δ \delta

Leeson's_equation.html

  1. L ( f m ) = 10 log [ 1 2 ( ( f 0 2 Q l f m ) 2 + 1 ) ( f c f m + 1 ) ( F k T P s ) ] L(f_{m})=10\log\bigg[\frac{1}{2}\bigg(\bigg(\frac{f_{0}}{2Q_{l}f_{m}}\bigg)^{2% }+1\bigg)\bigg(\frac{f_{c}}{f_{m}}+1\bigg)\bigg(\frac{FkT}{P_{s}}\bigg)\bigg]

Leigh_Mercer.html

  1. 12 + 144 + 20 + 3 4 7 + ( 5 × 11 ) = 9 2 + 0 \frac{12+144+20+3\sqrt{4}}{7}+(5\times 11)=9^{2}+0

Length_measurement.html

  1. λ e = h 2 m e e V , \lambda_{e}=\frac{h}{\sqrt{2m_{e}eV}}\ ,

Leon_Mirsky.html

  1. D ( n ) = ( 1 + o ( 1 ) ) exp ( π 8 log n 3 log log n ) . D(n)=\bigl(1+o(1)\bigr)\exp\left(\frac{\pi\sqrt{8\log n}}{\sqrt{3}\log\log n}% \right).
  2. n 2 - 2 n + 2 n^{2}-2n+2
  3. n × n n\times n

LeRoy_Apker.html

  1. 10 - 8 10^{-}8

Lethargy_theorem.html

  1. V 1 V 2 V_{1}\subset V_{2}\subset\ldots
  2. ϵ 1 ϵ 2 \epsilon_{1}\geq\epsilon_{2}\geq\ldots
  3. ϵ i \epsilon_{i}

LH_(complexity).html

  1. i i
  2. i - 1 i-1

Liblzg.html

  1. 1 256 \tfrac{1}{256}
  2. 4 256 \tfrac{4}{256}

Lie_conformal_algebra.html

  1. L L
  2. \mathbb{C}
  3. [ , ] : L L L [\cdot,\cdot]:L\otimes L\rightarrow L
  4. R R
  5. [ ] \mathbb{C}[\partial]
  6. [ λ ] : R R [ λ ] R [\cdot_{\lambda}\cdot]:R\otimes R\rightarrow\mathbb{C}[\lambda]\otimes R
  7. [ a λ b ] = - λ [ a λ b ] , [ a λ b ] = ( λ + ) [ a λ b ] , [\partial a_{\lambda}b]=-\lambda[a_{\lambda}b],[a_{\lambda}\partial b]=(% \lambda+\partial)[a_{\lambda}b],
  8. [ a λ b ] = - [ b - λ - a ] , [a_{\lambda}b]=-[b_{-\lambda-\partial}a],\,
  9. [ a λ [ b μ c ] ] - [ b μ [ a λ c ] ] = [ [ a λ b ] λ + μ c ] . [a_{\lambda}[b_{\mu}c]]-[b_{\mu}[a_{\lambda}c]]=[[a_{\lambda}b]_{\lambda+\mu}c% ].\,
  10. [ ] \mathbb{C}[\partial]
  11. L L
  12. [ L λ L ] = ( 2 λ + ) L . [L_{\lambda}L]=(2\lambda+\partial)L.\,
  13. [ ] \mathbb{C}[\partial]
  14. 𝔤 𝔠 n \mathfrak{gc}_{n}
  15. 𝔠 𝔢 𝔫 𝔡 n \mathfrak{cend}_{n}