wpmath0000005_0

't_Hooft–Polyakov_monopole.html

  1. r = 0 r=0
  2. H i ( i = 1 , 2 , 3 ) H_{i}\qquad(i=1,2,3)\,
  3. x i f ( | x | ) x_{i}f(|x|)\,

1::N_expansion.html

  1. 1 / N 1/N
  2. N N
  3. = 1 2 μ ϕ a μ ϕ a - m 2 2 ϕ a ϕ a - λ 8 N ( ϕ a ϕ a ) 2 \mathcal{L}={1\over 2}\partial^{\mu}\phi_{a}\partial_{\mu}\phi_{a}-{m^{2}\over 2% }\phi_{a}\phi_{a}-{\lambda\over 8N}(\phi_{a}\phi_{a})^{2}
  4. a a
  5. = 1 2 μ ϕ a μ ϕ a - m 2 2 ϕ a ϕ a + 1 2 F 2 - λ / N 2 F ϕ a ϕ a \mathcal{L}={1\over 2}\partial^{\mu}\phi_{a}\partial_{\mu}\phi_{a}-{m^{2}\over 2% }\phi_{a}\phi_{a}+{1\over 2}F^{2}-{\sqrt{\lambda/N}\over 2}F\phi_{a}\phi_{a}

134_(number).html

  1. C 1 8 + C 3 8 + C 4 8 {}_{8}C_{1}+{}_{8}C_{3}+{}_{8}C_{4}

136_(number).html

  1. 1 3 + 3 3 + 6 3 = 244 1^{3}+3^{3}+6^{3}=244
  2. 2 3 + 4 3 + 4 3 = 136 2^{3}+4^{3}+4^{3}=136

138_(number).html

  1. 2 3 23 2\cdot 3\cdot 23

14:9.html

  1. ( 16 / 9 ) × ( 4 / 3 ) 1.5396 13.8 : 9 \sqrt{(16/9)\times(4/3)}\approx 1.5396\approx 13.8:9

143_(number).html

  1. 3 2 + 4 2 = 5 2 3^{2}+4^{2}=5^{2}
  2. 3 3 + 4 3 + 5 3 = 6 3 3^{3}+4^{3}+5^{3}=6^{3}
  3. 3 4 + 4 4 + 5 4 + 6 4 = 7 4 - 143 3^{4}+4^{4}+5^{4}+6^{4}=7^{4}-143

149_(number).html

  1. 3 n - 1 3n-1

154_(number).html

  1. 0 ! 0!
  2. 0 ! = 1 0!=1

173_(number).html

  1. 2 2 + 13 2 2^{2}+13^{2}

1989_Newcastle_earthquake.html

  1. M b M_{b}

30-day_yield.html

  1. Yield = 2 [ ( a - b c d + 1 ) 6 - 1 ] . \mathrm{Yield}=2\left[\left(\frac{a-b}{cd}+1\right)^{6}-1\right].

5-alpha_reductase.html

  1. \rightleftharpoons

5.5_Metre_(keelboat).html

  1. 5.500 metres 0.9 ( L S 2 12 D 3 + L + S 2 4 ) 5.500\mbox{ metres}~{}\geq 0.9\cdot\left(\frac{L\cdot\sqrt[2]{S}}{12\cdot\sqrt% [3]{D}}+\frac{L+\sqrt[2]{S}}{4}\right)
  2. L L
  3. S S
  4. D D

5040_(number).html

  1. σ ( n ) \sigma(n)
  2. γ \gamma
  3. σ ( n ) e γ n log log n \sigma(n)\geq e^{\gamma}n\log\log n
  4. lim sup n σ ( n ) n log log n = e γ . \limsup_{n\rightarrow\infty}\frac{\sigma(n)}{n\ \log\log n}=e^{\gamma}.

A_major.html

  1. 2 ^ \hat{2}

Abel_transform.html

  1. F ( y ) = 2 y f ( r ) r d r r 2 - y 2 . F(y)=2\int_{y}^{\infty}\frac{f(r)r\,dr}{\sqrt{r^{2}-y^{2}}}.
  2. f ( r ) = - 1 π r d F d y d y y 2 - r 2 . f(r)=-\frac{1}{\pi}\int_{r}^{\infty}\frac{dF}{dy}\,\frac{dy}{\sqrt{y^{2}-r^{2}% }}.
  3. F ( y ) = - f ( x 2 + y 2 ) d x F(y)=\int_{-\infty}^{\infty}f\!\left(\sqrt{x^{2}+y^{2}}\right)\,dx
  4. d x = r d r r 2 - y 2 . dx=\frac{r\,dr}{\sqrt{r^{2}-y^{2}}}.
  5. - f ( r ) d x = 2 0 f ( r ) d x , r = x 2 + y 2 . \int_{-\infty}^{\infty}f(r)\,dx=2\int_{0}^{\infty}f(r)\,dx,\;\;r=\sqrt{x^{2}+y% ^{2}}.
  6. F ( y , z ) = - f ( ρ , z ) d x = 2 y f ( ρ , z ) ρ d ρ ρ 2 - y 2 F(y,z)=\int_{-\infty}^{\infty}f(\rho,z)\,dx=2\int_{y}^{\infty}\frac{f(\rho,z)% \rho\,d\rho}{\sqrt{\rho^{2}-y^{2}}}
  7. F ( s ) = - f ( r ) d x = 2 s f ( r ) r d r r 2 - s 2 F(s)=\int_{-\infty}^{\infty}f(r)\,dx=2\int_{s}^{\infty}\frac{f(r)r\,dr}{\sqrt{% r^{2}-s^{2}}}
  8. u = f ( r ) u=f(r)
  9. v = r 2 - y 2 v=\sqrt{r^{2}-y^{2}}
  10. F ( y ) = - 2 y f ( r ) r 2 - y 2 d r . F(y)=-2\int_{y}^{\infty}f^{\prime}(r)\sqrt{r^{2}-y^{2}}\,dr.
  11. F ( y ) = 2 y y f ( r ) r 2 - y 2 d r . F^{\prime}(y)=2y\int_{y}^{\infty}\frac{f^{\prime}(r)}{\sqrt{r^{2}-y^{2}}}\,dr.
  12. - 1 π r F ( y ) y 2 - r 2 d y = r y - 2 y π ( y 2 - r 2 ) ( s 2 - y 2 ) f ( s ) d s d y . -\frac{1}{\pi}\int_{r}^{\infty}\frac{F^{\prime}(y)}{\sqrt{y^{2}-r^{2}}}\,dy=% \int_{r}^{\infty}\int_{y}^{\infty}\frac{-2y}{\pi\sqrt{(y^{2}-r^{2})(s^{2}-y^{2% })}}f^{\prime}(s)\,dsdy.
  13. r r s - 2 y π ( y 2 - r 2 ) ( s 2 - y 2 ) d y f ( s ) d s = r ( - 1 ) f ( s ) d s = f ( r ) . \int_{r}^{\infty}\int_{r}^{s}\frac{-2y}{\pi\sqrt{(y^{2}-r^{2})(s^{2}-y^{2})}}% \,dyf^{\prime}(s)\,ds=\int_{r}^{\infty}(-1)f^{\prime}(s)\,ds=f(r).
  14. F ( y ) F(y)
  15. y = y Δ y=y_{\Delta}
  16. Δ F \Delta F
  17. y Δ y_{\Delta}
  18. Δ F \Delta F
  19. Δ F lim ϵ 0 [ F ( y Δ - ϵ ) - F ( y Δ + ϵ ) ] \Delta F\equiv\lim_{\epsilon\rightarrow 0}[F(y_{\Delta}-\epsilon)-F(y_{\Delta}% +\epsilon)]
  20. F ( y ) F(y)
  21. f ( r ) f(r)
  22. F ( y ) = 2 y f ( r ) r d r r 2 - y 2 . F(y)=2\int_{y}^{\infty}\frac{f(r)r\,dr}{\sqrt{r^{2}-y^{2}}}.
  23. f ( r ) = [ 1 2 δ ( r - y Δ ) 1 - ( y Δ / r ) 2 - 1 π H ( y Δ - r ) y Δ 2 - r 2 ] Δ F - 1 π r d F d y d y y 2 - r 2 . f(r)=\left[\frac{1}{2}\delta(r-y_{\Delta})\sqrt{1-(y_{\Delta}/r)^{2}}-\frac{1}% {\pi}\frac{H(y_{\Delta}-r)}{\sqrt{y_{\Delta}^{2}-r^{2}}}\right]\Delta F-\frac{% 1}{\pi}\int_{r}^{\infty}\frac{dF}{dy}\frac{dy}{\sqrt{y^{2}-r^{2}}}.
  24. δ \delta
  25. H ( x ) H(x)
  26. F ( y ) F(y)
  27. Δ F = 0 \Delta F=0
  28. F ( y ) F(y)
  29. F A = H . FA=H.\,

Abelian_von_Neumann_algebra.html

  1. ψ f ψ . \psi\mapsto f\psi.
  2. ( { 1 , 2 , , n } ) , n 1 \ell^{\infty}(\{1,2,\ldots,n\}),\quad n\geq 1
  3. ( 𝐍 ) \ell^{\infty}(\mathbf{N})
  4. L ( [ 0 , 1 ] ) L^{\infty}([0,1])
  5. L ( [ 0 , 1 ] { 1 , 2 , , n } ) , n 1 L^{\infty}([0,1]\cup\{1,2,\ldots,n\}),\quad n\geq 1
  6. L ( [ 0 , 1 ] 𝐍 ) . L^{\infty}([0,1]\cup\mathbf{N}).
  7. σ \sigma
  8. σ \sigma
  9. A = L ( X , 𝔄 , μ ) A=L^{\infty}(X,\mathfrak{A},\mu)
  10. σ \sigma
  11. 𝔄 / { A μ ( A ) = 0 } \mathfrak{A}/\{A\mid\mu(A)=0\}
  12. σ \sigma
  13. ϕ : X N Y M , \phi:X\setminus N\rightarrow Y\setminus M,\quad
  14. U A U * = B . UAU^{*}=B.
  15. X H ( x ) d μ ( x ) \int_{X}^{\oplus}H(x)\,d\mu(x)
  16. Φ : L ( X , μ ) L ( Y , ν ) \Phi:L^{\infty}(X,\mu)\rightarrow L^{\infty}(Y,\nu)
  17. η : X M Y N \eta:X\setminus M\rightarrow Y\setminus N
  18. Φ ( f ) = f η - 1 . \Phi(f)=f\circ\eta^{-1}.
  19. H = X H x d μ ( x ) , K = Y K y d ν ( y ) H=\int_{X}^{\oplus}H_{x}d\mu(x),\quad K=\int_{Y}^{\oplus}K_{y}d\nu(y)
  20. U L ( X , μ ) U * = L ( Y , ν ) U\,L^{\infty}(X,\mu)\,U^{*}=L^{\infty}(Y,\nu)
  21. U x : H x K η ( x ) U_{x}:H_{x}\rightarrow K_{\eta(x)}
  22. U ( X ψ x d μ ( x ) ) = Y d ( μ η - 1 ) d ν ( y ) U η - 1 ( y ) ( ψ η - 1 ( y ) ) d ν ( y ) , U\bigg(\int_{X}^{\oplus}\psi_{x}d\mu(x)\bigg)=\int_{Y}^{\oplus}\sqrt{\frac{d(% \mu\circ\eta^{-1})}{d\nu}(y)}\ U_{\eta^{-1}(y)}\bigg(\psi_{\eta^{-1}(y)}\bigg)% d\nu(y),

Abnormal_return.html

  1. Abnormal Return = Actual Return - Expected Return \textrm{Abnormal\ Return}=\textrm{Actual\ Return}-\textrm{Expected\ Return}

Absolute_neutrophil_count.html

  1. ( % n e u t r o p h i l s + % b a n d s ) × ( W B C ) ( 100 ) (\%neutrophils+\%bands)\times(WBC)\over(100)

Absolutely_convex_set.html

  1. C C
  2. x 1 , x 2 x_{1},\,x_{2}
  3. C C
  4. λ 1 , λ 2 \lambda_{1},\,\lambda_{2}
  5. | λ 1 | + | λ 2 | 1 |\lambda_{1}|+|\lambda_{2}|\leq 1
  6. λ 1 x 1 + λ 2 x 2 \lambda_{1}x_{1}+\lambda_{2}x_{2}
  7. C C
  8. absconv A = { i = 1 n λ i x i : n 𝒩 , x i A , i = 1 n | λ i | 1 } \mbox{absconv}~{}A=\left\{\sum_{i=1}^{n}\lambda_{i}x_{i}:n\in\mathcal{N},\,x_{% i}\in A,\,\sum_{i=1}^{n}|\lambda_{i}|\leq 1\right\}

Absorbing_set.html

  1. x X x\in X
  2. α 𝔽 : | α | r x α S \forall\alpha\in\mathbb{F}:|\alpha|\geq r\Rightarrow x\in\alpha S
  3. α S := { α s s S } \alpha S:=\{\alpha s\mid s\in S\}

Absorption_(chemistry).html

  1. [ x ] 1 [ x ] 2 = constant = K N ( x , 12 ) \frac{[x]_{1}}{[x]_{2}}=\,\text{constant}=K_{N(x,12)}

Abstract_index_notation.html

  1. h V * V * \scriptstyle h\in V^{*}\otimes V^{*}
  2. h = h ( - , - ) . h=h(-,-).\,
  3. h = h a b . h=h_{ab}.\,
  4. t a b b {t_{ab}}^{b}
  5. V V * V * V V * . V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}.
  6. V a V b V c V d V e V^{a}V_{b}V_{c}V^{d}V_{e}\,
  7. V a b c d e . {{{V^{a}}_{bc}}^{d}}_{e}.
  8. h a b c d e V a b c d e = V V * V * V V * . {{{h^{a}}_{bc}}^{d}}_{e}\in{{{V^{a}}_{bc}}^{d}}_{e}=V\otimes V^{*}\otimes V^{*% }\otimes V\otimes V^{*}.
  9. Tr 12 : V V * V * V V * V * V V * \mathrm{Tr}_{12}:V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}\to V^{*}% \otimes V\otimes V^{*}
  10. Tr 15 : V V * V * V V * V * V * V \mathrm{Tr}_{15}:V\otimes V^{*}\otimes V^{*}\otimes V\otimes V^{*}\to V^{*}% \otimes V^{*}\otimes V
  11. Tr 12 : h a b c d e h a a c d e \mathrm{Tr}_{12}:{{{h^{a}}_{bc}}^{d}}_{e}\mapsto{{{h^{a}}_{ac}}^{d}}_{e}
  12. Tr 15 : h a b c d e h a b c d a . \mathrm{Tr}_{15}:{{{h^{a}}_{bc}}^{d}}_{e}\mapsto{{{h^{a}}_{bc}}^{d}}_{a}.
  13. τ ( 12 ) : V V V V \tau_{(12)}:V\otimes V\rightarrow V\otimes V
  14. τ ( 12 ) ( v w ) = w v \tau_{(12)}(v\otimes w)=w\otimes v
  15. τ σ \tau_{\sigma}
  16. σ \sigma
  17. R R
  18. V * V * V * V V^{*}\otimes V^{*}\otimes V^{*}\otimes V
  19. R + τ ( 123 ) R + τ ( 132 ) R = 0. R+\tau_{(123)}R+\tau_{(132)}R=0.
  20. R = R a b c d V a b c d = V * V * V * V , R={R_{abc}}^{d}\in{V_{abc}}^{d}=V^{*}\otimes V^{*}\otimes V^{*}\otimes V,
  21. R a b c d + R c a b d + R b c a d = 0. {R_{abc}}^{d}+{R_{cab}}^{d}+{R_{bca}}^{d}=0.

Abstract_polytope.html

  1. \infty
  2. I = ( I i j ) I=(I_{ij})
  3. I i i I i j = I j i I j j ( i < j ) . I_{ii}\cdot I_{ij}=I_{ji}\cdot I_{jj}\ \ (i<j).

Abuse_of_notation.html

  1. 3 \mathbb{R}^{3}
  2. X X
  3. 𝒯 \mathcal{T}
  4. ( X , 𝒯 ) (X,\mathcal{T})
  5. ( X , 𝒯 ) (X,\mathcal{T}^{\prime})
  6. X X
  7. X X
  8. ( G , ) (G,\star)
  9. G G
  10. d y d x \frac{dy}{dx}
  11. d y d x \frac{dy}{dx}
  12. d y d x = d y d u d u d x \frac{dy}{dx}=\frac{dy}{du}\frac{du}{dx}
  13. d y d x = g ( x ) h ( y ) \frac{dy}{dx}={g(x)\over h(y)}
  14. h ( y ) d y = g ( x ) d x h(y)dy={g(x)dx}
  15. 1 x d x \int{1\over x}\,dx
  16. d x x \int{dx\over x}
  17. d x dx
  18. 1 x 1\over x
  19. \nabla
  20. f \nabla f
  21. v \nabla\cdot\vec{v}
  22. × v \nabla\times\vec{v}
  23. \nabla
  24. \nabla
  25. \nabla
  26. det [ a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ] = a 1 det [ b 2 b 3 c 2 c 3 ] - a 2 det [ b 1 b 3 c 1 c 3 ] + a 3 det [ b 1 b 2 c 1 c 2 ] \det\begin{bmatrix}a_{1}&a_{2}&a_{3}\\ b_{1}&b_{2}&b_{3}\\ c_{1}&c_{2}&c_{3}\end{bmatrix}=a_{1}\det\begin{bmatrix}b_{2}&b_{3}\\ c_{2}&c_{3}\end{bmatrix}-a_{2}\det\begin{bmatrix}b_{1}&b_{3}\\ c_{1}&c_{3}\end{bmatrix}+a_{3}\det\begin{bmatrix}b_{1}&b_{2}\\ c_{1}&c_{2}\end{bmatrix}
  27. det [ a 2 a 3 b 2 b 3 ] 𝐢 - det [ a 1 a 3 b 1 b 3 ] 𝐣 + det [ a 1 a 2 b 1 b 2 ] 𝐤 \det\begin{bmatrix}a_{2}&a_{3}\\ b_{2}&b_{3}\end{bmatrix}\mathbf{i}-\det\begin{bmatrix}a_{1}&a_{3}\\ b_{1}&b_{3}\end{bmatrix}\mathbf{j}+\det\begin{bmatrix}a_{1}&a_{2}\\ b_{1}&b_{2}\end{bmatrix}\mathbf{k}
  28. 𝐚 × 𝐛 = det [ 𝐢 𝐣 𝐤 a 1 a 2 a 3 b 1 b 2 b 3 ] \mathbf{a}\times\mathbf{b}=\det\begin{bmatrix}\mathbf{i}&\mathbf{j}&\mathbf{k}% \\ a_{1}&a_{2}&a_{3}\\ b_{1}&b_{2}&b_{3}\\ \end{bmatrix}
  29. f x = f ( x ) f\,x=f(x)
  30. f 2 x = f f x = f ( f ( x ) ) f^{2}x=ff\,x=f(f(x))
  31. ( E × F ) × G = E × ( F × G ) = E × F × G (E\times F)\times G=E\times(F\times G)=E\times F\times G
  32. x E x\in E
  33. y F y\in F
  34. z G z\in G
  35. ( ( x , y ) , z ) = ( x , ( y , z ) ) ((x,y),z)=(x,(y,z))
  36. ( x , y ) = x (x,y)=x
  37. z = ( y , z ) z=(y,z)
  38. ( ( x , y ) , z ) = ( x , y , z ) ((x,y),z)=(x,y,z)
  39. sin ( x ) \sin(x)
  40. sin 2 ( x ) \sin^{2}(x)
  41. sin - 1 ( x ) \sin^{-1}(x)
  42. f n ( x ) f^{n}(x)
  43. f 2 ( x ) = f ( f ( x ) ) f^{2}(x)=f(f(x))
  44. f - 1 ( x ) f^{-1}(x)
  45. f 2 ( x ) = ( f ( x ) ) 2 f^{2}(x)=(f(x))^{2}
  46. f - 1 ( x ) = 1 f ( x ) f^{-1}(x)=\frac{1}{f(x)}
  47. sin x 2 \sin x^{2}
  48. ( sin x ) 2 (\sin x)^{2}
  49. sin ( x 2 ) \sin(x^{2})
  50. sin 2 ( x ) = sin ( sin x ) \sin^{2}(x)=\sin(\sin x)
  51. sin 1 2 ( x ) , \sin^{\frac{1}{2}}(x),
  52. sin ( x ) \sin(x)
  53. f ( x ) f(x)
  54. O ( g ( x ) ) O(g(x))
  55. O ( n 2 ) O(n^{2})
  56. T ( n ) = O ( n 2 ) T(n)=O(n^{2})
  57. f ( n ) O ( g ( n ) ) f(n)\in O(g(n))
  58. f ( n ) = O ( g ( n ) ) f(n)=O(g(n))
  59. O ( n log n ) O ( n 2 ) O(n\cdot\log n)\subset O(n^{2})
  60. O ( 2 n ) O ( n 2 ) O(2^{n})\bigcup O(n^{2})
  61. g g
  62. f ( n ) O ( g ( n ) ) f(n)\in O(g(n))
  63. O ( n m ) O(n^{m})
  64. O O
  65. O ( 2 m ) O(2^{m})
  66. n n
  67. O ( n 3 ) O(n^{3})
  68. m m
  69. O ( c ) O(c)
  70. O ( 1 ) O(1)
  71. c c
  72. ( n n log n ) O ( n n 2 ) (n\mapsto n\cdot\log n)\in O(n\mapsto n^{2})
  73. O ( n n log n ) O ( n n 2 ) O(n\mapsto n\cdot\log n)\subset O(n\mapsto n^{2})
  74. T graph O ( ( v , e ) v log e ) T_{\mbox{graph}~{}}\in O((v,e)\mapsto v\cdot\log e)
  75. O ( f ) O(f)
  76. r r
  77. r r
  78. P ( x ) P(x)
  79. P ( X = x ) P(X=x)
  80. X X
  81. P ( X = x ) P(X=x)
  82. X = x X=x
  83. P P
  84. { T , F } \{T,F\}
  85. P P
  86. X X
  87. X X
  88. Ω \Omega
  89. P P
  90. i = m i n m a x f ( i ) \sum_{i=min}^{max}f(i)
  91. F ( f , i , m i n , m a x ) F(f,i,min,max)
  92. F ( f , t r u t h v a l u e , m a x ) F(f,truthvalue,max)
  93. P ( X , { x } ) P(X,\{x\})
  94. X X
  95. x x
  96. P ( X | x ) P(\left.X\right|_{x})
  97. P ( X = x ) P(X=x)
  98. P ( x ) P(x)
  99. P ( ϕ ) P(\phi)
  100. P ( ϕ ) P(\phi)
  101. ϕ \phi
  102. P ( ϕ ) = P μ ( ϕ ) = P Ω , μ ( ϕ ) = μ ( { w ϕ ( w ) and w Ω } ) μ ( { w w Ω } ) P(\phi)=P_{\mu}(\phi)=P_{\Omega,\mu}(\phi)=\frac{\mu(\{w\mid\phi(w)\,\text{ % and }w\in\Omega\})}{\mu(\{w\mid w\in\Omega\})}
  103. P ( ϕ ) P(\phi)
  104. P ( X = x ) P(X=x)
  105. P ( X ) P(X)
  106. X X
  107. P ( X = x ) P(X=x)
  108. P ( ) P()
  109. P ( Z ) P(Z)
  110. Z Z
  111. Z = ( X = x ) Z=(X=x)
  112. Z Z
  113. X X
  114. x x
  115. Z = ( X = x ) Z=(X=x)
  116. P ( Z ) = P ( X = x ) P(Z)=P(X=x)
  117. P ( X = x ) P(X=x)
  118. P ( X ) P(X)
  119. X X

AC_power.html

  1. cos ϕ \cos\phi
  2. p f = P a + P b + P c | S a | + | S b | + | S c | pf={P_{a}+P_{b}+P_{c}\over|S_{a}|+|S_{b}|+|S_{c}|}
  3. p f = P a + P b + P c | P a + P b + P c + j ( Q a + Q b + Q c ) | pf={P_{a}+P_{b}+P_{c}\over|P_{a}+P_{b}+P_{c}+j(Q_{a}+Q_{b}+Q_{c})|}
  4. P = S P=S
  5. P = S = V RMS I RMS = I RMS 2 R = V RMS 2 R P=S=V_{\mathrm{RMS}}I_{\mathrm{RMS}}=I_{\mathrm{RMS}}^{2}R=\frac{V_{\mathrm{% RMS}}^{2}}{R}\,\!
  6. P \displaystyle P
  7. Q = I RMS 2 X = V RMS 2 X Q=I_{\mathrm{RMS}}^{2}X=\frac{V_{\mathrm{RMS}}^{2}}{X}
  8. P inst ( t ) = V ( t ) I ( t ) P\text{inst}(t)=V(t)I(t)
  9. P avg = 1 t 2 - t 1 t 1 t 2 V ( t ) I ( t ) d t P\text{avg}=\frac{1}{t_{2}-t_{1}}\int_{t_{1}}^{t_{2}}V(t)I(t)\,\operatorname{d}t
  10. P avg = 1 n k = 1 n V [ k ] I [ k ] P\text{avg}=\frac{1}{n}\sum_{k=1}^{n}V[k]I[k]
  11. A cos ( ω 1 t + k 1 ) cos ( ω 2 t + k 2 ) = A 2 cos [ ( ω 1 t + k 1 ) + ( ω 2 t + k 2 ) ] + A 2 cos [ ( ω 1 t + k 1 ) - ( ω 2 t + k 2 ) ] = A 2 cos [ ( ω 1 + ω 2 ) t + k 1 + k 2 ] + A 2 cos [ ( ω 1 - ω 2 ) t + k 1 - k 2 ] \begin{aligned}&\displaystyle A\cos(\omega_{1}t+k_{1})\cos(\omega_{2}t+k_{2})% \\ \displaystyle=&\displaystyle\frac{A}{2}\cos\left[\left(\omega_{1}t+k_{1}\right% )+\left(\omega_{2}t+k_{2}\right)\right]+\frac{A}{2}\cos\left[\left(\omega_{1}t% +k_{1}\right)-\left(\omega_{2}t+k_{2}\right)\right]\\ \displaystyle=&\displaystyle\frac{A}{2}\cos\left[\left(\omega_{1}+\omega_{2}% \right)t+k_{1}+k_{2}\right]+\frac{A}{2}\cos\left[\left(\omega_{1}-\omega_{2}% \right)t+k_{1}-k_{2}\right]\end{aligned}

Accumulated_cyclone_energy.html

  1. ACE = 10 - 4 v max 2 \,\text{ACE}=10^{-4}\sum v_{\max}^{2}

Acetic_acid_(data_page).html

  1. P m m H g = 10 7.80307 - 1651.2 225 + T \scriptstyle P_{mmHg}=10^{7.80307-\frac{1651.2}{225+T}}
  2. P m m H g = 10 7.18807 - 1416.7 211 + T \scriptstyle P_{mmHg}=10^{7.18807-\frac{1416.7}{211+T}}
  3. log 10 P m m H g = 7.80307 - 1651.2 225 + T \scriptstyle\log_{10}P_{mmHg}=7.80307-\frac{1651.2}{225+T}
  4. log 10 P m m H g = 7.18807 - 1416.7 211 + T \scriptstyle\log_{10}P_{mmHg}=7.18807-\frac{1416.7}{211+T}

Acid_value.html

  1. A N = ( V e q - b e q ) N 56.1 W o i l AN=(V_{eq}-b_{eq})N\frac{56.1}{W_{oil}}
  2. N = 1000 W K H P 204.23 V e q N=\frac{1000W_{KHP}}{204.23V_{eq}}

Acousto-optic_modulator.html

  1. sin θ = m λ Λ \sin\theta=\frac{m\lambda}{\Lambda}
  2. sin θ = m n λ 0 Λ \sin\theta=\frac{mn\lambda_{0}}{\Lambda}
  3. f f + m F f\rightarrow f+mF

Active_set_method.html

  1. g 1 ( x ) 0 , , g k ( x ) 0 g_{1}(x)\geq 0,\dots,g_{k}(x)\geq 0
  2. x x
  3. g i ( x ) 0 g_{i}(x)\geq 0
  4. x x
  5. g i ( x ) = 0 g_{i}(x)=0
  6. x x
  7. g i ( x ) > 0. g_{i}(x)>0.
  8. x x
  9. g i ( x ) g_{i}(x)

Activity_coefficient.html

  1. μ B \mu_{B}
  2. μ B = μ B + R T ln x B \mu_{B}=\mu_{B}^{\ominus}+RT\ln x_{B}\,
  3. μ B \mu_{B}^{\ominus}
  4. μ B = μ B + R T ln a B \mu_{B}=\mu_{B}^{\ominus}+RT\ln a_{B}\,
  5. a B a_{B}
  6. a B = x B γ B a_{B}=x_{B}\gamma_{B}
  7. γ B \gamma_{B}
  8. x B x_{B}
  9. γ B \gamma_{B}
  10. γ B 1 \gamma_{B}\approx 1
  11. γ B > 1 \gamma_{B}>1
  12. γ B < 1 \gamma_{B}<1
  13. x B x_{B}
  14. γ ± \gamma_{\pm}
  15. γ ± = γ + γ - \gamma_{\pm}=\sqrt{\gamma_{+}\gamma_{-}}
  16. γ ± = γ A p γ B q p + q \gamma_{\pm}=\sqrt[p+q]{\gamma_{A}^{p}\gamma_{B}^{q}}
  17. log 10 ( γ 0 ) = b I \log_{10}(\gamma_{0})=bI
  18. ln ( a w ) = - ν m 55.51 \ln(a_{w})=\frac{-\nu m}{55.51}
  19. l g ( γ i ) = - A z i 2 I 1 + B a I lg(\gamma_{i})=-\frac{Az_{i}^{2}\sqrt{I}}{1+Ba\sqrt{I}}
  20. V E ¯ i = R T ( ln ( γ i ) ) P \bar{V^{E}}_{i}=RT\frac{\partial(\ln(\gamma_{i}))}{\partial P}
  21. H E ¯ i = - R T 2 ( ln ( γ i ) ) T \bar{H^{E}}_{i}=-RT^{2}\frac{\partial(\ln(\gamma_{i}))}{\partial T}
  22. Δ r G \Delta_{r}G
  23. α A + β B σ S + τ T \alpha A+\beta B\rightleftharpoons\sigma S+\tau T
  24. Δ r G = σ μ S + τ μ T - ( α μ A + β μ B ) = 0 \Delta_{r}G=\sigma\mu_{S}+\tau\mu_{T}-(\alpha\mu_{A}+\beta\mu_{B})=0\,
  25. Δ r G = σ μ S + σ R T ln a S + τ μ T + τ R T ln a T - ( α μ A + α R T ln a A + β μ B + β R T ln a B ) = 0 \Delta_{r}G=\sigma\mu_{S}^{\ominus}+\sigma RT\ln a_{S}+\tau\mu_{T}^{\ominus}+% \tau RT\ln a_{T}-(\alpha\mu_{A}^{\ominus}+\alpha RT\ln a_{A}+\beta\mu_{B}^{% \ominus}+\beta RT\ln a_{B})=0
  26. Δ r G = ( σ μ S + τ μ T - α μ A - β μ B ) + R T ln a S σ a T τ a A α a B β = 0 \Delta_{r}G=\left(\sigma\mu_{S}^{\ominus}+\tau\mu_{T}^{\ominus}-\alpha\mu_{A}^% {\ominus}-\beta\mu_{B}^{\ominus}\right)+RT\ln\frac{a_{S}^{\sigma}a_{T}^{\tau}}% {a_{A}^{\alpha}a_{B}^{\beta}}=0
  27. ( σ μ S + τ μ T - α μ A - β μ B ) \left(\sigma\mu_{S}^{\ominus}+\tau\mu_{T}^{\ominus}-\alpha\mu_{A}^{\ominus}-% \beta\mu_{B}^{\ominus}\right)
  28. Δ r G \Delta_{r}G^{\ominus}
  29. Δ r G = - R T ln K \Delta_{r}G^{\ominus}=-RT\ln K
  30. K = [ S ] σ [ T ] τ [ A ] α [ B ] β × γ S σ γ T τ γ A α γ B β K=\frac{[S]^{\sigma}[T]^{\tau}}{[A]^{\alpha}[B]^{\beta}}\times\frac{\gamma_{S}% ^{\sigma}\gamma_{T}^{\tau}}{\gamma_{A}^{\alpha}\gamma_{B}^{\beta}}
  31. K = [ S ] σ [ T ] τ [ A ] α [ B ] β K=\frac{[S]^{\sigma}[T]^{\tau}}{[A]^{\alpha}[B]^{\beta}}

AdaBoost.html

  1. F T ( x ) = t = 1 T f t ( x ) F_{T}(x)=\sum_{t=1}^{T}f_{t}(x)\,\!
  2. f t f_{t}
  3. x x
  4. T T
  5. h ( x i ) h(x_{i})
  6. t t
  7. α t \alpha_{t}
  8. E t E_{t}
  9. t t
  10. E t = i E [ F t - 1 ( x i ) + α t h ( x i ) ] E_{t}=\sum_{i}E[F_{t-1}(x_{i})+\alpha_{t}h(x_{i})]
  11. F t - 1 ( x ) F_{t-1}(x)
  12. E ( F ) E(F)
  13. f t ( x ) = α t h ( x ) f_{t}(x)=\alpha_{t}h(x)
  14. E ( F t - 1 ( x i ) ) E(F_{t-1}(x_{i}))
  15. { ( x 1 , y 1 ) , , ( x N , y N ) } \{(x_{1},y_{1}),\ldots,(x_{N},y_{N})\}
  16. x i x_{i}
  17. y i { - 1 , 1 } y_{i}\in\{-1,1\}
  18. { k 1 , , k L } \{k_{1},\ldots,k_{L}\}
  19. k j ( x i ) { - 1 , 1 } k_{j}(x_{i})\in\{-1,1\}
  20. m - 1 m-1
  21. C ( m - 1 ) ( x i ) = α 1 k 1 ( x i ) + + α m - 1 k m - 1 ( x i ) C_{(m-1)}(x_{i})=\alpha_{1}k_{1}(x_{i})+\cdots+\alpha_{m-1}k_{m-1}(x_{i})
  22. m m
  23. C m ( x i ) = C ( m - 1 ) ( x i ) + α m k m ( x i ) C_{m}(x_{i})=C_{(m-1)}(x_{i})+\alpha_{m}k_{m}(x_{i})
  24. k m k_{m}
  25. α m \alpha_{m}
  26. E E
  27. C m C_{m}
  28. E = i = 1 N e - y i C m ( x i ) E=\sum_{i=1}^{N}e^{-y_{i}C_{m}(x_{i})}
  29. w i ( 1 ) = 1 w_{i}^{(1)}=1
  30. w i ( m ) = e - y i C m - 1 ( x i ) w_{i}^{(m)}=e^{-y_{i}C_{m-1}(x_{i})}
  31. m > 1 m>1
  32. E = i = 1 N w i ( m ) e - y i α m k m ( x i ) E=\sum_{i=1}^{N}w_{i}^{(m)}e^{-y_{i}\alpha_{m}k_{m}(x_{i})}
  33. k m k_{m}
  34. y i k m ( x i ) = 1 y_{i}k_{m}(x_{i})=1
  35. y i k m ( x i ) = - 1 y_{i}k_{m}(x_{i})=-1
  36. E = y i = k m ( x i ) w i ( m ) e - α m + y i k m ( x i ) w i ( m ) e α m = i = 1 N w i ( m ) e - α m + y i k m ( x i ) w i ( m ) ( e α m - e - α m ) E=\sum_{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}e^{-\alpha_{m}}+\sum_{y_{i}\neq k_{m}(x_% {i})}w_{i}^{(m)}e^{\alpha_{m}}=\sum_{i=1}^{N}w_{i}^{(m)}e^{-\alpha_{m}}+\sum_{% y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}(e^{\alpha_{m}}-e^{-\alpha_{m}})
  37. k m k_{m}
  38. y i k m ( x i ) w i ( m ) \sum_{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}
  39. k m k_{m}
  40. E E
  41. y i k m ( x i ) w i ( m ) \sum_{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}
  42. w i ( m ) = e - y i C m - 1 ( x i ) w_{i}^{(m)}=e^{-y_{i}C_{m-1}(x_{i})}
  43. α m \alpha_{m}
  44. E E
  45. k m k_{m}
  46. d E d α m = y i k m ( x i ) w i ( m ) e α m - y i = k m ( x i ) w i ( m ) e - α m \frac{dE}{d\alpha_{m}}=\sum_{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}e^{\alpha_{m}}-% \sum_{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}e^{-\alpha_{m}}
  47. α m \alpha_{m}
  48. α m = 1 2 ln ( y i = k m ( x i ) w i ( m ) y i k m ( x i ) w i ( m ) ) \alpha_{m}=\frac{1}{2}\ln(\frac{\sum_{y_{i}=k_{m}(x_{i})}w_{i}^{(m)}}{\sum_{y_% {i}\neq k_{m}(x_{i})}w_{i}^{(m)}})
  49. ϵ m = y i k m ( x i ) w i ( m ) / i = 1 N w i ( m ) \epsilon_{m}=\sum_{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}/\sum_{i=1}^{N}w_{i}^{(m)}
  50. α m = 1 2 ln ( 1 - ϵ m ϵ m ) \alpha_{m}=\frac{1}{2}\ln(\frac{1-\epsilon_{m}}{\epsilon_{m}})
  51. k m k_{m}
  52. y i k m ( x i ) w i ( m ) \sum_{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}
  53. ϵ m = y i k m ( x i ) w i ( m ) / i = 1 N w i ( m ) \epsilon_{m}=\sum_{y_{i}\neq k_{m}(x_{i})}w_{i}^{(m)}/\sum_{i=1}^{N}w_{i}^{(m)}
  54. α m = 1 2 ln ( 1 - ϵ m ϵ m ) \alpha_{m}=\frac{1}{2}\ln(\frac{1-\epsilon_{m}}{\epsilon_{m}})
  55. C m - 1 C_{m-1}
  56. C m = C ( m - 1 ) + α m k m C_{m}=C_{(m-1)}+\alpha_{m}k_{m}
  57. x i x_{i}
  58. h h
  59. x i x_{i}
  60. i = 1 n e - Y i μ ^ ( x i ) + x 1 x n μ ^ ′′ ( x ) 2 d x \sum_{i=1}^{n}e^{-Y_{i}\hat{\mu}(x_{i})}+\infty\int_{x_{1}}^{x_{n}}\hat{\mu}^{% \prime\prime}(x)^{2}\,dx
  61. μ ^ i \hat{\mu}_{i}
  62. F ( x ) F(x)
  63. y ( x ) y(x)
  64. E ( f ) = ( y ( x ) - f ( x ) ) 2 E(f)=(y(x)-f(x))^{2}
  65. E ( f ) = e - y ( x ) f ( x ) E(f)=e^{-y(x)f(x)}
  66. | F ( x ) | |F(x)|
  67. x i x_{i}
  68. - y ( x i ) f ( x i ) -y(x_{i})f(x_{i})
  69. e i - y i f ( x i ) = i e - y i f ( x i ) e^{\sum_{i}-y_{i}f(x_{i})}=\prod_{i}e^{-y_{i}f(x_{i})}
  70. F t ( x ) F_{t}(x)
  71. ϕ ( y , f ( x ) ) = { - 4 y f ( x ) if y f ( x ) < - 1 , ( y f ( x ) - 1 ) 2 if - 1 y f ( x ) 1 , 0 if y f ( x ) > 1 \phi(y,f(x))=\begin{cases}-4yf(x)&\mbox{if }~{}yf(x)<-1,\\ (yf(x)-1)^{2}&\mbox{if }~{}-1\leq yf(x)\leq 1,\\ 0&\mbox{if }~{}yf(x)>1\end{cases}
  72. f ( x ) f(x)
  73. y f ( x ) > 1 yf(x)>1
  74. i ϕ ( i , y , f ) = i e - y i f ( x i ) \sum_{i}\phi(i,y,f)=\sum_{i}e^{-y_{i}f(x_{i})}
  75. i ϕ ( i , y , f ) = i ln ( 1 + e - y i f ( x i ) ) \sum_{i}\phi(i,y,f)=\sum_{i}\ln\left(1+e^{-y_{i}f(x_{i})}\right)
  76. ( F t ( x 1 ) , , F t ( x n ) ) \left(F_{t}(x_{1}),\dots,F_{t}(x_{n})\right)
  77. h ( x ) h(x)
  78. ( y 1 , , y n ) (y_{1},\dots,y_{n})
  79. E T ( x 1 , , x n ) E_{T}(x_{1},\dots,x_{n})
  80. h ( x ) h(x)
  81. α \alpha
  82. α h ( x ) \alpha h(x)
  83. F t F_{t}
  84. x 1 x n x_{1}\dots x_{n}
  85. y 1 y n , y { - 1 , 1 } y_{1}\dots y_{n},y\in\{-1,1\}
  86. w 1 , 1 w n , 1 w_{1,1}\dots w_{n,1}
  87. 1 n \frac{1}{n}
  88. E ( f ( x ) , y , i ) = e - y i f ( x i ) E(f(x),y,i)=e^{-y_{i}f(x_{i})}
  89. h : x [ - 1 , 1 ] h\colon x\rightarrow[-1,1]
  90. t t
  91. 1 T 1\dots T
  92. f t ( x ) f_{t}(x)
  93. h t ( x ) h_{t}(x)
  94. ϵ t \epsilon_{t}
  95. ϵ t = i w i , t E ( h t ( x ) , y , i ) \epsilon_{t}=\sum_{i}w_{i,t}E(h_{t}(x),y,i)
  96. α t = 1 2 ln ( 1 - ϵ t ϵ t ) \alpha_{t}=\frac{1}{2}\ln\left(\frac{1-\epsilon_{t}}{\epsilon_{t}}\right)
  97. F t ( x ) = F t - 1 ( x ) + α t h t ( x ) F_{t}(x)=F_{t-1}(x)+\alpha_{t}h_{t}(x)
  98. w i , t + 1 = w i , t e - y i α t h t ( x i ) w_{i,t+1}=w_{i,t}e^{-y_{i}\alpha_{t}h_{t}(x_{i})}
  99. w i , t + 1 w_{i,t+1}
  100. i w i , t + 1 = 1 \sum_{i}w_{i,t+1}=1
  101. h t + 1 ( x i ) = y i w i , t + 1 h t + 1 ( x i ) y i w i , t + 1 = h t ( x i ) = y i w i , t h t ( x i ) y i w i , t \frac{\sum_{h_{t+1}(x_{i})=y_{i}}w_{i,t+1}}{\sum_{h_{t+1}(x_{i})\neq y_{i}}w_{% i,t+1}}=\frac{\sum_{h_{t}(x_{i})=y_{i}}w_{i,t}}{\sum_{h_{t}(x_{i})\neq y_{i}}w% _{i,t}}
  102. α t \alpha_{t}
  103. α t \alpha_{t}
  104. i w i e - y i h i α t \sum_{i}w_{i}e^{-y_{i}h_{i}\alpha_{t}}
  105. i , h i [ - 1 , 1 ] \forall i,h_{i}\in[-1,1]
  106. i w i e - y i h i α t \displaystyle\sum_{i}w_{i}e^{-y_{i}h_{i}\alpha_{t}}
  107. α t \alpha_{t}
  108. ( 1 + ϵ t 2 ) e α t - ( 1 - ϵ t 2 ) e - α t \displaystyle\left(\frac{1+\epsilon_{t}}{2}\right)e^{\alpha_{t}}-\left(\frac{1% -\epsilon_{t}}{2}\right)e^{-\alpha_{t}}
  109. h i { - 1 , 1 } h_{i}\in\{-1,1\}
  110. h ( x ) { a , b } , a - b h(x)\in\{a,b\},a\neq-b
  111. h ( x ) { a , b , , n } h(x)\in\{a,b,\dots,n\}
  112. h ( x ) h(x)\in\mathbb{R}
  113. f t = α t h t ( x ) f_{t}=\alpha_{t}h_{t}(x)
  114. α , h \alpha,h
  115. i w i , t e - y i f t ( x i ) \sum_{i}w_{i,t}e^{-y_{i}f_{t}(x_{i})}
  116. p ( x ) = P ( y = 1 | x ) p(x)=P(y=1|x)
  117. x x
  118. e - y ( F t - 1 ( x ) + f t ( p ( x ) ) ) e^{-y\left(F_{t-1}(x)+f_{t}(p(x))\right)}
  119. p ( x ) p(x)
  120. f t ( x ) = 1 2 ln ( x 1 - x ) f_{t}(x)=\frac{1}{2}\ln\left(\frac{x}{1-x}\right)
  121. f t ( x ) f_{t}(x)
  122. z t = y * - p t ( x ) 2 p t ( x ) ( 1 - p t ( x ) ) z_{t}=\frac{y^{*}-p_{t}(x)}{2p_{t}(x)(1-p_{t}(x))}
  123. p t ( x ) = e F t - 1 ( x ) e F t - 1 ( x ) + e - F t - 1 ( x ) p_{t}(x)=\frac{e^{F_{t-1}(x)}}{e^{F_{t-1}(x)}+e^{-F_{t-1}(x)}}
  124. w t = p t ( x ) ( 1 - p t ( x ) ) w_{t}=p_{t}(x)(1-p_{t}(x))
  125. y * = y + 1 2 y^{*}=\frac{y+1}{2}
  126. z t z_{t}
  127. t t
  128. f t f_{t}
  129. z t z_{t}
  130. p t ( x i ) ( 1 - p t ( x i ) ) p_{t}(x_{i})(1-p_{t}(x_{i}))
  131. f t f_{t}
  132. f t f_{t}
  133. i w t , i ( y i - f t ( x i ) ) 2 \sum_{i}w_{t,i}(y_{i}-f_{t}(x_{i}))^{2}
  134. f t ( x ) = α t h t ( x ) f_{t}(x)=\alpha_{t}h_{t}(x)
  135. y y
  136. α t = \alpha_{t}=\infty
  137. α \alpha
  138. α t \alpha_{t}

Adams_operation.html

  1. χ ψ k ( ρ ) ( g ) = χ ρ ( g k ) . \chi_{\psi^{k}(\rho)}(g)=\chi_{\rho}(g^{k})\ .

Adaptive_equalizer.html

  1. x x
  2. d ( n ) d(n)
  3. x x

Additive_polynomial.html

  1. P ( a + b ) = P ( a ) + P ( b ) P(a+b)=P(a)+P(b)\,
  2. ( a + b ) p = n = 0 p ( p n ) a n b p - n . (a+b)^{p}=\sum_{n=0}^{p}{p\choose n}a^{n}b^{p-n}.
  3. ( p n ) \scriptstyle{p\choose n}
  4. ( a + b ) p a p + b p mod p (a+b)^{p}\equiv a^{p}+b^{p}\mod p
  5. τ p n ( x ) = x p n \tau_{p}^{n}(x)=x^{p^{n}}
  6. τ p n ( x ) \scriptstyle\tau_{p}^{n}(x)
  7. k { τ p } . k\{\tau_{p}\}.\,
  8. 𝔽 p = 𝐙 / p 𝐙 \scriptstyle\mathbb{F}_{p}=\mathbf{Z}/p\mathbf{Z}
  9. ( a x ) p = a x p , (ax)^{p}=ax^{p},\,
  10. 𝔽 p . \scriptstyle\mathbb{F}_{p}.
  11. { w 1 , , w m } k \scriptstyle\{w_{1},\dots,w_{m}\}\subset k
  12. { w 1 , , w m } \scriptstyle\{w_{1},\dots,w_{m}\}

Adiabatic_invariant.html

  1. d W = P d V = N k B T V d V dW=PdV={Nk_{B}T\over V}dV
  2. d T = 1 N C v d E dT={1\over NC_{v}}dE
  3. C v C_{v}
  4. N C v d T = - d W = - N k B T V d V NC_{v}dT=-dW=-{N{k_{B}}T\over V}dV
  5. k B k_{B}
  6. d ( C v N log T ) = - d ( N log V ) \,d(C_{v}N\log T)=-d(N\log V)
  7. C v N log T + N log V \,C_{v}N\log T+N\log V
  8. S = C v N log T + N log V - N log N = N log ( T C v V / N ) \,S=C_{v}N\log T+N\log V-N\log N=N\log(T^{C_{v}}V/N)
  9. E = 1 2 m k p k 1 2 + p k 2 2 + p k 3 2 E={1\over 2m}\sum_{k}p_{k1}^{2}+p_{k2}^{2}+p_{k3}^{2}
  10. 2 m E \scriptstyle\sqrt{2mE}
  11. 2 π 3 N / 2 ( 2 m E ) 3 N - 1 2 Γ ( 3 N / 2 ) {2\pi^{3N/2}(2mE)^{{3N-1}\over 2}}\over{\Gamma(3N/2)}
  12. Γ \Gamma
  13. 2 π 3 N / 2 ( 2 m E ) 3 N - 1 2 V N Γ ( 3 N / 2 ) {2\pi^{3N/2}(2mE)^{{3N-1}\over 2}}V^{N}\over{\Gamma(3N/2)}
  14. N ! = Γ ( N + 1 ) N!=\Gamma(N+1)
  15. S = N ( 3 / 2 l o g ( E ) - 3 / 2 l o g ( 3 N / 2 ) + l o g ( V ) - l o g ( N ) ) S=N\big(3/2log(E)-3/2log(3N/2)+log(V)-log(N)\big)
  16. = N ( 3 / 2 l o g ( 2 3 E / N ) + l o g ( V / N ) ) =N\big(3/2log(\scriptstyle{\frac{2}{3}}\displaystyle E/N)+log(V/N)\big)
  17. Δ f = 2 v c f \Delta f={2v\over c}f
  18. Δ E = v 2 E c \,\Delta E=v{2E\over c}
  19. Δ f f = Δ E E {\Delta f\over f}={\Delta E\over E}
  20. 1 / 2 β 1/2\beta
  21. E f = e - β h f \,\langle E_{f}\rangle=e^{-\beta hf}
  22. H t ( p , x ) = p 2 2 m + m ω ( t ) 2 x 2 2 H_{t}(p,x)={p^{2}\over 2m}+{m\omega(t)^{2}x^{2}\over 2}\,
  23. J = 0 T p ( t ) d x d t d t J=\int_{0}^{T}p(t){dx\over dt}dt\,
  24. θ \theta
  25. d θ d t = H J = H ( J ) {d\theta\over dt}={\partial H\over\partial J}=H^{\prime}(J)\,
  26. H H^{\prime}
  27. θ \theta
  28. H : H^{\prime}:
  29. d J d J = 1 = 0 T ( p J d x d t + p J d x d t ) d t = H 0 T ( p J x θ - p θ x J ) d t {dJ\over dJ}=1=\int_{0}^{T}\bigg({\partial p\over\partial J}{dx\over dt}+p{% \partial\over\partial J}{dx\over dt}\bigg)dt=H^{\prime}\int_{0}^{T}\bigg({% \partial p\over\partial J}{\partial x\over\partial\theta}-{\partial p\over% \partial\theta}{\partial x\over\partial J}\bigg)dt\,
  30. 1 = H 0 T { x , p } d t = H T 1=H^{\prime}\int_{0}^{T}\{x,p\}dt=H^{\prime}T\,
  31. H H^{\prime}
  32. θ \theta
  33. H = ω J \,H=\omega J\,
  34. J = 0 2 π p x θ d θ J=\int_{0}^{2\pi}p{\partial x\over\partial\theta}d\theta\,
  35. d J d t = 0 2 π ( d p d t x θ + p d d t x θ ) d θ {dJ\over dt}=\int_{0}^{2\pi}\bigg({dp\over dt}{\partial x\over\partial\theta}+% p{d\over dt}{\partial x\over\partial\theta}\bigg)d\theta\,
  36. d J d t = 0 2 π ( p θ x θ + p θ x θ ) d θ {dJ\over dt}=\int_{0}^{2\pi}\bigg({\partial p\over\partial\theta}{\partial x% \over\partial\theta}+p{\partial\over\partial\theta}{\partial x\over\partial% \theta}\bigg)d\theta\,
  37. θ \theta
  38. E = p 2 2 m + m ω 2 x 2 2 E={p^{2}\over 2m}+{m\omega^{2}x^{2}\over 2}\,
  39. 2 E / ω 2 m \scriptstyle\sqrt{2E/\omega^{2}m}
  40. 2 m E \scriptstyle\sqrt{2mE}
  41. 2 π E / ω 2\pi E/\omega
  42. E = h f = ω E=hf=\hbar\omega\,
  43. 3 k B 3k_{B}
  44. p d q = n h \int pdq=nh\,
  45. μ = m v 2 2 B \mu=\frac{mv_{\perp}^{2}}{2B}
  46. ω / ω c \omega/\omega_{c}
  47. ω \omega
  48. J = a b p d s J=\int_{a}^{b}p_{\parallel}ds

Adjoint_representation_of_a_Lie_algebra.html

  1. x x
  2. 𝔤 \mathfrak{g}
  3. x x
  4. 𝔤 \mathfrak{g}
  5. ad x : 𝔤 𝔤 with ad x ( y ) = [ x , y ] \operatorname{ad}_{x}:\mathfrak{g}\to\mathfrak{g}\qquad\,\text{with}\qquad% \operatorname{ad}_{x}(y)=[x,y]
  6. y y
  7. 𝔤 \mathfrak{g}
  8. 𝔤 \mathfrak{g}
  9. k k
  10. ad : 𝔤 End ( 𝔤 ) \operatorname{ad}:\mathfrak{g}\to\operatorname{End}(\mathfrak{g})
  11. ( 𝔤 ) (\mathfrak{g})
  12. ( 𝔤 ) (\mathfrak{g})
  13. [ ad x , ad y ] = ad x ad y - ad y ad x [\operatorname{ad}_{x},\operatorname{ad}_{y}]=\operatorname{ad}_{x}\circ% \operatorname{ad}_{y}-\operatorname{ad}_{y}\circ\operatorname{ad}_{x}
  14. 𝔤 \mathfrak{g}
  15. ( 𝔤 ) (\mathfrak{g})
  16. 𝔤 𝔩 ( 𝔤 ) \mathfrak{gl}(\mathfrak{g})
  17. 𝔤 \mathfrak{g}
  18. [ x , [ y , z ] ] + [ y , [ z , x ] ] + [ z , [ x , y ] ] = 0 [x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0
  19. ( [ ad x , ad y ] ) ( z ) = ( ad [ x , y ] ) ( z ) \left([\operatorname{ad}_{x},\operatorname{ad}_{y}]\right)(z)=\left(% \operatorname{ad}_{[x,y]}\right)(z)
  20. x x
  21. y y
  22. z z
  23. 𝔤 \mathfrak{g}
  24. 𝔤 \mathfrak{g}
  25. 𝔤 \mathfrak{g}
  26. δ : 𝔤 𝔤 \delta:\mathfrak{g}\rightarrow\mathfrak{g}
  27. δ ( [ x , y ] ) = [ δ ( x ) , y ] + [ x , δ ( y ) ] \delta([x,y])=[\delta(x),y]+[x,\delta(y)]
  28. x x
  29. y y
  30. 𝔤 \mathfrak{g}
  31. ( 𝔤 ) (\mathfrak{g})
  32. 𝔤 \mathfrak{g}
  33. [ e i , e j ] = k c i j k e k . [e^{i},e^{j}]=\sum_{k}{c^{ij}}_{k}e^{k}.
  34. [ ad e i ] k j = c i j k . {\left[\operatorname{ad}_{e^{i}}\right]_{k}}^{j}={c^{ij}}_{k}~{}.
  35. G G
  36. Ψ : G A u t ( G ) Ψ:G→Aut(G)
  37. Ψ g ( h ) = g h g - 1 . \Psi_{g}(h)=ghg^{-1}~{}.
  38. Ad g = ( d Ψ g ) e : T e G T e G \operatorname{Ad}_{g}=(d\Psi_{g})_{e}:T_{e}G\rightarrow T_{e}G
  39. d d
  40. e e
  41. e e
  42. G G
  43. G G
  44. 𝔤 \mathfrak{g}
  45. ( 𝔤 ) (\mathfrak{g})
  46. G G
  47. A u t ( V ) Aut(V)
  48. E n d ( V ) End(V)
  49. ad = d ( Ad ) e : T e G End ( T e G ) . \operatorname{ad}=d(\operatorname{Ad})_{e}:T_{e}G\rightarrow\operatorname{End}% (T_{e}G).
  50. x x
  51. 𝔤 \mathfrak{g}
  52. X X
  53. G G
  54. 𝔤 \mathfrak{g}
  55. [ - F o r m u l a E r r o r - ] [-FormulaError-]

Adjunction_space.html

  1. X f Y = ( X Y ) / { f ( A ) A } . X\cup_{f}Y=(X\amalg Y)/\{f(A)\sim A\}.
  2. X + f Y X+\!_{f}\,Y

Admissible_decision_rule.html

  1. Θ \Theta\,
  2. 𝒳 \mathcal{X}
  3. 𝒜 \mathcal{A}
  4. Θ \Theta\,
  5. 𝒳 \mathcal{X}
  6. 𝒜 \mathcal{A}
  7. x 𝒳 x\in\mathcal{X}\,\!
  8. F ( x θ ) F(x\mid\theta)\,\!
  9. θ Θ \theta\in\Theta\,\!
  10. δ : 𝒳 𝒜 \delta:{\mathcal{X}}\rightarrow{\mathcal{A}}
  11. x 𝒳 x\in\mathcal{X}
  12. δ ( x ) 𝒜 \delta(x)\in\mathcal{A}\,\!
  13. L : Θ × 𝒜 L:\Theta\times\mathcal{A}\rightarrow\mathbb{R}
  14. a 𝒜 a\in\mathcal{A}
  15. θ Θ \theta\in\Theta
  16. x 𝒳 x\in\mathcal{X}
  17. L ( θ , δ ( x ) ) L(\theta,\delta(x))\,\!
  18. R ( θ , δ ) = E F ( x θ ) [ L ( θ , δ ( x ) ) ] . R(\theta,\delta)=\operatorname{E}_{F(x\mid\theta)}[{L(\theta,\delta(x))]}.\,\!
  19. δ \delta\,\!
  20. θ \theta\,\!
  21. δ * \delta^{*}\,\!
  22. δ \delta\,\!
  23. R ( θ , δ * ) R ( θ , δ ) R(\theta,\delta^{*})\leq R(\theta,\delta)
  24. θ \theta\,\!
  25. θ \theta\,\!
  26. θ \theta\,\!
  27. δ \delta\,\!
  28. θ \theta\,\!
  29. θ \theta\,\!
  30. π ( θ ) \pi(\theta)\,\!
  31. Θ \Theta\,\!
  32. δ \delta\,\!
  33. π ( θ ) \pi(\theta)\,\!
  34. r ( π , δ ) = E π ( θ ) [ R ( θ , δ ) ] . r(\pi,\delta)=\operatorname{E}_{\pi(\theta)}[R(\theta,\delta)].\,\!
  35. δ \delta\,\!
  36. r ( π , δ ) r(\pi,\delta)\,\!
  37. π ( θ ) \pi(\theta)\,\!
  38. δ \delta\,\!
  39. x x\,\!
  40. x 𝒳 x\in\mathcal{X}\,\!
  41. x x\,\!
  42. θ Θ \theta\in\Theta\,\!
  43. x x\,\!
  44. ρ ( π , δ x ) = E π ( θ x ) [ L ( θ , δ ( x ) ) ] . \rho(\pi,\delta\mid x)=\operatorname{E}_{\pi(\theta\mid x)}[L(\theta,\delta(x)% )].\,\!
  45. θ \theta\,\!
  46. x x\,\!
  47. π ( θ ) \pi(\theta)\,\!
  48. F ( x θ ) F(x\mid\theta)\,\!
  49. x x\,\!
  50. δ \delta\,\!
  51. x x\,\!
  52. δ ( x ) \delta(x)\,\!
  53. π ( θ ) \pi(\theta)\,\!
  54. δ ( x ) \delta(x)\,\!
  55. Θ \Theta\,\!
  56. 𝒳 \mathcal{X}
  57. x θ x\sim\theta\,\!
  58. θ π \theta\sim\pi\,\!
  59. δ \delta\,\!
  60. x 𝒳 x\in\mathcal{X}
  61. x x\,\!
  62. δ \delta\,\!
  63. δ \delta\,\!
  64. δ ( x ) \delta(x)\!\,
  65. x x\,\!
  66. δ ( x ) \delta(x)\,\!
  67. x x\,\!
  68. X 𝒳 X\subseteq\mathcal{X}
  69. π ( θ ) \pi(\theta)\,\!
  70. x x\,\!
  71. π ( θ x ) \pi(\theta\mid x)\,\!
  72. x x\,\!
  73. π ( θ ) \pi(\theta)\,\!
  74. θ \theta\,\!

Advanced_Z-transform.html

  1. F ( z , m ) = k = 0 f ( k T + m ) z - k F(z,m)=\sum_{k=0}^{\infty}f(kT+m)z^{-k}
  2. [ 0 , T ) . [0,T).
  3. 𝒵 { k = 1 n c k f k ( t ) } = k = 1 n c k F ( z , m ) . \mathcal{Z}\left\{\sum_{k=1}^{n}c_{k}f_{k}(t)\right\}=\sum_{k=1}^{n}c_{k}F(z,m).
  4. 𝒵 { u ( t - n T ) f ( t - n T ) } = z - n F ( z , m ) . \mathcal{Z}\left\{u(t-nT)f(t-nT)\right\}=z^{-n}F(z,m).
  5. 𝒵 { f ( t ) e - a t } = e - a m F ( e a T z , m ) . \mathcal{Z}\left\{f(t)e^{-a\,t}\right\}=e^{-a\,m}F(e^{a\,T}z,m).
  6. 𝒵 { t y f ( t ) } = ( - T z d d z + m ) y F ( z , m ) . \mathcal{Z}\left\{t^{y}f(t)\right\}=\left(-Tz\frac{d}{dz}+m\right)^{y}F(z,m).
  7. lim k f ( k T + m ) = lim z 1 ( 1 - z - 1 ) F ( z , m ) . \lim_{k\to\infty}f(kT+m)=\lim_{z\to 1}(1-z^{-1})F(z,m).
  8. f ( t ) = cos ( ω t ) f(t)=\cos(\omega t)
  9. F ( z , m ) = \displaystyle F(z,m)=
  10. m = 0 m=0
  11. F ( z , m ) F(z,m)
  12. F ( z , 0 ) = z 2 - z cos ( ω T ) z 2 - 2 z cos ( ω T ) + 1 F(z,0)=\frac{z^{2}-z\cos(\omega T)}{z^{2}-2z\cos(\omega T)+1}
  13. f ( t ) f(t)

Aether_drag_hypothesis.html

  1. v n v_{n}
  2. v n = c n v_{n}=\frac{c}{n}
  3. v d v_{d}
  4. v d = v ( 1 - ρ e ρ g ) v_{d}=v(1-\frac{\rho_{e}}{\rho_{g}})
  5. ρ e \rho_{e}
  6. ρ g \rho_{g}
  7. v v
  8. ( 1 - ρ e ρ g ) (1-\frac{\rho_{e}}{\rho_{g}})
  9. ( 1 - 1 n 2 ) (1-\frac{1}{n^{2}})
  10. V = c n + v ( 1 - 1 n 2 ) V=\frac{c}{n}+v(1-\frac{1}{n^{2}})
  11. α \alpha
  12. tan ( α ) = v δ t c δ t . \tan(\alpha)=\frac{v\delta t}{c\delta t}.
  13. tan ( α ) = v c \tan(\alpha)=\frac{v}{c}
  14. α \alpha
  15. 1 - v 2 / c 2 \sqrt{1-v^{2}/c^{2}}
  16. V V
  17. U U
  18. v v
  19. U = c n U=\frac{c}{n}
  20. V = c / n + v 1 + v / n c V=\frac{c/n+v}{1+v/nc}
  21. V c n + v ( 1 - 1 n 2 ) V\approx\frac{c}{n}+v\left(1-\frac{1}{n^{2}}\right)

AEX_index.html

  1. I t = i = 1 N Q i , t F i , t f i , t C i , t d t I_{t}=\frac{\sum_{i=1}^{N}Q_{i,t}\,F_{i,t}\,f_{i,t}\,C_{i,t}\,}{d_{t}\,}

Affine_Lie_algebra.html

  1. 𝔤 \mathfrak{g}
  2. L 𝔤 L\mathfrak{g}
  3. 𝔤 \mathfrak{g}
  4. 𝔤 ^ \hat{\mathfrak{g}}
  5. 𝔤 \mathfrak{g}
  6. L σ 𝔤 L_{\sigma}\mathfrak{g}
  7. 𝔤 \mathfrak{g}
  8. 𝔤 \mathfrak{g}
  9. 𝔤 ^ \hat{\mathfrak{g}}
  10. 𝔤 [ t , t - 1 ] \mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]
  11. c . \mathbb{C}c.
  12. 𝔤 ^ = 𝔤 [ t , t - 1 ] c , \widehat{\mathfrak{g}}=\mathfrak{g}\otimes\mathbb{C}[t,t^{-1}]\oplus\mathbb{C}c,
  13. [ t , t - 1 ] \mathbb{C}[t,t^{-1}]
  14. [ a t n + α c , b t m + β c ] = [ a , b ] t n + m + a | b n δ m + n , 0 c [a\otimes t^{n}+\alpha c,b\otimes t^{m}+\beta c]=[a,b]\otimes t^{n+m}+\langle a% |b\rangle n\delta_{m+n,0}c
  15. a , b 𝔤 , α , β a,b\in\mathfrak{g},\alpha,\beta\in\mathbb{C}
  16. n , m n,m\in\mathbb{Z}
  17. [ a , b ] [a,b]
  18. 𝔤 \mathfrak{g}
  19. | \langle\cdot|\cdot\rangle
  20. 𝔤 . \mathfrak{g}.
  21. δ ( a t m + α c ) = t d d t ( a t m ) . \delta(a\otimes t^{m}+\alpha c)=t{d\over dt}(a\otimes t^{m}).
  22. n \mathbb{C}^{n}

Affine_pricing.html

  1. T = p * q + k T=p*q+k

Afshar_experiment.html

  1. 𝒪 \mathcal{O}

Aggregate_expenditure.html

  1. - - A E = C + I --~{}~{}~{}~{}AE=C+I
  2. - - A E = C + I + G + N X --~{}~{}~{}~{}AE=C+I+G+NX

Air_quality_index.html

  1. 10 {}_{10}
  2. 2.5 {}_{2.5}
  3. 2 {}_{2}
  4. 2 {}_{2}
  5. 3 {}_{3}
  6. 3 {}_{3}
  7. 10 {}_{10}
  8. 2.5 {}_{2.5}
  9. 2 {}_{2}
  10. 3 {}_{3}
  11. 2 {}_{2}
  12. 3 {}_{3}
  13. I = I h i g h - I l o w C h i g h - C l o w ( C - C l o w ) + I l o w I=\frac{I_{high}-I_{low}}{C_{high}-C_{low}}(C-C_{low})+I_{low}
  14. I I
  15. C C
  16. C l o w C_{low}
  17. C C
  18. C h i g h C_{high}
  19. C C
  20. I l o w I_{low}
  21. C l o w C_{low}
  22. I h i g h I_{high}
  23. C h i g h C_{high}
  24. 50 - 0 12.0 - 0 ( 12.0 - 0 ) + 0 = 50 \frac{50-0}{12.0-0}(12.0-0)+0=50

Air_shower_(physics).html

  1. π 0 \scriptstyle\pi^{0}
  2. γ \scriptstyle\gamma
  3. π 0 γ + γ \scriptstyle\pi^{0}\rightarrow\gamma+\gamma
  4. π ± \scriptstyle\pi^{\pm}
  5. π + μ + + ν \scriptstyle\pi^{+}\rightarrow\mu^{+}+\nu
  6. π - μ - + ν \scriptstyle\pi^{-}\rightarrow\mu^{-}+\nu
  7. K + / - μ + / - + ν \scriptstyle K^{+/-}\rightarrow\mu^{+/-}+\nu
  8. K + / - π + / - + π 0 \scriptstyle K^{+/-}\rightarrow\pi^{+/-}+\pi^{0}

Airglow.html

  1. S 0 ( V ) = 4.0 × 10 - 12 S_{0}(V)=4.0\times 10^{-12}
  2. B = 0.2 B=0.2
  3. ν 6 × 10 14 \nu\sim 6\times 10^{14}
  4. N s N_{s}
  5. N s = 10 - 28 / 2.5 × S 0 ( V ) × B h ν N_{s}=10^{-28/2.5}\times\frac{S_{0}(V)\times B}{h\nu}
  6. h h
  7. h ν h\nu
  8. ν \nu
  9. N a N_{a}
  10. N a = 10 - 23 / 2.5 × S 0 ( V ) × B h ν N_{a}=10^{-23/2.5}\times\frac{S_{0}(V)\times B}{h\nu}
  11. A A
  12. S / N = A × N s N s + N a S/N=\sqrt{A}\times\frac{N_{s}}{\sqrt{N_{s}+N_{a}}}
  13. 1.3 × 10 7 ± 3500 1.3\times 10^{7}\pm 3500
  14. 1.3 × 10 5 1.3\times 10^{5}
  15. 1.3 × 10 5 1.3 × 10 7 = 36 \frac{1.3\times 10^{5}}{\sqrt{1.3\times 10^{7}}}=36

Air–fuel_ratio.html

  1. 25 2 O 2 + C 8 H 18 8 C O 2 + 9 H 2 O \frac{25}{2}O_{2}+C_{8}H_{18}\to 8CO_{2}+9H_{2}O
  2. A F R = m a i r m f u e l AFR=\frac{m_{air}}{m_{fuel}}
  3. m f u e l m_{fuel}
  4. F A R = 1 A F R FAR=\frac{1}{AFR}
  5. λ = A F R A F R s t o i c h \lambda=\frac{AFR}{AFR_{stoich}}
  6. ϕ = fuel-to-oxidizer ratio ( fuel-to-oxidizer ratio ) s t = m f u e l / m o x ( m f u e l / m o x ) s t = n f u e l / n o x ( n f u e l / n o x ) s t \phi=\frac{\mbox{fuel-to-oxidizer ratio}~{}}{(\mbox{fuel-to-oxidizer ratio}~{}% )_{st}}=\frac{m_{fuel}/m_{ox}}{(m_{fuel}/m_{ox})_{st}}=\frac{n_{fuel}/n_{ox}}{% (n_{fuel}/n_{ox})_{st}}
  7. m C 2 H 6 m O 2 = 1 ( 2 12 + 6 1 ) 1 ( 2 16 ) = 30 32 = 0.9375 \frac{m_{\rm C_{2}H_{6}}}{m_{\rm O_{2}}}=\frac{1\cdot(2\cdot 12+6\cdot 1)}{1% \cdot(2\cdot 16)}=\frac{30}{32}=0.9375
  8. n C 2 H 6 n O 2 = 1 1 = 1 \frac{n_{\rm C_{2}H_{6}}}{n_{\rm O_{2}}}=\tfrac{1}{1}=1
  9. C 2 H 6 + 7 2 O 2 2 C O 2 + 3 H 2 O {\rm C_{2}H_{6}}+\tfrac{7}{2}{\rm O_{2}}\rightarrow 2{\rm CO_{2}}+3{\rm H_{2}O}
  10. ( fuel-to-oxidizer ratio based on mass ) s t = ( m C 2 H 6 m O 2 ) s t = 1 ( 2 12 + 6 1 ) 3.5 ( 2 16 ) = 30 112 = 0.268 (\mbox{fuel-to-oxidizer ratio based on mass}~{})_{st}=\left(\frac{m_{\rm C_{2}% H_{6}}}{m_{\rm O_{2}}}\right)_{st}=\frac{1\cdot(2\cdot 12+6\cdot 1)}{3.5\cdot(% 2\cdot 16)}=\frac{30}{112}=0.268
  11. ( fuel-to-oxidizer ratio based on number of moles ) s t = ( n C 2 H 6 n O 2 ) s t = 1 3.5 = 0.286 (\mbox{fuel-to-oxidizer ratio based on number of moles}~{})_{st}=\left(\frac{n% _{\rm C_{2}H_{6}}}{n_{\rm O_{2}}}\right)_{st}=\tfrac{1}{3.5}=0.286
  12. ϕ = m C 2 H 6 / m O 2 ( m C 2 H 6 / m O 2 ) s t = 0.938 0.268 = 3.5 \phi=\frac{m_{\rm C_{2}H_{6}}/m_{\rm O_{2}}}{(m_{\rm C_{2}H_{6}}/m_{\rm O_{2}}% )_{st}}=\tfrac{0.938}{0.268}=3.5
  13. ϕ = n C 2 H 6 / n O 2 ( n C 2 H 6 / n O 2 ) s t = 1 0.286 = 3.5 \phi=\frac{n_{\rm C_{2}H_{6}}/n_{\rm O_{2}}}{(n_{\rm C_{2}H_{6}}/n_{\rm O_{2}}% )_{st}}=\tfrac{1}{0.286}=3.5
  14. ϕ = 1 λ \phi=\frac{1}{\lambda}
  15. Z = [ s Y F - Y O + Y O , 0 s Y F , 0 + Y O , 0 ] Z=\left[\frac{sY_{F}-Y_{O}+Y_{O,0}}{sY_{F,0}+Y_{O,0}}\right]
  16. s = A F R s t o i c h = W O v O W F v F s=AFR_{stoich}=\frac{W_{O}\cdot v_{O}}{W_{F}\cdot v_{F}}
  17. Y F , 0 Y_{F,0}
  18. Y O , 0 Y_{O,0}
  19. W F W_{F}
  20. W O W_{O}
  21. v F v_{F}
  22. v O v_{O}
  23. Z s t = [ 1 1 + Y F , 0 W O v O Y O , 0 W F v F ] Z_{st}=\left[\frac{1}{1+\frac{Y_{F,0}\cdot W_{O}\cdot v_{O}}{Y_{O,0}\cdot W_{F% }\cdot v_{F}}}\right]
  24. λ \lambda
  25. ϕ \phi
  26. Z s t = λ 1 + λ = 1 1 + ϕ Z_{st}=\frac{\lambda}{1+\lambda}=\frac{1}{1+\phi}
  27. A F R = Y O , 0 Y F , 0 AFR=\frac{Y_{O,0}}{Y_{F,0}}
  28. Mass % O 2 in propane combustion gas = - 0.1433 ( % excess O 2 ) 2 + 0.214 ( % excess O 2 ) \mathrm{Mass\%\ O_{2}\ in\ propane\ combustion\ gas}=-0.1433(\mathrm{\%\ % excess\ O_{2}})^{2}+0.214(\mathrm{\%\ excess\ O_{2}})
  29. Volume % O 2 in propane combustion gas = - 0.1208 ( % excess O 2 ) 2 + 0.186 ( % excess O 2 ) \mathrm{Volume\%\ O_{2}\ in\ propane\ combustion\ gas}=-0.1208(\mathrm{\%\ % excess\ O_{2}})^{2}+0.186(\mathrm{\%\ excess\ O_{2}})

Al-Mahani.html

  1. x 3 + c 2 b = c x 2 x^{3}+c^{2}b=cx^{2}

Albanese_variety.html

  1. V Alb ( V ) V\to\operatorname{Alb}(V)
  2. H 1 ( X , O X ) H^{1}(X,O_{X})
  3. dim X h 1 , 0 \dim X\leq h^{1,0}
  4. Alb V = ( Pic 0 V ) . \operatorname{Alb}\,V=(\operatorname{Pic}_{0}\,V)^{\vee}.

Albert_Einstein_Memorial.html

  1. R μ ν - 1 2 g μ ν R = κ T μ ν R_{\mu\nu}-{1\over 2}g_{\mu\nu}R=\kappa T_{\mu\nu}
  2. e V = h ν - A eV=h\nu-A\,
  3. E = m c 2 E=mc^{2}\,

Algebraic_graph_theory.html

  1. S 5 S_{5}
  2. t ( t - 1 ) ( t - 2 ) ( t 7 - 12 t 6 + 67 t 5 - 230 t 4 + 529 t 3 - 814 t 2 + 775 t - 352 ) t(t-1)(t-2)(t^{7}-12t^{6}+67t^{5}-230t^{4}+529t^{3}-814t^{2}+775t-352)

Algebraic_solution.html

  1. x = - b ± b 2 - 4 a c 2 a , x=\frac{-b\pm\sqrt{b^{2}-4ac\ }}{2a},
  2. a x 2 + b x + c = 0 ax^{2}+bx+c=0\,
  3. x 10 = a x^{10}=a
  4. x = a 1 / 10 . x=a^{1/10}.

All-pairs_testing.html

  1. N N
  2. { P i } = { P 1 , P 2 , , P N } \{P_{i}\}=\{P_{1},P_{2},...,P_{N}\}
  3. R ( P i ) = R i R(P_{i})=R_{i}
  4. | R i | = n i |R_{i}|=n_{i}
  5. P ( X , Y , Z ) P(X,Y,Z)
  6. p ( u , v ) p(u,v)
  7. P ( X , Y , Z ) P(X,Y,Z)
  8. p x y ( X , Y ) , p y z ( Y , Z ) , p z x ( Z , X ) p_{xy}(X,Y),p_{yz}(Y,Z),p_{zx}(Z,X)
  9. X = { n i } X=\{n_{i}\}
  10. m a x ( S ) max(S)
  11. S S
  12. T = m a x ( X ) × m a x ( X m a x ( X ) ) T=max(X)\times max(X\setminus max(X))
  13. n = m a x ( X ) n=max(X)
  14. m = m a x ( X m a x ( X ) ) m=max(X\setminus max(X))
  15. n i \prod n_{i}
  16. X = { n i } X=\{n_{i}\}
  17. P = { P i } P=\{P_{i}\}
  18. N N
  19. P s = < P i > ; i < j | R ( P i ) | < | R ( P j ) | P_{s}=<P_{i}>\;;\;i<j\implies|R(P_{i})|<|R(P_{j})|
  20. X ( 2 ) = { P N - 1 , P N - 2 } X(2)=\{P_{N-1},P_{N-2}\}
  21. X ( 3 ) = { P N - 1 , P N - 2 , P N - 3 } X(3)=\{P_{N-1},P_{N-2},P_{N-3}\}
  22. X ( T ) = { P N - 1 , P N - 2 , , P N - T } X(T)=\{P_{N-1},P_{N-2},...,P_{N-T}\}

Allan_Joseph_Champneys_Cunningham.html

  1. 2 p - 1 2^{p}-1
  2. 2 2 n + 1 2^{2^{n}}+1

Allee_effect.html

  1. d N d t = r N ( N A - 1 ) ( 1 - N K ) , \frac{dN}{dt}=rN\left(\frac{N}{A}-1\right)\left(1-\frac{N}{K}\right),
  2. 0 < N < A 0<N<A
  3. A < N < K A<N<K
  4. 0 < A < K 0<A<K
  5. d N d t = r N ( 1 - N K ) \frac{dN}{dt}=rN\left(1-\frac{N}{K}\right)
  6. N t = D 2 N x 2 + r N ( N A - 1 ) ( 1 - N K ) , \frac{\partial N}{\partial t}=D\frac{\partial^{2}N}{\partial x^{2}}+rN\left(% \frac{N}{A}-1\right)\left(1-\frac{N}{K}\right),
  7. 2 x 2 \frac{\partial^{2}}{\partial x^{2}}

Allen_Telescope_Array.html

  1. N N
  2. B B
  3. N N
  4. F F
  5. 2 B N log 2 ( F ) ( 10 O P s ) + ( N N + 1 2 ) × 4 ( 8 O P s ) 2B\langle N\log_{2}(F)(10OPs)+(N\frac{N+1}{2})\times 4(8OPs)\rangle
  6. O p s Ops
  7. 2 B 2B

Allometry.html

  1. y = k x a y=kx^{a}\,\!
  2. log y = a log x + log k \log y=a\log x+\log k\,\!
  3. a a
  4. - 4 {}^{-4}
  5. - 1 {}^{-1}
  6. - 2 {}^{-2}
  7. + 1 {}^{+1}
  8. - 1 {}^{-1}
  9. + 2 {}^{+2}
  10. - 2 {}^{-2}
  11. + 2 {}^{+2}
  12. - 5 {}^{-5}
  13. 0 {}^{0}
  14. Metabolic Rate = 70 M 0.75 \mathrm{Metabolic\ Rate}=70M^{0.75}
  15. M M
  16. freq = 1 mass 1 / 3 \mathrm{freq}=\frac{1}{\mathrm{mass}^{1/3}}
  17. k leg = peak force peak displacement \mathrm{k_{leg}}=\frac{\mathrm{peak\ force}}{\mathrm{peak\ displacement}}
  18. M 0.67 M^{0.67}
  19. M M
  20. M 0.97 M^{0.97}
  21. M 0.30 M^{0.30}
  22. M - 0.034 M^{-0.034}
  23. M 0.11 M^{0.11}
  24. q 0 q_{0}
  25. M M
  26. 3 / 4 3/4
  27. q 0 M 3 4 q_{0}\sim M^{\frac{3}{4}}
  28. t t
  29. M M
  30. 1 / 4 1/4
  31. t M 1 4 t\sim M^{\frac{1}{4}}
  32. A A
  33. M M
  34. A M 7 8 A\sim M^{\frac{7}{8}}
  35. V o p t V_{opt}
  36. M M
  37. 1 / 6 1/6
  38. V o p t 30 M 1 6 [ m s - 1 ] V_{opt}\sim 30\cdot M^{\frac{1}{6}}[m\cdot s^{-1}]

Alpha_(finance).html

  1. α i \alpha_{i}
  2. SCL : R i , t - R f = α i + β i ( R M , t - R f ) + ϵ i , t \mathrm{SCL}:R_{i,t}-R_{f}=\alpha_{i}+\beta_{i}\,(R_{M,t}-R_{f})+\epsilon_{i,t% }\frac{}{}
  3. α i < 0 \alpha_{i}<0
  4. α i = 0 \alpha_{i}=0
  5. α i > 0 \alpha_{i}>0

Alpha_Cassiopeiae.html

  1. δ = d S D S \begin{smallmatrix}{\delta}=\frac{d_{S}}{D_{S}}\end{smallmatrix}
  2. δ {\delta}
  3. d S {d_{S}}
  4. D S {D_{S}}
  5. d S {d_{S}}
  6. d S = δ D S = 0.00562 70.0 = 0.393 A U \begin{smallmatrix}d_{S}=\delta\cdot D_{S}={0.00562}\cdot 70.0=0.393AU\end{smallmatrix}
  7. R S = ( d S 2 ) = ( 0.393 2 ) = 0.197 A U \begin{smallmatrix}R_{S}={\left({\frac{d_{S}}{2}}\right)}={\left({\frac{0.393}% {2}}\right)}=0.197AU\end{smallmatrix}
  8. d S = ( 0.197 A U ) ( 149 , 597 , 871 k m 696 , 000 k m ) = 42.31 R ( r o u n d e d ) \begin{smallmatrix}d_{S}={\left(0.197AU\right)}{\left({\frac{149,597,871km}{69% 6,000km}}\right)}=42.31R_{\odot}(rounded)\end{smallmatrix}
  9. L S L = ( R S R ) 2 ( T S T ) 4 \begin{smallmatrix}\frac{L_{\rm S}}{L_{\odot}}={\left(\frac{R_{\rm S}}{R_{% \odot}}\right)}^{2}{\left(\frac{T_{\rm S}}{T_{\odot}}\right)}^{4}\end{smallmatrix}
  10. L S L = ( 42.3 1 ) 2 ( 4 , 530 5 , 778 ) 4 = 676 L \begin{smallmatrix}\frac{L_{\rm S}}{L_{\odot}}={\left({\frac{42.3}{1}}\right)}% ^{2}{\left({\frac{4,530}{5,778}}\right)}^{4}=676L_{\odot}\end{smallmatrix}

Alternating_series_test.html

  1. n = 1 ( - 1 ) n - 1 a n = a 1 - a 2 + a 3 - \sum_{n=1}^{\infty}(-1)^{n-1}a_{n}=a_{1}-a_{2}+a_{3}-\cdots\!
  2. n = 1 ( - 1 ) n a n = - a 1 + a 2 - a 3 + \sum_{n=1}^{\infty}(-1)^{n}a_{n}=-a_{1}+a_{2}-a_{3}+\cdots\!
  3. a n a_{n}
  4. lim n a n = 0 \lim_{n\to\infty}a_{n}=0
  5. S k = n = 1 k ( - 1 ) n - 1 a n S_{k}=\sum_{n=1}^{k}(-1)^{n-1}a_{n}\!
  6. | S k - L | | S k - S k + 1 | = a k + 1 . \left|S_{k}-L\right|\leq\left|S_{k}-S_{k+1}\right|=a_{k+1}.\!
  7. n = 1 ( - 1 ) n - 1 a n \sum_{n=1}^{\infty}(-1)^{n-1}a_{n}\!
  8. lim n a n = 0 \lim_{n\rightarrow\infty}a_{n}=0
  9. a n a n + 1 a_{n}\geq a_{n+1}
  10. n = 1 ( - 1 ) n a n \sum_{n=1}^{\infty}(-1)^{n}a_{n}\!
  11. S 2 m + 1 = n = 1 2 m + 1 ( - 1 ) n - 1 a n S_{2m+1}=\sum_{n=1}^{2m+1}(-1)^{n-1}a_{n}
  12. S 2 m = n = 1 2 m ( - 1 ) n - 1 a n S_{2m}=\sum_{n=1}^{2m}(-1)^{n-1}a_{n}
  13. S k = n = 1 k ( - 1 ) n - 1 a n S_{k}=\sum_{n=1}^{k}(-1)^{n-1}a_{n}
  14. S 2 ( m + 1 ) + 1 = S 2 m + 1 - a 2 m + 2 + a 2 m + 3 S 2 m + 1 S_{2(m+1)+1}=S_{2m+1}-a_{2m+2}+a_{2m+3}\leq S_{2m+1}
  15. S 2 ( m + 1 ) = S 2 m + a 2 m + 1 - a 2 m + 2 S 2 m S_{2(m+1)}=S_{2m}+a_{2m+1}-a_{2m+2}\geq S_{2m}
  16. S 2 m + 1 - S 2 m = a 2 m + 1 0 S_{2m+1}-S_{2m}=a_{2m+1}\geq 0
  17. a 1 - a 2 = S 2 S 2 m < S 2 m + 1 S 1 = a 1 . a_{1}-a_{2}=S_{2}\leq S_{2m}<S_{2m+1}\leq S_{1}=a_{1}.
  18. lim n S 2 m + 1 - S 2 m = lim n a 2 m + 1 = 0. \lim_{n\to\infty}S_{2m+1}-S_{2m}=\lim_{n\to\infty}a_{2m+1}=0.
  19. S 2 m L S 2 m + 1 S_{2m}\leq L\leq S_{2m+1}
  20. | S k - L | a k + 1 \left|S_{k}-L\right|\leq a_{k+1}\!
  21. | S 2 m + 1 - L | = S 2 m + 1 - L S 2 m + 1 - S 2 m + 2 = a ( 2 m + 1 ) + 1 \left|S_{2m+1}-L\right|=S_{2m+1}-L\leq S_{2m+1}-S_{2m+2}=a_{(2m+1)+1}
  22. | S 2 m - L | = L - S 2 m S 2 m + 1 - S 2 m = a 2 m + 1 \left|S_{2m}-L\right|=L-S_{2m}\leq S_{2m+1}-S_{2m}=a_{2m+1}

Alternating_sign_matrix.html

  1. [ 0 0 1 0 1 0 0 0 0 1 - 1 1 0 0 1 0 ] . \begin{bmatrix}0&0&1&0\\ 1&0&0&0\\ 0&1&-1&1\\ 0&0&1&0\end{bmatrix}.
  2. n × n n\times n
  3. k = 0 n - 1 ( 3 k + 1 ) ! ( n + k ) ! = 1 ! 4 ! 7 ! ( 3 n - 2 ) ! n ! ( n + 1 ) ! ( 2 n - 1 ) ! . \prod_{k=0}^{n-1}\frac{(3k+1)!}{(n+k)!}=\frac{1!4!7!\cdots(3n-2)!}{n!(n+1)!% \cdots(2n-1)!}.
  4. 1 , 2 , 7 , 42 , 429 , 7436 , 1,2,7,42,429,7436,\cdots

Ambient_occlusion.html

  1. A p ¯ A_{\bar{p}}
  2. p ¯ \bar{p}
  3. n ^ \hat{n}
  4. Ω \Omega
  5. A p ¯ = 1 π Ω V p ¯ , ω ^ ( n ^ ω ^ ) d ω A_{\bar{p}}=\frac{1}{\pi}\int_{\Omega}V_{\bar{p},\hat{\omega}}(\hat{n}\cdot% \hat{\omega})\,\operatorname{d}\omega
  6. V p ¯ , ω ^ V_{\bar{p},\hat{\omega}}
  7. p ¯ \bar{p}
  8. p ¯ \bar{p}
  9. ω ^ \hat{\omega}
  10. d ω \operatorname{d}\omega
  11. ω ^ \hat{\omega}
  12. p ¯ \bar{p}
  13. p ¯ \bar{p}
  14. n ^ b \hat{n}_{b}

Ambipolar_diffusion.html

  1. v e k B T e / m e v_{e}\approx\sqrt{k_{B}T_{e}/m_{e}}
  2. c s k B T e / m i c_{s}\approx\sqrt{k_{B}T_{e}/m_{i}}

Amenable_number.html

  1. n = i = 1 n a i = i = 1 n a i n=\sum_{i=1}^{n}a_{i}=\prod_{i=1}^{n}a_{i}

Amine_gas_treating.html

  1. \Leftrightarrow

Amortization_(business).html

  1. P = A 1 - ( 1 1 + r ) n r P\,=\,A\cdot\frac{1-\left(\frac{1}{1+r}\right)^{n}}{r}
  2. A = P r ( 1 + r ) n ( 1 + r ) n - 1 A\,=\,P\cdot\frac{r(1+r)^{n}}{(1+r)^{n}-1}

Amplitude_versus_offset.html

  1. R = Z 1 - Z 0 Z 1 + Z 0 R=\frac{Z_{1}-Z_{0}}{Z_{1}+Z_{0}}
  2. Z 0 Z_{0}
  3. Z 1 Z_{1}
  4. R ( θ ) = R ( 0 ) + G sin 2 θ + F ( tan 2 θ sin 2 θ ) R(\theta)=R(0)+G\sin^{2}\theta+F(\tan^{2}\theta\sin^{2}\theta)
  5. R ( 0 ) = 1 2 ( Δ V P V P + Δ ρ ρ ) R(0)=\frac{1}{2}\left(\frac{\Delta V_{\mathrm{P}}}{V_{\mathrm{P}}}+\frac{% \Delta\rho}{\rho}\right)
  6. G = 1 2 Δ V P V P - 2 V S 2 V P 2 ( Δ ρ ρ + 2 Δ V S V S ) G=\frac{1}{2}\frac{\Delta V_{\mathrm{P}}}{V_{\mathrm{P}}}-2\frac{V^{2}_{% \mathrm{S}}}{V^{2}_{\mathrm{P}}}\left(\frac{\Delta\rho}{\rho}+2\frac{\Delta V_% {\mathrm{S}}}{V_{\mathrm{S}}}\right)
  7. F = 1 2 Δ V P V P F=\frac{1}{2}\frac{\Delta V_{\mathrm{P}}}{V_{\mathrm{P}}}
  8. θ {\theta}
  9. V p {V_{p}}
  10. Δ V p {{\Delta}V_{p}}
  11. V s {V_{s}}
  12. Δ V s {{\Delta}V_{s}}
  13. < m t p l > ρ <mtpl>{{\rho}}
  14. Δ ρ {{\Delta}{\rho}}
  15. R ( θ ) = R ( 0 ) + G sin 2 θ R(\theta)=R(0)+G\sin^{2}\theta

Anaplerotic_reactions.html

  1. \longrightarrow
  2. \longrightarrow

AND_gate.html

  1. C = A B C=A\cdot B

Andalusian_cadence.html

  1. 4 - 3 5 {}^{5}_{4-3}
  2. 4 - 3 5 {}^{5}_{4-3}

Anderson_Gray_McKendrick.html

  1. n t + n a = - μ ( t , a ) n . \frac{\partial n}{\partial t}+\frac{\partial n}{\partial a}=-\mu(t,a)n.

Angle_of_parallelism.html

  1. Π ( a ) \Pi(a)
  2. Π ( a ) \Pi(a)
  3. lim a 0 Π ( a ) = 1 2 π and lim a Π ( a ) = 0. \lim_{a\to 0}\Pi(a)=\tfrac{1}{2}\pi\quad\,\text{ and }\quad\lim_{a\to\infty}% \Pi(a)=0.
  4. Π ( a ) \Pi(a)
  5. sin Π ( a ) = sech a = 1 cosh a , \sin\Pi(a)=\operatorname{sech}a=\frac{1}{\cosh a},
  6. cos Π ( a ) = tanh a , \cos\Pi(a)=\tanh a,
  7. tan Π ( a ) = csch a = 1 sinh a , \tan\Pi(a)=\operatorname{csch}a=\frac{1}{\sinh a},
  8. tan ( 1 2 Π ( a ) ) = e - a , \tan(\tfrac{1}{2}\Pi(a))=e^{-a},
  9. Π ( a ) = 1 2 π - gd ( a ) , \Pi(a)=\tfrac{1}{2}\pi-\operatorname{gd}(a),
  10. x = 1 2 ( 1 - y 2 ) . x=\tfrac{1}{2}(1-y^{2}).
  11. tan ϕ = y - x = 2 y y 2 - 1 = 2 e a e 2 a - 1 = 1 sinh a . \tan\phi=\frac{y}{-x}=\frac{2y}{y^{2}-1}=\frac{2e^{a}}{e^{2a}-1}=\frac{1}{% \sinh a}.

Angular_mil.html

  1. sin θ θ \sin\theta\simeq\theta
  2. D = S m i l D=\frac{S}{mil}
  3. D = S m i l m a g 10 D=\frac{S}{mil}\cdot\frac{mag}{10}
  4. 1 / 6400 {1}/{6400}
  5. 1 / 6283 {1}/{6283}
  6. 1 / 6000 {1}/{6000}
  7. 1 / 6300 {1}/{6300}

Angular_momentum_coupling.html

  1. 𝐉 = 𝐋 + 𝐒 , \mathbf{J}=\mathbf{L}+\mathbf{S},\,
  2. 𝐋 = i s y m b o l i , 𝐒 = i 𝐬 i . \mathbf{L}=\sum_{i}symbol{\ell}_{i},\ \mathbf{S}=\sum_{i}\mathbf{s}_{i}.\,
  3. 𝐉 = i 𝐣 i = i ( s y m b o l i + 𝐬 i ) . \mathbf{J}=\sum_{i}\mathbf{j}_{i}=\sum_{i}(symbol{\ell}_{i}+\mathbf{s}_{i}).

Antenna_aperture.html

  1. A e f f = P o P F D A_{eff}=\frac{P_{o}}{PFD}\,
  2. e a = A e f f A p h y s e_{a}=\frac{A_{eff}}{A_{phys}}\,
  3. A e f f = λ 2 4 π A_{eff}=\frac{\lambda^{2}}{4\pi}\,
  4. G = 4 π A e f f λ 2 = 4 π A p h y s e a λ 2 G=\frac{4\pi A_{eff}}{\lambda^{2}}=\frac{4\pi A_{phys}e_{a}}{\lambda^{2}}\,
  5. P r = A t A r r 2 λ 2 P t P_{r}=\frac{A_{t}A_{r}}{r^{2}\lambda^{2}}P_{t}\,
  6. λ \lambda
  7. λ \lambda
  8. λ \lambda
  9. l e f f = V 0 / E s l_{eff}=V_{0}/E_{s}\,

Anti-reflective_coating.html

  1. R = ( n 0 - n S n 0 + n S ) 2 R=\left(\frac{n_{0}-n_{S}}{n_{0}+n_{S}}\right)^{2}
  2. n 1 = n 0 n S n_{1}=\sqrt{n_{0}n_{S}}

Apéry's_theorem.html

  1. ζ ( 3 ) = n = 1 1 n 3 = 1 1 3 + 1 2 3 + 1 3 3 + = 1.2020569 \zeta(3)=\sum_{n=1}^{\infty}\frac{1}{n^{3}}=\frac{1}{1^{3}}+\frac{1}{2^{3}}+% \frac{1}{3^{3}}+\ldots=1.2020569\ldots
  2. 1 1 2 n + 1 2 2 n + 1 3 2 n + 1 4 2 n + = p q π 2 n \frac{1}{1^{2n}}+\frac{1}{2^{2n}}+\frac{1}{3^{2n}}+\frac{1}{4^{2n}}+\ldots=% \frac{p}{q}\pi^{2n}
  3. ζ ( 2 n ) = ( - 1 ) n + 1 B 2 n ( 2 π ) 2 n 2 ( 2 n ) ! \zeta(2n)=(-1)^{n+1}\frac{B_{2n}(2\pi)^{2n}}{2(2n)!}
  4. ζ ( 2 n + 1 ) π 2 n + 1 , \frac{\zeta(2n+1)}{\pi^{2n+1}},
  5. | ξ - p q | < c q 1 + δ \left|\xi-\frac{p}{q}\right|<\frac{c}{q^{1+\delta}}
  6. ζ ( 3 ) = 5 2 n = 1 ( - 1 ) n - 1 n 3 ( 2 n n ) . \zeta(3)=\frac{5}{2}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{3}{\left({{2n}% \atop{n}}\right)}}.
  7. c n , k = m = 1 n 1 m 3 + m = 1 k ( - 1 ) m - 1 2 m 3 ( n m ) ( n + m m ) . c_{n,k}=\sum_{m=1}^{n}\frac{1}{m^{3}}+\sum_{m=1}^{k}\frac{(-1)^{m-1}}{2m^{3}{% \left({{n}\atop{m}}\right)}{\left({{n+m}\atop{m}}\right)}}.
  8. a n = k = 0 n c n , k ( n k ) 2 ( n + k k ) 2 a_{n}=\sum_{k=0}^{n}c_{n,k}{\left({{n}\atop{k}}\right)}^{2}{\left({{n+k}\atop{% k}}\right)}^{2}
  9. b n = k = 0 n ( n k ) 2 ( n + k k ) 2 . b_{n}=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}^{2}{\left({{n+k}\atop{k}}% \right)}^{2}.
  10. P n ~ ( x ) \tilde{P_{n}}(x)
  11. 0 1 0 1 - log ( x y ) 1 - x y P n ~ ( x ) P n ~ ( y ) d x d y = A n + B n ζ ( 3 ) lcm [ 1 , , n ] 3 \int_{0}^{1}\int_{0}^{1}\frac{-\log(xy)}{1-xy}\tilde{P_{n}}(x)\tilde{P_{n}}(y)% dxdy=\frac{A_{n}+B_{n}\zeta(3)}{\operatorname{lcm}\left[1,\ldots,n\right]^{3}}
  12. 0 < 1 b | A n + B n ζ ( 3 ) | 4 ( 4 5 ) n , 0<\frac{1}{b}\leq\left|A_{n}+B_{n}\zeta(3)\right|\leq 4\left(\frac{4}{5}\right% )^{n},
  13. ζ ( 2 ) = 3 n = 1 1 n 2 ( 2 n n ) . \zeta(2)=3\sum_{n=1}^{\infty}\frac{1}{n^{2}{\left({{2n}\atop{n}}\right)}}.
  14. ζ ( 5 ) = ξ 5 n = 1 ( - 1 ) n - 1 n 5 ( 2 n n ) . \zeta(5)=\xi_{5}\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{5}{\left({{2n}\atop{n}% }\right)}}.

API_gravity.html

  1. API gravity = 141.5 SG - 131.5 \,\text{API gravity}=\frac{141.5}{\,\text{SG}}-131.5
  2. SG at 60 F = 141.5 API gravity + 131.5 \,\text{SG at}~{}60^{\circ}\,\text{F}=\frac{141.5}{\,\text{API gravity}+131.5}
  3. 141.5 1.0 - 131.5 = 10.0 API \frac{141.5}{1.0}-131.5=10.0^{\circ}{\,\text{API}}
  4. barrels of crude oil per metric ton = API gravity + 131.5 141.5 × 0.159 \,\text{barrels of crude oil per metric ton}=\frac{\,\text{API gravity}+131.5}% {141.5\times 0.159}
  5. SG oil = ρ oil ρ H 2 O \mbox{SG oil}~{}=\frac{\rho\text{oil}}{\rho_{\,\text{H}_{2}\,\text{O}}}

Apothem.html

  1. A = n s a 2 = p a 2 . A=\frac{nsa}{2}=\frac{pa}{2}.
  2. A = p a 2 = ( 2 π r ) r 2 = π r 2 A=\frac{pa}{2}=\frac{(2\pi r)r}{2}=\pi r^{2}
  3. a = s 2 tan ( 180 / n ) = R cos ( 180 / n ) . a=\frac{s}{2\tan(180/n)}=R\cos(180/n).
  4. a = 1 2 s / tan ( 180 n ) . a=\frac{1}{2}s/\tan\!\left(\frac{180}{n}\right).
  5. s = p n . s=\frac{p}{n}.

Appell_sequence.html

  1. d d x p n ( x ) = n p n - 1 ( x ) , {d\over dx}p_{n}(x)=np_{n-1}(x),
  2. d d x p n ( x ) = n p n - 1 ( x ) {d\over dx}p_{n}(x)=np_{n-1}(x)
  3. p n ( x ) = k = 0 n ( n k ) c k x n - k ; p_{n}(x)=\sum_{k=0}^{n}{n\choose k}c_{k}x^{n-k};
  4. p n ( x ) = ( k = 0 c k k ! D k ) x n , p_{n}(x)=\left(\sum_{k=0}^{\infty}{c_{k}\over k!}D^{k}\right)x^{n},
  5. D = d d x ; D={d\over dx};
  6. p n ( x + y ) = k = 0 n ( n k ) p k ( x ) y n - k . p_{n}(x+y)=\sum_{k=0}^{n}{n\choose k}p_{k}(x)y^{n-k}.
  7. p n ( x ) = ( k = 0 c k k ! D k ) x n = S x n , p_{n}(x)=\left(\sum_{k=0}^{\infty}{c_{k}\over k!}D^{k}\right)x^{n}=Sx^{n},
  8. T = S - 1 = ( k = 0 c k k ! D k ) - 1 = k = 1 a k k ! D k T=S^{-1}=\left(\sum_{k=0}^{\infty}{c_{k}\over k!}D^{k}\right)^{-1}=\sum_{k=1}^% {\infty}{a_{k}\over k!}D^{k}
  9. T p n ( x ) = x n . Tp_{n}(x)=x^{n}.\,
  10. log T = log ( k = 0 a k k ! D k ) \log T=\log\left(\sum_{k=0}^{\infty}{a_{k}\over k!}D^{k}\right)
  11. p n + 1 ( x ) = ( x - ( log T ) ) p n ( x ) . p_{n+1}(x)=(x-(\log T)^{\prime})p_{n}(x).\,
  12. p n ( x ) = k = 0 n a n , k x k and q n ( x ) = k = 0 n b n , k x k . p_{n}(x)=\sum_{k=0}^{n}a_{n,k}x^{k}\ \mbox{and}~{}\ q_{n}(x)=\sum_{k=0}^{n}b_{% n,k}x^{k}.
  13. ( p n q ) ( x ) = k = 0 n a n , k q k ( x ) = 0 k n a n , k b k , x (p_{n}\circ q)(x)=\sum_{k=0}^{n}a_{n,k}q_{k}(x)=\sum_{0\leq k\leq\ell\leq n}a_% {n,k}b_{k,\ell}x^{\ell}
  14. p n ( x ) = ( k = 0 c k k ! D k ) x n , p_{n}(x)=\left(\sum_{k=0}^{\infty}{c_{k}\over k!}D^{k}\right)x^{n},
  15. d d x p n ( x ) = p n - 1 ( x ) {d\over dx}p_{n}(x)=p_{n-1}(x)

Aptamer.html

  1. K d K_{d}
  2. k o f f k_{off}
  3. k o n k_{on}

APX.html

  1. f ( n ) f(n)
  2. n n
  3. f ( n ) f(n)
  4. f ( n ) f(n)
  5. f ( n ) f(n)
  6. c c
  7. f ( n ) f(n)
  8. f ( n ) f(n)
  9. f ( n ) f(n)
  10. f ( n ) f(n)
  11. f ( n ) f(n)
  12. O ( f ( n ) ) O(f(n))
  13. f ( n ) f(n)

Arc_elasticity.html

  1. E x , y = % change in x % change in y E_{x,y}=\frac{\%\mbox{ change in }~{}x}{\%\mbox{ change in }~{}y}
  2. % change in x = x 2 - x 1 ( x 2 + x 1 ) / 2 ; \%\mbox{ change in }~{}x=\frac{x_{2}-x_{1}}{(x_{2}+x_{1})/2};
  3. % change in y = y 2 - y 1 ( y 2 + y 1 ) / 2 . \%\mbox{ change in }~{}y=\frac{y_{2}-y_{1}}{(y_{2}+y_{1})/2}.
  4. E x , y = x y y x = ln x ln y E_{x,y}=\frac{\partial x}{\partial y}\cdot\frac{y}{x}=\frac{\partial\ln x}{% \partial\ln y}
  5. ( % change in Q ) / ( % change in P ) (\%\mbox{ change in }~{}Q)/(\%\mbox{ change in }~{}P)
  6. ( Q 1 , P 1 ) (Q_{1},P_{1})
  7. ( Q 2 , P 2 ) (Q_{2},P_{2})
  8. E p = Q 2 - Q 1 ( Q 1 + Q 2 ) / 2 P 2 - P 1 ( P 1 + P 2 ) / 2 . E_{p}=\frac{\frac{Q_{2}-Q_{1}}{(Q_{1}+Q_{2})/2}}{\frac{P_{2}-P_{1}}{(P_{1}+P_{% 2})/2}}.

Arc_length.html

  1. L ( C ) = sup a = t 0 < t 1 < < t n = b i = 0 n - 1 d ( f ( t i ) , f ( t i + 1 ) ) L(C)=\sup_{a=t_{0}<t_{1}<\cdots<t_{n}=b}\sum_{i=0}^{n-1}d(f(t_{i}),f(t_{i+1}))
  2. i = 0 n - 1 d ( f ( t i ) , f ( t i + 1 ) ) \sum_{i=0}^{n-1}d(f(t_{i}),f(t_{i+1}))
  3. i = 0 n - 1 d ( g ( u i ) , g ( u i + 1 ) ) \sum_{i=0}^{n-1}d(g(u_{i}),g(u_{i+1}))
  4. u i = S ( t i ) u_{i}=S(t_{i})
  5. f ( x ) = d y d x f^{\prime}(x)=\frac{dy}{dx}
  6. ( d s ) 2 = ( d x ) 2 + ( d y ) 2 (ds)^{2}=(dx)^{2}+(dy)^{2}
  7. ( d s ) 2 = ( d x ) 2 + ( d y ) 2 (ds)^{2}=(dx)^{2}+(dy)^{2}
  8. ( d s ) 2 ( d x ) 2 = 1 + ( d y ) 2 ( d x ) 2 \frac{(ds)^{2}}{(dx)^{2}}=1+\frac{(dy)^{2}}{(dx)^{2}}
  9. ( d s ) 2 ( d x ) 2 = 1 + ( d y ) 2 ( d x ) 2 \sqrt{\frac{(ds)^{2}}{(dx)^{2}}}=\sqrt{1+\frac{(dy)^{2}}{(dx)^{2}}}
  10. d s d x = 1 + ( d y ) 2 ( d x ) 2 \frac{ds}{dx}=\sqrt{1+\frac{(dy)^{2}}{(dx)^{2}}}
  11. d s = 1 + ( d y d x ) 2 d x ds=\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}dx
  12. s = a b 1 + [ f ( x ) ] 2 d x . s=\int_{a}^{b}\sqrt{1+[f^{\prime}(x)]^{2}}\,dx.
  13. s = a b [ X ( t ) ] 2 + [ Y ( t ) ] 2 d t . s=\int_{a}^{b}\sqrt{[X^{\prime}(t)]^{2}+[Y^{\prime}(t)]^{2}}\,dt.
  14. Δ x \Delta x
  15. Δ y \Delta y
  16. s = lim a b ( Δ x ) 2 + ( Δ y ) 2 = a b ( d x ) 2 + ( d y ) 2 = a b ( d x d t ) 2 + ( d y d t ) 2 d t . s=\lim\sum_{a}^{b}\sqrt{(\Delta x)^{2}+(\Delta y)^{2}}=\int_{a}^{b}\sqrt{(dx)^% {2}+(dy)^{2}}=\int_{a}^{b}\sqrt{\left(\frac{dx}{dt}\right)^{2}+\left(\frac{dy}% {dt}\right)^{2}}\,dt.
  17. y = f ( x ) y=f(x)
  18. x = t x=t
  19. y = f ( t ) y=f(t)
  20. s = a b 1 + ( d y d x ) 2 d x . s=\int_{a}^{b}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dx.
  21. r = f ( θ ) r=f(\theta)
  22. s = a b r 2 + ( d r d θ ) 2 d θ . s=\int_{a}^{b}\sqrt{r^{2}+\left(\frac{dr}{d\theta}\right)^{2}}\,d\theta.
  23. d s = ( d x ) 2 + ( d y ) 2 . ds=\sqrt{(dx)^{2}+(dy)^{2}}.\,
  24. a b ( d x d t ) 2 + ( d y d t ) 2 d t , \int_{a}^{b}\sqrt{\bigg(\frac{dx}{dt}\bigg)^{2}+\bigg(\frac{dy}{dt}\bigg)^{2}}% \,dt,
  25. a b 1 + ( d y d x ) 2 d x , \int_{a}^{b}\sqrt{1+\bigg(\frac{dy}{dx}\bigg)^{2}}\,dx,
  26. { y = t 5 , x = t 3 . \begin{cases}y=t^{5},\\ x=t^{3}.\end{cases}
  27. - 1 1 ( 3 t 2 ) 2 + ( 5 t 4 ) 2 d t = - 1 1 9 t 4 + 25 t 8 d t . \int_{-1}^{1}\sqrt{(3t^{2})^{2}+(5t^{4})^{2}}\,dt=\int_{-1}^{1}\sqrt{9t^{4}+25% t^{8}}\,dt.
  28. Δ x \Delta x
  29. Δ y \Delta y
  30. ( Δ x ) 2 + ( Δ y ) 2 \sqrt{(\Delta x)^{2}+(\Delta y)^{2}}
  31. S i = 1 n ( Δ x i ) 2 + ( Δ y i ) 2 S\sim\sum_{i=1}^{n}\sqrt{(\Delta x_{i})^{2}+(\Delta y_{i})^{2}}
  32. ( Δ x ) 2 ( Δ x ) 2 \frac{(\Delta x)^{2}}{(\Delta x)^{2}}
  33. ( Δ x ) 2 + ( Δ y ) 2 = ( Δ x ) 2 + ( ( Δ y ) 2 ) ( Δ x ) 2 ( Δ x ) 2 = 1 + ( Δ y ) 2 ( Δ x ) 2 Δ x = 1 + ( Δ y Δ x ) 2 Δ x \sqrt{(\Delta x)^{2}+(\Delta y)^{2}}=\sqrt{{(\Delta x)^{2}+((\Delta y)^{2}})\,% \frac{(\Delta x)^{2}}{(\Delta x)^{2}}}=\sqrt{1+\frac{(\Delta y)^{2}}{(\Delta x% )^{2}}}\,\Delta x=\sqrt{1+\left(\frac{\Delta y}{\Delta x}\right)^{2}}\,\Delta x
  34. S i = 1 n 1 + ( Δ y i Δ x i ) 2 Δ x i S\sim\sum_{i=1}^{n}\sqrt{1+\left(\frac{\Delta y_{i}}{\Delta x_{i}}\right)^{2}}% \,\Delta x_{i}
  35. Δ x \Delta x
  36. Δ x \Delta x
  37. S S
  38. S = lim Δ x i 0 i = 1 1 + ( Δ y i Δ x i ) 2 Δ x i = a b 1 + ( d y d x ) 2 d x = a b 1 + [ f ( x ) ] 2 d x . S=\lim_{\Delta x_{i}\to 0}\sum_{i=1}^{\infty}\sqrt{1+\left(\frac{\Delta y_{i}}% {\Delta x_{i}}\right)^{2}}\,\Delta x_{i}=\int_{a}^{b}\sqrt{1+\left(\frac{dy}{% dx}\right)^{2}}\,dx=\int_{a}^{b}\sqrt{1+\left[f^{\prime}\left(x\right)\right]^% {2}}\,dx.
  39. a b f ( x , y ) x [ t ] 2 + y [ t ] 2 d t \int_{a}^{b}f(x,y)\sqrt{x^{\prime}[t]^{2}+y^{\prime}[t]^{2}}\,dt
  40. a b x [ t ] 2 + y [ t ] 2 d t \int_{a}^{b}\sqrt{x^{\prime}[t]^{2}+y^{\prime}[t]^{2}}dt
  41. a b 1 + f ( x ) 2 d x \int_{a}^{b}\sqrt{1+f^{\prime}(x)^{2}}\,dx
  42. t 1 t 2 r 2 + ( d r d θ ) 2 d θ \int_{t_{1}}^{t_{2}}\sqrt{r^{2}+\left(\frac{dr}{d\theta}\right)^{2}}\,d\theta
  43. t 1 t 2 ( d r d t ) 2 + r 2 ( d θ d t ) 2 + ( d z d t ) 2 d t \int_{t_{1}}^{t_{2}}\sqrt{\left(\frac{dr}{dt}\right)^{2}+r^{2}\left(\frac{d% \theta}{dt}\right)^{2}+\left(\frac{dz}{dt}\right)^{2}}\,dt
  44. t 1 t 2 ( d ρ d t ) 2 + ρ 2 sin 2 φ ( d θ d t ) 2 + ρ 2 ( d φ d t ) 2 d t \int_{t_{1}}^{t_{2}}\sqrt{\left(\frac{d\rho}{dt}\right)^{2}+\rho^{2}\sin^{2}% \varphi\left(\frac{d\theta}{dt}\right)^{2}+\rho^{2}\left(\frac{d\varphi}{dt}% \right)^{2}}\,dt
  45. r r
  46. d d
  47. C C
  48. s s
  49. θ \theta
  50. r , d , C , r,d,C,
  51. s s
  52. C = 2 π r , C=2\pi r,
  53. C = π d . C=\pi d.
  54. π \pi
  55. s = π r . s=\pi r.
  56. θ \theta
  57. s = r θ . s=r\theta.
  58. θ \theta
  59. s = π r θ 180 , s=\frac{\pi r\theta}{180},
  60. s = C θ 360 . s=\frac{C\theta}{360}.
  61. θ \theta
  62. s = π r θ 200 , s=\frac{\pi r\theta}{200},
  63. s = C θ 400 . s=\frac{C\theta}{400}.
  64. θ \theta
  65. 2 π 2\pi
  66. s = C θ . s=C\theta.
  67. s = θ s=\theta
  68. s s
  69. θ \theta
  70. 1 / 60 {1}/{60}
  71. s s
  72. θ \theta
  73. 1 / 100 {1}/{100}
  74. y = x 3 / 2 y=x^{3/2}\,
  75. 3 2 a 1 / 2 \textstyle{3\over 2}a^{1/2}
  76. y = 3 2 a 1 / 2 ( x - a ) + f ( a ) . y=\textstyle{3\over 2}{a^{1/2}}(x-a)+f(a).
  77. A C 2 \displaystyle AC^{2}
  78. A C = ε 1 + 9 4 a . AC=\textstyle\varepsilon\sqrt{1+{9\over 4}a\ }.
  79. ( γ ) = 0 1 ± g ( γ ( t ) , γ ( t ) ) d t , \ell(\gamma)=\int_{0}^{1}\sqrt{\pm g(\gamma^{\prime}(t),\gamma^{\prime}(t))}\,dt,

Area_of_a_disk.html

  1. π \pi
  2. 1 / 2 {1}/{2}
  3. π \pi
  4. π \pi
  5. π \pi
  6. E = C - T > G n P n = C - G n > C - E P n > T \begin{aligned}\displaystyle E&\displaystyle{}=C-T\\ &\displaystyle{}>G_{n}\\ \displaystyle P_{n}&\displaystyle{}=C-G_{n}\\ &\displaystyle{}>C-E\\ \displaystyle P_{n}&\displaystyle{}>T\end{aligned}
  7. D = T - C > G n P n = C + G n < C + D P n < T \begin{aligned}\displaystyle D&\displaystyle{}=T-C\\ &\displaystyle{}>G_{n}\\ \displaystyle P_{n}&\displaystyle{}=C+G_{n}\\ &\displaystyle{}<C+D\\ \displaystyle P_{n}&\displaystyle{}<T\end{aligned}
  8. π \pi
  9. π \pi
  10. π \pi
  11. π \pi
  12. π \pi
  13. Area ( r ) = 0 r 2 π t d t = [ ( 2 π ) t 2 2 ] t = 0 r = π r 2 . \begin{aligned}\displaystyle\mathrm{Area}(r)&\displaystyle{}=\int_{0}^{r}2\pi t% \,dt\\ &\displaystyle{}=\left[(2\pi)\frac{t^{2}}{2}\right]_{t=0}^{r}\\ &\displaystyle{}=\pi r^{2}.\end{aligned}
  14. Area ( r ) \displaystyle\mathrm{Area}(r)
  15. - r r r 2 - x 2 d x \int_{-r}^{r}\sqrt{r^{2}-x^{2}}\,dx
  16. x = r sin θ x=r\sin\theta
  17. d x = r cos θ d θ dx=r\cos\theta d\theta
  18. θ = arcsin ( x r ) \theta=\arcsin\left(\frac{x}{r}\right)
  19. - r r r 2 - x 2 d x \int_{-r}^{r}\sqrt{r^{2}-x^{2}}\,dx
  20. = - π 2 π 2 r 2 ( 1 - sin 2 θ ) r cos θ d θ =\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sqrt{r^{2}(1-\sin^{2}\theta)}\cdot r% \cos\theta\,d\theta
  21. = - π 2 π 2 r 2 cos 2 θ d θ =\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}r^{2}\cos^{2}\theta\,d\theta
  22. = r 2 2 - π 2 π 2 ( 1 + cos 2 θ ) d θ =\frac{r^{2}}{2}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}(1+\cos 2\theta)\,d\theta
  23. = r 2 2 ( [ θ ] - π 2 π 2 + [ 1 2 sin 2 θ ] - π 2 π 2 ) =\frac{r^{2}}{2}\left(\left[\theta\right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}}+% \left[\frac{1}{2}\sin 2\theta\right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\right)
  24. = π r 2 2 =\frac{\pi r^{2}}{2}
  25. 2 π r 2 2 = π r 2 2\cdot\frac{\pi r^{2}}{2}=\pi r^{2}
  26. u 2 n = U 2 n u n u_{2n}=\sqrt{U_{2n}u_{n}}
  27. U 2 n = 2 U n u n U n + u n U_{2n}=\frac{2U_{n}u_{n}}{U_{n}+u_{n}}
  28. π \pi
  29. π \pi
  30. π \pi
  31. n 3 sin π n 2 + cos π n < π < n [ 2 sin π 3 n + tan π 3 n ] . n\frac{3\sin\frac{\pi}{n}}{2+\cos\frac{\pi}{n}}<\pi<n[2\sin\frac{\pi}{3n}+\tan% \frac{\pi}{3n}].
  32. c 2 n 2 = ( r + 1 2 c n ) 2 r c 2 n = s n s 2 n . \begin{aligned}\displaystyle c_{2n}^{2}&\displaystyle{}=\left(r+\frac{1}{2}c_{% n}\right)2r\\ \displaystyle c_{2n}&\displaystyle{}=\frac{s_{n}}{s_{2n}}.\end{aligned}
  33. c 2 n = 2 + c n . c_{2n}=\sqrt{2+c_{n}}.\,\!
  34. c n = 2 s n S n . c_{n}=2\frac{s_{n}}{S_{n}}.\,\!
  35. c 2 n = s n s 2 n = 2 s 2 n S 2 n , c_{2n}=\frac{s_{n}}{s_{2n}}=2\frac{s_{2n}}{S_{2n}},
  36. u 2 n 2 = u n U 2 n . u_{2n}^{2}=u_{n}U_{2n}.\,\!
  37. 2 s 2 n S 2 n s n s 2 n = 2 + 2 s n S n , 2\frac{s_{2n}}{S_{2n}}\frac{s_{n}}{s_{2n}}=2+2\frac{s_{n}}{S_{n}},
  38. 2 U 2 n = 1 u n + 1 U n . \frac{2}{U_{2n}}=\frac{1}{u_{n}}+\frac{1}{U_{n}}.
  39. π \pi
  40. π \pi
  41. π \pi
  42. π \pi
  43. Area = 1 2 * base * height = 1 2 * 2 π r * r = π r 2 \begin{aligned}\displaystyle\,\text{Area}&\displaystyle{}=\frac{1}{2}*\,\text{% base}*\,\text{height}\\ &\displaystyle{}=\frac{1}{2}*2\pi r*r\\ &\displaystyle{}=\pi r^{2}\end{aligned}

Argon–argon_dating.html

  1. t = 1 λ ln ( J × R + 1 ) t=\frac{1}{\lambda}\ln(J\times R+1)

Arithmetic_genus.html

  1. 𝒪 M \mathcal{O}_{M}
  2. p a = ( - 1 ) n ( χ ( 𝒪 M ) - 1 ) . p_{a}=(-1)^{n}(\chi(\mathcal{O}_{M})-1).\,

Arithmetica.html

  1. x n + y n = z n x^{n}+y^{n}=z^{n}
  2. x x
  3. y y
  4. z z
  5. n n
  6. n n
  7. 4 n + 3 4n+3

Array_processing.html

  1. x ( t ) = K = 1 q a ( θ k ) s k ( t ) + n ( t ) \textstyle x(t)=\sum_{K=1}^{q}a(\theta_{k})s_{k}(t)+n(t)
  2. x ( t ) = A ( θ ) s ( t ) + n ( t ) \textstyle x(t)=A(\theta)s(t)+n(t)
  3. X = A ( θ ) S + N X=A(\theta)S+N
  4. X = [ x ( t 1 ) , , x ( t M ) ] X=[x(t_{1}),......,x(t_{M})]
  5. N = [ n ( t 1 ) , , n ( t M ) ] N=[n(t_{1}),......,n(t_{M})]
  6. S = [ s ( t 1 ) , , s ( t M ) ] S=[s(t_{1}),......,s(t_{M})]
  7. 1. R x = 1 M t = 1 M x ( t ) X * ( t ) \textstyle 1.\ R_{x}=\frac{1}{M}\sum_{t=1}^{M}x(t)X^{*}(t)
  8. 2. C a l c u l a t e B ( W i ) = F * R x F ( W i ) \textstyle 2.\ Calculate\ B(W_{i})=F^{*}R_{x}F(W_{i})
  9. 3. F i n d P e a k s o f B ( W i ) f o r a l l p o s s i b l e w i s . \textstyle 3.\ Find\ Peaks\ of\ B(W_{i})\ for\ all\ possible\ w_{i}^{\prime}s.
  10. 4. C a l c u l a t e θ k , i = 1 , . q . \textstyle 4.\ Calculate\ \theta_{k},\ i=1,....q.
  11. 1. S u b s p a c e d e c o m p o s i t i o n b y p e r f o r m i n g e i g e n v a l u e d e c o m p o s i t i o n : \textstyle 1.\ Subspace\ decomposition\ by\ performing\ eigenvalue\ decomposition:
  12. R x = A R s A * + σ 2 I = k = 1 M λ k e k r k * \textstyle R_{x}=AR_{s}A^{*}+\sigma^{2}I=\sum_{k=1}^{M}\lambda_{k}e_{k}r_{k}^{*}
  13. 2. s p a n { A } = s p a n e { e 1 , . , e d } = s p a n { E s } . \textstyle 2.\ span\{A\}=spane\{e1,....,e_{d}\}=span\{E_{s}\}.
  14. 3. C h e c k w h i c h a ( θ ) ϵ s p a n { E s } o r P A a ( θ ) o r P A a ( θ ) , w h e r e P A i s a p r o j e c t i o n m a t r i x . \textstyle 3.\ Check\ which\ a(\theta)\ \epsilon span\{E_{s}\}\ or\ P_{A}a(% \theta)\ or\ P_{A}^{\perp}a(\theta),\ where\ P_{A}\ is\ a\ projection\ matrix.
  15. 4. S e a r c h f o r a l l p o s s i b l e θ s u c h t h a t : | P A a ( θ ) | 2 = 0 o r M ( θ ) = 1 P A a ( θ ) = \textstyle 4.\ Search\ for\ all\ possible\ \theta\ such\ that:\left|P_{A}^{% \perp}a(\theta)\right|^{2}=0\ or\ M(\theta)=\frac{1}{P_{A}a(\theta)}=\infty
  16. 5. A f t e r E V D o f R x : \textstyle 5.\ After\ EVD\ of\ R_{x}:
  17. P A = I - E s E s * = E n E n * \textstyle P_{A}^{\perp}=I-E_{s}E_{s}^{*}=E_{n}E_{n}^{*}
  18. w h e r e t h e n o i s e e i g e n v e c t o r m a t r i x E n = [ e d + 1 , . , e M ] \textstyle where\ the\ noise\ eigenvector\ matrix\ E_{n}=[e_{d}+1,....,e_{M}]
  19. 1. F i n d W K t o m i n i m i z e : \textstyle 1.\ Find\ W_{K}\ to\ minimize:
  20. m i n a * ( θ k w k = 1 ) E { | W k X ( t ) | 2 } \textstyle min_{a^{*}(\theta_{k}w_{k}=1)}\ E\{\left|W_{k}X(t)\right|^{2}\}
  21. = m i n a * ( θ k w k = 1 ) W k * R k W k \textstyle=min_{a^{*}(\theta_{k}w_{k}=1)}\ W_{k}^{*}R_{k}W_{k}
  22. 2. U s e t h e l a n g r a n g e m e t h o d : \textstyle 2.\ Use\ the\ langrange\ method:
  23. m i n a * ( θ k w k = 1 ) E { | W k X ( t ) | 2 } \textstyle min_{a^{*}(\theta_{k}w_{k}=1)}\ E\{\left|W_{k}X(t)\right|^{2}\}
  24. = m i n a * ( θ k w k = 1 ) W k * R k W k + 2 μ ( a * ( θ k ) w k 1 ) \textstyle=min_{a^{*}(\theta_{k}w_{k}=1)}\ W_{k}^{*}R_{k}W_{k}+2\mu(a^{*}(% \theta_{k})w_{k}\Leftrightarrow 1)
  25. 3. D i f f e r e n t i a t i n g i t , w e o b t a i n \textstyle 3.\ Differentiating\ it,\ we\ obtain
  26. R x w k = μ a ( θ k ) , o r W k = μ R x - 1 a ( θ k ) \textstyle R_{x}w_{k}=\mu a(\theta_{k}),\ or\ W_{k}=\mu R_{x}^{-1}a(\theta_{k})
  27. 4. s i n c e \textstyle 4.\ since
  28. a * ( θ k ) W k = μ a ( θ k ) * R x - 1 a ( θ k ) = 1 \textstyle a^{*}(\theta_{k})W_{k}=\mu a(\theta_{k})^{*}R_{x}^{-1}a(\theta_{k})=1
  29. T h e n \textstyle Then
  30. μ = a ( θ k ) * R x - 1 a ( θ k ) \textstyle\mu=a(\theta_{k})^{*}R_{x}^{-1}a(\theta_{k})
  31. 5. C a p o n s B e a m f o r m e r \textstyle 5.\ Capon^{\prime}s\ Beamformer
  32. W k = R x - 1 a ( θ k ) / ( a * ( θ k ) R x - 1 a ( θ k ) ) \textstyle W_{k}=R_{x}^{-1}a(\theta_{k})/(a^{*}(\theta_{k})R_{x}^{-1}a(\theta_% {k}))
  33. S XX ( f ) S_{\,\text{XX}}(f)
  34. S XX ( f ) = - R XX ( τ ) cos ( 2 π f τ ) , d τ S_{\,\text{XX}}(f)=\int_{-}\infty^{\infty}R_{\,\text{XX}}(\tau)\cos(2\pi f\tau% ),\mathrm{d}\tau
  35. R XX ( τ ) R_{\,\text{XX}}(\tau)
  36. τ \tau
  37. R XX ( τ ) = ( V X ( t ) V X ( t + τ ) ) R_{\,\text{XX}}(\tau)=\left(V_{X}(t)V_{X}(t+\tau)\right)
  38. V Y ( t ) V_{Y}(t)
  39. R XY ( τ ) R_{\,\text{XY}}(\tau)
  40. S XY ( f ) S_{\,\text{XY}}(f)
  41. 𝐚 \mathbf{a}
  42. 𝐑 = 𝐑 v + σ s 2 𝐚𝐚 + σ n 2 𝐈 \mathbf{R}=\mathbf{R}_{v}+\sigma_{s}^{2}\mathbf{a}\mathbf{a}^{\dagger}+\sigma_% {n}^{2}\mathbf{I}
  43. 𝐑 v \mathbf{R}_{v}
  44. σ s 2 \sigma_{s}^{2}
  45. σ n 2 \sigma_{n}^{2}
  46. \dagger
  47. 𝐏 a \mathbf{P}_{a}^{\perp}
  48. 𝐏 a = 𝐈 - 𝐚 ( 𝐚 𝐚 ) - 1 𝐚 \mathbf{P}_{a}^{\perp}=\mathbf{I}-\mathbf{a}(\mathbf{a}^{\dagger}\mathbf{a})^{% -1}\mathbf{a}^{\dagger}
  49. 𝐑 ~ = 𝐏 a 𝐑𝐏 a = 𝐏 a 𝐑 v 𝐏 a + σ n 2 𝐏 a \tilde{\mathbf{R}}=\mathbf{P}_{a}^{\perp}\mathbf{R}\mathbf{P}_{a}^{\perp}=% \mathbf{P}_{a}^{\perp}\mathbf{R}_{v}\mathbf{P}_{a}^{\perp}+\sigma_{n}^{2}% \mathbf{P}_{a}^{\perp}
  50. 𝐚 \mathbf{a}
  51. 𝐏 a \mathbf{P}_{a}^{\perp}
  52. 𝐑 \mathbf{R}
  53. 𝐚 \mathbf{a}
  54. 𝐑 \mathbf{R}
  55. 𝐑 ~ = ( σ s 2 𝐚𝐚 + σ n 2 𝐈 ) - 1 2 𝐑 ( σ s 2 𝐚𝐚 + σ n 2 𝐈 ) - 1 2 \tilde{\mathbf{R}}=(\sigma_{s}^{2}\mathbf{a}\mathbf{a}^{\dagger}+\sigma_{n}^{2% }\mathbf{I})^{-{\frac{1}{2}}}\mathbf{R}(\sigma_{s}^{2}\mathbf{a}\mathbf{a}^{% \dagger}+\sigma_{n}^{2}\mathbf{I})^{-{\frac{1}{2}}}
  56. 𝐑 ~ = ( ) - 1 2 𝐑 v ( ) - 1 2 + 𝐈 \tilde{\mathbf{R}}=(\cdot)^{-{\frac{1}{2}}}\mathbf{R}_{v}(\cdot)^{-{\frac{1}{2% }}}+\mathbf{I}
  57. 𝐚 \mathbf{a}
  58. 𝐮 1 \mathbf{u}_{1}
  59. 𝐑 = 𝐔 𝚲 𝐔 \mathbf{R}=\mathbf{U}\mathbf{\Lambda}\mathbf{U}^{\dagger}
  60. σ s 2 λ 1 - σ n 2 \sigma_{s}^{2}\approx\lambda_{1}-\sigma_{n}^{2}
  61. λ 1 \lambda_{1}
  62. 𝐑 \mathbf{R}
  63. 𝐑 ~ = 𝐑 - σ s 2 𝐚𝐚 \tilde{\mathbf{R}}=\mathbf{R}-\sigma_{s}^{2}\mathbf{a}\mathbf{a}^{\dagger}
  64. 𝐑 \mathbf{R}
  65. 𝐑 ~ ( 𝐈 - α 𝐮 1 𝐮 1 ) 𝐑 ( 𝐈 - α 𝐮 1 𝐮 1 ) = 𝐑 - 𝐮 1 𝐮 1 λ 1 ( 2 α - α 2 ) \tilde{\mathbf{R}}\approx(\mathbf{I}-\alpha\mathbf{u}_{1}\mathbf{u}_{1}^{% \dagger})\mathbf{R}(\mathbf{I}-\alpha\mathbf{u}_{1}\mathbf{u}_{1}^{\dagger})=% \mathbf{R}-\mathbf{u}_{1}\mathbf{u}_{1}^{\dagger}\lambda_{1}(2\alpha-\alpha^{2})
  66. λ 1 ( 2 α - α 2 ) σ s 2 \lambda_{1}(2\alpha-\alpha^{2})\approx\sigma_{s}^{2}
  67. α \alpha

Art_gallery_problem.html

  1. S S
  2. p p
  3. q S q\in S
  4. p p
  5. q q
  6. n / 3 \left\lfloor n/3\right\rfloor
  7. n n
  8. n / 3 \lfloor n/3\rfloor
  9. n / 4 \lfloor n/4\rfloor
  10. n / 2 \lceil n/2\rceil
  11. n / 3 \left\lfloor n/3\right\rfloor

Artin_group.html

  1. x 1 , x 2 , , x n | x 1 , x 2 m 1 , 2 = x 2 , x 1 m 2 , 1 , , x n - 1 , x n m n - 1 , n = x n , x n - 1 m n , n - 1 \Big\langle x_{1},x_{2},\ldots,x_{n}\Big|\langle x_{1},x_{2}\rangle^{m_{1,2}}=% \langle x_{2},x_{1}\rangle^{m_{2,1}},\ldots,\langle x_{n-1},x_{n}\rangle^{m_{n% -1,n}}=\langle x_{n},x_{n-1}\rangle^{m_{n,n-1}}\Big\rangle
  2. m i , j = m j , i { 2 , 3 , , } m_{i,j}=m_{j,i}\in\{2,3,\ldots,\infty\}
  3. m < m<\infty
  4. x i , x j m \langle x_{i},x_{j}\rangle^{m}
  5. x i x_{i}
  6. x j x_{j}
  7. m m
  8. x i x_{i}
  9. x i , x j 3 = x i x j x i \langle x_{i},x_{j}\rangle^{3}=x_{i}x_{j}x_{i}
  10. x i , x j 4 = x i x j x i x j \langle x_{i},x_{j}\rangle^{4}=x_{i}x_{j}x_{i}x_{j}
  11. m = m=\infty
  12. x i x_{i}
  13. x j x_{j}
  14. m i , j m_{i,j}
  15. x i 2 = 1 {x_{i}}^{2}=1
  16. m i , i + 1 = 3 m_{i,i+1}=3
  17. m i , j = 2 m_{i,j}=2
  18. | i - j | > 1. |i-j|>1.
  19. x i x j = x j x i x_{i}x_{j}=x_{j}x_{i}\quad
  20. Γ . \Gamma.

Artin–Mazur_zeta_function.html

  1. ζ f ( z ) = exp ( n = 1 card ( Fix ( f n ) ) z n n ) , \zeta_{f}(z)=\exp\left(\sum_{n=1}^{\infty}\textrm{card}\left(\textrm{Fix}(f^{n% })\right)\frac{z^{n}}{n}\right),

Asaṃkhyeya.html

  1. 10 ( a 2 b ) 10^{(a\cdot 2^{b})}
  2. 10 ( 5 2 103 ) 10^{(5\cdot 2^{103})}
  3. 10 ( 7 2 103 ) 10^{(7\cdot 2^{103})}
  4. 10 ( 10 2 104 ) 10^{(10\cdot 2^{104})}

Astronomical_constant.html

  1. τ A ( SI ) = ( 1 + L B ) 1 3 τ A ( TDB ) \tau_{A}({\rm SI})=(1+L_{\rm B})^{\frac{1}{3}}\tau_{A}({\rm TDB})\,
  2. G E ( SI ) = ( 1 + L B ) G E ( TDB ) GE({\rm SI})=(1+L_{\rm B})GE({\rm TDB})\,
  3. G S ( SI ) = ( 1 + L B ) G S ( TDB ) GS({\rm SI})=(1+L_{\rm B})GS({\rm TDB})\,
  4. × 10 10 \times 10^{−}10
  5. × 10 8 \times 10^{−}8
  6. × 10 8 \times 10^{−}8
  7. × 10 9 \times 10^{−}9
  8. × 10 11 \times 10^{−}11
  9. × 10 6 \times 10^{6}
  10. × 10 8 \times 10^{−}8
  11. × 10 7 \times 10^{7}
  12. × 10 9 \times 10^{−}9
  13. × 10 8 \times 10^{−}8
  14. × 10 8 \times 10^{−}8
  15. × 10 1 4 \times 10^{1}4
  16. × 10 9 \times 10^{−}9
  17. × 10 11 \times 10^{−}11
  18. × 10 4 \times 10^{−}4
  19. × 10 8 \times 10^{−}8
  20. × 10 11 \times 10^{−}11
  21. × 10 8 \times 10^{−}8
  22. × 10 2 0 \times 10^{2}0
  23. × 10 10 \times 10^{−}10
  24. × 10 3 0 \times 10^{3}0
  25. × 10 4 \times 10^{−}4
  26. × 10 11 \times 10^{−}11
  27. × 10 1 5 \times 10^{1}5
  28. × 10 2 6 \times 10^{2}6
  29. × 10 15 \times 10^{−}15

Asymmetric_relation.html

  1. a , b X , a R b ¬ ( b R a ) \forall a,b\in X,\ aRb\;\Rightarrow\lnot(bRa)

Asymptotically_flat_spacetime.html

  1. M ~ \tilde{M}
  2. M ~ \tilde{M}
  3. M ~ \tilde{M}
  4. M ~ \tilde{M}
  5. M ~ \tilde{M}
  6. t , x , y , z t,x,y,z
  7. g a b = η a b + h a b g_{ab}=\eta_{ab}+h_{ab}
  8. r 2 = x 2 + y 2 + z 2 r^{2}=x^{2}+y^{2}+z^{2}
  9. lim r h a b = O ( 1 / r ) \lim_{r\rightarrow\infty}h_{ab}=O(1/r)
  10. lim r h a b , p = O ( 1 / r 2 ) \lim_{r\rightarrow\infty}h_{ab,p}=O(1/r^{2})
  11. lim r h a b , p q = O ( 1 / r 3 ) \lim_{r\rightarrow\infty}h_{ab,pq}=O(1/r^{3})
  12. O ( 1 / r 4 ) O(1/r^{4})
  13. O ( 1 / r 4 ) O(1/r^{4})

Atmospheric_refraction.html

  1. R = cot ( h a + 7.31 h a + 4.4 ) ; R=\cot\left(h_{\mathrm{a}}+\frac{7.31}{h_{\mathrm{a}}+4.4}\right)\,;
  2. R = 1.02 cot ( h + 10.3 h + 5.11 ) ; R=1.02\cot\left(h+\frac{10.3}{h+5.11}\right)\,;
  3. P 101 283 273 + T \frac{P}{101}\,\frac{283}{273+T}

Atom_economy.html

  1. atom economy = molecular mass of desired product molecular mass of all reactants × 100 % \,\text{atom economy}=\frac{\,\text{molecular mass of desired product}}{\,% \text{molecular mass of all reactants}}\times 100\%

Atom_interferometer.html

  1. 0.99 ± 0.022 0.99\pm 0.022
  2. 0.3 r a d / s H z 0.3rad/s\sqrt{Hz}
  3. 3 10 - 10 3\cdot 10^{-10}
  4. 2 10 - 8 / s / H z 2\cdot 10^{-8}/s/\sqrt{Hz}
  5. α ± 1.5 10 - 9 \alpha\pm 1.5\cdot 10^{-9}

Atoroidal.html

  1. × \mathbb{Z}\times\mathbb{Z}

Attenuation_length.html

  1. λ \lambda
  2. 1 / e 1/e
  3. 1 / e 1/e
  4. P ( x ) = e - x / λ P(x)=e^{-x/\lambda}\!\,
  5. λ \lambda

AU_Microscopii.html

  1. b b
  2. b 15 A U b\approx 15AU
  3. b - α b^{-\alpha}
  4. α 1.8 \alpha\approx 1.8
  5. b 43 A U b\approx 43AU
  6. b - α b^{-\alpha}
  7. α 4.8 \alpha\approx 4.8

Autocode.html

  1. f ( t ) = | t | + 5 t 3 f(t)=\sqrt{|t|}+5t^{3}

Autocovariance.html

  1. X = ( X t ) X=(X_{t})
  2. μ t = E [ X t ] \mu_{t}=E[X_{t}]
  3. C X X ( t , s ) = c o v ( X t , X s ) = E [ ( X t - μ t ) ( X s - μ s ) ] = E [ X t X s ] - μ t μ s . C_{XX}(t,s)=cov(X_{t},X_{s})=E[(X_{t}-\mu_{t})(X_{s}-\mu_{s})]=E[X_{t}X_{s}]-% \mu_{t}\mu_{s}.\,
  4. X = ( X 1 , X 2 , , X n ) X=(X_{1},X_{2},...,X_{n})
  5. C X X C_{XX}
  6. C X X ( j , k ) = c o v ( X j , X k ) . C_{XX}(j,k)=cov(X_{j},X_{k}).\,
  7. μ t = μ s = μ \mu_{t}=\mu_{s}=\mu\,
  8. C X X ( t , s ) = C X X ( s - t ) = C X X ( τ ) C_{XX}(t,s)=C_{XX}(s-t)=C_{XX}(\tau)\,
  9. τ = | s - t | \tau=|s-t|\,
  10. C X X ( τ ) = E [ ( X ( t ) - μ ) ( X ( t + τ ) - μ ) ] C_{XX}(\tau)=E[(X(t)-\mu)(X(t+\tau)-\mu)]\,
  11. = E [ X ( t ) X ( t + τ ) ] - μ 2 =E[X(t)X(t+\tau)]-\mu^{2}\,
  12. = σ 2 R X X ( τ ) - μ 2 , =\sigma^{2}R_{XX}(\tau)-\mu^{2},\,
  13. R X X ( τ ) R_{XX}(\tau)
  14. σ 2 \sigma^{2}
  15. C X X ( τ ) = R X X ( τ ) - μ 2 , C_{XX}(\tau)=R_{XX}(\tau)-\mu^{2},\,
  16. c X X ( τ ) = C X X ( τ ) σ 2 . c_{XX}(\tau)=\frac{C_{XX}(\tau)}{\sigma^{2}}.\,
  17. Y t Y_{t}
  18. Y t = k = - a k X t + k Y_{t}=\sum_{k=-\infty}^{\infty}a_{k}X_{t+k}\,
  19. C Y Y ( τ ) = k , l = - a k a l * C X X ( τ + k - l ) . C_{YY}(\tau)=\sum_{k,l=-\infty}^{\infty}a_{k}a^{*}_{l}C_{XX}(\tau+k-l).\,

Autonomous_consumption.html

  1. C = c 0 + c 1 Y d C=c_{0}+c_{1}Y_{d}

Autoregressive_integrated_moving_average.html

  1. A R I M A ( p , d , q ) ARIMA(p,d,q)
  2. p p
  3. d d
  4. q q
  5. A R I M A ( p , d , q ) ( P , D , Q ) m ARIMA(p,d,q)(P,D,Q)_{m}
  6. m m
  7. P , D , Q P,D,Q
  8. X t X_{t}
  9. t t
  10. X t X_{t}
  11. ( 1 - i = 1 p α i L i ) X t = ( 1 + i = 1 q θ i L i ) ε t \left(1-\sum_{i=1}^{p^{\prime}}\alpha_{i}L^{i}\right)X_{t}=\left(1+\sum_{i=1}^% {q}\theta_{i}L^{i}\right)\varepsilon_{t}\,
  12. L L
  13. α i \alpha_{i}
  14. θ i \theta_{i}
  15. ε t \varepsilon_{t}
  16. ε t \varepsilon_{t}
  17. ( 1 - i = 1 p α i L i ) \left(1-\sum_{i=1}^{p^{\prime}}\alpha_{i}L^{i}\right)
  18. ( 1 - i = 1 p α i L i ) = ( 1 - i = 1 p - d ϕ i L i ) ( 1 - L ) d . \left(1-\sum_{i=1}^{p^{\prime}}\alpha_{i}L^{i}\right)=\left(1-\sum_{i=1}^{p^{% \prime}-d}\phi_{i}L^{i}\right)\left(1-L\right)^{d}.
  19. ( 1 - i = 1 p ϕ i L i ) ( 1 - L ) d X t = ( 1 + i = 1 q θ i L i ) ε t \left(1-\sum_{i=1}^{p}\phi_{i}L^{i}\right)\left(1-L\right)^{d}X_{t}=\left(1+% \sum_{i=1}^{q}\theta_{i}L^{i}\right)\varepsilon_{t}\,
  20. ( 1 - i = 1 p ϕ i L i ) ( 1 - L ) d X t = δ + ( 1 + i = 1 q θ i L i ) ε t \left(1-\sum_{i=1}^{p}\phi_{i}L^{i}\right)\left(1-L\right)^{d}X_{t}=\delta+% \left(1+\sum_{i=1}^{q}\theta_{i}L^{i}\right)\varepsilon_{t}\,
  21. ( 1 - L s ) \left(1-L^{s}\right)
  22. ( 1 - 3 L + L 2 ) \left(1-\sqrt{3}L+L^{2}\right)
  23. Y t = ( 1 - L ) d X t Y_{t}=\left(1-L\right)^{d}X_{t}
  24. ( 1 - i = 1 p ϕ i L i ) Y t = ( 1 + i = 1 q θ i L i ) ε t . \left(1-\sum_{i=1}^{p}\phi_{i}L^{i}\right)Y_{t}=\left(1+\sum_{i=1}^{q}\theta_{% i}L^{i}\right)\varepsilon_{t}\,.
  25. Y t Y_{t}
  26. X t = X t - 1 + ε t X_{t}=X_{t-1}+\varepsilon_{t}
  27. X t = c + X t - 1 + ε t X_{t}=c+X_{t-1}+\varepsilon_{t}
  28. X t = X t - 1 + X t - 2 + ( α + β - 2 ) ε t - 1 + ( 1 - α ) ε t - 2 + ε t X_{t}=X_{t-1}+X_{t-2}+(\alpha+\beta-2)\varepsilon_{t-1}+(1-\alpha)\varepsilon_% {t-2}+\varepsilon_{t}
  29. X t X_{t}

Autoregressive_model.html

  1. A R ( p ) AR(p)
  2. X t = c + i = 1 p φ i X t - i + ε t X_{t}=c+\sum_{i=1}^{p}\varphi_{i}X_{t-i}+\varepsilon_{t}\,
  3. φ 1 , , φ p \varphi_{1},\ldots,\varphi_{p}
  4. c c
  5. ε t \varepsilon_{t}
  6. X t = c + i = 1 p φ i B i X t + ε t X_{t}=c+\sum_{i=1}^{p}\varphi_{i}B^{i}X_{t}+\varepsilon_{t}
  7. ϕ ( B ) X t = c + ε t . \phi(B)X_{t}=c+\varepsilon_{t}\,.
  8. | φ 1 | 1 |\varphi_{1}|\geq 1
  9. z p - i = 1 p φ i z p - i \textstyle z^{p}-\sum_{i=1}^{p}\varphi_{i}z^{p-i}
  10. z i z_{i}
  11. | z i | < 1 |z_{i}|<1
  12. X t = c + φ 1 X t - 1 + ε t X_{t}=c+\varphi_{1}X_{t-1}+\varepsilon_{t}
  13. ε t \varepsilon_{t}
  14. X 1 X_{1}
  15. ε 1 \varepsilon_{1}
  16. X 2 X_{2}
  17. X 1 X_{1}
  18. X 2 X_{2}
  19. φ 1 ε 1 \varphi_{1}\varepsilon_{1}
  20. X 3 X_{3}
  21. X 2 X_{2}
  22. X 3 X_{3}
  23. φ 1 2 ε 1 \varphi_{1}^{2}\varepsilon_{1}
  24. ε 1 \varepsilon_{1}
  25. ϕ ( B ) X t = ε t \phi(B)X_{t}=\varepsilon_{t}\,
  26. X t = 1 ϕ ( B ) ε t . X_{t}=\frac{1}{\phi(B)}\varepsilon_{t}\,.
  27. ε t \varepsilon_{t}
  28. ε t \varepsilon_{t}
  29. ρ ( τ ) = k = 1 p a k y k - | τ | , \rho(\tau)=\sum_{k=1}^{p}a_{k}y_{k}^{-|\tau|},
  30. y k y_{k}
  31. ϕ ( B ) = 1 - k = 1 p φ k B k \phi(B)=1-\sum_{k=1}^{p}\varphi_{k}B^{k}
  32. ϕ ( . ) \phi(.)
  33. φ k \varphi_{k}
  34. φ \varphi
  35. φ \varphi
  36. φ \varphi
  37. φ 1 \varphi_{1}
  38. φ 2 \varphi_{2}
  39. φ 1 \varphi_{1}
  40. φ 2 \varphi_{2}
  41. X t = c + φ X t - 1 + ε t X_{t}=c+\varphi X_{t-1}+\varepsilon_{t}\,
  42. ε t \varepsilon_{t}
  43. σ ε 2 \sigma_{\varepsilon}^{2}
  44. φ 1 \varphi_{1}
  45. | φ | < 1 |\varphi|<1
  46. φ = 1 \varphi=1
  47. X t X_{t}
  48. | φ | < 1 |\varphi|<1
  49. E ( X t ) \operatorname{E}(X_{t})
  50. μ \mu
  51. E ( X t ) = E ( c ) + φ E ( X t - 1 ) + E ( ε t ) , \operatorname{E}(X_{t})=\operatorname{E}(c)+\varphi\operatorname{E}(X_{t-1})+% \operatorname{E}(\varepsilon_{t}),
  52. μ = c + φ μ + 0 , \mu=c+\varphi\mu+0,
  53. μ = c 1 - φ . \mu=\frac{c}{1-\varphi}.
  54. c = 0 c=0
  55. var ( X t ) = E ( X t 2 ) - μ 2 = σ ε 2 1 - φ 2 , \textrm{var}(X_{t})=\operatorname{E}(X_{t}^{2})-\mu^{2}=\frac{\sigma_{% \varepsilon}^{2}}{1-\varphi^{2}},
  56. σ ε \sigma_{\varepsilon}
  57. ε t \varepsilon_{t}
  58. var ( X t ) = φ 2 var ( X t - 1 ) + σ ε 2 , \textrm{var}(X_{t})=\varphi^{2}\textrm{var}(X_{t-1})+\sigma_{\varepsilon}^{2},
  59. B n = E ( X t + n X t ) - μ 2 = σ ε 2 1 - φ 2 φ | n | . B_{n}=\operatorname{E}(X_{t+n}X_{t})-\mu^{2}=\frac{\sigma_{\varepsilon}^{2}}{1% -\varphi^{2}}\,\,\varphi^{|n|}.
  60. τ = - 1 / ln ( φ ) \tau=-1/\ln(\varphi)
  61. B n = K ϕ | n | B_{n}=K\phi^{|n|}
  62. K K
  63. n n
  64. ϕ | n | = e | n | ln ϕ \phi^{|n|}=e^{|n|\ln\phi}
  65. e - n / τ e^{-n/\tau}
  66. Φ ( ω ) = 1 2 π n = - B n e - i ω n = 1 2 π ( σ ε 2 1 + φ 2 - 2 φ cos ( ω ) ) . \Phi(\omega)=\frac{1}{\sqrt{2\pi}}\,\sum_{n=-\infty}^{\infty}B_{n}e^{-i\omega n% }=\frac{1}{\sqrt{2\pi}}\,\left(\frac{\sigma_{\varepsilon}^{2}}{1+\varphi^{2}-2% \varphi\cos(\omega)}\right).
  67. X j X_{j}
  68. Δ t = 1 \Delta t=1
  69. τ \tau
  70. B n B_{n}
  71. B ( t ) σ ε 2 1 - φ 2 φ | t | B(t)\approx\frac{\sigma_{\varepsilon}^{2}}{1-\varphi^{2}}\,\,\varphi^{|t|}
  72. Φ ( ω ) = 1 2 π σ ε 2 1 - φ 2 γ π ( γ 2 + ω 2 ) \Phi(\omega)=\frac{1}{\sqrt{2\pi}}\,\frac{\sigma_{\varepsilon}^{2}}{1-\varphi^% {2}}\,\frac{\gamma}{\pi(\gamma^{2}+\omega^{2})}
  73. γ = 1 / τ \gamma=1/\tau
  74. τ \tau
  75. X t X_{t}
  76. c + φ X t - 2 + ε t - 1 c+\varphi X_{t-2}+\varepsilon_{t-1}
  77. X t - 1 X_{t-1}
  78. X t = c k = 0 N - 1 φ k + φ N X t - N + k = 0 N - 1 φ k ε t - k . X_{t}=c\sum_{k=0}^{N-1}\varphi^{k}+\varphi^{N}X_{t-N}+\sum_{k=0}^{N-1}\varphi^% {k}\varepsilon_{t-k}.
  79. φ N \varphi^{N}
  80. X t = c 1 - φ + k = 0 φ k ε t - k . X_{t}=\frac{c}{1-\varphi}+\sum_{k=0}^{\infty}\varphi^{k}\varepsilon_{t-k}.
  81. X t X_{t}
  82. φ k \varphi^{k}
  83. ε t \varepsilon_{t}
  84. X t X_{t}
  85. X t X_{t}
  86. φ \varphi
  87. X t + 1 = X t + θ ( μ - X t ) + ϵ t + 1 X_{t+1}=X_{t}+\theta(\mu-X_{t})+\epsilon_{t+1}\,
  88. | θ | < 1 |\theta|<1\,
  89. μ \mu
  90. X t + 1 = c + ϕ X t X_{t+1}=c+\phi X_{t}\,
  91. X t + n X_{t+n}\,
  92. E ( X t + n | X t ) = μ [ 1 - ( 1 - θ ) n ] + X t ( 1 - θ ) n \operatorname{E}(X_{t+n}|X_{t})=\mu\left[1-\left(1-\theta\right)^{n}\right]+X_% {t}(1-\theta)^{n}
  93. Var ( X t + n | X t ) = σ 2 [ 1 - ( 1 - θ ) 2 n ] 1 - ( 1 - θ ) 2 \operatorname{Var}(X_{t+n}|X_{t})=\sigma^{2}\frac{\left[1-(1-\theta)^{2n}% \right]}{1-(1-\theta)^{2}}
  94. X t = i = 1 p φ i X t - i + ε t . X_{t}=\sum_{i=1}^{p}\varphi_{i}X_{t-i}+\varepsilon_{t}.\,
  95. φ i \varphi_{i}
  96. γ m = k = 1 p φ k γ m - k + σ ε 2 δ m , 0 , \gamma_{m}=\sum_{k=1}^{p}\varphi_{k}\gamma_{m-k}+\sigma_{\varepsilon}^{2}% \delta_{m,0},
  97. γ m \gamma_{m}
  98. σ ε \sigma_{\varepsilon}
  99. δ m , 0 \delta_{m,0}
  100. [ γ 1 γ 2 γ 3 γ p ] = [ γ 0 γ - 1 γ - 2 γ 1 γ 0 γ - 1 γ 2 γ 1 γ 0 γ p - 1 γ p - 2 γ p - 3 ] [ φ 1 φ 2 φ 3 φ p ] \begin{bmatrix}\gamma_{1}\\ \gamma_{2}\\ \gamma_{3}\\ \vdots\\ \gamma_{p}\\ \end{bmatrix}=\begin{bmatrix}\gamma_{0}&\gamma_{-1}&\gamma_{-2}&\dots\\ \gamma_{1}&\gamma_{0}&\gamma_{-1}&\dots\\ \gamma_{2}&\gamma_{1}&\gamma_{0}&\dots\\ \vdots&\vdots&\vdots&\ddots\\ \gamma_{p-1}&\gamma_{p-2}&\gamma_{p-3}&\dots\\ \end{bmatrix}\begin{bmatrix}\varphi_{1}\\ \varphi_{2}\\ \varphi_{3}\\ \vdots\\ \varphi_{p}\\ \end{bmatrix}
  101. { φ m ; m = 1 , 2 , , p } . \{\varphi_{m};m=1,2,\cdots,p\}.
  102. γ 0 = k = 1 p φ k γ - k + σ ε 2 , \gamma_{0}=\sum_{k=1}^{p}\varphi_{k}\gamma_{-k}+\sigma_{\varepsilon}^{2},
  103. { φ m ; m = 1 , 2 , , p } \{\varphi_{m};m=1,2,\cdots,p\}
  104. σ ε 2 . \sigma_{\varepsilon}^{2}.
  105. ρ ( τ ) \rho(\tau)
  106. ρ ( τ ) = k = 1 p φ k ρ ( k - τ ) \rho(\tau)=\sum_{k=1}^{p}\varphi_{k}\rho(k-\tau)
  107. γ 1 = φ 1 γ 0 \gamma_{1}=\varphi_{1}\gamma_{0}
  108. ρ 1 = γ 1 / γ 0 = φ 1 \rho_{1}=\gamma_{1}/\gamma_{0}=\varphi_{1}
  109. γ 1 = φ 1 γ 0 + φ 2 γ - 1 \gamma_{1}=\varphi_{1}\gamma_{0}+\varphi_{2}\gamma_{-1}
  110. γ 2 = φ 1 γ 1 + φ 2 γ 0 \gamma_{2}=\varphi_{1}\gamma_{1}+\varphi_{2}\gamma_{0}
  111. γ - k = γ k \gamma_{-k}=\gamma_{k}
  112. ρ 1 = γ 1 / γ 0 = φ 1 1 - φ 2 \rho_{1}=\gamma_{1}/\gamma_{0}=\frac{\varphi_{1}}{1-\varphi_{2}}
  113. ρ 2 = γ 2 / γ 0 = φ 1 2 - φ 2 2 + φ 2 1 - φ 2 \rho_{2}=\gamma_{2}/\gamma_{0}=\frac{\varphi_{1}^{2}-\varphi_{2}^{2}+\varphi_{% 2}}{1-\varphi_{2}}
  114. X t = c + i = 1 p φ i X t - i + ε t * . X_{t}=c+\sum_{i=1}^{p}\varphi_{i}X_{t-i}+\varepsilon^{*}_{t}\,.
  115. Var ( Z t ) = σ Z 2 \mathrm{Var}(Z_{t})=\sigma_{Z}^{2}
  116. S ( f ) = σ Z 2 | 1 - k = 1 p φ k e - 2 π i k f | 2 . S(f)=\frac{\sigma_{Z}^{2}}{|1-\sum_{k=1}^{p}\varphi_{k}e^{-2\pi ikf}|^{2}}.
  117. S ( f ) = σ Z 2 . S(f)=\sigma_{Z}^{2}.
  118. S ( f ) = σ Z 2 | 1 - φ 1 e - 2 π i f | 2 = σ Z 2 1 + φ 1 2 - 2 φ 1 cos 2 π f S(f)=\frac{\sigma_{Z}^{2}}{|1-\varphi_{1}e^{-2\pi if}|^{2}}=\frac{\sigma_{Z}^{% 2}}{1+\varphi_{1}^{2}-2\varphi_{1}\cos 2\pi f}
  119. φ 1 > 0 \varphi_{1}>0
  120. φ 1 \varphi_{1}
  121. φ 1 < 0 \varphi_{1}<0
  122. z 1 , z 2 = 1 2 ( φ 1 ± φ 1 2 + 4 φ 2 ) z_{1},z_{2}=\frac{1}{2}\left(\varphi_{1}\pm\sqrt{\varphi_{1}^{2}+4\varphi_{2}}\right)
  123. φ 1 2 + 4 φ 2 < 0 \varphi_{1}^{2}+4\varphi_{2}<0
  124. f * = 1 2 π cos - 1 ( φ 1 ( φ 2 - 1 ) 4 φ 2 ) f^{*}=\frac{1}{2\pi}\cos^{-1}\left(\frac{\varphi_{1}(\varphi_{2}-1)}{4\varphi_% {2}}\right)
  125. φ 1 > 0 \varphi_{1}>0
  126. f = 0 f=0
  127. φ 1 < 0 \varphi_{1}<0
  128. f = 1 / 2 f=1/2
  129. - 1 φ 2 1 - | φ 1 | -1\leq\varphi_{2}\leq 1-|\varphi_{1}|
  130. S ( f ) = σ Z 2 1 + φ 1 2 + φ 2 2 - 2 φ 1 ( 1 - φ 2 ) cos ( 2 π f ) - 2 φ 2 cos ( 4 π f ) S(f)=\frac{\sigma_{Z}^{2}}{1+\varphi_{1}^{2}+\varphi_{2}^{2}-2\varphi_{1}(1-% \varphi_{2})\cos(2\pi f)-2\varphi_{2}\cos(4\pi f)}
  131. X t = c + i = 1 p φ i X t - i + ε t X_{t}=c+\sum_{i=1}^{p}\varphi_{i}X_{t-i}+\varepsilon_{t}\,
  132. ε t \varepsilon_{t}
  133. ε t \varepsilon_{t}\,

Axial_compressor.html

  1. r 1 r_{1}\,
  2. V w 1 V_{w1}\,
  3. r 2 r_{2}\,
  4. V w 2 V_{w2}\,
  5. V 1 V_{1}\,
  6. V 2 V_{2}\,
  7. V f 1 V_{f1}\,
  8. V f 2 V_{f2}\,
  9. V w 1 V_{w1}\,
  10. V w 2 V_{w2}\,
  11. V r 1 V_{r1}\,
  12. V r 2 V_{r2}\,
  13. U U\,
  14. α \alpha
  15. β \beta
  16. F = m ˙ ( V w 2 - V w 1 ) = m ˙ ( V f 2 tan α 2 - V f 1 tan α 1 ) F=\dot{m}(V_{w2}-V_{w1})=\dot{m}(V_{f2}\tan\alpha_{2}-V_{f1}\tan\alpha_{1})\,
  17. P = m ˙ U ( V f 2 tan α 2 - V f 1 tan α 1 ) P=\dot{m}U(V_{f2}\tan\alpha_{2}-V_{f1}\tan\alpha_{1})\,
  18. P = m ˙ ( h 02 - h 01 ) = m ˙ c p ( T 02 - T 01 ) P=\dot{m}(h_{02}-h_{01})=\dot{m}c_{p}(T_{02}-T_{01})\,
  19. P = m ˙ U ( V f 2 tan α 2 - V f 1 tan α 1 ) = m ˙ c p ( T 02 - T 01 ) P=\dot{m}U(V_{f2}\tan\alpha_{2}-V_{f1}\tan\alpha_{1})=\dot{m}c_{p}(T_{02}-T_{0% 1})\,
  20. δ ( T 0 ) isentropic = U ( V f 2 tan α 2 - V f 1 tan α 1 ) c p \delta(T_{0})\text{isentropic}=U\frac{(V_{f2}\tan\alpha_{2}-V_{f1}\tan\alpha_{% 1})}{c_{p}}\,
  21. p 2 - p 1 p_{2}-p_{1}\,
  22. p 1 ( ( T 2 T 1 ) γ γ - 1 - 1 ) p_{1}\left(\left(\frac{T_{2}}{T_{1}}\right)^{\frac{\gamma}{\gamma-1}}-1\right)\,
  23. ( p 02 ) actual p 01 = ( 1 + η stage δ ( T 0 ) isentropic T 01 ) γ γ - 1 \frac{(p_{02})\text{actual}}{p_{01}}=\left(1+\frac{\eta\text{stage}\delta(T_{0% })\text{isentropic}}{T_{01}}\right)^{\frac{\gamma}{\gamma-1}}\,
  24. ( p 02 ) actual p 01 = ( 1 + η stage U ( V f 2 tan α 2 - V f 1 tan α 1 ) c p T 01 ) γ γ - 1 \frac{(p_{02})\text{actual}}{p_{01}}=\left(1+\frac{\eta\text{stage}U\frac{(V_{% f2}\tan\alpha_{2}-V_{f1}\tan\alpha_{1})}{c_{p}}}{T_{01}}\right)^{\frac{\gamma}% {\gamma-1}}\,
  25. R = h 2 - h 1 h 02 - h 01 R=\frac{h_{2}-h_{1}}{h_{02}-h_{01}}\,
  26. P = m ˙ c p ( T 2 + V 2 2 2 c p - [ T 1 + V 1 2 2 c p ] ) P=\dot{m}c_{p}\left(T_{2}+\frac{V_{2}^{2}}{2c_{p}}-\left[T_{1}+\frac{V_{1}^{2}% }{2c_{p}}\right]\right)\,
  27. P = m ˙ ( h 2 - h 1 + [ V 2 2 2 - V 1 2 2 ] ) P=\dot{m}\left(h_{2}-h_{1}+\left[\frac{V_{2}^{2}}{2}-\frac{V_{1}^{2}}{2}\right% ]\right)\,
  28. h 2 - h 1 = [ V r 1 2 2 - V r 2 2 2 ] h_{2}-h_{1}=\left[\frac{V_{r1}^{2}}{2}-\frac{V_{r2}^{2}}{2}\right]\,
  29. T 2 - T 1 = [ V r 1 2 2 c p - V r 2 2 2 c p ] T_{2}-T_{1}=\left[\frac{V_{r1}^{2}}{2c_{p}}-\frac{V_{r2}^{2}}{2c_{p}}\right]\,
  30. V r 1 2 - V r 2 2 V r 1 2 - V r 2 2 + V 1 2 - V 2 2 \frac{V_{r1}^{2}-V_{r2}^{2}}{V_{r1}^{2}-V_{r2}^{2}+V_{1}^{2}-V_{2}^{2}}\,
  31. m ˙ T 01 P 01 \frac{\dot{m}\sqrt{T_{01}}}{P_{01}}\,
  32. ϕ \phi\,
  33. ψ = V U 2 \psi=\frac{V}{U^{2}}\,
  34. P 1 P 2 \frac{P_{1}}{P_{2}}\,
  35. ψ \psi\,
  36. ϕ \phi\,
  37. ψ = ϕ ( tan α 2 - tan α 1 ) \psi=\phi(\tan\alpha_{2}-\tan\alpha_{1})\,
  38. tan α 2 = 1 ϕ - tan β 2 \tan\alpha_{2}=\frac{1}{\phi}-\tan\beta_{2}\,
  39. ψ = 1 - ϕ ( tan β 2 + tan α 1 ) \psi=1-\phi(\tan\beta_{2}+\tan\alpha_{1})\,
  40. ( tan β 2 + tan α 1 ) (\tan\beta_{2}+\tan\alpha_{1})\,
  41. α 1 = α 3 \alpha_{1}=\alpha_{3}\,
  42. α 3 \alpha_{3}\,
  43. J = tan β 2 + tan α 3 ) J=\tan\beta_{2}+\tan\alpha_{3})\,
  44. ψ = 1 - J ( ϕ ) \psi^{^{\prime}}=1-J(\phi^{^{\prime}})\,
  45. J = 1 - ψ ϕ J=\frac{1-\psi^{^{\prime}}}{\phi^{^{\prime}}}\,
  46. ψ = 1 - J ( ϕ ) \psi=1-J(\phi)\,
  47. ψ = 1 - ϕ ( 1 - ψ ϕ ) \psi=1-\phi(\frac{1-\psi^{^{\prime}}}{\phi^{^{\prime}}})\,
  48. m ˙ , P D \dot{m},P_{D}\,
  49. P H P_{H}\,

Axial_ratio.html

  1. τ \tau
  2. 2 × τ 2\times\tau
  3. 2 × ϵ 2\times\epsilon
  4. ϵ = \arccot ( ± A R ) \epsilon=\arccot(\pm AR)
  5. \arccot \arccot

Axiom_S5.html

  1. \Box
  2. \Diamond
  3. φ \Box\varphi
  4. φ \Diamond\varphi
  5. ( φ ψ ) ( φ ψ ) \Box(\varphi\implies\psi)\implies(\Box\varphi\implies\Box\psi)
  6. φ φ \frac{\varphi}{\ \Box\varphi\ }
  7. φ φ \Box\varphi\implies\varphi
  8. φ φ \Diamond\varphi\implies\Box\Diamond\varphi
  9. R R
  10. ( w R v w R u ) v R u (wRv\land wRu)\implies vRu
  11. O O O φ OOO\ldots\Box\varphi
  12. φ \Box\varphi
  13. O O O OOO\ldots
  14. \Box
  15. \Diamond
  16. O O O φ OOO\ldots\Diamond\varphi
  17. φ \Diamond\varphi

AZE.html

  1. E Z A {}^{A}_{Z}E

B+_tree.html

  1. b / 2 m b \lceil b/2\rceil\leq m\leq b
  2. b / 2 \lceil b/2\rceil
  3. b - 1 b-1
  4. b / 2 \lceil b/2\rceil
  5. b / 2 \lceil b/2\rceil
  6. n m a x = b h - b h - 1 n_{max}=b^{h}-b^{h-1}
  7. n m i n = 2 b 2 h - 1 n_{min}=2\left\lceil\frac{b}{2}\right\rceil^{h-1}
  8. n k m i n = 2 b 2 h - 1 n_{kmin}=2\left\lceil\frac{b}{2}\right\rceil^{h}-1
  9. n k m a x = n h n_{kmax}=n^{h}
  10. O ( n ) O(n)
  11. O ( log b n ) O(\log_{b}n)
  12. O ( log b n ) O(\log_{b}n)
  13. O ( log b n ) O(\log_{b}n)
  14. O ( log b n + k ) O(\log_{b}n+k)
  15. b = ( B / k ) - 1 b=(B/k)-1
  16. i i
  17. i , i + 1... i + k i,i+1...i+k

Back-face_culling.html

  1. ( V 0 - P ) N 0 \left(V_{0}-P\right)\cdot N\geq 0
  2. 𝐏 \mathbf{P}
  3. 𝐍 \mathbf{N}
  4. N = ( V 1 - V 0 ) × ( V 2 - V 0 ) N=\left(V_{1}-V_{0}\right)\times\left(V_{2}-V_{0}\right)
  5. ( V 1 - V 0 ) × ( V 2 - V 0 ) = - ( V 2 - V 0 ) × ( V 1 - V 0 ) \left(V_{1}-V_{0}\right)\times\left(V_{2}-V_{0}\right)=-\left(V_{2}-V_{0}% \right)\times\left(V_{1}-V_{0}\right)
  6. 𝐏 \mathbf{P}
  7. ( 𝟎 , 𝟎 , 𝟎 ) (\mathbf{0},\mathbf{0},\mathbf{0})
  8. - V 0 N 0 -V_{0}\cdot N\geq 0
  9. U 0 = [ 0 0 1 ] , U 1 = [ 1 0 1 ] , U 2 = [ 0 1 1 ] U_{0}=\begin{bmatrix}0\\ 0\\ 1\end{bmatrix},U_{1}=\begin{bmatrix}1\\ 0\\ 1\end{bmatrix},U_{2}=\begin{bmatrix}0\\ 1\\ 1\end{bmatrix}
  10. V 0 = [ x 0 y 0 1 ] , V 1 = [ x 1 y 1 1 ] , V 2 = [ x 2 y 2 1 ] V_{0}=\begin{bmatrix}x_{0}\\ y_{0}\\ 1\end{bmatrix},V_{1}=\begin{bmatrix}x_{1}\\ y_{1}\\ 1\end{bmatrix},V_{2}=\begin{bmatrix}x_{2}\\ y_{2}\\ 1\end{bmatrix}
  11. M = [ x 1 - x 0 x 2 - x 0 x 0 y 1 - y 0 y 2 - y 0 y 0 0 0 1 ] M=\begin{bmatrix}x_{1}-x_{0}&x_{2}-x_{0}&x_{0}\\ y_{1}-y_{0}&y_{2}-y_{0}&y_{0}\\ 0&0&1\end{bmatrix}
  12. M U 0 = V 0 MU_{0}=V_{0}
  13. M U 1 = V 1 MU_{1}=V_{1}
  14. M U 2 = V 2 MU_{2}=V_{2}
  15. M M
  16. | M | < 0 \left|M\right|<0
  17. M M

Back-stripping.html

  1. ϕ = ϕ 0 e - c z \phi=\phi_{0}e^{-cz}
  2. ϕ \phi
  3. z z
  4. ϕ 0 \phi_{0}
  5. c c
  6. Y = S ( ρ m - ρ s ) ( ρ m - ρ w ) + W d - Δ S L ρ m ( ρ m - ρ w ) Y=S\cdot\frac{(\rho_{m}-\rho_{s})}{(\rho_{m}-\rho_{w})}+W_{d}-\Delta_{SL}\cdot% \frac{\rho_{m}}{(\rho_{m}-\rho_{w})}
  7. Y Y
  8. S S
  9. ρ s \rho_{s}
  10. W d W_{d}
  11. ρ w \rho_{w}
  12. ρ m \rho_{m}
  13. Δ S L \Delta_{SL}
  14. W d W_{d}
  15. c c
  16. S S
  17. g g
  18. ρ w \rho_{w}
  19. ρ s \rho_{s}
  20. ρ c \rho_{c}
  21. Y Y
  22. ρ m \rho_{m}
  23. b b
  24. b = S + W d - Δ S L - Y b=S+W_{d}-\Delta_{SL}-Y
  25. L * L^{*}
  26. ρ L \rho_{L}
  27. l l
  28. l l
  29. L * = j = 1 l L j L^{*}=\sum_{j=1}^{l}L_{j}
  30. l l
  31. L * L^{*}
  32. ρ L * = j = 1 l L j ( ϕ j ρ w + ( 1 - ϕ j ) ρ g ) L * \rho_{L^{*}}=\frac{\sum_{j=1}^{l}L_{j}(\phi_{j}\rho_{w}+(1-\phi_{j})\rho_{g})}% {L^{*}}
  33. L * L^{*}
  34. ρ L * \rho_{L^{*}}
  35. L L
  36. ρ L \rho_{L}

Backpropagation.html

  1. Δ w h \Delta w_{h}
  2. Δ w i \Delta w_{i}
  3. x 1 x_{1}
  4. x 2 x_{2}
  5. t t
  6. x 1 x_{1}
  7. x 2 x_{2}
  8. t t
  9. x 1 x_{1}
  10. x 2 x_{2}
  11. y y
  12. t t
  13. t t
  14. y y
  15. E = ( t - y ) 2 E=(t-y)^{2}\,
  16. E E
  17. ( 1 , 1 , 0 ) (1,1,0)
  18. x 1 x_{1}
  19. x 2 x_{2}
  20. t t
  21. y y
  22. E E
  23. y y
  24. y y
  25. E E
  26. x x
  27. y y
  28. t t
  29. y = x 1 w 1 + x 2 w 2 y=x_{1}w_{1}+x_{2}w_{2}
  30. w 1 w_{1}
  31. w 2 w_{2}
  32. k k
  33. k + 1 k+1
  34. k + 1 k+1
  35. E = 1 2 ( t - y ) 2 E=\tfrac{1}{2}(t-y)^{2}
  36. E E
  37. t t
  38. y y
  39. 1 2 \textstyle\frac{1}{2}
  40. j j
  41. o j o_{j}
  42. o j = φ ( net ) j = φ ( k = 1 n w k j x k ) o_{j}=\varphi(\mbox{net}~{}_{j})=\varphi\left(\sum_{k=1}^{n}w_{kj}x_{k}\right)
  43. net j \mbox{net}~{}_{j}
  44. o k o_{k}
  45. o k o_{k}
  46. x k x_{k}
  47. n n
  48. w i j w_{ij}
  49. i i
  50. j j
  51. φ \varphi
  52. φ ( z ) = 1 1 + e - z \varphi(z)=\frac{1}{1+e^{-z}}
  53. φ z = φ ( 1 - φ ) \frac{\partial\varphi}{\partial z}=\varphi(1-\varphi)
  54. w i j w_{ij}
  55. E w i j = E o j o j net j net j w i j \frac{\partial E}{\partial w_{ij}}=\frac{\partial E}{\partial o_{j}}\frac{% \partial o_{j}}{\partial\mathrm{net_{j}}}\frac{\partial\mathrm{net_{j}}}{% \partial w_{ij}}
  56. net j \mathrm{net_{j}}
  57. w i j w_{ij}
  58. net j w i j = w i j ( k = 1 n w k j x k ) = x i \frac{\partial\mathrm{net_{j}}}{\partial w_{ij}}=\frac{\partial}{\partial w_{% ij}}\left(\sum_{k=1}^{n}w_{kj}x_{k}\right)=x_{i}
  59. j j
  60. o j net j = net j φ ( net j ) = φ ( net j ) ( 1 - φ ( net j ) ) \frac{\partial o_{j}}{\partial\mathrm{net_{j}}}=\frac{\partial}{\partial% \mathrm{net_{j}}}\varphi(\mathrm{net_{j}})=\varphi(\mathrm{net_{j}})(1-\varphi% (\mathrm{net_{j}}))
  61. o j = y o_{j}=y
  62. E o j = E y = y 1 2 ( t - y ) 2 = y - t \frac{\partial E}{\partial o_{j}}=\frac{\partial E}{\partial y}=\frac{\partial% }{\partial y}\frac{1}{2}(t-y)^{2}=y-t
  63. j j
  64. E E
  65. o j o_{j}
  66. E E
  67. L = u , v , , w L={u,v,\dots,w}
  68. j j
  69. E ( o j ) o j = E ( net u , net v , , net w ) o j \frac{\partial E(o_{j})}{\partial o_{j}}=\frac{\partial E(\mathrm{net}_{u},% \mathrm{net}_{v},\dots,\mathrm{net}_{w})}{\partial o_{j}}
  70. o j o_{j}
  71. E o j = l L ( E net l net l o j ) = l L ( E o l o l net l w j l ) \frac{\partial E}{\partial o_{j}}=\sum_{l\in L}\left(\frac{\partial E}{% \partial\mathrm{net}_{l}}\frac{\partial\mathrm{net}_{l}}{\partial o_{j}}\right% )=\sum_{l\in L}\left(\frac{\partial E}{\partial o_{l}}\frac{\partial o_{l}}{% \partial\mathrm{net}_{l}}w_{jl}\right)
  72. o j o_{j}
  73. o l o_{l}
  74. E w i j = δ j x i \dfrac{\partial E}{\partial w_{ij}}=\delta_{j}x_{i}
  75. δ j = E o j o j net j = { ( o j - t j ) φ ( net ) j ( 1 - φ ( net ) j ) if j is an output neuron, ( l L δ l w j l ) φ ( net ) j ( 1 - φ ( net ) j ) if j is an inner neuron. \delta_{j}=\frac{\partial E}{\partial o_{j}}\frac{\partial o_{j}}{\partial% \mathrm{net_{j}}}=\begin{cases}(o_{j}-t_{j})\varphi(\mbox{net}~{}_{j})(1-% \varphi(\mbox{net}~{}_{j}))&\mbox{if }~{}j\mbox{ is an output neuron,}\\ (\sum_{l\in L}\delta_{l}w_{jl})\varphi(\mbox{net}~{}_{j})(1-\varphi(\mbox{net}% ~{}_{j}))&\mbox{if }~{}j\mbox{ is an inner neuron.}\end{cases}
  76. w i j w_{ij}
  77. α \alpha
  78. - 1 -1
  79. Δ w i j = - α E w i j \Delta w_{ij}=-\alpha\frac{\partial E}{\partial w_{ij}}
  80. - 1 \textstyle-1

Backtracking_line_search.html

  1. 𝐱 \mathbf{x}
  2. 𝐩 \mathbf{p}
  3. α \alpha
  4. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  5. α \alpha
  6. f ( 𝐱 + α 𝐩 ) f(\mathbf{x}+\alpha\,\mathbf{p})
  7. f ( 𝐱 ) f(\mathbf{x})
  8. α \alpha
  9. f f
  10. α \alpha
  11. α \alpha
  12. α \alpha
  13. f ( 𝐱 ) . \nabla f(\mathbf{x})\,.
  14. α \alpha
  15. 𝐩 \mathbf{p}
  16. m = 𝐩 T f ( 𝐱 ) . m=\mathbf{p}^{\mathrm{T}}\,\nabla f(\mathbf{x})\,.
  17. 𝐩 \mathbf{p}
  18. m < 0 m<0
  19. c ( 0 , 1 ) c\,\in\,(0,1)
  20. 𝐱 \mathbf{x}
  21. 𝐱 + α 𝐩 \mathbf{x}+\alpha\,\mathbf{p}
  22. f ( 𝐱 + α 𝐩 ) f ( 𝐱 ) + α c m . f(\mathbf{x}+\alpha\,\mathbf{p})\leq f(\mathbf{x})+\alpha\,c\,m\,.
  23. α \displaystyle\alpha
  24. τ ( 0 , 1 ) \tau\,\in\,(0,1)
  25. c c
  26. τ \tau
  27. c c
  28. τ \tau
  29. α 0 > 0 \alpha_{0}>0\,
  30. τ ( 0 , 1 ) \tau\,\in\,(0,1)
  31. c ( 0 , 1 ) c\,\in\,(0,1)
  32. t = - c m t=-c\,m
  33. j = 0 j\,=\,0
  34. f ( 𝐱 ) - f ( 𝐱 + α j 𝐩 ) α j t , f(\mathbf{x})-f(\mathbf{x}+\alpha_{j}\,\mathbf{p})\geq\alpha_{j}\,t,
  35. j j
  36. α j = τ α j - 1 . \alpha_{j}=\tau\,\alpha_{j-1}\,.
  37. α j \alpha_{j}
  38. α 0 \alpha_{0}
  39. τ \tau\,

Baker's_map.html

  1. S baker-folded ( x , y ) = { ( 2 x , y / 2 ) for 0 x < 1 2 ( 2 - 2 x , 1 - y / 2 ) for 1 2 x < 1. S\text{baker-folded}(x,y)=\begin{cases}(2x,y/2)&\,\text{for }0\leq x<\frac{1}{% 2}\\ (2-2x,1-y/2)&\,\text{for }\frac{1}{2}\leq x<1.\end{cases}
  2. S baker-unfolded ( x , y ) = ( 2 x - 2 x , y + 2 x 2 ) . S\text{baker-unfolded}(x,y)=\left(2x-\left\lfloor 2x\right\rfloor\,,\,\frac{y+% \left\lfloor 2x\right\rfloor}{2}\right).
  3. S tent ( x ) = { 2 x for 0 x < 1 2 2 ( 1 - x ) for 1 2 x < 1 S_{\mathrm{tent}}(x)=\begin{cases}2x&\,\text{for }0\leq x<\frac{1}{2}\\ 2(1-x)&\,\text{for }\frac{1}{2}\leq x<1\end{cases}
  4. U U
  5. [ U f ] ( x , y ) = ( f S - 1 ) ( x , y ) . \left[Uf\right](x,y)=(f\circ S^{-1})(x,y).
  6. 𝒫 x L y 2 \mathcal{P}_{x}\otimes L^{2}_{y}
  7. σ = ( , σ - 2 , σ - 1 , σ 0 , σ 1 , σ 2 , ) \sigma=\left(\ldots,\sigma_{-2},\sigma_{-1},\sigma_{0},\sigma_{1},\sigma_{2},% \ldots\right)
  8. σ k { 0 , 1 } \sigma_{k}\in\{0,1\}
  9. τ ( , σ k , σ k + 1 , σ k + 2 , ) = ( , σ k - 1 , σ k , σ k + 1 , ) \tau(\ldots,\sigma_{k},\sigma_{k+1},\sigma_{k+2},\ldots)=(\ldots,\sigma_{k-1},% \sigma_{k},\sigma_{k+1},\ldots)
  10. 0 x , y 1 0\leq x,y\leq 1
  11. x ( σ ) = k = 0 σ k 2 - ( k + 1 ) x(\sigma)=\sum_{k=0}^{\infty}\sigma_{k}2^{-(k+1)}
  12. y ( σ ) = k = 0 σ - k - 1 2 - ( k + 1 ) . y(\sigma)=\sum_{k=0}^{\infty}\sigma_{-k-1}2^{-(k+1)}.
  13. τ ( x , y ) = ( x + 2 y 2 , 2 y - 2 y ) \tau(x,y)=\left(\frac{x+\left\lfloor 2y\right\rfloor}{2}\,,\,2y-\left\lfloor 2% y\right\rfloor\right)

Balance_wheel.html

  1. T = 2 π I κ T=2\pi\sqrt{\frac{I}{\kappa}}\,
  2. l θ = l 0 ( 1 + α θ + β θ 2 ) l_{\theta}=l_{0}(1+\alpha\theta+\beta\theta^{2})\,
  3. l 0 \scriptstyle l_{0}
  4. θ \scriptstyle\theta
  5. l θ \scriptstyle l_{\theta}
  6. θ \scriptstyle\theta
  7. α \scriptstyle\alpha
  8. β \scriptstyle\beta

Balanced_budget.html

  1. Y 1 = c 0 + c 1 ( Y - T ) + I + G Y_{1}=c_{0}+c_{1}\left(Y-T\right)+I+G
  2. Y 1 = 1 1 - c 1 ( c 0 + I + G - c 1 T ) Y_{1}=\frac{1}{1-c_{1}}\left(c_{0}+I+G-c_{1}T\right)
  3. G = G + α G=G+\alpha\,
  4. T = T + α T=T+\alpha\,
  5. Y 2 = 1 1 - c 1 ( c 0 + I + ( G + α ) - c 1 ( T + α ) ) Y_{2}=\frac{1}{1-c_{1}}\left(c_{0}+I+\left(G+\alpha\right)-c_{1}\left(T+\alpha% \right)\right)
  6. Δ Y = Y 2 - Y 1 = α 1 - c 1 ( 1 - c 1 ) = α \Delta Y=Y_{2}-Y_{1}=\frac{\alpha}{1-c_{1}}\left(1-c_{1}\right)=\alpha
  7. Δ T - Δ G = α - α = 0 \Delta T-\Delta G=\alpha-\alpha=0\,
  8. Y = C + I + G Y=C+I+G
  9. C = b ( Y - T ) C=b(Y-T)
  10. T = t Y T=tY
  11. G = t Y G=tY
  12. Y = b ( Y - t Y ) + I + t Y Y=b(Y-tY)+I+tY
  13. Y = b Y - b t Y + t Y + I Y=bY-btY+tY+I
  14. Y ( 1 - ( b - b t + t ) ) = I Y(1-(b-bt+t))=I
  15. Y = 1 1 - b + b t - t ( I ) Y=\frac{1}{1-b+bt-t}(I)
  16. Y = 1 1 - b + b t - t Y^{\prime}=\frac{1}{1-b+bt-t}

Balanced_set.html

  1. α S S \alpha S\subseteq S
  2. α S := { α x x S } . \alpha S:=\{\alpha x\mid x\in S\}.

Band_matrix.html

  1. a i , j = 0 if j < i - k 1 or j > i + k 2 ; k 1 , k 2 0. a_{i,j}=0\quad\mbox{if}~{}\quad j<i-k_{1}\quad\mbox{ or }~{}\quad j>i+k_{2};% \quad k_{1},k_{2}\geq 0.\,
  2. a i , j = 0 a_{i,j}=0
  3. | i - j | > k |i-j|>k
  4. [ B 11 B 12 0 0 B 21 B 22 B 23 0 B 32 B 33 B 34 B 43 B 44 B 45 0 B 54 B 55 B 56 0 0 B 65 B 66 ] \begin{bmatrix}B_{11}&B_{12}&0&\cdots&\cdots&0\\ B_{21}&B_{22}&B_{23}&\ddots&\ddots&\vdots\\ 0&B_{32}&B_{33}&B_{34}&\ddots&\vdots\\ \vdots&\ddots&B_{43}&B_{44}&B_{45}&0\\ \vdots&\ddots&\ddots&B_{54}&B_{55}&B_{56}\\ 0&\cdots&\cdots&0&B_{65}&B_{66}\end{bmatrix}
  5. [ 0 B 11 B 12 B 21 B 22 B 23 B 32 B 33 B 34 B 43 B 44 B 45 B 54 B 55 B 56 B 65 B 66 0 ] . \begin{bmatrix}0&B_{11}&B_{12}\\ B_{21}&B_{22}&B_{23}\\ B_{32}&B_{33}&B_{34}\\ B_{43}&B_{44}&B_{45}\\ B_{54}&B_{55}&B_{56}\\ B_{65}&B_{66}&0\end{bmatrix}.
  6. [ A 11 A 12 A 13 0 0 A 22 A 23 A 24 A 33 A 34 A 35 0 A 44 A 45 A 46 s y m A 55 A 56 A 66 ] . \begin{bmatrix}A_{11}&A_{12}&A_{13}&0&\cdots&0\\ &A_{22}&A_{23}&A_{24}&\ddots&\vdots\\ &&A_{33}&A_{34}&A_{35}&0\\ &&&A_{44}&A_{45}&A_{46}\\ &sym&&&A_{55}&A_{56}\\ &&&&&A_{66}\end{bmatrix}.
  7. [ A 11 A 12 A 13 A 22 A 23 A 24 A 33 A 34 A 35 A 44 A 45 A 46 A 55 A 56 0 A 66 0 0 ] . \begin{bmatrix}A_{11}&A_{12}&A_{13}\\ A_{22}&A_{23}&A_{24}\\ A_{33}&A_{34}&A_{35}\\ A_{44}&A_{45}&A_{46}\\ A_{55}&A_{56}&0\\ A_{66}&0&0\end{bmatrix}.

Band_sum.html

  1. p 1 K 1 p_{1}\in K_{1}
  2. p 2 K 2 p_{2}\in K_{2}
  3. h h
  4. K 1 K 2 K_{1}\sqcup K_{2}
  5. p 1 p 2 p_{1}\sqcup p_{2}

Banks–Zaks_fixed_point.html

  1. β ( g ) = - b 0 g 3 + b 1 g 5 + 𝒪 ( g 7 ) \beta(g)=-b_{0}g^{3}+b_{1}g^{5}+\mathcal{O}(g^{7})\,
  2. b 0 b_{0}
  3. b 1 b_{1}
  4. g = g g=g_{\ast}
  5. β ( g ) = 0 \beta(g_{\ast})=0
  6. g 2 = b 0 b 1 . g_{\ast}^{2}=\frac{b_{0}}{b_{1}}.
  7. b 0 b_{0}
  8. b 1 b_{1}
  9. g 2 < 1 g^{2}_{\ast}<1
  10. g g_{\ast}
  11. S U ( N c ) SU(N_{c})
  12. b 0 = 1 16 π 2 1 3 ( 11 N c - 2 N f ) and b 1 = - 1 ( 16 π 2 ) 2 ( 34 3 N c 2 - 1 2 N f ( 2 N c 2 - 1 N c + 20 3 N c ) ) b_{0}=\frac{1}{16\pi^{2}}\frac{1}{3}(11N_{c}-2N_{f})\;\;\;\;\,\text{ and }\;\;% \;\;b_{1}=-\frac{1}{(16\pi^{2})^{2}}\left(\frac{34}{3}N_{c}^{2}-\frac{1}{2}N_{% f}\left(2\frac{N_{c}^{2}-1}{N_{c}}+\frac{20}{3}N_{c}\right)\right)
  13. N c N_{c}
  14. N f N_{f}
  15. N f N_{f}
  16. 11 2 N c \tfrac{11}{2}N_{c}
  17. N f N_{f}
  18. 11 2 N c > N f > 68 N c 2 ( 16 + 20 N c ) \frac{11}{2}N_{c}>N_{f}>\frac{68N_{c}^{2}}{(16+20N_{c})}
  19. b 1 > 0 b_{1}>0
  20. b 1 b_{1}
  21. - b 0 -b_{0}
  22. β ( g ) = 0 \beta(g)=0
  23. g g

Barnett_effect.html

  1. M = χ ω / γ , M=\chi\omega/\gamma\ ,

Barrelled_space.html

  1. X X
  2. X X^{\prime}
  3. σ ( X , X ) \sigma(X^{\prime},X)
  4. β ( X , X ) \beta(X,X^{\prime})
  5. X X
  6. E β E_{\beta}^{\prime}
  7. X X
  8. 0
  9. X X
  10. X X
  11. X X^{\prime}
  12. X X
  13. X X
  14. β ( X , X ) \beta(X^{\prime},X)
  15. X X^{\prime}

Barrett–Crane_model.html

  1. B B
  2. s o ( 3 , 1 ) so(3,1)
  3. B i j B k l B^{ij}\wedge B^{kl}
  4. B i j B^{ij}
  5. ± e i e j \pm e^{i}\wedge e^{j}
  6. ± ϵ i j k l e k e l \pm\epsilon^{ijkl}e_{k}\wedge e_{l}
  7. e i e^{i}
  8. ϵ i j k l \epsilon^{ijkl}
  9. s o ( 3 , 1 ) so(3,1)
  10. B B
  11. B i j = e i e j B^{ij}=e^{i}\wedge e^{j}

Barrier_option.html

  1. C = C i n + C o u t C=C_{in}+C_{out}

Bas_van_Fraassen.html

  1. x P x x P x \forall x\,Px\Rightarrow\exists x\,Px
  2. ( x P x and x ( x = a ) ) x P x (\forall x\,Px\and\exists x\,(x=a))\Rightarrow\exists x\,Px

Base_change.html

  1. f : Y X f:Y\rightarrow X
  2. g : X X g:X^{\prime}\rightarrow X
  3. f f^{\prime}
  4. g g^{\prime}
  5. Y := Y × X X Y^{\prime}:=Y\times_{X}X^{\prime}
  6. X X^{\prime}
  7. Y Y
  8. \mathcal{F}
  9. f * R i g * R i g * f * . f^{*}R^{i}g_{*}\mathcal{F}\rightarrow R^{i}g^{\prime}_{*}f^{\prime*}\mathcal{F}.
  10. R i g * R^{i}g_{*}\mathcal{F}
  11. \mathcal{F}
  12. \mathcal{F}
  13. X X^{\prime}

Battle_of_the_sexes_(game_theory).html

  1. M F = W O + 3 W F W O + 3 W F + 2 W O + 0 W F = W O + 3 W F 3 W O + 3 W F M_{F}=\frac{W_{O}+3W_{F}}{W_{O}+3W_{F}+2W_{O}+0W_{F}}=\frac{W_{O}+3W_{F}}{3W_{% O}+3W_{F}}
  2. W O + W F = 1 W_{O}+W_{F}=1
  3. M F = 1 3 ( W O + 3 W F ) M_{F}=\tfrac{1}{3}(W_{O}+3W_{F})
  4. M O \displaystyle M_{O}
  5. M F M_{F}
  6. M F \displaystyle M_{F}
  7. M F M_{F}
  8. M F = 3 5 M_{F}=\tfrac{3}{5}
  9. M F + M O = 1 M_{F}+M_{O}=1
  10. M O = 1 - 3 5 = 2 5 M_{O}=1-\tfrac{3}{5}=\tfrac{2}{5}
  11. W F = 2 3 M F = 2 5 W_{F}=\tfrac{2}{3}M_{F}=\tfrac{2}{5}
  12. W O = 1 - 2 5 = 3 5 W_{O}=1-\tfrac{2}{5}=\tfrac{3}{5}
  13. P c Pc
  14. P c = M F W F + M O W O = 3 5 2 5 + 2 5 3 5 = 12 25 Pc=M_{F}W_{F}+M_{O}W_{O}=\tfrac{3}{5}\tfrac{2}{5}+\tfrac{2}{5}\tfrac{3}{5}=% \tfrac{12}{25}
  15. P m Pm
  16. P m = M F W O + M O W F = 3 5 3 5 + 2 5 2 5 = 13 25 Pm=M_{F}W_{O}+M_{O}W_{F}=\tfrac{3}{5}\tfrac{3}{5}+\tfrac{2}{5}\tfrac{2}{5}=% \tfrac{13}{25}
  17. P c + P m = 12 25 + 13 25 = 25 25 = 1 Pc+Pm=\tfrac{12}{25}+\tfrac{13}{25}=\tfrac{25}{25}=1
  18. 13 25 \tfrac{13}{25}
  19. E m Em
  20. E w Ew
  21. E m m f w f Emmfwf
  22. E m = M F W F E m m f w f + M F W O E m m f w o + M O W O E m m o w o + M O W F E m m o w f Em=M_{F}W_{F}Emmfwf+M_{F}W_{O}Emmfwo+M_{O}W_{O}Emmowo+M_{O}W_{F}Emmowf
  23. E m = 3 5 2 5 3 + 3 5 3 5 1 + 2 5 3 5 2 + 2 5 2 5 0 = 39 25 Em=\tfrac{3}{5}\tfrac{2}{5}3+\tfrac{3}{5}\tfrac{3}{5}1+\tfrac{2}{5}\tfrac{3}{5% }2+\tfrac{2}{5}\tfrac{2}{5}0=\tfrac{39}{25}
  24. 6 5 \tfrac{6}{5}
  25. M F = 1 M_{F}=1
  26. M O = 0 M_{O}=0
  27. W F = 2 3 M F = 2 3 W_{F}=\tfrac{2}{3}M_{F}=\tfrac{2}{3}
  28. W O = 1 - 2 3 = 1 3 W_{O}=1-\tfrac{2}{3}=\tfrac{1}{3}
  29. E m = M F W F E m m f w f + M F W O E m m f w o + M O W O E m m o w o + M O W F E m m o w f Em=M_{F}W_{F}Emmfwf+M_{F}W_{O}Emmfwo+M_{O}W_{O}Emmowo+M_{O}W_{F}Emmowf
  30. E m = 1 2 3 3 + 1 1 3 1 + 0 1 3 2 + 0 2 3 0 = 7 3 Em=1\tfrac{2}{3}3+1\tfrac{1}{3}1+0\tfrac{1}{3}2+0\tfrac{2}{3}0=\tfrac{7}{3}
  31. E w Ew

Bayesian_experimental_design.html

  1. θ \theta\,
  2. y y\,
  3. ξ \xi\,
  4. p ( y | θ , ξ ) p(y|\theta,\xi)\,
  5. y y
  6. θ \theta
  7. ξ \xi
  8. p ( θ ) p(\theta)\,
  9. p ( y | ξ ) p(y|\xi)\,
  10. p ( θ | y , ξ ) p(\theta|y,\xi)\,
  11. U ( ξ ) U(\xi)\,
  12. ξ \xi
  13. U ( y , ξ ) U(y,\xi)\,
  14. y y
  15. ξ \xi
  16. θ \theta
  17. p ( θ ) p(\theta)
  18. p ( y | θ , ξ ) p(y|\theta,\xi)
  19. y y
  20. θ \theta
  21. ξ \xi
  22. p ( θ | y , ξ ) = p ( y | θ , ξ ) p ( θ ) p ( y | ξ ) , p(\theta|y,\xi)=\frac{p(y|\theta,\xi)p(\theta)}{p(y|\xi)}\,,
  23. p ( y | ξ ) p(y|\xi)
  24. p ( y | ξ ) = p ( θ ) p ( y | θ , ξ ) d θ . p(y|\xi)=\int{p(\theta)p(y|\theta,\xi)d\theta}\,.
  25. ξ \xi
  26. U ( ξ ) = p ( y | ξ ) U ( y , ξ ) d y , U(\xi)=\int{p(y|\xi)U(y,\xi)dy}\,,
  27. U ( y , ξ ) U(y,\xi)
  28. p ( θ | y , ξ ) p(\theta|y,\xi)
  29. y y
  30. ξ \xi
  31. U ( y , ξ ) = log ( p ( θ | y , ξ ) ) p ( θ | y , ξ ) d θ - log ( p ( θ ) ) p ( θ ) d θ . U(y,\xi)=\int{\log(p(\theta|y,\xi))p(\theta|y,\xi)d\theta}-\int{\log(p(\theta)% )p(\theta)d\theta}\,.
  32. U ( y , ξ ) = D K L ( p ( θ | y , ξ ) p ( θ | ξ ) ) , U(y,\xi)=D_{KL}(p(\theta|y,\xi)\|p(\theta|\xi))\,,
  33. U ( ξ ) = log ( p ( θ | y , ξ ) ) p ( θ , y | ξ ) d θ d y - log ( p ( θ ) ) p ( θ ) d θ = log ( p ( y | θ , ξ ) ) p ( θ , y | ξ ) d y d θ - log ( p ( y | ξ ) ) p ( y | ξ ) d y , \begin{aligned}\displaystyle U(\xi)&\displaystyle=\int{\int{\log(p(\theta|y,% \xi))p(\theta,y|\xi)d\theta}dy}-\int{\log(p(\theta))p(\theta)d\theta}\\ &\displaystyle=\int{\int{\log(p(y|\theta,\xi))p(\theta,y|\xi)dy}d\theta}-\int{% \log(p(y|\xi))p(y|\xi)dy},\end{aligned}\,
  34. p ( θ | y , ξ ) p(\theta|y,\xi)
  35. y y
  36. ξ \xi
  37. p ( θ ) log p ( θ ) p(\theta)\log p(\theta)
  38. ξ \xi
  39. U ( ξ ) = I ( θ ; y ) , U(\xi)=I(\theta;y)\,,

Bayesian_game.html

  1. p ( types of other players | type of this player ) p(\mbox{types of other players}~{}|\mbox{type of this player}~{})
  2. U ( x , y ) U(x,y)
  3. U ( x * , t ) U(x^{*},t)
  4. x * x^{*}
  5. G = N , Ω , A i , u i , T i , τ i , p i , C i i N G=\langle N,\Omega,\langle A_{i},u_{i},T_{i},\tau_{i},p_{i},C_{i}\rangle_{i\in N}\rangle
  6. N N
  7. Ω \Omega
  8. A i A_{i}
  9. i i
  10. A = A 1 × A 2 × × A N A=A_{1}\times A_{2}\times\cdots\times A_{N}
  11. T i T_{i}
  12. i i
  13. τ i : Ω T i \tau_{i}\colon\Omega\rightarrow T_{i}
  14. C i A i × T i C_{i}\subseteq A_{i}\times T_{i}
  15. i i
  16. T i T_{i}
  17. u i : Ω × A R u_{i}\colon\Omega\times A\rightarrow R
  18. i i
  19. L = { ( ω , a 1 , , a N ) ω Ω , i , ( a i , τ i ( ω ) ) C i } L=\{(\omega,a_{1},\ldots,a_{N})\mid\omega\in\Omega,\forall i,(a_{i},\tau_{i}(% \omega))\in C_{i}\}
  20. u i : L R u_{i}\colon L\rightarrow R
  21. p i p_{i}
  22. Ω \Omega
  23. i i
  24. s i : T i A i s_{i}\colon T_{i}\rightarrow A_{i}
  25. ( s i ( t i ) , t i ) C i (s_{i}(t_{i}),t_{i})\in C_{i}
  26. t i t_{i}
  27. i i
  28. u i ( S ) = E ω p i [ u i ( ω , s 1 ( τ 1 ( ω ) ) , , s N ( τ N ( ω ) ) ) ] u_{i}(S)=E_{\omega\sim p_{i}}[u_{i}(\omega,s_{1}(\tau_{1}(\omega)),\ldots,s_{N% }(\tau_{N}(\omega)))]
  29. S i S_{i}
  30. S i = { s i : T i A i ( s i ( t i ) , t i ) C i , t i } . S_{i}=\{s_{i}\colon T_{i}\rightarrow A_{i}\mid(s_{i}(t_{i}),t_{i})\in C_{i},% \forall t_{i}\}.
  31. G G
  32. G ^ = N , A ^ = S 1 × S 2 × × S N , u ^ = u \hat{G}=\langle N,\hat{A}=S_{1}\times S_{2}\times\cdots\times S_{N},\hat{u}=u\rangle
  33. G G

Bayesian_search_theory.html

  1. p = p ( 1 - q ) ( 1 - p ) + p ( 1 - q ) = p 1 - q 1 - p q < p . p^{\prime}=\frac{p(1-q)}{(1-p)+p(1-q)}=p\frac{1-q}{1-pq}<p.
  2. r = r 1 1 - p q > r . r^{\prime}=r\frac{1}{1-pq}>r.

Bead_sort.html

  1. n \sqrt{n}

Beamforming.html

  1. L L
  2. P P
  3. 1 σ n 2 P L \frac{1}{\sigma_{n}^{2}}P\cdot L
  4. σ n 2 \sigma_{n}^{2}

Beer_can_pyramid.html

  1. n = 1 N n 2 \!\ \sum_{n=1}^{N}n^{2}
  2. n = 1 N i = 1 n i \!\ \sum_{n=1}^{N}\sum_{i=1}^{n}i

Bejan_number.html

  1. Be = S ˙ gen , Δ T S ˙ gen , Δ T + S ˙ gen , Δ p \mathrm{Be}=\frac{\dot{S}^{\prime}_{\mathrm{gen},\,\Delta T}}{\dot{S}^{\prime}% _{\mathrm{gen},\,\Delta T}+\dot{S}^{\prime}_{\mathrm{gen},\,\Delta p}}
  2. S ˙ gen , Δ T \dot{S}^{\prime}_{\mathrm{gen},\,\Delta T}
  3. S ˙ gen , Δ p \dot{S}^{\prime}_{\mathrm{gen},\,\Delta p}
  4. L L
  5. Be = Δ P L 2 μ ν \mathrm{Be}=\frac{\Delta PL^{2}}{\mu\nu}
  6. μ \mu
  7. ν \nu
  8. L L
  9. Be = Δ P L 2 μ α \mathrm{Be}=\frac{\Delta PL^{2}}{\mu\alpha}
  10. μ \mu
  11. α \alpha
  12. L L
  13. Be = Δ P L 2 μ D \mathrm{Be}=\frac{\Delta PL^{2}}{\mu D}
  14. μ \mu
  15. D D
  16. Be = Δ P L 2 ρ δ 2 \mathrm{Be}=\frac{\Delta PL^{2}}{\rho\delta^{2}}
  17. ρ \rho
  18. δ \delta
  19. Be = 32 R e L 3 d 3 \mathrm{Be}={{32ReL^{3}}\over{d^{3}}}
  20. R e Re
  21. L L
  22. d d

Bekenstein_bound.html

  1. S 2 π k R E c S\leq\frac{2\pi kRE}{\hbar c}
  2. I 2 π R E c ln 2 I\leq\frac{2\pi RE}{\hbar c\ln 2}
  3. I 2 π c R m ln 2 2.577 × 10 43 m R I\leq\frac{2\pi cRm}{\hbar\ln 2}\approx 2.577\times 10^{43}mR
  4. m m
  5. R R
  6. S = k A 4 S=\frac{kA}{4}
  7. G / c 3 \hbar G/c^{3}
  8. 2.6 × 10 42 \approx 2.6\times 10^{42}
  9. O = 2 I O=2^{I}
  10. 10 7.8 × 10 41 \approx 10^{7.8\times 10^{41}}

Bell_series.html

  1. f f
  2. p p
  3. f p ( x ) f_{p}(x)
  4. f f
  5. p p
  6. f p ( x ) = n = 0 f ( p n ) x n . f_{p}(x)=\sum_{n=0}^{\infty}f(p^{n})x^{n}.
  7. f f
  8. g g
  9. f = g f=g
  10. f p ( x ) = g p ( x ) f_{p}(x)=g_{p}(x)
  11. p p
  12. f f
  13. g g
  14. h = f * g h=f*g
  15. p p
  16. h p ( x ) = f p ( x ) g p ( x ) . h_{p}(x)=f_{p}(x)g_{p}(x).\,
  17. f f
  18. f p ( x ) = 1 1 - f ( p ) x . f_{p}(x)=\frac{1}{1-f(p)x}.
  19. μ \mu
  20. μ p ( x ) = 1 - x . \mu_{p}(x)=1-x.
  21. φ \varphi
  22. φ p ( x ) = 1 - x 1 - p x . \varphi_{p}(x)=\frac{1-x}{1-px}.
  23. δ \delta
  24. δ p ( x ) = 1. \delta_{p}(x)=1.
  25. λ \lambda
  26. λ p ( x ) = 1 1 + x . \lambda_{p}(x)=\frac{1}{1+x}.
  27. ( Id k ) p ( x ) = 1 1 - p k x . (\textrm{Id}_{k})_{p}(x)=\frac{1}{1-p^{k}x}.
  28. Id k ( n ) = n k \operatorname{Id}_{k}(n)=n^{k}
  29. σ k \sigma_{k}
  30. ( σ k ) p ( x ) = 1 1 - ( 1 + p k ) x + p k x 2 . (\sigma_{k})_{p}(x)=\frac{1}{1-(1+p^{k})x+p^{k}x^{2}}.

Bellman_equation.html

  1. c ( W ) c(W)
  2. H ( W ) H(W)
  3. t t
  4. x t x_{t}
  5. x 0 x_{0}
  6. a t Γ ( x t ) a_{t}\in\Gamma(x_{t})
  7. a t a_{t}
  8. x x
  9. T ( x , a ) T(x,a)
  10. a a
  11. a a
  12. x x
  13. F ( x , a ) F(x,a)
  14. 0 < β < 1 0<\beta<1
  15. V ( x 0 ) = max { a t } t = 0 t = 0 β t F ( x t , a t ) , V(x_{0})\;=\;\max_{\left\{a_{t}\right\}_{t=0}^{\infty}}\sum_{t=0}^{\infty}% \beta^{t}F(x_{t},a_{t}),
  16. a t Γ ( x t ) , x t + 1 = T ( x t , a t ) , t = 0 , 1 , 2 , a_{t}\in\Gamma(x_{t}),\;x_{t+1}=T(x_{t},a_{t}),\;\forall t=0,1,2,\dots
  17. V ( x 0 ) V(x_{0})
  18. x 0 x_{0}
  19. x 1 x_{1}
  20. max a 0 { F ( x 0 , a 0 ) + β [ max { a t } t = 1 t = 1 β t - 1 F ( x t , a t ) : a t Γ ( x t ) , x t + 1 = T ( x t , a t ) , t 1 ] } \max_{a_{0}}\left\{F(x_{0},a_{0})+\beta\left[\max_{\left\{a_{t}\right\}_{t=1}^% {\infty}}\sum_{t=1}^{\infty}\beta^{t-1}F(x_{t},a_{t}):a_{t}\in\Gamma(x_{t}),\;% x_{t+1}=T(x_{t},a_{t}),\;\forall t\geq 1\right]\right\}
  21. a 0 Γ ( x 0 ) , x 1 = T ( x 0 , a 0 ) . a_{0}\in\Gamma(x_{0}),\;x_{1}=T(x_{0},a_{0}).
  22. a 0 a_{0}
  23. x 1 = T ( x 0 , a 0 ) x_{1}=T(x_{0},a_{0})
  24. x 1 = T ( x 0 , a 0 ) x_{1}=T(x_{0},a_{0})
  25. V ( x 0 ) = max a 0 { F ( x 0 , a 0 ) + β V ( x 1 ) } V(x_{0})=\max_{a_{0}}\{F(x_{0},a_{0})+\beta V(x_{1})\}
  26. a 0 Γ ( x 0 ) , x 1 = T ( x 0 , a 0 ) . a_{0}\in\Gamma(x_{0}),\;x_{1}=T(x_{0},a_{0}).
  27. V ( x ) = max a Γ ( x ) { F ( x , a ) + β V ( T ( x , a ) ) } . V(x)=\max_{a\in\Gamma(x)}\{F(x,a)+\beta V(T(x,a))\}.
  28. max t = 0 β t u ( c t ) \max\sum_{t=0}^{\infty}\beta^{t}u(c_{t})
  29. a t + 1 = ( 1 + r ) ( a t - c t ) , c t 0 , a_{t+1}=(1+r)(a_{t}-c_{t}),\;c_{t}\geq 0,
  30. lim t a t 0. \lim_{t\rightarrow\infty}a_{t}\geq 0.
  31. V ( a ) = max 0 c a { u ( c ) + β V ( ( 1 + r ) ( a - c ) ) } , V(a)=\max_{0\leq c\leq a}\{u(c)+\beta V((1+r)(a-c))\},
  32. max E ( t = 0 β t u ( c t ) ) . \max E(\sum_{t=0}^{\infty}\beta^{t}u(c_{t})).
  33. V ( a , r ) = max 0 c a { u ( c ) + β V ( ( 1 + r ) ( a - c ) , r ) Q ( r , d μ r ) } . V(a,r)=\max_{0\leq c\leq a}\{u(c)+\beta\int V((1+r)(a-c),r^{\prime})Q(r,d\mu_{% r})\}.
  34. V ( x , z ) = max c Γ ( x , z ) F ( x , c , z ) + β V ( T ( x , c ) , z ) d μ z ( z ) . V(x,z)=\max_{c\in\Gamma(x,z)}F(x,c,z)+\beta\int V(T(x,c),z^{\prime})d\mu_{z}(z% ^{\prime}).
  35. π \pi
  36. V π ( s ) = R ( s , π ( s ) ) + γ s P ( s | s , π ( s ) ) V π ( s ) . V^{\pi}(s)=R(s,\pi(s))+\gamma\sum_{s^{\prime}}P(s^{\prime}|s,\pi(s))V^{\pi}(s^% {\prime}).
  37. π \pi
  38. V * ( s ) = max a R ( s , a ) + γ s P ( s | s , a ) V * ( s ) . V^{*}(s)=\max_{a}R(s,a)+\gamma\sum_{s^{\prime}}P(s^{\prime}|s,a)V^{*}(s^{% \prime}).

Belt_(mechanical).html

  1. P = ( T 1 - T 2 ) v P=(T_{1}-T_{2})v
  2. T 1 T 2 = e μ α \frac{T_{1}}{T_{2}}=e^{\mu\alpha}

Bending.html

  1. w w
  2. d 2 w ( x ) d x 2 = M ( x ) E ( x ) I ( x ) \cfrac{\mathrm{d}^{2}w(x)}{\mathrm{d}x^{2}}=\frac{M(x)}{E(x)I(x)}
  3. x x
  4. E E
  5. I I
  6. M M
  7. q ( x ) q(x)
  8. E I d 4 w ( x ) d x 4 = q ( x ) EI~{}\cfrac{\mathrm{d}^{4}w(x)}{\mathrm{d}x^{4}}=q(x)
  9. M M
  10. Q Q
  11. M ( x ) = - E I d 2 w d x 2 ; Q ( x ) = d M d x . M(x)=-EI~{}\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2}}~{};~{}~{}Q(x)=\cfrac{% \mathrm{d}M}{\mathrm{d}x}.
  12. σ = M y I x {\sigma}=\frac{My}{I_{x}}
  13. σ {\sigma}
  14. σ = M y I x \sigma=\tfrac{My}{I_{x}}
  15. σ x ( y , z ) = - ( M z I y + M y I y z ) I y I z - I y z 2 y + ( M y I z + M z I y z ) I y I z - I y z 2 z {\sigma_{x}}(y,z)=-\frac{(M_{z}~{}I_{y}+M_{y}~{}I_{yz})}{I_{y}~{}I_{z}-I_{yz}^% {2}}y+\frac{(M_{y}~{}I_{z}+M_{z}~{}I_{yz})}{I_{y}~{}I_{z}-I_{yz}^{2}}z
  16. y , z y,z
  17. M y M_{y}
  18. M z M_{z}
  19. I y I_{y}
  20. I z I_{z}
  21. I y z I_{yz}
  22. M y , M z , I y , I z , I y z M_{y},M_{z},I_{y},I_{z},I_{yz}
  23. ρ \rho
  24. ρ < 10 h . \rho<10h.
  25. σ = F A + M ρ A + M I x y ρ ρ + y \sigma=\frac{F}{A}+\frac{M}{\rho A}+{\frac{M}{{I_{x}}^{\prime}}}y{\frac{\rho}{% \rho+y}}
  26. F F
  27. A A
  28. M M
  29. ρ \rho
  30. I x {{I_{x}}^{\prime}}
  31. y y
  32. y y
  33. σ \sigma
  34. ρ \rho
  35. y ρ y\ll\rho
  36. σ = F A ± M y I \sigma={F\over A}\pm\frac{My}{I}
  37. E I d 4 w d x 4 = q ( x ) - E I k A G d 2 q d x 2 EI~{}\cfrac{\mathrm{d}^{4}w}{\mathrm{d}x^{4}}=q(x)-\cfrac{EI}{kAG}~{}\cfrac{% \mathrm{d}^{2}q}{\mathrm{d}x^{2}}
  38. I I
  39. A A
  40. G G
  41. k k
  42. ν \nu
  43. k = 5 + 5 ν 6 + 5 ν k=\cfrac{5+5\nu}{6+5\nu}
  44. φ ( x ) \varphi(x)
  45. d φ d x = - d 2 w d x 2 - q ( x ) k A G \cfrac{\mathrm{d}\varphi}{\mathrm{d}x}=-\cfrac{\mathrm{d}^{2}w}{\mathrm{d}x^{2% }}-\cfrac{q(x)}{kAG}
  46. M M
  47. Q Q
  48. M ( x ) = - E I d φ d x ; Q ( x ) = k A G ( d w d x - φ ) = - E I d 2 φ d x 2 = d M d x M(x)=-EI~{}\cfrac{\mathrm{d}\varphi}{\mathrm{d}x}~{};~{}~{}Q(x)=kAG\left(% \cfrac{\mathrm{d}w}{\mathrm{d}x}-\varphi\right)=-EI~{}\cfrac{\mathrm{d}^{2}% \varphi}{\mathrm{d}x^{2}}=\cfrac{\mathrm{d}M}{\mathrm{d}x}
  49. q ( x , t ) q(x,t)
  50. E I 4 w x 4 + m 2 w t 2 = q ( x , t ) EI~{}\cfrac{\partial^{4}w}{\partial x^{4}}+m~{}\cfrac{\partial^{2}w}{\partial t% ^{2}}=q(x,t)
  51. E E
  52. I I
  53. w ( x , t ) w(x,t)
  54. m m
  55. E I 4 w x 4 + m 2 w t 2 = 0 EI~{}\cfrac{\partial^{4}w}{\partial x^{4}}+m~{}\cfrac{\partial^{2}w}{\partial t% ^{2}}=0
  56. w ( x , t ) = Re [ w ^ ( x ) e - i ω t ] 2 w t 2 = - ω 2 w ( x , t ) w(x,t)=\,\text{Re}[\hat{w}(x)~{}e^{-i\omega t}]\quad\implies\quad\cfrac{% \partial^{2}w}{\partial t^{2}}=-\omega^{2}~{}w(x,t)
  57. E I d 4 w ^ d x 4 - m ω 2 w ^ = 0 EI~{}\cfrac{\mathrm{d}^{4}\hat{w}}{\mathrm{d}x^{4}}-m\omega^{2}\hat{w}=0
  58. w ^ = A 1 cosh ( β x ) + A 2 sinh ( β x ) + A 3 cos ( β x ) + A 4 sin ( β x ) \hat{w}=A_{1}\cosh(\beta x)+A_{2}\sinh(\beta x)+A_{3}\cos(\beta x)+A_{4}\sin(% \beta x)
  59. A 1 , A 2 , A 3 , A 4 A_{1},A_{2},A_{3},A_{4}
  60. β := ( m E I ω 2 ) 1 / 4 \beta:=\left(\cfrac{m}{EI}~{}\omega^{2}\right)^{1/4}
  61. E I 4 w x 4 + m 2 w t 2 - ( J + E I m k A G ) 4 w x 2 t 2 + J m k A G 4 w t 4 = q ( x , t ) + J k A G 2 q t 2 - E I k A G 2 q x 2 EI~{}\cfrac{\partial^{4}w}{\partial x^{4}}+m~{}\cfrac{\partial^{2}w}{\partial t% ^{2}}-\left(J+\cfrac{EIm}{kAG}\right)\cfrac{\partial^{4}w}{\partial x^{2}~{}% \partial t^{2}}+\cfrac{Jm}{kAG}~{}\cfrac{\partial^{4}w}{\partial t^{4}}=q(x,t)% +\cfrac{J}{kAG}~{}\cfrac{\partial^{2}q}{\partial t^{2}}-\cfrac{EI}{kAG}~{}% \cfrac{\partial^{2}q}{\partial x^{2}}
  62. J = m I A J=\tfrac{mI}{A}
  63. m = ρ A m=\rho A
  64. ρ \rho
  65. A A
  66. G G
  67. k k
  68. ν \nu
  69. k = 5 + 5 ν 6 + 5 ν rectangular cross-section = 6 + 12 ν + 6 ν 2 7 + 12 ν + 4 ν 2 circular cross-section \begin{array}[]{rcl}k&=&\tfrac{5+5\nu}{6+5\nu}\quad\,\text{rectangular cross-% section}\\ &=&\tfrac{6+12\nu+6\nu^{2}}{7+12\nu+4\nu^{2}}\quad\,\text{circular cross-% section}\end{array}
  70. E I d 4 w ^ d x 4 + m ω 2 ( J m + E I k A G ) d 2 w ^ d x 2 + m ω 2 ( ω 2 J k A G - 1 ) w ^ = 0 EI~{}\cfrac{\mathrm{d}^{4}\hat{w}}{\mathrm{d}x^{4}}+m\omega^{2}\left(\cfrac{J}% {m}+\cfrac{EI}{kAG}\right)\cfrac{\mathrm{d}^{2}\hat{w}}{\mathrm{d}x^{2}}+m% \omega^{2}\left(\cfrac{\omega^{2}J}{kAG}-1\right)~{}\hat{w}=0
  71. w w
  72. e k x e^{kx}
  73. α k 4 + β k 2 + γ = 0 ; α := E I , β := m ω 2 ( J m + E I k A G ) , γ := m ω 2 ( ω 2 J k A G - 1 ) \alpha~{}k^{4}+\beta~{}k^{2}+\gamma=0~{};~{}~{}\alpha:=EI~{},~{}~{}\beta:=m% \omega^{2}\left(\cfrac{J}{m}+\cfrac{EI}{kAG}\right)~{},~{}~{}\gamma:=m\omega^{% 2}\left(\cfrac{\omega^{2}J}{kAG}-1\right)
  74. k 1 = + z + , k 2 = - z + , k 3 = + z - , k 4 = - z - k_{1}=+\sqrt{z_{+}}~{},~{}~{}k_{2}=-\sqrt{z_{+}}~{},~{}~{}k_{3}=+\sqrt{z_{-}}~% {},~{}~{}k_{4}=-\sqrt{z_{-}}
  75. z + := - β + β 2 - 4 α γ 2 α , z - := - β - β 2 - 4 α γ 2 α z_{+}:=\cfrac{-\beta+\sqrt{\beta^{2}-4\alpha\gamma}}{2\alpha}~{},~{}~{}z_{-}:=% \cfrac{-\beta-\sqrt{\beta^{2}-4\alpha\gamma}}{2\alpha}
  76. w ^ = A 1 e k 1 x + A 2 e - k 1 x + A 3 e k 3 x + A 4 e - k 3 x \hat{w}=A_{1}~{}e^{k_{1}x}+A_{2}~{}e^{-k_{1}x}+A_{3}~{}e^{k_{3}x}+A_{4}~{}e^{-% k_{3}x}
  77. u α ( 𝐱 ) = - x 3 w 0 x α = - x 3 w , α 0 ; α = 1 , 2 u 3 ( 𝐱 ) = w 0 ( x 1 , x 2 ) \begin{aligned}\displaystyle u_{\alpha}(\mathbf{x})&\displaystyle=-x_{3}~{}% \frac{\partial w^{0}}{\partial x_{\alpha}}=-x_{3}~{}w^{0}_{,\alpha}~{};~{}~{}% \alpha=1,2\\ \displaystyle u_{3}(\mathbf{x})&\displaystyle=w^{0}(x_{1},x_{2})\end{aligned}
  78. 𝐮 \mathbf{u}
  79. w 0 w^{0}
  80. ε α β = - x 3 w , α β 0 ε α 3 = 0 ε 33 = 0 \begin{aligned}\displaystyle\varepsilon_{\alpha\beta}&\displaystyle=-x_{3}~{}w% ^{0}_{,\alpha\beta}\\ \displaystyle\varepsilon_{\alpha 3}&\displaystyle=0\\ \displaystyle\varepsilon_{33}&\displaystyle=0\end{aligned}
  81. M α β , α β + q ( x ) = 0 ; M α β := - h h x 3 σ α β d x 3 M_{\alpha\beta,\alpha\beta}+q(x)=0~{};~{}~{}M_{\alpha\beta}:=\int_{-h}^{h}x_{3% }~{}\sigma_{\alpha\beta}~{}dx_{3}
  82. q ( x ) q(x)
  83. w , 1111 0 + 2 w , 1212 0 + w , 2222 0 = 0 w^{0}_{,1111}+2~{}w^{0}_{,1212}+w^{0}_{,2222}=0
  84. 2 2 w = 0 \nabla^{2}\nabla^{2}w=0
  85. u α ( 𝐱 ) = - x 3 φ α ; α = 1 , 2 u 3 ( 𝐱 ) = w 0 ( x 1 , x 2 ) \begin{aligned}\displaystyle u_{\alpha}(\mathbf{x})&\displaystyle=-x_{3}~{}% \varphi_{\alpha}~{};~{}~{}\alpha=1,2\\ \displaystyle u_{3}(\mathbf{x})&\displaystyle=w^{0}(x_{1},x_{2})\end{aligned}
  86. φ α \varphi_{\alpha}
  87. ε α β = - x 3 φ α , β ε α 3 = 1 2 κ ( w , α 0 - φ α ) ε 33 = 0 \begin{aligned}\displaystyle\varepsilon_{\alpha\beta}&\displaystyle=-x_{3}~{}% \varphi_{\alpha,\beta}\\ \displaystyle\varepsilon_{\alpha 3}&\displaystyle=\cfrac{1}{2}~{}\kappa\left(w% ^{0}_{,\alpha}-\varphi_{\alpha}\right)\\ \displaystyle\varepsilon_{33}&\displaystyle=0\end{aligned}
  88. κ \kappa
  89. M α β , β - Q α = 0 Q α , α + q = 0 \begin{aligned}&\displaystyle M_{\alpha\beta,\beta}-Q_{\alpha}=0\\ &\displaystyle Q_{\alpha,\alpha}+q=0\end{aligned}
  90. Q α := κ - h h σ α 3 d x 3 Q_{\alpha}:=\kappa~{}\int_{-h}^{h}\sigma_{\alpha 3}~{}dx_{3}
  91. M α β , α β - q ( x , t ) = J 1 w ¨ 0 - J 3 w ¨ , α α 0 M_{\alpha\beta,\alpha\beta}-q(x,t)=J_{1}~{}\ddot{w}^{0}-J_{3}~{}\ddot{w}^{0}_{% ,\alpha\alpha}
  92. ρ = ρ ( x ) \rho=\rho(x)
  93. J 1 := - h h ρ d x 3 ; J 3 := - h h x 3 2 ρ d x 3 J_{1}:=\int_{-h}^{h}\rho~{}dx_{3}~{};~{}~{}J_{3}:=\int_{-h}^{h}x_{3}^{2}~{}% \rho~{}dx_{3}
  94. w ¨ 0 = 2 w 0 t 2 ; w ¨ , α β 0 = 2 w ¨ 0 x α x β \ddot{w}^{0}=\frac{\partial^{2}w^{0}}{\partial t^{2}}~{};~{}~{}\ddot{w}^{0}_{,% \alpha\beta}=\frac{\partial^{2}\ddot{w}^{0}}{\partial x_{\alpha}\partial x_{% \beta}}

Bernoulli_differential_equation.html

  1. y + P ( x ) y = Q ( x ) y n y^{\prime}+P(x)y=Q(x)y^{n}\,
  2. x 0 ( a , b ) x_{0}\in(a,b)
  3. { z : ( a , b ) ( 0 , ) , if α { 1 , 2 } , z : ( a , b ) { 0 } , if α = 2 , \left\{\begin{array}[]{ll}z:(a,b)\rightarrow(0,\infty)\ ,&\textrm{if}\ \alpha% \in\mathbb{R}\setminus\{1,2\},\\ z:(a,b)\rightarrow\mathbb{R}\setminus\{0\}\ ,&\textrm{if}\ \alpha=2,\\ \end{array}\right.
  4. z ( x ) = ( 1 - α ) P ( x ) z ( x ) + ( 1 - α ) Q ( x ) . z^{\prime}(x)=(1-\alpha)P(x)z(x)+(1-\alpha)Q(x).
  5. y ( x ) := [ z ( x ) ] 1 1 - α y(x):=[z(x)]^{\frac{1}{1-\alpha}}
  6. y ( x ) = P ( x ) y ( x ) + Q ( x ) y α ( x ) , y ( x 0 ) = y 0 := [ z ( x 0 ) ] 1 1 - α . y^{\prime}(x)=P(x)y(x)+Q(x)y^{\alpha}(x)\ ,\ y(x_{0})=y_{0}:=[z(x_{0})]^{\frac% {1}{1-\alpha}}.
  7. α > 0 \alpha>0
  8. y 0 y\equiv 0
  9. y 0 = 0 y_{0}=0
  10. y - 2 y x = - x 2 y 2 y^{\prime}-\frac{2y}{x}=-x^{2}y^{2}
  11. y = 0 y=0
  12. y 2 y^{2}
  13. y y - 2 - 2 x y - 1 = - x 2 y^{\prime}y^{-2}-\frac{2}{x}y^{-1}=-x^{2}
  14. w = 1 y w=\frac{1}{y}
  15. w = - y y 2 . w^{\prime}=\frac{-y^{\prime}}{y^{2}}.
  16. w + 2 x w = x 2 w^{\prime}+\frac{2}{x}w=x^{2}
  17. M ( x ) = e 2 1 x d x = e 2 ln x = x 2 . M(x)=e^{2\int\frac{1}{x}dx}=e^{2\ln x}=x^{2}.
  18. M ( x ) M(x)
  19. w x 2 + 2 x w = x 4 , w^{\prime}x^{2}+2xw=x^{4},\,
  20. w x 2 wx^{2}
  21. d [ w x 2 ] = x 4 d x \int d[wx^{2}]=\int x^{4}dx
  22. w x 2 = 1 5 x 5 + C wx^{2}=\frac{1}{5}x^{5}+C
  23. 1 y x 2 = 1 5 x 5 + C \frac{1}{y}x^{2}=\frac{1}{5}x^{5}+C
  24. y y
  25. y = x 2 1 5 x 5 + C y=\frac{x^{2}}{\frac{1}{5}x^{5}+C}

Bernstein–Sato_polynomial.html

  1. P ( s ) f ( x ) s + 1 = b ( s ) f ( x ) s . P(s)f(x)^{s+1}=b(s)f(x)^{s}.\,
  2. f ( x ) = x 1 2 + + x n 2 f(x)=x_{1}^{2}+\cdots+x_{n}^{2}\,
  3. i = 1 n i 2 f ( x ) s + 1 = 4 ( s + 1 ) ( s + n 2 ) f ( x ) s \sum_{i=1}^{n}\partial_{i}^{2}f(x)^{s+1}=4(s+1)\left(s+\frac{n}{2}\right)f(x)^% {s}
  4. b ( s ) = ( s + 1 ) ( s + n 2 ) . b(s)=(s+1)\left(s+\frac{n}{2}\right).
  5. f ( x ) = x 1 n 1 x 2 n 2 x r n r f(x)=x_{1}^{n_{1}}x_{2}^{n_{2}}\cdots x_{r}^{n_{r}}
  6. j = 1 r x j n j f ( x ) s + 1 = j = 1 r i = 1 n j ( n j s + i ) f ( x ) s \prod_{j=1}^{r}\partial_{x_{j}}^{n_{j}}\quad f(x)^{s+1}=\prod_{j=1}^{r}\prod_{% i=1}^{n_{j}}(n_{j}s+i)\quad f(x)^{s}
  7. b ( s ) = j = 1 r i = 1 n j ( s + i n j ) . b(s)=\prod_{j=1}^{r}\prod_{i=1}^{n_{j}}\left(s+\frac{i}{n_{j}}\right).
  8. ( s + 1 ) ( s + 5 6 ) ( s + 7 6 ) . (s+1)\left(s+\frac{5}{6}\right)\left(s+\frac{7}{6}\right).
  9. s ( s + 1 ) ( s + n - 1 ) s(s+1)\cdots(s+n-1)
  10. Ω ( det ( t i j ) s ) = s ( s + 1 ) ( s + n - 1 ) det ( t i j ) s - 1 \Omega(\det(t_{ij})^{s})=s(s+1)\cdots(s+n-1)\det(t_{ij})^{s-1}
  11. f ( x ) s = 1 b ( s ) P ( s ) f ( x ) s + 1 . f(x)^{s}={1\over b(s)}P(s)f(x)^{s+1}.
  12. f ¯ ( x ) \bar{f}(x)
  13. f ¯ ( x ) f ( x ) . \bar{f}(x)f(x).
  14. ( f 1 ( x ) ) s 1 ( f 2 ( x ) ) s 2 (f_{1}(x))^{s_{1}}(f_{2}(x))^{s_{2}}
  15. P ( s 1 , s 2 ) P(s_{1},s_{2})
  16. b ( s 1 , s 2 ) b(s_{1},s_{2})

Bessel's_inequality.html

  1. x x
  2. H H
  3. e 1 , e 2 , e_{1},e_{2},...
  4. H H
  5. x x
  6. H H
  7. k = 1 | x , e k | 2 x 2 \sum_{k=1}^{\infty}\left|\left\langle x,e_{k}\right\rangle\right|^{2}\leq\left% \|x\right\|^{2}
  8. H H
  9. x = k = 1 x , e k e k , x^{\prime}=\sum_{k=1}^{\infty}\left\langle x,e_{k}\right\rangle e_{k},
  10. x x
  11. e k e_{k}
  12. x H x^{\prime}\in H
  13. e 1 , e 2 , e_{1},e_{2},...
  14. x x^{\prime}
  15. x x
  16. 0 x - k = 1 n x , e k e k 2 = x 2 - 2 k = 1 n | x , e k | 2 + k = 1 n | x , e k | 2 = x 2 - k = 1 n | x , e k | 2 , 0\leq\left\|x-\sum_{k=1}^{n}\langle x,e_{k}\rangle e_{k}\right\|^{2}=\|x\|^{2}% -2\sum_{k=1}^{n}|\langle x,e_{k}\rangle|^{2}+\sum_{k=1}^{n}|\langle x,e_{k}% \rangle|^{2}=\|x\|^{2}-\sum_{k=1}^{n}|\langle x,e_{k}\rangle|^{2},

BEST_theorem.html

  1. ec ( G ) = t w ( G ) v V ( deg ( v ) - 1 ) ! . \operatorname{ec}(G)=t_{w}(G)\prod_{v\in V}\bigl(\deg(v)-1\bigr)!.

Beta-dual_space.html

  1. β β
  2. X X
  3. β β
  4. X X
  5. X β := { x X : i = 1 x i y i < y X } . X^{\beta}:=\left\{x\in X\ :\ \sum_{i=1}^{\infty}x_{i}y_{i}<\infty\quad\forall y% \in X\right\}.
  6. X X
  7. y y
  8. X X
  9. f y ( x ) := i = 1 x i y i x X . f_{y}(x):=\sum_{i=1}^{\infty}x_{i}y_{i}\qquad x\in X.
  10. c 0 β = 1 c_{0}^{\beta}=\ell^{1}
  11. ( 1 ) β = (\ell^{1})^{\beta}=\ell^{\infty}
  12. ω β = \omega^{\beta}=\emptyset
  13. E E
  14. E E
  15. E E

Beta_normal_form.html

  1. ( ( λ x . A ( x ) ) t ) ((\mathbf{\lambda}x.A(x))t)
  2. A ( x ) A(x)
  3. x x
  4. ( ( λ x . A ( x ) ) t ) A ( t ) ((\mathbf{\lambda}x.A(x))t)\rightarrow A(t)
  5. A ( t ) A(t)
  6. t t
  7. x x
  8. A ( x ) A(x)
  9. λ x 0 λ x i - 1 . ( λ x i . A ( x i ) ) M 1 M 2 M n λ x 0 λ x i - 1 . A ( M 1 ) M 2 M n \lambda x_{0}\ldots\lambda x_{i-1}.(\lambda x_{i}.A(x_{i}))M_{1}M_{2}\ldots M_% {n}\rightarrow\lambda x_{0}\ldots\lambda x_{i-1}.A(M_{1})M_{2}\ldots M_{n}
  10. i 0 , n 1 i\geq 0,n\geq 1
  11. ( λ x . z ) ( ( λ w . w w w ) ( λ w . w w w ) ) (\mathbf{\lambda}x.z)((\lambda w.www)(\lambda w.www))
  12. ( λ x . z ) ( ( λ w . w w w ) ( λ w . w w w ) ( λ w . w w w ) ) \rightarrow(\lambda x.z)((\lambda w.www)(\lambda w.www)(\lambda w.www))
  13. ( λ x . z ) ( ( λ w . w w w ) ( λ w . w w w ) ( λ w . w w w ) ( λ w . w w w ) ) \rightarrow(\lambda x.z)((\lambda w.www)(\lambda w.www)(\lambda w.www)(\lambda w% .www))
  14. ( λ x . z ) ( ( λ w . w w w ) ( λ w . w w w ) ( λ w . w w w ) ( λ w . w w w ) ( λ w . w w w ) ) \rightarrow(\lambda x.z)((\lambda w.www)(\lambda w.www)(\lambda w.www)(\lambda w% .www)(\lambda w.www))
  15. \ldots
  16. ( λ x . z ) ( ( λ w . w w w ) ( λ w . w w w ) ) (\mathbf{\lambda}x.z)((\lambda w.www)(\lambda w.www))
  17. z \rightarrow z

Betatron.html

  1. θ 0 = 2 π r 0 2 H 0 , \theta_{0}=2\pi r_{0}^{2}H_{0},
  2. θ 0 \theta_{0}
  3. r 0 r_{0}
  4. H 0 H_{0}
  5. r 0 r_{0}
  6. H 0 = 1 2 θ 0 π r 0 2 . \Leftrightarrow H_{0}=\frac{1}{2}\frac{\theta_{0}}{\pi r_{0}^{2}}.

Bethe_lattice.html

  1. N k = z ( z - 1 ) k - 1 for k > 0. \,N_{k}=z(z-1)^{k-1}\,\text{ for }k>0.

BF_model.html

  1. S = M K [ 𝐁 𝐅 ] S=\int_{M}K[\mathbf{B}\wedge\mathbf{F}]
  2. 𝔤 \mathfrak{g}
  3. 𝐅 d 𝐀 + 𝐀 𝐀 \mathbf{F}\equiv d\mathbf{A}+\mathbf{A}\wedge\mathbf{A}
  4. 𝐅 = 0 \mathbf{F}=0
  5. d 𝐀 B = 0 d_{\mathbf{A}}B=0

Biclustering.html

  1. m m
  2. n n
  3. m × n m\times n
  4. ( m × n ) / t (m\times n)/t

Bicubic_interpolation.html

  1. f f
  2. f x f_{x}
  3. f y f_{y}
  4. f x y f_{xy}
  5. ( 0 , 0 ) (0,0)
  6. ( 1 , 0 ) (1,0)
  7. ( 0 , 1 ) (0,1)
  8. ( 1 , 1 ) (1,1)
  9. p ( x , y ) = i = 0 3 j = 0 3 a i j x i y j . p(x,y)=\sum_{i=0}^{3}\sum_{j=0}^{3}a_{ij}x^{i}y^{j}.
  10. a i j a_{ij}
  11. p ( x , y ) p(x,y)
  12. f ( 0 , 0 ) = p ( 0 , 0 ) = a 00 f(0,0)=p(0,0)=a_{00}
  13. f ( 1 , 0 ) = p ( 1 , 0 ) = a 00 + a 10 + a 20 + a 30 f(1,0)=p(1,0)=a_{00}+a_{10}+a_{20}+a_{30}
  14. f ( 0 , 1 ) = p ( 0 , 1 ) = a 00 + a 01 + a 02 + a 03 f(0,1)=p(0,1)=a_{00}+a_{01}+a_{02}+a_{03}
  15. f ( 1 , 1 ) = p ( 1 , 1 ) = i = 0 3 j = 0 3 a i j f(1,1)=p(1,1)=\textstyle\sum_{i=0}^{3}\sum_{j=0}^{3}a_{ij}
  16. x x
  17. y y
  18. f x ( 0 , 0 ) = p x ( 0 , 0 ) = a 10 f_{x}(0,0)=p_{x}(0,0)=a_{10}
  19. f x ( 1 , 0 ) = p x ( 1 , 0 ) = a 10 + 2 a 20 + 3 a 30 f_{x}(1,0)=p_{x}(1,0)=a_{10}+2a_{20}+3a_{30}
  20. f x ( 0 , 1 ) = p x ( 0 , 1 ) = a 10 + a 11 + a 12 + a 13 f_{x}(0,1)=p_{x}(0,1)=a_{10}+a_{11}+a_{12}+a_{13}
  21. f x ( 1 , 1 ) = p x ( 1 , 1 ) = i = 1 3 j = 0 3 a i j i f_{x}(1,1)=p_{x}(1,1)=\textstyle\sum_{i=1}^{3}\sum_{j=0}^{3}a_{ij}i
  22. f y ( 0 , 0 ) = p y ( 0 , 0 ) = a 01 f_{y}(0,0)=p_{y}(0,0)=a_{01}
  23. f y ( 1 , 0 ) = p y ( 1 , 0 ) = a 01 + a 11 + a 21 + a 31 f_{y}(1,0)=p_{y}(1,0)=a_{01}+a_{11}+a_{21}+a_{31}
  24. f y ( 0 , 1 ) = p y ( 0 , 1 ) = a 01 + 2 a 02 + 3 a 03 f_{y}(0,1)=p_{y}(0,1)=a_{01}+2a_{02}+3a_{03}
  25. f y ( 1 , 1 ) = p y ( 1 , 1 ) = i = 0 3 j = 1 3 a i j j f_{y}(1,1)=p_{y}(1,1)=\textstyle\sum_{i=0}^{3}\sum_{j=1}^{3}a_{ij}j
  26. x y xy
  27. f x y ( 0 , 0 ) = p x y ( 0 , 0 ) = a 11 f_{xy}(0,0)=p_{xy}(0,0)=a_{11}
  28. f x y ( 1 , 0 ) = p x y ( 1 , 0 ) = a 11 + 2 a 21 + 3 a 31 f_{xy}(1,0)=p_{xy}(1,0)=a_{11}+2a_{21}+3a_{31}
  29. f x y ( 0 , 1 ) = p x y ( 0 , 1 ) = a 11 + 2 a 12 + 3 a 13 f_{xy}(0,1)=p_{xy}(0,1)=a_{11}+2a_{12}+3a_{13}
  30. f x y ( 1 , 1 ) = p x y ( 1 , 1 ) = i = 1 3 j = 1 3 a i j i j f_{xy}(1,1)=p_{xy}(1,1)=\textstyle\sum_{i=1}^{3}\sum_{j=1}^{3}a_{ij}ij
  31. p x ( x , y ) = i = 1 3 j = 0 3 a i j i x i - 1 y j p_{x}(x,y)=\textstyle\sum_{i=1}^{3}\sum_{j=0}^{3}a_{ij}ix^{i-1}y^{j}
  32. p y ( x , y ) = i = 0 3 j = 1 3 a i j x i j y j - 1 p_{y}(x,y)=\textstyle\sum_{i=0}^{3}\sum_{j=1}^{3}a_{ij}x^{i}jy^{j-1}
  33. p x y ( x , y ) = i = 1 3 j = 1 3 a i j i x i - 1 j y j - 1 p_{xy}(x,y)=\textstyle\sum_{i=1}^{3}\sum_{j=1}^{3}a_{ij}ix^{i-1}jy^{j-1}
  34. p ( x , y ) p(x,y)
  35. [ 0 , 1 ] × [ 0 , 1 ] [0,1]\times[0,1]
  36. a i j a_{ij}
  37. α = [ a 00 a 10 a 20 a 30 a 01 a 11 a 21 a 31 a 02 a 12 a 22 a 32 a 03 a 13 a 23 a 33 ] T \alpha=\left[\begin{smallmatrix}a_{00}&a_{10}&a_{20}&a_{30}&a_{01}&a_{11}&a_{2% 1}&a_{31}&a_{02}&a_{12}&a_{22}&a_{32}&a_{03}&a_{13}&a_{23}&a_{33}\end{% smallmatrix}\right]^{T}
  38. x = [ f ( 0 , 0 ) f ( 1 , 0 ) f ( 0 , 1 ) f ( 1 , 1 ) f x ( 0 , 0 ) f x ( 1 , 0 ) f x ( 0 , 1 ) f x ( 1 , 1 ) f y ( 0 , 0 ) f y ( 1 , 0 ) f y ( 0 , 1 ) f y ( 1 , 1 ) f x y ( 0 , 0 ) f x y ( 1 , 0 ) f x y ( 0 , 1 ) f x y ( 1 , 1 ) ] T x=\left[\begin{smallmatrix}f(0,0)&f(1,0)&f(0,1)&f(1,1)&f_{x}(0,0)&f_{x}(1,0)&f% _{x}(0,1)&f_{x}(1,1)&f_{y}(0,0)&f_{y}(1,0)&f_{y}(0,1)&f_{y}(1,1)&f_{xy}(0,0)&f% _{xy}(1,0)&f_{xy}(0,1)&f_{xy}(1,1)\end{smallmatrix}\right]^{T}
  39. A α = x A\alpha=x
  40. A - 1 x = α A^{-1}x=\alpha
  41. α \alpha
  42. A - 1 = [ 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 - 3 3 0 0 - 2 - 1 0 0 0 0 0 0 0 0 0 0 2 - 2 0 0 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 - 3 3 0 0 - 2 - 1 0 0 0 0 0 0 0 0 0 0 2 - 2 0 0 1 1 0 0 - 3 0 3 0 0 0 0 0 - 2 0 - 1 0 0 0 0 0 0 0 0 0 - 3 0 3 0 0 0 0 0 - 2 0 - 1 0 9 - 9 - 9 9 6 3 - 6 - 3 6 - 6 3 - 3 4 2 2 1 - 6 6 6 - 6 - 3 - 3 3 3 - 4 4 - 2 2 - 2 - 2 - 1 - 1 2 0 - 2 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 2 0 - 2 0 0 0 0 0 1 0 1 0 - 6 6 6 - 6 - 4 - 2 4 2 - 3 3 - 3 3 - 2 - 1 - 2 - 1 4 - 4 - 4 4 2 2 - 2 - 2 2 - 2 2 - 2 1 1 1 1 ] A^{-1}=\left[\begin{smallmatrix}1&0&0&0&0&0&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0&0&0&0&0&0&0&0\\ -3&3&0&0&-2&-1&0&0&0&0&0&0&0&0&0&0\\ 2&-2&0&0&1&1&0&0&0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&1&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0&0&0&0&0&1&0&0&0\\ 0&0&0&0&0&0&0&0&-3&3&0&0&-2&-1&0&0\\ 0&0&0&0&0&0&0&0&2&-2&0&0&1&1&0&0\\ -3&0&3&0&0&0&0&0&-2&0&-1&0&0&0&0&0\\ 0&0&0&0&-3&0&3&0&0&0&0&0&-2&0&-1&0\\ 9&-9&-9&9&6&3&-6&-3&6&-6&3&-3&4&2&2&1\\ -6&6&6&-6&-3&-3&3&3&-4&4&-2&2&-2&-2&-1&-1\\ 2&0&-2&0&0&0&0&0&1&0&1&0&0&0&0&0\\ 0&0&0&0&2&0&-2&0&0&0&0&0&1&0&1&0\\ -6&6&6&-6&-4&-2&4&2&-3&3&-3&3&-2&-1&-2&-1\\ 4&-4&-4&4&2&2&-2&-2&2&-2&2&-2&1&1&1&1\end{smallmatrix}\right]
  43. [ f ( 0 , 0 ) f ( 0 , 1 ) f y ( 0 , 0 ) f y ( 0 , 1 ) f ( 1 , 0 ) f ( 1 , 1 ) f y ( 1 , 0 ) f y ( 1 , 1 ) f x ( 0 , 0 ) f x ( 0 , 1 ) f x y ( 0 , 0 ) f x y ( 0 , 1 ) f x ( 1 , 0 ) f x ( 1 , 1 ) f x y ( 1 , 0 ) f x y ( 1 , 1 ) ] = [ 1 0 0 0 1 1 1 1 0 1 0 0 0 1 2 3 ] [ a 00 a 01 a 02 a 03 a 10 a 11 a 12 a 13 a 20 a 21 a 22 a 23 a 30 a 31 a 32 a 33 ] [ 1 1 0 0 0 1 1 1 0 1 0 2 0 1 0 3 ] \begin{bmatrix}f(0,0)&f(0,1)&f_{y}(0,0)&f_{y}(0,1)\\ f(1,0)&f(1,1)&f_{y}(1,0)&f_{y}(1,1)\\ f_{x}(0,0)&f_{x}(0,1)&f_{xy}(0,0)&f_{xy}(0,1)\\ f_{x}(1,0)&f_{x}(1,1)&f_{xy}(1,0)&f_{xy}(1,1)\end{bmatrix}=\begin{bmatrix}1&0&% 0&0\\ 1&1&1&1\\ 0&1&0&0\\ 0&1&2&3\end{bmatrix}\begin{bmatrix}a_{00}&a_{01}&a_{02}&a_{03}\\ a_{10}&a_{11}&a_{12}&a_{13}\\ a_{20}&a_{21}&a_{22}&a_{23}\\ a_{30}&a_{31}&a_{32}&a_{33}\end{bmatrix}\begin{bmatrix}1&1&0&0\\ 0&1&1&1\\ 0&1&0&2\\ 0&1&0&3\end{bmatrix}
  44. [ a 00 a 01 a 02 a 03 a 10 a 11 a 12 a 13 a 20 a 21 a 22 a 23 a 30 a 31 a 32 a 33 ] = [ 1 0 0 0 0 0 1 0 - 3 3 - 2 - 1 2 - 2 1 1 ] [ f ( 0 , 0 ) f ( 0 , 1 ) f y ( 0 , 0 ) f y ( 0 , 1 ) f ( 1 , 0 ) f ( 1 , 1 ) f y ( 1 , 0 ) f y ( 1 , 1 ) f x ( 0 , 0 ) f x ( 0 , 1 ) f x y ( 0 , 0 ) f x y ( 0 , 1 ) f x ( 1 , 0 ) f x ( 1 , 1 ) f x y ( 1 , 0 ) f x y ( 1 , 1 ) ] [ 1 0 - 3 2 0 0 3 - 2 0 1 - 2 1 0 0 - 1 1 ] \begin{bmatrix}a_{00}&a_{01}&a_{02}&a_{03}\\ a_{10}&a_{11}&a_{12}&a_{13}\\ a_{20}&a_{21}&a_{22}&a_{23}\\ a_{30}&a_{31}&a_{32}&a_{33}\end{bmatrix}=\begin{bmatrix}1&0&0&0\\ 0&0&1&0\\ -3&3&-2&-1\\ 2&-2&1&1\end{bmatrix}\begin{bmatrix}f(0,0)&f(0,1)&f_{y}(0,0)&f_{y}(0,1)\\ f(1,0)&f(1,1)&f_{y}(1,0)&f_{y}(1,1)\\ f_{x}(0,0)&f_{x}(0,1)&f_{xy}(0,0)&f_{xy}(0,1)\\ f_{x}(1,0)&f_{x}(1,1)&f_{xy}(1,0)&f_{xy}(1,1)\end{bmatrix}\begin{bmatrix}1&0&-% 3&2\\ 0&0&3&-2\\ 0&1&-2&1\\ 0&0&-1&1\end{bmatrix}
  45. f x f_{x}
  46. f y f_{y}
  47. f x f_{x}
  48. f ( x , y ) f(x,y)
  49. f y f_{y}
  50. f x y f_{xy}
  51. f x f_{x}
  52. x x
  53. f y f_{y}
  54. f f
  55. f x y ( x , y ) f_{xy}(x,y)
  56. f y f_{y}
  57. f x f_{x}
  58. W ( x ) = { ( a + 2 ) | x | 3 - ( a + 3 ) | x | 2 + 1 for | x | 1 a | x | 3 - 5 a | x | 2 + 8 a | x | - 4 a for 1 < | x | < 2 0 otherwise W(x)=\begin{cases}(a+2)|x|^{3}-(a+3)|x|^{2}+1&\,\text{for }|x|\leq 1\\ a|x|^{3}-5a|x|^{2}+8a|x|-4a&\,\text{for }1<|x|<2\\ 0&\,\text{otherwise}\end{cases}
  59. a a
  60. W ( 0 ) = 1 W(0)=1
  61. W ( n ) = 0 W(n)=0
  62. n n
  63. a = - 0.5 a=-0.5
  64. a = - 0.5 a=-0.5
  65. p ( t ) = 1 2 [ 1 t t 2 t 3 ] [ 0 2 0 0 - 1 0 1 0 2 - 5 4 - 1 - 1 3 - 3 1 ] [ a - 1 a 0 a 1 a 2 ] p(t)=\tfrac{1}{2}\begin{bmatrix}1&t&t^{2}&t^{3}\\ \end{bmatrix}\begin{bmatrix}0&2&0&0\\ -1&0&1&0\\ 2&-5&4&-1\\ -1&3&-3&1\\ \end{bmatrix}\begin{bmatrix}a_{-1}\\ a_{0}\\ a_{1}\\ a_{2}\\ \end{bmatrix}
  66. t t
  67. x x
  68. y y
  69. b - 1 = p ( t x , a ( - 1 , - 1 ) , a ( 0 , - 1 ) , a ( 1 , - 1 ) , a ( 2 , - 1 ) ) \textstyle b_{-1}=p(t_{x},a_{(-1,-1)},a_{(0,-1)},a_{(1,-1)},a_{(2,-1)})
  70. b 0 = p ( t x , a ( - 1 , 0 ) , a ( 0 , 0 ) , a ( 1 , 0 ) , a ( 2 , 0 ) ) \textstyle b_{0}=p(t_{x},a_{(-1,0)},a_{(0,0)},a_{(1,0)},a_{(2,0)})
  71. b 1 = p ( t x , a ( - 1 , 1 ) , a ( 0 , 1 ) , a ( 1 , 1 ) , a ( 2 , 1 ) ) \textstyle b_{1}=p(t_{x},a_{(-1,1)},a_{(0,1)},a_{(1,1)},a_{(2,1)})
  72. b 2 = p ( t x , a ( - 1 , 2 ) , a ( 0 , 2 ) , a ( 1 , 2 ) , a ( 2 , 2 ) ) \textstyle b_{2}=p(t_{x},a_{(-1,2)},a_{(0,2)},a_{(1,2)},a_{(2,2)})
  73. p ( x , y ) = p ( t y , b - 1 , b 0 , b 1 , b 2 ) \textstyle p(x,y)=p(t_{y},b_{-1},b_{0},b_{1},b_{2})

Bicyclic_semigroup.html

  1. 𝒞 \mathcal{C}

Bidiagonal_matrix.html

  1. ( 1 4 0 0 0 4 1 0 0 0 3 4 0 0 0 3 ) \begin{pmatrix}1&4&0&0\\ 0&4&1&0\\ 0&0&3&4\\ 0&0&0&3\\ \end{pmatrix}
  2. ( 1 0 0 0 2 4 0 0 0 3 3 0 0 0 4 3 ) . \begin{pmatrix}1&0&0&0\\ 2&4&0&0\\ 0&3&3&0\\ 0&0&4&3\\ \end{pmatrix}.

Bidirectional_reflectance_distribution_function.html

  1. ω i \omega_{\,\text{i}}
  2. ω r \omega_{\,\text{r}}
  3. n n
  4. f r ( ω i , ω r ) f_{\,\text{r}}(\omega_{\,\text{i}},\,\omega_{\,\text{r}})
  5. ω i \omega_{\,\text{i}}
  6. ω r \omega_{\,\text{r}}
  7. 𝐧 \mathbf{n}
  8. ω r \omega_{\,\text{r}}
  9. ω i \omega_{\,\text{i}}
  10. ω \omega
  11. ϕ \phi
  12. θ \theta
  13. f r ( ω i , ω r ) = d L r ( ω r ) d E i ( ω i ) = d L r ( ω r ) L i ( ω i ) cos θ i d ω i f_{\,\text{r}}(\omega_{\,\text{i}},\,\omega_{\,\text{r}})\,=\,\frac{% \operatorname{d}L_{\,\text{r}}(\omega_{\,\text{r}})}{\operatorname{d}E_{\,% \text{i}}(\omega_{\,\text{i}})}\,=\,\frac{\operatorname{d}L_{\,\text{r}}(% \omega_{\,\text{r}})}{L_{\,\text{i}}(\omega_{\,\text{i}})\cos\theta_{\,\text{i% }}\,\operatorname{d}\omega_{\,\text{i}}}
  14. L L
  15. E E
  16. θ i \theta_{\,\text{i}}
  17. ω i \omega_{\,\text{i}}
  18. 𝐧 \mathbf{n}
  19. i \,\text{i}
  20. r \,\text{r}
  21. d E i ( ω i ) \operatorname{d}E_{\,\text{i}}(\omega_{\,\text{i}})
  22. f r ( ω i , ω r ) f_{\,\text{r}}(\omega_{\,\text{i}},\,\omega_{\,\text{r}})
  23. L r ( ω r ) L_{\,\text{r}}(\omega_{\,\text{r}})
  24. d L r ( ω r ) \operatorname{d}L_{\,\text{r}}(\omega_{\,\text{r}})
  25. d E i ( ω i ) \operatorname{d}E_{\,\text{i}}(\omega_{\,\text{i}})
  26. f r ( ω i , ω r , 𝐱 ) f_{\,\text{r}}(\omega_{\,\text{i}},\,\omega_{\,\text{r}},\,\mathbf{x})
  27. 𝐱 \mathbf{x}
  28. S ( 𝐱 i , ω i , 𝐱 r , ω r ) S(\mathbf{x}_{\,\text{i}},\,\omega_{\,\text{i}},\,\mathbf{x}_{\,\text{r}},\,% \omega_{\,\text{r}})
  29. f r ( λ i , ω i , λ r , ω r ) f_{\,\text{r}}(\lambda_{\,\text{i}},\,\omega_{\,\text{i}},\,\lambda_{\,\text{r% }},\,\omega_{\,\text{r}})
  30. f r ( ω i , ω r ) 0 f_{\,\text{r}}(\omega_{\,\text{i}},\,\omega_{\,\text{r}})\geq 0
  31. f r ( ω i , ω r ) = f r ( ω r , ω i ) f_{\,\text{r}}(\omega_{\,\text{i}},\,\omega_{\,\text{r}})=f_{\,\text{r}}(% \omega_{\,\text{r}},\,\omega_{\,\text{i}})
  32. ω r , Ω f r ( ω i , ω r ) cos θ i d ω i 1 \forall\omega_{\,\text{r}},\,\int_{\Omega}f_{\,\text{r}}(\omega_{\,\text{i}},% \,\omega_{\,\text{r}})\,\cos{\theta_{\,\text{i}}}d\omega_{\,\text{i}}\leq 1

Bijection,_injection_and_surjection.html

  1. f : A B f:\;A\to B
  2. x , y A , f ( x ) = f ( y ) x = y . \forall x,y\in A,f(x)=f(y)\Rightarrow x=y.
  3. x , y A , x y f ( x ) f ( y ) . \forall x,y\in A,x\neq y\Rightarrow f(x)\neq f(y).
  4. y B , x A such that y = f ( x ) . \forall y\in B,\exists x\in A\,\text{ such that }y=f(x).
  5. f : A B f:A\to B
  6. a , b A a,b\in A
  7. f ( a ) = f ( b ) \Rarr a = b . f(a)=f(b)\Rarr a=b.
  8. f : A B f:A\to B
  9. b B b\in B
  10. a A a\in A
  11. f ( a ) = b . f(a)=b.
  12. f : A B f:A\to B
  13. b B b\in B
  14. a A a\in A
  15. f ( a ) = b . f(a)=b.
  16. A A
  17. B B
  18. A A
  19. B B
  20. A A
  21. B B
  22. A A
  23. B B
  24. A A
  25. B B
  26. 𝐑 𝐑 : x x \mathbf{R}\to\mathbf{R}:x\mapsto x
  27. 𝐑 + 𝐑 + : x x 2 \mathbf{R}^{+}\to\mathbf{R}^{+}:x\mapsto x^{2}
  28. 𝐑 + 𝐑 + : x x \mathbf{R}^{+}\to\mathbf{R}^{+}:x\mapsto\sqrt{x}
  29. exp : 𝐑 𝐑 + : x e x \exp:\mathbf{R}\to\mathbf{R}^{+}:x\mapsto\mathrm{e}^{x}
  30. ln : 𝐑 + 𝐑 : x ln x \ln:\mathbf{R}^{+}\to\mathbf{R}:x\mapsto\ln{x}
  31. exp : 𝐑 𝐑 : x e x \exp:\mathbf{R}\to\mathbf{R}:x\mapsto\mathrm{e}^{x}
  32. 𝐑 𝐑 : x ( x - 1 ) x ( x + 1 ) = x 3 - x \mathbf{R}\to\mathbf{R}:x\mapsto(x-1)x(x+1)=x^{3}-x
  33. 𝐑 [ - 1 , 1 ] : x sin ( x ) \mathbf{R}\to[-1,1]:x\mapsto\sin(x)
  34. 𝐑 𝐑 : x sin ( x ) \mathbf{R}\to\mathbf{R}:x\mapsto\sin(x)

Binary_Ordered_Compression_for_Unicode.html

  1. 256 - 13 = 243 256-13=243

Binary_quadratic_form.html

  1. q ( x , y ) = a x 2 + b x y + c y 2 , q(x,y)=ax^{2}+bxy+cy^{2},\,
  2. D ( f ) = b 2 - 4 a c , D ( f ) 0 , 1 ( mod 4 ) . D(f)=b^{2}-4ac,\quad D(f)\equiv 0,1\;\;(\mathop{{\rm mod}}4).
  3. 𝐐 ( D ) \mathbf{Q}(\sqrt{D})

Biquaternion.html

  1. ( h 0 0 - h ) ( 0 1 - 1 0 ) = ( 0 h h 0 ) \begin{pmatrix}h&0\\ 0&-h\end{pmatrix}\begin{pmatrix}0&1\\ -1&0\end{pmatrix}=\begin{pmatrix}0&h\\ h&0\end{pmatrix}
  2. ( u + h v w + h x - w + h x u - h v ) \begin{pmatrix}u+hv&w+hx\\ -w+hx&u-hv\end{pmatrix}
  3. ( h i ) 2 = h 2 i 2 = ( - 1 ) ( - 1 ) = + 1. (hi)^{2}\ =\ h^{2}i^{2}\ =\ (-1)(-1)\ =\ +1.
  4. { x + y ( h i ) : x , y R } \{x+y(hi):x,y\in R\}
  5. { x + y j : x , y C } \{x+yj:x,y\in C\}
  6. q * = w - x i - y j - z k , q^{*}=w-xi-yj-zk\!\ ,
  7. q = w + x i + y j + z k q^{\star}=w^{\star}+x^{\star}i+y^{\star}j+z^{\star}k\!
  8. z = a - b h z^{\star}=a-bh
  9. z = a + b h , a , b R , h 2 = - 1. z=a+bh,\quad a,b\in R,\quad h^{2}=-1.
  10. ( p q ) * = q * p * , ( p q ) = p q , ( q * ) = ( q ) * . (pq)^{*}=q^{*}p^{*},\quad(pq)^{\star}=p^{\star}q^{\star},\quad(q^{*})^{\star}=% (q^{\star})^{*}.
  11. q q * = 0 qq^{*}=0\!
  12. { q q * } - 1 \{qq^{*}\}^{-1}\!
  13. q q * = q * q qq^{*}=q^{*}q\!
  14. q - 1 = q * { q q * } - 1 q^{-1}=q^{*}\{qq^{*}\}^{-1}\!
  15. q q * 0. qq^{*}\neq 0.
  16. M = { q : q * = q } = { t + x ( h i ) + y ( h j ) + z ( h k ) : t , x , y , z R } . M=\{q:q^{*}=q^{\star}\}=\{t+x(hi)+y(hj)+z(hk):t,x,y,z\in R\}.
  17. ( h i ) ( h j ) = h 2 i j = - k M . (hi)(hj)=h^{2}ij=-k\notin M.
  18. q q * = t 2 - x 2 - y 2 - z 2 qq^{*}=t^{2}-x^{2}-y^{2}-z^{2}\!
  19. q q * = ( t + x h i + y h j + z h k ) ( t - x h i - y h j - z h k ) qq^{*}=(t+xhi+yhj+zhk)(t-xhi-yhj-zhk)\!
  20. = t 2 - x 2 ( h i ) 2 - y 2 ( h j ) 2 - z 2 ( h k ) 2 = t 2 - x 2 - y 2 - z 2 . =t^{2}-x^{2}(hi)^{2}-y^{2}(hj)^{2}-z^{2}(hk)^{2}=t^{2}-x^{2}-y^{2}-z^{2}.\!
  21. T ( q ) = g * q g . T(q)=g^{*}qg^{\star}.
  22. ( g * q g ) * = ( g ) * q * g = ( g * ) q g = ( g * q g ) . (g^{*}qg^{\star})^{*}=(g^{\star})^{*}q^{*}g=(g^{*})^{\star}q^{\star}g=(g^{*}qg% ^{\star})^{\star}.
  23. T ( q ) ( T ( q ) ) * = q q * \quad T(q)(T(q))^{*}=qq^{*}\!
  24. g ( g ) * = 1. g^{\star}(g^{\star})^{*}=1.
  25. ( g * q g ) ( g * q g ) * = g * q g ( g ) * q * g = g * q q * g = q q * . (g^{*}qg^{\star})(g^{*}qg^{\star})^{*}=g^{*}qg^{\star}(g^{\star})^{*}q^{*}g=g^% {*}qq^{*}g=qq^{*}.
  26. G = { g : g g * = 1 } G=\{g:gg^{*}=1\}\!
  27. G H G\cap H
  28. G M . G\cap M.
  29. g = g g=g^{\star}
  30. T ( q ) = g - 1 q g T(q)=g^{-1}qg\!
  31. g * = g - 1 . g^{*}=g^{-1}.\!
  32. G H . \cong G\cap H.
  33. G M G\cap M
  34. D r = { z = x + y h r : x , y R } D_{r}=\{z=x+yhr:x,y\in R\}
  35. D r D_{r}\!
  36. e x p ( a h r ) = cosh ( a ) + h r sinh ( a ) , a R . exp(ahr)=\cosh(a)+hr\ \sinh(a),\quad a\in R.\!
  37. exp ( a h r ) exp ( b h r ) = exp ( ( a + b ) h r ) . \exp(ahr)\exp(bhr)=\exp((a+b)hr).\!
  38. G D r . G\cap D_{r}.
  39. A = { q : q * = - q } A=\{q:q^{*}=-q\}\!
  40. exp : A G \exp:A\to G
  41. G H G\cap H
  42. G M . G\cap M.
  43. g = exp ( - 0.5 a h r ) = g * g^{\star}=\exp(-0.5ahr)=g^{*}
  44. T ( exp ( a h r ) ) = 1. T(\exp(ahr))=1.
  45. G M , G\cap M,
  46. { q : q q * = 0 } = { w + x i + y j + z k : w 2 + x 2 + y 2 + z 2 = 0 } \{q\ :\ qq^{*}=0\}=\{w+xi+yj+zk\ :\ w^{2}+x^{2}+y^{2}+z^{2}=0\}