wpmath0000002_2

Boundary_(topology).html

  1. ( - , a ) (-\infty,a)

Boundary_layer.html

  1. u x + υ y = 0 {\partial u\over\partial x}+{\partial\upsilon\over\partial y}=0
  2. u u x + υ u y = - 1 ρ p x + ν ( 2 u x 2 + 2 u y 2 ) u{\partial u\over\partial x}+\upsilon{\partial u\over\partial y}=-{1\over\rho}% {\partial p\over\partial x}+{\nu}\left({\partial^{2}u\over\partial x^{2}}+{% \partial^{2}u\over\partial y^{2}}\right)
  3. u υ x + υ υ y = - 1 ρ p y + ν ( 2 υ x 2 + 2 υ y 2 ) u{\partial\upsilon\over\partial x}+\upsilon{\partial\upsilon\over\partial y}=-% {1\over\rho}{\partial p\over\partial y}+{\nu}\left({\partial^{2}\upsilon\over% \partial x^{2}}+{\partial^{2}\upsilon\over\partial y^{2}}\right)
  4. u u
  5. υ \upsilon
  6. ρ \rho
  7. p p
  8. ν \nu
  9. u u
  10. υ \upsilon
  11. u u x + υ u y = - 1 ρ p x + ν 2 u y 2 u{\partial u\over\partial x}+\upsilon{\partial u\over\partial y}=-{1\over\rho}% {\partial p\over\partial x}+{\nu}{\partial^{2}u\over\partial y^{2}}
  12. 1 ρ p y = 0 {1\over\rho}{\partial p\over\partial y}=0
  13. u x + υ y = 0 {\partial u\over\partial x}+{\partial\upsilon\over\partial y}=0
  14. υ \upsilon
  15. u u
  16. p p
  17. y y
  18. u 0 u_{0}
  19. u u
  20. u 0 u_{0}
  21. p p
  22. u u x + υ u y = u 0 u 0 x + ν 2 u y 2 u{\partial u\over\partial x}+\upsilon{\partial u\over\partial y}=u_{0}{% \partial u_{0}\over\partial x}+{\nu}{\partial^{2}u\over\partial y^{2}}
  23. p p
  24. p x = 0 {\partial p\over\partial x}=0
  25. u 0 u_{0}
  26. u u x + υ u y = ν 2 u y 2 u{\partial u\over\partial x}+\upsilon{\partial u\over\partial y}={\nu}{% \partial^{2}u\over\partial y^{2}}
  27. u ¯ x + v ¯ y = 0 {\partial\overline{u}\over\partial x}+{\partial\overline{v}\over\partial y}=0
  28. u ¯ u ¯ x + v ¯ u ¯ y = - 1 ρ p ¯ x + ν ( 2 u ¯ x 2 + 2 u ¯ y 2 ) - y ( u v ¯ ) - x ( u 2 ¯ ) \overline{u}{\partial\overline{u}\over\partial x}+\overline{v}{\partial% \overline{u}\over\partial y}=-{1\over\rho}{\partial\overline{p}\over\partial x% }+\nu\left({\partial^{2}\overline{u}\over\partial x^{2}}+{\partial^{2}% \overline{u}\over\partial y^{2}}\right)-\frac{\partial}{\partial y}(\overline{% u^{\prime}v^{\prime}})-\frac{\partial}{\partial x}(\overline{u^{\prime 2}})
  29. u ¯ v ¯ x + v ¯ v ¯ y = - 1 ρ p ¯ y + ν ( 2 v ¯ x 2 + 2 v ¯ y 2 ) - x ( u v ¯ ) - y ( v 2 ¯ ) \overline{u}{\partial\overline{v}\over\partial x}+\overline{v}{\partial% \overline{v}\over\partial y}=-{1\over\rho}{\partial\overline{p}\over\partial y% }+\nu\left({\partial^{2}\overline{v}\over\partial x^{2}}+{\partial^{2}% \overline{v}\over\partial y^{2}}\right)-\frac{\partial}{\partial x}(\overline{% u^{\prime}v^{\prime}})-\frac{\partial}{\partial y}(\overline{v^{\prime 2}})
  30. δ \delta
  31. L L
  32. δ L \delta<<L
  33. u ¯ u ¯ x + v ¯ u ¯ y = - 1 ρ p ¯ x - y ( u v ¯ ) \overline{u}{\partial\overline{u}\over\partial x}+\overline{v}{\partial% \overline{u}\over\partial y}=-{1\over\rho}{\partial\overline{p}\over\partial x% }-\frac{\partial}{\partial y}(\overline{u^{\prime}v^{\prime}})
  34. η δ \eta<<\delta
  35. 0 = - 1 ρ p ¯ x + ν 2 u ¯ y 2 - y ( u v ¯ ) 0=-{1\over\rho}{\partial\overline{p}\over\partial x}+{\nu}{\partial^{2}% \overline{u}\over\partial y^{2}}-\frac{\partial}{\partial y}(\overline{u^{% \prime}v^{\prime}})
  36. η \eta
  37. ν u * \frac{\nu}{u_{*}}
  38. u * u_{*}
  39. u v ¯ \overline{u^{\prime}v^{\prime}}
  40. u ( y ) = u 0 [ 1 - ( y - h ) 2 h 2 ] = u 0 y h [ 2 - y h ] , u(y)=u_{0}\left[1-\frac{(y-h)^{2}}{h^{2}}\right]=u_{0}\frac{y}{h}\left[2-\frac% {y}{h}\right]\;,
  41. y 0 y\rightarrow 0
  42. u ( y ) 2 u 0 y h = θ y u(y)\approx 2u_{0}\frac{y}{h}=\theta y
  43. u ( y ) = θ ( x ) y u(y)=\theta(x)y
  44. P r Pr
  45. P r Pr
  46. x = x 0 x=x_{0}
  47. δ \delta
  48. δ 5.0 × x R e \delta\approx{5.0\times x\over\sqrt{Re}}
  49. δ \delta
  50. v v_{\infty}
  51. x x
  52. R e Re
  53. ρ v x / μ \rho v_{\infty}x/\mu
  54. ρ = \rho=
  55. μ = \mu=
  56. v x - v S v - v S = v x v = v y v = 0 {v_{x}-v_{S}\over v_{\infty}-v_{S}}={v_{x}\over v_{\infty}}={v_{y}\over v_{% \infty}}=0
  57. y = 0 y=0
  58. v x - v S v - v S = v x v = 1 {v_{x}-v_{S}\over v_{\infty}-v_{S}}={v_{x}\over v_{\infty}}=1
  59. y = y=\infty
  60. x = 0 x=0
  61. v S v_{S}
  62. v S = 0 v_{S}=0
  63. v x T x + v y T y = k ρ C p 2 T y 2 v_{x}{\partial T\over\partial x}+v_{y}{\partial T\over\partial y}={k\over\rho Cp% }{\partial^{2}T\over\partial y^{2}}
  64. v x c A x + v y c A y = D A B 2 c A y 2 v_{x}{\partial c_{A}\over\partial x}+v_{y}{\partial c_{A}\over\partial y}=D_{% AB}{\partial^{2}c_{A}\over\partial y^{2}}
  65. ν \nu
  66. α = k / ρ C P \alpha={k/\rho C_{P}}
  67. D A B D_{AB}
  68. k k
  69. ρ \rho
  70. C P C_{P}
  71. α = D A B = ν \alpha=D_{AB}=\nu
  72. P r = ν / α = 1 Pr=\nu/\alpha=1
  73. S c = ν / D A B = 1 Sc=\nu/D_{AB}=1
  74. v v
  75. T T
  76. c A c_{A}
  77. v x - v S v - v S = T - T S T - T S = c A - c A S c A - c A S = 0 {v_{x}-v_{S}\over v_{\infty}-v_{S}}={T-T_{S}\over T_{\infty}-T_{S}}={c_{A}-c_{% AS}\over c_{A\infty}-c_{AS}}=0
  78. y = 0 y=0
  79. v x - v S v - v S = T - T S T - T S = c A - c A S c A - c A S = 1 {v_{x}-v_{S}\over v_{\infty}-v_{S}}={T-T_{S}\over T_{\infty}-T_{S}}={c_{A}-c_{% AS}\over c_{A\infty}-c_{AS}}=1
  80. y = y=\infty
  81. x = 0 x=0
  82. τ 0 = ( v x y ) y = 0 = 0.332 v x R e 1 / 2 \tau_{0}=\left({\partial v_{x}\over\partial y}\right)_{y=0}=0.332{v_{\infty}% \over x}Re^{1/2}
  83. v x - v S v - v S = T - T S T - T S = c A - c A S c A - c A S {v_{x}-v_{S}\over v_{\infty}-v_{S}}={T-T_{S}\over T_{\infty}-T_{S}}={c_{A}-c_{% AS}\over c_{A\infty}-c_{AS}}
  84. ( T y ) y = 0 = 0.332 T - T S x R e 1 / 2 \left({\partial T\over\partial y}\right)_{y=0}=0.332{T_{\infty}-T_{S}\over x}% Re^{1/2}
  85. ( c A y ) y = 0 = 0.332 c A - c A S x R e 1 / 2 \left({\partial c_{A}\over\partial y}\right)_{y=0}=0.332{c_{A\infty}-c_{AS}% \over x}Re^{1/2}
  86. P r = S c = 1 Pr=Sc=1
  87. δ = δ T = δ c = 5.0 * x R e \delta=\delta_{T}=\delta_{c}={5.0*x\over\sqrt{Re}}
  88. δ T , δ c \delta_{T},\delta_{c}
  89. T T
  90. c A c_{A}
  91. P r 1 Pr\neq 1
  92. S c Sc
  93. P r = 1 Pr=1
  94. δ δ T = P r 1 / 3 {\delta\over\delta_{T}}=Pr^{1/3}
  95. δ δ c = S c 1 / 3 {\delta\over\delta_{c}}=Sc^{1/3}
  96. q A = - k ( T y ) y = 0 = h x ( T S - T ) {q\over A}=-k\left({\partial T\over\partial y}\right)_{y=0}=h_{x}(T_{S}-T_{% \infty})
  97. h x = 0.332 k x R e x 1 / 2 P r 1 / 3 h_{x}=0.332{k\over x}Re^{1/2}_{x}Pr^{1/3}
  98. h x h_{x}
  99. h L = 0.664 k x R e L 1 / 2 P r 1 / 3 h_{L}=0.664{k\over x}Re^{1/2}_{L}Pr^{1/3}
  100. k k
  101. D A B D_{AB}
  102. S c = ν / D A B Sc=\nu/D_{AB}
  103. k x = 0.332 D A B x R e x 1 / 2 S c 1 / 3 k^{\prime}_{x}=0.332{D_{AB}\over x}Re^{1/2}_{x}Sc^{1/3}
  104. k L = 0.664 D A B x R e L 1 / 2 S c 1 / 3 k^{\prime}_{L}=0.664{D_{AB}\over x}Re^{1/2}_{L}Sc^{1/3}
  105. N w N_{w}
  106. ρ v w \rho vw
  107. μ v δ 1 {\mu v\over\delta_{1}}
  108. ρ v w = μ v δ 1 \rho vw={\mu v\over\delta_{1}}
  109. δ 1 = μ ρ w = v w \delta_{1}=\sqrt{\mu\over\rho w}=\sqrt{\ v\over\ w}
  110. L δ 1 = L w v = N w {L\over\delta_{1}}={L\sqrt{w\over\ v}}=N_{w}
  111. N w N_{w}
  112. ρ \rho
  113. v v
  114. δ 1 \delta_{1}
  115. μ \mu
  116. L L
  117. N R N_{R}
  118. ρ v 2 L \rho v^{2}\over\ L
  119. μ v δ 2 2 {\mu v\over\delta_{2}^{2}}
  120. ρ v 2 L = μ v δ 2 2 {\rho v^{2}\over\ L}={\mu v\over\delta_{2}^{2}}
  121. δ 2 = μ L ρ v \delta_{2}=\sqrt{\mu L\over\rho v}
  122. L δ 2 = ρ v L μ = N R {L\over\delta_{2}}={\sqrt{\rho vL\over\mu}}=\sqrt{N_{R}}
  123. N R N_{R}
  124. ρ \rho
  125. v v
  126. δ 2 \delta_{2}
  127. μ \mu
  128. L L

Bounded_function.html

  1. | f ( x ) | M |f(x)|\leq M
  2. | a n | M |a_{n}|\leq M
  3. d ( f ( x ) , a ) M d(f(x),a)\leq M
  4. f ( x ) = 1 x 2 - 1 f(x)=\frac{1}{x^{2}-1}
  5. f ( x ) = 1 x 2 + 1 f(x)=\frac{1}{x^{2}+1}

Bounded_rationality.html

  1. U * U^{*}
  2. U ( s ) U * - ϵ U(s)\geq U^{*}-\epsilon

Bowen_ratio.html

  1. B = Q h Q e B={\frac{Q_{h}}{Q_{e}}}
  2. Q h Q_{h}
  3. Q e Q_{e}
  4. B B
  5. B B
  6. Q e 0 {Q_{e}\rightarrow 0}
  7. B B
  8. E F EF
  9. E F = Q e Q e + Q h = 1 1 + B {EF=\frac{Q_{e}}{Q_{e}+Q_{h}}=\frac{1}{1+B}}
  10. B {B}

Box_plot.html

  1. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  2. 1.5 × I Q R × e 3 M C , 1.5 × I Q R × e - 4 M C 1.5\times IQR\times e^{3MC},~{}\qquad~{}1.5\times IQR\times e^{-4MC}
  3. M C 0 MC\geq 0
  4. 1.5 × I Q R × e 4 M C , 1.5 × I Q R × e - 3 M C 1.5\times IQR\times e^{4MC},~{}\quad~{}1.5\times IQR\times e^{-3MC}
  5. M C 0 MC\leq 0
  6. 1.5 × I Q R 1.5\times IQR

Brachistochrone_curve.html

  1. v = 2 g y v=\sqrt{2gy}
  2. sin θ v = 1 v d x d s = 1 v m \frac{\sin{\theta}}{v}=\frac{1}{v}\frac{dx}{ds}=\frac{1}{v_{m}}
  3. θ \theta
  4. v m = 2 g D v_{m}=\sqrt{2gD}
  5. v m 2 d x 2 = v 2 d s 2 = v 2 ( d x 2 + d y 2 ) v_{m}^{2}dx^{2}=v^{2}ds^{2}=v^{2}(dx^{2}+dy^{2})
  6. d x = v d y v m 2 - v 2 dx=\frac{v\,dy}{\sqrt{v_{m}^{2}-v^{2}}}
  7. d x = y D - y d y dx=\sqrt{\frac{y}{D-y}}dy
  8. d s 2 = d x 2 + d y 2 ds^{2}=dx^{2}+dy^{2}
  9. 2 d s d 2 s = 2 d x d 2 x 2ds\ d^{2}s=2dx\ d^{2}x
  10. d x d s d 2 x = d 2 s = v d 2 t \frac{dx}{ds}d^{2}x=d^{2}s=v\ d^{2}t
  11. d 2 t 1 = 1 v 1 d x 1 d s 1 d 2 x d^{2}t_{1}=\frac{1}{v_{1}}\frac{dx_{1}}{ds_{1}}d^{2}x
  12. d 2 t 2 = 1 v 2 d x 2 d s 2 d 2 x d^{2}t_{2}=\frac{1}{v_{2}}\frac{dx_{2}}{ds_{2}}d^{2}x
  13. d 2 t 2 - d 2 t 1 = 0 = ( 1 v 2 d x 2 d s 2 - 1 v 1 d x 1 d s 1 ) d 2 x d^{2}t_{2}-d^{2}t_{1}=0=\bigg(\frac{1}{v_{2}}\frac{dx_{2}}{ds_{2}}-\frac{1}{v_% {1}}\frac{dx_{1}}{ds_{1}}\bigg)d^{2}x
  14. 1 v 2 d x 2 d s 2 = 1 v 1 d x 1 d s 1 \frac{1}{v_{2}}\frac{dx_{2}}{ds_{2}}=\frac{1}{v_{1}}\frac{dx_{1}}{ds_{1}}

Brahmagupta's_formula.html

  1. K = ( s - a ) ( s - b ) ( s - c ) ( s - d ) K=\sqrt{(s-a)(s-b)(s-c)(s-d)}
  2. s = a + b + c + d 2 . s=\frac{a+b+c+d}{2}.
  3. K = 1 4 ( - a + b + c + d ) ( a - b + c + d ) ( a + b - c + d ) ( a + b + c - d ) . K=\frac{1}{4}\sqrt{(-a+b+c+d)(a-b+c+d)(a+b-c+d)(a+b+c-d)}.
  4. K = ( a 2 + b 2 + c 2 + d 2 ) 2 + 8 a b c d - 2 ( a 4 + b 4 + c 4 + d 4 ) 4 K=\frac{\sqrt{(a^{2}+b^{2}+c^{2}+d^{2})^{2}+8abcd-2(a^{4}+b^{4}+c^{4}+d^{4})}}% {4}\cdot
  5. A D B \triangle ADB
  6. B D C \triangle BDC
  7. = 1 2 p q sin A + 1 2 r s sin C . =\frac{1}{2}pq\sin A+\frac{1}{2}rs\sin C.
  8. A B C D ABCD
  9. D A B = 180 - D C B . \angle DAB=180^{\circ}-\angle DCB.
  10. sin A = sin C . \sin A=\sin C.
  11. K = 1 2 p q sin A + 1 2 r s sin A K=\frac{1}{2}pq\sin A+\frac{1}{2}rs\sin A
  12. K 2 = 1 4 ( p q + r s ) 2 sin 2 A K^{2}=\frac{1}{4}(pq+rs)^{2}\sin^{2}A
  13. 4 K 2 = ( p q + r s ) 2 ( 1 - cos 2 A ) = ( p q + r s ) 2 - ( p q + r s ) 2 cos 2 A . 4K^{2}=(pq+rs)^{2}(1-\cos^{2}A)=(pq+rs)^{2}-(pq+rs)^{2}\cos^{2}A.\,
  14. \triangle
  15. \triangle
  16. p 2 + q 2 - 2 p q cos A = r 2 + s 2 - 2 r s cos C . p^{2}+q^{2}-2pq\cos A=r^{2}+s^{2}-2rs\cos C.\,
  17. cos C = - cos A \cos C=-\cos A
  18. A A
  19. C C
  20. 2 ( p q + r s ) cos A = p 2 + q 2 - r 2 - s 2 . 2(pq+rs)\cos A=p^{2}+q^{2}-r^{2}-s^{2}.\,
  21. 4 K 2 = ( p q + r s ) 2 - 1 4 ( p 2 + q 2 - r 2 - s 2 ) 2 4K^{2}=(pq+rs)^{2}-\frac{1}{4}(p^{2}+q^{2}-r^{2}-s^{2})^{2}
  22. 16 K 2 = 4 ( p q + r s ) 2 - ( p 2 + q 2 - r 2 - s 2 ) 2 . 16K^{2}=4(pq+rs)^{2}-(p^{2}+q^{2}-r^{2}-s^{2})^{2}.
  23. a 2 - b 2 = ( a - b ) ( a + b ) a^{2}-b^{2}=(a-b)(a+b)
  24. [ 2 ( p q + r s ) - p 2 - q 2 + r 2 + s 2 ] [ 2 ( p q + r s ) + p 2 + q 2 - r 2 - s 2 ] [2(pq+rs)-p^{2}-q^{2}+r^{2}+s^{2}][2(pq+rs)+p^{2}+q^{2}-r^{2}-s^{2}]\,
  25. = [ ( r + s ) 2 - ( p - q ) 2 ] [ ( p + q ) 2 - ( r - s ) 2 ] =[(r+s)^{2}-(p-q)^{2}][(p+q)^{2}-(r-s)^{2}]\,
  26. = ( q + r + s - p ) ( p + r + s - q ) ( p + q + s - r ) ( p + q + r - s ) . =(q+r+s-p)(p+r+s-q)(p+q+s-r)(p+q+r-s).\,
  27. S = p + q + r + s 2 , S=\frac{p+q+r+s}{2},
  28. 16 K 2 = 16 ( S - p ) ( S - q ) ( S - r ) ( S - s ) . 16K^{2}=16(S-p)(S-q)(S-r)(S-s).\,
  29. K = ( S - p ) ( S - q ) ( S - r ) ( S - s ) . K=\sqrt{(S-p)(S-q)(S-r)(S-s)}.
  30. K = ( s - a ) ( s - b ) ( s - c ) ( s - d ) - a b c d cos 2 θ K=\sqrt{(s-a)(s-b)(s-c)(s-d)-abcd\cos^{2}\theta}
  31. a b c d cos 2 θ = a b c d cos 2 ( 90 ) = a b c d 0 = 0 , abcd\cos^{2}\theta=abcd\cos^{2}\left(90^{\circ}\right)=abcd\cdot 0=0,\,
  32. K = ( s - a ) ( s - b ) ( s - c ) ( s - d ) - 1 4 ( a c + b d + p q ) ( a c + b d - p q ) K=\sqrt{(s-a)(s-b)(s-c)(s-d)-\textstyle{1\over 4}(ac+bd+pq)(ac+bd-pq)}\,
  33. p q = a c + b d pq=ac+bd

Brahmagupta.html

  1. b x + c = d x + e bx+c=dx+e
  2. x = e - c b - d x=\tfrac{e-c}{b-d}
  3. a x 2 + b x = c ax^{2}+bx=c
  4. x = 4 a c + b 2 - b 2 a x=\frac{\sqrt{4ac+b^{2}}-b}{2a}
  5. x = a c + b 2 4 - b 2 a . x=\frac{\sqrt{ac+\tfrac{b^{2}}{4}}-\tfrac{b}{2}}{a}.
  6. a c + b c \tfrac{a}{c}+\tfrac{b}{c}
  7. a c b d \tfrac{a}{c}\cdot\tfrac{b}{d}
  8. a 1 + b d \tfrac{a}{1}+\tfrac{b}{d}
  9. a c + b d a c = a ( d + b ) c d \tfrac{a}{c}+\tfrac{b}{d}\cdot\tfrac{a}{c}=\tfrac{a(d+b)}{cd}
  10. a c - b d a c = a ( d - b ) c d \tfrac{a}{c}-\tfrac{b}{d}\cdot\tfrac{a}{c}=\tfrac{a(d-b)}{cd}
  11. 0 0 = 0 \tfrac{0}{0}=0
  12. a 0 \tfrac{a}{0}
  13. a 0 a\neq 0
  14. N x 2 + 1 = y 2 Nx^{2}+1=y^{2}
  15. ( x 1 2 - N y 1 2 ) ( x 2 2 - N y 2 2 ) = ( x 1 x 2 + N y 1 y 2 ) 2 - N ( x 1 y 2 + x 2 y 1 ) 2 (x^{2}_{1}-Ny^{2}_{1})(x^{2}_{2}-Ny^{2}_{2})=(x_{1}x_{2}+Ny_{1}y_{2})^{2}-N(x_% {1}y_{2}+x_{2}y_{1})^{2}
  16. ( x 1 2 - y 1 2 ) ( x 2 2 - y 2 2 ) = ( x 1 x 2 + y 1 y 2 ) 2 - ( x 1 y 2 + x 2 y 1 ) 2 . (x^{2}_{1}-y^{2}_{1})(x^{2}_{2}-y^{2}_{2})=(x_{1}x_{2}+y_{1}y_{2})^{2}-(x_{1}y% _{2}+x_{2}y_{1})^{2}.
  17. ( x 1 , (x_{1},
  18. y 1 ) y_{1})
  19. ( x 2 , (x_{2},
  20. y 2 ) y_{2})
  21. x 2 - N y 2 = k 1 x^{2}-Ny^{2}=k_{1}
  22. x 2 - N y 2 = k 2 x^{2}-Ny^{2}=k_{2}
  23. ( x 1 x 2 + N y 1 y 2 , (x_{1}x_{2}+Ny_{1}y_{2},
  24. x 1 y 2 + x 2 y 1 ) x_{1}y_{2}+x_{2}y_{1})
  25. x 2 - N y 2 = k 1 k 2 x^{2}-Ny^{2}=k_{1}k_{2}
  26. x 2 - N y 2 = k i x^{2}-Ny^{2}=k_{i}
  27. x 2 - N y 2 = k x^{2}-Ny^{2}=k
  28. x 2 - N y 2 = 1 x^{2}-Ny^{2}=1
  29. ( p + r 2 ) ( q + s 2 ) (\tfrac{p+r}{2})(\tfrac{q+s}{2})
  30. t = p + q + r + s 2 t=\tfrac{p+q+r+s}{2}
  31. ( t - p ) ( t - q ) ( t - r ) ( t - s ) . \sqrt{(t-p)(t-q)(t-r)(t-s)}.
  32. 1 2 ( b ± c 2 - a 2 b ) \frac{1}{2}(b\pm\frac{c^{2}-a^{2}}{b})
  33. a = 1 2 ( u 2 v + v ) , b = 1 2 ( u 2 w + w ) , c = 1 2 ( u 2 v - v + u 2 w - w ) a=\frac{1}{2}\left(\frac{u^{2}}{v}+v\right),\ \ b=\frac{1}{2}\left(\frac{u^{2}% }{w}+w\right),\ \ c=\frac{1}{2}\left(\frac{u^{2}}{v}-v+\frac{u^{2}}{w}-w\right)
  34. p r + q s \sqrt{pr+qs}
  35. 10 \sqrt{10}
  36. f f
  37. f ( a + x h ) f ( a ) + x ( Δ f ( a ) + Δ f ( a - h ) 2 ) + x 2 Δ 2 f ( a - h ) 2 ! . f(a+xh)\approx f(a)+x\left(\frac{\Delta f(a)+\Delta f(a-h)}{2}\right)+\frac{x^% {2}\Delta^{2}f(a-h)}{2!}.
  38. Δ f ( a ) = def f ( a + h ) - f ( a ) . \Delta f(a)\ \stackrel{\mathrm{def}}{=}\ f(a+h)-f(a).

Breadth-first_search.html

  1. G G
  2. v v
  3. G G
  4. v v
  5. O ( | V | + | E | ) O(|V|+|E|)
  6. | V | |V|
  7. | E | |E|
  8. O ( | E | ) O(|E|)
  9. O ( 1 ) O(1)
  10. O ( | V | 2 ) O(|V|^{2})
  11. O ( | V | ) O(|V|)
  12. | V | |V|
  13. Θ ( | V | + | E | ) \Theta(|V|+|E|)
  14. Θ ( | V | 2 ) \Theta(|V|^{2})
  15. d d
  16. O ( b < s u p > d + 1 ) O(b<sup>d+1)

Break-even_(economics).html

  1. TR \displaystyle\,\text{TR}
  2. ( P - V ) \left(\,\text{P}-\,\text{V}\right)
  3. Total Contribution \displaystyle\,\text{Total Contribution}
  4. Break-even(in Sales) = Fixed Costs C / P . \,\text{Break-even(in Sales)}=\frac{\,\text{Fixed Costs}}{\,\text{C}/\,\text{P% }}.

Breakthrough_Propulsion_Physics_Program.html

  1. V = ( - m ) [ - G ( x + d ) 2 + y 2 ] V=(-m)\begin{bmatrix}\frac{-G}{\sqrt{(x+d)^{2}+y^{2}}}\end{bmatrix}
  2. + ( + m ) [ - G ( x + d ) 2 + y 2 ] +(+m)\begin{bmatrix}\frac{-G}{\sqrt{(x+d)^{2}+y^{2}}}\end{bmatrix}
  3. V V\!
  4. G G\!
  5. m m\!
  6. d d\!
  7. V = - G m S ( x - d ) 2 + y 2 V=\frac{-Gm_{S}}{\sqrt{(x-d)^{2}+y^{2}}}
  8. V = ( x B e . . r 2 + 1 ) [ - G ( m / r ) ] V=(xBe^{..r^{2}}+1)[-G(m/r)]
  9. V = [ ( - G m ) / r ] + ( - x A e . . r 2 ) V=[(-Gm)/r]+(-xAe^{..r^{2}})
  10. e . . r 2 e^{..r^{2}}
  11. r r\!
  12. A A\!
  13. B B\!

Brute-force_search.html

  1. \leftarrow
  2. \neq
  3. \leftarrow
  4. \geq
  5. < <
  6. \geq

Bruun's_FFT_algorithm.html

  1. X k = n = 0 N - 1 x n e - 2 π i N n k k = 0 , , N - 1. X_{k}=\sum_{n=0}^{N-1}x_{n}e^{-\frac{2\pi i}{N}nk}\qquad k=0,\dots,N-1.
  2. ω N n = e - 2 π i N n \omega_{N}^{n}=e^{-\frac{2\pi i}{N}n}
  3. x ( z ) = n = 0 N - 1 x n z n . x(z)=\sum_{n=0}^{N-1}x_{n}z^{n}.
  4. X k = x ( ω N k ) = x ( z ) mod ( z - ω N k ) X_{k}=x(\omega_{N}^{k})=x(z)\mod(z-\omega_{N}^{k})
  5. x ( z ) x(z)
  6. x ( z ) x(z)
  7. U ( z ) U(z)
  8. V ( z ) V(z)
  9. U ( z ) U(z)
  10. V ( z ) V(z)
  11. x ( z ) x(z)
  12. ( z - ω N k ) (z-\omega_{N}^{k})
  13. z N - 1 z^{N}-1
  14. z N - 1 z^{N}-1
  15. x ( z ) x(z)
  16. z N - 1 = F 1 ( z ) F 2 ( z ) F 3 ( z ) z^{N}-1=F_{1}(z)F_{2}(z)F_{3}(z)
  17. F k ( z ) = F k , 1 ( z ) F k , 2 ( z ) F_{k}(z)=F_{k,1}(z)F_{k,2}(z)
  18. z N - 1 z^{N}-1
  19. F 1 = ( z N / 2 - 1 ) F_{1}=(z^{N/2}-1)
  20. F 2 = ( z N / 2 + 1 ) F_{2}=(z^{N/2}+1)
  21. x ( z ) x(z)
  22. F 2 F_{2}
  23. F 1 F_{1}
  24. z 2 M - 1 = ( z M - 1 ) ( z M + 1 ) z^{2M}-1=(z^{M}-1)(z^{M}+1)\,
  25. z 4 M + a z 2 M + 1 = ( z 2 M + 2 - a z M + 1 ) ( z 2 M - 2 - a z M + 1 ) z^{4M}+az^{2M}+1=(z^{2M}+\sqrt{2-a}z^{M}+1)(z^{2M}-\sqrt{2-a}z^{M}+1)
  26. a = 2 cos ( ϕ ) a=2\cos(\phi)
  27. ϕ ( 0 , π ) \phi\in(0,\pi)
  28. 2 + a = 2 cos ϕ 2 \sqrt{2+a}=2\cos\tfrac{\phi}{2}
  29. 2 - a = 2 cos ( π - ϕ 2 ) \sqrt{2-a}=2\cos(\pi-\tfrac{\phi}{2})
  30. p s , 0 , , p s , 2 s - 1 p_{s,0},\dots,p_{s,2^{s}-1}
  31. p s , 0 ( z ) = p ( z ) mod ( z 2 n - s - 1 ) and p s , m ( z ) = p ( z ) mod ( z 2 n - s - 2 cos ( m 2 s π ) z 2 n - 1 - s + 1 ) m = 1 , 2 , , 2 s - 1 \begin{aligned}\displaystyle p_{s,0}(z)&\displaystyle=p(z)\mod\left(z^{2^{n-s}% }-1\right)&&\displaystyle\,\text{and}\\ \displaystyle p_{s,m}(z)&\displaystyle=p(z)\mod\left(z^{2^{n-s}}-2\cos\left(% \tfrac{m}{2^{s}}\pi\right)z^{2^{n-1-s}}+1\right)&\displaystyle m&\displaystyle% =1,2,\dots,2^{s}-1\end{aligned}
  32. X k = p ( e 2 π i k 2 n ) X_{k}=p(e^{2\pi i\tfrac{k}{2^{n}}})
  33. p s , ( z ) p_{s,\ell}(z)
  34. p s + 1 , ( z ) p_{s+1,\ell}(z)
  35. p s + 1 , 2 s - ( z ) p_{s+1,2^{s}-\ell}(z)
  36. ϕ N , α ( z ) = { z 2 N - 2 cos ( 2 π α ) z N + 1 if 0 < α < 1 z 2 N - 1 if α = 0 \phi_{N,\alpha}(z)=\left\{\begin{matrix}z^{2N}-2\cos(2\pi\alpha)z^{N}+1&\mbox{% if }~{}0<\alpha<1\\ \\ z^{2N}-1&\mbox{if }~{}\alpha=0\end{matrix}\right.
  37. e 2 π i ( ± α + k ) / N e^{2\pi i(\pm\alpha+k)/N}
  38. k = 0 , 1 , , N - 1 k=0,1,\dots,N-1
  39. α 0 \alpha\neq 0
  40. e 2 π i k / 2 N e^{2\pi ik/2N}
  41. k = 0 , 1 , , 2 N - 1 k=0,1,\dots,2N-1
  42. α = 0 \alpha=0
  43. ϕ r M , α ( z ) = { = 0 r - 1 ϕ M , ( α + ) / r if 0 < α 0.5 = 0 r - 1 ϕ M , ( 1 - α + ) / r if 0.5 < α < 1 = 0 r - 1 ϕ M , / ( 2 r ) if α = 0 \phi_{rM,\alpha}(z)=\left\{\begin{array}[]{ll}\prod_{\ell=0}^{r-1}\phi_{M,(% \alpha+\ell)/r}&\mbox{if }~{}0<\alpha\leq 0.5\\ \\ \prod_{\ell=0}^{r-1}\phi_{M,(1-\alpha+\ell)/r}&\mbox{if }~{}0.5<\alpha<1\\ \\ \prod_{\ell=0}^{r-1}\phi_{M,\ell/(2r)}&\mbox{if }~{}\alpha=0\end{array}\right.

Budget_constraint.html

  1. P x x + P y y = m P_{x}x+P_{y}y=m
  2. m = m=
  3. P x = P_{x}=
  4. P y = P_{y}=
  5. x = x=
  6. y = y=
  7. y = ( m / P y ) - ( P x / P y ) x y=(m/P_{y})-(P_{x}/P_{y})x
  8. ( m / P y ) (m/P_{y})
  9. ( - P x / P y ) (-P_{x}/P_{y})
  10. P x P_{x}
  11. P y P_{y}
  12. n n\,
  13. x i x_{i}\,
  14. i = 1 , , n i=1,\dots,n\,
  15. x i x_{i}\,
  16. p i p_{i}\,
  17. W \,W\,
  18. i = 1 n p i x i W . \sum_{i=1}^{n}p_{i}x_{i}\leq W.
  19. i = 1 n p i x i = W . \sum_{i=1}^{n}p_{i}x_{i}=W.
  20. x i x_{i}\,
  21. x i x_{i}\,
  22. p i / p j p_{i}/p_{j}\,
  23. x j . x_{j}.\,

Building_(mathematics).html

  1. A ~ 1 {\scriptstyle\tilde{A}_{1}}
  2. \cap
  3. \subset
  4. \subset
  5. \subset
  6. \subset
  7. \oplus
  8. \oplus
  9. R = { x : x p 1 } . R=\{x:\|x\|_{p}\leq 1\}.
  10. \oplus
  11. \oplus
  12. \subset
  13. \subset
  14. \subset
  15. \subset
  16. \subset
  17. \rightarrow

Buoyancy.html

  1. apparent immersed weight = weight - weight of displaced fluid \,\text{apparent immersed weight}=\,\text{weight}-\,\text{weight of displaced % fluid}\,
  2. density density of fluid = weight weight of displaced fluid , \frac{\,\text{density}}{\,\text{density of fluid}}=\frac{\,\text{weight}}{\,% \text{weight of displaced fluid}},\,
  3. density of object density of fluid = weight weight - apparent immersed weight \frac{\text{density of object}}{\,\text{density of fluid}}=\frac{\,\text{% weight}}{\,\text{weight}-\,\text{apparent immersed weight}}\,
  4. 𝐟 + div σ = 0 \mathbf{f}+\operatorname{div}\,\sigma=0
  5. σ i j = - p δ i j . \sigma_{ij}=-p\delta_{ij}.\,
  6. 𝐟 = p . \mathbf{f}=\nabla p.\,
  7. 𝐟 = - Φ . \mathbf{f}=-\nabla\Phi.\,
  8. ( p + Φ ) = 0 p + Φ = constant . \nabla(p+\Phi)=0\Longrightarrow p+\Phi=\,\text{constant}.\,
  9. p = ρ f g z . p=\rho_{f}gz.\,
  10. 𝐁 = σ d 𝐀 . \mathbf{B}=\oint\sigma\,d\mathbf{A}.
  11. 𝐁 = div σ d V = - 𝐟 d V = - ρ f 𝐠 d V = - ρ f 𝐠 V \mathbf{B}=\int\operatorname{div}\sigma\,dV=-\int\mathbf{f}\,dV=-\rho_{f}% \mathbf{g}\int\,dV=-\rho_{f}\mathbf{g}V
  12. B = ρ f V disp g , B=\rho_{f}V\text{disp}\,g,\,
  13. B = ρ f V g . B=\rho_{f}Vg.\,
  14. F net = 0 = m g - ρ f V disp g F\text{net}=0=mg-\rho_{f}V\text{disp}g\,
  15. m g = ρ f V disp g , mg=\rho_{f}V\text{disp}g,\,
  16. m = ρ f V disp . m=\rho_{f}V\text{disp}.\,
  17. T = ρ f V g - m g . T=\rho_{f}Vg-mg.\,
  18. N = m g - ρ f V g . N=mg-\rho_{f}Vg.\,

Burnside's_lemma.html

  1. | X / G | = 1 | G | g G | X g | . |X/G|=\frac{1}{|G|}\sum_{g\in G}|X^{g}|.
  2. | G | | X / G | = g G | X g | . |G||X/G|=\sum_{g\in G}|X^{g}|.
  3. 1 24 ( 3 6 + 6 3 3 + 3 3 4 + 8 3 2 + 6 3 3 ) = 57. \frac{1}{24}\left(3^{6}+6\cdot 3^{3}+3\cdot 3^{4}+8\cdot 3^{2}+6\cdot 3^{3}% \right)=57.
  4. 1 24 ( n 6 + 3 n 4 + 12 n 3 + 8 n 2 ) . \frac{1}{24}\left(n^{6}+3n^{4}+12n^{3}+8n^{2}\right).
  5. g G | X g | = | { ( g , x ) G × X g . x = x } | = x X | G x | . \sum_{g\in G}|X^{g}|=|\{(g,x)\in G\times X\mid g.x=x\}|=\sum_{x\in X}|G_{x}|.
  6. | G . x | = [ G : G x ] = | G | / | G x | . |G.x|=[G\,:\,G_{x}]=|G|/|G_{x}|.
  7. x X | G x | = x X | G | | G . x | = | G | x X 1 | G . x | . \sum_{x\in X}|G_{x}|=\sum_{x\in X}\frac{|G|}{|G.x|}=|G|\sum_{x\in X}\frac{1}{|% G.x|}.
  8. x X 1 | G . x | = A X / G x A 1 | A | = A X / G 1 = | X / G | . \sum_{x\in X}\frac{1}{|G.x|}=\sum_{A\in X/G}\sum_{x\in A}\frac{1}{|A|}=\sum_{A% \in X/G}1=|X/G|.
  9. g G | X g | = | G | | X / G | . \sum_{g\in G}|X^{g}|=|G|\cdot|X/G|.

Burr_puzzle.html

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

Burrows–Abadi–Needham_logic.html

  1. A S : A , { T A , K a b , B } K a s A\rightarrow S:A,\{T_{A},K_{ab},B\}_{K_{as}}
  2. S B : { T S , K a b , A } K b s S\rightarrow B:\{T_{S},K_{ab},A\}_{K_{bs}}

Byzantine_text-type.html

  1. 𝔐 \mathfrak{M}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}

C-symmetry.html

  1. ψ - i ( ψ ¯ γ 0 γ 2 ) T \psi\rightarrow-i(\bar{\psi}\gamma^{0}\gamma^{2})^{T}
  2. ψ ¯ - i ( γ 0 γ 2 ψ ) T \bar{\psi}\rightarrow-i(\gamma^{0}\gamma^{2}\psi)^{T}
  3. A μ - A μ A^{\mu}\rightarrow-A^{\mu}
  4. C ψ ( q ) = C ψ ( - q ) C\psi(q)=C\psi(-q)
  5. C ψ ( q ) = - C ψ ( - q ) C\psi(q)=-C\psi(-q)
  6. ψ = def ϕ + i χ 2 \psi\ \stackrel{\mathrm{def}}{=}\ {\phi+i\chi\over\sqrt{2}}

CA.html

  1. c a , c_{\mathrm{a}}\,,
  2. c l c_{\mathrm{l}}\,
  3. c z \!\ c_{\mathrm{z}}

CAC_40.html

  1. I t = 1000 × i = 1 N Q i , t F i , t f i , t C i , t K t i = 1 N Q i , 0 C i , 0 I_{t}=1000\times\frac{\sum_{i=1}^{N}Q_{i,t}\,F_{i,t}\,f_{i,t}\,C_{i,t}\,}{K_{t% }\,\sum_{i=1}^{N}Q_{i,0}\,C_{i,0}\,}

Calabi–Yau_manifold.html

  1. ( Ω Ω ¯ - ω n / n ! ) (\Omega\wedge\bar{\Omega}-\omega^{n}/n!)
  2. ω \omega
  3. g g

Calculus_of_variations.html

  1. y y
  2. J y Jy
  3. f f
  4. Δ J = J y - J f ΔJ=Jy-Jf
  5. y y
  6. f . f.

Call_option.html

  1. 𝒪 \mathcal{O}
  2. Π \Pi
  3. 0
  4. T + T\in\mathbb{R}^{+}
  5. K K\in\mathbb{R}
  6. S : [ 0 , T ] S:[0,T]\to\mathbb{R}
  7. Π \Pi
  8. 𝒪 \mathcal{O}
  9. 𝒪 \mathcal{O}
  10. - K + S T -K+S_{T}
  11. 0
  12. S T - K 0 S_{T}-K\geq 0
  13. S T - K < 0 S_{T}-K<0
  14. ( S T - K ) 0 (S_{T}-K)\vee 0
  15. ( S T - K ) + (S_{T}-K)^{+}

Campanology.html

  1. n n
  2. n n

Cantilever.html

  1. δ = 3 σ ( 1 - ν ) E L 2 t 2 \delta=\frac{3\sigma\left(1-\nu\right)}{E}\frac{L^{2}}{t^{2}}
  2. E E
  3. L L
  4. t t
  5. k k
  6. k = F δ = E w t 3 4 L 3 k=\frac{F}{\delta}=\frac{Ewt^{3}}{4L^{3}}
  7. F F
  8. w w
  9. ω 0 \omega_{0}
  10. ω 0 = k / m equivalent \omega_{0}=\sqrt{k/m\text{equivalent}}

Capacitance.html

  1. C = q V . C=\frac{q}{V}.
  2. I ( t ) = C d V ( t ) d t . I(t)=C\frac{\mathrm{d}V(t)}{\mathrm{d}t}.
  3. d W = q C d q \mathrm{d}W=\frac{q}{C}\,\mathrm{d}q
  4. W charging = 0 Q q C d q = 1 2 Q 2 C = 1 2 Q V = 1 2 C V 2 = W stored . W\text{charging}=\int_{0}^{Q}\frac{q}{C}\,\mathrm{d}q=\frac{1}{2}\frac{Q^{2}}{% C}=\frac{1}{2}QV=\frac{1}{2}CV^{2}=W\text{stored}.
  5. C = ε r ε 0 A d \ C=\varepsilon_{r}\varepsilon_{0}\frac{A}{d}
  6. C = ε r A 4 π d C=\varepsilon_{r}\frac{A}{4\pi d}
  7. W stored = 1 2 C V 2 = 1 2 ε r ε 0 A d V 2 . W\text{stored}=\frac{1}{2}CV^{2}=\frac{1}{2}\varepsilon_{r}\varepsilon_{0}% \frac{A}{d}V^{2}.
  8. d Q = C ( V ) d V \ dQ=C(V)\,dV
  9. Q = 0 V C ( V ) d V Q=\int_{0}^{V}C(V)\,dV
  10. d W = Q d V = [ 0 V d V C ( V ) ] d V . dW=Q\,dV=\left[\int_{0}^{V}\ dV^{\prime}\ C(V^{\prime})\right]\ dV\ .
  11. W = 0 V d V 0 V d V C ( V ) = 0 V d V V V d V C ( V ) = 0 V d V ( V - V ) C ( V ) , W=\int_{0}^{V}\ dV\ \int_{0}^{V}\ dV^{\prime}\ C(V^{\prime})=\int_{0}^{V}\ dV^% {\prime}\ \int_{V^{\prime}}^{V}\ dV\ C(V^{\prime})=\int_{0}^{V}\ dV^{\prime}% \left(V-V^{\prime}\right)C(V^{\prime})\ ,
  12. s y m b o l D ( t ) = ε 0 - t ε r ( t - t ) s y m b o l E ( t ) d t , symbol{D(t)}=\varepsilon_{0}\int_{-\infty}^{t}\ \varepsilon_{r}(t-t^{\prime})% symbolE(t^{\prime})\ dt^{\prime},
  13. s y m b o l D ( ω ) = ε 0 ε r ( ω ) s y m b o l E ( ω ) , symbolD(\omega)=\varepsilon_{0}\varepsilon_{r}(\omega)symbolE(\omega)\ ,
  14. I ( ω ) = j ω Q ( ω ) = j ω Σ s y m b o l D ( s y m b o l r , ω ) d s y m b o l Σ I(\omega)=j\omega Q(\omega)=j\omega\oint_{\Sigma}symbolD(symbolr,\ \omega)% \cdot dsymbol{\Sigma}
  15. = [ G ( ω ) + j ω C ( ω ) ] V ( ω ) = V ( ω ) Z ( ω ) , =\left[G(\omega)+j\omega C(\omega)\right]V(\omega)=\frac{V(\omega)}{Z(\omega)}\ ,
  16. ε r ( ω ) = ε r ( ω ) - j ε r ′′ ( ω ) = 1 j ω Z ( ω ) C 0 = C cmplx ( ω ) C 0 , \varepsilon_{r}(\omega)=\varepsilon^{\prime}_{r}(\omega)-j\varepsilon^{\prime% \prime}_{r}(\omega)=\frac{1}{j\omega Z(\omega)C_{0}}=\frac{C_{\,\text{cmplx}}(% \omega)}{C_{0}}\ ,
  17. C = Q / V C=Q/V
  18. Q 1 , Q 2 , Q 3 Q_{1},Q_{2},Q_{3}
  19. V 1 = P 11 Q 1 + P 12 Q 2 + P 13 Q 3 , V_{1}=P_{11}Q_{1}+P_{12}Q_{2}+P_{13}Q_{3},
  20. P 12 = P 21 P_{12}=P_{21}
  21. P i j = V i Q j P_{ij}=\frac{\partial V_{i}}{\partial Q_{j}}
  22. C m C_{m}
  23. C m = Q / V C_{m}=Q/V
  24. C m = 1 ( P 11 + P 22 ) - ( P 12 + P 21 ) C_{m}=\frac{1}{(P_{11}+P_{22})-(P_{12}+P_{21})}
  25. C i j = Q i V j C_{ij}=\frac{\partial Q_{i}}{\partial V_{j}}
  26. C = 4 π ε 0 R C=4\pi\varepsilon_{0}R\,
  27. ε A / d \varepsilon A/d
  28. 2 π ε l ln ( R 2 / R 1 ) \frac{2\pi\varepsilon l}{\ln\left(R_{2}/R_{1}\right)}
  29. π ε l arcosh ( d 2 a ) = π ε l ln ( d 2 a + d 2 4 a 2 - 1 ) \frac{\pi\varepsilon l}{\operatorname{arcosh}\left(\frac{d}{2a}\right)}=\frac{% \pi\varepsilon l}{\ln\left(\frac{d}{2a}+\sqrt{\frac{d^{2}}{4a^{2}}-1}\right)}
  30. 2 π ε l arcosh ( d a ) = 2 π ε l ln ( d a + d 2 a 2 - 1 ) \frac{2\pi\varepsilon l}{\operatorname{arcosh}\left(\frac{d}{a}\right)}=\frac{% 2\pi\varepsilon l}{\ln\left(\frac{d}{a}+\sqrt{\frac{d^{2}}{a^{2}}-1}\right)}
  31. ε l K ( 1 - k 2 ) K ( k ) \varepsilon l\frac{K\left(\sqrt{1-k^{2}}\right)}{K\left(k\right)}
  32. 4 π ε 1 R 1 - 1 R 2 \frac{4\pi\varepsilon}{\frac{1}{R_{1}}-\frac{1}{R_{2}}}
  33. 2 π ε a n = 1 sinh ( ln ( D + D 2 - 1 ) ) sinh ( n ln ( D + D 2 - 1 ) ) 2\pi\varepsilon a\sum_{n=1}^{\infty}\frac{\sinh\left(\ln\left(D+\sqrt{D^{2}-1}% \right)\right)}{\sinh\left(n\ln\left(D+\sqrt{D^{2}-1}\right)\right)}
  34. = 2 π ε a { 1 + 1 2 D + 1 4 D 2 + 1 8 D 3 + 1 8 D 4 + 3 32 D 5 + O ( 1 D 6 ) } =2\pi\varepsilon a\left\{1+\frac{1}{2D}+\frac{1}{4D^{2}}+\frac{1}{8D^{3}}+% \frac{1}{8D^{4}}+\frac{3}{32D^{5}}+O\left(\frac{1}{D^{6}}\right)\right\}
  35. = 2 π ε a { ln 2 + γ - 1 2 ln ( 2 D - 2 ) + O ( 2 D - 2 ) } =2\pi\varepsilon a\left\{\ln 2+\gamma-\frac{1}{2}\ln\left(2D-2\right)+O\left(2% D-2\right)\right\}
  36. 4 π ε a n = 1 sinh ( ln ( D + D 2 - 1 ) ) sinh ( n ln ( D + D 2 - 1 ) ) 4\pi\varepsilon a\sum_{n=1}^{\infty}\frac{\sinh\left(\ln\left(D+\sqrt{D^{2}-1}% \right)\right)}{\sinh\left(n\ln\left(D+\sqrt{D^{2}-1}\right)\right)}
  37. 4 π ε a 4\pi\varepsilon a
  38. 8 ε a 8\varepsilon a
  39. 2 π ε l Λ { 1 + 1 Λ ( 1 - ln 2 ) + 1 Λ 2 [ 1 + ( 1 - ln 2 ) 2 - π 2 12 ] + O ( 1 Λ 3 ) } \frac{2\pi\varepsilon l}{\Lambda}\left\{1+\frac{1}{\Lambda}\left(1-\ln 2\right% )+\frac{1}{\Lambda^{2}}\left[1+\left(1-\ln 2\right)^{2}-\frac{\pi^{2}}{12}% \right]+O\left(\frac{1}{\Lambda^{3}}\right)\right\}
  40. μ ( N ) = U ( N ) - U ( N - 1 ) \mu(N)=U(N)-U(N-1)
  41. 1 C Δ V Δ Q {1\over C}\equiv{\Delta\,V\over\Delta\,Q}
  42. Δ V = Δ μ e = μ ( N + Δ N ) - μ ( N ) e \Delta\,V={\Delta\,\mu\,\over e}={\mu(N+\Delta\,N)-\mu(N)\over e}
  43. Δ N = 1 \Delta\,N=1
  44. Δ Q = e \Delta\,Q=e
  45. C Q ( N ) = e 2 μ ( N + 1 ) - μ ( N ) = e 2 E ( N ) C_{Q}(N)={e^{2}\over\mu(N+1)-\mu(N)}={e^{2}\over E(N)}
  46. C Q ( N ) = e 2 U ( N ) C_{Q}(N)={e^{2}\over U(N)}
  47. W stored = U W\text{stored}=U
  48. C = Q 2 2 U C={Q^{2}\over 2U}
  49. Q = N e Q=Ne
  50. W charging = U = 0 Q q C d q W\text{charging}=U=\int_{0}^{Q}\frac{q}{C}\,\mathrm{d}q
  51. d q = 0 \mathrm{d}q=0
  52. d q Δ Q = e \mathrm{d}q\to\Delta\,Q=e
  53. Q = C V Q=CV
  54. U = Q V U=QV
  55. C = Q 1 V = Q Q U = Q 2 U C=Q{1\over V}=Q{Q\over U}={Q^{2}\over U}
  56. U ( N ) U(N)
  57. U ( N ) U U(N)\to U
  58. C ( N ) = ( N e ) 2 U ( N ) C(N)={(Ne)^{2}\over U(N)}

Capillary_action.html

  1. Ψ m \Psi_{m}
  2. h = 2 γ cos θ ρ g r , h={{2\gamma\cos{\theta}}\over{\rho gr}},
  3. γ \scriptstyle\gamma
  4. h 1.48 × 10 - 5 r m . h\approx{{1.48\times 10^{-5}}\over r}\ \mbox{m}~{}.
  5. V = A S t , V=AS\sqrt{t},
  6. i = V A i=\frac{V}{A}
  7. x = i f = S f t . x=\frac{i}{f}=\frac{S}{f}\sqrt{t}.

Capital_asset_pricing_model.html

  1. E ( R i ) - R f β i = E ( R m ) - R f \frac{E(R_{i})-R_{f}}{\beta_{i}}=E(R_{m})-R_{f}
  2. E ( R i ) E(R_{i})~{}~{}
  3. E ( R i ) = R f + β i ( E ( R m ) - R f ) E(R_{i})=R_{f}+\beta_{i}(E(R_{m})-R_{f})\,
  4. E ( R i ) E(R_{i})~{}~{}
  5. R f R_{f}~{}
  6. β i \beta_{i}~{}~{}
  7. β i = Cov ( R i , R m ) Var ( R m ) \beta_{i}=\frac{\mathrm{Cov}(R_{i},R_{m})}{\mathrm{Var}(R_{m})}
  8. E ( R m ) E(R_{m})~{}
  9. E ( R m ) - R f E(R_{m})-R_{f}~{}
  10. E ( R i ) - R f E(R_{i})-R_{f}~{}
  11. E ( R i ) - R f = β i ( E ( R m ) - R f ) E(R_{i})-R_{f}=\beta_{i}(E(R_{m})-R_{f})\,
  12. E ( R i ) = R f + β ( R P m ) + R P s + R P u E(R_{i})=R_{f}+\beta(RP_{m})+RP_{s}+RP_{u}
  13. E ( R i ) E(R_{i})
  14. R f R_{f}
  15. R P m RP_{m}
  16. R P s RP_{s}
  17. R P u RP_{u}
  18. SML : E ( R i ) = R f + β i ( E ( R M ) - R f ) . \mathrm{SML}:E(R_{i})=R_{f}+\beta_{i}(E(R_{M})-R_{f}).~{}
  19. E ( R i ) E(R_{i})
  20. t t
  21. E ( R t ) = E ( P t + 1 ) - P t P t E(R_{t})=\frac{E(P_{t+1})-P_{t}}{P_{t}}
  22. P t P_{t}
  23. t + 1 t+1
  24. P 0 P_{0}
  25. P 0 = 1 1 + R f [ E ( P T ) - Cov ( P T , R M ) ( E ( R M ) - R f ) Var ( R M ) ] P_{0}=\frac{1}{1+R_{f}}\left[E(P_{T})-\frac{\mathrm{Cov}(P_{T},R_{M})(E(R_{M})% -R_{f})}{\mathrm{Var}(R_{M})}\right]
  26. P T P_{T}

Carbon-14.html

  1. C 6 14 N 7 14 + e - + ν ¯ e \mathrm{~{}^{14}_{6}C}\rightarrow\mathrm{~{}^{14}_{7}N}+e^{-}+\bar{\nu}_{e}

Carbon-burning_process.html

  1. M \begin{smallmatrix}M_{\odot}\end{smallmatrix}

Card.html

  1. c a r d card

Cardiac_output.html

  1. Q Q
  2. Q ˙ c \dot{Q}_{c}
  3. C O [ L / m i n ] = S V [ L / b e a t ] × H R [ b e a t s / m i n ] CO_{[L/min]}=SV_{[L/beat]}\times HR_{[beats/min]}
  4. S V = std ( A P ) χ SV=\mathrm{std}(AP)\cdot\chi
  5. B P k ( constant ) BP\cdot k\mathrm{\ (constant)}
  6. S V = E D V - E S V E F = S V E D V × 100 % Q = S V × H R = E F × E D V × H R 100 % \begin{aligned}\displaystyle SV&\displaystyle=EDV-ESV\\ \displaystyle EF&\displaystyle=\frac{SV}{EDV}\times 100\%\\ \displaystyle Q&\displaystyle=SV\times HR\\ &\displaystyle=\frac{EF\times EDV\times HR}{100\%}\end{aligned}
  7. B S A [ m 2 ] = W [ k g ] 0.425 × H [ c m ] 0725 × 0.007184 BSA_{[m^{2}]}=W^{0.425}_{[kg]}\times H^{0725}_{[cm]}\times 0.007184
  8. S I [ m l / b e a t / m 2 ] = S V [ m l ] B S A [ m 2 ] SI_{[ml/beat/{m}^{2}]}=\frac{SV_{[ml]}}{BSA_{[m^{2}]}}
  9. C I [ l / m i n / m 2 ] = C O [ l / m i n ] B S A [ m 2 ] CI_{[l/min/{m}^{2}]}=\frac{CO_{[l/min]}}{BSA_{[{m}^{2}]}}
  10. C I [ l / m i n / m 2 ] = ( S I [ m l / b e a t / m 2 ] × H R [ b p m ] ) / 1000 CI_{[l/min/{m}^{2}]}=(SI_{[ml/beat/{m}^{2}]}\times HR_{[bpm]})/1000
  11. V O 2 = ( Q × C A ) - ( Q × C V ) V_{O_{2}}=(Q\times C_{A})-(Q\times C_{V})
  12. Q = V O 2 C A - C V Q\ =\frac{{{V}_{O}}_{2}}{{C}_{A}-{C}_{V}}
  13. Oxygen content of blood \displaystyle\mathrm{Oxygen\ content\ of\ blood}

Carl_Gustav_Jacob_Jacobi.html

  1. g g
  2. 𝐂 g {\mathbf{C}}^{g}

Carnot's_theorem.html

  1. D F + D G + D H = R + r , DF+DG+DH=R+r,

Carnot's_theorem_(thermodynamics).html

  1. η max = η Carnot = 1 - T C T H \eta_{\,\text{max}}=\eta_{\,\text{Carnot}}=1-\frac{T_{C}}{T_{H}}
  2. η \eta
  3. η M \eta_{M}
  4. η L \eta_{L}
  5. η M > η L \eta_{M}>\eta_{L}
  6. Q outhot = Q < η M η L Q = Q inhot Q\text{out}\text{hot}=Q<\frac{\eta_{M}}{\eta_{L}}Q=Q\text{in}\text{hot}
  7. η L \eta_{L}
  8. η L \eta_{L}
  9. η L \eta_{L}
  10. η M \eta_{M}
  11. Q Q
  12. W W
  13. η L η M \eta_{L}\leqslant\eta_{M}
  14. E i n E_{in}
  15. E o u t E_{out}
  16. E i n M = Q = ( 1 - η M ) Q + η M Q = E o u t M E_{in}^{M}=Q=(1-\eta_{M})Q+\eta_{M}Q=E_{out}^{M}
  17. E i n L = η M Q + η M Q ( 1 η L - 1 ) = η M η L Q = E o u t L E_{in}^{L}=\eta_{M}Q+\eta_{M}Q\left(\frac{1}{\eta_{L}}-1\right)=\frac{\eta_{M}% }{\eta_{L}}Q=E_{out}^{L}
  18. η = W / Q h \eta=W/Q_{h}
  19. η M = W M Q h M = η M Q Q = η M \eta_{M}=\frac{W_{M}}{Q^{M}_{h}}=\frac{\eta_{M}Q}{Q}=\eta_{M}
  20. η L = W L Q h L = η M Q η M η L Q = η L \eta_{L}=\frac{W_{L}}{Q^{L}_{h}}=\frac{\eta_{M}Q}{\frac{\eta_{M}}{\eta_{L}}Q}=% \eta_{L}
  21. W / Q h W/Q_{h}
  22. Q c / W Q_{c}/W
  23. W / Q h W/Q_{h}
  24. T 1 T_{1}
  25. T 2 T_{2}
  26. η \eta
  27. Δ S = a b d Q r e v T \Delta S=\int_{a}^{b}\frac{dQ_{rev}}{T}
  28. η M > η L \eta_{M}>\eta_{L}
  29. η = w c y q H = q H - q C q H = 1 - q C q H \eta=\frac{w_{cy}}{q_{H}}=\frac{q_{H}-q_{C}}{q_{H}}=1-\frac{q_{C}}{q_{H}}
  30. q C q H = f ( T H , T C ) \frac{q_{C}}{q_{H}}=f(T_{H},T_{C})
  31. f ( T 1 , T 3 ) = q 3 q 1 = q 2 q 3 q 1 q 2 = f ( T 1 , T 2 ) f ( T 2 , T 3 ) . f(T_{1},T_{3})=\frac{q_{3}}{q_{1}}=\frac{q_{2}q_{3}}{q_{1}q_{2}}=f(T_{1},T_{2}% )f(T_{2},T_{3}).
  32. T 1 T_{1}
  33. f ( T 2 , T 3 ) = f ( T 1 , T 3 ) f ( T 1 , T 2 ) = 273.16 f ( T 1 , T 3 ) 273.16 f ( T 1 , T 2 ) . f(T_{2},T_{3})=\frac{f(T_{1},T_{3})}{f(T_{1},T_{2})}=\frac{273.16\cdot f(T_{1}% ,T_{3})}{273.16\cdot f(T_{1},T_{2})}.
  34. T = 273.16 f ( T 1 , T ) T=273.16\cdot f(T_{1},T)\,
  35. f ( T 2 , T 3 ) = T 3 T 2 , f(T_{2},T_{3})=\frac{T_{3}}{T_{2}},
  36. q C q H = f ( T H , T C ) = T C T H \frac{q_{C}}{q_{H}}=f(T_{H},T_{C})=\frac{T_{C}}{T_{H}}
  37. η = 1 - q C q H = 1 - T C T H \eta=1-\frac{q_{C}}{q_{H}}=1-\frac{T_{C}}{T_{H}}
  38. T = T H = T C T=T_{H}=T_{C}
  39. T H = T C T_{H}=T_{C}

Caron.html

  1. x ˇ \check{x}

Cartesian_closed_category.html

  1. Hom ( X × Y , Z ) Hom ( X , Z Y ) \mathrm{Hom}(X\times Y,Z)\cong\mathrm{Hom}(X,Z^{Y})

Catagenesis_(geology).html

  1. X 0 H c + X ( t ) X_{0}\rightarrow Hc+X(t)
  2. d X d t = - κ X \frac{dX}{dt}=-\kappa X

Catalan_number.html

  1. C n = 1 n + 1 ( 2 n n ) = ( 2 n ) ! ( n + 1 ) ! n ! = k = 2 n n + k k for n 0. C_{n}=\frac{1}{n+1}{2n\choose n}=\frac{(2n)!}{(n+1)!\,n!}=\prod\limits_{k=2}^{% n}\frac{n+k}{k}\qquad\mbox{ for }~{}n\geq 0.
  2. C n = ( 2 n n ) - ( 2 n n + 1 ) = 1 n + 1 ( 2 n n ) for n 0 , C_{n}={2n\choose n}-{2n\choose n+1}={1\over n+1}{2n\choose n}\quad\,\text{ for% }n\geq 0,
  3. ( 2 n n + 1 ) = n n + 1 ( 2 n n ) {\textstyle\left({{2n}\atop{n+1}}\right)}=\tfrac{n}{n+1}{\textstyle\left({{2n}% \atop{n}}\right)}
  4. C 0 = 1 and C n + 1 = i = 0 n C i C n - i for n 0 ; C_{0}=1\quad\mbox{and}~{}\quad C_{n+1}=\sum_{i=0}^{n}C_{i}\,C_{n-i}\quad\,% \text{for }n\geq 0;
  5. C n = 1 n + 1 i = 0 n ( n i ) 2 . C_{n}=\frac{1}{n+1}\sum_{i=0}^{n}{n\choose i}^{2}.
  6. ( 2 n n ) = i = 0 n ( n i ) 2 , {\textstyle\left({{2n}\atop{n}}\right)}=\sum_{i=0}^{n}{\textstyle\left({{n}% \atop{i}}\right)}^{2},
  7. C 0 = 1 and C n + 1 = 2 ( 2 n + 1 ) n + 2 C n , C_{0}=1\quad\mbox{and}~{}\quad C_{n+1}=\frac{2(2n+1)}{n+2}C_{n},
  8. C n 4 n n 3 / 2 π C_{n}\sim\frac{4^{n}}{n^{3/2}\sqrt{\pi}}
  9. C n 4 n n 3 / 2 C_{n}\sim\frac{4^{n}}{n^{3/2}}
  10. C n = 0 4 x n ρ ( x ) d x C_{n}=\int_{0}^{4}x^{n}\rho(x)dx
  11. ρ ( x ) = 1 2 π 4 - x x . \rho(x)=\tfrac{1}{2\pi}\sqrt{\tfrac{4-x}{x}}.
  12. ρ ( x ) \rho(x)
  13. [ 0 , 4 ] [0,4]
  14. H n ( x ) = k = 0 n ( n + k n - k ) ( - x ) k . H_{n}(x)=\sum_{k=0}^{n}{n+k\choose n-k}(-x)^{k}.
  15. C n = 1 n + 1 ( 2 n n ) C_{n}=\frac{1}{n+1}{2n\choose n}
  16. C 0 = 1 and C n + 1 = i = 0 n C i C n - i for n 0. C_{0}=1\quad\,\text{and}\quad C_{n+1}=\sum_{i=0}^{n}C_{i}\,C_{n-i}\quad\,\text% {for }n\geq 0.
  17. c ( x ) = n = 0 C n x n . c(x)=\sum_{n=0}^{\infty}C_{n}x^{n}.
  18. c ( x ) = 1 + x c ( x ) 2 ; c(x)=1+xc(x)^{2};\,
  19. c ( x ) = 1 - 1 - 4 x 2 x = 2 1 + 1 - 4 x c(x)=\frac{1-\sqrt{1-4x}}{2x}=\frac{2}{1+\sqrt{1-4x}}
  20. lim x 0 + c ( x ) = C 0 = 1 \lim_{x\to 0^{+}}c(x)=C_{0}=1
  21. 1 + y = n = 0 ( 1 2 n ) y n = 1 - 2 n = 1 ( 2 n - 2 n - 1 ) ( - 1 4 ) n y n n . \sqrt{1+y}=\sum_{n=0}^{\infty}{\frac{1}{2}\choose n}y^{n}=1-2\sum_{n=1}^{% \infty}{2n-2\choose n-1}\left(\frac{-1}{4}\right)^{n}\frac{y^{n}}{n}.
  22. c ( x ) = n = 0 ( 2 n n ) x n n + 1 . c(x)=\sum_{n=0}^{\infty}{2n\choose n}\frac{x^{n}}{n+1}.
  23. 0 x t n d t \int_{0}^{x}\!t^{n}\,dt
  24. ( 2 n n ) {\textstyle\left({{2n}\atop{n}}\right)}
  25. ( n - 1 + n + 1 n - 1 ) = ( 2 n n - 1 ) = ( 2 n n + 1 ) {n-1+n+1\choose n-1}={2n\choose n-1}={2n\choose n+1}
  26. C n = ( 2 n n ) - ( 2 n n + 1 ) . C_{n}={2n\choose n}-{2n\choose n+1}.
  27. ( 2 n n ) {2n\choose n}
  28. C n = 1 n + 1 ( 2 n n ) . C_{n}=\frac{1}{n+1}{2n\choose n}.
  29. ( 4 n + 2 ) C n = ( n + 2 ) C n + 1 . (4n+2)C_{n}=(n+2)C_{n+1}.
  30. C 0 = 1 and C n + 1 = i = 0 n C i C n - i for n 0. C_{0}=1\quad\,\text{and}\quad C_{n+1}=\sum_{i=0}^{n}C_{i}\,C_{n-i}\quad\,\text% {for }n\geq 0.
  31. B n = ( 2 n n ) = d n C n \textstyle B_{n}={2n\choose n}=d_{n}C_{n}
  32. B n + 1 = 2 i = 0 n B i C n - i . B_{n+1}=2\sum_{i=0}^{n}B_{i}C_{n-i}.
  33. B n + 1 - C n + 1 = i = 0 n ( 2 i + 1 i ) C n - i = i = 0 n 2 i + 1 i + 1 B i C n - i . B_{n+1}-C_{n+1}=\sum_{i=0}^{n}{2i+1\choose i}C_{n-i}=\sum_{i=0}^{n}\frac{2i+1}% {i+1}B_{i}C_{n-i}.
  34. C n + 1 = 2 i = 0 n d i C i C n - i - i = 0 n 2 i + 1 i + 1 d i C i C n - i = i = 0 n d i i + 1 C i C n - i . C_{n+1}=2\sum_{i=0}^{n}d_{i}C_{i}C_{n-i}-\sum_{i=0}^{n}\frac{2i+1}{i+1}d_{i}C_% {i}C_{n-i}=\sum_{i=0}^{n}\frac{d_{i}}{i+1}C_{i}C_{n-i}.
  35. C n = 1 n + 1 ( 2 n n ) . C_{n}=\frac{1}{n+1}{2n\choose n}.
  36. det [ 1 1 2 5 1 2 5 14 2 5 14 42 5 14 42 132 ] = 1. \det\begin{bmatrix}1&1&2&5\\ 1&2&5&14\\ 2&5&14&42\\ 5&14&42&132\end{bmatrix}=1.
  37. det [ 1 2 5 14 2 5 14 42 5 14 42 132 14 42 132 429 ] = 1. \det\begin{bmatrix}1&2&5&14\\ 2&5&14&42\\ 5&14&42&132\\ 14&42&132&429\end{bmatrix}=1.
  38. ( 2 m ) ! ( 2 n ) ! ( m + n ) ! m ! n ! \frac{(2m)!(2n)!}{(m+n)!m!n!}
  39. m = 1 m=1
  40. m = n m=n
  41. m = 2 m=2
  42. m = 3 m=3

Catenoid.html

  1. x = c cosh v c cos u x=c\cosh\frac{v}{c}\cos u
  2. y = c cosh v c sin u y=c\cosh\frac{v}{c}\sin u
  3. z = v z=v
  4. ρ = c cosh z c \rho=c\cosh\frac{z}{c}
  5. x ( u , v ) = cos θ sinh v sin u + sin θ cosh v cos u x(u,v)=\cos\theta\,\sinh v\,\sin u+\sin\theta\,\cosh v\,\cos u
  6. y ( u , v ) = - cos θ sinh v cos u + sin θ cosh v sin u y(u,v)=-\cos\theta\,\sinh v\,\cos u+\sin\theta\,\cosh v\,\sin u
  7. z ( u , v ) = u cos θ + v sin θ z(u,v)=u\cos\theta+v\sin\theta\,
  8. ( u , v ) ( - π , π ] × ( - , ) (u,v)\in(-\pi,\pi]\times(-\infty,\infty)
  9. - π < θ π -\pi<\theta\leq\pi
  10. θ = π \theta=\pi
  11. θ = ± π / 2 \theta=\pm\pi/2
  12. θ = 0 \theta=0

Cauchy's_integral_formula.html

  1. γ \gamma
  2. f ( a ) = 1 2 π i γ f ( z ) z - a d z f(a)=\frac{1}{2\pi i}\oint_{\gamma}\frac{f(z)}{z-a}\,dz
  3. f ( n ) ( a ) = n ! 2 π i γ f ( z ) ( z - a ) n + 1 d z . f^{(n)}(a)=\frac{n!}{2\pi i}\oint_{\gamma}\frac{f(z)}{(z-a)^{n+1}}\,dz.
  4. C 1 z - a d z = 2 π i , \oint_{C}\frac{1}{z-a}\,dz=2\pi i,
  5. z ( t ) = a + ε e i t z(t)=a+\varepsilon e^{it}
  6. | 1 2 π i C f ( z ) z - a d z - f ( a ) | \displaystyle\left|\frac{1}{2\pi i}\oint_{C}\frac{f(z)}{z-a}\,dz-f(a)\right|
  7. g ( z ) = z 2 z 2 + 2 z + 2 g(z)=\frac{z^{2}}{z^{2}+2z+2}
  8. g ( z ) = z 2 ( z - z 1 ) ( z - z 2 ) g(z)=\frac{z^{2}}{(z-z_{1})(z-z_{2})}
  9. z 1 = - 1 + i , z_{1}=-1+i,
  10. z 2 = - 1 - i . z_{2}=-1-i.
  11. z 1 z_{1}
  12. z 2 z_{2}
  13. f 1 ( z ) = z 2 z - z 2 f_{1}(z)=\frac{z^{2}}{z-z_{2}}
  14. g ( z ) = f 1 ( z ) z - z 1 g(z)=\frac{f_{1}(z)}{z-z_{1}}
  15. C f 1 ( z ) z - a d z = 2 π i f 1 ( a ) \oint_{C}\frac{f_{1}(z)}{z-a}\,dz=2\pi i\cdot f_{1}(a)
  16. C 1 g ( z ) d z = C 1 f 1 ( z ) z - z 1 d z = 2 π i z 1 2 z 1 - z 2 . \oint_{C_{1}}g(z)\,dz=\oint_{C_{1}}\frac{f_{1}(z)}{z-z_{1}}\,dz=2\pi i\frac{z_% {1}^{2}}{z_{1}-z_{2}}.
  17. f 2 ( z ) = z 2 z - z 1 , f_{2}(z)=\frac{z^{2}}{z-z_{1}},
  18. C 2 g ( z ) d z = C 2 f 2 ( z ) z - z 2 d z = 2 π i z 2 2 z 2 - z 1 . \oint_{C_{2}}g(z)\,dz=\oint_{C_{2}}\frac{f_{2}(z)}{z-z_{2}}\,dz=2\pi i\frac{z_% {2}^{2}}{z_{2}-z_{1}}.
  19. C g ( z ) d z \displaystyle\oint_{C}g(z)\,dz
  20. C g ( z ) d z = C ( 1 - 1 z - z 1 - 1 z - z 2 ) d z = 0 - 2 π i - 2 π i = - 4 π i \oint_{C}g(z)dz=\oint_{C}\left(1-\frac{1}{z-z_{1}}-\frac{1}{z-z_{2}}\right)dz=% 0-2\pi i-2\pi i=-4\pi i
  21. f ( ζ ) = 1 2 π i C f ( z ) z - ζ d z . f(\zeta)=\frac{1}{2\pi i}\int_{C}\frac{f(z)}{z-\zeta}\,dz.
  22. | a n | r - n sup | z | = r | f ( z ) | . \displaystyle{|a_{n}|\leq r^{-n}\sup_{|z|=r}|f(z)|.}
  23. f ( ζ ) = 1 2 π i D f ( z ) d z z - ζ + 1 2 π i D f z ¯ ( z ) d z d z ¯ z - ζ . f(\zeta)=\frac{1}{2\pi i}\int_{\partial D}\frac{f(z)dz}{z-\zeta}+\frac{1}{2\pi i% }\iint_{D}\frac{\partial f}{\partial\bar{z}}(z)\frac{dz\wedge d\bar{z}}{z-% \zeta}.
  24. d μ = 1 2 π i ϕ d z d z ¯ d\mu=\frac{1}{2\pi i}\phi\,dz\wedge d\bar{z}
  25. f ( ζ , ζ ¯ ) f(\zeta,\bar{\zeta})
  26. f z ¯ = ϕ ( z , z ¯ ) . \frac{\partial f}{\partial\bar{z}}=\phi(z,\bar{z}).
  27. k ( z ) = p . v . 1 z k(z)=\operatorname{p.v.}\frac{1}{z}
  28. f ( ζ ) = 1 2 π i f z ¯ d z d z ¯ z - ζ , f(\zeta)=\frac{1}{2\pi i}\iint\frac{\partial f}{\partial\bar{z}}\frac{dz\wedge d% \bar{z}}{z-\zeta},
  29. ( π z ) - 1 (\pi z)^{-1}
  30. / z ¯ \partial/\partial\overline{z}
  31. χ X z ¯ = i 2 X d z , {\partial\chi_{X}\over\partial\overline{z}}={i\over 2}\oint_{\partial X}dz,
  32. D = i = 1 n D i . D=\prod_{i=1}^{n}D_{i}.
  33. f ( ζ ) = 1 ( 2 π i ) n D 1 × × D n f ( z 1 , , z n ) ( z 1 - ζ 1 ) ( z n - ζ n ) d z 1 d z n f(\zeta)=\frac{1}{(2\pi i)^{n}}\int\cdots\iint_{\partial D_{1}\times\dots% \times\partial D_{n}}\frac{f(z_{1},\dots,z_{n})}{(z_{1}-\zeta_{1})\dots(z_{n}-% \zeta_{n})}\,dz_{1}\cdots dz_{n}
  34. = e ^ i i \nabla=\hat{e}_{i}\partial_{i}
  35. k k
  36. ψ ( r ) \psi(\vec{r})
  37. ψ \nabla\psi
  38. k + 1 k+1
  39. k - 1 k-1
  40. k = 1 k=1
  41. k = 0 k=0
  42. k = 2 k=2
  43. G ( r , r ) = 1 S n r - r | r - r | n G(\vec{r},\vec{r}^{\prime})=\frac{1}{S_{n}}\frac{\vec{r}-\vec{r}^{\prime}}{|% \vec{r}-\vec{r}^{\prime}|^{n}}
  44. S n S_{n}
  45. S 2 = 2 π S_{2}=2\pi
  46. S 3 = 4 π S_{3}=4\pi
  47. G ( r , r ) = δ ( r - r ) \nabla G(\vec{r},\vec{r}^{\prime})=\delta(\vec{r}-\vec{r}^{\prime})
  48. V d S f ( r ) = V d V f ( r ) \oint_{\partial V}d\vec{S}\;f(\vec{r})=\int_{V}d\vec{V}\;\nabla f(\vec{r})
  49. n n
  50. d S d\vec{S}
  51. ( n - 1 ) (n-1)
  52. d V d\vec{V}
  53. n n
  54. f ( r ) f(\vec{r})
  55. G ( r , r ) f ( r ) G(\vec{r},\vec{r}^{\prime})f(\vec{r}^{\prime})
  56. V G ( r , r ) d S f ( r ) = V ( [ G ( r , r ) ] f ( r ) + G ( r , r ) f ( r ) ) d V \oint_{\partial V^{\prime}}G(\vec{r},\vec{r}^{\prime})\;d\vec{S}^{\prime}\;f(% \vec{r}^{\prime})=\int_{V}\left([\nabla^{\prime}G(\vec{r},\vec{r}^{\prime})]f(% \vec{r}^{\prime})+G(\vec{r},\vec{r}^{\prime})\nabla^{\prime}f(\vec{r}^{\prime}% )\right)\;d\vec{V}
  57. f = 0 \nabla\vec{f}=0
  58. f ( r ) f(\vec{r})
  59. V G ( r , r ) d S f ( r ) = V [ G ( r , r ) ] f ( r ) = - V δ ( r - r ) f ( r ) d V = - i n f ( r ) \oint_{\partial V^{\prime}}G(\vec{r},\vec{r}^{\prime})\;d\vec{S}^{\prime}\;f(% \vec{r}^{\prime})=\int_{V}[\nabla^{\prime}G(\vec{r},\vec{r}^{\prime})]f(\vec{r% }^{\prime})=-\int_{V}\delta(\vec{r}-\vec{r}^{\prime})f(\vec{r}^{\prime})\;d% \vec{V}=-i_{n}f(\vec{r})
  60. i n i_{n}
  61. n n
  62. f ( r ) = - 1 i n V G ( r , r ) d S f ( r ) = - 1 i n V r - r S n | r - r | n d S f ( r ) f(\vec{r})=-\frac{1}{i_{n}}\oint_{\partial V}G(\vec{r},\vec{r}^{\prime})\;d% \vec{S}\;f(\vec{r}^{\prime})=-\frac{1}{i_{n}}\oint_{\partial V}\frac{\vec{r}-% \vec{r}^{\prime}}{S_{n}|\vec{r}-\vec{r}^{\prime}|^{n}}\;d\vec{S}\;f(\vec{r}^{% \prime})

Causality_(physics).html

  1. m 1 d 2 𝐫 1 d t 2 = - m 1 m 2 g ( 𝐫 1 - 𝐫 2 ) | 𝐫 1 - 𝐫 2 | 3 ; m 2 d 2 𝐫 2 d t 2 = - m 1 m 2 g ( 𝐫 2 - 𝐫 1 ) | 𝐫 2 - 𝐫 1 | 3 , m_{1}\frac{d^{2}{\mathbf{r}}_{1}}{dt^{2}}=-\frac{m_{1}m_{2}g({\mathbf{r}}_{1}-% {\mathbf{r}}_{2})}{|{\mathbf{r}}_{1}-{\mathbf{r}}_{2}|^{3}};\;m_{2}\frac{d^{2}% {\mathbf{r}}_{2}}{dt^{2}}=-\frac{m_{1}m_{2}g({\mathbf{r}}_{2}-{\mathbf{r}}_{1}% )}{|{\mathbf{r}}_{2}-{\mathbf{r}}_{1}|^{3}},
  2. 𝐫 1 ( t ) \scriptstyle{\mathbf{r}}_{1}(t)
  3. 𝐫 2 ( t ) \scriptstyle{\mathbf{r}}_{2}(t)

Cayley's_theorem.html

  1. S 3 S_{3}
  2. S 6 S_{6}
  3. f g - 1 f_{g^{-1}}
  4. ( f g f h ) ( x ) = f g ( f h ( x ) ) = f g ( h * x ) = g * ( h * x ) = ( g * h ) * x = f g * h ( x ) , (f_{g}\cdot f_{h})(x)=f_{g}(f_{h}(x))=f_{g}(h*x)=g*(h*x)=(g*h)*x=f_{g*h}(x),
  5. T ( g ) T ( h ) = f g f h = f g * h = T ( g * h ) . T(g)\cdot T(h)=f_{g}\cdot f_{h}=f_{g*h}=T(g*h).
  6. G G
  7. ϕ \phi
  8. G = G / H G=G/H
  9. H = { e } H=\{e\}
  10. g . e g.e
  11. ϕ \phi
  12. ϕ \phi
  13. g ker ϕ g\in\ker\phi
  14. g = g . e = ϕ ( g ) . e g=g.e=\phi(g).e
  15. g ker ϕ g\in\ker\phi
  16. ϕ ( g ) = e \phi(g)=e
  17. ker ϕ \ker\phi
  18. Im ϕ < G \mathrm{Im}\phi<G

Cayley–Dickson_construction.html

  1. ( a , b ) ( c , d ) = ( a c - b d , a d + b c ) . (a,b)(c,d)=(ac-bd,ad+bc).\,
  2. ( a , b ) * = ( a , - b ) . (a,b)^{*}=(a,-b).\,
  3. ( a , b ) * ( a , b ) = ( a a + b b , a b - b a ) = ( a 2 + b 2 , 0 ) , (a,b)^{*}(a,b)=(aa+bb,ab-ba)=(a^{2}+b^{2},0),\,
  4. | z | = ( z * z ) 1 / 2 . |z|=(z^{*}z)^{1/2}.\,
  5. z - 1 = z * / | z | 2 . z^{-1}={z^{*}/|z|^{2}}.\,
  6. ( a , b ) (a,b)
  7. a a
  8. b b
  9. ( a , b ) ( c , d ) = ( a c - d * b , d a + b c * ) . (a,b)(c,d)=(ac-d^{*}b,da+bc^{*}).\,
  10. ( a , b ) * (a,b)^{*}\,
  11. ( a , b ) (a,b)
  12. ( a , b ) * = ( a * , - b ) . (a,b)^{*}=(a^{*},-b).\,
  13. a a
  14. b b
  15. ( a , b ) * ( a , b ) = ( a * , - b ) ( a , b ) = ( a * a + b * b , b a * - b a * ) = ( | a | 2 + | b | 2 , 0 ) . (a,b)^{*}(a,b)=(a^{*},-b)(a,b)=(a^{*}a+b^{*}b,ba^{*}-ba^{*})=(|a|^{2}+|b|^{2},% 0).\,
  16. p p
  17. q q
  18. p q = q p pq=qp
  19. p q = ( q p ) pq=(qp)^{\prime}
  20. ( a , b ) = ( a , - b ) (a,b)^{\prime}=(a,-b)
  21. ( p , q ) (p,q)
  22. p p
  23. q q
  24. ( p , q ) ( r , s ) = ( p r - s * q , s p + q r * ) . (p,q)(r,s)=(pr-s^{*}q,sp+qr^{*}).\,
  25. r * q r^{*}q
  26. q r * qr^{*}
  27. p p
  28. q q
  29. r r
  30. ( p q ) r = p ( q r ) . (pq)r=p(qr).
  31. s s
  32. s n s m = s n + m s^{n}s^{m}=s^{n+m}
  33. ( p , q ) ( r , s ) = ( p r - γ s * q , s p + q r * ) (p,q)(r,s)=(pr-\gamma s^{*}q,sp+qr^{*})\,
  34. ( p , q ) * = ( p * , - q ) (p,q)^{*}=(p^{*},-q)

Cayley–Hamilton_theorem.html

  1. 3 3
  2. n × n n×n
  3. A A
  4. n × n n×n
  5. n × n n×n
  6. A A
  7. p ( λ ) = det ( λ I n - A ) , p(\lambda)=\det(\lambda I_{n}-A)~{},
  8. d e t det
  9. λ λ
  10. n n
  11. λ λ
  12. A A
  13. λ λ
  14. p ( A ) = 0. p(A)=0.
  15. A A
  16. λ λ
  17. p ( λ ) p(λ)
  18. A A
  19. A A
  20. n n
  21. A A
  22. 4 × 4 4×4
  23. 2 × 2 2×2
  24. 3 × 3 3×3
  25. 2 × 2 2×2
  26. A = ( 1 2 3 4 ) A=\begin{pmatrix}1&2\\ 3&4\end{pmatrix}
  27. p ( λ ) = det ( λ I 2 - A ) = det ( λ - 1 - 2 - 3 λ - 4 ) = ( λ - 1 ) ( λ - 4 ) - ( - 2 ) ( - 3 ) = λ 2 - 5 λ - 2. p(\lambda)=\det(\lambda I_{2}-A)=\det\begin{pmatrix}\lambda-1&-2\\ -3&\lambda-4\end{pmatrix}=(\lambda-1)(\lambda-4)-(-2)(-3)=\lambda^{2}-5\lambda% -2.
  28. p ( X ) = X 2 - 5 X - 2 I 2 , p(X)=X^{2}-5X-2I_{2},
  29. p ( A ) = A 2 - 5 A - 2 I 2 = ( 0 0 0 0 ) , p(A)=A^{2}-5A-2I_{2}=\begin{pmatrix}0&0\\ 0&0\\ \end{pmatrix},
  30. 1 × 1 1×1
  31. A = ( a ) A=(a)
  32. p ( λ ) = λ a p(λ)=λ−a
  33. p ( A ) = ( a ) a ( 1 ) = 0 p(A)=(a) −a(1)=0
  34. 2 × 2 2×2
  35. A = ( a b c d ) , A=\begin{pmatrix}a&b\\ c&d\\ \end{pmatrix},
  36. p ( A ) = A 2 - ( a + d ) A + ( a d - b c ) I 2 = ( 0 0 0 0 ) ; p(A)=A^{2}-(a+d)A+(ad-bc)I_{2}=\begin{pmatrix}0&0\\ 0&0\\ \end{pmatrix};
  37. A A
  38. n × n n×n
  39. A A
  40. A A
  41. A A
  42. t r A trA
  43. A A
  44. - ( - 1 ) n det ( A ) I n = A ( A n - 1 + c n - 1 A n - 2 + + c 1 I n ) , -(-1)^{n}\det(A)I_{n}=A(A^{n-1}+c_{n-1}A^{n-2}+\cdots+c_{1}I_{n}),
  45. A - 1 = ( - 1 ) n - 1 det A ( A n - 1 + c n - 1 A n - 2 + + c 1 I n ) . A^{-1}=\frac{(-1)^{n-1}}{\det A}(A^{n-1}+c_{n-1}A^{n-2}+\cdots+c_{1}I_{n}).
  46. A A
  47. A A
  48. A - 1 = 1 det A s = 0 n - 1 A s k 1 , k 2 , , k n - 1 l = 1 n - 1 ( - 1 ) k l + 1 l k l k l ! tr ( A l ) k l , A^{-1}=\frac{1}{\det A}\sum_{s=0}^{n-1}A^{s}\sum_{k_{1},k_{2},\ldots,k_{n-1}}% \prod_{l=1}^{n-1}\frac{(-1)^{k_{l}+1}}{l^{k_{l}}k_{l}!}\mathrm{tr}(A^{l})^{k_{% l}},
  49. s s
  50. s + l = 1 n - 1 l k l = n - 1. s+\sum_{l=1}^{n-1}lk_{l}=n-1.
  51. A A
  52. 2 × 2 2×2
  53. λ λ
  54. A A
  55. t r A trA
  56. A A
  57. n × n n×n
  58. 3 × 3 3×3
  59. A A
  60. A 3 - ( tr A ) A 2 + 1 2 ( ( tr A ) 2 - tr ( A 2 ) ) A - det ( A ) I 3 = 0 , A^{3}-(\operatorname{tr}A)A^{2}+\frac{1}{2}\left((\operatorname{tr}A)^{2}-% \operatorname{tr}(A^{2})\right)A-\det(A)I_{3}=0,
  61. 3 × 3 3×3
  62. n = 3 n=3
  63. 1 6 ( ( tr A ) 3 - 3 tr ( A 2 ) ( tr A ) + 2 tr ( A 3 ) ) , \tfrac{1}{6}\left((\operatorname{tr}A)^{3}-3\operatorname{tr}(A^{2})(% \operatorname{tr}A)+2\operatorname{tr}(A^{3})\right),
  64. 4 × 4 4×4
  65. A A
  66. A 4 - ( tr A ) A 3 + 1 2 ( ( tr A ) 2 - tr ( A 2 ) ) A 2 - 1 6 ( ( tr A ) 3 - 3 tr ( A 2 ) ( tr A ) + 2 tr ( A 3 ) ) A + det ( A ) I 4 = 0 , A^{4}-(\operatorname{tr}A)A^{3}+\tfrac{1}{2}\bigl((\operatorname{tr}A)^{2}-% \operatorname{tr}(A^{2})\bigr)A^{2}-\tfrac{1}{6}\bigl((\operatorname{tr}A)^{3}% -3\operatorname{tr}(A^{2})(\operatorname{tr}A)+2\operatorname{tr}(A^{3})\bigr)% A+\det(A)I_{4}=0,
  67. 1 24 ( ( tr A ) 4 - 6 tr ( A 2 ) ( tr A ) 2 + 3 ( tr ( A 2 ) ) 2 + 8 tr ( A 3 ) tr ( A ) - 6 tr ( A 4 ) ) \tfrac{1}{24}\left((\operatorname{tr}A)^{4}-6\operatorname{tr}(A^{2})(% \operatorname{tr}A)^{2}+3(\operatorname{tr}(A^{2}))^{2}+8\operatorname{tr}(A^{% 3})\operatorname{tr}(A)-6\operatorname{tr}(A^{4})\right)
  68. n × n n×n
  69. p ( λ ) = det ( λ I n - A ) = λ n exp ( tr ( log ( I n - A / λ ) ) ) . p(\lambda)=\det~{}(\lambda I_{n}-A)=\lambda^{n}\exp(\operatorname{tr}(\log(I_{% n}-A/\lambda))).
  70. p ( λ ) = λ n exp ( - tr m = 1 ( A λ ) m m ) , p(\lambda)=\lambda^{n}\exp\left(-\operatorname{tr}\sum_{m=1}^{\infty}{({A\over% \lambda})^{m}\over m}\right),
  71. p ( λ ) p(λ)
  72. n n
  73. λ λ
  74. λ λ
  75. n n
  76. m × m m×m
  77. p ( λ ) / λ = p ( λ ) m = 0 λ - ( m + 1 ) tr A m . \partial p(\lambda)/\partial\lambda=p(\lambda)\sum^{\infty}_{m=0}\lambda^{-(m+% 1)}\operatorname{tr}A^{m}.
  78. M 0 0 c n = 1 ( k = 0 ) M k A M k - 1 + c n - k + 1 I c n - k = - 1 k tr ( A M k ) k = 1 , , n . \begin{aligned}\displaystyle M_{0}&\displaystyle\equiv 0&\displaystyle c_{n}&% \displaystyle=1&\displaystyle(k=0)\\ \displaystyle M_{k}&\displaystyle\equiv AM_{k-1}+c_{n-k+1}I&\displaystyle c_{n% -k}&\displaystyle=-\frac{1}{k}\mathrm{tr}(AM_{k})&\displaystyle k=1,\ldots,n~{% }.\end{aligned}
  79. B < s u b > k M n k B<sub>k≡M_{n−k}
  80. c n - m = ( - ) m m ! | tr A m - 1 0 tr A 2 tr A m - 2 tr A m - 1 tr A m - 2 1 tr A m tr A m - 1 tr A | . c_{n-m}=\frac{(-)^{m}}{m!}\begin{vmatrix}\operatorname{tr}A&m-1&0&\cdots\\ \operatorname{tr}A^{2}&\operatorname{tr}A&m-2&\cdots\\ \vdots&\vdots&&&\vdots\\ \operatorname{tr}A^{m-1}&\operatorname{tr}A^{m-2}&\cdots&\cdots&1\\ \operatorname{tr}A^{m}&\operatorname{tr}A^{m-1}&\cdots&\cdots&\operatorname{tr% }A\end{vmatrix}~{}.
  81. A A
  82. A A
  83. 2 × 2 2×2
  84. A 2 = 5 A + 2 I 2 . A^{2}=5A+2I_{2}\,.
  85. A 3 = ( 5 A + 2 I 2 ) A = 5 A 2 + 2 A = 5 ( 5 A + 2 I 2 ) + 2 A = 27 A + 10 I 2 A^{3}=(5A+2I_{2})A=5A^{2}+2A=5(5A+2I_{2})+2A=27A+10I_{2}\,
  86. A 4 = A 3 A = ( 27 A + 10 I 2 ) A = 27 A 2 + 10 A = 27 ( 5 A + 2 I 2 ) + 10 A = 145 A + 54 I 2 . A^{4}=A^{3}A=(27A+10I_{2})A=27A^{2}+10A=27(5A+2I_{2})+10A=145A+54I_{2}\,.
  87. A - 1 = A - 5 I 2 2 . A^{-1}=\frac{A-5I_{2}}{2}~{}.
  88. exp : 𝔤 G ; t X e t X = n = 0 t n X n n ! = I + t X + t 2 X 2 2 + , t , X 𝔤 . \exp:\mathfrak{g}\rightarrow G;tX\mapsto e^{tX}=\sum_{n=0}^{\infty}\frac{t^{n}% X^{n}}{n!}=I+tX+\frac{t^{2}X^{2}}{2}+\cdots,t\in\mathbb{R},X\in\mathfrak{g}.
  89. X X
  90. n 1 n−1
  91. X X
  92. S U ( 2 ) SU(2)
  93. e i ( θ / 2 ) ( n ^ σ ) = I 2 cos θ / 2 + i ( n ^ σ ) sin θ / 2 , e^{i(\theta/2)(\hat{n}\cdot\sigma)}=I_{2}\cos\theta/2+i(\hat{n}\cdot\sigma)% \sin\theta/2,
  94. σ σ
  95. S O ( 3 ) SO(3)
  96. e i θ ( n ^ 𝐉 ) = I 3 + i ( n ^ 𝐉 ) sin θ + ( n ^ 𝐉 ) 2 ( cos θ - 1 ) , e^{i\theta(\hat{n}\cdot\mathbf{J})}=I_{3}+i(\hat{n}\cdot\mathbf{J})\sin\theta+% (\hat{n}\cdot\mathbf{J})^{2}(\cos\theta-1),
  97. S O ( 3 , 1 ) SO(3,1)
  98. O ( 4 , 2 ) O(4,2)
  99. S U ( 2 , 2 ) SU(2,2)
  100. G L ( n , 𝐑 ) GL(n,\mathbf{R})
  101. O ( 4 , 2 ) O(4,2)
  102. S U ( 2 , 2 ) SU(2,2)
  103. O ( 4 , 2 ) O(4,2)
  104. S U ( 2 ) SU(2)
  105. S O ( 3 ) SO(3)
  106. n × n n×n
  107. A = ( a i j ) i , j = 1 n A=(a_{ij})_{i,j=1}^{n}
  108. t t
  109. p ( t ) = det ( t I n - A ) = | t - a 1 , 1 - a 1 , 2 - a 1 , n - a 2 , 1 t - a 2 , 2 - a 2 , n - a n , 1 - a n , 2 t - a n , n | = t n + c n - 1 t n - 1 + + c 1 t + c 0 , p(t)=\det(tI_{n}-A)=\begin{vmatrix}t-a_{1,1}&-a_{1,2}&\cdots&-a_{1,n}\\ -a_{2,1}&t-a_{2,2}&\cdots&-a_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ -a_{n,1}&-a_{n,2}&\cdots&t-a_{n,n}\\ \end{vmatrix}=t^{n}+c_{n-1}t^{n-1}+\cdots+c_{1}t+c_{0},
  110. A A
  111. n × n n×n
  112. A n + c n - 1 A n - 1 + + c 1 A + c 0 I n = ( 0 0 0 0 ) . A^{n}+c_{n-1}A^{n-1}+\cdots+c_{1}A+c_{0}I_{n}=\begin{pmatrix}0&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&0\end{pmatrix}.
  113. n × n n×n
  114. A A
  115. 0
  116. n n
  117. n n
  118. n n
  119. v v
  120. n n
  121. A A
  122. λ λ
  123. A v = λ v A⋅v=λv
  124. p ( A ) v = A n v + c n - 1 A n - 1 v + + c 1 A v + c 0 I n v = λ n v + c n - 1 λ n - 1 v + + c 1 λ v + c 0 v = p ( λ ) v , \begin{aligned}\displaystyle p(A)\cdot v&\displaystyle=A^{n}\cdot v+c_{n-1}A^{% n-1}\cdot v+\cdots+c_{1}A\cdot v+c_{0}I_{n}\cdot v\\ &\displaystyle=\lambda^{n}v+c_{n-1}\lambda^{n-1}v+\cdots+c_{1}\lambda v+c_{0}v% =p(\lambda)v,\end{aligned}
  125. p ( λ ) = 0 p(λ)=0
  126. A A
  127. p ( t ) p(t)
  128. λ λ
  129. A A
  130. A A
  131. A A
  132. 0
  133. 0
  134. a d j ( M ) adj(M)
  135. n × n n×n
  136. M M
  137. M M
  138. ( n 1 ) × ( n 1 ) (n− 1)×(n− 1)
  139. adj ( M ) M = det ( M ) I n = M adj ( M ) . \operatorname{adj}(M)\cdot M=\det(M)I_{n}=M\cdot\operatorname{adj}(M)~{}.
  140. ( i , j ) (i,j)
  141. j j
  142. M M
  143. i i
  144. j j
  145. d e t ( M ) det(M)
  146. i = j i=j
  147. A A
  148. B = adj ( t I n - A ) . B=\operatorname{adj}(tI_{n}-A).
  149. ( t I n - A ) B = det ( t I n - A ) I n = p ( t ) I n . (tI_{n}-A)\cdot B=\det(tI_{n}-A)I_{n}=p(t)I_{n}.
  150. B = i = 0 n - 1 t i B i B=\sum_{i=0}^{n-1}t^{i}B_{i}
  151. n n
  152. p ( t ) I n = ( t I n - A ) B = ( t I n - A ) i = 0 n - 1 t i B i = i = 0 n - 1 t I n t i B i - i = 0 n - 1 A t i B i = i = 0 n - 1 t i + 1 B i - i = 0 n - 1 t i A B i = t n B n - 1 + i = 1 n - 1 t i ( B i - 1 - A B i ) - A B 0 . \begin{aligned}\displaystyle p(t)I_{n}&\displaystyle=(tI_{n}-A)\cdot B\\ &\displaystyle=(tI_{n}-A)\cdot\sum_{i=0}^{n-1}t^{i}B_{i}\\ &\displaystyle=\sum_{i=0}^{n-1}tI_{n}\cdot t^{i}B_{i}-\sum_{i=0}^{n-1}A\cdot t% ^{i}B_{i}\\ &\displaystyle=\sum_{i=0}^{n-1}t^{i+1}B_{i}-\sum_{i=0}^{n-1}t^{i}A\cdot B_{i}% \\ &\displaystyle=t^{n}B_{n-1}+\sum_{i=1}^{n-1}t^{i}(B_{i-1}-A\cdot B_{i})-A\cdot B% _{0}.\end{aligned}
  153. p ( t ) I n = t n I n + t n - 1 c n - 1 I n + + t c 1 I n + c 0 I n , p(t)I_{n}=t^{n}I_{n}+t^{n-1}c_{n-1}I_{n}+\cdots+tc_{1}I_{n}+c_{0}I_{n},
  154. B n - 1 = I n , B i - 1 - A B i = c i I n for 1 i n - 1 , - A B 0 = c 0 I n . B_{n-1}=I_{n},\qquad B_{i-1}-A\cdot B_{i}=c_{i}I_{n}\quad\,\text{for }1\leq i% \leq n-1,\qquad-AB_{0}=c_{0}I_{n}.
  155. 0 = A n + c n - 1 A n - 1 + + c 1 A + c 0 I n = p ( A ) . 0=A^{n}+c_{n-1}A^{n-1}+\cdots+c_{1}A+c_{0}I_{n}=p(A).
  156. ( f + g ) ( x ) = i ( f i + g i ) x i = i f i x i + i g i x i = f ( x ) + g ( x ) (f+g)(x)=\sum_{i}\left(f_{i}+g_{i}\right)x^{i}=\sum_{i}{f_{i}x^{i}}+\sum_{i}{g% _{i}x^{i}}=f(x)+g(x)
  157. t I n - A tI_{n}-A
  158. ( i M i t i ) ( j N j t j ) = i , j ( M i N j ) t i + j , \left(\sum_{i}M_{i}t^{i}\right)\cdot\left(\sum_{j}N_{j}t^{j}\right)=\sum_{i,j}% (M_{i}\cdot N_{j})t^{i+j},
  159. ( t I n - A ) B = p ( t ) I n . (tI_{n}-A)\cdot B=p(t)I_{n}.
  160. M t i N t j = ( M N ) t i + j Mt^{i}Nt^{j}=(M\cdot N)t^{i+j}
  161. I n I_{n}
  162. B i B_{i}
  163. I n I_{n}
  164. B i B_{i}
  165. ( i = 0 m B i t i ) ( t I n - A ) \displaystyle\left(\sum_{i=0}^{m}B_{i}t^{i}\right)(tI_{n}-A)
  166. ev A ( p ( t ) I n ) \displaystyle\operatorname{ev}_{A}\bigl(p(t)I_{n}\bigr)
  167. p ( t ) I n p(t)I_{n}
  168. I n t - A I_{n}t-A
  169. P Q + r = P Q + r PQ+r=PQ^{\prime}+r^{\prime}
  170. P ( Q - Q ) = r - r P(Q-Q^{\prime})=r^{\prime}-r
  171. P ( Q - Q ) P(Q-Q^{\prime})
  172. Q = Q Q=Q^{\prime}
  173. p ( t ) I n p(t)I_{n}
  174. I n t - A I_{n}t-A
  175. p ( t ) I n = ( I n t - A ) B p(t)I_{n}=(I_{n}t-A)B
  176. ev A : ( R [ A ] ) [ t ] R [ A ] \operatorname{ev}_{A}:(R[A])[t]\to R[A]
  177. p ( A ) = 0 ev A ( B ) = 0 p(A)=0\cdot\operatorname{ev}_{A}(B)=0
  178. B 0 = adj ( - A ) B_{0}=\operatorname{adj}(-A)
  179. B n - 1 , , B 1 , B 0 B_{n-1},\ldots,B_{1},B_{0}
  180. adj ( - A ) = i = 1 n c i A i - 1 , \operatorname{adj}(-A)=\sum_{i=1}^{n}c_{i}A^{i-1},
  181. t n + c n - 1 t n - 1 + + c 1 t + c 0 t^{n}+c_{n-1}t^{n-1}+\cdots+c_{1}t+c_{0}
  182. - A adj ( - A ) = adj ( - A ) - A = det ( - A ) I n = c 0 I n . -A\cdot\operatorname{adj}(-A)=\operatorname{adj}(-A)\cdot-A=\det(-A)I_{n}=c_{0% }I_{n}.
  183. t I n - A tI_{n}-A
  184. A i , j A_{i,j}
  185. A i , j A_{i,j}
  186. A i , j I n A_{i,j}I_{n}
  187. A i , j A_{i,j}
  188. φ I n - A \varphi I_{n}-A
  189. det ( φ I n - A ) \det(\varphi I_{n}-A)
  190. φ ( e i ) = j = 1 n A j , i e j for i = 1 , , n . \varphi(e_{i})=\sum_{j=1}^{n}A_{j,i}e_{j}\quad\,\text{for }i=1,\ldots,n.
  191. ψ ( v ) \psi(v)
  192. φ I n E = A tr E , \varphi I_{n}\cdot E=A^{\mathrm{tr}}\cdot E,
  193. E V n E\in V^{n}
  194. ( φ I n - A tr ) E = 0 V n (\varphi I_{n}-A^{\mathrm{tr}})\cdot E=0\in V^{n}
  195. φ I n - A \varphi I_{n}-A
  196. φ I n - A tr \varphi I_{n}-A^{\mathrm{tr}}
  197. 0 = adj ( φ I n - A tr ) ( ( φ I n - A tr ) E ) = ( adj ( φ I n - A tr ) ( φ I n - A tr ) ) E = ( det ( φ I n - A tr ) I n ) E = ( p ( φ ) I n ) E ; \begin{aligned}\displaystyle 0&\displaystyle=\operatorname{adj}(\varphi I_{n}-% A^{\mathrm{tr}})\cdot((\varphi I_{n}-A^{\mathrm{tr}})\cdot E)\\ &\displaystyle=(\operatorname{adj}(\varphi I_{n}-A^{\mathrm{tr}})\cdot(\varphi I% _{n}-A^{\mathrm{tr}}))\cdot E\\ &\displaystyle=(\det(\varphi I_{n}-A^{\mathrm{tr}})I_{n})\cdot E\\ &\displaystyle=(p(\varphi)I_{n})\cdot E;\end{aligned}
  198. φ ( e i ) = j A j , i e j \varphi(e_{i})=\sum_{j}A_{j,i}e_{j}
  199. p ( λ ) = det ( λ I n - A ) p(\lambda)=\det(\lambda I_{n}-A)
  200. p ( A ) = det ( A I n - A ) = det ( A - A ) = 0. p(A)=\det(AI_{n}-A)=\det(A-A)=0.
  201. det ( λ I n - A ) \det(\lambda I_{n}-A)
  202. λ I n - A \lambda I_{n}-A
  203. det ( λ - 1 - 2 - 3 λ - 4 ) . \det\begin{pmatrix}\lambda-1&-2\\ -3&\lambda-4\end{pmatrix}.
  204. det ( ( 1 2 3 4 ) - 1 - 2 - 3 ( 1 2 3 4 ) - 4 ) , \det\begin{pmatrix}\begin{pmatrix}1&2\\ 3&4\end{pmatrix}-1&-2\\ -3&\begin{pmatrix}1&2\\ 3&4\end{pmatrix}-4\end{pmatrix},
  205. det ( ( 1 2 3 4 ) - I 2 - 2 I 2 - 3 I 2 ( 1 2 3 4 ) - 4 I 2 ) , \det\begin{pmatrix}\begin{pmatrix}1&2\\ 3&4\end{pmatrix}-I_{2}&-2I_{2}\\ -3I_{2}&\begin{pmatrix}1&2\\ 3&4\end{pmatrix}-4I_{2}\end{pmatrix},
  206. A I n - A AI_{n}-A
  207. p ( A ) = det ( A I n - A ) = 0 p(A)=\det(AI_{n}-A)=0
  208. q ( λ ) = perm ( λ I n - A ) q(\lambda)=\operatorname{perm}(\lambda I_{n}-A)
  209. perm ( a b c d ) = a d + b c . \operatorname{perm}\begin{pmatrix}a&b\\ c&d\end{pmatrix}=ad+bc.
  210. q ( λ ) = perm ( λ I 2 - A ) = perm ( λ - 1 - 2 - 3 λ - 4 ) = ( λ - 1 ) ( λ - 4 ) + ( - 2 ) ( - 3 ) = λ 2 - 5 λ + 10. q(\lambda)=\operatorname{perm}(\lambda I_{2}-A)=\operatorname{perm}\begin{% pmatrix}\lambda-1&-2\\ -3&\lambda-4\end{pmatrix}=(\lambda-1)(\lambda-4)+(-2)(-3)=\lambda^{2}-5\lambda% +10.
  211. q ( A ) = A 2 - 5 A + 10 I 2 = 12 I 2 0. q(A)=A^{2}-5A+10I_{2}=12I_{2}\not=0.
  212. p ( A ) = det ( A I n - A ) = 0 p(A)=\det(AI_{n}-A)=0
  213. A I n AI_{n}
  214. φ ( e j ) = a i j e i , j = 1 , , n . \varphi(e_{j})=\sum a_{ij}e_{i},\qquad j=1,\cdots,n.
  215. 2 × 2 2×2
  216. S U ( 2 ) SU(2)
  217. S U ( 1 , 1 ) SU(1,1)

Ceilometer.html

  1. d i s t a n c e = c δ t 2 distance=\frac{c\delta t}{2}

Cent_(music).html

  1. n = 1200 log 2 ( b a ) 3986 log 10 ( b a ) n=1200\cdot\log_{2}\left(\frac{b}{a}\right)\approx 3986\cdot\log_{10}\left(% \frac{b}{a}\right)
  2. b = a × 2 n / 1200 b=a\times 2^{n/1200}

Center_of_mass.html

  1. i = 1 n m i ( 𝐫 i - 𝐑 ) = 0. \sum_{i=1}^{n}m_{i}(\mathbf{r}_{i}-\mathbf{R})=0.
  2. 𝐑 = 1 M i = 1 n m i 𝐫 i , \mathbf{R}=\frac{1}{M}\sum_{i=1}^{n}m_{i}\mathbf{r}_{i},
  3. M M
  4. V ρ ( 𝐫 ) ( 𝐫 - 𝐑 ) d V = 0. \int_{V}\rho(\mathbf{r})(\mathbf{r}-\mathbf{R})dV=0.
  5. 𝐑 = 1 M V ρ ( 𝐫 ) 𝐫 d V , \mathbf{R}=\frac{1}{M}\int_{V}\rho(\mathbf{r})\mathbf{r}dV,
  6. 𝐑 = 1 m 1 + m 2 ( m 1 𝐫 1 + m 2 𝐫 2 ) . \mathbf{R}=\frac{1}{m_{1}+m_{2}}(m_{1}\mathbf{r}_{1}+m_{2}\mathbf{r}_{2}).
  7. θ i = x i x m a x 2 π \theta_{i}=\frac{x_{i}}{x_{max}}2\pi
  8. ( ξ i , ζ i ) (\xi_{i},\zeta_{i})
  9. ξ i = cos ( θ i ) \xi_{i}=\cos(\theta_{i})
  10. ζ i = sin ( θ i ) \zeta_{i}=\sin(\theta_{i})
  11. ( ξ , ζ ) (\xi,\zeta)
  12. ξ i \xi_{i}
  13. ζ i \zeta_{i}
  14. ξ ¯ \overline{\xi}
  15. ζ ¯ \overline{\zeta}
  16. θ ¯ \overline{\theta}
  17. θ ¯ = atan2 ( - ζ ¯ , - ξ ¯ ) + π \overline{\theta}=\mathrm{atan2}(-\overline{\zeta},-\overline{\xi})+\pi
  18. x c o m = x m a x θ ¯ 2 π x_{com}=x_{max}\frac{\overline{\theta}}{2\pi}
  19. ( ξ ¯ , ζ ¯ ) = ( 0 , 0 ) (\overline{\xi},\overline{\zeta})=(0,0)
  20. θ ¯ \overline{\theta}
  21. 𝐟 ( 𝐫 ) = - d m g k = - ρ ( 𝐫 ) d V g k , \mathbf{f}(\mathbf{r})=-dm\,g\vec{k}=-\rho(\mathbf{r})dV\,g\vec{k},
  22. 𝐅 = V 𝐟 ( 𝐫 ) = V ρ ( 𝐫 ) d V ( - g k ) = - M g k , \mathbf{F}=\int_{V}\mathbf{f}(\mathbf{r})=\int_{V}\rho(\mathbf{r})dV(-g\vec{k}% )=-Mg\vec{k},
  23. 𝐓 = V ( 𝐫 - 𝐑 ) × 𝐟 ( 𝐫 ) = V ( 𝐫 - 𝐑 ) × ( - g ρ ( 𝐫 ) d V k ) = ( V ρ ( 𝐫 ) ( 𝐫 - 𝐑 ) d V ) × ( - g k ) . \mathbf{T}=\int_{V}(\mathbf{r}-\mathbf{R})\times\mathbf{f}(\mathbf{r})=\int_{V% }(\mathbf{r}-\mathbf{R})\times(-g\rho(\mathbf{r})dV\vec{k})=\left(\int_{V}\rho% (\mathbf{r})(\mathbf{r}-\mathbf{R})dV\right)\times(-g\vec{k}).
  24. V ρ ( 𝐫 ) ( 𝐫 - 𝐑 ) d V = 0 , \int_{V}\rho(\mathbf{r})(\mathbf{r}-\mathbf{R})dV=0,
  25. 𝐫 i = ( 𝐫 i - 𝐑 ) + 𝐑 , 𝐯 i = d d t ( 𝐫 i - 𝐑 ) + 𝐯 . \mathbf{r}_{i}=(\mathbf{r}_{i}-\mathbf{R})+\mathbf{R},\quad\mathbf{v}_{i}=% \frac{d}{dt}(\mathbf{r}_{i}-\mathbf{R})+\mathbf{v}.
  26. 𝐩 = d d t ( i = 1 n m i ( 𝐫 i - 𝐑 ) ) + ( i = 1 n m i ) 𝐯 , \mathbf{p}=\frac{d}{dt}\left(\sum_{i=1}^{n}m_{i}(\mathbf{r}_{i}-\mathbf{R})% \right)+\left(\sum_{i=1}^{n}m_{i}\right)\mathbf{v},
  27. 𝐋 = i = 1 n m i ( 𝐫 i - 𝐑 ) × d d t ( 𝐫 i - 𝐑 ) + ( i = 1 n m i ( 𝐫 i - 𝐑 ) ) × 𝐯 . \mathbf{L}=\sum_{i=1}^{n}m_{i}(\mathbf{r}_{i}-\mathbf{R})\times\frac{d}{dt}(% \mathbf{r}_{i}-\mathbf{R})+\left(\sum_{i=1}^{n}m_{i}(\mathbf{r}_{i}-\mathbf{R}% )\right)\times\mathbf{v}.
  28. 𝐩 = m 𝐯 , 𝐋 = i = 1 n m i ( 𝐫 i - 𝐑 ) × d d t ( 𝐫 i - 𝐑 ) , \mathbf{p}=m\mathbf{v},\quad\mathbf{L}=\sum_{i=1}^{n}m_{i}(\mathbf{r}_{i}-% \mathbf{R})\times\frac{d}{dt}(\mathbf{r}_{i}-\mathbf{R}),
  29. 𝐓 = ( 𝐫 1 - 𝐑 ) × 𝐅 1 + ( 𝐫 2 - 𝐑 ) × 𝐅 2 + ( 𝐫 3 - 𝐑 ) × 𝐅 3 = 0 , \mathbf{T}=(\mathbf{r}_{1}-\mathbf{R})\times\mathbf{F}_{1}+(\mathbf{r}_{2}-% \mathbf{R})\times\mathbf{F}_{2}+(\mathbf{r}_{3}-\mathbf{R})\times\mathbf{F}_{3% }=0,
  30. 𝐑 × ( - W k ) = 𝐫 1 × 𝐅 1 + 𝐫 2 × 𝐅 2 + 𝐫 3 × 𝐅 3 . \mathbf{R}\times(-W\vec{k})=\mathbf{r}_{1}\times\mathbf{F}_{1}+\mathbf{r}_{2}% \times\mathbf{F}_{2}+\mathbf{r}_{3}\times\mathbf{F}_{3}.
  31. 𝐑 * = - 1 W k × ( 𝐫 1 × 𝐅 1 + 𝐫 2 × 𝐅 2 + 𝐫 3 × 𝐅 3 ) . \mathbf{R}^{*}=-\frac{1}{W}\vec{k}\times(\mathbf{r}_{1}\times\mathbf{F}_{1}+% \mathbf{r}_{2}\times\mathbf{F}_{2}+\mathbf{r}_{3}\times\mathbf{F}_{3}).
  32. L ( t ) = 𝐑 * + t k . L(t)=\mathbf{R}^{*}+t\vec{k}.

Centers_of_gravity_in_non-uniform_fields.html

  1. 𝐫 cg × 𝐅 = s y m b o l τ , \mathbf{r}_{\mathrm{cg}}\times\mathbf{F}=symbol{\tau},
  2. 𝐅 \mathbf{F}
  3. τ \mathbf{τ}
  4. 𝐅 \mathbf{F}
  5. τ \mathbf{τ}
  6. 𝐅 \mathbf{F}
  7. τ \mathbf{τ}
  8. 𝐅 \mathbf{F}
  9. 𝐠 ( 𝐫 ) = g ( 𝐫 ) 𝐧 \mathbf{g}(\mathbf{r})=g(\mathbf{r})\mathbf{n}
  10. 𝐧 \mathbf{n}
  11. 𝐫 cg = 1 W i w i 𝐫 i , \mathbf{r}_{\mathrm{cg}}=\frac{1}{W}\sum_{i}w_{i}\mathbf{r}_{i},
  12. i i
  13. W W
  14. M M
  15. 𝐫 \mathbf{r}
  16. G m M ( 𝐫 cg - 𝐫 ) | 𝐫 cg - 𝐫 | 3 = 𝐅 , \frac{GmM(\mathbf{r}_{\mathrm{cg}}-\mathbf{r})}{|\mathbf{r}_{\mathrm{cg}}-% \mathbf{r}|^{3}}=\mathbf{F},
  17. G G
  18. m m
  19. 𝐫 < s u b > c g \mathbf{r}<sub>cg

Centralizer_and_normalizer.html

  1. C G ( S ) = { g G s g = g s for all s S } \mathrm{C}_{G}(S)=\{g\in G\mid sg=gs\,\text{ for all }s\in S\}
  2. N G ( S ) = { g G g S = S g } \mathrm{N}_{G}(S)=\{g\in G\mid gS=Sg\}
  3. 𝔏 \mathfrak{L}
  4. 𝔏 \mathfrak{L}
  5. C 𝔏 ( S ) = { x 𝔏 [ x , s ] = 0 for all s S } \mathrm{C}_{\mathfrak{L}}(S)=\{x\in\mathfrak{L}\mid[x,s]=0\,\text{ for all }s% \in S\}
  6. 𝔏 \mathfrak{L}
  7. N 𝔏 ( S ) = { x 𝔏 [ x , s ] S for all s S } \mathrm{N}_{\mathfrak{L}}(S)=\{x\in\mathfrak{L}\mid[x,s]\in S\,\text{ for all % }s\in S\}
  8. 𝔏 \mathfrak{L}
  9. 𝔏 \mathfrak{L}
  10. N 𝔏 ( S ) \mathrm{N}_{\mathfrak{L}}(S)
  11. S = { x A : s x = x s for every s S } . S^{\prime}=\{x\in A:sx=xs\ \mbox{for}~{}\ \mbox{every}~{}\ s\in S\}.
  12. S = S ′′′ = S ′′′′′ S^{\prime}=S^{\prime\prime\prime}=S^{\prime\prime\prime\prime\prime}

Centrifuge.html

  1. ω \omega
  2. RCF = r ω 2 g \,\text{RCF}=\frac{r\omega^{2}}{g}
  3. g \textstyle g
  4. r \textstyle r
  5. ω \omega
  6. RCF = 1.11824396 × 10 - 6 r mm N RPM 2 \,\text{RCF}=1.11824396\,\times 10^{-6}\,r\text{mm}\,N\text{RPM}^{2}
  7. r mm \textstyle r\text{mm}
  8. N RPM \textstyle N\text{RPM}

Centroid.html

  1. M A 2 + M B 2 + M C 2 = G A 2 + G B 2 + G C 2 + 3 M G 2 . MA^{2}+MB^{2}+MC^{2}=GA^{2}+GB^{2}+GC^{2}+3MG^{2}.
  2. A B 2 + B C 2 + C A 2 = 3 ( G A 2 + G B 2 + G C 2 ) . AB^{2}+BC^{2}+CA^{2}=3(GA^{2}+GB^{2}+GC^{2}).
  3. k {k}
  4. 𝐱 1 , 𝐱 2 , , 𝐱 k \mathbf{x}_{1},\mathbf{x}_{2},\ldots,\mathbf{x}_{k}
  5. n \mathbb{R}^{n}
  6. 𝐂 = 𝐱 1 + 𝐱 2 + + 𝐱 k k \mathbf{C}=\frac{\mathbf{x}_{1}+\mathbf{x}_{2}+\cdots+\mathbf{x}_{k}}{k}
  7. X X
  8. X 1 , X 2 , , X n X_{1},X_{2},\dots,X_{n}
  9. C i C_{i}
  10. A i A_{i}
  11. C x = C i x A i A i , C y = C i y A i A i C_{x}=\frac{\sum C_{i_{x}}A_{i}}{\sum A_{i}},C_{y}=\frac{\sum C_{i_{y}}A_{i}}{% \sum A_{i}}
  12. X X
  13. A i A_{i}
  14. A i A_{i}
  15. A i A_{i}
  16. p p
  17. p p
  18. X X
  19. x = 5 × 10 2 + 13.33 × 1 2 10 2 - 3 × π 2.5 2 10 2 + 1 2 10 2 - π 2.5 2 8.5 units . x=\frac{5\times 10^{2}+13.33\times\frac{1}{2}10^{2}-3\times\pi 2.5^{2}}{10^{2}% +\frac{1}{2}10^{2}-\pi 2.5^{2}}\approx 8.5\mbox{ units}~{}.
  20. A i A_{i}
  21. X i X_{i}
  22. \R d \R^{d}
  23. d d
  24. d d
  25. \R n \R^{n}
  26. C = x g ( x ) d x g ( x ) d x C=\frac{\int xg(x)\;dx}{\int g(x)\;dx}
  27. \R n \R^{n}
  28. C k = z S k ( z ) d z S k ( z ) d z C_{k}=\frac{\int zS_{k}(z)\;dz}{\int S_{k}(z)\;dz}
  29. C x = x S y ( x ) d x A C_{\mathrm{x}}=\frac{\int xS_{\mathrm{y}}(x)\;dx}{A}
  30. C y = y S x ( y ) d y A C_{\mathrm{y}}=\frac{\int yS_{\mathrm{x}}(y)\;dy}{A}
  31. ( x ¯ , y ¯ ) (\bar{x},\;\bar{y})
  32. f f
  33. g g
  34. f ( x ) g ( x ) f(x)\geq g(x)
  35. [ a , b ] [a,b]
  36. a x b a\leq x\leq b
  37. x ¯ = 1 A a b x [ f ( x ) - g ( x ) ] d x \bar{x}=\frac{1}{A}\int_{a}^{b}x[f(x)-g(x)]\;dx
  38. y ¯ = 1 A a b [ f ( x ) + g ( x ) 2 ] [ f ( x ) - g ( x ) ] d x , \bar{y}=\frac{1}{A}\int_{a}^{b}\left[\frac{f(x)+g(x)}{2}\right][f(x)-g(x)]\;dx,
  39. A A
  40. a b [ f ( x ) - g ( x ) ] d x \int_{a}^{b}[f(x)-g(x)]\;dx
  41. a = ( x a , y a ) , a=(x_{a},y_{a}),
  42. b = ( x b , y b ) , b=(x_{b},y_{b}),
  43. c = ( x c , y c ) , c=(x_{c},y_{c}),
  44. C = 1 3 ( a + b + c ) = ( 1 3 ( x a + x b + x c ) , 1 3 ( y a + y b + y c ) ) . C=\frac{1}{3}(a+b+c)=\left(\frac{1}{3}(x_{a}+x_{b}+x_{c}),\;\;\frac{1}{3}(y_{a% }+y_{b}+y_{c})\right).
  45. 1 3 : 1 3 : 1 3 \tfrac{1}{3}:\tfrac{1}{3}:\tfrac{1}{3}
  46. C = 1 a : 1 b : 1 c = b c : c a : a b = csc A : csc B : csc C C=\frac{1}{a}:\frac{1}{b}:\frac{1}{c}=bc:ca:ab=\csc A:\csc B:\csc C
  47. = cos A + cos B cos C : cos B + cos C cos A : cos C + cos A cos B =\cos A+\cos B\cdot\cos C:\cos B+\cos C\cdot\cos A:\cos C+\cos A\cdot\cos B
  48. = sec A + sec B sec C : sec B + sec C sec A : sec C + sec A sec B . =\sec A+\sec B\cdot\sec C:\sec B+\sec C\cdot\sec A:\sec C+\sec A\cdot\sec B.
  49. C H = 2 C O . CH=2CO.
  50. C H = 4 C N , CH=4CN,
  51. C O = 2 C N , CO=2CN,
  52. I C < H C , IC<HC,
  53. I H < H C , IH<HC,
  54. I C < I O . IC<IO.
  55. v 0 , , v n {v_{0},\ldots,v_{n}}
  56. C = 1 n + 1 i = 0 n v i . C=\frac{1}{n+1}\sum_{i=0}^{n}v_{i}.
  57. C x = 1 6 A i = 0 n - 1 ( x i + x i + 1 ) ( x i y i + 1 - x i + 1 y i ) C_{\mathrm{x}}=\frac{1}{6A}\sum_{i=0}^{n-1}(x_{i}+x_{i+1})(x_{i}\ y_{i+1}-x_{i% +1}\ y_{i})
  58. C y = 1 6 A i = 0 n - 1 ( y i + y i + 1 ) ( x i y i + 1 - x i + 1 y i ) C_{\mathrm{y}}=\frac{1}{6A}\sum_{i=0}^{n-1}(y_{i}+y_{i+1})(x_{i}\ y_{i+1}-x_{i% +1}\ y_{i})
  59. A = 1 2 i = 0 n - 1 ( x i y i + 1 - x i + 1 y i ) A=\frac{1}{2}\sum_{i=0}^{n-1}(x_{i}\ y_{i+1}-x_{i+1}\ y_{i})\;

Ceva's_theorem.html

  1. A F F B B D D C C E E A = 1. \frac{AF}{FB}\cdot\frac{BD}{DC}\cdot\frac{CE}{EA}=1.
  2. A F F B B D D C C E E A = 1 , \frac{AF}{FB}\cdot\frac{BD}{DC}\cdot\frac{CE}{EA}=1,
  3. | B O D | | C O D | = B D D C = | B A D | | C A D | . \frac{|\triangle BOD|}{|\triangle COD|}=\frac{BD}{DC}=\frac{|\triangle BAD|}{|% \triangle CAD|}.
  4. B D D C = | B A D | - | B O D | | C A D | - | C O D | = | A B O | | C A O | . \frac{BD}{DC}=\frac{|\triangle BAD|-|\triangle BOD|}{|\triangle CAD|-|% \triangle COD|}=\frac{|\triangle ABO|}{|\triangle CAO|}.
  5. C E E A = | B C O | | A B O | , \frac{CE}{EA}=\frac{|\triangle BCO|}{|\triangle ABO|},
  6. A F F B = | C A O | | B C O | . \frac{AF}{FB}=\frac{|\triangle CAO|}{|\triangle BCO|}.
  7. | A F F B B D D C C E E A | = 1 , \left|\frac{AF}{FB}\cdot\frac{BD}{DC}\cdot\frac{CE}{EA}\right|=1,
  8. A B B F F O O C C E E A = - 1 \frac{AB}{BF}\cdot\frac{FO}{OC}\cdot\frac{CE}{EA}=-1
  9. B A A F F O O C C D D B = - 1. \frac{BA}{AF}\cdot\frac{FO}{OC}\cdot\frac{CD}{DB}=-1.
  10. A F F B = A F F B \frac{AF}{FB}=\frac{AF^{\prime}}{F^{\prime}B}

Chain_(algebraic_topology).html

  1. c = t 1 + t 2 + t 3 c=t_{1}+t_{2}+t_{3}\,
  2. v 1 v_{1}\,
  3. v 4 v_{4}\,
  4. t 1 = [ v 1 , v 2 ] t_{1}=[v_{1},v_{2}]\,
  5. t 2 = [ v 2 , v 3 ] t_{2}=[v_{2},v_{3}]\,
  6. t 3 = [ v 3 , v 4 ] t_{3}=[v_{3},v_{4}]\,
  7. 1 c = 1 ( t 1 + t 2 + t 3 ) = 1 ( t 1 ) + 1 ( t 2 ) + 1 ( t 3 ) = 1 ( [ v 1 , v 2 ] ) + 1 ( [ v 2 , v 3 ] ) + 1 ( [ v 3 , v 4 ] ) = ( [ v 2 ] - [ v 1 ] ) + ( [ v 3 ] - [ v 2 ] ) + ( [ v 4 ] - [ v 3 ] ) = [ v 4 ] - [ v 1 ] . \begin{aligned}\displaystyle\partial_{1}c&\displaystyle=\partial_{1}(t_{1}+t_{% 2}+t_{3})\\ &\displaystyle=\partial_{1}(t_{1})+\partial_{1}(t_{2})+\partial_{1}(t_{3})\\ &\displaystyle=\partial_{1}([v_{1},v_{2}])+\partial_{1}([v_{2},v_{3}])+% \partial_{1}([v_{3},v_{4}])\\ &\displaystyle=([v_{2}]-[v_{1}])+([v_{3}]-[v_{2}])+([v_{4}]-[v_{3}])\\ &\displaystyle=[v_{4}]-[v_{1}].\end{aligned}

Chain_complex.html

  1. ( A , d ) (A_{\bullet},d_{\bullet})
  2. A n + 1 d n + 1 A n d n A n - 1 d n - 1 A n - 2 d 2 A 1 d 1 A 0 d 0 A - 1 d - 1 A - 2 d - 2 \cdots\to A_{n+1}\xrightarrow{d_{n+1}}A_{n}\xrightarrow{d_{n}}A_{n-1}% \xrightarrow{d_{n-1}}A_{n-2}\to\cdots\xrightarrow{d_{2}}A_{1}\xrightarrow{d_{1% }}A_{0}\xrightarrow{d_{0}}A_{-1}\xrightarrow{d_{-1}}A_{-2}\xrightarrow{d_{-2}}\cdots
  3. ( A , d ) (A^{\bullet},d^{\bullet})
  4. A - 2 A^{-2}
  5. A - 1 A^{-1}
  6. A 0 A^{0}
  7. A 1 A^{1}
  8. A 2 A^{2}
  9. d n : A n A n + 1 d^{n}\colon A^{n}\to A^{n+1}
  10. d n + 1 d n = 0 d^{n+1}d^{n}=0
  11. A - 2 d - 2 A - 1 d - 1 A 0 d 0 A 1 d 1 A 2 A n - 1 d n - 1 A n d n A n + 1 . \cdots\to A^{-2}\xrightarrow{d^{-2}}A^{-1}\xrightarrow{d^{-1}}A^{0}% \xrightarrow{d^{0}}A^{1}\xrightarrow{d^{1}}A^{2}\to\cdots\to A^{n-1}% \xrightarrow{d^{n-1}}A^{n}\xrightarrow{d^{n}}A^{n+1}\to\cdots.
  12. n n
  13. A n A_{n}
  14. A n A^{n}
  15. d d = 0. dd=0.
  16. V W V\otimes W
  17. ( V W ) i = { j , k | j + k = i } V j W k (V\otimes W)_{i}=\bigoplus_{\{j,k|j+k=i\}}V_{j}\otimes W_{k}
  18. ( a b ) = a b + ( - 1 ) | a | a b \partial(a\otimes b)=\partial a\otimes b+(-1)^{|a|}a\otimes\partial b
  19. | a | |a|
  20. Ch K \,\text{Ch}_{K}
  21. a b ( - 1 ) | a | | b | b a a\otimes b\mapsto(-1)^{|a||b|}b\otimes a
  22. Π i Hom K ( V i , W i + n ) \Pi_{i}\,\text{Hom}_{K}(V_{i},W_{i+n})
  23. ( f ) ( v ) = ( f ( v ) ) - ( - 1 ) | f | f ( ( v ) ) (\partial f)(v)=\partial(f(v))-(-1)^{|f|}f(\partial(v))
  24. Hom ( A B , C ) Hom ( A , Hom ( B , C ) ) \,\text{Hom}(A\otimes B,C)\cong\,\text{Hom}(A,\,\text{Hom}(B,C))
  25. n : C n ( X ) C n - 1 ( X ) : ( σ : [ v 0 , , v n ] X ) ( n σ = i = 0 n ( - 1 ) i σ ( [ v 0 , , v ^ i , , v n ] ) , \partial_{n}:C_{n}(X)\to C_{n-1}(X):\,(\sigma:[v_{0},\ldots,v_{n}]\to X)% \mapsto(\partial_{n}\sigma=\sum_{i=0}^{n}(-1)^{i}\sigma([v_{0},\ldots,\hat{v}_% {i},\ldots,v_{n}]),
  26. ( C , ) (C_{\bullet},\partial_{\bullet})
  27. H ( X ) H_{\bullet}(X)
  28. H n ( X ) = ker n / im n + 1 . H_{n}(X)=\ker\partial_{n}/\mbox{im }~{}\partial_{n+1}.
  29. Ω 0 ( M ) d 0 Ω 1 ( M ) Ω 2 ( M ) Ω 3 ( M ) . \Omega^{0}(M)\ \stackrel{d_{0}}{\to}\ \Omega^{1}(M)\to\Omega^{2}(M)\to\Omega^{% 3}(M)\to\cdots.
  30. H DR 0 ( M ) = ker d 0 = H^{0}_{\mathrm{DR}}(M)=\ker d_{0}=
  31. \cong\mathbb{R}
  32. H DR k ( M ) = ker d k / im d k - 1 . H^{k}_{\mathrm{DR}}(M)=\ker d_{k}/\mathrm{im}\,d_{k-1}.
  33. ( A , d A , ) (A_{\bullet},d_{A,\bullet})
  34. ( B , d B , ) (B_{\bullet},d_{B,\bullet})
  35. f f_{\bullet}
  36. f n : A n B n f_{n}:A_{n}\rightarrow B_{n}
  37. d B , n f n = f n - 1 d A , n d_{B,n}\circ f_{n}=f_{n-1}\circ d_{A,n}
  38. ( f ) * : H ( A , d A , ) H ( B , d B , ) (f_{\bullet})_{*}:H_{\bullet}(A_{\bullet},d_{A,\bullet})\rightarrow H_{\bullet% }(B_{\bullet},d_{B,\bullet})

Chamber_music.html

  1. 3 + 2 + 2 + 3 8 \tfrac{3+2+2+3}{8}
  2. 3 4 \tfrac{3}{4}
  3. 4 4 \tfrac{4}{4}

Change_ringing.html

  1. n n
  2. n ! n!
  3. n n

Channel_capacity.html

  1. X X
  2. Y Y
  3. p Y | X ( y | x ) p_{Y|X}(y|x)
  4. Y Y
  5. X X
  6. p X ( x ) p_{X}(x)
  7. p X , Y ( x , y ) p_{X,Y}(x,y)
  8. p X , Y ( x , y ) = p Y | X ( y | x ) p X ( x ) \ p_{X,Y}(x,y)=p_{Y|X}(y|x)\,p_{X}(x)
  9. I ( X ; Y ) I(X;Y)
  10. C = sup p X ( x ) I ( X ; Y ) \ C=\sup_{p_{X}(x)}I(X;Y)\,
  11. p X ( x ) p_{X}(x)
  12. C = B log 2 ( 1 + S N ) C=B\log_{2}\left(1+\frac{S}{N}\right)
  13. 10 30 / 10 = 10 3 = 1000 10^{30/10}=10^{3}=1000
  14. P ¯ \bar{P}
  15. N 0 N_{0}
  16. C a w g n = W log 2 ( 1 + P ¯ N 0 W ) C_{awgn}=W\log_{2}\left(1+\frac{\bar{P}}{N_{0}W}\right)
  17. P ¯ N 0 W \frac{\bar{P}}{N_{0}W}
  18. C W log 2 P ¯ N 0 W C\approx W\log_{2}\frac{\bar{P}}{N_{0}W}
  19. P ¯ N o = 10 6 \frac{\bar{P}}{N_{o}}=10^{6}
  20. C N c = n = 0 N c - 1 log 2 ( 1 + P n * | h ¯ n | 2 N 0 ) , C_{N_{c}}=\sum_{n=0}^{N_{c}-1}\log_{2}\left(1+\frac{P_{n}^{*}|\bar{h}_{n}|^{2}% }{N_{0}}\right),
  21. P n * = max ( ( 1 λ - N 0 | h ¯ n | 2 ) , 0 ) P_{n}^{*}=\max\left(\left(\frac{1}{\lambda}-\frac{N_{0}}{|\bar{h}_{n}|^{2}}% \right),0\right)
  22. | h ¯ n | 2 |\bar{h}_{n}|^{2}
  23. n n
  24. λ \lambda
  25. log 2 ( 1 + | h | 2 S N R ) \log_{2}(1+|h|^{2}SNR)
  26. | h | 2 |h|^{2}
  27. R R
  28. p o u t = ( log ( 1 + | h | 2 S N R ) < R ) p_{out}=\mathbb{P}(\log(1+|h|^{2}SNR)<R)
  29. R R
  30. p o u t p_{out}
  31. ϵ \epsilon
  32. ϵ \epsilon
  33. 𝔼 ( log 2 ( 1 + | h | 2 S N R ) ) \mathbb{E}(\log_{2}(1+|h|^{2}SNR))

Character_(mathematics).html

  1. χ 1 , χ 2 , , χ n \chi_{1},\chi_{2},\ldots,\chi_{n}
  2. a 1 χ 1 + a 2 χ 2 + + a n χ n = 0 a_{1}\chi_{1}+a_{2}\chi_{2}+\ldots+a_{n}\chi_{n}=0
  3. a 1 = a 2 = = a n = 0 a_{1}=a_{2}=\cdots=a_{n}=0

Characteristic_function.html

  1. 𝟏 A : X { 0 , 1 } , \mathbf{1}_{A}\colon X\to\{0,1\},
  2. φ X ( t ) = E ( e i t X ) \varphi_{X}(t)=\operatorname{E}\left(e^{itX}\right)
  3. χ A ( x ) := { 0 , x A ; + , x A . \chi_{A}(x):=\begin{cases}0,&x\in A;\\ +\infty,&x\not\in A.\end{cases}

Characteristic_polynomial.html

  1. ( t - a 1 ) ( t - a 2 ) ( t - a 3 ) . (t-a_{1})(t-a_{2})(t-a_{3})\cdots.\,
  2. A 𝐯 = λ 𝐯 , A\mathbf{v}=\lambda\mathbf{v},\,
  3. ( λ I - A ) 𝐯 = 0 (\lambda I-A)\mathbf{v}=0\,
  4. p A ( t ) = det ( t s y m b o l I - A ) p_{A}(t)=\det\left(tsymbol{I}-A\right)
  5. A = ( 2 1 - 1 0 ) . A=\begin{pmatrix}2&1\\ -1&0\end{pmatrix}.
  6. t I - A = ( t - 2 - 1 1 t - 0 ) tI-A=\begin{pmatrix}t-2&-1\\ 1&t-0\end{pmatrix}
  7. ( t - 2 ) t - 1 ( - 1 ) = t 2 - 2 t + 1 , (t-2)t-1(-1)=t^{2}-2t+1\,\!,
  8. A = ( cosh ( ϕ ) sinh ( ϕ ) sinh ( ϕ ) cosh ( ϕ ) ) . A=\begin{pmatrix}\cosh(\phi)&\sinh(\phi)\\ \sinh(\phi)&\cosh(\phi)\end{pmatrix}.
  9. det ( t I - A ) = ( t - cosh ( ϕ ) ) 2 - sinh 2 ( ϕ ) = t 2 - 2 t cosh ( ϕ ) + 1 = ( t - e ϕ ) ( t - e - ϕ ) . \det(tI-A)=(t-\cosh(\phi))^{2}-\sinh^{2}(\phi)=t^{2}-2t\ \cosh(\phi)+1=(t-e^{% \phi})(t-e^{-\phi}).
  10. t r ( A ) tr(A)
  11. t 2 - tr ( A ) t + det ( A ) . t^{2}-\operatorname{tr}(A)t+\det(A).
  12. p A ( t ) = k = 0 n t n - k ( - 1 ) k tr ( Λ k A ) p_{A}(t)=\sum_{k=0}^{n}t^{n-k}(-1)^{k}\operatorname{tr}(\Lambda^{k}A)
  13. ( n k ) {\textstyle\left({{n}\atop{k}}\right)}
  14. k × k k×k
  15. 1 k ! | tr A k - 1 0 tr A 2 tr A k - 2 tr A k - 1 tr A k - 2 1 tr A k tr A k - 1 tr A | . \frac{1}{k!}\begin{vmatrix}\operatorname{tr}A&k-1&0&\cdots&\\ \operatorname{tr}A^{2}&\operatorname{tr}A&k-2&\cdots&\\ \vdots&\vdots&&\ddots&\vdots\\ \operatorname{tr}A^{k-1}&\operatorname{tr}A^{k-2}&&\cdots&1\\ \operatorname{tr}A^{k}&\operatorname{tr}A^{k-1}&&\cdots&\operatorname{tr}A\end% {vmatrix}~{}.
  16. p A B ( t ) = p B A ( t ) . p_{AB}(t)=p_{BA}(t).\,
  17. B A = A - 1 ( A B ) A . BA=A^{-1}(AB)A.
  18. p B A ( t ) = t n - m p A B ( t ) . p_{BA}(t)=t^{n-m}p_{AB}(t).\,

Charles's_law.html

  1. V T V\propto T\,
  2. V T = k \frac{V}{T}=k
  3. V 1 T 1 = V 2 T 2 or V 2 V 1 = T 2 T 1 or V 1 T 2 = V 2 T 1 . \frac{V_{1}}{T_{1}}=\frac{V_{2}}{T_{2}}\qquad\mathrm{or}\qquad\frac{V_{2}}{V_{% 1}}=\frac{T_{2}}{T_{1}}\qquad\mathrm{or}\qquad V_{1}T_{2}=V_{2}T_{1}.
  4. V 100 - V 0 = k V 0 V_{100}-V_{0}=kV_{0}\,
  5. 1 / 2.6666 {1}/{2.6666}
  6. 1 / 2.7315 {1}/{2.7315}
  7. T E k ¯ . T\propto\bar{E_{\rm k}}.\,
  8. p V = 2 3 N E k ¯ pV=\frac{2}{3}N\bar{E_{\rm k}}\,

Chebyshev's_inequality.html

  1. Pr ( | X - μ | k σ ) 1 k 2 . \Pr(|X-\mu|\geq k\sigma)\leq\frac{1}{k^{2}}.
  2. 1 k 2 1. \frac{1}{k^{2}}\geq 1.
  3. 2 \sqrt{2}
  4. μ ( { x X : | f ( x ) | t } ) 1 t p | f | > t | f | p d μ . \mu(\{x\in X\,:\,\,|f(x)|\geq t\})\leq{1\over t^{p}}\int_{|f|>t}|f|^{p}\,d\mu.
  5. μ ( { x X : f ( x ) t } ) 1 g ( t ) X g f d μ . \mu(\{x\in X\,:\,\,f(x)\geq t\})\leq{1\over g(t)}\int_{X}g\circ f\,d\mu.
  6. g ( t ) g(t)
  7. t 2 t^{2}
  8. t 0 t\geq 0
  9. 0
  10. | f | |f|
  11. f f
  12. X = { - 1 , with probability 1 2 k 2 0 , with probability 1 - 1 k 2 1 , with probability 1 2 k 2 X=\begin{cases}-1,&\,\text{with probability }\frac{1}{2k^{2}}\\ 0,&\,\text{with probability }1-\frac{1}{k^{2}}\\ 1,&\,\text{with probability }\frac{1}{2k^{2}}\end{cases}
  13. Pr ( | X - μ | k σ ) = Pr ( | Z | 1 ) = 1 k 2 . \Pr(|X-\mu|\geq k\sigma)=\Pr(|Z|\geq 1)=\frac{1}{k^{2}}.
  14. Pr ( | X - μ | k σ ) = E ( I | X - μ | k σ ) = E ( I ( X - μ k σ ) 2 1 ) E ( ( X - μ k σ ) 2 ) = 1 k 2 E ( ( X - μ ) 2 ) σ 2 = 1 k 2 . \begin{aligned}\displaystyle\Pr(|X-\mu|\geq k\sigma)&\displaystyle=% \operatorname{E}\left(I_{|X-\mu|\geq k\sigma}\right)\\ &\displaystyle=\operatorname{E}\left(I_{\left(\frac{X-\mu}{k\sigma}\right)^{2}% \geq 1}\right)\\ &\displaystyle\leq\operatorname{E}\left(\left({X-\mu\over k\sigma}\right)^{2}% \right)\\ &\displaystyle={1\over k^{2}}{\operatorname{E}((X-\mu)^{2})\over\sigma^{2}}\\ &\displaystyle={1\over k^{2}}.\end{aligned}
  15. t t
  16. A t A_{t}
  17. A t = { x X f ( x ) t } A_{t}=\{x\in X\mid f(x)\geq t\}
  18. 1 A t 1_{A_{t}}
  19. A t A_{t}
  20. x x
  21. 0 g ( t ) 1 A t g ( f ( x ) ) 1 A t , 0\leq g(t)1_{A_{t}}\leq g(f(x))\,1_{A_{t}},
  22. g ( t ) μ ( A t ) = X g ( t ) 1 A t d μ A t g f d μ X g f d μ . \begin{aligned}\displaystyle g(t)\mu(A_{t})&\displaystyle=\int_{X}g(t)1_{A_{t}% }\,d\mu\\ &\displaystyle\leq\int_{A_{t}}g\circ f\,d\mu\\ &\displaystyle\leq\int_{X}g\circ f\,d\mu.\end{aligned}
  23. k 1 + k 2 = 0 k_{1}+k_{2}=0
  24. Pr ( k 1 < X < k 2 ) 1 - 4 σ 2 ( k 2 - k 1 ) 2 \Pr(k_{1}<X<k_{2})\geq 1-\frac{4\sigma^{2}}{(k_{2}-k_{1})^{2}}
  25. k 1 + k 2 = 2 μ k_{1}+k_{2}=2\mu
  26. Pr ( k 1 < X < k 2 ) 4 [ ( μ - k 1 ) ( k 2 - μ ) - σ 2 ] ( k 2 - k 1 ) 2 , \Pr(k_{1}<X<k_{2})\geq\frac{4[(\mu-k_{1})(k_{2}-\mu)-\sigma^{2}]}{(k_{2}-k_{1}% )^{2}},
  27. μ μ
  28. Pr ( k 11 X 1 k 12 , k 21 X 2 k 22 ) 1 - T i \Pr(k_{11}\leq X_{1}\leq k_{12},k_{21}\leq X_{2}\leq k_{22})\geq 1-\sum T_{i}
  29. T i = 4 σ i 2 + [ 2 μ i - ( k i 1 + k i 2 ) ] 2 ( k i 2 - k i 1 ) 2 . T_{i}=\frac{4\sigma_{i}^{2}+[2\mu_{i}-(k_{i1}+k_{i2})]^{2}}{(k_{i2}-k_{i1})^{2% }}.
  30. ρ ρ
  31. Pr ( i = 1 2 [ | X i - μ i | σ i < k ] ) 1 - 1 + 1 - ρ 2 k 2 . \Pr\left(\bigcap_{i=1}^{2}\left[\frac{|X_{i}-\mu_{i}|}{\sigma_{i}}<k\right]% \right)\geq 1-\frac{1+\sqrt{1-\rho^{2}}}{k^{2}}.
  32. Pr ( i = 1 2 [ | X i - μ i | σ i k i ] ) 1 - k 1 2 + k 2 2 + ( k 1 2 + k 2 2 ) 2 - 4 k 1 2 k 2 2 ρ 2 ( k 1 k 2 ) 2 \Pr\left(\bigcap_{i=1}^{2}\left[\frac{|X_{i}-\mu_{i}|}{\sigma_{i}}\leq k_{i}% \right]\right)\geq 1-\frac{k_{1}^{2}+k_{2}^{2}+\sqrt{(k_{1}^{2}+k_{2}^{2})^{2}% -4k_{1}^{2}k_{2}^{2}\rho}}{2(k_{1}k_{2})^{2}}
  33. Z = Pr ( ( - k 1 < X 1 < k 2 ) ( - k 1 < X 2 < k 2 ) ) , 0 < k 1 k 2 . Z=\Pr\left(\left(-k_{1}<X_{1}<k_{2}\right)\cap\left(-k_{1}<X_{2}<k_{2}\right)% \right),\qquad 0<k_{1}\leq k_{2}.
  34. λ = k 1 ( 1 + ρ ) + ( 1 - ρ 2 ) ( k 1 2 + ρ ) 2 k 1 - 1 + ρ \lambda=\frac{k_{1}(1+\rho)+\sqrt{(1-\rho^{2})(k_{1}^{2}+\rho)}}{2k_{1}-1+\rho}
  35. 2 k 1 2 > 1 - ρ 2k_{1}^{2}>1-\rho
  36. k 2 - k 1 2 λ k_{2}-k_{1}\geq 2\lambda
  37. Z 2 λ 2 2 λ 2 + 1 + ρ . Z\leq\frac{2\lambda^{2}}{2\lambda^{2}+1+\rho}.
  38. 2 ( k 1 k 2 - 1 ) 2 2 ( 1 - ρ 2 ) + ( 1 - ρ ) ( k 2 - k 1 ) 2 2(k_{1}k_{2}-1)^{2}\geq 2(1-\rho^{2})+(1-\rho)(k_{2}-k_{1})^{2}
  39. Z ( k 2 - k 1 ) 2 + 4 + 16 ( 1 - ρ 2 ) + 8 ( 1 - ρ ) ( k 2 - k 1 ) ( k 1 + k 2 ) 2 . Z\leq\frac{(k_{2}-k_{1})^{2}+4+\sqrt{16(1-\rho^{2})+8(1-\rho)(k_{2}-k_{1})}}{(% k_{1}+k_{2})^{2}}.
  40. Pr ( i = 1 n ( X i - μ i ) 2 σ i 2 t i 2 k 2 ) 1 k 2 i = 1 n 1 t i 2 \Pr\left(\sum_{i=1}^{n}\frac{(X_{i}-\mu_{i})^{2}}{\sigma_{i}^{2}t_{i}^{2}}\geq k% ^{2}\right)\leq\frac{1}{k^{2}}\sum_{i=1}^{n}\frac{1}{t_{i}^{2}}
  41. i i
  42. i i
  43. i i
  44. Pr ( i = 1 n | X i - μ i | σ i k i ) ( 1 - 1 k i 2 ) \Pr\left(\bigcap_{i=1}^{n}\frac{|X_{i}-\mu_{i}|}{\sigma_{i}}\leq k_{i}\right)% \geq\prod\left(1-\frac{1}{k_{i}^{2}}\right)
  45. n n
  46. Pr ( i = 1 n | X i - μ i | σ i < k i ) 1 - 1 n 2 ( U + n - 1 n i 1 k i 2 - u ) 2 \Pr\left(\bigcap_{i=1}^{n}\frac{|X_{i}-\mu_{i}|}{\sigma_{i}}<k_{i}\right)\geq 1% -\frac{1}{n^{2}}\left(\sqrt{U}+\sqrt{n-1}\sqrt{n\sum_{i}{\frac{1}{k_{i}^{2}}-u% }}\right)^{2}
  47. u = i = 1 n 1 k i 2 + 2 i = 1 n j < i ρ i j k i k j u=\sum_{i=1}^{n}\frac{1}{k_{i}^{2}}+2\sum_{i=1}^{n}\sum_{j<i}\frac{\rho_{ij}}{% k_{i}k_{j}}
  48. [ u ! ! ] [ u ! ! ] [u^{\prime}!!^{\prime}]⋅[u^{\prime}!!^{\prime}]
  49. Pr ( X - μ k σ ) 1 k 2 . \Pr(\|X-\mu\|\geq k\|\sigma\|)\leq\frac{1}{k^{2}}.
  50. N N
  51. X X
  52. E ( X ) E(X)
  53. X X
  54. S S
  55. k > 0 k>0
  56. Pr ( ( X - E ( X ) ) T S - 1 ( X - E ( X ) ) < k ) 1 - N k \Pr\left((X-\operatorname{E}(X))^{T}S^{-1}(X-\operatorname{E}(X))<k\right)\geq 1% -\frac{N}{k}
  57. Y Y
  58. X X
  59. 𝒳 \mathcal{X}
  60. 𝒳 \mathcal{X}
  61. X X
  62. E ( X α 2 ) < \operatorname{E}\left(\|X\|_{\alpha}^{2}\right)<\infty
  63. X X
  64. μ 𝒳 \mu\in\mathcal{X}
  65. X X
  66. σ a := E X - μ α 2 \sigma_{a}:=\sqrt{\operatorname{E}\|X-\mu\|_{\alpha}^{2}}
  67. k > 0 : Pr ( X - μ α k σ α ) 1 k 2 . \forall k>0:\quad\Pr\left(\|X-\mu\|_{\alpha}\geq k\sigma_{\alpha}\right)\leq% \frac{1}{k^{2}}.
  68. X X
  69. μ μ
  70. X - μ α k σ α 2 \|X-\mu\|_{\alpha}\geq k\sigma_{\alpha}^{2}
  71. 1 = X - μ α 2 X - μ α 2 1=\tfrac{\|X-\mu\|_{\alpha}^{2}}{\|X-\mu\|_{\alpha}^{2}}
  72. Pr ( X - μ α k σ α ) \displaystyle\Pr\left(\|X-\mu\|_{\alpha}\geq k\sigma_{\alpha}\right)
  73. Pr ( | X - E ( X ) | k ) E ( | X - E ( X ) | n ) k n , k > 0 , n 2. \Pr(|X-\operatorname{E}(X)|\geq k)\leq\frac{\operatorname{E}(|X-\operatorname{% E}(X)|^{n})}{k^{n}},\qquad k>0,n\geq 2.
  74. Pr ( X ε ) e - t ε E ( e t X ) , t > 0. \Pr(X\geq\varepsilon)\leq e^{-t\varepsilon}\operatorname{E}\left(e^{tX}\right)% ,\qquad t>0.
  75. K ( x , t ) K(x,t)
  76. K ( x , t ) = log ( E ( e t x ) ) . K(x,t)=\log\left(\operatorname{E}\left(e^{tx}\right)\right).
  77. K ( x , t ) K(x,t)
  78. - log ( Pr ( X ε ) ) sup t ( t ε - K ( x , t ) ) . -\log(\Pr(X\geq\varepsilon))\leq\sup_{t}(t\varepsilon-K(x,t)).
  79. a a , b aa,b
  80. M = m a x ( | a | , | b | ) M=max(|a|,|b|)
  81. x x
  82. k > 0 k>0
  83. E ( | X | r ) - k r M r Pr ( | X | k ) E ( | X | r ) k r . \frac{\operatorname{E}(|X|^{r})-k^{r}}{M^{r}}\leq\Pr(|X|\geq k)\leq\frac{% \operatorname{E}(|X|^{r})}{k^{r}}.
  84. r = 2 r=2
  85. 0 X M 0≤X≤M
  86. M > 0 M>0
  87. Pr ( X < k ) = 0 if E ( X ) > k and E ( X 2 ) < k E ( X ) + M E ( X ) - k M \Pr(X<k)=0\qquad\,\text{if}\qquad\operatorname{E}(X)>k\quad\,\text{and}\quad% \operatorname{E}(X^{2})<k\operatorname{E}(X)+M\operatorname{E}(X)-kM
  88. Pr ( X < k ) 1 - k E ( X ) + M E ( X ) - E ( X 2 ) k M if { E ( X ) > k and E ( X 2 ) k E ( X ) + M E ( X ) - k M or E ( X ) k and E ( X 2 ) k E ( X ) \Pr(X<k)\geq 1-\frac{k\operatorname{E}(X)+M\operatorname{E}(X)-\operatorname{E% }(X^{2})}{kM}\qquad\,\text{if}\qquad\begin{cases}\operatorname{E}(X)>k\quad\,% \text{and}\quad E(X^{2})\geq k\operatorname{E}(X)+M\operatorname{E}(X)-kM\\ \qquad\qquad\qquad\,\text{or}\\ \operatorname{E}(X)\leq k\quad\,\text{and}\quad\operatorname{E}(X^{2})\geq k% \operatorname{E}(X)\end{cases}
  89. Pr ( X < k ) E ( X ) 2 - 2 k E ( X ) + k 2 E ( X 2 ) - 2 k E ( X ) + k 2 if E ( X ) k and E ( X 2 ) < k E ( X ) \Pr(X<k)\geq\frac{\operatorname{E}(X)^{2}-2k\operatorname{E}(X)+k^{2}}{% \operatorname{E}(X^{2})-2k\operatorname{E}(X)+k^{2}}\qquad\,\text{if}\qquad% \operatorname{E}(X)\leq k\quad\,\text{and}\quad\operatorname{E}(X^{2})<k% \operatorname{E}(X)
  90. P ( | X - m | k s ) g N + 1 ( N k 2 N - 1 + k 2 ) N + 1 ( N N + 1 ) 1 / 2 P(|X-m|\geq ks)\leq\frac{g_{N+1}\left(\frac{Nk^{2}}{N-1+k^{2}}\right)}{N+1}% \left(\frac{N}{N+1}\right)^{1/2}
  91. a 2 = Q ( Q - R ) 1 + R ( Q - R ) . a^{2}=\frac{Q(Q-R)}{1+R(Q-R)}.
  92. g Q ( x ) = R if R is even g_{Q}(x)=R\quad\,\text{if }R\,\text{ is even}
  93. g Q ( x ) = R if R is odd and x < a 2 g_{Q}(x)=R\quad\,\text{if }R\,\text{ is odd and }x<a^{2}
  94. g Q ( x ) = R - 1 if R is odd and x a 2 . g_{Q}(x)=R-1\quad\,\text{if }R\,\text{ is odd and }x\geq a^{2}.
  95. P ( | X - m | k s ) 1 [ N ( N + 1 ) ] 1 / 2 [ ( N - 1 k 2 + 1 ) ] P(|X-m|\geq ks)\leq\frac{1}{[N(N+1)]^{1/2}}\left[\left(\frac{N-1}{k^{2}}+1% \right)\right]
  96. P ( | X - m | k s ) N - 1 N 1 k 2 s 2 m 2 + 1 N . P(|X-m|\geq ks)\leq\frac{N-1}{N}\frac{1}{k^{2}}\frac{s^{2}}{m^{2}}+\frac{1}{N}.
  97. P ( | X - m | k s ) 1 N + 1 . P(|X-m|\geq ks)\leq\frac{1}{N+1}.
  98. P ( | X - m | k s ) 1 k 2 ( N + 1 ) . P(|X-m|\geq ks)\leq\frac{1}{k^{2}(N+1)}.
  99. Z = X - E ( X ) Var ( X ) 1 / 2 . Z=\frac{X-\operatorname{E}(X)}{\operatorname{Var}(X)^{1/2}}.
  100. P ( Z k ) 1 1 + k 2 . P(Z\geq k)\leq\frac{1}{1+k^{2}}.
  101. P ( Z k ) 1 2 k 2 . P(Z\geq k)\leq\frac{1}{2k^{2}}.
  102. P ( Z - u or Z v ) 4 + ( u - v ) 2 ( u + v ) 2 . P(Z\leq-u\,\text{ or }Z\geq v)\leq\frac{4+(u-v)^{2}}{(u+v)^{2}}.
  103. σ + 2 = ( x - m ) 2 n - 1 \sigma_{+}^{2}=\frac{\sum(x-m)^{2}}{n-1}
  104. σ - 2 = ( m - x ) 2 n - 1 \sigma_{-}^{2}=\frac{\sum(m-x)^{2}}{n-1}
  105. σ 2 = σ + 2 + σ - 2 . \sigma^{2}=\sigma_{+}^{2}+\sigma_{-}^{2}.
  106. Pr ( x m - a σ - ) 1 a 2 . \Pr(x\leq m-a\sigma_{-})\leq\frac{1}{a^{2}}.
  107. a = k σ σ - . a=\frac{k\sigma}{\sigma_{-}}.
  108. Pr ( x m - k σ ) 1 k 2 σ - 2 σ 2 . \Pr(x\leq m-k\sigma)\leq\frac{1}{k^{2}}\frac{\sigma_{-}^{2}}{\sigma^{2}}.
  109. σ u 2 = max ( σ - 2 , σ + 2 ) , \sigma_{u}^{2}=\max(\sigma_{-}^{2},\sigma_{+}^{2}),
  110. Pr ( | x m - k σ | ) 1 k 2 σ u 2 σ 2 . \Pr(|x\leq m-k\sigma|)\leq\frac{1}{k^{2}}\frac{\sigma_{u}^{2}}{\sigma^{2}}.
  111. σ + 2 = σ - 2 = 1 2 σ 2 \sigma_{+}^{2}=\sigma_{-}^{2}=\frac{1}{2}\sigma^{2}
  112. Pr ( x m - k σ ) 1 2 k 2 . \Pr(x\leq m-k\sigma)\leq\frac{1}{2k^{2}}.
  113. Y = α X + β Y=\alpha X+\beta
  114. α = 2 k b - a \alpha=\frac{2k}{b-a}
  115. β = - ( b + a ) k b - a . \beta=\frac{-(b+a)k}{b-a}.
  116. μ Y = α μ X + β \mu_{Y}=\alpha\mu_{X}+\beta
  117. σ Y 2 = α 2 σ X 2 . \sigma_{Y}^{2}=\alpha^{2}\sigma_{X}^{2}.
  118. P ( | Y | < k ) ( k - μ Y ) 2 ( k - μ Y ) 2 + σ Y 2 if σ Y 2 μ Y ( k - μ Y ) P(|Y|<k)\geq\frac{(k-\mu_{Y})^{2}}{(k-\mu_{Y})^{2}+\sigma_{Y}^{2}}\quad\,\text% { if }\quad\sigma_{Y}^{2}\leq\mu_{Y}(k-\mu_{Y})
  119. P ( | Y | < k ) 1 - σ Y 2 + μ Y 2 k 2 if μ Y ( k - μ Y ) σ Y 2 k 2 - μ Y 2 P(|Y|<k)\geq 1-\frac{\sigma_{Y}^{2}+\mu_{Y}^{2}}{k^{2}}\quad\,\text{ if }\quad% \mu_{Y}(k-\mu_{Y})\leq\sigma_{Y}^{2}\leq k^{2}-\mu_{Y}^{2}
  120. P ( | Y | < k ) 0 if k 2 - μ Y 2 σ Y 2 . P(|Y|<k)\geq 0\quad\,\text{ if }\quad k^{2}-\mu_{Y}^{2}\leq\sigma_{Y}^{2}.
  121. P ( X - μ a ) σ 2 σ 2 + a 2 P(X-\mu\geq a)\leq\frac{\sigma^{2}}{\sigma^{2}+a^{2}}
  122. Pr ( X - μ k σ ) 1 1 + k 2 . \Pr(X-\mu\geq k\sigma)\leq\frac{1}{1+k^{2}}.
  123. X = 1 X=1
  124. σ 2 1 + σ 2 \frac{\sigma^{2}}{1+\sigma^{2}}
  125. X = - σ 2 X=-\sigma^{2}
  126. 1 1 + σ 2 . \frac{1}{1+\sigma^{2}}.
  127. | μ - ν | σ . \left|\mu-\nu\right|\leq\sigma.
  128. Pr ( X - μ σ ) 1 2 Pr ( X μ + σ ) 1 2 . \Pr(X-\mu\geq\sigma)\leq\frac{1}{2}\implies\Pr(X\geq\mu+\sigma)\leq\frac{1}{2}.
  129. Pr ( X μ - σ ) 1 2 . \Pr(X\leq\mu-\sigma)\leq\frac{1}{2}.
  130. P ( X > k σ ) κ - γ 2 - 1 ( κ - γ 2 - 1 ) ( 1 + k 2 ) + ( k 2 - k γ - 1 ) . P(X>k\sigma)\leq\frac{\kappa-\gamma^{2}-1}{(\kappa-\gamma^{2}-1)(1+k^{2})+(k^{% 2}-k\gamma-1)}.
  131. [ X - E ( X ) ] 2 k > 0 [X-E(X)]^{2k}>0
  132. E ( [ X - E ( X ) ] 2 k ) E([X-E(X)]^{2k})
  133. P ( | X - E ( X ) | > t [ E ( X - E ( X ) ) 2 k ] 1 / 2 k ) min [ 1 , 1 t 2 k ] . P(|X-E(X)|>t[E(X-E(X))^{2k}]^{1/2k})\leq\min\left[1,\frac{1}{t^{2k}}\right].
  134. Pr ( X - μ k σ ) [ 1 + k 2 + ( k 2 - k θ 3 - 1 ) 2 θ 4 - θ 3 2 - 1 ] - 1 \Pr(X-\mu\geq k\sigma)\leq\left[1+k^{2}+\frac{\left(k^{2}-k\theta_{3}-1\right)% ^{2}}{\theta_{4}-\theta_{3}^{2}-1}\right]^{-1}
  135. k θ 3 + θ 3 2 + 4 2 , θ m = M m σ m k\geq\frac{\theta_{3}+\sqrt{\theta_{3}^{2}+4}}{2},\qquad\theta_{m}=\frac{M_{m}% }{\sigma_{m}}
  136. m m
  137. σ σ
  138. n n
  139. Pr ( i = 1 n X i n - 1 1 n ) 7 8 . \Pr\left(\frac{\sum_{i=1}^{n}X_{i}}{n}-1\geq\frac{1}{n}\right)\leq\frac{7}{8}.
  140. X X
  141. a X b a≤X≤b
  142. E X X = 0 EXX=0
  143. s > 0 s>0
  144. E [ e s X ] e 1 8 s 2 ( b - a ) 2 . E\left[e^{sX}\right]\leq e^{\frac{1}{8}s^{2}(b-a)^{2}}.
  145. Pr ( | i = 1 n X i n | 1 ) 0.5. \Pr\left(\left|\frac{\sum_{i=1}^{n}X_{i}}{\sqrt{n}}\right|\leq 1\right)\geq 0.5.
  146. P r > 0.31 Pr>0.31
  147. P ( | X | k ) 4 E ( X 2 ) 9 k 2 if k 2 4 3 E ( X 2 ) , P(|X|\geq k)\leq\frac{4\operatorname{E}(X^{2})}{9k^{2}}\quad\,\text{if}\quad k% ^{2}\geq\frac{4}{3}\operatorname{E}(X^{2}),
  148. P ( | X | k ) 1 - k 2 3 E ( X 2 ) if k 2 4 3 E ( X 2 ) . P(|X|\geq k)\leq 1-\frac{k^{2}}{3\operatorname{E}(X^{2})}\quad\,\text{if}\quad k% ^{2}\leq\frac{4}{3}\operatorname{E}(X^{2}).
  149. σ ω 2 σ , \sigma\leq\omega\leq 2\sigma,
  150. | ν - μ | 3 4 ω . |\nu-\mu|\leq\sqrt{\frac{3}{4}}\omega.
  151. P ( | X | k ) ( r r + 1 ) r E ( | X | ) r k r if k r r r ( r + 1 ) r + 1 E ( | X | r ) , P(|X|\geq k)\leq\left(\frac{r}{r+1}\right)^{r}\frac{\operatorname{E}(|X|)^{r}}% {k^{r}}\quad\,\text{if}\quad k^{r}\geq\frac{r^{r}}{(r+1)^{r+1}}\operatorname{E% }(|X|^{r}),
  152. P ( | X | k ) ( 1 - [ k r ( r + 1 ) E ( | X | ) r ] 1 / r ) if k r r r ( r + 1 ) r + 1 E ( | X | r ) . P(|X|\geq k)\leq\left(1-\left[\frac{k^{r}}{(r+1)\operatorname{E}(|X|)^{r}}% \right]^{1/r}\right)\quad\,\text{if}\quad k^{r}\leq\frac{r^{r}}{(r+1)^{r+1}}% \operatorname{E}(|X|^{r}).
  153. P ( | X | > k ) max ( [ r ( r + 1 ) k ] r E | X r | , s ( s - 1 ) k r E | X r | - 1 s - 1 ) P(|X|>k)\leq\max\left(\left[\frac{r}{(r+1)k}\right]^{r}E|X^{r}|,\frac{s}{(s-1)% k^{r}}E|X^{r}|-\frac{1}{s-1}\right)
  154. P ( X k ) 1 2 - k 2 3 if 0 k 2 3 , P(X\geq k)\leq\frac{1}{2}-\frac{k}{2\sqrt{3}}\quad\,\text{if}\quad 0\leq k\leq% \frac{2}{\sqrt{3}},
  155. P ( X k ) 2 9 k 2 if 2 3 k 2 N 3 . P(X\geq k)\leq\frac{2}{9k^{2}}\quad\,\text{if}\quad\frac{2}{\sqrt{3}}\leq k% \leq\frac{2N}{3}.
  156. f ( x ) = 1 2 3 if | x | < 3 f(x)=\frac{1}{2\sqrt{3}}\quad\,\text{if}\quad|x|<\sqrt{3}
  157. f ( x ) = 0 if | x | 3 . f(x)=0\quad\,\text{if}\quad|x|\geq\sqrt{3}.
  158. f k ( x ) = 1 3 k if | x | < 3 k 2 , f_{k}(x)=\frac{1}{3k}\quad\,\text{if}\quad|x|<\frac{3k}{2},
  159. f k ( x ) = 0 if | x | 3 k 2 . f_{k}(x)=0\quad\,\text{if}\quad|x|\geq\frac{3k}{2}.
  160. P ( | X - μ | k σ ) 1 - k 3 if k 2 3 P(|X-\mu|\geq k\sigma)\leq 1-\frac{k}{\sqrt{3}}\quad\,\text{if}\quad k\leq% \frac{2}{\sqrt{3}}
  161. P ( | X - μ | k σ ) 4 9 k 2 if k > 2 3 P(|X-\mu|\geq k\sigma)\leq\frac{4}{9k^{2}}\quad\,\text{if}\quad k>\frac{2}{% \sqrt{3}}
  162. P ( | X - μ | k σ ) 1 3 k 2 P(|X-\mu|\geq k\sigma)\leq\frac{1}{3k^{2}}
  163. P ( | X | 1 ) min ( 1 , σ 2 ) P(|X|\geq 1)\leq\min(1,\sigma^{2})
  164. P ( X 1 ) σ 2 1 + σ 2 P(X\geq 1)\leq\frac{\sigma^{2}}{1+\sigma^{2}}
  165. P ( | X | > ϵ ) ( 1 - ϵ 2 ) 2 ψ - 1 + ( 1 - ϵ 2 ) 2 ( 1 - ϵ 2 ) 2 ψ P(|X|>\epsilon)\geq\frac{(1-\epsilon^{2})^{2}}{\psi-1+(1-\epsilon^{2})^{2}}% \geq\frac{(1-\epsilon^{2})^{2}}{\psi}
  166. P ( X ϵ ) C 0 ψ - C 1 ψ ϵ + C 2 ψ ψ ϵ P(X\geq\epsilon)\geq\frac{C_{0}}{\psi}-\frac{C_{1}}{\sqrt{\psi}}\epsilon+\frac% {C_{2}}{\psi\sqrt{\psi}}\epsilon
  167. C 0 = 2 3 - 3 ( 0.464 ) C_{0}=2\sqrt{3}-3\quad(\approxeq 0.464)
  168. C 1 = 1.397 C_{1}=1.397
  169. C 2 = 0.0231 C_{2}=0.0231
  170. P ( X > 0 ) C 0 ψ P(X>0)\geq\frac{C_{0}}{\psi}
  171. ψ 3 3 + 1 ( 1.098 ) \psi\geq\frac{3}{\sqrt{3}+1}\quad(\approxeq 1.098)
  172. ψ 3 3 + 1 \psi\leq\frac{3}{\sqrt{3}+1}
  173. P ( X > 0 ) 2 3 + ψ + ( 1 + ψ ) 2 - 4 P(X>0)\geq\frac{2}{3+\psi+\sqrt{(1+\psi)^{2}-4}}
  174. 1 b - a a b f ( x ) g ( x ) d x [ 1 b - a a b f ( x ) d x ] [ 1 b - a a b g ( x ) d x ] \frac{1}{b-a}\int_{a}^{b}\!f(x)g(x)\,dx\geq\left[\frac{1}{b-a}\int_{a}^{b}\!f(% x)\,dx\right]\left[\frac{1}{b-a}\int_{a}^{b}\!g(x)\,dx\right]
  175. z = x - γ 6 ( x 2 - 1 ) + x 72 [ 2 γ 2 ( 4 x 2 - 7 ) - 3 κ ( x 2 - 3 ) ] + z=x-\frac{\gamma}{6}(x^{2}-1)+\frac{x}{72}[2\gamma^{2}(4x^{2}-7)-3\kappa(x^{2}% -3)]+\cdots
  176. δ - ( 1 + δ ) log ( 1 + δ ) < - δ 2 2 + δ . \delta-(1+\delta)\log(1+\delta)<\frac{-\delta^{2}}{2+\delta}.
  177. P ( X > ( 1 + δ ) μ ) e - δ 2 μ 2 + δ , P(X>(1+\delta)\mu)\leq e^{\frac{-\delta^{2}\mu}{2+\delta}},
  178. P ( X < ( 1 - δ ) μ ) e - δ 2 μ 2 + δ . P(X<(1-\delta)\mu)\leq e^{\frac{-\delta^{2}\mu}{2+\delta}}.

Chebyshev's_theorem.html

  1. π ( x ) ln x / x \scriptstyle\pi(x)\ln x/x

Chebyshev_polynomials.html

  1. n n
  2. [ - 1 , 1 ] [-1,1]
  3. ( 1 - x 2 ) y ′′ - x y + n 2 y = 0 (1-x^{2})\,y^{\prime\prime}-x\,y^{\prime}+n^{2}\,y=0\,\!
  4. ( 1 - x 2 ) y ′′ - 3 x y + n ( n + 2 ) y = 0 (1-x^{2})\,y^{\prime\prime}-3x\,y^{\prime}+n(n+2)\,y=0\,\!
  5. T 0 ( x ) = 1 T 1 ( x ) = x T n + 1 ( x ) = 2 x T n ( x ) - T n - 1 ( x ) . \begin{aligned}\displaystyle T_{0}(x)&\displaystyle=1\\ \displaystyle T_{1}(x)&\displaystyle=x\\ \displaystyle T_{n+1}(x)&\displaystyle=2xT_{n}(x)-T_{n-1}(x).\end{aligned}
  6. n = 0 T n ( x ) t n = 1 - t x 1 - 2 t x + t 2 ; \sum_{n=0}^{\infty}T_{n}(x)t^{n}=\frac{1-tx}{1-2tx+t^{2}};\,\!
  7. n = 0 T n ( x ) t n n ! = 1 2 ( e ( x - x 2 - 1 ) t + e ( x + x 2 - 1 ) t ) = e t x cosh ( t x 2 - 1 ) . \sum_{n=0}^{\infty}T_{n}(x)\frac{t^{n}}{n!}=\tfrac{1}{2}\left(e^{(x-\sqrt{x^{2% }-1})t}+e^{(x+\sqrt{x^{2}-1})t}\right)=e^{tx}\cosh(t\sqrt{x^{2}-1}).\,\!
  8. n = 1 T n ( x ) t n n = ln 1 1 - 2 t x + t 2 . \sum\limits_{n=1}^{\infty}T_{n}\left(x\right)\frac{t^{n}}{n}=\ln\frac{1}{\sqrt% {1-2tx+t^{2}}}.
  9. U 0 ( x ) = 1 U 1 ( x ) = 2 x U n + 1 ( x ) = 2 x U n ( x ) - U n - 1 ( x ) . \begin{aligned}\displaystyle U_{0}(x)&\displaystyle=1\\ \displaystyle U_{1}(x)&\displaystyle=2x\\ \displaystyle U_{n+1}(x)&\displaystyle=2xU_{n}(x)-U_{n-1}(x).\end{aligned}
  10. n = 0 U n ( x ) t n = 1 1 - 2 t x + t 2 ; \sum_{n=0}^{\infty}U_{n}(x)t^{n}=\frac{1}{1-2tx+t^{2}};\,\!
  11. n = 0 U n ( x ) t n n ! = e t x ( cosh ( t x 2 - 1 ) + x x 2 - 1 sinh ( t x 2 - 1 ) ) . \sum_{n=0}^{\infty}U_{n}(x)\frac{t^{n}}{n!}=e^{tx}\left(\cosh(t\sqrt{x^{2}-1})% +\frac{x}{\sqrt{x^{2}-1}}\sinh(t\sqrt{x^{2}-1})\right).\,\!
  12. T n ( x ) = cos ( n arccos x ) = cosh ( n arcosh x ) T_{n}(x)=\cos(n\arccos x)=\cosh(n\,\operatorname{arcosh}\,x)\,\!
  13. T n ( cos ( ϑ ) ) = cos ( n ϑ ) T_{n}(\cos(\vartheta))=\cos(n\vartheta)\,\!
  14. U n ( cos ( ϑ ) ) = sin ( ( n + 1 ) ϑ ) sin ϑ , U_{n}(\cos(\vartheta))=\frac{\sin((n+1)\vartheta)}{\sin\vartheta}\,,
  15. D n ( x ) D_{n}(x)\,\!
  16. D n ( x ) = sin ( ( 2 n + 1 ) ( x / 2 ) ) sin ( x / 2 ) = U 2 n ( cos ( x / 2 ) ) . D_{n}(x)=\frac{\sin((2n+1)(x/2))}{\sin(x/2)}=U_{2n}(\cos(x/2))\,.
  17. T 0 ( x ) = cos ( 0 x ) = 1 T_{0}(x)=\cos(0x)=1
  18. T 1 ( cos ( x ) ) = cos ( x ) , T_{1}(\cos(x))=\cos(x)\,,
  19. cos ( 2 ϑ ) = 2 cos ϑ cos ϑ - cos ( 0 ϑ ) = 2 cos 2 ϑ - 1 \cos(2\vartheta)=2\cos\vartheta\cos\vartheta-\cos(0\vartheta)=2\cos^{2}\,% \vartheta-1\,\!
  20. cos ( 3 ϑ ) = 2 cos ϑ cos ( 2 ϑ ) - cos ϑ = 4 cos 3 ϑ - 3 cos ϑ , \cos(3\vartheta)=2\cos\vartheta\cos(2\vartheta)-\cos\vartheta=4\cos^{3}\,% \vartheta-3\cos\vartheta\,,
  21. T n ( T m ( x ) ) = T n m ( x ) ; T_{n}(T_{m}(x))=T_{nm}(x)\,;
  22. z n = | z | n ( cos ( n arccos a | z | ) + i sin ( n arccos a | z | ) ) = | z | n T n ( a | z | ) + i b | z | n - 1 U n - 1 ( a | z | ) . \begin{aligned}\displaystyle z^{n}&\displaystyle=|z|^{n}\left(\cos\left(n% \arccos\frac{a}{|z|}\right)+i\sin\left(n\arccos\frac{a}{|z|}\right)\right)\\ &\displaystyle=|z|^{n}T_{n}\left(\frac{a}{|z|}\right)+ib\ |z|^{n-1}\ U_{n-1}% \left(\frac{a}{|z|}\right).\end{aligned}
  23. T n ( x ) 2 - ( x 2 - 1 ) U n - 1 ( x ) 2 = 1 T_{n}(x)^{2}-(x^{2}-1)U_{n-1}(x)^{2}=1\,\!
  24. T n ( x ) + U n - 1 ( x ) x 2 - 1 = ( x + x 2 - 1 ) n . T_{n}(x)+U_{n-1}(x)\sqrt{x^{2}-1}=(x+\sqrt{x^{2}-1})^{n}.\,\!
  25. 2 T m ( x ) T n ( x ) = T m + n ( x ) + T | m - n | ( x ) 2T_{m}(x)T_{n}(x)=T_{m+n}(x)+T_{|m-n|}(x)
  26. 2 cos α cos β = cos ( α + β ) + cos ( α - β ) 2\cos\alpha\cos\beta=\cos(\alpha+\beta)+\cos(\alpha-\beta)
  27. α = m arccos x , β = n arccos x . \alpha=m\arccos x,\beta=n\arccos x.
  28. T 2 n ( x ) = 2 T n 2 ( x ) - T 0 ( x ) = 2 T n 2 ( x ) - 1 T_{2n}(x)=2T_{n}^{2}(x)-T_{0}(x)=2T_{n}^{2}(x)-1
  29. T 2 n + 1 ( x ) = 2 T n + 1 ( x ) T n ( x ) - T 1 ( x ) = 2 T n + 1 ( x ) T n ( x ) - x T_{2n+1}(x)=2T_{n+1}(x)T_{n}(x)-T_{1}(x)=2T_{n+1}(x)T_{n}(x)-x
  30. T 2 n - 1 ( x ) = 2 T n - 1 ( x ) T n ( x ) - T 1 ( x ) = 2 T n - 1 ( x ) T n ( x ) - x T_{2n-1}(x)=2T_{n-1}(x)T_{n}(x)-T_{1}(x)=2T_{n-1}(x)T_{n}(x)-x
  31. U m ( x ) U n ( x ) = k = 0 n U m - n + 2 k ( x ) = p = m - n , s t e p 2 m + n U p ( x ) . U_{m}(x)U_{n}(x)=\sum_{k=0}^{n}U_{m-n+2k}(x)=\sum_{p=m-n,step2}^{m+n}U_{p}(x).
  32. U m + 2 ( x ) = U 2 ( x ) U m ( x ) - U m ( x ) - U m - 2 ( x ) = U m ( x ) ( U 2 ( x ) - 1 ) - U m - 2 ( x ) , U_{m+2}(x)=U_{2}(x)U_{m}(x)-U_{m}(x)-U_{m-2}(x)=U_{m}(x)(U_{2}(x)-1)-U_{m-2}(x),
  33. d d x T n ( x ) = n U n - 1 ( x ) , n = 1 , \tfrac{d}{dx}\,T_{n}(x)=nU_{n-1}(x)\mbox{ , }~{}n=1,\ldots
  34. T n ( x ) = 1 2 ( U n ( x ) - U n - 2 ( x ) ) . T_{n}(x)=\tfrac{1}{2}(U_{n}(x)-\,U_{n-2}(x)).
  35. T n + 1 ( x ) = x T n ( x ) - ( 1 - x 2 ) U n - 1 ( x ) T_{n+1}(x)=xT_{n}(x)-(1-x^{2})U_{n-1}(x)\,
  36. T n ( x ) = U n ( x ) - x U n - 1 ( x ) , T_{n}(x)=U_{n}(x)-x\,U_{n-1}(x),
  37. U n ( x ) = 2 j odd n T j ( x ) U_{n}(x)=2\sum_{j\,\,\,\text{odd}}^{n}T_{j}(x)
  38. U n ( x ) = 2 j even n T j ( x ) - 1 U_{n}(x)=2\sum_{j\,\,\text{even}}^{n}T_{j}(x)-1
  39. 2 T n ( x ) = 1 n + 1 d d x T n + 1 ( x ) - 1 n - 1 d d x T n - 1 ( x ) , n = 2 , 3 , 2T_{n}(x)=\frac{1}{n+1}\;\frac{d}{dx}T_{n+1}(x)-\frac{1}{n-1}\;\frac{d}{dx}T_{% n-1}(x)\mbox{ , }~{}\quad n=2,3,\ldots
  40. T 0 ( x ) = 1 T_{0}(x)=1\,\!
  41. U - 1 ( x ) = 0 U_{-1}(x)=0\,\!
  42. T n + 1 ( x ) = x T n ( x ) - ( 1 - x 2 ) U n - 1 ( x ) T_{n+1}(x)=xT_{n}(x)-(1-x^{2})U_{n-1}(x)\,
  43. U n ( x ) = x U n - 1 ( x ) + T n ( x ) U_{n}(x)=xU_{n-1}(x)+T_{n}(x)\,
  44. x = cos ϑ x=\cos\vartheta
  45. T n + 1 ( x ) = T n + 1 ( cos ( ϑ ) ) = cos ( ( n + 1 ) ϑ ) = cos ( n ϑ ) cos ( ϑ ) - sin ( n ϑ ) sin ( ϑ ) = T n ( cos ( ϑ ) ) cos ( ϑ ) - U n - 1 ( cos ( ϑ ) ) sin 2 ( ϑ ) = x T n ( x ) - ( 1 - x 2 ) U n - 1 ( x ) . \begin{aligned}\displaystyle T_{n+1}(x)&\displaystyle=T_{n+1}(\cos(\vartheta))% \\ &\displaystyle=\cos((n+1)\vartheta)\\ &\displaystyle=\cos(n\vartheta)\cos(\vartheta)-\sin(n\vartheta)\sin(\vartheta)% \\ &\displaystyle=T_{n}(\cos(\vartheta))\cos(\vartheta)-U_{n-1}(\cos(\vartheta))% \sin^{2}(\vartheta)\\ &\displaystyle=xT_{n}(x)-(1-x^{2})U_{n-1}(x).\\ \end{aligned}
  46. T n ( x ) 2 - T n - 1 ( x ) T n + 1 ( x ) = 1 - x 2 > 0 for - 1 < x < 1 T_{n}(x)^{2}-T_{n-1}(x)T_{n+1}(x)=1-x^{2}>0\,\text{ for }-1<x<1\!
  47. U n ( x ) 2 - U n - 1 ( x ) U n + 1 ( x ) = 1 > 0. U_{n}(x)^{2}-U_{n-1}(x)U_{n+1}(x)=1>0.\!
  48. - 1 1 T n ( y ) d y ( y - x ) 1 - y 2 = π U n - 1 ( x ) , \int_{-1}^{1}\frac{T_{n}(y)dy}{(y-x)\sqrt{1-y^{2}}}=\pi U_{n-1}(x),
  49. - 1 1 1 - y 2 U n - 1 ( y ) d y y - x = - π T n ( x ) \int_{-1}^{1}\frac{\sqrt{1-y^{2}}U_{n-1}(y)dy}{y-x}=-\pi T_{n}(x)
  50. T n ( x ) = { cos ( n arccos ( x ) ) | x | 1 cosh ( n arcosh ( x ) ) x 1 ( - 1 ) n cosh ( n arcosh ( - x ) ) x - 1 T_{n}(x)=\begin{cases}\cos(n\arccos(x))&\ |x|\leq 1\\ \cosh(n\,\operatorname{arcosh}(x))&\ x\geq 1\\ (-1)^{n}\cosh(n\,\operatorname{arcosh}(-x))&\ x\leq-1\\ \end{cases}
  51. T n ( x ) = ( x - x 2 - 1 ) n + ( x + x 2 - 1 ) n 2 = k = 0 n 2 ( n 2 k ) ( x 2 - 1 ) k x n - 2 k = x n k = 0 n 2 ( n 2 k ) ( 1 - x - 2 ) k = n 2 k = 0 n 2 ( - 1 ) k ( n - k - 1 ) ! k ! ( n - 2 k ) ! ( 2 x ) n - 2 k n > 0 = n k = 0 n ( - 2 ) k ( n + k - 1 ) ! ( n - k ) ! ( 2 k ) ! ( 1 - x ) k n > 0 = F 1 2 ( - n , n ; 1 2 ; 1 2 ( 1 - x ) ) \begin{aligned}\displaystyle T_{n}(x)&\displaystyle=\frac{\left(x-\sqrt{x^{2}-% 1}\right)^{n}+\left(x+\sqrt{x^{2}-1}\right)^{n}}{2}\\ &\displaystyle=\sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}{\left({{n}% \atop{2k}}\right)}\left(x^{2}-1\right)^{k}x^{n-2k}\\ &\displaystyle=x^{n}\sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}{\left({{% n}\atop{2k}}\right)}\left(1-x^{-2}\right)^{k}\\ &\displaystyle=\tfrac{n}{2}\sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}(-% 1)^{k}\frac{(n-k-1)!}{k!(n-2k)!}~{}(2x)^{n-2k}&&\displaystyle n>0\\ &\displaystyle=n\sum_{k=0}^{n}(-2)^{k}\frac{(n+k-1)!}{(n-k)!(2k)!}(1-x)^{k}&&% \displaystyle n>0\\ &\displaystyle={}_{2}F_{1}\left(-n,n;\tfrac{1}{2};\tfrac{1}{2}(1-x)\right)\\ \end{aligned}
  52. x n = 2 1 - n j = 0 , n - j even n ( n ( n - j ) / 2 ) T j ( x ) , x^{n}=2^{1-n}\mathop{{\sum}^{\prime}}_{j=0,n-j\mathrm{even}}^{n}{\left({{n}% \atop{(n-j)/2}}\right)}T_{j}(x),
  53. j = 0 j=0
  54. U n ( x ) = ( x + x 2 - 1 ) n + 1 - ( x - x 2 - 1 ) n + 1 2 x 2 - 1 = k = 0 n 2 ( n + 1 2 k + 1 ) ( x 2 - 1 ) k x n - 2 k = x n k = 0 n 2 ( n + 1 2 k + 1 ) ( 1 - x - 2 ) k = k = 0 n 2 ( 2 k - ( n + 1 ) k ) ( 2 x ) n - 2 k n > 0 = k = 0 n 2 ( - 1 ) k ( n - k k ) ( 2 x ) n - 2 k n > 0 = k = 0 n ( - 2 ) k ( n + k + 1 ) ! ( n - k ) ! ( 2 k + 1 ) ! ( 1 - x ) k n > 0 = ( n + 1 ) F 1 2 ( - n , n + 2 ; 3 2 ; 1 2 ( 1 - x ) ) \begin{aligned}\displaystyle U_{n}(x)&\displaystyle=\frac{\left(x+\sqrt{x^{2}-% 1}\right)^{n+1}-\left(x-\sqrt{x^{2}-1}\right)^{n+1}}{2\sqrt{x^{2}-1}}\\ &\displaystyle=\sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}{\left({{n+1}% \atop{2k+1}}\right)}\left(x^{2}-1\right)^{k}x^{n-2k}\\ &\displaystyle=x^{n}\sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}{\left({{% n+1}\atop{2k+1}}\right)}\left(1-x^{-2}\right)^{k}\\ &\displaystyle=\sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}{\left({{2k-(n% +1)}\atop{k}}\right)}~{}(2x)^{n-2k}&&\displaystyle n>0\\ &\displaystyle=\sum_{k=0}^{\left\lfloor\frac{n}{2}\right\rfloor}(-1)^{k}{\left% ({{n-k}\atop{k}}\right)}~{}(2x)^{n-2k}&&\displaystyle n>0\\ &\displaystyle=\sum_{k=0}^{n}(-2)^{k}\frac{(n+k+1)!}{(n-k)!(2k+1)!}(1-x)^{k}&&% \displaystyle n>0\\ &\displaystyle=(n+1)\ {}_{2}F_{1}\left(-n,n+2;\tfrac{3}{2};\tfrac{1}{2}(1-x)% \right)\\ \end{aligned}
  55. cos ( ( 2 k + 1 ) π 2 ) = 0 \cos\left((2k+1)\,\tfrac{\pi}{2}\right)=0
  56. x k = cos ( 2 k - 1 2 n π ) , k = 1 , , n . x_{k}=\cos\left(\frac{2k-1}{2n}\pi\right),\quad k=1,\ldots,n.
  57. x k = cos ( k n + 1 π ) , k = 1 , , n . x_{k}=\cos\left(\frac{k}{n+1}\pi\right),\quad k=1,\ldots,n.
  58. x k = cos ( k n π ) , k = 0 , , n . x_{k}=\cos\left(\tfrac{k}{n}\pi\right),\quad k=0,\ldots,n.
  59. T n ( 1 ) = 1 T_{n}(1)=1\,
  60. T n ( - 1 ) = ( - 1 ) n T_{n}(-1)=(-1)^{n}\,
  61. U n ( 1 ) = n + 1 U_{n}(1)=n+1\,
  62. U n ( - 1 ) = ( n + 1 ) ( - 1 ) n . U_{n}(-1)=(n+1)(-1)^{n}.\,
  63. d T n d x = n U n - 1 \frac{dT_{n}}{dx}=nU_{n-1}\,
  64. d U n d x = ( n + 1 ) T n + 1 - x U n x 2 - 1 \frac{dU_{n}}{dx}=\frac{(n+1)T_{n+1}-xU_{n}}{x^{2}-1}\,
  65. d 2 T n d x 2 = n n T n - x U n - 1 x 2 - 1 = n ( n + 1 ) T n - U n x 2 - 1 . \frac{d^{2}T_{n}}{dx^{2}}=n\frac{nT_{n}-xU_{n-1}}{x^{2}-1}=n\frac{(n+1)T_{n}-U% _{n}}{x^{2}-1}.\,
  66. d 2 T n d x 2 | x = 1 = n 4 - n 2 3 , \frac{d^{2}T_{n}}{dx^{2}}\Bigg|_{x=1}\!\!=\frac{n^{4}-n^{2}}{3},
  67. d 2 T n d x 2 | x = - 1 = ( - 1 ) n n 4 - n 2 3 . \frac{d^{2}T_{n}}{dx^{2}}\Bigg|_{x=-1}\!\!=(-1)^{n}\frac{n^{4}-n^{2}}{3}.
  68. T n ′′ = n n T n - x U n - 1 x 2 - 1 T^{\prime\prime}_{n}=n\frac{nT_{n}-xU_{n-1}}{x^{2}-1}
  69. T n ′′ ( 1 ) = lim x 1 n n T n - x U n - 1 x 2 - 1 T^{\prime\prime}_{n}(1)=\lim_{x\to 1}n\frac{nT_{n}-xU_{n-1}}{x^{2}-1}
  70. x = 1 x=1
  71. T n ′′ ( 1 ) = lim x 1 n n T n - x U n - 1 ( x + 1 ) ( x - 1 ) = lim x 1 n n T n - x U n - 1 x - 1 x + 1 . T^{\prime\prime}_{n}(1)=\lim_{x\to 1}n\frac{nT_{n}-xU_{n-1}}{(x+1)(x-1)}=\lim_% {x\to 1}n\frac{\frac{nT_{n}-xU_{n-1}}{x-1}}{x+1}.
  72. T n ′′ ( 1 ) = n lim x 1 n T n - x U n - 1 x - 1 lim x 1 ( x + 1 ) = n 2 lim x 1 n T n - x U n - 1 x - 1 . T^{\prime\prime}_{n}(1)=n\frac{\lim_{x\to 1}\frac{nT_{n}-xU_{n-1}}{x-1}}{\lim_% {x\to 1}(x+1)}=\frac{n}{2}\lim_{x\to 1}\frac{nT_{n}-xU_{n-1}}{x-1}.
  73. U n - 1 ( 1 ) = n T n ( 1 ) = n U_{n-1}(1)=nT_{n}(1)=n
  74. T n ′′ ( 1 ) \displaystyle T^{\prime\prime}_{n}(1)
  75. x = - 1 x=-1
  76. T n ( - 1 ) = ( - 1 ) n T_{n}(-1)=(-1)^{n}
  77. d p T n d x p | x = ± 1 = ( ± 1 ) n + p k = 0 p - 1 n 2 - k 2 2 k + 1 . \frac{d^{p}T_{n}}{dx^{p}}\Bigg|_{x=\pm 1}\!\!=(\pm 1)^{n+p}\prod_{k=0}^{p-1}% \frac{n^{2}-k^{2}}{2k+1}.
  78. d p d x p T n ( x ) = 2 p n k 0 , n - p - k even n - p ( ( n + p - k ) / 2 - 1 ( n - p - k ) / 2 ) [ ( n + p + k ) / 2 - 1 ] ! [ ( n - p + k ) / 2 ] ! T k ( x ) , p 1 , \frac{d^{p}}{dx^{p}}T_{n}(x)=2^{p}n\mathop{{\sum}^{\prime}}_{k\geq 0,n-p-k% \mathrm{even}}^{n-p}{\left({{(n+p-k)/2-1}\atop{(n-p-k)/2}}\right)}\frac{[(n+p+% k)/2-1]!}{[(n-p+k)/2]!}T_{k}(x),\quad p\geq 1,
  79. k = 0 k=0
  80. U n d x = T n + 1 n + 1 \int U_{n}\,dx=\frac{T_{n+1}}{n+1}\,
  81. T n d x = 1 2 ( T n + 1 n + 1 - T n - 1 n - 1 ) = n T n + 1 n 2 - 1 - x T n n - 1 . \int T_{n}\,dx=\frac{1}{2}\left(\frac{T_{n+1}}{n+1}-\frac{T_{n-1}}{n-1}\right)% =\frac{nT_{n+1}}{n^{2}-1}-\frac{xT_{n}}{n-1}.\,
  82. 1 1 - x 2 , \frac{1}{\sqrt{1-x^{2}}},\,\!
  83. - 1 1 T n ( x ) T m ( x ) d x 1 - x 2 = { 0 : n m π : n = m = 0 π / 2 : n = m 0 \int_{-1}^{1}T_{n}(x)T_{m}(x)\,\frac{dx}{\sqrt{1-x^{2}}}=\begin{cases}0&:n\neq m% \\ \pi&:n=m=0\\ \pi/2&:n=m\neq 0\end{cases}
  84. x = c o s ( θ ) x=cos(θ)
  85. 1 - x 2 \sqrt{1-x^{2}}\,\!
  86. - 1 1 U n ( x ) U m ( x ) 1 - x 2 d x = { 0 : n m , π / 2 : n = m . \int_{-1}^{1}U_{n}(x)U_{m}(x)\sqrt{1-x^{2}}\,dx=\begin{cases}0&:n\neq m,\\ \pi/2&:n=m.\end{cases}
  87. 1 - x 2 d x \sqrt{1-x^{2}}\,dx
  88. k = 0 N - 1 T i ( x k ) T j ( x k ) = { 0 : i j N : i = j = 0 N / 2 : i = j 0 \sum_{k=0}^{N-1}{T_{i}(x_{k})T_{j}(x_{k})}=\begin{cases}0&:i\neq j\\ N&:i=j=0\\ N/2&:i=j\neq 0\end{cases}\,\!
  89. x k = cos ( π 2 k + 1 2 N ) . x_{k}=\cos\left(\pi\frac{2k+1}{2N}\right).
  90. k = 0 N - 1 U i ( x k ) U j ( x k ) ( 1 - x k 2 ) = { 0 : i j N / 2 : i = j \sum_{k=0}^{N-1}{U_{i}(x_{k})U_{j}(x_{k})(1-x_{k}^{2})}=\begin{cases}0&:i\neq j% \\ N/2&:i=j\end{cases}\,\!
  91. k = 0 N - 1 U i ( x k ) U j ( x k ) = { 0 : p a r i t y ( i ) p a r i t y ( j ) N + N * m i n ( i , j ) : p a r i t y ( i ) = p a r i t y ( j ) \sum_{k=0}^{N-1}{U_{i}(x_{k})U_{j}(x_{k})}=\begin{cases}0&:parity(i)\neq parity% (j)\\ N+N*min(i,j)&:parity(i)=parity(j)\\ \end{cases}\,\!
  92. x k = cos ( π k + 1 N + 1 ) x_{k}=\cos\left(\pi\frac{k+1}{N+1}\right)
  93. k = 0 N - 1 U i ( x k ) U j ( x k ) ( 1 - x k 2 ) = { 0 : i j N + 1 2 : i = j \sum_{k=0}^{N-1}{U_{i}(x_{k})U_{j}(x_{k})(1-x_{k}^{2})}=\begin{cases}0&:i\neq j% \\ \frac{N+1}{2}&:i=j\end{cases}\,\!
  94. k = 0 N - 1 U i ( x k ) U j ( x k ) = { 0 : p a r i t y ( i ) p a r i t y ( j ) ( m i n ( i , j ) + 1 ) ( N - m a x ( i , j ) ) : p a r i t y ( i ) = p a r i t y ( j ) \sum_{k=0}^{N-1}{U_{i}(x_{k})U_{j}(x_{k})}=\begin{cases}0&:parity(i)\neq parity% (j)\\ (min(i,j)+1)(N-max(i,j))&:parity(i)=parity(j)\\ \end{cases}\,\!
  95. f ( x ) = 1 2 n - 1 T n ( x ) f(x)=\frac{1}{2^{n-1}}T_{n}(x)
  96. 1 2 n - 1 \frac{1}{2^{n-1}}
  97. x = cos k π n for 0 k n . x=\cos\frac{k\pi}{n}\,\text{ for }0\leq k\leq n.
  98. w n ( x ) w_{n}(x)
  99. 1 2 n - 1 \frac{1}{2^{n-1}}
  100. f n ( x ) = 1 2 n - 1 T n ( x ) - w n ( x ) f_{n}(x)=\frac{1}{2^{n-1}}T_{n}(x)-w_{n}(x)
  101. T n T_{n}
  102. | w n ( x ) | < | 1 2 n - 1 T n ( x ) | |w_{n}(x)|<\left|\frac{1}{2^{n-1}}T_{n}(x)\right|
  103. f n ( x ) > 0 for x = cos 2 k π n where 0 2 k n f_{n}(x)>0\,\text{ for }x=\cos\frac{2k\pi}{n}\,\text{ where }0\leq 2k\leq n
  104. f n ( x ) < 0 for x = cos ( 2 k + 1 ) π n where 0 2 k + 1 n f_{n}(x)<0\,\text{ for }x=\cos\frac{(2k+1)\pi}{n}\,\text{ where }0\leq 2k+1\leq n
  105. f n ( x ) f_{n}(x)
  106. f n ( x ) f_{n}(x)
  107. T n ( x ) = 1 < m t p l > ( n - 1 2 n ) T_{n}(x)=\frac{1}{<}mtpl>{{n-\frac{1}{2}\choose n}}
  108. U n ( x ) = 1 2 ( n + 1 2 n ) P n 1 2 , 1 2 ( x ) = C n 1 ( x ) . U_{n}(x)=\frac{1}{2{n+\frac{1}{2}\choose n}}P_{n}^{\frac{1}{2},\frac{1}{2}}(x)% =C_{n}^{1}(x).
  109. T 2 n ( x ) = T n ( 2 x 2 - 1 ) = 2 T n ( x ) 2 - 1 T_{2n}(x)=T_{n}\left(2x^{2}-1\right)=2T_{n}(x)^{2}-1
  110. 2 x U n ( 1 - 2 x 2 ) = ( - 1 ) n U 2 n + 1 ( x ) . 2xU_{n}\left(1-2x^{2}\right)=(-1)^{n}U_{2n+1}(x).
  111. T j ( x ) T k ( x ) = 1 2 ( T j + k ( x ) + T | k - j | ( x ) ) , j , k 0 , T_{j}(x)T_{k}(x)=\tfrac{1}{2}\left(T_{j+k}(x)+T_{|k-j|}(x)\right),\quad\forall j% ,k\geq 0,\,
  112. T j ( x ) U k ( x ) = { 1 2 ( U j + k ( x ) + U k - j ( x ) ) , if k j - 1. 1 2 ( U j + k ( x ) - U j - k - 2 ( x ) ) , if k j - 2. T_{j}(x)U_{k}(x)=\begin{cases}\tfrac{1}{2}\left(U_{j+k}(x)+U_{k-j}(x)\right),&% \,\text{if }k\geq j-1.\\ \tfrac{1}{2}\left(U_{j+k}(x)-U_{j-k-2}(x)\right),&\,\text{if }k\leq j-2.\end{cases}
  113. U - 1 0 U_{-1}\equiv 0
  114. T n ( cos θ ) = cos ( n θ ) , T_{n}\left(\cos\theta\right)=\cos(n\theta),
  115. T 2 n + 1 ( sin θ ) = ( - 1 ) n sin ( ( 2 n + 1 ) θ ) T_{2n+1}\left(\sin\theta\right)=(-1)^{n}\sin((2n+1)\theta)
  116. x 0 x\neq 0
  117. T n ( x + x - 1 2 ) = x n + x - n 2 T_{n}\left(\dfrac{x+x^{-1}}{2}\right)=\dfrac{x^{n}+x^{-n}}{2}
  118. x n = T n ( x + x - 1 2 ) + x - x - 1 2 U n - 1 ( x + x - 1 2 ) x^{n}=T_{n}\left(\dfrac{x+x^{-1}}{2}\right)+\dfrac{x-x^{-1}}{2}U_{n-1}\left(% \dfrac{x+x^{-1}}{2}\right)
  119. x = e i θ x=e^{i\theta}
  120. C n ( x ) = 2 T n ( x 2 ) C_{n}(x)=2T_{n}\left(\frac{x}{2}\right)
  121. C n ( x ) C_{n}(x)
  122. C m ( x ) C_{m}(x)
  123. C n ( C m ( x ) ) = C m ( C n ( x ) ) C_{n}\left(C_{m}(x)\right)=C_{m}(C_{n}(x))
  124. T a T_{a}
  125. T a ( cos x ) = F 1 2 ( a , - a ; 1 2 ; 1 2 ( 1 - cos x ) ) = cos ( a x ) , T_{a}(\cos x)={}_{2}F_{1}\left(a,-a;\tfrac{1}{2};\tfrac{1}{2}(1-\cos x)\right)% =\cos(ax),
  126. a a
  127. T a ( x ) = cos ( π a 2 ) + a j = 1 ( 2 x ) j 2 j cos ( π ( a - j ) 2 ) ( a + j - 2 2 j - 1 ) . T_{a}(x)=\cos\left(\frac{\pi a}{2}\right)+a\sum_{j=1}\frac{(2x)^{j}}{2j}\cos% \left(\frac{\pi(a-j)}{2}\right){\frac{a+j-2}{2}\choose j-1}.
  128. U 0 ( x ) = 1 U_{0}(x)=1\,
  129. U 1 ( x ) = 2 x U_{1}(x)=2x\,
  130. U 2 ( x ) = 4 x 2 - 1 U_{2}(x)=4x^{2}-1\,
  131. U 3 ( x ) = 8 x 3 - 4 x U_{3}(x)=8x^{3}-4x\,
  132. U 4 ( x ) = 16 x 4 - 12 x 2 + 1 U_{4}(x)=16x^{4}-12x^{2}+1\,
  133. U 5 ( x ) = 32 x 5 - 32 x 3 + 6 x U_{5}(x)=32x^{5}-32x^{3}+6x\,
  134. U 6 ( x ) = 64 x 6 - 80 x 4 + 24 x 2 - 1 U_{6}(x)=64x^{6}-80x^{4}+24x^{2}-1\,
  135. U 7 ( x ) = 128 x 7 - 192 x 5 + 80 x 3 - 8 x U_{7}(x)=128x^{7}-192x^{5}+80x^{3}-8x\,
  136. U 8 ( x ) = 256 x 8 - 448 x 6 + 240 x 4 - 40 x 2 + 1 U_{8}(x)=256x^{8}-448x^{6}+240x^{4}-40x^{2}+1\,
  137. U 9 ( x ) = 512 x 9 - 1024 x 7 + 672 x 5 - 160 x 3 + 10 x . U_{9}(x)=512x^{9}-1024x^{7}+672x^{5}-160x^{3}+10x.\,
  138. f ( x ) = n = 0 a n T n ( x ) . f(x)=\sum_{n=0}^{\infty}a_{n}T_{n}(x).
  139. log ( 1 + x ) \log(1+x)
  140. log ( 1 + x ) = n = 0 a n T n ( x ) . \log(1+x)=\sum_{n=0}^{\infty}a_{n}T_{n}(x).
  141. a n a_{n}
  142. - 1 + 1 T m ( x ) log ( 1 + x ) 1 - x 2 d x = n = 0 a n - 1 + 1 T m ( x ) T n ( x ) 1 - x 2 d x , \int_{-1}^{+1}\frac{T_{m}(x)\log(1+x)}{\sqrt{1-x^{2}}}dx=\sum_{n=0}^{\infty}a_% {n}\int_{-1}^{+1}\frac{T_{m}(x)T_{n}(x)}{\sqrt{1-x^{2}}}dx,
  143. a n = { - log ( 2 ) : n = 0 - π ( - 1 ) n n : n > 0. a_{n}=\begin{cases}-\log(2)&:n=0\\ \frac{-\pi(-1)^{n}}{n}&:n>0.\end{cases}
  144. a n = 2 - δ 0 n N k = 0 N - 1 T n ( x k ) log ( 1 + x k ) , a_{n}=\frac{2-\delta_{0n}}{N}\sum_{k=0}^{N-1}T_{n}(x_{k})\log(1+x_{k}),
  145. δ i j \delta_{ij}
  146. x k x_{k}
  147. T N ( x ) T_{N}(x)
  148. x k = cos ( π ( k + 1 2 ) N ) . x_{k}=\cos\left(\frac{\pi\left(k+\frac{1}{2}\right)}{N}\right).
  149. a n a_{n}
  150. a n = 2 - δ 0 n N k = 0 N - 1 cos ( n π ( k + 1 2 ) N ) log ( 1 + x k ) . a_{n}=\frac{2-\delta_{0n}}{N}\sum_{k=0}^{N-1}\cos\left(\frac{n\pi\left(k+\frac% {1}{2}\right)}{N}\right)\log(1+x_{k}).
  151. ( 1 - x 2 ) α = \displaystyle(1-x^{2})^{\alpha}=
  152. f ( x ) = n = 0 a n T n ( x ) f(x)=\sum_{n=0}^{\infty}a_{n}T_{n}(x)
  153. x k = - cos ( k π N - 1 ) ; k = 0 , 1 , , N - 1. x_{k}=-\cos\left(\frac{k\pi}{N-1}\right);\qquad\ k=0,1,\dots,N-1.
  154. p ( x ) = n = 0 N a n T n ( x ) . p(x)=\sum_{n=0}^{N}a_{n}T_{n}(x).

Chemical_equation.html

  1. \rightarrow
  2. = =
  3. \rightarrow
  4. \rightleftarrows
  5. \rightleftharpoons
  6. Δ \Delta
  7. h ν h\nu

Chemical_potential.html

  1. d G = - S d T + V d P + i = 1 n μ i d N i dG=-SdT+VdP+\sum_{i=1}^{n}\mu_{i}dN_{i}\,
  2. d G = i = 1 n μ i d N i = μ 1 d N 1 + μ 2 d N 2 + dG=\sum_{i=1}^{n}\mu_{i}dN_{i}=\mu_{1}dN_{1}+\mu_{2}dN_{2}+...\,
  3. μ i = ( G N i ) T , P , N j i \mu_{i}=\left(\frac{\partial G}{\partial N_{i}}\right)_{T,P,N_{j\neq i}}
  4. μ i = ( A N i ) T , V , N j i \mu_{i}=\left(\frac{\partial A}{\partial N_{i}}\right)_{T,V,N_{j\neq i}}
  5. μ 1 d N 1 + μ 2 d N 2 + = 0 \mu_{1}dN_{1}+\mu_{2}dN_{2}+...=0\,
  6. μ i = ( H N i ) S , P , N j i or μ i = ( U N i ) S , V , N j i \mu_{i}=\left(\frac{\partial H}{\partial N_{i}}\right)_{S,P,N_{j\neq i}}\,% \text{or}\ \mu_{i}=\left(\frac{\partial U}{\partial N_{i}}\right)_{S,V,N_{j% \neq i}}
  7. μ i = μ i std + R T ln a i , \mu_{i}=\mu_{i}^{\,\text{std}}+RT\ln a_{i},
  8. μ i = μ i std + R T ln f i , \mu_{i}=\mu_{i}^{\,\text{std}}+RT\ln f_{i},
  9. ( μ T ) p , n = - ( S n ) T , p \left(\frac{\partial\mu}{\partial T}\right)_{p,n}=-\left(\frac{\partial S}{% \partial n}\right)_{T,p}
  10. ( μ p ) T , n = ( V n ) T , p \left(\frac{\partial\mu}{\partial p}\right)_{T,n}=\left(\frac{\partial V}{% \partial n}\right)_{T,p}
  11. d μ B = - n A n B d μ A d\mu_{B}=-\frac{n_{A}}{n_{B}}d\mu_{A}
  12. μ tot = μ int + μ ext \mu_{\mathrm{tot}}=\mu_{\mathrm{int}}+\mu_{\mathrm{ext}}\,
  13. μ ext = q V + m g h + \mu_{\mathrm{ext}}=qV+mgh+\cdots
  14. μ Mulliken = - χ Mulliken = - I P + E A 2 = [ δ E [ N ] δ N ] N = N 0 \mu_{\mathrm{Mulliken}}=-\chi_{\mathrm{Mulliken}}=-\frac{IP+EA}{2}=\left[\frac% {\delta E[N]}{\delta N}\right]_{N=N_{0}}

Chemiluminescence.html

  1. Δ H r x n \Delta H_{rxn}

Chern_class.html

  1. c 1 ( L L ) = c 1 ( L ) + c 1 ( L ) c_{1}(L\otimes L^{\prime})=c_{1}(L)+c_{1}(L^{\prime})
  2. det ( i t Ω 2 π + I ) = k c k ( V ) t k \det\left(\frac{it\Omega}{2\pi}+I\right)=\sum_{k}c_{k}(V)t^{k}
  3. Ω = d ω + 1 2 [ ω , ω ] \Omega=d\omega+\tfrac{1}{2}[\omega,\omega]
  4. k c k ( V ) t k = [ I + i tr ( Ω ) 2 π t + tr ( Ω 2 ) - tr ( Ω ) 2 8 π 2 t 2 + i - 2 t r ( Ω 3 ) + 3 t r ( Ω 2 ) tr ( Ω ) - tr ( Ω ) 3 48 π 3 t 3 + ] . \sum_{k}c_{k}(V)t^{k}=\left[I+i\frac{\mathrm{tr}(\Omega)}{2\pi}t+\frac{\mathrm% {tr}(\Omega^{2})-\mathrm{tr}(\Omega)^{2}}{8\pi^{2}}t^{2}+i\frac{-2\mathrm{tr}(% \Omega^{3})+3\mathrm{tr}(\Omega^{2})\mathrm{tr}(\Omega)-\mathrm{tr}(\Omega)^{3% }}{48\pi^{3}}t^{3}+\cdots\right].
  5. G L n ( ) GL_{n}(\mathbb{C})
  6. B = E - B B^{\prime}=E-B
  7. E B E^{\prime}\to B^{\prime}
  8. π | B : B B \pi|_{B^{\prime}}:B^{\prime}\to B
  9. H k ( B ; ) π | B * H k ( B ; ) , \cdots\to\operatorname{H}^{k}(B;\mathbb{Z})\overset{\pi|_{B^{\prime}}^{*}}{\to% }\operatorname{H}^{k}(B^{\prime};\mathbb{Z})\to\cdots,
  10. π | B * \pi|_{B^{\prime}}^{*}
  11. c 1 ( 𝐂𝐏 1 × 𝐂 ) = 0. c_{1}({\mathbf{C}\mathbf{P}}^{1}\times{\mathbf{C}})=0.
  12. c 1 ( V ) 0. c_{1}(V)\not=0.
  13. h = d z d z ¯ ( 1 + | z | 2 ) 2 . h=\frac{dzd\bar{z}}{(1+|z|^{2})^{2}}.
  14. Ω = 2 d z d z ¯ ( 1 + | z | 2 ) 2 . \Omega=\frac{2dz\wedge d\bar{z}}{(1+|z|^{2})^{2}}.
  15. c 1 = [ i 2 π tr Ω ] . c_{1}=\left[\frac{i}{2\pi}\mathrm{tr}\ \Omega\right].
  16. c 1 d z d z ¯ = i π d z d z ¯ ( 1 + | z | 2 ) 2 = 2 \int c_{1}dz\wedge d\bar{z}=\frac{i}{\pi}\int\frac{dz\wedge d\bar{z}}{(1+|z|^{% 2})^{2}}=2
  17. 0 𝒪 𝐏 n 𝒪 𝐏 n ( 1 ) ( n + 1 ) T 𝐏 n 0 0\to\mathcal{O}_{\mathbb{C}\mathbf{P}^{n}}\to\mathcal{O}_{\mathbb{C}\mathbf{P}% ^{n}}(1)^{\oplus(n+1)}\to T\mathbb{C}\mathbf{P}^{n}\to 0
  18. 𝒪 𝐏 n \mathcal{O}_{\mathbb{C}\mathbf{P}^{n}}
  19. 𝒪 𝐏 n ( 1 ) \mathcal{O}_{\mathbb{C}\mathbf{P}^{n}}(1)
  20. c ( 𝐏 n ) = def c ( T 𝐏 n ) = c ( 𝒪 𝐏 n ( 1 ) ) n + 1 = ( 1 + a ) n + 1 c(\mathbb{C}\mathbf{P}^{n})\overset{\mathrm{def}}{=}c(T\mathbb{C}\mathbf{P}^{n% })=c(\mathcal{O}_{\mathbb{C}\mathbf{P}^{n}}(1))^{n+1}=(1+a)^{n+1}
  21. H 2 ( 𝐏 n , ) H^{2}(\mathbb{C}\mathbf{P}^{n},\mathbb{Z})
  22. 𝒪 𝐏 n ( - 1 ) \mathcal{O}_{\mathbb{C}\mathbf{P}^{n}}(-1)
  23. c 1 ( E * ) = - c 1 ( E ) c_{1}(E^{*})=-c_{1}(E)
  24. c k ( 𝐏 n ) = ( n + 1 k ) a k . c_{k}(\mathbb{C}\mathbf{P}^{n})={\left({{n+1}\atop{k}}\right)}a^{k}.
  25. c t ( E ) = 1 + c 1 ( E ) t + + c n ( E ) t n . c_{t}(E)=1+c_{1}(E)t+\cdots+c_{n}(E)t^{n}.
  26. c t ( E ) c_{t}(E)
  27. c ( E ) = 1 + c 1 ( E ) + + c n ( E ) c(E)=1+c_{1}(E)+\cdots+c_{n}(E)
  28. c t ( E E ) = c t ( E ) c t ( E ) . c_{t}(E\oplus E^{\prime})=c_{t}(E)c_{t}(E^{\prime}).
  29. E = L 1 L n E=L_{1}\oplus...\oplus L_{n}
  30. c t ( E ) = ( 1 + a 1 ( E ) t ) ( 1 + a n ( E ) t ) c_{t}(E)=(1+a_{1}(E)t)\cdots(1+a_{n}(E)t)
  31. a i ( E ) = c 1 ( L i ) a_{i}(E)=c_{1}(L_{i})
  32. a i ( E ) a_{i}(E)
  33. c k ( E ) = σ k ( a 1 ( E ) , , a n ( E ) ) c_{k}(E)=\sigma_{k}(a_{1}(E),\cdots,a_{n}(E))
  34. c t ( E ) c_{t}(E)
  35. t 1 k + + t n k = s k ( σ 1 ( t 1 , , t n ) , , σ k ( t 1 , , t n ) ) t_{1}^{k}+\cdots+t_{n}^{k}=s_{k}(\sigma_{1}(t_{1},\cdots,t_{n}),\cdots,\sigma_% {k}(t_{1},\cdots,t_{n}))
  36. s 1 = σ 1 , s 2 = σ 1 2 - 2 σ 2 s_{1}=\sigma_{1},s_{2}=\sigma_{1}^{2}-2\sigma_{2}
  37. ch ( E ) = e a 1 ( E ) + + e a n ( E ) = s k ( c 1 ( E ) , , c n ( E ) ) / k ! \operatorname{ch}(E)=e^{a_{1}(E)}+\cdots+e^{a_{n}(E)}=\sum s_{k}(c_{1}(E),% \cdots,c_{n}(E))/k!
  38. ch ( E ) = rk + c 1 + 1 2 ( c 1 2 - 2 c 2 ) + 1 6 ( c 1 3 - 3 c 1 c 2 + 3 c 3 ) + . \operatorname{ch}(E)=\operatorname{rk}+c_{1}+\frac{1}{2}(c_{1}^{2}-2c_{2})+% \frac{1}{6}(c_{1}^{3}-3c_{1}c_{2}+3c_{3})+\dots.
  39. td ( E ) \displaystyle\operatorname{td}(E)
  40. f E : X G n f_{E}:X\to G_{n}
  41. f E * : [ σ 1 , , σ n ] H * ( X , ) . f_{E}^{*}:\mathbb{Z}[\sigma_{1},\cdots,\sigma_{n}]\to H^{*}(X,\mathbb{Z}).
  42. c k ( E ) = f E * ( σ k ) . c_{k}(E)=f_{E}^{*}(\sigma_{k}).
  43. Vect n \operatorname{Vect}_{n}^{\mathbb{C}}
  44. Vect n = [ - , G n ] \operatorname{Vect}_{n}^{\mathbb{C}}=[-,G_{n}]
  45. H * ( - , ) . H^{*}(-,\mathbb{Z}).
  46. Nat ( [ - , G n ] , H * ( - , ) ) = H * ( G n , ) = [ σ 1 , , σ n ] . \operatorname{Nat}([-,G_{n}],H^{*}(-,\mathbb{Z}))=H^{*}(G_{n},\mathbb{Z})=% \mathbb{Z}[\sigma_{1},\cdots,\sigma_{n}].
  47. c ( E ) = c 0 ( E ) + c 1 ( E ) + c 2 ( E ) + . c(E)=c_{0}(E)+c_{1}(E)+c_{2}(E)+\cdots.
  48. c 0 ( E ) = 1 c_{0}(E)=1
  49. f : Y X f:Y\to X
  50. c k ( f * E ) = f * c k ( E ) c_{k}(f^{*}E)=f^{*}c_{k}(E)
  51. F X F\to X
  52. E F E\oplus F
  53. c ( E F ) = c ( E ) c ( F ) ; c(E\oplus F)=c(E)\smile c(F);
  54. c k ( E F ) = i = 0 k c i ( E ) c k - i ( F ) . c_{k}(E\oplus F)=\sum_{i=0}^{k}c_{i}(E)\smile c_{k-i}(F).
  55. 𝐂𝐏 k - 1 𝐂𝐏 k \mathbf{CP}^{k-1}\subseteq\mathbf{CP}^{k}
  56. 0 E E E ′′ 0 \ 0\to E^{\prime}\to E\to E^{\prime\prime}\to 0
  57. c ( E ) = c ( E ) c ( E ′′ ) c(E)=c(E^{\prime})\smile c(E^{\prime\prime})
  58. c ( E ) = 1 + e ( E 𝐑 ) c(E)=1+e(E_{\mathbf{R}})
  59. e ( E 𝐑 ) e(E_{\mathbf{R}})
  60. b B b\in B
  61. c 1 ( τ ) = : - a c_{1}(\tau)=:-a
  62. 1 , a , a 2 , , a n - 1 H * ( 𝐏 ( E ) ) 1,a,a^{2},\ldots,a^{n-1}\in H^{*}(\mathbf{P}(E))
  63. c 1 ( E ) , c n ( E ) c_{1}(E),\ldots c_{n}(E)
  64. - a n -a^{n}
  65. - a n = c 1 ( E ) . a n - 1 + c n - 1 ( E ) . a + c n ( E ) . -a^{n}=c_{1}(E).a^{n-1}+\ldots c_{n-1}(E).a+c_{n}(E).
  66. c k ( V ) = 0 c_{k}(V)=0
  67. c n ( V ) c_{n}(V)
  68. ch ( L ) = exp ( c 1 ( L ) ) := m = 0 c 1 ( L ) m m ! . \operatorname{ch}(L)=\exp(c_{1}(L)):=\sum_{m=0}^{\infty}\frac{c_{1}(L)^{m}}{m!}.
  69. V = L 1 L n V=L_{1}\oplus...\oplus L_{n}
  70. x i = c 1 ( L i ) , x_{i}=c_{1}(L_{i}),
  71. ch ( V ) = e x 1 + + e x n := m = 0 1 m ! ( x 1 m + + x n m ) . \operatorname{ch}(V)=e^{x_{1}}+\dots+e^{x_{n}}:=\sum_{m=0}^{\infty}\frac{1}{m!% }(x_{1}^{m}+...+x_{n}^{m}).
  72. x i x_{i}
  73. c i ( V ) = e i ( x 1 , , x n ) . c_{i}(V)=e_{i}(x_{1},...,x_{n}).
  74. c ( V ) := i = 0 n c i ( V ) , c(V):=\sum_{i=0}^{n}c_{i}(V),
  75. c ( V ) = c ( L 1 L n ) = i = 1 n c ( L i ) = i = 1 n ( 1 + x i ) = i = 0 n e i ( x 1 , , x n ) . c(V)=c(L_{1}\oplus\dots\oplus L_{n})=\prod_{i=1}^{n}c(L_{i})=\prod_{i=1}^{n}(1% +x_{i})=\sum_{i=0}^{n}e_{i}(x_{1},\dots,x_{n}).
  76. ch ( V ) = dim ( V ) + c 1 ( V ) + 1 2 ( c 1 ( V ) 2 - 2 c 2 ( V ) ) + 1 6 ( c 1 ( V ) 3 - 3 c 1 ( V ) c 2 ( V ) + 3 c 3 ( V ) ) + . \operatorname{ch}(V)=\operatorname{dim}(V)+c_{1}(V)+\frac{1}{2}(c_{1}(V)^{2}-2% c_{2}(V))+\frac{1}{6}(c_{1}(V)^{3}-3c_{1}(V)c_{2}(V)+3c_{3}(V))+....
  77. ch ( V ) = [ tr ( exp ( i Ω 2 π ) ) ] \hbox{ch}(V)=\left[\hbox{tr}\left(\exp\left(\frac{i\Omega}{2\pi}\right)\right)\right]
  78. ch ( V W ) = ch ( V ) + ch ( W ) \hbox{ch}(V\oplus W)=\hbox{ch}(V)+\hbox{ch}(W)
  79. ch ( V W ) = ch ( V ) ch ( W ) . \hbox{ch}(V\otimes W)=\hbox{ch}(V)\hbox{ch}(W).
  80. H 2 * ( M , ) k η ( H 2 * ( M , ) ) [ t ] , x x t | x | / 2 H^{2*}(M,\mathbb{Z})\to\oplus_{k}^{\infty}\eta(H^{2*}(M,\mathbb{Z}))[t],x% \mapsto xt^{|x|/2}

Chern–Simons_form.html

  1. A {A}
  2. Tr [ A ] . {\rm Tr}[{A}].
  3. Tr [ F A - 1 3 A A A ] . {\rm Tr}\left[{F}\wedge{A}-\frac{1}{3}{A}\wedge{A}\wedge{A}\right].
  4. Tr [ F F A - 1 2 F A A A + 1 10 A A A A A ] {\rm Tr}\left[{F}\wedge{F}\wedge{A}-\frac{1}{2}{F}\wedge{A}\wedge{A}\wedge{A}+% \frac{1}{10}{A}\wedge{A}\wedge{A}\wedge{A}\wedge{A}\right]
  5. F = d A + A A . {F}=d{A}+{A}\wedge{A}.
  6. ω 2 k - 1 \omega_{2k-1}
  7. d ω 2 k - 1 = Tr ( F k ) , d\omega_{2k-1}={\rm Tr}\left(F^{k}\right),
  8. A {A}

Chi-squared_distribution.html

  1. 1 Γ ( k 2 ) γ ( k 2 , x 2 ) \frac{1}{\Gamma\left(\frac{k}{2}\right)}\;\gamma\left(\frac{k}{2},\,\frac{x}{2% }\right)
  2. k ( 1 - 2 9 k ) 3 \approx k\bigg(1-\frac{2}{9k}\bigg)^{3}
  3. 8 / k \scriptstyle\sqrt{8/k}\,
  4. k 2 + ln ( 2 Γ ( k / 2 ) ) + ( 1 - k / 2 ) ψ ( k / 2 ) \begin{aligned}\displaystyle\frac{k}{2}&\displaystyle+\ln(2\Gamma(k/2))\\ &\displaystyle\!+(1-k/2)\psi(k/2)\end{aligned}
  5. Q = i = 1 k Z i 2 , Q\ =\sum_{i=1}^{k}Z_{i}^{2},
  6. Q χ 2 ( k ) or Q χ k 2 . Q\ \sim\ \chi^{2}(k)\ \ \,\text{or}\ \ Q\ \sim\ \chi^{2}_{k}.
  7. f ( x ; k ) = { x ( k / 2 - 1 ) e - x / 2 2 k / 2 Γ ( k 2 ) , x 0 ; 0 , otherwise . f(x;\,k)=\begin{cases}\frac{x^{(k/2-1)}e^{-x/2}}{2^{k/2}\Gamma\left(\frac{k}{2% }\right)},&x\geq 0;\\ 0,&\,\text{otherwise}.\end{cases}
  8. { 2 x f ( x ) + f ( x ) ( - k + x + 2 ) = 0 , f ( 1 ) = 2 - k / 2 e Γ ( k 2 ) } \left\{\begin{array}[]{l}2xf^{\prime}(x)+f(x)(-k+x+2)=0,\\ f(1)=\frac{2^{-k/2}}{\sqrt{e}\Gamma\left(\frac{k}{2}\right)}\end{array}\right\}
  9. F ( x ; k ) = γ ( k 2 , x 2 ) Γ ( k 2 ) = P ( k 2 , x 2 ) , F(x;\,k)=\frac{\gamma(\frac{k}{2},\,\frac{x}{2})}{\Gamma(\frac{k}{2})}=P\left(% \frac{k}{2},\,\frac{x}{2}\right),
  10. F ( x ; 2 ) = 1 - e - x 2 F(x;\,2)=1-e^{-\frac{x}{2}}
  11. z x / k z\equiv x/k
  12. 0 < z < 1 0<z<1
  13. F ( z k ; k ) ( z e 1 - z ) k / 2 . F(zk;\,k)\leq(ze^{1-z})^{k/2}.
  14. z > 1 z>1
  15. 1 - F ( z k ; k ) ( z e 1 - z ) k / 2 . 1-F(zk;\,k)\leq(ze^{1-z})^{k/2}.
  16. X ¯ = 1 n i = 1 n X i Gamma ( α = n k / 2 , θ = 2 / n ) where X i χ 2 ( k ) \bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i}\sim\mathrm{Gamma}\left(\alpha=n\,k/2,% \theta=2/n\right)\qquad\mathrm{where}\quad X_{i}\sim\chi^{2}(k)
  17. α \alpha
  18. μ = α θ \mu=\alpha\cdot\theta
  19. σ 2 = α θ 2 \sigma^{2}=\alpha\,\theta^{2}
  20. X ¯ n N ( μ = k , σ 2 = 2 k / n ) \bar{X}\xrightarrow{n\to\infty}N(\mu=k,\sigma^{2}=2\,k/n)
  21. k k
  22. k k
  23. 2 k 2\,k
  24. X ¯ \bar{X}
  25. σ 2 = 2 k / n \sigma^{2}=2\,k/n
  26. h = - f ( x ; k ) ln f ( x ; k ) d x = k 2 + ln [ 2 Γ ( k 2 ) ] + ( 1 - k 2 ) ψ [ k 2 ] , h=\int_{-\infty}^{\infty}f(x;\,k)\ln f(x;\,k)\,dx=\frac{k}{2}+\ln\!\left[2\,% \Gamma\!\left(\frac{k}{2}\right)\right]+\left(1-\frac{k}{2}\right)\,\psi\!% \left[\frac{k}{2}\right],
  27. E ( X ) = k E(X)=k
  28. E ( ln ( X ) ) = ψ ( k / 2 ) + l o g ( 2 ) E(\ln(X))=\psi\left(k/2\right)+log(2)
  29. E ( X m ) = k ( k + 2 ) ( k + 4 ) ( k + 2 m - 2 ) = 2 m Γ ( m + k 2 ) Γ ( k 2 ) . \operatorname{E}(X^{m})=k(k+2)(k+4)\cdots(k+2m-2)=2^{m}\frac{\Gamma(m+\frac{k}% {2})}{\Gamma(\frac{k}{2})}.
  30. κ n = 2 n - 1 ( n - 1 ) ! k \kappa_{n}=2^{n-1}(n-1)!\,k
  31. ( X - k ) / 2 k (X-k)/\sqrt{2k}
  32. 8 / k \sqrt{8/k}
  33. 2 X \scriptstyle\sqrt{2X}
  34. 2 k - 1 \scriptstyle\sqrt{2k-1}
  35. X / k 3 \scriptstyle\sqrt[3]{X/k}
  36. 1 - 2 / ( 9 k ) \scriptstyle 1-2/(9k)
  37. 2 / ( 9 k ) . \scriptstyle 2/(9k).
  38. k k\to\infty
  39. ( χ k 2 - k ) / 2 k 𝑑 N ( 0 , 1 ) (\chi^{2}_{k}-k)/\sqrt{2k}~{}\xrightarrow{d}\ N(0,1)\,
  40. χ k 2 χ k 2 ( 0 ) \chi_{k}^{2}\sim{\chi^{\prime}}^{2}_{k}(0)
  41. λ = 0 \lambda=0
  42. X F ( ν 1 , ν 2 ) X\sim\mathrm{F}(\nu_{1},\nu_{2})
  43. Y = lim ν 2 ν 1 X Y=\lim_{\nu_{2}\to\infty}\nu_{1}X
  44. χ ν 1 2 \chi^{2}_{\nu_{1}}
  45. X F ( 1 , ν 2 ) X\sim\mathrm{F}(1,\nu_{2})\,
  46. Y = lim ν 2 X Y=\lim_{\nu_{2}\to\infty}X\,
  47. χ 1 2 \chi^{2}_{1}
  48. s y m b o l N i = 1 , , k ( 0 , 1 ) 2 χ k 2 \|symbol{N}_{i=1,...,k}{(0,1)}\|^{2}\sim\chi^{2}_{k}
  49. X χ 2 ( ν ) X\sim{\chi}^{2}(\nu)\,
  50. c > 0 c>0\,
  51. c X Γ ( k = ν / 2 , θ = 2 c ) cX\sim{\Gamma}(k=\nu/2,\theta=2c)\,
  52. X χ k 2 X\sim\chi^{2}_{k}
  53. X χ k \sqrt{X}\sim\chi_{k}
  54. X χ 2 ( 2 ) X\sim\chi^{2}\left(2\right)
  55. X Exp ( 1 / 2 ) X\sim\mathrm{Exp(1/2)}
  56. X Rayleigh ( 1 ) X\sim\mathrm{Rayleigh}(1)\,
  57. X 2 χ 2 ( 2 ) X^{2}\sim\chi^{2}(2)\,
  58. X Maxwell ( 1 ) X\sim\mathrm{Maxwell}(1)\,
  59. X 2 χ 2 ( 3 ) X^{2}\sim\chi^{2}(3)\,
  60. X χ 2 ( ν ) X\sim\chi^{2}(\nu)
  61. 1 X Inv- χ 2 ( ν ) \tfrac{1}{X}\sim\mbox{Inv-}~{}\chi^{2}(\nu)\,
  62. X χ 2 ( ν 1 ) X\sim\chi^{2}(\nu_{1})\,
  63. Y χ 2 ( ν 2 ) Y\sim\chi^{2}(\nu_{2})\,
  64. X X + Y Beta ( ν 1 2 , ν 2 2 ) \tfrac{X}{X+Y}\sim{\rm Beta}(\tfrac{\nu_{1}}{2},\tfrac{\nu_{2}}{2})\,
  65. X U ( 0 , 1 ) X\sim{\rm U}(0,1)\,
  66. - 2 log ( X ) χ 2 ( 2 ) -2\log{(X)}\sim\chi^{2}(2)\,
  67. χ 2 ( 6 ) \chi^{2}(6)\,
  68. X i Laplace ( μ , β ) X_{i}\sim\mathrm{Laplace}(\mu,\beta)\,
  69. i = 1 n 2 | X i - μ | β χ 2 ( 2 n ) \sum_{i=1}^{n}{\frac{2|X_{i}-\mu|}{\beta}}\sim\chi^{2}(2n)\,
  70. Y = X 1 / k 1 X 2 / k 2 \scriptstyle Y=\frac{X_{1}/k_{1}}{X_{2}/k_{2}}
  71. X \scriptstyle\sqrt{X}
  72. X 1 , , X n X_{1},...,X_{n}
  73. a 1 , , a n > 0 a_{1},...,a_{n}\in\mathbb{R}_{>0}
  74. X = i = 1 n a i X i X=\sum_{i=1}^{n}a_{i}X_{i}
  75. i = 1 n ( X i - X ¯ ) 2 σ 2 χ n - 1 2 \sum_{i=1}^{n}(X_{i}-\bar{X})^{2}\sim\sigma^{2}\chi^{2}_{n-1}
  76. X ¯ = 1 n i = 1 n X i \bar{X}=\frac{1}{n}\sum_{i=1}^{n}X_{i}
  77. i = 1 k ( X i - μ i σ i ) 2 \sum_{i=1}^{k}\left(\frac{X_{i}-\mu_{i}}{\sigma_{i}}\right)^{2}
  78. i = 1 k ( X i σ i ) 2 \sum_{i=1}^{k}\left(\frac{X_{i}}{\sigma_{i}}\right)^{2}
  79. i = 1 k ( X i - μ i σ i ) 2 \sqrt{\sum_{i=1}^{k}\left(\frac{X_{i}-\mu_{i}}{\sigma_{i}}\right)^{2}}
  80. i = 1 k ( X i σ i ) 2 \sqrt{\sum_{i=1}^{k}\left(\frac{X_{i}}{\sigma_{i}}\right)^{2}}

Chi-squared_test.html

  1. χ 2 \chi^{2}
  2. χ 2 \chi^{2}
  3. A B C D total Blue collar 90 60 104 95 349 White collar 30 50 51 20 151 Service 30 40 45 35 150 total 150 150 200 150 650 \begin{array}[]{l|c|c|c|c|c|c}&\,\text{A}&\,\text{B}&\,\text{C}&\,\text{D}&&\,% \text{total}\\ \hline\,\text{Blue collar}&90&60&104&95&&349\\ \hline\,\text{White collar}&30&50&51&20&&151\\ \hline\,\text{Service}&30&40&45&35&&150\\ \hline\,\text{total}&150&150&200&150&&650\end{array}
  4. 150 650 × 349 650 × 650 80.54. \frac{150}{650}\times\frac{349}{650}\times 650\approx 80.54.
  5. ( observed - expected ) 2 expected = ( 90 - 80.54 ) 2 80.54 . \frac{(\,\text{observed}-\,\text{expected})^{2}}{\,\text{expected}}=\frac{(90-% 80.54)^{2}}{80.54}.
  6. ( number of rows - 1 ) ( number of columns - 1 ) = ( 3 - 1 ) ( 4 - 1 ) = 6. (\,\text{number of rows}-1)(\,\text{number of columns}-1)=(3-1)(4-1)=6.\,

Chi_(letter).html

  1. χ 2 \chi^{2}
  2. χ \chi

Chimney.html

  1. Q = C A 2 g H T i - T e T e Q=C\;A\;\sqrt{2\;g\;H\;\frac{T_{i}-T_{e}}{T_{e}}}

Chiral_anomaly.html

  1. 𝒜 \mathcal{A}
  2. μ \,\mu
  3. 𝒵 = d μ exp ( i 𝒜 / ) \mathcal{Z}=\int{d\mu\,\exp(i\mathcal{A}/\hbar)}
  4. \hbar
  5. 2 π 2\pi
  6. d ψ \!d\psi
  7. d ψ ¯ d\bar{\psi}
  8. ψ \!\psi
  9. ψ e i α γ 5 ψ \psi\rightarrow e^{i\alpha\gamma^{5}}\psi
  10. α \!\alpha
  11. ψ \!\psi
  12. 𝒵 \mathcal{Z}
  13. 0 , \hbar\to 0,
  14. 𝒜 \mathcal{A}
  15. π 0 γ γ \pi^{0}\to\gamma\gamma
  16. π 0 e + e - γ \pi^{0}\to e^{+}e^{-}\gamma
  17. C C
  18. C P CP
  19. U ( 1 ) U(1)
  20. q q ¯ q\bar{q}
  21. J μ B J_{\mu}^{B}
  22. μ J μ B = j μ ( q ¯ j γ μ q j ) = 0. \partial^{\mu}J_{\mu}^{B}=\sum_{j}\partial^{\mu}(\bar{q}_{j}\gamma_{\mu}q_{j})% =0.
  23. μ J μ B = g 2 C 16 π 2 G μ ν a G ~ μ ν a , \partial^{\mu}J_{\mu}^{B}=\frac{g^{2}C}{16\pi^{2}}G^{\mu\nu a}\tilde{G}_{\mu% \nu}^{a},
  24. C C
  25. G ~ μ ν a = 1 2 ϵ μ ν α β G α β a , \tilde{G}_{\mu\nu}^{a}=\frac{1}{2}\epsilon_{\mu\nu\alpha\beta}G^{\alpha\beta a},
  26. G μ ν a G_{\mu\nu}^{a}
  27. G μ ν a = μ A ν a - ν A μ a + g f b c a A μ b A ν c . G_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+gf^{a}_{bc}% A_{\mu}^{b}A_{\nu}^{c}.
  28. G μ ν a G ~ μ ν a = μ K μ G^{\mu\nu a}\tilde{G}_{\mu\nu}^{a}=\partial^{\mu}K_{\mu}
  29. K μ K_{\mu}
  30. K μ = 2 ϵ μ ν α β ( A ν a α A β a + 1 3 f a b c A ν a A α b A β c ) , K_{\mu}=2\epsilon_{\mu\nu\alpha\beta}\left(A^{\nu a}\partial^{\alpha}A^{\beta a% }+\frac{1}{3}f^{abc}A^{\nu a}A^{\alpha b}A^{\beta c}\right),

Chirp_Z-transform.html

  1. z k = A W - k , k = 0 , 1 , , M - 1 z_{k}=A\cdot W^{-k},k=0,1,\dots,M-1
  2. X k = n = 0 N - 1 x n e - 2 π i N n k k = 0 , , N - 1. X_{k}=\sum_{n=0}^{N-1}x_{n}e^{-\frac{2\pi i}{N}nk}\qquad k=0,\dots,N-1.
  3. n k = - ( k - n ) 2 2 + n 2 2 + k 2 2 nk=\frac{-(k-n)^{2}}{2}+\frac{n^{2}}{2}+\frac{k^{2}}{2}
  4. X k = e - π i N k 2 n = 0 N - 1 ( x n e - π i N n 2 ) e π i N ( k - n ) 2 k = 0 , , N - 1. X_{k}=e^{-\frac{\pi i}{N}k^{2}}\sum_{n=0}^{N-1}\left(x_{n}e^{-\frac{\pi i}{N}n% ^{2}}\right)e^{\frac{\pi i}{N}(k-n)^{2}}\qquad k=0,\dots,N-1.
  5. a n = x n e - π i N n 2 a_{n}=x_{n}e^{-\frac{\pi i}{N}n^{2}}
  6. b n = e π i N n 2 , b_{n}=e^{\frac{\pi i}{N}n^{2}},
  7. X k = b k * n = 0 N - 1 a n b k - n k = 0 , , N - 1. X_{k}=b_{k}^{*}\sum_{n=0}^{N-1}a_{n}b_{k-n}\qquad k=0,\dots,N-1.
  8. b n + N = e π i N ( n + N ) 2 = b n e π i N ( 2 N n + N 2 ) = ( - 1 ) N b n . b_{n+N}=e^{\frac{\pi i}{N}(n+N)^{2}}=b_{n}e^{\frac{\pi i}{N}(2Nn+N^{2})}=(-1)^% {N}b_{n}.
  9. X k = n = 0 N - 1 x n z n k k = 0 , , M - 1 , X_{k}=\sum_{n=0}^{N-1}x_{n}z^{nk}\qquad k=0,\dots,M-1,

Cholesky_decomposition.html

  1. 𝐀 = 𝐋𝐋 * \mathbf{A=LL}^{*}
  2. 𝐀 = 𝐋𝐃𝐋 * \mathbf{A=LDL}^{*}
  3. 𝐀 = 𝐋𝐃𝐋 * = 𝐋𝐃 1 2 𝐃 1 2 * 𝐋 * = 𝐋𝐃 1 2 ( 𝐋𝐃 1 2 ) * \mathbf{A=LDL}^{*}=\mathbf{L}\mathbf{D}^{\frac{1}{2}}\mathbf{D}^{{\frac{1}{2}}% {*}}\mathbf{L}^{*}=\mathbf{L}\mathbf{D}^{\frac{1}{2}}(\mathbf{L}\mathbf{D}^{% \frac{1}{2}})^{*}
  4. ( 4 12 - 16 12 37 - 43 - 16 - 43 98 ) \displaystyle\left(\begin{array}[]{*{3}{r}}4&12&-16\\ 12&37&-43\\ -16&-43&98\\ \end{array}\right)
  5. ( 4 12 - 16 12 37 - 43 - 16 - 43 98 ) \displaystyle\left(\begin{array}[]{*{3}{r}}4&12&-16\\ 12&37&-43\\ -16&-43&98\\ \end{array}\right)
  6. x 1 x_{1}
  7. x 2 x_{2}
  8. z 1 z_{1}
  9. z 2 z_{2}
  10. x 1 = z 1 x_{1}=z_{1}
  11. x 2 = ρ z 1 + 1 - ρ 2 z 2 x_{2}=\rho z_{1}+\sqrt{1-\rho^{2}}z_{2}
  12. n 3 \textstyle{n^{3}}
  13. 1 2 n 3 \textstyle\frac{1}{2}n^{3}
  14. 𝐁 - 1 = 𝐁 * ( 𝐁𝐁 * ) - 1 \mathbf{B}^{-1}=\mathbf{B}^{*}\mathbf{(BB^{*})}^{-1}
  15. 𝐀 ( i ) = ( 𝐈 i - 1 0 0 0 a i , i 𝐛 i * 0 𝐛 i 𝐁 ( i ) ) , \mathbf{A}^{(i)}=\begin{pmatrix}\mathbf{I}_{i-1}&0&0\\ 0&a_{i,i}&\mathbf{b}_{i}^{*}\\ 0&\mathbf{b}_{i}&\mathbf{B}^{(i)}\end{pmatrix},
  16. 𝐋 i := ( 𝐈 i - 1 0 0 0 a i , i 0 0 1 a i , i 𝐛 i 𝐈 n - i ) , \mathbf{L}_{i}:=\begin{pmatrix}\mathbf{I}_{i-1}&0&0\\ 0&\sqrt{a_{i,i}}&0\\ 0&\frac{1}{\sqrt{a_{i,i}}}\mathbf{b}_{i}&\mathbf{I}_{n-i}\end{pmatrix},
  17. 𝐀 ( i ) = 𝐋 i 𝐀 ( i + 1 ) 𝐋 i * \mathbf{A}^{(i)}=\mathbf{L}_{i}\mathbf{A}^{(i+1)}\mathbf{L}_{i}^{*}
  18. 𝐀 ( i + 1 ) = ( 𝐈 i - 1 0 0 0 1 0 0 0 𝐁 ( i ) - 1 a i , i 𝐛 i 𝐛 i * ) . \mathbf{A}^{(i+1)}=\begin{pmatrix}\mathbf{I}_{i-1}&0&0\\ 0&1&0\\ 0&0&\mathbf{B}^{(i)}-\frac{1}{a_{i,i}}\mathbf{b}_{i}\mathbf{b}_{i}^{*}\end{% pmatrix}.
  19. 𝐋 := 𝐋 1 𝐋 2 𝐋 n . \mathbf{L}:=\mathbf{L}_{1}\mathbf{L}_{2}\dots\mathbf{L}_{n}.
  20. 𝐀 = 𝐋𝐋 𝐓 = ( L 11 0 0 L 21 L 22 0 L 31 L 32 L 33 ) ( L 11 L 21 L 31 0 L 22 L 32 0 0 L 33 ) = ( L 11 2 ( symmetric ) L 21 L 11 L 21 2 + L 22 2 L 31 L 11 L 31 L 21 + L 32 L 22 L 31 2 + L 32 2 + L 33 2 ) \begin{aligned}\displaystyle{\mathbf{A=LL^{T}}}&\displaystyle=\begin{pmatrix}L% _{11}&0&0\\ L_{21}&L_{22}&0\\ L_{31}&L_{32}&L_{33}\\ \end{pmatrix}\begin{pmatrix}L_{11}&L_{21}&L_{31}\\ 0&L_{22}&L_{32}\\ 0&0&L_{33}\end{pmatrix}\\ &\displaystyle=\begin{pmatrix}L_{11}^{2}&&(\,\text{symmetric})\\ L_{21}L_{11}&L_{21}^{2}+L_{22}^{2}&\\ L_{31}L_{11}&L_{31}L_{21}+L_{32}L_{22}&L_{31}^{2}+L_{32}^{2}+L_{33}^{2}\end{% pmatrix}\end{aligned}
  21. L j , j = A j , j - k = 1 j - 1 L j , k 2 . L_{j,j}=\sqrt{A_{j,j}-\sum_{k=1}^{j-1}L_{j,k}^{2}}.
  22. L i , j = 1 L j , j ( A i , j - k = 1 j - 1 L i , k L j , k ) , for i > j . L_{i,j}=\frac{1}{L_{j,j}}\left(A_{i,j}-\sum_{k=1}^{j-1}L_{i,k}L_{j,k}\right),% \qquad\,\text{for }i>j.
  23. L j , j = A j , j - k = 1 j - 1 L j , k L j , k * . L_{j,j}=\sqrt{A_{j,j}-\sum_{k=1}^{j-1}L_{j,k}L_{j,k}^{*}}.
  24. L i , j = 1 L j , j ( A i , j - k = 1 j - 1 L i , k L j , k * ) , for i > j . L_{i,j}=\frac{1}{L_{j,j}}\left(A_{i,j}-\sum_{k=1}^{j-1}L_{i,k}L_{j,k}^{*}% \right),\qquad\,\text{for }i>j.
  25. 𝐄 2 c n ε 𝐀 2 . \|\mathbf{E}\|_{2}\leq c_{n}\varepsilon\|\mathbf{A}\|_{2}.
  26. 𝐀 = 𝐋𝐃𝐋 T = ( 1 0 0 L 21 1 0 L 31 L 32 1 ) ( D 1 0 0 0 D 2 0 0 0 D 3 ) ( 1 L 21 L 31 0 1 L 32 0 0 1 ) = ( D 1 ( symmetric ) L 21 D 1 L 21 2 D 1 + D 2 L 31 D 1 L 31 L 21 D 1 + L 32 D 2 L 31 2 D 1 + L 32 2 D 2 + D 3 . ) \begin{aligned}\displaystyle{\mathbf{A=LDL}^{\mathrm{T}}}&\displaystyle=\begin% {pmatrix}1&0&0\\ L_{21}&1&0\\ L_{31}&L_{32}&1\\ \end{pmatrix}\begin{pmatrix}D_{1}&0&0\\ 0&D_{2}&0\\ 0&0&D_{3}\\ \end{pmatrix}\begin{pmatrix}1&L_{21}&L_{31}\\ 0&1&L_{32}\\ 0&0&1\\ \end{pmatrix}\\ &\displaystyle=\begin{pmatrix}D_{1}&&(\mathrm{symmetric})\\ L_{21}D_{1}&L_{21}^{2}D_{1}+D_{2}&\\ L_{31}D_{1}&L_{31}L_{21}D_{1}+L_{32}D_{2}&L_{31}^{2}D_{1}+L_{32}^{2}D_{2}+D_{3% }.\end{pmatrix}\end{aligned}
  27. D j = A j j - k = 1 j - 1 L j k 2 D k D_{j}=A_{jj}-\sum_{k=1}^{j-1}L_{jk}^{2}D_{k}
  28. L i j = 1 D j ( A i j - k = 1 j - 1 L i k L j k D k ) , for i > j . L_{ij}=\frac{1}{D_{j}}\left(A_{ij}-\sum_{k=1}^{j-1}L_{ik}L_{jk}D_{k}\right),% \qquad\,\text{for }i>j.
  29. D j = A j j - k = 1 j - 1 L j k L j k * D k D_{j}=A_{jj}-\sum_{k=1}^{j-1}L_{jk}L_{jk}^{*}D_{k}
  30. L i j = 1 D j ( A i j - k = 1 j - 1 L i k L j k * D k ) , for i > j . L_{ij}=\frac{1}{D_{j}}\left(A_{ij}-\sum_{k=1}^{j-1}L_{ik}L_{jk}^{*}D_{k}\right% ),\qquad\,\text{for }i>j.
  31. 𝐀 = 𝐋𝐃𝐋 T = ( 𝐈 0 0 𝐋 21 𝐈 0 𝐋 31 𝐋 32 𝐈 ) ( 𝐃 1 0 0 0 𝐃 2 0 0 0 𝐃 3 ) ( 𝐈 𝐋 21 T 𝐋 31 T 0 𝐈 𝐋 32 T 0 0 𝐈 ) = ( 𝐃 1 ( symmetric ) 𝐋 21 𝐃 1 𝐋 21 𝐃 1 𝐋 21 T + 𝐃 2 𝐋 31 𝐃 1 𝐋 31 𝐃 1 𝐋 21 T + 𝐋 32 𝐃 2 𝐋 31 𝐃 1 𝐋 31 T + 𝐋 32 𝐃 2 𝐋 32 T + 𝐃 3 ) \begin{aligned}\displaystyle{\mathbf{A=LDL}^{\mathrm{T}}}&\displaystyle=\begin% {pmatrix}\mathbf{I}&0&0\\ \mathbf{L}_{21}&\mathbf{I}&0\\ \mathbf{L}_{31}&\mathbf{L}_{32}&\mathbf{I}\\ \end{pmatrix}\begin{pmatrix}\mathbf{D}_{1}&0&0\\ 0&\mathbf{D}_{2}&0\\ 0&0&\mathbf{D}_{3}\\ \end{pmatrix}\begin{pmatrix}\mathbf{I}&\mathbf{L}_{21}^{\mathrm{T}}&\mathbf{L}% _{31}^{\mathrm{T}}\\ 0&\mathbf{I}&\mathbf{L}_{32}^{\mathrm{T}}\\ 0&0&\mathbf{I}\\ \end{pmatrix}\\ &\displaystyle=\begin{pmatrix}\mathbf{D}_{1}&&(\mathrm{symmetric})\\ \mathbf{L}_{21}\mathbf{D}_{1}&\mathbf{L}_{21}\mathbf{D}_{1}\mathbf{L}_{21}^{% \mathrm{T}}+\mathbf{D}_{2}&\\ \mathbf{L}_{31}\mathbf{D}_{1}&\mathbf{L}_{31}\mathbf{D}_{1}\mathbf{L}_{21}^{% \mathrm{T}}+\mathbf{L}_{32}\mathbf{D}_{2}&\mathbf{L}_{31}\mathbf{D}_{1}\mathbf% {L}_{31}^{\mathrm{T}}+\mathbf{L}_{32}\mathbf{D}_{2}\mathbf{L}_{32}^{\mathrm{T}% }+\mathbf{D}_{3}\end{pmatrix}\end{aligned}
  32. 𝐃 j = 𝐀 j j - k = 1 j - 1 𝐋 j k 𝐃 k 𝐋 j k T \mathbf{D}_{j}=\mathbf{A}_{jj}-\sum_{k=1}^{j-1}\mathbf{L}_{jk}\mathbf{D}_{k}% \mathbf{L}_{jk}^{\mathrm{T}}
  33. 𝐋 i j = ( 𝐀 i j - k = 1 j - 1 𝐋 i k 𝐃 k 𝐋 j k T ) 𝐃 j - 1 \mathbf{L}_{ij}=\left(\mathbf{A}_{ij}-\sum_{k=1}^{j-1}\mathbf{L}_{ik}\mathbf{D% }_{k}\mathbf{L}_{jk}^{\mathrm{T}}\right)\mathbf{D}_{j}^{-1}
  34. 𝐀 ~ \tilde{\mathbf{A}}
  35. 𝐀 ~ = 𝐋 ~ 𝐋 ~ * \tilde{\mathbf{A}}=\tilde{\mathbf{L}}\tilde{\mathbf{L}}^{*}
  36. 𝐀 ~ \tilde{\mathbf{A}}
  37. 𝐀 ~ \tilde{\mathbf{A}}
  38. 𝐀 ~ = 𝐀 + 𝐱𝐱 * \tilde{\mathbf{A}}=\mathbf{A}+\mathbf{x}\mathbf{x}^{*}
  39. 𝐀 ~ = 𝐀 - 𝐱𝐱 * \tilde{\mathbf{A}}=\mathbf{A}-\mathbf{x}\mathbf{x}^{*}
  40. 𝐀 ~ \tilde{\mathbf{A}}
  41. 𝐋 k 2 𝐋 k 𝐋 k * = 𝐀 k . \|\mathbf{L}_{k}\|^{2}\leq\|\mathbf{L}_{k}\mathbf{L}_{k}^{*}\|=\|\mathbf{A}_{k% }\|.
  42. 𝐀 x , y = lim 𝐀 k x , y = lim 𝐋 k 𝐋 k * x , y = 𝐋𝐋 * x , y . \langle\mathbf{A}x,y\rangle=\langle\lim\mathbf{A}_{k}x,y\rangle=\langle\lim% \mathbf{L}_{k}\mathbf{L}_{k}^{*}x,y\rangle=\langle\mathbf{L}\mathbf{L}^{*}x,y\rangle.
  43. { n } \{\mathcal{H}_{n}\}
  44. 𝐀 = [ 𝐀 11 𝐀 12 𝐀 13 𝐀 12 * 𝐀 22 𝐀 23 𝐀 13 * 𝐀 23 * 𝐀 33 ] \mathbf{A}=\begin{bmatrix}\mathbf{A}_{11}&\mathbf{A}_{12}&\mathbf{A}_{13}&\\ \mathbf{A}_{12}^{*}&\mathbf{A}_{22}&\mathbf{A}_{23}&\\ \mathbf{A}_{13}^{*}&\mathbf{A}_{23}^{*}&\mathbf{A}_{33}&\\ &&&\ddots\end{bmatrix}
  45. = n n , \mathcal{H}=\oplus_{n}\mathcal{H}_{n},
  46. 𝐀 i j : j i \mathbf{A}_{ij}:\mathcal{H}_{j}\rightarrow\mathcal{H}_{i}
  47. h n = 1 k k , h\in\oplus_{n=1}^{k}\mathcal{H}_{k},
  48. h , 𝐀 h 0 \langle h,\mathbf{A}h\rangle\geq 0

Chord_(geometry).html

  1. θ \theta
  2. θ \theta
  3. crd θ = ( 1 - cos θ ) 2 + sin 2 θ = 2 - 2 cos θ = 2 sin ( θ 2 ) . \mathrm{crd}\ \theta=\sqrt{(1-\cos\theta)^{2}+\sin^{2}\theta}=\sqrt{2-2\cos% \theta}=2\sin\left(\frac{\theta}{2}\right).
  4. sin 2 θ + cos 2 θ = 1 \sin^{2}\theta+\cos^{2}\theta=1\,
  5. crd 2 θ + crd 2 ( 180 - θ ) = 4 \mathrm{crd}^{2}\theta+\mathrm{crd}^{2}(180^{\circ}-\theta)=4\,
  6. sin θ 2 = ± 1 - cos θ 2 \sin\frac{\theta}{2}=\pm\sqrt{\frac{1-\cos\theta}{2}}\,
  7. crd θ 2 = ± 2 - crd ( 180 - θ ) \mathrm{crd}\ \frac{\theta}{2}=\pm\sqrt{2-\mathrm{crd}(180^{\circ}-\theta)}\,
  8. c = 2 r 2 - a 2 c=2\sqrt{r^{2}-a^{2}}
  9. c = D 2 - 4 a 2 c=\sqrt{D^{2}-4a^{2}}
  10. c = 2 r sin ( θ 2 ) c=2r\sin\left(\frac{\theta}{2}\right)
  11. c = D 2 crd θ c=\frac{D}{2}\mathrm{crd}\ \theta

CHSH_inequality.html

  1. 2 \sqrt{2}
  2. E = N + + - N + - - N - + + N - - N + + + N + - + N - + + N - - ( 3 ) E=\frac{N_{++}-N_{+-}-N_{-+}+N_{--}}{N_{++}+N_{+-}+N_{-+}+N_{--}}\qquad(3)
  3. E ( a , b ) = A ¯ ( a , λ ) B ¯ ( b , λ ) ρ ( λ ) d λ ( 4 ) E(a,b)=\int\underline{A}(a,\lambda)\underline{B}(b,\lambda)\rho(\lambda)d% \lambda\qquad(4)
  4. | A ¯ | 1 , | B ¯ | 1 ( 5 ) |\underline{A}|\leq 1,|\underline{B}|\leq 1\qquad(5)
  5. E ( a , b ) - E ( a , b ) = [ A ¯ ( a , λ ) B ¯ ( b , λ ) - A ¯ ( a , λ ) B ¯ ( b , λ ) ] ρ ( λ ) d λ E(a,b)-E(a,b^{\prime})=\int[\underline{A}(a,\lambda)\underline{B}(b,\lambda)-% \underline{A}(a,\lambda)\underline{B}(b^{\prime},\lambda)]\rho(\lambda)d\lambda
  6. = A ¯ ( a , λ ) B ¯ ( b , λ ) [ 1 ± A ¯ ( a , λ ) B ¯ ( b , λ ) ] ρ ( λ ) d λ =\int\underline{A}(a,\lambda)\underline{B}(b,\lambda)[1\pm\underline{A}(a^{% \prime},\lambda)\underline{B}(b^{\prime},\lambda)]\rho(\lambda)d\lambda
  7. - A ¯ ( a , λ ) B ¯ ( b , λ ) [ 1 ± A ¯ ( a , λ ) B ¯ ( b , λ ) ] ρ ( λ ) d λ ( 6 ) -\int\underline{A}(a,\lambda)\underline{B}(b^{\prime},\lambda)[1\pm\underline{% A}(a^{\prime},\lambda)\underline{B}(b,\lambda)]\rho(\lambda)d\lambda\qquad(6)
  8. [ 1 ± A ¯ ( a , λ ) B ¯ ( b , λ ) ] ρ ( λ ) [1\pm\underline{A}(a^{\prime},\lambda)\underline{B}(b^{\prime},\lambda)]\rho(\lambda)
  9. [ 1 ± A ¯ ( a , λ ) B ¯ ( b , λ ) ] ρ ( λ ) [1\pm\underline{A}(a^{\prime},\lambda)\underline{B}(b,\lambda)]\rho(\lambda)
  10. | E ( a , b ) - E ( a , b ) | [ 1 ± A ¯ ( a , λ ) B ¯ ( b , λ ) ] ρ ( λ ) d λ |E(a,b)-E(a,b^{\prime})|\leq\int[1\pm\underline{A}(a^{\prime},\lambda)% \underline{B}(b^{\prime},\lambda)]\rho(\lambda)d\lambda
  11. + [ 1 ± A ¯ ( a , λ ) B ¯ ( b , λ ) ] ρ ( λ ) d λ , +\int[1\pm\underline{A}(a^{\prime},\lambda)\underline{B}(b,\lambda)]\rho(% \lambda)d\lambda,
  12. | E ( a , b ) - E ( a , b ) | 2 ± [ E ( a , b ) + E ( a , b ) ] , |E(a,b)-E(a,b^{\prime})|\leq 2\pm[E(a^{\prime},b^{\prime})+E(a^{\prime},b)],

Ciphertext.html

  1. m m\!
  2. E k E_{k}\!
  3. k {}_{k}\!
  4. c c\!
  5. c = E k ( m ) c=E_{k}(m)\!
  6. k k\!
  7. E k - 1 {E_{k}}^{-1}\!
  8. D k D_{k}\!
  9. D k ( c ) = D k ( E k ( m ) ) = m D_{k}(c)=D_{k}(E_{k}(m))=m\!

Circle_of_confusion.html

  1. DoF = 2 N c ( m + 1 ) m 2 - ( N c f ) 2 , \mathrm{DoF}=\frac{2Nc\left(m+1\right)}{m^{2}-\left(\frac{Nc}{f}\right)^{2}}\,,
  2. C = A | S 2 - S 1 | S 2 . C=A{|S_{2}-S_{1}|\over S_{2}}\,.
  3. c = C m , c=Cm\,,
  4. m = f 1 S 1 . m={f_{1}\over S_{1}}\,.
  5. 1 f = 1 f 1 + 1 S 1 , {1\over f}={1\over f_{1}}+{1\over S_{1}}\,,
  6. f 1 = f S 1 S 1 - f . f_{1}={fS_{1}\over S_{1}-f}\,.
  7. m = f S 1 - f , m={f\over S_{1}-f}\,,
  8. c = A | S 2 - S 1 | S 2 f S 1 - f . c=A{|S_{2}-S_{1}|\over S_{2}}{f\over S_{1}-f}\,.
  9. c = | S 2 - S 1 | S 2 f 2 N ( S 1 - f ) . c={|S_{2}-S_{1}|\over S_{2}}{f^{2}\over N(S_{1}-f)}\,.
  10. c = A m | S 2 - S 1 | S 2 . c=Am{|S_{2}-S_{1}|\over S_{2}}\,.
  11. c = f A S 2 = f 2 N S 2 . c={fA\over S_{2}}={f^{2}\over NS_{2}}\,.
  12. c = f A S 1 - f = f 2 N ( S 1 - f ) . c={fA\over S_{1}-f}={f^{2}\over N(S_{1}-f)}\,.
  13. c = f A S c={fA\over S}
  14. f f

Circle_of_fifths.html

  1. / 12 \mathbb{Z}/12\mathbb{Z}
  2. 12 \mathbb{Z}_{12}

Circular_dichroism.html

  1. R e x p = 3 h c 10 3 ln ( 10 ) 32 π 3 N A Δ ϵ ν d ν R_{exp}=\frac{3hc10^{3}\ln(10)}{32\pi^{3}N_{A}}\int\frac{\Delta\epsilon}{\nu}d% {\nu}
  2. R t h e o = 1 2 m c I m Ψ g M ^ ( e l e c . d i p o l e ) Ψ e d τ Ψ g M ^ ( m a g . d i p o l e ) Ψ e d τ R_{theo}=\frac{1}{2mc}Im\int\Psi_{g}\widehat{M}_{(elec.dipole)}\Psi_{e}d\tau% \bullet\int\Psi_{g}\widehat{M}_{(mag.dipole)}\Psi_{e}d\tau
  3. Δ ϵ \Delta\epsilon
  4. M ^ ( e l e c . d i p o l e ) \widehat{M}_{(elec.dipole)}
  5. M ^ ( m a g . d i p o l e ) \widehat{M}_{(mag.dipole)}
  6. C n C_{n}
  7. D n D_{n}
  8. Δ A = A L - A R \Delta A=A_{L}-A_{R}\,
  9. Δ A = ( ϵ L - ϵ R ) C l \Delta A=(\epsilon_{L}-\epsilon_{R})Cl\,
  10. Δ ϵ = ϵ L - ϵ R \Delta\epsilon=\epsilon_{L}-\epsilon_{R}\,
  11. Δ ϵ \Delta\epsilon
  12. Δ ϵ \Delta\epsilon
  13. [ θ ] = 3298.2 Δ ε . [\theta]=3298.2\,\Delta\varepsilon.\,
  14. tan θ = E R - E L E R + E L \tan\theta=\frac{E_{R}-E_{L}}{E_{R}+E_{L}}\,
  15. θ ( radians ) = ( I R 1 / 2 - I L 1 / 2 ) ( I R 1 / 2 + I L 1 / 2 ) \theta(\,\text{radians})=\frac{(I_{R}^{1/2}-I_{L}^{1/2})}{(I_{R}^{1/2}+I_{L}^{% 1/2})}\,
  16. I = I 0 e - A ln 10 I=I_{0}e^{-A\ln 10}\,
  17. θ ( radians ) = ( e - A R 2 ln 10 - e - A L 2 ln 10 ) ( e - A R 2 ln 10 + e - A L 2 ln 10 ) = e Δ A ln 10 2 - 1 e Δ A ln 10 2 + 1 \theta(\,\text{radians})=\frac{(e^{\frac{-A_{R}}{2}\ln 10}-e^{\frac{-A_{L}}{2}% \ln 10})}{(e^{\frac{-A_{R}}{2}\ln 10}+e^{\frac{-A_{L}}{2}\ln 10})}=\frac{e^{% \Delta A\frac{\ln 10}{2}}-1}{e^{\Delta A\frac{\ln 10}{2}}+1}\,
  18. [ θ ] = 100 θ Cl [\theta]=\frac{100\theta}{\,\text{Cl}}\,
  19. [ θ ] = 100 Δ ε ( ln 10 4 ) ( 180 π ) = 3298.2 Δ ε [\theta]=100\,\Delta\varepsilon\left(\frac{\ln 10}{4}\right)\left(\frac{180}{% \pi}\right)=3298.2\,\Delta\varepsilon\,

Circumflex.html

  1. f ^ \hat{f}
  2. ı ^ \hat{\mathbf{\imath}}
  3. 𝐱 ^ \hat{\mathbf{x}}
  4. 𝐞 ^ 1 \hat{\mathbf{e}}_{1}
  5. ε ^ \hat{\varepsilon}
  6. ε \varepsilon

Citizen's_dividend.html

  1. 2 {}_{2}

Class_field_theory.html

  1. \mathbb{Q}
  2. \mathbb{Q}
  3. μ \C × \mu_{\infty}\subset\C^{\times}
  4. < ^ m t p l > \Z × G \Q ab = Gal ( \Q ( μ ) / \Q ) , x ( ζ ζ x ) , \hat{<}mtpl>{{\Z}}^{\times}\to G_{\Q}^{\rm ab}={\rm Gal}(\Q(\mu_{\infty})/\Q),% \quad x\mapsto(\zeta\mapsto\zeta^{x}),
  5. < ^ m t p l > \Z × G \Q ab = Gal ( \Q ( μ ) / \Q ) , x ( ζ ζ - x ) , \hat{<}mtpl>{{\Z}}^{\times}\to G_{\Q}^{\rm ab}={\rm Gal}(\Q(\mu_{\infty})/\Q),% \quad x\mapsto(\zeta\mapsto\zeta^{-x}),
  6. K 1 K_{1}
  7. p p
  8. p p

Clique_problem.html

  1. n o ( k ) n^{o(k)}

Clopen_set.html

  1. 2 \sqrt{2}

Closed_timelike_curve.html

  1. x = 0 x=0
  2. t t
  3. c t ct
  4. t t
  5. t t

Coalgebra.html

  1. ( id C Δ ) Δ = ( Δ id C ) Δ (\mathrm{id}_{C}\otimes\Delta)\circ\Delta=(\Delta\otimes\mathrm{id}_{C})\circ\Delta
  2. ( id C ϵ ) Δ = id C = ( ϵ id C ) Δ (\mathrm{id}_{C}\otimes\epsilon)\circ\Delta=\mathrm{id}_{C}=(\epsilon\otimes% \mathrm{id}_{C})\circ\Delta
  3. Δ ( X n ) = k = 0 n ( n k ) X k X n - k , \Delta(X^{n})=\sum_{k=0}^{n}{\displaystyle\left({{n}\atop{k}}\right)}X^{k}% \otimes X^{n-k},
  4. ϵ ( X n ) = { 1 if n = 0 0 if n > 0 \epsilon(X^{n})=\begin{cases}1&\mbox{if }~{}n=0\\ 0&\mbox{if }~{}n>0\end{cases}
  5. Δ ( c ) = i c ( 1 ) ( i ) c ( 2 ) ( i ) . \Delta(c)=\sum_{i}c_{(1)}^{(i)}\otimes c_{(2)}^{(i)}.
  6. Δ ( c ) = ( c ) c ( 1 ) c ( 2 ) . \Delta(c)=\sum_{(c)}c_{(1)}\otimes c_{(2)}.
  7. c = ( c ) ε ( c ( 1 ) ) c ( 2 ) = ( c ) c ( 1 ) ε ( c ( 2 ) ) . c=\sum_{(c)}\varepsilon(c_{(1)})c_{(2)}=\sum_{(c)}c_{(1)}\varepsilon(c_{(2)}).\;
  8. ( c ) c ( 1 ) ( ( c ( 2 ) ) ( c ( 2 ) ) ( 1 ) ( c ( 2 ) ) ( 2 ) ) = ( c ) ( ( c ( 1 ) ) ( c ( 1 ) ) ( 1 ) ( c ( 1 ) ) ( 2 ) ) c ( 2 ) . \sum_{(c)}c_{(1)}\otimes\left(\sum_{(c_{(2)})}(c_{(2)})_{(1)}\otimes(c_{(2)})_% {(2)}\right)=\sum_{(c)}\left(\sum_{(c_{(1)})}(c_{(1)})_{(1)}\otimes(c_{(1)})_{% (2)}\right)\otimes c_{(2)}.
  9. ( c ) c ( 1 ) c ( 2 ) c ( 3 ) . \sum_{(c)}c_{(1)}\otimes c_{(2)}\otimes c_{(3)}.
  10. Δ ( c ) = c ( 1 ) c ( 2 ) \Delta(c)=c_{(1)}\otimes c_{(2)}
  11. c = ε ( c ( 1 ) ) c ( 2 ) = c ( 1 ) ε ( c ( 2 ) ) . c=\varepsilon(c_{(1)})c_{(2)}=c_{(1)}\varepsilon(c_{(2)}).\;
  12. σ Δ = Δ \sigma\circ\Delta=\Delta
  13. c ( 1 ) c ( 2 ) = c ( 2 ) c ( 1 ) c_{(1)}\otimes c_{(2)}=c_{(2)}\otimes c_{(1)}\;
  14. ( f f ) Δ 1 = Δ 2 f (f\otimes f)\circ\Delta_{1}=\Delta_{2}\circ f
  15. ϵ 2 f = ϵ 1 \epsilon_{2}\circ f=\epsilon_{1}
  16. f ( c ( 1 ) ) f ( c ( 2 ) ) = f ( c ) ( 1 ) f ( c ) ( 2 ) . f(c_{(1)})\otimes f(c_{(2)})=f(c)_{(1)}\otimes f(c)_{(2)}.

Cocktail_sort.html

  1. O ( n 2 ) O(n^{2})
  2. O ( n ) O(n)
  3. O ( k * n ) . O(k*n).

Codex_Vaticanus.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}

Coenzyme_Q_–_cytochrome_c_reductase.html

  1. \rightleftharpoons

Coherence_(physics).html

  1. τ c Δ f 1 \tau_{c}\Delta f\lesssim 1
  2. τ c \tau_{c}
  3. τ c \tau_{c}
  4. Δ t \Delta t
  5. Δ f \Delta f
  6. Δ f Δ t 1 \Delta f\Delta t\geq 1
  7. θ ( f ) f \theta(f)\propto f
  8. ψ ( 𝐫 ) \psi(\mathbf{r})

Coherent_states.html

  1. n n
  2. α α
  3. a ^ | α = α | α . \hat{a}|\alpha\rangle=\alpha|\alpha\rangle~{}.
  4. α α
  5. α = | α | e i θ , \alpha=|\alpha|e^{i\theta}~{}~{}~{},
  6. α α
  7. θ θ
  8. X X
  9. P P
  10. m m
  11. k k
  12. P = 1 2 m ω p ^ , X = m ω 2 x ^ , where ω k / m . {P}=\sqrt{\frac{1}{2\hbar m\omega}}\ \hat{p}\,\text{,}\quad{X}=\sqrt{\frac{m% \omega}{2\hbar}}\ \hat{x}\,\text{,}\quad\quad\,\text{where }\omega\equiv\sqrt{% k/m}~{}.
  13. E R = ( ω 2 ϵ 0 V ) 1 / 2 cos ( θ ) X and E I = ( ω 2 ϵ 0 V ) 1 / 2 sin ( θ ) X ~{}E_{\rm R}=\left(\frac{\hbar\omega}{2\epsilon_{0}V}\right)^{1/2}\!\!\!\cos(% \theta)X\qquad\,\text{and}\qquad~{}E_{\rm I}=\left(\frac{\hbar\omega}{2% \epsilon_{0}V}\right)^{1/2}\!\!\!\sin(\theta)X~{}
  14. H = ω ( P 2 + X 2 ) , with [ X , P ] X P - P X = i 2 I . {H}=\hbar\omega\left({P}^{2}+{X}^{2}\right)\,\text{,}\qquad\,\text{with}\qquad% \left[{X},{P}\right]\equiv{XP}-{PX}=\frac{i}{2}\,{I}.
  15. X X
  16. P P
  17. ( X - X ) | α = - i ( P - P ) | α , \left({X}-\langle{X}\rangle\right)\,|\alpha\rangle=-i\left({P}-\langle{P}% \rangle\right)\,|\alpha\rangle\,\text{,}
  18. ( X + i P ) | α = X + i P | α . \left({X}+i{P}\right)\,\left|\alpha\right\rangle=\left\langle{X}+i{P}\right% \rangle\,\left|\alpha\right\rangle.
  19. ( X + i P ) (X+iP)
  20. ( X + i P ) (X+iP)
  21. X X
  22. P P
  23. θ θ
  24. | 0 |0\rangle
  25. α α
  26. α α
  27. α α
  28. α α
  29. D ( α ) D(α)
  30. | α = e α a ^ - α * a ^ | 0 = D ( α ) | 0 |\alpha\rangle=e^{\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}}|0\rangle=D(\alpha% )|0\rangle
  31. = X + i P â=X+iP
  32. | α = e - | α | 2 2 n = 0 α n n ! | n = e - | α | 2 2 e α a ^ | 0 , |\alpha\rangle=e^{-{|\alpha|^{2}\over 2}}\sum_{n=0}^{\infty}{\alpha^{n}\over% \sqrt{n!}}|n\rangle=e^{-{|\alpha|^{2}\over 2}}e^{\alpha\hat{a}^{\dagger}}|0% \rangle~{},
  33. H = a ^ a ^ + 1 2 H=\hat{a}^{\dagger}\hat{a}+\frac{1}{2}
  34. n n
  35. P ( n ) = | n | α | 2 = e - n n n n ! . P(n)=|\langle n|\alpha\rangle|^{2}=e^{-\langle n\rangle}\frac{\langle n\rangle% ^{n}}{n!}~{}.
  36. n = a ^ a ^ = | α | 2 ~{}\langle n\rangle=\langle\hat{a}^{\dagger}\hat{a}\rangle=|\alpha|^{2}~{}
  37. ( Δ n ) 2 = Var ( a ^ a ^ ) = | α | 2 ~{}(\Delta n)^{2}={\rm Var}\left(\hat{a}^{\dagger}\hat{a}\right)=|\alpha|^{2}~{}
  38. n n
  39. α 1 α≫1
  40. a ( t ) | α = e - i ω t a ( 0 ) | α ~{}a(t)|\alpha\rangle=e^{-i\omega t}a(0)|\alpha\rangle
  41. α ( t ) = e - i ω t α ( 0 ) ~{}\alpha(t)=e^{-i\omega t}\alpha(0)~{}
  42. a | α ( t ) = α ( t ) | α ( t ) ~{}a|\alpha(t)\rangle=\alpha(t)|\alpha(t)\rangle
  43. m ω 2 ( x + m ω x ) ψ α ( x , t ) = α ( t ) ψ α ( x , t ) , ~{}\sqrt{\frac{m\omega}{2\hbar}}\left(x+\frac{\hbar}{m\omega}\frac{\partial}{% \partial x}\right)\psi^{\alpha}(x,t)=\alpha(t)\psi^{\alpha}(x,t)~{},
  44. ψ ( α ) ( x , t ) = ( m ω π ) 1 / 4 e - m ω 2 ( x - 2 m ω [ α ( t ) ] ) 2 + i 2 m ω [ α ( t ) ] x + i δ ( t ) , ~{}\psi^{(\alpha)}(x,t)=\left(\frac{m\omega}{\pi\hbar}\right)^{1/4}e^{-\frac{m% \omega}{2\hbar}\left(x-\sqrt{\frac{2\hbar}{m\omega}}\Re[\alpha(t)]\right)^{2}+% i\sqrt{\frac{2m\omega}{\hbar}}\Im[\alpha(t)]x+i\delta(t)}~{},
  45. δ ( t ) δ(t)
  46. δ ( t ) = - ω t 2 + | α ( 0 ) | 2 sin ( 2 ω t - 2 σ ) 2 , where α ( 0 ) | α ( 0 ) | exp ( i σ ) , ~{}\delta(t)=-\frac{\omega t}{2}+\frac{|\alpha(0)|^{2}\sin(2\omega t-2\sigma)}% {2}~{},\,\text{where}\qquad\alpha(0)\equiv|\alpha(0)|\exp(i\sigma)~{},
  47. σ σ
  48. | ψ ( α ) ( x , t ) | 2 = m ω π e - m ω ( x - x ^ ( t ) ) 2 . |\psi^{(\alpha)}(x,t)|^{2}=\sqrt{\frac{m\omega}{\pi\hbar}}e^{-\frac{m\omega}{% \hbar}\left(x-\langle\hat{x}(t)\rangle\right)^{2}}.
  49. | α |\alpha\rangle
  50. a ^ | α = α | α \hat{a}|\alpha\rangle=\alpha|\alpha\rangle\,
  51. | α = e α a ^ - α * a ^ | 0 = D ( α ) | 0 |\alpha\rangle=e^{\alpha\hat{a}^{\dagger}-\alpha^{*}\hat{a}}|0\rangle=D(\alpha% )|0\rangle\,
  52. Δ X = Δ P = 1 / 2 \Delta X=\Delta P=1/\sqrt{2}\,
  53. β | α = e - 1 2 ( | β | 2 + | α | 2 - 2 β * α ) δ ( α - β ) \langle\beta|\alpha\rangle=e^{-{1\over 2}(|\beta|^{2}+|\alpha|^{2}-2\beta^{*}% \alpha)}\neq\delta(\alpha-\beta)
  54. | α |\alpha\rangle
  55. | β |\beta\rangle
  56. I I
  57. 1 π | α α | d 2 α = I d 2 α d ( α ) d ( α ) . \frac{1}{\pi}\int|\alpha\rangle\langle\alpha|d^{2}\alpha=I\qquad d^{2}\alpha% \equiv d\Re(\alpha)\,d\Im(\alpha)~{}.
  58. a ^ \hat{a}^{\dagger}
  59. a | α = ( α + α * 2 ) | α . a^{\dagger}|\alpha\rangle=\left({\partial\over\partial\alpha}+{\alpha^{*}\over 2% }\right)|\alpha\rangle~{}.
  60. | α , 1 . |\alpha,1\rangle.
  61. n n
  62. | α , n = [ a ^ ] n | α / [ a ^ ] n | α . |\alpha,n\rangle=[{\hat{a}^{\dagger}]}^{n}|\alpha\rangle/\|[{\hat{a}^{\dagger}% ]}^{n}|\alpha\rangle\|~{}.
  63. x | | y \langle x|\cdots|y\rangle
  64. δ ( x - y ) δ(x-y)
  65. ψ α ( x , t ) = x | α ( t ) \psi^{\alpha}(x,t)=\langle x|\alpha(t)\rangle
  66. ( α ) \Im(\alpha)
  67. ( α ) \Re(\alpha)
  68. x x
  69. | α | x , p x x ^ p p ^ |\alpha\rangle\equiv|x,p\rangle\qquad\qquad x\equiv\langle\hat{x}\rangle\qquad% \qquad p\equiv\langle\hat{p}\rangle
  70. I = | x , p x , p | d x d p 2 π . I=\int|x,p\rangle\,\langle x,p|~{}\frac{\mathrm{d}x\,\mathrm{d}p}{2\pi\hbar}~{}.
  71. ( 2 π ) - 1 (2\pi\hbar)^{-1}
  72. x ^ | x , p x | x , p p ^ | x , p p | x , p \hat{x}|x,p\rangle\approx x|x,p\rangle\qquad\qquad\hat{p}|x,p\rangle\approx p|% x,p\rangle
  73. x , p | ( x ^ - x ) 2 | x , p = ( Δ x ) 2 x , p | ( p ^ - p ) 2 | x , p = ( Δ p ) 2 . \langle x,p|\left(\hat{x}-x\right)^{2}|x,p\rangle=\left(\Delta x\right)^{2}% \qquad\qquad\langle x,p|\left(\hat{p}-p\right)^{2}|x,p\rangle=\left(\Delta p% \right)^{2}~{}.
  74. ρ ( α , β ) = 1 Z D ( α ) e - β ω a a D ( α ) , \rho(\alpha,\beta)=\frac{1}{Z}D(\alpha)e^{-\hbar\beta\omega a^{\dagger}a}D^{% \dagger}(\alpha),
  75. D ( α ) D(\alpha)
  76. D ( α ) | 0 = | α D(\alpha)|0\rangle=|\alpha\rangle
  77. α \alpha
  78. β = 1 / ( k B T ) \beta=1/(k_{B}T)
  79. Z = tr { e - β ω a a } = n = 0 e - n β ω = 1 1 - e - β ω . Z=\,\text{tr}\left\{\displaystyle e^{-\hbar\beta\omega a^{\dagger}a}\right\}=% \sum_{n=0}^{\infty}e^{-n\beta\hbar\omega}=\frac{1}{1-e^{-\hbar\beta\omega}}.
  80. I n = 0 | n n | I\equiv\sum_{n=0}^{\infty}|n\rangle\langle n|
  81. ρ ( α , β ) = 1 Z n = 0 e - n β ω D ( α ) | n n | D ( α ) = 1 Z n = 0 e - n β ω | α , n α , n | , \rho(\alpha,\beta)=\frac{1}{Z}\sum_{n=0}^{\infty}e^{-n\hbar\beta\omega}D(% \alpha)|n\rangle\langle n|D^{\dagger}(\alpha)=\frac{1}{Z}\sum_{n=0}^{\infty}e^% {-n\hbar\beta\omega}|\alpha,n\rangle\langle\alpha,n|,
  82. | α , n |\alpha,n\rangle
  83. lim β ρ ( α , β ) = lim β n = 0 e - n β ω ( 1 - e - β ω ) | α , n α , n | = n = 0 δ n , 0 | α , n α , n | = | α , 0 α , 0 | , \lim_{\beta\to\infty}\rho(\alpha,\beta)=\lim_{\beta\to\infty}\sum_{n=0}^{% \infty}e^{-n\hbar\beta\omega}(1-e^{-\hbar\beta\omega})|\alpha,n\rangle\langle% \alpha,n|=\sum_{n=0}^{\infty}\delta_{n,0}|\alpha,n\rangle\langle\alpha,n|=|% \alpha,0\rangle\langle\alpha,0|,
  84. n = Tr { ρ a a } = 1 Z Tr { D ( α ) a D ( α ) D ( α ) a D ( α ) e - β ω a a } = 1 Z Tr { ( a + α * ) ( a + α ) e - β ω a a } = \langle n\rangle=\,\text{Tr}\{\rho a^{\dagger}a\}=\frac{1}{Z}\,\text{Tr}\{D^{% \dagger}(\alpha)a^{\dagger}D({\alpha})D^{\dagger}(\alpha)aD(\alpha)e^{-\beta% \hbar\omega a^{\dagger}a}\}=\frac{1}{Z}\,\text{Tr}\{(a^{\dagger}+\alpha^{*})(a% +\alpha)e^{-\beta\hbar\omega a^{\dagger}a}\}=
  85. = | α | 2 1 Z Tr { e - β ω a a } + 1 Z Tr { a a e - β ω a a } = | α | 2 + 1 Z n = 0 n e - n β ω , =|\alpha|^{2}\frac{1}{Z}\,\text{Tr}\{e^{-\beta\hbar\omega a^{\dagger}a}\}+% \frac{1}{Z}\,\text{Tr}\{a^{\dagger}ae^{-\beta\hbar\omega a^{\dagger}a}\}=|% \alpha|^{2}+\frac{1}{Z}\sum_{n=0}^{\infty}ne^{-n\beta\hbar\omega},
  86. n = 0 n e - n β ω = - ( β ω ) ( n = 0 e - n β ω ) = e - β ω ( 1 - e - β ω ) 2 . \sum_{n=0}^{\infty}ne^{-n\beta\hbar\omega}=-\frac{\partial}{\partial(\beta% \hbar\omega)}\left(\sum_{n=0}^{\infty}e^{-n\beta\hbar\omega}\right)=\frac{e^{-% \beta\hbar\omega}}{(1-e^{-\beta\hbar\omega})^{2}}.
  87. n = | α | 2 + n th , \langle n\rangle=|\alpha|^{2}+\langle n\rangle_{\,\text{th}},
  88. n th \langle n\rangle_{\,\text{th}}
  89. O th = 1 Z tr { e - β ω a a O } , \langle O\rangle_{\,\text{th}}=\frac{1}{Z}\,\text{tr}\{e^{-\beta\hbar\omega a^% {\dagger}a}O\},
  90. n th = 1 e β ω - 1 . \langle n\rangle_{\,\text{th}}=\frac{1}{e^{\beta\hbar\omega}-1}.
  91. β \beta\to\infty
  92. n = | α | 2 \langle n\rangle=|\alpha|^{2}
  93. σ 2 = n 2 - n 2 = σ th 2 + | α | 2 ( 1 + 2 a a th ) , \sigma^{2}=\langle n^{2}\rangle-\langle n\rangle^{2}=\sigma_{\,\text{th}}^{2}+% |\alpha|^{2}\left(1+2\langle a^{\dagger}a\rangle_{\,\text{th}}\right),
  94. σ th 2 = n 2 th - n th 2 \sigma_{\,\text{th}}^{2}=\langle n^{2}\rangle_{\,\text{th}}-\langle n\rangle_{% \,\text{th}}^{2}

Cohomology.html

  1. C n n C n - 1 \cdots\rightarrow C_{n}\stackrel{\partial_{n}}{\rightarrow}\ C_{n-1}\rightarrow\cdots
  2. δ n : C n - 1 * C n * \delta^{n}:C_{n-1}^{*}\rightarrow C_{n}^{*}
  3. C n * δ n C n - 1 * \cdots\leftarrow C_{n}^{*}\stackrel{\delta^{n}}{\leftarrow}\ C_{n-1}^{*}\leftarrow\cdots
  4. δ n ( φ ) = φ n \delta^{n}(\varphi)=\varphi\circ\partial_{n}
  5. h : H n ( C ; G ) Hom ( H n ( C ) , G ) . h:H^{n}(C;G)\rightarrow\,\text{Hom}(H_{n}(C),G).
  6. 0 ker h H n ( C ; G ) h Hom ( H n ( C ) , G ) 0. 0\rightarrow\ker h\rightarrow H^{n}(C;G)\stackrel{h}{\rightarrow}\,\text{Hom}(% H_{n}(C),G)\rightarrow 0.
  7. H * ( X ; A ) H_{*}(X;A)
  8. H k ( X ; A ) = A r k H_{k}(X;A)=A^{r_{k}}
  9. H k ( X ; A ) H_{k}(X;A)
  10. π k S ( X ) \pi^{S}_{k}(X)
  11. M O * ( X ) , M S O * ( X ) , M U * ( X ) MO_{*}(X),MSO_{*}(X),MU_{*}(X)
  12. K O * ( X ) KO_{*}(X)
  13. k O * ( X ) kO_{*}(X)
  14. K U * ( X ) KU_{*}(X)
  15. k U * ( X ) kU_{*}(X)
  16. E * ( X ) E^{*}(X)

Coincidence_circuit.html

  1. P P
  2. P 2 P^{2}
  3. P = 0.1 P=0.1
  4. P 2 = 0.01 P^{2}=0.01

Cokernel.html

  1. coeq ( f , g ) = coker ( g - f ) \mathrm{coeq}(f,g)=\mathrm{coker}(g-f)
  2. im ( f ) = ker ( coker f ) coim ( f ) = coker ( ker f ) \begin{aligned}\displaystyle\mathrm{im}(f)&\displaystyle=\ker(\mathrm{coker}f)% \\ \displaystyle\mathrm{coim}(f)&\displaystyle=\mathrm{coker}(\ker f)\end{aligned}
  3. m = ker ( coker ( m ) ) m=\ker(\mathrm{coker}(m))
  4. 0 ker T V W coker T 0. 0\to\ker T\to V\to W\to\mathrm{coker}\,T\to 0.

Collision_detection.html

  1. I 1 × I 2 × I 3 = [ a 1 , b 1 ] × [ a 2 , b 2 ] × [ a 3 , b 3 ] I_{1}\times I_{2}\times I_{3}=[a_{1},b_{1}]\times[a_{2},b_{2}]\times[a_{3},b_{% 3}]
  2. I 1 × I 2 × I 3 I_{1}\times I_{2}\times I_{3}
  3. J 1 × J 2 × J 3 J_{1}\times J_{2}\times J_{3}
  4. I 1 I_{1}
  5. J 1 J_{1}
  6. I 2 I_{2}
  7. J 2 J_{2}
  8. I 3 I_{3}
  9. J 3 J_{3}
  10. I k I_{k}
  11. J k J_{k}
  12. n n
  13. n × n n\times n
  14. M = ( m i j ) M=(m_{ij})
  15. m i j m_{ij}
  16. i i
  17. j j
  18. M M
  19. M M
  20. S = S 1 , S 2 , , S n S={S_{1},S_{2},\dots,S_{n}}
  21. T = T 1 , T 2 , , T n T={T_{1},T_{2},\dots,T_{n}}
  22. S j S_{j}
  23. T k T_{k}
  24. n 2 n^{2}
  25. S S
  26. T T
  27. S S
  28. T T
  29. E E
  30. B ( E ) B(E)
  31. B ( E ) B(E)
  32. B ( E ) B(E)
  33. E E
  34. B ( S ) B(S)
  35. B ( T ) B(T)
  36. S S
  37. T T
  38. E = E 1 , E 2 , , E m E={E_{1},E_{2},\dots,E_{m}}
  39. L ( E ) := E 1 , E 2 , , E m / 2 L(E):={E_{1},E_{2},\dots,E_{m/2}}
  40. R ( E ) := E m / 2 + 1 , , E m - 1 , E m R(E):={E_{m/2+1},\dots,E_{m-1},E_{m}}
  41. S S
  42. T T
  43. B ( L ( S ) ) , B ( R ( S ) ) B(L(S)),B(R(S))
  44. B ( L ( T ) ) , B ( R ( T ) ) B(L(T)),B(R(T))
  45. B ( S ) B(S)
  46. B ( T ) B(T)
  47. B ( S ) B(S)
  48. B ( L ( T ) ) B(L(T))
  49. S S
  50. L ( T ) L(T)
  51. N N
  52. L ( N ) L(N)
  53. R ( N ) R(N)
  54. B ( N ) B(N)
  55. B ( T ) B(T)
  56. v 1 , v 2 , v 3 {v_{1},v_{2},v_{3}}
  57. v 4 , v 5 , v 6 {v_{4},v_{5},v_{6}}
  58. v j v_{j}
  59. 3 \mathbb{R}^{3}
  60. v i , v j , v k v_{i},v_{j},v_{k}
  61. v 1 ( t ) , v 2 ( t ) , v 3 ( t ) {v_{1}(t),v_{2}(t),v_{3}(t)}
  62. v 4 ( t ) , v 5 ( t ) , v 6 ( t ) {v_{4}(t),v_{5}(t),v_{6}(t)}
  63. P ( u , v , w ) P(u,v,w)
  64. u , v , w u,v,w
  65. 3 \mathbb{R}^{3}
  66. P ( v i ( t ) , v j ( t ) , v k ( t ) ) P(v_{i}(t),v_{j}(t),v_{k}(t))
  67. t t
  68. n n

Color_charge.html

  1. ϕ ( x ) ϕ(x)
  2. A μ A μ + μ ϕ ( x ) A_{\mu}\to A_{\mu}+\partial_{\mu}\phi(x)
  3. ψ exp [ i Q ϕ ( x ) ] ψ \psi\to\exp[iQ\phi(x)]\psi
  4. ψ ¯ exp [ - i Q ϕ ( x ) ] ψ ¯ \overline{\psi}\to\exp[-iQ\phi(x)]\overline{\psi}
  5. A μ A_{\mu}
  6. ψ ψ
  7. Q = 1 Q=−1
  8. ψ ψ
  9. ψ ψ
  10. ψ = ( ψ 1 ψ 2 ψ 3 ) \psi=\begin{pmatrix}\psi_{1}\\ \psi_{2}\\ \psi_{3}\end{pmatrix}
  11. ψ ¯ = ( ψ ¯ 1 * ψ ¯ 2 * ψ ¯ 3 * ) . \overline{\psi}=\begin{pmatrix}{\overline{\psi}}^{*}_{1}\\ {\overline{\psi}}^{*}_{2}\\ {\overline{\psi}}^{*}_{3}\end{pmatrix}.
  12. 𝐀 μ = A μ a λ a . {\mathbf{A}}_{\mu}=A_{\mu}^{a}\lambda_{a}.
  13. ψ U ψ ψ→Uψ
  14. U U
  15. 3 × 3 3 × 3
  16. ψ < s u b > i ψ<sub>i

Color_of_chemicals.html

  1. E = h f = h c λ E=hf=\frac{hc}{\lambda}\,\!
  2. λ \lambda

Column_space.html

  1. c 1 𝐯 1 + c 2 𝐯 2 + + c n 𝐯 n , c_{1}\mathbf{v}_{1}+c_{2}\mathbf{v}_{2}+\cdots+c_{n}\mathbf{v}_{n},
  2. A [ c 1 c n ] = [ a 11 a 1 n a m 1 a m n ] [ c 1 c n ] = [ c 1 a 11 + + c n a 1 n c 1 a m 1 + + c n a m n ] = c 1 [ a 11 a m 1 ] + + c n [ a 1 n a m n ] = c 1 𝐯 1 + + c n 𝐯 n \begin{array}[]{rcl}A\begin{bmatrix}c_{1}\\ \vdots\\ c_{n}\end{bmatrix}&=&\begin{bmatrix}a_{11}&\cdots&a_{1n}\\ \vdots&\ddots&\vdots\\ a_{m1}&\cdots&a_{mn}\end{bmatrix}\begin{bmatrix}c_{1}\\ \vdots\\ c_{n}\end{bmatrix}=\begin{bmatrix}c_{1}a_{11}+&\cdots&+c_{n}a_{1n}\\ \vdots&\vdots&\vdots\\ c_{1}a_{m1}+&\cdots&+c_{n}a_{mn}\end{bmatrix}=c_{1}\begin{bmatrix}a_{11}\\ \vdots\\ a_{m1}\end{bmatrix}+\cdots+c_{n}\begin{bmatrix}a_{1n}\\ \vdots\\ a_{mn}\end{bmatrix}\\ &=&c_{1}\mathbf{v}_{1}+\cdots+c_{n}\mathbf{v}_{n}\end{array}
  3. A = [ 1 0 0 1 2 0 ] A=\begin{bmatrix}1&0\\ 0&1\\ 2&0\end{bmatrix}
  4. c 1 [ 1 0 2 ] + c 2 [ 0 1 0 ] = [ c 1 c 2 2 c 1 ] c_{1}\begin{bmatrix}1\\ 0\\ 2\end{bmatrix}+c_{2}\begin{bmatrix}0\\ 1\\ 0\end{bmatrix}=\begin{bmatrix}c_{1}\\ c_{2}\\ 2c_{1}\end{bmatrix}\,
  5. A = [ 1 3 1 4 2 7 3 9 1 5 3 1 1 2 0 8 ] . A=\begin{bmatrix}1&3&1&4\\ 2&7&3&9\\ 1&5&3&1\\ 1&2&0&8\end{bmatrix}\,\text{.}
  6. [ 1 3 1 4 2 7 3 9 1 5 3 1 1 2 0 8 ] [ 1 3 1 4 0 1 1 1 0 2 2 - 3 0 - 1 - 1 4 ] [ 1 0 - 2 1 0 1 1 1 0 0 0 - 5 0 0 0 5 ] [ 1 0 - 2 0 0 1 1 0 0 0 0 1 0 0 0 0 ] . \begin{bmatrix}1&3&1&4\\ 2&7&3&9\\ 1&5&3&1\\ 1&2&0&8\end{bmatrix}\sim\begin{bmatrix}1&3&1&4\\ 0&1&1&1\\ 0&2&2&-3\\ 0&-1&-1&4\end{bmatrix}\sim\begin{bmatrix}1&0&-2&1\\ 0&1&1&1\\ 0&0&0&-5\\ 0&0&0&5\end{bmatrix}\sim\begin{bmatrix}1&0&-2&0\\ 0&1&1&0\\ 0&0&0&1\\ 0&0&0&0\end{bmatrix}\,\text{.}
  7. [ 1 2 1 1 ] , [ 3 7 5 2 ] , [ 4 9 1 8 ] . \begin{bmatrix}1\\ 2\\ 1\\ 1\end{bmatrix},\;\;\begin{bmatrix}3\\ 7\\ 5\\ 2\end{bmatrix},\;\;\begin{bmatrix}4\\ 9\\ 1\\ 8\end{bmatrix}\,\text{.}
  8. rank ( A ) + nullity ( A ) = n . \,\text{rank}(A)+\,\text{nullity}(A)=n.\,
  9. A 𝖳 𝐱 = [ 𝐯 1 𝐱 𝐯 2 𝐱 𝐯 n 𝐱 ] , A^{\mathsf{T}}\mathbf{x}=\begin{bmatrix}\mathbf{v}_{1}\cdot\mathbf{x}\\ \mathbf{v}_{2}\cdot\mathbf{x}\\ \vdots\\ \mathbf{v}_{n}\cdot\mathbf{x}\end{bmatrix},
  10. k = 1 n 𝐯 k c k \sum\limits_{k=1}^{n}\mathbf{v}_{k}c_{k}

Column_vector.html

  1. 𝐱 = [ x 1 x 2 x m ] \mathbf{x}=\begin{bmatrix}x_{1}\\ x_{2}\\ \vdots\\ x_{m}\end{bmatrix}
  2. [ x 1 x 2 x m ] T = [ x 1 x 2 x m ] \begin{bmatrix}x_{1}\\ x_{2}\\ \vdots\\ x_{m}\end{bmatrix}^{\rm T}=\begin{bmatrix}x_{1}\;x_{2}\;\dots\;x_{m}\end{bmatrix}
  3. 𝐱 = [ x 1 x 2 x m ] T \mathbf{x}=\begin{bmatrix}x_{1}\;x_{2}\;\dots\;x_{m}\end{bmatrix}^{\rm T}
  4. 𝐱 = [ x 1 , x 2 , , x m ] T \mathbf{x}=\begin{bmatrix}x_{1},x_{2},\dots,x_{m}\end{bmatrix}^{\rm T}
  5. [ x 1 x 2 x m ] \begin{bmatrix}x_{1}\;x_{2}\;\dots\;x_{m}\end{bmatrix}
  6. [ x 1 x 2 x m ] or [ x 1 x 2 x m ] T \begin{bmatrix}x_{1}\\ x_{2}\\ \vdots\\ x_{m}\end{bmatrix}\,\text{ or }\begin{bmatrix}x_{1}\;x_{2}\;\dots\;x_{m}\end{% bmatrix}^{\rm T}
  7. [ x 1 , x 2 , , x m ] \begin{bmatrix}x_{1},x_{2},\dots,x_{m}\end{bmatrix}\qquad
  8. [ x 1 , x 2 , , x m ] T \begin{bmatrix}x_{1},x_{2},\dots,x_{m}\end{bmatrix}^{\rm T}
  9. [ x 1 , x 2 , , x m ] \begin{bmatrix}x_{1},x_{2},\dots,x_{m}\end{bmatrix}\qquad
  10. [ x 1 ; x 2 ; ; x m ] \begin{bmatrix}x_{1};x_{2};\dots;x_{m}\end{bmatrix}
  11. 𝐚 𝐛 = 𝐚 T 𝐛 = [ a 1 a 2 a 3 ] [ b 1 b 2 b 3 ] . \mathbf{a}\cdot\mathbf{b}=\mathbf{a}^{\mathrm{T}}\mathbf{b}=\begin{bmatrix}a_{% 1}&a_{2}&a_{3}\end{bmatrix}\begin{bmatrix}b_{1}\\ b_{2}\\ b_{3}\end{bmatrix}.

Combination_tone.html

  1. f f
  2. 2 f 2f
  3. 4 f , 6 f , 8 f , 4f,6f,8f,
  4. 3 f , 6 f , 9 f , 3f,6f,9f,
  5. 2 f , 3 f , 4 f , 6 f , 8 f , 9 f , 2f,3f,4f,6f,8f,9f,
  6. f 2 - f 1 f_{2}-f_{1}
  7. 2 f 1 - f 2 , 3 f 1 - 2 f 2 , , f 1 - k ( f 2 - f 1 ) 2f_{1}-f_{2},3f_{1}-2f_{2},\ldots,f_{1}-k(f_{2}-f_{1})

Combinational_logic.html

  1. A A
  2. B B
  3. C C
  4. ¬ A ¬ B ¬ C \neg A\cdot\neg B\cdot\neg C
  5. ¬ A ¬ B C \neg A\cdot\neg B\cdot C
  6. ¬ A B ¬ C \neg A\cdot B\cdot\neg C
  7. ¬ A B C \neg A\cdot B\cdot C
  8. A ¬ B ¬ C A\cdot\neg B\cdot\neg C
  9. A ¬ B C A\cdot\neg B\cdot C
  10. A B ¬ C A\cdot B\cdot\neg C
  11. A B C A\cdot B\cdot C
  12. A ¬ B ¬ C + A B C A\cdot\neg B\cdot\neg C+A\cdot B\cdot C\,
  13. A ( ¬ B ¬ C + B C ) A\cdot(\neg B\cdot\neg C+B\cdot C)\,
  14. ( A + B ) ( A + C ) = A + ( B C ) (A+B)\cdot(A+C)=A+(B\cdot C)
  15. ( A B ) + ( A C ) = A ( B + C ) \quad(A\cdot B)+(A\cdot C)=A\cdot(B+C)
  16. A + ( A B ) = A A+(A\cdot B)=A
  17. A ( A + B ) = A \quad A\cdot(A+B)=A
  18. A + ( ¬ A B ) = A + B A+(\lnot A\cdot B)=A+B
  19. A ( ¬ A + B ) = A B \quad A\cdot(\lnot A+B)=A\cdot B
  20. ( A + B ) ( ¬ A + B ) = B (A+B)\cdot(\lnot A+B)=B
  21. ( A B ) + ( ¬ A B ) = B \quad(A\cdot B)+(\lnot A\cdot B)=B
  22. ( A B ) + ( ¬ A C ) + ( B C ) = ( A B ) + ( ¬ A C ) (A\cdot B)+(\lnot A\cdot C)+(B\cdot C)=(A\cdot B)+(\lnot A\cdot C)
  23. ( A + B ) ( ¬ A + C ) ( B + C ) = ( A + B ) ( ¬ A + C ) (A+B)\cdot(\lnot A+C)\cdot(B+C)=(A+B)\cdot(\lnot A+C)

Combinatorial_game_theory.html

  1. L L
  2. R R
  3. { ( A 1 , A 2 ) , ( B 1 , B 2 ) , | ( A 1 , B 1 ) , ( A 2 , B 2 ) , } . \{(A1,A2),(B1,B2),\dots|(A1,B1),(A2,B2),\dots\}.
  4. { ( A 1 , A 2 ) | ( A 1 , B 1 ) } = { { | } | { | } } . \{(A1,A2)|(A1,B1)\}=\{\{|\}|\{|\}\}.

Combinatorial_species.html

  1. \mathcal{B}
  2. F : ; F\colon\mathcal{B}\to\mathcal{B};\,
  3. \mathcal{B}
  4. F ( x ) = n 0 f n x n n ! F(x)=\sum_{n\geq 0}f_{n}\frac{x^{n}}{n!}
  5. f n f_{n}
  6. f n = 1 f_{n}=1
  7. E ( x ) = e x E(x)=e^{x}
  8. f n = n ! f_{n}=n!
  9. S ( x ) = 1 / ( 1 - x ) S(x)=1/(1-x)
  10. f n = n 2 f_{n}=n^{2}
  11. T 2 ( x ) = x ( x + 1 ) e x T_{2}(x)=x(x+1)e^{x}
  12. ( F G ) [ A ] = A = B + C F [ B ] × G [ C ] . (F\cdot G)[A]=\sum_{A=B+C}F[B]\times G[C].
  13. ( F G ) [ A ] = π P [ A ] ( F [ π ] × Π B π G [ B ] ) . (F\circ G)[A]=\sum_{\pi\in P[A]}(F[\pi]\times\Pi_{B\in\pi}G[B]).
  14. ( F G ) ( x ) = F ( G ( x ) ) (F\circ G)(x)=F(G(x))
  15. P ( x ) = e ( e x - 1 ) P(x)=e^{(e^{x}-1)}
  16. ( F ) [ A ] = F [ A { } ] , (F^{\prime})[A]=F[A\uplus\{\star\}],
  17. \star
  18. A A
  19. d d x ( 1 - x ) - 1 = ( 1 - x ) - 2 . \frac{d}{dx}{(1-x)}^{-1}={(1-x)}^{-2}.
  20. C ( x ) = 0 x d t 1 - t = log 1 1 - x . C(x)=\int_{0}^{x}\frac{dt}{1-t}=\log\frac{1}{1-x}.
  21. Z Fano ( x 1 , x 2 , x 3 , x 4 , x 7 ) = 1 168 [ x 1 7 + 21 x 1 3 x 2 2 + 42 x 1 x 2 x 4 + 56 x 1 x 3 2 + 48 x 7 ] Z_{\mathrm{Fano}}(x_{1},x_{2},x_{3},x_{4},x_{7})={1\over 168}[x_{1}^{7}+21x_{1% }^{3}x_{2}^{2}+42x_{1}x_{2}x_{4}+56x_{1}x_{3}^{2}+48x_{7}]
  22. Z Fano ′′ = x 1 . 1 4 [ x 1 4 + 3 x 2 2 ] = x 1 Z Klein Z_{\mathrm{Fano}}^{\prime\prime}=x_{1}.{1\over 4}[x_{1}^{4}+3x_{2}^{2}]=x_{1}Z% _{\mathrm{Klein}}
  23. x 6 x 4 4 ! = 30 x 7 7 ! \int\int x\cdot 6\cdot{x^{4}\over 4!}=30\cdot{x^{7}\over 7!}
  24. ( F G ) [ A ] = F [ G [ A ] ] (F\Box G)[A]=F[G[A]]
  25. F ~ ( x ) = n 0 | T ( F n ) | x n . \tilde{F}(x)=\sum_{n\geq 0}|T(F_{n})|x^{n}.
  26. Z F ( x 1 , x 2 , ) = n 0 1 n ! ( σ S n | Fix ( F [ σ ] ) | x 1 σ 1 x 2 σ 2 ) . Z_{F}(x_{1},x_{2},\dots)=\sum_{n\geq 0}\frac{1}{n!}\left(\sum_{\sigma\in S_{n}% }|\mathrm{Fix}(F[\sigma])|x_{1}^{\sigma_{1}}x_{2}^{\sigma_{2}}\cdots\right).
  27. F ( x ) = Z F ( x , 0 , 0 , ) F(x)=Z_{F}(x,0,0,\dots)\,
  28. F ~ ( x ) = Z F ( x , x 2 , x 3 , ) \tilde{F}(x)=Z_{F}(x,x^{2},x^{3},\dots)\,
  29. Z F + G = Z F + Z G Z_{F+G}=Z_{F}+Z_{G}\,
  30. Z F G ( x 1 , x 2 , ) = Z F ( x 1 , x 2 , ) Z G ( x 1 , x 2 , ) Z_{F\cdot G}(x_{1},x_{2},\dots)=Z_{F}(x_{1},x_{2},\dots)\,Z_{G}(x_{1},x_{2},\dots)
  31. Z F G ( x 1 , x 2 , x 3 , ) = Z F ( Z G ( x 1 , x 2 , x 3 , ) , Z G ( x 2 , x 4 , x 6 , ) , Z G ( x 3 , x 6 , x 9 , ) , ) Z_{F\circ G}(x_{1},x_{2},x_{3},\dots)=Z_{F}(Z_{G}(x_{1},x_{2},x_{3},\dots),Z_{% G}(x_{2},x_{4},x_{6},\dots),Z_{G}(x_{3},x_{6},x_{9},\dots),\dots)
  32. Z F ( x 1 , x 2 , ) = ( x 1 Z F ) ( x 1 , x 2 , ) Z_{F^{\prime}}(x_{1},x_{2},\dots)=\left(\frac{\partial}{\partial x_{1}}Z_{F}% \right)(x_{1},x_{2},\dots)
  33. Z F ( x 1 , x 2 , ) = x 1 Z F ( x 1 , x 2 , ) Z_{F^{\bullet}}(x_{1},x_{2},\dots)=x_{1}Z_{F^{\prime}}(x_{1},x_{2},\dots)
  34. Z F × G ( x 1 , x 2 , ) = Z F ( x 1 , x 2 , ) × Z G ( x 1 , x 2 , ) . Z_{F\times G}(x_{1},x_{2},\dots)=Z_{F}(x_{1},x_{2},\dots)\times Z_{G}(x_{1},x_% {2},\dots).\,
  35. ( F G ) ~ ( x ) = Z F ( G ~ ( x ) , G ~ ( x 2 ) , G ~ ( x 3 ) , ) \tilde{(F\circ G)}(x)=Z_{F}(\tilde{G}(x),\tilde{G}(x^{2}),\tilde{G}(x^{3}),\dots)
  36. F ~ ( x ) = ( x 1 Z F ) ( x , x 2 , x 3 , ) \tilde{F^{\prime}}(x)=\left(\frac{\partial}{\partial x_{1}}Z_{F}\right)(x,x^{2% },x^{3},\dots)
  37. F ~ ( x ) = x F ~ ( x ) \tilde{F^{\bullet}}(x)=x\tilde{F^{\prime}}(x)
  38. F × G ~ ( x ) = ( Z F × Z G ) ( x , x 2 , x 3 , ) . \tilde{F\times G}(x)=(Z_{F}\times Z_{G})(x,x^{2},x^{3},\dots).
  39. \mathcal{B}
  40. \varnothing
  41. { } \{\,\varnothing\,\}
  42. \mathcal{B}
  43. \mathcal{B}
  44. k \mathcal{B}^{k}\rightarrow\mathcal{B}
  45. R \mathcal{B}_{R}

Combinatory_logic.html

  1. W , \langle W,\subseteq\rangle
  2. \Vdash
  3. X A A X . X\Vdash A\iff A\in X.
  4. X A B X\Vdash A\to B
  5. Y W Y\in W
  6. Y X Y\supseteq X
  7. Y A Y\Vdash A
  8. Y B Y\Vdash B
  9. X ⊮ A B X\not\Vdash A\to B
  10. X , A ⊬ B X,A\not\vdash B
  11. X { A } X\cup\{A\}
  12. Y W Y\in W
  13. Y X Y\supseteq X
  14. Y A Y\Vdash A
  15. Y ⊮ B Y\not\Vdash B
  16. X ⊮ A X\not\Vdash A

Common_base.html

  1. A v = v o v i | R L = {A_{v}}=\begin{matrix}{v_{\mathrm{o}}\over v_{\mathrm{i}}}\end{matrix}\Big|_{R% _{L}=\infty}
  2. ( g m r o + 1 ) R C R C + r o \begin{matrix}\frac{(g_{m}r_{\mathrm{o}}+1)R_{C}}{R_{C}+r_{o}}\end{matrix}
  3. g m R C \begin{matrix}g_{m}R_{C}\end{matrix}
  4. r o R C r_{o}\gg R_{C}
  5. A i = i o i i | R L = 0 A_{i}=\begin{matrix}{i_{\mathrm{o}}\over i_{\mathrm{i}}}\end{matrix}\Big|_{R_{% L}=0}
  6. r π + β r o r π + ( β + 1 ) r o \begin{matrix}\frac{r_{\pi}+\beta r_{o}}{r_{\pi}+(\beta+1)r_{o}}\end{matrix}% \begin{matrix}\end{matrix}
  7. 1 1
  8. β 1 \beta\gg 1
  9. R in = v i i i R_{\mathrm{in}}=\begin{matrix}\frac{v_{i}}{i_{i}}\end{matrix}
  10. ( r o + R C R L ) r E r o + r E + R C R L β + 1 \begin{matrix}\frac{(r_{o}+R_{C}\|R_{L})r_{E}}{r_{o}+r_{E}+\frac{R_{C}\|R_{L}}% {\beta+1}}\end{matrix}
  11. r e ( 1 g m ) r_{e}\left(\approx\frac{1}{g_{m}}\right)
  12. r o R C R L ( β 1 ) r_{o}\gg R_{C}\|R_{L}\ \ \left(\beta\gg 1\right)
  13. R out = v o - i o | v s = 0 R_{\mathrm{out}}=\begin{matrix}\frac{v_{o}}{-i_{o}}\end{matrix}\Big|_{v_{s}=0}
  14. R C { [ 1 + g m ( r π R S ) ] r o + ( r π R S ) } R_{C}\|\{[1+g_{m}(r_{\pi}\|R_{S})]r_{o}+(r_{\pi}\|R_{S})\}
  15. R C | | r o R_{C}||r_{o}
  16. R C | | [ r o ( 1 + g m ( r π | | R S ) ) ] R_{C}||\left[r_{o}(1+g_{m}(r_{\pi}||R_{S}))\right]
  17. R S r E \ \ R_{S}\ll r_{E}
  18. R S r E \ \ R_{S}\gg r_{E}
  19. R i n = r E = V T I E R_{in}=r_{E}=\begin{matrix}\frac{V_{T}}{I_{E}}\end{matrix}
  20. v o u t = i i n R L = v s R L R S v_{out}=i_{in}R_{L}=v_{s}\begin{matrix}\frac{R_{L}}{R_{S}}\end{matrix}
  21. \
  22. \rightarrow
  23. A v = v o u t v S = R L R S A_{v}=\begin{matrix}\frac{v_{out}}{v_{S}}=\frac{R_{L}}{R_{S}}\end{matrix}
  24. A v = v o u t v S = R L r E g m R L A_{v}=\begin{matrix}\frac{v_{out}}{v_{S}}=\frac{R_{L}}{r_{E}}\approx g_{m}R_{L% }\end{matrix}

Common_logarithm.html

  1. log 10 120 = log 10 ( 10 2 × 1.2 ) = 2 + log 10 1.2 2 + 0.079181. \log_{10}120=\log_{10}(10^{2}\times 1.2)=2+\log_{10}1.2\approx 2+0.079181.
  2. log 10 0.012 = log 10 ( 10 - 2 × 1.2 ) = - 2 + log 10 1.2 - 2 + 0.079181 = - 1.920819 \log_{10}0.012=\log_{10}(10^{-2}\times 1.2)=-2+\log_{10}1.2\approx-2+0.079181=% -1.920819
  3. log 10 0.012 - 2 + 0.079181 = 2 ¯ .079181 \log_{10}0.012\approx-2+0.079181=\bar{2}.079181
  4. n ¯ \bar{n}
  5. 2 ¯ .079181 \bar{2}.079181
  6. 1 ¯ \overline{1}
  7. 6 ¯ \overline{6}
  8. x x
  9. log 10 ( x × 10 i ) = log 10 ( x ) + log 10 ( 10 i ) = log 10 ( x ) + i \log_{10}(x\times 10^{i})=\log_{10}(x)+\log_{10}(10^{i})=\log_{10}(x)+i
  10. i i
  11. log 10 ( x ) \log_{10}(x)
  12. x x
  13. As found above, log 10 0.012 2 ¯ .079181 Since log 10 0.85 = log 10 ( 10 - 1 × 8.5 ) = - 1 + log 10 8.5 - 1 + 0.929419 = 1 ¯ .929419 , log 10 ( 0.012 × 0.85 ) = log 10 0.012 + log 10 0.85 2 ¯ .079181 + 1 ¯ .929419 = ( - 2 + 0.079181 ) + ( - 1 + 0.929419 ) = - ( 2 + 1 ) + ( 0.079181 + 0.929419 ) = - 3 + 1.008600 = - 2 + 0.008600 * log 10 ( 10 - 2 ) + log 10 ( 1.02 ) = log 10 ( 0.01 × 1.02 ) = log 10 ( 0.0102 ) \begin{array}[]{rll}\,\text{As found above,}&\log_{10}0.012\approx\bar{2}.0791% 81\\ \,\text{Since}\;\;\log_{10}0.85&=\log_{10}(10^{-1}\times 8.5)=-1+\log_{10}8.5&% \approx-1+0.929419=\bar{1}.929419\;,\\ \log_{10}(0.012\times 0.85)&=\log_{10}0.012+\log_{10}0.85&\approx\bar{2}.07918% 1+\bar{1}.929419\\ &=(-2+0.079181)+(-1+0.929419)&=-(2+1)+(0.079181+0.929419)\\ &=-3+1.008600&=-2+0.008600\;^{*}\\ &\approx\log_{10}(10^{-2})+\log_{10}(1.02)&=\log_{10}(0.01\times 1.02)\\ &=\log_{10}(0.0102)\end{array}
  14. log 10 ( x ) = ln ( x ) ln ( 10 ) or log 10 ( x ) = log 2 ( x ) log 2 ( 10 ) \log_{10}(x)=\frac{\ln(x)}{\ln(10)}\qquad\,\text{ or }\qquad\log_{10}(x)=\frac% {\log_{2}(x)}{\log_{2}(10)}

Communicating_sequential_processes.html

  1. a P a\rightarrow P
  2. a \mathit{a}
  3. a \mathit{a}
  4. P \mathit{P}
  5. ( a P ) ( b Q ) \left(a\rightarrow P\right)\Box\left(b\rightarrow Q\right)
  6. a \mathit{a}
  7. b \mathit{b}
  8. P \mathit{P}
  9. Q \mathit{Q}
  10. a \mathit{a}
  11. b \mathit{b}
  12. ( a P ) ( b Q ) \left(a\rightarrow P\right)\sqcap\left(b\rightarrow Q\right)
  13. ( a P ) \left(a\rightarrow P\right)
  14. ( b Q ) \left(b\rightarrow Q\right)
  15. a \mathit{a}
  16. b \mathit{b}
  17. a \mathit{a}
  18. b \mathit{b}
  19. ( a a S T O P ) ( a b S T O P ) \left(a\rightarrow a\rightarrow STOP\right)\Box\left(a\rightarrow b\rightarrow STOP\right)
  20. a ( ( a S T O P ) ( b S T O P ) ) a\rightarrow\left(\left(a\rightarrow STOP\right)\sqcap\left(b\rightarrow STOP% \right)\right)
  21. P | | | Q P\;|||\;Q
  22. P \mathit{P}
  23. Q \mathit{Q}
  24. P | [ { a } ] | Q P\left|\left[\left\{a\right\}\right]\right|Q
  25. P \mathit{P}
  26. Q \mathit{Q}
  27. a \mathit{a}
  28. ( a P ) | [ { a } ] | ( a Q ) \left(a\rightarrow P\right)\left|\left[\left\{a\right\}\right]\right|\left(a% \rightarrow Q\right)
  29. a \mathit{a}
  30. P | [ { a } ] | Q P\left|\left[\left\{a\right\}\right]\right|Q
  31. ( a P ) | [ { a , b } ] | ( b Q ) \left(a\rightarrow P\right)\left|\left[\left\{a,b\right\}\right]\right|\left(b% \rightarrow Q\right)
  32. ( a P ) { a } \left(a\rightarrow P\right)\setminus\left\{a\right\}
  33. a \mathit{a}
  34. P \mathit{P}
  35. P \mathit{P}
  36. 𝑉𝑒𝑛𝑑𝑖𝑛𝑔𝑀𝑎𝑐ℎ𝑖𝑛𝑒 = 𝑐𝑜𝑖𝑛 𝑐ℎ𝑜𝑐 𝑆𝑇𝑂𝑃 \,\textit{VendingMachine}=\,\textit{coin}\rightarrow\,\textit{choc}\rightarrow% \,\textit{STOP}
  37. 𝑃𝑒𝑟𝑠𝑜𝑛 = ( 𝑐𝑜𝑖𝑛 𝑆𝑇𝑂𝑃 ) ( 𝑐𝑎𝑟𝑑 𝑆𝑇𝑂𝑃 ) \,\textit{Person}=(\,\textit{coin}\rightarrow\,\textit{STOP})\Box(\,\textit{% card}\rightarrow\,\textit{STOP})
  38. 𝑉𝑒𝑛𝑑𝑖𝑛𝑔𝑀𝑎𝑐ℎ𝑖𝑛𝑒 | [ { 𝑐𝑜𝑖𝑛 , 𝑐𝑎𝑟𝑑 } ] | 𝑃𝑒𝑟𝑠𝑜𝑛 𝑐𝑜𝑖𝑛 𝑐ℎ𝑜𝑐 𝑆𝑇𝑂𝑃 \,\textit{VendingMachine}\left|\left[\left\{\,\textit{coin},\,\textit{card}% \right\}\right]\right|\,\textit{Person}\equiv\,\textit{coin}\rightarrow\,% \textit{choc}\rightarrow\,\textit{STOP}
  39. 𝑉𝑒𝑛𝑑𝑖𝑛𝑔𝑀𝑎𝑐ℎ𝑖𝑛𝑒 | [ { 𝑐𝑜𝑖𝑛 } ] | 𝑃𝑒𝑟𝑠𝑜𝑛 ( 𝑐𝑜𝑖𝑛 𝑐ℎ𝑜𝑐 𝑆𝑇𝑂𝑃 ) ( 𝑐𝑎𝑟𝑑 𝑆𝑇𝑂𝑃 ) \,\textit{VendingMachine}\left|\left[\left\{\,\textit{coin}\right\}\right]% \right|\,\textit{Person}\equiv\left(\,\textit{coin}\rightarrow\,\textit{choc}% \rightarrow\,\textit{STOP}\right)\Box\left(\,\textit{card}\rightarrow\,\textit% {STOP}\right)
  40. ( ( 𝑐𝑜𝑖𝑛 𝑐ℎ𝑜𝑐 𝑆𝑇𝑂 P ) ( 𝑐𝑎𝑟𝑑 𝑆𝑇𝑂𝑃 ) ) { 𝑐𝑜𝑖𝑛 , c a r d } \left(\left(\,\textit{coin}\rightarrow\,\textit{choc}\rightarrow\,\textit{STO}% P\right)\Box\left(\,\textit{card}\rightarrow\,\textit{STOP}\right)\right)% \setminus\left\{\,\textit{coin},card\right\}
  41. ( 𝑐ℎ𝑜𝑐 𝑆𝑇𝑂𝑃 ) 𝑆𝑇𝑂𝑃 \left(\,\textit{choc}\rightarrow\,\textit{STOP}\right)\sqcap\,\textit{STOP}
  42. e \mathit{e}
  43. X \mathit{X}
  44. P r o c : := 𝑆𝑇𝑂𝑃 | 𝑆𝐾𝐼𝑃 | e 𝑃𝑟𝑜𝑐 ( prefixing ) | 𝑃𝑟𝑜𝑐 𝑃𝑟𝑜𝑐 ( external choice ) | 𝑃𝑟𝑜𝑐 𝑃𝑟𝑜𝑐 ( nondeterministic choice ) | 𝑃𝑟𝑜𝑐 | | | 𝑃𝑟𝑜𝑐 ( interleaving ) | 𝑃𝑟𝑜𝑐 | [ { X } ] | 𝑃𝑟𝑜𝑐 ( interface parallel ) | 𝑃𝑟𝑜𝑐 X ( hiding ) | 𝑃𝑟𝑜𝑐 ; 𝑃𝑟𝑜𝑐 ( sequential composition ) | if b then 𝑃𝑟𝑜𝑐 else P r o c ( boolean conditional ) | 𝑃𝑟𝑜𝑐 𝑃𝑟𝑜𝑐 ( timeout ) | 𝑃𝑟𝑜𝑐 𝑃𝑟𝑜𝑐 ( interrupt ) \begin{matrix}Proc&::=&\,\textit{STOP}&\\ &|&\,\textit{SKIP}&\\ &|&e\rightarrow\,\textit{Proc}&(\,\text{prefixing})\\ &|&\,\textit{Proc}\;\Box\;\,\textit{Proc}&(\,\text{external}\;\,\text{choice})% \\ &|&\,\textit{Proc}\;\sqcap\;\,\textit{Proc}&(\,\text{nondeterministic}\;\,% \text{choice})\\ &|&\,\textit{Proc}\;|||\;\,\textit{Proc}&(\,\text{interleaving})\\ &|&\,\textit{Proc}\;|[\{X\}]|\;\,\textit{Proc}&(\,\text{interface}\;\,\text{% parallel})\\ &|&\,\textit{Proc}\setminus X&(\,\text{hiding})\\ &|&\,\textit{Proc};\,\textit{Proc}&(\,\text{sequential}\;\,\text{composition})% \\ &|&\mathrm{if}\;b\;\mathrm{then}\;\,\textit{Proc}\;\mathrm{else}\;Proc&(\,% \text{boolean}\;\,\text{conditional})\\ &|&\,\textit{Proc}\;\triangleright\;\,\textit{Proc}&(\,\text{timeout})\\ &|&\,\textit{Proc}\;\triangle\;\,\textit{Proc}&(\,\text{interrupt})\end{matrix}
  45. 𝐝𝐢𝐯 \mathbf{div}
  46. 𝑡𝑟𝑎𝑐𝑒𝑠 ( 𝑆𝑇𝑂𝑃 ) = { } \mathit{traces}\left(\mathit{STOP}\right)=\left\{\langle\rangle\right\}
  47. 𝑆𝑇𝑂𝑃 \mathit{STOP}
  48. 𝑡𝑟𝑎𝑐𝑒𝑠 ( a b 𝑆𝑇𝑂𝑃 ) = { , a , a , b } \mathit{traces}\left(a\rightarrow b\rightarrow\mathit{STOP}\right)=\left\{% \langle\rangle,\langle a\rangle,\langle a,b\rangle\right\}
  49. ( a b 𝑆𝑇𝑂𝑃 ) (a\rightarrow b\rightarrow\mathit{STOP})
  50. a a
  51. a a
  52. b b
  53. P P
  54. 𝑡𝑟𝑎𝑐𝑒𝑠 ( P ) Σ \mathit{traces}\left(P\right)\subseteq\Sigma^{\ast}
  55. 𝑡𝑟𝑎𝑐𝑒𝑠 ( P ) \langle\rangle\in\mathit{traces}\left(P\right)
  56. 𝑡𝑟𝑎𝑐𝑒𝑠 ( P ) \mathit{traces}\left(P\right)
  57. s 1 s 2 𝑡𝑟𝑎𝑐𝑒𝑠 ( P ) s 1 𝑡𝑟𝑎𝑐𝑒𝑠 ( P ) s_{1}\smallfrown s_{2}\in\mathit{traces}\left(P\right)\implies s_{1}\in\mathit% {traces}\left(P\right)
  58. 𝑡𝑟𝑎𝑐𝑒𝑠 ( P ) \mathit{traces}\left(P\right)
  59. Σ \Sigma^{\ast}
  60. X Σ X\subseteq\Sigma
  61. ( s , X ) \left(s,X\right)
  62. s s
  63. X X
  64. s s
  65. ( 𝑡𝑟𝑎𝑐𝑒𝑠 ( P ) , 𝑓𝑎𝑖𝑙𝑢𝑟𝑒𝑠 ( P ) ) \left(\mathit{traces}\left(P\right),\mathit{failures}\left(P\right)\right)
  66. 𝑓𝑎𝑖𝑙𝑢𝑟𝑒𝑠 ( ( a 𝑆𝑇𝑂𝑃 ) ( b 𝑆𝑇𝑂𝑃 ) ) = { ( , ) , ( a , { a , b } ) , ( b , { a , b } ) } \mathit{failures}\left(\left(a\rightarrow\mathit{STOP}\right)\Box\left(b% \rightarrow\mathit{STOP}\right)\right)=\left\{\left(\langle\rangle,\emptyset% \right),\left(\langle a\rangle,\left\{a,b\right\}\right),\left(\langle b% \rangle,\left\{a,b\right\}\right)\right\}
  67. 𝑓𝑎𝑖𝑙𝑢𝑟𝑒𝑠 ( ( a 𝑆𝑇𝑂𝑃 ) ( b 𝑆𝑇𝑂𝑃 ) ) = { ( , { a } ) , ( , { b } ) , ( a , { a , b } ) , ( b , { a , b } ) } \mathit{failures}\left(\left(a\rightarrow\mathit{STOP}\right)\sqcap\left(b% \rightarrow\mathit{STOP}\right)\right)=\left\{\left(\langle\rangle,\left\{a% \right\}\right),\left(\langle\rangle,\left\{b\right\}\right),\left(\langle a% \rangle,\left\{a,b\right\}\right),\left(\langle b\rangle,\left\{a,b\right\}% \right)\right\}
  68. ( 𝑓𝑎𝑖𝑙𝑢𝑟𝑒𝑠 ( P ) , 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒𝑠 ( P ) ) \left(\mathit{failures}_{\perp}\left(P\right),\mathit{divergences}\left(P% \right)\right)
  69. 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒𝑠 ( P ) \mathit{divergences}\left(P\right)
  70. 𝑓𝑎𝑖𝑙𝑢𝑟𝑒𝑠 ( P ) = 𝑓𝑎𝑖𝑙𝑢𝑟𝑒𝑠 ( P ) { ( s , X ) s 𝑑𝑖𝑣𝑒𝑟𝑔𝑒𝑛𝑐𝑒𝑠 ( P ) } \mathit{failures}_{\perp}\left(P\right)=\mathit{failures}\left(P\right)\cup% \left\{\left(s,X\right)\mid s\in\mathit{divergences}\left(P\right)\right\}

Commutative_algebra.html

  1. \mathbb{Z}
  2. I 1 I k - 1 I k I k + 1 I_{1}\subseteq\cdots I_{k-1}\subseteq I_{k}\subseteq I_{k+1}\subseteq\cdots
  3. I n = I n + 1 = I_{n}=I_{n+1}=\cdots
  4. R [ X 0 , , X n - 1 ] R[X_{0},\ldots,X_{n-1}]
  5. R n R^{n}
  6. 𝔞 R [ X 0 , , X n - 1 ] \mathfrak{a}\subset R[X_{0},\ldots,X_{n-1}]
  7. A A
  8. R R
  9. A R [ X 0 , , X n - 1 ] / 𝔞 A\simeq R[X_{0},\ldots,X_{n-1}]/\mathfrak{a}
  10. 𝔞 \mathfrak{a}
  11. 𝔞 \mathfrak{a}
  12. 𝔞 = ( p 0 , , p N - 1 ) \mathfrak{a}=(p_{0},\ldots,p_{N-1})
  13. A A
  14. I = i = 1 t Q i I=\bigcap_{i=1}^{t}Q_{i}
  15. I = i = 1 k P i I=\bigcap_{i=1}^{k}P_{i}
  16. m s \frac{m}{s}
  17. V ( I ) = { P Spec ( A ) I P } V(I)=\{P\in\operatorname{Spec}\,(A)\mid I\subseteq P\}
  18. \mathbb{Z}
  19. R [ x 1 , , x n ] R[x_{1},...,x_{n}]

Commutative_diagram.html

  1. f = f ~ π f=\tilde{f}\circ\pi
  2. h f = k g h\circ f=k\circ g
  3. \hookrightarrow
  4. \twoheadrightarrow
  5. \overset{\sim}{\rightarrow}
  6. \exists
  7. ! !
  8. ! \exists!
  9. f : X X f\colon X\to X
  10. \bullet\rightrightarrows\bullet
  11. f , g : X Y f,g\colon X\to Y

Commutative_property.html

  1. " 3 5 5 3 " "3−5≠5−3"
  2. y + z = z + y for all y , z y+z=z+y\qquad\mbox{for all }~{}y,z\in\mathbb{R}
  3. y z = z y for all y , z yz=zy\qquad\mbox{for all }~{}y,z\in\mathbb{R}
  4. E A + T = E A T T E A = T + E A EA+T=EAT\neq TEA=T+EA
  5. 0 - 1 1 - 0 0-1\neq 1-0
  6. 1 / 2 2 / 1 1/2\neq 2/1
  7. [ 0 2 0 1 ] = [ 1 1 0 1 ] [ 0 1 0 1 ] [ 0 1 0 1 ] [ 1 1 0 1 ] = [ 0 1 0 1 ] \begin{bmatrix}0&2\\ 0&1\end{bmatrix}=\begin{bmatrix}1&1\\ 0&1\end{bmatrix}\cdot\begin{bmatrix}0&1\\ 0&1\end{bmatrix}\neq\begin{bmatrix}0&1\\ 0&1\end{bmatrix}\cdot\begin{bmatrix}1&1\\ 0&1\end{bmatrix}=\begin{bmatrix}0&1\\ 0&1\end{bmatrix}
  8. ( P Q ) ( Q P ) (PQ)\Leftrightarrow(QP)
  9. ( P and Q ) ( Q and P ) (P\and Q)\Leftrightarrow(Q\and P)
  10. \Leftrightarrow
  11. ( P and Q ) ( Q and P ) (P\and Q)\leftrightarrow(Q\and P)
  12. ( P Q ) ( Q P ) (PQ)\leftrightarrow(QP)
  13. ( P ( Q R ) ) ( Q ( P R ) ) (P\to(Q\to R))\leftrightarrow(Q\to(P\to R))
  14. ( P Q ) ( Q P ) (P\leftrightarrow Q)\leftrightarrow(Q\leftrightarrow P)
  15. f ( x , y ) = x + y 2 , f(x,y)=\frac{x+y}{2},
  16. f ( - 4 , f ( 0 , + 4 ) ) = - 1 f(-4,f(0,+4))=-1
  17. f ( f ( - 4 , 0 ) , + 4 ) = + 1 f(f(-4,0),+4)=+1
  18. a R b b R a aRb\Leftrightarrow bRa
  19. d d x \frac{d}{dx}
  20. x d d x x\frac{d}{dx}
  21. d d x x \frac{d}{dx}x
  22. ψ ( x ) \psi(x)
  23. x d d x ψ = x ψ d d x x ψ = ψ + x ψ x{d\over dx}\psi=x\psi^{\prime}\neq{d\over dx}x\psi=\psi+x\psi^{\prime}
  24. x x
  25. - i x -i\hbar\frac{\partial}{\partial x}
  26. \hbar
  27. - i -i\hbar

Comoving_distance.html

  1. χ = t e t c d t a ( t ) \chi=\int_{t_{e}}^{t}c\;{\mbox{d}~{}t^{\prime}\over a(t^{\prime})}
  2. χ \!\chi
  3. χ \!\chi
  4. d s 2 = - c 2 d τ 2 = - c 2 d t 2 + a ( t ) 2 ( d r 2 1 - k r 2 + r 2 ( d θ 2 + sin 2 θ d ϕ 2 ) ) \!ds^{2}=-c^{2}d\tau^{2}=-c^{2}dt^{2}+a(t)^{2}\left(\frac{dr^{2}}{1-kr^{2}}+r^% {2}\left(d\theta^{2}+\sin^{2}\theta d\phi^{2}\right)\right)
  5. r \!r
  6. χ \!\chi
  7. χ = r \!\chi=r
  8. χ = sin - 1 r \!\chi=\sin^{-1}r
  9. χ = sinh - 1 r \!\chi=\sinh^{-1}r
  10. χ \!\chi
  11. d ( t ) \!d(t)
  12. t \!t
  13. d ( t ) = a ( t ) χ \!d(t)=a(t)\chi
  14. a ( t ) \!a(t)
  15. d ( t ) \!d(t)
  16. v t o t = v r e c + v p e c \!v_{tot}=v_{rec}+v_{pec}
  17. v r e c \!v_{rec}
  18. v p e c \!v_{pec}
  19. v r e c = a ˙ ( t ) χ ( t ) \!v_{rec}=\dot{a}(t)\chi(t)
  20. v p e c = a ( t ) χ ˙ ( t ) \!v_{pec}=a(t)\dot{\chi}(t)
  21. v p e c \!v_{pec}
  22. v t o t \!v_{tot}

Compact_Muon_Solenoid.html

  1. ( 3.0 < | η | < 5.0 ) \scriptstyle(3.0\;<\;|\eta|\;<\;5.0)

CompactFlash.html

  1. R = K 150 kByte/s R={K\cdot 150}\ \,\text{kByte/s}

Compactness_theorem.html

  1. i \mathcal{M}_{i}
  2. i Σ i \prod_{i\subseteq\Sigma}\mathcal{M}_{i}
  3. j \mathcal{M}_{j}
  4. j \mathcal{M}_{j}
  5. i Σ i / U \prod_{i\subseteq\Sigma}\mathcal{M}_{i}/U