wpmath0000009_7

List_of_disproved_mathematical_ideas.html

  1. 2 2 m + 1 2^{2^{m}}+1
  2. 1 \beth_{1}
  3. 0 \aleph_{0}
  4. π ( x ) > li ( x ) \scriptstyle\pi(x)\,>\,\mathrm{li}(x)

List_of_Indian_inventions_and_discoveries.html

  1. arctan x \arctan x
  2. 104348 / 33215 104348/33215
  3. 3.14159265359 3.14159265359
  4. x 2 - N y 2 = 1 , \ x^{2}-Ny^{2}=1,
  5. π \pi

List_of_NBA_teams_by_single_season_win_percentage.html

  1. Average point differential = Total points for - Total points against Total games played \mbox{Average point differential}~{}=\frac{\mbox{Total points for}~{}-\mbox{% Total points against}~{}}{\mbox{Total games played}~{}}

List_of_New_Testament_papyri.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}
  7. 𝔓 \mathfrak{P}
  8. 𝔓 \mathfrak{P}
  9. 𝔓 \mathfrak{P}
  10. 𝔓 \mathfrak{P}
  11. 𝔓 \mathfrak{P}
  12. 𝔓 \mathfrak{P}
  13. 𝔓 \mathfrak{P}
  14. 𝔓 \mathfrak{P}
  15. 𝔓 \mathfrak{P}
  16. 𝔓 \mathfrak{P}
  17. 𝔓 \mathfrak{P}
  18. 𝔓 \mathfrak{P}
  19. 𝔓 \mathfrak{P}
  20. 𝔓 \mathfrak{P}
  21. 𝔓 \mathfrak{P}
  22. 𝔓 \mathfrak{P}
  23. 𝔓 \mathfrak{P}
  24. 𝔓 52 \mathfrak{P}^{52}
  25. 𝔓 1 \mathfrak{P}^{1}
  26. 𝔓 2 \mathfrak{P}^{2}
  27. 𝔓 3 \mathfrak{P}^{3}
  28. 𝔓 4 \mathfrak{P}^{4}
  29. 𝔓 5 \mathfrak{P}^{5}
  30. 𝔓 6 \mathfrak{P}^{6}
  31. 𝔓 7 \mathfrak{P}^{7}
  32. 𝔓 8 \mathfrak{P}^{8}
  33. 𝔓 9 \mathfrak{P}^{9}
  34. 𝔓 10 \mathfrak{P}^{10}
  35. 𝔓 11 \mathfrak{P}^{11}
  36. 𝔓 12 \mathfrak{P}^{12}
  37. 𝔓 13 \mathfrak{P}^{13}
  38. 𝔓 14 \mathfrak{P}^{14}
  39. 𝔓 15 \mathfrak{P}^{15}
  40. 𝔓 16 \mathfrak{P}^{16}
  41. 𝔓 17 \mathfrak{P}^{17}
  42. 𝔓 18 \mathfrak{P}^{18}
  43. 𝔓 19 \mathfrak{P}^{19}
  44. 𝔓 20 \mathfrak{P}^{20}
  45. 𝔓 21 \mathfrak{P}^{21}
  46. 𝔓 22 \mathfrak{P}^{22}
  47. 𝔓 23 \mathfrak{P}^{23}
  48. 𝔓 24 \mathfrak{P}^{24}
  49. 𝔓 25 \mathfrak{P}^{25}
  50. 𝔓 26 \mathfrak{P}^{26}
  51. 𝔓 27 \mathfrak{P}^{27}
  52. 𝔓 28 \mathfrak{P}^{28}
  53. 𝔓 29 \mathfrak{P}^{29}
  54. 𝔓 30 \mathfrak{P}^{30}
  55. 𝔓 31 \mathfrak{P}^{31}
  56. 𝔓 32 \mathfrak{P}^{32}
  57. 𝔓 33 \mathfrak{P}^{33}
  58. 𝔓 58 \mathfrak{P}^{58}
  59. 𝔓 34 \mathfrak{P}^{34}
  60. 𝔓 35 \mathfrak{P}^{35}
  61. 𝔓 36 \mathfrak{P}^{36}
  62. 𝔓 37 \mathfrak{P}^{37}
  63. 𝔓 38 \mathfrak{P}^{38}
  64. 𝔓 39 \mathfrak{P}^{39}
  65. 𝔓 40 \mathfrak{P}^{40}
  66. 𝔓 41 \mathfrak{P}^{41}
  67. 𝔓 42 \mathfrak{P}^{42}
  68. 𝔓 43 \mathfrak{P}^{43}
  69. 𝔓 44 \mathfrak{P}^{44}
  70. 𝔓 45 \mathfrak{P}^{45}
  71. 𝔓 46 \mathfrak{P}^{46}
  72. 𝔓 47 \mathfrak{P}^{47}
  73. 𝔓 48 \mathfrak{P}^{48}
  74. 𝔓 49 \mathfrak{P}^{49}
  75. 𝔓 50 \mathfrak{P}^{50}
  76. 𝔓 51 \mathfrak{P}^{51}
  77. 𝔓 52 \mathfrak{P}^{52}
  78. 𝔓 53 \mathfrak{P}^{53}
  79. 𝔓 54 \mathfrak{P}^{54}
  80. 𝔓 55 \mathfrak{P}^{55}
  81. 𝔓 56 \mathfrak{P}^{56}
  82. 𝔓 57 \mathfrak{P}^{57}
  83. 𝔓 58 \mathfrak{P}^{58}
  84. 𝔓 33 \mathfrak{P}^{33}
  85. 𝔓 59 \mathfrak{P}^{59}
  86. 𝔓 60 \mathfrak{P}^{60}
  87. 𝔓 61 \mathfrak{P}^{61}
  88. 𝔓 62 \mathfrak{P}^{62}
  89. 𝔓 63 \mathfrak{P}^{63}
  90. 𝔓 64 \mathfrak{P}^{64}
  91. 𝔓 67 \mathfrak{P}^{67}
  92. 𝔓 65 \mathfrak{P}^{65}
  93. 𝔓 66 \mathfrak{P}^{66}
  94. 𝔓 68 \mathfrak{P}^{68}
  95. 𝔓 69 \mathfrak{P}^{69}
  96. 𝔓 70 \mathfrak{P}^{70}
  97. 𝔓 71 \mathfrak{P}^{71}
  98. 𝔓 72 \mathfrak{P}^{72}
  99. 𝔓 73 \mathfrak{P}^{73}
  100. 𝔓 74 \mathfrak{P}^{74}
  101. 𝔓 75 \mathfrak{P}^{75}
  102. 𝔓 76 \mathfrak{P}^{76}
  103. 𝔓 77 \mathfrak{P}^{77}
  104. 𝔓 78 \mathfrak{P}^{78}
  105. 𝔓 79 \mathfrak{P}^{79}
  106. 𝔓 80 \mathfrak{P}^{80}
  107. 𝔓 81 \mathfrak{P}^{81}
  108. 𝔓 82 \mathfrak{P}^{82}
  109. 𝔓 83 \mathfrak{P}^{83}
  110. 𝔓 84 \mathfrak{P}^{84}
  111. 𝔓 85 \mathfrak{P}^{85}
  112. 𝔓 86 \mathfrak{P}^{86}
  113. 𝔓 87 \mathfrak{P}^{87}
  114. 𝔓 88 \mathfrak{P}^{88}
  115. 𝔓 89 \mathfrak{P}^{89}
  116. 𝔓 90 \mathfrak{P}^{90}
  117. 𝔓 91 \mathfrak{P}^{91}
  118. 𝔓 92 \mathfrak{P}^{92}
  119. 𝔓 93 \mathfrak{P}^{93}
  120. 𝔓 94 \mathfrak{P}^{94}
  121. 𝔓 95 \mathfrak{P}^{95}
  122. 𝔓 96 \mathfrak{P}^{96}
  123. 𝔓 97 \mathfrak{P}^{97}
  124. 𝔓 98 \mathfrak{P}^{98}
  125. 𝔓 99 \mathfrak{P}^{99}
  126. 𝔓 100 \mathfrak{P}^{100}
  127. 𝔓 101 \mathfrak{P}^{101}
  128. 𝔓 102 \mathfrak{P}^{102}
  129. 𝔓 103 \mathfrak{P}^{103}
  130. 𝔓 104 \mathfrak{P}^{104}
  131. 𝔓 105 \mathfrak{P}^{105}
  132. 𝔓 106 \mathfrak{P}^{106}
  133. 𝔓 107 \mathfrak{P}^{107}
  134. 𝔓 108 \mathfrak{P}^{108}
  135. 𝔓 109 \mathfrak{P}^{109}
  136. 𝔓 110 \mathfrak{P}^{110}
  137. 𝔓 111 \mathfrak{P}^{111}
  138. 𝔓 112 \mathfrak{P}^{112}
  139. 𝔓 113 \mathfrak{P}^{113}
  140. 𝔓 114 \mathfrak{P}^{114}
  141. 𝔓 115 \mathfrak{P}^{115}
  142. 𝔓 116 \mathfrak{P}^{116}
  143. 𝔓 117 \mathfrak{P}^{117}
  144. 𝔓 118 \mathfrak{P}^{118}
  145. 𝔓 119 \mathfrak{P}^{119}
  146. 𝔓 120 \mathfrak{P}^{120}
  147. 𝔓 121 \mathfrak{P}^{121}
  148. 𝔓 122 \mathfrak{P}^{122}
  149. 𝔓 123 \mathfrak{P}^{123}
  150. 𝔓 124 \mathfrak{P}^{124}
  151. 𝔓 125 \mathfrak{P}^{125}
  152. 𝔓 126 \mathfrak{P}^{126}
  153. 𝔓 127 \mathfrak{P}^{127}
  154. 𝔓 128 \mathfrak{P}^{128}
  155. 𝔓 129 \mathfrak{P}^{129}
  156. 𝔓 130 \mathfrak{P}^{130}
  157. 𝔓 131 \mathfrak{P}^{131}
  158. 𝔓 \mathfrak{P}
  159. 𝔓 \mathfrak{P}
  160. 𝔓 \mathfrak{P}
  161. 𝔓 \mathfrak{P}
  162. 𝔓 \mathfrak{P}
  163. 𝔓 \mathfrak{P}
  164. 𝔓 \mathfrak{P}
  165. 𝔓 \mathfrak{P}

List_of_operators.html

  1. L : 𝒢 L:\mathcal{F}\to\mathcal{G}
  2. y y\in\mathcal{F}
  3. L [ y ] 𝒢 L[y]\in\mathcal{G}
  4. \mathcal{F}
  5. 𝒢 \mathcal{G}
  6. L [ y ] = y ( n ) L[y]=y^{(n)}
  7. L [ y ] = a t y d t L[y]=\int_{a}^{t}y\,dt
  8. y = y ( x ) y=y(x)
  9. x = t x=t
  10. L [ y ] = y f L[y]=y\circ f
  11. L [ y ] = y t + y - t 2 L[y]=\frac{y\circ t+y\circ-t}{2}
  12. L [ y ] = y t - y - t 2 L[y]=\frac{y\circ t-y\circ-t}{2}
  13. L [ y ] = y ( t + 1 ) - y t = Δ y L[y]=y\circ(t+1)-y\circ t=\Delta y
  14. L [ y ] = y ( t ) - y ( t - 1 ) = y L[y]=y\circ(t)-y\circ(t-1)=\nabla y
  15. L [ y ] = y = Δ - 1 y L[y]=\sum y=\Delta^{-1}y
  16. L [ y ] = - ( p y ) + q y L[y]=-(py^{\prime})^{\prime}+qy\,
  17. F [ y ] = y [ - 1 ] F[y]=y^{[-1]}
  18. F [ y ] = t y [ - 1 ] - y y [ - 1 ] F[y]=t\,y^{\prime[-1]}-y\circ y^{\prime[-1]}
  19. F [ y ] = f y F[y]=f\circ y
  20. F [ y ] = y F[y]=\prod y
  21. F [ y ] = y y F[y]=\frac{y^{\prime}}{y}
  22. F [ y ] = t y y F[y]={\frac{ty^{\prime}}{y}}
  23. F [ y ] = y ′′′ y - 3 2 ( y ′′ y ) 2 F[y]={y^{\prime\prime\prime}\over y^{\prime}}-{3\over 2}\left({y^{\prime\prime% }\over y^{\prime}}\right)^{2}
  24. F [ y ] = a t | y | d t F[y]=\int_{a}^{t}|y^{\prime}|\,dt
  25. F [ y ] = 1 t - a a t y d t F[y]=\frac{1}{t-a}\int_{a}^{t}y\,dt
  26. F [ y ] = exp ( 1 t - a a t ln y d t ) F[y]=\exp\left(\frac{1}{t-a}\int_{a}^{t}\ln y\,dt\right)
  27. F [ y ] = - y y F[y]=-\frac{y}{y^{\prime}}
  28. y = y ( x ) y=y(x)
  29. x = t x=t
  30. F [ x , y ] = - y x y F[x,y]=-\frac{yx^{\prime}}{y^{\prime}}
  31. x = x ( t ) x=x(t)
  32. y = y ( t ) y=y(t)
  33. F [ r ] = - r 2 r F[r]=-\frac{r^{2}}{r^{\prime}}
  34. r = r ( ϕ ) r=r(\phi)
  35. ϕ = t \phi=t
  36. F [ r ] = 1 2 a t r 2 d t F[r]=\frac{1}{2}\int_{a}^{t}r^{2}dt
  37. r = r ( ϕ ) r=r(\phi)
  38. ϕ = t \phi=t
  39. F [ y ] = a t 1 + y 2 d t F[y]=\int_{a}^{t}\sqrt{1+y^{\prime 2}}\,dt
  40. y = y ( x ) y=y(x)
  41. x = t x=t
  42. F [ x , y ] = a t x 2 + y 2 d t F[x,y]=\int_{a}^{t}\sqrt{x^{\prime 2}+y^{\prime 2}}\,dt
  43. x = x ( t ) x=x(t)
  44. y = y ( t ) y=y(t)
  45. F [ r ] = a t r 2 + r 2 d t F[r]=\int_{a}^{t}\sqrt{r^{2}+r^{\prime 2}}\,dt
  46. r = r ( ϕ ) r=r(\phi)
  47. ϕ = t \phi=t
  48. F [ x , y ] = a t y ′′ 3 d t F[x,y]=\int_{a}^{t}\sqrt[3]{y^{\prime\prime}}\,dt
  49. y = y ( x ) y=y(x)
  50. x = t x=t
  51. F [ x , y ] = a t x y ′′ - x ′′ y 3 d t F[x,y]=\int_{a}^{t}\sqrt[3]{x^{\prime}y^{\prime\prime}-x^{\prime\prime}y^{% \prime}}\,dt
  52. x = x ( t ) x=x(t)
  53. y = y ( t ) y=y(t)
  54. F [ x , y , z ] = a t z ′′′ ( x y ′′ - y x ′′ ) + z ′′ ( x ′′′ y - x y ′′′ ) + z ( x ′′ y ′′′ - x ′′′ y ′′ ) 3 F[x,y,z]=\int_{a}^{t}\sqrt[3]{z^{\prime\prime\prime}(x^{\prime}y^{\prime\prime% }-y^{\prime}x^{\prime\prime})+z^{\prime\prime}(x^{\prime\prime\prime}y^{\prime% }-x^{\prime}y^{\prime\prime\prime})+z^{\prime}(x^{\prime\prime}y^{\prime\prime% \prime}-x^{\prime\prime\prime}y^{\prime\prime})}
  55. x = x ( t ) x=x(t)
  56. y = y ( t ) y=y(t)
  57. z = z ( t ) z=z(t)
  58. F [ y ] = y ′′ ( 1 + y 2 ) 3 / 2 F[y]=\frac{y^{\prime\prime}}{(1+y^{\prime 2})^{3/2}}
  59. y = y ( x ) y=y(x)
  60. x = t x=t
  61. F [ x , y ] = x y ′′ - y x ′′ ( x 2 + y 2 ) 3 / 2 F[x,y]=\frac{x^{\prime}y^{\prime\prime}-y^{\prime}x^{\prime\prime}}{(x^{\prime 2% }+y^{\prime 2})^{3/2}}
  62. x = x ( t ) x=x(t)
  63. y = y ( t ) y=y(t)
  64. F [ r ] = r 2 + 2 r 2 - r r ′′ ( r 2 + r 2 ) 3 / 2 F[r]=\frac{r^{2}+2r^{\prime 2}-rr^{\prime\prime}}{(r^{2}+r^{\prime 2})^{3/2}}
  65. r = r ( ϕ ) r=r(\phi)
  66. ϕ = t \phi=t
  67. F [ x , y , z ] = ( z ′′ y - z y ′′ ) 2 + ( x ′′ z - z ′′ x ) 2 + ( y ′′ x - x ′′ y ) 2 ( x 2 + y 2 + z 2 ) 3 / 2 F[x,y,z]=\frac{\sqrt{(z^{\prime\prime}y^{\prime}-z^{\prime}y^{\prime\prime})^{% 2}+(x^{\prime\prime}z^{\prime}-z^{\prime\prime}x^{\prime})^{2}+(y^{\prime% \prime}x^{\prime}-x^{\prime\prime}y^{\prime})^{2}}}{(x^{\prime 2}+y^{\prime 2}% +z^{\prime 2})^{3/2}}
  68. x = x ( t ) x=x(t)
  69. y = y ( t ) y=y(t)
  70. z = z ( t ) z=z(t)
  71. F [ y ] = 1 3 y ′′′′ ( y ′′ ) 5 / 3 - 5 9 y ′′′ 2 ( y ′′ ) 8 / 3 F[y]=\frac{1}{3}\frac{y^{\prime\prime\prime\prime}}{(y^{\prime\prime})^{5/3}}-% \frac{5}{9}\frac{y^{\prime\prime\prime 2}}{(y^{\prime\prime})^{8/3}}
  72. y = y ( x ) y=y(x)
  73. x = t x=t
  74. F [ x , y ] = x ′′ y ′′′ - x ′′′ y ′′ ( x y ′′ - x ′′ y ) 5 / 3 - 1 2 [ 1 ( x y ′′ - x ′′ y ) 2 / 3 ] ′′ F[x,y]=\frac{x^{\prime\prime}y^{\prime\prime\prime}-x^{\prime\prime\prime}y^{% \prime\prime}}{(x^{\prime}y^{\prime\prime}-x^{\prime\prime}y^{\prime})^{5/3}}-% \frac{1}{2}\left[\frac{1}{(x^{\prime}y^{\prime\prime}-x^{\prime\prime}y^{% \prime})^{2/3}}\right]^{\prime\prime}
  75. x = x ( t ) x=x(t)
  76. y = y ( t ) y=y(t)
  77. F [ x , y , z ] = z ′′′ ( x y ′′ - y x ′′ ) + z ′′ ( x ′′′ y - x y ′′′ ) + z ( x ′′ y ′′′ - x ′′′ y ′′ ) ( x 2 + y 2 + z 2 ) ( x ′′ 2 + y ′′ 2 + z ′′ 2 ) F[x,y,z]=\frac{z^{\prime\prime\prime}(x^{\prime}y^{\prime\prime}-y^{\prime}x^{% \prime\prime})+z^{\prime\prime}(x^{\prime\prime\prime}y^{\prime}-x^{\prime}y^{% \prime\prime\prime})+z^{\prime}(x^{\prime\prime}y^{\prime\prime\prime}-x^{% \prime\prime\prime}y^{\prime\prime})}{(x^{\prime 2}+y^{\prime 2}+z^{\prime 2})% (x^{\prime\prime 2}+y^{\prime\prime 2}+z^{\prime\prime 2})}
  78. x = x ( t ) x=x(t)
  79. y = y ( t ) y=y(t)
  80. z = z ( t ) z=z(t)
  81. X [ x , y ] = y y x - x y X[x,y]=\frac{y^{\prime}}{yx^{\prime}-xy^{\prime}}
  82. Y [ x , y ] = x x y - y x Y[x,y]=\frac{x^{\prime}}{xy^{\prime}-yx^{\prime}}
  83. x = x ( t ) x=x(t)
  84. y = y ( t ) y=y(t)
  85. X [ x , y ] = x + a y x 2 + y 2 X[x,y]=x+\frac{ay^{\prime}}{\sqrt{x^{\prime 2}+y^{\prime 2}}}
  86. Y [ x , y ] = y - a x x 2 + y 2 Y[x,y]=y-\frac{ax^{\prime}}{\sqrt{x^{\prime 2}+y^{\prime 2}}}
  87. x = x ( t ) x=x(t)
  88. y = y ( t ) y=y(t)
  89. X [ x , y ] = x + y x 2 + y 2 x ′′ y - y ′′ x X[x,y]=x+y^{\prime}\frac{x^{\prime 2}+y^{\prime 2}}{x^{\prime\prime}y^{\prime}% -y^{\prime\prime}x^{\prime}}
  90. Y [ x , y ] = y + x x 2 + y 2 y ′′ x - x ′′ y Y[x,y]=y+x^{\prime}\frac{x^{\prime 2}+y^{\prime 2}}{y^{\prime\prime}x^{\prime}% -x^{\prime\prime}y^{\prime}}
  91. x = x ( t ) x=x(t)
  92. y = y ( t ) y=y(t)
  93. F [ r ] = t ( r r [ - 1 ] ) F[r]=t(r^{\prime}\circ r^{[-1]})
  94. r = r ( s ) r=r(s)
  95. s = t s=t
  96. X [ x , y ] = x - x a t x 2 + y 2 d t x 2 + y 2 X[x,y]=x-\frac{x^{\prime}\int_{a}^{t}\sqrt{x^{\prime 2}+y^{\prime 2}}\,dt}{% \sqrt{x^{\prime 2}+y^{\prime 2}}}
  97. Y [ x , y ] = y - y a t x 2 + y 2 d t x 2 + y 2 Y[x,y]=y-\frac{y^{\prime}\int_{a}^{t}\sqrt{x^{\prime 2}+y^{\prime 2}}\,dt}{% \sqrt{x^{\prime 2}+y^{\prime 2}}}
  98. x = x ( t ) x=x(t)
  99. y = y ( t ) y=y(t)
  100. X [ x , y ] = ( x y - y x ) y x 2 + y 2 X[x,y]=\frac{(xy^{\prime}-yx^{\prime})y^{\prime}}{x^{\prime 2}+y^{\prime 2}}
  101. Y [ x , y ] = ( y x - x y ) x x 2 + y 2 Y[x,y]=\frac{(yx^{\prime}-xy^{\prime})x^{\prime}}{x^{\prime 2}+y^{\prime 2}}
  102. x = x ( t ) x=x(t)
  103. y = y ( t ) y=y(t)
  104. X [ x , y ] = ( x 2 - y 2 ) y + 2 x y x x y - y x X[x,y]=\frac{(x^{\prime 2}-y^{\prime 2})y^{\prime}+2xyx^{\prime}}{xy^{\prime}-% yx^{\prime}}
  105. Y [ x , y ] = ( x 2 - y 2 ) x + 2 x y y x y - y x Y[x,y]=\frac{(x^{\prime 2}-y^{\prime 2})x^{\prime}+2xyy^{\prime}}{xy^{\prime}-% yx^{\prime}}
  106. x = x ( t ) x=x(t)
  107. y = y ( t ) y=y(t)
  108. X [ y ] = a t cos [ a t 1 y d t ] d t X[y]=\int_{a}^{t}\cos\left[\int_{a}^{t}\frac{1}{y}\,dt\right]dt
  109. Y [ y ] = a t sin [ a t 1 y d t ] d t Y[y]=\int_{a}^{t}\sin\left[\int_{a}^{t}\frac{1}{y}\,dt\right]dt
  110. y = r ( s ) y=r(s)
  111. s = t s=t
  112. F [ y ] = || y || = E y 2 d t F[y]=||y||=\sqrt{\int_{E}y^{2}\,dt}
  113. F [ x , y ] = E x y d t F[x,y]=\int_{E}xy\,dt
  114. F [ x , y ] = arccos [ E x y d t E x 2 d t E y 2 d t ] F[x,y]=\arccos\left[\frac{\int_{E}xy\,dt}{\sqrt{\int_{E}x^{2}\,dt}\sqrt{\int_{% E}y^{2}\,dt}}\right]
  115. F [ x , y ] = x * y = E x ( s ) y ( t - s ) d s F[x,y]=x*y=\int_{E}x(s)y(t-s)\,ds
  116. F [ y ] = E y ln y d y F[y]=\int_{E}y\ln y\,dy
  117. F [ y ] = E y t d t F[y]=\int_{E}yt\,dt
  118. F [ y ] = E ( t - E y t d t ) 2 y d t F[y]=\int_{E}(t-\int_{E}yt\,dt)^{2}y\,dt

List_of_optimization_software.html

  1. \to

List_of_problems_in_loop_theory_and_quasigroup_theory.html

  1. G G
  2. M ( G , 2 ) M(G,2)
  3. G G
  4. C 2 C_{2}
  5. ( g , 0 ) ( h , 0 ) = ( g h , 0 ) (g,0)(h,0)=(gh,0)
  6. ( g , 0 ) ( h , 1 ) = ( h g , 1 ) (g,0)(h,1)=(hg,1)
  7. ( g , 1 ) ( h , 0 ) = ( g h - 1 , 1 ) (g,1)(h,0)=(gh^{-1},1)
  8. ( g , 1 ) ( h , 1 ) = ( h - 1 g , 0 ) (g,1)(h,1)=(h^{-1}g,0)
  9. M ( G , 2 ) M(G,2)
  10. G G
  11. M ( G , 2 ) M(G,2)
  12. G G
  13. M ( G , 2 ) M(G,2)
  14. G G
  15. ( Q , * ) (Q,*)
  16. ( Q , + ) (Q,+)
  17. Q Q
  18. d ( * , + ) d(*,+)
  19. ( a , b ) (a,b)
  20. Q × Q Q\times Q
  21. a * b a + b a*b\neq a+b
  22. α \alpha
  23. ( Q , * ) (Q,*)
  24. ( Q , + ) (Q,+)
  25. n n
  26. d ( * , + ) < α d(*,+)<\alpha
  27. n 2 n^{2}
  28. α = 1 / 9 \alpha=1/9
  29. α = 1 / 4 \alpha=1/4
  30. M ( q ) M(q)
  31. M ( q ) M(q)
  32. S 3 S_{3}
  33. G G
  34. G G

List_of_quantum-mechanical_systems_with_analytical_solutions.html

  1. 𝐫 \mathbf{r}
  2. H ψ ( 𝐫 , t ) = ( T + V ) ψ ( 𝐫 , t ) = [ - 2 2 m 2 + V ( 𝐫 ) ] ψ ( 𝐫 , t ) = i ψ ( 𝐫 , t ) t H\psi\left(\mathbf{r},t\right)=\left(T+V\right)\,\psi\left(\mathbf{r},t\right)% =\left[-\frac{\hbar^{2}}{2m}\nabla^{2}+V\left(\mathbf{r}\right)\right]\psi% \left(\mathbf{r},t\right)=i\hbar\frac{\partial\psi\left(\mathbf{r},t\right)}{% \partial t}
  3. ψ \psi
  4. [ - 2 2 m 2 + V ( 𝐫 ) ] ψ ( 𝐫 ) = E ψ ( 𝐫 ) \left[-\frac{\hbar^{2}}{2m}\nabla^{2}+V\left(\mathbf{r}\right)\right]\psi\left% (\mathbf{r}\right)=E\psi\left(\mathbf{r}\right)

List_of_representations_of_e.html

  1. e e
  2. e e
  3. e e
  4. e e
  5. e = [ 2 ; 1 , 𝟐 , 1 , 1 , 𝟒 , 1 , 1 , 𝟔 , 1 , 1 , 𝟖 , 1 , 1 , , 𝟐𝐧 , 1 , 1 , ] . e=[2;1,\,\textbf{2},1,1,\,\textbf{4},1,1,\,\textbf{6},1,1,\,\textbf{8},1,1,% \ldots,\,\textbf{2n},1,1,\ldots].\,
  6. e = [ 1 ; 0.5 , 12 , 5 , 28 , 9 , 44 , 13 , 60 , 17 , , 4(4n-1) , 4n+1 , ] . e=[1;\,\textbf{0.5},12,5,28,9,44,13,60,17,\ldots,\,\textbf{4(4n-1)},\,\textbf{% 4n+1},\ldots].\,
  7. e e
  8. e = 2 + 1 1 + 1 2 + 2 3 + 3 4 + 4 5 + = 2 + 2 2 + 3 3 + 4 4 + 5 5 + 6 6 + e=2+\cfrac{1}{1+\cfrac{1}{2+\cfrac{2}{3+\cfrac{3}{4+\cfrac{4}{5+\ddots}}}}}=2+% \cfrac{2}{2+\cfrac{3}{3+\cfrac{4}{4+\cfrac{5}{5+\cfrac{6}{6+\ddots\,}}}}}
  9. e = 2 + 1 1 + 2 5 + 1 10 + 1 14 + 1 18 + = 1 + 2 1 + 1 6 + 1 10 + 1 14 + 1 18 + e=2+\cfrac{1}{1+\cfrac{2}{5+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\ddots\,}}}% }}=1+\cfrac{2}{1+\cfrac{1}{6+\cfrac{1}{10+\cfrac{1}{14+\cfrac{1}{18+\ddots\,}}% }}}
  10. e x / y = 1 + 2 x 2 y - x + x 2 6 y + x 2 10 y + x 2 14 y + x 2 18 y + e^{x/y}=1+\cfrac{2x}{2y-x+\cfrac{x^{2}}{6y+\cfrac{x^{2}}{10y+\cfrac{x^{2}}{14y% +\cfrac{x^{2}}{18y+\ddots}}}}}
  11. e e
  12. e x = k = 0 x k k ! e^{x}=\sum_{k=0}^{\infty}\frac{x^{k}}{k!}
  13. e = k = 0 1 k ! e=\sum_{k=0}^{\infty}\frac{1}{k!}
  14. e - 1 = k = 0 ( - 1 ) k k ! e^{-1}=\sum_{k=0}^{\infty}\frac{(-1)^{k}}{k!}
  15. e = [ k = 0 1 - 2 k ( 2 k ) ! ] - 1 e=\left[\sum_{k=0}^{\infty}\frac{1-2k}{(2k)!}\right]^{-1}
  16. e = 1 2 k = 0 k + 1 k ! e=\frac{1}{2}\sum_{k=0}^{\infty}\frac{k+1}{k!}
  17. e = 2 k = 0 k + 1 ( 2 k + 1 ) ! e=2\sum_{k=0}^{\infty}\frac{k+1}{(2k+1)!}
  18. e = k = 0 3 - 4 k 2 ( 2 k + 1 ) ! e=\sum_{k=0}^{\infty}\frac{3-4k^{2}}{(2k+1)!}
  19. e = k = 0 ( 3 k ) 2 + 1 ( 3 k ) ! e=\sum_{k=0}^{\infty}\frac{(3k)^{2}+1}{(3k)!}
  20. e = [ k = 0 4 k + 3 2 2 k + 1 ( 2 k + 1 ) ! ] 2 e=\left[\sum_{k=0}^{\infty}\frac{4k+3}{2^{2k+1}\,(2k+1)!}\right]^{2}
  21. e = [ - 12 π 2 k = 1 1 k 2 cos ( 9 k π + k 2 π 2 - 9 ) ] - 1 / 3 e=\left[-\frac{12}{\pi^{2}}\sum_{k=1}^{\infty}\frac{1}{k^{2}}\ \cos\left(\frac% {9}{k\pi+\sqrt{k^{2}\pi^{2}-9}}\right)\right]^{-1/3}
  22. e = k = 1 k n B n ( k ! ) e=\sum_{k=1}^{\infty}\frac{k^{n}}{B_{n}(k!)}
  23. B n B_{n}
  24. n t h n^{th}
  25. e = k = 1 k k ! = k = 1 1 ( k - 1 ) ! = k = 0 1 k ! e=\sum_{k=1}^{\infty}\frac{k}{k!}=\sum_{k=1}^{\infty}\frac{1}{(k-1)!}=\sum_{k=% 0}^{\infty}\frac{1}{k!}
  26. e = k = 1 k 2 2 ( k ! ) e=\sum_{k=1}^{\infty}\frac{k^{2}}{2(k!)}
  27. e = k = 1 k 3 5 ( k ! ) e=\sum_{k=1}^{\infty}\frac{k^{3}}{5(k!)}
  28. e = k = 1 k 4 15 ( k ! ) e=\sum_{k=1}^{\infty}\frac{k^{4}}{15(k!)}
  29. e = k = 1 k 5 52 ( k ! ) e=\sum_{k=1}^{\infty}\frac{k^{5}}{52(k!)}
  30. e = k = 1 k 6 203 ( k ! ) e=\sum_{k=1}^{\infty}\frac{k^{6}}{203(k!)}
  31. e = k = 1 k 7 877 ( k ! ) e=\sum_{k=1}^{\infty}\frac{k^{7}}{877(k!)}
  32. e e
  33. e = 2 ( 2 1 ) 1 / 2 ( 2 3 4 3 ) 1 / 4 ( 4 5 6 5 6 7 8 7 ) 1 / 8 e=2\left(\frac{2}{1}\right)^{1/2}\left(\frac{2}{3}\;\frac{4}{3}\right)^{1/4}% \left(\frac{4}{5}\;\frac{6}{5}\;\frac{6}{7}\;\frac{8}{7}\right)^{1/8}\cdots
  34. e = ( 2 1 ) 1 / 1 ( 2 2 1 3 ) 1 / 2 ( 2 3 4 1 3 3 ) 1 / 3 ( 2 4 4 4 1 3 6 5 ) 1 / 4 , e=\left(\frac{2}{1}\right)^{1/1}\left(\frac{2^{2}}{1\cdot 3}\right)^{1/2}\left% (\frac{2^{3}\cdot 4}{1\cdot 3^{3}}\right)^{1/3}\left(\frac{2^{4}\cdot 4^{4}}{1% \cdot 3^{6}\cdot 5}\right)^{1/4}\cdots,
  35. k = 0 n ( k + 1 ) ( - 1 ) k + 1 ( n k ) , \prod_{k=0}^{n}(k+1)^{(-1)^{k+1}{n\choose k}},
  36. e = 2 2 ( ln ( 2 ) - 1 ) 2 2 ln ( 2 ) - 1 2 ( ln ( 2 ) - 1 ) 3 . e=\frac{2\cdot 2^{(\ln(2)-1)^{2}}\cdots}{2^{\ln(2)-1}\cdot 2^{(\ln(2)-1)^{3}}% \cdots}.
  37. e e
  38. e = lim n n ( 2 π n n ! ) 1 / n e=\lim_{n\to\infty}n\cdot\left(\frac{\sqrt{2\pi n}}{n!}\right)^{1/n}
  39. e = lim n n n ! n e=\lim_{n\to\infty}\frac{n}{\sqrt[n]{n!}}
  40. e = lim n [ ( n + 1 ) n + 1 n n - n n ( n - 1 ) n - 1 ] e=\lim_{n\to\infty}\left[\frac{(n+1)^{n+1}}{n^{n}}-\frac{n^{n}}{(n-1)^{n-1}}\right]
  41. e e
  42. e = lim n ( p n # ) 1 / p n e=\lim_{n\to\infty}(p_{n}\#)^{1/p_{n}}
  43. p n p_{n}
  44. p n # p_{n}\#
  45. e = lim n n π ( n ) / n e=\lim_{n\to\infty}n^{\pi(n)/n}
  46. π ( n ) \pi(n)
  47. e x = lim n ( 1 + x n ) n . e^{x}=\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^{n}.
  48. x = 1 x=1
  49. e = lim n ( 1 + 1 n ) n . e=\lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}.
  50. e e
  51. e x = sinh ( x ) + cosh ( x ) e^{x}=\sinh(x)+\cosh(x)\,

List_of_space_groups.html

  1. a a
  2. b b
  3. c c
  4. n n
  5. d d
  6. e e
  7. \color B l a c k 360 n \color{Black}\tfrac{360^{\circ}}{n}
  8. C 1 1 C_{1}^{1}
  9. 1 ¯ \overline{1}
  10. 1 ¯ \overline{1}
  11. 1 ¯ \overline{1}
  12. C i 1 C_{i}^{1}
  13. C 2 1 C_{2}^{1}
  14. C 2 2 C_{2}^{2}
  15. C 2 3 C_{2}^{3}
  16. C s 1 C_{s}^{1}
  17. C s 2 C_{s}^{2}
  18. C s 3 C_{s}^{3}
  19. C s 4 C_{s}^{4}
  20. C 2 h 1 C_{2h}^{1}
  21. C 2 h 2 C_{2h}^{2}
  22. C 2 h 3 C_{2h}^{3}
  23. C 2 h 4 C_{2h}^{4}
  24. C 2 h 5 C_{2h}^{5}
  25. C 2 h 6 C_{2h}^{6}
  26. D 2 1 D_{2}^{1}
  27. D 2 2 D_{2}^{2}
  28. D 2 3 D_{2}^{3}
  29. D 2 4 D_{2}^{4}
  30. D 2 5 D_{2}^{5}
  31. D 2 6 D_{2}^{6}
  32. D 2 7 D_{2}^{7}
  33. D 2 8 D_{2}^{8}
  34. D 2 9 D_{2}^{9}
  35. C 2 v 1 C_{2v}^{1}
  36. C 2 v 2 C_{2v}^{2}
  37. C 2 v 3 C_{2v}^{3}
  38. C 2 v 4 C_{2v}^{4}
  39. C 2 v 5 C_{2v}^{5}
  40. C 2 v 6 C_{2v}^{6}
  41. C 2 v 7 C_{2v}^{7}
  42. C 2 v 8 C_{2v}^{8}
  43. C 2 v 9 C_{2v}^{9}
  44. C 2 v 10 C_{2v}^{10}
  45. C 2 v 11 C_{2v}^{11}
  46. C 2 v 12 C_{2v}^{12}
  47. C 2 v 13 C_{2v}^{13}
  48. C 2 v 14 C_{2v}^{14}
  49. C 2 v 15 C_{2v}^{15}
  50. C 2 v 16 C_{2v}^{16}
  51. C 2 v 17 C_{2v}^{17}
  52. C 2 v 18 C_{2v}^{18}
  53. C 2 v 19 C_{2v}^{19}
  54. C 2 v 20 C_{2v}^{20}
  55. C 2 v 21 C_{2v}^{21}
  56. C 2 v 22 C_{2v}^{22}
  57. D 2 h 1 D_{2h}^{1}
  58. D 2 h 2 D_{2h}^{2}
  59. D 2 h 3 D_{2h}^{3}
  60. D 2 h 4 D_{2h}^{4}
  61. D 2 h 5 D_{2h}^{5}
  62. D 2 h 6 D_{2h}^{6}
  63. D 2 h 7 D_{2h}^{7}
  64. D 2 h 8 D_{2h}^{8}
  65. D 2 h 9 D_{2h}^{9}
  66. D 2 h 10 D_{2h}^{10}
  67. D 2 h 11 D_{2h}^{11}
  68. D 2 h 12 D_{2h}^{12}
  69. D 2 h 13 D_{2h}^{13}
  70. D 2 h 14 D_{2h}^{14}
  71. D 2 h 15 D_{2h}^{15}
  72. D 2 h 16 D_{2h}^{16}
  73. D 2 h 17 D_{2h}^{17}
  74. D 2 h 18 D_{2h}^{18}
  75. D 2 h 19 D_{2h}^{19}
  76. D 2 h 20 D_{2h}^{20}
  77. D 2 h 21 D_{2h}^{21}
  78. D 2 h 22 D_{2h}^{22}
  79. D 2 h 23 D_{2h}^{23}
  80. D 2 h 24 D_{2h}^{24}
  81. D 2 h 25 D_{2h}^{25}
  82. D 2 h 26 D_{2h}^{26}
  83. D 2 h 27 D_{2h}^{27}
  84. D 2 h 28 D_{2h}^{28}
  85. C 4 1 C_{4}^{1}
  86. C 4 2 C_{4}^{2}
  87. C 4 3 C_{4}^{3}
  88. C 4 4 C_{4}^{4}
  89. C 4 5 C_{4}^{5}
  90. C 4 6 C_{4}^{6}
  91. 4 ¯ \overline{4}
  92. 4 ¯ \overline{4}
  93. 4 ¯ \overline{4}
  94. S 4 1 S_{4}^{1}
  95. 4 ¯ \overline{4}
  96. 4 ¯ \overline{4}
  97. 4 ¯ \overline{4}
  98. S 4 2 S_{4}^{2}
  99. C 4 h 1 C_{4h}^{1}
  100. C 4 h 2 C_{4h}^{2}
  101. C 4 h 3 C_{4h}^{3}
  102. C 4 h 4 C_{4h}^{4}
  103. C 4 h 5 C_{4h}^{5}
  104. C 4 h 6 C_{4h}^{6}
  105. D 4 1 D_{4}^{1}
  106. D 4 2 D_{4}^{2}
  107. D 4 3 D_{4}^{3}
  108. D 4 4 D_{4}^{4}
  109. D 4 5 D_{4}^{5}
  110. D 4 6 D_{4}^{6}
  111. D 4 7 D_{4}^{7}
  112. D 4 8 D_{4}^{8}
  113. D 4 9 D_{4}^{9}
  114. D 4 10 D_{4}^{10}
  115. C 4 v 1 C_{4v}^{1}
  116. C 4 v 2 C_{4v}^{2}
  117. C 4 v 3 C_{4v}^{3}
  118. C 4 v 4 C_{4v}^{4}
  119. C 4 v 5 C_{4v}^{5}
  120. C 4 v 6 C_{4v}^{6}
  121. C 4 v 7 C_{4v}^{7}
  122. C 4 v 8 C_{4v}^{8}
  123. C 4 v 9 C_{4v}^{9}
  124. C 4 v 10 C_{4v}^{10}
  125. C 4 v 11 C_{4v}^{11}
  126. C 4 v 12 C_{4v}^{12}
  127. 4 ¯ \overline{4}
  128. 4 ¯ \overline{4}
  129. 4 ¯ \overline{4}
  130. D 2 d 1 D_{2d}^{1}
  131. 4 ¯ \overline{4}
  132. 4 ¯ \overline{4}
  133. 4 ¯ \overline{4}
  134. D 2 d 2 D_{2d}^{2}
  135. 4 ¯ \overline{4}
  136. 4 ¯ \overline{4}
  137. 4 ¯ \overline{4}
  138. D 2 d 3 D_{2d}^{3}
  139. 4 ¯ \overline{4}
  140. 4 ¯ \overline{4}
  141. 4 ¯ \overline{4}
  142. D 2 d 4 D_{2d}^{4}
  143. 4 ¯ \overline{4}
  144. 4 ¯ \overline{4}
  145. 4 ¯ \overline{4}
  146. D 2 d 5 D_{2d}^{5}
  147. 4 ¯ \overline{4}
  148. 4 ¯ \overline{4}
  149. 4 ¯ \overline{4}
  150. D 2 d 6 D_{2d}^{6}
  151. 4 ¯ \overline{4}
  152. 4 ¯ \overline{4}
  153. 4 ¯ \overline{4}
  154. D 2 d 7 D_{2d}^{7}
  155. 4 ¯ \overline{4}
  156. 4 ¯ \overline{4}
  157. 4 ¯ \overline{4}
  158. D 2 d 8 D_{2d}^{8}
  159. 4 ¯ \overline{4}
  160. 4 ¯ \overline{4}
  161. 4 ¯ \overline{4}
  162. D 2 d 9 D_{2d}^{9}
  163. 4 ¯ \overline{4}
  164. 4 ¯ \overline{4}
  165. 4 ¯ \overline{4}
  166. D 2 d 10 D_{2d}^{10}
  167. 4 ¯ \overline{4}
  168. 4 ¯ \overline{4}
  169. 4 ¯ \overline{4}
  170. D 2 d 11 D_{2d}^{11}
  171. 4 ¯ \overline{4}
  172. 4 ¯ \overline{4}
  173. 4 ¯ \overline{4}
  174. D 2 d 12 D_{2d}^{12}
  175. D 4 h 1 D_{4h}^{1}
  176. D 4 h 2 D_{4h}^{2}
  177. D 4 h 3 D_{4h}^{3}
  178. D 4 h 4 D_{4h}^{4}
  179. D 4 h 5 D_{4h}^{5}
  180. D 4 h 6 D_{4h}^{6}
  181. D 4 h 7 D_{4h}^{7}
  182. D 4 h 8 D_{4h}^{8}
  183. D 4 h 9 D_{4h}^{9}
  184. D 4 h 10 D_{4h}^{10}
  185. D 4 h 11 D_{4h}^{11}
  186. D 4 h 12 D_{4h}^{12}
  187. D 4 h 13 D_{4h}^{13}
  188. D 4 h 14 D_{4h}^{14}
  189. D 4 h 15 D_{4h}^{15}
  190. D 4 h 16 D_{4h}^{16}
  191. D 4 h 17 D_{4h}^{17}
  192. D 4 h 18 D_{4h}^{18}
  193. D 4 h 19 D_{4h}^{19}
  194. D 4 h 20 D_{4h}^{20}
  195. C 3 1 C_{3}^{1}
  196. C 3 2 C_{3}^{2}
  197. C 3 3 C_{3}^{3}
  198. C 3 4 C_{3}^{4}
  199. 3 ¯ \overline{3}
  200. 3 ¯ \overline{3}
  201. 3 ¯ \overline{3}
  202. C 3 i 1 C_{3i}^{1}
  203. 3 ¯ \overline{3}
  204. 3 ¯ \overline{3}
  205. 3 ¯ \overline{3}
  206. C 3 i 2 C_{3i}^{2}
  207. D 3 1 D_{3}^{1}
  208. D 3 2 D_{3}^{2}
  209. D 3 3 D_{3}^{3}
  210. D 3 4 D_{3}^{4}
  211. D 3 5 D_{3}^{5}
  212. D 3 6 D_{3}^{6}
  213. D 3 7 D_{3}^{7}
  214. C 3 v 1 C_{3v}^{1}
  215. C 3 v 2 C_{3v}^{2}
  216. C 3 v 3 C_{3v}^{3}
  217. C 3 v 4 C_{3v}^{4}
  218. C 3 v 5 C_{3v}^{5}
  219. C 3 v 6 C_{3v}^{6}
  220. 3 ¯ \overline{3}
  221. 3 ¯ \overline{3}
  222. 3 ¯ \overline{3}
  223. D 3 d 1 D_{3d}^{1}
  224. 3 ¯ \overline{3}
  225. 3 ¯ \overline{3}
  226. 3 ¯ \overline{3}
  227. D 3 d 2 D_{3d}^{2}
  228. 3 ¯ \overline{3}
  229. 3 ¯ \overline{3}
  230. 3 ¯ \overline{3}
  231. D 3 d 3 D_{3d}^{3}
  232. 3 ¯ \overline{3}
  233. 3 ¯ \overline{3}
  234. 3 ¯ \overline{3}
  235. D 3 d 4 D_{3d}^{4}
  236. 3 ¯ \overline{3}
  237. 3 ¯ \overline{3}
  238. 3 ¯ \overline{3}
  239. D 3 d 5 D_{3d}^{5}
  240. 3 ¯ \overline{3}
  241. 3 ¯ \overline{3}
  242. 3 ¯ \overline{3}
  243. D 3 d 6 D_{3d}^{6}
  244. C 6 1 C_{6}^{1}
  245. C 6 2 C_{6}^{2}
  246. C 6 3 C_{6}^{3}
  247. C 6 4 C_{6}^{4}
  248. C 6 5 C_{6}^{5}
  249. C 6 6 C_{6}^{6}
  250. 6 ¯ \overline{6}
  251. 6 ¯ \overline{6}
  252. 6 ¯ \overline{6}
  253. C 3 h 1 C_{3h}^{1}
  254. C 6 h 1 C_{6h}^{1}
  255. C 6 h 2 C_{6h}^{2}
  256. D 6 1 D_{6}^{1}
  257. D 6 2 D_{6}^{2}
  258. D 6 3 D_{6}^{3}
  259. D 6 4 D_{6}^{4}
  260. D 6 5 D_{6}^{5}
  261. D 6 6 D_{6}^{6}
  262. C 6 v 1 C_{6v}^{1}
  263. C 6 v 2 C_{6v}^{2}
  264. C 6 v 3 C_{6v}^{3}
  265. C 6 v 4 C_{6v}^{4}
  266. 6 ¯ \overline{6}
  267. 6 ¯ \overline{6}
  268. 6 ¯ \overline{6}
  269. D 3 h 1 D_{3h}^{1}
  270. 6 ¯ \overline{6}
  271. 6 ¯ \overline{6}
  272. 6 ¯ \overline{6}
  273. D 3 h 2 D_{3h}^{2}
  274. 6 ¯ \overline{6}
  275. 6 ¯ \overline{6}
  276. 6 ¯ \overline{6}
  277. D 3 h 3 D_{3h}^{3}
  278. 6 ¯ \overline{6}
  279. 6 ¯ \overline{6}
  280. 6 ¯ \overline{6}
  281. D 3 h 4 D_{3h}^{4}
  282. D 6 h 1 D_{6h}^{1}
  283. D 6 h 2 D_{6h}^{2}
  284. D 6 h 3 D_{6h}^{3}
  285. D 6 h 4 D_{6h}^{4}
  286. T 1 T^{1}
  287. T 2 T^{2}
  288. T 3 T^{3}
  289. T 4 T^{4}
  290. T 5 T^{5}
  291. 3 ¯ \overline{3}
  292. 3 ¯ \overline{3}
  293. 3 ¯ \overline{3}
  294. T h 1 T_{h}^{1}
  295. 3 ¯ \overline{3}
  296. 3 ¯ \overline{3}
  297. 3 ¯ \overline{3}
  298. T h 2 T_{h}^{2}
  299. 3 ¯ \overline{3}
  300. 3 ¯ \overline{3}
  301. 3 ¯ \overline{3}
  302. T h 3 T_{h}^{3}
  303. 3 ¯ \overline{3}
  304. 3 ¯ \overline{3}
  305. 3 ¯ \overline{3}
  306. T h 4 T_{h}^{4}
  307. 3 ¯ \overline{3}
  308. 3 ¯ \overline{3}
  309. 3 ¯ \overline{3}
  310. T h 5 T_{h}^{5}
  311. 3 ¯ \overline{3}
  312. 3 ¯ \overline{3}
  313. 3 ¯ \overline{3}
  314. T h 6 T_{h}^{6}
  315. 3 ¯ \overline{3}
  316. 3 ¯ \overline{3}
  317. 3 ¯ \overline{3}
  318. T h 7 T_{h}^{7}
  319. O 1 O^{1}
  320. O 2 O^{2}
  321. O 2 O^{2}
  322. O 4 O^{4}
  323. O 5 O^{5}
  324. O 6 O^{6}
  325. O 7 O^{7}
  326. O 8 O^{8}
  327. 4 ¯ \overline{4}
  328. 4 ¯ \overline{4}
  329. 4 ¯ \overline{4}
  330. T d 1 T_{d}^{1}
  331. 4 ¯ \overline{4}
  332. 4 ¯ \overline{4}
  333. 4 ¯ \overline{4}
  334. T d 2 T_{d}^{2}
  335. 4 ¯ \overline{4}
  336. 4 ¯ \overline{4}
  337. 4 ¯ \overline{4}
  338. T d 3 T_{d}^{3}
  339. 4 ¯ \overline{4}
  340. 4 ¯ \overline{4}
  341. 4 ¯ \overline{4}
  342. T d 4 T_{d}^{4}
  343. 4 ¯ \overline{4}
  344. 4 ¯ \overline{4}
  345. 4 ¯ \overline{4}
  346. T d 5 T_{d}^{5}
  347. 4 ¯ \overline{4}
  348. 4 ¯ \overline{4}
  349. 4 ¯ \overline{4}
  350. T d 6 T_{d}^{6}
  351. 3 ¯ \overline{3}
  352. 3 ¯ \overline{3}
  353. 3 ¯ \overline{3}
  354. O h 1 O_{h}^{1}
  355. 3 ¯ \overline{3}
  356. 3 ¯ \overline{3}
  357. 3 ¯ \overline{3}
  358. O h 2 O_{h}^{2}
  359. 3 ¯ \overline{3}
  360. 3 ¯ \overline{3}
  361. 3 ¯ \overline{3}
  362. O h 3 O_{h}^{3}
  363. 3 ¯ \overline{3}
  364. 3 ¯ \overline{3}
  365. 3 ¯ \overline{3}
  366. O h 4 O_{h}^{4}
  367. 3 ¯ \overline{3}
  368. 3 ¯ \overline{3}
  369. 3 ¯ \overline{3}
  370. O h 5 O_{h}^{5}
  371. 3 ¯ \overline{3}
  372. 3 ¯ \overline{3}
  373. 3 ¯ \overline{3}
  374. O h 6 O_{h}^{6}
  375. 3 ¯ \overline{3}
  376. 3 ¯ \overline{3}
  377. 3 ¯ \overline{3}
  378. O h 7 O_{h}^{7}
  379. 3 ¯ \overline{3}
  380. 3 ¯ \overline{3}
  381. 3 ¯ \overline{3}
  382. O h 8 O_{h}^{8}
  383. 3 ¯ \overline{3}
  384. 3 ¯ \overline{3}
  385. 3 ¯ \overline{3}
  386. O h 9 O_{h}^{9}
  387. 3 ¯ \overline{3}
  388. 3 ¯ \overline{3}
  389. 3 ¯ \overline{3}
  390. O h 10 O_{h}^{10}

List_of_taxa_named_by_Ruiz_and_Pavón.html

  1. \neq

List_of_thermal_conductivities.html

  1. \parallel
  2. \perp

Littelmann_path_model.html

  1. \otimes
  2. 𝔤 1 \mathfrak{g}_{1}
  3. 𝔤 \mathfrak{g}
  4. 𝔤 1 \mathfrak{g}_{1}
  5. 𝔤 𝔩 \mathfrak{gl}
  6. 𝔰 𝔩 \mathfrak{sl}
  7. 𝔤 𝔩 \mathfrak{gl}
  8. 𝔤 𝔩 \mathfrak{gl}
  9. \oplus
  10. 𝔤 𝔩 \mathfrak{gl}
  11. 𝔤 \mathfrak{g}
  12. 𝔤 \mathfrak{g}
  13. 𝔤 \mathfrak{g}
  14. π : [ 0 , 1 ] 𝐐 P 𝐙 𝐐 \pi:[0,1]\cap\mathbf{Q}\rightarrow P\otimes_{\mathbf{Z}}\mathbf{Q}
  15. 𝔥 \mathfrak{h}
  16. 𝔥 \mathfrak{h}
  17. h ( t ) = ( π ( t ) , H α ) h(t)=(\pi(t),H_{\alpha})
  18. \cap
  19. l ( t ) = min t s 1 ( 1 , h ( s ) - M ) , r ( t ) = 1 - min 0 s t ( 1 , h ( s ) - M ) . l(t)=\min_{t\leq s\leq 1}(1,h(s)-M),\,\,\,\,\,\,r(t)=1-\min_{0\leq s\leq t}(1,% h(s)-M).
  20. π r ( t ) = π ( t ) + r ( t ) α , π l ( t ) = π ( t ) - l ( t ) α \pi_{r}(t)=\pi(t)+r(t)\alpha,\,\,\,\,\,\,\pi_{l}(t)=\pi(t)-l(t)\alpha
  21. e α [ π ] = [ π r ] \displaystyle{e_{\alpha}[\pi]=[\pi_{r}]}
  22. f α [ π ] = [ π l ] \displaystyle{f_{\alpha}[\pi]=[\pi_{l}]}
  23. 𝒜 \mathcal{A}
  24. 𝒜 \mathcal{A}
  25. 𝒢 π \mathcal{G}_{\pi}
  26. 𝒢 π \mathcal{G}_{\pi}
  27. L ( λ ) L ( μ ) = η L ( λ + τ ( 1 ) ) , L(\lambda)\otimes L(\mu)=\bigoplus_{\eta}L(\lambda+\tau(1)),
  28. 𝒢 σ \mathcal{G}_{\sigma}
  29. \star
  30. \star
  31. 𝔤 1 \mathfrak{g}_{1}
  32. 𝔤 \mathfrak{g}
  33. \supset
  34. L ( λ ) | 𝔤 1 = σ L ( σ ( 1 ) ) , L(\lambda)|_{\mathfrak{g}_{1}}=\bigoplus_{\sigma}L(\sigma(1)),
  35. 𝒢 π \mathcal{G}_{\pi}
  36. 𝔤 1 \mathfrak{g}_{1}
  37. 𝔤 \mathfrak{g}
  38. 𝔤 1 \mathfrak{g}_{1}

Load_factor_(aeronautics).html

  1. n = L W n=\frac{L}{W}
  2. n = 1 cos θ n=\frac{1}{\cos\,\theta}

Load_regulation.html

  1. % Load Regulation = 100 % V m i n - l o a d - V m a x - l o a d V n o m - l o a d \%\,\text{Load Regulation}=100\%\,\frac{V_{min-load}-V_{max-load}}{V_{nom-load}}
  2. V m a x - l o a d V_{max-load}
  3. V m i n - l o a d V_{min-load}
  4. V n o m - l o a d V_{nom-load}
  5. Load Regulation ( V ) = V m i n - l o a d - V m a x - l o a d \,\text{Load Regulation}(V)=V_{min-load}-V_{max-load}
  6. R m a x - l o a d = 5 V 3 A = 1.67 Ω R_{max-load}=\frac{5\,V}{3\,A}=1.67\,\Omega
  7. R n o m - l o a d = 2 × R f u l l - l o a d = 3.33 Ω R_{nom-load}=2\times R_{full-load}=3.33\,\Omega

Local_volatility.html

  1. S t S_{t}
  2. t t
  3. S t S_{t}
  4. t t
  5. d S t = ( r t - d t ) S t d t + σ t S t d W t dS_{t}=(r_{t}-d_{t})S_{t}\,dt+\sigma_{t}S_{t}\,dW_{t}
  6. r t r_{t}
  7. W t W_{t}
  8. σ t \sigma_{t}
  9. σ t \sigma_{t}
  10. σ t = σ ( S t , t ) \sigma_{t}=\sigma(S_{t},t)
  11. C T = 1 2 σ 2 ( K , T ; S 0 ) K 2 2 C K 2 - ( r - d ) K C K - d C \frac{\partial C}{\partial T}=\frac{1}{2}\sigma^{2}(K,T;S_{0})K^{2}\frac{% \partial^{2}C}{\partial K^{2}}-(r-d)K\frac{\partial C}{\partial K}-dC
  12. S t S_{t}
  13. d S t = ( r - d ) S t d t + σ ( t , S t ) S t d W t dS_{t}=(r-d)S_{t}dt+\sigma(t,S_{t})S_{t}dW_{t}
  14. p ( t , S t ) p(t,S_{t})
  15. S 0 S_{0}
  16. p t = - [ ( r - d ) s p ] s + 1 2 [ ( σ s ) 2 p ] s s p_{t}=-[(r-d)s\,p]_{s}+\frac{1}{2}[(\sigma s)^{2}p]_{ss}
  17. T T
  18. K K
  19. C = e - r T 𝔼 Q [ ( S T - K ) + ] = e - r T K ( s - K ) p d s = e - r T K s p d s - K e - r T K p d s \begin{aligned}\displaystyle C&\displaystyle=e^{-rT}\mathbb{E}^{Q}[(S_{T}-K)^{% +}]\\ &\displaystyle=e^{-rT}\int_{K}^{\infty}(s-K)\,p\,ds\\ &\displaystyle=e^{-rT}\int_{K}^{\infty}s\,p\,ds-K\,e^{-rT}\int_{K}^{\infty}p\,% ds\end{aligned}
  20. K K
  21. C K = - e - r T K p d s C_{K}=-e^{-rT}\int_{K}^{\infty}pds
  22. e - r T K s p d s = C - K C K e^{-rT}\int_{K}^{\infty}s\,p\,ds=C-K\,C_{K}
  23. K K
  24. C K K = e - r T p C_{KK}=e^{-rT}p
  25. T T
  26. C T = - r C + e - r T K ( s - K ) p T d s C_{T}=-r\,C+e^{-rT}\int_{K}^{\infty}(s-K)p_{T}ds
  27. C T = - r C - e - r T K ( s - K ) [ ( r - d ) s p ] s d s + 1 2 e - r T K ( s - K ) [ ( σ s ) 2 p ] s s d s C_{T}=-r\,C-e^{-rT}\int_{K}^{\infty}(s-K)[(r-d)s\,p]_{s}\,ds+\frac{1}{2}e^{-rT% }\int_{K}^{\infty}(s-K)[(\sigma s)^{2}\,p]_{ss}\,ds
  28. C T = - r C + ( r - d ) e - r T K s p d s + 1 2 e - r T ( σ K ) 2 p C_{T}=-r\,C+(r-d)e^{-rT}\int_{K}^{\infty}s\,p\,ds+\frac{1}{2}e^{-rT}(\sigma K)% ^{2}\,p
  29. K K
  30. C T = - r C + ( r - d ) ( C - K C K ) + 1 2 σ 2 K 2 C K K = - ( r - d ) K C K - d C + 1 2 σ 2 K 2 C K K \begin{aligned}\displaystyle C_{T}&\displaystyle=-r\,C+(r-d)(C-K\,C_{K})+\frac% {1}{2}\sigma^{2}K^{2}C_{KK}\\ &\displaystyle=-(r-d)K\,C_{K}-d\,C+\frac{1}{2}\sigma^{2}K^{2}C_{KK}\end{aligned}

Locality-sensitive_hashing.html

  1. \mathcal{F}
  2. = ( M , d ) \mathcal{M}=(M,d)
  3. R > 0 R>0
  4. c > 1 c>1
  5. \mathcal{F}
  6. h : S h:{\mathcal{M}}\to S
  7. s S s\in S
  8. p , q p,q\in{\mathcal{M}}
  9. h h\in\mathcal{F}
  10. d ( p , q ) R d(p,q)\leq R
  11. h ( p ) = h ( q ) h(p)=h(q)
  12. p p
  13. q q
  14. P 1 P_{1}
  15. d ( p , q ) c R d(p,q)\geq cR
  16. h ( p ) = h ( q ) h(p)=h(q)
  17. P 2 P_{2}
  18. P 1 > P 2 P_{1}>P_{2}
  19. \mathcal{F}
  20. ( R , c R , P 1 , P 2 ) (R,cR,P_{1},P_{2})
  21. U U
  22. ϕ : U × U [ 0 , 1 ] \phi:U\times U\to[0,1]
  23. H H
  24. D D
  25. h H h\in H
  26. D D
  27. P r h H [ h ( a ) = h ( b ) ] = ϕ ( a , b ) Pr_{h\in H}[h(a)=h(b)]=\phi(a,b)
  28. a , b U a,b\in U
  29. ( d 1 , d 2 , p 1 , p 2 ) (d_{1},d_{2},p_{1},p_{2})
  30. \mathcal{F}
  31. 𝒢 \mathcal{G}
  32. \mathcal{F}
  33. 𝒢 \mathcal{G}
  34. g g
  35. g g
  36. k k
  37. h 1 , , h k h_{1},...,h_{k}
  38. \mathcal{F}
  39. g 𝒢 g\in\mathcal{G}
  40. g ( x ) = g ( y ) g(x)=g(y)
  41. h i ( x ) = h i ( y ) h_{i}(x)=h_{i}(y)
  42. i = 1 , 2 , , k i=1,2,...,k
  43. \mathcal{F}
  44. g 𝒢 g\in\mathcal{G}
  45. 𝒢 \mathcal{G}
  46. ( d 1 , d 2 , p 1 k , p 2 k ) (d_{1},d_{2},p_{1}^{k},p_{2}^{k})
  47. 𝒢 \mathcal{G}
  48. g g
  49. g g
  50. k k
  51. h 1 , , h k h_{1},...,h_{k}
  52. \mathcal{F}
  53. g 𝒢 g\in\mathcal{G}
  54. g ( x ) = g ( y ) g(x)=g(y)
  55. h i ( x ) = h i ( y ) h_{i}(x)=h_{i}(y)
  56. i i
  57. \mathcal{F}
  58. g 𝒢 g\in\mathcal{G}
  59. 𝒢 \mathcal{G}
  60. ( d 1 , d 2 , 1 - ( 1 - p 1 ) k , 1 - ( 1 - p 2 ) k ) (d_{1},d_{2},1-(1-p_{1})^{k},1-(1-p_{2})^{k})
  61. { 0 , 1 } d \{0,1\}^{d}
  62. \mathcal{F}
  63. d d
  64. = { h : { 0 , 1 } d { 0 , 1 } h ( x ) = x i for some i { 1 , , d } } {\mathcal{F}}=\{h:\{0,1\}^{d}\to\{0,1\}\mid h(x)=x_{i}\,\text{ for some }i\in% \{1,...,d\}\}
  65. x i x_{i}
  66. i i
  67. x x
  68. h h
  69. {\mathcal{F}}
  70. P 1 = 1 - R / d P_{1}=1-R/d
  71. P 2 = 1 - c R / d P_{2}=1-cR/d
  72. U U
  73. S S
  74. J J
  75. π \pi
  76. S S
  77. A S A\subseteq S
  78. h ( A ) = min a A { π ( a ) } h(A)=\min_{a\in A}\{\pi(a)\}
  79. π \pi
  80. h h
  81. S S
  82. H H
  83. D D
  84. A , B S A,B\subseteq S
  85. h ( A ) = h ( B ) h(A)=h(B)
  86. π \pi
  87. A B A\bigcup B
  88. A B A\bigcap B
  89. h h
  90. P r [ h ( A ) = h ( B ) ] = J ( A , B ) Pr[h(A)=h(B)]=J(A,B)\,
  91. ( H , D ) (H,D)\,
  92. π \pi
  93. l c m ( 1 , 2 , , n ) e n - o ( n ) lcm(1,2,...,n)\geq e^{n-o(n)}
  94. ϵ \epsilon
  95. 1 - θ π 1-\frac{\theta}{\pi}
  96. cos ( θ ) \cos(\theta)
  97. r r
  98. v v
  99. r r
  100. h ( v ) = s g n ( v r ) h(v)=sgn(v\cdot r)
  101. h ( v ) = ± 1 h(v)=\pm 1
  102. v v
  103. r r
  104. H H
  105. D D
  106. u , v u,v
  107. P r [ h ( u ) = h ( v ) ] = 1 - θ ( u , v ) π Pr[h(u)=h(v)]=1-\frac{\theta(u,v)}{\pi}
  108. θ ( u , v ) \theta(u,v)
  109. u u
  110. v v
  111. 1 - θ ( u , v ) π 1-\frac{\theta(u,v)}{\pi}
  112. cos ( θ ( u , v ) ) \cos(\theta(u,v))
  113. h 𝐚 , b ( s y m b o l υ ) : d 𝒩 h_{\mathbf{a},b}(symbol{\upsilon}):\mathcal{R}^{d}\to\mathcal{N}
  114. s y m b o l υ symbol{\upsilon}
  115. 𝐚 \mathbf{a}
  116. b b
  117. 𝐚 \mathbf{a}
  118. b b
  119. 𝐚 , b \mathbf{a},b
  120. h 𝐚 , b h_{\mathbf{a},b}
  121. h 𝐚 , b ( s y m b o l υ ) = 𝐚 s y m b o l υ + b r h_{\mathbf{a},b}(symbol{\upsilon})=\left\lfloor\frac{\mathbf{a}\cdot symbol{% \upsilon}+b}{r}\right\rfloor
  122. \mathcal{F}
  123. k k
  124. L L
  125. 𝒢 \mathcal{G}
  126. g g
  127. g g
  128. k k
  129. h 1 , , h k h_{1},...,h_{k}
  130. \mathcal{F}
  131. g ( p ) = [ h 1 ( p ) , , h k ( p ) ] g(p)=[h_{1}(p),...,h_{k}(p)]
  132. g g
  133. k k
  134. \mathcal{F}
  135. L L
  136. g g
  137. n n
  138. S S
  139. L L
  140. n n
  141. O ( n ) O(n)
  142. q q
  143. L L
  144. g g
  145. g g
  146. q q
  147. c R cR
  148. q q
  149. k k
  150. L L
  151. O ( n L k t ) O(nLkt)
  152. t t
  153. h h\in\mathcal{F}
  154. p p
  155. O ( n L ) O(nL)
  156. O ( L ( k t + d n P 2 k ) ) O(L(kt+dnP_{2}^{k}))
  157. c R cR
  158. q q
  159. R R
  160. 1 - ( 1 - P 1 k ) L 1-(1-P_{1}^{k})^{L}
  161. c = 1 + ϵ c=1+\epsilon
  162. P 1 P_{1}
  163. P 2 P_{2}
  164. k = log n log 1 / P 2 k={\log n\over\log 1/P_{2}}
  165. L = n ρ L=n^{\rho}
  166. ρ = log P 1 log P 2 \rho={\log P_{1}\over\log P_{2}}
  167. O ( n 1 + ρ k t ) O(n^{1+\rho}kt)
  168. O ( n 1 + ρ ) O(n^{1+\rho})
  169. O ( n ρ ( k t + d ) ) O(n^{\rho}(kt+d))

Localized_molecular_orbitals.html

  1. L ^ = i = 1 n ϕ i ϕ i | L ^ | ϕ i ϕ i \langle\hat{L}\rangle=\sum_{i=1}^{n}\langle\phi_{i}\phi_{i}|\hat{L}|\phi_{i}% \phi_{i}\rangle
  2. L ^ \hat{L}
  3. ϕ i \phi_{i}
  4. L ^ \hat{L}
  5. L ^ \langle\hat{L}\rangle
  6. L ^ = | r 1 - r 2 | 2 \hat{L}=|\vec{r}_{1}-\vec{r}_{2}|^{2}
  7. i > j n [ ϕ i | r | ϕ i - ϕ j | r | ϕ j ] 2 \sum_{i>j}^{n}[\langle\phi_{i}|\vec{r}|\phi_{i}\rangle-\langle\phi_{j}|\vec{r}% |\phi_{j}\rangle]^{2}
  8. L ^ \langle\hat{L}\rangle
  9. L ^ = | r 1 - r 2 | - 1 \hat{L}=|\vec{r}_{1}-\vec{r}_{2}|^{-1}
  10. L ^ PM = A atoms i orbitals | q i A | 2 \langle\hat{L}\rangle_{\textrm{PM}}=\sum_{A}^{\textrm{atoms}}\sum_{i}^{\textrm% {orbitals}}|q_{i}^{A}|^{2}

Loewner's_torus_inequality.html

  1. 𝕋 2 \mathbb{T}^{2}
  2. sys 2 2 3 area ( 𝕋 2 ) , \operatorname{sys}^{2}\leq\frac{2}{\sqrt{3}}\;\operatorname{area}(\mathbb{T}^{% 2}),
  3. γ 2 \gamma_{2}
  4. sys 2 γ 2 area ( 𝕋 2 ) . \operatorname{sys}^{2}\leq\gamma_{2}\;\operatorname{area}(\mathbb{T}^{2}).
  5. \mathbb{C}
  6. 2 \mathbb{R}^{2}
  7. 3 \mathbb{R}^{3}
  8. 2 \mathbb{Z}^{2}
  9. g 2 g\in\mathbb{Z}^{2}
  10. p 2 p\in\mathbb{R}^{2}
  11. dist ( p , g . p ) 2 2 3 area ( F ) \operatorname{dist}(p,g.p)^{2}\leq\frac{2}{\sqrt{3}}\operatorname{area}(F)
  12. F F
  13. dist \operatorname{dist}
  14. p p
  15. g . p g.p
  16. E ( X 2 ) - ( E ( X ) ) 2 = var ( X ) . E(X^{2})-(E(X))^{2}=\mathrm{var}(X).\,
  17. area - 3 2 ( sys ) 2 var ( f ) , \mathrm{area}-\frac{\sqrt{3}}{2}(\mathrm{sys})^{2}\geq\mathrm{var}(f),
  18. ( sys ) 2 γ 2 area (\mathrm{sys})^{2}\leq\gamma_{2}\,\mathrm{area}

Log-Laplace_distribution.html

  1. f ( x | μ , b ) = 1 2 b x exp ( - | ln x - μ | b ) f(x|\mu,b)=\frac{1}{2bx}\exp\left(-\frac{|\ln x-\mu|}{b}\right)\,\!
  2. = 1 2 b x { exp ( - μ - ln x b ) if x < μ exp ( - ln x - μ b ) if x μ =\frac{1}{2bx}\left\{\begin{matrix}\exp\left(-\frac{\mu-\ln x}{b}\right)&\mbox% {if }~{}x<\mu\\ \exp\left(-\frac{\ln x-\mu}{b}\right)&\mbox{if }~{}x\geq\mu\end{matrix}\right.
  3. F ( y ) = 0.5 [ 1 + sgn ( log ( y ) - μ ) ( 1 - exp ( - | log ( y ) - μ | / b ) ) ] . F(y)=0.5\,[1+\operatorname{sgn}(\log(y)-\mu)\,(1-\exp(-|\log(y)-\mu|/b))].
  4. { { b x f ( x ) + ( b - 1 ) f ( x ) = 0 , f ( 1 ) = e - μ b 2 b } if x < μ { b x f ( x ) + ( b + 1 ) f ( x ) = 0 , f ( 1 ) = e μ b 2 b } if x μ \left\{\begin{matrix}\left\{bxf^{\prime}(x)+(b-1)f(x)=0,f(1)=\frac{e^{-\frac{% \mu}{b}}}{2b}\right\}&\mbox{if }~{}x<\mu\\ \left\{bxf^{\prime}(x)+(b+1)f(x)=0,f(1)=\frac{e^{\frac{\mu}{b}}}{2b}\right\}&% \mbox{if }~{}x\geq\mu\end{matrix}\right.

Log-rank_test.html

  1. j j
  2. N 1 j N_{1j}
  3. N 2 j N_{2j}
  4. j j
  5. N j = N 1 j + N 2 j N_{j}=N_{1j}+N_{2j}
  6. O 1 j O_{1j}
  7. O 2 j O_{2j}
  8. j j
  9. O j = O 1 j + O 2 j O_{j}=O_{1j}+O_{2j}
  10. O j O_{j}
  11. j j
  12. O 1 j O_{1j}
  13. N j N_{j}
  14. N 1 j N_{1j}
  15. O j O_{j}
  16. E 1 j = O j N j N 1 j E_{1j}=\frac{O_{j}}{N_{j}}N_{1j}
  17. V j = O j ( N 1 j / N j ) ( 1 - N 1 j / N j ) ( N j - O j ) N j - 1 V_{j}=\frac{O_{j}(N_{1j}/N_{j})(1-N_{1j}/N_{j})(N_{j}-O_{j})}{N_{j}-1}
  18. O 1 j O_{1j}
  19. E 1 j E_{1j}
  20. Z = j = 1 J ( O 1 j - E 1 j ) j = 1 J V j . Z=\frac{\sum_{j=1}^{J}(O_{1j}-E_{1j})}{\sqrt{\sum_{j=1}^{J}V_{j}}}.
  21. α \alpha
  22. Z > z α Z>z_{\alpha}
  23. z α z_{\alpha}
  24. α \alpha
  25. λ \lambda
  26. n n
  27. d d
  28. n d nd
  29. ( log λ ) n d 4 (\log{\lambda})\,\sqrt{\frac{n\,d}{4}}
  30. α \alpha
  31. 1 - β 1-\beta
  32. n = 4 ( z α + z β ) 2 d log 2 λ n=\frac{4\,(z_{\alpha}+z_{\beta})^{2}}{d\log^{2}{\lambda}}
  33. z α z_{\alpha}
  34. z β z_{\beta}
  35. Z 1 Z_{1}
  36. Z 2 Z_{2}
  37. Z 1 Z_{1}
  38. λ \lambda
  39. d 1 d_{1}
  40. d 2 d_{2}
  41. d 1 d 2 d_{1}\leq d_{2}
  42. Z 1 Z_{1}
  43. Z 2 Z_{2}
  44. log λ n d 1 4 \log{\lambda}\,\sqrt{\frac{n\,d_{1}}{4}}
  45. log λ n d 2 4 \log{\lambda}\,\sqrt{\frac{n\,d_{2}}{4}}
  46. d 1 d 2 \sqrt{\frac{d_{1}}{d_{2}}}
  47. Z Z
  48. D D
  49. λ ^ \hat{\lambda}
  50. log λ ^ Z 4 / D \log{\hat{\lambda}}\approx Z\,\sqrt{4/D}

Logarithmic_number_system.html

  1. s b ( z ) s_{b}(z)
  2. d b ( z ) d_{b}(z)
  3. X X
  4. x x
  5. X { s , x = log b ( | X | ) } , X\rightarrow\{s,x=\log_{b}(|X|)\},
  6. s s
  7. X X
  8. s = 0 s=0
  9. X > 0 X>0
  10. s = 1 s=1
  11. X < 0 X<0
  12. x x
  13. log b ( | X | + | Y | ) = x + s b ( z ) \log_{b}(|X|+|Y|)=x+s_{b}(z)
  14. log b ( | | X | - | Y | | ) = x + d b ( z ) , \log_{b}(||X|-|Y||)=x+d_{b}(z),
  15. z = y - x z=y-x
  16. s b ( z ) = log b ( 1 + b z ) s_{b}(z)=\log_{b}(1+b^{z})
  17. d b ( z ) = log b ( | 1 - b z | ) d_{b}(z)=\log_{b}(|1-b^{z}|)
  18. s b ( z ) s_{b}(z)
  19. d b ( z ) d_{b}(z)

Logarithmically_concave_measure.html

  1. μ ( λ A + ( 1 - λ ) B ) μ ( A ) λ μ ( B ) 1 - λ , \mu(\lambda A+(1-\lambda)B)\geq\mu(A)^{\lambda}\mu(B)^{1-\lambda},

Logic_alphabet.html

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

Logic_redundancy.html

  1. Y = A B + A ¯ C + B C . Y=AB+\overline{A}C+BC.
  2. B C BC
  3. A A
  4. B = 1 B=1
  5. C = 1 C=1
  6. Y Y
  7. A A
  8. A A
  9. f ( A , B , C , D ) = E ( 6 , 8 , 9 , 10 , 11 , 12 , 13 , 14 ) . f(A,B,C,D)=E(6,8,9,10,11,12,13,14).
  10. F = A C ¯ + A B ¯ + B C D ¯ . F=A\overline{C}+A\overline{B}+BC\overline{D}.
  11. A B C D = 1110 ABCD=1110
  12. A B C D = 1010 ABCD=1010
  13. B C D ¯ BC\overline{D}
  14. A B ¯ A\overline{B}
  15. A B C D = 1110 ABCD=1110
  16. A B C D = 1100 ABCD=1100
  17. A D ¯ A\overline{D}
  18. A D ¯ A\overline{D}
  19. A + D ¯ A+\overline{D}

Long_short_term_memory.html

  1. y = s ( w i x i ) y=s(\sum w_{i}x_{i})
  2. Π \Pi
  3. y = Π x i y=\Pi x_{i}
  4. Σ \Sigma
  5. y = w i x i y=\sum w_{i}x_{i}

Loss_given_default.html

  1. L G D * L_{GD}^{*}
  2. L G D * = L G D E * E L_{GD}^{*}=L_{GD}\cdot\frac{E^{*}}{E}

Loss–DiVincenzo_quantum_computer.html

  1. H s ( t ) = J ( t ) S L S R . H_{s}(t)=J(t)\vec{S}_{L}\cdot\vec{S}_{R}.
  2. Δ E \Delta E
  3. k T \;kT
  4. τ s \tau_{s}
  5. / Δ E \hbar/\Delta E
  6. Γ - 1 \Gamma^{-1}
  7. τ s \tau_{s}
  8. U s ( t ) = 𝐓 e x p { - i 0 t d t H s ( t ) } . U_{s}(t)=\mathbf{T}exp\{-i\int_{0}^{t}dt^{\prime}H_{s}(t^{\prime})\}.
  9. J ( t ) J(t)
  10. J 0 τ s = π ( m o d 2 π ) , J_{0}\tau_{s}=\pi(mod2\pi),
  11. U s U_{s}
  12. U s ( J 0 τ s = π ) U s w U_{s}(J_{0}\tau_{s}=\pi)\equiv U_{sw}
  13. s w a p \sqrt{swap}
  14. U X O R = e i π 2 S L z e - i π 2 S R z U s w 1 / 2 e i π S L z U s w 1 / 2 . U_{XOR}=e^{i\frac{\pi}{2}S_{L}^{z}}e^{-i\frac{\pi}{2}S_{R}^{z}}U_{sw}^{1/2}e^{% i\pi S_{L}^{z}}U_{sw}^{1/2}.
  15. S L + S R \vec{S}_{L}+\vec{S}_{R}

Low-energy_electron_microscopy.html

  1. λ = h 2 m E , λ [ A ] = 150 E [ eV ] \displaystyle\lambda=\frac{h}{\sqrt{2mE}},\qquad\lambda[\textrm{A}]=\sqrt{% \frac{150}{E[\textrm{eV}]}}
  2. 𝐤 0 = 2 π / λ 0 \,\textbf{k}_{0}=2\pi/\lambda_{0}
  3. 𝐤 = 2 π / λ \begin{aligned}\displaystyle\,\textbf{k}=2\pi/\lambda\end{aligned}
  4. 𝐤 - 𝐤 0 = 𝐆 hkl \,\textbf{k}-\,\textbf{k}_{0}=\,\textbf{G}_{\textrm{hkl}}
  5. 𝐆 hkl = h 𝐚 * + k 𝐛 * + l 𝐜 * \,\textbf{G}_{\textrm{hkl}}=h\,\textbf{a}^{*}+k\,\textbf{b}^{*}+l\,\textbf{c}^% {*}

Low_(computability).html

  1. Π 1 0 \Pi^{0}_{1}
  2. 2 ω 2^{\omega}

Low_basis_theorem.html

  1. Π 1 0 \Pi^{0}_{1}
  2. 2 ω 2^{\omega}
  3. Π 1 0 \Pi^{0}_{1}
  4. Π 1 0 \Pi^{0}_{1}

LPBoost.html

  1. f : 𝒳 { - 1 , 1 } , f:\mathcal{X}\to\{-1,1\},
  2. 𝒳 \mathcal{X}
  3. f ( s y m b o l x ) = j = 1 J α j h j ( s y m b o l x ) , f(symbol{x})=\sum_{j=1}^{J}\alpha_{j}h_{j}(symbol{x}),
  4. α j \alpha_{j}
  5. h j : 𝒳 { - 1 , 1 } h_{j}:\mathcal{X}\to\{-1,1\}
  6. h j h_{j}
  7. f f
  8. s y m b o l α symbol{\alpha}
  9. = { h ( ; ω ) | ω Ω } \mathcal{H}=\{h(\cdot;\omega)|\omega\in\Omega\}
  10. s y m b o l α symbol{\alpha}
  11. s y m b o l ξ symbol{\xi}
  12. ρ \rho
  13. min s y m b o l α , s y m b o l ξ , ρ - ρ + D n = 1 ξ n sb.t. ω Ω y n α ω h ( s y m b o l x n ; ω ) + ξ n ρ , n = 1 , , , ω Ω α ω = 1 , ξ n 0 , n = 1 , , , α ω 0 , ω Ω , ρ . \begin{array}[]{cl}\underset{symbol{\alpha},symbol{\xi},\rho}{\min}&-\rho+D% \sum_{n=1}^{\ell}\xi_{n}\\ \textrm{sb.t.}&\sum_{\omega\in\Omega}y_{n}\alpha_{\omega}h(symbol{x}_{n};% \omega)+\xi_{n}\geq\rho,\qquad n=1,\dots,\ell,\\ &\sum_{\omega\in\Omega}\alpha_{\omega}=1,\\ &\xi_{n}\geq 0,\qquad n=1,\dots,\ell,\\ &\alpha_{\omega}\geq 0,\qquad\omega\in\Omega,\\ &\rho\in{\mathbb{R}}.\end{array}
  14. s y m b o l ξ 0 symbol{\xi}\geq 0
  15. D D
  16. Ω \Omega
  17. ω Ω \omega\in\Omega
  18. h ( ; ω ) : 𝒳 { - 1 , 1 } h(\cdot;\omega):\mathcal{X}\to\{-1,1\}
  19. s y m b o l α symbol{\alpha}
  20. max s y m b o l λ , γ γ sb.t. n = 1 y n h ( s y m b o l x n ; ω ) λ n + γ 0 , ω Ω , 0 λ n D , n = 1 , , , n = 1 λ n = 1 , γ . \begin{array}[]{cl}\underset{symbol{\lambda},\gamma}{\max}&\gamma\\ \textrm{sb.t.}&\sum_{n=1}^{\ell}y_{n}h(symbol{x}_{n};\omega)\lambda_{n}+\gamma% \leq 0,\qquad\omega\in\Omega,\\ &0\leq\lambda_{n}\leq D,\qquad n=1,\dots,\ell,\\ &\sum_{n=1}^{\ell}\lambda_{n}=1,\\ &\gamma\in\mathbb{R}.\end{array}
  21. γ * \gamma^{*}
  22. ω * Ω \omega^{*}\in\Omega
  23. ω * = argmax ω Ω n = 1 y n h ( s y m b o l x n ; ω ) λ n . \omega^{*}=\underset{\omega\in\Omega}{\textrm{argmax}}\sum_{n=1}^{\ell}y_{n}h(% symbol{x}_{n};\omega)\lambda_{n}.
  24. \mathcal{H}
  25. h ( ; ω * ) h(\cdot;\omega^{*})
  26. D D
  27. D D
  28. D = 1 ν D=\frac{1}{\ell\nu}
  29. \ell
  30. 0 < ν < 1 0<\nu<1
  31. ν \nu
  32. ν \nu
  33. k k
  34. k ν \frac{k}{\ell}\leq\nu
  35. ν \nu
  36. X = { s y m b o l x 1 , , s y m b o l x } X=\{symbol{x}_{1},\dots,symbol{x}_{\ell}\}
  37. s y m b o l x i 𝒳 symbol{x}_{i}\in\mathcal{X}
  38. Y = { y 1 , , y } Y=\{y_{1},\dots,y_{\ell}\}
  39. y i { - 1 , 1 } y_{i}\in\{-1,1\}
  40. θ 0 \theta\geq 0
  41. f : 𝒳 { - 1 , 1 } f:\mathcal{X}\to\{-1,1\}
  42. λ n 1 , n = 1 , , \lambda_{n}\leftarrow\frac{1}{\ell},\quad n=1,\dots,\ell
  43. γ 0 \gamma\leftarrow 0
  44. J 1 J\leftarrow 1
  45. h ^ argmax ω Ω n = 1 y n h ( s y m b o l x n ; ω ) λ n \hat{h}\leftarrow\underset{\omega\in\Omega}{\textrm{argmax}}\sum_{n=1}^{\ell}y% _{n}h(symbol{x}_{n};\omega)\lambda_{n}
  46. n = 1 y n h ^ ( s y m b o l x n ) λ n + γ θ \sum_{n=1}^{\ell}y_{n}\hat{h}(symbol{x}_{n})\lambda_{n}+\gamma\leq\theta
  47. h J h ^ h_{J}\leftarrow\hat{h}
  48. J J + 1 J\leftarrow J+1
  49. ( s y m b o l λ , γ ) (symbol{\lambda},\gamma)\leftarrow
  50. s y m b o l α symbol{\alpha}\leftarrow
  51. f ( s y m b o l x ) := sign ( j = 1 J α j h j ( s y m b o l x ) ) f(symbol{x}):=\textrm{sign}\left(\sum_{j=1}^{J}\alpha_{j}h_{j}(symbol{x})\right)
  52. θ = 0 \theta=0
  53. θ \theta
  54. ρ ( s y m b o l α ) := min n = 1 , , y n α ω Ω α ω h ( s y m b o l x n ; ω ) . \rho(symbol{\alpha}):=\min_{n=1,\dots,\ell}y_{n}\sum_{\alpha_{\omega}\in\Omega% }\alpha_{\omega}h(symbol{x}_{n};\omega).
  55. \mathcal{H}
  56. 𝒳 n \mathcal{X}\subseteq{\mathbb{R}}^{n}
  57. h ( s y m b o l x ; ω { 1 , - 1 } , p { 1 , , n } , t ) := { ω if s y m b o l x p t - ω otherwise . h(symbol{x};\omega\in\{1,-1\},p\in\{1,\dots,n\},t\in{\mathbb{R}}):=\left\{% \begin{array}[]{cl}\omega&\textrm{if~{}}symbol{x}_{p}\leq t\\ -\omega&\textrm{otherwise}\end{array}\right..
  58. p p
  59. t t
  60. ω \omega
  61. p p
  62. t t
  63. ω \omega

Lubrication_theory.html

  1. H H
  2. L L
  3. ϵ = H / L \epsilon=H/L
  4. ϵ 1 \epsilon\ll 1
  5. p z = 0 \frac{\partial p}{\partial z}=0
  6. p x = μ 2 u z 2 \frac{\partial p}{\partial x}=\mu\frac{\partial^{2}u}{\partial z^{2}}
  7. x x
  8. z z
  9. p p
  10. u u
  11. μ \mu

Lumer–Phillips_theorem.html

  1. u , A u = 0 1 u ( x ) u ( x ) d x = - 1 2 u ( 0 ) 2 0 , \langle u,Au\rangle=\int_{0}^{1}u(x)u^{\prime}(x)\,\mathrm{d}x=-\frac{1}{2}u(0% )^{2}\leq 0,
  2. u ( x ) = e λ x x 1 e - λ t f ( t ) d t u(x)={\rm e}^{\lambda x}\int_{x}^{1}{\rm e}^{-\lambda t}f(t)\,dt

Lyman-alpha_emitter.html

  1. 1 + z = λ 1215.67 Å 1+z=\frac{\lambda}{1215.67\mathrm{\AA}}
  2. λ \lambda

Lyman–Werner_photons.html

  1. 2 {}_{2}
  2. 2 {}_{2}

M._Riesz_extension_theorem.html

  1. ϕ ( x ) 0 for x F K . \phi(x)\geq 0\quad\,\text{for}\quad x\in F\cap K.
  2. ψ | F = ϕ and ψ ( x ) 0 for x K . \psi|_{F}=\phi\quad\,\text{and}\quad\psi(x)\geq 0\quad\,\text{for}\quad x\in K.
  3. K K
  4. ψ | F = ϕ , ψ ( y ) = sup { ϕ ( x ) x F , y - x K } , \psi|_{F}=\phi,\quad\psi(y)=\sup\left\{\phi(x)\,\mid\,x\in F,\,y-x\in K\right\},
  5. ψ ( y ) ψ ( - x ) = - ψ ( x ) , \psi(y)\geq\psi(-x)=-\psi(x),
  6. ψ ( x ) - ψ ( x 1 ) = ψ ( x - x 1 ) = ψ ( z 1 + z / a ) = ϕ ( z 1 + z / a ) 0 , \psi(x)-\psi(x_{1})=\psi(x-x_{1})=\psi(z_{1}+z/a)=\phi(z_{1}+z/a)\geq 0~{},
  7. ϕ ( x ) N ( x ) , x U . \phi(x)\leq N(x),\quad x\in U.
  8. K = { ( a , x ) N ( x ) a } . K=\left\{(a,x)\,\mid\,N(x)\leq a\right\}.
  9. ϕ 1 ( a , x ) = a - ϕ ( x ) . \phi_{1}(a,x)=a-\phi(x).
  10. ψ ( x ) = - ψ 1 ( 0 , x ) \psi(x)=-\psi_{1}(0,x)
  11. ψ 1 ( N ( x ) , x ) = N ( x ) - ψ ( x ) < 0 , \psi_{1}(N(x),x)=N(x)-\psi(x)<0,

MA_plot.html

  1. M = log 2 ( R / G ) = log 2 ( R ) - log 2 ( G ) M=\log_{2}(R/G)=\log_{2}(R)-\log_{2}(G)
  2. A = 1 2 log 2 ( R G ) = 1 2 ( log 2 ( R ) + log 2 ( G ) ) A=\frac{1}{2}\log_{2}(RG)=\frac{1}{2}(\log_{2}(R)+\log_{2}(G))

Macaulay's_method.html

  1. ± E I d 2 w d x 2 = M \pm EI\dfrac{d^{2}w}{dx^{2}}=M
  2. w w
  3. M M
  4. w w
  5. M M
  6. x x
  7. M M
  8. M = M 1 ( x ) + P 1 x - a 1 + P 2 x - a 2 + P 3 x - a 3 + M=M_{1}(x)+P_{1}\langle x-a_{1}\rangle+P_{2}\langle x-a_{2}\rangle+P_{3}% \langle x-a_{3}\rangle+\dots
  9. P i x - a i P_{i}\langle x-a_{i}\rangle
  10. x - a i \langle x-a_{i}\rangle
  11. x - a i = { 0 if x < a i x - a i if x > a i \langle x-a_{i}\rangle=\begin{cases}0&\mathrm{if}~{}x<a_{i}\\ x-a_{i}&\mathrm{if}~{}x>a_{i}\end{cases}
  12. P ( x - a ) P(x-a)
  13. P ( x - a ) d x = P [ x 2 2 - a x ] + C \int P(x-a)~{}dx=P\left[\cfrac{x^{2}}{2}-ax\right]+C
  14. P x - a d x = P x - a 2 2 + C m \int P\langle x-a\rangle~{}dx=P\cfrac{\langle x-a\rangle^{2}}{2}+C_{m}
  15. C m C_{m}
  16. M M
  17. R A + R C = P , L R C = P a R_{A}+R_{C}=P,~{}~{}LR_{C}=Pa
  18. R A = P b / L R_{A}=Pb/L
  19. 0 < x < a 0<x<a
  20. M = R A x = P b x / L M=R_{A}x=Pbx/L
  21. E I d 2 w d x 2 = P b x L EI\dfrac{d^{2}w}{dx^{2}}=\dfrac{Pbx}{L}
  22. 0 < x < a 0<x<a
  23. E I d w d x = P b x 2 2 L + C 1 ( i ) E I w = P b x 3 6 L + C 1 x + C 2 ( ii ) \begin{aligned}\displaystyle EI\dfrac{dw}{dx}&\displaystyle=\dfrac{Pbx^{2}}{2L% }+C_{1}&&\displaystyle\quad\mathrm{(i)}\\ \displaystyle EIw&\displaystyle=\dfrac{Pbx^{3}}{6L}+C_{1}x+C_{2}&&% \displaystyle\quad\mathrm{(ii)}\end{aligned}
  24. x = a - x=a_{-}
  25. E I d w d x ( a - ) = P b a 2 2 L + C 1 ( iii ) E I w ( a - ) = P b a 3 6 L + C 1 a + C 2 ( iv ) \begin{aligned}\displaystyle EI\dfrac{dw}{dx}(a_{-})&\displaystyle=\dfrac{Pba^% {2}}{2L}+C_{1}&&\displaystyle\quad\mathrm{(iii)}\\ \displaystyle EIw(a_{-})&\displaystyle=\dfrac{Pba^{3}}{6L}+C_{1}a+C_{2}&&% \displaystyle\quad\mathrm{(iv)}\end{aligned}
  26. a < x < L a<x<L
  27. M = R A x - P ( x - a ) = P b x / L - P ( x - a ) M=R_{A}x-P(x-a)=Pbx/L-P(x-a)
  28. M = P b x L - P x - a M=\frac{Pbx}{L}-P\langle x-a\rangle
  29. E I d 2 w d x 2 = P b x L - P x - a EI\dfrac{d^{2}w}{dx^{2}}=\dfrac{Pbx}{L}-P\langle x-a\rangle
  30. a < x < L a<x<L
  31. E I d w d x = P b x 2 2 L - P x - a 2 2 + D 1 ( v ) E I w = P b x 3 6 L - P x - a 3 6 + D 1 x + D 2 ( vi ) \begin{aligned}\displaystyle EI\dfrac{dw}{dx}&\displaystyle=\dfrac{Pbx^{2}}{2L% }-P\cfrac{\langle x-a\rangle^{2}}{2}+D_{1}&&\displaystyle\quad\mathrm{(v)}\\ \displaystyle EIw&\displaystyle=\dfrac{Pbx^{3}}{6L}-P\cfrac{\langle x-a\rangle% ^{3}}{6}+D_{1}x+D_{2}&&\displaystyle\quad\mathrm{(vi)}\end{aligned}
  32. x = a + x=a_{+}
  33. E I d w d x ( a + ) = P b a 2 2 L + D 1 ( vii ) E I w ( a + ) = P b a 3 6 L + D 1 a + D 2 ( viii ) \begin{aligned}\displaystyle EI\dfrac{dw}{dx}(a_{+})&\displaystyle=\dfrac{Pba^% {2}}{2L}+D_{1}&&\displaystyle\quad\mathrm{(vii)}\\ \displaystyle EIw(a_{+})&\displaystyle=\dfrac{Pba^{3}}{6L}+D_{1}a+D_{2}&&% \displaystyle\quad\mathrm{(viii)}\end{aligned}
  34. C 1 = D 1 C_{1}=D_{1}
  35. C 2 = D 2 C_{2}=D_{2}
  36. x - a n , x - b n , x - c n \langle x-a\rangle^{n},\langle x-b\rangle^{n},\langle x-c\rangle^{n}
  37. E I d 2 w d x 2 = P b x L - P x - a EI\dfrac{d^{2}w}{dx^{2}}=\dfrac{Pbx}{L}-P\langle x-a\rangle
  38. x < a x<a
  39. x > a x>a
  40. E I d w d x = [ P b x 2 2 L + C 1 ] - P x - a 2 2 E I w = [ P b x 3 6 L + C 1 x + C 2 ] - P x - a 3 6 \begin{aligned}\displaystyle EI\dfrac{dw}{dx}&\displaystyle=\left[\dfrac{Pbx^{% 2}}{2L}+C_{1}\right]-\cfrac{P\langle x-a\rangle^{2}}{2}\\ \displaystyle EIw&\displaystyle=\left[\dfrac{Pbx^{3}}{6L}+C_{1}x+C_{2}\right]-% \cfrac{P\langle x-a\rangle^{3}}{6}\end{aligned}
  41. x < a x<a
  42. x > a x>a
  43. x < a x<a
  44. w = 0 w=0
  45. x = 0 x=0
  46. C 2 = 0 C2=0
  47. w = 0 w=0
  48. x = L x=L
  49. [ P b L 2 6 + C 1 L ] - P ( L - a ) 3 6 = 0 \left[\dfrac{PbL^{2}}{6}+C_{1}L\right]-\cfrac{P(L-a)^{3}}{6}=0
  50. C 1 = - P b 6 L ( L 2 - b 2 ) . C_{1}=-\cfrac{Pb}{6L}(L^{2}-b^{2})~{}.
  51. E I d w d x = [ P b x 2 2 L - P b 6 L ( L 2 - b 2 ) ] - P x - a 2 2 E I w = [ P b x 3 6 L - P b x 6 L ( L 2 - b 2 ) ] - P x - a 3 6 \begin{aligned}\displaystyle EI\dfrac{dw}{dx}&\displaystyle=\left[\dfrac{Pbx^{% 2}}{2L}-\cfrac{Pb}{6L}(L^{2}-b^{2})\right]-\cfrac{P\langle x-a\rangle^{2}}{2}% \\ \displaystyle EIw&\displaystyle=\left[\dfrac{Pbx^{3}}{6L}-\cfrac{Pbx}{6L}(L^{2% }-b^{2})\right]-\cfrac{P\langle x-a\rangle^{3}}{6}\end{aligned}
  52. w w
  53. d w / d x = 0 dw/dx=0
  54. x < a x<a
  55. P b x 2 2 L - P b 6 L ( L 2 - b 2 ) = 0 \dfrac{Pbx^{2}}{2L}-\cfrac{Pb}{6L}(L^{2}-b^{2})=0
  56. x = ± ( L 2 - b 2 ) 1 / 2 3 x=\pm\cfrac{(L^{2}-b^{2})^{1/2}}{\sqrt{3}}
  57. x < 0 x<0
  58. E I w max = 1 3 [ P b ( L 2 - b 2 ) 3 / 2 6 3 L ] - P b ( L 2 - b 2 ) 3 / 2 6 3 L EIw_{\mathrm{max}}=\cfrac{1}{3}\left[\dfrac{Pb(L^{2}-b^{2})^{3/2}}{6\sqrt{3}L}% \right]-\cfrac{Pb(L^{2}-b^{2})^{3/2}}{6\sqrt{3}L}
  59. w max = - P b ( L 2 - b 2 ) 3 / 2 9 3 E I L . w_{\mathrm{max}}=-\dfrac{Pb(L^{2}-b^{2})^{3/2}}{9\sqrt{3}EIL}~{}.
  60. x = a x=a
  61. E I w B = P b a 3 6 L - P b a 6 L ( L 2 - b 2 ) = P b a 6 L ( a 2 + b 2 - L 2 ) EIw_{B}=\dfrac{Pba^{3}}{6L}-\cfrac{Pba}{6L}(L^{2}-b^{2})=\frac{Pba}{6L}(a^{2}+% b^{2}-L^{2})
  62. w B = - P a 2 b 2 3 L E I w_{B}=-\cfrac{Pa^{2}b^{2}}{3LEI}
  63. w max / w ( L / 2 ) w_{\mathrm{max}}/w(L/2)
  64. x = L / 2 x=L/2
  65. E I w ( L / 2 ) = P b L 2 48 - P b 12 ( L 2 - b 2 ) = - P b 12 [ 3 L 2 4 - b 2 ] EIw(L/2)=\dfrac{PbL^{2}}{48}-\cfrac{Pb}{12}(L^{2}-b^{2})=-\frac{Pb}{12}\left[% \frac{3L^{2}}{4}-b^{2}\right]
  66. w max w ( L / 2 ) = 4 ( L 2 - b 2 ) 3 / 2 3 3 L [ 3 L 2 4 - b 2 ] = 4 ( 1 - b 2 L 2 ) 3 / 2 3 3 [ 3 4 - b 2 L 2 ] = 16 ( 1 - k 2 ) 3 / 2 3 3 ( 3 - 4 k 2 ) \frac{w_{\mathrm{max}}}{w(L/2)}=\frac{4(L^{2}-b^{2})^{3/2}}{3\sqrt{3}L\left[% \frac{3L^{2}}{4}-b^{2}\right]}=\frac{4(1-\frac{b^{2}}{L^{2}})^{3/2}}{3\sqrt{3}% \left[\frac{3}{4}-\frac{b^{2}}{L^{2}}\right]}=\frac{16(1-k^{2})^{3/2}}{3\sqrt{% 3}\left(3-4k^{2}\right)}
  67. k = B / L k=B/L
  68. a < b ; 0 < k < 0.5 a<b;0<k<0.5
  69. a = b = L / 2 a=b=L/2
  70. w w
  71. x = [ L 2 - ( L / 2 ) 2 ] 1 / 2 3 = L 2 x=\cfrac{[L^{2}-(L/2)^{2}]^{1/2}}{\sqrt{3}}=\frac{L}{2}
  72. w max = - P ( L / 2 ) b [ L 2 - ( L / 2 ) 2 ] 3 / 2 9 3 E I L = - P L 3 48 E I = w ( L / 2 ) . w_{\mathrm{max}}=-\dfrac{P(L/2)b[L^{2}-(L/2)^{2}]^{3/2}}{9\sqrt{3}EIL}=-\frac{% PL^{3}}{48EI}=w(L/2)~{}.

Macaulay_brackets.html

  1. { x } = { 0 , x < 0 x , x 0. \{x\}=\begin{cases}0,&x<0\\ x,&x\geq 0.\end{cases}
  2. x \langle x\rangle
  3. x x
  4. ( x ) (x)
  5. x x
  6. { } \{...\}
  7. { x - a } n = { 0 , x < a ( x - a ) n , x a . \{x-a\}^{n}=\begin{cases}0,&x<a\\ (x-a)^{n},&x\geq a.\end{cases}
  8. ( n 0 ) (n\geq 0)\,\!
  9. ( x - a ) n (x-a)^{n}

Macdonald_polynomials.html

  1. μ λ \mu\leq\lambda
  2. λ - μ \lambda-\mu
  3. ( a ; q ) = r 0 ( 1 - a q r ) (a;q)_{\infty}=\prod_{r\geq 0}(1-aq^{r})
  4. Δ = α R ( e α ; q ) ( t e α ; q ) . \Delta=\prod_{\alpha\in R}{(e^{\alpha};q)_{\infty}\over(te^{\alpha};q)_{\infty% }}.
  5. f , g = ( constant term of f g ¯ Δ ) / | W | \langle f,g\rangle=(\,\text{constant term of }f\overline{g}\Delta)/|W|
  6. P λ = μ λ u λ μ m μ P_{\lambda}=\sum_{\mu\leq\lambda}u_{\lambda\mu}m_{\mu}
  7. P λ , P λ = α R , α > 0 0 < i < k 1 - q ( λ + k ρ , 2 α / ( α , α ) ) + i 1 - q ( λ + k ρ , 2 α / ( α , α ) ) - i . \langle P_{\lambda},P_{\lambda}\rangle=\prod_{\alpha\in R,\alpha>0}\prod_{0<i<% k}{1-q^{(\lambda+k\rho,2\alpha/(\alpha,\alpha))+i}\over 1-q^{(\lambda+k\rho,2% \alpha/(\alpha,\alpha))-i}}.
  8. P λ ( , q μ i t ρ i , ) P λ ( t ρ ) = P μ ( , q λ i t ρ i , ) P μ ( t ρ ) . \frac{P_{\lambda}(\dots,q^{\mu_{i}}t^{\rho_{i}},\dots)}{P_{\lambda}(t^{\rho})}% =\frac{P_{\mu}(\dots,q^{\lambda_{i}}t^{\rho_{i}},\dots)}{P_{\mu}(t^{\rho})}.
  9. H ~ μ \widetilde{H}_{\mu}
  10. ( q , t ) \mathbb{Q}(q,t)
  11. s λ s_{\lambda}
  12. D μ = C [ x , y ] Δ μ D_{\mu}=C[\partial x,\partial y]\,\Delta_{\mu}
  13. Δ μ = det ( x i p j y i q j ) 1 i , j , n \Delta_{\mu}=\det(x_{i}^{p_{j}}y_{i}^{q_{j}})_{1\leq i,j,\leq n}
  14. Δ μ = x 1 y 2 + x 2 y 3 + x 3 y 1 - x 2 y 1 - x 3 y 2 - x 1 y 3 \Delta_{\mu}=x_{1}y_{2}+x_{2}y_{3}+x_{3}y_{1}-x_{2}y_{1}-x_{3}y_{2}-x_{1}y_{3}
  15. y 2 - y 3 y_{2}-y_{3}
  16. y 3 - y 1 y_{3}-y_{1}
  17. x 3 - x 2 x_{3}-x_{2}
  18. x 1 - x 3 x_{1}-x_{3}
  19. 1 1
  20. H ~ μ \widetilde{H}_{\mu}
  21. P λ P_{\lambda}
  22. H ~ μ ( x ; q , t ) = σ : μ + q i n v ( σ ) t m a j ( σ ) x σ \widetilde{H}_{\mu}(x;q,t)=\sum_{\sigma:\mu\to\mathbb{Z}_{+}}q^{inv(\sigma)}t^% {maj(\sigma)}x^{\sigma}
  23. x 1 σ 1 x 2 σ 2 x_{1}^{\sigma_{1}}x_{2}^{\sigma_{2}}\cdots
  24. H ~ μ ( x ; q , t ) \widetilde{H}_{\mu}(x;q,t)
  25. P λ P_{\lambda}
  26. J λ ( x ; q , t ) J_{\lambda}(x;q,t)
  27. P λ ( x ; q , t ) P_{\lambda}(x;q,t)
  28. J λ ( x ; q , t ) = s D ( λ ) ( 1 - q a ( s ) t 1 + l ( s ) ) P λ ( x ; q , t ) J_{\lambda}(x;q,t)=\prod_{s\in D(\lambda)}(1-q^{a(s)}t^{1+l(s)})\cdot P_{% \lambda}(x;q,t)
  29. D ( λ ) D(\lambda)
  30. λ \lambda
  31. a ( s ) a(s)
  32. l ( s ) l(s)
  33. s s
  34. H ~ μ ( x ; q , t ) \widetilde{H}_{\mu}(x;q,t)
  35. J μ J_{\mu}
  36. H ~ μ ( x ; q , t ) = t - n ( μ ) J μ [ X 1 - t - 1 ; q , t - 1 ] \widetilde{H}_{\mu}(x;q,t)=t^{-n(\mu)}J_{\mu}\left[\frac{X}{1-t^{-1}};q,t^{-1}\right]
  37. n ( μ ) = i μ i ( i - 1 ) . n(\mu)=\sum_{i}\mu_{i}\cdot(i-1).

Machmeter.html

  1. M = 5 [ ( p t p ) 2 7 - 1 ] {M}=\sqrt{5\left[\left(\frac{p_{t}}{p}\right)^{\frac{2}{7}}-1\right]}\,
  2. M \ M\,
  3. p t \ p_{t}\,
  4. p \ p
  5. M = 0.88128485 [ ( p t p + 1 ) ( 1 - 1 [ 7 M 2 ] ) 5 2 ] {M}=0.88128485\sqrt{\left[\left(\frac{p_{t}}{p}+1\right)\left(1-\frac{1}{[7M^{% 2}]}\right)^{\frac{5}{2}}\right]}
  6. p t \ p_{t}

Maclaurin's_inequality.html

  1. S k = 1 i 1 < < i k n a i 1 a i 2 a i k ( n k ) . S_{k}=\frac{\displaystyle\sum_{1\leq i_{1}<\cdots<i_{k}\leq n}a_{i_{1}}a_{i_{2% }}\cdots a_{i_{k}}}{\displaystyle{n\choose k}}.
  2. ( n k ) . \scriptstyle{n\choose k}.
  3. S 1 S 2 S 3 3 S n n S_{1}\geq\sqrt{S_{2}}\geq\sqrt[3]{S_{3}}\geq\cdots\geq\sqrt[n]{S_{n}}
  4. a 1 + a 2 + a 3 + a 4 4 \displaystyle{}\quad\frac{a_{1}+a_{2}+a_{3}+a_{4}}{4}

Magnetic_impurity.html

  1. χ i m p = C T + θ \chi_{imp}=\frac{C}{T+\theta}

Magnetic_trap_(atoms).html

  1. Δ E = - μ B \Delta E=-\vec{\mu}\cdot\vec{B}

Magnetic_tweezers.html

  1. m ( B ) \overrightarrow{m}(\overrightarrow{B})
  2. B \overrightarrow{B}
  3. m ( B ) = V χ B μ 0 \overrightarrow{m}(\overrightarrow{B})=\frac{V\chi\overrightarrow{B}}{\mu_{0}}
  4. μ 0 \mu_{0}
  5. V V
  6. χ \chi
  7. χ = 3 μ r - 1 μ r + 2 \chi=3\frac{\mu_{r}-1}{\mu_{r}+2}
  8. μ r \mu_{r}
  9. m s a t \overrightarrow{m}_{sat}
  10. F \overrightarrow{F}
  11. U = - 1 2 m ( B ) B U=-\frac{1}{2}\overrightarrow{m}(\overrightarrow{B})\cdot\overrightarrow{B}
  12. F = - U = { V χ 2 μ 0 | B | 2 in a weak magnetic field 1 2 ( m s a t B ) in a strong magnetic field \overrightarrow{F}=-\overrightarrow{\nabla}U=\begin{cases}\frac{V\chi}{2\mu_{0% }}\overrightarrow{\nabla}\left|\overrightarrow{B}\right|^{2}&\qquad\,\text{in % a weak magnetic field}\\ \frac{1}{2}\overrightarrow{\nabla}\left(\overrightarrow{m}_{sat}\cdot% \overrightarrow{B}\right)&\qquad\,\text{in a strong magnetic field}\end{cases}
  13. Γ \overrightarrow{\Gamma}
  14. m \overrightarrow{m}
  15. B \overrightarrow{B}
  16. Γ = m × B \overrightarrow{\Gamma}=\overrightarrow{m}\times\overrightarrow{B}
  17. 10 3 pNnm 10^{3}\mathrm{pNnm}
  18. x x
  19. x x
  20. δ x \delta x
  21. F χ F_{\chi}
  22. δ x \delta x
  23. F F
  24. l l
  25. F χ = F l δ x F_{\chi}=\frac{F}{l}\delta x
  26. 1 2 k B T \frac{1}{2}k_{B}T
  27. E p = 1 2 F l δ x 2 = 1 2 k B T \langle E_{p}\rangle=\frac{1}{2}\frac{F}{l}\langle\delta x^{2}\rangle=\frac{1}% {2}k_{B}T
  28. F = l k B T δ x 2 F=\frac{lk_{B}T}{\langle\delta x^{2}\rangle}
  29. P ( ω ) P(\omega)
  30. X ( ω ) X(\omega)
  31. Δ f \Delta f
  32. P ( ω ) = | X ( ω ) | 2 Δ f P(\omega)=\frac{\left|X(\omega)\right|^{2}}{\Delta f}
  33. X ( ω ) X(\omega)
  34. m m
  35. m 2 x ( t ) t 2 = - 6 π R η x ( t ) t - F l x ( t ) + f ( t ) m\frac{\partial^{2}x(t)}{\partial t^{2}}=-6\pi R\eta\frac{\partial x(t)}{% \partial t}-\frac{F}{l}x(t)+f(t)
  36. 6 π R η x ( t ) t 6\pi R\eta\frac{\partial x(t)}{\partial t}
  37. R R
  38. η \eta
  39. F l x ( t ) \frac{F}{l}x(t)
  40. f ( t ) f(t)
  41. m 2 x ( t ) t 2 m\frac{\partial^{2}x(t)}{\partial t^{2}}
  42. ( Re < 10 - 5 ) \left(\mathrm{Re}<10^{-5}\right)
  43. f ( t ) = \displaystyle f(t)=
  44. X ( ω ) = F ( ω ) 6 π i R η ω + F l X(\omega)=\frac{F(\omega)}{6\pi iR\eta\omega+\frac{F}{l}}
  45. F ( ω ) F(\omega)
  46. | F ( ω ) | 2 Δ f = 4 k B T 6 π η R \frac{\left|F(\omega)\right|^{2}}{\Delta f}=4k_{B}T\cdot 6\pi\eta R
  47. P ( ω ) = 24 π k B T η R 36 π 2 R 2 η 2 ω 2 + ( F l ) 2 P(\omega)=\frac{24\pi k_{B}T\eta R}{36\pi^{2}R^{2}\eta^{2}\omega^{2}+\left(% \frac{F}{l}\right)^{2}}
  48. F F
  49. F = 6 π η R v F=6\pi\eta Rv
  50. v v

Magnetorotational_instability.html

  1. s y m b o l J \timessymbol B , symbolJ\timessymbol B\ ,
  2. s y m b o l J symbolJ
  3. s y m b o l B symbolB
  4. Ω \Omega
  5. R , R\ ,
  6. Ω . \Omega\ .
  7. Ω \Omega
  8. R , R\ ,
  9. r = R 0 . r=R_{0}\ .
  10. - R Ω 2 ( R ) , -R\Omega^{2}(R)\ ,
  11. - G M / R 2 , -GM/R^{2},
  12. G G
  13. M M
  14. Ω ( R 0 ) = Ω 0 , \Omega(R_{0})=\Omega_{0}\ ,
  15. - 2 s y m b o l Ω 0 \timessymbol v -2symbol\Omega_{0}\timessymbol v
  16. R Ω 0 2 . R\Omega_{0}^{2}\ .
  17. v v
  18. R 0 , R_{0}\ ,
  19. R 0 + x , R_{0}+x\ ,
  20. x x
  21. R 0 . R_{0}\ .
  22. R [ Ω 0 2 - Ω 2 ( R 0 + x ) ] - x R d Ω 2 d R R[\Omega_{0}^{2}-\Omega^{2}(R_{0}+x)]\simeq-xR{d\Omega^{2}\over dR}
  23. x . x\ .
  24. x x
  25. y y
  26. x x
  27. y y
  28. R = R 0 R=R_{0}
  29. x ¨ - 2 Ω 0 y ˙ = - x R d Ω 2 d R + f x \ddot{x}-2\Omega_{0}\dot{y}=-xR{d\Omega^{2}\over dR}+f_{x}
  30. y ¨ + 2 Ω 0 x ˙ = f y \ddot{y}+2\Omega_{0}\dot{x}=f_{y}
  31. f x f_{x}
  32. f y f_{y}
  33. x x
  34. y y
  35. x ˙ \dot{x}
  36. x x
  37. x ¨ \ddot{x}
  38. x x
  39. f x f_{x}
  40. f y f_{y}
  41. f x = 0 f_{x}=0
  42. f y = 0 f_{y}=0
  43. e i ω t , e^{i\omega t}\ ,
  44. ω \omega
  45. ω 2 = 4 Ω 0 2 + R d Ω 2 d R κ 2 \omega^{2}=4\Omega_{0}^{2}+R{d\Omega^{2}\over dR}\equiv\kappa^{2}
  46. κ 2 \kappa^{2}
  47. ( 1 / R 3 ) ( d R 4 Ω 2 / d R ) , (1/R^{3})(dR^{4}\Omega^{2}/dR)\ ,
  48. R 1 / 2 , R^{1/2}\ ,
  49. f x = - K x , f_{x}=-Kx\ ,
  50. f y = - K y f_{y}=-Ky
  51. K K
  52. x x
  53. y y
  54. e i ω t , e^{i\omega t}\ ,
  55. ω : \omega\ :
  56. K K
  57. ω 2 , \omega^{2}\ ,
  58. ω . \omega\ .
  59. s y m b o l B , symbolB\ ,
  60. s y m b o l E = - v × B . symbol{E=-{v\times B}}\ .
  61. s y m b o l E + v × B symbol{E+v\times B}
  62. s y m b o l B symbolB
  63. - s y m b o l × E = \partialsymbol B t or \quadsymbol × ( v × B ) = \partialsymbol B t -symbol{\nabla\times E}={\partialsymbol B\over\partial t}\quad{\rm or}% \quadsymbol{\nabla\times(v\times B)}={\partialsymbol B\over\partial t}
  64. δ t \delta t
  65. s y m b o l ξ = s y m b o l v δ t , symbol\xi=symbolv\delta t\ ,
  66. δ s y m b o l B = s y m b o l × ( ξ × B ) \delta symbolB=symbol{\nabla\times(\xi\times B)}
  67. s y m b o l B symbolB
  68. ξ \xi
  69. δ s y m b o l B = s y m b o l ( B ) ξ , \delta symbolB=symbol{\ {(B\cdot\nabla)\xi}},
  70. × ( ξ × 𝐁 ) = ξ ( 𝐁 ) - 𝐁 ( ξ ) + ( 𝐁 ) ξ - ( ξ ) 𝐁 . \nabla\times(\mathbf{\xi}\times\mathbf{B})=\mathbf{\xi}(\nabla\cdot\mathbf{B})% -\mathbf{B}(\nabla\cdot\mathbf{\xi})+(\mathbf{B}\cdot\nabla)\mathbf{\xi}-(% \mathbf{\xi}\cdot\nabla)\mathbf{B}\ .
  71. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  72. ξ = 0 \nabla\cdot\mathbf{\xi}=0
  73. ( ξ ) 𝐁 = 0 (\mathbf{\xi}\cdot\nabla)\mathbf{B}=0
  74. s y m b o l B symbolB
  75. s y m b o l B symbolB
  76. z z
  77. ξ \xi
  78. e i k z . e^{ikz}\ .
  79. s y m b o l ξ symbol\xi
  80. cos ( k z ) , \cos(kz)\ ,
  81. \deltasymbol B \deltasymbol B
  82. - sin ( k z ) . -\sin(kz)\ .
  83. s y m b o l J \timessymbol B . symbolJ\timessymbol B\ .
  84. μ 0 s y m b o l J = × B , \mu_{0}symbol{J=\nabla\times B}\ ,
  85. ( 1 μ 0 ) s y m b o l ( × B ) × B = - s y m b o l ( B 2 2 μ 0 ) + ( 1 μ 0 ) s y m b o l ( B ) B \left(\frac{1}{\mu_{0}}\right)symbol{(\nabla\times B)\times B}=-symbol{\nabla}% \left(\frac{B^{2}}{2\mu_{0}}\right)+\left(\frac{1}{\mu_{0}}\right)symbol{(B% \cdot\nabla)B}
  86. z . z\ .
  87. \deltasymbol B , \deltasymbol B\ ,
  88. ( 1 μ 0 ρ ) s y m b o l ( B ) δ B = ( i k B s y m b o l δ B μ 0 ρ ) = - k 2 B 2 μ 0 ρ s y m b o l ( ξ ) \left(\frac{1}{\mu_{0}\rho}\right)symbol{(B\cdot\nabla)\delta B}=\left(\frac{% ikBsymbol{\delta B}}{\mu_{0}\rho}\right)=-{k^{2}B^{2}\over\mu_{0}\rho}symbol(\xi)
  89. ω \omega
  90. K = k 2 B 2 / μ 0 ρ : K={k^{2}B^{2}/\mu_{0}\rho}\ :
  91. d Ω 2 / d R < 0 , d\Omega^{2}/dR<0\ ,
  92. k k
  93. ( k 2 B 2 / μ 0 ρ ) < - R d Ω 2 / d R . (k^{2}B^{2}/\mu_{0}\rho)<-Rd\Omega^{2}/dR\ .
  94. k B . kB\ .
  95. B B
  96. k k
  97. k . k\ .
  98. Ω 2 = G M R 3 \Omega^{2}={GM\over R^{3}}
  99. G G
  100. M M
  101. R R
  102. R d Ω 2 / d R = - 3 Ω 2 < 0 , Rd\Omega^{2}/dR=-3\Omega^{2}<0\ ,
  103. γ = 3 Ω / 4 , \gamma=3\Omega/4\ ,
  104. ( k 2 B 2 / μ 0 ρ ) = 15 Ω 2 / 16 . (k^{2}B^{2}/\mu_{0}\rho)=15\Omega^{2}/16\ .
  105. γ \gamma

Majority_problem_(cellular_automaton).html

  1. j i + j < ρ \tfrac{j}{i+j}<\rho
  2. j i + j > ρ \tfrac{j}{i+j}>\rho
  3. j i + j = ρ \tfrac{j}{i+j}=\rho
  4. 2 i + j 2^{i+j}
  5. ρ \rho
  6. ρ \rho
  7. ρ \rho
  8. ρ \rho

Mallows's_Cp.html

  1. E j ( Y ^ j - E ( Y j X j ) ) 2 / σ 2 , E\sum_{j}(\hat{Y}_{j}-E(Y_{j}\mid X_{j}))^{2}/\sigma^{2},
  2. Y ^ j \hat{Y}_{j}
  3. C p = S S E p S 2 - N + 2 P , C_{p}={SSE_{p}\over S^{2}}-N+2P,
  4. S S E p = i = 1 N ( Y i - Y p i ) 2 SSE_{p}=\sum_{i=1}^{N}(Y_{i}-Y_{pi})^{2}

Mapping_cone_(homological_algebra).html

  1. A , B A,B
  2. d A , d B ; d_{A},d_{B};
  3. A = A n - 1 d A n - 1 A n d A n A n + 1 A=\dots\to A^{n-1}\xrightarrow{d_{A}^{n-1}}A^{n}\xrightarrow{d_{A}^{n}}A^{n+1}\to\cdots
  4. B . B.
  5. f : A B , f:A\to B,
  6. Cone ( f ) \operatorname{Cone}(f)
  7. C ( f ) , C(f),
  8. C ( f ) = A [ 1 ] B = A n B n - 1 A n + 1 B n A n + 2 B n + 1 C(f)=A[1]\oplus B=\dots\to A^{n}\oplus B^{n-1}\to A^{n+1}\oplus B^{n}\to A^{n+% 2}\oplus B^{n+1}\to\cdots
  9. d C ( f ) = ( d A [ 1 ] 0 f [ 1 ] d B ) d_{C(f)}=\begin{pmatrix}d_{A[1]}&0\\ f[1]&d_{B}\end{pmatrix}
  10. A [ 1 ] A[1]
  11. A [ 1 ] n = A n + 1 A[1]^{n}=A^{n+1}
  12. d A [ 1 ] n = - d A n + 1 d^{n}_{A[1]}=-d^{n+1}_{A}
  13. C ( f ) C(f)
  14. A [ 1 ] B A[1]\oplus B
  15. d C ( f ) n ( a n + 1 , b n ) = ( d A [ 1 ] n 0 f [ 1 ] n d B n ) ( a n + 1 b n ) = ( - d A n + 1 0 f n + 1 d B n ) ( a n + 1 b n ) = ( - d A n + 1 ( a n + 1 ) f n + 1 ( a n + 1 ) + d B n ( b n ) ) = ( - d A n + 1 ( a n + 1 ) , f n + 1 ( a n + 1 ) + d B n ( b n ) ) . \begin{array}[]{ccl}d^{n}_{C(f)}(a^{n+1},b^{n})&=&\begin{pmatrix}d^{n}_{A[1]}&% 0\\ f[1]^{n}&d^{n}_{B}\end{pmatrix}\begin{pmatrix}a^{n+1}\\ b^{n}\end{pmatrix}\\ &=&\begin{pmatrix}-d^{n+1}_{A}&0\\ f^{n+1}&d^{n}_{B}\end{pmatrix}\begin{pmatrix}a^{n+1}\\ b^{n}\end{pmatrix}\\ &=&\begin{pmatrix}-d^{n+1}_{A}(a^{n+1})\\ f^{n+1}(a^{n+1})+d^{n}_{B}(b^{n})\end{pmatrix}\\ &=&\left(-d^{n+1}_{A}(a^{n+1}),f^{n+1}(a^{n+1})+d^{n}_{B}(b^{n})\right).\end{array}
  16. A 𝑓 B C ( f ) A\xrightarrow{f}B\to C(f)\to
  17. B C ( f ) , C ( f ) A [ 1 ] B\to C(f),C(f)\to A[1]
  18. H i - 1 ( C ( f ) ) H i ( A ) f * H i ( B ) H i ( C ( f ) ) \dots\to H^{i-1}(C(f))\to H^{i}(A)\xrightarrow{f^{*}}H^{i}(B)\to H^{i}(C(f))\to\cdots
  19. C ( f ) C(f)
  20. f * f^{*}
  21. A , B A,B
  22. A = 0 A 0 0 , A=\dots\to 0\to A^{0}\to 0\to\cdots,
  23. B = 0 B 0 0 , B=\dots\to 0\to B^{0}\to 0\to\cdots,
  24. f : A B f\colon A\to B
  25. f 0 : A 0 B 0 f^{0}\colon A^{0}\to B^{0}
  26. C ( f ) = 0 A 0 [ - 1 ] f 0 B 0 [ 0 ] 0 . C(f)=\dots\to 0\to\underset{[-1]}{A^{0}}\xrightarrow{f^{0}}\underset{[0]}{B^{0% }}\to 0\to\cdots.
  27. H - 1 ( C ( f ) ) = ker ( f 0 ) , H^{-1}(C(f))=\operatorname{ker}(f^{0}),
  28. H 0 ( C ( f ) ) = coker ( f 0 ) , H^{0}(C(f))=\operatorname{coker}(f^{0}),
  29. H i ( C ( f ) ) = 0 for i - 1 , 0. H^{i}(C(f))=0\,\text{ for }i\neq-1,0.
  30. ϕ : X Y \phi:X\rightarrow Y
  31. c o n e ( ϕ ) cone(\phi)

Marangoni_number.html

  1. Ma = - d γ d T L Δ T η α \mathrm{Ma}=-{\frac{d\gamma}{dT}}\frac{L\Delta T}{\eta\alpha}
  2. α \alpha

Markov_brothers'_inequality.html

  1. max - 1 x 1 | P ( k ) ( x ) | n 2 ( n 2 - 1 2 ) ( n 2 - 2 2 ) ( n 2 - ( k - 1 ) 2 ) 1 3 5 ( 2 k - 1 ) max - 1 x 1 | P ( x ) | . \max_{-1\leq x\leq 1}|P^{(k)}(x)|\leq\frac{n^{2}(n^{2}-1^{2})(n^{2}-2^{2})% \cdots(n^{2}-(k-1)^{2})}{1\cdot 3\cdot 5\cdots(2k-1)}\max_{-1\leq x\leq 1}|P(x% )|.

Markov_partition.html

  1. { E 1 , E 2 , E r } \{E_{1},E_{2},\cdots E_{r}\}
  2. x , y E i x,y\in E_{i}
  3. W s ( x ) W u ( y ) E i W_{s}(x)\cap W_{u}(y)\in E_{i}
  4. Int E i Int E j = \operatorname{Int}E_{i}\cap\operatorname{Int}E_{j}=\emptyset
  5. i j i\neq j
  6. x Int E i x\in\operatorname{Int}E_{i}
  7. ϕ ( x ) Int E j \phi(x)\in\operatorname{Int}E_{j}
  8. ϕ [ W u ( x ) E i ] W u ( ϕ x ) E j \phi\left[W_{u}(x)\cap E_{i}\right]\supset W_{u}(\phi x)\cap E_{j}
  9. ϕ [ W s ( x ) E i ] W s ( ϕ x ) E j \phi\left[W_{s}(x)\cap E_{i}\right]\subset W_{s}(\phi x)\cap E_{j}
  10. W u ( x ) W_{u}(x)
  11. W s ( x ) W_{s}(x)
  12. Int E i \operatorname{Int}E_{i}
  13. E i E_{i}
  14. E i E_{i}

Markovian_arrival_process.html

  1. Q = [ D 0 D 1 0 0 0 D 0 D 1 0 0 0 D 0 D 1 ] . Q=\left[\begin{matrix}D_{0}&D_{1}&0&0&\dots\\ 0&D_{0}&D_{1}&0&\dots\\ 0&0&D_{0}&D_{1}&\dots\\ \vdots&\vdots&\ddots&\ddots&\ddots\end{matrix}\right]\;.
  2. 0 [ D 1 ] i , j < 0 [ D 0 ] i , j < i j [ D 0 ] i , i < 0 ( D 0 + D 1 ) s y m b o l 1 = s y m b o l 0 \begin{aligned}\displaystyle 0\leq[D_{1}]_{i,j}&\displaystyle<\infty\\ \displaystyle 0\leq[D_{0}]_{i,j}&\displaystyle<\infty\quad i\neq j\\ \displaystyle\,[D_{0}]_{i,i}&\displaystyle<0\\ \displaystyle(D_{0}+D_{1})symbol{1}&\displaystyle=symbol{0}\end{aligned}
  3. D 1 = diag { λ 1 , , λ m } D 0 = R - D 1 . \begin{aligned}\displaystyle D_{1}&\displaystyle=\operatorname{diag}\{\lambda_% {1},\dots,\lambda_{m}\}\\ \displaystyle D_{0}&\displaystyle=R-D_{1}.\end{aligned}
  4. ( s y m b o l α , S ) (symbol{\alpha},S)
  5. s y m b o l S 0 = - S s y m b o l 1 symbol{S}^{0}=-Ssymbol{1}
  6. Q = [ S s y m b o l S 0 s y m b o l α 0 0 0 S s y m b o l S 0 s y m b o l α 0 0 0 S s y m b o l S 0 s y m b o l α ] Q=\left[\begin{matrix}S&symbol{S}^{0}symbol{\alpha}&0&0&\dots\\ 0&S&symbol{S}^{0}symbol{\alpha}&0&\dots\\ 0&0&S&symbol{S}^{0}symbol{\alpha}&\dots\\ \vdots&\vdots&\ddots&\ddots&\ddots\\ \end{matrix}\right]
  7. Q = [ D 0 D 1 D 2 D 3 0 D 0 D 1 D 2 0 0 D 0 D 1 ] . Q=\left[\begin{matrix}D_{0}&D_{1}&D_{2}&D_{3}&\dots\\ 0&D_{0}&D_{1}&D_{2}&\dots\\ 0&0&D_{0}&D_{1}&\dots\\ \vdots&\vdots&\ddots&\ddots&\ddots\end{matrix}\right]\;.
  8. k k
  9. D k D_{k}
  10. D k D_{k}
  11. λ i , j \lambda_{i,j}
  12. 0 [ D k ] i , j < 1 k 0\leq[D_{k}]_{i,j}<\infty\;\;\;\;1\leq k
  13. 0 [ D 0 ] i , j < i j 0\leq[D_{0}]_{i,j}<\infty\;\;\;\;i\neq j
  14. [ D 0 ] i , i < 0 [D_{0}]_{i,i}<0\;
  15. k = 0 D k s y m b o l 1 = s y m b o l 0 \sum^{\infty}_{k=0}D_{k}symbol{1}=symbol{0}

Marsaglia_polar_method.html

  1. x - 2 ln ( s ) s , y - 2 ln ( s ) s . x\sqrt{\frac{-2\ln(s)}{s}}\,,\ \ y\sqrt{\frac{-2\ln(s)}{s}}.
  2. Pr ( ρ < a ) = 0 a r e - r 2 / 2 d r \Pr(\rho<a)=\int_{0}^{a}re^{-r^{2}/2}\,dr
  3. ( ρ cos ( 2 π u ) , ρ sin ( 2 π u ) ) \left(\rho\cos(2\pi u),\rho\sin(2\pi u)\right)
  4. I = - e - x 2 / 2 d x I=\int_{-\infty}^{\infty}e^{-x^{2}/2}\,dx
  5. I 2 = - - e - ( x 2 + y 2 ) / 2 d x d y = 0 2 π 0 r e - r 2 / 2 d r d θ . I^{2}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-(x^{2}+y^{2})/2}\,dx\,% dy=\int_{0}^{2\pi}\int_{0}^{\infty}re^{-r^{2}/2}\,dr\,d\theta.
  6. r e - r 2 / 2 . re^{-r^{2}/2}.\,
  7. x = - 2 ln ( u 1 ) cos ( 2 π u 2 ) , y = - 2 ln ( u 1 ) sin ( 2 π u 2 ) x=\sqrt{-2\ln(u_{1})}\cos(2\pi u_{2}),\quad y=\sqrt{-2\ln(u_{1})}\sin(2\pi u_{% 2})
  8. - 2 ln ( u 1 ) ; \sqrt{-2\ln(u_{1})};
  9. expi ( z ) = e i z = cos ( z ) + i sin ( z ) , \,\text{expi}(z)=e^{iz}=\cos(z)+i\sin(z),\,
  10. ( x s , y s ) , \left(\frac{x}{\sqrt{s}},\frac{y}{\sqrt{s}}\right),\,
  11. π / 4 79 % \pi/4\approx 79\%
  12. - 2 ln ( s ) , \sqrt{-2\ln(s)},\,

Martin_measure.html

  1. D D
  2. [ X ] D [X]\in D
  3. [ X ] [X]
  4. [ Y ] [Y]
  5. X T Y X\leq_{T}Y
  6. X X
  7. A A
  8. A A
  9. A A
  10. A A
  11. A A
  12. ω 1 \omega_{1}
  13. ω 1 \omega_{1}

Mason's_invariant.html

  1. [ Z 11 Z 12 Z 21 Z 22 ] = [ Z 11 + j x 11 Z 12 + j x 12 Z 21 + j x 21 Z 22 + j x 22 ] \begin{bmatrix}Z^{\prime}_{11}&Z^{\prime}_{12}\\ Z^{\prime}_{21}&Z^{\prime}_{22}\end{bmatrix}=\begin{bmatrix}Z_{11}+jx_{11}&Z_{% 12}+jx_{12}\\ Z_{21}+jx_{21}&Z_{22}+jx_{22}\end{bmatrix}
  2. [ Z 11 Z 12 Z 21 Z 22 ] = [ n 11 n 12 n 21 n 22 ] [ Z 11 Z 12 Z 21 Z 22 ] [ n 11 n 12 n 21 n 22 ] \begin{bmatrix}Z^{\prime}_{11}&Z^{\prime}_{12}\\ Z^{\prime}_{21}&Z^{\prime}_{22}\end{bmatrix}=\begin{bmatrix}n_{11}&n_{12}\\ n_{21}&n_{22}\end{bmatrix}\begin{bmatrix}Z_{11}&Z_{12}\\ Z_{21}&Z_{22}\end{bmatrix}\begin{bmatrix}n_{11}&n_{12}\\ n_{21}&n_{22}\end{bmatrix}
  3. [ Z 11 Z 12 Z 21 Z 22 ] = [ Z 11 Z 12 Z 21 Z 22 ] - 1 \begin{bmatrix}Z^{\prime}_{11}&Z^{\prime}_{12}\\ Z^{\prime}_{21}&Z^{\prime}_{22}\end{bmatrix}=\begin{bmatrix}Z_{11}&Z_{12}\\ Z_{21}&Z_{22}\end{bmatrix}^{-1}
  4. [ Z - Z t ] \left[Z-Z_{t}\right]
  5. [ Z + Z * ] \left[Z+Z^{*}\right]
  6. [ Z - Z t ] [ Z + Z * ] \left[Z-Z_{t}\right]\left[Z+Z^{*}\right]
  7. det [ Z - Z t ] det [ Z + Z * ] \dfrac{\det{\left[Z-Z_{t}\right]}}{\det{\left[Z+Z^{*}\right]}}
  8. U \displaystyle U
  9. U ( f max ) = 1 U(f_{\max})=1

Mass_diffusivity.html

  1. D = D 0 e - E A / ( k T ) D=D_{0}\,{\mathrm{e}}^{-E_{\mathrm{A}}/(kT)}
  2. D T 1 D T 2 = T 1 T 2 μ T 2 μ T 1 \frac{D_{T_{1}}}{D_{T_{2}}}=\frac{T_{1}}{T_{2}}\frac{\mu_{T_{2}}}{\mu_{T_{1}}}
  3. D = 1.858 10 - 3 T 3 / 2 1 / M 1 + 1 / M 2 p σ 12 2 Ω D=\frac{1.858\cdot 10^{-3}T^{3/2}\sqrt{1/M_{1}+1/M_{2}}}{p\sigma_{12}^{2}\Omega}
  4. σ 12 = 1 2 ( σ 1 + σ 2 ) \sigma_{12}=\frac{1}{2}(\sigma_{1}+\sigma_{2})
  5. D P 1 D P 2 = ρ P 2 ρ P 1 \frac{D_{P1}}{D_{P2}}=\frac{\rho_{P2}}{\rho_{P1}}
  6. D e = D ε t δ τ D_{e}=\frac{D\varepsilon_{t}\delta}{\tau}

Mass_flux.html

  1. j m = lim A 0 I m A j_{m}=\lim\limits_{A\rightarrow 0}\frac{I_{m}}{A}
  2. I m = lim Δ t 0 Δ m Δ t = d m d t I_{m}=\lim\limits_{\Delta t\rightarrow 0}\frac{\Delta m}{\Delta t}=\frac{dm}{dt}
  3. m = t 1 t 2 S j m n ^ d A d t m=\int_{t_{1}}^{t_{2}}\iint_{S}{j}_{m}\cdot{\hat{n}}{\rm d}A{\rm d}t
  4. n ^ {\hat{n}}
  5. A = A n ^ {A}=A{\hat{n}}
  6. n ^ {\hat{n}}
  7. j m n ^ = j m cos θ {j}_{m}\cdot{\hat{n}}=j_{m}\cos\theta
  8. A = π r 2 A=\pi r^{2}
  9. Δ m = ρ Δ V \displaystyle\Delta m=\rho\Delta V
  10. j m = Δ m A Δ t = ρ V π r 2 t j_{m}=\frac{\Delta m}{A\Delta t}=\frac{\rho V}{\pi r^{2}t}
  11. j m = 1000 × ( 1.5 × 10 - 3 ) π × ( 2 × 10 - 2 ) 2 × 2 = 3 16 π × 10 4 j_{m}=\frac{1000\times(1.5\times 10^{-3})}{\pi\times(2\times 10^{-2})^{2}% \times 2}=\frac{3}{16\pi}\times 10^{4}
  12. j m = ρ 𝐮 {j}_{\rm m}=\rho\mathbf{u}
  13. j m , i = ρ i 𝐮 i {j}_{{\rm m},\,i}=\rho_{i}\mathbf{u}_{i}
  14. j m , i = ρ ( 𝐮 i - 𝐮 ) {j}_{{\rm m},\,i}=\rho\left(\mathbf{u}_{i}-\langle\mathbf{u}\rangle\right)
  15. 𝐮 \langle\mathbf{u}\rangle
  16. 𝐮 = 1 ρ i ρ i 𝐮 i = 1 ρ i 𝐣 m , i \langle\mathbf{u}\rangle=\frac{1}{\rho}\sum_{i}\rho_{i}\mathbf{u}_{i}=\frac{1}% {\rho}\sum_{i}\mathbf{j}_{{\rm m},\,i}
  17. j n = n 𝐮 {j}_{\rm n}=n\mathbf{u}
  18. j n , i = n i 𝐮 i {j}_{{\rm n},\,i}=n_{i}\mathbf{u}_{i}
  19. j n , i = n ( 𝐮 i - 𝐮 ) {j}_{{\rm n},\,i}=n\left(\mathbf{u}_{i}-\langle\mathbf{u}\rangle\right)
  20. 𝐮 \langle\mathbf{u}\rangle
  21. 𝐮 = 1 n i n i 𝐮 i = 1 n i 𝐣 n , i \langle\mathbf{u}\rangle=\frac{1}{n}\sum_{i}n_{i}\mathbf{u}_{i}=\frac{1}{n}% \sum_{i}\mathbf{j}_{{\rm n},\,i}
  22. j m + ρ t = 0 \nabla\cdot{j}_{\rm m}+\frac{\partial\rho}{\partial t}=0
  23. j n = - D c \nabla\cdot{j}_{\rm n}=-\nabla\cdot D\nabla c

Material_nonimplication.html

  1. ~{}\nrightarrow

Mathematical_descriptions_of_the_electromagnetic_field.html

  1. 𝐄 = ρ ε 0 \nabla\cdot\mathbf{E}=\frac{\rho}{\varepsilon_{0}}
  2. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  3. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}
  4. × 𝐁 = μ 0 𝐉 + μ 0 ε 0 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\frac{\partial% \mathbf{E}}{\partial t}
  5. φ \varphi
  6. 𝐄 = - φ - 𝐀 t \mathbf{E}=-\mathbf{\nabla}\varphi-\frac{\partial\mathbf{A}}{\partial t}
  7. 𝐁 = × 𝐀 \mathbf{B}=\mathbf{\nabla}\times\mathbf{A}
  8. 𝐀 = 𝐀 + λ \mathbf{A}^{\prime}=\mathbf{A}+\mathbf{\nabla}\lambda
  9. φ = φ - λ t \varphi^{\prime}=\varphi-\frac{\partial\lambda}{\partial t}
  10. 𝐀 = 0 \mathbf{\nabla}\cdot\mathbf{A}^{\prime}=0
  11. 2 λ = - 𝐀 \nabla^{2}\lambda=-\mathbf{\nabla}\cdot\mathbf{A}
  12. 2 φ = - ρ ε 0 \nabla^{2}\varphi^{\prime}=-\frac{\rho}{\varepsilon_{0}}
  13. 2 𝐀 - μ 0 ε 0 2 𝐀 t 2 = - μ 0 𝐉 + μ 0 ε 0 ( φ t ) \nabla^{2}\mathbf{A}^{\prime}-\mu_{0}\varepsilon_{0}\frac{\partial^{2}\mathbf{% A}^{\prime}}{\partial t^{2}}=-\mu_{0}\mathbf{J}+\mu_{0}\varepsilon_{0}\nabla% \left(\frac{\partial\varphi^{\prime}}{\partial t}\right)
  14. 𝐀 = - μ 0 ε 0 φ t \mathbf{\nabla}\cdot\mathbf{A}^{\prime}=-\mu_{0}\varepsilon_{0}\frac{\partial% \varphi^{\prime}}{\partial t}
  15. 2 λ - μ 0 ε 0 2 λ t 2 = - 𝐀 - μ 0 ε 0 φ t \nabla^{2}\lambda-\mu_{0}\varepsilon_{0}\frac{\partial^{2}\lambda}{\partial t^% {2}}=-\mathbf{\nabla}\cdot\mathbf{A}-\mu_{0}\varepsilon_{0}\frac{\partial% \varphi}{\partial t}
  16. 2 𝐀 - μ 0 ε 0 2 𝐀 t 2 = 2 𝐀 = - μ 0 𝐉 \nabla^{2}\mathbf{A}^{\prime}-\mu_{0}\varepsilon_{0}\frac{\partial^{2}\mathbf{% A}^{\prime}}{\partial t^{2}}=\Box^{2}\mathbf{A}^{\prime}=-\mu_{0}\mathbf{J}
  17. 2 φ - μ 0 ε 0 2 φ t 2 = 2 φ = - ρ ε 0 \nabla^{2}\varphi^{\prime}-\mu_{0}\varepsilon_{0}\frac{\partial^{2}\varphi^{% \prime}}{\partial t^{2}}=\Box^{2}\varphi^{\prime}=-\frac{\rho}{\varepsilon_{0}}
  18. 2 \Box^{2}
  19. \Box
  20. 2 𝐀 - 1 c 2 2 𝐀 t 2 = - μ 0 𝐉 \nabla^{2}\mathbf{A}-\frac{1}{c^{2}}\frac{\partial^{2}\mathbf{A}}{\partial t^{% 2}}=-\mu_{0}\mathbf{J}
  21. 2 φ - 1 c 2 2 φ t 2 = - ρ ε 0 \nabla^{2}\varphi-\frac{1}{c^{2}}\frac{\partial^{2}\varphi}{\partial t^{2}}=-% \frac{\rho}{\varepsilon_{0}}
  22. 𝐉 = - e ψ s y m b o l α ψ ρ = - e ψ ψ , \mathbf{J}=-e\psi^{\dagger}symbol{\alpha}\psi\,\quad\rho=-e\psi^{\dagger}\psi\,,
  23. F = E + I c B = E k σ k + I c B k σ k {F}={E}+Ic{B}=E^{k}\sigma_{k}+IcB^{k}\sigma_{k}
  24. c ρ - J = c ρ - J k σ k c\rho-{J}=c\rho-J^{k}\sigma_{k}
  25. C 3 , 0 ( ) C\ell_{3,0}(\mathbb{R})
  26. { σ k } \{\sigma_{k}\}
  27. I = σ 1 σ 2 σ 3 I=\sigma_{1}\sigma_{2}\sigma_{3}
  28. s y m b o l = σ k k symbol{\nabla}=\sigma^{k}\partial_{k}
  29. s y m b o l F = s y m b o l F + s y m b o l F = s y m b o l F + I s y m b o l × F symbol{\nabla}{F}=symbol{\nabla}\cdot{F}+symbol{\nabla}\wedge{F}=symbol{\nabla% }\cdot{F}+Isymbol{\nabla}\times{F}
  30. ( s y m b o l 𝐄 - ρ ϵ 0 ) - c ( s y m b o l × 𝐁 - μ 0 ϵ 0 𝐄 t - μ 0 𝐉 ) + I ( s y m b o l × 𝐄 + 𝐁 t ) + I c ( s y m b o l 𝐁 ) = 0 \left(symbol{\nabla}\cdot\mathbf{E}-\frac{\rho}{\epsilon_{0}}\right)-c\left(% symbol{\nabla}\times\mathbf{B}-\mu_{0}\epsilon_{0}\frac{\partial{\mathbf{E}}}{% \partial{t}}-\mu_{0}\mathbf{J}\right)+I\left(symbol{\nabla}\times\mathbf{E}+% \frac{\partial{\mathbf{B}}}{\partial{t}}\right)+Ic\left(symbol{\nabla}\cdot% \mathbf{B}\right)=0
  31. C 1 , 3 ( ) C\ell_{1,3}(\mathbb{R})
  32. σ k = γ k γ 0 \sigma_{k}=\gamma_{k}\gamma_{0}
  33. I = γ 0 γ 1 γ 2 γ 3 I=\gamma_{0}\gamma_{1}\gamma_{2}\gamma_{3}
  34. = γ μ μ . \nabla=\gamma^{\mu}\partial_{\mu}.
  35. F = E + I c B = E k γ k γ 0 - c ( B 1 γ 2 γ 3 + B 2 γ 3 γ 1 + B 3 γ 1 γ 2 ) , F={E}+Ic{B}=E^{k}\gamma_{k}\gamma_{0}-c(B^{1}\gamma_{2}\gamma_{3}+B^{2}\gamma_% {3}\gamma_{1}+B^{3}\gamma_{1}\gamma_{2}),
  36. J = J μ γ μ = c ρ γ 0 + J k γ k = γ 0 ( c ρ - J k σ k ) . J=J^{\mu}\gamma_{\mu}=c\rho\gamma_{0}+J^{k}\gamma_{k}=\gamma_{0}(c\rho-J^{k}% \sigma_{k}).
  37. γ 0 = γ 0 γ 0 0 + γ 0 γ k k = 0 + σ k k = 1 c t + s y m b o l , \gamma_{0}\nabla=\gamma_{0}\gamma^{0}\partial_{0}+\gamma_{0}\gamma^{k}\partial% _{k}=\partial_{0}+\sigma^{k}\partial_{k}=\frac{1}{c}\dfrac{\partial}{\partial t% }+symbol{\nabla},
  38. F μ ν F_{\mu\nu}
  39. F 1 2 F μ ν d x μ d x ν = B x d y d z + B y d z d x + B z d x d y + E x d x d t + E y d y d t + E z d z d t \begin{aligned}\displaystyle{F}\equiv&\displaystyle\frac{1}{2}F_{\mu\nu}dx^{% \mu}\wedge dx^{\nu}\\ \displaystyle=&\displaystyle B_{x}dy\wedge dz+B_{y}dz\wedge dx+B_{z}dx\wedge dy% +E_{x}dx\wedge dt+E_{y}dy\wedge dt+E_{z}dz\wedge dt\end{aligned}
  40. F = d A = ( μ A ν ) d x μ d x ν {F}=\mathrm{d}{A}=(\partial_{\mu}A_{\nu})dx^{\mu}\wedge dx^{\nu}
  41. d x d y = - d z d t , d x d t = d y d z , {\star dx}\wedge dy=-dz\wedge dt,\quad{\star dx}\wedge dt=dy\wedge dz,
  42. F = - B x d x d t - B y d y d t - B z d z d t + E x d y d z + E y d z d x + E z d x d y {\star{F}}=-B_{x}dx\wedge dt-B_{y}dy\wedge dt-B_{z}dz\wedge dt+E_{x}dy\wedge dz% +E_{y}dz\wedge dx+E_{z}dx\wedge dy
  43. J = ρ d x d y d z - j x d t d y d z - j y d t d z d x - j z d t d x d y {J}=\rho dx\wedge dy\wedge dz-j_{x}dt\wedge dy\wedge dz-j_{y}dt\wedge dz\wedge dx% -j_{z}dt\wedge dx\wedge dy
  44. J = - ρ d t + j x d x + j y d y + j z d z {\star{J}}=-\rho dt+j_{x}dx+j_{y}dy+j_{z}dz
  45. \star
  46. d J = 0. \mathrm{d}{{J}}=0.
  47. C : Λ 2 F G Λ ( 4 - 2 ) C:\Lambda^{2}\ni{F}\mapsto{G}\in\Lambda^{(4-2)}
  48. d F = 0 \mathrm{d}{F}=0
  49. d G = J \mathrm{d}{G}={J}
  50. F = 1 2 F p q θ p θ q . {F}=\frac{1}{2}F_{pq}{\theta}^{p}\wedge{\theta}^{q}.
  51. G p q = C p q m n F m n G_{pq}=C_{pq}^{mn}F_{mn}
  52. C p q m n = 1 2 g m a g n b ϵ a b p q - g C_{pq}^{mn}=\frac{1}{2}g^{ma}g^{nb}\epsilon_{abpq}\sqrt{-g}
  53. J = - ρ d t + j x d x + j y d y + j z d z {J}=-\rho dt+j_{x}dx+j_{y}dy+j_{z}dz
  54. J = ρ d x d y d z - j x d t d y d z - j y d t d z d x - j z d t d x d y {\star{J}}=\rho dx\wedge dy\wedge dz-j_{x}dt\wedge dy\wedge dz-j_{y}dt\wedge dz% \wedge dx-j_{z}dt\wedge dx\wedge dy
  55. d J = d 2 F = 0 \mathrm{d}\,{\star{J}}=\mathrm{d}^{2}\,{\star{F}}=0
  56. 4 π c j β = α F α β + Γ α μ α F μ β + Γ β μ α F α μ = def α F α β = def F α β ; α {4\pi\over c}j^{\beta}=\partial_{\alpha}F^{\alpha\beta}+{\Gamma^{\alpha}}_{\mu% \alpha}F^{\mu\beta}+{\Gamma^{\beta}}_{\mu\alpha}F^{\alpha\mu}\ \stackrel{% \mathrm{def}}{=}\ \nabla_{\alpha}F^{\alpha\beta}\ \stackrel{\mathrm{def}}{=}\ % {F^{\alpha\beta}}_{;\alpha}\,\!
  57. 0 = γ F α β + β F γ α + α F β γ = γ F α β + β F γ α + α F β γ . 0=\partial_{\gamma}F_{\alpha\beta}+\partial_{\beta}F_{\gamma\alpha}+\partial_{% \alpha}F_{\beta\gamma}=\nabla_{\gamma}F_{\alpha\beta}+\nabla_{\beta}F_{\gamma% \alpha}+\nabla_{\alpha}F_{\beta\gamma}.\,
  58. Γ α μ β {\Gamma^{\alpha}}_{\mu\beta}\!
  59. F = 1 2 F α β d x α d x β . {F}=\frac{1}{2}F_{\alpha\beta}\,\mathrm{d}\,x^{\alpha}\wedge\mathrm{d}\,x^{% \beta}.
  60. J = 4 π c ( 1 6 j α - g ϵ α β γ δ d x β d x γ d x δ . ) {J}={4\pi\over c}\left(\frac{1}{6}j^{\alpha}\sqrt{-g}\,\epsilon_{\alpha\beta% \gamma\delta}\mathrm{d}\,x^{\beta}\wedge\mathrm{d}\,x^{\gamma}\wedge\mathrm{d}% \,x^{\delta}.\right)
  61. d F = 2 ( γ F α β + β F γ α + α F β γ ) d x α d x β d x γ = 0 , \mathrm{d}{F}=2(\partial_{\gamma}F_{\alpha\beta}+\partial_{\beta}F_{\gamma% \alpha}+\partial_{\alpha}F_{\beta\gamma})\mathrm{d}\,x^{\alpha}\wedge\mathrm{d% }\,x^{\beta}\wedge\mathrm{d}\,x^{\gamma}=0,
  62. d F = 1 6 F α β ; α - g ϵ β γ δ η d x γ d x δ d x η = J , \mathrm{d}\,{\star{F}}=\frac{1}{6}{F^{\alpha\beta}}_{;\alpha}\sqrt{-g}\,% \epsilon_{\beta\gamma\delta\eta}\mathrm{d}\,x^{\gamma}\wedge\mathrm{d}\,x^{% \delta}\wedge\mathrm{d}\,x^{\eta}={J},
  63. d J = 4 π c j α ; α - g ϵ α β γ δ d x α d x β d x γ d x δ = 0. \mathrm{d}{J}={4\pi\over c}{j^{\alpha}}_{;\alpha}\sqrt{-g}\,\epsilon_{\alpha% \beta\gamma\delta}\mathrm{d}\,x^{\alpha}\wedge\mathrm{d}\,x^{\beta}\wedge% \mathrm{d}\,x^{\gamma}\wedge\mathrm{d}\,x^{\delta}=0.
  64. φ = φ - λ t , 𝐀 = 𝐀 + λ \varphi^{\prime}=\varphi-\frac{\partial\lambda}{\partial t},\quad\mathbf{A}^{% \prime}=\mathbf{A}+\mathbf{\nabla}\lambda
  65. 𝐀 = 0 \mathbf{\nabla}\cdot\mathbf{A}=0
  66. 𝐀 + 1 c 2 φ t = 0 . \mathbf{\nabla}\cdot\mathbf{A}+\frac{1}{c^{2}}\frac{\partial\varphi}{\partial t% }=0\,.

Mathematical_markup_language.html

  1. ¬ \neg
  2. a 2 a^{2}
  3. k = 1 N k 2 \sum_{k=1}^{N}k^{2}
  4. ¬ ( a > 2 ) a 2 \neg(a>2)\Rightarrow a\leq 2

Mathematics_of_cyclic_redundancy_checks.html

  1. x n x^{n}
  2. n n
  3. M ( x ) x n = Q ( x ) G ( x ) + R ( x ) . M(x)\cdot x^{n}=Q(x)\cdot G(x)+R(x).
  4. M ( x ) M(x)
  5. G ( x ) G(x)
  6. n n
  7. M ( x ) x n M(x)\cdot x^{n}
  8. n n
  9. R ( x ) R(x)
  10. n n
  11. Q ( x ) Q(x)
  12. R ( x ) = M ( x ) x n mod G ( x ) . R(x)=M(x)\cdot x^{n}\,\bmod\,G(x).
  13. n n
  14. M ( x ) x n - R ( x ) M(x)\cdot x^{n}-R(x)
  15. G ( x ) G(x)
  16. n n
  17. ( x 3 + x ) + ( x + 1 ) = x 3 + 2 x + 1 x 3 + 1 ( mod 2 ) . (x^{3}+x)+(x+1)=x^{3}+2x+1\equiv x^{3}+1\;\;(\mathop{{\rm mod}}2).
  18. 2 x 2x
  19. 2 x = x + x = x × ( 1 + 1 ) x × 0 = 0 ( mod 2 ) . 2x=x+x=x\times(1+1)\equiv x\times 0=0\;\;(\mathop{{\rm mod}}2).
  20. ( x 2 + x ) ( x + 1 ) = x 3 + 2 x 2 + x x 3 + x ( mod 2 ) . (x^{2}+x)(x+1)=x^{3}+2x^{2}+x\equiv x^{3}+x\;\;(\mathop{{\rm mod}}2).
  21. x 3 + x 2 + x x^{3}+x^{2}+x
  22. x + 1 x+1
  23. x 3 + x 2 + x x + 1 = ( x 2 + 1 ) - 1 x + 1 . \frac{x^{3}+x^{2}+x}{x+1}=(x^{2}+1)-\frac{1}{x+1}.
  24. ( x 3 + x 2 + x ) = ( x 2 + 1 ) ( x + 1 ) - 1 ( x 2 + 1 ) ( x + 1 ) + 1 ( mod 2 ) . (x^{3}+x^{2}+x)=(x^{2}+1)(x+1)-1\equiv(x^{2}+1)(x+1)+1\;\;(\mathop{{\rm mod}}2).
  25. x 2 + x + 1 x^{2}+x+1
  26. x + 1 x+1
  27. 1 1
  28. x 0 x^{0}
  29. x 1 x^{1}
  30. x 3 + x 2 + x x^{3}+x^{2}+x
  31. M ( x ) M(x)
  32. R ( x ) R(x)
  33. M ( x ) x n - R ( x ) = Q ( x ) G ( x ) M(x)\cdot x^{n}-R(x)=Q(x)\cdot G(x)
  34. G ( x ) G(x)
  35. n n
  36. M ( x ) x n + i = m m + n - 1 x i = Q ( x ) G ( x ) + R ( x ) M(x)\cdot x^{n}+\sum_{i=m}^{m+n-1}x^{i}=Q(x)\cdot G(x)+R(x)
  37. m > deg ( M ( x ) ) m>\deg(M(x))
  38. R ( x ) R(x)
  39. ( i = m m + n - 1 x i ) mod G ( x ) \left(\sum_{i=m}^{m+n-1}x^{i}\right)\,\bmod\,G(x)
  40. M ( x ) M(x)
  41. M ( x ) M(x)
  42. G ( x ) G(x)
  43. x x
  44. n n
  45. C ( x ) C(x)
  46. C ( x ) = ( i = n 2 n - 1 x i ) mod G ( x ) C(x)=\left(\sum_{i=n}^{2n-1}x^{i}\right)\,\bmod\,G(x)
  47. n n
  48. M ( x ) = i = 0 n - 1 x i M(x)=\sum_{i=0}^{n-1}x^{i}
  49. n n
  50. x n x^{n}
  51. n n
  52. x n - 1 x^{n-1}
  53. x 0 x^{0}
  54. n n
  55. x 0 x^{0}
  56. x n - 1 x^{n-1}
  57. x n - 1 x^{n-1}
  58. x n x^{n}
  59. x 0 x^{0}
  60. x i x^{i}
  61. x 0 x^{0}
  62. x 0 x^{0}
  63. x i x^{i}
  64. n n
  65. G ( x ) G(x)
  66. x n G ( x - 1 ) x^{n}G(x^{-1})
  67. n n
  68. n n
  69. E ( x ) E(x)
  70. x k x^{k}
  71. x k x^{k}
  72. x i x^{i}
  73. i k i\leq k
  74. E ( x ) = x i + x k = x k ( x i - k + 1 ) , i > k E(x)=x^{i}+x^{k}=x^{k}\cdot(x^{i-k}+1),\;i>k
  75. x k x^{k}
  76. x i - k + 1 x^{i-k}+1
  77. i - k {i-k}
  78. x i - k + 1 x^{i-k}+1
  79. n n
  80. 2 n - 1 2^{n}-1
  81. x + 1 x+1
  82. x + 1 x+1
  83. n - i n-i
  84. x + 1 x+1
  85. n n
  86. n n
  87. x 0 x^{0}
  88. x 0 x^{0}
  89. x 0 x^{0}
  90. x x
  91. K ( x ) K(x)
  92. K ( x ) = x K ( x ) K(x)=x\cdot K^{\prime}(x)
  93. M ( x ) x n - 1 = Q ( x ) K ( x ) + R ( x ) M(x)\cdot x^{n-1}=Q(x)\cdot K^{\prime}(x)+R(x)
  94. M ( x ) x n = Q ( x ) x K ( x ) + x R ( x ) M(x)\cdot x^{n}=Q(x)\cdot x\cdot K^{\prime}(x)+x\cdot R(x)
  95. M ( x ) x n = Q ( x ) K ( x ) + x R ( x ) M(x)\cdot x^{n}=Q(x)\cdot K(x)+x\cdot R(x)
  96. K ( x ) K(x)
  97. K ( x ) K^{\prime}(x)
  98. n - 1 n-1
  99. x + 1 x+1
  100. n n
  101. 1 + + X n 1+\cdots+X^{n}
  102. n + 1 n+1
  103. n n
  104. n + 1 n+1
  105. 2 n - 1 - 1 2^{n-1}-1
  106. n = 16 n=16
  107. 2 n - 1 2^{n}-1

Mathieu_wavelet.html

  1. a , q a\in\mathbb{R},q\in\mathbb{C}
  2. d 2 y d w 2 + ( a - 2 q cos 2 w ) y = 0. \frac{d^{2}y}{dw^{2}}+(a-2q\cos 2w)y=0.
  3. ν \nu
  4. π \pi
  5. 2 π 2\pi
  6. π \pi
  7. 2 π 2\pi
  8. q 0 q\neq 0
  9. a = a r ( q ) a=a_{r}(q)
  10. a = b r ( q ) a=b_{r}(q)
  11. c e r ( ω , q ) = m A r , m cos m ω for a = a r ( q ) ce_{r}(\omega,q)=\sum_{m}A_{r,m}\cos{m\omega}\,\text{ for }a=a_{r}(q)
  12. s e r ( ω , q ) = m A r , m sin m ω for a = b r ( q ) se_{r}(\omega,q)=\sum_{m}A_{r,m}\sin{m\omega}\,\text{ for }a=b_{r}(q)
  13. π \pi
  14. 2 π 2\pi
  15. A r , m A_{r,m}
  16. A m A_{m}
  17. q 0 q\to 0
  18. r 0 r\neq 0
  19. lim q 0 c e r ( ω , q ) = cos r ω \lim_{q\to 0}ce_{r}(\omega,q)=\cos{r\omega}
  20. lim q 0 s e r ( ω , q ) = sin r ω \lim_{q\to 0}se_{r}(\omega,q)=\sin{r\omega}
  21. ν \nu
  22. ψ ( t ) \psi(t)
  23. ϕ ( t ) \phi(t)
  24. Ψ ( ω ) \Psi(\omega)
  25. Φ ( ω ) \Phi(\omega)
  26. ϕ ( t ) = 2 n Z h n ϕ ( 2 t - n ) \phi(t)=\sqrt{2}\sum_{n\in Z}h_{n}\phi(2t-n)
  27. H ( ω ) = 1 2 k Z h k e j ω k H(\omega)=\frac{1}{\sqrt{2}}\sum_{k\in Z}h_{k}e^{j\omega k}
  28. G ( ω ) = 1 2 k Z g k e j ω k G(\omega)=\frac{1}{\sqrt{2}}\sum_{k\in Z}g_{k}e^{j\omega k}
  29. G ν ( ω ) = e j ( ν - 2 ) [ ω - π 2 ] . c e ν ( ω - π 2 , q ) c e ν ( 0 , q ) . G_{\nu}(\omega)=e^{j(\nu-2)[\frac{\omega-\pi}{2}]}.\frac{ce_{\nu}(\frac{\omega% -\pi}{2},q)}{{ce_{\nu}(0,q)}}.
  30. H ν ( ω ) = - e j ν [ ω 2 ] . c e ν ( ω 2 , q ) c e ν ( 0 , q ) . H_{\nu}(\omega)=-e^{j\nu[\frac{\omega}{2}]}.\frac{ce_{\nu}(\frac{\omega}{2},q)% }{{ce_{\nu}(0,q)}}.
  31. ν \nu
  32. G ν ( 0 ) = 0 G_{\nu}(0)=0
  33. G ν ( π ) = 1 G_{\nu}(\pi)=1
  34. ν \nu
  35. ν \nu
  36. 0 | ω | π 0\leq|\omega|\leq\pi
  37. H ν ( ω ) H_{\nu}(\omega)
  38. G ν ( ω ) G_{\nu}(\omega)
  39. ν = 1 \nu=1
  40. ν = 1 \nu=1
  41. ν = 5 \nu=5
  42. ν = 5 \nu=5
  43. { A 2 l + 1 } l Z \{A_{2l+1}\}_{l\in Z}
  44. h l 2 = - A | 2 l + 1 | / 2 c e ν ( 0 , q ) \frac{h_{l}}{\sqrt{2}}=-\frac{A_{|2l+1|}/2}{ce_{\nu}(0,q)}
  45. g l 2 = ( - 1 ) l A | 2 l - 3 | / 2 c e ν ( 0 , q ) \frac{g_{l}}{\sqrt{2}}=(-1)^{l}\frac{A_{|2l-3|}/2}{ce_{\nu}(0,q)}
  46. ( a - 1 - q ) A 1 - q A 3 = 0 (a-1-q)A_{1}-qA_{3}=0
  47. ( a - m 2 ) A m - q ( A m - 2 + A m + 2 = 0 (a-m^{2})A_{m}-q(A_{m-2}+A_{m+2}=0
  48. m 3 m\geq 3
  49. h - l = h | l | - 1 h_{-l}=h_{|l|-1}
  50. l > 0 \forall l>0
  51. k = - k = + h k = - 1 \sum_{k=-\infty}^{k=+\infty}{h_{k}=-1}
  52. k = - k = + ( - 1 ) k h k = 0 \sum_{k=-\infty}^{k=+\infty}{(-1)^{k}h_{k}=0}

Matrix_coefficient.html

  1. ρ ρ
  2. G G
  3. V V
  4. f v , η ( g ) = η ( ρ ( g ) v ) f_{v,\eta}(g)=\eta(\rho(g)v)
  5. v v
  6. V V
  7. η η
  8. V V
  9. g g
  10. G G
  11. G G
  12. V V
  13. f v , w ( g ) = ρ ( g ) v , w f_{v,w}(g)=\langle\rho(g)v,w\rangle
  14. v v
  15. w w
  16. V V
  17. V V
  18. v v
  19. w w

Matrix_determinant_lemma.html

  1. det ( 𝐀 + 𝐮𝐯 T ) = ( 1 + 𝐯 T 𝐀 - 1 𝐮 ) det ( 𝐀 ) . \det(\mathbf{A}+\mathbf{uv}^{\mathrm{T}})=(1+\mathbf{v}^{\mathrm{T}}\mathbf{A}% ^{-1}\mathbf{u})\,\det(\mathbf{A})\,.
  2. det ( 𝐀 + 𝐮𝐯 T ) = det ( 𝐀 ) + 𝐯 T adj ( 𝐀 ) 𝐮 , \det(\mathbf{A}+\mathbf{uv}^{\mathrm{T}})=\det(\mathbf{A})+\mathbf{v}^{\mathrm% {T}}\mathrm{adj}(\mathbf{A})\mathbf{u}\,,
  3. ( 𝐈 0 𝐯 T 1 ) ( 𝐈 + 𝐮𝐯 T 𝐮 0 1 ) ( 𝐈 0 - 𝐯 T 1 ) = ( 𝐈 𝐮 0 1 + 𝐯 T 𝐮 ) . \begin{pmatrix}\mathbf{I}&0\\ \mathbf{v}^{\mathrm{T}}&1\end{pmatrix}\begin{pmatrix}\mathbf{I}+\mathbf{uv}^{% \mathrm{T}}&\mathbf{u}\\ 0&1\end{pmatrix}\begin{pmatrix}\mathbf{I}&0\\ -\mathbf{v}^{\mathrm{T}}&1\end{pmatrix}=\begin{pmatrix}\mathbf{I}&\mathbf{u}\\ 0&1+\mathbf{v}^{\mathrm{T}}\mathbf{u}\end{pmatrix}.
  4. det ( 𝐈 + 𝐮𝐯 T ) = ( 1 + 𝐯 T 𝐮 ) . \det(\mathbf{I}+\mathbf{uv}^{\mathrm{T}})=(1+\mathbf{v}^{\mathrm{T}}\mathbf{u}).
  5. det ( 𝐀 + 𝐮𝐯 T ) \displaystyle\det(\mathbf{A}+\mathbf{uv}^{\mathrm{T}})
  6. det ( 𝐀 + 𝐔𝐕 T ) = det ( 𝐈 𝐦 + 𝐕 T 𝐀 - 1 𝐔 ) det ( 𝐀 ) . \operatorname{det}(\mathbf{A}+\mathbf{UV}^{\mathrm{T}})=\operatorname{det}(% \mathbf{I_{m}}+\mathbf{V}^{\mathrm{T}}\mathbf{A}^{-1}\mathbf{U})\operatorname{% det}(\mathbf{A}).
  7. 𝐀 = 𝐈 𝐧 \mathbf{A}=\mathbf{I_{n}}
  8. det ( 𝐀 + 𝐔𝐖𝐕 T ) = det ( 𝐖 - 1 + 𝐕 T 𝐀 - 1 𝐔 ) det ( 𝐖 ) det ( 𝐀 ) . \operatorname{det}(\mathbf{A}+\mathbf{UWV}^{\mathrm{T}})=\det(\mathbf{W}^{-1}+% \mathbf{V}^{\mathrm{T}}\mathbf{A}^{-1}\mathbf{U})\det(\mathbf{W})\det(\mathbf{% A}).

Matter-dominated_era.html

  1. a ( t ) t 2 / 3 a(t)\propto t^{2/3}

Matthews_correlation_coefficient.html

  1. | MCC | = χ 2 n |\,\text{MCC}|=\sqrt{\frac{\chi^{2}}{n}}
  2. MCC = T P × T N - F P × F N ( T P + F P ) ( T P + F N ) ( T N + F P ) ( T N + F N ) \,\text{MCC}=\frac{TP\times TN-FP\times FN}{\sqrt{(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}
  3. N = T N + T P + F N + F P \,\text{N}=TN+TP+FN+FP
  4. S = T P + F N N \,\text{S}=\frac{TP+FN}{N}
  5. P = T P + F P N \,\text{P}=\frac{TP+FP}{N}
  6. MCC = T P / N - S × P P S ( 1 - S ) ( 1 - P ) \,\text{MCC}=\frac{TP/N-S\times P}{\sqrt{PS(1-S)(1-P)}}
  7. 𝑇𝑃𝑅 = 𝑇𝑃 / P = 𝑇𝑃 / ( 𝑇𝑃 + 𝐹𝑁 ) \mathit{TPR}=\mathit{TP}/P=\mathit{TP}/(\mathit{TP}+\mathit{FN})
  8. 𝑆𝑃𝐶 = 𝑇𝑁 / N = 𝑇𝑁 / ( 𝐹𝑃 + 𝑇𝑁 ) \mathit{SPC}=\mathit{TN}/N=\mathit{TN}/(\mathit{FP}+\mathit{TN})
  9. 𝑃𝑃𝑉 = 𝑇𝑃 / ( 𝑇𝑃 + 𝐹𝑃 ) \mathit{PPV}=\mathit{TP}/(\mathit{TP}+\mathit{FP})
  10. 𝑁𝑃𝑉 = 𝑇𝑁 / ( 𝑇𝑁 + 𝐹𝑁 ) \mathit{NPV}=\mathit{TN}/(\mathit{TN}+\mathit{FN})
  11. 𝐹𝑃𝑅 = 𝐹𝑃 / N = 𝐹𝑃 / ( 𝐹𝑃 + 𝑇𝑁 ) \mathit{FPR}=\mathit{FP}/N=\mathit{FP}/(\mathit{FP}+\mathit{TN})
  12. 𝐹𝐷𝑅 = 𝐹𝑃 / ( 𝐹𝑃 + 𝑇𝑃 ) = 1 - 𝑃𝑃𝑉 \mathit{FDR}=\mathit{FP}/(\mathit{FP}+\mathit{TP})=1-\mathit{PPV}
  13. 𝐹𝑁𝑅 = 𝐹𝑁 / ( 𝐹𝑁 + 𝑇𝑃 ) \mathit{FNR}=\mathit{FN}/(\mathit{FN}+\mathit{TP})
  14. 𝐴𝐶𝐶 = ( 𝑇𝑃 + 𝑇𝑁 ) / ( P + N ) \mathit{ACC}=(\mathit{TP}+\mathit{TN})/(P+N)
  15. F1 = 2 𝑇𝑃 / ( 2 𝑇𝑃 + 𝐹𝑃 + 𝐹𝑁 ) \mathit{F1}=2\mathit{TP}/(2\mathit{TP}+\mathit{FP}+\mathit{FN})
  16. T P × T N - F P × F N ( T P + F P ) ( T P + F N ) ( T N + F P ) ( T N + F N ) \frac{TP\times TN-FP\times FN}{\sqrt{(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}
  17. T P R + S P C - 1 TPR+SPC-1
  18. P P V + N P V - 1 PPV+NPV-1

Max-plus_algebra.html

  1. - -\infty
  2. \oplus
  3. \otimes
  4. a b = max ( a , b ) a\oplus b=\max(a,b)
  5. a b = a + b a\otimes b=a+b
  6. \oplus
  7. \otimes
  8. \oplus
  9. \oplus
  10. [ A B ] i j = [ A ] i j [ B ] i j = max ( [ A ] i j , [ B ] i j ) [A\oplus B]_{ij}=[A]_{ij}\oplus[B]_{ij}=\max([A]_{ij},[B]_{ij})
  11. \otimes
  12. \oplus
  13. \cdot
  14. \otimes
  15. \otimes
  16. [ A B ] i j = k = 1 p ( [ A ] i k [ B ] k j ) = max ( [ A ] i 1 + [ B ] 1 j , , [ A ] i p + [ B ] p j ) [A\otimes B]_{ij}=\bigoplus_{k=1}^{p}([A]_{ik}\otimes[B]_{kj})=\max([A]_{i1}+[% B]_{1j},\dots,[A]_{ip}+[B]_{pj})
  17. - -\infty
  18. = - =-\infty
  19. \oplus
  20. \otimes
  21. e = 0 e=0
  22. \otimes
  23. \oplus
  24. \otimes
  25. ( a b ) c = a ( b c ) (a\oplus b)\oplus c=a\oplus(b\oplus c)
  26. ( a b ) c = a ( b c ) (a\otimes b)\otimes c=a\otimes(b\otimes c)
  27. a b = b a a\oplus b=b\oplus a
  28. a b = b a a\otimes b=b\otimes a
  29. ( a b ) c = a c b c (a\oplus b)\otimes c=a\otimes c\oplus b\otimes c
  30. a ε = a a\oplus\varepsilon=a
  31. a e = a a\otimes e=a
  32. \oplus
  33. a a = a a\oplus a=a

Maximal_ergodic_theorem.html

  1. ( X , , μ ) (X,\mathcal{B},\mu)
  2. T : X X T:X\to X
  3. f L 1 ( μ ) f\in L^{1}(\mu)
  4. f * f^{*}
  5. f * = sup N 1 1 N i = 0 N - 1 f T i . f^{*}=\sup_{N\geq 1}\frac{1}{N}\sum_{i=0}^{N-1}f\circ T^{i}.
  6. f * > λ f d μ λ μ { f * > λ } \int_{f^{*}>\lambda}f\,d\mu\geq\lambda\cdot\mu\{f^{*}>\lambda\}

Maximal_semilattice_quotient.html

  1. x y x\propto y
  2. x y x\asymp y
  3. x y x\propto y
  4. y x y\propto x
  5. \asymp
  6. M / M/{\asymp}
  7. M / M/{\asymp}
  8. g : M / S g\colon M/{\asymp}\to S
  9. M / M/{\asymp}

Maximum_entropy_spectral_estimation.html

  1. x i x_{i}
  2. R x x ( k ) , k = 0 , 1 , P R_{xx}(k),k=0,1,\dots P
  3. R x x ( k ) = α k R_{xx}(k)=\alpha_{k}
  4. α k \alpha_{k}
  5. P P
  6. x i = - k = 1 P a k x i - k + y i x_{i}=-\sum_{k=1}^{P}a_{k}x_{i-k}+y_{i}
  7. y i y_{i}
  8. σ 2 \sigma^{2}
  9. a k a_{k}
  10. x i x_{i}

Maximum_subarray_problem.html

  1. O ( n ) O(n)
  2. O ( n + k ) O(n+k)

Mayo–Lewis_equation.html

  1. M 1 M_{1}\,
  2. M 2 M_{2}\,
  3. M * M^{*}\,
  4. k k\,
  5. M 1 * + M 1 k 11 M 1 M 1 * M_{1}^{*}+M_{1}\xrightarrow{k_{11}}M_{1}M_{1}^{*}\,
  6. M 1 * + M 2 k 12 M 1 M 2 * M_{1}^{*}+M_{2}\xrightarrow{k_{12}}M_{1}M_{2}^{*}\,
  7. M 2 * + M 2 k 22 M 2 M 2 * M_{2}^{*}+M_{2}\xrightarrow{k_{22}}M_{2}M_{2}^{*}\,
  8. M 2 * + M 1 k 21 M 2 M 1 * M_{2}^{*}+M_{1}\xrightarrow{k_{21}}M_{2}M_{1}^{*}\,
  9. r 1 = k 11 k 12 r_{1}=\frac{k_{11}}{k_{12}}\,
  10. r 2 = k 22 k 21 r_{2}=\frac{k_{22}}{k_{21}}\,
  11. d [ M 1 ] d [ M 2 ] = [ M 1 ] ( r 1 [ M 1 ] + [ M 2 ] ) [ M 2 ] ( [ M 1 ] + r 2 [ M 2 ] ) \frac{d\left[M_{1}\right]}{d\left[M_{2}\right]}=\frac{\left[M_{1}\right]\left(% r_{1}\left[M_{1}\right]+\left[M_{2}\right]\right)}{\left[M_{2}\right]\left(% \left[M_{1}\right]+r_{2}\left[M_{2}\right]\right)}
  12. r 1 = r 2 1 r_{1}=r_{2}>>1\,
  13. r 1 = r 2 > 1 r_{1}=r_{2}>1\,
  14. r 1 = r 2 1 r_{1}=r_{2}\approx 1\,
  15. r 1 = r 2 0 r_{1}=r_{2}\approx 0\,
  16. r 1 1 r 2 r_{1}>>1>>r_{2}\,
  17. r 1 r_{1}\,
  18. r 2 , r_{2},
  19. r 1 r_{1}\,
  20. r 2 , r_{2},
  21. F 1 = 1 - F 2 = r 1 f 1 2 + f 1 f 2 r 1 f 1 2 + 2 f 1 f 2 + r 2 f 2 2 F_{1}=1-F_{2}=\frac{r_{1}f_{1}^{2}+f_{1}f_{2}}{r_{1}f_{1}^{2}+2f_{1}f_{2}+r_{2% }f_{2}^{2}}\,
  22. F F\,
  23. F 1 = 1 - F 2 = d M 1 d ( M 1 + M 2 ) F_{1}=1-F_{2}=\frac{dM_{1}}{d(M_{1}+M_{2})}\,
  24. f f\,
  25. f 1 = 1 - f 2 = M 1 ( M 1 + M 2 ) f_{1}=1-f_{2}=\frac{M_{1}}{(M_{1}+M_{2})}\,
  26. f ( 1 - F ) F = r 2 - r 1 ( f 2 F ) \frac{f(1-F)}{F}=r_{2}-r_{1}\left(\frac{f^{2}}{F}\right)\,
  27. f = [ M 1 ] [ M 2 ] f=\frac{[M_{1}]}{[M_{2}]}\,
  28. F = d [ M 1 ] d [ M 2 ] F=\frac{d[M_{1}]}{d[M_{2}]}\,
  29. f ( 1 - F ) F \frac{f(1-F)}{F}\,
  30. f 2 F \frac{f^{2}}{F}\,
  31. r 1 r_{1}\,
  32. r 2 r_{2}\,
  33. - d [ M 1 ] d t = k 11 [ M 1 ] [ M 1 * ] + k 21 [ M 1 ] [ M 2 * ] \frac{-d[M_{1}]}{dt}=k_{11}[M_{1}]\sum[M_{1}^{*}]+k_{21}[M_{1}]\sum[M_{2}^{*}]\,
  34. [ M x * ] \sum[M_{x}^{*}]
  35. - d [ M 2 ] d t = k 12 [ M 2 ] [ M 1 * ] + k 22 [ M 2 ] [ M 2 * ] \frac{-d[M_{2}]}{dt}=k_{12}[M_{2}]\sum[M_{1}^{*}]+k_{22}[M_{2}]\sum[M_{2}^{*}]\,
  36. d [ M 1 ] d [ M 2 ] = [ M 1 ] [ M 2 ] ( k 11 [ M 1 * ] [ M 2 * ] + k 21 k 12 [ M 1 * ] [ M 2 * ] + k 22 ) \frac{d[M_{1}]}{d[M_{2}]}=\frac{[M_{1}]}{[M_{2}]}\left(\frac{k_{11}\frac{\sum[% M_{1}^{*}]}{\sum[M_{2}^{*}]}+k_{21}}{k_{12}\frac{\sum[M_{1}^{*}]}{\sum[M_{2}^{% *}]}+k_{22}}\right)\,
  37. d [ M 1 * ] d t = d [ M 2 * ] d t 0 \frac{d\sum[M_{1}^{*}]}{dt}=\frac{d\sum[M_{2}^{*}]}{dt}\approx 0\,
  38. k 21 [ M 1 ] [ M 2 * ] = k 12 [ M 2 ] [ M 1 * ] k_{21}[M_{1}]\sum[M_{2}^{*}]=k_{12}[M_{2}]\sum[M_{1}^{*}]\,
  39. [ M 1 * ] [ M 2 * ] = k 21 [ M 1 ] k 12 [ M 2 ] \frac{\sum[M_{1}^{*}]}{\sum[M_{2}^{*}]}=\frac{k_{21}[M_{1}]}{k_{12}[M_{2}]}\,

Mazur's_lemma.html

  1. u n u 0 as n . u_{n}\rightharpoonup u_{0}\mbox{ as }~{}n\to\infty.
  2. f ( u n ) f ( u 0 ) as n . f(u_{n})\to f(u_{0})\mbox{ as }~{}n\to\infty.
  3. { α ( n ) k | k = n , , N ( n ) } \{\alpha(n)_{k}|k=n,\dots,N(n)\}
  4. k = n N ( n ) α ( n ) k = 1 \sum_{k=n}^{N(n)}\alpha(n)_{k}=1
  5. v n = k = n N ( n ) α ( n ) k u k v_{n}=\sum_{k=n}^{N(n)}\alpha(n)_{k}u_{k}
  6. v n - u 0 0 as n . \|v_{n}-u_{0}\|\to 0\mbox{ as }~{}n\to\infty.

McShane's_identity.html

  1. 𝕋 \mathbb{T}
  2. γ 1 1 + e ( γ ) = 1 2 \sum_{\gamma}\frac{1}{1+e^{\ell(\gamma)}}=\frac{1}{2}

MDS_matrix.html

  1. A ~ = ( Id n A ) \tilde{A}=\left(\begin{array}[]{c}{\rm Id}_{n}\\ \hline{\rm A}\end{array}\right)
  2. A ~ \tilde{A}

Mean_absolute_error.html

  1. MAE = 1 n i = 1 n | f i - y i | = 1 n i = 1 n | e i | . \mathrm{MAE}=\frac{1}{n}\sum_{i=1}^{n}\left|f_{i}-y_{i}\right|=\frac{1}{n}\sum% _{i=1}^{n}\left|e_{i}\right|.
  2. | e i | = | f i - y i | |e_{i}|=|f_{i}-y_{i}|
  3. f i f_{i}
  4. y i y_{i}

Mean_of_circular_quantities.html

  1. α \alpha
  2. ( cos α , sin α ) (\cos\alpha,\sin\alpha)
  3. α 1 , , α n \alpha_{1},\dots,\alpha_{n}
  4. α ¯ = atan2 ( j = 1 n sin α j n , j = 1 n cos α j n ) \bar{\alpha}=\operatorname{atan2}\left(\frac{\sum_{j=1}^{n}\sin\alpha_{j}}{n},% \frac{\sum_{j=1}^{n}\cos\alpha_{j}}{n}\right)
  5. α ¯ = arg ( 1 n j = 1 n exp ( i α j ) ) \bar{\alpha}=\arg\left(\frac{1}{n}\cdot\sum_{j=1}^{n}\exp(i\cdot\alpha_{j})\right)
  6. α ¯ \bar{\alpha}
  7. α ¯ = argmin 𝛽 j = 1 n d ( α j , β ) \bar{\alpha}=\underset{\beta}{\operatorname{argmin}}\sum_{j=1}^{n}d(\alpha_{j}% ,\beta)
  8. d ( φ , β ) = 1 - cos ( φ - β ) . d(\varphi,\beta)=1-\cos(\varphi-\beta).
  9. d ( φ , β ) d(\varphi,\beta)
  10. φ \varphi
  11. β \beta

Mean_reciprocal_rank.html

  1. MRR = 1 | Q | i = 1 | Q | 1 rank i . \,\text{MRR}=\frac{1}{|Q|}\sum_{i=1}^{|Q|}\frac{1}{\,\text{rank}_{i}}.\!

Mean_shift.html

  1. x x
  2. K ( x i - x ) K(x_{i}-x)
  3. K ( x i - x ) = e - c || x i - x || 2 K(x_{i}-x)=e^{-c||x_{i}-x||^{2}}
  4. K K
  5. m ( x ) = x i N ( x ) K ( x i - x ) x i x i N ( x ) K ( x i - x ) m(x)=\frac{\sum_{x_{i}\in N(x)}K(x_{i}-x)x_{i}}{\sum_{x_{i}\in N(x)}K(x_{i}-x)}
  6. N ( x ) N(x)
  7. x x
  8. K ( x ) 0 K(x)\neq 0
  9. x m ( x ) x\leftarrow m(x)
  10. m ( x ) m(x)
  11. λ \lambda
  12. K ( x ) = { 1 if x λ 0 if x > λ K(x)=\begin{cases}1&\,\text{if}\ \|x\|\leq\lambda\\ 0&\,\text{if}\ \|x\|>\lambda\\ \end{cases}
  13. m ( x ) - x m(x)-x
  14. s m ( s ) s\leftarrow m(s)
  15. s S s\in S
  16. h h
  17. f ( x ) = i K ( x - x i ) = i k ( x - x i 2 h 2 ) f(x)=\sum_{i}K(x-x_{i})=\sum_{i}k\left(\frac{\|x-x_{i}\|^{2}}{h^{2}}\right)
  18. x i x_{i}
  19. k ( r ) k(r)
  20. f ( x ) f(x)
  21. f ( x ) f(x)
  22. y k y_{k}
  23. x 1 x_{1}
  24. f ( x ) f(x)
  25. y k y_{k}
  26. R n R^{n}
  27. x i x_{i}
  28. x 2 = x T x \|x\|^{2}=x^{T}x
  29. X R X\leftarrow R
  30. k : [ 0 , ] R k:[0,\infty]\rightarrow R
  31. K ( x ) = k ( x 2 ) K(x)=k(\|x\|^{2})
  32. k ( a ) k ( b ) k(a)\geq k(b)
  33. a < b a<b
  34. 0 k ( r ) d r < \int_{0}^{\infty}k(r)\,dr<\infty
  35. F ( x ) = { 1 if x λ 0 if x > λ F(x)=\begin{cases}1&\,\text{if}\ \|x\|\leq\lambda\\ 0&\,\text{if}\ \|x\|>\lambda\\ \end{cases}
  36. G ( x ) = c k , d k ( x 2 ) G(x)=c_{k,d}k(\|x\|^{2})
  37. c k , d c_{k,d}
  38. k ( x ) k(x)
  39. e - 1 / 2 x 2 e^{-1/2\|x\|^{2}}
  40. σ \sigma
  41. G ( x ) = 1 2 π σ d e - 1 / 2 x 2 σ 2 G(x)=\frac{1}{\sqrt{2\pi}\sigma^{d}}e^{-1/2\frac{\|x\|^{2}}{\sigma^{2}}}
  42. h h
  43. x i x_{i}
  44. z i , i = 1 , , n , z_{i},i=1,...,n,
  45. j = 1 j=1
  46. y i , 1 = x i y_{i,1}=x_{i}
  47. y i , j + 1 y_{i,j+1}
  48. m ( ) m(\cdot)
  49. y = y i , c y=y_{i,c}
  50. z i = ( x i s , y i , c r ) z_{i}=(x_{i}^{s},y_{i,c}^{r})
  51. y i , c r y_{i,c}^{r}
  52. O ( k N 2 ) O(kN^{2})
  53. O ( k N ) O(kN)

Mean_square_quantization_error.html

  1. q i q_{i}
  2. t i - 1 t_{i-1}
  3. t i t_{i}
  4. k k
  5. p ( x ) p(x)
  6. x ^ \hat{x}
  7. x x
  8. x ^ \hat{x}
  9. q i q_{i}
  10. t i - 1 x < t i t_{i}-1\leq x<t_{i}
  11. MSQE = E [ ( x - x ^ ) 2 ] = t 0 t k ( x - x ^ ) 2 p ( x ) d x = i = 1 k t i - 1 t i ( x - q i ) 2 p ( x ) d x . \begin{aligned}\displaystyle\operatorname{MSQE}&\displaystyle=\operatorname{E}% [(x-\hat{x})^{2}]\\ &\displaystyle=\int_{t_{0}}^{t_{k}}(x-\hat{x})^{2}p(x)\,dx\\ &\displaystyle=\sum_{i=1}^{k}\int_{t_{i-1}}^{t_{i}}(x-q_{i})^{2}p(x)\,dx.\end{aligned}

Mechanical_amplifier.html

  1. m x ¨ + c x ˙ + k x = F ( t ) m\ddot{x}+c\dot{x}+kx=F(t)
  2. f n = 1 2 π k m f_{n}={1\over 2\pi}\sqrt{k\over m}
  3. x ¨ + c x ˙ + [ δ - 2 ε cos 2 t ] x = 0 \ddot{x}+c\dot{x}+[\delta-2\varepsilon\cos{2t}]x=0
  4. F A F B = a b \frac{F_{A}}{F_{B}}=\frac{a}{b}
  5. M A = F B F A MA=\frac{F_{B}}{F_{A}}
  6. M A = T B T A = N B N A . MA=\frac{T_{B}}{T_{A}}=\frac{N_{B}}{N_{A}}.
  7. T A T B = ω B ω A . \frac{T_{A}}{T_{B}}=\frac{\omega_{B}}{\omega_{A}}.

Median_absolute_deviation.html

  1. MAD = median i ( | X i - median j ( X j ) | ) , \operatorname{MAD}=\operatorname{median}_{i}\left(\ \left|X_{i}-\operatorname{% median}_{j}(X_{j})\right|\ \right),\,
  2. σ ^ = K MAD , \hat{\sigma}=K\cdot\operatorname{MAD},\,
  3. 1 / ( Φ - 1 ( 3 / 4 ) ) 1.4826 1/\left(\Phi^{-1}(3/4)\right)\approx 1.4826
  4. Φ - 1 \Phi^{-1}
  5. 1 2 = P ( | X - μ | MAD ) = P ( | X - μ σ | MAD σ ) = P ( | Z | MAD σ ) . \frac{1}{2}=P(|X-\mu|\leq\operatorname{MAD})=P\left(\left|\frac{X-\mu}{\sigma}% \right|\leq\frac{\operatorname{MAD}}{\sigma}\right)=P\left(|Z|\leq\frac{% \operatorname{MAD}}{\sigma}\right).
  6. Φ ( MAD / σ ) - Φ ( - MAD / σ ) = 1 / 2 \Phi\left(\operatorname{MAD}/\sigma\right)-\Phi\left(-\operatorname{MAD}/% \sigma\right)=1/2
  7. Φ ( - MAD / σ ) = 1 - Φ ( MAD / σ ) \Phi\left(-\operatorname{MAD}/\sigma\right)=1-\Phi\left(\operatorname{MAD}/% \sigma\right)
  8. MAD / σ = Φ - 1 ( 3 / 4 ) \operatorname{MAD}/\sigma=\Phi^{-1}\left(3/4\right)
  9. K = 1 / ( Φ - 1 ( 3 / 4 ) ) K=1/\left(\Phi^{-1}(3/4)\right)
  10. σ 1.4826 MAD . \sigma\approx 1.4826\ \operatorname{MAD}.\,
  11. 1.4826 1 / ( Φ - 1 ( 3 / 4 ) ) 1.4826\ \approx 1/\left(\Phi^{-1}(3/4)\right)
  12. Φ - 1 ( P ) \Phi^{-1}(P)
  13. P = 3 / 4 P=3/4

Medical_test.html

  1. b n = Δ p × r i × ( b i - h i ) - h t b_{n}=\Delta p\times r_{i}\times(b_{i}-h_{i})-h_{t}

Melnikov_distance.html

  1. x ¨ = f ( x ) + ϵ g ( t ) \ddot{x}=f(x)+\epsilon g(t)
  2. ϵ 0 \epsilon\geq 0
  3. g ( t ) g(t)
  4. T T
  5. ϵ = 0 \epsilon=0
  6. ϕ ( t ) \phi(t)
  7. ϵ 0 \epsilon\neq 0
  8. ϕ ( t ) \phi(t)
  9. d ( t ) d(t)
  10. d ( t ) = ϵ ( M ( t ) + O ( ϵ ) ) d(t)=\epsilon(M(t)+O(\epsilon))
  11. M ( t ) M(t)

Membrane_analogy.html

  1. 3 T / b t 2 3T/bt^{2}

MEMO_Model.html

  1. ρ \rho
  2. p p
  3. θ \theta
  4. u u
  5. v v
  6. w w
  7. p < m t p l > n h p_{<}mtpl>{{nh}}
  8. p < m t p l > n h p_{<}mtpl>{{nh}}
  9. u u
  10. v v
  11. w w
  12. θ \theta
  13. p p
  14. p h p_{h}
  15. u u
  16. v v
  17. w w

Menger_curvature.html

  1. c ( x , y , z ) = 1 R . c(x,y,z)=\frac{1}{R}.
  2. c ( x , y , z ) = 1 R = 4 A | x - y | | y - z | | z - x | , c(x,y,z)=\frac{1}{R}=\frac{4A}{|x-y||y-z||z-x|},
  3. c ( x , y , z ) = 2 sin x y z | x - z | c(x,y,z)=\frac{2\sin\angle xyz}{|x-z|}
  4. x y z \angle xyz
  5. { x , y , z } \{x,y,z\}
  6. 2 \mathbb{R}^{2}
  7. c X ( x , y , z ) = c ( f ( x ) , f ( y ) , f ( z ) ) . c_{X}(x,y,z)=c(f(x),f(y),f(z)).
  8. n \mathbb{R}^{n}
  9. μ \mu
  10. n \mathbb{R}^{n}
  11. c p ( μ ) = c ( x , y , z ) p d μ ( x ) d μ ( y ) d μ ( z ) . c^{p}(\mu)=\int\int\int c(x,y,z)^{p}d\mu(x)d\mu(y)d\mu(z).
  12. E n E\subseteq\mathbb{R}^{n}
  13. c 2 ( H 1 | E ) < c^{2}(H^{1}|_{E})<\infty
  14. H 1 | E H^{1}|_{E}
  15. E E
  16. c ( x , y , z ) max { | x - y | , | y - z | , | z - y | } c(x,y,z)\max\{|x-y|,|y-z|,|z-y|\}
  17. p > 3 p>3
  18. f : S 1 n f:S^{1}\rightarrow\mathbb{R}^{n}
  19. Γ = f ( S 1 ) \Gamma=f(S^{1})
  20. f C 1 , 1 - 3 p ( S 1 ) f\in C^{1,1-\frac{3}{p}}(S^{1})
  21. c p ( H 1 | Γ ) < c^{p}(H^{1}|_{\Gamma})<\infty
  22. 0 < H s ( E ) < 0<H^{s}(E)<\infty
  23. 0 < s 1 2 0<s\leq\frac{1}{2}
  24. c 2 s ( H s | E ) < c^{2s}(H^{s}|_{E})<\infty
  25. E E
  26. C 1 C^{1}
  27. Γ i \Gamma_{i}
  28. H s ( E \ Γ i ) = 0 H^{s}(E\backslash\bigcup\Gamma_{i})=0
  29. 1 2 < s < 1 \frac{1}{2}<s<1
  30. c 2 s ( H s | E ) = c^{2s}(H^{s}|_{E})=\infty
  31. 1 < s n 1<s\leq n
  32. E Γ 2 E\subseteq\Gamma\subseteq\mathbb{R}^{2}
  33. H 1 ( E ) > 0 H^{1}(E)>0
  34. Γ \Gamma
  35. μ \mu
  36. E E
  37. μ B ( x , r ) r \mu B(x,r)\leq r
  38. x E x\in E
  39. r > 0 r>0
  40. c 2 ( μ ) < c^{2}(\mu)<\infty
  41. H 1 ( B ( x , r ) Γ ) C r H^{1}(B(x,r)\cap\Gamma)\leq Cr
  42. x Γ x\in\Gamma
  43. c 2 ( H 1 | E ) < c^{2}(H^{1}|_{E})<\infty

Mersenne's_laws.html

  1. f 1 1 L . f_{1}\propto\tfrac{1}{L}.
  2. f 1 F . f_{1}\propto\sqrt{F}.
  3. f 1 1 μ . f_{1}\propto\frac{1}{\sqrt{\mu}}.
  4. f 1 = ν λ = 1 2 L F μ . f_{1}=\frac{\nu}{\lambda}=\frac{1}{2L}\sqrt{\frac{F}{\mu}}.
  5. f 0 = 1 2 L F μ , f_{0}=\frac{1}{2L}\sqrt{\frac{F}{\mu}},

Mersenne_conjectures.html

  1. 2 n - 1 2^{n}-1
  2. e γ log log ( x ) / log ( 2 ) , e^{\gamma}\cdot\log\log(x)/\log(2),
  3. e γ log ( y ) / log ( 2 ) . e^{\gamma}\cdot\log(y)/\log(2).
  4. e γ log ( 10 ) / log ( 2 ) e^{\gamma}\cdot\log(10)/\log(2)
  5. M p M_{p}

Method_of_simulated_moments.html

  1. β ^ G M M = argmin m ( x , β ) W m ( x , β ) \hat{\beta}_{GMM}=\operatorname{argmin}\,m(x,\beta)^{\prime}Wm(x,\beta)
  2. m ( x , β ) m(x,\beta)
  3. β ^ M S M = argmin m ^ ( x , β ) W m ^ ( x , β ) \hat{\beta}_{MSM}=\operatorname{argmin}\,\hat{m}(x,\beta)^{\prime}W\hat{m}(x,\beta)
  4. m ^ ( x , β ) \hat{m}(x,\beta)
  5. E [ m ^ ( x , β ) ] = m ( x , β ) E[\hat{m}(x,\beta)]=m(x,\beta)

Methylmalonyl_CoA_epimerase.html

  1. \rightleftharpoons

Metric_outer_measure.html

  1. μ ( A B ) = μ ( A ) + μ ( B ) \mu(A\cup B)=\mu(A)+\mu(B)
  2. μ ( E ) = lim δ 0 μ δ ( E ) , \mu(E)=\lim_{\delta\to 0}\mu_{\delta}(E),
  3. μ δ ( E ) = inf { i = 1 τ ( C i ) | C i Σ , diam ( C i ) δ , i = 1 C i E } , \mu_{\delta}(E)=\inf\left\{\left.\sum_{i=1}^{\infty}\tau(C_{i})\right|C_{i}\in% \Sigma,\mathrm{diam}(C_{i})\leq\delta,\bigcup_{i=1}^{\infty}C_{i}\supseteq E% \right\},
  4. τ ( C ) = diam ( C ) s , \tau(C)=\mathrm{diam}(C)^{s},\,
  5. A 1 A 2 A = n = 1 A n , A_{1}\subseteq A_{2}\subseteq\dots\subseteq A=\bigcup_{n=1}^{\infty}A_{n},
  6. μ ( A ) = sup n μ ( A n ) . \mu(A)=\sup_{n\in\mathbb{N}}\mu(A_{n}).
  7. μ ( A B ) = μ ( A ) + μ ( B ) . \mu(A\cup B)=\mu(A)+\mu(B).

Metrical_task_system.html

  1. ( S , d ) (S,d)
  2. S = { s 1 , s 2 , , s n } S=\{s_{1},s_{2},\ldots,s_{n}\}
  3. d : S × S d:S\times S\rightarrow\mathbb{R}
  4. d d
  5. ( S , d ) (S,d)
  6. σ = T 1 , T 2 , , T l \sigma=T_{1},T_{2},\ldots,T_{l}
  7. i i
  8. T i T_{i}
  9. n n
  10. n n
  11. i i
  12. π \pi
  13. π ( i ) = s j \pi(i)=s_{j}
  14. i i
  15. T i T_{i}
  16. s j s_{j}
  17. cost ( π , σ ) = i = 1 l d ( π ( i - 1 ) , π ( i ) ) + T i ( π ( i ) ) . \mathrm{cost}(\pi,\sigma)=\sum_{i=1}^{l}d(\pi(i-1),\pi(i))+T_{i}(\pi(i)).
  18. 2 n - 1 2n-1
  19. Ω ( log n / log log n ) \Omega(\log n/\log\log n)
  20. O ( log 2 n log log n ) O(\log^{2}n\log\log n)

Mian–Chowla_sequence.html

  1. a 1 = 1. a_{1}=1.
  2. n > 1 n>1
  3. a n a_{n}
  4. a i + a j a_{i}+a_{j}
  5. i i
  6. j j
  7. n n
  8. a 1 a_{1}
  9. a 2 a_{2}
  10. a 3 a_{3}
  11. a 3 = 4 a_{3}=4
  12. a 1 = 0 a_{1}=0

Microscopic_traffic_flow_model.html

  1. x α x_{\alpha}
  2. v α v_{\alpha}
  3. v α v_{\alpha}
  4. s α = x α - 1 - x α - l α - 1 s_{\alpha}=x_{\alpha-1}-x_{\alpha}-l_{\alpha-1}
  5. α - 1 \alpha-1
  6. l α - 1 l_{\alpha-1}
  7. v α - 1 v_{\alpha-1}
  8. x ¨ α ( t ) = v ˙ α ( t ) = F ( v α ( t ) , s α ( t ) , v α - 1 ( t ) ) \ddot{x}_{\alpha}(t)=\dot{v}_{\alpha}(t)=F(v_{\alpha}(t),s_{\alpha}(t),v_{% \alpha-1}(t))
  9. α \alpha
  10. α - 1 \alpha-1
  11. n a n_{a}
  12. v ˙ α ( t ) = f ( x α ( t ) , v α ( t ) , x α - 1 ( t ) , v α - 1 ( t ) , , x α - n a ( t ) , v α - n a ( t ) ) \dot{v}_{\alpha}(t)=f(x_{\alpha}(t),v_{\alpha}(t),x_{\alpha-1}(t),v_{\alpha-1}% (t),\ldots,x_{\alpha-n_{a}}(t),v_{\alpha-n_{a}}(t))
  13. Δ x \Delta x
  14. Δ t \Delta t
  15. v α t + 1 = f ( s α t , v α t , v α - 1 t , ) v_{\alpha}^{t+1}=f(s_{\alpha}^{t},v_{\alpha}^{t},v_{\alpha-1}^{t},\ldots)
  16. x α t + 1 = x α t + v α t + 1 x_{\alpha}^{t+1}=x_{\alpha}^{t}+v_{\alpha}^{t+1}
  17. t t
  18. Δ t \Delta t
  19. x α x_{\alpha}
  20. Δ x \Delta x
  21. Δ t = 1 s \Delta t=1\,\text{s}
  22. Δ t \Delta t
  23. Δ x \Delta x
  24. 5 Δ x / Δ t = 135 km/h 5\Delta x/\Delta t=135\,\text{km/h}
  25. Δ x / ( Δ t ) 2 = 7.5 m / s 2 \Delta x/(\Delta t)^{2}=7.5\,\text{m}/\,\text{s}^{2}
  26. Δ x = 1.5 m \Delta x=1.5\,\text{m}
  27. 1.5 m / s 2 1.5\,\text{m}/\,\text{s}^{2}

MIDI_usage_and_applications.html

  1. p = 69 + 12 × log 2 ( f 440 ) p=69+12\times\log_{2}{\left(\frac{f}{440}\right)}

Midpoint_circle_algorithm.html

  1. x 2 + y 2 = r 2 x^{2}+y^{2}=r^{2}
  2. x 2 + y 2 r 2 x^{2}+y^{2}<=r^{2}
  3. x 2 + y 2 x^{2}+y^{2}
  4. ( 0 , 0 ) (0,0)
  5. ( r , 0 ) (r,0)
  6. y y
  7. y y
  8. x x
  9. x 2 + y 2 - r 2 = 0 x^{2}+y^{2}-r^{2}=0
  10. r 2 r^{2}
  11. n = 1 n=1
  12. ( r , 0 ) (r,0)
  13. x n 2 + y n 2 = r 2 \displaystyle x_{n}^{2}+y_{n}^{2}=r^{2}
  14. x n 2 = r 2 - y n 2 \displaystyle x_{n}^{2}=r^{2}-y_{n}^{2}
  15. x n + 1 2 = r 2 - y n + 1 2 \displaystyle x_{n+1}^{2}=r^{2}-y_{n+1}^{2}
  16. y n + 1 2 \displaystyle y_{n+1}^{2}
  17. x n + 1 2 = r 2 - y n 2 - 2 y n - 1 \begin{aligned}\displaystyle x_{n+1}^{2}=r^{2}-y_{n}^{2}-2y_{n}-1\end{aligned}
  18. x n 2 = r 2 - y n 2 x_{n}^{2}=r^{2}-y_{n}^{2}
  19. x n + 1 2 = x n 2 - 2 y n - 1 \displaystyle x_{n+1}^{2}=x_{n}^{2}-2y_{n}-1
  20. r r
  21. ( x i , y i ) (x_{i},y_{i})
  22. R E ( x i , y i ) = | x i 2 + y i 2 - r 2 | RE(x_{i},y_{i})=\left|x_{i}^{2}+y_{i}^{2}-r^{2}\right|
  23. ( r , 0 ) (r,0)
  24. R E ( x i , y i ) = | r 2 + 0 2 - r 2 | = 0 RE(x_{i},y_{i})=\left|r^{2}+0^{2}-r^{2}\right|=0
  25. y i + 1 = y i + 1 y_{i+1}=y_{i}+1
  26. R E ( x i - 1 , y i + 1 ) < R E ( x i , y i + 1 ) RE(x_{i}-1,y_{i}+1)<RE(x_{i},y_{i}+1)
  27. ( x i - 1 , y i + 1 ) (x_{i}-1,y_{i}+1)
  28. ( x i , y i + 1 ) (x_{i},y_{i}+1)
  29. R E ( x i - 1 , y i + 1 ) < R E ( x i , y i + 1 ) | ( x i - 1 ) 2 + ( y i + 1 ) 2 - r 2 | < | x i 2 + ( y i + 1 ) 2 - r 2 | | ( x i 2 - 2 x i + 1 ) + ( y i 2 + 2 y i + 1 ) - r 2 | < | x i 2 + ( y i 2 + 2 y i + 1 ) - r 2 | \begin{array}[]{lcl}RE(x_{i}-1,y_{i}+1)&<&RE(x_{i},y_{i}+1)\\ \left|(x_{i}-1)^{2}+(y_{i}+1)^{2}-r^{2}\right|&<&\left|x_{i}^{2}+(y_{i}+1)^{2}% -r^{2}\right|\\ \left|(x_{i}^{2}-2x_{i}+1)+(y_{i}^{2}+2y_{i}+1)-r^{2}\right|&<&\left|x_{i}^{2}% +(y_{i}^{2}+2y_{i}+1)-r^{2}\right|\\ \end{array}
  30. [ ( x i 2 - 2 x i + 1 ) + ( y i 2 + 2 y i + 1 ) - r 2 ] 2 < [ x i 2 + ( y i 2 + 2 y i + 1 ) - r 2 ] 2 [ ( x i 2 + y i 2 - r 2 + 2 y i + 1 ) + ( 1 - 2 x i ) ] 2 < [ x i 2 + y i 2 - r 2 + 2 y i + 1 ] 2 ( x i 2 + y i 2 - r 2 + 2 y i + 1 ) 2 + 2 ( 1 - 2 x i ) ( x i 2 + y i 2 - r 2 + 2 y i + 1 ) + ( 1 - 2 x i ) 2 < [ x i 2 + y i 2 - r 2 + 2 y i + 1 ] 2 2 ( 1 - 2 x i ) ( x i 2 + y i 2 - r 2 + 2 y i + 1 ) + ( 1 - 2 x i ) 2 < 0 \begin{array}[]{lcl}\left[(x_{i}^{2}-2x_{i}+1)+(y_{i}^{2}+2y_{i}+1)-r^{2}% \right]^{2}&<&\left[x_{i}^{2}+(y_{i}^{2}+2y_{i}+1)-r^{2}\right]^{2}\\ \left[(x_{i}^{2}+y_{i}^{2}-r^{2}+2y_{i}+1)+(1-2x_{i})\right]^{2}&<&\left[x_{i}% ^{2}+y_{i}^{2}-r^{2}+2y_{i}+1\right]^{2}\\ \left(x_{i}^{2}+y_{i}^{2}-r^{2}+2y_{i}+1\right)^{2}+2(1-2x_{i})(x_{i}^{2}+y_{i% }^{2}-r^{2}+2y_{i}+1)+(1-2x_{i})^{2}&<&\left[x_{i}^{2}+y_{i}^{2}-r^{2}+2y_{i}+% 1\right]^{2}\\ 2(1-2x_{i})(x_{i}^{2}+y_{i}^{2}-r^{2}+2y_{i}+1)+(1-2x_{i})^{2}&<&0\\ \end{array}
  31. ( 1 - 2 x i ) < 0 (1-2x_{i})<0
  32. 2 [ ( x i 2 + y i 2 - r 2 ) + ( 2 y i + 1 ) ] + ( 1 - 2 x i ) > 0 2 [ R E ( x i , y i ) + Y C h a n g e ] + X C h a n g e > 0 \begin{array}[]{lcl}2\left[(x_{i}^{2}+y_{i}^{2}-r^{2})+(2y_{i}+1)\right]+(1-2x% _{i})&>&0\\ 2\left[RE(x_{i},y_{i})+YChange\right]+XChange&>&0\\ \end{array}
  33. α \alpha
  34. β \beta
  35. x x
  36. y y
  37. y / x y/x

Millioctave.html

  1. n = 1000 log 2 ( a b ) 3322 log 10 ( a b ) n=1000\log_{2}\left(\frac{a}{b}\right)\approx 3322\log_{10}\left(\frac{a}{b}\right)
  2. a = b × 2 n 1000 a=b\times 2^{\frac{n}{1000}}

Milman's_reverse_Brunn–Minkowski_inequality.html

  1. vol ( K + L ) 1 / n vol ( K ) 1 / n + vol ( L ) 1 / n , \mathrm{vol}(K+L)^{1/n}\geq\mathrm{vol}(K)^{1/n}+\mathrm{vol}(L)^{1/n}~{},
  2. vol ( s φ K + t ψ L ) 1 / n C ( s vol ( φ K ) 1 / n + t vol ( ψ L ) 1 / n ) . \mathrm{vol}(s\,\varphi K+t\,\psi L)^{1/n}\leq C\left(s\,\mathrm{vol}(\varphi K% )^{1/n}+t\,\mathrm{vol}(\psi L)^{1/n}\right)~{}.

Milner_Baily_Schaefer.html

  1. H ( E , X ) = q E X H(E,X)=qEX\!
  2. X ˙ = 0 \dot{X}=0
  3. H ( E ) = q K E ( 1 - q E r ) H(E)=qKE(1-\frac{qE}{r})

Milnor_conjecture_(topology).html

  1. ( p , q ) (p,q)
  2. ( p - 1 ) ( q - 1 ) / 2. (p-1)(q-1)/2.

Minimal_polynomial_(linear_algebra).html

  1. n × n n×n
  2. A A
  3. 𝐅 \mathbf{F}
  4. P P
  5. 𝐅 \mathbf{F}
  6. P ( A ) = 0 P(A)=0
  7. Q Q
  8. Q ( A ) = 0 Q(A)=0
  9. λ λ
  10. λ λ
  11. A A
  12. λ λ
  13. A A
  14. λ λ
  15. m m
  16. m m
  17. m m
  18. 𝐅 \mathbf{F}
  19. 𝐅 \mathbf{F}
  20. 1 1
  21. P P
  22. P P
  23. P P
  24. P ( A ) P(A)
  25. 1 1
  26. P ( A ) P(A)
  27. d e g ( P ) deg(P)
  28. A A
  29. A A
  30. X a X−a
  31. a a
  32. T T
  33. V V
  34. 𝐅 \mathbf{F}
  35. I T = { p 𝐅 [ t ] | p ( T ) = 0 } \mathit{I}_{T}=\{p\in\mathbf{F}[t]\;|\;p(T)=0\}
  36. 𝐅 t t \mathbf{F}tt
  37. 𝐅 \mathbf{F}
  38. 𝐅 t t \mathbf{F}tt
  39. 𝐅 \mathbf{F}
  40. 𝐅 t t \mathbf{F}tt
  41. 𝐅 \mathbf{F}
  42. φ φ
  43. 𝐅 \mathbf{F}
  44. 𝐅 \mathbf{F}
  45. X λ X−λ
  46. λ λ
  47. λ λ
  48. λ λ
  49. 1 1
  50. φ φ
  51. P ( φ ) = 0 P(φ)=0
  52. P P
  53. 𝐅 \mathbf{F}
  54. P P
  55. k = 2 k=2
  56. 2 2
  57. 0
  58. 1 1
  59. k 2 k≥2
  60. φ φ
  61. 0
  62. v v
  63. V V
  64. I T , v = { p 𝐅 [ t ] | p ( T ) ( v ) = 0 } . \mathit{I}_{T,v}=\{p\in\mathbf{F}[t]\;|\;p(T)(v)=0\}.
  65. Q Q
  66. T T
  67. ( 1 - 1 - 1 1 - 2 1 0 1 - 3 ) . \begin{pmatrix}1&-1&-1\\ 1&-2&1\\ 0&1&-3\end{pmatrix}.
  68. T T
  69. e 1 = [ 1 0 0 ] , T e 1 = [ 1 1 0 ] . T 2 e 1 = [ 0 - 1 1 ] and T 3 e 1 = [ 0 3 - 4 ] e_{1}=\begin{bmatrix}1\\ 0\\ 0\end{bmatrix},\quad T\cdot e_{1}=\begin{bmatrix}1\\ 1\\ 0\end{bmatrix}.\quad T^{2}\cdot e_{1}=\begin{bmatrix}0\\ -1\\ 1\end{bmatrix}\mbox{ and}~{}\quad T^{3}\cdot e_{1}=\begin{bmatrix}0\\ 3\\ -4\end{bmatrix}
  70. T T
  71. v v
  72. T v T⋅v
  73. T T
  74. v v
  75. T T
  76. v v
  77. [ 0 1 - 3 3 - 13 23 - 4 19 - 36 ] + 4 [ 0 0 1 - 1 4 - 6 1 - 5 10 ] + [ 1 - 1 - 1 1 - 2 1 0 1 - 3 ] + [ - 1 0 0 0 - 1 0 0 0 - 1 ] = [ 0 0 0 0 0 0 0 0 0 ] \begin{bmatrix}0&1&-3\\ 3&-13&23\\ -4&19&-36\end{bmatrix}+4\begin{bmatrix}0&0&1\\ -1&4&-6\\ 1&-5&10\end{bmatrix}+\begin{bmatrix}1&-1&-1\\ 1&-2&1\\ 0&1&-3\end{bmatrix}+\begin{bmatrix}-1&0&0\\ 0&-1&0\\ 0&0&-1\end{bmatrix}=\begin{bmatrix}0&0&0\\ 0&0&0\\ 0&0&0\end{bmatrix}

Minimal_subtraction_scheme.html

  1. MS ¯ \overline{\,\text{MS}}
  2. d 4 p μ 4 - d d d p d^{4}p\to\mu^{4-d}d^{d}p
  3. μ 2 μ 2 e γ E 4 π \mu^{2}\to\mu^{2}\frac{e^{\gamma_{E}}}{4\pi}
  4. γ E \gamma_{E}

Minimal_volume.html

  1. M n M^{n}
  2. g g
  3. V o l ( M , g ) Vol({\it M,g})
  4. M M
  5. g g
  6. K g K_{g}
  7. M M
  8. M i n V o l ( M ) := inf g { V o l ( M , g ) : | K g | 1 } MinVol(M):=\inf_{g}\{Vol(M,g):|K_{g}|\leq 1\}
  9. M M
  10. M M
  11. g g
  12. λ g \lambda g
  13. V o l ( M , λ g ) = λ n / 2 V o l ( M , g ) Vol(M,\lambda g)=\lambda^{n/2}Vol(M,g)
  14. K λ g = 1 λ K g \textstyle K_{\lambda g}=\frac{1}{\lambda}K_{g}
  15. M i n V o l ( M ) = 0 MinVol(M)=0
  16. M n M^{n}
  17. n n
  18. M M

Minimax_eversion.html

  1. S 2 S^{2}
  2. \R 3 \R^{3}
  3. S 2 S^{2}
  4. 𝐑 3 \mathbf{R}^{3}

Minimum_deviation.html

  1. n λ = sin ( A + D λ 2 ) sin ( A 2 ) n_{\lambda}={\sin({A+D_{\lambda}\over 2})\over\sin({A\over 2})}

Minkowski's_bound.html

  1. \mathbb{Q}
  2. 2 r 2 = n - r 1 2r_{2}=n-r_{1}
  3. r 1 r_{1}
  4. M K = | D | ( 4 π ) r 2 n ! n n . M_{K}=\sqrt{|D|}\left(\frac{4}{\pi}\right)^{r_{2}}\frac{n!}{n^{n}}\ .
  5. | D | ( π 4 ) r 2 n n n ! ( π 4 ) n / 2 n n n ! . \sqrt{|D|}\geq\left(\frac{\pi}{4}\right)^{r_{2}}\frac{n^{n}}{n!}\geq\left(% \frac{\pi}{4}\right)^{n/2}\frac{n^{n}}{n!}\ .

Minkowski's_first_inequality_for_convex_bodies.html

  1. n V 1 ( K , L ) = lim ε 0 V ( K + ε L ) - V ( K ) ε , nV_{1}(K,L)=\lim_{\varepsilon\downarrow 0}\frac{V(K+\varepsilon L)-V(K)}{% \varepsilon},
  2. V 1 ( K , L ) V ( K ) ( n - 1 ) / n V ( L ) 1 / n , V_{1}(K,L)\geq V(K)^{(n-1)/n}V(L)^{1/n},
  3. ( V ( K ) V ( B ) ) 1 / n ( S ( K ) S ( B ) ) 1 / ( n - 1 ) , \left(\frac{V(K)}{V(B)}\right)^{1/n}\leq\left(\frac{S(K)}{S(B)}\right)^{1/(n-1% )},

Minkowski_diagram.html

  1. tan ( α ) = v c = β \tan(\alpha)=\frac{v}{c}=\beta
  2. U = U 1 + β 2 1 - β 2 U^{\prime}=U\cdot\sqrt{\frac{1+\beta^{2}}{1-\beta^{2}}}
  3. t 2 - x 2 = 1 \scriptstyle t^{2}-x^{2}=1
  4. β = v / c \beta=v/c
  5. γ = 1 / 1 - β 2 \scriptstyle\gamma=1/\sqrt{1-\beta^{2}}
  6. S 0 S_{0}
  7. ( 1 ) β = 2 β 0 1 + β 0 2 , ( 2 ) β 0 = γ - 1 β γ . \begin{aligned}\displaystyle(1)&&\displaystyle\beta&\displaystyle=\frac{2\beta% _{0}}{1+\beta_{0}^{2}},\\ \displaystyle(2)&&\displaystyle\beta_{0}&\displaystyle=\frac{\gamma-1}{\beta% \gamma}.\end{aligned}
  8. β = 0.5 \beta=0.5
  9. S 0 S_{0}
  10. β 0 = 0.5 \beta_{0}=0.5
  11. S 0 S_{0}
  12. tan α = β 0 \tan\alpha=\beta_{0}
  13. β 0 \beta_{0}
  14. β = v / c \beta=v/c
  15. φ \varphi
  16. θ \theta
  17. sin φ = cos θ = β , cos φ = sin θ = 1 / γ , tan φ = cot θ = β γ . \begin{aligned}\displaystyle\sin\varphi=\cos\theta&\displaystyle=\beta,\\ \displaystyle\cos\varphi=\sin\theta&\displaystyle=1/\gamma,\\ \displaystyle\tan\varphi=\cot\theta&\displaystyle=\beta\cdot\gamma.\end{aligned}
  18. φ \varphi
  19. θ \theta
  20. ( x , t ; x , t ) (x,t;x^{\prime},t^{\prime})
  21. R R
  22. ( ξ , τ ; ξ , τ ) (\xi,\tau;\xi^{\prime},\tau^{\prime})
  23. ( 1 + β 2 ) 1 / 2 ( 1 - β 2 ) - 1 / 2 \scriptstyle(1+\beta^{2})^{1/2}(1-\beta^{2})^{-1/2}