wpmath0000001_12

Interquartile_mean.html

  1. x IQM = 2 n i = n 4 + 1 3 n 4 x i x_{\mathrm{IQM}}={2\over n}\sum_{i=\frac{n}{4}+1}^{\frac{3n}{4}}{x_{i}}

Interquartile_range.html

  1. Q 1 = CDF - 1 ( 0.25 ) , Q_{1}=\,\text{CDF}^{-1}(0.25),
  2. Q 3 = CDF - 1 ( 0.75 ) , Q_{3}=\,\text{CDF}^{-1}(0.75),
  3. Q 1 = ( σ z 1 ) + X Q_{1}=(\sigma\,z_{1})+X
  4. Q 3 = ( σ z 3 ) + X Q_{3}=(\sigma\,z_{3})+X

Interstellar_medium.html

  1. n ( r ) r - 3.5 n(r)\propto r^{-3.5}
  2. r 2 n r - 1.5 r^{2}n\propto r^{-1.5}
  3. E p h o t o n - E i o n i z a t i o n E_{photon}-E_{ionization}
  4. H 2 H_{2}
  5. H 2 H_{2}
  6. α = T 2 - T T d - T \alpha=\frac{T_{2}-T}{T_{d}-T}
  7. T T
  8. T d T_{d}
  9. T 2 T_{2}
  10. α = 0.35 \alpha=0.35
  11. α \alpha

Interstellar_travel.html

  1. 1 / 2 {1}/{2}
  2. × 10 1 7 \times 10^{1}7
  3. p e + + π 0 p\rightarrow e^{+}+\pi^{0}
  4. G μ ν = 8 π G T μ ν G_{\mu\nu}=8\pi\,GT_{\mu\nu}\,

Interval_(mathematics).html

  1. x x
  2. 0 x 1 0≤x≤1
  3. 0
  4. 1 1
  5. \R \R
  6. a a
  7. b b
  8. a a
  9. b b
  10. [ a , b ] [a,b]
  11. ( a , b ) = ] a , b [ \displaystyle(a,b)=\mathopen{]}a,b\mathclose{[}
  12. ( a , a ) (a,a)
  13. [ a , a ) [a,a)
  14. ( a , a ] (a,a]
  15. [ a , a ] [a,a]
  16. a > b a>b
  17. ( a , b ) (a,b)
  18. a a , b aa,b
  19. a a , b aa,b
  20. a a , b aa,b
  21. ( a , b ) (a,b)
  22. a a
  23. b b
  24. a = a=−∞
  25. b = + b=+∞
  26. ( 0 , + ) (0, +∞)
  27. ( , + ) (−∞, +∞)
  28. −∞
  29. + +∞
  30. [ , b ] [−∞,b]
  31. [ , b ) [−∞,b)
  32. [ a , + ] [a, +∞]
  33. ( a , + ] (a, +∞]
  34. ( , + ] (−∞, +∞]
  35. −∞
  36. [ a . . b ] [a..b]
  37. a a
  38. b b
  39. a . . b a..b
  40. a a
  41. b b
  42. a . . b 1 a..b− 1
  43. a + 1.. b a+ 1..b
  44. a + 1.. b 1 a+ 1..b− 1
  45. [ a . . b ) [a..b)
  46. a a . . b aa..b
  47. ( 0 , 1 ) (0,1)
  48. 0
  49. 1 1
  50. [ 0 , 1 ] [0,1]
  51. 0
  52. 1 1
  53. + +∞
  54. 0
  55. a a
  56. b b
  57. ( a + b ) / 2 (a+b)/2
  58. | a b | / 2 |a−b|/2
  59. [ 0 , 1 ) [0,1)
  60. x 0 x Align l t ; 1 {x0≤x<1}
  61. I I
  62. I I
  63. I I
  64. I I
  65. I I
  66. I I
  67. I I
  68. X X
  69. X X
  70. X X
  71. X X
  72. a a
  73. b b
  74. a < b a<b
  75. [ b , a ] = ( a , a ) = [ a , a ) = ( a , a ] = { } = [b,a]=(a,a)=[a,a)=(a,a]=\{\}=\emptyset
  76. [ a , a ] = { a } [a,a]=\{a\}
  77. ( a , b ) = { x | a < x < b } (a,b)=\{x\,|\,a<x<b\}
  78. [ a , b ] = { x | a x b } [a,b]=\{x\,|\,a\leq x\leq b\}
  79. [ a , b ) = { x | a x < b } [a,b)=\{x\,|\,a\,\leq x<b\}
  80. ( a , b ] = { x | a < x b } (a,b]=\{x\,|\,a<x\leq b\}
  81. ( a , ) = { x | x > a } (a,\infty)=\{x\,|\,x>a\}
  82. [ a , ) = { x | x a } [a,\infty)=\{x\,|\,x\geq a\}
  83. ( - , b ) = { x | x < b } (-\infty,b)=\{x\,|\,x<b\}
  84. ( - , b ] = { x | x b } (-\infty,b]=\{x\,|\,x\leq b\}
  85. ( - , + ) = \R (-\infty,+\infty)=\R
  86. −∞
  87. + +∞
  88. [ , b ] [−∞,b]
  89. [ , b ) [−∞,b)
  90. [ a , + ] [a, +∞]
  91. ( a , + ] (a, +∞]
  92. ( , + ) (−∞, +∞)
  93. [ , + ] [−∞, +∞]
  94. ( , + ) (−∞, +∞)
  95. \R \R
  96. \R \R
  97. \R \R
  98. X \R X\subseteq\R
  99. X X
  100. ( a , b ) [ b , c ] = ( a , c ] (a,b)\cup[b,c]=(a,c]
  101. \R \R
  102. ( c + r , c r ) (c+r,c−r)
  103. [ c + r , c r ] [c+r,c−r]
  104. x x
  105. I I
  106. I I
  107. I I
  108. I I
  109. I I
  110. I I
  111. x x
  112. [ x , x ] = { x } [x,x]=\{x\}
  113. x x
  114. I I
  115. I I
  116. x x
  117. I I
  118. j 2 n \frac{j}{2^{n}}
  119. j + 1 2 n \frac{j+1}{2^{n}}
  120. j j
  121. n n
  122. p = 2 p=2
  123. n n
  124. \R n \R^{n}
  125. n n
  126. I = I 1 × I 2 × × I n I=I_{1}\times I_{2}\times\cdots\times I_{n}
  127. n = 2 n=2
  128. n = 3 n=3
  129. I I
  130. I k I_{k}
  131. I k I_{k}
  132. I I
  133. I I
  134. I I
  135. \R n \R^{n}
  136. R R R\oplus R

Interval_(music).html

  1. n = 1200 log 2 ( f 2 f 1 ) . n=1200\cdot\log_{2}\left(\frac{f_{2}}{f_{1}}\right).
  2. D N c = 1 + ( D N 1 - 1 ) + ( D N 2 - 1 ) + + ( D N n - 1 ) , DN_{c}=1+(DN_{1}-1)+(DN_{2}-1)+...+(DN_{n}-1),
  3. D N c = D N 1 + D N 2 + + D N n - ( n - 1 ) , DN_{c}=DN_{1}+DN_{2}+...+DN_{n}-(n-1),

Inverse-square_law.html

  1. Intensity 1 distance 2 \mbox{Intensity}~{}\ \propto\ \frac{1}{\mbox{distance}~{}^{2}}\,
  2. Intensity 1 Intensity 2 = distance 2 2 distance 2 1 \frac{\mbox{Intensity}~{}_{1}}{\mbox{Intensity}~{}_{2}}=\frac{{\mbox{distance}% ~{}_{2}}^{2}}{{\mbox{distance}~{}_{1}}^{2}}
  3. I = P A = P 4 π r 2 . I=\frac{P}{A}=\frac{P}{4\pi r^{2}}.\,
  4. p 1 r p\ \propto\ \frac{1}{r}\,
  5. v v\,
  6. p p\,
  7. v 1 r v\ \propto\frac{1}{r}\ \,
  8. I = p v 1 r 2 . I\ =\ pv\ \propto\ \frac{1}{r^{2}}.\,
  9. I 1 r n - 1 I\propto\frac{1}{r^{n-1}}

Inverse_element.html

  1. S S
  2. * *
  3. e e
  4. ( S , * ) (S,*)
  5. a * b = e a*b=e
  6. a a
  7. b b
  8. b b
  9. a a
  10. x x
  11. y y
  12. x x
  13. y y
  14. S S
  15. S S
  16. ( S , * ) (S,*)
  17. e e
  18. * *
  19. S S
  20. U ( S ) U(S)
  21. S S
  22. \langle
  23. \rangle
  24. x x
  25. - x -x
  26. x x
  27. 1 x \frac{1}{x}
  28. x - 1 x^{-1}
  29. g g
  30. f f
  31. g f g\circ f
  32. f g f\circ g
  33. f f
  34. f f
  35. f - 1 f^{-1}
  36. M M
  37. K K
  38. M M
  39. R R
  40. R R
  41. A : m × n m > n A:m\times n\mid m>n
  42. ( A T A ) - 1 A T A - 1 left A = I n \underbrace{(A^{T}A)^{-1}A^{T}}_{A^{-1}\text{left}}A=I_{n}
  43. A : m × n m < n A:m\times n\mid m<n
  44. A A T ( A A T ) - 1 A - 1 right = I m A\underbrace{A^{T}(AA^{T})^{-1}}_{A^{-1}\text{right}}=I_{m}
  45. A x = b Ax=b
  46. x = ( A T A ) - 1 A T b . x=(A^{T}A)^{-1}A^{T}b.
  47. A : 2 × 3 = [ 1 2 3 4 5 6 ] A:2\times 3=\begin{bmatrix}1&2&3\\ 4&5&6\end{bmatrix}
  48. A A T \displaystyle AA^{T}
  49. A T A = [ 1 4 2 5 3 6 ] [ 1 2 3 4 5 6 ] = [ 17 22 27 22 29 36 27 36 45 ] A^{T}A=\begin{bmatrix}1&4\\ 2&5\\ 3&6\end{bmatrix}\cdot\begin{bmatrix}1&2&3\\ 4&5&6\end{bmatrix}=\begin{bmatrix}17&22&27\\ 22&29&36\\ 27&36&45\end{bmatrix}

Inverse_function.html

  1. f f
  2. x x
  3. y y
  4. f f
  5. y y
  6. x x
  7. f f
  8. X X
  9. Y Y
  10. f f
  11. g g
  12. Y Y
  13. X X
  14. f ( x ) = y g ( y ) = x . f(x)=y\,\,\Leftrightarrow\,\,g(y)=x.
  15. f f
  16. g g
  17. g g
  18. g g
  19. f f
  20. Y Y
  21. y Y y∈Y
  22. x X x∈X
  23. f f
  24. f f
  25. X X
  26. Y Y
  27. y Y y∈Y
  28. X X
  29. f : X Y f\colon X\to Y
  30. f f
  31. X X
  32. Y Y
  33. X X
  34. f f
  35. Y Y
  36. f f
  37. f f
  38. f : X Y f:X→Y
  39. Y Y
  40. X X
  41. Y Y
  42. Y Y
  43. f f
  44. y Y y∈Y
  45. x X x∈X
  46. f f
  47. X X
  48. Y Y
  49. f - 1 ( f ( x ) ) = x f^{-1}\left(\,f(x)\,\right)=x
  50. x X . x\in X.
  51. f - 1 f = id X , f^{-1}\circ f=\mathrm{id}_{X},
  52. X X
  53. f f
  54. n n
  55. x x
  56. f f
  57. f ( x ) f(x)
  58. f f
  59. s i n x sinx
  60. x x
  61. f f
  62. f f
  63. X X
  64. Y Y
  65. Y Y
  66. X X
  67. f f
  68. f : X Y f:X→Y
  69. g : Y X g:Y→X
  70. g f = id X f g = id Y . g\circ f=\mathrm{id}_{X}\Rightarrow f\circ g=\mathrm{id}_{Y}.
  71. f f
  72. ( f - 1 ) - 1 = f . \left(f^{-1}\right)^{-1}=f.
  73. ( g f ) - 1 = f - 1 g - 1 (g\circ f)^{-1}=f^{-1}\circ g^{-1}
  74. g g
  75. f f
  76. f f
  77. g g
  78. g g
  79. f f
  80. f ( x ) = 3 x f(x)=3x
  81. g ( x ) = x + 5 g(x)=x+5
  82. g f g∘f
  83. ( g f ) ( x ) = 3 x + 5 (g\circ f)(x)=3x+5
  84. ( g f ) - 1 ( y ) = 1 3 ( y - 5 ) (g\circ f)^{-1}(y)=\tfrac{1}{3}(y-5)
  85. X X
  86. X X
  87. id X - 1 = id X {\mathrm{id}_{X}}^{-1}=\mathrm{id}_{X}
  88. f : X X f:X→X
  89. f f f∘f
  90. f ( x ) = ( 2 x + 8 ) 3 . f(x)=(2x+8)^{3}.
  91. f f
  92. y = f ( x ) y=f(x)
  93. y y
  94. x x
  95. f ( x ) f(x)
  96. f < s u p > 1 ( y ) f<sup> −1(y)
  97. y = f ( x ) y=f(x)
  98. x x
  99. f f
  100. f ( x ) = ( 2 x + 8 ) 3 f(x)=(2x+8)^{3}
  101. x x
  102. y \displaystyle y
  103. f - 1 ( y ) = y 3 - 8 2 . f^{-1}(y)=\dfrac{\sqrt[3]{y}-8}{2}.
  104. f f
  105. f ( x ) = x - sin x , f(x)=x-\sin x,
  106. f f
  107. f - 1 ( y ) = n = 1 y n 3 n ! lim θ 0 ( d n - 1 d θ n - 1 ( θ θ - sin ( θ ) 3 n ) ) f^{-1}(y)=\displaystyle\sum_{n=1}^{\infty}{\frac{y^{\frac{n}{3}}}{n!}}\lim_{% \theta\to 0}\left(\frac{\mathrm{d}^{\,n-1}}{\mathrm{d}\theta^{\,n-1}}\left(% \frac{\theta}{\sqrt[3]{\theta-\sin(\theta)}}^{n}\right)\right)
  108. f f
  109. y = f - 1 ( x ) y=f^{-1}(x)
  110. x = f ( y ) . x=f(y).
  111. y = f ( x ) y=f(x)
  112. f f
  113. x x
  114. y y
  115. f f
  116. x x
  117. y y
  118. y = x y=x
  119. f f
  120. f ( x ) = x 3 + x f(x)=x^{3}+x
  121. f f
  122. f ( x ) 0 f′(x)≠0
  123. ( f - 1 ) ( y ) = 1 f ( f - 1 ( y ) ) . \left(f^{-1}\right)^{\prime}(y)=\frac{1}{f^{\prime}\left(f^{-1}(y)\right)}.
  124. y = f ( x ) y=f(x)
  125. d x d y = 1 d y / d x . \frac{dx}{dy}=\frac{1}{dy/dx}.
  126. p p
  127. f f
  128. p p
  129. f ( p ) f(p)
  130. f f
  131. p p
  132. f f
  133. F = f ( C ) = 9 5 C + 32 ; F=f(C)=\tfrac{9}{5}C+32;
  134. C = f - 1 ( F ) = 5 9 ( F - 32 ) , C=f^{-1}(F)=\tfrac{5}{9}(F-32),
  135. f - 1 ( f ( C ) ) = f - 1 ( 9 5 C + 32 ) = 5 9 ( ( 9 5 C + 32 ) - 32 ) = C , for every C . f^{-1}\left(\,f(C)\,\right)=f^{-1}\left(\,\tfrac{9}{5}C+32\,\right)=\tfrac{5}{% 9}\left(\left(\,\tfrac{9}{5}C+32\,\right)-32\right)=C\,\text{, for every }C\,% \text{.}
  136. f f
  137. f ( Allan ) = 2005 , f ( Brad ) = 2007 , f ( Cary ) = 2001 f - 1 ( 2005 ) = Allan , f - 1 ( 2007 ) = Brad , f - 1 ( 2001 ) = Cary \begin{aligned}\displaystyle f(\,\text{Allan})&\displaystyle=2005,&% \displaystyle f(\,\text{Brad})&\displaystyle=2007,&\displaystyle f(\,\text{% Cary})&\displaystyle=2001\\ \displaystyle f^{-1}(2005)&\displaystyle=\,\text{Allan},&\displaystyle f^{-1}(% 2007)&\displaystyle=\,\text{Brad},&\displaystyle f^{-1}(2001)&\displaystyle=\,% \text{Cary}\end{aligned}
  138. R R
  139. x x
  140. F F
  141. x x
  142. x x
  143. f f
  144. f f
  145. f ( x ) = x 2 f(x)=x^{2}
  146. x 0 x≥0
  147. f - 1 ( y ) = y . f^{-1}(y)=\sqrt{y}.
  148. x 0 x≤0
  149. y y
  150. f - 1 ( y ) = ± y . f^{-1}(y)=\pm\sqrt{y}.
  151. f f
  152. x \sqrt{x}
  153. x \sqrt{x}
  154. y y
  155. sin ( x + 2 π ) = sin ( x ) \sin(x+2\pi)=\sin(x)
  156. x x
  157. s i n ( x + 2 π n ) = s i n ( x ) sin(x+2πn)=sin(x)
  158. n n
  159. [ π 2 , π 2 ] [−\frac{π}{2},\frac{π}{2}]
  160. π 2 \frac{π}{2}
  161. π 2 \frac{π}{2}
  162. π 2 s i n < s u p > 1 ( x ) π 2 −\frac{π}{2}≤sin<sup>−1(x)≤\frac{π}{2}
  163. f : X Y f:X→Y
  164. f f
  165. f f
  166. g : Y X g:Y→X
  167. g f = id X . g\circ f=\mathrm{id}_{X}.
  168. g g
  169. f ( x ) = y \displaystyle f(x)=y
  170. g ( y ) = x . \displaystyle g(y)=x.
  171. g g
  172. f f
  173. f f
  174. Y Y
  175. f f
  176. f f
  177. f f
  178. f f
  179. h : Y X h:Y→X
  180. f h = id Y . f\circ h=\mathrm{id}_{Y}.
  181. h h
  182. h ( y ) = x \displaystyle h(y)=x
  183. f ( x ) = y . \displaystyle f(x)=y.
  184. h ( y ) h(y)
  185. X X
  186. y y
  187. f f
  188. f f
  189. g g
  190. f f
  191. g g
  192. f f
  193. g g
  194. f f
  195. g g
  196. f f
  197. f : 𝐑 [ 0 , ) f:\mathbf{R}→[0, ∞)
  198. x x
  199. 𝐑 \mathbf{R}
  200. g : [ 0 , ) 𝐑 g:[0, ∞)→\mathbf{R}
  201. g ( x ) = g(x)=
  202. x \sqrt{x}
  203. x 0 x≥0
  204. f ( g ( x ) ) = x f(g(x))=x
  205. x x
  206. [ 0 , ) [0, ∞)
  207. g g
  208. f f
  209. g g
  210. f f
  211. g ( f ( 1 ) ) = 1 1 g(f(−1))=1≠−1
  212. f : X Y f:X→Y
  213. y Y y∈Y
  214. X X
  215. y y
  216. f - 1 ( { y } ) = { x X : f ( x ) = y } . f^{-1}(\{y\})=\left\{x\in X:f(x)=y\right\}.
  217. y y
  218. y y
  219. f f
  220. S S
  221. Y Y
  222. S S
  223. X X
  224. S S
  225. f - 1 ( S ) = { x X : f ( x ) S } . f^{-1}(S)=\left\{x\in X:f(x)\in S\right\}.
  226. f : 𝐑 𝐑 f:\mathbf{R}→\mathbf{R}
  227. f - 1 ( { 1 , 4 , 9 , 16 } ) = { - 4 , - 3 , - 2 , - 1 , 1 , 2 , 3 , 4 } f^{-1}(\left\{1,4,9,16\right\})=\left\{-4,-3,-2,-1,1,2,3,4\right\}
  228. y Y y∈Y
  229. y y
  230. Y Y

Inverse_limit.html

  1. lim i I A i = { a i I A i | a i = f i j ( a j ) for all i j in I } . \underleftarrow{\lim}_{i\in I}A_{i}=\Big\{\vec{a}\in\prod_{i\in I}A_{i}\;\Big|% \;a_{i}=f_{ij}(a_{j})\mbox{ for all }~{}i\leq j\mbox{ in }~{}I\Big\}.
  2. X = lim X i X=\underleftarrow{\lim}X_{i}
  3. lim : C I o p C \underleftarrow{\lim}:C^{I^{op}}\rightarrow C
  4. R [ [ t ] ] \textstyle R[[t]]
  5. R [ t ] / t n R [ t ] \textstyle R[t]/t^{n}R[t]
  6. R [ t ] / t n + j R [ t ] \textstyle R[t]/t^{n+j}R[t]
  7. R [ t ] / t n R [ t ] \textstyle R[t]/t^{n}R[t]
  8. lim : C I C \underleftarrow{\lim}:C^{I}\rightarrow C
  9. lim \underleftarrow{\lim}
  10. lim : 1 Ab I Ab \underleftarrow{\lim}{}^{1}:\operatorname{Ab}^{I}\rightarrow\operatorname{Ab}
  11. 0 A i B i C i 0 0\rightarrow A_{i}\rightarrow B_{i}\rightarrow C_{i}\rightarrow 0
  12. 0 lim A i lim B i lim C i lim A i 1 0\rightarrow\underleftarrow{\lim}A_{i}\rightarrow\underleftarrow{\lim}B_{i}% \rightarrow\underleftarrow{\lim}C_{i}\rightarrow\underleftarrow{\lim}{}^{1}A_{i}
  13. f k j ( A j ) = f k i ( A i ) f_{kj}(A_{j})=f_{ki}(A_{i})
  14. lim A i 1 = 0. \underleftarrow{\lim}{}^{1}A_{i}=0.
  15. lim A i 1 = 𝐙 p / 𝐙 \underleftarrow{\lim}{}^{1}A_{i}=\mathbf{Z}_{p}/\mathbf{Z}
  16. R n lim : C I C . R^{n}\underleftarrow{\lim}:C^{I}\rightarrow C.
  17. lim n R n lim . \underleftarrow{\lim}{}^{n}\cong R^{n}\underleftarrow{\lim}.
  18. d \aleph_{d}

Inverse_transform_sampling.html

  1. u u
  2. x x
  3. p ( X ) p(X)
  4. p ( - < X < x ) u p(-\infty<X<x)\leq u
  5. p ( X ) p(X)
  6. u = 0.5 u=0.5
  7. X 0 X\leq 0
  8. u = 0.975 u=0.975
  9. u = 0.995 u=0.995
  10. u = 0.999999 u=0.999999
  11. u = 0.9999995 u=0.9999995
  12. u = 0.9999999999999997779553951 = 1 - 2 - 52 u=0.9999999999999997779553951=1-2^{-52}
  13. u = 0.9999999999999998889776975 = 1 - 2 - 53 u=0.9999999999999998889776975=1-2^{-53}
  14. X X
  15. F X F_{X}
  16. Y = F X ( X ) Y=F_{X}(X)
  17. Y Y
  18. X X
  19. F X F_{X}
  20. F X - 1 ( Y ) F_{X}^{-1}(Y)
  21. X X
  22. F - 1 ( u ) = inf { x F ( x ) u } ( 0 < u < 1 ) . F^{-1}(u)=\inf\;\{x\mid F(x)\geq u\}\qquad(0<u<1).
  23. F - 1 ( U ) F^{-1}(U)
  24. Pr ( F - 1 ( U ) x ) = Pr ( U F ( x ) ) (applying F , which is monotonic, to both sides) = F ( x ) (because Pr ( U y ) = y , since U is uniform on the unit interval) . \begin{aligned}&\displaystyle\Pr(F^{-1}(U)\leq x)\\ &\displaystyle{}=\Pr(U\leq F(x))&\displaystyle\,\text{(applying }F,\,\text{ % which is monotonic, to both sides)}\\ &\displaystyle{}=F(x)&\displaystyle\,\text{(because }\Pr(U\leq y)=y,\,\text{ % since }U\,\text{ is uniform on the unit interval)}.\end{aligned}
  25. F X - 1 ( u ) F^{-1}_{X}(u)
  26. F X F_{X}

Ionization.html

  1. E i E_{i}
  2. ω \omega
  3. W P P T = | C n * l * | 2 6 π f l m E i ( 2 ( 2 E i ) 3 2 / F ) n * 2 - | m | - 3 / 2 ( 1 + γ ) 2 ) | m / 2 | + 3 / 4 A m ( ω , γ ) e - ( 2 ( 2 E i ) 3 2 / F ) g ( γ ) W_{PPT}=|C_{n^{*}l^{*}}|^{2}\sqrt{\frac{6}{\pi}}f_{lm}E_{i}(2(2E_{i})^{\frac{3% }{2}}/F)^{n^{*2}-|m|-3/2}(1+\gamma)^{2})^{|m/2|+3/4}A_{m}(\omega,\gamma)e^{-(2% (2E_{i})^{\frac{3}{2}}/F)g(\gamma)}
  4. γ = ω F 2 E i \gamma=\frac{\omega F}{\sqrt{2E_{i}}}
  5. n * = 2 E i / Z 2 n^{*}=\sqrt{2E_{i}}/Z^{2}
  6. F F
  7. l * = n * - 1 l^{*}=n^{*}-1
  8. f l m f_{lm}
  9. g ( γ ) g(\gamma)
  10. C n * l * C_{n^{*}l^{*}}
  11. f l m = ( 2 l + 1 ) ( l + | m | ) ! 2 m | m | ! ( l - | m | ) ! f_{lm}=\frac{(2l+1)(l+|m|)^{!}}{2^{m}|m|^{!}(l-|m|)^{!}}
  12. g ( γ ) = 3 2 γ ( 1 + 1 2 γ 2 sinh - 1 ( γ ) - 1 + γ 2 2 γ ) g(\gamma)=\frac{3}{2\gamma}(1+\frac{1}{2\gamma^{2}}\sinh^{-1}(\gamma)-\frac{% \sqrt{1+\gamma^{2}}}{2\gamma})
  13. | C n * l * | 2 = 2 2 n * n * Γ ( n * + l * + 1 ) Γ ( n * l * ) |C_{n^{*}l^{*}}|^{2}=\frac{2^{2n^{*}}}{n^{*}\Gamma(n^{*}+l^{*}+1)\Gamma(n^{*}l% ^{*})}
  14. A m ( ω , γ ) A_{m}(\omega,\gamma)
  15. A m ( ω , γ ) = 4 3 π 1 | m | ! γ 2 1 + γ 2 n > v w m ( 2 γ 1 + γ 2 ( n - v ) e - ( n - v ) α ( γ ) ) A_{m}(\omega,\gamma)=\frac{4}{3\pi}\frac{1}{|m|^{!}}\frac{\gamma^{2}}{1+\gamma% ^{2}}\sum_{n>v}^{\infty}w_{m}(\sqrt{\frac{2\gamma}{\sqrt{1+\gamma^{2}}}(n-v)}e% ^{-(n-v)\alpha(\gamma)})
  16. w m ( x ) = e - x 2 0 x ( x 2 - y 2 ) m e y 2 d y w_{m}(x)=e^{-x^{2}}\int_{0}^{x}(x^{2}-y^{2})^{m}e^{y^{2}}\,dy
  17. α ( γ ) = 2 ( sinh - 1 ( γ ) - γ 1 + γ 2 ) \alpha(\gamma)=2(\sinh^{-1}(\gamma)-\frac{\gamma}{\sqrt{1+\gamma^{2}}})
  18. v = E i ω ( 1 + 2 γ 2 ) v=\frac{E_{i}}{\omega}(1+\frac{2}{\gamma^{2}})
  19. γ \gamma
  20. W A D K = | C n * l * | 2 6 π f l m E i ( 2 ( 2 E i ) 3 2 / F ) n * 2 - | m | - 3 / 2 e - ( 2 ( 2 E i ) 3 2 / 3 F ) W_{ADK}=|C_{n^{*}l^{*}}|^{2}\sqrt{\frac{6}{\pi}}f_{lm}E_{i}(2(2E_{i})^{\frac{3% }{2}}/F)^{n^{*2}-|m|-3/2}e^{-(2(2E_{i})^{\frac{3}{2}}/3F)}
  21. W P P T W_{PPT}
  22. W K R A = n = N n = 2 π ω 2 p ( n - n osc ) 2 d Ω | F T ( I K A R Ψ ( 𝐫 ) ) | 2 J n 2 ( n f , n osc 2 ) W_{KRA}=\sum_{n=N}^{n=\infty}2\pi\omega^{2}p(n-n_{\mathrm{osc}})^{2}\int d% \Omega|FT(I_{KAR}\Psi(\mathbf{r}))|^{2}J_{n}^{2}(n_{f},\frac{n_{\mathrm{osc}}}% {2})
  23. N = [ n i + n osc ] N=[n_{i}+n_{\mathrm{osc}}]
  24. n i = E i / ω n_{i}=E_{i}/\omega
  25. n osc = U p / ω n_{\mathrm{osc}}=U_{p}/\omega
  26. U p U_{p}
  27. J n ( u , v ) J_{n}(u,v)
  28. p = 2 ω ( n - n osc - n i ) p=\sqrt{2\omega(n-n_{\mathrm{osc}}-n_{i})}
  29. n f = 2 n osc / ω p c o s ( θ ) n_{f}=2\sqrt{n_{\mathrm{osc}}/\omega}pcos(\theta)
  30. θ \theta
  31. I K A R = ( 2 Z 2 n 2 F r ) n I_{KAR}=(\frac{2Z^{2}}{n^{2}Fr})^{n}
  32. m m
  33. I r I_{r}
  34. V m V_{m}
  35. Γ ( t ) = Γ m I ( t ) m / 2 \Gamma(t)=\Gamma_{m}I(t)^{m/2}
  36. P g P_{g}
  37. P g = exp ( - 2 π W m 2 d W / d t ) P_{g}=\exp\left(-\frac{2\pi W_{m}^{2}}{dW/dt}\right)
  38. W W
  39. d W / d t = 0 dW/dt=0
  40. A + L - > A + + L - > A + + A+L->A^{+}+L->A^{++}
  41. A + L - > A + + A+L->A^{++}
  42. W N S ( A n + ) = i = 1 n - 1 α n ( λ ) W A D K ( A i + ) W_{NS}(A^{n+})=\sum_{i=1}^{n-1}\alpha_{n}(\lambda)W_{ADK}(A^{i+})
  43. W A D K ( A i + ) W_{ADK}(A^{i+})
  44. α n ( λ ) \alpha_{n}(\lambda)
  45. U p U_{p}
  46. U p U_{p}
  47. d 2 x d t 2 = F sin ω t \frac{{\rm d}^{2}x}{{\rm d}t^{2}}=F\sin\omega t
  48. x ( t ) = - F ω 2 sin ω t = - a sin ω t x(t)=-\frac{F}{\omega^{2}}\sin\omega t=-a\sin\omega t
  49. x x + a sin ω t x\to x+a\sin\omega t
  50. V ( x ) = - 1 | x + a sin ω t | V(x)=-\frac{1}{|x+a\sin\omega t|}
  51. V A V = - 1 2 | x + a / 2 | - 1 2 | x - a / 2 | V_{AV}=-\frac{1}{2|x+a/\sqrt{2}|}-\frac{1}{2|x-a/\sqrt{2}|}
  52. x x
  53. x = 0 x=0
  54. x ( t ) = 0 x(t)=0
  55. d x d t = - F ω cos ω t \frac{{\rm d}x}{{\rm d}t}=-\frac{F}{\omega}\cos\omega t
  56. r / ( a ω ) r/(a\omega)

Ionization_energy.html

  1. V = Z e a V=\frac{Ze}{a}\,\!
  2. E = e V = Z e 2 a E=eV=\frac{Ze^{2}}{a}\,\!
  3. L = | 𝐫 × 𝐩 | = r m v = n L=|\mathbf{r}\times\mathbf{p}|=rmv=n\hbar
  4. E = T + U = p 2 2 m e - Z e 2 r = m e v 2 2 - Z e 2 r E=T+U=\frac{p^{2}}{2m_{e}}-\frac{Ze^{2}}{r}=\frac{m_{e}v^{2}}{2}-\frac{Ze^{2}}% {r}
  5. T = Z e 2 2 r T=\frac{Ze^{2}}{2r}
  6. n 2 2 r m e = Z e 2 \frac{n^{2}\hbar^{2}}{rm_{e}}=Ze^{2}
  7. r ( n ) = n 2 2 Z m e e 2 r(n)=\frac{n^{2}\hbar^{2}}{Zm_{e}e^{2}}
  8. E = - Z e 2 2 r E=-\frac{Ze^{2}}{2r}
  9. 2 m e 2 \frac{\hbar^{2}}{me^{2}}
  10. E = - 1 n 2 Z 2 e 2 2 a 0 = - Z 2 13.6 e V n 2 E=-\frac{1}{n^{2}}\frac{Z^{2}e^{2}}{2a_{0}}=-\frac{Z^{2}13.6eV}{n^{2}}
  11. Z - n + 1 Z-n+1
  12. Z - n Z-n

Ionosphere.html

  1. f critical = 9 × N f_{\,\text{critical}}=9\times\sqrt{N}
  2. f muf = f critical sin α f\text{muf}=\frac{f\text{critical}}{\sin\alpha}
  3. α \alpha

Irreducible_element.html

  1. a a
  2. R R
  3. a | b c a|bc
  4. b b
  5. c c
  6. R , R,
  7. a | b a|b
  8. a | c . ) a|c.)
  9. D D
  10. x x
  11. D D
  12. x x
  13. D D
  14. 𝐙 [ - 5 ] , \mathbf{Z}[\sqrt{-5}],
  15. 3 ( 2 + - 5 ) ( 2 - - 5 ) = 9 , 3\mid\left(2+\sqrt{-5}\right)\left(2-\sqrt{-5}\right)=9,
  16. 3 3
  17. p p
  18. p = a b . p=ab.
  19. p | a b p | a p|ab\Rightarrow p|a
  20. p | b . p|b.
  21. p | a a = p c , p|a\Rightarrow a=pc,
  22. p = a b = p c b p ( 1 - c b ) = 0. p=ab=pcb\Rightarrow p(1-cb)=0.
  23. R R
  24. c b = 1. cb=1.
  25. b b
  26. p p

Irreducible_fraction.html

  1. 120 90 = 12 9 = 4 3 . \frac{120}{90}=\frac{12}{9}=\frac{4}{3}\,.
  2. 120 90 = 4 3 . \frac{120}{90}=\frac{4}{3}\,.
  3. 2 3 = - 2 - 3 \tfrac{2}{3}=\tfrac{-2}{-3}
  4. a b = c d \tfrac{a}{b}=\tfrac{c}{d}
  5. a a
  6. b b
  7. a a
  8. c c
  9. a = c a=c
  10. b = d b=d
  11. a b \tfrac{a}{b}
  12. a b \tfrac{a}{b}
  13. 2 b - a a - b \tfrac{2b-a}{a-b}
  14. a b \tfrac{a}{b}

ISO_216.html

  1. 1 : 2 1.414 1:\sqrt{2}\approx 1.414
  2. x / y x/y
  3. y / ( x / 2 ) y/(x/2)
  4. x / y = y / ( x / 2 ) \ x/y=y/(x/2)
  5. x / y = 2 x/y=\sqrt{2}
  6. 1 : 2 1:\sqrt{2}
  7. n n
  8. a n = 2 1 4 - n 2 a_{n}=2^{\frac{1}{4}-\frac{n}{2}}
  9. n n
  10. a n a_{n}
  11. a n + 1 a_{n+1}
  12. 2 - n m 2 2^{-n}m^{2}
  13. n n
  14. 1000 / ( 2 2 n - 1 4 ) + 0.2 \left\lfloor 1000/\left(2^{\frac{2n-1}{4}}\right)+0.2\right\rfloor
  15. 1 : 2 1:\sqrt{2}
  16. 1.5 1.22 \sqrt{1.5}\approx 1.22
  17. 2 4 1.19 \sqrt[4]{2}\approx 1.19
  18. n n
  19. 1000 / ( 2 n - 1 2 ) + 0.2 \left\lfloor 1000/\left(2^{\frac{n-1}{2}}\right)+0.2\right\rfloor
  20. 2 8 \sqrt[8]{2}
  21. n n
  22. 1000 / ( 2 4 n - 3 8 ) + 0.2 \left\lfloor 1000/\left(2^{\frac{4n-3}{8}}\right)+0.2\right\rfloor
  23. 1 : 2 1:\sqrt{2}
  24. 1 : 2 1:\sqrt{2}
  25. 1 : ( 1 + 2 ) 1:(1+\sqrt{2})

Isoelectric_point.html

  1. pI = p K a1 + p K a2 2 \mathrm{pI}=\frac{\mathrm{p}K_{\mathrm{a1}}+\mathrm{p}K_{\mathrm{a2}}}{2}
  2. p K - - p K + = Δ p K = log [ MOH ] 2 [ MOH ] 2 + [ MO - ] \mathrm{p}K^{-}-\mathrm{p}K^{+}=\Delta\mathrm{p}K=\log{\frac{\left[\mathrm{MOH% }\right]^{2}}{\left[\mathrm{MOH}{{}_{2}^{+}}\right]\left[\mathrm{MO}^{-}\right% ]}}

Isomorphism.html

  1. + \mathbb{R}^{+}
  2. \mathbb{R}
  3. log : + \log\colon\mathbb{R}^{+}\to\mathbb{R}
  4. log ( x y ) = log x + log y \log(xy)=\log x+\log y
  5. x , y + x,y\in\mathbb{R}^{+}
  6. exp : + \exp\colon\mathbb{R}\to\mathbb{R}^{+}
  7. exp ( x + y ) = ( exp x ) ( exp y ) \exp(x+y)=(\exp x)(\exp y)
  8. x , y x,y\in\mathbb{R}
  9. log exp x = x \log\exp x=x
  10. exp log y = y \exp\log y=y
  11. log \log
  12. exp \exp
  13. log \log
  14. log \log
  15. log \log
  16. ( 6 , + ) (\mathbb{Z}_{6},+)
  17. ( 2 × 3 , + ) (\mathbb{Z}_{2}\times\mathbb{Z}_{3},+)
  18. m \mathbb{Z}_{m}
  19. n \mathbb{Z}_{n}
  20. ( m n , + ) (\mathbb{Z}_{mn},+)
  21. S ( f ( u ) , f ( v ) ) R ( u , v ) \operatorname{S}(f(u),f(v))\iff\operatorname{R}(u,v)
  22. \scriptstyle\sqsubseteq
  23. f ( u ) f ( v ) u v . f(u)\sqsubseteq f(v)\iff u\leq v.
  24. A = { x x 2 < 2 } A=\{x\in\mathbb{Z}\mid x^{2}<2\}
  25. B = { - 1 , 0 , 1 } B=\{-1,0,1\}\,
  26. A 1 , B 2 , C 3 , \,\text{A}\mapsto 1,\,\text{B}\mapsto 2,\,\text{C}\mapsto 3,
  27. A 3 , B 2 , C 1 , \,\text{A}\mapsto 3,\,\text{B}\mapsto 2,\,\text{C}\mapsto 1,
  28. A 1 , B 2 , C 3 \,\text{A}\mapsto 1,\,\text{B}\mapsto 2,\,\text{C}\mapsto 3
  29. V V * \scriptstyle V\,\overset{\sim}{\to}\,V^{*}
  30. v ϕ v V * such that ϕ v ( u ) = v T u v\ \overset{\sim}{\mapsto}\ \phi_{v}\in V^{*}\quad\,\text{such that}\quad\phi_% {v}(u)=v^{\mathrm{T}}u
  31. v x v V * * such that x v ( ϕ ) = ϕ ( v ) v\ \overset{\sim}{\mapsto}\ x_{v}\in V^{**}\quad\,\text{such that}\quad x_{v}(% \phi)=\phi(v)
  32. V V * * \scriptstyle V\,\overset{\sim}{\to}\,V^{**}
  33. S 2 := { ( x , y , z ) 3 x 2 + y 2 + z 2 = 1 } S^{2}:=\{(x,y,z)\in\mathbb{R}^{3}\mid x^{2}+y^{2}+z^{2}=1\}
  34. ^ \widehat{\mathbb{C}}
  35. 𝐏 1 := ( 2 { ( 0 , 0 ) } ) / ( * ) \mathbf{P}_{\mathbb{C}}^{1}:=(\mathbb{C}^{2}\setminus\{(0,0)\})/(\mathbb{C}^{*})
  36. 𝐂 𝐑 1 𝐑 i = 𝐑 2 \mathbf{C}\cong\mathbf{R}\cdot 1\oplus\mathbf{R}\cdot i=\mathbf{R}^{2}
  37. i ; i;
  38. ( - i ) , (-i),
  39. 3 ! = 6 3!=6
  40. Iso ( A , B ) , \operatorname{Iso}(A,B),
  41. Aut ( A ) \operatorname{Aut}(A)

Isomorphism_class.html

  1. X X
  2. p p
  3. π 1 ( X , p ) \pi_{1}(X,p)
  4. π 1 ( X ) \pi_{1}(X)
  5. X X
  6. π 1 ( X , p ) \pi_{1}(X,p)
  7. π 1 ( X , p ) \pi_{1}(X,p)

Isomorphism_theorem.html

  1. K K
  2. G G
  3. N K G N\subseteq K\subseteq G
  4. K / N K/N
  5. G / N G/N
  6. G / N G/N
  7. K / N K/N
  8. K K
  9. G G
  10. N K G N\subseteq K\subseteq G
  11. K K
  12. G G
  13. N K G N\subseteq K\subseteq G
  14. K / N K/N
  15. G / N G/N
  16. G / N G/N
  17. K / N K/N
  18. K K
  19. G G
  20. N K G N\subseteq K\subseteq G
  21. K K
  22. G G
  23. N K G N\subseteq K\subseteq G
  24. ( G / N ) / ( K / N ) (G/N)/(K/N)
  25. G / K G/K
  26. ι π \iota\circ\pi
  27. ker f \ker\,f
  28. κ : ker f G \kappa:\ker\,f\rightarrow G
  29. ker f \ker\,f
  30. G / ker f G/\ker\,f
  31. G / ker f G/\ker\,f
  32. im κ \operatorname{im}\,\kappa
  33. im σ \operatorname{im}\,\sigma
  34. ρ : G ker f \rho:G\rightarrow\ker\,f
  35. ρ κ = id ker f \rho\circ\kappa=\operatorname{id}_{\ker\,f}
  36. im κ × im σ \operatorname{im}\,\kappa\times\operatorname{im}\,\sigma
  37. im κ im σ \operatorname{im}\,\kappa\oplus\operatorname{im}\,\sigma
  38. 0 G / ker f H coker f 0 0\rightarrow G/\ker\,f\rightarrow H\rightarrow\operatorname{coker}\,f\rightarrow 0
  39. A A
  40. R R
  41. B A R B\subseteq A\subseteq R
  42. A / B A/B
  43. R / B R/B
  44. R / B R/B
  45. A / B A/B
  46. A A
  47. R R
  48. B A R B\subseteq A\subseteq R
  49. A A
  50. R R
  51. B A R B\subseteq A\subseteq R
  52. A / B A/B
  53. R / B R/B
  54. R / B R/B
  55. A / B A/B
  56. A A
  57. R R
  58. B A R B\subseteq A\subseteq R
  59. A A
  60. R R
  61. B A R B\subseteq A\subseteq R
  62. ( R / B ) / ( A / B ) (R/B)/(A/B)
  63. R / A R/A
  64. S S
  65. M M
  66. T S M T\subseteq S\subseteq M
  67. S / T S/T
  68. M / T M/T
  69. M / T M/T
  70. S / T S/T
  71. S S
  72. M M
  73. T S M T\subseteq S\subseteq M
  74. S S
  75. M M
  76. T S M T\subseteq S\subseteq M
  77. ( M / T ) / ( S / T ) (M/T)/(S/T)
  78. M / S M/S
  79. A A
  80. Φ \Phi
  81. A × A A\times A
  82. A / Φ A/\Phi
  83. Φ \Phi
  84. A × A A\times A
  85. f : A B f:A\rightarrow B
  86. f f
  87. B B
  88. Φ : f ( x ) = f ( y ) \Phi:f(x)=f(y)
  89. A A
  90. A / Φ \ A/\Phi
  91. im f \,\text{im}\ f
  92. A A
  93. B B
  94. A A
  95. Φ \Phi
  96. A A
  97. Φ B = Φ ( B × B ) \Phi_{B}=\Phi\cap(B\times B)
  98. Φ \Phi
  99. B B
  100. [ B ] Φ = { K A / Φ : K B } [B]^{\Phi}=\{K\in A/\Phi:K\cap B\neq\emptyset\}
  101. B B
  102. Φ B \Phi_{B}
  103. B B
  104. [ B ] Φ \ [B]^{\Phi}
  105. A / Φ A/\Phi
  106. [ B ] Φ [B]^{\Phi}
  107. B / Φ B B/\Phi_{B}
  108. A A
  109. Φ , Ψ \Phi,\Psi
  110. A A
  111. Ψ Φ \Psi\subseteq\Phi
  112. Φ / Ψ = { ( [ a ] Ψ , [ a ′′ ] Ψ ) : ( a , a ′′ ) Φ } = [ ] Ψ Φ [ ] Ψ - 1 \Phi/\Psi=\{([a^{\prime}]_{\Psi},[a^{\prime\prime}]_{\Psi}):(a^{\prime},a^{% \prime\prime})\in\Phi\}=[\ ]_{\Psi}\circ\Phi\circ[\ ]_{\Psi}^{-1}
  113. A / Ψ A/\Psi
  114. A / Φ A/\Phi
  115. ( A / Ψ ) / ( Φ / Ψ ) (A/\Psi)/(\Phi/\Psi)

Isotope_separation.html

  1. π \pi

Isotropy.html

  1. 𝒢 \mathcal{G}
  2. G 𝒢 G\in\mathcal{G}
  3. 𝒢 ( G , G ) \mathcal{G}(G,G)
  4. G G
  5. G G

IS–LM_model.html

  1. Y = C ( Y - T ( Y ) ) + I ( r ) + G + N X ( Y ) , Y=C\left({Y}-{T(Y)}\right)+I\left({r}\right)+G+NX(Y),
  2. C ( Y - T ( Y ) ) C(Y-T(Y))
  3. I ( r ) I(r)
  4. M / P = L ( i , Y ) M/P=L(i,Y)

Iterative_method.html

  1. A x = b A{x}={b}

Jacob_Bernoulli.html

  1. 1 n \sum{\frac{1}{n}}
  2. 1 n 2 \sum{\frac{1}{n^{2}}}
  3. y = p ( x ) y + q ( x ) y n . y^{\prime}=p(x)y+q(x)y^{n}.
  4. e e
  5. lim n ( 1 + 1 n ) n \lim_{n\to\infty}\left(1+\frac{1}{n}\right)^{n}
  6. n n
  7. n n
  8. n n
  9. e e
  10. R R
  11. e e
  12. R R

Jacobson_radical.html

  1. J ( R ) J(R)
  2. x R x\in R
  3. 1 + R x R 1+RxR
  4. J ( R ) = { x R 1 + R x R R × } J(R)=\{\,x\in R\,\mid\,1+RxR\subset R^{\times}\,\}
  5. { 0 } = T 0 T 1 T k = R \left\{0\right\}=T_{0}\subseteq T_{1}\subseteq\cdots\subseteq T_{k}=R
  6. ( J ( R ) ) k = 0 \left(J\left(R\right)\right)^{k}=0
  7. T u / T u - 1 T_{u}/T_{u-1}
  8. ( T u / T u - 1 ) J ( R ) = 0 \left(T_{u}/T_{u-1}\right)\cdot J\left(R\right)=0
  9. T u J ( R ) T u - 1 T_{u}\cdot J\left(R\right)\subseteq T_{u-1}
  10. T u ( J ( R ) ) i T u - i T_{u}\cdot\left(J\left(R\right)\right)^{i}\subseteq T_{u-i}

Jerk_(physics).html

  1. j ( t ) = d a ( t ) d t = a ˙ ( t ) = d 2 v ( t ) d t 2 = v ¨ ( t ) = d 3 r ( t ) d t 3 = r ( t ) \vec{j}(t)=\frac{\mathrm{d}\vec{a}(t)}{\mathrm{d}t}=\dot{\vec{a}}(t)=\frac{% \mathrm{d}^{2}\vec{v}(t)}{\mathrm{d}t^{2}}=\ddot{\vec{v}}(t)=\frac{\mathrm{d}^% {3}\vec{r}(t)}{\mathrm{d}t^{3}}=\overset{...}{\vec{r}}(t)
  2. a \vec{a}
  3. v \vec{v}
  4. r \vec{r}
  5. t \mathit{t}
  6. j j
  7. a ˙ , v ¨ , r \dot{a},\;\ddot{v},\;\overset{...}{r}
  8. J ( x , x ¨ , x ˙ , x ) = 0 J\left(\overset{...}{x},\ddot{x},\dot{x},x\right)=0
  9. x x
  10. F F
  11. k r k_{r}
  12. F = - k r x F=-k_{r}x\,
  13. v v
  14. F D F_{D}
  15. v v
  16. v 2 v^{2}
  17. F D = 1 2 ρ v 2 C D A F_{D}\,=\,\tfrac{1}{2}\,\rho\,v^{2}\,C_{D}\,A
  18. ρ \rho
  19. v v
  20. A A
  21. C D C_{D}
  22. a a
  23. a a
  24. F = m a F=m\cdot a
  25. F F
  26. m m
  27. r r
  28. v 2 / r v^{2}/r
  29. v 2 / r v^{2}/r
  30. θ \theta
  31. ω ( t ) = θ ˙ ( t ) = d θ ( t ) d t \omega(t)=\dot{\theta}(t)=\frac{\mathrm{d}\theta(t)}{\mathrm{d}t}
  32. θ ( t ) \theta(t)
  33. α ( t ) = ω ˙ ( t ) = d ω ( t ) d t \alpha(t)=\dot{\omega}(t)=\frac{\mathrm{d}\omega(t)}{\mathrm{d}t}
  34. ω ( t ) \omega(t)
  35. α ( t ) \alpha(t)
  36. ζ ( t ) \zeta(t)
  37. ζ ( t ) = α ˙ ( t ) = ω ¨ ( t ) = θ ( t ) \zeta(t)=\dot{\alpha}(t)=\ddot{\omega}(t)=\overset{...}{\theta}(t)
  38. Ω ( t ) \vec{\Omega}(t)
  39. v ( t ) \vec{v}(t)
  40. α ( t ) = d d t Ω ( t ) = Ω ˙ ( t ) \vec{\alpha}(t)=\frac{\mathrm{d}}{\mathrm{d}t}\vec{\Omega}(t)=\dot{\vec{\Omega% }}(t)
  41. ζ ( t ) = d d t α ( t ) = α ˙ ( t ) \vec{\zeta}(t)=\frac{\mathrm{d}}{\mathrm{d}t}\vec{\alpha}(t)=\dot{\vec{\alpha}% }(t)
  42. θ \theta
  43. α \alpha
  44. ζ \zeta

Jet_engine.html

  1. F N = m ˙ g 0 I s p - v a c - A e p F_{N}=\dot{m}\,g_{0}\,I_{sp-vac}-A_{e}\,p\;
  2. F N F_{N}
  3. I s p ( v a c ) I_{sp(vac)}
  4. g 0 g_{0}
  5. m ˙ \dot{m}
  6. A e A_{e}
  7. p p
  8. F N = ( m ˙ a i r + m ˙ f u e l ) v e - m ˙ a i r v F_{N}=(\dot{m}_{air}+\dot{m}_{fuel})v_{e}-\dot{m}_{air}v
  9. F N = m ˙ a i r ( v e - v ) F_{N}=\dot{m}_{air}(v_{e}-v)
  10. η \eta
  11. η p \eta_{p}
  12. η v e \eta_{v_{e}}
  13. η \eta
  14. η = η p η v e \eta=\eta_{p}\eta_{v_{e}}
  15. v v
  16. v e v_{e}
  17. η p = 2 1 + v e v \eta_{p}=\frac{2}{1+\frac{v_{e}}{v}}
  18. η p = 2 ( v v e ) 1 + ( v v e ) 2 \eta_{p}=\frac{2\,(\frac{v}{v_{e}})}{1+(\frac{v}{v_{e}})^{2}}
  19. F N = m ˙ e v e - m ˙ o v o + B P R ( m ˙ c v f ) F_{N}=\dot{m}_{e}v_{e}-\dot{m}_{o}v_{o}+BPR\,(\dot{m}_{c}v_{f})

JFET.html

  1. q N d μ n qN_{d}\mu_{n}
  2. I D = b W L q N d μ n V D S I_{D}=\frac{bW}{L}qN_{d}\mu_{n}V_{DS}
  3. I D = b W L q N d μ n V D S = a W L q N d μ n ( 1 - V G S V P ) V D S I_{D}=\frac{bW}{L}qN_{d}{{\mu}_{n}}V_{DS}=\frac{aW}{L}qN_{d}{{\mu}_{n}}\left(1% -\sqrt{\frac{V_{GS}}{V_{P}}}\right)V_{DS}
  4. I D S S I_{DSS}
  5. I D = 2 I D S S V P 2 ( V G S - V P - V D S 2 ) V D S I_{D}=\frac{2I_{DSS}}{V_{P}^{2}}\left(V_{GS}-V_{P}-\frac{V_{DS}}{2}\right)V_{DS}
  6. I D S = I D S S ( 1 - V G S V P ) 2 I_{DS}=I_{DSS}\left(1-\frac{V_{GS}}{V_{P}}\right)^{2}
  7. b = a ( 1 - V G S V P ) b=a\left(1-\sqrt{\frac{V_{GS}}{V_{P}}}\right)

John_von_Neumann.html

  1. ( A B ) ( B A ) (A\land B)\neq(B\land A)
  2. P ( Q R ) = ( P Q ) ( P R ) P\lor(Q\land R)=(P\lor Q)\land(P\lor R)
  3. P ( Q R ) = ( P Q ) ( P R ) P\land(Q\lor R)=(P\land Q)\lor(P\land R)
  4. A ( B C ) = A 1 = A A\land(B\lor C)=A\land 1=A
  5. ( A B ) ( A C ) = 0 0 = 0 (A\land B)\lor(A\land C)=0\lor 0=0

Jones_calculus.html

  1. ( E x ( t ) E y ( t ) 0 ) = ( E 0 x e i ( k z - ω t + ϕ x ) E 0 y e i ( k z - ω t + ϕ y ) 0 ) = ( E 0 x e i ϕ x E 0 y e i ϕ y 0 ) e i ( k z - ω t ) \begin{pmatrix}E_{x}(t)\\ E_{y}(t)\\ 0\end{pmatrix}=\begin{pmatrix}E_{0x}e^{i(kz-\omega t+\phi_{x})}\\ E_{0y}e^{i(kz-\omega t+\phi_{y})}\\ 0\end{pmatrix}=\begin{pmatrix}E_{0x}e^{i\phi_{x}}\\ E_{0y}e^{i\phi_{y}}\\ 0\end{pmatrix}e^{i(kz-\omega t)}
  2. i i
  3. i 2 = - 1 i^{2}=-1
  4. ( E 0 x e i ϕ x E 0 y e i ϕ y ) . \begin{pmatrix}E_{0x}e^{i\phi_{x}}\\ E_{0y}e^{i\phi_{y}}\end{pmatrix}\;.
  5. ϕ = k z - ω t \phi=kz-\omega t
  6. ϕ x \phi_{x}
  7. ϕ y \phi_{y}
  8. i i
  9. = e i π / 2 =e^{i\pi/2}
  10. π / 2 \pi/2
  11. = e 0 =e^{0}
  12. ϕ = ω t - k z \phi=\omega t-kz
  13. ( 1 0 ) \begin{pmatrix}1\\ 0\end{pmatrix}
  14. | H |H\rangle
  15. ( 0 1 ) \begin{pmatrix}0\\ 1\end{pmatrix}
  16. | V |V\rangle
  17. 1 2 ( 1 1 ) \frac{1}{\sqrt{2}}\begin{pmatrix}1\\ 1\end{pmatrix}
  18. | D = 1 2 ( | H + | V ) |D\rangle=\frac{1}{\sqrt{2}}(|H\rangle+|V\rangle)
  19. 1 2 ( 1 - 1 ) \frac{1}{\sqrt{2}}\begin{pmatrix}1\\ -1\end{pmatrix}
  20. | A = 1 2 ( | H - | V ) |A\rangle=\frac{1}{\sqrt{2}}(|H\rangle-|V\rangle)
  21. 1 2 ( 1 - i ) \frac{1}{\sqrt{2}}\begin{pmatrix}1\\ -i\end{pmatrix}
  22. | R = 1 2 ( | H - i | V ) |R\rangle=\frac{1}{\sqrt{2}}(|H\rangle-i|V\rangle)
  23. 1 2 ( 1 + i ) \frac{1}{\sqrt{2}}\begin{pmatrix}1\\ +i\end{pmatrix}
  24. | L = 1 2 ( | H + i | V ) |L\rangle=\frac{1}{\sqrt{2}}(|H\rangle+i|V\rangle)
  25. | ψ |\psi\rangle
  26. | 0 |0\rangle
  27. | 1 |1\rangle
  28. | 0 |0\rangle
  29. | H |H\rangle
  30. | 1 |1\rangle
  31. | V |V\rangle
  32. | H |H\rangle
  33. | V |V\rangle
  34. | D |D\rangle
  35. | A |A\rangle
  36. | R |R\rangle
  37. | L |L\rangle
  38. | R |R\rangle
  39. | L |L\rangle
  40. | H , | D , | V , | A |H\rangle,|D\rangle,|V\rangle,|A\rangle
  41. ( 1 0 0 0 ) \begin{pmatrix}1&0\\ 0&0\end{pmatrix}
  42. ( 0 0 0 1 ) \begin{pmatrix}0&0\\ 0&1\end{pmatrix}
  43. 1 2 ( 1 ± 1 ± 1 1 ) \frac{1}{2}\begin{pmatrix}1&\pm 1\\ \pm 1&1\end{pmatrix}
  44. 1 2 ( 1 i - i 1 ) \frac{1}{2}\begin{pmatrix}1&i\\ -i&1\end{pmatrix}
  45. 1 2 ( 1 - i i 1 ) \frac{1}{2}\begin{pmatrix}1&-i\\ i&1\end{pmatrix}
  46. ( e i ϕ x 0 0 e i ϕ y ) \begin{pmatrix}e^{i\phi_{x}}&0\\ 0&e^{i\phi_{y}}\end{pmatrix}
  47. ϕ x \phi_{x}
  48. ϕ y \phi_{y}
  49. x x
  50. y y
  51. ϕ = k z - ω t \phi=kz-\omega t
  52. ϵ = ϕ y - ϕ x \epsilon=\phi_{y}-\phi_{x}
  53. ϵ \epsilon
  54. ϕ y \phi_{y}
  55. ϕ x \phi_{x}
  56. E y E_{y}
  57. E x E_{x}
  58. E x E_{x}
  59. E y E_{y}
  60. ϵ < 0 \epsilon<0
  61. E y E_{y}
  62. E x E_{x}
  63. E x E_{x}
  64. E y E_{y}
  65. ϕ x < ϕ y \phi_{x}<\phi_{y}
  66. ϕ y = ϕ x + π / 2 \phi_{y}=\phi_{x}+\pi/2
  67. ϕ = ω t - k z \phi=\omega t-kz
  68. ϵ = ϕ x - ϕ y \epsilon=\phi_{x}-\phi_{y}
  69. ϵ > 0 \epsilon>0
  70. E y E_{y}
  71. E x E_{x}
  72. E x E_{x}
  73. E y E_{y}
  74. e i π / 4 ( 1 0 0 - i ) e^{i\pi/4}\begin{pmatrix}1&0\\ 0&-i\end{pmatrix}
  75. e i π / 4 ( 1 0 0 i ) e^{i\pi/4}\begin{pmatrix}1&0\\ 0&i\end{pmatrix}
  76. θ \theta
  77. ( cos 2 θ sin 2 θ sin 2 θ - cos 2 θ ) \begin{pmatrix}\cos 2\theta&\sin 2\theta\\ \sin 2\theta&-\cos 2\theta\end{pmatrix}
  78. ( e i ϕ x cos 2 θ + e i ϕ y sin 2 θ ( e i ϕ x - e i ϕ y ) e - i ϕ cos θ sin θ ( e i ϕ x - e i ϕ y ) e i ϕ cos θ sin θ e i ϕ x sin 2 θ + e i ϕ y cos 2 θ ) \begin{pmatrix}e^{i\phi_{x}}\cos^{2}\theta+e^{i\phi_{y}}\sin^{2}\theta&(e^{i% \phi_{x}}-e^{i\phi_{y}})e^{-i\phi}\cos\theta\sin\theta\\ (e^{i\phi_{x}}-e^{i\phi_{y}})e^{i\phi}\cos\theta\sin\theta&e^{i\phi_{x}}\sin^{% 2}\theta+e^{i\phi_{y}}\cos^{2}\theta\end{pmatrix}
  79. ϕ y - ϕ x \phi_{y}-\phi_{x}
  80. θ \theta
  81. ϕ \phi
  82. ϕ \phi
  83. ϕ \phi
  84. π \pi
  85. θ \theta
  86. π \pi
  87. ϕ \phi
  88. π \pi
  89. π \pi
  90. M ( θ ) = R ( θ ) M R ( - θ ) , M(\theta)=R(\theta)\,M\,R(-\theta),
  91. R ( θ ) = ( cos θ - sin θ sin θ cos θ ) . R(\theta)=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}.
  92. R ( θ ) = ( r t t r ) R(\theta)=\begin{pmatrix}r&t^{\prime}\\ t&r^{\prime}\end{pmatrix}
  93. θ t - θ r + θ t’ - θ r’ = ± π \theta\text{t}-\theta\text{r}+\theta\text{t'}-\theta\text{r'}=\pm\pi
  94. r * t + t * r = 0. r^{*}t^{\prime}+t^{*}r^{\prime}=0.

Joseph_Stefan.html

  1. j = σ T 4 j^{\star}=\sigma T^{4}
  2. j = 0 ( d j d λ ) d λ j^{\star}=\int_{0}^{\infty}\left({dj^{\star}\over d\lambda}\right)d\lambda

Joseph_Stiglitz.html

  1. u h ( x h , z h ) u^{h}(x^{h},z^{h})
  2. x h x^{h}
  3. z h z^{h}
  4. x 1 h + q x ¯ h I h + a h f π f x^{h}_{1}+q\cdot\bar{x}^{h}\leq I^{h}+\sum a^{hf}\cdot\pi^{f}
  5. x h = ( x 1 h , x ¯ h ) x^{h}=\left(x^{h}_{1},\bar{x}^{h}\right)
  6. π f = y 1 f + p y ¯ 1 \pi^{f}=y^{f}_{1}+p\cdot\bar{y}_{1}
  7. y 1 f - G f ( y ¯ f , z f ) 0 y^{f}_{1}-G^{f}(\bar{y}^{f},z^{f})\leq 0
  8. y f = ( y 1 f , y ¯ f ) y^{f}=\left(y^{f}_{1},\bar{y}^{f}\right)
  9. R = t x ¯ - I h R=t\cdot\bar{x}-\sum I^{h}
  10. x ¯ h ( q , I , z ) - y ¯ f ( p , z ) = x ¯ ( q , I , z ) - y ¯ f ( p , z ) = 0 \sum\bar{x}^{h}(q,I,z)-\sum\bar{y}^{f}(p,z)=\bar{x}(q,I,z)-\sum\bar{y}^{f}(p,z% )=0
  11. E h q = E q h \frac{\partial E^{h}}{\partial q}=E^{h}_{q}
  12. E h ( q , z h , u h ) E^{h}\left(q,z^{h},u^{h}\right)
  13. maximize t , I \displaystyle\underset{t,I}{\operatorname{maximize}}
  14. d I h d t + a h f ( π z f d z f d t + π P f d p d t ) = E q h d q d t + E z h d z h d t \frac{dI^{h}}{dt}+\sum a^{hf}\left(\pi^{f}_{z}\frac{dz^{f}}{dt}+\pi^{f}_{P}% \frac{dp}{dt}\right)=E^{h}_{q}\frac{dq}{dt}+E^{h}_{z}\frac{dz^{h}}{dt}
  15. π z f = π * f z f \pi^{f}_{z}=\frac{\partial\pi^{f}_{*}}{\partial z^{f}}
  16. π * f ( p , z f ) \pi^{f}_{*}(p,z^{f})
  17. E q h + ( E q h - a h f π P f ) d p d t = d I h d t + { a h f π z f d z f d t - E z h d z h d t } E^{h}_{q}+\left(E^{h}_{q}-\sum a^{hf}\pi^{f}_{P}\right)\frac{dp}{dt}=\frac{dI^% {h}}{dt}+\left\{\sum a^{hf}\pi^{f}_{z}\frac{dz^{f}}{dt}-E^{h}_{z}\frac{dz^{h}}% {dt}\right\}
  18. a h f = 1 \sum a^{hf}=1
  19. E q h + ( E q h - π P f ) d p d t = d I h d t + { π z f d z f d t - E z h d z h d t } \sum E^{h}_{q}+\left(\sum E^{h}_{q}-\sum\pi^{f}_{P}\right)\frac{dp}{dt}=\sum% \frac{dI^{h}}{dt}+\left\{\sum\pi^{f}_{z}\frac{dz^{f}}{dt}-\sum E^{h}_{z}\frac{% dz^{h}}{dt}\right\}
  20. x ^ k h ( q ; z h , u h ) = E h q | z h , u h \widehat{x}^{h}_{k}(q;z^{h},u^{h})=\left.\frac{\partial E^{h}}{\partial q}% \right|_{z^{h},u^{h}}
  21. π * f p k 1 | z f = y k f \left.\frac{\partial\pi^{f}_{*}}{\partial p_{k_{1}}}\right|_{z^{f}}=y^{f}_{k}
  22. x ¯ + ( x ¯ - y ¯ ) d p d t = d I h d t + ( π z f d z f d t - E z h d z h d t ) \bar{x}+\left(\bar{x}-\bar{y}\right)\frac{dp}{dt}=\sum\frac{dI^{h}}{dt}+\left(% \sum\pi^{f}_{z}\frac{dz^{f}}{dt}-\sum E^{h}_{z}\frac{dz^{h}}{dt}\right)
  23. x ¯ = y ¯ \bar{x}=\bar{y}
  24. d I h d t = x ¯ - ( π z f d z f d t - E z h d z h d t ) \sum\frac{dI^{h}}{dt}=\bar{x}-\left(\sum\pi^{f}_{z}\frac{dz^{f}}{dt}-\sum E^{h% }_{z}\frac{dz^{h}}{dt}\right)
  25. d R d t = x ¯ + d x ¯ d t t - d I h d t \frac{dR}{dt}=\bar{x}+\frac{d\bar{x}}{dt}\cdot t-\sum\frac{dI^{h}}{dt}
  26. d I h d t \sum\frac{dI^{h}}{dt}
  27. d R d t = d x ¯ d t t + ( π z f d z f d t - E z h d z h d t ) = d x ¯ d t t + ( Π t - B t ) \frac{dR}{dt}=\frac{d\bar{x}}{dt}\cdot t+(\sum\pi^{f}_{z}\frac{dz^{f}}{dt}-% \sum E^{h}_{z}\frac{dz^{h}}{dt})=\frac{d\bar{x}}{dt}\cdot t+(\Pi^{t}-B^{t})
  28. d R d t = ( Π t - B t ) = 0 \frac{dR}{dt}=\left(\Pi^{t}-B^{t}\right)=0

Josiah_Willard_Gibbs.html

  1. μ \mu
  2. d U = T d S - p d V + i μ i d N i \mathrm{d}U=T\mathrm{d}S-p\,\mathrm{d}V+\sum_{i}\mu_{i}\,\mathrm{d}N_{i}\,
  3. F = C - P + 2 F\;=\;C\;-\;P\;+\;2

Josip_Plemelj.html

  1. f + ( z ) = 1 2 i π Γ ϕ ( t ) - ϕ ( z ) t - z d t + ϕ ( z ) f_{+}(z)={1\over 2i\pi}\int_{\Gamma}{\phi(t)-\phi(z)\over{t-z}}\,dt+\phi(z)
  2. f ( z ) = 1 2 i π Γ ϕ ( t ) - ϕ ( z ) t - z d t + 1 2 ϕ ( z ) f(z)={1\over 2i\pi}\int_{\Gamma}{\phi(t)-\phi(z)\over{t-z}}\,dt+{1\over 2}\phi% (z)
  3. f - ( z ) = 1 2 i π Γ ϕ ( t ) - ϕ ( z ) t - z d t z Γ f_{-}(z)={1\over 2i\pi}\int_{\Gamma}{\phi(t)-\phi(z)\over{t-z}}\,dt\quad z\in\Gamma

Joule.html

  1. J = kg m 2 s 2 = N m = Pa m 3 = W s = C V \rm J={}\rm\frac{kg\cdot m^{2}}{s^{2}}=N\cdot m=\rm Pa\cdot m^{3}={}\rm W\cdot s% =C\cdot V
  2. E = τ θ E=\tau\theta

JPEG.html

  1. [ 52 55 61 66 70 61 64 73 63 59 55 90 109 85 69 72 62 59 68 113 144 104 66 73 63 58 71 122 154 106 70 69 67 61 68 104 126 88 68 70 79 65 60 70 77 68 58 75 85 71 64 59 55 61 65 83 87 79 69 68 65 76 78 94 ] . \left[\begin{array}[]{rrrrrrrr}52&55&61&66&70&61&64&73\\ 63&59&55&90&109&85&69&72\\ 62&59&68&113&144&104&66&73\\ 63&58&71&122&154&106&70&69\\ 67&61&68&104&126&88&68&70\\ 79&65&60&70&77&68&58&75\\ 85&71&64&59&55&61&65&83\\ 87&79&69&68&65&76&78&94\end{array}\right].
  2. [ 0 , 255 ] [0,255]
  3. [ - 128 , 127 ] [-128,127]
  4. g = x [ - 76 - 73 - 67 - 62 - 58 - 67 - 64 - 55 - 65 - 69 - 73 - 38 - 19 - 43 - 59 - 56 - 66 - 69 - 60 - 15 16 - 24 - 62 - 55 - 65 - 70 - 57 - 6 26 - 22 - 58 - 59 - 61 - 67 - 60 - 24 - 2 - 40 - 60 - 58 - 49 - 63 - 68 - 58 - 51 - 60 - 70 - 53 - 43 - 57 - 64 - 69 - 73 - 67 - 63 - 45 - 41 - 49 - 59 - 60 - 63 - 52 - 50 - 34 ] y . g=\begin{array}[]{c}x\\ \longrightarrow\\ \left[\begin{array}[]{rrrrrrrr}-76&-73&-67&-62&-58&-67&-64&-55\\ -65&-69&-73&-38&-19&-43&-59&-56\\ -66&-69&-60&-15&16&-24&-62&-55\\ -65&-70&-57&-6&26&-22&-58&-59\\ -61&-67&-60&-24&-2&-40&-60&-58\\ -49&-63&-68&-58&-51&-60&-70&-53\\ -43&-57&-64&-69&-73&-67&-63&-45\\ -41&-49&-59&-60&-63&-52&-50&-34\end{array}\right]\end{array}\Bigg\downarrow y.
  5. u u
  6. v v
  7. G u , v = 1 4 α ( u ) α ( v ) x = 0 7 y = 0 7 g x , y cos [ ( 2 x + 1 ) u π 16 ] cos [ ( 2 y + 1 ) v π 16 ] \ G_{u,v}=\frac{1}{4}\alpha(u)\alpha(v)\sum_{x=0}^{7}\sum_{y=0}^{7}g_{x,y}\cos% \left[\frac{(2x+1)u\pi}{16}\right]\cos\left[\frac{(2y+1)v\pi}{16}\right]
  8. u \ u
  9. 0 u < 8 \ 0\leq u<8
  10. v \ v
  11. 0 v < 8 \ 0\leq v<8
  12. α ( u ) = { 1 2 , if u = 0 1 , otherwise \alpha(u)=\begin{cases}\frac{1}{\sqrt{2}},&\mbox{if }~{}u=0\\ 1,&\mbox{otherwise}\end{cases}
  13. g x , y \ g_{x,y}
  14. ( x , y ) \ (x,y)
  15. G u , v \ G_{u,v}
  16. ( u , v ) . \ (u,v).
  17. G = u [ - 415.38 - 30.19 - 61.20 27.24 56.12 - 20.10 - 2.39 0.46 4.47 - 21.86 - 60.76 10.25 13.15 - 7.09 - 8.54 4.88 - 46.83 7.37 77.13 - 24.56 - 28.91 9.93 5.42 - 5.65 - 48.53 12.07 34.10 - 14.76 - 10.24 6.30 1.83 1.95 12.12 - 6.55 - 13.20 - 3.95 - 1.87 1.75 - 2.79 3.14 - 7.73 2.91 2.38 - 5.94 - 2.38 0.94 4.30 1.85 - 1.03 0.18 0.42 - 2.42 - 0.88 - 3.02 4.12 - 0.66 - 0.17 0.14 - 1.07 - 4.19 - 1.17 - 0.10 0.50 1.68 ] v . G=\begin{array}[]{c}u\\ \longrightarrow\\ \left[\begin{array}[]{rrrrrrrr}-415.38&-30.19&-61.20&27.24&56.12&-20.10&-2.39&% 0.46\\ 4.47&-21.86&-60.76&10.25&13.15&-7.09&-8.54&4.88\\ -46.83&7.37&77.13&-24.56&-28.91&9.93&5.42&-5.65\\ -48.53&12.07&34.10&-14.76&-10.24&6.30&1.83&1.95\\ 12.12&-6.55&-13.20&-3.95&-1.87&1.75&-2.79&3.14\\ -7.73&2.91&2.38&-5.94&-2.38&0.94&4.30&1.85\\ -1.03&0.18&0.42&-2.42&-0.88&-3.02&4.12&-0.66\\ -0.17&0.14&-1.07&-4.19&-1.17&-0.10&0.50&1.68\end{array}\right]\end{array}\Bigg% \downarrow v.
  18. Q = [ 16 11 10 16 24 40 51 61 12 12 14 19 26 58 60 55 14 13 16 24 40 57 69 56 14 17 22 29 51 87 80 62 18 22 37 56 68 109 103 77 24 35 55 64 81 104 113 92 49 64 78 87 103 121 120 101 72 92 95 98 112 100 103 99 ] . Q=\begin{bmatrix}16&11&10&16&24&40&51&61\\ 12&12&14&19&26&58&60&55\\ 14&13&16&24&40&57&69&56\\ 14&17&22&29&51&87&80&62\\ 18&22&37&56&68&109&103&77\\ 24&35&55&64&81&104&113&92\\ 49&64&78&87&103&121&120&101\\ 72&92&95&98&112&100&103&99\end{bmatrix}.
  19. B j , k = round ( G j , k Q j , k ) for j = 0 , 1 , 2 , , 7 ; k = 0 , 1 , 2 , , 7 B_{j,k}=\mathrm{round}\left(\frac{G_{j,k}}{Q_{j,k}}\right)\mbox{ for }~{}j=0,1% ,2,\ldots,7;k=0,1,2,\ldots,7
  20. G G
  21. Q Q
  22. B B
  23. B = [ - 26 - 3 - 6 2 2 - 1 0 0 0 - 2 - 4 1 1 0 0 0 - 3 1 5 - 1 - 1 0 0 0 - 3 1 2 - 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] . B=\left[\begin{array}[]{rrrrrrrr}-26&-3&-6&2&2&-1&0&0\\ 0&-2&-4&1&1&0&0&0\\ -3&1&5&-1&-1&0&0&0\\ -3&1&2&-1&0&0&0&0\\ 1&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\end{array}\right].
  24. round ( - 415.37 16 ) = round ( - 25.96 ) = - 26. \mathrm{round}\left(\frac{-415.37}{16}\right)=\mathrm{round}\left(-25.96\right% )=-26.
  25. B i B_{i}
  26. ( p , q ) (p,q)
  27. p = 0 , 1 , , 7 p=0,1,...,7
  28. q = 0 , 1 , , 7 q=0,1,...,7
  29. B i ( p , q ) B_{i}(p,q)
  30. i i
  31. B i ( 0 , 0 ) B_{i}(0,0)
  32. B i ( 0 , 1 ) B_{i}(0,1)
  33. B i ( 1 , 0 ) B_{i}(1,0)
  34. B i ( 2 , 0 ) B_{i}(2,0)
  35. B i ( 1 , 1 ) B_{i}(1,1)
  36. B i ( 0 , 2 ) B_{i}(0,2)
  37. B i ( 0 , 3 ) B_{i}(0,3)
  38. B i ( 1 , 2 ) B_{i}(1,2)
  39. B 0 , B 1 , B 2 , , B n - 1 B_{0},B_{1},B_{2},...,B_{n-1}
  40. B i ( 0 , 0 ) B_{i}(0,0)
  41. i = 0 , 1 , 2 , , N - 1 i=0,1,2,...,N-1
  42. B i ( 0 , 1 ) B_{i}(0,1)
  43. B i ( 1 , 0 ) B_{i}(1,0)
  44. B i ( 2 , 0 ) B_{i}(2,0)
  45. x x
  46. x x
  47. x x
  48. x x
  49. x x
  50. [ - 26 - 3 - 6 2 2 - 1 0 0 0 - 2 - 4 1 1 0 0 0 - 3 1 5 - 1 - 1 0 0 0 - 3 1 2 - 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] \left[\begin{array}[]{rrrrrrrr}-26&-3&-6&2&2&-1&0&0\\ 0&-2&-4&1&1&0&0&0\\ -3&1&5&-1&-1&0&0&0\\ -3&1&2&-1&0&0&0&0\\ 1&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\end{array}\right]
  51. [ - 416 - 33 - 60 32 48 - 40 0 0 0 - 24 - 56 19 26 0 0 0 - 42 13 80 - 24 - 40 0 0 0 - 42 17 44 - 29 0 0 0 0 18 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] \left[\begin{array}[]{rrrrrrrr}-416&-33&-60&32&48&-40&0&0\\ 0&-24&-56&19&26&0&0&0\\ -42&13&80&-24&-40&0&0&0\\ -42&17&44&-29&0&0&0&0\\ 18&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\\ 0&0&0&0&0&0&0&0\end{array}\right]
  52. f x , y = 1 4 u = 0 7 v = 0 7 α ( u ) α ( v ) F u , v cos [ ( 2 x + 1 ) u π 16 ] cos [ ( 2 y + 1 ) v π 16 ] f_{x,y}=\frac{1}{4}\sum_{u=0}^{7}\sum_{v=0}^{7}\alpha(u)\alpha(v)F_{u,v}\cos% \left[\frac{(2x+1)u\pi}{16}\right]\cos\left[\frac{(2y+1)v\pi}{16}\right]
  53. x \ x
  54. 0 x < 8 \ 0\leq x<8
  55. y \ y
  56. 0 y < 8 \ 0\leq y<8
  57. α ( u ) \ \alpha(u)
  58. 0 u < 8 \ 0\leq u<8
  59. F u , v \ F_{u,v}
  60. ( u , v ) . \ (u,v).
  61. f x , y \ f_{x,y}
  62. ( x , y ) \ (x,y)
  63. [ - 66 - 63 - 71 - 68 - 56 - 65 - 68 - 46 - 71 - 73 - 72 - 46 - 20 - 41 - 66 - 57 - 70 - 78 - 68 - 17 20 - 14 - 61 - 63 - 63 - 73 - 62 - 8 27 - 14 - 60 - 58 - 58 - 65 - 61 - 27 - 6 - 40 - 68 - 50 - 57 - 57 - 64 - 58 - 48 - 66 - 72 - 47 - 53 - 46 - 61 - 74 - 65 - 63 - 62 - 45 - 47 - 34 - 53 - 74 - 60 - 47 - 47 - 41 ] \left[\begin{array}[]{rrrrrrrr}-66&-63&-71&-68&-56&-65&-68&-46\\ -71&-73&-72&-46&-20&-41&-66&-57\\ -70&-78&-68&-17&20&-14&-61&-63\\ -63&-73&-62&-8&27&-14&-60&-58\\ -58&-65&-61&-27&-6&-40&-68&-50\\ -57&-57&-64&-58&-48&-66&-72&-47\\ -53&-46&-61&-74&-65&-63&-62&-45\\ -47&-34&-53&-74&-60&-47&-47&-41\end{array}\right]
  64. [ 62 65 57 60 72 63 60 82 57 55 56 82 108 87 62 71 58 50 60 111 148 114 67 65 65 55 66 120 155 114 68 70 70 63 67 101 122 88 60 78 71 71 64 70 80 62 56 81 75 82 67 54 63 65 66 83 81 94 75 54 68 81 81 87 ] . \left[\begin{array}[]{rrrrrrrr}62&65&57&60&72&63&60&82\\ 57&55&56&82&108&87&62&71\\ 58&50&60&111&148&114&67&65\\ 65&55&66&120&155&114&68&70\\ 70&63&67&101&122&88&60&78\\ 71&71&64&70&80&62&56&81\\ 75&82&67&54&63&65&66&83\\ 81&94&75&54&68&81&81&87\end{array}\right].
  65. [ 0 , 255 ] [0,255]
  66. [ - 10 - 10 4 6 - 2 - 2 4 - 9 6 4 - 1 8 1 - 2 7 1 4 9 8 2 - 4 - 10 - 1 8 - 2 3 5 2 - 1 - 8 2 - 1 - 3 - 2 1 3 4 0 8 - 8 8 - 6 - 4 - 0 - 3 6 2 - 6 10 - 11 - 3 5 - 8 - 4 - 1 - 0 6 - 15 - 6 14 - 3 - 5 - 3 7 ] \left[\begin{array}[]{rrrrrrrr}-10&-10&4&6&-2&-2&4&-9\\ 6&4&-1&8&1&-2&7&1\\ 4&9&8&2&-4&-10&-1&8\\ -2&3&5&2&-1&-8&2&-1\\ -3&-2&1&3&4&0&8&-8\\ 8&-6&-4&-0&-3&6&2&-6\\ 10&-11&-3&5&-8&-4&-1&-0\\ 6&-15&-6&14&-3&-5&-3&7\end{array}\right]
  67. 1 64 x = 0 7 y = 0 7 | e ( x , y ) | = 4.9197 \frac{1}{64}\sum_{x=0}^{7}\sum_{y=0}^{7}|e(x,y)|=4.9197

JPEG_2000.html

  1. Y = R + 2 G + B 4 ; C B = B - G ; C R = R - G ; Y=\left\lfloor\frac{R+2G+B}{4}\right\rfloor;C_{B}=B-G;C_{R}=R-G;
  2. G = Y - C B + C R 4 ; R = C R + G ; B = C B + G . G=Y-\left\lfloor\frac{C_{B}+C_{R}}{4}\right\rfloor;R=C_{R}+G;B=C_{B}+G.

Julia_set.html

  1. f ( z ) = p ( z ) / q ( z ) f(z)=p(z)/q(z)
  2. f ( z ) = 0 f^{\prime}(z)=0
  3. f ( z ) = 1 / g ( z ) + c f(z)=1/g(z)+c
  4. | f ( z ) - f ( w ) | > | z - w | |f(z)-f(w)|>|z-w|
  5. n f - n ( z ) \bigcup_{n}f^{-n}(z)
  6. f - 1 ( J ( f ) ) = f ( J ( f ) ) = J ( f ) f^{-1}(J(f))=f(J(f))=J(f)
  7. f - 1 ( F ( f ) ) = f ( F ( f ) ) = F ( f ) f^{-1}(F(f))=f(F(f))=F(f)
  8. f ( z ) = z 2 f(z)=z^{2}
  9. 2 π 2\pi
  10. f ( z ) = z 2 - 2 f(z)=z^{2}-2
  11. x 4 ( x - 1 2 ) 2 x\to 4(x-\tfrac{1}{2})^{2}
  12. z 2 + c z^{2}+c
  13. f ( z ) = z - f ( z ) f ( z ) = 1 + ( n - 1 ) z n n z n - 1 . f(z)=z-\frac{f(z)}{f^{\prime}(z)}=\frac{1+(n-1)z^{n}}{nz^{n-1}}.
  14. f c ( z ) = z 2 + c f_{c}(z)=z^{2}+c
  15. J ( f c ) J(f_{c})
  16. J ( f c ) J(f_{c})
  17. f ( z ) = z 2 f(z)=z^{2}
  18. | f ( z ) | = | z | 2 |f(z)|=|z|^{2}
  19. φ ( z ) = lim k log | z k | 2 k , \varphi(z)=\lim_{k\to\infty}\frac{\log|z_{k}|}{2^{k}},
  20. z k z_{k}
  21. f ( z ) = z 2 + c f(z)=z^{2}+c
  22. | ψ ( f ( z ) ) | = | ψ ( z ) | 2 |\psi(f(z))|=|\psi(z)|^{2}
  23. φ ( z ) = lim k log | z k | 2 k . \varphi(z)=\lim_{k\to\infty}\frac{\log|z_{k}|}{2^{k}}.
  24. φ ( z ) = lim k log | z k | d k , \varphi(z)=\lim_{k\to\infty}\frac{\log|z_{k}|}{d^{k}},
  25. | z k | > N |z_{k}|>N
  26. log | z k | d k = log ( N ) d ν ( z ) , \frac{\log|z_{k}|}{d^{k}}=\frac{\log(N)}{d^{\nu(z)}},
  27. ν ( z ) \nu(z)
  28. ν ( z ) = k - log ( log | z k | / log ( N ) ) log ( d ) , \nu(z)=k-\frac{\log(\log|z_{k}|/\log(N))}{\log(d)},
  29. f ( f ( f ( z * ) ) ) = z * f(f(...f(z^{*})))=z^{*}
  30. α = 1 | ( d ( f ( f ( f ( z ) ) ) ) / d z ) z = z * | ( > 1 ) \alpha=\frac{1}{\left|(d(f(f(\cdots f(z))))/dz)_{z=z^{*}}\right|}\qquad(>1)
  31. α = lim k | w - z * | | w - z * | . \alpha=\lim_{k\to\infty}\frac{|w-z^{*}|}{|w^{\prime}-z^{*}|}.
  32. | z k r - z * | α k |z_{kr}-z^{*}|\alpha^{k}
  33. φ ( z ) = lim k 1 ( | z k r - z * | α k ) . \varphi(z)=\lim_{k\to\infty}\frac{1}{(|z_{kr}-z^{*}|\alpha^{k})}.
  34. | z k - z * | < ϵ |z_{k}-z^{*}|<\epsilon
  35. φ ( z ) = 1 ( ε α ν ( z ) ) \varphi(z)=\frac{1}{(\varepsilon\alpha^{\nu(z)})}
  36. ν ( z ) \nu(z)
  37. ν ( z ) = k - log ( ε / | z k - z * | ) log ( α ) . \nu(z)=k-\frac{\log(\varepsilon/|z_{k}-z^{*}|)}{\log(\alpha)}.
  38. α = lim k log | w - z * | log | w - z * | , \alpha=\lim_{k\to\infty}\frac{\log|w^{\prime}-z^{*}|}{\log|w-z^{*}|},
  39. φ ( z ) = lim k log ( 1 / | z k r - z * | ) α k . \varphi(z)=\lim_{k\to\infty}\frac{\log(1/|z_{kr}-z^{*}|)}{\alpha^{k}}.
  40. ν ( z ) = k - log ( log | z k - z * | / log ( ε ) ) log ( α ) . \nu(z)=k-\frac{\log(\log|z_{k}-z^{*}|/\log(\varepsilon))}{\log(\alpha)}.
  41. ν ( z ) \nu(z)
  42. z 2 + c z^{2}+c
  43. f ( f ( f ( z * ) ) ) = z * f(f(\dots f(z^{*})))=z^{*}
  44. α = ( d ( f ( f ( f ( z ) ) ) ) / d z ) z = z * . \alpha=(d(f(f(\dots f(z))))/dz)_{z=z^{*}}.
  45. z i , i = 1 , , r ( z 1 = z * ) z_{i},i=1,\dots,r(z_{1}=z^{*})
  46. f ( z i ) f^{\prime}(z_{i})
  47. f ( f ( f ( z ) ) ) f(f(\dots f(z)))
  48. α = | α | e β i \alpha=|\alpha|e^{\beta i}
  49. z k ( k = 0 , 1 , 2 , , z 0 = z ) z_{k}(k=0,1,2,\dots,z_{0}=z)
  50. | z k - z * | < ϵ |z_{k}-z^{*}|<\epsilon
  51. z k - z * z_{k}-z^{*}
  52. ψ - k β = θ mod π . \psi-k\beta=\theta\mod\pi.\,
  53. ψ - k β mod π \psi-k\beta\mod\pi
  54. z 2 + c z^{2}+c
  55. | φ ( z ) | |\varphi^{\prime}(z)|
  56. δ ( z ) = φ ( z ) / | φ ( z ) | \delta(z)=\varphi(z)/|\varphi^{\prime}(z)|
  57. f ( z ) = p ( z ) / q ( z ) f(z)=p(z)/q(z)
  58. z k z_{k}
  59. z k z^{\prime}_{k}
  60. z k z_{k}
  61. z k = f ( f ( f ( z ) ) ) z_{k}=f(f(\cdots f(z)))
  62. z k z^{\prime}_{k}
  63. f ( z k ) f^{\prime}(z_{k})
  64. z k + 1 = f ( z k ) z k z^{\prime}_{k+1}=f^{\prime}(z_{k})z^{\prime}_{k}
  65. z 0 = 1 z^{\prime}_{0}=1
  66. z k + 1 = f ( z k ) z_{k+1}=f(z_{k})
  67. | φ ( z ) | = lim k | z k | | z k | d k , |\varphi^{\prime}(z)|=\lim_{k\to\infty}\frac{|z^{\prime}_{k}|}{|z_{k}|d^{k}},
  68. δ ( z ) = φ ( z ) / | φ ( z ) | = lim k log | z k | | z k | / | z k | . \delta(z)=\varphi(z)/|\varphi^{\prime}(z)|=\lim_{k\to\infty}\log|z_{k}||z_{k}|% /|z^{\prime}_{k}|.\,
  69. | φ ( z ) | = lim k | z k r | / ( | z k r - z * | 2 α k ) , |\varphi^{\prime}(z)|=\lim_{k\to\infty}|z^{\prime}_{kr}|/(|z_{kr}-z^{*}|^{2}% \alpha^{k}),\,
  70. δ ( z ) = φ ( z ) / | φ ( z ) | = lim k | z k r - z * | / | z k r | . \delta(z)=\varphi(z)/|\varphi^{\prime}(z)|=\lim_{k\to\infty}|z_{kr}-z^{*}|/|z^% {\prime}_{kr}|.\,
  71. δ ( z ) = lim k log | z k r - z * | 2 / | z k r | . \delta(z)=\lim_{k\to\infty}\log|z_{kr}-z^{*}|^{2}/|z^{\prime}_{kr}|.\,
  72. f n f^{n}
  73. z n - 1 = z n - c . z_{n-1}=\sqrt{z_{n}-c}.

Jupiter_trojan.html

  1. F = 1 2 π σ P 2 exp ( - ( P - P 0 ) 2 / σ 2 ) F=\begin{smallmatrix}\frac{1}{\sqrt{2\pi}\sigma}P^{2}\exp(-(P-P_{0})^{2}/% \sigma^{2})\end{smallmatrix}
  2. P 0 P_{0}
  3. σ \sigma

Jurij_Vega.html

  1. π 4 = 4 arctan ( 1 5 ) - arctan ( 1 239 ) {\pi\over 4}=4\arctan\left({1\over 5}\right)-\arctan\left({1\over 239}\right)
  2. π 4 = 5 arctan ( 1 7 ) + 2 arctan ( 3 79 ) , {\pi\over 4}=5\arctan\left({1\over 7}\right)+2\arctan\left({3\over 79}\right)\;,
  3. π 4 = 2 arctan ( 1 3 ) + arctan ( 1 7 ) . {\pi\over 4}=2\arctan\left({1\over 3}\right)+\arctan\left({1\over 7}\right)\;.

Just_intonation.html

  1. S = 8 : 5 5 / 4 , S={8:5^{5/4}},
  2. T = 5 : 2 , T=\sqrt{5}:2,
  3. P = 5 4 . P=\sqrt[4]{5}.
  4. 2 3 5 4 = 10 12 = 5 6 . {2\over 3}\cdot{5\over 4}={10\over 12}={5\over 6}.
  5. 5 6 2 1 = 10 6 = 5 3 . {5\over 6}\cdot{2\over 1}={10\over 6}={5\over 3}.
  6. 1 1 \frac{1}{1}
  7. 256 243 \frac{256}{243}
  8. 16 15 \frac{16}{15}
  9. 10 9 \frac{10}{9}
  10. 9 8 \frac{9}{8}
  11. 32 27 \frac{32}{27}
  12. 6 5 \frac{6}{5}
  13. 5 4 \frac{5}{4}
  14. 81 64 \frac{81}{64}
  15. 4 3 \frac{4}{3}
  16. 27 20 \frac{27}{20}
  17. 45 32 \frac{45}{32}
  18. 729 512 \frac{729}{512}
  19. 3 2 \frac{3}{2}
  20. 128 81 \frac{128}{81}
  21. 8 5 \frac{8}{5}
  22. 5 3 \frac{5}{3}
  23. 27 16 \frac{27}{16}
  24. 16 9 \frac{16}{9}
  25. 9 5 \frac{9}{5}
  26. 15 8 \frac{15}{8}
  27. 243 128 \frac{243}{128}
  28. 2 1 \frac{2}{1}

Kaluza–Klein_theory.html

  1. g ~ a b \widetilde{g}_{ab}
  2. g μ ν {g}_{\mu\nu}
  3. A μ A^{\mu}
  4. ϕ \phi
  5. g ~ a b [ g μ ν + ϕ 2 A μ A ν ϕ 2 A μ ϕ 2 A ν ϕ 2 ] . \widetilde{g}_{ab}\equiv\begin{bmatrix}g_{\mu\nu}+\phi^{2}A_{\mu}A_{\nu}&\phi^% {2}A_{\mu}\\ \phi^{2}A_{\nu}&\phi^{2}\end{bmatrix}.
  6. g ~ μ ν g μ ν + ϕ 2 A μ A ν , g ~ 5 ν g ~ ν 5 ϕ 2 A ν , g ~ 55 ϕ 2 \widetilde{g}_{\mu\nu}\equiv g_{\mu\nu}+\phi^{2}A_{\mu}A_{\nu},\qquad% \widetilde{g}_{5\nu}\equiv\widetilde{g}_{\nu 5}\equiv\phi^{2}A_{\nu},\qquad% \widetilde{g}_{55}\equiv\phi^{2}
  7. 5 5
  8. g ~ a b [ g μ ν - A μ - A ν g α β A α A β + 1 ϕ 2 ] . \widetilde{g}^{ab}\equiv\begin{bmatrix}g^{\mu\nu}&-A^{\mu}\\ -A^{\nu}&g_{\alpha\beta}A^{\alpha}A^{\beta}+{1\over\phi^{2}}\end{bmatrix}.
  9. d s ds
  10. d s 2 g ~ a b d x a d x b = g μ ν d x μ d x ν + ϕ 2 ( A ν d x ν + d x 5 ) 2 ds^{2}\equiv\widetilde{g}_{ab}dx^{a}dx^{b}=g_{\mu\nu}dx^{\mu}dx^{\nu}+\phi^{2}% (A_{\nu}dx^{\nu}+dx^{5})^{2}
  11. Γ ~ b c a \widetilde{\Gamma}^{a}_{bc}
  12. g ~ a b \widetilde{g}_{ab}
  13. R ~ a b \widetilde{R}_{ab}
  14. g ~ a b x 5 = 0 {\partial\widetilde{g}_{ab}\over\partial x^{5}}=0
  15. R ~ a b = 0 \widetilde{R}_{ab}=0
  16. R ~ a b c Γ ~ a b c - b Γ ~ c a c + Γ ~ c d c Γ ~ a b d - Γ ~ b d c Γ ~ a c d \widetilde{R}_{ab}\equiv\partial_{c}\widetilde{\Gamma}^{c}_{ab}-\partial_{b}% \widetilde{\Gamma}^{c}_{ca}+\widetilde{\Gamma}^{c}_{cd}\widetilde{\Gamma}^{d}_% {ab}-\widetilde{\Gamma}^{c}_{bd}\widetilde{\Gamma}^{d}_{ac}
  17. Γ ~ b c a 1 2 g ~ a d ( b g ~ d c + c g ~ d b - d g ~ b c ) \widetilde{\Gamma}^{a}_{bc}\equiv{1\over 2}\widetilde{g}^{ad}(\partial_{b}% \widetilde{g}_{dc}+\partial_{c}\widetilde{g}_{db}-\partial_{d}\widetilde{g}_{% bc})
  18. ϕ \phi
  19. R ~ 55 = 0 ϕ = 1 4 ϕ 3 F α β F α β \widetilde{R}_{55}=0\Rightarrow\Box\phi={1\over 4}\phi^{3}F^{\alpha\beta}F_{% \alpha\beta}
  20. F α β α A β - β A α F_{\alpha\beta}\equiv\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha}
  21. g μ ν μ ν \Box\equiv g^{\mu\nu}\nabla_{\mu}\nabla_{\nu}
  22. μ \nabla_{\mu}
  23. A ν A^{\nu}
  24. R ~ 5 α = 0 = 1 2 g β μ μ ( ϕ 3 F α β ) \widetilde{R}_{5\alpha}=0={1\over 2}g^{\beta\mu}\nabla_{\mu}(\phi^{3}F_{\alpha% \beta})
  25. R μ ν R_{\mu\nu}
  26. R ~ μ ν - 1 2 g ~ μ ν R ~ = 0 R μ ν - 1 2 g μ ν R = 1 2 ϕ 2 ( g α β F μ α F ν β - 1 4 g μ ν F α β F α β ) + 1 ϕ ( μ ν ϕ - g μ ν ϕ ) \widetilde{R}_{\mu\nu}-{1\over 2}\widetilde{g}_{\mu\nu}\widetilde{R}=0% \Rightarrow R_{\mu\nu}-{1\over 2}g_{\mu\nu}R={1\over 2}\phi^{2}\left(g^{\alpha% \beta}F_{\mu\alpha}F_{\nu\beta}-{1\over 4}g_{\mu\nu}F_{\alpha\beta}F^{\alpha% \beta}\right)+{1\over\phi}\left(\nabla_{\mu}\nabla_{\nu}\phi-g_{\mu\nu}\Box% \phi\right)
  27. R R
  28. A μ A^{\mu}
  29. k k
  30. A μ k A μ A^{\mu}\rightarrow kA^{\mu}
  31. k 2 2 = 8 π G c 4 1 μ 0 {k^{2}\over 2}={8\pi G\over c^{4}}{1\over\mu_{0}}
  32. G G
  33. μ 0 \mu_{0}
  34. ϕ 2 \phi^{2}
  35. G ~ a b R ~ a b - 1 2 g ~ a b R ~ \widetilde{G}_{ab}\equiv\widetilde{R}_{ab}-{1\over 2}\widetilde{g}_{ab}% \widetilde{R}
  36. G ~ a b \widetilde{G}_{ab}
  37. R ~ a b \widetilde{R}_{ab}
  38. U ~ a d x a / d s \widetilde{U}^{a}\equiv dx^{a}/ds
  39. U ~ b ~ b U ~ a = d U ~ a d s + Γ ~ b c a U ~ b U ~ c = 0 \widetilde{U}^{b}\widetilde{\nabla}_{b}\widetilde{U}^{a}={d\widetilde{U}^{a}% \over ds}+\widetilde{\Gamma}^{a}_{bc}\widetilde{U}^{b}\widetilde{U}^{c}=0
  40. c 2 d τ 2 g μ ν d x μ d x ν c^{2}d\tau^{2}\equiv g_{\mu\nu}dx^{\mu}dx^{\nu}
  41. d s ds
  42. d s 2 = c 2 d τ 2 + ϕ 2 ( k A ν d x ν + d x 5 ) 2 ds^{2}=c^{2}d\tau^{2}+\phi^{2}(kA_{\nu}dx^{\nu}+dx^{5})^{2}
  43. U ν d x ν / d τ U^{\nu}\equiv dx^{\nu}/d\tau
  44. d U ν d τ + Γ ~ α β μ U α U β + 2 Γ ~ 5 α μ U α U 5 + Γ ~ 55 μ ( U 5 ) 2 + U μ d d τ ln ( c d τ d s ) = 0 {dU^{\nu}\over d\tau}+\widetilde{\Gamma}^{\mu}_{\alpha\beta}U^{\alpha}U^{\beta% }+2\widetilde{\Gamma}^{\mu}_{5\alpha}U^{\alpha}U^{5}+\widetilde{\Gamma}^{\mu}_% {55}(U^{5})^{2}+U^{\mu}{d\over d\tau}\ln\left({cd\tau\over ds}\right)=0
  45. U ν U^{\nu}
  46. Γ ~ α β μ = Γ α β μ + 1 2 g μ ν k 2 ϕ 2 ( A α F β ν + A β F α ν + A α A β ν ln ϕ 2 ) \widetilde{\Gamma}^{\mu}_{\alpha\beta}=\Gamma^{\mu}_{\alpha\beta}+{1\over 2}g^% {\mu\nu}k^{2}\phi^{2}(A_{\alpha}F_{\beta\nu}+A_{\beta}F_{\alpha\nu}+A_{\alpha}% A_{\beta}\partial_{\nu}\ln\phi^{2})
  47. U ν U^{\nu}
  48. Γ ~ 5 α μ = 1 2 g μ ν k ϕ 2 ( F α ν - A α ν ln ϕ 2 ) \widetilde{\Gamma}^{\mu}_{5\alpha}={1\over 2}g^{\mu\nu}k\phi^{2}(F_{\alpha\nu}% -A_{\alpha}\partial_{\nu}\ln\phi^{2})
  49. k U 5 = k d x 5 d τ q m c kU^{5}=k{dx^{5}\over d\tau}\rightarrow{q\over mc}
  50. m m
  51. q q
  52. U 5 U^{5}
  53. Γ ~ 55 μ = - 1 2 g μ α α ϕ 2 \widetilde{\Gamma}^{\mu}_{55}=-{1\over 2}g^{\mu\alpha}\partial_{\alpha}\phi^{2}
  54. U 5 U^{5}
  55. U 5 c q / m G 1 / 2 U^{5}\sim c{q/m\over G^{1/2}}
  56. U 5 > 10 20 c U^{5}>{\rm 10}^{20}c
  57. U 5 U^{5}
  58. U 5 U^{5}
  59. d U ~ a d s = 1 2 U ~ b U ~ c g ~ b c x a {d\widetilde{U}_{a}\over ds}={1\over 2}\widetilde{U}^{b}\widetilde{U}^{c}{% \partial\widetilde{g}_{bc}\over\partial x^{a}}
  60. U ~ 5 \widetilde{U}_{5}
  61. U ~ 5 = g ~ 5 a U ~ a = ϕ 2 c d τ d s ( k A ν U ν + U 5 ) = constant \widetilde{U}_{5}=\widetilde{g}_{5a}\widetilde{U}^{a}=\phi^{2}{cd\tau\over ds}% (kA_{\nu}U^{\nu}+U^{5})={\rm constant}
  62. T ~ M a b \widetilde{T}_{M}^{ab}
  63. T ~ M a b = ρ d x a d s d x b d s \widetilde{T}_{M}^{ab}=\rho{dx^{a}\over ds}{dx^{b}\over ds}
  64. ρ \rho
  65. d s ds
  66. T ~ M μ ν = ρ d x μ d s d x ν d s \widetilde{T}_{M}^{\mu\nu}=\rho{dx^{\mu}\over ds}{dx^{\nu}\over ds}
  67. T ~ M 5 μ = ρ d x μ d s d x 5 d s = ρ U μ q k m c \widetilde{T}_{M}^{5\mu}=\rho{dx^{\mu}\over ds}{dx^{5}\over ds}=\rho U^{\mu}{q% \over kmc}
  68. λ 5 \lambda^{5}
  69. U 5 U^{5}
  70. p 5 = h / λ 5 p^{5}=h/\lambda^{5}
  71. m U 5 = c q G 1 / 2 = h λ 5 λ 5 h G 1 / 2 c q mU^{5}={cq\over G^{1/2}}={h\over\lambda^{5}}\rightarrow\lambda^{5}\sim{hG^{1/2% }\over cq}
  72. h h
  73. λ 5 10 - 30 \lambda^{5}\sim{\rm 10}^{-30}
  74. R ~ a b = 0 \widetilde{R}_{ab}=0
  75. G μ ν = 8 π T μ ν G_{\mu\nu}=8\pi T_{\mu\nu}\,
  76. R ~ a b = 0 \widetilde{R}_{ab}=0
  77. S ( g ) = M R ( g ) vol ( g ) S(g)=\int_{M}R(g)\mathrm{vol}(g)\,
  78. δ S ( g ) δ g = 0 \frac{\delta S(g)}{\delta g}=0
  79. R i j - 1 2 g i j R = 0 R_{ij}-\frac{1}{2}g_{ij}R=0
  80. g ^ \widehat{g}
  81. g ^ \widehat{g}
  82. Vert P p T p P \mbox{Vert}~{}_{p}P\subset T_{p}P
  83. g ^ \widehat{g}
  84. Hor P p T p P \mbox{Hor}~{}_{p}P\subset T_{p}P
  85. S ( g ^ ) = P R ( g ^ ) vol ( g ^ ) S(\widehat{g})=\int_{P}R(\widehat{g})\;\mbox{vol}~{}(\widehat{g})\,
  86. R ( g ^ ) = π * ( R ( g ) - Λ 2 2 | F | 2 ) R(\widehat{g})=\pi^{*}\left(R(g)-\frac{\Lambda^{2}}{2}|F|^{2}\right)
  87. π * F = d A \pi^{*}F=\mathrm{d}A
  88. S ( g ^ ) = Λ M ( R ( g ) - 1 Λ 2 | F | 2 ) vol ( g ) S(\widehat{g})=\Lambda\int_{M}\left(R(g)-\frac{1}{\Lambda^{2}}|F|^{2}\right)\;% \mbox{vol}~{}(g)
  89. R i j - 1 2 g i j R = 1 Λ 2 T i j R_{ij}-\frac{1}{2}g_{ij}R=\frac{1}{\Lambda^{2}}T_{ij}
  90. T i j = F i k F j l g k l - 1 4 g i j | F | 2 , T^{ij}=F^{ik}F^{jl}g_{kl}-\frac{1}{4}g^{ij}|F|^{2},

Karl_Popper.html

  1. PS 1 TT 1 EE 1 PS 2 . \mathrm{PS}_{1}\rightarrow\mathrm{TT}_{1}\rightarrow\mathrm{EE}_{1}\rightarrow% \mathrm{PS}_{2}.\,
  2. PS 1 \mathrm{PS}_{1}
  3. TT \mathrm{TT}
  4. EE \mathrm{EE}
  5. PS 1 \mathrm{PS}_{1}
  6. PS 2 \mathrm{PS}_{2}
  7. 𝑉𝑠 ( a ) = 𝐶𝑇 v ( a ) - 𝐶𝑇 f ( a ) \mathit{Vs}(a)=\mathit{CT}_{v}(a)-\mathit{CT}_{f}(a)\,
  8. 𝑉𝑠 ( a ) \mathit{Vs}(a)
  9. 𝐶𝑇 v ( a ) \mathit{CT}_{v}(a)
  10. 𝐶𝑇 f ( a ) \mathit{CT}_{f}(a)

Karplus–Strong_string_synthesis.html

  1. - 2 π -2\pi

Kate_Bush.html

  1. π \pi

Keno.html

  1. P n ( k ) P_{n}(k)
  2. P n ( k ) = ( n k ) ( 80 - n 20 - k ) < m t p l > ( 80 20 ) P_{n}(k)=\frac{{n\choose k}{80-n\choose 20-k}}{<}mtpl>{{80\choose 20}}
  3. W n ( k ) . W_{n}(k).
  4. i = 0 o r 1 n P n ( i ) W n ( i ) \sum\limits_{i=0or1}^{n}P_{n}(i)W_{n}(i)

Kepler's_laws_of_planetary_motion.html

  1. ε π 4 186 - 179 186 + 179 0.015 , \varepsilon\approx\frac{\pi}{4}\frac{186-179}{186+179}\approx 0.015,
  2. r = p 1 + ε cos θ , r=\frac{p}{1+\varepsilon\,\cos\theta},
  3. p p
  4. p p
  5. r max = p 1 - ε r_{\max}=\frac{p}{1-\varepsilon}
  6. r max - a = a - r min \,r_{\max}-a=a-r_{\min}
  7. a = p 1 - ε 2 a=\frac{p}{1-\varepsilon^{2}}
  8. r max b = b r min \frac{r_{\max}}{b}=\frac{b}{r_{\min}}
  9. b = p 1 - ε 2 b=\frac{p}{\sqrt{1-\varepsilon^{2}}}
  10. 1 r min - 1 p = 1 p - 1 r max \frac{1}{r_{\min}}-\frac{1}{p}=\frac{1}{p}-\frac{1}{r_{\max}}
  11. p a = r max r min = b 2 pa=r_{\max}r_{\min}=b^{2}\,
  12. ε = r max - r min r max + r min . \varepsilon=\frac{r_{\max}-r_{\min}}{r_{\max}+r_{\min}}.
  13. A = π a b . A=\pi ab\,.
  14. d t dt\,
  15. r r\,
  16. r d θ r\,d\theta
  17. d A = 1 2 r r d θ dA=\tfrac{1}{2}\cdot r\cdot rd\theta
  18. d A d t = 1 2 r 2 d θ d t . \frac{dA}{dt}=\tfrac{1}{2}r^{2}\frac{d\theta}{dt}.
  19. π a b . \pi ab.\,
  20. P P\,
  21. P 1 2 r 2 d θ d t = π a b P\cdot\tfrac{1}{2}r^{2}\frac{d\theta}{dt}=\pi ab
  22. n = 2 π / P n=2\pi/P
  23. r 2 d θ = a b n d t . r^{2}\,d\theta=abn\,dt.
  24. P 2 / a 3 P^{2}/a^{3}
  25. 𝐫 = r 𝐫 ^ \mathbf{r}=r\hat{\mathbf{r}}
  26. r r
  27. 𝐫 ^ \hat{\mathbf{r}}
  28. d 𝐫 ^ d t = 𝐫 ^ ˙ = θ ˙ s y m b o l θ ^ , d s y m b o l θ ^ d t = s y m b o l θ ^ ˙ = - θ ˙ 𝐫 ^ \frac{d\hat{\mathbf{r}}}{dt}=\dot{\hat{\mathbf{r}}}=\dot{\theta}\hat{symbol% \theta},\qquad\frac{d\hat{symbol\theta}}{dt}=\dot{\hat{symbol\theta}}=-\dot{% \theta}\hat{\mathbf{r}}
  29. s y m b o l θ ^ \hat{symbol\theta}
  30. 𝐫 ^ \hat{\mathbf{r}}
  31. θ \theta
  32. 𝐫 ˙ = r ˙ 𝐫 ^ + r 𝐫 ^ ˙ = r ˙ 𝐫 ^ + r θ ˙ s y m b o l θ ^ , \dot{\mathbf{r}}=\dot{r}\hat{\mathbf{r}}+r\dot{\hat{\mathbf{r}}}=\dot{r}\hat{% \mathbf{r}}+r\dot{\theta}\hat{symbol{\theta}},
  33. 𝐫 ¨ = ( r ¨ 𝐫 ^ + r ˙ 𝐫 ^ ˙ ) + ( r ˙ θ ˙ s y m b o l θ ^ + r θ ¨ s y m b o l θ ^ + r θ ˙ s y m b o l θ ^ ˙ ) = ( r ¨ - r θ ˙ 2 ) 𝐫 ^ + ( r θ ¨ + 2 r ˙ θ ˙ ) s y m b o l θ ^ . \ddot{\mathbf{r}}=(\ddot{r}\hat{\mathbf{r}}+\dot{r}\dot{\hat{\mathbf{r}}})+(% \dot{r}\dot{\theta}\hat{symbol{\theta}}+r\ddot{\theta}\hat{symbol{\theta}}+r% \dot{\theta}\dot{\hat{symbol{\theta}}})=(\ddot{r}-r\dot{\theta}^{2})\hat{% \mathbf{r}}+(r\ddot{\theta}+2\dot{r}\dot{\theta})\hat{symbol{\theta}}.
  34. 𝐫 ¨ = a r s y m b o l r ^ + a θ s y m b o l θ ^ \ddot{\mathbf{r}}=a_{r}\hat{symbol{r}}+a_{\theta}\hat{symbol{\theta}}
  35. a r = r ¨ - r θ ˙ 2 a_{r}=\ddot{r}-r\dot{\theta}^{2}
  36. a θ = r θ ¨ + 2 r ˙ θ ˙ . a_{\theta}=r\ddot{\theta}+2\dot{r}\dot{\theta}.
  37. r 2 θ ˙ = n a b r^{2}\dot{\theta}=nab
  38. a θ a_{\theta}
  39. d ( r 2 θ ˙ ) d t = r ( 2 r ˙ θ ˙ + r θ ¨ ) = r a θ = 0. \frac{d(r^{2}\dot{\theta})}{dt}=r(2\dot{r}\dot{\theta}+r\ddot{\theta})=ra_{% \theta}=0.
  40. a r a_{r}
  41. a r = r ¨ - r θ ˙ 2 = r ¨ - r ( n a b r 2 ) 2 = r ¨ - n 2 a 2 b 2 r 3 . a_{r}=\ddot{r}-r\dot{\theta}^{2}=\ddot{r}-r\left(\frac{nab}{r^{2}}\right)^{2}=% \ddot{r}-\frac{n^{2}a^{2}b^{2}}{r^{3}}.
  42. p r = 1 + ε cos θ . \frac{p}{r}=1+\varepsilon\cos\theta.
  43. - p r ˙ r 2 = - ε sin θ θ ˙ -\frac{p\dot{r}}{r^{2}}=-\varepsilon\sin\theta\,\dot{\theta}
  44. p r ˙ = n a b ε sin θ . p\dot{r}=nab\,\varepsilon\sin\theta.
  45. p r ¨ = n a b ε cos θ θ ˙ = n a b ε cos θ n a b r 2 = n 2 a 2 b 2 r 2 ε cos θ . p\ddot{r}=nab\varepsilon\cos\theta\,\dot{\theta}=nab\varepsilon\cos\theta\,% \frac{nab}{r^{2}}=\frac{n^{2}a^{2}b^{2}}{r^{2}}\varepsilon\cos\theta.
  46. a r a_{r}
  47. p a r = n 2 a 2 b 2 r 2 ε cos θ - p n 2 a 2 b 2 r 3 = n 2 a 2 b 2 r 2 ( ε cos θ - p r ) . pa_{r}=\frac{n^{2}a^{2}b^{2}}{r^{2}}\varepsilon\cos\theta-p\frac{n^{2}a^{2}b^{% 2}}{r^{3}}=\frac{n^{2}a^{2}b^{2}}{r^{2}}\left(\varepsilon\cos\theta-\frac{p}{r% }\right).
  48. p a r = n 2 a 2 b 2 r 2 ( p r - 1 - p r ) = - n 2 a 2 r 2 b 2 . pa_{r}=\frac{n^{2}a^{2}b^{2}}{r^{2}}\left(\frac{p}{r}-1-\frac{p}{r}\right)=-% \frac{n^{2}a^{2}}{r^{2}}b^{2}.
  49. b 2 = p a b^{2}=pa
  50. a r = - n 2 a 3 r 2 . a_{r}=-\frac{n^{2}a^{3}}{r^{2}}.
  51. 𝐫 ¨ \mathbf{\ddot{r}}
  52. 𝐫 ¨ = - α r 2 𝐫 ^ \mathbf{\ddot{r}}=-\frac{\alpha}{r^{2}}\hat{\mathbf{r}}
  53. α = n 2 a 3 \alpha=n^{2}a^{3}\,
  54. 𝐫 ^ \hat{\mathbf{r}}
  55. r r\,
  56. α \alpha
  57. 𝐅 = m Planet 𝐫 ¨ = - m Planet α r - 2 𝐫 ^ \mathbf{F}=m\text{Planet}\mathbf{\ddot{r}}=-m\text{Planet}\alpha r^{-2}\hat{% \mathbf{r}}
  58. m Planet m\text{Planet}
  59. α \alpha
  60. m Sun m\text{Sun}
  61. α = G m Sun \alpha=Gm\text{Sun}
  62. G G
  63. 𝐫 ¨ 𝐢 = G j i m j r i j - 2 𝐫 ^ i j \mathbf{\ddot{r}_{i}}=G\sum_{j\neq i}m_{j}r_{ij}^{-2}\hat{\mathbf{r}}_{ij}
  64. m j m_{j}
  65. r i j r_{ij}
  66. 𝐫 ^ i j \hat{\mathbf{r}}_{ij}
  67. 𝐫 ¨ Earth = G m Sun r Earth , Sun - 2 𝐫 ^ Earth , Sun \mathbf{\ddot{r}}\text{Earth}=Gm\text{Sun}r_{\,\text{Earth},\,\text{Sun}}^{-2}% \hat{\mathbf{r}}_{\,\text{Earth},\,\text{Sun}}
  68. 𝐫 ¨ Moon = G m Earth r Moon , Earth - 2 𝐫 ^ Moon , Earth \mathbf{\ddot{r}}\text{Moon}=Gm\text{Earth}r_{\,\text{Moon},\,\text{Earth}}^{-% 2}\hat{\mathbf{r}}_{\,\text{Moon},\,\text{Earth}}
  69. 𝐫 ¨ Sun = G m Earth r Sun , Earth - 2 𝐫 ^ Sun , Earth + G m Moon r Sun , Moon - 2 𝐫 ^ Sun , Moon \mathbf{\ddot{r}}\text{Sun}=Gm\text{Earth}r_{\,\text{Sun},\,\text{Earth}}^{-2}% \hat{\mathbf{r}}_{\,\text{Sun},\,\text{Earth}}+Gm\text{Moon}r_{\,\text{Sun},\,% \text{Moon}}^{-2}\hat{\mathbf{r}}_{\,\text{Sun},\,\text{Moon}}
  70. 𝐫 ¨ Earth = G m Sun r Earth , Sun - 2 𝐫 ^ Earth , Sun + G m Moon r Earth , Moon - 2 𝐫 ^ Earth , Moon \mathbf{\ddot{r}}\text{Earth}=Gm\text{Sun}r_{\,\text{Earth},\,\text{Sun}}^{-2}% \hat{\mathbf{r}}_{\,\text{Earth},\,\text{Sun}}+Gm\text{Moon}r_{\,\text{Earth},% \,\text{Moon}}^{-2}\hat{\mathbf{r}}_{\,\text{Earth},\,\text{Moon}}
  71. 𝐫 ¨ Moon = G m Sun r Moon , Sun - 2 𝐫 ^ Moon , Sun + G m Earth r Moon , Earth - 2 𝐫 ^ Moon , Earth \mathbf{\ddot{r}}\text{Moon}=Gm\text{Sun}r_{\,\text{Moon},\,\text{Sun}}^{-2}% \hat{\mathbf{r}}_{\,\text{Moon},\,\text{Sun}}+Gm\text{Earth}r_{\,\text{Moon},% \,\text{Earth}}^{-2}\hat{\mathbf{r}}_{\,\text{Moon},\,\text{Earth}}
  72. n P = 2 π n\cdot P=2\pi
  73. M = E - ε sin E \ M=E-\varepsilon\sin E
  74. ( 1 - ε ) tan 2 θ 2 = ( 1 + ε ) tan 2 E 2 (1-\varepsilon)\tan^{2}\frac{\theta}{2}=(1+\varepsilon)\tan^{2}\frac{E}{2}
  75. r = a ( 1 - ε cos E ) . r=a(1-\varepsilon\cos E).
  76. a = | c z | , \ a=|cz|,
  77. ε = | c s | a , \ \varepsilon={|cs|\over a},
  78. b = a 1 - ε 2 , \ b=a\sqrt{1-\varepsilon^{2}},
  79. r = | s p | , \ r=|sp|,
  80. θ = z s p , \theta=\angle zsp,
  81. x , \ x,
  82. y , \ y,
  83. M = z c y , M=\angle zcy,
  84. | z s p | = b a | z s x | . |zsp|=\frac{b}{a}\cdot|zsx|.
  85. | z c y | = a 2 M 2 . \ |zcy|=\frac{a^{2}M}{2}.
  86. | z s p | = b a | z s x | = b a | z c y | = b a a 2 M 2 = a b M 2 , |zsp|=\frac{b}{a}\cdot|zsx|=\frac{b}{a}\cdot|zcy|=\frac{b}{a}\cdot\frac{a^{2}M% }{2}=\frac{abM}{2},
  87. M = n t , M=nt,\,
  88. E = z c x E=\angle zcx
  89. | z c y | = | z s x | = | z c x | - | s c x | \ |zcy|=|zsx|=|zcx|-|scx|
  90. a 2 M 2 = a 2 E 2 - a ε a sin E 2 \frac{a^{2}M}{2}=\frac{a^{2}E}{2}-\frac{a\varepsilon\cdot a\sin E}{2}
  91. M = E - ε sin E . M=E-\varepsilon\cdot\sin E.
  92. c d = c s + s d \overrightarrow{cd}=\overrightarrow{cs}+\overrightarrow{sd}
  93. a cos E = a ε + r cos θ . a\cdot\cos E=a\cdot\varepsilon+r\cdot\cos\theta.
  94. a a
  95. r a = 1 - ε 2 1 + ε cos θ \ \frac{r}{a}=\frac{1-\varepsilon^{2}}{1+\varepsilon\cdot\cos\theta}
  96. cos E = ε + 1 - ε 2 1 + ε cos θ cos θ \cos E=\varepsilon+\frac{1-\varepsilon^{2}}{1+\varepsilon\cdot\cos\theta}\cdot\cos\theta
  97. = ε ( 1 + ε cos θ ) + ( 1 - ε 2 ) cos θ 1 + ε cos θ =\frac{\varepsilon\cdot(1+\varepsilon\cdot\cos\theta)+(1-\varepsilon^{2})\cdot% \cos\theta}{1+\varepsilon\cdot\cos\theta}
  98. = ε + cos θ 1 + ε cos θ . =\frac{\varepsilon+\cos\theta}{1+\varepsilon\cdot\cos\theta}.
  99. tan 2 x 2 = 1 - cos x 1 + cos x . \tan^{2}\frac{x}{2}=\frac{1-\cos x}{1+\cos x}.
  100. tan 2 E 2 = 1 - cos E 1 + cos E \tan^{2}\frac{E}{2}=\frac{1-\cos E}{1+\cos E}
  101. = 1 - ε + cos θ 1 + ε cos θ 1 + ε + cos θ 1 + ε cos θ =\frac{1-\frac{\varepsilon+\cos\theta}{1+\varepsilon\cdot\cos\theta}}{1+\frac{% \varepsilon+\cos\theta}{1+\varepsilon\cdot\cos\theta}}
  102. = ( 1 + ε cos θ ) - ( ε + cos θ ) ( 1 + ε cos θ ) + ( ε + cos θ ) =\frac{(1+\varepsilon\cdot\cos\theta)-(\varepsilon+\cos\theta)}{(1+\varepsilon% \cdot\cos\theta)+(\varepsilon+\cos\theta)}
  103. = 1 - ε 1 + ε 1 - cos θ 1 + cos θ = 1 - ε 1 + ε tan 2 θ 2 . =\frac{1-\varepsilon}{1+\varepsilon}\cdot\frac{1-\cos\theta}{1+\cos\theta}=% \frac{1-\varepsilon}{1+\varepsilon}\cdot\tan^{2}\frac{\theta}{2}.
  104. ( 1 - ε ) tan 2 θ 2 = ( 1 + ε ) tan 2 E 2 (1-\varepsilon)\cdot\tan^{2}\frac{\theta}{2}=(1+\varepsilon)\cdot\tan^{2}\frac% {E}{2}
  105. r ( 1 + ε cos θ ) = a ( 1 - ε 2 ) \ r\cdot(1+\varepsilon\cdot\cos\theta)=a\cdot(1-\varepsilon^{2})
  106. r = a ( 1 - ε cos E ) . \ r=a\cdot(1-\varepsilon\cdot\cos E).

Kepler–Poinsot_polyhedron.html

  1. χ = V - E + F = 2 \chi=V-E+F=2
  2. d v d_{v}
  3. d f d_{f}
  4. d v V - E + d f F = 2 D . d_{v}V-E+d_{f}F=2D.

Kernel_(algebra).html

  1. k e r T kerT
  2. ker T := { 𝐯 V : T 𝐯 = 𝟎 W } . \operatorname{ker}T:=\{\mathbf{v}\in V:T\mathbf{v}=\mathbf{0}_{W}\}\,\text{.}
  3. k e r f kerf
  4. ker f := { g G : f ( g ) = e H } . \operatorname{ker}f:=\{g\in G:f(g)=e_{H}\}\mbox{.}~{}
  5. k e r f kerf
  6. ker f := { r R : f ( r ) = 0 S } . \operatorname{ker}f:=\{r\in R:f(r)=0_{S}\}\mbox{.}~{}\!
  7. k e r f kerf
  8. ker f := { ( m , m ) M × M : f ( m ) = f ( m ) } . \operatorname{ker}f:=\{(m,m^{\prime})\in M\times M:f(m)=f(m^{\prime})\}\mbox{.% }~{}\!
  9. k e r f kerf
  10. ker f := { ( a , a ) A × A : f ( a ) = f ( a ) } . \operatorname{ker}f:=\{(a,a^{\prime})\in A\times A:f(a)=f(a^{\prime})\}\mbox{.% }~{}\!
  11. k e r f kerf
  12. ker f := { a A : f ( a ) = e B } . \operatorname{ker}f:=\{a\in A:f(a)=e_{B}\}\mbox{.}~{}\!

Ket.html

  1. | ψ |\psi\rangle

Kilogram_per_cubic_metre.html

  1. kg m 3 \tfrac{\,\text{kg}}{\,\text{m}^{3}}\!

Kin_selection.html

  1. r B > C rB>C

Kinematics.html

  1. 𝐏 = ( x P , y P , z P ) = x P i ^ + y P j ^ + z P k ^ , \mathbf{P}=(x_{P},y_{P},z_{P})=x_{P}\hat{i}+y_{P}\hat{j}+z_{P}\hat{k},
  2. | 𝐏 | = x P 2 + y P 2 + z P 2 . |\mathbf{P}|=\sqrt{x_{P}^{\ 2}+y_{P}^{\ 2}+z_{P}^{\ 2}}.
  3. 𝐏 ( t ) = x P ( t ) i ^ + y P ( t ) j ^ + z P ( t ) k ^ , \mathbf{P}(t)=x_{P}(t)\hat{i}+y_{P}(t)\hat{j}+z_{P}(t)\hat{k},
  4. 𝐕 ¯ = Δ 𝐏 Δ t , \overline{\mathbf{V}}=\frac{\Delta\mathbf{P}}{\Delta t}\ ,
  5. 𝐕 = lim Δ t 0 Δ 𝐏 Δ t = d 𝐏 d t = 𝐏 ˙ = x ˙ p i ^ + y ˙ P j ^ + z ˙ P k ^ . \mathbf{V}=\lim_{\Delta t\rightarrow 0}\frac{\Delta\mathbf{P}}{\Delta t}=\frac% {d\mathbf{P}}{dt}=\dot{\mathbf{P}}=\dot{x}_{p}\hat{i}+\dot{y}_{P}\hat{j}+\dot{% z}_{P}\hat{k}.
  6. | 𝐕 | = | 𝐏 ˙ | = d s d t , |\mathbf{V}|=|\dot{\mathbf{P}}|=\frac{ds}{dt},
  7. 𝐀 ¯ = Δ 𝐕 Δ t , \overline{\mathbf{A}}=\frac{\Delta\mathbf{V}}{\Delta t}\ ,
  8. 𝐀 = lim Δ t 0 Δ 𝐕 Δ t = d 𝐕 d t = 𝐕 ˙ = 𝐏 ¨ = x ¨ p i ^ + y ¨ P j ^ + z ¨ P k ^ . \mathbf{A}=\lim_{\Delta t\rightarrow 0}\frac{\Delta\mathbf{V}}{\Delta t}=\frac% {d\mathbf{V}}{dt}=\dot{\mathbf{V}}=\ddot{\mathbf{P}}=\ddot{x}_{p}\hat{i}+\ddot% {y}_{P}\hat{j}+\ddot{z}_{P}\hat{k}.
  9. 𝐑 B / A = 𝐏 B - 𝐏 A = ( x B - x A , y B - y A , z B - z A ) . \mathbf{R}_{B/A}=\mathbf{P}_{B}-\mathbf{P}_{A}=(x_{B}-x_{A},y_{B}-y_{A},z_{B}-% z_{A}).
  10. 𝐏 B = 𝐏 A + ( 𝐏 B - 𝐏 A ) = 𝐏 A + 𝐑 B / A . \mathbf{P}_{B}=\mathbf{P}_{A}+(\mathbf{P}_{B}-\mathbf{P}_{A})=\mathbf{P}_{A}+% \mathbf{R}_{B/A}.
  11. 𝐕 B / A = 𝐕 B - 𝐕 A . \mathbf{V}_{B/A}=\mathbf{V}_{B}-\mathbf{V}_{A}\,\!.
  12. 𝐕 B = 𝐕 A + 𝐕 B / A . \mathbf{V}_{B}=\mathbf{V}_{A}+\mathbf{V}_{B/A}\,\!.
  13. 𝐕 ( t ) = 0 t 𝐀 d t = 𝐀 t + 𝐕 0 . \mathbf{V}(t)=\int_{0}^{t}\mathbf{A}dt=\mathbf{A}t+\mathbf{V}_{0}.
  14. 𝐏 ( t ) = 0 t 𝐕 ( t ) d t = ( 𝐀 t + 𝐕 0 ) d t = 1 2 𝐀 t 2 + 𝐕 0 t + 𝐏 0 . \mathbf{P}(t)=\int_{0}^{t}\mathbf{V}(t)dt=\int(\mathbf{A}t+\mathbf{V}_{0})dt=% \tfrac{1}{2}\mathbf{A}t^{2}+\mathbf{V}_{0}t+\mathbf{P}_{0}.
  15. 𝐏 ( t ) = 𝐏 0 + ( 𝐕 + 𝐕 0 2 ) t . \mathbf{P}(t)=\mathbf{P}_{0}+\left(\frac{\mathbf{V}+\mathbf{V}_{0}}{2}\right)t.
  16. ( 𝐏 - 𝐏 0 ) 𝐀 t = ( 𝐕 - 𝐕 0 ) 𝐕 + 𝐕 0 2 t , (\mathbf{P}-\mathbf{P}_{0})\cdot\mathbf{A}t=\left(\mathbf{V}-\mathbf{V}_{0}% \right)\cdot\frac{\mathbf{V}+\mathbf{V}_{0}}{2}t\ ,
  17. 2 ( 𝐏 - 𝐏 0 ) 𝐀 = | 𝐕 | 2 - | 𝐕 0 | 2 . 2(\mathbf{P}-\mathbf{P}_{0})\cdot\mathbf{A}=|\mathbf{V}|^{2}-|\mathbf{V}_{0}|^% {2}.
  18. | 𝐕 | 2 = | 𝐕 0 | 2 + 2 | 𝐀 | ( | 𝐏 - 𝐏 0 | ) . |\mathbf{V}|^{2}=|\mathbf{V}_{0}|^{2}+2|\mathbf{A}|(|\mathbf{P}-\mathbf{P}_{0}% |).
  19. v 2 = v 0 2 + 2 a ( r - r 0 ) . v^{2}=v_{0}^{2}+2a(r-r_{0}).
  20. 𝐏 ( t ) = X ( t ) i ^ + Y ( t ) j ^ + Z ( t ) k ^ , \,\textbf{P}(t)=X(t)\hat{i}+Y(t)\hat{j}+Z(t)\hat{k},
  21. 𝐏 ( t ) = R cos θ ( t ) i ^ + R sin θ ( t ) j ^ + Z ( t ) k ^ . \,\textbf{P}(t)=R\cos\theta(t)\hat{i}+R\sin\theta(t)\hat{j}+Z(t)\hat{k}.
  22. 𝐞 r = cos θ ( t ) i ^ + sin θ ( t ) j ^ , 𝐞 t = - sin θ ( t ) i ^ + cos θ ( t ) j ^ . \,\textbf{e}_{r}=\cos\theta(t)\hat{i}+\sin\theta(t)\hat{j},\quad\,\textbf{e}_{% t}=-\sin\theta(t)\hat{i}+\cos\theta(t)\hat{j}.
  23. 𝐏 ( t ) = R 𝐞 r + Z ( t ) k ^ , \,\textbf{P}(t)=R\,\textbf{e}_{r}+Z(t)\hat{k},
  24. 𝐏 ( t ) = R ( t ) 𝐞 r + Z ( t ) k ^ . \,\textbf{P}(t)=R(t)\,\textbf{e}_{r}+Z(t)\hat{k}.
  25. 𝐕 P = d d t ( R ( t ) 𝐞 r + Z ( t ) k ^ ) = R ˙ 𝐞 r + R θ ˙ 𝐞 t + Z ˙ k ^ , \,\textbf{V}_{P}=\frac{d}{dt}(R(t)\,\textbf{e}_{r}+Z(t)\hat{k})=\dot{R}\,% \textbf{e}_{r}+R\dot{\theta}\,\textbf{e}_{t}+\dot{Z}\hat{k},
  26. d d t 𝐞 r = θ ˙ 𝐞 t . \frac{d}{dt}\,\textbf{e}_{r}=\dot{\theta}\,\textbf{e}_{t}.
  27. 𝐀 P = d d t ( R ˙ 𝐞 r + R θ ˙ 𝐞 t + Z ˙ ( t ) k ^ ) = ( R ¨ - R θ ˙ 2 ) 𝐞 r + ( R θ ¨ + 2 R ˙ θ ˙ ) 𝐞 t + Z ¨ ( t ) k ^ . \,\textbf{A}_{P}=\frac{d}{dt}(\dot{R}\,\textbf{e}_{r}+R\dot{\theta}\,\textbf{e% }_{t}+\dot{Z}(t)\hat{k})=(\ddot{R}-R\dot{\theta}^{2})\,\textbf{e}_{r}+(R\ddot{% \theta}+2\dot{R}\dot{\theta})\,\textbf{e}_{t}+\ddot{Z}(t)\hat{k}.
  28. 𝐕 P = d d t ( R 𝐞 r + Z ( t ) k ^ ) = R θ ˙ 𝐞 t + Z ˙ k ^ . \,\textbf{V}_{P}=\frac{d}{dt}(R\,\textbf{e}_{r}+Z(t)\hat{k})=R\dot{\theta}\,% \textbf{e}_{t}+\dot{Z}\hat{k}.
  29. 𝐀 P = d d t ( R θ ˙ 𝐞 t + Z ˙ k ^ ) = - R θ ˙ 2 𝐞 r + R θ ¨ 𝐞 t + Z ¨ k ^ . \,\textbf{A}_{P}=\frac{d}{dt}(R\dot{\theta}\,\textbf{e}_{t}+\dot{Z}\hat{k})=-R% \dot{\theta}^{2}\,\textbf{e}_{r}+R\ddot{\theta}\,\textbf{e}_{t}+\ddot{Z}\hat{k}.
  30. 𝐏 ( t ) = R 𝐞 r + Z 0 k ^ , \,\textbf{P}(t)=R\,\textbf{e}_{r}+Z_{0}\hat{k},
  31. 𝐕 P = d d t ( R 𝐞 r + Z 0 k ^ ) = R θ ˙ 𝐞 t = R ω 𝐞 t , \,\textbf{V}_{P}=\frac{d}{dt}(R\,\textbf{e}_{r}+Z_{0}\hat{k})=R\dot{\theta}\,% \textbf{e}_{t}=R\omega\,\textbf{e}_{t},
  32. ω = θ ˙ , \omega=\dot{\theta},
  33. 𝐀 P = d d t ( R θ ˙ 𝐞 t ) = - R θ ˙ 2 𝐞 r + R θ ¨ 𝐞 t . \,\textbf{A}_{P}=\frac{d}{dt}(R\dot{\theta}\,\textbf{e}_{t})=-R\dot{\theta}^{2% }\,\textbf{e}_{r}+R\ddot{\theta}\,\textbf{e}_{t}.
  34. a r = - R θ ˙ 2 , a t = R θ ¨ , a_{r}=-R\dot{\theta}^{2},\quad a_{t}=R\ddot{\theta},
  35. ω = θ ˙ , α = θ ¨ , \omega=\dot{\theta},\quad\alpha=\ddot{\theta},
  36. a r = - R ω 2 , a t = R α . a_{r}=-R\omega^{2},\quad a_{t}=R\alpha.
  37. [ T ( ϕ , 𝐝 ) ] = [ A ( ϕ ) 𝐝 0 , 0 1 ] = [ cos ϕ - sin ϕ d x sin ϕ cos ϕ d y 0 0 1 ] . [T(\phi,\mathbf{d})]=\begin{bmatrix}A(\phi)&\mathbf{d}\\ 0,0&1\end{bmatrix}=\begin{bmatrix}\cos\phi&-\sin\phi&d_{x}\\ \sin\phi&\cos\phi&d_{y}\\ 0&0&1\end{bmatrix}.
  38. 𝐏 = [ T ( ϕ , 𝐝 ) ] 𝐩 = [ cos ϕ - sin ϕ d x sin ϕ cos ϕ d y 0 0 1 ] { x y 1 } . \,\textbf{P}=[T(\phi,\mathbf{d})]\,\textbf{p}=\begin{bmatrix}\cos\phi&-\sin% \phi&d_{x}\\ \sin\phi&\cos\phi&d_{y}\\ 0&0&1\end{bmatrix}\begin{Bmatrix}x\\ y\\ 1\end{Bmatrix}.
  39. 𝐏 ( t ) = [ T ( 0 , 𝐝 ( t ) ) ] 𝐩 = 𝐝 ( t ) + 𝐩 . \,\textbf{P}(t)=[T(0,\,\textbf{d}(t))]\,\textbf{p}=\,\textbf{d}(t)+\,\textbf{p}.
  40. 𝐕 P = 𝐏 ˙ ( t ) = 𝐝 ˙ ( t ) = 𝐕 O , 𝐀 P = 𝐏 ¨ ( t ) = 𝐝 ¨ ( t ) = 𝐀 O , \,\textbf{V}_{P}=\dot{\,\textbf{P}}(t)=\dot{\,\textbf{d}}(t)=\,\textbf{V}_{O},% \quad\,\textbf{A}_{P}=\ddot{\,\textbf{P}}(t)=\ddot{\,\textbf{d}}(t)=\,\textbf{% A}_{O},
  41. 𝐏 ( t ) = [ A ( t ) ] 𝐩 , \mathbf{P}(t)=[A(t)]\mathbf{p},
  42. [ A ( t ) ] = [ cos θ ( t ) - sin θ ( t ) sin θ ( t ) cos θ ( t ) ] , [A(t)]=\begin{bmatrix}\cos\theta(t)&-\sin\theta(t)\\ \sin\theta(t)&\cos\theta(t)\end{bmatrix},
  43. 𝐕 P = 𝐏 ˙ = [ A ˙ ( t ) ] 𝐩 . \mathbf{V}_{P}=\dot{\mathbf{P}}=[\dot{A}(t)]\mathbf{p}.
  44. 𝐕 P = [ A ˙ ( t ) ] [ A ( t ) - 1 ] 𝐏 = [ Ω ] 𝐏 , \mathbf{V}_{P}=[\dot{A}(t)][A(t)^{-1}]\mathbf{P}=[\Omega]\mathbf{P},
  45. [ Ω ] = [ 0 - ω ω 0 ] , [\Omega]=\begin{bmatrix}0&-\omega\\ \omega&0\end{bmatrix},
  46. ω = d θ d t . \omega=\frac{d\theta}{dt}.
  47. 𝐀 P = P ¨ ( t ) = [ Ω ˙ ] 𝐏 + [ Ω ] 𝐏 ˙ , \mathbf{A}_{P}=\ddot{P}(t)=[\dot{\Omega}]\mathbf{P}+[\Omega]\dot{\mathbf{P}},
  48. 𝐀 P = [ Ω ˙ ] 𝐏 + [ Ω ] [ Ω ] 𝐏 , \mathbf{A}_{P}=[\dot{\Omega}]\mathbf{P}+[\Omega][\Omega]\mathbf{P},
  49. [ Ω ˙ ] = [ 0 - α α 0 ] , [\dot{\Omega}]=\begin{bmatrix}0&-\alpha\\ \alpha&0\end{bmatrix},
  50. α = d 2 θ d t 2 . \alpha=\frac{d^{2}\theta}{dt^{2}}.
  51. ω = d θ d t \omega=\frac{d\theta}{dt}
  52. α = d ω d t \alpha=\frac{d\omega}{dt}
  53. ω f = ω i + α t \omega_{\mathrm{f}}=\omega_{\mathrm{i}}+\alpha t\!
  54. θ f - θ i = ω i t + 1 2 α t 2 \theta_{\mathrm{f}}-\theta_{\mathrm{i}}=\omega_{\mathrm{i}}t+\tfrac{1}{2}% \alpha t^{2}
  55. θ f - θ i = 1 2 ( ω f + ω i ) t \theta_{\mathrm{f}}-\theta_{\mathrm{i}}=\tfrac{1}{2}(\omega_{\mathrm{f}}+% \omega_{\mathrm{i}})t
  56. ω f 2 = ω i 2 + 2 α ( θ f - θ i ) . \omega_{\mathrm{f}}^{2}=\omega_{\mathrm{i}}^{2}+2\alpha(\theta_{\mathrm{f}}-% \theta_{\mathrm{i}}).
  57. 𝐏 ( t ) = [ T ( t ) ] 𝐩 = { 𝐏 1 } = [ A ( t ) 𝐝 ( t ) 0 1 ] { 𝐩 1 } . \,\textbf{P}(t)=[T(t)]\,\textbf{p}=\begin{Bmatrix}\,\textbf{P}\\ 1\end{Bmatrix}=\begin{bmatrix}A(t)&\,\textbf{d}(t)\\ 0&1\end{bmatrix}\begin{Bmatrix}\,\textbf{p}\\ 1\end{Bmatrix}.
  58. 𝐩 = [ T ( t ) ] - 1 𝐏 ( t ) = { 𝐩 1 } = [ A ( t ) T - A ( t ) T 𝐝 ( t ) 0 1 ] { 𝐏 ( t ) 1 } . \,\textbf{p}=[T(t)]^{-1}\,\textbf{P}(t)=\begin{Bmatrix}\,\textbf{p}\\ 1\end{Bmatrix}=\begin{bmatrix}A(t)^{T}&-A(t)^{T}\,\textbf{d}(t)\\ 0&1\end{bmatrix}\begin{Bmatrix}\,\textbf{P}(t)\\ 1\end{Bmatrix}.
  59. [ A ( t ) ] T [ A ( t ) ] = I . [A(t)]^{T}[A(t)]=I.\!
  60. 𝐕 P = [ T ˙ ( t ) ] 𝐩 = { 𝐕 P 0 } = [ A ˙ ( t ) 𝐝 ˙ ( t ) 0 0 ] { 𝐩 1 } . \,\textbf{V}_{P}=[\dot{T}(t)]\,\textbf{p}=\begin{Bmatrix}\,\textbf{V}_{P}\\ 0\end{Bmatrix}=\begin{bmatrix}\dot{A}(t)&\dot{\,\textbf{d}}(t)\\ 0&0\end{bmatrix}\begin{Bmatrix}\,\textbf{p}\\ 1\end{Bmatrix}.
  61. 𝐕 P = [ T ˙ ( t ) ] [ T ( t ) ] - 1 𝐏 ( t ) = { 𝐕 P 0 } = [ A ˙ A T - A ˙ A T 𝐝 + 𝐝 ˙ 0 0 ] { 𝐏 ( t ) 1 } = [ S ] 𝐏 . \,\textbf{V}_{P}=[\dot{T}(t)][T(t)]^{-1}\,\textbf{P}(t)=\begin{Bmatrix}\,% \textbf{V}_{P}\\ 0\end{Bmatrix}=\begin{bmatrix}\dot{A}A^{T}&-\dot{A}A^{T}\,\textbf{d}+\dot{\,% \textbf{d}}\\ 0&0\end{bmatrix}\begin{Bmatrix}\,\textbf{P}(t)\\ 1\end{Bmatrix}=[S]\,\textbf{P}.
  62. [ S ] = [ Ω - Ω 𝐝 + 𝐝 ˙ 0 0 ] [S]=\begin{bmatrix}\Omega&-\Omega\,\textbf{d}+\dot{\,\textbf{d}}\\ 0&0\end{bmatrix}
  63. [ Ω ] = A ˙ A T , [\Omega]=\dot{A}A^{T},
  64. 𝐕 P = [ Ω ] ( 𝐏 - 𝐝 ) + 𝐝 ˙ = ω × 𝐑 P / O + 𝐕 O , \,\textbf{V}_{P}=[\Omega](\,\textbf{P}-\,\textbf{d})+\dot{\,\textbf{d}}=\omega% \times\,\textbf{R}_{P/O}+\,\textbf{V}_{O},
  65. 𝐑 P / O = 𝐏 - 𝐝 , \,\textbf{R}_{P/O}=\,\textbf{P}-\,\textbf{d},
  66. 𝐕 O = 𝐝 ˙ , \,\textbf{V}_{O}=\dot{\,\textbf{d}},
  67. 𝐀 P = d d t 𝐕 P = d d t ( [ S ] 𝐏 ) = [ S ˙ ] 𝐏 + [ S ] 𝐏 ˙ = [ S ˙ ] 𝐏 + [ S ] [ S ] 𝐏 . \,\textbf{A}_{P}=\frac{d}{dt}\,\textbf{V}_{P}=\frac{d}{dt}\big([S]\,\textbf{P}% \big)=[\dot{S}]\,\textbf{P}+[S]\dot{\,\textbf{P}}=[\dot{S}]\,\textbf{P}+[S][S]% \,\textbf{P}.
  68. [ S ˙ ] = [ Ω ˙ - Ω ˙ 𝐝 - Ω 𝐝 ˙ + 𝐝 ¨ 0 0 ] = [ Ω ˙ - Ω ˙ 𝐝 - Ω 𝐕 O + 𝐀 O 0 0 ] [\dot{S}]=\begin{bmatrix}\dot{\Omega}&-\dot{\Omega}\,\textbf{d}-\Omega\dot{\,% \textbf{d}}+\ddot{\,\textbf{d}}\\ 0&0\end{bmatrix}=\begin{bmatrix}\dot{\Omega}&-\dot{\Omega}\,\textbf{d}-\Omega% \,\textbf{V}_{O}+\,\textbf{A}_{O}\\ 0&0\end{bmatrix}
  69. [ S ] 2 = [ Ω - Ω 𝐝 + 𝐕 O 0 0 ] 2 = [ Ω 2 - Ω 2 𝐝 + Ω 𝐕 O 0 0 ] . [S]^{2}=\begin{bmatrix}\Omega&-\Omega\,\textbf{d}+\,\textbf{V}_{O}\\ 0&0\end{bmatrix}^{2}=\begin{bmatrix}\Omega^{2}&-\Omega^{2}\,\textbf{d}+\Omega% \,\textbf{V}_{O}\\ 0&0\end{bmatrix}.
  70. 𝐀 P = Ω ˙ ( 𝐏 - 𝐝 ) + 𝐀 O + Ω 2 ( 𝐏 - 𝐝 ) , \,\textbf{A}_{P}=\dot{\Omega}(\,\textbf{P}-\,\textbf{d})+\,\textbf{A}_{O}+% \Omega^{2}(\,\textbf{P}-\,\textbf{d}),
  71. 𝐀 P = α × 𝐑 P / O + ω × ω × 𝐑 P / O + 𝐀 O , \,\textbf{A}_{P}=\alpha\times\,\textbf{R}_{P/O}+\omega\times\omega\times\,% \textbf{R}_{P/O}+\,\textbf{A}_{O},
  72. 𝐑 P / O = 𝐏 - 𝐝 , \,\textbf{R}_{P/O}=\,\textbf{P}-\,\textbf{d},
  73. 𝐀 O = 𝐝 ¨ \,\textbf{A}_{O}=\ddot{\,\textbf{d}}
  74. s y m b o l v G ( t ) = s y m b o l Ω × s y m b o l r G / O . symbol{v}_{G}(t)=symbol{\Omega}\times symbol{r}_{G/O}.
  75. v = r ω v=r\omega

Kinetic_energy.html

  1. E k = 1 2 m v 2 E\text{k}=\tfrac{1}{2}mv^{2}
  2. m m
  3. v v
  4. E k = 1 2 80 kg ( 18 m/s ) 2 = 12960 J = 12.96 kJ E\text{k}=\frac{1}{2}\cdot 80\,\,\text{kg}\cdot\left(18\,\,\text{m/s}\right)^{% 2}=12960\,\,\text{J}=12.96\,\,\text{kJ}
  5. F s = 1 2 m v 2 Fs=\tfrac{1}{2}mv^{2}
  6. E k = p 2 2 m E\text{k}=\frac{p^{2}}{2m}
  7. p p\;
  8. m m\;
  9. m m\;
  10. v v\;
  11. E t = 1 2 m v 2 E\text{t}=\tfrac{1}{2}mv^{2}
  12. m m\;
  13. v v\;
  14. 𝐅 d 𝐱 = 𝐅 𝐯 d t = d 𝐩 d t 𝐯 d t = 𝐯 d 𝐩 = 𝐯 d ( m 𝐯 ) , \mathbf{F}\cdot d\mathbf{x}=\mathbf{F}\cdot\mathbf{v}dt=\frac{d\mathbf{p}}{dt}% \cdot\mathbf{v}dt=\mathbf{v}\cdot d\mathbf{p}=\mathbf{v}\cdot d(m\mathbf{v})\,,
  15. d ( 𝐯 𝐯 ) = ( d 𝐯 ) 𝐯 + 𝐯 ( d 𝐯 ) = 2 ( 𝐯 d 𝐯 ) . d(\mathbf{v}\cdot\mathbf{v})=(d\mathbf{v})\cdot\mathbf{v}+\mathbf{v}\cdot(d% \mathbf{v})=2(\mathbf{v}\cdot d\mathbf{v}).
  16. 𝐯 d ( m 𝐯 ) = m 2 d ( 𝐯 𝐯 ) = m 2 d v 2 = d ( m v 2 2 ) . \mathbf{v}\cdot d(m\mathbf{v})=\frac{m}{2}d(\mathbf{v}\cdot\mathbf{v})=\frac{m% }{2}dv^{2}=d\left(\frac{mv^{2}}{2}\right).
  17. E k = 0 t 𝐅 d 𝐱 = 0 t 𝐯 d ( m 𝐯 ) = 0 v d ( m v 2 2 ) = m v 2 2 . E\text{k}=\int_{0}^{t}\mathbf{F}\cdot d\mathbf{x}=\int_{0}^{t}\mathbf{v}\cdot d% (m\mathbf{v})=\int_{0}^{v}d\left(\frac{mv^{2}}{2}\right)=\frac{mv^{2}}{2}.
  18. E r E\text{r}\,
  19. E r = Q v 2 d m 2 = Q ( r ω ) 2 d m 2 = ω 2 2 Q r 2 d m = ω 2 2 I = 1 2 I ω 2 E\text{r}=\int_{Q}\frac{v^{2}dm}{2}=\int_{Q}\frac{(r\omega)^{2}dm}{2}=\frac{% \omega^{2}}{2}\int_{Q}{r^{2}}dm=\frac{\omega^{2}}{2}I=\begin{matrix}\frac{1}{2% }\end{matrix}I\omega^{2}
  20. I I\,
  21. Q r 2 d m \int_{Q}{r^{2}}dm
  22. 𝐕 \textstyle\mathbf{V}
  23. v 2 = ( v i + V ) 2 = ( 𝐯 i + 𝐕 ) ( 𝐯 i + 𝐕 ) = 𝐯 i 𝐯 i + 2 𝐯 i 𝐕 + 𝐕 𝐕 = v i 2 + 2 𝐯 i 𝐕 + V 2 \textstyle v^{2}=(v_{i}+V)^{2}=(\mathbf{v}_{i}+\mathbf{V})\cdot(\mathbf{v}_{i}% +\mathbf{V})=\mathbf{v}_{i}\cdot\mathbf{v}_{i}+2\mathbf{v}_{i}\cdot\mathbf{V}+% \mathbf{V}\cdot\mathbf{V}=v_{i}^{2}+2\mathbf{v}_{i}\cdot\mathbf{V}+V^{2}
  24. E k = v 2 2 d m = v i 2 2 d m + 𝐕 𝐯 i d m + V 2 2 d m . E\text{k}=\int\frac{v^{2}}{2}dm=\int\frac{v_{i}^{2}}{2}dm+\mathbf{V}\cdot\int% \mathbf{v}_{i}dm+\frac{V^{2}}{2}\int dm.
  25. v i 2 2 d m = E i \int\frac{v_{i}^{2}}{2}dm=E_{i}
  26. 𝐯 i d m \int\mathbf{v}_{i}dm
  27. d m = M \int dm=M
  28. E k = E i + M V 2 2 . E\text{k}=E_{i}+\frac{MV^{2}}{2}.
  29. E k = E t + E r E\text{k}=E_{t}+E\text{r}\,
  30. 𝐩 = m γ 𝐯 \mathbf{p}=m\gamma\mathbf{v}
  31. γ = 1 / 1 - v 2 / c 2 \gamma=1/\sqrt{1-v^{2}/c^{2}}
  32. E k = 𝐯 d 𝐩 = 𝐯 d ( m γ 𝐯 ) = m γ 𝐯 𝐯 - m γ 𝐯 d 𝐯 = m γ v 2 - m 2 γ d ( v 2 ) E\text{k}=\int\mathbf{v}\cdot d\mathbf{p}=\int\mathbf{v}\cdot d(m\gamma\mathbf% {v})=m\gamma\mathbf{v}\cdot\mathbf{v}-\int m\gamma\mathbf{v}\cdot d\mathbf{v}=% m\gamma v^{2}-\frac{m}{2}\int\gamma d(v^{2})
  33. γ = ( 1 - v 2 / c 2 ) - 1 / 2 \gamma=(1-v^{2}/c^{2})^{-1/2}\!
  34. E k = m γ v 2 - - m c 2 2 γ d ( 1 - v 2 / c 2 ) = m γ v 2 + m c 2 ( 1 - v 2 / c 2 ) 1 / 2 - E 0 \begin{aligned}\displaystyle E\text{k}&\displaystyle=m\gamma v^{2}-\frac{-mc^{% 2}}{2}\int\gamma d(1-v^{2}/c^{2})\\ &\displaystyle=m\gamma v^{2}+mc^{2}(1-v^{2}/c^{2})^{1/2}-E_{0}\end{aligned}
  35. E 0 E_{0}
  36. E k = m γ ( v 2 + c 2 ( 1 - v 2 / c 2 ) ) - E 0 = m γ ( v 2 + c 2 - v 2 ) - E 0 = m γ c 2 - E 0 \begin{aligned}\displaystyle E\text{k}&\displaystyle=m\gamma(v^{2}+c^{2}(1-v^{% 2}/c^{2}))-E_{0}\\ &\displaystyle=m\gamma(v^{2}+c^{2}-v^{2})-E_{0}\\ &\displaystyle=m\gamma c^{2}-E_{0}\end{aligned}
  37. E 0 E_{0}
  38. 𝐯 = 0 , γ = 1 \mathbf{v}=0,\ \gamma=1\!
  39. E k = 0 E\text{k}=0\!
  40. E 0 = m c 2 E_{0}=mc^{2}\,
  41. E k = m γ c 2 - m c 2 = m c 2 1 - v 2 / c 2 - m c 2 E\text{k}=m\gamma c^{2}-mc^{2}=\frac{mc^{2}}{\sqrt{1-v^{2}/c^{2}}}-mc^{2}
  42. E rest = E 0 = m c 2 E\text{rest}=E_{0}=mc^{2}\!
  43. v v
  44. E k m c 2 ( 1 + 1 2 v 2 / c 2 ) - m c 2 = 1 2 m v 2 E\text{k}\approx mc^{2}\left(1+\frac{1}{2}v^{2}/c^{2}\right)-mc^{2}=\frac{1}{2% }mv^{2}
  45. E k E_{k}
  46. E k m c 2 ( 1 + 1 2 v 2 / c 2 + 3 8 v 4 / c 4 ) - m c 2 = 1 2 m v 2 + 3 8 m v 4 / c 2 E\text{k}\approx mc^{2}\left(1+\frac{1}{2}v^{2}/c^{2}+\frac{3}{8}v^{4}/c^{4}% \right)-mc^{2}=\frac{1}{2}mv^{2}+\frac{3}{8}mv^{4}/c^{2}
  47. E k = p 2 c 2 + m 2 c 4 - m c 2 E\text{k}=\sqrt{p^{2}c^{2}+m^{2}c^{4}}-mc^{2}
  48. g α β u α u β = - c 2 g_{\alpha\beta}\,u^{\alpha}\,u^{\beta}\,=\,-c^{2}
  49. u α = d x α d τ u^{\alpha}\,=\,\frac{dx^{\alpha}}{d\tau}
  50. τ \tau\,
  51. p β = m g β α u α p_{\beta}\,=\,m\,g_{\beta\alpha}\,u^{\alpha}
  52. E = - p β u obs β E\,=\,-\,p_{\beta}\,u_{\,\text{obs}}^{\beta}
  53. E k = - p β u obs β - m c 2 . E_{k}\,=\,-\,p_{\beta}\,u_{\,\text{obs}}^{\beta}\,-\,m\,c^{2}\,.
  54. u α = d x α d t d t d τ = v α u t u^{\alpha}=\frac{dx^{\alpha}}{dt}\frac{dt}{d\tau}=v^{\alpha}u^{t}\,
  55. - c 2 = g α β u α u β = g t t ( u t ) 2 + g s s v 2 ( u t ) 2 . -c^{2}=g_{\alpha\beta}u^{\alpha}u^{\beta}=g_{tt}(u^{t})^{2}+g_{ss}v^{2}(u^{t})% ^{2}\,.
  56. u t = c - 1 g t t + g s s v 2 . u^{t}=c\sqrt{\frac{-1}{g_{tt}+g_{ss}v^{2}}}\,.
  57. u obs t = c - 1 g t t u_{\,\text{obs}}^{t}=c\sqrt{\frac{-1}{g_{tt}}}\,
  58. E k = - m g t t u t u obs t - m c 2 = m c 2 g t t g t t + g s s v 2 - m c 2 . E\text{k}=-mg_{tt}u^{t}u_{\,\text{obs}}^{t}-mc^{2}=mc^{2}\sqrt{\frac{g_{tt}}{g% _{tt}+g_{ss}v^{2}}}-mc^{2}\,.
  59. E k = m c 2 ( g t t g t t + g s s v 2 - 1 ) . E\text{k}=mc^{2}\left(\sqrt{\frac{g_{tt}}{g_{tt}+g_{ss}v^{2}}}-1\right)\,.
  60. g t t = - c 2 g_{tt}=-c^{2}\,
  61. g s s = 1 . g_{ss}=1\,.
  62. g t t = - ( c 2 + 2 Φ ) g_{tt}=-\left(c^{2}+2\Phi\right)\,
  63. g s s = 1 - 2 Φ c 2 g_{ss}=1-\frac{2\Phi}{c^{2}}\,
  64. p ^ \hat{p}
  65. T ^ = p ^ 2 2 m . \hat{T}=\frac{\hat{p}^{2}}{2m}.
  66. p p
  67. p ^ \hat{p}
  68. E k = p 2 2 m . E\text{k}=\frac{p^{2}}{2m}.
  69. p ^ \hat{p}
  70. - i -i\hbar\nabla
  71. T ^ = - 2 2 m 2 . \hat{T}=-\frac{\hbar^{2}}{2m}\nabla^{2}.
  72. T ^ \langle\hat{T}\rangle
  73. | ψ |\psi\rangle
  74. T ^ = ψ | i = 1 N - 2 2 m e i 2 | ψ = - 2 2 m e i = 1 N ψ | i 2 | ψ \langle\hat{T}\rangle=\bigg\langle\psi\bigg|\sum_{i=1}^{N}\frac{-\hbar^{2}}{2m% \text{e}}\nabla^{2}_{i}\bigg|\psi\bigg\rangle=-\frac{\hbar^{2}}{2m\text{e}}% \sum_{i=1}^{N}\bigg\langle\psi\bigg|\nabla^{2}_{i}\bigg|\psi\bigg\rangle
  75. m e m\text{e}
  76. i 2 \nabla^{2}_{i}
  77. ρ ( 𝐫 ) \rho(\mathbf{r})
  78. T [ ρ ] = 1 8 ρ ( 𝐫 ) ρ ( 𝐫 ) ρ ( 𝐫 ) d 3 r T[\rho]=\frac{1}{8}\int\frac{\nabla\rho(\mathbf{r})\cdot\nabla\rho(\mathbf{r})% }{\rho(\mathbf{r})}d^{3}r
  79. T [ ρ ] T[\rho]

Kinetic_energy_penetrator.html

  1. ( 1 2 m v 2 ) \left(\frac{1}{2}mv^{2}\right)
  2. π r 2 \pi r^{2}

Kinetic_theory.html

  1. Δ p = p i , x - p f , x = p i , x - ( - p i , x ) = 2 p i , x = 2 m v x \Delta p=p_{i,x}-p_{f,x}=p_{i,x}-(-p_{i,x})=2p_{i,x}=2mv_{x}\,
  2. Δ t = 2 L v x \Delta t=\frac{2L}{v_{x}}
  3. F = Δ p Δ t = m v x 2 L . F=\frac{\Delta p}{\Delta t}=\frac{mv_{x}^{2}}{L}.
  4. F = N m v x 2 ¯ L F=\frac{Nm\overline{v_{x}^{2}}}{L}
  5. v x 2 ¯ = v 2 ¯ / 3 \overline{v_{x}^{2}}=\overline{v^{2}}/3
  6. F = N m v 2 ¯ 3 L . F=\frac{Nm\overline{v^{2}}}{3L}.
  7. P = F L 2 = N m v 2 ¯ 3 V P=\frac{F}{L^{2}}=\frac{Nm\overline{v^{2}}}{3V}
  8. P = n m v 2 ¯ 3 . P=\frac{nm\overline{v^{2}}}{3}.
  9. 1 2 m v 2 ¯ {1\over 2}m\overline{v^{2}}
  10. P V = N m v 2 ¯ 3 PV={Nm\overline{v^{2}}\over 3}
  11. k B \displaystyle k_{B}
  12. T \displaystyle T
  13. k B T = m v 2 ¯ 3 k_{B}T={m\overline{v^{2}}\over 3}
  14. 1 2 m v 2 ¯ = 3 2 k B T \displaystyle\frac{1}{2}m\overline{v^{2}}=\frac{3}{2}k_{B}T
  15. K = 1 2 N m v 2 ¯ K=\frac{1}{2}Nm\overline{v^{2}}
  16. T \displaystyle T
  17. T = m v 2 ¯ 3 k B \displaystyle T={m\overline{v^{2}}\over 3k_{B}}
  18. T = 2 3 K N k B . \displaystyle T=\frac{2}{3}\frac{K}{Nk_{B}}.
  19. P V = 2 3 K . \displaystyle PV=\frac{2}{3}K.
  20. 3 N \displaystyle 3N
  21. N \displaystyle N
  22. A = 1 4 N V v a v g = n 4 8 k B T π m . A=\frac{1}{4}\frac{N}{V}v_{avg}=\frac{n}{4}\sqrt{\frac{8k_{B}T}{\pi m}}.\,
  23. v rms = 3 k B T m v_{\mathrm{rms}}=\sqrt{{3k_{B}T}\over{m}}

Kirkendall_effect.html

  1. v v
  2. v = ( D 1 - D 2 ) d N 2 d x , v=(D_{1}-D_{2})\frac{dN_{2}}{dx},
  3. D 1 D_{1}
  4. D 2 D_{2}
  5. N 2 N_{2}
  6. D D
  7. D = N 1 D 2 + N 2 D 1 . D=N_{1}D_{2}+N_{2}D_{1}.
  8. X K = ( a 1 Δ C 1 + a 2 Δ C 2 + + a n - 1 Δ C n - 1 ) t X^{K}=(a_{1}\Delta C_{1}^{\circ}+a_{2}\Delta C_{2}^{\circ}+\dots+a_{n-1}\Delta C% _{n-1}^{\circ})\sqrt{t}
  9. X K X^{K}
  10. a a
  11. Δ C \Delta C^{\circ}

Kleene_star.html

  1. V * = i 𝒩 V i = { ε } V V 2 V 3 V 4 . V^{*}=\bigcup_{i\in\mathcal{N}}V_{i}=\{\varepsilon\}\cup V\cup V_{2}\cup V_{3}% \cup V_{4}\cup\ldots.
  2. V + = i 𝒩 { 0 } V i = V 1 V 2 V 3 . V^{+}=\bigcup_{i\in\mathcal{N}\setminus\{0\}}V_{i}=V_{1}\cup V_{2}\cup V_{3}% \cup\ldots.

Klein_bottle.html

  1. x \displaystyle x
  2. w \displaystyle w
  3. x = R ( cos θ 2 cos v - sin θ 2 sin 2 v ) y = R ( sin θ 2 cos v + cos θ 2 sin 2 v ) z = P cos θ ( 1 + ϵ sin v ) w = P sin θ ( 1 + ϵ sin v ) \ \begin{aligned}\displaystyle x&\displaystyle=R\left(\cos\frac{\theta}{2}\cos v% -\sin\frac{\theta}{2}\sin 2v\right)\\ \displaystyle y&\displaystyle=R\left(\sin\frac{\theta}{2}\cos v+\cos\frac{% \theta}{2}\sin 2v\right)\\ \displaystyle z&\displaystyle=P\cos\theta\left(1+{\epsilon}\sin v\right)\\ \displaystyle w&\displaystyle=P\sin\theta\left(1+{\epsilon}\sin v\right)\end{aligned}
  4. x ( θ , φ ) = ( R + r cos θ ) cos φ x(\theta,\varphi)=(R+r\cos\theta)\cos{\varphi}
  5. y ( θ , φ ) = ( R + r cos θ ) sin φ y(\theta,\varphi)=(R+r\cos\theta)\sin{\varphi}
  6. z ( θ , φ ) = r sin θ cos ( φ / 2 ) z(\theta,\varphi)=r\sin\theta\cos({\varphi/2})
  7. w ( θ , φ ) = r sin θ sin ( φ / 2 ) w(\theta,\varphi)=r\sin\theta\sin({\varphi/2})
  8. x ( u , v ) = - 2 15 cos u ( 3 cos v - 30 sin u + 90 cos 4 u sin u - 60 cos 6 u sin u + 5 cos u cos v sin u ) y ( u , v ) = - 1 15 sin u ( 3 cos v - 3 cos 2 u cos v - 48 cos 4 u cos v + 48 cos 6 u cos v - 60 sin u + 5 cos u cos v sin u - 5 cos 3 u cos v sin u - 80 cos 5 u cos v sin u + 80 cos 7 u cos v sin u ) z ( u , v ) = 2 15 ( 3 + 5 cos u sin u ) sin v \begin{aligned}\displaystyle x(u,v)&\displaystyle=-\frac{2}{15}\cos u(3\cos{v}% -30\sin{u}+90\cos^{4}{u}\sin{u}\\ &\displaystyle\quad-60\cos^{6}{u}\sin{u}+5\cos{u}\cos{v}\sin{u})\\ \displaystyle y(u,v)&\displaystyle=-\frac{1}{15}\sin u(3\cos{v}-3\cos^{2}{u}% \cos{v}-48\cos^{4}{u}\cos{v}+48\cos^{6}{u}\\ &\displaystyle\quad\cos{v}-60\sin{u}+5\cos{u}\cos{v}\sin{u}-5\cos^{3}{u}\cos{v% }\sin{u}-80\\ &\displaystyle\quad\cos^{5}{u}\cos{v}\sin{u}+80\cos^{7}{u}\cos{v}\sin{u})\\ \displaystyle z(u,v)&\displaystyle=\frac{2}{15}(3+5\cos{u}\sin{u})\sin{v}\end{aligned}
  9. D 2 × S 1 \scriptstyle D^{2}\times S^{1}

Klein_four-group.html

  1. V = a , b a 2 = b 2 = ( a b ) 2 = 1 . \mathrm{V}=\langle a,b\mid a^{2}=b^{2}=(ab)^{2}=1\rangle.
  2. { , { α } , { β } , { α , β } } \{\emptyset,\{\alpha\},\{\beta\},\{\alpha,\beta\}\}

Knapsack_problem.html

  1. i = 1 n v i x i \sum_{i=1}^{n}v_{i}x_{i}
  2. i = 1 n w i x i W \sum_{i=1}^{n}w_{i}x_{i}\leq W
  3. x i { 0 , 1 } x_{i}\in\{0,1\}
  4. x i x_{i}
  5. c i c_{i}
  6. i = 1 n v i x i \sum_{i=1}^{n}v_{i}x_{i}
  7. i = 1 n w i x i W \sum_{i=1}^{n}w_{i}x_{i}\leq W
  8. x i { 0 , , c i } x_{i}\in\{0,\ldots,c_{i}\}
  9. x i x_{i}
  10. i = 1 n v i x i \sum_{i=1}^{n}v_{i}x_{i}
  11. i = 1 n w i x i W \sum_{i=1}^{n}w_{i}x_{i}\leq W
  12. x i 0 x_{i}\geq 0
  13. w 1 , , w n w_{1},\ldots,w_{n}
  14. m [ 0 ] = 0 m[0]=0\,\!
  15. m [ w ] = max w i w ( v i + m [ w - w i ] ) m[w]=\max_{w_{i}\leq w}(v_{i}+m[w-w_{i}])
  16. v i v_{i}
  17. m [ 0 ] m[0]
  18. m [ W ] m[W]
  19. m [ w ] m[w]
  20. n n
  21. W W
  22. m [ w ] m[w]
  23. O ( n W ) O(nW)
  24. w 1 , w 2 , , w n , W w_{1},\,w_{2},\,\ldots,\,w_{n},\,W
  25. O ( n W ) O(nW)
  26. W W
  27. n n
  28. W W
  29. W W
  30. log W \log W
  31. W W
  32. w 1 , w 2 , , w n , W w_{1},\,w_{2},\,\ldots,\,w_{n},\,W
  33. m [ i , w ] m[i,w]
  34. w w
  35. i i
  36. i i
  37. m [ i , w ] m[i,w]
  38. m [ i , w ] = m [ i - 1 , w ] m[i,\,w]=m[i-1,\,w]
  39. w i > w w_{i}>w\,\!
  40. m [ i , w ] = max ( m [ i - 1 , w ] , m [ i - 1 , w - w i ] + v i ) m[i,\,w]=\max(m[i-1,\,w],\,m[i-1,w-w_{i}]+v_{i})
  41. w i w w_{i}\leqslant w
  42. m [ n , W ] m[n,W]
  43. O ( n W ) O(nW)
  44. m [ w ] m[w]
  45. i + 1 i+1
  46. m [ W ] m[W]
  47. m [ 1 ] m[1]
  48. O ( W ) O(W)
  49. W W
  50. w i w_{i}
  51. d d
  52. W W
  53. 10 d 10^{d}
  54. O ( W * 10 d ) O(W*10^{d})
  55. O ( n W * 10 d ) O(nW*10^{d})
  56. O ( 2 n / 2 ) O(2^{n/2})
  57. O ( n * 2 n / 2 ) O(n*2^{n/2})
  58. O ( n * 2 n ) O(n*2^{n})
  59. v i / w i v_{i}/w_{i}
  60. m m
  61. m / 2 m/2
  62. v i v_{i}
  63. { v i 1 i n } \{v_{i}\mid 1\leq i\leq n\}
  64. v i v^{\prime}_{i}
  65. v i v_{i}
  66. v i v^{\prime}_{i}
  67. S S^{\prime}
  68. profit ( S ) ( 1 - ε ) profit ( S * ) \mathrm{profit}(S^{\prime})\geq(1-\varepsilon)\cdot\mathrm{profit}(S^{*})
  69. S * S^{*}
  70. j J w j x j α w i \qquad\sum_{j\in J}w_{j}\,x_{j}\ \leq\alpha\,w_{i}
  71. j J v j x j α v i \qquad\sum_{j\in J}v_{j}\,x_{j}\ \geq\alpha\,v_{i}\,
  72. x Z + n x\in Z_{+}^{n}
  73. α Z + , J N \alpha\in Z_{+}\,,J\subseteq N
  74. i J i\not\in J
  75. x x
  76. i J i\ll J
  77. j J w j x j w i \qquad\sum_{j\in J}w_{j}\,x_{j}\ \leq w_{i}
  78. j J v j x j v i \qquad\sum_{j\in J}v_{j}\,x_{j}\ \geq v_{i}
  79. x Z + n x\in Z_{+}^{n}
  80. α = 1 \alpha=1
  81. i J i\prec\prec J
  82. j J w j x j α w i \qquad\sum_{j\in J}w_{j}\,x_{j}\ \leq\alpha\,w_{i}
  83. j J v j x j α v i \qquad\sum_{j\in J}v_{j}\,x_{j}\ \geq\alpha\,v_{i}\,
  84. x Z + n x\in Z_{+}^{n}
  85. α 1 \alpha\geq 1
  86. α \alpha
  87. t i = ( α - 1 ) w i t_{i}=(\alpha-1)w_{i}
  88. α - 1 \alpha-1
  89. i m j i\ll_{m}j
  90. w j x j w i \qquad w_{j}\,x_{j}\ \leq w_{i}
  91. v j x j v i \qquad v_{j}\,x_{j}\ \geq v_{i}
  92. x j Z + x_{j}\in Z_{+}
  93. J = { j } , α = 1 , x j = w i w j J=\{j\},\alpha=1,x_{j}=\lfloor\frac{w_{i}}{w_{j}}\rfloor
  94. v b w b v i w i \frac{v_{b}}{w_{b}}\geq\frac{v_{i}}{w_{i}}\,
  95. i j i\ll_{\equiv}j
  96. w j + t w b w i w_{j}+tw_{b}\leq w_{i}
  97. v j + t v b v i v_{j}+tv_{b}\geq v_{i}
  98. J = { b , j } , α = 1 , x b = t , x j = 1 J=\{b,j\},\alpha=1,x_{b}=t,x_{j}=1
  99. i i
  100. w i ¯ = ( w i 1 , , w i D ) \overline{w_{i}}=(w_{i1},\ldots,w_{iD})
  101. ( W 1 , , W D ) (W_{1},\ldots,W_{D})
  102. d d
  103. W d W_{d}
  104. D = 2 D=2
  105. = =
  106. J = { 1 , 2 , , m } J=\{1,2,\ldots,m\}
  107. m < D m<D
  108. i i
  109. z > m \exists z>m
  110. j J { z } , w i j 0 \forall j\in J\cup\{z\},\ w_{ij}\geq 0
  111. y J { z } , w i y = 0 \forall y\notin J\cup\{z\},w_{iy}=0
  112. w i = v i w_{i}=v_{i}

Knight's_tour.html

  1. n × n n\times n
  2. U t + 1 ( N i , j ) = U t ( N i , j ) + 2 - N G ( N i , j ) V t ( N ) U_{t+1}(N_{i,j})=U_{t}(N_{i,j})+2-\sum_{N\in G(N_{i,j})}V_{t}(N)
  3. V t + 1 ( N i , j ) = { 1 if U t + 1 ( N i , j ) > 3 0 if U t + 1 ( N i , j ) < 0 V t ( N i , j ) otherwise , V_{t+1}(N_{i,j})=\left\{\begin{array}[]{ll}1&\mbox{if}~{}\,\,U_{t+1}(N_{i,j})>% 3\\ 0&\mbox{if}~{}\,\,U_{t+1}(N_{i,j})<0\\ V_{t}(N_{i,j})&\mbox{otherwise}~{},\end{array}\right.
  4. t t
  5. U ( N i , j ) U(N_{i,j})
  6. i i
  7. j j
  8. V ( N i , j ) V(N_{i,j})
  9. i i
  10. j j
  11. G ( N i , j ) G(N_{i,j})
  12. t t
  13. t + 1 t+1

Koch_snowflake.html

  1. N n = N n - 1 4 = 3 4 n . N_{n}=N_{n-1}\cdot 4=3\cdot 4^{n}\,.
  2. S n = S n - 1 3 = s 3 n . S_{n}=\frac{S_{n-1}}{3}=\frac{s}{3^{n}}\,.
  3. P n = N n S n = 3 s ( 4 3 ) n . P_{n}=N_{n}\cdot S_{n}=3\cdot s\cdot{\left(\frac{4}{3}\right)}^{n}\,.
  4. lim n P n = lim n 3 s ( 4 3 ) n , \lim_{n\rightarrow\infty}P_{n}=\lim_{n\rightarrow\infty}3\cdot s\cdot\left(% \frac{4}{3}\right)^{n}\rightarrow\infty\,,
  5. | 4 3 | > 1 \left|\frac{4}{3}\right|>1
  6. log 4 log 3 \frac{\log 4}{\log 3}
  7. T n = N n - 1 = 3 4 n - 1 = 3 4 4 n . T_{n}=N_{n-1}=3\cdot 4^{n-1}=\frac{3}{4}\cdot 4^{n}\,.
  8. a n = a n - 1 9 = a 0 9 n . a_{n}=\frac{a_{n-1}}{9}=\frac{a_{0}}{9^{n}}\,.
  9. b n = T n a n = 3 4 ( 4 9 ) n a 0 b_{n}=T_{n}\cdot a_{n}=\frac{3}{4}\cdot{\left(\frac{4}{9}\right)}^{n}\cdot a_{0}
  10. A n = a 0 + k = 1 n b k = a 0 ( 1 + 3 4 k = 1 n ( 4 9 ) k ) = a 0 ( 1 + 1 3 k = 0 n - 1 ( 4 9 ) k ) . A_{n}=a_{0}+\sum_{k=1}^{n}b_{k}=a_{0}\left(1+\frac{3}{4}\sum_{k=1}^{n}\left(% \frac{4}{9}\right)^{k}\right)=a_{0}\left(1+\frac{1}{3}\sum_{k=0}^{n-1}\left(% \frac{4}{9}\right)^{k}\right)\,.
  11. A n = a 0 ( 1 + 3 5 ( 1 - ( 4 9 ) n ) ) = a 0 5 ( 8 - 3 ( 4 9 ) n ) . A_{n}=a_{0}\left(1+\frac{3}{5}\left(1-\left(\frac{4}{9}\right)^{n}\right)% \right)=\frac{a_{0}}{5}\left(8-3\left(\frac{4}{9}\right)^{n}\right)\,.
  12. lim n A n = lim n a 0 5 ( 8 - 3 ( 4 9 ) n ) = 8 5 a 0 , \lim_{n\rightarrow\infty}A_{n}=\lim_{n\rightarrow\infty}\frac{a_{0}}{5}\cdot% \left(8-3\left(\frac{4}{9}\right)^{n}\right)=\frac{8}{5}\cdot a_{0}\,,
  13. | 4 9 | < 1 \left|\frac{4}{9}\right|<1
  14. 2 s 2 3 5 \frac{2s^{2}\sqrt{3}}{5}
  15. A n = 1 5 + 4 5 k = 0 n ( 5 9 ) k giving lim n A n = 2 \begin{aligned}\displaystyle A_{n}&\displaystyle=\frac{1}{5}+\frac{4}{5}\sum_{% k=0}^{n}\left(\frac{5}{9}\right)^{k}\mbox{giving}~{}\lim_{n\rightarrow\infty}A% _{n}=2\\ \end{aligned}
  16. P n = 4 ( 5 3 ) n \begin{aligned}\displaystyle P_{n}&\displaystyle=4\left(\frac{5}{3}\right)^{n}% \\ \end{aligned}

Kolmogorov–Smirnov_test.html

  1. F n ( x ) = 1 n i = 1 n I [ - , x ] ( X i ) F_{n}(x)={1\over n}\sum_{i=1}^{n}I_{[-\infty,x]}(X_{i})
  2. I [ - , x ] ( X i ) I_{[-\infty,x]}(X_{i})
  3. X i x X_{i}\leq x
  4. D n = sup x | F n ( x ) - F ( x ) | D_{n}=\sup_{x}|F_{n}(x)-F(x)|
  5. n n
  6. K = sup t [ 0 , 1 ] | B ( t ) | K=\sup_{t\in[0,1]}|B(t)|
  7. Pr ( K x ) = 1 - 2 k = 1 ( - 1 ) k - 1 e - 2 k 2 x 2 = 2 π x k = 1 e - ( 2 k - 1 ) 2 π 2 / ( 8 x 2 ) . \operatorname{Pr}(K\leq x)=1-2\sum_{k=1}^{\infty}(-1)^{k-1}e^{-2k^{2}x^{2}}=% \frac{\sqrt{2\pi}}{x}\sum_{k=1}^{\infty}e^{-(2k-1)^{2}\pi^{2}/(8x^{2})}.
  8. n D n n sup t | B ( F ( t ) ) | \sqrt{n}D_{n}\xrightarrow{n\to\infty}\sup_{t}|B(F(t))|
  9. n D n \sqrt{n}D_{n}
  10. α \alpha
  11. n D n > K α , \sqrt{n}D_{n}>K_{\alpha},\,
  12. Pr ( K K α ) = 1 - α . \operatorname{Pr}(K\leq K_{\alpha})=1-\alpha.\,
  13. f ( x ) f(x)
  14. g ( x ) g(x)
  15. x i x_{i}
  16. lim x x i g ( x ) \lim_{x\rightarrow x_{i}}g(x)
  17. sup x | g ( x ) - f ( x ) | = max i [ max ( | g ( x i ) - f ( x i ) | , lim x x i | g ( x ) - f ( x i - 1 ) ) ] , \sup_{x}|g(x)-f(x)|=\max_{i}\left[\max\left(|g(x_{i})-f(x_{i})|,\lim_{x% \rightarrow x_{i}}|g(x)-f(x_{i-1})\right)\right],
  18. D n , n = sup x | F 1 , n ( x ) - F 2 , n ( x ) | , D_{n,n^{\prime}}=\sup_{x}|F_{1,n}(x)-F_{2,n^{\prime}}(x)|,
  19. F 1 , n F_{1,n}
  20. F 2 , n F_{2,n^{\prime}}
  21. s u p sup
  22. α \alpha
  23. D n , n > c ( α ) n + n n n . D_{n,n^{\prime}}>c(\alpha)\sqrt{\frac{n+n^{\prime}}{nn^{\prime}}}.
  24. c ( α ) c({\alpha})
  25. α \alpha
  26. α \alpha
  27. c ( α ) c({\alpha})
  28. Pr ( x < X and y < Y ) \Pr(x<X\and y<Y)
  29. Pr ( X < x and Y > y ) \Pr(X<x\and Y>y)

Konstantin_Tsiolkovsky.html

  1. δ V \delta V
  2. I 0 I_{0}
  3. δ V = I 0 ln ( M 0 M 1 ) \delta V=I_{0}\ln\left({M_{0}\over M_{1}}\right)

Krull_dimension.html

  1. 𝔭 0 𝔭 1 𝔭 n \mathfrak{p}_{0}\subsetneq\mathfrak{p}_{1}\subsetneq\ldots\subsetneq\mathfrak{% p}_{n}
  2. R R
  3. R R
  4. 𝔭 \mathfrak{p}
  5. 𝔭 \mathfrak{p}
  6. ht ( 𝔭 ) \operatorname{ht}(\mathfrak{p})
  7. 𝔭 \mathfrak{p}
  8. 𝔭 0 𝔭 1 𝔭 n 𝔭 \mathfrak{p}_{0}\subsetneq\mathfrak{p}_{1}\subsetneq\ldots\subsetneq\mathfrak{% p}_{n}\subseteq\mathfrak{p}
  9. 𝔭 \mathfrak{p}
  10. 𝔭 \mathfrak{p}
  11. 𝔭 𝔮 \mathfrak{p}\subset\mathfrak{q}
  12. 𝔭 \mathfrak{p}
  13. 𝔮 \mathfrak{q}
  14. 𝔭 \mathfrak{p}
  15. 𝔮 \mathfrak{q}
  16. R R
  17. 𝔭 \mathfrak{p}
  18. 𝔭 \mathfrak{p}
  19. - -\infty
  20. - 1 -1
  21. gr I ( R ) = 0 I k / I k + 1 \operatorname{gr}_{I}(R)=\oplus_{0}^{\infty}I^{k}/I^{k+1}
  22. dim gr I ( R ) \operatorname{dim}\operatorname{gr}_{I}(R)
  23. dim R M := dim ( R / Ann R ( M ) ) \operatorname{dim}_{R}M:=\operatorname{dim}(R/\operatorname{Ann}_{R}(M))

Kruskal's_algorithm.html

  1. log V 2 = 2 log V \log V^{2}=2\log V
  2. \;
  3. P P
  4. Y Y
  5. P P
  6. Y Y
  7. Y Y
  8. Y Y
  9. Y Y
  10. P P

Kuiper's_test.html

  1. z i = F ( x i ) , z_{i}=F(x_{i}),
  2. D + = max [ i / n - z i ] , D^{+}=\mathrm{max}\left[i/n-z_{i}\right],
  3. D - = max [ z i - ( i - 1 ) / n ] , D^{-}=\mathrm{max}\left[z_{i}-(i-1)/n\right],
  4. V = D + + D - . V=D^{+}+D^{-}.

Kuiper_belt.html

  1. 11 / 2 1{1}/{2}
  2. d N d D D - q \frac{dN}{dD}\propto D^{-q}
  3. N D 1 - q + a constant . N\propto D^{1-q}+\,\text{a constant}.

Kurtosis.html

  1. β 2 = E [ ( X - μ ) 4 ] ( E [ ( X - μ ) 2 ] ) 2 = μ 4 σ 4 , {\beta_{2}=}\frac{\operatorname{E}[(X-{\mu})^{4}]}{(\operatorname{E}[(X-{\mu})% ^{2}])^{2}}{=}\frac{\mu_{4}}{\sigma^{4}},
  2. μ 4 σ 4 ( μ 3 σ 3 ) 2 + 1. \frac{\mu_{4}}{\sigma^{4}}\geq\left(\frac{\mu_{3}}{\sigma^{3}}\right)^{2}+1.
  3. μ 4 σ 4 1 2 n - 3 n - 2 ( μ 3 σ 3 ) 2 + n 2 . \frac{\mu_{4}}{\sigma^{4}}\leq\frac{1}{2}\frac{n-3}{n-2}\left(\frac{\mu_{3}}{% \sigma^{3}}\right)^{2}+\frac{n}{2}.
  4. γ 2 = κ 4 κ 2 2 = μ 4 σ 4 - 3 , \gamma_{2}=\frac{\kappa_{4}}{\kappa_{2}^{2}}=\frac{\mu_{4}}{\sigma^{4}}-3,
  5. Kurt [ Y ] = κ 4 ( Y ) κ 2 ( Y ) 2 = n κ 4 ( X ) ( n κ 2 ( X ) ) 2 = 1 n κ 4 ( X ) κ 2 ( X ) 2 = 1 n Kurt [ X ] . \operatorname{Kurt}[Y]=\frac{\kappa_{4}(Y)}{\kappa_{2}(Y)^{2}}=\frac{n\kappa_{% 4}(X)}{(n\kappa_{2}(X))^{2}}=\frac{1}{n}\frac{\kappa_{4}(X)}{\kappa_{2}(X)^{2}% }=\frac{1}{n}\operatorname{Kurt}[X].
  6. Kurt ( i = 1 n X i ) = 1 n 2 i = 1 n Kurt ( X i ) , \operatorname{Kurt}\left(\sum_{i=1}^{n}X_{i}\right)={1\over n^{2}}\sum_{i=1}^{% n}\operatorname{Kurt}(X_{i}),
  7. X i X_{i}
  8. Kurt ( i = 1 n X i ) = i = 1 n σ i 4 Kurt ( X i ) ( j = 1 n σ j 2 ) 2 , \operatorname{Kurt}\left(\sum_{i=1}^{n}X_{i}\right)=\sum_{i=1}^{n}\frac{\sigma% _{i}^{\,4}\cdot\operatorname{Kurt}(X_{i})}{\left(\sum_{j=1}^{n}\sigma_{j}^{\,2% }\right)^{2}},
  9. σ i \sigma_{i}
  10. X i X_{i}
  11. Z = X - μ σ Z=\frac{X-\mu}{\sigma}
  12. κ = E ( Z 4 ) = v a r ( Z 2 ) + [ E ( Z 2 ) ] 2 = v a r ( Z 2 ) + 1 \kappa=E(Z^{4})=var(Z^{2})+[E(Z^{2})]^{2}=var(Z^{2})+1
  13. p = 1 / 2 ± 1 / 12 p=1/2\pm\sqrt{1/12}
  14. f ( x ; a , m ) = Γ ( m ) a π Γ ( m - 1 / 2 ) [ 1 + ( x a ) 2 ] - m , f(x;a,m)=\frac{\Gamma(m)}{a\,\sqrt{\pi}\,\Gamma(m-1/2)}\left[1+\left(\frac{x}{% a}\right)^{2}\right]^{-m},\!
  15. m = 5 / 2 + 3 / γ 2 m=5/2+3/\gamma_{2}
  16. γ 2 \gamma_{2}
  17. g ( x ; γ 2 ) = f ( x ; a = 2 + 6 / γ 2 , m = 5 / 2 + 3 / γ 2 ) . g(x;\gamma_{2})=f(x;\;a=\sqrt{2+6/\gamma_{2}},\;m=5/2+3/\gamma_{2}).\!
  18. γ 2 \gamma_{2}\to\infty
  19. g ( x ) = 3 ( 2 + x 2 ) - 5 / 2 , g(x)=3\left(2+x^{2}\right)^{-5/2},\!
  20. γ 2 0 \gamma_{2}\to 0
  21. x g ( x ; 2 ) x\mapsto g(x;2)
  22. γ 2 = \gamma_{2}=\infty
  23. g 2 = m 4 m 2 2 - 3 = 1 n i = 1 n ( x i - x ¯ ) 4 ( 1 n i = 1 n ( x i - x ¯ ) 2 ) 2 - 3 g_{2}=\frac{m_{4}}{m_{2}^{2}}-3=\frac{\tfrac{1}{n}\sum_{i=1}^{n}(x_{i}-% \overline{x})^{4}}{\left(\tfrac{1}{n}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}% \right)^{2}}-3
  24. x ¯ \overline{x}
  25. 24 n ( n - 1 ) 2 ( n - 3 ) ( n - 2 ) ( n + 3 ) ( n + 5 ) \frac{24n(n-1)^{2}}{(n-3)(n-2)(n+3)(n+5)}
  26. G 2 = k 4 k 2 2 G_{2}=\frac{k_{4}}{k_{2}^{2}}
  27. = n 2 ( ( n + 1 ) m 4 - 3 ( n - 1 ) m 2 2 ) ( n - 1 ) ( n - 2 ) ( n - 3 ) ( n - 1 ) 2 n 2 m 2 2 =\frac{n^{2}\,((n+1)\,m_{4}-3\,(n-1)\,m_{2}^{2})}{(n-1)\,(n-2)\,(n-3)}\;\frac{% (n-1)^{2}}{n^{2}\,m_{2}^{2}}
  28. = n - 1 ( n - 2 ) ( n - 3 ) ( ( n + 1 ) m 4 m 2 2 - 3 ( n - 1 ) ) =\frac{n-1}{(n-2)\,(n-3)}\left((n+1)\,\frac{m_{4}}{m_{2}^{2}}-3\,(n-1)\right)
  29. = n - 1 ( n - 2 ) ( n - 3 ) ( ( n + 1 ) g 2 + 6 ) =\frac{n-1}{(n-2)\,(n-3)}\left((n+1)\,g_{2}+6\right)
  30. = ( n + 1 ) n ( n - 1 ) ( n - 2 ) ( n - 3 ) i = 1 n ( x i - x ¯ ) 4 ( i = 1 n ( x i - x ¯ ) 2 ) 2 - 3 ( n - 1 ) 2 ( n - 2 ) ( n - 3 ) =\frac{(n+1)\,n\,(n-1)}{(n-2)\,(n-3)}\;\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{4}% }{\left(\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}\right)^{2}}-3\,\frac{(n-1)^{2}}{(n-2% )\,(n-3)}
  31. = ( n + 1 ) n ( n - 1 ) ( n - 2 ) ( n - 3 ) i = 1 n ( x i - x ¯ ) 4 k 2 2 - 3 ( n - 1 ) 2 ( n - 2 ) ( n - 3 ) =\frac{(n+1)\,n}{(n-1)\,(n-2)\,(n-3)}\;\frac{\sum_{i=1}^{n}(x_{i}-\bar{x})^{4}% }{k_{2}^{2}}-3\,\frac{(n-1)^{2}}{(n-2)(n-3)}
  32. x ¯ \bar{x}
  33. G 2 G_{2}

L'Hôpital's_rule.html

  1. f f
  2. g g
  3. I I
  4. c c
  5. I I
  6. lim x c f ( x ) = lim x c g ( x ) = 0 or ± \lim_{x\to c}f(x)=\lim_{x\to c}g(x)=0\,\text{ or }\pm\infty
  7. lim x c f ( x ) g ( x ) \lim_{x\to c}\frac{f^{\prime}(x)}{g^{\prime}(x)}
  8. g ( x ) 0 g^{\prime}(x)\neq 0
  9. x x
  10. I I
  11. x c x≠c
  12. lim x c f ( x ) g ( x ) = lim x c f ( x ) g ( x ) \lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f^{\prime}(x)}{g^{\prime}(x)}
  13. c c
  14. L L
  15. g ( x ) 0 g^{\prime}(x)\neq 0
  16. lim x c f ( x ) g ( x ) = L . \lim_{x\to c}{\frac{f^{\prime}(x)}{g^{\prime}(x)}}=L.
  17. lim x c f ( x ) = lim x c g ( x ) = 0 \lim_{x\to c}{f(x)}=\lim_{x\to c}g(x)=0
  18. lim x c | f ( x ) | = lim x c | g ( x ) | = , \lim_{x\to c}{|f(x)|}=\lim_{x\to c}{|g(x)|}=\infty,
  19. lim x c f ( x ) g ( x ) = L . \lim_{x\to c}{\frac{f(x)}{g(x)}}=L.
  20. lim x c | g ( x ) | = . \lim_{x\to c}{|g(x)|}=\infty.
  21. g ( x ) 0 g^{\prime}(x)\neq 0
  22. lim x c f ( x ) g ( x ) \lim_{x\to c}\frac{f^{\prime}(x)}{g^{\prime}(x)}
  23. f f\,^{\prime}
  24. g g\,^{\prime}
  25. f ( x ) = x + sin ( x ) f(x)=x+\sin(x)
  26. g ( x ) = x g(x)=x
  27. lim x f ( x ) g ( x ) = lim x 1 + cos x 1 ; \lim_{x\to\infty}\frac{f^{\prime}(x)}{g^{\prime}(x)}=\lim_{x\to\infty}\frac{1+% \cos x}{1};
  28. lim x f ( x ) g ( x ) \lim_{x\to\infty}\frac{f(x)}{g(x)}
  29. lim x f ( x ) g ( x ) = lim x ( 1 + sin x x ) = 1 \lim_{x\to\infty}\frac{f(x)}{g(x)}=\lim_{x\to\infty}\left(1+\frac{\sin x}{x}% \right)=1
  30. sin π x π x \frac{\sin\pi x}{\pi x}
  31. 0 / 0 {0}/{0}
  32. x = 0 x=0
  33. lim x 0 sinc ( x ) \displaystyle\lim_{x\to 0}\operatorname{sinc}(x)
  34. 1 \displaystyle 1
  35. 0 / 0 {0}/{0}
  36. lim x 0 2 sin x - sin 2 x x - sin x \displaystyle\lim_{x\to 0}{\frac{2\sin x-\sin 2x}{x-\sin x}}
  37. 0 / 0 {0}/{0}
  38. b > 0 b>0
  39. lim x 0 b x - 1 x = lim x 0 b x ln b 1 = ln b lim x 0 b x = ln b . \lim_{x\to 0}{\frac{b^{x}-1}{x}}=\lim_{x\to 0}{\frac{b^{x}\ln b}{1}}=\ln b\lim% _{x\to 0}{b^{x}}=\ln b.
  40. 0 / 0 {0}/{0}
  41. lim x 0 e x - 1 - x x 2 = lim x 0 e x - 1 2 x = lim x 0 e x 2 = 1 2 . \lim_{x\to 0}{\frac{e^{x}-1-x}{x^{2}}}=\lim_{x\to 0}{\frac{e^{x}-1}{2x}}=\lim_% {x\to 0}{\frac{e^{x}}{2}}={\frac{1}{2}}.
  42. / {∞}/{∞}
  43. n n
  44. lim x x n e - x = lim x x n e x = lim x n x n - 1 e x = n lim x x n - 1 e x . \lim_{x\to\infty}x^{n}e^{-x}=\lim_{x\to\infty}{\frac{x^{n}}{e^{x}}}=\lim_{x\to% \infty}{\frac{nx^{n-1}}{e^{x}}}=n\lim_{x\to\infty}{\frac{x^{n-1}}{e^{x}}}.
  45. / {∞}/{∞}
  46. lim x 0 + x ln x = lim x 0 + ln x 1 / x = lim x 0 + 1 / x - 1 / x 2 = lim x 0 + - x = 0. \lim_{x\to 0^{+}}x\ln x=\lim_{x\to 0^{+}}{\frac{\ln x}{1/x}}=\lim_{x\to 0^{+}}% {\frac{1/x}{-1/x^{2}}}=\lim_{x\to 0^{+}}-x=0.
  47. 0 / 0 {0}/{0}
  48. lim t 1 / 2 sinc ( t ) cos π t 1 - ( 2 t ) 2 \displaystyle\lim_{t\to 1/2}\operatorname{sinc}(t)\frac{\cos\pi t}{1-(2t)^{2}}
  49. f f′′
  50. x x
  51. lim h 0 f ( x + h ) + f ( x - h ) - 2 f ( x ) h 2 \displaystyle\lim_{h\to 0}\frac{f(x+h)+f(x-h)-2f(x)}{h^{2}}
  52. f ( x ) + f ( x ) f(x)+f′(x)
  53. x x→∞
  54. e x f ( x ) e^{x}f(x)
  55. lim x f ( x ) = lim x e x f ( x ) e x = lim x e x ( f ( x ) + f ( x ) ) e x = lim x ( f ( x ) + f ( x ) ) \lim_{x\to\infty}f(x)=\lim_{x\to\infty}{e^{x}f(x)\over e^{x}}=\lim_{x\to\infty% }{e^{x}(f(x)+f^{\prime}(x))\over e^{x}}=\lim_{x\to\infty}(f(x)+f^{\prime}(x))
  56. lim x f ( x ) \lim_{x\to\infty}f(x)
  57. lim x f ( x ) = 0. \ \lim_{x\to\infty}f^{\prime}(x)=0.
  58. e x f ( x ) e^{x}f(x)
  59. u ( x ) = x 1 - ρ - 1 1 - ρ u(x)=\frac{x^{1-\rho}-1}{1-\rho}
  60. ρ > 0 \rho>0
  61. ρ \rho
  62. ρ \rho
  63. lim ρ 1 x 1 - ρ - 1 1 - ρ = lim ρ 1 - x 1 - ρ ln x - 1 = ln x . \lim_{\rho\to 1}\frac{x^{1-\rho}-1}{1-\rho}=\lim_{\rho\to 1}\frac{-x^{1-\rho}% \ln x}{-1}=\ln x.
  64. lim x e x + e - x e x - e - x = lim x e x - e - x e x + e - x = lim x e x + e - x e x - e - x = . \lim_{x\to\infty}\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=\lim_{x\to\infty}\frac{e^{x% }-e^{-x}}{e^{x}+e^{-x}}=\lim_{x\to\infty}\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=\dots.
  65. y = e x y=e^{x}
  66. lim x e x + e - x e x - e - x = lim y y + y - 1 y - y - 1 = lim y 1 - y - 2 1 + y - 2 = 1 1 = 1. \lim_{x\to\infty}\frac{e^{x}+e^{-x}}{e^{x}-e^{-x}}=\lim_{y\to\infty}\frac{y+y^% {-1}}{y-y^{-1}}=\lim_{y\to\infty}\frac{1-y^{-2}}{1+y^{-2}}=\frac{1}{1}=1.
  67. lim x x 1 / 2 + x - 1 / 2 x 1 / 2 - x - 1 / 2 = lim x 1 2 x - 1 / 2 - 1 2 x - 3 / 2 1 2 x - 1 / 2 + 1 2 x - 3 / 2 = lim x - 1 4 x - 3 / 2 + 3 4 x - 5 / 2 - 1 4 x - 3 / 2 - 3 4 x - 5 / 2 . \lim_{x\to\infty}\frac{x^{1/2}+x^{-1/2}}{x^{1/2}-x^{-1/2}}=\lim_{x\to\infty}% \frac{\tfrac{1}{2}x^{-1/2}-\tfrac{1}{2}x^{-3/2}}{\tfrac{1}{2}x^{-1/2}+\tfrac{1% }{2}x^{-3/2}}=\lim_{x\to\infty}\frac{-\tfrac{1}{4}x^{-3/2}+\tfrac{3}{4}x^{-5/2% }}{-\tfrac{1}{4}x^{-3/2}-\tfrac{3}{4}x^{-5/2}}\dots.
  68. y = x 1 / 2 y=x^{1/2}
  69. lim x x 1 / 2 + x - 1 / 2 x 1 / 2 - x - 1 / 2 = lim y y + y - 1 y - y - 1 = lim y 1 - y - 2 1 + y - 2 = 1 1 = 1. \lim_{x\to\infty}\frac{x^{1/2}+x^{-1/2}}{x^{1/2}-x^{-1/2}}=\lim_{y\to\infty}% \frac{y+y^{-1}}{y-y^{-1}}=\lim_{y\to\infty}\frac{1-y^{-2}}{1+y^{-2}}=\frac{1}{% 1}=1.
  70. lim h 0 ( x + h ) n - x n h = n x n - 1 . \lim_{h\to 0}\frac{(x+h)^{n}-x^{n}}{h}=nx^{n-1}.
  71. 0 × 0 × ∞
  72. ∞ − ∞
  73. ∞ − ∞
  74. lim x 1 ( x x - 1 - 1 ln x ) \displaystyle\lim_{x\to 1}\left(\frac{x}{x-1}-\frac{1}{\ln x}\right)
  75. lim x 0 + x x = lim x 0 + ( e ln x ) x = lim x 0 + e x ln x = e lim x 0 + ( x ln x ) . \lim_{x\to 0^{+}}x^{x}=\lim_{x\to 0^{+}}(e^{\ln x})^{x}=\lim_{x\to 0^{+}}e^{x% \ln x}=e^{\lim_{x\to 0^{+}}(x\ln x)}.
  76. x x
  77. lim x 0 + x ln x = 0. \lim_{x\to 0^{+}}x\ln x=0.
  78. lim x 0 + x x = e 0 = 1. \lim_{x\to 0^{+}}x^{x}=e^{0}=1.
  79. lim | x | x sin 1 x . \lim_{|x|\to\infty}x\sin\frac{1}{x}.
  80. lim | x | x sin 1 x \displaystyle\lim_{|x|\to\infty}x\sin\frac{1}{x}
  81. y = 1 / x y=1/x
  82. x x
  83. y y
  84. lim | x | x sin 1 x = lim y 0 sin y y = 1. \lim_{|x|\to\infty}x\sin\frac{1}{x}=\lim_{y\to 0}\frac{\sin y}{y}=1.
  85. lim | x | x sin 1 x \displaystyle\lim_{|x|\to\infty}x\sin\frac{1}{x}
  86. x 1 x≥1
  87. x x
  88. g ( t ) g(t)
  89. y y
  90. f ( t ) f(t)
  91. [ g ( t ) , f ( t ) ] . [g(t),f(t)].
  92. f ( c ) = g ( c ) = 0 f(c)=g(c)=0
  93. f ( t ) / g ( t ) f(t)/g(t)
  94. t c t→c
  95. f f ( c ) , g ( c ) = 0 , 00 ff(c),g(c)=0,00
  96. g g ( t ) , f ( t ) gg(t),f(t)
  97. g g ( t ) , f ( t ) gg′(t),f′(t)
  98. t = c t=c
  99. f f
  100. g g
  101. c c
  102. f f
  103. g g
  104. c c
  105. f ( c ) = g ( c ) = 0 f(c)=g(c)=0\,
  106. g ( c ) 0 g^{\prime}(c)\neq 0
  107. lim x c f ( x ) g ( x ) = lim x c f ( x ) - 0 g ( x ) - 0 = lim x c f ( x ) - f ( c ) g ( x ) - g ( c ) = lim x c ( f ( x ) - f ( c ) x - c ) ( g ( x ) - g ( c ) x - c ) = lim x c ( f ( x ) - f ( c ) x - c ) lim x c ( g ( x ) - g ( c ) x - c ) = f ( c ) g ( c ) = lim x c f ( x ) g ( x ) . \lim_{x\to c}\frac{f(x)}{g(x)}=\lim_{x\to c}\frac{f(x)-0}{g(x)-0}=\lim_{x\to c% }\frac{f(x)-f(c)}{g(x)-g(c)}=\lim_{x\to c}\frac{\left(\frac{f(x)-f(c)}{x-c}% \right)}{\left(\frac{g(x)-g(c)}{x-c}\right)}=\frac{\lim\limits_{x\to c}\left(% \frac{f(x)-f(c)}{x-c}\right)}{\lim\limits_{x\to c}\left(\frac{g(x)-g(c)}{x-c}% \right)}=\frac{f^{\prime}(c)}{g^{\prime}(c)}=\lim_{x\to c}\frac{f^{\prime}(x)}% {g^{\prime}(x)}.
  108. c c
  109. g ( c ) 0 g^{\prime}(c)\neq 0
  110. \mathcal{I}
  111. g ( x ) 0 g^{\prime}(x)\neq 0
  112. \mathcal{I}
  113. \mathcal{I}
  114. m ( x ) = inf f ( ξ ) g ( ξ ) m(x)=\inf\frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}
  115. M ( x ) = sup f ( ξ ) g ( ξ ) M(x)=\sup\frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}
  116. ξ \xi
  117. \mathcal{I}
  118. \mathcal{I}
  119. ξ \xi
  120. f ( x ) - f ( y ) g ( x ) - g ( y ) = f ( ξ ) g ( ξ ) \frac{f(x)-f(y)}{g(x)-g(y)}=\frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}
  121. m ( x ) f ( x ) - f ( y ) g ( x ) - g ( y ) M ( x ) m(x)\leq\frac{f(x)-f(y)}{g(x)-g(y)}\leq M(x)
  122. lim x c f ( x ) = lim x c g ( x ) = 0 \lim_{x\rightarrow c}f(x)=\lim_{x\rightarrow c}g(x)=0
  123. \mathcal{I}
  124. m ( x ) f ( x ) - f ( y ) g ( x ) - g ( y ) = f ( x ) g ( x ) - f ( y ) g ( x ) 1 - g ( y ) g ( x ) M ( x ) m(x)\leq\frac{f(x)-f(y)}{g(x)-g(y)}=\frac{\frac{f(x)}{g(x)}-\frac{f(y)}{g(x)}}% {1-\frac{g(y)}{g(x)}}\leq M(x)
  125. f ( y ) g ( x ) \frac{f(y)}{g(x)}
  126. g ( y ) g ( x ) \frac{g(y)}{g(x)}
  127. m ( x ) f ( x ) g ( x ) M ( x ) m(x)\leq\frac{f(x)}{g(x)}\leq M(x)
  128. lim x c | g ( x ) | = \lim_{x\rightarrow c}|g(x)|=\infty
  129. \mathcal{I}
  130. S x = { y y is between x and c } S_{x}=\{y\mid y\,\text{ is between }x\,\text{ and }c\}
  131. m ( x ) f ( y ) - f ( x ) g ( y ) - g ( x ) = f ( y ) g ( y ) - f ( x ) g ( y ) 1 - g ( x ) g ( y ) M ( x ) m(x)\leq\frac{f(y)-f(x)}{g(y)-g(x)}=\frac{\frac{f(y)}{g(y)}-\frac{f(x)}{g(y)}}% {1-\frac{g(x)}{g(y)}}\leq M(x)
  132. f ( x ) g ( y ) \frac{f(x)}{g(y)}
  133. g ( x ) g ( y ) \frac{g(x)}{g(y)}
  134. m ( x ) lim inf y S x f ( y ) g ( y ) lim sup y S x f ( y ) g ( y ) M ( x ) m(x)\leq\liminf_{y\in S_{x}}\frac{f(y)}{g(y)}\leq\limsup_{y\in S_{x}}\frac{f(y% )}{g(y)}\leq M(x)
  135. lim x c m ( x ) = lim x c M ( x ) = lim x c f ( x ) g ( x ) = L \lim_{x\rightarrow c}m(x)=\lim_{x\rightarrow c}M(x)=\lim_{x\rightarrow c}\frac% {f^{\prime}(x)}{g^{\prime}(x)}=L
  136. lim x c m ( x ) \lim_{x\rightarrow c}m(x)
  137. lim x c M ( x ) \lim_{x\rightarrow c}M(x)
  138. x i c x_{i}\rightarrow c
  139. lim i m ( x i ) lim i f ( x i ) g ( x i ) lim i M ( x i ) \lim_{i}m(x_{i})\leqslant\lim_{i}\frac{f^{\prime}(x_{i})}{g^{\prime}(x_{i})}% \leqslant\lim_{i}M(x_{i})
  140. lim x c m ( x ) lim x c f ( x ) g ( x ) lim x c M ( x ) \lim_{x\rightarrow c}m(x)\leqslant\lim_{x\rightarrow c}\frac{f^{\prime}(x)}{g^% {\prime}(x)}\leqslant\lim_{x\rightarrow c}M(x)
  141. lim x c M ( x ) lim x c f ( x ) g ( x ) \lim_{x\rightarrow c}M(x)\leqslant\lim_{x\rightarrow c}\frac{f^{\prime}(x)}{g^% {\prime}(x)}
  142. ϵ i > 0 \epsilon_{i}>0
  143. lim i ϵ i = 0 \lim_{i}\epsilon_{i}=0
  144. x i c x_{i}\rightarrow c
  145. x i < y i < c x_{i}<y_{i}<c
  146. f ( y i ) g ( y i ) + ϵ i sup x i < ξ < c f ( ξ ) g ( ξ ) \frac{f^{\prime}(y_{i})}{g^{\prime}(y_{i})}+\epsilon_{i}\geqslant\sup_{x_{i}<% \xi<c}\frac{f^{\prime}(\xi)}{g^{\prime}(\xi)}
  147. sup \sup
  148. lim i M ( x i ) lim i f ( y i ) g ( y i ) + ϵ i \lim_{i}M(x_{i})\leqslant\lim_{i}\frac{f^{\prime}(y_{i})}{g^{\prime}(y_{i})}+% \epsilon_{i}
  149. = lim i f ( y i ) g ( y i ) + lim i ϵ i = lim i f ( y i ) g ( y i ) =\lim_{i}\frac{f^{\prime}(y_{i})}{g^{\prime}(y_{i})}+\lim_{i}\epsilon_{i}=\lim% _{i}\frac{f^{\prime}(y_{i})}{g^{\prime}(y_{i})}
  150. lim x c m ( x ) lim x c f ( x ) g ( x ) \lim_{x\rightarrow c}m(x)\geqslant\lim_{x\rightarrow c}\frac{f^{\prime}(x)}{g^% {\prime}(x)}
  151. lim x c ( lim inf y S x f ( y ) g ( y ) ) = lim inf x c f ( x ) g ( x ) \lim_{x\rightarrow c}\left(\liminf_{y\in S_{x}}\frac{f(y)}{g(y)}\right)=% \liminf_{x\rightarrow c}\frac{f(x)}{g(x)}
  152. lim x c ( lim sup y S x f ( y ) g ( y ) ) = lim sup x c f ( x ) g ( x ) \lim_{x\rightarrow c}\left(\limsup_{y\in S_{x}}\frac{f(y)}{g(y)}\right)=% \limsup_{x\rightarrow c}\frac{f(x)}{g(x)}
  153. lim x c f ( x ) g ( x ) \lim_{x\rightarrow c}\frac{f(x)}{g(x)}
  154. lim inf x c f ( x ) g ( x ) = lim sup x c f ( x ) g ( x ) = L \liminf_{x\rightarrow c}\frac{f(x)}{g(x)}=\limsup_{x\rightarrow c}\frac{f(x)}{% g(x)}=L
  155. lim x c f ( x ) g ( x ) \lim_{x\rightarrow c}\frac{f(x)}{g(x)}
  156. f ( x ) f^{\prime}(x)
  157. x = a x=a
  158. lim x a f ( x ) \lim_{x\rightarrow a}f^{\prime}(x)
  159. f ( a ) f^{\prime}(a)
  160. f ( a ) = lim x a f ( x ) f^{\prime}(a)=\lim_{x\rightarrow a}f^{\prime}(x)
  161. h ( x ) = f ( x ) - f ( a ) h(x)=f(x)-f(a)
  162. g ( x ) = x - a g(x)=x-a
  163. lim x a h ( x ) = 0 \lim_{x\rightarrow a}h(x)=0
  164. lim x a g ( x ) = 0 \lim_{x\rightarrow a}g(x)=0
  165. f ( a ) := lim x a f ( x ) - f ( a ) x - a = lim x a h ( x ) g ( x ) = lim x a f ( x ) f^{\prime}(a):=\lim_{x\rightarrow a}\frac{f(x)-f(a)}{x-a}=\lim_{x\rightarrow a% }\frac{h(x)}{g(x)}=\lim_{x\rightarrow a}f^{\prime}(x)

L-system.html

  1. ( k - 1 , k ) (k-1,k)
  2. G ( n ) = G ( n - 1 ) G ( n - 2 ) G(n)=G(n-1)G(n-2)
  3. G ( n ) G(n)

Labor_theory_of_value.html

  1. c + L = W c+L=W
  2. c c
  3. L L
  4. W W
  5. w w
  6. v u v_{u}
  7. w i = W v u \begin{matrix}w_{i}=\frac{W}{\sum v_{u}}\end{matrix}
  8. v u \sum v_{u}
  9. L L
  10. N L NL
  11. S L SL
  12. v v
  13. s s
  14. c + v + s = W c+v+s=W
  15. c + N L + S L = W c+NL+SL=W
  16. c + v c+v
  17. s s
  18. N L + S L NL+SL

Labour_economics.html

  1. maximize U ( w L + π , A ) subject to L + A k . \,\text{maximize}\quad U(wL+\pi,A)\quad\,\text{subject to}\quad L+A\leq k.
  2. M U L M U Y = d Y d L , {{MU^{L}}\over{MU^{Y}}}={{dY}\over{dL}},
  3. M P P L . P Q = V M P P L MPP_{L}.P_{Q}=VMPP_{L}
  4. V M P P L VMPP_{L}

Lagged_Fibonacci_generator.html

  1. S n = S n - 1 + S n - 2 S_{n}=S_{n-1}+S_{n-2}
  2. S n S n - j S n - k ( mod m ) , 0 < j < k S_{n}\equiv S_{n-j}\star S_{n-k}\;\;(\mathop{{\rm mod}}m),0<j<k
  3. \star
  4. x n x_{n}
  5. x n - a x_{n-a}
  6. x n - b x_{n-b}
  7. p = m a x ( a , b , c , ) p=max(a,b,c,\ldots)

Lagrange's_theorem_(group_theory).html

  1. f - 1 ( y ) = a b - 1 y . f^{-1}(y)=ab^{-1}y\mbox{.}~{}
  2. | G | = [ G : H ] | H | , \left|G\right|=\left[G:H\right]\cdot\left|H\right|\mbox{,}~{}
  3. a n = e . \displaystyle a^{n}=e\mbox{.}~{}
  4. ( p - 1 ) ! + 1 (p-1)!+1
  5. 2 p - 1 2^{p}-1
  6. ( ( / q ) \ 0 ) , ((\mathbb{Z}/q)\backslash 0),\cdot
  7. q - 1 q-1
  8. p < q p<q

Lagrangian_point.html

  1. M 1 ( R - r ) 2 = M 2 r 2 + ( M 1 M 1 + M 2 R - r ) M 1 + M 2 R 3 \frac{M_{1}}{(R-r)^{2}}=\frac{M_{2}}{r^{2}}+\left(\frac{M_{1}}{M_{1}+M_{2}}R-r% \right)\frac{M_{1}+M_{2}}{R^{3}}
  2. r R M 2 3 M 1 3 r\approx R\sqrt[3]{\frac{M_{2}}{3M_{1}}}
  3. 3 1.73 \sqrt{3}\approx 1.73
  4. T s , M 2 ( r ) = T M 2 , M 1 ( R ) 3 . T_{s,M_{2}}(r)=\frac{T_{M_{2},M_{1}}(R)}{\sqrt{3}}.
  5. M 1 ( R + r ) 2 + M 2 r 2 = ( M 1 M 1 + M 2 R + r ) M 1 + M 2 R 3 \frac{M_{1}}{(R+r)^{2}}+\frac{M_{2}}{r^{2}}=\left(\frac{M_{1}}{M_{1}+M_{2}}R+r% \right)\frac{M_{1}+M_{2}}{R^{3}}
  6. r R M 2 3 M 1 3 r\approx R\sqrt[3]{\frac{M_{2}}{3M_{1}}}
  7. M 1 ( R - r ) 2 + M 2 ( 2 R - r ) 2 = ( M 2 M 1 + M 2 R + R - r ) M 1 + M 2 R 3 \frac{M_{1}}{(R-r)^{2}}+\frac{M_{2}}{(2R-r)^{2}}=\left(\frac{M_{2}}{M_{1}+M_{2% }}R+R-r\right)\frac{M_{1}+M_{2}}{R^{3}}
  8. r R 7 M 2 12 M 1 r\approx R\frac{7M_{2}}{12M_{1}}
  9. 25 + 621 2 \tfrac{25+\sqrt{621}}{2}

LALR_parser.html

  1. k > 0 k>0

Lambda_calculus.html

  1. sqsum ( x , y ) = x × x + y × y \operatorname{sqsum}(x,y)=x\times x+y\times y
  2. ( x , y ) x × x + y × y (x,y)\mapsto x\times x+y\times y
  3. x x
  4. y y
  5. x × x + y × y x\times x+y\times y
  6. id ( x ) = x \operatorname{id}(x)=x
  7. x x x\mapsto x
  8. sqsum \operatorname{sqsum}
  9. ( x , y ) x × x + y × y (x,y)\mapsto x\times x+y\times y
  10. x ( y x × x + y × y ) x\mapsto(y\mapsto x\times x+y\times y)
  11. sqsum \operatorname{sqsum}
  12. ( ( x , y ) x × x + y × y ) ( 5 , 2 ) ((x,y)\mapsto x\times x+y\times y)(5,2)
  13. = 5 × 5 + 2 × 2 =5\times 5+2\times 2
  14. = 29 =29
  15. ( ( x ( y x × x + y × y ) ) ( 5 ) ) ( 2 ) ((x\mapsto(y\mapsto x\times x+y\times y))(5))(2)
  16. = ( y 5 × 5 + y × y ) ( 2 ) =(y\mapsto 5\times 5+y\times y)(2)
  17. = 5 × 5 + 2 × 2 =5\times 5+2\times 2
  18. = 29 =29
  19. x x
  20. t t
  21. x x
  22. ( λ x . t ) (\lambda x.t)
  23. t t
  24. s s
  25. ( t s ) (ts)
  26. λ x . t \lambda x.t
  27. x x
  28. t t
  29. λ x . x 2 + 2 \lambda x.x^{2}+2
  30. f ( x ) = x 2 + 2 f(x)=x^{2}+2
  31. x 2 + 2 x^{2}+2
  32. t t
  33. x x
  34. t t
  35. t s ts
  36. t t
  37. s s
  38. t t
  39. s s
  40. t ( s ) t(s)
  41. λ x . x + y \lambda x.x+y
  42. f ( x ) = x + y f(x)=x+y
  43. y y
  44. λ x . x + y \lambda x.x+y
  45. y y
  46. λ x . ( ( λ x . x ) x ) \lambda x.((\lambda x.x)x)
  47. ( λ x . ( λ x . x ) ) x (\lambda x.(\lambda x.x))x
  48. λ x . x \lambda x.x
  49. x x x\rightarrow x
  50. ( λ x . x ) y (\lambda x.x)y
  51. y y
  52. ( λ x . y ) (\lambda x.y)
  53. x y x\rightarrow y
  54. y y
  55. s t x stx
  56. ( s t ) x (st)x
  57. λ x . x \lambda x.x
  58. λ y . y \lambda y.y
  59. x x
  60. y y
  61. x x
  62. x x
  63. λ x . t \lambda x.t
  64. t t
  65. x x
  66. t s ts
  67. t t
  68. s s
  69. λ x . x \lambda x.x
  70. λ x . x + y \lambda x.x+y
  71. y y
  72. t t
  73. s s
  74. r r
  75. x x
  76. y y
  77. t [ x := r ] t[x:=r]
  78. r r
  79. x x
  80. t t
  81. x [ x := r ] = r x[x:=r]=r
  82. y [ x := r ] = y y[x:=r]=y
  83. x y x\neq y
  84. ( t s ) [ x := r ] = ( t [ x := r ] ) ( s [ x := r ] ) (ts)[x:=r]=(t[x:=r])(s[x:=r])
  85. ( λ x . t ) [ x := r ] = λ x . t (\lambda x.t)[x:=r]=\lambda x.t
  86. ( λ y . t ) [ x := r ] = λ y . ( t [ x := r ] ) (\lambda y.t)[x:=r]=\lambda y.(t[x:=r])
  87. x y x\neq y
  88. y y
  89. r r
  90. y y
  91. r r
  92. ( λ x . x ) [ y := y ] = λ x . ( x [ y := y ] ) = λ x . x (\lambda x.x)[y:=y]=\lambda x.(x[y:=y])=\lambda x.x
  93. ( ( λ x . y ) x ) [ x := y ] = ( ( λ x . y ) [ x := y ] ) ( x [ x := y ] ) = ( λ x . y ) y ((\lambda x.y)x)[x:=y]=((\lambda x.y)[x:=y])(x[x:=y])=(\lambda x.y)y
  94. y y
  95. r r
  96. ( λ x . y ) [ y := x ] = λ x . ( y [ y := x ] ) = λ x . x (\lambda x.y)[y:=x]=\lambda x.(y[y:=x])=\lambda x.x
  97. λ x . y \lambda x.y
  98. λ x . x \lambda x.x
  99. ( λ x . y ) [ y := x ] (\lambda x.y)[y:=x]
  100. z z
  101. ( λ z . y ) [ y := x ] = λ z . ( y [ y := x ] ) = λ z . x (\lambda z.y)[y:=x]=\lambda z.(y[y:=x])=\lambda z.x
  102. ( λ x . t ) s (\lambda x.t)s
  103. t [ x := s ] t[x:=s]
  104. ( λ x . t ) s t [ x := s ] (\lambda x.t)s\to t[x:=s]
  105. ( λ x . t ) s (\lambda x.t)s
  106. t [ x := s ] t[x:=s]
  107. s s
  108. ( λ x . x ) s x [ x := s ] = s (\lambda x.x)s\to x[x:=s]=s
  109. λ x . x \lambda x.x
  110. ( λ x . y ) s y [ x := s ] = y (\lambda x.y)s\to y[x:=s]=y
  111. λ x . y \lambda x.y
  112. ( λ x . x x ) ( λ x . x x ) (\lambda x.xx)(\lambda x.xx)
  113. ( λ x . x x ) ( λ x . x x ) ( x x ) [ x := λ x . x x ] = ( x [ x := λ x . x x ] ) ( x [ x := λ x . x x ] ) = ( λ x . x x ) ( λ x . x x ) (\lambda x.xx)(\lambda x.xx)\to(xx)[x:=\lambda x.xx]=(x[x:=\lambda x.xx])(x[x:% =\lambda x.xx])=(\lambda x.xx)(\lambda x.xx)
  114. λ \lambda

Lambert's_cosine_law.html

  1. I 0 = I d Ω d A d Ω 0 d A 0 I_{0}=\frac{I\,d\Omega\,dA}{d\Omega_{0}\,dA_{0}}
  2. I 0 = I cos ( θ ) d Ω d A d Ω 0 cos ( θ ) d A 0 = I d Ω d A d Ω 0 d A 0 I_{0}=\frac{I\cos(\theta)\,d\Omega\,dA}{d\Omega_{0}\,\cos(\theta)\,dA_{0}}=% \frac{I\,d\Omega\,dA}{d\Omega_{0}\,dA_{0}}
  3. F t o t F_{tot}
  4. I m a x I_{max}
  5. F t o t = 0 π / 2 0 2 π cos ( θ ) I m a x sin ( θ ) d ϕ d θ F_{tot}=\int\limits_{0}^{\pi/2}\,\int\limits_{0}^{2\pi}\cos(\theta)I_{max}\,% \sin(\theta)\,\operatorname{d}\phi\,\operatorname{d}\theta
  6. = 2 π I m a x 0 π / 2 cos ( θ ) sin ( θ ) d θ =2\pi\cdot I_{max}\int\limits_{0}^{\pi/2}\cos(\theta)\sin(\theta)\,% \operatorname{d}\theta
  7. = 2 π I m a x 0 π / 2 sin ( 2 θ ) 2 d θ =2\pi\cdot I_{max}\int\limits_{0}^{\pi/2}\frac{\sin(2\theta)}{2}\,% \operatorname{d}\theta
  8. F t o t = π sr I m a x F_{tot}=\pi\,\mathrm{sr}\cdot I_{max}
  9. sin ( θ ) \sin(\theta)
  10. I m a x I_{max}
  11. 1 / ( π sr ) 1/(\pi\,\mathrm{sr})
  12. π sr \pi\,\mathrm{sr}

Langmuir_probe.html

  1. n i n_{i}
  2. T i T_{i}
  3. f e ( v ) f_{e}(v)
  4. c s = k B ( Z T e + γ i T i ) / m i c_{s}=\sqrt{k_{B}(ZT_{e}+\gamma_{i}T_{i})/m_{i}}
  5. γ i \gamma_{i}
  6. γ i \gamma_{i}
  7. γ i = 1 \gamma_{i}=1
  8. γ i = 3 \gamma_{i}=3
  9. Z = 1 Z=1
  10. T i = T e T_{i}=T_{e}
  11. 2 \sqrt{2}
  12. e n e en_{e}
  13. j i m a x = e n e c s j_{i}^{max}=en_{e}c_{s}
  14. f ( v x ) d v x e - 1 2 m e v x 2 / k B T e f(v_{x})\,dv_{x}\propto e^{-\frac{1}{2}m_{e}v_{x}^{2}/k_{B}T_{e}}
  15. v e = v e 0 f ( v x ) v x d v x - f ( v x ) d v x \langle v_{e}\rangle=\frac{\int_{v_{e0}}^{\infty}f(v_{x})\,v_{x}\,dv_{x}}{\int% _{-\infty}^{\infty}f(v_{x})\,dv_{x}}
  16. v e 0 = 2 e Δ V / m e v_{e0}=\sqrt{2e\Delta V/m_{e}}
  17. Δ V \Delta V
  18. v e = k B T e 2 π m e e - e Δ V / k B T e \langle v_{e}\rangle=\sqrt{\frac{k_{B}T_{e}}{2\pi m_{e}}}\,e^{-e\Delta V/k_{B}% T_{e}}
  19. j e = j i m a x m i / 2 π m e e - e Δ V / k B T e j_{e}=j_{i}^{max}\sqrt{m_{i}/2\pi m_{e}}\,e^{-e\Delta V/k_{B}T_{e}}
  20. j = j i m a x ( - 1 + m i / 2 π m e e - e Δ V / k B T e ) j=j_{i}^{max}\left(-1+\sqrt{m_{i}/2\pi m_{e}}\,e^{-e\Delta V/k_{B}T_{e}}\right)
  21. Δ V = ( k B T e / e ) ( 1 / 2 ) ln ( m i / 2 π m e ) \Delta V=(k_{B}T_{e}/e)\,(1/2)\ln(m_{i}/2\pi m_{e})
  22. μ i = m i / m e \mu_{i}=m_{i}/m_{e}
  23. Δ V = ( k B T e / e ) ( 2.8 + 0.5 ln μ i ) \Delta V=(k_{B}T_{e}/e)\,(2.8+0.5\ln\mu_{i})
  24. j = j i m a x ( - 1 + e e ( V 0 - Δ V ) / k B T e ) j=j_{i}^{max}\left(-1+\,e^{e(V_{0}-\Delta V)/k_{B}T_{e}}\right)
  25. v e 0 = 0 v_{e0}=0
  26. j e m a x = j i m a x m i / π m e = j i m a x ( 24.2 μ i ) j_{e}^{max}=j_{i}^{max}\sqrt{m_{i}/\pi m_{e}}=j_{i}^{max}\left(24.2\,\sqrt{\mu% _{i}}\right)
  27. j e m a x = e n e k B ( γ e T e + T i ) / m e = j i m a x m i / m e = j i m a x ( 42.8 μ i ) j_{e}^{max}=en_{e}\sqrt{k_{B}(\gamma_{e}T_{e}+T_{i})/m_{e}}=j_{i}^{max}\sqrt{m% _{i}/m_{e}}=j_{i}^{max}\left(42.8\,\sqrt{\mu_{i}}\right)
  28. Φ p r e = 1 2 m i c s 2 Z e = k B ( T e + Z γ i T i ) / ( 2 Z e ) \Phi_{pre}=\frac{\frac{1}{2}m_{i}c_{s}^{2}}{Ze}=k_{B}(T_{e}+Z\gamma_{i}T_{i})/% (2Ze)
  29. A e f f A_{eff}
  30. T i = T e T_{i}=T_{e}
  31. Z = 1 Z=1
  32. γ i = 3 \gamma_{i}=3
  33. n e , s h = 0.5 n e n_{e,sh}=0.5\,n_{e}
  34. I = I i m a x ( - 1 + e ( V p r - V f l ) / ( k B T e / e ) ) I=I_{i}^{max}(-1+e^{(V_{pr}-V_{fl})/(k_{B}T_{e}/e)})
  35. I i m a x = e n e k B T e / m i A e f f I_{i}^{max}=en_{e}\sqrt{k_{B}T_{e}/m_{i}}\,A_{eff}
  36. V b i a s V_{bias}
  37. I = I i m a x ( - 1 + e e ( V 2 - V f l ) / k B T e ) = - I i m a x ( - 1 + e e ( V 1 - V f l ) / k B T e ) I=I_{i}^{max}\left(-1+\,e^{e(V_{2}-V_{fl})/k_{B}T_{e}}\right)=-I_{i}^{max}% \left(-1+\,e^{e(V_{1}-V_{fl})/k_{B}T_{e}}\right)
  38. V b i a s = V 2 - V 1 V_{bias}=V_{2}-V_{1}
  39. I = I i m a x tanh ( 1 2 e V b i a s k B T e ) I=I_{i}^{max}\tanh\left(\frac{1}{2}\,\frac{eV_{bias}}{k_{B}T_{e}}\right)
  40. A 1 A_{1}
  41. I = A 1 J i m a x [ coth ( e V b i a s 2 k B T e ) + ( A 1 A 2 - 1 ) e - e V b i a s / 2 k B T e 2 sinh ( e V b i a s 2 k B T e ) ] - 1 I=A_{1}J_{i}^{max}\left[\coth\left(\frac{eV_{bias}}{2k_{B}T_{e}}\right)+\frac{% \left(\frac{A_{1}}{A_{2}}-1\right)\,e^{-eV_{bias}/2k_{B}T_{e}}}{2\sinh\left(% \frac{eV_{bias}}{2k_{B}T_{e}}\right)}\right]^{-1}
  42. - I + = I - = I i m a x -I_{+}=I_{-}=I_{i}^{max}
  43. I f l = 0 I_{fl}=0
  44. I p r o b e = - I e e - e V p r o b e / ( k T e ) + I i m a x I_{probe}=-I_{e}e^{-eV_{probe}/(kT_{e})}+I_{i}^{max}
  45. I e = S J e = S n e e k T e / 2 π m e I_{e}=SJ_{e}=Sn_{e}e\sqrt{kT_{e}/2\pi m_{e}}
  46. I + = - I e e - e V + / ( k T e ) + I i m a x I_{+}=-I_{e}e^{-eV_{+}/(kT_{e})}+I_{i}^{max}
  47. I - = - I e e - e V - / ( k T e ) + I i m a x I_{-}=-I_{e}e^{-eV_{-}/(kT_{e})}+I_{i}^{max}
  48. I f l = - I e e - e V f l / ( k T e ) + I i m a x I_{fl}=-I_{e}e^{-eV_{fl}/(kT_{e})}+I_{i}^{max}
  49. ( I + - I f l ) / ( I + - I - ) = ( 1 - e - e ( V f l - V + ) / ( k T e ) ) / ( 1 - e - e ( V - - V + ) / ( k T e ) ) \left(I_{+}-I_{fl})/(I_{+}-I_{-}\right)=\left(1-e^{-e(V_{fl}-V_{+})/(kT_{e})}% \right)/\left(1-e^{-e(V_{-}-V_{+})/(kT_{e})}\right)
  50. 1 / 2 = ( 1 - e - e ( V f l - V + ) / ( k T e ) ) / ( 1 - e - e ( V - - V + ) / ( k T e ) ) 1/2=\left(1-e^{-e(V_{fl}-V_{+})/(kT_{e})}\right)/\left(1-e^{-e(V_{-}-V_{+})/(% kT_{e})}\right)
  51. ( V + - V f l ) = ( k B T e / e ) ln 2 (V_{+}-V_{fl})=(k_{B}T_{e}/e)\ln 2
  52. Φ t u r b = n ~ e v ~ E × B I ~ i m a x ( V ~ f l , 2 - V ~ f l , 1 ) \Phi_{turb}=\langle\tilde{n}_{e}\tilde{v}_{E\times B}\rangle\propto\langle% \tilde{I}_{i}^{max}(\tilde{V}_{fl,2}-\tilde{V}_{fl,1})\rangle
  53. i ( V ) i(V)
  54. i ( V ) i(V)
  55. V V
  56. S z S_{z}
  57. O O
  58. h h
  59. λ D λ T e \lambda_{D}\ll\lambda_{Te}
  60. λ D \lambda_{D}
  61. λ T e \lambda_{Te}
  62. O O
  63. Δ S \Delta S
  64. d i di
  65. Δ S \Delta S
  66. d i = e Δ S d n ( v , ϑ ) v cos ϑ di=e\Delta Sdn(v,\vartheta)v\cos\vartheta
  67. v v
  68. v \vec{v}
  69. d n ( v , ϑ ) = n f ( v ) 2 π sin ϑ 4 π d v d ϑ dn(v,\vartheta)=nf(v)\frac{2\pi\sin\vartheta}{4\pi}dvd\vartheta
  70. 2 π sin ϑ d ϑ 2\pi\sin\vartheta d\vartheta
  71. 2 π sin ϑ d ϑ / 4 π 2\pi\sin\vartheta d\vartheta/4\pi
  72. ϑ \vartheta
  73. O O
  74. v \vec{v}
  75. d v dv
  76. f ( v ) f(v)
  77. 0 f ( v ) d v = 1 \int\limits_{0}^{\infty}f(v)dv=1
  78. Δ S S z \Delta S\rightarrow S_{z}
  79. ϑ \vartheta
  80. v v
  81. i ( v ) = e n S z 1 4 π 2 e V / m f ( v ) d v 0 ζ v cos ϑ 2 π sin ϑ d ϑ i(v)=enS_{z}\frac{1}{4\pi}\int\limits_{\sqrt{2eV/m}}^{\infty}f(v)dv\int\limits% _{0}^{\zeta}v\cos\vartheta 2\pi\sin\vartheta d\vartheta
  82. V V
  83. V = 0 V=0
  84. 2 e V / m \sqrt{2eV/m}
  85. V V
  86. ζ \zeta
  87. ϑ \vartheta
  88. v v
  89. ζ \zeta
  90. v cos ζ = 2 e V / m v\cos\zeta=\sqrt{2eV/m}
  91. ζ \zeta
  92. - < V 0 -\infty<V\leq 0
  93. i ( V ) = e n S z 4 2 e V / m f ( v ) ( 1 - 2 e V m v 2 ) v d v i(V)=\frac{enS_{z}}{4}\int\limits_{\sqrt{2eV/m}}^{\infty}f(v)\left(1-\frac{2eV% }{mv^{2}}\right)vdv
  94. V V
  95. i ′′ ( V ) = e 2 n S z 4 m 1 V f ( 2 e V / m ) i^{\prime\prime}(V)=\frac{e^{2}nS_{z}}{4m}\frac{1}{V}f\left(\sqrt{2eV/m}\right)
  96. f ( 2 e V / m ) f\left(\sqrt{2eV/m}\right)
  97. f ( 0 ) ( v ) = 4 π v 2 v p 3 exp ( - v 2 / v p 2 ) f^{(0)}(v)=\frac{4}{\sqrt{\pi}}\frac{v^{2}}{v_{p}^{3}}\exp\left(-v^{2}/v_{p}^{% 2}\right)
  98. v p = v π / 2 v_{p}=\langle v\rangle\sqrt{\pi}/2
  99. i ( 0 ) ( V ) = e n v 4 S z exp ( - e V / p ) i^{(0)}(V)=\frac{en\langle v\rangle}{4}S_{z}\exp\left(-eV/\mathcal{E}_{p}\right)
  100. ln ( i ( 0 ) ( V ) / i ( 0 ) ( 0 ) ) = - e V / p \ln\left(i^{(0)}(V)/i^{(0)}(0)\right)=-eV/\mathcal{E}_{p}
  101. p = k B T \mathcal{E}_{p}=k_{B}T
  102. i t h ( 0 ) i_{th}(0)
  103. S z = 2 π r z l z S_{z}=2\pi r_{z}l_{z}
  104. V = 0 V=0
  105. v \langle v\rangle
  106. V = 0 V=0
  107. i t h ( 0 ) = e n v 1 4 × 2 π r z l z i_{th}(0)=en\langle v\rangle\frac{1}{4}\times 2\pi r_{z}l_{z}
  108. n n
  109. r z r_{z}
  110. l z l_{z}
  111. v d v v_{d}\gg\langle v\rangle
  112. i d = e n v d × 2 r z l z i_{d}=env_{d}\times 2r_{z}l_{z}
  113. M ( 0 ) = v d / v = ( π / 2 ) α 1 M^{(0)}=v_{d}/\langle v\rangle=(\sqrt{\pi}/2)\alpha\gtrsim 1
  114. α \alpha
  115. ( π / 2 ) v = v p (\sqrt{\pi}/2)\langle v\rangle=v_{p}
  116. v p v_{p}
  117. α = v d / v p \alpha=v_{d}/v_{p}
  118. α 1 \alpha\gtrsim 1
  119. v d v_{d}
  120. V = 0 V=0
  121. i ( 0 ) e n S z = v 4 exp ( - α 2 / 2 ) I 0 ( α 2 / 2 ) ( 1 + α 2 ( 1 + I 1 ( α 2 / 2 ) / I 0 ( α 2 / 2 ) ) ) \frac{i(0)}{enS_{z}}=\frac{\langle v\rangle}{4}\exp(-\alpha^{2}/2)I_{0}(\alpha% ^{2}/2)\left(1+\alpha^{2}\left(1+I_{1}(\alpha^{2}/2)/I_{0}(\alpha^{2}/2)\right% )\right)
  122. I 0 I_{0}
  123. I 1 I_{1}
  124. α 0 \alpha\rightarrow 0
  125. α \alpha\rightarrow\infty
  126. i ′′ ( V ) i^{\prime\prime}(V)
  127. V V
  128. i ′′ ( x ) = e n S z v p 2 π 3 / 2 ( p / e ) 2 1 x 0 π ( x - cos φ ) exp ( - α 2 ( x - cos φ ) ) d φ i^{\prime\prime}(x)=enS_{z}\frac{v_{p}}{2\pi^{3/2}(\mathcal{E}_{p}/e)^{2}}% \frac{1}{\sqrt{x}}\int\limits_{0}^{\pi}(\sqrt{x}-\cos\varphi)\exp\left(-\alpha% ^{2}(\sqrt{x}-\cos\varphi)\right)d\varphi
  129. x = 1 α 2 V p / e x=\frac{1}{\alpha^{2}}\frac{V}{\mathcal{E}_{p}/e}
  130. p / e \mathcal{E}_{p}/e
  131. n n
  132. α \alpha
  133. v \langle v\rangle
  134. v p v_{p}
  135. i ′′ ( V ) i^{\prime\prime}(V)

Laplace's_equation.html

  1. 2 φ = 0 or Δ φ = 0 \nabla^{2}\varphi=0\qquad\mbox{or}~{}\qquad\Delta\varphi=0
  2. φ \varphi
  3. Δ f = 2 f x 2 + 2 f y 2 + 2 f z 2 = 0. \Delta f=\frac{\partial^{2}f}{\partial x^{2}}+\frac{\partial^{2}f}{\partial y^% {2}}+\frac{\partial^{2}f}{\partial z^{2}}=0.
  4. Δ f = 1 r r ( r f r ) + 1 r 2 2 f ϕ 2 + 2 f z 2 = 0 \Delta f=\frac{1}{r}\frac{\partial}{\partial r}\left(r\frac{\partial f}{% \partial r}\right)+\frac{1}{r^{2}}\frac{\partial^{2}f}{\partial\phi^{2}}+\frac% {\partial^{2}f}{\partial z^{2}}=0
  5. Δ f = 1 ρ 2 ρ ( ρ 2 f ρ ) + 1 ρ 2 sin θ θ ( sin θ f θ ) + 1 ρ 2 sin 2 θ 2 f φ 2 = 0. \Delta f=\frac{1}{\rho^{2}}\frac{\partial}{\partial\rho}\left(\rho^{2}\frac{% \partial f}{\partial\rho}\right)+\frac{1}{\rho^{2}\sin\theta}\frac{\partial}{% \partial\theta}\left(\sin\theta\frac{\partial f}{\partial\theta}\right)+\frac{% 1}{\rho^{2}\sin^{2}\theta}\frac{\partial^{2}f}{\partial\varphi^{2}}=0.
  6. Δ f = ξ j ( f ξ k g k j ) + f ξ j g j m Γ m n n = 0 , \Delta f=\frac{\partial}{\partial\xi^{j}}\left(\frac{\partial f}{\partial\xi^{% k}}g^{kj}\right)+\frac{\partial f}{\partial\xi^{j}}g^{jm}\Gamma^{n}_{mn}=0,
  7. Δ f = 1 | g | ξ i ( | g | g i j f ξ j ) = 0 , ( g = det { g i j } ) . \Delta f=\frac{1}{\sqrt{|g|}}\frac{\partial}{\partial\xi^{i}}\!\left(\sqrt{|g|% }g^{ij}\frac{\partial f}{\partial\xi^{j}}\right)=0,\qquad(g=\mathrm{det}\{g_{% ij}\}).
  8. 2 f = 0 \nabla^{2}f=0
  9. Δ f = 0 , \Delta f=0,
  10. Δ f = 2 f = f = div grad f , \Delta f=\nabla^{2}f=\nabla\cdot\nabla f=\operatorname{div}\operatorname{grad}f,
  11. Δ f = h \Delta f=h
  12. 2 ψ x 2 + 2 ψ y 2 ψ x x + ψ y y = 0. \frac{\partial^{2}\psi}{\partial x^{2}}+\frac{\partial^{2}\psi}{\partial y^{2}% }\equiv\psi_{xx}+\psi_{yy}=0.
  13. f ( z ) = u ( x , y ) + i v ( x , y ) , f(z)=u(x,y)+iv(x,y),
  14. u x = v y , v x = - u y . u_{x}=v_{y},\quad v_{x}=-u_{y}.
  15. u y y = ( - v x ) y = - ( v y ) x = - ( u x ) x . u_{yy}=(-v_{x})_{y}=-(v_{y})_{x}=-(u_{x})_{x}.
  16. f ( z ) = φ ( x , y ) + i ψ ( x , y ) , f(z)=\varphi(x,y)+i\psi(x,y),
  17. ψ x = - φ y , ψ y = φ x . \psi_{x}=-\varphi_{y},\quad\psi_{y}=\varphi_{x}.
  18. d ψ = - φ y d x + φ x d y . d\psi=-\varphi_{y}\,dx+\varphi_{x}\,dy.
  19. ψ x y = ψ y x , \psi_{xy}=\psi_{yx},
  20. φ = log r , \varphi=\log r,
  21. f ( z ) = log z = log r + i θ . f(z)=\log z=\log r+i\theta.
  22. f ( z ) = n = 0 c n z n , f(z)=\sum_{n=0}^{\infty}c_{n}z^{n},
  23. c n = a n + i b n . c_{n}=a_{n}+ib_{n}.
  24. f ( z ) = n = 0 [ a n r n cos n θ - b n r n sin n θ ] + i n = 1 [ a n r n sin n θ + b n r n cos n θ ] , f(z)=\sum_{n=0}^{\infty}\left[a_{n}r^{n}\cos n\theta-b_{n}r^{n}\sin n\theta% \right]+i\sum_{n=1}^{\infty}\left[a_{n}r^{n}\sin n\theta+b_{n}r^{n}\cos n% \theta\right],
  25. u x + v y = 0 , u_{x}+v_{y}=0,
  26. × 𝐕 = v x - u y = 0. \nabla\times\mathbf{V}=v_{x}-u_{y}=0.
  27. d ψ = v d x - u d y , d\psi=vdx-udy,
  28. ψ x = v , ψ y = - u , \psi_{x}=v,\quad\psi_{y}=-u,
  29. φ x = - u , φ y = - v . \varphi_{x}=-u,\quad\varphi_{y}=-v.
  30. × ( u , v , 0 ) = ( v x - u y ) 𝐤 ^ = 𝟎 , \nabla\times(u,v,0)=(v_{x}-u_{y})\hat{\mathbf{k}}=\mathbf{0},
  31. ( u , v ) = ρ , \nabla\cdot(u,v)=\rho,
  32. d φ = - u d x - v d y , d\varphi=-u\,dx-v\,dy,
  33. φ x = - u , φ y = - v . \varphi_{x}=-u,\quad\varphi_{y}=-v.
  34. φ x x + φ y y = - ρ , \varphi_{xx}+\varphi_{yy}=-\rho,
  35. Δ u = u x x + u y y + u z z = - δ ( x - x , y - y , z - z ) , \Delta u=u_{xx}+u_{yy}+u_{zz}=-\delta(x-x^{\prime},y-y^{\prime},z-z^{\prime}),
  36. V u d V = - 1. \iiint_{V}\nabla\cdot\nabla u\,dV=-1.
  37. - 1 = V u d V = S d u d r d S = 4 π a 2 d u d r | r = a . -1=\iiint_{V}\nabla\cdot\nabla u\,dV=\iint_{S}\frac{du}{dr}\,dS=\left.4\pi a^{% 2}\frac{du}{dr}\right|_{r=a}.
  38. d u d r = - 1 4 π r 2 , \frac{du}{dr}=-\frac{1}{4\pi r^{2}},
  39. u = 1 4 π r . u=\frac{1}{4\pi r}.
  40. u = - log ( r ) 2 π . u=-\frac{\log(r)}{2\pi}.
  41. G ( x , y , z ; x , y , z ) G(x,y,z;x^{\prime},y^{\prime},z^{\prime})
  42. G = - δ ( x - x , y - y , z - z ) in V , \nabla\cdot\nabla G=-\delta(x-x^{\prime},y-y^{\prime},z-z^{\prime})\qquad\hbox% {in }V,
  43. G = 0 if ( x , y , z ) on S . G=0\quad\hbox{if}\quad(x,y,z)\qquad\hbox{on }S.
  44. u = - f , \nabla\cdot\nabla u=-f,
  45. V [ G u - u G ] d V = V [ G u - u G ] d V = S [ G u n - u G n ] d S . \iiint_{V}\left[G\,\nabla\cdot\nabla u-u\,\nabla\cdot\nabla G\right]\,dV=% \iiint_{V}\nabla\cdot\left[G\nabla u-u\nabla G\right]\,dV=\iint_{S}\left[Gu_{n% }-uG_{n}\right]\,dS.\,
  46. u ( x , y , z ) = V G f d V + S G n g d S . u(x^{\prime},y^{\prime},z^{\prime})=\iiint_{V}Gf\,dV+\iint_{S}G_{n}g\,dS.\,
  47. ρ = a 2 ρ . \rho^{\prime}=\frac{a^{2}}{\rho}.\,
  48. 1 4 π R - a 4 π ρ R , \frac{1}{4\pi R}-\frac{a}{4\pi\rho R^{\prime}},\,
  49. u ( P ) = 1 4 π a 3 ( 1 - ρ 2 a 2 ) g ( θ , φ ) sin φ ( a 2 + ρ 2 - 2 a ρ cos Θ ) 3 2 d θ d φ , u(P)=\frac{1}{4\pi}a^{3}\left(1-\frac{\rho^{2}}{a^{2}}\right)\iint\frac{g(% \theta^{\prime},\varphi^{\prime})\sin\varphi^{\prime}}{(a^{2}+\rho^{2}-2a\rho% \cos\Theta)^{\frac{3}{2}}}d\theta^{\prime}\,d\varphi^{\prime},
  50. cos Θ = cos φ cos φ + sin φ sin φ cos ( θ - θ ) . \cos\Theta=\cos\varphi\cos\varphi^{\prime}+\sin\varphi\sin\varphi^{\prime}\cos% (\theta-\theta^{\prime}).
  51. E = - V E=-\nabla V
  52. 2 V = - ρ ε 0 \nabla^{2}V=-\frac{\rho}{\varepsilon_{0}}

Laplace_transform.html

  1. z = X ( x ) e a x d x and z = X ( x ) x A d x z=\int X(x)e^{ax}\,dx\quad\,\text{ and }\quad z=\int X(x)x^{A}\,dx
  2. X ( x ) e - a x a x d x , \int X(x)e^{-ax}a^{x}\,dx,
  3. x s ϕ ( x ) d x , \int x^{s}\phi(x)\,dx,
  4. F ( s ) = 0 e - s t f ( t ) d t . F(s)=\int_{0}^{\infty}e^{-st}f(t)\,dt.
  5. s = σ + i ω , s=\sigma+i\omega,
  6. σ \sigma
  7. { f } \mathcal{L}\{f\}
  8. { f ( t ) } \mathcal{L}\{f(t)\}
  9. [ 0 , ) [0,∞)
  10. { μ } ( s ) = [ 0 , ) e - s t d μ ( t ) . \mathcal{L}\{\mu\}(s)=\int_{[0,\infty)}e^{-st}\,d\mu(t).
  11. { f } ( s ) = 0 - e - s t f ( t ) d t , \mathcal{L}\{f\}(s)=\int_{0^{-}}^{\infty}e^{-st}f(t)\,dt,
  12. lim ε 0 - ε . \lim_{\varepsilon\downarrow 0}\int_{-\varepsilon}^{\infty}.
  13. { f } ( s ) = E [ e - s X ] . \mathcal{L}\{f\}(s)=E\!\left[e^{-sX}\right]\!.
  14. F X ( x ) = - 1 { E [ e - s X ] s } ( x ) = - 1 { { f } ( s ) s } ( x ) . F_{X}(x)=\mathcal{L}^{-1}\!\left\{\frac{E\left[e^{-sX}\right]}{s}\right\}\!(x)% =\mathcal{L}^{-1}\!\left\{\frac{\mathcal{L}\{f\}(s)}{s}\right\}\!(x).
  15. { f } ( s ) = - e - s t f ( t ) d t . \mathcal{B}\{f\}(s)=\int_{-\infty}^{\infty}e^{-st}f(t)\,dt.
  16. f ( t ) = - 1 { F } ( t ) = 1 2 π i lim T γ - i T γ + i T e s t F ( s ) d s , f(t)=\mathcal{L}^{-1}\{F\}(t)=\frac{1}{2\pi i}\lim_{T\to\infty}\int_{\gamma-iT% }^{\gamma+iT}e^{st}F(s)\,ds,
  17. lim R 0 R f ( t ) e - s t d t \lim_{R\to\infty}\int_{0}^{R}f(t)e^{-st}\,dt
  18. 0 | f ( t ) e - s t | d t \int_{0}^{\infty}\left|f(t)e^{-st}\right|\,dt
  19. F ( s ) = ( s - s 0 ) 0 e - ( s - s 0 ) t β ( t ) d t , β ( u ) = 0 u e - s 0 t f ( t ) d t . F(s)=(s-s_{0})\int_{0}^{\infty}e^{-(s-s_{0})t}\beta(t)\,dt,\quad\beta(u)=\int_% {0}^{u}e^{-s_{0}t}f(t)\,dt.
  20. f ( t ) \displaystyle f(t)
  21. a f ( t ) + b g ( t ) af(t)+bg(t)
  22. a F ( s ) + b G ( s ) aF(s)+bG(s)
  23. t f ( t ) tf(t)
  24. - F ( s ) -F^{\prime}(s)
  25. t n f ( t ) t^{n}f(t)
  26. ( - 1 ) n F ( n ) ( s ) (-1)^{n}F^{(n)}(s)
  27. f ( t ) f^{\prime}(t)
  28. s F ( s ) - f ( 0 ) sF(s)-f(0)
  29. f ′′ ( t ) f^{\prime\prime}(t)
  30. s 2 F ( s ) - s f ( 0 ) - f ( 0 ) s^{2}F(s)-sf(0)-f^{\prime}(0)
  31. f ( n ) ( t ) f^{(n)}(t)
  32. s n F ( s ) - k = 1 n s n - k f ( k - 1 ) ( 0 ) s^{n}F(s)-\sum_{k=1}^{n}s^{n-k}f^{(k-1)}(0)
  33. 1 t f ( t ) \frac{1}{t}f(t)
  34. s F ( σ ) d σ \int_{s}^{\infty}F(\sigma)\,d\sigma
  35. 0 t f ( τ ) d τ = ( u * f ) ( t ) \int_{0}^{t}f(\tau)\,d\tau=(u*f)(t)
  36. 1 s F ( s ) {1\over s}F(s)
  37. f ( a t ) f(at)
  38. 1 a F ( s a ) \frac{1}{a}F\left({s\over a}\right)
  39. a > 0 a>0
  40. e a t f ( t ) e^{at}f(t)
  41. F ( s - a ) F(s-a)
  42. f ( t - a ) u ( t - a ) f(t-a)u(t-a)
  43. e - a s F ( s ) e^{-as}F(s)
  44. f ( t ) g ( t ) f(t)g(t)
  45. 1 2 π i lim T c - i T c + i T F ( σ ) G ( s - σ ) d σ \frac{1}{2\pi i}\lim_{T\to\infty}\int_{c-iT}^{c+iT}F(\sigma)G(s-\sigma)\,d\sigma
  46. ( f * g ) ( t ) = 0 t f ( τ ) g ( t - τ ) d τ (f*g)(t)=\int_{0}^{t}f(\tau)g(t-\tau)\,d\tau
  47. F ( s ) G ( s ) F(s)\cdot G(s)
  48. f * ( t ) f^{*}(t)
  49. F * ( s * ) F^{*}(s^{*})
  50. f ( t ) g ( t ) f(t)\star g(t)
  51. F * ( - s * ) G ( s ) F^{*}(-s^{*})\cdot G(s)
  52. f ( t ) f(t)
  53. 1 1 - e - T s 0 T e - s t f ( t ) d t {1\over 1-e^{-Ts}}\int_{0}^{T}e^{-st}f(t)\,dt
  54. f ( 0 + ) = lim s s F ( s ) . f(0^{+})=\lim_{s\to\infty}{sF(s)}.
  55. f ( ) = lim s 0 s F ( s ) f(\infty)=\lim_{s\to 0}{sF(s)}
  56. F ( s ) F(s)
  57. f ( t ) = e t f(t)=e^{t}
  58. f ( t ) = sin ( t ) f(t)=\sin(t)
  59. n = 0 a ( n ) x n \sum_{n=0}^{\infty}a(n)x^{n}
  60. 0 f ( t ) x t d t \int_{0}^{\infty}f(t)x^{t}\,dt
  61. 0 f ( t ) ( e log x ) t d t \int_{0}^{\infty}f(t)\left(e^{\log{x}}\right)^{t}\,dt
  62. log x < 0 \log{x}<0
  63. s = l o g x −s=logx
  64. 0 f ( t ) e - s t d t \int_{0}^{\infty}f(t)e^{-st}\,dt
  65. μ n = 0 t n f ( t ) d t \mu_{n}=\int_{0}^{\infty}t^{n}f(t)\,dt
  66. ( - 1 ) n ( f ) ( n ) ( 0 ) = μ n (-1)^{n}(\mathcal{L}f)^{(n)}(0)=\mu_{n}
  67. μ n = E [ X n ] \mu_{n}=E[X^{n}]
  68. μ n = ( - 1 ) n d n d s n E [ e - s X ] . \mu_{n}=(-1)^{n}\frac{d^{n}}{ds^{n}}E\left[e^{-sX}\right].
  69. { f ( t ) } \displaystyle\mathcal{L}\left\{f(t)\right\}
  70. { f ( t ) } = s { f ( t ) } - f ( 0 - ) , \mathcal{L}\left\{f^{\prime}(t)\right\}=s\cdot\mathcal{L}\left\{f(t)\right\}-f% (0^{-}),
  71. { f ( t ) } = s - e - s t f ( t ) d t = s { f ( t ) } . \mathcal{L}\left\{{f^{\prime}(t)}\right\}=s\int_{-\infty}^{\infty}e^{-st}f(t)% \,dt=s\cdot\mathcal{L}\{f(t)\}.
  72. { f ( n ) ( t ) } = s n { f ( t ) } - s n - 1 f ( 0 - ) - - f ( n - 1 ) ( 0 - ) , \mathcal{L}\left\{f^{(n)}(t)\right\}=s^{n}\cdot\mathcal{L}\left\{f(t)\right\}-% s^{n-1}f(0^{-})-\cdots-f^{(n-1)}(0^{-}),
  73. { f ( t ) } = F ( s ) \mathcal{L}\left\{f(t)\right\}=F(s)
  74. { f ( t ) t } = s F ( p ) d p , \mathcal{L}\left\{\frac{f(t)}{t}\right\}=\int_{s}^{\infty}F(p)\,dp,
  75. 0 f ( t ) t e - s t d t = s F ( p ) d p . \int_{0}^{\infty}\frac{f(t)}{t}e^{-st}\,dt=\int_{s}^{\infty}F(p)\,dp.
  76. 0 f ( t ) t d t = 0 F ( p ) d p . \int_{0}^{\infty}\frac{f(t)}{t}\,dt=\int_{0}^{\infty}F(p)\,dp.
  77. 0 cos a t - cos b t t d t = 0 ( p p 2 + a 2 - p p 2 + b 2 ) d p = 1 2 ln p 2 + a 2 p 2 + b 2 | 0 = ln b - ln a . \int_{0}^{\infty}\frac{\cos at-\cos bt}{t}\,dt=\int_{0}^{\infty}\left(\frac{p}% {p^{2}+a^{2}}-\frac{p}{p^{2}+b^{2}}\right)\,dp=\frac{1}{2}\left.\ln\frac{p^{2}% +a^{2}}{p^{2}+b^{2}}\right|_{0}^{\infty}=\ln b-\ln a.
  78. { * g } ( s ) = 0 e - s t d g ( t ) . \{\mathcal{L}^{*}g\}(s)=\int_{0}^{\infty}e^{-st}dg(t).
  79. g ( x ) = 0 x f ( t ) d t g(x)=\int_{0}^{x}f(t)\,dt
  80. f ^ ( ω ) = { f ( t ) } = { f ( t ) } | s = i ω = F ( s ) | s = i ω = - e - i ω t f ( t ) d t . \begin{aligned}\displaystyle\hat{f}(\omega)&\displaystyle=\mathcal{F}\{f(t)\}% \\ &\displaystyle=\mathcal{L}\{f(t)\}|_{s=i\omega}=F(s)|_{s=i\omega}\\ &\displaystyle=\int_{-\infty}^{\infty}e^{-i\omega t}f(t)\,dt.\\ \end{aligned}
  81. lim σ 0 + F ( σ + i ω ) = f ^ ( ω ) \lim_{\sigma\to 0^{+}}F(\sigma+i\omega)=\hat{f}(\omega)
  82. G ( s ) = { g ( θ ) } = 0 θ s g ( θ ) d θ θ G(s)=\mathcal{M}\{g(\theta)\}=\int_{0}^{\infty}\theta^{s}g(\theta)\frac{d% \theta}{\theta}
  83. z = def e s T z\stackrel{\mathrm{def}}{{}={}}e^{sT}
  84. Δ T ( t ) = def n = 0 δ ( t - n T ) \Delta_{T}(t)\ \stackrel{\mathrm{def}}{=}\ \sum_{n=0}^{\infty}\delta(t-nT)
  85. x q ( t ) = def x ( t ) Δ T ( t ) = x ( t ) n = 0 δ ( t - n T ) = n = 0 x ( n T ) δ ( t - n T ) = n = 0 x [ n ] δ ( t - n T ) \begin{aligned}\displaystyle x_{q}(t)&\displaystyle\stackrel{\mathrm{def}}{=}% \ x(t)\Delta_{T}(t)=x(t)\sum_{n=0}^{\infty}\delta(t-nT)\\ &\displaystyle=\sum_{n=0}^{\infty}x(nT)\delta(t-nT)=\sum_{n=0}^{\infty}x[n]% \delta(t-nT)\end{aligned}
  86. x [ n ] = def x ( n T ) x[n]\stackrel{\mathrm{def}}{{}={}}x(nT)
  87. x q ( t ) x_{q}(t)
  88. X q ( s ) \displaystyle X_{q}(s)
  89. X ( z ) = n = 0 x [ n ] z - n X(z)=\sum_{n=0}^{\infty}x[n]z^{-n}
  90. X q ( s ) = X ( z ) | z = e s T . X_{q}(s)=X(z)\Big|_{z=e^{sT}}.
  91. F ( s ) = 0 f ( z ) e - s z d z F(s)=\int_{0}^{\infty}f(z)e^{-sz}\,dz
  92. | f ( z ) | A e B | z | |f(z)|\leq Ae^{B|z|}
  93. { f ( t ) + g ( t ) } = { f ( t ) } + { g ( t ) } \mathcal{L}\{f(t)+g(t)\}=\mathcal{L}\{f(t)\}+\mathcal{L}\{g(t)\}
  94. { a f ( t ) } = a { f ( t ) } \mathcal{L}\{af(t)\}=a\mathcal{L}\{f(t)\}
  95. f ( t ) = - 1 { F ( s ) } f(t)=\mathcal{L}^{-1}\{F(s)\}
  96. F ( s ) = { f ( t ) } F(s)=\mathcal{L}\{f(t)\}
  97. δ ( t ) \delta(t)
  98. 1 1
  99. δ ( t - τ ) \delta(t-\tau)
  100. e - τ s e^{-\tau s}
  101. u ( t ) u(t)
  102. 1 s {1\over s}
  103. u ( t - τ ) u(t-\tau)
  104. 1 s e - τ s \frac{1}{s}e^{-\tau s}
  105. t u ( t ) t\cdot u(t)
  106. 1 s 2 \frac{1}{s^{2}}
  107. t n u ( t ) t^{n}\cdot u(t)
  108. n ! s n + 1 {n!\over s^{n+1}}
  109. t q u ( t ) t^{q}\cdot u(t)
  110. Γ ( q + 1 ) s q + 1 {\Gamma(q+1)\over s^{q+1}}
  111. t n u ( t ) \sqrt[n]{t}\cdot u(t)
  112. 1 s 1 n + 1 Γ ( 1 n + 1 ) {1\over s^{\frac{1}{n}+1}}\Gamma\left(\frac{1}{n}+1\right)
  113. t n e - α t u ( t ) t^{n}e^{-\alpha t}\cdot u(t)
  114. n ! ( s + α ) n + 1 \frac{n!}{(s+\alpha)^{n+1}}
  115. ( t - τ ) n e - α ( t - τ ) u ( t - τ ) (t-\tau)^{n}e^{-\alpha(t-\tau)}\cdot u(t-\tau)
  116. n ! e - τ s ( s + α ) n + 1 \frac{n!\cdot e^{-\tau s}}{(s+\alpha)^{n+1}}
  117. e - α t u ( t ) e^{-\alpha t}\cdot u(t)
  118. 1 s + α {1\over s+\alpha}
  119. e - α | t | e^{-\alpha|t|}
  120. 2 α α 2 - s 2 {2\alpha\over\alpha^{2}-s^{2}}
  121. ( 1 - e - α t ) u ( t ) (1-e^{-\alpha t})\cdot u(t)
  122. α s ( s + α ) \frac{\alpha}{s(s+\alpha)}
  123. sin ( ω t ) u ( t ) \sin(\omega t)\cdot u(t)
  124. ω s 2 + ω 2 {\omega\over s^{2}+\omega^{2}}
  125. cos ( ω t ) u ( t ) \cos(\omega t)\cdot u(t)
  126. s s 2 + ω 2 {s\over s^{2}+\omega^{2}}
  127. sinh ( α t ) u ( t ) \sinh(\alpha t)\cdot u(t)
  128. α s 2 - α 2 {\alpha\over s^{2}-\alpha^{2}}
  129. cosh ( α t ) u ( t ) \cosh(\alpha t)\cdot u(t)
  130. s s 2 - α 2 {s\over s^{2}-\alpha^{2}}
  131. e - α t sin ( ω t ) u ( t ) e^{-\alpha t}\sin(\omega t)\cdot u(t)
  132. ω ( s + α ) 2 + ω 2 {\omega\over(s+\alpha)^{2}+\omega^{2}}
  133. e - α t cos ( ω t ) u ( t ) e^{-\alpha t}\cos(\omega t)\cdot u(t)
  134. s + α ( s + α ) 2 + ω 2 {s+\alpha\over(s+\alpha)^{2}+\omega^{2}}
  135. ln ( t ) u ( t ) \ln(t)\cdot u(t)
  136. - 1 s [ ln ( s ) + γ ] -{1\over s}\,\left[\ln(s)+\gamma\right]
  137. J n ( ω t ) u ( t ) J_{n}(\omega t)\cdot u(t)
  138. ( s 2 + ω 2 - s ) n ω n s 2 + ω 2 \frac{\left(\sqrt{s^{2}+\omega^{2}}-s\right)^{n}}{\omega^{n}\sqrt{s^{2}+\omega% ^{2}}}
  139. erf ( t ) u ( t ) \mathrm{erf}(t)\cdot u(t)
  140. 1 s e 1 4 s 2 ( 1 - erf s 2 ) \frac{1}{s}e^{\frac{1}{4}s^{2}}\left(1-\operatorname{erf}\frac{s}{2}\right)
  141. δ ( t ) \delta(t)\,
  142. d N d t = - λ N , \frac{dN}{dt}=-\lambda N,
  143. d N d t + λ N = 0. \frac{dN}{dt}+\lambda N=0.
  144. ( s N ~ ( s ) - N o ) + λ N ~ ( s ) = 0 , \left(s\tilde{N}(s)-N_{o}\right)+\lambda\tilde{N}(s)=0,
  145. N ~ ( s ) = { N ( t ) } \tilde{N}(s)=\mathcal{L}\{N(t)\}
  146. N o = N ( 0 ) . N_{o}=N(0).
  147. N ~ ( s ) = N o s + λ . \tilde{N}(s)=\frac{N_{o}}{s+\lambda}.
  148. N ( t ) \displaystyle N(t)
  149. i = C d v d t i=C{dv\over dt}
  150. I ( s ) = C ( s V ( s ) - V o ) , I(s)=C(sV(s)-V_{o}),
  151. I ( s ) \displaystyle I(s)
  152. V o = v ( t ) | t = 0 . V_{o}=v(t)|_{t=0}.\,
  153. V ( s ) = I ( s ) s C + V o s . V(s)={I(s)\over sC}+{V_{o}\over s}.
  154. Z ( s ) = V ( s ) I ( s ) | V o = 0 . Z(s)={V(s)\over I(s)}\bigg|_{V_{o}=0}.
  155. Z ( s ) = 1 s C , Z(s)=\frac{1}{sC},
  156. H ( s ) = 1 ( s + α ) ( s + β ) . H(s)=\frac{1}{(s+\alpha)(s+\beta)}.
  157. h ( t ) = - 1 { H ( s ) } . h(t)=\mathcal{L}^{-1}\{H(s)\}.
  158. 1 ( s + α ) ( s + β ) = P s + α + R s + β . \frac{1}{(s+\alpha)(s+\beta)}={P\over s+\alpha}+{R\over s+\beta}.
  159. 1 s + β = P + R ( s + α ) s + β . \frac{1}{s+\beta}=P+{R(s+\alpha)\over s+\beta}.
  160. P = 1 s + β | s = - α = 1 β - α . P=\left.{1\over s+\beta}\right|_{s=-\alpha}={1\over\beta-\alpha}.
  161. R = 1 s + α | s = - β = 1 α - β . R=\left.{1\over s+\alpha}\right|_{s=-\beta}={1\over\alpha-\beta}.
  162. R = - 1 β - α = - P R={-1\over\beta-\alpha}=-P
  163. H ( s ) = ( 1 β - α ) ( 1 s + α - 1 s + β ) . H(s)=\left(\frac{1}{\beta-\alpha}\right)\cdot\left({1\over s+\alpha}-{1\over s% +\beta}\right).
  164. h ( t ) = - 1 { H ( s ) } = 1 β - α ( e - α t - e - β t ) , h(t)=\mathcal{L}^{-1}\{H(s)\}=\frac{1}{\beta-\alpha}\left(e^{-\alpha t}-e^{-% \beta t}\right),
  165. H ( s ) = 1 ( s + a ) ( s + b ) = 1 s + a 1 s + b H(s)=\frac{1}{(s+a)(s+b)}=\frac{1}{s+a}\cdot\frac{1}{s+b}
  166. - 1 { 1 s + a } * - 1 { 1 s + b } = e - a t * e - b t = 0 t e - a x e - b ( t - x ) d x = e - a t - e - b t b - a . \mathcal{L}^{-1}\!\left\{\frac{1}{s+a}\right\}*\mathcal{L}^{-1}\!\left\{\frac{% 1}{s+b}\right\}=e^{-at}*e^{-bt}=\int_{0}^{t}e^{-ax}e^{-b(t-x)}\,dx=\frac{e^{-% at}-e^{-bt}}{b-a}.
  167. e - α t [ cos ( ω t ) + ( β - α ω ) sin ( ω t ) ] u ( t ) e^{-\alpha t}[\cos{(\omega t)}+(\frac{\beta-\alpha}{\omega})\sin{(\omega t)}]u% (t)
  168. s + β ( s + α ) 2 + ω 2 \frac{s+\beta}{(s+\alpha)^{2}+\omega^{2}}
  169. X ( s ) = s + β ( s + α ) 2 + ω 2 , X(s)=\frac{s+\beta}{(s+\alpha)^{2}+\omega^{2}},
  170. X ( s ) = s + α ( s + α ) 2 + ω 2 + β - α ( s + α ) 2 + ω 2 . X(s)=\frac{s+\alpha}{(s+\alpha)^{2}+\omega^{2}}+\frac{\beta-\alpha}{(s+\alpha)% ^{2}+\omega^{2}}.
  171. x ( t ) \displaystyle x(t)
  172. x ( t ) \displaystyle x(t)
  173. sin ( ω t + ϕ ) \sin{(\omega t+\phi)}
  174. s sin ϕ + ω cos ϕ s 2 + ω 2 \frac{s\sin\phi+\omega\cos\phi}{s^{2}+\omega^{2}}
  175. cos ( ω t + ϕ ) \cos{(\omega t+\phi)}
  176. s cos ϕ - ω sin ϕ s 2 + ω 2 . \frac{s\cos\phi-\omega\sin\phi}{s^{2}+\omega^{2}}.
  177. X ( s ) = s sin ϕ + ω cos ϕ s 2 + ω 2 X(s)=\frac{s\sin\phi+\omega\cos\phi}{s^{2}+\omega^{2}}
  178. X ( s ) \displaystyle X(s)
  179. x ( t ) \displaystyle x(t)
  180. x ( t ) = sin ( ω t + ϕ ) . x(t)=\sin(\omega t+\phi).
  181. - 1 { s cos ϕ - ω sin ϕ s 2 + ω 2 } = cos ( ω t + ϕ ) . \mathcal{L}^{-1}\left\{\frac{s\cos\phi-\omega\sin\phi}{s^{2}+\omega^{2}}\right% \}=\cos{(\omega t+\phi)}.