wpmath0000007_14

Uncompetitive_inhibitor.html

  1. 1 v = K m V m a x [ S ] + 1 V max \ \frac{1}{v}=\frac{K_{m}}{V_{max}[S]}+{1\over V_{\max}}
  2. [ I ] K i \ \frac{[I]}{K_{i}}
  3. 1 v = K m V m a x [ S ] + 1 + [ I ] K i V m a x \ \frac{1}{v}=\frac{K_{m}}{V_{max}[S]}+\frac{1+\frac{[I]}{K_{i}}}{V_{max}}

Underactuation.html

  1. q ¨ = f ( q , q ˙ , u , t ) \ddot{q}=f(q,\dot{q},u,t)
  2. q n q\in\mathbb{R}^{n}
  3. u m u\in\mathbb{R}^{m}
  4. t t
  5. q ¨ = f 1 ( q , q ˙ , t ) + f 2 ( q , q ˙ , t ) u \ddot{q}=f_{1}(q,\dot{q},t)+f_{2}(q,\dot{q},t)u
  6. r a n k [ f 2 ( q , q ˙ , t ) ] < d i m [ q ] rank[{f_{2}(q,\dot{q},t)}]<dim[q]
  7. f 2 ( q , q ˙ , t ) f_{2}(q,\dot{q},t)
  8. f 2 ( q , q ˙ , t ) f_{2}(q,\dot{q},t)
  9. q , q ˙ q,\dot{q}

Uniform_algebra.html

  1. \in
  2. \in
  3. \neq
  4. M x M_{x}
  5. || a 2 || = || a || 2 ||a^{2}||=||a||^{2}

Unit-weighted_regression.html

  1. y ^ = f ^ ( 𝐱 ) = b ^ + i x i \hat{y}=\hat{f}(\mathbf{x})=\hat{b}+\sum_{i}x_{i}
  2. x i x_{i}
  3. y ^ = f ^ ( 𝐱 ) = b ^ + i w ^ i x i \hat{y}=\hat{f}(\mathbf{x})=\hat{b}+\sum_{i}\hat{w}_{i}x_{i}
  4. y ^ < 0 \hat{y}<0
  5. y ^ 0 \hat{y}\geq 0

Unit_dummy_force_method.html

  1. 𝐪 M × 1 \mathbf{q}_{M\times 1}
  2. 𝐫 N × 1 \mathbf{r}_{N\times 1}
  3. 𝐑 N × 1 * \mathbf{R}^{*}_{N\times 1}
  4. 𝐐 M × 1 * \mathbf{Q}^{*}_{M\times 1}
  5. 𝐑 N × 1 * \mathbf{R}^{*}_{N\times 1}
  6. 𝐐 M × 1 * = 𝐁 M × N 𝐑 N × 1 * ( 1 ) \mathbf{Q}^{*}_{M\times 1}=\mathbf{B}_{M\times N}\mathbf{R}^{*}_{N\times 1}% \qquad\qquad\qquad\mathrm{(1)}
  7. 𝐐 M × 1 * \mathbf{Q}^{*}_{M\times 1}
  8. 𝐑 * T 𝐫 \mathbf{R}^{*T}\mathbf{r}
  9. 𝐐 * T 𝐪 \mathbf{Q}^{*T}\mathbf{q}
  10. 𝐑 * T 𝐫 = 𝐐 * T 𝐪 \mathbf{R}^{*T}\mathbf{r}=\mathbf{Q}^{*T}\mathbf{q}
  11. 𝐑 * T 𝐫 = 𝐑 * T 𝐁 T 𝐪 \mathbf{R}^{*T}\mathbf{r}=\mathbf{R}^{*T}\mathbf{B}^{T}\mathbf{q}
  12. 𝐑 * \mathbf{R}^{*}
  13. 𝐫 = 𝐁 T 𝐪 ( 2 ) \mathbf{r}=\mathbf{B}^{T}\mathbf{q}\qquad\qquad\qquad\mathrm{(2)}
  14. B i , j B_{i,j}
  15. R j * = 1 R^{*}_{j}=1
  16. s y m b o l ϵ symbol{\epsilon}
  17. R * × r = 1 × r R^{*}\times r=1\times r
  18. V s y m b o l ϵ T s y m b o l σ * d V \int_{V}symbol{\epsilon}^{T}symbol{\sigma}^{*}dV
  19. s y m b o l σ * symbol{\sigma}^{*}
  20. 1 × r = V s y m b o l ϵ T s y m b o l σ * d V 1\times r=\int_{V}symbol{\epsilon}^{T}symbol{\sigma}^{*}dV

Units_of_energy.html

  1. 1 J = 1 kg ( m s ) 2 = 1 kg m 2 s 2 1\ \mathrm{J}=1\ \mathrm{kg}\left(\frac{\mathrm{m}}{\mathrm{s}}\right)^{2}=1\ % \frac{\mathrm{kg}\cdot\mathrm{m}^{2}}{\mathrm{s}^{2}}
  2. E = h ν = h c / λ E=h\nu=hc/\lambda
  3. h c 2 10 - 23 J c m \ hc\sim 2\cdot 10^{-23}\ J\ cm

Universal_extra_dimension.html

  1. M K K R - 1 . M_{KK}\approx R^{-1}.

Universal_hashing.html

  1. U U
  2. m m
  3. [ m ] = { 0 , , m - 1 } [m]=\{0,\dots,m-1\}
  4. S U S\subseteq U
  5. | S | = n |S|=n
  6. S S
  7. U U
  8. m n m\cdot n
  9. S S
  10. H = { h : U [ m ] } H=\{h:U\to[m]\}
  11. x , y U , x y : Pr h H [ h ( x ) = h ( y ) ] 1 m \forall x,y\in U,~{}x\neq y:~{}~{}\Pr_{h\in H}[h(x)=h(y)]\leq\frac{1}{m}
  12. 1 / m 1/m
  13. h h
  14. H H
  15. O ( 1 / m ) O(1/m)
  16. ϵ < 1 \epsilon<1
  17. ϵ \epsilon
  18. x , y U , x y \forall x,y\in U,~{}x\neq y
  19. h h
  20. H H
  21. h ( x ) - h ( y ) mod m h(x)-h(y)~{}\bmod~{}m
  22. [ m ] [m]
  23. h ( x ) - h ( y ) = 0 h(x)-h(y)=0
  24. x , y U , x y \forall x,y\in U,~{}x\neq y
  25. h ( x ) h ( y ) mod m h(x)\oplus h(y)~{}\bmod~{}m
  26. [ m ] [m]
  27. \oplus
  28. m m
  29. x , y U , x y \forall x,y\in U,~{}x\neq y
  30. x , y x,y
  31. z 1 , z 2 z_{1},z_{2}
  32. P ( h ( x ) = z 1 h ( y ) = z 2 ) = 1 / m 2 P(h(x)=z_{1}\land h(y)=z_{2})=1/m^{2}
  33. P ( h ( x ) = z ) = 1 / m P(h(x)=z)=1/m
  34. z z
  35. [ m ] [m]
  36. m m
  37. H H
  38. m = 2 L m=2^{L}
  39. h mod 2 L h\bmod{2^{L^{\prime}}}
  40. h H h\in H
  41. L L L^{\prime}\leq L
  42. h ( x ) = x h(x)=x
  43. h ( x ) = x mod 2 L h(x)=x\bmod{2^{L^{\prime}}}
  44. S S
  45. n n
  46. x x
  47. S S
  48. h ( x ) h(x)
  49. n / m n/m
  50. x x
  51. x , y x,y
  52. S S
  53. x y x\neq y
  54. h ( x ) = h ( y ) h(x)=h(y)
  55. n ( n - 1 ) / 2 m n(n-1)/2m
  56. O ( n 2 / m ) O(n^{2}/m)
  57. m m
  58. O ( n ) O(n)
  59. O ( n ) O(n)
  60. n 2 n^{2}
  61. t t
  62. 2 n / ( t - 2 ( n / m ) + 1 ) 2n/(t-2(n/m)+1)
  63. t = 3 n / m t=3n/m
  64. O ( m ) O(m)
  65. 1 / m 1/m
  66. O ( 1 / m ) O(1/m)
  67. S S
  68. O ( n ) O(n)
  69. h h
  70. U = { 0 , , u - 1 } U=\{0,\dots,u-1\}
  71. p u p\geq u
  72. h a , b ( x ) = ( ( a x + b ) mod p ) mod m h_{a,b}(x)=((ax+b)~{}\bmod~{}p)~{}\bmod~{}m
  73. a , b a,b
  74. p p
  75. a 0 a\neq 0
  76. H = { h a , b } H=\{h_{a,b}\}
  77. h ( x ) = h ( y ) h(x)=h(y)
  78. a x + b a y + b + i m ( mod p ) ax+b\equiv ay+b+i\cdot m\;\;(\mathop{{\rm mod}}p)
  79. i i
  80. 0
  81. p / m p/m
  82. x y x\neq y
  83. x - y x-y
  84. p p
  85. a a
  86. a i m ( x - y ) - 1 ( mod p ) a\equiv i\cdot m\cdot(x-y)^{-1}\;\;(\mathop{{\rm mod}}p)
  87. p - 1 p-1
  88. a a
  89. a = 0 a=0
  90. i i
  91. p / m \lfloor p/m\rfloor
  92. p / m / ( p - 1 ) \lfloor p/m\rfloor/(p-1)
  93. 1 / m 1/m
  94. p p
  95. H H
  96. h ( x ) - h ( y ) h(x)-h(y)
  97. h ( x ) - h ( y ) ( a ( x - y ) mod p ) ( mod m ) h(x)-h(y)\equiv(a(x-y)~{}\bmod~{}p)\;\;(\mathop{{\rm mod}}m)
  98. x - y x-y
  99. a a
  100. { 1 , , p } \{1,\dots,p\}
  101. a ( x - y ) a(x-y)
  102. p p
  103. { 1 , , p } \{1,\dots,p\}
  104. ( h ( x ) - h ( y ) ) mod m (h(x)-h(y))~{}\bmod~{}m
  105. ± 1 / p \pm 1/p
  106. O ( m / p ) O(m/p)
  107. p m p\gg m
  108. h a ( x ) = ( a x mod p ) mod m h_{a}(x)=(ax~{}\bmod~{}p)~{}\bmod~{}m
  109. Pr { h a ( x ) = h a ( y ) } 2 / m \Pr\{h_{a}(x)=h_{a}(y)\}\leq 2/m
  110. x y x\neq y
  111. Pr { h a ( 1 ) = h a ( m + 1 ) } 2 / ( m - 1 ) \Pr\{h_{a}(1)=h_{a}(m+1)\}\geq 2/(m-1)
  112. ( p - 1 ) mod m = 1 (p-1)~{}\bmod~{}m=1
  113. m = 2 M m=2^{M}
  114. w w
  115. a < 2 w a<2^{w}
  116. w w
  117. h a ( x ) h_{a}(x)
  118. x x
  119. a a
  120. 2 w 2^{w}
  121. M M
  122. h a ( x ) = ( a x mod 2 w ) div 2 w - M h_{a}(x)=(a\cdot x\,\,\bmod\,2^{w})\,\,\mathrm{div}\,\,2^{w-M}
  123. h a ( x ) = h_{a}(x)=
  124. 2 / m 2/m
  125. x y x\neq y
  126. Pr { h a ( x ) = h a ( y ) } 2 / m \Pr\{h_{a}(x)=h_{a}(y)\}\leq 2/m
  127. a x mod 2 w ax\bmod 2^{w}
  128. a y mod 2 w ay\bmod 2^{w}
  129. a ( x - y ) mod 2 w a(x-y)\bmod 2^{w}
  130. a x mod 2 w ax\bmod 2^{w}
  131. a y mod 2 w ay\bmod 2^{w}
  132. x - y x-y
  133. w - c w-c
  134. a a
  135. Z 2 w Z_{2^{w}}
  136. a ( x - y ) mod 2 w a(x-y)\bmod 2^{w}
  137. w w
  138. w - c w-c
  139. 2 / 2 M = 2 / m 2/2^{M}=2/m
  140. c < M c<M
  141. a ( x - y ) mod 2 w a(x-y)\bmod 2^{w}
  142. h ( x ) h ( y ) h(x)\neq h(y)
  143. c = M c=M
  144. w - M w-M
  145. a ( x - y ) mod 2 w a(x-y)\bmod 2^{w}
  146. h a ( x ) = h a ( y ) h_{a}(x)=h_{a}(y)
  147. w - 1 , , w - M + 1 w-1,\ldots,w-M+1
  148. 1 / 2 M - 1 = 2 / m 1/2^{M-1}=2/m
  149. x = 2 w - M - 2 x=2^{w-M-2}
  150. y = 3 x y=3x
  151. h a , b ( x ) = ( ( a x + b ) mod 2 w ) div 2 w - M h_{a,b}(x)=((ax+b)\bmod 2^{w})\,\mathrm{div}\,2^{w-M}
  152. h a , b ( x ) = h_{a,b}(x)=
  153. a a
  154. a < 2 w a<2^{w}
  155. b b
  156. b < 2 w - M b<2^{w-M}
  157. a a
  158. b b
  159. Pr { h a , b ( x ) = h a , b ( y ) } 1 / m \Pr\{h_{a,b}(x)=h_{a,b}(y)\}\leq 1/m
  160. x y ( mod 2 w ) x\not\equiv y\;\;(\mathop{{\rm mod}}2^{w})
  161. x ¯ = ( x 0 , , x k - 1 ) \bar{x}=(x_{0},\dots,x_{k-1})
  162. k k
  163. w w
  164. H H
  165. h ( x ¯ ) = ( i = 0 k - 1 h i ( x i ) ) mod m h(\bar{x})=\left(\sum_{i=0}^{k-1}h_{i}(x_{i})\right)\,\bmod~{}m
  166. h i H h_{i}\in H
  167. m m
  168. a ¯ = ( a 0 , , a k - 1 ) \bar{a}=(a_{0},\dots,a_{k-1})
  169. 2 w 2w
  170. m = 2 M m=2^{M}
  171. M w M\leq w
  172. h a ¯ ( x ¯ ) = ( ( i = 0 k - 1 x i a i ) mod 2 2 w ) div 2 2 w - M h_{\bar{a}}(\bar{x})=\left(\big(\sum_{i=0}^{k-1}x_{i}\cdot a_{i}\big)~{}\bmod~% {}2^{2w}\right)\,\,\mathrm{div}\,\,2^{2w-M}
  173. a ¯ = ( a 0 , , a k - 1 ) \bar{a}=(a_{0},\dots,a_{k-1})
  174. 2 w 2w
  175. h a ¯ ( x ¯ ) = ( ( i = 0 k / 2 ( x 2 i + a 2 i ) ( x 2 i + 1 + a 2 i + 1 ) ) mod 2 2 w ) div 2 2 w - M h_{\bar{a}}(\bar{x})=\left(\Big(\sum_{i=0}^{\lceil k/2\rceil}(x_{2i}+a_{2i})% \cdot(x_{2i+1}+a_{2i+1})\Big)\bmod~{}2^{2w}\right)\,\,\mathrm{div}\,\,2^{2w-M}
  176. w / 2 w/2
  177. k / 2 \lceil k/2\rceil
  178. k k
  179. a ¯ = ( a 0 , , a k ) \bar{a}=(a_{0},\dots,a_{k})
  180. 2 w 2w
  181. h a ¯ ( x ¯ ) strong = ( a 0 + i = 0 k - 1 a i + 1 x i mod 2 2 w ) div 2 w h_{\bar{a}}(\bar{x})^{\mathrm{strong}}=(a_{0}+\sum_{i=0}^{k-1}a_{i+1}x_{i}% \bmod~{}2^{2w})\,\,\mathrm{div}\,\,2^{w}
  182. w w
  183. w = 32 w=32
  184. h ( s ) h(s)
  185. s s
  186. x ¯ = ( x 0 , , x ) \bar{x}=(x_{0},\dots,x_{\ell})
  187. \ell
  188. x x
  189. x i [ u ] x_{i}\in[u]
  190. p max { u , m } p\geq\max\{u,m\}
  191. h a ( x ¯ ) = h int ( ( i = 0 x i a i ) mod p ) h_{a}(\bar{x})=h_{\mathrm{int}}\left(\big(\sum_{i=0}^{\ell}x_{i}\cdot a^{i}% \big)\bmod~{}p\right)
  192. a [ p ] a\in[p]
  193. h int h_{\mathrm{int}}
  194. [ p ] [ m ] [p]\mapsto[m]
  195. x ¯ , y ¯ \bar{x},\bar{y}
  196. \ell
  197. \ell
  198. h int h_{\mathrm{int}}
  199. a a
  200. x ¯ - y ¯ \bar{x}-\bar{y}
  201. \ell
  202. p p
  203. / p \ell/p
  204. h int h_{\mathrm{int}}
  205. 1 m + p \frac{1}{m}+\frac{\ell}{p}
  206. p p
  207. p p
  208. p p
  209. p = 2 61 - 1 p=2^{61}-1
  210. x i x_{i}
  211. k / 16 \lceil k/16\rceil
  212. 2 w 2^{w}

Universal_one-way_hash_function.html

  1. A A
  2. A A
  3. x x
  4. H H
  5. A A
  6. H H
  7. y x y\neq x
  8. H ( x ) = H ( y ) H(x)=H(y)
  9. A A
  10. A A

Unrestricted_grammar.html

  1. G = ( N , Σ , P , S ) G=(N,\Sigma,P,S)
  2. N N
  3. Σ \Sigma
  4. N N
  5. Σ \Sigma
  6. P P
  7. α β \alpha\to\beta
  8. α \alpha
  9. β \beta
  10. N Σ N\cup\Sigma
  11. α \alpha
  12. S N S\in N
  13. G G
  14. L ( G ) L(G)
  15. w w
  16. G G
  17. β γ \beta\to\gamma
  18. G G
  19. β \beta
  20. β \beta
  21. γ \gamma
  22. β \beta
  23. γ \gamma
  24. β \beta
  25. γ \gamma
  26. G G
  27. L ( G ) L(G)
  28. s s

Unrestricted_Hartree–Fock.html

  1. 𝐅 α 𝐂 α = 𝐒𝐂 α ϵ α \mathbf{F}^{\alpha}\ \mathbf{C}^{\alpha}\ =\mathbf{S}\mathbf{C}^{\alpha}\ % \mathbf{\epsilon}^{\alpha}
  2. 𝐅 β 𝐂 β = 𝐒𝐂 β ϵ β \mathbf{F}^{\beta}\ \mathbf{C}^{\beta}\ =\mathbf{S}\mathbf{C}^{\beta}\ \mathbf% {\epsilon}^{\beta}
  3. 𝐅 α \mathbf{F}^{\alpha}
  4. 𝐅 β \mathbf{F}^{\beta}
  5. α \alpha
  6. β \beta
  7. 𝐂 α \mathbf{C}^{\alpha}
  8. 𝐂 β \mathbf{C}^{\beta}
  9. α \alpha
  10. β \beta
  11. 𝐒 \mathbf{S}
  12. ϵ α \mathbf{\epsilon}^{\alpha}
  13. ϵ β \mathbf{\epsilon}^{\beta}
  14. α \alpha
  15. β \beta
  16. 𝐒 2 \mathbf{S}^{2}
  17. 𝐒 2 \mathbf{S}^{2}
  18. 𝐒 2 \langle\mathbf{S}^{2}\rangle
  19. 1 2 ( 1 2 + 1 ) = 0.75 \tfrac{1}{2}(\tfrac{1}{2}+1)=0.75
  20. 𝐒 2 \langle\mathbf{S}^{2}\rangle
  21. 𝐒 2 \langle\mathbf{S}^{2}\rangle

Upper-convected_Maxwell_model.html

  1. 𝐓 + λ 𝐓 = 2 η 0 𝐃 \mathbf{T}+\lambda\stackrel{\nabla}{\mathbf{T}}=2\eta_{0}\mathbf{D}
  2. 𝐓 \mathbf{T}
  3. λ \lambda
  4. 𝐓 \stackrel{\nabla}{\mathbf{T}}
  5. 𝐓 = t 𝐓 + 𝐯 𝐓 - ( ( 𝐯 ) T 𝐓 + 𝐓 ( 𝐯 ) ) \stackrel{\nabla}{\mathbf{T}}=\frac{\partial}{\partial t}\mathbf{T}+\mathbf{v}% \cdot\nabla\mathbf{T}-((\nabla\mathbf{v})^{T}\cdot\mathbf{T}+\mathbf{T}\cdot(% \nabla\mathbf{v}))
  6. 𝐯 \mathbf{v}
  7. η 0 \eta_{0}
  8. 𝐃 \mathbf{D}
  9. T 12 = η 0 γ ˙ T_{12}=\eta_{0}\dot{\gamma}\,
  10. T 11 = 2 η 0 λ γ ˙ 2 T_{11}=2\eta_{0}\lambda{\dot{\gamma}}^{2}\,
  11. γ ˙ \dot{\gamma}
  12. T 11 - T 22 T_{11}-T_{22}
  13. T 22 - T 33 T_{22}-T_{33}
  14. T 12 = η 0 γ ˙ ( 1 - exp ( - t λ ) ) T_{12}=\eta_{0}\dot{\gamma}\left(1-\exp\left(-\frac{t}{\lambda}\right)\right)
  15. T 11 = 2 η 0 λ γ ˙ 2 ( 1 - exp ( - t λ ) ( 1 + t λ ) ) T_{11}=2\eta_{0}\lambda{\dot{\gamma}}^{2}\left(1-\exp\left(-\frac{t}{\lambda}% \right)\left(1+\frac{t}{\lambda}\right)\right)
  16. σ = T 11 - T 22 = T 11 - T 33 \sigma=T_{11}-T_{22}=T_{11}-T_{33}
  17. σ = 2 η 0 ϵ ˙ 1 - 2 λ ϵ ˙ + η 0 ϵ ˙ 1 + λ ϵ ˙ \sigma=\frac{2\eta_{0}\dot{\epsilon}}{1-2\lambda\dot{\epsilon}}+\frac{\eta_{0}% \dot{\epsilon}}{1+\lambda\dot{\epsilon}}
  18. ϵ ˙ \dot{\epsilon}
  19. 3 η 0 3\eta_{0}
  20. ϵ ˙ 1 λ \dot{\epsilon}\ll\frac{1}{\lambda}
  21. ϵ ˙ = 1 2 λ \dot{\epsilon}_{\infty}=\frac{1}{2\lambda}
  22. ϵ ˙ - = - 1 λ \dot{\epsilon}_{-\infty}=-\frac{1}{\lambda}

Upper-convected_time_derivative.html

  1. 𝐀 = D D t 𝐀 - ( 𝐯 ) T 𝐀 - 𝐀 ( 𝐯 ) \stackrel{\triangledown}{\mathbf{A}}=\frac{D}{Dt}\mathbf{A}-(\nabla\mathbf{v})% ^{T}\cdot\mathbf{A}-\mathbf{A}\cdot(\nabla\mathbf{v})
  2. 𝐀 {\stackrel{\triangledown}{\mathbf{A}}}
  3. 𝐀 \mathbf{A}
  4. D D t \frac{D}{Dt}
  5. 𝐯 = v j x i \nabla\mathbf{v}=\frac{\partial v_{j}}{\partial x_{i}}
  6. A i , j = A i , j t + v k A i , j x k - v i x k A k , j - v j x k A i , k {\stackrel{\triangledown}{A}}_{i,j}=\frac{\partial A_{i,j}}{\partial t}+v_{k}% \frac{\partial A_{i,j}}{\partial x_{k}}-\frac{\partial v_{i}}{\partial x_{k}}A% _{k,j}-\frac{\partial v_{j}}{\partial x_{k}}A_{i,k}
  7. 𝐯 = ( 0 0 0 γ ˙ 0 0 0 0 0 ) \nabla\mathbf{v}=\begin{pmatrix}0&0&0\\ {\dot{\gamma}}&0&0\\ 0&0&0\end{pmatrix}
  8. 𝐀 = D D t 𝐀 - γ ˙ ( 2 A 12 A 22 A 23 A 22 0 0 A 23 0 0 ) \stackrel{\triangledown}{\mathbf{A}}=\frac{D}{Dt}\mathbf{A}-\dot{\gamma}\begin% {pmatrix}2A_{12}&A_{22}&A_{23}\\ A_{22}&0&0\\ A_{23}&0&0\end{pmatrix}
  9. 𝐯 = ( ϵ ˙ 0 0 0 - ϵ ˙ 2 0 0 0 - ϵ ˙ 2 ) \nabla\mathbf{v}=\begin{pmatrix}\dot{\epsilon}&0&0\\ 0&-\frac{\dot{\epsilon}}{2}&0\\ 0&0&-\frac{\dot{\epsilon}}{2}\end{pmatrix}
  10. 𝐀 = D D t 𝐀 - ϵ ˙ 2 ( 4 A 11 A 12 A 13 A 12 - 2 A 22 - 2 A 23 A 13 - 2 A 23 - 2 A 33 ) \stackrel{\triangledown}{\mathbf{A}}=\frac{D}{Dt}\mathbf{A}-\frac{\dot{% \epsilon}}{2}\begin{pmatrix}4A_{11}&A_{12}&A_{13}\\ A_{12}&-2A_{22}&-2A_{23}\\ A_{13}&-2A_{23}&-2A_{33}\end{pmatrix}

Uranium_trioxide.html

  1. s = e - R - R O B s=e^{-\frac{R-R_{O}}{B}}

Uranyl.html

  1. d z 2 d_{z^{2}}
  2. f z 3 f_{z^{3}}
  3. d x z d_{xz}
  4. d y z d_{yz}
  5. f x z 2 f_{xz^{2}}
  6. f y z 2 f_{yz^{2}}

Useful_conversions_and_formulas_for_air_dispersion_modeling.html

  1. ppmv = mg / m 3 ( 0.08205 T ) M \mathrm{ppmv}=\mathrm{mg}/\mathrm{m}^{3}\cdot\frac{(0.08205\cdot T)}{M}
  2. mg / m 3 = ppmv M ( 0.08205 T ) \mathrm{mg}/\mathrm{m}^{3}=\mathrm{ppmv}\cdot\frac{M}{(0.08205\cdot T)}
  3. T T
  4. M M
  5. P a = 0.9877 a P_{a}=0.9877^{a}
  6. C a = C 0.9877 a C_{a}=C\cdot 0.9877^{a}
  7. a a
  8. P P
  9. a a
  10. C C
  11. C a C_{a}
  12. a a
  13. d r y b a s i s c o n c e n t r a t i o n = ( w e t b a s i s c o n c e n t r a t i o n ) / ( 1 - w ) dry\;basis\;concentration=(wet\;basis\;concentration)/(1-w)
  14. w w
  15. C r = C m ( 20.9 - r ) ( 20.9 - m ) C_{r}=C_{m}\cdot\frac{(20.9-r)}{(20.9-m)}
  16. C r C_{r}
  17. r r
  18. C m C_{m}
  19. m m
  20. C r = C m r m C_{r}=C_{m}\cdot\frac{r}{m}
  21. C r C_{r}
  22. r r
  23. C m C_{m}
  24. m m

UV_mapping.html

  1. P P
  2. d ^ \hat{d}
  3. P P
  4. [ 0 , 1 ] [0,1]
  5. u = 0.5 + arctan 2 ( d z , d x ) 2 π u=0.5+\frac{\arctan 2(d_{z},d_{x})}{2\pi}
  6. v = 0.5 - arcsin ( d y ) π v=0.5-\frac{\arcsin(d_{y})}{\pi}

Vacuum_permeability.html

  1. | s y m b o l F m | L = μ 0 2 π | s y m b o l I | 2 | s y m b o l r | . \frac{|symbol{F}_{m}|}{L}={\mu_{0}\over 2\pi}{|symbol{I}|^{2}\over|symbol{r}|}.
  2. μ 0 = 4 π × 10 - 7 ( N / A 2 ) 1.2566370614 × 10 - 6 ( N / A 2 ) \mu_{0}=4\pi\times 10^{-7}(\rm{N/A^{2}})\approx 1.2566370614\cdots\times 10^{-% 6}(\rm{N/A^{2}})
  3. F m I 2 r . F_{\mathrm{m}}\propto\frac{I^{2}}{r}.\;
  4. F m = k m I 2 r . F_{\mathrm{m}}=k_{\mathrm{m}}\frac{I^{2}}{r}.\;
  5. s y m b o l H = s y m b o l B μ 0 - s y m b o l M , symbol{H}={symbol{B}\over\mu_{0}}-symbol{M},
  6. c 0 = 1 μ 0 ε 0 . c_{0}={1\over\sqrt{\mu_{0}\varepsilon_{0}}}.

Valuation_using_discounted_cash_flows.html

  1. t = 1 n F C F F t ( 1 + W A C C g ) t + [ F C F F n + 1 ( W A C C s t - g n ) ] ( 1 + W A C C g ) n \sum_{t=1}^{n}\frac{FCFF_{t}}{(1+WACC_{g})^{t}}+\frac{\left[\frac{FCFF_{n+1}}{% (WACC_{st}-g_{n})}\right]}{(1+WACC_{g})^{n}}

Value_(mathematics).html

  1. f f
  2. f ( x ) = 2 x 2 - 3 x + 1 f(x)=2x^{2}-3x+1
  3. f ( 3 ) = 10. f(3)=10.

Van_'t_Hoff_equation.html

  1. d ln K e q d < m t p l > 1 T = - Δ H R . \frac{d\ln K_{eq}}{d{\frac{<}{m}tpl>{{1}}{{T}}}}=-\frac{\Delta H^{\ominus}}{R}.
  2. ln ( < m t p l > K 2 K 1 ) = - Δ H R ( 1 T 2 - 1 T 1 ) . \ln\left({\frac{<}{m}tpl>{{K_{2}}}{{K_{1}}}}\right)=\frac{{\ -\Delta H^{% \ominus}}}{R}\left({\frac{1}{{T_{2}}}-\frac{1}{{T_{1}}}}\right).
  3. Δ G = Δ H - T Δ S \Delta G^{\ominus}=\Delta H^{\ominus}-T\Delta S^{\ominus}
  4. Δ G = - R T ln K e q \Delta G^{\ominus}=-RT\ln K_{eq}
  5. ln K e q = - < m t p l > Δ H R T + Δ S R . \ln K_{eq}=-\frac{<}{m}tpl>{{\Delta H^{\ominus}}}{RT}+\frac{{\Delta S^{\ominus% }}}{R}.
  6. ( d G d ξ ) T , p = Δ r G + R T ln Q r \left(\frac{dG}{d\xi}\right)_{T,p}=\Delta_{r}G+RT\ln Q_{r}~{}
  7. Δ r G \Delta_{r}G
  8. Q r Q_{r}~{}
  9. Q r = K e q Q_{r}~{}=K_{eq}
  10. Δ r G < 0 \Delta_{r}G<0
  11. Δ r G > 0 \Delta_{r}G>0
  12. ln K e q \ln K_{eq}
  13. 1 / T 1/T
  14. ln K e q = - < m t p l > Δ H R T + Δ S R . \ln K_{eq}=-\frac{<}{m}tpl>{{\Delta H^{\ominus}}}{RT}+\frac{{\Delta S^{\ominus% }}}{R}.
  15. - Δ H R -\frac{\Delta H}{R}
  16. Δ S R \frac{\Delta S}{R}
  17. Δ H = - R * s l o p e , \Delta H=-R*slope,
  18. Δ S = R * i n t e r c e p t . \Delta S=R*intercept.
  19. S l o p e = - Δ H R Slope=-\frac{\Delta H}{R}
  20. Δ H > 0 \Delta H>0
  21. S l o p e = - Δ H R < 0 Slope=-\frac{\Delta H}{R}<0
  22. S l o p e = - Δ H R Slope=-\frac{\Delta H}{R}
  23. Δ H < 0 \Delta H<0
  24. S l o p e = - Δ H R > 0 Slope=-\frac{\Delta H}{R}>0
  25. a A + d D b B \mathrm{a\ A+d\ D\longrightarrow b\ B}
  26. a A + d D c C \mathrm{a\ A+d\ D\longrightarrow c\ C}
  27. K e q K_{eq}
  28. B C > 1 \frac{B}{C}>1
  29. B C < 1 \frac{B}{C}<1
  30. Δ H 1 = - R * s l o p e 1 , \Delta H_{1}=-R*slope_{1},
  31. Δ S 1 = R * i n t e r c e p t i o n 1 ; \Delta S_{1}=R*interception_{1};
  32. Δ H 2 = - R * s l o p e 2 , \Delta H_{2}=-R*slope_{2},
  33. Δ S 2 = R * i n t e r c e p t i o n 2 ; \Delta S_{2}=R*interception_{2};
  34. c T 2 \frac{c}{T^{2}}
  35. l n K e q = a + b T + c T 2 , lnK_{eq}=a+\frac{b}{T}+\frac{c}{T^{2}},
  36. Δ H = - R * ( b + 2 c T ) , \Delta H=-R*(b+2\frac{c}{T}),
  37. Δ S = R * ( a - c T 2 ) . \Delta S=R*(a-\frac{c}{T^{2}}).

Van_der_Corput_sequence.html

  1. n = k = 0 L - 1 d k ( n ) b k , n=\sum_{k=0}^{L-1}d_{k}(n)b^{k},
  2. 1 2 , 1 4 , 3 4 , 1 8 , 5 8 , 3 8 , 7 8 , 1 16 , 9 16 , 5 16 , 13 16 , 3 16 , 11 16 , 7 16 , 15 16 , \tfrac{1}{2},\tfrac{1}{4},\tfrac{3}{4},\tfrac{1}{8},\tfrac{5}{8},\tfrac{3}{8},% \tfrac{7}{8},\tfrac{1}{16},\tfrac{9}{16},\tfrac{5}{16},\tfrac{13}{16},\tfrac{3% }{16},\tfrac{11}{16},\tfrac{7}{16},\tfrac{15}{16},\ldots

Van_der_Waerden_notation.html

  1. Σ left = ( ψ α 0 ) \Sigma_{\mathrm{left}}=\begin{pmatrix}\psi_{\alpha}\\ 0\end{pmatrix}
  2. Σ right = ( 0 χ ¯ α ˙ ) \Sigma_{\mathrm{right}}=\begin{pmatrix}0\\ \bar{\chi}^{\dot{\alpha}}\\ \end{pmatrix}
  3. α = 1 , 2 , α ˙ = 1 ˙ , 2 ˙ \alpha=1,2\,,\dot{\alpha}=\dot{1},\dot{2}
  4. Σ α ^ = ( ψ α χ ¯ α ˙ ) \Sigma_{\hat{\alpha}}=\begin{pmatrix}\psi_{\alpha}\\ \bar{\chi}^{\dot{\alpha}}\\ \end{pmatrix}
  5. α ^ = ( α , α ˙ ) = 1 , 2 , 1 ˙ , 2 ˙ \hat{\alpha}=(\alpha,\dot{\alpha})=1,2,\dot{1},\dot{2}
  6. Σ α ^ = ( χ α ψ ¯ α ˙ ) \Sigma^{\hat{\alpha}}=\begin{pmatrix}\chi^{\alpha}&\bar{\psi}_{\dot{\alpha}}% \end{pmatrix}

Van_Hove_singularity.html

  1. k = 2 π λ = n 2 π L k=\frac{2\pi}{\lambda}=n\frac{2\pi}{L}
  2. λ \lambda
  3. k m a x = π / a k_{max}=\pi/a
  4. n m a x = L / 2 a n_{max}=L/2a
  5. g ( k ) d k = d n = L 2 π d k g(k)dk=dn=\frac{L}{2\pi}\,dk
  6. g ( k ) d 3 k = d 3 n = L 3 ( 2 π ) 3 d 3 k g(\vec{k})d^{3}k=d^{3}n=\frac{L^{3}}{(2\pi)^{3}}\,d^{3}k
  7. d 3 k d^{3}k
  8. d E = E k x d k x + E k y d k y + E k z d k z = E d k dE=\frac{\partial E}{\partial k_{x}}dk_{x}+\frac{\partial E}{\partial k_{y}}dk% _{y}+\frac{\partial E}{\partial k_{z}}dk_{z}=\vec{\nabla}E\cdot d\vec{k}
  9. \vec{\nabla}
  10. g ( E ) d E = E g ( k ) d 3 k = L 3 ( 2 π ) 3 E d k x d k y d k z g(E)dE=\iint_{\partial E}g(\vec{k})\,d^{3}k=\frac{L^{3}}{(2\pi)^{3}}\iint_{% \partial E}dk_{x}\,dk_{y}\,dk_{z}
  11. E \partial E
  12. k x , k y , k z k^{\prime}_{x},k^{\prime}_{y},k^{\prime}_{z}\,
  13. k z k^{\prime}_{z}\,
  14. d k x d k y d k z = d k x d k y d k z dk^{\prime}_{x}\,dk^{\prime}_{y}\,dk^{\prime}_{z}=dk_{x}\,dk_{y}\,dk_{z}
  15. d E = | E | d k z dE=|\vec{\nabla}E|\,dk^{\prime}_{z}
  16. g ( E ) = L 3 ( 2 π ) 3 d k x d k y | E | g(E)=\frac{L^{3}}{(2\pi)^{3}}\iint\frac{dk^{\prime}_{x}\,dk^{\prime}_{y}}{|% \vec{\nabla}E|}
  17. d k x d k y dk^{\prime}_{x}\,dk^{\prime}_{y}
  18. g ( E ) g(E)
  19. k k
  20. E ( k ) E(\vec{k})
  21. k k
  22. E = 2 k 2 / 2 m E=\hbar^{2}k^{2}/2m
  23. | E | = 2 k / m = 2 E m |\vec{\nabla}E|=\hbar^{2}k/m=\hbar\sqrt{\frac{2E}{m}}
  24. E \vec{\nabla}E
  25. A p \vec{A}\cdot\vec{p}
  26. A \vec{A}
  27. p \vec{p}
  28. ϕ ( x + L ) = ϕ ( x ) \phi(x+L)=\phi(x)
  29. k L = 2 n π kL=2n\pi

Vapour_Pressure_Deficit.html

  1. v p s a t = e A / T + B + C T + D T 2 + E T 3 + F ln T vp_{sat}=e^{A/T+B+CT+DT^{2}+ET^{3}+F\ln T}
  2. v p s a t vp_{sat}
  3. A = - 1.88 × 10 4 A=-1.88\times 10^{4}
  4. B = - 13.1 B=-13.1
  5. C = - 1.5 × 10 - 2 C=-1.5\times 10^{-2}
  6. D = 8 × 10 - 7 D=8\times 10^{-7}
  7. E = - 1.69 × 10 - 11 E=-1.69\times 10^{-11}
  8. F = 6.456 F=6.456
  9. T = T=
  10. T ( K ) = T ( C ) + 273.15 T(K)=T(C)+273.15
  11. v p a i r = v p s a t * vp_{air}=vp_{sat}*
  12. v p s a t - v p a i r vp_{sat}-vp_{air}
  13. v p vp
  14. - v p a i r -vp_{air}

Variable_(mathematics).html

  1. y = f ( x ) y=f(x)
  2. y y
  3. x x
  4. y = f ( x ) y=f(x)
  5. f f
  6. x x
  7. y y
  8. x x
  9. a a
  10. f ( x ) f(x)
  11. L L
  12. ( ϵ > 0 ) ( η > 0 ) ( x ) | x - a | < η | L - f ( x ) | < ϵ , (\forall\epsilon>0)(\exists\eta>0)(\forall x)\;|x-a|<\eta\Rightarrow|L-f(x)|<\epsilon,
  13. a x 3 + b x 2 + c x + d = 0 , ax^{3}+bx^{2}+cx+d=0,
  14. a , b , c , d a,b,c,d
  15. x , x,
  16. x x
  17. x x
  18. f : x f ( x ) f:x↦f(x)
  19. f f
  20. x x
  21. x x
  22. x x
  23. π π
  24. y y
  25. x x
  26. y y
  27. x x
  28. y y
  29. x x
  30. y y
  31. f ( x , y , z ) f(x,y,z)
  32. y y
  33. z z
  34. x x
  35. x x
  36. f ( x ) = x 2 + sin ( x + 4 ) f(x)=x^{2}+\sin(x+4)
  37. i = 1 n i = n 2 + n 2 \sum_{i=1}^{n}i=\frac{n^{2}+n}{2}
  38. a x 2 + b x + c , ax^{2}+bx+c\,,
  39. x a x 2 + b x + c , x\mapsto ax^{2}+bx+c\,,

Variance_decomposition_of_forecast_errors.html

  1. y t = ν + A 1 y t - 1 + + A p y t - p + u t y_{t}=\nu+A_{1}y_{t-1}+\dots+A_{p}y_{t-p}+u_{t}
  2. Y t = V + A Y t - 1 + U t Y_{t}=V+AY_{t-1}+U_{t}
  3. A = [ A 1 A 2 A p - 1 A p 𝐈 k 0 0 0 0 𝐈 k 0 0 0 0 𝐈 k 0 ] A=\begin{bmatrix}A_{1}&A_{2}&\dots&A_{p-1}&A_{p}\\ \mathbf{I}_{k}&0&\dots&0&0\\ 0&\mathbf{I}_{k}&&0&0\\ \vdots&&\ddots&\vdots&\vdots\\ 0&0&\dots&\mathbf{I}_{k}&0\\ \end{bmatrix}
  4. Y = [ y 1 y p ] Y=\begin{bmatrix}y_{1}\\ \vdots\\ y_{p}\end{bmatrix}
  5. V = [ ν 0 0 ] V=\begin{bmatrix}\nu\\ 0\\ \vdots\\ 0\end{bmatrix}
  6. U t = [ u t 0 0 ] U_{t}=\begin{bmatrix}u_{t}\\ 0\\ \vdots\\ 0\end{bmatrix}
  7. y t y_{t}
  8. ν \nu
  9. u u
  10. k k
  11. A A
  12. k p kp
  13. k p kp
  14. Y Y
  15. V V
  16. U U
  17. k p kp
  18. 𝐌𝐒𝐄 [ y j , t ( h ) ] = i = 0 h - 1 k = 1 K ( e j Θ i e k ) 2 = ( i = 0 h - 1 Θ i Θ i ) j j = ( i = 0 h - 1 Φ i Σ u Φ i ) j j , \mathbf{MSE}[y_{j,t}(h)]=\sum_{i=0}^{h-1}\sum_{k=1}^{K}(e_{j}^{\prime}\Theta_{% i}e_{k})^{2}=\bigg(\sum_{i=0}^{h-1}\Theta_{i}\Theta_{i}^{\prime}\bigg)_{jj}=% \bigg(\sum_{i=0}^{h-1}\Phi_{i}\Sigma_{u}\Phi_{i}^{\prime}\bigg)_{jj},
  19. e j e_{j}
  20. I K I_{K}
  21. j j jj
  22. Θ i = Φ i P , \Theta_{i}=\Phi_{i}P,
  23. P P
  24. Σ u \Sigma_{u}
  25. Σ u = P P \Sigma_{u}=PP^{\prime}
  26. Σ u \Sigma_{u}
  27. u t u_{t}
  28. Φ i = J A i J , \Phi_{i}=JA^{i}J^{\prime},
  29. J = [ 𝐈 k 0 0 ] , J=\begin{bmatrix}\mathbf{I}_{k}&0&\dots&0\end{bmatrix},
  30. J J
  31. k k
  32. k p kp
  33. j j
  34. k k
  35. ω j k , h , \omega_{jk,h},
  36. ω j k , h = i = 0 h - 1 ( e j Θ i e k ) 2 / M S E [ y j , t ( h ) ] . \omega_{jk,h}=\sum_{i=0}^{h-1}(e_{j}^{\prime}\Theta_{i}e_{k})^{2}/MSE[y_{j,t}(% h)].

Variation_ratio.html

  1. 𝐯 := 1 - f m N , \mathbf{v}:=1-\frac{f_{m}}{N},

Vasicek_model.html

  1. d r t = a ( b - r t ) d t + σ d W t dr_{t}=a(b-r_{t})\,dt+\sigma\,dW_{t}
  2. σ \sigma
  3. b , a b,a
  4. σ \sigma
  5. r 0 r_{0}
  6. a a
  7. b b
  8. r r
  9. a a
  10. a a
  11. b b
  12. σ \sigma
  13. σ \sigma
  14. σ 2 / ( 2 a ) {\sigma^{2}}/(2a)
  15. r r
  16. a a
  17. σ \sigma
  18. σ \sigma
  19. a a
  20. b b
  21. a a
  22. σ 2 2 a \frac{\sigma^{2}}{2a}
  23. σ \sigma
  24. a a
  25. a ( b - r t ) a(b-r_{t})
  26. d W t = 0 dW_{t}=0
  27. a ( b - r t ) a(b-r_{t})
  28. r ( t ) = r ( 0 ) e - a t + b ( 1 - e - a t ) + σ e - a t 0 t e a s d W s . r(t)=r(0)e^{-at}+b\left(1-e^{-at}\right)+\sigma e^{-at}\int_{0}^{t}e^{as}\,dW_% {s}.\,\!
  29. E [ r t ] = r 0 e - a t + b ( 1 - e - a t ) \mathrm{E}[r_{t}]=r_{0}e^{-at}+b(1-e^{-at})
  30. Var [ r t ] = σ 2 2 a ( 1 - e - 2 a t ) . \mathrm{Var}[r_{t}]=\frac{\sigma^{2}}{2a}(1-e^{-2at}).
  31. lim t E [ r t ] = b \lim_{t\to\infty}\mathrm{E}[r_{t}]=b
  32. lim t Var [ r t ] = σ 2 2 a . \lim_{t\to\infty}\mathrm{Var}[r_{t}]=\frac{\sigma^{2}}{2a}.

Vector-valued_differential_form.html

  1. Ω p ( M , E ) = Γ ( E Λ p T * M ) . \Omega^{p}(M,E)=\Gamma(E\otimes\Lambda^{p}T^{*}M).
  2. Γ ( E Λ p T * M ) = Γ ( E ) Ω 0 ( M ) Γ ( Λ p T * M ) = Γ ( E ) Ω 0 ( M ) Ω p ( M ) , \Gamma(E\otimes\Lambda^{p}T^{*}M)=\Gamma(E)\otimes_{\Omega^{0}(M)}\Gamma(% \Lambda^{p}T^{*}M)=\Gamma(E)\otimes_{\Omega^{0}(M)}\Omega^{p}(M),
  3. Ω 0 ( M , E ) = Γ ( E ) . \Omega^{0}(M,E)=\Gamma(E).\,
  4. T M T M E TM\otimes\cdots\otimes TM\to E
  5. Ω p ( M ) V Ω p ( M , V ) , \Omega^{p}(M)\otimes_{\mathbb{R}}V\to\Omega^{p}(M,V),
  6. Ω p ( M , V ) = Ω 0 ( M , V ) Ω 0 ( M ) Ω p ( M ) , \Omega^{p}(M,V)=\Omega^{0}(M,V)\otimes_{\Omega^{0}(M)}\Omega^{p}(M),
  7. \mathbb{R}
  8. Ω 0 ( M , V ) Ω 0 ( M ) Ω p ( M ) = ( V Ω 0 ( M ) ) Ω 0 ( M ) Ω p ( M ) = V ( Ω 0 ( M ) Ω 0 ( M ) Ω p ( M ) ) = V Ω p ( M ) . \Omega^{0}(M,V)\otimes_{\Omega^{0}(M)}\Omega^{p}(M)=(V\otimes_{\mathbb{R}}% \Omega^{0}(M))\otimes_{\Omega^{0}(M)}\Omega^{p}(M)=V\otimes_{\mathbb{R}}(% \Omega^{0}(M)\otimes_{\Omega^{0}(M)}\Omega^{p}(M))=V\otimes_{\mathbb{R}}\Omega% ^{p}(M).
  9. ( φ * ω ) x ( v 1 , , v p ) = ω φ ( x ) ( d φ x ( v 1 ) , , d φ x ( v p ) ) . (\varphi^{*}\omega)_{x}(v_{1},\cdots,v_{p})=\omega_{\varphi(x)}(\mathrm{d}% \varphi_{x}(v_{1}),\cdots,\mathrm{d}\varphi_{x}(v_{p})).
  10. : Ω p ( M , E 1 ) × Ω q ( M , E 2 ) Ω p + q ( M , E 1 E 2 ) . \wedge:\Omega^{p}(M,E_{1})\times\Omega^{q}(M,E_{2})\to\Omega^{p+q}(M,E_{1}% \otimes E_{2}).
  11. ( ω η ) ( v 1 , , v p + q ) = 1 ( p + q ) ! σ S p + q sgn ( σ ) ω ( v σ ( 1 ) , , v σ ( p ) ) η ( v σ ( p + 1 ) , , v σ ( p + q ) ) . (\omega\wedge\eta)(v_{1},\cdots,v_{p+q})=\frac{1}{(p+q)!}\sum_{\sigma\in S_{p+% q}}\operatorname{sgn}(\sigma)\omega(v_{\sigma(1)},\cdots,v_{\sigma(p)})\otimes% \eta(v_{\sigma(p+1)},\cdots,v_{\sigma(p+q)}).
  12. ω η = ( - 1 ) p q η ω . \omega\wedge\eta=(-1)^{pq}\eta\wedge\omega.
  13. Ω ( M , E ) = p = 0 dim M Ω p ( M , E ) \Omega(M,E)=\bigoplus_{p=0}^{\dim M}\Omega^{p}(M,E)
  14. d ω = ( d ω α ) e α . d\omega=(d\omega^{\alpha})e_{\alpha}.\,
  15. d ( ω + η ) = d ω + d η \displaystyle d(\omega+\eta)=d\omega+d\eta
  16. : Ω 0 ( M , E ) Ω 1 ( M , E ) . \nabla:\Omega^{0}(M,E)\to\Omega^{1}(M,E).
  17. d : Ω p ( M , E ) Ω p + 1 ( M , E ) d_{\nabla}:\Omega^{p}(M,E)\to\Omega^{p+1}(M,E)
  18. d ( ω η ) = d ω η + ( - 1 ) p ω d η d_{\nabla}(\omega\wedge\eta)=d_{\nabla}\omega\wedge\eta+(-1)^{p}\,\omega\wedge d\eta
  19. ( R g ) * ω = ρ ( g - 1 ) ω (R_{g})^{*}\omega=\rho(g^{-1})\omega\,
  20. ω ( v 1 , , v p ) = 0 \omega(v_{1},\ldots,v_{p})=0
  21. ϕ ¯ \overline{\phi}
  22. ϕ = u - 1 π * ϕ ¯ \phi=u^{-1}\pi^{*}\overline{\phi}
  23. V E π ( u ) = ( π * E ) u , v [ u , v ] V\overset{\simeq}{\to}E_{\pi(u)}=(\pi^{*}E)_{u},v\mapsto[u,v]
  24. ϕ ¯ \overline{\phi}
  25. Γ ( M , E ) { f : P V | f ( u g ) = ρ ( g ) - 1 f ( u ) } , f ¯ f \Gamma(M,E)\simeq\{f:P\to V|f(ug)=\rho(g)^{-1}f(u)\},\,\overline{f}\leftrightarrow f
  26. ϕ ¯ = D ϕ ¯ . \nabla\overline{\phi}=\overline{D\phi}.
  27. : Γ ( M , E ) Γ ( M , T * M E ) \nabla:\Gamma(M,E)\to\Gamma(M,T^{*}M\otimes E)
  28. D ( f ϕ ) = D f ϕ + f D ϕ D(f\phi)=Df\otimes\phi+fD\phi

Vector-valued_function.html

  1. 𝐫 ( t ) = f ( t ) 𝐢 + g ( t ) 𝐣 \mathbf{r}(t)=f(t)\mathbf{i}+g(t)\mathbf{j}
  2. 𝐫 ( t ) = f ( t ) 𝐢 + g ( t ) 𝐣 + h ( t ) 𝐤 \mathbf{r}(t)=f(t)\mathbf{i}+g(t)\mathbf{j}+h(t)\mathbf{k}
  3. 𝐫 ( t ) = f ( t ) , g ( t ) \mathbf{r}(t)=\langle f(t),g(t)\rangle
  4. 𝐫 ( t ) = f ( t ) , g ( t ) , h ( t ) \mathbf{r}(t)=\langle f(t),g(t),h(t)\rangle
  5. 𝐫 ( t ) = f ( t ) 𝐢 + g ( t ) 𝐣 + h ( t ) 𝐤 \mathbf{r}(t)=f(t)\mathbf{i}+g(t)\mathbf{j}+h(t)\mathbf{k}
  6. d 𝐫 ( t ) d t = f ( t ) 𝐢 + g ( t ) 𝐣 + h ( t ) 𝐤 . \frac{d\mathbf{r}(t)}{dt}=f^{\prime}(t)\mathbf{i}+g^{\prime}(t)\mathbf{j}+h^{% \prime}(t)\mathbf{k}.
  7. 𝐯 ( t ) = d 𝐫 ( t ) d t . \mathbf{v}(t)=\frac{d\mathbf{r}(t)}{dt}.
  8. d v ( t ) d t = a ( t ) . \frac{d{v}(t)}{dt}={a}(t).
  9. 𝐚 q = i = 1 n a i q 𝐞 i \frac{\partial\mathbf{a}}{\partial q}=\sum_{i=1}^{n}\frac{\partial a_{i}}{% \partial q}\mathbf{e}_{i}
  10. d 𝐚 d t = i = 1 3 d a i d t 𝐞 i . \frac{d\mathbf{a}}{dt}=\sum_{i=1}^{3}\frac{da_{i}}{dt}\mathbf{e}_{i}.
  11. d 𝐚 d t = r = 1 n 𝐚 q r d q r d t + 𝐚 t . \frac{d\mathbf{a}}{dt}=\sum_{r=1}^{n}\frac{\partial\mathbf{a}}{\partial q_{r}}% \frac{dq_{r}}{dt}+\frac{\partial\mathbf{a}}{\partial t}.
  12. d N 𝐚 d t = i = 1 3 d a i d t 𝐞 i + i = 1 3 a i d N 𝐞 i d t \frac{{}^{\mathrm{N}}d\mathbf{a}}{dt}=\sum_{i=1}^{3}\frac{da_{i}}{dt}\mathbf{e% }_{i}+\sum_{i=1}^{3}a_{i}\frac{{}^{\mathrm{N}}d\mathbf{e}_{i}}{dt}
  13. d N 𝐚 d t = d E 𝐚 d t + ω E N × 𝐚 \frac{{}^{\mathrm{N}}d\mathbf{a}}{dt}=\frac{{}^{\mathrm{E}}d\mathbf{a}}{dt}+{}% ^{\mathrm{N}}\mathbf{\omega}^{\mathrm{E}}\times\mathbf{a}
  14. d N d t ( 𝐫 R ) = d E d t ( 𝐫 R ) + ω E N × 𝐫 R . \frac{{}^{\mathrm{N}}d}{dt}(\mathbf{r}^{\mathrm{R}})=\frac{{}^{\mathrm{E}}d}{% dt}(\mathbf{r}^{\mathrm{R}})+{}^{\mathrm{N}}\mathbf{\omega}^{\mathrm{E}}\times% \mathbf{r}^{\mathrm{R}}.
  15. 𝐯 R N = 𝐯 R E + ω E N × 𝐫 R {}^{\mathrm{N}}\mathbf{v}^{\mathrm{R}}={}^{\mathrm{E}}\mathbf{v}^{\mathrm{R}}+% {}^{\mathrm{N}}\mathbf{\omega}^{\mathrm{E}}\times\mathbf{r}^{\mathrm{R}}
  16. q ( p 𝐚 ) = p q 𝐚 + p 𝐚 q . \frac{\partial}{\partial q}(p\mathbf{a})=\frac{\partial p}{\partial q}\mathbf{% a}+p\frac{\partial\mathbf{a}}{\partial q}.
  17. q ( 𝐚 𝐛 ) = 𝐚 q 𝐛 + 𝐚 𝐛 q . \frac{\partial}{\partial q}(\mathbf{a}\cdot\mathbf{b})=\frac{\partial\mathbf{a% }}{\partial q}\cdot\mathbf{b}+\mathbf{a}\cdot\frac{\partial\mathbf{b}}{% \partial q}.
  18. q ( 𝐚 × 𝐛 ) = 𝐚 q × 𝐛 + 𝐚 × 𝐛 q . \frac{\partial}{\partial q}(\mathbf{a}\times\mathbf{b})=\frac{\partial\mathbf{% a}}{\partial q}\times\mathbf{b}+\mathbf{a}\times\frac{\partial\mathbf{b}}{% \partial q}.
  19. R n R^{n}
  20. f ( t ) = ( f 1 ( t ) , f 2 ( t ) , , f n ( t ) ) f(t)=(f_{1}(t),f_{2}(t),\ldots,f_{n}(t))
  21. f ( t ) = ( f 1 ( t ) , f 2 ( t ) , , f n ( t ) ) f^{\prime}(t)=(f_{1}^{\prime}(t),f_{2}^{\prime}(t),\ldots,f_{n}^{\prime}(t))
  22. t R m t\in R^{m}
  23. n × m n\times m
  24. f ( t ) = lim h 0 f ( t + h ) - f ( t ) h . f^{\prime}(t)=\lim_{h\rightarrow 0}\frac{f(t+h)-f(t)}{h}.
  25. t R n t\in R^{n}
  26. t Y t\in Y
  27. f = ( f 1 , f 2 , f 3 , ) f=(f_{1},f_{2},f_{3},\ldots)
  28. f = f 1 e 1 + f 2 e 2 + f 3 e 3 + f=f_{1}e_{1}+f_{2}e_{2}+f_{3}e_{3}+\cdots
  29. e 1 , e 2 , e 3 , e_{1},e_{2},e_{3},\ldots
  30. f ( t ) f^{\prime}(t)
  31. f ( t ) = ( f 1 ( t ) , f 2 ( t ) , f 3 ( t ) , ) f^{\prime}(t)=(f_{1}^{\prime}(t),f_{2}^{\prime}(t),f_{3}^{\prime}(t),\ldots)

Verhoeff_algorithm.html

  1. ( 0 1 2 3 4 5 6 7 8 9 1 5 7 6 2 8 3 0 9 4 ) = ( 1 5 8 9 4 2 7 0 ) ( 3 6 ) \begin{pmatrix}0&1&2&3&4&5&6&7&8&9\\ 1&5&7&6&2&8&3&0&9&4\end{pmatrix}=\begin{pmatrix}1&5&8&9&4&2&7&0\end{pmatrix}% \begin{pmatrix}3&6\end{pmatrix}

Verifiable_secret_sharing.html

  1. g v = c 0 c 1 i c 2 i 2 c t i t = j = 0 t c j i j = j = 0 t g a j i j = g j = 0 t a j i j = g p ( i ) g^{v}=c_{0}c_{1}^{i}c_{2}^{i^{2}}\cdots c_{t}^{i^{t}}=\prod_{j=0}^{t}c_{j}^{i^% {j}}=\prod_{j=0}^{t}g^{a_{j}i^{j}}=g^{\sum_{j=0}^{t}a_{j}i^{j}}=g^{p(i)}
  2. f 1 = 3 x f_{1}=3x
  3. f 2 = 11 x 6 f_{2}=11x^{6}
  4. t = 6 t=6
  5. f 1 = 18 x 7 f_{1}=18x^{7}
  6. f 2 = - 18 x 7 f_{2}=-18x^{7}
  7. t = 6 t=6
  8. f 1 = 2 x 2 + 3 x 3 f_{1}=2x^{2}+3x^{3}
  9. f 2 = x + x 7 f_{2}=x+x^{7}
  10. t = 6 t=6
  11. P 1 , , P k P_{1},...,P_{k}
  12. P i 1 , , P i m P_{i_{1}},...,P_{i_{m}}
  13. P + j = m + 1 k P j P+\textstyle\sum_{j={m+1}}^{k}P_{j}
  14. P + j = m + 1 k P j P+\textstyle\sum_{j={m+1}}^{k}P_{j}
  15. P i 1 , , P i m P_{i_{1}},...,P_{i_{m}}

Vertex_model.html

  1. V V V\otimes V
  2. 2 4 2^{4}
  3. ε i j k \varepsilon_{ij}^{k\ell}
  4. exp ( - β ε ( state ) ) = vertices exp ( - β ε i j k ) \exp(-\beta\varepsilon(\mbox{state}~{}))=\prod_{\mbox{vertices}}~{}\exp(-\beta% \varepsilon_{ij}^{k\ell})
  5. R i j k = exp ( - β ε i j k ) R_{ij}^{k\ell}=\exp(-\beta\varepsilon_{ij}^{k\ell})
  6. = states exp ( - β ε ( state ) ) \mathbb{Z}=\sum_{\mbox{states}}~{}\exp(-\beta\varepsilon(\mbox{state}~{}))
  7. exp ( - β ε ( state ) ) \frac{\exp(-\beta\varepsilon(\mbox{state}~{}))}{\mathbb{Z}}
  8. ε = states ε exp ( - β ε ) states exp ( - β ε ) = k T 2 T ln \langle\varepsilon\rangle=\frac{\sum_{\mbox{states}}~{}\varepsilon\exp(-\beta% \varepsilon)}{\sum_{\mbox{states}}~{}\exp(-\beta\varepsilon)}=kT^{2}\frac{% \partial}{\partial T}\ln\mathbb{Z}
  9. T i 1 k 1 k N i 1 1 l N = r 1 , , r N - 1 R i 1 k 1 r 1 1 R r 1 k 2 r 2 2 R r N - 1 k N i 1 N T_{i_{1}k_{1}\dots k_{N}}^{i^{\prime}_{1}\ell_{1}\dots l_{N}}=\sum_{r_{1},% \dots,r_{N-1}}R_{i_{1}k_{1}}^{r_{1}\ell_{1}}R_{r_{1}k_{2}}^{r_{2}\ell_{2}}% \cdots R_{r_{N-1}k_{N}}^{i^{\prime}_{1}\ell_{N}}
  10. { v 1 , , v n } \{v_{1},\ldots,v_{n}\}
  11. R E n d ( V V ) R\in End(V\otimes V)
  12. R ( v i v j ) = k , R i j k v k v R(v_{i}\otimes v_{j})=\sum_{k,\ell}R_{ij}^{k\ell}v_{k}\otimes v_{\ell}
  13. T E n d ( V V N ) T\in End(V\otimes V^{\otimes N})
  14. T ( v i 1 v k 1 v k N ) = i 1 , 1 , N T i 1 k 1 k N i 1 1 N v i 1 v 1 v N T(v_{i_{1}}\otimes v_{k_{1}}\otimes\cdots\otimes v_{k_{N}})=\sum_{i^{\prime}_{% 1},\ell_{1},\dots\ell_{N}}T_{i_{1}k_{1}\dots k_{N}}^{i^{\prime}_{1}\ell_{1}% \dots\ell_{N}}v_{i^{\prime}_{1}}\otimes v_{\ell_{1}}\otimes\cdots\otimes v_{% \ell_{N}}
  15. T = R 01 R 02 R 0 N T=R_{01}R_{02}\cdots R_{0N}
  16. V V N V\otimes V^{\otimes N}
  17. i 1 = i 1 i_{1}=i^{\prime}_{1}
  18. ( trace V ( T ) ) k 1 k N 1 N (\operatorname{trace}_{V}(T))_{k_{1}\dots k_{N}}^{\ell_{1}\dots\ell_{N}}
  19. τ = trace V ( T ) \tau=\operatorname{trace}_{V}(T)
  20. ( trace V ( T ) ) k 1 k N 1 N ( trace V ( T ) ) j 1 j N k 1 k N (\operatorname{trace}_{V}(T))_{k_{1}\dots k_{N}}^{\ell_{1}\dots\ell_{N}}(% \operatorname{trace}_{V}(T))_{j_{1}\dots j_{N}}^{k_{1}\dots k_{N}}
  21. ( ( trace V ( T ) ) 2 ) j 1 j N 1 N ((\operatorname{trace}_{V}(T))^{2})_{j_{1}\dots j_{N}}^{\ell_{1}\dots\ell_{N}}
  22. ( ( trace V ( T ) ) M ) 1 N 1 N ((\operatorname{trace}_{V}(T))^{M})_{\ell^{\prime}_{1}\dots\ell^{\prime}_{N}}^% {\ell_{1}\dots\ell_{N}}
  23. τ \tau
  24. = trace V N ( τ M ) λ m a x M \mathbb{Z}=\operatorname{trace}_{V^{\otimes N}}(\tau^{M})\sim\lambda_{max}^{M}
  25. λ m a x \lambda_{max}
  26. τ \tau
  27. τ M \tau^{M}
  28. τ \tau
  29. M M\rightarrow\infty
  30. \mathbb{Z}
  31. τ \tau
  32. τ \tau
  33. τ \tau
  34. μ , ν , λ \forall\mu,\nu,\exists\lambda
  35. R 12 ( λ ) R 13 ( μ ) R 23 ( ν ) = R 23 ( ν ) R 13 ( μ ) R 12 ( λ ) R_{12}(\lambda)R_{13}(\mu)R_{23}(\nu)=R_{23}(\nu)R_{13}(\mu)R_{12}(\lambda)
  36. λ , μ \lambda,\mu
  37. ν \nu
  38. R ( λ ) ( 1 T ( μ ) ) ( T ( ν ) 1 ) = ( T ( ν ) 1 ) ( 1 T ( μ ) ) R ( λ ) R(\lambda)(1\otimes T(\mu))(T(\nu)\otimes 1)=(T(\nu)\otimes 1)(1\otimes T(\mu)% )R(\lambda)
  39. V V V N V\otimes V\otimes V^{\otimes N}
  40. R ( λ ) R(\lambda)
  41. R ( λ ) - 1 R(\lambda)^{-1}
  42. R ( λ ) R(\lambda)
  43. λ \forall\lambda
  44. τ ( μ ) \tau(\mu)
  45. τ ( ν ) , μ , ν \tau(\nu),\ \forall\mu,\nu
  46. τ \tau
  47. λ , ν \lambda,\nu
  48. τ \tau
  49. { 1 , , n } \{1,\ldots,n\}
  50. { | a | b } , 1 a , b n \{|a\rangle\otimes|b\rangle\},1\leq a,b\leq n

Vibrational_partition_function.html

  1. E j , i = ω j ( i + 1 2 ) E_{j,i}=\hbar\omega_{j}(i+\frac{1}{2})
  2. Q v i b = j i e - E j , i k T Q_{vib}=\prod_{j}{\sum_{i}{e^{-\frac{E_{j,i}}{kT}}}}

Vinogradov's_theorem.html

  1. r ( N ) = 1 2 G ( N ) N 2 + O ( N 2 log - A N ) , r(N)={1\over 2}G(N)N^{2}+O\left(N^{2}\log^{-A}N\right),
  2. r ( N ) = k 1 + k 2 + k 3 = N Λ ( k 1 ) Λ ( k 2 ) Λ ( k 3 ) , r(N)=\sum_{k_{1}+k_{2}+k_{3}=N}\Lambda(k_{1})\Lambda(k_{2})\Lambda(k_{3}),
  3. Λ \Lambda
  4. G ( N ) = ( p N ( 1 - 1 ( p - 1 ) 2 ) ) ( p N ( 1 + 1 ( p - 1 ) 3 ) ) . G(N)=\left(\prod_{p\mid N}\left(1-{1\over{\left(p-1\right)}^{2}}\right)\right)% \left(\prod_{p\nmid N}\left(1+{1\over{\left(p-1\right)}^{3}}\right)\right).
  5. N 2 r ( N ) N^{2}\ll r(N)
  6. O ( N 3 2 log 2 N ) O\left(N^{3\over 2}\log^{2}N\right)
  7. N 2 log - 3 N ( number of ways N can be written as a sum of three primes ) . N^{2}\log^{-3}N\ll\left(\hbox{number of ways N can be written as a sum of % three primes}\right).
  8. S ( α ) = n = 1 N Λ ( n ) e ( α n ) S(\alpha)=\sum_{n=1}^{N}\Lambda(n)e(\alpha n)
  9. S ( α ) 3 = n 1 , n 2 , n 3 N Λ ( n 1 ) Λ ( n 2 ) Λ ( n 3 ) e ( α ( n 1 + n 2 + n 3 ) ) = n 3 N r ~ ( n ) e ( α n ) S(\alpha)^{3}=\sum_{n_{1},n_{2},n_{3}\leq N}\Lambda(n_{1})\Lambda(n_{2})% \Lambda(n_{3})e(\alpha(n_{1}+n_{2}+n_{3}))=\sum_{n\leq 3N}\tilde{r}(n)e(\alpha n)
  10. r ~ \tilde{r}
  11. N \leq N
  12. r ( N ) = 0 1 S ( α ) 3 e ( - α N ) d α r(N)=\int_{0}^{1}S(\alpha)^{3}e(-\alpha N)\;d\alpha
  13. α \alpha
  14. p q \frac{p}{q}
  15. S ( α ) S(\alpha)
  16. q q
  17. S ( α ) S(\alpha)
  18. α \alpha
  19. | S ( α ) | |S(\alpha)|
  20. | α - a q | < 1 q 2 |\alpha-\frac{a}{q}|<\frac{1}{q^{2}}
  21. | S ( α ) | ( N q + N 4 / 5 + N q ) log 4 N |S(\alpha)|\ll\left(\frac{N}{\sqrt{q}}+N^{4/5}+\sqrt{Nq}\right)\log^{4}N
  22. q q
  23. log N \log N
  24. | S ( α ) | N log A N |S(\alpha)|\ll\frac{N}{\log^{A}N}
  25. C N log A N 0 1 | S ( α ) | 2 d α N 2 log A - 1 N \frac{CN}{\log^{A}N}\int_{0}^{1}|S(\alpha)|^{2}\;d\alpha\ll\frac{N^{2}}{\log^{% A-1}N}

Voltage_droop.html

  1. V m a x - V n o m V_{max}-V_{nom}
  2. V m a x - V m i n V_{max}-V_{min}

Volterra_operator.html

  1. V ( f ) ( t ) = 0 t f ( s ) d s . V(f)(t)=\int_{0}^{t}f(s)\,ds.
  2. V * ( f ) ( t ) = t 1 f ( s ) d s . V^{*}(f)(t)=\int_{t}^{1}f(s)\,ds.

Volume_fraction.html

  1. ϕ i = V i j V j \phi_{i}=\frac{V_{i}}{\sum_{j}V_{j}}
  2. i = 1 N V i = V ; i = 1 N ϕ i = 1 \sum_{i=1}^{N}V_{i}=V;\qquad\sum_{i=1}^{N}\phi_{i}=1

Volume–price_trend.html

  1. VPT = VPTprev + volume × closetoday - closeprev closeprev \,\text{VPT}=\,\text{VPT}\text{prev}+\,\text{volume}\times{\,\text{close}\text% {today}-\,\text{close}\text{prev}\over\,\text{close}\text{prev}}

Vortex-induced_vibration.html

  1. S t = f s t D / U St=f_{st}D/U

W-algebra.html

  1. ( 𝔤 , e ) (\mathfrak{g},e)
  2. 𝔤 \mathfrak{g}
  3. \mathbb{Z}
  4. 𝔤 = 𝔤 ( i ) . \mathfrak{g}=\bigoplus\mathfrak{g}(i).
  5. χ \chi
  6. χ ( x ) = κ ( e , x ) \chi(x)=\kappa(e,x)
  7. κ \kappa
  8. ω χ ( x , y ) = χ ( [ x , y ] ) . \omega_{\chi}(x,y)=\chi([x,y]).
  9. l l
  10. 𝔪 = l + i - 2 𝔤 ( i ) . \mathfrak{m}=l+\bigoplus_{i\leq-2}\mathfrak{g}(i).
  11. I I
  12. U ( 𝔤 ) U(\mathfrak{g})
  13. { x - χ ( x ) : x 𝔪 } \{x-\chi(x):x\in\mathfrak{m}\}
  14. U ( 𝔤 ) / I U(\mathfrak{g})/I
  15. ( 𝔪 ) (\mathfrak{m})
  16. U ( 𝔤 ) U(\mathfrak{g})
  17. ( U ( 𝔤 ) / I ) ad ( 𝔪 ) (U(\mathfrak{g})/I)^{\,\text{ad}(\mathfrak{m})}
  18. U ( 𝔤 , e ) U(\mathfrak{g},e)

W_state.html

  1. | W = 1 3 ( | 001 + | 010 + | 100 ) |W\rangle=\frac{1}{\sqrt{3}}(|001\rangle+|010\rangle+|100\rangle)
  2. | W |W\rangle
  3. | G H Z |GHZ\rangle
  4. n n
  5. | 1 |1\rangle
  6. | 0 |0\rangle
  7. | W = 1 n ( | 100...0 + | 010...0 + + | 00...01 ) |W\rangle=\frac{1}{\sqrt{n}}(|100...0\rangle+|010...0\rangle+...+|00...01\rangle)
  8. n n
  9. | 00...0 |00...0\rangle
  10. | ψ |\psi\rangle
  11. N N
  12. A A
  13. B B
  14. A B = { 1 , , N } A\cup B=\{1,...,N\}
  15. | ψ = | ϕ A | γ B |\psi\rangle=|\phi\rangle_{A}\otimes|\gamma\rangle_{B}
  16. | ψ |\psi\rangle
  17. A | B A|B
  18. 1 , 2 , 3 1,2,3
  19. ( 12 ) 3 , 1 ( 23 ) (12)3,1(23)
  20. ( 13 ) 2 (13)2
  21. 4 2 \mathbb{C}^{4}\otimes\mathbb{C}^{2}

Wadge_hierarchy.html

  1. A A
  2. B B
  3. A A
  4. B B
  5. A A
  6. B B
  7. f f
  8. A = f - 1 [ B ] A=f^{-1}[B]
  9. A A
  10. A A
  11. A A
  12. B B
  13. B B
  14. A A
  15. G ( A , B ) G(A,B)
  16. x x
  17. A A
  18. y y
  19. B B
  20. B B
  21. A A
  22. A A
  23. B B
  24. G ( A , B ) G(A,B)
  25. A A
  26. B B
  27. A , B A,B
  28. A A
  29. B B
  30. B B
  31. A A
  32. A A
  33. A A
  34. A A
  35. A A
  36. W W
  37. A A
  38. B B
  39. A = f - 1 [ B ] A=f^{-1}[B]
  40. f f

Wage–fund_doctrine.html

  1. W a g e = C a p i t a l P o p u l a t i o n Wage=\frac{Capital}{Population}

Wagner_model.html

  1. σ ( t ) = - p 𝐈 + - t M ( t - t ) h ( I 1 , I 2 ) 𝐁 ( t ) d t \mathbf{\sigma}(t)=-p\mathbf{I}+\int_{-\infty}^{t}M(t-t^{\prime})h(I_{1},I_{2}% )\mathbf{B}(t^{\prime})\,dt^{\prime}
  2. σ ( t ) \mathbf{\sigma}(t)
  3. 𝐈 \mathbf{I}
  4. M ( x ) = k = 1 m g i θ i exp ( - x θ i ) M(x)=\sum_{k=1}^{m}\frac{g_{i}}{\theta_{i}}\exp(\frac{-x}{\theta_{i}})
  5. g i g_{i}
  6. θ i \theta_{i}
  7. h ( I 1 , I 2 ) h(I_{1},I_{2})
  8. 𝐁 \mathbf{B}
  9. h ( I 1 , I 2 ) = m * exp ( - n 1 I 1 - 3 ) + ( 1 - m * ) exp ( - n 2 I 2 - 3 ) h(I_{1},I_{2})=m^{*}\exp(-n_{1}\sqrt{I_{1}-3})+(1-m^{*})\exp(-n_{2}\sqrt{I_{2}% -3})

Wald–Wolfowitz_runs_test.html

  1. μ = 2 N + N - N + 1 \mu=\frac{2\ N_{+}\ N_{-}}{N}+1\,
  2. σ 2 = 2 N + N - ( 2 N + N - - N ) N 2 ( N - 1 ) = ( μ - 1 ) ( μ - 2 ) N - 1 . \sigma^{2}=\frac{2\ N_{+}\ N_{-}\ (2\ N_{+}\ N_{-}-N)}{N^{2}\ (N-1)}=\frac{(% \mu-1)(\mu-2)}{N-1}\,.

Wallace_tree.html

  1. n 2 n^{2}
  2. a 2 b 3 a_{2}b_{3}
  3. O ( log n ) O(\log n)
  4. O ( 1 ) O(1)
  5. O ( 1 ) O(1)
  6. O ( log n ) O(\log n)
  7. O ( log n ) O(\log n)
  8. O ( log 2 n ) O(\log^{2}n)
  9. a n b m a_{n}b_{m}
  10. n n
  11. m m
  12. 2 n 2 m = 2 n + m 2^{n}2^{m}=2^{n+m}
  13. a n b m a_{n}b_{m}
  14. 2 n + m 2^{n+m}
  15. n = 4 n=4
  16. a 3 a 2 a 1 a 0 a_{3}a_{2}a_{1}a_{0}
  17. b 3 b 2 b 1 b 0 b_{3}b_{2}b_{1}b_{0}
  18. a 0 b 0 a_{0}b_{0}
  19. a 0 b 1 a_{0}b_{1}
  20. a 1 b 0 a_{1}b_{0}
  21. a 0 b 2 a_{0}b_{2}
  22. a 1 b 1 a_{1}b_{1}
  23. a 2 b 0 a_{2}b_{0}
  24. a 0 b 3 a_{0}b_{3}
  25. a 1 b 2 a_{1}b_{2}
  26. a 2 b 1 a_{2}b_{1}
  27. a 3 b 0 a_{3}b_{0}
  28. a 1 b 3 a_{1}b_{3}
  29. a 2 b 2 a_{2}b_{2}
  30. a 3 b 1 a_{3}b_{1}
  31. a 2 b 3 a_{2}b_{3}
  32. a 3 b 2 a_{3}b_{2}
  33. a 3 b 3 a_{3}b_{3}

Walras'_law.html

  1. i = 1 n P i D i - i = 1 n P i S i = 0 , \sum_{i=1}^{n}P_{i}\cdot D_{i}-\sum_{i=1}^{n}P_{i}\cdot S_{i}=0,
  2. P i P_{i}
  3. D i D_{i}
  4. S i S_{i}

Water_balance.html

  1. P = Q + E + Δ S P=Q+E+\Delta S
  2. P P
  3. Q Q
  4. E E
  5. Δ S \Delta S

Water_retention_curve.html

  1. Ψ m \Psi_{m}
  2. θ ( ψ ) = θ r + θ s - θ r [ 1 + ( α | ψ | ) n ] 1 - 1 / n \theta(\psi)=\theta_{r}+\frac{\theta_{s}-\theta_{r}}{\left[1+(\alpha|\psi|)^{n% }\right]^{1-1/n}}
  3. θ ( ψ ) \theta(\psi)
  4. | ψ | |\psi|
  5. θ s \theta_{s}
  6. θ r \theta_{r}
  7. α \alpha
  8. α > 0 \alpha>0
  9. n n
  10. n > 1 n>1

Wave_field_synthesis.html

  1. s y m b o l P ( w , z ) = d A ( G ( w , z | z ) n P ( w , z ) - P ( w , z ) n G ( w , z | z ) ) d z symbol{P}(w,z)=\iint_{dA}\left(G(w,z|z^{\prime})\frac{\partial}{\partial n}P(w% ,z^{\prime})-P(w,z^{\prime})\frac{\partial}{\partial n}G(w,z|z^{\prime})\right% )dz^{\prime}
  2. f alias = c Δ x | sin Θ sec - sin Θ v | f_{\,\text{alias}}=\frac{c}{\Delta x\left|\sin\Theta^{\,\text{sec}}-\sin\Theta% ^{\,\text{v}}\right|}

Wave_function_renormalization.html

  1. \neq
  2. i p 2 - m 0 2 + i ε i Z p 2 - m 2 + i ε \frac{i}{p^{2}-m_{0}^{2}+i\varepsilon}\rightarrow\frac{iZ}{p^{2}-m^{2}+i\varepsilon}
  3. Z \sqrt{Z}

Wave_propagation.html

  1. v p = ω k , v_{p}=\frac{\omega}{k},
  2. ω = Ω ( k ) . \omega=\Omega(k).\,
  3. v g = ω k v_{g}=\frac{\partial\omega}{\partial k}

Waveguide_(electromagnetism).html

  1. 2 Φ = 0 \nabla^{2}\Phi=0
  2. W \scriptstyle W
  3. λ = 2 W \scriptstyle\lambda\;=\;2W
  4. f = c / λ = c / 2 W \scriptstyle f\;=\;c/\lambda\;=\;c/2W

Wavelet_transform.html

  1. ψ L 2 ( ) \scriptstyle\psi\,\in\,L^{2}(\mathbb{R})
  2. L 2 ( ) \scriptstyle L^{2}\left(\mathbb{R}\right)
  3. { ψ j k : j , k \Z } \scriptstyle\{\psi_{jk}:\,j,\,k\,\in\,\Z\}
  4. ψ \scriptstyle\psi\,
  5. ψ j k ( x ) = 2 j 2 ψ ( 2 j x - k ) \psi_{jk}(x)=2^{\frac{j}{2}}\psi\left(2^{j}x-k\right)\,
  6. j , k \scriptstyle j,\,k\,\in\,\mathbb{Z}
  7. L 2 ( ) \scriptstyle L^{2}\left(\mathbb{R}\right)
  8. f , g = - f ( x ) g ( x ) ¯ d x \langle f,g\rangle=\int_{-\infty}^{\infty}f(x)\overline{g(x)}dx
  9. ψ j k , ψ l m \displaystyle\langle\psi_{jk},\psi_{lm}\rangle
  10. δ j l \scriptstyle\delta_{jl}\,
  11. h L 2 ( ) \scriptstyle h\,\in\,L^{2}\left(\mathbb{R}\right)
  12. h ( x ) = j , k = - c j k ψ j k ( x ) h(x)=\sum_{j,k=-\infty}^{\infty}c_{jk}\psi_{jk}(x)
  13. [ W ψ f ] ( a , b ) = 1 | a | - ψ ( x - b a ) ¯ f ( x ) d x \left[W_{\psi}f\right](a,b)=\frac{1}{\sqrt{|a|}}\int_{-\infty}^{\infty}% \overline{\psi\left(\frac{x-b}{a}\right)}f(x)dx\,
  14. c j k \scriptstyle c_{jk}
  15. c j k = [ W ψ f ] ( 2 - j , k 2 - j ) c_{jk}=\left[W_{\psi}f\right]\left(2^{-j},k2^{-j}\right)
  16. a = 2 - j \scriptstyle a\;=\;2^{-j}
  17. b = k 2 - j \scriptstyle b\;=\;k2^{-j}
  18. Δ t Δ ω 1 2 \Delta t\Delta\omega\geqq\frac{1}{2}
  19. Δ t \scriptstyle\Delta t
  20. y ( t ) = sin ( 2 π f 0 t ) + sin ( 4 π f 0 t ) + sin ( 8 π f 0 t ) \scriptstyle y(t)\;=\;\sin(2\pi f_{0}t)\;+\;\sin(4\pi f_{0}t)\;+\;\sin(8\pi f_% {0}t)
  21. f ( ξ ) = - f ( x ) e - 2 π i x ξ d x f(\xi)=\int_{-\infty}^{\infty}f(x)e^{-2\pi ix\xi}\,dx
  22. X ( t , f ) X(t,f)
  23. X ( a , b ) = 1 a - Ψ ( t - b a ) ¯ x ( t ) d t X(a,b)=\frac{1}{\sqrt{a}}\int_{-\infty}^{\infty}\overline{\Psi\left(\frac{t-b}% {a}\right)}x(t)\,dt
  24. Y D W ( n , m ) = 1 c 0 n k = 0 K - 1 y ( k ) Ψ [ ( k c 0 n - m ) T ] Y_{DW}(n,m)=\frac{1}{\sqrt{c_{0}^{n}}}\cdot\sum_{k=0}^{K-1}y(k)\cdot\Psi\left[% \left(\frac{k}{c_{0}^{n}}-m\right)T\right]
  25. Y W ( c , τ ) = 1 c - y ( k ) Ψ ( t - τ c ) d t Y_{W}(c,\tau)=\frac{1}{\sqrt{c}}\cdot\int_{-\infty}^{\infty}y(k)\cdot\Psi\left% (\frac{t-\tau}{c}\right)\,dt
  26. c n c_{n}
  27. c n c_{n}
  28. ( c , τ ) (c,\tau)
  29. c n c_{n}
  30. c n c_{n}

Weak_formulation.html

  1. V V
  2. u V u\in V
  3. A u = f Au=f
  4. A : V V A:V\to V^{\prime}
  5. f V f\in V^{\prime}
  6. V V^{\prime}
  7. V V
  8. u V u\in V
  9. v V v\in V
  10. [ A u ] ( v ) = f ( v ) [Au](v)=f(v)
  11. v v
  12. u V u\in V
  13. a ( u , v ) = f ( v ) v V , a(u,v)=f(v)\quad\forall v\in V,
  14. a ( u , v ) := [ A u ] ( v ) . a(u,v):=[Au](v).
  15. V = n V=\mathbb{R}^{n}
  16. A : V V A:V\to V
  17. A u = f Au=f
  18. u V u\in V
  19. v V v\in V
  20. A u , v = f , v , \langle Au,v\rangle=\langle f,v\rangle,\,
  21. , \langle\cdot,\cdot\rangle
  22. A A
  23. A u , e i = f , e i i = 1 , , n . \langle Au,e_{i}\rangle=\langle f,e_{i}\rangle\quad i=1,\ldots,n.\,
  24. u = j = 1 n u j e j u=\sum_{j=1}^{n}u_{j}e_{j}
  25. 𝐀𝐮 = 𝐟 , \mathbf{A}\mathbf{u}=\mathbf{f},
  26. a i j = A e j , e i a_{ij}=\langle Ae_{j},e_{i}\rangle
  27. f i = f , e i f_{i}=\langle f,e_{i}\rangle
  28. a ( u , v ) = 𝐯 T 𝐀𝐮 . a(u,v)=\mathbf{v}^{T}\mathbf{A}\mathbf{u}.
  29. - 2 u = f , -\nabla^{2}u=f,\,
  30. Ω d \Omega\subset\mathbb{R}^{d}
  31. u = 0 u=0
  32. V V
  33. L 2 L^{2}
  34. u , v = Ω u v d x \langle u,v\rangle=\int_{\Omega}uv\,dx
  35. v v
  36. - Ω ( 2 u ) v d x = Ω f v d x . -\int_{\Omega}(\nabla^{2}u)v\,dx=\int_{\Omega}fv\,dx.
  37. Ω u v d x = Ω f v d x . \int_{\Omega}\nabla u\cdot\nabla v\,dx=\int_{\Omega}fv\,dx.
  38. V V
  39. H 0 1 ( Ω ) H^{1}_{0}(\Omega)
  40. L 2 ( Ω ) L^{2}(\Omega)
  41. a ( u , v ) = Ω u v d x a(u,v)=\int_{\Omega}\nabla u\cdot\nabla v\,dx
  42. f ( v ) = Ω f v d x . f(v)=\int_{\Omega}fv\,dx.
  43. V V
  44. a ( , ) a(\cdot,\cdot)
  45. V V
  46. | a ( u , v ) | C u v |a(u,v)|\leq C\|u\|\|v\|
  47. a ( u , u ) c u 2 . a(u,u)\geq c\|u\|^{2}.
  48. f V f\in V^{\prime}
  49. u V u\in V
  50. a ( u , v ) = f ( v ) a(u,v)=f(v)
  51. u 1 c f V . \|u\|\leq\frac{1}{c}\|f\|_{V^{\prime}}.
  52. n \mathbb{R}^{n}
  53. | a ( u , v ) | A u v |a(u,v)|\leq\|A\|\,\|u\|\,\|v\|\,
  54. A A
  55. c c
  56. u 1 c f , \|u\|\leq\frac{1}{c}\|f\|,\,
  57. c c
  58. A A
  59. V = H 0 1 ( Ω ) V=H^{1}_{0}(\Omega)
  60. v V := v , \|v\|_{V}:=\|\nabla v\|,
  61. L 2 L^{2}
  62. Ω \Omega
  63. V V
  64. | a ( u , u ) | = u 2 |a(u,u)|=\|\nabla u\|^{2}
  65. | a ( u , v ) | u v |a(u,v)|\leq\|\nabla u\|\,\|\nabla v\|
  66. f [ H 0 1 ( Ω ) ] f\in[H^{1}_{0}(\Omega)]^{\prime}
  67. u V u\in V
  68. u f [ H 0 1 ( Ω ) ] . \|\nabla u\|\leq\|f\|_{[H^{1}_{0}(\Omega)]^{\prime}}.

Weak_solution.html

  1. u t + u x = 0 ( 1 ) \frac{\partial u}{\partial t}+\frac{\partial u}{\partial x}=0\quad\quad(1)
  2. φ \varphi\,\!
  3. - - u ( t , x ) t φ ( t , x ) d t d x + - - u ( t , x ) x φ ( t , x ) d t d x = 0. \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\frac{\partial u(t,x)}{\partial t% }\varphi(t,x)\,\mathrm{d}t\mathrm{d}x+\int_{-\infty}^{\infty}\int_{-\infty}^{% \infty}\frac{\partial u(t,x)}{\partial x}\varphi(t,x)\,\mathrm{d}t\mathrm{d}x=0.
  4. - - - u ( t , x ) φ ( t , x ) t d t d x - - - u ( t , x ) φ ( t , x ) x d t d x = 0. ( 2 ) -\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}u(t,x)\frac{\partial\varphi(t,x% )}{\partial t}\,\mathrm{d}t\mathrm{d}x-\int_{-\infty}^{\infty}\int_{-\infty}^{% \infty}u(t,x)\frac{\partial\varphi(t,x)}{\partial x}\,\mathrm{d}t\mathrm{d}x=0% .\quad\quad(2)
  5. φ \varphi\,\!
  6. φ \varphi\,\!
  7. φ \varphi\,\!
  8. φ \varphi\,\!
  9. P ( x , ) u ( x ) = a α 1 , α 2 , , α n ( x ) α 1 α 2 α n u ( x ) P(x,\partial)u(x)=\sum a_{\alpha_{1},\alpha_{2},\dots,\alpha_{n}}(x)\partial^{% \alpha_{1}}\partial^{\alpha_{2}}\cdots\partial^{\alpha_{n}}u(x)
  10. a α 1 , α 2 , , α n a_{\alpha_{1},\alpha_{2},\dots,\alpha_{n}}
  11. φ \varphi\,\!
  12. W u ( x ) Q ( x , ) φ ( x ) d x = 0 \int_{W}u(x)Q(x,\partial)\varphi(x)\,\mathrm{d}x=0
  13. Q ( x , ) φ ( x ) = ( - 1 ) | α | α 1 α 2 α n [ a α 1 , α 2 , , α n ( x ) φ ( x ) ] . Q(x,\partial)\varphi(x)=\sum(-1)^{|\alpha|}\partial^{\alpha_{1}}\partial^{% \alpha_{2}}\cdots\partial^{\alpha_{n}}\left[a_{\alpha_{1},\alpha_{2},\dots,% \alpha_{n}}(x)\varphi(x)\right].
  14. ( - 1 ) | α | = ( - 1 ) α 1 + α 2 + + α n (-1)^{|\alpha|}=(-1)^{\alpha_{1}+\alpha_{2}+\cdots+\alpha_{n}}
  15. φ \varphi\,\!
  16. P ( x , ) u ( x ) = 0 for all x W P(x,\partial)u(x)=0\mbox{ for all }~{}x\in W
  17. W u ( x ) Q ( x , ) φ ( x ) d x = 0 \int_{W}u(x)Q(x,\partial)\varphi(x)\,\mathrm{d}x=0
  18. φ \varphi\,\!

Weakly_normal_subgroup.html

  1. H H
  2. G G
  3. H g N G ( H ) H^{g}\leq N_{G}(H)
  4. g N G ( H ) g\in N_{G}(H)

Wedge.html

  1. \land

Weeks_manifold.html

  1. 3 23 3 / 2 ζ k ( 2 ) 4 π 4 , \frac{3\cdot 23^{3/2}\zeta_{k}(2)}{4\pi^{4}},

Weibull_fading.html

  1. α \alpha
  2. μ \mu

Weighing_matrix.html

  1. W W T = w I n WW^{T}=wI_{n}
  2. W T W^{T}
  3. W W
  4. I n I_{n}
  5. n n
  6. W T W = w I W^{T}W=wI
  7. W - 1 = w - 1 W T W^{-1}=w^{-1}W^{T}
  8. W - 1 W^{-1}
  9. W W
  10. det ( W ) = ± w n / 2 \operatorname{det}(W)=\pm w^{n/2}
  11. det ( W ) \operatorname{det}(W)
  12. W W
  13. - -
  14. ( 1 1 1 - ) \begin{pmatrix}1&1\\ 1&-\end{pmatrix}
  15. ( 1 1 1 1 0 0 0 1 - 0 0 1 1 0 1 0 - 0 - 0 1 1 0 0 - 0 - - 0 1 - 0 0 1 - 0 1 0 - 1 0 1 0 0 1 - - 1 0 ) \begin{pmatrix}1&1&1&1&0&0&0\\ 1&-&0&0&1&1&0\\ 1&0&-&0&-&0&1\\ 1&0&0&-&0&-&-\\ 0&1&-&0&0&1&-\\ 0&1&0&-&1&0&1\\ 0&0&1&-&-&1&0\end{pmatrix}

Weight-balanced_tree.html

  1. n n
  2. α α
  3. α α
  4. α < 1 - 1 2 \alpha<1-\frac{1}{\sqrt{2}}
  5. 2 / 11 {2}/{11}
  6. α α
  7. n n
  8. h log 1 1 - α n = log 2 n log 2 ( 1 1 - α ) = O ( log n ) h\leq\log_{\frac{1}{1-\alpha}}n=\frac{\log_{2}n}{\log_{2}\left(\frac{1}{1-% \alpha}\right)}=O(\log n)
  9. n n
  10. n n

Weighted_random_early_detection.html

  1. a v g = o * ( 1 - 2 - n ) + c * ( 2 - n ) avg=o*(1-2^{-n})+c*(2^{-n})\,\!
  2. n n
  3. o o
  4. c c
  5. n n

Werner_syndrome_ATP-dependent_helicase.html

  1. μ m 2 s \tfrac{\mathrm{\mu m}^{2}}{\mathrm{s}}
  2. μ m 2 s \textstyle\tfrac{\mathrm{\mu m}^{2}}{\mathrm{s}}

Wet-bulb_temperature.html

  1. ( H sat - H 0 ) λ = ( T 0 - T sat ) c s (H_{\mathrm{sat}}-H_{0})\cdot\lambda=(T_{0}-T_{\mathrm{sat}})\cdot c_{\mathrm{% s}}
  2. H sat H_{\mathrm{sat}}
  3. H 0 H_{0}
  4. λ \lambda
  5. T 0 T_{0}
  6. T sat T_{\mathrm{sat}}
  7. c s c_{s}
  8. ( H sat - H 0 ) λ k = ( T 0 - T eq ) h c (H_{\mathrm{sat}}-H_{0})\cdot\lambda\cdot k^{\prime}=(T_{0}-T_{\mathrm{eq}})% \cdot h_{\mathrm{c}}
  9. H sat H_{\mathrm{sat}}
  10. H 0 H_{0}
  11. k k^{\prime}
  12. T 0 T_{0}
  13. T eq T_{\mathrm{eq}}
  14. h c h_{\mathrm{c}}
  15. ( H - H 0 ) (H-H_{0})
  16. H sat - H 0 H_{\mathrm{sat}}-H_{0}
  17. ( T 0 - T ) (T_{0}-T)
  18. T T
  19. T eq T_{\mathrm{eq}}
  20. ( H sat - H 0 ) λ = ( T 0 - T eq ) h c k (H_{\mathrm{sat}}-H_{0})\cdot\lambda=(T_{0}-T_{\mathrm{eq}})\cdot\frac{h_{% \mathrm{c}}}{k^{\prime}}
  21. H 0 H_{0}
  22. T 0 T_{0}
  23. ( T 0 - T sat ) c s = ( T 0 - T eq ) h c k (T_{0}-T_{\mathrm{sat}})\cdot c_{\mathrm{s}}=(T_{0}-T_{\mathrm{eq}})\cdot\frac% {h_{\mathrm{c}}}{k^{\prime}}
  24. T 0 - T sat = ( T 0 - T eq ) h c k c s T_{0}-T_{\mathrm{sat}}=(T_{0}-T_{\mathrm{eq}})\cdot\frac{h_{\mathrm{c}}}{k^{% \prime}\cdot c_{\mathrm{s}}}
  25. h c k c s = 1 \dfrac{h_{\mathrm{c}}}{k^{\prime}c_{\mathrm{s}}}=1

Wetted_perimeter.html

  1. P = i = 0 l i P=\sum_{i=0}^{\infty}{l_{i}}
  2. l i l_{i}

Weyl's_inequality.html

  1. | c - a / q | t q - 2 , |c-a/q|\leq tq^{-2},\,
  2. ε \scriptstyle\varepsilon
  3. x = M M + N exp ( 2 π i f ( x ) ) = O ( N 1 + ε ( t q + 1 N + t N k - 1 + q N k ) 2 1 - k ) as N . \sum_{x=M}^{M+N}\exp(2\pi if(x))=O\left(N^{1+\varepsilon}\left({t\over q}+{1% \over N}+{t\over N^{k-1}}+{q\over N^{k}}\right)^{2^{1-k}}\right)\,\text{ as }N% \to\infty.
  4. q < N k , q<N^{k},\,
  5. N \scriptstyle\leq\,N
  6. M = H + P \scriptstyle M\,=\,H\,+\,P
  7. μ 1 μ n \mu_{1}\geq\cdots\geq\mu_{n}\,
  8. ν 1 ν n \nu_{1}\geq\cdots\geq\nu_{n}\,
  9. ρ 1 ρ n \rho_{1}\geq\cdots\geq\rho_{n}\,
  10. i = 1 , , n \scriptstyle i\,=\,1,\dots,n
  11. ν i + ρ n μ i ν i + ρ 1 \nu_{i}+\rho_{n}\leq\mu_{i}\leq\nu_{i}+\rho_{1}\,
  12. j + k - n i r + s - 1 , , n \scriptstyle j+k-n\,\geq\,i\,\geq\,r+s-1,\dots,n
  13. ν j + ρ k μ i ν r + ρ s \nu_{j}+\rho_{k}\leq\mu_{i}\leq\nu_{r}+\rho_{s}\,
  14. ρ n > 0 \scriptstyle\rho_{n}\,>\,0
  15. μ i > ν i i = 1 , , n . \mu_{i}>\nu_{i}\quad\forall i=1,\dots,n.\,
  16. P 2 ϵ \|P\|_{2}\leq\epsilon
  17. ϵ \epsilon
  18. | μ i - ν i | ϵ i = 1 , , n . |\mu_{i}-\nu_{i}|\leq\epsilon\qquad\forall i=1,\ldots,n.

Weyl_character_formula.html

  1. V V
  2. 𝔤 \mathfrak{g}
  3. ch ( V ) = w W ε ( w ) e w ( λ + ρ ) e ρ α Δ + ( 1 - e - α ) \operatorname{ch}(V)=\frac{\sum_{w\in W}\varepsilon(w)e^{w(\lambda+\rho)}}{e^{% \rho}\prod_{\alpha\in\Delta^{+}}(1-e^{-\alpha})}
  4. W W
  5. Δ + \Delta^{+}
  6. Δ \Delta
  7. ρ \rho
  8. λ \lambda
  9. V V
  10. ε ( w ) \varepsilon(w)
  11. w w
  12. 𝔥 𝔤 \mathfrak{h}\subset\mathfrak{g}
  13. ( - 1 ) ( w ) (-1)^{\ell(w)}
  14. ( w ) \ell(w)
  15. w w
  16. V V
  17. G G
  18. ch ( V ) = w W ε ( w ) ξ w ( λ + ρ ) - ρ α Δ + ( 1 - ξ - α ) \operatorname{ch}(V)=\frac{\sum_{w\in W}\varepsilon(w)\xi_{w(\lambda+\rho)-% \rho}}{\prod_{\alpha\in\Delta^{+}}(1-\xi_{-\alpha})}
  19. ξ α \xi_{\alpha}
  20. T T
  21. α \alpha
  22. 𝔱 0 \mathfrak{t}_{0}
  23. T T
  24. ρ \rho
  25. T T
  26. G G
  27. ch ( V ) = w W ε ( w ) ξ w ( λ + ρ ) ξ ρ α Δ + ( 1 - ξ - α ) = w W ε ( w ) ξ w ( λ + ρ ) w W ε ( w ) ξ w ( ρ ) \operatorname{ch}(V)=\frac{\sum_{w\in W}\varepsilon(w)\xi_{w(\lambda+\rho)}}{% \xi_{\rho}\prod_{\alpha\in\Delta^{+}}(1-\xi_{-\alpha})}=\frac{\sum_{w\in W}% \varepsilon(w)\xi_{w(\lambda+\rho)}}{\sum_{w\in W}\varepsilon(w)\xi_{w(\rho)}}
  28. w W ε ( w ) e w ( ρ ) = e ρ α Δ + ( 1 - e - α ) . {\sum_{w\in W}\varepsilon(w)e^{w(\rho)}=e^{\rho}\prod_{\alpha\in\Delta^{+}}(1-% e^{-\alpha})}.
  29. σ S n sgn ( σ ) X 1 σ ( 1 ) - 1 X n σ ( n ) - 1 = 1 i < j n ( X j - X i ) \sum_{\sigma\in S_{n}}\operatorname{sgn}(\sigma)\,X_{1}^{\sigma(1)-1}\cdots X_% {n}^{\sigma(n)-1}=\prod_{1\leq i<j\leq n}(X_{j}-X_{i})
  30. dim ( V Λ ) = α Δ + ( Λ + ρ , α ) α Δ + ( ρ , α ) \dim(V_{\Lambda})={\prod_{\alpha\in\Delta^{+}}(\Lambda+\rho,\alpha)\over\prod_% {\alpha\in\Delta^{+}}(\rho,\alpha)}
  31. ( Λ + ρ 2 - λ + ρ 2 ) dim V λ = 2 α Δ + j 1 ( λ + j α , α ) dim V λ + j α (\|\Lambda+\rho\|^{2}-\|\lambda+\rho\|^{2})\dim V_{\lambda}=2\sum_{\alpha\in% \Delta^{+}}\sum_{j\geq 1}(\lambda+j\alpha,\alpha)\dim V_{\lambda+j\alpha}
  32. m = 1 ( 1 - x 2 m ) ( 1 - x 2 m - 1 y ) ( 1 - x 2 m - 1 y - 1 ) = n = - ( - 1 ) n x n 2 y n . \prod_{m=1}^{\infty}\left(1-x^{2m}\right)\left(1-x^{2m-1}y\right)\left(1-x^{2m% -1}y^{-1}\right)=\sum_{n=-\infty}^{\infty}(-1)^{n}x^{n^{2}}y^{n}.
  33. w W ( - 1 ) ( w ) w ( e λ + ρ S ) e ρ α Δ + ( 1 - e - α ) . {\sum_{w\in W}(-1)^{\ell(w)}w(e^{\lambda+\rho}S)\over e^{\rho}\prod_{\alpha\in% \Delta^{+}}(1-e^{-\alpha})}.
  34. S = I ( - 1 ) | I | e Σ I S=\sum_{I}(-1)^{|I|}e^{\Sigma I}\,
  35. j ( p ) - j ( q ) = ( 1 p - 1 q ) n , m = 1 ( 1 - p n q m ) c n m j(p)-j(q)=\left({1\over p}-{1\over q}\right)\prod_{n,m=1}^{\infty}(1-p^{n}q^{m% })^{c_{nm}}
  36. ( β , β - 2 ρ ) c β = γ + δ = β ( γ , δ ) c γ c δ (\beta,\beta-2\rho)c_{\beta}=\sum_{\gamma+\delta=\beta}(\gamma,\delta)c_{% \gamma}c_{\delta}\,
  37. c β = n 1 mult ( β / n ) n . c_{\beta}=\sum_{n\geq 1}{\operatorname{mult}(\beta/n)\over n}.
  38. π \pi
  39. λ \lambda
  40. Θ π \Theta_{\pi}
  41. π \pi
  42. Θ π | H = w W / W λ a w e w λ e ρ α Δ + ( 1 - e - α ) . \Theta_{\pi}|_{H^{\prime}}={\sum_{w\in W/W_{\lambda}}a_{w}e^{w\lambda}\over e^% {\rho}\prod_{\alpha\in\Delta^{+}}(1-e^{-\alpha})}.
  43. H H_{\mathbb{C}}
  44. G G_{\mathbb{C}}
  45. W λ W_{\lambda}
  46. λ \lambda
  47. a w a_{w}

Wheel_factorization.html

  1. lim inf φ ( n ) n log log n = e - γ . \lim\inf\frac{\varphi(n)}{n}\log\log n=e^{-\gamma}.
  2. n = p 1 p 2 p i < x n=p_{1}p_{2}...p_{i}<x
  3. n * p i + 1 x n*p_{i+1}>=x
  4. S 1 = { 1 } S_{1}=\{1\}
  5. n = 1 n=1
  6. S 2 = { 1 } S_{2}=\{1\}
  7. S 6 = { 1 , 5 } S_{6}=\{1,5\}
  8. S n + k S_{n}+k
  9. S n S_{n}
  10. S n p i + 1 = F p i + 1 [ S n S n + n S n + 2 n S n + n ( p i + 1 - 1 ) ] S_{np_{i+1}}=F_{p_{i+1}}[S_{n}\cup S_{n}+n\cup S_{n}+2n\cup...\cup S_{n}+n(p_{% i+1}-1)]
  11. F x F_{x}
  12. p i + 1 p_{i+1}
  13. S n S_{n}
  14. n > 2 n>2
  15. n / 2 n/2

White_test.html

  1. L M = n R 2 . \ LM=n\cdot R^{2}.

Whitehead's_lemma.html

  1. [ u 0 0 u - 1 ] \begin{bmatrix}u&0\\ 0&u^{-1}\end{bmatrix}
  2. [ u 0 0 u - 1 ] = e 21 ( u - 1 ) e 12 ( 1 - u ) e 21 ( - 1 ) e 12 ( 1 - u - 1 ) . \begin{bmatrix}u&0\\ 0&u^{-1}\end{bmatrix}=e_{21}(u^{-1})e_{12}(1-u)e_{21}(-1)e_{12}(1-u^{-1}).
  3. e i j ( s ) e_{ij}(s)
  4. 1 1
  5. i j t h ij^{th}
  6. s s
  7. E ( A ) = [ GL ( A ) , GL ( A ) ] \operatorname{E}(A)=[\operatorname{GL}(A),\operatorname{GL}(A)]
  8. GL ( 2 , / 2 ) \operatorname{GL}(2,\mathbb{Z}/2\mathbb{Z})
  9. Alt ( 3 ) [ GL 2 ( / 2 ) , GL 2 ( / 2 ) ] < E 2 ( / 2 ) = SL 2 ( / 2 ) = GL 2 ( / 2 ) Sym ( 3 ) , \operatorname{Alt}(3)\cong[\operatorname{GL}_{2}(\mathbb{Z}/2\mathbb{Z}),% \operatorname{GL}_{2}(\mathbb{Z}/2\mathbb{Z})]<\operatorname{E}_{2}(\mathbb{Z}% /2\mathbb{Z})=\operatorname{SL}_{2}(\mathbb{Z}/2\mathbb{Z})=\operatorname{GL}_% {2}(\mathbb{Z}/2\mathbb{Z})\cong\operatorname{Sym}(3),

Whitehead_group.html

  1. K 1 ( A ) K_{1}(A)

Wien_approximation.html

  1. I ( ν , T ) = 2 h ν 3 c 2 e - h ν k T I(\nu,T)=\frac{2h\nu^{3}}{c^{2}}e^{-\frac{h\nu}{kT}}
  2. I ( ν , T ) I(\nu,T)
  3. T T
  4. h h
  5. c c
  6. k k
  7. I ( λ , T ) = 2 h c 2 λ 5 e - h c λ k T I(\lambda,T)=\frac{2hc^{2}}{\lambda^{5}}e^{-\frac{hc}{\lambda kT}}
  8. I ( λ , T ) I(\lambda,T)
  9. λ m a x T = 0.290 cm K \lambda_{max}\cdot T\ =\ 0.290\ \mathrm{cm\cdot K}
  10. ν m a x = 5.88 × 10 10 T \nu_{max}\ =\ 5.88\times 10^{10}\cdot T
  11. I ( ν , T ) = 2 h ν 3 c 2 1 e h ν k T - 1 I(\nu,T)=\frac{2h\nu^{3}}{c^{2}}\frac{1}{e^{\frac{h\nu}{kT}}-1}
  12. h ν k T h\nu\gg kT
  13. 1 e h ν k T - 1 e - h ν k T \frac{1}{e^{\frac{h\nu}{kT}}-1}\approx e^{-\frac{h\nu}{kT}}

Wiener's_tauberian_theorem.html

  1. f f
  2. Z Z
  3. f f
  4. Z Z
  5. f f
  6. Z Z
  7. f f
  8. f f
  9. f ( x + a ) f(x+a)
  10. f f
  11. f * h f*h
  12. g * h g*h
  13. lim x ( f * h ) ( x ) = A f ( x ) d x \lim_{x\to\infty}(f*h)(x)=A\int f(x)\,dx
  14. lim x ( g * h ) ( x ) = A g ( x ) d x \lim_{x\to\infty}(g*h)(x)=A\int g(x)\,dx
  15. φ ( θ ) = n f ( n ) e - i n θ \varphi(\theta)=\sum_{n\in\mathbb{Z}}f(n)e^{-in\theta}\,
  16. f * h f*h
  17. h h
  18. g * h g*h
  19. φ φ
  20. 1 / φ 1/φ
  21. φ φ
  22. A ( 𝐓 ) A(\mathbf{T})
  23. A ( 𝐓 ) A(\mathbf{T})
  24. M x = { f A ( 𝕋 ) f ( x ) = 0 } , x 𝕋 . M_{x}=\left\{f\in A(\mathbb{T})\,\mid\,f(x)=0\right\},\quad x\in\mathbb{T}.\,
  25. f ( x + a ) f(x+a)
  26. f f
  27. φ ( θ ) = n f ( n ) e - i n θ \varphi(\theta)=\sum_{n\in\mathbb{Z}}f(n)e^{-in\theta}\,

Wiener–Hopf_method.html

  1. Φ \Phi
  2. Φ ± \Phi_{\pm}
  3. Φ + ( α ) = 1 2 π i C 1 Φ ( z ) d z z - α \Phi_{+}(\alpha)=\frac{1}{2\pi i}\int_{C_{1}}\Phi(z)\frac{dz}{z-\alpha}
  4. Φ - ( α ) = - 1 2 π i C 2 Φ ( z ) d z z - α , \Phi_{-}(\alpha)=-\frac{1}{2\pi i}\int_{C_{2}}\Phi(z)\frac{dz}{z-\alpha},
  5. C 1 C_{1}
  6. C 2 C_{2}
  7. z = α z=\alpha
  8. K ( α ) = K + ( α ) K - ( α ) K(\alpha)=K_{+}(\alpha)K_{-}(\alpha)
  9. s y m b o l L x y f ( x , y ) = 0 , symbol{L}_{xy}f(x,y)=0,
  10. s y m b o l L x y symbol{L}_{xy}
  11. x x
  12. y y
  13. y = 0 y=0
  14. g ( x ) g(x)
  15. f = g ( x ) for x 0 , f y = 0 when x > 0 f=g(x)\,\text{ for }x\leq 0,\quad f_{y}=0\,\text{ when }x>0
  16. f 0 f\rightarrow 0
  17. s y m b o l x symbol{x}\rightarrow\infty
  18. x x
  19. s y m b o l L y f ^ ( k , y ) - P ( k , y ) f ^ ( k , y ) = 0 , symbol{L}_{y}\hat{f}(k,y)-P(k,y)\hat{f}(k,y)=0,
  20. s y m b o l L y symbol{L}_{y}
  21. y y
  22. P ( k , y ) P(k,y)
  23. y y
  24. k k
  25. f ^ ( k , y ) = - f ( x , y ) e - i k x d x . \hat{f}(k,y)=\int_{-\infty}^{\infty}f(x,y)e^{-ikx}\textrm{d}x.
  26. F ( k , y ) F(k,y)
  27. f ^ ( k , y ) = C ( k ) F ( k , y ) , \hat{f}(k,y)=C(k)F(k,y),
  28. C ( k ) C(k)
  29. y = 0 y=0
  30. f ^ \hat{f}
  31. f ^ + \hat{f}_{+}
  32. f ^ - \hat{f}_{-}
  33. f ^ + ( k , y ) = 0 f ( x , y ) e - i k x d x , \hat{f}_{+}(k,y)=\int_{0}^{\infty}f(x,y)e^{-ikx}\textrm{d}x,
  34. f ^ - ( k , y ) = - 0 f ( x , y ) e - i k x d x . \hat{f}_{-}(k,y)=\int_{-\infty}^{0}f(x,y)e^{-ikx}\textrm{d}x.
  35. g ^ ( k ) + f ^ + ( k , 0 ) = f ^ - ( k , 0 ) + f ^ + ( k , 0 ) = f ^ ( k , 0 ) = C ( k ) F ( k , 0 ) \hat{g}(k)+\hat{f}_{+}(k,0)=\hat{f}_{-}(k,0)+\hat{f}_{+}(k,0)=\hat{f}(k,0)=C(k% )F(k,0)
  36. y y
  37. f ^ - ( k , 0 ) = f ^ - ( k , 0 ) + f ^ + ( k , 0 ) = f ^ ( k , 0 ) = C ( k ) F ( k , 0 ) . \hat{f}^{\prime}_{-}(k,0)=\hat{f}^{\prime}_{-}(k,0)+\hat{f}^{\prime}_{+}(k,0)=% \hat{f}^{\prime}(k,0)=C(k)F^{\prime}(k,0).
  38. C ( k ) C(k)
  39. g ^ ( k ) + f ^ + ( k , 0 ) = f ^ - ( k , 0 ) / K ( k ) , \hat{g}(k)+\hat{f}_{+}(k,0)=\hat{f}^{\prime}_{-}(k,0)/K(k),
  40. K ( k ) = F ( k , 0 ) F ( k , 0 ) . K(k)=\frac{F^{\prime}(k,0)}{F(k,0)}.
  41. K ( k ) K(k)
  42. K - K^{-}
  43. K + K^{+}
  44. K ( k ) = K + ( k ) K - ( k ) , K(k)=K^{+}(k)K^{-}(k),
  45. log K - = 1 2 π i - log ( K ( z ) ) z - k d z , Im k > 0 , \hbox{log}K^{-}=\frac{1}{2\pi i}\int_{-\infty}^{\infty}\frac{\hbox{log}(K(z))}% {z-k}\textrm{d}z,\quad\hbox{Im}k>0,
  46. log K + = - 1 2 π i - log ( K ( z ) ) z - k d z , Im k < 0. \hbox{log}K^{+}=-\frac{1}{2\pi i}\int_{-\infty}^{\infty}\frac{\hbox{log}(K(z))% }{z-k}\textrm{d}z,\quad\hbox{Im}k<0.
  47. K K
  48. 1 1
  49. k k\rightarrow\infty
  50. K + g ^ K^{+}\hat{g}
  51. G + G^{+}
  52. G - G^{-}
  53. K + ( k ) g ^ ( k ) = G + ( k ) + G - ( k ) . K^{+}(k)\hat{g}(k)=G^{+}(k)+G^{-}(k).
  54. K ( k ) . K(k).
  55. G + ( k ) + K + ( k ) f ^ + ( k , 0 ) = f ^ - ( k , 0 ) / K - ( k ) - G - ( k ) . G^{+}(k)+K_{+}(k)\hat{f}_{+}(k,0)=\hat{f}^{\prime}_{-}(k,0)/K_{-}(k)-G^{-}(k).
  56. k k
  57. f ^ + ( k , 0 ) = - G + ( k ) K + ( k ) , \hat{f}_{+}(k,0)=-\frac{G^{+}(k)}{K^{+}(k)},
  58. C ( k ) = K + ( k ) g ^ ( k ) - G + ( k ) K + ( k ) F ( k , 0 ) . C(k)=\frac{K^{+}(k)\hat{g}(k)-G^{+}(k)}{K^{+}(k)F(k,0)}.

Wiener–Khinchin_theorem.html

  1. x x
  2. r x x ( τ ) = E [ x ( t ) x * ( t - τ ) ] r_{xx}(\tau)=\operatorname{E}\big[\,x(t)x^{*}(t-\tau)\,\big]
  3. τ \tau
  4. F ( f ) F(f)
  5. - < f < -\infty<f<\infty
  6. r x x ( τ ) = - e 2 π i τ f d F ( f ) r_{xx}(\tau)=\int_{-\infty}^{\infty}e^{2\pi i\tau f}dF(f)
  7. x ( t ) x(t)\,
  8. r x x r_{xx}
  9. F ( f ) F(f)
  10. F F
  11. S ( f ) S(f)
  12. x ( t ) x(t)\,
  13. F F
  14. F F
  15. S ( f ) = 1 2 ( lim ϵ 0 1 ϵ ( F ( f + ϵ ) - F ( f ) ) + lim ϵ 0 1 ϵ ( F ( f + ϵ ) - F ( f ) ) ) S(f)=\frac{1}{2}(\lim_{\epsilon\downarrow 0}\frac{1}{\epsilon}(F(f+\epsilon)-F% (f))+\lim_{\epsilon\uparrow 0}\frac{1}{\epsilon}(F(f+\epsilon)-F(f)))
  16. r x x ( τ ) = - S ( f ) e 2 π i τ f d f . r_{xx}(\tau)=\int_{-\infty}^{\infty}S(f)e^{2\pi i\tau f}df.
  17. S ( f ) = - r x x ( τ ) e - 2 π i f τ d τ . S(f)=\int_{-\infty}^{\infty}r_{xx}(\tau)e^{-2\pi if\tau}d\tau.
  18. x [ n ] x[n]\,
  19. S ( f ) = k = - r x x [ k ] e - i ( 2 π f ) k , S(f)=\sum_{k=-\infty}^{\infty}r_{xx}[k]e^{-i(2\pi f)k},
  20. r x x [ k ] = E [ x [ n ] x * [ n - k ] ] r_{xx}[k]=\operatorname{E}\big[\,x[n]x^{*}[n-k]\,\big]
  21. x [ n ] x[n]\,
  22. S S

Wigner's_theorem.html

  1. Ψ ¯ = { e i α Ψ | α } , Ψ , \underline{\Psi}=\{e^{i\alpha}\Psi|\alpha\in\mathbb{R}\},\Psi\in\mathcal{H},
  2. Φ [ u u n d e r l i n e , u 3 a 8 ] Φ∈[u^{\prime}underline^{\prime},u^{\prime}\u{0}3a8^{\prime}]
  3. Φ Φ
  4. [ u u n d e r l i n e , u 3 a 8 ] [u^{\prime}underline^{\prime},u^{\prime}\u{0}3a8^{\prime}]
  5. H H
  6. B B
  7. B = { Ψ : || Ψ || = 1 } B=\{\Psi\in\mathcal{H}:||\Psi||=1\}
  8. H H
  9. N N
  10. B B
  11. 2 N 1 2N−1
  12. B B
  13. Ψ Φ Ψ = e i α Φ , α . \Psi\cong\Phi\Leftrightarrow\Psi=e^{i\alpha}\Phi,\quad\alpha\in\mathbb{R}.
  14. B B
  15. S S
  16. S = B / . S=B/\cong.
  17. N N
  18. S S
  19. 2 N 2 2N−2
  20. N 1 N−1
  21. H H
  22. Ψ Φ Ψ = z Φ , z * , \Psi\approx\Phi\Leftrightarrow\Psi=z\Phi,\quad z\in\mathbb{C}^{*},
  23. S = H / . S=H/\approx.
  24. R = H / , R=H/\cong,
  25. H H
  26. R R
  27. 2 N 1 2N−1
  28. N N
  29. R R
  30. S S
  31. T T
  32. T : S Ψ ¯ S Ψ ¯ = T Ψ ¯ . T:S\ni\underline{\Psi}\subset\mathcal{H}\mapsto S^{\prime}\ni\underline{\Psi^{% \prime}}=T\underline{\Psi}\subset\mathcal{H^{\prime}}.
  33. S S
  34. R R
  35. T : R R ; T ( λ Ψ ¯ ) λ T Ψ ¯ , Ψ ¯ S , λ . T:R\rightarrow R^{\prime};T(\lambda\underline{\Psi})\equiv\lambda T\underline{% \Psi},\quad\underline{\Psi}\in S,\lambda\in\mathbb{R}.
  36. Ψ ¯ Φ ¯ = | ( Ψ , Φ ) | , \underline{\Psi}\cdot\underline{\Phi}=|(\Psi,\Phi)|,
  37. ( Ψ , Φ ) (Ψ,Φ)
  38. T Ψ ¯ T Φ ¯ = Ψ ¯ Φ ¯ , Ψ , Φ . T\underline{\Psi}\cdot T\underline{\Phi}=\underline{\Psi}\cdot\underline{\Phi}% ,\quad\forall\Psi,\Phi\in\mathcal{H}.
  39. R R
  40. P ( Ψ Φ ) = | ( Ψ , Φ ) | 2 = [ Ψ ¯ Φ ¯ ] 2 = [ T Ψ ¯ T Φ ¯ ] 2 = | ( Ψ , Φ ) | 2 = P ( Ψ Φ ) , Ψ T Ψ ¯ , Φ T Φ ¯ . P(\Psi\rightarrow\Phi)=|(\Psi,\Phi)|^{2}=[\underline{\Psi}\cdot\underline{\Phi% }]^{2}=[T\underline{\Psi}\cdot T\underline{\Phi}]^{2}=|(\Psi^{\prime},\Phi^{% \prime})|^{2}=P(\Psi^{\prime}\rightarrow\Phi^{\prime}),\quad\Psi^{\prime}\in T% \underline{\Psi},\Phi^{\prime}\in T\underline{\Phi}.
  41. U U
  42. ( U Ψ , U Φ ) = ( Ψ , Φ ) , (U\Psi,U\Phi)=(\Psi,\Phi),
  43. ( A Ψ , A Φ ) = ( Ψ , Φ ) * = ( Φ , Ψ ) . (A\Psi,A\Phi)=(\Psi,\Phi)^{*}=(\Phi,\Psi).
  44. U U
  45. T : Ψ ¯ = { e i α Ψ | α } Ψ ¯ = { e i β U Ψ | β } . T:\underline{\Psi}=\{e^{i\alpha}\Psi|\alpha\in\mathbb{R}\}\mapsto\underline{% \Psi^{\prime}}=\{e^{i\beta}U\Psi|\beta\in\mathbb{R}\}.
  46. T Ψ ¯ T Φ ¯ = Ψ ¯ Φ ¯ = | ( e i α U Ψ , e i β U Φ ) | = | ( Ψ , Φ ) | = Ψ ¯ Φ ¯ . T\underline{\Psi}\cdot T\underline{\Phi}=\underline{\Psi^{\prime}}\cdot% \underline{\Phi^{\prime}}=|(e^{i\alpha}U\Psi,e^{i\beta}U\Phi)|=|(\Psi,\Phi)|=% \underline{\Psi}\cdot\underline{\Phi}.
  47. U U
  48. T T
  49. Ψ Ψ
  50. T Ψ ¯ = { e i α U Ψ | α } , T\underline{\Psi}=\{e^{i\alpha}U\Psi|\alpha\in\mathbb{R}\},
  51. U Ψ T Ψ ¯ . U\Psi\in T\underline{\Psi}.
  52. H H
  53. K K
  54. T : Ψ ¯ Ψ ¯ 𝒦 T:\underline{\Psi}\subset\mathcal{H}\mapsto\underline{\Psi^{\prime}}\subset% \mathcal{K}
  55. V : H K V:H→K
  56. T T
  57. V V
  58. d i m H 2 dimH≥2
  59. d i m H = 1 dimH=1
  60. U : H K U:H→K
  61. A : H K A:H→K
  62. T T
  63. U U
  64. V V
  65. H H
  66. K K
  67. T T
  68. d i m H 2 dimH≥2
  69. V = U e i α , α . V=Ue^{i\alpha},\alpha\in\mathbb{R}.
  70. H H
  71. V h = U e i α ( h ) h , α , h ( wrong unless α ( h ) = const ) Vh=Ue^{i\alpha(h)}h,\alpha\in\mathbb{R},h\in\mathcal{H}\quad(\,\text{wrong % unless }\alpha(h)=\,\text{const})
  72. α ( h ) α ( k ) α(h)≠α(k)
  73. h | k = 0 ⟨h|k⟩=0
  74. G G
  75. f , g , h G f,g,h∈G
  76. f g = h fg=h
  77. T ( f ) T ( g ) = T ( h ) , T(f)T(g)=T(h),
  78. T T
  79. U U
  80. U ( f ) U ( g ) = ω ( f , g ) U ( f g ) = e i ξ ( f , g ) U ( f g ) , U(f)U(g)=\omega(f,g)U(fg)=e^{i\xi(f,g)}U(fg),
  81. ω ( f , g ) ω(f,g)
  82. ω ω
  83. 2 2
  84. U : G G L ( V ) U:G→GL(V)
  85. V V
  86. ω ( f , g ) = 1 ω(f,g)=1
  87. g T ( g ) g→T(g)
  88. S = P H S=PH
  89. G P G L ( H ) G→PGL(H)
  90. G G L ( H ) G→GL(H)
  91. f g h fgh
  92. H H
  93. ω ( f , g ) ω ( f g , h ) = ω ( g , h ) ω ( f , g h ) , ξ ( f , g ) + ξ ( f g , h ) = ξ ( g , h ) + ξ ( f , g h ) ( mod 2 π ) . \begin{aligned}\displaystyle\omega(f,g)\omega(fg,h)&\displaystyle=\omega(g,h)% \omega(f,gh),\\ \displaystyle\xi(f,g)+\xi(fg,h)&\displaystyle=\xi(g,h)+\xi(f,gh)\quad(% \operatorname{mod}2\pi).\end{aligned}
  94. ω ( g , e ) = ω ( e , g ) = 1 , ξ ( g , e ) = ξ ( e , g ) = 0 ( mod 2 π ) , ω ( g , g - 1 ) = ω ( g - 1 , g ) , ξ ( g , g - 1 ) = ξ ( g - 1 , g ) = 0 ( mod 2 π ) . \begin{aligned}\displaystyle\omega(g,e)&\displaystyle=\omega(e,g)=1,\\ \displaystyle\xi(g,e)&\displaystyle=\xi(e,g)=0\quad(\operatorname{mod}2\pi),\\ \displaystyle\omega(g,g^{-1})&\displaystyle=\omega(g^{-1},g),\\ \displaystyle\xi(g,g^{-1})&\displaystyle=\xi(g^{-1},g)=0\quad(\operatorname{% mod}2\pi).\\ \end{aligned}
  95. U ( g ) U ^ ( g ) = η ( g ) U ( g ) = e i ζ ( g ) U ( g ) , U(g)\mapsto\hat{U}(g)=\eta(g)U(g)=e^{i\zeta(g)}U(g),
  96. ω ^ ( g , h ) = ω ( g , h ) η ( g ) η ( h ) η ( g h ) - 1 , ξ ^ ( g , h ) = ξ ( g , h ) + ζ ( g ) + ζ ( h ) - ζ ( g h ) ( mod 2 π ) , \begin{aligned}\displaystyle\hat{\omega}(g,h)&\displaystyle=\omega(g,h)\eta(g)% \eta(h)\eta(gh)^{-1},\\ \displaystyle\hat{\xi}(g,h)&\displaystyle=\xi(g,h)+\zeta(g)+\zeta(h)-\zeta(gh)% \quad(\operatorname{mod}2\pi),\end{aligned}
  97. U ^ ( f ) U ^ ( g ) = ω ^ ( f , g ) U ^ ( f g ) = e i ξ ^ ( f , g ) U ^ ( f g ) . \hat{U}(f)\hat{U}(g)=\hat{\omega}(f,g)\hat{U}(fg)=e^{i\hat{\xi}(f,g)}\hat{U}(% fg).
  98. G G
  99. ω ( g , h ) = 1 ω(g,h)=1
  100. ω ( g , h ) = ± 1 ω(g,h)=±1
  101. ω ( g , h ) = 1 ω(g,h)=1
  102. ω ω
  103. H < s u p > 2 ( G ) H<sup>2(G)
  104. R R
  105. S S
  106. H H
  107. h h
  108. H H
  109. h h

Wigner_distribution_function.html

  1. x [ t ] x[t]
  2. C x ( t 1 , t 2 ) = ( x [ t 1 ] - μ [ t 1 ] ) ( x [ t 2 ] - μ [ t 2 ] ) * , C_{x}(t_{1},t_{2})=\left\langle\left(x[t_{1}]-\mu[t_{1}]\right)\left(x[t_{2}]-% \mu[t_{2}]\right)^{*}\right\rangle,
  3. \langle\cdots\rangle
  4. μ ( t ) \mu(t)
  5. W x ( t , f ) W_{x}(t,f)
  6. t = ( t 1 + t 2 ) / 2 t=(t_{1}+t_{2})/2
  7. τ = t 1 - t 2 \tau=t_{1}-t_{2}
  8. W x ( t , f ) = - C x ( t + τ 2 , t - τ 2 ) e - 2 π i τ f d τ . W_{x}(t,f)=\int_{-\infty}^{\infty}C_{x}\left(t+\frac{\tau}{2},t-\frac{\tau}{2}% \right)\,e^{-2\pi i\tau f}\,d\tau.
  9. W x ( t , f ) = - x ( t + τ 2 ) x ( t - τ 2 ) * e - 2 π i τ f d τ . W_{x}(t,f)=\int_{-\infty}^{\infty}x\left(t+\frac{\tau}{2}\right)\,x\left(t-% \frac{\tau}{2}\right)^{*}\,e^{-2\pi i\tau f}\,d\tau.
  10. t t
  11. W x ( t , f ) = - e - i 2 π τ f d τ = δ ( f ) . W_{x}(t,f)=\int_{-\infty}^{\infty}e^{-i2\pi\tau\,f}\,d\tau=\delta(f).
  12. W x ( t , f ) = - e i 2 π k ( t + τ 2 ) e - i 2 π k ( t - τ 2 ) e - i 2 π τ f d τ = - e - i 2 π τ ( f - k ) d τ = δ ( f - k ) . \begin{aligned}\displaystyle W_{x}(t,f)&\displaystyle=\int_{-\infty}^{\infty}e% ^{i2\pi k\left(t+\frac{\tau}{2}\right)}e^{-i2\pi k\left(t-\frac{\tau}{2}\right% )}e^{-i2\pi\tau\,f}\,d\tau\\ &\displaystyle=\int_{-\infty}^{\infty}e^{-i2\pi\tau\left(f-k\right)}\,d\tau\\ &\displaystyle=\delta(f-k).\end{aligned}
  13. x ( t ) = e i 2 π k t 2 x(t)=e^{i2\pi kt^{2}}
  14. 1 2 π d ( 2 π k t 2 ) d t = 2 k t , \frac{1}{2\pi}\frac{d(2\pi kt^{2})}{dt}=2kt~{},
  15. W x ( t , f ) = - e i 2 π k ( t + τ 2 ) 2 e - i 2 π k ( t - τ 2 ) 2 e - i 2 π τ f d τ = - e i 4 π k t τ e - i 2 π τ f d τ = - e - i 2 π τ ( f - 2 k t ) d τ = δ ( f - 2 k t ) . \begin{aligned}\displaystyle W_{x}(t,f)&\displaystyle=\int_{-\infty}^{\infty}e% ^{i2\pi k\left(t+\frac{\tau}{2}\right)^{2}}e^{-i2\pi k\left(t-\frac{\tau}{2}% \right)^{2}}e^{-i2\pi\tau\,f}\,d\tau\\ &\displaystyle=\int_{-\infty}^{\infty}e^{i4\pi kt\tau}e^{-i2\pi\tau f}\,d\tau% \\ &\displaystyle=\int_{-\infty}^{\infty}e^{-i2\pi\tau(f-2kt)}\,d\tau\\ &\displaystyle=\delta(f-2kt)~{}.\end{aligned}
  16. W x ( t , f ) = - δ ( t + τ 2 ) δ ( t - τ 2 ) e - i 2 π τ f d τ = 4 - δ ( 2 t + τ ) δ 2 t - τ ) e - i 2 π τ f d τ = 4 δ ( 4 t ) e i 4 π t f = δ ( t ) e i 4 π t f = δ ( t ) . \begin{aligned}\displaystyle W_{x}(t,f)&\displaystyle=\int_{-\infty}^{\infty}% \delta\left(t+\frac{\tau}{2}\right)\delta\left(t-\frac{\tau}{2}\right)e^{-i2% \pi\tau\,f}\,d\tau\\ &\displaystyle=4\int_{-\infty}^{\infty}\delta(2t+\tau)\delta 2t-\tau)e^{-i2\pi% \tau f}\,d\tau\\ &\displaystyle=4\delta(4t)e^{i4\pi tf}\\ &\displaystyle=\delta(t)e^{i4\pi tf}\\ &\displaystyle=\delta(t).\end{aligned}
  17. x ( t ) = { 1 | t | < 1 / 2 0 otherwise x(t)=\begin{cases}1&|t|<1/2\\ 0&\,\text{otherwise}\end{cases}\qquad
  18. W x ( t , f ) = 1 π f sin ( f [ 1 - 2 | t | ] ) W_{x}(t,f)=\frac{1}{\pi f}\sin(f[1-2|t|])
  19. x ( t ) = { cos ( 2 π t ) t - 2 cos ( 4 π t ) - 2 < t 2 cos ( 3 π t ) t > 2 x(t)=\begin{cases}\cos(2\pi t)&t\leq-2\\ \cos(4\pi t)&-2<t\leq 2\\ \cos(3\pi t)&t>2\end{cases}
  20. x ( t ) = e i t 3 x(t)=e^{it^{3}}
  21. | x ( t ) | 2 = - W x ( t , f ) d f | X ( f ) | 2 = - W x ( t , f ) d t \begin{aligned}\displaystyle|x(t)|^{2}&\displaystyle=\int_{-\infty}^{\infty}W_% {x}(t,f)\,df\\ \displaystyle|X(f)|^{2}&\displaystyle=\int_{-\infty}^{\infty}W_{x}(t,f)\,dt% \end{aligned}
  22. - - W x ( t , f ) d f d t = - | x ( t ) | 2 d t = - | X ( f ) | 2 d f \int_{-\infty}^{\infty}\int_{-\infty}^{\infty}W_{x}(t,f)\,df\,dt=\int_{-\infty% }^{\infty}|x(t)|^{2}\,dt=\int_{-\infty}^{\infty}|X(f)|^{2}\,df
  23. - W x ( t 2 , f ) e i 2 π f t d f = x ( t ) x * ( 0 ) - W x ( t , f 2 ) e i 2 π f t d t = X ( f ) X * ( 0 ) \begin{aligned}\displaystyle\int_{-\infty}^{\infty}W_{x}\left(\frac{t}{2},f% \right)e^{i2\pi ft}\,df&\displaystyle=x(t)x^{*}(0)\\ \displaystyle\int_{-\infty}^{\infty}W_{x}\left(t,\frac{f}{2}\right)e^{i2\pi ft% }\,dt&\displaystyle=X(f)X^{*}(0)\end{aligned}
  24. X ( f ) = | X ( f ) | e i 2 π ψ ( f ) , x ( t ) = | x ( t ) | e i 2 π ϕ ( t ) , if ϕ ( t ) = | x ( t ) | - 2 - f W x ( t , f ) d f and - ψ ( f ) = | X ( f ) | - 2 - t W x ( t , f ) d t \begin{aligned}\displaystyle X(f)&\displaystyle=|X(f)|e^{i2\pi\psi(f)},\quad x% (t)=|x(t)|e^{i2\pi\phi(t)},\\ \displaystyle\,\text{if }\phi^{\prime}(t)&\displaystyle=|x(t)|^{-2}\int_{-% \infty}^{\infty}fW_{x}(t,f)\,df\\ \displaystyle\,\text{ and }-\psi^{\prime}(f)&\displaystyle=|X(f)|^{-2}\int_{-% \infty}^{\infty}tW_{x}(t,f)\,dt\end{aligned}
  25. - - t n W x ( t , f ) d t d f = - t n | x ( t ) | 2 d t - - f n W x ( t , f ) d t d f = - f n | X ( f ) | 2 d f \begin{aligned}\displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}t^{n% }W_{x}(t,f)\,dt\,df&\displaystyle=\int_{-\infty}^{\infty}t^{n}|x(t)|^{2}\,dt\\ \displaystyle\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f^{n}W_{x}(t,f)\,dt% \,df&\displaystyle=\int_{-\infty}^{\infty}f^{n}|X(f)|^{2}\,df\end{aligned}
  26. W x * ( t , f ) = W x ( t , f ) W^{*}_{x}(t,f)=W_{x}(t,f)
  27. If x ( t ) = 0 for t > t 0 then W x ( t , f ) = 0 for t > t 0 If x ( t ) = 0 for t < t 0 then W x ( t , f ) = 0 for t < t 0 \begin{aligned}\displaystyle\,\text{If }x(t)&\displaystyle=0\,\text{ for }t>t_% {0}\,\text{ then }W_{x}(t,f)=0\,\text{ for }t>t_{0}\\ \displaystyle\,\text{If }x(t)&\displaystyle=0\,\text{ for }t<t_{0}\,\text{ % then }W_{x}(t,f)=0\,\text{ for }t<t_{0}\end{aligned}
  28. If y ( t ) = x ( t ) h ( t ) then W y ( t , f ) = - W x ( t , r h o ) W h ( t , f - ρ ) d ρ \begin{aligned}\displaystyle\,\text{If }y(t)&\displaystyle=x(t)h(t)\\ \displaystyle\,\text{then }W_{y}(t,f)&\displaystyle=\int_{-\infty}^{\infty}W_{% x}(t,\ rho)W_{h}(t,f-\rho)\,d\rho\end{aligned}
  29. If y ( t ) = - x ( t - τ ) h ( τ ) d τ then W y ( t , f ) = - W x ( ρ , f ) W h ( t - ρ , f ) d ρ \begin{aligned}\displaystyle\,\text{If }y(t)&\displaystyle=\int_{-\infty}^{% \infty}x(t-\tau)h(\tau)\,d\tau\\ \displaystyle\,\text{then }W_{y}(t,f)&\displaystyle=\int_{-\infty}^{\infty}W_{% x}(\rho,f)W_{h}(t-\rho,f)\,d\rho\end{aligned}
  30. If y ( t ) = - x ( t + τ ) h * ( τ ) d τ then W y ( t , ω ) = - W x ( ρ , ω ) W h ( - t + ρ , ω ) d ρ \begin{aligned}\displaystyle\,\text{If }y(t)&\displaystyle=\int_{-\infty}^{% \infty}x(t+\tau)h^{*}(\tau)\,d\tau\,\text{ then }\\ \displaystyle W_{y}(t,\omega)&\displaystyle=\int_{-\infty}^{\infty}W_{x}(\rho,% \omega)W_{h}(-t+\rho,\omega)\,d\rho\end{aligned}
  31. If y ( t ) = x ( t - t 0 ) then W y ( t , f ) = W x ( t - t 0 , f ) \begin{aligned}\displaystyle\,\text{If }y(t)&\displaystyle=x(t-t_{0})\\ \displaystyle\,\text{then }W_{y}(t,f)&\displaystyle=W_{x}(t-t_{0},f)\end{aligned}
  32. If y ( t ) = e i 2 π f 0 t x ( t ) then W y ( t , f ) = W x ( t , f - f 0 ) \begin{aligned}\displaystyle\,\text{If }y(t)&\displaystyle=e^{i2\pi f_{0}t}x(t% )\\ \displaystyle\,\text{then }W_{y}(t,f)&\displaystyle=W_{x}(t,f-f_{0})\end{aligned}
  33. If y ( t ) = a x ( a t ) for some a > 0 then then W y ( t , f ) = W x ( a t , f a ) \begin{aligned}\displaystyle\,\text{If }y(t)&\displaystyle=\sqrt{a}x(at)\,% \text{ for some }a>0\,\text{ then }\\ \displaystyle\,\text{then }W_{y}(t,f)&\displaystyle=W_{x}(at,\frac{f}{a})\end{aligned}

Wigner–Weyl_transform.html

  1. [ P , Q ] = P Q - Q P = - i , [P,Q]=PQ-QP=-i\hbar,\,
  2. g = e i a Q + i b P + i c , \,g=e^{iaQ+ibP+ic},
  3. ρ ( e i a q + i b p + i c ) \rho(e^{iaq+ibp+ic})\,
  4. x | Φ [ f ] | y = - d p h e i p ( x - y ) / f ( x + y 2 , p ) . \langle x|\Phi[f]|y\rangle=\int_{-\infty}^{\infty}{\,\text{d}p\over h}~{}e^{ip% (x-y)/\hbar}~{}f\left({x+y\over 2},p\right).
  5. \scriptstyle\mathcal{H}
  6. [ P , Q ] = P Q - Q P = - i Id , [P,Q]=PQ-QP=-i\hbar~{}\operatorname{Id}_{\mathcal{H}},
  7. Φ [ f 1 f 2 ] = Φ [ f 1 ] Φ [ f 2 ] . \Phi[f_{1}\star f_{2}]=\Phi[f_{1}]\Phi[f_{2}].\,
  8. ħ 0 ħ→0
  9. f 1 f 2 = n = 0 1 n ! ( i 2 ) n Π n ( f 1 , f 2 ) . f_{1}\star f_{2}=\sum_{n=0}^{\infty}\frac{1}{n!}\left(\frac{i\hbar}{2}\right)^% {n}\Pi^{n}(f_{1},f_{2}).
  10. Π 0 ( f 1 , f 2 ) = f 1 f 2 \Pi^{0}(f_{1},f_{2})=f_{1}f_{2}
  11. Π 1 ( f 1 , f 2 ) = { f 1 , f 2 } = f 1 q f 2 p - f 1 p f 2 q , \Pi^{1}(f_{1},f_{2})=\{f_{1},f_{2}\}=\frac{\partial f_{1}}{\partial q}\frac{% \partial f_{2}}{\partial p}-\frac{\partial f_{1}}{\partial p}\frac{\partial f_% {2}}{\partial q}~{},
  12. Π n ( f 1 , f 2 ) = k = 0 n ( - 1 ) k ( n k ) ( k p k n - k q n - k f 1 ) × ( n - k p n - k k q k f 2 ) \Pi^{n}(f_{1},f_{2})=\sum_{k=0}^{n}(-1)^{k}{n\choose k}\left(\frac{\partial^{k% }}{\partial p^{k}}\frac{\partial^{n-k}}{\partial q^{n-k}}f_{1}\right)\times% \left(\frac{\partial^{n-k}}{\partial p^{n-k}}\frac{\partial^{k}}{\partial q^{k% }}f_{2}\right)
  13. ( n k ) {n\choose k}
  14. exp ( - a ( q 2 + p 2 ) ) exp ( - b ( q 2 + p 2 ) ) = 1 1 + 2 a b exp ( - a + b 1 + 2 a b ( q 2 + p 2 ) ) , \exp\left(-{a}(q^{2}+p^{2})\right)~{}\star~{}\exp\left(-{b}(q^{2}+p^{2})\right% )={1\over 1+\hbar^{2}ab}\exp\left(-{a+b\over 1+\hbar^{2}ab}(q^{2}+p^{2})\right),
  15. δ ( q ) δ ( p ) = 2 h exp ( 2 i q p ) , \delta(q)~{}\star~{}\delta(p)={2\over h}\exp\left(2i{qp\over\hbar}\right),
  16. Φ Φ
  17. f f
  18. ħ / S ħ/S
  19. ħ ħ

Wilks_Coefficient.html

  1. 𝐶𝑜𝑒𝑓𝑓 = 500 a + b x + c x 2 + d x 3 + e x 4 + f x 5 \,\textit{Coeff}=\frac{500}{a+bx+cx^{2}+dx^{3}+ex^{4}+fx^{5}}

William_Shanks.html

  1. π 4 = 4 arctan ( 1 5 ) - arctan ( 1 239 ) \frac{\pi}{4}=4\arctan\left(\frac{1}{5}\right)-\arctan\left(\frac{1}{239}\right)

Wilson_current_mirror.html

  1. i C 1 = i C 2 {{i}_{C1}}={{i}_{C2}}
  2. i B 1 = i B 2 {{i}_{B1}}={{i}_{B2}}
  3. i E 3 {{i}_{E3}}
  4. i C 1 {{i}_{C1}}
  5. i I N {{i}_{IN}}
  6. i I N {{i}_{IN}}
  7. i B = i B 1 = i B 2 i B 3 {{i}_{B}}={{i}_{B1}}={{i}_{B2}}\approx{{i}_{B3}}
  8. i I N - i B {{i}_{IN}}-{{i}_{B}}
  9. i E 3 = i C 2 + 2 i B = i I N - i B + 2 i B = i I N + i B {{i}_{E3}}={{i}_{C2}}+2{{i}_{B}}={{i}_{IN}}-{{i}_{B}}+2{{i}_{B}}={{i}_{IN}}+{{% i}_{B}}
  10. i O U T = i I N + i B - i B = i I N {{i}_{OUT}}={{i}_{IN}}+{{i}_{B}}-{{i}_{B}}={{i}_{IN}}
  11. i C 1 = i C 2 i C {{i}_{C1}}={{i}_{C2}}\equiv{{i}_{C}}
  12. i B 1 = i B 2 i B {{i}_{B1}}={{i}_{B2}}\equiv{{i}_{B}}
  13. i B 3 = i C 3 β {{i}_{B3}}=\frac{{{i}_{C3}}}{\beta}
  14. i E 3 = β + 1 β i C 3 {{i}_{E3}}=\frac{\beta+1}{\beta}{{i}_{C3}}
  15. i E 3 = i C 2 + i B 1 + i B 2 = i C + 2 i B = β + 2 β i C {{i}_{E3}}={{i}_{C2}}+{{i}_{B1}}+{{i}_{B2}}={{i}_{C}}+2{{i}_{B}}=\frac{\beta+2% }{\beta}{{i}_{C}}
  16. i E 3 {{i}_{E3}}
  17. i C = ( β + 1 β + 2 ) i C 3 {{i}_{C}}=\left(\frac{\beta+1}{\beta+2}\right){{i}_{C3}}
  18. i I N = i C 1 + i B 3 = i C + i C 3 β {{i}_{IN}}={{i}_{C1}}+{{i}_{B3}}={{i}_{C}}+\frac{{{i}_{C3}}}{\beta}
  19. i C {{i}_{C}}
  20. i I N = ( β + 1 β + 2 + 1 β ) i C 3 {{i}_{IN}}=\left(\frac{\beta+1}{\beta+2}+\frac{1}{\beta}\right){{i}_{C3}}
  21. i C 3 = ( β ( β + 2 ) β ( β + 2 ) + 2 ) i I N {{i}_{C3}}=\left(\frac{\beta\left(\beta+2\right)}{\beta\left(\beta+2\right)+2}% \right){{i}_{IN}}
  22. i C 3 {{i}_{C3}}
  23. i I N - i O U T = 2 i I N β ( β + 2 ) + 2 2 i I N β 2 {{i}_{IN}}-{{i}_{OUT}}=\frac{2{{i}_{IN}}}{\beta\left(\beta+2\right)+2}\approx% \frac{2{{i}_{IN}}}{{{\beta}^{2}}}
  24. β \beta
  25. I R 1 = V C C - V B E 2 - V B E 3 R 1 {{I}_{R1}}=\frac{{{V}_{CC}}-{{V}_{BE2}}-{{V}_{BE3}}}{R1}
  26. V B E {{V}_{BE}}
  27. I O U T V C C - 1.4 R 1 {{I}_{OUT}}\approx\frac{{{V}_{CC}}-1.4}{R1}
  28. v C E 2 = v B E 2 {{v}_{CE2}}={{v}_{BE2}}
  29. v C E {{v}_{CE}}
  30. β \beta
  31. β 3 {{\beta}_{3}}
  32. β {\beta}
  33. i I N - i O U T = 2 ( β 12 ¯ - β 3 ) + 2 β 12 ¯ β 3 + 2 β 12 ¯ + 2 {{i}_{IN}}-{{i}_{OUT}}=\frac{2\left(\overline{{{\beta}_{12}}}-{{\beta}_{3}}% \right)+2}{\overline{{{\beta}_{12}}}{{\beta}_{3}}+2\overline{{{\beta}_{12}}}+2}
  34. β 12 ¯ \overline{{{\beta}_{12}}}
  35. β 12 ¯ = 2 [ 1 β 1 + 1 β 2 ] - 1 \overline{{{\beta}_{12}}}=2{{\left[\frac{1}{{{\beta}_{1}}}+\frac{1}{{{\beta}_{% 2}}}\right]}^{-1}}
  36. i C = I S C exp ( v B E V T ) {{i}_{C}}={{I}_{SC}}\exp\left(\frac{{{v}_{BE}}}{{{V}_{T}}}\right)
  37. V T = k T q {{V}_{T}}=\frac{kT}{q}
  38. I S C {{I}_{SC}}
  39. I S C {{I}_{SC}}
  40. ± 1 to ± 10 \pm 1\,\text{ to }\pm 10
  41. v t e s t {{v}_{test}}
  42. z o u t v t e s t i t e s t {{z}_{out}}\equiv\frac{{{v}_{test}}}{{{i}_{test}}}
  43. i c 1 {{i}_{c1}}
  44. i e 3 {{i}_{e3}}
  45. i c 1 i e 3 {{i}_{c1}}\approx{{i}_{e3}}
  46. i t e s t = i e 3 + i c 1 = 2 i c 1 or i c 1 = 1 2 i t e s t {{i}_{test}}={{i}_{e3}}+{{i}_{c1}}=2{{i}_{c1}}\,\text{ or }{{i}_{c1}}=\tfrac{1% }{2}{{i}_{test}}
  47. r O 3 {{r}_{O3}}
  48. v t e s t {{v}_{test}}
  49. i b 3 = - i c 1 {{i}_{b3}}=-{{i}_{c1}}
  50. i c 1 {{i}_{c1}}
  51. i t e s t = v t e s t r O 3 - β 2 i t e s t {{i}_{test}}=\frac{{{v}_{test}}}{{{r}_{O3}}}-\frac{\beta}{2}{{i}_{test}}
  52. z o u t = v t e s t i t e s t = ( 1 + β 2 ) r O 3 β 2 r O 3 {{z}_{out}}=\frac{{{v}_{test}}}{{{i}_{test}}}=\left(1+\frac{\beta}{2}\right){{% r}_{O3}}\approx\frac{\beta}{2}{{r}_{O3}}
  53. r O 3 {{r}_{O3}}
  54. β 2 \frac{\beta}{2}
  55. Δ V B E 3 = Δ I I N g m 3 \Delta{{V}_{BE3}}=\frac{\Delta{{I}_{IN}}}{{{g}_{m3}}}
  56. Δ V B E 2 Δ I I N g m 2 \Delta{{V}_{BE2}}\approx\frac{\Delta{{I}_{IN}}}{{{g}_{m2}}}
  57. g m 2 = g m 3 {{g}_{m2}}={{g}_{m3}}
  58. z i n = 2 g m 3 {{z}_{in}}=\frac{2}{{{g}_{m3}}}
  59. g m = I C V T {{g}_{m}}=\frac{{{I}_{C}}}{{{V}_{T}}}
  60. z i n = 2 k T q I I N {{z}_{in}}=\frac{2kT}{q{{I}_{IN}}}
  61. k T q = V T \frac{kT}{q}={{V}_{T}}
  62. z i n {{z}_{in}}
  63. C μ 3 {{C}_{\mu 3}}
  64. C μ 3 {{C}_{\mu 3}}
  65. 2 C μ 3 2{{C}_{\mu 3}}
  66. C μ 3 {{C}_{\mu 3}}
  67. f T {{f}_{T}}
  68. β ( f ) \beta\left(f\right)
  69. f T {{f}_{T}}
  70. f T 10 \tfrac{{{f}_{T}}}{10}
  71. β ( f T 10 ) - j 10 \beta\left(\tfrac{{{f}_{T}}}{10}\right)\approx-j10
  72. V T β I I N = k T q β I I N \frac{{{V}_{T}}}{\beta{{I}_{IN}}}=\frac{kT}{q\beta{{I}_{IN}}}
  73. i E 3 {{i}_{E3}}
  74. H W C M ( s ) i o u t ( s ) i i n ( s ) {{H}_{WCM}}\left(s\right)\equiv\frac{{{i}_{out}}\left(s\right)}{{{i}_{in}}% \left(s\right)}
  75. f T 3 \tfrac{{{f}_{T}}}{3}
  76. f T = 3.0 {{f}_{T}}=3.0
  77. β = 100 \beta=100
  78. ( | H W C M ( s ) | = 2.4 ) \left(\left|{{H}_{WCM}}\left(s\right)\right|=2.4\right)
  79. ( | H W C M ( s ) | = 1.08 ) \left(\left|{{H}_{WCM}}\left(s\right)\right|=1.08\right)
  80. V C C {{V}_{CC}}
  81. v M I R R O R _ O U T = v B E 2 + v C E 3 {{v}_{MIRROR\_OUT}}={{v}_{BE2}}+{{v}_{CE3}}
  82. V C C - v M I R R O R _ O U T {{V}_{CC}}-{{v}_{MIRROR\_OUT}}
  83. v M I R R O R _ O U T {{v}_{MIRROR\_OUT}}
  84. v M I R R O R _ O U T {{v}_{MIRROR\_OUT}}
  85. V B E 0.7 {{V}_{BE}}\geq 0.7
  86. v I N = v B E 2 + v B E 3 {{v}_{IN}}={{v}_{BE2}}+{{v}_{BE3}}
  87. 2 V B E 1.4 2{{V}_{BE}}\approx 1.4
  88. V B E {{V}_{BE}}
  89. β \beta
  90. i C = I S exp ( v B E V T ) {{i}_{C}}={{I}_{S}}\exp\left(\frac{{{v}_{BE}}}{{{V}_{T}}}\right)
  91. β 2 \tfrac{\beta}{2}
  92. f T 10 \tfrac{{{f}_{T}}}{10}
  93. f T {{f}_{T}}
  94. V D S {{V}_{DS}}
  95. i D W L ( v G S - V T H ) 2 {{i}_{D}}\propto\frac{W}{L}{{\left({{v}_{GS}}-{{V}_{TH}}\right)}^{2}}
  96. W W
  97. L L
  98. V T H {{V}_{TH}}
  99. W L \tfrac{W}{L}
  100. V T H {{V}_{TH}}
  101. Δ W L W 2 L 2 - W 1 L 1 \Delta\tfrac{W}{L}\equiv\frac{{{W}_{2}}}{{{L}_{2}}}-\frac{{{W}_{1}}}{{{L}_{1}}}
  102. Δ V T H = V T H 2 - V T H 1 \Delta{{V}_{TH}}={{V}_{TH2}}-{{V}_{TH1}}
  103. i D 1 {{i}_{D1}}
  104. i I N - i O U T = ( 2 Δ V T H V G S 1 - V T H 1 - Δ W L W 1 L 1 ) i I N {{i}_{IN}}-{{i}_{OUT}}=\left(\frac{2\,\Delta{{V}_{TH}}}{{{V}_{GS1}}-{{V}_{TH1}% }}-\frac{\Delta\tfrac{W}{L}}{\frac{{{W}_{1}}}{{{L}_{1}}}}\right){{i}_{IN}}
  105. V G S - V T H {{V}_{GS}}-{{V}_{TH}}
  106. Δ W L \Delta\tfrac{W}{L}
  107. z O ( 1 + g m 4 r O 1 ) r O 4 {{z}_{O}}\approx\left(1+{{g}_{m4}}{{r}_{O1}}\right){{r}_{O4}}
  108. z O ( 2 + g m 4 r O 1 ) r O 4 {{z}_{O}}\approx\left(2+{{g}_{m4}}{{r}_{O1}}\right){{r}_{O4}}
  109. r O 4 {{r}_{O4}}
  110. 2 + g m 4 r O 1 2+{{g}_{m4}}{{r}_{O1}}
  111. v G S 1 + v G S 4 {{v}_{GS1}}+{{v}_{GS4}}
  112. v G S {{v}_{GS}}
  113. v G S 2 + v G S 4 - V T H 4 {{v}_{GS2}}+{{v}_{GS4}}-{{V}_{TH4}}

Wind_profile_power_law.html

  1. u u r = ( z z r ) α \frac{u}{u_{r}}=\bigg(\frac{z}{z_{r}}\bigg)^{\alpha}
  2. u u
  3. z z
  4. u r u_{r}
  5. z r z_{r}
  6. α \alpha
  7. α \alpha
  8. u = u r ( z z r ) α u=u_{r}\bigg(\frac{z}{z_{r}}\bigg)^{\alpha}

Witness_set.html

  1. w C ( c ) w_{C}(c)
  2. max { w C ( c ) : c C } \max\{w_{C}(c):c\in C\}

Witt's_theorem.html

  1. ( V 1 , q 1 ) ( V , q ) ( V 2 , q 2 ) ( V , q ) . (V_{1},q_{1})\oplus(V,q)\simeq(V_{2},q_{2})\oplus(V,q).
  2. ( V 1 , q 1 ) ( V 2 , q 2 ) . (V_{1},q_{1})\simeq(V_{2},q_{2}).
  3. ( V , q ) ( V 0 , 0 ) ( V a , q a ) ( V h , q h ) , (V,q)\simeq(V_{0},0)\oplus(V_{a},q_{a})\oplus(V_{h},q_{h}),

Wold's_decomposition.html

  1. V = ( α A S ) U V=(\oplus_{\alpha\in A}S)\oplus U
  2. H = H V ( H ) V 2 ( H ) = H 0 H 1 H 2 , H=H\supset V(H)\supset V^{2}(H)\supset\cdots=H_{0}\supset H_{1}\supset H_{2}% \supset\cdots,
  3. H i = V i ( H ) H_{i}=V^{i}(H)
  4. M i = H i H i + 1 = V i ( H V ( H ) ) for i 0 , M_{i}=H_{i}\ominus H_{i+1}=V^{i}(H\ominus V(H))\quad\,\text{for}\quad i\geq 0\;,
  5. H = ( i 0 M i ) ( i 0 H i ) = K 1 K 2 . H=(\oplus_{i\geq 0}M_{i})\oplus(\cap_{i\geq 0}H_{i})=K_{1}\oplus K_{2}.
  6. K 1 = H α K_{1}=\oplus H_{\alpha}
  7. V = V | K 1 V | K 2 = ( α A S ) U , V=V|_{K_{1}}\oplus V|_{K_{2}}=(\oplus_{\alpha\in A}S)\oplus U,
  8. V = 1 α N S . V=\oplus_{1\leq\alpha\leq N}S.
  9. T f + T g = T f + g . T_{f}+T_{g}=T_{f+g}.\,
  10. T f * = T < m t p l > f ¯ . T_{f}^{*}=T_{<}mtpl>{{\bar{f}}}.
  11. T f T g - T f g T_{f}T_{g}-T_{fg}\,
  12. V = ( α A T z ) U . V=(\oplus_{\alpha\in A}T_{z})\oplus U.
  13. Φ : C * ( S ) C * ( V ) by Φ ( T f + K ) = α A ( T f + K ) f ( U ) . \Phi:C^{*}(S)\rightarrow C^{*}(V)\quad\,\text{by}\quad\Phi(T_{f}+K)=\oplus_{% \alpha\in A}(T_{f}+K)\oplus f(U).

Wold's_theorem.html

  1. Y t Y_{t}
  2. Y t = j = 0 b j ε t - j + η t , Y_{t}=\sum_{j=0}^{\infty}b_{j}\varepsilon_{t-j}+\eta_{t},
  3. Y t Y_{t}\,
  4. ε t \varepsilon_{t}
  5. Y t Y_{t}\,
  6. { b j } \{b_{j}\}
  7. b b\,
  8. η t \eta_{t}\,
  9. j = 1 | b j | 2 \sum_{j=1}^{\infty}|b_{j}|^{2}
  10. b j b_{j}
  11. b 0 = 1 b_{0}=1
  12. b j b_{j}
  13. Y t Y_{t}
  14. ε t \varepsilon_{t}
  15. Y t Y_{t}
  16. ε t \varepsilon_{t}
  17. Y t Y_{t}

Wood_drying.html

  1. moisture content = m g - m o d m o d × 100 % \mathrm{moisture~{}content}=\frac{m_{g}-m_{od}}{m_{od}}\times 100\%
  2. m g m_{g}\;
  3. m o d m_{od}\;
  4. moisture content = Weight when cut - Oven dry weight Oven dry weight × 100 % \mathrm{moisture~{}content}=\frac{\mathrm{Weight~{}when~{}cut}-\mathrm{Oven~{}% dry~{}weight}}{\mathrm{Oven~{}dry~{}weight}}\times 100\%
  5. X f s p X_{fsp}\;
  6. X f s p = 0.30 - 0.001 ( T - 20 ) X_{fsp}=0.30-0.001(T-20)\;
  7. M e M_{e}
  8. d M d t = - M - M e τ \frac{dM}{dt}=-\frac{M-M_{e}}{\tau}
  9. τ \tau
  10. L r , L t , L L / 10 L_{r},\,L_{t},\,L_{L}/10
  11. M - M e M 0 - M e = e - t / τ \frac{M-M_{e}}{M_{0}-M_{e}}=e^{-t/\tau}
  12. M 0 M_{0}
  13. τ \tau
  14. τ = L n a + b p sat ( T ) \tau=\frac{L^{n}}{a+bp\text{sat}(T)}
  15. p sat ( T ) p\text{sat}(T)
  16. p sat p\text{sat}
  17. t = - τ ln ( M - M e M 0 - M e ) = - L n a + b p sat ( T ) ln ( M - M e M 0 - M e ) t=-\tau\,\ln\left(\frac{M-M_{e}}{M_{0}-M_{e}}\right)=\frac{-L^{n}}{a+bp\text{% sat}(T)}\,\ln\left(\frac{M-M_{e}}{M_{0}-M_{e}}\right)
  18. τ = 3.03 \tau=3.03

Woodward–Hoffmann_rules.html

  1. π \pi
  2. Ψ 3 \Psi_{3}
  3. 4 n + 2 = a ( 4 q + 2 ) s + b ( 4 p + 2 ) a + c ( 4 t ) s + d ( 4 r ) a 4n+2=a(4q+2)_{s}+b(4p+2)_{a}+c(4t)_{s}+d(4r)_{a}
  4. a ( 4 q + 2 ) = 2 k ( 4 q + 2 ) = 4 ( 2 k q + k ) a(4q+2)=2k(4q+2)=4(2kq+k)
  5. b ( 4 p + 2 ) = ( 2 k + 1 ) ( 4 p + 2 ) = 4 ( 2 p k + p + k ) + 2 b(4p+2)=(2k^{\prime}+1)(4p+2)=4(2pk^{\prime}+p+k^{\prime})+2
  6. a ( 4 q + 2 ) + b ( 4 p + 2 ) = ( 2 k + 1 ) ( 4 q + 2 ) + ( 2 k + 1 ) ( 4 p + 2 ) = 4 ( 2 k q + 2 k p + q + p + k + k + 1 ) a(4q+2)+b(4p+2)=(2k+1)(4q+2)+(2k^{\prime}+1)(4p+2)=4(2kq+2k^{\prime}p+q+p+k+k^% {\prime}+1)
  7. 4 n = a ( 4 q + 2 ) s + b ( 4 p + 2 ) a + c ( 4 t ) s + d ( 4 r ) a 4n=a(4q+2)_{s}+b(4p+2)_{a}+c(4t)_{s}+d(4r)_{a}
  8. a = 2 k b = 2 k a=2k\Leftrightarrow b=2k^{\prime}
  9. f ( 2 ) ( r ) = 2 ρ ( r ) N 2 f^{(2)}(r)=\frac{\partial^{2}\rho(r)}{\partial N^{2}}
  10. ρ ( r ) \rho(r)
  11. N N
  12. f ( r ) = ρ ( r ) N f(r)=\frac{\partial\rho(r)}{\partial N}
  13. f ( 2 ) > 0 f^{(2)}>0
  14. f ( 2 ) < 0 f^{(2)}<0
  15. f ( 2 ) ( r ) = f ( r ) N | ϕ LUMO ( r ) | 2 - | ϕ HOMO ( r ) | 2 f^{(2)}(r)=\frac{\partial f(r)}{\partial N}\cong|\phi_{\,\text{LUMO}}(r)|^{2}-% |\phi_{\,\text{HOMO}}(r)|^{2}

Word_problem_(mathematics).html

  1. 2 + 3 = ? 8 + ( - 3 ) 2+3\stackrel{?}{=}8+(-3)
  2. 2 + x = ? 8 + ( - x ) 2+x\stackrel{?}{=}8+(-x)
  3. { x 3 } \{x\mapsto 3\}
  4. ( ( a - 1 a ) ( b b - 1 ) ) - 1 R 2 ( 1 ( b b - 1 ) ) - 1 R 13 ( 1 1 ) - 1 R 1 1 - 1 R 8 1 ((a^{-1}\cdot a)\cdot(b\cdot b^{-1}))^{-1}\stackrel{R2}{\rightsquigarrow}(1% \cdot(b\cdot b^{-1}))^{-1}\stackrel{R13}{\rightsquigarrow}(1\cdot 1)^{-1}% \stackrel{R1}{\rightsquigarrow}1^{-1}\stackrel{R8}{\rightsquigarrow}1
  5. b ( ( a b ) - 1 a ) R 17 b ( ( b - 1 a - 1 ) a ) R 3 b ( b - 1 ( a - 1 a ) ) R 2 b ( b - 1 1 ) R 11 b b - 1 R 13 1 b\cdot((a\cdot b)^{-1}\cdot a)\stackrel{R17}{\rightsquigarrow}b\cdot((b^{-1}% \cdot a^{-1})\cdot a)\stackrel{R3}{\rightsquigarrow}b\cdot(b^{-1}\cdot(a^{-1}% \cdot a))\stackrel{R2}{\rightsquigarrow}b\cdot(b^{-1}\cdot 1)\stackrel{R11}{% \rightsquigarrow}b\cdot b^{-1}\stackrel{R13}{\rightsquigarrow}1
  6. 1 1
  7. 1 ( a b ) 1\cdot(a\cdot b)
  8. b ( 1 a ) b\cdot(1\cdot a)
  9. a b a\cdot b
  10. b a b\cdot a
  11. 1 x 1\cdot x
  12. = x =x
  13. x - 1 x x^{-1}\cdot x
  14. = 1 =1
  15. ( x y ) z (x\cdot y)\cdot z
  16. = x ( y z ) =x\cdot(y\cdot z)
  17. 1 x 1\cdot x
  18. x \rightsquigarrow x
  19. x - 1 x x^{-1}\cdot x
  20. 1 \rightsquigarrow 1
  21. ( x y ) z (x\cdot y)\cdot z
  22. x ( y z ) \rightsquigarrow x\cdot(y\cdot z)
  23. x - 1 ( x y ) x^{-1}\cdot(x\cdot y)
  24. y \rightsquigarrow y
  25. 1 - 1 1^{-1}
  26. 1 \rightsquigarrow 1
  27. x 1 x\cdot 1
  28. x \rightsquigarrow x
  29. ( x - 1 ) - 1 (x^{-1})^{-1}
  30. x \rightsquigarrow x
  31. x x - 1 x\cdot x^{-1}
  32. 1 \rightsquigarrow 1
  33. x ( x - 1 y ) x\cdot(x^{-1}\cdot y)
  34. y \rightsquigarrow y
  35. ( x y ) - 1 (x\cdot y)^{-1}
  36. y - 1 x - 1 \rightsquigarrow y^{-1}\cdot x^{-1}

Work_(thermodynamics).html

  1. d U = δ Q - δ W . dU=\delta Q-\delta W.\;
  2. δ W \delta W
  3. d U = δ Q + δ W dU=\delta Q+\delta W\,
  4. V V
  5. δ W = P d V \delta W=PdV\,
  6. δ W \delta W
  7. P P
  8. d V dV
  9. W = V i V f P d V . W=\int_{V_{i}}^{V_{f}}P\,dV.
  10. W W
  11. d U = δ Q - P d V . dU=\delta Q-PdV\,.
  12. δ W = - P d V \delta W=-PdV\,
  13. d U = δ Q - P d V dU=\delta Q-PdV\,
  14. d U = δ Q - δ W dU=\delta Q-\delta W
  15. δ Q = 0 \delta Q=0
  16. δ W \delta W
  17. δ W δW
  18. đ W đW
  19. đ W đW
  20. W W
  21. đ W đW
  22. W W
  23. đ W đW
  24. W = F s W=Fs\,
  25. W = 1 2 F d s . W=\int_{1}^{2}F\,ds.
  26. T = F r T=Fr\,
  27. F = T r F=\frac{T}{r}
  28. s = 2 ( r π n ) s=2\left(r\pi n\right)
  29. W s = F s = 2 π n T W_{s}=Fs=2\pi nT
  30. W ˙ s = 2 π T n ˙ \dot{W}_{s}=2\pi T\dot{n}
  31. w s = F d x \partial w_{s}=Fdx
  32. F = K x F=Kx
  33. W s = 0.5 k ( x 1 2 - x 2 2 ) W_{s}=0.5k\left(x_{1}^{2}-x_{2}^{2}\right)
  34. W = 1 2 F d x . W=\int_{1}^{2}F\,dx.
  35. W = 1 2 A σ d x . W=\int_{1}^{2}A\sigma\,dx.
  36. W s = 1 2 σ s d A . W_{s}=\int_{1}^{2}\sigma_{s}\,dA.

World_Football_Elo_Ratings.html

  1. R n = R o + K G ( W - W e ) R_{n}=R_{o}+KG(W-W_{e})
  2. P = K G ( W - W e ) P=KG(W-W_{e})
  3. R n R_{n}
  4. R o R_{o}
  5. K K
  6. G G
  7. W W
  8. W e W_{e}
  9. P P
  10. G = 1 G=1
  11. G = 3 2 G=\frac{3}{2}
  12. G = 11 + N 8 G=\frac{11+N}{8}
  13. W e = 1 10 - d r / 400 + 1 W_{e}=\frac{1}{10^{-dr/400}+1}
  14. K K
  15. G G
  16. W W
  17. W e W_{e}
  18. K K
  19. G G
  20. W W
  21. W e W_{e}

Wythoff_symbol.html

  1. { p , q } \begin{Bmatrix}p,q\end{Bmatrix}
  2. t { p , q } t\begin{Bmatrix}p,q\end{Bmatrix}
  3. { p q } \begin{Bmatrix}p\\ q\end{Bmatrix}
  4. t { q , p } t\begin{Bmatrix}q,p\end{Bmatrix}
  5. { q , p } \begin{Bmatrix}q,p\end{Bmatrix}
  6. r { p q } r\begin{Bmatrix}p\\ q\end{Bmatrix}
  7. t { p q } t\begin{Bmatrix}p\\ q\end{Bmatrix}
  8. s { p q } s\begin{Bmatrix}p\\ q\end{Bmatrix}
  9. { p , q } \begin{Bmatrix}p,q\end{Bmatrix}
  10. t { p , q } t\begin{Bmatrix}p,q\end{Bmatrix}
  11. { p q } \begin{Bmatrix}p\\ q\end{Bmatrix}
  12. t { q , p } t\begin{Bmatrix}q,p\end{Bmatrix}
  13. { q , p } \begin{Bmatrix}q,p\end{Bmatrix}
  14. r { p q } r\begin{Bmatrix}p\\ q\end{Bmatrix}
  15. t { p q } t\begin{Bmatrix}p\\ q\end{Bmatrix}
  16. s { p q } s\begin{Bmatrix}p\\ q\end{Bmatrix}
  17. { p , 2 } \begin{Bmatrix}p,2\end{Bmatrix}
  18. t { p , 2 } t\begin{Bmatrix}p,2\end{Bmatrix}
  19. t { 2 , p } t\begin{Bmatrix}2,p\end{Bmatrix}
  20. { 2 , p } \begin{Bmatrix}2,p\end{Bmatrix}
  21. t { p 2 } t\begin{Bmatrix}p\\ 2\end{Bmatrix}
  22. s { p 2 } s\begin{Bmatrix}p\\ 2\end{Bmatrix}
  23. { p , q } \begin{Bmatrix}p,q\end{Bmatrix}
  24. t { p , q } t\begin{Bmatrix}p,q\end{Bmatrix}
  25. { p q } \begin{Bmatrix}p\\ q\end{Bmatrix}
  26. t { q , p } t\begin{Bmatrix}q,p\end{Bmatrix}
  27. { q , p } \begin{Bmatrix}q,p\end{Bmatrix}
  28. r { p q } r\begin{Bmatrix}p\\ q\end{Bmatrix}
  29. t { p q } t\begin{Bmatrix}p\\ q\end{Bmatrix}
  30. s { p q } s\begin{Bmatrix}p\\ q\end{Bmatrix}

X-ray_absorption_near_edge_structure.html

  1. λ \lambda
  2. λ \lambda
  3. E kinetic = h ν - E binding = 2 k 2 / ( 2 m ) = ( 2 π ) 2 2 / ( 2 m λ 2 ) , E\text{kinetic}=h\nu-E\text{binding}=\hbar^{2}k^{2}/(2m)=(2\pi)^{2}\hbar^{2}/(% 2m\lambda^{2}),
  4. λ \lambda

X-ray_crystal_truncation_rod.html

  1. K 0 K_{0}
  2. 𝐐 \mathbf{Q}
  3. x x
  4. y y
  5. z z
  6. 𝐐 \mathbf{Q}
  7. I ( 𝐐 ) = sin 2 ( 1 2 N x Q x a x ) sin 2 ( 1 2 Q x a x ) sin 2 ( 1 2 N y Q y a y ) sin 2 ( 1 2 Q y a y ) 1 + α 2 N z - 2 α N z cos ( N z Q z c ) 1 + α 2 - 2 α cos ( Q z c ) I(\mathbf{Q})=\frac{\sin^{2}\left(\tfrac{1}{2}N_{x}Q_{x}a_{x}\right)}{\sin^{2}% \left(\tfrac{1}{2}Q_{x}a_{x}\right)}\ \frac{\sin^{2}\left(\tfrac{1}{2}N_{y}Q_{% y}a_{y}\right)}{\sin^{2}\left(\tfrac{1}{2}Q_{y}a_{y}\right)}\ \frac{1+\alpha^{% 2N_{z}}-2\alpha^{N_{z}}\cos\left(N_{z}Q_{z}c\right)}{1+\alpha^{2}-2\alpha\cos% \left(Q_{z}c\right)}
  8. α \alpha
  9. a x a_{x}
  10. a y a_{y}
  11. c c
  12. α = 0 \alpha=0
  13. Q z Q_{z}
  14. 𝐐 \mathbf{Q_{\parallel}}
  15. 𝐐 \mathbf{Q}
  16. 𝐐 = 𝐆 h k = h 𝐚 x * + k 𝐚 y * \mathbf{Q}_{\parallel}=\mathbf{G}_{hk}=h\mathbf{a}^{*}_{x}+k\mathbf{a}^{*}_{y}
  17. h h
  18. k k
  19. α \alpha
  20. α \alpha
  21. 10 - 7 10^{-7}
  22. 10 9 p h o t o n s m m 2 s 10^{9}\tfrac{photons}{mm^{2}s}
  23. Z Z
  24. Z = 79 Z=79
  25. α \alpha
  26. α \alpha

X-ray_reflectivity.html

  1. ρ e ( z ) \rho_{e}(z)
  2. R ( Q ) / R F ( Q ) = | 1 ρ - e i Q z ( d ρ e d z ) d z | 2 R(Q)/R_{F}(Q)=\left|\frac{1}{\rho_{\infty}}{\int\limits_{-\infty}^{\infty}{e^{% iQz}\left(\frac{d\rho_{e}}{dz}\right)dz}}\right|^{2}
  3. R ( Q ) R(Q)
  4. Q = 4 π sin ( θ ) / λ Q=4\pi\sin(\theta)/\lambda
  5. λ \lambda
  6. ρ \rho_{\infty}
  7. θ \theta

XNOR_gate.html

  1. A B + A ¯ B ¯ A\cdot B+\overline{A}\cdot\overline{B}

XOR_gate.html

  1. A B ¯ + A ¯ B A\cdot\overline{B}+\overline{A}\cdot B
  2. ( A + B ) (A+B)\cdot
  3. A ¯ + B ¯ \overline{A}+\overline{B}
  4. A B ¯ + A ¯ B A\cdot\overline{B}+\overline{A}\cdot B
  5. A B ¯ + A ¯ B ( A + B ) A\cdot\overline{B}+\overline{A}\cdot B\equiv(A+B)\cdot
  6. A ¯ + B ¯ \overline{A}+\overline{B}
  7. ( A + B ) (A+B)\cdot
  8. ( A B ) ¯ \overline{(A\cdot B)}

XTR.html

  1. G F ( p 2 ) GF(p^{2})
  2. G F ( p 6 ) * GF(p^{6})^{*}
  3. g g
  4. q q
  5. G F ( p 6 ) * GF(p^{6})^{*}
  6. q q
  7. g g
  8. G F ( p 6 ) * GF(p^{6})^{*}
  9. G F ( p 2 ) GF(p^{2})
  10. G F ( p 6 ) GF(p^{6})
  11. G F ( p 6 ) GF(p^{6})
  12. p 6 p^{6}
  13. p 2 - p + 1 p^{2}-p+1
  14. p 2 - p + 1 p^{2}-p+1
  15. G F ( p 6 ) * GF(p^{6})^{*}
  16. g \langle g\rangle
  17. G F ( p 2 ) GF(p^{2})
  18. G F ( p 6 ) GF(p^{6})
  19. G F ( p 2 ) GF(p^{2})
  20. G F ( p 6 ) GF(p^{6})
  21. G F ( p 2 ) GF(p^{2})
  22. ( / 3 ) * (\mathbb{Z}/3\mathbb{Z})^{*}
  23. Φ 3 ( x ) = x 2 + x + 1 \Phi_{3}(x)=x^{2}+x+1
  24. G F ( p ) GF(p)
  25. α \alpha
  26. α p \alpha^{p}
  27. G F ( p 2 ) GF(p^{2})
  28. G F ( p ) GF(p)
  29. G F ( p 2 ) { x 1 α + x 2 α p : x 1 , x 2 G F ( p ) } . GF(p^{2})\cong\{x_{1}\alpha+x_{2}\alpha^{p}:x_{1},x_{2}\in GF(p)\}.
  30. G F ( p 2 ) { y 1 α + y 2 α 2 : α 2 + α + 1 = 0 , y 1 , y 2 G F ( p ) } . GF(p^{2})\cong\{y_{1}\alpha+y_{2}\alpha^{2}:\alpha^{2}+\alpha+1=0,y_{1},y_{2}% \in GF(p)\}.
  31. G F ( p ) GF(p)
  32. G F ( p ) GF(p)
  33. G F ( p ) GF(p)
  34. G F ( p 2 ) GF(p^{2})
  35. G F ( p 2 ) GF(p^{2})
  36. h G F ( p 6 ) h\in GF(p^{6})
  37. G F ( p 2 ) GF(p^{2})
  38. h , h p 2 h,h^{p^{2}}
  39. h p 4 h^{p^{4}}
  40. h h
  41. T r ( h ) = h + h p 2 + h p 4 . Tr(h)=h+h^{p^{2}}+h^{p^{4}}.
  42. T r ( h ) G F ( p 2 ) Tr(h)\in GF(p^{2})
  43. T r ( h ) p 2 = h p 2 + h p 4 + h p 6 = h + h p 2 + h p 4 = T r ( h ) \begin{aligned}\displaystyle Tr(h)^{p^{2}}&\displaystyle=h^{p^{2}}+h^{p^{4}}+h% ^{p^{6}}\\ &\displaystyle=h+h^{p^{2}}+h^{p^{4}}\\ &\displaystyle=Tr(h)\end{aligned}
  44. g g
  45. q q
  46. g \langle g\rangle
  47. p 2 - p + 1 p^{2}-p+1
  48. q p 2 - p + 1 q\mid p^{2}-p+1
  49. p p
  50. q q
  51. q > 3 q>3
  52. g g
  53. p 2 - p + 1 p^{2}-p+1
  54. p 2 = p - 1 p^{2}=p-1
  55. p 4 = ( p - 1 ) 2 = p 2 - 2 p + 1 = - p p^{4}=(p-1)^{2}=p^{2}-2p+1=-p
  56. T r ( g ) = g + g p 2 + g p 4 = g + g p - 1 + g - p . \begin{aligned}\displaystyle Tr(g)&\displaystyle=g+g^{p^{2}}+g^{p^{4}}\\ &\displaystyle=g+g^{p-1}+g^{-p}.\end{aligned}
  57. g g
  58. 1 1
  59. g g
  60. g g
  61. G F ( p 2 ) GF(p^{2})
  62. ( x - g ) ( x - g p - 1 ) ( x - g - p ) (x-g)\!\ (x-g^{p-1})(x-g^{-p})
  63. x 3 - T r ( g ) x 2 + T r ( g ) p x - 1 x^{3}-Tr(g)\!\ x^{2}+Tr(g)^{p}x-1
  64. T r ( g ) Tr(g)
  65. g g
  66. g g
  67. G F ( p 2 ) GF(p^{2})
  68. g g
  69. g g
  70. g n g^{n}
  71. x 3 - T r ( g n ) x 2 + T r ( g n ) p x - 1 x^{3}-Tr(g^{n})\!\ x^{2}+Tr(g^{n})^{p}x-1
  72. T r ( g n ) Tr(g^{n})
  73. g n G F ( p 6 ) g^{n}\in GF(p^{6})
  74. T r ( g n ) G F ( p 2 ) Tr(g^{n})\in GF(p^{2})
  75. T r ( g n ) Tr(g^{n})
  76. T r ( g ) Tr(g)
  77. T r ( g n ) Tr(g^{n})
  78. T r ( g n ) Tr(g^{n})
  79. T r ( g ) Tr(g)
  80. g n g^{n}
  81. g g
  82. T r ( g n ) Tr(g^{n})
  83. T r ( g ) Tr(g)
  84. G F ( p 2 ) GF(p^{2})
  85. F ( c , X ) = X 3 - c X 2 + c p X - 1 G F ( p 2 ) [ X ] . F(c,X)=X^{3}-cX^{2}+c^{p}X-1\in GF(p^{2})[X].
  86. h 0 , h 1 , h 2 h_{0},\!\ h_{1},h_{2}
  87. F ( c , X ) F(c,X)
  88. G F ( p 6 ) GF(p^{6})
  89. n n
  90. \mathbb{Z}
  91. c n = h 0 n + h 1 n + h 2 n . c_{n}=h_{0}^{n}+h_{1}^{n}+h_{2}^{n}.
  92. c n c_{n}
  93. F ( c , X ) F(c,X)
  94. c = c 1 c=c_{1}
  95. c - n = c n p = c n p c_{-n}=c_{np}=c_{n}^{p}
  96. c n G F ( p 2 ) for n c_{n}\in GF(p^{2})\,\text{ for }n\in\mathbb{Z}
  97. c u + v = c u c v - c v p c u - v + c u - 2 v for u , v c_{u+v}=c_{u}c_{v}-c_{v}^{p}c_{u-v}+c_{u-2v}\,\text{ for }u,v\in\mathbb{Z}
  98. h j h_{j}
  99. p 2 - p + 1 p^{2}-p+1
  100. > 3 >3
  101. h j h_{j}
  102. G F ( p 2 ) GF(p^{2})
  103. F ( c , X ) F(c,X)
  104. p 2 - p + 1 p^{2}-p+1
  105. > 3 >3
  106. F ( c , X ) F(c,X)
  107. G F ( p 2 ) GF(p^{2})
  108. c p + 1 G F ( p ) c_{p+1}\in GF(p)
  109. c , c n - 1 , c n , c n + 1 c,\!\ c_{n-1},c_{n},c_{n+1}
  110. c 2 n = c n 2 - 2 c n p c_{2n}=c_{n}^{2}-2c_{n}^{p}
  111. G F ( p ) GF(p)
  112. c n + 2 = c n + 1 c - c p c n + c n - 1 c_{n+2}=c_{n+1}\cdot c-c^{p}\cdot c_{n}+c_{n-1}
  113. G F ( p ) GF(p)
  114. c 2 n - 1 = c n - 1 c n - c p c n p + c n + 1 p c_{2n-1}=c_{n-1}\cdot c_{n}-c^{p}\cdot c_{n}^{p}+c_{n+1}^{p}
  115. G F ( p ) GF(p)
  116. c 2 n + 1 = c n + 1 c n - c c n p + c n - 1 p c_{2n+1}=c_{n+1}\cdot c_{n}-c\cdot c_{n}^{p}+c_{n-1}^{p}
  117. G F ( p ) GF(p)
  118. S n ( c ) = ( c n - 1 , c n , c n + 1 ) G F ( p 2 ) 3 S_{n}(c)=(c_{n-1},c_{n},c_{n+1})\in GF(p^{2})^{3}
  119. S n ( c ) S_{n}(c)
  120. n n
  121. c c
  122. n < 0 n<0
  123. - n -n
  124. c c
  125. n = 0 n=0
  126. S 0 ( c ) = ( c p , 3 , c ) S_{0}(c)\!\ =(c^{p},3,c)
  127. n = 1 n=1
  128. S 1 ( c ) = ( 3 , c , c 2 - 2 c p ) S_{1}(c)\!\ =(3,c,c^{2}-2c^{p})
  129. n = 2 n=2
  130. c n + 2 = c n + 1 c - c p c n + c n - 1 c_{n+2}=c_{n+1}\cdot c-c^{p}\cdot c_{n}+c_{n-1}
  131. S 1 ( c ) S_{1}(c)
  132. c 3 c_{3}
  133. S 2 ( c ) S_{2}(c)
  134. n > 2 n>2
  135. S n ( c ) S_{n}(c)
  136. S ¯ i ( c ) = S 2 i + 1 ( c ) \bar{S}_{i}(c)=S_{2i+1}(c)
  137. m ¯ = n \bar{m}=n
  138. m ¯ = n - 1 \bar{m}=n-1
  139. m ¯ = 2 m + 1 , k = 1 \bar{m}=2m+1,k=1
  140. S ¯ k ( c ) = S 3 ( c ) \bar{S}_{k}(c)=S_{3}(c)
  141. S 2 ( c ) S_{2}(c)
  142. m = j = 0 r m j 2 j m=\sum_{j=0}^{r}m_{j}2^{j}
  143. m j 0 , 1 m_{j}\in{0,1}
  144. m r = 1 m_{r}=1
  145. j = r - 1 , r - 2 , , 0 j=r-1,r-2,...,0
  146. m j = 0 m_{j}=0
  147. S ¯ k ( c ) \bar{S}_{k}(c)
  148. S ¯ 2 k ( c ) \bar{S}_{2k}(c)
  149. m j = 1 m_{j}=1
  150. S ¯ k ( c ) \bar{S}_{k}(c)
  151. S ¯ 2 k + 1 ( c ) \bar{S}_{2k+1}(c)
  152. k k
  153. 2 k + m j 2k+m_{j}
  154. k = m k=m
  155. S m ¯ ( c ) = S ¯ m ( c ) S_{\bar{m}}(c)=\bar{S}_{m}(c)
  156. S m ¯ ( c ) S_{\bar{m}}(c)
  157. S ¯ m + 1 ( c ) \bar{S}_{m+1}(c)
  158. p p
  159. q q
  160. p p
  161. G F ( p 6 ) GF(p^{6})
  162. p 2 mod 3 p\equiv 2\ \,\text{mod}\ 3
  163. q q
  164. q q
  165. p 2 - p + 1 p^{2}-p+1
  166. P P
  167. Q Q
  168. p p
  169. q q
  170. 6 P 6P
  171. P 170 P\approx 170
  172. Q Q
  173. p p
  174. q q
  175. r r\in\mathbb{Z}
  176. q = r 2 - r + 1 q=r^{2}-r+1
  177. Q Q
  178. k k\in\mathbb{Z}
  179. p = r + k q p=r+k\cdot q
  180. P P
  181. p 2 mod 3 p\equiv 2\ \,\text{mod}\ 3
  182. q p 2 - p + 1 q\mid p^{2}-p+1
  183. p p
  184. q q
  185. p 2 - p + 1 = r 2 + 2 r k q + k 2 q 2 - r - k q + 1 = r 2 - r + 1 + q ( 2 r k + k 2 q - k ) = q ( 1 + 2 r k + k 2 q - k ) p^{2}-p+1=r^{2}+2rkq+k^{2}q^{2}-r-kq+1=r^{2}-r+1+q(2rk+k^{2}q-k)=q(1+2rk+k^{2}% q-k)
  186. q p 2 - p + 1 q\mid p^{2}-p+1
  187. p p
  188. p p
  189. G F ( p ) GF(p)
  190. k k
  191. k = 1 k=1
  192. r r
  193. r 2 - r + 1 and r 2 + 1 r^{2}-r+1\,\text{ and }r^{2}+1
  194. r 2 + 1 2 mod 3 r^{2}+1\equiv 2\,\text{ mod }3
  195. p p
  196. r r
  197. r 1 mod 4 r\equiv 1\,\text{ mod }4
  198. p p
  199. p p
  200. Q Q
  201. q q
  202. q 7 mod 12 q\equiv 7\ \,\text{mod}\ 12
  203. r 1 r_{1}
  204. r 2 r_{2}
  205. X 2 - X + 1 mod q X^{2}-X+1\ \,\text{mod}\ q
  206. k k\in\mathbb{Z}
  207. p = r i + k q p=r_{i}+k\cdot q
  208. P P
  209. p 2 mod 3 p\equiv 2\ \,\text{mod}\ 3
  210. i { 1 , 2 } i\in\{1,2\}
  211. q 7 mod 12 q\equiv 7\ \,\text{mod}\ 12
  212. q 1 mod 3 q\equiv 1\ \,\text{mod}\ 3
  213. 7 1 mod 3 7\equiv 1\ \,\text{mod}\ 3
  214. 3 12 3\mid 12
  215. r 1 r_{1}
  216. r 2 r_{2}
  217. q p 2 - p + 1 q\mid p^{2}-p+1
  218. p 2 - p + 1 p^{2}-p+1
  219. r i { 1 , 2 } r_{i}\in\{1,2\}
  220. p 2 - p + 1 = r i 2 + 2 r i k q + k 2 q 2 - r i - k q + 1 = r i 2 - r i + 1 + q ( 2 r k + k 2 q - k ) = q ( 2 r k + k 2 q - k ) p^{2}-p+1=r_{i}^{2}+2r_{i}kq+k^{2}q^{2}-r_{i}-kq+1=r_{i}^{2}-r_{i}+1+q(2rk+k^{% 2}q-k)=q(2rk+k^{2}q-k)
  221. r 1 r_{1}
  222. r 2 r_{2}
  223. X 2 - X + 1 X^{2}-X+1
  224. q p 2 - p + 1 q\mid p^{2}-p+1
  225. p p
  226. q q
  227. G F ( p 6 ) GF(p^{6})
  228. G F ( p 6 ) * GF(p^{6})^{*}
  229. g \langle g\rangle
  230. G F ( p 6 ) * GF\!\ (p^{6})^{*}
  231. g G F ( p 6 ) g\in GF(p^{6})
  232. g = q \mid\!\!\langle g\rangle\!\!\mid=q
  233. g G F ( p 6 ) g\in GF(p^{6})
  234. c G F ( p 2 ) c\in GF(p^{2})
  235. c = T r ( g ) c=Tr(g)
  236. g G F ( p 6 ) g\in GF(p^{6})
  237. q q
  238. T r ( g ) Tr(g)
  239. g g
  240. F ( T r ( g ) , X ) F(Tr(g),\ X)
  241. c c
  242. F ( c , X ) F(c,\ X)
  243. F ( c , X ) F(c,\ X)
  244. p 2 - p + 1 p^{2}-p+1
  245. F ( c , X ) F(c,\ X)
  246. c c
  247. q q
  248. c G F ( p 2 ) c\in GF(p^{2})
  249. F ( c , X ) F(c,\ X)
  250. c G F ( p 2 ) \ G F ( p ) c\in GF(p^{2})\backslash GF(p)
  251. c G F ( p 2 ) c\in GF(p^{2})
  252. F ( c , X ) = X 3 - c X 2 + c p X - 1 G F ( p 2 ) [ X ] F(c,\ X)=X^{3}-cX^{2}+c^{p}X-1\in GF(p^{2})[X]
  253. T r ( g ) Tr(g)
  254. c G F ( p 2 ) \ G F ( p ) c\in GF(p^{2})\backslash GF(p)
  255. F ( c , X ) F(c,\ X)
  256. d = c ( p 2 - p + 1 ) / q d=c_{(p^{2}-p+1)/q}
  257. d d
  258. q q
  259. T r ( g ) = d Tr(g)=d
  260. G F ( p 2 ) GF(p^{2})
  261. T r ( g ) Tr(g)
  262. g G F ( p 6 ) g\in GF(p^{6})
  263. q q
  264. ( p , q , T r ( g ) ) \left(p,q,Tr(g)\right)
  265. K K
  266. a a\in\mathbb{Z}
  267. 1 < a < q - 2 1<a<q-2
  268. S a ( T r ( g ) ) = ( T r ( g a - 1 ) , T r ( g a ) , T r ( g a + 1 ) ) G F ( p 2 ) 3 S_{a}(Tr(g))=\left(Tr(g^{a-1}),Tr(g^{a}),Tr(g^{a+1})\right)\in GF(p^{2})^{3}
  269. T r ( g a ) G F ( p 2 ) Tr(g^{a})\in GF(p^{2})
  270. T r ( g a ) Tr(g^{a})
  271. b b\in\mathbb{Z}
  272. 1 < b < q - 2 1<b<q-2
  273. S b ( T r ( g ) ) = ( T r ( g b - 1 ) , T r ( g b ) , T r ( g b + 1 ) ) G F ( p 2 ) 3 S_{b}(Tr(g))=\left(Tr(g^{b-1}),Tr(g^{b}),Tr(g^{b+1})\right)\in GF(p^{2})^{3}
  274. T r ( g b ) G F ( p 2 ) Tr(g^{b})\in GF(p^{2})
  275. T r ( g b ) Tr(g^{b})
  276. S a ( T r ( g b ) ) = ( T r ( g ( a - 1 ) b ) , T r ( g a b ) , T r ( g ( a + 1 ) b ) ) G F ( p 2 ) 3 S_{a}(Tr(g^{b}))=\left(Tr(g^{(a-1)b}),Tr(g^{ab}),Tr(g^{(a+1)b})\right)\in GF(p% ^{2})^{3}
  277. K K
  278. T r ( g a b ) G F ( p 2 ) Tr(g^{ab})\in GF(p^{2})
  279. S b ( T r ( g a ) ) = ( T r ( g a ( b - 1 ) ) , T r ( g a b ) , T r ( g a ( b + 1 ) ) ) G F ( p 2 ) 3 S_{b}(Tr(g^{a}))=\left(Tr(g^{a(b-1)}),Tr(g^{ab}),Tr(g^{a(b+1)})\right)\in GF(p% ^{2})^{3}
  280. K K
  281. T r ( g a b ) G F ( p 2 ) Tr(g^{ab})\in GF(p^{2})
  282. ( p , q , T r ( g ) ) (p,q,Tr(g))
  283. k k
  284. T r ( g k ) Tr(g^{k})
  285. ( p , q , T r ( g ) , T r ( g k ) ) \left(p,q,Tr(g),Tr(g^{k})\right)
  286. M M
  287. b b\in\mathbb{Z}
  288. 1 < b < q - 2 1<b<q-2
  289. S b ( T r ( g ) ) = ( T r ( g b - 1 ) , T r ( g b ) , T r ( g b + 1 ) ) G F ( p 2 ) 3 S_{b}(Tr(g))=\left(Tr(g^{b-1}),Tr(g^{b}),Tr(g^{b+1})\right)\in GF(p^{2})^{3}
  290. S b ( T r ( g k ) ) = ( T r ( g ( b - 1 ) k ) , T r ( g b k ) , T r ( g ( b + 1 ) k ) ) G F ( p 2 ) 3 S_{b}(Tr(g^{k}))=\left(Tr(g^{(b-1)k}),Tr(g^{bk}),Tr(g^{(b+1)k})\right)\in GF(p% ^{2})^{3}
  291. K K
  292. T r ( g b k ) G F ( p 2 ) Tr(g^{bk})\in GF(p^{2})
  293. K K
  294. M M
  295. E E
  296. ( T r ( g b ) , E ) (Tr(g^{b}),\ E)
  297. ( T r ( g b ) , E ) (Tr(g^{b}),\ E)
  298. S k ( T r ( g b ) ) = ( T r ( g b ( k - 1 ) ) , T r ( g b k ) , T r ( g b ( k + 1 ) ) ) G F ( p 2 ) 3 S_{k}(Tr(g^{b}))=\left(Tr(g^{b(k-1)}),Tr(g^{bk}),Tr(g^{b(k+1)})\right)\in GF(p% ^{2})^{3}
  299. K K
  300. T r ( g b k ) G F ( p 2 ) Tr(g^{bk})\in GF(p^{2})
  301. K K
  302. E E
  303. M M
  304. K K
  305. G F ( p 2 ) GF(p^{2})
  306. T r ( g ) G F ( p 2 ) p G F ( p 6 ) * Tr(g)\in GF(p^{2})\ \forall p\in GF(p^{6})^{*}
  307. G F ( p 2 ) GF(p^{2})
  308. M M\!\ ^{\prime}
  309. M M
  310. G F ( p 2 ) GF(p^{2})
  311. E E
  312. E = K M G F ( p 2 ) E=K\cdot M\in GF(p^{2})
  313. E E
  314. M M
  315. M = E K - 1 M=E\cdot K^{-1}
  316. K - 1 K^{-1}
  317. K K
  318. G F ( p 2 ) GF(p^{2})
  319. G F ( p t ) GF\left(p^{t}\right)
  320. γ \langle\gamma\rangle
  321. ω \omega
  322. γ \langle\gamma\rangle
  323. γ x y given γ , γ x and γ y \gamma^{xy}\,\text{ given }\gamma,\gamma^{x}\,\text{ and }\gamma^{y}
  324. D H ( γ x , γ y ) = γ x y DH(\gamma^{x},\ \gamma^{y})=\gamma^{xy}
  325. c = D H ( a , b ) c=DH(a,b)
  326. a , b , c γ a,b,c\in\langle\gamma\rangle
  327. x = D L ( a ) x=DL(a)
  328. a = γ x γ with 0 x < ω a=\gamma^{x}\in\langle\gamma\rangle\,\text{ with }0\leq x<\omega
  329. γ \langle\gamma\rangle
  330. ω \omega
  331. γ \langle\gamma\rangle
  332. γ \langle\gamma\rangle
  333. ω \omega
  334. γ \langle\gamma\rangle
  335. ω \omega
  336. G F ( p t ) * GF\left(p^{t}\right)^{*}
  337. G F ( p t ) GF(p^{t})
  338. G F ( p ) GF(p)
  339. t t
  340. 𝒪 ( ω ) \mathcal{O}(\sqrt{\omega})
  341. γ \langle\gamma\rangle
  342. γ \langle\gamma\rangle
  343. γ \langle\gamma\rangle
  344. ω \omega
  345. G F ( p t ) GF\left(p^{t}\right)
  346. γ \langle\gamma\rangle
  347. ω \omega
  348. γ \langle\gamma\rangle
  349. G F ( p t ) GF\left(p^{t}\right)
  350. p p
  351. q q
  352. g \langle g\rangle
  353. G F ( p 6 ) GF(p^{6})
  354. q p 2 - p + 1 q\mid p^{2}-p+1
  355. p 2 - p + 1 p^{2}-p+1
  356. G F ( p 6 ) * = p 6 - 1 \mid\!GF(p^{6})^{*}\!\mid=p^{6}-1
  357. p s - 1 for s { 1 , 2 , 3 } p^{s}-1\,\text{ for }s\in\{1,2,3\}
  358. g \langle g\rangle
  359. G F ( p s ) * GF\!\ (p^{s})^{*}
  360. s { 1 , 2 , 3 } s\in\{1,2,3\}
  361. G F ( p 6 ) GF(p^{6})
  362. T r ( g x y ) Tr(g^{xy})
  363. T r ( g x ) Tr(g^{x})
  364. T r ( g y ) Tr(g^{y})
  365. X D H ( g x , g y ) = g x y XDH(g^{x},\ g^{y})=g^{xy}
  366. X D H ( a , b ) = c XDH(a,b)=c
  367. a , b , c T r ( g ) a,b,c\in Tr(\langle g\rangle)
  368. a T r ( g ) a\in Tr(\langle g\rangle)
  369. x = X D L ( a ) x=XDL(a)
  370. 0 x < q 0\leq x<q
  371. a = T r ( g x ) a=Tr(g^{x})
  372. 𝒜 \mathcal{A}
  373. \mathcal{B}
  374. 𝒜 \mathcal{A}
  375. \mathcal{B}
  376. \mathcal{B}
  377. 𝒜 \mathcal{A}
  378. g \langle g\rangle
  379. g \langle g\rangle
  380. g \langle g\rangle
  381. G F ( p 2 ) GF(p^{2})
  382. G F ( p 6 ) GF(p^{6})
  383. g \langle g\rangle

X–Y–Z_matrix.html

  1. A A
  2. A i , j , k A_{i,j,k}
  3. 1 i M , 1 j N , 1 k P 1\leq i\leq M,\ 1\leq j\leq N,\ 1\leq k\leq P
  4. M , N , P . M,N,P.

Yang–Baxter_equation.html

  1. R R
  2. ( R 𝟏 ) ( 𝟏 R ) ( R 𝟏 ) = ( 𝟏 R ) ( R 𝟏 ) ( 𝟏 R ) (R\otimes\mathbf{1})(\mathbf{1}\otimes R)(R\otimes\mathbf{1})=(\mathbf{1}% \otimes R)(R\otimes\mathbf{1})(\mathbf{1}\otimes R)
  3. R R
  4. R R
  5. A A
  6. R ( u ) R(u)
  7. A A A\otimes A
  8. u u
  9. R 12 ( u ) R 13 ( u + v ) R 23 ( v ) = R 23 ( v ) R 13 ( u + v ) R 12 ( u ) , R_{12}(u)\ R_{13}(u+v)\ R_{23}(v)=R_{23}(v)\ R_{13}(u+v)\ R_{12}(u),
  10. u u
  11. v v
  12. R ( u ) R(u)
  13. R 12 ( u ) R 13 ( u v ) R 23 ( v ) = R 23 ( v ) R 13 ( u v ) R 12 ( u ) , R_{12}(u)\ R_{13}(uv)\ R_{23}(v)=R_{23}(v)\ R_{13}(uv)\ R_{12}(u),
  14. u u
  15. v v
  16. R 12 ( w ) = ϕ 12 ( R ( w ) ) R_{12}(w)=\phi_{12}(R(w))
  17. R 13 ( w ) = ϕ 13 ( R ( w ) ) R_{13}(w)=\phi_{13}(R(w))
  18. R 23 ( w ) = ϕ 23 ( R ( w ) ) R_{23}(w)=\phi_{23}(R(w))
  19. w w
  20. ϕ 12 : A A A A A \phi_{12}:A\otimes A\to A\otimes A\otimes A
  21. ϕ 13 : A A A A A \phi_{13}:A\otimes A\to A\otimes A\otimes A
  22. ϕ 23 : A A A A A \phi_{23}:A\otimes A\to A\otimes A\otimes A
  23. ϕ 12 ( a b ) = a b 1 , \phi_{12}(a\otimes b)=a\otimes b\otimes 1,
  24. ϕ 13 ( a b ) = a 1 b , \phi_{13}(a\otimes b)=a\otimes 1\otimes b,
  25. ϕ 23 ( a b ) = 1 a b . \phi_{23}(a\otimes b)=1\otimes a\otimes b.
  26. R ( u ) R(u)
  27. u = u 0 u=u_{0}
  28. R R
  29. u = u 0 u=u_{0}
  30. A A
  31. R R
  32. A A A\otimes A
  33. R 12 R 13 R 23 = R 23 R 13 R 12 , R_{12}\ R_{13}\ R_{23}=R_{23}\ R_{13}\ R_{12},
  34. R 12 = ϕ 12 ( R ) R_{12}=\phi_{12}(R)
  35. R 13 = ϕ 13 ( R ) R_{13}=\phi_{13}(R)
  36. R 23 = ϕ 23 ( R ) R_{23}=\phi_{23}(R)
  37. V V
  38. A A
  39. T : V V V V T:V\otimes V\to V\otimes V
  40. T ( x y ) = y x T(x\otimes y)=y\otimes x
  41. x , y V x,y\in V
  42. B n B_{n}
  43. V n V^{\otimes n}
  44. σ i = 1 i - 1 R ˇ 1 n - i - 1 \sigma_{i}=1^{\otimes i-1}\otimes\check{R}\otimes 1^{\otimes n-i-1}
  45. i = 1 , , n - 1 i=1,\dots,n-1
  46. R ˇ = T R \check{R}=T\circ R
  47. V V V\otimes V

Yao's_principle.html

  1. 𝒳 \mathcal{X}
  2. 𝒜 \mathcal{A}
  3. a 𝒜 a\in\mathcal{A}
  4. x 𝒳 x\in\mathcal{X}
  5. c ( a , x ) 0 c(a,x)\geq 0
  6. a a
  7. x x
  8. p p
  9. 𝒜 \mathcal{A}
  10. A A
  11. p p
  12. q q
  13. 𝒳 \mathcal{X}
  14. X X
  15. q q
  16. max x 𝒳 E [ c ( A , x ) ] min a 𝒜 E [ c ( a , X ) ] \underset{x\in\mathcal{X}}{\max}\ {E}[c(A,x)]\geq\underset{a\in\mathcal{A}}{% \min}\ {E}[c(a,X)]
  17. q q
  18. C = max x 𝒳 E [ c ( A , x ) ] C=\underset{x\in\mathcal{X}}{\max}\ {E}[c(A,x)]
  19. x x
  20. a p a c ( a , x ) C \sum_{a}p_{a}c(a,x)\leq C
  21. x q x a p a c ( a , x ) C \sum_{x}q_{x}\sum_{a}p_{a}c(a,x)\leq C
  22. a p a x q x c ( a , x ) C \sum_{a}p_{a}\sum_{x}q_{x}c(a,x)\leq C
  23. a a
  24. x q x c ( a , x ) C \sum_{x}q_{x}c(a,x)\leq C

Yegor_Ivanovich_Zolotarev.html

  1. x + A x 4 + a x 3 + b x 2 + c x + d d x \int\frac{x+A}{\sqrt{x^{4}+ax^{3}+bx^{2}+cx+d}}\,dx

Yield_(engineering).html

  1. σ 1 , σ 2 , σ 3 \sigma_{1},\sigma_{2},\sigma_{3}
  2. σ 1 \sigma_{1}\,\!
  3. σ 2 \sigma_{2}\,\!
  4. σ 3 \sigma_{3}\,\!
  5. σ 1 σ y \ \sigma_{1}\leq\sigma_{y}\,\!
  6. σ 1 - ν ( σ 2 + σ 3 ) σ y . \ \sigma_{1}-\nu(\sigma_{2}+\sigma_{3})\leq\sigma_{y}.\,\!
  7. τ \tau\!
  8. τ y \tau_{y}\!
  9. τ = σ 1 - σ 3 2 τ y . \ \tau=\frac{\sigma_{1}-\sigma_{3}}{2}\leq\tau_{y}.\,\!
  10. σ 1 2 + σ 2 2 + σ 3 2 - 2 ν ( σ 1 σ 2 + σ 2 σ 3 + σ 1 σ 3 ) σ y 2 . \ \sigma_{1}^{2}+\sigma_{2}^{2}+\sigma_{3}^{2}-2\nu(\sigma_{1}\sigma_{2}+% \sigma_{2}\sigma_{3}+\sigma_{1}\sigma_{3})\leq\sigma_{y}^{2}.\,\!
  11. σ y = C ( ϵ ˙ ) m \sigma_{y}=C(\dot{\epsilon})^{m}\,\!
  12. m = l n σ ( ϵ ) l n ( ϵ ˙ ) m=\frac{\partial ln\sigma(\epsilon)}{\partial ln(\dot{\epsilon})}
  13. σ y = 1 α sinh - 1 [ Z A ] ( 1 / n ) \sigma_{y}=\frac{1}{\alpha}\sinh^{-1}\left[\frac{Z}{A}\right]^{(1/n)}\,\!
  14. Z = ( ϵ ˙ ) exp ( Q H W R T ) Z=(\dot{\epsilon})\exp\left(\frac{Q_{HW}}{RT}\right)\,\!
  15. Δ σ y = G b ρ \Delta\sigma_{y}=Gb\sqrt{\rho}
  16. σ y \sigma_{y}
  17. ρ \rho
  18. Δ τ = G b C s ϵ 3 / 2 \Delta\tau=Gb\sqrt{C_{s}}\epsilon^{3/2}
  19. τ \tau
  20. G G
  21. b b
  22. C s C_{s}
  23. ϵ \epsilon
  24. Δ τ = r particle l interparticle γ particle - matrix \Delta\tau=\cfrac{r_{\rm{particle}}}{l_{\rm{interparticle}}}\gamma_{\rm{% particle-matrix}}
  25. Δ τ = G b l interparticle - 2 r particle \Delta\tau=\cfrac{Gb}{l_{\rm{interparticle}}-2r_{\rm{particle}}}
  26. r particle r_{\rm{particle}}\,
  27. γ particle - matrix \gamma_{\rm{particle-matrix}}\,
  28. l interparticle l_{\rm{interparticle}}\,
  29. σ y = σ 0 + k d - 1 / 2 \sigma_{y}=\sigma_{0}+kd^{-1/2}\,
  30. σ 0 \sigma_{0}

Z-factor.html

  1. μ \mu
  2. σ \sigma
  3. μ p \mu_{p}
  4. σ p \sigma_{p}
  5. μ n \mu_{n}
  6. σ n \sigma_{n}
  7. Z-factor = 1 - 3 ( σ p + σ n ) | μ p - μ n | . \,\text{Z-factor}=1-{3(\sigma_{p}+\sigma_{n})\over|\mu_{p}-\mu_{n}|}.
  8. Estimated Z-factor = 1 - 3 ( σ ^ p + σ ^ n ) | μ ^ p - μ ^ n | . \,\text{Estimated Z-factor}=1-{3(\hat{\sigma}_{p}+\hat{\sigma}_{n})\over|\hat{% \mu}_{p}-\hat{\mu}_{n}|}.
  9. σ p = σ n \sigma_{p}=\sigma_{n}
  10. μ p \mu_{p}
  11. μ n \mu_{n}

Z-spread.html

  1. P \displaystyle P

Z_(disambiguation).html

  1. \mathbb{Z}

Zar_Points.html

  1. ( a + b ) + ( a - d ) - 8 2 = ( a - 4 ) + 0.5 ( b - d ) \frac{(a+b)+(a-d)-8}{2}=(a-4)+0.5(b-d)

Zarankiewicz_problem.html

  1. z ( m , n ; s , t ) < ( s - 1 ) 1 / t ( n - t + 1 ) m 1 - 1 / t + ( t - 1 ) m . z(m,n;s,t)<(s-1)^{1/t}(n-t+1)m^{1-1/t}+(t-1)m.
  2. z ( n , t ) < ( t - 1 ) 1 / t n 2 - 1 / t + 1 2 ( t - 1 ) n . z(n,t)<(t-1)^{1/t}n^{2-1/t}+\frac{1}{2}(t-1)n.
  3. z ( m , n ; s , t ) = O ( n m 1 - 1 / t + m ) z(m,n;s,t)=O(nm^{1-1/t}+m)
  4. z ( n ; t ) = O ( n 2 - 1 / t ) . z(n;t)=O(n^{2-1/t}).
  5. z ( n ; 2 ) = n 3 / 2 ( 1 + o ( 1 ) ) . z(n;2)=n^{3/2}(1+o(1)).
  6. z ( n , n ; 2 , t ) = n 3 / 2 t 1 / 2 ( 1 + o ( 1 ) ) . z(n,n;2,t)=n^{3/2}t^{1/2}(1+o(1)).
  7. z ( n ; t ) = Θ ( n 2 - 1 / t ) z(n;t)=\Theta(n^{2-1/t})
  8. z ( n , n ; s , t ) = Θ ( n 2 - 1 / t ) , z(n,n;s,t)=\Theta(n^{2-1/t}),

Zeiss_formula.html

  1. c = f 2 / ( N H ) c=f\,^{2}/(NH)
  2. c = d / 1730 c=d/1730
  3. d d
  4. c c

Zener_pinning.html

  1. F = 2 π r γ cos θ sin θ . F=2\pi\ r\gamma\ \cos\theta\ \sin\theta.\ \,\!
  2. N t o t a l = 3 F v 4 π r 3 . N_{total}=\frac{3F_{v}}{4\pi\ r^{3}}.\,\!
  3. N i n t e r a c t = 2 r N t o t a l = 3 F v 2 π r 2 . N_{interact}=2rN_{total}=\frac{3F_{v}}{2\pi\ r^{2}}.\,\!
  4. P s = N i n t e r a c t F m a x = 3 F v γ 2 r . P_{s}=N_{interact}F_{max}=\frac{3F_{v}\gamma\ }{2r}.\,\!

Zero-crossing_rate.html

  1. z c r = 1 T - 1 t = 1 T - 1 𝕀 { s t s t - 1 < 0 } zcr=\frac{1}{T-1}\sum_{t=1}^{T-1}{{\mathbb{I}}\left\{{s_{t}s_{t-1}<0}\right\}}
  2. s s
  3. T T
  4. 𝕀 { A } {{\mathbb{I}}\left\{{A}\right\}}
  5. A A

Zero-lift_drag_coefficient.html

  1. C D , 0 C_{D,0}
  2. C D , 0 = C D - C D , i C_{D,0}=C_{D}-C_{D,i}
  3. C D C_{D}
  4. C D , i C_{D,i}
  5. C D , 0 C_{D,0}
  6. f f
  7. C D , 0 × S C_{D,0}\times S
  8. S S
  9. V m a x p o w e r / f 3 V_{max}\ \propto\ \sqrt[3]{power/f}
  10. C D , 0 = C D - C D , i C_{D,0}=C_{D}-C_{D,i}
  11. C D = 550 η P 1 2 ρ 0 [ σ S ( 1.47 V ) 3 ] C_{D}=\frac{550\eta P}{\frac{1}{2}\rho_{0}[\sigma S(1.47V)^{3}]}
  12. η \eta
  13. ρ 0 \rho_{0}
  14. σ \sigma
  15. ρ 0 \rho_{0}
  16. C D = 1.456 × 10 5 ( η P σ S V 3 ) C_{D}=1.456\times 10^{5}(\frac{\eta P}{\sigma SV^{3}})
  17. C D , i = C L 2 π A ϵ C_{D,i}=\frac{C_{L}^{2}}{\pi A\epsilon}
  18. C L C_{L}
  19. ϵ \epsilon
  20. C L C_{L}
  21. C D , i = 4.822 × 10 4 A ϵ σ 2 V 4 ( W / S ) 2 C_{D,i}=\frac{4.822\times 10^{4}}{A\epsilon\sigma^{2}V^{4}}(W/S)^{2}

Zero-phonon_line_and_phonon_sideband.html

  1. Ω i \hbar\Omega_{i}
  2. Ω m \hbar\Omega_{m}

Zero-product_property.html

  1. a b = 0 ab=0
  2. a = 0 a=0
  3. b = 0 b=0
  4. \mathbb{Z}
  5. \mathbb{Q}
  6. \mathbb{R}
  7. \mathbb{C}
  8. A A
  9. A A
  10. A A
  11. A A
  12. { 0 , 1 , 2 , } \{0,1,2,\ldots\}
  13. A A
  14. B B
  15. A A
  16. B B
  17. a a
  18. b b
  19. B B
  20. a b = 0 ab=0
  21. a = 0 a=0
  22. b = 0 b=0
  23. a a
  24. b b
  25. A A
  26. p p
  27. p p
  28. { 0 , 1 , 2 , } \{0,1,2,\ldots\}
  29. n \mathbb{Z}_{n}
  30. n n
  31. 6 \mathbb{Z}_{6}
  32. 2 3 0 ( mod 6 ) 2\cdot 3\equiv 0\;\;(\mathop{{\rm mod}}6)
  33. n n
  34. n \mathbb{Z}_{n}
  35. n = q m n=qm
  36. 0 < q , m < n 0<q,m<n
  37. m m
  38. q q
  39. n n
  40. q m 0 ( mod n ) qm\equiv 0\;\;(\mathop{{\rm mod}}n)
  41. 2 × 2 \mathbb{Z}^{2\times 2}
  42. M = ( 1 - 1 0 0 ) M=\begin{pmatrix}1&-1\\ 0&0\end{pmatrix}
  43. N = ( 0 1 0 1 ) N=\begin{pmatrix}0&1\\ 0&1\end{pmatrix}
  44. M N = ( 1 - 1 0 0 ) ( 0 1 0 1 ) = ( 0 0 0 0 ) = 0 MN=\begin{pmatrix}1&-1\\ 0&0\end{pmatrix}\begin{pmatrix}0&1\\ 0&1\end{pmatrix}=\begin{pmatrix}0&0\\ 0&0\end{pmatrix}=0
  45. M M
  46. N N
  47. f : [ 0 , 1 ] f:[0,1]\to\mathbb{R}
  48. f 1 , , f n f_{1},\ldots,f_{n}
  49. f i f j f_{i}\,f_{j}
  50. i j i\neq j
  51. P P
  52. Q Q
  53. x x
  54. P ( x ) Q ( x ) = 0 P(x)Q(x)=0
  55. x x
  56. P ( x ) = 0 P(x)=0
  57. Q ( x ) = 0 Q(x)=0
  58. P Q PQ
  59. P P
  60. Q Q
  61. x 3 - 2 x 2 - 5 x + 6 x^{3}-2x^{2}-5x+6
  62. ( x - 3 ) ( x - 1 ) ( x + 2 ) (x-3)(x-1)(x+2)
  63. R R
  64. f f
  65. d 1 d\geq 1
  66. R R
  67. f f
  68. d d
  69. r 1 , , r d R r_{1},\ldots,r_{d}\in R
  70. f f
  71. f ( x ) = ( x - r 1 ) ( x - r d ) f(x)=(x-r_{1})\cdots(x-r_{d})
  72. r 1 , , r d r_{1},\ldots,r_{d}
  73. f f
  74. f f
  75. ( x - r i ) (x-r_{i})
  76. i i
  77. f f
  78. d d
  79. R R
  80. x 3 + 3 x 2 + 2 x x^{3}+3x^{2}+2x
  81. 6 \mathbb{Z}_{6}
  82. \mathbb{Z}

Zero_differential_overlap.html

  1. 𝚽 i \mathbf{\Phi}_{i}
  2. χ μ A \mathbf{\chi}_{\mu}^{A}
  3. 𝚽 i = μ = 1 N 𝐂 i μ χ μ A \mathbf{\Phi}_{i}\ =\sum_{\mu=1}^{N}\mathbf{C}_{i\mu}\ \mathbf{\chi}_{\mu}^{A}\,
  4. 𝐂 i μ \mathbf{C}_{i\mu}
  5. μ ν | λ σ = χ μ A ( 1 ) χ ν B ( 1 ) 1 r 12 χ λ C ( 2 ) χ σ D ( 2 ) d τ 1 d τ 2 \langle\mu\nu|\lambda\sigma\rangle=\iint\mathbf{\chi}_{\mu}^{A}(1)\mathbf{\chi% }_{\nu}^{B}(1)\frac{1}{r_{12}}\mathbf{\chi}_{\lambda}^{C}(2)\mathbf{\chi}_{% \sigma}^{D}(2)d\tau_{1}\,d\tau_{2}
  6. χ μ A ( 1 ) χ ν B ( 1 ) \mathbf{\chi}_{\mu}^{A}(1)\mathbf{\chi}_{\nu}^{B}(1)
  7. μ ν | λ σ = δ μ ν δ λ σ μ μ | λ λ \langle\mu\nu|\lambda\sigma\rangle=\delta_{\mu\nu}\delta_{\lambda\sigma}% \langle\mu\mu|\lambda\lambda\rangle
  8. δ μ ν = { 0 μ ν 1 μ = ν \delta_{\mu\nu}=\begin{cases}0&\mu\neq\nu\\ 1&\mu=\nu\end{cases}
  9. μ μ | λ λ \langle\mu\mu|\lambda\lambda\rangle

Zero_element.html

  1. R R
  2. { 0 } \{0\}
  3. 0 1 , 1 = [ 0 ] , 0 2 , 2 = [ 0 0 0 0 ] , 0 2 , 3 = [ 0 0 0 0 0 0 ] , 0_{1,1}=\begin{bmatrix}0\end{bmatrix},\ 0_{2,2}=\begin{bmatrix}0&0\\ 0&0\end{bmatrix},\ 0_{2,3}=\begin{bmatrix}0&0&0\\ 0&0&0\end{bmatrix},
  4. K m , n K_{m,n}
  5. 0 K m , n 0_{K_{m,n}}
  6. K m , n K_{m,n}
  7. 0 K 0_{K}
  8. 0 K 0_{K}
  9. 0 K m , n = [ 0 K 0 K 0 K 0 K 0 K 0 K 0 K 0 K 0 K ] 0_{K_{m,n}}=\begin{bmatrix}0_{K}&0_{K}&\cdots&0_{K}\\ 0_{K}&0_{K}&\cdots&0_{K}\\ \vdots&\vdots&&\vdots\\ 0_{K}&0_{K}&\cdots&0_{K}\end{bmatrix}
  10. K m , n K_{m,n}
  11. A K m , n A\in K_{m,n}
  12. 0 K m , n + A = A + 0 K m , n = A 0_{K_{m,n}}+A=A+0_{K_{m,n}}=A
  13. ( M , 0 , + ) (M,0,+)
  14. ( a , b M ) a + b = 0 a = 0 = b (\forall a,b\in M)\ a+b=0\implies a=0=b\!
  15. 0 + 0 0+0

Zimm–Bragg_model.html

  1. θ \theta
  2. θ = i N \theta=\frac{\left\langle i\right\rangle}{N}
  3. i \left\langle i\right\rangle
  4. N N
  5. C C ...CC...
  6. 1 1
  7. C H ...CH...
  8. σ s \sigma s
  9. H C ...HC...
  10. σ s \sigma s
  11. H H ...HH...
  12. σ s 2 \sigma s^{2}
  13. σ s \sigma s
  14. σ \sigma
  15. s = [ H ] [ C ] s=\frac{[H]}{[C]}
  16. s s
  17. σ 1 < s \sigma\ll 1<s
  18. θ \theta
  19. i \left\langle i\right\rangle
  20. i = ( s q ) d q d s \left\langle i\right\rangle=\left(\frac{s}{q}\right)\frac{dq}{ds}
  21. s s
  22. q q
  23. θ = 1 N ( s q ) d q d s \theta=\frac{1}{N}\left(\frac{s}{q}\right)\frac{dq}{ds}
  24. 𝒵 = ( 0 , 1 ) { j = 1 N 𝐖 j } ( 1 , 1 ) \mathcal{Z}=\left(0,1\right)\cdot\left\{\prod_{j=1}^{N}\mathbf{W}_{j}\right\}% \cdot\left(1,1\right)
  25. 𝐖 j = [ s j 1 σ j s j 1 ] \mathbf{W}_{j}=\begin{bmatrix}s_{j}&1\\ \sigma_{j}s_{j}&1\end{bmatrix}

Zinc–bromine_battery.html

  1. Z n ( s ) Z n ( a q ) 2 + + 2 e - Zn_{(s)}\leftrightarrow Zn^{2+}_{(aq)}+2e^{-}
  2. B r 2 ( a q ) + 2 e - 2 B r ( a q ) - Br_{2(aq)}+2e^{-}\leftrightarrow 2Br^{-}_{(aq)}
  3. Z n ( s ) + B r 2 ( a q ) 2 B r ( a q ) - + Z n ( a q ) 2 + Zn_{(s)}+Br_{2(aq)}\leftrightarrow 2Br^{-}_{(aq)}+Zn^{2+}_{(aq)}