wpmath0000003_16

Stretch_rule.html

  1. I z = V d 3 r ρ ( 𝐫 ) r 2 I_{z}=\int_{V}d^{3}r\,\rho(\mathbf{r})\,r^{2}
  2. r r
  3. I z = 0 L d z x , y d x d y ρ ( x , y , z ) r 2 I_{z}=\int_{0}^{L}dz\int_{x,y}dxdy\,\rho(x,y,z)\,r^{2}
  4. L L
  5. a a
  6. a a
  7. ρ ( x , y , z ) = ρ ( x , y , z / a ) / a \rho^{\prime}(x,y,z)=\rho(x,y,z/a)/a
  8. 0
  9. a L aL
  10. I z = 0 a L d z x , y d x d y ρ ( x , y , z ) r 2 I_{z}^{\prime}=\int_{0}^{aL}dz\int_{x,y}dxdy\,\rho^{\prime}(x,y,z)\,r^{2}
  11. = 0 L a d z x , y d x d y ρ ( x , y , z / a ) a r 2 =\int_{0}^{L}adz^{\prime}\int_{x,y}dxdy\,\frac{\rho(x,y,z/a)}{a}\,r^{2}
  12. = 0 L d z x , y d x d y ρ ( x , y , z ) r 2 = I z =\int_{0}^{L}dz^{\prime}\int_{x,y}dxdy\,\rho(x,y,z^{\prime})\,r^{2}=I_{z}

Strict_function.html

  1. f ( ) = f\left(\perp\right)=\perp
  2. \perp

Stroke_volume.html

  1. S V = E D V - E S V SV=EDV-ESV
  2. S V I = S V B S A = ( C O / H R ) B S A = C O H R × B S A SVI={SV\over BSA}={(CO/HR)\over BSA}={CO\over{HR\times BSA}}

Strong_operator_topology.html

  1. T T x T\mapsto\|Tx\|
  2. T T x T\mapsto Tx
  3. U ( T 0 , x , ϵ ) = { T : T x - T 0 x < ϵ } U(T_{0},x,\epsilon)=\{T:\|Tx-T_{0}x\|<\epsilon\}

Student's_t-test.html

  1. t = Z ( s / n ) = ( X ¯ - μ ) / ( σ / n ) ( s / n ) t=\frac{Z}{(s/\sqrt{n})}=\frac{(\bar{X}-\mu)/(\sigma/\sqrt{n})}{(s/\sqrt{n})}
  2. X ¯ \bar{X}
  3. X 1 , X 2 , , X n X_{1},X_{2},...,X_{n}
  4. n n
  5. s s
  6. σ \sigma
  7. μ \mu
  8. X X
  9. μ \mu
  10. σ 2 \sigma^{2}
  11. t = x ¯ - μ 0 s / n t=\frac{\overline{x}-\mu_{0}}{s/\sqrt{n}}
  12. x ¯ \overline{x}
  13. x ¯ \overline{x}
  14. Y = α + β x + ε , Y=\alpha+\beta x+\varepsilon,
  15. α ^ , β ^ \displaystyle\widehat{\alpha},\widehat{\beta}
  16. t score = β ^ - β 0 S E β ^ 𝒯 n - 2 t\text{score}=\frac{\widehat{\beta}-\beta_{0}}{SE_{\widehat{\beta}}}\sim% \mathcal{T}_{n-2}
  17. S E β ^ = 1 n - 2 i = 1 n ( y i - y ^ i ) 2 i = 1 n ( x i - x ¯ ) 2 SE_{\widehat{\beta}}=\frac{\sqrt{\frac{1}{n-2}\sum_{i=1}^{n}(y_{i}-\widehat{y}% _{i})^{2}}}{\sqrt{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}}
  18. ε ^ i \displaystyle\widehat{\varepsilon}_{i}
  19. t score t\text{score}
  20. t score = ( β ^ - β 0 ) n - 2 SSR / i = 1 n ( x i - x ¯ ) 2 . t\text{score}=\frac{(\widehat{\beta}-\beta_{0})\sqrt{n-2}}{\sqrt{\,\text{SSR}/% \sum_{i=1}^{n}\left(x_{i}-\overline{x}\right)^{2}}}.
  21. t = X ¯ 1 - X ¯ 2 s X 1 X 2 1 n t=\frac{\bar{X}_{1}-\bar{X}_{2}}{s_{X_{1}X_{2}}\cdot\sqrt{\frac{1}{n}}}
  22. s X 1 X 2 = ( s X 1 2 + s X 2 2 ) \ s_{X_{1}X_{2}}=\sqrt{(s_{X_{1}}^{2}+s_{X_{2}}^{2})}
  23. s X 1 X 2 s_{X_{1}X_{2}}
  24. s X 1 2 s_{X_{1}}^{2}
  25. s X 2 2 s_{X_{2}}^{2}
  26. t = X ¯ 1 - X ¯ 2 s X 1 X 2 1 n 1 + 1 n 2 t=\frac{\bar{X}_{1}-\bar{X}_{2}}{s_{X_{1}X_{2}}\cdot\sqrt{\frac{1}{n_{1}}+% \frac{1}{n_{2}}}}
  27. s X 1 X 2 = ( n 1 - 1 ) s X 1 2 + ( n 2 - 1 ) s X 2 2 n 1 + n 2 - 2 . s_{X_{1}X_{2}}=\sqrt{\frac{(n_{1}-1)s_{X_{1}}^{2}+(n_{2}-1)s_{X_{2}}^{2}}{n_{1% }+n_{2}-2}}.
  28. s X 1 X 2 s_{X_{1}X_{2}}
  29. t = X ¯ 1 - X ¯ 2 s X ¯ 1 - X ¯ 2 t={\overline{X}_{1}-\overline{X}_{2}\over s_{\overline{X}_{1}-\overline{X}_{2}}}
  30. s X ¯ 1 - X ¯ 2 = s 1 2 n 1 + s 2 2 n 2 . s_{\overline{X}_{1}-\overline{X}_{2}}=\sqrt{{s_{1}^{2}\over n_{1}}+{s_{2}^{2}% \over n_{2}}}.
  31. s X ¯ 1 - X ¯ 2 2 {s_{\overline{X}_{1}-\overline{X}_{2}}}^{2}
  32. d . f . = ( s 1 2 / n 1 + s 2 2 / n 2 ) 2 ( s 1 2 / n 1 ) 2 / ( n 1 - 1 ) + ( s 2 2 / n 2 ) 2 / ( n 2 - 1 ) . \mathrm{d.f.}=\frac{(s_{1}^{2}/n_{1}+s_{2}^{2}/n_{2})^{2}}{(s_{1}^{2}/n_{1})^{% 2}/(n_{1}-1)+(s_{2}^{2}/n_{2})^{2}/(n_{2}-1)}.
  33. t = X ¯ D - μ 0 s D / n . t=\frac{\overline{X}_{D}-\mu_{0}}{s_{D}/\sqrt{n}}.
  34. A 1 = { 30.02 , 29.99 , 30.11 , 29.97 , 30.01 , 29.99 } A_{1}=\{30.02,\ 29.99,\ 30.11,\ 29.97,\ 30.01,\ 29.99\}
  35. A 2 = { 29.89 , 29.93 , 29.72 , 29.98 , 30.02 , 29.98 } A_{2}=\{29.89,\ 29.93,\ 29.72,\ 29.98,\ 30.02,\ 29.98\}
  36. X ¯ i \overline{X}_{i}
  37. X ¯ 1 - X ¯ 2 = 0.095. \overline{X}_{1}-\overline{X}_{2}=0.095.
  38. s 1 2 n 1 + s 2 2 n 2 0.0485 \sqrt{{s_{1}^{2}\over n_{1}}+{s_{2}^{2}\over n_{2}}}\approx 0.0485
  39. df 7.03. \,\text{df}\approx 7.03.\,
  40. S X 1 X 2 0.084 S_{X_{1}X_{2}}\approx 0.084\,
  41. d f = 10. df=10.\,
  42. μ {\mathbf{\mu}}
  43. μ 0 {\mathbf{\mu}_{0}}
  44. T 2 = n ( 𝐱 ¯ - μ 0 ) 𝐒 - 1 ( 𝐱 ¯ - μ 0 ) T^{2}=n(\overline{\mathbf{x}}-{\mathbf{\mu}_{0}})^{\prime}{\mathbf{S}}^{-1}(% \overline{\mathbf{x}}-{\mathbf{\mu}_{0}})
  45. 𝐱 ¯ \overline{\mathbf{x}}
  46. 𝐒 {\mathbf{S}}
  47. m × m m\times m
  48. μ 1 {\mathbf{\mu}}_{1}
  49. μ 2 {\mathbf{\mu}}_{2}
  50. T 2 = n 1 n 2 n 1 + n 2 ( 𝐱 ¯ 1 - 𝐱 ¯ 2 ) 𝐒 pooled - 1 ( 𝐱 ¯ 1 - 𝐱 ¯ 2 ) . T^{2}=\frac{n_{1}n_{2}}{n_{1}+n_{2}}(\overline{\mathbf{x}}_{1}-\overline{% \mathbf{x}}_{2})^{\prime}{\mathbf{S}\text{pooled}}^{-1}(\overline{\mathbf{x}}_% {1}-\overline{\mathbf{x}}_{2}).

Studentized_residual.html

  1. Y = α 0 + α 1 X + ε . Y=\alpha_{0}+\alpha_{1}X+\varepsilon.\,
  2. Y i = α 0 + α 1 X i + ε i , Y_{i}=\alpha_{0}+\alpha_{1}X_{i}+\varepsilon_{i},\,
  3. ε ^ \scriptstyle\widehat{\varepsilon}
  4. ε \scriptstyle\varepsilon
  5. i = 1 n ε ^ i = 0 \sum_{i=1}^{n}\widehat{\varepsilon}_{i}=0
  6. i = 1 n ε ^ i x i = 0. \sum_{i=1}^{n}\widehat{\varepsilon}_{i}x_{i}=0.
  7. ε ^ i \scriptstyle\widehat{\varepsilon}_{i}
  8. X = [ 1 x 1 1 x n ] X=\left[\begin{matrix}1&x_{1}\\ \vdots&\vdots\\ 1&x_{n}\end{matrix}\right]
  9. H = X ( X T X ) - 1 X T . H=X(X^{T}X)^{-1}X^{T}.\,
  10. var ( ε ^ i ) = σ 2 ( 1 - h i i ) . \operatorname{var}(\widehat{\varepsilon}_{i})=\sigma^{2}(1-h_{ii}).
  11. var ( ε ^ i ) = σ 2 ( 1 - 1 n - ( x i - x ¯ ) 2 j = 1 n ( x j - x ¯ ) 2 ) . \operatorname{var}(\widehat{\varepsilon}_{i})=\sigma^{2}\left(1-\frac{1}{n}-% \frac{(x_{i}-\bar{x})^{2}}{\sum_{j=1}^{n}(x_{j}-\bar{x})^{2}}\right).
  12. t i = ε ^ i σ ^ 1 - h i i t_{i}={\widehat{\varepsilon}_{i}\over\widehat{\sigma}\sqrt{1-h_{ii}\ }}
  13. σ ^ \widehat{\sigma}
  14. σ ^ 2 = 1 n - m j = 1 n ε ^ j 2 . \widehat{\sigma}^{2}={1\over n-m}\sum_{j=1}^{n}\widehat{\varepsilon}_{j}^{\,2}.
  15. σ ^ ( i ) 2 = 1 n - m - 1 j = 1 j i n ε ^ j 2 , \widehat{\sigma}_{(i)}^{2}={1\over n-m-1}\sum_{\begin{smallmatrix}j=1\\ j\neq i\end{smallmatrix}}^{n}\widehat{\varepsilon}_{j}^{\,2},
  16. t i t_{i}
  17. σ ^ ( i ) 2 \widehat{\sigma}_{(i)}^{2}
  18. t i ( i ) t_{i(i)}
  19. t i ( i ) t_{i(i)}
  20. - \scriptstyle-\infty
  21. + \scriptstyle+\infty
  22. 0 ± ν \scriptstyle 0\,\pm\,\sqrt{\nu}
  23. t i ν t t 2 + ν - 1 t_{i}\sim\sqrt{\nu}{t\over\sqrt{t^{2}+\nu-1}}
  24. - 3 \scriptstyle-\sqrt{3}
  25. + 3 \scriptstyle+\sqrt{3}
  26. 2 , - 5 / 5 , - 5 / 5 \sqrt{2},\ -\sqrt{5}/5,\ -\sqrt{5}/5

Sturm–Liouville_theory.html

  1. d d x [ p ( x ) d y d x ] + q ( x ) y = - λ w ( x ) y , \frac{\mathrm{d}}{\mathrm{d}x}\left[p(x)\frac{\mathrm{d}y}{\mathrm{d}x}\right]% +q(x)y=-\lambda w(x)y,
  2. α 1 y ( a ) + α 2 y ( a ) = 0 ( α 1 2 + α 2 2 > 0 ) , \alpha_{1}y(a)+\alpha_{2}y^{\prime}(a)=0\qquad\qquad\qquad(\alpha_{1}^{2}+% \alpha_{2}^{2}>0),
  3. β 1 y ( b ) + β 2 y ( b ) = 0 ( β 1 2 + β 2 2 > 0 ) , \beta_{1}y(b)+\beta_{2}y^{\prime}(b)=0\qquad\qquad\qquad(\beta_{1}^{2}+\beta_{% 2}^{2}>0),
  4. λ 1 < λ 2 < λ 3 < < λ n < ; \lambda_{1}<\lambda_{2}<\lambda_{3}<\cdots<\lambda_{n}<\cdots\to\infty;
  5. a b y n ( x ) y m ( x ) w ( x ) d x = δ m n , \int_{a}^{b}y_{n}(x)y_{m}(x)w(x)\,\mathrm{d}x=\delta_{mn},
  6. x 2 y ′′ + x y + ( x 2 - ν 2 ) y = 0 x^{2}y^{\prime\prime}+xy^{\prime}+\left(x^{2}-\nu^{2}\right)y=0
  7. ( x y ) + ( x - ν 2 x ) y = 0. (xy^{\prime})^{\prime}+\left(x-\frac{\nu^{2}}{x}\right)y=0.
  8. ( 1 - x 2 ) y ′′ - 2 x y + ν ( ν + 1 ) y = 0 (1-x^{2})y^{\prime\prime}-2xy^{\prime}+\nu(\nu+1)y=0
  9. [ ( 1 - x 2 ) y ] + ν ( ν + 1 ) y = 0 [(1-x^{2})y^{\prime}]^{\prime}+\nu(\nu+1)y=0
  10. x 3 y ′′ - x y + 2 y = 0. x^{3}y^{\prime\prime}-xy^{\prime}+2y=0.
  11. y ′′ - 1 x 2 y + 2 x 3 y = 0 y^{\prime\prime}-\frac{1}{x^{2}}y^{\prime}+\frac{2}{x^{3}}y=0
  12. μ ( x ) = e - 1 x 2 d x = e 1 x , \mu(x)=e^{\int-\frac{1}{x^{2}}\,\mathrm{d}x}=e^{\frac{1}{x}},
  13. e 1 x y ′′ - e 1 x x 2 y + 2 e 1 x x 3 y = 0 e^{\frac{1}{x}}y^{\prime\prime}-\frac{e^{\frac{1}{x}}}{x^{2}}y^{\prime}+\frac{% 2e^{\frac{1}{x}}}{x^{3}}y=0
  14. D e 1 x = - e 1 x x 2 De^{\frac{1}{x}}=-\frac{e^{\frac{1}{x}}}{x^{2}}
  15. ( e 1 x y ) + 2 e 1 x x 3 y = 0. (e^{\frac{1}{x}}y^{\prime})^{\prime}+\frac{2e^{\frac{1}{x}}}{x^{3}}y=0.
  16. P ( x ) y ′′ + Q ( x ) y + R ( x ) y = 0 P(x)y^{\prime\prime}+Q(x)y^{\prime}+R(x)y=0
  17. μ ( x ) = 1 P ( x ) e Q ( x ) P ( x ) d x , \mu(x)=\frac{1}{P(x)}e^{\int\frac{Q(x)}{P(x)}\mathrm{d}x},
  18. d d x ( μ ( x ) P ( x ) y ) + μ ( x ) R ( x ) y = 0 \frac{d}{dx}(\mu(x)P(x)y^{\prime})+\mu(x)R(x)y=0
  19. d d x ( e Q ( x ) P ( x ) d x y ) + R ( x ) P ( x ) e Q ( x ) P ( x ) d x y = 0 \frac{d}{dx}\left(e^{\int\frac{Q(x)}{P(x)}\mathrm{d}x}y^{\prime}\right)+\frac{% R(x)}{P(x)}e^{\int\frac{Q(x)}{P(x)}\,\mathrm{d}x}y=0
  20. L u = - 1 w ( x ) ( d d x [ p ( x ) d u d x ] + q ( x ) u ) Lu=-\frac{1}{w(x)}\left(\frac{\mathrm{d}}{\mathrm{d}x}\left[p(x)\frac{\mathrm{% d}u}{\mathrm{d}x}\right]+q(x)u\right)
  21. L u = λ u . Lu=\lambda u.
  22. f , g = a b f ( x ) ¯ g ( x ) w ( x ) d x . \langle f,g\rangle=\int_{a}^{b}\overline{f(x)}g(x)w(x)\,\mathrm{d}x.
  23. ( L - z ) - 1 , z , (L-z)^{-1},\qquad z\in\mathbb{C},
  24. ( L - z ) - 1 u = α u , L u = ( z + α - 1 ) u , (L-z)^{-1}u=\alpha u,\qquad Lu=\left(z+\alpha^{-1}\right)u,
  25. L u = - d 2 u d x 2 = λ u Lu=-\frac{\mathrm{d}^{2}u}{\mathrm{d}x^{2}}=\lambda u
  26. u ( 0 ) = u ( π ) = 0. u(0)=u(\pi)=0.
  27. u ( x ) = sin k x u(x)=\sin kx
  28. L u = x , x ( 0 , π ) Lu=x,\qquad x\in(0,\pi)
  29. L u = k = 1 - 2 ( - 1 ) k k sin k x . Lu=\sum_{k=1}^{\infty}-2\frac{(-1)^{k}}{k}\sin kx.
  30. u = k = 1 2 ( - 1 ) k k 3 sin k x . u=\sum_{k=1}^{\infty}2\frac{(-1)^{k}}{k^{3}}\sin kx.
  31. u = 1 6 ( x 3 - π 2 x ) , u=\tfrac{1}{6}\left(x^{3}-\pi^{2}x\right),
  32. 2 W x 2 + 2 W y 2 = 1 c 2 2 W t 2 . \frac{\partial^{2}W}{\partial x^{2}}+\frac{\partial^{2}W}{\partial y^{2}}=% \frac{1}{c^{2}}\frac{\partial^{2}W}{\partial t^{2}}.
  33. W m n ( x , y , t ) = A m n sin ( m π x L 1 ) sin ( n π y L 2 ) cos ( ω m n t ) W_{mn}(x,y,t)=A_{mn}\sin\left(\frac{m\pi x}{L_{1}}\right)\sin\left(\frac{n\pi y% }{L_{2}}\right)\cos\left(\omega_{mn}t\right)
  34. ω m n 2 = c 2 ( m 2 π 2 L 1 2 + n 2 π 2 L 2 2 ) . \omega^{2}_{mn}=c^{2}\left(\frac{m^{2}\pi^{2}}{L_{1}^{2}}+\frac{n^{2}\pi^{2}}{% L_{2}^{2}}\right).
  35. ω m n \omega_{mn}
  36. y ( x ) a x d t p ( t ) y ( t ) 2 y(x)\int_{a}^{x}\frac{\mathrm{d}t}{p(t)y(t)^{2}}
  37. u 0 = y 0 k = 0 ( λ - λ 0 * ) k X ~ ( 2 k ) , u_{0}=y_{0}\sum_{k=0}^{\infty}\left(\lambda-\lambda_{0}^{*}\right)^{k}% \widetilde{X}^{(2k)},
  38. u 1 = y 0 k = 0 ( λ - λ 0 * ) k X ( 2 k + 1 ) . u_{1}=y_{0}\sum_{k=0}^{\infty}\left(\lambda-\lambda_{0}^{*}\right)^{k}X^{(2k+1% )}.
  39. f ( x ) 2 u x 2 + g ( x ) u x + h ( x ) u = u t + k ( t ) u f(x)\frac{\partial^{2}u}{\partial x^{2}}+g(x)\frac{\partial u}{\partial x}+h(x% )u=\frac{\partial u}{\partial t}+k(t)u
  40. u ( a , t ) = u ( b , t ) = 0 u(a,t)=u(b,t)=0
  41. u ( x , 0 ) = s ( x ) u(x,0)=s(x)
  42. u ( x , t ) = X ( x ) T ( t ) u(x,t)=X(x)T(t)
  43. L ^ X ( x ) X ( x ) = M ^ T ( t ) T ( t ) \frac{\hat{L}X(x)}{X(x)}=\frac{\hat{M}T(t)}{T(t)}
  44. L ^ = f ( x ) d 2 d x 2 + g ( x ) d d x + h ( x ) , M ^ = d d t + k ( t ) \hat{L}=f(x)\frac{\mathrm{d}^{2}}{\mathrm{d}x^{2}}+g(x)\frac{\mathrm{d}}{% \mathrm{d}x}+h(x),\qquad\hat{M}=\frac{\mathrm{d}}{\mathrm{d}t}+k(t)
  45. L ^ \hat{L}
  46. X ( x ) X(x)
  47. M ^ \hat{M}
  48. T ( t ) T(t)
  49. L ^ X ( x ) = λ X ( x ) \hat{L}X(x)=\lambda X(x)
  50. X ( a ) = X ( b ) = 0 X(a)=X(b)=0\,
  51. M ^ T ( t ) = λ T ( t ) \hat{M}T(t)=\lambda T(t)\,
  52. X n ( x ) X_{n}(x)
  53. λ n \lambda_{n}
  54. d d t T n ( t ) = ( λ n - k ( t ) ) T n ( t ) \frac{\mathrm{d}}{\mathrm{d}t}T_{n}(t)=(\lambda_{n}-k(t))T_{n}(t)
  55. T n ( t ) = a n e - ( λ n t - 0 t k ( τ ) d τ ) T_{n}(t)=a_{n}e^{-\left(\lambda_{n}t-\int_{0}^{t}k(\tau)\mathrm{d}\tau\right)}
  56. u ( x , t ) = n a n X n ( x ) e - ( λ n t - 0 t k ( τ ) d τ ) u(x,t)=\sum_{n}a_{n}X_{n}(x)e^{-\left(\lambda_{n}t-\int_{0}^{t}k(\tau)\mathrm{% d}\tau\right)}
  57. a n = X n ( x ) , s ( x ) X n ( x ) , X n ( x ) a_{n}=\frac{\langle X_{n}(x),s(x)\rangle}{\langle X_{n}(x),X_{n}(x)\rangle}
  58. y ( x ) , z ( x ) = a b y ( x ) z ( x ) w ( x ) d x \langle y(x),z(x)\rangle=\int_{a}^{b}y(x)z(x)w(x)\mathrm{d}x
  59. w ( x ) = e g ( x ) f ( x ) d x f ( x ) w(x)=\frac{e^{\int\frac{g(x)}{f(x)}\mathrm{d}x}}{f(x)}

Sub-orbital_spaceflight.html

  1. ϵ \epsilon
  2. ε = - μ 2 a > - μ R \varepsilon=-{\mu\over{2a}}>-{\mu\over{R}}\,\!
  3. μ \mu\,\!
  4. - μ 2 R -{\mu\over{2R}}\,\!
  5. μ 2 R \mu\over{2R}\,\!
  6. semi-major axis = ( 1 + sin θ ) R \,\text{semi-major axis}=(1+\sin\theta)R
  7. semi-minor axis = R 2 ( sin θ + sin 2 θ ) = ( R sin θ ) semi-major axis \,\text{semi-minor axis}=R\sqrt{2(\sin\theta+\sin^{2}\theta)}=\sqrt{(R\sin% \theta)\,\text{semi-major axis}}
  8. distance of apogee from centre of earth = ( 1 + sin θ + cos θ ) R / 2 \,\text{distance of apogee from centre of earth}=(1+\sin\theta+\cos\theta)R/2
  9. altitude of apogee above surface = ( sin θ 2 - sin 2 θ 2 ) R = ( sin ( θ + π / 4 ) 2 - 1 2 ) R \,\text{altitude of apogee above surface}=\left(\frac{\sin\theta}{2}-\sin^{2}% \frac{\theta}{2}\right)R=\left(\frac{\sin(\theta+\pi/4)}{\sqrt{}}{2}-\frac{1}{% 2}\right)R
  10. specific kinetic energy at launch = μ R - μ major axis = μ R sin θ 1 + sin θ \,\text{specific kinetic energy at launch}=\frac{\mu}{R}-\frac{\mu}{\,}\text{% major axis}=\frac{\mu}{R}\frac{\sin\theta}{1+\sin\theta}
  11. Δ v = speed at launch = 2 μ R sin θ 1 + sin θ = 2 g R sin θ 1 + sin θ \Delta v=\,\text{speed at launch}=\sqrt{2\frac{\mu}{R}\frac{\sin\theta}{1+\sin% \theta}}=\sqrt{2gR\frac{\sin\theta}{1+\sin\theta}}
  12. period = ( semi-major axis / R ) 3 / 2 × period of low earth orbit = ( 1 + sin θ 2 ) 3 / 2 2 π R / g \,\text{period}=(\,\text{semi-major axis}/R)^{3/2}\times\,\text{period of low % earth orbit}=\left(\frac{1+\sin\theta}{2}\right)^{3/2}2\pi\sqrt{R/g}
  13. area fraction = arcsin 2 sin θ 1 + sin θ π + 2 cos θ sin θ π (major axis)(minor axis) \,\text{area fraction}=\frac{\arcsin\sqrt{\frac{2\sin\theta}{1+\sin\theta}}}{% \pi}+\frac{2\cos\theta\sin\theta}{\pi\,\text{(major axis)(minor axis)}}
  14. time of flight = ( ( 1 + sin θ 2 ) 3 / 2 arcsin 2 sin θ 1 + sin θ + 1 2 cos θ sin θ ) 2 R g = ( ( 1 + sin θ 2 ) 3 / 2 arccos cos θ 1 + sin θ + 1 2 cos θ sin θ ) 2 R g \begin{aligned}\displaystyle\,\text{time of flight}&\displaystyle=\left(\left(% \frac{1+\sin\theta}{2}\right)^{3/2}\arcsin\sqrt{\frac{2\sin\theta}{1+\sin% \theta}}+\frac{1}{2}\cos\theta\sqrt{\sin\theta}\right)2\sqrt{\frac{R}{g}}\\ &\displaystyle=\left(\left(\frac{1+\sin\theta}{2}\right)^{3/2}\arccos\frac{% \cos\theta}{1+\sin\theta}+\frac{1}{2}\cos\theta\sqrt{\sin\theta}\right)2\sqrt{% \frac{R}{g}}\\ \end{aligned}
  15. 2 d / g \sqrt{2d/g}

Subbase.html

  1. X X
  2. T T
  3. B B
  4. T T
  5. T T
  6. T T
  7. B B
  8. X X
  9. T T
  10. T T
  11. B B
  12. T T
  13. B B
  14. T T
  15. T T
  16. B B
  17. T T
  18. X X
  19. B B
  20. T T
  21. B B
  22. X X
  23. T T
  24. T T
  25. B B
  26. x x
  27. U X U⊆X
  28. B B
  29. x x
  30. U U
  31. X X
  32. S S
  33. P ( X ) P(X)
  34. S S
  35. X X
  36. S S
  37. P ( X ) P(X)
  38. B B
  39. X X
  40. X X
  41. B B
  42. X X
  43. T T
  44. B B
  45. T T
  46. T T
  47. B B
  48. B B
  49. X X
  50. 𝐑 \mathbf{R}
  51. ( , a ) (−∞,a)
  52. ( b , ) (b,∞)
  53. a a
  54. b b
  55. ( a , b ) = ( , b ) ( a , ) (a,b)=(−∞,b)∩(a,∞)
  56. a Align l t ; b a&lt;b
  57. a a
  58. b b
  59. ( a , b ) (a,b)
  60. a a
  61. b b
  62. ( , a ) (−∞,a)
  63. a a
  64. X X
  65. X X
  66. X X
  67. U U
  68. X X
  69. Y Y
  70. V ( K , U ) = { f : X Y f ( K ) U } V(K,U)=\{f\colon X\to Y\mid f(K)\subseteq U\}
  71. K X K⊆X
  72. U U
  73. Y Y
  74. B B
  75. Y Y
  76. f : X Y f:X→Y
  77. f < s u p > 1 ( U ) f<sup>−1(U)

Subgraph_isomorphism_problem.html

  1. G = ( V , E ) G=(V,E)
  2. H = ( V , E ) H=(V^{\prime},E^{\prime})
  3. G 0 = ( V 0 , E 0 ) : V 0 V , E 0 = E ( V 0 × V 0 ) G_{0}=(V_{0},E_{0}):V_{0}\subseteq V,E_{0}=E\cap(V_{0}\times V_{0})
  4. G 0 H G_{0}\cong H
  5. f : V 0 V f\colon V_{0}\rightarrow V^{\prime}
  6. ( v 1 , v 2 ) E 0 ( f ( v 1 ) , f ( v 2 ) ) E (v_{1},v_{2})\in E_{0}\Leftrightarrow(f(v_{1}),f(v_{2}))\in E^{\prime}

Sublimation_(phase_transition).html

  1. Δ H s u b l i m a t i o n = - U l a t t i c e e n e r g y - 2 R T \Delta H_{sublimation}=-U_{lattice~{}energy}-2RT

Submanifold.html

  1. i : T p S T p M . i_{\ast}:T_{p}S\to T_{p}M.

Submersion_(mathematics).html

  1. D f p : T p M T f ( p ) N Df_{p}:T_{p}M\to T_{f(p)}N\,
  2. π : m + n n m + n \pi:\mathbb{R}^{m+n}\rightarrow\mathbb{R}^{n}\subset\mathbb{R}^{m+n}
  3. f ( x 1 , , x n , x n + 1 , , x m ) = ( x 1 , , x n ) . f(x_{1},\ldots,x_{n},x_{n+1},\ldots,x_{m})=(x_{1},\ldots,x_{n}).

Subobject.html

  1. u = v w u=v\circ w

Subobject_classifier.html

  1. χ A ( x ) = { 0 , if x A 1 , if x A \chi_{A}(x)=\begin{cases}0,&\mbox{if }~{}x\notin A\\ 1,&\mbox{if }~{}x\in A\end{cases}
  2. 𝒫 ( S ) \mathcal{P}(S)
  3. y : 𝒫 ( S ) 2 S y:\mathcal{P}(S)\rightarrow 2^{S}
  4. 𝒫 ( S ) \mathcal{P}(S)
  5. Ω ( U ) \Omega(U)
  6. C C
  7. Set C o p \mathrm{Set}^{C^{op}}
  8. c C c\in C
  9. Ω ( c ) \Omega(c)
  10. c c

Substitute_good.html

  1. X i X_{i}
  2. Y Y
  3. X i P Y > 0 \frac{\partial X_{i}}{\partial P_{Y}}>0
  4. X i P Y | U = c o n s t > 0 \left.\frac{\partial X_{i}}{\partial P_{Y}}\right|_{U=const}>0
  5. U = U ( X , Y ) U=U(X,Y)

Substitution_matrix.html

  1. ( i , j ) (i,j)
  2. i i
  3. j j
  4. [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ] \begin{bmatrix}1&0&\cdots&0&0\\ 0&1&&0&0\\ \vdots&&\ddots&&\vdots\\ 0&0&&1&0\\ 0&0&\cdots&0&1\end{bmatrix}
  5. S i , j = log p i M i , j p i p j = log M i , j p j = log o b s e r v e d f r e q u e n c y e x p e c t e d f r e q u e n c y S_{i,j}=\log\frac{p_{i}\cdot M_{i,j}}{p_{i}\cdot p_{j}}=\log\frac{M_{i,j}}{p_{% j}}=\log\frac{observed\;frequency}{expected\;frequency}
  6. M i , j M_{i,j}
  7. i i
  8. j j
  9. p i p_{i}
  10. p j p_{j}
  11. W 2 = W 1 2 W_{2}=W_{1}^{2}
  12. W 1 W_{1}
  13. W 2 W_{2}

Substructural_logic.html

  1. Γ Σ \Gamma\vdash\Sigma
  2. 𝒜 , 𝒞 \mathcal{A},\mathcal{B}\vdash\mathcal{C}
  3. \vdash
  4. , 𝒜 𝒞 \mathcal{B},\mathcal{A}\vdash\mathcal{C}
  5. 𝒜 , 𝒞 \mathcal{A},\mathcal{B}\vdash\mathcal{C}
  6. Γ , 𝒜 , 𝒜 , Δ 𝒞 \Gamma,\mathcal{A},\mathcal{A},\Delta\vdash\mathcal{C}
  7. Γ , 𝒜 , Δ 𝒞 \Gamma,\mathcal{A},\Delta\vdash\mathcal{C}
  8. Γ , 𝒜 , Δ 𝒞 \Gamma,\mathcal{A},\Delta\vdash\mathcal{C}
  9. Γ , 𝒜 , , Δ 𝒞 \Gamma,\mathcal{A},\mathcal{B},\Delta\vdash\mathcal{C}

Subtangent.html

  1. tan φ = d y d x = A P T A = A N A P . \tan\varphi=\frac{dy}{dx}=\frac{AP}{TA}=\frac{AN}{AP}.
  2. y cot φ = y d y d x , y\cot\varphi=\frac{y}{\tfrac{dy}{dx}},
  3. y tan φ = y d y d x . y\tan\varphi=y\frac{dy}{dx}.
  4. y sec φ = y 1 + ( d y d x ) 2 , y\sec\varphi=y\sqrt{1+\left(\frac{dy}{dx}\right)^{2}},
  5. y csc φ = y d y d x 1 + ( d y d x ) 2 . y\csc\varphi=\frac{y}{\tfrac{dy}{dx}}\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}.
  6. tan ψ = r d r d θ = O P O N = O T O P . \tan\psi=\frac{r}{\tfrac{dr}{d\theta}}=\frac{OP}{ON}=\frac{OT}{OP}.
  7. r tan ψ = r 2 d r d θ , r\tan\psi=\frac{r^{2}}{\tfrac{dr}{d\theta}},
  8. r cot ψ = d r d θ . r\cot\psi=\frac{dr}{d\theta}.

Subtonic.html

  1. 7 ^ \hat{7}
  2. 7 ^ \hat{7}
  3. 1 ^ \hat{1}

Successor_cardinal.html

  1. κ + = | inf { λ O N | κ < | λ | } | \kappa^{+}=|\inf\{\lambda\in ON\ |\ \kappa<|\lambda|\}|
  2. 0 = ω \aleph_{0}=\omega
  3. α + 1 = α + \aleph_{\alpha+1}=\aleph_{\alpha}^{+}
  4. λ = β < λ β \aleph_{\lambda}=\bigcup_{\beta<\lambda}\aleph_{\beta}
  5. β \aleph_{\beta}
  6. λ \aleph_{\lambda}
  7. κ + = | inf { λ O N | | λ | κ } | \kappa^{+}=|\inf\{\lambda\in ON\ |\ |\lambda|\nleq\kappa\}|

Successor_ordinal.html

  1. S ( α ) = α { α } . S(\alpha)=\alpha\cup\{\alpha\}.
  2. α + S ( β ) = S ( α + β ) \alpha+S(\beta)=S(\alpha+\beta)\!
  3. α + λ = β < λ ( α + β ) \alpha+\lambda=\bigcup_{\beta<\lambda}(\alpha+\beta)

Sufficiently_large.html

  1. P P
  2. x x
  3. a \exists~{}a\in\mathbb{R}
  4. P P
  5. x a \forall~{}x\geq a
  6. a a
  7. a a

Superalgebra.html

  1. A = A 0 A 1 A=A_{0}\oplus A_{1}
  2. A i A j \sube A i + j A_{i}A_{j}\sube A_{i+j}
  3. | x y | = | x | + | y | . |xy|=|x|+|y|.
  4. y x = ( - 1 ) | x | | y | x y yx=(-1)^{|x||y|}xy\,
  5. μ : A 1 A 0 A 1 A 0 \mu:A_{1}\otimes_{A_{0}}A_{1}\to A_{0}
  6. μ ( x y ) z = x μ ( y z ) \mu(x\otimes y)\cdot z=x\cdot\mu(y\otimes z)
  7. x ^ = ( - 1 ) | x | x \hat{x}=(-1)^{|x|}x
  8. x ^ = x 0 - x 1 \hat{x}=x_{0}-x_{1}
  9. A i = { x A : x ^ = ( - 1 ) i x } . A_{i}=\{x\in A:\hat{x}=(-1)^{i}x\}.
  10. [ x , y ] = x y - ( - 1 ) | x | | y | y x [x,y]=xy-(-1)^{|x||y|}yx\,
  11. Z ( A ) = { a A : [ a , x ] = 0 for all x A } . Z(A)=\{a\in A:[a,x]=0\,\text{ for all }x\in A\}.
  12. ( a 1 b 1 ) ( a 2 b 2 ) = ( - 1 ) | b 1 | | a 2 | ( a 1 a 2 b 1 b 2 ) . (a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=(-1)^{|b_{1}||a_{2}|}(a_{1}a_{2}% \otimes b_{1}b_{2}).
  13. r ( x y ) = ( r x ) y = ( - 1 ) | r | | x | x ( r y ) r\cdot(xy)=(r\cdot x)y=(-1)^{|r||x|}x(r\cdot y)
  14. μ : A A A η : R A \begin{aligned}\displaystyle\mu&\displaystyle:A\otimes A\to A\\ \displaystyle\eta&\displaystyle:R\to A\end{aligned}

Superconducting_magnet.html

  1. H ( t ) = H 0 e - ( R / L ) t H(t)=H_{0}e^{-(R/L)t}\,
  2. R R\,

Supergravity.html

  1. E 8 × E 8 E_{8}\times E_{8}
  2. α ˙ \dot{\alpha}
  3. β ˙ \dot{\beta}
  4. M ^ , α ^ \hat{M},\hat{\alpha}
  5. M = ( μ , α , α ˙ ) M=(\mu,\alpha,\dot{\alpha})
  6. e N M ^ e^{\hat{M}}_{N}
  7. ω M ^ N ^ P \omega_{\hat{M}\hat{N}P}
  8. E M ^ N E^{N}_{\hat{M}}
  9. e N M ^ ( x , θ ¯ , θ ) * = e N * M ^ * ( x , θ , θ ¯ ) e^{\hat{M}}_{N}(x,\overline{\theta},\theta)^{*}=e^{\hat{M}^{*}}_{N^{*}}(x,% \theta,\overline{\theta})
  10. μ * = μ \mu^{*}=\mu
  11. α * = α ˙ \alpha^{*}=\dot{\alpha}
  12. α ˙ * = α \dot{\alpha}^{*}=\alpha
  13. ω ( x , θ ¯ , θ ) * = ω ( x , θ , θ ¯ ) \omega(x,\overline{\theta},\theta)^{*}=\omega(x,\theta,\overline{\theta})
  14. D M ^ f = E M ^ N ( N f + ω N [ f ] ) D_{\hat{M}}f=E^{N}_{\hat{M}}\left(\partial_{N}f+\omega_{N}[f]\right)
  15. D ¯ α ˙ ^ X = 0 \overline{D}_{\hat{\dot{\alpha}}}X=0
  16. { D ¯ α ˙ ^ , D ¯ β ˙ ^ } = c α ˙ ^ β ˙ ^ γ ˙ ^ D ¯ γ ˙ ^ \left\{\overline{D}_{\hat{\dot{\alpha}}},\overline{D}_{\hat{\dot{\beta}}}% \right\}=c_{\hat{\dot{\alpha}}\hat{\dot{\beta}}}^{\hat{\dot{\gamma}}}\overline% {D}_{\hat{\dot{\gamma}}}
  17. D α ^ e α ˙ ^ + D ¯ α ˙ ^ e α ^ 0 D_{\hat{\alpha}}e_{\hat{\dot{\alpha}}}+\overline{D}_{\hat{\dot{\alpha}}}e_{% \hat{\alpha}}\neq 0
  18. T α ¯ ^ β ¯ ^ γ ¯ ^ = 0 T^{\hat{\underline{\gamma}}}_{\hat{\underline{\alpha}}\hat{\underline{\beta}}}=0
  19. T α ^ β ^ μ ^ = 0 T^{\hat{\mu}}_{\hat{\alpha}\hat{\beta}}=0
  20. T α ˙ ^ β ˙ ^ μ ^ = 0 T^{\hat{\mu}}_{\hat{\dot{\alpha}}\hat{\dot{\beta}}}=0
  21. T α ^ β ˙ ^ μ ^ = 2 i σ α ^ β ˙ ^ μ ^ T^{\hat{\mu}}_{\hat{\alpha}\hat{\dot{\beta}}}=2i\sigma^{\hat{\mu}}_{\hat{% \alpha}\hat{\dot{\beta}}}
  22. T μ ^ α ¯ ^ ν ^ = 0 T^{\hat{\nu}}_{\hat{\mu}\hat{\underline{\alpha}}}=0
  23. T μ ^ ν ^ ρ ^ = 0 T^{\hat{\rho}}_{\hat{\mu}\hat{\nu}}=0
  24. α ¯ \underline{\alpha}
  25. | e | \left|e\right|
  26. e μ ^ = 0 e μ ^ = 3 e α ^ = 1 e α ^ = 2 e α ˙ ^ = 1 e α ˙ ^ = 2 e^{\hat{\mu}=0}\wedge\cdots\wedge e^{\hat{\mu}=3}\wedge e^{\hat{\alpha}=1}% \wedge e^{\hat{\alpha}=2}\wedge e^{\hat{\dot{\alpha}}=1}\wedge e^{\hat{\dot{% \alpha}}=2}
  27. E α ˙ ^ μ = 0 E^{\mu}_{\hat{\dot{\alpha}}}=0
  28. E α ˙ ^ β = 0 E^{\beta}_{\hat{\dot{\alpha}}}=0
  29. E α ˙ ^ β ˙ = δ α ˙ β ˙ E^{\dot{\beta}}_{\hat{\dot{\alpha}}}=\delta^{\dot{\beta}}_{\dot{\alpha}}
  30. ( D ¯ 2 - 8 R ) f \left(\bar{D}^{2}-8R\right)f
  31. S = d 4 x d 2 Θ 2 [ 3 8 ( D ¯ 2 - 8 R ) e - K ( X ¯ , X ) / 3 + W ( X ) ] + c . c . S=\int d^{4}xd^{2}\Theta 2\mathcal{E}\left[\frac{3}{8}\left(\bar{D}^{2}-8R% \right)e^{-K(\bar{X},X)/3}+W(X)\right]+c.c.
  32. \mathcal{E}

Superluminal_motion.html

  1. t 1 t_{1}
  2. t 2 t_{2}
  3. t 1 t_{1}^{\prime}
  4. t 2 t_{2}^{\prime}
  5. ϕ \phi
  6. D L D_{L}
  7. A B = v δ t AB=v\delta t
  8. A C = v δ t cos θ AC=v\delta t\cos\theta
  9. B C = v δ t sin θ BC=v\delta t\sin\theta
  10. t 2 - t 1 = δ t t_{2}-t_{1}=\delta t
  11. t 1 = t 1 + D L + v δ t cos θ c t_{1}^{\prime}=t_{1}+\frac{D_{L}+v\delta t\cos\theta}{c}
  12. t 2 = t 2 + D L c t_{2}^{\prime}=t_{2}+\frac{D_{L}}{c}
  13. δ t = t 2 - t 1 = t 2 - t 1 - v δ t cos θ c = δ t - v δ t cos θ c = δ t ( 1 - β cos θ ) \delta t^{\prime}=t_{2}^{\prime}-t_{1}^{\prime}=t_{2}-t_{1}-\frac{v\delta t% \cos\theta}{c}=\delta t-\frac{v\delta t\cos\theta}{c}=\delta t(1-\beta\cos\theta)
  14. β = v / c \beta=v/c
  15. δ t = δ t 1 - β cos θ \delta t=\frac{\delta t^{\prime}}{1-\beta\cos\theta}
  16. B C = D L sin ϕ ϕ D L = v δ t sin θ ϕ D L = v sin θ δ t 1 - β cos θ BC=D_{L}\sin\phi\approx\phi D_{L}=v\delta t\sin\theta\Rightarrow\phi D_{L}=v% \sin\theta\frac{\delta t^{\prime}}{1-\beta\cos\theta}
  17. C B CB
  18. v T = ϕ D L δ t = v sin θ 1 - β cos θ v\text{T}=\frac{\phi D_{L}}{\delta t^{\prime}}=\frac{v\sin\theta}{1-\beta\cos\theta}
  19. β T = v T c = β sin θ 1 - β cos θ . \beta\text{T}=\frac{v\text{T}}{c}=\frac{\beta\sin\theta}{1-\beta\cos\theta}.
  20. β T θ = θ [ β sin θ 1 - β cos θ ] = β cos θ 1 - β cos θ - ( β sin θ ) 2 ( 1 - β cos θ ) 2 = 0 \frac{\partial\beta\text{T}}{\partial\theta}=\frac{\partial}{\partial\theta}% \left[\frac{\beta\sin\theta}{1-\beta\cos\theta}\right]=\frac{\beta\cos\theta}{% 1-\beta\cos\theta}-\frac{(\beta\sin\theta)^{2}}{(1-\beta\cos\theta)^{2}}=0
  21. β cos θ ( 1 - β cos θ ) 2 = ( 1 - β cos θ ) ( β sin θ ) 2 \Rightarrow\beta\cos\theta(1-\beta\cos\theta)^{2}=(1-\beta\cos\theta)(\beta% \sin\theta)^{2}
  22. β cos θ ( 1 - β cos θ ) = ( β sin θ ) 2 β cos θ - β 2 cos 2 θ = β 2 s i n 2 θ cos θ max = β \Rightarrow\beta\cos\theta(1-\beta\cos\theta)=(\beta\sin\theta)^{2}\Rightarrow% \beta\cos\theta-\beta^{2}\cos^{2}\theta=\beta^{2}sin^{2}\theta\Rightarrow\cos% \theta\text{max}=\beta
  23. sin θ max = 1 - cos 2 θ max = 1 - β 2 = 1 γ \Rightarrow\sin\theta\text{max}=\sqrt{1-\cos^{2}\theta\text{max}}=\sqrt{1-% \beta^{2}}=\frac{1}{\gamma}
  24. γ = 1 1 - β 2 \gamma=\frac{1}{\sqrt{1-\beta^{2}}}
  25. β Tmax = β sin θ max 1 - β cos θ max = β / γ 1 - β 2 = β γ \therefore\beta\text{T}\text{max}=\frac{\beta\sin\theta\text{max}}{1-\beta\cos% \theta\text{max}}=\frac{\beta/\gamma}{1-\beta^{2}}=\beta\gamma
  26. γ 1 \gamma\gg 1
  27. β Tmax > 1 \beta\text{T}\text{max}>1
  28. β < 1 \beta<1
  29. β T > 1 \beta\text{T}>1
  30. C B CB

Superspace.html

  1. [ t , t ] = [ t , θ ] = [ t , θ * ] = { θ , θ } = { θ , θ * } = { θ * , θ * } = 0 \left[t,t\right]=\left[t,\theta\right]=\left[t,\theta^{*}\right]=\left\{\theta% ,\theta\right\}=\left\{\theta,\theta^{*}\right\}=\left\{\theta^{*},\theta^{*}% \right\}=0
  2. [ a , b ] [a,b]
  3. { a , b } \{a,b\}
  4. Φ ( t , Θ , Θ * ) = ϕ ( t ) + Θ Ψ ( t ) - Θ * Φ * ( t ) + Θ Θ * F ( t ) \Phi\left(t,\Theta,\Theta^{*}\right)=\phi(t)+\Theta\Psi(t)-\Theta^{*}\Phi^{*}(% t)+\Theta\Theta^{*}F(t)
  5. { θ , Θ } = { θ * , Θ * } = 1 \left\{\frac{\partial}{\partial\theta}\,,\Theta\right\}=\left\{\frac{\partial}% {\partial\theta^{*}}\,,\Theta^{*}\right\}=1
  6. Q = θ + i Θ * t and Q = θ * + i Θ t Q=\frac{\partial}{\partial\theta}+i\Theta^{*}\frac{\partial}{\partial t}\quad% \,\text{and}\quad Q^{\dagger}=\frac{\partial}{\partial\theta^{*}}+i\Theta\frac% {\partial}{\partial t}
  7. { Q , Q } = 2 i t \left\{Q,Q^{\dagger}\,\right\}=2i\frac{\partial}{\partial t}
  8. δ ϵ Φ = ( ϵ * Q + ϵ Q ) Φ . \delta_{\epsilon}\Phi=(\epsilon^{*}Q+\epsilon Q^{\dagger})\Phi.
  9. [ Q , Φ ] = ( θ + i θ * t ) Φ = ψ + θ * ( F + i ϕ ˙ ) - i θ θ * ψ ˙ . \left[Q,\Phi\right]=\left(\frac{\partial}{\partial\theta}\,+i\theta^{*}\frac{% \partial}{\partial t}\right)\Phi=\psi+\theta^{*}\left(F+i\dot{\phi}\right)-i% \theta\theta^{*}\dot{\psi}.
  10. D = θ - i θ t and D = θ * - i θ t D=\frac{\partial}{\partial\theta}-i\theta\frac{\partial}{\partial t}\quad\,% \text{and}\quad D^{\dagger}=\frac{\partial}{\partial\theta^{*}}-i\theta\frac{% \partial}{\partial t}
  11. { D , D } = - 2 i t \left\{D,D^{\dagger}\right\}=-2i\frac{\partial}{\partial t}
  12. θ ¯ = def i θ γ 0 = - θ C \overline{\theta}\ \stackrel{\mathrm{def}}{=}\ i\theta^{\dagger}\gamma^{0}=-% \theta^{\perp}C
  13. Q = - θ ¯ + γ μ θ μ Q=-\frac{\partial}{\partial\overline{\theta}}+\gamma^{\mu}\theta\partial_{\mu}
  14. { Q , Q } = { Q ¯ , Q } C = 2 γ μ μ C = - 2 i γ μ P μ C \left\{Q,Q\right\}=\left\{\overline{Q},Q\right\}C=2\gamma^{\mu}\partial_{\mu}C% =-2i\gamma^{\mu}P_{\mu}C
  15. P = i μ P=i\partial_{\mu}

Supervenience.html

  1. x y ( X B ( X x X y ) Y A ( Y x Y y ) ) \forall x\forall y(\forall X_{\in B}(Xx\leftrightarrow Xy)\rightarrow\forall Y% _{\in A}(Yx\leftrightarrow Yy))
  2. x X A ( X x Y B ( Y x and y ( Y y X y ) ) ) \forall x\forall X_{\in A}(Xx\rightarrow\exists Y_{\in B}(Yx\and\forall y(Yy% \rightarrow Xy)))
  3. ( X ( X x X y ) ) (\forall X(Xx\leftrightarrow Xy))

Support_(mathematics).html

  1. supp ( f ) = { x X | f ( x ) 0 } . \operatorname{supp}(f)=\{x\in X\,|\,f(x)\neq 0\}.
  2. supp ( f ) := { x X | f ( x ) 0 } ¯ . \operatorname{supp}(f):=\overline{\{x\in X\,|\,f(x)\neq 0\}}.
  3. f ( x ) = { 1 - x 2 if | x | < 1 0 if | x | 1 f(x)=\begin{cases}1-x^{2}&\,\text{if }|x|<1\\ 0&\,\text{if }|x|\geq 1\end{cases}
  4. X X
  5. X X
  6. X X
  7. n n
  8. n \mathbb{R}^{n}
  9. f : f:\mathbb{R}\rightarrow\mathbb{R}
  10. f : f:\mathbb{R}\rightarrow\mathbb{R}
  11. f ( x ) = x 1 + x 2 f(x)=\frac{x}{1+x^{2}}
  12. f ( x ) 0 f(x)\rightarrow 0
  13. | x | |x|\rightarrow\infty
  14. \mathbb{R}
  15. ε > 0 \varepsilon>0
  16. f f
  17. \mathbb{R}
  18. C C
  19. \mathbb{R}
  20. | f ( x ) - I C ( x ) f ( x ) | < ε |f(x)-I_{C}(x)f(x)|<\varepsilon
  21. x X x\in X
  22. I C I_{C}
  23. C C
  24. ess supp ( f ) := X { Ω X | Ω is open and f = 0 μ -almost everywhere in Ω } . \operatorname{ess\,supp}(f):=X\setminus\bigcup\left\{\Omega\subset X\,|\,% \Omega\,\,\text{is open and}\,f=0\,\mu\,\text{-almost everywhere in}\,\Omega% \right\}.
  25. ϕ \phi
  26. ϕ \phi
  27. f ( ϕ ) = 0 f(\phi)=0
  28. U α U_{\alpha}
  29. ϕ \phi
  30. U α \bigcup U_{\alpha}
  31. ϕ \phi
  32. f ( ϕ ) = 0 f(\phi)=0
  33. { 0 } \{0\}

Surface-wave-sustained_mode.html

  1. δ \delta
  2. δ c / ω p e 2 - ω 2 . \delta\simeq c\,\big/\sqrt{\omega_{p_{e}}^{2}-\omega^{2}}.

Surface_energy.html

  1. r r
  2. l l
  3. P P
  4. δ G = - P δ l + γ δ A = 0 γ = P δ l δ A \delta G=-P~{}\delta l+\gamma~{}\delta A=0\qquad\implies\qquad\gamma=P\cfrac{% \delta l}{\delta A}
  5. G G
  6. A A
  7. A = 2 π r 2 + 2 π r l δ A = 4 π r δ r + 2 π l δ r + 2 π r δ l A=2\pi r^{2}+2\pi rl\qquad\implies\qquad\delta A=4\pi r\delta r+2\pi l\delta r% +2\pi r\delta l
  8. V V
  9. δ V \delta V
  10. V = π r 2 l = constant δ V = 2 π r l δ r + π r 2 δ l = 0 δ r = - r 2 l δ l . V=\pi r^{2}l=\,\text{constant}\qquad\implies\qquad\delta V=2\pi rl\delta r+\pi r% ^{2}\delta l=0\implies\delta r=-\cfrac{r}{2l}\delta l~{}.
  11. γ = P l π r ( l - 2 r ) . \gamma=\cfrac{Pl}{\pi r(l-2r)}~{}.
  12. P P
  13. r r
  14. l l
  15. γ = 1 A ( E 1 - E 0 ) \gamma=\frac{1}{A}(E_{1}-E_{0})
  16. E s u r f a c e n - l a y e r s = E n - k E b u l k 2 A E^{n-layers}_{surface}=\frac{E_{n}-k\cdot E_{bulk}}{2A}
  17. E n E_{n}
  18. E b u l k E_{bulk}
  19. γ = ( z σ - z β ) W AA 2 a 0 \gamma=\frac{(z_{\sigma}-z_{\beta})\frac{W_{\,\text{AA}}}{2}}{a_{0}}
  20. z σ z_{\sigma}
  21. z β z_{\beta}
  22. a 0 a_{0}
  23. W AA W_{\,\text{AA}}
  24. a 0 = V molecule 2/3 = ( M ¯ ρ N A ) 2/3 a_{0}=V_{\,\text{molecule}}^{\,\text{2/3}}=\left(\frac{\bar{M}}{\rho N_{A}}% \right)\text{2/3}
  25. M ¯ \bar{M}
  26. ρ \rho
  27. N A N_{A}
  28. Δ sub H = - W AA N A z b 2 \Delta_{\,\text{sub}}H=\frac{-W_{\,\text{AA}}N_{A}z_{b}}{2}
  29. γ ( z σ - z β ) ( - Δ sub H ) a 0 N A z β \gamma\approx\frac{(z_{\sigma}-z_{\beta})(-\Delta_{\,\text{sub}}H)}{a_{0}N_{A}% z_{\beta}}
  30. V α V_{\alpha}
  31. V β V_{\beta}
  32. V = V α + V β V=V_{\alpha}+V_{\beta}
  33. U U
  34. n i n_{i}
  35. S S
  36. U = U α + U β + U σ U=U_{\alpha}+U_{\beta}+U_{\sigma}
  37. N i = N i α + N i β + N i σ N_{i}=N_{\,\text{i}\alpha}+N_{\,\text{i}\beta}+N_{\,\text{i}\sigma}
  38. S = S α + S β + S σ S=S_{\alpha}+S_{\beta}+S_{\sigma}
  39. N i σ = N i - c i α V α - c i β V β N_{\,\text{i}\sigma}=N_{i}-c_{\,\text{i}\alpha}V_{\alpha}-c_{\,\text{i}\beta}V% _{\beta}
  40. c i α c_{\,\text{i}\alpha}
  41. c i β c_{\,\text{i}\beta}
  42. i i
  43. α \alpha
  44. β \beta
  45. γ \gamma
  46. Γ i = N i α A \Gamma_{i}=\frac{N_{\,\text{i}\alpha}}{A}
  47. S = γ s - γ l - γ s - l S=\gamma_{s}-\gamma_{l}-\gamma_{\,\text{s - l}}
  48. S S
  49. γ s \gamma_{\,\text{s}}
  50. γ l \gamma_{l}
  51. γ s - l \gamma_{\,\text{s - l}}
  52. S < 0 S<0
  53. S > 0 S>0
  54. θ = 0 \theta=0
  55. 0 < θ < 90 0<\theta<90
  56. 90 θ < 180 90<<\theta<180
  57. θ = 180 \theta=180
  58. γ s - g = γ s - l + γ l - g c o s θ \gamma_{\,\text{s - g}}=\gamma_{\,\text{s - l}}+\gamma_{\,\text{l - g}}cos\theta
  59. γ s - g \gamma_{\,\text{s - g}}
  60. γ s - l \gamma_{\,\text{s - l}}
  61. γ l - g \gamma_{\,\text{l - g}}
  62. θ \theta
  63. R T × l n P 0 K P 0 = γ V m × ( 1 R 1 + 1 R 2 ) RT\times ln\frac{P_{0}^{K}}{P_{0}}=\gamma V_{m}\times\left(\frac{1}{R_{1}}+% \frac{1}{R_{2}}\right)
  64. P 0 K P_{0}^{K}
  65. P 0 P_{0}
  66. γ \gamma
  67. V m V_{\rm{m}}
  68. R R
  69. T T
  70. R 1 R_{1}
  71. R 2 R_{2}

Surface_integral.html

  1. S f d S = T f ( 𝐱 ( s , t ) ) 𝐱 s × 𝐱 t d s d t \iint_{S}f\,\mathrm{d}S=\iint_{T}f(\mathbf{x}(s,t))\left\|{\partial\mathbf{x}% \over\partial s}\times{\partial\mathbf{x}\over\partial t}\right\|\mathrm{d}s\,% \mathrm{d}t
  2. z = f ( x , y ) z=f\,(x,y)
  3. A = S d S = T 𝐫 x × 𝐫 y d x d y A=\iint_{S}\,\mathrm{d}S=\iint_{T}\left\|{\partial\mathbf{r}\over\partial x}% \times{\partial\mathbf{r}\over\partial y}\right\|\mathrm{d}x\,\mathrm{d}y
  4. 𝐫 = ( x , y , z ) = ( x , y , f ( x , y ) ) \mathbf{r}=(x,y,z)=(x,y,f(x,y))
  5. 𝐫 x = ( 1 , 0 , f x ( x , y ) ) {\partial\mathbf{r}\over\partial x}=(1,0,f_{x}(x,y))
  6. 𝐫 y = ( 0 , 1 , f y ( x , y ) ) {\partial\mathbf{r}\over\partial y}=(0,1,f_{y}(x,y))
  7. A = T ( 1 , 0 , f x ) × ( 0 , 1 , f y ) d x d y = T ( - f x , - f y , 1 ) d x d y = T ( f x ) 2 + ( f y ) 2 + 1 d x d y \begin{aligned}\displaystyle A&\displaystyle{}=\iint_{T}\left\|\left(1,0,{% \partial f\over\partial x}\right)\times\left(0,1,{\partial f\over\partial y}% \right)\right\|\mathrm{d}x\,\mathrm{d}y\\ &\displaystyle{}=\iint_{T}\left\|\left(-{\partial f\over\partial x},-{\partial f% \over\partial y},1\right)\right\|\mathrm{d}x\,\mathrm{d}y\\ &\displaystyle{}=\iint_{T}\sqrt{\left({\partial f\over\partial x}\right)^{2}+% \left({\partial f\over\partial y}\right)^{2}+1}\,\,\mathrm{d}x\,\mathrm{d}y% \end{aligned}
  8. S 𝐯 d 𝐒 = S 𝐯 𝐧 d S = T 𝐯 ( 𝐱 ( s , t ) ) ( 𝐱 s × 𝐱 t ) d s d t . \iint_{S}{\mathbf{v}}\cdot\mathrm{d}{\mathbf{S}}=\iint_{S}{\mathbf{v}}\cdot{% \mathbf{n}}\,\mathrm{d}S=\iint_{T}{\mathbf{v}}(\mathbf{x}(s,t))\cdot\left({% \partial\mathbf{x}\over\partial s}\times{\partial\mathbf{x}\over\partial t}% \right)\mathrm{d}s\,\mathrm{d}t.
  9. 𝐯 , 𝐧 d S \langle\mathbf{v},\mathbf{n}\rangle\;\mathrm{d}S
  10. d S \mathrm{d}S
  11. f = f z d x d y + f x d y d z + f y d z d x f=f_{z}\,\mathrm{d}x\wedge\mathrm{d}y+f_{x}\,\mathrm{d}y\wedge\mathrm{d}z+f_{y% }\,\mathrm{d}z\wedge\mathrm{d}x
  12. 𝐱 ( s , t ) = ( x ( s , t ) , y ( s , t ) , z ( s , t ) ) \mathbf{x}(s,t)=(x(s,t),y(s,t),z(s,t))\!
  13. ( s , t ) (s,t)
  14. ( x , y ) (x,y)
  15. ( s , t ) (s,t)
  16. d x = d x d s d s + d x d t d t \mathrm{d}x=\frac{\mathrm{d}x}{\mathrm{d}s}\mathrm{d}s+\frac{\mathrm{d}x}{% \mathrm{d}t}\mathrm{d}t
  17. d y = d y d s d s + d y d t d t \mathrm{d}y=\frac{\mathrm{d}y}{\mathrm{d}s}\mathrm{d}s+\frac{\mathrm{d}y}{% \mathrm{d}t}\mathrm{d}t
  18. d x d y \mathrm{d}x\wedge\mathrm{d}y
  19. ( x , y ) ( s , t ) d s d t \frac{\partial(x,y)}{\partial(s,t)}\mathrm{d}s\wedge\mathrm{d}t
  20. ( x , y ) ( s , t ) \frac{\partial(x,y)}{\partial(s,t)}
  21. ( s , t ) (s,t)
  22. ( x , y ) (x,y)
  23. D [ f z ( 𝐱 ( s , t ) ) ( x , y ) ( s , t ) + f x ( 𝐱 ( s , t ) ) ( y , z ) ( s , t ) + f y ( 𝐱 ( s , t ) ) ( z , x ) ( s , t ) ] d s d t \iint_{D}\left[f_{z}(\mathbf{x}(s,t))\frac{\partial(x,y)}{\partial(s,t)}+f_{x}% (\mathbf{x}(s,t))\frac{\partial(y,z)}{\partial(s,t)}+f_{y}(\mathbf{x}(s,t))% \frac{\partial(z,x)}{\partial(s,t)}\right]\,\mathrm{d}s\,\mathrm{d}t
  24. 𝐱 s × 𝐱 t = ( ( y , z ) ( s , t ) , ( z , x ) ( s , t ) , ( x , y ) ( s , t ) ) {\partial\mathbf{x}\over\partial s}\times{\partial\mathbf{x}\over\partial t}=% \left(\frac{\partial(y,z)}{\partial(s,t)},\frac{\partial(z,x)}{\partial(s,t)},% \frac{\partial(x,y)}{\partial(s,t)}\right)
  25. f x f_{x}
  26. f y f_{y}
  27. f z f_{z}

Surface_layer.html

  1. x x
  2. u ¯ \overline{u}
  3. u \ u^{\prime}
  4. u = u ¯ + u u=\overline{u}+u^{\prime}
  5. w \ w
  6. w = w ¯ + w w=\overline{w}+w^{\prime}
  7. u * \ u_{*}
  8. u w \ u^{\prime}w^{\prime}
  9. u * 2 = | ( u w ) s ¯ | u_{*}^{2}=\left|\overline{(u^{\prime}w^{\prime})_{s}}\right|
  10. K m \ K_{m}
  11. u * 2 \ u_{*}^{2}
  12. K m u ¯ z = u * 2 \ K_{m}\frac{\partial\overline{u}}{\partial z}=u_{*}^{2}
  13. K m \ K_{m}
  14. K m = ξ 2 ¯ | u ¯ z | \ K_{m}=\overline{\xi^{\prime 2}}\left|\frac{\partial\overline{u}}{\partial z}\right|
  15. ξ \ \xi^{\prime}
  16. u * \ u_{*}
  17. u ¯ z = u * ξ ¯ \frac{\partial\overline{u}}{\partial z}=\frac{u_{*}}{\overline{\xi^{\prime}}}
  18. ξ \ \xi^{\prime}
  19. ξ = k z \ \xi^{\prime}=kz
  20. z \ z
  21. k \ k
  22. u ¯ \ \overline{u}
  23. u ¯ = u * k ln z z o \overline{u}=\frac{u_{*}}{k}\ln\frac{z}{z_{o}}

Surface_of_revolution.html

  1. x ( t ) x(t)
  2. y ( t ) y(t)
  3. t t
  4. [ a , b ] [a,b]
  5. y y
  6. A y A_{y}
  7. A y = 2 π a b x ( t ) ( d x d t ) 2 + ( d y d t ) 2 d t , A_{y}=2\pi\int_{a}^{b}x(t)\ \sqrt{\left({dx\over dt}\right)^{2}+\left({dy\over dt% }\right)^{2}}\,dt,
  8. x ( t ) x(t)
  9. ( d x d t ) 2 + ( d y d t ) 2 \sqrt{\left({dx\over dt}\right)^{2}+\left({dy\over dt}\right)^{2}}
  10. 2 π x ( t ) 2\pi x(t)
  11. x x
  12. y ( t ) y(t)
  13. A x = 2 π a b y ( t ) ( d x d t ) 2 + ( d y d t ) 2 d t . A_{x}=2\pi\int_{a}^{b}y(t)\ \sqrt{\left({dx\over dt}\right)^{2}+\left({dy\over dt% }\right)^{2}}\,dt.
  14. A x = 2 π a b y 1 + ( d y d x ) 2 d x = 2 π a b f ( x ) 1 + ( f ( x ) ) 2 d x A_{x}=2\pi\int_{a}^{b}y\sqrt{1+\left(\frac{dy}{dx}\right)^{2}}\,dx=2\pi\int_{a% }^{b}f(x)\sqrt{1+\left(f^{\prime}(x)\right)^{2}}\,dx
  15. A y = 2 π a b x 1 + ( d x d y ) 2 d y A_{y}=2\pi\int_{a}^{b}x\sqrt{1+\left(\frac{dx}{dy}\right)^{2}}\,dy
  16. [ 0 , π ] [0,\pi]
  17. A = 2 π 0 π sin ( t ) ( cos ( t ) ) 2 + ( sin ( t ) ) 2 d t = 2 π 0 π sin ( t ) d t = 4 π . \begin{aligned}\displaystyle A&\displaystyle{}=2\pi\int_{0}^{\pi}\sin(t)\sqrt{% \left(\cos(t)\right)^{2}+\left(\sin(t)\right)^{2}}\,dt\\ &\displaystyle{}=2\pi\int_{0}^{\pi}\sin(t)\,dt\\ &\displaystyle{}=4\pi.\end{aligned}
  18. r r\,
  19. y ( x ) = r 2 - x 2 y(x)=\sqrt{r^{2}-x^{2}}
  20. A = 2 π - r r r 2 - x 2 1 + x 2 r 2 - x 2 d x = 2 π r - r r r 2 - x 2 1 r 2 - x 2 d x = 2 π r - r r d x = 4 π r 2 \begin{aligned}\displaystyle A&\displaystyle{}=2\pi\int_{-r}^{r}\sqrt{r^{2}-x^% {2}}\,\sqrt{1+\frac{x^{2}}{r^{2}-x^{2}}}\,dx\\ &\displaystyle{}=2\pi r\int_{-r}^{r}\,\sqrt{r^{2}-x^{2}}\,\sqrt{\frac{1}{r^{2}% -x^{2}}}\,dx\\ &\displaystyle{}=2\pi r\int_{-r}^{r}\,dx\\ &\displaystyle{}=4\pi r^{2}\end{aligned}
  21. y = f ( x ) y=f(x)
  22. u u
  23. u u
  24. v v
  25. f ( u ) sin v f(u)\sin v
  26. f ( u ) cos v f(u)\cos v
  27. y = f ( x ) y=f(x)
  28. x z xz
  29. r ( u , v ) = u , f ( u ) sin v , f ( u ) cos v \vec{r}(u,v)=\langle u,f(u)\sin v,f(u)\cos v\rangle
  30. u = x u=x
  31. v [ 0 , 2 π ] v\in[0,2\pi]

Survival_analysis.html

  1. S ( t ) = Pr ( T > t ) S(t)=\Pr(T>t)
  2. F ( t ) = Pr ( T t ) = 1 - S ( t ) . F(t)=\Pr(T\leq t)=1-S(t).
  3. f ( t ) = F ( t ) = d d t F ( t ) . f(t)=F^{\prime}(t)=\frac{d}{dt}F(t).
  4. S ( t ) = Pr ( T > t ) = t f ( u ) d u = 1 - F ( t ) . S(t)=\Pr(T>t)=\int_{t}^{\infty}f(u)\,du=1-F(t).
  5. s ( t ) = S ( t ) = d d t S ( t ) = d d t t f ( u ) d u = d d t [ 1 - F ( t ) ] = - f ( t ) . s(t)=S^{\prime}(t)=\frac{d}{dt}S(t)=\frac{d}{dt}\int_{t}^{\infty}f(u)\,du=% \frac{d}{dt}[1-F(t)]=-f(t).
  6. λ \lambda
  7. λ ( t ) = lim d t 0 Pr ( t T < t + d t ) d t S ( t ) = f ( t ) S ( t ) = - S ( t ) S ( t ) . \lambda(t)=\lim_{dt\rightarrow 0}\frac{\Pr(t\leq T<t+dt)}{dt\cdot S(t)}=\frac{% f(t)}{S(t)}=-\frac{S^{\prime}(t)}{S(t)}.
  8. μ \mu
  9. [ 0 , ] [0,\infty]
  10. Λ \Lambda
  11. Λ ( t ) = - log S ( t ) \,\Lambda(t)=-\log S(t)
  12. S ( t ) = exp ( - Λ ( t ) ) \,S(t)=\exp(-\Lambda(t))
  13. d d t Λ ( t ) = - S ( t ) S ( t ) = λ ( t ) . \frac{d}{dt}\Lambda(t)=-\frac{S^{\prime}(t)}{S(t)}=\lambda(t).
  14. Λ ( t ) = 0 t λ ( u ) d u \Lambda(t)=\int_{0}^{t}\lambda(u)\,du
  15. Λ ( t ) \Lambda(t)
  16. λ ( t ) \lambda(t)
  17. exp ( - t ) \exp(-t)
  18. t 0 t_{0}
  19. t 0 t_{0}
  20. T - t 0 T-t_{0}
  21. t 0 + t t_{0}+t
  22. t 0 t_{0}
  23. P ( T t 0 + t T > t 0 ) = P ( t 0 < T t 0 + t ) P ( T > t 0 ) = F ( t 0 + t ) - F ( t 0 ) S ( t 0 ) . P(T\leq t_{0}+t\mid T>t_{0})=\frac{P(t_{0}<T\leq t_{0}+t)}{P(T>t_{0})}=\frac{F% (t_{0}+t)-F(t_{0})}{S(t_{0})}.
  24. d d t F ( t 0 + t ) - F ( t 0 ) S ( t 0 ) = f ( t 0 + t ) S ( t 0 ) \frac{d}{dt}\frac{F(t_{0}+t)-F(t_{0})}{S(t_{0})}=\frac{f(t_{0}+t)}{S(t_{0})}
  25. 1 S ( t 0 ) 0 t f ( t 0 + t ) d t = 1 S ( t 0 ) t 0 S ( t ) d t , \frac{1}{S(t_{0})}\int_{0}^{\infty}t\,f(t_{0}+t)\,dt=\frac{1}{S(t_{0})}\int_{t% _{0}}^{\infty}S(t)\,dt,
  26. t 0 = 0 t_{0}=0
  27. L ( θ ) = T i u n c . Pr ( T = T i θ ) i l . c . Pr ( T < T i θ ) i r . c . Pr ( T > T i θ ) i i . c . Pr ( T i , l < T < T i , r θ ) . L(\theta)=\prod_{T_{i}\in unc.}\Pr(T=T_{i}\mid\theta)\prod_{i\in l.c.}\Pr(T<T_% {i}\mid\theta)\prod_{i\in r.c.}\Pr(T>T_{i}\mid\theta)\prod_{i\in i.c.}\Pr(T_{i% ,l}<T<T_{i,r}\mid\theta).
  28. T i T_{i}
  29. Pr ( T = T i θ ) = f ( T i θ ) . \Pr(T=T_{i}\mid\theta)=f(T_{i}\mid\theta).
  30. T i T_{i}
  31. Pr ( T < T i θ ) = F ( T i θ ) = 1 - S ( T i θ ) . \Pr(T<T_{i}\mid\theta)=F(T_{i}\mid\theta)=1-S(T_{i}\mid\theta).
  32. T i T_{i}
  33. Pr ( T > T i θ ) = 1 - F ( T i θ ) = S ( T i θ ) . \Pr(T>T_{i}\mid\theta)=1-F(T_{i}\mid\theta)=S(T_{i}\mid\theta).
  34. T i , r T_{i,r}
  35. T i , l T_{i,l}
  36. Pr ( T i , l < T < T i , r θ ) = S ( T i , l θ ) - S ( T i , r θ ) . \Pr(T_{i,l}<T<T_{i,r}\mid\theta)=S(T_{i,l}\mid\theta)-S(T_{i,r}\mid\theta).
  37. T i T_{i}

Suspension_(vehicle).html

  1. F = - k x F=-kx\,
  2. k = d 4 G 8 N D 3 k=\frac{d^{4}G}{8ND^{3}}\,

Swap_(finance).html

  1. V swap = B fixed - B floating V_{\mathrm{swap}}=B_{\mathrm{fixed}}-B_{\mathrm{floating}}\,
  2. V swap = B floating - B fixed V_{\mathrm{swap}}=B_{\mathrm{floating}}-B_{\mathrm{fixed}}\,
  3. V swap = B domestic - S 0 B foreign V_{\mathrm{swap}}=B_{\mathrm{domestic}}-S_{0}B_{\mathrm{foreign}}
  4. B domestic B_{\mathrm{domestic}}
  5. B foreign B_{\mathrm{foreign}}

Symbolic_method.html

  1. f ( x ) = A 0 x 1 2 + 2 A 1 x 1 x 2 + A 2 x 2 2 \displaystyle f(x)=A_{0}x_{1}^{2}+2A_{1}x_{1}x_{2}+A_{2}x_{2}^{2}
  2. Δ = A 0 A 2 - A 1 2 . \displaystyle\Delta=A_{0}A_{2}-A_{1}^{2}.
  3. 2 Δ = ( a b ) 2 \displaystyle 2\Delta=(ab)^{2}
  4. ( a b ) = a 1 b 2 - a 2 b 1 . \displaystyle(ab)=a_{1}b_{2}-a_{2}b_{1}.
  5. ( a b ) 2 = a 1 2 b 2 2 - 2 a 1 a 2 b 1 b 2 + a 2 2 b 1 2 . \displaystyle(ab)^{2}=a_{1}^{2}b_{2}^{2}-2a_{1}a_{2}b_{1}b_{2}+a_{2}^{2}b_{1}^% {2}.
  6. f ( x ) = ( a 1 x 1 + a 2 x 2 ) 2 = ( b 1 x 1 + b 2 x 2 ) 2 \displaystyle f(x)=(a_{1}x_{1}+a_{2}x_{2})^{2}=(b_{1}x_{1}+b_{2}x_{2})^{2}
  7. A i = a 1 2 - i a 2 i = b 1 2 - i b 2 i \displaystyle A_{i}=a_{1}^{2-i}a_{2}^{i}=b_{1}^{2-i}b_{2}^{i}
  8. ( a b ) 2 = A 2 A 0 - 2 A 1 A 1 + A 0 A 2 = 2 Δ . \displaystyle(ab)^{2}=A_{2}A_{0}-2A_{1}A_{1}+A_{0}A_{2}=2\Delta.
  9. f ( x ) = A 0 x 1 n + ( n 1 ) A 1 x 1 n - 1 x 2 + + A n x 2 n \displaystyle f(x)=A_{0}x_{1}^{n}+{\left({{n}\atop{1}}\right)}A_{1}x_{1}^{n-1}% x_{2}+\cdots+A_{n}x_{2}^{n}
  10. f ( x ) = ( a 1 x 1 + a 2 x 2 ) n = ( b 1 x 1 + b 2 x 2 ) n = ( c 1 x 1 + c 2 x 2 ) n = . f(x)=(a_{1}x_{1}+a_{2}x_{2})^{n}=(b_{1}x_{1}+b_{2}x_{2})^{n}=(c_{1}x_{1}+c_{2}% x_{2})^{n}=\cdots.
  11. f ( x ) = ( a 1 x 1 + a 2 x 2 + a 3 x 3 + ) n = ( b 1 x 1 + b 2 x 2 + b 3 x 3 + ) n = ( c 1 x 1 + c 2 x 2 + c 3 x 3 + ) n = . f(x)=(a_{1}x_{1}+a_{2}x_{2}+a_{3}x_{3}+\cdots)^{n}=(b_{1}x_{1}+b_{2}x_{2}+b_{3% }x_{3}+\cdots)^{n}=(c_{1}x_{1}+c_{2}x_{2}+c_{3}x_{3}+\cdots)^{n}=\cdots.

Symmetry_of_second_derivatives.html

  1. f ( x 1 , x 2 , , x n ) f(x_{1},x_{2},\dots,x_{n})
  2. x i x_{i}
  3. i i
  4. f i j f_{ij}
  5. f i j = f j i f_{ij}=f_{ji}
  6. x ( f y ) = y ( f x ) . \frac{\partial}{\partial x}\left(\frac{\partial f}{\partial y}\right)=\frac{% \partial}{\partial y}\left(\frac{\partial f}{\partial x}\right).
  7. x y f = y x f . \partial_{xy}f=\partial_{yx}f.
  8. f : n f\colon\mathbb{R}^{n}\to\mathbb{R}
  9. n \mathbb{R}^{n}
  10. ( a 1 , , a n ) , (a_{1},\dots,a_{n}),
  11. i , j { 1 , 2 , , n } , \forall i,j\in\{1,2,\ldots,n\},
  12. 2 f x i x j ( a 1 , , a n ) = 2 f x j x i ( a 1 , , a n ) . \frac{\partial^{2}f}{\partial x_{i}\,\partial x_{j}}(a_{1},\dots,a_{n})=\frac{% \partial^{2}f}{\partial x_{j}\,\partial x_{i}}(a_{1},\dots,a_{n}).\,\!
  13. ( D 1 D 2 f ) [ ϕ ] = - ( D 2 f ) [ D 1 ϕ ] = f [ D 2 D 1 ϕ ] = f [ D 1 D 2 ϕ ] = - ( D 1 f ) [ D 2 ϕ ] = ( D 2 D 1 f ) [ ϕ ] . (D_{1}D_{2}f)[\phi]=-(D_{2}f)[D_{1}\phi]=f[D_{2}D_{1}\phi]=f[D_{1}D_{2}\phi]=-% (D_{1}f)[D_{2}\phi]=(D_{2}D_{1}f)[\phi].
  14. f ( x , y ) = { x y ( x 2 - y 2 ) x 2 + y 2 for ( x , y ) ( 0 , 0 ) 0 for ( x , y ) = ( 0 , 0 ) . f(x,y)=\begin{cases}\frac{xy(x^{2}-y^{2})}{x^{2}+y^{2}}&\mbox{ for }~{}(x,y)% \neq(0,0)\\ 0&\mbox{ for }~{}(x,y)=(0,0).\end{cases}
  15. x f | ( 0 , 0 ) = y f | ( 0 , 0 ) = 0 \partial_{x}f|_{(0,0)}=\partial_{y}f|_{(0,0)}=0
  16. x f , y f \partial_{x}f,\partial_{y}f
  17. y f | ( x , 0 ) = x \partial_{y}f|_{(x,0)}=x
  18. x y f | ( 0 , 0 ) = lim ϵ 0 y f | ( ϵ , 0 ) - y f | ( 0 , 0 ) ϵ = 1. \partial_{x}\partial_{y}f|_{(0,0)}=\lim_{\epsilon\rightarrow 0}\frac{\partial_% {y}f|_{(\epsilon,0)}-\partial_{y}f|_{(0,0)}}{\epsilon}=1.
  19. x f | ( 0 , y ) = - y \partial_{x}f|_{(0,y)}=-y
  20. y x f | ( 0 , 0 ) = - 1 \partial_{y}\partial_{x}f|_{(0,0)}=-1
  21. x y f y x f \partial_{xy}f\neq\partial_{yx}f
  22. f ( h , k ) - f ( h , 0 ) - f ( 0 , k ) + f ( 0 , 0 ) f(h,k)-f(h,0)-f(0,k)+f(0,0)
  23. 0
  24. ( 0 , 0 ) (0,0)

Synaptic_plasticity.html

  1. d W i ( t ) d t = 1 τ ( [ C a 2 + ] i ) ( Ω ( [ C a 2 + ] i ) - W i ) , \frac{dW_{i}(t)}{dt}=\frac{1}{\tau([Ca^{2+}]_{i})}\left(\Omega([Ca^{2+}]_{i})-% W_{i}\right),
  2. W i W_{i}
  3. i i
  4. τ \tau
  5. [ C a 2 + ] [Ca^{2+}]
  6. Ω = β A m fp \Omega=\beta A_{m}^{\rm fp}
  7. Ω \Omega
  8. τ \tau

Synchronous_motor.html

  1. N s = 120 f p N_{s}=120\frac{f}{p}
  2. ω s = 4 π f p \omega_{s}=4\pi\frac{f}{p}
  3. f f
  4. p p
  5. p p
  6. N s = 120 × 50 4 = 1500 rpm N_{s}=120\times\frac{50}{4}=1500\,\,\,\text{rpm}
  7. 𝐓 = 𝐓 𝐦𝐚𝐱 sin δ \mathbf{T}={\mathbf{T_{max}}}{\sin\delta}
  8. 𝐓 \mathbf{T}
  9. δ \delta
  10. 𝐓 𝐦𝐚𝐱 \mathbf{T_{max}}
  11. 𝐓 𝐦𝐚𝐱 = 𝟑 𝐕 𝐄 𝐗 𝐬 ω s \mathbf{T_{max}}=\frac{{\mathbf{3}}{\mathbf{V}}{\mathbf{E}}}{{\mathbf{X_{s}}}{% \omega_{s}}}
  12. δ \delta
  13. δ \delta

Synchrotron_function.html

  1. F ( x ) = x x K 5 3 ( t ) d t F(x)=x\int_{x}^{\infty}K_{\frac{5}{3}}(t)\,dt
  2. G ( x ) = x K 2 3 ( x ) G(x)=xK_{\frac{2}{3}}(x)

Synchrotron_light_source.html

  1. brilliance = photons second mrad 2 mm 2 0.1 % BW \,\text{brilliance}=\frac{\,\text{photons}}{\,\text{second}\cdot\,\text{mrad}^% {2}\cdot\,\text{mm}^{2}\cdot 0.1\%~{}\,\text{BW}}
  2. γ \gamma
  3. γ \gamma

Syntax_(logic).html

  1. 𝒮 \mathcal{FS}
  2. 𝒮 \mathcal{FS}
  3. Γ F S A \Gamma\vdash_{\mathrm{F}S}A
  4. 𝒮 \mathcal{S}
  5. 𝒮 \mathcal{S}

Syrian_Arab_Air_Force.html

  1. } \Big\}
  2. } \Bigg\}
  3. } \Bigg\}

Systolic_array.html

  1. x x
  2. a j a_{j}

Szemerédi's_theorem.html

  1. lim sup n | A { 1 , 2 , 3 , , n } | n > 0 \limsup_{n\to\infty}\frac{|A\cap\{1,2,3,\ldots,n\}|}{n}>0
  2. δ ( 0 , 1 ] \delta\in(0,1]
  3. N = N ( k , δ ) N=N(k,\delta)\,
  4. r k ( N ) = o ( N ) r_{k}(N)=o(N)
  5. C N exp ( - n 2 ( n - 1 ) / 2 log N n + 1 2 n log log N ) r k ( N ) N ( log log N ) 2 - 2 k + 9 CN\exp(-n2^{(n-1)/2}\sqrt[n]{\log N}+\frac{1}{2n}\log\log N)\leq r_{k}(N)\leq% \frac{N}{(\log\log N)^{2^{-2^{k+9}}}}
  6. N 2 - 8 log N r 3 ( N ) C ( log log N ) 4 log N N N2^{-\sqrt{8\log N}}\leq r_{3}(N)\leq C\frac{(\log\log N)^{4}}{\log N}N
  7. r 4 ( N ) C N e c log log N r_{4}(N)\leq C\frac{N}{e^{c\sqrt{\log\log N}}}
  8. A A\subset\mathbb{N}
  9. p 1 ( n ) , p 2 ( n ) , , p k ( n ) p_{1}(n),p_{2}(n),\ldots,p_{k}(n)
  10. p i ( k ) = 0 p_{i}(k)=0
  11. u , n u,n\in\mathbb{Z}
  12. u + p i ( n ) A u+p_{i}(n)\in A
  13. 1 i k 1\leq i\leq k

T-duality.html

  1. R R
  2. 1 / R 1/R
  3. R R
  4. 1 / R 1/R
  5. \cdots
  6. \cdots
  7. R R
  8. S R 1 S_{R}^{1}
  9. 2 π R 2\pi R
  10. φ ( θ ) \varphi(\theta)
  11. θ \theta
  12. φ ( θ ) = m R θ + x + n 0 c n e i n θ \varphi(\theta)=mR\theta+x+\sum_{n\neq 0}c_{n}e^{in\theta}
  13. m m
  14. x = c 0 x=c_{0}
  15. x x
  16. c n c_{n}
  17. x ˙ \dot{x}
  18. x x
  19. x ˙ = n / R \dot{x}=n/R
  20. n n
  21. H = ( m R ) 2 + x ˙ 2 + n | c ˙ n | 2 + n 2 | c n | 2 H=(mR)^{2}+\dot{x}^{2}+\sum_{n}|\dot{c}_{n}|^{2}+n^{2}|c_{n}|^{2}
  22. ( m R ) 2 + ( n / R ) 2 (mR)^{2}+(n/R)^{2}
  23. R R
  24. 1 / R 1/R
  25. m m
  26. n n
  27. H H
  28. R R
  29. 1 / R 1/R
  30. R R
  31. 1 / R 1/R

Tabu_search.html

  1. x x
  2. x x^{\prime}
  3. x x
  4. N * ( x ) N^{*}(x)
  5. x x
  6. x x^{\prime}
  7. N * ( x ) N^{*}(x)
  8. N * ( x ) N^{*}(x)
  9. n n
  10. n n
  11. N * ( x ) N^{*}(x)

Tagged_union.html

  1. A + B A+B
  2. inj 1 : A A + B \,\text{inj}_{1}:A\to A+B
  3. inj 2 : B A + B \,\text{inj}_{2}:B\to A+B
  4. e e
  5. A + B A+B
  6. e 1 e_{1}
  7. e 2 e_{2}
  8. τ \tau
  9. x : A x:A
  10. y : B y:B
  11. 𝖼𝖺𝗌𝖾 e 𝗈𝖿 x e 1 | y e 2 \mathsf{case}\ e\ \mathsf{of}\ x\Rightarrow e_{1}|y\Rightarrow e_{2}
  12. τ \tau
  13. return : A ( A + E ) = a value a \,\text{return}\colon A\to\left(A+E\right)=a\mapsto\,\text{value}\,a
  14. bind : ( A + E ) ( A ( B + E ) ) ( B + E ) = a f { err e if a = err e f a if a = value a \,\text{bind}\colon\left(A+E\right)\to\left(A\to\left(B+E\right)\right)\to% \left(B+E\right)=a\mapsto f\mapsto\begin{cases}\,\text{err}\,e&\,\text{if}\ a=% \,\text{err}\,e\\ f\,a^{\prime}&\,\text{if}\ a=\,\text{value}\,a^{\prime}\end{cases}
  15. fmap : ( A B ) ( ( A + E ) ( B + E ) ) = f a { err e if a = err e value f a if a = value a \,\text{fmap}\colon(A\to B)\to\left(\left(A+E\right)\to\left(B+E\right)\right)% =f\mapsto a\mapsto\begin{cases}\,\text{err}\,e&\,\text{if}\ a=\,\text{err}\,e% \\ \,\text{value}\,f\,a^{\prime}&\,\text{if}\ a=\,\text{value}\,a^{\prime}\end{cases}
  16. join : ( ( A + E ) + E ) ( A + E ) = a { err e if a = err e err e if a = value err e value a if a = value value a \,\text{join}\colon((A+E)+E)\to(A+E)=a\mapsto\begin{cases}\,\text{err}\,e&% \mbox{if}~{}\ a=\,\text{err}\,e\\ \,\text{err}\,e&\,\text{if}\ a=\,\text{value}\,\,\text{err}\,e\\ \,\text{value}\,a^{\prime}&\,\text{if}\ a=\,\text{value}\,\,\text{value}\,a^{% \prime}\end{cases}

Tarski's_theorem_about_choice.html

  1. A A
  2. A A
  3. A × A A\times A
  4. A A
  5. | A | = | A × A | |A|=|A\times A|
  6. B B
  7. B B
  8. B B
  9. β \beta
  10. B B
  11. β \beta
  12. B B
  13. β \beta
  14. | B β | = | ( B β ) × ( B β ) | |B\cup\beta|=|(B\cup\beta)\times(B\cup\beta)|
  15. f : B β ( B β ) × ( B β ) f:B\cup\beta\to(B\cup\beta)\times(B\cup\beta)
  16. x B x\in B
  17. β × { x } f [ B ] \beta\times\{x\}\subseteq f[B]
  18. B B
  19. β \beta
  20. γ β \gamma\in\beta
  21. f ( γ ) β × { x } f(\gamma)\in\beta\times\{x\}
  22. S x = { γ | f ( γ ) β × { x } } S_{x}=\{\gamma|f(\gamma)\in\beta\times\{x\}\}
  23. g ( x ) = min S x g(x)=\min S_{x}
  24. S x S_{x}
  25. x , y B , x y x,y\in B,x\neq y
  26. S x S_{x}
  27. S y S_{y}
  28. B B
  29. x , y B x,y\in B
  30. x y g ( x ) g ( y ) x\leq y\iff g(x)\leq g(y)
  31. g g
  32. g [ B ] g[B]

Tarski's_undefinability_theorem.html

  1. Σ n 0 \Sigma^{0}_{n}
  2. Σ k 0 \Sigma^{0}_{k}

Taxicab_geometry.html

  1. 1 \ell_{1}
  2. d 1 d_{1}
  3. 𝐩 , 𝐪 \mathbf{p},\mathbf{q}
  4. d 1 ( 𝐩 , 𝐪 ) = 𝐩 - 𝐪 1 = i = 1 n | p i - q i | , d_{1}(\mathbf{p},\mathbf{q})=\|\mathbf{p}-\mathbf{q}\|_{1}=\sum_{i=1}^{n}|p_{i% }-q_{i}|,
  5. ( 𝐩 , 𝐪 ) (\mathbf{p},\mathbf{q})
  6. 𝐩 = ( p 1 , p 2 , , p n ) and 𝐪 = ( q 1 , q 2 , , q n ) \mathbf{p}=(p_{1},p_{2},\dots,p_{n})\,\text{ and }\mathbf{q}=(q_{1},q_{2},% \dots,q_{n})\,
  7. ( p 1 , p 2 ) (p_{1},p_{2})
  8. ( q 1 , q 2 ) (q_{1},q_{2})
  9. | p 1 - q 1 | + | p 2 - q 2 | . |p_{1}-q_{1}|+|p_{2}-q_{2}|.
  10. 2 \sqrt{2}
  11. π \pi
  12. | x | + | y | = 1 |x|+|y|=1
  13. r = 1 | sin θ | + | cos θ | r=\frac{1}{|\sin\theta|+|\cos\theta|}

Taxicab_number.html

  1. Ta ( 1 ) = 2 = 1 3 + 1 3 \begin{matrix}\operatorname{Ta}(1)&=&2&=&1^{3}+1^{3}\end{matrix}
  2. Ta ( 2 ) = 1729 = 1 3 + 12 3 = 9 3 + 10 3 \begin{matrix}\operatorname{Ta}(2)&=&1729&=&1^{3}&+&12^{3}\\ &&&=&9^{3}&+&10^{3}\end{matrix}
  3. Ta ( 3 ) = 87539319 = 167 3 + 436 3 = 228 3 + 423 3 = 255 3 + 414 3 \begin{matrix}\operatorname{Ta}(3)&=&87539319&=&167^{3}&+&436^{3}\\ &&&=&228^{3}&+&423^{3}\\ &&&=&255^{3}&+&414^{3}\end{matrix}
  4. Ta ( 4 ) = 6963472309248 = 2421 3 + 19083 3 = 5436 3 + 18948 3 = 10200 3 + 18072 3 = 13322 3 + 16630 3 \begin{matrix}\operatorname{Ta}(4)&=&6963472309248&=&2421^{3}&+&19083^{3}\\ &&&=&5436^{3}&+&18948^{3}\\ &&&=&10200^{3}&+&18072^{3}\\ &&&=&13322^{3}&+&16630^{3}\end{matrix}
  5. Ta ( 5 ) = 48988659276962496 = 38787 3 + 365757 3 = 107839 3 + 362753 3 = 205292 3 + 342952 3 = 221424 3 + 336588 3 = 231518 3 + 331954 3 \begin{matrix}\operatorname{Ta}(5)&=&48988659276962496&=&38787^{3}&+&365757^{3% }\\ &&&=&107839^{3}&+&362753^{3}\\ &&&=&205292^{3}&+&342952^{3}\\ &&&=&221424^{3}&+&336588^{3}\\ &&&=&231518^{3}&+&331954^{3}\end{matrix}
  6. Ta ( 6 ) = 24153319581254312065344 = 582162 3 + 28906206 3 = 3064173 3 + 28894803 3 = 8519281 3 + 28657487 3 = 16218068 3 + 27093208 3 = 17492496 3 + 26590452 3 = 18289922 3 + 26224366 3 \begin{matrix}\operatorname{Ta}(6)&=&24153319581254312065344&=&582162^{3}&+&28% 906206^{3}\\ &&&=&3064173^{3}&+&28894803^{3}\\ &&&=&8519281^{3}&+&28657487^{3}\\ &&&=&16218068^{3}&+&27093208^{3}\\ &&&=&17492496^{3}&+&26590452^{3}\\ &&&=&18289922^{3}&+&26224366^{3}\end{matrix}

Taylor_rule.html

  1. i t = π t + r t * + a π ( π t - π t * ) + a y ( y t - y ¯ t ) . i_{t}=\pi_{t}+r_{t}^{*}+a_{\pi}(\pi_{t}-\pi_{t}^{*})+a_{y}(y_{t}-\bar{y}_{t}).
  2. i t \,i_{t}\,
  3. π t \,\pi_{t}\,
  4. π t * \pi^{*}_{t}
  5. r t * r_{t}^{*}
  6. y t \,y_{t}\,
  7. y ¯ t \bar{y}_{t}
  8. a π a_{\pi}
  9. a y a_{y}
  10. a π = a y = 0.5 a_{\pi}=a_{y}=0.5
  11. a π > 0 a_{\pi}>0
  12. 1 + a π 1+a_{\pi}
  13. π t \pi_{t}
  14. a π > 0 a_{\pi}>0
  15. a y = 0 a_{y}=0

Telephone_numbering_plan.html

  1. ( V 1 - V 2 ) 2 + ( H 1 - H 2 ) 2 10 \sqrt{\frac{(V1-V2)^{2}+(H1-H2)^{2}}{10}}

Telescoping_series.html

  1. n = 1 1 n ( n + 1 ) \sum_{n=1}^{\infty}\frac{1}{n(n+1)}
  2. n = 1 1 n ( n + 1 ) = n = 1 ( 1 n - 1 n + 1 ) = lim N n = 1 N ( 1 n - 1 n + 1 ) = lim N [ ( 1 - 1 2 ) + ( 1 2 - 1 3 ) + + ( 1 N - 1 N + 1 ) ] = lim N [ 1 + ( - 1 2 + 1 2 ) + ( - 1 3 + 1 3 ) + + ( - 1 N + 1 N ) - 1 N + 1 ] = 1. \begin{aligned}\displaystyle\sum_{n=1}^{\infty}\frac{1}{n(n+1)}&\displaystyle{% }=\sum_{n=1}^{\infty}\left(\frac{1}{n}-\frac{1}{n+1}\right)\\ &\displaystyle{}=\lim_{N\to\infty}\sum_{n=1}^{N}\left(\frac{1}{n}-\frac{1}{n+1% }\right)\\ &\displaystyle{}=\lim_{N\to\infty}\left[{\left(1-\frac{1}{2}\right)+\left(% \frac{1}{2}-\frac{1}{3}\right)+\cdots+\left(\frac{1}{N}-\frac{1}{N+1}\right)}% \right]\\ &\displaystyle{}=\lim_{N\to\infty}\left[{1+\left(-\frac{1}{2}+\frac{1}{2}% \right)+\left(-\frac{1}{3}+\frac{1}{3}\right)+\cdots+\left(-\frac{1}{N}+\frac{% 1}{N}\right)-\frac{1}{N+1}}\right]=1.\end{aligned}
  3. a n a_{n}
  4. n = 1 N ( a n - a n - 1 ) = a N - a 0 , \sum_{n=1}^{N}\left(a_{n}-a_{n-1}\right)=a_{N}-a_{0},
  5. a n 0 a_{n}\rightarrow 0
  6. n = 1 ( a n - a n - 1 ) = - a 0 . \sum_{n=1}^{\infty}\left(a_{n}-a_{n-1}\right)=-a_{0}.
  7. 0 = n = 1 0 = n = 1 ( 1 - 1 ) = 1 + n = 1 ( - 1 + 1 ) = 1 0=\sum_{n=1}^{\infty}0=\sum_{n=1}^{\infty}(1-1)=1+\sum_{n=1}^{\infty}(-1+1)=1\,
  8. n = 1 N 1 n ( n + 1 ) \displaystyle\sum_{n=1}^{N}\frac{1}{n(n+1)}
  9. n = 1 N sin ( n ) \displaystyle\sum_{n=1}^{N}\sin\left(n\right)
  10. n = 1 N f ( n ) g ( n ) , \sum_{n=1}^{N}{f(n)\over g(n)},
  11. n = 0 2 n + 3 ( n + 1 ) ( n + 2 ) \displaystyle\sum^{\infty}_{n=0}\frac{2n+3}{(n+1)(n+2)}
  12. n = 1 1 n ( n + k ) = H k k \sum^{\infty}_{n=1}{\frac{1}{n(n+k)}}=\frac{H_{k}}{k}
  13. Pr ( X t = x ) = ( λ t ) x e - λ t x ! , \Pr(X_{t}=x)=\frac{(\lambda t)^{x}e^{-\lambda t}}{x!},
  14. f ( t ) \displaystyle f(t)
  15. f ( t ) = λ x t x - 1 e - λ t ( x - 1 ) ! . f(t)=\frac{\lambda^{x}t^{x-1}e^{-\lambda t}}{(x-1)!}.

Temporal_logic.html

  1. ¬ , , and , \neg,,\and,\rightarrow
  2. ϕ \phi
  3. ψ \psi
  4. ϕ 𝒰 ψ \phi~{}\mathcal{U}~{}\psi
  5. ( B 𝒰 C ) ( ϕ ) = ( i : C ( ϕ i ) ( j < i : B ( ϕ j ) ) ) \begin{matrix}(B\,\mathcal{U}\,C)(\phi)=\\ (\exists i:C(\phi_{i})\land(\forall j<i:B(\phi_{j})))\end{matrix}
  6. ψ \psi
  7. ϕ \phi
  8. ϕ \phi
  9. ϕ \phi
  10. ψ \psi
  11. ϕ ψ \phi~{}\mathcal{R}~{}\psi
  12. ( B C ) ( ϕ ) = ( i : C ( ϕ i ) ( j < i : B ( ϕ j ) ) ) \begin{matrix}(B\,\mathcal{R}\,C)(\phi)=\\ (\forall i:C(\phi_{i})\lor(\exists j<i:B(\phi_{j})))\end{matrix}
  13. ϕ \phi
  14. ψ \psi
  15. ψ \psi
  16. ϕ \phi
  17. ϕ \phi
  18. ϕ \bigcirc\phi
  19. 𝒩 B ( ϕ i ) = B ( ϕ i + 1 ) \mathcal{N}B(\phi_{i})=B(\phi_{i+1})
  20. ϕ \phi
  21. ϕ \phi
  22. ϕ \Diamond\phi
  23. B ( ϕ ) = ( t r u e 𝒰 B ) ( ϕ ) \mathcal{F}B(\phi)=(true\,\mathcal{U}\,B)(\phi)
  24. ϕ \phi
  25. ϕ \phi
  26. ϕ \Box\phi
  27. 𝒢 B ( ϕ ) = ¬ ¬ B ( ϕ ) \mathcal{G}B(\phi)=\neg\mathcal{F}\neg B(\phi)
  28. ϕ \phi
  29. ϕ \phi
  30. ϕ \forall\phi
  31. ( 𝒜 B ) ( ψ ) = ( ϕ : ϕ 0 = ψ B ( ϕ ) ) \begin{matrix}(\mathcal{A}B)(\psi)=\\ (\forall\phi:\phi_{0}=\psi\to B(\phi))\end{matrix}
  32. ϕ \phi
  33. ϕ \phi
  34. ϕ \exists\phi
  35. ( B ) ( ψ ) = ( ϕ : ϕ 0 = ψ B ( ϕ ) ) \begin{matrix}(\mathcal{E}B)(\psi)=\\ (\exists\phi:\phi_{0}=\psi\land B(\phi))\end{matrix}
  36. ϕ \phi
  37. f W g fWg
  38. f U g G f fUgGf
  39. ϕ \phi
  40. ϕ \phi
  41. ϕ \phi

Temporal_logic_in_finite-state_verification.html

  1. ( ( c a l l o p e n ) \displaystyle\Box((call\lor\Diamond open)\to

Tensor_algebra.html

  1. T k V = V k = V V V . T^{k}V=V^{\otimes k}=V\otimes V\otimes\cdots\otimes V.
  2. T ( V ) = k = 0 T k V = K V ( V V ) ( V V V ) . T(V)=\bigoplus_{k=0}^{\infty}T^{k}V=K\oplus V\oplus(V\otimes V)\oplus(V\otimes V% \otimes V)\oplus\cdots.
  3. T k V T V T k + V T^{k}V\otimes T^{\ell}V\to T^{k+\ell}V
  4. T k V = { 0 } T^{k}V=\{0\}
  5. T ( V ) T(V)
  6. T ( V * ) T(V^{*})
  7. V * , V^{*},
  8. x 1 , , x n x^{1},\dots,x^{n}
  9. Δ ( v 1 v m ) := i = 0 m ( v 1 v i ) ( v i + 1 v m ) \Delta(v_{1}\otimes\dots\otimes v_{m}):=\sum_{i=0}^{m}(v_{1}\otimes\dots% \otimes v_{i})\otimes(v_{i+1}\otimes\dots\otimes v_{m})
  10. ε ( v ) = v \varepsilon\left(v\right)=v
  11. v T 0 ( V ) v\in T^{0}\left(V\right)
  12. ε ( v ) = 0 \varepsilon\left(v\right)=0
  13. v T k ( V ) v\in T^{k}\left(V\right)
  14. k > 0 k>0
  15. T m V i + j = m T i V T j V T^{m}V\to\bigoplus_{i+j=m}T^{i}V\otimes T^{j}V
  16. Δ ( x 1 x m ) = p = 0 m σ Sh p , m - p ( x σ ( 1 ) x σ ( p ) ) ( x σ ( p + 1 ) x σ ( m ) ) \Delta(x_{1}\otimes\dots\otimes x_{m})=\sum_{p=0}^{m}\sum_{\sigma\in\mathrm{Sh% }_{p,m-p}}\left(x_{\sigma(1)}\otimes\dots\otimes x_{\sigma(p)}\right)\otimes% \left(x_{\sigma(p+1)}\otimes\dots\otimes x_{\sigma(m)}\right)
  17. S ( x 1 x m ) = ( - 1 ) m x m x 1 S(x_{1}\otimes\dots\otimes x_{m})=(-1)^{m}x_{m}\otimes\dots\otimes x_{1}
  18. T 1 ( V ) = V T^{1}(V)=V
  19. Δ ( x ) = x 1 + 1 x \Delta(x)=x\otimes 1+1\otimes x
  20. T m ( V ) T^{m}(V)
  21. Δ ( x 1 x m ) = Δ ( x 1 ) Δ ( x 2 ) Δ ( x m ) . \Delta(x_{1}\otimes\dots\otimes x_{m})=\Delta(x_{1})\Delta(x_{2})\cdots\Delta(% x_{m}).
  22. T 1 ( V ) = V T^{1}(V)=V
  23. S ( x ) = - x S(x)=-x
  24. T ( V ) T(V)
  25. T m ( V ) T^{m}(V)
  26. S ( x 1 x m ) = S ( x m ) S ( x m - 1 ) S ( x 2 ) S ( x 1 ) . S(x_{1}\otimes\dots\otimes x_{m})=S(x_{m})S(x_{m-1})\cdots S(x_{2})S(x_{1}).

Tensor_product_of_algebras.html

  1. A R B A\otimes_{R}B
  2. ( a 1 b 1 ) ( a 2 b 2 ) = a 1 a 2 b 1 b 2 (a_{1}\otimes b_{1})(a_{2}\otimes b_{2})=a_{1}a_{2}\otimes b_{1}b_{2}
  3. a a 1 B a\mapsto a\otimes 1_{B}
  4. b 1 A b b\mapsto 1_{A}\otimes b
  5. H o m ( A B , X ) { ( f , g ) H o m ( A , X ) × H o m ( B , X ) a A , b B : [ f ( a ) , g ( b ) ] = 0 } Hom(A\otimes B,X)\cong\{(f,g)\in Hom(A,X)\times Hom(B,X)\mid\forall a\in A,b% \in B:[f(a),g(b)]=0\}
  6. ϕ : A B X \phi:A\otimes B\to X
  7. ( f , g ) (f,g)
  8. f ( a ) := ϕ ( a 1 ) f(a):=\phi(a\otimes 1)
  9. g ( b ) := ϕ ( 1 b ) g(b):=\phi(1\otimes b)

Tensor_product_of_fields.html

  1. K L = k ( K L ) KL=k(K\cup L)
  2. K L K\otimes_{\mathbb{Q}}L
  3. K N L K\otimes_{N}L
  4. K N L K\otimes_{N}L
  5. ( a b ) ( c d ) (a\otimes b)(c\otimes d)
  6. a c b d ac\otimes bd
  7. K N L K\otimes_{N}L
  8. K N L K\otimes_{N}L
  9. γ ( a b ) = ( α ( a ) 1 ) ( 1 β ( b ) ) = α ( a ) . β ( b ) . \gamma(a\otimes b)=(\alpha(a)\otimes 1)\star(1\otimes\beta(b))=\alpha(a).\beta% (b).
  10. K N L K\otimes_{N}L
  11. ( K N L ) / R (K\otimes_{N}L)/R
  12. K K K\otimes_{\mathbb{Q}}K
  13. L K L L\otimes_{K}L
  14. T 1 / p 1 - 1 T 1 / p T^{1/p}\otimes 1-1\otimes T^{1/p}
  15. K K\otimes_{\mathbb{Q}}\mathbb{R}
  16. K p , K\otimes_{\mathbb{Q}}\mathbb{Q}_{p},

Terms_of_trade.html

  1. p x c q x 0 p x 0 q x 0 / p m c q m 0 p m 0 q m 0 {{p_{x}^{c}\,q_{x}^{0}}\over{p_{x}^{0}\,q_{x}^{0}}}\left/{{p_{m}^{c}\,q_{m}^{0% }}\over{p_{m}^{0}\,q_{m}^{0}}}\right.
  2. p x c = p_{x}^{c}=
  3. q x 0 = q_{x}^{0}=
  4. p x 0 = p_{x}^{0}=
  5. p m c = p_{m}^{c}=
  6. q m 0 = q_{m}^{0}=
  7. p m 0 = p_{m}^{0}=

Tetrahedral_number.html

  1. T n = n ( n + 1 ) ( n + 2 ) 6 = n 3 ¯ 3 ! T_{n}={n(n+1)(n+2)\over 6}={n^{\overline{3}}\over 3!}
  2. T n = ( n + 2 3 ) . T_{n}={n+2\choose 3}.
  3. n = 1 6 n ( n + 1 ) ( n + 2 ) = 3 2 . \!\ \sum_{n=1}^{\infty}\frac{6}{n(n+1)(n+2)}=\frac{3}{2}.
  4. T r n = ( n + 1 2 ) = ( m + 2 3 ) = T e m . Tr_{n}={n+1\choose 2}={m+2\choose 3}=Te_{m}.

Tetration.html

  1. a = a + 1 a^{\prime}=a+1
  2. a a
  3. a a
  4. a + n = a + 1 + 1 + + 1 n a+n=a+\underbrace{1+1+\cdots+1}_{n}
  5. a × n = a + a + + a n a\times n=\underbrace{a+a+\cdots+a}_{n}
  6. a n = a × a × × a n a^{n}=\underbrace{a\times a\times\cdots\times a}_{n}
  7. a n = a a a n {{}^{n}a}=\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}_{n}
  8. a = a + 1 a^{\prime}=a+1
  9. a + n a+n
  10. a n an
  11. a n a^{n}
  12. a n {}^{n}a
  13. a > 0 a>0
  14. n 0 n\geq 0
  15. a n \,\!{{}^{n}a}
  16. a n := { 1 if n = 0 a [ a ( n - 1 ) ] if n > 0 {{}^{n}a}:=\begin{cases}1&\,\text{if }n=0\\ a^{\left[{}^{(n-1)}a\right]}&\,\text{if }n>0\end{cases}
  17. 2 4 = 2 2 2 2 = 2 [ 2 ( 2 2 ) ] = 2 ( 2 4 ) = 2 16 = 65 , 536 \,\!\ {}^{4}2=2^{2^{2^{2}}}=2^{\left[2^{\left(2^{2}\right)}\right]}=2^{\left(2% ^{4}\right)}=2^{16}=65,\!536
  18. 2 2 2 2 [ ( 2 2 ) 2 ] 2 = 2 2 2 2 = 256 \,\!2^{2^{2^{2}}}\neq\left[{\left(2^{2}\right)}^{2}\right]^{2}=2^{2\cdot 2% \cdot 2}=256
  19. a n \,\!\ {}^{n}a
  20. a a a n {\ \atop{\ }}{{\underbrace{a^{a^{\cdot^{\cdot^{a}}}}}}\atop n}
  21. a a a a a^{a^{\cdot^{\cdot^{a^{a}}}}}
  22. a a a x a^{a^{\cdot^{\cdot^{a^{x}}}}}
  23. a 1 a 2 a n a_{1}^{a_{2}^{\cdot^{\cdot^{a_{n}}}}}
  24. a 1 a 2 a 3 a_{1}^{a_{2}^{a_{3}^{\cdot^{\cdot^{\cdot}}}}}
  25. a n \,{}^{n}a
  26. a n a{\uparrow\uparrow}n
  27. a n 2 a\rightarrow n\rightarrow 2
  28. 2 n = A ( 4 , n - 3 ) + 3 {}^{n}2=\operatorname{A}(4,n-3)+3
  29. a = 2 a=2
  30. a n = exp a n ( 1 ) {}^{n}a=\exp_{a}^{n}(1)
  31. uxp a n \operatorname{uxp}_{a}n
  32. a n a^{\frac{n}{}}
  33. a [ 4 ] n a[4]n
  34. H 4 ( a , n ) H_{4}(a,n)
  35. exp a n ( x ) = a a a x \exp_{a}^{n}(x)=a^{a^{\cdot^{\cdot^{a^{x}}}}}
  36. exp a n ( x ) \exp_{a}^{n}(x)
  37. exp a ( x ) = a x \exp_{a}(x)=a^{x}
  38. f n ( x ) f^{n}(x)
  39. ( a ) n ( x ) (a{\uparrow})^{n}(x)
  40. ( a , x ) n \,{}^{n}(a,x)
  41. x x
  42. x 2 {}^{2}x
  43. x 3 {}^{3}x
  44. x 4 {}^{4}x
  45. x 5 {}^{5}x
  46. exp 10 2 ( 4.295 ) \exp_{10}^{2}(4.295)
  47. exp 10 3 ( 1.09902 ) \exp_{10}^{3}(1.09902)
  48. exp 10 3 ( 2.18726 ) \exp_{10}^{3}(2.18726)
  49. exp 10 2 ( 3.33931 ) \exp_{10}^{2}(3.33931)
  50. exp 10 3 ( 3.33928 ) \exp_{10}^{3}(3.33928)
  51. exp 10 4 ( 3.33928 ) \exp_{10}^{4}(3.33928)
  52. exp 10 2 ( 4.55997 ) \exp_{10}^{2}(4.55997)
  53. exp 10 3 ( 4.55997 ) \exp_{10}^{3}(4.55997)
  54. exp 10 2 ( 5.84259 ) \exp_{10}^{2}(5.84259)
  55. exp 10 3 ( 5.84259 ) \exp_{10}^{3}(5.84259)
  56. exp 10 2 ( 7.18045 ) \exp_{10}^{2}(7.18045)
  57. exp 10 3 ( 7.18045 ) \exp_{10}^{3}(7.18045)
  58. exp 10 2 ( 8.56784 ) \exp_{10}^{2}(8.56784)
  59. exp 10 3 ( 8.56784 ) \exp_{10}^{3}(8.56784)
  60. exp 10 2 ( 10 ) \exp_{10}^{2}(10)
  61. exp 10 3 ( 10 ) \exp_{10}^{3}(10)
  62. 0 n {{}^{n}0}
  63. 0 0 0^{0}
  64. 0 n \,{{}^{n}0}
  65. lim x 0 x n \lim_{x\rightarrow 0}{}^{n}x
  66. lim x 0 x n = { 1 , n even 0 , n odd \lim_{x\rightarrow 0}{}^{n}x=\begin{cases}1,&n\,\text{ even}\\ 0,&n\,\text{ odd}\end{cases}
  67. 0 n = lim x 0 x n {}^{n}0=\lim_{x\rightarrow 0}{}^{n}x
  68. 0 0 = 1 0^{0}=1
  69. 0 0 = 1 {}^{0}0=1
  70. a 0 = 1 {{}^{0}a}=1
  71. z = a + b i \scriptstyle z\;=\;a+bi
  72. i 2 = - 1 \scriptstyle i^{2}\;=\;-1
  73. z n \scriptstyle{}^{n}z
  74. z = i \scriptstyle z\;=\;i
  75. i a + b i = e 1 2 π i ( a + b i ) = e - 1 2 π b ( cos π a 2 + i sin π a 2 ) i^{a+bi}=e^{\frac{1}{2}{\pi i}(a+bi)}=e^{-\frac{1}{2}{\pi b}}\left(\cos{\frac{% \pi a}{2}}+i\sin{\frac{\pi a}{2}}\right)
  76. i ( n + 1 ) = a + b i \scriptstyle{}^{(n+1)}i\;=\;a^{\prime}+b^{\prime}i
  77. i n = a + b i \scriptstyle{}^{n}i\;=\;a+bi
  78. a \displaystyle a^{\prime}
  79. i n {}^{n}i
  80. i 1 = i {}^{1}i=i
  81. i i
  82. i 2 = i ( i 1 ) {}^{2}i=i^{\left({}^{1}i\right)}
  83. 0.2079 0.2079
  84. i 3 = i ( i 2 ) {}^{3}i=i^{\left({}^{2}i\right)}
  85. 0.9472 + 0.3208 i 0.9472+0.3208i
  86. i 4 = i ( i 3 ) {}^{4}i=i^{\left({}^{3}i\right)}
  87. 0.0501 + 0.6021 i 0.0501+0.6021i
  88. i 5 = i ( i 4 ) {}^{5}i=i^{\left({}^{4}i\right)}
  89. 0.3872 + 0.0305 i 0.3872+0.0305i
  90. i 6 = i ( i 5 ) {}^{6}i=i^{\left({}^{5}i\right)}
  91. 0.7823 + 0.5446 i 0.7823+0.5446i
  92. i 7 = i ( i 6 ) {}^{7}i=i^{\left({}^{6}i\right)}
  93. 0.1426 + 0.4005 i 0.1426+0.4005i
  94. i 8 = i ( i 7 ) {}^{8}i=i^{\left({}^{7}i\right)}
  95. 0.5198 + 0.1184 i 0.5198+0.1184i
  96. i 9 = i ( i 8 ) {}^{9}i=i^{\left({}^{8}i\right)}
  97. 0.5686 + 0.6051 i 0.5686+0.6051i
  98. i 0 = 1 \scriptstyle\,{}^{0}i\;=\;1
  99. i ( - 1 ) = 0 \scriptstyle\,{}^{(-1)}i\;=\;0
  100. 0.4383 + 0.3606 i 0.4383+0.3606i
  101. | W ( - ln z ) - ln z | \left|\frac{\mathrm{W}(-\ln{z})}{-\ln{z}}\right|
  102. a n {}^{n}a
  103. 2 2 2 \sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\cdot^{\cdot^{\cdot}}}}}
  104. 2 2 2 2 2 1.414 \displaystyle\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{\sqrt{2}^{1.414}}}}}
  105. x x x^{x^{\cdot^{\cdot^{\cdot}}}}
  106. x n {}^{n}x
  107. z = z z = W ( - ln z ) - ln z , {}^{\infty}z=z^{z^{\cdot^{\cdot^{\cdot}}}}=\frac{\mathrm{W}(-\ln{z})}{-\ln{z}}% ~{},
  108. a ( k + 1 ) = a ( a k ) {{}^{(k+1)}a}=a^{({{}^{k}a})}
  109. k k
  110. a k = log a ( a ( k + 1 ) ) {{}^{k}a}=\log_{a}\left({{}^{(k+1)}a}\right)
  111. a ( - 1 ) = log a ( a 0 ) = log a 1 = 0 {}^{(-1)}a=\log_{a}\left({}^{0}a\right)=\log_{a}1=0
  112. a ( - 2 ) = log a ( a - 1 ) = log a 0 {}^{(-2)}a=\log_{a}\left({}^{-1}a\right)=\log_{a}0
  113. n = 1 n=1
  114. 1 ( - 1 ) \,\!{{}^{(-1)}1}
  115. 1 0 = 1 = 1 n {{}^{0}1}=1=1^{n}
  116. n = 1 ( - 1 ) \,\!n={{}^{(-1)}1}
  117. n n
  118. f ( x ) = a x \,f(x)={}^{x}a
  119. a ( - 1 ) = 0 \,{}^{(-1)}a=0
  120. a 0 = 1 \,{}^{0}a=1
  121. a x = a ( a ( x - 1 ) ) for all real x > - 1. \,{}^{x}a=a^{\left({}^{(x-1)}a\right)}\,\text{ for all real }x>-1.
  122. a x {}^{x}a
  123. x > 0 x>0
  124. ( d 2 d x 2 f ( x ) > 0 ) \left(\frac{d^{2}}{dx^{2}}f(x)>0\right)
  125. x > 0 x>0
  126. a x \,{}^{x}a
  127. a x { log a ( x + 1 a ) x - 1 1 + x - 1 < x 0 a ( a x - 1 ) 0 < x {}^{x}a\approx\begin{cases}\log_{a}(^{x+1}a)&x\leq-1\\ 1+x&-1<x\leq 0\\ a^{\left({}^{x-1}a\right)}&0<x\end{cases}
  128. a x x + 1 \,{}^{x}a\approx x+1
  129. - 1 < x < 0 -1<x<0
  130. a x a x \,{}^{x}a\approx a^{x}
  131. 0 < x < 1 0<x<1
  132. a x a a ( x - 1 ) \,{}^{x}a\approx a^{a^{(x-1)}}
  133. 1 < x < 2 1<x<2
  134. ln a \ln{a}
  135. e 1 2 π 5.868... , 0.5 - 4.3 4.03335... \begin{aligned}\displaystyle{}^{\frac{1}{2}\pi}e&\displaystyle\approx 5.868...% ,\\ \displaystyle{}^{-4.3}0.5&\displaystyle\approx 4.03335...\end{aligned}
  136. 0 < a 1 0<a\neq 1
  137. f : ( - 2 , + ) f:(-2,+\infty)\rightarrow\mathbb{R}
  138. f ( x ) = a f ( x - 1 ) for all x > - 1 , f ( 0 ) = 1 , f(x)=a^{f(x-1)}\;\;\,\text{for all}\;\;x>-1,\;f(0)=1,
  139. f f
  140. ( - 1 , 0 ) , (-1,0),
  141. f f^{\prime}
  142. ( - 1 , 0 ) , (-1,0),
  143. f ( 0 + ) = ( ln a ) f ( 0 - ) or f ( - 1 + ) = f ( 0 - ) . f^{\prime}(0^{+})=(\ln a)f^{\prime}(0^{-})\,\text{ or }f^{\prime}(-1^{+})=f^{% \prime}(0^{-}).
  144. f f
  145. f ( x ) = exp a [ x ] ( a x ) = exp a [ x + 1 ] ( x ) for all x > - 2 , f(x)=\exp^{[x]}_{a}(a^{x})=\exp^{[x+1]}_{a}(x)\quad\,\text{for all}\;\;x>-2,
  146. ( x ) = x - [ x ] (x)=x-[x]
  147. exp a [ x ] \exp^{[x]}_{a}
  148. [ x ] [x]
  149. exp a \exp_{a}
  150. e x {}^{x}e
  151. f : ( - 2 , + ) f:(-2,+\infty)\rightarrow\mathbb{R}
  152. f ( x ) = e f ( x - 1 ) for all x > - 1 , f ( 0 ) = 1 , f(x)=e^{f(x-1)}\;\;\,\text{for all}\;\;x>-1,\;f(0)=1,
  153. f f
  154. ( - 1 , 0 ) , (-1,0),
  155. f ( 0 - ) f ( 0 + ) . f^{\prime}(0^{-})\leq f^{\prime}(0^{+}).
  156. f = uxp f=\,\text{uxp}
  157. f = uxp f=\,\text{uxp}
  158. f ( - 1 + ) = f ( 0 + ) , f^{\prime}(-1^{+})=f^{\prime}(0^{+}),
  159. f f
  160. f ( x ) = e f ( x - 1 ) ( x > - 1 ) f(x)=e^{f(x-1)}\;\;(x>-1)
  161. f ( 0 ) = 1 f(0)=1
  162. ( - 1 , + ) (-1,+\infty)
  163. a x { log a ( a x + 1 ) x - 1 1 + 2 ln ( a ) 1 + ln ( a ) x - 1 - ln ( a ) 1 + ln ( a ) x 2 - 1 < x 0 a ( a x - 1 ) 0 < x {}^{x}a\approx\begin{cases}\log_{a}({}^{x+1}a)&x\leq-1\\ 1+\frac{2\ln(a)}{1\;+\;\ln(a)}x-\frac{1\;-\;\ln(a)}{1\;+\;\ln(a)}x^{2}&-1<x% \leq 0\\ a^{\left({}^{x-1}a\right)}&0<x\end{cases}
  164. x > 0 x>0
  165. a = e a=e
  166. ( a 1 / n ) n = a (a^{1/n})^{n}=a
  167. ( a 1 / n ) n = ( a 1 / n ) ( a 1 / n ) ( a 1 / n ) n a {}^{n}({}^{1/n}a)=\underbrace{({}^{1/n}a)^{({}^{1/n}a)^{\cdot^{\cdot^{\cdot^{% \cdot^{({}^{1/n}a)}}}}}}}_{n}\neq a
  168. f = F ( x + i y ) f=F(x+{\rm i}y)
  169. | f | = 1 , e ± 1 , e ± 2 , |f|=1,e^{\pm 1},e^{\pm 2},\ldots
  170. arg ( f ) = 0 , ± 1 , ± 2 , \arg(f)=0,\pm 1,\pm 2,\ldots
  171. S S
  172. S ( z ) = F ( z + n = 1 sin ( 2 π n z ) α n + n = 1 ( 1 - cos ( 2 π n z ) ) β n ) S(z)=F\!\left(~{}z~{}+\sum_{n=1}^{\infty}\sin(2\pi nz)~{}\alpha_{n}+\sum_{n=1}% ^{\infty}\Big(1-\cos(2\pi nz)\Big)~{}\beta_{n}\right)
  173. α \alpha
  174. β \beta
  175. ( z ) \Im(z)
  176. π \pi
  177. e e
  178. π \pi
  179. y n = x {}^{n}y=x
  180. 2 4 = 2 2 2 2 = 65 , 536 {}^{4}2=2^{2^{2^{2}}}=65,536
  181. 3 3 = 3 3 3 = 7 , 625 , 597 , 484 , 987 {}^{3}3=3^{3^{3}}=7,625,597,484,987
  182. ssrt ( x ) \mathrm{ssrt}(x)
  183. x s \sqrt{x}_{s}
  184. x 2 = x x {}^{2}x=x^{x}
  185. ssrt ( x ) = e W ( ln ( x ) ) = ln ( x ) W ( ln ( x ) ) \mathrm{ssrt}(x)=e^{W(\mathrm{ln}(x))}=\frac{\mathrm{ln}(x)}{W(\mathrm{ln}(x))}
  186. y = ssrt ( x ) y=\mathrm{ssrt}(x)
  187. x y = log y x \sqrt[y]{x}=\log_{y}x
  188. e - 1 / e < x < 1 e^{-1/e}<x<1
  189. x > 1 x>1
  190. e - 1 / e e^{-1/e}
  191. x n s \sqrt[n]{x}_{s}
  192. x = y y y x=y^{y^{y}}
  193. x 3 s \sqrt[3]{x}_{s}
  194. x 4 s \sqrt[4]{x}_{s}
  195. x n s \sqrt[n]{x}_{s}
  196. x n s \sqrt[n]{x}_{s}
  197. n = n=\infty
  198. x = y x={{}^{\infty}y}
  199. x = y x , x=y^{x},
  200. y = x 1 / x y=x^{1/x}
  201. x s = x 1 / x \sqrt[\infty]{x}_{s}=x^{1/x}
  202. 2 s = 2 1 / 2 = 2 \sqrt[\infty]{2}_{s}=2^{1/2}=\sqrt{2}
  203. n s \sqrt{n}_{s}
  204. n 3 s \sqrt[3]{n}_{s}
  205. slog a x \mathrm{slog}_{a}{x}
  206. slog a a x = x \mathrm{slog}_{a}{{}^{x}a}=x
  207. slog a a x = 1 + slog a x \mathrm{slog}_{a}a^{x}=1+\mathrm{slog}_{a}x
  208. slog a x = 1 + slog a log a x \mathrm{slog}_{a}x=1+\mathrm{slog}_{a}\log_{a}x
  209. slog a x > - 2 \mathrm{slog}_{a}x>-2
  210. y = x [ x [ x ( ) ] ] y=x^{[x^{[x(\cdots)]}]}
  211. a n \ {{}^{n}a}
  212. f ( x ) 2 2 x c f(x)\leq\underbrace{2^{2^{\cdot^{\cdot^{x}}}}}_{c}
  213. ( c , x ) 2 2 x c (c,x)\mapsto\underbrace{2^{2^{\cdot^{\cdot^{x}}}}}_{c}

Theoretical_computer_science.html

  1. P Q P\rightarrow Q\,
  2. Γ x : I n t \Gamma\vdash x:Int

Therapeutic_index.html

  1. Therapeutic ratio = LD 50 ED 50 \mbox{Therapeutic ratio}~{}=\frac{\mathrm{LD}_{50}}{\mathrm{ED}_{50}}
  2. Therapeutic ratio = TD 50 ED 50 \mbox{Therapeutic ratio}~{}=\frac{\mathrm{TD}_{50}}{\mathrm{ED}_{50}}

Thermal_energy.html

  1. U t h e r m a l = C ( T ) T . U_{thermal}=C(T)\cdot T.
  2. E t h e r m a l = f 1 2 k T E_{thermal}=f\cdot\tfrac{1}{2}kT\,\!
  3. U t h e r m a l = N f 1 2 k T . U_{thermal}=N\cdot f\cdot\tfrac{1}{2}kT.
  4. E k i n e t i c = 1 2 m v 2 E_{kinetic}=\tfrac{1}{2}mv^{2}\,\!
  5. U t h e r m a l = 1 2 N m v 2 ¯ = 3 2 N k T , U_{thermal}=\tfrac{1}{2}Nm\overline{v^{2}}=\tfrac{3}{2}NkT,
  6. U = U t h e r m a l . U=U_{thermal}.\;

Thermodynamic_system.html

  1. δ Q δQ
  2. δ W δW
  3. d U = d U i n + δ Q - d U o u t - δ W \mathrm{d}U=\mathrm{d}U_{in}+\delta Q-\mathrm{d}U_{out}-\delta W\,
  4. P V PV
  5. δ W = d ( P o u t V o u t ) - d ( P i n V i n ) + δ W s h a f t \delta W=\mathrm{d}(P_{out}V_{out})-\mathrm{d}(P_{in}V_{in})+\delta W_{shaft}\,
  6. d U c v = d U i n + d ( P i n V i n ) - d U o u t - d ( P o u t V o u t ) + δ Q - δ W s h a f t \mathrm{d}U_{cv}=\mathrm{d}U_{in}+\mathrm{d}(P_{in}V_{in})-\mathrm{d}U_{out}-% \mathrm{d}(P_{out}V_{out})+\delta Q-\delta W_{shaft}\,
  7. H = U + P V H=U+PV
  8. U U
  9. P V PV
  10. d U c v = d H i n - d H o u t + δ Q - δ W s h a f t \mathrm{d}U_{cv}=\mathrm{d}H_{in}-\mathrm{d}H_{out}+\delta Q-\delta W_{shaft}\,
  11. δ W s h a f t d t = d H i n d t - d H o u t d t + δ Q d t \frac{\delta W_{shaft}}{\mathrm{d}t}=\frac{\mathrm{d}H_{in}}{\mathrm{d}t}-% \frac{\mathrm{d}H_{out}}{\mathrm{d}t}+\frac{\delta Q}{\mathrm{d}t}\,
  12. i i
  13. Δ U = Q - W + m i ( h + 1 2 v 2 + g z ) i - m e ( h + 1 2 v 2 + g z ) e \Delta U=Q-W+m_{i}(h+\frac{1}{2}v^{2}+gz)_{i}-m_{e}(h+\frac{1}{2}v^{2}+gz)_{e}
  14. d U = δ Q - δ W . \mathrm{d}U=\delta Q-\delta W.
  15. δ W = P d V . \delta W=P\mathrm{d}V.
  16. δ Q = T d S \delta Q=T\mathrm{d}S
  17. d U = T d S - P d V . \mathrm{d}U=T\mathrm{d}S-P\mathrm{d}V.
  18. j = 1 m a i j N j = b i 0 \sum_{j=1}^{m}a_{ij}N_{j}=b_{i}^{0}

Thermoelectric_effect.html

  1. 𝐉 = σ ( - s y m b o l V + 𝐄 emf ) \mathbf{J}=\sigma(-symbol\nabla V+\mathbf{E}_{\rm emf})
  2. V \scriptstyle V
  3. σ \scriptstyle\sigma
  4. 𝐄 emf = - S s y m b o l T \mathbf{E}_{\rm emf}=-Ssymbol\nabla T
  5. S \scriptstyle S
  6. s y m b o l T \scriptstyle symbol\nabla T
  7. T \scriptstyle T
  8. 𝐉 = 0 \scriptstyle\mathbf{J}\;=\;0
  9. - s y m b o l V = S s y m b o l T \scriptstyle-symbol\nabla V\;=\;Ssymbol\nabla T
  10. Q ˙ \scriptstyle\dot{Q}
  11. Q ˙ = ( Π A - Π B ) I \dot{Q}=\left(\Pi_{\mathrm{A}}-\Pi_{\mathrm{B}}\right)I
  12. Π A \scriptstyle\Pi_{A}
  13. Π B \scriptstyle\Pi_{B}
  14. I \scriptstyle I
  15. Π A \scriptstyle\Pi_{A}
  16. Π B \scriptstyle\Pi_{B}
  17. Π = T S \scriptstyle\Pi\;=\;TS
  18. 𝐉 \scriptstyle\mathbf{J}
  19. q ˙ \scriptstyle\dot{q}
  20. q ˙ = - 𝒦 𝐉 s y m b o l T \dot{q}=-\mathcal{K}\mathbf{J}\cdot symbol\nabla T
  21. s y m b o l T \scriptstyle symbol\nabla T
  22. 𝒦 \scriptstyle\mathcal{K}
  23. 𝒦 = T d S d T \scriptstyle\mathcal{K}\;=\;T\,\frac{dS}{dT}
  24. 𝐉 = σ ( - s y m b o l V - S s y m b o l T ) \mathbf{J}=\sigma(-symbol\nabla V-Ssymbol\nabla T)
  25. e ˙ , \scriptstyle\dot{e},
  26. e ˙ = s y m b o l ( κ s y m b o l T ) - s y m b o l ( V + Π ) 𝐉 + q ˙ ext \dot{e}=symbol\nabla\cdot(\kappa symbol\nabla T)-symbol\nabla\cdot(V+\Pi)% \mathbf{J}+\dot{q}_{\rm ext}
  27. κ \scriptstyle\kappa
  28. q ˙ ext \scriptstyle\dot{q}_{\rm ext}
  29. e ˙ = 0 \scriptstyle\dot{e}\;=\;0
  30. s y m b o l 𝐉 = 0 \scriptstyle symbol\nabla\,\cdot\,\mathbf{J}\;=\;0
  31. - q ˙ ext = s y m b o l ( κ s y m b o l T ) + 𝐉 ( σ - 1 𝐉 ) - T 𝐉 \cdotsymbol S -\dot{q}_{\rm ext}=symbol\nabla\cdot(\kappa symbol\nabla T)+\mathbf{J}\cdot% \left(\sigma^{-1}\mathbf{J}\right)-T\mathbf{J}\cdotsymbol\nabla S
  32. s y m b o l S \scriptstyle symbol\nabla S
  33. s y m b o l S \scriptstyle symbol\nabla S
  34. 𝐉 \scriptstyle\mathbf{J}
  35. 𝒦 d Π d T - S \mathcal{K}\equiv{d\Pi\over dT}-S
  36. T \scriptstyle T
  37. 𝒦 \scriptstyle\mathcal{K}
  38. Π \scriptstyle\Pi
  39. S \scriptstyle S
  40. 𝒦 = T d S d T \scriptstyle\mathcal{K}\;=\;T\frac{dS}{dT}
  41. Π = T S \Pi=TS
  42. V = - μ / e \scriptstyle V\;=\;-\mu/e
  43. μ \scriptstyle\mu

Thermoelectric_materials.html

  1. Power factor = σ S 2 . \mathrm{Power~{}factor}=\sigma S^{2}.
  2. η \eta
  3. η = energy provided to the load heat energy absorbed at hot junction . \eta={\,\text{energy provided to the load}\over\,\text{heat energy absorbed at% hot junction}}.
  4. Z T = σ S 2 T λ ZT={\sigma S^{2}T\over\lambda}
  5. η max \eta_{\mathrm{max}}
  6. η max = T H - T C T H 1 + Z T ¯ - 1 1 + Z T ¯ + T C T H , \eta_{\mathrm{max}}={T_{H}-T_{C}\over T_{H}}{\sqrt{1+Z\bar{T}}-1\over\sqrt{1+Z% \bar{T}}+{T_{C}\over T_{H}}},
  7. T H T_{H}
  8. T C T_{C}
  9. Z T ¯ Z\bar{T}
  10. Z T ¯ = ( S p - S n ) 2 T ¯ [ ( ρ n κ n ) 1 / 2 + ( ρ p κ p ) 1 / 2 ] 2 Z\bar{T}={(S_{p}-S_{n})^{2}\bar{T}\over[(\rho_{n}\kappa_{n})^{1/2}+(\rho_{p}% \kappa_{p})^{1/2}]^{2}}
  11. ρ \rho
  12. T ¯ \bar{T}
  13. T H T_{H}
  14. T C T_{C}
  15. η max \eta_{\mathrm{max}}

Theta_function.html

  1. θ 1 \theta_{1}
  2. u = i π z u=i\pi z
  3. q = e i π τ = 0.1 e 0.1 i π q=e^{i\pi\tau}=0.1e^{0.1i\pi}
  4. θ 1 ( u ; q ) = 2 q 1 / 4 n = 0 ( - 1 ) n q n ( n + 1 ) sin ( 2 n + 1 ) u \theta_{1}(u;q)=2q^{1/4}\sum_{n=0}^{\infty}(-1)^{n}q^{n(n+1)}\sin(2n+1)u
  5. θ 1 ( u ; q ) = n = - n = ( - 1 ) n - 1 / 2 q ( n + 1 / 2 ) 2 e ( 2 n + 1 ) i u \theta_{1}(u;q)=\sum_{n=-\infty}^{n=\infty}(-1)^{n-1/2}q^{(n+1/2)^{2}}e^{(2n+1% )iu}
  6. ϑ ( z ; τ ) = n = - exp ( π i n 2 τ + 2 π i n z ) = 1 + 2 n = 1 ( e π i τ ) n 2 cos ( 2 π n z ) = n = - q n 2 η n \vartheta(z;\tau)=\sum_{n=-\infty}^{\infty}\exp(\pi in^{2}\tau+2\pi inz)=1+2% \sum_{n=1}^{\infty}\left(e^{\pi i\tau}\right)^{n^{2}}\cos(2\pi nz)=\sum_{n=-% \infty}^{\infty}q^{n^{2}}\eta^{n}
  7. ϑ ( z + 1 ; τ ) = ϑ ( z ; τ ) . \vartheta(z+1;\tau)=\vartheta(z;\tau).
  8. ϑ ( z + a + b τ ; τ ) = exp ( - π i b 2 τ - 2 π i b z ) ϑ ( z ; τ ) \vartheta(z+a+b\tau;\tau)=\exp(-\pi ib^{2}\tau-2\pi ibz)\,\vartheta(z;\tau)
  9. θ 1 \theta_{1}
  10. q = e i π τ q=e^{i\pi\tau}
  11. q q
  12. τ \tau
  13. θ 1 \theta_{1}
  14. q = e i π τ q=e^{i\pi\tau}
  15. q q
  16. τ \tau
  17. ϑ 00 ( z ; τ ) = ϑ ( z ; τ ) \vartheta_{00}(z;\tau)=\vartheta(z;\tau)
  18. ϑ 01 ( z ; τ ) = ϑ ( z + 1 2 ; τ ) ϑ 10 ( z ; τ ) = exp ( 1 4 π i τ + π i z ) ϑ ( z + 1 2 τ ; τ ) ϑ 11 ( z ; τ ) = exp ( 1 4 π i τ + π i ( z + 1 2 ) ) ϑ ( z + 1 2 τ + 1 2 ; τ ) . \begin{aligned}\displaystyle\vartheta_{01}(z;\tau)&\displaystyle=\vartheta\!% \left(z+{\textstyle\frac{1}{2}};\tau\right)\\ \displaystyle\vartheta_{10}(z;\tau)&\displaystyle=\exp\!\left({\textstyle\frac% {1}{4}}\pi i\tau+\pi iz\right)\vartheta\!\left(z+{\textstyle\frac{1}{2}}\tau;% \tau\right)\\ \displaystyle\vartheta_{11}(z;\tau)&\displaystyle=\exp\!\left({\textstyle\frac% {1}{4}}\pi i\tau+\pi i\!\left(z+{\textstyle\frac{1}{2}}\right)\right)\vartheta% \!\left(z+{\textstyle\frac{1}{2}}\tau+{\textstyle\frac{1}{2}};\tau\right).\end% {aligned}
  19. q = e π i τ q=e^{\pi i\tau}
  20. θ 1 ( z ; q ) \displaystyle\theta_{1}(z;q)
  21. ϑ 00 ( 0 ; τ ) 4 = ϑ 01 ( 0 ; τ ) 4 + ϑ 10 ( 0 ; τ ) 4 \vartheta_{00}(0;\tau)^{4}=\vartheta_{01}(0;\tau)^{4}+\vartheta_{10}(0;\tau)^{4}
  22. α = ( - i τ ) 1 2 exp ( π τ i z 2 ) . \alpha=(-i\tau)^{\frac{1}{2}}\exp\!\left(\frac{\pi}{\tau}iz^{2}\right).\,
  23. ϑ 00 ( z τ ; - 1 τ ) = α ϑ 00 ( z ; τ ) ϑ 01 ( z τ ; - 1 τ ) = α ϑ 10 ( z ; τ ) ϑ 10 ( z τ ; - 1 τ ) = α ϑ 01 ( z ; τ ) ϑ 11 ( z τ ; - 1 τ ) = - i α ϑ 11 ( z ; τ ) . \begin{aligned}\displaystyle\vartheta_{00}\!\left({\textstyle\frac{z}{\tau};% \frac{-1}{\tau}}\right)&\displaystyle=\alpha\,\vartheta_{00}(z;\tau)&% \displaystyle\vartheta_{01}\!\left({\textstyle\frac{z}{\tau};\frac{-1}{\tau}}% \right)&\displaystyle=\alpha\,\vartheta_{10}(z;\tau)\\ \displaystyle\vartheta_{10}\!\left({\textstyle\frac{z}{\tau};\frac{-1}{\tau}}% \right)&\displaystyle=\alpha\,\vartheta_{01}(z;\tau)&\displaystyle\vartheta_{1% 1}\!\left({\textstyle\frac{z}{\tau};\frac{-1}{\tau}}\right)&\displaystyle=-i% \alpha\,\vartheta_{11}(z;\tau).\end{aligned}
  24. z z\,
  25. τ \tau\,
  26. w w\,
  27. w = e π i z w=e^{\pi{\mathrm{i}}z}\,
  28. q = e π i τ q=e^{\pi{\mathrm{i}}\tau}\,
  29. ϑ 00 ( w , q ) = n = - ( w 2 ) n q n 2 ϑ 01 ( w , q ) = n = - ( - 1 ) n ( w 2 ) n q n 2 ϑ 10 ( w , q ) = n = - ( w 2 ) ( n + 1 / 2 ) q ( n + 1 / 2 ) 2 ϑ 11 ( w , q ) = i n = - ( - 1 ) n ( w 2 ) ( n + 1 / 2 ) q ( n + 1 / 2 ) 2 . \begin{aligned}\displaystyle\vartheta_{00}(w,q)&\displaystyle=\sum_{n=-\infty}% ^{\infty}(w^{2})^{n}q^{n^{2}}&\displaystyle\vartheta_{01}(w,q)&\displaystyle=% \sum_{n=-\infty}^{\infty}(-1)^{n}(w^{2})^{n}q^{n^{2}}\\ \displaystyle\vartheta_{10}(w,q)&\displaystyle=\sum_{n=-\infty}^{\infty}(w^{2}% )^{\left(n+1/2\right)}q^{\left(n+1/2\right)^{2}}&\displaystyle\vartheta_{11}(w% ,q)&\displaystyle=i\sum_{n=-\infty}^{\infty}(-1)^{n}(w^{2})^{\left(n+1/2\right% )}q^{\left(n+1/2\right)^{2}}.\end{aligned}
  30. m = 1 ( 1 - q 2 m ) ( 1 + w 2 q 2 m - 1 ) ( 1 + w - 2 q 2 m - 1 ) = n = - w 2 n q n 2 . \prod_{m=1}^{\infty}\left(1-q^{2m}\right)\left(1+w^{2}q^{2m-1}\right)\left(1+w% ^{-2}q^{2m-1}\right)=\sum_{n=-\infty}^{\infty}w^{2n}q^{n^{2}}.
  31. q = exp ( π i τ ) q=\exp(\pi i\tau)
  32. w = exp ( π i z ) w=\exp(\pi iz)
  33. ϑ ( z ; τ ) = n = - exp ( π i τ n 2 ) exp ( π i z 2 n ) = n = - w 2 n q n 2 . \vartheta(z;\tau)=\sum_{n=-\infty}^{\infty}\exp(\pi i\tau n^{2})\exp(\pi iz2n)% =\sum_{n=-\infty}^{\infty}w^{2n}q^{n^{2}}.
  34. ϑ ( z ; τ ) = m = 1 ( 1 - exp ( 2 m π i τ ) ) ( 1 + exp ( ( 2 m - 1 ) π i τ + 2 π i z ) ) ( 1 + exp ( ( 2 m - 1 ) π i τ - 2 π i z ) ) . \vartheta(z;\tau)=\prod_{m=1}^{\infty}\left(1-\exp(2m\pi i\tau)\right)\left(1+% \exp((2m-1)\pi i\tau+2\pi iz)\right)\left(1+\exp((2m-1)\pi i\tau-2\pi iz)% \right).
  35. ϑ ( z ; τ ) = m = 1 ( 1 - q 2 m ) ( 1 + q 2 m - 1 w 2 ) ( 1 + q 2 m - 1 / w 2 ) \vartheta(z;\tau)=\prod_{m=1}^{\infty}\left(1-q^{2m}\right)\left(1+q^{2m-1}w^{% 2}\right)\left(1+q^{2m-1}/w^{2}\right)
  36. = ( q 2 ; q 2 ) ( - w 2 q ; q 2 ) ( - q / w 2 ; q 2 ) =(q^{2};q^{2})_{\infty}\,(-w^{2}q;q^{2})_{\infty}\,(-q/w^{2};q^{2})_{\infty}
  37. = ( q 2 ; q 2 ) θ ( - w 2 q ; q 2 ) =(q^{2};q^{2})_{\infty}\,\theta(-w^{2}q;q^{2})
  38. ( ) (\cdot\cdot)_{\infty}
  39. θ ( ) \theta(\cdot\cdot)
  40. m = 1 ( 1 - q 2 m ) ( 1 + ( w 2 + w - 2 ) q 2 m - 1 + q 4 m - 2 ) , \prod_{m=1}^{\infty}\left(1-q^{2m}\right)\left(1+(w^{2}+w^{-2})q^{2m-1}+q^{4m-% 2}\right),
  41. ϑ ( z | q ) = m = 1 ( 1 - q 2 m ) ( 1 + 2 cos ( 2 π z ) q 2 m - 1 + q 4 m - 2 ) . \vartheta(z|q)=\prod_{m=1}^{\infty}\left(1-q^{2m}\right)\left(1+2\cos(2\pi z)q% ^{2m-1}+q^{4m-2}\right).
  42. ϑ 01 ( z | q ) = m = 1 ( 1 - q 2 m ) ( 1 - 2 cos ( 2 π z ) q 2 m - 1 + q 4 m - 2 ) . \vartheta_{01}(z|q)=\prod_{m=1}^{\infty}\left(1-q^{2m}\right)\left(1-2\cos(2% \pi z)q^{2m-1}+q^{4m-2}\right).
  43. ϑ 10 ( z | q ) = 2 q 1 / 4 cos ( π z ) m = 1 ( 1 - q 2 m ) ( 1 + 2 cos ( 2 π z ) q 2 m + q 4 m ) . \vartheta_{10}(z|q)=2q^{1/4}\cos(\pi z)\prod_{m=1}^{\infty}\left(1-q^{2m}% \right)\left(1+2\cos(2\pi z)q^{2m}+q^{4m}\right).
  44. ϑ 11 ( z | q ) = - 2 q 1 / 4 sin ( π z ) m = 1 ( 1 - q 2 m ) ( 1 - 2 cos ( 2 π z ) q 2 m + q 4 m ) . \vartheta_{11}(z|q)=-2q^{1/4}\sin(\pi z)\prod_{m=1}^{\infty}\left(1-q^{2m}% \right)\left(1-2\cos(2\pi z)q^{2m}+q^{4m}\right).
  45. ϑ 00 ( z ; τ ) = - i i - i + e i π τ u 2 cos ( 2 u z + π u ) sin ( π u ) d u \vartheta_{00}(z;\tau)=-i\int_{i-\infty}^{i+\infty}{e^{i\pi\tau u^{2}}\cos(2uz% +\pi u)\over\sin(\pi u)}du
  46. ϑ 01 ( z ; τ ) = - i i - i + e i π τ u 2 cos ( 2 u z ) sin ( π u ) d u . \vartheta_{01}(z;\tau)=-i\int_{i-\infty}^{i+\infty}{e^{i\pi\tau u^{2}}\cos(2uz% )\over\sin(\pi u)}du.
  47. ϑ 10 ( z ; τ ) = - i e i z + i π τ / 4 i - i + e i π τ u 2 cos ( 2 u z + π u + π τ u ) sin ( π u ) d u \vartheta_{10}(z;\tau)=-ie^{iz+i\pi\tau/4}\int_{i-\infty}^{i+\infty}{e^{i\pi% \tau u^{2}}\cos(2uz+\pi u+\pi\tau u)\over\sin(\pi u)}du
  48. ϑ 11 ( z ; τ ) = e i z + i π τ / 4 i - i + e i π τ u 2 cos ( 2 u z + π τ u ) sin ( π u ) d u \vartheta_{11}(z;\tau)=e^{iz+i\pi\tau/4}\int_{i-\infty}^{i+\infty}{e^{i\pi\tau u% ^{2}}\cos(2uz+\pi\tau u)\over\sin(\pi u)}du
  49. φ ( e - π x ) = ϑ ( 0 ; i x ) = θ 3 ( 0 ; e - π x ) = n = - e - x π n 2 \varphi(e^{-\pi x})=\vartheta(0;{\mathrm{i}}x)=\theta_{3}(0;e^{-\pi x})=\sum_{% n=-\infty}^{\infty}e^{-x\pi n^{2}}
  50. φ ( e - π ) = π 4 Γ ( 3 4 ) \varphi\left(e^{-\pi}\right)=\frac{\sqrt[4]{\pi}}{\Gamma(\frac{3}{4})}
  51. φ ( e - 2 π ) = 6 π + 4 2 π 4 2 Γ ( 3 4 ) \varphi\left(e^{-2\pi}\right)=\frac{\sqrt[4]{6\pi+4\sqrt{2}\pi}}{2\Gamma(\frac% {3}{4})}
  52. φ ( e - 3 π ) = 27 π + 18 3 π 4 3 Γ ( 3 4 ) \varphi\left(e^{-3\pi}\right)=\frac{\sqrt[4]{27\pi+18\sqrt{3}\pi}}{3\Gamma(% \frac{3}{4})}
  53. φ ( e - 4 π ) = 8 π 4 + 2 π 4 4 Γ ( 3 4 ) \varphi\left(e^{-4\pi}\right)=\frac{\sqrt[4]{8\pi}+2\sqrt[4]{\pi}}{4\Gamma(% \frac{3}{4})}
  54. φ ( e - 5 π ) = 225 π + 100 5 π 4 5 Γ ( 3 4 ) \varphi\left(e^{-5\pi}\right)=\frac{\sqrt[4]{225\pi+100\sqrt{5}\pi}}{5\Gamma(% \frac{3}{4})}
  55. φ ( e - 6 π ) = 3 2 + 3 3 4 + 2 3 - 27 4 + 1728 4 - 4 3 243 π 2 8 6 1 + 6 - 2 - 3 6 Γ ( 3 4 ) \varphi\left(e^{-6\pi}\right)=\frac{\sqrt[3]{3\sqrt{2}+3\sqrt[4]{3}+2\sqrt{3}-% \sqrt[4]{27}+\sqrt[4]{1728}-4}\cdot\sqrt[8]{243{\pi}^{2}}}{6\sqrt[6]{1+\sqrt{6% }-\sqrt{2}-\sqrt{3}}{\Gamma(\frac{3}{4})}}
  56. ϑ 4 2 ( q ) = i q 1 4 k = - q 2 k 2 - k ϑ 1 ( 2 k - 1 2 i ln q , q ) , \vartheta_{4}^{2}(q)=iq^{\frac{1}{4}}\sum_{k=-\infty}^{\infty}q^{2k^{2}-k}% \vartheta_{1}\left(\frac{2k-1}{2i}\ln q,q\right),
  57. ϑ 4 2 ( q ) = k = - q 2 k 2 ϑ 4 ( k ln q i , q ) . \vartheta_{4}^{2}(q)=\sum_{k=-\infty}^{\infty}q^{2k^{2}}\vartheta_{4}\left(% \frac{k\ln q}{i},q\right).
  58. π 2 1 Γ 2 ( 3 4 ) = k = - ϑ 4 ( i k π , e - π ) e 2 π k 2 \sqrt{\frac{\pi}{2}}\frac{1}{\Gamma^{2}\left(\frac{3}{4}\right)}=\sum_{k=-% \infty}^{\infty}\frac{\vartheta_{4}(ik\pi,e^{-\pi})}{e^{2\pi k^{2}}}
  59. ϑ ( z , τ ) = ϑ 3 ( z , τ ) = 0 z = m + n τ + 1 2 + τ 2 \vartheta(z,\tau)=\vartheta_{3}(z,\tau)=0\quad\Longleftrightarrow\quad z=m+n% \tau+\frac{1}{2}+\frac{\tau}{2}
  60. ϑ 1 ( z , τ ) = 0 z = m + n τ \vartheta_{1}(z,\tau)=0\quad\Longleftrightarrow\quad z=m+n\tau
  61. ϑ 2 ( z , τ ) = 0 z = m + n τ + 1 2 \vartheta_{2}(z,\tau)=0\quad\Longleftrightarrow\quad z=m+n\tau+\frac{1}{2}
  62. ϑ 4 ( z , τ ) = 0 z = m + n τ + τ 2 \vartheta_{4}(z,\tau)=0\quad\Longleftrightarrow\quad z=m+n\tau+\frac{\tau}{2}
  63. ϑ ( 0 ; - 1 / τ ) = ( - i τ ) 1 / 2 ϑ ( 0 ; τ ) \vartheta(0;-1/\tau)=(-i\tau)^{1/2}\vartheta(0;\tau)
  64. Γ ( s 2 ) π - s / 2 ζ ( s ) = 1 2 0 [ ϑ ( 0 ; i t ) - 1 ] t s / 2 d t t \Gamma\left(\frac{s}{2}\right)\pi^{-s/2}\zeta(s)=\frac{1}{2}\int_{0}^{\infty}% \left[\vartheta(0;it)-1\right]t^{s/2}\frac{dt}{t}
  65. ( z ; τ ) = - ( log ϑ 11 ( z ; τ ) ) ′′ + c \wp(z;\tau)=-(\log\vartheta_{11}(z;\tau))^{\prime\prime}+c
  66. ( z ) \wp(z)
  67. ( Γ q 2 ( x ) Γ q 2 ( 1 - x ) ) - 1 = q 2 x ( 1 - x ) ( q - 2 ; q - 2 ) 3 ( q 2 - 1 ) ϑ 4 ( 1 2 i ( 1 - 2 x ) log q , 1 q ) . \left(\Gamma_{q^{2}}(x)\Gamma_{q^{2}}(1-x)\right)^{-1}=\frac{q^{2x(1-x)}}{(q^{% -2};q^{-2})^{3}_{\infty}(q^{2}-1)}\vartheta_{4}\left(\frac{1}{2i}(1-2x)\log q,% \frac{1}{q}\right).
  68. q = e π i τ q=e^{\pi i\tau}
  69. θ 2 ( 0 , q ) = ϑ 10 ( 0 ; τ ) = 2 η 2 ( 2 τ ) η ( τ ) \theta_{2}(0,q)=\vartheta_{10}(0;\tau)=\frac{2\eta^{2}(2\tau)}{\eta(\tau)}
  70. θ 3 ( 0 , q ) = ϑ 00 ( 0 ; τ ) = η 5 ( τ ) η 2 ( 1 2 τ ) η 2 ( 2 τ ) = η 2 ( 1 2 ( τ + 1 ) ) η ( τ + 1 ) \theta_{3}(0,q)=\vartheta_{00}(0;\tau)=\frac{\eta^{5}(\tau)}{\eta^{2}(\tfrac{1% }{2}\tau)\eta^{2}(2\tau)}=\frac{\eta^{2}\left(\tfrac{1}{2}(\tau+1)\right)}{% \eta(\tau+1)}
  71. θ 4 ( 0 , q ) = ϑ 01 ( 0 ; τ ) = η 2 ( 1 2 τ ) η ( τ ) \theta_{4}(0,q)=\vartheta_{01}(0;\tau)=\frac{\eta^{2}(\tfrac{1}{2}\tau)}{\eta(% \tau)}
  72. ϑ ( x , i t ) = 1 + 2 n = 1 exp ( - π n 2 t ) cos ( 2 π n x ) \vartheta(x,it)=1+2\sum_{n=1}^{\infty}\exp(-\pi n^{2}t)\cos(2\pi nx)
  73. t ϑ ( x , i t ) = 1 4 π 2 x 2 ϑ ( x , i t ) . \frac{\partial}{\partial t}\vartheta(x,it)=\frac{1}{4\pi}\frac{\partial^{2}}{% \partial x^{2}}\vartheta(x,it).
  74. lim t 0 ϑ ( x , i t ) = n = - δ ( x - n ) \lim_{t\rightarrow 0}\vartheta(x,it)=\sum_{n=-\infty}^{\infty}\delta(x-n)
  75. θ F ( z ) = m Z n exp ( 2 π i z F ( m ) ) \theta_{F}(z)=\sum_{m\in Z^{n}}\exp(2\pi izF(m))
  76. θ ^ F ( z ) = k = 0 R F ( k ) exp ( 2 π i k z ) , \widehat{\theta}_{F}(z)=\sum_{k=0}^{\infty}R_{F}(k)\exp(2\pi ikz),
  77. n = { F M ( n , ) s . t . F = F T and Im F > 0 } \mathbb{H}_{n}=\{F\in M(n,\mathbb{C})\;\mathrm{s.t.}\,F=F^{T}\;\textrm{and}\;% \mbox{Im}~{}F>0\}
  78. Ker { Sp ( 2 n , ) Sp ( 2 n , / k ) } \textrm{Ker}\{\textrm{Sp}(2n,\mathbb{Z})\rightarrow\textrm{Sp}(2n,\mathbb{Z}/k% \mathbb{Z})\}
  79. τ n \tau\in\mathbb{H}_{n}
  80. θ ( z , τ ) = m Z n exp ( 2 π i ( 1 2 m T τ m + m T z ) ) . \theta(z,\tau)=\sum_{m\in Z^{n}}\exp\left(2\pi i\left(\frac{1}{2}m^{T}\tau m+m% ^{T}z\right)\right).
  81. z n z\in\mathbb{C}^{n}
  82. τ \tau\in\mathbb{H}
  83. \mathbb{H}
  84. n × n . \mathbb{C}^{n}\times\mathbb{H}_{n}.
  85. θ ( z + a + τ b , τ ) = exp 2 π i ( - b T z - 1 2 b T τ b ) θ ( z , τ ) \theta(z+a+\tau b,\tau)=\exp 2\pi i\left(-b^{T}z-\frac{1}{2}b^{T}\tau b\right)% \theta(z,\tau)
  86. a , b n a,b\in\mathbb{Z}^{n}
  87. z n z\in\mathbb{C}^{n}
  88. τ n \tau\in\mathbb{H}_{n}

Thomas_Muir_(mathematician).html

  1. det ( 𝐁 ) = pf 2 ( 𝐁 ) \det(\mathbf{B})=\operatorname{pf}^{2}(\mathbf{B})\,

Thompson_groups.html

  1. F T V F\subseteq T\subseteq V
  2. A , B [ A B - 1 , A - 1 B A ] = [ A B - 1 , A - 2 B A 2 ] = id \langle A,B\mid\ [AB^{-1},A^{-1}BA]=[AB^{-1},A^{-2}BA^{2}]=\mathrm{id}\rangle
  3. x 0 , x 1 , x 2 , x k - 1 x n x k = x n + 1 for k < n . \langle x_{0},x_{1},x_{2},\dots\ \mid\ x_{k}^{-1}x_{n}x_{k}=x_{n+1}\ \mathrm{% for}\ k<n\rangle.

Thompson_sporadic_group.html

  1. × 10 1 6 \times 10^{1}6
  2. T 3 C ( τ ) T_{3C}(\tau)
  3. T 3 C ( τ ) = ( j ( 3 τ ) ) 1 / 3 = 1 q + 248 q 2 + 4124 q 5 + 34752 q 8 + 213126 q 11 + 1057504 q 14 + T_{3C}(\tau)=\Big(j(3\tau)\Big)^{1/3}=\frac{1}{q}\,+\,248q^{2}\,+\,4124q^{5}\,% +\,34752q^{8}\,+\,213126q^{11}\,+\,1057504q^{14}+\cdots\,

Thomson_scattering.html

  1. ν m c 2 / h \nu\ll mc^{2}/h
  2. d V dV
  3. ϵ t = π σ t 2 I n \epsilon_{t}=\frac{\pi\sigma_{t}}{2}~{}I\,n
  4. ϵ r = π σ t 2 I n cos 2 χ \epsilon_{r}=\frac{\pi\sigma_{t}}{2}~{}I\,n\,\cos^{2}\chi
  5. σ t \sigma_{t}
  6. d V dV
  7. ϵ d Ω = 0 2 π d ϕ 0 π d χ ( ϵ t + ϵ r ) sin χ = I σ t n ( 8 / 3 ) ( π ) 2 \int\epsilon d\Omega=\int_{0}^{2\pi}d\phi\int_{0}^{\pi}d\chi\left(\epsilon_{t}% +\epsilon_{r}\right)\sin\chi=I\,\sigma_{t}\,n\,(8/3)(\pi)^{2}
  8. d σ t d Ω ( q 2 m c 2 ) 2 1 + cos 2 χ 2 = ( q 2 4 π ϵ 0 m c 2 ) 2 1 + cos 2 χ 2 \frac{d\sigma_{t}}{d\Omega}\equiv\left(\frac{q^{2}}{mc^{2}}\right)^{2}\frac{1+% \cos^{2}\chi}{2}=\left(\frac{q^{2}}{4\pi\epsilon_{0}mc^{2}}\right)^{2}\frac{1+% \cos^{2}\chi}{2}
  9. ϵ 0 \epsilon_{0}
  10. σ t = 8 π 3 ( q 2 m c 2 ) 2 = 8 π 3 ( q 2 4 π ϵ 0 m c 2 ) 2 \sigma_{t}=\frac{8\pi}{3}\left(\frac{q^{2}}{mc^{2}}\right)^{2}=\frac{8\pi}{3}% \left(\frac{q^{2}}{4\pi\epsilon_{0}mc^{2}}\right)^{2}
  11. σ t 8 π 3 r e 2 \sigma_{t}\equiv\frac{8\pi}{3}r_{e}^{2}
  12. λ c \lambda_{c}
  13. σ t = 8 π 3 ( α λ c 2 π ) 2 \sigma_{t}=\frac{8\pi}{3}\left(\frac{\alpha\lambda_{c}}{2\pi}\right)^{2}
  14. σ t = 8 π 3 ( α c m c 2 ) 2 = 6.652458734 × 10 - 29 m 2 = 66.52458734 (fm) 2 \sigma_{t}=\frac{8\pi}{3}\left(\frac{\alpha\hbar c}{mc^{2}}\right)^{2}=6.65245% 8734\ldots\times 10^{-29}~{}\textrm{m}^{2}=66.52458734\ldots~{}\textrm{(fm)}^{2}

Thue–Morse_sequence.html

  1. i = 0 ( 1 - x 2 i ) = j = 0 ( - 1 ) t j x j , \prod_{i=0}^{\infty}(1-x^{2^{i}})=\sum_{j=0}^{\infty}(-1)^{t_{j}}x^{j}\mbox{,}% ~{}\!
  2. X X
  3. X = A , A ¯ , A A ¯ A , X=A,\,\overline{A},\,A\overline{A}A,
  4. A ¯ A A ¯ \overline{A}A\overline{A}
  5. A = T k A=T_{k}
  6. k 0 k\geq 0
  7. A ¯ \overline{A}
  8. A A
  9. k = 0 k=0
  10. A = T 0 = 0 \ A=T_{0}=0
  11. A A ¯ A A A ¯ A = 010010 A\overline{A}AA\overline{A}A=010010
  12. T T
  13. T T
  14. t ( Z ) = n = 0 T ( n ) Z n . t(Z)=\sum_{n=0}^{\infty}T(n)Z^{n}\ .
  15. Z + ( 1 + Z ) 2 t + ( 1 + Z ) 3 t 2 = 0 Z+(1+Z)^{2}t+(1+Z)^{3}t^{2}=0
  16. n n
  17. t n = 0 t_{n}=0
  18. x S 0 x i = x S 1 x i \sum_{x\in S_{0}}x^{i}=\sum_{x\in S_{1}}x^{i}

Tiger_muskellunge.html

  1. W = c L b W=cL^{b}\!\,

Time_of_flight.html

  1. u u
  2. a a
  3. s = v t - 1 2 a t 2 s=vt-\begin{matrix}\frac{1}{2}\end{matrix}at^{2}
  4. t = 2 v sin θ a t=\frac{2v\sin\theta}{a}

Time_series.html

  1. Y t = α 0 + α 1 Y t - 1 + α 2 Y t - 2 + + α p Y t - p + ε t Y_{t}=\alpha_{0}+\alpha_{1}Y_{t-1}+\alpha_{2}Y_{t-2}+\cdots+\alpha_{p}Y_{t-p}+% \varepsilon_{t}\,
  2. E [ ε t ] = 0 , E[\varepsilon_{t}]=0\,,
  3. E [ ε t 2 ] = σ 2 , E[\varepsilon^{2}_{t}]=\sigma^{2}\,,
  4. E [ ε t ε s ] = 0 for all t s . E[\varepsilon_{t}\varepsilon_{s}]=0\quad\,\text{ for all }t\not=s\,.

Timothy_Gowers.html

  1. A { 1 , , N } A\subset\{1,\dots,N\}
  2. O ( N ( log log N ) - c k ) O(N(\log\log N)^{-c_{k}})
  3. c k > 0 c_{k}>0

Titration_curve.html

  1. p H = p K a + log ( [ base ] [ acid ] ) pH=pK_{a}+\log\left(\frac{[\mbox{base}~{}]}{[\mbox{acid}~{}]}\right)
  2. p H = p K a + log ( 1 ) pH=pK_{a}+\log(1)\,
  3. p H = p K a pH=pK_{a}\,

Tits_group.html

  1. × 10 7 \times 10^{7}
  2. a 2 = b 3 = ( a b ) 13 = [ a , b ] 5 = [ a , b a b ] 4 = ( a b a b a b a b a b - 1 ) 6 = 1 , a^{2}=b^{3}=(ab)^{13}=[a,b]^{5}=[a,bab]^{4}=(ababababab^{-1})^{6}=1,\,

Toffoli_gate.html

  1. [ 1 0 0 0 0 1 0 0 0 0 0 1 0 0 1 0 ] \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&0&1\\ 0&0&1&0\\ \end{bmatrix}
  2. [ 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 ] \begin{bmatrix}1&0&0&0&0&0&0&0\\ 0&1&0&0&0&0&0&0\\ 0&0&1&0&0&0&0&0\\ 0&0&0&1&0&0&0&0\\ 0&0&0&0&1&0&0&0\\ 0&0&0&0&0&1&0&0\\ 0&0&0&0&0&0&0&1\\ 0&0&0&0&0&0&1&0\\ \end{bmatrix}

Tolerant_sequence.html

  1. T 1 T_{1}
  2. T n T_{n}
  3. S 1 S_{1}
  4. S n S_{n}
  5. S i + 1 S_{i+1}
  6. S i S_{i}
  7. Π 1 \Pi_{1}

Tomography.html

  1. l [ θ , s ] ( t ) {l}_{[\theta,s]}(t)
  2. l [ θ , s ] ( t ) = t [ - sin θ cos θ ] + [ s cos θ s sin θ ] {l}_{[\theta,s]}(t)=t\begin{bmatrix}-\sin\theta\\ \cos\theta\\ \end{bmatrix}+\begin{bmatrix}s\cos\theta\\ s\sin\theta\\ \end{bmatrix}
  3. p θ ( s ) = - - μ ( s cos θ - t sin θ , s sin θ + t cos θ ) d t p_{\theta}(s)=-{\int}_{-\infty}^{\infty}\mu(s\cos\theta-t\sin\theta,s\sin% \theta+t\cos\theta)\,dt
  4. p ( s , θ ) p(s,\theta)

Toom–Cook_multiplication.html

  1. i = max { log b m / k , log b n / k } + 1. i=\max\{{\lfloor}{\lfloor}\log_{b}m{\rfloor}/k{\rfloor},{\lfloor}{\lfloor}\log% _{b}n{\rfloor}/k{\rfloor}\}+1.
  2. m 2 \displaystyle m_{2}
  3. p ( x ) = m 2 x 2 + m 1 x + m 0 = 123456 x 2 + 78901234 x + 56789012 p(x)=m_{2}x^{2}+m_{1}x+m_{0}=123456x^{2}+78901234x+56789012\,
  4. q ( x ) = n 2 x 2 + n 1 x + n 0 = 98765 x 2 + 43219876 x + 54321098 q(x)=n_{2}x^{2}+n_{1}x+n_{0}=98765x^{2}+43219876x+54321098\,
  5. i = max { log b m / k m , log b n / k n } . i=\max\{{\lfloor}{\lceil}\log_{b}m{\rceil}/k_{m}{\rfloor},{\lfloor}{\lceil}% \log_{b}n{\rceil}/k_{n}{\rfloor}\}.
  6. p ( 0 ) \displaystyle p(0)
  7. ( p ( 0 ) p ( 1 ) p ( - 1 ) p ( - 2 ) p ( ) ) = ( 0 0 0 1 0 2 1 0 1 1 1 2 ( - 1 ) 0 ( - 1 ) 1 ( - 1 ) 2 ( - 2 ) 0 ( - 2 ) 1 ( - 2 ) 2 0 0 1 ) ( m 0 m 1 m 2 ) = ( 1 0 0 1 1 1 1 - 1 1 1 - 2 4 0 0 1 ) ( m 0 m 1 m 2 ) . \left(\begin{matrix}p(0)\\ p(1)\\ p(-1)\\ p(-2)\\ p(\infty)\end{matrix}\right)=\left(\begin{matrix}0^{0}&0^{1}&0^{2}\\ 1^{0}&1^{1}&1^{2}\\ (-1)^{0}&(-1)^{1}&(-1)^{2}\\ (-2)^{0}&(-2)^{1}&(-2)^{2}\\ 0&0&1\end{matrix}\right)\left(\begin{matrix}m_{0}\\ m_{1}\\ m_{2}\end{matrix}\right)=\left(\begin{matrix}1&0&0\\ 1&1&1\\ 1&-1&1\\ 1&-2&4\\ 0&0&1\end{matrix}\right)\left(\begin{matrix}m_{0}\\ m_{1}\\ m_{2}\end{matrix}\right).
  8. ( r ( 0 ) r ( 1 ) r ( - 1 ) r ( - 2 ) r ( ) ) \displaystyle\left(\begin{matrix}r(0)\\ r(1)\\ r(-1)\\ r(-2)\\ r(\infty)\end{matrix}\right)
  9. ( r 0 r 1 r 2 r 3 r 4 ) \displaystyle\left(\begin{matrix}r_{0}\\ r_{1}\\ r_{2}\\ r_{3}\\ r_{4}\end{matrix}\right)
  10. r ( x ) \displaystyle r(x)
  11. ( 1 ) - 1 = ( 1 ) . \left(\begin{matrix}1\end{matrix}\right)^{-1}=\left(\begin{matrix}1\end{matrix% }\right).
  12. ( 1 0 0 1 ) - 1 = ( 1 0 0 1 ) . \left(\begin{matrix}1&0\\ 0&1\end{matrix}\right)^{-1}=\left(\begin{matrix}1&0\\ 0&1\end{matrix}\right).
  13. ( 1 0 0 1 1 1 0 0 1 ) - 1 = ( 1 0 0 - 1 1 - 1 0 0 1 ) . \left(\begin{matrix}1&0&0\\ 1&1&1\\ 0&0&1\end{matrix}\right)^{-1}=\left(\begin{matrix}1&0&0\\ -1&1&-1\\ 0&0&1\end{matrix}\right).
  14. ( 1 0 0 0 1 1 1 1 1 - 1 1 - 1 0 0 0 1 ) - 1 = ( 1 0 0 0 0 1 / 2 - 1 / 2 - 1 - 1 1 / 2 1 / 2 0 0 0 0 1 ) . \left(\begin{matrix}1&0&0&0\\ 1&1&1&1\\ 1&-1&1&-1\\ 0&0&0&1\end{matrix}\right)^{-1}=\left(\begin{matrix}1&0&0&0\\ 0&1/2&-1/2&-1\\ -1&1/2&1/2&0\\ 0&0&0&1\end{matrix}\right).

Top-down_parsing.html

  1. A a B C A\rightarrow aBC
  2. B c c d B\rightarrow c\mid cd
  3. C d f e g C\rightarrow df\mid eg
  4. A a B C A\rightarrow aBC
  5. B c c d B\rightarrow c\mid cd
  6. C d f e g C\rightarrow df\mid eg

Topological_quantum_field_theory.html

  1. S = M B F S=\int_{M}BF\,
  2. M A d A . \int_{M}A\wedge dA.
  3. S S
  4. δ \delta
  5. δ S = 0 \delta S=0
  6. δ 2 = 0 \delta^{2}=0
  7. O 1 , , O n O_{1},\dots,O_{n}
  8. δ O i = 0 \delta O_{i}=0
  9. i { 1 , , n } i\in\{1,\dots,n\}
  10. T α β = δ G α β T^{\alpha\beta}=\delta G^{\alpha\beta}
  11. G α β G^{\alpha\beta}
  12. B B
  13. δ \delta
  14. δ 2 = 0 \delta^{2}=0
  15. S = M B δ B S=\int_{M}B\wedge\delta B
  16. δ S = M δ ( B δ B ) = M δ B δ B + M B δ 2 B = 0 \delta S=\int_{M}\delta(B\wedge\delta B)=\int_{M}\delta B\wedge\delta B+\int_{% M}B\wedge\delta^{2}B=0
  17. δ \delta
  18. B B
  19. δ δ B α β S = M δ δ B α β B δ B + M B δ δ δ B α β B = M δ δ B α β B δ B - M δ B δ δ B α β B = 2 M δ B δ δ B α β B \frac{\delta}{\delta B^{\alpha\beta}}S=\int_{M}\frac{\delta}{\delta B^{\alpha% \beta}}B\wedge\delta B+\int_{M}B\wedge\delta\frac{\delta}{\delta B^{\alpha% \beta}}B=\int_{M}\frac{\delta}{\delta B^{\alpha\beta}}B\wedge\delta B-\int_{M}% \delta B\wedge\frac{\delta}{\delta B^{\alpha\beta}}B=2\int_{M}\delta B\wedge% \frac{\delta}{\delta B^{\alpha\beta}}B
  20. δ δ B α β S \frac{\delta}{\delta B^{\alpha\beta}}S
  21. δ G \delta G
  22. G G
  23. < O i > := d μ O i e i S <O_{i}>:=\int d\mu O_{i}e^{iS}
  24. μ \mu
  25. B B
  26. δ δ B < O i d μ O i i δ δ B S e i S d μ O i δ G e i S = δ ( d μ O i G e i S ) = 0 \frac{\delta}{\delta B}<O_{i}>=\int d\mu O_{i}i\frac{\delta}{\delta B}Se^{iS}% \propto\int d\mu O_{i}\delta Ge^{iS}=\delta(\int d\mu O_{i}Ge^{iS})=0
  27. δ O i = δ S = 0 \delta O_{i}=\delta S=0
  28. d μ O i G e i S \int d\mu O_{i}Ge^{iS}
  29. Z ¯ ( M ) \overline{Z}{(}{M}{)}
  30. Z ( Σ f ) = Trace Σ ( f ) Z(\Sigma_{f})=\,\text{Trace}\ \Sigma(f)
  31. M Σ M * M\cup_{\Sigma}M^{*}
  32. Z ( M Σ M * ) = | Z ( M ) | 2 Z(M\cup_{\Sigma}M^{*})=|Z(M)|^{2}
  33. M = Σ 0 * Σ 1 \partial M=\Sigma^{*}_{0}\cup\Sigma_{1}

Torque_converter.html

  1. r N 2 D 5 r\,N^{2}D^{5}
  2. r r
  3. N N
  4. D D

Torsion_spring.html

  1. τ = - κ θ \tau=-\kappa\theta\,
  2. τ \tau\,
  3. θ \theta\,
  4. κ \kappa\,
  5. U = 1 2 κ θ 2 U=\frac{1}{2}\kappa\theta^{2}
  6. θ \theta\,
  7. I I\,
  8. kg m 2 \mathrm{kg\,m^{2}}\,
  9. C C\,
  10. kg m 2 s - 1 rad - 1 \mathrm{kg\,m^{2}\,s^{-1}\,{rad}^{-1}}\,
  11. κ \kappa\,
  12. N m rad - 1 \mathrm{N\,m\,{rad}^{-1}}\,
  13. τ \tau\,
  14. N m \mathrm{N\,m}\,
  15. f n f_{n}\,
  16. T n T_{n}\,
  17. ω n \omega_{n}\,
  18. rad s - 1 \mathrm{rad\,s^{-1}}\,
  19. f f\,
  20. ω \omega\,
  21. rad s - 1 \mathrm{rad\,s^{-1}}\,
  22. α \alpha\,
  23. s - 1 \mathrm{s^{-1}}\,
  24. ϕ \phi\,
  25. L L\,
  26. I d 2 θ d t 2 + C d θ d t + κ θ = τ ( t ) I\frac{d^{2}\theta}{dt^{2}}+C\frac{d\theta}{dt}+\kappa\theta=\tau(t)
  27. C κ I C\ll\sqrt{\frac{\kappa}{I}}\,
  28. f n = ω n 2 π = 1 2 π κ I f_{n}=\frac{\omega_{n}}{2\pi}=\frac{1}{2\pi}\sqrt{\frac{\kappa}{I}}\,
  29. T n = 1 f n = 2 π ω n = 2 π I κ T_{n}=\frac{1}{f_{n}}=\frac{2\pi}{\omega_{n}}=2\pi\sqrt{\frac{I}{\kappa}}\,
  30. τ = 0 \tau=0\,
  31. θ = A e - α t cos ( ω t + ϕ ) \theta=Ae^{-\alpha t}\cos{(\omega t+\phi)}\,
  32. α = C / 2 I \alpha=C/2I\,
  33. ω = ω n 2 - α 2 = κ / I - ( C / 2 I ) 2 \omega=\sqrt{\omega_{n}^{2}-\alpha^{2}}=\sqrt{\kappa/I-(C/2I)^{2}}\,
  34. f n f_{n}\,
  35. I I\,
  36. κ \kappa\,
  37. F F\,
  38. L L\,
  39. τ ( t ) = F L \tau(t)=FL\,
  40. θ = F L / κ \theta=FL/\kappa\,
  41. F F\,
  42. κ \kappa\,
  43. κ = ( 2 π f n ) 2 I \kappa=(2\pi f_{n})^{2}I\,
  44. C c C_{c}\,
  45. C c = 2 κ I C_{c}=2\sqrt{\kappa I}\,

Touchard_polynomials.html

  1. T n ( x ) = k = 0 n S ( n , k ) x k = k = 0 n { n k } x k , T_{n}(x)=\sum_{k=0}^{n}S(n,k)x^{k}=\sum_{k=0}^{n}\left\{{n\atop k}\right\}x^{k},
  2. S ( n , k ) = { n k } S(n,k)=\left\{{n\atop k}\right\}
  3. T n ( 1 ) = B n . T_{n}(1)=B_{n}.
  4. T n ( x ) = e - x k = 0 x k k n k ! . T_{n}(x)=e^{-x}\sum_{k=0}^{\infty}\frac{x^{k}k^{n}}{k!}.
  5. T n ( λ + μ ) = k = 0 n ( n k ) T k ( λ ) T n - k ( μ ) . T_{n}(\lambda+\mu)=\sum_{k=0}^{n}{n\choose k}T_{k}(\lambda)T_{n-k}(\mu).
  6. T n ( e x ) = e - e x d n d x n ( e e x ) T_{n}\left(e^{x}\right)=e^{-e^{x}}\frac{d^{n}}{dx^{n}}\left(e^{e^{x}}\right)
  7. T n + 1 ( x ) = x ( 1 + d d x ) T n ( x ) T_{n+1}(x)=x\left(1+\frac{d}{dx}\right)T_{n}(x)
  8. T n + 1 ( x ) = x k = 0 n ( n k ) T k ( x ) . T_{n+1}(x)=x\sum_{k=0}^{n}{n\choose k}T_{k}(x).
  9. T n ( λ + μ ) = ( T ( λ ) + T ( μ ) ) n , T_{n}(\lambda+\mu)=\left(T(\lambda)+T(\mu)\right)^{n},
  10. T n + 1 ( x ) = x ( 1 + T ( x ) ) n . T_{n+1}(x)=x\left(1+T(x)\right)^{n}.
  11. n = 0 T n ( x ) n ! t n = e x ( e t - 1 ) , \sum_{n=0}^{\infty}{T_{n}(x)\over n!}t^{n}=e^{x\left(e^{t}-1\right)},
  12. T n ( x ) = n ! 2 π i e x ( e t - 1 ) t n + 1 d t . T_{n}(x)=\frac{n!}{2\pi i}\oint\frac{e^{x({e^{t}}-1)}}{t^{n+1}}\,\mathrm{d}t.
  13. B n ( x 1 , x 2 , , x n ) B_{n}(x_{1},x_{2},\dots,x_{n})
  14. T n ( x ) T_{n}(x)
  15. T n ( x ) = B n ( x , x , , x ) . T_{n}(x)=B_{n}(x,x,\dots,x).
  16. T n ( x ) = n ! π 0 π e x ( e cos ( θ ) cos ( sin ( θ ) ) - 1 ) cos ( x e cos ( θ ) sin ( sin ( θ ) ) - n θ ) d θ T_{n}(x)=\frac{n!}{\pi}\int^{\pi}_{0}e^{x\bigl(e^{\cos(\theta)}\cos(\sin(% \theta))-1\bigr)}\cos\bigl(xe^{\cos(\theta)}\sin(\sin(\theta))-n\theta\bigr)\,% \mathrm{d}\theta

Trachtenberg_system.html

  1. a × b a\times b
  2. a a
  3. b b
  4. a a
  5. b b
  6. n n
  7. i i
  8. a (digit at i ) × b (digit at ( n - i ) ) . a\,\text{ (digit at }i\,\text{ )}\times b\,\text{ (digit at }(n-i)\,\text{)}.
  9. 123456 × 789 123456\times 789
  10. 9 × 6 = 4 9\times 6=4
  11. 9 × 5 9\times 5
  12. 9 × 6 9\times 6
  13. 8 × 6 8\times 6
  14. 5 + 5 + 8 = 18 5+5+8=18
  15. 8 8
  16. 1 1
  17. 9 × 3 9\times 3
  18. 9 × 4 9\times 4
  19. 8 × 4 8\times 4
  20. 8 × 5 8\times 5
  21. 7 × 5 7\times 5
  22. 7 × 6 7\times 6
  23. 7 + 3 + 2 + 4 + 5 + 4 = 25 + 1 7+3+2+4+5+4=25+1
  24. 6 6
  25. 2 2

Tractive_force.html

  1. t = c P d 2 s D t=\frac{cPd^{2}s}{D}

Tractography.html

  1. d 𝐫 ( s ) d s = 𝐓 ( s ) \frac{d{\mathbf{r}}(s)}{ds}={\mathbf{T}}(s)
  2. 𝐓 ( s ) {\mathbf{T}}(s)
  3. D D
  4. λ 1 , λ 2 , λ 3 \lambda_{1},\lambda_{2},\lambda_{3}
  5. 𝐮 1 , 𝐮 2 , 𝐮 3 {\mathbf{u}}_{1},{\mathbf{u}}_{2},{\mathbf{u}}_{3}
  6. d 𝐫 ( s ) d s = 𝐮 1 ( 𝐫 ( s ) ) \frac{d{\mathbf{r}}(s)}{ds}={\mathbf{u}}_{1}({\mathbf{r}}(s))
  7. 𝐫 ( s ) {\mathbf{r}}(s)
  8. 𝐮 1 ( s ) {\mathbf{u}}_{1}(s)

Transferase.html

  1. X g r o u p + Y t r a n s f e r a s e X + Y g r o u p Xgroup+Y\xrightarrow[transferase]{}X+Ygroup
  2. \rightleftharpoons
  3. \rightarrow
  4. \rightleftharpoons
  5. \rightleftharpoons
  6. \rightleftharpoons
  7. \rightleftharpoons
  8. \rightleftharpoons
  9. \rightarrow
  10. \rightleftharpoons
  11. \rightarrow

Transmittance.html

  1. T = Φ e t Φ e i , T=\frac{\Phi_{\mathrm{e}}^{\mathrm{t}}}{\Phi_{\mathrm{e}}^{\mathrm{i}}},
  2. T ν = Φ e , ν t Φ e , ν i , T_{\nu}=\frac{\Phi_{\mathrm{e},\nu}^{\mathrm{t}}}{\Phi_{\mathrm{e},\nu}^{% \mathrm{i}}},
  3. T λ = Φ e , λ t Φ e , λ i , T_{\lambda}=\frac{\Phi_{\mathrm{e},\lambda}^{\mathrm{t}}}{\Phi_{\mathrm{e},% \lambda}^{\mathrm{i}}},
  4. T Ω = L e , Ω t L e , Ω i , T_{\Omega}=\frac{L_{\mathrm{e},\Omega}^{\mathrm{t}}}{L_{\mathrm{e},\Omega}^{% \mathrm{i}}},
  5. T ν , Ω = L e , Ω , ν t L e , Ω , ν i , T_{\nu,\Omega}=\frac{L_{\mathrm{e},\Omega,\nu}^{\mathrm{t}}}{L_{\mathrm{e},% \Omega,\nu}^{\mathrm{i}}},
  6. T λ , Ω = L e , Ω , λ t L e , Ω , λ i , T_{\lambda,\Omega}=\frac{L_{\mathrm{e},\Omega,\lambda}^{\mathrm{t}}}{L_{% \mathrm{e},\Omega,\lambda}^{\mathrm{i}}},
  7. T = e - τ = 10 - A , T=e^{-\tau}=10^{-A},
  8. T = e - i = 1 N σ i 0 n i ( z ) d z = 10 - i = 1 N ε i 0 c i ( z ) d z , T=e^{-\sum_{i=1}^{N}\sigma_{i}\int_{0}^{\ell}n_{i}(z)\mathrm{d}z}=10^{-\sum_{i% =1}^{N}\varepsilon_{i}\int_{0}^{\ell}c_{i}(z)\mathrm{d}z},
  9. τ = i = 1 N τ i = i = 1 N σ i 0 n i ( z ) d z , \tau=\sum_{i=1}^{N}\tau_{i}=\sum_{i=1}^{N}\sigma_{i}\int_{0}^{\ell}n_{i}(z)\,% \mathrm{d}z,
  10. A = i = 1 N A i = i = 1 N ε i 0 c i ( z ) d z , A=\sum_{i=1}^{N}A_{i}=\sum_{i=1}^{N}\varepsilon_{i}\int_{0}^{\ell}c_{i}(z)\,% \mathrm{d}z,
  11. ε i = N A ln 10 σ i , \varepsilon_{i}=\frac{\mathrm{N_{A}}}{\ln{10}}\,\sigma_{i},
  12. c i = n i N A , c_{i}=\frac{n_{i}}{\mathrm{N_{A}}},
  13. T = e - i = 1 N σ i n i = 10 - i = 1 N ε i c i , T=e^{-\sum_{i=1}^{N}\sigma_{i}n_{i}\ell}=10^{-\sum_{i=1}^{N}\varepsilon_{i}c_{% i}\ell},
  14. τ = i = 1 N σ i n i , \tau=\sum_{i=1}^{N}\sigma_{i}n_{i}\ell,
  15. A = i = 1 N ε i c i . A=\sum_{i=1}^{N}\varepsilon_{i}c_{i}\ell.

Transparency_(graphic).html

  1. ( G 1 + G 2 ) / 2 (G1+G2)/2
  2. ( G 1 + G 1 ) / 2 = G 1 (G1+G1)/2=G1
  3. G 2 / 2 G2/2
  4. ( G 2 + 1 ) / 2 (G2+1)/2
  5. ( G 1 + G 2 ) / 2 = ( G 2 + G 1 ) / 2 (G1+G2)/2=(G2+G1)/2
  6. r e d = ( R 1 + R 2 ) / 2 red=(R1+R2)/2
  7. G 2 G2
  8. T 2 T2
  9. G 1 G1
  10. T 1 T1
  11. ( 1 - T 2 ) * G 1 + G 2 (1-T2)*G1+G2
  12. 1 - ( 1 - T 2 ) * ( 1 - T 1 ) 1-(1-T2)*(1-T1)

Transport_function.html

  1. J ( n , x ) = 0 x t n e t ( e t - 1 ) 2 d t . J(n,x)=\int_{0}^{x}t^{n}\frac{e^{t}}{(e^{t}-1)^{2}}\,dt.

Transverse_mode.html

  1. I p l ( ρ , φ ) = I 0 ρ l [ L p l ( ρ ) ] 2 cos 2 ( l φ ) e - ρ I_{pl}(\rho,\varphi)=I_{0}\rho^{l}\left[L_{p}^{l}(\rho)\right]^{2}\cos^{2}(l% \varphi)e^{-\rho}
  2. E m n ( x , y , z ) = E 0 w 0 w H m ( 2 x w ) H n ( 2 y w ) exp [ - ( x 2 + y 2 ) ( 1 w 2 + j k 2 R ) - j k z - j ( m + n + 1 ) ζ ( z ) ] E_{mn}(x,y,z)=E_{0}\frac{w_{0}}{w}H_{m}\left(\frac{\sqrt{2}x}{w}\right)H_{n}% \left(\frac{\sqrt{2}y}{w}\right)\exp\left[-(x^{2}+y^{2})\left(\frac{1}{w^{2}}+% \frac{jk}{2R}\right)-jkz-j(m+n+1)\zeta(z)\right]
  3. w 0 w_{0}
  4. w w
  5. R R
  6. ζ ( z ) \zeta(z)
  7. E 0 E_{0}
  8. H k H_{k}
  9. I m n ( x , y , z ) = I 0 [ H m ( 2 x w ) exp ( - x 2 w 2 ) ] 2 [ H n ( 2 y w ) exp ( - y 2 w 2 ) ] 2 I_{mn}(x,y,z)=I_{0}\left[H_{m}\left(\frac{\sqrt{2}x}{w}\right)\exp\left(\frac{% -x^{2}}{w^{2}}\right)\right]^{2}\left[H_{n}\left(\frac{\sqrt{2}y}{w}\right)% \exp\left(\frac{-y^{2}}{w^{2}}\right)\right]^{2}
  10. V = k 0 a n 1 2 - n 2 2 V=k_{0}a\sqrt{n_{1}^{2}-n_{2}^{2}}
  11. k 0 k_{0}
  12. a a
  13. n 1 n_{1}
  14. n 2 n_{2}

Trapezoidal_rule.html

  1. a b f ( x ) d x . \int_{a}^{b}f(x)\,dx.
  2. f ( x ) f(x)
  3. a b f ( x ) d x ( b - a ) [ f ( a ) + f ( b ) 2 ] . \int_{a}^{b}f(x)\,dx\approx(b-a)\left[\frac{f(a)+f(b)}{2}\right].
  4. a b f ( x ) d x h 2 k = 1 N ( f ( x k + 1 ) + f ( x k ) ) \int_{a}^{b}f(x)\,dx\approx\frac{h}{2}\sum_{k=1}^{N}\left(f(x_{k+1})+f(x_{k})\right)
  5. = b - a 2 N ( f ( x 1 ) + 2 f ( x 2 ) + 2 f ( x 3 ) + 2 f ( x 4 ) + + 2 f ( x N ) + f ( x N + 1 ) ) . {}=\frac{b-a}{2N}(f(x_{1})+2f(x_{2})+2f(x_{3})+2f(x_{4})+\cdots+2f(x_{N})+f(x_% {N+1})).
  6. a b f ( x ) d x 1 2 k = 1 N ( x k + 1 - x k ) ( f ( x k + 1 ) + f ( x k ) ) . \int_{a}^{b}f(x)\,dx\approx\frac{1}{2}\sum_{k=1}^{N}\left(x_{k+1}-x_{k}\right)% \left(f(x_{k+1})+f(x_{k})\right).
  7. error = a b f ( x ) d x - b - a N [ f ( a ) + f ( b ) 2 + k = 1 N - 1 f ( a + k b - a N ) ] \,\text{error}=\int_{a}^{b}f(x)\,dx-\frac{b-a}{N}\left[{f(a)+f(b)\over 2}+\sum% _{k=1}^{N-1}f\left(a+k\frac{b-a}{N}\right)\right]
  8. error = - ( b - a ) 3 12 N 2 f ′′ ( ξ ) \,\text{error}=-\frac{(b-a)^{3}}{12N^{2}}f^{\prime\prime}(\xi)
  9. error = - ( b - a ) 2 12 N 2 [ f ( b ) - f ( a ) ] + O ( N - 3 ) . \,\text{error}=-\frac{(b-a)^{2}}{12N^{2}}\big[f^{\prime}(b)-f^{\prime}(a)\big]% +O(N^{-3}).

Tree-adjoining_grammar.html

  1. α \alpha
  2. β \beta
  3. { a n b n c n d n | 1 n } \{a^{n}b^{n}c^{n}d^{n}|1\leq n\}

Tree_traversal.html

  1. ( ) , (),
  2. ( 1 ) , ( 2 ) , , (1),(2),\dots,
  3. ( 1 , 1 ) , ( 1 , 2 ) , , ( 2 , 1 ) , ( 2 , 2 ) , , (1,1),(1,2),\ldots,(2,1),(2,2),\ldots,

Trefoil_knot.html

  1. x = sin t + 2 sin 2 t x=\sin t+2\sin 2t
  2. y = cos t - 2 cos 2 t \qquad y=\cos t-2\cos 2t
  3. z = - sin 3 t \qquad z=-\sin 3t
  4. ( r - 2 ) 2 + z 2 = 1 (r-2)^{2}+z^{2}=1
  5. x = ( 2 + cos 3 t ) cos 2 t x=(2+\cos 3t)\cos 2t
  6. y = ( 2 + cos 3 t ) sin 2 t \qquad y=(2+\cos 3t)\sin 2t
  7. z = sin 3 t \qquad z=\sin 3t
  8. S 3 S^{3}
  9. S 1 S^{1}
  10. ( z , w ) (z,w)
  11. | z | 2 + | w | 2 = 1 |z|^{2}+|w|^{2}=1
  12. z 2 + w 3 = 0 z^{2}+w^{3}=0
  13. ϕ ( z , w ) = ( z 2 + w 3 ) / | z 2 + w 3 | \phi(z,w)=(z^{2}+w^{3})/|z^{2}+w^{3}|
  14. Δ ( t ) = t - 1 + t - 1 , \Delta(t)=t-1+t^{-1},\,
  15. ( z ) = z 2 + 1. \nabla(z)=z^{2}+1.
  16. V ( q ) = q - 1 + q - 3 - q - 4 , V(q)=q^{-1}+q^{-3}-q^{-4},\,
  17. L ( a , z ) = z a 5 + z 2 a 4 - a 4 + z a 3 + z 2 a 2 - 2 a 2 . L(a,z)=za^{5}+z^{2}a^{4}-a^{4}+za^{3}+z^{2}a^{2}-2a^{2}.\,
  18. x , y x 2 = y 3 \langle x,y\mid x^{2}=y^{3}\rangle\,
  19. x , y x y x = y x y . \langle x,y\mid xyx=yxy\rangle.\,

Trend_estimation.html

  1. t t
  2. y t y_{t}
  3. a a
  4. b b
  5. t { [ ( a t + b ) - y t ] 2 } \sum_{t}\left\{[(at+b)-y_{t}]^{2}\right\}
  6. y t = a t + b + e t y_{t}=at+b+e_{t}\,
  7. a a
  8. b b
  9. e e
  10. e e
  11. e e
  12. a t + b at+b
  13. y t y_{t}
  14. e t e_{t}
  15. e t e_{t}
  16. e t e_{t}
  17. a a
  18. a a
  19. V V
  20. S S

TRIAC.html

  1. d v d t dv\over dt
  2. ( d v d t ) s \left(\frac{\operatorname{d}v}{\operatorname{d}t}\right)_{s}
  3. d i d t \frac{\operatorname{d}i}{\operatorname{d}t}
  4. ( d v d t ) c \left(\frac{\operatorname{d}v}{\operatorname{d}t}\right)_{c}
  5. ( d i d t ) c \left(\frac{\operatorname{d}i}{\operatorname{d}t}\right)_{c}
  6. V g t V_{gt}
  7. I g t I_{gt}
  8. V d r m V_{drm}
  9. V r r m V_{rrm}
  10. I t I_{t}
  11. I t s m I_{tsm}
  12. V t V_{t}

Trial_division.html

  1. n = 12 n=12
  2. 12 = 3 × 4 12=3×4
  3. n \scriptstyle\sqrt{n}
  4. π ( 2 n / 2 ) 2 n / 2 ( n 2 ) ln 2 \pi(2^{n/2})\approx{2^{n/2}\over\left(\frac{n}{2}\right)\ln 2}
  5. π ( x ) \scriptstyle\pi(x)
  6. 2 n / 2 2^{n/2}
  7. a = 1000 a=1000
  8. n = 10 n=10

Triangular_matrix.html

  1. L = [ 1 , 1 0 2 , 1 2 , 2 3 , 1 3 , 2 n , 1 n , 2 n , n - 1 n , n ] L=\begin{bmatrix}\ell_{1,1}&&&&0\\ \ell_{2,1}&\ell_{2,2}&&&\\ \ell_{3,1}&\ell_{3,2}&\ddots&&\\ \vdots&\vdots&\ddots&\ddots&\\ \ell_{n,1}&\ell_{n,2}&\ldots&\ell_{n,n-1}&\ell_{n,n}\end{bmatrix}
  2. U = [ u 1 , 1 u 1 , 2 u 1 , 3 u 1 , n u 2 , 2 u 2 , 3 u 2 , n u n - 1 , n 0 u n , n ] U=\begin{bmatrix}u_{1,1}&u_{1,2}&u_{1,3}&\ldots&u_{1,n}\\ &u_{2,2}&u_{2,3}&\ldots&u_{2,n}\\ &&\ddots&\ddots&\vdots\\ &&&\ddots&u_{n-1,n}\\ 0&&&&u_{n,n}\end{bmatrix}
  3. [ 1 4 2 0 3 4 0 0 1 ] \begin{bmatrix}1&4&2\\ 0&3&4\\ 0&0&1\\ \end{bmatrix}
  4. [ 1 0 0 2 8 0 4 9 7 ] \begin{bmatrix}1&0&0\\ 2&8&0\\ 4&9&7\\ \end{bmatrix}
  5. 𝔫 . \mathfrak{n}.
  6. 𝔟 \mathfrak{b}
  7. 𝔫 = [ 𝔟 , 𝔟 ] . \mathfrak{n}=[\mathfrak{b},\mathfrak{b}].
  8. 𝔫 \mathfrak{n}
  9. 𝐋 i = [ 1 0 0 0 1 0 0 1 0 i + 1 , i 1 0 i + 2 , i 0 1 0 0 n , i 0 0 1 ] . \mathbf{L}_{i}=\begin{bmatrix}1&&&&&&&0\\ 0&\ddots&&&&&&\\ 0&\ddots&1&&&&&\\ 0&\ddots&0&1&&&&\\ &&0&\ell_{i+1,i}&1&&&\\ \vdots&&0&\ell_{i+2,i}&0&\ddots&&\\ &&\vdots&\vdots&\vdots&\ddots&1&\\ 0&\dots&0&\ell_{n,i}&0&\dots&0&1\\ \end{bmatrix}.
  10. 𝐋 i - 1 = [ 1 0 0 0 1 0 0 1 0 - i + 1 , i 1 0 - i + 2 , i 0 1 0 0 - n , i 0 0 1 ] , \mathbf{L}_{i}^{-1}=\begin{bmatrix}1&&&&&&&0\\ 0&\ddots&&&&&&\\ 0&\ddots&1&&&&&\\ 0&\ddots&0&1&&&&\\ &&0&-\ell_{i+1,i}&1&&&\\ \vdots&&0&-\ell_{i+2,i}&0&\ddots&&\\ &&\vdots&\vdots&\vdots&\ddots&1&\\ 0&\dots&0&-\ell_{n,i}&0&\dots&0&1\\ \end{bmatrix},
  11. [ 1 0 0 0 0 1 0 0 0 4 1 0 0 2 0 1 ] \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&4&1&0\\ 0&2&0&1\\ \end{bmatrix}
  12. [ 1 0 0 0 0 1 0 0 0 - 4 1 0 0 - 2 0 1 ] . \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&-4&1&0\\ 0&-2&0&1\\ \end{bmatrix}.
  13. x I - A xI-A
  14. ( e 1 , , e n ) (e_{1},\ldots,e_{n})
  15. 0 < e 1 < e 1 , e 2 < < e 1 , , e n = K n . 0<\left\langle e_{1}\right\rangle<\left\langle e_{1},e_{2}\right\rangle<\cdots% <\left\langle e_{1},\ldots,e_{n}\right\rangle=K^{n}.
  16. A 1 , , A k A_{1},\ldots,A_{k}
  17. A i , A_{i},
  18. K [ A 1 , , A k ] . K[A_{1},\ldots,A_{k}].
  19. A , B A,B
  20. A 1 , , A k A_{1},\ldots,A_{k}
  21. K [ A 1 , , A k ] K[A_{1},\ldots,A_{k}]
  22. K [ x 1 , , x k ] K[x_{1},\ldots,x_{k}]
  23. A 1 , , A k A_{1},\ldots,A_{k}
  24. p ( A 1 , , A k ) [ A i , A j ] p(A_{1},\ldots,A_{k})[A_{i},A_{j}]
  25. [ A i , A j ] [A_{i},A_{j}]
  26. A i A_{i}
  27. [ A i , A j ] [A_{i},A_{j}]
  28. A k A_{k}
  29. 2 n 2^{n}
  30. ± 1 \pm 1
  31. 𝔟 \mathfrak{b}
  32. 𝐋𝐱 = 𝐛 \mathbf{L}\mathbf{x}=\mathbf{b}
  33. 𝐔𝐱 = 𝐛 \mathbf{U}\mathbf{x}=\mathbf{b}
  34. x 1 x_{1}
  35. x 2 x_{2}
  36. x n x_{n}
  37. x n x_{n}
  38. x n - 1 x_{n-1}
  39. x 1 x_{1}
  40. 1 , 1 x 1 = b 1 2 , 1 x 1 + 2 , 2 x 2 = b 2 m , 1 x 1 + m , 2 x 2 + + m , m x m = b m \begin{matrix}\ell_{1,1}x_{1}&&&&&=&b_{1}\\ \ell_{2,1}x_{1}&+&\ell_{2,2}x_{2}&&&=&b_{2}\\ \vdots&&\vdots&\ddots&&&\vdots\\ \ell_{m,1}x_{1}&+&\ell_{m,2}x_{2}&+\cdots+&\ell_{m,m}x_{m}&=&b_{m}\\ \end{matrix}
  41. 1 , 1 x 1 = b 1 \ell_{1,1}x_{1}=b_{1}
  42. x 1 x_{1}
  43. x 1 x_{1}
  44. x 1 x_{1}
  45. x 2 x_{2}
  46. x 1 x_{1}
  47. k k
  48. x 1 , , x k x_{1},\dots,x_{k}
  49. x k x_{k}
  50. x 1 , , x k - 1 x_{1},\dots,x_{k-1}
  51. x 1 = b 1 1 , 1 , x_{1}=\frac{b_{1}}{\ell_{1,1}},
  52. x 2 = b 2 - 2 , 1 x 1 2 , 2 , x_{2}=\frac{b_{2}-\ell_{2,1}x_{1}}{\ell_{2,2}},
  53. \vdots
  54. x m = b m - i = 1 m - 1 m , i x i m , m . x_{m}=\frac{b_{m}-\sum_{i=1}^{m-1}\ell_{m,i}x_{i}}{\ell_{m,m}}.

Trichotomy_(mathematics).html

  1. x = y x=y
  2. x > y x>y
  3. x X y X ( ( x < y ¬ ( y < x ) ¬ ( x = y ) ) ( ¬ ( x < y ) y < x ¬ ( x = y ) ) ( ¬ ( x < y ) ¬ ( y < x ) x = y ) ) . \forall x\in X\,\forall y\in X\,((x<y\,\land\,\lnot(y<x)\,\land\,\lnot(x=y)\,)% \lor\,(\lnot(x<y)\,\land\,y<x\,\land\,\lnot(x=y)\,)\lor\,(\lnot(x<y)\,\land\,% \lnot(y<x)\,\land\,x=y\,\,))\,.
  4. x X y X ( ( x < y ) ( y < x ) ( x = y ) ) . \forall x\in X\,\forall y\in X\,((x<y)\,\lor\,(y<x)\,\lor\,(x=y))\,.

Tridiagonal_matrix.html

  1. ( 1 4 0 0 3 4 1 0 0 2 3 4 0 0 1 3 ) . \begin{pmatrix}1&4&0&0\\ 3&4&1&0\\ 0&2&3&4\\ 0&0&1&3\\ \end{pmatrix}.
  2. f n = | a 1 b 1 c 1 a 2 b 2 c 2 b n - 1 c n - 1 a n | . f_{n}=\begin{vmatrix}a_{1}&b_{1}\\ c_{1}&a_{2}&b_{2}\\ &c_{2}&\ddots&\ddots\\ &&\ddots&\ddots&b_{n-1}\\ &&&c_{n-1}&a_{n}\end{vmatrix}.
  3. f n = a n f n - 1 - c n - 1 b n - 1 f n - 2 f_{n}=a_{n}f_{n-1}-c_{n-1}b_{n-1}f_{n-2}
  4. T = ( a 1 b 1 c 1 a 2 b 2 c 2 b n - 1 c n - 1 a n ) T=\begin{pmatrix}a_{1}&b_{1}\\ c_{1}&a_{2}&b_{2}\\ &c_{2}&\ddots&\ddots\\ &&\ddots&\ddots&b_{n-1}\\ &&&c_{n-1}&a_{n}\end{pmatrix}
  5. ( T - 1 ) i j = { ( - 1 ) i + j b i b j - 1 θ i - 1 ϕ j + 1 / θ n if i j ( - 1 ) i + j c j c i - 1 θ j - 1 ϕ i + 1 / θ n if i > j (T^{-1})_{ij}=\begin{cases}(-1)^{i+j}b_{i}\cdots b_{j-1}\theta_{i-1}\phi_{j+1}% /\theta_{n}&\,\text{ if }i\leq j\\ (-1)^{i+j}c_{j}\cdots c_{i-1}\theta_{j-1}\phi_{i+1}/\theta_{n}&\,\text{ if }i>% j\\ \end{cases}
  6. θ i = a i θ i - 1 - b i - 1 c i - 1 θ i - 2 for i = 2 , 3 , , n \theta_{i}=a_{i}\theta_{i-1}-b_{i-1}c_{i-1}\theta_{i-2}\quad\,\text{ for }i=2,% 3,\ldots,n
  7. ϕ i = a i ϕ i + 1 - b i c i ϕ i + 2 for i = n - 1 , , 1 \phi_{i}=a_{i}\phi_{i+1}-b_{i}c_{i}\phi_{i+2}\quad\,\text{ for }i=n-1,\ldots,1
  8. b \reals n \scriptstyle b\in\reals^{n}
  9. a + 2 b c cos ( k π / ( n + 1 ) ) a+2\sqrt{bc}\,\cos(k\pi/{(n+1)})
  10. k = 1 , , n . k=1,...,n.
  11. a - 2 b c cos ( k π / ( n + 1 ) ) a-2\sqrt{bc}\,\cos(k\pi/{(n+1)})
  12. cos ( x ) = - cos ( π - x ) \cos(x)=-\cos(\pi-x)

Trigonometric_polynomial.html

  1. T ( x ) = a 0 + n = 1 N a n cos ( n x ) + i n = 1 N b n sin ( n x ) ( x 𝐑 ) T(x)=a_{0}+\sum_{n=1}^{N}a_{n}\cos(nx)+\mathrm{i}\sum_{n=1}^{N}b_{n}\sin(nx)% \qquad(x\in\mathbf{R})
  2. T ( x ) = n = - N N c n e i n x ( x 𝐑 ) . T(x)=\sum_{n=-N}^{N}c_{n}\mathrm{e}^{\mathrm{i}nx}\qquad(x\in\mathbf{R}).
  3. t ( x ) = a 0 + n = 1 N a n cos ( n x ) + n = 1 N b n sin ( n x ) ( x 𝐑 ) t(x)=a_{0}+\sum_{n=1}^{N}a_{n}\cos(nx)+\sum_{n=1}^{N}b_{n}\sin(nx)\qquad(x\in% \mathbf{R})

Trigonometric_substitution.html

  1. x = a sin ( θ ) x=a\sin(\theta)
  2. 1 - sin 2 ( θ ) = cos 2 ( θ ) . 1-\sin^{2}(\theta)=\cos^{2}(\theta).
  3. x = a tan ( θ ) x=a\tan(\theta)
  4. 1 + tan 2 ( θ ) = sec 2 ( θ ) . 1+\tan^{2}(\theta)=\sec^{2}(\theta).
  5. x = a sec ( θ ) x=a\sec(\theta)
  6. sec 2 ( θ ) - 1 = tan 2 ( θ ) . \sec^{2}(\theta)-1=\tan^{2}(\theta).
  7. d x a 2 - x 2 \int\frac{\mathrm{d}x}{\sqrt{a^{2}-x^{2}}}
  8. x = a sin ( θ ) , d x = a cos ( θ ) d θ , θ = arcsin ( x a ) x=a\sin(\theta),\quad\mathrm{d}x=a\cos(\theta)\,\mathrm{d}\theta,\quad\theta=% \arcsin\left(\frac{x}{a}\right)
  9. d x a 2 - x 2 \displaystyle\int\frac{\mathrm{d}x}{\sqrt{a^{2}-x^{2}}}
  10. d x < m t p l > a 2 + x 2 \int\frac{\mathrm{d}x}{<}mtpl>{{a^{2}+x^{2}}}
  11. x = a tan ( θ ) , d x = a sec 2 ( θ ) d θ , θ = arctan ( x a ) x=a\tan(\theta),\quad\mathrm{d}x=a\sec^{2}(\theta)\,\mathrm{d}\theta,\quad% \theta=\arctan\left(\tfrac{x}{a}\right)
  12. d x < m t p l > a 2 + x 2 \displaystyle\int\frac{\mathrm{d}x}{<}mtpl>{{a^{2}+x^{2}}}
  13. d x x 2 - a 2 \int\frac{\mathrm{d}x}{x^{2}-a^{2}}
  14. x 2 - a 2 d x \int\sqrt{x^{2}-a^{2}}\,\mathrm{d}x
  15. x = a sec ( θ ) , d x = a sec ( θ ) tan ( θ ) d θ , θ = \arcsec ( x a ) x=a\sec(\theta),\quad\mathrm{d}x=a\sec(\theta)\tan(\theta)\,\mathrm{d}\theta,% \quad\theta=\arcsec\left(\tfrac{x}{a}\right)
  16. x 2 - a 2 d x \displaystyle\int\sqrt{x^{2}-a^{2}}\,\mathrm{d}x
  17. f ( sin ( x ) , cos ( x ) ) d x \displaystyle\int f(\sin(x),\cos(x))\,\mathrm{d}x
  18. 1 a 2 + x 2 d x \int\frac{1}{\sqrt{a^{2}+x^{2}}}\,\mathrm{d}x
  19. x = a sinh u x=a\sinh{u}
  20. d x = a cosh u d u \mathrm{d}x=a\cosh{u}\,\mathrm{d}u
  21. cosh 2 ( x ) - sinh 2 ( x ) = 1 \cosh^{2}(x)-\sinh^{2}(x)=1
  22. sinh - 1 x = ln ( x + x 2 + 1 ) \sinh^{-1}{x}=\ln(x+\sqrt{x^{2}+1})
  23. 1 a 2 + x 2 d x = a cosh u a 2 + a 2 sinh 2 u d u = a cosh u a 1 + sinh 2 u d u = a cosh u a cosh u d u = u + C = sinh - 1 x a + C = ln ( x 2 a 2 + 1 + x a ) + C = ln ( x 2 + a 2 + x a ) + C \begin{aligned}\displaystyle\int\frac{1}{\sqrt{a^{2}+x^{2}}}\,\mathrm{d}x&% \displaystyle=\int\frac{a\cosh{u}}{\sqrt{a^{2}+a^{2}\sinh^{2}{u}}}\,\mathrm{d}% u\\ &\displaystyle=\int\frac{a\cosh{u}}{a\sqrt{1+\sinh^{2}{u}}}\,\mathrm{d}u\\ &\displaystyle=\int\frac{a\cosh{u}}{a\cosh{u}}\,\mathrm{d}u\\ &\displaystyle=u+C\\ &\displaystyle=\sinh^{-1}{\frac{x}{a}}+C\\ &\displaystyle=\ln\left(\sqrt{\frac{x^{2}}{a^{2}}+1}+\frac{x}{a}\right)+C\\ &\displaystyle=\ln\left(\frac{\sqrt{x^{2}+a^{2}}+x}{a}\right)+C\end{aligned}

True_airspeed.html

  1. TAS = EAS ρ 0 ρ \mathrm{TAS}=\mathrm{EAS}\sqrt{\frac{\rho_{0}}{\rho}}
  2. TAS \mathrm{TAS}
  3. EAS \mathrm{EAS}
  4. ρ 0 \rho_{0}
  5. ρ \rho
  6. TAS = a 0 M T T 0 \mathrm{TAS}={a_{0}}M\sqrt{T\over T_{0}}
  7. a 0 {a_{0}}
  8. M M
  9. T T
  10. T 0 T_{0}
  11. TAS = 39 M T \mathrm{TAS}=39M\sqrt{T}
  12. TAS = a 0 5 T T 0 [ ( q c P + 1 ) 2 7 - 1 ] \mathrm{TAS}={a_{0}}\sqrt{{5T\over T_{0}}\left[\left(\frac{q_{c}}{P}+1\right)^% {\frac{2}{7}}-1\right]}
  13. q c {q_{c}}
  14. P P
  15. T = T t 1 + 0.2 M 2 T={\frac{T_{t}}{1+0.2M^{2}}}
  16. T t = T_{t}=

Truncated_dodecahedron.html

  1. A = 5 ( 3 + 6 5 + 2 5 ) a 2 100.99076 a 2 A=5\left(\sqrt{3}+6\sqrt{5+2\sqrt{5}}\right)a^{2}\approx 100.99076a^{2}
  2. V = 5 12 ( 99 + 47 5 ) a 3 85.0396646 a 3 V=\frac{5}{12}\left(99+47\sqrt{5}\right)a^{3}\approx 85.0396646a^{3}

Truncated_mean.html

  1. { 92 , 19 , 𝟏𝟎𝟏 , 58 , 𝟏𝟎𝟓𝟑 , 91 , 26 , 78 , 10 , 13 , - 𝟒𝟎 , 𝟏𝟎𝟏 , 86 , 85 , 15 , 89 , 89 , 28 , - 𝟓 , 41 } ( N = 20 , m e a n = 101.5 ) \{92,19,\mathbf{101},58,\mathbf{1053},91,26,78,10,13,\mathbf{-40},\mathbf{101}% ,86,85,15,89,89,28,\mathbf{-5},41\}\qquad(N=20,mean=101.5)
  2. { 92 , 19 , 101 , 58 , 91 , 26 , 78 , 10 , 13 , 101 , 86 , 85 , 15 , 89 , 89 , 28 , - 5 , 41 } ( N = 18 , m e a n = 56.5 ) \{92,19,101,58,91,26,78,10,13,101,86,85,15,89,89,28,-5,41\}\qquad(N=18,mean=56% .5)

Truss.html

  1. m 2 j - r ( a ) m\geq 2j-r\qquad\qquad\mathrm{(a)}
  2. m = 2 j - 3 m=2j-3

Truth_function.html

  1. ¬ P Q ¬P∨Q
  2. P Q P→Q
  3. \vee
  4. \wedge
  5. \bot
  6. \bot
  7. ↮ \not\leftrightarrow
  8. ↮ \not\leftrightarrow
  9. ↛ \not\to
  10. ↚ \not\leftarrow
  11. ↛ \not\to
  12. ↚ \not\leftarrow
  13. ↛ \not\to
  14. ↚ \not\leftarrow
  15. ↛ \not\to
  16. \top
  17. ↚ \not\leftarrow
  18. \top
  19. ↛ \not\to
  20. \leftrightarrow
  21. ↚ \not\leftarrow
  22. \leftrightarrow
  23. \lor
  24. \leftrightarrow
  25. \bot
  26. \lor
  27. \leftrightarrow
  28. ↮ \not\leftrightarrow
  29. \lor
  30. ↮ \not\leftrightarrow
  31. \top
  32. \land
  33. \leftrightarrow
  34. \bot
  35. \land
  36. \leftrightarrow
  37. ↮ \not\leftrightarrow
  38. \land
  39. ↮ \not\leftrightarrow
  40. \top
  41. \land
  42. \lor
  43. a ( a b ) = a a\land(a\lor b)=a
  44. \vee
  45. \wedge
  46. \top
  47. \bot
  48. ¬ \neg
  49. \leftrightarrow
  50. ↮ \not\leftrightarrow
  51. \top
  52. \bot
  53. ¬ \neg
  54. \vee
  55. \wedge
  56. \top
  57. \rightarrow
  58. \leftrightarrow
  59. \vee
  60. \wedge
  61. ↮ \not\leftrightarrow
  62. \bot
  63. 2 2 n 2^{2^{n}}
  64. 3 3 n 3^{3^{n}}
  65. k k k^{k}
  66. k k 2 k^{k^{2}}
  67. k k 3 k^{k^{3}}
  68. k k n k^{k^{n}}
  69. k n k \mathbb{Z}_{k}^{n}\to\mathbb{Z}_{k}
  70. | k | | k n | = k k n |\mathbb{Z}_{k}|^{|\mathbb{Z}_{k}^{n}|}=k^{k^{n}}
  71. ( 3 2 ) 16 - ( 3 1 ) 4 + ( 3 0 ) 2 {\left({{3}\atop{2}}\right)}\cdot 16-{\left({{3}\atop{1}}\right)}\cdot 4+{% \left({{3}\atop{0}}\right)}\cdot 2
  72. f ( x , y , z ) = ¬ x f(x,y,z)=\lnot x
  73. \wedge
  74. \vee
  75. Ω \Omega\!
  76. Ω = Ω 0 Ω 1 Ω j Ω m . \Omega=\Omega_{0}\cup\Omega_{1}\cup\ldots\cup\Omega_{j}\cup\ldots\cup\Omega_{m% }\,.
  77. Ω j \Omega_{j}\!
  78. j j\!
  79. Ω \Omega\!
  80. Ω 0 = { , } \Omega_{0}=\{\bot,\top\}\,
  81. Ω 1 = { ¬ } \Omega_{1}=\{\lnot\}\,
  82. Ω 2 { , , , } \Omega_{2}\subseteq\{\land,\lor,\rightarrow,\leftrightarrow\}\,
  83. ↚ \not\leftarrow
  84. ↛ \not\rightarrow

Tsirelson_space.html

  1. { x n } n = 1 N \{x_{n}\}_{n=1}^{N}
  2. { a n , b n } n = 1 N \textstyle\{a_{n},b_{n}\}_{n=1}^{N}
  3. a 1 b 1 < a 2 b 2 < b N a_{1}\leq b_{1}<a_{2}\leq b_{2}<\ldots\leq b_{N}
  4. ( x n ) i = 0 (x_{n})_{i}=0
  5. i < a n i<a_{n}
  6. i > b n i>b_{n}
  7. λ e j \lambda e_{j}
  8. ( x 1 , , x N ) \textstyle(x_{1},\dots,x_{N})
  9. 1 2 P N ( x 1 + + x N ) \textstyle{{1\over 2}P_{N}(x_{1}+\cdots+x_{N})}
  10. T * T*
  11. C C
  12. X X
  13. Y Y
  14. X X
  15. C C
  16. C C
  17. T * T*
  18. T T
  19. T * T*
  20. n 1 \scriptstyle{\ell^{1}_{n}}
  21. n n
  22. n n
  23. T T
  24. T * T*
  25. T * T*
  26. T * T*
  27. T * T*
  28. c c
  29. T T
  30. T * T*
  31. n n
  32. C C
  33. N N
  34. C C
  35. N N
  36. n n
  37. n n

Tunable_diode_laser_absorption_spectroscopy.html

  1. ( ν ~ ) (\tilde{\nu})
  2. I ( ν ~ ) = I 0 ( ν ~ ) exp ( - α ( ν ~ ) L ) = I 0 ( ν ~ ) exp ( - σ ( ν ~ ) N L ) I(\tilde{\nu})=I_{0}(\tilde{\nu})\exp(-\alpha(\tilde{\nu})L)=I_{0}(\tilde{\nu}% )\exp(-\sigma(\tilde{\nu})NL)
  3. I ( ν ~ ) I(\tilde{\nu})
  4. L L
  5. I 0 ( ν ~ ) I_{0}(\tilde{\nu})
  6. α ( ν ~ ) = σ ( ν ~ ) N = S ( T ) ϕ ( ν ~ - ν ~ 0 ) \alpha(\tilde{\nu})=\sigma(\tilde{\nu})N=S(T)\phi(\tilde{\nu}-\tilde{\nu}_{0})
  7. σ ( ν ~ ) \sigma(\tilde{\nu})
  8. N N\!
  9. S ( T ) S(T)\!
  10. T T
  11. ϕ ( ν ~ - ν ~ 0 ) \phi(\tilde{\nu}-\tilde{\nu}_{0})
  12. g ( ν ~ - ν ~ 0 ) g(\tilde{\nu}-\tilde{\nu}_{0})
  13. ν ~ 0 \tilde{\nu}_{0}
  14. T T\!
  15. S ( T ) S(T)\!
  16. R = ( S 1 S 2 ) T = ( S 1 S 2 ) T 0 exp [ - h c ( E 1 - E 2 ) k ( 1 T - 1 T 0 ) ] R=\left(\frac{S_{1}}{S_{2}}\right)_{T}=\left(\frac{S_{1}}{S_{2}}\right)_{T_{0}% }\exp\left[-\frac{hc(E_{1}-E_{2})}{k}\left(\frac{1}{T}-\frac{1}{T_{0}}\right)\right]
  17. T 0 T_{0}\!
  18. Δ E = ( E 1 - E 2 ) \Delta E=(E_{1}-E_{2})\!
  19. F W H M ( Δ ν ~ D ) = ν ~ 0 8 k T ln 2 m c 2 = ν ~ 0 ( 7.1623 x 10 - 7 ) T M FWHM(\Delta\tilde{\nu}_{D})=\tilde{\nu}_{0}\sqrt{\frac{8kT\ln 2}{mc^{2}}}=% \tilde{\nu}_{0}(7.1623\mbox{x}~{}10^{-7})\sqrt{\frac{T}{M}}
  20. m m
  21. M M
  22. T T
  23. M M
  24. Δ ν ~ D = V c ν ~ 0 cos θ \Delta\tilde{\nu}_{D}=\frac{V}{c}\tilde{\nu}_{0}\cos\theta
  25. θ \theta
  26. Δ ν ~ D \Delta\tilde{\nu}_{D}

Turán's_theorem.html

  1. n n
  2. ( r + 1 ) (r+1)
  3. r r
  4. T ( n , r ) T(n,r)
  5. n n
  6. G G
  7. n n
  8. G G
  9. G G
  10. r - 1 r n 2 2 = ( 1 - 1 r ) n 2 2 . \frac{r-1}{r}\cdot\frac{n^{2}}{2}=\left(1-\frac{1}{r}\right)\cdot\frac{n^{2}}{% 2}.
  11. n n
  12. ( r + 1 ) (r+1)
  13. T ( n , r ) T(n,r)
  14. G G
  15. n n
  16. ( r + 1 ) (r+1)
  17. G G
  18. G G
  19. r r
  20. S 1 , S 2 , , S r S_{1},S_{2},\ldots,S_{r}
  21. S i S_{i}
  22. S j S_{j}
  23. S i S_{i}
  24. S 1 S_{1}
  25. S 2 S_{2}
  26. | S 1 | = 12 |S_{1}|=12
  27. | S 2 | = 11 |S_{2}|=11
  28. G G
  29. 1 2 23 2 2 = 23 2 4 = 132.25 \frac{1}{2}\frac{23^{2}}{2}=\frac{23^{2}}{4}=132.25
  30. G G
  31. u , v , w u,v,w
  32. G G
  33. u v uv
  34. u w uw
  35. v w vw
  36. u , v , w u,v,w
  37. u w u~{}w
  38. w v w~{}v
  39. u v u~{}v
  40. n n
  41. G G′
  42. ( r + 1 ) (r+1)
  43. G G
  44. d ( w ) < d ( u ) or d ( w ) < d ( v ) . d(w)<d(u)\,\text{ or }d(w)<d(v).
  45. d ( w ) < d ( u ) d(w)<d(u)
  46. w w
  47. u u
  48. u u
  49. u u′
  50. { u , u } \{u,u^{\prime}\}
  51. ( r + 1 ) (r+1)
  52. | E ( G ) | = | E ( G ) | - d ( w ) + d ( u ) > | E ( G ) | . |E(G^{\prime})|=|E(G)|-d(w)+d(u)>|E(G)|.
  53. d ( w ) d ( u ) d(w)\geq d(u)
  54. d ( w ) d ( v ) d(w)\geq d(v)
  55. u u
  56. v v
  57. w w
  58. ( r + 1 ) (r+1)
  59. | E ( G ) | = | E ( G ) | - ( d ( u ) + d ( v ) - 1 ) + 2 d ( w ) | E ( G ) | + 1. |E(G^{\prime})|=|E(G)|-(d(u)+d(v)-1)+2d(w)\geq|E(G)|+1.
  60. G G
  61. G G
  62. k k
  63. G G
  64. k k
  65. | A | > | B | + 1 |A|>|B|+1
  66. G G
  67. B B
  68. | B | |B|
  69. | A | - 1 |A|-1
  70. | A | - 1 - | B | 1 |A|-1-|B|\geq 1
  71. n n
  72. n 2 / 4 . \lfloor n^{2}/4\rfloor.
  73. n n
  74. n 2 / 4 \lfloor n^{2}/4\rfloor
  75. n 2 / 4 \lfloor n^{2}/4\rfloor

Turán_graph.html

  1. ( n mod r ) (n\,\bmod\,r)
  2. n / r \lceil n/r\rceil
  3. r - ( n mod r ) r-(n\,\bmod\,r)
  4. n / r \lfloor n/r\rfloor
  5. K n / r , n / r , , n / r , n / r . K_{\lceil n/r\rceil,\lceil n/r\rceil,\ldots,\lfloor n/r\rfloor,\lfloor n/r% \rfloor}.
  6. n - n / r n-\lceil n/r\rceil
  7. n - n / r n-\lfloor n/r\rfloor
  8. 1 2 ( n 2 - ( n mod r ) n / r 2 - ( r - ( n mod r ) ) n / r 2 ) ( 1 - 1 r ) n 2 2 . \frac{1}{2}(n^{2}-(n\,\bmod\,r)\lceil n/r\rceil^{2}-(r-(n\,\bmod\,r))\lfloor n% /r\rfloor^{2})\leq\left(1-\frac{1}{r}\right)\frac{n^{2}}{2}.
  9. r - 1 2 r ( 2 α - 1 ) n 2 \frac{r\,{-}\,1}{2r}(2\alpha-1)n^{2}
  10. T ( n , n / 3 ) T(n,\lceil n/3\rceil)

Turbo_code.html

  1. R 1 = n 1 + n 2 2 n 1 + n 2 R 2 = n 1 + n 2 n 1 + 2 n 2 \begin{aligned}\displaystyle~{}R_{1}&\displaystyle=\frac{n_{1}+n_{2}}{2n_{1}+n% _{2}}\\ \displaystyle~{}R_{2}&\displaystyle=\frac{n_{1}+n_{2}}{n_{1}+2n_{2}}\end{aligned}
  2. D E C 1 \scriptstyle DEC_{1}
  3. R 1 \scriptstyle R_{1}
  4. C 1 \scriptstyle C_{1}
  5. D E C 2 \scriptstyle DEC_{2}
  6. C 2 \scriptstyle C_{2}
  7. D E C 1 \scriptstyle DEC_{1}
  8. L 1 \scriptstyle L_{1}
  9. D E C 2 \scriptstyle DEC_{2}
  10. L 2 \scriptstyle L_{2}
  11. D E C 1 \scriptstyle DEC_{1}
  12. D E C 1 \scriptstyle DEC_{1}
  13. D E C 2 \scriptstyle DEC_{2}
  14. y 1 k \scriptstyle y_{1k}
  15. y 2 k \scriptstyle y_{2k}
  16. x k \displaystyle~{}x_{k}
  17. a k \scriptstyle a_{k}
  18. b k \scriptstyle b_{k}
  19. σ 2 \scriptstyle\sigma^{2}
  20. Y k \scriptstyle Y_{k}
  21. y k \scriptstyle y_{k}
  22. D E C 1 \scriptstyle DEC_{1}
  23. y k = y 1 k \scriptstyle y_{k}\;=\;y_{1k}
  24. D E C 2 \scriptstyle DEC_{2}
  25. y k = y 2 k \scriptstyle y_{k}\;=\;y_{2k}
  26. D E C 1 \scriptstyle DEC_{1}
  27. Λ ( d k ) = log p ( d k = 1 ) p ( d k = 0 ) \Lambda(d_{k})=\log\frac{p(d_{k}=1)}{p(d_{k}=0)}
  28. D E C 2 \scriptstyle DEC_{2}
  29. Λ ( d k ) \scriptstyle\Lambda(d_{k})
  30. p ( d k = i ) , i { 0 , 1 } \scriptstyle p(d_{k}\;=\;i),\,i\,\in\,\{0,\,1\}
  31. d k \scriptstyle d_{k}
  32. d k \scriptstyle d_{k}
  33. i \scriptstyle i
  34. D E C 2 \scriptstyle DEC_{2}
  35. D E C 1 \scriptstyle DEC_{1}
  36. D E C 2 \scriptstyle DEC_{2}
  37. D E C 1 \scriptstyle DEC_{1}
  38. n / 2 {n}/{2}

Twin-lead.html

  1. Z = R + j ω L G + j ω C Z=\sqrt{{R+j\omega L}\over{G+j\omega C}}
  2. R = 2 R s π d R=2{R_{s}\over\pi d}
  3. L = μ π arccosh ( D d ) L={\mu\over\pi}\,\operatorname{arccosh}\left({D\over d}\right)
  4. G = π σ arccosh ( D d ) G={\pi\sigma\over\operatorname{arccosh}({D\over d})}
  5. C = π ϵ arccosh ( D d ) C={\pi\epsilon\over\operatorname{arccosh}({D\over d})}
  6. R s = π f μ c / σ c R_{s}=\sqrt{\pi f\mu_{c}/\sigma_{c}}
  7. Z = Z 0 π ϵ r arccosh ( D d ) Z=\frac{Z_{0}}{\pi\sqrt{\epsilon_{r}}}\,\operatorname{arccosh}\left(\frac{D}{d% }\right)
  8. Z 276 log 10 ( 2 D d ) Z\approx 276\log_{10}\left(2\frac{D}{d}\right)
  9. D = d cosh ( π Z ϵ r Z 0 ) D=d\cosh\left(\pi\frac{Z\sqrt{\epsilon_{r}}}{Z_{0}}\right)

Two-level_grammar.html

  1. { a n b n a n | n 1 } . \{a^{n}b^{n}a^{n}|n\geq 1\}.
  2. a N b N a N \langle a^{N}\rangle\langle b^{N}\rangle\langle a^{N}\rangle
  3. X N 1 \langle X^{N1}\rangle
  4. X N X \langle X^{N}\rangle X
  5. X 1 \langle X^{1}\rangle

Type_rule.html

  1. e e
  2. τ \tau
  3. e : τ e\!:\!\tau
  4. Γ \Gamma
  5. Γ 1 e 1 : τ 1 Γ n e n : τ n Γ e : τ \frac{\Gamma_{1}\vdash e_{1}\!:\!\tau_{1}\quad\cdots\quad\Gamma_{n}\vdash e_{n% }\!:\!\tau_{n}}{\Gamma\vdash e\!:\!\tau}
  6. e i e_{i}
  7. τ i \tau_{i}
  8. Γ i \Gamma_{i}
  9. i = 1.. n i=1..n
  10. e e
  11. Γ \Gamma
  12. τ \tau
  13. Γ e 1 : r e a l Γ e 2 : r e a l Γ e 1 + e 2 : r e a l Γ e 1 : i n t e g e r Γ e 2 : i n t e g e r Γ e 1 + e 2 : i n t e g e r \frac{\Gamma\vdash e_{1}\!:\!real\quad\Gamma\vdash e_{2}\!:\!real}{\Gamma% \vdash e_{1}+e_{2}\!:\!real}\qquad\frac{\Gamma\vdash e_{1}\!:\!integer\quad% \Gamma\vdash e_{2}:integer}{\Gamma\vdash e_{1}+e_{2}\!:\!integer}\qquad\cdots
  14. i d id
  15. τ \tau^{\prime}
  16. Γ \Gamma
  17. Γ e : τ Γ , i d : τ e : τ Γ let id = e in e end : τ \frac{\Gamma\vdash e^{\prime}\!:\!\tau^{\prime}\quad\Gamma,id\!:\!\tau^{\prime% }\vdash e\!:\!\tau}{\Gamma\vdash\,\text{let id = }e^{\prime}\,\text{ in }e\,% \text{ end}:\!\tau}

Type_safety.html

  1. t t
  2. t t
  3. t t
  4. t t

Typed_lambda_calculus.html

  1. λ \lambda
  2. \to
  3. σ τ \sigma\to\tau
  4. A A
  5. B B
  6. A A
  7. B B

Ubiquitin_ligase.html

  1. \rightleftharpoons

Ultrafiltration.html

  1. J = T M P μ R t J={TMP\over\mu R_{t}}

Ultrametric_space.html

  1. d ( x , y ) max { d ( x , z ) , d ( z , y ) } d(x,y)\leq\max\left\{d(x,z),d(z,y)\right\}
  2. M M
  3. d : M × M d\colon M\times M\rightarrow\mathbb{R}
  4. \mathbb{R}
  5. x , y , z M x,y,z\in M
  6. d ( x , y ) 0 d(x,y)\geq 0
  7. d ( x , y ) = 0 d(x,y)=0
  8. x = y x=y
  9. d ( x , y ) = d ( y , x ) d(x,y)=d(y,x)
  10. d ( x , z ) max { d ( x , y ) , d ( y , z ) } d(x,z)\leq\max\left\{d(x,y),d(y,z)\right\}
  11. M M
  12. d d
  13. \|\cdot\|
  14. d ( x , y ) = x - y d(x,y)=\|x-y\|
  15. x + y max { x , y } \|x+y\|\leq\max\left\{\|x\|,\|y\|\right\}
  16. x y \|x\|\neq\|y\|
  17. x + y max { x , y } \|x+y\|\leq\max\left\{\|x\|,\|y\|\right\}
  18. x y \|x\|\neq\|y\|
  19. x > y \|x\|>\|y\|
  20. x + y x \|x+y\|\leq\|x\|
  21. x = ( x + y ) - y max { x + y , y } \|x\|=\|(x+y)-y\|\leq\max\left\{\|x+y\|,\|y\|\right\}
  22. max { x + y , y } \max\left\{\|x+y\|,\|y\|\right\}
  23. y \|y\|
  24. x y \|x\|\leq\|y\|
  25. max { x + y , y } = x + y \max\left\{\|x+y\|,\|y\|\right\}=\|x+y\|
  26. x x + y \|x\|\leq\|x+y\|
  27. x x + y x \|x\|\leq\|x+y\|\leq\|x\|
  28. x + y = x \|x+y\|=\|x\|
  29. x , y , z M x,y,z\in M
  30. r , s r,s\in\mathbb{R}
  31. d ( x , y ) = d ( y , z ) d(x,y)=d(y,z)
  32. d ( x , z ) = d ( y , z ) d(x,z)=d(y,z)
  33. d ( x , y ) = d ( z , x ) d(x,y)=d(z,x)
  34. B ( x ; r ) = { y M | d ( x , y ) < r } B(x;r)=\{y\in M|d(x,y)<r\}
  35. d ( x , y ) < r d(x,y)<r
  36. B ( x ; r ) = B ( y ; r ) B(x;r)=B(y;r)
  37. B ( x ; r ) B ( y ; s ) B(x;r)\cap B(y;s)
  38. B ( x ; r ) B ( y ; s ) B(x;r)\subseteq B(y;s)
  39. B ( y ; s ) B ( x ; r ) B(y;s)\subseteq B(x;r)
  40. < <
  41. \leq
  42. r > 0 r>0
  43. r r

Undersampling.html

  1. 2 f H n f s 2 f L n - 1 \frac{2f_{H}}{n}\leq f_{s}\leq\frac{2f_{L}}{n-1}
  2. 1 n f H f H - f L 1\leq n\leq\left\lfloor\frac{f_{H}}{f_{H}-f_{L}}\right\rfloor
  3. W = f H - f L = 108 MHz - 88 MHz = 20 MHz W=f_{H}-f_{L}=108\ \mathrm{MHz}-88\ \mathrm{MHz}=20\ \mathrm{MHz}
  4. 1 n 5.4 = 108 MHz 20 MHz 1\leq n\leq\lfloor 5.4\rfloor=\left\lfloor{108\ \mathrm{MHz}\over 20\ \mathrm{% MHz}}\right\rfloor
  5. 43.2 MHz < f s < 44 MHz 43.2\ \mathrm{MHz}<f_{\mathrm{s}}<44\ \mathrm{MHz}
  6. ( - 1 2 f s , 1 2 f s ) , \scriptstyle\left(-\frac{1}{2}f_{\mathrm{s}},\frac{1}{2}f_{\mathrm{s}}\right),
  7. sinc ( t / T ) . \scriptstyle\operatorname{sinc}\left(t/T\right).
  8. ( - n 2 f s , - n - 1 2 f s ) ( n - 1 2 f s , n 2 f s ) \left(-\frac{n}{2}f_{\mathrm{s}},-\frac{n-1}{2}f_{\mathrm{s}}\right)\cup\left(% \frac{n-1}{2}f_{\mathrm{s}},\frac{n}{2}f_{\mathrm{s}}\right)
  9. n n\,
  10. n sinc ( n t T ) - ( n - 1 ) sinc ( ( n - 1 ) t T ) n\operatorname{sinc}\left(\frac{nt}{T}\right)-(n-1)\operatorname{sinc}\left(% \frac{(n-1)t}{T}\right)

Unicoherent_space.html

  1. X X
  2. A , B X A,B\subset X
  3. X = A B X=A\cup B
  4. A B A\cap B

Uniform_norm.html

  1. f = f , S = sup { | f ( x ) | : x S } . \|f\|_{\infty}=\|f\|_{\infty,S}=\sup\left\{\,\left|f(x)\right|:x\in S\,\right\}.
  2. { f n } \{f_{n}\}
  3. f n f_{n}
  4. f f
  5. x = ( x 1 , , x n ) x=(x_{1},\dots,x_{n})
  6. x = max { | x 1 | , , | x n | } . \|x\|_{\infty}=\max\{|x_{1}|,\dots,|x_{n}|\}.
  7. lim p f p = f , \lim_{p\rightarrow\infty}\|f\|_{p}=\|f\|_{\infty},
  8. f p = ( D | f | p d μ ) 1 / p \|f\|_{p}=\left(\int_{D}\left|f\right|^{p}\,d\mu\right)^{1/p}
  9. d ( f , g ) = f - g d(f,g)=\|f-g\|_{\infty}
  10. lim n f n - f = 0. \lim_{n\rightarrow\infty}\|f_{n}-f\|_{\infty}=0.\,
  11. [ a , b ] [a,b]
  12. [ a , b ] [a,b]

Unimodular_matrix.html

  1. G L n ( ) GL_{n}(\mathbb{Z})
  2. det ( A B ) = ( det A ) q ( det B ) p , \det(A\otimes B)=(\det A)^{q}(\det B)^{p},
  3. { min c x A x b , x 0 } \{\min cx\mid Ax\geq b,x\geq 0\}
  4. { max c x A x b } \{\max cx\mid Ax\leq b\}
  5. { x A x b } \{x\mid Ax\geq b\}
  6. A A
  7. B B
  8. C C
  9. A A
  10. A A
  11. A A
  12. B B
  13. C C
  14. A A
  15. B B
  16. C C
  17. s : R ± 1 s:R\to\pm 1
  18. r R s ( r ) r \sum_{r\in R}s(r)r
  19. { 0 , ± 1 } \{0,\pm 1\}
  20. G G
  21. P P
  22. G G
  23. A A
  24. V ( G ) V(G)
  25. P P
  26. A A
  27. G G
  28. ± \pm
  29. 0 , ± 1 0,\pm 1
  30. A = [ - 1 - 1 0 0 0 + 1 + 1 0 - 1 - 1 0 0 0 + 1 + 1 0 - 1 0 0 0 0 + 1 + 1 - 1 ] . A=\begin{bmatrix}-1&-1&0&0&0&+1\\ +1&0&-1&-1&0&0\\ 0&+1&+1&0&-1&0\\ 0&0&0&+1&+1&-1\\ \end{bmatrix}.
  31. A = [ + 1 + 1 + 1 - 1 ] . A=\begin{bmatrix}\vdots&\vdots&\vdots&\vdots&\vdots\\ \cdots&+1&\cdots&+1&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \cdots&+1&\cdots&-1&\cdots\\ \vdots&\vdots&\vdots&\vdots&\vdots\\ \end{bmatrix}.
  32. G L n R GL_{n}R\,

Unique_prime.html

  1. Φ n ( 10 ) gcd ( Φ n ( 10 ) , n ) = p c \frac{\Phi_{n}(10)}{\gcd(\Phi_{n}(10),n)}=p^{c}
  2. Φ 47498 ( 10 ) \Phi_{47498}(10)
  3. Φ 28 L ( 2 ) \Phi_{28L}(2)
  4. Φ 28 M ( 2 ) \Phi_{28M}(2)
  5. Φ 36 L ( 2 ) \Phi_{36L}(2)
  6. Φ 36 M ( 2 ) \Phi_{36M}(2)
  7. Φ 44 L ( 2 ) \Phi_{44L}(2)
  8. Φ 44 M ( 2 ) \Phi_{44M}(2)
  9. Φ n ( 2 ) gcd ( Φ n ( 2 ) , n ) = p c \frac{\Phi_{n}(2)}{\gcd(\Phi_{n}(2),n)}=p^{c}
  10. Φ n ( 2 ) \Phi_{n}(2)
  11. Φ n ( 2 ) gcd ( Φ n ( 2 ) , n ) \frac{\Phi_{n}(2)}{\gcd(\Phi_{n}(2),n)}
  12. 1 p \frac{1}{p}
  13. Φ n ( 2 ) \Phi_{n}(2)
  14. Φ n ( 2 ) gcd ( Φ n ( 2 ) , n ) \frac{\Phi_{n}(2)}{\gcd(\Phi_{n}(2),n)}
  15. Φ n ( 2 ) \Phi_{n}(2)
  16. Φ n ( 2 ) \Phi_{n}(2)
  17. Φ n ( 2 ) \Phi_{n}(2)
  18. Φ n ( 2 ) \Phi_{n}(2)
  19. 2 13372531 + 1 3 \frac{2^{13372531}+1}{3}
  20. 2 42737 + 1 3 \frac{2^{42737}+1}{3}
  21. 16 1025393 + 1 17 \frac{16^{1025393}+1}{17}
  22. Φ n ( 2 ) gcd ( Φ n ( 2 ) , n ) \frac{\Phi_{n}(2)}{\gcd(\Phi_{n}(2),n)}

Unit_cube.html

  1. n \sqrt{n}

Unit_fraction.html

  1. 1 x × 1 y = 1 x y . \frac{1}{x}\times\frac{1}{y}=\frac{1}{xy}.
  2. 1 x + 1 y = x + y x y \frac{1}{x}+\frac{1}{y}=\frac{x+y}{xy}
  3. 1 x - 1 y = y - x x y \frac{1}{x}-\frac{1}{y}=\frac{y-x}{xy}
  4. 1 x ÷ 1 y = y x . \frac{1}{x}\div\frac{1}{y}=\frac{y}{x}.
  5. a x + b y = 1 , \displaystyle ax+by=1,
  6. a x 1 ( mod y ) , \displaystyle ax\equiv 1\;\;(\mathop{{\rm mod}}y),
  7. a 1 x ( mod y ) . a\equiv\frac{1}{x}\;\;(\mathop{{\rm mod}}y).
  8. 4 5 = 1 2 + 1 4 + 1 20 = 1 3 + 1 5 + 1 6 + 1 10 . \frac{4}{5}=\frac{1}{2}+\frac{1}{4}+\frac{1}{20}=\frac{1}{3}+\frac{1}{5}+\frac% {1}{6}+\frac{1}{10}.
  9. 1 1 + 1 2 + 1 3 + + 1 n \frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\cdots+\frac{1}{n}
  10. B i , j = 1 i + j - 1 . B_{i,j}=\frac{1}{i+j-1}.
  11. C i , j = 1 F i + j - 1 , C_{i,j}=\frac{1}{F_{i+j-1}},

Unit_of_length.html

  1. a 0 0.529 177 210 92 × 10 - 10 m a_{0}\approx 0.529\;177\;210\;92\times 10^{-10}\mbox{ m}~{}
  2. λ ¯ C 386.159 268 00 × 10 - 15 m \bar{\lambda}\text{C}\approx 386.159\;268\;00\times 10^{-15}\mbox{m}~{}
  3. P 1.616 199 × 10 - 35 m \ell\text{P}\approx 1.616\;199\times 10^{-35}\mbox{ m}~{}

United_States_congressional_apportionment.html

  1. A n = P n ( n + 1 ) A_{n}=\frac{P}{\sqrt{n(n+1)}}
  2. A n + 1 = n n + 2 A n A_{n+1}=\sqrt{\frac{n}{n+2}}\ A_{n}
  3. A 1 = P 2 A_{1}=\frac{P}{\sqrt{2}}

United_States_Treasury_security.html

  1. discount yield ( % ) = face value - purchase price face value × 360 days till maturity X 100 % \,\text{discount yield}\,(\%)=\frac{\,\text{face value}-\,\text{purchase price% }}{\,\text{face value}}\times\frac{360}{\,\text{days till maturity}}\ X100\,\%

Universal_enveloping_algebra.html

  1. L L
  2. U ( L ) U(L)
  3. L L
  4. L L
  5. A A
  6. K K
  7. K K
  8. [ a , b ] = a b - b a [a,b]=ab-ba
  9. L L
  10. K K
  11. K K
  12. A A
  13. L L
  14. A A
  15. U ( L ) U(L)
  16. L L
  17. U ( L ) U(L)
  18. L L
  19. U ( L ) U(L)
  20. ρ ρ
  21. x x
  22. ρ ( x ) ρ(x)
  23. ρ ( x ) ρ ( y ) ρ(x)ρ(y)
  24. ρ ( x ) ρ ( y ) = 0 ρ(x)ρ(y)=0
  25. X X
  26. K K
  27. K K
  28. U U
  29. U U
  30. X X
  31. K K
  32. A A
  33. g : U A g:U→A
  34. X X
  35. A A
  36. L L
  37. T ( L ) T(L)
  38. L L
  39. U ( L ) U(L)
  40. T ( L ) T(L)
  41. a b - b a = [ a , b ] a\otimes b-b\otimes a=[a,b]
  42. a a
  43. b b
  44. T ( L ) T(L)
  45. L L
  46. L L
  47. U ( L ) = T ( L ) / I U(L)=T(L)/I
  48. I I
  49. T ( L ) T(L)
  50. a b - b a - [ a , b ] , a , b L . a\otimes b-b\otimes a-[a,b],\quad a,b\in L.
  51. L T ( L ) L→T(L)
  52. h : L U ( L ) h:L→U(L)
  53. L L
  54. 0
  55. U ( L ) U(L)
  56. L L
  57. U ( L ) U(L)
  58. K K
  59. L L
  60. G , U ( L ) G,U(L)
  61. G G
  62. L L
  63. L L
  64. V V
  65. U ( L ) U(L)
  66. Z ( L ) Z(L)
  67. G G
  68. U ( L ) U(L)
  69. e e
  70. G G
  71. n n
  72. U ( L ) U(L)
  73. L L
  74. U ( L ) U(L)
  75. h : L U ( L ) h:L→U(L)
  76. U ( L ) U(L)
  77. L L
  78. L L
  79. L L
  80. U ( L ) U(L)
  81. L L
  82. T ( L ) T(L)
  83. U ( L ) U(L)
  84. U ( L ) U(L)
  85. L L
  86. L L
  87. U U
  88. K K
  89. K K
  90. A A
  91. A < s u b > L A<sub>L

UPGMA.html

  1. 1 | 𝒜 | | | x 𝒜 y d ( x , y ) {1\over{|\mathcal{A}|\cdot|\mathcal{B}|}}\sum_{x\in\mathcal{A}}\sum_{y\in% \mathcal{B}}d(x,y)
  2. O ( n 2 ) O(n^{2})

Urysohn_and_completely_Hausdorff_spaces.html

  1. \Rightarrow
  2. \Downarrow
  3. \Downarrow
  4. \Rightarrow
  5. \Rightarrow
  6. \Rightarrow

Utility_frequency.html

  1. N = 120 f P N=\frac{120f}{P}\,
  2. 1 / 3 {1}/{3}
  3. 2 / 3 {2}/{3}
  4. 1 / 3 {1}/{3}
  5. 2 / 3 {2}/{3}
  6. 2 / 3 {2}/{3}
  7. 2 / 3 {2}/{3}
  8. 2 / 3 {2}/{3}
  9. 1 / 3 {1}/{3}
  10. 2 / 3 {2}/{3}

Valence_electron.html

  1. n n
  2. ( n + 1 ) (n+1)

Valuation_(algebra).html

  1. K K
  2. ( Γ , + , ) (Γ,+,≥)
  3. ( Γ , · , ) (Γ,·,≥)
  4. Γ Γ
  5. α ∞≥α
  6. α α
  7. Γ Γ
  8. + α = α + = ∞+α=α+∞=∞
  9. Γ Γ
  10. K K
  11. v ( a ) = v(a)=∞
  12. a = 0 a=0
  13. v ( a b ) = v ( a ) + v ( b ) v(ab)=v(a)+v(b)
  14. v ( a + b ) m i n ( v ( a ) , v ( b ) ) v(a+b)≥min(v(a),v(b))
  15. K K
  16. O α O≤α
  17. α α
  18. Γ Γ
  19. O · α = α · O = O O·α=α·O=O
  20. Γ Γ
  21. v ( a ) = O v(a)=O
  22. a = 0 a=0
  23. v ( a b ) = v ( a ) · v ( b ) v(ab)=v(a)·v(b)
  24. v ( a + b ) m a x ( v ( a ) , v ( b ) ) v(a+b)≤max(v(a),v(b))
  25. Γ Γ
  26. K K
  27. K K
  28. K K
  29. 𝐐 \mathbf{Q}
  30. 𝐐 \mathbf{Q}
  31. K K
  32. K K
  33. K K
  34. Γ Γ
  35. K K
  36. K K
  37. K K
  38. K K
  39. Γ = 𝐙 Γ=\mathbf{Z}
  40. π π
  41. R R
  42. K K
  43. π π
  44. R R
  45. R R
  46. a = π e a p 1 e 1 p 2 e 2 p n e n a=\pi^{e_{a}}p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{n}^{e_{n}}
  47. R R
  48. π π
  49. v π ( 0 ) = v_{\pi}(0)=\infty
  50. v π ( a / b ) = e a - e b , for a , b R , a , b 0. v_{\pi}(a/b)=e_{a}-e_{b},\,\text{ for }a,b\in R,a,b\neq 0.
  51. R R
  52. R = 𝐙 R=\mathbf{Z}
  53. K = 𝐐 K=\mathbf{Q}
  54. π π
  55. 𝐐 \mathbf{Q}
  56. R R
  57. K K
  58. R R
  59. R R
  60. K K
  61. P P
  62. K K
  63. 𝐂 x x , y , 𝐂 ( x , y ) \mathbf{C}xx,y,\mathbf{C}(x,y)
  64. f ( x , y ) = y - n = 3 x n 𝐂 { x , y } f(x,y)=y-\sum_{n=3}^{\infty}x^{n}\in\mathbf{C}\{x,y\}
  65. t t
  66. V f = { ( x , y ) 𝐂 2 : f ( x , y ) = 0 } = { ( x , y ) 𝐂 2 : ( x , y ) = ( t , n = 3 t n ) } V_{f}=\left\{(x,y)\in\mathbf{C}^{2}\ :\ f(x,y)=0\right\}=\left\{(x,y)\in% \mathbf{C}^{2}\ :\ (x,y)=\left(t,\sum_{n=3}^{\infty}t^{n}\right)\right\}
  67. v : 𝐂 x x , y 𝐙 v:\mathbf{C}xx,y→\mathbf{Z}
  68. t t
  69. P P
  70. 𝐂 x x , y \mathbf{C}xx,y
  71. v ( P ) = ord t ( P | V f ) = ord t ( P ( t , n = 3 t n ) ) , P 𝐂 [ x , y ] v(P)=\mathrm{ord}_{t}\left(P|_{V_{f}}\right)=\mathrm{ord}_{t}\left(P\left(t,% \sum_{n=3}^{\infty}t^{n}\right)\right),\qquad\forall P\in\mathbf{C}[x,y]
  72. v v
  73. 𝐂 ( x , y ) \mathbf{C}(x,y)
  74. v ( P / Q ) = { v ( P ) - v ( Q ) P / Q 𝐂 ( x , y ) * P 0 𝐂 ( x , y ) v(P/Q)=\begin{cases}v(P)-v(Q)&P/Q\in{\mathbf{C}(x,y)}^{*}\\ \infty&P\equiv 0\in\mathbf{C}(x,y)\end{cases}
  75. f f
  76. v v
  77. v ( P ) v(P)
  78. v ( x ) \displaystyle v(x)
  79. Γ Γ
  80. f - 1 ( B ) f^{-1}(B)
  81. Γ Γ

Van_der_Pauw_method.html

  1. R 12 , 34 R_{12,34}
  2. R 12 , 34 = V 34 I 12 R_{12,34}=\frac{V_{34}}{I_{12}}
  3. R 12 , 34 R_{12,34}
  4. R 23 , 41 R_{23,41}
  5. e - π R 12 , 34 / R s + e - π R 23 , 41 / R s = 1 e^{-\pi R_{12,34}/R_{s}}+e^{-\pi R_{23,41}/R_{s}}=1
  6. R A B , C D = R C D , A B R_{AB,CD}=R_{CD,AB}
  7. R 12 , 34 R_{12,34}
  8. R 23 , 41 R_{23,41}
  9. R 34 , 12 R_{34,12}
  10. R 41 , 23 R_{41,23}
  11. R vertical = R 12 , 34 + R 34 , 12 2 R_{\,\text{vertical}}=\frac{R_{12,34}+R_{34,12}}{2}
  12. R horizontal = R 23 , 41 + R 41 , 23 2 R_{\,\text{horizontal}}=\frac{R_{23,41}+R_{41,23}}{2}
  13. e - π R vertical / R S + e - π R horizontal / R S = 1 e^{-\pi R_{\,\text{vertical}}/R_{S}}+e^{-\pi R_{\,\text{horizontal}}/R_{S}}=1
  14. R vertical = R 12 , 34 + R 34 , 12 + R 21 , 43 + R 43 , 21 4 R_{\,\text{vertical}}=\frac{R_{12,34}+R_{34,12}+R_{21,43}+R_{43,21}}{4}
  15. R horizontal = R 23 , 41 + R 41 , 23 + R 32 , 14 + R 14 , 32 4 R_{\,\text{horizontal}}=\frac{R_{23,41}+R_{41,23}+R_{32,14}+R_{14,32}}{4}
  16. R s = π R ln 2 R_{s}=\frac{\pi R}{\ln 2}
  17. F L = q v B F_{L}=qvB\,\!
  18. q q
  19. v v
  20. B B
  21. V H V_{H}
  22. v = I n A q v=\frac{I}{nAq}
  23. n n
  24. A A
  25. q q
  26. q q
  27. F L = I B n A F_{L}=\frac{IB}{nA}
  28. V H V_{H}
  29. ϵ \epsilon
  30. q ϵ q\epsilon
  31. ϵ = I B q n A \epsilon=\frac{IB}{qnA}
  32. V H = w ϵ = w I B q n A = I B q n d \begin{aligned}\displaystyle V_{H}&\displaystyle=w\epsilon\\ &\displaystyle=\frac{wIB}{qnA}\\ &\displaystyle=\frac{IB}{qnd}\end{aligned}
  33. d d
  34. n s n_{s}
  35. V H = I B q n s V_{H}=\frac{IB}{qn_{s}}
  36. V H = V 13 + V 24 + V 31 + V 42 8 V_{H}=\frac{V_{13}+V_{24}+V_{31}+V_{42}}{8}
  37. n s = I B q | V H | n_{s}=\frac{IB}{q|V_{H}|}
  38. ρ = 1 q ( n μ n + p μ p ) \rho=\frac{1}{q(n\mu_{n}+p\mu_{p})}
  39. ρ = 1 q n m μ m \rho=\frac{1}{qn_{m}\mu_{m}}
  40. R s = 1 q n s μ m R_{s}=\frac{1}{qn_{s}\mu_{m}}
  41. μ m = 1 q n s R s \mu_{m}=\frac{1}{qn_{s}R_{s}}