wpmath0000013_13

Synchronous_frame.html

  1. d s 2 = c 2 d τ 2 = g 00 ( d x 0 ) 2 , ds^{2}=c^{2}d\tau^{2}=g_{00}\left(dx^{0}\right)^{2},\,
  2. d τ = 1 c g 00 d x 0 , d\tau=\frac{1}{c}\sqrt{g_{00}}\,dx^{0},
  3. τ = 1 c g 00 d x 0 . \tau=\frac{1}{c}\int\sqrt{g_{00}}\,dx^{0}.
  4. g 00 > 0. g_{00}>0.\,
  5. d s 2 = g α β d x α d x β + 2 g 0 α d x 0 d x α + g 00 ( d x 0 ) 2 , ds^{2}=g_{\alpha\beta}\,dx^{\alpha}\,dx^{\beta}+2g_{0\alpha}\,dx^{0}\,dx^{% \alpha}+g_{00}\left(dx^{0}\right)^{2},
  6. d x 0 ( 1 ) = 1 g 00 ( - g 0 α d x α - ( g 0 α g 0 β - g α β g 00 ) d x α d x β ) , dx^{0(1)}=\frac{1}{g_{00}}\left(-g_{0\alpha}\,dx^{\alpha}-\sqrt{\left(g_{0% \alpha}g_{0\beta}-g_{\alpha\beta}g_{00}\right)\,dx^{\alpha}\,dx^{\beta}}\right),
  7. d x 0 ( 2 ) = 1 g 00 ( - g 0 α d x α + ( g 0 α g 0 β - g α β g 00 ) d x α d x β ) , dx^{0(2)}=\frac{1}{g_{00}}\left(-g_{0\alpha}\,dx^{\alpha}+\sqrt{\left(g_{0% \alpha}g_{0\beta}-g_{\alpha\beta}g_{00}\right)\,dx^{\alpha}\,dx^{\beta}}\right),
  8. d x 0 ( 2 ) - d x 0 ( 1 ) = 2 g 00 ( g 0 α g 0 β - g α β g 00 ) d x α d x β . dx^{0(2)}-dx^{0(1)}=\frac{2}{g_{00}}\sqrt{\left(g_{0\alpha}g_{0\beta}-g_{% \alpha\beta}g_{00}\right)\,dx^{\alpha}\,dx^{\beta}}.
  9. g 00 / c \scriptstyle{\sqrt{g_{00}}/c}
  10. d l 2 = ( - g α β + g 0 α g 0 β g 00 ) d x α d x β . dl^{2}=\left(-g_{\alpha\beta}+\frac{g_{0\alpha}g_{0\beta}}{g_{00}}\right)\,dx^% {\alpha}\,dx^{\beta}.
  11. d l 2 = γ α β d x α d x β , dl^{2}=\gamma_{\alpha\beta}\,dx^{\alpha}\,dx^{\beta},\,
  12. γ α β = - g α β + g 0 α g 0 β g 00 \gamma_{\alpha\beta}=-g_{\alpha\beta}+\frac{g_{0\alpha}g_{0\beta}}{g_{00}}
  13. δ l i \scriptstyle{\delta_{l}^{i}}
  14. g α β g β γ + g α 0 g 0 γ = δ γ α , g^{\alpha\beta}g_{\beta\gamma}+g^{\alpha 0}g_{0\gamma}=\delta_{\gamma}^{\alpha},
  15. g α β g β 0 + g α 0 g 00 = 0 , g^{\alpha\beta}g_{\beta 0}+g^{\alpha 0}g_{00}=0,\,
  16. g 0 β g β 0 + g 00 g 00 = 1. g^{0\beta}g_{\beta 0}+g^{00}g_{00}=1.\,
  17. - g α β γ β γ = δ γ α , -g^{\alpha\beta}\gamma_{\beta\gamma}=\delta_{\gamma}^{\alpha},
  18. γ α β = - g α β . \gamma^{\alpha\beta}=-g^{\alpha\beta}.\,
  19. - g = g 00 γ . -g=g_{00}\gamma.\,
  20. g α = - g 0 α g 00 . g_{\alpha}=-\frac{g_{0\alpha}}{g_{00}}.
  21. g α = γ α β g β = - g 0 α . g^{\alpha}=\gamma^{\alpha\beta}g_{\beta}=-g^{0\alpha}.\,
  22. g 00 = 1 g 00 - g α g α . g^{00}=\frac{1}{g_{00}}-g_{\alpha}g^{\alpha}.
  23. x 0 + Δ x 0 = x 0 + 1 2 ( d x 0 ( 2 ) + d x 0 ( 1 ) ) . x^{0}+\Delta x^{0}=x^{0}+\tfrac{1}{2}\left(dx^{0(2)}+dx^{0(1)}\right).
  24. Δ x 0 = - g 0 α d x α g 00 g α d x α . \Delta x^{0}=-\frac{g_{0\alpha}\,dx^{\alpha}}{g_{00}}\equiv g_{\alpha}\,dx^{% \alpha}.
  25. Δ x 0 = g 0 i d x i = 0 \Delta x_{0}=g_{0i}dx^{i}=0\,
  26. g 00 = 1 , g 0 α = 0 , g_{00}=1,\quad g_{0\alpha}=0,\,
  27. d s 2 = d t 2 - g α β d x α d x β , ds^{2}=dt^{2}-g_{\alpha\beta}\,dx^{\alpha}\,dx^{\beta},\,
  28. γ α β = - g α β . \gamma_{\alpha\beta}=-g_{\alpha\beta}.\,

Syntaxin.html

  1. α \alpha
  2. α \alpha
  3. C a 2 + Ca^{2+}
  4. C a 2 + Ca^{2+}

Synthetic_position.html

  1. K e - r T Ke^{-rT}
  2. K K
  3. r r
  4. T T
  5. S S
  6. - S -S
  7. K K
  8. m a x ( 0 , S - K ) max(0,S-K)
  9. K - S + m a x ( 0 , S - K ) = m a x ( 0 , K - S ) K-S+max(0,S-K)=max(0,K-S)

System_of_polynomial_equations.html

  1. sin ( x ) 3 + cos ( 3 x ) = 0 \sin(x)^{3}+\cos(3x)=0\,
  2. { s 3 + 4 c 3 - 3 c = 0 s 2 + c 2 - 1 = 0 \begin{cases}s^{3}+4c^{3}-3c&=0\\ s^{2}+c^{2}-1&=0\end{cases}
  3. 2 \sqrt{2}
  4. 2 \sqrt{2}
  5. i i
  6. 1 i n 1≤i≤n
  7. { f 1 ( x 1 ) = 0 f 2 ( x 1 , x 2 ) = 0 f n ( x 1 , x 2 , , x n ) = 0. \begin{cases}f_{1}(x_{1})=0\\ f_{2}(x_{1},x_{2})=0\\ \cdots\\ f_{n}(x_{1},x_{2},\ldots,x_{n})=0.\end{cases}
  8. { x 2 - 1 = 0 ( x - 1 ) ( y - 1 ) = 0 y 2 - 1 = 0 \begin{cases}x^{2}-1=0\\ (x-1)(y-1)=0\\ y^{2}-1=0\end{cases}
  9. 1 1
  10. 0
  11. { h ( x 0 ) = 0 x 1 = g 1 ( x 0 ) / g 0 ( x 0 ) x n = g n ( x 0 ) / g 0 ( x 0 ) \begin{cases}h(x_{0})=0\\ x_{1}=g_{1}(x_{0})/g_{0}(x_{0})\\ \cdots\\ x_{n}=g_{n}(x_{0})/g_{0}(x_{0})\end{cases}
  12. h h
  13. D D
  14. D D
  15. h h
  16. h h
  17. h h
  18. h h
  19. h h
  20. h h
  21. x x
  22. y y
  23. x + y x+y
  24. t = x y t=x\frac{–}{y}
  25. { t 3 - t = 0 x = t 2 + 2 t - 1 3 t 2 - 1 y = t 2 - 2 t - 1 3 t 2 - 1 \begin{cases}t^{3}-t=0\\ x=\frac{t^{2}+2t-1}{3t^{2}-1}\\ y=\frac{t^{2}-2t-1}{3t^{2}-1}\\ \end{cases}
  26. g 1 = 0 , , g n = 0 g_{1}=0,\ldots,g_{n}=0
  27. f 1 = 0 , , f n = 0 f_{1}=0,\ldots,f_{n}=0
  28. ( 1 - t ) g 1 + t f 1 = 0 , , ( 1 - t ) g n + t f n = 0 (1-t)g_{1}+tf_{1}=0,\ldots,(1-t)g_{n}+tf_{n}=0
  29. t 1 < t 2 t_{1}<t_{2}
  30. t = t 2 t=t_{2}
  31. t = t 1 t=t_{1}
  32. t 2 - t 1 : t_{2}-t_{1}:

Szegő_inequality.html

  1. 1 p < + 1\leq p<+\infty
  2. u : n + in W 1 , p ( n ) , u:\mathbb{R}^{n}\rightarrow\mathbb{R}^{+}\,\text{ in }W^{1,p}(\mathbb{R}^{n}),
  3. n | u * | p d n n | u | p d n , \int_{\mathbb{R}^{n}}|\nabla u^{*}|^{p}\,d\mathcal{H}^{n}\leq\int_{\mathbb{R}^% {n}}|\nabla u|^{p}\,d\mathcal{H}^{n},
  4. u * u^{*}
  5. u u

Szegő_kernel.html

  1. Ω ¯ \overline{\Omega}
  2. f P f ( z ) f\mapsto Pf(z)
  3. P f ( z ) = Ω f ( ζ ) k z ( ζ ) ¯ d σ ( ζ ) . Pf(z)=\int_{\partial\Omega}f(\zeta)\overline{k_{z}(\zeta)}\,d\sigma(\zeta).
  4. S ( z , ζ ) = k z ( ζ ) ¯ , z Ω , ζ Ω . S(z,\zeta)=\overline{k_{z}(\zeta)},\quad z\in\Omega,\zeta\in\partial\Omega.
  5. S ( z , ζ ) = i = 1 ϕ i ( z ) ϕ i ( ζ ) ¯ . S(z,\zeta)=\sum_{i=1}^{\infty}\phi_{i}(z)\overline{\phi_{i}(\zeta)}.

T-statistic.html

  1. β ^ \scriptstyle\hat{\beta}
  2. t β ^ = β ^ - β 0 s . e . ( β ^ ) , t_{\hat{\beta}}=\frac{\hat{\beta}-\beta_{0}}{\mathrm{s.e.}(\hat{\beta})},
  3. s . e . ( β ^ ) \scriptstyle s.e.(\hat{\beta})
  4. β ^ \scriptstyle\hat{\beta}
  5. β ^ \scriptstyle\hat{\beta}
  6. β ^ \scriptstyle\hat{\beta}
  7. s . e . ( β ^ ) \scriptstyle s.e.(\hat{\beta})
  8. g ( x , X ) = x - X ¯ s g(x,X)=\frac{x-\overline{X}}{s}
  9. N ( μ , σ 2 ) N(\mu,\sigma^{2})
  10. X n + 1 , X_{n+1},
  11. X n + 1 - X ¯ n s n 1 + n - 1 T n - 1 \frac{X_{n+1}-\overline{X}_{n}}{s_{n}\sqrt{1+n^{-1}}}\sim T^{n-1}
  12. X n + 1 X_{n+1}
  13. X ¯ n + s n 1 + n - 1 T n - 1 \overline{X}_{n}+s_{n}\sqrt{1+n^{-1}}\cdot T^{n-1}
  14. X n + 1 X_{n+1}

Table_of_costs_of_operations_in_elliptic_curves.html

  1. P , [ 2 ] P = P + P , [ 3 ] P = [ 2 ] P + P , , [ n ] P = [ n - 1 ] P + P P,\quad[2]P=P+P,\quad[3]P=[2]P+P,\dots,[n]P=[n-1]P+P
  2. k = i l k i 2 i k=\sum_{i\leq l}k_{i}2^{i}
  3. l = [ l o g 2 k ] l=[log_{2}k]
  4. [ 2 ] ( . ( [ 2 ] ( [ 2 ] ( [ 2 ] ( [ 2 ] ( [ 2 ] P + [ k ( l - 1 ) ] P ) + [ k ( l - 2 ) ] P ) + [ k ( l - 3 ) ] P ) + ) + [ k 1 ] P ) + [ k 0 ] P = [ 2 l ] P + [ k ( l - 1 ) 2 l - 1 ] P + + [ k 1 2 ] P + [ k 0 ] P [2](....([2]([2]([2]([2]([2]P+[k_{(l-1)}]P)+[k_{(l-2)}]P)+[k_{(l-3)}]P)+\dots)% \dots+[k_{1}]P)+[k_{0}]P=[2^{l}]P+[k_{(l-1)}2^{l-1}]P+\dots+[k_{1}2]P+[k_{0}]P

Taguchi_loss_function.html

  1. y ¯ \bar{y}
  2. y ¯ \bar{y}

Talagrand's_concentration_inequality.html

  1. Ω = Ω 1 × Ω 2 × × Ω n \Omega=\Omega_{1}\times\Omega_{2}\times\cdots\times\Omega_{n}
  2. A A
  3. t 0 t\geq 0
  4. Pr [ A ] Pr [ A t ¯ ] e - t 2 / 4 , \Pr[A]\cdot\Pr\left[\overline{A_{t}}\right]\leq e^{-t^{2}/4}\,,
  5. A t ¯ \overline{A_{t}}
  6. A t = { x Ω : ρ ( A , x ) t } A_{t}=\{x\in\Omega~{}:~{}\rho(A,x)\leq t\}
  7. ρ \rho
  8. ρ ( A , x ) = max α , α 2 1 min y A i : x i y i α i \rho(A,x)=\max_{\alpha,\|\alpha\|_{2}\leq 1}\ \min_{y\in A}\ \sum_{i~{}:~{}x_{% i}\neq y_{i}}\alpha_{i}
  9. α 𝐑 n \alpha\in\mathbf{R}^{n}
  10. x , y Ω x,y\in\Omega
  11. n n
  12. α i , x i , y i \alpha_{i},x_{i},y_{i}
  13. 2 \|\cdot\|_{2}
  14. 2 \ell^{2}
  15. α 2 = ( i α i 2 ) 1 / 2 \|\alpha\|_{2}=\left(\sum_{i}\alpha_{i}^{2}\right)^{1/2}

Tangent_indicatrix.html

  1. γ ( t ) \gamma(t)\,
  2. γ ˙ \dot{\gamma}
  3. T ( t ) T(t)\,
  4. γ \gamma\,
  5. T = γ ˙ | γ ˙ | T=\frac{\dot{\gamma}}{|\dot{\gamma}|}
  6. γ \gamma\,
  7. T T\,

Tape_correction_(surveying).html

  1. C L = M L ± C o r r × M L N L C_{L}=M_{L}\pm Corr\times\frac{M_{L}}{N_{L}}
  2. m < 20 % m<20\%
  3. C h = h 2 2 s C_{h}=\frac{h^{2}}{2s}
  4. 20 % m 30 % 20\%\leq m\leq 30\%
  5. C h = h 2 2 s + h 4 8 s 3 C_{h}=\frac{h^{2}}{2s}+\frac{h^{4}}{8s^{3}}
  6. m > 30 % m>30\%
  7. C h = s ( 1 - cos θ ) C_{h}=s(1-\cos\theta)
  8. C h C_{h}
  9. θ \theta
  10. C h C_{h}
  11. s s
  12. d = s - C h d=s-C_{h}
  13. C t = C L ( T m - T s ) C_{t}=C\cdot L(T_{m}-T_{s})
  14. C t C_{t}
  15. T m T_{m}
  16. T s T_{s}
  17. C t C_{t}
  18. L L
  19. d = L + C t d=L+C_{t}
  20. C p = ( P m - P s ) L A E C_{p}=\frac{(P_{m}-P_{s})L}{AE}
  21. C p C_{p}
  22. P m P_{m}
  23. P s P_{s}
  24. C p C_{p}
  25. L L
  26. d = L + C p d=L+C_{p}
  27. A = W ( L ) ( U w ) A=\frac{W}{(L)(U_{w})}
  28. U w U_{w}
  29. U w U_{w}
  30. 7.866 × 10 - 3 k g / c m 3 7.866\times 10^{-3}kg/cm^{3}
  31. C s = ω 2 L 3 24 P 2 C_{s}=\frac{{\omega}^{2}L^{3}}{24P^{2}}
  32. C s C_{s}
  33. ω \omega
  34. C s C_{s}
  35. L L
  36. d = L - C s d=L-C_{s}
  37. ω = W L \omega=\frac{W}{L}
  38. W = ω L W=\omega L
  39. C s = W 2 L 24 P 2 C_{s}=\frac{W^{2}L}{24P^{2}}
  40. y = P ω g cosh ( x ω g P ) y=\frac{P}{\omega g}\cosh\left(\frac{x\omega g}{P}\right)
  41. L = - k / 2 + k / 2 1 + ( d y / d x ) 2 d x L=\int_{-k/2}^{+k/2}\sqrt{1+\left(dy/dx\right)^{2}}\,dx
  42. a = P ω g a=\frac{P}{\omega g}
  43. 1 + ( d y / d x ) 2 = 1 + ( d d x ( a cosh ( x a ) ) ) 2 = 1 + sinh 2 ( x a ) = cosh ( x a ) \sqrt{1+\left(dy/dx\right)^{2}}=\sqrt{1+\left(\frac{d}{dx}\left(a\cosh\left(% \frac{x}{a}\right)\right)\right)^{2}}=\sqrt{1+\sinh^{2}\left(\frac{x}{a}\right% )}=\cosh\left(\frac{x}{a}\right)
  44. L = - k / 2 + k / 2 cosh ( x a ) d x = [ a sinh ( x a ) ] x = - k / 2 x = + k / 2 = ( 2 a ) sinh ( k 2 a ) L=\int_{-k/2}^{+k/2}\cosh\left(\frac{x}{a}\right)dx=\left[a\sinh\left(\frac{x}% {a}\right)\right]_{x=-k/2}^{x=+k/2}=\left(2a\right)\sinh\left(\frac{k}{2a}\right)
  45. δ = k - L \delta=k-L
  46. δ \delta
  47. C s C_{s}
  48. δ \delta
  49. δ \delta
  50. δ \delta
  51. L 2 a = sinh ( k 2 a ) \frac{L}{2a}=\sinh\left(\frac{k}{2a}\right)
  52. sinh - 1 ( L 2 a ) = k 2 a \sinh^{-1}\left(\frac{L}{2a}\right)=\frac{k}{2a}
  53. k = ( 2 a ) sinh - 1 ( L 2 a ) k=\left(2a\right)\sinh^{-1}\left(\frac{L}{2a}\right)
  54. δ = k - L = ( 2 a ) sinh - 1 ( L 2 a ) - L \delta=k-L=\left(2a\right)\sinh^{-1}\left(\frac{L}{2a}\right)-L
  55. d δ d L = 1 L 2 4 a 2 + 1 - 1 \frac{d\delta}{dL}=\frac{1}{\sqrt{\frac{L^{2}}{4a^{2}}+1}}-1
  56. δ \delta
  57. d 2 δ d L 2 = - L 4 a 2 ( L 2 4 a 2 + 1 ) 3 / 2 \frac{d^{2}\delta}{{dL}^{2}}=-\frac{L}{4a^{2}\left(\frac{L^{2}}{4a^{2}}+1% \right)^{3/2}}
  58. d 3 δ d L 3 = - ( 8 a 3 - 4 aL 2 ) ( 4 a 2 + L 2 ) 5 / 2 \frac{d^{3}\delta}{{dL}^{3}}=-\frac{\left(8a^{3}-4\,\text{aL}^{2}\right)}{% \left(4a^{2}+L^{2}\right)^{5/2}}
  59. δ [ d 3 δ d L 3 ] L = 0 L 3 3 ! = - 1 4 a 2 L 3 6 = - L 3 24 a 2 = - L 3 ω 2 g 2 24 P 2 \delta\cong\left[\frac{d^{3}\delta}{{dL}^{3}}\right]_{L=0}\frac{L^{3}}{3!}=-% \frac{1}{4a^{2}}\frac{L^{3}}{6}=\frac{-L^{3}}{24a^{2}}=\frac{-L^{3}\omega^{2}g% ^{2}}{24P^{2}}

Tapering_(mathematics).html

  1. q = [ a 0 0 0 b 0 0 0 1 ] p , q=\begin{bmatrix}a&0&0\\ 0&b&0\\ 0&0&1\end{bmatrix}p,
  2. q = [ a ( p z ) 0 0 0 b ( p z ) 0 0 0 1 ] p . q=\begin{bmatrix}a(p_{z})&0&0\\ 0&b(p_{z})&0\\ 0&0&1\end{bmatrix}p.
  3. a ( z ) = α 0 + α 1 z a(z)=\alpha_{0}+\alpha_{1}z
  4. a ( z ) = α 0 + α 1 z + α 2 z 2 a(z)={\alpha}_{0}+{\alpha}_{1}z+{\alpha}_{2}z^{2}

Tate–Shafarevich_group.html

  1. v ker ( H 1 ( G K , A ) H 1 ( G K v , A v ) ) . \bigcap_{v}\mathrm{ker}(H^{1}(G_{K},A)\rightarrow H^{1}(G_{K_{v}},A_{v})).
  2. x 4 - 17 = 2 y 2 x^{4}-17=2y^{2}
  3. 3 x 3 + 4 y 3 + 5 z 3 = 0. 3x^{3}+4y^{3}+5z^{3}=0.

Taylor_microscale.html

  1. R e - 1 / 2 Re^{-1/2}
  2. R e - 3 / 4 Re^{-3/4}
  3. R e Re

TC_(complexity).html

  1. O ( log i n ) O(\log^{i}n)
  2. TC = i 0 TC . i \mbox{TC}~{}=\bigcup_{i\geq 0}\mbox{TC}~{}^{i}.
  3. NC i AC i TC i NC . i + 1 \mbox{NC}~{}^{i}\subseteq\mbox{AC}~{}^{i}\subseteq\mbox{TC}~{}^{i}\subseteq% \mbox{NC}~{}^{i+1}.
  4. NC 0 AC 0 TC 0 NC . 1 \mbox{NC}~{}^{0}\subsetneq\mbox{AC}~{}^{0}\subsetneq\mbox{TC}~{}^{0}\subseteq% \mbox{NC}~{}^{1}.

TD-Gammon.html

  1. w t + 1 - w t = α ( Y t + 1 - Y t ) k = 1 t λ t - k w Y k w_{t+1}-w_{t}=\alpha(Y_{t+1}-Y_{t})\sum_{k=1}^{t}\lambda^{t-k}\nabla_{w}Y_{k}
  2. w t + 1 - w t w_{t+1}-w_{t}
  3. Y t + 1 - Y t Y_{t+1}-Y_{t}
  4. α \alpha
  5. λ \lambda
  6. λ = 0 \lambda=0
  7. λ = 1 \lambda=1
  8. λ \lambda
  9. w Y k \nabla_{w}Y_{k}

Teichmüller–Tukey_lemma.html

  1. A A\in\mathcal{F}
  2. A A
  3. \mathcal{F}
  4. A A
  5. \mathcal{F}
  6. A A
  7. \mathcal{F}
  8. \mathcal{F}

Telescoping_Markov_chain.html

  1. N > 1 N>1
  2. { 𝒮 } = 1 N \{\mathcal{S}^{\ell}\}_{\ell=1}^{N}
  3. θ k \theta_{k}
  4. θ k = ( θ k 1 , . . , θ k N ) 𝒮 1 × × 𝒮 N \theta_{k}=(\theta_{k}^{1},.....,\theta_{k}^{N})\in\mathcal{S}^{1}\times......% \times\mathcal{S}^{N}
  5. { Λ n } n = 1 N \{\Lambda^{n}\}_{n=1}^{N}
  6. θ k 1 \theta_{k}^{1}
  7. Λ 1 \Lambda^{1}
  8. ( θ k 1 = s | θ k - 1 1 = r ) = Λ 1 ( s | r ) \mathbb{P}(\theta_{k}^{1}=s|\theta_{k-1}^{1}=r)=\Lambda^{1}(s|r)
  9. ( θ k n = s | θ k - 1 n = r , θ k n - 1 = t ) = Λ n ( s | r , t ) \mathbb{P}(\theta_{k}^{n}=s|\theta_{k-1}^{n}=r,\theta_{k}^{n-1}=t)=\Lambda^{n}% (s|r,t)
  10. n 2. n\geq 2.
  11. θ k \theta_{k}
  12. Λ \Lambda
  13. ( θ k + 1 = s | θ k = r ) = Λ 1 ( s 1 | r 1 ) = 2 N Λ ( s | r , s - 1 ) \mathbb{P}(\theta_{k+1}=\vec{s}|\theta_{k}=\vec{r})=\Lambda^{1}(s_{1}|r_{1})% \prod_{\ell=2}^{N}\Lambda^{\ell}(s_{\ell}|r_{\ell},s_{\ell-1})
  14. s = ( s 1 , , s N ) 𝒮 1 × × 𝒮 N \vec{s}=(s_{1},\ldots,s_{N})\in\mathcal{S}^{1}\times\cdots\times\mathcal{S}^{N}
  15. r = ( r 1 , , r N ) 𝒮 1 × × 𝒮 N . \vec{r}=(r_{1},\ldots,r_{N})\in\mathcal{S}^{1}\times\cdots\times\mathcal{S}^{N}.

Template:Base_Planck_units.html

  1. l P = G c 3 l\text{P}=\sqrt{\frac{\hbar G}{c^{3}}}
  2. m P = c G m\text{P}=\sqrt{\frac{\hbar c}{G}}
  3. t P = l P c = m P c 2 = G c 5 t\text{P}=\frac{l\text{P}}{c}=\frac{\hbar}{m\text{P}c^{2}}=\sqrt{\frac{\hbar G% }{c^{5}}}
  4. q P = 4 π ε 0 c q\text{P}=\sqrt{4\pi\varepsilon_{0}\hbar c}
  5. T P = m P c 2 k B = c 5 G k B 2 T\text{P}=\frac{m\text{P}c^{2}}{k\text{B}}=\sqrt{\frac{\hbar c^{5}}{Gk\text{B}% ^{2}}}

Template:F::::doc.html

  1. f / f/

Template:Graph_families_defined_by_their_automorphisms.html

  1. s y m b o l symbol{\rightarrow}
  2. s y m b o l symbol{\leftarrow}
  3. s y m b o l symbol{\downarrow}
  4. s y m b o l symbol{\leftarrow}
  5. s y m b o l symbol{\downarrow}
  6. s y m b o l symbol{\rightarrow}
  7. s y m b o l symbol{\rightarrow}
  8. s y m b o l symbol{\downarrow}
  9. s y m b o l symbol{\downarrow}
  10. s y m b o l symbol{\downarrow}
  11. s y m b o l symbol{\rightarrow}
  12. s y m b o l symbol{\rightarrow}
  13. s y m b o l symbol{\uparrow}
  14. s y m b o l symbol{\leftarrow}

Template:Infobox_control_chart::doc.html

  1. n p ¯ ± 3 n p ¯ ( 1 - p ¯ ) n\bar{p}\pm 3\sqrt{n\bar{p}(1-\bar{p})}
  2. n p ¯ i = j = 1 n { 1 if x i j defective 0 otherwise n\bar{p}_{i}=\sum_{j=1}^{n}\begin{cases}1&\mbox{if }~{}x_{ij}\mbox{ defective}% \\ 0&\mbox{otherwise}\end{cases}

Template:Infobox_probability_distribution::doc.html

  1. 𝒩 ( μ , σ 2 ) \scriptstyle\mathcal{N}(\mu,\sigma^{2})
  2. F ( ) F()
  3. Q ( ) Q()
  4. Q ( F ( x ) ) = x Q(F(x))=x

Template:List_of_mesons.html

  1. u u ¯ - d d ¯ 2 \mathrm{\tfrac{u\bar{u}-d\bar{d}}{\sqrt{2}}}\,
  2. u u ¯ + d d ¯ + s s ¯ 3 \mathrm{\tfrac{u\bar{u}+d\bar{d}+s\bar{s}}{\sqrt{3}}}\,
  3. 1 / 2 {1}/{2}
  4. 1 / 2 {1}/{2}
  5. d s ¯ - s d ¯ 2 \mathrm{\tfrac{d\bar{s}-s\bar{d}}{\sqrt{2}}}\,
  6. 1 / 2 {1}/{2}
  7. d s ¯ + s d ¯ 2 \mathrm{\tfrac{d\bar{s}+s\bar{d}}{\sqrt{2}}}\,
  8. 1 / 2 {1}/{2}
  9. 1 / 2 {1}/{2}
  10. 1 / 2 {1}/{2}
  11. ħ / Γ {ħ}/{Γ}
  12. u u ¯ - d d ¯ 2 \mathrm{\tfrac{u\bar{u}-d\bar{d}}{\sqrt{2}}}\,
  13. u u ¯ + d d ¯ 2 \mathrm{\tfrac{u\bar{u}+d\bar{d}}{\sqrt{2}}}\,
  14. 1 / 2 {1}/{2}
  15. 1 / 2 {1}/{2}
  16. ħ / Γ {ħ}/{Γ}

Template:MEst::doc.html

  1. M = π * ρ * d 3 6 . M=\frac{\pi*\rho*d^{3}}{6}.

Template:Negative_binomial_distribution.html

  1. 6 r + ( 1 - p ) 2 p r \frac{6}{r}+\frac{(1-p)^{2}}{pr}
  2. ( 1 - p 1 - p e t ) r for t < - log p \biggl(\frac{1-p}{1-pe^{t}}\biggr)^{\!r}\,\text{ for }t<-\log p
  3. ( 1 - p 1 - p e i t ) r with t \biggl(\frac{1-p}{1-pe^{i\,t}}\biggr)^{\!r}\,\text{ with }t\in\mathbb{R}
  4. ( 1 - p 1 - p z ) r for | z | < 1 p \biggl(\frac{1-p}{1-pz}\biggr)^{\!r}\,\text{ for }|z|<\frac{1}{p}
  5. r p 2 ( 1 - p ) \frac{r}{p^{2}(1-p)}

Template:Odd_polygon_db.html

  1. 128.571 \approx 128.571
  2. 147.273 \approx 147.273
  3. 152.308 \approx 152.308
  4. 158.82 \approx 158.82
  5. 161.052 \approx 161.052

Template:Quantum_Variables.html

  1. V 0 V_{0}
  2. V V
  3. λ \lambda
  4. τ 0 \tau_{0}
  5. τ H \tau_{H}
  6. τ L \tau_{L}

Template:Rail-interchange::doc.html

  1. f ( w h e r e , h o w ) b r a n d , l i n k , i c o n ( s ) f(where,how)\to brand,link,icon(s)

Template:Repeat::doc.html

  1. 10 x = 2 x 10x=2^{x}

Template:Surface_temperature::doc.html

  1. T = T eff ( 1 - q p ν ) 1 / 4 2 52 / r , \begin{smallmatrix}T\ =\ \frac{T_{\textrm{eff}}(1-qp_{\nu})^{1/4}}{\sqrt{2}}% \sqrt{52/r},\end{smallmatrix}

Template:Table_of_atomic_and_nuclear_constants.html

  1. a 0 = α / 4 π R a_{0}=\alpha/4\pi R_{\infty}\,
  2. r e = e 2 / 4 π ε 0 m e c 2 r_{\mathrm{e}}=e^{2}/4\pi\varepsilon_{0}m_{e}c^{2}\,
  3. m e m_{\mathrm{e}}\,
  4. G F / ( c ) 3 G_{\mathrm{F}}/(\hbar c)^{3}
  5. α = μ 0 e 2 c / 2 h = e 2 / 4 π ε 0 c \alpha=\mu_{0}e^{2}c/2h=e^{2}/4\pi\varepsilon_{0}\hbar c\,
  6. E h = 2 R h c E_{\mathrm{h}}=2R_{\infty}hc\,
  7. m p m_{\mathrm{p}}\,
  8. h / 2 m e h/2m_{\mathrm{e}}\,
  9. R = α 2 m e c / 2 h R_{\infty}=\alpha^{2}m_{\mathrm{e}}c/2h\,
  10. ( 8 π / 3 ) r e 2 (8\pi/3)r_{\mathrm{e}}^{2}
  11. sin 2 θ W = 1 - ( m W / m Z ) 2 \sin^{2}\theta_{\mathrm{W}}=1-(m_{\mathrm{W}}/m_{\mathrm{Z}})^{2}\,
  12. [ - F o r m u l a E r r o r - ] [-FormulaError-]

Template:Table_of_electromagnetic_constants.html

  1. μ 0 \mu_{0}\,
  2. ε 0 = 1 / μ 0 c 2 \varepsilon_{0}=1/\mu_{0}c^{2}\,
  3. Z 0 = μ 0 c Z_{0}=\mu_{0}c\,
  4. k e = 1 / 4 π ε 0 k_{\mathrm{e}}=1/4\pi\varepsilon_{0}\,
  5. e e\,
  6. μ B = e / 2 m e \mu_{\mathrm{B}}=e\hbar/2m_{e}
  7. G 0 = 2 e 2 / h G_{0}=2e^{2}/h\,
  8. G 0 - 1 = h / 2 e 2 G_{0}^{-1}=h/2e^{2}\,
  9. K J = 2 e / h K_{\mathrm{J}}=2e/h\,
  10. ϕ 0 = h / 2 e \phi_{0}=h/2e\,
  11. μ N = e / 2 m p \mu_{\mathrm{N}}=e\hbar/2m_{p}
  12. R K = h / e 2 R_{\mathrm{K}}=h/e^{2}\,

Template:Table_of_physico-chemical_constants.html

  1. m u = 1 u m_{\mathrm{u}}=1\,\mathrm{u}\,
  2. N A , L N_{\mathrm{A}},L\,
  3. k = k B = R / N A k=k_{\mathrm{B}}=R/N_{\mathrm{A}}\,
  4. F = N A e F=N_{\mathrm{A}}e\,
  5. c 1 = 2 π h c 2 c_{1}=2\pi hc^{2}\,
  6. c 1 L = c 1 / π c_{\mathrm{1L}}=c_{1}/\pi\,
  7. T T
  8. p p
  9. n 0 = N A / V m n_{0}=N_{\mathrm{A}}/V_{\mathrm{m}}\,
  10. R R\,
  11. N A h N_{\mathrm{A}}h\,
  12. T T
  13. p p
  14. V m = R T / p V_{\mathrm{m}}=RT/p\,
  15. T T
  16. p p
  17. T T
  18. p p
  19. S 0 / R = 5 2 S_{0}/R=\frac{5}{2}
  20. + ln [ ( 2 π m u k T / h 2 ) 3 / 2 k T / p ] +\ln\left[(2\pi m_{\mathrm{u}}kT/h^{2})^{3/2}kT/p\right]
  21. T T
  22. p p
  23. c 2 = h c / k c_{2}=hc/k\,
  24. σ = π 2 k 4 / 60 3 c 2 \sigma=\pi^{2}k^{4}/60\hbar^{3}c^{2}
  25. b = h c k - 1 / b=hck^{-1}/\,

Template:Table_of_universal_constants.html

  1. c c\,
  2. G G\,
  3. h h\,
  4. = h / ( 2 π ) \hbar=h/(2\pi)

Tensor_product_model_transformation.html

  1. f ( 𝐱 ) f({\mathbf{x}})
  2. 𝐱 R N \mathbf{x}\in R^{N}
  3. f ( 𝐱 ) = i 1 = 1 I 1 i 2 = 1 I 2 i N = 1 I N n = 1 N w n , i n ( x n ) s i 1 , i 2 , , i N , f(\mathbf{x})=\sum_{i_{1}=1}^{I_{1}}\sum_{i_{2}=1}^{I_{2}}\ldots\sum_{i_{N}=1}% ^{I_{N}}\prod_{n=1}^{N}w_{n,i_{n}}(x_{n})s_{i_{1},i_{2},\ldots,i_{N}},
  4. \otimes
  5. f ( 𝐱 ) = 𝒮 n = 1 N 𝐰 n ( x n ) , f(\mathbf{x})=\mathcal{S}\mathop{\otimes}_{n=1}^{N}\mathbf{w}_{n}(x_{n}),
  6. 𝒮 I 1 × I 2 × × I N \mathcal{S}\in\mathcal{R}^{I_{1}\times I_{2}\times\ldots\times I_{N}}
  7. s i 1 i 2 i N s_{i_{1}i_{2}\ldots i_{N}}
  8. 𝐰 n ( x n ) , ( n = 1 N ) \mathbf{w}_{n}(x_{n}),(n=1\ldots N)
  9. w n , i n ( x n ) , ( i n = 1 I n ) w_{n,i_{n}}(x_{n}),(i_{n}=1\ldots I_{n})
  10. w n , i n ( x n ) w_{n,i_{n}}(x_{n})
  11. i n i_{n}
  12. n n
  13. x n x_{n}
  14. n n
  15. 𝐱 \mathbf{x}
  16. I n I_{n}
  17. n n
  18. ( 𝐱 ) = 𝒮 n = 1 N 𝐰 n ( x n ) . \mathcal{F}(\mathbf{x})=\mathcal{S}\boxtimes_{n=1}^{N}\mathbf{w}_{n}(x_{n}).
  19. 𝒴 = ( 𝐱 ) \mathcal{Y}=\mathcal{F}({\mathbf{x}})
  20. 𝒴 L 1 × L 2 × L O \mathcal{Y}\in\mathcal{R}^{L_{1}\times L_{2}\times\ldots L_{O}}
  21. 𝒮 I 1 × I 2 × × I N × L 1 × L 2 × × L O \mathcal{S}\in\mathcal{R}^{I_{1}\times I_{2}\times\ldots\times I_{N}\times L_{% 1}\times L_{2}\times...\times L_{O}}
  22. \boxtimes
  23. \otimes
  24. L 1 × L 2 × × L O L_{1}\times L_{2}\times...\times L_{O}
  25. 𝒮 \mathcal{S}
  26. 𝐱 \mathbf{x}
  27. Ω = [ a 1 , b 1 ] × [ a 2 , b 2 ] × × [ a N , b N ] R N \Omega=[a_{1},b_{1}]\times[a_{2},b_{2}]\times...\times[a_{N},b_{N}]\subset R^{N}
  28. n : i n = 1 I n w n , i n ( x n ) = 1 \forall n:\sum_{i_{n}=1}^{I_{n}}w_{n,i_{n}}(x_{n})=1
  29. w n , i n ( x n ) [ 0 , 1 ] . w_{n,i_{n}}(x_{n})\in[0,1].
  30. f ( 𝐱 ) f(\mathbf{x})
  31. 𝐱 Ω \mathbf{x}\in\Omega
  32. 𝒴 = ( 𝐱 ) \mathcal{Y}=\mathcal{F}(\mathbf{x})
  33. 𝐱 Ω R N \mathbf{x}\in\Omega\subset R^{N}
  34. ( 𝐱 ) = 𝒮 n = 1 N 𝐰 n ( x n ) \mathcal{F}(\mathbf{x})=\mathcal{S}\boxtimes_{n=1}^{N}\mathbf{w}_{n}(x_{n})
  35. 𝒮 \mathcal{S}
  36. 𝐰 n ( x n ) \mathbf{w}_{n}(x_{n})
  37. n = 1 N n=1\ldots N
  38. ( 𝐱 ) \mathcal{F}(\mathbf{x})
  39. ( 𝐱 ) 𝒮 n = 1 N 𝐰 n ( x n ) , \mathcal{F}(\mathbf{x})\approx\mathcal{S}\boxtimes_{n=1}^{N}\mathbf{w}_{n}(x_{% n}),
  40. N + O N+O
  41. n = 1 N n=1\ldots N
  42. O O

Term_(logic).html

  1. n ( n + 1 ) 2 \frac{n(n+1)}{2}
  2. θ h + 1 = i = 0 f i θ h i \theta_{h+1}=\sum_{i=0}^{\infty}f_{i}\cdot\theta_{h}^{i}
  3. a * ( ( a + 1 ) * ( a + 2 ) ) 1 * ( 2 * 3 ) \frac{a*((a+1)*(a+2))}{1*(2*3)}
  4. a * ( b + 1 ) 1 * ( 2 * 3 ) \frac{a*(b+1)}{1*(2*3)}
  5. a * ( . ) 1 * ( 2 * 3 ) \frac{a*(\;.\;)}{1*(2*3)}
  6. 0 \vec{0}
  7. v , w \vec{v},\vec{w}
  8. ( v + 0 ) * a , w * b \langle(\vec{v}+\vec{0})*a,\vec{w}*b\rangle
  9. v + a \vec{v}+a
  10. a * v a*\vec{v}
  11. i = 1 n i 2 \sum_{i=1}^{n}i^{2}
  12. a b sin ( k t ) d t \int_{a}^{b}\sin(k\cdot t)dt
  13. t a b sin ( k t ) d t t\cdot\int_{a}^{b}\sin(k\cdot t)\;dt
  14. k a b sin ( k t ) d t k\cdot\int_{a}^{b}\sin(k\cdot t)\;dt

Terminal_sliding_mode.html

  1. x 1 ( t ) = x 2 ( t ) \overset{\cdot}{x}_{1}(t)=x_{2}(t)
  2. x n - 1 ( t ) = x n ( t ) \overset{\cdot}{x}_{n-1}(t)=x_{n}(t)
  3. x n ( t ) = a ( x ) + b ( x ) u ( t ) \overset{\cdot}{x}_{n}(t)=a(x)+b(x)u(t)
  4. x ( t ) R n x(t)\in R^{n}
  5. u R u\in R
  6. a ( x ) a(x)
  7. b ( x ) b(x)
  8. x ( t ) x(t)
  9. s 1 ( t ) = s 0 ( t ) + α 1 ( t ) s 0 γ 1 ( t ) s_{1}(t)=\overset{\cdot}{s}_{0}(t)+\alpha_{1}(t)s_{0}^{\gamma_{1}}(t)
  10. s 2 ( t ) = s 1 ( t ) + α 2 ( t ) s 1 γ 2 ( t ) s_{2}(t)=\overset{\cdot}{s}_{1}(t)+\alpha_{2}(t)s_{1}^{\gamma_{2}}(t)
  11. s n - 1 ( t ) = s n - 2 ( t ) + α n - 1 ( t ) s n - 2 γ n - 1 ( t ) s_{n-1}(t)=\overset{\cdot}{s}_{n-2}(t)+\alpha_{n-1}(t)s_{n-2}^{\gamma_{n-1}}(t)
  12. s 0 ( t ) = x 1 ( t ) s_{0}(t)=x_{1}(t)
  13. γ i = p i q i , i = 1 , 2 , , n - 1 \gamma_{i}=\frac{p_{i}}{q_{i}},i=1,2,...,n-1
  14. p i , q i p_{i},q_{i}
  15. p i q i p_{i}\leq q_{i}

Ternary_tree.html

  1. h h
  2. M ( h ) M(h)
  3. M ( h ) = 1 + 3 + 9 + + 3 h = i = 0 h 3 i = 3 h + 1 - 1 2 M(h)=1+3+9+\cdots+3^{h}=\sum_{i=0}^{h}3^{i}=\frac{3^{h+1}-1}{2}
  4. 3 h + 1 - 1 2 \frac{3^{h+1}-1}{2}
  5. N N
  6. [ k ] [k]
  7. [ 3 k - 1 ] [3k-1]
  8. [ 3 k ] [3k]
  9. [ 3 k + 1 ] [3k+1]

Tertiary_ideal.html

  1. t ( I ) = { r R | s I , x ( s ) x I and ( x ) ( r ) I } . t(I)=\{r\in R\mbox{ }~{}|\mbox{ }~{}\forall s\notin I,\mbox{ }~{}\exists x\in(% s)\mbox{ }~{}x\notin I\,\text{ and }(x)(r)\subset I\}.\,
  2. I = T 1 T n I=T_{1}\cap\dots\cap T_{n}

Test_Template_Framework.html

  1. O p Op
  2. x 1 x n x_{1}\dots x_{n}
  3. O p Op
  4. T 1 T n T_{1}\dots T_{n}
  5. O p Op
  6. I S O p IS_{Op}
  7. [ x 1 : T 1 x n : T n ] [x_{1}:T_{1}\dots x_{n}:T_{n}]
  8. O p Op
  9. pre O p \,\text{pre }Op
  10. O p Op
  11. O p Op
  12. V I S O p VIS_{Op}
  13. [ I S O p | pre O p ] [IS_{Op}|\,\text{pre }Op]
  14. O p Op
  15. P P
  16. V I S O p VIS_{Op}
  17. [ V I S O p | P ] [VIS_{Op}|P]
  18. O p Op
  19. [ I S O p | pre O p P ] [IS_{Op}|\,\text{pre }Op\land P]
  20. C O p C_{Op}
  21. O p Op
  22. [ C O p | P ] [C_{Op}|P]
  23. O p Op
  24. C O p C_{Op}
  25. O p Op
  26. P P
  27. C O p = = [ C O p | P ] C^{\prime}_{Op}==[C_{Op}|P]
  28. C O p C^{\prime}_{Op}
  29. C O p C^{\prime}_{Op}
  30. P P
  31. S T S\spadesuit T
  32. \spadesuit
  33. \cup
  34. \cap
  35. \setminus
  36. S = , T = S , T , S T S = , T S , T , T S S , T = S , T , T = S S , T , S T = S , T , S T , ¬ ( S T ) , ¬ ( T S ) , S T \begin{array}[]{l|l}S=\emptyset,T=\emptyset&S\neq\emptyset,T\neq\emptyset,S% \subset T\\ \hline S=\emptyset,T\neq\emptyset&S\neq\emptyset,T\neq\emptyset,T\subset S\\ \hline S\neq\emptyset,T=\emptyset&S\neq\emptyset,T\neq\emptyset,T=S\\ \hline S\neq\emptyset,T\neq\emptyset,S\cap T=\emptyset&S\neq\emptyset,T\neq% \emptyset,S\cap T\neq\emptyset,\lnot(S\subseteq T),\lnot(T\subseteq S),S\neq T% \end{array}
  37. R G = ( dom G R ) G R\oplus G=(\,\text{dom }G\ntriangleleft R)\cup G
  38. \ntriangleleft
  39. \ntriangleleft
  40. \cup
  41. \oplus
  42. e x p r { e x p r 1 , , e x p r n } expr\in\{expr_{1},\dots,expr_{n}\}
  43. n n
  44. e x p r = e x p r i expr=expr_{i}
  45. v a r = v a l var=val
  46. v a r var
  47. v a l val
  48. \mathbb{Z}
  49. \mathbb{N}
  50. n n
  51. \mathbb{Z}
  52. [ i , j ] [i,j]
  53. n < i n<i
  54. n = i n=i
  55. i < n n < j i<n\land n<j
  56. n = j n=j
  57. n > j n>j
  58. C O L O U R : := r e d | b l u e | g r e e n COLOUR::=red|blue|green
  59. c c
  60. C O L O U R COLOUR
  61. c c
  62. r e d red
  63. c c
  64. b l u e blue
  65. c c
  66. g r e e n green
  67. e x p r { e x p r 1 , , e x p r n } expr\subset\{expr_{1},\dots,expr_{n}\}
  68. 2 n - 1 2^{n}-1
  69. e x p r = A i expr=A_{i}
  70. i [ 1 , 2 n - 1 ] i\in[1,2^{n}-1]
  71. A i { e x p r 1 , , e x p r n } { { e x p r 1 , , e x p r n } } A_{i}\in\mathbb{P}\{expr_{1},\dots,expr_{n}\}\setminus\{\{expr_{1},\dots,expr_% {n}\}\}
  72. { e x p r 1 , , e x p r n } \{expr_{1},\dots,expr_{n}\}
  73. { e x p r 1 , , e x p r n } \mathbb{P}\{expr_{1},\dots,expr_{n}\}
  74. e x p r expr
  75. { e x p r 1 , , e x p r n } \{expr_{1},\dots,expr_{n}\}
  76. e x p r { e x p r 1 , , e x p r n } expr\subseteq\{expr_{1},\dots,expr_{n}\}
  77. 2 n 2^{n}
  78. { e x p r 1 , , e x p r n } \{expr_{1},\dots,expr_{n}\}
  79. V I S = = [ I S | P ] T C L T 1 1 = = [ V I S | P T 1 1 ] T C L T 1 n = = [ V I S | P T 1 n ] T C L T 2 1 = = [ T C L T 1 i | P T 2 1 ] T C L T 2 m = = [ T C L T 1 i | P T 2 m ] T C L T 3 1 = = [ T C L T 2 j | P T 3 1 ] T C L T 3 k = = [ T C L T 2 j | P T 3 k ] \begin{aligned}\displaystyle VIS&\displaystyle==[IS|P]\\ \displaystyle TCL_{T_{1}}^{1}&\displaystyle==[VIS|P_{T_{1}}^{1}]\\ &\displaystyle\dots\\ \displaystyle TCL_{T_{1}}^{n}&\displaystyle==[VIS|P_{T_{1}}^{n}]\\ \displaystyle TCL_{T_{2}}^{1}&\displaystyle==[TCL_{T_{1}}^{i}|P_{T_{2}}^{1}]\\ &\displaystyle\dots\\ \displaystyle TCL_{T_{2}}^{m}&\displaystyle==[TCL_{T_{1}}^{i}|P_{T_{2}}^{m}]\\ &\displaystyle\dots\\ \displaystyle TCL_{T_{3}}^{1}&\displaystyle==[TCL_{T_{2}}^{j}|P_{T_{3}}^{1}]\\ &\displaystyle\dots\\ \displaystyle TCL_{T_{3}}^{k}&\displaystyle==[TCL_{T_{2}}^{j}|P_{T_{3}}^{k}]\\ &\displaystyle\dots\\ &\displaystyle\dots\\ &\displaystyle\dots\end{aligned}
  80. O p Op
  81. V I S O p VIS_{Op}
  82. O p Op
  83. x 1 : T 1 x n : T n x_{1}:T_{1}\dots x_{n}:T_{n}
  84. V I S O p VIS_{Op}
  85. C O p C_{Op}
  86. O p Op
  87. P 1 P m P_{1}\dots P_{m}
  88. C O p C_{Op}
  89. V I S O p VIS_{Op}
  90. v 1 : T 1 v n : T n v_{1}:T_{1}\dots v_{n}:T_{n}
  91. n n
  92. P 1 P m P_{1}\land\dots\land P_{m}
  93. C O p C_{Op}
  94. [ C O p | x 1 = v 1 x n = v n ] [C_{Op}|x_{1}=v_{1}\land\dots\land x_{n}=v_{n}]

Tetrahedral_hypothesis.html

  1. V = k × A × A V=k\times A\times\sqrt{A}

Textual_variants_in_the_New_Testament.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}
  3. 𝔓 \mathfrak{P}
  4. 𝔓 \mathfrak{P}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}
  7. 𝔓 \mathfrak{P}
  8. 𝔓 \mathfrak{P}
  9. 𝔓 \mathfrak{P}
  10. 𝔓 \mathfrak{P}
  11. 𝔓 \mathfrak{P}
  12. 𝔓 \mathfrak{P}
  13. 𝔓 \mathfrak{P}
  14. 𝔓 \mathfrak{P}
  15. 𝔓 \mathfrak{P}
  16. 𝔓 \mathfrak{P}
  17. 𝔓 \mathfrak{P}
  18. 𝔓 \mathfrak{P}
  19. 𝔓 \mathfrak{P}
  20. 𝔓 \mathfrak{P}
  21. 𝔓 \mathfrak{P}
  22. 𝔓 \mathfrak{P}
  23. 𝔓 \mathfrak{P}
  24. 𝔓 \mathfrak{P}
  25. 𝔓 \mathfrak{P}
  26. 𝔓 \mathfrak{P}
  27. 𝔓 \mathfrak{P}
  28. 𝔓 \mathfrak{P}
  29. 𝔓 \mathfrak{P}
  30. 𝔓 \mathfrak{P}
  31. 𝔓 \mathfrak{P}
  32. 𝔓 \mathfrak{P}
  33. 𝔓 \mathfrak{P}
  34. 𝔓 \mathfrak{P}
  35. 𝔓 \mathfrak{P}
  36. 𝔓 \mathfrak{P}
  37. 𝔓 \mathfrak{P}
  38. 𝔓 \mathfrak{P}
  39. 𝔓 \mathfrak{P}
  40. 𝔓 \mathfrak{P}
  41. 𝔓 \mathfrak{P}
  42. 𝔓 \mathfrak{P}
  43. 𝔓 \mathfrak{P}
  44. 𝔓 \mathfrak{P}
  45. 𝔓 \mathfrak{P}
  46. 𝔓 \mathfrak{P}
  47. 𝔓 \mathfrak{P}
  48. 𝔓 \mathfrak{P}
  49. 𝔓 \mathfrak{P}
  50. 𝔓 \mathfrak{P}
  51. 𝔓 \mathfrak{P}
  52. 𝔓 \mathfrak{P}
  53. 𝔓 \mathfrak{P}
  54. 𝔓 \mathfrak{P}
  55. 𝔓 \mathfrak{P}
  56. 𝔓 \mathfrak{P}
  57. 𝔓 \mathfrak{P}
  58. 𝔓 \mathfrak{P}
  59. 𝔓 \mathfrak{P}
  60. 𝔓 \mathfrak{P}
  61. 𝔓 \mathfrak{P}
  62. 𝔓 \mathfrak{P}
  63. 𝔓 \mathfrak{P}
  64. 𝔓 66 \mathfrak{P}^{66}
  65. 𝔓 \mathfrak{P}
  66. 𝔓 \mathfrak{P}
  67. 𝔓 \mathfrak{P}
  68. 𝔓 \mathfrak{P}
  69. 𝔓 \mathfrak{P}
  70. 𝔓 \mathfrak{P}
  71. 𝔓 \mathfrak{P}
  72. 𝔓 \mathfrak{P}
  73. 𝔓 \mathfrak{P}
  74. 𝔓 \mathfrak{P}
  75. 𝔓 \mathfrak{P}
  76. 𝔓 \mathfrak{P}
  77. 𝔓 \mathfrak{P}
  78. 𝔓 \mathfrak{P}
  79. 𝔓 \mathfrak{P}
  80. 𝔓 \mathfrak{P}
  81. 𝔓 \mathfrak{P}
  82. 𝔓 \mathfrak{P}
  83. 𝔓 \mathfrak{P}
  84. 𝔓 \mathfrak{P}
  85. 𝔓 \mathfrak{P}
  86. 𝔓 \mathfrak{P}
  87. 𝔓 \mathfrak{P}
  88. 𝔓 \mathfrak{P}
  89. 𝔓 \mathfrak{P}
  90. 𝔓 \mathfrak{P}
  91. 𝔓 \mathfrak{P}
  92. 𝔓 \mathfrak{P}
  93. 𝔓 \mathfrak{P}
  94. 𝔓 \mathfrak{P}
  95. 𝔓 \mathfrak{P}
  96. 𝔓 \mathfrak{P}
  97. 𝔓 \mathfrak{P}
  98. 𝔓 46 \mathfrak{P}^{46}
  99. 𝔓 \mathfrak{P}
  100. 𝔓 \mathfrak{P}
  101. 𝔓 \mathfrak{P}
  102. 𝔓 \mathfrak{P}
  103. 𝔓 \mathfrak{P}
  104. 𝔓 \mathfrak{P}
  105. 𝔓 \mathfrak{P}
  106. 𝔓 \mathfrak{P}
  107. 𝔓 \mathfrak{P}
  108. 𝔓 \mathfrak{P}
  109. 𝔓 \mathfrak{P}
  110. 𝔓 \mathfrak{P}
  111. 𝔓 \mathfrak{P}
  112. 𝔓 72 \mathfrak{P}^{72}
  113. 𝔓 \mathfrak{P}
  114. 𝔓 \mathfrak{P}
  115. 𝔓 \mathfrak{P}
  116. 𝔓 \mathfrak{P}
  117. 𝔓 \mathfrak{P}
  118. 𝔓 \mathfrak{P}
  119. 𝔓 \mathfrak{P}

Themistocles_M._Rassias.html

  1. ( ψ , γ ) (\psi,\gamma)

Theorem_of_Bertini.html

  1. 𝐏 n \mathbf{P}^{n}
  2. | H | |H|
  3. 𝐏 n \mathbf{P}^{n}
  4. ( 𝐏 n ) (\mathbf{P}^{n})^{\star}
  5. 𝐏 n \mathbf{P}^{n}
  6. 𝐏 n \mathbf{P}^{n}
  7. | H | |H|
  8. ( 𝐏 n ) (\mathbf{P}^{n})^{\star}
  9. X × | H | X\times|H|
  10. x X x\in X
  11. X 𝐏 n X\subset\mathbf{P}^{n}
  12. n n
  13. | H | |H|
  14. \mathbb{C}
  15. Y σ × X Z Y^{\sigma}\times_{X}Z
  16. Y σ × X Z Y^{\sigma}\times_{X}Z
  17. σ H \sigma\in H
  18. X = n X=\mathbb{P}^{n}

Theorem_of_three_moments.html

  1. l l^{\prime}
  2. w w^{\prime}
  3. M A , M B , M C M_{A},\,M_{B},\,M_{C}
  4. M A l + 2 M B ( l + l ) + M C l = 1 4 w l 3 + 1 4 w ( l ) 3 . M_{A}l+2M_{B}(l+l^{\prime})+M_{C}l^{\prime}=\frac{1}{4}wl^{3}+\frac{1}{4}w^{% \prime}(l^{\prime})^{3}.
  5. M A l + 2 M B ( l + l ) + M C l = 6 a 1 x 1 l + 6 a 2 x 2 l M_{A}l+2M_{B}(l+l^{\prime})+M_{C}l^{\prime}=\frac{6a_{1}x_{1}}{l}+\frac{6a_{2}% x_{2}}{l^{\prime}}
  6. P R R B = S Q B S , \dfrac{PR}{RB^{\prime}}=\dfrac{SQ}{B^{\prime}S},
  7. P R L 1 = S Q L 2 \dfrac{PR}{L1}=\dfrac{SQ}{L2}
  8. Δ B - Δ A + P A L 1 = Δ C - Δ B - Q C L 2 \dfrac{\Delta B-\Delta A+PA^{\prime}}{L1}=\dfrac{\Delta C-\Delta B-QC^{\prime}% }{L2}
  9. P A L 1 + Q C L 2 = Δ A - Δ B L 1 + Δ C - Δ B L 2 \dfrac{PA^{\prime}}{L1}+\dfrac{QC^{\prime}}{L2}=\dfrac{\Delta A-\Delta B}{L1}+% \dfrac{\Delta C-\Delta B}{L2}
  10. P A = ( 1 2 × M 1 E 1 I 1 × L 1 ) × L 1 × 1 3 + ( 1 2 × M 2 E 2 I 2 × L 1 ) × L 1 × 2 3 + A 1 X 1 E 1 I 1 PA^{\prime}=\left(\frac{1}{2}\times\frac{M_{1}}{E_{1}I_{1}}\times L_{1}\right)% \times L_{1}\times\frac{1}{3}+\left(\frac{1}{2}\times\frac{M_{2}}{E_{2}I_{2}}% \times L_{1}\right)\times L_{1}\times\frac{2}{3}+\frac{A_{1}X_{1}}{E_{1}I_{1}}
  11. Q C = ( 1 2 × M 3 E 2 I 2 × L 2 ) × L 2 × 1 3 + ( 1 2 × M 2 E 2 I 2 × L 2 ) × L 2 × 2 3 + A 2 X 2 E 2 I 2 QC^{\prime}=\left(\frac{1}{2}\times\frac{M_{3}}{E_{2}I_{2}}\times L_{2}\right)% \times L_{2}\times\frac{1}{3}+\left(\frac{1}{2}\times\frac{M_{2}}{E_{2}I_{2}}% \times L_{2}\right)\times L_{2}\times\frac{2}{3}+\frac{A_{2}X_{2}}{E_{2}I_{2}}
  12. M 1 L 1 E 1 I 1 + 2 M 2 ( L 1 E 1 I 1 + L 2 E 2 I 2 ) + M 3 L 2 E 2 I 2 = 6 [ Δ A - Δ B L 1 + Δ C - Δ B L 2 ] - 6 [ A 1 X 1 E 1 I 1 L 1 + A 2 X 2 E 2 I 2 L 2 ] \frac{M_{1}L_{1}}{E_{1}I_{1}}+2M_{2}\left(\frac{L_{1}}{E_{1}I_{1}}+\frac{L_{2}% }{E_{2}I_{2}}\right)+\frac{M_{3}L_{2}}{E_{2}I_{2}}=6[\frac{\Delta A-\Delta B}{% L_{1}}+\frac{\Delta C-\Delta B}{L_{2}}]-6[\frac{A_{1}X_{1}}{E_{1}I_{1}L_{1}}+% \frac{A_{2}X_{2}}{E_{2}I_{2}L_{2}}]

Theory_of_sonics.html

  1. I e f f 2 = 1 T 0 T i 2 d t I_{eff}^{2}=\frac{1}{T}\int\limits_{0}^{T}i^{2}\,dt
  2. v e f f = I e f f ω v_{eff}=\frac{I_{eff}}{\omega}
  3. δ = 2 r Ω = 2 I a \delta=2r\Omega=2\frac{I}{a}
  4. p = H sin ( a t + Φ ) + p m p=H\sin{(at+\Phi)}+p_{m}
  5. Φ = \Phi=
  6. p m p_{m}
  7. P m i n = P m - H P_{min}=P_{m}-H
  8. P m a x = P m + H P_{max}=P_{m}+H
  9. h = p 1 - p 2 = H sin ( a t + Φ ) h=p_{1}-p_{2}=H\sin{(at+\Phi)}
  10. H e f f = H 2 H_{eff}=\frac{H}{\sqrt{2}}
  11. H = R i H=Ri
  12. k g . s e c . c m . 5 \frac{kg.sec.}{cm.^{5}}
  13. R = ϵ γ l v e f f 2 g ω d R=\epsilon\frac{\gamma lv_{eff}}{2g\omega d}
  14. γ \gamma
  15. ω \omega
  16. ϵ = 0.02 + 0.18 v e f f d \epsilon=0.02+\frac{0.18}{\sqrt{v_{eff}d}}
  17. ϵ \epsilon
  18. R = γ l g ω ( 0.01 v d + 0.09 d v e f f d ) R=\frac{\gamma l}{g\omega}\big(0.01\frac{v}{d}+\frac{0.09}{d}\sqrt{\frac{v_{% eff}}{d}}\big)
  19. 100 k = v e f f d + 9 d v e f f d = v e f f d ( 1 + 9 v e f f v e f f d ) 100k=\frac{v_{eff}}{d}+\frac{9}{d}\sqrt{\frac{v_{eff}}{d}}=\frac{v_{eff}}{d}% \big(1+\frac{9}{v_{eff}}\sqrt{\frac{v_{eff}}{d}}\big)
  20. R = k γ l g ω R=k\frac{\gamma l}{g\omega}
  21. W = 1 T 0 T h i d t W=\frac{1}{T}\int_{0}^{T}hi\,dt
  22. W = 1 T 0 T R i 2 d t = R T 0 T i 2 d t = R I 2 2 W=\frac{1}{T}\int_{0}^{T}Ri^{2}\,dt=\frac{R}{T}\int_{0}^{T}i^{2}\,dt=\frac{RI^% {2}}{2}
  23. W = R I 2 2 = H I 2 = H e f f × I e f f W=\frac{RI^{2}}{2}=\frac{HI}{2}=H_{eff}\times I_{eff}
  24. F = f A = f ω C F=\frac{f}{A}=\frac{f\omega}{C}
  25. B = F f B=Ff
  26. F = 0.4 d 3 D σ F=0.4\frac{d^{3}}{D}\sigma
  27. d = F D 0.4 σ 3 d=\sqrt[3]{\frac{FD}{0.4\sigma}}
  28. n = 1 0.4 σ 3 n=\sqrt[3]{\frac{1}{0.4\sigma}}
  29. d = n F D 3 d=n\sqrt[3]{FD}

Thermal_resistance.html

  1. ...
  2. Q Q
  3. q q
  4. X X
  5. P P
  6. T T
  7. k B k_{B}
  8. V V
  9. F F
  10. Q Q
  11. Q ˙ \dot{Q}
  12. I I
  13. σ \sigma
  14. q \overrightarrow{q}
  15. j j
  16. 1 / k 1/k
  17. R R
  18. R R
  19. R R
  20. k k
  21. 1 / R 1/R
  22. 1 / R 1/R
  23. Δ X = F / k \Delta X=F/k
  24. Δ P = Q R \Delta P=QR
  25. Δ T = Q ˙ R \Delta T=\dot{Q}R
  26. Δ V = I R \Delta V=IR
  27. ...
  28. q = - k T \overrightarrow{q}=-k{\nabla}T
  29. 𝐉 = σ 𝐄 \mathbf{J}=\sigma\mathbf{E}
  30. R θ J C R_{\theta JC}
  31. T J M A X T_{JMAX}
  32. Δ T H S \Delta T_{HS}
  33. Δ T H S \Delta T_{HS}
  34. R θ J C + R θ B R_{\theta JC}+R_{\theta B}
  35. R θ B R_{\theta B}
  36. T J M A X - ( T A M B + Δ T H S ) T_{JMAX}-(T_{AMB}+\Delta T_{HS})
  37. Δ T \Delta T
  38. R θ R_{\theta}
  39. Q Q
  40. Δ T = Q × R θ \Delta T=Q\times R_{\theta}\,
  41. T J M A X - ( T A M B + Δ T H S ) = Q M A X × ( R θ J C + R θ B + R θ H A ) T_{JMAX}-(T_{AMB}+\Delta T_{HS})=Q_{MAX}\times(R_{\theta JC}+R_{\theta B}+R_{% \theta HA})\,
  42. Q M A X = T J M A X - ( T A M B + Δ T H S ) R θ J C + R θ B + R θ H A Q_{MAX}={{T_{JMAX}-(T_{AMB}+\Delta T_{HS})}\over{R_{\theta JC}+R_{\theta B}+R_% {\theta HA}}}
  43. Q M A X Q_{MAX}
  44. T J M A X = 125 C T_{JMAX}=125\ ^{\circ}\mbox{C}~{}
  45. T A M B = 21 C T_{AMB}=21\ ^{\circ}\mbox{C}~{}
  46. R θ J C = 1.5 C / W R_{\theta JC}=1.5\ ^{\circ}\mathrm{C}/\mathrm{W}\,
  47. R θ B = 0.1 C / W R_{\theta B}=0.1\ ^{\circ}\mathrm{C}/\mathrm{W}\,
  48. R θ H A = 4 C / W R_{\theta HA}=4\ ^{\circ}\mathrm{C}/\mathrm{W}\,
  49. Q = 125 - ( 21 ) 1.5 + 0.1 + 4 = 18.6 W Q={{125-(21)}\over{1.5+0.1+4}}=18.6\ \mathrm{W}
  50. R θ = x A × k R_{\theta}=\frac{x}{A\times k}
  51. R θ R_{\theta}

Thermal_simulations_for_integrated_circuits.html

  1. q = - κ T q=-\kappa\nabla T
  2. κ \kappa
  3. Q = j 2 ρ Q=j^{2}\rho
  4. j j
  5. ρ \rho
  6. Ω {\Omega}
  7. Q Q
  8. ( κ ( T ) T ) + g = ρ C T t \nabla\left(\kappa\left(T\right)\nabla T\right)+g=\rho C\frac{\partial T}{% \partial t}
  9. κ \kappa
  10. ρ \rho
  11. C C
  12. k = κ ρ C k=\frac{\kappa}{\rho C}
  13. g g
  14. κ \kappa
  15. θ = T s + 1 κ s T s T κ ( T ) d T \theta=T_{s}+\frac{1}{\kappa_{s}}\int_{T_{s}}^{T}\kappa(T)dT
  16. κ s = κ ( T s ) \kappa_{s}=\kappa\left(T_{s}\right)
  17. T s T_{s}
  18. k 2 θ + k κ s g = θ t k\nabla^{2}\theta+\frac{k}{\kappa_{s}}g=\frac{\partial\theta}{\partial t}
  19. k = κ ρ C k=\frac{\kappa}{\rho C}
  20. k s τ = 0 t k ( θ ) d t k_{s}\tau=\int_{0}^{t}k(\theta)dt
  21. 2 θ - 1 k s θ τ = - g κ s \nabla^{2}\theta-\frac{1}{k_{s}}\frac{\partial\theta}{\partial\tau}=-\frac{g}{% \kappa_{s}}
  22. 2 Θ ¯ - s k s Θ ¯ = 0 \nabla^{2}\bar{\Theta}-\frac{s}{k_{s}}\bar{\Theta}=0
  23. Θ ¯ = s θ - θ ( τ = 0 ) \overline{\Theta}=s\theta-\theta\left(\tau=0\right)
  24. Θ ¯ \overline{\Theta}
  25. x x
  26. y y
  27. z z
  28. Δ θ i = j = 1 N R T H i j ( t ) P j ( t ) \Delta\theta_{i}=\sum_{j=1}^{N}R_{TH_{ij}}(t)P_{j}(t)
  29. j j
  30. i i
  31. N i e = 1 2 ξ ( 1 ζ ) , i = 1 , 4 N_{i}^{e}=\frac{1}{2}\xi\left(1\mp\zeta\right),\qquad i=1,4
  32. N i e = 1 2 η ( 1 ζ ) , i = 2 , 5 N_{i}^{e}=\frac{1}{2}\eta\left(1\mp\zeta\right),\qquad i=2,5
  33. N i e = 1 2 ( 1 - ξ - η ) ( 1 ζ ) , i = 3 , 6 N_{i}^{e}=\frac{1}{2}\left(1-\xi-\eta\right)\left(1\mp\zeta\right),\qquad i=3,6
  34. ζ = 2 ( z - z c ) / h z \zeta=2(z-z_{c})/h_{z}
  35. z c = 0 z_{c}=0
  36. ζ = 2 z / h z \zeta=2z/h_{z}
  37. [ S ] { θ } + [ R ] d d t { θ } = { B } \left[S\right]\left\{\theta\right\}+\left[R\right]\frac{d}{dt}\left\{\theta% \right\}=\left\{B\right\}
  38. R i j = v N j N i d V R_{ij}=\int_{v}N_{j}N_{i}dV
  39. S i j = k v N j . N i d V S_{ij}=k\int_{v}\nabla N_{j}.\nabla N_{i}dV
  40. B i = k κ s Ω 1 N i p ( x , y ) d Ω + k κ s v N i g d V - k T o j = 0 N D v N j D . N i d V B_{i}=\frac{k}{\kappa_{s}}\int_{\Omega_{1}}N_{i}p(x,y)d\Omega+\frac{k}{\kappa_% {s}}\int_{v}N_{i}gdV-kT_{o}\sum_{j=0}^{N_{D}}\int_{v}\nabla N_{j}^{D}.\nabla N% _{i}dV
  41. 1 m s 1ms
  42. C { x } + K { x } = F { u } C\left\{x\right\}^{\prime}+K\left\{x\right\}=F\left\{u\right\}
  43. V T C V { z } + V T K V { z } = V T F { u } V^{T}CV\left\{z\right\}^{\prime}+V^{T}KV\left\{z\right\}=V^{T}F\left\{u\right\}
  44. 0.3 W 0.3W
  45. 3.333 e 8 W / m 2 3.333e8W/m_{2}

Thermodynamic_square.html

  1. U U
  2. G G
  3. H H
  4. U U
  5. F F
  6. U U
  7. d U = - p [ Differential ] + T [ Differential ] dU=-p[\,\text{Differential}]+T[\,\text{Differential}]
  8. p p
  9. V V
  10. T T
  11. S S
  12. d U = - p d V + T d S dU=-pdV+TdS
  13. μ d N \mu dN
  14. μ \mu
  15. d U = - p d V + T d S + μ d N dU=-pdV+TdS+\mu dN
  16. \sqcup
  17. ( S p ) T = - ( V T ) p \left({\partial S\over\partial p}\right)_{T}=-\left({\partial V\over\partial T% }\right)_{p}
  18. \sqcup
  19. \sqsupset
  20. ( p T ) V = ( S V ) T \left({\partial p\over\partial T}\right)_{V}=\left({\partial S\over\partial V}% \right)_{T}

Thermodynamics_of_nanostructures.html

  1. L L
  2. L L
  3. L L
  4. L L
  5. k = 1 3 C v g Λ = 1 3 C v g 2 τ k=\frac{1}{3}Cv_{g}\Lambda=\frac{1}{3}Cv_{g}^{2}\tau
  6. τ \tau
  7. ( C v g ) i = k B 4 T 3 2 3 π 2 1 v p , i 2 [ x 4 e x p ( x ) ( e x p ( x ) - 1 ) 2 ] d x (Cv_{g})_{i}=\frac{k_{B}^{4}T^{3}}{2\hbar^{3}\pi^{2}}\int\frac{1}{v_{p,i}^{2}}% \left[\frac{x^{4}exp(x)}{(exp(x)-1)^{2}}\right]\,dx
  8. x = h ω / k B T x=h\omega/k_{B}T
  9. v p , i v_{p,i}
  10. G ( T ) α λ a ( k z ) L ω α ( k z ) 2 π d f b d T v z ( α , k z ) d k z G(T)\simeq\sum_{\alpha}{\int{\frac{\lambda_{a}(k_{z})}{L}\frac{\hbar\omega_{% \alpha}(k_{z})}{2\pi}\frac{df_{b}}{dT}v_{z}(\alpha,k_{z})dk_{z}}}
  11. k ( T ) = 1 S α 0 π a z λ α ( k z ) ω α ( k z ) 2 π d f B d T v z ( α , k z ) d k z k(T)=\frac{1}{S}\sum{\alpha}{\int_{0}^{\frac{\pi}{a_{z}}}\lambda_{\alpha}(k_{z% })\frac{\hbar\omega_{\alpha}(k_{z})}{2\pi}\frac{df_{B}}{dT}v_{z}(\alpha,k_{z})% \,dk_{z}}
  12. l - 1 = B ( h d ) 2 1 d ( ω ω D ) 2 N ( ω ) l^{-1}=B\left(\frac{h}{d}\right)^{2}\frac{1}{d}\left(\frac{\omega}{\omega_{D}}% \right)^{2}N(\omega)
  13. G ( T ) = 1 2 π 0 ( N 1 ( ω ) 1 + L / l ( ω ) + N 2 ( ω ) 1 + L / d ) 3 ω 2 k B T 2 × e ω k B T ( e ω k B T - 1 ) 2 G(T)=\frac{1}{2\pi\hbar}\int_{0}^{\infty}{\left(\frac{N_{1}(\omega)}{1+L/l(% \omega)}+\frac{N_{2}(\omega)}{1+L/d}\right)\frac{\hbar^{3}\omega^{2}}{k_{B}T^{% 2}}\times\frac{e^{\frac{\hbar\omega}{k_{B}T}}}{{(e^{\frac{\hbar\omega}{k_{B}T}% }-1)}^{2}}}
  14. k z z = C v z 2 τ k_{zz}=\sum C{v_{z}}^{2}\tau
  15. τ \tau
  16. G 0 = 2 e 2 h G_{0}=\frac{2e^{2}}{h}
  17. G t h = π 2 k B 2 T 3 h G_{th}=\frac{\pi^{2}{k_{B}}^{2}T}{3h}
  18. k n = ( C 1 v 1 C 2 v 2 C 1 v 1 + C 2 v 2 ) ( d 1 + d 2 2 ) k_{n}=\left(\frac{C_{1}v_{1}C_{2}v_{2}}{C_{1}v_{1}+C_{2}v_{2}}\right)\left(% \frac{d_{1}+d_{2}}{2}\right)
  19. τ G - 1 = B v d G \tau_{G}^{-1}=B\frac{v}{d_{G}}
  20. τ G - 1 = 2 ν π d G [ 1 - e x p ( - π 2 4 ν G ) ] {\tau_{G}}^{-1}=\frac{2\nu}{\pi d_{G}}\left[1-exp\left(-\frac{\pi^{2}}{4}\nu_{% G}\right)\right]
  21. ν G = j σ j ν j \nu_{G}=\sum_{j}\sigma_{j}\nu_{j}
  22. σ j \sigma_{j}

Theta_function_(disambiguation).html

  1. ϑ ( z ; τ ) \vartheta(z;\tau)
  2. θ ( z ; q ) \theta(z;q)
  3. Θ Λ ( τ ) \Theta_{\Lambda}(\tau)
  4. μ ( q ) , f ( q ) , ϕ ( q ) , ψ ( q ) , χ ( q ) \mu(q),f(q),\phi(q),\psi(q),\chi(q)
  5. f ( a , b ) f(a,b)
  6. θ ( t ) \theta(t)
  7. ϑ ( x ) \vartheta(x)
  8. θ α ( β ) \theta_{\alpha}(\beta)
  9. θ ( x ) \theta(x)
  10. ϑ i j ( z ; τ ) , ϑ j ( z ) , θ i ( z ; q ) \vartheta_{ij}(z;\tau),\vartheta_{j}(z),\theta_{i}(z;q)

Theta_function_of_a_lattice.html

  1. Θ Λ ( τ ) = x Λ e i π τ x 2 Im τ > 0. \Theta_{\Lambda}(\tau)=\sum_{x\in\Lambda}e^{i\pi\tau\|x\|^{2}}\qquad\mathrm{Im% }\,\tau>0.
  2. q = e 2 i π τ q=e^{2i\pi\tau}

Thin-film_interference.html

  1. π \pi
  2. n 1 < n 2 n_{1}<n_{2}
  3. O P D = n 2 ( A B ¯ + B C ¯ ) - n 1 ( A D ¯ ) OPD=n_{2}(\overline{AB}+\overline{BC})-n_{1}(\overline{AD})
  4. A B ¯ = B C ¯ = d cos ( θ 2 ) \overline{AB}=\overline{BC}=\frac{d}{\cos(\theta_{2})}
  5. A D ¯ = 2 d tan ( θ 2 ) sin ( θ 1 ) \overline{AD}=2d\tan(\theta_{2})\sin(\theta_{1})
  6. n 1 s i n ( θ 1 ) = n 2 s i n ( θ 2 ) n_{1}sin(\theta_{1})=n_{2}sin(\theta_{2})
  7. O P D = n 2 ( 2 d cos ( θ 2 ) ) - 2 d tan ( θ 2 ) n 2 sin ( θ 2 ) OPD=n_{2}\left(\frac{2d}{\cos(\theta_{2})}\right)-2d\tan(\theta_{2})n_{2}\sin(% \theta_{2})
  8. O P D = 2 n 2 d ( 1 - sin 2 ( θ 2 ) cos ( θ 2 ) ) OPD=2n_{2}d\left(\frac{1-\sin^{2}(\theta_{2})}{\cos(\theta_{2})}\right)
  9. O P D = 2 n 2 d cos ( θ 2 ) OPD=2n_{2}d\cos\big(\theta_{2})
  10. λ \lambda
  11. 2 n 2 d cos ( θ 2 ) = m λ 2n_{2}d\cos\big(\theta_{2})=m\lambda
  12. n air = 1 n_{\rm air}=1
  13. n film > 1 n_{\rm film}>1
  14. n air < n film n_{\rm air}<n_{\rm film}
  15. n film > n air n_{\rm film}>n_{\rm air}
  16. 2 n film d cos ( θ 2 ) = ( m - 1 2 ) λ 2n_{\rm film}d\cos(\theta_{2})=\left(m-\frac{1}{2}\right)\lambda
  17. 2 n film d cos ( θ 2 ) = m λ 2n_{\rm film}d\cos\big(\theta_{2})=m\lambda
  18. d d
  19. n film n_{\rm film}
  20. θ 2 \theta_{2}
  21. m m
  22. λ \lambda
  23. n air < n water < n oil n_{\rm air}<n_{\rm water}<n_{\rm oil}
  24. n air < n oil n_{\rm air}<n_{\rm oil}
  25. n oil > n water n_{\rm oil}>n_{\rm water}
  26. 2 n oil d cos ( θ 2 ) = ( m - 1 2 ) λ 2n_{\rm oil}d\cos(\theta_{2})=\left(m-\frac{1}{2}\right)\lambda
  27. 2 n oil d cos ( θ 2 ) = m λ 2n_{\rm oil}d\cos\big(\theta_{2})=m\lambda
  28. d n c o a t i n g dn_{coating}
  29. n air < n coating < n glass n_{\rm air}<n_{\rm coating}<n_{\rm glass}
  30. d = λ / ( 4 n coating ) d=\lambda/(4n_{\rm coating})
  31. n air < n coating n_{\rm air}<n_{\rm coating}
  32. n coating < n glass n_{\rm coating}<n_{\rm glass}
  33. 2 n coating d cos ( θ 2 ) = m λ 2n_{\rm coating}d\cos\big(\theta_{2})=m\lambda
  34. 2 n coating d cos ( θ 2 ) = ( m - 1 2 ) λ 2n_{\rm coating}d\cos(\theta_{2})=\left(m-\frac{1}{2}\right)\lambda
  35. d n coating dn_{\rm coating}
  36. ( θ 2 = 0 ) (\theta_{2}=0)
  37. ρ \rho

Threaded_rod.html

  1. P = 0.33 F u π d 2 4 P=\frac{0.33F_{u}\pi d^{2}}{4}
  2. P = 0.6 F y π d s 2 4 P=\frac{0.6F_{y}\pi d_{s}^{2}}{4}

Three-detector_problem_and_Newell's_method.html

  1. v f v_{f}
  2. k j k_{j}
  3. v f v_{f}
  4. τ 1 = L U / v f \tau_{1}=L_{U}/v_{f}
  5. τ 1 \tau_{1}
  6. τ 1 \tau_{1}
  7. τ 1 \tau_{1}
  8. P 2 P_{2}
  9. - w < 0 -w<0
  10. k j ( - w ) k_{j}(-w)
  11. τ 2 = L D / ( - w ) \tau_{2}=L_{D}/(-w)
  12. δ = k j ( - w ) τ 2 = k j L D \delta=k_{j}(-w)\tau_{2}=k_{j}L_{D}
  13. τ 2 \tau_{2}
  14. δ \delta
  15. x u x_{u}
  16. x m x_{m}
  17. x d x_{d}
  18. N ( t , x m ) = min { N ( t - L U / v f , x u ) , N ( t + L D / w , x d ) + k j L D } ( 1 ) N(t,x_{m})=\min\{\ N(t-L_{U}/v_{f},x_{u})\ ,\ N(t+L_{D}/w,x_{d})+k_{j}L_{D}\ % \}\qquad(1)
  19. L U = x m - x u L_{U}=x_{m}-x_{u}
  20. L D = x d - x m L_{D}=x_{d}-x_{m}
  21. N p N_{p}
  22. N L N_{L}
  23. C A B C_{AB}
  24. C A B = r ( v 0 ) Δ t C_{AB}=r(v^{0})\Delta{t}
  25. v A B v_{AB}
  26. C A B C_{AB}
  27. C A B = r ( v A B ) Δ t = q 0 Δ t - k 0 Δ x = q 0 ( t B - t A ) - k 0 ( x B - x A ) ; f o r v A B [ - w , v f ] ( 2 ) C_{AB}=r(v_{AB})\Delta{t}=q_{0}\Delta{t}-k_{0}\Delta{x}=q_{0}(t_{B}-t_{A})-k_{% 0}(x_{B}-x_{A});for\ v_{AB}\in[-w,v_{f}]\qquad(2)
  28. C A B C_{AB}
  29. N p N_{p}
  30. [ - w , v f ] [-w,v_{f}]
  31. N P N L + L ( C A B ) , v A B [ - w , v f ] ( 3 ) N_{P}\leq N_{L}+\sum_{L}(C_{AB}),\ v_{AB}\in[-w,v_{f}]\qquad(3)
  32. N P N_{P}
  33. N P = min L { N L + L ( C A B ) } ( 4 ) N_{P}=\min_{L}\{N_{L}+\sum_{L}(C_{AB})\}\qquad(4)
  34. C A B = r ( v A B ) C_{AB}=r(v_{AB})
  35. N U N_{U}
  36. N D N_{D}
  37. N P = min { N U , N D } N_{P}=\min\{N_{U}\ ,\ N_{D}\}
  38. C Q P = q 0 Δ t - k 0 Δ x = q 0 ( t P - t Q ) - k 0 ( x P - x Q ) ( 5 ) C_{QP}=q_{0}\Delta{t}-k_{0}\Delta{x}=q_{0}(t_{P}-t_{Q})-k_{0}(x_{P}-x_{Q})% \qquad(5)
  39. Δ t = t P - t Q = ( x M - x U ) v Q P \Delta{t}=t_{P}-t_{Q}=\frac{(x_{M}-x_{U})}{v_{QP}}
  40. v Q P [ 0 , v f ] v_{QP}\in[0,v_{f}]
  41. Δ x = x M - x U \Delta{x}=x_{M}-x_{U}
  42. N Q + q 0 ( t P - t Q ) - k 0 ( x M - x U ) {N_{Q}+q_{0}(t_{P}-t_{Q})-k_{0}(x_{M}-x_{U})}
  43. N U = min t Q { N Q + q 0 ( t P - t Q ) - k 0 ( x M - x U ) } ( 6 ) N_{U}=\min_{t_{Q}}\{N_{Q}+q_{0}(t_{P}-t_{Q})-k_{0}(x_{M}-x_{U})\}\qquad(6)
  44. d N Q / d t q 0 dN_{Q}/dt\leq q_{0}
  45. t Q * = t P 1 t_{Q}^{*}=t_{P_{1}}
  46. P 1 P_{1}
  47. C Q P = C P 1 P = q 0 ( x M - x U v f ) - k 0 ( x M - x U ) = 0 ( 7 ) C_{QP}=C_{P_{1}P}=q_{0}\left(\frac{x_{M}-x_{U}}{v_{f}}\right)-k_{0}(x_{M}-x_{U% })=0\qquad(7)
  48. N U = N P 1 N_{U}=N_{P_{1}}
  49. N D = min Q { N Q + C Q P } = min Q { N Q + q 0 Δ t - k 0 Δ x } N_{D}=\min_{Q^{^{\prime}}}\{N_{Q^{^{\prime}}}+C_{Q^{^{\prime}}P}\}=\min_{Q^{^{% \prime}}}\{N_{Q^{^{\prime}}}+q_{0}\Delta{t}-k_{0}\Delta{x}\}
  50. N D N_{D}
  51. Q = P 2 Q^{^{\prime}}=P_{2}
  52. P 2 P_{2}
  53. N D = N P 2 + q 0 ( x D - x M w ) - k 0 ( x D - x M ) ( 8 ) N_{D}=N_{P_{2}}+q_{0}(\frac{x_{D}-x_{M}}{w})-k_{0}(x_{D}-x_{M})\qquad(8)
  54. q 0 w + k 0 = k j \frac{q_{0}}{w}+k_{0}=k_{j}
  55. N D = N P 2 + ( x D - x M ) k j ( 9 ) N_{D}=N_{P_{2}}+(x_{D}-x_{M})k_{j}\qquad(9)
  56. N U N_{U}
  57. N D N_{D}
  58. N P = min { N U , N D } = min { N P 1 , N P 2 + ( x D - x M ) k j } ( 10 ) N_{P}=\min\{N_{U}\ ,\ N_{D}\}=\min\{N_{P_{1}}\ ,\ N_{P_{2}}+(x_{D}-x_{M})k_{j}% \}\qquad(10)

Three-twist_knot.html

  1. Δ ( t ) = 2 t - 3 + 2 t - 1 , \Delta(t)=2t-3+2t^{-1},\,
  2. ( z ) = 2 z 2 + 1 , \nabla(z)=2z^{2}+1,\,
  3. V ( q ) = q - 1 - q - 2 + 2 q - 3 - q - 4 + q - 5 - q - 6 . V(q)=q^{-1}-q^{-2}+2q^{-3}-q^{-4}+q^{-5}-q^{-6}.\,

Three_subgroups_lemma.html

  1. x y x^{y}
  2. [ X , Y , Z ] = 1 [X,Y,Z]=1
  3. [ Y , Z , X ] = 1 [Y,Z,X]=1
  4. [ Z , X , Y ] = 1 [Z,X,Y]=1
  5. N G N\triangleleft G
  6. [ X , Y , Z ] N [X,Y,Z]\subseteq N
  7. [ Y , Z , X ] N [Y,Z,X]\subseteq N
  8. [ Z , X , Y ] N [Z,X,Y]\subseteq N
  9. x , y , z G x,y,z\in G
  10. [ x , y - 1 , z ] y [ y , z - 1 , x ] z [ z , x - 1 , y ] x = 1 [x,y^{-1},z]^{y}\cdot[y,z^{-1},x]^{z}\cdot[z,x^{-1},y]^{x}=1
  11. x X x\in X
  12. y Y y\in Y
  13. z Z z\in Z
  14. [ x , y - 1 , z ] = 1 = [ y , z - 1 , x ] [x,y^{-1},z]=1=[y,z^{-1},x]
  15. [ z , x - 1 , y ] x = 1 [z,x^{-1},y]^{x}=1
  16. [ z , x - 1 , y ] = 1 [z,x^{-1},y]=1
  17. [ z , x - 1 ] C G ( Y ) [z,x^{-1}]\subseteq{C}_{G}(Y)
  18. z Z z\in Z
  19. x X x\in X
  20. [ Z , X ] [Z,X]
  21. [ Z , X ] C G ( Y ) [Z,X]\subseteq{C}_{G}(Y)
  22. [ Z , X , Y ] = 1 [Z,X,Y]=1

Thyristor_controlled_reactor.html

  1. I t c r - m a x = V s v c 2 π f L t c r I_{tcr-max}={V_{svc}\over{2\pi fL_{tcr}}}
  2. ω t < π - α : I ( ω t ) = I t c r - m a x 2 [ - c o s ( α ) - c o s ( ω t ) ] \omega t<{\pi-\alpha}:I(\omega t)=I_{tcr-max}\sqrt{2}[-cos(\alpha)-cos(\omega t)]
  3. α < ω t < 2 π - α : I ( ω t ) = I t c r - m a x 2 [ c o s ( α ) - c o s ( ω t ) ] \alpha<\omega t<2\pi-\alpha:I(\omega t)=I_{tcr-max}\sqrt{2}[cos(\alpha)-cos(% \omega t)]
  4. ω t > π + α : I ( ω t ) = I t c r - m a x 2 [ - c o s ( α ) - c o s ( ω t ) ] {\omega t>{\pi+\alpha}}:I(\omega t)=I_{tcr-max}\sqrt{2}[-cos(\alpha)-cos(% \omega t)]

Tide-predicting_machine.html

  1. A 1 cos ( ω 1 t + ϕ 1 ) + A 2 cos ( ω 2 t + ϕ 2 ) + A 3 cos ( ω 3 t + ϕ 3 ) + A_{1}\cos(\omega_{1}t+\phi_{1})+A_{2}\cos(\omega_{2}t+\phi_{2})+A_{3}\cos(% \omega_{3}t+\phi_{3})+\ldots
  2. t t
  3. t t
  4. A 1 cos ( ω 1 t + ϕ 1 ) A_{1}\cos(\omega_{1}t+\phi_{1})
  5. A 1 A_{1}
  6. ω 1 \omega_{1}
  7. ϕ 1 \phi_{1}

Time_deviation.html

  1. σ x ( τ ) \sigma_{x}(\tau)
  2. σ x 2 ( τ ) \sigma_{x}^{2}(\tau)
  3. σ x 2 ( τ ) = τ 2 3 mod σ y 2 ( n τ 0 ) \sigma_{x}^{2}(\tau)=\frac{\tau^{2}}{3}\operatorname{mod}\sigma_{y}^{2}(n\tau_% {0})
  4. τ = n τ o \tau=n\tau_{o}
  5. σ x ( τ ) = τ 3 mod σ y ( n τ 0 ) \sigma_{x}(\tau)=\frac{\tau}{\sqrt{3}}\operatorname{mod}\sigma_{y}(n\tau_{0})

Timsort.html

  1. Θ ( n log n ) \Theta(n\log n)
  2. Θ ( n log n ) \Theta(n\log n)
  3. Θ ( n ) \Theta(n)
  4. Θ ( n log n ) \Theta(n\log n)
  5. Θ ( n log n ) \Theta(n\log n)
  6. Θ ( n ) \Theta(n)
  7. Θ ( n 2 ) \Theta(n^{2})
  8. Θ ( n ) \Theta(n)
  9. Θ ( n log n ) \Theta(n\log n)
  10. Θ ( n log n ) \Theta(n\log n)
  11. Θ ( n log n ) \Theta(n\log n)
  12. Θ ( n log n ) \Theta(n\log n)
  13. Θ ( n 2 ) \Theta(n^{2})
  14. Θ ( n 2 ) \Theta(n^{2})
  15. Θ ( n log n ) \Theta(n\log n)
  16. Θ ( n log n ) \Theta(n\log n)
  17. Θ ( n log n ) \Theta(n\log n)
  18. Θ ( n log n ) \Theta(n\log n)
  19. Θ ( n 2 ) \Theta(n^{2})
  20. Θ ( n 2 ) \Theta(n^{2})
  21. Θ ( n 2 ) \Theta(n^{2})
  22. Θ ( n log n ) \Theta(n\log n)
  23. O ( n ) O(n)
  24. O ( 1 ) O(1)
  25. O ( n ) O(n)
  26. O ( n ) O(n)
  27. O ( log n ) O(\log n)
  28. O ( 1 ) O(1)
  29. O ( 1 ) O(1)
  30. O ( 1 ) O(1)
  31. log n \log n

Toads_and_Frogs.html

  1. T T
  2. F F
  3. \square
  4. T F T\square\square F
  5. P P
  6. L 1 L_{1}
  7. L 2 L_{2}
  8. R 1 R_{1}
  9. R 2 R_{2}
  10. P P
  11. P = { L 1 , L 2 , | R 1 , R 2 , } . P=\{L_{1},L_{2},\dots|R_{1},R_{2},\dots\}.
  12. T F = { T F | T F } T\square\square F=\{\square T\square F|T\square F\square\}
  13. T F = 0 T\square\square F=0
  14. T F = 1 TF\square\square=1
  15. T F = 1 2 T\square F\square=\frac{1}{2}
  16. T F T F = { 0 | 0 } = TFT\square F=\{0|0\}=\star
  17. T T F F = { 0 | } = T\square TFF=\{0|\star\}=\uparrow
  18. T a b F T^{a}\square^{b}F
  19. T a b F a T^{a}\square^{b}F^{a}

Ton_class.html

  1. < m t p l > T h a m e s T o n n a g e = ( l e n g t h - b e a m ) × b e a m 2 188 <mtpl>{{ThamesTonnage}}=\frac{({length}-{beam})\times{beam}^{2}}{188}
  2. T = ( L - 0.25 P ) P S 130 T=\frac{(L-0.25P)\cdot P\cdot\sqrt{S}}{130}

Topological_excitations.html

  1. M M
  2. G G
  3. M = R 2 M=R^{2}
  4. G = U ( 1 ) G=U(1)
  5. M = R 3 M=R^{3}
  6. G = S O ( 3 ) / U ( 1 ) G=SO(3)/U(1)

Toronto_function.html

  1. T ( m , n , r ) = r 2 n - m + 1 e - r 2 Γ ( 1 2 m + 1 2 ) Γ ( n + 1 ) F 1 1 ( 1 2 m + 1 2 ; n + 1 ; r 2 ) . T(m,n,r)=r^{2n-m+1}e^{-r^{2}}\frac{\Gamma(\frac{1}{2}m+\frac{1}{2})}{\Gamma(n+% 1)}{}_{1}F_{1}({\textstyle\frac{1}{2}}m+{\textstyle\frac{1}{2}};n+1;r^{2}).

Total_internal_reflection_microscopy.html

  1. θ c \theta_{c}
  2. θ c = sin - 1 ( n 2 / n 1 ) \theta_{c}=\sin^{-1}(n_{2}/n_{1})
  3. n 1 n_{1}
  4. n 2 n_{2}
  5. θ c \theta_{c}
  6. I ( z ) I(z)
  7. I ( z ) = I 0 e - β z I(z)=I_{0}e^{-\beta z}
  8. β = 4 π λ ( n 1 sin ( θ ) ) 2 - n 2 2 \beta=\frac{4\pi}{\lambda}\sqrt{(n_{1}\sin(\theta))^{2}-n_{2}^{2}}
  9. z z
  10. p ( z ) = 1 Z e - V ( z ) k T p(z)=\frac{1}{Z}e^{-\frac{V(z)}{kT}}
  11. Z Z
  12. k k

Total_variation_denoising.html

  1. y n y_{n}
  2. V ( y ) = n | y n + 1 - y n | V(y)=\sum\limits_{n}\left|y_{n+1}-y_{n}\right|
  3. x n x_{n}
  4. y n y_{n}
  5. x n x_{n}
  6. x n x_{n}
  7. E ( x , y ) = 1 2 n ( x n - y n ) 2 E(x,y)=\frac{1}{2}\sum\limits_{n}\left(x_{n}-y_{n}\right)^{2}
  8. y n y_{n}
  9. E ( x , y ) + λ V ( y ) E(x,y)+\lambda V(y)
  10. y n y_{n}
  11. x n x_{n}
  12. y n y_{n}
  13. λ \lambda
  14. λ = 0 \lambda=0
  15. λ \lambda\to\infty
  16. V ( y ) = i , j | y i + 1 , j - y i , j | 2 + | y i , j + 1 - y i , j | 2 V(y)=\sum_{i,j}\sqrt{|y_{i+1,j}-y_{i,j}|^{2}+|y_{i,j+1}-y_{i,j}|^{2}}
  17. V aniso ( y ) = i , j | y i + 1 , j - y i , j | 2 + | y i , j + 1 - y i , j | 2 = i , j | y i + 1 , j - y i , j | + | y i , j + 1 - y i , j | . V\text{aniso}(y)=\sum_{i,j}\sqrt{|y_{i+1,j}-y_{i,j}|^{2}}+\sqrt{|y_{i,j+1}-y_{% i,j}|^{2}}=\sum_{i,j}|y_{i+1,j}-y_{i,j}|+|y_{i,j+1}-y_{i,j}|.
  18. min y E ( x , y ) + λ V ( y ) \min_{y}\;E(x,y)+\lambda V(y)

Track_significance.html

  1. S = a o / E r r o r ( a o ) S=a_{o}/Error(a_{o})

Tracy–Widom_distribution.html

  1. F 2 F_{2}
  2. F 1 F_{1}
  3. F 2 ( s ) = lim n Prob ( ( λ max - 2 n ) ( 2 ) n 1 / 6 s ) F_{2}(s)=\lim\limits_{n\rightarrow\infty}{\rm Prob}\left((\lambda_{\rm max}-% \sqrt{2n})(\sqrt{2})n^{1/6}\leq s\right)
  4. 2 n \sqrt{2n}
  5. ( 2 ) n 1 / 6 (\sqrt{2})n^{1/6}
  6. n - 1 / 6 n^{-1/6}
  7. F 2 ( s ) = det ( I - A s ) F_{2}(s)=\det(I-A_{s})\,
  8. Ai ( x ) Ai ( y ) - Ai ( x ) Ai ( y ) x - y . \frac{\mathrm{Ai}(x)\mathrm{Ai}^{\prime}(y)-\mathrm{Ai}^{\prime}(x)\mathrm{Ai}% (y)}{x-y}.\,
  9. F 2 ( s ) = exp ( - s ( x - s ) q 2 ( x ) d x ) F_{2}(s)=\exp\left(-\int_{s}^{\infty}(x-s)q^{2}(x)\,dx\right)
  10. q ′′ ( s ) = s q ( s ) + 2 q ( s ) 3 q^{\prime\prime}(s)=sq(s)+2q(s)^{3}\,
  11. q ( s ) Ai ( s ) , s . \displaystyle q(s)\sim\textrm{Ai}(s),s\rightarrow\infty.
  12. F 1 ( s ) = exp ( - 1 2 s q ( x ) d x ) ( F 2 ( s ) ) 1 / 2 F_{1}(s)=\exp\left(-\frac{1}{2}\int_{s}^{\infty}q(x)\,dx\right)\,\left(F_{2}(s% )\right)^{1/2}
  13. F 4 ( s / 2 ) = cosh ( 1 2 s q ( x ) d x ) ( F 2 ( s ) ) 1 / 2 . F_{4}(s/\sqrt{2})=\cosh\left(\frac{1}{2}\int_{s}^{\infty}q(x)\,dx\right)\,% \left(F_{2}(s)\right)^{1/2}.

Traditional_Balsamic_Vinegar.html

  1. Y i e l d = m T B V m R E F I L L I N G Yield=\frac{mTBV}{mREFILLING}

Trammel_of_Archimedes.html

  1. x = ( p + q ) cos θ x=(p+q)\cos\theta\,
  2. y = q sin θ y=q\sin\theta\,
  3. x 2 ( p + q ) 2 + y 2 q 2 = 1 \frac{x^{2}}{(p+q)^{2}}+\frac{y^{2}}{q^{2}}=1

Transfinite_interpolation.html

  1. c 1 ( u ) \vec{c}_{1}(u)
  2. c 3 ( u ) \vec{c}_{3}(u)
  3. c 2 ( v ) \vec{c}_{2}(v)
  4. c 4 ( v ) \vec{c}_{4}(v)
  5. S ( u , v ) = ( 1 - v ) c 1 ( u ) + v c 3 ( u ) + ( 1 - u ) c 2 ( v ) + u c 4 ( v ) - [ ( 1 - u ) ( 1 - v ) P 1 , 2 + u v P 3 , 4 + u ( 1 - v ) P 1 , 4 + ( 1 - u ) v P 3 , 2 ] \begin{array}[]{rcl}\vec{S}(u,v)&=&(1-v)\vec{c}_{1}(u)+v\vec{c}_{3}(u)+(1-u)% \vec{c}_{2}(v)+u\vec{c}_{4}(v)\\ &&-\left[(1-u)(1-v)\vec{P}_{1,2}+uv\vec{P}_{3,4}+u(1-v)\vec{P}_{1,4}+(1-u)v% \vec{P}_{3,2}\right]\end{array}
  6. P 1 , 2 \vec{P}_{1,2}
  7. c 1 \vec{c}_{1}
  8. c 2 \vec{c}_{2}

Transformation_between_distributions_in_time–frequency_analysis.html

  1. C ( t , ω ) = 1 4 π 2 s * ( u - 1 2 τ ) s ( u + 1 2 τ ) ϕ ( θ , τ ) e - j θ t - j τ ω + j θ u d u d τ d θ , C(t,\omega)=\dfrac{1}{4\pi^{2}}\iiint s^{*}\left(u-\dfrac{1}{2}\tau\right)s% \left(u+\dfrac{1}{2}\tau\right)\phi(\theta,\tau)e^{-j\theta t-j\tau\omega+j% \theta u}\,du\,d\tau\,d\theta,
  2. ϕ ( θ , τ ) \phi(\theta,\tau)
  3. C ( t , ω ) = 1 4 π 2 M ( θ , τ ) e - j θ t - j τ ω d θ d τ C(t,\omega)=\dfrac{1}{4\pi^{2}}\iint M(\theta,\tau)e^{-j\theta t-j\tau\omega}% \,d\theta\,d\tau
  4. M ( θ , τ ) \displaystyle M(\theta,\tau)
  5. A ( θ , τ ) A(\theta,\tau)
  6. C 1 C_{1}
  7. C 2 C_{2}
  8. ϕ 1 \phi_{1}
  9. ϕ 2 \phi_{2}
  10. M 1 ( ϕ , τ ) = ϕ 1 ( θ , τ ) s * ( u - 1 2 τ ) s ( u + 1 2 τ ) e j θ u d u M_{1}(\phi,\tau)=\phi_{1}(\theta,\tau)\int s^{*}\left(u-\dfrac{1}{2}\tau\right% )s\left(u+\dfrac{1}{2}\tau\right)e^{j\theta u}\,du
  11. M 2 ( ϕ , τ ) = ϕ 2 ( θ , τ ) s * ( u - 1 2 τ ) s ( u + 1 2 τ ) e j θ u d u M_{2}(\phi,\tau)=\phi_{2}(\theta,\tau)\int s^{*}\left(u-\dfrac{1}{2}\tau\right% )s\left(u+\dfrac{1}{2}\tau\right)e^{j\theta u}\,du
  12. M 1 ( ϕ , τ ) = ϕ 1 ( θ , τ ) ϕ 2 ( θ , τ ) M 2 ( ϕ , τ ) M_{1}(\phi,\tau)=\dfrac{\phi_{1}(\theta,\tau)}{\phi_{2}(\theta,\tau)}M_{2}(% \phi,\tau)
  13. C 1 ( t , ω ) = 1 4 π 2 ϕ 1 ( θ , τ ) ϕ 2 ( θ , τ ) M 2 ( θ , τ ) e - j θ t - j τ ω d θ d τ C_{1}(t,\omega)=\dfrac{1}{4\pi^{2}}\iint\dfrac{\phi_{1}(\theta,\tau)}{\phi_{2}% (\theta,\tau)}M_{2}(\theta,\tau)e^{-j\theta t-j\tau\omega}\,d\theta\,d\tau
  14. M 2 M_{2}
  15. C 2 C_{2}
  16. C 1 ( t , ω ) = 1 4 π 2 ϕ 1 ( θ , τ ) ϕ 2 ( θ , τ ) C 2 ( t , ω ) e j θ ( t - t ) + j τ ( ω - ω ) d θ d τ d t d ω C_{1}(t,\omega)=\dfrac{1}{4\pi^{2}}\iiiint\dfrac{\phi_{1}(\theta,\tau)}{\phi_{% 2}(\theta,\tau)}C_{2}(t,\omega^{^{\prime}})e^{j\theta(t^{^{\prime}}-t)+j\tau(% \omega^{^{\prime}}-\omega)}\,d\theta\,d\tau\,dt^{^{\prime}}\,d\omega^{^{\prime}}
  17. C 1 ( t , ω ) = g 12 ( t - t , ω - ω ) C 2 ( t , ω ) d t d ω C_{1}(t,\omega)=\iint g_{12}(t^{^{\prime}}-t,\omega^{\prime}-\omega)C_{2}(t,% \omega^{\prime})\,dt^{^{\prime}}\,d\omega^{\prime}
  18. g 12 ( t , ω ) = 1 4 π 2 ϕ 1 ( θ , τ ) ϕ 2 ( θ , τ ) e j θ t + j τ ω d θ d τ g_{12}(t,\omega)=\dfrac{1}{4\pi^{2}}\iint\dfrac{\phi_{1}(\theta,\tau)}{\phi_{2% }(\theta,\tau)}e^{j\theta t+j\tau\omega}\,d\theta\,d\tau
  19. C 1 C_{1}
  20. C 2 C_{2}
  21. ϕ S P = ϕ 1 \phi_{SP}=\phi_{1}
  22. ϕ = ϕ 2 \phi=\phi_{2}
  23. g S P = g 12 g_{SP}=g_{12}
  24. C S P ( t , ω ) = g S P ( t - t , ω - ω ) C ( t , ω ) d t d ω C_{SP}(t,\omega)=\iint g_{SP}(t^{^{\prime}}-t,\omega^{^{\prime}}-\omega)C(t,% \omega^{^{\prime}})\,dt^{^{\prime}}\,d\omega^{^{\prime}}
  25. h ( t ) h(t)
  26. A h ( - θ , τ ) A_{h}(-\theta,\tau)
  27. g S P ( t , ω ) \displaystyle g_{SP}(t,\omega)
  28. ϕ ( - θ , τ ) ϕ ( θ , τ ) = 1 \phi(-\theta,\tau)\phi(\theta,\tau)=1
  29. g S P ( t , ω ) g_{SP}(t,\omega)
  30. - ω -\omega
  31. g S P ( t , ω ) = C h ( t , - ω ) g_{SP}(t,\omega)=C_{h}(t,-\omega)
  32. ϕ ( - θ , τ ) ϕ ( θ , τ ) = 1 \phi(-\theta,\tau)\phi(\theta,\tau)=1
  33. C S P ( t , ω ) = C s ( t , ω ) C h ( t - t , ω - ω ) d t d ω C_{SP}(t,\omega)=\iint C_{s}(t^{^{\prime}},\omega^{^{\prime}})C_{h}(t^{^{% \prime}}-t,\omega^{^{\prime}}-\omega)\,dt^{^{\prime}}\,d\omega^{^{\prime}}
  34. ϕ ( - θ , τ ) ϕ ( θ , τ ) = 1 \phi(-\theta,\tau)\phi(\theta,\tau)=1
  35. ϕ ( - θ , τ ) ϕ ( θ , τ ) \phi(-\theta,\tau)\phi(\theta,\tau)
  36. C S P ( t , ω ) = G ( t ′′ , ω ′′ ) C s ( t , ω ) C h ( t ′′ + t - t , - ω ′′ + ω - ω ) d t d t ′′ d ω d ω ′′ C_{SP}(t,\omega)=\iiiint G(t^{{}^{\prime\prime}},\omega^{{}^{\prime\prime}})C_% {s}(t^{^{\prime}},\omega^{^{\prime}})C_{h}(t^{{}^{\prime\prime}}+t^{^{\prime}}% -t,-\omega^{{}^{\prime\prime}}+\omega-\omega^{^{\prime}})\,dt^{^{\prime}}\,dt^% {{}^{\prime\prime}}\,d\omega d\omega^{{}^{\prime\prime}}
  37. G ( t , ω ) = 1 4 π 2 e - j θ t - j τ ω ϕ ( θ , τ ) ϕ ( - θ , τ ) d θ d τ G(t,\omega)=\dfrac{1}{4\pi^{2}}\iint\dfrac{e^{-j\theta t-j\tau\omega}}{\phi(% \theta,\tau)\phi(-\theta,\tau)}\,d\theta\,d\tau

Transition_metal_dioxygen_complex.html

  1. \overrightarrow{\leftarrow}

Transition_path_sampling.html

  1. C ( t ) = h A ( 0 ) h B ( t ) h A C(t)=\frac{\langle h_{A}(0)h_{B}(t)\rangle}{\langle h_{A}\rangle}
  2. k A B T P S ( t ) = d d t C ( t ) = h B ( t ) ˙ A B h B ( t ) A B C ( t ) k_{AB}^{TPS}(t)=\frac{d}{dt}C(t)=\frac{\langle\dot{h_{B}(t)}\rangle_{AB}}{% \langle h_{B}(t^{\prime})\rangle_{AB}}C(t^{\prime})
  3. k A B = Φ 1 , 0 i = 1 n - 1 P A ( i + 1 | i ) k_{AB}=\Phi_{1,0}\prod_{i=1}^{n-1}P_{A}(i+1|i)

Transmission-line_matrix_method.html

  1. Z Z
  2. Z / 3 Z/3
  3. R = Z / 3 - Z Z / 3 + Z = - 0.5 R=\frac{Z/3-Z}{Z/3+Z}=-0.5
  4. T = 2 ( Z / 3 ) Z / 3 + Z = 0.5 T=\frac{2(Z/3)}{Z/3+Z}=0.5
  5. E I = v i Δ t = 1 ( 1 / Z ) Δ t = Δ t / Z E_{I}=vi\,\Delta t=1\left(1/Z\right)\Delta t=\Delta t/Z
  6. E S = [ 0.5 2 + 0.5 2 + 0.5 2 + ( - 0.5 ) 2 ] ( Δ t / Z ) = Δ t / Z E_{S}=\left[0.5^{2}+0.5^{2}+0.5^{2}+(-0.5)^{2}\right](\Delta t/Z)=\Delta t/Z
  7. Δ t \Delta t
  8. Δ x \Delta x
  9. Δ y \Delta y
  10. Δ z \Delta z
  11. t = k Δ t t=k\,\Delta t
  12. x = l Δ x x=l\,\Delta x
  13. y = m Δ y y=m\,\Delta y
  14. z = n Δ z z=n\,\Delta z
  15. k = 0 , 1 , 2 , k=0,1,2,\ldots
  16. m , n , l m,n,l
  17. Δ x = Δ y = Δ z \Delta x=\Delta y=\Delta z
  18. Δ l \Delta l
  19. Δ t = Δ l c 0 , \Delta t=\frac{\Delta l}{c_{0}},
  20. c 0 c_{0}
  21. E x E_{x}
  22. E y E_{y}
  23. H z H_{z}
  24. H z y = ε E x t \frac{\partial{H_{z}}}{\partial{y}}=\varepsilon\frac{\partial{E_{x}}}{\partial% {t}}
  25. - H z x = ε E y t -\frac{\partial{H_{z}}}{\partial{x}}=\varepsilon\frac{\partial{E_{y}}}{% \partial{t}}
  26. E y x - E x y = - μ H z t \frac{\partial{E_{y}}}{\partial{x}}-\frac{\partial{E_{x}}}{\partial{y}}=-\mu% \frac{\partial{H_{z}}}{\partial{t}}
  27. 2 H z x 2 + 2 H z y 2 = μ ε 2 H z t 2 \frac{\partial^{2}H_{z}}{\partial{x}^{2}}+\frac{\partial^{2}{H_{z}}}{\partial{% y}^{2}}=\mu\varepsilon\frac{\partial^{2}{H_{z}}}{\partial{t}^{2}}
  28. Δ x \Delta x
  29. Δ y \Delta y
  30. Δ z \Delta z
  31. L L^{\prime}
  32. C C^{\prime}
  33. H z H_{z}
  34. E x E_{x}
  35. E y E_{y}
  36. L x = L y L_{x}=L_{y}
  37. - V 1 + V 2 + V 3 - V 4 = 2 L Δ l I t -V_{1}+V_{2}+V_{3}-V_{4}=2L^{\prime}\,\Delta l\frac{\partial{I}}{\partial{t}}
  38. Δ x = Δ y = Δ l \Delta x=\Delta y=\Delta l
  39. ( V 3 - V 1 ) - ( V 4 - V 2 ) = 2 L Δ l I t \left(V_{3}-V_{1}\right)-\left(V_{4}-V_{2}\right)=2L^{\prime}\,\Delta l\frac{% \partial I}{\partial t}
  40. [ E x ( y + Δ y ) - E x ( y ) ] Δ x - [ E y ( x + Δ x ) - E y ( x ) ] Δ y = 2 L Δ l I t \left[E_{x}(y+\Delta y)-E_{x}(y)\right]\,\Delta x-[E_{y}(x+\Delta x)-E_{y}(x)]% \Delta y=2L^{\prime}\,\Delta l\frac{\partial{I}}{\partial{t}}
  41. Δ x Δ y \Delta x\Delta y
  42. E x ( y + Δ y ) - E x ( y ) Δ y - E y ( x + Δ x ) - E y ( x ) Δ x = 2 L Δ l I t 1 Δ x Δ y \frac{E_{x}(y+\Delta y)-E_{x}(y)}{\Delta y}-\frac{E_{y}(x+\Delta x)-E_{y}(x)}{% \Delta x}=2L^{\prime}\,\Delta l\frac{\partial{I}}{\partial{t}}\frac{1}{\Delta x% \,\Delta y}
  43. Δ x = Δ y = Δ z = Δ l \Delta x=\Delta y=\Delta z=\Delta l
  44. I = H z Δ z I=H_{z}\,\Delta z
  45. Δ E x Δ y - Δ E y Δ x = 2 L H z t \frac{\Delta E_{x}}{\Delta y}-\frac{\Delta E_{y}}{\Delta x}=2L^{\prime}\frac{% \partial H_{z}}{\partial t}
  46. Δ l 0 \Delta l\rightarrow 0
  47. H z y = C E x t \frac{\partial{H_{z}}}{\partial{y}}=C^{\prime}\frac{\partial{E_{x}}}{\partial{% t}}
  48. - H z x = C E y t -\frac{\partial{H_{z}}}{\partial{x}}=C^{\prime}\frac{\partial{E_{y}}}{\partial% {t}}
  49. V 1 i k {}_{k}V^{i}_{1}
  50. V 1 r k = 0.5 ( V 1 i k + k V 2 i + k V 3 i - k V 4 i ) {}_{k}V^{r}_{1}=0.5\left({}_{k}V^{i}_{1}+_{k}V^{i}_{2}+_{k}V^{i}_{3}-_{k}V^{i}% _{4}\right)
  51. 𝐕 r k = 𝐒 k 𝐕 i {}_{k}\mathbf{V}^{r}=\mathbf{S}_{k}\mathbf{V}^{i}
  52. 𝐕 i k {}_{k}\mathbf{V}^{i}
  53. 𝐕 r k {}_{k}\mathbf{V}^{r}
  54. 𝐒 = 1 2 [ 1 1 1 - 1 1 1 - 1 1 1 - 1 1 1 - 1 1 1 1 ] \mathbf{S}=\frac{1}{2}\left[\begin{array}[]{cccc}1&1&1&-1\\ 1&1&-1&1\\ 1&-1&1&1\\ -1&1&1&1\end{array}\right]
  55. V 1 i k + 1 ( x , y ) = k V 3 r ( x , y - 1 ) {}_{k+1}V^{i}_{1}(x,y)=_{k}V^{r}_{3}(x,y-1)
  56. V 2 i k + 1 ( x , y ) = k V 4 r ( x - 1 , y ) {}_{k+1}V^{i}_{2}(x,y)=_{k}V^{r}_{4}(x-1,y)
  57. V 3 i k + 1 ( x , y ) = k V 1 r ( x , y + 1 ) {}_{k+1}V^{i}_{3}(x,y)=_{k}V^{r}_{1}(x,y+1)
  58. V 4 i k + 1 ( x , y ) = k V 2 r ( x + 1 , y ) {}_{k+1}V^{i}_{4}(x,y)=_{k}V^{r}_{2}(x+1,y)
  59. 𝐒 \,\textbf{S}
  60. H x H_{x}
  61. H y H_{y}
  62. E z E_{z}
  63. 𝐄 l , m , n k = k [ E 1 , E 2 , , E 11 , E 12 ] l , m , n T {}_{k}\mathbf{E}_{l,m,n}=_{k}\left[E_{1},E_{2},\ldots,E_{11},E_{12}\right]^{T}% _{l,m,n}
  64. 𝐇 l , m , n k = k [ H 1 , H 2 , , H 11 , H 12 ] l , m , n T {}_{k}\mathbf{H}_{l,m,n}=_{k}\left[H_{1},H_{2},\ldots,H_{11},H_{12}\right]^{T}% _{l,m,n}
  65. 𝐚 l , m , n k = 1 2 Z F 𝐄 l , m , n k + Z F 2 𝐇 l , m , n k {}_{k}\mathbf{a}_{l,m,n}=\frac{1}{2\sqrt{Z_{F}}}{{}_{k}\mathbf{E}}_{l,m,n}+% \frac{\sqrt{Z_{F}}}{2}{{}_{k}\mathbf{H}}_{l,m,n}
  66. 𝐛 l , m , n k = 1 2 Z F 𝐄 l , m , n k - Z F 2 𝐇 l , m , n k {}_{k}\mathbf{b}_{l,m,n}=\frac{1}{2\sqrt{Z_{F}}}{{}_{k}\mathbf{E}}_{l,m,n}-% \frac{\sqrt{Z_{F}}}{2}{{}_{k}\mathbf{H}}_{l,m,n}
  67. Z F = μ ε Z_{F}=\sqrt{\frac{\mu}{\varepsilon}}
  68. 𝐚 l , m , n k {}_{k}\mathbf{a}_{l,m,n}
  69. 𝐛 l , m , n k {}_{k}\mathbf{b}_{l,m,n}
  70. 𝐛 l , m , n k = 𝐒 k 𝐚 l , m , n {}_{k}\mathbf{b}_{l,m,n}=\mathbf{S}_{k}\mathbf{a}_{l,m,n}
  71. 𝐒 = [ 0 𝐒 0 𝐒 0 T 𝐒 0 T 0 𝐒 0 𝐒 0 𝐒 0 T 0 ] \mathbf{S}=\left[\begin{array}[]{ccc}0&\mathbf{S}_{0}&\mathbf{S}^{T}_{0}\\ \mathbf{S}^{T}_{0}&0&\mathbf{S}_{0}\\ \mathbf{S}_{0}&\mathbf{S}^{T}_{0}&0\end{array}\right]
  72. 𝐒 0 = 1 2 [ 0 0 1 - 1 0 0 - 1 1 1 1 0 0 1 1 0 0 ] \mathbf{S}_{0}=\frac{1}{2}\left[\begin{array}[]{cccc}0&0&1&-1\\ 0&0&-1&1\\ 1&1&0&0\\ 1&1&0&0\end{array}\right]

Transparent_conducting_film.html

  1. O O x V O + 0.5 O 2 ( g ) + 2 e O_{O}^{x}\leftrightarrow V_{O}^{\bullet\bullet}+0.5O_{2}(g)+2e^{\prime}
  2. n c 1 / 3 a H * = 0.26 ± 0.05 n_{c}^{1/3}a_{H}^{*}=0.26\pm 0.05

Transport_length.html

  1. l * = l 1 - g l^{*}=\frac{l}{1-g}
  2. g = < c o s ( θ ) Align g t ; g=<cos(\theta)&gt;

Transport_phenomena.html

  1. τ = - ν ρ υ x \tau=-\nu\frac{\partial\rho\upsilon}{\partial x}
  2. q A = - k d T d x \frac{q}{A}=-k\frac{dT}{dx}
  3. J = - D C x J=-D\frac{\partial C}{\partial x}
  4. τ z x = - ν ρ υ x z \tau_{zx}=-\nu\frac{\partial\rho\upsilon_{x}}{\partial z}
  5. J A y = - D A B C a y J_{Ay}=-D_{AB}\frac{\partial Ca}{\partial y}
  6. q = - k d T d x q=-k\frac{dT}{dx}
  7. Q = h A Δ T Q=h\cdot A\cdot{\Delta T}
  8. Δ T {\Delta T}
  9. N u a = h a D k Nu_{a}=\frac{h_{a}D}{k}

Transposable_integer.html

  1. 1 / 143 {1}/{143}
  2. gcd ( N , 10 m - 1 ) = gcd ( N c , 10 m - 1 ) , \gcd\left(N,10^{m}-1\right)=\gcd\left(N_{c},10^{m}-1\right),
  3. N c = 10 N - d ( 10 m - 1 ) , N_{c}=10N-d\left(10^{m}-1\right),\,
  4. 091575 / 999999 {091575}/{999999}
  5. 915750 / 999999 {915750}/{999999}
  6. 157509 / 999999 {157509}/{999999}
  7. 575091 / 999999 {575091}/{999999}
  8. 750915 / 999999 {750915}/{999999}
  9. 509157 / 999999 {509157}/{999999}
  10. 25 / 273 {25}/{273}
  11. 250 / 273 {250}/{273}
  12. 43 / 273 {43}/{273}
  13. 157 / 273 {157}/{273}
  14. 205 / 273 {205}/{273}
  15. 139 / 273 {139}/{273}
  16. 1 / F {1}/{F}
  17. 1 / F {1}/{F}
  18. 1 / F {1}/{F}
  19. n j / F {n j}/{F}
  20. 1 / F {1}/{F}
  21. j / F {j}/{F}
  22. 1 / 10 {1}/{10}
  23. F / 10 {F}/{10}
  24. n / s {n}/{s}
  25. s / F {s}/{F}
  26. n j / F {n j}/{F}
  27. j s / F {j s}/{F}
  28. 1 / 10 {1}/{10}
  29. F 10 / s F{10}/{s}
  30. j s / F {j s}/{F}
  31. X = D 10 m - 1 n 10 k - 1 , X=D\frac{10^{m}-1}{n10^{k}-1},
  32. X = D 10 m - 1 10 k - n , X=D\frac{10^{m}-1}{10^{k}-n},
  33. D < 10 k n - 1 , D<\frac{10^{k}}{n}-1,
  34. gcd ( 10 , t ) . \operatorname{gcd}\left(10^{\infty},t\right).
  35. 1 / 7 {1}/{7}
  36. 3 / 7 {3}/{7}
  37. 2 / 7 {2}/{7}
  38. 6 / 7 {6}/{7}
  39. 4 / 7 {4}/{7}
  40. 5 / 7 {5}/{7}
  41. 5 / 6 {5}/{6}
  42. 123456 / 999999 {123456}/{999999}
  43. 234561 / 123456 {234561}/{123456}
  44. 234561 / 123456 {234561}/{123456}
  45. 1 / F {1}/{F}
  46. n / F {n}/{F}
  47. 1 / F {1}/{F}
  48. n / F {n}/{F}
  49. 1 / F {1}/{F}
  50. n / F {n}/{F}
  51. 1 / F {1}/{F}
  52. 1 / F {1}/{F}
  53. n / s {n}/{s}
  54. 4 / 39 {4}/{39}
  55. 16 / 39 {16}/{39}
  56. 1 / F {1}/{F}
  57. F F
  58. 1 / F {1}/{F}
  59. F F
  60. F F
  61. 1 / 199 {1}/{199}
  62. 2 / 199 {2}/{199}
  63. 2 / 199 {2}/{199}
  64. 3 / 199 {3}/{199}
  65. 99 / 199 {99}/{199}
  66. 1 / 299 {1}/{299}
  67. 3 / 299 {3}/{299}
  68. 2 / 299 {2}/{299}
  69. 3 / 299 {3}/{299}
  70. 99 / 299 {99}/{299}
  71. 1 / 13 {1}/{13}
  72. 3 / 13 {3}/{13}
  73. 2 / 13 {2}/{13}
  74. 3 / 13 {3}/{13}
  75. 4 / 13 {4}/{13}
  76. 1 / 23 {1}/{23}
  77. 3 / 23 {3}/{23}
  78. 2 / 23 {2}/{23}
  79. 3 / 23 {3}/{23}
  80. 7 / 23 {7}/{23}
  81. 1 / 399 {1}/{399}
  82. 4 / 399 {4}/{399}
  83. 2 / 399 {2}/{399}
  84. 3 / 399 {3}/{399}
  85. 99 / 399 {99}/{399}
  86. 1 / 7 {1}/{7}
  87. 4 / 7 {4}/{7}
  88. 1 / 19 {1}/{19}
  89. 4 / 19 {4}/{19}
  90. 2 / 19 {2}/{19}
  91. 3 / 19 {3}/{19}
  92. 4 / 19 {4}/{19}
  93. 1 / 499 {1}/{499}
  94. 5 / 499 {5}/{499}
  95. 2 / 499 {2}/{499}
  96. 3 / 499 {3}/{499}
  97. 99 / 499 {99}/{499}
  98. 1 / 299 {1}/{299}
  99. 1 / 399 {1}/{399}
  100. 1 / F {1}/{F}
  101. 3 / F {3}/{F}
  102. 1 / F {1}/{F}
  103. F F
  104. 1 / 7 {1}/{7}
  105. 3 / 7 {3}/{7}
  106. 2 / 7 {2}/{7}
  107. 6 / 7 {6}/{7}
  108. 1 / F {1}/{F}
  109. 1 / F {1}/{F}
  110. 1 / 8 {1}/{8}
  111. 7 / 3 {7}/{3}
  112. 3 / 2 {3}/{2}
  113. 2 / F {2}/{F}
  114. 2 / 17 {2}/{17}
  115. n / s {n}/{s}
  116. 1 / 2 {1}/{2}
  117. 2 / 19 {2}/{19}
  118. 1 / 2 {1}/{2}
  119. 1 / 19 {1}/{19}
  120. 4 / 19 {4}/{19}
  121. 6 / 19 {6}/{19}
  122. 8 / 19 {8}/{19}
  123. 10 / 19 {10}/{19}
  124. 12 / 19 {12}/{19}
  125. 14 / 19 {14}/{19}
  126. 16 / 19 {16}/{19}
  127. 18 / 19 {18}/{19}
  128. 3 / 2 {3}/{2}
  129. 2 / 17 {2}/{17}
  130. 3 / 2 {3}/{2}
  131. 3 / 17 {3}/{17}
  132. 4 / 17 {4}/{17}
  133. 6 / 17 {6}/{17}
  134. 8 / 17 {8}/{17}
  135. 10 / 17 {10}/{17}
  136. 7 / 2 {7}/{2}
  137. 2 / 13 {2}/{13}
  138. 7 / 2 {7}/{2}
  139. 7 / 13 {7}/{13}
  140. 9 / 2 {9}/{2}
  141. 2 / 11 {2}/{11}
  142. 9 / 2 {9}/{2}
  143. 9 / 11 {9}/{11}
  144. 7 / 3 {7}/{3}
  145. 3 / 23 {3}/{23}
  146. 7 / 3 {7}/{3}
  147. 7 / 23 {7}/{23}
  148. 6 / 23 {6}/{23}
  149. 9 / 23 {9}/{23}
  150. 12 / 23 {12}/{23}
  151. 15 / 23 {15}/{23}
  152. 18 / 23 {18}/{23}
  153. 21 / 23 {21}/{23}
  154. 19 / 4 {19}/{4}
  155. 4 / 21 {4}/{21}
  156. 19 / 4 {19}/{4}
  157. 19 / 21 {19}/{21}
  158. 1 / F {1}/{F}
  159. F F
  160. R R
  161. R R
  162. F F
  163. 1 / F {1}/{F}
  164. F F
  165. 1 / 7 {1}/{7}
  166. 2 / 7 {2}/{7}
  167. 2 / 7 {2}/{7}
  168. 3 / 7 {3}/{7}
  169. 1 / 97 {1}/{97}
  170. 3 / 97 {3}/{97}
  171. 2 / 97 {2}/{97}
  172. 3 / 97 {3}/{97}
  173. 4 / 97 {4}/{97}
  174. 5 / 97 {5}/{97}
  175. 31 / 97 {31}/{97}
  176. 32 / 97 {32}/{97}
  177. 1 / 19 {1}/{19}
  178. 5 / 19 {5}/{19}
  179. 2 / 19 {2}/{19}
  180. 3 / 19 {3}/{19}
  181. 1 / 47 {1}/{47}
  182. 6 / 47 {6}/{47}
  183. 2 / 47 {2}/{47}
  184. 3 / 47 {3}/{47}
  185. 4 / 47 {4}/{47}
  186. 5 / 47 {5}/{47}
  187. 6 / 47 {6}/{47}
  188. 7 / 47 {7}/{47}
  189. 1 / 31 {1}/{31}
  190. 7 / 31 {7}/{31}
  191. 2 / 31 {2}/{31}
  192. 3 / 31 {3}/{31}
  193. 4 / 31 {4}/{31}
  194. 1 / 93 {1}/{93}
  195. 2 / 93 {2}/{93}
  196. 4 / 93 {4}/{93}
  197. 5 / 93 {5}/{93}
  198. 7 / 93 {7}/{93}
  199. 8 / 93 {8}/{93}
  200. 10 / 93 {10}/{93}
  201. 11 / 93 {11}/{93}
  202. 13 / 93 {13}/{93}
  203. 1 / 23 {1}/{23}
  204. 8 / 23 {8}/{23}
  205. 2 / 23 {2}/{23}
  206. 1 / 13 {1}/{13}
  207. 9 / 13 {9}/{13}
  208. 1 / 91 {1}/{91}
  209. 2 / 91 {2}/{91}
  210. 3 / 91 {3}/{91}
  211. 4 / 91 {4}/{91}
  212. 5 / 91 {5}/{91}
  213. 6 / 91 {6}/{91}
  214. 8 / 91 {8}/{91}
  215. 9 / 91 {9}/{91}
  216. 10 / 91 {10}/{91}
  217. 1 / 89 {1}/{89}
  218. 11 / 89 {11}/{89}
  219. 2 / 89 {2}/{89}
  220. 3 / 89 {3}/{89}
  221. 4 / 89 {4}/{89}
  222. 5 / 89 {5}/{89}
  223. 6 / 89 {6}/{89}
  224. 7 / 89 {7}/{89}
  225. 8 / 89 {8}/{89}
  226. 1 / 29 {1}/{29}
  227. 13 / 29 {13}/{29}
  228. 2 / 29 {2}/{29}
  229. 1 / 87 {1}/{87}
  230. 2 / 87 {2}/{87}
  231. 4 / 87 {4}/{87}
  232. 5 / 87 {5}/{87}
  233. 6 / 87 {6}/{87}
  234. 1 / 43 {1}/{43}
  235. 14 / 43 {14}/{43}
  236. 2 / 43 {2}/{43}
  237. 3 / 43 {3}/{43}
  238. 1 / 17 {1}/{17}
  239. 15 / 17 {15}/{17}

Transversality_theorem.html

  1. f : X Y f:X\rightarrow Y
  2. Z Y Z\subseteq Y
  3. f : X Y f:X\rightarrow Y
  4. Z Z
  5. Y Y
  6. f f
  7. Z Z
  8. f Z f\pitchfork Z
  9. x f - 1 ( Z ) x\in f^{-1}\left(Z\right)
  10. I m ( d f x ) + T f ( x ) Z = T f ( x ) Y Im\left(df_{x}\right)+T_{f\left(x\right)}Z=T_{f\left(x\right)}Y
  11. f f
  12. Z Z
  13. f - 1 ( Z ) f^{-1}\left(Z\right)
  14. X X
  15. X X
  16. f f
  17. f : X Y \partial f:\partial X\rightarrow Y
  18. f \partial f
  19. f Z f\pitchfork Z
  20. f Z \partial f\pitchfork Z
  21. f - 1 ( Z ) f^{-1}\left(Z\right)
  22. X X
  23. f - 1 ( Z ) = f - 1 ( Z ) X \partial f^{-1}\left(Z\right)=f^{-1}\left(Z\right)\cap\partial X
  24. F : X × S Y F:X\times S\rightarrow Y
  25. f s ( x ) = F ( x , s ) f_{s}\left(x\right)=F\left(x,s\right)
  26. f s : X Y f_{s}:X\rightarrow Y
  27. S S
  28. F F
  29. F : X × S Y F:X\times S\rightarrow Y
  30. X X
  31. Z Z
  32. Y Y
  33. F F
  34. F \partial F
  35. Z Z
  36. s S s\in S
  37. f s f_{s}
  38. f s \partial f_{s}
  39. Z Z
  40. G δ G_{\delta}
  41. F : X × S Y F:X\times S\rightarrow Y
  42. C k C^{k}
  43. C C^{\infty}
  44. X X
  45. S S
  46. Y Y
  47. C C^{\infty}
  48. 𝕂 \mathbb{K}
  49. C k C^{k}
  50. F : X × S Y F:X\times S\rightarrow Y
  51. k 1 k\geq 1
  52. y y
  53. s S s\in S
  54. f s ( x ) = F ( x , s ) f_{s}\left(x\right)=F\left(x,s\right)
  55. ind D f s ( x ) < k \mathop{\mathrm{ind}}Df_{s}\left(x\right)<k
  56. x f s - 1 ( { y } ) x\in f_{s}^{-1}\left(\left\{y\right\}\right)
  57. s n s s_{n}\rightarrow s
  58. S S
  59. n n\rightarrow\infty
  60. F ( x n , s n ) = y F\left(x_{n},s_{n}\right)=y
  61. n n
  62. x n x x_{n}\rightarrow x
  63. n n\rightarrow\infty
  64. x X x\in X
  65. S 0 S_{0}
  66. S S
  67. y y
  68. f s f_{s}
  69. s S 0 s\in S_{0}
  70. s S 0 s\in S_{0}
  71. n 0 n\geq 0
  72. ind D f s ( x ) = n \mathrm{ind}Df_{s}\left(x\right)=n
  73. x X x\in X
  74. f s ( x ) = y f_{s}\left(x\right)=y
  75. f s - 1 ( { y } ) f_{s}^{-1}\left(\left\{y\right\}\right)
  76. n n
  77. C k C^{k}
  78. ind D f s ( x ) = 0 \mathrm{ind}Df_{s}\left(x\right)=0
  79. f s ( x ) = y f_{s}\left(x\right)=y
  80. S 0 S_{0}
  81. S S
  82. s S 0 s\in S_{0}

Tree-depth.html

  1. t d ( G ) = { 1 , if | G | = 1 ; 1 + min v V t d ( G - v ) , if G is connected and | G | > 1 ; max i t d ( G i ) , otherwise ; td(G)=\begin{cases}1,&\,\text{if }|G|=1;\\ 1+\min_{v\in V}td(G-v),&\,\text{if }G\,\text{ is connected and }|G|>1;\\ \max_{i}td(G_{i}),&\,\text{otherwise};\end{cases}
  2. G i G_{i}
  3. n n
  4. log 2 ( n + 1 ) \lceil\log_{2}(n+1)\rceil
  5. log 2 ( n + 1 ) \lceil\log_{2}(n+1)\rceil
  6. O ( 1 ) O(1)
  7. O ( ( log n ) 2 ) O((\log n)^{2})
  8. f ( d ) n O ( 1 ) f(d)n^{O(1)}

Treynor–Black_model.html

  1. σ M \sigma_{M}
  2. r i = R F + β i ( R M - R F ) + α i + ϵ i r_{i}=R_{F}+\beta_{i}(R_{M}-R_{F})+\alpha_{i}+\epsilon_{i}
  3. ϵ i \epsilon_{i}
  4. σ i \sigma_{i}
  5. w i = α i / σ i 2 j = 1 N α j / σ j 2 w_{i}=\frac{\alpha_{i}/\sigma_{i}^{2}}{\sum_{j=1}^{N}\alpha_{j}/\sigma_{j}^{2}}
  6. w M = 1 - w A w_{M}=1-w_{A}
  7. w A = w 0 1 + ( 1 - β A ) w 0 w_{A}=\frac{w_{0}}{1+(1-\beta_{A})w_{0}}
  8. w 0 = α A / σ A 2 ( R M - R F ) / σ M 2 w_{0}=\frac{\alpha_{A}/\sigma_{A}^{2}}{(R_{M}-R_{F})/\sigma_{M}^{2}}
  9. α A = w i α i \alpha_{A}=\sum w_{i}\alpha_{i}
  10. β A = w i β i \beta_{A}=\sum w_{i}\beta_{i}
  11. σ A 2 = w i 2 σ i 2 \sigma_{A}^{2}=\sum w_{i}^{2}\sigma_{i}^{2}

Triad_method.html

  1. R 1 \vec{R}_{1}
  2. R 2 \vec{R}_{2}
  3. r 1 , r 2 \vec{r}_{1},\vec{r}_{2}
  4. i = 1 , 2 i=1,2
  5. A A
  6. A T A = I , d e t ( A ) = + 1 A^{T}A=I,det(A)=+1
  7. A A
  8. A A
  9. A A
  10. \vdots
  11. r 1 , r 2 \vec{r}_{1},\vec{r}_{2}
  12. Γ := [ S ^ M ^ S ^ × M ^ ] \Gamma:=\left[\hat{S}~{}\vdots~{}\hat{M}~{}\vdots~{}\hat{S}\times\hat{M}\right]
  13. Δ = [ s ^ m ^ s ^ × m ^ ] . \Delta=\left[\hat{s}~{}\vdots~{}\hat{m}~{}\vdots~{}\hat{s}\times\hat{m}\right].
  14. Γ \Gamma
  15. Δ \Delta
  16. d e t ( Γ ) det\left(\Gamma\right)
  17. 1 1
  18. - 1 -1
  19. Δ = ± 1 \Delta=\pm 1

Triangle_graph.html

  1. C 3 C_{3}
  2. K 3 K_{3}
  3. ( x - 3 ) ( x - 2 ) x (x-3)(x-2)x

Triatomic_hydrogen.html

  1. H 3 H 3 + + e - H_{3}\quad\longrightarrow\quad H_{3}^{+}\ +\ e^{-}
  2. H 3 * H + H 2 H_{3}^{*}\quad\longrightarrow\quad H\ +\ H_{2}
  3. H 3 * H + H + H H_{3}^{*}\quad\longrightarrow\quad H\ +\ H\ +\ H

Triatomic_molecule.html

  1. ω a = k 1 M m A m B \omega_{a}=\sqrt{\frac{k_{1}M}{m_{A}m_{B}}}
  2. ω s 1 = k 1 m A \omega_{s1}=\sqrt{\frac{k_{1}}{m_{A}}}
  3. ω s 2 = 2 k 2 M m A m B \omega_{s2}=\sqrt{\frac{2k_{2}M}{m_{A}m_{B}}}

Tripling-oriented_Doche–Icart–Kohel_curve.html

  1. K K
  2. T a : y 2 = x 3 + 3 a ( x + 1 ) 2 T_{a}\ :\ y^{2}=x^{3}+3a(x+1)^{2}
  3. a K a\in K
  4. T a T_{a}
  5. ( x , y ) (x,y)
  6. T a : y 2 = x 3 + 3 a ( x + 1 ) 2 T_{a}\ :\ y^{2}=x^{3}+3a(x+1)^{2}
  7. a 0 , 9 4 a\neq 0,\frac{9}{4}
  8. P 1 = ( x 1 , y 1 ) P_{1}=(x_{1},y_{1})
  9. P 2 = ( x 2 , y 2 ) P_{2}=(x_{2},y_{2})
  10. T a T_{a}
  11. P 3 = ( x 3 , y 3 ) = P 1 + P 2 P_{3}=(x_{3},y_{3})=P_{1}+P_{2}
  12. x 3 = ( - x 1 3 + ( x 2 - 3 a ) x 1 2 + ( x 2 2 + 6 a x 2 ) x 1 + ( y 1 2 - 2 y 2 y 1 + ( - x 2 3 - 3 a x 2 2 + y 2 2 ) ) ) ( x 1 2 - 2 x 2 x 1 + x 2 2 ) x_{3}=\frac{(-{x_{1}}^{3}+(x_{2}-3a){x_{1}}^{2}+({x_{2}}^{2}+6ax_{2})x_{1}+({y% _{1}}^{2}-2{y_{2}}{y_{1}}+(-{x_{2}}^{3}-3a{x_{2}}^{2}+{y_{2}}^{2})))}{({x_{1}}% ^{2}-2{x_{2}}{x_{1}}+{x_{2}}^{2})}
  13. y 3 = ( ( - y 1 + 2 y 2 ) x 1 3 + ( - 3 a y 1 + ( - 3 y 2 x 2 + 3 a y 2 ) ) x 1 2 + ( ( 3 x 2 2 + 6 a x 2 ) y 1 - 6 a y 2 x 2 ) x 1 + ( y 1 3 - 3 y 2 y 1 2 + ( - 2 x 2 3 - 3 a x 2 2 + 3 y 2 2 ) y 1 + ( y 2 x 2 3 + 3 a y 2 x 2 2 - y 2 3 ) ) ) ( - x 1 3 + 3 x 2 x 1 2 - 3 x 2 2 x 1 + x 2 3 ) y_{3}=\frac{((-y_{1}+2y_{2}){x_{1}}^{3}+(-3ay_{1}+(-3y_{2}x_{2}+3ay_{2})){x_{1% }}^{2}+((3{x_{2}}^{2}+6ax_{2})y_{1}-6ay_{2}x_{2})x_{1}+({y_{1}}^{3}-3y_{2}{y_{% 1}}^{2}+(-2{x_{2}}^{3}-3a{x_{2}}^{2}+3{y_{2}}^{2})y_{1}+(y_{2}{x_{2}}^{3}+3ay_% {2}{x_{2}}^{2}-{y_{2}}^{3})))}{(-{x_{1}}^{3}+3{x_{2}}{x_{1}}^{2}-3{x_{2}}^{2}x% _{1}+{x_{2}}^{3})}
  14. P 1 = ( x 1 , y 1 ) P_{1}=(x_{1},y_{1})
  15. T a T_{a}
  16. P 3 = ( x 3 , y 3 ) = 2 P 1 P_{3}=(x_{3},y_{3})=2P_{1}
  17. x 3 = 9 4 y 1 2 x 1 4 + 9 y 1 2 a x 1 3 + ( 9 y 1 2 a 2 + 9 y 1 2 a ) x 1 2 + ( 18 y 1 2 a 2 - 2 ) x 1 + 9 y 1 2 a 2 - 3 a x_{3}=\frac{9}{4{y_{1}}^{2}{x_{1}}^{4}}+\frac{9}{{y_{1}}^{2}a{x_{1}}^{3}}+(% \frac{9}{{y_{1}}^{2}a^{2}}+\frac{9}{{y_{1}}^{2}a}){x_{1}}^{2}+(\frac{18}{{y_{1% }}^{2}a^{2}}-2)x_{1}+\frac{9}{{y_{1}}^{2}a^{2}-3a}
  18. y 3 = - 27 8 y 1 3 x 1 6 - 81 4 y 1 3 a x 1 5 + ( - 81 2 y 1 3 a 2 - 81 4 y 1 3 a ) x 1 4 + ( - 27 y 1 3 a 3 - 81 y 1 3 a 2 + 9 2 y 1 ) x 1 3 + ( - 81 y 1 3 a 3 - 81 2 y 1 3 a 2 + 27 2 y 1 a ) x 1 2 + ( - 81 y 1 3 a 3 + 9 y 1 a 2 + 9 y 1 a ) x 1 + ( - 27 y 1 3 a 3 + 9 y 1 a 2 - y 1 ) y_{3}=-\frac{27}{8{y_{1}}^{3}{x_{1}}^{6}}-\frac{81}{4{y_{1}}^{3}a{x_{1}}^{5}}+% (-\frac{81}{2{y_{1}}^{3}a^{2}}-\frac{81}{4{y_{1}}^{3}a}){x_{1}}^{4}+(-\frac{27% }{{y_{1}}^{3}a^{3}}-\frac{81}{{y_{1}}^{3}a^{2}}+\frac{9}{2y_{1}}){x_{1}}^{3}+(% -\frac{81}{{y_{1}}^{3}a^{3}}-\frac{81}{2{y_{1}}^{3}}a^{2}+\frac{27}{2y_{1}a}){% x_{1}}2+(-\frac{81}{{y_{1}}^{3}a^{3}}+\frac{9}{y_{1}a^{2}}+\frac{9}{y_{1}a})x_% {1}+(-\frac{27}{{y_{1}}^{3}a^{3}}+\frac{9}{y_{1}a^{2}}-y_{1})
  19. P 1 = ( x 1 , y 1 ) P_{1}=(x_{1},y_{1})
  20. T a T_{a}
  21. ( 0 : 1 : 0 ) (0:1:0)
  22. - P 1 = ( x 1 , - y 1 ) -P_{1}=(x_{1},-y_{1})
  23. P = ( X : Y : Z : Z 2 ) P=(X:Y:Z:Z^{2})
  24. P = ( X : Y : Z : Z 2 ) = ( λ 2 X : λ 3 Y : λ Z : λ 2 Z 2 ) , P=(X:Y:Z:Z^{2})=(\lambda^{2}X:\lambda^{3}Y:\lambda Z:\lambda^{2}Z^{2}),
  25. λ K \lambda\in K
  26. P = ( X : Y : Z : Z 2 ) P=(X:Y:Z:Z^{2})
  27. Q = ( 4 X : 8 Y : 2 Z : 4 Z 2 ) Q=(4X:8Y:2Z:4Z^{2})
  28. λ = 2 \lambda=2
  29. P = ( x , y ) P=(x,y)
  30. T a T_{a}
  31. P = ( X : Y : Z : Z 2 ) P=(X:Y:Z:Z^{2})
  32. x = X / Z 2 x=X/Z^{2}
  33. y = Y / Z 3 y=Y/Z^{3}
  34. T a T_{a}
  35. T a : Y 2 = X 3 + 3 a Z 2 ( X + Z 2 ) 2 . T_{a}\ :\ Y^{2}=X^{3}+3aZ^{2}(X+Z^{2})^{2}.
  36. Z 2 Z^{2}
  37. ( 1 : 1 : 0 : 0 ) (1:1:0:0)
  38. P 1 P_{1}
  39. P 2 P_{2}
  40. P 3 = ( X 3 , Y 3 , Z 3 , Z Z 3 ) P_{3}=(X_{3},Y_{3},Z_{3},ZZ_{3})
  41. Z 2 = 1 Z_{2}=1
  42. a 3 = 3 a a_{3}=3a
  43. A = X 2 Z Z 1 A=X_{2}ZZ_{1}
  44. B = Y 2 Z Z 1 Z 1 B=Y_{2}ZZ_{1}Z_{1}
  45. C = X 1 - A C=X_{1}-A
  46. D = 2 ( Y 1 - B ) D=2(Y_{1}-B)
  47. F = C 2 F=C^{2}
  48. F 4 = 4 F F_{4}=4F
  49. Z 3 = ( Z 1 + C ) 2 - Z Z 1 - F Z_{3}=(Z_{1}+C)^{2}-ZZ_{1}-F
  50. E = Z 3 2 E={Z_{3}}^{2}
  51. G = C F 4 G=CF_{4}
  52. H = A F 4 H=AF_{4}
  53. X 3 = D 2 - G - 2 H - a 3 E X_{3}=D^{2}-G-2H-a_{3}E
  54. Y 3 = D ( H - X 3 ) - 2 B G Y_{3}=D(H-X_{3})-2BG
  55. Z Z 3 = E ZZ_{3}=E
  56. P 1 = ( 1 , 13 ) P_{1}=(1,\sqrt{13})
  57. P 2 = ( 0 , 3 ) P_{2}=(0,\sqrt{3})
  58. \mathbb{R}
  59. y 2 = x 3 + 3 ( x + 1 ) 2 y^{2}=x^{3}+3(x+1)^{2}
  60. A = X 2 Z Z 1 = 0 A=X_{2}ZZ_{1}=0
  61. B = Y 2 Z Z 1 Z 1 = 3 B=Y_{2}ZZ_{1}Z_{1}=\sqrt{3}
  62. C = X 1 - A = 1 C=X_{1}-A=1
  63. D = 2 ( Y 1 - B ) = 2 ( 13 - 3 ) D=2(Y_{1}-B)=2(\sqrt{13}-\sqrt{3})
  64. F = C 2 = 1 F=C^{2}=1
  65. F 4 = 4 F = 4 F_{4}=4F=4
  66. Z 3 = ( Z 1 + C ) 2 - Z Z 1 - F = 2 Z_{3}=(Z_{1}+C)^{2}-ZZ_{1}-F=2
  67. E = Z 3 2 = 4 E={Z_{3}}^{2}=4
  68. G = C F 4 = 4 G=CF_{4}=4
  69. H = A F 4 = 0 H=AF_{4}=0
  70. X 3 = D 2 - G - 2 H - a 3 E = 48 - 8 39 X_{3}=D^{2}-G-2H-a_{3}E=48-8\sqrt{39}
  71. Y 3 = D ( H - X 3 ) - 2 B G = 296 3 - 144 13 Y_{3}=D(H-X_{3})-2BG=296\sqrt{3}-144\sqrt{13}
  72. Z Z 3 = E = 4 ZZ_{3}=E=4
  73. Z 1 = Z 2 = 1 Z_{1}=Z_{2}=1
  74. P 3 = ( X 3 , Y 3 , Z 3 , Z Z 3 ) = ( 48 - 8 39 , 296 3 - 144 13 , 2 , 4 ) P_{3}=(X_{3},Y_{3},Z_{3},ZZ_{3})=(48-8\sqrt{39},296\sqrt{3}-144\sqrt{13},2,4)
  75. P 3 = ( 12 - 2 39 , 37 3 - 18 13 ) P_{3}=(12-2\sqrt{39},37\sqrt{3}-18\sqrt{13})
  76. P 1 P_{1}
  77. a 3 = 3 a a_{3}=3a
  78. a 2 = 2 a a_{2}=2a
  79. A = X 1 2 A={X_{1}}^{2}
  80. B = a 2 Z Z 1 ( X 1 + Z Z 1 ) B=a_{2}ZZ_{1}(X_{1}+ZZ_{1})
  81. C = 3 ( A + B ) C=3(A+B)
  82. D = Y 1 2 D={Y_{1}}^{2}
  83. E = D 2 E=D^{2}
  84. Z 3 = ( Y 1 + Z 1 ) 2 - D - Z Z 1 Z_{3}=(Y_{1}+Z_{1})^{2}-D-ZZ_{1}
  85. Z Z 3 = Z 3 2 ZZ_{3}=Z_{3}^{2}
  86. F = 2 ( ( X 1 + D ) 2 - A - E ) F=2((X_{1}+D)^{2}-A-E)
  87. X 3 = C 2 - a 3 Z Z 3 - 2 F X_{3}=C^{2}-a_{3}ZZ_{3}-2F
  88. Y 3 = C ( F - X 3 ) - 8 E Y_{3}=C(F-X_{3})-8E
  89. P 1 = ( 0 , 3 ) P_{1}=(0,\sqrt{3})
  90. y 2 = x 3 + 3 ( x + 1 ) 2 y^{2}=x^{3}+3(x+1)^{2}
  91. A = X 1 2 = 0 A={X_{1}}^{2}=0
  92. B = a 2 Z Z 1 ( X 1 + Z Z 1 ) = 2 B=a_{2}ZZ_{1}(X_{1}+ZZ_{1})=2
  93. C = 3 ( A + B ) = 6 C=3(A+B)=6
  94. D = Y 1 2 = 3 D={Y_{1}}^{2}=3
  95. E = D 2 = 9 E=D^{2}=9
  96. Z 3 = ( Y 1 + Z 1 ) 2 - D - Z Z 1 = 2 3 Z_{3}=(Y_{1}+Z_{1})^{2}-D-ZZ_{1}=2\sqrt{3}
  97. Z Z 3 = Z 3 2 = 12 ZZ_{3}=Z_{3}^{2}=12
  98. F = 2 ( ( X 1 + D ) 2 - A - E ) = 0 F=2((X_{1}+D)^{2}-A-E)=0
  99. X 3 = C 2 - a 3 Z Z 3 - 2 F = 0 X_{3}=C^{2}-a_{3}ZZ_{3}-2F=0
  100. Y 3 = C ( F - X 3 ) - 8 E = - 72 Y_{3}=C(F-X_{3})-8E=-72
  101. Z 1 = 1 Z_{1}=1
  102. P 3 = ( 0 , - 72 , 2 3 , 12 ) P_{3}=(0,-72,2\sqrt{3},12)
  103. P 3 = ( 0 , - 3 ) P_{3}=(0,-\sqrt{3})
  104. T a , λ T_{a,\lambda}
  105. y 2 = x 3 + 3 λ a ( x + λ ) 2 y^{2}=x^{3}+3\lambda a(x+\lambda)^{2}
  106. ( x , y ) ( x - λ a , y ) (x,y)\mapsto(x-\lambda a,y)
  107. T a , λ T_{a,\lambda}
  108. y 2 = x 3 - 3 λ 2 a ( a - 2 ) x + λ 3 a ( 2 a 2 - 6 a + 3 ) y^{2}=x^{3}-3{\lambda}^{2}a(a-2)x+{\lambda}^{3}a(2a^{2}-6a+3)
  109. E c , d E_{c,d}
  110. y 2 = x 3 + c x 2 + d y^{2}=x^{3}+cx^{2}+d
  111. λ = - 3 d ( a - 2 ) a ( 2 a 2 - 6 a + 3 ) \lambda=\frac{-3d(a-2)}{a(2a^{2}-6a+3)}
  112. a a
  113. 6912 a ( a - 2 ) 3 - j ( 4 a - 9 ) 6912a(a-2)^{3}-j(4a-9)
  114. j = 6912 c 3 4 c 3 + 27 d 2 j=\frac{6912c^{3}}{4c^{3}+27d^{2}}
  115. E c , d E_{c,d}

True_arithmetic.html

  1. 𝒩 \mathcal{N}
  2. \mathbb{N}
  3. \mathbb{N}
  4. \mathbb{N}
  5. 𝒩 \mathcal{N}
  6. 𝒩 φ \mathcal{N}\models\varphi
  7. 𝒩 . \mathcal{N}.
  8. 𝒩 \mathcal{N}
  9. 𝒩 \mathcal{N}
  10. φ ( x ) \varphi(x)
  11. 𝒩 θ \mathcal{N}\models\theta\qquad
  12. 𝒩 φ ( # ( θ ) ¯ ) . \mathcal{N}\models\varphi(\underline{\#(\theta)}).
  13. # ( θ ) ¯ \underline{\#(\theta)}
  14. Σ n 0 \Sigma^{0}_{n}
  15. Σ n 0 \Sigma^{0}_{n}
  16. Th ( 𝒩 ) = n Th ( 𝒩 ) n \mbox{Th}~{}(\mathcal{N})=\bigcup_{n\in\mathbb{N}}\mbox{Th}~{}_{n}(\mathcal{N})
  17. 2 κ 2^{\kappa}
  18. κ \kappa
  19. 2 0 2^{\aleph_{0}}
  20. 𝒩 \mathcal{N}
  21. \mathbb{N}

TrueAchievements.html

  1. S c o r e T A = S c o r e G S * T g a m e T a c h Score_{TA}=Score_{GS}*\sqrt{T_{game}\over T_{ach}}
  2. S c o r e G S Score_{GS}
  3. T g a m e T_{game}
  4. T a c h T_{ach}
  5. 40 47115 / 25314 = 40 1.86 = 40 * 1.36 54 40\sqrt{47115/25314}=40\sqrt{1.86}=40*1.36\approx 54

Truncated_5-cubes.html

  1. 1 / ( 2 + 2 ) 1/(\sqrt{2}+2)
  2. ( ± 1 , ± ( 1 + 2 ) , ± ( 1 + 2 ) , ± ( 1 + 2 ) , ± ( 1 + 2 ) ) \left(\pm 1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+% \sqrt{2})\right)
  3. 2 \sqrt{2}
  4. ( 0 , ± 1 , ± 2 , ± 2 , ± 2 ) \left(0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2\right)

Truncated_6-cubes.html

  1. 1 / ( 2 + 2 ) 1/(\sqrt{2}+2)
  2. ( ± 1 , ± ( 1 + 2 ) , ± ( 1 + 2 ) , ± ( 1 + 2 ) , ± ( 1 + 2 ) , ± ( 1 + 2 ) ) \left(\pm 1,\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+\sqrt{2}),\ \pm(1+% \sqrt{2}),\ \pm(1+\sqrt{2})\right)
  3. ( 0 , ± 1 , ± 2 , ± 2 , ± 2 , ± 2 ) \left(0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2,\ \pm 2\right)
  4. ( 0 , 0 , ± 1 , ± 2 , ± 2 , ± 2 ) \left(0,\ 0,\ \pm 1,\ \pm 2,\ \pm 2,\ \pm 2\right)

Truncation_error_(numerical_integration).html

  1. y = f ( t , y ) , y ( t 0 ) = y 0 , t t 0 y^{\prime}=f(t,y),\qquad y(t_{0})=y_{0},\qquad t\geq t_{0}
  2. y n y_{n}
  3. y ( t n ) y(t_{n})
  4. t 1 , t 2 , , t N t_{1},t_{2},\ldots,t_{N}
  5. h = t n - t n - 1 , n = 1 , 2 , , N . h=t_{n}-t_{n-1},\qquad n=1,2,\ldots,N.
  6. y n y_{n}
  7. y n = y n - 1 + h A ( t n - 1 , y n - 1 , h , f ) . y_{n}=y_{n-1}+hA(t_{n-1},y_{n-1},h,f).
  8. A A
  9. y n y_{n}
  10. τ n \tau_{n}
  11. A A
  12. τ n \tau_{n}
  13. n n
  14. y n y n - 1 + h A ( t n - 1 , y n - 1 , h , f ) y_{n}\approx y_{n-1}+hA(t_{n-1},y_{n-1},h,f)
  15. τ n = y ( t n ) - y ( t n - 1 ) - h A ( t n - 1 , y ( t n - 1 ) , h , f ) . \tau_{n}=y(t_{n})-y(t_{n-1})-hA(t_{n-1},y(t_{n-1}),h,f).
  16. o ( h ) o(h)
  17. ε > 0 \varepsilon>0
  18. H H
  19. | τ n | < ε h |\tau_{n}|<\varepsilon h
  20. h < H h<H
  21. A A
  22. A ( t , y , 0 , f ) = f ( t , y ) A(t,y,0,f)=f(t,y)
  23. p p
  24. O ( h p + 1 ) O(h^{p+1})
  25. C C
  26. H H
  27. | τ n | < C h p + 1 |\tau_{n}|<Ch^{p+1}
  28. h < H h<H
  29. e n e_{n}
  30. t n t_{n}
  31. e n \displaystyle e_{n}
  32. lim h 0 max n | e n | = 0 \lim_{h\to 0}\max_{n}|e_{n}|=0
  33. e n + 1 = e n + h ( A ( t n , y ( t n ) , h , f ) - A ( t n , y n , h , f ) ) + τ n + 1 . e_{n+1}=e_{n}+h\Big(A(t_{n},y(t_{n}),h,f)-A(t_{n},y_{n},h,f)\Big)+\tau_{n+1}.
  34. L L
  35. t t
  36. y 1 y_{1}
  37. y 2 y_{2}
  38. | A ( t , y 1 , h , f ) - A ( t , y 2 , h , f ) | L | y 1 - y 2 | . |A(t,y_{1},h,f)-A(t,y_{2},h,f)|\leq L|y_{1}-y_{2}|.
  39. | e n | max j τ j h L ( e L ( t n - t 0 ) - 1 ) . |e_{n}|\leq\frac{\max_{j}\tau_{j}}{hL}\left(\mathrm{e}^{L(t_{n}-t_{0})}-1% \right).
  40. f f
  41. A A
  42. h h
  43. y n + s + a s - 1 y n + s - 1 + a s - 2 y n + s - 2 + + a 0 y n \displaystyle y_{n+s}+a_{s-1}y_{n+s-1}+a_{s-2}y_{n+s-2}+\cdots+a_{0}y_{n}
  44. y n + s = - k = 0 s - 1 a n + k y n + k + h k = 0 s b k f ( t n + k , y n + k ) . y_{n+s}=-\sum_{k=0}^{s-1}a_{n+k}y_{n+k}+h\sum_{k=0}^{s}b_{k}f(t_{n+k},y_{n+k}).
  45. τ n = y ( t n + s ) + k = 0 s - 1 a n + k y ( t n + k ) - h k = 0 s b k f ( t n + k , y ( t n + k ) ) . \tau_{n}=y(t_{n+s})+\sum_{k=0}^{s-1}a_{n+k}y(t_{n+k})-h\sum_{k=0}^{s}b_{k}f(t_% {n+k},y(t_{n+k})).
  46. τ n = o ( h ) \tau_{n}=o(h)
  47. τ n = O ( h p + 1 ) \tau_{n}=O(h^{p+1})
  48. τ n = O ( h p + 1 ) \tau_{n}=O(h^{p+1})
  49. e n = O ( h p ) e_{n}=O(h^{p})
  50. τ n / h \tau_{n}/h

Truth_table.html

  1. \nleftarrow
  2. \nleftarrow
  3. \nleftarrow
  4. \nleftarrow
  5. \nleftarrow
  6. \nrightarrow
  7. \nrightarrow
  8. \rightarrow
  9. \leftarrow
  10. \cdot
  11. P P
  12. Q Q
  13. P Q P\land Q
  14. P Q P\lor Q
  15. P ¯ Q P\underline{\lor}Q
  16. P ¯ Q P\underline{\land}Q
  17. P Q P\Rightarrow Q
  18. P Q P\Leftarrow Q
  19. P Q P\Leftrightarrow Q
  20. \land
  21. \lor
  22. ¯ \underline{\lor}
  23. ¯ \underline{\land}
  24. \rightarrow
  25. \leftarrow
  26. \iff
  27. and ¯ \underline{\and}

Tukey's_test_of_additivity.html

  1. μ ^ = Y ¯ \hat{\mu}=\bar{Y}_{\cdot\cdot}
  2. α ^ i = Y ¯ i - Y ¯ \hat{\alpha}_{i}=\bar{Y}_{i\cdot}-\bar{Y}_{\cdot\cdot}
  3. β ^ j = Y ¯ j - Y ¯ . \hat{\beta}_{j}=\bar{Y}_{\cdot j}-\bar{Y}_{\cdot\cdot}.
  4. γ ^ i j = Y i j - ( μ ^ + α ^ i + β ^ j ) , \hat{\gamma}_{ij}=Y_{ij}-(\hat{\mu}+\hat{\alpha}_{i}+\hat{\beta}_{j}),
  5. Y ^ i j = μ ^ + α ^ i + β ^ j + γ ^ i j Y i j \hat{Y}_{ij}=\hat{\mu}+\hat{\alpha}_{i}+\hat{\beta}_{j}+\hat{\gamma}_{ij}% \equiv Y_{ij}
  6. E Y i j = μ + α i + β j + λ α i β j EY_{ij}=\mu+\alpha_{i}+\beta_{j}+\lambda\alpha_{i}\beta_{j}
  7. S S A n i ( Y ¯ i - Y ¯ ) 2 SS_{A}\equiv n\sum_{i}(\bar{Y}_{i\cdot}-\bar{Y}_{\cdot\cdot})^{2}
  8. S S B m j ( Y ¯ j - Y ¯ ) 2 SS_{B}\equiv m\sum_{j}(\bar{Y}_{\cdot j}-\bar{Y}_{\cdot\cdot})^{2}
  9. S S A B ( i j Y i j ( Y ¯ i - Y ¯ ) ( Y ¯ j - Y ¯ ) ) 2 i ( Y ¯ i - Y ¯ ) 2 j ( Y ¯ j - Y ¯ ) 2 SS_{AB}\equiv\frac{(\sum_{ij}Y_{ij}(\bar{Y}_{i\cdot}-\bar{Y}_{\cdot\cdot})(% \bar{Y}_{\cdot j}-\bar{Y}_{\cdot\cdot}))^{2}}{\sum_{i}(\bar{Y}_{i\cdot}-\bar{Y% }_{\cdot\cdot})^{2}\sum_{j}(\bar{Y}_{\cdot j}-\bar{Y}_{\cdot\cdot})^{2}}
  10. S S T i j ( Y i j - Y ¯ ) 2 SS_{T}\equiv\sum_{ij}(Y_{ij}-\bar{Y}_{\cdot\cdot})^{2}
  11. S S E S S T - S S A - S S B - S S A B SS_{E}\equiv SS_{T}-SS_{A}-SS_{B}-SS_{AB}
  12. S S A B / 1 M S E . \frac{SS_{AB}/1}{MS_{E}}.

Turán–Kubilius_inequality.html

  1. ν ν
  2. A ( x ) = p ν x f ( p ν ) p - ν ( 1 - p - 1 ) A(x)=\sum_{p^{\nu}\leq x}f(p^{\nu})p^{-\nu}(1-p^{-1})
  3. B ( x ) 2 = p ν x | f ( p ν ) | 2 p - ν . B(x)^{2}=\sum_{p^{\nu}\leq x}\left|f(p^{\nu})\right|^{2}p^{-\nu}.
  4. 1 x n x | f ( n ) - A ( x ) | 2 ( 2 + ε ( x ) ) B ( x ) 2 . \frac{1}{x}\sum_{n\leq x}|f(n)-A(x)|^{2}\leq(2+\varepsilon(x))B(x)^{2}.

Turbo_equalizer.html

  1. a a
  2. b b
  3. b b
  4. b b
  5. c c
  6. c c
  7. x x
  8. a ^ \hat{a}
  9. y y
  10. y y
  11. x x
  12. c ^ \hat{c}
  13. c ^ \hat{c}
  14. a ^ \hat{a}
  15. b ^ \hat{b}
  16. a a
  17. b b
  18. b ~ \tilde{b}
  19. x x
  20. x ~ \tilde{x}
  21. y y
  22. a ^ \hat{a}
  23. x ^ \hat{x}
  24. b b

Turbulent_diffusion.html

  1. c i t + x j ( u j , c i ) = D i 2 c i x j x j + R i ( c 1 , , c N , T ) + S i ( x , t ) {\partial c_{i}\over\partial t}+{\partial\over\partial x_{j}}(u_{j},c_{i})=D_{% i}{\partial^{2}c_{i}\over\partial x_{j}\partial x_{j}}+R_{i}(c_{1},...,c_{N},T% )+S_{i}(x,t)
  2. i = 1 , 2 , , N {i=1,2,...,N}
  3. c i c_{i}
  4. u j u_{j}
  5. x j x_{j}
  6. D i D_{i}
  7. R i R_{i}
  8. c i c_{i}
  9. S i S_{i}
  10. c i c_{i}
  11. c i c_{i}
  12. k g / k g kg/kg
  13. u j , c = - K j j ( c ) x j \langle u_{j}^{\prime},c^{\prime}\rangle=-K_{jj}{\partial(c)\over\partial x_{j}}
  14. c i t + u ¯ j ( c ) x j = x j ( K j j ( c ) x j ) {\partial c_{i}\over\partial t}+\overline{u}_{j}{\partial(c)\over\partial x_{j% }}={\partial\over\partial x_{j}}\bigg(K_{jj}{\partial(c)\over\partial x_{j}}\bigg)
  15. s y m b o l ψ ( 𝐱 , t ) = - - - Q ( 𝐱 , t | 𝐱 , t ) s y m b o l ψ ( 𝐱 , t ) d 𝐱 symbol{\psi}(\mathbf{x},\mathit{t})=\int_{-\infty}^{\infty}\int_{-\infty}^{% \infty}\int_{-\infty}^{\infty}\mathit{Q}(\mathbf{x},\mathit{t}|\mathbf{x}^{% \prime},\mathit{t}^{\prime})symbol{\psi}(\mathbf{x}^{\prime},\mathit{t}^{% \prime})d\mathbf{x}^{\prime}
  16. c ( 𝐱 , t ) = i = 1 m s y m b o l ψ i ( 𝐱 , t ) \langle c(\mathbf{x},\mathit{t})\rangle=\sum_{i=1}^{m}symbol{\psi}_{i}(\mathbf% {x},\mathit{t})
  17. c ( 𝐱 , t ) = = - - - Q ( 𝐱 , t | 𝐱 0 , t 0 ) c ( 𝐱 0 , t 0 ) d 𝐱 0 + - - - t 0 t Q ( 𝐱 , t | 𝐱 , t ) S ( 𝐱 , t ) d t d 𝐱 \langle c(\mathbf{x},\mathit{t})\rangle==\int_{-\infty}^{\infty}\int_{-\infty}% ^{\infty}\int_{-\infty}^{\infty}\mathit{Q}(\mathbf{x},\mathit{t}|\mathbf{x}_{0% },\mathit{t}_{0})\langle c(\mathbf{x}_{0},\mathit{t}_{0})\rangle d\mathbf{x}_{% 0}+\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}\int_{% t_{0}}^{t}\mathit{Q}(\mathbf{x},\mathit{t}|\mathbf{x}^{\prime},\mathit{t}^{% \prime})\mathit{S}(\mathbf{x}^{\prime},\mathit{t}^{\prime})d\mathit{t}d\mathbf% {x}^{\prime}
  18. c ( x , y , z ) = q 2 π σ y σ z e x p [ - ( y 2 σ y 2 + z 2 σ z 2 ) ] \langle c(x,y,z)\rangle=\frac{q}{2\pi\sigma_{y}\sigma_{z}}exp\bigg[-\bigg(% \frac{y^{2}}{\sigma_{y}^{2}}+\frac{z^{2}}{\sigma_{z}^{2}}\bigg)\bigg]
  19. σ y 2 = 2 K y y x u ¯ σ z 2 = 2 K z z x u ¯ \sigma_{y}^{2}=\frac{2K_{yy}x}{\overline{u}}\sigma_{z}^{2}=\frac{2K_{zz}x}{% \overline{u}}

Turkish_Identification_Number.html

  1. d 1 d_{1}
  2. { 10 - [ ( 3 i = 1 5 d 2 i - 1 + i = 1 4 d 2 i ) ( mod 10 ) ] } ( mod 10 ) \left\{10-\left[(3\cdot\sum_{i=1}^{5}d_{2i-1}+\sum_{i=1}^{4}d_{2i})\;\;(% \mathop{{\rm mod}}10)\right]\right\}\;\;(\mathop{{\rm mod}}10)
  3. { 10 - [ ( i = 1 5 d 2 i - 1 + 3 i = 1 5 d 2 i ) ( mod 10 ) ] } ( mod 10 ) \left\{10-\left[(\sum_{i=1}^{5}d_{2i-1}+3\cdot\sum_{i=1}^{5}d_{2i})\;\;(% \mathop{{\rm mod}}10)\right]\right\}\;\;(\mathop{{\rm mod}}10)

Tutte_12-cage.html

  1. ( x - 3 ) x 28 ( x + 3 ) ( x 2 - 6 ) 21 ( x 2 - 2 ) 27 . (x-3)x^{28}(x+3)(x^{2}-6)^{21}(x^{2}-2)^{27}.

Tutte_graph.html

  1. ( x - 3 ) ( x 15 - 22 x 13 + x 12 + 184 x 11 - 26 x 10 - 731 x 9 + 199 x 8 + 1383 x 7 - 576 x 6 - 1061 x 5 + 561 x 4 + 233 x 3 - 151 x 2 + 4 x + 4 ) 2 (x-3)(x^{15}-22x^{13}+x^{12}+184x^{11}-26x^{10}-731x^{9}+199x^{8}+1383x^{7}-57% 6x^{6}-1061x^{5}+561x^{4}+233x^{3}-151x^{2}+4x+4)^{2}
  2. ( x 15 + 3 x 14 - 16 x 13 - 50 x 12 + 94 x 11 + 310 x 10 - 257 x 9 - 893 x 8 + 366 x 7 + 1218 x 6 - 347 x 5 - 717 x 4 + 236 x 3 + 128 x 2 - 56 x + 4 ) . (x^{15}+3x^{14}-16x^{13}-50x^{12}+94x^{11}+310x^{10}-257x^{9}-893x^{8}+366x^{7% }+1218x^{6}-347x^{5}-717x^{4}+236x^{3}+128x^{2}-56x+4).

Twist_knot.html

  1. n n
  2. n + 2 n+2
  3. n n
  4. Δ ( t ) = { n + 1 2 t - n + n + 1 2 t - 1 if n is odd - n 2 t + ( n + 1 ) - n 2 t - 1 if n is even, \Delta(t)=\begin{cases}\frac{n+1}{2}t-n+\frac{n+1}{2}t^{-1}&\,\text{if }n\,% \text{ is odd}\\ -\frac{n}{2}t+(n+1)-\frac{n}{2}t^{-1}&\,\text{if }n\,\text{ is even,}\\ \end{cases}
  5. ( z ) = { n + 1 2 z 2 + 1 if n is odd 1 - n 2 z 2 if n is even. \nabla(z)=\begin{cases}\frac{n+1}{2}z^{2}+1&\,\text{if }n\,\text{ is odd}\\ 1-\frac{n}{2}z^{2}&\,\text{if }n\,\text{ is even.}\\ \end{cases}
  6. n n
  7. V ( q ) = 1 + q - 2 + q - n - q - n - 3 q + 1 , V(q)=\frac{1+q^{-2}+q^{-n}-q^{-n-3}}{q+1},
  8. n n
  9. V ( q ) = q 3 + q - q 3 - n + q - n q + 1 . V(q)=\frac{q^{3}+q-q^{3-n}+q^{-n}}{q+1}.

Twisted_Edwards_curve.html

  1. a x 2 + y 2 = 1 + d x 2 y 2 ax^{2}+y^{2}=1+dx^{2}y^{2}
  2. ( x 1 , y 1 ) (x_{1},y_{1})
  3. ( x 2 , y 2 ) (x_{2},y_{2})
  4. a x 2 + y 2 = 1 + d x 2 y 2 ax^{2}+y^{2}=1+dx^{2}y^{2}
  5. ( x 1 , y 1 ) , ( x 2 , y 2 ) (x_{1},y_{1}),(x_{2},y_{2})
  6. ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 y 2 + y 1 x 2 1 + d x 1 x 2 y 1 y 2 , y 1 y 2 - a x 1 x 2 1 - d x 1 x 2 y 1 y 2 ) (x_{1},y_{1})+(x_{2},y_{2})=\left(\frac{x_{1}y_{2}+y_{1}x_{2}}{1+dx_{1}x_{2}y_% {1}y_{2}},\frac{y_{1}y_{2}-ax_{1}x_{2}}{1-dx_{1}x_{2}y_{1}y_{2}}\right)
  7. ( x 1 , y 1 ) (x_{1},y_{1})
  8. - x 1 , y 1 ) -x_{1},y_{1})
  9. 3 x 2 + y 2 = 1 + 2 x 2 y 2 3x^{2}+y^{2}=1+2x^{2}y^{2}
  10. P 1 = ( 1 , 2 ) P_{1}=(1,\sqrt{2})
  11. P 2 = ( 1 , - 2 ) P_{2}=(1,-\sqrt{2})
  12. x 3 = x 1 y 2 + y 1 x 2 1 + d x 1 x 2 y 1 y 2 = 0 x_{3}=\frac{x_{1}y_{2}+y_{1}x_{2}}{1+dx_{1}x_{2}y_{1}y_{2}}=0
  13. y 3 = y 1 y 2 - a x 1 x 2 1 - d x 1 x 2 y 1 y 2 = - 1 y_{3}=\frac{y_{1}y_{2}-ax_{1}x_{2}}{1-dx_{1}x_{2}y_{1}y_{2}}=-1
  14. x 3 = x 1 y 1 + y 1 x 1 1 + d x 1 x 1 y 1 y 1 = 2 x 1 y 1 a x 1 2 + y 1 2 x_{3}=\frac{x_{1}y_{1}+y_{1}x_{1}}{1+dx_{1}x_{1}y_{1}y_{1}}=\frac{2x_{1}y_{1}}% {ax_{1}^{2}+y_{1}^{2}}
  15. y 3 = y 1 y 1 - a x 1 x 1 1 - d x 1 x 1 y 1 y 1 = y 1 2 - a x 1 2 2 - a x 1 2 - y 1 2 . y_{3}=\frac{y_{1}y_{1}-ax_{1}x_{1}}{1-dx_{1}x_{1}y_{1}y_{1}}=\frac{y_{1}^{2}-% ax_{1}^{2}}{2-ax_{1}^{2}-y_{1}^{2}}.
  16. P 1 = ( 1 , 2 ) P_{1}=(1,\sqrt{2})
  17. x 3 = x 1 y 1 + y 1 x 1 1 + d x 1 x 1 y 1 y 1 = 2 2 5 x_{3}=\frac{x_{1}y_{1}+y_{1}x_{1}}{1+dx_{1}x_{1}y_{1}y_{1}}=\frac{2\sqrt{2}}{5}
  18. y 3 = y 1 y 1 - a x 1 x 1 1 - d x 1 x 1 y 1 y 1 = 1 3 . y_{3}=\frac{y_{1}y_{1}-ax_{1}x_{1}}{1-dx_{1}x_{1}y_{1}y_{1}}=\frac{1}{3}.
  19. P 3 = ( 2 2 5 , 1 3 ) P_{3}=(\frac{2\sqrt{2}}{5},\frac{1}{3})
  20. 3 x 2 + y 2 = 1 + 2 x 2 y 2 3x^{2}+y^{2}=1+2x^{2}y^{2}
  21. ( x , y , z ) (x,y,z)
  22. a x 2 + y 2 = 1 + d x 2 y 2 ax^{2}+y^{2}=1+dx^{2}y^{2}
  23. ( X 1 : Y 1 : Z 1 ) (X_{1}:Y_{1}:Z_{1})
  24. ( X 2 + a Y 2 ) Z 2 = X 2 Y 2 + d Z 4 (X^{2}+aY^{2})Z^{2}=X^{2}Y^{2}+dZ^{4}
  25. X 1 Y 1 Z 1 X_{1}Y_{1}Z_{1}
  26. ( Z 1 / X 1 , Z 1 / Y 1 ) (Z_{1}/X_{1},Z_{1}/Y_{1})
  27. ( a X 2 + Y 2 ) Z 2 = Z 4 + d X 2 Y 2 (aX^{2}+Y^{2})Z^{2}=Z^{4}+dX^{2}Y^{2}

Twisted_Hessian_curves.html

  1. a x 3 + y 3 + 1 = d x y a\cdot x^{3}+y^{3}+1=d\cdot x\cdot y
  2. a X 3 + Y 3 + Z 3 = d X Y Z a\cdot X^{3}+Y^{3}+Z^{3}=d\cdot X\cdot Y\cdot Z
  3. x = X Z x=\frac{X}{Z}
  4. y = Y Z y=\frac{Y}{Z}
  5. t 3 + y 3 + 1 = d x y t^{3}+y^{3}+1=d\cdot x\cdot y
  6. x 3 = x 1 - y 1 3 x 1 a y 1 x 1 3 - y 1 x_{3}=\frac{x_{1}-y_{1}^{3}\cdot x_{1}}{a\cdot y_{1}\cdot x_{1}^{3}-y_{1}}
  7. y 3 = y 1 3 - a x 1 3 a y 1 x 1 3 - y 1 y_{3}=\frac{y_{1}^{3}-a\cdot x_{1}^{3}}{a\cdot y_{1}\cdot x_{1}^{3}-y_{1}}
  8. x 1 = x - y 3 x a y x 3 - y x_{1}=\frac{x-y^{3}\cdot x}{a\cdot y\cdot x^{3}-y}
  9. y 1 = y 3 - a x 3 a y x 3 - y y_{1}=\frac{y^{3}-a\cdot x^{3}}{a\cdot y\cdot x^{3}-y}
  10. A = X 1 Z 2 A=X_{1}\cdot Z_{2}
  11. B = Z 1 Z 2 B=Z_{1}\cdot Z_{2}
  12. C = Y 1 X 2 C=Y_{1}X_{2}
  13. D = Y 1 Y 2 D=Y_{1}\cdot Y_{2}
  14. E = Z 1 Y 2 E=Z_{1}\cdot Y_{2}
  15. F = a X 1 X 2 F=a\cdot X_{1}\cdot X_{2}
  16. X 3 = A B - C D X_{3}=A\cdot B-C\cdot D
  17. Y 3 = D E - F A Y_{3}=D\cdot E-F\cdot A
  18. Z 3 = F C - B E Z_{3}=F\cdot C-B\cdot E
  19. A = - 1 ; B = - 1 ; C = - 1 ; D = - 1 ; E = 1 ; F = 2 ; A=-1;B=-1;C=-1;D=-1;E=1;F=2;
  20. x 3 = 0 x_{3}=0
  21. y 3 = - 3 y_{3}=-3
  22. z 3 = - 3 z_{3}=-3
  23. D = X 1 3 D=X_{1}^{3}
  24. E = Y 1 3 E=Y_{1}^{3}
  25. F = Z 1 3 F=Z_{1}^{3}
  26. G = a D G=a\cdot D
  27. X 3 = X 1 ( E - F ) X_{3}=X_{1}\cdot(E-F)
  28. Y 3 = Z 1 ( G - E ) Y_{3}=Z_{1}\cdot(G-E)
  29. Z 3 = Y 1 ( F - G ) Z_{3}=Y_{1}\cdot(F-G)
  30. D = 1 ; E = - 1 ; F = 1 ; G = - 4 ; D=1;E=-1;F=1;G=-4;
  31. x 3 = - 2 x_{3}=-2
  32. y 3 = - 3 y_{3}=-3
  33. z 3 = - 5 z_{3}=-5

Twists_of_curves.html

  1. y 2 = x 3 + a 2 x 2 + a 4 x + a 6 . y^{2}=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.\,
  2. d K K 2 d\in K\setminus K^{2}
  3. d 0 d\neq 0
  4. E E
  5. E d E^{d}
  6. d y 2 = x 3 + a 2 x 2 + a 4 x + a 6 . dy^{2}=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.\,
  7. y 2 = x 3 + d a 2 x 2 + d 2 a 4 x + d 3 a 6 . y^{2}=x^{3}+da_{2}x^{2}+d^{2}a_{4}x+d^{3}a_{6}.\,
  8. E E
  9. E d E^{d}
  10. K K
  11. K ( d ) K(\sqrt{d})
  12. y 2 + a 1 x y + a 3 y = x 3 + a 2 x 2 + a 4 x + a 6 . y^{2}+a_{1}xy+a_{3}y=x^{3}+a_{2}x^{2}+a_{4}x+a_{6}.\,
  13. d K d\in K
  14. X 2 + X + d X^{2}+X+d
  15. y 2 + a 1 x y + a 3 y = x 3 + ( a 2 + d a 1 2 ) x 2 + a 4 x + a 6 + d a 3 2 . y^{2}+a_{1}xy+a_{3}y=x^{3}+(a_{2}+da_{1}^{2})x^{2}+a_{4}x+a_{6}+da_{3}^{2}.\,
  16. E E
  17. E d E^{d}
  18. K K
  19. K [ X ] / ( X 2 + X + d ) K[X]/(X^{2}+X+d)
  20. K K
  21. q q
  22. x x
  23. y y
  24. ( x , y ) (x,y)
  25. E E
  26. E d E^{d}
  27. y y
  28. 2 2
  29. | E ( K ) | + | E d ( K ) | = 2 q + 2 |E(K)|+|E^{d}(K)|=2q+2
  30. t E d = - t E t_{E^{d}}=-t_{E}
  31. t E t_{E}
  32. K K

Two-dimensional_correlation_analysis.html

  1. ϕ ( ν 1 , ν 2 ) = 1 n - 1 y T ( ν 1 ) . y ( ν 2 ) \phi(\nu_{1},\nu_{2})=\frac{1}{n-1}y^{T}(\nu_{1}).y(\nu_{2})
  2. ψ ( ν 1 , ν 2 ) = 1 n - 1 y T ( ν 1 ) . N . y ( ν 2 ) \psi(\nu_{1},\nu_{2})=\frac{1}{n-1}y^{T}(\nu_{1}).N.y(\nu_{2})
  3. 1 π ( k - j ) \frac{1}{\pi(k-j)}

Two-electron_atom.html

  1. E ψ = - 2 [ 1 2 μ ( 1 2 + 2 2 ) + 1 M 1 2 ] ψ + e 2 4 π ϵ 0 [ 1 r 12 - Z ( 1 r 1 + 1 r 2 ) ] ψ E\psi=-\hbar^{2}\left[\frac{1}{2\mu}\left(\nabla_{1}^{2}+\nabla_{2}^{2}\right)% +\frac{1}{M}\nabla_{1}\cdot\nabla_{2}\right]\psi+\frac{e^{2}}{4\pi\epsilon_{0}% }\left[\frac{1}{r_{12}}-Z\left(\frac{1}{r_{1}}+\frac{1}{r_{2}}\right)\right]\psi
  2. | r 12 | = | r 2 - r 1 | |{r}_{12}|=|{r}_{2}-{r}_{1}|\,\!
  3. μ = m e M m e + M \mu=\frac{m_{e}M}{m_{e}+M}\,\!
  4. 1 M 1 2 \frac{1}{M}\nabla_{1}\cdot\nabla_{2}\,\!
  5. ψ = ψ ( r 1 , r 2 ) \psi=\psi({r}_{1},{r}_{2})

Type-1.5_superconductor.html

  1. λ \lambda
  2. ξ \xi
  3. ξ > λ \xi>\lambda
  4. ξ < λ \xi<\lambda
  5. 2 1 / 2 2^{1/2}
  6. ξ > 2 λ \xi>\sqrt{2}\lambda
  7. ξ < 2 λ \xi<\sqrt{2}\lambda
  8. s s
  9. s + i s s+is
  10. U ( 1 ) U(1)
  11. U ( 1 ) × U ( 1 ) U(1)\times U(1)
  12. 2 λ < ξ \sqrt{2}\lambda<\xi
  13. 2 λ > ξ \sqrt{2}\lambda>\xi
  14. ξ 1 \xi_{1}
  15. ξ 2 \xi_{2}
  16. ξ 1 < 2 λ < ξ 2 \xi_{1}<\sqrt{2}\lambda<\xi_{2}
  17. A A
  18. F = i , j = 1 , 2 1 2 m | ( - i e A ) ψ i | 2 + α i | ψ i | 2 + β i | ψ i | 4 + 1 2 ( × A ) 2 F=\sum_{i,j=1,2}\frac{1}{2m}|(\nabla-ieA)\psi_{i}|^{2}+\alpha_{i}|\psi_{i}|^{2% }+\beta_{i}|\psi_{i}|^{4}+\frac{1}{2}(\nabla\times A)^{2}
  19. ψ i = | ψ i | e i ϕ i , i = 1 , 2 \psi_{i}=|\psi_{i}|e^{i\phi_{i}},i=1,2
  20. A A
  21. λ = 1 e | ψ 1 | 2 + | ψ 2 | 2 \lambda=\frac{1}{e\sqrt{|\psi_{1}|^{2}+|\psi_{2}|^{2}}}
  22. ξ 1 = 1 2 α 1 , ξ 2 = 1 2 α 2 \xi_{1}=\frac{1}{\sqrt{2\alpha_{1}}},\xi_{2}=\frac{1}{\sqrt{2\alpha_{2}}}
  23. A A
  24. ξ 1 > λ > ξ 2 \xi_{1}>\lambda>\xi_{2}
  25. F = i , j = 1 , 2 1 2 m | ( - i e A ) ψ i | 2 + α i | ψ i | 2 + β i | ψ i | 4 - η ( ψ 1 ψ 2 * + ψ 1 * ψ 2 ) + γ [ ( - i e A ) ψ 1 ( + i e A ) ψ 2 * + ( + i e A ) ψ 1 * ( - i e A ) ψ 2 ] + ν | ψ 1 | 2 | ψ 2 | 2 + 1 2 ( × A ) 2 F=\sum_{i,j=1,2}\frac{1}{2m}|(\nabla-ieA)\psi_{i}|^{2}+\alpha_{i}|\psi_{i}|^{2% }+\beta_{i}|\psi_{i}|^{4}-\eta(\psi_{1}\psi_{2}^{*}+\psi_{1}^{*}\psi_{2})+% \gamma[(\nabla-ieA)\psi_{1}\cdot(\nabla+ieA)\psi_{2}^{*}+(\nabla+ieA)\psi_{1}^% {*}\cdot(\nabla-ieA)\psi_{2}]+\nu|\psi_{1}|^{2}|\psi_{2}|^{2}+\frac{1}{2}(% \nabla\times A)^{2}
  26. ψ i = | ψ i | e i ϕ i , i = 1 , 2 \psi_{i}=|\psi_{i}|e^{i\phi_{i}},i=1,2
  27. η 0 , γ 0 \eta\neq 0,\gamma\neq 0
  28. η 0 , γ 0 , ν 0 \eta\neq 0,\gamma\neq 0,\nu\neq 0
  29. ξ ~ 1 ( α 1 , β 1 , α 2 , β 2 , η , γ , ν ) , ξ ~ 2 ( α 1 , β 1 , α 2 , β 2 , η , γ , ν ) \tilde{\xi}_{1}(\alpha_{1},\beta_{1},\alpha_{2},\beta_{2},\eta,\gamma,\nu),% \tilde{\xi}_{2}(\alpha_{1},\beta_{1},\alpha_{2},\beta_{2},\eta,\gamma,\nu)

Type_constructor.html

  1. * *

Tyrolean_Zugspitze_Cable_Car.html

  1. \mathrm{\nearrow}

Udwadia–Kalaba_equation.html

  1. 𝐪 := [ q 1 , q 2 , , q n ] T . \mathbf{q}:=[q_{1},q_{2},\ldots,q_{n}]^{\mathrm{T}}.
  2. q ˙ i = d q i d t . \dot{q}_{i}=\frac{dq_{i}}{dt}\,.
  3. 𝐪 ˙ ( 0 ) \dot{\mathbf{q}}(0)
  4. 𝐪 ˙ ( 0 ) \dot{\mathbf{q}}(0)
  5. ( 𝐌 > 0 ) (\mathbf{M}>0)
  6. ( 𝐌 0 ) (\mathbf{M}\geq 0)
  7. 𝐀 ( q , q ˙ , t ) 𝐪 ¨ = 𝐛 ( q , q ˙ , t ) , \mathbf{A}(q,\dot{q},t)\ddot{\mathbf{q}}=\mathbf{b}(q,\dot{q},t),
  8. φ ( q , t ) = 0 \varphi(q,t)=0
  9. ψ ( q , q ˙ , t ) = 0 \psi(q,\dot{q},t)=0
  10. q ˙ ( t ) \dot{q}(t)
  11. 𝐀 q ¨ = 𝐛 \mathbf{A}\ddot{q}=\mathbf{b}
  12. 𝐌 𝐪 ¨ = 𝐐 + 𝐌 1 / 2 ( 𝐀𝐌 - 1 / 2 ) + ( 𝐛 - 𝐀𝐌 - 1 𝐐 ) , \mathbf{M}\ddot{\mathbf{q}}=\mathbf{Q}+\mathbf{M}^{1/2}\left(\mathbf{A}\mathbf% {M}^{-1/2}\right)^{+}(\mathbf{b}-\mathbf{A}\mathbf{M}^{-1}\mathbf{Q}),
  13. 𝐀𝐌 - 1 / 2 \mathbf{A}\mathbf{M}^{-1/2}
  14. 𝐐 c = 𝐌 1 / 2 ( 𝐀𝐌 - 1 / 2 ) + ( 𝐛 - 𝐀𝐌 - 1 𝐐 ) , \mathbf{Q}_{c}=\mathbf{M}^{1/2}\left(\mathbf{A}\mathbf{M}^{-1/2}\right)^{+}(% \mathbf{b}-\mathbf{A}\mathbf{M}^{-1}\mathbf{Q}),
  15. 𝐪 ¨ = 𝐌 - 1 𝐐 + 𝐌 - 1 / 2 ( 𝐀𝐌 - 1 / 2 ) + ( 𝐛 - 𝐀𝐌 - 1 𝐐 ) . \ddot{\mathbf{q}}=\mathbf{M}^{-1}\mathbf{Q}+\mathbf{M}^{-1/2}\left(\mathbf{A}% \mathbf{M}^{-1/2}\right)^{+}(\mathbf{b}-\mathbf{A}\mathbf{M}^{-1}\mathbf{Q}).
  16. ( 𝐌 0 ) (\mathbf{M}\geq 0)
  17. 𝐌 ^ = [ 𝐌 𝐀 ] \hat{\mathbf{M}}=\left[\begin{array}[]{c}\mathbf{M}\\ \mathbf{A}\end{array}\right]
  18. 𝐌 ^ \hat{\mathbf{M}}
  19. 𝐌 𝐀 𝐪 ¨ := ( 𝐌 + 𝐀 + 𝐀 ) 𝐪 ¨ = 𝐐 + 𝐀 + 𝐛 := 𝐐 𝐛 , \mathbf{M}_{\mathbf{A}}\ddot{\mathbf{q}}:=(\mathbf{M}+\mathbf{A}^{+}\mathbf{A}% )\ddot{\mathbf{q}}=\mathbf{Q}+\mathbf{A}^{+}\mathbf{b}:=\mathbf{Q}_{\mathbf{b}},
  20. 𝐌 𝐀 𝐪 ¨ = 𝐐 𝐛 + 𝐌 𝐀 1 / 2 ( 𝐀𝐌 𝐀 - 1 / 2 ) + ( 𝐛 - 𝐀𝐌 𝐀 - 1 𝐐 𝐛 ) . \mathbf{M}_{\mathbf{A}}\ddot{\mathbf{q}}=\mathbf{Q}_{\mathbf{b}}+\mathbf{M}_{% \mathbf{A}}^{1/2}(\mathbf{A}\mathbf{M}_{\mathbf{A}}^{-1/2})^{+}(\mathbf{b}-% \mathbf{A}\mathbf{M}_{\mathbf{A}}^{-1}\mathbf{Q}_{\mathbf{b}}).
  21. 𝐌 ^ \hat{\mathbf{M}}
  22. 𝐌 𝐀 \mathbf{M}_{\mathbf{A}}
  23. 𝐪 ¨ = 𝐌 𝐀 - 1 𝐐 𝐛 + 𝐌 𝐀 - 1 / 2 ( 𝐀𝐌 𝐀 - 1 / 2 ) + ( 𝐛 - 𝐀𝐌 𝐀 - 1 𝐐 𝐛 ) . \ddot{\mathbf{q}}=\mathbf{M}_{\mathbf{A}}^{-1}\mathbf{Q}_{\mathbf{b}}+\mathbf{% M}_{\mathbf{A}}^{-1/2}(\mathbf{A}\mathbf{M}_{\mathbf{A}}^{-1/2})^{+}(\mathbf{b% }-\mathbf{A}\mathbf{M}_{\mathbf{A}}^{-1}\mathbf{Q}_{\mathbf{b}}).
  24. 𝐐 c = 𝐌 𝐀 1 / 2 ( 𝐀𝐌 𝐀 - 1 / 2 ) + ( 𝐛 - 𝐀𝐌 𝐀 - 1 𝐐 𝐛 ) . \mathbf{Q}_{c}=\mathbf{M}_{\mathbf{A}}^{1/2}(\mathbf{A}\mathbf{M}_{\mathbf{A}}% ^{-1/2})^{+}(\mathbf{b}-\mathbf{A}\mathbf{M}_{\mathbf{A}}^{-1}\mathbf{Q}_{% \mathbf{b}}).
  25. W c ( t ) = 𝐂 T ( q , q ˙ , t ) δ 𝐫 ( t ) W_{c}(t)=\mathbf{C}^{\mathrm{T}}(q,\dot{q},t)\delta\mathbf{r}(t)
  26. 𝐂 ( q , q ˙ , t ) \mathbf{C}(q,\dot{q},t)
  27. 𝐂 ( q , q ˙ , t ) = 0 \mathbf{C}(q,\dot{q},t)=0
  28. 𝐌 𝐪 ¨ = 𝐐 + 𝐌 1 / 2 ( 𝐀𝐌 - 1 / 2 ) + ( 𝐛 - 𝐀𝐌 - 1 𝐐 ) + 𝐌 1 / 2 [ 𝐈 - ( 𝐀𝐌 - 1 / 2 ) + 𝐀𝐌 - 1 / 2 ] 𝐌 1 / 2 𝐂 \mathbf{M}\ddot{\mathbf{q}}=\mathbf{Q}+\mathbf{M}^{1/2}\left(\mathbf{A}\mathbf% {M}^{-1/2}\right)^{+}(\mathbf{b}-\mathbf{A}\mathbf{M}^{-1}\mathbf{Q})+\mathbf{% M}^{1/2}\left[\mathbf{I}-\left(\mathbf{A}\mathbf{M}^{-1/2}\right)^{+}\mathbf{A% }\mathbf{M}^{-1/2}\right]\mathbf{M}^{1/2}\mathbf{C}
  29. r = ϵ x + l r=\epsilon x+l
  30. r = x 2 + y 2 r=\sqrt{x^{2}+y^{2}}
  31. ϵ \epsilon
  32. ( x - r ϵ ) x ¨ + y y ¨ = - ( x y ˙ - y x ˙ ) 2 r 2 (x-r\epsilon)\ddot{x}+y\ddot{y}=-\frac{(x\dot{y}-y\dot{x})^{2}}{r^{2}}
  33. m ( x y ˙ - y x ˙ ) = L m(x\dot{y}-y\dot{x})=L
  34. x y ¨ - y x ¨ = 0 x\ddot{y}-y\ddot{x}=0
  35. ( x - r ϵ y y - x ) ( x ¨ y ¨ ) = ( - L 2 m 2 r 2 0 ) \begin{pmatrix}x-r\epsilon&y\\ y&-x\end{pmatrix}\begin{pmatrix}\ddot{x}\\ \ddot{y}\end{pmatrix}=\begin{pmatrix}-\frac{L^{2}}{m^{2}r^{2}}\\ 0\end{pmatrix}
  36. ( x - r ϵ y y - x ) - 1 = 1 l r ( x y y - ( x - r ϵ ) ) \begin{pmatrix}x-r\epsilon&y\\ y&-x\end{pmatrix}^{-1}=\frac{1}{lr}\begin{pmatrix}x&y\\ y&-(x-r\epsilon)\end{pmatrix}
  37. 𝐅 c = m 𝐀 - 1 𝐛 = m l r ( x y y - ( x - r ϵ ) ) ( - L 2 m 2 r 2 0 ) = - L 2 m l r 2 ( cos θ sin θ ) \mathbf{F}_{c}=m\mathbf{A}^{-1}\mathbf{b}=\frac{m}{lr}\begin{pmatrix}x&y\\ y&-(x-r\epsilon)\end{pmatrix}\begin{pmatrix}-\frac{L^{2}}{m^{2}r^{2}}\\ 0\end{pmatrix}=-\frac{L^{2}}{mlr^{2}}\begin{pmatrix}\cos{\theta}\\ \sin{\theta}\end{pmatrix}
  38. α \alpha
  39. y = x tan α y=x\tan{\alpha}
  40. ( - tan α 1 ) ( x ¨ y ¨ ) = 0 \begin{pmatrix}-\tan{\alpha}&1\end{pmatrix}\begin{pmatrix}\ddot{x}\\ \ddot{y}\end{pmatrix}=0
  41. ( - tan α 1 ) + = cos 2 α ( - tan α 1 ) \begin{pmatrix}-\tan{\alpha}&1\end{pmatrix}^{+}=\cos^{2}{\alpha}\begin{pmatrix% }-\tan{\alpha}\\ 1\end{pmatrix}
  42. 𝐂 = - μ m g cos α sgn y ˙ ( cos α sin α ) \mathbf{C}=-\mu mg\cos{\alpha}\operatorname{sgn}{\dot{y}}\begin{pmatrix}\cos{% \alpha}\\ \sin{\alpha}\end{pmatrix}
  43. 𝐅 e x t = 𝐐 = - m g ( 0 y ) \mathbf{F}_{ext}=\mathbf{Q}=-mg\begin{pmatrix}0\\ y\end{pmatrix}
  44. 𝐅 c , i = - 𝐀 + 𝐀𝐐 = m g cos 2 α ( - tan α 1 ) ( - tan α 1 ) ( 0 y ) = m g ( - sin α cos α cos 2 α ) \mathbf{F}_{c,i}=-\mathbf{A}^{+}\mathbf{A}\mathbf{Q}=mg\cos^{2}{\alpha}\begin{% pmatrix}-\tan{\alpha}\\ 1\end{pmatrix}\begin{pmatrix}-\tan{\alpha}&1\end{pmatrix}\begin{pmatrix}0\\ y\end{pmatrix}=mg\begin{pmatrix}-\sin{\alpha}\cos{\alpha}\\ \cos^{2}{\alpha}\end{pmatrix}
  45. 𝐅 c , n i = ( 𝐈 - 𝐀 + 𝐀 ) 𝐂 = - μ m g cos α sgn y ˙ [ ( 1 0 0 1 ) - cos 2 α ( - tan α 1 ) ( - tan α 1 ) ] = - μ m g cos α sgn y ˙ ( cos 2 α sin α cos α ) \mathbf{F}_{c,ni}=(\mathbf{I}-\mathbf{A}^{+}\mathbf{A})\mathbf{C}=-\mu mg\cos{% \alpha}\operatorname{sgn}{\dot{y}}\left[\begin{pmatrix}1&0\\ 0&1\end{pmatrix}-\cos^{2}{\alpha}\begin{pmatrix}-\tan{\alpha}\\ 1\end{pmatrix}\begin{pmatrix}-\tan{\alpha}&1\end{pmatrix}\right]=-\mu mg\cos{% \alpha}\operatorname{sgn}{\dot{y}}\begin{pmatrix}\cos^{2}{\alpha}\\ \sin{\alpha}\cos{\alpha}\end{pmatrix}
  46. ( x ¨ y ¨ ) = 1 m ( 𝐅 e x t + 𝐅 c , i + 𝐅 c , n i ) = - g ( sin α + μ cos α sgn y ˙ ) ( cos α sin α ) \begin{pmatrix}\ddot{x}\\ \ddot{y}\end{pmatrix}=\frac{1}{m}\left(\mathbf{F}_{ext}+\mathbf{F}_{c,i}+% \mathbf{F}_{c,ni}\right)=-g\left(\sin{\alpha}+\mu\cos{\alpha}\operatorname{sgn% }{\dot{y}}\right)\begin{pmatrix}\cos{\alpha}\\ \sin{\alpha}\end{pmatrix}

Ultrasonic_thickness_measurement.html

  1. l m = c t / 2 l_{m}=ct/2
  2. l m l_{m}
  3. c c
  4. t t

Umkehr_effect.html

  1. N ( θ ) = 100 log 10 I ( λ , θ ) I ( λ , θ ) + K N(\theta)=100\log_{10}\frac{I(\lambda^{\prime},\theta)}{I(\lambda,\theta)}+K

Uncertain_inference.html

  1. P ( d q ) P(d\to q)
  2. d q d\to q
  3. d q d\to q
  4. q = A B C q=A\wedge B\wedge C
  5. P ( D ( A B C ) ) P(D\to(A\wedge B\wedge C))
  6. P ( ( A B C ) | D ) P((A\wedge B\wedge C)|D)

Uncertainty_coefficient.html

  1. H ( X ) = - x P X ( x ) log P X ( x ) , H(X)=-\sum_{x}P_{X}(x)\log P_{X}(x),
  2. H ( X | Y ) = - x , y P X , Y ( x , y ) log P X | Y ( x | y ) . H(X|Y)=-\sum_{x,~{}y}P_{X,Y}(x,~{}y)\log P_{X|Y}(x|y).
  3. U ( X | Y ) = H ( X ) - H ( X | Y ) H ( X ) = I ( X ; Y ) H ( X ) , U(X|Y)=\frac{H(X)-H(X|Y)}{H(X)}=\frac{I(X;Y)}{H(X)},
  4. U ( X , Y ) = H ( X ) U ( X | Y ) + H ( Y ) U ( Y | X ) H ( X ) + H ( Y ) = 2 [ H ( X ) + H ( Y ) - H ( X , Y ) H ( X ) + H ( Y ) ] . \begin{aligned}\displaystyle U(X,~{}Y)&\displaystyle=\frac{H(X)U(X|Y)+H(Y)U(Y|% X)}{H(X)+H(Y)}\\ &\displaystyle=2\left[\frac{H(X)+H(Y)-H(X,~{}Y)}{H(X)+H(Y)}\right].\end{aligned}

Uncertainty_theory.html

  1. { Γ } = 1 for the universal set Γ \mathcal{M}\{\Gamma\}=1\,\text{ for the universal set }\Gamma
  2. { Λ 1 } { Λ 2 } whenever Λ 1 Λ 2 \mathcal{M}\{\Lambda_{1}\}\leq\mathcal{M}\{\Lambda_{2}\}\,\text{ whenever }% \Lambda_{1}\subset\Lambda_{2}
  3. { Λ } + { Λ c } = 1 for any event Λ \mathcal{M}\{\Lambda\}+\mathcal{M}\{\Lambda^{c}\}=1\,\text{ for any event }\Lambda
  4. { i = 1 Λ i } i = 1 { Λ i } \mathcal{M}\left\{\bigcup_{i=1}^{\infty}\Lambda_{i}\right\}\leq\sum_{i=1}^{% \infty}\mathcal{M}\{\Lambda_{i}\}
  5. ( Γ k , k , k ) (\Gamma_{k},\mathcal{L}_{k},\mathcal{M}_{k})
  6. k = 1 , 2 , , n k=1,2,\cdots,n
  7. \mathcal{M}
  8. { i = 1 n Λ i } = min 1 i n i { Λ i } \mathcal{M}\left\{\prod_{i=1}^{n}\Lambda_{i}\right\}=\underset{1\leq i\leq n}{% \operatorname{min}}\mathcal{M}_{i}\{\Lambda_{i}\}
  9. ( Γ , L , M ) (\Gamma,L,M)
  10. { ξ B } = { γ Γ | ξ ( γ ) B } \{\xi\in B\}=\{\gamma\in\Gamma|\xi(\gamma)\in B\}
  11. Φ ( x ) : R [ 0 , 1 ] \Phi(x):R\rightarrow[0,1]
  12. Φ ( x ) = M { ξ x } \Phi(x)=M\{\xi\leq x\}
  13. Φ ( x ) : R [ 0 , 1 ] \Phi(x):R\rightarrow[0,1]
  14. Φ ( x ) 0 \Phi(x)\equiv 0
  15. Φ ( x ) 1 \Phi(x)\equiv 1
  16. ξ 1 , ξ 2 , , ξ m \xi_{1},\xi_{2},\ldots,\xi_{m}
  17. M { i = 1 m ( ξ B i ) } = min M 1 i m { ξ i B i } M\{\cap_{i=1}^{m}(\xi\in B_{i})\}=\mbox{min}~{}_{1\leq i\leq m}M\{\xi_{i}\in B% _{i}\}
  18. B 1 , B 2 , , B m B_{1},B_{2},\ldots,B_{m}
  19. ξ 1 , ξ 2 , , ξ m \xi_{1},\xi_{2},\ldots,\xi_{m}
  20. M { i = 1 m ( ξ B i ) } = max M 1 i m { ξ i B i } M\{\cup_{i=1}^{m}(\xi\in B_{i})\}=\mbox{max}~{}_{1\leq i\leq m}M\{\xi_{i}\in B% _{i}\}
  21. B 1 , B 2 , , B m B_{1},B_{2},\ldots,B_{m}
  22. ξ 1 , ξ 2 , , ξ m \xi_{1},\xi_{2},\ldots,\xi_{m}
  23. f 1 , f 2 , , f m f_{1},f_{2},\ldots,f_{m}
  24. f 1 ( ξ 1 ) , f 2 ( ξ 2 ) , , f m ( ξ m ) f_{1}(\xi_{1}),f_{2}(\xi_{2}),\ldots,f_{m}(\xi_{m})
  25. Φ i \Phi_{i}
  26. ξ i , i = 1 , 2 , , m \xi_{i},\quad i=1,2,\ldots,m
  27. Φ \Phi
  28. ( ξ 1 , ξ 2 , , ξ m ) (\xi_{1},\xi_{2},\ldots,\xi_{m})
  29. ξ 1 , ξ 2 , , ξ m \xi_{1},\xi_{2},\ldots,\xi_{m}
  30. Φ ( x 1 , x 2 , , x m ) = min Φ i 1 i m ( x i ) \Phi(x_{1},x_{2},\ldots,x_{m})=\mbox{min}~{}_{1\leq i\leq m}\Phi_{i}(x_{i})
  31. x 1 , x 2 , , x m x_{1},x_{2},\ldots,x_{m}
  32. ξ 1 , ξ 2 , , ξ m \xi_{1},\xi_{2},\ldots,\xi_{m}
  33. f : R n R f:R^{n}\rightarrow R
  34. ξ = f ( ξ 1 , ξ 2 , , ξ m ) \xi=f(\xi_{1},\xi_{2},\ldots,\xi_{m})
  35. { ξ B } = { sup f ( B 1 , B 2 , , B n ) B min 1 k n k { ξ k B k } , if sup f ( B 1 , B 2 , , B n ) B min 1 k n k { ξ k B k } > 0.5 1 - sup f ( B 1 , B 2 , , B n ) B c min 1 k n k { ξ k B k } , if sup f ( B 1 , B 2 , , B n ) B c min 1 k n k { ξ k B k } > 0.5 0.5 , otherwise \mathcal{M}\{\xi\in B\}=\begin{cases}\underset{f(B_{1},B_{2},\cdots,B_{n})% \subset B}{\operatorname{sup}}\;\underset{1\leq k\leq n}{\operatorname{min}}% \mathcal{M}_{k}\{\xi_{k}\in B_{k}\},&\,\text{if }\underset{f(B_{1},B_{2},% \cdots,B_{n})\subset B}{\operatorname{sup}}\;\underset{1\leq k\leq n}{% \operatorname{min}}\mathcal{M}_{k}\{\xi_{k}\in B_{k}\}>0.5\\ 1-\underset{f(B_{1},B_{2},\cdots,B_{n})\subset B^{c}}{\operatorname{sup}}\;% \underset{1\leq k\leq n}{\operatorname{min}}\mathcal{M}_{k}\{\xi_{k}\in B_{k}% \},&\,\text{if }\underset{f(B_{1},B_{2},\cdots,B_{n})\subset B^{c}}{% \operatorname{sup}}\;\underset{1\leq k\leq n}{\operatorname{min}}\mathcal{M}_{% k}\{\xi_{k}\in B_{k}\}>0.5\\ 0.5,&\,\text{otherwise}\end{cases}
  36. B , B 1 , B 2 , , B m B,B_{1},B_{2},\ldots,B_{m}
  37. f ( B 1 , B 2 , , B m ) B f(B_{1},B_{2},\ldots,B_{m})\subset B
  38. f ( x 1 , x 2 , , x m ) B f(x_{1},x_{2},\ldots,x_{m})\in B
  39. x 1 B 1 , x 2 B 2 , , x m B m x_{1}\in B_{1},x_{2}\in B_{2},\ldots,x_{m}\in B_{m}
  40. ξ \xi
  41. ξ \xi
  42. E [ ξ ] = 0 + M { ξ r } d r - - 0 M { ξ r } d r E[\xi]=\int_{0}^{+\infty}M\{\xi\geq r\}dr-\int_{-\infty}^{0}M\{\xi\leq r\}dr
  43. ξ \xi
  44. Φ \Phi
  45. E [ ξ ] = 0 + ( 1 - Φ ( x ) ) d x - - 0 Φ ( x ) d x E[\xi]=\int_{0}^{+\infty}(1-\Phi(x))dx-\int_{-\infty}^{0}\Phi(x)dx
  46. ξ \xi
  47. Φ \Phi
  48. E [ ξ ] = 0 1 Φ - 1 ( α ) d α E[\xi]=\int_{0}^{1}\Phi^{-1}(\alpha)d\alpha
  49. ξ \xi
  50. η \eta
  51. a a
  52. b b
  53. E [ a ξ + b η ] = a E [ ξ ] + b [ η ] E[a\xi+b\eta]=aE[\xi]+b[\eta]
  54. ξ \xi
  55. e e
  56. ξ \xi
  57. V [ ξ ] = E [ ( ξ - e ) 2 ] V[\xi]=E[(\xi-e)^{2}]
  58. ξ \xi
  59. a a
  60. b b
  61. V [ a ξ + b ] = a 2 V [ ξ ] V[a\xi+b]=a^{2}V[\xi]
  62. ξ \xi
  63. α ( 0 , 1 ] \alpha\in(0,1]
  64. ξ s u p ( α ) = sup { r | M { ξ r } α } \xi_{sup}(\alpha)=\mbox{sup}~{}\{r|M\{\xi\geq r\}\geq\alpha\}
  65. ξ \xi
  66. ξ i n f ( α ) = inf { r | M { ξ r } α } \xi_{inf}(\alpha)=\mbox{inf}~{}\{r|M\{\xi\leq r\}\geq\alpha\}
  67. ξ \xi
  68. ξ \xi
  69. Φ \Phi
  70. ξ s u p ( α ) = Φ - 1 ( 1 - α ) \xi_{sup}(\alpha)=\Phi^{-1}(1-\alpha)
  71. ξ i n f ( α ) = Φ - 1 ( α ) \xi_{inf}(\alpha)=\Phi^{-1}(\alpha)
  72. ξ \xi
  73. α ( 0 , 1 ] \alpha\in(0,1]
  74. α > 0.5 \alpha>0.5
  75. ξ i n f ( α ) ξ s u p ( α ) \xi_{inf}(\alpha)\geq\xi_{sup}(\alpha)
  76. α 0.5 \alpha\leq 0.5
  77. ξ i n f ( α ) ξ s u p ( α ) \xi_{inf}(\alpha)\leq\xi_{sup}(\alpha)
  78. ξ \xi
  79. η \eta
  80. α ( 0 , 1 ] \alpha\in(0,1]
  81. ( ξ + η ) s u p ( α ) = ξ s u p ( α ) + η s u p α (\xi+\eta)_{sup}(\alpha)=\xi_{sup}(\alpha)+\eta_{sup}{\alpha}
  82. ( ξ + η ) i n f ( α ) = ξ i n f ( α ) + η i n f α (\xi+\eta)_{inf}(\alpha)=\xi_{inf}(\alpha)+\eta_{inf}{\alpha}
  83. ( ξ η ) s u p ( α ) = ξ s u p ( α ) η s u p α (\xi\vee\eta)_{sup}(\alpha)=\xi_{sup}(\alpha)\vee\eta_{sup}{\alpha}
  84. ( ξ η ) i n f ( α ) = ξ i n f ( α ) η i n f α (\xi\vee\eta)_{inf}(\alpha)=\xi_{inf}(\alpha)\vee\eta_{inf}{\alpha}
  85. ( ξ η ) s u p ( α ) = ξ s u p ( α ) η s u p α (\xi\wedge\eta)_{sup}(\alpha)=\xi_{sup}(\alpha)\wedge\eta_{sup}{\alpha}
  86. ( ξ η ) i n f ( α ) = ξ i n f ( α ) η i n f α (\xi\wedge\eta)_{inf}(\alpha)=\xi_{inf}(\alpha)\wedge\eta_{inf}{\alpha}
  87. ξ \xi
  88. Φ \Phi
  89. H [ ξ ] = - + S ( Φ ( x ) ) d x H[\xi]=\int_{-\infty}^{+\infty}S(\Phi(x))dx
  90. S ( x ) = - t ln ( t ) - ( 1 - t ) ln ( 1 - t ) S(x)=-t\mbox{ln}~{}(t)-(1-t)\mbox{ln}~{}(1-t)
  91. ξ \xi
  92. Φ \Phi
  93. H [ ξ ] = 0 1 Φ - 1 ( α ) ln α 1 - α d α H[\xi]=\int_{0}^{1}\Phi^{-1}(\alpha)\mbox{ln}~{}\frac{\alpha}{1-\alpha}d\alpha
  94. ξ \xi
  95. η \eta
  96. a a
  97. b b
  98. H [ a ξ + b η ] = | a | E [ ξ ] + | b | E [ η ] H[a\xi+b\eta]=|a|E[\xi]+|b|E[\eta]
  99. ξ \xi
  100. e e
  101. σ 2 \sigma^{2}
  102. H [ ξ ] π σ 3 H[\xi]\leq\frac{\pi\sigma}{\sqrt{3}}
  103. ξ \xi
  104. t > 0 t>0
  105. p > 0 p>0
  106. M { | ξ | t } E [ | ξ | p ] t p M\{|\xi|\geq t\}\leq\frac{E[|\xi|^{p}]}{t^{p}}
  107. ξ \xi
  108. V [ ξ ] V[\xi]
  109. t > 0 t>0
  110. M { | ξ - E [ ξ ] | t } V [ ξ ] t 2 M\{|\xi-E[\xi]|\geq t\}\leq\frac{V[\xi]}{t^{2}}
  111. p p
  112. q q
  113. 1 / p + 1 / q = 1 1/p+1/q=1
  114. ξ \xi
  115. η \eta
  116. E [ | ξ | p ] < E[|\xi|^{p}]<\infty
  117. E [ | η | q ] < E[|\eta|^{q}]<\infty
  118. E [ | ξ η | ] E [ | ξ | p ] p E [ η | p ] p E[|\xi\eta|]\leq\sqrt[p]{E[|\xi|^{p}]}\sqrt[p]{E[\eta|^{p}]}
  119. p p
  120. p 1 p\leq 1
  121. ξ \xi
  122. η \eta
  123. E [ | ξ | p ] < E[|\xi|^{p}]<\infty
  124. E [ | η | q ] < E[|\eta|^{q}]<\infty
  125. E [ | ξ + η | p ] p E [ | ξ | p ] p + E [ η | p ] p \sqrt[p]{E[|\xi+\eta|^{p}]}\leq\sqrt[p]{E[|\xi|^{p}]}+\sqrt[p]{E[\eta|^{p}]}
  126. ξ , ξ 1 , ξ 2 , \xi,\xi_{1},\xi_{2},\ldots
  127. ( Γ , L , M ) (\Gamma,L,M)
  128. { ξ i } \{\xi_{i}\}
  129. ξ \xi
  130. Λ \Lambda
  131. M { Λ } = 1 M\{\Lambda\}=1
  132. lim | i ξ i ( γ ) - ξ ( γ ) | = 0 \mbox{lim}~{}_{i\rightarrow\infty}|\xi_{i}(\gamma)-\xi(\gamma)|=0
  133. γ Λ \gamma\in\Lambda
  134. ξ i ξ \xi_{i}\rightarrow\xi
  135. ξ , ξ 1 , ξ 2 , \xi,\xi_{1},\xi_{2},\ldots
  136. { ξ i } \{\xi_{i}\}
  137. ξ \xi
  138. lim M i { | ξ i - ξ | ε } = 0 \mbox{lim}~{}_{i\rightarrow\infty}M\{|\xi_{i}-\xi|\leq\varepsilon\}=0
  139. ε > 0 \varepsilon>0
  140. ξ , ξ 1 , ξ 2 , \xi,\xi_{1},\xi_{2},\ldots
  141. { ξ i } \{\xi_{i}\}
  142. ξ \xi
  143. lim E i [ | ξ i - ξ | ] = 0 \mbox{lim}~{}_{i\rightarrow\infty}E[|\xi_{i}-\xi|]=0
  144. Φ , ϕ 1 , Φ 2 , \Phi,\phi_{1},\Phi_{2},\ldots
  145. ξ , ξ 1 , ξ 2 , \xi,\xi_{1},\xi_{2},\ldots
  146. { ξ i } \{\xi_{i}\}
  147. ξ \xi
  148. Φ i Φ \Phi_{i}\rightarrow\Phi
  149. Φ \Phi
  150. \Rightarrow
  151. \Rightarrow
  152. \nLeftrightarrow
  153. \nLeftrightarrow
  154. ( Γ , L , M ) (\Gamma,L,M)
  155. A , B L A,B\in L
  156. { A | B } = { { A B } { B } , if { A B } { B } < 0.5 1 - { A c B } { B } , if { A c B } { B } < 0.5 0.5 , otherwise \mathcal{M}\{A|B\}=\begin{cases}\displaystyle\frac{\mathcal{M}\{A\cap B\}}{% \mathcal{M}\{B\}},&\displaystyle\,\text{if }\frac{\mathcal{M}\{A\cap B\}}{% \mathcal{M}\{B\}}<0.5\\ \displaystyle 1-\frac{\mathcal{M}\{A^{c}\cap B\}}{\mathcal{M}\{B\}},&% \displaystyle\,\text{if }\frac{\mathcal{M}\{A^{c}\cap B\}}{\mathcal{M}\{B\}}<0% .5\\ 0.5,&\,\text{otherwise}\end{cases}
  157. provided that { B } > 0 \,\text{provided that }\mathcal{M}\{B\}>0
  158. ( Γ , L , M ) (\Gamma,L,M)
  159. M { B } > 0 M\{B\}>0
  160. ( Γ , L , M { · | B } ) (\Gamma,L,M\{\mbox{·}~{}|B\})
  161. ξ \xi
  162. ( Γ , L , M ) (\Gamma,L,M)
  163. ξ \xi
  164. ξ | B \xi|_{B}
  165. ( Γ , L , M { · | B } ) (\Gamma,L,M\{\mbox{·}~{}|_{B}\})
  166. ξ | B ( γ ) = ξ ( γ ) , γ Γ \xi|_{B}(\gamma)=\xi(\gamma),\forall\gamma\in\Gamma
  167. Φ [ 0 , 1 ] \Phi\rightarrow[0,1]
  168. ξ \xi
  169. Φ ( x | B ) = M { ξ x | B } \Phi(x|B)=M\{\xi\leq x|B\}
  170. M { B } > 0 M\{B\}>0
  171. ξ \xi
  172. Φ ( x ) \Phi(x)
  173. t t
  174. Φ ( t ) < 1 \Phi(t)<1
  175. ξ \xi
  176. ξ > t \xi>t
  177. Φ ( x | ( t , + ) ) = { 0 , if Φ ( x ) Φ ( t ) Φ ( x ) 1 - Φ ( t ) and 0.5 , if Φ ( t ) < Φ ( x ) ( 1 + Φ ( t ) ) / 2 Φ ( x ) - Φ ( t ) 1 - Φ ( t ) , if ( 1 + Φ ( t ) ) / 2 Φ ( x ) \Phi(x|(t,+\infty))=\begin{cases}0,&\,\text{if }\Phi(x)\leq\Phi(t)\\ \displaystyle\frac{\Phi(x)}{1-\Phi(t)}\and 0.5,&\,\text{if }\Phi(t)<\Phi(x)% \leq(1+\Phi(t))/2\\ \displaystyle\frac{\Phi(x)-\Phi(t)}{1-\Phi(t)},&\,\text{if }(1+\Phi(t))/2\leq% \Phi(x)\end{cases}
  178. ξ \xi
  179. Φ ( x ) \Phi(x)
  180. t t
  181. Φ ( t ) > 0 \Phi(t)>0
  182. ξ \xi
  183. ξ t \xi\leq t
  184. Φ ( x | ( - , t ] ) = { Φ ( x ) Φ ( t ) , if Φ ( x ) Φ ( t ) / 2 Φ ( x ) + Φ ( t ) - 1 Φ ( t ) 0.5 , if Φ ( t ) / 2 Φ ( x ) < Φ ( t ) 1 , if Φ ( t ) Φ ( x ) \Phi(x|(-\infty,t])=\begin{cases}\displaystyle\frac{\Phi(x)}{\Phi(t)},&\,\text% {if }\Phi(x)\leq\Phi(t)/2\\ \displaystyle\frac{\Phi(x)+\Phi(t)-1}{\Phi(t)}0.5,&\,\text{if }\Phi(t)/2\leq% \Phi(x)<\Phi(t)\\ 1,&\,\text{if }\Phi(t)\leq\Phi(x)\end{cases}
  185. ξ \xi
  186. ξ \xi
  187. E [ ξ | B ] = 0 + M { ξ r | B } d r - - 0 M { ξ r | B } d r E[\xi|B]=\int_{0}^{+\infty}M\{\xi\geq r|B\}dr-\int_{-\infty}^{0}M\{\xi\leq r|B% \}dr

Undertow_(water_waves).html

  1. M w M_{w}
  2. E k E_{k}
  3. c c
  4. M w = 2 E k c M_{w}=\frac{2E_{k}}{c}
  5. u ¯ = - 2 E k ρ c h , \bar{u}=-\frac{2E_{k}}{\rho ch},
  6. h h
  7. ρ \rho
  8. u ¯ \bar{u}
  9. E k E_{k}
  10. E p E_{p}
  11. E w = E k + E p 2 E k 2 E p , E_{w}=E_{k}+E_{p}\approx 2E_{k}\approx 2E_{p},
  12. E w E_{w}
  13. E p E_{p}
  14. E w 1 8 ρ g H 2 {E_{w}\approx\tfrac{1}{8}\rho gH^{2}}
  15. H H
  16. u ¯ - 1 8 g H 2 c h . \bar{u}\approx-\frac{1}{8}\frac{gH^{2}}{ch}.
  17. H rms 8 σ , H\text{rms}\approx\sqrt{8}\;\sigma,
  18. σ \sigma
  19. E p = 1 2 ρ g σ 2 E_{p}=\tfrac{1}{2}\rho g\sigma^{2}
  20. E w ρ g σ 2 . E_{w}\approx\rho g\sigma^{2}.

Unibranch_local_ring.html

  1. f : X Y f\colon X\to Y
  2. y Y y\in Y
  3. f - 1 ( y ) f^{-1}(y)

Uniform_convergence_(combinatorics).html

  1. H H\,\!
  2. X X\,\!
  3. x = ( x 1 , x 2 , , x m ) x=(x_{1},x_{2},\dots,x_{m})\,\!
  4. x i X x_{i}\in X\,\!
  5. h H h\in H\,\!
  6. x x\,\!
  7. Q x ^ ( h ) = 1 m | { i : 1 i m , h ( x i ) = 1 } | \widehat{Q_{x}}(h)=\frac{1}{m}|\{i:1\leq i\leq m,h(x_{i})=1\}|\,\!
  8. H H\,\!
  9. X X\,\!
  10. P P\,\!
  11. Q P ( h ) = P { y X : h ( y ) = 1 } Q_{P}(h)=P\{y\in X:h(y)=1\}\,\!
  12. H H\,\!
  13. H H\,\!
  14. { 0 , 1 } \{0,1\}\,\!
  15. X X\,\!
  16. P P\,\!
  17. X X\,\!
  18. ϵ > 0 \epsilon>0\,\!
  19. m m\,\!
  20. P m { | Q P ( h ) - Q x ^ ( h ) | ϵ P^{m}\{|Q_{P}(h)-\widehat{Q_{x}}(h)|\geq\epsilon\,\!
  21. h H } 4 Π H ( 2 m ) e - ϵ 2 m 8 . h\in H\}\leq 4\Pi_{H}(2m)e^{-\frac{\epsilon^{2}m}{8}}.\,\!
  22. x X m x\in X^{m}\,\!
  23. Q P ( h ) = P { ( y X : h ( y ) = 1 } Q_{P}(h)=P\{(y\in X:h(y)=1\}\,\!
  24. Q x ^ ( h ) = 1 m | { i : 1 i m , h ( x i ) = 1 } | \widehat{Q_{x}}(h)=\frac{1}{m}|\{i:1\leq i\leq m,h(x_{i})=1\}|\,\!
  25. | x | = m |x|=m\,\!
  26. P m P^{m}\,\!
  27. x x\,\!
  28. m m\,\!
  29. P P\,\!
  30. Π H \Pi_{H}\,\!
  31. { 0 , 1 } \{0,1\}\,\!
  32. H H\,\!
  33. X X\,\!
  34. D X D\subseteq X\,\!
  35. Π H ( D ) = { h D : h H } \Pi_{H}(D)=\{h\cap D:h\in H\}\,\!
  36. m m\,\!
  37. Π H ( m ) \Pi_{H}(m)\,\!
  38. Π H ( m ) = m a x | { h D : | D | = m , h H } | \Pi_{H}(m)=max|\{h\cap D:|D|=m,h\in H\}|\,\!
  39. H H\,\!
  40. X X\,\!
  41. Π h ( m ) \Pi_{h}(m)\,\!
  42. Π H ( m ) ( e m d ) d \Pi_{H}(m)\leq\left(\frac{em}{d}\right)^{d}\,\!
  43. d d\,\!
  44. H H\,\!
  45. Π H ( m ) m d \Pi_{H}(m)\leq m^{d}\,\!
  46. | Q P ( h ) - Q ^ x ( h ) | ϵ |Q_{P}(h)-\widehat{Q}_{x}(h)|\geq\epsilon\,\!
  47. | Q ^ r ( h ) - Q ^ s ( h ) | ϵ / 2 |\widehat{Q}_{r}(h)-\widehat{Q}_{s}(h)|\geq\epsilon/2\,\!
  48. r r\,\!
  49. s s\,\!
  50. m m\,\!
  51. P P\,\!
  52. r r\,\!
  53. m m\,\!
  54. s s\,\!
  55. Q P ( h ) Q_{P}(h)\,\!
  56. r r\,\!
  57. s s\,\!
  58. r r\,\!
  59. s s\,\!
  60. | Q ^ r ( h ) - Q ^ s ( h ) | ϵ / 2 |\widehat{Q}_{r}(h)-\widehat{Q}_{s}(h)|\geq\epsilon/2\,\!
  61. h H h\in H\,\!
  62. x = r | | s x=r||s\,\!
  63. σ ( x ) \sigma(x)\,\!
  64. x i x_{i}\,\!
  65. x m + i x_{m+i}\,\!
  66. 1 , 2 , , m {1,2,...,m}\,\!
  67. r | | s r||s\,\!
  68. r r\,\!
  69. s s\,\!
  70. H H\,\!
  71. H H\,\!
  72. V = { x X m : | Q P ( h ) - Q x ^ ( h ) | ϵ V=\{x\in X^{m}:|Q_{P}(h)-\widehat{Q_{x}}(h)|\geq\epsilon\,\!
  73. h H } h\in H\}\,\!
  74. R = { ( r , s ) X m × X m : | Q r ^ ( h ) - Q s ^ ( h ) | ϵ / 2 R=\{(r,s)\in X^{m}\times X^{m}:|\widehat{Q_{r}}(h)-\widehat{Q_{s}}(h)|\geq% \epsilon/2\,\!
  75. h H } h\in H\}\,\!
  76. m 2 ϵ 2 m\geq\frac{2}{\epsilon^{2}}\,\!
  77. P m ( V ) 2 P 2 m ( R ) P^{m}(V)\leq 2P^{2m}(R)\,\!
  78. | Q P ( h ) - Q r ^ ( h ) | ϵ |Q_{P}(h)-\widehat{Q_{r}}(h)|\geq\epsilon\,\!
  79. | Q P ( h ) - Q s ^ ( h ) | ϵ / 2 |Q_{P}(h)-\widehat{Q_{s}}(h)|\leq\epsilon/2\,\!
  80. | Q r ^ ( h ) - Q s ^ ( h ) | ϵ / 2 |\widehat{Q_{r}}(h)-\widehat{Q_{s}}(h)|\geq\epsilon/2\,\!
  81. P 2 m ( R ) P 2 m { h H , | Q P ( h ) - Q r ^ ( h ) | ϵ P^{2m}(R)\geq P^{2m}\{\exists h\in H,|Q_{P}(h)-\widehat{Q_{r}}(h)|\geq\epsilon\,\!
  82. | Q P ( h ) - Q s ^ ( h ) | ϵ / 2 } |Q_{P}(h)-\widehat{Q_{s}}(h)|\leq\epsilon/2\}\,\!
  83. = V P m { s : h H , | Q P ( h ) - Q r ^ ( h ) | ϵ =\int_{V}P^{m}\{s:\exists h\in H,|Q_{P}(h)-\widehat{Q_{r}}(h)|\geq\epsilon\,\!
  84. | Q P ( h ) - Q s ^ ( h ) | ϵ / 2 } d P m ( r ) = A |Q_{P}(h)-\widehat{Q_{s}}(h)|\leq\epsilon/2\}dP^{m}(r)=A\,\!
  85. r r\,\!
  86. s s\,\!
  87. r V r\in V\,\!
  88. h H h\in H\,\!
  89. | Q P ( h ) - Q r ^ ( h ) | ϵ |Q_{P}(h)-\widehat{Q_{r}}(h)|\geq\epsilon\,\!
  90. h h\,\!
  91. P m { | Q P ( h ) - Q s ^ ( h ) | ϵ 2 } 1 2 P^{m}\{|Q_{P}(h)-\widehat{Q_{s}}(h)|\leq\frac{\epsilon}{2}\}\geq\frac{1}{2}\,\!
  92. r V r\in V\,\!
  93. A P m ( V ) 2 A\geq\frac{P^{m}(V)}{2}\,\!
  94. P 2 m ( R ) P m ( V ) 2 P^{2m}(R)\geq\frac{P^{m}(V)}{2}\,\!
  95. m Q s ^ ( h ) m\cdot\widehat{Q_{s}}(h)\,\!
  96. m Q P ( h ) m\cdot Q_{P}(h)\,\!
  97. m Q P ( h ) ( 1 - Q P ( h ) ) m\cdot Q_{P}(h)(1-Q_{P}(h))\,\!
  98. P m { | Q P ( h ) - Q s ( h ) ^ | > ϵ 2 } m Q P ( h ) ( 1 - Q P ( h ) ) ( ϵ m / 2 ) 2 1 ϵ 2 m 1 2 P^{m}\{|Q_{P}(h)-\widehat{Q_{s}(h)}|>\frac{\epsilon}{2}\}\leq\frac{m\cdot Q_{P% }(h)(1-Q_{P}(h))}{(\epsilon m/2)^{2}}\leq\frac{1}{\epsilon^{2}m}\leq\frac{1}{2% }\,\!
  99. m m\,\!
  100. x ( 1 - x ) 1 / 4 x(1-x)\leq 1/4\,\!
  101. x x\,\!
  102. Γ m \Gamma_{m}\,\!
  103. { 1 , 2 , 3 , , 2 m } \{1,2,3,\dots,2m\}\,\!
  104. i i\,\!
  105. m + i m+i\,\!
  106. i \forall i\,\!
  107. { 1 , 2 , 3 , , 2 m } \{1,2,3,...,2m\}\,\!
  108. R R\,\!
  109. X 2 m X^{2m}\,\!
  110. P P\,\!
  111. X X\,\!
  112. P 2 m ( R ) = E [ P r [ σ ( x ) R ] ] m a x x X 2 m ( P r [ σ ( x ) R ] ) , P^{2m}(R)=E[Pr[\sigma(x)\in R]]\leq max_{x\in X^{2m}}(Pr[\sigma(x)\in R]),\,\!
  113. x x\,\!
  114. P 2 m P^{2m}\,\!
  115. σ \sigma\,\!
  116. Γ m \Gamma_{m}\,\!
  117. σ Γ m , \sigma\in\Gamma_{m},\,\!
  118. P 2 m ( R ) = P 2 m { x : σ ( x ) R } P^{2m}(R)=P^{2m}\{x:\sigma(x)\in R\}\,\!
  119. P 2 m P^{2m}\,\!
  120. P 2 m ( R ) = X 2 m 1 R ( x ) d P 2 m ( x ) \therefore P^{2m}(R)=\int_{X^{2m}}1_{R}(x)dP^{2m}(x)\,\!
  121. = 1 | Γ m | σ Γ m X 2 m 1 R ( σ ( x ) ) d P 2 m ( x ) =\frac{1}{|\Gamma_{m}|}\sum_{\sigma\in\Gamma_{m}}\int_{X^{2m}}1_{R}(\sigma(x))% dP^{2m}(x)\,\!
  122. = X 2 m 1 | Γ m | σ Γ m 1 R ( σ ( x ) ) d P 2 m ( x ) =\int_{X^{2m}}\frac{1}{|\Gamma_{m}|}\sum_{\sigma\in\Gamma_{m}}1_{R}(\sigma(x))% dP^{2m}(x)\,\!
  123. | Γ m | |\Gamma_{m}|\,\!
  124. = X 2 m P r [ σ ( x ) R ] d P 2 m ( x ) =\int_{X^{2m}}Pr[\sigma(x)\in R]dP^{2m}(x)
  125. m a x x X 2 m ( P r [ σ ( x ) R ] ) \leq max_{x\in X^{2m}}(Pr[\sigma(x)\in R])\,\!
  126. m a x x X 2 m ( P r [ σ ( x ) R ] ) 4 Π H ( 2 m ) e - ϵ 2 m 8 max_{x\in X^{2m}}(Pr[\sigma(x)\in R])\leq 4\Pi_{H}(2m)e^{-\frac{\epsilon^{2}m}% {8}}\,\!
  127. x = ( x 1 , x 2 , , x 2 m ) x=(x_{1},x_{2},...,x_{2m})\,\!
  128. t = | H | x | t=|H|_{x}|\,\!
  129. Π H ( 2 m ) \Pi_{H}(2m)\,\!
  130. h 1 , h 2 , , h t H h_{1},h_{2},...,h_{t}\in H\,\!
  131. h H , i h\in H,\exists i\,\!
  132. 1 1\,\!
  133. t t\,\!
  134. h i ( x k ) = h ( x k ) h_{i}(x_{k})=h(x_{k})\,\!
  135. 1 k 2 m 1\leq k\leq 2m\,\!
  136. σ ( x ) R \sigma(x)\in R\,\!
  137. h h\,\!
  138. H H\,\!
  139. | 1 m | { 1 i m : h ( x σ i ) = 1 } | - 1 m | { m + 1 i 2 m : h ( x σ i ) = 1 } | | ϵ 2 |\frac{1}{m}|\{1\leq i\leq m:h(x_{\sigma_{i}})=1\}|-\frac{1}{m}|\{m+1\leq i% \leq 2m:h(x_{\sigma_{i}})=1\}||\geq\frac{\epsilon}{2}\,\!
  140. w i j = 1 w^{j}_{i}=1\,\!
  141. h j ( x i ) = 1 h_{j}(x_{i})=1\,\!
  142. w i j = 0 w^{j}_{i}=0\,\!
  143. 1 i m 1\leq i\leq m\,\!
  144. 1 j t 1\leq j\leq t\,\!
  145. σ ( x ) R \sigma(x)\in R\,\!
  146. j j\,\!
  147. 1 , , t {1,...,t}\,\!
  148. | 1 m ( i w σ ( i ) j - i w σ ( m + i ) j ) | ϵ 2 |\frac{1}{m}\left(\sum_{i}w^{j}_{\sigma(i)}-\sum_{i}w^{j}_{\sigma(m+i)}\right)% |\geq\frac{\epsilon}{2}\,\!
  149. P r [ σ ( x ) R ] t m a x ( P r [ | 1 m ( i w σ i j - i w σ m + i j ) | ϵ 2 ] ) Pr[\sigma(x)\in R]\leq t\cdot max\left(Pr[|\frac{1}{m}\left(\sum_{i}w^{j}_{% \sigma_{i}}-\sum_{i}w^{j}_{\sigma_{m+i}}\right)|\geq\frac{\epsilon}{2}]\right)
  150. Π H ( 2 m ) m a x ( P r [ | 1 m ( i w σ i j - i w σ m + i j ) | ϵ 2 ] ) \leq\Pi_{H}(2m)\cdot max\left(Pr[|\frac{1}{m}\left(\sum_{i}w^{j}_{\sigma_{i}}-% \sum_{i}w^{j}_{\sigma_{m+i}}\right)|\geq\frac{\epsilon}{2}]\right)\,\!
  151. σ \sigma\,\!
  152. i i\,\!
  153. w σ i j - w σ m + i j w^{j}_{\sigma_{i}}-w^{j}_{\sigma_{m+i}}\,\!
  154. ± | w i j - w m + i j | \pm|w^{j}_{i}-w^{j}_{m+i}|\,\!
  155. P r [ | 1 m ( i ( w σ i j - w σ m + i j ) ) | ϵ 2 ] = P r [ | 1 m ( i | w i j - w m + i j | β i ) | ϵ 2 ] Pr[|\frac{1}{m}\left(\sum_{i}\left(w^{j}_{\sigma_{i}}-w^{j}_{\sigma_{m+i}}% \right)\right)|\geq\frac{\epsilon}{2}]=Pr[|\frac{1}{m}\left(\sum_{i}|w^{j}_{i}% -w^{j}_{m+i}|\beta_{i}\right)|\geq\frac{\epsilon}{2}]\,\!
  156. β i \beta_{i}\,\!
  157. 2 e - m ϵ 2 8 2e^{-\frac{m\epsilon^{2}}{8}}\,\!

Uniform_honeycombs_in_hyperbolic_space.html

  1. B H ¯ 3 {\bar{BH}}_{3}
  2. J ¯ 3 {\bar{J}}_{3}
  3. D H ¯ 3 {\bar{DH}}_{3}
  4. A B ^ 3 {\widehat{AB}}_{3}
  5. K ¯ 3 {\bar{K}}_{3}
  6. A H ^ 3 {\widehat{AH}}_{3}
  7. B B ^ 3 {\widehat{BB}}_{3}
  8. B H ^ 3 {\widehat{BH}}_{3}
  9. H H ^ 3 {\widehat{HH}}_{3}
  10. B H ¯ 3 {\bar{BH}}_{3}
  11. J ¯ 3 {\bar{J}}_{3}
  12. D H ¯ 3 {\bar{DH}}_{3}
  13. A B ^ 3 {\widehat{AB}}_{3}
  14. K ¯ 3 {\bar{K}}_{3}
  15. A H ^ 3 {\widehat{AH}}_{3}
  16. B B ^ 3 {\widehat{BB}}_{3}
  17. B H ^ 3 {\widehat{BH}}_{3}
  18. H H ^ 3 {\widehat{HH}}_{3}

Uniformly_distributed_measure.html

  1. 0 < μ ( 𝐁 r ( x ) ) = μ ( 𝐁 r ( y ) ) < + 0<\mu(\mathbf{B}_{r}(x))=\mu(\mathbf{B}_{r}(y))<+\infty
  2. 𝐁 r ( x ) := { z X | d ( x , z ) < r } . \mathbf{B}_{r}(x):=\{z\in X|d(x,z)<r\}.

Unimolecular_ion_decomposition.html

  1. A B C D + A D + + B C ABCD^{+}\to AD^{+}+BC\,
  2. A B C D + A B + + C D ABCD^{+}\to AB^{+}+CD\,

Unisolvent_point_set.html

  1. X R n X\subset R^{n}
  2. W W
  3. w W w\in W
  4. X X
  5. X X
  6. Π n m \Pi^{m}_{n}
  7. Π n m \Pi^{m}_{n}
  8. X X
  9. R R
  10. R R
  11. Π k \Pi^{k}

Unit_circle.html

  1. x 2 + y 2 = 1. x^{2}+y^{2}=1.
  2. z = e i t = cos ( t ) + i sin ( t ) z=\,\mathrm{e}^{it}\,=\cos(t)+i\sin(t)\,
  3. cos ( t ) = x \cos(t)=x\,\!
  4. sin ( t ) = y . \sin(t)=y.\,\!
  5. cos 2 ( t ) + sin 2 ( t ) = 1. \cos^{2}(t)+\sin^{2}(t)=1.\,\!
  6. cos t = cos ( 2 π k + t ) \cos t=\cos(2\pi k+t)\,\!
  7. sin t = sin ( 2 π k + t ) \sin t=\sin(2\pi k+t)\,\!
  8. \perp
  9. \perp
  10. cos θ + i sin θ \cos\theta+i\sin\theta
  11. f 0 ( x ) = x 2 f_{0}(x)=x^{2}\,