wpmath0000007_13

Størmer's_theorem.html

  1. S = { p 1 e 1 p 2 e 2 p k e k e i { 0 , 1 , 2 , } } S=\left\{p_{1}^{e_{1}}p_{2}^{e_{2}}\cdots p_{k}^{e_{k}}\mid e_{i}\in\{0,1,2,% \ldots\}\right\}
  2. x 2 - 2 q y 2 = 1 x^{2}-2qy^{2}=1

Subderivative.html

  1. f ( x ) - f ( x 0 ) c ( x - x 0 ) f(x)-f(x_{0})\geq c(x-x_{0})
  2. a = lim x x 0 - f ( x ) - f ( x 0 ) x - x 0 a=\lim_{x\to x_{0}^{-}}\frac{f(x)-f(x_{0})}{x-x_{0}}
  3. b = lim x x 0 + f ( x ) - f ( x 0 ) x - x 0 b=\lim_{x\to x_{0}^{+}}\frac{f(x)-f(x_{0})}{x-x_{0}}
  4. x 0 x_{0}
  5. x 0 x_{0}
  6. f ( x ) - f ( x 0 ) v ( x - x 0 ) f(x)-f(x_{0})\geq v\cdot(x-x_{0})
  7. f ( x ) - f ( x 0 ) v * ( x - x 0 ) . f(x)-f(x_{0})\geq v^{*}(x-x_{0}).

Subharmonic_function.html

  1. G G
  2. n {\mathbb{R}}^{n}
  3. φ : G { - } \varphi\colon G\to{\mathbb{R}}\cup\{-\infty\}
  4. φ \varphi
  5. B ( x , r ) ¯ \overline{B(x,r)}
  6. x x
  7. r r
  8. G G
  9. h h
  10. B ( x , r ) ¯ \overline{B(x,r)}
  11. B ( x , r ) B(x,r)
  12. φ ( y ) h ( y ) \varphi(y)\leq h(y)
  13. y y
  14. B ( x , r ) \partial B(x,r)
  15. B ( x , r ) B(x,r)
  16. φ ( y ) h ( y ) \varphi(y)\leq h(y)
  17. y B ( x , r ) . y\in B(x,r).
  18. u u
  19. - u -u
  20. ϕ \phi\,
  21. G G
  22. n {\mathbb{R}}^{n}
  23. ϕ \phi\,
  24. Δ ϕ 0 \Delta\phi\geq 0
  25. G G
  26. Δ \Delta
  27. - -\infty
  28. φ \varphi
  29. G G\subset\mathbb{C}
  30. D ( z , r ) G D(z,r)\subset G
  31. z z
  32. r r
  33. φ ( z ) 1 2 π 0 2 π φ ( z + r e i θ ) d θ . \varphi(z)\leq\frac{1}{2\pi}\int_{0}^{2\pi}\varphi(z+r\mathrm{e}^{i\theta})\,d\theta.
  34. f f
  35. φ ( z ) = log | f ( z ) | \varphi(z)=\log\left|f(z)\right|
  36. φ ( z ) \varphi(z)
  37. f f
  38. ψ α ( z ) = | f ( z ) | α \psi_{\alpha}(z)=\left|f(z)\right|^{\alpha}
  39. G G\subset\mathbb{C}
  40. u u
  41. Ω \Omega
  42. h h
  43. Ω \Omega
  44. h h
  45. u u
  46. Ω \Omega
  47. u u
  48. h h
  49. Ω \Omega
  50. u u
  51. ( M φ ) ( e i θ ) = sup 0 r < 1 φ ( r e i θ ) . (M\varphi)(\mathrm{e}^{\mathrm{i}\theta})=\sup_{0\leq r<1}\varphi(r\mathrm{e}^% {\mathrm{i}\theta}).
  52. 0 φ ( r e i θ ) 1 2 π 0 2 π P r ( θ - t ) φ ( e i t ) d t , r < 1. 0\leq\varphi(r\mathrm{e}^{\mathrm{i}\theta})\leq\frac{1}{2\pi}\int_{0}^{2\pi}P% _{r}\left(\theta-t\right)\varphi\left(\mathrm{e}^{\mathrm{i}t}\right)\,\mathrm% {d}t,\ \ \ r<1.
  53. φ * ( e i θ ) = sup 0 < α π 1 2 α θ - α θ + α φ ( e i t ) d t , \varphi^{*}(\mathrm{e}^{\mathrm{i}\theta})=\sup_{0<\alpha\leq\pi}\frac{1}{2% \alpha}\int_{\theta-\alpha}^{\theta+\alpha}\varphi\left(\mathrm{e}^{\mathrm{i}% t}\right)\,\mathrm{d}t,
  54. 0 2 π ( sup 0 r < 1 | F ( r e i θ ) | ) p d θ C 2 sup 0 r < 1 0 2 π | F ( r e i θ ) | p d θ . \int_{0}^{2\pi}\Bigl(\sup_{0\leq r<1}|F(r\mathrm{e}^{\mathrm{i}\theta})|\Bigr)% ^{p}\,\mathrm{d}\theta\leq C^{2}\,\sup_{0\leq r<1}\int_{0}^{2\pi}|F(r\mathrm{e% }^{\mathrm{i}\theta})|^{p}\,\mathrm{d}\theta.
  55. f : M f:\;M\to{\mathbb{R}}
  56. U M U\subset M
  57. f 1 f f_{1}\geq f
  58. f 1 f f_{1}\geq f
  59. Δ f 0 \Delta f\geq 0
  60. Δ \Delta

Subsonic_and_transonic_wind_tunnel.html

  1. A V = c o n s t a n t d A A = - d V V AV=constant\Rightarrow\frac{dA}{A}=-\frac{dV}{V}
  2. P t o t a l = P s t a t i c + P d y n a m i c = P s + 1 2 ρ V 2 P_{total}=P_{static}+P_{dynamic}=P_{s}+\frac{1}{2}\rho V^{2}
  3. V m 2 = 2 C 2 C 2 - 1 P s e t t l - p m ρ 2 Δ p ρ V_{m}^{2}=2\frac{C^{2}}{C^{2}-1}\frac{P_{settl}-p_{m}}{\rho}\approx 2\frac{% \Delta p}{\rho}
  4. C = A s e t t l A m C=\frac{A_{settl}}{A_{m}}
  5. - d ρ ρ = - 1 a 2 d p ρ = - 1 a 2 - ρ V d V ρ = V a 2 d V -\frac{d\rho}{\rho}=-\frac{1}{a^{2}}\frac{dp}{\rho}=-\frac{1}{a^{2}}\frac{-% \rho VdV}{\rho}=\frac{V}{a^{2}}dV
  6. d A A = ( M 2 - 1 ) d V V \frac{dA}{A}=(M^{2}-1)\frac{dV}{V}
  7. ( A A t h r o a t ) 2 = 1 M 2 ( 2 γ + 1 ( 1 + γ - 1 2 M 2 ) ) γ + 1 γ - 1 \left(\frac{A}{A_{throat}}\right)^{2}=\frac{1}{M^{2}}\left(\frac{2}{\gamma+1}% \left(1+\frac{\gamma-1}{2}M^{2}\right)\right)^{\frac{\gamma+1}{\gamma-1}}
  8. d A d x = 0 \frac{dA}{dx}=0
  9. d A d x > 0 \frac{dA}{dx}>0\Rightarrow
  10. A A t h r o a t \frac{A}{A_{throat}}

Successive_over-relaxation.html

  1. A 𝐱 = 𝐛 A\mathbf{x}=\mathbf{b}
  2. A = [ a 11 a 12 a 1 n a 21 a 22 a 2 n a n 1 a n 2 a n n ] , 𝐱 = [ x 1 x 2 x n ] , 𝐛 = [ b 1 b 2 b n ] . A=\begin{bmatrix}a_{11}&a_{12}&\cdots&a_{1n}\\ a_{21}&a_{22}&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\cdots&a_{nn}\end{bmatrix},\qquad\mathbf{x}=\begin{bmatrix}x_{1}% \\ x_{2}\\ \vdots\\ x_{n}\end{bmatrix},\qquad\mathbf{b}=\begin{bmatrix}b_{1}\\ b_{2}\\ \vdots\\ b_{n}\end{bmatrix}.
  3. A = D + L + U , A=D+L+U,
  4. D = [ a 11 0 0 0 a 22 0 0 0 a n n ] , L = [ 0 0 0 a 21 0 0 a n 1 a n 2 0 ] , U = [ 0 a 12 a 1 n 0 0 a 2 n 0 0 0 ] . D=\begin{bmatrix}a_{11}&0&\cdots&0\\ 0&a_{22}&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&a_{nn}\end{bmatrix},\quad L=\begin{bmatrix}0&0&\cdots&0\\ a_{21}&0&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ a_{n1}&a_{n2}&\cdots&0\end{bmatrix},\quad U=\begin{bmatrix}0&a_{12}&\cdots&a_{% 1n}\\ 0&0&\cdots&a_{2n}\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&0\end{bmatrix}.
  5. ( D + ω L ) 𝐱 = ω 𝐛 - [ ω U + ( ω - 1 ) D ] 𝐱 (D+\omega L)\mathbf{x}=\omega\mathbf{b}-[\omega U+(\omega-1)D]\mathbf{x}
  6. 𝐱 ( k + 1 ) = ( D + ω L ) - 1 ( ω 𝐛 - [ ω U + ( ω - 1 ) D ] 𝐱 ( k ) ) = L w 𝐱 ( k ) + 𝐜 , \mathbf{x}^{(k+1)}=(D+\omega L)^{-1}\big(\omega\mathbf{b}-[\omega U+(\omega-1)% D]\mathbf{x}^{(k)}\big)=L_{w}\mathbf{x}^{(k)}+\mathbf{c},
  7. 𝐱 ( k ) \mathbf{x}^{(k)}
  8. 𝐱 \mathbf{x}
  9. 𝐱 ( k + 1 ) \mathbf{x}^{(k+1)}
  10. 𝐱 \mathbf{x}
  11. x i ( k + 1 ) = ( 1 - ω ) x i ( k ) + ω a i i ( b i - j < i a i j x j ( k + 1 ) - j > i a i j x j ( k ) ) , i = 1 , 2 , , n . x^{(k+1)}_{i}=(1-\omega)x^{(k)}_{i}+\frac{\omega}{a_{ii}}\left(b_{i}-\sum_{j<i% }a_{ij}x^{(k+1)}_{j}-\sum_{j>i}a_{ij}x^{(k)}_{j}\right),\quad i=1,2,\ldots,n.
  12. A A
  13. ρ ( L ω ) < 1 \rho(L_{\omega})<1
  14. 0 < ω < 2 0<\omega<2
  15. A A
  16. b b
  17. ω ω
  18. ϕ \phi
  19. ϕ \phi
  20. i i
  21. n n
  22. σ 0 \sigma\leftarrow 0
  23. j j
  24. n n
  25. j j
  26. i i
  27. σ σ + a i j ϕ j \sigma\leftarrow\sigma+a_{ij}\phi_{j}
  28. j j
  29. ϕ i ( 1 - ω ) ϕ i + ω a i i ( b i - σ ) \phi_{i}\leftarrow(1-\omega)\phi_{i}+\frac{\omega}{a_{ii}}(b_{i}-\sigma)
  30. i i
  31. ( 1 - ω ) ϕ i + ω a i i ( b i - σ ) (1-\omega)\phi_{i}+\frac{\omega}{a_{ii}}(b_{i}-\sigma)
  32. ϕ i + ω ( b i - σ a i i - ϕ i ) \phi_{i}+\omega\left(\frac{b_{i}-\sigma}{a_{ii}}-\phi_{i}\right)
  33. U = L T , U=L^{T},\,
  34. P = ( D ω + L ) 1 2 - ω D - 1 ( D ω + U ) , P=\left(\frac{D}{\omega}+L\right)\frac{1}{2-\omega}D^{-1}\left(\frac{D}{\omega% }+U\right),
  35. 𝐱 k + 1 = 𝐱 k - γ k P - 1 ( A 𝐱 k - 𝐛 ) , k 0. \mathbf{x}^{k+1}=\mathbf{x}^{k}-\gamma^{k}P^{-1}(A\mathbf{x}^{k}-\mathbf{b}),% \ k\geq 0.
  36. x n + 1 = f ( x n ) x_{n+1}=f(x_{n})
  37. x n + 1 SOR = ( 1 - ω ) x n SOR + ω f ( x n SOR ) . x^{\mathrm{SOR}}_{n+1}=(1-\omega)x^{\mathrm{SOR}}_{n}+\omega f(x^{\mathrm{SOR}% }_{n}).
  38. x x
  39. 𝐱 ( k + 1 ) = ( 1 - ω ) 𝐱 ( k ) + ω L * - 1 ( 𝐛 - U 𝐱 ( k ) ) . \mathbf{x}^{(k+1)}=(1-\omega)\mathbf{x}^{(k)}+\omega L_{*}^{-1}(\mathbf{b}-U% \mathbf{x}^{(k)}).
  40. ω > 1 \omega>1
  41. ω < 1 \omega<1
  42. ω \omega

Suction_cup.html

  1. F = A P F=AP
  2. P = F / A P=F/A
  3. π \pi

Sudan_function.html

  1. F 0 ( x , y ) = x + y , F_{0}(x,y)=x+y,\,
  2. F n + 1 ( x , 0 ) = x , n 0 F_{n+1}(x,0)=x,\ n\geq 0\,
  3. F n + 1 ( x , y + 1 ) = F n ( F n + 1 ( x , y ) , F n + 1 ( x , y ) + y + 1 ) , n 0. F_{n+1}(x,y+1)=F_{n}(F_{n+1}(x,y),F_{n+1}(x,y)+y+1),\ n\geq 0.\,
  4. × 10 1 0 \times 10^{1}0
  5. × 10 2 4 \times 10^{2}4
  6. × 10 5 8 \times 10^{5}8
  7. × 10 1 35 \times 10^{1}35

Sugawara_theory.html

  1. T z z ( z ) a j z a ( z ) j z a ( z ) T_{zz}(z)\sim\sum_{a}j^{a}_{z}(z)j^{a}_{z}(z)

Sum-frequency_generation.html

  1. ω 1 \omega_{1}
  2. ω 2 \omega_{2}
  3. ω 3 \omega_{3}
  4. χ ( 2 ) \chi^{(2)}
  5. ω 3 = ω 1 + ω 2 \hbar\omega_{3}=\hbar\omega_{1}+\hbar\omega_{2}
  6. k 3 k 1 + k 2 \hbar k_{3}\approx\hbar k_{1}+\hbar k_{2}
  7. k 1 , k 2 , k 3 k_{1},k_{2},k_{3}

Sum-product_number.html

  1. n = ( i = 1 l d i ) ( j = 1 l d j ) n=(\sum_{i=1}^{l}d_{i})(\prod_{j=1}^{l}d_{j})
  2. 2 i 3 j 7 k 2^{i}3^{j}7^{k}
  3. 3 i 5 j 7 k 3^{i}5^{j}7^{k}
  4. n b l - 1 n\geq b^{l-1}
  5. l ( b - 1 ) l(b-1)
  6. ( b - 1 ) l (b-1)^{l}
  7. l ( b - 1 ) l + 1 l(b-1)^{l+1}
  8. l ( b - 1 ) l + 1 n b l - 1 l(b-1)^{l+1}\geq n\geq b^{l-1}
  9. l ( b - 1 ) 2 ( b / ( b - 1 ) ) l - 1 l(b-1)^{2}\geq(b/(b-1))^{l-1}

Sum_of_normally_distributed_random_variables.html

  1. X N ( μ X , σ X 2 ) X\sim N(\mu_{X},\sigma_{X}^{2})
  2. Y N ( μ Y , σ Y 2 ) Y\sim N(\mu_{Y},\sigma_{Y}^{2})
  3. Z = X + Y , Z=X+Y,
  4. Z N ( μ X + μ Y , σ X 2 + σ Y 2 ) . Z\sim N(\mu_{X}+\mu_{Y},\sigma_{X}^{2}+\sigma_{Y}^{2}).
  5. φ X + Y ( t ) = E ( e i t ( X + Y ) ) \varphi_{X+Y}(t)=\operatorname{E}\left(e^{it(X+Y)}\right)
  6. φ X ( t ) = E ( e i t X ) , φ Y ( t ) = E ( e i t Y ) \varphi_{X}(t)=\operatorname{E}\left(e^{itX}\right),\qquad\varphi_{Y}(t)=% \operatorname{E}\left(e^{itY}\right)
  7. φ ( t ) = exp ( i t μ - σ 2 t 2 2 ) . \varphi(t)=\exp\left(it\mu-{\sigma^{2}t^{2}\over 2}\right).
  8. φ X + Y ( t ) = φ X ( t ) φ Y ( t ) = exp ( i t μ X - σ X 2 t 2 2 ) exp ( i t μ Y - σ Y 2 t 2 2 ) = exp ( i t ( μ X + μ Y ) - ( σ X 2 + σ Y 2 ) t 2 2 ) . \varphi_{X+Y}(t)=\varphi_{X}(t)\varphi_{Y}(t)=\exp\left(it\mu_{X}-{\sigma_{X}^% {2}t^{2}\over 2}\right)\exp\left(it\mu_{Y}-{\sigma_{Y}^{2}t^{2}\over 2}\right)% =\exp\left(it(\mu_{X}+\mu_{Y})-{(\sigma_{X}^{2}+\sigma_{Y}^{2})t^{2}\over 2}% \right).
  9. μ X + μ Y \mu_{X}+\mu_{Y}
  10. σ X 2 + σ Y 2 \sigma_{X}^{2}+\sigma_{Y}^{2}
  11. f Z ( z ) = - f Y ( z - x ) f X ( x ) d x f_{Z}(z)=\int_{-\infty}^{\infty}f_{Y}(z-x)f_{X}(x)dx
  12. f X ( x ) = 1 2 π σ X e - ( x - μ X ) 2 2 σ X 2 f_{X}(x)=\frac{1}{\sqrt{2\pi}\sigma_{X}}e^{-{(x-\mu_{X})^{2}\over 2\sigma_{X}^% {2}}}
  13. f Y ( y ) = 1 2 π σ Y e - ( y - μ Y ) 2 2 σ Y 2 f_{Y}(y)=\frac{1}{\sqrt{2\pi}\sigma_{Y}}e^{-{(y-\mu_{Y})^{2}\over 2\sigma_{Y}^% {2}}}
  14. f Z ( z ) = - 1 2 π σ Y e - ( z - x - μ Y ) 2 2 σ Y 2 1 2 π σ X e - ( x - μ X ) 2 2 σ X 2 d x f_{Z}(z)=\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}\sigma_{Y}}e^{-{(z-x-\mu_{% Y})^{2}\over 2\sigma_{Y}^{2}}}\frac{1}{\sqrt{2\pi}\sigma_{X}}e^{-{(x-\mu_{X})^% {2}\over 2\sigma_{X}^{2}}}dx
  15. = - 1 2 π σ X 2 + σ Y 2 exp [ - ( z - ( μ X + μ Y ) ) 2 2 ( σ X 2 + σ Y 2 ) ] 1 2 π σ X σ Y σ X 2 + σ Y 2 exp [ - ( x - σ X 2 ( z - μ Y ) + σ Y 2 μ X σ X 2 + σ Y 2 ) 2 2 ( σ X σ Y σ X 2 + σ Y 2 ) 2 ] d x =\int_{-\infty}^{\infty}\frac{1}{\sqrt{2\pi}\sqrt{\sigma_{X}^{2}+\sigma_{Y}^{2% }}}\exp\left[-{(z-(\mu_{X}+\mu_{Y}))^{2}\over 2(\sigma_{X}^{2}+\sigma_{Y}^{2})% }\right]\frac{1}{\sqrt{2\pi}\frac{\sigma_{X}\sigma_{Y}}{\sqrt{\sigma_{X}^{2}+% \sigma_{Y}^{2}}}}\exp\left[-\frac{\left(x-\frac{\sigma_{X}^{2}(z-\mu_{Y})+% \sigma_{Y}^{2}\mu_{X}}{\sigma_{X}^{2}+\sigma_{Y}^{2}}\right)^{2}}{2\left(\frac% {\sigma_{X}\sigma_{Y}}{\sqrt{\sigma_{X}^{2}+\sigma_{Y}^{2}}}\right)^{2}}\right% ]dx
  16. = 1 2 π ( σ X 2 + σ Y 2 ) exp [ - ( z - ( μ X + μ Y ) ) 2 2 ( σ X 2 + σ Y 2 ) ] - 1 2 π σ X σ Y σ X 2 + σ Y 2 exp [ - ( x - σ X 2 ( z - μ Y ) + σ Y 2 μ X σ X 2 + σ Y 2 ) 2 2 ( σ X σ Y σ X 2 + σ Y 2 ) 2 ] d x =\frac{1}{\sqrt{2\pi(\sigma_{X}^{2}+\sigma_{Y}^{2})}}\exp\left[-{(z-(\mu_{X}+% \mu_{Y}))^{2}\over 2(\sigma_{X}^{2}+\sigma_{Y}^{2})}\right]\int_{-\infty}^{% \infty}\frac{1}{\sqrt{2\pi}\frac{\sigma_{X}\sigma_{Y}}{\sqrt{\sigma_{X}^{2}+% \sigma_{Y}^{2}}}}\exp\left[-\frac{\left(x-\frac{\sigma_{X}^{2}(z-\mu_{Y})+% \sigma_{Y}^{2}\mu_{X}}{\sigma_{X}^{2}+\sigma_{Y}^{2}}\right)^{2}}{2\left(\frac% {\sigma_{X}\sigma_{Y}}{\sqrt{\sigma_{X}^{2}+\sigma_{Y}^{2}}}\right)^{2}}\right% ]dx
  17. f Z ( z ) = 1 2 π ( σ X 2 + σ Y 2 ) exp [ - ( z - ( μ X + μ Y ) ) 2 2 ( σ X 2 + σ Y 2 ) ] f_{Z}(z)=\frac{1}{\sqrt{2\pi(\sigma_{X}^{2}+\sigma_{Y}^{2})}}\exp\left[-{(z-(% \mu_{X}+\mu_{Y}))^{2}\over 2(\sigma_{X}^{2}+\sigma_{Y}^{2})}\right]
  18. f ( x ) = 1 / 2 π e - x 2 / 2 f(x)=\sqrt{1/2\pi\,}e^{-x^{2}/2}
  19. g ( y ) = 1 / 2 π e - y 2 / 2 . g(y)=\sqrt{1/2\pi\,}e^{-y^{2}/2}.
  20. z x + y z f ( x ) g ( y ) d x d y . z\mapsto\int_{x+y\leq z}f(x)g(y)\,dx\,dy.
  21. f ( x ) g ( y ) = ( 1 / 2 π ) e - ( x 2 + y 2 ) / 2 f(x)g(y)=(1/2\pi)e^{-(x^{2}+y^{2})/2}\,
  22. x , y x^{\prime},y^{\prime}
  23. x = c x^{\prime}=c
  24. c = c ( z ) c=c(z)
  25. f ( x ) g ( y ) = f ( x ) g ( y ) f(x)g(y)=f(x^{\prime})g(y^{\prime})
  26. x c , y \reals f ( x ) g ( y ) d x d y . \int_{x^{\prime}\leq c,y^{\prime}\in\reals}f(x^{\prime})g(y^{\prime})\,dx^{% \prime}\,dy^{\prime}.
  27. - c ( z ) f ( x ) d x = Φ ( c ( z ) ) . \int_{-\infty}^{c(z)}f(x^{\prime})\,dx^{\prime}=\Phi(c(z)).
  28. c ( z ) c(z)
  29. ( z / 2 , z / 2 ) (z/2,z/2)\,
  30. c = ( z / 2 ) 2 + ( z / 2 ) 2 = z / 2 c=\sqrt{(z/2)^{2}+(z/2)^{2}}=z/\sqrt{2}\,
  31. Φ ( z / 2 ) \Phi(z/\sqrt{2})
  32. Z = X + Y N ( 0 , 2 ) . Z=X+Y\sim N(0,2).
  33. a X + b Y z aX+bY\leq z
  34. a x + b y = z ax+by=z
  35. z a 2 + b 2 \frac{z}{\sqrt{a^{2}+b^{2}}}
  36. a X + b Y N ( 0 , a 2 + b 2 ) . aX+bY\sim N(0,a^{2}+b^{2}).
  37. X i N ( 0 , σ i 2 ) , i = 1 , , n , X_{i}\sim N(0,\sigma_{i}^{2}),\qquad i=1,\dots,n,
  38. X 1 + + X n N ( 0 , σ 1 2 + + σ n 2 ) . X_{1}+\cdots+X_{n}\sim N(0,\sigma_{1}^{2}+\cdots+\sigma_{n}^{2}).
  39. X N ( μ , σ 2 ) 1 σ ( X - μ ) N ( 0 , 1 ) . X\sim N(\mu,\sigma^{2})\Leftrightarrow\frac{1}{\sigma}(X-\mu)\sim N(0,1).
  40. X i N ( μ i , σ i 2 ) , i = 1 , , n , X_{i}\sim N(\mu_{i},\sigma_{i}^{2}),\qquad i=1,\dots,n,
  41. i = 1 n a i X i N ( i = 1 n a i μ i , i = 1 n ( a i σ i ) 2 ) . \sum_{i=1}^{n}a_{i}X_{i}\sim N\left(\sum_{i=1}^{n}a_{i}\mu_{i},\sum_{i=1}^{n}(% a_{i}\sigma_{i})^{2}\right).
  42. σ X + Y = σ X 2 + σ Y 2 + 2 ρ σ X σ Y , \sigma_{X+Y}=\sqrt{\sigma_{X}^{2}+\sigma_{Y}^{2}+2\rho\sigma_{X}\sigma_{Y}},
  43. y z - x y\rightarrow z-x
  44. f Z ( z ) = 1 2 π σ + exp ( - z 2 2 σ + 2 ) f_{Z}(z)=\frac{1}{\sqrt{2\pi}\sigma_{+}}\exp\left(-\frac{z^{2}}{2\sigma_{+}^{2% }}\right)
  45. σ + = σ x 2 + σ y 2 + 2 ρ σ x σ y . \sigma_{+}=\sqrt{\sigma_{x}^{2}+\sigma_{y}^{2}+2\rho\sigma_{x}\sigma_{y}}.
  46. f Z ( z ) = 1 2 π ( σ x 2 + σ y 2 - 2 ρ σ x σ y ) exp ( - z 2 2 ( σ x 2 + σ y 2 - 2 ρ σ x σ y ) ) f_{Z}(z)=\frac{1}{\sqrt{2\pi(\sigma_{x}^{2}+\sigma_{y}^{2}-2\rho\sigma_{x}% \sigma_{y})}}\exp\left(-\frac{z^{2}}{2(\sigma_{x}^{2}+\sigma_{y}^{2}-2\rho% \sigma_{x}\sigma_{y})}\right)
  47. σ - = σ x 2 + σ y 2 - 2 ρ σ x σ y . \sigma_{-}=\sqrt{\sigma_{x}^{2}+\sigma_{y}^{2}-2\rho\sigma_{x}\sigma_{y}}.

Sunrise_equation.html

  1. cos ω = - tan ϕ × tan δ \cos\omega_{\circ}=-\tan\phi\times\tan\delta
  2. ω \omega_{\circ}
  3. ϕ \phi
  4. δ \delta
  5. ω × hour 15 \omega_{\circ}\times\frac{\mathrm{hour}}{{15}^{\circ}}
  6. ϕ \phi
  7. δ \delta
  8. - 90 + δ < ϕ < 90 - δ -90^{\circ}+\delta<\phi<90^{\circ}-\delta
  9. - 90 - δ < ϕ < 90 + δ -90^{\circ}-\delta<\phi<90^{\circ}+\delta
  10. cos ω = sin a - sin ϕ × sin δ cos ϕ × cos δ \cos\omega_{\circ}=\dfrac{\sin a-\sin\phi\times\sin\delta}{\cos\phi\times\cos\delta}
  11. n = J d a t e - 2451545.0009 - l w 360 n^{\star}=J_{date}-2451545.0009-\dfrac{l_{w}}{360^{\circ}}
  12. n = n + 1 2 n=\left\lfloor n^{\star}+\frac{1}{2}\right\rfloor
  13. J d a t e J_{date}
  14. l ω l_{\omega}
  15. n n
  16. J = 2451545.0009 + l w 360 + n J^{\star}=2451545.0009+\dfrac{l_{w}}{360^{\circ}}+n
  17. J J^{\star}
  18. l w l_{w}
  19. M = [ 357.5291 + 0.98560028 × ( J - 2451545 ) ] mod 360 M=[357.5291+0.98560028\times(J^{\star}-2451545)]\mod 360
  20. C = 1.9148 sin ( M ) + 0.0200 sin ( 2 M ) + 0.0003 sin ( 3 M ) C=1.9148\sin(M)+0.0200\sin(2M)+0.0003\sin(3M)
  21. λ = ( M + 102.9372 + C + 180 ) mod 360 \lambda=(M+102.9372+C+180)\mod 360
  22. J t r a n s i t = J + 0.0053 sin M - 0.0069 sin ( 2 λ ) J_{transit}=J^{\star}+0.0053\sin M-0.0069\sin\left(2\lambda\right)
  23. sin δ = sin λ × sin 23.45 \sin\delta=\sin\lambda\times\sin 23.45^{\circ}
  24. δ \delta
  25. cos ω = sin ( - 0.83 ) - sin ϕ × sin δ cos ϕ × cos δ \cos\omega_{\circ}=\dfrac{\sin(-0.83^{\circ})-\sin\phi\times\sin\delta}{\cos% \phi\times\cos\delta}
  26. ϕ \phi
  27. - 1.15 elevation in feet / 60 -1.15^{\circ}\sqrt{\,\text{elevation in feet}}/60^{\circ}
  28. - 2.076 elevation in metres / 60 -2.076^{\circ}\sqrt{\,\text{elevation in metres}}/60^{\circ}
  29. J s e t = J t r a n s i t + ω 360 J_{set}=J_{transit}+\dfrac{\omega_{\circ}}{360^{\circ}}
  30. J r i s e = J t r a n s i t - ω 360 J_{rise}=J_{transit}-\dfrac{\omega_{\circ}}{360^{\circ}}

Super_vector_space.html

  1. V = V 0 V 1 . V=V_{0}\oplus V_{1}.
  2. | x | = { 0 x V 0 1 x V 1 |x|=\begin{cases}0&x\in V_{0}\\ 1&x\in V_{1}\end{cases}
  3. ( Π V ) 0 = V 1 ( Π V ) 1 = V 0 . \begin{aligned}\displaystyle(\Pi V)_{0}&\displaystyle=V_{1}\\ \displaystyle(\Pi V)_{1}&\displaystyle=V_{0}.\end{aligned}
  4. f ( V i ) \sub W i f(V_{i})\sub W_{i}
  5. f ( V i ) \sub W 1 - i f(V_{i})\sub W_{1-i}
  6. ( V W ) 0 = V 0 W 0 (V\oplus W)_{0}=V_{0}\oplus W_{0}
  7. ( V W ) 1 = V 1 W 1 . (V\oplus W)_{1}=V_{1}\oplus W_{1}.
  8. ( V W ) i = j + k = i V j W k (V\otimes W)_{i}=\bigoplus_{j+k=i}V_{j}\otimes W_{k}
  9. ( V W ) 0 = ( V 0 W 0 ) ( V 1 W 1 ) , (V\otimes W)_{0}=(V_{0}\otimes W_{0})\oplus(V_{1}\otimes W_{1}),
  10. ( V W ) 1 = ( V 0 W 1 ) ( V 1 W 0 ) . (V\otimes W)_{1}=(V_{0}\otimes W_{1})\oplus(V_{1}\otimes W_{0}).
  11. R = K [ θ 1 , , θ N ] R=K[\theta_{1},\cdots,\theta_{N}]
  12. K [ θ 1 , , θ N ] V . K[\theta_{1},\cdots,\theta_{N}]\otimes V.
  13. τ V , W : V W W V , \tau_{V,W}:V\otimes W\rightarrow W\otimes V,
  14. τ V , W ( x y ) = ( - 1 ) | x | | y | y x \tau_{V,W}(x\otimes y)=(-1)^{|x||y|}y\otimes x
  15. Hom ( V , W ) = 𝐇𝐨𝐦 ( V , W ) 0 . \mathrm{Hom}(V,W)=\mathbf{Hom}(V,W)_{0}.
  16. Hom ( U V , W ) Hom ( U , 𝐇𝐨𝐦 ( V , W ) ) . \mathrm{Hom}(U\otimes V,W)\cong\mathrm{Hom}(U,\mathbf{Hom}(V,W)).
  17. μ : A A A \mu:A\otimes A\to A

Superconducting_coherence_length.html

  1. ξ \xi
  2. ξ = 2 2 m | α | \xi=\sqrt{\frac{\hbar^{2}}{2m|\alpha|}}
  3. ξ = v f π Δ \xi=\frac{\hbar v_{f}}{\pi\Delta}
  4. \hbar
  5. m m
  6. v f v_{f}
  7. Δ \Delta

Superconductor_Insulator_Transition.html

  1. Ψ = Δ exp ( i θ ) \Psi=\Delta\exp(i\theta)
  2. Δ \Delta
  3. θ \theta
  4. Ψ ( 0 ) Ψ ( r ) r - γ \langle\Psi(0)\Psi(r)\rangle\propto r^{-\gamma}
  5. τ \tau
  6. T c 0 T_{c0}
  7. τ G L = π 8 k B ( T c 0 - T ) \tau_{GL}=\frac{\pi\hbar}{8k_{B}(T_{c0}-T)}
  8. Δ \Delta
  9. T c 0 T_{c0}
  10. T c T_{c}
  11. T = 0 T=0
  12. B < B c 1 B<B_{c1}
  13. B c 1 B_{c1}

Supersolvable_group.html

  1. { 1 } = H 0 H 1 H s - 1 H s = G \{1\}=H_{0}\triangleleft H_{1}\triangleleft\cdots\triangleleft H_{s-1}% \triangleleft H_{s}=G
  2. H i + 1 / H i H_{i+1}/H_{i}\;
  3. H i H_{i}
  4. G G
  5. H i H_{i}
  6. G G
  7. A 4 A_{4}

Supersonic_wind_tunnel.html

  1. P t P a m b ( P t 1 P t 2 ) M 1 = M m \frac{P_{t}}{P_{amb}}\leq\left(\frac{P_{t_{1}}}{P_{t_{2}}}\right)_{M_{1}=M_{m}}
  2. T m T t = ( 1 + γ - 1 2 M m 2 ) - 1 \frac{T_{m}}{T_{t}}=\left(1+\frac{\gamma-1}{2}M_{m}^{2}\right)^{-1}
  3. T t T_{t}
  4. T m T_{m}
  5. M m M_{m}

Supply_(economics).html

  1. Q s = f ( P ; P rg ) Q_{\,\text{s}}=f(P;P_{\,\text{rg}})
  2. P P
  3. P rg P_{\,\text{rg}}
  4. Q s = 325 + P - 30 P rg Q_{\,\text{s}}=325+P-30P_{\,\text{rg}}
  5. 325 325
  6. P P
  7. P rg P_{\,\text{rg}}
  8. P = f ( Q ) P=f(Q)
  9. Q = 40 P - 2 P r g Q=40P-2P_{rg}
  10. P = Q 40 + P r g 20 P=\tfrac{Q}{40}+\tfrac{P_{rg}}{20}
  11. ( Δ Q Δ P ) × P Q \left(\tfrac{\Delta Q}{\Delta P}\right)\times\tfrac{P}{Q}
  12. ( Q P ) × P Q \left(\tfrac{\partial Q}{\partial P}\right)\times\tfrac{P}{Q}

Supporting_hyperplane.html

  1. S S
  2. n \mathbb{R}^{n}
  3. S S
  4. S S
  5. S S
  6. X = n , X=\mathbb{R}^{n},
  7. x 0 x_{0}
  8. S , S,
  9. x 0 . x_{0}.
  10. x * X * \ { 0 } x^{*}\in X^{*}\backslash\{0\}
  11. X * X^{*}
  12. X X
  13. x * x^{*}
  14. x * ( x 0 ) x * ( x ) x^{*}\left(x_{0}\right)\geq x^{*}(x)
  15. x S x\in S
  16. H = { x X : x * ( x ) = x * ( x 0 ) } H=\{x\in X:x^{*}(x)=x^{*}\left(x_{0}\right)\}
  17. S S
  18. S S
  19. S S
  20. S , S,
  21. S S
  22. S S

Surface_states.html

  1. Ψ n s y m b o l k = e i s y m b o l k \cdotsymbol r u n s y m b o l k ( s y m b o l r ) . \begin{aligned}\displaystyle\Psi_{nsymbol{k}}&\displaystyle=\mathrm{e}^{% isymbol{k}\cdotsymbol{r}}u_{nsymbol{k}}(symbol{r}).\end{aligned}
  2. u n s y m b o l k ( s y m b o l r ) u_{nsymbol{k}}(symbol{r})
  3. [ - 2 2 m d 2 d z 2 + V ( z ) ] Ψ ( z ) = E Ψ ( z ) , \begin{aligned}\displaystyle\left[-\frac{\hbar^{2}}{2m}\frac{d^{2}}{dz^{2}}+V(% z)\right]\Psi(z)&\displaystyle=&\displaystyle E\Psi(z),\end{aligned}
  4. V ( z ) = { V ( z + l a ) , for z < 0 V 0 , for z > 0 , \begin{aligned}\displaystyle V(z)=\left\{\begin{array}[]{cc}V(z+la),&\textrm{% for}\quad z<0\\ V_{0},&\textrm{for}\quad z>0\end{array}\right.,\end{aligned}
  5. Ψ ( z ) = { B u - k e - i k z + C u k e i k z , for z < 0 A exp [ - 2 m ( V 0 - E ) z ] , for z > 0 , \begin{aligned}\displaystyle\Psi(z)&\displaystyle=&\displaystyle\left\{\begin{% array}[]{cc}Bu_{-k}e^{-ikz}+Cu_{k}e^{ikz},&\textrm{for}\quad z<0\\ A\exp\left[-\sqrt{2m(V_{0}-E)}\frac{z}{\hbar}\right],&\textrm{for}\quad z>0% \end{array}\right.,\end{aligned}
  6. V ( z ) = V [ exp ( i 2 π z a ) + exp ( - i 2 π z a ) ] = 2 V cos ( 2 π z a ) , \begin{aligned}\displaystyle V(z)&\displaystyle=V\left[\exp\left(i\frac{2\pi z% }{a}\right)+\exp\left(-i\frac{2\pi z}{a}\right)\right]\\ &\displaystyle=2V\cos\left(\frac{2\pi z}{a}\right),\\ \end{aligned}
  7. k = π / a k=\pi/a
  8. k = - π / a k=-\pi/a
  9. Ψ ( z ) = A e i k z + B e i [ k - ( 2 π / a ) ] z . \begin{aligned}\displaystyle\Psi(z)&\displaystyle=&\displaystyle Ae^{ikz}+Be^{% i[k-(2\pi/a)]z}.\end{aligned}
  10. G = 2 π / a G=2\pi/a
  11. k = π / a + κ k_{\perp}=\pi/a+\kappa
  12. E = 2 2 m ( π a + κ ) 2 ± | V | [ - 2 π κ m a | V | ± ( 2 π κ m a | V | ) 2 + 1 ] \begin{aligned}\displaystyle E&\displaystyle=&\displaystyle\frac{\hbar^{2}}{2m% }\left(\frac{\pi}{a}+\kappa\right)^{2}\pm|V|\left[-\frac{\hbar^{2}\pi\kappa}{% ma|V|}\pm\sqrt{\left(\frac{\hbar^{2}\pi\kappa}{ma|V|}\right)^{2}+1}\right]\end% {aligned}
  13. Ψ i = C e i κ z ( e i π z / a + [ - 2 π κ m a | V | ± ( 2 π κ m a | V | ) 2 + 1 ] e - i π z / a ) \begin{aligned}\displaystyle\Psi_{i}&\displaystyle=&\displaystyle Ce^{i\kappa z% }\left(e^{i\pi z/a}+\left[-\frac{\hbar^{2}\pi\kappa}{ma|V|}\pm\sqrt{\left(% \frac{\hbar^{2}\pi\kappa}{ma|V|}\right)^{2}+1}\right]e^{-i\pi z/a}\right)\end{aligned}
  14. Ψ 0 = D exp [ - 2 m 2 ( V 0 - E ) z ] \begin{aligned}\displaystyle\Psi_{0}&\displaystyle=&\displaystyle D\exp\left[-% \sqrt{\frac{2m}{\hbar^{2}}(V_{0}-E)}z\right]\end{aligned}
  15. i sin ( 2 δ ) = - i 2 π q m a V \begin{aligned}\displaystyle i\sin(2\delta)&\displaystyle=&\displaystyle-i% \frac{\hbar^{2}\pi q}{maV}\end{aligned}
  16. Ψ i ( z 0 ) = F e q z [ exp [ i ( π a z ± δ ) ] ± exp [ - i ( π a z ± δ ) ] ] e i δ \begin{aligned}\displaystyle\Psi_{i}(z\leq 0)&\displaystyle=&\displaystyle Fe^% {qz}\left[\exp\left[i\left(\frac{\pi}{a}z\pm\delta\right)\right]\pm\exp\left[-% i\left(\frac{\pi}{a}z\pm\delta\right)\right]\right]e^{\mp i\delta}\end{aligned}
  17. E = 2 2 m [ ( π a ) 2 - q 2 ] ± V 1 - ( 2 π q m a V ) 2 \begin{aligned}\displaystyle E&\displaystyle=&\displaystyle\frac{\hbar^{2}}{2m% }\left[\left(\frac{\pi}{a}\right)^{2}-q^{2}\right]\pm V\sqrt{1-\left(\frac{% \hbar^{2}\pi q}{maV}\right)^{2}}\end{aligned}
  18. 0 q q m a x = m a V 2 π 0\leq q\leq q_{max}=maV\hbar^{2}\pi
  19. 𝐤 | | = ( k x , k y ) \,\textbf{k}_{||}=(k_{x},k_{y})
  20. Ψ 0 ( 𝐫 ) = ψ 0 ( z ) u 𝐤 | | ( 𝐫 | | ) e - i 𝐫 | | 𝐤 | | \begin{aligned}\displaystyle\Psi_{0}(\,\textbf{r})&\displaystyle=&% \displaystyle\psi_{0}(z)u_{\,\textbf{k}_{||}}(\,\textbf{r}_{||})e^{-i\,\textbf% {r}_{||}\cdot\,\textbf{k}_{||}}\end{aligned}
  21. E | | E_{||}
  22. E s = E 0 + 2 𝐤 | | 2 2 m * , \begin{aligned}\displaystyle E_{s}=E_{0}+\frac{\hbar^{2}\,\textbf{k}^{2}_{||}}% {2m^{*}},\end{aligned}
  23. 𝐤 | | \,\textbf{k}_{||}
  24. 𝐤 | | \,\textbf{k}_{||}
  25. E s E_{s}
  26. 𝐤 | | \,\textbf{k}_{||}
  27. 𝐤 | | \mathbf{k}_{||}
  28. 𝐤 \mathbf{k}_{\perp}
  29. 𝐤 | | \mathbf{k}_{||}
  30. 𝐤 \mathbf{k}_{\perp}

Surya_Siddhanta.html

  1. s = g sin θ cos θ = g tan θ s=\frac{g\sin\theta}{\cos\theta}=g\tan\theta
  2. h = g r cos θ = g r 1 cos θ = g r sec θ h=\frac{gr}{\cos\theta}=gr\frac{1}{\cos\theta}=gr\sec\theta
  3. g \ g
  4. r \ r
  5. s \ s
  6. h \ h

Sverdrup_balance.html

  1. V = 𝐤 ^ × τ β V=\hat{\mathbf{k}}\cdot\frac{\nabla\times\mathbf{\tau}}{\beta}
  2. β \beta
  3. τ \tau
  4. β v = f w / z \beta v=f\,\partial{w}/\partial{z}
  5. β V = ρ f w E \beta V=\rho fw_{E}
  6. ρ \rho
  7. w E w_{E}
  8. w E w_{E}
  9. ρ w E = 𝐤 ^ ( × τ ) / f \rho w_{E}=\hat{\mathbf{k}}\cdot(\nabla\times\tau)/f

Switched_capacitor.html

  1. 1 {}_{1}
  2. 2 {}_{2}
  3. q q
  4. f f
  5. q = C V q=CV
  6. 1 {}_{1}
  7. 2 {}_{2}
  8. S {}_{S}
  9. q IN = C S V IN . q_{\,\text{IN}}=C_{S}V_{\,\text{IN}}.
  10. 2 {}_{2}
  11. S {}_{S}
  12. q OUT = C S V OUT . q_{\,\text{OUT}}=C_{S}V_{\,\text{OUT}}.
  13. q = q IN - q OUT = C S ( V IN - V OUT ) q=q_{\,\text{IN}}-q_{\,\text{OUT}}=C_{S}(V_{\,\text{IN}}-V_{\,\text{OUT}})
  14. I = q f . I=qf.
  15. I = C S ( V IN - V OUT ) f I=C_{S}(V_{\,\text{IN}}-V_{\,\text{OUT}})f
  16. V = V IN - V OUT . V=V_{\,\text{IN}}-V_{\,\text{OUT}}.
  17. R = V I = 1 C S f . R={V\over I}={1\over{C_{S}f}}.
  18. S {}_{S}
  19. q = C V q=CV
  20. C f b C_{fb}
  21. T = 1 / f T=1/f
  22. charge = capacitance × voltage \,\text{charge}=\,\text{capacitance}\times\,\text{voltage}
  23. C s C_{s}
  24. Q s ( t ) = C s V s ( t ) Q_{s}(t)=C_{s}\cdot V_{s}(t)\,
  25. C f b C_{fb}
  26. Q f b ( t ) = Q s ( t - T ) + Q f b ( t - T ) Q_{fb}(t)=Q_{s}(t-T)+Q_{fb}(t-T)\,
  27. C f b C_{fb}
  28. V f b ( t ) = Q s ( t - T ) C f b + V f b ( t - T ) V_{fb}(t)=\frac{Q_{s}(t-T)}{C_{fb}}+V_{fb}(t-T)\,
  29. V f b ( t ) = C s C f b V s ( t - T ) + V f b ( t - T ) V_{fb}(t)=\frac{C_{s}}{C_{fb}}\cdot V_{s}(t-T)+V_{fb}(t-T)\,
  30. C f b C_{fb}
  31. C s C_{s}
  32. T T
  33. d t T dt\leftarrow T
  34. d V f b V f b ( t ) - V f b ( t - d t ) dV_{fb}\leftarrow V_{fb}(t)-V_{fb}(t-dt)
  35. d V f b ( t ) d t = f C s C f b V s ( t ) \frac{dV_{fb}(t)}{dt}=f\frac{C_{s}}{C_{fb}}\cdot V_{s}(t)\,
  36. V O U T ( t ) = - V f b ( t ) = - 1 1 f C s C f b V s ( t ) d t V_{OUT}(t)=-V_{fb}(t)=-\frac{1}{\frac{1}{fC_{s}}C_{fb}}\int V_{s}(t)dt\,
  37. R e q = 1 f C s R_{eq}=\frac{1}{fC_{s}}
  38. H ( z ) = 1 z - 1 H(z)=\frac{1}{z-1}
  39. V O u t = V i ( C 1 + C 2 ) - ( d - 1 ) V r C 2 + V o s ( C 1 + C 2 + C p ) C 1 + ( C 1 + C 2 + C p ) A V_{Out}=\frac{V_{i}\cdot(C_{1}+C_{2})-(d-1)\cdot V_{r}\cdot C_{2}+V_{os}\cdot(% C_{1}+C_{2}+C_{p})}{C_{1}+\frac{(C_{1}+C_{2}+C_{p})}{A}}

SWR_meter.html

  1. Γ = V r e v V f w d \Gamma=\frac{V_{rev}}{V_{fwd}}
  2. V S W R = 1 + Γ 1 - Γ VSWR=\frac{1+\Gamma}{1-\Gamma}

Sylvester's_sequence.html

  1. s n = 1 + i = 0 n - 1 s i . s_{n}=1+\prod_{i=0}^{n-1}s_{i}.
  2. s i = s i - 1 ( s i - 1 - 1 ) + 1 , \displaystyle s_{i}=s_{i-1}(s_{i-1}-1)+1,
  3. s n = E 2 n + 1 + 1 2 , s_{n}=\left\lfloor E^{2^{n+1}}+\frac{1}{2}\right\rfloor,
  4. 2 2 n + 1 2^{2^{n}}+1
  5. F n = 2 + i = 0 n - 1 F i . F_{n}=2+\prod_{i=0}^{n-1}F_{i}.
  6. i = 0 1 s i = 1 2 + 1 3 + 1 7 + 1 43 + 1 1807 + . \sum_{i=0}^{\infty}\frac{1}{s_{i}}=\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1% }{43}+\frac{1}{1807}+\cdots.
  7. i = 0 j - 1 1 s i = 1 - 1 s j - 1 = s j - 2 s j - 1 . \sum_{i=0}^{j-1}\frac{1}{s_{i}}=1-\frac{1}{s_{j}-1}=\frac{s_{j}-2}{s_{j}-1}.
  8. 1 s i - 1 - 1 s i + 1 - 1 = 1 s i , \frac{1}{s_{i}-1}-\frac{1}{s_{i+1}-1}=\frac{1}{s_{i}},
  9. i = 0 j - 1 1 s i = i = 0 j - 1 ( 1 s i - 1 - 1 s i + 1 - 1 ) = 1 s 0 - 1 - 1 s j - 1 = 1 - 1 s j - 1 . \sum_{i=0}^{j-1}\frac{1}{s_{i}}=\sum_{i=0}^{j-1}\left(\frac{1}{s_{i}-1}-\frac{% 1}{s_{i+1}-1}\right)=\frac{1}{s_{0}-1}-\frac{1}{s_{j}-1}=1-\frac{1}{s_{j}-1}.
  10. 1 = 1 2 + 1 3 + 1 7 + 1 43 + 1 1807 + . 1=\frac{1}{2}+\frac{1}{3}+\frac{1}{7}+\frac{1}{43}+\frac{1}{1807}+\cdots.
  11. 1 = 1 2 + 1 3 + 1 6 , 1 = 1 2 + 1 3 + 1 7 + 1 42 , 1 = 1 2 + 1 3 + 1 7 + 1 43 + 1 1806 , . 1=\tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{6},\quad 1=\tfrac{1}{2}+\tfrac{1}{3}+% \tfrac{1}{7}+\tfrac{1}{42},\quad 1=\tfrac{1}{2}+\tfrac{1}{3}+\tfrac{1}{7}+% \tfrac{1}{43}+\tfrac{1}{1806},\quad\dots.
  12. a n a_{n}
  13. a n a n - 1 2 - a n - 1 + 1 , a_{n}\geq a_{n-1}^{2}-a_{n-1}+1,
  14. A = 1 a i A=\sum\frac{1}{a_{i}}
  15. a n = a n - 1 2 - a n - 1 + 1 a_{n}=a_{n-1}^{2}-a_{n-1}+1
  16. lim n a n a n - 1 2 = 1. \lim_{n\rightarrow\infty}\frac{a_{n}}{a_{n-1}^{2}}=1.
  17. O ( π ( x ) / log log log x ) O(\pi(x)/\log\log\log x)
  18. × 10 7 \times 10^{7}

Symmetric_derivative.html

  1. lim h 0 f ( x + h ) - f ( x - h ) 2 h . \lim_{h\to 0}\frac{f(x+h)-f(x-h)}{2h}.
  2. f ( x ) = | x | f(x)=\left|x\right|
  3. x = 0 x=0
  4. f s ( 0 ) = lim h 0 f ( 0 + h ) - f ( 0 - h ) 2 h f s ( 0 ) = lim h 0 f ( h ) - f ( - h ) 2 h f s ( 0 ) = lim h 0 | h | - | - h | 2 h f s ( 0 ) = lim h 0 h - ( - ( - h ) ) 2 h f s ( 0 ) = 0 \begin{matrix}\\ f_{s}(0)=\lim_{h\to 0}\frac{f(0+h)-f(0-h)}{2h}\\ \\ f_{s}(0)=\lim_{h\to 0}\frac{f(h)-f(-h)}{2h}\\ \\ f_{s}(0)=\lim_{h\to 0}\frac{\left|h\right|-\left|-h\right|}{2h}\\ \\ f_{s}(0)=\lim_{h\to 0}\frac{h-(-(-h))}{2h}\\ \\ f_{s}(0)=0\\ \end{matrix}
  5. h > 0 h>0
  6. h 0 h\longrightarrow 0
  7. | - h | \left|-h\right|
  8. - ( - h ) -(-h)
  9. x = 0 x=0
  10. x = 0 x=0
  11. f ( x ) = 1 / x 2 f(x)=1/x^{2}
  12. x = 0 x=0
  13. f s ( 0 ) = lim h 0 f ( 0 + h ) - f ( 0 - h ) 2 h f s ( 0 ) = lim h 0 f ( h ) - f ( - h ) 2 h f s ( 0 ) = lim h 0 1 / h 2 - 1 / ( - h ) 2 2 h f s ( 0 ) = lim h 0 1 / h 2 - 1 / h 2 2 h f s ( 0 ) = 0 \begin{matrix}\\ f_{s}(0)=\lim_{h\to 0}\frac{f(0+h)-f(0-h)}{2h}\\ \\ f_{s}(0)=\lim_{h\to 0}\frac{f(h)-f(-h)}{2h}\\ \\ f_{s}(0)=\lim_{h\to 0}\frac{1/h^{2}-1/(-h)^{2}}{2h}\\ \\ f_{s}(0)=\lim_{h\to 0}\frac{1/h^{2}-1/h^{2}}{2h}\\ \\ f_{s}(0)=0\\ \end{matrix}
  14. h > 0 h>0
  15. h 0 h\longrightarrow 0
  16. x = 0 x=0
  17. x = 0 x=0
  18. x = 0 x=0
  19. f ( x ) = { 1 , if x is rational 0 , if x is irrational f(x)=\begin{cases}1,&\,\text{if }x\,\text{ is rational}\\ 0,&\,\text{if }x\,\text{ is irrational}\end{cases}
  20. x \forall x\in\mathbb{Q}
  21. x - \forall x\in\mathbb{R}-\mathbb{Q}
  22. | 2 | - | - 1 | 2 - ( - 1 ) = 1 3 \frac{|2|-|-1|}{2-(-1)}=\frac{1}{3}
  23. f s ( x ) f ( b ) - f ( a ) b - a f s ( y ) f_{s}(x)\leq\frac{f(b)-f(a)}{b-a}\leq f_{s}(y)
  24. f s ( z ) = f ( b ) - f ( a ) b - a f_{s}(z)=\frac{f(b)-f(a)}{b-a}
  25. lim h 0 f ( x + h ) - 2 f ( x ) + f ( x - h ) h 2 . \lim_{h\to 0}\frac{f(x+h)-2f(x)+f(x-h)}{h^{2}}.
  26. sgn ( x ) \operatorname{sgn}(x)
  27. sgn ( x ) = { - 1 if x < 0 , 0 if x = 0 , 1 if x > 0. \operatorname{sgn}(x)=\begin{cases}-1&\,\text{if }x<0,\\ 0&\,\text{if }x=0,\\ 1&\,\text{if }x>0.\end{cases}
  28. x = 0 x=0
  29. x = 0 x=0
  30. lim h 0 sgn ( 0 + h ) - 2 sgn ( 0 ) + sgn ( 0 - h ) h 2 \displaystyle\lim_{h\to 0}\frac{\operatorname{sgn}(0+h)-2\operatorname{sgn}(0)% +\operatorname{sgn}(0-h)}{h^{2}}

Symmetric_monoidal_category.html

  1. s A B s_{AB}
  2. s B A s A B = 1 A B s_{BA}\circ s_{AB}=1_{A\otimes B}
  3. E E_{\infty}
  4. s A B : A B B A s_{AB}:A\otimes B\simeq B\otimes A

Symmetrically_continuous_function.html

  1. f : f:\mathbb{R}\to\mathbb{R}
  2. lim h 0 f ( x + h ) - f ( x - h ) = 0. \lim_{h\to 0}f(x+h)-f(x-h)=0.
  3. x - 2 x^{-2}
  4. x = 0 x=0

Symmetry_set.html

  1. I I\subseteq\mathbb{R}
  2. γ : I 2 \gamma:I\to\mathbb{R}^{2}
  3. γ ( I ) 2 \gamma(I)\subset\mathbb{R}^{2}
  4. m m
  5. n \mathbb{R}^{n}
  6. m < n m<n
  7. U m U\subseteq\mathbb{R}^{m}
  8. ( u 1 , u m ) := u ¯ U (u_{1}\ldots,u_{m}):=\underline{u}\in U
  9. X ¯ : U \R n \underline{X}:U\to\R^{n}
  10. n n
  11. F : n × U , where F ( x ¯ , u ¯ ) = ( x ¯ - X ¯ ) ( x ¯ - X ¯ ) . F:\mathbb{R}^{n}\times U\to\mathbb{R}\ ,\quad\mbox{where}~{}\quad F(\underline% {x},\underline{u})=(\underline{x}-\underline{X})\cdot(\underline{x}-\underline% {X})\ .
  12. x ¯ 0 n \underline{x}_{0}\in\mathbb{R}^{n}
  13. F ( x ¯ 0 , u ¯ ) F(\underline{x}_{0},\underline{u})
  14. x ¯ 0 \underline{x}_{0}
  15. X ¯ \underline{X}
  16. X ¯ ( u 1 , u m ) . \underline{X}(u_{1}\ldots,u_{m}).
  17. x ¯ \R n \underline{x}\in\R^{n}
  18. F ( x ¯ , - ) F(\underline{x},-)
  19. u ¯ U . \underline{u}\in U.
  20. 𝓇 F = 0 ¯ \mathcal{r}F=\underline{0}
  21. x ¯ n \underline{x}\in\mathbb{R}^{n}
  22. ( u ¯ 1 , u ¯ 2 ) U × U (\underline{u}_{1},\underline{u}_{2})\in U\times U
  23. u ¯ 1 u ¯ 2 \underline{u}_{1}\neq\underline{u}_{2}
  24. 𝓇 F ( x ¯ , u ¯ 1 ) = 𝓇 F ( x ¯ , u ¯ 2 ) = 0 ¯ \mathcal{r}F(\underline{x},\underline{u}_{1})=\mathcal{r}F(\underline{x},% \underline{u}_{2})=\underline{0}

Symplectic_cut.html

  1. ( X , ω ) (X,\omega)
  2. μ : X \mu:X\to\mathbb{R}
  3. X X
  4. ϵ \epsilon
  5. μ \mu
  6. μ - 1 ( ϵ ) \mu^{-1}(\epsilon)
  7. μ - 1 ( ϵ ) \mu^{-1}(\epsilon)
  8. μ - 1 ( [ ϵ , ) ) \mu^{-1}([\epsilon,\infty))
  9. μ - 1 ( ϵ ) \mu^{-1}(\epsilon)
  10. X ¯ μ ϵ \overline{X}_{\mu\geq\epsilon}
  11. X ¯ μ ϵ \overline{X}_{\mu\geq\epsilon}
  12. X X
  13. μ - 1 ( ( - , ϵ ) ) \mu^{-1}((-\infty,\epsilon))
  14. X ¯ μ ϵ \overline{X}_{\mu\geq\epsilon}
  15. V V
  16. μ - 1 ( ( - , ϵ ] ) \mu^{-1}((-\infty,\epsilon])
  17. X ¯ μ ϵ \overline{X}_{\mu\leq\epsilon}
  18. V V
  19. X ¯ μ ϵ \overline{X}_{\mu\leq\epsilon}
  20. X ¯ μ ϵ \overline{X}_{\mu\geq\epsilon}
  21. V V
  22. X ¯ μ ϵ V X ¯ μ ϵ . \overline{X}_{\mu\leq\epsilon}\cup_{V}\overline{X}_{\mu\geq\epsilon}.
  23. ( X , ω ) (X,\omega)
  24. U ( 1 ) U(1)
  25. X X
  26. μ : X . \mu:X\to\mathbb{R}.
  27. X × X\times\mathbb{C}
  28. z z
  29. \mathbb{C}
  30. ω ( - i d z d z ¯ ) . \omega\oplus(-idz\wedge d\bar{z}).
  31. U ( 1 ) U(1)
  32. e i θ ( x , z ) = ( e i θ x , e - i θ z ) e^{i\theta}\cdot(x,z)=(e^{i\theta}\cdot x,e^{-i\theta}z)
  33. ν ( x , z ) = μ ( x ) - | z | 2 . \nu(x,z)=\mu(x)-|z|^{2}.
  34. ϵ \epsilon
  35. μ - 1 ( ϵ ) \mu^{-1}(\epsilon)
  36. ϵ \epsilon
  37. ν \nu
  38. ν - 1 ( ϵ ) \nu^{-1}(\epsilon)
  39. ν - 1 ( ϵ ) \nu^{-1}(\epsilon)
  40. ( x , z ) (x,z)
  41. μ ( x ) = ϵ \mu(x)=\epsilon
  42. | z | 2 = 0 |z|^{2}=0
  43. μ - 1 ( ϵ ) \mu^{-1}(\epsilon)
  44. ( x , z ) (x,z)
  45. μ ( x ) > ϵ \mu(x)>\epsilon
  46. X > ϵ := μ - 1 ( ( ϵ , ) ) X_{>\epsilon}:=\mu^{-1}((\epsilon,\infty))
  47. ν - 1 ( ϵ ) \nu^{-1}(\epsilon)
  48. X ¯ μ ϵ := ν - 1 ( ϵ ) / U ( 1 ) . \overline{X}_{\mu\geq\epsilon}:=\nu^{-1}(\epsilon)/U(1).
  49. X μ > ϵ X_{\mu>\epsilon}
  50. V := μ - 1 ( ϵ ) / U ( 1 ) , V:=\mu^{-1}(\epsilon)/U(1),
  51. X ¯ μ ϵ \overline{X}_{\mu\geq\epsilon}
  52. X X
  53. X ¯ μ ϵ \overline{X}_{\mu\geq\epsilon}
  54. X μ > ϵ X_{\mu>\epsilon}
  55. X ¯ μ ϵ \overline{X}_{\mu\leq\epsilon}
  56. V V
  57. X ¯ μ ϵ \overline{X}_{\mu\geq\epsilon}
  58. X ¯ μ ϵ \overline{X}_{\mu\leq\epsilon}
  59. V V
  60. X X
  61. X X
  62. μ - 1 ( ϵ ) \mu^{-1}(\epsilon)
  63. X X
  64. Z Z
  65. Z Z
  66. E E
  67. ϵ \epsilon
  68. ( X , ω ) (X,\omega)
  69. U ( 1 ) U(1)
  70. μ \mu
  71. m m
  72. Z Z
  73. X X
  74. U ( 1 ) U(1)
  75. N X Z N_{X}Z
  76. 1 1
  77. ϵ \epsilon
  78. X μ > m - ϵ X_{\mu>m-\epsilon}
  79. Z Z
  80. X ¯ μ m - ϵ \overline{X}_{\mu\leq m-\epsilon}
  81. ϵ \epsilon
  82. Z Z
  83. X X
  84. Z Z

Synchronous_coordinates.html

  1. d s 2 = - d t 2 + h a b d x a d x b ds^{2}=-dt^{2}+h_{ab}dx^{a}dx^{b}
  2. h a b h_{ab}
  3. t = 0 t=0

Syndetic_set.html

  1. S \sub S\sub\mathbb{N}
  2. \mathbb{N}
  3. n F ( S - n ) = \bigcup_{n\in F}(S-n)=\mathbb{N}
  4. S - n = { m : m + n S } S-n=\{m\in\mathbb{N}:m+n\in S\}
  5. S S
  6. p = p ( S ) p=p(S)
  7. [ a , a + 1 , a + 2 , , a + p ] S [a,a+1,a+2,...,a+p]\bigcap S\neq\emptyset
  8. a a\in\mathbb{N}

Table_of_Newtonian_series.html

  1. a n a_{n}
  2. f ( s ) = n = 0 ( - 1 ) n ( s n ) a n = n = 0 ( - s ) n n ! a n f(s)=\sum_{n=0}^{\infty}(-1)^{n}{s\choose n}a_{n}=\sum_{n=0}^{\infty}\frac{(-s% )_{n}}{n!}a_{n}
  3. ( s n ) {s\choose n}
  4. ( s ) n (s)_{n}
  5. ( 1 + z ) s = n = 0 ( s n ) z n = 1 + ( s 1 ) z + ( s 2 ) z 2 + . (1+z)^{s}=\sum_{n=0}^{\infty}{s\choose n}z^{n}=1+{s\choose 1}z+{s\choose 2}z^{% 2}+\cdots.
  6. ( 1 + z ) d ( 1 + z ) s d z = s ( 1 + z ) s . (1+z)\frac{d(1+z)^{s}}{dz}=s(1+z)^{s}.
  7. ψ ( s + 1 ) = - γ - n = 1 ( - 1 ) n n ( s n ) . \psi(s+1)=-\gamma-\sum_{n=1}^{\infty}\frac{(-1)^{n}}{n}{s\choose n}.
  8. { n k } = 1 k ! j = 0 k ( - 1 ) k - j ( k j ) j n . \left\{\begin{matrix}n\\ k\end{matrix}\right\}=\frac{1}{k!}\sum_{j=0}^{k}(-1)^{k-j}{k\choose j}j^{n}.
  9. Δ k x n = j = 0 k ( - 1 ) k - j ( k j ) ( x + j ) n . \Delta^{k}x^{n}=\sum_{j=0}^{k}(-1)^{k-j}{k\choose j}(x+j)^{n}.
  10. k = 0 n ( n k ) ( - 1 ) k s - k = n ! s ( s - 1 ) ( s - 2 ) ( s - n ) = Γ ( n + 1 ) Γ ( s - n ) Γ ( s + 1 ) = B ( n + 1 , s - n ) \sum_{k=0}^{n}{n\choose k}\frac{(-1)^{k}}{s-k}=\frac{n!}{s(s-1)(s-2)\cdots(s-n% )}=\frac{\Gamma(n+1)\Gamma(s-n)}{\Gamma(s+1)}=B(n+1,s-n)
  11. Γ ( x ) \Gamma(x)
  12. B ( x , y ) B(x,y)
  13. n = 0 ( - 1 ) n ( s 2 n ) = 2 s / 2 cos π s 4 \sum_{n=0}^{\infty}(-1)^{n}{s\choose 2n}=2^{s/2}\cos\frac{\pi s}{4}
  14. n = 0 ( - 1 ) n ( s 2 n + 1 ) = 2 s / 2 sin π s 4 \sum_{n=0}^{\infty}(-1)^{n}{s\choose 2n+1}=2^{s/2}\sin\frac{\pi s}{4}
  15. ( s ) n (s)_{n}
  16. s - ( s ) 3 3 ! + ( s ) 5 5 ! - ( s ) 7 7 ! + s-\frac{(s)_{3}}{3!}+\frac{(s)_{5}}{5!}-\frac{(s)_{7}}{7!}+\cdots\,
  17. k = 0 B k z k , \!\sum_{k=0}B_{k}z^{k},
  18. k = 0 B k z k = 0 e - t t z e t z - 1 d t = k = 1 z ( k z + 1 ) 2 . \sum_{k=0}B_{k}z^{k}=\int_{0}^{\infty}e^{-t}\frac{tz}{e^{tz}-1}dt=\sum_{k=1}% \frac{z}{(kz+1)^{2}}.
  19. k = 0 B k ( x ) z k < m t p l > ( 1 - s k ) s - 1 = z s - 1 ζ ( s , x + z ) , \sum_{k=0}\frac{B_{k}(x)}{z^{k}}\frac{<}{m}tpl>{{1-s\choose k}}{s-1}=z^{s-1}% \zeta(s,x+z),
  20. ζ \zeta
  21. B k ( x ) B_{k}(x)
  22. 1 Γ ( x ) = k = 0 ( x - a k ) j = 0 k ( - 1 ) k - j Γ ( a + j ) ( k j ) , \frac{1}{\Gamma(x)}=\sum_{k=0}^{\infty}{x-a\choose k}\sum_{j=0}^{k}\frac{(-1)^% {k-j}}{\Gamma(a+j)}{k\choose j},
  23. x > a x>a
  24. f ( x ) = k = 0 ( x - a h k ) j = 0 k ( - 1 ) k - j ( k j ) f ( a + j h ) . f(x)=\sum_{k=0}{\frac{x-a}{h}\choose k}\sum_{j=0}^{k}(-1)^{k-j}{k\choose j}f(a% +jh).

Tagged_pointer.html

  1. 2 4 = 16 2^{4}=16

Tak_(function).html

  1. τ ( x , y , z ) = { τ ( τ ( x - 1 , y , z ) , τ ( y - 1 , z , x ) , τ ( z - 1 , x , y ) ) if y < x z otherwise \tau(x,y,z)=\begin{cases}\tau(\tau(x-1,y,z),\tau(y-1,z,x),\tau(z-1,x,y))&\,% \text{if }y<x\\ z&\,\text{otherwise}\end{cases}

Takt_time.html

  1. T = T a D T=\frac{T_{a}}{D}

Tau_Zero.html

  1. τ \tau
  2. τ = 1 - v 2 / c 2 \tau=\sqrt{1-v^{2}/c^{2}}
  3. τ \tau
  4. τ \tau

Tautology_(logic).html

  1. S \vDash S
  2. \top
  3. \bot
  4. \lor
  5. \land
  6. ¬ \lnot
  7. ( A B ) ( ¬ A ) ( ¬ B ) (A\land B)\lor(\lnot A)\lor(\lnot B)
  8. ( A B ) (A\land B)
  9. ( A ¬ A ) (A\lor\lnot A)
  10. ¬ \lnot
  11. ( A B ) ( ¬ B ¬ A ) (A\to B)\Leftrightarrow(\lnot B\to\lnot A)
  12. ( ( ¬ A B ) ( ¬ A ¬ B ) ) A ((\lnot A\to B)\land(\lnot A\to\lnot B))\to A
  13. ¬ ( A B ) ( ¬ A ¬ B ) \lnot(A\land B)\Leftrightarrow(\lnot A\lor\lnot B)
  14. ( ( A B ) ( B C ) ) ( A C ) ((A\to B)\land(B\to C))\to(A\to C)
  15. ( ( A B ) ( A C ) ( B C ) ) C ((A\lor B)\land(A\to C)\land(B\to C))\to C
  16. ( A B ) ( A B ) (AB)\to(AB)
  17. C C C\to C
  18. ( ( A B ) C ) ( A ( B C ) ) . ((A\land B)\to C)\Leftrightarrow(A\to(B\to C)).
  19. A B A\land B
  20. ( A B ) C (A\land B)\to C
  21. B C B\to C
  22. A ( B C ) A\to(B\to C)
  23. ( ( A B ) C ) ( A ( B C ) ) ((A\land B)\to C)\Leftrightarrow(A\to(B\to C))
  24. R S R\models S
  25. R S R\to S
  26. A ( B ¬ B ) A\land(B\lor\lnot B)
  27. B ¬ B B\lor\lnot B
  28. A C A\land C
  29. R S R\models S
  30. ( A B ) ( ¬ A ) ( ¬ B ) (A\land B)\lor(\lnot A)\lor(\lnot B)
  31. C D C\lor D
  32. C E C\to E
  33. ( ( C D ) ( C E ) ) ( ¬ ( C D ) ) ( ¬ ( C E ) ) ((C\lor D)\land(C\to E))\lor(\lnot(C\lor D))\lor(\lnot(C\to E))
  34. ¬ S \lnot S
  35. A ¬ A A\lor\lnot A
  36. ( x ( x = x ) ) ( ¬ x ( x = x ) ) (\forall x(x=x))\lor(\lnot\forall x(x=x))
  37. ( ( ( x R x ) ¬ ( x S x ) ) x T x ) ( ( x R x ) ( ( ¬ x S x ) x T x ) ) . (((\exists xRx)\land\lnot(\exists xSx))\to\forall xTx)\Leftrightarrow((\exists xRx% )\to((\lnot\exists xSx)\to\forall xTx)).
  38. A A
  39. x R x \exists xRx
  40. B B
  41. ¬ x S x \lnot\exists xSx
  42. C C
  43. x T x \forall xTx
  44. ( ( A B ) C ) ( A ( B C ) ) ((A\land B)\to C)\Leftrightarrow(A\to(B\to C))
  45. ( x R x ) ¬ x ¬ R x (\forall xRx)\to\lnot\exists x\lnot Rx
  46. A B A\to B

Tay_Bridge_disaster.html

  1. P t = 0.005 ( V t ) 2 P_{t}=0.005(V_{t})^{2}\,
  2. P t P_{t}
  3. V t V_{t}
  4. P m = 0.01 ( V h ) 2 P_{m}=0.01(V_{h})^{2}\,
  5. P m P_{m}
  6. V h V_{h}

Taylor_expansions_for_the_moments_of_functions_of_random_variables.html

  1. μ \mu
  2. = E [ X ] =\operatorname{E}\left[X\right]
  3. σ 2 \sigma^{2}
  4. = var [ X ] =\operatorname{var}\left[X\right]
  5. E [ f ( X ) ] \displaystyle\operatorname{E}\left[f(X)\right]
  6. E [ X - μ X ] = 0 E[X-\mu_{X}]=0
  7. E [ ( X - μ X ) 2 ] E[(X-\mu_{X})^{2}]
  8. σ X 2 \sigma_{X}^{2}
  9. E [ f ( X ) ] f ( μ X ) + f ′′ ( μ X ) 2 σ X 2 \operatorname{E}\left[f(X)\right]\approx f(\mu_{X})+\frac{f^{\prime\prime}(\mu% _{X})}{2}\sigma_{X}^{2}
  10. μ X \mu_{X}
  11. σ X 2 \sigma^{2}_{X}
  12. E [ X Y ] E [ X ] E [ Y ] - cov [ X , Y ] E [ Y ] 2 + E [ X ] E [ Y ] 3 var [ Y ] \operatorname{E}\left[\frac{X}{Y}\right]\approx\frac{\operatorname{E}\left[X% \right]}{\operatorname{E}\left[Y\right]}-\frac{\operatorname{cov}\left[X,Y% \right]}{\operatorname{E}\left[Y\right]^{2}}+\frac{\operatorname{E}\left[X% \right]}{\operatorname{E}\left[Y\right]^{3}}\operatorname{var}\left[Y\right]
  13. var [ f ( X ) ] ( f ( E [ X ] ) ) 2 var [ X ] = ( f ( μ X ) ) 2 σ X 2 . \operatorname{var}\left[f(X)\right]\approx\left(f^{\prime}(\operatorname{E}% \left[X\right])\right)^{2}\operatorname{var}\left[X\right]=\left(f^{\prime}(% \mu_{X})\right)^{2}\sigma^{2}_{X}.
  14. f ( X ) f(X)
  15. var [ X Y ] var [ X ] E [ Y ] 2 - 2 E [ X ] E [ Y ] 3 cov [ X , Y ] + E [ X ] 2 E [ Y ] 4 var [ Y ] . \operatorname{var}\left[\frac{X}{Y}\right]\approx\frac{\operatorname{var}\left% [X\right]}{\operatorname{E}\left[Y\right]^{2}}-\frac{2\operatorname{E}\left[X% \right]}{\operatorname{E}\left[Y\right]^{3}}\operatorname{cov}\left[X,Y\right]% +\frac{\operatorname{E}\left[X\right]^{2}}{\operatorname{E}\left[Y\right]^{4}}% \operatorname{var}\left[Y\right].

Taylor_KO_Factor.html

  1. TKOF = m bullet v bullet d bullet 7000 \mathrm{TKOF}=\frac{m_{\mathrm{bullet}}\cdot v_{\mathrm{bullet}}\cdot d_{% \mathrm{bullet}}}{7000}
  2. m B u l l e t m_{Bullet}
  3. v B u l l e t v_{Bullet}
  4. d B u l l e t d_{Bullet}
  5. \Rightarrow
  6. \Rightarrow
  7. \Rightarrow
  8. TKOF = 0.30 150 2820 7000 = 18.1 \mathrm{TKOF}=\frac{0.30\cdot 150\cdot 2820}{7000}=18.1

Teaching_dimension.html

  1. max c C { w C ( c ) } \max_{c\in C}\{w_{C}(c)\}
  2. w C ( c ) {w_{C}(c)}
  3. 2 k ( | X | k ) , 2^{k}{|X|\choose k},

Technos_acxel.html

  1. π \pi
  2. π \pi

Temperature_jump.html

  1. Δ H o = R T 2 . d ln K d T {\Delta H^{o}}={RT^{2}}.\frac{d\ln K}{dT}
  2. ( τ ) (\tau)
  3. A B A\rightleftharpoons B
  4. 1 / τ = k a + k b 1/\tau=k_{a}+k_{b}
  5. ( τ ) (\tau)
  6. K = k a / k b K=k_{a}/k_{b}

Template:Reg_polyhedra_db.html

  1. { 3 3 } \begin{Bmatrix}3\\ 3\end{Bmatrix}
  2. s { 3 3 } s\begin{Bmatrix}3\\ 3\end{Bmatrix}

Template:Semireg_dual_polyhedra_db.html

  1. arccos ( - 7 11 ) \arccos(-\frac{7}{11})
  2. arccos ( - 3 + 8 2 17 ) \arccos(-\frac{3+8\sqrt{2}}{17})
  3. arccos ( - 4 5 ) \arccos(-\frac{4}{5})
  4. arccos ( - 24 + 15 5 61 ) \arccos(-\frac{24+15\sqrt{5}}{61})
  5. arccos ( - 80 + 9 5 109 ) \arccos(-\frac{80+9\sqrt{5}}{109})

Template:Semireg_polyhedra_db.html

  1. { 4 3 } \begin{Bmatrix}4\\ 3\end{Bmatrix}
  2. r { 3 3 } r\begin{Bmatrix}3\\ 3\end{Bmatrix}
  3. sec - 1 ( - 3 ) \sec^{-1}\left(-\sqrt{3}\right)
  4. cos - 1 ( - 1 15 ( 5 + 2 5 ) ) \cos^{-1}\left(-\sqrt{\frac{1}{15}\left(5+2\sqrt{5}\right)}\right)
  5. t { 4 3 } t\begin{Bmatrix}4\\ 3\end{Bmatrix}
  6. t { 5 3 } t\begin{Bmatrix}5\\ 3\end{Bmatrix}
  7. r { 4 3 } r\begin{Bmatrix}4\\ 3\end{Bmatrix}
  8. r { 5 3 } r\begin{Bmatrix}5\\ 3\end{Bmatrix}
  9. s { 4 3 } s\begin{Bmatrix}4\\ 3\end{Bmatrix}
  10. s { 5 3 } s\begin{Bmatrix}5\\ 3\end{Bmatrix}

Temporal_Process_Language.html

  1. a . P a.P
  2. a . P a a . P a.P\rightarrow^{a}a.P
  3. E ( F ) σ F \lfloor E\rfloor(F)\rightarrow^{\sigma}F
  4. E ( F ) σ E \lfloor E\rfloor(F)\rightarrow^{\sigma}E^{\prime}
  5. a . P + Ω a.P+\Omega
  6. P r o c : := α . P r o c | P r o c ( P r o c ) | P r o c + P r o c | P r o c | P r o c | r e c X . P r o c | X | Ω | P r o c a | 0 \begin{matrix}Proc::=&\alpha.Proc\\ |&\lfloor Proc\rfloor(Proc)\\ |&Proc+Proc\\ |&Proc\;|\;Proc\\ |&recX.Proc\\ |&X\\ |&\Omega\\ |&Proc\setminus a\\ |&0\\ \end{matrix}

Tension_(physics).html

  1. = τ ( x ) =\tau(x)
  2. x x
  3. - d d x [ τ ( x ) d ρ ( x ) d x ] + v ( x ) ρ ( x ) = ω 2 σ ( x ) ρ ( x ) -\frac{d}{dx}\bigg[\tau(x)\frac{d\rho(x)}{dx}\bigg]+v(x)\rho(x)=\omega^{2}% \sigma(x)\rho(x)
  4. v ( x ) v(x)
  5. ω 2 \omega^{2}
  6. ρ ( x ) \rho(x)
  7. σ 11 \sigma_{11}
  8. F = 0 \sum\vec{F}=0
  9. F = T + m g = 0 \sum\vec{F}=\vec{T}+m\vec{g}=0
  10. F 0 \sum\vec{F}\neq 0
  11. | m g | > | T | |mg|>|T|
  12. F = T - m g 0 \sum\vec{F}=T-mg\neq 0
  13. m 1 m_{1}
  14. m 2 m_{2}
  15. w 1 = m 1 g w_{1}=m_{1}g
  16. T T
  17. F 1 F_{1}
  18. w 1 - T w_{1}-T
  19. m 1 a = m 1 g - T m_{1}a=m_{1}g-T

Ternary_relation.html

  1. a b ( mod m ) a\equiv b\;\;(\mathop{{\rm mod}}m)
  2. Γ e : σ \Gamma\vdash e\!:\!\sigma
  3. e e
  4. σ \sigma
  5. Γ \Gamma

Tetragonal_disphenoid_honeycomb.html

  1. 3 ¯ \overline{3}
  2. C ~ 3 {\tilde{C}}_{3}
  3. 3 * {}^{*}_{3}
  4. 3 * {}^{*}_{3}
  5. x = y x=y
  6. x = z x=z
  7. y = z y=z
  8. 3 ¯ \overline{3}
  9. C ~ 3 {\tilde{C}}_{3}
  10. 3 ¯ \overline{3}
  11. C ~ 3 {\tilde{C}}_{3}
  12. 3 ¯ \overline{3}
  13. C ~ 3 {\tilde{C}}_{3}

Tetrahedral-octahedral_honeycomb.html

  1. 3 ¯ \overline{3}
  2. C ~ 3 {\tilde{C}}_{3}
  3. B ~ 3 {\tilde{B}}_{3}
  4. A ~ 3 {\tilde{A}}_{3}
  5. B ~ 3 {\tilde{B}}_{3}
  6. C ~ 3 {\tilde{C}}_{3}
  7. A ~ 3 {\tilde{A}}_{3}
  8. B ~ 3 {\tilde{B}}_{3}
  9. 3 ¯ \overline{3}
  10. 4 ¯ \overline{4}
  11. 4 ¯ \overline{4}
  12. 4 ¯ \overline{4}
  13. A ~ 3 {\tilde{A}}_{3}
  14. C ~ 2 {\tilde{C}}_{2}
  15. 3 + {}^{+}_{3}
  16. n + {}^{+}_{n}
  17. 3 * {}^{*}_{3}
  18. 3 * {}^{*}_{3}
  19. 3 4 {}^{4}_{3}
  20. 3 4 {}^{4}_{3}
  21. 3 * {}^{*}_{3}
  22. A ~ 3 {\tilde{A}}_{3}
  23. B ~ 3 {\tilde{B}}_{3}
  24. A ~ 3 {\tilde{A}}_{3}
  25. 3 ¯ \overline{3}
  26. A ~ 3 {\tilde{A}}_{3}
  27. B ~ 3 {\tilde{B}}_{3}
  28. A ~ 3 {\tilde{A}}_{3}
  29. 3 ¯ \overline{3}
  30. 4 ¯ \overline{4}
  31. B ~ 4 {\tilde{B}}_{4}
  32. 3 ¯ \overline{3}
  33. B ~ 4 {\tilde{B}}_{4}
  34. 3 ¯ \overline{3}

TFNP.html

  1. \cap

Thames_Measurement.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 × b e a m 2 94 <mtpl>{{Thames\ Tonnage}}=\frac{({length}-{beam})\times{beam}\times\frac{beam}% {2}}{94}
  2. < 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>{{Thames\ Tonnage}}=\frac{({length}-{beam})\times{beam}^{2}}{188}

The_Book_of_Squares.html

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

The_Complexity_of_Songs.html

  1. N \sqrt{N}
  2. log N \log N
  3. S 0 = ϵ , S k = V k S k - 1 , k 1 , S_{0}=\epsilon,S_{k}=V_{k}S_{k-1},\,k\geq 1,
  4. V k = V_{k}=
  5. U U
  6. U U
  7. k 1 k\geq 1
  8. U = U=
  9. S k = C 2 S k - 1 S_{k}=C_{2}S_{k-1}
  10. C 2 = l a C_{2}=^{\prime}la^{\prime}
  11. S k = C 1 S k - 1 S_{k}=C_{1}S_{k-1}
  12. C 1 = i C_{1}=^{\prime}i^{\prime}

Theoretical_motivation_for_general_relativity.html

  1. π \pi
  2. u = ( γ , γ 𝐯 c ) u=\left(\gamma,\gamma{\mathbf{v}\over c}\right)
  3. 𝐯 \mathbf{v}
  4. γ \gamma
  5. γ = 1 1 - 𝐯 𝐯 c 2 \gamma={1\over\sqrt{{1-{{\mathbf{v}\cdot\mathbf{v}}\over c^{2}}}}}
  6. u α u α = - 1 u_{\alpha}u^{\alpha}=-1
  7. η μ ν = η μ ν = ( - 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 1 ) \eta^{\mu\nu}=\eta_{\mu\nu}=\begin{pmatrix}-1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}
  8. a d u d τ = d d τ ( γ , γ 𝐯 c ) = ( 0 , γ 2 𝐚 c 2 ) = ( 0 , - γ 2 𝐯 𝐯 c 2 𝐫 r 2 ) a\equiv{{du}\over{d\tau}}={d\over{d\tau}}{\left(\gamma,\gamma{\mathbf{v}\over c% }\right)}={\left(0,\gamma^{2}{\mathbf{a}\over c^{2}}\right)}={\left(0,-\gamma^% {2}{{\mathbf{v}\cdot\mathbf{v}}\over c^{2}}{{\mathbf{r}}\over r^{2}}\right)}
  9. d τ d\tau
  10. c d t = γ d τ cdt=\gamma d\tau
  11. 𝐚 = - ω 2 𝐫 = - 𝐯 𝐯 𝐫 r 2 \mathbf{a}=-\omega^{2}\mathbf{r}=-{\mathbf{v}\cdot\mathbf{v}}{{\mathbf{r}}% \over r^{2}}
  12. ω \omega
  13. 𝐫 \mathbf{r}
  14. d u μ d τ - a μ = 0 {{du^{\mu}}\over{d\tau}}-a^{\mu}=0
  15. d u μ d τ + R μ α ν β u α x ν u β = 0 {{du^{\mu}}\over{d\tau}}+{R^{\mu}}_{\alpha\nu\beta}u^{\alpha}x^{\nu}u^{\beta}=0
  16. x μ x^{\mu}
  17. R μ α ν β {R^{\mu}}_{\alpha\nu\beta}
  18. R μ α ν β = 1 r 2 η α β δ μ ν {R^{\mu}}_{\alpha\nu\beta}={1\over r^{2}}\eta_{\alpha\beta}{\delta^{\mu}}_{\nu}
  19. δ μ ν {\delta^{\mu}}_{\nu}
  20. u α u α = - 1 u_{\alpha}u^{\alpha}=-1
  21. a α u α = 0 a_{\alpha}u^{\alpha}=0
  22. R α β R ν α ν β R_{\alpha\beta}\equiv{R^{\nu}}_{\alpha\nu\beta}
  23. R R α α R\equiv{R^{\alpha}}_{\alpha}
  24. r r
  25. v v
  26. 𝐳 ´ \acute{\mathbf{z}}
  27. 𝐫 \mathbf{r}
  28. 𝐱 ´ \acute{\mathbf{x}}
  29. 𝐲 ´ \acute{\mathbf{y}}
  30. x ´ μ \acute{x}^{\mu}
  31. d 2 x ´ μ d τ 2 + R μ ´ α ν β u ´ α x ´ ν u ´ β = 0 {{d^{2}\acute{x}^{\mu}}\over{d\tau^{2}}}+\acute{{R}^{\mu}}_{\alpha\nu\beta}% \acute{u}^{\alpha}\acute{x}^{\nu}\acute{u}^{\beta}=0
  32. u ´ μ = d x ´ μ d τ \acute{u}^{\mu}={{d\acute{x}^{\mu}}\over{d\tau}}
  33. R μ ´ α ν β \acute{{R}^{\mu}}_{\alpha\nu\beta}
  34. d u μ d τ = 0 {{d{u}^{\mu}}\over{d\tau}}=0
  35. D u ´ μ D τ = d u ´ μ d τ + Γ μ α β u ´ α u ´ β = 0 {{D\acute{u}^{\mu}}\over{D\tau}}={{d\acute{u}^{\mu}}\over{d\tau}}+{\Gamma^{\mu% }}_{\alpha\beta}\acute{u}^{\alpha}\acute{u}^{\beta}=0
  36. Γ μ α β {\Gamma^{\mu}}_{\alpha\beta}
  37. R μ ´ α ν β = Γ μ α β x ν - Γ μ α ν x β + Γ μ γ ν Γ γ α β - Γ μ γ β Γ γ α ν \acute{{R}^{\mu}}_{\alpha\nu\beta}={{\partial{{\Gamma}^{\mu}}}_{\alpha\beta}% \over{\partial x^{\nu}}}-{{\partial{{\Gamma}^{\mu}}}_{\alpha\nu}\over{\partial x% ^{\beta}}}+{{\Gamma}^{\mu}}_{\gamma\nu}{{\Gamma}^{\gamma}}_{\alpha\beta}-{{% \Gamma}^{\mu}}_{\gamma\beta}{{\Gamma}^{\gamma}}_{\alpha\nu}
  38. d s 2 = d x 2 + d y 2 + d z 2 - c 2 d t 2 g μ ν d x ´ μ d x ´ ν ds^{2}=dx^{2}+dy^{2}+dz^{2}-c^{2}dt^{2}\equiv g_{\mu\nu}d\acute{x}^{\mu}d% \acute{x}^{\nu}
  39. = d x ´ 2 + d y ´ 2 + d z ´ 2 - c 2 d t ´ 2 + 2 γ cos ( θ ) cos ( ϕ ) v d t ´ d x ´ + 2 γ cos ( θ ) sin ( ϕ ) v d t ´ d y ´ - 2 γ sin ( θ ) v d t ´ d z ´ =d\acute{x}^{2}+d\acute{y}^{2}+d\acute{z}^{2}-c^{2}d\acute{t}^{2}+2\gamma\cos(% \theta)\cos(\phi)\,v\,d\acute{t}\,d\acute{x}+2\gamma\cos(\theta)\sin(\phi)v\,d% \acute{t}\,d\acute{y}-2\gamma\sin(\theta)v\,d\acute{t}\,d\acute{z}
  40. θ \theta
  41. z z
  42. ϕ \phi
  43. x x
  44. g μ ν = ( - 1 γ cos ( θ ) cos ( ϕ ) v c γ cos ( θ ) sin ( ϕ ) v c - γ sin ( θ ) v c γ cos ( θ ) cos ( ϕ ) v c 1 0 0 γ cos ( θ ) sin ( ϕ ) v c 0 1 0 - γ sin ( θ ) v c 0 0 1 ) g_{\mu\nu}=\begin{pmatrix}-1&\gamma\cos(\theta)\cos(\phi)\frac{v}{c}&\gamma% \cos(\theta)\sin(\phi)\frac{v}{c}&-\gamma\sin(\theta)\frac{v}{c}\\ \gamma\cos(\theta)\cos(\phi){\frac{v}{c}}&1&0&0\\ \gamma\cos(\theta)\sin(\phi){\frac{v}{c}}&0&1&0\\ -\gamma\sin(\theta)\frac{v}{c}&0&0&1\end{pmatrix}
  45. g μ ν g^{\mu\nu}
  46. g μ α g α ν = δ μ ν g_{\mu\alpha}g^{\alpha\nu}=\delta_{\mu}^{\nu}
  47. d Ω d\Omega
  48. d Ω ´ = - g d Ω d\acute{\Omega}=\sqrt{-g}d{\Omega}
  49. d g = g g μ ν d g μ ν = - g g μ ν d g μ ν dg=gg^{\mu\nu}dg_{\mu\nu}=-gg_{\mu\nu}dg^{\mu\nu}
  50. Γ α μ ν = g α β Γ β μ ν {{\Gamma}^{\alpha}}_{\mu\nu}=g^{\alpha\beta}{\Gamma}_{\beta\mu\nu}
  51. Γ β μ ν = 1 2 ( g β ν x μ + g β μ x ν - g μ ν x β ) {\Gamma}_{\beta\mu\nu}={\frac{1}{2}}\left({{\partial{g}}_{\beta\nu}\over{% \partial x^{\mu}}}+{{\partial{g}}_{\beta\mu}\over{\partial x^{\nu}}}-{{% \partial{g}}_{\mu\nu}\over{\partial x^{\beta}}}\right)
  52. S = 1 2 d Ω S=\int_{1}^{2}\mathcal{L}\,d\Omega
  53. \mathcal{L}
  54. p \mathcal{L}_{p}
  55. e \mathcal{L}_{e}
  56. = p + e \mathcal{L}=\mathcal{L}_{p}+\mathcal{L}_{e}
  57. S p = C 1 2 R ´ d Ω ´ = C 1 2 R ´ - g d Ω = C 1 2 g α β R ´ α β - g d Ω S_{p}=C\int_{1}^{2}\acute{R}\,d\acute{\Omega}=C\int_{1}^{2}{\acute{R}}\sqrt{-g% }\,d{\Omega}=C\int_{1}^{2}g^{\alpha\beta}\acute{R}_{\alpha\beta}\sqrt{-g}\,d{\Omega}
  58. C C
  59. p = C g α β R ´ α β - g \mathcal{L}_{p}=Cg^{\alpha\beta}\acute{R}_{\alpha\beta}\sqrt{-g}
  60. S = 1 2 C g α β R ´ α β - g d Ω + 1 2 e d Ω S=\int_{1}^{2}Cg^{\alpha\beta}\acute{R}_{\alpha\beta}\sqrt{-g}\,d\Omega+\int_{% 1}^{2}\mathcal{L}_{e}\,d\Omega
  61. 0 = δ S = 1 2 C ( R ´ α β - 1 2 R ´ g α β ) δ g α β - g d Ω - 1 2 T ´ α β δ g α β - g d Ω 0=\delta S=\int_{1}^{2}C\left(\acute{R}_{\alpha\beta}-{1\over 2}\acute{R}g^{% \alpha\beta}\right)\delta g^{\alpha\beta}\sqrt{-g}\,d\Omega-\int_{1}^{2}\acute% {T}_{\alpha\beta}\delta g^{\alpha\beta}\sqrt{-g}\,d\Omega
  62. T ´ α β = 1 - g ( d d x ν e ( d g α β d x ν ) - e g α β ) \acute{T}_{\alpha\beta}={1\over\sqrt{-g}}\left({d\over{dx^{\nu}}}{{\partial% \mathcal{L}_{e}}\over{\partial\left({{dg^{\alpha\beta}}\over{dx^{\nu}}}\right)% }}-{{\partial\mathcal{L}_{e}}\over{\partial g^{\alpha\beta}}}\right)
  63. T ´ α β = C ( R ´ α β - 1 2 R ´ g α β ) \acute{T}_{\alpha\beta}=C\left(\acute{R}_{\alpha\beta}-{1\over 2}\acute{R}\,g_% {\alpha\beta}\right)
  64. m m
  65. M M
  66. 𝐟 = d 2 𝐫 d τ 2 = - G M c 2 r 3 𝐫 \mathbf{f}={d^{2}\mathbf{r}\over d\tau^{2}}=-{GM\over{c^{2}r^{3}}}\mathbf{r}
  67. G G
  68. 𝐫 \mathbf{r}
  69. M M
  70. m m
  71. r r
  72. τ c t \tau\equiv ct
  73. 𝐟 \mathbf{f}
  74. m m
  75. m m
  76. M M
  77. 𝐟 = - 4 π G 3 c 2 ρ ( r ) 𝐫 \mathbf{f}=-{4\pi G\over{3c^{2}}}\rho(r)\mathbf{r}
  78. ρ ( r ) \rho(r)
  79. r r
  80. 𝐟 = - 4 π G 3 c 4 ( M c 2 V ) 𝐫 \mathbf{f}=-{4\pi G\over{3c^{4}}}\left({Mc^{2}\over V}\right)\mathbf{r}
  81. V V
  82. r r
  83. M c 2 Mc^{2}
  84. r r
  85. T 00 = - T 0 0 = i = 1 N ( γ i m i c 2 V ) T_{00}=-{T^{0}}_{0}=\sum_{i=1}^{N}\left({\gamma_{i}m_{i}c^{2}\over V}\right)
  86. γ i 1 1 - 𝐯 i 𝐯 i c 2 \gamma_{i}\equiv{1\over{\sqrt{1-{{\mathbf{v}_{i}\cdot\mathbf{v}_{i}}\over c^{2% }}}}}
  87. 𝐯 𝐢 \mathbf{v_{i}}
  88. m i m_{i}
  89. u α T α β u β T 00 u^{\alpha}T_{\alpha\beta}u^{\beta}\rightarrow T_{00}
  90. T T α α = - u α u α T = - u α T η α β u β - T 00 T\equiv{T^{\alpha}}_{\alpha}=-u_{\alpha}u^{\alpha}T=-u^{\alpha}T\eta_{\alpha% \beta}u^{\beta}\rightarrow-T_{00}
  91. u α u^{\alpha}
  92. T 00 u α ( A T α β + B T η α β ) u β T_{00}\rightarrow u^{\alpha}\left(AT_{\alpha\beta}+BT\eta_{\alpha\beta}\right)% u^{\beta}
  93. A + B = 1 A+B=1
  94. f μ = - 8 π G 3 c 4 ( A 2 T α β + B 2 T η α β ) δ ν μ u α x ν u β f^{\mu}=-8\pi{G\over{3c^{4}}}\left({A\over 2}T_{\alpha\beta}+{B\over 2}T\eta_{% \alpha\beta}\right)\delta^{\mu}_{\nu}u^{\alpha}x^{\nu}u^{\beta}
  95. μ = 0 \mu=0
  96. f μ f μ + u μ u ν f ν f^{\mu}\rightarrow f^{\mu}+u^{\mu}u_{\nu}f^{\nu}
  97. f μ = - 8 π G 3 c 4 ( A 2 T α β + B 2 T η α β ) ( δ ν μ + u μ u ν ) u α x ν u β f^{\mu}=-8\pi{G\over{3c^{4}}}\left({A\over 2}T_{\alpha\beta}+{B\over 2}T\eta_{% \alpha\beta}\right)\left(\delta^{\mu}_{\nu}+u^{\mu}u_{\nu}\right)u^{\alpha}x^{% \nu}u^{\beta}
  98. f ´ μ = - 8 π G 3 c 4 ( A 2 T ´ α β + B 2 T ´ g α β ) ( δ ν μ + u ´ μ u ´ ν ) u ´ α x ´ ν u ´ β \acute{f}^{\mu}=-8\pi{G\over{3c^{4}}}\left({A\over 2}\acute{T}_{\alpha\beta}+{% B\over 2}\acute{T}g_{\alpha\beta}\right)\left(\delta^{\mu}_{\nu}+\acute{u}^{% \mu}\acute{u}_{\nu}\right)\acute{u}^{\alpha}\acute{x}^{\nu}\acute{u}^{\beta}
  99. a μ = f μ a^{\mu}=f^{\mu}
  100. R μ ´ α ν β u ´ α x ´ ν u ´ β = - f ´ μ \acute{{R}^{\mu}}_{\alpha\nu\beta}\acute{u}^{\alpha}\acute{x}^{\nu}\acute{u}^{% \beta}=-\acute{f}^{\mu}
  101. R ´ α β = 8 π G c 4 ( A 2 T ´ α β + B 2 T ´ g α β ) \acute{R}_{\alpha\beta}=8\pi{G\over{c^{4}}}\left({A\over 2}\acute{T}_{\alpha% \beta}+{B\over 2}\acute{T}g_{\alpha\beta}\right)
  102. R ´ = R ´ α α = 8 π G c 4 ( A 2 T ´ α α + B 2 T ´ δ α α ) = 8 π G c 4 ( A 2 + 2 B ) T ´ \acute{R}=\acute{R}_{\alpha}^{\alpha}=8\pi{G\over{c^{4}}}\left({A\over 2}% \acute{T}_{\alpha}^{\alpha}+{B\over 2}\acute{T}\delta_{\alpha}^{\alpha}\right)% =8\pi{G\over{c^{4}}}\left({A\over 2}+2B\right)\acute{T}
  103. A = 2 A=2
  104. B = - 1 B=-1
  105. C = ( 8 π G c 4 ) - 1 C=\left(8\pi{G\over{c^{4}}}\right)^{-1}
  106. R ´ = - 8 π G c 4 T ´ \acute{R}=-8\pi{G\over{c^{4}}}\acute{T}
  107. 𝒢 α β = 8 π G c 4 T ´ α β \mathcal{G}_{\alpha\beta}=8\pi{G\over{c^{4}}}\acute{T}_{\alpha\beta}
  108. 𝒢 α β R ´ α β - 1 2 R ´ g α β \mathcal{G}_{\alpha\beta}\equiv\acute{R}_{\alpha\beta}-{1\over 2}\acute{R}g_{% \alpha\beta}
  109. g μ ν g_{\mu\nu}
  110. 4 π c J b = a F a b + Γ a μ a F μ b + Γ b μ a F a μ D a F a b F a b ; a {4\pi\over c}J^{b}=\partial_{a}F^{ab}+{\Gamma^{a}}_{\mu a}F^{\mu b}+{\Gamma^{b% }}_{\mu a}F^{a\mu}\equiv D_{a}F^{ab}\equiv{F^{ab}}_{;a}\,\!
  111. 0 = c F a b + b F c a + a F b c = D c F a b + D b F c a + D a F b c 0=\partial_{c}F_{ab}+\partial_{b}F_{ca}+\partial_{a}F_{bc}=D_{c}F_{ab}+D_{b}F_% {ca}+D_{a}F_{bc}
  112. J a \,J^{a}
  113. F a b \,F^{ab}
  114. ϵ a b c d \,\epsilon_{abcd}
  115. x a a , a ( / c t , ) {\partial\over{\partial x^{a}}}\equiv\partial_{a}\equiv{}_{,a}\equiv(\partial/% \partial ct,\nabla)
  116. - A α ; β ; β + R α β A β = 4 π c J α -{A^{\alpha;\beta}}_{;\beta}+{R^{\alpha}}_{\beta}A^{\beta}={4\pi\over c}J^{\alpha}
  117. F a b = b A a - a A b F^{ab}=\partial^{b}A^{a}-\partial^{a}A^{b}\,\!
  118. A μ ; μ = 0 {A^{\mu}}_{;\mu}=0

Theory_(mathematical_logic).html

  1. ϕ T \phi\in T
  2. T T
  3. T ϕ T\vdash\phi
  4. \mathcal{E}
  5. 𝒯 \mathcal{T}
  6. 𝒯 \mathcal{T}
  7. 𝒯 \mathcal{T}
  8. \mathcal{E}
  9. 𝒯 \mathcal{T}
  10. 𝒯 \mathcal{T}
  11. \cup
  12. \cup
  13. 𝒬 𝒮 \mathcal{QS}
  14. 𝒬 \mathcal{Q}
  15. 𝒬 𝒮 \mathcal{QS}
  16. 𝒬 𝒮 \mathcal{QS}
  17. 𝒬 𝒮 \mathcal{QS}
  18. 𝒬 𝒮 A \mathcal{QS}\vdash A
  19. 𝒬 𝒮 \mathcal{QS}
  20. 𝒬 𝒮 \mathcal{QS}
  21. 𝒬 𝒮 \mathcal{QS}
  22. 𝒬 𝒮 \mathcal{QS}
  23. 𝒬 𝒮 \mathcal{QS}

Theory_of_reasoned_action.html

  1. B I < m t p l ( A B ) W 1 + ( S N ) W 2 BI<mtpl>{{=}}(AB)W_{1}+(SN)W_{2}\,\!

Thermal_efficiency.html

  1. η t h \eta_{th}\,
  2. Q i n Q_{in}\,
  3. W o u t W_{out}\,
  4. Q o u t Q_{out}\,
  5. η t h output input . \eta_{th}\equiv\frac{\,\text{output}}{\,\text{input}}.
  6. 0 η t h 1 0\leq\eta_{th}\leq 1
  7. Q i n = W o u t + Q o u t Q_{in}=W_{out}+Q_{out}\,
  8. η t h W o u t Q i n = 1 - Q o u t Q i n \eta_{th}\equiv\frac{W_{out}}{Q_{in}}=1-\frac{Q_{out}}{Q_{in}}
  9. T H T_{H}\,
  10. T C T_{C}\,
  11. η t h 1 - T C T H \eta_{th}\leq 1-\frac{T_{C}}{T_{H}}\,
  12. T H T_{H}\,
  13. T C T_{C}\,
  14. T H = 816 C = 1500 F = 1089 K T_{H}=816^{\circ}\,\text{C}=1500^{\circ}\,\text{F}=1089\,\text{K}\,
  15. T C = 21 C = 70 F = 294 K T_{C}=21^{\circ}\,\text{C}=70^{\circ}\,\text{F}=294\,\text{K}\,
  16. η t h 1 - 294 K 1089 K = 73.0 % \eta_{th}\leq 1-\frac{294K}{1089K}=73.0\%
  17. T C T_{C}\,
  18. T H T_{H}\,
  19. T H T_{H}\,
  20. T C T_{C}\,
  21. T H T_{H}\,
  22. T C T_{C}\,
  23. η t h = 1 - 1 r γ - 1 \eta_{th}=1-\frac{1}{r^{\gamma-1}}\,
  24. η t h = 1 - r 1 - γ ( r c γ - 1 ) γ ( r c - 1 ) \eta_{th}=1-\frac{r^{1-\gamma}(r_{c}^{\gamma}-1)}{\gamma(r_{c}-1)}\,
  25. η t h = 1 - ( p 2 p 1 ) 1 - γ γ \eta_{th}=1-\bigg(\frac{p_{2}}{p_{1}}\bigg)^{\frac{1-\gamma}{\gamma}}\,
  26. η t h Q o u t Q i n \eta_{th}\equiv\frac{Q_{out}}{Q_{in}}
  27. Q Q
  28. Q H = Q C + W i n Q_{H}=Q_{C}+W_{in}\,
  29. COP heating Q H W i n \mathrm{COP}_{\mathrm{heating}}\equiv\frac{Q_{H}}{W_{in}}\,
  30. COP cooling Q C W i n \mathrm{COP}_{\mathrm{cooling}}\equiv\frac{Q_{C}}{W_{in}}\,
  31. COP heating T H T H - T C \mathrm{COP}_{\mathrm{heating}}\leq\frac{T_{H}}{T_{H}-T_{C}}\,
  32. COP cooling T C T H - T C \mathrm{COP}_{\mathrm{cooling}}\leq\frac{T_{C}}{T_{H}-T_{C}}\,
  33. COP heating - COP cooling = 1 \mathrm{COP}_{\mathrm{heating}}-\mathrm{COP}_{\mathrm{cooling}}=1\,

Thermal_quantum_field_theory.html

  1. A = Tr [ exp ( - β H ) A ] Tr [ exp ( - β H ) ] \langle A\rangle=\frac{\mbox{Tr}~{}\,[\exp(-\beta H)A]}{\mbox{Tr}~{}\,[\exp(-% \beta H)]}
  2. τ = - i t ( 0 τ β ) \tau=-it(0\leq\tau\leq\beta)
  3. β = 1 / ( k T ) \beta=1/(kT)
  4. = 1 \hbar=1
  5. v n = n / β v_{n}=n/\beta
  6. E n = n K T E_{n}=nKT
  7. t i t_{i}
  8. t i - i β t_{i}-i\beta
  9. t f t_{f}
  10. t i - i β t_{i}-i\beta
  11. t i - i β t_{i}-i\beta

Thermodynamic_cycle.html

  1. Δ E = E o u t - E i n = 0 \Delta E=E_{out}-E_{in}=0
  2. (1) W = P d V \,\text{(1)}\qquad W=\oint P\ dV
  3. (2) W = Q = Q i n - Q o u t \,\text{(2)}\qquad W=Q=Q_{in}-Q_{out}
  4. (3) W n e t = W 1 2 + W 2 3 + W 3 4 + W 4 1 \,\text{(3)}\qquad W_{net}=W_{1\to 2}+W_{2\to 3}+W_{3\to 4}+W_{4\to 1}
  5. W 1 2 = V 1 V 2 P d V , negative, work done on system W_{1\to 2}=\int_{V_{1}}^{V_{2}}P\,dV,\,\,\,\text{negative, work done on system}
  6. W 2 3 = V 2 V 3 P d V , zero work if V2 equal V3 W_{2\to 3}=\int_{V_{2}}^{V_{3}}P\,dV,\,\,\,\text{zero work if V2 equal V3}
  7. W 3 4 = V 3 V 4 P d V , positive, work done by system W_{3\to 4}=\int_{V_{3}}^{V_{4}}P\,dV,\,\,\,\text{positive, work done by system}
  8. W 4 1 = V 4 V 1 P d V , zero work if V4 equal V1 W_{4\to 1}=\int_{V_{4}}^{V_{1}}P\,dV,\,\,\,\text{zero work if V4 equal V1}
  9. (4) W n e t = W 1 2 + W 3 4 \,\text{(4)}\qquad W_{net}=W_{1\to 2}+W_{3\to 4}
  10. Δ U = R T ln V 2 V 1 - R T ln V 2 V 1 = 0 (Note: U of an isothermal process has to equal 0) \Delta U=RT\ln\frac{V_{2}}{V_{1}}-RT\ln\frac{V_{2}}{V_{1}}=0\,\text{ (Note: U % of an isothermal process has to equal 0)}
  11. Δ U = C v Δ T - 0 = C v Δ T \Delta U=C_{v}\Delta T-0=C_{v}\Delta T
  12. Δ U = C p Δ T - R Δ T ( or P Δ V ) = C v Δ T \Delta U=C_{p}\Delta T-R\Delta T(\,\text{ or }P\Delta V)=C_{v}\Delta T
  13. η = 1 - T L T H \eta=1-\frac{T_{L}}{T_{H}}
  14. T L {T_{L}}
  15. T H {T_{H}}
  16. C O P = 1 + T L T H - T L \ COP=1+\frac{T_{L}}{T_{H}-T_{L}}
  17. C O P = T L T H - T L \ COP=\frac{T_{L}}{T_{H}-T_{L}}
  18. T H T_{H}
  19. T C T_{C}
  20. d Z = 0 \oint dZ=0
  21. S = Q T S={Q\over T}
  22. Δ S = Δ Q T \Delta S={\Delta Q\over T}
  23. d S = d Q T = 0 \oint dS=\oint{dQ\over T}=0

Thermodynamic_instruments.html

  1. P V = N k T PV=NkT\,

Thermodynamic_process.html

  1. δ Q = d U \delta Q=dU
  2. P V n = C , PV^{\,n}=C,

Theta_representation.html

  1. H 3 ( ) H_{3}(\mathbb{R})
  2. τ \tau
  3. τ \tau
  4. ( S a f ) ( z ) = f ( z + a ) = exp ( a z ) f ( z ) (S_{a}f)(z)=f(z+a)=\exp(a\partial_{z})~{}f(z)
  5. ( T b f ) ( z ) = exp ( i π b 2 τ + 2 π i b z ) f ( z + b τ ) = exp ( 2 π i b z + b τ z ) f ( z ) . (T_{b}f)(z)=\exp(i\pi b^{2}\tau+2\pi ibz)f(z+b\tau)=\exp(2\pi ibz+b\tau% \partial_{z})~{}f(z).
  6. S a 1 ( S a 2 f ) = ( S a 1 S a 2 ) f = S a 1 + a 2 f S_{a_{1}}(S_{a_{2}}f)=(S_{a_{1}}\circ S_{a_{2}})f=S_{a_{1}+a_{2}}f
  7. T b 1 ( T b 2 f ) = ( T b 1 T b 2 ) f = T b 1 + b 2 f . T_{b_{1}}(T_{b_{2}}f)=(T_{b_{1}}\circ T_{b_{2}})f=T_{b_{1}+b_{2}}f.
  8. S a T b = exp ( 2 π i a b ) T b S a . S_{a}\circ T_{b}=\exp(2\pi iab)\;T_{b}\circ S_{a}.
  9. H = U ( 1 ) × × H=U(1)\times\mathbb{R}\times\mathbb{R}
  10. U τ ( λ , a , b ) H U_{\tau}(\lambda,a,b)\in H
  11. U τ ( λ , a , b ) f ( z ) = λ ( S a T b f ) ( z ) = λ exp ( i π b 2 τ + 2 π i b z ) f ( z + a + b τ ) U_{\tau}(\lambda,a,b)\;f(z)=\lambda(S_{a}\circ T_{b}f)(z)=\lambda\exp(i\pi b^{% 2}\tau+2\pi ibz)f(z+a+b\tau)
  12. λ U ( 1 ) \lambda\in U(1)
  13. U ( 1 ) = Z ( H ) U(1)=Z(H)
  14. [ H , H ] [H,H]
  15. τ \tau
  16. U τ ( λ , a , b ) U_{\tau}(\lambda,a,b)
  17. τ \tau
  18. U τ ( λ , a , b ) U_{\tau}(\lambda,a,b)
  19. f τ 2 = exp ( - 2 π y 2 τ ) | f ( x + i y ) | 2 d x d y . \|f\|_{\tau}^{2}=\int_{\mathbb{C}}\exp\left(\frac{-2\pi y^{2}}{\Im\tau}\right)% |f(x+iy)|^{2}\ dx\ dy.
  20. τ \Im\tau
  21. τ \tau
  22. τ \mathcal{H}_{\tau}
  23. τ \tau
  24. τ \tau
  25. τ \mathcal{H}_{\tau}
  26. U τ ( λ , a , b ) U_{\tau}(\lambda,a,b)
  27. τ \mathcal{H}_{\tau}
  28. U τ ( λ , a , b ) U_{\tau}(\lambda,a,b)
  29. U τ ( λ , a , b ) U_{\tau}(\lambda,a,b)
  30. τ \mathcal{H}_{\tau}
  31. τ \mathcal{H}_{\tau}
  32. M ( a , b , c ) = [ 1 a c 0 1 b 0 0 1 ] \operatorname{M}(a,b,c)=\begin{bmatrix}1&a&c\\ 0&1&b\\ 0&0&1\end{bmatrix}
  33. H 3 ( ) H_{3}(\mathbb{R})
  34. ρ h \rho_{h}
  35. ρ h ( M ( a , b , c ) ) ψ ( x ) = exp ( i b x + i h c ) ψ ( x + h a ) \rho_{h}(M(a,b,c))\;\psi(x)=\exp(ibx+ihc)\psi(x+ha)
  36. x x\in\mathbb{R}
  37. ψ L 2 ( ) \psi\in L^{2}(\mathbb{R})
  38. M ( a , 0 , 0 ) S a h M(a,0,0)\to S_{ah}
  39. M ( 0 , b , 0 ) T b / 2 π M(0,b,0)\to T_{b/2\pi}
  40. M ( 0 , 0 , c ) e i h c M(0,0,c)\to e^{ihc}
  41. Γ τ H τ \Gamma_{\tau}\subset H_{\tau}
  42. Γ τ = { U τ ( 1 , a , b ) H τ : a , b } . \Gamma_{\tau}=\{U_{\tau}(1,a,b)\in H_{\tau}:a,b\in\mathbb{Z}\}.
  43. ϑ ( z ; τ ) = n = - exp ( π i n 2 τ + 2 π i n z ) . \vartheta(z;\tau)=\sum_{n=-\infty}^{\infty}\exp(\pi in^{2}\tau+2\pi inz).
  44. Γ τ \Gamma_{\tau}
  45. ϑ ( z + 1 ; τ ) = ϑ ( z ; τ ) \vartheta(z+1;\tau)=\vartheta(z;\tau)
  46. ϑ ( z + a + b τ ; τ ) = exp ( - π i b 2 τ - 2 π i b z ) ϑ ( z ; τ ) \vartheta(z+a+b\tau;\tau)=\exp(-\pi ib^{2}\tau-2\pi ibz)\vartheta(z;\tau)

Thick_set.html

  1. T T
  2. p p\in\mathbb{N}
  3. n n\in\mathbb{N}
  4. { n , n + 1 , n + 2 , , n + p } T \{n,n+1,n+2,...,n+p\}\subset T

Thiele's_interpolation_formula.html

  1. f ( x ) f(x)
  2. x i x_{i}
  3. f ( x i ) f(x_{i})
  4. f ( x ) = f ( x 1 ) + x - x 1 ρ ( x 1 , x 2 ) + x - x 2 ρ 2 ( x 1 , x 2 , x 3 ) - f ( x 1 ) + x - x 3 ρ 3 ( x 1 , x 2 , x 3 , x 4 ) - ρ ( x 1 , x 2 ) + f(x)=f(x_{1})+\cfrac{x-x_{1}}{\rho(x_{1},x_{2})+\cfrac{x-x_{2}}{\rho_{2}(x_{1}% ,x_{2},x_{3})-f(x_{1})+\cfrac{x-x_{3}}{\rho_{3}(x_{1},x_{2},x_{3},x_{4})-\rho(% x_{1},x_{2})+\cdots}}}

Thin_lens.html

  1. 1 f = ( n - 1 ) [ 1 R 1 - 1 R 2 + ( n - 1 ) d n R 1 R 2 ] , \frac{1}{f}=(n-1)\left[\frac{1}{R_{1}}-\frac{1}{R_{2}}+\frac{(n-1)d}{nR_{1}R_{% 2}}\right],
  2. 1 f ( n - 1 ) [ 1 R 1 - 1 R 2 ] . \frac{1}{f}\approx\left(n-1\right)\left[\frac{1}{R_{1}}-\frac{1}{R_{2}}\right].
  3. 1 s + 1 s = 1 f {1\over s}+{1\over s^{\prime}}={1\over f}
  4. exp ( 2 π i λ r 2 2 f ) \exp\left(\frac{2\pi i}{\lambda}\frac{r^{2}}{2f}\right)

Thirring_model.html

  1. = ψ ¯ ( i / - m ) ψ - g 2 ( ψ ¯ γ μ ψ ) ( ψ ¯ γ μ ψ ) \mathcal{L}=\overline{\psi}(i\partial\!\!\!/-m)\psi-\frac{g}{2}\left(\overline% {\psi}\gamma^{\mu}\psi\right)\left(\overline{\psi}\gamma_{\mu}\psi\right)
  2. ψ = ( ψ + , ψ - ) \psi=(\psi_{+},\psi_{-})
  3. γ μ \gamma^{\mu}
  4. μ = 0 , 1 \mu=0,1
  5. ( ψ ¯ / ψ ) 2 (\bar{\psi}\partial\!\!\!/\psi)^{2}
  6. n n

Thom_conjecture.html

  1. C C
  2. d d
  3. g = ( d - 1 ) ( d - 2 ) / 2 g=(d-1)(d-2)/2
  4. Σ \Sigma
  5. C C
  6. g g
  7. Σ \Sigma
  8. g ( d - 1 ) ( d - 2 ) / 2 g\geq(d-1)(d-2)/2
  9. Σ \Sigma

Thomae's_function.html

  1. f ( x ) = { 1 q if x is rational, x = p q in lowest terms and q > 0 0 if x is irrational. f(x)=\begin{cases}\frac{1}{q}&\,\text{if }x\,\text{ is rational, }x=\tfrac{p}{% q}\,\text{ in lowest terms and }q>0\\ 0&\,\text{if }x\,\text{ is irrational.}\end{cases}
  2. x \scriptstyle\lfloor x\rfloor
  3. x \scriptstyle\lceil x\rceil
  4. A = n = 1 F n \scriptstyle A\;=\;\bigcup_{n=1}^{\infty}F_{n}
  5. F n \scriptstyle F_{n}
  6. f A ( x ) = { 1 n if x is rational and n is minimal so that x F n - 1 n if x is irrational and n is minimal so that x F n 0 if x A f_{A}(x)=\begin{cases}\frac{1}{n}&\,\text{if }x\,\text{ is rational and }n\,% \text{ is minimal so that }x\in F_{n}\\ -\frac{1}{n}&\,\text{if }x\,\text{ is irrational and }n\,\text{ is minimal so % that }x\in F_{n}\\ 0&\,\text{if }x\notin A\end{cases}
  7. f A \scriptstyle f_{A}
  8. m , n m,n
  9. f ( n , m ) f(n,m)
  10. q = n / ( n + m ) q=n/(n+m)
  11. g ( q ) g(q)
  12. g ( a / ( a + b ) ) = t = 1 f ( t a ) f ( t b ) g(a/(a+b))=\sum_{t=1}^{\infty}f(ta)f(tb)
  13. f ( k ) = k - α e - β k / Li α ( e - β ) f(k)=k^{-\alpha}e^{-\beta k}/\mathrm{Li}_{\alpha}(e^{-\beta})
  14. Li α \mathrm{Li}_{\alpha}
  15. g ( a / ( a + b ) ) = ( a b ) - α Li 2 α ( e - ( a + b ) β ) / Li α 2 ( e - β ) g(a/(a+b))=(ab)^{-\alpha}\mathrm{Li}_{2\alpha}(e^{-(a+b)\beta})/\mathrm{Li}^{2% }_{\alpha}(e^{-\beta})
  16. { 1 , 2 , , L } \{1,2,\ldots,L\}
  17. g ( a / ( a + b ) ) = ( 1 / L 2 ) L / max ( a , b ) g(a/(a+b))=(1/L^{2})\lfloor L/\max(a,b)\rfloor

Thomas_Banchoff.html

  1. R 3 R^{3}

Thomson_(unit).html

  1. 1 Th = 1 u e = 1 Da e = 1.036426 × 10 - 8 kg C - 1 1~{}\mathrm{Th}=1~{}\frac{\mathrm{u}}{e}=1~{}\frac{\mathrm{Da}}{e}=1.036426% \times 10^{-8}\,\mathrm{kg\,C^{-1}}

Thorium_fuel_cycle.html

  1. n + Th 90 232 Th 90 233 β - Pa 91 233 β - U 92 233 \mathrm{n}+{}_{\ 90}^{232}\mathrm{Th}\rightarrow{}_{\ 90}^{233}\mathrm{Th}% \xrightarrow{\beta^{-}}{}_{\ 91}^{233}\mathrm{Pa}\xrightarrow{\beta^{-}}{}_{\ % 92}^{233}\mathrm{U}
  2. n + Th 90 232 Th 90 233 β - Pa 91 233 β - U 92 233 + n U 92 232 + 2 n \mathrm{n}+{}_{\ 90}^{232}\mathrm{Th}\rightarrow{}_{\ 90}^{233}\mathrm{Th}% \xrightarrow{\beta^{-}}{}_{\ 91}^{233}\mathrm{Pa}\xrightarrow{\beta^{-}}{}_{\ % 92}^{233}\mathrm{U}+\mathrm{n}\rightarrow{}_{\ 92}^{232}\mathrm{U}+2\mathrm{n}
  3. n + Th 90 232 Th 90 233 β - Pa 91 233 + n Pa 91 232 + 2 n β - U 92 232 \mathrm{n}+{}_{\ 90}^{232}\mathrm{Th}\rightarrow{}_{\ 90}^{233}\mathrm{Th}% \xrightarrow{\beta^{-}}{}_{\ 91}^{233}\mathrm{Pa}+\mathrm{n}\rightarrow{}_{\ 9% 1}^{232}\mathrm{Pa}+2\mathrm{n}\xrightarrow{\beta^{-}}{}_{\ 92}^{232}\mathrm{U}
  4. n + Th 90 232 Th 90 231 + 2 n β - Pa 91 231 + n Pa 91 232 β - U 92 232 \mathrm{n}+{}_{\ 90}^{232}\mathrm{Th}\rightarrow{}_{\ 90}^{231}\mathrm{Th}+2% \mathrm{n}\xrightarrow{\beta^{-}}{}_{\ 91}^{231}\mathrm{Pa}+\mathrm{n}% \rightarrow{}_{\ 91}^{232}\mathrm{Pa}\xrightarrow{\beta^{-}}{}_{\ 92}^{232}% \mathrm{U}
  5. U 92 232 𝛼 Th 90 228 ( 68.9 years ) {}_{\ 92}^{232}\mathrm{U}\xrightarrow{\ \alpha\ }{}_{\ 90}^{228}\mathrm{Th}\ % \mathrm{(68.9\ years)}
  6. Th 90 228 𝛼 Ra 88 224 ( 1.9 year ) {}_{\ 90}^{228}\mathrm{Th}\xrightarrow{\ \alpha\ }{}_{\ 88}^{224}\mathrm{Ra}\ % \mathrm{(1.9\ year)}
  7. Ra 88 224 𝛼 Rn 86 220 ( 3.6 day , 0.24 MeV ) {}_{\ 88}^{224}\mathrm{Ra}\xrightarrow{\ \alpha\ }{}_{\ 86}^{220}\mathrm{Rn}\ % \mathrm{(3.6\ day,\ 0.24\ MeV)}
  8. Rn 86 220 𝛼 Po 84 216 ( 55 s , 0.54 MeV ) {}_{\ 86}^{220}\mathrm{Rn}\xrightarrow{\ \alpha\ }{}_{\ 84}^{216}\mathrm{Po}\ % \mathrm{(55\ s,\ 0.54\ MeV)}
  9. Po 84 216 𝛼 Pb 82 212 ( 0.15 s ) {}_{\ 84}^{216}\mathrm{Po}\xrightarrow{\ \alpha\ }{}_{\ 82}^{212}\mathrm{Pb}\ % \mathrm{(0.15\ s)}
  10. Pb 82 212 β - Bi 83 212 ( 10.64 h ) {}_{\ 82}^{212}\mathrm{Pb}\xrightarrow{\beta^{-}\ }{}_{\ 83}^{212}\mathrm{Bi}% \ \mathrm{(10.64\ h)}
  11. Bi 83 212 𝛼 Tl 81 208 ( 61 m , 0.78 MeV ) {}_{\ 83}^{212}\mathrm{Bi}\xrightarrow{\ \alpha\ }{}_{\ 81}^{208}\mathrm{Tl}\ % \mathrm{(61\ m,\ 0.78\ MeV)}
  12. Tl 81 208 β - Pb 82 208 ( 3 m , 2.6 MeV ) {}_{\ 81}^{208}\mathrm{Tl}\xrightarrow{\beta^{-}\ }{}_{\ 82}^{208}\mathrm{Pb}% \ \mathrm{(3\ m,\ 2.6\ MeV)}

Three_point_flexural_test.html

  1. E f E_{f}
  2. σ f \sigma_{f}
  3. ϵ f \epsilon_{f}
  4. σ f \sigma_{f}
  5. σ f = 3 F L 2 b d 2 \sigma_{f}=\frac{3FL}{2bd^{2}}
  6. σ f = F L π R 3 \sigma_{f}=\frac{FL}{\pi R^{3}}
  7. ϵ f \epsilon_{f}
  8. ϵ f = 6 D d L 2 \epsilon_{f}=\frac{6Dd}{L^{2}}
  9. E f E_{f}
  10. E f = L 3 m 4 b d 3 E_{f}=\frac{L^{3}m}{4bd^{3}}
  11. σ f \sigma_{f}
  12. ϵ f \epsilon_{f}
  13. E f E_{f}
  14. F F
  15. L L
  16. b b
  17. d d
  18. D D
  19. m m
  20. R R
  21. K I = 4 P B π W [ 1.6 ( a W ) 1 / 2 - 2.6 ( a W ) 3 / 2 + 12.3 ( a W ) 5 / 2 - 21.2 ( a W ) 7 / 2 + 21.8 ( a W ) 9 / 2 ] \begin{aligned}\displaystyle K_{\rm I}&\displaystyle=\frac{4P}{B}\sqrt{\frac{% \pi}{W}}\left[1.6\left(\frac{a}{W}\right)^{1/2}-2.6\left(\frac{a}{W}\right)^{3% /2}+12.3\left(\frac{a}{W}\right)^{5/2}\right.\\ &\displaystyle\qquad\left.-21.2\left(\frac{a}{W}\right)^{7/2}+21.8\left(\frac{% a}{W}\right)^{9/2}\right]\end{aligned}
  22. P P
  23. B B
  24. a a
  25. W W
  26. K I c K_{Ic}
  27. K I = 6 P B W a 1 / 2 Y K_{\rm I}=\cfrac{6P}{BW}\,a^{1/2}\,Y
  28. Y = 1.99 - a / W ( 1 - a / W ) ( 2.15 - 3.93 a / W + 2.7 ( a / W ) 2 ) ( 1 + 2 a / W ) ( 1 - a / W ) 3 / 2 . Y=\cfrac{1.99-a/W\,(1-a/W)(2.15-3.93a/W+2.7(a/W)^{2})}{(1+2a/W)(1-a/W)^{3/2}}\,.
  29. K I K_{\rm I}
  30. W W

Three_Prisoners_problem.html

  1. A A
  2. B B
  3. C C
  4. b b
  5. P ( A | b ) = P ( b | A ) P ( A ) P ( b | A ) P ( A ) + P ( b | B ) P ( B ) + P ( b | C ) P ( C ) = P(A|b)=\frac{P(b|A)P(A)}{P(b|A)P(A)+P(b|B)P(B)+P(b|C)P(C)}=
  6. = 1 2 × 1 3 1 2 × 1 3 + 0 × 1 3 + 1 × 1 3 = 1 3 . =\frac{\tfrac{1}{2}\times\tfrac{1}{3}}{\tfrac{1}{2}\times\tfrac{1}{3}+0\times% \tfrac{1}{3}+1\times\tfrac{1}{3}}=\tfrac{1}{3}.
  7. B B
  8. A A
  9. C C

Thyroid_function_tests.html

  1. G ^ T = β T ( D T + [ T S H ] ) ( 1 + K 41 [ T B G ] + K 42 [ T B P A ] ) [ F T 4 ] α T [ T S H ] \hat{G}_{T}={{\beta_{T}(D_{T}+[TSH])(1+K_{41}[TBG]+K_{42}[TBPA])[FT_{4}]}\over% {\alpha_{T}[TSH]}}
  2. G ^ T = β T ( D T + [ T S H ] ) [ T T 4 ] α T [ T S H ] \hat{G}_{T}={{\beta_{T}(D_{T}+[TSH])[TT_{4}]}\over{\alpha_{T}[TSH]}}
  3. α T \alpha_{T}
  4. β T \beta_{T}
  5. G ^ D = β 31 ( K M 1 + [ F T 4 ] ) ( 1 + K 30 [ T B G ] ) [ F T 3 ] α 31 [ F T 4 ] \hat{G}_{D}={{\beta_{31}(K_{M1}+[FT_{4}])(1+K_{30}[TBG])[FT_{3}]}\over{\alpha_% {31}[FT_{4}]}}
  6. G ^ D = β 31 ( K M 1 + [ F T 4 ] ) [ T T 3 ] α 31 [ F T 4 ] \hat{G}_{D}={{\beta_{31}(K_{M1}+[FT_{4}])[TT_{3}]}\over{\alpha_{31}[FT_{4}]}}
  7. α 31 \alpha_{31}
  8. β 31 \beta_{31}
  9. T S H I = L N ( T S H ) + 0.1345 * F T 4 TSHI=LN(TSH)+0.1345*FT4
  10. s T S H I = ( T S H I - 2.7 ) / 0.676 sTSHI=(TSHI-2.7)/0.676

Tijdeman's_theorem.html

  1. y m = x n + 1 , y^{m}=x^{n}+1,
  2. y m = x n + k y^{m}=x^{n}+k
  3. A y m = B x n + k Ay^{m}=Bx^{n}+k

Tilt_(optics).html

  1. a 1 ρ cos ( θ ) a_{1}\rho\cos(\theta)
  2. a 2 ρ sin ( θ ) a_{2}\rho\sin(\theta)
  3. ρ \rho
  4. 0 ρ 1 0\leq\rho\leq 1
  5. θ \theta
  6. 0 θ 2 π 0\leq\theta\leq 2\pi
  7. a 1 a_{1}
  8. a 2 a_{2}

Time_dependent_vector_field.html

  1. Ω × M \Omega\subset\mathbb{R}\times M
  2. T M TM
  3. X : Ω × M T M X:\Omega\subset\mathbb{R}\times M\longrightarrow TM
  4. ( t , x ) X ( t , x ) = X t ( x ) T x M (t,x)\longmapsto X(t,x)=X_{t}(x)\in T_{x}M
  5. ( t , x ) Ω (t,x)\in\Omega
  6. X t ( x ) X_{t}(x)
  7. T x M T_{x}M
  8. t t\in\mathbb{R}
  9. Ω t = { x M | ( t , x ) Ω } M \Omega_{t}=\{x\in M|(t,x)\in\Omega\}\subset M
  10. X t X_{t}
  11. Ω t M \Omega_{t}\subset M
  12. d x d t = X ( t , x ) \frac{dx}{dt}=X(t,x)
  13. α : I M \alpha:I\subset\mathbb{R}\longrightarrow M
  14. t 0 I \forall t_{0}\in I
  15. ( t 0 , α ( t 0 ) ) (t_{0},\alpha(t_{0}))
  16. d α d t | t = t 0 = X ( t 0 , α ( t 0 ) ) \frac{d\alpha}{dt}\left.{\!\!\frac{}{}}\right|_{t=t_{0}}=X(t_{0},\alpha(t_{0}))
  17. × M \mathbb{R}\times M
  18. ( t , x ) (t,x)
  19. t t\in\mathbb{R}
  20. Ω × M \Omega\subset\mathbb{R}\times M
  21. X ~ \tilde{X}
  22. Ω \Omega
  23. X ~ \tilde{X}
  24. X ~ ( t , x ) = ( 1 , X ( t , x ) ) \tilde{X}(t,x)=(1,X(t,x))
  25. ( t , x ) Ω (t,x)\in\Omega
  26. T ( t , x ) ( × M ) T_{(t,x)}(\mathbb{R}\times M)
  27. × T x M \mathbb{R}\times T_{x}M
  28. X ~ = t + X \tilde{X}=\frac{\partial{}}{\partial{t}}+X
  29. X ~ \tilde{X}
  30. F : D ( X ) × Ω M F:D(X)\subset\mathbb{R}\times\Omega\longrightarrow M
  31. ( t 0 , x ) Ω (t_{0},x)\in\Omega
  32. t F ( t , t 0 , x ) t\longrightarrow F(t,t_{0},x)
  33. α \alpha
  34. α ( t 0 ) = x \alpha(t_{0})=x
  35. F t , s F_{t,s}
  36. F t , s ( p ) = F ( t , s , p ) F_{t,s}(p)=F(t,s,p)
  37. ( t 1 , t 0 , p ) D ( X ) (t_{1},t_{0},p)\in D(X)
  38. ( t 2 , t 1 , F t 1 , t 0 ( p ) ) D ( X ) (t_{2},t_{1},F_{t_{1},t_{0}}(p))\in D(X)
  39. F t 2 , t 1 F t 1 , t 0 ( p ) = F t 2 , t 0 ( p ) F_{t_{2},t_{1}}\circ F_{t_{1},t_{0}}(p)=F_{t_{2},t_{0}}(p)
  40. t , s \forall t,s
  41. F t , s F_{t,s}
  42. F s , t F_{s,t}
  43. F F
  44. d d t | t = t 1 ( F t , t 0 * Y t ) p = ( F t 1 , t 0 * ( [ X t 1 , Y t 1 ] + d d t | t = t 1 Y t ) ) p \frac{d}{dt}\left.{\!\!\frac{}{}}\right|_{t=t_{1}}(F^{*}_{t,t_{0}}Y_{t})_{p}=% \left(F^{*}_{t_{1},t_{0}}\left([X_{t_{1}},Y_{t_{1}}]+\frac{d}{dt}\left.{\!\!% \frac{}{}}\right|_{t=t_{1}}Y_{t}\right)\right)_{p}
  45. η \eta
  46. d d t | t = t 1 ( F t , t 0 * η t ) p = ( F t 1 , t 0 * ( X t 1 η t 1 + d d t | t = t 1 η t ) ) p \frac{d}{dt}\left.{\!\!\frac{}{}}\right|_{t=t_{1}}(F^{*}_{t,t_{0}}\eta_{t})_{p% }=\left(F^{*}_{t_{1},t_{0}}\left(\mathcal{L}_{X_{t_{1}}}\eta_{t_{1}}+\frac{d}{% dt}\left.{\!\!\frac{}{}}\right|_{t=t_{1}}\eta_{t}\right)\right)_{p}

Tirukkannapuram_Vijayaraghavan.html

  1. a n > 0 a_{n}>0
  2. a 1 + a 2 + a 3 + a 4 + \sqrt{a_{1}+\sqrt{a_{2}+\sqrt{a_{3}+\sqrt{a_{4}+\cdots}}}}
  3. lim ¯ ( log a n ) / 2 n < + , \overline{\lim}(\log a_{n})/2^{n}<+\infty,
  4. lim ¯ \overline{\lim}

Tisserand's_Criterion.html

  1. 1 2 a 1 + a 1 ( 1 - e 1 2 ) cos i 1 = 1 2 a 2 + a 2 ( 1 - e 2 2 ) cos i 2 \frac{1}{2a_{1}}+\sqrt{a_{1}(1-e_{1}^{2})}\cos i_{1}=\frac{1}{2a_{2}}+\sqrt{a_% {2}(1-e_{2}^{2})}\cos i_{2}
  2. 1 2 a + a ( 1 - e 2 ) cos i const \frac{1}{2a}+\sqrt{a(1-e^{2})}\cos i\approx{\rm const}
  3. G ( μ 1 + μ 2 ) 1 G ( μ 1 + μ 3 ) G(\mu_{1}+\mu_{2})\approx 1\approx G(\mu_{1}+\mu_{3})
  4. C J = 2 ( μ 1 r 1 + μ 2 r 2 ) + 2 n ( ξ η ˙ - η ζ ˙ ) - ( ξ ˙ 2 + η ˙ 2 + ζ ˙ 2 ) C_{J}=2\cdot(\frac{\mu_{1}}{r_{1}}+\frac{\mu_{2}}{r_{2}})+2n(\xi\dot{\eta}-% \eta\dot{\zeta})-(\dot{\xi}^{2}+\dot{\eta}^{2}+\dot{\zeta}^{2})
  5. ( ξ ˙ 2 + η ˙ 2 + ζ ˙ 2 ) = v 2 = μ ( 2 r - 1 a ) (\dot{\xi}^{2}+\dot{\eta}^{2}+\dot{\zeta}^{2})=v^{2}=\mu\left({{2\over{r}}-{1% \over{a}}}\right)
  6. ζ \zeta
  7. 𝐡 = 𝐫 × 𝐫 ˙ \mathbf{h}=\mathbf{r}\times\mathbf{\dot{r}}
  8. ξ η ˙ - η ξ ˙ = h cos I \xi\dot{\eta}-\eta\dot{\xi}=h\cos I
  9. I I\,\!
  10. h = | 𝐡 | = a ( 1 - e 2 ) h=|\mathbf{h}|=\sqrt{a(1-e^{2})}
  11. 1 2 a + a ( 1 - e 2 ) cos i const \frac{1}{2a}+\sqrt{a(1-e^{2})}\cos i\approx{\rm const}

Tobit_model.html

  1. y i y_{i}
  2. x i x_{i}
  3. y i * y_{i}^{*}
  4. x i x_{i}
  5. β \beta
  6. x i x_{i}
  7. y i * y_{i}^{*}
  8. u i u_{i}
  9. y i y_{i}
  10. y i = { y i * if y i * > 0 0 if y i * 0 y_{i}=\begin{cases}y_{i}^{*}&\textrm{if}\;y_{i}^{*}>0\\ 0&\textrm{if}\;y_{i}^{*}\leq 0\end{cases}
  11. y i * y_{i}^{*}
  12. y i * = β x i + u i , u i N ( 0 , σ 2 ) y_{i}^{*}=\beta x_{i}+u_{i},u_{i}\sim N(0,\sigma^{2})\,
  13. β \beta
  14. y i y_{i}
  15. x i x_{i}
  16. β \beta
  17. x i x_{i}
  18. y i y_{i}
  19. y i y_{i}
  20. y i y_{i}
  21. y i * y_{i}^{*}
  22. x i x_{i}
  23. y L y_{L}
  24. y i = { y i * if y i * > y L y L if y i * y L . y_{i}=\begin{cases}y_{i}^{*}&\textrm{if}\;y_{i}^{*}>y_{L}\\ y_{L}&\textrm{if}\;y_{i}^{*}\leq y_{L}.\end{cases}
  25. y U y_{U}
  26. y i = { y i * if y i * < y U y U if y i * y U . y_{i}=\begin{cases}y_{i}^{*}&\textrm{if}\;y_{i}^{*}<y_{U}\\ y_{U}&\textrm{if}\;y_{i}^{*}\geq y_{U}.\end{cases}
  27. y i y_{i}
  28. y i = { y i * if y L < y i * < y U y L if y i * y L y U if y i * y U . y_{i}=\begin{cases}y_{i}^{*}&\textrm{if}\;y_{L}<y_{i}^{*}<y_{U}\\ y_{L}&\textrm{if}\;y_{i}^{*}\leq y_{L}\\ y_{U}&\textrm{if}\;y_{i}^{*}\geq y_{U}.\end{cases}
  29. y 2 i = { y 2 i * if y 1 i * > 0 0 if y 1 i * 0. y_{2i}=\begin{cases}y_{2i}^{*}&\textrm{if}\;y_{1i}^{*}>0\\ 0&\textrm{if}\;y_{1i}^{*}\leq 0.\end{cases}
  30. y 1 i = { y 1 i * if y 1 i * > 0 0 if y 1 i * 0. y_{1i}=\begin{cases}y_{1i}^{*}&\textrm{if}\;y_{1i}^{*}>0\\ 0&\textrm{if}\;y_{1i}^{*}\leq 0.\end{cases}
  31. y 2 i = { y 2 i * if y 1 i * > 0 0 if y 1 i * 0. y_{2i}=\begin{cases}y_{2i}^{*}&\textrm{if}\;y_{1i}^{*}>0\\ 0&\textrm{if}\;y_{1i}^{*}\leq 0.\end{cases}
  32. y 1 i = { y 1 i * if y 1 i * > 0 0 if y 1 i * 0. y_{1i}=\begin{cases}y_{1i}^{*}&\textrm{if}\;y_{1i}^{*}>0\\ 0&\textrm{if}\;y_{1i}^{*}\leq 0.\end{cases}
  33. y 2 i = { y 2 i * if y 1 i * > 0 0 if y 1 i * 0. y_{2i}=\begin{cases}y_{2i}^{*}&\textrm{if}\;y_{1i}^{*}>0\\ 0&\textrm{if}\;y_{1i}^{*}\leq 0.\end{cases}
  34. y 3 i = { y 3 i * if y 1 i * > 0 0 if y 1 i * 0. y_{3i}=\begin{cases}y_{3i}^{*}&\textrm{if}\;y_{1i}^{*}>0\\ 0&\textrm{if}\;y_{1i}^{*}\leq 0.\end{cases}
  35. y 1 i * y_{1i}^{*}
  36. y 2 i = { y 2 i * if y 1 i * > 0 0 if y 1 i * 0. y_{2i}=\begin{cases}y_{2i}^{*}&\textrm{if}\;y_{1i}^{*}>0\\ 0&\textrm{if}\;y_{1i}^{*}\leq 0.\end{cases}
  37. y 3 i = { y 3 i * if y 1 i * > 0 0 if y 1 i * 0. y_{3i}=\begin{cases}y_{3i}^{*}&\textrm{if}\;y_{1i}^{*}>0\\ 0&\textrm{if}\;y_{1i}^{*}\leq 0.\end{cases}
  38. y L y_{L}
  39. y j * y L y_{j}^{*}\leq y_{L}
  40. I ( y j ) I(y_{j})
  41. I ( y j ) = { 0 if y j = y L 1 if y j y L . I(y_{j})=\begin{cases}0&\textrm{if}\;y_{j}=y_{L}\\ 1&\textrm{if}\;y_{j}\neq y_{L}.\end{cases}
  42. Φ \Phi
  43. ϕ \phi
  44. ( β , σ ) = j = 1 N ( 1 σ ϕ ( y j - X j β σ ) ) I ( y j ) ( 1 - Φ ( X j β - y L σ ) ) 1 - I ( y j ) \mathcal{L}(\beta,\sigma)=\prod_{j=1}^{N}\left(\frac{1}{\sigma}\phi\left(\frac% {y_{j}-X_{j}\beta}{\sigma}\right)\right)^{I\left(y_{j}\right)}\left(1-\Phi% \left(\frac{X_{j}\beta-y_{L}}{\sigma}\right)\right)^{1-I\left(y_{j}\right)}
  45. log ( β , σ ) = j = 1 n I ( y j ) log ( 1 σ ϕ ( y j - X j β σ ) ) + ( 1 - I ( y j ) ) log ( 1 - Φ ( X j β - y L σ ) ) \log\mathcal{L}(\beta,\sigma)=\sum^{n}_{j=1}I(y_{j})\log\left(\frac{1}{\sigma}% \phi\left(\frac{y_{j}-X_{j}\beta}{\sigma}\right)\right)+(1-I(y_{j}))\log\left(% 1-\Phi\left(\frac{X_{j}\beta-y_{L}}{\sigma}\right)\right)

Toda_lattice.html

  1. d d t p ( n , t ) = e - ( q ( n , t ) - q ( n - 1 , t ) ) - e - ( q ( n + 1 , t ) - q ( n , t ) ) , d d t q ( n , t ) = p ( n , t ) , \begin{aligned}\displaystyle\frac{d}{dt}p(n,t)&\displaystyle=e^{-(q(n,t)-q(n-1% ,t))}-e^{-(q(n+1,t)-q(n,t))},\\ \displaystyle\frac{d}{dt}q(n,t)&\displaystyle=p(n,t),\end{aligned}
  2. q ( n , t ) q(n,t)
  3. n n
  4. p ( n , t ) p(n,t)
  5. m = 1 m=1
  6. a ( n , t ) = 1 2 e - ( q ( n + 1 , t ) - q ( n , t ) ) / 2 , b ( n , t ) = - 1 2 p ( n , t ) a(n,t)=\frac{1}{2}{\rm e}^{-(q(n+1,t)-q(n,t))/2},\qquad b(n,t)=-\frac{1}{2}p(n% ,t)
  7. a ˙ ( n , t ) = a ( n , t ) ( b ( n + 1 , t ) - b ( n , t ) ) , b ˙ ( n , t ) = 2 ( a ( n , t ) 2 - a ( n - 1 , t ) 2 ) . \begin{aligned}\displaystyle\dot{a}(n,t)&\displaystyle=a(n,t)\Big(b(n+1,t)-b(n% ,t)\Big),\\ \displaystyle\dot{b}(n,t)&\displaystyle=2\Big(a(n,t)^{2}-a(n-1,t)^{2}\Big).% \end{aligned}
  8. d d t L ( t ) = [ P ( t ) , L ( t ) ] \frac{d}{dt}L(t)=[P(t),L(t)]
  9. 2 ( ) \ell^{2}(\mathbb{Z})
  10. L ( t ) f ( n ) = a ( n , t ) f ( n + 1 ) + a ( n - 1 , t ) f ( n - 1 ) + b ( n , t ) f ( n ) , P ( t ) f ( n ) = a ( n , t ) f ( n + 1 ) - a ( n - 1 , t ) f ( n - 1 ) . \begin{aligned}\displaystyle L(t)f(n)&\displaystyle=a(n,t)f(n+1)+a(n-1,t)f(n-1% )+b(n,t)f(n),\\ \displaystyle P(t)f(n)&\displaystyle=a(n,t)f(n+1)-a(n-1,t)f(n-1).\end{aligned}
  11. L ( t ) L(t)

Tolman_length.html

  1. δ \delta
  2. 1 / R 1/R
  3. R = R e R=R_{e}
  4. Δ p = 2 σ R ( 1 - δ R + ) \Delta p=\frac{2\sigma}{R}\left(1-\frac{\delta}{R}+\ldots\right)
  5. Δ p = p l - p v \Delta p=p_{l}-p_{v}
  6. σ \sigma
  7. R = R=\infty
  8. δ \delta
  9. 1 / R 1/R
  10. σ ( R ) \sigma(R)
  11. 1 / R 1/R
  12. σ ( R ) = σ ( 1 - 2 δ R + ) \sigma(R)=\sigma\left(1-\frac{2\delta}{R}+\ldots\right)
  13. σ ( R ) \sigma(R)
  14. σ \sigma
  15. 1 / R 1/R
  16. δ σ = 2 k R 0 \delta\sigma=\frac{2k}{R_{0}}
  17. R 0 R_{0}
  18. R 0 R_{0}
  19. R = R s R=R_{s}
  20. Δ p = 2 σ s R s \Delta p=\frac{2\sigma_{s}}{R_{s}}
  21. σ s = σ ( R = R s ) \sigma_{s}=\sigma(R=R_{s})
  22. δ = Γ s Δ ρ 0 \delta=\frac{\Gamma_{s}}{\Delta\rho_{0}}
  23. Δ ρ 0 = ρ l , 0 - ρ v , 0 \Delta\rho_{0}=\rho_{l,0}-\rho_{v,0}
  24. δ = lim R s ( R e - R s ) = z e - z s \delta=\lim_{R_{s}\rightarrow\infty}(R_{e}-R_{s})=z_{e}-z_{s}
  25. z e , z s z_{e},z_{s}

Tolman–Oppenheimer–Volkoff_limit.html

  1. λ \lambda
  2. \hbar
  3. c c
  4. G G
  5. P = 1 / λ 4 P=1/\lambda^{4}
  6. M M
  7. P 3 = M 2 ρ 4 P^{3}=M^{2}\rho^{4}
  8. ρ \rho
  9. ρ = m / λ 3 \rho=m/\lambda^{3}
  10. m m
  11. M = 1 / m 2 , M=1/m^{2},
  12. m m

Tonality_diamond.html

  1. 1 r < 2 1\leq r<2
  2. 1 r < 2 1\leq r<2
  3. 1 r < 2 1\leq r<2
  4. d ( n ) = m < n o d d ϕ ( m ) . d(n)=\sum_{m<n\ odd}\phi(m).
  5. 2 π 2 n 2 \frac{2}{\pi^{2}}n^{2}

Tonelli–Shanks_algorithm.html

  1. x 2 n ( mod p ) x^{2}\equiv n\;\;(\mathop{{\rm mod}}p)
  2. \equiv
  3. ( mod p ) \;\;(\mathop{{\rm mod}}p)
  4. ( n p ) = 1 \bigl(\tfrac{n}{p}\bigr)=1
  5. R 2 n R^{2}\equiv n
  6. p - 1 = Q 2 S p-1=Q2^{S}
  7. S = 1 S=1
  8. p 3 ( mod 4 ) p\equiv 3\;\;(\mathop{{\rm mod}}4)
  9. R ± n p + 1 4 R\equiv\pm n^{\frac{p+1}{4}}
  10. ( z p ) = - 1 \bigl(\tfrac{z}{p}\bigr)=-1
  11. c z Q c\equiv z^{Q}
  12. R n Q + 1 2 , t n Q , M = S . R\equiv n^{\frac{Q+1}{2}},t\equiv n^{Q},M=S.
  13. t 1 t\equiv 1
  14. 0 < i < M 0<i<M
  15. t 2 i 1 t^{2^{i}}\equiv 1
  16. b c 2 M - i - 1 b\equiv c^{2^{M-i-1}}
  17. R R b , t t b 2 , c b 2 R\equiv Rb,\;t\equiv tb^{2},c\equiv b^{2}
  18. M = i M=\;i
  19. x 2 10 ( mod 13 ) x^{2}\equiv 10\;\;(\mathop{{\rm mod}}13)
  20. 13 13
  21. 10 13 - 1 2 = 10 6 1 ( mod 13 ) 10^{\frac{13-1}{2}}=10^{6}\equiv 1\;\;(\mathop{{\rm mod}}13)
  22. p - 1 = 12 = 3 2 2 p-1=12=3\cdot 2^{2}
  23. Q = 3 Q=3
  24. S = 2 S=2
  25. z = 2 z=2
  26. 2 13 - 1 2 = - 1 ( mod 13 ) 2^{\frac{13-1}{2}}=-1\;\;(\mathop{{\rm mod}}13)
  27. c = 2 3 8 ( mod 13 ) . c=2^{3}\equiv 8\;\;(\mathop{{\rm mod}}13).
  28. R = 10 2 - 4 , t 10 3 - 1 ( mod 13 ) , M = 2. R=10^{2}\equiv-4,\;t\equiv 10^{3}\equiv-1\;\;(\mathop{{\rm mod}}13),M=2.
  29. t 1 ( mod 13 ) t\not\equiv 1\;\;(\mathop{{\rm mod}}13)
  30. 0 < i < 2 0<i<\;2
  31. i = 1. i=\;1.
  32. b 8 2 2 - 1 - 1 8 ( mod 13 ) b\equiv 8^{2^{2-1-1}}\equiv 8\;\;(\mathop{{\rm mod}}13)
  33. b 2 8 2 - 1 ( mod 13 ) b^{2}\equiv 8^{2}\equiv-1\;\;(\mathop{{\rm mod}}13)
  34. R = - 4 8 7 ( mod 13 ) R=-4\cdot 8\equiv 7\;\;(\mathop{{\rm mod}}13)
  35. t - 1 - 1 1 ( mod 13 ) t\equiv-1\cdot-1\equiv 1\;\;(\mathop{{\rm mod}}13)
  36. M = 1. M=\;1.
  37. t 1 ( mod 13 ) t\equiv 1\;\;(\mathop{{\rm mod}}13)
  38. R 7 ( mod 13 ) . R\equiv 7\;\;(\mathop{{\rm mod}}13).
  39. 7 2 = 49 10 ( mod 13 ) 7^{2}=49\equiv 10\;\;(\mathop{{\rm mod}}13)
  40. ( - 7 ) 2 6 2 10 ( mod 13 ) (-7)^{2}\equiv 6^{2}\equiv 10\;\;(\mathop{{\rm mod}}13)
  41. p - 1 = Q 2 S p-1=Q2^{S}
  42. r n Q + 1 2 ( mod p ) r\equiv n^{\frac{Q+1}{2}}\;\;(\mathop{{\rm mod}}p)
  43. t n Q ( mod p ) t\equiv n^{Q}\;\;(\mathop{{\rm mod}}p)
  44. r 2 n t ( mod p ) r^{2}\equiv nt\;\;(\mathop{{\rm mod}}p)
  45. t 1 ( mod p ) t\equiv 1\;\;(\mathop{{\rm mod}}p)
  46. r 2 n ( mod p ) r^{2}\equiv n\;\;(\mathop{{\rm mod}}p)
  47. R ± r ( mod p ) R\equiv\pm r\;\;(\mathop{{\rm mod}}p)
  48. t 1 ( mod p ) t\not\equiv 1\;\;(\mathop{{\rm mod}}p)
  49. z z
  50. p p
  51. c z Q ( mod p ) c\equiv z^{Q}\;\;(\mathop{{\rm mod}}p)
  52. c 2 S ( z Q ) 2 S z 2 S Q z p - 1 1 ( mod p ) c^{2^{S}}\equiv(z^{Q})^{2^{S}}\equiv z^{2^{S}Q}\equiv z^{p-1}\equiv 1\;\;(% \mathop{{\rm mod}}p)
  53. c 2 S - 1 z p - 1 2 - 1 ( mod p ) c^{2^{S-1}}\equiv z^{\frac{p-1}{2}}\equiv-1\;\;(\mathop{{\rm mod}}p)
  54. c c
  55. 2 S 2^{S}
  56. t 2 S 1 ( mod p ) t^{2^{S}}\equiv 1\;\;(\mathop{{\rm mod}}p)
  57. t t
  58. 2 S 2^{S}
  59. t t
  60. 2 S 2^{S^{\prime}}
  61. n n
  62. p p
  63. t n Q ( mod p ) t\equiv n^{Q}\;\;(\mathop{{\rm mod}}p)
  64. S S - 1 S^{\prime}\leq S-1
  65. b c 2 S - S - 1 ( mod p ) b\equiv c^{2^{S-S^{\prime}-1}}\;\;(\mathop{{\rm mod}}p)
  66. r b r ( mod p ) r^{\prime}\equiv br\;\;(\mathop{{\rm mod}}p)
  67. c b 2 ( mod p ) c^{\prime}\equiv b^{2}\;\;(\mathop{{\rm mod}}p)
  68. t c t ( mod p ) t^{\prime}\equiv c^{\prime}t\;\;(\mathop{{\rm mod}}p)
  69. r 2 n t ( mod p ) r^{\prime 2}\equiv nt^{\prime}\;\;(\mathop{{\rm mod}}p)
  70. t t
  71. c c^{\prime}
  72. 2 S 2^{S^{\prime}}
  73. t t^{\prime}
  74. 2 S ′′ 2^{S^{\prime\prime}}
  75. S ′′ < S S^{\prime\prime}<S^{\prime}
  76. S ′′ = 0 S^{\prime\prime}=0
  77. t 1 ( mod p ) t^{\prime}\equiv 1\;\;(\mathop{{\rm mod}}p)
  78. R ± r ( mod p ) R\equiv\pm r^{\prime}\;\;(\mathop{{\rm mod}}p)
  79. b b^{\prime}
  80. r ′′ r^{\prime\prime}
  81. c ′′ c^{\prime\prime}
  82. t ′′ t^{\prime\prime}
  83. S ( j ) S^{(j)^{\prime}}
  84. 2 m + 2 k + S ( S - 1 ) 4 + 1 2 S - 1 - 9 2m+2k+\frac{S(S-1)}{4}+\frac{1}{2^{S-1}}-9
  85. m m
  86. p p
  87. k k
  88. p p
  89. z z
  90. y y
  91. 2 2
  92. y y
  93. p + 1 2 p = 1 + 1 p 2 \frac{\frac{p+1}{2}}{p}=\frac{1+\frac{1}{p}}{2}
  94. 1 1
  95. 1 2 \geq\frac{1}{2}
  96. y y
  97. p p
  98. S S
  99. p p
  100. S ( S - 1 ) > 8 m + 20 S(S-1)>8m+20
  101. 𝔽 p \mathbb{F}_{p}
  102. S ( S - 1 ) S(S-1)
  103. O ( S log S / log log S ) O(S\log S/\log\log S)
  104. e e
  105. c e n Q c^{e}\equiv n^{Q}
  106. R c - e / 2 n ( Q + 1 ) / 2 R\equiv c^{-e/2}n^{(Q+1)/2}
  107. R 2 n R^{2}\equiv n
  108. e e
  109. n n
  110. z z
  111. z z
  112. z < 2 ln 2 p z<2\ln^{2}{p}
  113. z z
  114. z z
  115. z z
  116. / p * \mathbb{Z}/p\mathbb{Z}^{*}
  117. p - 1 = Q 2 S p-1=Q2^{S}
  118. R n Q + 1 2 , t n Q R 2 / n R\equiv n^{\frac{Q+1}{2}},t\equiv n^{Q}\equiv R^{2}/n
  119. b b
  120. b 2 t b^{2}\equiv t
  121. R R / b R\equiv R/b

Topkis's_theorem.html

  1. x * ( θ ) = arg max x D f ( x , θ ) x^{*}(\theta)=\arg\max_{x\in D}f(x,\theta)
  2. max s U ( s , p ) \max_{s}U(s,p)
  3. s ( p ) p . \frac{\partial s^{\ast}(p)}{\partial p}.
  4. s ( p ) s^{\ast}(p)
  5. s ( p ) s^{\ast}(p)
  6. U s ( s ( p ) , p ) = 0 U_{s}(s^{\ast}(p),p)=0
  7. U s s ( s ( p ) , p ) ( s ( p ) / ( p ) ) + U s p ( s ( p ) , p ) = 0 U_{ss}(s^{\ast}(p),p)(\partial s^{\ast}(p)/(\partial p))+U_{sp}(s^{\ast}(p),p)=0
  8. s ( p ) p = - U s p ( s ( p ) , p ) U s s ( s ( p ) , p ) negative since we assumed U ( . ) was concave in s . \frac{\partial s^{\ast}(p)}{\partial p}=\underset{\,\text{negative since we % assumed }U(.)\,\text{ was concave in }s}{\underbrace{\frac{-U_{sp}(s^{\ast}(p)% ,p)}{U_{ss}(s^{\ast}(p),p)}}}.
  9. s ( p ) p = sign U s p ( s ( p ) , p ) . \frac{\partial s^{\ast}(p)}{\partial p}\overset{\,\text{sign}}{=}U_{sp}(s^{% \ast}(p),p).
  10. U s p ( s ( p ) , p ) < 0 U_{sp}(s^{\ast}(p),p)<0
  11. s ( p ) p < 0 \frac{\partial s^{\ast}(p)}{\partial p}<0
  12. U ( s , p ) U(s,p)
  13. ( s , p ) \left(s,p\right)
  14. 2 U s p < 0 \frac{\partial^{2}U}{\partial s\,\partial p}<0
  15. 2 U s p < 0 , \frac{\partial^{2}U}{\partial s\,\partial p}<0,
  16. s ( p ) p < 0 \frac{\partial s^{\ast}(p)}{\partial p}<0

Topological_string_theory.html

  1. θ ¯ ± \overline{\theta}^{\pm}
  2. θ - \theta^{-}
  3. θ ¯ + \overline{\theta}^{+}
  4. θ ¯ ± \overline{\theta}^{\pm}

Toroidal_coordinates.html

  1. F 1 F_{1}
  2. F 2 F_{2}
  3. a a
  4. x y xy
  5. z z
  6. ( σ , τ , ϕ ) (\sigma,\tau,\phi)
  7. x = a sinh τ cosh τ - cos σ cos ϕ x=a\ \frac{\sinh\tau}{\cosh\tau-\cos\sigma}\cos\phi
  8. y = a sinh τ cosh τ - cos σ sin ϕ y=a\ \frac{\sinh\tau}{\cosh\tau-\cos\sigma}\sin\phi
  9. z = a sin σ cosh τ - cos σ z=a\ \frac{\sin\sigma}{\cosh\tau-\cos\sigma}
  10. σ \sigma
  11. P P
  12. F 1 P F 2 F_{1}PF_{2}
  13. τ \tau
  14. d 1 d_{1}
  15. d 2 d_{2}
  16. τ = ln d 1 d 2 . \tau=\ln\frac{d_{1}}{d_{2}}.
  17. - π < σ π -\pi<\sigma\leq\pi
  18. τ 0 \tau\geq 0
  19. 0 ϕ < 2 π . 0\leq\phi<2\pi.
  20. σ \sigma
  21. ( x 2 + y 2 ) + ( z - a cot σ ) 2 = a 2 sin 2 σ \left(x^{2}+y^{2}\right)+\left(z-a\cot\sigma\right)^{2}=\frac{a^{2}}{\sin^{2}\sigma}
  22. τ \tau
  23. z 2 + ( x 2 + y 2 - a coth τ ) 2 = a 2 sinh 2 τ z^{2}+\left(\sqrt{x^{2}+y^{2}}-a\coth\tau\right)^{2}=\frac{a^{2}}{\sinh^{2}\tau}
  24. σ \sigma
  25. z z
  26. τ \tau
  27. x y xy
  28. tan ϕ = y x \tan\phi=\frac{y}{x}
  29. ρ 2 = x 2 + y 2 \rho^{2}=x^{2}+y^{2}
  30. d 1 2 = ( ρ + a ) 2 + z 2 d_{1}^{2}=(\rho+a)^{2}+z^{2}
  31. d 2 2 = ( ρ - a ) 2 + z 2 d_{2}^{2}=(\rho-a)^{2}+z^{2}
  32. τ = ln d 1 d 2 \tau=\ln\frac{d_{1}}{d_{2}}
  33. cos σ = - 4 a 2 - d 1 2 - d 2 2 2 d 1 d 2 \cos\sigma=-\frac{4a^{2}-d_{1}^{2}-d_{2}^{2}}{2d_{1}d_{2}}
  34. σ \sigma
  35. τ \tau
  36. h σ = h τ = a cosh τ - cos σ h_{\sigma}=h_{\tau}=\frac{a}{\cosh\tau-\cos\sigma}
  37. h ϕ = a sinh τ cosh τ - cos σ h_{\phi}=\frac{a\sinh\tau}{\cosh\tau-\cos\sigma}
  38. d V = a 3 sinh τ ( cosh τ - cos σ ) 3 d σ d τ d ϕ dV=\frac{a^{3}\sinh\tau}{\left(\cosh\tau-\cos\sigma\right)^{3}}\,d\sigma\,d% \tau\,d\phi
  39. 2 Φ = ( cosh τ - cos σ ) 3 a 2 sinh τ [ sinh τ σ ( 1 cosh τ - cos σ Φ σ ) + τ ( sinh τ cosh τ - cos σ Φ τ ) + 1 sinh τ ( cosh τ - cos σ ) 2 Φ ϕ 2 ] \begin{aligned}\displaystyle\nabla^{2}\Phi=\frac{\left(\cosh\tau-\cos\sigma% \right)^{3}}{a^{2}\sinh\tau}&\displaystyle\left[\sinh\tau\frac{\partial}{% \partial\sigma}\left(\frac{1}{\cosh\tau-\cos\sigma}\frac{\partial\Phi}{% \partial\sigma}\right)\right.\\ &\displaystyle{}\quad+\left.\frac{\partial}{\partial\tau}\left(\frac{\sinh\tau% }{\cosh\tau-\cos\sigma}\frac{\partial\Phi}{\partial\tau}\right)+\frac{1}{\sinh% \tau\left(\cosh\tau-\cos\sigma\right)}\frac{\partial^{2}\Phi}{\partial\phi^{2}% }\right]\end{aligned}
  40. 𝐅 \nabla\cdot\mathbf{F}
  41. × 𝐅 \nabla\times\mathbf{F}
  42. ( σ , τ , ϕ ) (\sigma,\tau,\phi)
  43. 2 Φ = 0 \nabla^{2}\Phi=0
  44. Φ = U cosh τ - cos σ \Phi=U\sqrt{\cosh\tau-\cos\sigma}
  45. Φ = cosh τ - cos σ S ν ( σ ) T μ ν ( τ ) V μ ( ϕ ) \Phi=\sqrt{\cosh\tau-\cos\sigma}\,\,S_{\nu}(\sigma)T_{\mu\nu}(\tau)V_{\mu}(% \phi)\,
  46. S ν ( σ ) = e i ν σ and e - i ν σ S_{\nu}(\sigma)=e^{i\nu\sigma}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\nu\sigma}
  47. T μ ν ( τ ) = P ν - 1 / 2 μ ( cosh τ ) and Q ν - 1 / 2 μ ( cosh τ ) T_{\mu\nu}(\tau)=P_{\nu-1/2}^{\mu}(\cosh\tau)\,\,\,\,\mathrm{and}\,\,\,\,Q_{% \nu-1/2}^{\mu}(\cosh\tau)
  48. V μ ( ϕ ) = e i μ ϕ and e - i μ ϕ V_{\mu}(\phi)=e^{i\mu\phi}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\mu\phi}
  49. 1 < z = cosh η \,\!1<z=\cosh\eta\,
  50. n = 0 \,\!n=0
  51. Q - 1 2 ( z ) = 2 1 + z K ( 2 1 + z ) Q_{-\frac{1}{2}}(z)=\sqrt{\frac{2}{1+z}}K\left(\sqrt{\frac{2}{1+z}}\right)
  52. P - 1 2 ( z ) = 2 π 2 1 + z K ( z - 1 z + 1 ) P_{-\frac{1}{2}}(z)=\frac{2}{\pi}\sqrt{\frac{2}{1+z}}K\left(\sqrt{\frac{z-1}{z% +1}}\right)
  53. K \,\!K
  54. E \,\!E
  55. Φ = U ρ \Phi=\frac{U}{\sqrt{\rho}}
  56. ρ = x 2 + y 2 = a sinh τ cosh τ - cos σ . \rho=\sqrt{x^{2}+y^{2}}=\frac{a\sinh\tau}{\cosh\tau-\cos\sigma}.
  57. Φ = a ρ S ν ( σ ) T μ ν ( τ ) V μ ( ϕ ) \Phi=\frac{a}{\rho}\,\,S_{\nu}(\sigma)T_{\mu\nu}(\tau)V_{\mu}(\phi)\,
  58. S ν ( σ ) = e i ν σ and e - i ν σ S_{\nu}(\sigma)=e^{i\nu\sigma}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\nu\sigma}
  59. T μ ν ( τ ) = P μ - 1 / 2 ν ( coth τ ) and Q μ - 1 / 2 ν ( coth τ ) T_{\mu\nu}(\tau)=P_{\mu-1/2}^{\nu}(\coth\tau)\,\,\,\,\mathrm{and}\,\,\,\,Q_{% \mu-1/2}^{\nu}(\coth\tau)
  60. V μ ( ϕ ) = e i μ ϕ and e - i μ ϕ . V_{\mu}(\phi)=e^{i\mu\phi}\,\,\,\,\mathrm{and}\,\,\,\,e^{-i\mu\phi}.
  61. coth τ \coth\tau
  62. cosh τ \cosh\tau
  63. μ \mu
  64. ν \nu
  65. θ \theta

Torricelli's_law.html

  1. v = 2 g h v=\sqrt{2gh}
  2. 1 2 m v 2 \frac{1}{2}mv^{2}
  3. v 2 2 + g z + p ρ = constant {v^{2}\over 2}+gz+{p\over\rho}=\,\text{constant}
  4. g z + p a t m ρ = v 2 2 + p a t m ρ gz+{p_{atm}\over\rho}={v^{2}\over 2}+{p_{atm}\over\rho}
  5. v 2 = 2 g z \Rightarrow v^{2}=2gz\,
  6. v = 2 g z \Rightarrow v=\sqrt{2gz}
  7. v = 2 g h v=\sqrt{2gh}
  8. v = 2 g h = d x d t v=\sqrt{2gh}\ ={dx\over dt}
  9. A . d h a . d t = 2 g h {A.dh\over\ a.dt}=\sqrt{2gh}
  10. A . d h a . 2 g h = d t \Rightarrow{A.dh\over\ a.\sqrt{2gh}}=dt
  11. A . d h a . 2 g h = d t \Rightarrow\int\ {A.dh\over\ a.\sqrt{2gh}}=\int\ dt
  12. 2 A h a 2 g = t \Rightarrow{2A\sqrt{h}\ \over\ a\sqrt{2g}}=t
  13. t = 2 A h a 2 g \Rightarrow t={2A\sqrt{h}\ \over\ a\sqrt{2g}}
  14. t = 2 A a 2 g ( h 1 - h 2 ) \Rightarrow t={2A\over\ a\sqrt{2g}}(\sqrt{h1}-\sqrt{h2})

Torsion-free_abelian_groups_of_rank_1.html

  1. n a + m b = 0 na+mb=0\;
  2. p n a p n = a p^{n}a_{p^{n}}=a\;
  3. T ( a ) = { t 2 , t 3 , t 5 , } T(a)=\{t_{2},t_{3},t_{5},\ldots\}\;
  4. n a + m b = 0 na+mb=0\;

Torsion_group.html

  1. x . ( ( x = e ) ( x x = e ) ( ( x x ) x = e ) ) \forall x.\,((x=e)\lor(x\circ x=e)\lor((x\circ x)\circ x=e)\lor\cdots)

Torsion_tensor.html

  1. T ( X , Y ) = X Y - Y X - [ X , Y ] T(X,Y)=\nabla_{X}Y-\nabla_{Y}X-[X,Y]
  2. T ( X , Y ) := X Y - Y X - [ X , Y ] T(X,Y):=\nabla_{X}Y-\nabla_{Y}X-[X,Y]
  3. R ( X , Y ) Z = X Y Z - Y X Z - [ X , Y ] Z . R(X,Y)Z=\nabla_{X}\nabla_{Y}Z-\nabla_{Y}\nabla_{X}Z-\nabla_{[X,Y]}Z.
  4. 𝔖 \mathfrak{S}
  5. 𝔖 ( R ( X , Y ) Z ) := R ( X , Y ) Z + R ( Y , Z ) X + R ( Z , X ) Y . \mathfrak{S}\left(R(X,Y)Z\right):=R(X,Y)Z+R(Y,Z)X+R(Z,X)Y.
  6. 𝔖 ( R ( X , Y ) Z ) = 𝔖 ( T ( T ( X , Y ) , Z ) + ( X T ) ( Y , Z ) ) \mathfrak{S}\left(R(X,Y)Z\right)=\mathfrak{S}\left(T(T(X,Y),Z)+(\nabla_{X}T)(Y% ,Z)\right)
  7. 𝔖 ( ( X R ) ( Y , Z ) + R ( T ( X , Y ) , Z ) ) = 0 \mathfrak{S}\left((\nabla_{X}R)(Y,Z)+R(T(X,Y),Z)\right)=0
  8. T c a b T^{c}{}_{ab}
  9. T k := i j Γ k - i j Γ k - j i γ k , i j i , j , k = 1 , 2 , , n . T^{k}{}_{ij}:=\Gamma^{k}{}_{ij}-\Gamma^{k}{}_{ji}-\gamma^{k}{}_{ij},\quad i,j,% k=1,2,\ldots,n.
  10. γ k = i j 0 \gamma^{k}{}_{ij}=0
  11. T k = i j 2 Γ k [ i j ] T^{k}{}_{ij}=2\Gamma^{k}{}_{[ij]}
  12. θ ( X ) = u - 1 ( d π ( X ) ) \theta(X)=u^{-1}(d\pi(X))
  13. Θ = d θ + ω θ . \Theta=d\theta+\omega\wedge\theta.
  14. R g * Θ = g - 1 Θ R_{g}^{*}\Theta=g^{-1}\cdot\Theta
  15. Ω = D ω = d ω + ω ω \Omega=D\omega=d\omega+\omega\wedge\omega
  16. D Θ = Ω θ D\Theta=\Omega\wedge\theta
  17. D Ω = 0. D\Omega=0.\,
  18. R ( X , Y ) Z = u ( 2 Ω ( π - 1 ( X ) , π - 1 ( Y ) ) ) ( u - 1 ( Z ) ) , R(X,Y)Z=u\left(2\Omega(\pi^{-1}(X),\pi^{-1}(Y))\right)(u^{-1}(Z)),
  19. T ( X , Y ) = u ( 2 Θ ( π - 1 ( X ) , π - 1 ( Y ) ) ) , T(X,Y)=u\left(2\Theta(\pi^{-1}(X),\pi^{-1}(Y))\right),
  20. D 𝐞 i = 𝐞 j ω j . i D{\mathbf{e}}_{i}={\mathbf{e}}_{j}\omega^{j}{}_{i}.
  21. Θ k = d θ k + ω k j θ j = T k θ i i j θ j . \Theta^{k}=d\theta^{k}+\omega^{k}{}_{j}\wedge\theta^{j}=T^{k}{}_{ij}\theta^{i}% \wedge\theta^{j}.
  22. T k = i j θ k ( 𝐞 i 𝐞 j - 𝐞 j 𝐞 i - [ 𝐞 i , 𝐞 j ] ) T^{k}{}_{ij}=\theta^{k}(\nabla_{\mathbf{e}_{i}}\mathbf{e}_{j}-\nabla_{\mathbf{% e}_{j}}\mathbf{e}_{i}-[\mathbf{e}_{i},\mathbf{e}_{j}])
  23. 𝐞 ~ i = 𝐞 j g j i \tilde{\mathbf{e}}_{i}=\mathbf{e}_{j}g^{j}{}_{i}
  24. Θ ~ i = ( g - 1 ) i Θ j j . \tilde{\Theta}^{i}=(g^{-1})^{i}{}_{j}\Theta^{j}.
  25. Θ Hom ( 2 T M , T M ) \Theta\in\,\text{Hom}(\wedge^{2}TM,TM)
  26. Θ = D θ , \Theta=D\theta,\,
  27. a i = T k , i k a_{i}=T^{k}{}_{ik},
  28. B i = j k T i + j k 1 n - 1 δ i a k j - 1 n - 1 δ i a j k B^{i}{}_{jk}=T^{i}{}_{jk}+\frac{1}{n-1}\delta^{i}{}_{j}a_{k}-\frac{1}{n-1}% \delta^{i}{}_{k}a_{j}
  29. T Hom ( 2 T M , T M ) . T\in\operatorname{Hom}\left(\wedge^{2}TM,TM\right).
  30. T ( X ) : Y T ( X Y ) . T(X):Y\mapsto T(X\wedge Y).
  31. ( tr T ) ( X ) = def tr ( T ( X ) ) . (\operatorname{tr}\,T)(X)\stackrel{\,\text{def}}{=}\operatorname{tr}(T(X)).
  32. T 0 = T - 1 n - 1 ι ( tr T ) T_{0}=T-\frac{1}{n-1}\iota(\operatorname{tr}\,T)
  33. / t x | x = 0 = 0. \left.\nabla_{\partial/\partial t}\frac{\partial}{\partial x}\right|_{x=0}=0.
  34. T ( x , t ) | x = 0 = x t | x = 0 . \left.T\left(\frac{\partial}{\partial x},\frac{\partial}{\partial t}\right)% \right|_{x=0}=\left.\nabla_{\frac{\partial}{\partial x}}\frac{\partial}{% \partial t}\right|_{x=0}.
  35. γ ( t ) \gamma(t)
  36. γ ˙ ( t ) γ ˙ ( t ) = 0 \nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)=0
  37. γ ˙ ( 0 ) \dot{\gamma}(0)
  38. Δ ( X , Y ) = X Y ~ - X Y ~ \Delta(X,Y)=\nabla_{X}\tilde{Y}-\nabla^{\prime}_{X}\tilde{Y}
  39. S ( X , Y ) = 1 2 ( Δ ( X , Y ) + Δ ( Y , X ) ) S(X,Y)=\tfrac{1}{2}\left(\Delta(X,Y)+\Delta(Y,X)\right)
  40. A ( X , Y ) = 1 2 ( Δ ( X , Y ) - Δ ( Y , X ) ) A(X,Y)=\tfrac{1}{2}\left(\Delta(X,Y)-\Delta(Y,X)\right)
  41. A ( X , Y ) = 1 2 ( T ( X , Y ) - T ( X , Y ) ) A(X,Y)=\tfrac{1}{2}\left(T(X,Y)-T^{\prime}(X,Y)\right)

Total_air_temperature.html

  1. T total T s = 1 + γ - 1 2 M a 2 \frac{T_{\mathrm{total}}}{T_{s}}={1+\frac{\gamma-1}{2}M_{a}^{2}}
  2. T s = T_{s}=
  3. T total = T_{\mathrm{total}}=
  4. M a = M_{a}=
  5. γ = \gamma\ =\,
  6. T total T s = 1 + γ - 1 2 e M a 2 \frac{T_{\mathrm{total}}}{T_{s}}={1+\frac{\gamma-1}{2}eM_{a}^{2}}
  7. R R total = T A T - S A T RR_{\mathrm{total}}=TAT-SAT\,
  8. e {e}
  9. R R total = T s γ - 1 2 e M a 2 RR_{\mathrm{total}}={T_{s}\frac{\gamma-1}{2}eM_{a}^{2}}
  10. M a = V a M_{a}={\frac{V}{a}}
  11. a = γ R s p T s a={\sqrt{\gamma R_{sp}T_{s}}}
  12. R R total = e V 2 γ - 1 γ 2 R s p RR_{\mathrm{total}}={eV^{2}\frac{\gamma-1}{\gamma 2R_{sp}}}
  13. R R t o t a l = V 2 2 C p e RR_{total}={\frac{V^{2}}{2C_{p}}}e
  14. R s p = C p - C v R_{sp}={C_{p}-C_{v}}
  15. γ = C p C v \gamma={\frac{C_{p}}{C_{v}}}
  16. a = a=
  17. γ = \gamma=
  18. R s p = R_{sp}=
  19. R s p R_{sp}
  20. C p = C_{p}=
  21. C v = C_{v}=
  22. T s = T_{s}=
  23. V = V=
  24. e = e=
  25. R R total = V 2 87 2 RR_{\mathrm{total}}=\frac{V^{2}}{87^{2}}

Total_average.html

  1. T A = T B + H B P + B B + S B A B - H + C S + G I D P TA=\frac{TB+HBP+BB+SB}{AB-H+CS+GIDP}

Total_ionic_strength_adjustment_buffer.html

  1. E = K + R T n F ln ( c ) E=K^{\prime}+\frac{RT}{nF}\ln(c)
  2. E E
  3. R R
  4. T T
  5. F F
  6. n n
  7. c c

Total_ring_of_fractions.html

  1. R R
  2. S S
  3. R R
  4. S S
  5. R R
  6. S S
  7. S - 1 R = Q ( R ) S^{-1}R=Q(R)
  8. R R
  9. S = R - { 0 } S=R-\{0\}
  10. Q ( R ) Q(R)
  11. S S
  12. R Q ( R ) R\to Q(R)
  13. R R
  14. Q ( A × B ) Q(A\times B)
  15. Q ( A ) × Q ( B ) Q(A)\times Q(B)
  16. R × R^{\times}
  17. Q ( R ) = ( R × ) - 1 R Q(R)=(R^{\times})^{-1}R
  18. Q ( R ) = R Q(R)=R
  19. Q ( R ) = R Q(R)=R
  20. 𝔭 i Q ( A ) \mathfrak{p}_{i}Q(A)
  21. 𝔭 i Q ( A ) \mathfrak{p}_{i}Q(A)
  22. 𝔭 i \mathfrak{p}_{i}
  23. \square
  24. R R
  25. S S
  26. R R
  27. S - 1 R S^{-1}R
  28. R R
  29. S - 1 R S^{-1}R
  30. 0 S 0\in S
  31. S - 1 R S^{-1}R

Totally_disconnected_group.html

  1. α \alpha
  2. U + = n 0 α n ( U ) U_{+}=\bigcap_{n\geq 0}\alpha^{n}(U)
  3. U - = n 0 α - n ( U ) U_{-}=\bigcap_{n\geq 0}\alpha^{-n}(U)
  4. U + + = n 0 α n ( U + ) U_{++}=\bigcup_{n\geq 0}\alpha^{n}(U_{+})
  5. U - - = n 0 α - n ( U - ) U_{--}=\bigcup_{n\geq 0}\alpha^{-n}(U_{-})
  6. α \alpha
  7. U = U + U - = U - U + U=U_{+}U_{-}=U_{-}U_{+}
  8. U + + U_{++}
  9. U - - U_{--}
  10. α ( U + ) \alpha(U_{+})
  11. U + U_{+}
  12. α \alpha
  13. s ( α ) s(\alpha)
  14. s s
  15. s ( x ) := s ( α x ) s(x):=s(\alpha_{x})
  16. α x \alpha_{x}
  17. x x
  18. s s
  19. s ( x ) = 1 s(x)=1
  20. s ( x n ) = s ( x ) n s(x^{n})=s(x)^{n}
  21. n n
  22. Δ ( x ) = s ( x ) s ( x - 1 ) - 1 \Delta(x)=s(x)s(x^{-1})^{-1}

Tracking_error.html

  1. T E = ω = Var ( r p - r b ) = E [ ( r p - r b ) 2 ] - ( E [ r p - r b ] ) 2 TE=\omega=\sqrt{\operatorname{Var}(r_{p}-r_{b})}=\sqrt{{E}[(r_{p}-r_{b})^{2}]-% ({E}[r_{p}-r_{b}])^{2}}

Traian_Lalescu.html

  1. L n = ( n + 1 ) ! n + 1 - n ! n L_{n}=\sqrt[n+1]{(n+1)!}-\sqrt[n]{n!}
  2. lim n L n = 1 e \lim_{n\to\infty}L_{n}=\frac{1}{e}

Trakhtenbrot's_theorem.html

  1. t ! s ( q Q H q ( s , t ) ) ¬ s t ( q , q Q , q q H q ( s , t ) H q ( s , t ) ) \forall t\exists!s(\bigvee_{q\in Q}H_{q}(s,t))\land\neg\exists s\exists t(% \bigvee_{q,q^{\prime}\in Q,q\neq q}H_{q}(s,t)\land H_{q^{\prime}}(s,t))
  2. s t q Q a Q r H q ( s , t ) \exists s\exists t\bigvee_{q\in Q_{a}\cup Q_{r}}H_{q}(s,t)
  3. θ 0 s t ( s m i n ¯ T 0 ( s , t ) H q ( s , t ) ) θ 2 \theta_{0}\equiv\forall s\forall t(s\neq\underline{min}\land T_{0}(s,t)\land H% _{q}(s,t))\to\theta_{2}
  4. T 1 ( s , t + 1 ) H q ( s - 1 , t + 1 ) s ( s s ( i = 0 , 1 T i ( s , t + 1 ) T i ( s , t ) ) ) T_{1}(s,t+1)\land H_{q^{\prime}}(s-1,t+1)\land\forall s^{\prime}(s\neq s^{% \prime}\to(\bigwedge_{i=0,1}T_{i}(s^{\prime},t+1)\leftrightarrow T_{i}(s^{% \prime},t)))
  5. θ 1 s t ( s = m i n ¯ T 0 ( s , t ) H q ( s , t ) ) θ 3 \theta_{1}\equiv\forall s\forall t(s=\underline{min}\land T_{0}(s,t)\land H_{q% }(s,t))\to\theta_{3}
  6. T 1 ( s , t + 1 ) H q ( s , t + 1 ) s ( s s ( i = 0 , 1 T i ( s , t + 1 ) T i ( s , t ) ) ) T_{1}(s,t+1)\land H_{q^{\prime}}(s,t+1)\land\forall s^{\prime}(s\neq s^{\prime% }\to(\bigwedge_{i=0,1}T_{i}(s^{\prime},t+1)\leftrightarrow T_{i}(s^{\prime},t)))
  7. 𝒜 \mathcal{A}
  8. ( 𝒜 , ϕ ) (\mathcal{A},\phi)
  9. 𝒜 \mathcal{A}
  10. 𝒜 ϕ \mathcal{A}\models\phi

Transcritical_bifurcation.html

  1. d x d t = r x - x 2 . \frac{dx}{dt}=rx-x^{2}.\,
  2. r r
  3. x x
  4. x x
  5. r r
  6. x = 0 x=0
  7. x = r x=r
  8. r r
  9. x = 0 x=0
  10. x = r x=r
  11. r > 0 r>0
  12. x = 0 x=0
  13. x = r x=r
  14. r = 0 r=0
  15. d x d t = r x ( 1 - x ) - p x , \frac{dx}{dt}=rx(1-x)-px,
  16. r x ( 1 - x ) rx(1-x)
  17. p x px
  18. x x

Transfer_matrix.html

  1. h h
  2. a a
  3. b b
  4. h h
  5. T h T_{h}
  6. ( T h ) j , k = h 2 j - k . (T_{h})_{j,k}=h_{2\cdot j-k}.
  7. T h = ( h a h a + 2 h a + 1 h a h a + 4 h a + 3 h a + 2 h a + 1 h a h b h b - 1 h b - 2 h b - 3 h b - 4 h b h b - 1 h b - 2 h b ) . T_{h}=\begin{pmatrix}h_{a}&&&&&\\ h_{a+2}&h_{a+1}&h_{a}&&&\\ h_{a+4}&h_{a+3}&h_{a+2}&h_{a+1}&h_{a}&\\ \ddots&\ddots&\ddots&\ddots&\ddots&\ddots\\ &h_{b}&h_{b-1}&h_{b-2}&h_{b-3}&h_{b-4}\\ &&&h_{b}&h_{b-1}&h_{b-2}\\ &&&&&h_{b}\end{pmatrix}.
  8. T h T_{h}
  9. \downarrow
  10. T h x = ( h * x ) 2. T_{h}\cdot x=(h*x)\downarrow 2.
  11. T h x = T x h T_{h}\cdot x=T_{x}\cdot h
  12. h e h_{\mathrm{e}}
  13. h h
  14. ( h e ) k = h 2 k (h_{\mathrm{e}})_{k}=h_{2k}
  15. h o h_{\mathrm{o}}
  16. h h
  17. ( h o ) k = h 2 k + 1 (h_{\mathrm{o}})_{k}=h_{2k+1}
  18. det T h = ( - 1 ) b - a + 1 4 h a h b res ( h e , h o ) \det T_{h}=(-1)^{\lfloor\frac{b-a+1}{4}\rfloor}\cdot h_{a}\cdot h_{b}\cdot% \mathrm{res}(h_{\mathrm{e}},h_{\mathrm{o}})
  19. res \mathrm{res}
  20. tr T g * h = tr T g tr T h \mathrm{tr}~{}T_{g*h}=\mathrm{tr}~{}T_{g}\cdot\mathrm{tr}~{}T_{h}
  21. det T g * h = det T g det T h res ( g - , h ) \det T_{g*h}=\det T_{g}\cdot\det T_{h}\cdot\mathrm{res}(g_{-},h)
  22. g - g_{-}
  23. ( g - ) k = ( - 1 ) k g k (g_{-})_{k}=(-1)^{k}\cdot g_{k}
  24. T h x = 0 T_{h}\cdot x=0
  25. T g * h ( g - * x ) = 0 T_{g*h}\cdot(g_{-}*x)=0
  26. T g * h T_{g*h}
  27. T h T_{h}
  28. T h T_{h}
  29. T g * h T_{g*h}
  30. x x
  31. T h T_{h}
  32. λ \lambda
  33. T h x = λ x T_{h}\cdot x=\lambda\cdot x
  34. x * ( 1 , - 1 ) x*(1,-1)
  35. T h * ( 1 , 1 ) T_{h*(1,1)}
  36. T h * ( 1 , 1 ) ( x * ( 1 , - 1 ) ) = λ ( x * ( 1 , - 1 ) ) T_{h*(1,1)}\cdot(x*(1,-1))=\lambda\cdot(x*(1,-1))
  37. λ a , , λ b \lambda_{a},\dots,\lambda_{b}
  38. T h T_{h}
  39. λ a + + λ b = tr T h \lambda_{a}+\dots+\lambda_{b}=\mathrm{tr}~{}T_{h}
  40. λ a n + + λ b n = tr ( T h n ) \lambda_{a}^{n}+\dots+\lambda_{b}^{n}=\mathrm{tr}(T_{h}^{n})
  41. T h T_{h}
  42. n n
  43. C k h C_{k}h
  44. h h
  45. 2 k - 1 2^{k}-1
  46. C k h C_{k}h
  47. 2 k - 1 2^{k}-1
  48. \uparrow
  49. tr ( T h n ) = ( C k h * ( C k h 2 ) * ( C k h 2 2 ) * * ( C k h 2 n - 1 ) ) [ 0 ] 2 n - 1 \mathrm{tr}(T_{h}^{n})=\left(C_{k}h*(C_{k}h\uparrow 2)*(C_{k}h\uparrow 2^{2})*% \cdots*(C_{k}h\uparrow 2^{n-1})\right)_{[0]_{2^{n}-1}}
  50. n - 2 n-2
  51. 2 log 2 n 2\cdot\log_{2}n
  52. ϱ ( T h ) \varrho(T_{h})
  53. ϱ ( T h ) a # h 1 3 # h \varrho(T_{h})\geq\frac{a}{\sqrt{\#h}}\geq\frac{1}{\sqrt{3\cdot\#h}}
  54. # h \#h
  55. ϱ ( T h ) a \varrho(T_{h})\leq a
  56. a = C 2 h 2 a=\|C_{2}h\|_{2}

Transformity.html

  1. Transformity = emergy input energy output \mbox{Transformity}~{}=\frac{\mbox{emergy input}~{}}{\mbox{energy output}~{}}
  2. Transformity = emergy input exergy output \mbox{Transformity}~{}=\frac{\mbox{emergy input}~{}}{\mbox{exergy output}~{}}
  3. T r = E m E x Tr=\frac{E_{m}}{E_{x}}
  4. T r = t = - t 0 P x d t E x Tr=\frac{\int_{t=-\infty}^{t_{0}}P_{x}\,dt}{E_{x}}
  5. T r e x Tr_{ex}
  6. T r t h e t a Tr_{theta}
  7. T r e x Tr_{ex}
  8. T r t h e t a Tr_{theta}
  9. E m U = T r t h e t a . T r e x . E x U Em_{U}=Tr_{theta}.Tr_{ex}.Ex_{U}

Transitively_normal_subgroup.html

  1. H H
  2. G G
  3. K K
  4. H H
  5. K K
  6. G G

Transmission_disequilibrium_test.html

  1. M 1 M_{1}
  2. M 2 M_{2}
  3. χ 2 = [ b - ( b + c ) / 2 ] 2 ( b + c ) / 2 + [ c - ( b + c ) / 2 ] 2 ( b + c ) / 2 = ( b - c ) 2 b + c \chi^{2}=\frac{[b-(b+c)/2]^{2}}{(b+c)/2}+\frac{[c-(b+c)/2]^{2}}{(b+c)/2}=\frac% {(b-c)^{2}}{b+c}
  4. a , b , c a,b,c
  5. d d
  6. b b
  7. c c
  8. χ 2 \chi^{2}
  9. a , b , c a,b,c
  10. d d
  11. ( a + b ) / 2 n (a+b)/2n
  12. ( c + d ) / 2 n (c+d)/2n
  13. ( a + c ) / 2 n (a+c)/2n
  14. ( b + d ) / 2 n (b+d)/2n
  15. χ 2 \chi^{2}
  16. h h
  17. i i
  18. M 1 M_{1}
  19. h - i - j h-i-j
  20. M 1 M_{1}
  21. M 2 M_{2}
  22. j j
  23. M 2 M_{2}
  24. b = 2 i + ( h - i - j ) = h + i - j b=2i+(h-i-j)=h+i-j\,
  25. c = 2 j + ( h - i - j ) = h - i + j c=2j+(h-i-j)=h-i+j\,
  26. χ t d t 2 = 4 ( i - j ) 2 h . \chi_{tdt}^{2}=\frac{4(i-j)^{2}}{h}.
  27. χ h s 2 = ( 2 i + 2 j - h ) 2 h . \chi^{2}_{hs}=\frac{(2i+2j-h)^{2}}{h}.
  28. χ t o t a l 2 = ( i - h / 4 ) 2 h / 4 + ( h - i - j - h / 2 ) 2 h / 2 + ( j - h / 4 ) 2 h / 4 = χ t d t 2 + χ h s 2 . \chi^{2}_{total}=\frac{(i-h/4)^{2}}{h/4}+\frac{(h-i-j-h/2)^{2}}{h/2}+\frac{(j-% h/4)^{2}}{h/4}=\chi^{2}_{tdt}+\chi^{2}_{hs}.
  29. χ 2 = [ [ n P Q - n Q Q ] P Q Q Q + 2 × [ n P P - n Q Q ] P Q P Q + [ n P P - n P Q ] P P P Q ] 2 [ n P Q + n Q Q ] P Q Q Q + 4 × [ n P P + n Q Q ] P Q P Q + [ n P Q + n P P ] P P P Q \chi^{2}=\frac{\left[[n_{PQ}-n_{QQ}]_{PQ\sim QQ}+2\times[n_{PP}-n_{QQ}]_{PQ% \sim PQ}+[n_{PP}-n_{PQ}]_{PP\sim PQ}\right]^{2}}{[n_{PQ}+n_{QQ}]_{PQ\sim QQ}+4% \times[n_{PP}+n_{QQ}]_{PQ\sim PQ}+[n_{PQ}+n_{PP}]_{PP\sim PQ}}
  30. [ n P Q ] P Q Q Q [n_{PQ}]_{PQ\sim QQ}

Transversality_(mathematics).html

  1. L 1 L 2 L_{1}\pitchfork L_{2}
  2. L 1 L 2 p L 1 L 2 , T p M = T p L 1 + T p L 2 . L_{1}\pitchfork L_{2}\iff\forall p\in L_{1}\cap L_{2},\mathrm{T}_{p}M=\mathrm{% T}_{p}L_{1}+\mathrm{T}_{p}L_{2}.
  3. f 1 : L 1 M f_{1}:L_{1}\to M
  4. f 2 : L 2 M f_{2}:L_{2}\to M
  5. L 1 , L 2 and M L_{1},L_{2}\,\text{ and }M
  6. 1 , 2 and m \ell_{1},\ell_{2}\,\text{ and }m
  7. M , L 1 M,L_{1}
  8. L 2 L_{2}
  9. 1 + 2 = m \ell_{1}+\ell_{2}=m
  10. 1 + 2 < m \ell_{1}+\ell_{2}<m
  11. L 1 L_{1}
  12. L 2 L_{2}
  13. M M
  14. f 1 f_{1}
  15. f 2 f_{2}
  16. 1 + 2 = m \ell_{1}+\ell_{2}=m
  17. L 1 L_{1}
  18. L 2 L_{2}
  19. M M
  20. 1 + 2 > m \ell_{1}+\ell_{2}>m
  21. f 1 f_{1}
  22. f 2 f_{2}
  23. 1 + 2 - m \ell_{1}+\ell_{2}-m
  24. G G
  25. 𝔤 \mathfrak{g}
  26. e 𝔤 e\in\mathfrak{g}
  27. 𝔰 𝔩 2 \mathfrak{sl_{2}}
  28. ( e , h , f ) (e,h,f)
  29. 𝔰 𝔩 2 \mathfrak{sl_{2}}
  30. 𝔤 = [ 𝔤 , e ] 𝔤 f \mathfrak{g}=[\mathfrak{g},e]\oplus\mathfrak{g}_{f}
  31. [ 𝔤 , e ] [\mathfrak{g},e]
  32. e e
  33. Ad ( G ) e \rm{Ad}(G)e
  34. e + 𝔤 f e+\mathfrak{g}_{f}
  35. e e
  36. e + 𝔤 f e+\mathfrak{g}_{f}
  37. F ( x , y , y ) d x \int{F(x,y,y^{\prime})}dx

Trapezoidal_thread_forms.html

  1. 1 / 4 {1}/{4}
  2. 1 / 16 {1}/{16}
  3. 5 / 16 {5}/{16}
  4. 1 / 14 {1}/{14}
  5. 3 / 8 {3}/{8}
  6. 1 / 12 {1}/{12}
  7. 1 / 2 {1}/{2}
  8. 1 / 10 {1}/{10}
  9. 5 / 8 {5}/{8}
  10. 1 / 8 {1}/{8}
  11. 3 / 4 {3}/{4}
  12. 7 / 8 {7}/{8}
  13. 1 / 6 {1}/{6}
  14. 1 / 4 {1}/{4}
  15. 1 / 5 {1}/{5}
  16. 1 / 2 {1}/{2}
  17. 3 / 4 {3}/{4}
  18. 1 / 4 {1}/{4}
  19. 1 / 2 {1}/{2}
  20. 1 / 3 {1}/{3}
  21. 1 / 2 {1}/{2}
  22. T r 60 × 9 Tr\,60\times 9
  23. T r 60 × 18 ( P 9 ) L H Tr\,60\times 18(P9)LH

Treadwear_rating.html

  1. μ = 2.25 T W 0.15 \mu=\frac{2.25}{TW^{0.15}}

Tree_alignment.html

  1. S S
  2. T T
  3. S S
  4. d d
  5. T T
  6. Σ e T d ( e ) \Sigma_{e\in T}d(e)
  7. d ( e ) d(e)
  8. e e
  9. S S
  10. s 1 s_{1}
  11. s n s_{n}
  12. P P
  13. p 1 p_{1}
  14. p 2 p_{2}
  15. p z p_{z}
  16. P i P_{i}
  17. P P
  18. P i P_{i}
  19. P i P_{i}
  20. P i P_{i}
  21. P i P_{i}
  22. P P
  23. p 1 p_{1}
  24. p 2 p_{2}
  25. p z p_{z}
  26. P i P_{i}
  27. v v
  28. v v
  29. P i P_{i}
  30. P P
  31. L ( v ) L(v)
  32. v v
  33. L p ( v ) Lp(v)
  34. P P
  35. n v n_{v}
  36. L p ( v ) Lp(v)
  37. n v n_{v}
  38. v v
  39. n v n_{v}
  40. P P
  41. L ( v ) L(v)
  42. L p ( v ) Lp(v)
  43. v v
  44. n v n_{v}
  45. n v n_{v}
  46. v v
  47. n v n_{v}
  48. v v
  49. v v^{\prime}
  50. v v
  51. v v^{\prime}
  52. n v n_{v}^{\prime}
  53. w w
  54. n v n_{v}
  55. w w
  56. n v n_{v}^{\prime}
  57. L ( n v ) L(n_{v})
  58. L ( n v ) L(n_{v}^{\prime})
  59. n v n_{v}^{\prime}
  60. P i P_{i}

Tree_spanner.html

  1. t > 3 t>3
  2. O ( m × n + n 2 × l o g ( n ) ) O(m\times n+n^{2}\times log(n))
  3. t > 1 t>1
  4. O ( m + n ) O(m+n)

Triangular_function.html

  1. tri ( t ) = and ( t ) = def max ( 1 - | t | , 0 ) = { 1 - | t | , | t | < 1 0 , otherwise \begin{aligned}\displaystyle\operatorname{tri}(t)=\and(t)&\displaystyle% \overset{\underset{\mathrm{def}}{}}{=}\ \max(1-|t|,0)\\ &\displaystyle=\begin{cases}1-|t|,&|t|<1\\ 0,&\mbox{otherwise}\end{cases}\end{aligned}
  2. tri ( t ) = rect ( t ) * rect ( t ) = def - rect ( τ ) rect ( t - τ ) d τ = - rect ( τ ) rect ( τ - t ) d τ . \begin{aligned}\displaystyle\operatorname{tri}(t)=\operatorname{rect}(t)*% \operatorname{rect}(t)&\displaystyle\overset{\underset{\mathrm{def}}{}}{=}\int% _{-\infty}^{\infty}\mathrm{rect}(\tau)\cdot\mathrm{rect}(t-\tau)\ d\tau\\ &\displaystyle=\int_{-\infty}^{\infty}\mathrm{rect}(\tau)\cdot\mathrm{rect}(% \tau-t)\ d\tau.\end{aligned}
  3. tri ( t ) = rect ( t / 2 ) ( 1 - | t | ) \operatorname{tri}(t)=\operatorname{rect}(t/2)\left(1-\left|t\right|\right)
  4. tri ( t ) = and ( t ) \displaystyle\operatorname{tri}(t)=\and(t)
  5. a 0 a\neq 0\,
  6. tri ( t / a ) = - rect ( τ ) rect ( τ - t / a ) d τ = { 1 - | t / a | , | t | < | a | 0 , otherwise . \begin{aligned}\displaystyle\operatorname{tri}(t/a)&\displaystyle=\int_{-% \infty}^{\infty}\mathrm{rect}(\tau)\cdot\mathrm{rect}(\tau-t/a)\ d\tau\\ &\displaystyle=\begin{cases}1-|t/a|,&|t|<|a|\\ 0,&\mbox{otherwise}~{}.\end{cases}\end{aligned}
  7. { tri ( t ) } = { rect ( t ) * rect ( t ) } = { rect ( t ) } { rect ( t ) } = { rect ( t ) } 2 = sinc 2 ( f ) , \begin{aligned}\displaystyle\mathcal{F}\{\operatorname{tri}(t)\}&\displaystyle% =\mathcal{F}\{\operatorname{rect}(t)*\operatorname{rect}(t)\}\\ &\displaystyle=\mathcal{F}\{\operatorname{rect}(t)\}\cdot\mathcal{F}\{% \operatorname{rect}(t)\}\\ &\displaystyle=\mathcal{F}\{\operatorname{rect}(t)\}^{2}\\ &\displaystyle=\mathrm{sinc}^{2}(f),\end{aligned}
  8. sinc ( x ) = sin ( π x ) / ( π x ) \operatorname{sinc}(x)=\sin(\pi x)/(\pi x)

Tridiagonal_matrix_algorithm.html

  1. a i x i - 1 + b i x i + c i x i + 1 = d i , a_{i}x_{i-1}+b_{i}x_{i}+c_{i}x_{i+1}=d_{i},\,\!
  2. a 1 = 0 a_{1}=0\,
  3. c n = 0 c_{n}=0\,
  4. [ b 1 c 1 0 a 2 b 2 c 2 a 3 b 3 c n - 1 0 a n b n ] [ x 1 x 2 x 3 x n ] = [ d 1 d 2 d 3 d n ] . \begin{bmatrix}{b_{1}}&{c_{1}}&&&{0}\\ {a_{2}}&{b_{2}}&{c_{2}}&&\\ &{a_{3}}&{b_{3}}&\ddots&\\ &&\ddots&\ddots&{c_{n-1}}\\ {0}&&&{a_{n}}&{b_{n}}\\ \end{bmatrix}\begin{bmatrix}{x_{1}}\\ {x_{2}}\\ {x_{3}}\\ \vdots\\ {x_{n}}\\ \end{bmatrix}=\begin{bmatrix}{d_{1}}\\ {d_{2}}\\ {d_{3}}\\ \vdots\\ {d_{n}}\\ \end{bmatrix}.
  5. O ( n ) O(n)
  6. O ( n 3 ) O(n^{3})
  7. a i a_{i}
  8. c i = { c i b i ; i = 1 c i b i - a i c i - 1 ; i = 2 , 3 , , n - 1 c^{\prime}_{i}=\begin{cases}\begin{array}[]{lcl}\cfrac{c_{i}}{b_{i}}&;&i=1\\ \cfrac{c_{i}}{b_{i}-a_{i}c^{\prime}_{i-1}}&;&i=2,3,\dots,n-1\\ \end{array}\end{cases}\,
  9. d i = { d i b i ; i = 1 d i - a i d i - 1 b i - a i c i - 1 ; i = 2 , 3 , , n . d^{\prime}_{i}=\begin{cases}\begin{array}[]{lcl}\cfrac{d_{i}}{b_{i}}&;&i=1\\ \cfrac{d_{i}-a_{i}d^{\prime}_{i-1}}{b_{i}-a_{i}c^{\prime}_{i-1}}&;&i=2,3,\dots% ,n.\\ \end{array}\end{cases}\,
  10. x n = d n x_{n}=d^{\prime}_{n}\,
  11. x i = d i - c i x i + 1 ; i = n - 1 , n - 2 , , 1. x_{i}=d^{\prime}_{i}-c^{\prime}_{i}x_{i+1}\qquad;\ i=n-1,n-2,\ldots,1.
  12. x 1 , , x n x_{1},\ldots,x_{n}
  13. b 1 x 1 + c 1 x 2 \displaystyle b_{1}x_{1}+c_{1}x_{2}
  14. i = 2 i=2
  15. ( equation 2 ) b 1 - ( equation 1 ) a 2 (\mbox{equation 2}~{})\cdot b_{1}-(\mbox{equation 1}~{})\cdot a_{2}
  16. ( a 2 x 1 + b 2 x 2 + c 2 x 3 ) b 1 - ( b 1 x 1 + c 1 x 2 ) a 2 = d 2 b 1 - d 1 a 2 (a_{2}x_{1}+b_{2}x_{2}+c_{2}x_{3})b_{1}-(b_{1}x_{1}+c_{1}x_{2})a_{2}=d_{2}b_{1% }-d_{1}a_{2}\,
  17. ( b 2 b 1 - c 1 a 2 ) x 2 + c 2 b 1 x 3 = d 2 b 1 - d 1 a 2 (b_{2}b_{1}-c_{1}a_{2})x_{2}+c_{2}b_{1}x_{3}=d_{2}b_{1}-d_{1}a_{2}\,
  18. x 1 x_{1}
  19. ( a 3 x 2 + b 3 x 3 + c 3 x 4 ) ( b 2 b 1 - c 1 a 2 ) - ( ( b 2 b 1 - c 1 a 2 ) x 2 + c 2 b 1 x 3 ) a 3 = d 3 ( b 2 b 1 - c 1 a 2 ) - ( d 2 b 1 - d 1 a 2 ) a 3 (a_{3}x_{2}+b_{3}x_{3}+c_{3}x_{4})(b_{2}b_{1}-c_{1}a_{2})-((b_{2}b_{1}-c_{1}a_% {2})x_{2}+c_{2}b_{1}x_{3})a_{3}=d_{3}(b_{2}b_{1}-c_{1}a_{2})-(d_{2}b_{1}-d_{1}% a_{2})a_{3}\,
  20. ( b 3 ( b 2 b 1 - c 1 a 2 ) - c 2 b 1 a 3 ) x 3 + c 3 ( b 2 b 1 - c 1 a 2 ) x 4 = d 3 ( b 2 b 1 - c 1 a 2 ) - ( d 2 b 1 - d 1 a 2 ) a 3 . (b_{3}(b_{2}b_{1}-c_{1}a_{2})-c_{2}b_{1}a_{3})x_{3}+c_{3}(b_{2}b_{1}-c_{1}a_{2% })x_{4}=d_{3}(b_{2}b_{1}-c_{1}a_{2})-(d_{2}b_{1}-d_{1}a_{2})a_{3}.\,
  21. x 2 x_{2}
  22. n t h n^{th}
  23. n t h n^{th}
  24. x n x_{n}
  25. ( n - 1 ) t h (n-1)^{th}
  26. a ~ i = 0 \tilde{a}_{i}=0\,
  27. b ~ 1 = b 1 \tilde{b}_{1}=b_{1}\,
  28. b ~ i = b i b ~ i - 1 - c ~ i - 1 a i \tilde{b}_{i}=b_{i}\tilde{b}_{i-1}-\tilde{c}_{i-1}a_{i}\,
  29. c ~ 1 = c 1 \tilde{c}_{1}=c_{1}\,
  30. c ~ i = c i b ~ i - 1 \tilde{c}_{i}=c_{i}\tilde{b}_{i-1}\,
  31. d ~ 1 = d 1 \tilde{d}_{1}=d_{1}\,
  32. d ~ i = d i b ~ i - 1 - d ~ i - 1 a i . \tilde{d}_{i}=d_{i}\tilde{b}_{i-1}-\tilde{d}_{i-1}a_{i}.\,
  33. b ~ i \tilde{b}_{i}
  34. a i = 0 a^{\prime}_{i}=0\,
  35. b i = 1 b^{\prime}_{i}=1\,
  36. c 1 = c 1 b 1 c^{\prime}_{1}=\frac{c_{1}}{b_{1}}\,
  37. c i = c i b i - c i - 1 a i c^{\prime}_{i}=\frac{c_{i}}{b_{i}-c^{\prime}_{i-1}a_{i}}\,
  38. d 1 = d 1 b 1 d^{\prime}_{1}=\frac{d_{1}}{b_{1}}\,
  39. d i = d i - d i - 1 a i b i - c i - 1 a i . d^{\prime}_{i}=\frac{d_{i}-d^{\prime}_{i-1}a_{i}}{b_{i}-c^{\prime}_{i-1}a_{i}}.\,
  40. b i x i + c i x i + 1 = d i ; i = 1 , , n - 1 b n x n = d n ; i = n . \begin{array}[]{lcl}b^{\prime}_{i}x_{i}+c^{\prime}_{i}x_{i+1}=d^{\prime}_{i}&;% &\ i=1,\ldots,n-1\\ b^{\prime}_{n}x_{n}=d^{\prime}_{n}&;&\ i=n.\\ \end{array}\,
  41. x n = d n / b n x_{n}=d^{\prime}_{n}/b^{\prime}_{n}\,
  42. x i = ( d i - c i x i + 1 ) / b i ; i = n - 1 , n - 2 , , 1. x_{i}=(d^{\prime}_{i}-c^{\prime}_{i}x_{i+1})/b^{\prime}_{i}\qquad;\ i=n-1,n-2,% \ldots,1.
  43. a 1 x n + b 1 x 1 + c 1 x 2 \displaystyle a_{1}x_{n}+b_{1}x_{1}+c_{1}x_{2}

Trigenus.html

  1. ( g 1 , g 2 , g 3 ) (g_{1},g_{2},g_{3})
  2. M = V 1 V 2 V 3 M=V_{1}\cup V_{2}\cup V_{3}
  3. int V i int V j = {\rm int}V_{i}\cap{\rm int}V_{j}=\varnothing
  4. i , j = 1 , 2 , 3 i,j=1,2,3
  5. g i g_{i}
  6. V i V_{i}
  7. trig ( M ) = ( 0 , 0 , h ) {\rm trig}(M)=(0,0,h)
  8. h h
  9. M M
  10. trig {\rm trig}
  11. trig ( M ) = ( 0 , g 2 , g 3 ) or ( 1 , g 2 , g 3 ) {\rm trig}(M)=(0,g_{2},g_{3})\quad\mbox{or}~{}\quad(1,g_{2},g_{3})
  12. w 1 w_{1}
  13. β ( w 1 ) = 0 or 0. \beta(w_{1})=0\quad\mbox{or}~{}\quad\neq 0.
  14. g 2 g_{2}
  15. G G
  16. M M
  17. D : H 1 ( M ; 2 ) H 2 ( M ; 2 ) , D\colon H^{1}(M;{\mathbb{Z}}_{2})\to H_{2}(M;{\mathbb{Z}}_{2}),
  18. D w 1 ( M ) = [ G ] Dw_{1}(M)=[G]
  19. β ( w 1 ) = 0 \beta(w_{1})=0\,
  20. trig ( M ) = ( 0 , 2 g , g 3 ) {\rm trig}(M)=(0,2g,g_{3})\,
  21. β ( w 1 ) 0. \beta(w_{1})\neq 0.\,
  22. trig ( M ) = ( 1 , 2 g - 1 , g 3 ) {\rm trig}(M)=(1,2g-1,g_{3})\,
  23. S 1 S^{1}

Trigonometric_series.html

  1. A 0 + n = 1 ( A n cos n x + B n sin n x ) . A_{0}+\displaystyle\sum_{n=1}^{\infty}(A_{n}\cos{nx}+B_{n}\sin{nx}).
  2. A n A_{n}
  3. B n B_{n}
  4. A n = 1 π 0 2 π f ( x ) cos n x d x ( n = 0 , 1 , 2 , 3 ) A_{n}=\frac{1}{\pi}\displaystyle\int^{2\pi}_{0}\!f(x)\cos{nx}\,dx\qquad(n=0,1,% 2,3\dots)
  5. B n = 1 π 0 2 π f ( x ) sin n x d x ( n = 1 , 2 , 3 , ) B_{n}=\frac{1}{\pi}\displaystyle\int^{2\pi}_{0}\!f(x)\sin{nx}\,dx\qquad(n=1,2,% 3,\dots)
  6. f f
  7. f ( x ) f(x)
  8. [ 0 , 2 π ] [0,2\pi]
  9. f f

Trilinear_coordinates.html

  1. ( 0 , Δ b , Δ c ) , (0,\frac{\Delta}{b},\frac{\Delta}{c}),
  2. 0 : c a : a b . 0:ca:ab.
  3. ( 0 , 2 Δ a cos C , 2 Δ a cos B ) (0,\frac{2\Delta}{a}\cos C,\frac{2\Delta}{a}\cos B)
  4. Δ , \Delta,
  5. 0 : cos C : cos B . 0:\cos C:\cos B.
  6. D = | p q r u v w x y z | D=\begin{vmatrix}p&q&r\\ u&v&w\\ x&y&z\end{vmatrix}
  7. l x + m y + n z = 0 lx+my+nz=0
  8. l x + m y + n z = 0 l^{\prime}x+m^{\prime}y+n^{\prime}z=0
  9. | l m n l m n a b c | = 0 , \begin{vmatrix}l&m&n\\ l^{\prime}&m^{\prime}&n^{\prime}\\ a&b&c\end{vmatrix}=0,
  10. l x + m y + n z = 0 lx+my+nz=0
  11. l x + m y + n z = 0 l^{\prime}x+m^{\prime}y+n^{\prime}z=0
  12. l l + m m + n n - ( m n + m n ) cos A - ( n l + n l ) cos B - ( l m + l m ) cos C = 0. ll^{\prime}+mm^{\prime}+nn^{\prime}-(mn^{\prime}+m^{\prime}n)\cos A-(nl^{% \prime}+n^{\prime}l)\cos B-(lm^{\prime}+l^{\prime}m)\cos C=0.
  13. y cos B - z cos C = 0. y\cos B-z\cos C=0.
  14. a p x + b q y + c r z = 0. apx+bqy+crz=0.
  15. a a + b b + c c = 2 Δ aa^{\prime}+bb^{\prime}+cc^{\prime}=2\Delta
  16. Δ \Delta
  17. d 2 sin 2 C = ( a 1 - a 2 ) 2 + ( b 1 - b 2 ) 2 + 2 ( a 1 - a 2 ) ( b 1 - b 2 ) cos C . d^{2}\sin^{2}C=(a^{\prime}_{1}-a^{\prime}_{2})^{2}+(b^{\prime}_{1}-b^{\prime}_% {2})^{2}+2(a^{\prime}_{1}-a^{\prime}_{2})(b^{\prime}_{1}-b^{\prime}_{2})\cos C.
  18. d = l a 0 + m b 0 + n c 0 l 2 + m 2 + n 2 - 2 m n cos A - 2 n l cos B - 2 l m cos C . d=\frac{la^{\prime}_{0}+mb^{\prime}_{0}+nc^{\prime}_{0}}{\sqrt{l^{2}+m^{2}+n^{% 2}-2mn\cos A-2nl\cos B-2lm\cos C}}.
  19. r x 2 + s y 2 + t z 2 + 2 u y z + 2 v z x + 2 w x y = 0. rx^{2}+sy^{2}+tz^{2}+2uyz+2vzx+2wxy=0.
  20. ( x - a ) 2 sin 2 A + ( y - b ) 2 sin 2 B + ( z - c ) 2 sin 2 C = 2 r 2 sin A sin B sin C . (x-a^{\prime})^{2}\sin 2A+(y-b^{\prime})^{2}\sin 2B+(z-c^{\prime})^{2}\sin 2C=% 2r^{2}\sin A\sin B\sin C.
  21. l y z + m z x + n x y = 0. lyz+mzx+nxy=0.
  22. y z ( x - y - z ) + z x ( y - z - x ) + x y ( z - x - y ) = 0. yz(x^{\prime}-y^{\prime}-z^{\prime})+zx(y^{\prime}-z^{\prime}-x^{\prime})+xy(z% ^{\prime}-x^{\prime}-y^{\prime})=0.
  23. l 2 x 2 + m 2 y 2 + n 2 z 2 ± 2 m n y z ± 2 n l z x ± 2 l m x y = 0 , l^{2}x^{2}+m^{2}y^{2}+n^{2}z^{2}\pm 2mnyz\pm 2nlzx\pm 2lmxy=0,
  24. ± x cos A 2 ± y cos B 2 ± z cos C 2 = 0 , \pm\sqrt{x}\cos\frac{A}{2}\pm\sqrt{y}\cos\frac{B}{2}\pm\sqrt{z}\cos\frac{C}{2}% =0,
  25. ± - x cos A 2 ± y cos B 2 ± z cos C 2 = 0. \pm\sqrt{-x}\cos\frac{A}{2}\pm\sqrt{y}\cos\frac{B}{2}\pm\sqrt{z}\cos\frac{C}{2% }=0.
  26. | x y z q r y z r p z x p q x y u v w | = 0. \begin{vmatrix}x&y&z\\ qryz&rpzx&pqxy\\ u&v&w\end{vmatrix}=0.
  27. k = 2 Δ a x + b y + c z k=\frac{2\Delta}{ax+by+cz}
  28. x : y : z = k 1 a : k 2 b : 1 - k 1 - k 2 c , x:y:z=\frac{k_{1}}{a}:\frac{k_{2}}{b}:\frac{1-k_{1}-k_{2}}{c},
  29. k 1 = a x a x + b y + c z , k 2 = b y a x + b y + c z . k_{1}=\frac{ax}{ax+by+cz},\quad k_{2}=\frac{by}{ax+by+cz}.
  30. P ¯ = a x a x + b y + c z A ¯ + b y a x + b y + c z B ¯ + c z a x + b y + c z C ¯ , \underline{P}=\frac{ax}{ax+by+cz}\underline{A}+\frac{by}{ax+by+cz}\underline{B% }+\frac{cz}{ax+by+cz}\underline{C},

Truncated_24-cells.html

  1. F 3 {F}_{3}
  2. C 3 {C}_{3}
  3. D 3 {D}_{3}
  4. 1 \scriptstyle 1
  5. 2 - 2 \scriptstyle\sqrt{2-\sqrt{2}}
  6. 2 \sqrt{2}
  7. 2 2 \sqrt{2–\sqrt{2}}
  8. 4 + 8 \sqrt{4+\sqrt{8}}
  9. 2 \sqrt{2}
  10. 2 \sqrt{2}
  11. 2 \sqrt{2}

Truncated_5-cell.html

  1. ( 3 10 , 3 2 , ± 3 , ± 1 ) \left(\frac{3}{\sqrt{10}},\ \sqrt{3\over 2},\ \pm\sqrt{3},\ \pm 1\right)
  2. ( 3 10 , 3 2 , 0 , ± 2 ) \left(\frac{3}{\sqrt{10}},\ \sqrt{3\over 2},\ 0,\ \pm 2\right)
  3. ( 3 10 , - 1 6 , 2 3 , ± 2 ) \left(\frac{3}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ \pm 2\right)
  4. ( 3 10 , - 1 6 , 4 3 , 0 ) \left(\frac{3}{\sqrt{10}},\ \frac{-1}{\sqrt{6}},\ \frac{4}{\sqrt{3}},\ 0\right)
  5. ( 3 10 , - 5 6 , 1 3 , ± 1 ) \left(\frac{3}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm 1\right)
  6. ( 3 10 , - 5 6 , - 2 3 , 0 ) \left(\frac{3}{\sqrt{10}},\ \frac{-5}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0\right)
  7. ( - 2 5 , 2 3 , 2 3 , ± 2 ) \left(-\sqrt{2\over 5},\ \sqrt{2\over 3},\ \frac{2}{\sqrt{3}},\ \pm 2\right)
  8. ( - 2 5 , 2 3 , - 4 3 , 0 ) \left(-\sqrt{2\over 5},\ \sqrt{2\over 3},\ \frac{-4}{\sqrt{3}},\ 0\right)
  9. ( - 2 5 , - 6 , 0 , 0 ) \left(-\sqrt{2\over 5},\ -\sqrt{6},\ 0,\ 0\right)
  10. ( - 7 10 , 1 6 , 1 3 , ± 1 ) \left(\frac{-7}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{1}{\sqrt{3}},\ \pm 1\right)
  11. ( - 7 10 , 1 6 , - 2 3 , 0 ) \left(\frac{-7}{\sqrt{10}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ 0\right)
  12. ( - 7 10 , - 3 2 , 0 , 0 ) \left(\frac{-7}{\sqrt{10}},\ -\sqrt{3\over 2},\ 0,\ 0\right)
  13. n n
  14. ± ( 5 2 , 5 6 , 2 3 , 0 ) \pm\left(\sqrt{\frac{5}{2}},\ \frac{5}{\sqrt{6}},\ \frac{2}{\sqrt{3}},\ 0\right)
  15. ± ( 5 2 , 5 6 , - 1 3 , ± 1 ) \pm\left(\sqrt{\frac{5}{2}},\ \frac{5}{\sqrt{6}},\ \frac{-1}{\sqrt{3}},\ \pm 1\right)
  16. ± ( 5 2 , 1 6 , 4 3 , 0 ) \pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\ \frac{4}{\sqrt{3}},\ 0\right)
  17. ± ( 5 2 , 1 6 , - 2 3 , ± 2 ) \pm\left(\sqrt{\frac{5}{2}},\ \frac{1}{\sqrt{6}},\ \frac{-2}{\sqrt{3}},\ \pm 2\right)
  18. ± ( 5 2 , - 3 2 , ± 3 , ± 1 ) \pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ \pm\sqrt{3},\ \pm 1\right)
  19. ± ( 5 2 , - 3 2 , 0 , ± 2 ) \pm\left(\sqrt{\frac{5}{2}},\ -\sqrt{\frac{3}{2}},\ 0,\ \pm 2\right)
  20. ± ( 0 , 2 2 3 , 4 3 , 0 ) \pm\left(0,\ 2\sqrt{\frac{2}{3}},\ \frac{4}{\sqrt{3}},\ 0\right)
  21. ± ( 0 , 2 2 3 , - 2 3 , ± 2 ) \pm\left(0,\ 2\sqrt{\frac{2}{3}},\ \frac{-2}{\sqrt{3}},\ \pm 2\right)

Truncated_dodecadodecahedron.html

  1. φ = 1 + 5 2 \varphi=\frac{1+\sqrt{5}}{2}
  2. ( 1 , 1 , 3 ) ; ( 1 φ , 1 φ 2 , 2 φ ) ; ( φ , 2 φ , φ 2 ) ; ( φ 2 , 1 φ 2 , 2 ) ; ( 2 φ - 1 , 1 , 2 φ - 1 ) . (1,1,3);\quad(\frac{1}{\varphi},\frac{1}{\varphi^{2}},2\varphi);\quad(\varphi,% \frac{2}{\varphi},\varphi^{2});\quad(\varphi^{2},\frac{1}{\varphi^{2}},2);% \quad(2\varphi-1,1,2\varphi-1).

Truncated_tesseract.html

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

Truncation_(geometry).html

  1. { 5 3 } \begin{Bmatrix}5\\ 3\end{Bmatrix}
  2. t { 5 3 } t\begin{Bmatrix}5\\ 3\end{Bmatrix}

Tsallis_entropy.html

  1. { p i } \{p_{i}\}
  2. i p i = 1 \sum_{i}p_{i}=1
  3. q q
  4. S q ( p i ) = k q - 1 ( 1 - i p i q ) , S_{q}({p_{i}})={k\over q-1}\left(1-\sum_{i}p_{i}^{q}\right),
  5. q q
  6. q 1 q\to 1
  7. S B G = S 1 ( p ) = - k i p i ln p i . S_{BG}=S_{1}(p)=-k\sum_{i}p_{i}\ln p_{i}.
  8. S q [ p ] = 1 q - 1 ( 1 - ( p ( x ) ) q d x ) , S_{q}[p]={1\over q-1}\left(1-\int(p(x))^{q}\,dx\right),
  9. p ( x ) p(x)
  10. S q = - lim x 1 D q i p i x S_{q}=-\lim_{x\rightarrow 1}D_{q}\sum_{i}p_{i}^{x}
  11. S = - lim x 1 d d x i p i x S=-\lim_{x\rightarrow 1}\frac{d}{dx}\sum_{i}p_{i}^{x}
  12. p ( A , B ) = p ( A ) p ( B ) , p(A,B)=p(A)p(B),\,
  13. S q ( A , B ) = S q ( A ) + S q ( B ) + ( 1 - q ) S q ( A ) S q ( B ) . S_{q}(A,B)=S_{q}(A)+S_{q}(B)+(1-q)S_{q}(A)S_{q}(B).\,
  14. | 1 - q | |1-q|
  15. S ( A , B ) = S ( A ) + S ( B ) , S(A,B)=S(A)+S(B),\,
  16. H q T ( p F ( x ; θ ) ) = 1 1 - q ( ( e F ( q θ ) - q F ( θ ) ) E p [ e ( q - 1 ) k ( x ) ] - 1 ) H^{T}_{q}(p_{F}(x;\theta))=\frac{1}{1-q}\left((e^{F(q\theta)-qF(\theta)})E_{p}% [e^{(q-1)k(x)}]-1\right)

Tsallis_statistics.html

  1. e q ( x ) = { exp ( x ) if q = 1 , [ 1 + ( 1 - q ) x ] 1 / ( 1 - q ) if q 1 and 1 + ( 1 - q ) x > 0 , 0 1 / ( 1 - q ) if q 1 and 1 + ( 1 - q ) x 0 , e_{q}(x)=\begin{cases}\exp(x)&\,\text{if }q=1,\\ [1+(1-q)x]^{1/(1-q)}&\,\text{if }q\neq 1\,\text{ and }1+(1-q)x>0,\\ 0^{1/(1-q)}&\,\text{if }q\neq 1\,\text{ and }1+(1-q)x\leq 0,\\ \end{cases}
  2. ln q ( x ) = { ln ( x ) if x 0 and q = 1 x 1 - q - 1 1 - q if x 0 and q 1 Undefined if x 0 \ln_{q}(x)=\begin{cases}\ln(x)&\,\text{if }x\geq 0\,\text{ and }q=1\\ \dfrac{x^{1-q}-1}{1-q}&\,\text{if }x\geq 0\,\text{ and }q\neq 1\\ \,\text{Undefined }&\,\text{if }x\leq 0\\ \end{cases}
  3. { e q ( ln q ( x ) ) = x ( x > 0 ) ln q ( e q ( x ) ) = x ( 0 < e q ( x ) < ) \begin{cases}e_{q}(\ln_{q}(x))=x&(x>0)\\ \ln_{q}(e_{q}(x))=x&(0<e_{q}(x)<\infty)\\ \end{cases}

Tube_lemma.html

  1. A × B U × V N A\times B\subset U\times V\subset N
  2. A = { x } A=\{x\}
  3. B = Y B=Y
  4. ( a , b ) A × B (a,b)\in A\times B
  5. U a , b X U_{a,b}\subset X
  6. V a , b Y V_{a,b}\subset Y
  7. ( a , b ) U a , b × V a , b N (a,b)\in U_{a,b}\times V_{a,b}\subset N
  8. a A a\in A
  9. ( V a , b : b B ) (V_{a,b}:b\in B)
  10. B B
  11. B B
  12. B 0 ( a ) B B_{0}(a)\subset B
  13. V a := b B 0 ( a ) V a , b B V^{\prime}_{a}:=\bigcup_{b\in B_{0}(a)}V_{a,b}\supset B
  14. U a := b B 0 ( a ) U a , b U^{\prime}_{a}:=\bigcap_{b\in B_{0}(a)}U_{a,b}
  15. B 0 ( a ) B_{0}(a)
  16. U a U^{\prime}_{a}
  17. V a V^{\prime}_{a}
  18. U a U^{\prime}_{a}
  19. V a V^{\prime}_{a}
  20. { a } × B U a × V a N \{a\}\times B\subset U^{\prime}_{a}\times V^{\prime}_{a}\subset N
  21. a a
  22. A 0 A A_{0}\subset A
  23. U ′′ := a A 0 U a A U^{\prime\prime}:=\bigcup_{a\in A_{0}}U^{\prime}_{a}\supset A
  24. V ′′ := a A 0 V a V^{\prime\prime}:=\bigcap_{a\in A_{0}}V^{\prime}_{a}
  25. A × B U ′′ × V ′′ N A\times B\subset U^{\prime\prime}\times V^{\prime\prime}\subset N
  26. U ′′ X U^{\prime\prime}\subset X
  27. V ′′ Y V^{\prime\prime}\subset Y

Tubular_neighborhood.html

  1. i : N 0 S i:N_{0}\rightarrow S
  2. π : E S \pi:E\rightarrow S
  3. J : E M J:E\rightarrow M
  4. J 0 E = i J\circ 0_{E}=i
  5. S M S\hookrightarrow M
  6. 0 E 0_{E}
  7. U E , V M \exists\ U\subseteq E,V\subseteq M
  8. 0 E [ S ] U 0_{E}[S]\subseteq U
  9. S V S\subseteq V
  10. J | U : U V J|_{U}:U\rightarrow V

Tutte_polynomial.html

  1. x 4 + x 3 + x 2 y x^{4}+x^{3}+x^{2}y
  2. y = 0 y=0
  3. T G T_{G}
  4. G = ( V , E ) G=(V,E)
  5. T G ( x , y ) = A E ( x - 1 ) k ( A ) - k ( E ) ( y - 1 ) k ( A ) + | A | - | V | , T_{G}(x,y)=\sum\nolimits_{A\subseteq E}(x-1)^{k(A)-k(E)}(y-1)^{k(A)+|A|-|V|},
  6. k ( A ) k(A)
  7. ( V , A ) (V,A)
  8. T G T_{G}
  9. r ( A ) = | V | - k ( A ) r(A)=|V|-k(A)
  10. ( V , A ) (V,A)
  11. R G ( u , v ) = A E u r ( E ) - r ( A ) v | A | - r ( A ) . R_{G}(u,v)=\sum\nolimits_{A\subseteq E}u^{r(E)-r(A)}v^{|A|-r(A)}.
  12. T G ( x , y ) = R G ( x - 1 , y - 1 ) . T_{G}(x,y)=R_{G}(x-1,y-1).
  13. Q G Q_{G}
  14. T G ( x , y ) = ( x - 1 ) - k ( G ) Q G ( x - 1 , y - 1 ) . T_{G}(x,y)=(x-1)^{-k(G)}Q_{G}(x-1,y-1).
  15. T G T_{G}
  16. T G ( x , y ) = i , j t i j x i y j , T_{G}(x,y)=\sum\nolimits_{i,j}t_{ij}x^{i}y^{j},
  17. t i j t_{ij}
  18. T G = T G - e + T G / e , T_{G}=T_{G-e}+T_{G/e},
  19. T G ( x , y ) = x i y j , T_{G}(x,y)=x^{i}y^{j},
  20. T G = 1 T_{G}=1
  21. Z G ( q , w ) = F E q k ( F ) w | F | Z_{G}(q,w)=\sum\nolimits_{F\subseteq E}q^{k(F)}w^{|F|}
  22. T G T_{G}
  23. T G ( x , y ) = ( x - 1 ) - k ( E ) ( y - 1 ) - | V | Z G ( ( x - 1 ) ( y - 1 ) , y - 1 ) . T_{G}(x,y)=(x-1)^{-k(E)}(y-1)^{-|V|}\cdot Z_{G}\Big((x-1)(y-1),\;y-1\Big).
  24. H H^{\prime}
  25. T G = T H T H T_{G}=T_{H}\cdot T_{H^{\prime}}
  26. G * G^{*}
  27. T G ( x , y ) = T G * ( y , x ) T_{G}(x,y)=T_{G^{*}}(y,x)
  28. x m x^{m}
  29. t i j t_{ij}
  30. x i y j x^{i}y^{j}
  31. 36 x \displaystyle 36x
  32. 12 y 2 x 2 + 11 x + 11 y + 40 y 3 + 32 y 2 + 46 y x + 24 x y 3 + 52 x y 2 \displaystyle 12\,{y}^{2}{x}^{2}+11\,x+11\,y+40\,{y}^{3}+32\,{y}^{2}+46\,yx+24% \,x{y}^{3}+52\,x{y}^{2}
  33. ( x , y ) (x,y)
  34. y = 0 y=0
  35. χ G ( λ ) = ( - 1 ) | V | - k ( G ) λ k ( G ) T G ( 1 - λ , 0 ) , \chi_{G}(\lambda)=(-1)^{|V|-k(G)}\lambda^{k(G)}T_{G}(1-\lambda,0),
  36. k ( G ) k(G)
  37. χ G ( λ ) \chi_{G}(\lambda)
  38. χ G ( λ ) \chi_{G}(\lambda)
  39. χ G ( λ ) = λ n \chi_{G}(\lambda)=\lambda^{n}
  40. χ G ( λ ) = 0 \chi_{G}(\lambda)=0
  41. χ G ( λ ) = χ G e ( λ ) - χ G / e ( λ ) . \chi_{G}(\lambda)=\chi_{G\setminus e}(\lambda)-\chi_{G/e}(\lambda).
  42. χ G ( λ ) \chi_{G}(\lambda)
  43. χ G ( λ ) \chi_{G}(\lambda)
  44. T G ( 2 , 0 ) = ( - 1 ) | V | χ G ( - 1 ) T_{G}(2,0)=(-1)^{|V|}\chi_{G}(-1)
  45. x y = 1 xy=1
  46. T G ( 2 , 1 ) T_{G}(2,1)
  47. T G ( 1 , 1 ) T_{G}(1,1)
  48. T G ( 1 , 2 ) T_{G}(1,2)
  49. T G ( 2 , 0 ) T_{G}(2,0)
  50. T G ( 0 , 2 ) T_{G}(0,2)
  51. ( - 1 ) | V | + k ( G ) T G ( 0 , - 2 ) (-1)^{|V|+k(G)}T_{G}(0,-2)
  52. k ( G ) k(G)
  53. 2 T G ( 3 , 3 ) 2T_{G}(3,3)
  54. 2 T G ( 3 , 3 ) 2T_{G}(3,3)
  55. H 2 : ( x - 1 ) ( y - 1 ) = 2 , H_{2}:\quad(x-1)(y-1)=2,
  56. Z ( ) , Z(\cdot),
  57. H 2 H_{2}
  58. Z ( G ) = 2 ( e - α ) | E | - r ( E ) ( 4 sinh α ) r ( E ) T G ( coth α , e 2 α ) . Z(G)=2\left(e^{-\alpha}\right)^{|E|-r(E)}\left(4\sinh\alpha\right)^{r(E)}T_{G}% \left(\coth\alpha,e^{2\alpha}\right).
  59. ( coth α - 1 ) ( e 2 α - 1 ) = 2 (\coth\alpha-1)\left(e^{2\alpha}-1\right)=2
  60. H q : ( x - 1 ) ( y - 1 ) = q , H_{q}:\quad(x-1)(y-1)=q,
  61. H q H_{q}
  62. H q H_{q}
  63. H q H_{q}
  64. y > 0 y>0
  65. H q H_{q}
  66. y = 1 y=1
  67. H q H_{q}
  68. x = 1 x=1
  69. x = 1 - q , y = 0 x=1-q,y=0
  70. x = 0 x=0
  71. 1 , 2 , , k - 1 1,2,\dots,k-1
  72. C G ( k ) C_{G}(k)
  73. G * G^{*}
  74. C G ( k ) = k - 1 χ G * ( k ) . C_{G}(k)=k^{-1}\chi_{G^{*}}(k).
  75. C G ( k ) = ( - 1 ) | E | + | V | + k ( G ) T G ( 0 , 1 - k ) . C_{G}(k)=(-1)^{|E|+|V|+k(G)}T_{G}(0,1-k).
  76. x = 1 x=1
  77. R G ( p ) R_{G}(p)
  78. R G ( p ) = ( 1 - p ) | V | - k ( G ) p | E | - | V | + k ( G ) T G ( 1 , 1 p ) . R_{G}(p)=(1-p)^{|V|-k(G)}p^{|E|-|V|+k(G)}T_{G}\left(1,\tfrac{1}{p}\right).
  79. Q G ( u , v ) = A E u k ( A ) v | A | - | V | + k ( A ) , Q_{G}(u,v)=\sum\nolimits_{A\subseteq E}u^{k(A)}v^{|A|-|V|+k(A)},
  80. k ( A ) k(A)
  81. Q G ( u , v ) = u k ( G ) R G ( u , v ) . Q_{G}(u,v)=u^{k(G)}\,R_{G}(u,v).
  82. m G ( x ) m_{\vec{G}}(x)
  83. G \vec{G}
  84. G m \vec{G}_{m}
  85. T G ( x , x ) = m G m ( x ) . T_{G}(x,x)=m_{\vec{G}_{m}}(x).
  86. x 3 + 2 x 2 + y 2 + 2 x y + x + y x^{3}+2x^{2}+y^{2}+2xy+x+y
  87. T G ( x , y ) = T G e ( x , y ) + T G / e ( x , y ) , e not a loop nor a bridge. T_{G}(x,y)=T_{G\setminus e}(x,y)+T_{G/e}(x,y),\qquad e\,\text{ not a loop nor % a bridge.}
  88. t ( n + m ) = t ( n + m - 1 ) + t ( n + m - 2 ) , t(n+m)=t(n+m-1)+t(n+m-2),
  89. t ( n + m ) = ( 1 + 5 2 ) n + m = O ( 1.6180 n + m ) . t(n+m)=\left(\frac{1+\sqrt{5}}{2}\right)^{n+m}=O\left(1.6180^{n+m}\right).
  90. τ ( G ) \tau(G)
  91. m = O ( n ) m=O(n)
  92. O ( exp ( n ) ) O(\exp(n))
  93. τ ( G ) = O ( ν k n n - 1 log n ) , \tau(G)=O\left(\nu_{k}^{n}n^{-1}\log n\right),
  94. ν k = ( k - 1 ) k - 1 ( k 2 - 2 k ) k 2 - 1 . \nu_{k}=\frac{(k-1)^{k-1}}{(k^{2}-2k)^{\frac{k}{2}-1}}.
  95. ν 5 4.4066. \nu_{5}\approx 4.4066.
  96. T G ( 1 , 1 ) T_{G}(1,1)
  97. τ ( G ) \tau(G)
  98. T G ( - 1 , - 1 ) T_{G}(-1,-1)
  99. H 2 H_{2}
  100. H 2 H_{2}
  101. T G T_{G}
  102. T G ( - 2 , 0 ) T_{G}(-2,0)
  103. ( x , y ) (x,y)
  104. ( x , y ) (x,y)
  105. T G ( x , y ) T_{G}(x,y)
  106. ( x , y ) (x,y)
  107. ( x , y ) (x,y)
  108. T G ( x , y ) T_{G}(x,y)
  109. T G ( x , y ) T_{G}(x,y)
  110. ( x , y ) (x,y)
  111. T G ( x , y ) T_{G}(x,y)
  112. x , y x,y\in\mathbb{C}
  113. ( x , y ) (x,y)
  114. H 1 H_{1}
  115. { ( 1 , 1 ) , ( - 1 , - 1 ) , ( 0 , - 1 ) , ( - 1 , 0 ) , ( i , - i ) , ( - i , i ) , ( j , j 2 ) , ( j 2 , j ) } , j = e 2 π i 3 . \left\{(1,1),(-1,-1),(0,-1),(-1,0),(i,-i),(-i,i),\left(j,j^{2}\right),\left(j^% {2},j\right)\right\},\qquad j=e^{\frac{2\pi i}{3}}.
  116. H 2 H_{2}
  117. T G ( 0 , - 2 ) T_{G}(0,-2)
  118. T G ( x , y ) T_{G}(x,y)
  119. H 2 H_{2}
  120. Ω ( n ) \Omega(n)

Twelvefold_way.html

  1. N N
  2. X X
  3. n = | N | n=|N|
  4. x = | X | x=|X|
  5. N N
  6. n n
  7. X X
  8. x x
  9. f : N X f:N\to X
  10. a a
  11. N N
  12. f f
  13. b b
  14. X X
  15. b b
  16. f f
  17. f ( a ) f(a)
  18. a a
  19. N N
  20. b b
  21. X X
  22. f f
  23. f f
  24. b b
  25. X X
  26. a a
  27. N N
  28. f ( a ) = b f(a)=b
  29. b b
  30. f f
  31. f f
  32. n = x n=x
  33. f f
  34. f f
  35. f f
  36. N N
  37. X X
  38. N N
  39. X X
  40. N N
  41. X X
  42. f f
  43. x n x^{n}\,
  44. x n ¯ x^{\underline{n}}
  45. x ! { n x } \textstyle x!\{{n\atop x}\}\,
  46. ( x + n - 1 n ) {\left({{x+n-1}\atop{n}}\right)}
  47. ( x n ) {\left({{x}\atop{n}}\right)}
  48. ( n - 1 n - x ) {\left({{n-1}\atop{n-x}}\right)}
  49. k = 0 x { n k } \sum_{k=0}^{x}\left\{{n\atop k}\right\}
  50. [ n x ] [n\leq x]
  51. { n x } \left\{{n\atop x}\right\}
  52. p x ( n + x ) p_{x}(n+x)\,
  53. [ n x ] [n\leq x]
  54. p x ( n ) p_{x}(n)\,
  55. x n ¯ = x ! ( x - n ) ! = x ( x - 1 ) ( x - 2 ) ( x - n + 1 ) x^{\underline{n}}=\frac{x!}{(x-n)!}=x(x-1)(x-2)\cdots(x-n+1)
  56. x n ¯ = ( x + n - 1 ) ! ( x - 1 ) ! = x ( x + 1 ) ( x + 2 ) ( x + n - 1 ) x^{\overline{n}}=\frac{(x+n-1)!}{(x-1)!}=x(x+1)(x+2)\cdots(x+n-1)
  57. n ! = n n ¯ = n ( n - 1 ) ( n - 2 ) 1 n!=n^{\underline{n}}=n(n-1)(n-2)\cdots 1
  58. { n k } \textstyle\{{n\atop k}\}
  59. ( n k ) = n k ¯ k ! \textstyle{\left({{n}\atop{k}}\right)}=\frac{n^{\underline{k}}}{k!}
  60. p k ( n ) p_{k}(n)
  61. X = { a , b , c } , N = { p , q } , then | { ( a , a ) , ( a , b ) , ( a , c ) , ( b , a ) , ( b , b ) , ( b , c ) , ( c , a ) , ( c , b ) , ( c , c ) } | = 3 2 = 9 X=\{a,b,c\},N=\{p,q\}\,\text{, then }\left|\{(a,a),(a,b),(a,c),(b,a),(b,b),(b,% c),(c,a),(c,b),(c,c)\}\right|=3^{2}=9
  62. x n ¯ = x ( x - 1 ) ( x - n + 1 ) . x^{\underline{n}}=x(x-1)\cdots(x-n+1).
  63. X = { a , b , c , d } , N = { 1 , 2 } , then X=\{a,b,c,d\},N=\{1,2\}\,\text{, then }
  64. | { ( a , b ) , ( a , c ) , ( a , d ) , ( b , a ) , ( b , c ) , ( b , d ) , ( c , a ) , ( c , b ) , ( c , d ) , ( d , a ) , ( d , b ) , ( d , c ) } | = 4 2 ¯ = 4 × 3 = 12 \left|\{(a,b),(a,c),(a,d),(b,a),(b,c),(b,d),(c,a),(c,b),(c,d),(d,a),(d,b),(d,c% )\}\right|=4^{\underline{2}}=4\times 3=12
  65. ( x n ) {\textstyle\left({{x}\atop{n}}\right)}
  66. ( x n ) = x n ¯ n ! = x ( x - 1 ) ( x - n + 2 ) ( x - n + 1 ) n ( n - 1 ) 2 1 . {\left({{x}\atop{n}}\right)}=\frac{x^{\underline{n}}}{n!}=\frac{x(x-1)\cdots(x% -n+2)(x-n+1)}{n(n-1)\cdots 2\cdot 1}.
  67. ( x + n - 1 n ) = ( x + n - 1 ) ( x + n - 2 ) ( x + 1 ) x n ( n - 1 ) 2 1 = x n ¯ n ! . {\left({{x+n-1}\atop{n}}\right)}=\frac{(x+n-1)(x+n-2)\cdots(x+1)x}{n(n-1)% \cdots 2\cdot 1}=\frac{x^{\overline{n}}}{n!}.
  68. ( n - 1 n - x ) . {\left({{n-1}\atop{n-x}}\right)}.
  69. ( n - 1 x - 1 ) , {\left({{n-1}\atop{x-1}}\right)},
  70. ( - 1 0 ) = 1 {\textstyle\left({{-1}\atop{0}}\right)}=1
  71. ( - 1 - 1 ) = 0 {\textstyle\left({{-1}\atop{-1}}\right)}=0
  72. [ n x ] [n\leq x]
  73. [ n x ] [n\leq x]
  74. { n x } \textstyle\{{n\atop x}\}
  75. x ! { n x } . \textstyle x!\{{n\atop x}\}.
  76. X = { a , b } , N = { p , q , r } , then | { ( a , a , b ) , ( a , b , a ) , ( a , b , b ) , ( b , b , a ) , ( b , a , b ) , ( b , a , a ) } | = 2 ! { 3 2 } = 2 × 3 = 6 X=\{a,b\},N=\{p,q,r\}\,\text{, then }\left|\{(a,a,b),(a,b,a),(a,b,b),\ (b,b,a)% ,(b,a,b),(b,a,a)\}\right|=\textstyle 2!\{{3\atop 2}\}=2\times 3=6
  77. k = 0 x { n k } \textstyle\sum_{k=0}^{x}\{{n\atop k}\}
  78. k = 0 n { n k } \textstyle\sum_{k=0}^{n}\{{n\atop k}\}
  79. { n x } \textstyle\{{n\atop x}\}
  80. k = 0 x p k ( n ) \textstyle\sum_{k=0}^{x}p_{k}(n)
  81. p x ( n + x ) p_{x}(n+x)
  82. 0 0 = 0 0 ¯ = 0 ! = ( 0 0 ) = ( - 1 0 ) = { 0 0 } = p 0 ( 0 ) = 1 0^{0}=0^{\underline{0}}=0!={\left({{0}\atop{0}}\right)}={\left({{-1}\atop{0}}% \right)}=\left\{{0\atop 0}\right\}=p_{0}(0)=1
  83. ( - 1 0 ) = ( - 1 ) 0 ¯ 0 ! = 1 \scriptstyle{\textstyle\left({{-1}\atop{0}}\right)}=\frac{(-1)^{\underline{0}}% }{0!}=1
  84. { n x } = p x ( n ) = 0 whenever either n > 0 = x or 0 n < x . \left\{{n\atop x}\right\}=p_{x}(n)=0\quad\hbox{whenever either }n>0=x\hbox{ or% }0\leq n<x.
  85. ( n + x - 1 n ) {\textstyle\left({{n+x-1}\atop{n}}\right)}
  86. ( n + x - 1 x - 1 ) {\textstyle\left({{n+x-1}\atop{x-1}}\right)}
  87. ( n - 1 n - x ) {\textstyle\left({{n-1}\atop{n-x}}\right)}
  88. ( n - 1 x - 1 ) {\textstyle\left({{n-1}\atop{x-1}}\right)}
  89. f : N X f\colon N\rightarrow X
  90. f f
  91. F F
  92. g G , h H g\in G,h\in H
  93. F = h f g F=h\circ f\circ g

Twisted_K-theory.html

  1. F r e d ( ) , Fred(\mathcal{H}),
  2. \mathcal{H}
  3. [ M F r e d ( ) ] [M\rightarrow Fred(\mathcal{H})]
  4. F r e d ( ) . Fred(\mathcal{H}).
  5. F r e d ( ) Fred(\mathcal{H})
  6. F r e d ( ) Fred(\mathcal{H})
  7. P U ( ) PU(\mathcal{H})
  8. P P
  9. P U ( ) PU(\mathcal{H})
  10. \mathcal{H}
  11. [ P F r e d ( ) ] P U ( ) [P\rightarrow Fred(\mathcal{H})]_{PU(\mathcal{H})}
  12. P P
  13. F r e d ( ) Fred(\mathcal{H})
  14. P U ( ) PU(\mathcal{H})
  15. [ M F r e d ( ) ] . [M\rightarrow Fred(\mathcal{H})].
  16. P U ( ) PU(\mathcal{H})
  17. P U ( ) PU(\mathcal{H})
  18. K ( 𝐙 , 2 ) . K(\mathbf{Z},2).
  19. F r e d ( ) Fred(\mathcal{H})
  20. P U ( ) PU(\mathcal{H})
  21. P H P_{H}
  22. K H ( M ) = [ P H F r e d ( ) ] P U ( ) . K_{H}(M)=[P_{H}\rightarrow Fred(\mathcal{H})]_{PU(\mathcal{H})}.
  23. F r e d ( ) Fred(\mathcal{H})
  24. P U ( ) PU(\mathcal{H})

Twisted_sector.html

  1. g G g\in G
  2. X ( σ + 2 π , τ ) = g [ X ( σ , τ ) ] X(\sigma+2\pi,\tau)=g[X(\sigma,\tau)]

Two-point_tensor.html

  1. 𝐐 = Q p q ( 𝐞 p 𝐞 q ) \mathbf{Q}=Q_{pq}(\mathbf{e}_{p}\otimes\mathbf{e}_{q})
  2. 𝐯 = 𝐐𝐮 \mathbf{v}=\mathbf{Q}\mathbf{u}
  3. 𝐆 = G p q ( 𝐞 p 𝐄 q ) \mathbf{G}=G_{pq}(\mathbf{e}_{p}\otimes\mathbf{E}_{q})
  4. 𝐯 = 𝐆𝐔 \mathbf{v}=\mathbf{GU}
  5. v p = Q p q v q v^{\prime}_{p}=Q_{pq}v_{q}
  6. T p q ( e p e q ) T_{pq}(e_{p}\otimes e_{q})
  7. e i e_{i}
  8. T p q ( e p e q ) T^{\prime}_{pq}(e^{\prime}_{p}\otimes e^{\prime}_{q})
  9. T i j = Q i p Q j r T p r T^{\prime}_{ij}=Q_{ip}Q_{jr}T_{pr}
  10. T = Q T Q T T^{\prime}=QTQ^{T}
  11. F p q ( e p e q ) F_{pq}(e^{\prime}_{p}\otimes e_{q})
  12. F = Q F F^{\prime}=QF
  13. v p = Q p q u q v^{\prime}_{p}=Q_{pq}u_{q}
  14. u = u q e q u=u_{q}e_{q}
  15. v = v p e p v=v^{\prime}_{p}e_{p}
  16. Q p q ( e p e q ) Q_{pq}(e^{\prime}_{p}\otimes e_{q})
  17. ( e p e q ) e q = ( e q . e q ) e p = e p ( 1 ) (e^{\prime}_{p}\otimes e_{q})e_{q}=(e_{q}.e_{q})e^{\prime}_{p}=e^{\prime}_{p}% \qquad(1)
  18. u p e p = ( Q p q ( e p e q ) ) ( v q e q ) u_{p}e_{p}=(Q_{pq}(e^{\prime}_{p}\otimes e_{q}))(v_{q}e_{q})
  19. u p e p = Q p q v q ( e p e q ) e q u_{p}e_{p}=Q_{pq}v_{q}(e^{\prime}_{p}\otimes e_{q})e_{q}
  20. u p e p = Q p q v q e p u_{p}e_{p}=Q_{pq}v_{q}e_{p}

Two-way_deterministic_finite_automaton.html

  1. M = ( Q , Σ , L , R , δ , s , t , r ) M=(Q,\Sigma,L,R,\delta,s,t,r)
  2. Q Q
  3. Σ \Sigma
  4. L L
  5. R R
  6. δ : Q × ( Σ { L , R } ) Q × { l e f t , r i g h t } \delta:Q\times(\Sigma\cup\{L,R\})\rightarrow Q\times\{left,right\}
  7. s s
  8. t t
  9. r r
  10. q Q q\in Q
  11. δ ( q , L ) = ( q , r i g h t ) \delta(q,L)=(q^{\prime},right)
  12. q Q q^{\prime}\in Q
  13. δ ( q , R ) = ( q , l e f t ) \delta(q,R)=(q^{\prime},left)
  14. q Q q^{\prime}\in Q
  15. σ Σ { L } \sigma\in\Sigma\cup\{L\}
  16. δ ( t , σ ) = ( t , R ) \delta(t,\sigma)=(t,R)
  17. δ ( r , σ ) = ( r , R ) \delta(r,\sigma)=(r,R)
  18. δ ( t , R ) = ( t , L ) \delta(t,R)=(t,L)
  19. δ ( r , R ) = ( r , L ) \delta(r,R)=(r,L)

Type_(model_theory).html

  1. \mathcal{M}
  2. \mathcal{M}
  3. \mathcal{M}
  4. \mathcal{M}
  5. L ( A ) = L { c a : a A } . L(A)=L\cup\{c_{a}:a\in A\}.
  6. \mathcal{M}
  7. p 0 ( b ) \mathcal{M}\models p_{0}(b)
  8. \mathcal{M}
  9. \mathcal{M}
  10. p 0 ( b 1 , , b n ) \mathcal{M}\models p_{0}(b_{1},\ldots,b_{n})
  11. ϕ ( s y m b o l x ) L ( A , s y m b o l x ) \phi(symbol{x})\in L(A,symbol{x})
  12. ϕ ( s y m b o l x ) p ( s y m b o l x ) \phi(symbol{x})\in p(symbol{x})
  13. ¬ ϕ ( s y m b o l x ) p ( s y m b o l x ) \lnot\phi(symbol{x})\in p(symbol{x})
  14. \mathcal{M}
  15. p ( s y m b o l b ) \mathcal{M}\models p(symbol{b})
  16. \mathcal{M}
  17. \mathcal{M}
  18. \mathcal{M}
  19. t p n ( s y m b o l b / A ) tp_{n}^{\mathcal{M}}(symbol{b}/A)
  20. ψ ( s y m b o l x ) p ( s y m b o l x ) , φ ( s y m b o l x ) ψ ( s y m b o l x ) \forall\psi(symbol{x})\in p(symbol{x}),\varphi(symbol{x})\rightarrow\psi(% symbol{x})
  21. \mathcal{M}
  22. \mathcal{M}
  23. φ ( s y m b o l b ) \mathcal{M}\models\varphi(symbol{b})
  24. \in
  25. \mathcal{M}
  26. ω , ω \langle\omega,\in_{\omega}\rangle
  27. ω \omega
  28. 𝒯 \mathcal{T}
  29. p ( x ) := { n x n ω ω } p(x):=\{n\in x\mid n\in_{\omega}\omega\}
  30. p 0 ( x ) p ( x ) p_{0}(x)\subseteq p(x)
  31. p ( x ) p(x)
  32. n ω n\in\omega
  33. p 0 p_{0}
  34. p 0 ( x ) p_{0}(x)
  35. p 0 ( x ) p_{0}(x)
  36. p ( x ) p(x)
  37. p ( x ) p(x)
  38. \mathcal{M}
  39. n ω n\in\omega
  40. ω \omega
  41. ω + 1 , ω + 1 \langle\omega+1,\in_{\omega+1}\rangle
  42. \mathcal{M}
  43. 𝒯 \mathcal{T}
  44. x y ( y x ) \exists x\forall y(y\in x)
  45. \mathcal{M}
  46. \mathcal{M}^{\prime}
  47. ω \omega\cup\mathbb{Z}^{\prime}
  48. \mathbb{Z}^{\prime}
  49. ω = \mathbb{Z}^{\prime}\cap\omega=\emptyset
  50. < <
  51. \mathbb{Z}^{\prime}
  52. \in
  53. = ω < ( ω × ) \in_{\mathcal{M}^{\prime}}=\in_{\omega}\cup<\cup\,(\omega\times\mathbb{Z}^{% \prime})
  54. \mathbb{Z}^{\prime}
  55. p ( x ) p(x)
  56. x 1 + 1 + 1 \,\!x\neq 1+1+1
  57. x 1 + 1 + 1 + 1 + 1 x\leq 1+1+1+1+1
  58. y ( y < x ) \exists y(y<x)
  59. x = 1 + 1 x=1+1
  60. y ( y 2 < 2 y < x ) \forall y(y^{2}<2\implies y<x)
  61. y ( ( y > 0 and y 2 > 2 ) y > x ) \forall y((y>0\and y^{2}>2)\implies y>x)
  62. { 0 < x < r : r } \{0<x<r:r\in\mathbb{R}\}
  63. x 1 x\neq 1
  64. x π x\neq\pi
  65. ψ ϕ x 1 , , x n ( ψ ( x 1 , , x n ) ϕ ( x 1 , , x n ) ) . \psi\equiv\phi\Leftrightarrow\mathcal{M}\models\forall x_{1},\ldots,x_{n}(\psi% (x_{1},\ldots,x_{n})\leftrightarrow\phi(x_{1},\ldots,x_{n})).
  66. ψ ϕ \psi\equiv\phi

Type_class.html

  1. * *

Ultrarelativistic_limit.html

  1. c c
  2. m m
  3. p p
  4. E 2 = m 2 c 4 + p 2 c 2 . E^{2}=m^{2}c^{4}+p^{2}c^{2}.
  5. p c m c < s u p > 2 pc≫mc<sup>2

Umbilic_torus.html

  1. a x 3 + 3 b x 2 y + 3 c x y 2 + d y 3 ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3}
  2. x = sin u ( 7 + cos ( u 3 - 2 v ) + 2 cos ( u 3 + v ) ) x=\sin u\left(7+\cos\left({u\over 3}-2v\right)+2\cos\left({u\over 3}+v\right)\right)
  3. y = cos u ( 7 + cos ( u 3 - 2 v ) + 2 cos ( u 3 + v ) ) y=\cos u\left(7+\cos\left({u\over 3}-2v\right)+2\cos\left({u\over 3}+v\right)\right)
  4. z = sin ( u 3 - 2 v ) + 2 sin ( u 3 + v ) z=\sin\left({u\over 3}-2v\right)+2\sin\left({u\over 3}+v\right)
  5. for - π u π , - π v π \mbox{for }~{}-\pi\leq u\leq\pi,\quad-\pi\leq v\leq\pi\,