wpmath0000004_15

Topological_sorting.html

  1. O ( | V | + | E | ) O(\left|{V}\right|+\left|{E}\right|)
  2. V V
  3. s s
  4. d d
  5. V V
  6. s s
  7. d s s = 0 dss=0
  8. d u u = duu=∞
  9. p p
  10. V V
  11. p u u puu
  12. u u
  13. s s
  14. u u
  15. u u
  16. T T
  17. v v
  18. u u
  19. u u
  20. v v
  21. w w
  22. u u
  23. v v
  24. d v v > d u u + w dvv>duu+w
  25. d v v d u u + w dvv←duu+w
  26. p v v u pvv←u
  27. n n
  28. m m
  29. Θ ( n + m ) Θ(n+m)
  30. V V
  31. V V
  32. V V
  33. V V
  34. d d
  35. V V
  36. s s
  37. d s s = 0 dss=0
  38. d u u = duu=∞
  39. p p
  40. V V
  41. p u u puu
  42. u u
  43. s s
  44. u u
  45. v v
  46. T T
  47. u u
  48. v v
  49. u u
  50. v v
  51. w w
  52. u u
  53. v v
  54. d v v > d u u + w dvv>duu+w
  55. d v v d u u + w dvv←duu+w
  56. p v v u pvv←u

Topology_optimization.html

  1. min ρ Ω ϕ ( ρ ) d Ω \min_{\rho}\;\int_{\Omega}\phi(\rho)\,\mathrm{d}\Omega
  2. ρ { 0 , 1 } \scriptstyle\rho\,\in\,\{0,\,1\}
  3. ( Ω ϕ ( ρ ) d Ω ) \scriptstyle\left(\int_{\Omega}\phi(\rho)\,\mathrm{d}\Omega\right)
  4. ( Ω ) \scriptstyle(\Omega)
  5. ( ρ ) \scriptstyle(\rho)
  6. Ω \scriptstyle\Omega
  7. ρ \scriptstyle\rho
  8. ( 0 , 1 ) \scriptstyle(0,\,1)
  9. ρ \scriptstyle\rho
  10. E = E 0 + ρ n E 1 \scriptstyle E\;=\;E_{0}\,+\,\rho^{n}E_{1}
  11. n n
  12. [ 0 , 3 ] \scriptstyle[0,\,3]
  13. min ρ Ω 1 2 σ : ε d Ω \min_{\rho}\;\int_{\Omega}\frac{1}{2}\mathbf{\sigma}:\mathbf{\varepsilon}\,% \mathrm{d}\Omega
  14. ρ [ 0 , 1 ] \scriptstyle\rho\,\in\,[0,\,1]
  15. Ω ρ d Ω V * \scriptstyle\int_{\Omega}\rho\,\mathrm{d}\Omega\;\leq\;V^{*}
  16. σ + 𝐅 = 0 \scriptstyle\mathbf{\nabla}\cdot\mathbf{\sigma}\,+\,\mathbf{F}\;=\;{\mathbf{0}}
  17. σ = 𝖢 : ε \scriptstyle\mathbf{\sigma}\;=\;\mathsf{C}:\mathbf{\varepsilon}

Tor_functor.html

  1. Tor n R ( A , B ) = ( L n T ) ( A ) \mathrm{Tor}_{n}^{R}(A,B)=(L_{n}T)(A)
  2. P 2 P 1 P 0 A 0 \cdots\rightarrow P_{2}\rightarrow P_{1}\rightarrow P_{0}\rightarrow A\rightarrow 0
  3. P 2 R B P 1 R B P 0 R B 0 \cdots\rightarrow P_{2}\otimes_{R}B\rightarrow P_{1}\otimes_{R}B\rightarrow P_% {0}\otimes_{R}B\rightarrow 0
  4. Tor 2 R ( M , B ) Tor 1 R ( K , B ) Tor 1 R ( L , B ) Tor 1 R ( M , B ) K B L B M B 0 \cdots\rightarrow\mathrm{Tor}_{2}^{R}(M,B)\rightarrow\mathrm{Tor}_{1}^{R}(K,B)% \rightarrow\mathrm{Tor}_{1}^{R}(L,B)\rightarrow\mathrm{Tor}_{1}^{R}(M,B)% \rightarrow K\otimes B\rightarrow L\otimes B\rightarrow M\otimes B\rightarrow 0
  5. Tor 1 R ( R / ( r ) , B ) = { b B : r b = 0 } , \mathrm{Tor}_{1}^{R}(R/(r),B)=\{b\in B:rb=0\},
  6. Tor n R ( i A i , j B j ) i j Tor n R ( A i , B j ) \mathrm{Tor}_{n}^{R}\left(\bigoplus_{i}A_{i},\bigoplus_{j}B_{j}\right)\simeq% \bigoplus_{i}\bigoplus_{j}\mathrm{Tor}_{n}^{R}(A_{i},B_{j})
  7. \simeq
  8. 0 M i K i L i , 0\to M_{i}\to K_{i}\to L_{i},
  9. \in
  10. β 03 \beta_{03}
  11. \in
  12. β 13 \beta_{13}
  13. \in
  14. \otimes
  15. α 12 \alpha_{12}
  16. β 03 \beta_{03}
  17. α 22 β 12 ( x 12 ) = β 13 α 21 ( x 12 ) = β 13 β 03 ( x ) = 0 , \alpha_{22}\circ\beta_{12}(x_{12})=\beta_{13}\circ\alpha_{21}(x_{12})=\beta_{1% 3}\circ\beta_{03}(x)=0,
  18. β 12 \beta_{12}
  19. \in
  20. α 22 \alpha_{22}
  21. β 12 \beta_{12}
  22. α 21 \alpha_{21}
  23. \in
  24. \otimes
  25. α 31 β 21 ( x 21 ) = β 22 α 21 ( x 21 ) = β 22 β 12 ( x 12 ) = 0 , \alpha_{31}\circ\beta_{21}(x_{21})=\beta_{22}\circ\alpha_{21}(x_{21})=\beta_{2% 2}\circ\beta_{12}(x_{12})=0,
  26. β 21 \beta_{21}
  27. \in
  28. α 31 \alpha_{31}
  29. β 21 \beta_{21}
  30. α 30 \alpha_{30}
  31. \in
  32. \to
  33. \to

Toric_variety.html

  1. | z 1 | 2 + | z 2 | 2 + | z 3 | 2 = 1 , |z_{1}|^{2}+|z_{2}|^{2}+|z_{3}|^{2}=1,\,\!
  2. ( z 1 , z 2 , z 3 ) e i ϕ ( z 1 , z 2 , z 3 ) . (z_{1},z_{2},z_{3})\approx e^{i\phi}(z_{1},z_{2},z_{3}).\,\!
  3. ( x , y , z ) = ( | z 1 | 2 , | z 2 | 2 , | z 3 | 2 ) . (x,y,z)=(|z_{1}|^{2},|z_{2}|^{2},|z_{3}|^{2}).\,\!
  4. x , y , z x,y,z
  5. x + y + z = 1 ; x+y+z=1;\,\!
  6. z = 1 - x - y . \quad z=1-x-y.\,\!
  7. z 1 , z 2 z_{1},z_{2}
  8. z 3 z_{3}
  9. U ( 1 ) U(1)
  10. x = 0 x=0
  11. y = 0 y=0
  12. z = 0 z=0
  13. z 1 , z 2 , z 3 z_{1},z_{2},z_{3}

Toroidal_reflector.html

  1. a b a\neq b

Torus_bundle.html

  1. f f
  2. T T
  3. M ( f ) M(f)
  4. T T
  5. f f
  6. M ( f ) M(f)
  7. f f
  8. f f
  9. M ( f ) M(f)
  10. f f
  11. M ( f ) M(f)
  12. f f
  13. M ( f ) M(f)
  14. f f
  15. f f

Torus_knot.html

  1. x = r cos ( p ϕ ) y = r sin ( p ϕ ) z = - sin ( q ϕ ) \begin{aligned}\displaystyle x&\displaystyle=r\cos(p\phi)\\ \displaystyle y&\displaystyle=r\sin(p\phi)\\ \displaystyle z&\displaystyle=-\sin(q\phi)\end{aligned}
  2. r = cos ( q ϕ ) + 2 r=\cos(q\phi)+2
  3. 0 < ϕ < 2 π 0<\phi<2\pi
  4. ( r - 2 ) 2 + z 2 = 1 (r-2)^{2}+z^{2}=1
  5. r = cos ( q ϕ ) + 4 r=\cos(q\phi)+4
  6. 3 cos ( ( p - q ) ϕ ) 3\cos((p-q)\phi)
  7. 3 sin ( ( p - q ) ϕ ) 3\sin((p-q)\phi)
  8. p < q < 2 p p<q<2p
  9. ( σ 1 σ 2 σ p - 1 ) q . (\sigma_{1}\sigma_{2}\cdots\sigma_{p-1})^{q}.
  10. g = 1 2 ( p - 1 ) ( q - 1 ) . g=\frac{1}{2}(p-1)(q-1).
  11. ( t p q - 1 ) ( t - 1 ) ( t p - 1 ) ( t q - 1 ) . \frac{(t^{pq}-1)(t-1)}{(t^{p}-1)(t^{q}-1)}.
  12. t ( p - 1 ) ( q - 1 ) / 2 1 - t p + 1 - t q + 1 + t p + q 1 - t 2 . t^{(p-1)(q-1)/2}\frac{1-t^{p+1}-t^{q+1}+t^{p+q}}{1-t^{2}}.
  13. x , y x p = y q . \langle x,y\mid x^{p}=y^{q}\rangle.
  14. x p = y q x^{p}=y^{q}
  15. f : \C 2 \C f:\C^{2}\to\C
  16. f ( w , z ) := w p + z q . f(w,z):=w^{p}+z^{q}.
  17. V f \C 2 V_{f}\subset\C^{2}
  18. ( w , z ) \C 2 (w,z)\in\C^{2}
  19. f ( w , z ) = 0. f(w,z)=0.
  20. 0 < ε 1 , 0<\varepsilon\ll 1,
  21. 𝕊 ε 3 \R 4 \C 2 \mathbb{S}^{3}_{\varepsilon}\subset\R^{4}\hookrightarrow\C^{2}
  22. | w | 2 + | z | 2 = ε 2 . |w|^{2}+|z|^{2}=\varepsilon^{2}.
  23. f f
  24. ( 0 , 0 ) \C 2 (0,0)\in\C^{2}
  25. f / w = f / z = 0 \partial f/\partial w=\partial f/\partial z=0
  26. w = z = 0. w=z=0.
  27. V f V_{f}
  28. ( 0 , 0 ) \C 2 . (0,0)\in\C^{2}.
  29. V f 𝕊 ε 3 𝕊 ε 3 . V_{f}\cap\mathbb{S}^{3}_{\varepsilon}\subset\mathbb{S}^{3}_{\varepsilon}.
  30. f ( w , z ) = w p + z q . f(w,z)=w^{p}+z^{q}.
  31. f ( w , z ) = w p + z q f(w,z)=w^{p}+z^{q}

Total_derivative.html

  1. f f
  2. t t
  3. x x
  4. y y
  5. t t
  6. \partial
  7. f f
  8. t t
  9. t t
  10. t t
  11. f f
  12. t . t.
  13. f ( t , x , y ) f(t,x,y)
  14. t t
  15. d f d t = f t d t d t + f x d x d t + f y d y d t \frac{\operatorname{d}f}{\operatorname{d}t}=\frac{\partial f}{\partial t}\frac% {\operatorname{d}t}{\operatorname{d}t}+\frac{\partial f}{\partial x}\frac{% \operatorname{d}x}{\operatorname{d}t}+\frac{\partial f}{\partial y}\frac{% \operatorname{d}y}{\operatorname{d}t}
  16. d f d t = f t + f x d x d t + f y d y d t . \frac{\operatorname{d}f}{\operatorname{d}t}=\frac{\partial f}{\partial t}+% \frac{\partial f}{\partial x}\frac{\operatorname{d}x}{\operatorname{d}t}+\frac% {\partial f}{\partial y}\frac{\operatorname{d}y}{\operatorname{d}t}.
  17. d t \operatorname{d}t
  18. d f = f t d t + f x d x + f y d y . {\operatorname{d}f}=\frac{\partial f}{\partial t}\operatorname{d}t+\frac{% \partial f}{\partial x}\operatorname{d}x+\frac{\partial f}{\partial y}% \operatorname{d}y.
  19. d f \operatorname{d}f
  20. f f
  21. f f
  22. t t
  23. f f
  24. t t
  25. f f
  26. x x
  27. y y
  28. d t \operatorname{d}t
  29. x x
  30. y y
  31. d x \operatorname{d}x
  32. d y \operatorname{d}y
  33. d f \operatorname{d}f
  34. D 𝐮 D t , \frac{D\mathbf{u}}{Dt},
  35. 2 \mathbb{R}^{2}
  36. f ( x , y ) = x y f(x,y)=xy
  37. f x = y \frac{\partial f}{\partial x}=y
  38. y = x ; y=x;
  39. f ( x , y ) = f ( x , x ) = x 2 f(x,y)=f(x,x)=x^{2}
  40. d f d x = 2 x \frac{\mathrm{d}f}{\mathrm{d}x}=2x
  41. d f d x = f x + f y d y d x = y + x 1 = x + y . \frac{\mathrm{d}f}{\mathrm{d}x}=\frac{\partial f}{\partial x}+\frac{\partial f% }{\partial y}\frac{\mathrm{d}y}{\mathrm{d}x}=y+x\cdot 1=x+y.
  42. d f d x = 2 x f x = y = x \frac{\mathrm{d}f}{\mathrm{d}x}=2x\neq\frac{\partial f}{\partial x}=y=x
  43. p i p_{i}
  44. d M d t = d d t M ( t , p 1 ( t ) , , p n ( t ) ) . {\operatorname{d}M\over\operatorname{d}t}=\frac{\operatorname{d}}{% \operatorname{d}t}M\bigl(t,p_{1}(t),\ldots,p_{n}(t)\bigr).
  45. d M d t = M t + i = 1 n M p i d p i d t = ( t + i = 1 n d p i d t p i ) ( M ) . {\operatorname{d}M\over\operatorname{d}t}=\frac{\partial M}{\partial t}+\sum_{% i=1}^{n}\frac{\partial M}{\partial p_{i}}\frac{\operatorname{d}p_{i}}{% \operatorname{d}t}=\biggl(\frac{\partial}{\partial t}+\sum_{i=1}^{n}\frac{% \operatorname{d}p_{i}}{\operatorname{d}t}\frac{\partial}{\partial p_{i}}\biggr% )(M).
  46. d f d t = f x d x d t + f y d y d t . \frac{\operatorname{d}f}{\operatorname{d}t}={\partial f\over\partial x}{% \operatorname{d}x\over\operatorname{d}t}+{\partial f\over\partial y}{% \operatorname{d}y\over\operatorname{d}t}.
  47. M ( t , p 1 , , p n ) M(t,p_{1},\dots,p_{n})
  48. p i p_{i}
  49. d M = M t d t + i = 1 n M p i d p i . \operatorname{d}M=\frac{\partial M}{\partial t}\operatorname{d}t+\sum_{i=1}^{n% }\frac{\partial M}{\partial p_{i}}\operatorname{d}p_{i}.
  50. p i p_{i}
  51. M ( t , p 1 , , p n ) M(t,p_{1},\dots,p_{n})
  52. p 1 2 = p 2 p 3 p_{1}^{2}=p_{2}p_{3}
  53. 2 p 1 d p 1 = p 3 d p 2 + p 2 d p 3 2p_{1}\operatorname{d}p_{1}=p_{3}\operatorname{d}p_{2}+p_{2}\operatorname{d}p_% {3}
  54. p i p_{i}
  55. d M = M t d t + i = 1 n M p i p i t d t . \operatorname{d}M=\frac{\partial M}{\partial t}\operatorname{d}t+\sum_{i=1}^{n% }\frac{\partial M}{\partial p_{i}}\frac{\partial p_{i}}{\partial t}\,% \operatorname{d}t.
  56. U n U\subseteq\mathbb{R}^{n}
  57. f : U m f:U\rightarrow\mathbb{R}^{m}
  58. p U p\in U
  59. d f p : n m \operatorname{d}f_{p}:\mathbb{R}^{n}\rightarrow\mathbb{R}^{m}
  60. lim x p f ( x ) - f ( p ) - d f p ( x - p ) x - p = 0. \lim_{x\rightarrow p}\frac{\|f(x)-f(p)-\operatorname{d}f_{p}(x-p)\|}{\|x-p\|}=0.
  61. d f p \operatorname{d}f_{p}
  62. f f
  63. p p
  64. f i : U f_{i}:U\rightarrow\mathbb{R}
  65. q = D ( p , I ) , q=D(p,I),
  66. q = S ( p , r , w ) , q=S(p,r,w),
  67. d p d r , \frac{dp}{dr},

Total_least_squares.html

  1. S = 𝐫 𝐓 𝐖𝐫 , S=\mathbf{r^{T}Wr},
  2. s y m b o l β symbol\beta
  3. 𝐫 = 𝐲 - 𝐗𝐬𝐲𝐦𝐛𝐨𝐥 β . \mathbf{r=y-Xsymbol\beta}.
  4. 𝐌 y \mathbf{M}_{y}
  5. 𝐗 𝐓 𝐖𝐗𝐬𝐲𝐦𝐛𝐨𝐥 β = 𝐗 𝐓 𝐖𝐲 \mathbf{X^{T}WXsymbol\beta=X^{T}Wy}
  6. 𝐌 x \mathbf{M}_{x}
  7. 𝐌 y \mathbf{M}_{y}
  8. S = 𝐫 𝐱 𝐓 𝐌 𝐱 - 𝟏 𝐫 𝐱 + 𝐫 𝐲 𝐓 𝐌 𝐲 - 𝟏 𝐫 𝐲 S=\mathbf{r_{x}^{T}M_{x}^{-1}r_{x}+r_{y}^{T}M_{y}^{-1}r_{y}}
  9. 𝐫 x \mathbf{r}_{x}\,
  10. 𝐫 y \mathbf{r}_{y}\,
  11. 𝐟 ( 𝐫 𝐱 , 𝐫 𝐲 , 𝐬𝐲𝐦𝐛𝐨𝐥 β ) \mathbf{f(r_{x},r_{y},symbol\beta)}
  12. 𝐅 = 𝚫 𝐲 - 𝐟 𝐫 𝐱 𝐫 𝐱 - 𝐟 𝐫 𝐲 𝐫 𝐲 - 𝐗 \Deltasymbol β = 𝟎 \mathbf{F=\Delta y-\frac{\partial f}{\partial r_{x}}r_{x}-\frac{\partial f}{% \partial r_{y}}r_{y}-X\Deltasymbol\beta=0}
  13. 𝐗 𝐓 𝐌 - 𝟏 𝐗 𝚫 𝐬𝐲𝐦𝐛𝐨𝐥 β = 𝐗 𝐓 𝐌 - 𝟏 𝚫 𝐲 \mathbf{X^{T}M^{-1}X\Delta symbol\beta=X^{T}M^{-1}\Delta y}
  14. 𝐗 𝐓 𝐌 - 𝟏 𝐗𝐬𝐲𝐦𝐛𝐨𝐥 β = 𝐗 𝐓 𝐌 - 𝟏 𝐲 \mathbf{X^{T}M^{-1}Xsymbol\beta=X^{T}M^{-1}y}
  15. 𝐌 = 𝐊 𝐱 𝐌 𝐱 𝐊 𝐱 𝐓 + 𝐊 𝐲 𝐌 𝐲 𝐊 𝐲 𝐓 ; 𝐊 𝐱 = - 𝐟 𝐫 𝐱 , 𝐊 𝐲 = - 𝐟 𝐫 𝐲 \mathbf{M=K_{x}M_{x}K_{x}^{T}+K_{y}M_{y}K_{y}^{T};\ K_{x}=-\frac{\partial f}{% \partial r_{x}},\ K_{y}=-\frac{\partial f}{\partial r_{y}}}
  16. f ( x i , β ) = α + β x i f(x_{i},\beta)=\alpha+\beta x_{i}\!
  17. M i i = σ y , i 2 + β 2 σ x , i 2 M_{ii}=\sigma^{2}_{y,i}+\beta^{2}\sigma^{2}_{x,i}
  18. β \beta
  19. M i i = σ y , i 2 + ( d y d x ) i 2 σ x , i 2 M_{ii}=\sigma^{2}_{y,i}+\left(\frac{dy}{dx}\right)^{2}_{i}\sigma^{2}_{x,i}
  20. X B Y XB\approx Y
  21. argmin E , F [ E F ] F , ( X + E ) B = Y + F \mathrm{argmin}_{E,F}\|[E\;F]\|_{F},\qquad(X+E)B=Y+F
  22. [ E F ] [E\;F]
  23. F \|\cdot\|_{F}
  24. [ ( X + E ) ( Y + F ) ] [ B - I k ] = 0 [(X+E)\;(Y+F)]\begin{bmatrix}B\\ -I_{k}\end{bmatrix}=0
  25. I k I_{k}
  26. k × k k\times k
  27. [ E F ] [E\;F]
  28. [ X Y ] [X\;Y]
  29. [ U ] [ Σ ] [ V ] * [U][\Sigma][V]*
  30. [ X Y ] [X\;Y]
  31. [ X Y ] = [ U X U Y ] [ Σ X 0 0 Σ Y ] [ V X X V X Y V Y X V Y Y ] * = [ U X U Y ] [ Σ X 0 0 Σ Y ] [ V X X * V Y X * V X Y * V Y Y * ] [X\;Y]=[U_{X}\;U_{Y}]\begin{bmatrix}\Sigma_{X}&0\\ 0&\Sigma_{Y}\end{bmatrix}\begin{bmatrix}V_{XX}&V_{XY}\\ V_{YX}&V_{YY}\end{bmatrix}^{*}=[U_{X}\;U_{Y}]\begin{bmatrix}\Sigma_{X}&0\\ 0&\Sigma_{Y}\end{bmatrix}\begin{bmatrix}V_{XX}^{*}&V_{YX}^{*}\\ V_{XY}^{*}&V_{YY}^{*}\end{bmatrix}
  32. U U
  33. V V
  34. k k
  35. [ ( X + E ) ( Y + F ) ] = [ U X U Y ] [ Σ X 0 0 0 k × k ] [ V X X V X Y V Y X V Y Y ] * [(X+E)\;(Y+F)]=[U_{X}\;U_{Y}]\begin{bmatrix}\Sigma_{X}&0\\ 0&0_{k\times k}\end{bmatrix}\begin{bmatrix}V_{XX}&V_{XY}\\ V_{YX}&V_{YY}\end{bmatrix}^{*}
  36. [ E F ] = - [ U X U Y ] [ 0 n × n 0 0 Σ Y ] [ V X X V X Y V Y X V Y Y ] * [E\;F]=-[U_{X}\;U_{Y}]\begin{bmatrix}0_{n\times n}&0\\ 0&\Sigma_{Y}\end{bmatrix}\begin{bmatrix}V_{XX}&V_{XY}\\ V_{YX}&V_{YY}\end{bmatrix}^{*}
  37. [ E F ] = - U Y Σ Y [ V X Y V Y Y ] * = - [ X Y ] [ V X Y V Y Y ] [ V X Y V Y Y ] * [E\;F]=-U_{Y}\Sigma_{Y}\begin{bmatrix}V_{XY}\\ V_{YY}\end{bmatrix}^{*}=-[X\;Y]\begin{bmatrix}V_{XY}\\ V_{YY}\end{bmatrix}\begin{bmatrix}V_{XY}\\ V_{YY}\end{bmatrix}^{*}
  38. [ ( X + E ) ( Y + F ) ] [ V X Y V Y Y ] = 0 [(X+E)\;(Y+F)]\begin{bmatrix}V_{XY}\\ V_{YY}\end{bmatrix}=0
  39. V Y Y V_{YY}
  40. V Y Y V_{YY}
  41. - V Y Y - 1 -V_{YY}^{-1}
  42. [ ( X + E ) ( Y + F ) ] [ - V X Y V Y Y - 1 - V Y Y V Y Y - 1 ] = [ ( X + E ) ( Y + F ) ] [ B - I k ] = 0 , [(X+E)\;(Y+F)]\begin{bmatrix}-V_{XY}V_{YY}^{-1}\\ -V_{YY}V_{YY}^{-1}\end{bmatrix}=[(X+E)\;(Y+F)]\begin{bmatrix}B\\ -I_{k}\end{bmatrix}=0,
  43. B = - V X Y V Y Y - 1 . B=-V_{XY}V_{YY}^{-1}.
  44. V Y Y V_{YY}
  45. B B
  46. X X
  47. B B
  48. 𝐉 𝐓 𝐌 - 𝟏 𝐉 𝚫 𝐬𝐲𝐦𝐛𝐨𝐥 β = 𝐉 𝐓 𝐌 - 𝟏 𝚫 𝐲 . \mathbf{J^{T}M^{-1}J\Delta symbol\beta=J^{T}M^{-1}\Delta y}.
  49. 𝐗 𝐓 𝐖𝐗𝐬𝐲𝐦𝐛𝐨𝐥 𝚫 𝐬𝐲𝐦𝐛𝐨𝐥 β = 𝐗 𝐓 𝐖𝐬𝐲𝐦𝐛𝐨𝐥 𝚫 𝐲 \mathbf{X^{T}WXsymbol\Delta symbol\beta=X^{T}Wsymbol\Delta y}
  50. s y m b o l Δ s y m b o l β symbol\Delta symbol\beta
  51. s y m b o l β symbol\beta
  52. s y m b o l Δ 𝐲 symbol\Delta\mathbf{y}
  53. s y m b o l β symbol\beta

Total_shareholder_return.html

  1. P r i c e b e g i n Price_{begin}
  2. P r i c e e n d Price_{end}
  3. T S R = ( P r i c e e n d - P r i c e b e g i n + D i v i d e n d s ) / P r i c e b e g i n TSR={(Price_{end}-Price_{begin}+Dividends)}/{Price_{begin}}

Total_variation.html

  1. f f
  2. [ a , b ] [a,b]\subset\mathbb{R}
  3. V b a ( f ) = sup 𝒫 i = 0 n P - 1 | f ( x i + 1 ) - f ( x i ) | , V^{a}_{b}(f)=\sup_{\mathcal{P}}\sum_{i=0}^{n_{P}-1}|f(x_{i+1})-f(x_{i})|,\,
  4. 𝒫 = { P = { x 0 , , x n P } | P is a partition of [ a , b ] } \scriptstyle\mathcal{P}=\left\{P=\{x_{0},\dots,x_{n_{P}}\}|P\,\text{ is a % partition of }[a,b]\right\}
  5. V ( f , Ω ) := sup { Ω f ( x ) div ϕ ( x ) d x : ϕ C c 1 ( Ω , n ) , ϕ L ( Ω ) 1 } , V(f,\Omega):=\sup\left\{\int_{\Omega}f(x)\operatorname{div}\phi(x)\,\mathrm{d}% x\colon\phi\in C_{c}^{1}(\Omega,\mathbb{R}^{n}),\ \|\phi\|_{L^{\infty}(\Omega)% }\leq 1\right\},
  6. C c 1 ( Ω , n ) \scriptstyle C_{c}^{1}(\Omega,\mathbb{R}^{n})
  7. Ω \Omega
  8. L ( Ω ) \scriptstyle\|\;\|_{L^{\infty}(\Omega)}
  9. Ω n \Omega\subseteq\mathbb{R}^{n}
  10. μ \mu
  11. ( X , Σ ) (X,\Sigma)
  12. W ¯ ( μ , ) \scriptstyle\overline{\mathrm{W}}(\mu,\cdot)
  13. W ¯ ( μ , ) \scriptstyle\underline{\mathrm{W}}(\mu,\cdot)
  14. W ¯ ( μ , E ) = sup { μ ( A ) A Σ and A E } E Σ \overline{\mathrm{W}}(\mu,E)=\sup\left\{\mu(A)\mid A\in\Sigma\,\text{ and }A% \subset E\right\}\qquad\forall E\in\Sigma
  15. W ¯ ( μ , E ) = inf { μ ( A ) A Σ and A E } E Σ \underline{\mathrm{W}}(\mu,E)=\inf\left\{\mu(A)\mid A\in\Sigma\,\text{ and }A% \subset E\right\}\qquad\forall E\in\Sigma
  16. W ¯ ( μ , E ) 0 W ¯ ( μ , E ) E Σ \overline{\mathrm{W}}(\mu,E)\geq 0\geq\underline{\mathrm{W}}(\mu,E)\qquad% \forall E\in\Sigma
  17. μ \mu
  18. | μ | ( E ) = W ¯ ( μ , E ) + | W ¯ ( μ , E ) | E Σ |\mu|(E)=\overline{\mathrm{W}}(\mu,E)+\left|\underline{\mathrm{W}}(\mu,E)% \right|\qquad\forall E\in\Sigma
  19. μ = | μ | ( X ) \|\mu\|=|\mu|(X)
  20. μ + ( ) = W ¯ ( μ , ) , \mu^{+}(\cdot)=\overline{\mathrm{W}}(\mu,\cdot)\,,
  21. μ - ( ) = - W ¯ ( μ , ) , \mu^{-}(\cdot)=-\underline{\mathrm{W}}(\mu,\cdot)\,,
  22. μ + \mu^{+}
  23. μ - \mu^{-}
  24. μ = μ + - μ - \mu=\mu^{+}-\mu^{-}\,
  25. | μ | = μ + + μ - |\mu|=\mu^{+}+\mu^{-}\,
  26. μ \mu
  27. μ \mu
  28. μ \mu
  29. | μ | ( E ) = sup π A \isin π | μ ( A ) | E Σ |\mu|(E)=\sup_{\pi}\sum_{A\isin\pi}|\mu(A)|\qquad\forall E\in\Sigma
  30. π \pi
  31. E E
  32. | μ | = μ + + μ - |\mu|=\mu^{+}+\mu^{-}\,
  33. μ \mu
  34. μ \mu
  35. | μ | ( E ) = sup π A \isin π μ ( A ) E Σ |\mu|(E)=\sup_{\pi}\sum_{A\isin\pi}\|\mu(A)\|\qquad\forall E\in\Sigma
  36. X X
  37. μ - ν \|\mu-\nu\|
  38. ( μ - ν ) ( X ) = 0 (\mu-\nu)(X)=0
  39. μ - ν = | μ - ν | ( X ) = sup { | μ ( A ) - ν ( A ) | : A Σ } \|\mu-\nu\|=|\mu-\nu|(X)=\sup\left\{\,\left|\mu(A)-\nu(A)\right|:A\in\Sigma\,\right\}
  40. δ ( μ , ν ) = 1 2 x | μ ( x ) - ν ( x ) | . \delta(\mu,\nu)=\frac{1}{2}\sum_{x}\left|\mu(x)-\nu(x)\right|\;.
  41. 1 2 \scriptstyle\frac{1}{2}
  42. f f
  43. f f
  44. [ a , b ] [a,b]\subset\mathbb{R}
  45. f f^{\prime}
  46. V b a ( f ) = a b | f ( x ) | d x V^{a}_{b}(f)=\int_{a}^{b}|f^{\prime}(x)|\mathrm{d}x
  47. f f
  48. Ω n \Omega\subseteq\mathbb{R}^{n}
  49. f f
  50. V ( f , Ω ) = Ω | f ( x ) | d x V(f,\Omega)=\int\limits_{\Omega}\left|\nabla f(x)\right|\mathrm{d}x
  51. | . | |.|
  52. l 2 l_{2}
  53. Ω f div φ = - Ω f φ \int\limits_{\Omega}f\,\mathrm{div}\varphi=-\int_{\Omega}\nabla f\cdot\varphi
  54. Ω div 𝐑 = Ω 𝐑 𝐧 \int\limits_{\Omega}\,\text{div}\mathbf{R}=\int\limits_{\partial\Omega}\mathbf% {R}\cdot\mathbf{n}
  55. 𝐑 := f φ \mathbf{R}:=f\mathbf{\varphi}
  56. Ω div ( f φ ) = Ω ( f φ ) 𝐧 \int\limits_{\Omega}\,\text{div}\left(f\mathbf{\varphi}\right)=\int\limits_{% \partial\Omega}\left(f\mathbf{\varphi}\right)\cdot\mathbf{n}
  57. φ \mathbf{\varphi}
  58. Ω \Omega
  59. Ω div ( f φ ) = 0 \int\limits_{\Omega}\,\text{div}\left(f\mathbf{\varphi}\right)=0
  60. Ω x i ( f φ i ) = 0 \int\limits_{\Omega}\partial_{x_{i}}\left(f\mathbf{\varphi}_{i}\right)=0
  61. Ω φ i x i f + f x i φ i = 0 \int\limits_{\Omega}\mathbf{\varphi}_{i}\partial_{x_{i}}f+f\partial_{x_{i}}% \mathbf{\varphi}_{i}=0
  62. Ω f x i φ i = - Ω φ i x i f \int\limits_{\Omega}f\partial_{x_{i}}\mathbf{\varphi}_{i}=-\int\limits_{\Omega% }\mathbf{\varphi}_{i}\partial_{x_{i}}f
  63. Ω f div φ = - Ω φ f \int\limits_{\Omega}f\,\text{div}\mathbf{\varphi}=-\int\limits_{\Omega}\mathbf% {\varphi}\cdot\nabla f
  64. Ω f div φ = - Ω φ f | Ω φ f | Ω | φ | | f | Ω | f | \int\limits_{\Omega}f\,\text{div}\mathbf{\varphi}=-\int\limits_{\Omega}\mathbf% {\varphi}\cdot\nabla f\leq\left|\int\limits_{\Omega}\mathbf{\varphi}\cdot% \nabla f\right|\leq\int\limits_{\Omega}\left|\mathbf{\varphi}\right|\cdot\left% |\nabla f\right|\leq\int\limits_{\Omega}\left|\nabla f\right|
  65. φ \mathbf{\varphi}
  66. θ n := 𝕀 [ - N , N ] f | f | \theta_{n}:=\mathbb{I}_{\left[-N,N\right]}\frac{\nabla f}{\left|\nabla f\right|}
  67. θ n * \theta^{*}_{n}
  68. ε \varepsilon
  69. θ \theta
  70. C c 1 C^{1}_{c}
  71. C c 1 C^{1}_{c}
  72. L 1 L^{1}
  73. lim N Ω f div θ n * = lim N Ω 𝕀 [ - N , N ] f f | f | = lim N 𝕀 [ - N , N ] f f | f | = Ω | f | \lim\limits_{N\rightarrow\infty}\int\limits_{\Omega}f\,\text{div}\theta^{*}_{n% }=\lim\limits_{N\rightarrow\infty}\int\limits_{\Omega}\mathbb{I}_{\left[-N,N% \right]}\nabla f\cdot\frac{\nabla f}{\left|\nabla f\right|}=\lim\limits_{N% \rightarrow\infty}\int\limits_{\mathbb{I}_{\left[-N,N\right]}}\nabla f\cdot% \frac{\nabla f}{\left|\nabla f\right|}=\int\limits_{\Omega}\left|\nabla f\right|
  74. Ω f div φ \int\limits_{\Omega}f\,\text{div}\mathbf{\varphi}
  75. Ω | f | \int\limits_{\Omega}\left|\nabla f\right|
  76. Ω f div φ Ω | f | \int\limits_{\Omega}f\,\text{div}\mathbf{\varphi}\leq\int\limits_{\Omega}\left% |\nabla f\right|
  77. φ - f | f | . \varphi\to\frac{-\nabla f}{\left|\nabla f\right|}.
  78. f f
  79. φ : \scriptstyle\varphi\colon\mathbb{R}\to\mathbb{R}
  80. φ ( t ) = μ ( ( - , t ] ) . \varphi(t)=\mu((-\infty,t])~{}.
  81. μ T V = μ + ( X ) + μ - ( X ) , \|\mu\|_{TV}=\mu_{+}(X)+\mu_{-}(X)~{},
  82. ( X , Σ ) (X,\Sigma)

Total_variation_distance_of_probability_measures.html

  1. \mathcal{F}
  2. Ω \Omega
  3. δ ( P , Q ) = sup A | P ( A ) - Q ( A ) | . \delta(P,Q)=\sup_{A\in\mathcal{F}}\left|P(A)-Q(A)\right|.
  4. δ ( P , Q ) = 1 2 P - Q 1 = 1 2 x | P ( x ) - Q ( x ) | . \delta(P,Q)=\frac{1}{2}\|P-Q\|_{1}=\frac{1}{2}\sum_{x}\left|P(x)-Q(x)\right|\;.
  5. Ω \Omega
  6. μ \mu
  7. P P
  8. Q Q
  9. f P f_{P}
  10. f Q f_{Q}
  11. μ \mu
  12. δ ( P , Q ) = 1 2 f P - f Q L 1 ( μ ) = 1 2 Ω | f P - f Q | d μ . \delta(P,Q)=\frac{1}{2}\|f_{P}-f_{Q}\|_{L_{1}(\mu)}=\frac{1}{2}\int_{\Omega}% \left|f_{P}-f_{Q}\right|d\mu\;.

Toughness.html

  1. energy volume = 0 ϵ f σ d ϵ \tfrac{\mbox{energy}~{}}{\mbox{volume}~{}}=\int_{0}^{\epsilon_{f}}\sigma\,d\epsilon
  2. ϵ \epsilon
  3. ϵ f \epsilon_{f}
  4. σ \sigma
  5. Y i e l d s t r e s s < s u p > 2 2 ( Y o u n g s m o d u l u s ) Yieldstress<sup>2\frac{2(Young}{smodulus)}

Tractrix.html

  1. d y d x = - a 2 - x 2 x \frac{dy}{dx}=-\frac{\sqrt{a^{2}-x^{2}}}{x}\,\!
  2. y = x a a 2 - t 2 t d t = ± ( a ln a + a 2 - x 2 x - a 2 - x 2 ) . y=\int_{x}^{a}\frac{\sqrt{a^{2}-t^{2}}}{t}\,dt=\pm\left(a\ln{\frac{a+\sqrt{a^{% 2}-x^{2}}}{x}}-\sqrt{a^{2}-x^{2}}\right).\,\!
  3. a arsech x a , a\ \mathrm{arsech}\frac{x}{a},\,\!
  4. y ( x ) = a cosh ( x / a ) y(x)=a\,\operatorname{cosh}(x/a)
  5. r = a cot ( x / y ) r=a\,\operatorname{cot}(x/y)
  6. a a
  7. a ln x 1 x 2 a\ln\frac{x_{1}}{x_{2}}
  8. π a 2 / 2 \pi a^{2}/2
  9. y = a cosh x a y=a\cosh\frac{x}{a}

Transcendence_theory.html

  1. | α - p q | < 1 q d + ε \left|\alpha-\frac{p}{q}\right|<\frac{1}{q^{d+\varepsilon}}
  2. | β 1 log α 1 + β 2 log α 2 | |\beta_{1}\log\alpha_{1}+\beta_{2}\log\alpha_{2}|\,
  3. α 1 β 1 α 2 β 2 α n β n \alpha_{1}^{\beta_{1}}\alpha_{2}^{\beta_{2}}\cdots\alpha_{n}^{\beta_{n}}
  4. K = ( x 1 , , x n , e x 1 , , e x n ) K=\mathbb{Q}(x_{1},\ldots,x_{n},e^{x_{1}},\ldots,e^{x_{n}})
  5. | P ( a ) | > F ( A , d ) |P(a)|>F(A,d)
  6. | a x + b | |ax+b|

Transconductance.html

  1. g m = Δ I out Δ V in g_{m}={\Delta I_{\mathrm{out}}\over\Delta V_{\mathrm{in}}}
  2. g m = i out v in g_{m}={i_{\mathrm{out}}\over v_{\mathrm{in}}}
  3. r m = Δ V out Δ I in r_{m}={\Delta V\text{out}\over\Delta I\text{in}}
  4. g m = μ r p g_{m}={\mu\over r_{p}}
  5. g m = 2 I D V eff g_{m}=\begin{matrix}\frac{2I_{D}}{V_{\mathrm{eff}}}\end{matrix}\,
  6. g m = 2 I D S S | V P | ( 1 - V G S < m t p l > V P ) g_{m}=\frac{{2I_{DSS}}}{{\left|{V_{P}}\right|}}\left({1-\frac{{V_{GS}}}{<}mtpl% >{{V_{P}}}}\right)
  7. g m = I C V T g_{m}=\frac{I_{C}}{V_{T}}

Transfer_(group_theory).html

  1. x 1 , , x n , x_{1},\dots,x_{n},\,
  2. G = x i H . G=\cup\ x_{i}H.
  3. y x i = x j h i yx_{i}=x_{j}h_{i}
  4. i = 1 n h i \textstyle\prod_{i=1}^{n}h_{i}

Transfer_principle.html

  1. x [ x ] x\geq[x]
  2. x [ x ] * , x\geq{}^{*}\![x],
  3. \mathbb{N}
  4. \mathbb{Z}
  5. \mathbb{Q}
  6. \mathbb{R}
  7. \mathbb{Q}
  8. x y x < y . \forall x\in\mathbb{R}\quad\exists y\in\mathbb{R}\quad x<y.
  9. x y x < y . \forall x\in{}^{\star}\mathbb{R}\quad\exists y\in{}^{\star}\mathbb{R}\quad x<y.
  10. x x < x + 1 \forall x\in\mathbb{R}\quad x<x+1
  11. x x < x + 1. \forall x\in{}^{\star}\mathbb{R}\quad x<x+1.
  12. 1 < ω , 1 + 1 < ω , 1 + 1 + 1 < ω , 1 + 1 + 1 + 1 < ω , 1<\omega,\quad 1+1<\omega,\quad 1+1+1<\omega,\quad 1+1+1+1<\omega,\ldots
  13. | x | + + | x | n terms < 1 for every finite cardinal number n . \underbrace{\left|x\right|+\cdots+\left|x\right|}_{n\,\text{ terms}}<1\,\text{% for every finite cardinal number }n.\,
  14. 1 + + 1 n terms < | y | for every finite cardinal number n . \underbrace{1+\cdots+1}_{n\,\text{ terms}}<\left|y\right|\,\text{ for every % finite cardinal number }n.\,
  15. \mapsto
  16. A A * , A\subseteq{{}^{*}\!A},\,
  17. A \scriptstyle A\,\subseteq\,\mathbb{R}
  18. f : A f:A\rightarrow\mathbb{R}\,
  19. f * : A * * ; {{}^{*}\!f}:{{}^{*}\!A}\rightarrow{{}^{*}\mathbb{R}};\,
  20. \mapsto
  21. x and x . \forall x\in\mathbb{R}\,\text{ and }\exists x\in\mathbb{R}.\,
  22. x y x + y = 0. \forall x\in\mathbb{R}\ \exists y\in\mathbb{R}\ x+y=0.\,
  23. x * \forall x\in{{}^{*}\!\mathbb{R}}\,
  24. x , \forall x\in\mathbb{R},\,
  25. \exists
  26. A \scriptstyle A\,\subseteq\,\mathbb{R}
  27. [ 0 , 1 ] = { x : 0 x 1 } [0,1]^{\ast}=\{\,x\in\mathbb{R}:0\leq x\leq 1\,\}^{\ast}
  28. { x * : 0 x 1 } , \{\,x\in{{}^{*}\mathbb{R}}:0\leq x\leq 1\,\},
  29. x ( x [ 0 , 1 ] if and only if 0 x 1 ) \forall x\in\mathbb{R}\ (x\in[0,1]\,\text{ if and only if }0\leq x\leq 1)
  30. * {{}^{*}\mathbb{N}}\setminus\mathbb{N}\,
  31. A or A . \forall A\subseteq\mathbb{R}\dots\,\text{ or }\exists A\subseteq\mathbb{R}% \dots\ .
  32. [ internal A * ] [\forall\,\text{ internal }A\subseteq{{}^{*}\mathbb{R}}\dots]\,
  33. [ internal A * ] . [\exists\,\text{ internal }A\subseteq{{}^{*}\mathbb{R}}\dots]\ .
  34. \mathbb{N}
  35. * {{}^{*}\mathbb{N}}\setminus\mathbb{N}\,
  36. n A x [ x A iff x n ] . \forall n\in\mathbb{N}\ \exists A\subseteq\mathbb{N}\ \forall x\in\mathbb{N}\ % [x\in A\,\text{ iff }x\leq n].
  37. n * internal A * x * [ x A iff x n ] . \forall n\in{{}^{*}\mathbb{N}}\ \exists\,\text{ internal }A\subseteq{{}^{*}% \mathbb{N}}\ \forall x\in{{}^{*}\mathbb{N}}\ [x\in A\,\text{ iff }x\leq n].
  38. f : A \forall f:A\rightarrow\mathbb{R}\dots\,
  39. internal f : A * * \forall\,\text{ internal }f:{{}^{*}\!A}\rightarrow{{}^{*}\mathbb{R}}\dots
  40. \exists
  41. \forall

Transformation_matrix.html

  1. x \vec{x}
  2. T ( x ) = 𝐀 x T(\vec{x})=\mathbf{A}\vec{x}
  3. T ( x ) T(x)
  4. 𝐀 = [ T ( e 1 ) T ( e 2 ) T ( e n ) ] \mathbf{A}=\begin{bmatrix}T(\vec{e}_{1})&T(\vec{e}_{2})&\cdots&T(\vec{e}_{n})% \end{bmatrix}
  5. T ( x ) = 5 x T(x)=5x
  6. T ( x ) = 5 x = 5 𝐈 x = [ 5 0 0 5 ] x T(\vec{x})=5\vec{x}=5\mathbf{I}\vec{x}=\begin{bmatrix}5&&0\\ 0&&5\end{bmatrix}\vec{x}
  7. E = [ e 1 e 2 e n ] E=[\vec{e}_{1}\vec{e}_{2}\ldots\vec{e}_{n}]
  8. [ v ] E = [ v 1 v 2 v n ] T [v]_{E}=[v_{1}v_{2}\ldots v_{n}]^{T}
  9. v = v 1 e 1 + v 2 e 2 + + v n e n = v i e i = E [ v ] E \vec{v}=v_{1}\vec{e}_{1}+v_{2}\vec{e}_{2}+\ldots+v_{n}\vec{e}_{n}=\sum v_{i}% \vec{e}_{i}=E[v]_{E}
  10. v \vec{v}
  11. A ( v ) = A ( v i e i ) = v i A ( e i ) = [ A ( e 1 ) A ( e 2 ) A ( e n ) ] [ v ] E = A(\vec{v})=A(\sum{v_{i}\vec{e}_{i}})=\sum{v_{i}A(\vec{e}_{i})}=[A(\vec{e}_{1})% A(\vec{e}_{2})\ldots A(\vec{e}_{n})][v]_{E}=
  12. = A [ v ] E = [ e 1 e 2 e n ] [ a 1 , 1 a 1 , 2 a 1 , n a 2 , 1 a 2 , 2 a 2 , n a n , 1 a n , 2 a n , n ] [ v 1 v 2 v n ] \;=\;A\cdot[v]_{E}=[\vec{e}_{1}\vec{e}_{2}\ldots\vec{e}_{n}]\begin{bmatrix}a_{% 1,1}&a_{1,2}&\ldots&a_{1,n}\\ a_{2,1}&a_{2,2}&\ldots&a_{2,n}\\ \vdots&\vdots&\ddots&\vdots\\ a_{n,1}&a_{n,2}&\ldots&a_{n,n}\\ \end{bmatrix}\begin{bmatrix}v_{1}\\ v_{2}\\ \vdots\\ v_{n}\end{bmatrix}
  13. a i , j a_{i,j}
  14. e j = [ 00 ( v j = 1 ) 0 ] T \vec{e}_{j}=[00\ldots(v_{j}=1)\ldots 0]^{T}
  15. A e j = a 1 , j e 1 + a 2 , j e 2 + + a n , j e n = a i , j e i A\vec{e}_{j}=a_{1,j}\vec{e}_{1}+a_{2,j}\vec{e}_{2}+\ldots+a_{n,j}\vec{e}_{n}=% \sum a_{i,j}\vec{e}_{i}
  16. a i , j a_{i,j}
  17. a i , j a_{i,j}
  18. a i , i a_{i,i}
  19. a i , j e i \sum a_{i,j}\vec{e}_{i}
  20. a i , i a_{i,i}
  21. λ i \lambda_{i}
  22. A e i = λ i e i A\vec{e}_{i}=\lambda_{i}\vec{e}_{i}
  23. x = x cos θ + y sin θ x^{\prime}=x\cos\theta+y\sin\theta
  24. y = - x sin θ + y cos θ y^{\prime}=-x\sin\theta+y\cos\theta
  25. [ x y ] = [ cos θ sin θ - sin θ cos θ ] [ x y ] \begin{bmatrix}x^{\prime}\\ y^{\prime}\end{bmatrix}=\begin{bmatrix}\cos\theta&\sin\theta\\ -\sin\theta&\cos\theta\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}
  26. x = x cos θ - y sin θ x^{\prime}=x\cos\theta-y\sin\theta
  27. y = x sin θ + y cos θ y^{\prime}=x\sin\theta+y\cos\theta
  28. [ x y ] = [ cos θ - sin θ sin θ cos θ ] [ x y ] \begin{bmatrix}x^{\prime}\\ y^{\prime}\end{bmatrix}=\begin{bmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}
  29. x = x + k y x^{\prime}=x+ky
  30. y = y y^{\prime}=y
  31. [ x y ] = [ 1 k 0 1 ] [ x y ] \begin{bmatrix}x^{\prime}\\ y^{\prime}\end{bmatrix}=\begin{bmatrix}1&k\\ 0&1\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}
  32. x = x x^{\prime}=x
  33. y = y + k x y^{\prime}=y+kx
  34. [ x y ] = [ 1 0 k 1 ] [ x y ] \begin{bmatrix}x^{\prime}\\ y^{\prime}\end{bmatrix}=\begin{bmatrix}1&0\\ k&1\end{bmatrix}\begin{bmatrix}x\\ y\end{bmatrix}
  35. l = ( l x , l y ) \scriptstyle\vec{l}=(l_{x},l_{y})
  36. 𝐀 = 1 l 2 [ l x 2 - l y 2 2 l x l y 2 l x l y l y 2 - l x 2 ] \mathbf{A}=\frac{1}{\lVert\vec{l}\rVert^{2}}\begin{bmatrix}l_{x}^{2}-l_{y}^{2}% &2l_{x}l_{y}\\ 2l_{x}l_{y}&l_{y}^{2}-l_{x}^{2}\end{bmatrix}
  37. u = ( u x , u y ) \scriptstyle\vec{u}\,=\,(u_{x},u_{y})
  38. 𝐀 = 1 u 2 [ u x 2 u x u y u x u y u y 2 ] \mathbf{A}=\frac{1}{\lVert\vec{u}\rVert^{2}}\begin{bmatrix}u_{x}^{2}&u_{x}u_{y% }\\ u_{x}u_{y}&u_{y}^{2}\end{bmatrix}
  39. [ l l ( 1 - cos θ ) + cos θ m l ( 1 - cos θ ) - n sin θ n l ( 1 - cos θ ) + m sin θ l m ( 1 - cos θ ) + n sin θ m m ( 1 - cos θ ) + cos θ n m ( 1 - cos θ ) - l sin θ l n ( 1 - cos θ ) - m sin θ m n ( 1 - cos θ ) + l sin θ n n ( 1 - cos θ ) + cos θ ] . \begin{bmatrix}ll(1-\cos\theta)+\cos\theta&ml(1-\cos\theta)-n\sin\theta&nl(1-% \cos\theta)+m\sin\theta\\ lm(1-\cos\theta)+n\sin\theta&mm(1-\cos\theta)+\cos\theta&nm(1-\cos\theta)-l% \sin\theta\\ ln(1-\cos\theta)-m\sin\theta&mn(1-\cos\theta)+l\sin\theta&nn(1-\cos\theta)+% \cos\theta\end{bmatrix}.
  40. a x + b y + c z = 0 ax+by+cz=0
  41. 𝐀 = 𝐈 - 2 𝐍𝐍 T \mathbf{A}=\mathbf{I}-2\mathbf{NN}^{T}
  42. 𝐈 \mathbf{I}
  43. 𝐍 \mathbf{N}
  44. a , b , a,b,
  45. c c
  46. 𝐀 = [ 1 - 2 a 2 - 2 a b - 2 a c - 2 a b 1 - 2 b 2 - 2 b c - 2 a c - 2 b c 1 - 2 c 2 ] \mathbf{A}=\begin{bmatrix}1-2a^{2}&-2ab&-2ac\\ -2ab&1-2b^{2}&-2bc\\ -2ac&-2bc&1-2c^{2}\end{bmatrix}
  47. 𝐁 ( 𝐀 x ) = ( 𝐁𝐀 ) x \mathbf{B}(\mathbf{A}\vec{x})=(\mathbf{BA})\vec{x}
  48. x = x + t x ; y = y + t y x^{\prime}=x+t_{x};y^{\prime}=y+t_{y}
  49. [ x y 1 ] = [ 1 0 t x 0 1 t y 0 0 1 ] [ x y 1 ] . \begin{bmatrix}x^{\prime}\\ y^{\prime}\\ 1\end{bmatrix}=\begin{bmatrix}1&0&t_{x}\\ 0&1&t_{y}\\ 0&0&1\end{bmatrix}\begin{bmatrix}x\\ y\\ 1\end{bmatrix}.
  50. [ cos θ sin θ 0 - sin θ cos θ 0 0 0 1 ] \begin{bmatrix}\cos\theta&\sin\theta&0\\ -\sin\theta&\cos\theta&0\\ 0&0&1\end{bmatrix}
  51. ( t x , t y ) (t^{\prime}_{x},t^{\prime}_{y})
  52. ( s x , s y ) (s_{x},s_{y})
  53. ( t x , t y ) (t_{x},t_{y})
  54. [ s x cos θ - s y sin θ t x s x cos θ - t y s y sin θ + t x s x sin θ s y cos θ t x s x sin θ + t y s y cos θ + t y 0 0 1 ] \begin{bmatrix}s_{x}\cos\theta&-s_{y}\sin\theta&t_{x}s_{x}\cos\theta-t_{y}s_{y% }\sin\theta+t^{\prime}_{x}\\ s_{x}\sin\theta&s_{y}\cos\theta&t_{x}s_{x}\sin\theta+t_{y}s_{y}\cos\theta+t^{% \prime}_{y}\\ 0&0&1\end{bmatrix}
  55. x = x / z x^{\prime}=x/z
  56. y = y / z y^{\prime}=y/z
  57. [ x c y c z c w c ] = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 1 0 ] [ x y z w ] \begin{bmatrix}x_{c}\\ y_{c}\\ z_{c}\\ w_{c}\end{bmatrix}=\begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&1&0\end{bmatrix}\begin{bmatrix}x\\ y\\ z\\ w\end{bmatrix}
  58. [ x y z 1 ] = 1 w c [ x c y c z c w c ] \begin{bmatrix}x^{\prime}\\ y^{\prime}\\ z^{\prime}\\ 1\end{bmatrix}=\frac{1}{w_{c}}\begin{bmatrix}x_{c}\\ y_{c}\\ z_{c}\\ w_{c}\end{bmatrix}

Translational_symmetry.html

  1. f : \R 2 \R f:\R^{2}\rightarrow\R
  2. f ( A ) = f ( A + t ) f(A)=f(A+t)
  3. T δ T_{\delta}
  4. δ A f = A ( T δ f ) . \forall\delta\ Af=A(T_{\delta}f).\,

Transmission_risks_and_rates.html

  1. β = γ × p \beta=\gamma\times p\,
  2. p = x n . p=\frac{x}{n}.

Transposition_(music).html

  1. 9 + 4 = 13 9+4=13
  2. 9 + 4 = 13 1 ( mod 12 ) 9+4=13\equiv 1\;\;(\mathop{{\rm mod}}12)
  3. T n p ( x ) = x + n T^{p}_{n}(x)=x+n
  4. T n p ( x ) x + n T^{p}_{n}(x)\rightarrow x+n
  5. T n ( x ) = x + n ( mod 12 ) T_{n}(x)=x+n\;\;(\mathop{{\rm mod}}12)
  6. T o ( p i , j ) = p i , j + t o T_{o}(p_{i,j})=p_{i,j}+t_{o}
  7. t o t_{o}
  8. t o t_{o}
  9. T o T_{o}
  10. t o t_{o}
  11. o o
  12. p i , j p_{i,j}
  13. i i
  14. P P
  15. j j

Transversal_(combinatorics).html

  1. G = H × K G=H\times K
  2. f f
  3. ker f := { { y X f ( x ) = f ( y ) } x X } \operatorname{ker}f:=\left\{\,\left\{\,y\in X\mid f(x)=f(y)\,\right\}\mid x\in X% \,\right\}
  4. ker f \operatorname{ker}f
  5. ker f \operatorname{ker}f
  6. Im f \operatorname{Im}f
  7. g : ( Im f ) T g:(\operatorname{Im}f)\to T
  8. Im f , g ( z ) = x \operatorname{Im}f,g(z)=x
  9. f ( x ) = z f(x)=z
  10. f g f = f f\circ g\circ f=f
  11. g g
  12. g f g = g g\circ f\circ g=g
  13. h = g f g h=g\circ f\circ g
  14. h f h = h h\circ f\circ h=h
  15. S i 1 S i 2 S i k S_{i_{1}}\cup S_{i_{2}}\cup\dots\cup S_{i_{k}}
  16. i 1 , i 2 , , i k i_{1},i_{2},\ldots,i_{k}

Transverse_Mercator_projection.html

  1. x = a λ x=a\lambda
  2. y y
  3. ϕ \phi
  4. x = a λ , y = a ln [ tan ( π 4 + ϕ 2 ) ] = a 2 ln [ 1 + sin ϕ 1 - sin ϕ ] . x=a\lambda\,,\qquad y=a\ln\bigg[\tan\bigg(\frac{\pi}{4}+\frac{\phi}{2}\bigg)% \bigg]=\frac{a}{2}\ln\left[\frac{1+\sin\phi}{1-\sin\phi}\right].
  5. k k
  6. k ( ϕ ) = sec ϕ . k(\phi)=\sec\phi.\,
  7. k 0 k_{0}
  8. k 0 k_{0}
  9. x x
  10. y y
  11. ( ϕ , λ ) (\phi,\lambda)
  12. ϕ \phi^{\prime}
  13. - λ -\lambda^{\prime}
  14. ( ϕ , λ ) (\phi^{\prime},\lambda^{\prime})
  15. ( ϕ , λ ) (\phi,\lambda)
  16. ( x , y ) (x^{\prime},y^{\prime})
  17. ( x , y ) (x,y)
  18. ( x , y ) (x^{\prime},y^{\prime})
  19. - λ -\lambda^{\prime}
  20. ϕ \phi^{\prime}
  21. x = - a λ y = a 2 ln [ 1 + sin ϕ 1 - sin ϕ ] . x^{\prime}=-a\lambda^{\prime}\,\qquad y^{\prime}=\frac{a}{2}\ln\left[\frac{1+% \sin\phi^{\prime}}{1-\sin\phi^{\prime}}\right].
  22. sin ϕ \displaystyle\sin\phi^{\prime}
  23. ( x , y ) (x,y)
  24. x = y x=y^{\prime}
  25. y = - x y=-x^{\prime}
  26. k 0 k_{0}
  27. x ( λ , ϕ ) \displaystyle x(\lambda,\phi)
  28. λ ( x , y ) \displaystyle\lambda(x,y)
  29. k = sec ϕ k=\sec\phi^{\prime}
  30. k ( λ , ϕ ) \displaystyle k(\lambda,\phi)
  31. k 0 = 0.9996 k_{0}=0.9996
  32. k = 1 k=1
  33. x x
  34. x x
  35. k 0 = 1.0004 k_{0}=1.0004
  36. γ \gamma
  37. γ \gamma
  38. γ ( λ , ϕ ) \displaystyle\gamma(\lambda,\phi)

Tree_(set_theory).html

  1. { a , b } \left\{a,b\right\}
  2. { 0 , 1 , 2 , , ω 0 , ω 1 } \left\{0,1,2,\dots,\omega_{0},\omega_{1}\right\}
  3. ω 0 , ω 1 \omega_{0},\omega_{1}

Triacontagon.html

  1. A = 15 2 t 2 cot π 30 = 15 2 t 2 ( 23 + 10 5 + 2 3 ( 85 + 38 5 ) = 15 4 t 2 ( 15 + 3 3 + 2 25 + 11 5 ) A=\frac{15}{2}t^{2}\cot\frac{\pi}{30}=\frac{15}{2}t^{2}(\sqrt{23+10\sqrt{5}+2% \sqrt{3(85+38\sqrt{5})}}=\frac{15}{4}t^{2}(\sqrt{15}+3\sqrt{3}+\sqrt{2}\sqrt{2% 5+11\sqrt{5}})
  2. r = 1 2 t cot π 30 = 1 4 t ( 15 + 3 3 + 2 25 + 11 5 ) r=\frac{1}{2}t\cot\frac{\pi}{30}=\frac{1}{4}t(\sqrt{15}+3\sqrt{3}+\sqrt{2}% \sqrt{25+11\sqrt{5}})
  3. R = 1 2 t csc π 30 = 1 2 t ( 2 + 5 + 15 + 6 5 ) R=\frac{1}{2}t\csc\frac{\pi}{30}=\frac{1}{2}t(2+\sqrt{5}+\sqrt{15+6\sqrt{5}})

Triakis_octahedron.html

  1. A = 3 7 + 4 2 A=3\sqrt{7+4\sqrt{2}}
  2. V = 1 2 ( 3 + 2 2 ) . V=\frac{1}{2}(3+2\sqrt{2}).

Triakis_tetrahedron.html

  1. 5 3 11 \tfrac{5}{3}\scriptstyle{\sqrt{11}}
  2. 25 36 2 \tfrac{25}{36}\scriptstyle{\sqrt{2}}

Triality.html

  1. F F
  2. V 1 × V 2 F , V_{1}\times V_{2}\to F,
  3. v v
  4. v v
  5. F F
  6. V 1 × V 2 × V 3 F , V_{1}\times V_{2}\times V_{3}\to F,
  7. V V
  8. V × V V V\times V\to V
  9. e < s u b > i e<sub>i

Triangular_cupola.html

  1. V = ( 5 3 2 ) a 3 1.17851... a 3 V=(\frac{5}{3\sqrt{2}})a^{3}\approx 1.17851...a^{3}
  2. A = ( 3 + 5 3 2 ) a 2 7.33013... a 2 A=(3+\frac{5\sqrt{3}}{2})a^{2}\approx 7.33013...a^{2}

Triangular_distribution.html

  1. c c\,
  2. a 2 + b 2 + c 2 - a b - a c - b c 18 \frac{a^{2}+b^{2}+c^{2}-ab-ac-bc}{18}
  3. 2 ( a + b - 2 c ) ( 2 a - b - c ) ( a - 2 b + c ) 5 ( a 2 + b 2 + c 2 - a b - a c - b c ) 3 2 \frac{\sqrt{2}(a\!+\!b\!-\!2c)(2a\!-\!b\!-\!c)(a\!-\!2b\!+\!c)}{5(a^{2}\!+\!b^% {2}\!+\!c^{2}\!-\!ab\!-\!ac\!-\!bc)^{\frac{3}{2}}}
  4. - 3 5 -\frac{3}{5}
  5. 1 2 + ln ( b - a 2 ) \frac{1}{2}+\ln\left(\frac{b-a}{2}\right)
  6. 2 ( b - c ) e a t - ( b - a ) e c t + ( c - a ) e b t ( b - a ) ( c - a ) ( b - c ) t 2 2\frac{(b\!-\!c)e^{at}\!-\!(b\!-\!a)e^{ct}\!+\!(c\!-\!a)e^{bt}}{(b-a)(c-a)(b-c% )t^{2}}
  7. - 2 ( b - c ) e i a t - ( b - a ) e i c t + ( c - a ) e i b t ( b - a ) ( c - a ) ( b - c ) t 2 -2\frac{(b\!-\!c)e^{iat}\!-\!(b\!-\!a)e^{ict}\!+\!(c\!-\!a)e^{ibt}}{(b-a)(c-a)% (b-c)t^{2}}
  8. E ( X ) = 2 3 Var ( X ) = 1 18 \begin{aligned}\displaystyle E(X)&\displaystyle=\frac{2}{3}\\ \displaystyle\mathrm{Var}(X)&\displaystyle=\frac{1}{18}\end{aligned}
  9. f ( x ) = { 4 x for 0 x < 1 2 4 - 4 x for 1 2 x 1 f(x)=\begin{cases}4x&\,\text{for }0\leq x<\frac{1}{2}\\ 4-4x&\,\text{for }\frac{1}{2}\leq x\leq 1\end{cases}
  10. F ( x ) = { 2 x 2 for 0 x < 1 2 1 - 2 ( 1 - x ) 2 for 1 2 x 1 F(x)=\begin{cases}2x^{2}&\,\text{for }0\leq x<\frac{1}{2}\\ 1-2(1-x)^{2}&\,\text{for }\frac{1}{2}\leq x\leq 1\end{cases}
  11. E ( X ) = 1 2 Var ( X ) = 1 24 \begin{aligned}\displaystyle E(X)&\displaystyle=\frac{1}{2}\\ \displaystyle\operatorname{Var}(X)&\displaystyle=\frac{1}{24}\end{aligned}
  12. f ( x ) = 2 - 2 x for 0 x < 1 F ( x ) = 2 x - x 2 for 0 x < 1 E ( X ) = 1 3 Var ( X ) = 1 18 \begin{aligned}\displaystyle f(x)&\displaystyle=2-2x\,\text{ for }0\leq x<1\\ \displaystyle F(x)&\displaystyle=2x-x^{2}\,\text{ for }0\leq x<1\\ \displaystyle E(X)&\displaystyle=\frac{1}{3}\\ \displaystyle\operatorname{Var}(X)&\displaystyle=\frac{1}{18}\end{aligned}
  13. { X = a + U ( b - a ) ( c - a ) for 0 < U < F ( c ) X = b - ( 1 - U ) ( b - a ) ( b - c ) for F ( c ) U < 1 \begin{matrix}\begin{cases}X=a+\sqrt{U(b-a)(c-a)}&\,\text{ for }0<U<F(c)\\ &\\ X=b-\sqrt{(1-U)(b-a)(b-c)}&\,\text{ for }F(c)\leq U<1\end{cases}\end{matrix}

Triangular_hebesphenorotunda.html

  1. ( 0 , 2 3 , 2 ϕ 2 3 ) \left(0,\ \frac{2}{\sqrt{3}},\ \frac{2\phi^{2}}{\sqrt{3}}\right)
  2. ( ± 1 , - 1 3 , 2 ϕ 2 3 ) \left(\pm 1,\ -\frac{1}{\sqrt{3}},\ \frac{2\phi^{2}}{\sqrt{3}}\right)
  3. ( ± 1 , ϕ 3 3 , 2 ϕ 3 ) \left(\pm 1,\ \frac{\phi^{3}}{\sqrt{3}},\ \frac{2\phi}{\sqrt{3}}\right)
  4. ( ± ϕ 2 , - 1 ϕ 3 , 2 ϕ 3 ) \left(\pm\phi^{2},\ -\frac{1}{\phi\sqrt{3}},\ \frac{2\phi}{\sqrt{3}}\right)
  5. ( ± ϕ , - ϕ + 2 3 , 2 ϕ 3 ) \left(\pm\phi,\ -\frac{\phi+2}{\sqrt{3}},\ \frac{2\phi}{\sqrt{3}}\right)
  6. ( ± ϕ 2 , ϕ 2 3 , 2 3 ) \left(\pm\phi^{2},\ \frac{\phi^{2}}{\sqrt{3}},\ \frac{2}{\sqrt{3}}\right)
  7. ( 0 , - 2 ϕ 2 3 , 2 3 ) \left(0,\ -\frac{2\phi^{2}}{\sqrt{3}},\ \frac{2}{\sqrt{3}}\right)
  8. ( ± 1 , ± 3 , 0 ) \left(\pm 1,\ \pm\sqrt{3},\ 0\right)
  9. ( ± 2 , 0 , 0 ) \left(\pm 2,\ 0,\ 0\right)
  10. ϕ = 1 + 5 2 \phi=\frac{1+\sqrt{5}}{2}

Triangular_orthobicupola.html

  1. V = 5 2 3 a 3 2.35702... a 3 V=\frac{5\sqrt{2}}{3}a^{3}\approx 2.35702...a^{3}
  2. A = 2 ( 3 + 3 ) a 2 9.4641... a 2 A=2(3+\sqrt{3})a^{2}\approx 9.4641...a^{2}

Tridecagon.html

  1. A = 13 4 a 2 cot π 13 13.1858 a 2 . A=\frac{13}{4}a^{2}\cot\frac{\pi}{13}\simeq 13.1858\,a^{2}.
  2. \scriptstyle\angle{}
  3. \scriptstyle\angle{}

Trigonometric_interpolation.html

  1. p ( x ) = a 0 + k = 1 K a k cos ( k x ) + k = 1 K b k sin ( k x ) . p(x)=a_{0}+\sum_{k=1}^{K}a_{k}\cos(kx)+\sum_{k=1}^{K}b_{k}\sin(kx).\,
  2. p ( x n ) = y n , n = 0 , , N - 1. p(x_{n})=y_{n},\quad n=0,\ldots,N-1.\,
  3. 0 x 0 < x 1 < x 2 < < x N - 1 < 2 π . 0\leq x_{0}<x_{1}<x_{2}<\ldots<x_{N-1}<2\pi.\,
  4. p ( x ) = k = - K K c k e i k x , p(x)=\sum_{k=-K}^{K}c_{k}e^{ikx},\,
  5. q ( z ) = k = - K K c k z k , q(z)=\sum_{k=-K}^{K}c_{k}z^{k},\,
  6. q ( e i x ) p ( x ) . q(e^{ix})\triangleq p(x).\,
  7. p ( x ) = k = 0 2 K y k t k ( x ) , p(x)=\sum_{k=0}^{2K}y_{k}\,t_{k}(x),
  8. t k ( x ) = e - i K x + i K x k m = 0 , m k 2 K e i x - e i x m e i x k - e i x m . t_{k}(x)=e^{-iKx+iKx_{k}}\prod_{m=0,m\neq k}^{2K}\frac{e^{ix}-e^{ix_{m}}}{e^{% ix_{k}}-e^{ix_{m}}}.
  9. e - i K x + i K x k e^{-iKx+iKx_{k}}
  10. e i x e^{ix}
  11. e i x e^{ix}
  12. t k ( x k ) = 1 t_{k}(x_{k})=1
  13. t k ( x ) t_{k}(x)
  14. e i x e^{ix}
  15. e i z 1 - e i z 2 = 2 i sin ( z 1 - z 2 2 ) e i 1 2 z 1 + i 1 2 z 2 , e^{iz_{1}}-e^{iz_{2}}=2i\sin\left(\frac{z_{1}-z_{2}}{2}\right)e^{i\frac{1}{2}z% _{1}+i\frac{1}{2}z_{2}},
  16. t k ( x ) t_{k}(x)
  17. t k ( x ) = m = 0 , m k 2 K sin 1 2 ( x - x m ) sin 1 2 ( x k - x m ) . t_{k}(x)=\prod_{m=0,m\neq k}^{2K}\frac{\sin\frac{1}{2}(x-x_{m})}{\sin\frac{1}{% 2}(x_{k}-x_{m})}.
  18. p ( x ) = k = 0 2 K - 1 y k t k ( x ) , p(x)=\sum_{k=0}^{2K-1}y_{k}\,t_{k}(x),
  19. t k ( x ) = e - i K x + i K x k e i x - e i α k e i x k - e i α k m = 0 , m k 2 K - 1 e i x - e i x m e i x k - e i x m . t_{k}(x)=e^{-iKx+iKx_{k}}\frac{e^{ix}-e^{i\alpha_{k}}}{e^{ix_{k}}-e^{i\alpha_{% k}}}\prod_{m=0,m\neq k}^{2K-1}\frac{e^{ix}-e^{ix_{m}}}{e^{ix_{k}}-e^{ix_{m}}}.
  20. α k \alpha_{k}
  21. cos ( K x ) \cos(Kx)
  22. sin ( K x ) \sin(Kx)
  23. t k ( x ) = cos ( K x - 1 2 α k - m = 0 , m k 2 K - 1 x m ) + m = - ( K - 1 ) K - 1 c k e i m x 2 N sin ( x k - α k 2 ) m = 0 , m k 2 K - 1 sin ( x k - x m 2 ) . t_{k}(x)=\frac{\cos\left(Kx-\frac{1}{2}\alpha_{k}-\sum\limits_{m=0,m\neq k}^{2% K-1}x_{m}\right)+\sum\limits_{m=-(K-1)}^{K-1}c_{k}e^{imx}}{2^{N}\sin(\frac{x_{% k}-\alpha_{k}}{2})\prod\limits_{m=0,m\neq k}^{2K-1}\sin(\frac{x_{k}-x_{m}}{2})}.
  24. α k = m = 0 , m k 2 K - 1 x m \alpha_{k}=\sum_{m=0,m\neq k}^{2K-1}x_{m}
  25. t k ( x ) = sin 1 2 ( x - α k ) sin 1 2 ( x k - α k ) m = 0 , m k 2 K - 1 sin 1 2 ( x - x m ) sin 1 2 ( x k - x m ) . t_{k}(x)=\frac{\sin\frac{1}{2}(x-\alpha_{k})}{\sin\frac{1}{2}(x_{k}-\alpha_{k}% )}\prod_{m=0,m\neq k}^{2K-1}\frac{\sin\frac{1}{2}(x-x_{m})}{\sin\frac{1}{2}(x_% {k}-x_{m})}.
  26. x m x_{m}
  27. x m = 2 π m N , x_{m}=\frac{2\pi m}{N},
  28. D ( x , N ) = 1 N + 2 N k = 1 1 2 ( N - 1 ) cos ( k x ) = sin 1 2 N x N sin 1 2 x , D(x,N)=\frac{1}{N}+\frac{2}{N}\sum_{k=1}^{\frac{1}{2}(N-1)}\cos(kx)=\frac{\sin% \frac{1}{2}Nx}{N\sin\frac{1}{2}x},
  29. N > 0 N>0
  30. D ( x , N ) D(x,N)
  31. e i x e^{ix}
  32. D ( x m , N ) = { 0 for m 0 1 for m = 0 . D(x_{m},N)=\begin{cases}0\,\text{ for }m\neq 0\\ 1\,\text{ for }m=0\end{cases}.
  33. t k ( x ) t_{k}(x)
  34. t k ( x ) = D ( x - x k , N ) = { sin 1 2 N ( x - x k ) N sin 1 2 ( x - x k ) for x x k lim x 0 sin 1 2 N x N sin 1 2 x = 1 for x = x k = sinc 1 2 N ( x - x k ) sinc 1 2 ( x - x k ) . \begin{aligned}\displaystyle t_{k}(x)&\displaystyle=D(x-x_{k},N)=\begin{cases}% \frac{\sin\frac{1}{2}N(x-x_{k})}{N\sin\frac{1}{2}(x-x_{k})}\,\text{ for }x\neq x% _{k}\\ \lim\limits_{x\to 0}\frac{\sin\frac{1}{2}Nx}{N\sin\frac{1}{2}x}=1\,\text{ for % }x=x_{k}\end{cases}\\ &\displaystyle=\frac{\mathrm{sinc}\,\frac{1}{2}N(x-x_{k})}{\mathrm{sinc}\,% \frac{1}{2}(x-x_{k})}.\end{aligned}
  35. sinc x = sin x x . \mathrm{sinc}\,x=\frac{\sin x}{x}.
  36. N N
  37. D ( x , N ) = 1 N + 1 N cos 1 2 N x + 2 N k = 1 1 2 N - 1 cos ( k x ) = sin 1 2 N x N tan 1 2 x . D(x,N)=\frac{1}{N}+\frac{1}{N}\cos\frac{1}{2}Nx+\frac{2}{N}\sum_{k=1}^{\frac{1% }{2}N-1}\cos(kx)=\frac{\sin\frac{1}{2}Nx}{N\tan\frac{1}{2}x}.
  38. D ( x , N ) D(x,N)
  39. e i x e^{ix}
  40. sin 1 2 N x \sin\frac{1}{2}Nx
  41. D ( x m , N ) = { 0 for m 0 1 for m = 0 . D(x_{m},N)=\begin{cases}0\,\text{ for }m\neq 0\\ 1\,\text{ for }m=0\end{cases}.
  42. t k ( x ) t_{k}(x)
  43. t k ( x ) = D ( x - x k , N ) = { sin 1 2 N ( x - x k ) N tan 1 2 ( x - x k ) for x x k lim x 0 sin 1 2 N x N tan 1 2 x = 1 for x = x k . = sinc 1 2 N ( x - x k ) sinc 1 2 ( x - x k ) cos 1 2 ( x - x k ) \begin{aligned}\displaystyle t_{k}(x)&\displaystyle=D(x-x_{k},N)=\begin{cases}% \frac{\sin\frac{1}{2}N(x-x_{k})}{N\tan\frac{1}{2}(x-x_{k})}\,\text{ for }x\neq x% _{k}\\ \lim\limits_{x\to 0}\frac{\sin\frac{1}{2}Nx}{N\tan\frac{1}{2}x}=1\,\text{ for % }x=x_{k}.\end{cases}\\ &\displaystyle=\frac{\mathrm{sinc}\,\frac{1}{2}N(x-x_{k})}{\mathrm{sinc}\,% \frac{1}{2}(x-x_{k})}\cos\frac{1}{2}(x-x_{k})\end{aligned}
  44. t k ( x ) t_{k}(x)
  45. sin 1 2 N x \sin\frac{1}{2}Nx
  46. sin 1 2 N x \sin\frac{1}{2}Nx
  47. x m x_{m}
  48. x n = 2 π n N , 0 n < N . x_{n}=2\pi\frac{n}{N},\qquad 0\leq n<N.
  49. Y k = n = 0 N - 1 y n e - i 2 π n k N Y_{k}=\sum_{n=0}^{N-1}y_{n}\ e^{-i2\pi\frac{nk}{N}}\,
  50. y n = p ( x n ) = 1 N k = 0 N - 1 Y k e i 2 π n k N y_{n}=p(x_{n})=\frac{1}{N}\sum_{k=0}^{N-1}Y_{k}\ e^{i2\pi\frac{nk}{N}}\,

Trilateration.html

  1. r 1 2 = x 2 + y 2 + z 2 r_{1}^{2}=x^{2}+y^{2}+z^{2}\,
  2. r 2 2 = ( x - d ) 2 + y 2 + z 2 r_{2}^{2}=(x-d)^{2}+y^{2}+z^{2}\,
  3. r 3 2 = ( x - i ) 2 + ( y - j ) 2 + z 2 r_{3}^{2}=(x-i)^{2}+(y-j)^{2}+z^{2}\,
  4. r 1 2 = x 2 + y 2 + z 2 r_{1}^{2}=x^{2}+y^{2}+z^{2}\,
  5. r 2 2 = ( x - d ) 2 + y 2 + z 2 . r_{2}^{2}=(x-d)^{2}+y^{2}+z^{2}.
  6. r 1 2 - r 2 2 = x 2 - ( x - d ) 2 r_{1}^{2}-r_{2}^{2}=x^{2}-(x-d)^{2}\,
  7. r 1 2 - r 2 2 = x 2 - ( x 2 - 2 x d + d 2 ) r_{1}^{2}-r_{2}^{2}=x^{2}-(x^{2}-2xd+d^{2})\,
  8. r 1 2 - r 2 2 = 2 x d - d 2 r_{1}^{2}-r_{2}^{2}=2xd-d^{2}\,
  9. r 1 2 - r 2 2 + d 2 = 2 x d r_{1}^{2}-r_{2}^{2}+d^{2}=2xd\,
  10. x = r 1 2 - r 2 2 + d 2 2 d . x=\frac{r_{1}^{2}-r_{2}^{2}+d^{2}}{2d}.
  11. d - r 1 < r 2 < d + r 1 . d-r_{1}<r_{2}<d+r_{1}.\,
  12. y 2 + z 2 = r 1 2 - ( r 1 2 - r 2 2 + d 2 ) 2 4 d 2 . y^{2}+z^{2}=r_{1}^{2}-\frac{(r_{1}^{2}-r_{2}^{2}+d^{2})^{2}}{4d^{2}}.
  13. z 2 = r 1 2 - x 2 - y 2 z^{2}=r_{1}^{2}-x^{2}-y^{2}
  14. y = r 1 2 - r 3 2 - x 2 + ( x - i ) 2 + j 2 2 j = r 1 2 - r 3 2 + i 2 + j 2 2 j - i j x . y=\frac{r_{1}^{2}-r_{3}^{2}-x^{2}+(x-i)^{2}+j^{2}}{2j}=\frac{r_{1}^{2}-r_{3}^{% 2}+i^{2}+j^{2}}{2j}-\frac{i}{j}x.
  15. z = ± r 1 2 - x 2 - y 2 . z=\pm\sqrt{r_{1}^{2}-x^{2}-y^{2}}.
  16. e ^ x = P 2 - P 1 P 2 - P 1 \hat{e}_{x}=\frac{P2-P1}{\|P2-P1\|}
  17. i = e ^ x ( P 3 - P 1 ) i=\hat{e}_{x}\cdot(P3-P1)
  18. e ^ y = P 3 - P 1 - i e ^ x P 3 - P 1 - i e ^ x \hat{e}_{y}=\frac{P3-P1-i\;\hat{e}_{x}}{\|P3-P1-i\;\hat{e}_{x}\|}
  19. e ^ z = e ^ x × e ^ y \hat{e}_{z}=\hat{e}_{x}\times\hat{e}_{y}
  20. d = P 2 - P 1 d=\|P2-P1\|
  21. j = e ^ y ( P 3 - P 1 ) j=\hat{e}_{y}\cdot(P3-P1)
  22. i , d i,\;d
  23. j j
  24. p 1 , 2 = P 1 + x e ^ x + y e ^ y ± z e ^ z \vec{p}_{1,2}=P1+x\ \hat{e}_{x}+y\ \hat{e}_{y}\ \pm\ z\ \hat{e}_{z}
  25. e ^ x , e ^ y \hat{e}_{x},\;\hat{e}_{y}
  26. e ^ z \hat{e}_{z}

Trilinear_interpolation.html

  1. ( x , y , z ) (x,y,z)
  2. D = 1 D=1
  3. D = 2 D=2
  4. D = 3 D=3
  5. ( 1 + n ) D = 8 (1+n)^{D}=8
  6. x d x_{d}
  7. y d y_{d}
  8. z d z_{d}
  9. x x
  10. y y
  11. z z
  12. x d = ( x - x 0 ) / ( x 1 - x 0 ) \ x_{d}=(x-x_{0})/(x_{1}-x_{0})
  13. y d = ( y - y 0 ) / ( y 1 - y 0 ) \ y_{d}=(y-y_{0})/(y_{1}-y_{0})
  14. z d = ( z - z 0 ) / ( z 1 - z 0 ) \ z_{d}=(z-z_{0})/(z_{1}-z_{0})
  15. x 0 x_{0}
  16. x x
  17. x 1 x_{1}
  18. x x
  19. y 0 , y 1 , z 0 y_{0},y_{1},z_{0}
  20. z 1 z_{1}
  21. x x
  22. c 00 = V [ x 0 , y 0 , z 0 ] ( 1 - x d ) + V [ x 1 , y 0 , z 0 ] x d \ c_{00}=V[x_{0},y_{0},z_{0}](1-x_{d})+V[x_{1},y_{0},z_{0}]x_{d}
  23. c 10 = V [ x 0 , y 1 , z 0 ] ( 1 - x d ) + V [ x 1 , y 1 , z 0 ] x d \ c_{10}=V[x_{0},y_{1},z_{0}](1-x_{d})+V[x_{1},y_{1},z_{0}]x_{d}
  24. c 01 = V [ x 0 , y 0 , z 1 ] ( 1 - x d ) + V [ x 1 , y 0 , z 1 ] x d \ c_{01}=V[x_{0},y_{0},z_{1}](1-x_{d})+V[x_{1},y_{0},z_{1}]x_{d}
  25. c 11 = V [ x 0 , y 1 , z 1 ] ( 1 - x d ) + V [ x 1 , y 1 , z 1 ] x d \ c_{11}=V[x_{0},y_{1},z_{1}](1-x_{d})+V[x_{1},y_{1},z_{1}]x_{d}
  26. V [ x 0 , y 0 , z 0 ] V[x_{0},y_{0},z_{0}]
  27. ( x 0 , y 0 , z 0 ) . (x_{0},y_{0},z_{0}).
  28. y y
  29. c 0 = c 00 ( 1 - y d ) + c 10 y d \ c_{0}=c_{00}(1-y_{d})+c_{10}y_{d}
  30. c 1 = c 01 ( 1 - y d ) + c 11 y d \ c_{1}=c_{01}(1-y_{d})+c_{11}y_{d}
  31. z z
  32. c = c 0 ( 1 - z d ) + c 1 z d . \ c=c_{0}(1-z_{d})+c_{1}z_{d}.
  33. x x
  34. y y
  35. z z
  36. C l ( b ( C 000 , C 010 , C 100 , C 110 ) , b ( C 001 , C 011 , C 101 , C 111 ) ) C\approx\ l(b(C_{000},C_{010},C_{100},C_{110}),b(C_{001},C_{011},C_{101},C_{11% 1}))

Trinomial_expansion.html

  1. ( a + b + c ) n = i + j + k = n i , j , k ( n i , j , k ) a i b j c k , (a+b+c)^{n}=\sum_{\stackrel{i,j,k}{i+j+k=n}}{n\choose i,j,k}\,a^{i}\,b^{\;\!j}% \;\!c^{k},
  2. n n
  3. i , j , i,j,
  4. k k
  5. i + j + k = n i+j+k=n
  6. ( n i , j , k ) = n ! i ! j ! k ! . {n\choose i,j,k}=\frac{n!}{i!\,j!\,k!}\,.
  7. m = 3 m=3
  8. t n + 1 = ( n + 2 ) ( n + 1 ) 2 , t_{n+1}=\frac{(n+2)(n+1)}{2},
  9. n n

Trip_distribution.html

  1. T i j = T i A j f ( C i j ) K i j j = 1 n A j f ( C i j ) K i j T_{ij}=T_{i}\frac{{A_{j}f\left({C_{ij}}\right)K_{ij}}}{{\sum_{j=1}^{n}{A_{j}f% \left({C_{ij}}\right)K_{ij}}}}
  2. T i j T_{ij}
  3. T i T_{i}
  4. A j A_{j}
  5. f ( C i j ) f(C_{ij})
  6. C i j b C_{ij}^{b}
  7. K i j K_{ij}
  8. T i j = a < m t p l > P i P j C i j b T_{ij}=a\frac{<}{m}tpl>{{P_{i}P_{j}}}{{C_{ij}^{b}}}
  9. P i ; P j P_{i};P_{j}
  10. a ; b a;b
  11. T i j = a < m t p l > T i c T j d C i j b T_{ij}=a\frac{<}{m}tpl>{{T_{i}^{c}T_{j}^{d}}}{{C_{ij}^{b}}}
  12. T i j = < m t p l > c T i d T j C i j b T_{ij}=\frac{<}{m}tpl>{{cT_{i}dT_{j}}}{{C_{ij}^{b}}}
  13. C i j C_{ij}
  14. T j T_{j}
  15. T i T_{i}
  16. T i j = K i K j T i T j f ( C i j ) T_{ij}=K_{i}K_{j}T_{i}T_{j}f(C_{ij})
  17. j T i j = T i , i T i j = T j \sum_{j}{T_{ij}=T_{i}},\sum_{i}{T_{ij}=T_{j}}
  18. K i = 1 j K j T j f ( C i j ) , K j = 1 i K i T i f ( C i j ) K_{i}=\frac{1}{{\sum_{j}{K_{j}T_{j}f(C_{ij})}}},K_{j}=\frac{1}{{\sum_{i}{K_{i}% T_{i}f(C_{ij})}}}
  19. T i j T_{ij}
  20. T i T_{i}
  21. T j T_{j}
  22. C i j C_{ij}
  23. K i , K j K_{i},K_{j}
  24. f f
  25. w ( T i j ) = < m t p l > 7 ! 2 ! 1 ! 1 ! 0 ! 2 ! 1 ! = 1260 w\left({T_{ij}}\right)=\frac{<}{m}tpl>{{7!}}{{2!1!1!0!2!1!}}=1260
  26. 2 n 2^{n}
  27. 2 1 = 2 2^{1}=2
  28. 2 2 = 4 2^{2}=4
  29. 4 ! / ( 4 ! 0 ! ) = 1 4!/(4!0!)=1
  30. 4 ! / ( 2 ! 2 ! ) = 6 4!/(2!2!)=6
  31. w = < m t p l > n ! i = 1 n n i ! w=\frac{<}{m}tpl>{{n!}}{{\prod_{i=1}^{n}{n_{i}!}}}
  32. p j = p 0 e β e j p_{j}=p_{0}e^{\beta e_{j}}
  33. p 0 p_{0}
  34. e j e_{j}
  35. β \beta
  36. max w ( T i j ) = < m t p l > T ! i j T i j ! \max w\left({T_{ij}}\right)=\frac{<}{m}tpl>{{T!}}{{\prod_{ij}{Tij!}}}
  37. j T i j = T i ; i T i j = T j \sum_{j}{T_{ij}=T_{i}};\sum_{i}{T_{ij}=T_{j}}
  38. T = j i T i j = i T i = j T j T=\sum_{j}{\sum_{i}{T_{ij}}}=\sum_{i}{T_{i}}=\sum_{j}{T_{j}}
  39. i j T i j C i j = C \sum_{i}{\sum_{j}{T_{ij}C_{ij}=C}}
  40. C i j C_{ij}
  41. Λ \Lambda
  42. Λ ( T i j , λ i , λ j ) = < m t p l > T ! i j T i j ! + i λ i ( T i - j T i j ) + j λ j ( T j - i T i j ) + β ( C - i j T i j C i j ) \Lambda(T_{ij},\lambda_{i},\lambda_{j})=\frac{<}{m}tpl>{{T!}}{{\prod_{ij}{Tij!% }}}+\sum_{i}{\lambda_{i}\left({T_{i}-\sum_{j}{T_{ij}}}\right)}+\sum_{j}{% \lambda_{j}\left({T_{j}-\sum_{i}{T_{ij}}}\right)+\beta\left({C-\sum_{i}{\sum_{% j}{T_{ij}C_{ij}}}}\right)}
  43. λ i , λ j \lambda_{i},\lambda_{j}
  44. β \beta
  45. β \beta
  46. w ( T i j ) w(T_{ij})
  47. ln N ! N ln N - N \ln N!\approx N\ln N-N
  48. < m t p l > ln N ! N ln N \frac{<}{m}tpl>{{\partial\ln N!}}{{\partial N}}\approx\ln N
  49. Λ ( T i j , λ i , λ j ) T i j = - ln T i j - λ i - λ j - β C i j = 0 \frac{{\partial\Lambda(T_{ij},\lambda_{i},\lambda_{j})}}{{\partial T_{ij}}}=-% \ln T_{ij}-\lambda_{i}-\lambda_{j}-\beta C_{ij}=0
  50. ln T i j = - λ i - λ j - β C i j \ln T_{ij}=-\lambda_{i}-\lambda_{j}-\beta C_{ij}
  51. T i j = e - λ i - λ j - β C i j T_{ij}=e^{-\lambda_{i}-\lambda_{j}-\beta C_{ij}}
  52. T i j T_{ij}
  53. j e - λ i - λ j - β C i j = T i ; i e - λ i - λ j - β C i j = T j \sum_{j}{e^{-\lambda_{i}-\lambda_{j}-\beta C_{ij}}}=T_{i};\sum_{i}{e^{-\lambda% _{i}-\lambda_{j}-\beta C_{ij}}}=T_{j}
  54. e - λ i = < m t p l > T i j e - λ j - β C i j ; e - λ j = T j i e - λ i - β C i j e^{-\lambda_{i}}=\frac{<}{m}tpl>{{T_{i}}}{{\sum_{j}{e^{-\lambda_{j}-\beta C_{% ij}}}}};e^{-\lambda_{j}}=\frac{{T_{j}}}{{\sum_{i}{e^{-\lambda_{i}-\beta C_{ij}% }}}}
  55. e - λ i < m t p l > T i = A i ; e - λ j T j = B j \frac{{e^{-\lambda_{i}}}}{<}mtpl>{{T_{i}}}=A_{i};\frac{{e^{-\lambda_{j}}}}{{T_% {j}}}=B_{j}
  56. T i j = A i B j T i T j e - β C i j T_{ij}=A_{i}B_{j}T_{i}T_{j}e^{-\beta C_{ij}}
  57. T i j T_{ij}
  58. A i A_{i}
  59. B j B_{j}
  60. β \beta
  61. β \beta
  62. A i A_{i}
  63. B j B_{j}
  64. β \beta
  65. β \beta
  66. β \beta
  67. A i A_{i}
  68. B j B_{j}
  69. β \beta

Trip_generation.html

  1. T r i p s = a + b l n ( A r e a ) Trips=a+bln(Area)

Triple_product.html

  1. 𝐚 ( 𝐛 × 𝐜 ) \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})
  2. 𝐚 ( 𝐛 × 𝐜 ) = 𝐛 ( 𝐜 × 𝐚 ) = 𝐜 ( 𝐚 × 𝐛 ) \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=\mathbf{b}\cdot(\mathbf{c}\times% \mathbf{a})=\mathbf{c}\cdot(\mathbf{a}\times\mathbf{b})
  3. 𝐚 ( 𝐛 × 𝐜 ) = ( 𝐚 × 𝐛 ) 𝐜 \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=(\mathbf{a}\times\mathbf{b})\cdot% \mathbf{c}
  4. 𝐚 ( 𝐛 × 𝐜 ) = - 𝐚 ( 𝐜 × 𝐛 ) \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=-\mathbf{a}\cdot(\mathbf{c}\times% \mathbf{b})
  5. 𝐚 ( 𝐛 × 𝐜 ) = - 𝐛 ( 𝐚 × 𝐜 ) \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=-\mathbf{b}\cdot(\mathbf{a}\times% \mathbf{c})
  6. 𝐚 ( 𝐛 × 𝐜 ) = - 𝐜 ( 𝐛 × 𝐚 ) \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=-\mathbf{c}\cdot(\mathbf{b}\times% \mathbf{a})
  7. 𝐚 ( 𝐛 × 𝐜 ) = det [ a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 ] . \mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})=\det\begin{bmatrix}a_{1}&a_{2}&a_{% 3}\\ b_{1}&b_{2}&b_{3}\\ c_{1}&c_{2}&c_{3}\\ \end{bmatrix}.
  8. 𝐚 ( 𝐚 × 𝐛 ) = 𝐚 ( 𝐛 × 𝐚 ) = 𝐚 ( 𝐛 × 𝐛 ) = 𝐚 ( 𝐚 × 𝐚 ) = 0 \mathbf{a}\cdot(\mathbf{a}\times\mathbf{b})=\mathbf{a}\cdot(\mathbf{b}\times% \mathbf{a})=\mathbf{a}\cdot(\mathbf{b}\times\mathbf{b})=\mathbf{a}\cdot(% \mathbf{a}\times\mathbf{a})=0
  9. [ 𝐚 ( 𝐛 × 𝐜 ) ] 𝐚 = ( 𝐚 × 𝐛 ) × ( 𝐚 × 𝐜 ) [\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c})]\mathbf{a}=(\mathbf{a}\times% \mathbf{b})\times(\mathbf{a}\times\mathbf{c})
  10. ( ( 𝐚 × 𝐛 ) 𝐜 ) ( ( 𝐝 × 𝐞 ) 𝐟 ) = det [ ( 𝐚 𝐛 𝐜 ) ( 𝐝 𝐞 𝐟 ) ] = det [ 𝐚 𝐝 𝐚 𝐞 𝐚 𝐟 𝐛 𝐝 𝐛 𝐞 𝐛 𝐟 𝐜 𝐝 𝐜 𝐞 𝐜 𝐟 ] ((\mathbf{a}\times\mathbf{b})\cdot\mathbf{c})\;((\mathbf{d}\times\mathbf{e})% \cdot\mathbf{f})=\det\left[\begin{pmatrix}\mathbf{a}\\ \mathbf{b}\\ \mathbf{c}\end{pmatrix}\cdot\begin{pmatrix}\mathbf{d}&\mathbf{e}&\mathbf{f}% \end{pmatrix}\right]=\det\begin{bmatrix}\mathbf{a}\cdot\mathbf{d}&\mathbf{a}% \cdot\mathbf{e}&\mathbf{a}\cdot\mathbf{f}\\ \mathbf{b}\cdot\mathbf{d}&\mathbf{b}\cdot\mathbf{e}&\mathbf{b}\cdot\mathbf{f}% \\ \mathbf{c}\cdot\mathbf{d}&\mathbf{c}\cdot\mathbf{e}&\mathbf{c}\cdot\mathbf{f}% \end{bmatrix}
  11. 𝐓𝐚 ( 𝐓𝐛 × 𝐓𝐜 ) = 𝐚 ( 𝐛 × 𝐜 ) , \mathbf{Ta}\cdot(\mathbf{Tb}\times\mathbf{Tc})=\mathbf{a}\cdot(\mathbf{b}% \times\mathbf{c}),
  12. 𝐓𝐚 ( 𝐓𝐛 × 𝐓𝐜 ) = - 𝐚 ( 𝐛 × 𝐜 ) . \mathbf{Ta}\cdot(\mathbf{Tb}\times\mathbf{Tc})=-\mathbf{a}\cdot(\mathbf{b}% \times\mathbf{c}).
  13. 𝐚 𝐛 𝐜 \mathbf{a}\wedge\mathbf{b}\wedge\mathbf{c}
  14. 𝐚 × ( 𝐛 × 𝐜 ) = 𝐛 ( 𝐚 𝐜 ) - 𝐜 ( 𝐚 𝐛 ) \mathbf{a}\times(\mathbf{b}\times\mathbf{c})=\mathbf{b}(\mathbf{a}\cdot\mathbf% {c})-\mathbf{c}(\mathbf{a}\cdot\mathbf{b})
  15. ( 𝐚 × 𝐛 ) × 𝐜 = - 𝐜 × ( 𝐚 × 𝐛 ) = - ( 𝐜 𝐛 ) 𝐚 + ( 𝐜 𝐚 ) 𝐛 (\mathbf{a}\times\mathbf{b})\times\mathbf{c}=-\mathbf{c}\times(\mathbf{a}% \times\mathbf{b})=-(\mathbf{c}\cdot\mathbf{b})\mathbf{a}+(\mathbf{c}\cdot% \mathbf{a})\mathbf{b}
  16. 𝐚 × ( 𝐛 × 𝐜 ) + 𝐛 × ( 𝐜 × 𝐚 ) + 𝐜 × ( 𝐚 × 𝐛 ) = 0 \mathbf{a}\times(\mathbf{b}\times\mathbf{c})\;+\mathbf{b}\times(\mathbf{c}% \times\mathbf{a})\;+\mathbf{c}\times(\mathbf{a}\times\mathbf{b})=0
  17. ( 𝐚 × 𝐛 ) × 𝐜 = 𝐚 × ( 𝐛 × 𝐜 ) - 𝐛 × ( 𝐚 × 𝐜 ) (\mathbf{a}\times\mathbf{b})\times\mathbf{c}=\mathbf{a}\times(\mathbf{b}\times% \mathbf{c})\;-\mathbf{b}\times(\mathbf{a}\times\mathbf{c})
  18. s y m b o l × ( s y m b o l × 𝐟 ) = s y m b o l ( s y m b o l 𝐟 ) - ( s y m b o l s y m b o l ) 𝐟 symbol{\nabla}\times(symbol{\nabla}\times\mathbf{f})=symbol{\nabla}(symbol{% \nabla}\cdot\mathbf{f})-(symbol{\nabla}\cdot symbol{\nabla})\mathbf{f}
  19. Δ = d δ + δ d \Delta=d\delta+\delta d
  20. x x
  21. 𝐮 × ( 𝐯 × 𝐰 ) \mathbf{u}\times(\mathbf{v}\times\mathbf{w})
  22. 𝐮 y ( 𝐯 x 𝐰 y - 𝐯 y 𝐰 x ) - 𝐮 z ( 𝐯 z 𝐰 x - 𝐯 x 𝐰 z ) \mathbf{u}_{y}(\mathbf{v}_{x}\mathbf{w}_{y}-\mathbf{v}_{y}\mathbf{w}_{x})-% \mathbf{u}_{z}(\mathbf{v}_{z}\mathbf{w}_{x}-\mathbf{v}_{x}\mathbf{w}_{z})
  23. 𝐯 x ( 𝐮 y 𝐰 y + 𝐮 z 𝐰 z ) - 𝐰 x ( 𝐮 y 𝐯 y + 𝐮 z 𝐯 z ) \mathbf{v}_{x}(\mathbf{u}_{y}\mathbf{w}_{y}+\mathbf{u}_{z}\mathbf{w}_{z})-% \mathbf{w}_{x}(\mathbf{u}_{y}\mathbf{v}_{y}+\mathbf{u}_{z}\mathbf{v}_{z})
  24. 𝐮 x 𝐯 x 𝐰 x \mathbf{u}_{x}\mathbf{v}_{x}\mathbf{w}_{x}
  25. 𝐯 x ( 𝐮 x 𝐰 x + 𝐮 y 𝐰 y + 𝐮 z 𝐰 z ) - 𝐰 x ( 𝐮 x 𝐯 x + 𝐮 y 𝐯 y + 𝐮 z 𝐯 z ) = ( 𝐮 𝐰 ) 𝐯 x - ( 𝐮 𝐯 ) 𝐰 x \mathbf{v}_{x}(\mathbf{u}_{x}\mathbf{w}_{x}+\mathbf{u}_{y}\mathbf{w}_{y}+% \mathbf{u}_{z}\mathbf{w}_{z})-\mathbf{w}_{x}(\mathbf{u}_{x}\mathbf{v}_{x}+% \mathbf{u}_{y}\mathbf{v}_{y}+\mathbf{u}_{z}\mathbf{v}_{z})=(\mathbf{u}\cdot% \mathbf{w})\mathbf{v}_{x}-(\mathbf{u}\cdot\mathbf{v})\mathbf{w}_{x}
  26. y y
  27. z z
  28. 𝐮 × ( 𝐯 × 𝐰 ) \mathbf{u}\times(\mathbf{v}\times\mathbf{w})
  29. ( 𝐮 𝐰 ) 𝐯 y - ( 𝐮 𝐯 ) 𝐰 y (\mathbf{u}\cdot\mathbf{w})\mathbf{v}_{y}-(\mathbf{u}\cdot\mathbf{v})\mathbf{w% }_{y}
  30. ( 𝐮 𝐰 ) 𝐯 z - ( 𝐮 𝐯 ) 𝐰 z (\mathbf{u}\cdot\mathbf{w})\mathbf{v}_{z}-(\mathbf{u}\cdot\mathbf{v})\mathbf{w% }_{z}
  31. 𝐮 × ( 𝐯 × 𝐰 ) = ( 𝐮 𝐰 ) 𝐯 - ( 𝐮 𝐯 ) 𝐰 \mathbf{u}\times(\mathbf{v}\times\mathbf{w})=(\mathbf{u}\cdot\mathbf{w})\ % \mathbf{v}-(\mathbf{u}\cdot\mathbf{v})\ \mathbf{w}
  32. - 𝐚 ( 𝐛 𝐜 ) = 𝐛 ( 𝐚 𝐜 ) - ( 𝐚 𝐛 ) 𝐜 = ( 𝐚 𝐜 ) 𝐛 - ( 𝐚 𝐛 ) 𝐜 \begin{aligned}\displaystyle-\mathbf{a}\;\big\lrcorner\;(\mathbf{b}\wedge% \mathbf{c})&\displaystyle=\mathbf{b}\wedge(\mathbf{a}\;\big\lrcorner\;\mathbf{% c})-(\mathbf{a}\;\big\lrcorner\;\mathbf{b})\wedge\mathbf{c}\\ &\displaystyle=(\mathbf{a}\cdot\mathbf{c})\mathbf{b}-(\mathbf{a}\cdot\mathbf{b% })\mathbf{c}\end{aligned}
  33. ( 𝐚 ( 𝐛 × 𝐜 ) ) = ε i j k a i b j c k (\mathbf{a}\cdot(\mathbf{b}\times\mathbf{c}))=\varepsilon_{ijk}a^{i}b^{j}c^{k}
  34. ( 𝐚 × ( 𝐛 × 𝐜 ) ) i = ε i j k a j ε k m b c m = ε i j k ε k m a j b c m (\mathbf{a}\times(\mathbf{b}\times\mathbf{c}))_{i}=\varepsilon_{ijk}a^{j}% \varepsilon_{k\ell m}b^{\ell}c^{m}=\varepsilon_{ijk}\varepsilon_{k\ell m}a^{j}% b^{\ell}c^{m}
  35. i i
  36. ε i j k ε k m = - ε i j k ε m k = δ i δ j m - δ i m δ j \varepsilon_{ijk}\varepsilon_{k\ell m}=-\varepsilon_{ijk}\varepsilon_{m\ell k}% =\delta_{i\ell}\delta_{jm}-\delta_{im}\delta_{\ell j}
  37. δ i j = 0 \delta_{ij}=0
  38. i j i\neq j
  39. δ i j = 1 \delta_{ij}=1
  40. i = j i=j
  41. k k
  42. i i
  43. j j
  44. i = l i=l
  45. j = m j=m
  46. i = m i=m
  47. l = j l=j
  48. ( 𝐚 × ( 𝐛 × 𝐜 ) ) i = ( δ i δ j m - δ i m δ j ) a j b c m = a j b i c j - a j b j c i = 𝐛 i ( 𝐚 𝐜 ) - 𝐜 i ( 𝐚 𝐛 ) (\mathbf{a}\times(\mathbf{b}\times\mathbf{c}))_{i}=(\delta_{i\ell}\delta_{jm}-% \delta_{im}\delta_{\ell j})a^{j}b^{\ell}c^{m}=a^{j}b^{i}c^{j}-a^{j}b^{j}c^{i}=% \mathbf{b}_{i}(\mathbf{a}\cdot\mathbf{c})-\mathbf{c}_{i}(\mathbf{a}\cdot% \mathbf{b})

Tropical_geometry.html

  1. x y = min { x , y } , x\oplus y=\min\{x,y\},
  2. x y = x + y . x\otimes y=x+y.

True_anomaly.html

  1. ν \,\nu
  2. θ \,\theta
  3. f f
  4. ν \nu\,\!
  5. ν = arccos 𝐞 𝐫 | 𝐞 | | 𝐫 | \nu=\arccos{{\mathbf{e}\cdot\mathbf{r}}\over{\mathbf{\left|e\right|}\mathbf{% \left|r\right|}}}
  6. 𝐫 𝐯 < 0 \mathbf{r}\cdot\mathbf{v}<0
  7. ν \nu
  8. 2 π - ν 2\pi-\nu
  9. 𝐯 \mathbf{v}\,
  10. 𝐞 \mathbf{e}\,
  11. 𝐫 \mathbf{r}\,
  12. u u\,\!
  13. u = arccos 𝐧 𝐫 | 𝐧 | | 𝐫 | u=\arccos{{\mathbf{n}\cdot\mathbf{r}}\over{\mathbf{\left|n\right|}\mathbf{% \left|r\right|}}}
  14. 𝐧 𝐯 > 0 \mathbf{n}\cdot\mathbf{v}>0
  15. u u
  16. 2 π - u 2\pi-u
  17. 𝐧 \mathbf{n}
  18. 𝐧 \mathbf{n}
  19. l = arccos r x | 𝐫 | l=\arccos{r_{x}\over{\mathbf{\left|r\right|}}}
  20. v x > 0 v_{x}>0
  21. l l
  22. 2 π - l 2\pi-l
  23. r x r_{x}\,
  24. 𝐫 \mathbf{r}
  25. v x v_{x}\,
  26. 𝐯 \mathbf{v}
  27. ν \,\nu
  28. cos ν = cos E - e 1 - e cos E \cos{\nu}={{\cos{E}-e}\over{1-e\cdot\cos{E}}}
  29. sin ν = 1 - e 2 sin E 1 - e cos E \sin{\nu}={{\sqrt{1-e^{2}}\sin{E}}\over{1-e\cos{E}}}
  30. tan ν = sin ν cos ν = 1 - e 2 sin E cos E - e \tan{\nu}={{\sin{\nu}}\over{\cos{\nu}}}={{\sqrt{1-e^{2}}\sin{E}}\over{\cos{E}-% e}}
  31. tan ν 2 = 1 + e 1 - e tan E 2 . \tan{\nu\over 2}=\sqrt{{{1+e}\over{1-e}}}\tan{E\over 2}.
  32. ν = 2 arg ( 1 - e cos E 2 , 1 + e sin E 2 ) \nu=2\,\mathop{\mathrm{arg}}\left(\sqrt{1-e}\,\cos\frac{E}{2},\sqrt{1+e}\sin% \frac{E}{2}\right)
  33. arg ( x , y ) \operatorname{arg}(x,y)
  34. ( x , y ) \left(x,y\right)
  35. r = a 1 - e 2 1 + e cos ν r=a\cdot{1-e^{2}\over 1+e\cdot\cos\nu}\,\!

True_longitude.html

  1. l l\,
  2. l = ν + ϖ l=\nu+\varpi\,
  3. ν \nu\,
  4. ϖ ω + Ω \varpi\equiv\omega+\Omega\,
  5. ω \omega\,
  6. Ω \Omega\,

Truncated_power_function.html

  1. n n
  2. x + n = { x n : x > 0 0 : x 0. x_{+}^{n}=\begin{cases}x^{n}&:\ x>0\\ 0&:\ x\leq 0.\end{cases}
  3. x + = { x : x > 0 0 : x 0. x_{+}=\begin{cases}x&:\ x>0\\ 0&:\ x\leq 0.\end{cases}
  4. x x + 0 x\mapsto x_{+}^{0}
  5. χ [ a , b ) ( x ) = ( b - x ) + 0 - ( a - x ) + 0 \chi_{[a,b)}(x)=(b-x)_{+}^{0}-(a-x)_{+}^{0}
  6. χ \chi

Tsiolkovsky_rocket_equation.html

  1. Δ v = v e ln m 0 m 1 \Delta v=v\text{e}\ln\frac{m_{0}}{m_{1}}
  2. m 0 m_{0}
  3. m 1 m_{1}
  4. v e v\text{e}
  5. Δ v \Delta v
  6. ln \ln
  7. v e = I sp g 0 v\text{e}=I\text{sp}\cdot g_{0}
  8. I sp I\text{sp}
  9. g 0 g_{0}
  10. F i F_{i}\,
  11. F i = lim Δ t 0 P 2 - P 1 Δ t \sum F_{i}=\lim_{\Delta t\to 0}\frac{P_{2}-P_{1}}{\Delta t}
  12. P 1 P_{1}\,
  13. P 1 = ( m + Δ m ) V P_{1}=\left({m+\Delta m}\right)V
  14. P 2 P_{2}\,
  15. t = Δ t t=\Delta t\,
  16. P 2 = m ( V + Δ V ) + Δ m V e P_{2}=m\left(V+\Delta V\right)+\Delta mV_{e}
  17. V V\,
  18. V + Δ V V+\Delta V\,
  19. t = Δ t t=\Delta t\,
  20. V e V_{e}\,
  21. Δ t \Delta t\,
  22. m + Δ m m+\Delta m\,
  23. m m\,
  24. t = Δ t t=\Delta t\,
  25. V e V_{e}
  26. v e v_{e}
  27. V e = V - v e V_{e}=V-v_{e}
  28. P 2 - P 1 = m Δ V - v e Δ m P_{2}-P_{1}=m\Delta V-v_{e}\Delta m\,
  29. d m = - Δ m dm=-\Delta m
  30. Δ m \Delta m
  31. F i = m d V d t + v e d m d t \sum F_{i}=m\frac{dV}{dt}+v_{e}\frac{dm}{dt}
  32. F i = 0 \sum F_{i}=0
  33. m d V d t = - v e d m d t m\frac{dV}{dt}=-v_{e}\frac{dm}{dt}
  34. v e v_{e}\,
  35. Δ V = v e ln m 0 m 1 \Delta V\ =v_{e}\ln\frac{m_{0}}{m_{1}}
  36. m 1 = m 0 e - Δ V / v e m_{1}=m_{0}e^{-\Delta V\ /v_{e}}
  37. m 0 = m 1 e Δ V / v e m_{0}=m_{1}e^{\Delta V\ /v_{e}}
  38. m 0 - m 1 = m 1 ( e Δ V / v e - 1 ) m_{0}-m_{1}=m_{1}(e^{\Delta V\ /v_{e}}-1)
  39. m 0 m_{0}
  40. m 1 m_{1}
  41. v e v_{e}
  42. m 0 - m 1 m_{0}-m_{1}
  43. M f = 1 - m 1 m 0 = 1 - e - Δ V / v e M_{f}=1-\frac{m_{1}}{m_{0}}=1-e^{-\Delta V\ /v\text{e}}
  44. M f M_{f}
  45. Δ V \Delta V
  46. Δ v \Delta v
  47. m 1 m_{1}
  48. m 0 m_{0}
  49. c c
  50. m 0 m 1 = [ 1 + Δ v c 1 - Δ v c ] c 2 v e \frac{m_{0}}{m_{1}}=\left[\frac{1+{\frac{\Delta v}{c}}}{1-{\frac{\Delta v}{c}}% }\right]^{\frac{c}{2v_{e}}}
  51. m 0 m 1 \frac{m_{0}}{m_{1}}
  52. R R
  53. Δ v c = R 2 v e c - 1 R 2 v e c + 1 \frac{\Delta v}{c}=\frac{R^{\frac{2v_{e}}{c}}-1}{R^{\frac{2v_{e}}{c}}+1}
  54. R 2 v e c = exp [ 2 v e c ln R ] R^{\frac{2v_{e}}{c}}=\exp\left[\frac{2v_{e}}{c}\ln R\right]
  55. tanh x = e 2 x - 1 e 2 x + 1 \tanh x=\frac{e^{2x}-1}{e^{2x}+1}
  56. Δ v = c tanh ( v e c ln m 0 m 1 ) \Delta v=c\cdot\tanh\left(\frac{v_{e}}{c}\ln\frac{m_{0}}{m_{1}}\right)
  57. v e = g 0 I sp , v\text{e}=g_{0}I\text{sp},
  58. I sp I\text{sp}
  59. v e v\text{e}
  60. g 0 g_{0}
  61. Δ v \Delta v
  62. Δ v \Delta v
  63. 1 - e - 9.7 / 4.5 1-e^{-9.7/4.5}
  64. Δ v \Delta v
  65. 1 - e - 5.0 / 4.5 1-e^{-5.0/4.5}
  66. Δ v \Delta v
  67. 1 - e - 4.7 / 4.5 1-e^{-4.7/4.5}
  68. Δ v = v e ln 100 100 - 80 = v e ln 5 = 1.61 v e . \begin{aligned}\displaystyle\Delta v&\displaystyle=v\text{e}\ln{100\over 100-8% 0}\\ &\displaystyle=v\text{e}\ln 5\\ &\displaystyle=1.61v\text{e}.\\ \end{aligned}
  69. v e v_{e}
  70. Δ v = 3 v e ln 5 = 4.83 v e \Delta v\ =3v\text{e}\ln 5\ =4.83v\text{e}
  71. Δ v = v e ln ( 100 / 11.2 ) = 2.19 v e . \Delta v\ =v\text{e}\ln(100/11.2)\ =2.19v\text{e}.
  72. F = d p / d t = m d v / d t + v d m / d t F=dp/dt=m\;dv/dt+v\;dm/dt
  73. m ( t ) m(t)
  74. v e v_{e}

Tsirelson's_bound.html

  1. A 0 A_{0}
  2. A 1 A_{1}
  3. B 0 B_{0}
  4. B 1 B_{1}
  5. + 1 , - 1 +1,-1
  6. [ A i , B j ] = 0 [A_{i},B_{j}]=0
  7. i , j i,j
  8. A 0 B 0 + A 0 B 1 + A 1 B 0 - A 1 B 1 2 2 \langle A_{0}B_{0}\rangle+\langle A_{0}B_{1}\rangle+\langle A_{1}B_{0}\rangle-% \langle A_{1}B_{1}\rangle\leq 2\sqrt{2}
  9. + 1 , - 1 +1,-1
  10. = A 0 B 0 + A 0 B 1 + A 1 B 0 - A 1 B 1 \mathcal{B}=A_{0}B_{0}+A_{0}B_{1}+A_{1}B_{0}-A_{1}B_{1}
  11. A i 2 = B j 2 = 𝕀 A_{i}^{2}=B_{j}^{2}=\mathbb{I}
  12. 2 = 4 𝕀 - [ A 0 , A 1 ] [ B 0 , B 1 ] \mathcal{B}^{2}=4\mathbb{I}-[A_{0},A_{1}][B_{0},B_{1}]
  13. [ A 0 , A 1 ] = 0 [A_{0},A_{1}]=0
  14. [ B 0 , B 1 ] = 0 [B_{0},B_{1}]=0
  15. 2 \langle\mathcal{B}\rangle\leq 2
  16. [ A 0 , A 1 ] 2 A 0 A 1 2 \|[A_{0},A_{1}]\|\leq 2\|A_{0}\|\|A_{1}\|\leq 2
  17. 2 2 \langle\mathcal{B}\rangle\leq 2\sqrt{2}
  18. B C n n cos ( π / n ) \langle BC_{n}\rangle\leq n\cos(\pi/n)
  19. W W Z B n 2 n - 1 2 \langle WWZB_{n}\rangle\leq 2^{\frac{n-1}{2}}
  20. I 3322 I_{3322}

Tunnel_magnetoresistance.html

  1. TMR := R ap - R p R p \mathrm{TMR}:=\frac{R_{\mathrm{ap}}-R_{\mathrm{p}}}{R_{\mathrm{p}}}
  2. R ap R_{\mathrm{ap}}
  3. R p R_{\mathrm{p}}
  4. 𝒟 \mathcal{D}
  5. P = 𝒟 ( E F ) - 𝒟 ( E F ) 𝒟 ( E F ) + 𝒟 ( E F ) P=\frac{\mathcal{D}_{\uparrow}(E_{\mathrm{F}})-\mathcal{D}_{\downarrow}(E_{% \mathrm{F}})}{\mathcal{D}_{\uparrow}(E_{\mathrm{F}})+\mathcal{D}_{\downarrow}(% E_{\mathrm{F}})}
  6. TMR = 2 P 1 P 2 1 - P 1 P 2 \mathrm{TMR}=\frac{2P_{1}P_{2}}{1-P_{1}P_{2}}
  7. 𝐓 = Tr [ 𝐓 ^ ρ ^ neq ] \mathbf{T}=\mathrm{Tr}[\hat{\mathbf{T}}\hat{\rho}_{\mathrm{neq}}]
  8. ρ ^ neq \hat{\rho}_{\mathrm{neq}}
  9. 𝐓 ^ \hat{\mathbf{T}}
  10. 𝐓 ^ = d 𝐒 ^ d t = - i [ 2 s y m b o l σ , H ^ ] \hat{\mathbf{T}}=\frac{d\hat{\mathbf{S}}}{dt}=-\frac{i}{\hbar}\left[\frac{% \hbar}{2}symbol{\sigma},\hat{H}\right]
  11. H ^ = H ^ 0 - Δ ( s y m b o l σ 𝐦 ) / 2 \hat{H}=\hat{H}_{0}-\Delta(symbol{\sigma}\cdot\mathbf{m})/2
  12. 𝐦 \mathbf{m}
  13. 𝐩 , 𝐪 \mathbf{p},\mathbf{q}
  14. ( s y m b o l σ 𝐩 ) ( s y m b o l σ 𝐪 ) = 𝐩 𝐪 + i ( 𝐩 × 𝐪 ) s y m b o l σ (symbol{\sigma}\cdot\mathbf{p})(symbol{\sigma}\cdot\mathbf{q})=\mathbf{p}\cdot% \mathbf{q}+i(\mathbf{p}\times\mathbf{q})\cdot symbol{\sigma}
  15. ( s y m b o l σ 𝐩 ) s y m b o l σ = 𝐩 + i s y m b o l σ × 𝐩 (symbol{\sigma}\cdot\mathbf{p})symbol{\sigma}=\mathbf{p}+isymbol{\sigma}\times% \mathbf{p}
  16. s y m b o l σ ( s y m b o l σ 𝐪 ) = 𝐪 + i 𝐪 × s y m b o l σ symbol{\sigma}(symbol{\sigma}\cdot\mathbf{q})=\mathbf{q}+i\mathbf{q}\times symbol% {\sigma}
  17. 𝐓 ^ \hat{\mathbf{T}}
  18. Δ , 𝐦 \Delta,\mathbf{m}
  19. s y m b o l σ = ( σ x , σ y , σ z ) symbol{\sigma}=(\sigma_{x},\sigma_{y},\sigma_{z})
  20. T = T x 2 + T z 2 T_{\parallel}=\sqrt{T_{x}^{2}+T_{z}^{2}}
  21. T = T y T_{\perp}=T_{y}
  22. T 0 T_{\perp}\equiv 0
  23. T T_{\parallel}
  24. θ \theta

Turing_degree.html

  1. 𝒟 \mathcal{D}
  2. 𝒟 \mathcal{D}
  3. 𝒟 \mathcal{D}
  4. 𝒟 \mathcal{D}
  5. 0 \aleph_{0}
  6. 2 0 2^{\aleph_{0}}
  7. 2 0 2^{\aleph_{0}}
  8. 2 0 2^{\aleph_{0}}
  9. 𝒟 \mathcal{D}
  10. 𝒟 \mathcal{D}
  11. 𝒟 \mathcal{D}

Turing_reduction.html

  1. A , B A,B\subseteq\mathbb{N}
  2. A A
  3. B B
  4. A T B A\leq_{T}B
  5. A A
  6. B B
  7. A T B A\equiv_{T}B\,
  8. A T B A\leq_{T}B
  9. B T A . B\leq_{T}A.
  10. X X
  11. 𝐝𝐞𝐠 ( X ) \,\textbf{deg}(X)
  12. 𝒳 𝒫 ( ) \mathcal{X}\subseteq\mathcal{P}(\mathbb{N})
  13. A A\subseteq\mathbb{N}
  14. 𝒳 \mathcal{X}
  15. X T A X\leq_{T}A
  16. X 𝒳 X\in\mathcal{X}
  17. A 𝒳 A\in\mathcal{X}
  18. A A
  19. 𝒳 \mathcal{X}
  20. 𝒳 \mathcal{X}
  21. W e W_{e}
  22. A = { e e W e } A=\{e\mid e\in W_{e}\}
  23. B = { ( e , n ) n W e } B=\{(e,n)\mid n\in W_{e}\}
  24. ( e , n ) (e,n)
  25. A T B A\leq_{T}B
  26. e A ( e , e ) B e\in A\Leftrightarrow(e,e)\in B
  27. ( e , n ) (e,n)
  28. i ( e , n ) i(e,n)
  29. i ( e , n ) i(e,n)
  30. i ( e , n ) i(e,n)
  31. i ( e , n ) A ( e , n ) B i(e,n)\in A\Leftrightarrow(e,n)\in B
  32. B T A B\leq_{T}A
  33. T \leq_{T}
  34. A T B A\leq_{T}B
  35. B T C B\leq_{T}C
  36. A T C A\leq_{T}C
  37. A A A\leq A
  38. T \leq_{T}
  39. A T B A\leq_{T}B
  40. B T A B\leq_{T}A
  41. A = B A=B
  42. ( A , B ) (A,B)
  43. T \leq_{T}
  44. T \leq_{T}
  45. B ( α ) B^{(\alpha)}

Turn_(geometry).html

  1. π \pi
  2. π \pi
  3. π / ρ \pi/\rho
  4. δ / π \delta/\pi
  5. π \pi
  6. 1 24 \frac{1}{24}
  7. 1 12 \frac{1}{12}
  8. 1 10 \frac{1}{10}
  9. 1 8 \frac{1}{8}
  10. 1 6 \frac{1}{6}
  11. 1 5 \frac{1}{5}
  12. 1 4 \frac{1}{4}
  13. 1 3 \frac{1}{3}
  14. 2 5 \frac{2}{5}
  15. 1 2 \frac{1}{2}
  16. 3 4 \frac{3}{4}
  17. π 12 \frac{\pi}{12}
  18. π 6 \frac{\pi}{6}
  19. π 5 \frac{\pi}{5}
  20. π 4 \frac{\pi}{4}
  21. π 3 \frac{\pi}{3}
  22. 2 π 5 2\frac{\pi}{5}
  23. π 2 \frac{\pi}{2}
  24. 2 π 3 2\frac{\pi}{3}
  25. 4 π 5 4\frac{\pi}{5}
  26. π \pi
  27. 3 π 2 3\frac{\pi}{2}
  28. π \pi
  29. 16 2 3 16\frac{2}{3}
  30. 33 1 3 33\frac{1}{3}
  31. 66 2 3 66\frac{2}{3}
  32. 133 1 3 133\frac{1}{3}
  33. τ \tau
  34. π \pi
  35. π π = 2 π \pi\!\;\!\!\!\pi=2\pi
  36. τ \tau
  37. τ \tau
  38. 2 5 τ \tfrac{2}{5}\tau
  39. 2 5 \tfrac{2}{5}
  40. 4 5 π \tfrac{4}{5}\pi
  41. τ \tau
  42. π \pi
  43. z u z . z\mapsto uz.

Twelfth_root_of_two.html

  1. 2 12 \sqrt[12]{2}
  2. 18 / 17 {18}/{17}
  3. 107 / 101 {107}/{101}
  4. 11011 / 10393 {11011}/{10393}
  5. 0 / 12 {}^{0/12}
  6. 1 / 12 {}^{1/12}
  7. 2 / 12 {}^{2/12}
  8. 3 / 12 {}^{3/12}
  9. 4 / 12 {}^{4/12}
  10. 5 / 12 {}^{5/12}
  11. 6 / 12 {}^{6/12}
  12. 7 / 12 {}^{7/12}
  13. 8 / 12 {}^{8/12}
  14. 9 / 12 {}^{9/12}
  15. 10 / 12 {}^{10/12}
  16. 11 / 12 {}^{11/12}
  17. 12 / 12 {}^{12/12}

Type-1_Gumbel_distribution.html

  1. f ( x | a , b ) = a b e - ( b e - a x + a x ) f(x|a,b)=abe^{-(be^{-ax}+ax)}\,
  2. - < x < . -\infty<x<\infty.

Type-2_Gumbel_distribution.html

  1. e - b x - a e^{-bx^{-a}}\!
  2. f ( x | a , b ) = a b x - a - 1 e - b x - a f(x|a,b)=abx^{-a-1}e^{-bx^{-a}}\,
  3. 0 < x < 0<x<\infty
  4. b = λ - k b=\lambda^{-k}
  5. a = - k a=-k
  6. 0 < a 1 0<a\leq 1
  7. 0 < a 2 0<a\leq 2
  8. F ( x | a , b ) = e - b x - a F(x|a,b)=e^{-bx^{-a}}\,
  9. E [ X k ] E[X^{k}]\,
  10. k < a k<a\,

Type_91_torpedo.html

  1. d x / d t = V X ( E q .1 ) W / g × ( d V X / d t ) = - D cos φ - L sin φ ( E q .2 ) d z / d t = V Z ( E q .3 ) W / g × ( d V Z / d t ) = D sin φ - L cos φ + W ( E q .4 ) d θ / d t = ω ( E q .5 ) I × ( d ω / d t ) = 57.3 M - b V ω ( E q .6 ) \begin{array}[]{lcll}dx/dt&=&V_{X}&\cdots(Eq.1)\\ W/g\times\left(dV_{X}/dt\right)&=&-D\cos\varphi-L\sin\varphi&\cdots(Eq.2)\\ dz/dt&=&V_{Z}&\ldots(Eq.3)\\ W/g\times\left(dV_{Z}/dt\right)&=&D\sin\varphi-L\cos\varphi+W&\cdots(Eq.4)\\ d\theta/dt&=&\omega&\cdots(Eq.5)\\ I\times\left(d\omega/dt\right)&=&57.3M-bV\omega&\cdots(Eq.6)\end{array}
  2. b = - ( δ C m g H / δ α H ) × ( ρ S l / 2 ) × l H b=-(\delta C_{mgH}/\delta\alpha_{H})\times(\rho S\mathit{l}/2)\times\mathit{l}% _{H}\,
  3. ω l H / V = ( δ C m g H / δ α H ) × ( ρ S l / 2 ) × l H × V × ω \omega\mathit{l}_{H}/V=(\delta C_{mgH}/\delta\alpha_{H})\times(\rho S\mathit{l% }/2)\times\mathit{l}_{H}\times V\times\omega\,
  4. l H = ( - C m g H l + C t H d ) / C n H \mathit{l}_{H}=(-C_{mgH}\,\mathit{l}+C_{tHd})/C_{nH}\,
  5. X = λ 0 λ d λ g C + k φ ( x ) , Z = λ 0 λ λ d λ g C + k φ ( x ) X=\int_{\lambda_{0}}^{\lambda}\,\frac{d\lambda}{gC}\,+\,k\,\varphi(x),\,\,Z=% \int_{\lambda_{0}}^{\lambda}\,\lambda\,\frac{d\lambda}{gC}\,+\,k\,\varphi(x)
  6. - ( 1 / 2 ) × 57.3 C 1 ρ V 2 b S 1 ( b / V ) ( d θ / d t ) -(1/2)\times 57.3C_{1}\,\rho\,V^{2}\,b\,S_{1}\,(b/V)\,(d\theta/dt)\,
  7. I 2 ( d 2 θ / d t 2 ) = - K - M ( d θ / d t ) ( E q .7 ) ( d 2 θ / d t 2 ) + M / I 2 ( d θ / d t ) = 0 ( E q .8 ) \begin{array}[]{lcll}I_{2}(d^{2}\theta/dt^{2})&=&-K-M(d\theta/dt)&\cdots(Eq.7)% \\ (d^{2}\theta/dt^{2})+M/I_{2}(d\theta/dt)&=&0&\cdots(Eq.8)\end{array}
  8. d θ / d t = - ( I 2 / M ) ( K / I 2 ) + ( ω 0 + K / M ) e - ( M / I 2 ) t ( E q .9 ) θ = - ( K / M ) t + ( I 2 / M ) ( ω 0 + K / M ) ( 1 - e - ( M / I 2 ) t ) ( E q .10 ) \begin{array}[]{lcll}d\theta/dt&=&-(I_{2}\,/\,M)\,(K\,/\,I_{2})\,+\,(\omega_{0% }\,+\,K\,/\,M)\,e^{-(M/I_{2})t}&\cdots(Eq.9)\\ \theta&=&-(K/M)t\,+\,(I_{2}/M)\,(\omega_{0}\,+\,K/M)\,(1\,-\,e^{-(M/I_{2})t})&% \cdots(Eq.10)\end{array}

Type_I_string_theory.html

  1. g g
  2. 1 / g 1/g

Typing.html

  1. ( I N F / ( C + I N F ) ) * 100 % (INF/(C+INF))*100\%

Ultrafast_monochromator.html

  1. m λ = d ( sin θ i - sin θ m ) m\lambda=d(\sin{\theta_{i}}-\sin{\theta_{m}})

Undulation_of_the_geoid.html

  1. ζ \zeta
  2. h h
  3. H N H_{N}
  4. ζ = h - H N \zeta=h-H_{N}

Undulator.html

  1. λ u \lambda_{u}
  2. K = e B λ u 2 π m e c K=\frac{eB\lambda_{u}}{2\pi m_{e}c}
  3. m e m_{e}
  4. K 1 K\ll 1
  5. K 1 K\gg 1
  6. N 2 N^{2}

Unified_neutral_theory_of_biodiversity.html

  1. n 1 n_{1}
  2. n 2 n_{2}
  3. n S n_{S}
  4. Pr ( n 1 , n 2 , , n S | θ , J ) = J ! θ S 1 ϕ 1 2 ϕ 2 J ϕ J ϕ 1 ! ϕ 2 ! ϕ J ! Π k = 1 J ( θ + k - 1 ) \Pr(n_{1},n_{2},\ldots,n_{S}|\theta,J)=\frac{J!\theta^{S}}{1^{\phi_{1}}2^{\phi% _{2}}\cdots J^{\phi_{J}}\phi_{1}!\phi_{2}!\cdots\phi_{J}!\Pi_{k=1}^{J}(\theta+% k-1)}
  5. θ = 2 J ν \theta=2J\nu
  6. ν \nu
  7. ϕ i \phi_{i}
  8. ϕ 1 = ϕ 3 = ϕ 6 = 1 \phi_{1}=\phi_{3}=\phi_{6}=1
  9. ϕ \phi
  10. Pr ( 3 , 6 , 1 | θ , 10 ) = 10 ! θ 3 1 1 3 1 6 1 1 ! 1 ! 1 ! θ ( θ + 1 ) ( θ + 2 ) ( θ + 9 ) \Pr(3,6,1|\theta,10)=\frac{10!\theta^{3}}{1^{1}\cdot 3^{1}\cdot 6^{1}\cdot 1!1% !1!\cdot\theta(\theta+1)(\theta+2)\cdots(\theta+9)}
  11. Pr ( 3 ; 3 , 6 , 1 ) = Pr ( 3 ; 1 , 3 , 6 ) = Pr ( 3 ; 3 , 1 , 6 ) \Pr(3;3,6,1)=\Pr(3;1,3,6)=\Pr(3;3,1,6)
  12. Pr ( S ; r 1 , r 2 , , r s , 0 , , 0 ) \Pr(S;r_{1},r_{2},\ldots,r_{s},0,\ldots,0)
  13. r i r_{i}
  14. r 1 r_{1}
  15. r 2 r_{2}
  16. E ( r i ) = k = 1 C r i ( k ) Pr ( S ; r 1 , r 2 , , r s , 0 , , 0 ) E(r_{i})=\sum_{k=1}^{C}r_{i}(k)\cdot\Pr(S;r_{1},r_{2},\ldots,r_{s},0,\ldots,0)
  17. r i ( k ) r_{i}(k)
  18. P r ( ) Pr(\ldots)
  19. θ J ! n ! ( J - n ) ! Γ ( γ ) Γ ( J + γ ) y = 0 γ Γ ( n + y ) Γ ( 1 + y ) Γ ( J - n + γ - y ) Γ ( γ - y ) exp ( - y θ / γ ) d y \theta\frac{J!}{n!(J-n)!}\frac{\Gamma(\gamma)}{\Gamma(J+\gamma)}\int_{y=0}^{% \gamma}\frac{\Gamma(n+y)}{\Gamma(1+y)}\frac{\Gamma(J-n+\gamma-y)}{\Gamma(% \gamma-y)}\exp(-y\theta/\gamma)\,dy
  20. Γ \Gamma
  21. γ = ( J - 1 ) m / ( 1 - m ) \gamma=(J-1)m/(1-m)
  22. θ ( I ) J ( J n ) 0 1 ( I x ) n ( I ( 1 - x ) ) J - n ( 1 - x ) θ - 1 x d x \frac{\theta}{(I)_{J}}{J\choose n}\int_{0}^{1}(Ix)_{n}(I(1-x))_{J-n}\frac{(1-x% )^{\theta-1}}{x}\,dx
  23. I = ( J - 1 ) * m / ( 1 - m ) I=(J-1)*m/(1-m)
  24. ϕ n \langle\phi_{n}\rangle
  25. Pr ( n 1 , n 2 , , n S | θ , m , J ) = J ! i = 1 S n i j = 1 J Φ j ! θ S ( I ) J A = S J K ( D , A ) I A ( θ ) A \Pr(n_{1},n_{2},\ldots,n_{S}|\theta,m,J)=\frac{J!}{\prod_{i=1}^{S}n_{i}\prod_{% j=1}^{J}\Phi_{j}!}\frac{\theta^{S}}{(I)_{J}}\sum_{A=S}^{J}K(\overrightarrow{D}% ,A)\frac{I^{A}}{(\theta)_{A}}
  26. K ( D , A ) K(\overrightarrow{D},A)
  27. A = S , , J A=S,...,J
  28. K ( D , A ) := { a 1 , , a S | i = 1 S a i = A } i = 1 S s ¯ ( n i , a i ) s ¯ ( a i , 1 ) s ¯ ( n i , 1 ) K(\overrightarrow{D},A):=\sum_{\{a_{1},...,a_{S}|\sum_{i=1}^{S}a_{i}=A\}}\prod% _{i=1}^{S}\frac{\overline{s}\left(n_{i},a_{i}\right)\overline{s}\left(a_{i},1% \right)}{\overline{s}\left(n_{i},1\right)}
  29. J J
  30. r r
  31. μ \mu
  32. J M J_{M}
  33. ( 1 - ν ) (1-\nu)
  34. ν \nu
  35. J M J_{M}
  36. r r
  37. μ \mu
  38. ν \nu
  39. ν = μ / ( r + μ ) \nu=\mu/(r+\mu)
  40. S M ( n ) S_{M}(n)
  41. n n
  42. S M ( n ) = θ n Γ ( J M + 1 ) Γ ( J M + θ - n ) Γ ( J M + 1 - n ) Γ ( J M + θ ) S_{M}(n)=\frac{\theta}{n}\frac{\Gamma(J_{M}+1)\Gamma(J_{M}+\theta-n)}{\Gamma(J% _{M}+1-n)\Gamma(J_{M}+\theta)}
  43. θ = ( J M - 1 ) ν / ( 1 - ν ) J M ν \theta=(J_{M}-1)\nu/(1-\nu)\approx J_{M}\nu
  44. n J M n\ll J_{M}
  45. S M ( n ) θ n ( J M J M + θ ) n S_{M}(n)\approx\frac{\theta}{n}\left(\frac{J_{M}}{J_{M}+\theta}\right)^{n}
  46. J J
  47. ( 1 - m ) (1-m)
  48. m m
  49. J J
  50. J M J_{M}
  51. θ \theta
  52. m m
  53. I I
  54. ϕ n \langle\phi_{n}\rangle
  55. m m
  56. m = 1 m=1
  57. m = 0 m=0
  58. 0 < m < 1 0<m<1
  59. S = c A z S=cA^{z}
  60. S = k J z S=kJ^{z}
  61. E { S | θ , J } = θ θ + θ θ + 1 + θ θ + 2 + + θ θ + J - 1 E\left\{S|\theta,J\right\}=\frac{\theta}{\theta}+\frac{\theta}{\theta+1}+\frac% {\theta}{\theta+2}+\cdots+\frac{\theta}{\theta+J-1}
  62. θ / ( θ + J - 1 ) \theta/(\theta+J-1)
  63. J = ρ A J=\rho A
  64. Σ θ / ( θ + ρ A - 1 ) \Sigma\theta/(\theta+\rho A-1)
  65. S ( θ ) = 1 + θ ln ( 1 + J - 1 θ ) . S(\theta)=1+\theta\ln\left(1+\frac{J-1}{\theta}\right).
  66. x ˙ = b - x / τ + D x ξ ( t ) \dot{x}=b-x/\tau+\sqrt{Dx}\xi(t)
  67. - x / τ -x/\tau
  68. ξ ( t ) \xi(t)
  69. P ( r , t ) = A λ + 1 λ ( e t / τ ) b / 2 D 1 - e - t / τ ( sinh ( t 2 τ ) λ ) b D + 1 ( 4 λ 2 ( λ + 1 ) 2 e t / τ - 4 λ ) b D + 1 2 . P(r,t)=A\frac{\lambda+1}{\lambda}\frac{(e^{t/\tau})^{b/2D}}{1-e^{-t/\tau}}% \left(\frac{\sinh(\frac{t}{2\tau})}{\lambda}\right)^{\frac{b}{D}+1}\left(\frac% {4\lambda^{2}}{(\lambda+1)^{2}e^{t/\tau}-4\lambda}\right)^{\frac{b}{D}+\frac{1% }{2}}.
  70. τ \tau
  71. τ \tau

Unified_Thread_Standard.html

  1. H = cos ( 30 ) × P = 3 2 × P 0.866 × P \begin{aligned}\displaystyle H&\displaystyle=\cos(30^{\circ})\times P\\ &\displaystyle=\frac{{\sqrt{3}}}{2}\times P\\ &\displaystyle\approx 0.866\times P\end{aligned}
  2. D min = D maj - 2 5 8 H = D maj - 5 3 8 P D maj - 1.082532 × P D p = D maj - 2 3 8 H = D maj - 3 3 8 P D maj - 0.649519 × P \begin{aligned}\displaystyle D\text{min}&\displaystyle=D\text{maj}-2\cdot\frac% {5}{8}\cdot H\\ &\displaystyle=D\text{maj}-\frac{5\sqrt{3}}{8}\cdot P\\ &\displaystyle\approx D\text{maj}-1.082532\times P\\ \displaystyle D\text{p}&\displaystyle=D\text{maj}-2\cdot\frac{3}{8}\cdot H\\ &\displaystyle=D\text{maj}-\frac{3\sqrt{3}}{8}\cdot P\\ &\displaystyle\approx D\text{maj}-0.649519\times P\end{aligned}
  3. 1 / 4 {1}/{4}
  4. 3 / 64 {3}/{64}
  5. 3 / 16 {3}/{16}
  6. 1 / 4 {1}/{4}
  7. 7 / 32 {7}/{32}
  8. 5 / 16 {5}/{16}
  9. 9 / 32 {9}/{32}
  10. 3 / 8 {3}/{8}
  11. 5 / 16 {5}/{16}
  12. 11 / 32 {11}/{32}
  13. 7 / 16 {7}/{16}
  14. 25 / 64 {25}/{64}
  15. 1 / 2 {1}/{2}
  16. 27 / 64 {27}/{64}
  17. 29 / 64 {29}/{64}
  18. 15 / 32 {15}/{32}
  19. 9 / 16 {9}/{16}
  20. 31 / 64 {31}/{64}
  21. 1 / 2 {1}/{2}
  22. 33 / 64 {33}/{64}
  23. 5 / 8 {5}/{8}
  24. 17 / 32 {17}/{32}
  25. 9 / 16 {9}/{16}
  26. 37 / 64 {37}/{64}
  27. 3 / 4 {3}/{4}
  28. 21 / 32 {21}/{32}
  29. 11 / 16 {11}/{16}
  30. 45 / 64 {45}/{64}
  31. 7 / 8 {7}/{8}
  32. 49 / 64 {49}/{64}
  33. 51 / 64 {51}/{64}
  34. 53 / 64 {53}/{64}
  35. 7 / 8 {7}/{8}
  36. 59 / 64 {59}/{64}
  37. 61 / 64 {61}/{64}

Uniform_absolute_continuity.html

  1. \mathcal{F}
  2. ϵ > 0 \epsilon>0
  3. δ > 0 \delta>0
  4. E E
  5. μ ( E ) < δ \mu(E)<\delta
  6. E | f | d μ < ϵ \int_{E}\!|f|\,\mathrm{d}\mu<\epsilon
  7. f f\in\mathcal{F}

United_States_tort_law.html

  1. B < P L B<PL

Universal_C*-algebra.html

  1. e * = e , e^{*}=e,\quad
  2. ( x * ) * = x , (x^{*})^{*}=x,\quad
  3. ( x y ) * = y * x * . (xy)^{*}=y^{*}x^{*}.\quad
  4. 1 ( S ) = { φ : S : φ = x S | φ ( x ) | < } . \ell^{1}(S)=\{\varphi:S\rightarrow\mathbb{C}:\|\varphi\|=\sum_{x\in S}|\varphi% (x)|<\infty\}.
  5. [ φ ψ ] ( x ) = { u , v : u v = x } φ ( u ) ψ ( v ) [\varphi\star\psi](x)=\sum_{\{u,v:uv=x\}}\varphi(u)\psi(v)
  6. φ * ( x ) = φ ( x * ) ¯ \varphi^{*}(x)=\overline{\varphi(x^{*})}
  7. φ = sup f π f ( φ ) \|\varphi\|=\sup_{f}\|\pi_{f}(\varphi)\|

Universal_coefficient_theorem.html

  1. X X
  2. A A
  3. A A
  4. A A
  5. 𝐙 / 2 𝐙 \mathbf{Z}/2\mathbf{Z}
  6. X X
  7. F F
  8. F F
  9. p p
  10. p p
  11. 0 H i ( X ; 𝐙 ) A 𝜇 H i ( X ; A ) Tor ( H i - 1 ( X ; 𝐙 ) , A ) 0. 0\to H_{i}(X;\mathbf{Z})\otimes A\overset{\mu}{\to}H_{i}(X;A)\to\mbox{Tor}~{}(% H_{i-1}(X;\mathbf{Z}),A)\to 0.
  12. μ μ
  13. A A
  14. 𝐙 / p 𝐙 \mathbf{Z}/p\mathbf{Z}
  15. G G
  16. R R
  17. 𝐙 \mathbf{Z}
  18. 0 Ext R 1 ( H i - 1 ( X ; R ) , G ) H i ( X ; G ) Hom R ( H i ( X ; R ) , G ) 0. 0\to\operatorname{Ext}_{R}^{1}(\operatorname{H}_{i-1}(X;R),G)\to H^{i}(X;G)% \overset{h}{\to}\operatorname{Hom}_{R}(H_{i}(X;R),G)\to 0.
  19. H i ( X ; G ) = ker i G / im i + 1 G H_{i}(X;G)=\ker\partial_{i}\otimes G/\operatorname{im}\partial_{i+1}\otimes G
  20. H * ( X ; G ) = ker ( Hom ( , G ) ) / im ( Hom ( , G ) ) . H^{*}(X;G)=\ker(\operatorname{Hom}(\partial,G))/\operatorname{im}(% \operatorname{Hom}(\partial,G)).
  21. h h
  22. h ( [ f ] ) ( [ x ] ) = f ( x ) . h([f])([x])=f(x).
  23. h h
  24. X X
  25. K ( G , i ) K(G,i)
  26. X X
  27. R = 𝐙 / 2 𝐙 R=\mathbf{Z}/2\mathbf{Z}
  28. H i ( X ; 𝐙 ) = { 𝐙 i = 0 or i = n odd, 𝐙 / 2 𝐙 0 < i < n , i odd, 0 else. H_{i}(X;\mathbf{Z})=\begin{cases}\mathbf{Z}&i=0\mbox{ or }~{}i=n\mbox{ odd,}\\ \mathbf{Z}/2\mathbf{Z}&0<i<n,\ i\ \mbox{odd,}\\ 0&\mbox{else.}\end{cases}
  29. E x t ( R , R ) = R , E x t ( 𝐙 , R ) = 0 Ext(R,R)=R,Ext(\mathbf{Z},R)=0
  30. i = 0 , , n : H i ( X ; R ) = R . \forall i=0,\cdots,n:\qquad\ H^{i}(X;R)=R.
  31. H * ( X ; R ) = R [ w ] / w n + 1 . H^{*}(X;R)=R[w]/\left\langle w^{n+1}\right\rangle.
  32. X X
  33. H i ( X ; 𝐙 ) 𝐙 β i ( X ) T i , H_{i}(X;\mathbf{Z})\cong\mathbf{Z}^{\beta_{i}(X)}\oplus T_{i},
  34. X X
  35. T i T_{i}
  36. H i H_{i}
  37. Hom ( H i ( X ) , 𝐙 ) Hom ( 𝐙 β i ( X ) , 𝐙 ) Hom ( T i , 𝐙 ) 𝐙 β i ( X ) , \mbox{Hom}~{}(H_{i}(X),\mathbf{Z})\cong\mbox{Hom}~{}(\mathbf{Z}^{\beta_{i}(X)}% ,\mathbf{Z})\oplus\mbox{Hom}~{}(T_{i},\mathbf{Z})\cong\mathbf{Z}^{\beta_{i}(X)},
  38. Ext ( H i ( X ) , 𝐙 ) Ext ( 𝐙 β i ( X ) , 𝐙 ) Ext ( T i , 𝐙 ) T i . \mbox{Ext}~{}(H_{i}(X),\mathbf{Z})\cong\mbox{Ext}~{}(\mathbf{Z}^{\beta_{i}(X)}% ,\mathbf{Z})\oplus\mbox{Ext}~{}(T_{i},\mathbf{Z})\cong T_{i}.
  39. H i ( X ; 𝐙 ) 𝐙 β i ( X ) T i - 1 . H^{i}(X;\mathbf{Z})\cong\mathbf{Z}^{\beta_{i}(X)}\oplus T_{i-1}.
  40. X X
  41. n n
  42. β < s u b > i 1 ( X ) = β n i ( X ) β<sub>i−1(X)=β_{n−i}(X)

Universal_generalization.html

  1. P ( x ) \vdash P(x)
  2. x P ( x ) \vdash\forall x\,P(x)
  3. Γ φ ( y ) \Gamma\vdash\varphi(y)
  4. Γ x φ ( x ) \Gamma\vdash\forall x\varphi(x)
  5. x P ( x ) \forall xP(x)
  6. P ( y ) P(y)
  7. z w ( z w ) \exists z\exists w(z\not=w)
  8. w ( y w ) \exists w(y\not=w)
  9. y x y\not=x
  10. x ( x x ) \forall x(x\not=x)
  11. z w ( z w ) x ( x x ) , \exists z\exists w(z\not=w)\vdash\forall x(x\not=x),
  12. x ( P ( x ) Q ( x ) ) ( x P ( x ) x Q ( x ) ) \forall x\,(P(x)\rightarrow Q(x))\rightarrow(\forall x\,P(x)\rightarrow\forall x% \,Q(x))
  13. x ( P ( x ) Q ( x ) ) \forall x\,(P(x)\rightarrow Q(x))
  14. x P ( x ) \forall x\,P(x)
  15. ( x ( P ( x ) Q ( x ) ) ) ( P ( y ) Q ( y ) ) ) (\forall x\,(P(x)\rightarrow Q(x)))\rightarrow(P(y)\rightarrow Q(y)))
  16. P ( y ) Q ( y ) P(y)\rightarrow Q(y)
  17. ( x P ( x ) ) P ( y ) (\forall x\,P(x))\rightarrow P(y)
  18. P ( y ) P(y)
  19. Q ( y ) Q(y)
  20. x Q ( x ) \forall x\,Q(x)
  21. x ( P ( x ) Q ( x ) ) , x P ( x ) x Q ( x ) \forall x\,(P(x)\rightarrow Q(x)),\forall x\,P(x)\vdash\forall x\,Q(x)
  22. x ( P ( x ) Q ( x ) ) x P ( x ) x Q ( x ) \forall x\,(P(x)\rightarrow Q(x))\vdash\forall x\,P(x)\rightarrow\forall x\,Q(x)
  23. x ( P ( x ) Q ( x ) ) ( x P ( x ) x Q ( x ) ) \vdash\forall x\,(P(x)\rightarrow Q(x))\rightarrow(\forall x\,P(x)\rightarrow% \forall x\,Q(x))

Unrolled_linked_list.html

  1. ( v / m + s ) n (v/m+s)n
  2. ( 2 v / m + s ) n (2v/m+s)n
  3. ( v + s ) n (v+s)n
  4. s n sn

Unruh_effect.html

  1. T = a 2 π c k B , T=\frac{\hbar a}{2\pi ck\text{B}},
  2. a a
  3. k B k\text{B}
  4. \hbar
  5. c c
  6. T H = g / ( 2 π c k B ) T\text{H}=\hbar g/(2\pi ck\text{B})
  7. d s 2 = - ρ 2 d σ 2 + d ρ 2 , ds^{2}=-\rho^{2}d\sigma^{2}+d\rho^{2},
  8. ρ = 1 / a \rho=1/a
  9. σ \sigma
  10. τ \tau
  11. σ = a τ \sigma=a\tau
  12. x = ρ cosh σ x=\rho\cosh\sigma
  13. t = ρ sinh σ . t=\rho\sinh\sigma.
  14. ρ \rho
  15. ρ \rho
  16. σ \sigma
  17. σ \sigma
  18. σ , \sigma,
  19. 2 π 2\pi
  20. e 2 π i H = 1. e^{2\pi iH}=1.
  21. 2 π 2\pi
  22. ( 2 π ) - 1 \scriptstyle(2\pi)^{-1}
  23. σ \sigma
  24. σ \sigma
  25. ρ \rho
  26. ρ \rho
  27. ρ \rho
  28. β = 2 π ρ . \beta=2\pi\rho.
  29. ρ \rho
  30. 1 / a 1/a
  31. β = 2 π a . \beta={2\pi\over a}.
  32. k B T = a 2 π c . k\text{B}T=\frac{\hbar a}{2\pi c}.
  33. G M S ( 1 AU ) 2 = 0.005932 m s - 2 \frac{GM_{S}}{\mathrm{\left(1~{}AU\right)}^{2}}=0.005932~{}\mathrm{m\cdot s^{-% 2}}

Untouchable_number.html

  1. σ ( n ) - n \sigma(n)-n
  2. n - ϕ ( n ) n-\phi(n)

Upper_topology.html

  1. { a } \{a\}
  2. a ] = { x a } a]=\{x\leq a\}
  3. a X a\in X
  4. \leq
  5. ] - , + ] = { + } ]-\infty,+\infty]=\mathbb{R}\cup\{+\infty\}
  6. { ] a , + ] : a { ± } } \{]a,+\infty]:a\in\mathbb{R}\cup\{\pm\infty\}\}
  7. { [ - , a [ : a { ± } } \{[-\infty,a[:a\in\mathbb{R}\cup\{\pm\infty\}\}
  8. [ - , + [ = { - } [-\infty,+\infty[=\mathbb{R}\cup\{-\infty\}
  9. [ - , + [ {[-\infty,+\infty[}
  10. ] - , + ] {]-\infty,+\infty]}

Urelement.html

  1. X U , X\in U,
  2. U X , U\in X,
  3. X X\in\emptyset

Uses_of_trigonometry.html

  1. + cos θ + sin θ 1 + cos ( 2 θ ) + sin ( 2 θ ) 2 + cos ( 3 θ ) + sin ( 3 θ ) 3 + \square+\underbrace{\square\cos\theta+\square\sin\theta}_{1}+\underbrace{% \square\cos(2\theta)+\square\sin(2\theta)}_{2}+\underbrace{\square\cos(3\theta% )+\square\sin(3\theta)}_{3}+\cdots\,
  2. \square
  3. 1 42 , 2 42 , 3 42 , , 39 42 , 40 42 , 41 42 . \frac{1}{42},\qquad\frac{2}{42},\qquad\frac{3}{42},\qquad\dots\dots,\qquad% \frac{39}{42},\qquad\frac{40}{42},\qquad\frac{41}{42}.
  4. 1 42 , 5 42 , 11 42 , , 31 42 , 37 42 , 41 42 . \frac{1}{42},\qquad\frac{5}{42},\qquad\frac{11}{42},\qquad\dots,\qquad\frac{31% }{42},\qquad\frac{37}{42},\qquad\frac{41}{42}.
  5. cos ( 2 π 1 42 ) + cos ( 2 π 5 42 ) + + cos ( 2 π 37 42 ) + cos ( 2 π 41 42 ) \cos\left(2\pi\cdot\frac{1}{42}\right)+\cos\left(2\pi\cdot\frac{5}{42}\right)+% \cdots+\cos\left(2\pi\cdot\frac{37}{42}\right)+\cos\left(2\pi\cdot\frac{41}{42% }\right)

Utility_maximization_problem.html

  1. x + L . x\in\mathbb{R}^{L}_{+}\ .
  2. p + L , p\in\mathbb{R}^{L}_{+}\ ,
  3. B ( p , w ) = { x + L : p , x w } , B(p,w)=\{x\in\mathbb{R}^{L}_{+}:\langle p,x\rangle\leq w\}\ ,
  4. p , x \langle p,x\rangle
  5. p , x = i = 1 L p i x i . \langle p,x\rangle=\sum_{i=1}^{L}p_{i}x_{i}.
  6. u : + L + . u:\mathbb{R}^{L}_{+}\rightarrow\mathbb{R}_{+}\ .
  7. x ( p , w ) = argmax x * B ( p , w ) u ( x * ) x(p,w)=\operatorname{argmax}_{x^{*}\in B(p,w)}u(x^{*})

Vacuum_state.html

  1. | 0 |0\rangle
  2. | |\rangle
  3. 0 | ϕ | 0 \langle 0|\phi|0\rangle
  4. ϕ \langle\phi\rangle
  5. Δ E Δ t , \Delta E\Delta t\geq\hbar\ ,
  6. ħ ħ
  7. 2 π
  8. q q
  9. p p
  10. q q , p = i ħ qq,p=iħ
  11. α α

Valuation_ring.html

  1. ( A , 𝔪 A ) (A,\mathfrak{m}_{A})
  2. ( B , 𝔪 B ) (B,\mathfrak{m}_{B})
  3. A B A\supset B
  4. 𝔪 A B = 𝔪 B \mathfrak{m}_{A}\cap B=\mathfrak{m}_{B}
  5. g G f ( g ) x g \sum_{g\in G}f(g)x^{g}
  6. x g x h = x g + h . x^{g}\cdot x^{h}=x^{g+h}.
  7. ( S , 𝔪 S ) (S,\mathfrak{m}_{S})
  8. ( R , 𝔪 R ) (R,\mathfrak{m}_{R})
  9. S R S\supset R
  10. 𝔪 S R = 𝔪 R \mathfrak{m}_{S}\cap R=\mathfrak{m}_{R}
  11. R S R\subset S
  12. ( A , 𝔭 ) (A,\mathfrak{p})
  13. 1 𝔭 R 1\not\in\mathfrak{p}R
  14. R R
  15. 𝔭 R \mathfrak{p}R
  16. x R x\not\in R
  17. 𝔭 R [ x ] = R [ x ] \mathfrak{p}R[x]=R[x]
  18. 1 = r 0 + r 1 x + + r n x n , r i 𝔭 R 1=r_{0}+r_{1}x+\cdots+r_{n}x^{n},\quad r_{i}\in\mathfrak{p}R
  19. 1 - r 0 1-r_{0}
  20. x - 1 x^{-1}
  21. 𝔭 \mathfrak{p}
  22. f : A k f:A\to k
  23. g : D k g:D\to k
  24. g : R k g:R\to k
  25. S S
  26. R [ x ] R[x]
  27. S / 𝔪 S S/\mathfrak{m}_{S}
  28. R / 𝔪 R R/\mathfrak{m}_{R}
  29. S S / 𝔪 S k S\to S/\mathfrak{m}_{S}\hookrightarrow k
  30. 𝔭 \mathfrak{p}
  31. R = D 𝔭 R=D_{\mathfrak{p}}
  32. R R
  33. D 𝔭 D_{\mathfrak{p}}
  34. 𝔭 D 𝔭 , Spec ( D ) \mathfrak{p}\mapsto D_{\mathfrak{p}},\operatorname{Spec}(D)\to
  35. x - 1 A [ x - 1 ] x^{-1}A[x^{-1}]
  36. A [ x - 1 ] A[x^{-1}]
  37. 𝔭 \mathfrak{p}
  38. A [ x - 1 ] A[x^{-1}]
  39. 𝔭 \mathfrak{p}
  40. x - 1 𝔪 R x^{-1}\in\mathfrak{m}_{R}
  41. x R x\not\in R
  42. k ( X ) k(X)
  43. R R
  44. 𝒪 x , X \mathcal{O}_{x,X}
  45. Γ A \Gamma_{A}
  46. v ( A - 0 ) v(A-0)
  47. - v ( A - 0 ) -v(A-0)
  48. Γ \Gamma
  49. Γ I \Gamma_{I}
  50. Γ \Gamma
  51. I Γ I I\mapsto\Gamma_{I}
  52. Γ \Gamma
  53. p \mathbb{Z}_{p}
  54. \mathbb{Z}
  55. \mathbb{Z}
  56. ( p ) p (p)\subset\mathbb{Z}_{p}
  57. \mathbb{Z}
  58. dim 𝐐 ( Γ 𝐙 𝐐 ) \mathrm{dim}_{\mathbf{Q}}(\Gamma\otimes_{\mathbf{Z}}\mathbf{Q})
  59. x D x\not\in D
  60. p ( 1 / x ) = 0 p(1/x)=0
  61. D D / 𝔪 D D\to D/\mathfrak{m}_{D}
  62. 𝔭 \mathfrak{p}
  63. A 𝔭 k ( 𝔭 ) A_{\mathfrak{p}}\to k(\mathfrak{p})
  64. p p p\rightsquigarrow p^{\prime}
  65. 𝔭 \mathfrak{p}
  66. 𝔭 \mathfrak{p}^{\prime}
  67. 𝔭 𝔭 \mathfrak{p}\subset\mathfrak{p}^{\prime}
  68. p p p\rightsquigarrow p^{\prime}
  69. D D D\supset D^{\prime}
  70. D D^{\prime}
  71. p p p\rightsquigarrow p^{\prime}
  72. p = q p | D p^{\prime}=q\circ p|_{D^{\prime}}
  73. k ( p ) k(p)
  74. p ( D ) p(D^{\prime})
  75. k ( p ) k(p)
  76. tr . deg k k ( p ) + dim D tr . deg k K \operatorname{tr.deg}_{k}k(p)+\dim D\leq\operatorname{tr.deg}_{k}K
  77. ker ( p ) A \operatorname{ker}(p)\cap A
  78. [ x ] [ y ] [x]\geq[y]
  79. x y - 1 D xy^{-1}\in D
  80. x - 1 x^{-1}
  81. x - 1 𝔪 S x^{-1}\in\mathfrak{m}_{S}
  82. x - 1 x^{-1}
  83. x A [ x ] = A [ x ] . xA[x]=A[x].

Van_'t_Hoff_factor.html

  1. i i
  2. α \alpha
  3. α \alpha
  4. n n
  5. i = α n + ( 1 - α ) = 1 + α ( n - 1 ) i=\alpha n+(1-\alpha)=1+\alpha(n-1)
  6. n = 2 n=2
  7. i = 1 + α i=1+\alpha
  8. α \alpha
  9. n n
  10. i = 1 - ( 1 - 1 n ) α i=1-(1-\frac{1}{n})\alpha
  11. i = 1 - ( 1 - 1 2 ) α = 1 - α 2 i=1-(1-\frac{1}{2})\alpha=1-\frac{\alpha}{2}
  12. i i
  13. i i
  14. i i
  15. i i
  16. i i
  17. i i

Van_Emde_Boas_tree.html

  1. M \sqrt{M}
  2. M \sqrt{M}
  3. M \sqrt{M}
  4. i = x M i=\left\lfloor\frac{x}{\sqrt{}}{M}\right\rfloor
  5. M \sqrt{M}
  6. M \sqrt{M}
  7. M \sqrt{M}
  8. M \sqrt{M}
  9. M \sqrt{M}
  10. M \sqrt{M}
  11. M \sqrt{M}
  12. M \sqrt{M}
  13. M \sqrt{M}
  14. O ( M ) = O ( 2 m ) O(M)=O(2^{m})
  15. S ( M ) = O ( M ) + ( M + 1 ) S ( M ) S(M)=O(\sqrt{M})+(\sqrt{M}+1)\cdot S(\sqrt{M})
  16. S ( M ) ( 1 + M ) log log M + log log M O ( M ) S(M)\in(1+\sqrt{M})^{\log\log M}+\log\log M\cdot O(\sqrt{M})
  17. S ( M ) = M - 2 S(M)=M-2
  18. M \sqrt{M}

Vandermonde's_identity.html

  1. ( m + n r ) = k = 0 r ( m k ) ( n r - k ) , m , n , r 0 , {m+n\choose r}=\sum_{k=0}^{r}{m\choose k}{n\choose r-k},\qquad m,n,r\in\mathbb% {N}_{0},
  2. ( i = 0 m a i x i ) ( j = 0 n b j x j ) = r = 0 m + n ( k = 0 r a k b r - k ) x r , \biggl(\sum_{i=0}^{m}a_{i}x^{i}\biggr)\biggl(\sum_{j=0}^{n}b_{j}x^{j}\biggr)=% \sum_{r=0}^{m+n}\biggl(\sum_{k=0}^{r}a_{k}b_{r-k}\biggr)x^{r},
  3. ( 1 + x ) m + n = r = 0 m + n ( m + n r ) x r . (1+x)^{m+n}=\sum_{r=0}^{m+n}{m+n\choose r}x^{r}.
  4. r = 0 m + n ( m + n r ) x r = ( 1 + x ) m + n = ( 1 + x ) m ( 1 + x ) n = ( i = 0 m ( m i ) x i ) ( j = 0 n ( n j ) x j ) = r = 0 m + n ( k = 0 r ( m k ) ( n r - k ) ) x r , \begin{aligned}\displaystyle\sum_{r=0}^{m+n}{m+n\choose r}x^{r}&\displaystyle=% (1+x)^{m+n}\\ &\displaystyle=(1+x)^{m}(1+x)^{n}\\ &\displaystyle=\biggl(\sum_{i=0}^{m}{m\choose i}x^{i}\biggr)\biggl(\sum_{j=0}^% {n}{n\choose j}x^{j}\biggr)\\ &\displaystyle=\sum_{r=0}^{m+n}\biggl(\sum_{k=0}^{r}{m\choose k}{n\choose r-k}% \biggr)x^{r},\end{aligned}
  5. ( m + n r ) . {m+n\choose r}.
  6. k = 0 r ( m k ) ( n r - k ) . \sum_{k=0}^{r}{m\choose k}{n\choose r-k}.
  7. ( r + ( m + n - r ) r ) = ( m + n r ) {\left({{r+(m+n-r)}\atop{r}}\right)}={\left({{m+n}\atop{r}}\right)}
  8. ( m k ) {\left({{m}\atop{k}}\right)}
  9. ( n r - k ) {\left({{n}\atop{r-k}}\right)}
  10. ( m k ) ( n r - k ) {\left({{m}\atop{k}}\right)}{\left({{n}\atop{r-k}}\right)}
  11. k 1 + + k y = 0 x ( n k 1 ) ( n k 2 ) ( n k 3 ) ( n x - j = 1 y k j ) = ( ( y + 1 ) n x ) . \sum_{k_{1}+\dots+k_{y}=0}^{x}{n\choose k_{1}}{n\choose k_{2}}{n\choose k_{3}}% \cdots{n\choose x-\sum_{j=1}^{y}k_{j}}={\left(y+1\right)n\choose x}.
  12. ( s + t n ) = k = 0 n ( s k ) ( t n - k ) {s+t\choose n}=\sum_{k=0}^{n}{s\choose k}{t\choose n-k}
  13. ( s + t ) n = k = 0 n ( n k ) ( s ) k ( t ) n - k (s+t)_{n}=\sum_{k=0}^{n}{n\choose k}(s)_{k}(t)_{n-k}
  14. F 1 2 ( a , b ; c ; 1 ) = Γ ( c ) Γ ( c - a - b ) Γ ( c - a ) Γ ( c - b ) \;{}_{2}F_{1}(a,b;c;1)=\frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b)}
  15. F 1 2 \;{}_{2}F_{1}
  16. Γ ( n + 1 ) = n ! \Gamma(n+1)=n!
  17. ( n k ) = ( - 1 ) k ( k - n - 1 k ) {n\choose k}=(-1)^{k}{k-n-1\choose k}

Varāhamihira.html

  1. sin 2 x + cos 2 x = 1 \sin^{2}x+\cos^{2}x=1\;\!
  2. sin x = cos ( π 2 - x ) \sin x=\cos\left(\frac{\pi}{2}-x\right)
  3. 1 - cos 2 x 2 = sin 2 x \frac{1-\cos 2x}{2}=\sin^{2}x

Variation_of_parameters.html

  1. y ( n ) ( x ) + i = 0 n - 1 a i ( x ) y ( i ) ( x ) = b ( x ) . ( i ) y^{(n)}(x)+\sum_{i=0}^{n-1}a_{i}(x)y^{(i)}(x)=b(x).\quad\quad{\rm(i)}
  2. y 1 ( x ) , , y n ( x ) y_{1}(x),\ldots,y_{n}(x)
  3. y ( n ) ( x ) + i = 0 n - 1 a i ( x ) y ( i ) ( x ) = 0. ( ii ) y^{(n)}(x)+\sum_{i=0}^{n-1}a_{i}(x)y^{(i)}(x)=0.\quad\quad{\rm(ii)}
  4. y p ( x ) = i = 1 n c i ( x ) y i ( x ) ( iii ) y_{p}(x)=\sum_{i=1}^{n}c_{i}(x)y_{i}(x)\quad\quad{\rm(iii)}
  5. c i ( x ) c_{i}(x)
  6. i = 1 n c i ( x ) y i ( j ) ( x ) = 0 , j = 0 , , n - 2. ( iv ) \sum_{i=1}^{n}c_{i}^{\prime}(x)y_{i}^{(j)}(x)=0\,\mathrm{,}\quad j=0,\ldots,n-% 2.\quad\quad{\rm(iv)}
  7. y p ( j ) ( x ) = i = 1 n c i ( x ) y i ( j ) ( x ) , j = 0 , , n - 1 . ( v ) y_{p}^{(j)}(x)=\sum_{i=1}^{n}c_{i}(x)y_{i}^{(j)}(x)\,\mathrm{,}\quad j=0,% \ldots,n-1\,\mathrm{.}\quad\quad{\rm(v)}
  8. y p ( n ) ( x ) = i = 1 n c i ( x ) y i ( n - 1 ) ( x ) + i = 1 n c i ( x ) y i ( n ) ( x ) . ( vi ) y_{p}^{(n)}(x)=\sum_{i=1}^{n}c_{i}^{\prime}(x)y_{i}^{(n-1)}(x)+\sum_{i=1}^{n}c% _{i}(x)y_{i}^{(n)}(x)\,\mathrm{.}\quad\quad{\rm(vi)}
  9. i = 1 n c i ( x ) y i ( n - 1 ) ( x ) = b ( x ) . ( vii ) \sum_{i=1}^{n}c_{i}^{\prime}(x)y_{i}^{(n-1)}(x)=b(x).\quad\quad{\rm(vii)}
  10. c i ( x ) = W i ( x ) W ( x ) , i = 1 , , n c_{i}^{\prime}(x)=\frac{W_{i}(x)}{W(x)},\,\quad i=1,\ldots,n
  11. W ( x ) W(x)
  12. W i ( x ) W_{i}(x)
  13. ( 0 , 0 , , b ( x ) ) . (0,0,\ldots,b(x)).
  14. i = 1 n y i ( x ) W i ( x ) W ( x ) d x . \sum_{i=1}^{n}y_{i}(x)\,\int\frac{W_{i}(x)}{W(x)}\ \mathrm{d}x.
  15. y + p ( x ) y = q ( x ) y^{\prime}+p(x)y=q(x)
  16. y + p ( x ) y = 0 y^{\prime}+p(x)y=0
  17. d d x y + p ( x ) y = 0 \frac{d}{dx}y+p(x)y=0
  18. d y d x = - p ( x ) y \frac{dy}{dx}=-p(x)y
  19. d y y = - p ( x ) d x , {dy\over y}=-{p(x)dx},
  20. 1 y d y = - p ( x ) d x \int\frac{1}{y}\,dy=-\int p(x)\,dx
  21. ln | y | = - p ( x ) d x + C 0 \ln|y|=-\int p(x)\,dx+C_{0}
  22. y = ± e - p ( x ) d x + C 0 = C e - p ( x ) d x y=\pm e^{-\int p(x)\,dx+C_{0}}=Ce^{-\int p(x)\,dx}
  23. y g = C e - p ( x ) d x y_{g}=Ce^{-\int p(x)\,dx}
  24. y + p ( x ) y = q ( x ) y^{\prime}+p(x)y=q(x)
  25. y p = C ( x ) e - p ( x ) d x y_{p}=C(x)e^{-\int p(x)\,dx}
  26. C ( x ) e - p ( x ) d x - C ( x ) p ( x ) e - p ( x ) d x + p ( x ) C ( x ) e - p ( x ) d x = q ( x ) C^{\prime}(x)e^{-\int p(x)\,dx}-C(x)p(x)e^{-\int p(x)\,dx}+p(x)C(x)e^{-\int p(% x)\,dx}=q(x)
  27. C ( x ) e - p ( x ) d x = q ( x ) C^{\prime}(x)e^{-\int p(x)\,dx}=q(x)
  28. C ( x ) = q ( x ) e p ( x ) d x C^{\prime}(x)=q(x)e^{\int p(x)\,dx}
  29. C ( x ) = q ( x ) e p ( x ) d x d x + C C(x)=\int q(x)e^{\int p(x)\,dx}\,dx+C
  30. y p = C e - p ( x ) d x q ( x ) e p ( x ) d x d x y_{p}=Ce^{-\int p(x)\,dx}\int q(x)e^{\int p(x)\,dx}\,dx
  31. y = y g + y p y=y_{g}+y_{p}
  32. y = C e - p ( x ) d x q ( x ) e p ( x ) d x d x + C e - p ( x ) d x y=Ce^{-\int p(x)\,dx}\int q(x)e^{\int p(x)\,dx}\,dx+Ce^{-\int p(x)\,dx}
  33. y = C e - p ( x ) d x ( q ( x ) e p ( x ) d x d x + 1 ) y=Ce^{-\int p(x)\,dx}(\int q(x)e^{\int p(x)\,dx}\,dx+1)
  34. y ′′ + 4 y + 4 y = cosh x . y^{\prime\prime}+4y^{\prime}+4y=\cosh{x}.\;\!
  35. y ′′ + 4 y + 4 y = 0. y^{\prime\prime}+4y^{\prime}+4y=0.\;\!
  36. λ 2 + 4 λ + 4 = ( λ + 2 ) 2 = 0 \lambda^{2}+4\lambda+4=(\lambda+2)^{2}=0\;\!
  37. λ = - 2. \lambda=-2.\;\!
  38. | e - 2 x x e - 2 x - 2 e - 2 x - e - 2 x ( 2 x - 1 ) | = - e - 2 x e - 2 x ( 2 x - 1 ) + 2 x e - 2 x e - 2 x \begin{vmatrix}e^{-2x}&xe^{-2x}\\ -2e^{-2x}&-e^{-2x}(2x-1)\\ \end{vmatrix}=-e^{-2x}e^{-2x}(2x-1)+2xe^{-2x}e^{-2x}
  39. = - e - 4 x ( 2 x - 1 ) + 2 x e - 4 x = ( - 2 x + 1 + 2 x ) e - 4 x = e - 4 x . =-e^{-4x}(2x-1)+2xe^{-4x}=(-2x+1+2x)e^{-4x}=e^{-4x}.\;\!
  40. A ( x ) = - 1 W u 2 ( x ) b ( x ) d x , B ( x ) = 1 W u 1 ( x ) b ( x ) d x A(x)=-\int{1\over W}u_{2}(x)b(x)\,\mathrm{d}x,\;B(x)=\int{1\over W}u_{1}(x)b(x% )\,\mathrm{d}x
  41. b ( x ) = cosh x b(x)=\cosh{x}
  42. A ( x ) = - 1 e - 4 x x e - 2 x cosh x d x = - x e 2 x cosh x d x = - 1 18 e x ( 9 ( x - 1 ) + e 2 x ( 3 x - 1 ) ) + C 1 A(x)=-\int{1\over e^{-4x}}xe^{-2x}\cosh{x}\,\mathrm{d}x=-\int xe^{2x}\cosh{x}% \,\mathrm{d}x=-{1\over 18}e^{x}(9(x-1)+e^{2x}(3x-1))+C_{1}
  43. B ( x ) = 1 e - 4 x e - 2 x cosh x d x = e 2 x cosh x d x = 1 6 e x ( 3 + e 2 x ) + C 2 B(x)=\int{1\over e^{-4x}}e^{-2x}\cosh{x}\,\mathrm{d}x=\int e^{2x}\cosh{x}\,% \mathrm{d}x={1\over 6}e^{x}(3+e^{2x})+C_{2}
  44. C 1 C_{1}
  45. C 2 C_{2}
  46. u ′′ + p ( x ) u + q ( x ) u = f ( x ) u^{\prime\prime}+p(x)u^{\prime}+q(x)u=f(x)\,
  47. L = D 2 + p ( x ) D + q ( x ) L=D^{2}+p(x)D+q(x)\,
  48. L u ( x ) = f ( x ) Lu(x)=f(x)
  49. u ( x ) u(x)
  50. L L
  51. f ( x ) f(x)
  52. u ′′ + p ( x ) u + q ( x ) u = 0 u^{\prime\prime}+p(x)u^{\prime}+q(x)u=0\,
  53. u G ( x ) u_{G}(x)
  54. u G ( x ) = A ( x ) u 1 ( x ) + B ( x ) u 2 ( x ) . u_{G}(x)=A(x)u_{1}(x)+B(x)u_{2}(x).\,
  55. A ( x ) A(x)
  56. B ( x ) B(x)
  57. u 1 ( x ) u_{1}(x)
  58. u 2 ( x ) u_{2}(x)
  59. A ( x ) A(x)
  60. B ( x ) B(x)
  61. L u G ( x ) = 0 Lu_{G}(x)=0
  62. A ( x ) u 1 ( x ) + B ( x ) u 2 ( x ) = 0. A^{\prime}(x)u_{1}(x)+B^{\prime}(x)u_{2}(x)=0.\,
  63. u G ( x ) = ( A ( x ) u 1 ( x ) + B ( x ) u 2 ( x ) ) = ( A ( x ) u 1 ( x ) ) + ( B ( x ) u 2 ( x ) ) u_{G}^{\prime}(x)=(A(x)u_{1}(x)+B(x)u_{2}(x))^{\prime}=(A(x)u_{1}(x))^{\prime}% +(B(x)u_{2}(x))^{\prime}\,
  64. = A ( x ) u 1 ( x ) + A ( x ) u 1 ( x ) + B ( x ) u 2 ( x ) + B ( x ) u 2 ( x ) =A^{\prime}(x)u_{1}(x)+A(x)u_{1}^{\prime}(x)+B^{\prime}(x)u_{2}(x)+B(x)u_{2}^{% \prime}(x)\,
  65. = A ( x ) u 1 ( x ) + B ( x ) u 2 ( x ) + A ( x ) u 1 ( x ) + B ( x ) u 2 ( x ) =A^{\prime}(x)u_{1}(x)+B^{\prime}(x)u_{2}(x)+A(x)u_{1}^{\prime}(x)+B(x)u_{2}^{% \prime}(x)\,
  66. u G ( x ) = A ( x ) u 1 ( x ) + B ( x ) u 2 ( x ) . u_{G}^{\prime}(x)=A(x)u_{1}^{\prime}(x)+B(x)u_{2}^{\prime}(x).\,
  67. u G ′′ ( x ) = A ( x ) u 1 ′′ ( x ) + B ( x ) u 2 ′′ ( x ) + A ( x ) u 1 ( x ) + B ( x ) u 2 ( x ) . u_{G}^{\prime\prime}(x)=A(x)u_{1}^{\prime\prime}(x)+B(x)u_{2}^{\prime\prime}(x% )+A^{\prime}(x)u_{1}^{\prime}(x)+B^{\prime}(x)u_{2}^{\prime}(x).\,
  68. L u G = A ( x ) L u 1 ( x ) + B ( x ) L u 2 ( x ) + A ( x ) u 1 ( x ) + B ( x ) u 2 ( x ) . Lu_{G}=A(x)Lu_{1}(x)+B(x)Lu_{2}(x)+A^{\prime}(x)u_{1}^{\prime}(x)+B^{\prime}(x% )u_{2}^{\prime}(x).\,
  69. L u G = A ( x ) u 1 ( x ) + B ( x ) u 2 ( x ) . Lu_{G}=A^{\prime}(x)u_{1}^{\prime}(x)+B^{\prime}(x)u_{2}^{\prime}(x).\,
  70. ( u 1 ( x ) u 2 ( x ) u 1 ( x ) u 2 ( x ) ) ( A ( x ) B ( x ) ) = ( 0 f ) . \begin{pmatrix}u_{1}(x)&u_{2}(x)\\ u_{1}^{\prime}(x)&u_{2}^{\prime}(x)\end{pmatrix}\begin{pmatrix}A^{\prime}(x)\\ B^{\prime}(x)\end{pmatrix}=\begin{pmatrix}0\\ f\end{pmatrix}.
  71. ( A ( x ) u 1 ( x ) + B ( x ) u 2 ( x ) A ( x ) u 1 ( x ) + B ( x ) u 2 ( x ) ) = ( 0 f ) . \begin{pmatrix}A^{\prime}(x)u_{1}(x)+B^{\prime}(x)u_{2}(x)\\ A^{\prime}(x)u_{1}^{\prime}(x)+B^{\prime}(x)u_{2}^{\prime}(x)\end{pmatrix}=% \begin{pmatrix}0\\ f\end{pmatrix}.
  72. A ( x ) u 1 ( x ) + B ( x ) u 2 ( x ) = 0 A^{\prime}(x)u_{1}(x)+B^{\prime}(x)u_{2}(x)=0\,
  73. A ( x ) u 1 ( x ) + B ( x ) u 2 ( x ) = L u G = f . A^{\prime}(x)u_{1}^{\prime}(x)+B^{\prime}(x)u_{2}^{\prime}(x)=Lu_{G}=f.\,
  74. ( u 1 ( x ) u 2 ( x ) u 1 ( x ) u 2 ( x ) ) ( A ( x ) B ( x ) ) = ( 0 f ) \begin{pmatrix}u_{1}(x)&u_{2}(x)\\ u_{1}^{\prime}(x)&u_{2}^{\prime}(x)\end{pmatrix}\begin{pmatrix}A^{\prime}(x)\\ B^{\prime}(x)\end{pmatrix}=\begin{pmatrix}0\\ f\end{pmatrix}
  75. ( A ( x ) B ( x ) ) = ( u 1 ( x ) u 2 ( x ) u 1 ( x ) u 2 ( x ) ) - 1 ( 0 f ) \begin{pmatrix}A^{\prime}(x)\\ B^{\prime}(x)\end{pmatrix}=\begin{pmatrix}u_{1}(x)&u_{2}(x)\\ u_{1}^{\prime}(x)&u_{2}^{\prime}(x)\end{pmatrix}^{-1}\begin{pmatrix}0\\ f\end{pmatrix}
  76. = 1 W ( u 2 ( x ) - u 2 ( x ) - u 1 ( x ) u 1 ( x ) ) ( 0 f ) , ={1\over W}\begin{pmatrix}u_{2}^{\prime}(x)&-u_{2}(x)\\ -u_{1}^{\prime}(x)&u_{1}(x)\end{pmatrix}\begin{pmatrix}0\\ f\end{pmatrix},
  77. A ( x ) = - 1 W u 2 ( x ) f ( x ) , B ( x ) = 1 W u 1 ( x ) f ( x ) A^{\prime}(x)=-{1\over W}u_{2}(x)f(x),\;B^{\prime}(x)={1\over W}u_{1}(x)f(x)
  78. A ( x ) = - 1 W u 2 ( x ) f ( x ) d x , B ( x ) = 1 W u 1 ( x ) f ( x ) d x . A(x)=-\int{1\over W}u_{2}(x)f(x)\,\mathrm{d}x,\;B(x)=\int{1\over W}u_{1}(x)f(x% )\,\mathrm{d}x.
  79. A ( x ) A(x)
  80. B ( x ) B(x)
  81. A ( x ) A(x)
  82. B ( x ) B(x)
  83. L u G ( x ) Lu_{G}(x)
  84. L L

Variety_(universal_algebra).html

  1. x ( y z ) = ( x y ) z . x(yz)=(xy)z.
  2. x ( y z ) = ( x y ) z x(yz)=(xy)z
  3. 1 x = x 1 = x 1x=x1=x
  4. x x - 1 = x - 1 x = 1. xx^{-1}=x^{-1}x=1.
  5. x y = y x , xy=yx,
  6. A A
  7. U : A S e t U\colon A\to{Set}
  8. K : A S e t 𝕋 K\colon A\to{Set}^{\mathbb{T}}
  9. S e t 𝕋 {Set}^{\mathbb{T}}
  10. S e t {Set}
  11. S e t {Set}

Vascular_resistance.html

  1. R = Δ P / Q R=\Delta P/Q
  2. 80 ( m e a n a r t e r i a l p r e s s u r e - m e a n r i g h t a t r i a l p r e s s u r e ) c a r d i a c o u t p u t \frac{80\cdot(mean\ arterial\ pressure-mean\ right\ atrial\ pressure)}{cardiac% \ output}
  3. 80 ( m e a n p u l m o n a r y a r t e r i a l p r e s s u r e - m e a n p u l m o n a r y a r t e r y w e d g e p r e s s u r e ) c a r d i a c o u t p u t \frac{80\cdot(mean\ pulmonary\ arterial\ pressure-mean\ pulmonary\ artery\ % wedge\ pressure)}{cardiac\ output}
  4. R = Δ P Q R=\frac{\Delta P}{Q}
  5. R = 8 L η / ( π r 4 ) R=8L\eta/(\pi r^{4})

Vascular_smooth_muscle.html

  1. α 1 \alpha_{1}
  2. α 2 \alpha_{2}
  3. β 2 \beta_{2}
  4. α 1 \alpha_{1}
  5. α 1 \alpha_{1}
  6. α 1 \alpha_{1}
  7. α 1 \alpha_{1}
  8. α 2 \alpha_{2}
  9. α 2 \alpha_{2}
  10. α 2 \alpha_{2}
  11. α 2 \alpha_{2}
  12. α 2 \alpha_{2}
  13. β 2 \beta_{2}
  14. β 2 \beta_{2}
  15. α 1 \alpha_{1}
  16. α 2 \alpha_{2}
  17. β 2 \beta_{2}

Vector_fields_in_cylindrical_and_spherical_coordinates.html

  1. θ \theta
  2. ϕ \phi
  3. [ r θ z ] = [ x 2 + y 2 arctan ( y / x ) z ] , 0 θ < 2 π , \begin{bmatrix}r\\ \theta\\ z\end{bmatrix}=\begin{bmatrix}\sqrt{x^{2}+y^{2}}\\ \operatorname{arctan}(y/x)\\ z\end{bmatrix},\ \ \ 0\leq\theta<2\pi,
  4. [ x y z ] = [ r cos θ r sin θ z ] . \begin{bmatrix}x\\ y\\ z\end{bmatrix}=\begin{bmatrix}r\cos\theta\\ r\sin\theta\\ z\end{bmatrix}.
  5. 𝐀 = A x 𝐱 ^ + A y 𝐲 ^ + A z 𝐳 ^ = A r 𝐫 ^ + A θ s y m b o l θ ^ + A z 𝐳 ^ \mathbf{A}=A_{x}\mathbf{\hat{x}}+A_{y}\mathbf{\hat{y}}+A_{z}\mathbf{\hat{z}}=A% _{r}\mathbf{\hat{r}}+A_{\theta}symbol{\hat{\theta}}+A_{z}\mathbf{\hat{z}}
  6. [ 𝐫 ^ s y m b o l θ ^ 𝐳 ^ ] = [ cos θ sin θ 0 - sin θ cos θ 0 0 0 1 ] [ 𝐱 ^ 𝐲 ^ 𝐳 ^ ] \begin{bmatrix}\mathbf{\hat{r}}\\ symbol{\hat{\theta}}\\ \mathbf{\hat{z}}\end{bmatrix}=\begin{bmatrix}\cos\theta&\sin\theta&0\\ -\sin\theta&\cos\theta&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}\mathbf{\hat{x}}\\ \mathbf{\hat{y}}\\ \mathbf{\hat{z}}\end{bmatrix}
  7. 𝐀 ˙ \dot{\mathbf{A}}
  8. 𝐀 ˙ = A ˙ x 𝐱 ^ + A ˙ y 𝐲 ^ + A ˙ z 𝐳 ^ \dot{\mathbf{A}}=\dot{A}_{x}\hat{\mathbf{x}}+\dot{A}_{y}\hat{\mathbf{y}}+\dot{% A}_{z}\hat{\mathbf{z}}
  9. 𝐀 ˙ = A ˙ r s y m b o l r ^ + A r s y m b o l r ^ ˙ + A ˙ θ s y m b o l θ ^ + A θ s y m b o l θ ^ ˙ + A ˙ z s y m b o l z ^ + A z s y m b o l z ^ ˙ \dot{\mathbf{A}}=\dot{A}_{r}\hat{symbol{r}}+A_{r}\dot{\hat{symbol{r}}}+\dot{A}% _{\theta}\hat{symbol{\theta}}+A_{\theta}\dot{\hat{symbol{\theta}}}+\dot{A}_{z}% \hat{symbol{z}}+A_{z}\dot{\hat{symbol{z}}}
  10. 𝐫 ^ ˙ \displaystyle\dot{\hat{\mathbf{r}}}
  11. 𝐀 ˙ = s y m b o l r ^ ( A ˙ r - A θ θ ˙ ) + s y m b o l θ ^ ( A ˙ θ + A r θ ˙ ) + 𝐳 ^ A ˙ z \dot{\mathbf{A}}=\hat{symbol{r}}(\dot{A}_{r}-A_{\theta}\dot{\theta})+\hat{% symbol{\theta}}(\dot{A}_{\theta}+A_{r}\dot{\theta})+\hat{\mathbf{z}}\dot{A}_{z}
  12. 𝐀 ¨ = 𝐫 ^ ( A ¨ r - A θ θ ¨ - 2 A ˙ θ θ ˙ - A r θ ˙ 2 ) + s y m b o l θ ^ ( A ¨ θ + A r θ ¨ + 2 A ˙ r θ ˙ - A θ θ ˙ 2 ) + 𝐳 ^ A ¨ z \mathbf{\ddot{A}}=\mathbf{\hat{r}}(\ddot{A}_{r}-A_{\theta}\ddot{\theta}-2\dot{% A}_{\theta}\dot{\theta}-A_{r}\dot{\theta}^{2})+symbol{\hat{\theta}}(\ddot{A}_{% \theta}+A_{r}\ddot{\theta}+2\dot{A}_{r}\dot{\theta}-A_{\theta}\dot{\theta}^{2}% )+\mathbf{\hat{z}}\ddot{A}_{z}
  13. 𝐀 = 𝐏 = r 𝐫 ^ + z 𝐳 ^ \mathbf{A}=\mathbf{P}=r\mathbf{\hat{r}}+z\mathbf{\hat{z}}
  14. 𝐏 ¨ = 𝐫 ^ ( r ¨ - r θ ˙ 2 ) + s y m b o l θ ^ ( r θ ¨ + 2 r ˙ θ ˙ ) + 𝐳 ^ z ¨ \ddot{\mathbf{P}}=\mathbf{\hat{r}}(\ddot{r}-r\dot{\theta}^{2})+symbol{\hat{% \theta}}(r\ddot{\theta}+2\dot{r}\dot{\theta})+\mathbf{\hat{z}}\ddot{z}
  15. r ¨ 𝐫 ^ \displaystyle\ddot{r}\mathbf{\hat{r}}
  16. [ ρ θ ϕ ] = [ x 2 + y 2 + z 2 arccos ( z / ρ ) arctan ( y / x ) ] , 0 θ π , 0 ϕ < 2 π , \begin{bmatrix}\rho\\ \theta\\ \phi\end{bmatrix}=\begin{bmatrix}\sqrt{x^{2}+y^{2}+z^{2}}\\ \arccos(z/\rho)\\ \arctan(y/x)\end{bmatrix},\ \ \ 0\leq\theta\leq\pi,\ \ \ 0\leq\phi<2\pi,
  17. [ x y z ] = [ ρ sin θ cos ϕ ρ sin θ sin ϕ ρ cos θ ] . \begin{bmatrix}x\\ y\\ z\end{bmatrix}=\begin{bmatrix}\rho\sin\theta\cos\phi\\ \rho\sin\theta\sin\phi\\ \rho\cos\theta\end{bmatrix}.
  18. 𝐀 = A x 𝐱 ^ + A y 𝐲 ^ + A z 𝐳 ^ = A \rhosymbol ρ ^ + A \thetasymbol θ ^ + A \phisymbol ϕ ^ \mathbf{A}=A_{x}\mathbf{\hat{x}}+A_{y}\mathbf{\hat{y}}+A_{z}\mathbf{\hat{z}}=A% _{\rhosymbol}{\hat{\rho}}+A_{\thetasymbol}{\hat{\theta}}+A_{\phisymbol}{\hat{% \phi}}
  19. [ s y m b o l ρ ^ s y m b o l θ ^ s y m b o l ϕ ^ ] = [ sin θ cos ϕ sin θ sin ϕ cos θ cos θ cos ϕ cos θ sin ϕ - sin θ - sin ϕ cos ϕ 0 ] [ 𝐱 ^ 𝐲 ^ 𝐳 ^ ] \begin{bmatrix}symbol{\hat{\rho}}\\ symbol{\hat{\theta}}\\ symbol{\hat{\phi}}\end{bmatrix}=\begin{bmatrix}\sin\theta\cos\phi&\sin\theta% \sin\phi&\cos\theta\\ \cos\theta\cos\phi&\cos\theta\sin\phi&-\sin\theta\\ -\sin\phi&\cos\phi&0\end{bmatrix}\begin{bmatrix}\mathbf{\hat{x}}\\ \mathbf{\hat{y}}\\ \mathbf{\hat{z}}\end{bmatrix}
  20. 𝐀 ˙ = A ˙ x 𝐱 ^ + A ˙ y 𝐲 ^ + A ˙ z 𝐳 ^ \mathbf{\dot{A}}=\dot{A}_{x}\mathbf{\hat{x}}+\dot{A}_{y}\mathbf{\hat{y}}+\dot{% A}_{z}\mathbf{\hat{z}}
  21. 𝐀 ˙ = A ˙ ρ s y m b o l ρ ^ + A ρ s y m b o l ρ ^ ˙ + A ˙ θ s y m b o l θ ^ + A θ s y m b o l θ ^ ˙ + A ˙ ϕ s y m b o l ϕ ^ + A ϕ s y m b o l ϕ ^ ˙ \mathbf{\dot{A}}=\dot{A}_{\rho}symbol{\hat{\rho}}+A_{\rho}symbol{\dot{\hat{% \rho}}}+\dot{A}_{\theta}symbol{\hat{\theta}}+A_{\theta}symbol{\dot{\hat{\theta% }}}+\dot{A}_{\phi}symbol{\hat{\phi}}+A_{\phi}symbol{\dot{\hat{\phi}}}
  22. s y m b o l ρ ^ ˙ \displaystyle symbol{\dot{\hat{\rho}}}
  23. 𝐀 ˙ = s y m b o l ρ ^ ( A ˙ ρ - A θ θ ˙ - A ϕ ϕ ˙ sin θ ) + s y m b o l θ ^ ( A ˙ θ + A ρ θ ˙ - A ϕ ϕ ˙ cos θ ) + s y m b o l ϕ ^ ( A ˙ ϕ + A ρ ϕ ˙ sin θ + A θ ϕ ˙ cos θ ) \mathbf{\dot{A}}=symbol{\hat{\rho}}(\dot{A}_{\rho}-A_{\theta}\dot{\theta}-A_{% \phi}\dot{\phi}\sin\theta)+symbol{\hat{\theta}}(\dot{A}_{\theta}+A_{\rho}\dot{% \theta}-A_{\phi}\dot{\phi}\cos\theta)+symbol{\hat{\phi}}(\dot{A}_{\phi}+A_{% \rho}\dot{\phi}\sin\theta+A_{\theta}\dot{\phi}\cos\theta)

Velocity_factor.html

  1. v P v_{\mathrm{P}}
  2. κ \kappa
  3. VF = 1 κ \mathrm{VF}={\frac{1}{\sqrt{\kappa}}}
  4. VF = 1 c L C \mathrm{VF}={\frac{1}{c\sqrt{LC}}}

Verlet_integration.html

  1. x ¨ ( t ) = A ( x ( t ) ) \ddot{\vec{x}}(t)=\vec{A}(\vec{x}(t))
  2. x ( t 0 ) = x 0 \vec{x}(t_{0})=\vec{x}_{0}
  3. x ˙ ( t 0 ) = v 0 \dot{\vec{x}}(t_{0})=\vec{v}_{0}
  4. x n x ( t n ) \vec{x}_{n}\approx\vec{x}(t_{n})
  5. t n = t 0 + n Δ t t_{n}=t_{0}+n\,\Delta t
  6. Δ t > 0 \Delta t>0
  7. x 1 = x 0 + v 0 Δ t + 1 2 A ( x 0 ) Δ t 2 \vec{x}_{1}=\vec{x}_{0}+\vec{v}_{0}\,\Delta t+\frac{1}{2}A(\vec{x}_{0})\,% \Delta t^{2}
  8. x n + 1 = 2 x n - x n - 1 + A ( x n ) Δ t 2 . \vec{x}_{n+1}=2\vec{x}_{n}-\vec{x}_{n-1}+A(\vec{x}_{n})\,\Delta t^{2}.
  9. M x ¨ ( t ) = F ( x ( t ) ) = - V ( x ( t ) ) M\ddot{\vec{x}}(t)=F(\vec{x}(t))=-\nabla V(\vec{x}(t))
  10. m k x ¨ k ( t ) = F k ( x ( t ) ) = - x k V ( x ( t ) ) m_{k}\ddot{\vec{x}}_{k}(t)=F_{k}(\vec{x}(t))=-\nabla_{\vec{x}_{k}}V(\vec{x}(t))
  11. x ( t ) = ( x 1 ( t ) , , x N ( t ) ) \vec{x}(t)=(\vec{x}_{1}(t),\ldots,\vec{x}_{N}(t))
  12. m k m_{k}
  13. x ¨ ( t ) = A ( x ( t ) ) \ddot{\vec{x}}(t)=A(\vec{x}(t))
  14. x ( 0 ) = x 0 \vec{x}(0)=\vec{x}_{0}
  15. v ( 0 ) = x ˙ ( 0 ) = v 0 \vec{v}(0)=\dot{\vec{x}}(0)=\vec{v}_{0}
  16. Δ t > 0 \Delta t>0
  17. t n = n Δ t t_{n}=n\Delta t
  18. x n \vec{x}_{n}
  19. x ( t n ) \vec{x}(t_{n})
  20. Δ 2 x n Δ t 2 = x n + 1 - x n Δ t - x n - x n - 1 Δ t Δ t = x n + 1 - 2 x n + x n - 1 Δ t 2 = a n = A ( x n ) \frac{\Delta^{2}\vec{x}_{n}}{\Delta t^{2}}=\frac{\frac{\vec{x}_{n+1}-\vec{x}_{% n}}{\Delta t}-\frac{\vec{x}_{n}-\vec{x}_{n-1}}{\Delta t}}{\Delta t}=\frac{\vec% {x}_{n+1}-2\vec{x}_{n}+\vec{x}_{n-1}}{\Delta t^{2}}=\vec{a}_{n}=A(\vec{x}_{n})
  21. x n + 1 = 2 x n - x n - 1 + a n Δ t 2 , a n = A ( x n ) . \vec{x}_{n+1}=2\vec{x}_{n}-\vec{x}_{n-1}+\vec{a}_{n}\,\Delta t^{2},\qquad\vec{% a}_{n}=A(\vec{x}_{n}).
  22. Δ t \Delta t
  23. x ( t n - 1 ) , x ( t n ) , x ( t n + 1 ) \vec{x}(t_{n-1}),\vec{x}(t_{n}),\vec{x}(t_{n+1})
  24. t = t n t=t_{n}
  25. x ( t ± Δ t ) \vec{x}(t\pm\Delta t)
  26. x ( t + Δ t ) = x ( t ) + v ( t ) Δ t + a ( t ) Δ t 2 2 + b ( t ) Δ t 3 6 + 𝒪 ( Δ t 4 ) x ( t - Δ t ) = x ( t ) - v ( t ) Δ t + a ( t ) Δ t 2 2 - b ( t ) Δ t 3 6 + 𝒪 ( Δ t 4 ) , \begin{aligned}\displaystyle\vec{x}(t+\Delta t)&\displaystyle=\vec{x}(t)+\vec{% v}(t)\Delta t+\frac{\vec{a}(t)\Delta t^{2}}{2}+\frac{\vec{b}(t)\Delta t^{3}}{6% }+\mathcal{O}(\Delta t^{4})\\ \displaystyle\vec{x}(t-\Delta t)&\displaystyle=\vec{x}(t)-\vec{v}(t)\Delta t+% \frac{\vec{a}(t)\Delta t^{2}}{2}-\frac{\vec{b}(t)\Delta t^{3}}{6}+\mathcal{O}(% \Delta t^{4}),\end{aligned}
  27. x \vec{x}
  28. v = x ˙ \vec{v}=\dot{\vec{x}}
  29. a = x ¨ \vec{a}=\ddot{\vec{x}}
  30. b \vec{b}
  31. t t
  32. x ( t + Δ t ) = 2 x ( t ) - x ( t - Δ t ) + a ( t ) Δ t 2 + 𝒪 ( Δ t 4 ) . \vec{x}(t+\Delta t)=2\vec{x}(t)-\vec{x}(t-\Delta t)+\vec{a}(t)\Delta t^{2}+% \mathcal{O}(\Delta t^{4}).\,
  33. a ( t ) = A ( x ( t ) ) \vec{a}(t)=A(\vec{x}(t))
  34. a n = A ( x n ) \vec{a}_{n}=A(\vec{x}_{n})
  35. x ¨ ( t ) = w 2 x ( t ) \ddot{x}(t)=w^{2}x(t)
  36. e w t e^{wt}
  37. e - w t e^{-wt}
  38. x n + 1 - 2 x n + x n - 1 = h 2 w 2 x n x n + 1 - 2 ( 1 + 1 2 ( w h ) 2 ) x n + x n - 1 = 0. \begin{aligned}\displaystyle x_{n+1}-2x_{n}+x_{n-1}&\displaystyle=h^{2}w^{2}x_% {n}\\ \displaystyle\iff\quad x_{n+1}-2(1+\tfrac{1}{2}(wh)^{2})x_{n}+x_{n-1}&% \displaystyle=0.\end{aligned}
  39. q 2 - 2 ( 1 + 1 2 ( w h ) 2 ) q + 1 = 0 q^{2}-2(1+\tfrac{1}{2}(wh)^{2})q+1=0
  40. q ± = 1 + 1 2 ( w h ) 2 ± w h 1 + 1 4 ( w h ) 2 q_{\pm}=1+\tfrac{1}{2}(wh)^{2}\pm wh\sqrt{1+\tfrac{1}{4}(wh)^{2}}
  41. x n = q + n x_{n}=q_{+}^{\;n}
  42. x n = q - n x_{n}=q_{-}^{\;n}
  43. q + = 1 + 1 2 ( w h ) 2 + w h ( 1 + 1 8 ( w h ) 2 - 3 128 ( w h ) 4 + 𝒪 ( h 6 ) ) = 1 + ( w h ) + 1 2 ( w h ) 2 + 1 8 ( w h ) 3 - 3 128 ( w h ) 5 + 𝒪 ( h 7 ) . \begin{aligned}\displaystyle q_{+}&\displaystyle=1+\tfrac{1}{2}(wh)^{2}+wh(1+% \tfrac{1}{8}(wh)^{2}-\tfrac{3}{128}(wh)^{4}+\mathcal{O}(h^{6}))\\ &\displaystyle=1+(wh)+\tfrac{1}{2}(wh)^{2}+\tfrac{1}{8}(wh)^{3}-\tfrac{3}{128}% (wh)^{5}+\mathcal{O}(h^{7}).\end{aligned}
  44. e w h e^{wh}
  45. 1 - 1 24 ( w h ) 3 + 𝒪 ( h 5 ) 1-\tfrac{1}{24}(wh)^{3}+\mathcal{O}(h^{5})
  46. q + = ( 1 - 1 24 ( w h ) 3 + 𝒪 ( h 5 ) ) e w h = e - 1 24 ( w h ) 3 + 𝒪 ( h 5 ) e w h . \begin{aligned}\displaystyle q_{+}&\displaystyle=(1-\tfrac{1}{24}(wh)^{3}+% \mathcal{O}(h^{5}))e^{wh}\\ &\displaystyle=e^{-\frac{1}{24}(wh)^{3}+\mathcal{O}(h^{5})}\,e^{wh}.\end{aligned}
  47. x n = q + n = e - 1 24 ( w h ) 2 w t n + 𝒪 ( h 4 ) e w t n = e w t n ( 1 - 1 24 ( w h ) 2 w t n + 𝒪 ( h 4 ) ) = e w t n + 𝒪 ( h 2 t n e w t n ) . \begin{aligned}\displaystyle x_{n}=q_{+}^{\;n}&\displaystyle=e^{-\frac{1}{24}(% wh)^{2}\,wt_{n}+\mathcal{O}(h^{4})}\,e^{wt_{n}}\\ &\displaystyle=e^{wt_{n}}\left(1-\tfrac{1}{24}(wh)^{2}\,wt_{n}+\mathcal{O}(h^{% 4})\right)\\ &\displaystyle=e^{wt_{n}}+\mathcal{O}(h^{2}t_{n}e^{wt_{n}}).\end{aligned}
  48. n = 1 n=1
  49. t = t 1 = Δ t t=t_{1}=\Delta t
  50. x 2 \vec{x}_{2}
  51. x 1 \vec{x}_{1}
  52. t = t 1 t=t_{1}
  53. t 0 = 0 t_{0}=0
  54. a 0 = A ( x 0 ) \vec{a}_{0}=A(\vec{x}_{0})
  55. x 1 = x 0 + v 0 Δ t + 1 2 a 0 Δ t 2 x ( Δ t ) + 𝒪 ( Δ t 3 ) . \vec{x}_{1}=\vec{x}_{0}+\vec{v}_{0}\Delta t+\tfrac{1}{2}\vec{a}_{0}\Delta t^{2% }\approx\vec{x}(\Delta t)+\mathcal{O}(\Delta t^{3}).\,
  56. 𝒪 ( Δ t 3 ) \mathcal{O}(\Delta t^{3})
  57. t n t_{n}
  58. 𝒪 ( e L t n Δ t 2 ) \mathcal{O}(e^{Lt_{n}}\Delta t^{2})
  59. x n \vec{x}_{n}
  60. x ( t n ) \vec{x}(t_{n})
  61. x n + 1 - x n Δ t \tfrac{\vec{x}_{n+1}-\vec{x}_{n}}{\Delta t}
  62. x ( t n + 1 ) - x ( t n ) Δ t \tfrac{\vec{x}(t_{n+1})-\vec{x}(t_{n})}{\Delta t}
  63. Δ t \Delta t
  64. x i + 1 = x i + ( x i - x i - 1 ) ( Δ t i / Δ t i - 1 ) + a Δ t i 2 \vec{x}_{i+1}=\vec{x}_{i}+(\vec{x}_{i}-\vec{x}_{i-1})(\Delta t_{i}/\Delta t_{i% -1})+\vec{a}\Delta t_{i}^{2}
  65. t i t_{i}
  66. t i + 1 = t i + Δ t i t_{i+1}=t_{i}+\Delta t_{i}
  67. t i - 1 = t i - Δ t i - 1 t_{i-1}=t_{i}-\Delta t_{i-1}
  68. v i \vec{v}_{i}
  69. x i + 1 - x i Δ t i + x i - 1 - x i Δ t i - 1 = a i Δ t i + Δ t i - 1 2 \frac{\vec{x}_{i+1}-\vec{x}_{i}}{\Delta t_{i}}+\frac{\vec{x}_{i-1}-\vec{x}_{i}% }{\Delta t_{i-1}}=\vec{a}_{i}\,\frac{\Delta t_{i}+\Delta t_{i-1}}{2}
  70. x i + 1 = x i + ( x i - x i - 1 ) Δ t i Δ t i - 1 + a i Δ t i + Δ t i - 1 2 Δ t i \vec{x}_{i+1}=\vec{x}_{i}+(\vec{x}_{i}-\vec{x}_{i-1})\frac{\Delta t_{i}}{% \Delta t_{i-1}}+\vec{a}_{i}\,\frac{\Delta t_{i}+\Delta t_{i-1}}{2}\,\Delta t_{i}
  71. t t
  72. t + Δ t t+\Delta t
  73. v ( t ) = x ( t + Δ t ) - x ( t - Δ t ) 2 Δ t + 𝒪 ( Δ t 2 ) . \vec{v}(t)=\frac{\vec{x}(t+\Delta t)-\vec{x}(t-\Delta t)}{2\Delta t}+\mathcal{% O}(\Delta t^{2}).
  74. t t
  75. t + Δ t t+\Delta t
  76. v n = x n + 1 - x n - 1 2 Δ t \vec{v}_{n}=\tfrac{\vec{x}_{n+1}-\vec{x}_{n-1}}{2\Delta t}
  77. v ( t n ) \vec{v}(t_{n})
  78. v n + 1 / 2 = x n + 1 - x n Δ t \vec{v}_{n+1/2}=\tfrac{\vec{x}_{n+1}-\vec{x}_{n}}{\Delta t}
  79. v ( t n + 1 / 2 ) \vec{v}(t_{n+1/2})
  80. t n + 1 / 2 = t n + 1 2 Δ t t_{n+1/2}=t_{n}+\tfrac{1}{2}\Delta t
  81. t + Δ t t+\Delta t
  82. v ( t + Δ t ) = x ( t + Δ t ) - x ( t ) Δ t + 𝒪 ( Δ t ) . \vec{v}(t+\Delta t)=\frac{\vec{x}(t+\Delta t)-\vec{x}(t)}{\Delta t}+\mathcal{O% }(\Delta t).
  83. x ( t + Δ t ) = x ( t ) + v ( t ) Δ t + 1 2 a ( t ) Δ t 2 \vec{x}(t+\Delta t)=\vec{x}(t)+\vec{v}(t)\,\Delta t+\frac{1}{2}\,\vec{a}(t)% \Delta t^{2}\,
  84. v ( t + Δ t ) = v ( t ) + a ( t ) + a ( t + Δ t ) 2 Δ t \vec{v}(t+\Delta t)=\vec{v}(t)+\frac{\vec{a}(t)+\vec{a}(t+\Delta t)}{2}\Delta t\,
  85. v ( t + 1 2 Δ t ) = v ( t ) + 1 2 a ( t ) Δ t \vec{v}\left(t+\tfrac{1}{2}\,\Delta t\right)=\vec{v}(t)+\tfrac{1}{2}\,\vec{a}(% t)\,\Delta t\,
  86. x ( t + Δ t ) = x ( t ) + v ( t + 1 2 Δ t ) Δ t \vec{x}(t+\Delta t)=\vec{x}(t)+\vec{v}\left(t+\tfrac{1}{2}\,\Delta t\right)\,% \Delta t\,
  87. a ( t + Δ t ) \vec{a}(t+\Delta t)
  88. x ( t + Δ t ) \vec{x}(t+\Delta t)
  89. v ( t + Δ t ) = v ( t + 1 2 Δ t ) + 1 2 a ( t + Δ t ) Δ t , \vec{v}(t+\Delta t)=\vec{v}\left(t+\tfrac{1}{2}\,\Delta t\right)+\tfrac{1}{2}% \,\vec{a}(t+\Delta t)\Delta t,
  90. x ( t + Δ t ) = x ( t ) + v ( t ) Δ t + 1 2 a ( t ) Δ t 2 \vec{x}(t+\Delta t)=\vec{x}(t)+\vec{v}(t)\,\Delta t+\tfrac{1}{2}\,\vec{a}(t)\,% \Delta t^{2}
  91. a ( t + Δ t ) \vec{a}(t+\Delta t)
  92. x ( t + Δ t ) \vec{x}(t+\Delta t)
  93. v ( t + Δ t ) = v ( t ) + 1 2 ( a ( t ) + a ( t + Δ t ) ) Δ t \vec{v}(t+\Delta t)=\vec{v}(t)+\tfrac{1}{2}\,\left(\vec{a}(t)+\vec{a}(t+\Delta t% )\right)\Delta t\,
  94. a ( t + Δ t ) \vec{a}(t+\Delta t)
  95. x ( t + Δ t ) \vec{x}(t+\Delta t)
  96. v ( t + Δ t ) \vec{v}(t+\Delta t)
  97. β = 0 \beta=0
  98. γ = 1 / 2 \gamma=1/2
  99. O ( Δ t 4 ) O(\Delta t^{4})
  100. O ( Δ t 2 ) O(\Delta t^{2})
  101. O ( Δ t 2 ) O(\Delta t^{2})
  102. O ( Δ t 2 ) O(\Delta t^{2})
  103. error ( x ( t 0 + Δ t ) ) = O ( Δ t 4 ) \mathrm{error}\bigl(x(t_{0}+\Delta t)\bigr)=O(\Delta t^{4})
  104. x ( t 0 + 2 Δ t ) = 2 x ( t 0 + Δ t ) - x ( t 0 ) + Δ t 2 x ′′ ( t 0 + Δ t ) + O ( Δ t 4 ) x(t_{0}+2\Delta t)=2x(t_{0}+\Delta t)-x(t_{0})+\Delta t^{2}x^{\prime\prime}(t_% {0}+\Delta t)+O(\Delta t^{4})\,
  105. error ( x ( t 0 + 2 Δ t ) ) = 2 e r r o r ( x ( t 0 + Δ t ) ) + O ( Δ t 4 ) = 3 O ( Δ t 4 ) \mathrm{error}\bigl(x(t_{0}+2\Delta t)\bigr)=2\mathrm{error}\bigl(x(t_{0}+% \Delta t)\bigr)+O(\Delta t^{4})=3\,O(\Delta t^{4})
  106. error ( x ( t 0 + 3 Δ t ) ) = 6 O ( Δ t 4 ) \mathrm{error}\bigl(x(t_{0}+3\Delta t)\bigl)=6\,O(\Delta t^{4})
  107. error ( x ( t 0 + 4 Δ t ) ) = 10 O ( Δ t 4 ) \mathrm{error}\bigl(x(t_{0}+4\Delta t)\bigl)=10\,O(\Delta t^{4})
  108. error ( x ( t 0 + 5 Δ t ) ) = 15 O ( Δ t 4 ) \mathrm{error}\bigl(x(t_{0}+5\Delta t)\bigl)=15\,O(\Delta t^{4})
  109. error ( x ( t 0 + n Δ t ) ) = n ( n + 1 ) 2 O ( Δ t 4 ) \mathrm{error}\bigl(x(t_{0}+n\Delta t)\bigr)=\frac{n(n+1)}{2}\,O(\Delta t^{4})
  110. x ( t ) x(t)
  111. x ( t + T ) x(t+T)
  112. T = n Δ t T=n\Delta t
  113. error ( x ( t 0 + T ) ) = ( T 2 2 Δ t 2 + T 2 Δ t ) O ( Δ t 4 ) \mathrm{error}\bigl(x(t_{0}+T)\bigr)=\left(\frac{T^{2}}{2\Delta t^{2}}+\frac{T% }{2\Delta t}\right)O(\Delta t^{4})
  114. error ( x ( t 0 + T ) ) = O ( Δ t 2 ) \mathrm{error}\bigr(x(t_{0}+T)\bigl)=O(\Delta t^{2})
  115. O ( Δ t 2 ) O(\Delta t^{2})
  116. d 1 = x 2 ( t ) - x 1 ( t ) d_{1}=x_{2}^{(t)}-x_{1}^{(t)}\,
  117. d 2 = d 1 d_{2}=\|d_{1}\|\,
  118. d 3 = d 2 - r d 2 d_{3}=\frac{d_{2}-r}{d_{2}}\,
  119. x 1 ( t + Δ t ) = x ~ 1 ( t + Δ t ) + 1 2 d 1 d 3 x_{1}^{(t+\Delta t)}=\tilde{x}_{1}^{(t+\Delta t)}+\frac{1}{2}d_{1}d_{3}\,
  120. x 2 ( t + Δ t ) = x ~ 2 ( t + Δ t ) - 1 2 d 1 d 3 x_{2}^{(t+\Delta t)}=\tilde{x}_{2}^{(t+\Delta t)}-\frac{1}{2}d_{1}d_{3}\,
  121. x i ( t ) x_{i}^{(t)}
  122. x ~ i ( t ) \tilde{x}_{i}^{(t)}

Vertical_bar.html

  1. | x | |x|
  2. ( x 1 , x 2 ) \|(x_{1},x_{2})\|
  3. A B C D AB\parallel CD
  4. A B AB
  5. C D CD
  6. { x | x < 2 } \{x|x<2\}
  7. | S | |S|
  8. P ( X | Y ) P(X|Y)
  9. a | b a|b
  10. a | b a|b
  11. P | a b P|ab
  12. P P
  13. a b ab
  14. P | a b P|ab
  15. a b ab
  16. f ( x ) | x = 4 f(x)|_{x=4}
  17. f | A : A F f|_{A}:A\to F
  18. f f
  19. A A
  20. | A | |A|
  21. | a b c d e f g h i | \begin{vmatrix}a&b&c\\ d&e&f\\ g&h&i\end{vmatrix}
  22. | ψ |\psi\rangle
  23. ψ \psi
  24. ψ | \langle\psi|
  25. ψ | ρ \langle\psi|\rho\rangle
  26. ψ \psi
  27. ρ \rho
  28. a | b c a|b\|c
  29. a b c a\mid b\parallel c

Vibrating_string.html

  1. v v
  2. T T
  3. μ \mu
  4. v = T μ . v=\sqrt{T\over\mu}.
  5. Δ x \Delta x
  6. m m
  7. μ \mu
  8. T T
  9. T 1 x = T 1 cos ( α ) T . T_{1x}=T_{1}\cos(\alpha)\approx T.
  10. T 2 x = T 2 cos ( β ) T . T_{2x}=T_{2}\cos(\beta)\approx T.
  11. a a
  12. Σ F y = - T 2 y - T 1 y = - T 2 sin ( β ) - T 1 sin ( α ) = Δ m a μ Δ x 2 y t 2 . \Sigma F_{y}=-T_{2y}-T_{1y}=-T_{2}\sin(\beta)-T_{1}\sin(\alpha)=\Delta ma% \approx\mu\Delta x\frac{\partial^{2}y}{\partial t^{2}}.
  13. T T
  14. - μ Δ x T 2 y t 2 = T 2 sin ( β ) T 2 cos ( β ) + T 1 sin ( α ) T 1 cos ( α ) = tan ( β ) + tan ( α ) -\frac{\mu\Delta x}{T}\frac{\partial^{2}y}{\partial t^{2}}=\frac{T_{2}\sin(% \beta)}{T_{2}\cos(\beta)}+\frac{T_{1}\sin(\alpha)}{T_{1}\cos(\alpha)}=\tan(% \beta)+\tan(\alpha)
  15. 1 Δ x ( y x | x + Δ x - y x | x ) = μ T 2 y t 2 \frac{1}{\Delta x}\left(\left.\frac{\partial y}{\partial x}\right|^{x+\Delta x% }-\left.\frac{\partial y}{\partial x}\right|^{x}\right)=\frac{\mu}{T}\frac{% \partial^{2}y}{\partial t^{2}}
  16. Δ x \Delta x
  17. y y
  18. 2 y x 2 = μ T 2 y t 2 . \frac{\partial^{2}y}{\partial x^{2}}=\frac{\mu}{T}\frac{\partial^{2}y}{% \partial t^{2}}.
  19. y ( x , t ) y(x,t)
  20. v - 2 v^{-2}
  21. v = T μ , v=\sqrt{T\over\mu},
  22. v v
  23. Δ x \Delta x
  24. T T
  25. λ \lambda
  26. τ \tau
  27. f f
  28. v = λ τ = λ f . v=\frac{\lambda}{\tau}=\lambda f.
  29. L L
  30. L L
  31. f = v 2 L = 1 2 L T μ f=\frac{v}{2L}={1\over 2L}\sqrt{T\over\mu}
  32. T T
  33. μ \mu
  34. L L
  35. λ n = 2 L / n \lambda_{n}=2L/n
  36. f n = n v 2 L f_{n}=\frac{nv}{2L}
  37. μ \mu
  38. f n = n 2 L T μ f_{n}=\frac{n}{2L}\sqrt{\frac{T}{\mu}}

Vibrating_structure_gyroscope.html

  1. ω r \scriptstyle\omega_{r}
  2. a c = 2 ( v × Ω ) \scriptstyle a_{c}=2(v\times\Omega)
  3. v \scriptstyle v
  4. Ω \scriptstyle\Omega
  5. X i p ω r cos ( ω r t ) \scriptstyle X_{ip}\omega_{r}\cos(\omega_{r}t)
  6. X i p sin ( ω r t ) \scriptstyle X_{ip}\sin(\omega_{r}t)
  7. y o p \scriptstyle y_{op}
  8. y o p = F c k o p = 2 m Ω X i p ω r cos ( ω r t ) k o p y_{op}=\frac{F_{c}}{k_{op}}=\frac{2m\Omega X_{ip}\omega_{r}\cos(\omega_{r}t)}{% k_{op}}
  9. m \scriptstyle m
  10. k o p \scriptstyle k_{op}
  11. Ω \scriptstyle\Omega

Vickrey_auction.html

  1. v i v_{i}
  2. b i b_{i}
  3. { v i - max j i b j if b i > max j i b j 0 otherwise \begin{cases}v_{i}-\max_{j\neq i}b_{j}&\,\text{if }b_{i}>\max_{j\neq i}b_{j}\\ 0&\,\text{otherwise}\end{cases}
  4. b i > v i b_{i}>v_{i}
  5. max j i b j < v i \max_{j\neq i}b_{j}<v_{i}
  6. max j i b j > b i \max_{j\neq i}b_{j}>b_{i}
  7. v i < max j i b j < b i v_{i}<\max_{j\neq i}b_{j}<b_{i}
  8. b i < v i b_{i}<v_{i}
  9. max j i b j > v i \max_{j\neq i}b_{j}>v_{i}
  10. max j i b j < b i \max_{j\neq i}b_{j}<b_{i}
  11. b i < max j i b j < v i b_{i}<\max_{j\neq i}b_{j}<v_{i}
  12. B ( v ) = e ( v ) = 1 2 v B(v)=e(v)=\tfrac{1}{2}v
  13. e k \scriptstyle e_{k}
  14. p k = d k + M C F ( G - e k ) - M C F ( G ) p_{k}=d_{k}+MCF(G-e_{k})-MCF(G)
  15. G G ( n , p ) \scriptstyle G\in G(n,p)
  16. p 2 - p \frac{p}{2-p}
  17. \scriptstyle\infty
  18. n p = ω ( n log n ) np=\omega(\sqrt{n\log n})
  19. Ω ( 1 n p ) \Omega\left(\frac{1}{np}\right)
  20. O ( 1 ) O(1)\,
  21. n p = ω ( log n ) . np=\omega(\log n).\,

Vieta's_formulas.html

  1. P ( x ) = a n x n + a n - 1 x n - 1 + + a 1 x + a 0 P(x)=a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}\,
  2. { x 1 + x 2 + + x n - 1 + x n = - a n - 1 a n ( x 1 x 2 + x 1 x 3 + + x 1 x n ) + ( x 2 x 3 + x 2 x 4 + + x 2 x n ) + + x n - 1 x n = a n - 2 a n x 1 x 2 x n = ( - 1 ) n a 0 a n . \begin{cases}x_{1}+x_{2}+\dots+x_{n-1}+x_{n}=-\dfrac{a_{n-1}}{a_{n}}\\ (x_{1}x_{2}+x_{1}x_{3}+\cdots+x_{1}x_{n})+(x_{2}x_{3}+x_{2}x_{4}+\cdots+x_{2}x% _{n})+\cdots+x_{n-1}x_{n}=\dfrac{a_{n-2}}{a_{n}}\\ {}\quad\vdots\\ x_{1}x_{2}\dots x_{n}=(-1)^{n}\dfrac{a_{0}}{a_{n}}.\end{cases}
  3. 1 i 1 < i 2 < < i k n x i 1 x i 2 x i k = ( - 1 ) k a n - k a n \sum_{1\leq i_{1}<i_{2}<\cdots<i_{k}\leq n}x_{i_{1}}x_{i_{2}}\cdots x_{i_{k}}=% (-1)^{k}\frac{a_{n-k}}{a_{n}}
  4. a i / a n a_{i}/a_{n}
  5. a n a_{n}
  6. x i x_{i}
  7. a n a_{n}
  8. P ( x ) P(x)
  9. a n ( x - x 1 ) ( x - x 2 ) ( x - x n ) a_{n}(x-x_{1})(x-x_{2})\dots(x-x_{n})
  10. P ( x ) = x 2 - 1 P(x)=x^{2}-1
  11. x 1 = 1 x_{1}=1
  12. x 2 = 3 x_{2}=3
  13. P ( x ) ( x - 1 ) ( x - 3 ) P(x)\neq(x-1)(x-3)
  14. P ( x ) P(x)
  15. ( x - 1 ) ( x - 7 ) (x-1)(x-7)
  16. ( x - 3 ) ( x - 5 ) (x-3)(x-5)
  17. x 1 = 1 x_{1}=1
  18. x 2 = 7 x_{2}=7
  19. x 1 = 3 x_{1}=3
  20. x 2 = 5 x_{2}=5
  21. P ( x ) = a x 2 + b x + c P(x)=ax^{2}+bx+c
  22. x 1 , x 2 x_{1},x_{2}
  23. P ( x ) = 0 P(x)=0
  24. x 1 + x 2 = - b a , x 1 x 2 = c a . x_{1}+x_{2}=-\frac{b}{a},\quad x_{1}x_{2}=\frac{c}{a}.
  25. P ( x ) = a x 3 + b x 2 + c x + d P(x)=ax^{3}+bx^{2}+cx+d
  26. x 1 , x 2 , x 3 x_{1},x_{2},x_{3}
  27. P ( x ) = 0 P(x)=0
  28. x 1 + x 2 + x 3 = - b a , x 1 x 2 + x 1 x 3 + x 2 x 3 = c a , x 1 x 2 x 3 = - d a . x_{1}+x_{2}+x_{3}=-\frac{b}{a},\quad x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3}=\frac{c}% {a},\quad x_{1}x_{2}x_{3}=-\frac{d}{a}.
  29. a n x n + a n - 1 x n - 1 + + a 1 x + a 0 = a n ( x - x 1 ) ( x - x 2 ) ( x - x n ) a_{n}x^{n}+a_{n-1}x^{n-1}+\cdots+a_{1}x+a_{0}=a_{n}(x-x_{1})(x-x_{2})\cdots(x-% x_{n})
  30. x 1 , x 2 , , x n x_{1},x_{2},\dots,x_{n}
  31. x . x.
  32. ( x - x 1 ) ( x - x 2 ) ( x - x n ) , (x-x_{1})(x-x_{2})\cdots(x-x_{n}),
  33. ( - 1 ) n - k x 1 b 1 x n b n x k , (-1)^{n-k}x_{1}^{b_{1}}\cdots x_{n}^{b_{n}}x^{k},
  34. b i b_{i}
  35. x i x_{i}
  36. x i x_{i}
  37. x k x^{k}
  38. x i x_{i}
  39. 2 n 2^{n}
  40. x i x_{i}
  41. x i . x_{i}.

Vigorish.html

  1. 2 / 3 {2}/{3}
  2. v = o ( 1 + o ) v=\frac{o}{(1+o)}
  3. o = v ( 1 - v ) o=\frac{v}{(1-v)}
  4. v = 100 * ( 1 - p * q p + q ) v=100*\left(1-{p*q\over p+q}\right)
  5. v = 100 * ( 1 / p + 1 / q + 1 / t ) - 1 1 / p + 1 / q + 1 / t v=100*{(1/p+1/q+1/t)-1\over 1/p+1/q+1/t}

Vinculum_(symbol).html

  1. AB ¯ . \overline{\rm AB}.
  2. 1 / 7 {1}/{7}
  3. 142857 ¯ \overline{142857}
  4. ( a - b + c ¯ ) , (a-\overline{b+c}),
  5. a ( b + c ) a−(b+c)
  6. a b + 2 ab+2
  7. a b + 2 n . \sqrt[n]{ab+2}.
  8. π 10.011 1 ¯ 111 1 ¯ 000 1 ¯ 011 1 ¯ 1101 / ¯ 11111100 1 ¯ 0000 1 ¯ 1 1 ¯ 1 ¯ 1 ¯ 1 ¯ 0 1 ¯ \pi\approx 10.011\overline{1}111\overline{1}000\overline{1}011\overline{1}1101% \overline{/}11111100\overline{1}0000\overline{1}1\overline{1}\overline{1}% \overline{1}\overline{1}0\overline{1}
  9. A B ¯ . \overline{AB}.
  10. z ¯ = x + i y ¯ = x - i y . \bar{z}=\overline{x+iy}={x-iy}.
  11. A B \overrightarrow{AB}
  12. a \vec{a}
  13. a ¯ \overline{a}
  14. A B ¯ \underline{AB}
  15. 33. 3 ¯ 33.\overline{3}

Virial_coefficient.html

  1. B i B_{i}
  2. B 2 B_{2}
  3. B 3 B_{3}
  4. Ξ = n λ n Q n = e ( p V ) / ( k B T ) \Xi=\sum_{n}{\lambda^{n}Q_{n}}=e^{\left(pV\right)/\left(k_{B}T\right)}
  5. p p
  6. V V
  7. k B k_{B}
  8. T T
  9. λ = exp [ μ / ( k B T ) ] \lambda=\exp[\mu/(k_{B}T)]
  10. μ \mu
  11. Q n Q_{n}
  12. n n
  13. Q n = tr [ e - H ( 1 , 2 , , n ) / ( k B T ) ] . Q_{n}=\operatorname{tr}[e^{-H(1,2,\ldots,n)/(k_{B}T)}].
  14. H ( 1 , 2 , , n ) H(1,2,\ldots,n)
  15. n n
  16. n n
  17. Ξ \Xi
  18. ln Ξ \ln\Xi
  19. p V / ( k B T ) pV/(k_{B}T)
  20. B 2 = V ( 1 2 - Q 2 Q 1 2 ) B_{2}=V\left(\frac{1}{2}-\frac{Q_{2}}{Q_{1}^{2}}\right)
  21. B 3 = V 2 [ 2 Q 2 Q 1 2 ( 2 Q 2 Q 1 2 - 1 ) - 1 3 ( 6 Q 3 Q 1 3 - 1 ) ] B_{3}=V^{2}\left[\frac{2Q_{2}}{Q_{1}^{2}}\Big(\frac{2Q_{2}}{Q_{1}^{2}}-1\Big)-% \frac{1}{3}\Big(\frac{6Q_{3}}{Q_{1}^{3}}-1\Big)\right]
  22. Q 1 Q_{1}
  23. = 0 \hbar=0
  24. B 3 B_{3}
  25. f ( 1 , 2 ) = exp [ - u ( | r 1 - r 2 | ) k B T ] - 1 f(1,2)=\exp\left[-\frac{u(|\vec{r}_{1}-\vec{r}_{2}|)}{k_{B}T}\right]-1
  26. u ( | r 1 - r 2 | ) u(|\vec{r}_{1}-\vec{r}_{2}|)
  27. B i B_{i}
  28. β i \beta_{i}
  29. B i + 1 = - i i + 1 β i B_{i+1}=-\frac{i}{i+1}\beta_{i}
  30. β i = The sum of all connected, irreducible graphs with one white and i black vertices \beta_{i}=\mbox{The sum of all connected, irreducible graphs with one white % and}~{}\ i\ \mbox{black vertices}~{}
  31. k = 0 k=0
  32. k = 1 , . . , i k=1,..,i
  33. b 1 = b_{1}=
  34. = d 𝟏 f ( 𝟎 , 𝟏 ) =\int d\mathbf{1}f(\mathbf{0},\mathbf{1})
  35. b 2 = b_{2}=
  36. = 1 2 d 𝟏 d 𝟐 f ( 𝟎 , 𝟏 ) f ( 𝟎 , 𝟐 ) f ( 𝟏 , 𝟐 ) =\frac{1}{2}\int d\mathbf{1}\int d\mathbf{2}f(\mathbf{0},\mathbf{1})f(\mathbf{% 0},\mathbf{2})f(\mathbf{1},\mathbf{2})
  37. B 2 = - 2 π ( e - u ( | r 1 | ) / ( k B T ) - 1 ) r 2 d r 1 , B_{2}=-2\pi\int{\Big(e^{-u(|\vec{r}_{1}|)/(k_{B}T)}-1\Big)}\cdot r^{2}d\vec{r}% _{1},
  38. r 2 = 0 \vec{r}_{2}=\vec{0}
  39. B 2 B_{2}

Virial_expansion.html

  1. N N
  2. p k B T = ρ + B 2 ( T ) ρ 2 + B 3 ( T ) ρ 3 + , \frac{p}{k_{B}T}=\rho+B_{2}(T)\rho^{2}+B_{3}(T)\rho^{3}+\cdots,
  3. p p
  4. k B k_{B}
  5. T T
  6. ρ N / V \rho\equiv N/V
  7. n n
  8. N A N_{A}
  9. p V = n N A k B T = n R T pV=nN_{A}k_{B}T=nRT
  10. β = ( k B T ) - 1 \beta=(k_{B}T)^{-1}
  11. β p ρ = 1 + i = 1 B i + 1 ( T ) ρ i \frac{\beta p}{\rho}=1+\sum_{i=1}^{\infty}B_{i+1}(T)\rho^{i}
  12. B i ( T ) B_{i}(T)
  13. T T

Virtual_Router_Redundancy_Protocol.html

  1. ( 256 - P r i o r i t y ) / 256 ({256-Priority)}/{256}

Vis-viva_equation.html

  1. v 2 = G M ( 2 r - 1 a ) v^{2}=GM\left({{2\over{r}}-{1\over{a}}}\right)
  2. ϵ = v a 2 2 - G M r a = v p 2 2 - G M r p \epsilon=\frac{v_{a}^{2}}{2}-\frac{GM}{r_{a}}=\frac{v_{p}^{2}}{2}-\frac{GM}{r_% {p}}
  3. v a 2 2 - v p 2 2 = G M r a - G M r p \frac{v_{a}^{2}}{2}-\frac{v_{p}^{2}}{2}=\frac{GM}{r_{a}}-\frac{GM}{r_{p}}
  4. h = r p v p = r a v a = constant h=r_{p}v_{p}=r_{a}v_{a}=\,\text{constant}
  5. v p = r a r p v a v_{p}=\frac{r_{a}}{r_{p}}v_{a}
  6. 1 2 ( 1 - r a 2 r p 2 ) v a 2 = G M r a - G M r p \frac{1}{2}\left(1-\frac{r_{a}^{2}}{r_{p}^{2}}\right)v_{a}^{2}=\frac{GM}{r_{a}% }-\frac{GM}{r_{p}}
  7. 1 2 ( r p 2 - r a 2 r p 2 ) v a 2 = G M r a - G M r p \frac{1}{2}\left(\frac{r_{p}^{2}-r_{a}^{2}}{r_{p}^{2}}\right)v_{a}^{2}=\frac{% GM}{r_{a}}-\frac{GM}{r_{p}}
  8. 1 2 v a 2 = ( G M r a - G M r p ) ( r p 2 r p 2 - r a 2 ) \frac{1}{2}v_{a}^{2}=\left(\frac{GM}{r_{a}}-\frac{GM}{r_{p}}\right)\left(\frac% {r_{p}^{2}}{r_{p}^{2}-r_{a}^{2}}\right)
  9. 1 2 v a 2 = G M ( r p - r a r a r p ) ( r p 2 r p 2 - r a 2 ) \frac{1}{2}v_{a}^{2}=GM\left(\frac{r_{p}-r_{a}}{r_{a}r_{p}}\right)\left(\frac{% r_{p}^{2}}{r_{p}^{2}-r_{a}^{2}}\right)
  10. 1 2 v a 2 = G M ( r p r a ( r p + r a ) ) \frac{1}{2}v_{a}^{2}=GM\left(\frac{r_{p}}{r_{a}(r_{p}+r_{a})}\right)
  11. 2 a = r p + r a 2a=r_{p}+r_{a}
  12. 1 2 v a 2 = G M ( 2 a - r a r a ( 2 a ) ) \frac{1}{2}v_{a}^{2}=GM\left(\frac{2a-r_{a}}{r_{a}(2a)}\right)
  13. ϵ = v a 2 2 - G M r a = G M ( 2 a - r a 2 a r a ) - G M r a \epsilon=\frac{v_{a}^{2}}{2}-\frac{GM}{r_{a}}=GM\left(\frac{2a-r_{a}}{2ar_{a}}% \right)-\frac{GM}{r_{a}}
  14. ϵ = G M ( 2 a - r a 2 a r a - 1 r a ) = - G M 2 a \epsilon=GM\left(\frac{2a-r_{a}}{2ar_{a}}-\frac{1}{r_{a}}\right)=-\frac{GM}{2a}
  15. ϵ = - G M 2 a \epsilon=-\frac{GM}{2a}
  16. v 2 2 - G M r = - G M 2 a \frac{v^{2}}{2}-\frac{GM}{r}=-\frac{GM}{2a}
  17. v 2 = G M ( 2 r - 1 a ) v^{2}=GM\left(\frac{2}{r}-\frac{1}{a}\right)
  18. r a = a ( 1 + e ) r_{a}=a\left(1+e\right)
  19. r p = a ( 1 - e ) r_{p}=a\left(1-e\right)
  20. r a r p = b 2 r_{a}r_{p}=b^{2}
  21. v a 2 = G M ( 2 r a - 1 a ) = G M a ( 2 1 + e - 1 1 ) = G M a ( 1 - e 1 + e ) = G M a ( r p r a ) = G M a ( b 2 r a 2 ) v_{a}^{2}=GM\left(\frac{2}{r_{a}}-\frac{1}{a}\right)=\frac{GM}{a}\left(\frac{2% }{1+e}-\frac{1}{1}\right)=\frac{GM}{a}\left(\frac{1-e}{1+e}\right)=\frac{GM}{a% }\left(\frac{r_{p}}{r_{a}}\right)=\frac{GM}{a}\left(\frac{b^{2}}{r_{a}^{2}}\right)
  22. L = m h = v a r a m = m b G M a L=mh=v_{a}r_{a}m=mb\,\sqrt{\frac{GM}{a}}
  23. ϵ \epsilon\,\!

Viscoelasticity.html

  1. σ \sigma
  2. ε \varepsilon
  3. σ d ε \oint\sigma\,d\varepsilon
  4. σ \sigma
  5. ε \varepsilon
  6. ϵ ( t ) = σ ( t ) E inst,creep + 0 t K ( t - t ) σ ˙ ( t ) d t \epsilon(t)=\frac{\sigma(t)}{E\text{inst,creep}}+\int_{0}^{t}K(t-t^{\prime})% \dot{\sigma}(t^{\prime})dt^{\prime}
  7. σ ( t ) = E inst,relax ϵ ( t ) + 0 t F ( t - t ) ϵ ˙ ( t ) d t \sigma(t)=E\text{inst,relax}\epsilon(t)+\int_{0}^{t}F(t-t^{\prime})\dot{% \epsilon}(t^{\prime})dt^{\prime}
  8. σ ( t ) \sigma(t)
  9. ϵ ( t ) \epsilon(t)
  10. E inst,creep E\text{inst,creep}
  11. E inst,relax E\text{inst,relax}
  12. G = G + i G ′′ G=G^{\prime}+iG^{\prime\prime}
  13. i 2 = - 1 i^{2}=-1
  14. G G^{\prime}
  15. G ′′ G^{\prime\prime}
  16. G = σ 0 ε 0 cos δ G^{\prime}=\frac{\sigma_{0}}{\varepsilon_{0}}\cos\delta
  17. G ′′ = σ 0 ε 0 sin δ G^{\prime\prime}=\frac{\sigma_{0}}{\varepsilon_{0}}\sin\delta
  18. σ 0 \sigma_{0}
  19. ε 0 \varepsilon_{0}
  20. δ \delta
  21. σ = E ε \sigma=E\varepsilon
  22. σ = η d ε d t \sigma=\eta\frac{d\varepsilon}{dt}
  23. d ϵ d t = d ϵ D d t + d ϵ S d t = σ η + 1 E d σ d t \frac{d\epsilon}{dt}=\frac{d\epsilon_{D}}{dt}+\frac{d\epsilon_{S}}{dt}=\frac{% \sigma}{\eta}+\frac{1}{E}\frac{d\sigma}{dt}
  24. σ ( t ) = E ε ( t ) + η d ε ( t ) d t \sigma(t)=E\varepsilon(t)+\eta\frac{d\varepsilon(t)}{dt}
  25. d ε d t = E 2 η ( η E 2 d σ d t + σ - E 1 ε ) E 1 + E 2 \frac{d\varepsilon}{dt}=\frac{\frac{E_{2}}{\eta}\left(\frac{\eta}{E_{2}}\frac{% d\sigma}{dt}+\sigma-E_{1}\varepsilon\right)}{E_{1}+E_{2}}
  26. E E
  27. G G
  28. K K
  29. G ( t ) = G + Σ i = 1 N G i exp ( - t / τ i ) G(t)=G_{\infty}+\Sigma_{i=1}^{N}G_{i}\exp(-t/\tau_{i})
  30. G G_{\infty}
  31. τ i \tau_{i}
  32. τ i \tau_{i}
  33. G , G i , τ i G_{\infty},G_{i},\tau_{i}
  34. G ( t = 0 ) = G 0 = G + Σ i = 1 N G i G(t=0)=G_{0}=G_{\infty}+\Sigma_{i=1}^{N}G_{i}
  35. G ( t ) = G 0 - Σ i = 1 N G i [ 1 - exp ( - t / τ i ) ] G(t)=G_{0}-\Sigma_{i=1}^{N}G_{i}[1-\exp(-t/\tau_{i})]
  36. G 0 G_{0}
  37. t 0 t_{0}
  38. t 1 t_{1}
  39. t > t 1 t>t_{1}

Vitali–Hahn–Saks_theorem.html

  1. lim n μ n ( X ) = μ ( X ) \lim_{n\rightarrow\infty}\mu_{n}(X)=\mu(X)

VO2_max.html

  1. VO 2 max = Q × ( C a O 2 - C v O 2 ) \mathrm{VO_{2}\;max}=Q\times\ (\mathrm{C_{a}O_{2}}-\mathrm{C_{v}O_{2}})
  2. VO 2 max 15 m L k g * m i n * HR max HR rest \mathrm{VO_{2}\;max}\approx{15{mL\over{kg*min}}*{\mbox{HR}~{}_{\mathrm{max}}% \over\mbox{HR}~{}_{\mathrm{rest}}}}
  3. VO 2 max d 12 - 504.9 44.73 \mathrm{VO_{2}\;max}\approx{d_{12}-504.9\over 44.73}
  4. VO 2 max ( 35.97 * d m i l e s 12 ) - 11.29 \mathrm{VO_{2}\;max}\approx{(35.97*dmiles_{12})-11.29}
  5. VO 2 max 132.853 - ( 0.0769 × body weight (lbs) ) - ( 0.3877 × age ) + ( 6.3150 × gender [ female = 0 , male = 1 ] ) - ( 3.2649 × mile time ) - ( 0.1565 × heart rate on completion ) \mathrm{VO_{2}\;max}\approx 132.853-(0.0769\times\,\text{body weight (lbs)})-(% 0.3877\times\,\text{age})+(6.3150\times\,\text{gender}[\,\text{female}=0,\,% \text{male}=1])-(3.2649\times\,\text{mile time})-(0.1565\times\,\text{heart % rate on completion})

Voltage_multiplier.html

  1. ϕ 1 \phi_{1}
  2. ϕ 1 \phi_{1}
  3. ϕ 1 \phi_{1}
  4. ϕ 2 \phi_{2}
  5. ϕ 1 \phi_{1}
  6. ϕ 2 \phi_{2}

Voltage_regulation.html

  1. Percent V R = | V n l | - | V f l | | V f l | × 100 \,\text{Percent }VR=\frac{|V_{nl}|-|V_{fl}|}{|V_{fl}|}\times 100

Volterra's_function.html

  1. f ( x ) = x 2 sin ( 1 / x ) f(x)=x^{2}\sin(1/x)

Von_Mises_yield_criterion.html

  1. J 2 J_{2}
  2. J 2 J_{2}
  3. J 2 J_{2}
  4. σ v \sigma_{v}
  5. σ y \sigma_{y}
  6. I 1 I_{1}
  7. J 2 = k 2 J_{2}=k^{2}\,\!
  8. k k
  9. k = σ y 3 k=\frac{\sigma_{y}}{\sqrt{3}}
  10. σ y \sigma_{y}
  11. σ v = σ y = 3 J 2 \sigma_{v}=\sigma_{y}=\sqrt{3J_{2}}
  12. σ v 2 = 3 J 2 = 3 k 2 \sigma_{v}^{2}=3J_{2}=3k^{2}
  13. J 2 J_{2}
  14. σ v 2 = 1 2 [ ( σ 11 - σ 22 ) 2 + ( σ 22 - σ 33 ) 2 + ( σ 33 - σ 11 ) 2 + 6 ( σ 23 2 + σ 31 2 + σ 12 2 ) ] \sigma_{v}^{2}=\tfrac{1}{2}[(\sigma_{11}-\sigma_{22})^{2}+(\sigma_{22}-\sigma_% {33})^{2}+(\sigma_{33}-\sigma_{11})^{2}+6(\sigma_{23}^{2}+\sigma_{31}^{2}+% \sigma_{12}^{2})]
  15. 2 k \sqrt{2}k
  16. 2 3 σ y \sqrt{\tfrac{2}{3}}\sigma_{y}
  17. σ 1 0 , σ 3 = σ 2 = 0 \sigma_{1}\neq 0,\sigma_{3}=\sigma_{2}=0
  18. σ 1 = σ y \sigma_{1}=\sigma_{y}\,\!
  19. σ 1 \sigma_{1}
  20. σ y \sigma_{y}
  21. σ v \sigma_{v}
  22. σ v = 3 J 2 = ( σ 11 - σ 22 ) 2 + ( σ 22 - σ 33 ) 2 + ( σ 33 - σ 11 ) 2 + 6 ( σ 12 2 + σ 23 2 + σ 31 2 ) 2 = ( σ 1 - σ 2 ) 2 + ( σ 2 - σ 3 ) 2 + ( σ 3 - σ 1 ) 2 2 = 3 2 s i j s i j \begin{aligned}\displaystyle\sigma_{v}&\displaystyle=\sqrt{3J_{2}}\\ &\displaystyle=\sqrt{\frac{(\sigma_{11}-\sigma_{22})^{2}+(\sigma_{22}-\sigma_{% 33})^{2}+(\sigma_{33}-\sigma_{11})^{2}+6(\sigma_{12}^{2}+\sigma_{23}^{2}+% \sigma_{31}^{2})}{2}}\\ &\displaystyle=\sqrt{\frac{(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})% ^{2}+(\sigma_{3}-\sigma_{1})^{2}}{2}}\\ &\displaystyle=\sqrt{\textstyle{\frac{3}{2}}\;s_{ij}s_{ij}}\end{aligned}\,\!
  23. s i j s_{ij}
  24. s y m b o l σ d e v symbol{\sigma}^{dev}
  25. s y m b o l σ d e v = s y m b o l σ - 1 3 ( tr s y m b o l σ ) 𝐈 symbol{\sigma}^{dev}=symbol{\sigma}-\frac{1}{3}\left(\mbox{tr}~{}\ symbol{% \sigma}\right)\mathbf{I}\,\!
  26. σ v \sigma_{v}
  27. σ y \sigma_{y}
  28. σ 12 = σ 21 0 \sigma_{12}=\sigma_{21}\neq 0
  29. σ i j = 0 \sigma_{ij}=0
  30. σ 12 = k = σ y 3 \sigma_{12}=k=\frac{\sigma_{y}}{\sqrt{3}}\,\!
  31. 3 \sqrt{3}
  32. ( σ 1 - σ 2 ) 2 + ( σ 2 - σ 3 ) 2 + ( σ 1 - σ 3 ) 2 = 2 σ y 2 (\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{2}+(\sigma_{1}-\sigma_{3}% )^{2}=2\sigma_{y}^{2}\,\!
  33. σ 3 = 0 \sigma_{3}=0
  34. σ 1 2 - σ 1 σ 2 + σ 2 2 = 3 k 2 = σ y 2 \sigma_{1}^{2}-\sigma_{1}\sigma_{2}+\sigma_{2}^{2}=3k^{2}=\sigma_{y}^{2}\,\!
  35. σ 1 - σ 2 \sigma_{1}-\sigma_{2}
  36. σ v = 1 2 [ ( σ 11 - σ 22 ) 2 + ( σ 22 - σ 33 ) 2 + ( σ 33 - σ 11 ) 2 + 6 ( σ 12 2 + σ 23 2 + σ 31 2 ) ] \sigma_{v}=\sqrt{\tfrac{1}{2}[(\sigma_{11}-\sigma_{22})^{2}+(\sigma_{22}-% \sigma_{33})^{2}+(\sigma_{33}-\sigma_{11})^{2}+6(\sigma_{12}^{2}+\sigma_{23}^{% 2}+\sigma_{31}^{2})]}
  37. σ v = 1 2 [ ( σ 1 - σ 2 ) 2 + ( σ 2 - σ 3 ) 2 + ( σ 3 - σ 1 ) 2 ] \sigma_{v}=\sqrt{\tfrac{1}{2}[(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{% 3})^{2}+(\sigma_{3}-\sigma_{1})^{2}]}
  38. σ 3 = 0 \sigma_{3}=0\!
  39. σ 31 = σ 23 = 0 \sigma_{31}=\sigma_{23}=0\!
  40. σ v = σ 1 2 - σ 1 σ 2 + σ 2 2 + 3 σ 12 2 \sigma_{v}=\sqrt{\sigma_{1}^{2}-\sigma_{1}\sigma_{2}+\sigma_{2}^{2}+3\sigma_{1% 2}^{2}}\!
  41. σ 3 = 0 \sigma_{3}=0\!
  42. σ 12 = σ 31 = σ 23 = 0 \sigma_{12}=\sigma_{31}=\sigma_{23}=0\!
  43. σ v = σ 1 2 - σ 1 σ 2 + σ 2 2 \sigma_{v}=\sqrt{\sigma_{1}^{2}-\sigma_{1}\sigma_{2}+\sigma_{2}^{2}}\!
  44. σ 1 = σ 2 = σ 3 = 0 \sigma_{1}=\sigma_{2}=\sigma_{3}=0\!
  45. σ 31 = σ 23 = 0 \sigma_{31}=\sigma_{23}=0\!
  46. σ v = 3 | σ 12 | \sigma_{v}=\sqrt{3}|\sigma_{12}|\!
  47. σ 2 = σ 3 = 0 \sigma_{2}=\sigma_{3}=0\!
  48. σ 12 = σ 31 = σ 23 = 0 \sigma_{12}=\sigma_{31}=\sigma_{23}=0\!
  49. σ v = σ 1 \sigma_{v}=\sigma_{1}\!
  50. σ i j \sigma_{ij}
  51. τ i j \tau_{ij}
  52. J 2 J_{2}
  53. W D W_{D}
  54. W D = J 2 2 G W_{D}=\frac{J_{2}}{2G}\,\!
  55. G = E 2 ( 1 + ν ) G=\frac{E}{2(1+\nu)}\,\!
  56. J 2 J_{2}
  57. τ o c t \tau_{oct}
  58. τ o c t = 2 3 J 2 \tau_{oct}=\sqrt{\tfrac{2}{3}J_{2}}\,\!
  59. τ o c t = 2 3 σ y \tau_{oct}=\tfrac{\sqrt{2}}{3}\sigma_{y}\,\!

Von_Neumann_conjecture.html

  1. F F_{\infty}

Von_Neumann_regular_ring.html

  1. 𝔦 \mathfrak{i}
  2. 𝔦 \mathfrak{i}
  3. 𝔦 \mathfrak{i}
  4. A = U ( I r 0 0 0 ) V A=U\begin{pmatrix}I_{r}&0\\ 0&0\end{pmatrix}V
  5. A X A = U ( I r 0 0 0 ) ( I r 0 0 0 ) V = U ( I r 0 0 0 ) V = A . AXA=U\begin{pmatrix}I_{r}&0\\ 0&0\end{pmatrix}\begin{pmatrix}I_{r}&0\\ 0&0\end{pmatrix}V=U\begin{pmatrix}I_{r}&0\\ 0&0\end{pmatrix}V=A.

Von_Staudt–Clausen_theorem.html

  1. B 2 n + ( p - 1 ) | 2 n 1 p \Z . B_{2n}+\sum_{(p-1)|2n}\frac{1}{p}\in\Z.
  2. B 2 n = j = 0 2 n 1 j + 1 m = 0 j ( - 1 ) m ( j m ) m 2 n B_{2n}=\sum_{j=0}^{2n}{\frac{1}{j+1}}\sum_{m=0}^{j}{(-1)^{m}{j\choose m}m^{2n}}\!
  3. B 2 n = j = 0 2 n j ! j + 1 ( - 1 ) j S ( 2 n , j ) B_{2n}=\sum_{j=0}^{2n}{\frac{j!}{j+1}}(-1)^{j}S(2n,j)\!
  4. S ( n , j ) S(n,j)\!
  5. m = 0 p - 1 ( - 1 ) m ( p - 1 m ) m 2 n - 1 ( mod p ) \sum_{m=0}^{p-1}{(-1)^{m}{p-1\choose m}m^{2n}}\equiv{-1}\;\;(\mathop{{\rm mod}% }p)\!
  6. m = 0 p - 1 ( - 1 ) m ( p - 1 m ) m 2 n 0 ( mod p ) \sum_{m=0}^{p-1}{(-1)^{m}{p-1\choose m}m^{2n}}\equiv 0\;\;(\mathop{{\rm mod}}p)\!
  7. m p - 1 1 ( mod p ) m^{p-1}\equiv 1\;\;(\mathop{{\rm mod}}p)\!
  8. m = 1 , 2 , , p - 1 m=1,2,...,p-1\!
  9. m 2 n 1 ( mod p ) m^{2n}\equiv 1\;\;(\mathop{{\rm mod}}p)\!
  10. m = 1 , 2 , , p - 1 m=1,2,...,p-1\!
  11. m = 1 p - 1 ( - 1 ) m ( p - 1 m ) m 2 n m = 1 p - 1 ( - 1 ) m ( p - 1 m ) ( mod p ) \sum_{m=1}^{p-1}{(-1)^{m}{p-1\choose m}m^{2n}}\equiv\sum_{m=1}^{p-1}{(-1)^{m}{% p-1\choose m}}\;\;(\mathop{{\rm mod}}p)\!
  12. m 2 n m 2 n - ( p - 1 ) ( mod p ) m^{2n}\equiv m^{2n-(p-1)}\;\;(\mathop{{\rm mod}}p)\!
  13. = [ 2 n p - 1 ] \wp=[\frac{2n}{p-1}]\!
  14. m 2 n m 2 n - ( p - 1 ) ( mod p ) m^{2n}\equiv m^{2n-\wp(p-1)}\;\;(\mathop{{\rm mod}}p)\!
  15. m = 1 , 2 , , p - 1 m=1,2,...,p-1\!
  16. 0 < 2 n - ( p - 1 ) < p - 1 0<2n-\wp(p-1)<p-1\!
  17. m = 0 p - 1 ( - 1 ) m ( p - 1 m ) m 2 n m = 0 p - 1 ( - 1 ) m ( p - 1 m ) m 2 n - ( p - 1 ) ( mod p ) \sum_{m=0}^{p-1}{(-1)^{m}{p-1\choose m}m^{2n}}\equiv\sum_{m=0}^{p-1}{(-1)^{m}{% p-1\choose m}m^{2n-\wp(p-1)}}\;\;(\mathop{{\rm mod}}p)\!
  18. m = 0 3 ( - 1 ) m ( 3 m ) m 2 n = 3 2 2 n - 3 2 n - 3 0 ( mod 4 ) \sum_{m=0}^{3}{(-1)^{m}{3\choose m}m^{2n}}=3\cdot 2^{2n}-3^{2n}-3\equiv 0\;\;(% \mathop{{\rm mod}}4)\!
  19. B 2 n = I n - ( p - 1 ) | 2 n 1 p B_{2n}=I_{n}-\sum_{(p-1)|2n}{\frac{1}{p}}\!
  20. I n I_{n}\!

Vortex_tube.html

  1. T - v ω × r c p = const T-\frac{\vec{v}\cdot\vec{\omega}\times\vec{r}}{c_{p}}=\mbox{const}~{}
  2. T T
  3. r \vec{r}
  4. v \vec{v}
  5. ω \vec{\omega}
  6. c p c_{p}

W._G._Unruh.html

  1. k T = a 2 π c kT=\frac{\hbar a}{2\pi c}

Wagstaff_prime.html

  1. p = 2 q + 1 3 p={{2^{q}+1}\over 3}
  2. 3 = 2 3 + 1 3 , 11 = 2 5 + 1 3 , 43 = 2 7 + 1 3 . \begin{aligned}\displaystyle 3&\displaystyle={2^{3}+1\over 3},\\ \displaystyle 11&\displaystyle={2^{5}+1\over 3},\\ \displaystyle 43&\displaystyle={2^{7}+1\over 3}.\end{aligned}
  3. 2 4031399 + 1 3 \frac{2^{4031399}+1}{3}
  4. 2 13347311 + 1 3 \frac{2^{13347311}+1}{3}
  5. 2 13372531 + 1 3 \frac{2^{13372531}+1}{3}
  6. Q ( b , n ) = b n + 1 b + 1 Q(b,n)=\frac{b^{n}+1}{b+1}
  7. b 2 b\geq 2
  8. n n
  9. b n + 1 b + 1 = ( - b ) n - 1 ( - b ) - 1 = R n ( - b ) \frac{b^{n}+1}{b+1}=\frac{(-b)^{n}-1}{(-b)-1}=R_{n}^{(-b)}
  10. - b -b
  11. b b
  12. Q ( b , n ) Q(b,n)
  13. n n
  14. b b
  15. x m + 1 x^{m}+1
  16. m m
  17. x + 1 x+1
  18. Q ( a m , n ) Q(a^{m},n)
  19. a n + 1 a^{n}+1
  20. b = 4 k 4 b=4k^{4}
  21. b b
  22. n n
  23. Q ( b , n ) Q(b,n)
  24. b = 10 b=10
  25. Q ( b , p r i m e ( n ) ) Q(b,prime(n))

Wallace_Clement_Sabine.html

  1. T = V A 0.161 s T=\frac{V}{A}\cdot 0.161\,\mathrm{s}

Wannier_function.html

  1. ψ 𝐤 ( 𝐫 ) = e i 𝐤 𝐫 u 𝐤 ( 𝐫 ) \psi_{\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{\mathbf{k}}(% \mathbf{r})
  2. ϕ 𝐑 ( 𝐫 ) = 1 N 𝐤 e - i 𝐤 𝐑 ψ 𝐤 ( 𝐫 ) \phi_{\mathbf{R}}(\mathbf{r})=\frac{1}{\sqrt{N}}\sum_{\mathbf{k}}e^{-i\mathbf{% k}\cdot\mathbf{R}}\psi_{\mathbf{k}}(\mathbf{r})
  3. 𝐤 N Ω BZ d 3 𝐤 \sum_{\mathbf{k}}\longleftarrow\frac{N}{\Omega}\int\text{BZ}d^{3}\mathbf{k}
  4. ϕ 𝐑 ( 𝐫 ) = ϕ 𝐑 + 𝐑 ( 𝐫 + 𝐑 ) \phi_{\mathbf{R}}(\mathbf{r})=\phi_{\mathbf{R}+\mathbf{R}^{\prime}}(\mathbf{r}% +\mathbf{R}^{\prime})
  5. ϕ ( 𝐫 - 𝐑 ) := ϕ 𝐑 ( 𝐫 ) \phi(\mathbf{r}-\mathbf{R}):=\phi_{\mathbf{R}}(\mathbf{r})
  6. ψ 𝐤 ( 𝐫 ) = 1 N 𝐑 e i 𝐤 𝐑 ϕ 𝐑 ( 𝐫 ) \psi_{\mathbf{k}}(\mathbf{r})=\frac{1}{\sqrt{N}}\sum_{\mathbf{R}}e^{i\mathbf{k% }\cdot\mathbf{R}}\phi_{\mathbf{R}}(\mathbf{r})
  7. ϕ 𝐑 \phi_{\mathbf{R}}
  8. crystal ϕ 𝐑 ( 𝐫 ) * ϕ 𝐑 ( 𝐫 ) d 3 𝐫 = 1 N 𝐤 , 𝐤 crystal e i 𝐤 𝐑 ψ 𝐤 ( 𝐫 ) * e - i 𝐤 𝐑 ψ 𝐤 ( 𝐫 ) d 3 𝐫 = 1 N 𝐤 , 𝐤 e i 𝐤 𝐑 e - i 𝐤 𝐑 δ 𝐤 , 𝐤 = 1 N 𝐤 e i 𝐤 ( 𝐑 - 𝐑 ) = δ 𝐑 , 𝐑 \int\text{crystal}\phi_{\mathbf{R}}(\mathbf{r})^{*}\phi_{\mathbf{R^{\prime}}}(% \mathbf{r})d^{3}\mathbf{r}=\frac{1}{N}\sum_{\mathbf{k,k^{\prime}}}\int\text{% crystal}e^{i\mathbf{k}\cdot\mathbf{R}}\psi_{\mathbf{k}}(\mathbf{r})^{*}e^{-i% \mathbf{k^{\prime}}\cdot\mathbf{R^{\prime}}}\psi_{\mathbf{k^{\prime}}}(\mathbf% {r})d^{3}\mathbf{r}=\frac{1}{N}\sum_{\mathbf{k,k^{\prime}}}e^{i\mathbf{k}\cdot% \mathbf{R}}e^{-i\mathbf{k^{\prime}}\cdot\mathbf{R^{\prime}}}\delta_{\mathbf{k,% k^{\prime}}}=\frac{1}{N}\sum_{\mathbf{k}}e^{i\mathbf{k}\cdot\mathbf{(R^{\prime% }-R)}}=\delta_{\mathbf{R,R^{\prime}}}
  9. 𝐩 𝐜 = - e n d 3 r 𝐫 | W n ( 𝐫 ) | 2 , \mathbf{p_{c}}=-e\sum_{n}\int\ d^{3}r\,\,\mathbf{r}|W_{n}(\mathbf{r})|^{2}\ ,

Warped_geometry.html

  1. d s 2 = g a b ( y ) d y a d y b + f ( y ) g i j ( x ) d x i d x j . ds^{2}\,=g_{ab}(y)\,dy^{a}\,dy^{b}+f(y)g_{ij}(x)\,dx^{i}\,dx^{j}.

Water_potential.html

  1. Ψ \Psi
  2. Ψ = Ψ 0 + Ψ π + Ψ p + Ψ s + Ψ v + Ψ m \Psi=\Psi_{0}+\Psi_{\pi}+\Psi_{p}+\Psi_{s}+\Psi_{v}+\Psi_{m}
  3. Ψ 0 \Psi_{0}
  4. Ψ π \Psi_{\pi}
  5. Ψ p \Psi_{p}
  6. Ψ s \Psi_{s}
  7. Ψ v \Psi_{v}
  8. Ψ m \Psi_{m}
  9. Ψ π \Psi_{\pi}
  10. Ψ π = - M i R T \Psi_{\pi}=-MiRT
  11. M M
  12. i i
  13. R R
  14. T T
  15. Ψ w \Psi_{w}

Water_rocket.html

  1. h = ( M i M R ) 2 ( P i ρ g ) h=\left({M_{i}\over M_{R}}\right)^{2}\left({P_{i}\over\rho g}\right)
  2. h h
  3. M i M_{i}
  4. M R M_{R}
  5. P i P_{i}
  6. ρ \rho
  7. g g
  8. F = 2 P A t F=2PA_{t}
  9. F F
  10. P P
  11. A t A_{t}

Watt_balance.html

  1. × 10 8 \times 10^{−}8
  2. × 10 8 \times 10^{−}8
  3. w = m g = B L I . w=mg=BLI\,.
  4. U = B L v . U=BLv\,.
  5. U I = m g v . UI=mgv\,.
  6. = 2 e / h =2e/h
  7. = h / e 2 =h/e^{2}
  8. K J 2 R K = K J - 90 2 R K - 90 m g v U 90 I 90 K_{\rm J}^{2}R_{\rm K}=K_{\rm J-90}^{2}R_{\rm K-90}\frac{mgv}{U_{90}I_{90}}
  9. h = 4 K J 2 R K h=\frac{4}{K_{\rm J}^{2}R_{\rm K}}
  10. m = U I g v m=\frac{UI}{gv}

Wave_power.html

  1. P = ρ g 2 64 π H m 0 2 T e ( 0.5 kW m 3 s ) H m 0 2 T e , P=\frac{\rho g^{2}}{64\pi}H_{m0}^{2}T_{e}\approx\left(0.5\frac{\,\text{kW}}{\,% \text{m}^{3}\cdot\,\text{s}}\right)H_{m0}^{2}\;T_{e},
  2. P 0.5 kW m 3 s ( 3 m ) 2 ( 8 s ) 36 kW m , P\approx 0.5\frac{\,\text{kW}}{\,\text{m}^{3}\cdot\,\text{s}}(3\cdot\,\text{m}% )^{2}(8\cdot\,\text{s})\approx 36\frac{\,\text{kW}}{\,\text{m}},
  3. E = 1 8 ρ g H m 0 2 , E=\frac{1}{8}\rho gH_{m0}^{2},
  4. P = E c g , P=E\,c_{g},\,
  5. c p = λ T = ω k \displaystyle c_{p}=\frac{\lambda}{T}=\frac{\omega}{k}
  6. g 2 π T \frac{g}{2\pi}T
  7. g h \sqrt{gh}
  8. g λ 2 π tanh ( 2 π h λ ) \sqrt{\frac{g\lambda}{2\pi}\tanh\left(\frac{2\pi h}{\lambda}\right)}
  9. c g = c p 2 ( λ / c p ) λ = ω k \displaystyle c_{g}=c_{p}^{2}\frac{\partial\left(\lambda/c_{p}\right)}{% \partial\lambda}=\frac{\partial\omega}{\partial k}
  10. g 4 π T \frac{g}{4\pi}T
  11. g h \sqrt{gh}
  12. 1 2 c p ( 1 + 4 π h λ 1 sinh ( 4 π h λ ) ) \frac{1}{2}c_{p}\left(1+\frac{4\pi h}{\lambda}\frac{1}{\sinh\left(% \displaystyle\frac{4\pi h}{\lambda}\right)}\right)
  13. c g c p \displaystyle\frac{c_{g}}{c_{p}}
  14. 1 2 \displaystyle\frac{1}{2}
  15. 1 \displaystyle 1
  16. 1 2 ( 1 + 4 π h λ 1 sinh ( 4 π h λ ) ) \frac{1}{2}\left(1+\frac{4\pi h}{\lambda}\frac{1}{\sinh\left(\displaystyle% \frac{4\pi h}{\lambda}\right)}\right)
  17. λ \displaystyle\lambda
  18. g 2 π T 2 \frac{g}{2\pi}T^{2}
  19. T g h T\sqrt{gh}
  20. ( 2 π T ) 2 = 2 π g λ tanh ( 2 π h λ ) \displaystyle\left(\frac{2\pi}{T}\right)^{2}=\frac{2\pi g}{\lambda}\tanh\left(% \frac{2\pi h}{\lambda}\right)
  21. E \displaystyle E
  22. 1 16 ρ g H m 0 2 \frac{1}{16}\rho gH_{m0}^{2}
  23. P \displaystyle P
  24. E c g \displaystyle E\;c_{g}
  25. ω \displaystyle\omega
  26. 2 π T \frac{2\pi}{T}
  27. k \displaystyle k
  28. 2 π λ \frac{2\pi}{\lambda}
  29. P = 1 16 ρ g H m 0 2 c g , P=\tfrac{1}{16}\rho gH_{m0}^{2}c_{g},
  30. c g c_{g}
  31. c g = g 4 π T c_{g}=\tfrac{g}{4\pi}T
  32. 1 / 16 {1}/{16}
  33. 1 / 8 {1}/{8}
  34. η = a cos 2 π ( x λ - t T ) \scriptstyle\eta=a\,\cos\,2\pi\left(\frac{x}{\lambda}-\frac{t}{T}\right)
  35. a , \scriptstyle a,\,
  36. E = 1 2 ρ g a 2 , \scriptstyle E=\frac{1}{2}\rho ga^{2},
  37. E = 1 8 ρ g H 2 \scriptstyle E=\frac{1}{8}\rho gH^{2}
  38. H = 2 a \scriptstyle H\,=\,2\,a\,
  39. m 0 = σ η 2 = ( η - η ¯ ) 2 ¯ = 1 2 a 2 , \scriptstyle m_{0}=\sigma_{\eta}^{2}=\overline{(\eta-\bar{\eta})^{2}}=\frac{1}% {2}a^{2},
  40. E = ρ g m 0 \scriptstyle E=\rho gm_{0}\,
  41. m 0 \scriptstyle m_{0}\,
  42. H m 0 = 4 m 0 \scriptstyle H_{m0}=4\sqrt{m_{0}}
  43. 1 / 16 {1}/{16}

Wave_vector.html

  1. ψ ( x , t ) = A cos ( k x - ω t + φ ) \psi(x,t)=A\cos(kx-\omega t+\varphi)
  2. ψ \psi
  3. ψ \psi
  4. ψ \psi
  5. φ \varphi
  6. ω \omega
  7. T T
  8. ω = 2 π / T \omega=2\pi/T
  9. k k
  10. k = 2 π / λ k=2\pi/\lambda
  11. ω / k \omega/k
  12. ψ ( x , t ) = A cos ( 2 π ( k x - ν t ) + φ ) \psi(x,t)=A\cos(2\pi(kx-\nu t)+\varphi)
  13. ψ ( 𝐫 , t ) = A cos ( 2 π ( 𝐤 𝐫 - ν t ) + φ ) \psi\left({\mathbf{r}},t\right)=A\cos\left(2\pi({\mathbf{k}}\cdot{\mathbf{r}}-% \nu t)+\varphi\right)
  14. ν \nu
  15. ω \omega
  16. 2 π ν = ω 2\pi\nu=\omega
  17. k = | 𝐤 | = 1 / λ k=|{\mathbf{k}}|=1/\lambda
  18. k = | 𝐤 | = 2 π / λ k=|{\mathbf{k}}|=2\pi/\lambda
  19. k μ = ( ω c , k ) k^{\mu}=\left(\frac{\omega}{c},\vec{k}\right)\,
  20. k μ = ( ω c , k x , k y , k z ) k^{\mu}=\left(\frac{\omega}{c},k_{x},k_{y},k_{z}\right)\,
  21. k μ = ( ω c , - k x , - k y , - k z ) . k_{\mu}=\left(\frac{\omega}{c},-k_{x},-k_{y},-k_{z}\right).\,
  22. k μ k μ = ( ω c ) 2 - k x 2 - k y 2 - k z 2 = 0 k^{\mu}k_{\mu}=\left(\frac{\omega}{c}\right)^{2}-k_{x}^{2}-k_{y}^{2}-k_{z}^{2}% \ =0
  23. p μ = ( E / c , p x , p y , p z ) = ( ω / c , k x , k y , k z ) = k μ p^{\mu}=(E/c,p_{x},p_{y},p_{z})=(\hbar\omega/c,\hbar k_{x},\hbar k_{y},\hbar k% _{z})=\hbar k^{\mu}
  24. Λ = ( γ - β γ 0 0 - β γ γ 0 0 0 0 1 0 0 0 0 1 ) \Lambda=\begin{pmatrix}\gamma&-\beta\gamma&0&0\\ -\beta\gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}
  25. k s μ = Λ ν μ k obs ν k^{\mu}_{s}=\Lambda^{\mu}_{\nu}k^{\nu}_{\mathrm{obs}}\,
  26. μ = 0 \mu=0
  27. k s 0 = Λ 0 0 k obs 0 + Λ 1 0 k obs 1 + Λ 2 0 k obs 2 + Λ 3 0 k obs 3 k^{0}_{s}=\Lambda^{0}_{0}k^{0}_{\mathrm{obs}}+\Lambda^{0}_{1}k^{1}_{\mathrm{% obs}}+\Lambda^{0}_{2}k^{2}_{\mathrm{obs}}+\Lambda^{0}_{3}k^{3}_{\mathrm{obs}}\,
  28. ω s c \frac{\omega_{s}}{c}\,
  29. = γ ω obs c - β γ k obs 1 =\gamma\frac{\omega_{\mathrm{obs}}}{c}-\beta\gamma k^{1}_{\mathrm{obs}}\,
  30. = γ ω obs c - β γ ω obs c cos θ . \quad=\gamma\frac{\omega_{\mathrm{obs}}}{c}-\beta\gamma\frac{\omega_{\mathrm{% obs}}}{c}\cos\theta.\,
  31. cos θ \cos\theta\,
  32. k 1 k^{1}
  33. k 0 , k 1 = k 0 cos θ . k^{0},k^{1}=k^{0}\cos\theta.
  34. ω obs ω s = 1 γ ( 1 - β cos θ ) \frac{\omega_{\mathrm{obs}}}{\omega_{s}}=\frac{1}{\gamma(1-\beta\cos\theta)}\,
  35. θ = π \theta=\pi
  36. ω obs ω s = 1 γ ( 1 + β ) = 1 - β 2 1 + β = ( 1 + β ) ( 1 - β ) 1 + β = 1 - β 1 + β \frac{\omega_{\mathrm{obs}}}{\omega_{s}}=\frac{1}{\gamma(1+\beta)}=\frac{\sqrt% {1-\beta^{2}}}{1+\beta}=\frac{\sqrt{(1+\beta)(1-\beta)}}{1+\beta}=\frac{\sqrt{% 1-\beta}}{\sqrt{1+\beta}}\,
  37. θ = 0 \theta=0
  38. ω obs ω s = 1 + β 1 - β \frac{\omega_{\mathrm{obs}}}{\omega_{s}}=\frac{\sqrt{1+\beta}}{\sqrt{1-\beta}}\,

Weak_key.html

  1. E K 1 ( E K 2 ( M ) ) = M E_{K_{1}}(E_{K_{2}}(M))=M

Weak_ordering.html

  1. \lesssim
  2. \lesssim
  3. \lesssim
  4. \lesssim
  5. \lesssim
  6. \lesssim
  7. \lesssim
  8. \lesssim
  9. \lesssim
  10. {}\sim{}
  11. {}\lesssim{}
  12. {}\sim{}
  13. {}\sim{}
  14. \lesssim

Weather_radar.html

  1. v = h r 2 θ 2 \,{v=hr^{2}\theta^{2}}
  2. θ \,\theta
  3. Distance = c Δ t 2 n , \,\text{Distance}=c\frac{\Delta t}{2n},
  4. H = r 2 + ( k e a e ) 2 + 2 r k e a e sin ( θ e ) - k e a e + h a , H=\sqrt{r^{2}+(k_{e}a_{e})^{2}+2rk_{e}a_{e}\sin(\theta_{e})}-k_{e}a_{e}+h_{a},
  5. G t = A r ( o r G r ) = G \scriptstyle G_{t}=A_{r}(or\ G_{r})=G
  6. P r = P t G 2 λ 2 σ 0 ( 4 π ) 3 R 4 σ 0 R 4 P_{r}=P_{t}{{G^{2}\lambda^{2}\sigma_{0}}\over{{(4\pi)}^{3}R^{4}}}\propto\frac{% \sigma_{0}}{R^{4}}
  7. P r \scriptstyle P_{r}
  8. P t \scriptstyle P_{t}
  9. G \scriptstyle G
  10. λ \scriptstyle\lambda
  11. σ \scriptstyle\sigma
  12. R \scriptstyle R
  13. σ 0 = σ ¯ 0 = V σ 0 j = V η \sigma_{0}=\bar{\sigma}_{0}=V\sum\sigma_{0j}=V\eta
  14. { V = s c a n n e d v o l u m e = p u l s e l e n g t h X b e a m w i d t h = c τ 2 π R 2 θ 2 4 \begin{cases}V\quad=scanned\ volume\\ \qquad=pulse\ length\ \ X\ beam\ width\\ \qquad=\frac{c\tau}{2}\frac{\pi R^{2}\theta^{2}}{4}\end{cases}
  15. c \,c
  16. τ \,\tau
  17. θ \,\theta
  18. P r = P t G 2 λ 2 ( 4 π ) 3 R 4 c τ 2 π R 2 θ 2 4 η = P t τ G 2 λ 2 θ 2 c 512 ( π 2 ) η R 2 P_{r}=P_{t}{{G^{2}\lambda^{2}}\over{{(4\pi)}^{3}R^{4}}}\frac{c\tau}{2}\frac{% \pi R^{2}\theta^{2}}{4}\eta=P_{t}\tau G^{2}\lambda^{2}\theta^{2}\frac{c}{512(% \pi^{2})}\frac{\eta}{R^{2}}
  19. P r η R 2 P_{r}\propto\frac{\eta}{R^{2}}
  20. R 2 \,R^{2}
  21. R 4 \,R^{4}
  22. Z e = 0 D m a x | K | 2 N 0 e - Λ D D 6 d D Z_{e}=\int_{0}^{Dmax}|K|^{2}N_{0}e^{-\Lambda D}D^{6}dD
  23. R = 0 D m a x N 0 e - Λ D π D 3 6 v ( D ) d D R=\int_{0}^{Dmax}N_{0}e^{-\Lambda D}{\pi D^{3}\over 6}v(D)dD
  24. Λ \Lambda
  25. I = I 0 sin ( 4 π ( x 0 + v Δ t ) λ ) = I 0 sin ( Θ 0 + Δ Θ ) { x = distance from radar to target λ = radar wavelength Δ t = time between two pulses I=I_{0}\sin\left(\frac{4\pi(x_{0}+v\Delta t)}{\lambda}\right)=I_{0}\sin\left(% \Theta_{0}+\Delta\Theta\right)\quad\begin{cases}x=\,\text{distance from radar % to target}\\ \lambda=\,\text{radar wavelength}\\ \Delta t=\,\text{time between two pulses}\end{cases}
  26. Δ Θ = 4 π v Δ t λ \Delta\Theta=\frac{4\pi v\Delta t}{\lambda}
  27. λ Δ Θ 4 π Δ t \frac{\lambda\Delta\Theta}{4\pi\Delta t}
  28. π \pi
  29. π \pi
  30. ± \pm
  31. λ 4 Δ t \frac{\lambda}{4\Delta t}
  32. Δ t \Delta t
  33. c Δ t 2 \frac{c\Delta t}{2}

Weber_(unit).html

  1. Wb = kg m 2 s 2 A = V s = T m 2 = J A = 10 8 Mx \mathrm{Wb}=\dfrac{\mathrm{kg}\cdot\mathrm{m}^{2}}{\mathrm{s}^{2}\cdot\mathrm{% A}}=\mathrm{V}\cdot\mathrm{s}=\mathrm{T}\cdot\mathrm{m}^{2}=\dfrac{\mathrm{J}}% {\mathrm{A}}=10^{8}\mathrm{Mx}
  2. × 10 - 7 \times 10^{-}7

Wedderburn–Etherington_number.html

  1. x n x^{n}
  2. x 5 x^{5}
  3. x ( x ( x ( x x ) ) ) x(x(x(xx)))
  4. x ( ( x x ) ( x x ) ) x((xx)(xx))
  5. ( x x ) ( x ( x x ) ) (xx)(x(xx))
  6. x x
  7. x x
  8. x n x^{n}
  9. a 2 n - 1 = i = 1 n - 1 a i a 2 n - i - 1 a_{2n-1}=\sum_{i=1}^{n-1}a_{i}a_{2n-i-1}
  10. a 2 n = a n ( a n + 1 ) 2 + i = 1 n - 1 a i a 2 n - i a_{2n}=\frac{a_{n}(a_{n}+1)}{2}+\sum_{i=1}^{n-1}a_{i}a_{2n-i}
  11. a 1 = 1 a_{1}=1
  12. a n ρ + ρ 2 B ( ρ 2 ) 2 π ρ - n n 3 / 2 , a_{n}\approx\sqrt{\frac{\rho+\rho^{2}B^{\prime}(\rho^{2})}{2\pi}}\frac{\rho^{-% n}}{n^{3/2}},

Wedge_sum.html

  1. X Y = ( X Y ) / , X\vee Y=(X\amalg Y)\;/{\sim},\,
  2. i X i = i X i / , \bigvee_{i}X_{i}=\coprod_{i}X_{i}\;/{\sim},\,
  3. S n S^{n}
  4. S n / = S n S n S^{n}/{\sim}=S^{n}\vee S^{n}
  5. Ψ \Psi
  6. Ψ : S n S n S n \Psi:S^{n}\to S^{n}\vee S^{n}
  7. f , g π n ( X , x 0 ) f,g\in\pi_{n}(X,x_{0})
  8. π n ( X , x 0 ) \pi_{n}(X,x_{0})
  9. x 0 X x_{0}\in X
  10. f f
  11. g g
  12. Ψ \Psi
  13. f + g = ( f g ) Ψ f+g=(f\vee g)\circ\Psi
  14. f f
  15. g g
  16. f : S n X f:S^{n}\to X
  17. g g
  18. s 0 S n s_{0}\in S^{n}
  19. x 0 X x_{0}\in X
  20. f ( s 0 ) = g ( s 0 ) = x 0 f(s_{0})=g(s_{0})=x_{0}

Weight_gain.html

  1. w f = w i + e i - e b 3500 w_{f}=w_{i}+\frac{e_{i}-e_{b}}{3500}
  2. = =
  3. 4000 - 1500 = 2500 4000-1500=2500
  4. 2500 ÷ 3500 = [ u F r a c t i o n , u 5 , u 7 ] 2500÷3500=[u^{\prime}Fraction^{\prime},u^{\prime}5^{\prime},u^{\prime}7^{% \prime}]
  5. [ u F r a c t i o n , u 5 , u 7 ] [u^{\prime}Fraction^{\prime},u^{\prime}5^{\prime},u^{\prime}7^{\prime}]
  6. [ u F r a c t i o n , u 125 , u 5 , u 7 ] [u^{\prime}Fraction^{\prime},u^{\prime}125^{\prime},u^{\prime}5^{\prime},u^{% \prime}7^{\prime}]
  7. e b = e i + 3500 ( w i - w f ) e_{b}=e_{i}+3500(w_{i}-w_{f})

Weight_transfer.html

  1. Δ W e i g h t f r o n t = a h w m \Delta Weight_{front}=a\frac{h}{w}m
  2. Δ W e i g h t f r o n t \Delta Weight_{front}
  3. a a
  4. h h
  5. w w
  6. m m

Weighted_round_robin.html

  1. n u m b e r = n o r m a l i z e d ( w e i g h t / m e a n p a c k e t s i z e ) number=normalized(weight/meanpacketsize)

Weil_pairing.html

  1. E ( K ¯ ) E(\overline{K})
  2. w ( P , Q ) μ n w(P,Q)\in\mu_{n}
  3. P , Q E ( K ) [ n ] P,Q\in E(K)[n]
  4. E ( K ) [ n ] = { T E ( K ) n T = O } E(K)[n]=\{T\in E(K)\mid n\cdot T=O\}
  5. μ n = { x K x n = 1 } \mu_{n}=\{x\in K\mid x^{n}=1\}
  6. div ( F ) = 0 k < n ( P + k Q ) - 0 k < n ( k Q ) . \mathrm{div}(F)=\sum_{0\leq k<n}(P+k\cdot Q)-\sum_{0\leq k<n}(k\cdot Q).
  7. w ( P , Q ) := G F w(P,Q):=\frac{G}{F}
  8. A [ n ] × A [ n ] μ n A[n]\times A^{\vee}[n]\longrightarrow\mu_{n}
  9. A A^{\vee}
  10. λ : A A \lambda:A\longrightarrow A^{\vee}
  11. A [ n ] × A [ n ] μ n . A[n]\times A[n]\longrightarrow\mu_{n}.
  12. J [ n ] × J [ n ] μ n J[n]\times J[n]\longrightarrow\mu_{n}