wpmath0000005_14

Thomson's_lamp.html

  1. โˆ‘ i = 0 n ( - 1 ) i . \sum_{i=0}^{n}{(-1)^{i}}.
  2. S = 1 - 1 + 1 - 1 + 1 - 1 + โ‹ฏ S=1-1+1-1+1-1+\cdots
  3. S = 1 - ( 1 - 1 + 1 - 1 + 1 - 1 + โ‹ฏ ) S=1-(1-1+1-1+1-1+\cdots)

Three-body_problem.html

  1. ๐ฑ ๐ข \mathbf{x_{i}}
  2. m i m_{i}
  3. ๐ฑ ยจ ๐Ÿ = - G m 2 ( x 1 - x 2 ) 3 ( ๐ฑ ๐Ÿ - ๐ฑ ๐Ÿ ) - G m 3 ( x 1 - x 3 ) 3 ( ๐ฑ ๐Ÿ - ๐ฑ ๐Ÿ‘ ) {\ddot{\mathbf{x}}_{\mathbf{1}}}=-\frac{Gm_{2}}{\left(x_{1}-x_{2}\right)^{3}}% \left(\mathbf{x_{1}}-\mathbf{x_{2}}\right)-\frac{Gm_{3}}{\left(x_{1}-x_{3}% \right)^{3}}\left(\mathbf{x_{1}}-\mathbf{x_{3}}\right)
  4. ๐ฑ ยจ ๐Ÿ = - G m 3 ( x 2 - x 3 ) 3 ( ๐ฑ ๐Ÿ - ๐ฑ ๐Ÿ‘ ) - G m 1 ( x 2 - x 1 ) 3 ( ๐ฑ ๐Ÿ - ๐ฑ ๐Ÿ ) {\ddot{\mathbf{x}}_{\mathbf{2}}}=-\frac{Gm_{3}}{\left(x_{2}-x_{3}\right)^{3}}% \left(\mathbf{x_{2}}-\mathbf{x_{3}}\right)-\frac{Gm_{1}}{\left(x_{2}-x_{1}% \right)^{3}}\left(\mathbf{x_{2}}-\mathbf{x_{1}}\right)
  5. ๐ฑ ยจ ๐Ÿ‘ = - G m 1 ( x 3 - x 1 ) 3 ( ๐ฑ ๐Ÿ‘ - ๐ฑ ๐Ÿ ) - G m 2 ( x 3 - x 2 ) 3 ( ๐ฑ ๐Ÿ‘ - ๐ฑ ๐Ÿ ) {\ddot{\mathbf{x}}_{\mathbf{3}}}=-\frac{Gm_{1}}{\left(x_{3}-x_{1}\right)^{3}}% \left(\mathbf{x_{3}}-\mathbf{x_{1}}\right)-\frac{Gm_{2}}{\left(x_{3}-x_{2}% \right)^{3}}\left(\mathbf{x_{3}}-\mathbf{x_{2}}\right)
  6. | Ln s | โ‰ค ฮฒ |\mathop{\,\text{Ln}}\,s|\leq\beta
  7. ฯƒ = e ฯ€ s / ( 2 ฮฒ ) - 1 e ฯ€ s / ( 2 ฮฒ ) + 1 . \sigma=\frac{e^{\pi s/(2\beta)}-1}{e^{\pi s/(2\beta)}+1}\,.
  8. 10 8 , 000 , 000 10^{8,000,000}

Threshold_voltage.html

  1. V S B V_{SB}
  2. V T N = V T O + ฮณ ( | V S B + 2 ฯ• F | - | 2 ฯ• F | ) V_{TN}=V_{TO}+\gamma(\sqrt{|{V_{SB}+2\phi_{F}|}}-\sqrt{|2\phi_{F}|})
  3. V T N V_{TN}
  4. V S B V_{SB}
  5. 2 ฯ• F 2\phi_{F}
  6. V T O V_{TO}
  7. ฮณ = ( t o x / ฯต o x ) 2 q ฯต s i N A \gamma=(t_{ox}/\epsilon_{ox})\sqrt{2q\epsilon_{si}N_{A}}
  8. t o x t_{ox}
  9. ฯต o x \epsilon_{ox}
  10. ฯต s i \epsilon_{si}
  11. N A N_{A}
  12. q q
  13. V T N V_{TN}
  14. ฮณ \gamma
  15. t O X t_{OX}
  16. I f n = C 1 W L ( E o x ) 2 e - E 0 / E o x I_{fn}=C_{1}WL(E_{ox})^{2}e^{-E_{0}/E_{ox}}
  17. C 1 C_{1}
  18. E 0 E_{0}
  19. E o x E_{ox}
  20. ฯ• F = ( k T / q ) ln ( N A / N i ) \phi_{F}=(kT/q)\ln{(N_{A}/N_{i})}
  21. ฯ• F \phi_{F}
  22. k k
  23. T T
  24. q q
  25. N A N_{A}
  26. N i N_{i}

Thresholding_(image_processing).html

  1. I i , j I_{i,j}
  2. I i , j < T I_{i,j}<T

Time-invariant_system.html

  1. x ( t ) x(t)
  2. y ( t ) y(t)
  3. x ( t + ฮด ) x(t+\delta)
  4. y ( t + ฮด ) y(t+\delta)
  5. y ( t ) = t x ( t ) y(t)=t\,x(t)
  6. y ( t ) = 10 x ( t ) y(t)=10x(t)
  7. x ( t ) x(t)
  8. y ( t ) y(t)
  9. x d ( t ) = x ( t + ฮด ) x_{d}(t)=\,\!x(t+\delta)
  10. y ( t ) = t x ( t ) y(t)=t\,x(t)
  11. y 1 ( t ) = t x d ( t ) = t x ( t + ฮด ) y_{1}(t)=t\,x_{d}(t)=t\,x(t+\delta)
  12. ฮด \delta
  13. y ( t ) = t x ( t ) y(t)=t\,x(t)
  14. y 2 ( t ) = y ( t + ฮด ) = ( t + ฮด ) x ( t + ฮด ) y_{2}(t)=\,\!y(t+\delta)=(t+\delta)x(t+\delta)
  15. y 1 ( t ) โ‰  y 2 ( t ) y_{1}(t)\,\!\neq y_{2}(t)
  16. x d ( t ) = x ( t + ฮด ) x_{d}(t)=\,\!x(t+\delta)
  17. y ( t ) = 10 x ( t ) y(t)=10\,x(t)
  18. y 1 ( t ) = 10 x d ( t ) = 10 x ( t + ฮด ) y_{1}(t)=10\,x_{d}(t)=10\,x(t+\delta)
  19. ฮด \,\!\delta
  20. y ( t ) = 10 x ( t ) y(t)=10\,x(t)
  21. y 2 ( t ) = y ( t + ฮด ) = 10 x ( t + ฮด ) y_{2}(t)=y(t+\delta)=10\,x(t+\delta)
  22. y 1 ( t ) = y 2 ( t ) y_{1}(t)=\,\!y_{2}(t)
  23. ๐•‹ r \mathbb{T}_{r}
  24. r r
  25. x ( t + 1 ) = ฮด ( t + 1 ) * x ( t ) x(t+1)=\,\!\delta(t+1)*x(t)
  26. x ~ 1 = ๐•‹ 1 x ~ \tilde{x}_{1}=\mathbb{T}_{1}\,\tilde{x}
  27. x ~ \tilde{x}
  28. x ~ = x ( t ) โˆ€ t โˆˆ โ„ \tilde{x}=x(t)\,\forall\,t\in\mathbb{R}
  29. x ~ 1 = x ( t + 1 ) โˆ€ t โˆˆ โ„ \tilde{x}_{1}=x(t+1)\,\forall\,t\in\mathbb{R}
  30. ๐•‹ 1 \mathbb{T}_{1}
  31. โ„ \mathbb{H}
  32. ๐•‹ r โ„ = โ„ ๐•‹ r โˆ€ r \mathbb{T}_{r}\,\mathbb{H}=\mathbb{H}\,\mathbb{T}_{r}\,\,\forall\,r
  33. y ~ = โ„ x ~ \tilde{y}=\mathbb{H}\,\tilde{x}
  34. โ„ \mathbb{H}
  35. x ~ \tilde{x}
  36. ๐•‹ r \mathbb{T}_{r}
  37. ๐•‹ r \mathbb{T}_{r}
  38. โ„ \mathbb{H}
  39. ๐•‹ r โ„ x ~ = ๐•‹ r y ~ = y ~ r \mathbb{T}_{r}\,\mathbb{H}\,\tilde{x}=\mathbb{T}_{r}\,\tilde{y}=\tilde{y}_{r}
  40. โ„ ๐•‹ r x ~ = โ„ x ~ r \mathbb{H}\,\mathbb{T}_{r}\,\tilde{x}=\mathbb{H}\,\tilde{x}_{r}
  41. โ„ x ~ r = y ~ r \mathbb{H}\,\tilde{x}_{r}=\tilde{y}_{r}

Time-lapse_photography.html

  1. perceived speed = projection frame rate camera frame rate ร— actual speed \mathrm{perceived\ speed}=\frac{\mathrm{projection\ frame\ rate}}{\mathrm{% camera\ frame\ rate}}\times\mathrm{actual\ speed}
  2. exposure time = shutter angle 360 โˆ˜ ร— frame interval \mathrm{exposure\ time}=\frac{\mathrm{shutter\ angle}}{360^{\circ}}\times% \mathrm{frame\ interval}
  3. actual speed = camera frame rate projection frame rate ร— perceived speed \mathrm{actual\ speed}=\frac{\mathrm{camera\ frame\ rate}}{\mathrm{projection% \ frame\ rate}}\times\mathrm{perceived\ speed}

Time-variant_system.html

  1. y ( t ) = T ( x ( t ) , t ) y(t)=T(x(t),t)
  2. T ( x ( t - k ) , t ) T(x(t-k),t)
  3. y ( t - k ) = T ( x ( t - k ) , t - k ) y(t-k)=T(x(t-k),t-k)

Time_consistency.html

  1. t t
  2. t + 1 t+1
  3. ( ฯ t ) t = 0 T \left(\rho_{t}\right)_{t=0}^{T}
  4. L 0 ( โ„ฑ T ) L^{0}(\mathcal{F}_{T})
  5. โˆ€ X , Y โˆˆ L 0 ( โ„ฑ T ) \forall X,Y\in L^{0}(\mathcal{F}_{T})
  6. t โˆˆ { 0 , 1 , โ€ฆ , T - 1 } : ฯ t + 1 ( X ) โ‰ฅ ฯ t + 1 ( Y ) t\in\{0,1,...,T-1\}:\rho_{t+1}(X)\geq\rho_{t+1}(Y)
  7. ฯ t ( X ) โ‰ฅ ฯ t ( Y ) \rho_{t}(X)\geq\rho_{t}(Y)
  8. t โˆˆ { 0 , 1 , โ€ฆ , T - 1 } : ฯ t + 1 ( X ) = ฯ t + 1 ( Y ) โ‡’ ฯ t ( X ) = ฯ t ( Y ) t\in\{0,1,...,T-1\}:\rho_{t+1}(X)=\rho_{t+1}(Y)\Rightarrow\rho_{t}(X)=\rho_{t}% (Y)
  9. t โˆˆ { 0 , 1 , โ€ฆ , T - 1 } : ฯ t ( X ) = ฯ t ( - ฯ t + 1 ( X ) ) t\in\{0,1,...,T-1\}:\rho_{t}(X)=\rho_{t}(-\rho_{t+1}(X))
  10. t โˆˆ { 0 , 1 , โ€ฆ , T - 1 } : A t = A t , t + 1 + A t + 1 t\in\{0,1,...,T-1\}:A_{t}=A_{t,t+1}+A_{t+1}
  11. A t A_{t}
  12. t t
  13. A t , t + 1 = A t โˆฉ L p ( โ„ฑ t + 1 ) A_{t,t+1}=A_{t}\cap L^{p}(\mathcal{F}_{t+1})
  14. t โˆˆ { 0 , 1 , โ€ฆ , T - 1 } : ฮฑ t ( Q ) = ฮฑ t , t + 1 ( Q ) + ๐”ผ Q [ ฮฑ t + 1 ( Q ) โˆฃ โ„ฑ t ] t\in\{0,1,...,T-1\}:\alpha_{t}(Q)=\alpha_{t,t+1}(Q)+\mathbb{E}^{Q}[\alpha_{t+1% }(Q)\mid\mathcal{F}_{t}]
  15. ฮฑ t ( Q ) = esssup X โˆˆ A t ๐”ผ Q [ - X โˆฃ โ„ฑ t ] \alpha_{t}(Q)=\operatorname*{esssup}_{X\in A_{t}}\mathbb{E}^{Q}[-X\mid\mathcal% {F}_{t}]
  16. A t A_{t}
  17. esssup \operatorname*{esssup}
  18. t t
  19. ฮฑ t , t + 1 ( Q ) = esssup X โˆˆ A t , t + 1 ๐”ผ Q [ - X โˆฃ โ„ฑ t ] \alpha_{t,t+1}(Q)=\operatorname*{esssup}_{X\in A_{t,t+1}}\mathbb{E}^{Q}[-X\mid% \mathcal{F}_{t}]
  20. ฯ T - 1 c o m := ฯ T - 1 \rho^{com}_{T-1}:=\rho_{T-1}
  21. โˆ€ t < T - 1 : ฯ t c o m := ฯ t ( - ฯ t + 1 c o m ) \forall t<T-1:\rho^{com}_{t}:=\rho_{t}(-\rho^{com}_{t+1})
  22. ฮฑ t \alpha_{t}
  23. ฯ t ( X ) = ess sup Q โˆˆ ๐’ฌ E Q [ - X | โ„ฑ t ] \rho_{t}(X)=\,\text{ess}\sup_{Q\in\mathcal{Q}}E^{Q}[-X|\mathcal{F}_{t}]
  24. ๐’ฌ = { Q โˆˆ โ„ณ 1 : E [ d Q d P | โ„ฑ j ] โ‰ค ฮฑ j - 1 E [ d Q d P | โ„ฑ j - 1 ] โˆ€ j = 1 , โ€ฆ , T } \mathcal{Q}=\left\{Q\in\mathcal{M}_{1}:E\left[\frac{dQ}{dP}|\mathcal{F}_{j}% \right]\leq\alpha_{j-1}E\left[\frac{dQ}{dP}|\mathcal{F}_{j-1}\right]\forall j=% 1,...,T\right\}
  25. ฯ g ( X ) := ๐”ผ g [ - X ] \rho_{g}(X):=\mathbb{E}^{g}[-X]
  26. g g
  27. ๐”ผ g \mathbb{E}^{g}
  28. g g

Time_derivative.html

  1. t t\,
  2. d x d t \frac{dx}{dt}
  3. x ห™ \dot{x}
  4. d 2 x d t 2 \frac{d^{2}x}{dt^{2}}
  5. x ยจ \ddot{x}
  6. V โ†’ = [ v 1 , v 2 , v 3 , โ‹ฏ ] , \vec{V}=\left[v_{1},\ v_{2},\ v_{3},\cdots\right]\ ,
  7. d V โ†’ d t = [ d v 1 d t , d v 2 d t , d v 3 d t , โ‹ฏ ] . \frac{d\vec{V}}{dt}=\left[\frac{dv_{1}}{dt},\frac{dv_{2}}{dt},\frac{dv_{3}}{dt% },\cdots\right]\ .
  8. x x\,
  9. x ห™ \dot{x}
  10. x ยจ \ddot{x}
  11. r = x i ^ + y j ^ r=x\hat{i}+y\hat{j}
  12. x \displaystyle x
  13. ๐ซ ( t ) = r cos ( t ) i ^ + r sin ( t ) j ^ \mathbf{r}(t)=r\cos(t)\hat{i}+r\sin(t)\hat{j}
  14. | ๐ซ ( t ) | = ๐ซ ( t ) โ‹… ๐ซ ( t ) = x ( t ) 2 + y ( t ) 2 = r cos 2 ( t ) + sin 2 ( t ) = r |\mathbf{r}(t)|=\sqrt{\mathbf{r}(t)\cdot\mathbf{r}(t)}=\sqrt{x(t)^{2}+y(t)^{2}% }=r\,\sqrt{\cos^{2}(t)+\sin^{2}(t)}=r
  15. โ‹… \cdot
  16. ๐ฏ ( t ) = d ๐ซ ( t ) d t \displaystyle\mathbf{v}(t)=\frac{d\,\mathbf{r}(t)}{dt}
  17. ๐ฏ โ‹… ๐ซ = [ - y , x ] โ‹… [ x , y ] = - y x + x y = 0โ€‰. \mathbf{v}\cdot\mathbf{r}=[-y,x]\cdot[x,y]=-yx+xy=0\,.
  18. ๐š ( t ) = d ๐ฏ ( t ) d t = [ - x ( t ) , - y ( t ) ] = - ๐ซ ( t ) . \mathbf{a}(t)=\frac{d\,\mathbf{v}(t)}{dt}=[-x(t),-y(t)]=-\mathbf{r}(t)\,.

Time_of_flight_detector.html

  1. m 1 m_{1}
  2. m 2 m_{2}
  3. v 1 v_{1}
  4. v 2 v_{2}
  5. ฮด t = L ( 1 v 1 - 1 v 2 ) โ‰ˆ L c 2 p 2 ( m 1 2 - m 2 2 ) \delta t=L\left(\frac{1}{v_{1}}-\frac{1}{v_{2}}\right)\approx\frac{Lc}{2p^{2}}% (m_{1}^{2}-m_{2}^{2})
  6. L L
  7. p p
  8. c c
  9. ฮด t \delta t

Time_reversibility.html

  1. ฯ€ i p i j = ฯ€ j p j i , \pi_{i}p_{ij}=\pi_{j}p_{ji},\,

Time_value.html

  1. T v T_{v}

Toda_field_theory.html

  1. โ„’ = 1 2 [ ( โˆ‚ ฯ• โˆ‚ t , โˆ‚ ฯ• โˆ‚ t ) - ( โˆ‚ ฯ• โˆ‚ x , โˆ‚ ฯ• โˆ‚ x ) ] - m 2 ฮฒ 2 โˆ‘ i = 1 r n i e ฮฒ ฮฑ i โ‹… ฯ• . \mathcal{L}=\frac{1}{2}\left[\left({\partial\phi\over\partial t},{\partial\phi% \over\partial t}\right)-\left({\partial\phi\over\partial x},{\partial\phi\over% \partial x}\right)\right]-{m^{2}\over\beta^{2}}\sum_{i=1}^{r}n_{i}e^{\beta% \alpha_{i}\cdot\phi}.
  2. ๐”ฅ \mathfrak{h}
  3. ๐”ฅ \mathfrak{h}
  4. ( 2 - 2 - 2 2 ) \begin{pmatrix}2&-2\\ -2&2\end{pmatrix}

Todd_class.html

  1. Q ( x ) = x 1 - e - x = โˆ‘ i = 0 โˆž ( - 1 ) i B i i ! x i = 1 + x 2 + x 2 12 - x 4 720 + โ‹ฏ Q(x)=\frac{x}{1-e^{-x}}=\sum_{i=0}^{\infty}\frac{(-1)^{i}B_{i}}{i!}x^{i}=1+% \dfrac{x}{2}+\dfrac{x^{2}}{12}-\dfrac{x^{4}}{720}+\cdots
  2. โˆ i = 1 m Q ( ฮฒ i x ) \prod_{i=1}^{m}Q(\beta_{i}x)
  3. t d ( E ) = โˆ Q ( ฮฑ i ) td(E)=\prod Q(\alpha_{i})
  4. T d * ( E โŠ• F ) = T d * ( E ) โ‹… T d * ( F ) . Td^{*}(E\oplus F)=Td^{*}(E)\cdot Td^{*}(F).
  5. ฮพ โˆˆ H 2 ( โ„‚ P n ) \xi\in H^{2}({\mathbb{C}}P^{n})
  6. โ„‚ P n {\mathbb{C}}P^{n}
  7. 0 โ†’ ๐’ช โ†’ ๐’ช ( 1 ) n + 1 โ†’ T โ„‚ P n โ†’ 0 , 0\to{\mathcal{O}}\to{\mathcal{O}}(1)^{n+1}\to T{\mathbb{C}}P^{n}\to 0,
  8. T d * ( T โ„‚ P n ) = ( ฮพ 1 - e - ฮพ ) n + 1 . Td^{*}(T{\mathbb{C}}P^{n})=\left(\dfrac{\xi}{1-e^{-\xi}}\right)^{n+1}.
  9. ฯ‡ ( F ) = โˆซ M C h * ( F ) โˆง T d * ( T M ) , \chi(F)=\int_{M}Ch^{*}(F)\wedge Td^{*}(TM),
  10. ฯ‡ ( F ) \chi(F)
  11. ฯ‡ ( F ) := โˆ‘ i = 0 dim โ„‚ M ( - 1 ) i dim โ„‚ H i ( F ) , \chi(F):=\sum_{i=0}^{\,\text{dim}_{\mathbb{C}}M}(-1)^{i}\,\text{dim}_{\mathbb{% C}}H^{i}(F),

Topness.html

  1. t ยฏ \overline{t}
  2. T = n t - n t ยฏ T=n\text{t}-n_{\bar{\,\text{t}}}

Topological_property.html

  1. ฮบ \kappa
  2. ฮบ \kappa
  3. ฮบ \kappa
  4. X X
  5. ฮ” ( X ) \Delta(X)
  6. ฮ” ( X ) = min { | G | : G โ‰  โˆ… , G is open } \Delta(X)=\min\{|G|:G\neq\emptyset,G\mbox{ is open}~{}\}
  7. ฮ” ( X ) \Delta(X)
  8. X X
  9. D D
  10. X X
  11. D D
  12. X X
  13. X X

Topological_skeleton.html

  1. B โІ A B\subseteq A
  2. D โŠˆ A D\not\subseteq A

Torricelli's_equation.html

  1. v f 2 = v i 2 + 2 a ฮ” d v_{f}^{2}=v_{i}^{2}+2a\Delta d\,
  2. v f v_{f}
  3. v i v_{i}
  4. a a
  5. ฮ” d \Delta d\,
  6. v f = v i + a t v_{f}=v_{i}+at\,\!
  7. v f 2 = ( v i + a t ) 2 = v i 2 + 2 a v i t + a 2 t 2 v_{f}^{2}=(v_{i}+at)^{2}=v_{i}^{2}+2av_{i}t+a^{2}t^{2}\,\!
  8. t 2 t^{2}\,\!
  9. d = d i + v i t + a t 2 2 d=d_{i}+v_{i}t+a\frac{t^{2}}{2}
  10. d - d i - v i t = a t 2 2 d-d_{i}-v_{i}t=a\frac{t^{2}}{2}
  11. t 2 = 2 d - d i - v i t a = 2 ฮ” d - v i t a t^{2}=2\frac{d-d_{i}-v_{i}t}{a}=2\frac{\Delta d-v_{i}t}{a}
  12. v f 2 = v i 2 + 2 a v i t + a 2 ( 2 ฮ” d - v i t a ) v_{f}^{2}=v_{i}^{2}+2av_{i}t+a^{2}\left(2\frac{\Delta d-v_{i}t}{a}\right)
  13. v f 2 = v i 2 + 2 a v i t + 2 a ( ฮ” d - v i t ) v_{f}^{2}=v_{i}^{2}+2av_{i}t+2a(\Delta d-v_{i}t)
  14. v f 2 = v i 2 + 2 a v i t + 2 a ฮ” d - 2 a v i t v_{f}^{2}=v_{i}^{2}+2av_{i}t+2a\Delta d-2av_{i}t\,\!
  15. v f 2 = v i 2 + 2 a ฮ” d v_{f}^{2}=v_{i}^{2}+2a\Delta d\,\!

Total_angular_momentum_quantum_number.html

  1. ๐ฃ = ๐ฌ + s y m b o l โ„“ \mathbf{j}=\mathbf{s}+symbol{\ell}
  2. | โ„“ - s | โ‰ค j โ‰ค โ„“ + s |\ell-s|\leq j\leq\ell+s
  3. โˆฅ ๐ฃ โˆฅ = j ( j + 1 ) โ„ \|\mathbf{j}\|=\sqrt{j\,(j+1)}\,\hbar
  4. j z = m j โ„ j_{z}=m_{j}\,\hbar

Total_factor_productivity.html

  1. Y = A ร— K ฮฑ ร— L ฮฒ Y=A\times K^{\alpha}\times L^{\beta}

Total_organic_carbon.html

  1. S 2 O 8 2 - โŸถ h v 2 SO 4 - โˆ™ \mathrm{S}_{2}\mathrm{O}_{8}^{2-}\underset{hv}{\longrightarrow}2\ \mathrm{SO}_% {4}^{-\bullet}
  2. H 2 O โŸถ h v H + + OH โˆ™ \mathrm{H}_{2}\mathrm{O}\underset{hv}{\longrightarrow}\mathrm{H}^{+}+\mathrm{% OH}^{\bullet}
  3. SO 4 - โˆ™ + H 2 O โŸถ SO 4 2 - + OH โˆ™ + H + \mathrm{SO}_{4}^{-\bullet}+\mathrm{H}_{2}\mathrm{O}\longrightarrow\mathrm{SO}_% {4}^{2-}+\mathrm{OH}^{\bullet}+\mathrm{H}^{+}
  4. R โŸถ h v R * \mathrm{R}\underset{hv}{\longrightarrow}\mathrm{R}^{*}
  5. R * + SO 4 - โˆ™ + OH โˆ™ โŸถ n CO 2 + โ€ฆ \mathrm{R}^{*}+\mathrm{SO}_{4}^{-\bullet}+\mathrm{OH}^{\bullet}\longrightarrow n% \mathrm{CO}_{2}+\dots

Total_relation.html

  1. โˆ€ a , b โˆˆ X , a R b b R a . \forall a,b\in X,\ aRbbRa.
  2. โˆ€ a , b โˆˆ X , a R b b R a ( a = b ) . \forall a,b\in X,\ aRbbRa(a=b).

Totally_bounded_space.html

  1. ( M , d ) (M,d)
  2. ฯต > 0 \epsilon>0
  3. M M
  4. ฯต \epsilon
  5. M M
  6. M M
  7. ฯต > 0 \epsilon>0
  8. ฯต \epsilon
  9. โˆ€ E โˆƒ n โˆˆ โ„• , A 1 , A 2 , โ€ฆ , A n โІ X ( S โІ โ‹ƒ i = 1 n A i and โˆ€ i = 1 , โ€ฆ , n size ( A i ) โ‰ค E ) . \forall{E}\;\exists{n\in\mathbb{N}}\;,{A_{1},A_{2},\ldots,A_{n}\subseteq X}% \left(S\subseteq\bigcup_{i=1}^{n}A_{i}\;\mbox{ and }~{}\;\forall{i=1,\ldots,n}% \;\mathrm{size}(A_{i})\leq E\right).\!

Tournament_(graph_theory).html

  1. a a
  2. b b
  3. a a
  4. b b
  5. n n
  6. n n
  7. n n
  8. n n
  9. T T
  10. n + 1 n+1
  11. v 0 v_{0}
  12. T T
  13. v 1 , v 2 , โ€ฆ , v n v_{1},v_{2},\ldots,v_{n}
  14. T โˆ– { v 0 } T\setminus\{v_{0}\}
  15. i โˆˆ { 0 , โ€ฆ , n } i\in\{0,\ldots,n\}
  16. j โ‰ค i j\leq i
  17. v j v_{j}
  18. v 0 v_{0}
  19. v 1 , โ€ฆ , v i , v 0 , v i + 1 , โ€ฆ , v n v_{1},\ldots,v_{i},v_{0},v_{i+1},\ldots,v_{n}
  20. O ( n log n ) \ O(n\log n)
  21. ( ( a โ†’ b ) ((a\rightarrow b)
  22. ( b โ†’ c ) ) (b\rightarrow c))
  23. โ‡’ \Rightarrow
  24. ( a โ†’ c ) (a\rightarrow c)
  25. 1 + โŒŠ log 2 n โŒ‹ 1+\lfloor\log_{2}n\rfloor
  26. 2 + 2 โŒŠ log 2 n โŒ‹ 2+2\lfloor\log_{2}n\rfloor
  27. ( n k ) k ! 2 ( n 2 ) - ( k 2 ) , {\left({{n}\atop{k}}\right)}k!2^{{\left({{n}\atop{2}}\right)}-{\left({{k}\atop% {2}}\right)}},
  28. 2 + 2 โŒŠ log 2 n โŒ‹ 2+2\lfloor\log_{2}n\rfloor
  29. 2 ( n 2 ) 2^{{\left({{n}\atop{2}}\right)}}
  30. V โˆ– S V\setminus S
  31. v 0 โ†’ v v_{0}\rightarrow v
  32. v โˆˆ S v\in S
  33. ( s 1 , s 2 , โ‹ฏ , s n ) (s_{1},s_{2},\cdots,s_{n})
  34. 0 โ‰ค s 1 โ‰ค s 2 โ‰ค โ‹ฏ โ‰ค s n 0\leq s_{1}\leq s_{2}\leq\cdots\leq s_{n}
  35. s 1 + s 2 + โ‹ฏ + s i โ‰ฅ ( i 2 ) , for i = 1 , 2 , โ‹ฏ , n - 1 s_{1}+s_{2}+\cdots+s_{i}\geq{i\choose 2},\mbox{for }~{}i=1,2,\cdots,n-1
  36. s 1 + s 2 + โ‹ฏ + s n = ( n 2 ) . s_{1}+s_{2}+\cdots+s_{n}={n\choose 2}.
  37. s ( n ) s(n)
  38. n n
  39. s ( n ) s(n)
  40. s ( n ) > c 1 4 n n - 5 2 , s(n)>c_{1}4^{n}n^{-{5\over 2}},
  41. c 1 = 0.049. c_{1}=0.049.
  42. s ( n ) < c 2 4 n n - 5 2 , s(n)<c_{2}4^{n}n^{-{5\over 2}},
  43. c 2 < 4.858. c_{2}<4.858.
  44. s ( n ) โˆˆ ฮ˜ ( 4 n n - 5 2 ) . s(n)\in\Theta(4^{n}n^{-{5\over 2}}).
  45. ฮ˜ \Theta

Tractatus_Logico-Philosophicus_(6.5).html

  1. โ–ก \Box
  2. โ–ก \Box
  3. โ–ก \Box
  4. โ–ก \Box
  5. โ–ก \Box
  6. โ–ก \Box
  7. โ–ก \Box
  8. โ–ก \Box

Trajectory_of_a_projectile.html

  1. d = v cos ฮธ g ( v sin ฮธ + ( v sin ฮธ ) 2 + 2 g y 0 ) d=\frac{v\cos\theta}{g}\left(v\sin\theta+\sqrt{(v\sin\theta)^{2}+2gy_{0}}\right)
  2. d = v 2 sin ( 2 ฮธ ) g d=\frac{v^{2}\sin(2\theta)}{g}
  3. d = v 2 g d=\frac{v^{2}}{g}
  4. t = d v cos ฮธ = v sin ฮธ + ( v sin ฮธ ) 2 + 2 g y 0 g t=\frac{d}{v\cos\theta}=\frac{v\sin\theta+\sqrt{(v\sin\theta)^{2}+2gy_{0}}}{g}
  5. t = 2 โ‹… v g t=\frac{\sqrt{2}\cdot v}{g}
  6. sin ( 2 ฮธ ) = g d v 2 \sin(2\theta)=\frac{gd}{v^{2}}
  7. ฮธ = 1 2 arcsin ( g d v 2 ) \theta=\frac{1}{2}\arcsin\left(\frac{gd}{v^{2}}\right)
  8. y = y 0 + x tan ฮธ - g x 2 2 ( v cos ฮธ ) 2 y=y_{0}+x\tan\theta-\frac{gx^{2}}{2(v\cos\theta)^{2}}
  9. | v | , |v|,
  10. | v | = v 2 - 2 g x tan ฮธ + ( g x v cos ฮธ ) 2 |v|=\sqrt{v^{2}-2gx\tan\theta+\left(\frac{gx}{v\cos\theta}\right)^{2}}
  11. | v | = V x 2 + V y 2 |v|=\sqrt{V_{x}^{2}+V_{y}^{2}}
  12. v f = v i + a t v_{f}=v_{i}+at
  13. t = x v cos ฮธ t=\frac{x}{v\cos\theta}
  14. V y = v sin ฮธ - g x v cos ฮธ V_{y}=v\sin\theta-\frac{gx}{v\cos\theta}
  15. | v | = ( v cos ฮธ ) 2 + ( v sin ฮธ - g x v cos ฮธ ) 2 |v|=\sqrt{(v\cos\theta)^{2}+\left(v\sin\theta-\frac{gx}{v\cos\theta}\right)^{2}}
  16. ฮธ \theta
  17. ฮธ \theta
  18. ฮธ = arctan ( v 2 ยฑ v 4 - g ( g x 2 + 2 y v 2 ) g x ) \theta=\arctan{\left(\frac{v^{2}\pm\sqrt{v^{4}-g(gx^{2}+2yv^{2})}}{gx}\right)}
  19. x = v t cos ฮธ x=vt\cos\theta
  20. y = v t sin ฮธ - 1 2 g t 2 y=vt\sin\theta-\frac{1}{2}gt^{2}
  21. y = x tan ฮธ - g x 2 2 v 2 cos 2 ฮธ = x tan ฮธ - g x 2 2 v 2 ( 1 + tan 2 ฮธ ) y=x\tan\theta-\frac{gx^{2}}{2v^{2}\cos^{2}\theta}=x\tan\theta-\frac{gx^{2}}{2v% ^{2}}(1+\tan^{2}\theta)
  22. 0 = - g x 2 2 v 2 tan 2 ฮธ + x tan ฮธ - g x 2 2 v 2 - y 0=\frac{-gx^{2}}{2v^{2}}\tan^{2}\theta+x\tan\theta-\frac{gx^{2}}{2v^{2}}-y
  23. p = tan ฮธ p=\tan\theta
  24. 0 = - g x 2 2 v 2 p 2 + x p - g x 2 2 v 2 - y 0=\frac{-gx^{2}}{2v^{2}}p^{2}+xp-\frac{gx^{2}}{2v^{2}}-y
  25. p = - x ยฑ x 2 - 4 ( - g x 2 2 v 2 ) ( - g x 2 2 v 2 - y ) 2 ( - g x 2 2 v 2 ) p={\frac{-x\pm\sqrt{x^{2}-4(\frac{-gx^{2}}{2v^{2}})(\frac{-gx^{2}}{2v^{2}}-y)}% }{2(\frac{-gx^{2}}{2v^{2}})}}
  26. tan ฮธ = v 2 ยฑ v 4 - g ( g x 2 + 2 y v 2 ) g x \tan\theta=\frac{v^{2}\pm\sqrt{v^{4}-g(gx^{2}+2yv^{2})}}{gx}
  27. ฮธ = tan - 1 ( v 2 ยฑ v 4 - g ( g x 2 + 2 y v 2 ) g x ) \theta=\tan^{-1}{\left(\frac{v^{2}\pm\sqrt{v^{4}-g(gx^{2}+2yv^{2})}}{gx}\right)}
  28. ฯ• \phi
  29. x = r cos ฯ• x=r\cos\phi
  30. y = r sin ฯ• y=r\sin\phi
  31. ฮธ = tan - 1 ( v 2 ยฑ v 4 - g ( g r 2 cos 2 ฯ• + 2 v 2 r sin ฯ• ) g r cos ฯ• ) \theta=\tan^{-1}{\left(\frac{v^{2}\pm\sqrt{v^{4}-g(gr^{2}\cos^{2}\phi+2v^{2}r% \sin\phi)}}{gr\cos\phi}\right)}
  32. v y v_{y}
  33. v x v_{x}
  34. h = v y t - 1 2 g t 2 , h=v_{y}t-\frac{1}{2}gt^{2},
  35. t t
  36. h = 0 h=0
  37. โˆด T = 2 v y g . \therefore T=\frac{2v_{y}}{g}.
  38. t t
  39. ( 0 โ‰ค t โ‰ค T ) (0\leq t\leq T)
  40. d = v x t d=v_{x}t
  41. D = d ( T ) = 2 v x v y g D=d(T)=\frac{2v_{x}v_{y}}{g}
  42. t t
  43. c = D - d = 2 v x v y g - v x t c=D-d=\frac{2v_{x}v_{y}}{g}-v_{x}t
  44. tan ( e ) = h c \tan(e)=\frac{h}{c}
  45. = v y t - 1 2 g t 2 , 2 v x v y g - v x t =\frac{v_{y}t-\frac{1}{2}gt^{2},}{\frac{2v_{x}v_{y}}{g}-v_{x}t}
  46. = g t 2 v x =\frac{gt}{2v_{x}}
  47. tan ( e ) = ( g 2 v x ) t \tan(e)=\left(\frac{g}{2v_{x}}\right)t
  48. F a โˆ v โ†’ F_{a}\propto\vec{v}
  49. v 0 v_{0}
  50. v x v_{x}
  51. v y v_{y}
  52. 0 o โ‰ค ฮธ โ‰ค 180 o 0^{o}\leq\theta\leq 180^{o}
  53. F a i r = - k v F_{air}=-kv
  54. F a i r = - k v 2 F_{air}=-kv^{2}
  55. F = - k x F=-kx
  56. 4 m / s = 7 N 4\ \mathrm{m}/\mathrm{s}=7\ \mathrm{N}
  57. 4 m / s ร— ( 7 4 N ร— s m ) = 7 N 4\ \mathrm{m}/\mathrm{s}\times(\frac{7}{4}\ \mathrm{N}\times\frac{\mathrm{s}}{% \mathrm{m}})=7\ \mathrm{N}
  58. 4 N ร— 7 4 = 7 N 4\ \mathrm{N}\times\frac{7}{4}=7\ \mathrm{N}
  59. s m ร— m s \frac{\mathrm{s}}{\mathrm{m}}\times\frac{\mathrm{m}}{\mathrm{s}}
  60. 7 N = 7 N ( 4 ร— 7 4 ) = 7 7\ \mathrm{N}=7\ \mathrm{N}(4\times\frac{7}{4})=7
  61. ฮฃ F = - k v x = m a x \Sigma F=-kv_{x}=ma_{x}
  62. ฮฃ F = - k v y - m g = m a y \Sigma F=-kv_{y}-mg=ma_{y}
  63. a x = - k v x m = d v x d t a_{x}=\frac{-kv_{x}}{m}=\frac{dv_{x}}{dt}
  64. a y = 1 m ( - k v y - m g ) = - k v y m - g = d v y d t a_{y}=\frac{1}{m}(-kv_{y}-mg)=\frac{-kv_{y}}{m}-g=\frac{dv_{y}}{dt}
  65. v x v_{x}
  66. x x
  67. v x = v x o v_{x}=v_{xo}
  68. v x o v_{xo}
  69. s x = 0 s_{x}=0
  70. t = 0 t=0
  71. v x = v x o e - k m t v_{x}=v_{xo}e^{-\frac{k}{m}t}
  72. s x = m k v x o ( 1 - e - k m t ) s_{x}=\frac{m}{k}v_{xo}(1-e^{-\frac{k}{m}t})
  73. v y = v y o v_{y}=v_{yo}
  74. s y = 0 s_{y}=0
  75. t = 0 t=0
  76. d v y d t = - k m v y - g \frac{dv_{y}}{dt}=\frac{-k}{m}v_{y}-g
  77. d v y d t + k m v y = - g \frac{dv_{y}}{dt}+\frac{k}{m}v_{y}=-g
  78. e โˆซ k m d t e^{\int\frac{k}{m}\,dt}
  79. e k m t ( d v y d t + k m v y ) = e k m t ( - g ) e^{\frac{k}{m}t}(\frac{dv_{y}}{dt}+\frac{k}{m}v_{y})=e^{\frac{k}{m}t}(-g)
  80. ( e k m t v y ) โ€ฒ = e k m t ( - g ) (e^{\frac{k}{m}t}v_{y})^{\prime}=e^{\frac{k}{m}t}(-g)
  81. โˆซ ( e k m t v y ) โ€ฒ d t = e k m t v y = โˆซ e k m t ( - g ) d t \int{(e^{\frac{k}{m}t}v_{y})^{\prime}\,dt}=e^{\frac{k}{m}t}v_{y}=\int{e^{\frac% {k}{m}t}(-g)\,dt}
  82. e k m t v y = m k e k m t ( - g ) + C e^{\frac{k}{m}t}v_{y}=\frac{m}{k}e^{\frac{k}{m}t}(-g)+C
  83. v y = - m g k + C e - k m t v_{y}=\frac{-mg}{k}+Ce^{\frac{-k}{m}t}
  84. s y = - m g k t - m k ( v y o + m g k ) e - k m t + C s_{y}=-\frac{mg}{k}t-\frac{m}{k}(v_{yo}+\frac{mg}{k})e^{-\frac{k}{m}t}+C
  85. v y ( t ) = - m g k + ( v y o + m g k ) e - k m t v_{y}(t)=-\frac{mg}{k}+(v_{yo}+\frac{mg}{k})e^{-\frac{k}{m}t}
  86. s y ( t ) = - m g k t - m k ( v y o + m g k ) e - k m t + m k ( v y o + m g k ) s_{y}(t)=-\frac{mg}{k}t-\frac{m}{k}(v_{yo}+\frac{mg}{k})e^{-\frac{k}{m}t}+% \frac{m}{k}(v_{yo}+\frac{mg}{k})
  87. s y ( t ) = - m g k t + m k ( v y o + m g k ) ( 1 - e - k m t ) s_{y}(t)=-\frac{mg}{k}t+\frac{m}{k}(v_{yo}+\frac{mg}{k})(1-e^{-\frac{k}{m}t})
  88. k = m g v t = ( 0.145 kg ) ( - 9.81 m / s 2 ) - 33.0 m / s = 0.0431 kg / s , ฮธ = 45 o k=\frac{mg}{v_{t}}=\frac{(0.145\mbox{ kg}~{})(-9.81\ \mathrm{m}/\mathrm{s}^{2}% )}{-33.0\ \mathrm{m}/\mathrm{s}}=0.0431\mbox{ kg}~{}/\mbox{s}~{},\ \theta=45^{o}
  89. F a โˆ v 2 F_{a}\propto v^{2}
  90. p p
  91. p 2 p^{2}
  92. 2 2 = 4 2^{2}=4
  93. F a i r = - k v 2 F_{air}=-kv^{2}
  94. k = 1 2 ฯ ๐€ C D k=\frac{1}{2}\rho\mathbf{A}C_{D}
  95. s y m b o l ฯ symbol{\rho}
  96. ๐€ \mathbf{A}
  97. s y m b o l C D symbol{C_{D}}
  98. ๐ญ \mathbf{t}
  99. ฮธ \theta
  100. a x = - k v x 2 m = d v x d t a_{x}=\frac{-kv^{2}_{x}}{m}=\frac{dv_{x}}{dt}
  101. - k d t m = d v x v x 2 \frac{-kdt}{m}=\frac{dv_{x}}{v^{2}_{x}}
  102. - โˆซ k m d t = โˆซ v x - 2 d v x -\int\frac{k}{m}dt=\int v^{-2}_{x}dv_{x}
  103. - k t m = - 1 v x + c -\frac{kt}{m}=-\frac{1}{v_{x}}+c
  104. c c
  105. t = 0 t=0
  106. v x = v x o v_{x}=v_{x_{o}}
  107. c = 1 v x o c=\frac{1}{v_{x_{o}}}
  108. k t m = 1 v x - 1 v x o \frac{kt}{m}=\frac{1}{v_{x}}-\frac{1}{v_{x_{o}}}
  109. v x = 1 1 v x 0 + k t m v_{x}=\frac{1}{\frac{1}{v_{x_{0}}}+\frac{kt}{m}}
  110. v x = 1 1 v 0 cos ฮธ + k t m v_{x}=\frac{1}{\frac{1}{v_{0}\cos\theta}+\frac{kt}{m}}
  111. โˆซ v x d t = โˆซ ( 1 1 v 0 cos ฮธ + k t m ) d t \int v_{x}dt=\int\left(\frac{1}{\frac{1}{v_{0}\cos\theta}+\frac{kt}{m}}\right)dt
  112. s x ( t ) = m k ln ( k t v 0 cos ฮธ + m ) + c s_{x}(t)=\frac{m}{k}\ln(kt{v_{0}\cos\theta}+m)+c
  113. t = 0 t=0
  114. s x = 0 s_{x}=0
  115. c = - m k ln ( m ) c=-\frac{m}{k}\ln(m)
  116. s x ( t ) = m k ( ln ( k t v 0 cos ฮธ + m m ) ) s_{x}(t)=\frac{m}{k}(\ln(\frac{kt{v_{0}\cos\theta}+m}{m}))
  117. a y = - k v y 2 m - g = d v y d t a_{y}=\frac{-kv^{2}_{y}}{m}-g=\frac{dv_{y}}{dt}
  118. v y ( t ) = m g k tan ( c - g k m t ) v_{y}(t)=\sqrt{\frac{mg}{k}}\tan(c-\sqrt{\frac{gk}{m}}t)
  119. t = 0 t=0
  120. v y ( t ) = v y o = v 0 sin ฮธ v_{y}(t)=v_{y_{o}}=v_{0}\sin\theta
  121. c = arctan ( k m g v y o ) c=\arctan(\sqrt{\frac{k}{mg}}v_{y_{o}})
  122. v y ( t ) = m g k tan ( arctan ( k m g v y o ) - g k m t ) v_{y}(t)=\sqrt{\frac{mg}{k}}\tan(\arctan(\sqrt{\frac{k}{mg}}v_{y_{o}})-\sqrt{% \frac{gk}{m}}t)
  123. โˆซ d s y = โˆซ ( m g k tan ( arctan ( k m g v y o ) - g k m t ) ) d t \int ds_{y}=\int(\sqrt{\frac{mg}{k}}\tan(\arctan(\sqrt{\frac{k}{mg}}v_{y_{o}})% -\sqrt{\frac{gk}{m}}t))dt
  124. s y ( t ) = m k ln ( cos ( g k m t - arctan ( k m g v y o ) ) ) + c s_{y}(t)=\frac{m}{k}\ln(\cos(\sqrt{\frac{gk}{m}}t-\arctan(\sqrt{\frac{k}{mg}}v% _{y_{o}})))+c
  125. t = 0 t=0
  126. s y ( t ) = 0 s_{y}(t)=0
  127. c = - m k ln ( cos ( arctan ( k m g v y o ) ) ) c=-\frac{m}{k}\ln(\cos(\arctan(\sqrt{\frac{k}{mg}}v_{y_{o}})))
  128. s y ( t ) = m k ln ( cos ( g k m t - arctan ( k m g v y o ) ) ) - m k ln ( cos ( arctan ( k m g v y o ) ) ) s_{y}(t)=\frac{m}{k}\ln(\cos(\sqrt{\frac{gk}{m}}t-\arctan(\sqrt{\frac{k}{mg}}v% _{y_{o}})))-\frac{m}{k}\ln(\cos(\arctan(\sqrt{\frac{k}{mg}}v_{y_{o}})))
  129. ๐ก \mathbf{h}
  130. ๐ญ ๐š๐ฌ๐œ๐ž๐ง๐ \mathbf{t_{ascend}}
  131. t a s c e n d = m g k arctan ( k m g v y o ) t_{ascend}=\sqrt{\frac{m}{gk}}\arctan(\sqrt{\frac{k}{mg}}v_{y_{o}})
  132. h = - m k ln ( cos ( arctan ( k m g v y o ) ) ) h=-\frac{m}{k}\ln(\cos(\arctan(\sqrt{\frac{k}{mg}}v_{y_{o}})))
  133. ๐ญ \mathbf{t}
  134. ๐ญ \mathbf{t}
  135. a y = k v y 2 m - g = d v y d t a_{y}=\frac{kv^{2}_{y}}{m}-g=\frac{dv_{y}}{dt}
  136. v y ( t ) = - m g k tanh ( c + g k m t ) v_{y}(t)=-\sqrt{\frac{mg}{k}}\tanh(c+\sqrt{\frac{gk}{m}}t)
  137. t = 0 t=0
  138. v y ( t ) = 0 v_{y}(t)=0
  139. c = 0 c=0
  140. v y ( t ) = - m g k tanh ( g k m t ) v_{y}(t)=-\sqrt{\frac{mg}{k}}\tanh(\sqrt{\frac{gk}{m}}t)
  141. ๐ฌ ๐ฒ ( ๐ญ ) \mathbf{s_{y}(t)}
  142. ๐ญ ๐š๐ฌ๐œ๐ž๐ง๐ \mathbf{t_{ascend}}
  143. s y ( t ) = - m k ln ( cosh ( g k m t ) ) + c s_{y}(t)=-\frac{m}{k}\ln(\cosh(\sqrt{\frac{gk}{m}}t))+c
  144. t = 0 t=0
  145. s y ( t ) = h s_{y}(t)=h
  146. ๐ก \mathbf{h}
  147. c = h c=h
  148. s y ( t ) = h - m k ln ( cosh ( g k m t ) ) s_{y}(t)=h-\frac{m}{k}\ln(\cosh(\sqrt{\frac{gk}{m}}t))
  149. ๐ญ ๐๐ž๐ฌ๐œ๐ž๐ง๐ \mathbf{t_{descend}}
  150. t d e s c e n d = m g k arccosh ( e h k m ) t_{descend}=\sqrt{\frac{m}{gk}}\operatorname{arccosh}\,(e^{\frac{hk}{m}})
  151. t d e s c e n d = m g k arccosh ( 1 + k v y o 2 g m ) t_{descend}=\sqrt{\frac{m}{gk}}\operatorname{arccosh}\,(\sqrt{1+\frac{kv^{2}_{% y_{o}}}{gm}})
  152. ๐ฏ ( ๐ญ ) \mathbf{v(t)}
  153. ๐“ \mathbf{T}
  154. ๐‘ \mathbf{R}
  155. ๐• ๐ก๐ข๐ญ \mathbf{V_{hit}}
  156. v ( t ) = v x 2 ( t ) + v y 2 ( t ) v(t)=\sqrt{v^{2}_{x}(t)+v^{2}_{y}(t)}
  157. T = t a s c e n d + t d e s c e n d T=t_{ascend}+t_{descend}
  158. R = m k ( ln ( k T v 0 cos ฮธ + m m ) ) R=\frac{m}{k}(\ln(\frac{kT{v_{0}\cos\theta}+m}{m}))
  159. V h i t = ( 1 1 v 0 cos ฮธ + k T m ) 2 + m g k ( tanh ( arccosh ( e h k m ) ) ) 2 V_{hit}=\sqrt{(\frac{1}{\frac{1}{v_{0}\cos\theta}+\frac{kT}{m}})^{2}+\frac{mg}% {k}(\tanh(\operatorname{arccosh}\,(e^{\frac{hk}{m}})))^{2}}
  160. ฯ = M P o R ( T o - L h ) ร— [ 1 - L h T o ] g M R L \rho=\frac{MP_{o}}{R(T_{o}-Lh)}\times[1-\frac{Lh}{T_{o}}]^{\frac{gM}{RL}}
  161. ๐‹ \mathbf{L}
  162. ๐ ๐จ \mathbf{P_{o}}
  163. ๐  \mathbf{g}
  164. ๐“ ๐จ \mathbf{T_{o}}
  165. ๐ก \mathbf{h}
  166. ๐Œ \mathbf{M}
  167. ๐‘ \mathbf{R}

Transfer_operator.html

  1. f : X โ†’ X f:X\rightarrow X
  2. X X
  3. โ„’ \mathcal{L}
  4. { ฮฆ : X โ†’ โ„‚ } \{\Phi:X\rightarrow\mathbb{C}\}
  5. ( โ„’ ฮฆ ) ( x ) = โˆ‘ y โˆˆ f - 1 ( x ) g ( y ) ฮฆ ( y ) (\mathcal{L}\Phi)(x)=\sum_{y\in f^{-1}(x)}g(y)\Phi(y)
  6. g : X โ†’ โ„‚ g:X\rightarrow\mathbb{C}
  7. f f
  8. g g
  9. g = 1 / | J | g=1/|J|
  10. f f
  11. b ( x ) = 2 x - โŒŠ 2 x โŒ‹ b(x)=2x-\lfloor 2x\rfloor
  12. h ( x ) = 1 / x - โŒŠ 1 / x โŒ‹ h(x)=1/x-\lfloor 1/x\rfloor

Transition_radiation_detector.html

  1. ฮณ \scriptstyle\gamma
  2. ฮณ \scriptstyle\gamma
  3. ฮณ \scriptstyle\gamma
  4. ฮณ \scriptstyle\gamma
  5. ฮณ \scriptstyle\gamma

Transpose_of_a_linear_map.html

  1. f * ( ฯ† ) = ฯ† โˆ˜ f f^{*}(\varphi)=\varphi\circ f
  2. [ f * ( ฯ† ) , v ] V = [ ฯ† , f ( v ) ] W , [f^{*}(\varphi),\,v]_{V}=[\varphi,\,f(v)]_{W},
  3. Y โŸถ ๐ฝ Y * โŸถ u t X * โŸถ I - 1 X Y\overset{J}{\longrightarrow}Y^{*}\overset{{}^{\,\text{t}}u}{\longrightarrow}X% ^{*}\overset{I^{-1}}{\longrightarrow}X

Transverse_isotropy.html

  1. x 2 x_{2}
  2. s y m b o l K ยฏ ยฏ \underline{\underline{symbol{K}}}
  3. s y m b o l A symbol{A}
  4. s y m b o l A โ‹… ๐Ÿ = s y m b o l K โ‹… ( s y m b o l A \cdotsymbol d ) โŸน ๐Ÿ = ( s y m b o l A - 1 \cdotsymbol K \cdotsymbol A ) \cdotsymbol d symbol{A}\cdot\mathbf{f}=symbol{K}\cdot(symbol{A}\cdotsymbol{d})\implies% \mathbf{f}=(symbol{A}^{-1}\cdotsymbol{K}\cdotsymbol{A})\cdotsymbol{d}
  5. s y m b o l K = s y m b o l A - 1 \cdotsymbol K \cdotsymbol A = s y m b o l A T \cdotsymbol K \cdotsymbol A symbol{K}=symbol{A}^{-1}\cdotsymbol{K}\cdotsymbol{A}=symbol{A}^{T}\cdotsymbol{% K}\cdotsymbol{A}
  6. 3 ร— 3 3\times 3
  7. s y m b o l A ยฏ ยฏ \underline{\underline{symbol{A}}}
  8. s y m b o l A ยฏ ยฏ = [ A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 ] . \underline{\underline{symbol{A}}}=\begin{bmatrix}A_{11}&A_{12}&A_{13}\\ A_{21}&A_{22}&A_{23}\\ A_{31}&A_{32}&A_{33}\end{bmatrix}~{}.
  9. s y m b o l K ยฏ ยฏ = s y m b o l A T ยฏ ยฏ s y m b o l K ยฏ ยฏ s y m b o l A ยฏ ยฏ \underline{\underline{symbol{K}}}=\underline{\underline{symbol{A}^{T}}}~{}% \underline{\underline{symbol{K}}}~{}\underline{\underline{symbol{A}}}
  10. s y m b o l A ยฏ ยฏ \underline{\underline{symbol{A}}}
  11. s y m b o l A ยฏ ยฏ = [ cos ฮธ sin ฮธ 0 - sin ฮธ cos ฮธ 0 0 0 1 ] . \underline{\underline{symbol{A}}}=\begin{bmatrix}\cos\theta&\sin\theta&0\\ -\sin\theta&\cos\theta&0\\ 0&0&1\end{bmatrix}~{}.
  12. x 3 x_{3}
  13. ฮธ \theta
  14. x 3 x_{3}
  15. ๐Ÿ = s y m b o l K โ‹… ๐ \mathbf{f}=symbol{K}\cdot\mathbf{d}
  16. ๐ , ๐Ÿ \mathbf{d},\mathbf{f}
  17. s y m b o l K symbol{K}
  18. ๐Ÿ ยฏ ยฏ = s y m b o l K ยฏ ยฏ ๐ ยฏ ยฏ โŸน [ f 1 f 2 f 3 ] = [ K 11 K 12 K 13 K 21 K 22 K 23 K 31 K 32 K 33 ] [ d 1 d 2 d 3 ] \underline{\underline{\mathbf{f}}}=\underline{\underline{symbol{K}}}~{}% \underline{\underline{\mathbf{d}}}\implies\begin{bmatrix}f_{1}\\ f_{2}\\ f_{3}\end{bmatrix}=\begin{bmatrix}K_{11}&K_{12}&K_{13}\\ K_{21}&K_{22}&K_{23}\\ K_{31}&K_{32}&K_{33}\end{bmatrix}\begin{bmatrix}d_{1}\\ d_{2}\\ d_{3}\end{bmatrix}
  19. ๐Ÿ \mathbf{f}
  20. ๐ \mathbf{d}
  21. s y m b o l K symbol{K}
  22. ๐‰ \mathbf{J}
  23. ๐„ \mathbf{E}
  24. s y m b o l ฯƒ symbol{\sigma}
  25. ๐ƒ \mathbf{D}
  26. ๐„ \mathbf{E}
  27. s y m b o l ฮต symbol{\varepsilon}
  28. ๐ \mathbf{B}
  29. ๐‡ \mathbf{H}
  30. s y m b o l ฮผ symbol{\mu}
  31. ๐ช \mathbf{q}
  32. - s y m b o l โˆ‡ T -symbol{\nabla}T
  33. s y m b o l ฮบ symbol{\kappa}
  34. ๐‰ \mathbf{J}
  35. - s y m b o l โˆ‡ c -symbol{\nabla}c
  36. s y m b o l D symbol{D}
  37. ฮท ฮผ ๐ฏ \eta_{\mu}\mathbf{v}
  38. s y m b o l โˆ‡ P symbol{\nabla}P
  39. s y m b o l ฮบ symbol{\kappa}
  40. ฮธ = ฯ€ \theta=\pi
  41. s y m b o l A ยฏ ยฏ \underline{\underline{symbol{A}}}
  42. K 13 = K 31 = K 23 = K 32 = 0 K_{13}=K_{31}=K_{23}=K_{32}=0
  43. ฮธ = ฯ€ 2 \theta=\tfrac{\pi}{2}
  44. K 11 = K 22 K_{11}=K_{22}
  45. K 12 = - K 21 K_{12}=-K_{21}
  46. K 12 , K 21 โ‰ฅ 0 K_{12},K_{21}\geq 0
  47. K 12 = K 21 = 0 K_{12}=K_{21}=0
  48. s y m b o l K ยฏ ยฏ = [ K 11 0 0 0 K 11 0 0 0 K 33 ] \underline{\underline{symbol{K}}}=\begin{bmatrix}K_{11}&0&0\\ 0&K_{11}&0\\ 0&0&K_{33}\end{bmatrix}
  49. s y m b o l ฯƒ ยฏ ยฏ = ๐–ข ยฏ ยฏ s y m b o l ฮต ยฏ ยฏ \underline{\underline{symbol{\sigma}}}=\underline{\underline{\mathsf{C}}}~{}% \underline{\underline{symbol{\varepsilon}}}
  50. [ ฯƒ 1 ฯƒ 2 ฯƒ 3 ฯƒ 4 ฯƒ 5 ฯƒ 6 ] = [ C 11 C 12 C 13 C 14 C 15 C 16 C 12 C 22 C 23 C 24 C 25 C 26 C 13 C 23 C 33 C 34 C 35 C 36 C 14 C 24 C 34 C 44 C 45 C 46 C 15 C 25 C 35 C 45 C 55 C 56 C 16 C 26 C 36 C 46 C 56 C 66 ] [ ฮต 1 ฮต 2 ฮต 3 ฮต 4 ฮต 5 ฮต 6 ] \begin{bmatrix}\sigma_{1}\\ \sigma_{2}\\ \sigma_{3}\\ \sigma_{4}\\ \sigma_{5}\\ \sigma_{6}\end{bmatrix}=\begin{bmatrix}C_{11}&C_{12}&C_{13}&C_{14}&C_{15}&C_{1% 6}\\ C_{12}&C_{22}&C_{23}&C_{24}&C_{25}&C_{26}\\ C_{13}&C_{23}&C_{33}&C_{34}&C_{35}&C_{36}\\ C_{14}&C_{24}&C_{34}&C_{44}&C_{45}&C_{46}\\ C_{15}&C_{25}&C_{35}&C_{45}&C_{55}&C_{56}\\ C_{16}&C_{26}&C_{36}&C_{46}&C_{56}&C_{66}\end{bmatrix}\begin{bmatrix}% \varepsilon_{1}\\ \varepsilon_{2}\\ \varepsilon_{3}\\ \varepsilon_{4}\\ \varepsilon_{5}\\ \varepsilon_{6}\end{bmatrix}
  51. ๐–ข ยฏ ยฏ = ๐–  ฮต ยฏ ยฏ T ๐–ข ยฏ ยฏ ๐–  ฮต ยฏ ยฏ \underline{\underline{\mathsf{C}}}=\underline{\underline{\mathsf{A}_{% \varepsilon}}}^{T}~{}\underline{\underline{\mathsf{C}}}~{}\underline{% \underline{\mathsf{A}_{\varepsilon}}}
  52. ๐–  ฮต ยฏ ยฏ = [ A 11 2 A 12 2 A 13 2 A 12 A 13 A 11 A 13 A 11 A 12 A 21 2 A 22 2 A 23 2 A 22 A 23 A 21 A 23 A 21 A 22 A 31 2 A 32 2 A 33 2 A 32 A 33 A 31 A 33 A 31 A 32 2 A 21 A 31 2 A 22 A 32 2 A 23 A 33 A 22 A 33 + A 23 A 32 A 21 A 33 + A 23 A 31 A 21 A 32 + A 22 A 31 2 A 11 A 31 2 A 12 A 32 2 A 13 A 33 A 12 A 33 + A 13 A 32 A 11 A 33 + A 13 A 31 A 11 A 32 + A 12 A 31 2 A 11 A 21 2 A 12 A 22 2 A 13 A 23 A 12 A 23 + A 13 A 22 A 11 A 23 + A 13 A 21 A 11 A 22 + A 12 A 21 ] \underline{\underline{\mathsf{A}_{\varepsilon}}}=\begin{bmatrix}A_{11}^{2}&A_{% 12}^{2}&A_{13}^{2}&A_{12}A_{13}&A_{11}A_{13}&A_{11}A_{12}\\ A_{21}^{2}&A_{22}^{2}&A_{23}^{2}&A_{22}A_{23}&A_{21}A_{23}&A_{21}A_{22}\\ A_{31}^{2}&A_{32}^{2}&A_{33}^{2}&A_{32}A_{33}&A_{31}A_{33}&A_{31}A_{32}\\ 2A_{21}A_{31}&2A_{22}A_{32}&2A_{23}A_{33}&A_{22}A_{33}+A_{23}A_{32}&A_{21}A_{3% 3}+A_{23}A_{31}&A_{21}A_{32}+A_{22}A_{31}\\ 2A_{11}A_{31}&2A_{12}A_{32}&2A_{13}A_{33}&A_{12}A_{33}+A_{13}A_{32}&A_{11}A_{3% 3}+A_{13}A_{31}&A_{11}A_{32}+A_{12}A_{31}\\ 2A_{11}A_{21}&2A_{12}A_{22}&2A_{13}A_{23}&A_{12}A_{23}+A_{13}A_{22}&A_{11}A_{2% 3}+A_{13}A_{21}&A_{11}A_{22}+A_{12}A_{21}\end{bmatrix}
  53. ฮธ \theta
  54. s y m b o l A ยฏ ยฏ \underline{\underline{symbol{A}}}
  55. ๐–ข ยฏ ยฏ = [ C 11 C 12 C 13 0 0 0 C 12 C 11 C 13 0 0 0 C 13 C 13 C 33 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44 0 0 0 0 0 0 ( C 11 - C 12 ) / 2 ] = [ C 11 C 11 - 2 C 66 C 13 0 0 0 C 11 - 2 C 66 C 11 C 13 0 0 0 C 13 C 13 C 33 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 44 0 0 0 0 0 0 C 66 ] . \underline{\underline{\mathsf{C}}}=\begin{bmatrix}C_{11}&C_{12}&C_{13}&0&0&0\\ C_{12}&C_{11}&C_{13}&0&0&0\\ C_{13}&C_{13}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{44}&0\\ 0&0&0&0&0&(C_{11}-C_{12})/2\end{bmatrix}=\begin{bmatrix}C_{11}&C_{11}-2C_{66}&% C_{13}&0&0&0\\ C_{11}-2C_{66}&C_{11}&C_{13}&0&0&0\\ C_{13}&C_{13}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{44}&0\\ 0&0&0&0&0&C_{66}\end{bmatrix}.
  56. C i j C_{ij}
  57. ๐–ข ยฏ ยฏ - 1 = 1 ฮ” [ C 11 C 33 - C 13 2 C 13 2 - C 12 C 33 ( C 12 - C 11 ) C 13 0 0 0 C 13 2 - C 12 C 33 C 11 C 33 - C 13 2 ( C 12 - C 11 ) C 13 0 0 0 ( C 12 - C 11 ) C 13 ( C 12 - C 11 ) C 13 C 11 2 - C 12 2 0 0 0 0 0 0 ฮ” C 44 0 0 0 0 0 0 ฮ” C 44 0 0 0 0 0 0 2 ฮ” ( C 11 - C 12 ) ] \underline{\underline{\mathsf{C}}}^{-1}=\frac{1}{\Delta}\begin{bmatrix}C_{11}C% _{33}-C_{13}^{2}&C_{13}^{2}-C_{12}C_{33}&(C_{12}-C_{11})C_{13}&0&0&0\\ C_{13}^{2}-C_{12}C_{33}&C_{11}C_{33}-C_{13}^{2}&(C_{12}-C_{11})C_{13}&0&0&0\\ (C_{12}-C_{11})C_{13}&(C_{12}-C_{11})C_{13}&C_{11}^{2}-C_{12}^{2}&0&0&0\\ 0&0&0&\frac{\Delta}{C_{44}}&0&0\\ 0&0&0&0&\frac{\Delta}{C_{44}}&0\\ 0&0&0&0&0&\frac{2\Delta}{(C_{11}-C_{12})}\end{bmatrix}
  58. ฮ” := ( C 11 - C 12 ) [ ( C 11 + C 12 ) C 33 - 2 C 13 C 13 ] \Delta:=(C_{11}-C_{12})[(C_{11}+C_{12})C_{33}-2C_{13}C_{13}]
  59. ๐–ข ยฏ ยฏ - 1 = [ 1 E x - ฮฝ yx E x - ฮฝ zx E z 0 0 0 - ฮฝ xy E x 1 E x - ฮฝ zx E z 0 0 0 - ฮฝ xz E x - ฮฝ xz E x 1 E z 0 0 0 0 0 0 1 G yz 0 0 0 0 0 0 1 G yz 0 0 0 0 0 0 2 ( 1 + ฮฝ xy ) E x ] \underline{\underline{\mathsf{C}}}^{-1}=\begin{bmatrix}\tfrac{1}{E_{\rm x}}&-% \tfrac{\nu_{\rm yx}}{E_{\rm x}}&-\tfrac{\nu_{\rm zx}}{E_{\rm z}}&0&0&0\\ -\tfrac{\nu_{\rm xy}}{E_{\rm x}}&\tfrac{1}{E_{\rm x}}&-\tfrac{\nu_{\rm zx}}{E_% {\rm z}}&0&0&0\\ -\tfrac{\nu_{\rm xz}}{E_{\rm x}}&-\tfrac{\nu_{\rm xz}}{E_{\rm x}}&\tfrac{1}{E_% {\rm z}}&0&0&0\\ 0&0&0&\tfrac{1}{G_{\rm yz}}&0&0\\ 0&0&0&0&\tfrac{1}{G_{\rm yz}}&0\\ 0&0&0&0&0&\tfrac{2(1+\nu_{\rm xy})}{E_{\rm x}}\end{bmatrix}
  60. E L = E z = C 33 - 2 C 13 C 13 / ( C 11 + C 12 ) E_{L}=E_{\rm z}=C_{33}-2C_{13}C_{13}/(C_{11}+C_{12})
  61. E T = E x = E y = ( C 11 - C 12 ) ( C 11 C 33 + C 12 C 33 - 2 C 13 C 13 ) / ( C 11 C 33 - C 13 C 13 ) E_{T}=E_{\rm x}=E_{\rm y}=(C_{11}-C_{12})(C_{11}C_{33}+C_{12}C_{33}-2C_{13}C_{% 13})/(C_{11}C_{33}-C_{13}C_{13})
  62. G x y = ( C 11 - C 12 ) / 2 = C 66 G_{xy}=(C_{11}-C_{12})/2=C_{66}
  63. ฮฝ L T = ฮฝ x z = C 13 / ( C 11 + C 12 ) \nu_{LT}=\nu_{xz}=C_{13}/(C_{11}+C_{12})
  64. i i
  65. ( a i , b i , c i , d i , e i ) (a_{i},b_{i},c_{i},d_{i},e_{i})
  66. ๐–ข i ยฏ ยฏ = [ a i a i - 2 e i b i 0 0 0 a i - 2 e i a i b i 0 0 0 b i b i c i 0 0 0 0 0 0 d i 0 0 0 0 0 0 d i 0 0 0 0 0 0 e i ] \underline{\underline{\mathsf{C}_{i}}}=\begin{bmatrix}a_{i}&a_{i}-2e_{i}&b_{i}% &0&0&0\\ a_{i}-2e_{i}&a_{i}&b_{i}&0&0&0\\ b_{i}&b_{i}&c_{i}&0&0&0\\ 0&0&0&d_{i}&0&0\\ 0&0&0&0&d_{i}&0\\ 0&0&0&0&0&e_{i}\\ \end{bmatrix}
  67. ๐–ข eff ยฏ ยฏ = [ A A - 2 E B 0 0 0 A - 2 E A B 0 0 0 B B C 0 0 0 0 0 0 D 0 0 0 0 0 0 D 0 0 0 0 0 0 E ] \underline{\underline{\mathsf{C}_{\mathrm{eff}}}}=\begin{bmatrix}A&A-2E&B&0&0&% 0\\ A-2E&A&B&0&0&0\\ B&B&C&0&0&0\\ 0&0&0&D&0&0\\ 0&0&0&0&D&0\\ 0&0&0&0&0&E\end{bmatrix}
  68. A = โŸจ a - b 2 c - 1 โŸฉ + โŸจ c - 1 โŸฉ - 1 โŸจ b c - 1 โŸฉ 2 B = โŸจ c - 1 โŸฉ - 1 โŸจ b c - 1 โŸฉ C = โŸจ c - 1 โŸฉ - 1 D = โŸจ d - 1 โŸฉ - 1 E = โŸจ e โŸฉ \begin{aligned}\displaystyle A&\displaystyle=\langle a-b^{2}c^{-1}\rangle+% \langle c^{-1}\rangle^{-1}\langle bc^{-1}\rangle^{2}\\ \displaystyle B&\displaystyle=\langle c^{-1}\rangle^{-1}\langle bc^{-1}\rangle% \\ \displaystyle C&\displaystyle=\langle c^{-1}\rangle^{-1}\\ \displaystyle D&\displaystyle=\langle d^{-1}\rangle^{-1}\\ \displaystyle E&\displaystyle=\langle e\rangle\\ \end{aligned}
  69. โŸจ โ‹… โŸฉ \langle\cdot\rangle
  70. b i = a i - 2 e i b_{i}=a_{i}-2e_{i}
  71. a i = c i a_{i}=c_{i}
  72. d i = e i d_{i}=e_{i}
  73. ฮธ \theta
  74. V q P ( ฮธ ) = C 11 sin 2 ( ฮธ ) + C 33 cos 2 ( ฮธ ) + C 44 + M ( ฮธ ) 2 ฯ V q S ( ฮธ ) = C 11 sin 2 ( ฮธ ) + C 33 cos 2 ( ฮธ ) + C 44 - M ( ฮธ ) 2 ฯ V S = C 66 sin 2 ( ฮธ ) + C 44 cos 2 ( ฮธ ) ฯ M ( ฮธ ) = [ ( C 11 - C 44 ) sin 2 ( ฮธ ) - ( C 33 - C 44 ) cos 2 ( ฮธ ) ] 2 + ( C 13 + C 44 ) 2 sin 2 ( 2 ฮธ ) \begin{aligned}\displaystyle V_{qP}(\theta)&\displaystyle=\sqrt{\frac{C_{11}% \sin^{2}(\theta)+C_{33}\cos^{2}(\theta)+C_{44}+\sqrt{M(\theta)}}{2\rho}}\\ \displaystyle V_{qS}(\theta)&\displaystyle=\sqrt{\frac{C_{11}\sin^{2}(\theta)+% C_{33}\cos^{2}(\theta)+C_{44}-\sqrt{M(\theta)}}{2\rho}}\\ \displaystyle V_{S}&\displaystyle=\sqrt{\frac{C_{66}\sin^{2}(\theta)+C_{44}% \cos^{2}(\theta)}{\rho}}\\ \displaystyle M(\theta)&\displaystyle=\left[\left(C_{11}-C_{44}\right)\sin^{2}% (\theta)-\left(C_{33}-C_{44}\right)\cos^{2}(\theta)\right]^{2}+\left(C_{13}+C_% {44}\right)^{2}\sin^{2}(2\theta)\\ \end{aligned}
  75. ฮธ \begin{aligned}\displaystyle\theta\end{aligned}
  76. ฯ \rho
  77. C i j C_{ij}
  78. ฯต \displaystyle\epsilon
  79. ๐ž 3 \mathbf{e}_{3}
  80. ฮด , ฮณ , ฯต โ‰ช 1 \delta,\gamma,\epsilon\ll 1
  81. V q P ( ฮธ ) โ‰ˆ V P 0 ( 1 + ฮด sin 2 ฮธ cos 2 ฮธ + ฯต sin 4 ฮธ ) V q S ( ฮธ ) โ‰ˆ V S 0 [ 1 + ( V P 0 V S 0 ) 2 ( ฯต - ฮด ) sin 2 ฮธ cos 2 ฮธ ] V S ( ฮธ ) โ‰ˆ V S 0 ( 1 + ฮณ sin 2 ฮธ ) \begin{aligned}\displaystyle V_{qP}(\theta)&\displaystyle\approx V_{P0}(1+% \delta\sin^{2}\theta\cos^{2}\theta+\epsilon\sin^{4}\theta)\\ \displaystyle V_{qS}(\theta)&\displaystyle\approx V_{S0}\left[1+\left(\frac{V_% {P0}}{V_{S0}}\right)^{2}(\epsilon-\delta)\sin^{2}\theta\cos^{2}\theta\right]\\ \displaystyle V_{S}(\theta)&\displaystyle\approx V_{S0}(1+\gamma\sin^{2}\theta% )\end{aligned}
  82. V P 0 = C 33 / ฯ ; V S 0 = C 44 / ฯ V_{P0}=\sqrt{C_{33}/\rho}~{};~{}~{}V_{S0}=\sqrt{C_{44}/\rho}
  83. ๐ž 3 \mathbf{e}_{3}
  84. ฮด \delta
  85. ฮธ = ฯ€ \theta=\pi
  86. ฮธ = ฯ€ 2 \theta=\tfrac{\pi}{2}

Treaties_of_Velasco.html

  1. [ - F o r m u l a E r r o r - ] [-FormulaError-]

Trend_stationary.html

  1. Y t = f ( t ) + e t , Y_{t}=f(t)+e_{t},
  2. f ( t ) f(t)
  3. Y t = a โ‹… t + b + e t Y_{t}=a\cdot t+b+e_{t}
  4. a ^ \hat{a}
  5. a a
  6. b ^ \hat{b}
  7. a ^ \hat{a}
  8. e ^ t = Y - a ^ โ‹… t - b ^ . \hat{e}_{t}=Y-\hat{a}\cdot t-\hat{b}.
  9. GDP t = B e a t U t \,\text{GDP}_{t}=Be^{at}U_{t}
  10. a a
  11. ln ( GDP t ) = ln B + a t + ln ( U t ) . \ln(\,\text{GDP}_{t})=\ln B+at+\ln(U_{t}).
  12. ( ln U ) t (\ln U)_{t}
  13. ( ln GDP ) t (\ln\,\text{GDP})_{t}
  14. U t U_{t}
  15. GDP t \,\text{GDP}_{t}
  16. ( ln U ) t (\ln U)_{t}
  17. Y t = a โ‹… t + c โ‹… t 2 + b + e t . Y_{t}=a\cdot t+c\cdot t^{2}+b+e_{t}.
  18. Y t Y_{t}

Treynor_ratio.html

  1. T = r i - r f ฮฒ i T=\frac{r_{i}-r_{f}}{\beta_{i}}
  2. T โ‰ก T\equiv
  3. r i โ‰ก r_{i}\equiv
  4. r f โ‰ก r_{f}\equiv
  5. ฮฒ i โ‰ก \beta_{i}\equiv

Triangular_arbitrage.html

  1. S a / $ = S a / b S b / $ S_{a/\$}=S_{a/b}S_{b/\$}
  2. S a / $ S_{a/\$}
  3. S a / b S_{a/b}
  4. S b / $ S_{b/\$}
  5. S a / $ S_{a/\$}
  6. S a / b S b / $ S_{a/b}S_{b/\$}

Triangulated_category.html

  1. X [ n ] = T n X X[n]=T^{n}X
  2. X โ†’ ๐‘ข Y โ†’ ๐‘ฃ Z โ†’ ๐‘ค X [ 1 ] X\xrightarrow{u}Y\xrightarrow{v}Z\xrightarrow{w}X[1]
  3. X โ†’ ๐‘ข Y โ†’ ๐‘ฃ Z โ†’ ๐‘ค X\xrightarrow{u}Y\xrightarrow{v}Z\xrightarrow{w}
  4. X โ†’ id X โ†’ 0 โ†’ โ‹… X\overset{\,\text{id}}{\to}X\to 0\to\cdot
  5. X โ†’ ๐‘ข Y โ†’ Z โ†’ โ‹… X\xrightarrow{u}Y\to Z\to\cdot
  6. X โ†’ ๐‘ข Y โ†’ ๐‘ฃ Z โ†’ ๐‘ค X [ 1 ] X\xrightarrow{u}Y\xrightarrow{v}Z\xrightarrow{w}X[1]
  7. X โ€ฒ โ†’ g u f - 1 Y โ€ฒ โ†’ h v g - 1 Z โ€ฒ โ†’ f [ 1 ] w h - 1 X โ€ฒ [ 1 ] X^{\prime}\xrightarrow{guf^{-1}}Y^{\prime}\xrightarrow{hvg^{-1}}Z^{\prime}% \xrightarrow{f[1]wh^{-1}}X^{\prime}[1]
  8. X โ†’ ๐‘ข Y โ†’ ๐‘ฃ Z โ†’ ๐‘ค X [ 1 ] X\xrightarrow{u}Y\xrightarrow{v}Z\xrightarrow{w}X[1]
  9. Y โ†’ ๐‘ฃ Z โ†’ ๐‘ค X [ 1 ] โ†’ - u [ 1 ] Y [ 1 ] Y\xrightarrow{v}Z\xrightarrow{w}X[1]\xrightarrow{-u[1]}Y[1]
  10. Z [ - 1 ] โ†’ - w [ - 1 ] X โ†’ ๐‘ข Y โ†’ ๐‘ฃ Z . Z[-1]\xrightarrow{-w[-1]}X\xrightarrow{u}Y\xrightarrow{v}Z.
  11. X โ†’ ๐‘ข Y โ†’ ๐‘— Z โ€ฒ โ†’ ๐‘˜ X\xrightarrow{u\,}Y\xrightarrow{j}Z^{\prime}\xrightarrow{k}
  12. Y โ†’ ๐‘ฃ Z โ†’ ๐‘™ X โ€ฒ โ†’ ๐‘– Y\xrightarrow{v\,}Z\xrightarrow{l}X^{\prime}\xrightarrow{i}
  13. X โ†’ v u Z โ†’ ๐‘š Y โ€ฒ โ†’ ๐‘› X\xrightarrow{vu}Z\xrightarrow{m}Y^{\prime}\xrightarrow{n}
  14. Z โ€ฒ โ†’ ๐‘“ Y โ€ฒ โ†’ ๐‘” X โ€ฒ โ†’ โ„Ž Z^{\prime}\xrightarrow{f}Y^{\prime}\xrightarrow{g}X^{\prime}\xrightarrow{h}
  15. l = g m , k = n f , h = j [ 1 ] i , i g = u [ 1 ] n , f j = m v . l=gm,\quad k=nf,\quad h=j[1]i,\quad ig=u[1]n,\quad fj=mv.
  16. Z โ€ฒ = Y / X , Y โ€ฒ = Z / X Z^{\prime}=Y/X,Y^{\prime}=Z/X
  17. X โ€ฒ = Z / Y X^{\prime}=Z/Y
  18. Y โ†’ Z โ†’ X โ€ฒ โ†’ Y\to Z\to X^{\prime}\to
  19. X โ€ฒ = Y โ€ฒ / Z โ€ฒ X^{\prime}=Y^{\prime}/Z^{\prime}
  20. Z โ€ฒ โ†’ Y โ€ฒ โ†’ X โ€ฒ โ†’ Z^{\prime}\to Y^{\prime}\to X^{\prime}\to
  21. ( Z / X ) / ( Y / X ) = Z / Y . (Z/X)/(Y/X)=Z/Y.
  22. X โ†’ Y โ†’ Z โ†’ X โ†’ Y X\to Y\to Z\to X\to Y
  23. X โ†’ Y โ†’ Z โ†’ X [ 1 ] , X\to Y\to Z\to X[1],
  24. X โ€ฒ โ†’ Y โ€ฒ โ†’ Z โ€ฒ โ†’ X โ€ฒ [ 1 ] X^{\prime}\to Y^{\prime}\to Z^{\prime}\to X^{\prime}[1]
  25. X โ†’ X โ€ฒ , Y โ†’ Y โ€ฒ X\to X^{\prime},Y\to Y^{\prime}
  26. Z โ†’ Z โ€ฒ Z\to Z^{\prime}
  27. G G
  28. G G
  29. X โ†’ ๐‘ข Y โ†’ ๐‘ฃ Z โ†’ ๐‘ค X [ 1 ] X\xrightarrow{u}Y\xrightarrow{v}Z\xrightarrow{w}X[1]
  30. X โ†’ ๐‘ข Y โ†’ ๐‘ฃ Z โ†’ ๐‘ค , X\xrightarrow{u}Y\xrightarrow{v}Z\xrightarrow{w},
  31. โ‹ฏ โ†’ X โ†’ ๐‘ข Y โ†’ ๐‘ฃ Z โ†’ ๐‘ค X [ 1 ] โ†’ u [ 1 ] โ‹ฏ , \cdots\to X\xrightarrow{u}Y\xrightarrow{v}Z\xrightarrow{w}X[1]\xrightarrow{u[1% ]}\cdots,
  32. โ‹ฏ โ†’ F ( X ) โ†’ F ( Y ) โ†’ F ( Z ) โ†’ F ( X [ 1 ] ) โ†’ โ‹ฏ . \cdots\to F(X)\to F(Y)\to F(Z)\to F(X[1])\to\cdots.
  33. Hom ( A , โ€“ ) , Hom ( โ€“ , A ) , \operatorname{Hom}(A,\,\text{--}),\operatorname{Hom}(\,\text{--},A),
  34. โ‹ฏ โ†’ Hom ( A , X [ i ] ) โ†’ Hom ( A , Y [ i ] ) โ†’ Hom ( A , Z [ i ] ) โ†’ Hom ( A , X [ i + 1 ] ) โ†’ โ‹ฏ . \cdots\to\operatorname{Hom}(A,X[i])\to\operatorname{Hom}(A,Y[i])\to% \operatorname{Hom}(A,Z[i])\to\operatorname{Hom}(A,X[i+1])\to\cdots.
  35. Ext i ( A , X ) = Hom ( A , X [ i ] ) \operatorname{Ext}^{i}(A,X)=\operatorname{Hom}(A,X[i])
  36. โ‹ฏ โ†’ Ext i ( A , X ) โ†’ Ext i ( A , Y ) โ†’ Ext i ( A , Z ) โ†’ Ext i + 1 ( A , X ) โ†’ โ‹ฏ . \cdots\to\operatorname{Ext}^{i}(A,X)\to\operatorname{Ext}^{i}(A,Y)\to% \operatorname{Ext}^{i}(A,Z)\to\operatorname{Ext}^{i+1}(A,X)\to\cdots.
  37. X โ†’ ๐‘ข Y โ†’ ๐‘ฃ Z โ†’ ๐‘ค X [ 1 ] X\xrightarrow{u}Y\xrightarrow{v}Z\xrightarrow{w}X[1]
  38. F ( X ) โ†’ F ( u ) F ( Y ) โ†’ F ( v ) F ( Z ) โ†’ ฮท X F ( w ) F ( X ) [ 1 ] F(X)\xrightarrow{F(u)}F(Y)\xrightarrow{F(v)}F(Z)\xrightarrow{\eta_{X}F(w)}F(X)% [1]

Triangulation_(geometry).html

  1. โ„ n + 1 \mathbb{R}^{n+1}
  2. โ„ n + 1 \mathbb{R}^{n+1}
  3. โ„ n + 1 \mathbb{R}^{n+1}
  4. P โŠ‚ โ„ n + 1 P\subset\mathbb{R}^{n+1}
  5. P P

Triangulation_(topology).html

  1. X X
  2. โ†’ \to

Trifid_cipher.html

  1. 2 3 = 8 < 26 < 27 = 3 3 2^{3}=8<26<27=3^{3}
  2. 4 3 = 64 4^{3}=64
  3. 2 4 = 16 < 26 2^{4}=16<26
  4. 3 4 = 81 3^{4}=81
  5. 2 5 = 32 > 26 2^{5}=32>26

Triple_bar.html

  1. a โ‰ก b ( mod N ) a\equiv b\;\;(\mathop{{\rm mod}}N)
  2. f โ‰ก g f\equiv g
  3. f ( x ) = g ( x ) f(x)=g(x)

Triplet_state.html

  1. โ„ \hbar
  2. โ†‘ โ†‘ , โ†‘ โ†“ , โ†“ โ†‘ , โ†“ โ†“ \uparrow\uparrow,\uparrow\downarrow,\downarrow\uparrow,\downarrow\downarrow
  3. | s 1 , m 1 โŸฉ | s 2 , m 2 โŸฉ = | s 1 , m 1 โŸฉ โŠ— | s 2 , m 2 โŸฉ |s_{1},m_{1}\rangle|s_{2},m_{2}\rangle=|s_{1},m_{1}\rangle\otimes|s_{2},m_{2}\rangle
  4. | 1 / 2 , m โŸฉ |1/2,m\rangle
  5. | 1 / 2 , m 1 โŸฉ | 1 / 2 , m 2 โŸฉ |1/2,m_{1}\rangle|1/2,m_{2}\rangle
  6. | s , m โŸฉ = โˆ‘ m 1 + m 2 = m C m 1 m 2 m s 1 s 2 s | s 1 m 1 โŸฉ | s 2 m 2 โŸฉ |s,m\rangle=\sum_{m_{1}+m_{2}=m}C_{m_{1}m_{2}m}^{s_{1}s_{2}s}|s_{1}m_{1}% \rangle|s_{2}m_{2}\rangle
  7. | 1 / 2 , + 1 / 2 โŸฉ | 1 / 2 , + 1 / 2 โŸฉ ( โ†‘ โ†‘ ) |1/2,+1/2\rangle\;|1/2,+1/2\rangle\ (\uparrow\uparrow)
  8. | 1 / 2 , + 1 / 2 โŸฉ | 1 / 2 , - 1 / 2 โŸฉ ( โ†‘ โ†“ ) |1/2,+1/2\rangle\;|1/2,-1/2\rangle\ (\uparrow\downarrow)
  9. | 1 / 2 , - 1 / 2 โŸฉ | 1 / 2 , + 1 / 2 โŸฉ ( โ†“ โ†‘ ) |1/2,-1/2\rangle\;|1/2,+1/2\rangle\ (\downarrow\uparrow)
  10. | 1 / 2 , - 1 / 2 โŸฉ | 1 / 2 , - 1 / 2 โŸฉ ( โ†“ โ†“ ) |1/2,-1/2\rangle\;|1/2,-1/2\rangle\ (\downarrow\downarrow)
  11. | 1 / 2 , m 1 โŸฉ | 1 / 2 , m 2 โŸฉ |1/2,\ m_{1}\rangle|1/2,\ m_{2}\rangle
  12. | 1 , 1 โŸฉ = โ†‘ โ†‘ | 1 , 0 โŸฉ = ( โ†‘ โ†“ + โ†“ โ†‘ ) / 2 | 1 , - 1 โŸฉ = โ†“ โ†“ } s = 1 ( triplet ) \left.\begin{array}[]{ll}|1,1\rangle&=\;\uparrow\uparrow\\ |1,0\rangle&=\;(\uparrow\downarrow+\downarrow\uparrow)/\sqrt{2}\\ |1,-1\rangle&=\;\downarrow\downarrow\end{array}\right\}\quad s=1\quad\mathrm{(% triplet)}
  13. | 0 , 0 โŸฉ = ( โ†‘ โ†“ - โ†“ โ†‘ ) / 2 } s = 0 ( singlet ) \left.|0,0\rangle=(\uparrow\downarrow-\downarrow\uparrow)/\sqrt{2}\;\right\}% \quad s=0\quad\mathrm{(singlet)}
  14. R 3 R^{3}

Troland.html

  1. T = L ร— p \mathrm{T}=\mathrm{L}\times\mathrm{p}
  2. T โ€ฒ = L โ€ฒ ร— p \mathrm{T^{\prime}}=\mathrm{L^{\prime}}\times\mathrm{p}

Trouton's_ratio.html

  1. L v a p T b o i l i n g โ‰ˆ 87 - 88 J K m o l \frac{L_{vap}}{T_{boiling}}\approx 87-88\frac{J}{Kmol}

Trouton's_rule.html

  1. ฮ” S ยฏ v a p = 10.5 R \Delta\bar{S}_{vap}=10.5R
  2. ฮ” S ยฏ v a p = 4.5 R + R ln T \Delta\bar{S}_{vap}=4.5R+R\ln T

Tullio_Regge.html

  1. cos ( ฮธ ) โ†’ โˆž \cos(\theta)\rightarrow\infty

Tunnel_ionization.html

  1. E โ‰ช E a E<<E_{a}
  2. w = 4 ฯ‰ a E a | E | exp [ - 2 3 E a | E | ] w=4\omega_{a}\frac{E_{a}}{\left|E\right|}\exp\left[-\frac{2}{3}\frac{E_{a}}{% \left|E\right|}\right]
  3. m = e = โ„ = 1 m=e=\hbar=1
  4. E a = m 2 e 5 ( 4 ฯ€ ฯต 0 ) 3 โ„ 4 E_{a}=\frac{m^{2}e^{5}}{(4\pi\epsilon_{0})^{3}\hbar^{4}}
  5. ฯ‰ a = m e 4 ( 4 ฯ€ ฯต 0 ) 2 โ„ 3 \omega_{a}=\frac{me^{4}}{(4\pi\epsilon_{0})^{2}\hbar^{3}}
  6. i โˆ‚ โˆ‚ t ฮจ ( ๐ซ , t ) = - 1 2 m โˆ‡ 2 ฮจ ( ๐ซ , t ) + ( ๐„ ( t ) . ๐ซ + V ( ๐ซ ) ) ฮจ ( ๐ซ , t ) i\frac{\partial}{\partial t}\Psi(\mathbf{r},\,t)=-\frac{1}{2m}\nabla^{2}\Psi(% \mathbf{r},\,t)+(\mathbf{E}(t)\mathbf{.r}+V(\mathbf{r}))\Psi(\mathbf{r},\,t)
  7. ๐„ ( t ) \mathbf{E}(t)
  8. V ( r ) V(r)
  9. 2 E i . ฮด ( ๐ซ ) \sqrt{2E_{i}}.\delta(\mathbf{r})
  10. E i E_{i}
  11. ๐‰ ( ๐ซ , t ) \mathbf{J}(\mathbf{r},t)
  12. W ( ๐„ , ฯ‰ ) W(\mathbf{E},\omega)
  13. W ( ๐„ , ฯ‰ ) = lim x โ†’ โˆž โˆซ 0 2 ฯ€ ฯ‰ โˆซ - โˆž โˆž โˆซ - โˆž โˆž ๐‰ ( ๐ซ , t ) d z d y d t W(\mathbf{E},\omega)=\lim_{x\to\infty}\int_{0}^{\frac{2\pi}{\omega}}\int_{-% \infty}^{\infty}\int_{-\infty}^{\infty}\mathbf{J}(\mathbf{r},t)\,dz\,dy\,dt
  14. ฯ‰ \omega
  15. x x
  16. Z r \frac{Z}{r}
  17. Z Z
  18. I P P T = ( 2 ( 2 E i ) 3 2 / F ) n * 2 I_{PPT}=(2(2E_{i})^{\frac{3}{2}}/F)^{n^{*2}}
  19. n * = 2 E i / Z 2 n^{*}=\sqrt{2E_{i}}/Z^{2}
  20. F F
  21. l l
  22. m m
  23. W P P T = I P P T W ( ๐„ , ฯ‰ ) = | C n * l * | 2 6 ฯ€ f l m E i ( 2 ( 2 E i ) 3 2 / F ) n * 2 - | m | - 3 / 2 ( 1 + ฮณ ) 2 ) | m / 2 | + 3 / 4 A m ( ฯ‰ , ฮณ ) e - ( 2 ( 2 E i ) 3 2 / F ) g ( ฮณ ) W_{PPT}=I_{PPT}W(\mathbf{E},\omega)=|C_{n^{*}l^{*}}|^{2}\sqrt{\frac{6}{\pi}}f_% {lm}E_{i}(2(2E_{i})^{\frac{3}{2}}/F)^{n^{*2}-|m|-3/2}(1+\gamma)^{2})^{|m/2|+3/% 4}A_{m}(\omega,\gamma)e^{-(2(2E_{i})^{\frac{3}{2}}/F)g(\gamma)}
  24. ฮณ = ฯ‰ 2 E i F \gamma=\frac{\omega\sqrt{2E_{i}}}{F}
  25. l * = n * - 1 l^{*}=n^{*}-1
  26. f l m f_{lm}
  27. g ( ฮณ ) g(\gamma)
  28. C n * l * C_{n^{*}l^{*}}
  29. f l m = ( 2 l + 1 ) ( l + | m | ) ! 2 m | m | ! ( l - | m | ) ! f_{lm}=\frac{(2l+1)(l+|m|)^{!}}{2^{m}|m|^{!}(l-|m|)^{!}}
  30. g ( ฮณ ) = 3 2 ฮณ ( 1 + 1 2 ฮณ 2 s i n h - 1 ( ฮณ ) - 1 + ฮณ 2 2 ฮณ ) g(\gamma)=\frac{3}{2\gamma}(1+\frac{1}{2\gamma^{2}}sinh^{-1}(\gamma)-\frac{% \sqrt{1+\gamma^{2}}}{2\gamma})
  31. | C n * l * | 2 = 2 2 n * n * ฮ“ ( n * + l * + 1 ) ฮ“ ( n * - l * ) |C_{n^{*}l^{*}}|^{2}=\frac{2^{2n^{*}}}{n^{*}\Gamma(n^{*}+l^{*}+1)\Gamma(n^{*}-% l^{*})}
  32. A m ( ฯ‰ , ฮณ ) A_{m}(\omega,\gamma)
  33. A m ( ฯ‰ , ฮณ ) = 4 3 ฯ€ 1 | m | ! ฮณ 2 1 + ฮณ 2 โˆ‘ n > v โˆž w m ( 2 ฮณ 1 + ฮณ 2 ( n - v ) e - ( n - v ) ฮฑ ( ฮณ ) ) A_{m}(\omega,\gamma)=\frac{4}{3\pi}\frac{1}{|m|^{!}}\frac{\gamma^{2}}{1+\gamma% ^{2}}\sum_{n>v}^{\infty}w_{m}(\sqrt{\frac{2\gamma}{\sqrt{1+\gamma^{2}}}(n-v)}e% ^{-(n-v)\alpha(\gamma)})
  34. w m ( x ) = e - x 2 โˆซ 0 x ( x 2 - y 2 ) m e y 2 d y w_{m}(x)=e^{-x^{2}}\int_{0}^{x}(x^{2}-y^{2})^{m}e^{y^{2}}\,dy
  35. ฮฑ ( ฮณ ) = 2 ( s i n h - 1 ( ฮณ ) - ฮณ 1 + ฮณ 2 ) \alpha(\gamma)=2(sinh^{-1}(\gamma)-\frac{\gamma}{\sqrt{1+\gamma^{2}}})
  36. v = E i ฯ‰ ( 1 + 1 2 ฮณ 2 ) v=\frac{E_{i}}{\omega}(1+\frac{1}{2\gamma^{2}})
  37. ฮณ \gamma
  38. W A D K = | C n * l * | 2 6 ฯ€ f l m E i ( 2 ( 2 E i ) 3 2 / F ) n * 2 - | m | - 3 / 2 e - ( 2 ( 2 E i ) 3 2 / 3 F ) W_{ADK}=|C_{n^{*}l^{*}}|^{2}\sqrt{\frac{6}{\pi}}f_{lm}E_{i}(2(2E_{i})^{\frac{3% }{2}}/F)^{n^{*2}-|m|-3/2}e^{-(2(2E_{i})^{\frac{3}{2}}/3F)}
  39. ฮณ < 1 / 2 \gamma<1/2
  40. ฮณ \gamma
  41. ฮณ = ฯ„ T 1 2 ฯ„ L \gamma=\frac{\tau_{T}}{\frac{1}{2}\tau_{L}}
  42. ฯ„ T \tau_{T}
  43. ฯ„ L \tau_{L}
  44. Z e f f r \frac{Z_{eff}}{r}
  45. 1 r \frac{1}{r}
  46. Z e f f Z_{eff}

Turing_jump.html

  1. X X
  2. X โ€ฒ Xโ€ฒ
  3. X โ€ฒ Xโ€ฒ
  4. X X
  5. X X
  6. X โ€ฒ Xโ€ฒ
  7. X X
  8. X X
  9. X X
  10. X โ€ฒ Xโ€ฒ
  11. X X
  12. X โ€ฒ = { x โˆฃ ฯ† x X ( x ) is defined } . X^{\prime}=\{x\mid\varphi_{x}^{X}(x)\ \mbox{is defined}~{}\}.
  13. n n
  14. X ( 0 ) = X , X^{(0)}=X,\,
  15. X ( n + 1 ) = ( X ( n ) ) โ€ฒ . X^{(n+1)}=(X^{(n)})^{\prime}.\,
  16. ฯ‰ ฯ‰
  17. X X
  18. n โˆˆ ๐ nโˆˆ\mathbf{N}
  19. X ( ฯ‰ ) = { p i k โˆฃ k โˆˆ X ( i ) } , X^{(\omega)}=\{p_{i}^{k}\mid k\in X^{(i)}\},\,
  20. i i
  21. 0 โ€ฒ 0โ€ฒ
  22. โˆ… โ€ฒ โˆ…โ€ฒ
  23. n n
  24. n n
  25. 0 โ€ฒ 0โ€ฒ
  26. n n
  27. ฮฃ n 0 \Sigma^{0}_{n}
  28. X X
  29. X < s u p > ( ฯ‰ ) X<sup>(ฯ‰)

Tutte_theorem.html

  1. G = ( V , E ) G=(V,E)
  2. U U
  3. V V
  4. V โˆ’ U Vโˆ’U
  5. ( * ) โˆ€ U โІ V , o ( G - U ) โ‰ค | U | (*)\qquad\forall U\subseteq V,\quad o(G-U)\leq|U|
  6. o ( X ) o(X)
  7. X X
  8. G G
  9. U U
  10. V V
  11. U U
  12. C C
  13. G โˆ’ U Gโˆ’U
  14. G G
  15. C C
  16. U U
  17. U U
  18. U U
  19. o ( G โˆ’ U ) โ‰ค | U | o(Gโˆ’U)ย โ‰ค|U|
  20. G G
  21. G G
  22. G โˆ— Gโˆ—
  23. G G
  24. G โˆ— Gโˆ—
  25. G โˆ— Gโˆ—
  26. G โˆ— Gโˆ—
  27. G โˆ— Gโˆ—
  28. G G
  29. U U
  30. G โˆ— Gโˆ—
  31. | V | โˆ’ 1 |V|โˆ’ย 1
  32. U U
  33. | U | |U|
  34. U U
  35. U U
  36. G โˆ— Gโˆ—
  37. G G

Twisted_cubic.html

  1. ฮฝ : ๐ 1 โ†’ ๐ 3 \nu:\mathbf{P}^{1}\to\mathbf{P}^{3}
  2. [ S : T ] [S:T]
  3. ฮฝ : [ S : T ] โ†ฆ [ S 3 : S 2 T : S T 2 : T 3 ] . \nu:[S:T]\mapsto[S^{3}:S^{2}T:ST^{2}:T^{3}].
  4. ฮฝ : x โ†ฆ ( x , x 2 , x 3 ) \nu:x\mapsto(x,x^{2},x^{3})
  5. ( x , x 2 , x 3 ) (x,x^{2},x^{3})
  6. F 0 = X Z - Y 2 F_{0}=XZ-Y^{2}
  7. F 1 = Y W - Z 2 F_{1}=YW-Z^{2}
  8. F 2 = X W - Y Z . F_{2}=XW-YZ.
  9. { X Z - Y 2 , Y W - Z 2 , X W - Y Z } . \{XZ-Y^{2},YW-Z^{2},XW-YZ\}.
  10. Z ( Y W - Z 2 ) - W ( X W - Y Z ) Z(YW-Z^{2})-W(XW-YZ)
  11. ( Y W - Z 2 ) 2 (YW-Z^{2})^{2}
  12. Y W - Z 2 YW-Z^{2}

Two-port_network.html

  1. V 1 V_{1}
  2. I 1 I_{1}
  3. V 2 V_{2}
  4. I 2 I_{2}
  5. [ V 1 V 2 ] = [ z 11 z 12 z 21 z 22 ] [ I 1 I 2 ] \begin{bmatrix}V_{1}\\ V_{2}\end{bmatrix}=\begin{bmatrix}z_{11}&z_{12}\\ z_{21}&z_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{2}\end{bmatrix}
  6. z 11 = def V 1 I 1 | I 2 = 0 z 12 = def V 1 I 2 | I 1 = 0 z 21 = def V 2 I 1 | I 2 = 0 z 22 = def V 2 I 2 | I 1 = 0 \begin{aligned}\displaystyle z_{11}&\displaystyle\stackrel{\,\text{def}}{=}\,% \left.\frac{V_{1}}{I_{1}}\right|_{I_{2}=0}\qquad z_{12}\,\stackrel{\,\text{def% }}{=}\,\left.\frac{V_{1}}{I_{2}}\right|_{I_{1}=0}\\ \displaystyle z_{21}&\displaystyle\stackrel{\,\text{def}}{=}\,\left.\frac{V_{2% }}{I_{1}}\right|_{I_{2}=0}\qquad z_{22}\,\stackrel{\,\text{def}}{=}\,\left.% \frac{V_{2}}{I_{2}}\right|_{I_{1}=0}\end{aligned}
  7. z 12 = z 21 \textstyle z_{12}=z_{21}
  8. z 11 = z 22 \textstyle z_{11}=z_{22}
  9. z mn \textstyle z_{\mathrm{mn}}
  10. R 21 = V 2 I 1 | I 2 = 0 R_{21}=\left.\frac{V_{2}}{I_{1}}\right|_{I_{2}=0}
  11. - ( ฮฒ r O - R E ) r E + R E r ฯ€ + r E + 2 R E -(\beta r_{O}-R_{E})\frac{r_{E}+R_{E}}{r_{\pi}+r_{E}+2R_{E}}
  12. - ฮฒ r o r E + R E r ฯ€ + 2 R E -\beta r_{o}\frac{r_{E}+R_{E}}{r_{\pi}+2R_{E}}
  13. R 11 = V 1 I 1 | I 2 = 0 R_{11}=\left.\frac{V_{1}}{I_{1}}\right|_{I_{2}=0}
  14. ( r E + R E ) โˆฅ ( r ฯ€ + R E ) (r_{E}+R_{E})\|(r_{\pi}+R_{E})
  15. R 22 = V 2 I 2 | I 1 = 0 R_{22}=\left.\frac{V_{2}}{I_{2}}\right|_{I_{1}=0}
  16. ( 1 + ฮฒ R E r ฯ€ + r E + 2 R E ) r O + r ฯ€ + r E + R E r ฯ€ + r E + 2 R E R E \left(1+\beta\frac{R_{E}}{r_{\pi}+r_{E}+2R_{E}}\right)r_{O}+\frac{r_{\pi}+r_{E% }+R_{E}}{r_{\pi}+r_{E}+2R_{E}}R_{E}
  17. ( 1 + ฮฒ R E r ฯ€ + 2 R E ) r O \left(1+\beta\frac{R_{E}}{r_{\pi}+2R_{E}}\right)r_{O}
  18. R 12 = V 1 I 2 | I 1 = 0 R_{12}=\left.\frac{V_{1}}{I_{2}}\right|_{I_{1}=0}
  19. R E r E + R E r ฯ€ + r E + 2 R E R_{E}\frac{r_{E}+R_{E}}{r_{\pi}+r_{E}+2R_{E}}
  20. R E r E + R E r ฯ€ + 2 R E R_{E}\frac{r_{E}+R_{E}}{r_{\pi}+2R_{E}}
  21. r ฯ€ r ฯ€ + 2 R E \frac{r_{\pi}}{r_{\pi}+2R_{E}}
  22. ( r E + R E ) (r_{E}+R_{E})
  23. [ I 1 I 2 ] = [ y 11 y 12 y 21 y 22 ] [ V 1 V 2 ] \begin{bmatrix}I_{1}\\ I_{2}\end{bmatrix}=\begin{bmatrix}y_{11}&y_{12}\\ y_{21}&y_{22}\end{bmatrix}\begin{bmatrix}V_{1}\\ V_{2}\end{bmatrix}
  24. y 11 = def I 1 V 1 | V 2 = 0 y 12 = def I 1 V 2 | V 1 = 0 y 21 = def I 2 V 1 | V 2 = 0 y 22 = def I 2 V 2 | V 1 = 0 \begin{aligned}\displaystyle y_{11}&\displaystyle\stackrel{\,\text{def}}{=}\,% \left.\frac{I_{1}}{V_{1}}\right|_{V_{2}=0}\qquad y_{12}\,\stackrel{\,\text{def% }}{=}\,\left.\frac{I_{1}}{V_{2}}\right|_{V_{1}=0}\\ \displaystyle y_{21}&\displaystyle\stackrel{\,\text{def}}{=}\,\left.\frac{I_{2% }}{V_{1}}\right|_{V_{2}=0}\qquad y_{22}\,\stackrel{\,\text{def}}{=}\,\left.% \frac{I_{2}}{V_{2}}\right|_{V_{1}=0}\end{aligned}
  25. y 12 = y 21 \textstyle y_{12}=y_{21}
  26. y 11 = y 22 \textstyle y_{11}=y_{22}
  27. y mn \textstyle y_{\mathrm{mn}}
  28. [ V 1 I 2 ] = [ h 11 h 12 h 21 h 22 ] [ I 1 V 2 ] \begin{bmatrix}V_{1}\\ I_{2}\end{bmatrix}=\begin{bmatrix}h_{11}&h_{12}\\ h_{21}&h_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ V_{2}\end{bmatrix}
  29. h 11 = def V 1 I 1 | V 2 = 0 h 12 = def V 1 V 2 | I 1 = 0 h 21 = def I 2 I 1 | V 2 = 0 h 22 = def I 2 V 2 | I 1 = 0 \begin{aligned}\displaystyle h_{11}&\displaystyle\stackrel{\,\text{def}}{=}\,% \left.\frac{V_{1}}{I_{1}}\right|_{V_{2}=0}\qquad h_{12}\,\stackrel{\,\text{def% }}{=}\,\left.\frac{V_{1}}{V_{2}}\right|_{I_{1}=0}\\ \displaystyle h_{21}&\displaystyle\stackrel{\,\text{def}}{=}\,\left.\frac{I_{2% }}{I_{1}}\right|_{V_{2}=0}\qquad h_{22}\,\stackrel{\,\text{def}}{=}\,\left.% \frac{I_{2}}{V_{2}}\right|_{I_{1}=0}\end{aligned}
  30. h 21 = I 2 I 1 | V 2 = 0 h_{21}=\left.\frac{I_{2}}{I_{1}}\right|_{V_{2}=0}
  31. - ฮฒ ฮฒ + 1 r O + r E r O + r E -\frac{\frac{\beta}{\beta+1}r_{O}+r_{E}}{r_{O}+r_{E}}
  32. - ฮฒ ฮฒ + 1 -\frac{\beta}{\beta+1}
  33. h 11 = V 1 I 1 | V 2 = 0 h_{11}=\left.\frac{V_{1}}{I_{1}}\right|_{V_{2}=0}
  34. r E โˆฅ r O r_{E}\|r_{O}
  35. r E r_{E}
  36. h 22 = I 2 V 2 | I 1 = 0 h_{22}=\left.\frac{I_{2}}{V_{2}}\right|_{I_{1}=0}
  37. 1 ( ฮฒ + 1 ) ( r O + r E ) \frac{1}{(\beta+1)(r_{O}+r_{E})}
  38. 1 ( ฮฒ + 1 ) r O \frac{1}{(\beta+1)r_{O}}
  39. h 12 = V 1 V 2 | I 1 = 0 h_{12}=\left.\frac{V_{1}}{V_{2}}\right|_{I_{1}=0}
  40. r E r E + r O \frac{r_{E}}{r_{E}+r_{O}}
  41. r E r O โ‰ช 1 \frac{r_{E}}{r_{O}}\ll 1
  42. [ I 1 V 2 ] = [ g 11 g 12 g 21 g 22 ] [ V 1 I 2 ] \begin{bmatrix}I_{1}\\ V_{2}\end{bmatrix}=\begin{bmatrix}g_{11}&g_{12}\\ g_{21}&g_{22}\end{bmatrix}\begin{bmatrix}V_{1}\\ I_{2}\end{bmatrix}
  43. g 11 = def I 1 V 1 | I 2 = 0 g 12 = def I 1 I 2 | V 1 = 0 g 21 = def V 2 V 1 | I 2 = 0 g 22 = def V 2 I 2 | V 1 = 0 \begin{aligned}\displaystyle g_{11}&\displaystyle\stackrel{\,\text{def}}{=}\,% \left.\frac{I_{1}}{V_{1}}\right|_{I_{2}=0}\qquad g_{12}\,\stackrel{\,\text{def% }}{=}\,\left.\frac{I_{1}}{I_{2}}\right|_{V_{1}=0}\\ \displaystyle g_{21}&\displaystyle\stackrel{\,\text{def}}{=}\,\left.\frac{V_{2% }}{V_{1}}\right|_{I_{2}=0}\qquad g_{22}\,\stackrel{\,\text{def}}{=}\,\left.% \frac{V_{2}}{I_{2}}\right|_{V_{1}=0}\end{aligned}
  44. g 21 = V 2 V 1 | I 2 = 0 g_{21}=\left.\frac{V_{2}}{V_{1}}\right|_{I_{2}=0}
  45. r o r ฯ€ + g m r O + 1 \frac{r_{o}}{r_{\pi}}+g_{m}r_{O}+1
  46. g m r O g_{m}r_{O}
  47. g 11 = I 1 V 1 | I 2 = 0 g_{11}=\left.\frac{I_{1}}{V_{1}}\right|_{I_{2}=0}
  48. 1 r ฯ€ \frac{1}{r_{\pi}}
  49. 1 r ฯ€ \frac{1}{r_{\pi}}
  50. g 22 = V 2 I 2 | V 1 = 0 g_{22}=\left.\frac{V_{2}}{I_{2}}\right|_{V_{1}=0}
  51. r O r_{O}
  52. r O r_{O}
  53. g 12 = I 1 I 2 | V 1 = 0 g_{12}=\left.\frac{I_{1}}{I_{2}}\right|_{V_{1}=0}
  54. - ฮฒ + 1 ฮฒ -\frac{\beta+1}{\beta}
  55. - 1 -1
  56. [ V 1 I 1 ] = [ A B C D ] [ V 2 - I 2 ] \begin{bmatrix}V_{1}\\ I_{1}\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\begin{bmatrix}V_{2}\\ -I_{2}\end{bmatrix}
  57. A D - B C = 1 \scriptstyle AD-BC=1
  58. A = D \scriptstyle A=D
  59. [ V 2 I 2 โ€ฒ ] = [ A โ€ฒ B โ€ฒ C โ€ฒ D โ€ฒ ] [ V 1 I 1 ] \begin{bmatrix}V_{2}\\ I^{\prime}_{2}\end{bmatrix}=\begin{bmatrix}A^{\prime}&B^{\prime}\\ C^{\prime}&D^{\prime}\end{bmatrix}\begin{bmatrix}V_{1}\\ I_{1}\end{bmatrix}
  60. A โ€ฒ = def V 2 V 1 | I 1 = 0 B โ€ฒ = def V 2 I 1 | V 1 = 0 C โ€ฒ = def - I 2 V 1 | I 1 = 0 D โ€ฒ = def - I 2 I 1 | V 1 = 0 \begin{aligned}\displaystyle A^{\prime}&\displaystyle\stackrel{\,\text{def}}{=% }\,\left.\frac{V_{2}}{V_{1}}\right|_{I_{1}=0}&\displaystyle\qquad B^{\prime}&% \displaystyle\stackrel{\,\text{def}}{=}\,\left.\frac{V_{2}}{I_{1}}\right|_{V_{% 1}=0}\\ \displaystyle C^{\prime}&\displaystyle\stackrel{\,\text{def}}{=}\,\left.-\frac% {I_{2}}{V_{1}}\right|_{I_{1}=0}&\displaystyle\qquad D^{\prime}&\displaystyle% \stackrel{\,\text{def}}{=}\,\left.-\frac{I_{2}}{I_{1}}\right|_{V_{1}=0}\end{aligned}
  61. C โ€ฒ \scriptstyle C^{\prime}
  62. D โ€ฒ \scriptstyle D^{\prime}
  63. I 2 โ€ฒ \scriptstyle I^{\prime}_{2}
  64. I 2 \scriptstyle I_{2}
  65. I 2 โ€ฒ = - I 2 \scriptstyle I^{\prime}_{2}=-I_{2}
  66. A โ€ฒ B โ€ฒ C โ€ฒ D โ€ฒ \scriptstyle A^{\prime}B^{\prime}C^{\prime}D^{\prime}
  67. A B C D \scriptstyle ABCD
  68. A โ€ฒ B โ€ฒ C โ€ฒ D โ€ฒ \scriptstyle A^{\prime}B^{\prime}C^{\prime}D^{\prime}
  69. [ ๐š ] = [ a 11 a 12 a 21 a 22 ] = [ A B C D ] [ ๐› ] = [ b 11 b 12 b 21 b 22 ] = [ A โ€ฒ B โ€ฒ C โ€ฒ D โ€ฒ ] \begin{aligned}\displaystyle\left[\mathbf{a}\right]&\displaystyle=\begin{% bmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{bmatrix}=\begin{bmatrix}A&B\\ C&D\end{bmatrix}\\ \displaystyle\left[\mathbf{b}\right]&\displaystyle=\begin{bmatrix}b_{11}&b_{12% }\\ b_{21}&b_{22}\end{bmatrix}=\begin{bmatrix}A^{\prime}&B^{\prime}\\ C^{\prime}&D^{\prime}\end{bmatrix}\end{aligned}
  70. [ 1 R 0 1 ] \begin{bmatrix}1&R\\ 0&1\end{bmatrix}
  71. [ 1 - R 0 1 ] \begin{bmatrix}1&-R\\ 0&1\end{bmatrix}
  72. [ 1 0 1 R 1 ] \begin{bmatrix}1&0\\ \frac{1}{R}&1\end{bmatrix}
  73. [ 1 0 - 1 R 1 ] \begin{bmatrix}1&0\\ -\frac{1}{R}&1\end{bmatrix}
  74. [ 1 1 G 0 1 ] \begin{bmatrix}1&\frac{1}{G}\\ 0&1\end{bmatrix}
  75. [ 1 - 1 G 0 1 ] \begin{bmatrix}1&-\frac{1}{G}\\ 0&1\end{bmatrix}
  76. [ 1 0 G 1 ] \begin{bmatrix}1&0\\ G&1\end{bmatrix}
  77. [ 1 0 - G 1 ] \begin{bmatrix}1&0\\ -G&1\end{bmatrix}
  78. [ 1 s L 0 1 ] \begin{bmatrix}1&sL\\ 0&1\end{bmatrix}
  79. [ 1 - s L 0 1 ] \begin{bmatrix}1&-sL\\ 0&1\end{bmatrix}
  80. [ 1 0 s C 1 ] \begin{bmatrix}1&0\\ sC&1\end{bmatrix}
  81. [ 1 0 - s C 1 ] \begin{bmatrix}1&0\\ -sC&1\end{bmatrix}
  82. [ cosh ( ฮณ l ) Z 0 sinh ( ฮณ l ) 1 Z 0 sinh ( ฮณ l ) cosh ( ฮณ l ) ] \begin{bmatrix}\cosh\left(\gamma l\right)&Z_{0}\sinh\left(\gamma l\right)\\ \frac{1}{Z_{0}}\sinh\left(\gamma l\right)&\cosh\left(\gamma l\right)\end{bmatrix}
  83. [ cosh ( ฮณ l ) - Z 0 sinh ( ฮณ l ) - 1 Z 0 sinh ( ฮณ l ) cosh ( ฮณ l ) ] \begin{bmatrix}\cosh\left(\gamma l\right)&-Z_{0}\sinh\left(\gamma l\right)\\ -\frac{1}{Z_{0}}\sinh\left(\gamma l\right)&\cosh\left(\gamma l\right)\end{bmatrix}
  84. ฮณ ฮณ
  85. ฮณ = ฮฑ + i ฮฒ \gamma=\alpha+i\beta
  86. l l
  87. [ b 1 b 2 ] = [ S 11 S 12 S 21 S 22 ] [ a 1 a 2 ] \begin{bmatrix}b_{1}\\ b_{2}\end{bmatrix}=\begin{bmatrix}S_{11}&S_{12}\\ S_{21}&S_{22}\end{bmatrix}\begin{bmatrix}a_{1}\\ a_{2}\end{bmatrix}
  88. a k \scriptstyle a_{k}
  89. b k \scriptstyle b_{k}
  90. a k \scriptstyle a_{k}
  91. b k \scriptstyle b_{k}
  92. S 12 = S 21 \textstyle S_{12}=S_{21}
  93. S 11 = S 22 \textstyle S_{11}=S_{22}
  94. S 11 = - S 22 \textstyle S_{11}=-S_{22}
  95. | S 11 | = | S 22 | \textstyle|S_{11}|=|S_{22}|
  96. | S 11 | 2 + | S 12 | 2 = 1 \textstyle|S_{11}|^{2}+|S_{12}|^{2}=1
  97. [ a 1 b 1 ] = [ T 11 T 12 T 21 T 22 ] [ b 2 a 2 ] \begin{bmatrix}a_{1}\\ b_{1}\end{bmatrix}=\begin{bmatrix}T_{11}&T_{12}\\ T_{21}&T_{22}\end{bmatrix}\begin{bmatrix}b_{2}\\ a_{2}\end{bmatrix}
  98. [ ๐ณ ] = [ ๐ณ ] 1 + [ ๐ณ ] 2 [\mathbf{z}]=[\mathbf{z}]_{1}+[\mathbf{z}]_{2}
  99. [ ๐ณ ] 1 = [ R 1 + R 2 R 2 R 2 R 2 ] [\mathbf{z}]_{1}=\begin{bmatrix}R_{1}+R_{2}&R_{2}\\ R_{2}&R_{2}\end{bmatrix}
  100. [ ๐ณ ] = [ ๐ณ ] 1 + [ ๐ณ ] 2 = 2 [ ๐ณ ] 1 = [ 2 R 1 + 2 R 2 2 R 2 2 R 2 2 R 2 ] [\mathbf{z}]=[\mathbf{z}]_{1}+[\mathbf{z}]_{2}=2[\mathbf{z}]_{1}=\begin{% bmatrix}2R_{1}+2R_{2}&2R_{2}\\ 2R_{2}&2R_{2}\end{bmatrix}
  101. [ ๐ณ ] = [ R 1 + 2 R 2 2 R 2 2 R 2 2 R 2 ] [\mathbf{z}]=\begin{bmatrix}R_{1}+2R_{2}&2R_{2}\\ 2R_{2}&2R_{2}\end{bmatrix}
  102. [ ๐ฒ ] = [ ๐ฒ ] 1 + [ ๐ฒ ] 2 [\mathbf{y}]=[\mathbf{y}]_{1}+[\mathbf{y}]_{2}
  103. [ ๐ก ] = [ ๐ก ] 1 + [ ๐ก ] 2 [\mathbf{h}]=[\mathbf{h}]_{1}+[\mathbf{h}]_{2}
  104. [ ๐  ] = [ ๐  ] 1 + [ ๐  ] 2 [\mathbf{g}]=[\mathbf{g}]_{1}+[\mathbf{g}]_{2}
  105. [ ๐š ] = [ ๐š ] 1 โ‹… [ ๐š ] 2 [\mathbf{a}]=[\mathbf{a}]_{1}\cdot[\mathbf{a}]_{2}
  106. [ ๐› ] = [ ๐› ] 2 โ‹… [ ๐› ] 1 [\mathbf{b}]=[\mathbf{b}]_{2}\cdot[\mathbf{b}]_{1}
  107. [ ] [ ๐› ] 1 \displaystyle[]\left[\mathbf{b}\right]_{1}
  108. [ ๐› ] \scriptstyle[\mathbf{b}]
  109. [ ] [ ๐› ] \displaystyle[][\mathbf{b}]
  110. [ V 2 I 2 โ€ฒ ] = [ 1 - R - s C 1 + s C R ] [ V 1 I 1 ] \begin{bmatrix}V_{2}\\ I^{\prime}_{2}\end{bmatrix}=\begin{bmatrix}1&-R\\ -sC&1+sCR\end{bmatrix}\begin{bmatrix}V_{1}\\ I_{1}\end{bmatrix}
  111. [ ๐ณ ] \mathbf{[z]}
  112. [ ๐ฒ ] \mathbf{[y]}
  113. [ ๐ก ] \mathbf{[h]}
  114. [ ๐  ] \mathbf{[g]}
  115. [ ๐š ] \mathbf{[a]}
  116. [ ๐› ] \mathbf{[b]}
  117. [ ๐ณ ] \mathbf{[z]}
  118. [ z 11 z 12 z 21 z 22 ] \begin{bmatrix}z_{11}&z_{12}\\ z_{21}&z_{22}\end{bmatrix}
  119. [ y 22 ฮ” [ ๐ฒ ] - y 12 ฮ” [ ๐ฒ ] - y 21 ฮ” [ ๐ฒ ] y 11 ฮ” [ ๐ฒ ] ] \begin{bmatrix}\dfrac{y_{22}}{\Delta\mathbf{[y]}}&\dfrac{-y_{12}}{\Delta% \mathbf{[y]}}\\ \dfrac{-y_{21}}{\Delta\mathbf{[y]}}&\dfrac{y_{11}}{\Delta\mathbf{[y]}}\end{bmatrix}
  120. [ ฮ” [ ๐ก ] h 22 h 12 h 22 - h 21 h 22 1 h 22 ] \begin{bmatrix}\dfrac{\Delta\mathbf{[h]}}{h_{22}}&\dfrac{h_{12}}{h_{22}}\\ \dfrac{-h_{21}}{h_{22}}&\dfrac{1}{h_{22}}\end{bmatrix}
  121. [ 1 g 11 - g 12 g 11 g 21 g 11 ฮ” [ ๐  ] g 11 ] \begin{bmatrix}\dfrac{1}{g_{11}}&\dfrac{-g_{12}}{g_{11}}\\ \dfrac{g_{21}}{g_{11}}&\dfrac{\Delta\mathbf{[g]}}{g_{11}}\end{bmatrix}
  122. [ a 11 a 21 ฮ” [ ๐š ] a 21 1 a 21 a 22 a 21 ] \begin{bmatrix}\dfrac{a_{11}}{a_{21}}&\dfrac{\Delta\mathbf{[a]}}{a_{21}}\\ \dfrac{1}{a_{21}}&\dfrac{a_{22}}{a_{21}}\end{bmatrix}
  123. [ - b 22 b 21 - 1 b 21 - ฮ” [ ๐› ] b 21 - b 11 b 21 ] \begin{bmatrix}\dfrac{-b_{22}}{b_{21}}&\dfrac{-1}{b_{21}}\\ \dfrac{-\Delta\mathbf{[b]}}{b_{21}}&\dfrac{-b_{11}}{b_{21}}\end{bmatrix}
  124. [ ๐ฒ ] \mathbf{[y]}
  125. [ z 22 ฮ” [ ๐ณ ] - z 12 ฮ” [ ๐ณ ] - z 21 ฮ” [ ๐ณ ] z 11 ฮ” [ ๐ณ ] ] \begin{bmatrix}\dfrac{z_{22}}{\Delta\mathbf{[z]}}&\dfrac{-z_{12}}{\Delta% \mathbf{[z]}}\\ \dfrac{-z_{21}}{\Delta\mathbf{[z]}}&\dfrac{z_{11}}{\Delta\mathbf{[z]}}\end{bmatrix}
  126. [ y 11 y 12 y 21 y 22 ] \begin{bmatrix}y_{11}&y_{12}\\ y_{21}&y_{22}\end{bmatrix}
  127. [ 1 h 11 - h 12 h 11 h 21 h 11 ฮ” [ ๐ก ] h 11 ] \begin{bmatrix}\dfrac{1}{h_{11}}&\dfrac{-h_{12}}{h_{11}}\\ \dfrac{h_{21}}{h_{11}}&\dfrac{\Delta\mathbf{[h]}}{h_{11}}\end{bmatrix}
  128. [ ฮ” [ ๐  ] g 22 g 12 g 22 - g 21 g 22 1 g 22 ] \begin{bmatrix}\dfrac{\Delta\mathbf{[g]}}{g_{22}}&\dfrac{g_{12}}{g_{22}}\\ \dfrac{-g_{21}}{g_{22}}&\dfrac{1}{g_{22}}\end{bmatrix}
  129. [ a 22 a 12 - ฮ” [ ๐š ] a 12 - 1 a 12 a 11 a 12 ] \begin{bmatrix}\dfrac{a_{22}}{a_{12}}&\dfrac{-\Delta\mathbf{[a]}}{a_{12}}\\ \dfrac{-1}{a_{12}}&\dfrac{a_{11}}{a_{12}}\end{bmatrix}
  130. [ - b 11 b 12 1 b 12 ฮ” [ ๐› ] b 12 - b 22 b 12 ] \begin{bmatrix}\dfrac{-b_{11}}{b_{12}}&\dfrac{1}{b_{12}}\\ \dfrac{\Delta\mathbf{[b]}}{b_{12}}&\dfrac{-b_{22}}{b_{12}}\end{bmatrix}
  131. [ ๐ก ] \mathbf{[h]}
  132. [ ฮ” [ ๐ณ ] z 22 z 12 z 22 - z 21 z 22 1 z 22 ] \begin{bmatrix}\dfrac{\Delta\mathbf{[z]}}{z_{22}}&\dfrac{z_{12}}{z_{22}}\\ \dfrac{-z_{21}}{z_{22}}&\dfrac{1}{z_{22}}\end{bmatrix}
  133. [ 1 y 11 - y 12 Y 11 y 21 y 11 ฮ” [ ๐ฒ ] y 11 ] \begin{bmatrix}\dfrac{1}{y_{11}}&\dfrac{-y_{12}}{Y_{11}}\\ \dfrac{y_{21}}{y_{11}}&\dfrac{\Delta\mathbf{[y]}}{y_{11}}\end{bmatrix}
  134. [ h 11 h 12 h 21 h 22 ] \begin{bmatrix}h_{11}&h_{12}\\ h_{21}&h_{22}\end{bmatrix}
  135. [ g 22 ฮ” [ ๐  ] - g 12 ฮ” [ ๐  ] - g 21 ฮ” [ ๐  ] g 11 ฮ” [ ๐  ] ] \begin{bmatrix}\dfrac{g_{22}}{\Delta\mathbf{[g]}}&\dfrac{-g_{12}}{\Delta% \mathbf{[g]}}\\ \dfrac{-g_{21}}{\Delta\mathbf{[g]}}&\dfrac{g_{11}}{\Delta\mathbf{[g]}}\end{bmatrix}
  136. [ a 12 a 22 ฮ” [ ๐š ] a 22 - 1 a 22 a 21 a 22 ] \begin{bmatrix}\dfrac{a_{12}}{a_{22}}&\dfrac{\Delta\mathbf{[a]}}{a_{22}}\\ \dfrac{-1}{a_{22}}&\dfrac{a_{21}}{a_{22}}\end{bmatrix}
  137. [ - b 12 b 11 1 b 11 - ฮ” [ ๐› ] b 11 - b 21 b 11 ] \begin{bmatrix}\dfrac{-b_{12}}{b_{11}}&\dfrac{1}{b_{11}}\\ \dfrac{-\Delta\mathbf{[b]}}{b_{11}}&\dfrac{-b_{21}}{b_{11}}\end{bmatrix}
  138. [ ๐  ] \mathbf{[g]}
  139. [ 1 z 11 - z 12 z 11 z 21 z 11 ฮ” [ ๐ณ ] z 11 ] \begin{bmatrix}\dfrac{1}{z_{11}}&\dfrac{-z_{12}}{z_{11}}\\ \dfrac{z_{21}}{z_{11}}&\dfrac{\Delta\mathbf{[z]}}{z_{11}}\end{bmatrix}
  140. [ ฮ” [ ๐ฒ ] y 22 y 12 y 22 - y 21 y 22 1 y 22 ] \begin{bmatrix}\dfrac{\Delta\mathbf{[y]}}{y_{22}}&\dfrac{y_{12}}{y_{22}}\\ \dfrac{-y_{21}}{y_{22}}&\dfrac{1}{y_{22}}\end{bmatrix}
  141. [ h 22 ฮ” [ ๐ก ] - h 12 ฮ” [ ๐ก ] - h 21 ฮ” [ ๐ก ] h 11 ฮ” [ ๐ก ] ] \begin{bmatrix}\dfrac{h_{22}}{\Delta\mathbf{[h]}}&\dfrac{-h_{12}}{\Delta% \mathbf{[h]}}\\ \dfrac{-h_{21}}{\Delta\mathbf{[h]}}&\dfrac{h_{11}}{\Delta\mathbf{[h]}}\end{bmatrix}
  142. [ g 11 g 12 g 21 g 22 ] \begin{bmatrix}g_{11}&g_{12}\\ g_{21}&g_{22}\end{bmatrix}
  143. [ a 21 a 11 - ฮ” [ ๐š ] a 11 1 a 11 a 12 a 11 ] \begin{bmatrix}\dfrac{a_{21}}{a_{11}}&\dfrac{-\Delta\mathbf{[a]}}{a_{11}}\\ \dfrac{1}{a_{11}}&\dfrac{a_{12}}{a_{11}}\end{bmatrix}
  144. [ - b 21 b 22 - 1 b 22 ฮ” [ ๐› ] b 22 - b 12 b 22 ] \begin{bmatrix}\dfrac{-b_{21}}{b_{22}}&\dfrac{-1}{b_{22}}\\ \dfrac{\Delta\mathbf{[b]}}{b_{22}}&\dfrac{-b_{12}}{b_{22}}\end{bmatrix}
  145. [ ๐š ] \mathbf{[a]}
  146. [ z 11 z 21 ฮ” [ ๐ณ ] z 21 1 z 21 z 22 z 21 ] \begin{bmatrix}\dfrac{z_{11}}{z_{21}}&\dfrac{\Delta\mathbf{[z]}}{z_{21}}\\ \dfrac{1}{z_{21}}&\dfrac{z_{22}}{z_{21}}\end{bmatrix}
  147. [ - y 22 y 21 - 1 y 21 - ฮ” [ ๐ฒ ] y 21 - y 11 y 21 ] \begin{bmatrix}\dfrac{-y_{22}}{y_{21}}&\dfrac{-1}{y_{21}}\\ \dfrac{-\Delta\mathbf{[y]}}{y_{21}}&\dfrac{-y_{11}}{y_{21}}\end{bmatrix}
  148. [ - ฮ” [ ๐ก ] h 21 - h 11 h 21 - h 22 h 21 - 1 h 21 ] \begin{bmatrix}\dfrac{-\Delta\mathbf{[h]}}{h_{21}}&\dfrac{-h_{11}}{h_{21}}\\ \dfrac{-h_{22}}{h_{21}}&\dfrac{-1}{h_{21}}\end{bmatrix}
  149. [ 1 g 21 g 22 g 21 g 11 g 21 ฮ” [ ๐  ] g 21 ] \begin{bmatrix}\dfrac{1}{g_{21}}&\dfrac{g_{22}}{g_{21}}\\ \dfrac{g_{11}}{g_{21}}&\dfrac{\Delta\mathbf{[g]}}{g_{21}}\end{bmatrix}
  150. [ a 11 a 12 a 21 a 22 ] \begin{bmatrix}a_{11}&a_{12}\\ a_{21}&a_{22}\end{bmatrix}
  151. [ b 22 ฮ” [ ๐› ] - b 12 ฮ” [ ๐› ] - b 21 ฮ” [ ๐› ] b 11 ฮ” [ ๐› ] ] \begin{bmatrix}\dfrac{b_{22}}{\Delta\mathbf{[b]}}&\dfrac{-b_{12}}{\Delta% \mathbf{[b]}}\\ \dfrac{-b_{21}}{\Delta\mathbf{[b]}}&\dfrac{b_{11}}{\Delta\mathbf{[b]}}\end{bmatrix}
  152. [ ๐› ] \mathbf{[b]}
  153. [ z 22 z 12 - ฮ” [ ๐ณ ] z 12 - 1 z 12 z 11 z 12 ] \begin{bmatrix}\dfrac{z_{22}}{z_{12}}&\dfrac{-\Delta\mathbf{[z]}}{z_{12}}\\ \dfrac{-1}{z_{12}}&\dfrac{z_{11}}{z_{12}}\end{bmatrix}
  154. [ - y 11 y 12 1 y 12 ฮ” [ ๐ฒ ] y 12 - y 22 y 12 ] \begin{bmatrix}\dfrac{-y_{11}}{y_{12}}&\dfrac{1}{y_{12}}\\ \dfrac{\Delta\mathbf{[y]}}{y_{12}}&\dfrac{-y_{22}}{y_{12}}\end{bmatrix}
  155. [ 1 h 12 - h 11 h 12 - h 22 h 12 ฮ” [ ๐ก ] h 12 ] \begin{bmatrix}\dfrac{1}{h_{12}}&\dfrac{-h_{11}}{h_{12}}\\ \dfrac{-h_{22}}{h_{12}}&\dfrac{\Delta\mathbf{[h]}}{h_{12}}\end{bmatrix}
  156. [ - ฮ” [ ๐  ] g 12 g 22 g 12 g 11 g 12 - 1 g 12 ] \begin{bmatrix}\dfrac{-\Delta\mathbf{[g]}}{g_{12}}&\dfrac{g_{22}}{g_{12}}\\ \dfrac{g_{11}}{g_{12}}&\dfrac{-1}{g_{12}}\end{bmatrix}
  157. [ a 22 ฮ” [ ๐š ] - a 12 ฮ” [ ๐š ] - a 21 ฮ” [ ๐š ] a 11 ฮ” [ ๐š ] ] \begin{bmatrix}\dfrac{a_{22}}{\Delta\mathbf{[a]}}&\dfrac{-a_{12}}{\Delta% \mathbf{[a]}}\\ \dfrac{-a_{21}}{\Delta\mathbf{[a]}}&\dfrac{a_{11}}{\Delta\mathbf{[a]}}\end{bmatrix}
  158. [ b 11 b 12 b 21 b 22 ] \begin{bmatrix}b_{11}&b_{12}\\ b_{21}&b_{22}\end{bmatrix}
  159. ฮ” [ ๐ฑ ] \Delta\mathbf{[x]}
  160. [ ๐ฒ ] = [ ๐ณ ] - 1 [\mathbf{y}]=[\mathbf{z}]^{-1}
  161. [ ๐  ] = [ ๐ก ] - 1 [\mathbf{g}]=[\mathbf{h}]^{-1}
  162. [ ๐› ] = [ ๐š ] - 1 [\mathbf{b}]=[\mathbf{a}]^{-1}
  163. [ V 1 V 2 V 3 ] = [ Z 11 Z 12 Z 13 Z 21 Z 22 Z 23 Z 31 Z 32 Z 33 ] [ I 1 I 2 I 3 ] \begin{bmatrix}V_{1}\\ V_{2}\\ V_{3}\end{bmatrix}=\begin{bmatrix}Z_{11}&Z_{12}&Z_{13}\\ Z_{21}&Z_{22}&Z_{23}\\ Z_{31}&Z_{32}&Z_{33}\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{2}\\ I_{3}\end{bmatrix}
  164. [ V 1 V 2 ] = [ z 11 z 12 z 21 z 22 ] [ I 1 I 2 ] \begin{bmatrix}V_{1}\\ V_{2}\end{bmatrix}=\begin{bmatrix}z_{11}&z_{12}\\ z_{21}&z_{22}\end{bmatrix}\begin{bmatrix}I_{1}\\ I_{2}\end{bmatrix}
  165. V 2 = - Z L I 2 V_{2}=-Z_{L}I_{2}\,
  166. V 1 = Z 11 I 1 + Z 12 I 2 - Z L I 2 = Z 21 I 1 + Z 22 I 2 \begin{aligned}\displaystyle V_{1}&\displaystyle=Z_{11}I_{1}+Z_{12}I_{2}\\ \displaystyle-Z_{L}I_{2}&\displaystyle=Z_{21}I_{1}+Z_{22}I_{2}\end{aligned}
  167. I 2 = - Z 21 Z L + Z 22 I 1 V 1 = Z 11 I 1 - Z 12 Z 21 Z L + Z 22 I 1 = ( Z 11 - Z 12 Z 21 Z L + Z 22 ) I 1 = Z in I 1 \begin{aligned}\displaystyle I_{2}&\displaystyle=-\frac{Z_{21}}{Z_{L}+Z_{22}}I% _{1}\\ \displaystyle V_{1}&\displaystyle=Z_{11}I_{1}-\frac{Z_{12}Z_{21}}{Z_{L}+Z_{22}% }I_{1}\\ &\displaystyle=\left(Z_{11}-\frac{Z_{12}Z_{21}}{Z_{L}+Z_{22}}\right)I_{1}=Z_{% \mathrm{in}}I_{1}\end{aligned}
  168. Z in Z_{\mathrm{in}}\,
  169. R 1 โˆฅ R 2 = 1 / ( 1 / R 1 + 1 / R 2 ) R_{1}\|R_{2}=1/(1/R_{1}+1/R_{2})

Two-sided_Laplace_transform.html

  1. โ„ฌ { f ( t ) } = F ( s ) = โˆซ - โˆž โˆž e - s t f ( t ) d t . \mathcal{B}\{f(t)\}=F(s)=\int_{-\infty}^{\infty}e^{-st}f(t)\,dt.
  2. โˆซ 0 โˆž e - s t f ( t ) d t , โˆซ - โˆž 0 e - s t f ( t ) d t \int_{0}^{\infty}e^{-st}f(t)\,dt,\quad\int_{-\infty}^{0}e^{-st}f(t)\,dt
  3. โ„ฌ \mathcal{B}
  4. ๐’ฏ { f ( t ) } = s โ„ฌ { f ( t ) } = s F ( s ) = s โˆซ - โˆž โˆž e - s t f ( t ) d t . \mathcal{T}\{f(t)\}=s\mathcal{B}\{f(t)\}=sF(s)=s\int_{-\infty}^{\infty}e^{-st}% f(t)\,dt.
  5. โ„’ \mathcal{L}
  6. โ„’ { f ( t ) } = โ„ฌ { f ( t ) u ( t ) } . \mathcal{L}\{f(t)\}=\mathcal{B}\{f(t)u(t)\}.
  7. โ„ฌ { f ( t ) } ( s ) = โ„’ { f ( t ) } ( s ) + โ„’ { f ( - t ) } ( - s ) , \mathcal{B}\{f(t)\}(s)=\mathcal{L}\{f(t)\}(s)+\mathcal{L}\{f(-t)\}(-s),
  8. โ„ณ { f ( t ) } ( s ) = โ„ฌ { f ( e - x ) } ( s ) , \mathcal{M}\{f(t)\}(s)=\mathcal{B}\{f(e^{-x})\}(s),
  9. โ„ฌ { f ( t ) } ( s ) = โ„ณ { f ( - ln x ) } ( s ) . \mathcal{B}\{f(t)\}(s)=\mathcal{M}\{f(-\ln x)\}(s).
  10. โ„ฑ { f ( t ) } = F ( s = i ฯ‰ ) = F ( ฯ‰ ) . \mathcal{F}\{f(t)\}=F(s=i\omega)=F(\omega).
  11. โ„ฑ { f ( t ) } = F ( s = i ฯ‰ ) = 1 2 ฯ€ โ„ฌ { f ( t ) } ( s ) \mathcal{F}\{f(t)\}=F(s=i\omega)=\frac{1}{\sqrt{2\pi}}\mathcal{B}\{f(t)\}(s)
  12. โ„ฌ { f ( t ) } ( s ) = โ„ฑ { f ( t ) } ( - i s ) . \mathcal{B}\{f(t)\}(s)=\mathcal{F}\{f(t)\}(-is).
  13. a < โ„‘ ( s ) < b a<\Im(s)<b
  14. โ„ฌ { f } ( - s ) \mathcal{B}\{f\}(-s)
  15. f โ€ฒ ( t ) f^{\prime}(t)
  16. s F ( s ) - f ( 0 ) sF(s)-f(0)
  17. s F ( s ) sF(s)
  18. f โ€ฒโ€ฒ ( t ) f^{\prime\prime}(t)
  19. s 2 F ( s ) - s f ( 0 ) - f โ€ฒ ( 0 ) s^{2}F(s)-sf(0)-f^{\prime}(0)
  20. s 2 F ( s ) s^{2}F(s)
  21. lim R โ†’ โˆž โˆซ 0 R f ( t ) e - s t d t \lim_{R\to\infty}\int_{0}^{R}f(t)e^{-st}\,dt
  22. โˆซ 0 โˆž | f ( t ) e - s t | d t \int_{0}^{\infty}\left|f(t)e^{-st}\right|\,dt
  23. F ( s ) = ( s - s 0 ) โˆซ 0 โˆž e - ( s - s 0 ) t ฮฒ ( t ) d t , ฮฒ ( u ) = โˆซ 0 u e - s 0 t f ( t ) d t . F(s)=(s-s_{0})\int_{0}^{\infty}e^{-(s-s_{0})t}\beta(t)\,dt,\quad\beta(u)=\int_% {0}^{u}e^{-s_{0}t}f(t)\,dt.

Two-state_quantum_system.html

  1. | ฯˆ โŸฉ |\psi\rangle
  2. | ฯˆ โŸฉ = ( c 1 c 2 ) = c 1 ( 1 0 ) + c 2 ( 0 1 ) ; |\psi\rangle=\begin{pmatrix}c_{1}\\ c_{2}\end{pmatrix}=c_{1}\begin{pmatrix}1\\ 0\end{pmatrix}+c_{2}\begin{pmatrix}0\\ 1\end{pmatrix};
  3. c 1 c_{1}
  4. c 2 c_{2}
  5. c 1 c_{1}
  6. c 2 c_{2}
  7. | c 1 | 2 + | c 2 | 2 = 1 {|c_{1}|}^{2}+{|c_{2}|}^{2}=1
  8. | 0 โŸฉ = ( 1 0 ) |0\rangle=\begin{pmatrix}1\\ 0\end{pmatrix}
  9. | 1 โŸฉ = ( 0 1 ) |1\rangle=\begin{pmatrix}0\\ 1\end{pmatrix}
  10. ร— \times
  11. ๐‡ = ( a 1 c - i d c + i d a 2 ) \mathbf{H}=\begin{pmatrix}a_{1}&c-id\\ c+id&a_{2}\end{pmatrix}
  12. a 1 , a 2 , c a_{1},a_{2},c
  13. d d
  14. ๐‡ = a โ‹… ฯƒ 0 + c โ‹… ฯƒ 1 + d โ‹… ฯƒ 2 + b โ‹… ฯƒ 3 ; \mathbf{H}=a\cdot\sigma_{0}+c\cdot\sigma_{1}+d\cdot\sigma_{2}+b\cdot\sigma_{3};
  15. a = ( a 1 + a 2 ) 2 a=\frac{(a_{1}+a_{2})}{2}
  16. b = ( a 1 - a 2 ) 2 b=\frac{(a_{1}-a_{2})}{2}
  17. ฯƒ 0 \sigma_{0}
  18. ร— \times
  19. ฯƒ k ( k = 1 , 2 , 3 ) \sigma_{k}(k=1,2,3)
  20. a , b , c a,b,c
  21. d d
  22. ๐‡ = a โ‹… ฯƒ 0 + ๐ซ โ‹… ฯƒ ; \mathbf{H}=a\cdot\sigma_{0}+\mathbf{r}\cdot\mathbf{\sigma};
  23. r r
  24. ( c , d , b ) (c,d,b)
  25. ฯƒ \sigma
  26. ( ฯƒ 1 , ฯƒ 2 , ฯƒ 3 ) (\sigma_{1},\sigma_{2},\sigma_{3})
  27. H H
  28. E ยฑ = a ยฑ | ๐ซ | E_{\pm}=a\pm|\mathbf{r}|
  29. | E + โŸฉ |E_{+}\rangle
  30. | E - โŸฉ |E_{-}\rangle
  31. H = ( E + 0 0 E - ) ; H=\begin{pmatrix}E_{+}&0\\ 0&E_{-}\end{pmatrix};
  32. U U
  33. U ( t ) = e - i H t โ„ = e - i a t โ„ ( cos ( | ๐ซ | ) ฯƒ 0 + i sin ( | ๐ซ | ) r ^ โ‹… ฯƒ ) ; U(t)=e^{\frac{-iHt}{\hbar}}=e^{\frac{-iat}{\hbar}}(\cos(|\mathbf{r}|)\sigma_{0% }+i\sin(|\mathbf{r}|)\hat{r}\cdot\mathbf{\sigma});
  34. r ^ = ๐ซ | ๐ซ | . \hat{r}=\frac{\mathbf{r}}{|\mathbf{r}|}.
  35. | 0 โŸฉ , | 1 โŸฉ |0\rangle,|1\rangle
  36. U U
  37. U ( t ) = ( e - i E + t โ„ 0 0 e - i E - t โ„ ) U(t)=\begin{pmatrix}e^{\frac{-iE_{+}t}{\hbar}}&0\\ 0&e^{\frac{-iE{-}t}{\hbar}}\end{pmatrix}
  38. e - i a t โ„ e^{\frac{-iat}{\hbar}}
  39. H H
  40. | + โŸฉ |+\rangle
  41. | - โŸฉ |-\rangle
  42. E + E_{+}
  43. E - E_{-}
  44. | ฯˆ ( t ) โŸฉ |\psi(t)\rangle
  45. t = 0 t=0
  46. | ฯˆ ( 0 ) โŸฉ = c + | + โŸฉ + c - | - โŸฉ , |\psi(0)\rangle=c_{+}|+\rangle+c_{-}|-\rangle,
  47. | ฯˆ ( 0 ) โŸฉ |\psi(0)\rangle
  48. | ฯˆ ( t ) โŸฉ = U ( t ) | ฯˆ ( 0 ) โŸฉ = c + e - i E + t โ„ | + โŸฉ + c - e - i E - t โ„ | - โŸฉ , |\psi(t)\rangle=U(t)|\psi(0)\rangle=c_{+}e^{\frac{-iE_{+}t}{\hbar}}|+\rangle+c% _{-}e^{\frac{-iE_{-}t}{\hbar}}|-\rangle,
  49. e - i E + t โ„ e^{\frac{-iE_{+}t}{\hbar}}
  50. | ฯˆ ( t ) โŸฉ = c + | + โŸฉ + c - e - i ( E - - E + ) t โ„ | - โŸฉ , |\psi(t)\rangle=c_{+}|+\rangle+c_{-}e^{\frac{-i(E_{-}-E_{+})t}{\hbar}}|-\rangle,
  51. | + โŸฉ |+\rangle
  52. | - โŸฉ |-\rangle
  53. t t
  54. | โŸจ ฯˆ ( 0 ) | ฯˆ ( t ) โŸฉ | 2 = | | c + | 2 + | c - | 2 e - i ( E - - E + ) t โ„ | 2 = 1 - 4 | c + c - | 2 sin 2 ( ฯ‰ t 2 ) {|\langle\psi(0)|\psi(t)\rangle|}^{2}={||c_{+}|^{2}+|c_{-}|^{2}e^{\frac{-i(E_{% -}-E_{+})t}{\hbar}}|}^{2}=1-4|c_{+}c_{-}|^{2}{\sin}^{2}(\frac{\omega t}{2})
  55. ฯ‰ \omega
  56. ฯ‰ = E + - E - โ„ , \omega=\frac{E_{+}-E_{-}}{\hbar},
  57. E + โ‰ฅ E - E_{+}\geq E_{-}
  58. 1 - 4 | c + c - | 2 1-4|c_{+}c_{-}|^{2}
  59. 1 1
  60. E + = E - E_{+}=E_{-}
  61. ๐ = B ๐ง ^ \mathbf{B}=B\mathbf{\hat{n}}
  62. H = - s y m b o l ฮผ โ‹… ๐ = - \musymbol ฯƒ โ‹… ๐ H=-symbol{\mu}\cdot\mathbf{B}=-\musymbol{\sigma}\cdot\mathbf{B}
  63. ฮผ \mu
  64. s y m b o l ฯƒ symbol{\sigma}
  65. H ฯˆ = i โ„ โˆ‚ t ฯˆ H\psi=i\hbar\partial_{t}\psi
  66. ฯˆ ( t ) = e i ฯ‰ t s y m b o l ฯƒ โ‹… ๐ง ^ ฯˆ ( 0 ) , \psi(t)=e^{i\omega tsymbol{\sigma}\cdot\mathbf{\hat{n}}}\psi(0),
  67. ฯ‰ = ฮผ B / โ„ \omega=\mu B/\hbar
  68. e i ฯ‰ t s y m b o l ฯƒ โ‹… ๐ง ^ = cos ( ฯ‰ t ) I + i ๐ง ^ \cdotsymbol ฯƒ sin ( ฯ‰ t ) e^{i\omega tsymbol{\sigma}\cdot\mathbf{\hat{n}}}=\cos{\left(\omega t\right)}I+% i\mathbf{\hat{n}}\cdotsymbol{\sigma}\sin{\left(\omega t\right)}
  69. ๐ง ^ \mathbf{\hat{n}}
  70. 2 ฯ‰ 2\omega
  71. ๐ณ ^ \mathbf{\hat{z}}
  72. e i ฯ‰ t s y m b o l ฯƒ โ‹… ๐ง ^ = ( e i ฯ‰ t 0 0 e - i ฯ‰ t ) . e^{i\omega tsymbol{\sigma}\cdot\mathbf{\hat{n}}}=\begin{pmatrix}e^{i\omega t}&% 0\\ 0&e^{-i\omega t}\end{pmatrix}.
  73. ฯˆ ( 0 ) \psi(0)
  74. ๐‘ = ( โŸจ ฯƒ x โŸฉ , โŸจ ฯƒ y โŸฉ , โŸจ ฯƒ z โŸฉ ) \mathbf{R}=\left(\langle\sigma_{x}\rangle,\langle\sigma_{y}\rangle,\langle% \sigma_{z}\rangle\right)
  75. ฯˆ ( 0 ) \psi(0)
  76. | โ†‘ โŸฉ |\uparrow\rangle
  77. | โ†“ โŸฉ |\downarrow\rangle
  78. ฯƒ z \sigma_{z}
  79. ฯˆ ( 0 ) = 1 2 ( 1 1 ) \psi(0)=\frac{1}{\sqrt{2}}\begin{pmatrix}1\\ 1\end{pmatrix}
  80. ฯˆ ( t ) \psi(t)
  81. ๐‘ = ( cos 2 ฯ‰ t , - sin 2 ฯ‰ t , 0 ) \mathbf{R}=\left(\cos{2\omega t},-\sin{2\omega t},0\right)
  82. ๐ฑ ^ \mathbf{\hat{x}}
  83. ๐ณ ^ \mathbf{\hat{z}}
  84. ๐ณ ^ \mathbf{\hat{z}}
  85. ฯˆ ( 0 ) \psi(0)
  86. a | โ†‘ โŸฉ + b | โ†“ โŸฉ a|\uparrow\rangle+b|\downarrow\rangle
  87. a a
  88. b b
  89. tan ( ฮธ / 2 ) = b / a \tan(\theta/2)=b/a
  90. ๐ณ ^ \mathbf{\hat{z}}
  91. B 0 ๐ณ ^ B_{0}\mathbf{\hat{z}}
  92. ๐ = ( B 1 cos ฯ‰ r t B 1 sin ฯ‰ r t B 0 ) . \mathbf{B}=\begin{pmatrix}B_{1}\cos\omega_{\mathrm{r}}t\\ B_{1}\sin\omega_{\mathrm{r}}t\\ B_{0}\end{pmatrix}.
  93. H = - \musymbol ฯƒ โ‹… ๐ H=-\musymbol{\sigma}\cdot\mathbf{B}
  94. ฯˆ ( t ) \psi(t)
  95. H ฯˆ = i โ„ โˆ‚ ฯˆ / โˆ‚ t H\psi=i\hbar\,\partial\psi/\partial t
  96. โˆ‚ ฯˆ โˆ‚ t = i ( ฯ‰ 1 ฯƒ x + ( w 0 + ฯ‰ r 2 ) ฯƒ z ) ฯˆ , \frac{\partial\psi}{\partial t}=i\left(\omega_{1}\sigma_{x}+\left(w_{0}+\frac{% \omega_{r}}{2}\right)\sigma_{z}\right)\psi,
  97. ฯ‰ 0 = ฮผ B 0 / โ„ \omega_{0}=\mu B_{0}/\hbar
  98. ฯ‰ 1 = ฮผ B 1 / โ„ \omega_{1}=\mu B_{1}/\hbar
  99. ( ฯ‰ 1 , 0 , ฯ‰ 0 + ฯ‰ r / 2 ) (\omega_{1},0,\omega_{0}+\omega_{r}/2)
  100. ฯ‰ 0 \omega_{0}
  101. ฯ‰ r = - 2 ฯ‰ 0 \omega_{r}=-2\omega_{0}
  102. x ^ \hat{x}
  103. 2 ฯ‰ 1 2\omega_{1}
  104. ฯ‰ r \omega_{r}
  105. - \musymbol ฯƒ โ‹… ๐ ฯˆ = i โ„ โˆ‚ ฯˆ โˆ‚ t . -\musymbol{\sigma}\cdot\mathbf{B}\psi=i\hbar\frac{\partial\psi}{\partial t}.
  106. i โ„ i\hbar
  107. โˆ‚ ฯˆ โˆ‚ t = i ( ฯ‰ 1 ฯƒ x cos ฯ‰ r t + ฯ‰ 1 ฯƒ y sin ฯ‰ r t + ฯ‰ 0 ฯƒ z ) ฯˆ . \frac{\partial\psi}{\partial t}=i\left(\omega_{1}\sigma_{x}\cos{\omega_{r}t}+% \omega_{1}\sigma_{y}\sin{\omega_{r}t}+\omega_{0}\sigma_{z}\right)\psi.
  108. ฯˆ โ†’ e - i ฯƒ z ฯ‰ r t / 2 ฯˆ \psi\rightarrow e^{-i\sigma_{z}\omega_{r}t/2}\psi
  109. - i ฯƒ z ฯ‰ r 2 e - i ฯƒ z ฯ‰ r t / 2 ฯˆ + e - i ฯƒ z ฯ‰ r t / 2 โˆ‚ ฯˆ โˆ‚ t = i ( ฯ‰ 1 ฯƒ x cos ฯ‰ r t + ฯ‰ 1 ฯƒ y sin ฯ‰ r t + ฯ‰ 0 ฯƒ z ) e - i ฯƒ z ฯ‰ r t / 2 ฯˆ -i\sigma_{z}\frac{\omega_{r}}{2}e^{-i\sigma_{z}\omega_{r}t/2}\psi+e^{-i\sigma_% {z}\omega_{r}t/2}\frac{\partial\psi}{\partial t}=i\left(\omega_{1}\sigma_{x}% \cos{\omega_{r}t}+\omega_{1}\sigma_{y}\sin{\omega_{r}t}+\omega_{0}\sigma_{z}% \right)e^{-i\sigma_{z}\omega_{r}t/2}\psi
  110. โˆ‚ ฯˆ โˆ‚ t = i e i ฯƒ z ฯ‰ r t / 2 ( ฯ‰ 1 ฯƒ x cos ฯ‰ r t + ฯ‰ 1 ฯƒ y sin ฯ‰ r t + ( ฯ‰ 0 + ฯ‰ r 2 ) ฯƒ z ) e - i ฯƒ z ฯ‰ r t / 2 ฯˆ \frac{\partial\psi}{\partial t}=ie^{i\sigma_{z}\omega_{r}t/2}\left(\omega_{1}% \sigma_{x}\cos{\omega_{r}t}+\omega_{1}\sigma_{y}\sin{\omega_{r}t}+\left(\omega% _{0}+\frac{\omega_{r}}{2}\right)\sigma_{z}\right)e^{-i\sigma_{z}\omega_{r}t/2}\psi
  111. e i ฯƒ z ฯ‰ r t / 2 ฯƒ x e - i ฯƒ z ฯ‰ r t / 2 = ( e i ฯ‰ r t / 2 0 0 e - i ฯ‰ r t / 2 ) ( 0 1 1 0 ) ( e - i ฯ‰ r t / 2 0 0 e i ฯ‰ r t / 2 ) = ( 0 e i ฯ‰ r t e - i ฯ‰ r t 0 ) e^{i\sigma_{z}\omega_{r}t/2}\sigma_{x}e^{-i\sigma_{z}\omega_{r}t/2}=\begin{% pmatrix}e^{i\omega_{r}t/2}&0\\ 0&e^{-i\omega_{r}t/2}\end{pmatrix}\begin{pmatrix}0&1\\ 1&0\end{pmatrix}\begin{pmatrix}e^{-i\omega_{r}t/2}&0\\ 0&e^{i\omega_{r}t/2}\end{pmatrix}=\begin{pmatrix}0&e^{i\omega_{r}t}\\ e^{-i\omega_{r}t}&0\end{pmatrix}
  112. e i ฯƒ z ฯ‰ r t / 2 ฯƒ y e - i ฯƒ z ฯ‰ r t / 2 = ( e i ฯ‰ r t / 2 0 0 e - i ฯ‰ r t / 2 ) ( 0 - i i 0 ) ( e - i ฯ‰ r t / 2 0 0 e i ฯ‰ r t / 2 ) = ( 0 - i e i ฯ‰ r t i e - i ฯ‰ r t 0 ) e^{i\sigma_{z}\omega_{r}t/2}\sigma_{y}e^{-i\sigma_{z}\omega_{r}t/2}=\begin{% pmatrix}e^{i\omega_{r}t/2}&0\\ 0&e^{-i\omega_{r}t/2}\end{pmatrix}\begin{pmatrix}0&-i\\ i&0\end{pmatrix}\begin{pmatrix}e^{-i\omega_{r}t/2}&0\\ 0&e^{i\omega_{r}t/2}\end{pmatrix}=\begin{pmatrix}0&-ie^{i\omega_{r}t}\\ ie^{-i\omega_{r}t}&0\end{pmatrix}
  113. e i ฯƒ z ฯ‰ r t / 2 ฯƒ z e - i ฯƒ z ฯ‰ r t / 2 = ( e i ฯ‰ r t / 2 0 0 e - i ฯ‰ r t / 2 ) ( 1 0 0 - 1 ) ( e - i ฯ‰ r t / 2 0 0 e i ฯ‰ r t / 2 ) = ฯƒ z e^{i\sigma_{z}\omega_{r}t/2}\sigma_{z}e^{-i\sigma_{z}\omega_{r}t/2}=\begin{% pmatrix}e^{i\omega_{r}t/2}&0\\ 0&e^{-i\omega_{r}t/2}\end{pmatrix}\begin{pmatrix}1&0\\ 0&-1\end{pmatrix}\begin{pmatrix}e^{-i\omega_{r}t/2}&0\\ 0&e^{i\omega_{r}t/2}\end{pmatrix}=\sigma_{z}
  114. โˆ‚ ฯˆ โˆ‚ t = i ( ฯ‰ 1 ( 0 e i ฯ‰ r t ( cos ฯ‰ r t - i sin ฯ‰ r t ) e - i ฯ‰ r t ( cos ฯ‰ r t + i sin ฯ‰ r t ) 0 ) + ( w 0 + ฯ‰ r 2 ) ฯƒ z ) ฯˆ \frac{\partial\psi}{\partial t}=i\left(\omega_{1}\begin{pmatrix}0&e^{i\omega_{% r}t}\left(\cos{\omega_{r}t}-i\sin{\omega_{r}t}\right)\\ e^{-i\omega_{r}t}\left(\cos{\omega_{r}t}+i\sin{\omega_{r}t}\right)&0\end{% pmatrix}+\left(w_{0}+\frac{\omega_{r}}{2}\right)\sigma_{z}\right)\psi
  115. โˆ‚ ฯˆ โˆ‚ t = i ( ฯ‰ 1 ฯƒ x + ( w 0 + ฯ‰ r 2 ) ฯƒ z ) ฯˆ \frac{\partial\psi}{\partial t}=i\left(\omega_{1}\sigma_{x}+\left(w_{0}+\frac{% \omega_{r}}{2}\right)\sigma_{z}\right)\psi
  116. i โ„ โˆ‚ t ฯˆ = - ฮผ ฯƒ โ†’ โ‹… B โ†’ ฯˆ i\hbar\partial_{t}\psi=-\mu\vec{\sigma}\cdot\vec{B}\psi
  117. โˆ‚ ฯˆ โˆ‚ t = i ฮผ โ„ ฯƒ i B i ฯˆ \frac{\partial\psi}{\partial t}=i\frac{\mu}{\hbar}\sigma_{i}B_{i}\psi
  118. ฯƒ i \sigma_{i}
  119. ฯˆ โ€  ฯƒ j โˆ‚ ฯˆ โˆ‚ t = i ฮผ โ„ ฯˆ โ€  ฯƒ j ฯƒ i B i ฯˆ = i ฮผ โ„ ฯˆ โ€  ( I ฮด i j - i ฯƒ k ฮต i j k ) B i ฯˆ = ฮผ โ„ ฯˆ โ€  ( i I ฮด i j + ฯƒ k ฮต i j k ) B i ฯˆ \psi^{\dagger}\sigma_{j}\frac{\partial\psi}{\partial t}=i\frac{\mu}{\hbar}\psi% ^{\dagger}\sigma_{j}\sigma_{i}B_{i}\psi=i\frac{\mu}{\hbar}\psi^{\dagger}\left(% I\delta_{ij}-i\sigma_{k}\varepsilon_{ijk}\right)B_{i}\psi=\frac{\mu}{\hbar}% \psi^{\dagger}\left(iI\delta_{ij}+\sigma_{k}\varepsilon_{ijk}\right)B_{i}\psi
  120. ฯˆ โ€  ฯƒ j โˆ‚ ฯˆ โˆ‚ t + โˆ‚ ฯˆ โ€  โˆ‚ t ฯƒ j ฯˆ = โˆ‚ ( ฯˆ โ€  ฯƒ j ฯˆ ) โˆ‚ t \psi^{\dagger}\sigma_{j}\frac{\partial\psi}{\partial t}+\frac{\partial\psi^{% \dagger}}{\partial t}\sigma_{j}\psi=\frac{\partial\left(\psi^{\dagger}\sigma_{% j}\psi\right)}{\partial t}
  121. ฮผ โ„ ฯˆ โ€  ( i I ฮด i j + ฯƒ k ฮต i j k ) B i ฯˆ + ฮผ โ„ ฯˆ โ€  ( - i I ฮด i j + ฯƒ k ฮต i j k ) B i ฯˆ = 2 ฮผ โ„ ( ฯˆ โ€  ฯƒ k ฯˆ ) B i ฮต i j k \frac{\mu}{\hbar}\psi^{\dagger}\left(iI\delta_{ij}+\sigma_{k}\varepsilon_{ijk}% \right)B_{i}\psi+\frac{\mu}{\hbar}\psi^{\dagger}\left(-iI\delta_{ij}+\sigma_{k% }\varepsilon_{ijk}\right)B_{i}\psi=\frac{2\mu}{\hbar}\left(\psi^{\dagger}% \sigma_{k}\psi\right)B_{i}\varepsilon_{ijk}
  122. โŸจ ฯƒ i โŸฉ = ฯˆ โ€  ฯƒ i ฯˆ = R i \langle\sigma_{i}\rangle=\psi^{\dagger}\sigma_{i}\psi=R_{i}
  123. 2 ฮผ โ„ \frac{2\mu}{\hbar}
  124. ฮณ \gamma
  125. โˆ‚ R j โˆ‚ t = ฮณ R k B i ฮต k i j \frac{\partial R_{j}}{\partial t}=\gamma R_{k}B_{i}\varepsilon_{kij}
  126. ฮต i j k = ฮต k i j \varepsilon_{ijk}=\varepsilon_{kij}
  127. โˆ‚ R โ†’ โˆ‚ t = ฮณ R โ†’ ร— B โ†’ \frac{\partial\vec{R}}{\partial t}=\gamma\vec{R}\times\vec{B}
  128. M โ†’ \vec{M}
  129. R โ†’ \vec{R}
  130. i โ„ d ฯƒ j d t = [ ฯƒ j , H ] = [ ฯƒ j , - ฮผ ฯƒ i B i ] = - ฮผ ( ฯƒ j ฯƒ i B i - ฯƒ i ฯƒ j B i ) = ฮผ [ ฯƒ i , ฯƒ j ] B i = 2 ฮผ i ฮต i j k ฯƒ k B i i\hbar\frac{d\sigma_{j}}{dt}=[\sigma_{j},H]=[\sigma_{j},-\mu\sigma_{i}B_{i}]=-% \mu\left(\sigma_{j}\sigma_{i}B_{i}-\sigma_{i}\sigma_{j}B_{i}\right)=\mu[\sigma% _{i},\sigma_{j}]B_{i}=2\mu i\varepsilon_{ijk}\sigma_{k}B_{i}
  131. R i โ†’ = โŸจ ฯƒ i โŸฉ \vec{R_{i}}=\langle\sigma_{i}\rangle

Typographical_conventions_in_mathematical_formulae.html

  1. A X = ฮฉ e x + a b + c d AX=\Omega_{e^{x}}+\begin{matrix}\frac{a}{b+\frac{c}{d}}\end{matrix}
  2. A X = ฮฉ exp ( x ) + a b + c / d AX=\Omega_{\,\exp(x)}+\begin{matrix}\frac{a}{b+c/d}\end{matrix}
  3. โ„• \mathbb{N}
  4. โ„ค \mathbb{Z}
  5. โ„š \mathbb{Q}
  6. โ„ \mathbb{R}
  7. Z {Z}
  8. โ„• \mathbb{N}

Ultraparallel_theorem.html

  1. a < b < c < d a<b<c<d
  2. p p
  3. q q
  4. a b ab
  5. c d cd
  6. p p
  7. q q
  8. x โ†’ x - a x\to x-a\,
  9. inversion in the unit semicircle. \mbox{inversion in the unit semicircle.}~{}\,
  10. a โ†’ โˆž a\to\infty
  11. b โ†’ ( b - a ) - 1 , c โ†’ ( c - a ) - 1 , d โ†’ ( d - a ) - 1 . b\to(b-a)^{-1},\quad c\to(c-a)^{-1},\quad d\to(d-a)^{-1}.
  12. x โ†’ x - ( b - a ) - 1 x\to x-(b-a)^{-1}\,
  13. x โ†’ [ ( c - a ) - 1 - ( b - a ) - 1 ] - 1 x x\to\left[(c-a)^{-1}-(b-a)^{-1}\right]^{-1}x\,
  14. a a
  15. โˆž \infty
  16. b โ†’ 0 b\to 0
  17. c โ†’ 1 c\to 1
  18. d โ†’ z d\to z
  19. 1 z 1z
  20. 1 2 ( z + 1 ) \begin{matrix}\frac{1}{2}\end{matrix}(z+1)
  21. 1 2 ( z - 1 ) \begin{matrix}\frac{1}{2}\end{matrix}(z-1)
  22. 1 z 1z
  23. 1 4 [ ( z + 1 ) 2 - ( z - 1 ) 2 ] = z . \frac{1}{4}\left[(z+1)^{2}-(z-1)^{2}\right]=z.\,
  24. z z
  25. z \sqrt{z}
  26. p p
  27. q q

Ultrashort_pulse.html

  1. E ( t ) = I ( t ) e i ฯ‰ 0 t e i ฯˆ ( t ) E(t)=\sqrt{I(t)}e^{i\omega_{0}t}e^{i\psi(t)}
  2. E ( ฯ‰ ) = โ„ฑ ( E ( t ) ) E(\omega)=\mathcal{F}(E(t))
  3. e i ฯ‰ 0 t e^{i\omega_{0}t}
  4. E ( ฯ‰ ) = S ( ฯ‰ ) e i ฯ• ( ฯ‰ ) E(\omega)=\sqrt{S(\omega)}e^{i\phi(\omega)}
  5. ๐Š 0 \,\textbf{K}_{0}
  6. ฯ‰ 0 \omega_{0}
  7. ๐„ ( ๐ฑ , t ) = A ( ๐ฑ , t ) exp ( i ๐Š 0 ๐ฑ - i ฯ‰ 0 t ) \,\textbf{E}(\,\textbf{x},t)=\,\textbf{ A }(\,\textbf{x},t)\exp(i\,\textbf{K}_% {0}\,\textbf{x}-i\omega_{0}t)
  8. ๐€ \,\textbf{A}
  9. โˆ‚ ๐€ โˆ‚ z = - ฮฒ 1 โˆ‚ ๐€ โˆ‚ t - i 2 ฮฒ 2 โˆ‚ 2 ๐€ โˆ‚ t 2 + 1 6 ฮฒ 3 โˆ‚ 3 ๐€ โˆ‚ t 3 + ฮณ x โˆ‚ ๐€ โˆ‚ x + ฮณ y โˆ‚ ๐€ โˆ‚ y \frac{\partial\,\textbf{A}}{\partial z}=~{}-~{}\beta_{1}\frac{\partial\,% \textbf{A}}{\partial t}~{}-~{}\frac{i}{2}\beta_{2}\frac{\partial^{2}\,\textbf{% A}}{\partial t^{2}}~{}+~{}\frac{1}{6}\beta_{3}\frac{\partial^{3}\,\textbf{A}}{% \partial t^{3}}~{}+~{}\gamma_{x}\frac{\partial\,\textbf{A}}{\partial x}~{}+~{}% \gamma_{y}\frac{\partial\,\textbf{A}}{\partial y}
  10. + i ฮณ t x โˆ‚ 2 ๐€ โˆ‚ t โˆ‚ x + i ฮณ t y โˆ‚ 2 ๐€ โˆ‚ t โˆ‚ y - i 2 ฮณ x x โˆ‚ 2 ๐€ โˆ‚ x 2 - i 2 ฮณ y y โˆ‚ 2 ๐€ โˆ‚ y 2 + i ฮณ x y โˆ‚ 2 ๐€ โˆ‚ x โˆ‚ y + โ‹ฏ ~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}+~{}i\gamma_{tx}\frac{\partial^{2}\,% \textbf{A}}{\partial t\partial x}~{}+~{}i\gamma_{ty}\frac{\partial^{2}\,% \textbf{A}}{\partial t\partial y}~{}-~{}\frac{i}{2}\gamma_{xx}\frac{\partial^{% 2}\,\textbf{A}}{\partial x^{2}}~{}-~{}\frac{i}{2}\gamma_{yy}\frac{\partial^{2}% \,\textbf{A}}{\partial y^{2}}~{}+~{}i\gamma_{xy}\frac{\partial^{2}\,\textbf{A}% }{\partial x\partial y}+\cdots
  11. ฮฒ 1 \beta_{1}
  12. ฮฒ 2 \beta_{2}
  13. ฮฒ 3 \beta_{3}
  14. ฮฒ 2 \beta_{2}
  15. ฮณ x \gamma_{x}
  16. ฮณ y \gamma_{y}
  17. ฮณ x ( ฮณ y ) \gamma_{x}~{}(\gamma_{y})
  18. x ( y ) x~{}(y)
  19. ฮณ x x \gamma_{xx}
  20. ฮณ y y \gamma_{yy}
  21. ฮณ t x \gamma_{tx}
  22. ฮณ t y \gamma_{ty}
  23. y y
  24. x x
  25. x x
  26. y y
  27. ฮฒ 2 \beta_{2}
  28. ฮณ x x \gamma_{xx}
  29. ฮณ y y \gamma_{yy}
  30. ฮณ x y \gamma_{xy}
  31. x - y x-y
  32. โ‹ฏ + 1 3 ฮณ t x x โˆ‚ 3 ๐€ โˆ‚ x 2 โˆ‚ t + 1 3 ฮณ t y y โˆ‚ 3 ๐€ โˆ‚ y 2 โˆ‚ t + 1 3 ฮณ t t x โˆ‚ 3 ๐€ โˆ‚ t 2 โˆ‚ x + โ‹ฏ \cdots~{}+~{}\frac{1}{3}\gamma_{txx}\frac{\partial^{3}\,\textbf{A}}{\partial x% ^{2}\partial t}~{}+~{}\frac{1}{3}\gamma_{tyy}\frac{\partial^{3}\,\textbf{A}}{% \partial y^{2}\partial t}~{}+~{}\frac{1}{3}\gamma_{ttx}\frac{\partial^{3}\,% \textbf{A}}{\partial t^{2}\partial x}+\cdots
  33. ฮฒ 3 \beta_{3}
  34. ฮณ t x x \gamma_{txx}
  35. ฯ‰ \omega
  36. ฮณ t t x \gamma_{ttx}
  37. t t
  38. x x
  39. ฮณ n l | A | 2 A \gamma_{nl}|A|^{2}A

UMAC.html

  1. Pr h โˆˆ H [ h ( f ) = d | h ( a ) = c ] \Pr_{h\in H}[h(f)=d|h(a)=c]\,
  2. h ( a ) = h 0 + โˆ‘ i = 1 n h i a i h(a)=h_{0}+\sum_{i=1}^{n}h_{i}a_{i}\,
  3. Pr h โˆˆ H [ h ( f ) = d | h ( a ) = c ] = 1 | D | \Pr_{h\in H}[h(f)=d|h(a)=c]={1\over|D|}
  4. NH K ( M ) = ( โˆ‘ i = 0 ( n / 2 ) - 1 ( ( m 2 i + k 2 i ) mod 2 w ) โ‹… ( ( m 2 i + 1 + k 2 i + 1 ) mod 2 w ) ) mod 2 2 w \operatorname{NH}_{K}(M)=\left(\sum_{i=0}^{(n/2)-1}((m_{2i}+k_{2i})\bmod~{}2^{% w})\cdot((m_{2i+1}+k_{2i+1})\bmod~{}2^{w})\right)\bmod~{}2^{2w}
  5. w / 2 w/2
  6. โŒˆ k / 2 โŒ‰ \lceil k/2\rceil
  7. k k

Unbeatable_strategy.html

  1. 1 / 4 , 1/4,
  2. x * x^{*}
  3. x x
  4. x 0 x_{0}
  5. x * = ( 2 x 0 ) 1 / 2 - x 0 x^{*}=(2x_{0})^{1/2}-x_{0}
  6. 1 / 2 1/2

Unbounded_operator.html

  1. X , Y X,Y
  2. T : X โ†’ Y T:Xโ†’Y
  3. T T
  4. D ( T ) โІ X D(T)โІX
  5. T T
  6. Y Y
  7. T T
  8. X X
  9. T T
  10. ฮ“ ( T ) ฮ“(T)
  11. ฮ“ ( T ) ฮ“(T)
  12. X โŠ• Y XโŠ•Y
  13. ( x , T x ) (x,Tx)
  14. x x
  15. T T
  16. T T
  17. x x
  18. T T
  19. T x = y Tx=y
  20. T T
  21. D ( T ) D(T)
  22. โˆฅ x โˆฅ T = โˆฅ x โˆฅ 2 + โˆฅ T x โˆฅ 2 . \|x\|_{T}=\sqrt{\|x\|^{2}+\|Tx\|^{2}}.
  23. T T
  24. X X
  25. X X
  26. T : X โ†’ Y T:Xโ†’Y
  27. X X
  28. T T
  29. H H
  30. T + a T+a
  31. a a
  32. x x
  33. T T
  34. T T
  35. โˆ’ T โˆ’T
  36. T T
  37. C ( 0 , 11 ) ) C(0,11))
  38. ( d d x f ) ( x ) = lim h โ†’ 0 f ( x + h ) - f ( x ) h , โˆ€ x โˆˆ [ 0 , 1 ] . \left(\frac{d}{dx}f\right)(x)=\lim_{h\to 0}\frac{f(x+h)-f(x)}{h},\qquad\forall x% \in[0,1].
  39. d d x : C ( 0 , 11 ) โ†’ C ( 0 , 11 ) ) \frac{d}{dx}:C(0,11)โ†’C(0,11))
  40. a f + b g aโ€‰f+bg
  41. f , g f,g
  42. ( d d x ) ( a f + b g ) = a ( d d x f ) + b ( d d x g ) . \left(\tfrac{d}{dx}\right)(af+bg)=a\left(\tfrac{d}{dx}f\right)+b\left(\tfrac{d% }{dx}g\right).
  43. { f n : [ 0 , 1 ] โ†’ [ - 1 , 1 ] f n ( x ) = sin ( 2 ฯ€ n x ) \begin{cases}f_{n}:[0,1]\to[-1,1]\\ f_{n}(x)=\sin(2\pi nx)\end{cases}
  44. โˆฅ f n โˆฅ 2 = 1 2 , \left\|f_{n}\right\|_{2}=\frac{1}{\sqrt{2}},
  45. โˆฅ ( d d x f n ) โˆฅ 2 = 2 ฯ€ n 2 โ†’ โˆž . \left\|\left(\tfrac{d}{dx}f_{n}\right)\right\|_{2}=\frac{2\pi n}{\sqrt{2}}\to\infty.
  46. Z โ†’ Z Zโ†’Z
  47. Z Z
  48. X โ†’ Y Xโ†’Y
  49. X , Y X,Y
  50. Z โ†’ Z Zโ†’Z
  51. Z Z
  52. I โŠ‚ ๐‘ IโŠ‚\mathbf{R}
  53. d d x : ( C 1 ( I ) , โˆฅ โ‹… โˆฅ C 1 ) โ†’ ( C 0 ( I ) , โˆฅ โ‹… โˆฅ โˆž ) , \frac{d}{dx}:\left(C^{1}(I),\|\cdot\|_{C^{1}}\right)\to\left(C^{0}(I),\|\cdot% \|_{\infty}\right),
  54. โˆฅ f โˆฅ C 1 = โˆฅ f โˆฅ โˆž + โˆฅ f โ€ฒ โˆฅ โˆž . \|f\|_{C^{1}}=\|f\|_{\infty}+\|f^{\prime}\|_{\infty}.
  55. T T
  56. โŸจ T x โˆฃ y โŸฉ 2 = โŸจ x โˆฃ T * y โŸฉ 1 , x โˆˆ D ( T ) . \langle Tx\mid y\rangle_{2}=\left\langle x\mid T^{*}y\right\rangle_{1},\qquad x% \in D(T).
  57. x โ†ฆ โŸจ T x โˆฃ y โŸฉ x\mapsto\langle Tx\mid y\rangle
  58. T T
  59. โŸจ T x โˆฃ y โŸฉ 2 = โŸจ x โˆฃ z โŸฉ 1 , x โˆˆ D ( T ) , \langle Tx\mid y\rangle_{2}=\langle x\mid z\rangle_{1},\qquad x\in D(T),
  60. y , z y,z
  61. T T
  62. T T
  63. y y
  64. x โ†ฆ โŸจ T x โˆฃ y โŸฉ x\mapsto\langle Tx\mid y\rangle
  65. T T
  66. T T
  67. T T
  68. T T
  69. J J
  70. { J : H 1 โŠ• H 2 โ†’ H 2 โŠ• H 1 J ( x โŠ• y ) = - y โŠ• x \begin{cases}J:H_{1}\oplus H_{2}\to H_{2}\oplus H_{1}\\ J(x\oplus y)=-y\oplus x\end{cases}
  71. J J
  72. S S
  73. T T
  74. S S
  75. โŸจ T x โˆฃ y โŸฉ 2 = โŸจ x โˆฃ S y โŸฉ 1 , \langle Tx\mid y\rangle_{2}=\langle x\mid Sy\rangle_{1},
  76. x x
  77. T T
  78. S S
  79. T T
  80. T T
  81. ker ( T ) = ran ( T * ) โŠฅ . \operatorname{ker}(T)=\operatorname{ran}(T^{*})^{\bot}.
  82. T T
  83. T T
  84. K > 0 K>0
  85. f f
  86. T T
  87. S S
  88. โˆฅ f โˆฅ 2 2 = | โŸจ T S f โˆฃ f โŸฉ 2 | โ‰ค โˆฅ S โˆฅ โˆฅ f โˆฅ 2 โˆฅ T * f โˆฅ 1 \|f\|_{2}^{2}=\left|\langle TSf\mid f\rangle_{2}\right|\leq\|S\|\|f\|_{2}\left% \|T^{*}f\right\|_{1}
  89. r a n ( T ) ran(T)
  90. โˆฅ T * f j โˆฅ 1 2 = | โŸจ T * f j โˆฃ T * f j โŸฉ 1 | โ‰ค โˆฅ T T * f j โˆฅ 2 โˆฅ f j โˆฅ 2 . \|T^{*}f_{j}\|_{1}^{2}=|\langle T^{*}f_{j}\mid T^{*}f_{j}\rangle_{1}|\leq\|TT^% {*}f_{j}\|_{2}\|f_{j}\|_{2}.
  91. f < s u b > j โ†’ g f<sub>jโ†’g
  92. T T
  93. T T
  94. T T
  95. T T
  96. x x
  97. A , B A,B
  98. T = A + i B T=A+iB
  99. x x
  100. T T
  101. T โ€ฒ : B 2 * โ†’ B 1 * T^{\prime}:{B_{2}}^{*}\to{B_{1}}^{*}
  102. โŸจ T x , y โ€ฒ โŸฉ = โŸจ x , T โ€ฒ y โ€ฒ โŸฉ \langle Tx,y^{\prime}\rangle=\langle x,T^{\prime}y^{\prime}\rangle
  103. โŸจ x , x โ€ฒ โŸฉ = x โ€ฒ ( x ) \langle x,x^{\prime}\rangle=x^{\prime}(x)
  104. J : H * โ†’ H J:H^{*}\to H
  105. f ( x ) = โŸจ x โˆฃ y โŸฉ H , ( x โˆˆ H ) f(x)=\langle x\mid y\rangle_{H},(x\in H)
  106. T * = J 1 T โ€ฒ J 2 - 1 T^{*}=J_{1}T^{\prime}J_{2}^{-1}
  107. J j : H j * โ†’ H j J_{j}:H_{j}^{*}\to H_{j}
  108. X , Y X,Y
  109. A : D ( A ) โŠ‚ X โ†’ Y A:D(A)โŠ‚Xโ†’Y
  110. D ( A ) D(A)
  111. x x
  112. X X
  113. n โ†’ โˆž nโ†’โˆž
  114. x โˆˆ D ( A ) xโˆˆD(A)
  115. A x = y Ax=y
  116. A A
  117. X โŠ• Y XโŠ•Y
  118. A A
  119. X โŠ• Y XโŠ•Y
  120. A A
  121. A A
  122. A A
  123. A ยฏ \overline{A}
  124. A A
  125. A ยฏ \overline{A}
  126. D ( A ) D(A)
  127. C C
  128. D ( A ) D(A)
  129. A A
  130. C C
  131. A ยฏ \overline{A}
  132. X X
  133. A A
  134. A โˆ’ ฮป I Aโˆ’ฮปI
  135. ฮป ฮป
  136. I I
  137. A A
  138. X X
  139. A A
  140. A A
  141. D ( A ) D(A)
  142. x x
  143. A = d d x A=\frac{d}{dx}
  144. X = Y = C ( a a , b ) ) X=Y=C(aa,b))
  145. a a , b aa,b
  146. D ( A ) D(A)
  147. A A
  148. A A
  149. T T
  150. โŸจ T x โˆฃ y โŸฉ = โŸจ x โˆฃ T y โŸฉ \langle Tx\mid y\rangle=\langle x\mid Ty\rangle
  151. T T
  152. T T
  153. ฮ“ ( T ) ฮ“(T)
  154. J ( ฮ“ ( T ) ) J(ฮ“(T))
  155. T โ€“ i Tโ€“i
  156. T + i T+i
  157. T y โ€“ i y = x Tyโ€“iy=x
  158. T z + i z = x Tz+iz=x
  159. ฮ“ ( T ) ฮ“(T)
  160. J ( ฮ“ ( T ) ) J(ฮ“(T))
  161. H โŠ• H . H\oplus H.
  162. โŸจ T x โˆฃ x โŸฉ \langle Tx\mid x\rangle
  163. โŸจ T x โˆฃ x โŸฉ โ‰ฅ 0 \langle Tx\mid x\rangle\geq 0
  164. ฮ“ ( S ) โІ ฮ“ ( T ) ฮ“(S)โІฮ“(T)
  165. S x = T x Sx=Tx
  166. y = 0 y=0
  167. T ยฏ \overline{T}
  168. T ยฏ . \overline{T}.
  169. T ยฏ = T * * \overline{T}=T^{**}
  170. ( T ยฏ ) * = T * . (\overline{T})^{*}=T^{*}.
  171. T โ€“ i Tโ€“i
  172. T + i T+i
  173. T T
  174. g โˆˆ X gโˆˆX
  175. T T
  176. Y Y
  177. ( f < s u b > j , T f j ) (f<sub>j,Tโ€‰f_{j})
  178. T < s u p > โˆ— T<sup>โˆ—
  179. T < s u p > โˆ— T<sup>โˆ—
  180. T < s u p > โˆ— โˆ— = T T<sup>โˆ—โˆ—=T
  181. T T
  182. T < s u p > โˆ— T<sup>โˆ—

Uniform_4-polytope.html

  1. { p , q } \{p,q\}
  2. sin ( ฯ€ p ) sin ( ฯ€ r ) > cos ( ฯ€ q ) \sin\left(\frac{\pi}{p}\right)\sin\left(\frac{\pi}{r}\right)>\cos\left(\frac{% \pi}{q}\right)
  3. s { 3 3 2 } s\left\{\begin{array}[]{l}3\\ 3\\ 2\end{array}\right\}
  4. s { 4 3 2 } s\left\{\begin{array}[]{l}4\\ 3\\ 2\end{array}\right\}
  5. s { 5 3 2 } s\left\{\begin{array}[]{l}5\\ 3\\ 2\end{array}\right\}

Uniform_distribution_(continuous).html

  1. e i t b - e i t a i t ( b - a ) \frac{\mathrm{e}^{itb}-\mathrm{e}^{ita}}{it(b-a)}
  2. f ( x ) = { 1 b - a for a โ‰ค x โ‰ค b , 0 for x < a or x > b f(x)=\begin{cases}\frac{1}{b-a}&\mathrm{for}\ a\leq x\leq b,\\ 0&\mathrm{for}\ x<a\ \mathrm{or}\ x>b\end{cases}
  3. f ( x ) = { 1 2 ฯƒ 3 for - ฯƒ 3 โ‰ค x - ฮผ โ‰ค ฯƒ 3 0 otherwise f(x)=\begin{cases}\frac{1}{2\sigma\sqrt{3}}&\mbox{for }~{}-\sigma\sqrt{3}\leq x% -\mu\leq\sigma\sqrt{3}\\ 0&\,\text{otherwise}\end{cases}
  4. F ( x ) = { 0 for x < a x - a b - a for a โ‰ค x < b 1 for x โ‰ฅ b F(x)=\begin{cases}0&\,\text{for }x<a\\ \frac{x-a}{b-a}&\,\text{for }a\leq x<b\\ 1&\,\text{for }x\geq b\end{cases}
  5. F - 1 ( p ) = a + p ( b - a ) for 0 < p < 1 F^{-1}(p)=a+p(b-a)\,\,\,\text{ for }0<p<1
  6. F ( x ) = { 0 for x - ฮผ < - ฯƒ 3 1 2 ( x - ฮผ ฯƒ 3 + 1 ) for - ฯƒ 3 โ‰ค x - ฮผ < ฯƒ 3 1 for x - ฮผ โ‰ฅ ฯƒ 3 F(x)=\begin{cases}0&\,\text{for }x-\mu<-\sigma\sqrt{3}\\ \frac{1}{2}\left(\frac{x-\mu}{\sigma\sqrt{3}}+1\right)&\,\text{for }-\sigma% \sqrt{3}\leq x-\mu<\sigma\sqrt{3}\\ 1&\,\text{for }x-\mu\geq\sigma\sqrt{3}\end{cases}
  7. F - 1 ( p ) = ฯƒ 3 ( 2 p - 1 ) + ฮผ for 0 โ‰ค p โ‰ค 1 F^{-1}(p)=\sigma\sqrt{3}(2p-1)+\mu\,\,\,\text{ for }0\leq p\leq 1
  8. M x = E ( e t x ) = e t b - e t a t ( b - a ) M_{x}=E(e^{tx})=\frac{e^{tb}-e^{ta}}{t(b-a)}\,\!
  9. m 1 = a + b 2 , m_{1}=\frac{a+b}{2},\,\!
  10. m 2 = a 2 + a b + b 2 3 , m_{2}=\frac{a^{2}+ab+b^{2}}{3},\,\!
  11. m k = 1 k + 1 โˆ‘ i = 0 k a i b k - i . m_{k}=\frac{1}{k+1}\sum_{i=0}^{k}a^{i}b^{k-i}.\,\!
  12. E ( X ) = 1 2 ( a + b ) . E(X)=\frac{1}{2}(a+b).
  13. V ( X ) = 1 12 ( b - a ) 2 V(X)=\frac{1}{12}(b-a)^{2}
  14. a = E ( X ) - 3 V ( X ) a=E(X)-\sqrt{3V(X)}
  15. b = E ( X ) + 3 V ( X ) b=E(X)+\sqrt{3V(X)}
  16. E ( X ( k ) ) = k n + 1 . \operatorname{E}(X_{(k)})={k\over n+1}.
  17. V ( X ( k ) ) = k ( n - k + 1 ) ( n + 1 ) 2 ( n + 2 ) . \operatorname{V}(X_{(k)})={k(n-k+1)\over(n+1)^{2}(n+2)}.
  18. P ( X โˆˆ [ x , x + d ] ) = โˆซ x x + d d y b - a = d b - a P\left(X\in\left[x,x+d\right]\right)=\int_{x}^{x+d}\frac{\mathrm{d}y}{b-a}\,=% \frac{d}{b-a}\,\!
  19. a = 0 a=0
  20. b = 1 b=1
  21. f ( x ) = H ( x - a ) - H ( x - b ) b - a , f(x)=\frac{\operatorname{H}(x-a)-\operatorname{H}(x-b)}{b-a},\,\!
  22. f ( x ) = 1 b - a rect ( x - ( a + b 2 ) b - a ) . f(x)=\frac{1}{b-a}\,\operatorname{rect}\left(\frac{x-\left(\frac{a+b}{2}\right% )}{b-a}\right).
  23. f ( x ) = sgn ( x - a ) - sgn ( x - b ) 2 ( b - a ) . f(x)=\frac{\operatorname{sgn}{(x-a)}-\operatorname{sgn}{(x-b)}}{2(b-a)}.
  24. b ^ U M V U = k + 1 k m = m + m k \hat{b}_{UMVU}=\frac{k+1}{k}m=m+\frac{m}{k}
  25. b ^ M L = m \hat{b}_{ML}=m
  26. m = X ( n ) m=X_{(n)}
  27. b ^ M M = 2 X ยฏ \hat{b}_{MM}=2\bar{X}
  28. X ยฏ \bar{X}
  29. f n ( X ( n ) ) = n 1 L ( X ( n ) L ) n - 1 = n X ( n ) n - 1 L n , 0 < X n < L f_{n}(X_{(n)})=n\frac{1}{L}\left(\frac{X_{(n)}}{L}\right)^{n-1}=n\frac{X_{(n)}% ^{n-1}}{L^{n}},0<X_{n}<L
  30. X ( n ) โ‰ค L โ‰ค X ( n ) / ฮฑ 1 / n X_{(n)}\leq L\leq X_{(n)}/\alpha^{1/n}

Uniform_distribution_(discrete).html

  1. e i a t - e i ( b + 1 ) t n ( 1 - e i t ) \frac{e^{iat}-e^{i(b+1)t}}{n(1-e^{it})}
  2. F ( k ; a , b ) = โŒŠ k โŒ‹ - a + 1 b - a + 1 F(k;a,b)=\frac{\lfloor k\rfloor-a+1}{b-a+1}
  3. 1 , 2 , โ€ฆ , N 1,2,\dots,N
  4. N ^ = k + 1 k m - 1 = m + m k - 1 \hat{N}=\frac{k+1}{k}m-1=m+\frac{m}{k}-1
  5. 1 k ( N - k ) ( N + 1 ) ( k + 2 ) โ‰ˆ N 2 k 2 for small samples k โ‰ช N \frac{1}{k}\frac{(N-k)(N+1)}{(k+2)}\approx\frac{N^{2}}{k^{2}}\,\text{ for % small samples }k\ll N
  6. N / k N/k
  7. m k \frac{m}{k}

Uniformizable_space.html

  1. D f , ฮต = { ( x , y ) โˆˆ X ร— X : | f ( x ) - f ( y ) | < ฮต } D_{f,\varepsilon}=\{(x,y)\in X\times X:|f(x)-f(y)|<\varepsilon\}
  2. D n โˆ˜ D n โŠ‚ D n - 1 D_{n}\circ D_{n}\subset D_{n-1}

Unimodality.html

  1. { p n ; n = โ€ฆ , - 1 , 0 , 1 , โ€ฆ } \{p_{n};n=\dots,-1,0,1,\dots\}
  2. โ€ฆ , p - 2 - p - 1 , p - 1 - p 0 , p 0 - p 1 , p 1 - p 2 , โ€ฆ \dots,p_{-2}-p_{-1},p_{-1}-p_{0},p_{0}-p_{1},p_{1}-p_{2},\dots
  3. X ~ \tilde{X}
  4. X ยฏ \bar{X}
  5. | X ~ - X ยฏ | ฯƒ โ‰ค ( 3 / 5 ) 1 / 2 \frac{\left|\tilde{X}-\bar{X}\right|}{\sigma}\leq(3/5)^{1/2}
  6. | X ~ - mode | ฯƒ โ‰ค 3 1 / 2 . \frac{\left|\tilde{X}-\mathrm{mode}\right|}{\sigma}\leq 3^{1/2}.
  7. ฮณ 2 - ฮบ โ‰ค 6 5 \gamma^{2}-\kappa\leq\frac{6}{5}
  8. ฮณ 2 - ฮบ โ‰ค 186 125 \gamma^{2}-\kappa\leq\frac{186}{125}
  9. x โ‰  c \ x\neq c
  10. c c

Unique_negative_dimension.html

  1. C C
  2. D โІ C D\subseteq C
  3. c โˆˆ D c\in D
  4. โˆฉ ( D โˆ– { c } ) โˆ– c \cap(D\setminus\{c\})\setminus c

Uniquely_inversible_grammar.html

  1. A โ†’ ฮฑ and B โ†’ ฮฒ โ‡” ( A < > B โ‡’ ฮฑ < > ฮฒ ) A\to\alpha\and B\to\beta\iff(A<>B\Rightarrow\alpha<>\beta)
  2. A โ†’ a A | b B A\to aA|bB
  3. B โ†’ b | a B\to b|a
  4. A โ†’ a A | b B A\to aA|bB
  5. B โ†’ b B | a B\to bB|a

Univalent_function.html

  1. ฯ• a \phi_{a}
  2. ฯ• a ( z ) = z - a 1 - a ยฏ z , \phi_{a}(z)=\frac{z-a}{1-\bar{a}z},
  3. | a | < 1 , |a|<1,
  4. G G
  5. ฮฉ \Omega
  6. f : G โ†’ ฮฉ f:G\to\Omega
  7. f ( G ) = ฮฉ f(G)=\Omega
  8. f f
  9. f f
  10. f f
  11. f - 1 f^{-1}
  12. ( f - 1 ) โ€ฒ ( f ( z ) ) = 1 f โ€ฒ ( z ) (f^{-1})^{\prime}(f(z))=\frac{1}{f^{\prime}(z)}
  13. z z
  14. G . G.
  15. f : ( - 1 , 1 ) โ†’ ( - 1 , 1 ) f:(-1,1)\to(-1,1)\,

Universal_instantiation.html

  1. โˆ€ x A ( x ) โ‡’ A ( a / x ) , \forall x\,A(x)\Rightarrow A(a/x),
  2. A ( a / x ) A(a/x)

Universality_(dynamical_systems).html

  1. a = a 0 | ฮฒ - ฮฒ c | ฮฑ . a=a_{0}\left|\beta-\beta_{c}\right|^{\alpha}.\,

Upper_hybrid_oscillation.html

  1. ฯ‰ 2 = ฯ‰ p e 2 + ฯ‰ c e 2 + 3 k 2 v e , th 2 \omega^{2}=\omega_{pe}^{2}+\omega_{ce}^{2}+3k^{2}v_{\mathrm{e,th}}^{2}
  2. ฯ‰ p e = ( 4 ฯ€ n e e 2 / m e ) 1 / 2 \omega_{pe}=(4\pi n_{e}e^{2}/m_{e})^{1/2}
  3. ฯ‰ c e = e B / m e \omega_{ce}=eB/m_{e}
  4. ฯ‰ 2 = ( 1 / 2 ) ฯ‰ h 2 ( 1 ยฑ 1 - ( cos ฮธ ฯ‰ h 2 / 2 ฯ‰ c ฯ‰ p ) 2 ) \omega^{2}=(1/2)\omega_{h}^{2}\,\left(1\pm\sqrt{1-\left(\frac{\cos\theta}{% \omega_{h}^{2}/2\omega_{c}\omega_{p}}\right)^{2}}\right)

Upsampling.html

  1. x L [ n ] , \scriptstyle x_{L}[n],
  2. x [ n ] , \scriptstyle x[n],
  3. y [ j + n L ] = โˆ‘ k = 0 K x [ n - k ] โ‹… h [ j + k L ] , j = 0 , 1 , โ€ฆ L - 1 , y[j+nL]=\sum_{k=0}^{K}x[n-k]\cdot h[j+kL],\ \ j=0,1,...L-1,
  4. x L [ n ] , \scriptstyle x_{L}[n],
  5. โˆ‘ n = - โˆž โˆž x ( n T ) โž x [ n ] e - i 2 ฯ€ f n T โŸ DTFT = 1 T โˆ‘ k = - โˆž โˆž X ( f - k / T ) . \underbrace{\sum_{n=-\infty}^{\infty}\overbrace{x(nT)}^{x[n]}\ e^{-i2\pi fnT}}% _{\,\text{DTFT}}=\frac{1}{T}\sum_{k=-\infty}^{\infty}X(f-k/T).
  6. f \scriptstyle f
  7. L T โˆ‘ k = - โˆž โˆž X ( f - k โ‹… L T ) , \frac{L}{T}\sum_{k=-\infty}^{\infty}X\left(f-k\cdot\frac{L}{T}\right),
  8. 0.5 L . \tfrac{0.5}{L}.
  9. 0.5 T \tfrac{0.5}{T}
  10. z = e i ฯ‰ . z=e^{i\omega}.
  11. โˆ‘ n = - โˆž โˆž x [ n ] z - n = โˆ‘ n = - โˆž โˆž x [ n ] e - i ฯ‰ n = 1 T โˆ‘ k = - โˆž โˆž X ( ฯ‰ 2 ฯ€ T - k T ) โŸ X ( ฯ‰ - 2 ฯ€ k 2 ฯ€ T ) , \sum_{n=-\infty}^{\infty}x[n]\ z^{-n}=\sum_{n=-\infty}^{\infty}x[n]\ e^{-i% \omega n}=\frac{1}{T}\sum_{k=-\infty}^{\infty}\underbrace{X\left(\tfrac{\omega% }{2\pi T}-\tfrac{k}{T}\right)}_{X\left(\frac{\omega-2\pi k}{2\pi T}\right)},
  12. โˆ‘ n = - โˆž โˆž x [ n ] z - n L = โˆ‘ n = - โˆž โˆž x [ n ] e - i ฯ‰ L n = 1 T โˆ‘ k = - โˆž โˆž X ( ฯ‰ L 2 ฯ€ T - k T ) โŸ X ( ฯ‰ - 2 ฯ€ k / L 2 ฯ€ T / L ) , \sum_{n=-\infty}^{\infty}x[n]\ z^{-nL}=\sum_{n=-\infty}^{\infty}x[n]\ e^{-i% \omega Ln}=\frac{1}{T}\sum_{k=-\infty}^{\infty}\underbrace{X\left(\tfrac{% \omega L}{2\pi T}-\tfrac{k}{T}\right)}_{X\left(\frac{\omega-2\pi k/L}{2\pi T/L% }\right)},
  13. ฯ€ L โ‹… L 2 ฯ€ T = 0.5 T \tfrac{\pi}{L}\cdot\tfrac{L}{2\pi T}=\tfrac{0.5}{T}
  14. 0.5 L \tfrac{0.5}{L}

Upside_potential_ratio.html

  1. U = โˆ‘ min + โˆž ( R r - R min ) P r โˆ‘ - โˆž min ( R r - R min ) 2 P r = ๐”ผ [ ( R r - R min ) + ] ๐”ผ [ ( R r - R min ) - 2 ] , U={{{\sum_{\min}^{+\infty}{(R_{r}-R_{\min}})P_{r}}}\over\sqrt{{{\sum_{-\infty}% ^{\min}{(R_{r}-R_{\min}})^{2}P_{r}}}}}=\frac{\mathbb{E}[(R_{r}-R_{\min})_{+}]}% {\sqrt{\mathbb{E}[(R_{r}-R_{\min})_{-}^{2}]}},
  2. R r R_{r}
  3. P r P_{r}
  4. R r R_{r}
  5. R min R_{\min}
  6. r = min r=\min
  7. ( X ) + = { X if X โ‰ฅ 0 0 else (X)_{+}=\begin{cases}X&\,\text{if }X\geq 0\\ 0&\,\text{else}\end{cases}
  8. ( X ) - = ( - X ) + (X)_{-}=(-X)_{+}
  9. ๐”ผ [ ( R r - R min ) + ] \mathbb{E}[(R_{r}-R_{\min})_{+}]
  10. ๐”ผ [ ( R r - R min ) - 2 ] \mathbb{E}[(R_{r}-R_{\min})_{-}^{2}]

Vacuum_angle.html

  1. ฯ• \,\phi
  2. ฮจ [ U ๐€ U - 1 - ( d U ) U - 1 , U ฯ• ] = ฮจ [ ๐€ , ฯ• ] \Psi[U\mathbf{A}U^{-1}-(dU)U^{-1},U\phi]=\Psi[\mathbf{A},\phi]
  3. ฮจ [ U ๐€ U - 1 - ( d U ) U - 1 , U ฯ• ] = e i ฮธ ฮจ [ ๐€ , ฯ• ] \Psi[U\mathbf{A}U^{-1}-(dU)U^{-1},U\phi]=e^{i\theta}\Psi[\mathbf{A},\phi]

Vacuum_polarization.html

  1. ฮฑ eff ( p 2 ) = ฮฑ 1 - [ ฮ  2 ( p 2 ) - ฮ  2 ( 0 ) ] \alpha\text{eff}(p^{2})=\frac{\alpha}{1-[\Pi_{2}(p^{2})-\Pi_{2}(0)]}

Vacuum_solution.html

  1. J a = 0 J^{a}=0
  2. T a b = 0 T_{ab}=0

Vacuum_solution_(general_relativity).html

  1. G a b = R a b - R 2 g a b , R a b = G a b - G 2 g a b G_{ab}=R_{ab}-\frac{R}{2}\,g_{ab},\;\;R_{ab}=G_{ab}-\frac{G}{2}\,g_{ab}
  2. R = R a a , G = G a a = - R R={R^{a}}_{a},\;\;G={G^{a}}_{a}=-R
  3. R a b c d = C a b c d R_{abcd}=C_{abcd}
  4. T a b = 0 T^{ab}=0

Van_der_Pol_oscillator.html

  1. d 2 x d t 2 - ฮผ ( 1 - x 2 ) d x d t + x = 0 {d^{2}x\over dt^{2}}-\mu(1-x^{2}){dx\over dt}+x=0
  2. y = x - x 3 / 3 - x ห™ / ฮผ y=x-x^{3}/3-\dot{x}/\mu
  3. x ห™ = ฮผ ( x - 1 3 x 3 - y ) \dot{x}=\mu\left(x-\frac{1}{3}x^{3}-y\right)
  4. y ห™ = 1 ฮผ x . \dot{y}=\frac{1}{\mu}x.
  5. y = x ห™ y=\dot{x}
  6. x ห™ = y \dot{x}=y
  7. y ห™ = ฮผ ( 1 - x 2 ) y - x . \dot{y}=\mu(1-x^{2})y-x.
  8. d 2 x d t 2 + x = 0. {d^{2}x\over dt^{2}}+x=0.
  9. d 2 x d t 2 - ฮผ ( 1 - x 2 ) d x d t + x - A sin ( ฯ‰ t ) = 0 , {d^{2}x\over dt^{2}}-\mu(1-x^{2}){dx\over dt}+x-A\sin(\omega t)=0,

Variable_speed_of_light.html

  1. ฮป \lambda
  2. c = ฮฝ ฮป c=\nu\lambda
  3. ฮฝ \nu
  4. ฮฝ 1 = ฮฝ 2 ( 1 + G M r c 2 ) . \nu_{1}=\nu_{2}\left(1+\frac{GM}{rc^{2}}\right).
  5. c = ฮฝ ฮป c=\nu\lambda
  6. n = c c 0 = 1 + 2 G M r c 2 n=\frac{c}{c_{0}}=1+\frac{2GM}{rc^{2}}
  7. ฯ• \phi
  8. | u | = 1 + 2 ฯ• + O ( v 3 ) |u|=1+2\phi+O(v^{3})

Variance_swap.html

  1. N var ( ฯƒ realised 2 - ฯƒ strike 2 ) N_{\,\text{var}}(\sigma_{\,\text{realised}}^{2}-\sigma_{\,\text{strike}}^{2})
  2. N var N_{\,\text{var}}
  3. ฯƒ realised 2 \sigma_{\,\text{realised}}^{2}
  4. ฯƒ strike 2 \sigma_{\,\text{strike}}^{2}
  5. S 0 , S 1 , โ€ฆ , S n . S_{\,\text{0}},S_{\,\text{1}},...,S_{\,\text{n}}.
  6. R i = ln ( S i / S i-1 ) , R_{\,\text{i}}=\ln(S_{\,\text{i}}/S_{\,\text{i-1}}),
  7. ฯƒ realised 2 = A n โˆ‘ i=1 n R i 2 \sigma_{\,\text{realised}}^{2}=\frac{A}{n}\sum_{\,\text{i=1}}^{\,\text{n}}R_{% \,\text{i}}^{2}
  8. A A
  9. n - 1 n-1
  10. n n
  11. N var = N vol 2 ฯƒ strike N_{\,\text{var}}=\frac{N_{\,\text{vol}}}{2\sigma_{\,\text{strike}}}
  12. N vol N_{\,\text{vol}}
  13. d S t S t = ฮผ d t + ฯƒ d Z t \frac{dS_{t}}{S_{t}}\ =\mu dt+\sigma dZ_{t}
  14. d ( log S t ) = ( ฮผ - ฯƒ 2 2 ) d t + ฯƒ d Z t d(\log S_{t})=\left(\mu-\frac{\sigma^{2}}{2}\ \right)dt+\sigma dZ_{t}
  15. d S t S t - d ( log S t ) = ฯƒ 2 2 d t \frac{dS_{t}}{S_{t}}\ -d(\log S_{t})=\frac{\sigma^{2}}{2}\ dt
  16. Variance = 1 T โˆซ 0 T ฯƒ 2 d t = 2 T ( โˆซ 0 T d S t S t - ln ( S T S 0 ) ) \,\text{Variance}=\frac{1}{T}\ \int\limits_{0}^{T}\sigma^{2}dt\ =\frac{2}{T}\ % \left(\int\limits_{0}^{T}\frac{dS_{t}}{S_{t}}\ \ -\ln\left(\frac{S_{T}}{S_{0}}% \ \right)\right)
  17. 1 S t \frac{1}{S_{t}}
  18. - ln ( S T S * ) = - S T - S * S * + โˆซ K โ‰ค S * ( K - S T ) + d K K 2 + โˆซ K โ‰ฅ S * ( S T - K ) + d K K 2 -\ln\left(\frac{S_{T}}{S^{*}}\ \right)=-\frac{S_{T}-S^{*}}{S^{*}}\ +\int% \limits_{K\leq S^{*}}(K-S_{T})^{+}\frac{dK}{K^{2}}\ +\int\limits_{K\geq S^{*}}% (S_{T}-K)^{+}\frac{dK}{K^{2}}
  19. K v a r = 2 T ( r T - ( S 0 S * e r T - 1 ) - log ( S * S 0 ) + e r T โˆซ 0 S * 1 K 2 P ( K ) d K + e r T โˆซ S * โˆž 1 K 2 C ( K ) d K ) K_{var}=\frac{2}{T}\ \left(rT-\left(\frac{S_{0}}{S^{*}}\ e^{rT}-1\right)-\log% \left(\frac{S^{*}}{S_{0}}\ \right)+e^{rT}\int\limits_{0}^{S^{*}}\frac{1}{K^{2}% }\ P(K)dK+e^{rT}\int\limits_{S^{*}}^{\infty}\frac{1}{K^{2}}\ C(K)dK\right)
  20. S 0 S_{0}
  21. S * > 0 S^{*}>0
  22. K K
  23. S * S^{*}
  24. S * = F 0 = S 0 e r T S^{*}=F_{0}=S_{0}e^{rT}
  25. K v a r = 2 e r T T ( โˆซ 0 F 0 1 K 2 P ( K ) d K + โˆซ F 0 โˆž 1 K 2 C ( K ) d K ) K_{var}=\frac{2e^{rT}}{T}\ \left(\int\limits_{0}^{F_{0}}\frac{1}{K^{2}}\ P(K)% dK+\int\limits_{F_{0}}^{\infty}\frac{1}{K^{2}}\ C(K)dK\right)

Variational_Bayesian_methods.html

  1. ๐™ = { Z 1 โ€ฆ Z n } \mathbf{Z}=\{Z_{1}\dots Z_{n}\}
  2. ๐— \mathbf{X}
  3. Q ( ๐™ ) Q(\mathbf{Z})
  4. P ( ๐™ โˆฃ ๐— ) โ‰ˆ Q ( ๐™ ) . P(\mathbf{Z}\mid\mathbf{X})\approx Q(\mathbf{Z}).
  5. Q ( ๐™ ) Q(\mathbf{Z})
  6. P ( ๐™ โˆฃ ๐— ) P(\mathbf{Z}\mid\mathbf{X})
  7. Q ( ๐™ ) Q(\mathbf{Z})
  8. P ( ๐™ โˆฃ ๐— ) P(\mathbf{Z}\mid\mathbf{X})
  9. d ( Q ; P ) d(Q;P)
  10. Q ( ๐™ ) Q(\mathbf{Z})
  11. d ( Q ; P ) d(Q;P)
  12. D KL ( Q | | P ) = โˆ‘ ๐™ Q ( ๐™ ) log Q ( ๐™ ) P ( ๐™ โˆฃ ๐— ) . D_{\mathrm{KL}}(Q||P)=\sum_{\mathbf{Z}}Q(\mathbf{Z})\log\frac{Q(\mathbf{Z})}{P% (\mathbf{Z}\mid\mathbf{X})}.
  13. D KL ( Q | | P ) = โˆ‘ ๐™ Q ( ๐™ ) log Q ( ๐™ ) P ( ๐™ , ๐— ) + log P ( ๐— ) , D_{\mathrm{KL}}(Q||P)=\sum_{\mathbf{Z}}Q(\mathbf{Z})\log\frac{Q(\mathbf{Z})}{P% (\mathbf{Z},\mathbf{X})}+\log P(\mathbf{X}),
  14. log P ( ๐— ) = D KL ( Q | | P ) - โˆ‘ ๐™ Q ( ๐™ ) log Q ( ๐™ ) P ( ๐™ , ๐— ) = D KL ( Q | | P ) + โ„’ ( Q ) . \log P(\mathbf{X})=D_{\mathrm{KL}}(Q||P)-\sum_{\mathbf{Z}}Q(\mathbf{Z})\log% \frac{Q(\mathbf{Z})}{P(\mathbf{Z},\mathbf{X})}=D_{\mathrm{KL}}(Q||P)+\mathcal{% L}(Q).
  15. log P ( ๐— ) \log P(\mathbf{X})
  16. Q Q
  17. โ„’ ( Q ) \mathcal{L}(Q)
  18. P P
  19. Q Q
  20. Q Q
  21. โ„’ ( Q ) \mathcal{L}(Q)
  22. Q Q
  23. P ( ๐™ โˆฃ ๐— ) P(\mathbf{Z}\mid\mathbf{X})
  24. โ„’ ( Q ) \mathcal{L}(Q)
  25. log P ( ๐— ) \log P(\mathbf{X})
  26. โ„’ ( Q ) \mathcal{L}(Q)
  27. E Q [ log P ( ๐™ , ๐— ) ] \operatorname{E}_{Q}[\log P(\mathbf{Z},\mathbf{X})]
  28. Q Q
  29. Q ( ๐™ ) Q(\mathbf{Z})
  30. ๐™ \mathbf{Z}
  31. ๐™ 1 โ€ฆ ๐™ M \mathbf{Z}_{1}\dots\mathbf{Z}_{M}
  32. Q ( ๐™ ) = โˆ i = 1 M q i ( ๐™ i โˆฃ ๐— ) Q(\mathbf{Z})=\prod_{i=1}^{M}q_{i}(\mathbf{Z}_{i}\mid\mathbf{X})
  33. q j * q_{j}^{*}
  34. q j q_{j}
  35. q j * ( ๐™ j โˆฃ ๐— ) = e E i โ‰  j [ ln p ( ๐™ , ๐— ) ] โˆซ e E i โ‰  j [ ln p ( ๐™ , ๐— ) ] d ๐™ j q_{j}^{*}(\mathbf{Z}_{j}\mid\mathbf{X})=\frac{e^{\operatorname{E}_{i\neq j}[% \ln p(\mathbf{Z},\mathbf{X})]}}{\int e^{\operatorname{E}_{i\neq j}[\ln p(% \mathbf{Z},\mathbf{X})]}\,d\mathbf{Z}_{j}}
  36. E i โ‰  j [ ln p ( ๐™ , ๐— ) ] \operatorname{E}_{i\neq j}[\ln p(\mathbf{Z},\mathbf{X})]
  37. ln q j * ( ๐™ j โˆฃ ๐— ) = E i โ‰  j [ ln p ( ๐™ , ๐— ) ] + constant \ln q_{j}^{*}(\mathbf{Z}_{j}\mid\mathbf{X})=\operatorname{E}_{i\neq j}[\ln p(% \mathbf{Z},\mathbf{X})]+\,\text{constant}
  38. q j * q_{j}^{*}
  39. E i โ‰  j [ ln p ( ๐™ , ๐— ) ] \operatorname{E}_{i\neq j}[\ln p(\mathbf{Z},\mathbf{X})]
  40. ๐™ j \mathbf{Z}_{j}
  41. ฮผ \displaystyle\mu
  42. N N
  43. ๐— = { x 1 , โ€ฆ , x N } \mathbf{X}=\{x_{1},\dots,x_{N}\}
  44. q ( ฮผ , ฯ„ ) = p ( ฮผ , ฯ„ โˆฃ x 1 , โ€ฆ , x N ) q(\mu,\tau)=p(\mu,\tau\mid x_{1},\ldots,x_{N})
  45. ฮผ \mu
  46. ฯ„ \tau
  47. ฮผ 0 \mu_{0}
  48. ฮป 0 \lambda_{0}
  49. a 0 a_{0}
  50. b 0 b_{0}
  51. ฮผ \mu
  52. ฯ„ \tau
  53. p ( ๐— , ฮผ , ฯ„ ) = p ( ๐— โˆฃ ฮผ , ฯ„ ) p ( ฮผ โˆฃ ฯ„ ) p ( ฯ„ ) p(\mathbf{X},\mu,\tau)=p(\mathbf{X}\mid\mu,\tau)p(\mu\mid\tau)p(\tau)
  54. p ( ๐— โˆฃ ฮผ , ฯ„ ) = โˆ n = 1 N ๐’ฉ ( x n โˆฃ ฮผ , ฯ„ - 1 ) p ( ฮผ โˆฃ ฯ„ ) = ๐’ฉ ( ฮผ โˆฃ ฮผ 0 , ( ฮป 0 ฯ„ ) - 1 ) p ( ฯ„ ) = Gamma ( ฯ„ โˆฃ a 0 , b 0 ) \begin{aligned}\displaystyle p(\mathbf{X}\mid\mu,\tau)&\displaystyle=\prod_{n=% 1}^{N}\mathcal{N}(x_{n}\mid\mu,\tau^{-1})\\ \displaystyle p(\mu\mid\tau)&\displaystyle=\mathcal{N}(\mu\mid\mu_{0},(\lambda% _{0}\tau)^{-1})\\ \displaystyle p(\tau)&\displaystyle=\operatorname{Gamma}(\tau\mid a_{0},b_{0})% \end{aligned}
  55. ๐’ฉ ( x โˆฃ ฮผ , ฯƒ 2 ) = 1 2 ฯ€ ฯƒ 2 e - ( x - ฮผ ) 2 2 ฯƒ 2 Gamma ( ฯ„ โˆฃ a , b ) = 1 ฮ“ ( a ) b a ฯ„ a - 1 e - b ฯ„ \begin{aligned}\displaystyle\mathcal{N}(x\mid\mu,\sigma^{2})&\displaystyle=% \frac{1}{\sqrt{2\pi\sigma^{2}}}e^{\frac{-(x-\mu)^{2}}{2\sigma^{2}}}\\ \displaystyle\operatorname{Gamma}(\tau\mid a,b)&\displaystyle=\frac{1}{\Gamma(% a)}b^{a}\tau^{a-1}e^{-b\tau}\end{aligned}
  56. q ( ฮผ , ฯ„ ) = q ( ฮผ ) q ( ฯ„ ) q(\mu,\tau)=q(\mu)q(\tau)
  57. ฮผ \mu
  58. ฯ„ \tau
  59. ln q ฮผ * ( ฮผ ) = E ฯ„ [ ln p ( ๐— โˆฃ ฮผ , ฯ„ ) + ln p ( ฮผ โˆฃ ฯ„ ) + ln p ( ฯ„ ) ] + C = E ฯ„ [ ln p ( ๐— โˆฃ ฮผ , ฯ„ ) ] + E ฯ„ [ ln p ( ฮผ โˆฃ ฯ„ ) ] + E ฯ„ [ ln p ( ฯ„ ) ] + C = E ฯ„ [ ln โˆ n = 1 N ๐’ฉ ( x n โˆฃ ฮผ , ฯ„ - 1 ) ] + E ฯ„ [ ln ๐’ฉ ( ฮผ โˆฃ ฮผ 0 , ( ฮป 0 ฯ„ ) - 1 ) ] + C 2 = E ฯ„ [ ln โˆ n = 1 N ฯ„ 2 ฯ€ e - ( x n - ฮผ ) 2 ฯ„ 2 ] + E ฯ„ [ ln ฮป 0 ฯ„ 2 ฯ€ e - ( ฮผ - ฮผ 0 ) 2 ฮป 0 ฯ„ 2 ] + C 2 = E ฯ„ [ โˆ‘ n = 1 N ( 1 2 ( ln ฯ„ - ln 2 ฯ€ ) - ( x n - ฮผ ) 2 ฯ„ 2 ) ) ] + E ฯ„ [ 1 2 ( ln ฮป 0 + ln ฯ„ - ln 2 ฯ€ ) - ( ฮผ - ฮผ 0 ) 2 ฮป 0 ฯ„ 2 ] + C 2 = E ฯ„ [ โˆ‘ n = 1 N - ( x n - ฮผ ) 2 ฯ„ 2 ] + E ฯ„ [ - ( ฮผ - ฮผ 0 ) 2 ฮป 0 ฯ„ 2 ] + E ฯ„ [ โˆ‘ n = 1 N 1 2 ( ln ฯ„ - ln 2 ฯ€ ) ] + E ฯ„ [ 1 2 ( ln ฮป 0 + ln ฯ„ - ln 2 ฯ€ ) ] + C 2 = E ฯ„ [ โˆ‘ n = 1 N - ( x n - ฮผ ) 2 ฯ„ 2 ] + E ฯ„ [ - ( ฮผ - ฮผ 0 ) 2 ฮป 0 ฯ„ 2 ] + C 3 = - E ฯ„ [ ฯ„ ] 2 { โˆ‘ n = 1 N ( x n - ฮผ ) 2 + ฮป 0 ( ฮผ - ฮผ 0 ) 2 } + C 3 \begin{aligned}\displaystyle\ln q_{\mu}^{*}(\mu)&\displaystyle=\operatorname{E% }_{\tau}\left[\ln p(\mathbf{X}\mid\mu,\tau)+\ln p(\mu\mid\tau)+\ln p(\tau)% \right]+C\\ &\displaystyle=\operatorname{E}_{\tau}\left[\ln p(\mathbf{X}\mid\mu,\tau)% \right]+\operatorname{E}_{\tau}\left[\ln p(\mu\mid\tau)\right]+\operatorname{E% }_{\tau}\left[\ln p(\tau)\right]+C\\ &\displaystyle=\operatorname{E}_{\tau}\left[\ln\prod_{n=1}^{N}\mathcal{N}(x_{n% }\mid\mu,\tau^{-1})\right]+\operatorname{E}_{\tau}\left[\ln\mathcal{N}(\mu\mid% \mu_{0},(\lambda_{0}\tau)^{-1})\right]+C_{2}\\ &\displaystyle=\operatorname{E}_{\tau}\left[\ln\prod_{n=1}^{N}\sqrt{\frac{\tau% }{2\pi}}e^{-\frac{(x_{n}-\mu)^{2}\tau}{2}}\right]+\operatorname{E}_{\tau}\left% [\ln\sqrt{\frac{\lambda_{0}\tau}{2\pi}}e^{-\frac{(\mu-\mu_{0})^{2}\lambda_{0}% \tau}{2}}\right]+C_{2}\\ &\displaystyle=\operatorname{E}_{\tau}\left[\sum_{n=1}^{N}\left(\frac{1}{2}(% \ln\tau-\ln 2\pi)-\frac{(x_{n}-\mu)^{2}\tau}{2})\right)\right]+\operatorname{E% }_{\tau}\left[\frac{1}{2}(\ln\lambda_{0}+\ln\tau-\ln 2\pi)-\frac{(\mu-\mu_{0})% ^{2}\lambda_{0}\tau}{2}\right]+C_{2}\\ &\displaystyle=\operatorname{E}_{\tau}\left[\sum_{n=1}^{N}-\frac{(x_{n}-\mu)^{% 2}\tau}{2}\right]+\operatorname{E}_{\tau}\left[-\frac{(\mu-\mu_{0})^{2}\lambda% _{0}\tau}{2}\right]+\operatorname{E}_{\tau}\left[\sum_{n=1}^{N}\frac{1}{2}(\ln% \tau-\ln 2\pi)\right]+\operatorname{E}_{\tau}\left[\frac{1}{2}(\ln\lambda_{0}+% \ln\tau-\ln 2\pi)\right]+C_{2}\\ &\displaystyle=\operatorname{E}_{\tau}\left[\sum_{n=1}^{N}-\frac{(x_{n}-\mu)^{% 2}\tau}{2}\right]+\operatorname{E}_{\tau}\left[-\frac{(\mu-\mu_{0})^{2}\lambda% _{0}\tau}{2}\right]+C_{3}\\ &\displaystyle=-\frac{\operatorname{E}_{\tau}[\tau]}{2}\left\{\sum_{n=1}^{N}(x% _{n}-\mu)^{2}+\lambda_{0}(\mu-\mu_{0})^{2}\right\}+C_{3}\end{aligned}
  60. C C
  61. C 2 C_{2}
  62. C 3 C_{3}
  63. ฮผ \mu
  64. E ฯ„ [ ln p ( ฯ„ ) ] \operatorname{E}_{\tau}[\ln p(\tau)]
  65. ฮผ \mu
  66. ฮผ \mu
  67. ฮผ \mu
  68. q ฮผ * ( ฮผ ) q_{\mu}^{*}(\mu)
  69. q ฮผ * ( ฮผ ) q_{\mu}^{*}(\mu)
  70. ฮผ \mu
  71. ฮผ 2 \mu^{2}
  72. ฮผ \mu
  73. ln q ฮผ * ( ฮผ ) \displaystyle\ln q_{\mu}^{*}(\mu)
  74. q ฮผ * ( ฮผ ) \displaystyle q_{\mu}^{*}(\mu)
  75. q ฯ„ * ( ฯ„ ) q_{\tau}^{*}(\tau)
  76. ln q ฯ„ * ( ฯ„ ) = E ฮผ [ ln p ( ๐— โˆฃ ฮผ , ฯ„ ) + ln p ( ฮผ โˆฃ ฯ„ ) ] + ln p ( ฯ„ ) + constant = ( a 0 - 1 ) ln ฯ„ - b 0 ฯ„ + 1 2 ln ฯ„ + N 2 ln ฯ„ - ฯ„ 2 E ฮผ [ โˆ‘ n = 1 N ( x n - ฮผ ) 2 + ฮป 0 ( ฮผ - ฮผ 0 ) 2 ] + constant \begin{aligned}\displaystyle\ln q_{\tau}^{*}(\tau)&\displaystyle=\operatorname% {E}_{\mu}[\ln p(\mathbf{X}\mid\mu,\tau)+\ln p(\mu\mid\tau)]+\ln p(\tau)+\,% \text{constant}\\ &\displaystyle=(a_{0}-1)\ln\tau-b_{0}\tau+\frac{1}{2}\ln\tau+\frac{N}{2}\ln% \tau-\frac{\tau}{2}\operatorname{E}_{\mu}[\sum_{n=1}^{N}(x_{n}-\mu)^{2}+% \lambda_{0}(\mu-\mu_{0})^{2}]+\,\text{constant}\end{aligned}
  77. q ฯ„ * ( ฯ„ ) q_{\tau}^{*}(\tau)
  78. q ฯ„ * ( ฯ„ ) \displaystyle q_{\tau}^{*}(\tau)
  79. q ฮผ * ( ฮผ ) \displaystyle q_{\mu}^{*}(\mu)
  80. q ฯ„ * ( ฯ„ ) โˆผ Gamma ( ฯ„ โˆฃ a N , b N ) a N = a 0 + N + 1 2 b N = b 0 + 1 2 E ฮผ [ โˆ‘ n = 1 N ( x n - ฮผ ) 2 + ฮป 0 ( ฮผ - ฮผ 0 ) 2 ] \begin{aligned}\displaystyle q_{\tau}^{*}(\tau)&\displaystyle\sim\operatorname% {Gamma}(\tau\mid a_{N},b_{N})\\ \displaystyle a_{N}&\displaystyle=a_{0}+\frac{N+1}{2}\\ \displaystyle b_{N}&\displaystyle=b_{0}+\frac{1}{2}\operatorname{E}_{\mu}\left% [\sum_{n=1}^{N}(x_{n}-\mu)^{2}+\lambda_{0}(\mu-\mu_{0})^{2}\right]\end{aligned}
  81. E [ ฯ„ โˆฃ a N , b N ] \displaystyle\operatorname{E}[\tau\mid a_{N},b_{N}]
  82. b N b_{N}
  83. b N \displaystyle b_{N}
  84. ฮผ N \displaystyle\mu_{N}
  85. ฮผ N \mu_{N}
  86. ฮป N \lambda_{N}
  87. b N b_{N}
  88. โˆ‘ n = 1 N x n \sum_{n=1}^{N}x_{n}
  89. โˆ‘ n = 1 N x n 2 . \sum_{n=1}^{N}x_{n}^{2}.
  90. ฮผ N \mu_{N}
  91. a N . a_{N}.
  92. ฮป N \lambda_{N}
  93. ฮป N , \lambda_{N},
  94. b N b_{N}
  95. b N , b_{N},
  96. ฮป N \lambda_{N}
  97. ๐— \mathbf{X}
  98. s y m b o l ฮ˜ symbol\Theta
  99. ๐™ \mathbf{Z}
  100. p ( ๐™ , s y m b o l ฮ˜ โˆฃ ๐— ) p(\mathbf{Z},symbol\Theta\mid\mathbf{X})
  101. ๐™ 1 , โ€ฆ , ๐™ M \mathbf{Z}_{1},\ldots,\mathbf{Z}_{M}
  102. ๐™ j \mathbf{Z}_{j}
  103. q j * ( ๐™ j โˆฃ ๐— ) q_{j}^{*}(\mathbf{Z}_{j}\mid\mathbf{X})
  104. ln q j * ( ๐™ j โˆฃ ๐— ) = E i โ‰  j [ ln p ( ๐™ , ๐— ) ] + constant \ln q_{j}^{*}(\mathbf{Z}_{j}\mid\mathbf{X})=\operatorname{E}_{i\neq j}[\ln p(% \mathbf{Z},\mathbf{X})]+\,\text{constant}
  105. ๐™ j \mathbf{Z}_{j}
  106. ๐™ j \mathbf{Z}_{j}
  107. ฯ€ โˆผ SymDir ( K , ฮฑ 0 ) ๐šฒ i = 1 โ€ฆ K โˆผ ๐’ฒ ( ๐– 0 , ฮฝ 0 ) ฮผ i = 1 โ€ฆ K โˆผ ๐’ฉ ( ฮผ 0 , ( ฮฒ 0 ๐šฒ i ) - 1 ) ๐ณ [ i = 1 โ€ฆ N ] โˆผ Mult ( 1 , ฯ€ ) ๐ฑ i = 1 โ€ฆ N โˆผ ๐’ฉ ( ฮผ z i , ๐šฒ z i - 1 ) K = number of mixing components N = number of data points \begin{aligned}\displaystyle\mathbf{\pi}&\displaystyle\sim\operatorname{SymDir% }(K,\alpha_{0})\\ \displaystyle\mathbf{\Lambda}_{i=1\dots K}&\displaystyle\sim\mathcal{W}(% \mathbf{W}_{0},\nu_{0})\\ \displaystyle\mathbf{\mu}_{i=1\dots K}&\displaystyle\sim\mathcal{N}(\mathbf{% \mu}_{0},(\beta_{0}\mathbf{\Lambda}_{i})^{-1})\\ \displaystyle\mathbf{z}[i=1\dots N]&\displaystyle\sim\operatorname{Mult}(1,% \mathbf{\pi})\\ \displaystyle\mathbf{x}_{i=1\dots N}&\displaystyle\sim\mathcal{N}(\mathbf{\mu}% _{z_{i}},{\mathbf{\Lambda}_{z_{i}}}^{-1})\\ \displaystyle K&\displaystyle=\,\text{number of mixing components}\\ \displaystyle N&\displaystyle=\,\text{number of data points}\end{aligned}
  108. K K
  109. ฮฑ 0 \alpha_{0}
  110. ๐’ฒ ( ) \mathcal{W}()
  111. K K
  112. ๐’ฉ ( ) \mathcal{N}()
  113. ๐— = { ๐ฑ 1 , โ€ฆ , ๐ฑ N } \mathbf{X}=\{\mathbf{x}_{1},\dots,\mathbf{x}_{N}\}
  114. N N
  115. K K
  116. ๐™ = { ๐ณ 1 , โ€ฆ , ๐ณ N } \mathbf{Z}=\{\mathbf{z}_{1},\dots,\mathbf{z}_{N}\}
  117. z n k z_{nk}
  118. k = 1 โ€ฆ K k=1\dots K
  119. ฯ€ \mathbf{\pi}
  120. K K
  121. ฮผ i = 1 โ€ฆ K \mathbf{\mu}_{i=1\dots K}
  122. ๐šฒ i = 1 โ€ฆ K \mathbf{\Lambda}_{i=1\dots K}
  123. p ( ๐— , ๐™ , ฯ€ , ฮผ , ๐šฒ ) = p ( ๐— โˆฃ ๐™ , ฮผ , ๐šฒ ) p ( ๐™ โˆฃ ฯ€ ) p ( ฯ€ ) p ( ฮผ โˆฃ ๐šฒ ) p ( ๐šฒ ) p(\mathbf{X},\mathbf{Z},\mathbf{\pi},\mathbf{\mu},\mathbf{\Lambda})=p(\mathbf{% X}\mid\mathbf{Z},\mathbf{\mu},\mathbf{\Lambda})p(\mathbf{Z}\mid\mathbf{\pi})p(% \mathbf{\pi})p(\mathbf{\mu}\mid\mathbf{\Lambda})p(\mathbf{\Lambda})
  124. p ( ๐— โˆฃ ๐™ , ฮผ , ๐šฒ ) = โˆ n = 1 N โˆ k = 1 K ๐’ฉ ( ๐ฑ n โˆฃ ฮผ k , ๐šฒ k - 1 ) z n k p ( ๐™ โˆฃ ฯ€ ) = โˆ n = 1 N โˆ k = 1 K ฯ€ k z n k p ( ฯ€ ) = ฮ“ ( K ฮฑ 0 ) ฮ“ ( ฮฑ 0 ) K โˆ k = 1 K ฯ€ k ฮฑ 0 - 1 p ( ฮผ โˆฃ ๐šฒ ) = โˆ k = 1 K ๐’ฉ ( ฮผ k โˆฃ ฮผ 0 , ( ฮฒ 0 ๐šฒ k ) - 1 ) p ( ๐šฒ ) = โˆ k = 1 K ๐’ฒ ( ๐šฒ k โˆฃ ๐– 0 , ฮฝ 0 ) \begin{aligned}\displaystyle p(\mathbf{X}\mid\mathbf{Z},\mathbf{\mu},\mathbf{% \Lambda})&\displaystyle=\prod_{n=1}^{N}\prod_{k=1}^{K}\mathcal{N}(\mathbf{x}_{% n}\mid\mathbf{\mu}_{k},\mathbf{\Lambda}_{k}^{-1})^{z_{nk}}\\ \displaystyle p(\mathbf{Z}\mid\mathbf{\pi})&\displaystyle=\prod_{n=1}^{N}\prod% _{k=1}^{K}\pi_{k}^{z_{nk}}\\ \displaystyle p(\mathbf{\pi})&\displaystyle=\frac{\Gamma(K\alpha_{0})}{\Gamma(% \alpha_{0})^{K}}\prod_{k=1}^{K}\pi_{k}^{\alpha_{0}-1}\\ \displaystyle p(\mathbf{\mu}\mid\mathbf{\Lambda})&\displaystyle=\prod_{k=1}^{K% }\mathcal{N}(\mathbf{\mu}_{k}\mid\mathbf{\mu}_{0},(\beta_{0}\mathbf{\Lambda}_{% k})^{-1})\\ \displaystyle p(\mathbf{\Lambda})&\displaystyle=\prod_{k=1}^{K}\mathcal{W}(% \mathbf{\Lambda}_{k}\mid\mathbf{W}_{0},\nu_{0})\end{aligned}
  125. ๐’ฉ ( ๐ฑ โˆฃ ฮผ , ๐šบ ) = 1 ( 2 ฯ€ ) D / 2 1 | ๐šบ | 1 / 2 exp { - 1 2 ( ๐ฑ - ฮผ ) T ๐šบ - 1 ( ๐ฑ - ฮผ ) } ๐’ฒ ( ๐šฒ โˆฃ ๐– , ฮฝ ) = B ( ๐– , ฮฝ ) | ๐šฒ | ( ฮฝ - D - 1 ) / 2 exp ( - 1 2 Tr ( ๐– - 1 ๐šฒ ) ) B ( ๐– , ฮฝ ) = | ๐– | - ฮฝ / 2 { 2 ฮฝ D / 2 ฯ€ D ( D - 1 ) / 4 โˆ i = 1 D ฮ“ ( ฮฝ + 1 - i 2 ) } - 1 D = dimensionality of each data point \begin{aligned}\displaystyle\mathcal{N}(\mathbf{x}\mid\mathbf{\mu},\mathbf{% \Sigma})&\displaystyle=\frac{1}{(2\pi)^{D/2}}\frac{1}{|\mathbf{\Sigma}|^{1/2}}% \exp\left\{-\frac{1}{2}(\mathbf{x}-\mathbf{\mu})^{\rm T}\mathbf{\Sigma}^{-1}(% \mathbf{x}-\mathbf{\mu})\right\}\\ \displaystyle\mathcal{W}(\mathbf{\Lambda}\mid\mathbf{W},\nu)&\displaystyle=B(% \mathbf{W},\nu)|\mathbf{\Lambda}|^{(\nu-D-1)/2}\exp\left(-\frac{1}{2}% \operatorname{Tr}(\mathbf{W}^{-1}\mathbf{\Lambda})\right)\\ \displaystyle B(\mathbf{W},\nu)&\displaystyle=|\mathbf{W}|^{-\nu/2}\left\{2^{% \nu D/2}\pi^{D(D-1)/4}\prod_{i=1}^{D}\Gamma\left(\frac{\nu+1-i}{2}\right)% \right\}^{-1}\\ \displaystyle D&\displaystyle=\,\text{dimensionality of each data point}\end{aligned}
  126. q ( ๐™ , ฯ€ , ฮผ , ๐šฒ ) = q ( ๐™ ) q ( ฯ€ , ฮผ , ๐šฒ ) q(\mathbf{Z},\mathbf{\pi},\mathbf{\mu},\mathbf{\Lambda})=q(\mathbf{Z})q(% \mathbf{\pi},\mathbf{\mu},\mathbf{\Lambda})
  127. ln q * ( ๐™ ) = E ฯ€ , ฮผ , ๐šฒ [ ln p ( ๐— , ๐™ , ฯ€ , ฮผ , ๐šฒ ) ] + constant = E ฯ€ [ ln p ( ๐™ โˆฃ ฯ€ ) ] + E ฮผ , ๐šฒ [ ln p ( ๐— โˆฃ ๐™ , ฮผ , ๐šฒ ) ] + constant = โˆ‘ n = 1 N โˆ‘ k = 1 K z n k ln ฯ n k + constant \begin{aligned}\displaystyle\ln q^{*}(\mathbf{Z})&\displaystyle=\operatorname{% E}_{\mathbf{\pi},\mathbf{\mu},\mathbf{\Lambda}}[\ln p(\mathbf{X},\mathbf{Z},% \mathbf{\pi},\mathbf{\mu},\mathbf{\Lambda})]+\,\text{constant}\\ &\displaystyle=\operatorname{E}_{\mathbf{\pi}}[\ln p(\mathbf{Z}\mid\mathbf{\pi% })]+\operatorname{E}_{\mathbf{\mu},\mathbf{\Lambda}}[\ln p(\mathbf{X}\mid% \mathbf{Z},\mathbf{\mu},\mathbf{\Lambda})]+\,\text{constant}\\ &\displaystyle=\sum_{n=1}^{N}\sum_{k=1}^{K}z_{nk}\ln\rho_{nk}+\,\text{constant% }\end{aligned}
  128. ln ฯ n k = E [ ln ฯ€ k ] + 1 2 E [ ln | ๐šฒ k | ] - D 2 ln ( 2 ฯ€ ) - 1 2 E ฮผ k , ๐šฒ k [ ( ๐ฑ n - ฮผ k ) T ๐šฒ k ( ๐ฑ n - ฮผ k ) ] \ln\rho_{nk}=\operatorname{E}[\ln\pi_{k}]+\frac{1}{2}\operatorname{E}[\ln|% \mathbf{\Lambda}_{k}|]-\frac{D}{2}\ln(2\pi)-\frac{1}{2}\operatorname{E}_{% \mathbf{\mu}_{k},\mathbf{\Lambda}_{k}}[(\mathbf{x}_{n}-\mathbf{\mu}_{k})^{\rm T% }\mathbf{\Lambda}_{k}(\mathbf{x}_{n}-\mathbf{\mu}_{k})]
  129. ln q * ( ๐™ ) \ln q^{*}(\mathbf{Z})
  130. q * ( ๐™ ) โˆ โˆ n = 1 N โˆ k = 1 K ฯ n k z n k q^{*}(\mathbf{Z})\propto\prod_{n=1}^{N}\prod_{k=1}^{K}\rho_{nk}^{z_{nk}}
  131. ฯ n k \rho_{nk}
  132. k k
  133. q * ( ๐™ ) = โˆ n = 1 N โˆ k = 1 K r n k z n k q^{*}(\mathbf{Z})=\prod_{n=1}^{N}\prod_{k=1}^{K}r_{nk}^{z_{nk}}
  134. r n k = ฯ n k โˆ‘ j = 1 K ฯ n j r_{nk}=\frac{\rho_{nk}}{\sum_{j=1}^{K}\rho_{nj}}
  135. q * ( ๐™ ) q^{*}(\mathbf{Z})
  136. ๐ณ n \mathbf{z}_{n}
  137. r n k r_{nk}
  138. k = 1 โ€ฆ K k=1\dots K
  139. E [ z n k ] = r n k \operatorname{E}[z_{nk}]=r_{nk}\,
  140. q ( ฯ€ , ฮผ , ๐šฒ ) q(\mathbf{\pi},\mathbf{\mu},\mathbf{\Lambda})
  141. q ( ฯ€ ) โˆ k = 1 K q ( ฮผ k , ๐šฒ k ) q(\mathbf{\pi})\prod_{k=1}^{K}q(\mathbf{\mu}_{k},\mathbf{\Lambda}_{k})
  142. ln q * ( ฯ€ ) = ln p ( ฯ€ ) + E ๐™ [ ln p ( ๐™ โˆฃ ฯ€ ) ] + constant = ( ฮฑ 0 - 1 ) โˆ‘ k = 1 K ln ฯ€ k + โˆ‘ n = 1 N โˆ‘ k = 1 K r n k ln ฯ€ k + constant \begin{aligned}\displaystyle\ln q^{*}(\mathbf{\pi})&\displaystyle=\ln p(% \mathbf{\pi})+\operatorname{E}_{\mathbf{Z}}[\ln p(\mathbf{Z}\mid\mathbf{\pi})]% +\,\text{constant}\\ &\displaystyle=(\alpha_{0}-1)\sum_{k=1}^{K}\ln\pi_{k}+\sum_{n=1}^{N}\sum_{k=1}% ^{K}r_{nk}\ln\pi_{k}+\,\text{constant}\end{aligned}
  143. q * ( ฯ€ ) q^{*}(\mathbf{\pi})
  144. q * ( ฯ€ ) โˆผ Dir ( ฮฑ ) q^{*}(\mathbf{\pi})\sim\operatorname{Dir}(\mathbf{\alpha})\,
  145. ฮฑ k = ฮฑ 0 + N k \alpha_{k}=\alpha_{0}+N_{k}\,
  146. N k = โˆ‘ n = 1 N r n k N_{k}=\sum_{n=1}^{N}r_{nk}\,
  147. ln q * ( ฮผ k , ๐šฒ k ) = ln p ( ฮผ k , ๐šฒ k ) + โˆ‘ n = 1 N E [ z n k ] ln ๐’ฉ ( ๐ฑ n โˆฃ ฮผ k , ๐šฒ k - 1 ) + constant \ln q^{*}(\mathbf{\mu}_{k},\mathbf{\Lambda}_{k})=\ln p(\mathbf{\mu}_{k},% \mathbf{\Lambda}_{k})+\sum_{n=1}^{N}\operatorname{E}[z_{nk}]\ln\mathcal{N}(% \mathbf{x}_{n}\mid\mathbf{\mu}_{k},\mathbf{\Lambda}_{k}^{-1})+\,\text{constant}
  148. ฮผ k \mathbf{\mu}_{k}
  149. ๐šฒ k \mathbf{\Lambda}_{k}
  150. q * ( ฮผ k , ๐šฒ k ) = ๐’ฉ ( ฮผ k โˆฃ ๐ฆ k , ( ฮฒ k ๐šฒ k ) - 1 ) ๐’ฒ ( ๐šฒ k โˆฃ ๐– k , ฮฝ k ) q^{*}(\mathbf{\mu}_{k},\mathbf{\Lambda}_{k})=\mathcal{N}(\mathbf{\mu}_{k}\mid% \mathbf{m}_{k},(\beta_{k}\mathbf{\Lambda}_{k})^{-1})\mathcal{W}(\mathbf{% \Lambda}_{k}\mid\mathbf{W}_{k},\nu_{k})
  151. ฮฒ k = ฮฒ 0 + N k ๐ฆ k = 1 ฮฒ k ( ฮฒ 0 ฮผ 0 + N k ๐ฑ ยฏ k ) ๐– k - 1 = ๐– 0 - 1 + N k ๐’ k + ฮฒ 0 N k ฮฒ 0 + N k ( ๐ฑ ยฏ k - ฮผ 0 ) ( ๐ฑ ยฏ k - ฮผ 0 ) T ฮฝ k = ฮฝ 0 + N k N k = โˆ‘ n = 1 N r n k ๐ฑ ยฏ k = 1 N k โˆ‘ n = 1 N r n k ๐ฑ n ๐’ k = 1 N k โˆ‘ n = 1 N ( ๐ฑ n - ๐ฑ ยฏ k ) ( ๐ฑ n - ๐ฑ ยฏ k ) T \begin{aligned}\displaystyle\beta_{k}&\displaystyle=\beta_{0}+N_{k}\\ \displaystyle\mathbf{m}_{k}&\displaystyle=\frac{1}{\beta_{k}}(\beta_{0}\mathbf% {\mu}_{0}+N_{k}{\bar{\mathbf{x}}}_{k})\\ \displaystyle\mathbf{W}_{k}^{-1}&\displaystyle=\mathbf{W}_{0}^{-1}+N_{k}% \mathbf{S}_{k}+\frac{\beta_{0}N_{k}}{\beta_{0}+N_{k}}({\bar{\mathbf{x}}}_{k}-% \mathbf{\mu}_{0})({\bar{\mathbf{x}}}_{k}-\mathbf{\mu}_{0})^{\rm T}\\ \displaystyle\nu_{k}&\displaystyle=\nu_{0}+N_{k}\\ \displaystyle N_{k}&\displaystyle=\sum_{n=1}^{N}r_{nk}\\ \displaystyle{\bar{\mathbf{x}}}_{k}&\displaystyle=\frac{1}{N_{k}}\sum_{n=1}^{N% }r_{nk}\mathbf{x}_{n}\\ \displaystyle\mathbf{S}_{k}&\displaystyle=\frac{1}{N_{k}}\sum_{n=1}^{N}(% \mathbf{x}_{n}-{\bar{\mathbf{x}}}_{k})(\mathbf{x}_{n}-{\bar{\mathbf{x}}}_{k})^% {\rm T}\end{aligned}
  152. r n k r_{nk}
  153. ฯ n k \rho_{nk}
  154. E [ ln ฯ€ k ] \operatorname{E}[\ln\pi_{k}]
  155. E [ ln | ๐šฒ k | ] \operatorname{E}[\ln|\mathbf{\Lambda}_{k}|]
  156. E ฮผ k , ๐šฒ k [ ( ๐ฑ n - ฮผ k ) T ๐šฒ k ( ๐ฑ n - ฮผ k ) ] \operatorname{E}_{\mathbf{\mu}_{k},\mathbf{\Lambda}_{k}}[(\mathbf{x}_{n}-% \mathbf{\mu}_{k})^{\rm T}\mathbf{\Lambda}_{k}(\mathbf{x}_{n}-\mathbf{\mu}_{k})]
  157. E ฮผ k , ๐šฒ k [ ( ๐ฑ n - ฮผ k ) T ๐šฒ k ( ๐ฑ n - ฮผ k ) ] \displaystyle\operatorname{E}_{\mathbf{\mu}_{k},\mathbf{\Lambda}_{k}}[(\mathbf% {x}_{n}-\mathbf{\mu}_{k})^{\rm T}\mathbf{\Lambda}_{k}(\mathbf{x}_{n}-\mathbf{% \mu}_{k})]
  158. r n k โˆ ฯ€ ~ k ฮ› ~ k 1 / 2 exp { - D 2 ฮฒ k - ฮฝ k 2 ( ๐ฑ n - ๐ฆ k ) T ๐– k ( ๐ฑ n - ๐ฆ k ) } r_{nk}\propto{\tilde{\pi}}_{k}{\tilde{\Lambda}}_{k}^{1/2}\exp\left\{-\frac{D}{% 2\beta_{k}}-\frac{\nu_{k}}{2}(\mathbf{x}_{n}-\mathbf{m}_{k})^{\rm T}\mathbf{W}% _{k}(\mathbf{x}_{n}-\mathbf{m}_{k})\right\}
  159. k k
  160. ฮฒ k \beta_{k}
  161. ๐ฆ k \mathbf{m}_{k}
  162. ๐– k \mathbf{W}_{k}
  163. ฮฝ k \nu_{k}
  164. ฮผ k \mathbf{\mu}_{k}
  165. ๐šฒ k \mathbf{\Lambda}_{k}
  166. N k N_{k}
  167. ๐ฑ ยฏ k {\bar{\mathbf{x}}}_{k}
  168. ๐’ k \mathbf{S}_{k}
  169. r n k r_{nk}
  170. ฮฑ 1 โ€ฆ K \alpha_{1\dots K}
  171. ฯ€ \mathbf{\pi}
  172. N k N_{k}
  173. r n k r_{nk}
  174. r n k r_{nk}
  175. ฮฒ k \beta_{k}
  176. ๐ฆ k \mathbf{m}_{k}
  177. ๐– k \mathbf{W}_{k}
  178. ฮฝ k \nu_{k}
  179. ๐– k \mathbf{W}_{k}
  180. ฮฝ k \nu_{k}
  181. ฮฑ 1 โ€ฆ K \alpha_{1\dots K}
  182. ฯ€ ~ k {\tilde{\pi}}_{k}
  183. ฮ› ~ k {\tilde{\Lambda}}_{k}
  184. r n k r_{nk}
  185. r n k r_{nk}
  186. r n k r_{nk}
  187. p ( ๐™ โˆฃ ๐— ) p(\mathbf{Z}\mid\mathbf{X})
  188. N k N_{k}
  189. ๐ฑ ยฏ k {\bar{\mathbf{x}}}_{k}
  190. ๐’ k \mathbf{S}_{k}

Variational_inequality.html

  1. E E
  2. K K
  3. E E
  4. F : s y m b o l K \rightarrowsymbol E โˆ— \scriptstyle F:symbol{K}\rightarrowsymbol{E}^{\ast}
  5. K K
  6. E * E^{*}
  7. E E
  8. x x
  9. K K
  10. โŸจ F ( x ) , y - x โŸฉ โ‰ฅ 0 โˆ€ y โˆˆ s y m b o l K \langle F(x),y-x\rangle\geq 0\qquad\forall y\in symbol{K}
  11. โŸจ โ‹… , โ‹… โŸฉ : s y m b o l E * \timessymbol E โ†’ โ„ \scriptstyle\langle\cdot,\cdot\rangle:symbol{E}^{*}\timessymbol{E}\rightarrow% \mathbb{R}
  12. f f
  13. I = [ a , b ] I=[a,b]
  14. x * \scriptstyle x^{*}
  15. I I
  16. a < x * < b \scriptstyle a<x^{*}<b
  17. f โ€ฒ ( x * ) = 0 ; \scriptstyle f^{\prime}(x^{*})=0;
  18. x * = a \scriptstyle x^{*}=a
  19. f โ€ฒ ( x * ) โ‰ฅ 0 ; \scriptstyle f^{\prime}(x^{*})\geq 0;
  20. x * = b \scriptstyle x^{*}=b
  21. f โ€ฒ ( x * ) โ‰ค 0. \scriptstyle f^{\prime}(x^{*})\leq 0.
  22. x * โˆˆ I \scriptstyle x^{*}\in I
  23. f โ€ฒ ( x * ) ( y - x * ) โ‰ฅ 0 โˆ€ y โˆˆ I f^{\prime}(x^{*})(y-x^{*})\geq 0\qquad\forall y\in I
  24. โ„ n \scriptstyle\mathbb{R}^{n}
  25. K K
  26. โ„ n \scriptstyle\mathbb{R}^{n}
  27. F : K โ†’ โ„ n \scriptstyle F:K\rightarrow\mathbb{R}^{n}
  28. K K
  29. n n
  30. x x
  31. K K
  32. โŸจ F ( x ) , y - x โŸฉ โ‰ฅ 0 โˆ€ y โˆˆ K \langle F(x),y-x\rangle\geq 0\qquad\forall y\in K
  33. โŸจ โ‹… , โ‹… โŸฉ : โ„ n ร— โ„ n โ†’ โ„ \scriptstyle\langle\cdot,\cdot\rangle:\mathbb{R}^{n}\times\mathbb{R}^{n}% \rightarrow\mathbb{R}
  34. โ„ n \scriptstyle\mathbb{R}^{n}
  35. \scriptstylesymbol u ( s y m b o l x ) = ( u 1 ( s y m b o l x ) , u 2 ( s y m b o l x ) , u 3 ( s y m b o l x ) ) \scriptstylesymbol{u}(symbol{x})=\left(u_{1}(symbol{x}),u_{2}(symbol{x}),u_{3}% (symbol{x})\right)
  36. A A
  37. โˆ‚ A \scriptstyle\partial A
  38. u u
  39. ๐’ฐ ฮฃ \scriptstyle\mathcal{U}_{\Sigma}
  40. B ( s y m b o l u , s y m b o l v ) - F ( s y m b o l v ) โ‰ฅ 0 โˆ€ s y m b o l v โˆˆ ๐’ฐ ฮฃ B(symbol{u},symbol{v})-F(symbol{v})\geq 0\qquad\forall symbol{v}\in\mathcal{U}% _{\Sigma}
  41. B ( s y m b o l u , s y m b o l v ) \scriptstyle B(symbol{u},symbol{v})
  42. F ( s y m b o l v ) \scriptstyle F(symbol{v})
  43. B ( s y m b o l u , s y m b o l v ) = - โˆซ A ฯƒ i k ( s y m b o l u ) ฮต i k ( s y m b o l v ) d x B(symbol{u},symbol{v})=-\int_{A}\sigma_{ik}(symbol{u})\varepsilon_{ik}(symbol{% v})\mathrm{d}x
  44. F ( s y m b o l v ) = โˆซ A v i f i d x + โˆซ โˆ‚ A โˆ– ฮฃ v i g i d ฯƒ s y m b o l u , s y m b o l v โˆˆ ๐’ฐ ฮฃ F(symbol{v})=\int_{A}v_{i}f_{i}\mathrm{d}x+\int_{\partial A\setminus\Sigma}\!% \!\!\!\!v_{i}g_{i}\mathrm{d}\sigma\qquad symbol{u},symbol{v}\in\mathcal{U}_{\Sigma}
  45. \scriptstylesymbol x โˆˆ A \scriptstylesymbol{x}\in A
  46. ฮฃ \Sigma
  47. \scriptstylesymbol f ( s y m b o l x ) = ( f 1 ( s y m b o l x ) , f 2 ( s y m b o l x ) , f 3 ( s y m b o l x ) ) \scriptstylesymbol{f}(symbol{x})=\left(f_{1}(symbol{x}),f_{2}(symbol{x}),f_{3}% (symbol{x})\right)
  48. \scriptstylesymbol g ( s y m b o l x ) = ( g 1 ( s y m b o l x ) , g 2 ( s y m b o l x ) , g 3 ( s y m b o l x ) ) \scriptstylesymbol{g}(symbol{x})=\left(g_{1}(symbol{x}),g_{2}(symbol{x}),g_{3}% (symbol{x})\right)
  49. โˆ‚ A โˆ– ฮฃ \scriptstyle\partial A\setminus\Sigma
  50. \scriptstylesymbol ฮต = s y m b o l ฮต ( s y m b o l u ) = ( ฮต i k ( s y m b o l u ) ) = ( 1 2 ( โˆ‚ u i โˆ‚ x k + โˆ‚ u k โˆ‚ x i ) ) \scriptstylesymbol{\varepsilon}=symbol{\varepsilon}(symbol{u})=\left(% \varepsilon_{ik}(symbol{u})\right)=\left(\frac{1}{2}\left(\frac{\partial u_{i}% }{\partial x_{k}}+\frac{\partial u_{k}}{\partial x_{i}}\right)\right)
  51. \scriptstylesymbol ฯƒ = ( ฯƒ i k ) \scriptstylesymbol{\sigma}=\left(\sigma_{ik}\right)
  52. ฯƒ i k = - โˆ‚ W โˆ‚ ฮต i k โˆ€ i , k = 1 , 2 , 3 \sigma_{ik}=-\frac{\partial W}{\partial\varepsilon_{ik}}\qquad\forall i,k=1,2,3
  53. W ( s y m b o l ฮต ) = a i k j h ( s y m b o l x ) ฮต i k ฮต i k \scriptstyle W(symbol{\varepsilon})=a_{ikjh}(symbol{x})\varepsilon_{ik}% \varepsilon_{ik}
  54. \scriptstylesymbol a ( s y m b o l x ) = ( a i k j h ( s y m b o l x ) ) \scriptstylesymbol{a}(symbol{x})=\left(a_{ikjh}(symbol{x})\right)

Variational_perturbation_theory.html

  1. s = โˆ‘ n = 0 โˆž a n g n s=\sum_{n=0}^{\infty}a_{n}g^{n}
  2. s = โˆ‘ n = 0 โˆž b n / ( g ฯ‰ ) n s=\sum_{n=0}^{\infty}b_{n}/(g^{\omega})^{n}
  3. ฯ‰ \omega
  4. g g
  5. g g

Variogram.html

  1. 2 ฮณ ( x , y ) 2\gamma(x,y)
  2. Z ( x ) Z(x)
  3. x x
  4. y y
  5. 2 ฮณ ( x , y ) = var ( Z ( x ) - Z ( y ) ) = E [ ( ( Z ( x ) - ฮผ ( x ) ) - ( Z ( y ) - ฮผ ( y ) ) ) 2 ] . 2\gamma(x,y)=\,\text{var}\left(Z(x)-Z(y)\right)=E\left[((Z(x)-\mu(x))-(Z(y)-% \mu(y)))^{2}\right].
  6. ฮผ \mu
  7. x x
  8. y y
  9. x x
  10. y y
  11. 2 ฮณ ( x , y ) = E [ ( Z ( x ) - Z ( y ) ) 2 ] , 2\gamma(x,y)=E\left[\left(Z(x)-Z(y)\right)^{2}\right],
  12. ฮณ ( x , y ) \gamma(x,y)
  13. ฮณ s ( h ) = ฮณ ( 0 , 0 + h ) \gamma_{s}(h)=\gamma(0,0+h)
  14. h = y - x h=y-x
  15. ฮณ ( x , y ) = ฮณ s ( y - x ) . \gamma(x,y)=\gamma_{s}(y-x).
  16. ฮณ i ( h ) := ฮณ s ( h e 1 ) \gamma_{i}(h):=\gamma_{s}(he_{1})
  17. h = โˆฅ y - x โˆฅ h=\|y-x\|
  18. ฮณ ( x , y ) = ฮณ i ( h ) . \gamma(x,y)=\gamma_{i}(h).
  19. i i
  20. s s
  21. ฮณ \gamma
  22. ฮณ ( x , y ) โ‰ฅ 0 \gamma(x,y)\geq 0
  23. ฮณ ( x , x ) = ฮณ i ( 0 ) = E ( ( Z ( x ) - Z ( x ) ) 2 ) = 0 \gamma(x,x)=\gamma_{i}(0)=E\left((Z(x)-Z(x))^{2}\right)=0
  24. Z ( x ) - Z ( x ) = 0 Z(x)-Z(x)=0
  25. w 1 , โ€ฆ , w N w_{1},\ldots,w_{N}
  26. โˆ‘ i = 1 N w i = 0 \sum_{i=1}^{N}w_{i}=0
  27. x 1 , โ€ฆ , x N x_{1},\ldots,x_{N}
  28. โˆ‘ i = 1 N โˆ‘ j = 1 N w i ฮณ ( x i , x j ) w j โ‰ค 0 \sum_{i=1}^{N}\sum_{j=1}^{N}w_{i}\gamma(x_{i},x_{j})w_{j}\leq 0
  29. v a r ( X ) var(X)
  30. X = โˆ‘ i = 1 N w i Z ( x i ) X=\sum_{i=1}^{N}w_{i}Z(x_{i})
  31. 2 ฮณ ( x , y ) = C ( x , x ) + C ( y , y ) - 2 C ( x , y ) 2\gamma(x,y)=C(x,x)+C(y,y)-2C(x,y)
  32. 2 ฮณ ( x , y ) = C ( x , x ) + C ( y , y ) - 2 C ( x , y ) + ( E [ Z ( x ) ] - E [ Z ( y ) ] ) 2 2\gamma(x,y)=C(x,x)+C(y,y)-2C(x,y)+(E\left[Z(x)\right]-E\left[Z(y)\right])^{2}
  33. C ( h ) = 0 C(h)=0
  34. h โ‰  0 h\not=0
  35. v a r ( Z ( x ) ) var(Z(x))
  36. ฮณ ( x , y ) = E [ | Z ( x ) - Z ( y ) | 2 ] = ฮณ ( y , x ) \gamma(x,y)=E\left[|Z(x)-Z(y)|^{2}\right]=\gamma(y,x)
  37. ฮณ s ( h ) = ฮณ s ( - h ) \gamma_{s}(h)=\gamma_{s}(-h)
  38. lim h โ†’ โˆž ฮณ s ( h ) = v a r ( Z ( x ) ) \lim_{h\to\infty}\gamma_{s}(h)=var(Z(x))
  39. z i , i = 1 , โ€ฆ , k z_{i},\;i=1,\ldots,k
  40. x 1 , โ€ฆ , x k x_{1},\ldots,x_{k}
  41. ฮณ ^ ( h ) \hat{\gamma}(h)
  42. ฮณ ^ ( h ) := 1 2 | N ( h ) | โˆ‘ ( i , j ) โˆˆ N ( h ) | z i - z j | 2 \hat{\gamma}(h):=\frac{1}{2|N(h)|}\sum_{(i,j)\in N(h)}|z_{i}-z_{j}|^{2}
  43. N ( h ) N(h)
  44. i , j i,\;j
  45. | x i - x j | = h |x_{i}-x_{j}|=h
  46. | N ( h ) | |N(h)|
  47. h h
  48. z i = Z ( x i ) z_{i}=Z(x_{i})
  49. Z ( x ) Z(x)
  50. E [ ฮณ ^ ( h ) ] = 1 2 | N ( h ) | โˆ‘ ( i , j ) โˆˆ N ( h ) E [ | Z ( x i ) - Z ( x j ) | 2 ] = 1 2 | N ( h ) | โˆ‘ ( i , j ) โˆˆ N ( h ) 2 ฮณ ( x j - x i ) = 2 | N ( h ) | 2 | N ( h ) | ฮณ ( h ) E\left[\hat{\gamma}(h)\right]=\frac{1}{2|N(h)|}\sum_{(i,j)\in N(h)}E\left[|Z(x% _{i})-Z(x_{j})|^{2}\right]=\frac{1}{2|N(h)|}\sum_{(i,j)\in N(h)}2\gamma(x_{j}-% x_{i})=\frac{2|N(h)|}{2|N(h)|}\gamma(h)
  51. n n
  52. s s
  53. r r
  54. h h
  55. ฮณ ( h ) = ( s - n ) ( 1 - exp ( - h / ( r a ) ) ) + n 1 ( 0 , โˆž ) ( h ) \gamma(h)=(s-n)(1-\exp(-h/(ra)))+n1_{(0,\infty)}(h)
  56. ฮณ ( h ) = ( s - n ) ( ( 3 h 2 r - h 3 2 r 3 ) 1 ( 0 , r ) ( h ) + 1 [ r , โˆž ) ( h ) ) + n 1 ( 0 , โˆž ) ( h ) \gamma(h)=(s-n)\left(\left(\frac{3h}{2r}-\frac{h^{3}}{2r^{3}}\right)1_{(0,r)}(% h)+1_{[r,\infty)}(h)\right)+n1_{(0,\infty)}(h)
  57. ฮณ ( h ) = ( s - n ) ( 1 - exp ( - h 2 r 2 a ) ) + n 1 ( 0 , โˆž ) ( h ) \gamma(h)=(s-n)\left(1-\exp\left(-\frac{h^{2}}{r^{2}a}\right)\right)+n1_{(0,% \infty)}(h)
  58. a a
  59. a = 1 / 3 a=1/3
  60. 1 A ( h ) 1_{A}(h)
  61. h โˆˆ A h\in A
  62. ( Z ( x ) - Z ( y ) ) 2 (Z(x)-Z(y))^{2}
  63. | Z ( x ) - Z ( y ) | |Z(x)-Z(y)|
  64. | Z ( x ) - Z ( y ) | 0.5 |Z(x)-Z(y)|^{0.5}
  65. 2 ฮณ ( x , y ) = E [ | Z ( x ) - Z ( y ) | ฮฑ ] 2\gamma(x,y)=E\left[\left|Z(x)-Z(y)\right|^{\alpha}\right]

Vector_algebra.html

  1. ๐‘ 3 \mathbf{R}^{3}

Vector_clock.html

  1. V C ( x ) VC(x)
  2. x x
  3. V C ( x ) z VC(x)_{z}
  4. z z
  5. V C ( x ) < V C ( y ) โ‡” โˆ€ z [ V C ( x ) z โ‰ค V C ( y ) z ] and โˆƒ z โ€ฒ [ V C ( x ) z โ€ฒ < V C ( y ) z โ€ฒ ] VC(x)<VC(y)\iff\forall z[VC(x)_{z}\leq VC(y)_{z}]\and\exists z^{\prime}[VC(x)_% {z^{\prime}}<VC(y)_{z^{\prime}}]
  6. V C ( x ) VC(x)
  7. V C ( y ) VC(y)
  8. V C ( x ) z VC(x)_{z}
  9. V C ( y ) z VC(y)_{z}
  10. z z
  11. V C ( x ) z โ€ฒ < V C ( y ) z โ€ฒ VC(x)_{z^{\prime}}<VC(y)_{z^{\prime}}
  12. x โ†’ y x\to y\;
  13. x x
  14. y y
  15. x โ†’ y x\to y\;
  16. V C ( x ) < V C ( y ) VC(x)<VC(y)
  17. V C ( a ) < V C ( b ) VC(a)<VC(b)
  18. a โ†’ b a\to b\;
  19. V C ( a ) < V C ( b ) VC(a)<VC(b)
  20. V C ( b ) < V C ( a ) VC(b)<VC(a)
  21. V C ( a ) < V C ( b ) VC(a)<VC(b)
  22. V C ( b ) < V C ( c ) VC(b)<VC(c)
  23. V C ( a ) < V C ( c ) VC(a)<VC(c)
  24. a โ†’ b a\to b\;
  25. b โ†’ c b\to c\;
  26. a โ†’ c a\to c\;
  27. R T ( x ) RT(x)
  28. x x
  29. V C ( a ) < V C ( b ) VC(a)<VC(b)
  30. R T ( a ) < R T ( b ) RT(a)<RT(b)
  31. C ( x ) C(x)
  32. x x
  33. V C ( a ) < V C ( b ) VC(a)<VC(b)
  34. C ( a ) < C ( b ) C(a)<C(b)

Vector_meson.html

  1. e - e + โ†’ ฮณ โ†’ q q ยฏ e^{-}e^{+}\to\gamma\to q\bar{q}

Vector_projection.html

  1. ๐š 1 = a 1 ๐› ^ \mathbf{a}_{1}=a_{1}\mathbf{\hat{b}}\,
  2. a 1 a_{1}
  3. a 1 = | ๐š | cos ฮธ = ๐š โ‹… ๐› ^ = ๐š โ‹… ๐› | ๐› | a_{1}=|\mathbf{a}|\cos\theta=\mathbf{a}\cdot\mathbf{\hat{b}}=\mathbf{a}\cdot% \frac{\mathbf{b}}{|\mathbf{b}|}\,
  4. ๐š 2 = ๐š - ๐š 1 . \mathbf{a}_{2}=\mathbf{a}-\mathbf{a}_{1}.
  5. a โ†’ 1 \vec{a}_{1}
  6. a 1 = | ๐š | cos ฮธ a_{1}=|\mathbf{a}|\cos\theta
  7. ๐š 1 = a 1 ๐› ^ = ( | ๐š | cos ฮธ ) ๐› ^ \mathbf{a}_{1}=a_{1}\mathbf{\hat{b}}=(|\mathbf{a}|\cos\theta)\mathbf{\hat{b}}
  8. ๐› ^ = ๐› | ๐› | \mathbf{\hat{b}}=\frac{\mathbf{b}}{|\mathbf{b}|}\,
  9. ๐š 2 = ๐š - ๐š 1 \mathbf{a}_{2}=\mathbf{a}-\mathbf{a}_{1}
  10. ๐š 2 = ๐š - ( | ๐š | cos ฮธ ) ๐› ^ . \mathbf{a}_{2}=\mathbf{a}-(|\mathbf{a}|\cos\theta)\mathbf{\hat{b}}.
  11. ๐š โ‹… ๐› | ๐š | | ๐› | = cos ฮธ \frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{a}|\,|\mathbf{b}|}=\cos\theta\,
  12. a 1 = | ๐š | cos ฮธ = | ๐š | ๐š โ‹… ๐› | ๐š | | ๐› | = ๐š โ‹… ๐› | ๐› | a_{1}=|\mathbf{a}|\cos\theta=|\mathbf{a}|\frac{\mathbf{a}\cdot\mathbf{b}}{|% \mathbf{a}|\,|\mathbf{b}|}=\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|}\,
  13. ๐š 1 = a 1 ๐› ^ = ๐š โ‹… ๐› | ๐› | ๐› | ๐› | , \mathbf{a}_{1}=a_{1}\mathbf{\hat{b}}=\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf% {b}|}\frac{\mathbf{b}}{|\mathbf{b}|},
  14. ๐š 1 = ( ๐š โ‹… ๐› ^ ) ๐› ^ , \mathbf{a}_{1}=(\mathbf{a}\cdot\mathbf{\hat{b}})\mathbf{\hat{b}},
  15. ๐š 1 = ๐š โ‹… ๐› | ๐› | 2 ๐› = ๐š โ‹… ๐› ๐› โ‹… ๐› ๐› . \mathbf{a}_{1}=\frac{\mathbf{a}\cdot\mathbf{b}}{|\mathbf{b}|^{2}}{\mathbf{b}}=% \frac{\mathbf{a}\cdot\mathbf{b}}{\mathbf{b}\cdot\mathbf{b}}{\mathbf{b}}.
  16. ๐š 2 = ๐š - ๐š 1 \mathbf{a}_{2}=\mathbf{a}-\mathbf{a}_{1}
  17. ๐š 2 = ๐š - ๐š โ‹… ๐› ๐› โ‹… ๐› ๐› . \mathbf{a}_{2}=\mathbf{a}-\frac{\mathbf{a}\cdot\mathbf{b}}{\mathbf{b}\cdot% \mathbf{b}}{\mathbf{b}}.
  18. P a = a a T = [ a x a y a z ] [ a x a y a z ] = [ a x 2 a x a y a x a z a x a y a y 2 a y a z a x a z a y a z a z 2 ] P_{a}=aa^{\mathrm{T}}=\begin{bmatrix}a_{x}\\ a_{y}\\ a_{z}\end{bmatrix}\begin{bmatrix}a_{x}&a_{y}&a_{z}\end{bmatrix}=\begin{bmatrix% }a_{x}^{2}&a_{x}a_{y}&a_{x}a_{z}\\ a_{x}a_{y}&a_{y}^{2}&a_{y}a_{z}\\ a_{x}a_{z}&a_{y}a_{z}&a_{z}^{2}\\ \end{bmatrix}
  19. ๐› ^ = ๐› / | ๐› | \mathbf{\hat{b}}={\mathbf{b}}/{|\mathbf{b}|}

Velocity-addition_formula.html

  1. V / c {V}/{c}
  2. c + V c+V
  3. c c
  4. V V
  5. ๐ฌ = ๐ฏ + ๐ฎ , \mathbf{s}=\mathbf{v}+\mathbf{u},
  6. s = v + u 1 + ( v u / c 2 ) . s={v+u\over 1+(vu/c^{2})}.
  7. c - s c + s = ( c - u c + u ) ( c - v c + v ) . {c-s\over c+s}=\left({c-u\over c+u}\right)\left({c-v\over c+v}\right).
  8. d x = ฮณ V ( d x โ€ฒ + v d t โ€ฒ ) , d y = d y โ€ฒ , d z = d z โ€ฒ , d t = ฮณ V ( d t โ€ฒ + V c 2 d x โ€ฒ ) . dx=\gamma_{V}(dx^{\prime}+vdt^{\prime}),\quad dy=dy^{\prime},\quad dz=dz^{% \prime},\quad dt=\gamma_{V}\left(dt^{\prime}+\frac{V}{c^{2}}dx^{\prime}\right).
  9. d x d t = ฮณ V ( d x โ€ฒ + v d t โ€ฒ ) ฮณ V ( d t โ€ฒ + V c 2 d x โ€ฒ ) , d y d t = d y โ€ฒ ฮณ V ( d t โ€ฒ + V c 2 d x โ€ฒ ) , d z d t = d z โ€ฒ ฮณ V ( d t โ€ฒ + V c 2 d x โ€ฒ ) , \frac{dx}{dt}=\frac{\gamma_{V}(dx^{\prime}+vdt^{\prime})}{\gamma_{V}(dt^{% \prime}+\frac{V}{c^{2}}dx^{\prime})},\quad\frac{dy}{dt}=\frac{dy^{\prime}}{% \gamma_{V}(dt^{\prime}+\frac{V}{c^{2}}dx^{\prime})},\quad\frac{dz}{dt}=\frac{% dz^{\prime}}{\gamma_{V}(dt^{\prime}+\frac{V}{c^{2}}dx^{\prime})},
  10. d x d t = d x โ€ฒ + v d t โ€ฒ d t โ€ฒ ( 1 + V c 2 d x โ€ฒ d t โ€ฒ โ€ฒ ) , d y d t = d y โ€ฒ ฮณ V d t โ€ฒ ( 1 + V c 2 d x โ€ฒ d t โ€ฒ ) , d z d t = d z โ€ฒ ฮณ V d t โ€ฒ ( 1 + V c 2 d x โ€ฒ d t โ€ฒ ) , \frac{dx}{dt}=\frac{dx^{\prime}+vdt^{\prime}}{dt^{\prime}(1+\frac{V}{c^{2}}% \frac{dx^{\prime}}{dt^{\prime}}^{\prime})},\quad\frac{dy}{dt}=\frac{dy^{\prime% }}{\gamma_{V}dt^{\prime}(1+\frac{V}{c^{2}}\frac{dx^{\prime}}{dt^{\prime}})},% \quad\frac{dz}{dt}=\frac{dz^{\prime}}{\gamma_{V}dt^{\prime}(1+\frac{V}{c^{2}}% \frac{dx^{\prime}}{dt^{\prime}})},
  11. x - y x-y
  12. v x = v cos ฮธ , v y = v sin ฮธ , v x โ€ฒ = v โ€ฒ cos ฮธ โ€ฒ , v y โ€ฒ = v โ€ฒ sin ฮธ โ€ฒ , v_{x}=v\cos\theta,v_{y}=v\sin\theta,\quad v_{x}^{\prime}=v^{\prime}\cos\theta^% {\prime},\quad v_{y}^{\prime}=v^{\prime}\sin\theta^{\prime},
  13. v = v x 2 + v y 2 = ( v x โ€ฒ + v ) 2 + ( 1 - V 2 c 2 ) v y โ€ฒ 2 1 + V c 2 v x โ€ฒ = v x โ€ฒ 2 + V 2 + 2 v x โ€ฒ v + ( 1 - V 2 c 2 ) v y โ€ฒ 2 1 + V c 2 v x โ€ฒ = v โ€ฒ 2 cos 2 ฮธ โ€ฒ + V 2 + 2 V v โ€ฒ cos ฮธ โ€ฒ + v โ€ฒ 2 sin 2 ฮธ โ€ฒ - V 2 c 2 v โ€ฒ 2 sin 2 ฮธ โ€ฒ 1 + V c 2 v x โ€ฒ = v โ€ฒ 2 + V 2 + 2 V v โ€ฒ cos ฮธ โ€ฒ - ( V v โ€ฒ sin ฮธ โ€ฒ c ) 2 1 + V c 2 v โ€ฒ cos ฮธ โ€ฒ \begin{aligned}\displaystyle v&\displaystyle=\sqrt{v_{x}^{2}+v_{y}^{2}}=\frac{% \sqrt{(v_{x}^{\prime}+v)^{2}+(1-\frac{V^{2}}{c^{2}})v_{y}^{\prime 2}}}{1+\frac% {V}{c^{2}}v_{x}^{\prime}}=\frac{\sqrt{v_{x}^{\prime 2}+V^{2}+2v_{x}^{\prime}v+% (1-\frac{V^{2}}{c^{2}})v_{y}^{\prime 2}}}{1+\frac{V}{c^{2}}v_{x}^{\prime}}\\ &\displaystyle=\frac{\sqrt{v^{\prime 2}\cos^{2}\theta^{\prime}+V^{2}+2Vv^{% \prime}\cos\theta^{\prime}+v^{\prime 2}\sin^{2}\theta^{\prime}-\frac{V^{2}}{c^% {2}}v^{\prime 2}\sin^{2}\theta^{\prime}}}{1+\frac{V}{c^{2}}v_{x}^{\prime}}\\ &\displaystyle=\frac{\sqrt{v^{\prime 2}+V^{2}+2Vv^{\prime}\cos\theta^{\prime}-% (\frac{Vv^{\prime}\sin\theta^{\prime}}{c})^{2}}}{1+\frac{V}{c^{2}}v^{\prime}% \cos\theta^{\prime}}\end{aligned}
  14. x x
  15. y y
  16. 1 1
  17. 1 1
  18. x x
  19. x x
  20. y y
  21. x โ€“ y xโ€“y
  22. ๐ฏ = ( v 1 , v 2 , v 3 ) = ( V 1 / V 0 , 0 , 0 ) , ๐ฎ = ( u 1 , u 2 , u 3 ) = ( U 1 / U 0 , U 2 / U 0 , 0 ) . \begin{aligned}\displaystyle\mathbf{v}&\displaystyle=(v_{1},v_{2},v_{3})=(V_{1% }/V_{0},0,0),\\ \displaystyle\mathbf{u}&\displaystyle=(u_{1},u_{2},u_{3})=(U_{1}/U_{0},U_{2}/U% _{0},0).\end{aligned}
  23. V V
  24. 1 1
  25. V 0 2 - V 1 2 = 1 , V_{0}^{2}-V_{1}^{2}=1,
  26. V 0 = 1 / 1 - v 1 2 , V 1 = v 1 / 1 - v 1 2 . V_{0}=1/\sqrt{1-v_{1}^{2}}\ ,\quad V_{1}=v_{1}/\sqrt{1-v_{1}^{2}}.
  27. V V
  28. ( V 0 V 1 0 0 V 1 V 0 0 0 0 0 1 0 0 0 0 1 ) . \begin{pmatrix}V_{0}&V_{1}&0&0\\ V_{1}&V_{0}&0&0\\ 0&0&1&0\\ 0&0&0&1\end{pmatrix}.
  29. ( 1 , 0 , 0 , 0 ) (1,0,0,0)
  30. U U
  31. S 0 = V 0 U 0 + V 1 U 1 , S 1 = V 1 U 0 + V 0 U 1 , S 2 = U 2 , S 3 = U 3 . \begin{aligned}\displaystyle S_{0}&\displaystyle=V_{0}U_{0}+V_{1}U_{1},\\ \displaystyle S_{1}&\displaystyle=V_{1}U_{0}+V_{0}U_{1},\\ \displaystyle S_{2}&\displaystyle=U_{2},\\ \displaystyle S_{3}&\displaystyle=U_{3}.\end{aligned}
  32. U U
  33. V V
  34. ๐ฎ \mathbf{u}
  35. ๐ฏ \mathbf{v}
  36. s 1 = v 1 + u 1 1 + v 1 u 1 , s 2 = u 2 ( 1 + v 1 u 1 ) 1 V 0 = u 2 1 + v 1 u 1 1 - v 1 2 , s 3 = 0. \begin{aligned}\displaystyle s_{1}&\displaystyle={v_{1}+u_{1}\over 1+v_{1}u_{1% }},\\ \displaystyle s_{2}&\displaystyle={u_{2}\over(1+v_{1}u_{1})}{1\over V_{0}}={u_% {2}\over 1+v_{1}u_{1}}\sqrt{1-v_{1}^{2}},\\ \displaystyle s_{3}&\displaystyle=0.\end{aligned}
  37. ๐ฎ \mathbf{u}
  38. ๐ฏ \mathbf{v}
  39. ๐ฏ \mathbf{v}
  40. ๐ฏ โ€ฒ \mathbf{v}โ€ฒ
  41. V V
  42. ๐ฏ \mathbf{v}
  43. ๐ฏ โ€ฒ \mathbf{v}โ€ฒ
  44. ๐• \mathbf{V}
  45. ๐ฏ = ๐ฏ โˆฅ + ๐ฏ โŸ‚ , ๐ฏ โ€ฒ = ๐ฏ โˆฅ โ€ฒ + ๐ฏ โŸ‚ โ€ฒ , \mathbf{v}=\mathbf{v}_{\parallel}+\mathbf{v}_{\perp},\quad\mathbf{v}^{\prime}=% \mathbf{v}^{\prime}_{\parallel}+\mathbf{v}^{\prime}_{\perp},
  46. ๐ฏ โˆฅ = v x ๐ž x , ๐ฏ โŸ‚ = v y ๐ž y + v z ๐ž z , ๐• = V ๐ž x , \mathbf{v}_{\parallel}=v_{x}\mathbf{e}_{x},\quad\mathbf{v}_{\perp}=v_{y}% \mathbf{e}_{y}+v_{z}\mathbf{e}_{z},\quad\mathbf{V}=V\mathbf{e}_{x},
  47. ๐ฏ โˆฅ = ๐ฏ โˆฅ โ€ฒ + ๐• 1 + ๐• โ‹… ๐ฏ โˆฅ โ€ฒ c 2 , ๐ฏ โŸ‚ = 1 - V 2 c 2 ๐ฏ โŸ‚ โ€ฒ 1 + ๐• โ‹… ๐ฏ โˆฅ โ€ฒ c 2 . \mathbf{v}_{\parallel}=\frac{\mathbf{v}_{\parallel}^{\prime}+\mathbf{V}}{1+% \frac{\mathbf{V}\cdot\mathbf{v}_{\parallel}^{\prime}}{c^{2}}},\quad\mathbf{v}_% {\perp}=\frac{\sqrt{1-\frac{V^{2}}{c^{2}}}\mathbf{v}_{\perp}^{\prime}}{1+\frac% {\mathbf{V}\cdot\mathbf{v}_{\parallel}^{\prime}}{c^{2}}}.
  48. ๐• \mathbf{V}
  49. ๐ฏ = ๐ฏ โˆฅ + ๐ฏ โŸ‚ = 1 1 + ๐• โ‹… ๐ฏ โ€ฒ c 2 [ ฮฑ V ๐ฏ โ€ฒ + ๐• + ( 1 - ฮฑ V ) ( ๐• โ‹… ๐ฏ โ€ฒ ) V 2 ๐• ] โ‰ก ๐• โŠ• ๐ฏ โ€ฒ , \mathbf{v}=\mathbf{v}_{\parallel}+\mathbf{v}_{\perp}=\frac{1}{1+\frac{\mathbf{% V}\cdot\mathbf{v}^{\prime}}{c^{2}}}\left[\alpha_{V}\mathbf{v}^{\prime}+\mathbf% {V}+(1-\alpha_{V})\frac{(\mathbf{V}\cdot\mathbf{v}^{\prime})}{V^{2}}\mathbf{V}% \right]\equiv\mathbf{V}\oplus\mathbf{v}^{\prime},
  50. ๐ฏ โ€ฒ \mathbf{v}โ€ฒ
  51. ๐ฏ โˆฅ โ€ฒ = ๐• โ‹… ๐ฏ โ€ฒ V 2 ๐• , \mathbf{v}^{\prime}_{\parallel}=\frac{\mathbf{V}\cdot\mathbf{v}^{\prime}}{V^{2% }}\mathbf{V},
  52. ๐ฏ โ€ฒ \mathbf{v}โ€ฒ
  53. ๐ฏ โŸ‚ โ€ฒ = - ๐• ร— ( ๐• ร— ๐ฏ โ€ฒ ) V 2 . \mathbf{v}^{\prime}_{\perp}=-\frac{\mathbf{V}\times(\mathbf{V}\times\mathbf{v}% ^{\prime})}{V^{2}}.
  54. ๐• / V \mathbf{V}/V
  55. ๐ฏ = ๐ฏ โˆฅ โ€ฒ + ๐• 1 + ๐• โ‹… ๐ฏ โˆฅ โ€ฒ c 2 + 1 - V 2 c 2 ( ๐ฏ - ๐ฏ โˆฅ โ€ฒ ) 1 + ๐• โ‹… ๐ฏ โˆฅ โ€ฒ c 2 , \mathbf{v}=\frac{\mathbf{v}_{\parallel}^{\prime}+\mathbf{V}}{1+\frac{\mathbf{V% }\cdot\mathbf{v}_{\parallel}^{\prime}}{c^{2}}}+\frac{\sqrt{1-\frac{V^{2}}{c^{2% }}}(\mathbf{v}-\mathbf{v}_{\parallel}^{\prime})}{1+\frac{\mathbf{V}\cdot% \mathbf{v}_{\parallel}^{\prime}}{c^{2}}},
  56. ๐• \mathbf{V}
  57. ๐ฎ \mathbf{u}
  58. ๐ฏ โ€ฒ \mathbf{v}โ€ฒ
  59. ๐ฏ \mathbf{v}
  60. ๐ฎ \mathbf{u}
  61. ๐ฏ \mathbf{v}
  62. ๐ฎ ร— ( ๐ฎ ร— ๐ฏ ) = ( ๐ฎ โ‹… ๐ฏ ) ๐ฎ โˆ’ ( ๐ฎ โ‹… ๐ฎ ) ๐ฏ \mathbf{u}ร—(\mathbf{u}ร—\mathbf{v})=(\mathbf{u}โ‹…\mathbf{v})\mathbf{u}โˆ’(\mathbf{% u}โ‹…\mathbf{u})\mathbf{v}
  63. ๐ฎ โŠ• ๐ฏ โ‰  ๐ฏ โŠ• ๐ฎ , \mathbf{u}\oplus\mathbf{v}\neq\mathbf{v}\oplus\mathbf{u},
  64. ๐ฎ โŠ• ( ๐ฏ โŠ• ๐ฐ ) โ‰  ( ๐ฎ โŠ• ๐ฏ ) โŠ• ๐ฐ . \mathbf{u}\oplus(\mathbf{v}\oplus\mathbf{w})\neq(\mathbf{u}\oplus\mathbf{v})% \oplus\mathbf{w}.
  65. ๐ฎ \mathbf{u}
  66. ๐ฏ \mathbf{v}
  67. ๐ฎ โŠ• ๐ฏ \mathbf{u}โŠ•\mathbf{v}
  68. ๐ฏ โŠ• ๐ฎ \mathbf{v}โŠ•\mathbf{u}
  69. | ๐ฎ โŠ• ๐ฏ | 2 = 1 ( 1 + ๐ฎ โ‹… ๐ฏ c 2 ) 2 [ ( ๐ฎ + ๐ฏ ) 2 - 1 c 2 ( ๐ฎ ร— ๐ฏ ) 2 ] = | ๐ฏ โŠ• ๐ฎ | 2 . |\mathbf{u}\oplus\mathbf{v}|^{2}=\frac{1}{\left(1+\frac{\mathbf{u}\cdot\mathbf% {v}}{c^{2}}\right)^{2}}\left[\left(\mathbf{u}+\mathbf{v}\right)^{2}-\frac{1}{c% ^{2}}\left(\mathbf{u}\times\mathbf{v}\right)^{2}\right]=|\mathbf{v}\oplus% \mathbf{u}|^{2}.
  70. ( 1 + ๐ฎ โ‹… ๐ฏ c 2 ) 2 | ๐ฎ โŠ• ๐ฏ | 2 = [ ๐ฎ + ๐ฏ + 1 c 2 ฮณ u 1 + ฮณ u ๐ฎ ร— ( ๐ฎ ร— ๐ฏ ) ] 2 = ( ๐ฎ + ๐ฏ ) 2 + 2 1 c 2 ฮณ u ฮณ u + 1 [ ( ๐ฎ โ‹… ๐ฏ ) 2 - ( ๐ฎ โ‹… ๐ฎ ) ( ๐ฏ โ‹… ๐ฏ ) ] + 1 c 4 ( ฮณ u ฮณ u + 1 ) 2 [ ( ๐ฎ โ‹… ๐ฎ ) 2 ( ๐ฏ โ‹… ๐ฏ ) - ( ๐ฎ โ‹… ๐ฏ ) 2 ( ๐ฎ โ‹… ๐ฎ ) ] = ( ๐ฎ + ๐ฏ ) 2 + 2 1 c 2 ฮณ u ฮณ u + 1 [ ( ๐ฎ โ‹… ๐ฏ ) 2 - ( ๐ฎ โ‹… ๐ฎ ) ( ๐ฏ โ‹… ๐ฏ ) ] + u 2 c 4 ( ฮณ u ฮณ u + 1 ) 2 [ ( ๐ฎ โ‹… ๐ฎ ) ( ๐ฏ โ‹… ๐ฏ ) - ( ๐ฎ โ‹… ๐ฏ ) 2 ] = ( ๐ฎ + ๐ฏ ) 2 + 2 1 c 2 ฮณ u ฮณ u + 1 [ ( ๐ฎ โ‹… ๐ฏ ) 2 - ( ๐ฎ โ‹… ๐ฎ ) ( ๐ฏ โ‹… ๐ฏ ) ] + ( 1 - ฮฑ u ) ( 1 + ฮฑ u ) c 2 ( ฮณ u ฮณ u + 1 ) 2 [ ( ๐ฎ โ‹… ๐ฎ ) ( ๐ฏ โ‹… ๐ฏ ) - ( ๐ฎ โ‹… ๐ฏ ) 2 ] = ( ๐ฎ + ๐ฏ ) 2 + 2 1 c 2 ฮณ u ฮณ u + 1 [ ( ๐ฎ โ‹… ๐ฏ ) 2 - ( ๐ฎ โ‹… ๐ฎ ) ( ๐ฏ โ‹… ๐ฏ ) ] + ( ฮณ u - 1 ) c 2 ( ฮณ u + 1 ) [ ( ๐ฎ โ‹… ๐ฎ ) ( ๐ฏ โ‹… ๐ฏ ) - ( ๐ฎ โ‹… ๐ฏ ) 2 ] = ( ๐ฎ + ๐ฏ ) 2 + 2 1 c 2 ฮณ u ฮณ u + 1 [ ( ๐ฎ โ‹… ๐ฏ ) 2 - ( ๐ฎ โ‹… ๐ฎ ) ( ๐ฏ โ‹… ๐ฏ ) ] + ( 1 - ฮณ u ) c 2 ( ฮณ u + 1 ) [ ( ๐ฎ โ‹… ๐ฏ ) 2 - ( ๐ฎ โ‹… ๐ฎ ) ( ๐ฏ โ‹… ๐ฏ ) ] = ( ๐ฎ + ๐ฏ ) 2 + 1 c 2 ฮณ u + 1 ฮณ u + 1 [ ( ๐ฎ โ‹… ๐ฏ ) 2 - ( ๐ฎ โ‹… ๐ฎ ) ( ๐ฏ โ‹… ๐ฏ ) ] = ( ๐ฎ + ๐ฏ ) 2 - 1 c 2 | ๐ฎ ร— ๐ฏ | 2 \begin{aligned}\displaystyle\left(1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}}% \right)^{2}|\mathbf{u}\oplus\mathbf{v}|^{2}&\displaystyle=\left[\mathbf{u}+% \mathbf{v}+\frac{1}{c^{2}}\frac{\gamma_{u}}{1+\gamma_{u}}\mathbf{u}\times(% \mathbf{u}\times\mathbf{v})\right]^{2}\\ &\displaystyle=(\mathbf{u}+\mathbf{v})^{2}+2\frac{1}{c^{2}}\frac{\gamma_{u}}{% \gamma_{u}+1}\left[(\mathbf{u}\cdot\mathbf{v})^{2}-(\mathbf{u}\cdot\mathbf{u})% (\mathbf{v}\cdot\mathbf{v})\right]+\frac{1}{c^{4}}\left(\frac{\gamma_{u}}{% \gamma_{u}+1}\right)^{2}\left[(\mathbf{u}\cdot\mathbf{u})^{2}(\mathbf{v}\cdot% \mathbf{v})-(\mathbf{u}\cdot\mathbf{v})^{2}(\mathbf{u}\cdot\mathbf{u})\right]% \\ &\displaystyle=(\mathbf{u}+\mathbf{v})^{2}+2\frac{1}{c^{2}}\frac{\gamma_{u}}{% \gamma_{u}+1}\left[(\mathbf{u}\cdot\mathbf{v})^{2}-(\mathbf{u}\cdot\mathbf{u})% (\mathbf{v}\cdot\mathbf{v})\right]+\frac{u^{2}}{c^{4}}\left(\frac{\gamma_{u}}{% \gamma_{u}+1}\right)^{2}\left[(\mathbf{u}\cdot\mathbf{u})(\mathbf{v}\cdot% \mathbf{v})-(\mathbf{u}\cdot\mathbf{v})^{2}\right]\\ &\displaystyle=(\mathbf{u}+\mathbf{v})^{2}+2\frac{1}{c^{2}}\frac{\gamma_{u}}{% \gamma_{u}+1}\left[(\mathbf{u}\cdot\mathbf{v})^{2}-(\mathbf{u}\cdot\mathbf{u})% (\mathbf{v}\cdot\mathbf{v})\right]+\frac{(1-\alpha_{u})(1+\alpha_{u})}{c^{2}}% \left(\frac{\gamma_{u}}{\gamma_{u}+1}\right)^{2}\left[(\mathbf{u}\cdot\mathbf{% u})(\mathbf{v}\cdot\mathbf{v})-(\mathbf{u}\cdot\mathbf{v})^{2}\right]\\ &\displaystyle=(\mathbf{u}+\mathbf{v})^{2}+2\frac{1}{c^{2}}\frac{\gamma_{u}}{% \gamma_{u}+1}\left[(\mathbf{u}\cdot\mathbf{v})^{2}-(\mathbf{u}\cdot\mathbf{u})% (\mathbf{v}\cdot\mathbf{v})\right]+\frac{(\gamma_{u}-1)}{c^{2}(\gamma_{u}+1)}% \left[(\mathbf{u}\cdot\mathbf{u})(\mathbf{v}\cdot\mathbf{v})-(\mathbf{u}\cdot% \mathbf{v})^{2}\right]\\ &\displaystyle=(\mathbf{u}+\mathbf{v})^{2}+2\frac{1}{c^{2}}\frac{\gamma_{u}}{% \gamma_{u}+1}\left[(\mathbf{u}\cdot\mathbf{v})^{2}-(\mathbf{u}\cdot\mathbf{u})% (\mathbf{v}\cdot\mathbf{v})\right]+\frac{(1-\gamma_{u})}{c^{2}(\gamma_{u}+1)}% \left[(\mathbf{u}\cdot\mathbf{v})^{2}-(\mathbf{u}\cdot\mathbf{u})(\mathbf{v}% \cdot\mathbf{v})\right]\\ &\displaystyle=(\mathbf{u}+\mathbf{v})^{2}+\frac{1}{c^{2}}\frac{\gamma_{u}+1}{% \gamma_{u}+1}\left[(\mathbf{u}\cdot\mathbf{v})^{2}-(\mathbf{u}\cdot\mathbf{u})% (\mathbf{v}\cdot\mathbf{v})\right]\\ &\displaystyle=(\mathbf{u}+\mathbf{v})^{2}-\frac{1}{c^{2}}|\mathbf{u}\times% \mathbf{v}|^{2}\end{aligned}
  71. ฮณ ๐ฎ โŠ• ๐ฏ = [ 1 - 1 c 2 1 ( 1 + ๐ฎ โ‹… ๐ฏ c 2 ) 2 ( ( ๐ฎ + ๐ฏ ) 2 - 1 c 2 ( u 2 v 2 - ( ๐ฎ โ‹… ๐ฏ ) 2 ) ) ] - 1 2 = ฮณ u ฮณ v ( 1 + ๐ฎ โ‹… ๐ฏ c 2 ) . \gamma_{\mathbf{u}\oplus\mathbf{v}}=\left[1-\frac{1}{c^{2}}\frac{1}{(1+\frac{% \mathbf{u}\cdot\mathbf{v}}{c^{2}})^{2}}\left((\mathbf{u}+\mathbf{v})^{2}-\frac% {1}{c^{2}}(u^{2}v^{2}-(\mathbf{u}\cdot\mathbf{v})^{2})\right)\right]^{-\frac{1% }{2}}=\gamma_{u}\gamma_{v}\left(1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}}% \right).
  72. ฮณ ๐ฎ โŠ• ๐ฏ = [ c 2 ( 1 + ๐ฎ โ‹… ๐ฏ c 2 ) 2 c 2 ( 1 + ๐ฎ โ‹… ๐ฏ c 2 ) 2 - 1 c 2 ( ๐ฎ + ๐ฏ ) 2 - 1 c 2 ( u 2 v 2 - ( ๐ฎ โ‹… ๐ฏ ) 2 ) ( 1 + ๐ฎ โ‹… ๐ฏ c 2 ) 2 ] - 1 2 = [ c 2 ( 1 + ๐ฎ โ‹… ๐ฏ c 2 ) 2 - ( ๐ฎ + ๐ฏ ) 2 + 1 c 2 ( u 2 v 2 - ( ๐ฎ โ‹… ๐ฏ ) 2 ) c 2 ( 1 + ๐ฎ โ‹… ๐ฏ c 2 ) 2 ] - 1 2 = [ c 2 ( 1 + 2 ๐ฎ โ‹… ๐ฏ c 2 + ( ๐ฎ โ‹… ๐ฏ ) 2 c 4 ) - u 2 - v 2 - 2 ( ๐ฎ โ‹… ๐ฏ ) + 1 c 2 ( u 2 v 2 - ( ๐ฎ โ‹… ๐ฏ ) 2 ) c 2 ( 1 + ๐ฎ โ‹… ๐ฏ c 2 ) 2 ] - 1 2 = [ 1 + 2 ๐ฎ โ‹… ๐ฏ c 2 + ( ๐ฎ โ‹… ๐ฏ ) 2 c 4 - u 2 c 2 - v 2 c 2 - 2 c 2 ( ๐ฎ โ‹… ๐ฏ ) + 1 c 4 ( u 2 v 2 - ( ๐ฎ โ‹… ๐ฏ ) 2 ) ( 1 + ๐ฎ โ‹… ๐ฏ c 2 ) 2 ] - 1 2 = [ 1 + ( ๐ฎ โ‹… ๐ฏ ) 2 c 4 - u 2 c 2 - v 2 c 2 + 1 c 4 ( u 2 v 2 - ( ๐ฎ โ‹… ๐ฏ ) 2 ) ( 1 + ๐ฎ โ‹… ๐ฏ c 2 ) 2 ] - 1 2 = [ ( 1 - u 2 c 2 ) ( 1 - v 2 c 2 ) ( 1 + ๐ฎ โ‹… ๐ฏ c 2 ) 2 ] - 1 2 = [ 1 ฮณ u 2 ฮณ v 2 ( 1 + ๐ฎ โ‹… ๐ฏ c 2 ) 2 ] - 1 2 = ฮณ u ฮณ v ( 1 + ๐ฎ โ‹… ๐ฏ c 2 ) \begin{aligned}\displaystyle\gamma_{\mathbf{u}\oplus\mathbf{v}}&\displaystyle=% \left[\frac{c^{2}(1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}})^{2}}{c^{2}(1+% \frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}})^{2}}-\frac{1}{c^{2}}\frac{(\mathbf{u}% +\mathbf{v})^{2}-\frac{1}{c^{2}}(u^{2}v^{2}-(\mathbf{u}\cdot\mathbf{v})^{2})}{% (1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}})^{2}}\right]^{-\frac{1}{2}}\\ &\displaystyle=\left[\frac{c^{2}(1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}})^{2% }-(\mathbf{u}+\mathbf{v})^{2}+\frac{1}{c^{2}}(u^{2}v^{2}-(\mathbf{u}\cdot% \mathbf{v})^{2})}{c^{2}(1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}})^{2}}\right]% ^{-\frac{1}{2}}\\ &\displaystyle=\left[\frac{c^{2}(1+2\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}}+% \frac{(\mathbf{u}\cdot\mathbf{v})^{2}}{c^{4}})-u^{2}-v^{2}-2(\mathbf{u}\cdot% \mathbf{v})+\frac{1}{c^{2}}(u^{2}v^{2}-(\mathbf{u}\cdot\mathbf{v})^{2})}{c^{2}% (1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}})^{2}}\right]^{-\frac{1}{2}}\\ &\displaystyle=\left[\frac{1+2\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}}+\frac{(% \mathbf{u}\cdot\mathbf{v})^{2}}{c^{4}}-\frac{u^{2}}{c^{2}}-\frac{v^{2}}{c^{2}}% -\frac{2}{c^{2}}(\mathbf{u}\cdot\mathbf{v})+\frac{1}{c^{4}}(u^{2}v^{2}-(% \mathbf{u}\cdot\mathbf{v})^{2})}{(1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}})^{% 2}}\right]^{-\frac{1}{2}}\\ &\displaystyle=\left[\frac{1+\frac{(\mathbf{u}\cdot\mathbf{v})^{2}}{c^{4}}-% \frac{u^{2}}{c^{2}}-\frac{v^{2}}{c^{2}}+\frac{1}{c^{4}}(u^{2}v^{2}-(\mathbf{u}% \cdot\mathbf{v})^{2})}{(1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}})^{2}}\right]% ^{-\frac{1}{2}}\\ &\displaystyle=\left[\frac{(1-\frac{u^{2}}{c^{2}})(1-\frac{v^{2}}{c^{2}})}{(1+% \frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}})^{2}}\right]^{-\frac{1}{2}}\\ &\displaystyle=\left[\frac{1}{\gamma_{u}^{2}\gamma_{v}^{2}(1+\frac{\mathbf{u}% \cdot\mathbf{v}}{c^{2}})^{2}}\right]^{-\frac{1}{2}}\\ &\displaystyle=\gamma_{u}\gamma_{v}(1+\frac{\mathbf{u}\cdot\mathbf{v}}{c^{2}})% \end{aligned}
  73. m m
  74. V V
  75. x x
  76. m m
  77. n n
  78. c m = V + c m โ€ฒ 1 + V c m โ€ฒ c 2 = V + c n 1 + V c n c 2 = c n 1 + n V c 1 + V n c = c n ( 1 + n V c ) 1 1 + V n c = ( c n + V ) ( 1 - V n c + ( V n c ) 2 - โ‹ฏ ) . \begin{aligned}\displaystyle c_{m}&\displaystyle=\frac{V+c_{m}^{\prime}}{1+% \frac{Vc_{m}^{\prime}}{c^{2}}}=\frac{V+\frac{c}{n}}{1+\frac{Vc}{nc^{2}}}=\frac% {c}{n}\frac{1+\frac{nV}{c}}{1+\frac{V}{nc}}\\ &\displaystyle=\frac{c}{n}(1+\frac{nV}{c})\frac{1}{1+\frac{V}{nc}}=(\frac{c}{n% }+V)\left(1-\frac{V}{nc}+\left(\frac{V}{nc}\right)^{2}-\cdots\right).\end{aligned}
  79. c m = c n + V ( 1 - 1 n 2 - V n c + โ‹ฏ ) . c_{m}=\frac{c}{n}+V(1-\frac{1}{n^{2}}-\frac{V}{nc}+\cdots).
  80. v โ€ฒ = v = c vโ€ฒ=v=c
  81. t a n ฮธ tanฮธ
  82. tan ฮธ = 1 - V 2 c 2 c sin ฮธ โ€ฒ c cos ฮธ โ€ฒ + V = 1 - V 2 c 2 sin ฮธ โ€ฒ cos ฮธ โ€ฒ + V c . \tan\theta=\frac{\sqrt{1-\frac{V^{2}}{c^{2}}}c\sin\theta^{\prime}}{c\cos\theta% ^{\prime}+V}=\frac{\sqrt{1-\frac{V^{2}}{c^{2}}}\sin\theta^{\prime}}{\cos\theta% ^{\prime}+\frac{V}{c}}.
  83. s i n ฮธ sinฮธ
  84. c o s ฮธ cosฮธ
  85. sin ฮธ = 1 - V 2 c 2 sin ฮธ โ€ฒ 1 + V c cos ฮธ โ€ฒ , \begin{aligned}\displaystyle\sin\theta&\displaystyle=\frac{\sqrt{1-\frac{V^{2}% }{c^{2}}}\sin\theta^{\prime}}{1+\frac{V}{c}\cos\theta^{\prime}},\end{aligned}
  86. v y v = 1 - V 2 c 2 v y โ€ฒ 1 + V c 2 v x โ€ฒ v โ€ฒ 2 + V 2 + 2 V v โ€ฒ cos ฮธ โ€ฒ - ( V v โ€ฒ sin ฮธ โ€ฒ c ) 2 1 + V c 2 v โ€ฒ cos ฮธ โ€ฒ = c 1 - V 2 c 2 sin ฮธ โ€ฒ c 2 + V 2 + 2 V c cos ฮธ โ€ฒ - V 2 sin 2 ฮธ โ€ฒ = c 1 - V 2 c 2 sin ฮธ โ€ฒ c 2 + V 2 + 2 V c cos ฮธ โ€ฒ - V 2 ( 1 - cos 2 ฮธ โ€ฒ ) = c 1 - V 2 c 2 sin ฮธ โ€ฒ c 2 + 2 V c cos ฮธ โ€ฒ + V 2 cos 2 ฮธ โ€ฒ = 1 - V 2 c 2 sin ฮธ โ€ฒ 1 + V c cos ฮธ โ€ฒ , \begin{aligned}\displaystyle\frac{v_{y}}{v}&\displaystyle=\frac{\frac{\sqrt{1-% \frac{V^{2}}{c^{2}}}v_{y}^{\prime}}{1+\frac{V}{c^{2}}v_{x}^{\prime}}}{\frac{% \sqrt{v^{\prime 2}+V^{2}+2Vv^{\prime}\cos\theta^{\prime}-(\frac{Vv^{\prime}% \sin\theta^{\prime}}{c})^{2}}}{1+\frac{V}{c^{2}}v^{\prime}\cos\theta^{\prime}}% }\\ &\displaystyle=\frac{c\sqrt{1-\frac{V^{2}}{c^{2}}}\sin\theta^{\prime}}{\sqrt{c% ^{2}+V^{2}+2Vc\cos\theta^{\prime}-V^{2}\sin^{2}\theta^{\prime}}}\\ &\displaystyle=\frac{c\sqrt{1-\frac{V^{2}}{c^{2}}}\sin\theta^{\prime}}{\sqrt{c% ^{2}+V^{2}+2Vc\cos\theta^{\prime}-V^{2}(1-\cos^{2}\theta^{\prime})}}=\frac{c% \sqrt{1-\frac{V^{2}}{c^{2}}}\sin\theta^{\prime}}{\sqrt{c^{2}+2Vc\cos\theta^{% \prime}+V^{2}\cos^{2}\theta^{\prime}}}\\ &\displaystyle=\frac{\sqrt{1-\frac{V^{2}}{c^{2}}}\sin\theta^{\prime}}{1+\frac{% V}{c}\cos\theta^{\prime}},\end{aligned}
  87. cos ฮธ = V c + cos ฮธ โ€ฒ 1 + V c cos ฮธ โ€ฒ , \cos\theta=\frac{\frac{V}{c}+\cos\theta^{\prime}}{1+\frac{V}{c}\cos\theta^{% \prime}},
  88. c o s cos
  89. s i n sin
  90. sin ฮธ - sin ฮธ โ€ฒ = sin ฮธ โ€ฒ ( 1 - V 2 c 2 1 + V c cos ฮธ โ€ฒ - 1 ) โ‰ˆ sin ฮธ โ€ฒ ( 1 - V c cos ฮธ โ€ฒ - 1 ) = - V c sin ฮธ โ€ฒ cos ฮธ โ€ฒ , \begin{aligned}\displaystyle\sin\theta-\sin\theta^{\prime}&\displaystyle=\sin% \theta^{\prime}\left(\frac{\sqrt{1-\frac{V^{2}}{c^{2}}}}{1+\frac{V}{c}\cos% \theta^{\prime}}-1\right)\\ &\displaystyle\approx\sin\theta^{\prime}\left(1-\frac{V}{c}\cos\theta^{\prime}% -1\right)=-\frac{V}{c}\sin\theta^{\prime}\cos\theta^{\prime},\end{aligned}
  91. v / c {v}/{c}
  92. sin ฮธ โ€ฒ - sin ฮธ = 2 sin 1 2 ( ฮธ โ€ฒ - ฮธ ) cos 1 2 ( ฮธ + ฮธ โ€ฒ ) โ‰ˆ ( ฮธ โ€ฒ - ฮธ ) cos ฮธ โ€ฒ , \begin{aligned}\displaystyle\sin\theta^{\prime}-\sin\theta&\displaystyle=2\sin% \frac{1}{2}(\theta^{\prime}-\theta)\cos\frac{1}{2}(\theta+\theta^{\prime})% \approx(\theta^{\prime}-\theta)\cos\theta^{\prime},\end{aligned}
  93. c o s 1 2 ( ฮธ + ฮธ โ€ฒ ) โ‰ˆ c o s ฮธ โ€ฒ , s i n 1 2 ( ฮธ โˆ’ ฮธ โ€ฒ ) โ‰ˆ 1 2 ( ฮธ โˆ’ ฮธ โ€ฒ ) cos\frac{1}{2}(ฮธ+ฮธโ€ฒ)โ‰ˆcosฮธโ€ฒ,sin\frac{1}{2}(ฮธโˆ’ฮธโ€ฒ)โ‰ˆ\frac{1}{2}(ฮธโˆ’ฮธโ€ฒ)
  94. ฮ” ฮธ โ‰ก ฮธ โ€ฒ - ฮธ = V c sin ฮธ โ€ฒ , \Delta\theta\equiv\theta^{\prime}-\theta=\frac{V}{c}\sin\theta^{\prime},
  95. V / c โ†’ 0 {V}/{c}โ†’0
  96. ฮป = - s T + V T = ( - s + V ) T \lambda=-sT+VT=(-s+V)T
  97. T T
  98. ฯ„ ฯ„
  99. ฮฝ = 1 / ฯ„ ฮฝ={1}/{ฯ„}
  100. s = s โ€ฒ = - c , s=s^{\prime}=-c,
  101. ฮฝ = - s ฮป = - s ( V - s ) T = c ( V + c ) ฮณ V T โ€ฒ = ฮฝ โ€ฒ c 1 - V 2 c 2 c + V = ฮฝ โ€ฒ 1 - ฮฒ 1 + ฮฒ . \nu={-s\over\lambda}={-s\over(V-s)T}={c\over(V+c)\gamma_{V}T^{\prime}}=\nu^{% \prime}\frac{c\sqrt{1-{V^{2}\over c^{2}}}}{c+V}=\nu^{\prime}\sqrt{\frac{1-% \beta}{1+\beta}}\,.
  102. c c
  103. s โ€ฒ sโ€ฒ
  104. s โ€ฒ โ‰  s sโ€ฒโ‰ s
  105. s s
  106. s = s โ€ฒ + V 1 + s โ€ฒ V c 2 . s=\frac{s^{\prime}+V}{1+{s^{\prime}V\over c^{2}}}.
  107. ฯ„ ฯ„
  108. ฮฝ ฮฝ
  109. L - s = ( - s โ€ฒ - V 1 + s โ€ฒ V c 2 + V ) T - s โ€ฒ - V 1 + s โ€ฒ V c 2 = ฮณ V ฮฝ โ€ฒ - s โ€ฒ - V + V ( 1 + s โ€ฒ V c 2 ) - s โ€ฒ - V = ฮณ V ฮฝ โ€ฒ ( s โ€ฒ ( 1 - V 2 c 2 ) s โ€ฒ + V ) = ฮณ V ฮฝ โ€ฒ ( s โ€ฒ ฮณ - 2 s โ€ฒ + V ) = 1 ฮณ V ฮฝ โ€ฒ ( 1 1 + V s โ€ฒ ) . \begin{aligned}\displaystyle{L\over-s}&\displaystyle=\frac{\left(\frac{-s^{% \prime}-V}{1+{s^{\prime}V\over c^{2}}}+V\right)T}{\frac{-s^{\prime}-V}{1+{s^{% \prime}V\over c^{2}}}}\\ &\displaystyle={\gamma_{V}\over\nu^{\prime}}\frac{-s^{\prime}-V+V(1+{s^{\prime% }V\over c^{2}})}{-s^{\prime}-V}\\ &\displaystyle={\gamma_{V}\over\nu^{\prime}}\left(\frac{s^{\prime}\left(1-{V^{% 2}\over c^{2}}\right)}{s^{\prime}+V}\right)\\ &\displaystyle={\gamma_{V}\over\nu^{\prime}}\left(\frac{s^{\prime}\gamma^{-2}}% {s^{\prime}+V}\right)\\ &\displaystyle={1\over\gamma_{V}\nu^{\prime}}\left(\frac{1}{1+{V\over s^{% \prime}}}\right).\\ \end{aligned}
  110. s โ€ฒ sโ€ฒ
  111. s โ€ฒ sโ€ฒ
  112. โˆ’ s > V โˆ’s>V
  113. s = โˆ’ c s=โˆ’c
  114. x x
  115. x x
  116. s โ€ฒ = โˆ’ c sโ€ฒ=โˆ’c
  117. ฮฝ = ฮฝ โ€ฒ ฮณ V ( 1 - ฮฒ ) = ฮฝ โ€ฒ 1 - ฮฒ 1 - ฮฒ 1 + ฮฒ = ฮฝ โ€ฒ 1 - ฮฒ 1 + ฮฒ . \nu=\nu^{\prime}\gamma_{V}(1-\beta)=\nu^{\prime}\frac{1-\beta}{\sqrt{1-\beta}% \sqrt{1+\beta}}=\nu^{\prime}\sqrt{\frac{1-\beta}{1+\beta}}\,.
  118. s โ€ฒ sโ€ฒ
  119. ฮฝ = ฮณ V ฮฝ โ€ฒ ( 1 + V s โ€ฒ cos ฮธ ) \nu=\gamma_{V}\nu^{\prime}\left(1+\frac{V}{s^{\prime}}\cos\theta\right)
  120. ฮธ ฮธ
  121. ฮธ = 0 ฮธ=0
  122. ฮธ = ฯ€ / 2 ฮธ=ฯ€/2
  123. tanh ( a + b ) = tanh a + tanh b 1 + tanh a tanh b \tanh(a+b)={\tanh a+\tanh b\over 1+\tanh a\tanh b}
  124. v c = tanh a , u c = tanh b , s c = tanh ( a + b ) , {v\over c}=\tanh a\ ,\quad{u\over c}=\tanh b\ ,\quad\,{s\over c}=\tanh(a+b),
  125. b b
  126. b b
  127. c c
  128. x x
  129. x x
  130. a a
  131. a a
  132. ๐”ฐ ๐”ฌ ( 3 , 1 ) โŠƒ span { K 1 , K 2 , K 3 } โ‰ˆ โ„ 3 โˆ‹ s y m b o l ฮถ = s y m b o l ฮฒ ^ tanh - 1 ฮฒ , s y m b o l ฮฒ โˆˆ ๐”น 3 , \mathfrak{so}(3,1)\supset\mathrm{span}\{K_{1},K_{2},K_{3}\}\approx\mathbb{R}^{% 3}\ni symbol{\zeta}=symbol{\hat{\beta}}\tanh^{-1}\beta,\quad symbol{\beta}\in% \mathbb{B}^{3},
  133. ฮถ \mathbf{ฮถ}
  134. d l s y m b o l ฮถ 2 = d ฮถ 2 - ( s y m b o l ฮถ ร— d s y m b o l ฮถ ) 2 ( 1 - ฮถ 2 ) 2 = d ฮถ 2 1 - ฮถ 2 ( d ฮธ 2 + sin 2 ฮธ d ฯ† 2 ) . dl_{s}ymbol{\zeta}^{2}=\frac{d\mathbf{\zeta}^{2}-(symbol\zeta\times dsymbol% \zeta)^{2}}{(1-\zeta^{2})^{2}}=\frac{d\zeta^{2}}{1-\zeta^{2}}(d\theta^{2}+\sin% ^{2}\theta d\varphi^{2}).
  135. ฮฒ ฮฒ
  136. ฮถ = | s y m b o l ฮถ | = tanh - 1 ฮฒ , \zeta=|symbol\zeta|=\tanh^{-1}\beta,
  137. ฮธ ฮธ
  138. ฯ† ฯ†
  139. d l s y m b o l ฮถ 2 = d ฮฒ 2 + sinh 2 ฮฒ ( d ฮธ 2 + sin 2 ฮธ d ฯ† 2 ) . dl_{symbol\zeta}^{2}=d\beta^{2}+\sinh^{2}\beta(d\theta^{2}+\sin^{2}\theta d% \varphi^{2}).
  140. ฮฑ < s u b > u ฮฑ<sub>u

Velocity_of_money.html

  1. 2 / year 2/\,\text{year}
  2. V T = P T M V_{T}=\frac{PT}{M}
  3. V T V_{T}\,
  4. T T\,
  5. P P\,
  6. M M\,
  7. P T PT
  8. P T PT
  9. M M
  10. V T V_{T}
  11. V = P Q M V=\frac{PQ}{M}
  12. V V\,
  13. P Q PQ\,

Veronese_surface.html

  1. ฮฝ : โ„™ 2 โ†’ โ„™ 5 \nu:\mathbb{P}^{2}\to\mathbb{P}^{5}
  2. ฮฝ : [ x : y : z ] โ†ฆ [ x 2 : y 2 : z 2 : y z : x z : x y ] \nu:[x:y:z]\mapsto[x^{2}:y^{2}:z^{2}:yz:xz:xy]
  3. [ x : โ‹ฏ ] [x:\cdots]
  4. ฮฝ \nu
  5. A x 2 + B x y + C y 2 + D x z + E y z + F z 2 = 0. Ax^{2}+Bxy+Cy^{2}+Dxz+Eyz+Fz^{2}=0.
  6. ( A , B , C , D , E , F ) (A,B,C,D,E,F)
  7. ( x , y , z ) (x,y,z)
  8. [ x : y : z ] , [x:y:z],
  9. ฮฝ d : โ„™ n โ†’ โ„™ m \nu_{d}\colon\mathbb{P}^{n}\to\mathbb{P}^{m}
  10. m = ( ( n + 1 d ) ) - 1 = ( n + d d ) - 1 = 1 n ! ( d + 1 ) ( n ) - 1. m=\left(\!\!{n+1\choose d}\!\!\right)-1={n+d\choose d}-1=\frac{1}{n!}(d+1)^{(n% )}-1.
  11. [ x 0 : โ€ฆ : x n ] [x_{0}:\ldots:x_{n}]
  12. + 1 +1
  13. - 1 -1
  14. 1 / n ! . 1/n!.
  15. d = 0 d=0
  16. ๐ 0 , \mathbf{P}^{0},
  17. d = 1 d=1
  18. ๐ n , \mathbf{P}^{n},
  19. ฮฝ d : โ„™ V โ†’ โ„™ ( Sym d V ) \nu_{d}:\mathbb{P}V\to\mathbb{P}(\rm{Sym}^{d}V)
  20. Sym d V \rm{Sym}^{d}V
  21. n = 1 , n=1,
  22. n = 1 , d = 1 n=1,d=1
  23. n = 1 , d = 2 , n=1,d=2,
  24. [ x 2 : x y : y 2 ] , [x^{2}:xy:y^{2}],
  25. ( x , x 2 ) . (x,x^{2}).
  26. n = 1 , d = 3 , n=1,d=3,
  27. [ x 3 : x 2 y : x y 2 : y 3 ] , [x^{3}:x^{2}y:xy^{2}:y^{3}],
  28. ( x , x 2 , x 3 ) . (x,x^{2},x^{3}).

Vertex_function.html

  1. ฯˆ \psi~{}
  2. ฯˆ ยฏ \bar{\psi}
  3. ฮ“ ฮผ = - 1 e ฮด 3 S eff ฮด ฯˆ ยฏ ฮด ฯˆ ฮด A ฮผ \Gamma^{\mu}=-{1\over e}{\delta^{3}S_{\mathrm{eff}}\over\delta\bar{\psi}\delta% \psi\delta A_{\mu}}
  4. ฮ“ ฮผ = ฮณ ฮผ F 1 ( q 2 ) + i ฯƒ ฮผ ฮฝ q ฮฝ 2 m F 2 ( q 2 ) \Gamma^{\mu}=\gamma^{\mu}F_{1}(q^{2})+\frac{i\sigma^{\mu\nu}q_{\nu}}{2m}F_{2}(% q^{2})
  5. ฯƒ ฮผ ฮฝ = ( i / 2 ) [ ฮณ ฮผ , ฮณ ฮฝ ] \sigma^{\mu\nu}=(i/2)[\gamma^{\mu},\gamma^{\nu}]
  6. q ฮฝ q_{\nu}
  7. a = g - 2 2 = F 2 ( 0 ) a=\frac{g-2}{2}=F_{2}(0)

Vertex_operator_algebra.html

  1. V V
  2. | 0 โŸฉ |0\rangle
  3. ฮฉ ฮฉ
  4. T : V โ†’ V T:Vโ†’V
  5. T T
  6. Y : V โŠ— V โ†’ V ( ( z ) ) Y:VโŠ—Vโ†’V((z))
  7. V ( ( z ) ) V((z))
  8. V V
  9. u โˆˆ V uโˆˆV
  10. Y ( u , z ) Y(u,z)
  11. u โŠ— v โ†ฆ Y ( u , z ) v = โˆ‘ n โˆˆ ๐™ u n v z - n - 1 u\otimes v\mapsto Y(u,z)v=\sum_{n\in\mathbf{Z}}u_{n}vz^{-n-1}
  12. T ( 1 ) = 0 T(1)=0
  13. u , v โˆˆ V u,vโˆˆV
  14. [ T , Y ( u , z ) ] v = T Y ( u , z ) v - Y ( u , z ) T v = d d z Y ( u , z ) v [T,Y(u,z)]v=TY(u,z)v-Y(u,z)Tv=\frac{d}{dz}Y(u,z)v
  15. u , v โˆˆ V u,vโˆˆV
  16. N N
  17. ( z - x ) N Y ( u , z ) Y ( v , x ) = ( z - x ) N Y ( v , x ) Y ( u , z ) . (z-x)^{N}Y(u,z)Y(v,x)=(z-x)^{N}Y(v,x)Y(u,z).
  18. โˆ€ u , v , w โˆˆ V : z - 1 ฮด ( y - x z ) Y ( u , x ) Y ( v , y ) w - z - 1 ฮด ( - y + x z ) Y ( v , y ) Y ( u , x ) w = y - 1 ฮด ( x + z y ) Y ( Y ( u , z ) v , y ) w , \forall u,v,w\in V:\qquad z^{-1}\delta\left(\frac{y-x}{z}\right)Y(u,x)Y(v,y)w-% z^{-1}\delta\left(\frac{-y+x}{z}\right)Y(v,y)Y(u,x)w=y^{-1}\delta\left(\frac{x% +z}{y}\right)Y(Y(u,z)v,y)w,
  19. ฮด ( y - x z ) := โˆ‘ s โ‰ฅ 0 , r โˆˆ ๐™ ( r s ) ( - 1 ) s y r - s x s z - r . \delta\left(\frac{y-x}{z}\right):=\sum_{s\geq 0,r\in\mathbf{Z}}{\left({{r}% \atop{s}}\right)}(-1)^{s}y^{r-s}x^{s}z^{-r}.
  20. ( u m ( v ) ) n ( w ) = โˆ‘ i โ‰ฅ 0 ( - 1 ) i ( m i ) ( u m - i ( v n + i ( w ) ) - ( - 1 ) m v m + n - i ( u i ( w ) ) ) (u_{m}(v))_{n}(w)=\sum_{i\geq 0}(-1)^{i}{\left({{m}\atop{i}}\right)}\left(u_{m% -i}(v_{n+i}(w))-(-1)^{m}v_{m+n-i}(u_{i}(w))\right)
  21. โˆ‘ i โˆˆ ๐™ ( u q + i ( v ) ) m + n - i ( w ) = โˆ‘ i โˆˆ ๐™ ( - 1 ) i ( q i ) ( u m + q - i ( v n + i ( w ) ) - ( - 1 ) q v n + q - i ( u m + i ( w ) ) ) \sum_{i\in\mathbf{Z}}\left(u_{q+i}(v)\right)_{m+n-i}(w)=\sum_{i\in\mathbf{Z}}(% -1)^{i}{\left({{q}\atop{i}}\right)}\left(u_{m+q-i}\left(v_{n+i}(w)\right)-(-1)% ^{q}v_{n+q-i}\left(u_{m+i}(w)\right)\right)
  22. u , v , w โˆˆ V u,v,wโˆˆV
  23. X ( u , v , w ; z , x ) โˆˆ V [ [ z , x ] ] [ z - 1 , x - 1 , ( z - x ) - 1 ] X(u,v,w;z,x)\in V[[z,x]]\left[z^{-1},x^{-1},(z-x)^{-1}\right]
  24. Y ( u , z ) Y ( v , x ) w Y(u,z)Y(v,x)w
  25. Y ( v , x ) Y ( u , z ) w Y(v,x)Y(u,z)w
  26. X ( u , v , w ; z , x ) X(u,v,w;z,x)
  27. V ( ( z ) ) ( ( x ) ) V((z))((x))
  28. V ( ( x ) ) ( ( z ) ) V((x))((z))
  29. ฯ‰ ฯ‰
  30. Y ( ฯ‰ , z ) Y(ฯ‰,z)
  31. L ( z ) L(z)
  32. Y ( ฯ‰ , z ) = โˆ‘ n โˆˆ ๐™ ฯ‰ n z - n - 1 = L ( z ) = โˆ‘ n โˆˆ ๐™ L n z - n - 2 Y(\omega,z)=\sum_{n\in\mathbf{Z}}\omega_{n}{z^{-n-1}}=L(z)=\sum_{n\in\mathbf{Z% }}L_{n}z^{-n-2}
  33. V V
  34. V V
  35. V V
  36. V V
  37. d e g ( u ) + d e g ( v ) โˆ’ n โˆ’ 1 deg(u)+deg(v)โˆ’nโˆ’1
  38. ฯ‰ ฯ‰
  39. [ L m , Y ( u , z ) ] = โˆ‘ k = 0 m + 1 ( m + 1 k ) z k Y ( L m - k u , z ) [L_{m},Y(u,z)]=\sum_{k=0}^{m+1}{\left({{m+1}\atop{k}}\right)}z^{k}Y(L_{m-k}u,z)
  40. X ( u , v , w ; z , x ) โˆˆ V [ [ z , x ] ] [ z - 1 , x - 1 , ( z - x ) - 1 ] X(u,v,w;z,x)\in V[[z,x]][z^{-1},x^{-1},(z-x)^{-1}]
  41. Y ( J n 1 + 1 a 1 J n 2 + 1 a 2 โ€ฆ J n k + 1 a k 1 , z ) = : โˆ‚ n 1 โˆ‚ z n 1 J a 1 ( z ) n 1 ! โˆ‚ n 2 โˆ‚ z n 2 J a 2 ( z ) n 2 ! โ‹ฏ โˆ‚ n k โˆ‚ z n k J a k ( z ) n k ! : Y(J^{a_{1}}_{n_{1}+1}J^{a_{2}}_{n_{2}+1}...J^{a_{k}}_{n_{k}+1}1,z)=:\frac{% \partial^{n_{1}}}{\partial_{z}^{n_{1}}}\frac{J^{a_{1}}(z)}{n_{1}!}\frac{% \partial^{n_{2}}}{\partial_{z}^{n_{2}}}\frac{J^{a_{2}}(z)}{n_{2}!}\cdots\frac{% \partial^{n_{k}}}{\partial_{z}^{n_{k}}}\frac{J^{a_{k}}(z)}{n_{k}!}:
  42. Y ( x n 1 + 1 x n 2 + 1 x n 3 + 1 โ€ฆ x n k + 1 , z ) โ‰ก 1 n 1 ! n 2 ! . . n k ! : โˆ‚ n 1 b ( z ) โˆ‚ n 2 b ( z ) โ€ฆ โˆ‚ n k b ( z ) : Y(x_{n_{1}+1}x_{n_{2}+1}x_{n_{3}+1}...x_{n_{k}+1},z)\equiv\frac{1}{n_{1}!n_{2}% !..n_{k}!}:\partial^{n_{1}}b(z)\partial^{n_{2}}b(z)...\partial^{n_{k}}b(z):
  43. Y [ f , z ] โ‰ก : f ( b ( z ) 0 ! , b โ€ฒ ( z ) 1 ! , b โ€ฒโ€ฒ ( z ) 2 ! , โ€ฆ ) : Y[f,z]\equiv:f(\frac{b(z)}{0!},\frac{b^{\prime}(z)}{1!},\frac{b^{\prime\prime}% (z)}{2!},...):
  44. T r V q L 0 = โˆ‘ n โˆˆ ๐‘ dim V n q n = โˆ n โ‰ฅ 2 ( 1 - q n ) - 1 Tr_{V}q^{L_{0}}=\sum_{n\in\mathbf{R}}\dim V_{n}q^{n}=\prod_{n\geq 2}(1-q^{n})^% {-1}
  45. Y ( L - n 1 - 2 L - n 2 - 2 โ€ฆ L - n k - 2 | 0 โŸฉ , z ) โ‰ก 1 n 1 ! n 2 ! . . n k ! : โˆ‚ n 1 L ( z ) โˆ‚ n 2 L ( z ) โ€ฆ โˆ‚ n k L ( z ) : Y(L_{-n_{1}-2}L_{-n_{2}-2}...L_{-n_{k}-2}|0\rangle,z)\equiv\frac{1}{n_{1}!n_{2% }!..n_{k}!}:\partial^{n_{1}}L(z)\partial^{n_{2}}L(z)...\partial^{n_{k}}L(z):
  46. ฯ‰ = L - 2 | 0 โŸฉ \omega=L_{-2}|0\rangle
  47. [ L ( z ) , L ( x ) ] = ( โˆ‚ โˆ‚ x L ( x ) ) w - 1 ฮด ( z x ) - 2 L ( x ) x - 1 โˆ‚ โˆ‚ z ฮด ( z x ) - 1 12 c x - 1 ( โˆ‚ โˆ‚ z ) 3 ฮด ( z x ) [L(z),L(x)]=\left(\frac{\partial}{\partial x}L(x)\right)w^{-1}\delta\left(% \frac{z}{x}\right)-2L(x)x^{-1}\frac{\partial}{\partial z}\delta\left(\frac{z}{% x}\right)-\frac{1}{12}cx^{-1}\left(\frac{\partial}{\partial z}\right)^{3}% \delta\left(\frac{z}{x}\right)
  48. 0 โ†’ โ„‚ โ†’ ๐”ค ^ โ†’ ๐”ค [ t , t - 1 ] โ†’ 0 0\to\mathbb{C}\to\hat{\mathfrak{g}}\to\mathfrak{g}[t,t^{-1}]\to 0
  49. ๐”ค [ t ] โ†’ ๐”ค [ t , t - 1 ] \mathfrak{g}[t]\to\mathfrak{g}[t,t^{-1}]
  50. ๐”ค \mathfrak{g}
  51. J a ( z ) = โˆ‘ n = - โˆž โˆž J n a z - n - 1 = โˆ‘ n = - โˆž โˆž ( J a t n ) z - n - 1 . J^{a}(z)=\sum_{n=-\infty}^{\infty}J^{a}_{n}z^{-n-1}=\sum_{n=-\infty}^{\infty}(% J^{a}t^{n})z^{-n-1}.
  52. ฯ‰ = 1 2 ( k + h โˆจ ) โˆ‘ a J a , - 1 J - 1 a 1 \omega=\frac{1}{2(k+h^{\vee})}\sum_{a}J_{a,-1}J^{a}_{-1}1
  53. k โ‹… dim ๐”ค / ( k + h โˆจ ) k\cdot\dim\mathfrak{g}/(k+h^{\vee})
  54. T r V q L 0 = โˆ‘ n โˆˆ ๐™ dim V n q n = โˆ n โ‰ฅ 1 ( 1 - q n ) - 1 Tr_{V}q^{L_{0}}=\sum_{n\in\mathbf{Z}}\dim V_{n}q^{n}=\prod_{n\geq 1}(1-q^{n})^% {-1}
  55. โ„‹ โ‰… โŠ• i โˆˆ I M i โŠ— M i ยฏ \mathcal{H}\cong\bigoplus_{i\in I}M_{i}\otimes\overline{M_{i}}
  56. X ( u , v , w ; z , x ) โˆˆ M [ [ z , x ] ] [ z - 1 , x - 1 , ( z - x ) - 1 ] X(u,v,w;z,x)\in M[[z,x]][z^{-1},x^{-1},(z-x)^{-1}]
  57. X ( u , v , w ; z , x ) X(u,v,w;z,x)
  58. z - 1 ฮด ( y - x z ) Y M ( u , x ) Y M ( v , y ) w - z - 1 ฮด ( - y + x z ) Y M ( v , y ) Y M ( u , x ) w = y - 1 ฮด ( x + z y ) Y M ( Y ( u , z ) v , y ) w . z^{-1}\delta\left(\frac{y-x}{z}\right)Y^{M}(u,x)Y^{M}(v,y)w-z^{-1}\delta\left(% \frac{-y+x}{z}\right)Y^{M}(v,y)Y^{M}(u,x)w=y^{-1}\delta\left(\frac{x+z}{y}% \right)Y^{M}(Y(u,z)v,y)w.
  59. ฮ› ฮ›
  60. V ฮ› โ‰… โŠ• ฮป โˆˆ ฮ› V ฮป V_{\Lambda}\cong\bigoplus_{\lambda\in\Lambda}V_{\lambda}
  61. ฮฑ โˆˆ ฮ› ฮฑโˆˆฮ›
  62. ฮ› ฮ›
  63. ฮ› ฮ›
  64. ฮต ( ฮฑ , ฮฒ ) ฮต(ฮฑ,ฮฒ)
  65. ยฑ 1 ยฑ1
  66. ฮฑ โˆˆ ฮ› ฮฑโˆˆฮ›
  67. ฮ› ฮ›
  68. ฮต ฮต
  69. Y ( v ฮป , z ) = e ฮป : exp โˆซ ฮป ( z ) := e ฮป z ฮป exp ( โˆ‘ n < 0 ฮป n z - n n ) exp ( โˆ‘ n > 0 ฮป n z - n n ) , Y(v_{\lambda},z)=e_{\lambda}:\exp\int\lambda(z):=e_{\lambda}z^{\lambda}\exp% \left(\sum_{n<0}\lambda_{n}\frac{z^{-n}}{n}\right)\exp\left(\sum_{n>0}\lambda_% {n}\frac{z^{-n}}{n}\right),
  70. ฮ› โŠ— ๐‚ ฮ›โŠ—\mathbf{C}
  71. ( s , ฮป ) โˆˆ ๐™ (s,ฮป)โˆˆ\mathbf{Z}
  72. ฮ› ฮ›
  73. V = V + โŠ• V - V=V_{+}\oplus V_{-}
  74. G ( z ) = โˆ‘ n G n z - n - 3 / 2 G(z)=\sum_{n}G_{n}z^{-n-3/2}
  75. [ G m , L n ] = ( m - n / 2 ) G m + n [G_{m},L_{n}]=(m-n/2)G_{m+n}
  76. [ G m , G n ] = ( m - n ) L m + n + ฮด m , - n 4 m 2 + 1 12 c [G_{m},G_{n}]=(m-n)L_{m+n}+\delta_{m,-n}\frac{4m^{2}+1}{12}c
  77. c ^ = 2 3 c = 1 - 8 m ( m + 2 ) m โ‰ฅ 3 \hat{c}=\frac{2}{3}c=1-\frac{8}{m(m+2)}\quad m\geq 3
  78. Y ( ฯ„ , z ) = G ( z ) = โˆ‘ m โˆˆ โ„ค + 1 / 2 G n z - n - 3 / 2 , Y(\tau,z)=G(z)=\sum_{m\in\mathbb{Z}+1/2}G_{n}z^{-n-3/2},
  79. V โ™ฎ V^{\natural}
  80. V โ™ฎ V^{\natural}
  81. j * j * ( A โŠ  A ) โ†’ ฮ” * A j_{*}j^{*}(A\boxtimes A)\to\Delta_{*}A

Vertex_separator.html

  1. S โŠ‚ V S\subset V
  2. a a
  3. b b
  4. S S
  5. a a
  6. b b
  7. G - S G-S
  8. C 1 C_{1}
  9. C 2 C_{2}
  10. C 1 C_{1}
  11. C 2 C_{2}
  12. S โŠ‘ a , b G T S\sqsubseteq_{a,b}^{G}T
  13. x โˆˆ S โˆ– T x\in S\setminus T

Vibronic_coupling.html

  1. ๐Ÿ k โ€ฒ k โ‰ก โŸจ ฯ‡ k โ€ฒ ( ๐ซ ; ๐‘ ) | โˆ‡ ^ ๐‘ ฯ‡ k ( ๐ซ ; ๐‘ ) โŸฉ ( ๐ซ ) \mathbf{f}_{k^{\prime}k}\equiv\langle\,\chi_{k^{\prime}}(\mathbf{r};\mathbf{R}% )\,|\,\hat{\nabla}_{\mathbf{R}}\chi_{k}(\mathbf{r};\mathbf{R})\rangle_{(% \mathbf{r})}
  2. ( ๐Ÿ k โ€ฒ k ) l โ‰ˆ 1 d [ ฮณ k โ€ฒ k ( ๐‘ | ๐‘ + d ๐ž l ) - ฮณ k โ€ฒ k ( ๐‘ | ๐‘ ) ] (\mathbf{f}_{k^{\prime}k})_{l}\approx\frac{1}{d}\left[\gamma^{k^{\prime}k}(% \mathbf{R}|\mathbf{R}+d\mathbf{e}_{l})-\gamma^{k^{\prime}k}(\mathbf{R}|\mathbf% {R})\right]
  3. ( ๐Ÿ k โ€ฒ k ) l โ‰ˆ 1 2 d [ ฮณ k โ€ฒ k ( ๐‘ | ๐‘ + d ๐ž l ) - ฮณ k โ€ฒ k ( ๐‘ | ๐‘ - d ๐ž l ) ] (\mathbf{f}_{k^{\prime}k})_{l}\approx\frac{1}{2d}\left[\gamma^{k^{\prime}k}(% \mathbf{R}|\mathbf{R}+d\mathbf{e}_{l})-\gamma^{k^{\prime}k}(\mathbf{R}|\mathbf% {R}-d\mathbf{e}_{l})\right]
  4. ๐ž l \mathbf{e}_{l}
  5. l l
  6. ฮณ k โ€ฒ k \gamma^{k^{\prime}k}
  7. ฮณ k โ€ฒ k ( ๐‘ 1 | ๐‘ 2 ) = โŸจ ฯ‡ k โ€ฒ ( ๐ซ ; ๐‘ 1 ) | ฯ‡ k ( ๐ซ ; ๐‘ 2 ) โŸฉ ( ๐ซ ) \gamma^{k^{\prime}k}(\mathbf{R}_{1}|\mathbf{R}_{2})=\langle\chi_{k^{\prime}}(% \mathbf{r};\mathbf{R}_{1})\,|\,\chi_{k}(\mathbf{r};\mathbf{R}_{2})\rangle_{(% \mathbf{r})}

Vienna_Standard_Mean_Ocean_Water.html

  1. 373.15 273.15 \textstyle\frac{373.15}{273.15}

Vieฬ€te's_formula.html

  1. 2 ฯ€ = 2 2 โ‹… 2 + 2 2 โ‹… 2 + 2 + 2 2 โ‹ฏ . \frac{2}{\pi}=\frac{\sqrt{2}}{2}\cdot\frac{\sqrt{2+\sqrt{2}}}{2}\cdot\frac{% \sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdots.
  2. ฯ€ \pi
  3. 223 71 < ฯ€ < 22 7 . \frac{223}{71}<\pi<\frac{22}{7}.
  4. ฯ€ \pi
  5. ฯ€ \pi
  6. ฯ€ \pi
  7. ฯ€ \pi
  8. ฯ€ \pi
  9. lim n โ†’ โˆž โˆ i = 1 n a i 2 = 2 ฯ€ \lim_{n\rightarrow\infty}\prod_{i=1}^{n}{a_{i}\over 2}=\frac{2}{\pi}
  10. 2 \sqrt{2}
  11. sin ( x ) x = cos ( x 2 ) โ‹… cos ( x 4 ) โ‹… cos ( x 8 ) โ‹ฏ . \frac{\sin(x)}{x}=\cos\left(\frac{x}{2}\right)\cdot\cos\left(\frac{x}{4}\right% )\cdot\cos\left(\frac{x}{8}\right)\cdots.
  12. cos x 2 = 1 + cos x 2 \cos\frac{x}{2}=\sqrt{\frac{1+\cos x}{2}}
  13. ฯ€ = 3 lim k โ†’ โˆž 2 k - 1 2 - 2 + 2 + 2 + 2 + โ‹ฏ + 2 + 1 โŸ k square roots \pi=3\lim_{k\to\infty}2^{k-1}\underbrace{\sqrt{2-\sqrt{2+\sqrt{2+\sqrt{2+\sqrt% {2+\cdots+\sqrt{2+1}}}}}}}_{k\ \mathrm{square}\ \mathrm{roots}}
  14. 2 n 2^{n}
  15. 2 n + 1 2^{n+1}
  16. 2 2 \tfrac{\sqrt{2}}{2}
  17. 2 n 2^{n}
  18. sin x = 2 sin x 2 cos x 2 , \sin x=2\sin\frac{x}{2}\cos\frac{x}{2},
  19. sin x = 2 n sin x 2 n ( โˆ i = 1 n cos x 2 i ) . \sin x=2^{n}\sin\frac{x}{2^{n}}\left(\prod_{i=1}^{n}\cos\frac{x}{2^{i}}\right).
  20. 2 n sin x 2 n 2^{n}\sin\tfrac{x}{2^{n}}

Villarceau_circles.html

  1. 0 = ( x 2 + y 2 + z 2 + 16 ) 2 - 100 ( x 2 + y 2 ) . 0=(x^{2}+y^{2}+z^{2}+16)^{2}-100(x^{2}+y^{2}).\,\!
  2. ( x , y , z ) = ( 4 cos ฯ‘ , + 3 + 5 sin ฯ‘ , 3 cos ฯ‘ ) (x,y,z)=(4\cos\vartheta,+3+5\sin\vartheta,3\cos\vartheta)\,\!
  3. ( x , y , z ) = ( 4 cos ฯ‘ , - 3 + 5 sin ฯ‘ , 3 cos ฯ‘ ) . (x,y,z)=(4\cos\vartheta,-3+5\sin\vartheta,3\cos\vartheta).\,\!
  4. 0 = ( x - R ) 2 + z 2 - r 2 0=(x-R)^{2}+z^{2}-r^{2}\,\!
  5. 0 = ( x 2 + y 2 + z 2 + R 2 - r 2 ) 2 - 4 R 2 ( x 2 + y 2 ) . 0=(x^{2}+y^{2}+z^{2}+R^{2}-r^{2})^{2}-4R^{2}(x^{2}+y^{2}).\,\!
  6. 0 = ( x + R ) 2 + z 2 - r 2 0=(x+R)^{2}+z^{2}-r^{2}\,\!
  7. 0 = x r - z R 2 - r 2 0=xr-z\sqrt{R^{2}-r^{2}}\,\!
  8. 0 = x r cos ฯ† + y r sin ฯ† - z R 2 - r 2 0=xr\cos\varphi+yr\sin\varphi-z\sqrt{R^{2}-r^{2}}\,\!
  9. ( - r sin ฯ† , r cos ฯ† , 0 ) . (-r\sin\varphi,r\cos\varphi,0).\,\!
  10. 0 = ( x 2 + y 2 + z 2 + R 2 w 2 - r 2 w 2 ) 2 - 4 R 2 w 2 ( x 2 + y 2 ) , 0=(x^{2}+y^{2}+z^{2}+R^{2}w^{2}-r^{2}w^{2})^{2}-4R^{2}w^{2}(x^{2}+y^{2}),\,\!
  11. 0 = ( x 2 + y 2 + z 2 ) 2 . 0=(x^{2}+y^{2}+z^{2})^{2}.\,\!

Virtual_work.html

  1. W = โˆซ ๐ซ ( t 0 ) = A ๐ซ ( t 1 ) = B ๐… โ‹… d ๐ซ = โˆซ t 0 t 1 ๐… โ‹… d ๐ซ d t d t = โˆซ t 0 t 1 ๐… โ‹… ๐ฏ d t , W=\int_{\mathbf{r}(t_{0})=A}^{\mathbf{r}(t_{1})=B}\mathbf{F}\cdot d\mathbf{r}=% \int_{t_{0}}^{t_{1}}\mathbf{F}\cdot\frac{d\mathbf{r}}{dt}~{}dt=\int_{t_{0}}^{t% _{1}}\mathbf{F}\cdot\mathbf{v}~{}dt,
  2. W ยฏ = โˆซ ๐ซ ( t 0 ) = A ๐ซ ( t 1 ) = B ๐… โ‹… d ( ๐ซ + ฯต ๐ก ) = โˆซ t 0 t 1 ๐… โ‹… d ( ๐ซ ( t ) + ฯต ๐ก ( t ) ) d t d t = โˆซ t 0 t 1 ๐… โ‹… ( ๐ฏ + ฯต ๐ก ห™ ) d t . \bar{W}=\int_{\mathbf{r}(t_{0})=A}^{\mathbf{r}(t_{1})=B}\mathbf{F}\cdot d(% \mathbf{r}+\epsilon\mathbf{h})=\int_{t_{0}}^{t_{1}}\mathbf{F}\cdot\frac{d(% \mathbf{r}(t)+\epsilon\mathbf{h}(t))}{dt}~{}dt=\int_{t_{0}}^{t_{1}}\mathbf{F}% \cdot(\mathbf{v}+\epsilon\dot{\mathbf{h}})~{}dt.
  3. ฮด W = W ยฏ - W = โˆซ t 0 t 1 ( ๐… โ‹… ฯต ๐ก ห™ ) d t . \delta W=\bar{W}-W=\int_{t_{0}}^{t_{1}}(\mathbf{F}\cdot\epsilon\dot{\mathbf{h}% })~{}dt.
  4. ๐ซ ( t ) = ๐ซ ( q 1 , q 2 , โ€ฆ , q n ; t ) \mathbf{r}(t)=\mathbf{r}(q_{1},q_{2},...,q_{n};t)
  5. ๐ก ( t ) = ๐ก ( q 1 , q 2 , โ€ฆ , q n ; t ) \mathbf{h}(t)=\mathbf{h}(q_{1},q_{2},...,q_{n};t)
  6. d d t ฮด ๐ซ = d d t ฯต ๐ก = โˆ‘ i = 1 n โˆ‚ ๐ก โˆ‚ q i ฯต q ห™ i , \frac{d}{dt}\delta\mathbf{r}=\frac{d}{dt}\epsilon\mathbf{h}=\sum_{i=1}^{n}% \frac{\partial\mathbf{h}}{\partial q_{i}}\epsilon\dot{q}_{i},
  7. ฮด W = โˆซ t 0 t 1 ( โˆ‘ i = 1 n ๐… โ‹… โˆ‚ ๐ก โˆ‚ q i ฯต q ห™ i ) d t = โˆ‘ i = 1 n ( โˆซ t 0 t 1 ๐… โ‹… โˆ‚ ๐ก โˆ‚ q i ฯต q ห™ i d t ) . \delta W=\int_{t_{0}}^{t_{1}}\left(\sum_{i=1}^{n}\mathbf{F}\cdot\frac{\partial% \mathbf{h}}{\partial q_{i}}\epsilon\dot{q}_{i}\right)dt=\sum_{i=1}^{n}\left(% \int_{t_{0}}^{t_{1}}\mathbf{F}\cdot\frac{\partial\mathbf{h}}{\partial q_{i}}% \epsilon\dot{q}_{i}~{}dt\right).
  8. Q i = ๐… โ‹… โˆ‚ ๐ก โˆ‚ q i = 0 , i = 1 , โ€ฆ , n . Q_{i}=\mathbf{F}\cdot\frac{\partial\mathbf{h}}{\partial q_{i}}=0,\quad i=1,% \ldots,n.
  9. ฮด W = โˆ‘ i = 1 m ๐… i โ‹… ฮด ๐ซ i + โˆ‘ j = 1 n ๐Œ j โ‹… ฮด ฯ• j = 0 , \delta W=\sum_{i=1}^{m}\mathbf{F}_{i}\cdot\delta\mathbf{r}_{i}+\sum_{j=1}^{n}% \mathbf{M}_{j}\cdot\delta\mathbf{\phi}_{j}=0,
  10. ฮด ๐ซ i ( q 1 , q 2 , โ€ฆ , q f ; t ) , i = 1 , 2 , โ€ฆ , m ; \delta\mathbf{r}_{i}(q_{1},q_{2},...,q_{f};t),\quad i=1,2,...,m;
  11. ฮด ฯ• j ( q 1 , q 2 , โ€ฆ , q f ; t ) , j = 1 , 2 , โ€ฆ , n . \delta\phi_{j}(q_{1},q_{2},...,q_{f};t),\quad j=1,2,...,n.
  12. ฮด W = โˆ‘ k = 1 f [ ( โˆ‘ i = 1 m ๐… i โ‹… โˆ‚ ๐ซ i โˆ‚ q k + โˆ‘ j = 1 n ๐Œ j โ‹… โˆ‚ ฯ• j โˆ‚ q k ) ฮด q k ] = โˆ‘ k = 1 f Q k ฮด q k , \delta W=\sum_{k=1}^{f}\left[\left(\sum_{i=1}^{m}\mathbf{F}_{i}\cdot\frac{% \partial\mathbf{r}_{i}}{\partial q_{k}}+\sum_{j=1}^{n}\mathbf{M}_{j}\cdot\frac% {\partial\mathbf{\phi}_{j}}{\partial q_{k}}\right)\delta q_{k}\right]=\sum_{k=% 1}^{f}Q_{k}\delta q_{k},
  13. Q k = โˆ‘ i = 1 m ๐… i โ‹… โˆ‚ ๐ซ i โˆ‚ q k + โˆ‘ j = 1 n ๐Œ j โ‹… โˆ‚ ฯ• j โˆ‚ q k , k = 1 , 2 , โ€ฆ , f . Q_{k}=\sum_{i=1}^{m}\mathbf{F}_{i}\cdot\frac{\partial\mathbf{r}_{i}}{\partial q% _{k}}+\sum_{j=1}^{n}\mathbf{M}_{j}\cdot\frac{\partial\mathbf{\phi}_{j}}{% \partial q_{k}},\quad k=1,2,...,f.
  14. Q k = โˆ‘ i = 1 m ๐… i โ‹… โˆ‚ ๐ฏ i โˆ‚ q ห™ k + โˆ‘ j = 1 n ๐Œ j โ‹… โˆ‚ ฯ‰ j โˆ‚ q ห™ k , k = 1 , 2 , โ€ฆ , f . Q_{k}=\sum_{i=1}^{m}\mathbf{F}_{i}\cdot\frac{\partial\mathbf{v}_{i}}{\partial% \dot{q}_{k}}+\sum_{j=1}^{n}\mathbf{M}_{j}\cdot\frac{\partial\mathbf{\omega}_{j% }}{\partial\dot{q}_{k}},\quad k=1,2,...,f.
  15. ฮด W = 0 โ‡’ Q k = 0 k = 1 , 2 , โ€ฆ , f . \delta W=0\quad\Rightarrow\quad Q_{k}=0\quad k=1,2,...,f.
  16. a = | ๐ซ A - ๐ซ P | , b = | ๐ซ B - ๐ซ P | , a=|\mathbf{r}_{A}-\mathbf{r}_{P}|,\quad b=|\mathbf{r}_{B}-\mathbf{r}_{P}|,
  17. ๐ซ A - ๐ซ P = a ๐ž A , ๐ซ B - ๐ซ P = b ๐ž B . \mathbf{r}_{A}-\mathbf{r}_{P}=a\mathbf{e}_{A},\quad\mathbf{r}_{B}-\mathbf{r}_{% P}=b\mathbf{e}_{B}.
  18. ๐ฏ A = ฮธ ห™ a ๐ž A โŸ‚ , ๐ฏ B = ฮธ ห™ b ๐ž B โŸ‚ , \mathbf{v}_{A}=\dot{\theta}a\mathbf{e}_{A}^{\perp},\quad\mathbf{v}_{B}=\dot{% \theta}b\mathbf{e}_{B}^{\perp},
  19. Q = ๐… A โ‹… โˆ‚ ๐ฏ A โˆ‚ ฮธ ห™ - ๐… B โ‹… โˆ‚ ๐ฏ B โˆ‚ ฮธ ห™ = a ( ๐… A โ‹… ๐ž A โŸ‚ ) - b ( ๐… B โ‹… ๐ž B โŸ‚ ) . Q=\mathbf{F}_{A}\cdot\frac{\partial\mathbf{v}_{A}}{\partial\dot{\theta}}-% \mathbf{F}_{B}\cdot\frac{\partial\mathbf{v}_{B}}{\partial\dot{\theta}}=a(% \mathbf{F}_{A}\cdot\mathbf{e}_{A}^{\perp})-b(\mathbf{F}_{B}\cdot\mathbf{e}_{B}% ^{\perp}).
  20. F A = ๐… A โ‹… ๐ž A โŸ‚ , F B = ๐… B โ‹… ๐ž B โŸ‚ . F_{A}=\mathbf{F}_{A}\cdot\mathbf{e}_{A}^{\perp},\quad F_{B}=\mathbf{F}_{B}% \cdot\mathbf{e}_{B}^{\perp}.
  21. Q = a F A - b F B = 0. Q=aF_{A}-bF_{B}=0.\,\!
  22. M A = F B F A = a b . MA=\frac{F_{B}}{F_{A}}=\frac{a}{b}.
  23. ฯ‰ A ฯ‰ B = R . \frac{\omega_{A}}{\omega_{B}}=R.
  24. ฯ‰ A = ฯ‰ , ฯ‰ B = ฯ‰ / R . \omega_{A}=\omega,\quad\omega_{B}=\omega/R.\!
  25. Q = T A โˆ‚ ฯ‰ A โˆ‚ ฯ‰ - T B โˆ‚ ฯ‰ B โˆ‚ ฯ‰ = T A - T B / R = 0. Q=T_{A}\frac{\partial\omega_{A}}{\partial\omega}-T_{B}\frac{\partial\omega_{B}% }{\partial\omega}=T_{A}-T_{B}/R=0.
  26. M A = T B T A = R . MA=\frac{T_{B}}{T_{A}}=R.
  27. Q * = - ( M ๐€ ) โ‹… โˆ‚ ๐• โˆ‚ q ห™ - ( [ I R ] ฮฑ + ฯ‰ ร— [ I R ] ฯ‰ ) โ‹… โˆ‚ ฯ‰ โ†’ โˆ‚ q ห™ . Q^{*}=-(M\mathbf{A})\cdot\frac{\partial\mathbf{V}}{\partial\dot{q}}-([I_{R}]% \alpha+\omega\times[I_{R}]\omega)\cdot\frac{\partial\vec{\omega}}{\partial\dot% {q}}.
  28. T = 1 2 M ๐• โ‹… ๐• + 1 2 ฯ‰ โ†’ โ‹… [ I R ] ฯ‰ โ†’ , T=\frac{1}{2}M\mathbf{V}\cdot\mathbf{V}+\frac{1}{2}\vec{\omega}\cdot[I_{R}]% \vec{\omega},
  29. Q * = - ( d d t โˆ‚ T โˆ‚ q ห™ - โˆ‚ T โˆ‚ q ) . Q^{*}=-\left(\frac{d}{dt}\frac{\partial T}{\partial\dot{q}}-\frac{\partial T}{% \partial q}\right).
  30. T = โˆ‘ i = 1 n ( 1 2 M ๐• i โ‹… ๐• i + 1 2 ฯ‰ โ†’ i โ‹… [ I R ] ฯ‰ โ†’ i ) , T=\sum_{i=1}^{n}(\frac{1}{2}M\mathbf{V}_{i}\cdot\mathbf{V}_{i}+\frac{1}{2}\vec% {\omega}_{i}\cdot[I_{R}]\vec{\omega}_{i}),
  31. Q j * = - ( d d t โˆ‚ T โˆ‚ q ห™ j - โˆ‚ T โˆ‚ q j ) , j = 1 , โ€ฆ , m . Q^{*}_{j}=-\left(\frac{d}{dt}\frac{\partial T}{\partial\dot{q}_{j}}-\frac{% \partial T}{\partial q_{j}}\right),\quad j=1,\ldots,m.
  32. ฮด W = ( F 1 + Q 1 * ) ฮด q 1 + โ€ฆ + ( F m + Q m * ) ฮด q m = 0 , \delta W=(F_{1}+Q^{*}_{1})\delta q_{1}+\ldots+(F_{m}+Q^{*}_{m})\delta q_{m}=0,
  33. F j + Q j * = 0 , j = 1 , โ€ฆ , m , F_{j}+Q^{*}_{j}=0,\quad j=1,\ldots,m,
  34. d d t โˆ‚ T โˆ‚ q ห™ j - โˆ‚ T โˆ‚ q j = Q j , j = 1 , โ€ฆ , m . \frac{d}{dt}\frac{\partial T}{\partial\dot{q}_{j}}-\frac{\partial T}{\partial q% _{j}}=Q_{j},\quad j=1,\ldots,m.
  35. d d t โˆ‚ T โˆ‚ q ห™ j - โˆ‚ T โˆ‚ q j = - โˆ‚ V โˆ‚ q j , j = 1 , โ€ฆ , m . \frac{d}{dt}\frac{\partial T}{\partial\dot{q}_{j}}-\frac{\partial T}{\partial q% _{j}}=-\frac{\partial V}{\partial q_{j}},\quad j=1,\ldots,m.
  36. d d t โˆ‚ L โˆ‚ q ห™ j - โˆ‚ L โˆ‚ q j = 0 j = 1 , โ€ฆ , m . \frac{d}{dt}\frac{\partial L}{\partial\dot{q}_{j}}-\frac{\partial L}{\partial q% _{j}}=0\quad j=1,\ldots,m.
  37. s y m b o l ฯƒ symbol{\sigma}
  38. s y m b o l ฯƒ symbol{\sigma}
  39. s y m b o l ฯต symbol{\epsilon}
  40. ๐ฎ * \mathbf{u}^{*}
  41. s y m b o l ฯต * symbol{\epsilon}^{*}
  42. s y m b o l ฯƒ symbol{\sigma}
  43. s y m b o l ฯต symbol{\epsilon}
  44. F A F_{A}
  45. F A F_{A}
  46. F B F_{B}
  47. F B ( u * + โˆ‚ u * โˆ‚ x d x ) - F A u * โ‰ˆ โˆ‚ u * โˆ‚ x ฯƒ d V + u * โˆ‚ ฯƒ โˆ‚ x d V = ฯต * ฯƒ d V - u * f d V F_{B}\big(u^{*}+\frac{\partial u^{*}}{\partial x}dx\big)-F_{A}u^{*}\approx% \frac{\partial u^{*}}{\partial x}\sigma dV+u^{*}\frac{\partial\sigma}{\partial x% }dV=\epsilon^{*}\sigma dV-u^{*}fdV
  48. โˆ‚ ฯƒ โˆ‚ x + f = 0 \frac{\partial\sigma}{\partial x}+f=0
  49. โˆซ V s y m b o l ฯต * T s y m b o l ฯƒ d V \int_{V}symbol{\epsilon}^{*T}symbol{\sigma}\,dV
  50. Total external virtual work = โˆซ V s y m b o l ฯต * T s y m b o l ฯƒ d V ( d ) \mbox{Total external virtual work}~{}=\int_{V}symbol{\epsilon}^{*T}symbol{% \sigma}dV\qquad\mathrm{(d)}
  51. โˆซ S ๐ฎ * T ๐“ d S + โˆซ V ๐ฎ * T ๐Ÿ d V = โˆซ V s y m b o l ฯต * T s y m b o l ฯƒ d V ( e ) \int_{S}\mathbf{u}^{*T}\mathbf{T}dS+\int_{V}\mathbf{u}^{*T}\mathbf{f}dV=\int_{% V}symbol{\epsilon}^{*T}symbol{\sigma}dV\qquad\mathrm{(e)}
  52. โˆซ S ๐ฎ โ‹… ๐“ d S = โˆซ S ๐ฎ โ‹… ๐ฌ๐ฒ๐ฆ๐›๐จ๐ฅ ฯƒ โ‹… ๐ง d S \int_{S}\mathbf{u\cdot T}dS=\int_{S}\mathbf{u\cdot symbol{\sigma}\cdot n}dS
  53. โˆซ S ๐ฎ โ‹… ๐ฌ๐ฒ๐ฆ๐›๐จ๐ฅ ฯƒ โ‹… ๐ง d S = โˆซ V โˆ‡ โ‹… ( ๐ฎ โ‹… s y m b o l ฯƒ ) d V \int_{S}\mathbf{u\cdot symbol\sigma\cdot n}dS=\int_{V}\nabla\cdot\left(\mathbf% {u}\cdot symbol{\sigma}\right)dV
  54. โˆซ V โˆ‡ โ‹… ( ๐ฎ โ‹… s y m b o l ฯƒ ) d V = โˆซ V โˆ‚ โˆ‚ x j ( u i ฯƒ i j ) d V = โˆซ V โˆ‚ u i โˆ‚ x j ฯƒ i j + u i โˆ‚ ฯƒ i j โˆ‚ x j d V \begin{aligned}\displaystyle\int_{V}\nabla\cdot\left(\mathbf{u}\cdot symbol{% \sigma}\right)dV&\displaystyle=\int_{V}\frac{\partial}{\partial x_{j}}\left(u_% {i}\sigma_{ij}\right)dV\\ &\displaystyle=\int_{V}\frac{\partial u_{i}}{\partial x_{j}}\sigma_{ij}+u_{i}% \frac{\partial\sigma_{ij}}{\partial x_{j}}dV\end{aligned}
  55. โˆ‚ ฯƒ i j โˆ‚ x j + f i = 0 \frac{\partial\sigma_{ij}}{\partial x_{j}}+f_{i}=0
  56. โˆซ V โˆ‚ u i โˆ‚ x j ฯƒ i j + u i โˆ‚ ฯƒ i j โˆ‚ x j d V = โˆซ V โˆ‚ u i โˆ‚ x j ฯƒ i j - u i f i d V \int_{V}\frac{\partial u_{i}}{\partial x_{j}}\sigma_{ij}+u_{i}\frac{\partial% \sigma_{ij}}{\partial x_{j}}dV=\int_{V}\frac{\partial u_{i}}{\partial x_{j}}% \sigma_{ij}-u_{i}f_{i}dV
  57. โˆซ V โˆ‚ u i โˆ‚ x j ฯƒ i j - u i f i d V \displaystyle\int_{V}\frac{\partial u_{i}}{\partial x_{j}}\sigma_{ij}-u_{i}f_{% i}dV
  58. s y m b o l ฯต symbol\epsilon
  59. โˆซ S ๐ฎ โ‹… ๐“ d S = โˆซ V s y m b o l ฯต : s y m b o l ฯƒ d V - โˆซ V ๐ฎ โ‹… ๐Ÿ d V \int_{S}\mathbf{u\cdot T}dS=\int_{V}symbol\epsilon:symbol\sigma dV-\int_{V}% \mathbf{u}\cdot\mathbf{f}dV
  60. โˆซ S ๐ฎ โ‹… ๐“ d S + โˆซ V ๐ฎ โ‹… ๐Ÿ d V = โˆซ V s y m b o l ฯต : s y m b o l ฯƒ d V \int_{S}\mathbf{u\cdot T}dS+\int_{V}\mathbf{u}\cdot\mathbf{f}dV=\int_{V}symbol% \epsilon:symbol\sigma dV
  61. ฮด ๐ฎ โ‰ก ๐ฎ * \delta\ \mathbf{u}\equiv\mathbf{u}^{*}
  62. ฮด s y m b o l ฯต โ‰ก s y m b o l ฯต * \delta\ symbol{\epsilon}\equiv symbol{\epsilon}^{*}
  63. S t S_{t}
  64. โˆซ S t ฮด ๐ฎ T ๐“ d S + โˆซ V ฮด ๐ฎ T ๐Ÿ d V = โˆซ V \deltasymbol ฯต T s y m b o l ฯƒ d V ( f ) \int_{S_{t}}\delta\ \mathbf{u}^{T}\mathbf{T}dS+\int_{V}\delta\ \mathbf{u}^{T}% \mathbf{f}dV=\int_{V}\deltasymbol{\epsilon}^{T}symbol{\sigma}dV\qquad\mathrm{(% f)}
  65. S t S_{t}
  66. S t S_{t}
  67. S t S_{t}
  68. S u S_{u}
  69. โˆซ S u ๐ฎ T ฮด ๐“ d S + โˆซ V ๐ฎ T ฮด ๐Ÿ d V = โˆซ V s y m b o l ฯต T ฮด s y m b o l ฯƒ d V ( g ) \int_{S_{u}}\mathbf{u}^{T}\delta\ \mathbf{T}dS+\int_{V}\mathbf{u}^{T}\delta\ % \mathbf{f}dV=\int_{V}symbol{\epsilon}^{T}\delta symbol{\sigma}dV\qquad\mathrm{% (g)}
  70. S u S_{u}

Vis_viva.html

  1. โˆ‘ i m i v i 2 \sum_{i}m_{i}v_{i}^{2}
  2. โˆ‘ i m i ๐ฏ i \,\!\sum_{i}m_{i}\mathbf{v}_{i}
  3. E = 1 2 โˆ‘ i m i v i 2 E=\frac{1}{2}\sum_{i}m_{i}v_{i}^{2}

Visibility.html

  1. C V ( x ) = F B ( x ) - F ( x ) F B ( x ) C\text{V}(x)=\frac{F\text{B}(x)-F(x)}{F\text{B}(x)}
  2. d F = - b ext F d x dF=-b\text{ext}Fdx
  3. d F ( x ) = [ b โ€ฒ F b ( x ) - b ext F ( x ) ] d x dF(x)=\left[b^{\prime}F\text{b}(x)-b\text{ext}F(x)\right]dx
  4. d F B ( x ) = 0 = [ b โ€ฒ F B ( x ) - b ext F b ( x ) ] d x dF\text{B}(x)=0=\left[b^{\prime}F\text{B}(x)-b\text{ext}F\text{b}(x)\right]dx
  5. d C V ( x ) d x = - b ext C V ( x ) \frac{dC\text{V}(x)}{dx}=-b\text{ext}C\text{V}(x)
  6. C V ( x ) = exp ( - b ext x ) C\text{V}(x)=\exp(-b\text{ext}x)
  7. x V = 3.912 b ext x\text{V}=\frac{3.912}{b\text{ext}}

Vital_capacity.html

  1. v c f e m a l e = ( 21.78 - 0.101 a ) โ‹… h v c m a l e = ( 27.63 - 0.112 a ) โ‹… h \begin{aligned}\displaystyle vc_{female}=(21.78-0.101a)\cdot h\\ \displaystyle vc_{male}=(27.63-0.112a)\cdot h\\ \end{aligned}
  2. v c vc
  3. a a
  4. h h

Viterbi_decoder.html

  1. 2 K - 1 2^{K-1}
  2. 2 K - 1 2^{K-1}
  3. โ‰ฅ 0 \geq 0
  4. T = N 0 / 2 k , \,\!T=\sqrt{N_{0}/2^{k}},
  5. N 0 N_{0}
  6. โ„“ \ell
  7. D = ( v r โ†’ - v i โ†’ ) 2 = v r โ†’ 2 - 2 v r โ†’ v i โ†’ + v i โ†’ 2 \,\!D=(\overrightarrow{v_{r}}-\overrightarrow{v_{i}})^{2}=\overrightarrow{v_{r% }}^{2}-2\overrightarrow{v_{r}}\overrightarrow{v_{i}}+\overrightarrow{v_{i}}^{2}
  8. v i โ†’ 2 = ( ยฑ 1 ) 2 + ( ยฑ 1 ) 2 = 2 \,\!\overrightarrow{v_{i}}^{2}=(\pm 1)^{2}+(\pm 1)^{2}=2
  9. D 0 = v r โ†’ 2 - 2 v r โ†’ v i 0 โ†’ + v i 0 โ†’ 2 \,\!D_{0}=\overrightarrow{v_{r}}^{2}-2\overrightarrow{v_{r}}\overrightarrow{v_% {i}^{0}}+\overrightarrow{v_{i}^{0}}^{2}
  10. D 1 = v r โ†’ 2 - 2 v r โ†’ v i 1 โ†’ + v i 1 โ†’ 2 \,\!D_{1}=\overrightarrow{v_{r}}^{2}-2\overrightarrow{v_{r}}\overrightarrow{v_% {i}^{1}}+\overrightarrow{v_{i}^{1}}^{2}
  11. m i n ( - 2 v r โ†’ v i 0 โ†’ , - 2 v r โ†’ v i 1 โ†’ ) = m a x ( v r โ†’ v i 0 โ†’ , v r โ†’ v i 1 โ†’ ) \,\!min(-2\overrightarrow{v_{r}}\overrightarrow{v_{i}^{0}},-2\overrightarrow{v% _{r}}\overrightarrow{v_{i}^{1}})=max(\overrightarrow{v_{r}}\overrightarrow{v_{% i}^{0}},\overrightarrow{v_{r}}\overrightarrow{v_{i}^{1}})
  12. m a x ( ยฑ r 0 ยฑ r 1 , ยฑ r 0 ยฑ r 1 ) \,\!max(\pm r_{0}\pm r_{1},\pm r_{0}\pm r_{1})

Vlasov_equation.html

  1. d f ( ๐ช , ๐ฉ , t ) d t = 0 , \frac{\operatorname{d}f(\mathbf{q},\mathbf{p},t)}{\operatorname{d}t}=0,
  2. โˆ‚ f โˆ‚ t + d ๐ช d t โ‹… โˆ‚ f โˆ‚ ๐ช + d ๐ฉ d t โ‹… โˆ‚ f โˆ‚ ๐ฉ = 0 , \frac{\partial f}{\partial t}+\frac{\operatorname{d}\mathbf{q}}{\operatorname{% d}t}\cdot\frac{\partial f}{\partial\mathbf{q}}+\frac{\operatorname{d}\mathbf{p% }}{\operatorname{d}t}\cdot\frac{\partial f}{\partial\mathbf{p}}=0,
  3. f e ( ๐ซ , ๐ฉ , t ) f_{e}(\mathbf{r},\mathbf{p},t)
  4. f i ( ๐ซ , ๐ฉ , t ) f_{i}(\mathbf{r},\mathbf{p},t)
  5. f ฮฑ ( ๐ซ , ๐ฉ , t ) f_{\alpha}(\mathbf{r},\mathbf{p},t)
  6. ฮฑ ฮฑ
  7. ฮฑ ฮฑ
  8. ๐ฉ \mathbf{p}
  9. ๐ซ \mathbf{r}
  10. t t
  11. โˆ‚ f e โˆ‚ t + ๐ฏ e โ‹… โˆ‡ f e \displaystyle\frac{\partial f_{e}}{\partial t}+\mathbf{v}_{e}\cdot\nabla f_{e}
  12. ฯ = e โˆซ ( Z i f i - f e ) d 3 p , ๐ฃ = e โˆซ ( Z i f i ๐ฏ i - f e ๐ฏ e ) d 3 p , ๐ฏ ฮฑ = ๐ฉ m ฮฑ ( 1 + p 2 ( m ฮฑ c ) 2 ) 1 / 2 \rho=e\int(Z_{i}f_{i}-f_{e})d^{3}p,\quad\mathbf{j}=e\int(Z_{i}f_{i}\mathbf{v}_% {i}-f_{e}\mathbf{v}_{e})d^{3}p,\quad\mathbf{v}_{\alpha}=\frac{\frac{\mathbf{p}% }{m_{\alpha}}}{\left(1+\frac{p^{2}}{(m_{\alpha}c)^{2}}\right)^{1/2}}
  13. e e
  14. c c
  15. ๐„ ( ๐ซ , t ) \mathbf{E}(\mathbf{r},t)
  16. ๐ ( ๐ซ , t ) \mathbf{B}(\mathbf{r},t)
  17. ๐ซ \mathbf{r}
  18. t t
  19. f e ( ๐ซ , ๐ฉ , t ) f_{e}(\mathbf{r},\mathbf{p},t)
  20. f i ( ๐ซ , ๐ฉ , t ) f_{i}(\mathbf{r},\mathbf{p},t)
  21. โˆ‚ f ฮฑ โˆ‚ t + ๐ฏ ฮฑ โ‹… โˆ‚ f ฮฑ โˆ‚ ๐ฑ + q ฮฑ ๐„ m ฮฑ โ‹… โˆ‚ f ฮฑ โˆ‚ ๐ฏ = 0 , \frac{\partial f_{\alpha}}{\partial t}+\mathbf{v}_{\alpha}\cdot\frac{\partial f% _{\alpha}}{\partial\mathbf{x}}+\frac{q_{\alpha}\mathbf{E}}{m_{\alpha}}\cdot% \frac{\partial f_{\alpha}}{\partial\mathbf{v}}=0,
  22. โˆ‡ 2 ฯ• + ฯ = 0. \nabla^{2}\phi+\rho=0.
  23. ๐„ ( ๐ฑ , t ) \mathbf{E}(\mathbf{x},t)
  24. ฯ• ( ๐ฑ , t ) \phi(\mathbf{x},t)
  25. ฯ ฯ
  26. f ( ๐ซ , ๐ฏ , t ) f(\mathbf{r},\mathbf{v},t)
  27. n n
  28. ๐ฎ \mathbf{u}
  29. ๐ฉ \mathbf{p}
  30. n n
  31. f f
  32. v n f v^{n}f
  33. โˆซ d d t f d 3 v = โˆซ ( โˆ‚ โˆ‚ t f + ( ๐ฏ โ‹… โˆ‡ r ) f + โˆ‡ v โ‹… ( ๐š f ) ) d 3 v = 0 \int\frac{\mathrm{d}}{\mathrm{d}t}fd^{3}v=\int\left(\frac{\partial}{\partial t% }f+(\mathbf{v}\cdot\nabla_{r})f+\nabla_{v}\cdot(\mathbf{a}f)\right)d^{3}v=0
  34. โˆ‚ โˆ‚ t n + โˆ‡ โ‹… ( n ๐ฎ ) = 0. \frac{\partial}{\partial t}n+\nabla\cdot(n\mathbf{u})=0.
  35. n n
  36. n ๐ฎ n\mathbf{u}
  37. n = โˆซ f d 3 v n=\int fd^{3}v
  38. n ๐ฎ = โˆซ ๐ฏ f d 3 v n\mathbf{u}=\int\mathbf{v}fd^{3}v
  39. m d ๐ฏ d t = q ( ๐„ + ๐ฏ ร— ๐ ) m\frac{\mathrm{d}\mathbf{v}}{\mathrm{d}t}=q(\mathbf{E}+\mathbf{v}\times\mathbf% {B})
  40. m n d d t ๐ฎ = - โˆ‡ โ‹… ๐ฉ + q n ๐„ + q n ๐ฎ ร— ๐ mn\frac{\mathrm{d}}{\mathrm{d}t}\mathbf{u}=-\nabla\cdot\mathbf{p}+qn\mathbf{E}% +qn\mathbf{u}\times\mathbf{B}
  41. ๐ฉ \mathbf{p}
  42. D D t = โˆ‚ โˆ‚ t + ๐ฎ โ‹… โˆ‡ . \frac{\mathrm{D}}{\mathrm{D}t}=\frac{\partial}{\partial t}+\mathbf{u}\cdot\nabla.
  43. p i j = m โˆซ ( v i - u i ) ( v j - u j ) f d 3 v . p_{ij}=m\int(v_{i}-u_{i})(v_{j}-u_{j})fd^{3}v.
  44. T , L T,L
  45. V V
  46. f f
  47. โˆ‚ f โˆ‚ t T โˆผ f | โˆ‚ f โˆ‚ ๐ซ | L โˆผ f | โˆ‚ f โˆ‚ ๐ฏ | V โˆผ f . \frac{\partial f}{\partial t}T\sim f\quad\left|\frac{\partial f}{\partial% \mathbf{r}}\right|L\sim f\quad\left|\frac{\partial f}{\partial\mathbf{v}}% \right|V\sim f.
  48. t โ€ฒ = t T ๐ซ โ€ฒ = ๐ซ L ๐ฏ โ€ฒ = ๐ฏ V . t^{\prime}=\frac{t}{T}\quad\mathbf{r}^{\prime}=\frac{\mathbf{r}}{L}\quad% \mathbf{v}^{\prime}=\frac{\mathbf{v}}{V}.
  49. 1 T โˆ‚ f โˆ‚ t โ€ฒ + V L ๐ฏ โ€ฒ โ‹… โˆ‚ f โˆ‚ ๐ซ โ€ฒ + q m V ( ๐„ + V ๐ฏ โ€ฒ ร— ๐ ) โ‹… โˆ‚ f โˆ‚ ๐ฏ โ€ฒ = 0. \frac{1}{T}\frac{\partial f}{\partial t^{\prime}}+\frac{V}{L}\mathbf{v}^{% \prime}\cdot\frac{\partial f}{\partial\mathbf{r}^{\prime}}+\frac{q}{mV}(% \mathbf{E}+V\mathbf{v}^{\prime}\times\mathbf{B})\cdot\frac{\partial f}{% \partial\mathbf{v}^{\prime}}=0.
  50. V = R ฯ‰ g V=R\omega_{g}
  51. ฯ‰ g = q B / m \omega_{g}=qB/m
  52. R R
  53. 1 ฯ‰ g T โˆ‚ f โˆ‚ t โ€ฒ + R L ๐ฏ โ€ฒ โ‹… โˆ‚ f โˆ‚ ๐ซ โ€ฒ + ( ๐„ V B + ๐ฏ โ€ฒ ร— ๐ B ) โ‹… โˆ‚ f โˆ‚ ๐ฏ โ€ฒ = 0 \frac{1}{\omega_{g}T}\frac{\partial f}{\partial t^{\prime}}+\frac{R}{L}\mathbf% {v}^{\prime}\cdot\frac{\partial f}{\partial\mathbf{r}^{\prime}}+\left(\frac{% \mathbf{E}}{VB}+\mathbf{v}^{\prime}\times\frac{\mathbf{B}}{B}\right)\cdot\frac% {\partial f}{\partial\mathbf{v}^{\prime}}=0
  54. 1 / ฯ‰ g โ‰ช T 1/\omega_{g}\ll T
  55. R โ‰ช L R\ll L
  56. f f
  57. โˆ‚ f / โˆ‚ t โ€ฒ โˆผ f , v โ€ฒ โ‰ฒ 1 \partial f/\partial t^{\prime}\sim f,v^{\prime}\lesssim 1
  58. โˆ‚ f / โˆ‚ ๐ซ โ€ฒ โˆผ f \partial f/\partial\mathbf{r}^{\prime}\sim f
  59. T , L T,L
  60. V V
  61. f f
  62. ( ๐„ V B + ๐ฏ โ€ฒ ร— ๐ B ) โ‹… โˆ‚ f โˆ‚ ๐ฏ โ€ฒ โ‰ˆ 0 โ‡’ ( ๐„ + ๐ฏ ร— ๐ ) โ‹… โˆ‚ f โˆ‚ ๐ฏ โ‰ˆ 0 \left(\frac{\mathbf{E}}{VB}+\mathbf{v}^{\prime}\times\frac{\mathbf{B}}{B}% \right)\cdot\frac{\partial f}{\partial\mathbf{v}^{\prime}}\approx 0\Rightarrow% (\mathbf{E}+\mathbf{v}\times\mathbf{B})\cdot\frac{\partial f}{\partial\mathbf{% v}}\approx 0
  63. ๐„ โˆฅ โˆ‚ f โˆ‚ ๐ฏ โˆฅ + ( ๐„ โŸ‚ + ๐ฏ ร— ๐ ) โ‹… โˆ‚ f โˆ‚ ๐ฏ โŸ‚ โ‰ˆ 0 \mathbf{E}_{\|}\frac{\partial f}{\partial\mathbf{v}_{\|}}+(\mathbf{E}_{\perp}+% \mathbf{v}\times\mathbf{B})\cdot\frac{\partial f}{\partial\mathbf{v}_{\perp}}\approx 0
  64. ๐ฏ = ๐ฏ 0 + ฮ” ๐ฏ \mathbf{v}=\mathbf{v}_{0}+\Delta\mathbf{v}
  65. ๐ฏ 0 ร— ๐ = - ๐„ โŸ‚ \mathbf{v}_{0}\times\mathbf{B}=-\mathbf{E}_{\perp}
  66. ๐„ โˆฅ โˆ‚ f โˆ‚ ๐ฏ โˆฅ + ( ฮ” ๐ฏ โŸ‚ ร— ๐ ) โ‹… โˆ‚ f โˆ‚ ๐ฏ โŸ‚ โ‰ˆ 0 \mathbf{E}_{\|}\frac{\partial f}{\partial\mathbf{v}_{\|}}+(\Delta\mathbf{v}_{% \perp}\times\mathbf{B})\cdot\frac{\partial f}{\partial\mathbf{v}_{\perp}}\approx 0
  67. ( ฮ” ๐ฏ โŸ‚ ร— ๐ ) โ‹… โˆ‚ f โˆ‚ ๐ฏ โŸ‚ โ‰ˆ 0 (\Delta\mathbf{v}_{\perp}\times\mathbf{B})\cdot\frac{\partial f}{\partial% \mathbf{v}_{\perp}}\approx 0
  68. ๐ฏ 0 \mathbf{v}_{0}
  69. ๐ฎ \mathbf{u}
  70. ๐„ + ๐ฎ ร— ๐ โ‰ˆ 0 \mathbf{E}+\mathbf{u}\times\mathbf{B}\approx 0
  71. V V
  72. R R

Voigt_notation.html

  1. โŸจ x 11 , x 22 , x 12 โŸฉ \langle x_{11},x_{22},x_{12}\rangle
  2. s y m b o l ฯƒ = [ ฯƒ x x ฯƒ x y ฯƒ x z ฯƒ y x ฯƒ y y ฯƒ y z ฯƒ z x ฯƒ z y ฯƒ z z ] . symbol{\sigma}=\left[{\begin{matrix}\sigma_{xx}&\sigma_{xy}&\sigma_{xz}\\ \sigma_{yx}&\sigma_{yy}&\sigma_{yz}\\ \sigma_{zx}&\sigma_{zy}&\sigma_{zz}\end{matrix}}\right].
  3. ฯƒ ~ = ( ฯƒ x x , ฯƒ y y , ฯƒ z z , ฯƒ y z , ฯƒ x z , ฯƒ x y ) โ‰ก ( ฯƒ 1 , ฯƒ 2 , ฯƒ 3 , ฯƒ 4 , ฯƒ 5 , ฯƒ 6 ) . \tilde{\sigma}=(\sigma_{xx},\sigma_{yy},\sigma_{zz},\sigma_{yz},\sigma_{xz},% \sigma_{xy})\equiv(\sigma_{1},\sigma_{2},\sigma_{3},\sigma_{4},\sigma_{5},% \sigma_{6}).
  4. s y m b o l ฯต = [ ฯต x x ฯต x y ฯต x z ฯต y x ฯต y y ฯต y z ฯต z x ฯต z y ฯต z z ] . symbol{\epsilon}=\left[{\begin{matrix}\epsilon_{xx}&\epsilon_{xy}&\epsilon_{xz% }\\ \epsilon_{yx}&\epsilon_{yy}&\epsilon_{yz}\\ \epsilon_{zx}&\epsilon_{zy}&\epsilon_{zz}\end{matrix}}\right].
  5. ฯต ~ = ( ฯต x x , ฯต y y , ฯต z z , ฮณ y z , ฮณ x z , ฮณ x y ) โ‰ก ( ฯต 1 , ฯต 2 , ฯต 3 , ฯต 4 , ฯต 5 , ฯต 6 ) , \tilde{\epsilon}=(\epsilon_{xx},\epsilon_{yy},\epsilon_{zz},\gamma_{yz},\gamma% _{xz},\gamma_{xy})\equiv(\epsilon_{1},\epsilon_{2},\epsilon_{3},\epsilon_{4},% \epsilon_{5},\epsilon_{6}),
  6. ฮณ x y = 2 ฯต x y \gamma_{xy}=2\epsilon_{xy}
  7. ฮณ y z = 2 ฯต y z \gamma_{yz}=2\epsilon_{yz}
  8. ฮณ z x = 2 ฯต z x \gamma_{zx}=2\epsilon_{zx}
  9. s y m b o l ฯƒ \cdotsymbol ฯต = ฯƒ i j ฯต i j = ฯƒ ~ โ‹… ฯต ~ symbol{\sigma}\cdotsymbol{\epsilon}=\sigma_{ij}\epsilon_{ij}=\tilde{\sigma}% \cdot\tilde{\epsilon}
  10. s y m b o l ฯƒ = [ ฯƒ 11 ฯƒ 12 ฯƒ 13 ฯƒ 21 ฯƒ 22 ฯƒ 23 ฯƒ 31 ฯƒ 32 ฯƒ 33 ] symbol{\sigma}=\left[{\begin{matrix}\sigma_{11}&\sigma_{12}&\sigma_{13}\\ \sigma_{21}&\sigma_{22}&\sigma_{23}\\ \sigma_{31}&\sigma_{32}&\sigma_{33}\end{matrix}}\right]
  11. ฯƒ ~ M = โŸจ ฯƒ 11 , ฯƒ 22 , ฯƒ 33 , 2 ฯƒ 23 , 2 ฯƒ 13 , 2 ฯƒ 12 โŸฉ . \tilde{\sigma}^{M}=\langle\sigma_{11},\sigma_{22},\sigma_{33},\sqrt{2}\sigma_{% 23},\sqrt{2}\sigma_{13},\sqrt{2}\sigma_{12}\rangle.
  12. ฯƒ ~ : ฯƒ ~ = ฯƒ ~ M โ‹… ฯƒ ~ M = ฯƒ 11 2 + ฯƒ 22 2 + ฯƒ 33 2 + 2 ฯƒ 23 2 + 2 ฯƒ 13 2 + 2 ฯƒ 12 2 . \tilde{\sigma}:\tilde{\sigma}=\tilde{\sigma}^{M}\cdot\tilde{\sigma}^{M}=\sigma% _{11}^{2}+\sigma_{22}^{2}+\sigma_{33}^{2}+2\sigma_{23}^{2}+2\sigma_{13}^{2}+2% \sigma_{12}^{2}.
  13. D i j k l = D j i k l D_{ijkl}=D_{jikl}
  14. D i j k l = D i j l k D_{ijkl}=D_{ijlk}
  15. D ~ M = ( D 1111 D 1122 D 1133 2 D 1123 2 D 1113 2 D 1112 D 2211 D 2222 D 2233 2 D 2223 2 D 2213 2 D 2212 D 3311 D 3322 D 3333 2 D 3323 2 D 3313 2 D 3312 2 D 2311 2 D 2322 2 D 2333 2 D 2323 2 D 2313 2 D 2312 2 D 1311 2 D 1322 2 D 1333 2 D 1323 2 D 1313 2 D 1312 2 D 1211 2 D 1222 2 D 1233 2 D 1223 2 D 1213 2 D 1212 ) . \tilde{D}^{M}=\begin{pmatrix}D_{1111}&D_{1122}&D_{1133}&\sqrt{2}D_{1123}&\sqrt% {2}D_{1113}&\sqrt{2}D_{1112}\\ D_{2211}&D_{2222}&D_{2233}&\sqrt{2}D_{2223}&\sqrt{2}D_{2213}&\sqrt{2}D_{2212}% \\ D_{3311}&D_{3322}&D_{3333}&\sqrt{2}D_{3323}&\sqrt{2}D_{3313}&\sqrt{2}D_{3312}% \\ \sqrt{2}D_{2311}&\sqrt{2}D_{2322}&\sqrt{2}D_{2333}&2D_{2323}&2D_{2313}&2D_{231% 2}\\ \sqrt{2}D_{1311}&\sqrt{2}D_{1322}&\sqrt{2}D_{1333}&2D_{1323}&2D_{1313}&2D_{131% 2}\\ \sqrt{2}D_{1211}&\sqrt{2}D_{1222}&\sqrt{2}D_{1233}&2D_{1223}&2D_{1213}&2D_{121% 2}\\ \end{pmatrix}.

Voigt_profile.html

  1. 0
  2. 0
  3. e - ฮณ | t | - ฯƒ 2 t 2 / 2 e^{-\gamma|t|-\sigma^{2}t^{2}/2}
  4. V ( x ; ฯƒ , ฮณ ) = โˆซ - โˆž โˆž G ( x โ€ฒ ; ฯƒ ) L ( x - x โ€ฒ ; ฮณ ) d x โ€ฒ V(x;\sigma,\gamma)=\int_{-\infty}^{\infty}G(x^{\prime};\sigma)L(x-x^{\prime};% \gamma)\,dx^{\prime}
  5. G ( x ; ฯƒ ) G(x;\sigma)
  6. G ( x ; ฯƒ ) โ‰ก e - x 2 / ( 2 ฯƒ 2 ) ฯƒ 2 ฯ€ G(x;\sigma)\equiv\frac{e^{-x^{2}/(2\sigma^{2})}}{\sigma\sqrt{2\pi}}
  7. L ( x ; ฮณ ) L(x;\gamma)
  8. L ( x ; ฮณ ) โ‰ก ฮณ ฯ€ ( x 2 + ฮณ 2 ) . L(x;\gamma)\equiv\frac{\gamma}{\pi(x^{2}+\gamma^{2})}.
  9. V ( x ; ฯƒ , ฮณ ) = Re [ w ( z ) ] ฯƒ 2 ฯ€ V(x;\sigma,\gamma)=\frac{\textrm{Re}[w(z)]}{\sigma\sqrt{2\pi}}
  10. z = x + i ฮณ ฯƒ 2 . z=\frac{x+i\gamma}{\sigma\sqrt{2}}.
  11. โˆซ - โˆž โˆž V ( x ; ฯƒ , ฮณ ) d x = 1 \int_{-\infty}^{\infty}V(x;\sigma,\gamma)\,dx=1
  12. ฯ† f ( t ; ฯƒ , ฮณ ) = E ( e i x t ) = e - ฯƒ 2 t 2 / 2 - ฮณ | t | . \varphi_{f}(t;\sigma,\gamma)=E(e^{ixt})=e^{-\sigma^{2}t^{2}/2-\gamma|t|}.
  13. F ( x 0 ; ฮผ , ฯƒ ) = โˆซ - โˆž x 0 Re ( w ( z ) ) ฯƒ 2 ฯ€ d x = Re ( 1 ฯ€ โˆซ z ( - โˆž ) z ( x 0 ) w ( z ) d z ) F(x_{0};\mu,\sigma)=\int_{-\infty}^{x_{0}}\frac{\mathrm{Re}(w(z))}{\sigma\sqrt% {2\pi}}\,dx=\mathrm{Re}\left(\frac{1}{\sqrt{\pi}}\int_{z(-\infty)}^{z(x_{0})}w% (z)\,dz\right)
  14. 1 ฯ€ โˆซ w ( z ) d z = 1 ฯ€ โˆซ e - z 2 [ 1 - erf ( - i z ) ] d z \frac{1}{\sqrt{\pi}}\int w(z)\,dz=\frac{1}{\sqrt{\pi}}\int e^{-z^{2}}\left[1-% \mathrm{erf}(-iz)\right]\,dz
  15. 1 ฯ€ โˆซ w ( z ) d z = erf ( z ) 2 + i z 2 ฯ€ 2 F 2 ( 1 , 1 ; 3 2 , 2 ; - z 2 ) \frac{1}{\sqrt{\pi}}\int w(z)\,dz=\frac{\mathrm{erf}(z)}{2}+\frac{iz^{2}}{\pi}% \,_{2}F_{2}\left(1,1;\frac{3}{2},2;-z^{2}\right)
  16. F 2 2 ( ) \,{}_{2}F_{2}()
  17. F ( x ; ฮผ , ฯƒ ) = Re [ 1 2 + erf ( z ) 2 + i z 2 ฯ€ 2 F 2 ( 1 , 1 ; 3 2 , 2 ; - z 2 ) ] F(x;\mu,\sigma)=\mathrm{Re}\left[\frac{1}{2}+\frac{\mathrm{erf}(z)}{2}+\frac{% iz^{2}}{\pi}\,_{2}F_{2}\left(1,1;\frac{3}{2},2;-z^{2}\right)\right]
  18. f G = 2 ฯƒ 2 ln ( 2 ) . f_{\mathrm{G}}=2\sigma\sqrt{2\ln(2)}.\,
  19. f L = 2 ฮณ f_{L}=2\gamma
  20. f L / f G f_{L}/f_{G}
  21. f V f_{V}
  22. f V โ‰ˆ f G ( 1 - c 0 c 1 + ฯ• 2 + 2 c 1 ฯ• + c 0 2 c 1 2 ) f_{\mathrm{V}}\approx f_{\mathrm{G}}\left(1-c_{0}c_{1}+\sqrt{\phi^{2}+2c_{1}% \phi+c_{0}^{2}c_{1}^{2}}\right)
  23. c 0 c_{0}
  24. c 1 c_{1}
  25. f V โ‰ˆ 0.5346 f L + 0.2166 f L 2 + f G 2 . f_{\mathrm{V}}\approx 0.5346f_{\mathrm{L}}+\sqrt{0.2166f_{\mathrm{L}}^{2}+f_{% \mathrm{G}}^{2}}.
  26. ฮผ G \mu_{G}
  27. ฮผ L \mu_{L}
  28. ฮผ G + ฮผ L \mu_{G}+\mu_{L}
  29. ฯ† f ( t ; ฯƒ , ฮณ , ฮผ G , ฮผ L ) = e i ( ฮผ G + ฮผ L ) t - ฯƒ 2 t 2 / 2 - ฮณ | t | . \varphi_{f}(t;\sigma,\gamma,\mu_{\mathrm{G}},\mu_{\mathrm{L}})=e^{i(\mu_{% \mathrm{G}}+\mu_{\mathrm{L}})t-\sigma^{2}t^{2}/2-\gamma|t|}.
  30. ฮผ G + ฮผ L \mu_{G}+\mu_{L}
  31. U ( x , t ) + i V ( x , t ) = ฯ€ 4 t e z 2 erfc ( z ) = ฯ€ 4 t w ( i z ) U(x,t)+iV(x,t)=\sqrt{\frac{\pi}{4t}}e^{z^{2}}\,\text{erfc}(z)=\sqrt{\frac{\pi}% {4t}}w(iz)
  32. H ( a , u ) = U ( u / a , 1 / 4 a 2 ) a ฯ€ H(a,u)=\frac{U(u/a,1/4a^{2})}{a\sqrt{\pi}}
  33. z = ( 1 - i x ) / 2 t , z=(1-ix)/2\sqrt{t},
  34. V ( x ; ฯƒ , ฮณ ) = H ( a , u ) / ( 2 ฯ€ ฯƒ ) V(x;\sigma,\gamma)=H(a,u)/(\sqrt{2}\sqrt{\pi}\sigma)
  35. a = ฮณ / ( 2 ฯƒ ) a=\gamma/(\sqrt{2}\sigma)
  36. u = x / ( 2 ฯƒ ) u=x/(\sqrt{2}\sigma)
  37. V p ( x ) = ฮท โ‹… L ( x ) + ( 1 - ฮท ) โ‹… G ( x ) V_{p}(x)=\eta\cdot L(x)+(1-\eta)\cdot G(x)
  38. 0 < ฮท < 1 0<\eta<1
  39. ฮท \eta
  40. ฮท = 1.36603 ( f L / f ) - 0.47719 ( f L / f ) 2 + 0.11116 ( f L / f ) 3 \eta=1.36603(f_{L}/f)-0.47719(f_{L}/f)^{2}+0.11116(f_{L}/f)^{3}
  41. f = [ f G 5 + 2.69269 f G 4 f L + 2.42843 f G 3 f L 2 + 4.47163 f G 2 f L 3 + 0.07842 f G f L 4 + f L 5 ] 1 / 5 f=[f_{G}^{5}+2.69269f_{G}^{4}f_{L}+2.42843f_{G}^{3}f_{L}^{2}+4.47163f_{G}^{2}f% _{L}^{3}+0.07842f_{G}f_{L}^{4}+f_{L}^{5}]^{1/5}

Volatility_clustering.html

  1. | r t | |r_{t}|
  2. t {}_{t}
  3. t + ฯ„ {}_{t+ฯ„}

Voltage_doubler.html

  1. ฯ• 1 \phi_{1}
  2. ฯ• 2 \phi_{2}
  3. ฯ• 1 \phi_{1}
  4. ฯ• 1 \phi_{1}
  5. ฯ• 2 \phi_{2}

Volume-weighted_average_price.html

  1. P VWAP = โˆ‘ j P j โ‹… Q j โˆ‘ j Q j P_{\mathrm{VWAP}}=\frac{\sum_{j}{P_{j}\cdot Q_{j}}}{\sum_{j}{Q_{j}}}\,
  2. P VWAP P_{\mathrm{VWAP}}
  3. P j P_{j}
  4. j j
  5. Q j Q_{j}
  6. j j
  7. j j

Volume_element.html

  1. d V = ฯ ( u 1 , u 2 , u 3 ) d u 1 d u 2 d u 3 dV=\rho(u_{1},u_{2},u_{3})\,du_{1}\,du_{2}\,du_{3}
  2. u i u_{i}
  3. B B
  4. Volume ( B ) = โˆซ B ฯ ( u 1 , u 2 , u 3 ) d u 1 d u 2 d u 3 . \operatorname{Volume}(B)=\int_{B}\rho(u_{1},u_{2},u_{3})\,du_{1}\,du_{2}\,du_{% 3}.
  5. d V = u 1 2 sin u 2 d u 1 d u 2 d u 3 dV=u_{1}^{2}\sin u_{2}\,du_{1}\,du_{2}\,du_{3}
  6. ฯ = u 1 2 sin u 2 \rho=u_{1}^{2}\sin u_{2}
  7. d V = d x d y d z . dV=dx\,dy\,dz.
  8. x = x ( u 1 , u 2 , u 3 ) , y = y ( u 1 , u 2 , u 3 ) , z = z ( u 1 , u 2 , u 3 ) x=x(u_{1},u_{2},u_{3}),y=y(u_{1},u_{2},u_{3}),z=z(u_{1},u_{2},u_{3})
  9. d V = | โˆ‚ ( x , y , z ) โˆ‚ ( u 1 , u 2 , u 3 ) | d u 1 d u 2 d u 3 . dV=\left|\frac{\partial(x,y,z)}{\partial(u_{1},u_{2},u_{3})}\right|\,du_{1}\,% du_{2}\,du_{3}.
  10. x = ฯ cos ฮธ sin ฯ• y = ฯ sin ฮธ sin ฯ• z = ฯ cos ฯ• \begin{aligned}\displaystyle x&\displaystyle=\rho\cos\theta\sin\phi\\ \displaystyle y&\displaystyle=\rho\sin\theta\sin\phi\\ \displaystyle z&\displaystyle=\rho\cos\phi\end{aligned}
  11. | โˆ‚ ( x , y , z ) โˆ‚ ( ฯ , ฮธ , ฯ• ) | = ฯ 2 sin ฯ• \left|\frac{\partial(x,y,z)}{\partial(\rho,\theta,\phi)}\right|=\rho^{2}\sin\phi
  12. d V = ฯ 2 sin ฯ• d ฯ d ฮธ d ฯ• . dV=\rho^{2}\sin\phi\,d\rho\,d\theta\,d\phi.
  13. F * F^{*}
  14. F * ( u d y 1 โˆง โ‹ฏ โˆง d y n ) = ( u โˆ˜ F ) det ( โˆ‚ F j โˆ‚ x i ) d x 1 โˆง โ‹ฏ โˆง d x n F^{*}(u\;dy^{1}\wedge\cdots\wedge dy^{n})=(u\circ F)\det\left(\frac{\partial F% ^{j}}{\partial x^{i}}\right)dx^{1}\wedge\cdots\wedge dx^{n}
  15. X 1 , โ€ฆ , X k . X_{1},\dots,X_{k}.
  16. X i X_{i}
  17. X i X_{i}
  18. det ( X i โ‹… X j ) i , j = 1 โ€ฆ k . \sqrt{\det(X_{i}\cdot X_{j})_{i,j=1\dots k}}.
  19. ( u 1 , u 2 , โ€ฆ , u k ) (u_{1},u_{2},\dots,u_{k})
  20. p = u 1 X 1 + โ‹ฏ + u k X k . p=u_{1}X_{1}+\cdots+u_{k}X_{k}.
  21. d u i du_{i}
  22. det ( ( d u i X i ) โ‹… ( d u j X j ) ) i , j = 1 โ€ฆ k = det ( X i โ‹… X j ) i , j = 1 โ€ฆ k d u 1 d u 2 โ‹ฏ d u k . \sqrt{\det\left((du_{i}X_{i})\cdot(du_{j}X_{j})\right)_{i,j=1\dots k}}=\sqrt{% \det(X_{i}\cdot X_{j})_{i,j=1\dots k}}\;du_{1}\,du_{2}\,\cdots\,du_{k}.
  23. f ( x ) = 1 f(x)=1
  24. ฯ‰ = โ‹† 1 \omega=\star 1
  25. ฯต \epsilon
  26. ฯ‰ = | det g | d x 1 โˆง โ‹ฏ โˆง d x n \omega=\sqrt{|\det g|}\,dx^{1}\wedge\cdots\wedge dx^{n}
  27. det g \det g
  28. U โŠ‚ ๐‘ 2 U\subset\mathbf{R}^{2}
  29. ฯ† : U โ†’ ๐‘ n \varphi:U\to\mathbf{R}^{n}
  30. ๐‘ n \mathbf{R}^{n}
  31. f ( u 1 , u 2 ) d u 1 d u 2 f(u_{1},u_{2})\,du_{1}\,du_{2}
  32. Area ( B ) = โˆซ B f ( u 1 , u 2 ) d u 1 d u 2 . \operatorname{Area}(B)=\int_{B}f(u_{1},u_{2})\,du_{1}\,du_{2}.
  33. ฮป i j = โˆ‚ ฯ† i โˆ‚ u j \lambda_{ij}=\frac{\partial\varphi_{i}}{\partial u_{j}}
  34. g = ฮป T ฮป g=\lambda^{T}\lambda
  35. g i j = โˆ‘ k = 1 n ฮป k i ฮป k j = โˆ‘ k = 1 n โˆ‚ ฯ† k โˆ‚ u i โˆ‚ ฯ† k โˆ‚ u j . g_{ij}=\sum_{k=1}^{n}\lambda_{ki}\lambda_{kj}=\sum_{k=1}^{n}\frac{\partial% \varphi_{k}}{\partial u_{i}}\frac{\partial\varphi_{k}}{\partial u_{j}}.
  36. det g = | โˆ‚ ฯ† โˆ‚ u 1 โˆง โˆ‚ ฯ† โˆ‚ u 2 | 2 = det ( ฮป T ฮป ) \det g=\left|\frac{\partial\varphi}{\partial u_{1}}\wedge\frac{\partial\varphi% }{\partial u_{2}}\right|^{2}=\det(\lambda^{T}\lambda)
  37. f : U โ†’ U , f\colon U\to U,\,\!
  38. ( u 1 , u 2 ) (u_{1},u_{2})
  39. ( v 1 , v 2 ) (v_{1},v_{2})
  40. ( u 1 , u 2 ) = f ( v 1 , v 2 ) (u_{1},u_{2})=f(v_{1},v_{2})
  41. F i j = โˆ‚ f i โˆ‚ v j . F_{ij}=\frac{\partial f_{i}}{\partial v_{j}}.
  42. โˆ‚ ฯ† i โˆ‚ v j = โˆ‘ k = 1 2 โˆ‚ ฯ† i โˆ‚ u k โˆ‚ f k โˆ‚ v j \frac{\partial\varphi_{i}}{\partial v_{j}}=\sum_{k=1}^{2}\frac{\partial\varphi% _{i}}{\partial u_{k}}\frac{\partial f_{k}}{\partial v_{j}}
  43. g ~ = F T g F \tilde{g}=F^{T}gF
  44. g ~ \tilde{g}
  45. det g ~ = det g ( det F ) 2 . \det\tilde{g}=\det g(\det F)^{2}.
  46. B โŠ‚ U B\subset U
  47. Area ( B ) = โˆฌ B det g d u 1 d u 2 = โˆฌ B det g | det F | d v 1 d v 2 = โˆฌ B det g ~ d v 1 d v 2 . \begin{aligned}\displaystyle\mbox{Area}~{}(B)&\displaystyle=\iint_{B}\sqrt{% \det g}\;du_{1}\;du_{2}\\ &\displaystyle=\iint_{B}\sqrt{\det g}\;|\det F|\;dv_{1}\;dv_{2}\\ &\displaystyle=\iint_{B}\sqrt{\det\tilde{g}}\;dv_{1}\;dv_{2}.\end{aligned}
  48. ฯ• ( u 1 , u 2 ) = ( r cos u 1 sin u 2 , r sin u 1 sin u 2 , r cos u 2 ) . \phi(u_{1},u_{2})=(r\cos u_{1}\sin u_{2},r\sin u_{1}\sin u_{2},r\cos u_{2}).
  49. g = ( r 2 sin 2 u 2 0 0 r 2 ) , g=\begin{pmatrix}r^{2}\sin^{2}u_{2}&0\\ 0&r^{2}\end{pmatrix},
  50. ฯ‰ = det g d u 1 d u 2 = r 2 sin u 2 d u 1 d u 2 . \omega=\sqrt{\det g}\;du_{1}du_{2}=r^{2}\sin u_{2}\,du_{1}du_{2}.

Volume_form.html

  1. d x 1 โˆง โ‹ฏ โˆง d x n dx^{1}\wedge\cdots\wedge dx^{n}
  2. ฯ‰ ( X 1 , X 2 , โ€ฆ , X n ) > 0. \omega(X_{1},X_{2},\dots,X_{n})>0.
  3. ฮฉ n ( M ) \Omega^{n}(M)
  4. ฮผ ฯ‰ ( U ) = โˆซ U ฯ‰ . \mu_{\omega}(U)=\int_{U}\omega.\,\!
  5. โˆซ b a f d x = - โˆซ a b f d x \int_{b}^{a}f\,dx=-\int_{a}^{b}f\,dx
  6. d x dx
  7. โˆซ b a \int_{b}^{a}
  8. [ a , b ] [a,b]
  9. [ a , b ] ยฏ \overline{[a,b]}
  10. ( div X ) ฯ‰ = L X ฯ‰ = d ( X โŒŸ ฯ‰ ) (\operatorname{div}X)\omega=L_{X}\omega=d(X\;\lrcorner\;\omega)
  11. โˆซ M ( div X ) ฯ‰ = โˆซ โˆ‚ M X โŒŸ ฯ‰ , \int_{M}(\operatorname{div}X)\omega=\int_{\partial M}X\;\lrcorner\;\omega,
  12. โ‹€ n T e * G \bigwedge^{n}T_{e}^{*}G
  13. ฯ‰ g = L g - 1 * ฯ‰ e \omega_{g}=L_{g^{-1}}^{*}\omega_{e}
  14. ฯ‰ = | g | d x 1 โˆง โ€ฆ โˆง d x n \omega=\sqrt{|g|}dx^{1}\wedge\dots\wedge dx^{n}
  15. d x i dx^{i}
  16. | g | |g|
  17. ฯ‰ = vol n = ฮต = * ( 1 ) . \omega=\mathrm{vol}_{n}=\varepsilon=*(1).
  18. ฯ‰ \omega
  19. f ฯ‰ f\omega
  20. ฯ‰ , ฯ‰ โ€ฒ \omega,\omega^{\prime}
  21. ฯ‰ โ€ฒ \omega^{\prime}
  22. ฯ‰ \omega
  23. d x 1 โˆง โ‹ฏ โˆง d x n dx^{1}\wedge\cdots\wedge dx^{n}
  24. ฯ‰ M , ฯ‰ N \omega_{M},\omega_{N}
  25. m โˆˆ M , n โˆˆ N m\in M,n\in N
  26. f : U โ†’ V f\colon U\to V
  27. f * ฯ‰ N | V = ฯ‰ M | U f^{*}\omega_{N}|_{V}=\omega_{M}|_{U}
  28. ฯ‰ \omega
  29. ๐‘ \mathbf{R}
  30. f ( x ) := โˆซ 0 x ฯ‰ . f(x):=\int_{0}^{x}\omega.
  31. d x dx
  32. ฯ‰ \omega
  33. ฯ‰ = f * d x \omega=f^{*}dx
  34. ฯ‰ = f d x \omega=f\,dx
  35. m โˆˆ M m\in M
  36. ๐‘ ร— ๐‘ n - 1 \mathbf{R}\times\mathbf{R}^{n-1}
  37. ฮผ ( M ) \mu(M)
  38. ๐‘ n \mathbf{R}^{n}
  39. f : M โ†’ N f\colon M\to N
  40. ฯ‰ N \omega_{N}
  41. ฯ‰ M \omega_{M}
  42. ฮผ ( N ) = โˆซ N ฯ‰ N = โˆซ f ( M ) ฯ‰ N = โˆซ M f * ฯ‰ N = โˆซ M ฯ‰ M = ฮผ ( M ) \mu(N)=\int_{N}\omega_{N}=\int_{f(M)}\omega_{N}=\int_{M}f^{*}\omega_{N}=\int_{% M}\omega_{M}=\mu(M)\,
  43. ๐‘ โ†’ S 1 \mathbf{R}\to S^{1}