wpmath0000012_13

Tellegen's_theorem.html

  1. G G
  2. b b
  3. n t n_{t}
  4. W k W_{k}
  5. F k F_{k}
  6. k = 1 , 2 , , b k=1,2,\dots,b
  7. W 1 , W 2 , , W b W_{1},W_{2},\dots,W_{b}
  8. F 1 , F 2 , , F b F_{1},F_{2},\dots,F_{b}
  9. k = 1 b W k F k = 0. \sum_{k=1}^{b}W_{k}F_{k}=0.
  10. W k W_{k}
  11. F k F_{k}
  12. W k W_{k}
  13. F k F_{k}
  14. n t × n f n_{t}\times n_{f}
  15. 𝐀 𝐚 \mathbf{A_{a}}
  16. a i j a_{ij}
  17. a i j = { 1 , if flow j leaves node i - 1 , if flow j enters node i 0 , if flow j is not incident with node i a_{ij}=\begin{cases}1,&\,\text{if flow }j\,\text{ leaves node }i\\ -1,&\,\text{if flow }j\,\text{ enters node }i\\ 0,&\,\text{if flow }j\,\text{ is not incident with node }i\end{cases}
  18. P 0 P_{0}
  19. ( n t - 1 ) × n f (n_{t}-1)\times n_{f}
  20. 𝐀 \mathbf{A}
  21. a 0 j a_{0j}
  22. P 0 P_{0}
  23. 𝐀𝐅 = 𝟎 \mathbf{A}\mathbf{F}=\mathbf{0}
  24. 𝐖 = 𝐀 𝐓 𝐰 \mathbf{W}=\mathbf{A^{T}}\mathbf{w}
  25. w k w_{k}
  26. P 0 P_{0}
  27. 𝐖 𝐓 𝐅 = ( 𝐀 𝐓 𝐰 ) 𝐓 𝐅 = ( 𝐰 𝐓 𝐀 ) 𝐅 = 𝐰 𝐓 𝐀𝐅 = 𝟎 \displaystyle\mathbf{W^{T}}\mathbf{F}=\mathbf{(A^{T}w)^{T}}\mathbf{F}=\mathbf{% (w^{T}A)}\mathbf{F}=\mathbf{w^{T}AF}=\mathbf{0}
  28. 𝐀𝐅 = 𝟎 \mathbf{AF}=\mathbf{0}
  29. k = 1 b W k F k = 𝐖 𝐓 𝐅 = 0 \sum_{k=1}^{b}W_{k}F_{k}=\mathbf{W^{T}}\mathbf{F}=0
  30. j = 1 n P W j d Z j d t = k = 1 n f W k f k + j = 1 n P w j p j + j = 1 n t w j t j , j = 1 , , n p + n t \sum_{j=1}^{n_{P}}W_{j}\frac{\operatorname{d}Z_{j}}{\operatorname{d}t}=\sum_{k% =1}^{n_{f}}W_{k}f_{k}+\sum_{j=1}^{n_{P}}w_{j}p_{j}+\sum_{j=1}^{n_{t}}w_{j}t_{j% },\quad j=1,\dots,n_{p}+n_{t}
  31. p j p_{j}
  32. t j t_{j}
  33. d Z j d t \frac{\operatorname{d}Z_{j}}{\operatorname{d}t}

Temperature-responsive_polymer.html

  1. Δ G m i x = Δ H m i x - T Δ S m i x \Delta G_{mix}=\Delta H_{mix}-T\cdot\Delta S_{mix}
  2. Δ G m i x R T = ϕ 1 m 1 l n ϕ 1 + ϕ 2 m 2 l n ϕ 2 + χ ϕ 1 ϕ 2 \frac{\Delta G_{mix}}{RT}=\frac{\phi_{1}}{m}_{1}ln\phi_{1}+\frac{\phi_{2}}{m}_% {2}ln\phi_{2}+\chi\phi_{1}\phi_{2}

Temperature.html

  1. T 1 / T 2 = - Q 1 / Q 2 . ( 1 ) T_{1}/T_{2}=-Q_{1}/Q_{2}\,\,\,.\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,(1)
  2. U U
  3. S S
  4. V , N V,N
  5. U = U ( S , V , N U=U(S,V,N
  6. T = ( U S ) V , N . ( 2 ) T=\left(\frac{\partial U}{\partial S}\right)_{V,N}\,.\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(2)
  7. S S
  8. U U
  9. V , N V,N
  10. S = S ( U , V , N ) S=S(U,V,N)
  11. 1 T = ( S U ) V , N . ( 3 ) \frac{1}{T}=\left(\frac{\partial S}{\partial U}\right)_{V,N}\,.\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(3)
  12. U U
  13. S S
  14. T T
  15. Δ Q ΔQ
  16. Δ T ΔT
  17. C V = Δ Q Δ T C_{V}=\frac{\Delta Q}{\Delta T}
  18. E = k B T E=k_{B}T
  19. p V = n R T pV=nRT\,\!
  20. E k = 1 2 m v rms 2 = 3 2 k T , E\text{k}=\frac{1}{2}mv_{\mathrm{rms}}^{2}=\frac{3}{2}kT\,,
  21. k k
  22. efficiency = w c y q H = q H - q C q H = 1 - q C q H ( 4 ) \textrm{efficiency}=\frac{w_{cy}}{q_{H}}=\frac{q_{H}-q_{C}}{q_{H}}=1-\frac{q_{% C}}{q_{H}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,(4)
  23. q C q H = f ( T H , T C ) ( 5 ) \frac{q_{C}}{q_{H}}=f(T_{H},T_{C})\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(5)
  24. q 13 = q 1 q 2 q 2 q 3 q_{13}=\frac{q_{1}q_{2}}{q_{2}q_{3}}
  25. q 13 = f ( T 1 , T 3 ) = f ( T 1 , T 2 ) f ( T 2 , T 3 ) q_{13}=f(T_{1},T_{3})=f(T_{1},T_{2})f(T_{2},T_{3})
  26. q C q H = T C T H ( 6 ) \frac{q_{C}}{q_{H}}=\frac{T_{C}}{T_{H}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(6)
  27. efficiency = 1 - q C q H = 1 - T C T H ( 7 ) \textrm{efficiency}=1-\frac{q_{C}}{q_{H}}=1-\frac{T_{C}}{T_{H}}\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(7)
  28. q H T H - q C T C = 0 \frac{q_{H}}{T_{H}}-\frac{q_{C}}{T_{C}}=0
  29. d S = d q rev T ( 8 ) dS=\frac{dq_{\mathrm{rev}}}{T}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,(8)
  30. T = d q rev d S ( 9 ) T=\frac{dq_{\mathrm{rev}}}{dS}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,(9)
  31. T - 1 = d d E S ( E ) ( 10 ) {T}^{-1}=\frac{d}{dE}S(E)\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,% \,(10)
  32. τ \tau
  33. τ \tau
  34. T = k - 1 ln 2 τ 2 τ 1 ( E - E F ( 1 + 3 2 N ) ) , T=k^{-1}\ln 2\frac{\tau_{\mathrm{2}}}{\tau_{\mathrm{1}}}\left(E-E_{F}\left(1+% \frac{3}{2N}\right)\right),

Template:Circlenotation.html

  1. \odot

Template:Detainee.html

  1. ( (
  2. ) )
  3. ( (
  4. ) )
  5. ( (
  6. ) )
  7. ( (
  8. ) )

Template:DoseConcentrationClearance.html

  1. A U C 0 - AUC_{0-\infty}
  2. A U C τ , ss AUC_{\tau,\,\text{ss}}
  3. C av , ss = 1 τ A U C τ , ss C_{\,\text{av},\,\text{ss}}=\frac{1}{\tau}AUC_{\tau,\,\text{ss}}

Template:Gr::doc.html

  1. g = G * M R 2 g=\frac{G*M}{R^{2}}

Template:Infobox_color::doc.html

  1. 1 / 12 {1}/{12}
  2. 1 / 8 {1}/{8}
  3. 1 / 6 {1}/{6}
  4. 1 / 4 {1}/{4}
  5. 1 / 2 {1}/{2}
  6. 3 / 4 {3}/{4}
  7. π 6 \tfrac{\pi}{6}
  8. π 4 \tfrac{\pi}{4}
  9. π 3 \tfrac{\pi}{3}
  10. π 2 \tfrac{\pi}{2}
  11. π \pi
  12. 3 π 2 \tfrac{3\pi}{2}
  13. 2 π 2\pi

Template:Logic_functions.html

  1. ↛ \not\rightarrow
  2. ↚ \not\leftarrow
  3. \leftarrow
  4. \rightarrow

Template:Math::doc.html

  1. s i n π = 0 sinπ=0
  2. 1 2 1 3 = 1 6 \frac{1}{2}−\frac{1}{3}=\frac{1}{6}
  3. 1 + 2 = 3 1+2=3
  4. 1 + 2 = 3 1+2=3
  5. 1 + 2 = 3 1+2=3
  6. b i g = 11 + 2 = 3 big=11+2=3
  7. f o r m a t . " , " p a r a m s " : " 1 " : " l a b e l " : " f o r m u l a " , " d e s c r i p t i o n " : " w r a p a n i n l i n e f o r m u l a i n w i k i t e x t . " , " t y p e " : " s t r i n g " , " r e q u i r e d " : t r u e , " b i g " : " l a b e l " : " b i g g e r f o n t s i z e " , " d e s c r i p t i o n " : " i f s e t t o 1 , t h i s w i l l r e n d e r t h e f o r m u l a i n a b i g g e r f o n t s i z e , i n c r e a s e d t o 165 % " , " t y p e " : " s t r i n g " , " r e q u i r e d " : f a l s e , " s i z e " : " l a b e l " : " c u s t o m f o n t s i z e " , " d e s c r i p t i o n " : " u s e t h i s t o s p e c i f y y o u r o w n f o n t s i z e " , " t y p e " : " s t r i n g " , " r e q u i r e d " : f a l s e {format.","params":{"1":{"label":"formula","description":"% wrapaninlineformulainwikitext.","type":"string","required":true},"big":{"label% ":"biggerfontsize","description":"ifsetto‘1’,% thiswillrendertheformulainabiggerfontsize,increasedto165\%","type":"string","% required":false},"size":{"label":"customfontsize","description":"% usethistospecifyyourownfontsize","type":"string","required":false}}}

Template:ND.html

  1. E m , n = 2 n - m ( n m ) E_{m,n}=2^{n-m}{n\choose m}

Template:NumBlk::doc.html

  1. ( f * g ) [ n ] (f*g)[n]\,
  2. = m = - f [ n - m ] g [ m ] . =\sum_{m=-\infty}^{\infty}f[n-m]\cdot g[m].\,

Template:Radic::doc.html

  1. 2 \sqrt{2}
  2. 1 + 2 x 4 \sqrt{1+2x4}

Template:Squarenotation.html

  1. \square

Template:V2::doc.html

  1. v 2 = 2 × G × M R v_{2}=\sqrt{\frac{2\times G\times M}{R}}

Template:Val::doc.html

  1. s t {}^{st}
  2. 0 {}^{0}
  3. n d {}^{nd}
  4. r d {}^{rd}

Template_modeling_score.html

  1. ( 0 , 1 ] (0,1]
  2. TM-score = max [ 1 L target i L aligned 1 1 + ( d i d 0 ( L target ) ) 2 ] \,\text{TM-score}=\max\left[\frac{1}{L\text{target}}\sum_{i}^{L\text{aligned}}% \frac{1}{1+\left(\frac{d_{i}}{d_{0}(L\text{target})}\right)^{2}}\right]
  3. L target L\text{target}
  4. L aligned L\text{aligned}
  5. d i d_{i}
  6. i i
  7. d 0 ( L target ) = 1.24 L target - 15 3 - 1.8 d_{0}(L\text{target})=1.24\sqrt[3]{L\text{target}-15}-1.8

Tensor_derivative_(continuum_mechanics).html

  1. f 𝐯 𝐮 = D f ( 𝐯 ) [ 𝐮 ] = [ d d α f ( 𝐯 + α 𝐮 ) ] α = 0 \frac{\partial f}{\partial\mathbf{v}}\cdot\mathbf{u}=Df(\mathbf{v})[\mathbf{u}% ]=\left[\frac{d}{d\alpha}~{}f(\mathbf{v}+\alpha~{}\mathbf{u})\right]_{\alpha=0}
  2. f ( 𝐯 ) = f 1 ( 𝐯 ) + f 2 ( 𝐯 ) f(\mathbf{v})=f_{1}(\mathbf{v})+f_{2}(\mathbf{v})
  3. f 𝐯 𝐮 = ( f 1 𝐯 + f 2 𝐯 ) 𝐮 \frac{\partial f}{\partial\mathbf{v}}\cdot\mathbf{u}=\left(\frac{\partial f_{1% }}{\partial\mathbf{v}}+\frac{\partial f_{2}}{\partial\mathbf{v}}\right)\cdot% \mathbf{u}
  4. f ( 𝐯 ) = f 1 ( 𝐯 ) f 2 ( 𝐯 ) f(\mathbf{v})=f_{1}(\mathbf{v})~{}f_{2}(\mathbf{v})
  5. f 𝐯 𝐮 = ( f 1 𝐯 𝐮 ) f 2 ( 𝐯 ) + f 1 ( 𝐯 ) ( f 2 𝐯 𝐮 ) \frac{\partial f}{\partial\mathbf{v}}\cdot\mathbf{u}=\left(\frac{\partial f_{1% }}{\partial\mathbf{v}}\cdot\mathbf{u}\right)~{}f_{2}(\mathbf{v})+f_{1}(\mathbf% {v})~{}\left(\frac{\partial f_{2}}{\partial\mathbf{v}}\cdot\mathbf{u}\right)
  6. f ( 𝐯 ) = f 1 ( f 2 ( 𝐯 ) ) f(\mathbf{v})=f_{1}(f_{2}(\mathbf{v}))
  7. f 𝐯 𝐮 = f 1 f 2 f 2 𝐯 𝐮 \frac{\partial f}{\partial\mathbf{v}}\cdot\mathbf{u}=\frac{\partial f_{1}}{% \partial f_{2}}~{}\frac{\partial f_{2}}{\partial\mathbf{v}}\cdot\mathbf{u}
  8. 𝐟 𝐯 𝐮 = D 𝐟 ( 𝐯 ) [ 𝐮 ] = [ d d α 𝐟 ( 𝐯 + α 𝐮 ) ] α = 0 \frac{\partial\mathbf{f}}{\partial\mathbf{v}}\cdot\mathbf{u}=D\mathbf{f}(% \mathbf{v})[\mathbf{u}]=\left[\frac{d}{d\alpha}~{}\mathbf{f}(\mathbf{v}+\alpha% ~{}\mathbf{u})\right]_{\alpha=0}
  9. 𝐟 ( 𝐯 ) = 𝐟 1 ( 𝐯 ) + 𝐟 2 ( 𝐯 ) \mathbf{f}(\mathbf{v})=\mathbf{f}_{1}(\mathbf{v})+\mathbf{f}_{2}(\mathbf{v})
  10. 𝐟 𝐯 𝐮 = ( 𝐟 1 𝐯 + 𝐟 2 𝐯 ) 𝐮 \frac{\partial\mathbf{f}}{\partial\mathbf{v}}\cdot\mathbf{u}=\left(\frac{% \partial\mathbf{f}_{1}}{\partial\mathbf{v}}+\frac{\partial\mathbf{f}_{2}}{% \partial\mathbf{v}}\right)\cdot\mathbf{u}
  11. 𝐟 ( 𝐯 ) = 𝐟 1 ( 𝐯 ) × 𝐟 2 ( 𝐯 ) \mathbf{f}(\mathbf{v})=\mathbf{f}_{1}(\mathbf{v})\times\mathbf{f}_{2}(\mathbf{% v})
  12. 𝐟 𝐯 𝐮 = ( 𝐟 1 𝐯 𝐮 ) × 𝐟 2 ( 𝐯 ) + 𝐟 1 ( 𝐯 ) × ( 𝐟 2 𝐯 𝐮 ) \frac{\partial\mathbf{f}}{\partial\mathbf{v}}\cdot\mathbf{u}=\left(\frac{% \partial\mathbf{f}_{1}}{\partial\mathbf{v}}\cdot\mathbf{u}\right)\times\mathbf% {f}_{2}(\mathbf{v})+\mathbf{f}_{1}(\mathbf{v})\times\left(\frac{\partial% \mathbf{f}_{2}}{\partial\mathbf{v}}\cdot\mathbf{u}\right)
  13. 𝐟 ( 𝐯 ) = 𝐟 1 ( 𝐟 2 ( 𝐯 ) ) \mathbf{f}(\mathbf{v})=\mathbf{f}_{1}(\mathbf{f}_{2}(\mathbf{v}))
  14. 𝐟 𝐯 𝐮 = 𝐟 1 𝐟 2 ( 𝐟 2 𝐯 𝐮 ) \frac{\partial\mathbf{f}}{\partial\mathbf{v}}\cdot\mathbf{u}=\frac{\partial% \mathbf{f}_{1}}{\partial\mathbf{f}_{2}}\cdot\left(\frac{\partial\mathbf{f}_{2}% }{\partial\mathbf{v}}\cdot\mathbf{u}\right)
  15. f ( s y m b o l S ) f(symbol{S})
  16. s y m b o l S symbol{S}
  17. f ( s y m b o l S ) f(symbol{S})
  18. s y m b o l S symbol{S}
  19. s y m b o l S symbol{S}
  20. s y m b o l T symbol{T}
  21. f s y m b o l S : s y m b o l T = D f ( s y m b o l S ) [ s y m b o l T ] = [ d d α f ( s y m b o l S + α s y m b o l T ) ] α = 0 \frac{\partial f}{\partial symbol{S}}:symbol{T}=Df(symbol{S})[symbol{T}]=\left% [\frac{d}{d\alpha}~{}f(symbol{S}+\alpha~{}symbol{T})\right]_{\alpha=0}
  22. s y m b o l T symbol{T}
  23. f ( s y m b o l S ) = f 1 ( s y m b o l S ) + f 2 ( s y m b o l S ) f(symbol{S})=f_{1}(symbol{S})+f_{2}(symbol{S})
  24. f s y m b o l S : s y m b o l T = ( f 1 s y m b o l S + f 2 s y m b o l S ) : s y m b o l T \frac{\partial f}{\partial symbol{S}}:symbol{T}=\left(\frac{\partial f_{1}}{% \partial symbol{S}}+\frac{\partial f_{2}}{\partial symbol{S}}\right):symbol{T}
  25. f ( s y m b o l S ) = f 1 ( s y m b o l S ) f 2 ( s y m b o l S ) f(symbol{S})=f_{1}(symbol{S})~{}f_{2}(symbol{S})
  26. f s y m b o l S : s y m b o l T = ( f 1 s y m b o l S : s y m b o l T ) f 2 ( s y m b o l S ) + f 1 ( s y m b o l S ) ( f 2 s y m b o l S : s y m b o l T ) \frac{\partial f}{\partial symbol{S}}:symbol{T}=\left(\frac{\partial f_{1}}{% \partial symbol{S}}:symbol{T}\right)~{}f_{2}(symbol{S})+f_{1}(symbol{S})~{}% \left(\frac{\partial f_{2}}{\partial symbol{S}}:symbol{T}\right)
  27. f ( s y m b o l S ) = f 1 ( f 2 ( s y m b o l S ) ) f(symbol{S})=f_{1}(f_{2}(symbol{S}))
  28. f s y m b o l S : s y m b o l T = f 1 f 2 ( f 2 s y m b o l S : s y m b o l T ) \frac{\partial f}{\partial symbol{S}}:symbol{T}=\frac{\partial f_{1}}{\partial f% _{2}}~{}\left(\frac{\partial f_{2}}{\partial symbol{S}}:symbol{T}\right)
  29. s y m b o l F ( s y m b o l S ) symbol{F}(symbol{S})
  30. s y m b o l S symbol{S}
  31. s y m b o l F ( s y m b o l S ) symbol{F}(symbol{S})
  32. s y m b o l S symbol{S}
  33. s y m b o l S symbol{S}
  34. s y m b o l T symbol{T}
  35. s y m b o l F s y m b o l S : s y m b o l T = D s y m b o l F ( s y m b o l S ) [ s y m b o l T ] = [ d d α s y m b o l F ( s y m b o l S + α s y m b o l T ) ] α = 0 \frac{\partial symbol{F}}{\partial symbol{S}}:symbol{T}=Dsymbol{F}(symbol{S})[% symbol{T}]=\left[\frac{d}{d\alpha}~{}symbol{F}(symbol{S}+\alpha~{}symbol{T})% \right]_{\alpha=0}
  36. s y m b o l T symbol{T}
  37. s y m b o l F ( s y m b o l S ) = s y m b o l F 1 ( s y m b o l S ) + s y m b o l F 2 ( s y m b o l S ) symbol{F}(symbol{S})=symbol{F}_{1}(symbol{S})+symbol{F}_{2}(symbol{S})
  38. s y m b o l F s y m b o l S : s y m b o l T = ( s y m b o l F 1 s y m b o l S + s y m b o l F 2 s y m b o l S ) : s y m b o l T \frac{\partial symbol{F}}{\partial symbol{S}}:symbol{T}=\left(\frac{\partial symbol% {F}_{1}}{\partial symbol{S}}+\frac{\partial symbol{F}_{2}}{\partial symbol{S}}% \right):symbol{T}
  39. s y m b o l F ( s y m b o l S ) = s y m b o l F 1 ( s y m b o l S ) \cdotsymbol F 2 ( s y m b o l S ) symbol{F}(symbol{S})=symbol{F}_{1}(symbol{S})\cdotsymbol{F}_{2}(symbol{S})
  40. s y m b o l F s y m b o l S : s y m b o l T = ( s y m b o l F 1 s y m b o l S : s y m b o l T ) \cdotsymbol F 2 ( s y m b o l S ) + s y m b o l F 1 ( s y m b o l S ) ( s y m b o l F 2 s y m b o l S : s y m b o l T ) \frac{\partial symbol{F}}{\partial symbol{S}}:symbol{T}=\left(\frac{\partial symbol% {F}_{1}}{\partial symbol{S}}:symbol{T}\right)\cdotsymbol{F}_{2}(symbol{S})+% symbol{F}_{1}(symbol{S})\cdot\left(\frac{\partial symbol{F}_{2}}{\partial symbol% {S}}:symbol{T}\right)
  41. s y m b o l F ( s y m b o l S ) = s y m b o l F 1 ( s y m b o l F 2 ( s y m b o l S ) ) symbol{F}(symbol{S})=symbol{F}_{1}(symbol{F}_{2}(symbol{S}))
  42. s y m b o l F s y m b o l S : s y m b o l T = s y m b o l F 1 s y m b o l F 2 : ( s y m b o l F 2 s y m b o l S : s y m b o l T ) \frac{\partial symbol{F}}{\partial symbol{S}}:symbol{T}=\frac{\partial symbol{% F}_{1}}{\partial symbol{F}_{2}}:\left(\frac{\partial symbol{F}_{2}}{\partial symbol% {S}}:symbol{T}\right)
  43. f ( s y m b o l S ) = f 1 ( s y m b o l F 2 ( s y m b o l S ) ) f(symbol{S})=f_{1}(symbol{F}_{2}(symbol{S}))
  44. f s y m b o l S : s y m b o l T = f 1 s y m b o l F 2 : ( s y m b o l F 2 s y m b o l S : s y m b o l T ) \frac{\partial f}{\partial symbol{S}}:symbol{T}=\frac{\partial f_{1}}{\partial symbol% {F}_{2}}:\left(\frac{\partial symbol{F}_{2}}{\partial symbol{S}}:symbol{T}\right)
  45. s y m b o l s y m b o l T symbol{\nabla}symbol{T}
  46. s y m b o l T ( 𝐱 ) symbol{T}(\mathbf{x})
  47. s y m b o l s y m b o l T 𝐜 = lim α 0 d d α s y m b o l T ( 𝐱 + α 𝐜 ) symbol{\nabla}symbol{T}\cdot\mathbf{c}=\lim_{\alpha\rightarrow 0}\quad\cfrac{d% }{d\alpha}~{}symbol{T}(\mathbf{x}+\alpha\mathbf{c})
  48. 𝐞 1 , 𝐞 2 , 𝐞 3 \mathbf{e}_{1},\mathbf{e}_{2},\mathbf{e}_{3}
  49. x 1 , x 2 , x 3 x_{1},x_{2},x_{3}
  50. s y m b o l T symbol{T}
  51. s y m b o l s y m b o l T = 𝐞 i s y m b o l T x i symbol{\nabla}symbol{T}=\mathbf{e}_{i}\otimes\cfrac{\partial{symbol{T}}}{% \partial x_{i}}
  52. 𝐱 = x i 𝐞 i \mathbf{x}=x_{i}~{}\mathbf{e}_{i}
  53. 𝐜 = c i 𝐞 i \mathbf{c}=c_{i}~{}\mathbf{e}_{i}
  54. s y m b o l s y m b o l T 𝐜 = d d α s y m b o l T ( x 1 + α c 1 , x 2 + α c 2 , x 3 + α c 3 ) | α = 0 d d α s y m b o l T ( y 1 , y 2 , y 3 ) | α = 0 = [ s y m b o l T y 1 y 1 α + s y m b o l T y 2 y 2 α + s y m b o l T y 3 y 3 α ] α = 0 = [ s y m b o l T y 1 c 1 + s y m b o l T y 2 c 2 + s y m b o l T y 3 c 3 ] α = 0 = s y m b o l T x 1 c 1 + s y m b o l T x 2 c 2 + s y m b o l T x 3 c 3 s y m b o l T x i c i = s y m b o l T x i ( 𝐞 i 𝐜 ) = [ 𝐞 i s y m b o l T x i ] 𝐜 \begin{aligned}\displaystyle symbol{\nabla}symbol{T}\cdot\mathbf{c}&% \displaystyle=\left.\cfrac{d}{d\alpha}~{}symbol{T}(x_{1}+\alpha c_{1},x_{2}+% \alpha c_{2},x_{3}+\alpha c_{3})\right|_{\alpha=0}\equiv\left.\cfrac{d}{d% \alpha}~{}symbol{T}(y_{1},y_{2},y_{3})\right|_{\alpha=0}\\ &\displaystyle=\left[\cfrac{\partial{symbol{T}}}{\partial y_{1}}~{}\cfrac{% \partial y_{1}}{\partial\alpha}+\cfrac{\partial{symbol{T}}}{\partial y_{2}}~{}% \cfrac{\partial y_{2}}{\partial\alpha}+\cfrac{\partial{symbol{T}}}{\partial y_% {3}}~{}\cfrac{\partial y_{3}}{\partial\alpha}\right]_{\alpha=0}=\left[\cfrac{% \partial{symbol{T}}}{\partial y_{1}}~{}c_{1}+\cfrac{\partial{symbol{T}}}{% \partial y_{2}}~{}c_{2}+\cfrac{\partial{symbol{T}}}{\partial y_{3}}~{}c_{3}% \right]_{\alpha=0}\\ &\displaystyle=\cfrac{\partial{symbol{T}}}{\partial x_{1}}~{}c_{1}+\cfrac{% \partial{symbol{T}}}{\partial x_{2}}~{}c_{2}+\cfrac{\partial{symbol{T}}}{% \partial x_{3}}~{}c_{3}\equiv\cfrac{\partial{symbol{T}}}{\partial x_{i}}~{}c_{% i}=\cfrac{\partial{symbol{T}}}{\partial x_{i}}~{}(\mathbf{e}_{i}\cdot\mathbf{c% })=\left[\mathbf{e}_{i}\otimes\cfrac{\partial{symbol{T}}}{\partial x_{i}}% \right]\cdot\mathbf{c}\qquad\square\end{aligned}
  55. ϕ \phi
  56. s y m b o l S symbol{S}
  57. s y m b o l ϕ = ϕ x i 𝐞 i s y m b o l 𝐯 = ( v j 𝐞 j ) x i 𝐞 i = v j x i 𝐞 j 𝐞 i s y m b o l s y m b o l S = ( S j k 𝐞 j 𝐞 k ) x i 𝐞 i = S j k x i 𝐞 j 𝐞 k 𝐞 i \begin{aligned}\displaystyle symbol{\nabla}\phi&\displaystyle=\cfrac{\partial% \phi}{\partial x_{i}}~{}\mathbf{e}_{i}\\ \displaystyle symbol{\nabla}\mathbf{v}&\displaystyle=\cfrac{\partial(v_{j}% \mathbf{e}_{j})}{\partial x_{i}}\otimes\mathbf{e}_{i}=\cfrac{\partial v_{j}}{% \partial x_{i}}~{}\mathbf{e}_{j}\otimes\mathbf{e}_{i}\\ \displaystyle symbol{\nabla}symbol{S}&\displaystyle=\cfrac{\partial(S_{jk}% \mathbf{e}_{j}\otimes\mathbf{e}_{k})}{\partial x_{i}}\otimes\mathbf{e}_{i}=% \cfrac{\partial S_{jk}}{\partial x_{i}}~{}\mathbf{e}_{j}\otimes\mathbf{e}_{k}% \otimes\mathbf{e}_{i}\end{aligned}
  58. 𝐠 1 , 𝐠 2 , 𝐠 3 \mathbf{g}^{1},\mathbf{g}^{2},\mathbf{g}^{3}
  59. ξ 1 , ξ 2 , ξ 3 \xi^{1},\xi^{2},\xi^{3}
  60. s y m b o l T symbol{T}
  61. s y m b o l s y m b o l T = s y m b o l T ξ i 𝐠 i symbol{\nabla}symbol{T}=\cfrac{\partial{symbol{T}}}{\partial\xi^{i}}\otimes% \mathbf{g}^{i}
  62. ϕ \phi
  63. s y m b o l S symbol{S}
  64. s y m b o l ϕ = ϕ ξ i 𝐠 i s y m b o l 𝐯 = ( v j 𝐠 j ) ξ i 𝐠 i = ( v j ξ i + v k Γ i k j ) 𝐠 j 𝐠 i = ( v j ξ i - v k Γ i j k ) 𝐠 j 𝐠 i s y m b o l s y m b o l S = ( S j k 𝐠 j 𝐠 k ) ξ i 𝐠 i = ( S j k ξ i - S l k Γ i j l - S j l Γ i k l ) 𝐠 j 𝐠 k 𝐠 i \begin{aligned}\displaystyle symbol{\nabla}\phi&\displaystyle=\cfrac{\partial% \phi}{\partial\xi^{i}}~{}\mathbf{g}^{i}\\ \displaystyle symbol{\nabla}\mathbf{v}&\displaystyle=\cfrac{\partial(v^{j}% \mathbf{g}_{j})}{\partial\xi^{i}}\otimes\mathbf{g}^{i}=\left(\cfrac{\partial v% ^{j}}{\partial\xi^{i}}+v^{k}~{}\Gamma_{ik}^{j}\right)~{}\mathbf{g}_{j}\otimes% \mathbf{g}^{i}=\left(\cfrac{\partial v_{j}}{\partial\xi^{i}}-v_{k}~{}\Gamma_{% ij}^{k}\right)~{}\mathbf{g}^{j}\otimes\mathbf{g}^{i}\\ \displaystyle symbol{\nabla}symbol{S}&\displaystyle=\cfrac{\partial(S_{jk}~{}% \mathbf{g}^{j}\otimes\mathbf{g}^{k})}{\partial\xi^{i}}\otimes\mathbf{g}^{i}=% \left(\cfrac{\partial S_{jk}}{\partial\xi_{i}}-S_{lk}~{}\Gamma_{ij}^{l}-S_{jl}% ~{}\Gamma_{ik}^{l}\right)~{}\mathbf{g}^{j}\otimes\mathbf{g}^{k}\otimes\mathbf{% g}^{i}\end{aligned}
  65. Γ i j k \Gamma_{ij}^{k}
  66. Γ i j k 𝐠 k = 𝐠 i ξ j Γ i j k = 𝐠 i ξ j 𝐠 k = - 𝐠 i 𝐠 k ξ j \Gamma_{ij}^{k}~{}\mathbf{g}_{k}=\cfrac{\partial\mathbf{g}_{i}}{\partial\xi^{j% }}\quad\implies\quad\Gamma_{ij}^{k}=\cfrac{\partial\mathbf{g}_{i}}{\partial\xi% ^{j}}\cdot\mathbf{g}_{k}=-\mathbf{g}_{i}\cdot\cfrac{\partial\mathbf{g}^{k}}{% \partial\xi^{j}}
  67. s y m b o l ϕ = ϕ r 𝐞 r + 1 r ϕ θ 𝐞 θ + ϕ z 𝐞 z s y m b o l 𝐯 = v r r 𝐞 r 𝐞 r + 1 r ( v r θ - v θ ) 𝐞 r 𝐞 θ + v r z 𝐞 r 𝐞 z + v θ r 𝐞 θ 𝐞 r + 1 r ( v θ θ + v r ) 𝐞 θ 𝐞 θ + v θ z 𝐞 θ 𝐞 z + v z r 𝐞 z 𝐞 r + 1 r v z θ 𝐞 z 𝐞 θ + v z z 𝐞 z 𝐞 z s y m b o l s y m b o l S = S r r r 𝐞 r 𝐞 r 𝐞 r + 1 r [ S r r θ - ( S θ r + S r θ ) ] 𝐞 r 𝐞 r 𝐞 θ + S r r z 𝐞 r 𝐞 r 𝐞 z + S r θ r 𝐞 r 𝐞 θ 𝐞 r + 1 r [ S r θ θ + ( S r r - S θ θ ) ] 𝐞 r 𝐞 θ 𝐞 θ + S r θ z 𝐞 r 𝐞 θ 𝐞 z + S r z r 𝐞 r 𝐞 z 𝐞 r + 1 r [ S r z θ - S θ z ] 𝐞 r 𝐞 z 𝐞 θ + S r z z 𝐞 r 𝐞 z 𝐞 z + S θ r r 𝐞 θ 𝐞 r 𝐞 r + 1 r [ S θ r θ + ( S r r - S θ θ ) ] 𝐞 θ 𝐞 r 𝐞 θ + S θ r z 𝐞 θ 𝐞 r 𝐞 z + S θ θ r 𝐞 θ 𝐞 θ 𝐞 r + 1 r [ S θ θ θ + ( S r θ + S θ r ) ] 𝐞 θ 𝐞 θ 𝐞 θ + S θ θ z 𝐞 θ 𝐞 θ 𝐞 z + S θ z r 𝐞 θ 𝐞 z 𝐞 r + 1 r [ S θ z θ + S r z ] 𝐞 θ 𝐞 z 𝐞 θ + S θ z z 𝐞 θ 𝐞 z 𝐞 z + S z r r 𝐞 z 𝐞 r 𝐞 r + 1 r [ S z r θ - S z θ ] 𝐞 z 𝐞 r 𝐞 θ + S z r z 𝐞 z 𝐞 r 𝐞 z + S z θ r 𝐞 z 𝐞 θ 𝐞 r + 1 r [ S z θ θ + S z r ] 𝐞 z 𝐞 θ 𝐞 θ + S z θ z 𝐞 z 𝐞 θ 𝐞 z + S z z r 𝐞 z 𝐞 z 𝐞 r + 1 r S z z θ 𝐞 z 𝐞 z 𝐞 θ + S z z z 𝐞 z 𝐞 z 𝐞 z \begin{aligned}\displaystyle symbol{\nabla}\phi&\displaystyle=\cfrac{\partial% \phi}{\partial r}~{}\mathbf{e}_{r}+\cfrac{1}{r}~{}\cfrac{\partial\phi}{% \partial\theta}~{}\mathbf{e}_{\theta}+\cfrac{\partial\phi}{\partial z}~{}% \mathbf{e}_{z}\\ \displaystyle symbol{\nabla}\mathbf{v}&\displaystyle=\cfrac{\partial v_{r}}{% \partial r}~{}\mathbf{e}_{r}\otimes\mathbf{e}_{r}+\cfrac{1}{r}\left(\cfrac{% \partial v_{r}}{\partial\theta}-v_{\theta}\right)~{}\mathbf{e}_{r}\otimes% \mathbf{e}_{\theta}+\cfrac{\partial v_{r}}{\partial z}~{}\mathbf{e}_{r}\otimes% \mathbf{e}_{z}+\cfrac{\partial v_{\theta}}{\partial r}~{}\mathbf{e}_{\theta}% \otimes\mathbf{e}_{r}+\cfrac{1}{r}\left(\cfrac{\partial v_{\theta}}{\partial% \theta}+v_{r}\right)~{}\mathbf{e}_{\theta}\otimes\mathbf{e}_{\theta}\\ &\displaystyle\quad+\cfrac{\partial v_{\theta}}{\partial z}~{}\mathbf{e}_{% \theta}\otimes\mathbf{e}_{z}+\cfrac{\partial v_{z}}{\partial r}~{}\mathbf{e}_{% z}\otimes\mathbf{e}_{r}+\cfrac{1}{r}\cfrac{\partial v_{z}}{\partial\theta}~{}% \mathbf{e}_{z}\otimes\mathbf{e}_{\theta}+\cfrac{\partial v_{z}}{\partial z}~{}% \mathbf{e}_{z}\otimes\mathbf{e}_{z}\\ \displaystyle symbol{\nabla}symbol{S}&\displaystyle=\frac{\partial S_{rr}}{% \partial r}~{}\mathbf{e}_{r}\otimes\mathbf{e}_{r}\otimes\mathbf{e}_{r}+\cfrac{% 1}{r}\left[\frac{\partial S_{rr}}{\partial\theta}-(S_{\theta r}+S_{r\theta})% \right]~{}\mathbf{e}_{r}\otimes\mathbf{e}_{r}\otimes\mathbf{e}_{\theta}+\frac{% \partial S_{rr}}{\partial z}~{}\mathbf{e}_{r}\otimes\mathbf{e}_{r}\otimes% \mathbf{e}_{z}+\frac{\partial S_{r\theta}}{\partial r}~{}\mathbf{e}_{r}\otimes% \mathbf{e}_{\theta}\otimes\mathbf{e}_{r}\\ &\displaystyle\quad+\cfrac{1}{r}\left[\frac{\partial S_{r\theta}}{\partial% \theta}+(S_{rr}-S_{\theta\theta})\right]~{}\mathbf{e}_{r}\otimes\mathbf{e}_{% \theta}\otimes\mathbf{e}_{\theta}+\frac{\partial S_{r\theta}}{\partial z}~{}% \mathbf{e}_{r}\otimes\mathbf{e}_{\theta}\otimes\mathbf{e}_{z}+\frac{\partial S% _{rz}}{\partial r}~{}\mathbf{e}_{r}\otimes\mathbf{e}_{z}\otimes\mathbf{e}_{r}+% \cfrac{1}{r}\left[\frac{\partial S_{rz}}{\partial\theta}-S_{\theta z}\right]~{% }\mathbf{e}_{r}\otimes\mathbf{e}_{z}\otimes\mathbf{e}_{\theta}\\ &\displaystyle\quad+\frac{\partial S_{rz}}{\partial z}~{}\mathbf{e}_{r}\otimes% \mathbf{e}_{z}\otimes\mathbf{e}_{z}+\frac{\partial S_{\theta r}}{\partial r}~{% }\mathbf{e}_{\theta}\otimes\mathbf{e}_{r}\otimes\mathbf{e}_{r}+\cfrac{1}{r}% \left[\frac{\partial S_{\theta r}}{\partial\theta}+(S_{rr}-S_{\theta\theta})% \right]~{}\mathbf{e}_{\theta}\otimes\mathbf{e}_{r}\otimes\mathbf{e}_{\theta}+% \frac{\partial S_{\theta r}}{\partial z}~{}\mathbf{e}_{\theta}\otimes\mathbf{e% }_{r}\otimes\mathbf{e}_{z}\\ &\displaystyle\quad+\frac{\partial S_{\theta\theta}}{\partial r}~{}\mathbf{e}_% {\theta}\otimes\mathbf{e}_{\theta}\otimes\mathbf{e}_{r}+\cfrac{1}{r}\left[% \frac{\partial S_{\theta\theta}}{\partial\theta}+(S_{r\theta}+S_{\theta r})% \right]~{}\mathbf{e}_{\theta}\otimes\mathbf{e}_{\theta}\otimes\mathbf{e}_{% \theta}+\frac{\partial S_{\theta\theta}}{\partial z}~{}\mathbf{e}_{\theta}% \otimes\mathbf{e}_{\theta}\otimes\mathbf{e}_{z}+\frac{\partial S_{\theta z}}{% \partial r}~{}\mathbf{e}_{\theta}\otimes\mathbf{e}_{z}\otimes\mathbf{e}_{r}\\ &\displaystyle\quad+\cfrac{1}{r}\left[\frac{\partial S_{\theta z}}{\partial% \theta}+S_{rz}\right]~{}\mathbf{e}_{\theta}\otimes\mathbf{e}_{z}\otimes\mathbf% {e}_{\theta}+\frac{\partial S_{\theta z}}{\partial z}~{}\mathbf{e}_{\theta}% \otimes\mathbf{e}_{z}\otimes\mathbf{e}_{z}+\frac{\partial S_{zr}}{\partial r}~% {}\mathbf{e}_{z}\otimes\mathbf{e}_{r}\otimes\mathbf{e}_{r}+\cfrac{1}{r}\left[% \frac{\partial S_{zr}}{\partial\theta}-S_{z\theta}\right]~{}\mathbf{e}_{z}% \otimes\mathbf{e}_{r}\otimes\mathbf{e}_{\theta}\\ &\displaystyle\quad+\frac{\partial S_{zr}}{\partial z}~{}\mathbf{e}_{z}\otimes% \mathbf{e}_{r}\otimes\mathbf{e}_{z}+\frac{\partial S_{z\theta}}{\partial r}~{}% \mathbf{e}_{z}\otimes\mathbf{e}_{\theta}\otimes\mathbf{e}_{r}+\cfrac{1}{r}% \left[\frac{\partial S_{z\theta}}{\partial\theta}+S_{zr}\right]~{}\mathbf{e}_{% z}\otimes\mathbf{e}_{\theta}\otimes\mathbf{e}_{\theta}+\frac{\partial S_{z% \theta}}{\partial z}~{}\mathbf{e}_{z}\otimes\mathbf{e}_{\theta}\otimes\mathbf{% e}_{z}\\ &\displaystyle\quad+\frac{\partial S_{zz}}{\partial r}~{}\mathbf{e}_{z}\otimes% \mathbf{e}_{z}\otimes\mathbf{e}_{r}+\cfrac{1}{r}~{}\frac{\partial S_{zz}}{% \partial\theta}~{}\mathbf{e}_{z}\otimes\mathbf{e}_{z}\otimes\mathbf{e}_{\theta% }+\frac{\partial S_{zz}}{\partial z}~{}\mathbf{e}_{z}\otimes\mathbf{e}_{z}% \otimes\mathbf{e}_{z}\end{aligned}
  68. s y m b o l T ( 𝐱 ) symbol{T}(\mathbf{x})
  69. ( s y m b o l \cdotsymbol T ) 𝐜 = s y m b o l ( 𝐜 \cdotsymbol T ) ; \qquadsymbol 𝐯 = tr ( s y m b o l 𝐯 ) (symbol{\nabla}\cdotsymbol{T})\cdot\mathbf{c}=symbol{\nabla}\cdot(\mathbf{c}% \cdotsymbol{T})~{};\qquadsymbol{\nabla}\cdot\mathbf{v}=\,\text{tr}(symbol{% \nabla}\mathbf{v})
  70. s y m b o l T symbol{T}
  71. s y m b o l S symbol{S}
  72. s y m b o l 𝐯 = v i x i s y m b o l \cdotsymbol S = S k i x k 𝐞 i \begin{aligned}\displaystyle symbol{\nabla}\cdot\mathbf{v}&\displaystyle=% \cfrac{\partial v_{i}}{\partial x_{i}}\\ \displaystyle symbol{\nabla}\cdotsymbol{S}&\displaystyle=\cfrac{\partial S_{ki% }}{\partial x_{k}}~{}\mathbf{e}_{i}\end{aligned}
  73. s y m b o l \cdotsymbol S div s y m b o l S = s y m b o l \cdotsymbol S T . symbol{\nabla}\cdotsymbol{S}\neq\operatorname{div}symbol{S}=symbol{\nabla}% \cdotsymbol{S}^{\mathrm{T}}.
  74. s y m b o l S symbol{S}
  75. s y m b o l 𝐯 = ( v i ξ i + v k Γ i k i ) s y m b o l \cdotsymbol S = ( S i k ξ i - S l k Γ i i l - S i l Γ i k l ) 𝐠 k \begin{aligned}\displaystyle symbol{\nabla}\cdot\mathbf{v}&\displaystyle=\left% (\cfrac{\partial v^{i}}{\partial\xi^{i}}+v^{k}~{}\Gamma_{ik}^{i}\right)\\ \displaystyle symbol{\nabla}\cdotsymbol{S}&\displaystyle=\left(\cfrac{\partial S% _{ik}}{\partial\xi_{i}}-S_{lk}~{}\Gamma_{ii}^{l}-S_{il}~{}\Gamma_{ik}^{l}% \right)~{}\mathbf{g}^{k}\end{aligned}
  76. s y m b o l 𝐯 = v r r + 1 r ( v θ θ + v r ) + v z z s y m b o l \cdotsymbol S = S r r r 𝐞 r + S r θ r 𝐞 θ + S r z r 𝐞 z + 1 r [ S θ r θ + ( S r r - S θ θ ) ] 𝐞 r + 1 r [ S θ θ θ + ( S r θ + S θ r ) ] 𝐞 θ + 1 r [ S θ z θ + S r z ] 𝐞 z + S z r z 𝐞 r + S z θ z 𝐞 θ + S z z z 𝐞 z \begin{aligned}\displaystyle symbol{\nabla}\cdot\mathbf{v}&\displaystyle=% \cfrac{\partial v_{r}}{\partial r}+\cfrac{1}{r}\left(\cfrac{\partial v_{\theta% }}{\partial\theta}+v_{r}\right)+\cfrac{\partial v_{z}}{\partial z}\\ \displaystyle symbol{\nabla}\cdotsymbol{S}&\displaystyle=\frac{\partial S_{rr}% }{\partial r}~{}\mathbf{e}_{r}+\frac{\partial S_{r\theta}}{\partial r}~{}% \mathbf{e}_{\theta}+\frac{\partial S_{rz}}{\partial r}~{}\mathbf{e}_{z}\\ &\displaystyle+\cfrac{1}{r}\left[\frac{\partial S_{\theta r}}{\partial\theta}+% (S_{rr}-S_{\theta\theta})\right]~{}\mathbf{e}_{r}+\cfrac{1}{r}\left[\frac{% \partial S_{\theta\theta}}{\partial\theta}+(S_{r\theta}+S_{\theta r})\right]~{% }\mathbf{e}_{\theta}+\cfrac{1}{r}\left[\frac{\partial S_{\theta z}}{\partial% \theta}+S_{rz}\right]~{}\mathbf{e}_{z}\\ &\displaystyle+\frac{\partial S_{zr}}{\partial z}~{}\mathbf{e}_{r}+\frac{% \partial S_{z\theta}}{\partial z}~{}\mathbf{e}_{\theta}+\frac{\partial S_{zz}}% {\partial z}~{}\mathbf{e}_{z}\end{aligned}
  77. s y m b o l T ( 𝐱 ) symbol{T}(\mathbf{x})
  78. ( s y m b o l \timessymbol T ) 𝐜 = s y m b o l × ( 𝐜 \cdotsymbol T ) ; ( s y m b o l × 𝐯 ) 𝐜 = s y m b o l ( 𝐯 × 𝐜 ) (symbol{\nabla}\timessymbol{T})\cdot\mathbf{c}=symbol{\nabla}\times(\mathbf{c}% \cdotsymbol{T})~{};\qquad(symbol{\nabla}\times\mathbf{v})\cdot\mathbf{c}=% symbol{\nabla}\cdot(\mathbf{v}\times\mathbf{c})
  79. 𝐯 × 𝐜 = e i j k v j c k 𝐞 i \mathbf{v}\times\mathbf{c}=e_{ijk}~{}v_{j}~{}c_{k}~{}\mathbf{e}_{i}
  80. e i j k e_{ijk}
  81. s y m b o l ( 𝐯 × 𝐜 ) = e i j k v j , i c k = ( e i j k v j , i 𝐞 k ) 𝐜 = ( s y m b o l × 𝐯 ) 𝐜 symbol{\nabla}\cdot(\mathbf{v}\times\mathbf{c})=e_{ijk}~{}v_{j,i}~{}c_{k}=(e_{% ijk}~{}v_{j,i}~{}\mathbf{e}_{k})\cdot\mathbf{c}=(symbol{\nabla}\times\mathbf{v% })\cdot\mathbf{c}
  82. s y m b o l × 𝐯 = e i j k v j , i 𝐞 k symbol{\nabla}\times\mathbf{v}=e_{ijk}~{}v_{j,i}~{}\mathbf{e}_{k}
  83. s y m b o l S symbol{S}
  84. 𝐜 \cdotsymbol S = c m S m j 𝐞 j \mathbf{c}\cdotsymbol{S}=c_{m}~{}S_{mj}~{}\mathbf{e}_{j}
  85. s y m b o l × ( 𝐜 \cdotsymbol S ) = e i j k c m S m j , i 𝐞 k = ( e i j k S m j , i 𝐞 k 𝐞 m ) 𝐜 = ( s y m b o l \timessymbol S ) 𝐜 symbol{\nabla}\times(\mathbf{c}\cdotsymbol{S})=e_{ijk}~{}c_{m}~{}S_{mj,i}~{}% \mathbf{e}_{k}=(e_{ijk}~{}S_{mj,i}~{}\mathbf{e}_{k}\otimes\mathbf{e}_{m})\cdot% \mathbf{c}=(symbol{\nabla}\timessymbol{S})\cdot\mathbf{c}
  86. s y m b o l \timessymbol S = e i j k S m j , i 𝐞 k 𝐞 m symbol{\nabla}\timessymbol{S}=e_{ijk}~{}S_{mj,i}~{}\mathbf{e}_{k}\otimes% \mathbf{e}_{m}
  87. s y m b o l T symbol{T}
  88. s y m b o l × ( s y m b o l s y m b o l T ) = s y m b o l 0 symbol{\nabla}\times(symbol{\nabla}symbol{T})=symbol{0}
  89. s y m b o l S symbol{S}
  90. s y m b o l \timessymbol S = s y m b o l 0 S m i , j - S m j , i = 0 symbol{\nabla}\timessymbol{S}=symbol{0}\quad\implies\quad S_{mi,j}-S_{mj,i}=0
  91. s y m b o l A symbol{A}
  92. s y m b o l A det ( s y m b o l A ) = det ( s y m b o l A ) [ s y m b o l A - 1 ] T . \frac{\partial}{\partial symbol{A}}\det(symbol{A})=\det(symbol{A})~{}[symbol{A% }^{-1}]^{T}~{}.
  93. s y m b o l A symbol{A}
  94. s y m b o l A symbol{A}
  95. f ( s y m b o l A ) = det ( s y m b o l A ) f(symbol{A})=\det(symbol{A})
  96. f s y m b o l A : s y m b o l T = d d α det ( s y m b o l A + α s y m b o l T ) | α = 0 = d d α det [ α s y m b o l A ( 1 α s y m b o l I + s y m b o l A - 1 \cdotsymbol T ) ] | α = 0 = d d α [ α 3 det ( s y m b o l A ) det ( 1 α s y m b o l I + s y m b o l A - 1 s y m b o l T ) ] | α = 0 . \begin{aligned}\displaystyle\frac{\partial f}{\partial symbol{A}}:symbol{T}&% \displaystyle=\left.\cfrac{d}{d\alpha}\det(symbol{A}+\alpha~{}symbol{T})\right% |_{\alpha=0}\\ &\displaystyle=\left.\cfrac{d}{d\alpha}\det\left[\alpha~{}symbol{A}\left(% \cfrac{1}{\alpha}~{}symbol{\mathit{I}}+symbol{A}^{-1}\cdotsymbol{T}\right)% \right]\right|_{\alpha=0}\\ &\displaystyle=\left.\cfrac{d}{d\alpha}\left[\alpha^{3}~{}\det(symbol{A})~{}% \det\left(\cfrac{1}{\alpha}~{}symbol{\mathit{I}}+symbol{A}^{-1}\cdot symbol{T}% \right)\right]\right|_{\alpha=0}.\end{aligned}
  97. I 1 , I 2 , I 3 I_{1},I_{2},I_{3}
  98. det ( λ s y m b o l I + s y m b o l A ) = λ 3 + I 1 ( s y m b o l A ) λ 2 + I 2 ( s y m b o l A ) λ + I 3 ( s y m b o l A ) . \det(\lambda~{}symbol{\mathit{I}}+symbol{A})=\lambda^{3}+I_{1}(symbol{A})~{}% \lambda^{2}+I_{2}(symbol{A})~{}\lambda+I_{3}(symbol{A}).
  99. f s y m b o l A : s y m b o l T = d d α [ α 3 det ( s y m b o l A ) ( 1 α 3 + I 1 ( s y m b o l A - 1 \cdotsymbol T ) 1 α 2 + I 2 ( s y m b o l A - 1 \cdotsymbol T ) 1 α + I 3 ( s y m b o l A - 1 \cdotsymbol T ) ) ] | α = 0 = det ( s y m b o l A ) d d α [ 1 + I 1 ( s y m b o l A - 1 \cdotsymbol T ) α + I 2 ( s y m b o l A - 1 \cdotsymbol T ) α 2 + I 3 ( s y m b o l A - 1 \cdotsymbol T ) α 3 ] | α = 0 = det ( s y m b o l A ) [ I 1 ( s y m b o l A - 1 \cdotsymbol T ) + 2 I 2 ( s y m b o l A - 1 \cdotsymbol T ) α + 3 I 3 ( s y m b o l A - 1 \cdotsymbol T ) α 2 ] | α = 0 = det ( s y m b o l A ) I 1 ( s y m b o l A - 1 \cdotsymbol T ) . \begin{aligned}\displaystyle\frac{\partial f}{\partial symbol{A}}:symbol{T}&% \displaystyle=\left.\cfrac{d}{d\alpha}\left[\alpha^{3}~{}\det(symbol{A})~{}% \left(\cfrac{1}{\alpha^{3}}+I_{1}(symbol{A}^{-1}\cdotsymbol{T})~{}\cfrac{1}{% \alpha^{2}}+I_{2}(symbol{A}^{-1}\cdotsymbol{T})~{}\cfrac{1}{\alpha}+I_{3}(% symbol{A}^{-1}\cdotsymbol{T})\right)\right]\right|_{\alpha=0}\\ &\displaystyle=\left.\det(symbol{A})~{}\cfrac{d}{d\alpha}\left[1+I_{1}(symbol{% A}^{-1}\cdotsymbol{T})~{}\alpha+I_{2}(symbol{A}^{-1}\cdotsymbol{T})~{}\alpha^{% 2}+I_{3}(symbol{A}^{-1}\cdotsymbol{T})~{}\alpha^{3}\right]\right|_{\alpha=0}\\ &\displaystyle=\left.\det(symbol{A})~{}\left[I_{1}(symbol{A}^{-1}\cdotsymbol{T% })+2~{}I_{2}(symbol{A}^{-1}\cdotsymbol{T})~{}\alpha+3~{}I_{3}(symbol{A}^{-1}% \cdotsymbol{T})~{}\alpha^{2}\right]\right|_{\alpha=0}\\ &\displaystyle=\det(symbol{A})~{}I_{1}(symbol{A}^{-1}\cdotsymbol{T})~{}.\end{aligned}
  100. I 1 I_{1}
  101. I 1 ( s y m b o l A ) = tr s y m b o l A . I_{1}(symbol{A})=\,\text{tr}{symbol{A}}.
  102. f s y m b o l A : s y m b o l T = det ( s y m b o l A ) tr ( s y m b o l A - 1 \cdotsymbol T ) = det ( s y m b o l A ) [ s y m b o l A - 1 ] T : s y m b o l T . \frac{\partial f}{\partial symbol{A}}:symbol{T}=\det(symbol{A})~{}\,\text{tr}(% symbol{A}^{-1}\cdotsymbol{T})=\det(symbol{A})~{}[symbol{A}^{-1}]^{T}:symbol{T}.
  103. s y m b o l T symbol{T}
  104. f s y m b o l A = det ( s y m b o l A ) [ s y m b o l A - 1 ] T . \frac{\partial f}{\partial symbol{A}}=\det(symbol{A})~{}[symbol{A}^{-1}]^{T}~{}.
  105. I 1 ( s y m b o l A ) = tr s y m b o l A I 2 ( s y m b o l A ) = 1 2 [ ( tr s y m b o l A ) 2 - tr s y m b o l A 2 ] I 3 ( s y m b o l A ) = det ( s y m b o l A ) \begin{aligned}\displaystyle I_{1}(symbol{A})&\displaystyle=\,\text{tr}{symbol% {A}}\\ \displaystyle I_{2}(symbol{A})&\displaystyle=\frac{1}{2}\left[(\,\text{tr}{% symbol{A}})^{2}-\,\text{tr}{symbol{A}^{2}}\right]\\ \displaystyle I_{3}(symbol{A})&\displaystyle=\det(symbol{A})\end{aligned}
  106. s y m b o l A symbol{A}
  107. I 1 s y m b o l A = s y m b o l 1 I 2 s y m b o l A = I 1 s y m b o l 1 - s y m b o l A T I 3 s y m b o l A = det ( s y m b o l A ) [ s y m b o l A - 1 ] T = I 2 s y m b o l 1 - s y m b o l A T ( I 1 s y m b o l 1 - s y m b o l A T ) = ( s y m b o l A 2 - I 1 s y m b o l A + I 2 s y m b o l 1 ) T \begin{aligned}\displaystyle\frac{\partial I_{1}}{\partial symbol{A}}&% \displaystyle=symbol{\mathit{1}}\\ \displaystyle\frac{\partial I_{2}}{\partial symbol{A}}&\displaystyle=I_{1}~{}% symbol{\mathit{1}}-symbol{A}^{T}\\ \displaystyle\frac{\partial I_{3}}{\partial symbol{A}}&\displaystyle=\det(% symbol{A})~{}[symbol{A}^{-1}]^{T}=I_{2}~{}symbol{\mathit{1}}-symbol{A}^{T}~{}(% I_{1}~{}symbol{\mathit{1}}-symbol{A}^{T})=(symbol{A}^{2}-I_{1}~{}symbol{A}+I_{% 2}~{}symbol{\mathit{1}})^{T}\end{aligned}
  108. I 3 s y m b o l A = det ( s y m b o l A ) [ s y m b o l A - 1 ] T . \frac{\partial I_{3}}{\partial symbol{A}}=\det(symbol{A})~{}[symbol{A}^{-1}]^{% T}~{}.
  109. det ( λ s y m b o l 1 + s y m b o l A ) = λ 3 + I 1 ( s y m b o l A ) λ 2 + I 2 ( s y m b o l A ) λ + I 3 ( s y m b o l A ) . \det(\lambda~{}symbol{\mathit{1}}+symbol{A})=\lambda^{3}+I_{1}(symbol{A})~{}% \lambda^{2}+I_{2}(symbol{A})~{}\lambda+I_{3}(symbol{A})~{}.
  110. s y m b o l A det ( λ s y m b o l 1 + s y m b o l A ) = det ( λ s y m b o l 1 + s y m b o l A ) [ ( λ s y m b o l 1 + s y m b o l A ) - 1 ] T . \frac{\partial}{\partial symbol{A}}\det(\lambda~{}symbol{\mathit{1}}+symbol{A}% )=\det(\lambda~{}symbol{\mathit{1}}+symbol{A})~{}[(\lambda~{}symbol{\mathit{1}% }+symbol{A})^{-1}]^{T}~{}.
  111. s y m b o l A det ( λ s y m b o l 1 + s y m b o l A ) = s y m b o l A [ λ 3 + I 1 ( s y m b o l A ) λ 2 + I 2 ( s y m b o l A ) λ + I 3 ( s y m b o l A ) ] = I 1 s y m b o l A λ 2 + I 2 s y m b o l A λ + I 3 s y m b o l A . \begin{aligned}\displaystyle\frac{\partial}{\partial symbol{A}}\det(\lambda~{}% symbol{\mathit{1}}+symbol{A})&\displaystyle=\frac{\partial}{\partial symbol{A}% }\left[\lambda^{3}+I_{1}(symbol{A})~{}\lambda^{2}+I_{2}(symbol{A})~{}\lambda+I% _{3}(symbol{A})\right]\\ &\displaystyle=\frac{\partial I_{1}}{\partial symbol{A}}~{}\lambda^{2}+\frac{% \partial I_{2}}{\partial symbol{A}}~{}\lambda+\frac{\partial I_{3}}{\partial symbol% {A}}~{}.\end{aligned}
  112. I 1 s y m b o l A λ 2 + I 2 s y m b o l A λ + I 3 s y m b o l A = det ( λ s y m b o l 1 + s y m b o l A ) [ ( λ s y m b o l 1 + s y m b o l A ) - 1 ] T \frac{\partial I_{1}}{\partial symbol{A}}~{}\lambda^{2}+\frac{\partial I_{2}}{% \partial symbol{A}}~{}\lambda+\frac{\partial I_{3}}{\partial symbol{A}}=\det(% \lambda~{}symbol{\mathit{1}}+symbol{A})~{}[(\lambda~{}symbol{\mathit{1}}+% symbol{A})^{-1}]^{T}
  113. ( λ s y m b o l 1 + s y m b o l A ) T [ I 1 s y m b o l A λ 2 + I 2 s y m b o l A λ + I 3 s y m b o l A ] = det ( λ s y m b o l 1 + s y m b o l A ) s y m b o l 1 . (\lambda~{}symbol{\mathit{1}}+symbol{A})^{T}\cdot\left[\frac{\partial I_{1}}{% \partial symbol{A}}~{}\lambda^{2}+\frac{\partial I_{2}}{\partial symbol{A}}~{}% \lambda+\frac{\partial I_{3}}{\partial symbol{A}}\right]=\det(\lambda~{}symbol% {\mathit{1}}+symbol{A})~{}symbol{\mathit{1}}~{}.
  114. ( λ s y m b o l 1 + s y m b o l A T ) [ I 1 s y m b o l A λ 2 + I 2 s y m b o l A λ + I 3 s y m b o l A ] = [ λ 3 + I 1 λ 2 + I 2 λ + I 3 ] s y m b o l 1 (\lambda~{}symbol{\mathit{1}}+symbol{A}^{T})\cdot\left[\frac{\partial I_{1}}{% \partial symbol{A}}~{}\lambda^{2}+\frac{\partial I_{2}}{\partial symbol{A}}~{}% \lambda+\frac{\partial I_{3}}{\partial symbol{A}}\right]=\left[\lambda^{3}+I_{% 1}~{}\lambda^{2}+I_{2}~{}\lambda+I_{3}\right]symbol{\mathit{1}}
  115. [ I 1 s y m b o l A λ 3 + I 2 s y m b o l A λ 2 + I 3 s y m b o l A λ ] s y m b o l 1 + s y m b o l A T I 1 s y m b o l A λ 2 + s y m b o l A T I 2 s y m b o l A λ + s y m b o l A T I 3 s y m b o l A = [ λ 3 + I 1 λ 2 + I 2 λ + I 3 ] s y m b o l 1 . \begin{aligned}\displaystyle\left[\frac{\partial I_{1}}{\partial symbol{A}}~{}% \lambda^{3}\right.&\displaystyle\left.+\frac{\partial I_{2}}{\partial symbol{A% }}~{}\lambda^{2}+\frac{\partial I_{3}}{\partial symbol{A}}~{}\lambda\right]% symbol{\mathit{1}}+symbol{A}^{T}\cdot\frac{\partial I_{1}}{\partial symbol{A}}% ~{}\lambda^{2}+symbol{A}^{T}\cdot\frac{\partial I_{2}}{\partial symbol{A}}~{}% \lambda+symbol{A}^{T}\cdot\frac{\partial I_{3}}{\partial symbol{A}}\\ &\displaystyle=\left[\lambda^{3}+I_{1}~{}\lambda^{2}+I_{2}~{}\lambda+I_{3}% \right]symbol{\mathit{1}}~{}.\end{aligned}
  116. I 0 := 1 I_{0}:=1
  117. I 4 := 0 I_{4}:=0
  118. [ I 1 s y m b o l A λ 3 + I 2 s y m b o l A λ 2 + I 3 s y m b o l A λ + I 4 s y m b o l A ] s y m b o l 1 + s y m b o l A T I 0 s y m b o l A λ 3 + s y m b o l A T I 1 s y m b o l A λ 2 + s y m b o l A T I 2 s y m b o l A λ + s y m b o l A T I 3 s y m b o l A = [ I 0 λ 3 + I 1 λ 2 + I 2 λ + I 3 ] s y m b o l 1 . \begin{aligned}\displaystyle\left[\frac{\partial I_{1}}{\partial symbol{A}}~{}% \lambda^{3}\right.&\displaystyle\left.+\frac{\partial I_{2}}{\partial symbol{A% }}~{}\lambda^{2}+\frac{\partial I_{3}}{\partial symbol{A}}~{}\lambda+\frac{% \partial I_{4}}{\partial symbol{A}}\right]symbol{\mathit{1}}+symbol{A}^{T}% \cdot\frac{\partial I_{0}}{\partial symbol{A}}~{}\lambda^{3}+symbol{A}^{T}% \cdot\frac{\partial I_{1}}{\partial symbol{A}}~{}\lambda^{2}+symbol{A}^{T}% \cdot\frac{\partial I_{2}}{\partial symbol{A}}~{}\lambda+symbol{A}^{T}\cdot% \frac{\partial I_{3}}{\partial symbol{A}}\\ &\displaystyle=\left[I_{0}~{}\lambda^{3}+I_{1}~{}\lambda^{2}+I_{2}~{}\lambda+I% _{3}\right]symbol{\mathit{1}}~{}.\end{aligned}
  119. λ 3 ( I 0 s y m b o l 1 - I 1 s y m b o l A s y m b o l 1 - s y m b o l A T I 0 s y m b o l A ) + λ 2 ( I 1 s y m b o l 1 - I 2 s y m b o l A s y m b o l 1 - s y m b o l A T I 1 s y m b o l A ) + λ ( I 2 s y m b o l 1 - I 3 s y m b o l A s y m b o l 1 - s y m b o l A T I 2 s y m b o l A ) + ( I 3 s y m b o l 1 - I 4 s y m b o l A s y m b o l 1 - s y m b o l A T I 3 s y m b o l A ) = 0 . \begin{aligned}\displaystyle\lambda^{3}&\displaystyle\left(I_{0}~{}symbol{% \mathit{1}}-\frac{\partial I_{1}}{\partial symbol{A}}~{}symbol{\mathit{1}}-% symbol{A}^{T}\cdot\frac{\partial I_{0}}{\partial symbol{A}}\right)+\lambda^{2}% \left(I_{1}~{}symbol{\mathit{1}}-\frac{\partial I_{2}}{\partial symbol{A}}~{}% symbol{\mathit{1}}-symbol{A}^{T}\cdot\frac{\partial I_{1}}{\partial symbol{A}}% \right)+\\ &\displaystyle\qquad\qquad\lambda\left(I_{2}~{}symbol{\mathit{1}}-\frac{% \partial I_{3}}{\partial symbol{A}}~{}symbol{\mathit{1}}-symbol{A}^{T}\cdot% \frac{\partial I_{2}}{\partial symbol{A}}\right)+\left(I_{3}~{}symbol{\mathit{% 1}}-\frac{\partial I_{4}}{\partial symbol{A}}~{}symbol{\mathit{1}}-symbol{A}^{% T}\cdot\frac{\partial I_{3}}{\partial symbol{A}}\right)=0~{}.\end{aligned}
  120. I 0 s y m b o l 1 - I 1 s y m b o l A s y m b o l 1 - s y m b o l A T I 0 s y m b o l A = 0 I 1 s y m b o l 1 - I 2 s y m b o l A s y m b o l 1 - I 2 s y m b o l 1 - I 3 s y m b o l A s y m b o l 1 - s y m b o l A T I 2 s y m b o l A = 0 I 3 s y m b o l 1 - I 4 s y m b o l A s y m b o l 1 - s y m b o l A T I 3 s y m b o l A = 0 . \begin{aligned}\displaystyle I_{0}~{}symbol{\mathit{1}}-\frac{\partial I_{1}}{% \partial symbol{A}}~{}symbol{\mathit{1}}-symbol{A}^{T}\cdot\frac{\partial I_{0% }}{\partial symbol{A}}&\displaystyle=0\\ \displaystyle I_{1}~{}symbol{\mathit{1}}-\frac{\partial I_{2}}{\partial symbol% {A}}~{}symbol{\mathit{1}}-I_{2}~{}symbol{\mathit{1}}-\frac{\partial I_{3}}{% \partial symbol{A}}~{}symbol{\mathit{1}}-symbol{A}^{T}\cdot\frac{\partial I_{2% }}{\partial symbol{A}}&\displaystyle=0\\ \displaystyle I_{3}~{}symbol{\mathit{1}}-\frac{\partial I_{4}}{\partial symbol% {A}}~{}symbol{\mathit{1}}-symbol{A}^{T}\cdot\frac{\partial I_{3}}{\partial symbol% {A}}&\displaystyle=0~{}.\end{aligned}
  121. I 1 s y m b o l A = s y m b o l 1 I 2 s y m b o l A = I 1 s y m b o l 1 - s y m b o l A T I 3 s y m b o l A = I 2 s y m b o l 1 - s y m b o l A T ( I 1 s y m b o l 1 - s y m b o l A T ) = ( s y m b o l A 2 - I 1 s y m b o l A + I 2 s y m b o l 1 ) T \begin{aligned}\displaystyle\frac{\partial I_{1}}{\partial symbol{A}}&% \displaystyle=symbol{\mathit{1}}\\ \displaystyle\frac{\partial I_{2}}{\partial symbol{A}}&\displaystyle=I_{1}~{}% symbol{\mathit{1}}-symbol{A}^{T}\\ \displaystyle\frac{\partial I_{3}}{\partial symbol{A}}&\displaystyle=I_{2}~{}% symbol{\mathit{1}}-symbol{A}^{T}~{}(I_{1}~{}symbol{\mathit{1}}-symbol{A}^{T})=% (symbol{A}^{2}-I_{1}~{}symbol{A}+I_{2}~{}symbol{\mathit{1}})^{T}\end{aligned}
  122. s y m b o l 1 symbol{\mathit{1}}
  123. s y m b o l A symbol{A}
  124. s y m b o l 1 s y m b o l A : s y m b o l T = s y m b o l 𝟢 : s y m b o l T = s y m b o l 0 \frac{\partial symbol{\mathit{1}}}{\partial symbol{A}}:symbol{T}=symbol{% \mathsf{0}}:symbol{T}=symbol{\mathit{0}}
  125. s y m b o l 1 symbol{\mathit{1}}
  126. s y m b o l A symbol{A}
  127. s y m b o l A symbol{A}
  128. s y m b o l A s y m b o l A : s y m b o l T = [ α ( s y m b o l A + α s y m b o l T ) ] α = 0 = s y m b o l T = s y m b o l 𝖨 : s y m b o l T \frac{\partial symbol{A}}{\partial symbol{A}}:symbol{T}=\left[\frac{\partial}{% \partial\alpha}(symbol{A}+\alpha~{}symbol{T})\right]_{\alpha=0}=symbol{T}=% symbol{\mathsf{I}}:symbol{T}
  129. s y m b o l A s y m b o l A = s y m b o l 𝖨 \frac{\partial symbol{A}}{\partial symbol{A}}=symbol{\mathsf{I}}
  130. s y m b o l 𝖨 symbol{\mathsf{I}}
  131. s y m b o l 𝖨 = δ i k δ j l 𝐞 i 𝐞 j 𝐞 k 𝐞 l symbol{\mathsf{I}}=\delta_{ik}~{}\delta_{jl}~{}\mathbf{e}_{i}\otimes\mathbf{e}% _{j}\otimes\mathbf{e}_{k}\otimes\mathbf{e}_{l}
  132. s y m b o l A T s y m b o l A : s y m b o l T = s y m b o l 𝖨 T : s y m b o l T = s y m b o l T T \frac{\partial symbol{A}^{T}}{\partial symbol{A}}:symbol{T}=symbol{\mathsf{I}}% ^{T}:symbol{T}=symbol{T}^{T}
  133. s y m b o l 𝖨 T = δ j k δ i l 𝐞 i 𝐞 j 𝐞 k 𝐞 l symbol{\mathsf{I}}^{T}=\delta_{jk}~{}\delta_{il}~{}\mathbf{e}_{i}\otimes% \mathbf{e}_{j}\otimes\mathbf{e}_{k}\otimes\mathbf{e}_{l}
  134. s y m b o l A symbol{A}
  135. s y m b o l A s y m b o l A = s y m b o l 𝖨 ( s ) = 1 2 ( s y m b o l 𝖨 + s y m b o l 𝖨 T ) \frac{\partial symbol{A}}{\partial symbol{A}}=symbol{\mathsf{I}}^{(s)}=\frac{1% }{2}~{}(symbol{\mathsf{I}}+symbol{\mathsf{I}}^{T})
  136. s y m b o l 𝖨 ( s ) = 1 2 ( δ i k δ j l + δ i l δ j k ) 𝐞 i 𝐞 j 𝐞 k 𝐞 l symbol{\mathsf{I}}^{(s)}=\frac{1}{2}~{}(\delta_{ik}~{}\delta_{jl}+\delta_{il}~% {}\delta_{jk})~{}\mathbf{e}_{i}\otimes\mathbf{e}_{j}\otimes\mathbf{e}_{k}% \otimes\mathbf{e}_{l}
  137. s y m b o l A symbol{A}
  138. s y m b o l T symbol{T}
  139. s y m b o l A ( s y m b o l A - 1 ) : s y m b o l T = - s y m b o l A - 1 \cdotsymbol T \cdotsymbol A - 1 \frac{\partial}{\partial symbol{A}}\left(symbol{A}^{-1}\right):symbol{T}=-% symbol{A}^{-1}\cdotsymbol{T}\cdotsymbol{A}^{-1}
  140. A i j - 1 A k l T k l = - A i k - 1 T k l A l j - 1 A i j - 1 A k l = - A i k - 1 A l j - 1 \frac{\partial A^{-1}_{ij}}{\partial A_{kl}}~{}T_{kl}=-A^{-1}_{ik}~{}T_{kl}~{}% A^{-1}_{lj}\implies\frac{\partial A^{-1}_{ij}}{\partial A_{kl}}=-A^{-1}_{ik}~{% }A^{-1}_{lj}
  141. s y m b o l A ( s y m b o l A - T ) : s y m b o l T = - s y m b o l A - T \cdotsymbol T T \cdotsymbol A - T \frac{\partial}{\partial symbol{A}}\left(symbol{A}^{-T}\right):symbol{T}=-% symbol{A}^{-T}\cdotsymbol{T}^{T}\cdotsymbol{A}^{-T}
  142. A j i - 1 A k l T k l = - A j k - 1 T l k A l i - 1 A j i - 1 A k l = - A l i - 1 A j k - 1 \frac{\partial A^{-1}_{ji}}{\partial A_{kl}}~{}T_{kl}=-A^{-1}_{jk}~{}T_{lk}~{}% A^{-1}_{li}\implies\frac{\partial A^{-1}_{ji}}{\partial A_{kl}}=-A^{-1}_{li}~{% }A^{-1}_{jk}
  143. s y m b o l A symbol{A}
  144. A i j - 1 A k l = - 1 2 ( A i k - 1 A j l - 1 + A i l - 1 A j k - 1 ) \frac{\partial A^{-1}_{ij}}{\partial A_{kl}}=-\cfrac{1}{2}\left(A^{-1}_{ik}~{}% A^{-1}_{jl}+A^{-1}_{il}~{}A^{-1}_{jk}\right)
  145. s y m b o l 1 s y m b o l A : s y m b o l T = s y m b o l 0 \frac{\partial symbol{\mathit{1}}}{\partial symbol{A}}:symbol{T}=symbol{% \mathit{0}}
  146. s y m b o l A - 1 \cdotsymbol A = s y m b o l 1 symbol{A}^{-1}\cdotsymbol{A}=symbol{\mathit{1}}
  147. s y m b o l A ( s y m b o l A - 1 \cdotsymbol A ) : s y m b o l T = s y m b o l 0 \frac{\partial}{\partial symbol{A}}(symbol{A}^{-1}\cdotsymbol{A}):symbol{T}=% symbol{\mathit{0}}
  148. s y m b o l S [ s y m b o l F 1 ( s y m b o l S ) \cdotsymbol F 2 ( s y m b o l S ) ] : s y m b o l T = ( s y m b o l F 1 s y m b o l S : s y m b o l T ) \cdotsymbol F 2 + s y m b o l F 1 ( s y m b o l F 2 s y m b o l S : s y m b o l T ) \frac{\partial}{\partial symbol{S}}[symbol{F}_{1}(symbol{S})\cdotsymbol{F}_{2}% (symbol{S})]:symbol{T}=\left(\frac{\partial symbol{F}_{1}}{\partial symbol{S}}% :symbol{T}\right)\cdotsymbol{F}_{2}+symbol{F}_{1}\cdot\left(\frac{\partial symbol% {F}_{2}}{\partial symbol{S}}:symbol{T}\right)
  149. s y m b o l A ( s y m b o l A - 1 \cdotsymbol A ) : s y m b o l T = ( s y m b o l A - 1 s y m b o l A : s y m b o l T ) \cdotsymbol A + s y m b o l A - 1 ( s y m b o l A s y m b o l A : s y m b o l T ) = s y m b o l 0 \frac{\partial}{\partial symbol{A}}(symbol{A}^{-1}\cdotsymbol{A}):symbol{T}=% \left(\frac{\partial symbol{A}^{-1}}{\partial symbol{A}}:symbol{T}\right)% \cdotsymbol{A}+symbol{A}^{-1}\cdot\left(\frac{\partial symbol{A}}{\partial symbol% {A}}:symbol{T}\right)=symbol{\mathit{0}}
  150. ( s y m b o l A - 1 s y m b o l A : s y m b o l T ) \cdotsymbol A = - s y m b o l A - 1 \cdotsymbol T \left(\frac{\partial symbol{A}^{-1}}{\partial symbol{A}}:symbol{T}\right)% \cdotsymbol{A}=-symbol{A}^{-1}\cdotsymbol{T}
  151. s y m b o l A ( s y m b o l A - 1 ) : s y m b o l T = - s y m b o l A - 1 \cdotsymbol T \cdotsymbol A - 1 \frac{\partial}{\partial symbol{A}}\left(symbol{A}^{-1}\right):symbol{T}=-% symbol{A}^{-1}\cdotsymbol{T}\cdotsymbol{A}^{-1}
  152. Ω s y m b o l F \otimessymbol s y m b o l G d Ω = Γ 𝐧 ( s y m b o l F \otimessymbol G ) d Γ - Ω s y m b o l G \otimessymbol s y m b o l F d Ω \int_{\Omega}symbol{F}\otimessymbol{\nabla}symbol{G}\,{\rm d}\Omega=\int_{% \Gamma}\mathbf{n}\otimes(symbol{F}\otimessymbol{G})\,{\rm d}\Gamma-\int_{% \Omega}symbol{G}\otimessymbol{\nabla}symbol{F}\,{\rm d}\Omega
  153. s y m b o l F symbol{F}
  154. s y m b o l G symbol{G}
  155. 𝐧 \mathbf{n}
  156. \otimes
  157. s y m b o l symbol{\nabla}
  158. s y m b o l F symbol{F}
  159. Ω s y m b o l s y m b o l G d Ω = Γ 𝐧 \otimessymbol G d Γ . \int_{\Omega}symbol{\nabla}symbol{G}\,{\rm d}\Omega=\int_{\Gamma}\mathbf{n}% \otimessymbol{G}\,{\rm d}\Gamma\,.
  160. Ω F i j k . G l m n , p d Ω = Γ n p F i j k G l m n d Γ - Ω G l m n F i j k , p d Ω . \int_{\Omega}F_{ijk....}\,G_{lmn...,p}\,{\rm d}\Omega=\int_{\Gamma}n_{p}\,F_{% ijk...}\,G_{lmn...}\,{\rm d}\Gamma-\int_{\Omega}G_{lmn...}\,F_{ijk...,p}\,{\rm d% }\Omega\,.
  161. s y m b o l F symbol{F}
  162. s y m b o l G symbol{G}
  163. Ω s y m b o l F ( s y m b o l \cdotsymbol G ) d Ω = Γ 𝐧 ( s y m b o l G \cdotsymbol F T ) d Γ - Ω ( s y m b o l s y m b o l F ) : s y m b o l G T d Ω . \int_{\Omega}symbol{F}\cdot(symbol{\nabla}\cdotsymbol{G})\,{\rm d}\Omega=\int_% {\Gamma}\mathbf{n}\cdot(symbol{G}\cdotsymbol{F}^{T})\,{\rm d}\Gamma-\int_{% \Omega}(symbol{\nabla}symbol{F}):symbol{G}^{T}\,{\rm d}\Omega\,.
  164. Ω F i j G p j , p d Ω = Γ n p F i j G p j d Γ - Ω G p j F i j , p d Ω . \int_{\Omega}F_{ij}\,G_{pj,p}\,{\rm d}\Omega=\int_{\Gamma}n_{p}\,F_{ij}\,G_{pj% }\,{\rm d}\Gamma-\int_{\Omega}G_{pj}\,F_{ij,p}\,{\rm d}\Omega\,.

Tensor_rank_decomposition.html

  1. 𝔽 n 1 × × n d 𝔽 n 1 𝔽 n d \mathbb{F}^{n_{1}\times\cdots\times n_{d}}\cong\mathbb{F}^{n_{1}}\otimes\cdots% \otimes\mathbb{F}^{n_{d}}
  2. 𝔽 \mathbb{F}
  3. \mathbb{R}
  4. \mathbb{C}
  5. d d
  6. r r
  7. r r
  8. 𝒜 = i = 1 r λ i 𝐚 i 1 𝐚 i 2 𝐚 i d , \mathcal{A}=\sum_{i=1}^{r}\lambda_{i}\mathbf{a}_{i}^{1}\otimes\mathbf{a}_{i}^{% 2}\otimes\cdots\otimes\mathbf{a}_{i}^{d},
  9. λ i 𝔽 \lambda_{i}\in\mathbb{F}
  10. 𝐚 i k 𝔽 n k \mathbf{a}_{i}^{k}\in\mathbb{F}^{n_{k}}
  11. k k
  12. r r
  13. r r
  14. r r
  15. 𝔽 m 𝔽 n 𝔽 2 \mathbb{F}^{m}\otimes\mathbb{F}^{n}\otimes\mathbb{F}^{2}
  16. 𝐚 1 𝐚 d \mathbf{a}_{1}\otimes\cdots\otimes\mathbf{a}_{d}
  17. 𝐚 k 𝔽 n k { 0 } \mathbf{a}_{k}\in\mathbb{F}^{n_{k}}\setminus\{0\}
  18. 𝒜 = 𝐱 1 𝐱 2 𝐱 3 + 𝐱 1 𝐲 2 𝐲 3 - 𝐲 1 𝐱 2 𝐲 3 + 𝐲 1 𝐲 2 𝐲 3 , \mathcal{A}=\mathbf{x}_{1}\otimes\mathbf{x}_{2}\otimes\mathbf{x}_{3}+\mathbf{x% }_{1}\otimes\mathbf{y}_{2}\otimes\mathbf{y}_{3}-\mathbf{y}_{1}\otimes\mathbf{x% }_{2}\otimes\mathbf{y}_{3}+\mathbf{y}_{1}\otimes\mathbf{y}_{2}\otimes\mathbf{y% }_{3},
  19. 𝒜 = 1 2 ( 𝐳 ¯ 1 𝐳 2 𝐳 ¯ 3 + 𝐳 1 𝐳 ¯ 2 𝐳 3 ) . \mathcal{A}=\frac{1}{2}(\bar{\mathbf{z}}_{1}\otimes\mathbf{z}_{2}\otimes\bar{% \mathbf{z}}_{3}+\mathbf{z}_{1}\otimes\bar{\mathbf{z}}_{2}\otimes\mathbf{z}_{3}).
  20. \mathbb{C}
  21. r ( n 1 , , n d ) r(n_{1},\ldots,n_{d})
  22. r r
  23. r r
  24. 𝔽 n 1 𝔽 n d \mathbb{F}^{n_{1}}\otimes\cdots\otimes\mathbb{F}^{n_{d}}
  25. r ( n 1 , , n d ) r(n_{1},\ldots,n_{d})
  26. S S
  27. S S
  28. r ( n 1 , , n d ) r(n_{1},\ldots,n_{d})
  29. 2 2 2 \mathbb{R}^{2}\otimes\mathbb{R}^{2}\otimes\mathbb{R}^{2}
  30. 2 2 2 \mathbb{C}^{2}\otimes\mathbb{C}^{2}\otimes\mathbb{C}^{2}
  31. 2 × 2 × 2 2\times 2\times 2
  32. 2 2 2 \mathbb{R}^{2}\otimes\mathbb{R}^{2}\otimes\mathbb{R}^{2}
  33. 𝔽 n 1 𝔽 n d \mathbb{F}^{n_{1}}\otimes\cdots\otimes\mathbb{F}^{n_{d}}
  34. n 1 n 2 n d n_{1}\geq n_{2}\geq\cdots\geq n_{d}
  35. n 1 > 1 + k = 2 d n k - k = 2 d ( n k - 1 ) , n_{1}>1+\prod_{k=2}^{d}n_{k}-\sum_{k=2}^{d}(n_{k}-1),
  36. r ( n 1 , , r d ) = min { n 1 , k = 2 d n k } r(n_{1},\ldots,r_{d})=\min\left\{n_{1},\prod_{k=2}^{d}n_{k}\right\}
  37. 𝔽 n 1 × × n d Z \mathbb{F}^{n_{1}\times\cdots\times n_{d}}\setminus Z
  38. Z Z
  39. r E ( n 1 , , n d ) = Π Σ + 1 r_{E}(n_{1},\ldots,n_{d})=\left\lceil\frac{\Pi}{\Sigma+1}\right\rceil
  40. Π = k = 1 d n k and Σ = k = 1 d ( n k - 1 ) . \Pi=\prod_{k=1}^{d}n_{k}\quad\,\text{and}\quad\Sigma=\sum_{k=1}^{d}(n_{k}-1).
  41. n 1 × × n d Z \mathbb{C}^{n_{1}\times\cdots\times n_{d}}\setminus Z
  42. Z Z
  43. r E ( n 1 , , n d ) r_{E}(n_{1},\ldots,n_{d})
  44. r E ( n 1 , , n d ) r_{E}(n_{1},\ldots,n_{d})
  45. 𝔽 n 1 × × n d \mathbb{F}^{n_{1}\times\cdots\times n_{d}}
  46. r ( n 1 , , n d ) r E ( n 1 , , n d ) . r(n_{1},\ldots,n_{d})\geq r_{E}(n_{1},\ldots,n_{d}).
  47. r ( n 1 , , n d ) = r E ( n 1 , , n d ) r(n_{1},\ldots,n_{d})=r_{E}(n_{1},\ldots,n_{d})
  48. 𝔽 4 × 4 × 3 \mathbb{F}^{4\times 4\times 3}
  49. 𝔽 ( 2 n + 1 ) × ( 2 n + 1 ) × 3 with n = 1 , 2 , \mathbb{F}^{(2n+1)\times(2n+1)\times 3}\,\text{ with }n=1,2,\ldots
  50. 𝔽 ( n + 1 ) × ( n + 1 ) × 2 × 2 with n = 1 , 2 , \mathbb{F}^{(n+1)\times(n+1)\times 2\times 2}\,\text{ with }n=1,2,\ldots
  51. r ( n 1 , , n d ) = r E ( n 1 , , n d ) + 1 r(n_{1},\ldots,n_{d})=r_{E}(n_{1},\ldots,n_{d})+1
  52. r ( n , n , n ) = r E ( n , n , n ) r(n,n,n)=r_{E}(n,n,n)
  53. n 3 n\neq 3
  54. r ( 2 , 2 , , 2 ) = r E ( 2 , 2 , , 2 ) r(2,2,\ldots,2)=r_{E}(2,2,\ldots,2)
  55. 𝔽 2 𝔽 2 𝔽 2 𝔽 2 \mathbb{F}^{2}\otimes\mathbb{F}^{2}\otimes\mathbb{F}^{2}\otimes\mathbb{F}^{2}
  56. r M ( n 1 , , n d ) r_{M}(n_{1},\ldots,n_{d})
  57. 𝔽 n 1 𝔽 n d \mathbb{F}^{n_{1}}\otimes\cdots\otimes\mathbb{F}^{n_{d}}
  58. n 1 n 2 n d n_{1}\geq n_{2}\geq\cdots\geq n_{d}
  59. r M ( n 1 , , n d ) min { k = 2 d n k , 2 r ( n 1 , , n d ) } , r_{M}(n_{1},\ldots,n_{d})\leq\min\left\{\prod_{k=2}^{d}n_{k},2\cdot r(n_{1},% \ldots,n_{d})\right\},
  60. r ( n 1 , , n d ) r(n_{1},\ldots,n_{d})
  61. 𝔽 n 1 𝔽 n d \mathbb{F}^{n_{1}}\otimes\cdots\otimes\mathbb{F}^{n_{d}}
  62. 2 × 2 × 2 \mathbb{R}^{2\times 2\times 2}
  63. r M ( 2 , 2 , 2 ) 4 r_{M}(2,2,2)\leq 4
  64. s s
  65. 𝒜 \mathcal{A}
  66. r < s r<s
  67. 𝒜 \mathcal{A}
  68. s s
  69. 𝒜 \mathcal{A}
  70. 3 \geq 3
  71. 𝒜 = 𝐮 𝐮 𝐯 + 𝐮 𝐯 𝐮 + 𝐯 𝐮 𝐮 , with 𝐮 = 𝐯 = 1 and 𝐮 , 𝐯 1. \mathcal{A}=\mathbf{u}\otimes\mathbf{u}\otimes\mathbf{v}+\mathbf{u}\otimes% \mathbf{v}\otimes\mathbf{u}+\mathbf{v}\otimes\mathbf{u}\otimes\mathbf{u},\quad% \,\text{with }\|\mathbf{u}\|=\|\mathbf{v}\|=1\,\text{ and }\langle\mathbf{u},% \mathbf{v}\rangle\neq 1.
  72. 𝒜 n \displaystyle\mathcal{A}_{n}
  73. n n\to\infty
  74. r r
  75. s s
  76. 𝒜 \mathcal{A}
  77. r < s r<s
  78. min 𝐚 i k 𝔽 n k 𝒜 - i = 1 r 𝐚 i 1 𝐚 i 2 𝐚 i d F , \min_{\mathbf{a}_{i}^{k}\in\mathbb{F}^{n_{k}}}\|\mathcal{A}-\sum_{i=1}^{r}% \mathbf{a}_{i}^{1}\otimes\mathbf{a}_{i}^{2}\otimes\cdots\otimes\mathbf{a}_{i}^% {d}\|_{F},
  79. F \|\cdot\|_{F}
  80. r r
  81. r r
  82. 𝒜 = 𝐮 𝐮 𝐯 + 𝐮 𝐯 𝐮 + 𝐯 𝐮 𝐮 , with 𝐮 = 𝐯 = 1 and 𝐮 , 𝐯 1 \mathcal{A}=\mathbf{u}\otimes\mathbf{u}\otimes\mathbf{v}+\mathbf{u}\otimes% \mathbf{v}\otimes\mathbf{u}+\mathbf{v}\otimes\mathbf{u}\otimes\mathbf{u},\quad% \,\text{with }\|\mathbf{u}\|=\|\mathbf{v}\|=1\,\text{ and }\langle\mathbf{u},% \mathbf{v}\rangle\neq 1
  83. 𝒜 n = n ( 𝐮 + 1 n 𝐯 ) ( 𝐮 + 1 n 𝐯 ) ( 𝐮 + 1 n 𝐯 ) - n 𝐮 𝐮 𝐮 \mathcal{A}_{n}=n(\mathbf{u}+\frac{1}{n}\mathbf{v})\otimes(\mathbf{u}+\frac{1}% {n}\mathbf{v})\otimes(\mathbf{u}+\frac{1}{n}\mathbf{v})-n\mathbf{u}\otimes% \mathbf{u}\otimes\mathbf{u}
  84. n n\to\infty
  85. r r
  86. 𝒜 n = i = 1 r 𝐚 i , n 1 𝐚 i , n 2 𝐚 i , n d \mathcal{A}_{n}=\sum_{i=1}^{r}\mathbf{a}^{1}_{i,n}\otimes\mathbf{a}^{2}_{i,n}% \otimes\cdots\otimes\mathbf{a}^{d}_{i,n}
  87. 𝒜 n 𝒜 \mathcal{A}_{n}\to\mathcal{A}
  88. n n\to\infty
  89. 1 i j r 1\leq i\neq j\leq r
  90. 𝐚 i , n 1 𝐚 i , n 2 𝐚 i , n d F and 𝐚 j , n 1 𝐚 j , n 2 𝐚 j , n d F \|\mathbf{a}^{1}_{i,n}\otimes\mathbf{a}^{2}_{i,n}\otimes\cdots\otimes\mathbf{a% }^{d}_{i,n}\|_{F}\to\infty\,\text{ and }\|\mathbf{a}^{1}_{j,n}\otimes\mathbf{a% }^{2}_{j,n}\otimes\cdots\otimes\mathbf{a}^{d}_{j,n}\|_{F}\to\infty
  91. n n\to\infty

Test_theories_of_special_relativity.html

  1. t = a T + e x t=aT+ex\,
  2. x = b ( X - v T ) x=b(X-vT)\,
  3. y = d Y y=dY\,
  4. z = d Z z=dZ\,
  5. 1 / a ( v ) 1/a(v)
  6. 1 / b ( v ) 1/b(v)
  7. 1 / a ( v ) = b ( v ) = 1 / 1 - v 2 / c 2 , 1/a(v)=b(v)=1/\sqrt{1-v^{2}/c^{2}}\,,
  8. d ( v ) = 1 , d(v)=1\,,
  9. e ( v ) = - v / c 2 , e(v)=-v/c^{2}\,,
  10. a ( v ) = b ( v ) = d ( v ) = 1 , and e ( v ) = 0 . a(v)=b(v)=d(v)=1,\,\text{ and }e(v)=0\,.
  11. a ( v ) 1 + α v 2 / c 2 a(v)\sim 1+\alpha v^{2}/c^{2}\,
  12. b ( v ) 1 + β v 2 / c 2 b(v)\sim 1+\beta v^{2}/c^{2}\,
  13. d ( v ) 1 + δ v 2 / c 2 d(v)\sim 1+\delta v^{2}/c^{2}\,
  14. c c 1 + ( β - δ - 1 2 ) v 2 c 2 sin 2 θ + ( α - β + 1 ) v 2 c 2 \frac{c}{c^{\prime}}\sim 1+\left(\beta-\delta-\frac{1}{2}\right)\frac{v^{2}}{c% ^{2}}\sin^{2}\theta+(\alpha-\beta+1)\frac{v^{2}}{c^{2}}
  15. c c\,
  16. c c^{\prime}\,
  17. θ \theta\,
  18. α = - 1 2 , β = 1 2 , δ = 0 \alpha=-\tfrac{1}{2},\ \beta=\tfrac{1}{2},\ \delta=0
  19. c / c = 1 c/c^{\prime}=1\,
  20. ( β - δ - 1 2 ) = ( 4 ± 8 ) × 10 - 12 \left(\beta-\delta-\tfrac{1}{2}\right)=(4\pm 8)\times 10^{-12}\,
  21. ( α - β + 1 ) = - 4.8 ( 3.7 ) × 10 - 8 (\alpha-\beta+1)=-4.8(3.7)\times 10^{-8}\,
  22. | α + 1 2 | 8.4 × 10 - 8 \left|\alpha+\tfrac{1}{2}\right|\leq 8.4\times 10^{-8}\,

Tethered_particle_motion.html

  1. N R r L l p / 3 < 1 N_{R}\equiv\frac{r}{\sqrt{L\cdot{l_{p}/3}}}<1
  2. r r
  3. L L
  4. l p l_{p}
  5. N R > 1 N_{R}>1
  6. x 0 x_{0}
  7. y 0 y_{0}
  8. R c m = 1 I t o t k = 1 N I k r k \vec{R}_{cm}=\frac{1}{I_{tot}}\cdot\sum_{k=1}^{N}I_{k}\cdot\vec{r}_{k}
  9. R c m \vec{R}_{cm}
  10. I t o t \textstyle I_{tot}
  11. I k \textstyle I_{k}
  12. r k \textstyle\vec{r}_{k}
  13. P 1 D ( x ) d x = 3 4 π L l p exp ( - 3 x 2 4 L l p ) d x ; P 2 D ( R ) d R = 2 π R 3 4 π L l p exp ( - 3 R 2 4 L l p ) d R P_{1D}(x)dx=\sqrt{\frac{3}{4{\pi}Ll_{p}}}\exp{\left(-\frac{3x^{2}}{4Ll_{p}}% \right)}dx\,\,\,;\,\,\,\,\,\,P_{2D}(R)dR=2{\pi}R\frac{3}{4{\pi}Ll_{p}}\exp{% \left(-\frac{3R^{2}}{4Ll_{p}}\right)}dR
  14. L L
  15. l p l_{p}
  16. P ( x ) exp ( - 1 2 k x 2 K B T ) P(x)\propto\exp{\left(-\frac{\frac{1}{2}kx^{2}}{K_{B}T}\right)}
  17. k k
  18. K B K_{B}
  19. T T
  20. P ( x ) P(x)
  21. k = 2 α K B T \displaystyle k=2{\alpha}K_{B}T
  22. α \alpha
  23. x 2 x^{2}

The_Magnificent_Seven_(neutron_stars).html

  1. R R
  2. T T
  3. D D
  4. F = σ T 4 ( R D ) 2 F=\sigma T^{4}(\tfrac{R}{D})^{2}
  5. D D
  6. T T

The_Whetstone_of_Witte.html

  1. p q = p × p × p × p p^{q}=p\times p\times p\cdots\times p

Theory_of_conjoint_measurement.html

  1. Q = r × [ Q ] Q=r\times[Q]
  2. ( a , y ) > ( b , x ) \left(a,y\right)>\left(b,x\right)
  3. ( b , z ) > ( c , y ) \left(b,z\right)>\left(c,y\right)
  4. ( a , y ) > ( b , x ) \left(a,y\right)>\left(b,x\right)
  5. a + y > b + x a+y>b+x
  6. ( b , z ) > ( c , y ) \left(b,z\right)>\left(c,y\right)
  7. b + z > c + y b+z>c+y
  8. a + y + b + z > b + x + c + y a+y+b+z>b+x+c+y
  9. ( a , z ) > ( c , x ) \left(a,z\right)>\left(c,x\right)
  10. ( a , z ) < ( c , x ) \left(a,z\right)<\left(c,x\right)
  11. ( a , z ) = ( c , x ) \left(a,z\right)=\left(c,x\right)
  12. ( n 3 ) {\textstyle\left({{n}\atop{3}}\right)}
  13. ( m 3 ) {\textstyle\left({{m}\atop{3}}\right)}
  14. ( a i , x ) = ( a i + 1 , y ) \left(a_{i},x\right)=\left(a_{i+1},y\right)
  15. ( a , x ) ( b , y ) ϕ A ( a ) + ϕ X ( x ) ϕ A ( b ) + ϕ X ( y ) \left(a,x\right)\succsim\left(b,y\right)\iff\phi_{A}\left(a\right)+\phi_{X}% \left(x\right)\geqslant\phi_{A}\left(b\right)+\phi_{X}\left(y\right)
  16. ϕ A \phi^{\prime}_{A}\,
  17. ϕ X \phi^{\prime}_{X}\,
  18. α > 0 , β A \alpha>0,\beta_{A}\,
  19. β X \beta_{X}\,
  20. ϕ A = α ϕ A + β A \phi^{\prime}_{A}=\alpha\phi_{A}+\beta_{A}\,
  21. ϕ X = α ϕ X + β X \phi^{\prime}_{X}=\alpha\phi_{X}+\beta_{X}\,
  22. ϕ A , ϕ A , ϕ X \phi^{\prime}_{A},\phi_{A},\phi^{\prime}_{X}\,
  23. ϕ X \phi_{X}\,

Thermal_capillary_wave.html

  1. h ( x , y , t ) h(x,y,t)
  2. E st = σ d x d y [ 1 + ( d h d x ) 2 + ( d h d y ) 2 - 1 ] σ 2 d x d y [ ( d h d x ) 2 + ( d h d y ) 2 ] , E_{\mathrm{st}}=\sigma\int dx\,dy\,\left[\sqrt{1+\left(\frac{dh}{dx}\right)^{2% }+\left(\frac{dh}{dy}\right)^{2}}-1\right]\approx\frac{\sigma}{2}\int dx\,dy\,% \left[\left(\frac{dh}{dx}\right)^{2}+\left(\frac{dh}{dy}\right)^{2}\right],
  3. σ \sigma
  4. k T / 2 kT/2

Thermal_conductivity_measurement.html

  1. T ( t , r ) T(t,r)
  2. r r
  3. t t
  4. T ( t , r ) = Q 4 π k Ei ( r 2 4 a t ) T(t,r)=\frac{Q}{4\pi k}\mathrm{Ei}\left(\frac{r^{2}}{4at}\right)
  5. Q Q
  6. k k
  7. Ei ( x ) \mathrm{Ei}(x)
  8. r r
  9. a a
  10. t t
  11. Ei ( x ) = - γ - ln ( x ) + O ( x 2 ) \mathrm{Ei}(x)=-\gamma-\ln(x)+O(x^{2})
  12. γ 0.577 \gamma\approx 0.577
  13. T ( t , r ) = Q 4 π k { - γ - ln ( r 2 4 a ) + ln ( t ) } T(t,r)=\frac{Q}{4\pi k}\left\{-\gamma-\ln\left(\frac{r^{2}}{4a}\right)+\ln(t)\right\}
  14. k ( T ) = a ( T ) c P ( T ) ρ ( T ) k(T)=a(T)\cdot c_{P}(T)\cdot\rho(T)
  15. k k
  16. a a
  17. c P c_{P}
  18. ρ \rho
  19. V = I R = I 0 e i ω t ( R 0 + R T Δ T ) = I 0 e i ω t ( R 0 + C 0 e i 2 ω t ) V=IR=I_{0}e^{i\omega t}\left(R_{0}+\frac{\partial R}{\partial T}\Delta T\right% )=I_{0}e^{i\omega t}\left(R_{0}+C_{0}e^{i2\omega t}\right)

Thermal_copper_pillar_bump.html

  1. Q m a x = S 2 T 2 2 R T o t a l = S 2 T 2 A 2 p c L Q_{max}=\frac{S^{2}T^{2}}{2\cdot R_{Total}}=\frac{S^{2}T^{2}A}{2p_{c}L}

Thermo-mechanical_fatigue.html

  1. 1 N f = 1 N f f a t i g u e + 1 N f o x i d a t i o n + 1 N f c r e e p \frac{1}{N_{f}}=\frac{1}{N_{f}^{fatigue}}+\frac{1}{N_{f}^{oxidation}}+\frac{1}% {N_{f}^{creep}}
  2. N f N_{f}
  3. Δ ϵ m 2 = C ( 2 N f f a t i g u e ) d \frac{\Delta\epsilon_{m}}{2}=C(2N_{f}^{fatigue})^{d}
  4. C C
  5. d d
  6. 1 N f o x i d a t i o n = ( h c r δ 0 B Φ o x i d a t i o n K p e f f ) - 1 β 2 ( Δ ϵ m ˙ ) 2 β + 1 ϵ 1 - α / β \frac{1}{N_{f}^{oxidation}}=\left(\frac{h_{cr}\delta_{0}}{B\Phi^{oxidation}K_{% p}^{eff}}\right)^{\frac{-1}{\beta}}\frac{2(\Delta\dot{\epsilon_{m}})^{\frac{2}% {\beta}+1}}{\epsilon^{1-\alpha/\beta}}
  7. K p e f f = 1 t c 0 t D 0 e x p ( - Q R T ( t ) ) d t K_{p}^{eff}=\frac{1}{t_{c}}\int_{0}^{t}D_{0}exp\left(\frac{-Q}{RT(t)}\right)dt
  8. Φ o x i d a t i o n = 1 t c 0 t e x p [ - 1 2 ( ( ϵ t h ˙ / ϵ m ˙ ) + 1 ζ ˙ o x i d a t i o n ) 2 ] d t \Phi^{oxidation}=\frac{1}{t_{c}}\int_{0}^{t}exp\left[-\frac{1}{2}\left(\frac{(% \dot{\epsilon_{th}}/\dot{\epsilon_{m}})+1}{\dot{\zeta}^{oxidation}}\right)^{2}% \right]dt
  9. D c r e e p = Φ c r e e p 0 t A e ( - Δ H / R T ( t ) ) ( α 1 σ ¯ + α 2 σ H K ) m d t D^{creep}=\Phi^{creep}\int_{0}^{t}Ae^{(-\Delta H/RT(t))}\left(\frac{\alpha_{1}% \bar{\sigma}+\alpha_{2}\sigma_{H}}{K}\right)^{m}dt
  10. Φ c r e e p = 1 t c 0 t e x p [ - 1 2 ( ( ϵ t h ˙ / ϵ m ˙ ) - 1 ζ ˙ c r e e p ) 2 ] d t \Phi^{creep}=\frac{1}{t_{c}}\int_{0}^{t}exp\left[-\frac{1}{2}\left(\frac{(\dot% {\epsilon_{th}}/\dot{\epsilon_{m}})-1}{\dot{\zeta}^{creep}}\right)^{2}\right]dt
  11. 1 N f = F p p N p p + F c c N c c + F p c N p c + F c p N c p \frac{1}{N_{f}}=\frac{F_{pp}}{N^{\prime}_{pp}}+\frac{F_{cc}}{N^{\prime}_{cc}}+% \frac{F_{pc}}{N^{\prime}_{pc}}+\frac{F_{cp}}{N^{\prime}_{cp}}
  12. F p p = Δ ϵ p p Δ ϵ i n e l a s t i c , F c c = Δ ϵ c c Δ ϵ i n e l a s t i c , F p c = Δ ϵ p c Δ ϵ i n e l a s t i c , F c p = Δ ϵ c p Δ ϵ i n e l a s t i c F_{pp}=\frac{\Delta\epsilon_{pp}}{\Delta\epsilon_{inelastic}},F_{cc}=\frac{% \Delta\epsilon_{cc}}{\Delta\epsilon_{inelastic}},F_{pc}=\frac{\Delta\epsilon_{% pc}}{\Delta\epsilon_{inelastic}},F_{cp}=\frac{\Delta\epsilon_{cp}}{\Delta% \epsilon_{inelastic}}
  13. N p p N^{\prime}_{pp}
  14. Δ ϵ i n e l a s t i c = A p p ( N p p ) C p p \Delta\epsilon_{inelastic}=A_{pp}(N^{\prime}_{pp})^{C_{pp}}
  15. N f = 2 l n [ a f / a 0 ] β D N_{f}=\frac{2ln[a_{f}/a_{0}]}{\beta D}
  16. a 0 a_{0}
  17. a f a_{f}
  18. β \beta

Thomas–Fermi_model.html

  1. n ( r ) n(\vec{r})
  2. V f = 4 3 π p f 3 ( r ) V_{f}=\frac{4}{3}\pi p_{f}^{3}(\vec{r})
  3. r \vec{r}
  4. Δ V p h = V f Δ V = 4 3 π p f 3 ( r ) Δ V . \Delta V_{ph}=V_{f}\ \Delta V=\frac{4}{3}\pi p_{f}^{3}(\vec{r})\ \Delta V.
  5. Δ N p h = 2 h 3 Δ V p h = 8 π 3 h 3 p f 3 ( r ) Δ V . \Delta N_{ph}=\frac{2}{h^{3}}\ \Delta V_{ph}=\frac{8\pi}{3h^{3}}p_{f}^{3}(\vec% {r})\ \Delta V.
  6. Δ N = n ( r ) Δ V \Delta N=n(\vec{r})\ \Delta V
  7. n ( r ) n(\vec{r})
  8. n ( r ) = 8 π 3 h 3 p f 3 ( r ) . n(\vec{r})=\frac{8\pi}{3h^{3}}p_{f}^{3}(\vec{r}).
  9. r \vec{r}
  10. F r ( p ) d p = 4 π p 2 d p 4 3 π p f 3 ( r ) p p f ( r ) = 0 otherwise \begin{aligned}\displaystyle F_{\vec{r}}(p)dp&\displaystyle=\frac{4\pi p^{2}dp% }{\frac{4}{3}\pi p_{f}^{3}(\vec{r})}\qquad\qquad p\leq p_{f}(\vec{r})\\ &\displaystyle=0\qquad\qquad\qquad\quad\,\text{otherwise}\\ \end{aligned}
  11. r \vec{r}
  12. t ( r ) = p 2 2 m e n ( r ) F r ( p ) d p = n ( r ) 0 p f ( r ) p 2 2 m e 4 π p 2 4 3 π p f 3 ( r ) d p = C F [ n ( r ) ] 5 / 3 \begin{aligned}\displaystyle t(\vec{r})&\displaystyle=\int\frac{p^{2}}{2m_{e}}% \ n(\vec{r})\ F_{\vec{r}}(p)\ dp\\ &\displaystyle=n(\vec{r})\int_{0}^{p_{f}(\vec{r})}\frac{p^{2}}{2m_{e}}\ \ % \frac{4\pi p^{2}}{\frac{4}{3}\pi p_{f}^{3}(\vec{r})}\ dp\\ &\displaystyle=C_{F}\ [n(\vec{r})]^{5/3}\end{aligned}
  13. n ( r ) n(\vec{r})
  14. p f ( r ) p_{f}(\vec{r})
  15. C F = 3 h 2 10 m e ( 3 8 π ) 2 3 . C_{F}=\frac{3h^{2}}{10m_{e}}\left(\frac{3}{8\pi}\right)^{\frac{2}{3}}.
  16. t ( r ) t(\vec{r})
  17. T = C F [ n ( r ) ] 5 / 3 d 3 r . T=C_{F}\int[n(\vec{r})]^{5/3}\ d^{3}r\ .
  18. n ( r ) , n(\vec{r}),
  19. U e N = n ( r ) V N ( r ) d 3 r U_{eN}=\int n(\vec{r})\ V_{N}(\vec{r})\ d^{3}r\,
  20. V N ( r ) V_{N}(\vec{r})\,
  21. r \vec{r}\,
  22. r = 0 \vec{r}=0
  23. V N ( r ) = - Z e 2 r . V_{N}(\vec{r})=\frac{-Ze^{2}}{r}.
  24. U e e = 1 2 e 2 n ( r ) n ( r ) | r - r | d 3 r d 3 r . U_{ee}=\frac{1}{2}\ e^{2}\int\frac{n(\vec{r})\ n(\vec{r}\,^{\prime})}{\left|% \vec{r}-\vec{r}\,^{\prime}\right|}\ d^{3}r\ d^{3}r^{\prime}.
  25. E = T + U e N + U e e = C F [ n ( r ) ] 5 / 3 d 3 r + n ( r ) V N ( r ) d 3 r + 1 2 e 2 n ( r ) n ( r ) | r - r | d 3 r d 3 r \begin{aligned}\displaystyle E&\displaystyle=T\ +\ U_{eN}\ +\ U_{ee}\\ &\displaystyle=C_{F}\int[n(\vec{r})]^{5/3}\ d^{3}r\ +\int n(\vec{r})\ V_{N}(% \vec{r})\ d^{3}r\ +\ \frac{1}{2}\ e^{2}\int\frac{n(\vec{r})\ n(\vec{r}\,^{% \prime})}{\left|\vec{r}-\vec{r}\,^{\prime}\right|}\ d^{3}r\ d^{3}r^{\prime}\\ \end{aligned}
  26. T W = 1 8 2 m | n ( r ) | 2 n ( r ) d r . T_{W}=\frac{1}{8}\frac{\hbar^{2}}{m}\int\frac{|\nabla n(\vec{r})|^{2}}{n(\vec{% r})}dr.

Three-dimensional_rotation_operator.html

  1. e ^ 1 , e ^ 2 , e ^ 3 \hat{e}_{1}\ ,\ \hat{e}_{2}\ ,\ \hat{e}_{3}
  2. 𝐀 e ^ 1 , 𝐀 e ^ 2 , 𝐀 e ^ 3 . \mathbf{A}\hat{e}_{1}\ ,\ \mathbf{A}\hat{e}_{2}\ ,\ \mathbf{A}\hat{e}_{3}.
  3. x ¯ = x 1 e ^ 1 + x 2 e ^ 2 + x 3 e ^ 3 \bar{x}\ =x_{1}\hat{e}_{1}+x_{2}\hat{e}_{2}+x_{3}\hat{e}_{3}
  4. 𝐀 x ¯ = x 1 𝐀 e ^ 1 + x 2 𝐀 e ^ 2 + x 3 𝐀 e ^ 3 \mathbf{A}\bar{x}\ =x_{1}\mathbf{A}\hat{e}_{1}+x_{2}\mathbf{A}\hat{e}_{2}+x_{3% }\mathbf{A}\hat{e}_{3}
  5. e ^ 1 , e ^ 2 , e ^ 3 \hat{e}_{1}\ ,\ \hat{e}_{2}\ ,\ \hat{e}_{3}
  6. [ A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 ] = [ e ^ 1 | 𝐀 e ^ 1 e ^ 1 | 𝐀 e ^ 2 e ^ 1 | 𝐀 e ^ 3 e ^ 2 | 𝐀 e ^ 1 e ^ 2 | 𝐀 e ^ 2 e ^ 2 | 𝐀 e ^ 3 e ^ 3 | 𝐀 e ^ 1 e ^ 3 | 𝐀 e ^ 2 e ^ 3 | 𝐀 e ^ 3 ] \begin{bmatrix}A_{11}&A_{12}&A_{13}\\ A_{21}&A_{22}&A_{23}\\ A_{31}&A_{32}&A_{33}\end{bmatrix}=\begin{bmatrix}\langle\hat{e}_{1}|\mathbf{A}% \hat{e}_{1}\rangle&\langle\hat{e}_{1}|\mathbf{A}\hat{e}_{2}\rangle&\langle\hat% {e}_{1}|\mathbf{A}\hat{e}_{3}\rangle\\ \langle\hat{e}_{2}|\mathbf{A}\hat{e}_{1}\rangle&\langle\hat{e}_{2}|\mathbf{A}% \hat{e}_{2}\rangle&\langle\hat{e}_{2}|\mathbf{A}\hat{e}_{3}\rangle\\ \langle\hat{e}_{3}|\mathbf{A}\hat{e}_{1}\rangle&\langle\hat{e}_{3}|\mathbf{A}% \hat{e}_{2}\rangle&\langle\hat{e}_{3}|\mathbf{A}\hat{e}_{3}\rangle\end{bmatrix}
  7. k = 1 3 A k i A k j = 𝐀 e ^ i | 𝐀 e ^ j = { 0 i j , 1 i = j , \sum_{k=1}^{3}A_{ki}A_{kj}=\langle\mathbf{A}\hat{e}_{i}|\mathbf{A}\hat{e}_{j}% \rangle=\begin{cases}0&i\neq j,\\ 1&i=j,\end{cases}
  8. [ A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 ] T [ A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 ] = [ 1 0 0 0 1 0 0 0 1 ] \begin{bmatrix}A_{11}&A_{12}&A_{13}\\ A_{21}&A_{22}&A_{23}\\ A_{31}&A_{32}&A_{33}\end{bmatrix}^{T}\begin{bmatrix}A_{11}&A_{12}&A_{13}\\ A_{21}&A_{22}&A_{23}\\ A_{31}&A_{32}&A_{33}\end{bmatrix}=\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix}
  9. e ^ 1 , e ^ 2 , e ^ 3 \hat{e}_{1}\ ,\ \hat{e}_{2}\ ,\ \hat{e}_{3}
  10. R 3 R^{3}
  11. θ \theta
  12. e ^ 3 \hat{e}_{3}
  13. [ Y 1 Y 2 Y 3 ] = [ cos θ - sin θ 0 sin θ cos θ 0 0 0 1 ] [ X 1 X 2 X 3 ] \begin{bmatrix}Y_{1}\\ Y_{2}\\ Y_{3}\end{bmatrix}=\begin{bmatrix}\cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1\end{bmatrix}\begin{bmatrix}X_{1}\\ X_{2}\\ X_{3}\end{bmatrix}
  14. x ¯ = [ e ^ 1 e ^ 2 e ^ 3 ] [ X 1 X 2 X 3 ] \bar{x}=\begin{bmatrix}\hat{e}_{1}&\hat{e}_{2}&\hat{e}_{3}\end{bmatrix}\begin{% bmatrix}X_{1}\\ X_{2}\\ X_{3}\end{bmatrix}
  15. y ¯ = [ e ^ 1 e ^ 2 e ^ 3 ] [ Y 1 Y 2 Y 3 ] \bar{y}=\begin{bmatrix}\hat{e}_{1}&\hat{e}_{2}&\hat{e}_{3}\end{bmatrix}\begin{% bmatrix}Y_{1}\\ Y_{2}\\ Y_{3}\end{bmatrix}
  16. det [ cos θ - sin θ 0 sin θ cos θ 0 0 0 1 ] = 1 \det\begin{bmatrix}\cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1\end{bmatrix}=1
  17. det [ cos θ - λ - sin θ 0 sin θ cos θ - λ 0 0 0 1 - λ ] = ( ( cos θ - λ ) 2 + sin θ 2 ) ( 1 - λ ) = - λ 3 + ( 2 cos θ + 1 ) λ 2 - ( 2 cos θ + 1 ) λ + 1 \begin{aligned}\displaystyle\det\begin{bmatrix}\cos\theta-\lambda&-\sin\theta&% 0\\ \sin\theta&\cos\theta-\lambda&0\\ 0&0&1-\lambda\end{bmatrix}&\displaystyle=\big({(\cos\theta-\lambda)}^{2}+{\sin% \theta}^{2}\big)(1-\lambda)\\ &\displaystyle=-\lambda^{3}+(2\ \cos\theta\ +\ 1)\ \lambda^{2}-(2\ \cos\theta% \ +\ 1)\ \lambda+1\\ \end{aligned}
  18. sin θ = 0 \sin\theta=0
  19. θ = 0 \theta=0
  20. θ = π \theta=\pi
  21. θ = 0 \theta=0
  22. θ = π \theta=\pi
  23. - ( λ - 1 ) ( λ + 1 ) 2 -(\lambda-1){(\lambda+1)}^{2}
  24. λ = 1 λ = - 1 \lambda=1\quad\lambda=-1
  25. λ = 1 \lambda=1
  26. x ¯ = α e ^ 3 - < α < \bar{x}=\alpha\ \hat{e}_{3}\quad-\infty<\alpha<\infty
  27. λ = - 1 \lambda=-1
  28. x ¯ = α e ^ 1 + β e ^ 2 - < α < - < β < \bar{x}=\alpha\ \hat{e}_{1}+\beta\ \hat{e}_{2}\quad-\infty<\alpha<\infty\quad-% \infty<\beta<\infty
  29. θ \theta
  30. sin θ 2 > 0 {\sin\theta}^{2}>0
  31. λ = 1 \lambda=1
  32. x ¯ = α e ^ 3 - < α < \bar{x}=\alpha\ \hat{e}_{3}\quad-\infty<\alpha<\infty
  33. θ \theta
  34. 𝐤 = [ k 1 k 2 k 3 ] \mathbf{k}=\left[\begin{array}[]{ccc}k_{1}\\ k_{2}\\ k_{3}\end{array}\right]
  35. R = I cos θ + [ 𝐤 ] × sin θ + ( 1 - cos θ ) 𝐤𝐤 𝖳 R=I\cos\theta+[\mathbf{k}]_{\times}\sin\theta+(1-\cos\theta)\mathbf{k}\mathbf{% k}^{\mathsf{T}}
  36. I I
  37. [ 𝐤 ] × [\mathbf{k}]_{\times}
  38. 𝐤 \mathbf{k}
  39. [ 𝐤 ] × = [ 0 - k 3 k 2 k 3 0 - k 1 - k 2 k 1 0 ] [\mathbf{k}]_{\times}=\left[\begin{array}[]{ccc}0&-k_{3}&k_{2}\\ k_{3}&0&-k_{1}\\ -k_{2}&k_{1}&0\end{array}\right]
  40. [ 𝐤 ] × [\mathbf{k}]_{\times}
  41. [ 𝐤 ] × 𝐯 = 𝐤 × 𝐯 [\mathbf{k}]_{\times}\mathbf{v}=\mathbf{k}\times\mathbf{v}
  42. 𝐯 \mathbf{v}
  43. θ \theta
  44. R 3 R^{3}
  45. e ^ 1 , e ^ 2 , e ^ 3 \hat{e}_{1}\ ,\ \hat{e}_{2}\ ,\ \hat{e}_{3}
  46. [ cos θ - sin θ 0 sin θ cos θ 0 0 0 1 ] \begin{bmatrix}\cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1\end{bmatrix}
  47. θ \theta
  48. [ A 11 A 12 A 13 A 21 A 22 A 23 A 31 A 32 A 33 ] \begin{bmatrix}A_{11}&A_{12}&A_{13}\\ A_{21}&A_{22}&A_{23}\\ A_{31}&A_{32}&A_{33}\end{bmatrix}
  49. f ^ 1 , f ^ 2 , f ^ 3 \hat{f}_{1}\ ,\ \hat{f}_{2}\ ,\ \hat{f}_{3}
  50. R n R^{n}
  51. e ^ 1 , e ^ 2 , e ^ 3 \hat{e}_{1}\ ,\ \hat{e}_{2}\ ,\ \hat{e}_{3}
  52. [ B 11 0 0 0 B 22 0 0 0 B 33 ] \begin{bmatrix}B_{11}&0&0\\ 0&B_{22}&0\\ 0&0&B_{33}\end{bmatrix}
  53. B i i B_{ii}
  54. θ = 0 \theta=0
  55. [ - 1 0 0 0 - 1 0 0 0 1 ] \begin{bmatrix}-1&0&0\\ 0&-1&0\\ 0&0&1\end{bmatrix}
  56. θ = π \theta=\pi
  57. E ¯ = α 1 f ^ 1 + α 2 f ^ 2 + α 3 f ^ 3 \bar{E}=\alpha_{1}\ \hat{f}_{1}+\alpha_{2}\ \hat{f}_{2}+\alpha_{3}\ \hat{f}_{3}
  58. α 1 = A 32 - A 23 2 \alpha_{1}=\frac{A_{32}-A_{23}}{2}
  59. α 2 = A 13 - A 31 2 \alpha_{2}=\frac{A_{13}-A_{31}}{2}
  60. α 3 = A 21 - A 12 2 \alpha_{3}=\frac{A_{21}-A_{12}}{2}
  61. λ = 1 \lambda=1
  62. e ^ 3 = E ¯ | E ¯ | \hat{e}_{3}=\frac{\bar{E}}{|\bar{E}|}
  63. e ^ 3 \hat{e}_{3}
  64. e ^ 1 , e ^ 2 \hat{e}_{1}\ ,\ \hat{e}_{2}
  65. e ^ 1 , e ^ 2 , e ^ 3 \hat{e}_{1}\ ,\ \hat{e}_{2},\ \hat{e}_{3}
  66. cos θ = A 11 + A 22 + A 33 - 1 2 \cos\theta=\frac{A_{11}+A_{22}+A_{33}-1}{2}
  67. sin θ = | E ¯ | \sin\theta=|\bar{E}|
  68. θ = 0 \theta=0
  69. θ = π \theta=\pi
  70. θ = π \theta=\pi
  71. E ¯ = α 1 f ^ 1 + α 2 f ^ 2 + α 3 f ^ 3 \bar{E}=\alpha_{1}\ \hat{f}_{1}+\alpha_{2}\ \hat{f}_{2}+\alpha_{3}\ \hat{f}_{3}
  72. E 4 E_{4}
  73. cos θ \cos\theta
  74. 1 - E 4 E 1 2 + E 2 2 + E 3 2 [ E 1 E 1 E 1 E 2 E 1 E 3 E 2 E 1 E 2 E 2 E 2 E 3 E 3 E 1 E 3 E 2 E 3 E 3 ] + [ E 4 - E 3 E 2 E 3 E 4 - E 1 - E 2 E 1 E 4 ] \frac{1-E_{4}}{{E_{1}}^{2}+{E_{2}}^{2}+{E_{3}}^{2}}\begin{bmatrix}E_{1}E_{1}&E% _{1}E_{2}&E_{1}E_{3}\\ E_{2}E_{1}&E_{2}E_{2}&E_{2}E_{3}\\ E_{3}E_{1}&E_{3}E_{2}&E_{3}E_{3}\end{bmatrix}+\begin{bmatrix}E_{4}&-E_{3}&E_{2% }\\ E_{3}&E_{4}&-E_{1}\\ -E_{2}&E_{1}&E_{4}\end{bmatrix}
  75. E 1 2 + E 2 2 + E 3 2 > 0 {E_{1}}^{2}+{E_{2}}^{2}+{E_{3}}^{2}>0
  76. θ = 0 \theta=0
  77. θ = π \theta=\pi
  78. E 1 , E 2 , E 3 , E 4 E_{1}\ ,\ E_{2}\ ,\ E_{3}\ ,\ E_{4}
  79. θ 2 \frac{\theta}{2}
  80. θ \theta
  81. q 1 , q 2 , q 3 q_{1}\ ,\ q_{2}\ ,\ q_{3}
  82. q 1 f 1 ^ + q 2 f 2 ^ + q 3 f 1 ^ = sin θ 2 e 3 ^ = sin θ 2 sin θ E ¯ q_{1}\ \hat{f_{1}}\ +\ q_{2}\ \hat{f_{2}}\ +\ \ q_{3}\ \hat{f_{1}}\ =\ \sin% \frac{\theta}{2}\quad\hat{e_{3}}=\frac{\sin\frac{\theta}{2}}{\sin\theta}\quad% \bar{E}
  83. q 4 = cos θ 2 q_{4}=\cos\frac{\theta}{2}
  84. θ \theta
  85. 0 θ π 0\leq\theta\leq\pi
  86. q 4 0 q_{4}\geq 0
  87. q 1 , q 2 , q 3 , q 4 q_{1}\ ,\ q_{2}\ ,\ q_{3}\ ,\ q_{4}
  88. - q 1 , - q 2 , - q 3 , - q 4 -q_{1}\ ,\ -q_{2}\ ,\ -q_{3}\ ,\ -q_{4}
  89. E k E_{k}
  90. E 1 = 2 q 4 q 1 E_{1}=2q_{4}q_{1}
  91. E 2 = 2 q 4 q 2 E_{2}=2q_{4}q_{2}
  92. E 3 = 2 q 4 q 3 E_{3}=2q_{4}q_{3}
  93. E 4 = q 4 2 - ( q 1 2 + q 2 2 + q 3 2 ) E_{4}={q_{4}}^{2}-({q_{1}}^{2}+{q_{2}}^{2}+{q_{3}}^{2})
  94. [ 2 ( q 1 2 + q 4 2 ) - 1 2 ( q 1 q 2 - q 3 q 4 ) 2 ( q 1 q 3 + q 2 q 4 ) 2 ( q 1 q 2 + q 3 q 4 ) 2 ( q 2 2 + q 4 2 ) - 1 2 ( q 2 q 3 - q 1 q 4 ) 2 ( q 1 q 3 - q 2 q 4 ) 2 ( q 2 q 3 + q 1 q 4 ) 2 ( q 3 2 + q 4 2 ) - 1 ] \begin{bmatrix}2({q_{1}}^{2}+{q_{4}}^{2})-1&2({q_{1}}{q_{2}}-{q_{3}}{q_{4}})&2% ({q_{1}}{q_{3}}+{q_{2}}{q_{4}})\\ 2({q_{1}}{q_{2}}+{q_{3}}{q_{4}})&2({q_{2}}^{2}+{q_{4}}^{2})-1&2({q_{2}}{q_{3}}% -{q_{1}}{q_{4}})\\ 2({q_{1}}{q_{3}}-{q_{2}}{q_{4}})&2({q_{2}}{q_{3}}+{q_{1}}{q_{4}})&2({q_{3}}^{2% }+{q_{4}}^{2})-1\\ \end{bmatrix}
  95. α = 10 β = 20 γ = 30 \alpha=10^{\circ}\quad\beta=20^{\circ}\quad\gamma=30^{\circ}\quad
  96. f ^ 1 , f ^ 2 , f ^ 3 \hat{f}_{1}\ ,\ \hat{f}_{2},\ \hat{f}_{3}
  97. [ 0.771281 - 0.633718 0.059391 0.613092 0.714610 - 0.336824 0.171010 0.296198 0.939693 ] \begin{bmatrix}0.771281&-0.633718&0.059391\\ 0.613092&0.714610&-0.336824\\ 0.171010&0.296198&0.939693\end{bmatrix}
  98. ( 0.171010 , - 0.030154 , 0.336824 , 0.925417 ) (0.171010,\ -0.030154,\ 0.336824,\ 0.925417)
  99. [ cos θ - sin θ 0 sin θ cos θ 0 0 0 1 ] \begin{bmatrix}\cos\theta&-\sin\theta&0\\ \sin\theta&\cos\theta&0\\ 0&0&1\end{bmatrix}
  100. θ = 44.537 \theta=44.537^{\circ}
  101. e ^ 3 = ( 0.451272 , - 0.079571 , 0.888832 ) \hat{e}_{3}=(0.451272,-0.079571,0.888832)
  102. ( 0 , 0 , 0.378951 , 0.925417 ) = ( 0 , 0 , sin θ 2 , cos θ 2 ) (0,\ 0,\ 0.378951,\ 0.925417)=(0,\ 0,\ \sin\frac{\theta}{2},\ \cos\frac{\theta% }{2})
  103. 10 , 20 , 30 10^{\circ},20^{\circ},30^{\circ}
  104. 44.537 44.537^{\circ}
  105. e ^ 3 \hat{e}_{3}

Three_dots.html

  1. \therefore\!\,

Threefish.html

  1. N w N_{w}
  2. 2 64 2^{64}
  3. N r 4 + 1 \frac{N_{r}}{4}+1
  4. N r N_{r}
  5. k 0 , k 1 , , k N w - 1 k_{0},k_{1},\dots,k_{N_{w}-1}
  6. k N w k_{N_{w}}
  7. t 0 , t 1 t_{0},t_{1}
  8. t 2 = t 0 t 1 t_{2}=t_{0}\oplus t_{1}
  9. k N w = C 240 k 0 k 1 k N w - 1 k_{N_{w}}=C_{240}\oplus k_{0}\oplus k_{1}\oplus\dots\oplus k_{N_{w}-1}
  10. C 240 = 0x1BD11BDAA9FC1A22 C_{240}=\,\text{0x1BD11BDAA9FC1A22}
  11. k s , i k_{s,i}
  12. k s , i = { k ( s + i ) mod ( N w + 1 ) i = 0 , , N w - 4 k ( s + i ) mod ( N w + 1 ) + t s mod 3 i = N w - 3 k ( s + i ) mod ( N w + 1 ) + t ( s + 1 ) mod 3 i = N w - 2 k ( s + i ) mod ( N w + 1 ) + s i = N w - 1 k_{s,i}=\begin{cases}k_{(s+i)\bmod(N_{w}+1)}&i=0,\dots,N_{w}-4\\ k_{(s+i)\bmod(N_{w}+1)}+t_{s\bmod 3}&i=N_{w}-3\\ k_{(s+i)\bmod(N_{w}+1)}+t_{(s+1)\bmod 3}&i=N_{w}-2\\ k_{(s+i)\bmod(N_{w}+1)}+s&i=N_{w}-1\end{cases}
  13. ( x 0 , x 1 ) (x_{0},x_{1})
  14. ( y 0 , y 1 ) (y_{0},y_{1})
  15. y 0 = x 0 + x 1 mod 2 64 y_{0}=x_{0}+x_{1}\mod 2^{64}
  16. y 1 = ( y 1 R ( d mod 8 ) , j ) y 0 y_{1}=(y_{1}\lll R_{(d\bmod 8),j})\oplus y_{0}
  17. R d , j R_{d,j}
  18. d mod 4 = = 0 d\;\bmod\;4==0
  19. k d 4 k_{\frac{d}{4}}
  20. 2 226 2^{226}
  21. 2 12 2^{12}
  22. 2 352.17 2^{352.17}
  23. 2 222 2^{222}
  24. 2 12 2^{12}
  25. 2 355.5 2^{355.5}

Threshold_displacement_energy.html

  1. T d T_{d}
  2. T d , m i n T_{d,min}
  3. T d , a v e T_{d,ave}
  4. T m a x T_{max}
  5. T m a x = 2 M E ( E + 2 m c 2 ) ( m + M ) 2 c 2 + 2 M E T_{max}={2ME(E+2mc^{2})\over(m+M)^{2}c^{2}+2ME}
  6. m c 2 mc^{2}
  7. T m a x = E 4 M m ( m + M ) 2 T_{max}=E{4Mm\over(m+M)^{2}}
  8. T m a x T_{max}
  9. T d ( θ , ϕ ) = T d ( [ h k l ] ) T_{d}(\theta,\phi)=T_{d}([hkl])
  10. T d , m i n = min ( T d ( θ , ϕ ) ) T_{d,min}=\min(T_{d}(\theta,\phi))
  11. T d , a v e = ave ( T d ( θ , ϕ ) ) T_{d,ave}={\rm ave}(T_{d}(\theta,\phi))
  12. T d l T^{l}_{d}
  13. T d u T^{u}_{d}
  14. T d , m i n l T^{l}_{d,min}
  15. T d l ( θ , ϕ ) T_{d}^{l}(\theta,\phi)
  16. N F P N_{FP}
  17. F D n F_{Dn}
  18. N F P = 0.8 F D n 2 T d , a v e N_{FP}=0.8{F_{Dn}\over 2T_{d,ave}}
  19. 2 T d , a v e / 0.8 2T_{d,ave}/0.8

Time-domain_thermoreflectance.html

  1. Δ T ( z ) = ( 1 - R ) Q C ( ζ A ) e x p ( - z / ζ ) \Delta T(z)=(1-R)\frac{Q}{C(\zeta A)}exp(-z/\zeta)
  2. g ( r ) = e x p ( - q r ) ( 2 π Λ r ) g(r)=\frac{exp(-qr)}{(2\pi\Lambda r)}
  3. q 2 = ( i w / d ) q^{2}=(iw/d)
  4. G ( k ) = 2 π 0 g ( r ) J 0 ( 2 π k r ) r d r = 1 Λ ( 4 π 2 k 2 + q 2 ) 1 / 2 G(k)=2\pi\int_{0}^{\infty}g(r)J_{0}(2\pi kr)r\,dr=\frac{1}{\Lambda(4\pi^{2}k^{% 2}+q^{2})^{1/2}}
  5. p ( r ) = 2 A π ω 0 2 e x p ( - 2 r 2 / ω 0 2 ) p(r)=\frac{2A}{\pi\omega_{0}^{2}}exp(-2r^{2}/\omega_{0}^{2})
  6. P ( k ) = A e x p ( - π 2 k 2 ω 0 2 / 2 ) P(k)=Aexp(-\pi^{2}k^{2}\omega_{0}^{2}/2)
  7. θ ( r ) = 2 π 0 P ( k ) G ( k ) J 0 ( 2 π k r ) k d k \theta(r)=2\pi\int_{0}^{\infty}P(k)G(k)J_{0}(2\pi kr)kdk
  8. Δ T = 2 π A 0 G ( k ) e x p ( - π 2 k 2 ( ω 0 2 + ω 1 2 ) / 2 ) k d k \Delta T=2\pi A\int_{0}^{\infty}G(k)exp(-\pi^{2}k^{2}(\omega_{0}^{2}+\omega_{1% }^{2})/2)kdk
  9. G ( k ) = ( B 1 + + B 1 - B 1 - - B 1 + ) 1 γ 1 G(k)=(\frac{B_{1}^{+}+B_{1}^{-}}{B_{1}^{-}-B_{1}^{+}})\frac{1}{\gamma_{1}}
  10. ( B + B - ) n = 1 2 γ n ( e x p ( - u n L n ) 0 0 e x p ( u n L n ) ) ( γ n + γ n + 1 γ n - γ n + 1 γ n - γ n + 1 γ n + γ n + 1 ) ( B + B - ) n + 1 \left(\begin{array}[]{c}B^{+}\\ B^{-}\end{array}\right)_{n}=\frac{1}{2\gamma_{n}}\left(\begin{array}[]{cc}exp(% -u_{n}L_{n})&0\\ 0&exp(u_{n}L_{n})\end{array}\right)\left(\begin{array}[]{cc}\gamma_{n}+\gamma_% {n+1}&\gamma_{n}-\gamma_{n+1}\\ \gamma_{n}-\gamma_{n+1}&\gamma_{n}+\gamma_{n+1}\end{array}\right)\left(\begin{% array}[]{c}B^{+}\\ B^{-}\end{array}\right)_{n+1}
  11. u n = ( 4 π 2 k 2 + q n 2 ) 1 / 2 , q n 2 = i w D n , γ n = Λ n u n u_{n}=(4\pi^{2}k^{2}+q_{n}^{2})^{1/2},q_{n}^{2}=\frac{iw}{D_{n}},\gamma_{n}=% \Lambda_{n}u_{n}
  12. R e [ Δ R M ( t ) ] = d R d T m = - M M ( Δ T ( m / τ + f ) + Δ T ( m / τ - f ) ) e x p ( i 2 π m t / τ ) Re[\Delta RM(t)]=\frac{dR}{dT}\sum_{m=-M}^{M}(\Delta T(m/\tau+f)+\Delta T(m/% \tau-f))exp(i2\pi mt/\tau)
  13. I m [ Δ R M ( t ) ] = - i d R d T m = - M M ( Δ T ( m / τ + f ) - Δ T ( m / τ - f ) ) e x p ( i 2 π m t / τ ) Im[\Delta RM(t)]=-i\frac{dR}{dT}\sum_{m=-M}^{M}(\Delta T(m/\tau+f)-\Delta T(m/% \tau-f))exp(i2\pi mt/\tau)
  14. V f ( t ) V 0 = Q 2 Δ R ( t ) R \frac{V_{f}(t)}{V_{0}}=\frac{Q}{\sqrt{2}}\frac{\Delta R(t)}{R}

Time-inhomogeneous_hidden_Bernoulli_model.html

  1. O ( N L ) O(NL)
  2. O ( N 2 L ) O(N^{2}L)
  3. N N
  4. L L

Time-of-flight_camera.html

  1. t D = 2 D c = 2 2.5 m 300 000 000 m s = 0.000 000 016 66 s = 16.66 ns t_{D}=2\cdot\frac{D}{c}=2\cdot\frac{2.5\;\mathrm{m}}{300\;000\;000\;\frac{% \mathrm{m}}{\mathrm{s}}}=0.000\;000\;016\;66\;\mathrm{s}=16.66\;\mathrm{ns}
  2. D max = 1 2 c t 0 = 1 2 300 000 000 m s 0.000 000 05 s = 7.5 m D_{\mathrm{max}}=\frac{1}{2}\cdot c\cdot t_{0}=\frac{1}{2}\cdot 300\;000\;000% \;\frac{\mathrm{m}}{\mathrm{s}}\cdot 0.000\;000\;05\;\mathrm{s}=\!\ 7.5\;% \mathrm{m}
  3. D = 1 2 c t 0 S 2 S 1 + S 2 D=\frac{1}{2}\cdot c\cdot t_{0}\cdot\frac{S2}{S1+S2}
  4. D = 7.5 m 0.33 0.33 + 0.66 = 2.5 m D=7.5\;\mathrm{m}\cdot\frac{0.33}{0.33+0.66}=2.5\;\mathrm{m}

Time_constant.html

  1. 1 - 1 / e 63.2 % 1-1/e\approx 63.2\,\%
  2. 1 / e 36.8 % 1/e\approx 36.8\,\%
  3. d V d t + 1 τ V = f ( t ) {dV\over dt}+\frac{1}{\tau}V=f(t)
  4. V = V ( t ) . V=V(t).
  5. u ( t ) = { 0 , t < 0 1 , t 0 u(t)=\begin{cases}0,&t<0\\ 1,&t\geq 0\end{cases}
  6. f ( t ) = A sin ( 2 π f t ) f(t)=A\sin(2\pi ft)
  7. f ( t ) = A e j ω t , f(t)=Ae^{j\omega t},
  8. V ( t ) = V o e - t / τ V(t)=V_{o}e^{-t/\tau}
  9. V o = V ( t = 0 ) V_{o}=V(t=0)
  10. V ( t ) = V 0 e - t τ . V(t)=V_{0}e^{-{t\over\tau}}.
  11. τ \tau
  12. t > 0 t>0
  13. t = 0 t=0
  14. V = V 0 e 0 V=V_{0}e^{0}
  15. V = V 0 V=V_{0}
  16. t = τ t=\tau
  17. V = V 0 e - 1 0.37 V 0 V=V_{0}e^{-1}\approx 0.37V_{0}
  18. V = f ( t ) = V 0 e - t τ V=f(t)=V_{0}e^{-{t\over\tau}}
  19. lim t f ( t ) = 0 \lim_{t\to\infty}f(t)=0
  20. t = 5 τ t=5\tau
  21. V = V 0 e - 5 0.0067 V 0 V=V_{0}e^{-5}\approx 0.0067V_{0}
  22. d V d t + 1 τ V = f ( t ) = A e j ω t . {dV\over dt}+\frac{1}{\tau}V=f(t)=Ae^{j\omega t}.
  23. V ( t ) = V 0 e - t / τ + A e - t / τ 0 t d t e t / τ e j ω t V(t)=V_{0}e^{-t/\tau}+Ae^{-t/\tau}\int_{0}^{t}\,dt^{\prime}\ e^{t^{\prime}/% \tau}e^{j\omega t^{\prime}}
  24. = V 0 e - t / τ + A 1 j ω + 1 / τ ( e j ω t - e - t / τ ) . =V_{0}e^{-t/\tau}+A\frac{1}{j\omega+1/\tau}\left(e^{j\omega t}-e^{-t/\tau}% \right).
  25. V ( t ) = A e j ω t j ω + 1 / τ . V_{\infty}(t)=A\frac{e^{j\omega t}}{j\omega+1/\tau}.
  26. | V ( t ) | = A 1 τ ( ω 2 + ( 1 / τ ) 2 ) 1 / 2 = A 1 1 + ( ω τ ) 2 . |V_{\infty}(t)|=A\frac{1}{\tau\left(\omega^{2}+(1/\tau)^{2}\right)^{1/2}}=A% \frac{1}{\sqrt{1+(\omega\tau)^{2}}}.
  27. f 3 d B = 1 2 π τ . f_{3dB}=\frac{1}{2\pi\tau}.
  28. τ \tau
  29. d V d t + 1 τ V = f ( t ) = A u ( t ) , {dV\over dt}+\frac{1}{\tau}V=f(t)=Au(t),
  30. V ( t ) = V 0 e - t / τ + A τ ( 1 - e - t / τ ) . V(t)=V_{0}e^{-t/\tau}+A\tau\left(1-e^{-t/\tau}\right).
  31. V = A τ . V_{\infty}=A\tau.
  32. τ \tau
  33. τ = L R \tau={L\over R}
  34. τ \tau
  35. τ = R C \tau=RC
  36. 5 τ = FO4 5\tau=\,\text{FO4}
  37. F = h A s ( T ( t ) - T a ) , F=hA_{s}\left(T(t)-T_{a}\right),
  38. ρ c p V d T d t = - F , \rho c_{p}V\frac{dT}{dt}=-F,
  39. ρ c p V d T d t = - h A s ( T ( t ) - T a ) . \rho c_{p}V\frac{dT}{dt}=-hA_{s}\left(T(t)-T_{a}\right).
  40. d T d t + 1 τ T = 1 τ T a , \frac{dT}{dt}+\frac{1}{\tau}T=\frac{1}{\tau}T_{a},
  41. τ = ρ c p V h A s . \tau=\frac{\rho c_{p}V}{hA_{s}}.
  42. d Δ T d t + 1 τ Δ T = 0. \frac{d\Delta T}{dt}+\frac{1}{\tau}\Delta T=0.
  43. Δ T ( t ) = Δ T 0 e - t / τ , \Delta T(t)=\Delta T_{0}e^{-t/\tau},
  44. τ \tau
  45. τ = r m c m \tau=r_{m}c_{m}
  46. V ( t ) = V max ( 1 - e - t / τ ) V(t)=V_{\textrm{max}}(1-e^{-t/\tau})
  47. V ( t ) = V max e - t / τ V(t)=V_{\textrm{max}}e^{-t/\tau}
  48. τ \tau
  49. V max = r m I V_{\textrm{max}}=r_{m}I
  50. τ \tau
  51. τ \tau
  52. τ \tau
  53. T H L = τ ln 2. T_{HL}=\tau\cdot\mathrm{ln}\,2.
  54. λ = 1 / τ . \lambda=1/\tau.

Time_in_physics.html

  1. s \mathrm{s}
  2. d d t L θ ˙ - L θ = 0. \frac{d}{dt}\frac{\partial L}{\partial\dot{\theta}}-\frac{\partial L}{\partial% \theta}=0.
  3. θ ˙ {\dot{\theta}}
  4. < m t p l > θ <mtpl>{{\theta}}
  5. θ ˙ {\dot{\theta}}
  6. < m t p l > θ <mtpl>{{\theta}}
  7. p ˙ = - H q = { p , H } = - { H , p } \dot{p}=-\frac{\partial H}{\partial q}=\{p,H\}=-\{H,p\}
  8. q ˙ = H p = { q , H } = - { H , q } \dot{q}=~{}~{}\frac{\partial H}{\partial p}=\{q,H\}=-\{H,q\}
  9. \nabla
  10. × 𝐄 = - 𝐁 t \nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial t}
  11. × 𝐁 = μ 0 ε 0 𝐄 t = 1 c 2 𝐄 t \nabla\times\mathbf{B}=\mu_{0}\varepsilon_{0}\frac{\partial\mathbf{E}}{% \partial t}=\frac{1}{c^{2}}\frac{\partial\mathbf{E}}{\partial t}
  12. 𝐄 = 0 \nabla\cdot\mathbf{E}=0
  13. 𝐁 = 0 \nabla\cdot\mathbf{B}=0
  14. 1 / ϵ 0 μ 0 1/\sqrt{\epsilon_{0}\mu_{0}}
  15. 𝐯 = d 𝐫 d t , \mathbf{v}={d\mathbf{r}\over dt}\,\text{,}
  16. { t = γ ( t - v x / c 2 ) where γ = 1 / 1 - v 2 / c 2 x = γ ( x - v t ) y = y z = z \begin{cases}t^{\prime}&=\gamma(t-vx/c^{2})\,\text{ where }\gamma=1/\sqrt{1-v^% {2}/c^{2}}\\ x^{\prime}&=\gamma(x-vt)\\ y^{\prime}&=y\\ z^{\prime}&=z\end{cases}
  17. ( c t x ) = ( cosh ϕ - sinh ϕ - sinh ϕ cosh ϕ ) ( c t x ) where ϕ = artanh v c , \begin{pmatrix}ct^{\prime}\\ x^{\prime}\end{pmatrix}=\begin{pmatrix}\cosh\phi&-\sinh\phi\\ -\sinh\phi&\cosh\phi\end{pmatrix}\begin{pmatrix}ct\\ x\end{pmatrix}\,\text{ where }\phi=\operatorname{artanh}\,\frac{v}{c}\,\text{,}
  18. ( x y ) = ( cos θ - sin θ sin θ cos θ ) ( x y ) \begin{pmatrix}x^{\prime}\\ y^{\prime}\end{pmatrix}=\begin{pmatrix}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{pmatrix}\begin{pmatrix}x\\ y\end{pmatrix}
  19. Δ t = Δ τ 1 - v 2 / c 2 \Delta t={{\Delta\tau}\over\sqrt{1-v^{2}/c^{2}}}
  20. [ ( d x 1 ) 2 + ( d x 2 ) 2 + ( d x 3 ) 2 - c ( d t ) 2 ) ] , \left[(dx^{1})^{2}+(dx^{2})^{2}+(dx^{3})^{2}-c(dt)^{2})\right],
  21. T = d t ( 1 - 2 G M r c 2 ) d t 2 - 1 c 2 ( 1 - 2 G M r c 2 ) - 1 d r 2 - r 2 c 2 d θ 2 - r 2 c 2 sin 2 θ d ϕ 2 T=\frac{dt}{\sqrt{\left(1-\frac{2GM}{rc^{2}}\right)dt^{2}-\frac{1}{c^{2}}\left% (1-\frac{2GM}{rc^{2}}\right)^{-1}dr^{2}-\frac{r^{2}}{c^{2}}d\theta^{2}-\frac{r% ^{2}}{c^{2}}\sin^{2}\theta\;d\phi^{2}}}
  22. T T
  23. r r
  24. d t dt
  25. G G
  26. M M
  27. ( 1 - 2 G M r c 2 ) d t 2 - 1 c 2 ( 1 - 2 G M r c 2 ) - 1 d r 2 - r 2 c 2 d θ 2 - r 2 c 2 sin 2 θ d ϕ 2 \sqrt{\left(1-\frac{2GM}{rc^{2}}\right)dt^{2}-\frac{1}{c^{2}}\left(1-\frac{2GM% }{rc^{2}}\right)^{-1}dr^{2}-\frac{r^{2}}{c^{2}}d\theta^{2}-\frac{r^{2}}{c^{2}}% \sin^{2}\theta\;d\phi^{2}}
  28. d τ d\tau
  29. d t d τ = 1 1 - ( 2 G M r c 2 ) . \frac{dt}{d\tau}=\frac{1}{\sqrt{1-\left(\frac{2GM}{rc^{2}}\right)}}.
  30. H ( t ) | ψ ( t ) = i t | ψ ( t ) H(t)\left|\psi(t)\right\rangle=i\hbar{\partial\over\partial t}\left|\psi(t)\right\rangle
  31. | ψ e ( t ) = e - i H t / | ψ e ( 0 ) |\psi_{e}(t)\rangle=e^{-iHt/\hbar}|\psi_{e}(0)\rangle
  32. e - i H t / e^{-iHt/\hbar}
  33. d d t A = ( i ) - 1 [ A , H ] + ( A t ) classical . \frac{d}{dt}A=(i\hbar)^{-1}[A,H]+\left(\frac{\partial A}{\partial t}\right)_{% \mathrm{classical}}.
  34. Δ E Δ T 2 \Delta E\Delta T\geq\frac{\hbar}{2}
  35. Δ E \Delta E
  36. Δ T \Delta T
  37. \hbar

Timed_event_system.html

  1. 𝒢 = < Z , Q , Q 0 , Q A , Δ Align g t ; \mathcal{G}=<Z,Q,Q_{0},Q_{A},\Delta&gt;
  2. Z \,Z
  3. Q \,Q
  4. Q 0 Q \,Q_{0}\subseteq Q
  5. Q A Q Q_{A}\subseteq Q
  6. Δ Q × Ω Z , [ t l , t u ] × Q \Delta\subseteq Q\times\Omega_{Z,[t_{l},t_{u}]}\times Q
  7. ( q , ω , q ) Δ (q,\omega,q^{\prime})\in\Delta
  8. q Q q\in Q
  9. q Q q^{\prime}\in Q
  10. ω Ω Z , [ t l , t u ] \omega\in\Omega_{Z,[t_{l},t_{u}]}
  11. ( q 1 , ω 1 , q 2 ) (q_{1},\omega_{1},q_{2})
  12. ( q 3 , ω 2 , q 4 ) Δ (q_{3},\omega_{2},q_{4})\in\Delta
  13. q 2 = q 3 q_{2}=q_{3}
  14. ω 1 \omega_{1}
  15. ω 2 \omega_{2}
  16. ( q , ω 1 , p ) (q,\omega_{1},p)
  17. ( p , ω 2 , q ) Δ (p,\omega_{2},q^{\prime})\in\Delta
  18. ( q , ω 1 ω 2 , q ) Δ (q,\omega_{1}\omega_{2},q^{\prime})\in\Delta
  19. 𝒢 = < Z , Q , Q 0 , Q A , Δ > \mathcal{G}=<Z,Q,Q_{0},Q_{A},\Delta>
  20. t t
  21. 0 t < 0\leq t<\infty
  22. t t
  23. 𝒢 \mathcal{G}
  24. L ( 𝒢 , t ) L(\mathcal{G},t)
  25. L ( 𝒢 , t ) = { ω Ω Z , [ 0 , t ] : ( q 0 , ω , q ) Δ , q 0 Q 0 , q Q A } . L(\mathcal{G},t)=\{\omega\in\Omega_{Z,[0,t]}:\exists(q_{0},\omega,q)\in\Delta,% q_{0}\in Q_{0},q\in Q_{A}\}.
  26. ω Ω Z , [ 0 , t ] \omega\in\Omega_{Z,[0,t]}
  27. t t
  28. 𝒢 \mathcal{G}
  29. ω L ( 𝒢 , t ) \omega\in L(\mathcal{G},t)
  30. t t
  31. 𝒢 \mathcal{G}
  32. L ( 𝒢 , ) L(\mathcal{G},\infty)
  33. L ( 𝒢 , ) = { ω lim t Ω Z , [ 0 , t ] : { q : ( q 0 , ω , q ) Δ , q 0 Q 0 } Q A } . L(\mathcal{G},\infty)=\{\omega\in\underset{t\rightarrow\infty}{\lim}\Omega_{Z,% [0,t]}:\exists\{q:(q_{0},\omega,q)\in\Delta,q_{0}\in Q_{0}\}\subseteq Q_{A}\}.
  34. ω lim t Ω Z , [ 0 , t ] \omega\in\underset{t\rightarrow\infty}{\lim}\Omega_{Z,[0,t]}
  35. 𝒢 \mathcal{G}
  36. ω L ( 𝒢 , ) \omega\in L(\mathcal{G},\infty)

Timeline_of_geometry.html

  1. x n x m = x m + n x^{n}\cdot x^{m}=x^{m+n}
  2. sin α = tan α / 1 + tan 2 α \sin\alpha=\tan\alpha/\sqrt{1+\tan^{2}\alpha}
  3. cos α = 1 / 1 + tan 2 α \cos\alpha=1/\sqrt{1+\tan^{2}\alpha}

Timeline_of_mathematics.html

  1. x n x m = x m + n x^{n}\cdot x^{m}=x^{m+n}
  2. x x
  3. x 2 x^{2}
  4. x 3 x^{3}
  5. 1 / x 1/x
  6. 1 / x 2 1/x^{2}
  7. 1 / x 3 1/x^{3}
  8. sin α = tan α / 1 + tan 2 α \sin\alpha=\tan\alpha/\sqrt{1+\tan^{2}\alpha}
  9. cos α = 1 / 1 + tan 2 α \cos\alpha=1/\sqrt{1+\tan^{2}\alpha}

Toda's_theorem.html

  1. Σ P 𝖡𝖯 𝖯 𝖡𝖯 𝖯 \Sigma^{P}\cdot\mathsf{BP}\cdot\oplus\mathsf{P}\subseteq\mathsf{BP}\cdot\oplus% \mathsf{P}
  2. 𝖡𝖯 𝖯 \mathsf{BP}\cdot\oplus\mathsf{P}
  3. 𝖯 \mathsf{P}
  4. 𝖯𝖧 𝖡𝖯 𝖯 \mathsf{PH}\subseteq\mathsf{BP}\cdot\oplus\mathsf{P}
  5. 𝖯 𝖯 𝖯 P \mathsf{P}\cdot\oplus\mathsf{P}\subseteq\mathsf{P}^{\sharp P}
  6. 𝖯𝖧 𝖡𝖯 𝖯 𝖯 𝖯 𝖯 P \mathsf{PH}\subseteq\mathsf{BP}\cdot\oplus\mathsf{P}\subseteq\mathsf{P}\cdot% \oplus\mathsf{P}\subseteq\mathsf{P}^{\sharp P}

Tollmien–Schlichting_wave.html

  1. D 2 U = 0 D^{2}U=0
  2. D D
  3. U U
  4. D E D t = - V u v ( d U d y ) - 1 R V ( v ) 2 \frac{DE}{Dt}=-\int_{V}u^{\prime}v^{\prime}\left(\frac{dU}{dy}\right)-\frac{1}% {R}\int_{V}\left(\nabla\vec{v}^{\prime}\right)^{2}
  5. u u^{\prime}
  6. v v^{\prime}

Tomasi–Kanade_factorization.html

  1. 𝐏 = ( x 11 x 1 P x F 1 x F P y 11 y 1 P y F 1 y F P ) \mathbf{P}=\left(\begin{array}[]{ccc}x_{11}&\cdots&x_{1P}\\ \vdots&\ddots&\vdots\\ x_{F1}&\cdots&x_{FP}\\ y_{11}&\cdots&y_{1P}\\ \vdots&\ddots&\vdots\\ y_{F1}&\cdots&y_{FP}\\ \end{array}\right)
  2. 𝐏 = 𝐌𝐒 . \mathbf{P}=\mathbf{M}\mathbf{S}.\,

Top-hat_filter.html

  1. ϵ \epsilon
  2. ϵ \epsilon

Top-hat_transform.html

  1. f : E R f:E\mapsto R
  2. b ( x ) b(x)
  3. T w ( f ) = f - f b T_{w}(f)=f-f\circ b
  4. \circ
  5. T b ( f ) = f b - f T_{b}(f)=f\bullet b-f
  6. \bullet
  7. b b

Topological_data_analysis.html

  1. C ε C_{\varepsilon}
  2. ϵ \epsilon
  3. S Y S\subseteq Y
  4. Y Y
  5. ϵ \epsilon
  6. K l K^{l}
  7. K l K^{l}
  8. H k l , p = Z k l / ( B k l + p Z k l ) H_{k}^{l,p}=Z_{k}^{l}/(B_{k}^{l+p}\cap Z_{k}^{l})
  9. z z
  10. k k
  11. I I
  12. σ \sigma
  13. z z z^{\prime}\sim z
  14. k k
  15. J J
  16. τ \tau
  17. z z
  18. ( I , J ) (I,J)
  19. σ \sigma
  20. z z
  21. τ \tau
  22. z z
  23. z z
  24. \infty
  25. K K
  26. K ρ = { σ i K ρ ( σ i ) ρ } K^{\rho}=\{\sigma^{i}\in K\mid\rho(\sigma^{i})\leq\rho\}
  27. ρ : S ( K ) \rho:S(K)\rightarrow\mathbb{R}
  28. π 0 \pi\geq 0
  29. π \pi
  30. K ρ K^{\rho}
  31. H k ρ , π = Z k ρ / ( B k ρ + π Z k ρ ) H_{k}^{\rho,\pi}=Z_{k}^{\rho}/(B_{k}^{\rho+\pi}\cap Z_{k}^{\rho})
  32. k k
  33. ρ i \rho_{i}
  34. ρ j \rho_{j}
  35. ρ j - ρ i \rho_{j}-\rho_{i}

Topological_pair.html

  1. ( X , A ) (X,A)
  2. i : A X i\colon A\hookrightarrow X
  3. i i
  4. ( X , A ) (X,A)
  5. ( X , A ) (X^{\prime},A^{\prime})
  6. f : X X f\colon X\rightarrow X^{\prime}
  7. g : A A g\colon A\rightarrow A^{\prime}
  8. i g = f i i^{\prime}\circ g=f\circ i
  9. A A
  10. X X
  11. ( X , A ) (X,A)
  12. X / A X/A
  13. X X
  14. ( X , ) (X,\varnothing)

Topology_(electrical_circuits).html

  1. t = n - 1 t=n-1
  2. b = l + t b=l+t
  3. N = b - n + s N=b-n+s
  4. R = n - s R=n-s
  5. R + N = b R+N=b

Toponogov's_theorem.html

  1. K δ . K\geq\delta\,.
  2. π / δ \pi/\sqrt{\delta}
  3. d ( q , r ) d ( q , r ) . d(q,r)\leq d(q^{\prime},r^{\prime}).\,

Topos.html

  1. R X × X / R X R\to X\times_{X/R}X\,\!
  2. X X
  3. ( P X , X ) (PX,\ni_{X})
  4. X P X × X {\ni_{X}}\subseteq PX\times X
  5. I I
  6. r : I P X r\colon I\to PX
  7. { ( i , x ) | x r ( i ) } I × X \{(i,x)~{}|~{}x\in r(i)\}\subseteq I\times X
  8. X \ni_{X}
  9. r × X : I × X P X × X r\times X:I\times X\to PX\times X
  10. R I × X R\subseteq I\times X
  11. r : I P X r\colon I\to PX

Total_curvature.html

  1. a b k ( s ) d s . \int_{a}^{b}k(s)\,ds.
  2. a b | γ ′′ ( s ) | sgn κ n - 1 ( s ) d s \int_{a}^{b}\left|\gamma^{\prime\prime}(s)\right|\operatorname{sgn}\kappa_{n-1% }(s)\,ds

Total_revenue.html

  1. T R ( Q ) = P ( Q ) × Q TR(Q)=P(Q)\times Q

Trace_Zero_Cryptography.html

  1. 𝔽 q \mathbb{F}_{q}
  2. C : y 2 + h ( x ) y = f ( x ) , C:~{}y^{2}+h(x)y=f(x),
  3. 𝔽 q \mathbb{F}_{q}
  4. J C ( 𝔽 q n ) J_{C}(\mathbb{F}_{q^{n}})
  5. 𝔽 q n \mathbb{F}_{q^{n}}
  6. Cl ( C / 𝔽 q n ) \operatorname{Cl}(C/\mathbb{F}_{q^{n}})
  7. J C ( 𝔽 q n ) J_{C}(\mathbb{F}_{q^{n}})
  8. 𝔽 q n [ x ] \mathbb{F}_{q^{n}}[x]
  9. J C ( 𝔽 q n ) J_{C}(\mathbb{F}_{q^{n}})
  10. χ ( T ) = T 2 g + a 1 T 2 g - 1 + + a g T g + + a 1 q g - 1 T + q g , \chi(T)=T^{2g}+a_{1}T^{2g-1}+\cdots+a_{g}T^{g}+\cdots+a_{1}q^{g-1}T+q^{g},
  11. 𝔽 q n \mathbb{F}_{q^{n}}
  12. | J C ( 𝔽 q n ) | = i = 1 2 g ( 1 - τ i n ) |J_{C}(\mathbb{F}_{q^{n}})|=\prod_{i=1}^{2g}(1-\tau_{i}^{n})
  13. J C ( 𝔽 q n ) J_{C}(\mathbb{F}_{q^{n}})
  14. J C ( 𝔽 q n ) J_{C}(\mathbb{F}_{q^{n}})
  15. Tr ( D ) = i = 0 n - 1 σ i ( D ) = D + σ ( D ) + + σ n - 1 ( D ) \operatorname{Tr}(D)=\sum_{i=0}^{n-1}\sigma^{i}(D)=D+\sigma(D)+\cdots+\sigma^{% n-1}(D)
  16. G = { D J C ( 𝔽 q n ) | Tr ( D ) = 0 } , ( 0 neutral element in J C ( 𝔽 q n ) G=\{D\in J_{C}(\mathbb{F}_{q^{n}})~{}|~{}\,\text{Tr}(D)=\,\textbf{\,{0}}\},~{}% ~{}~{}(\,\textbf{\,{0}}\,\text{ neutral element in }J_{C}(\mathbb{F}_{q^{n}})
  17. J C ( 𝔽 q n ) J_{C}(\mathbb{F}_{q^{n}})
  18. J C ( 𝔽 q ) J_{C}(\mathbb{F}_{q})
  19. J C ( 𝔽 q ) J_{C}(\mathbb{F}_{q})
  20. gcd ( n , | J C ( 𝔽 q ) | ) = 1 \gcd(n,|J_{C}(\mathbb{F}_{q})|)=1
  21. | G | = | J C ( 𝔽 q n ) | | J C ( 𝔽 q ) | = i = 1 2 g ( 1 - τ i n ) i = 1 2 g ( 1 - τ i ) |G|=\dfrac{|J_{C}(\mathbb{F}_{q^{n}})|}{|J_{C}(\mathbb{F}_{q})|}=\dfrac{\prod_% {i=1}^{2g}(1-\tau_{i}^{n})}{\prod_{i=1}^{2g}(1-\tau_{i})}
  22. J C ( 𝔽 q n ) J_{C}(\mathbb{F}_{q^{n}})
  23. s = q - 1 1 - a 1 mod s=\dfrac{q-1}{1-a_{1}}\bmod{\ell}
  24. s = q 2 - q - a 1 2 q + a 1 q + 1 q - 2 a 1 q + a 1 3 - a 1 2 + a 1 - 1 mod s=\dfrac{q^{2}-q-a_{1}^{2}q+a_{1}q+1}{q-2a_{1}q+a_{1}^{3}-a_{1}^{2}+a_{1}-1}% \bmod{\ell}
  25. s = - q 2 - a 2 + a 1 a 1 q - a 2 + 1 mod s=-\dfrac{q^{2}-a_{2}+a_{1}}{a_{1}q-a_{2}+1}\bmod{\ell}
  26. m 0 D + m 1 σ ( D ) + + m n - 1 σ n - 1 ( D ) , where m i = O ( 1 / ( n - 1 ) ) = O ( q g ) m_{0}D+m_{1}\sigma(D)+\cdots+m_{n-1}\sigma^{n-1}(D),~{}~{}~{}~{}\,\text{where % }m_{i}=O(\ell^{1/(n-1)})=O(q^{g})
  27. 𝔽 p \mathbb{F}_{p^{\prime}}

Train_track_map.html

  1. F k = H 1 H m U F_{k}=H_{1}\ast\dots H_{m}\ast U
  2. \mathbb{R}
  3. 1 C λ n ( ϕ ) || ϕ n ( w ) || X C λ n ( ϕ ) , \frac{1}{C}\lambda^{n}(\phi)\leq||\phi^{n}(w)||_{X}\leq C\lambda^{n}(\phi),

Transfer-matrix_method_(optics).html

  1. z z\,
  2. k k\,
  3. E ( z ) = E r e i k z + E l e - i k z E(z)=E_{r}e^{ikz}+E_{l}e^{-ikz}\,
  4. E E\,
  5. F = d E / d z F=dE/dz\,
  6. ( E ( z ) , F ( z ) ) (E(z),F(z))\,
  7. F ( z ) = i k E r e i k z - i k E l e - i k z F(z)=ikE_{r}e^{ikz}-ikE_{l}e^{-ikz}\,
  8. E E\,
  9. F F\,
  10. E r E_{r}\,
  11. E l E_{l}\,
  12. L L\,
  13. z z\,
  14. M = ( cos k L 1 k sin k L - k sin k L cos k L ) , M=\left(\begin{array}[]{cc}\cos kL&\frac{1}{k}\sin kL\\ -k\sin kL&\cos kL\end{array}\right),
  15. ( E ( z + L ) F ( z + L ) ) = M ( E ( z ) F ( z ) ) \left(\begin{array}[]{c}E(z+L)\\ F(z+L)\end{array}\right)=M\cdot\left(\begin{array}[]{c}E(z)\\ F(z)\end{array}\right)
  16. k k\,
  17. L L\,
  18. N N\,
  19. j j\,
  20. M j M_{j}\,
  21. j j\,
  22. z z\,
  23. M s = M N M 2 M 1 . M_{s}=M_{N}\cdot\ldots\cdot M_{2}\cdot M_{1}.
  24. z = 0 z=0\,
  25. z z\,
  26. E L ( z ) = E 0 e i k L z + r E 0 e - i k L z , z < 0 , E_{L}(z)=E_{0}e^{ik_{L}z}+rE_{0}e^{-ik_{L}z},\qquad z<0,
  27. E 0 E_{0}\,
  28. k L k_{L}\,
  29. r r\,
  30. E R ( z ) = t E 0 e i k R z , z > L , E_{R}(z)=tE_{0}e^{ik_{R}z},\qquad z>L^{\prime},
  31. t t\,
  32. k R k_{R}\,
  33. L L^{\prime}
  34. F L = d E L / d z F_{L}=dE_{L}/dz\,
  35. F R = d E R / d z F_{R}=dE_{R}/dz\,
  36. ( E ( z R ) F ( z R ) ) = M ( E ( 0 ) F ( 0 ) ) \left(\begin{array}[]{c}E(z_{R})\\ F(z_{R})\end{array}\right)=M\cdot\left(\begin{array}[]{c}E(0)\\ F(0)\end{array}\right)
  37. M m n M_{mn}\,
  38. M s M_{s}\,
  39. t = 2 i k L e - i k R L [ M 11 M 22 - M 12 M 21 - M 21 + k L k R M 12 + i ( k R M 11 + k L M 22 ) ] t=2ik_{L}e^{-ik_{R}L}\left[\frac{M_{11}M_{22}-M_{12}M_{21}}{-M_{21}+k_{L}k_{R}% M_{12}+i(k_{R}M_{11}+k_{L}M_{22})}\right]
  40. r = [ ( M 21 + k L k R M 12 ) + i ( k L M 22 - k R M 11 ) ( - M 21 + k L k R M 12 ) + i ( k L M 22 + k R M 11 ) ] r=\left[\frac{(M_{21}+k_{L}k_{R}M_{12})+i(k_{L}M_{22}-k_{R}M_{11})}{(-M_{21}+k% _{L}k_{R}M_{12})+i(k_{L}M_{22}+k_{R}M_{11})}\right]
  41. | E 0 | 2 \left|E_{0}\right|^{2}
  42. T = | t | 2 T=|t|^{2}\,
  43. R = | r | 2 R=|r|^{2}\,
  44. k = n k k^{\prime}=nk\,
  45. M = ( cos k d sin ( k d ) / k - k sin k d cos k d ) M=\left(\begin{array}[]{cc}\cos k^{\prime}d&\sin(k^{\prime}d)/k^{\prime}\\ -k^{\prime}\sin k^{\prime}d&\cos k^{\prime}d\end{array}\right)
  46. r = ( 1 / n - n ) sin k d ( n + 1 / n ) sin k d + 2 i cos ( k d ) r=\frac{(1/n-n)\sin k^{\prime}d}{(n+1/n)\sin k^{\prime}d+2i\cos(k^{\prime}d)}
  47. k d = 0 , π , 2 π , k^{\prime}d=0,\pi,2\pi,\cdots\,
  48. σ = C d u / d z \sigma=Cdu/dz
  49. C C
  50. Q z = 4 π λ sin θ = 2 k z Q_{z}=\frac{4\pi}{\lambda}\sin\theta=2k_{z}
  51. k n = k z 2 - 4 π ( ρ n - ρ 0 ) k_{n}=\sqrt{{k_{z}}^{2}-4\pi({\rho}_{n}-{\rho}_{0})}
  52. r n , n + 1 = k n - k n + 1 k n + k n + 1 r_{n,n+1}=\frac{k_{n}-k_{n+1}}{k_{n}+k_{n+1}}
  53. r n , n + 1 = k n - k n + 1 k n + k n + 1 exp ( - 2 k n k n + 1 σ n , n + 1 2 ) r_{n,n+1}=\frac{k_{n}-k_{n+1}}{k_{n}+k_{n+1}}\exp(-2k_{n}k_{n+1}{\sigma_{n,n+1% }}^{2})
  54. β 0 = 0 \beta_{0}=0
  55. β n = i k n d n \beta_{n}=ik_{n}d_{n}
  56. i 2 = - 1 i^{2}=-1
  57. c n = [ exp ( β n ) r n , n + 1 exp ( β n ) r n , n + 1 exp ( - β n ) exp ( - β n ) ] c_{n}=\left[\begin{array}[]{cc}\exp\left(\beta_{n}\right)&r_{n,n+1}\exp\left(% \beta_{n}\right)\\ r_{n,n+1}\exp\left(-\beta_{n}\right)&\exp\left(-\beta_{n}\right)\end{array}\right]
  58. M = n c n M=\prod_{n}c_{n}
  59. R = | M 10 M 00 | 2 R=\left|\frac{M_{10}}{M_{00}}\right|^{2}

Transformation_semigroup.html

  1. s x = s ( x ) for s S , x X . s\cdot x=s(x)\,\text{ for }s\in S,x\in X.
  2. s x = t x for all x X , s\cdot x=t\cdot x\,\text{ for all }x\in X,
  3. T s ( x ) = T ( s , x ) . T_{s}(x)=T(s,x).\,

Transient_equilibrium.html

  1. A d = ( [ A P ( 0 ) λ d λ d - λ P × ( e - λ P t - e - λ d t ) ] × B R ) + A d ( 0 ) e - λ d t A_{d}=([A_{P}(0)\frac{\lambda_{d}}{\lambda_{d}-\lambda_{P}}\times(e^{-\lambda_% {P}t}-e^{-\lambda_{d}t})]\times BR)+A_{d}(0)e^{-\lambda_{d}t}
  2. A P A_{P}
  3. A d A_{d}
  4. T P T_{P}
  5. T d T_{d}
  6. A d A P = T P T P - T d × B R \frac{A_{d}}{A_{P}}=\frac{T_{P}}{T_{P}-T_{d}}\times BR
  7. t m a x = 1.44 × T P T d T P - T d × l n ( T P / T d ) t_{max}=\frac{1.44\times T_{P}T_{d}}{T_{P}-T_{d}}\times ln(T_{P}/T_{d})
  8. T P T_{P}
  9. T d T_{d}
  10. T 99 m c {}^{99m}Tc
  11. M 99 o {}^{99}Mo
  12. t m a x t_{max}

Translinear_circuit.html

  1. n ϵ C W I n = n ϵ C C W I n \,\!\prod_{n\epsilon CW}I_{n}=\prod_{n\epsilon CCW}I_{n}
  2. I = λ I s e η V / U T I=\lambda I_{s}e^{\eta V/U_{T}}
  3. I s I_{s}
  4. λ \lambda
  5. I s I_{s}
  6. η \eta
  7. U T U_{T}
  8. k T / q kT/q
  9. V r e f V_{ref}
  10. V r e f V_{ref}
  11. n ϵ C W V n = n ϵ C C W V n \,\!\sum_{n\epsilon CW}V_{n}=\sum_{n\epsilon CCW}V_{n}
  12. n ϵ C W I n = n ϵ C C W I n \,\!\prod_{n\epsilon CW}I_{n}=\prod_{n\epsilon CCW}I_{n}
  13. l o g ( a ) + l o g ( b ) = l o g ( a b ) log(a)+log(b)=log(ab)
  14. n ϵ C W I n = n ϵ C C W I n \,\!\prod_{n\epsilon CW}I_{n}=\prod_{n\epsilon CCW}I_{n}
  15. I y I y = I x I u \,\!I_{y}I_{y}=I_{x}I_{u}
  16. I u I_{u}
  17. I x = I y 2 \,\!I_{x}=I_{y}^{2}
  18. z = x y \,\!z=xy
  19. x \,\!x
  20. z = z + - z - \,\!z=z^{+}-z^{-}
  21. x = x + - x - \,\!x=x^{+}-x^{-}
  22. x = I x / I u \,\!x=I_{x}/I_{u}
  23. x + = I x + / I u \,\!x^{+}=I_{x}^{+}/I_{u}
  24. ( I z + I u - I z - I u ) = ( I x + I u - I x - I u ) ( I y I u ) \,\!\left(\frac{I_{z}^{+}}{I_{u}}-\frac{I_{z}^{-}}{I_{u}}\right)=\left(\frac{I% _{x}^{+}}{I_{u}}-\frac{I_{x}^{-}}{I_{u}}\right)\left(\frac{I_{y}}{I_{u}}\right)
  25. ( I z + I u - I z - I u ) = ( I x + - I x - ) ( I y ) \,\!(I_{z}^{+}I_{u}-I_{z}^{-}I_{u})=(I_{x}^{+}-I_{x}^{-})(I_{y})
  26. I y I x + = I u I z + I_{y}I_{x}^{+}=I_{u}I_{z}^{+}
  27. I y I x - = I u I z - I_{y}I_{x}^{-}=I_{u}I_{z}^{-}

Traveling_wave_reactor.html

  1. U 92 238 + 0 1 n 92 239 U 93 239 Np + β 94 239 Pu + β \mathrm{{}^{238}_{\ 92}U+\,^{1}_{0}n\;\rightarrow\;^{239}_{\ 92}U\;\rightarrow% \;^{239}_{\ 93}Np+\beta\;\rightarrow\;^{239}_{\ 94}Pu+\beta}

Tree_taper.html

  1. d ( h ) 2 = D 2 ( H - h H - h b ) 1.6 d(h)^{2}={D^{2}}\left({H-h\over H-h_{b}}\right)^{1.6}
  2. d ( h ) d(h)
  3. D D
  4. H H
  5. h h
  6. h b h_{b}

Tree_walking_automaton.html

  1. A A
  2. Q = { q 0 , q 𝑙𝑒𝑓𝑡 , q 𝑟𝑖𝑔ℎ𝑡 } Q=\{q_{0},q_{\mathit{left}},q_{\mathit{right}}\}
  3. A A
  4. q 0 q_{0}
  5. A A
  6. v v
  7. q 𝑙𝑒𝑓𝑡 q_{\mathit{left}}
  8. v v
  9. v v
  10. A A
  11. v v
  12. q 𝑟𝑖𝑔ℎ𝑡 q_{\mathit{right}}
  13. v v
  14. A A
  15. v v
  16. q 𝑙𝑒𝑓𝑡 q_{\mathit{left}}
  17. q 𝑟𝑖𝑔ℎ𝑡 q_{\mathit{right}}
  18. v v
  19. 𝐷𝑇𝑊𝐴 𝑇𝑊𝐴 \mathit{DTWA}\subsetneq\mathit{TWA}
  20. 𝑇𝑊𝐴 𝑅𝐸𝐺 \mathit{TWA}\subsetneq\mathit{REG}

Tricritical_point.html

  1. κ \kappa
  2. κ = 0.76 / 2 \kappa=0.76/\sqrt{2}
  3. κ = 1 / 2 \kappa=1/\sqrt{2}

Trifocal_tensor.html

  1. 𝐓 1 , 𝐓 2 , 𝐓 3 {\mathbf{T}}_{1},\;{\mathbf{T}}_{2},\;{\mathbf{T}}_{3}
  2. 𝐏 = [ 𝐈 | 0 ] {\mathbf{P}}=[{\mathbf{I}}\;|\;{\mathbf{0}}]
  3. 𝐏 = [ 𝐀 | 𝐚 4 ] {\mathbf{P}}^{^{\prime}}=[{\mathbf{A}}\;|\;{\mathbf{a}}_{4}]
  4. 𝐏 ′′ = [ 𝐁 | 𝐛 4 ] {\mathbf{P}^{{}^{\prime\prime}}}=[{\mathbf{B}}\;|\;{\mathbf{b}}_{4}]
  5. 𝐓 i = 𝐚 i 𝐛 4 t - 𝐚 4 𝐛 i t , i = 1 3 {\mathbf{T}}_{i}={\mathbf{a}}_{i}{\mathbf{b}}_{4}^{t}-{\mathbf{a}}_{4}{\mathbf% {b}}_{i}^{t},\;i=1\ldots 3
  6. 𝐚 i , 𝐛 i {\mathbf{a}}_{i},\;{\mathbf{b}}_{i}
  7. 𝐱 𝐱 𝐱 ′′ {\mathbf{x}}\;\leftrightarrow\;{\mathbf{x}}^{{}^{\prime}}\;\leftrightarrow\;{% \mathbf{x}}^{{}^{\prime\prime}}
  8. 𝐥 𝐥 𝐥 ′′ {\mathbf{l}}\;\leftrightarrow\;{\mathbf{l}}^{{}^{\prime}}\;\leftrightarrow\;{% \mathbf{l}}^{{}^{\prime\prime}}
  9. ( 𝐥 t [ 𝐓 1 , 𝐓 2 , 𝐓 3 ] 𝐥 ′′ ) [ 𝐥 ] × = 𝟎 t ({\mathbf{l}}^{{}^{\prime}t}\left[{\mathbf{T}}_{1},\;{\mathbf{T}}_{2},\;{% \mathbf{T}}_{3}\right]{\mathbf{l}}^{{}^{\prime\prime}})[{\mathbf{l}}]_{\times}% ={\mathbf{0}}^{t}
  10. 𝐥 t ( i x i 𝐓 i ) 𝐥 ′′ = 0 {\mathbf{l}}^{{}^{\prime}t}\left(\sum_{i}x_{i}{\mathbf{T}}_{i}\right){\mathbf{% l}}^{{}^{\prime\prime}}=0
  11. 𝐥 t ( i x i 𝐓 i ) [ 𝐱 ′′ ] × = 𝟎 t {\mathbf{l}}^{{}^{\prime}t}\left(\sum_{i}x_{i}{\mathbf{T}}_{i}\right)[{\mathbf% {x}}^{{}^{\prime\prime}}]_{\times}={\mathbf{0}}^{t}
  12. [ 𝐱 ] × ( i x i 𝐓 i ) 𝐥 ′′ = 𝟎 [{\mathbf{x}}^{^{\prime}}]_{\times}\left(\sum_{i}x_{i}{\mathbf{T}}_{i}\right){% \mathbf{l}}^{{}^{\prime\prime}}={\mathbf{0}}
  13. [ 𝐱 ] × ( i x i 𝐓 i ) [ 𝐱 ′′ ] × = 𝟎 3 × 3 [{\mathbf{x}}^{^{\prime}}]_{\times}\left(\sum_{i}x_{i}{\mathbf{T}}_{i}\right)[% {\mathbf{x}}^{{}^{\prime\prime}}]_{\times}={\mathbf{0}}_{3\times 3}
  14. [ ] × [\cdot]_{\times}

Trigonometric_number.html

  1. cos ( π / 23 ) = - ( 1 / 2 ) ( - 1 ) 22 / 23 ( 1 + ( - 1 ) 2 / 23 ) . \cos(\pi/23)=-(1/2)(-1)^{22/23}(1+(-1)^{2/23}).
  2. θ = 2 π k / n \theta=2\pi k/n
  3. ( cos θ + i sin θ ) n = 1. (\cos\theta+i\sin\theta)^{n}=1.
  4. cos θ \cos\theta
  5. sin 2 θ ; \sin^{2}\theta;
  6. sin 2 θ = 1 - cos 2 θ \sin^{2}\theta=1-\cos^{2}\theta
  7. cos θ \cos\theta
  8. sin θ \sin\theta
  9. cos ( θ - π / 2 ) . \cos(\theta-\pi/2).
  10. tan θ , \tan\theta,
  11. θ \theta
  12. π , \pi,
  13. cos n θ \cos^{n}\theta
  14. tan θ . \tan\theta.

Trigonometry.html

  1. sin A = opposite hypotenuse = a c . \sin A=\frac{\textrm{opposite}}{\textrm{hypotenuse}}=\frac{a}{\,c\,}\,.
  2. cos A = adjacent hypotenuse = b c . \cos A=\frac{\textrm{adjacent}}{\textrm{hypotenuse}}=\frac{b}{\,c\,}\,.
  3. tan A = opposite adjacent = a b = a c * c b = a c / b c = sin A cos A . \tan A=\frac{\textrm{opposite}}{\textrm{adjacent}}=\frac{a}{\,b\,}=\frac{a}{\,% c\,}*\frac{c}{\,b\,}=\frac{a}{\,c\,}/\frac{b}{\,c\,}=\frac{\sin A}{\cos A}\,.
  4. csc A = 1 sin A = hypotenuse opposite = c a , \csc A=\frac{1}{\sin A}=\frac{\textrm{hypotenuse}}{\textrm{opposite}}=\frac{c}% {a},
  5. sec A = 1 cos A = hypotenuse adjacent = c b , \sec A=\frac{1}{\cos A}=\frac{\textrm{hypotenuse}}{\textrm{adjacent}}=\frac{c}% {b},
  6. cot A = 1 tan A = adjacent opposite = cos A sin A = b a . \cot A=\frac{1}{\tan A}=\frac{\textrm{adjacent}}{\textrm{opposite}}=\frac{\cos A% }{\sin A}=\frac{b}{a}.
  7. e x + i y = e x ( cos y + i sin y ) . e^{x+iy}=e^{x}(\cos y+i\sin y).
  8. sin 2 A + cos 2 A = 1 \sin^{2}A+\cos^{2}A=1
  9. sec 2 A - tan 2 A = 1 \sec^{2}A-\tan^{2}A=1
  10. csc 2 A - cot 2 A = 1 \csc^{2}A-\cot^{2}A=1
  11. sin ( A ± B ) = sin A cos B ± cos A sin B \sin(A\pm B)=\sin A\ \cos B\pm\cos A\ \sin B
  12. cos ( A ± B ) = cos A cos B sin A sin B \cos(A\pm B)=\cos A\ \cos B\mp\sin A\ \sin B
  13. tan ( A ± B ) = tan A ± tan B 1 tan A tan B \tan(A\pm B)=\frac{\tan A\pm\tan B}{1\mp\tan A\ \tan B}
  14. cot ( A ± B ) = cot A cot B 1 cot B ± cot A \cot(A\pm B)=\frac{\cot A\ \cot B\mp 1}{\cot B\pm\cot A}
  15. a sin A = b sin B = c sin C = 2 R = a b c 2 Δ , \frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}=2R=\frac{abc}{2\Delta},
  16. Δ \Delta
  17. R = a b c ( a + b + c ) ( a - b + c ) ( a + b - c ) ( b + c - a ) . R=\frac{abc}{\sqrt{(a+b+c)(a-b+c)(a+b-c)(b+c-a)}}.
  18. Area = Δ = 1 2 a b sin C . \mbox{Area}~{}=\Delta=\frac{1}{2}ab\sin C.
  19. c 2 = a 2 + b 2 - 2 a b cos C , c^{2}=a^{2}+b^{2}-2ab\cos C,\,
  20. cos C = a 2 + b 2 - c 2 2 a b . \cos C=\frac{a^{2}+b^{2}-c^{2}}{2ab}.\,
  21. s = 1 2 ( a + b + c ) , s=\frac{1}{2}(a+b+c),
  22. Area = Δ = s ( s - a ) ( s - b ) ( s - c ) = a b c 4 R \mbox{Area}~{}=\Delta=\sqrt{s(s-a)(s-b)(s-c)}=\frac{abc}{4R}
  23. a - b a + b = tan [ 1 2 ( A - B ) ] tan [ 1 2 ( A + B ) ] \frac{a-b}{a+b}=\frac{\tan\left[\tfrac{1}{2}(A-B)\right]}{\tan\left[\tfrac{1}{% 2}(A+B)\right]}
  24. e i x = cos x + i sin x e^{ix}=\cos x+i\sin x
  25. sin x = e i x - e - i x 2 i , cos x = e i x + e - i x 2 , tan x = i ( e - i x - e i x ) e i x + e - i x . \sin x=\frac{e^{ix}-e^{-ix}}{2i},\qquad\cos x=\frac{e^{ix}+e^{-ix}}{2},\qquad% \tan x=\frac{i(e^{-ix}-e^{ix})}{e^{ix}+e^{-ix}}.

Trimmed_estimator.html

  1. 2 2 erf - 1 ( 1 / 2 ) 1.349 2\sqrt{2}\operatorname{erf}^{-1}(1/2)\approx 1.349

TRIN_(finance).html

  1. T R I N = a d v a n c i n g i s s u e s / d e c l i n i n g i s s u e s a d v a n c i n g v o l u m e / d e c l i n i n g v o l u m e TRIN=\frac{advancing\ issues/declining\ issues}{advancing\ volume/declining\ volume}

Trinomial_tree.html

  1. u u\,
  2. d d\,
  3. m m\,
  4. u = e σ 2 Δ t u=e^{\sigma\sqrt{2\Delta t}}
  5. d = e - σ 2 Δ t = 1 u d=e^{-\sigma\sqrt{2\Delta t}}=\frac{1}{u}\,
  6. m = 1 m=1\,
  7. p u = ( e ( r - q ) Δ t / 2 - e - σ Δ t / 2 e σ Δ t / 2 - e - σ Δ t / 2 ) 2 p_{u}=\left(\frac{e^{(r-q)\Delta t/2}-e^{-\sigma\sqrt{\Delta t/2}}}{e^{\sigma% \sqrt{\Delta t/2}}-e^{-\sigma\sqrt{\Delta t/2}}}\right)^{2}\,
  8. p d = ( e σ Δ t / 2 - e ( r - q ) Δ t / 2 e σ Δ t / 2 - e - σ Δ t / 2 ) 2 p_{d}=\left(\frac{e^{\sigma\sqrt{\Delta t/2}}-e^{(r-q)\Delta t/2}}{e^{\sigma% \sqrt{\Delta t/2}}-e^{-\sigma\sqrt{\Delta t/2}}}\right)^{2}\,
  9. p m = 1 - ( p u + p d ) p_{m}=1-(p_{u}+p_{d})\,
  10. Δ t \Delta t\,
  11. r r\,
  12. σ \sigma\,
  13. q q\,
  14. p u p_{u}
  15. p d p_{d}
  16. p m p_{m}
  17. ( 0 , 1 ) (0,1)
  18. Δ t \Delta t
  19. Δ t < 2 σ 2 ( r - q ) 2 \Delta t<2\frac{\sigma^{2}}{(r-q)^{2}}
  20. Δ t \Delta t

Tripadi.html

  1. - or or - - or - -\smile\ \mathrm{or}\ \smile\smile\smile\ \mathrm{or}\ --\ \mathrm{or}\ \smile% \smile-\
  2. \smile
  3. - -
  4. - Foot1 | - Foot2 - Foot3 | - - Foot4 \overbrace{\smile\smile\smile-}^{\mathrm{Foot1}}|\overbrace{\smile\smile\smile% -}^{\mathrm{Foot2}}\star\overbrace{\smile\smile\smile-}^{\mathrm{Foot3}}|% \overbrace{-\smile-}^{\mathrm{Foot4}}
  5. Foot5 | - - Brahma FootVI | - Foot7 | Foot8 \overbrace{\smile\smile\smile\smile}^{\mathrm{Foot5}}|\overbrace{\underbrace{-% -}_{\mathrm{Brahma}}}^{\mathrm{FootVI}}|\overbrace{\smile\smile\smile-}^{% \mathrm{Foot7}}|\overbrace{\smile\smile\smile\smile}^{\mathrm{Foot8}}
  6. Foot9 | - - Brahma FootX | - Foot11 | | \overbrace{\smile\smile\smile\smile}^{\mathrm{Foot9}}|\overbrace{\underbrace{-% -}_{\mathrm{Brahma}}}^{\mathrm{FootX}}|\overbrace{\smile\smile\smile-}^{% \mathrm{Foot11}}||
  7. - Foot1 | - - Foot2 - Foot3 | - - Foot4 \overbrace{\smile\smile-\smile}^{\mathrm{Foot1}}|\overbrace{--\smile}^{\mathrm% {Foot2}}\star\overbrace{\smile\smile-\smile}^{\mathrm{Foot3}}|\overbrace{-% \smile-}^{\mathrm{Foot4}}
  8. Foot5 | - Brahma FootVI | - Foot7 | - Foot8 \overbrace{\smile\smile\smile\smile}^{\mathrm{Foot5}}|\overbrace{\underbrace{-% \smile}_{\mathrm{Brahma}}}^{\mathrm{FootVI}}|\overbrace{\smile\smile-\smile}^{% \mathrm{Foot7}}|\overbrace{-\smile\smile\smile}^{\mathrm{Foot8}}
  9. - Foot9 | - Brahma FootX | - Foot11 | | \overbrace{\smile\smile-\smile}^{\mathrm{Foot9}}|\overbrace{\underbrace{-% \smile}_{\mathrm{Brahma}}}^{\mathrm{FootX}}|\overbrace{\smile\smile\smile-}^{% \mathrm{Foot11}}||

Triple_system.html

  1. ( , , ) : V × V × V V . (\cdot,\cdot,\cdot)\colon V\times V\times V\to V.
  2. [ u , v , w ] = - [ v , u , w ] [u,v,w]=-[v,u,w]
  3. [ u , v , w ] + [ w , u , v ] + [ v , w , u ] = 0 [u,v,w]+[w,u,v]+[v,w,u]=0
  4. [ u , v , [ w , x , y ] ] = [ [ u , v , w ] , x , y ] + [ w , [ u , v , x ] , y ] + [ w , x , [ u , v , y ] ] . [u,v,[w,x,y]]=[[u,v,w],x,y]+[w,[u,v,x],y]+[w,x,[u,v,y]].
  5. 𝔤 := 𝔨 𝔪 \mathfrak{g}:=\mathfrak{k}\oplus\mathfrak{m}
  6. [ ( L , u ) , ( M , v ) ] = ( [ L , M ] + L u , v , L ( v ) - M ( u ) ) . [(L,u),(M,v)]=([L,M]+L_{u,v},L(v)-M(u)).
  7. { u , v , w } = { u , w , v } \{u,v,w\}=\{u,w,v\}
  8. { u , v , { w , x , y } } = { w , x , { u , v , y } } + { w , { u , v , x } , y } - { { v , u , w } , x , y } . \{u,v,\{w,x,y\}\}=\{w,x,\{u,v,y\}\}+\{w,\{u,v,x\},y\}-\{\{v,u,w\},x,y\}.
  9. [ L u , v , L w , x ] := L u , v L w , x - L w , x L u , v = L w , { u , v , x } - L { v , u , w } , x [L_{u,v},L_{w,x}]:=L_{u,v}\circ L_{w,x}-L_{w,x}\circ L_{u,v}=L_{w,\{u,v,x\}}-L% _{\{v,u,w\},x}
  10. [ u , v , w ] = { u , v , w } - { v , u , w } . [u,v,w]=\{u,v,w\}-\{v,u,w\}.
  11. V 𝔤 0 V * V\oplus\mathfrak{g}_{0}\oplus V^{*}
  12. { , , } + : V - × S 2 V + V + \{\cdot,\cdot,\cdot\}_{+}\colon V_{-}\times S^{2}V_{+}\to V_{+}
  13. { , , } - : V + × S 2 V - V - \{\cdot,\cdot,\cdot\}_{-}\colon V_{+}\times S^{2}V_{-}\to V_{-}
  14. { u , v , { w , x , y } + } + = { w , x , { u , v , y } + } + + { w , { u , v , x } + , y } + - { { v , u , w } - , x , y } + \{u,v,\{w,x,y\}_{+}\}_{+}=\{w,x,\{u,v,y\}_{+}\}_{+}+\{w,\{u,v,x\}_{+},y\}_{+}-% \{\{v,u,w\}_{-},x,y\}_{+}\,
  15. L u , v + : V + V + by L u , v + ( y ) = { u , v , y } + L^{+}_{u,v}:V_{+}\to V_{+}\quad\,\text{by}\quad L^{+}_{u,v}(y)=\{u,v,y\}_{+}
  16. [ L u , v ± , L w , x ± ] = L w , { u , v , x } ± ± - L { v , u , w } , x ± [L^{\pm}_{u,v},L^{\pm}_{w,x}]=L^{\pm}_{w,\{u,v,x\}_{\pm}}-L^{\pm}_{\{v,u,w\}_{% \mp},x}
  17. V + V - 𝔤 𝔩 ( V + ) 𝔤 𝔩 ( V - ) V_{+}\otimes V_{-}\to\mathfrak{gl}(V_{+})\oplus\mathfrak{gl}(V_{-})
  18. 𝔤 0 \mathfrak{g}_{0}
  19. V + 𝔤 0 V - , V_{+}\oplus\mathfrak{g}_{0}\oplus V_{-},
  20. 𝔤 = 𝔤 + 1 𝔤 0 𝔤 - 1 \mathfrak{g}=\mathfrak{g}_{+1}\oplus\mathfrak{g}_{0}\oplus\mathfrak{g}_{-1}
  21. ( 𝔤 + 1 , 𝔤 - 1 ) (\mathfrak{g}_{+1},\mathfrak{g}_{-1})
  22. { X , Y ± , Z ± } ± := [ [ X , Y ± ] , Z ± ] . \{X_{\mp},Y_{\pm},Z_{\pm}\}_{\pm}:=[[X_{\mp},Y_{\pm}],Z_{\pm}].
  23. End ( S 2 V + ) S 2 V + * S 2 V - * End ( S 2 V - ) . \mathrm{End}(S^{2}V_{+})\cong S^{2}V_{+}^{*}\otimes S^{2}V_{-}^{*}\cong\mathrm% {End}(S^{2}V_{-}).
  24. 𝔤 \mathfrak{g}
  25. 𝔤 + 1 \mathfrak{g}_{+1}
  26. 𝔤 - 1 \mathfrak{g}_{-1}

Trophic_state_index.html

  1. ( 1 z ) ( ln I 0 I z ) = k w + α C \left(\frac{1}{z}\right)\left(\ln\frac{I_{0}}{I_{z}}\right)=k_{w}+\alpha C

True_quantified_Boolean_formula.html

  1. x y z ( ( x z ) y ) \forall x\ \exists y\ \exists z\ ((x\lor z)\land y)
  2. ϕ \displaystyle\phi
  3. n \displaystyle n
  4. x 1 x 2 x 3 Q n x n ϕ ( x 1 , x 2 , x 3 , , x n ) \displaystyle\exists x_{1}\forall x_{2}\exists x_{3}\cdots Q_{n}x_{n}\phi(x_{1% },x_{2},x_{3},\dots,x_{n})
  5. y 1 \displaystyle y_{1}
  6. x 1 x 2 ϕ ( x 1 , x 2 ) x 1 y 1 x 2 ϕ ( x 1 , x 2 ) \displaystyle\exists x_{1}\exists x_{2}\phi(x_{1},x_{2})\quad\mapsto\quad% \exists x_{1}\forall y_{1}\exists x_{2}\phi(x_{1},x_{2})
  7. Q 1 x 1 Q 2 x 2 Q n x n ϕ ( x 1 , x 2 , , x n ) . Q_{1}x_{1}Q_{2}x_{2}\cdots Q_{n}x_{n}\phi(x_{1},x_{2},\dots,x_{n}).
  8. A = Q 2 x 2 Q n x n ϕ ( 0 , x 2 , , x n ) , A=Q_{2}x_{2}\cdots Q_{n}x_{n}\phi(0,x_{2},\dots,x_{n}),
  9. B = Q 2 x 2 Q n x n ϕ ( 1 , x 2 , , x n ) . B=Q_{2}x_{2}\cdots Q_{n}x_{n}\phi(1,x_{2},\dots,x_{n}).
  10. Q 1 = Q_{1}=\exists
  11. A B A\lor B
  12. Q 1 = Q_{1}=\forall
  13. A B A\land B
  14. L PSPACE , L p TQBF . \forall L\in\textrm{PSPACE},L\leq_{p}\textrm{TQBF}.
  15. x x
  16. f ( x ) f(x)
  17. f f
  18. x L f ( x ) TQBF . x\in L\iff f(x)\in\textrm{TQBF}.
  19. f f
  20. c 1 c_{1}
  21. c 2 c_{2}
  22. ϕ c 1 , c 2 , t \phi_{c_{1},c_{2},t}
  23. c 1 c_{1}
  24. c 2 c_{2}
  25. f f
  26. ϕ c s t a r t , c a c c e p t , T \phi_{c_{start},c_{accept},T}
  27. c s t a r t c_{start}
  28. c a c c e p t c_{accept}
  29. c accept c_{\mathrm{accept}}
  30. c accept c_{\mathrm{accept}}
  31. w w
  32. c s t a r t c_{start}
  33. ϕ c s t a r t , c a c c e p t , T \phi_{c_{start},c_{accept},T}
  34. f ( w ) f(w)
  35. f f
  36. t = 1 t=1
  37. ϕ c 1 , c 2 , t \phi_{c_{1},c_{2},t}
  38. t > 1 t>1
  39. ϕ c 1 , c 2 , t \phi_{c_{1},c_{2},t}
  40. m 1 m_{1}
  41. ϕ c 1 , c 2 , t = m 1 ( ϕ c 1 , m 1 , t / 2 ϕ m 1 , c 2 , t / 2 ) . \phi_{c_{1},c_{2},t}=\exists m_{1}(\phi_{c_{1},m_{1},\lceil t/2\rceil}\wedge% \phi_{m_{1},c_{2},\lceil t/2\rceil}).
  42. c 1 c_{1}
  43. c 2 c_{2}
  44. c 1 c_{1}
  45. m 1 m_{1}
  46. t / 2 t/2
  47. c 2 c_{2}
  48. t / 2 t/2
  49. m 1 m_{1}
  50. ϕ c 1 , c 2 , t \phi_{c_{1},c_{2},t}
  51. c 3 c_{3}
  52. c 4 c_{4}
  53. { ( c 1 , m 1 ) , ( m 1 , c 2 ) } \{(c_{1},m_{1}),(m_{1},c_{2})\}
  54. ϕ c 1 , c 2 , t \phi_{c_{1},c_{2},t}
  55. ϕ c 1 , c 2 , t = m 1 ( c 3 , c 4 ) { ( c 1 , m 1 ) , ( m 1 , c 2 ) } ( ϕ c 3 , c 4 , t / 2 ) . \phi_{c_{1},c_{2},t}=\exists m_{1}\forall(c_{3},c_{4})\in\{(c_{1},m_{1}),(m_{1% },c_{2})\}(\phi_{c_{3},c_{4},\lceil t/2\rceil}).
  56. ( c 3 , c 4 ) (c_{3},c_{4})
  57. ϕ c 1 , c 2 , t ϕ c 3 , c 4 , t / 2 \phi_{c_{1},c_{2},t}\iff\phi_{c_{3},c_{4},\lceil t/2\rceil}
  58. L PSPACE , L p TQBF \forall L\in\textrm{PSPACE},L\leq_{p}\textrm{TQBF}
  59. ϕ \phi
  60. x 1 x n ϕ ( x 1 , , x n ) \exists x_{1}\cdots\exists x_{n}\phi(x_{1},\ldots,x_{n})
  61. \subseteq
  62. x L y 1 y 2 Q i y i V ( x , y 1 , y 2 , , y i ) = 1 x\in L\Leftrightarrow\exists y_{1}\forall y_{2}\cdots Q_{i}y_{i}\ V(x,y_{1},y_% {2},\dots,y_{i})\ =\ 1
  63. ϕ \exists\phi
  64. x 1 x 2 Q i x i ϕ ( x 1 , x 2 , , x i ) = 1 \exists\vec{x_{1}}\forall\vec{x_{2}}\cdots Q_{i}\vec{x_{i}}\ \phi(\vec{x_{1}},% \vec{x_{2}},\dots,\vec{x_{i}})\ =\ 1
  65. x i \vec{x_{i}}

Truncated_octahedral_prism.html

  1. s { 3 3 2 } s\left\{\begin{array}[]{l}3\\ 3\\ 2\end{array}\right\}

Truncation_(statistics).html

  1. F Y ( y ) = F ( y ) - F ( a ) F ( b ) - F ( a ) F_{Y}(y)=\frac{F(y)-F(a)}{F(b)-F(a)}\,
  2. F Y ( y ) = F ( y ) - F ( a - ) F ( b ) - F ( a - ) F_{Y}(y)=\frac{F(y)-F(a-)}{F(b)-F(a-)}\,
  3. F ( b ) - F ( a ) {F(b)-F(a)}

Tsai–Wu_failure_criterion.html

  1. F i σ i + F i j σ i σ j 1 F_{i}~{}\sigma_{i}+F_{ij}~{}\sigma_{i}~{}\sigma_{j}\leq 1
  2. i , j = 1 6 i,j=1\dots 6
  3. F i , F i j F_{i},F_{ij}
  4. σ i \sigma_{i}
  5. F i j F_{ij}
  6. F i i F j j - F i j 2 0 F_{ii}F_{jj}-F_{ij}^{2}\geq 0
  7. F i i F_{ii}
  8. F i j = F j i F_{ij}=F_{ji}
  9. F 1 σ 1 + F 2 σ 2 + F 3 σ 3 + F 4 σ 4 + F 5 σ 5 + F 6 σ 6 + F 11 σ 1 2 + F 22 σ 2 2 + F 33 σ 3 2 + F 44 σ 4 2 + F 55 σ 5 2 + F 66 σ 6 2 + 2 F 12 σ 1 σ 2 + 2 F 13 σ 1 σ 3 + 2 F 23 σ 2 σ 3 1 \begin{aligned}\displaystyle F_{1}\sigma_{1}+&\displaystyle F_{2}\sigma_{2}+F_% {3}\sigma_{3}+F_{4}\sigma_{4}+F_{5}\sigma_{5}+F_{6}\sigma_{6}\\ &\displaystyle+F_{11}\sigma_{1}^{2}+F_{22}\sigma_{2}^{2}+F_{33}\sigma_{3}^{2}+% F_{44}\sigma_{4}^{2}+F_{55}\sigma_{5}^{2}+F_{66}\sigma_{6}^{2}\\ &\displaystyle\qquad+2F_{12}\sigma_{1}\sigma_{2}+2F_{13}\sigma_{1}\sigma_{3}+2% F_{23}\sigma_{2}\sigma_{3}\leq 1\end{aligned}
  10. σ 1 t , σ 1 c , σ 2 t , σ 2 c , σ 3 t , σ 3 c \sigma_{1t},\sigma_{1c},\sigma_{2t},\sigma_{2c},\sigma_{3t},\sigma_{3c}
  11. τ 23 , τ 12 , τ 31 \tau_{23},\tau_{12},\tau_{31}
  12. F 1 = 1 σ 1 t - 1 σ 1 c ; F 2 = 1 σ 2 t - 1 σ 2 c ; F 3 = 1 σ 3 t - 1 σ 3 c ; F 4 = F 5 = F 6 = 0 F 11 = 1 σ 1 c σ 1 t ; F 22 = 1 σ 2 c σ 2 t ; F 33 = 1 σ 3 c σ 3 t ; F 44 = 1 τ 23 2 ; F 55 = 1 τ 31 2 ; F 66 = 1 τ 12 2 \begin{aligned}\displaystyle F_{1}=&\displaystyle\cfrac{1}{\sigma_{1t}}-\cfrac% {1}{\sigma_{1c}}~{};~{}~{}F_{2}=\cfrac{1}{\sigma_{2t}}-\cfrac{1}{\sigma_{2c}}~% {};~{}~{}F_{3}=\cfrac{1}{\sigma_{3t}}-\cfrac{1}{\sigma_{3c}}~{};~{}~{}F_{4}=F_% {5}=F_{6}=0\\ \displaystyle F_{11}=&\displaystyle\cfrac{1}{\sigma_{1c}\sigma_{1t}}~{};~{}~{}% F_{22}=\cfrac{1}{\sigma_{2c}\sigma_{2t}}~{};~{}~{}F_{33}=\cfrac{1}{\sigma_{3c}% \sigma_{3t}}~{};~{}~{}F_{44}=\cfrac{1}{\tau_{23}^{2}}~{};~{}~{}F_{55}=\cfrac{1% }{\tau_{31}^{2}}~{};~{}~{}F_{66}=\cfrac{1}{\tau_{12}^{2}}\\ \end{aligned}
  13. F 12 , F 13 , F 23 F_{12},F_{13},F_{23}
  14. σ 1 = σ 2 = σ b 12 , σ 1 = σ 3 = σ b 13 , σ 2 = σ 3 = σ b 23 \sigma_{1}=\sigma_{2}=\sigma_{b12},\sigma_{1}=\sigma_{3}=\sigma_{b13},\sigma_{% 2}=\sigma_{3}=\sigma_{b23}
  15. F 12 = 1 2 σ b 12 2 [ 1 - σ b 12 ( F 1 + F 2 ) - σ b 12 2 ( F 11 + F 22 ) ] F 13 = 1 2 σ b 13 2 [ 1 - σ b 13 ( F 1 + F 3 ) - σ b 13 2 ( F 11 + F 33 ) ] F 23 = 1 2 σ b 23 2 [ 1 - σ b 23 ( F 2 + F 3 ) - σ b 23 2 ( F 22 + F 33 ) ] \begin{aligned}\displaystyle F_{12}&\displaystyle=\cfrac{1}{2\sigma_{b12}^{2}}% \left[1-\sigma_{b12}(F_{1}+F_{2})-\sigma_{b12}^{2}(F_{11}+F_{22})\right]\\ \displaystyle F_{13}&\displaystyle=\cfrac{1}{2\sigma_{b13}^{2}}\left[1-\sigma_% {b13}(F_{1}+F_{3})-\sigma_{b13}^{2}(F_{11}+F_{33})\right]\\ \displaystyle F_{23}&\displaystyle=\cfrac{1}{2\sigma_{b23}^{2}}\left[1-\sigma_% {b23}(F_{2}+F_{3})-\sigma_{b23}^{2}(F_{22}+F_{33})\right]\end{aligned}
  16. F 12 , F 13 , F 23 F_{12},F_{13},F_{23}
  17. F 1 = F 2 ; F 4 = F 5 = F 6 = 0 ; F 11 = F 22 ; F 44 = F 55 ; F 13 = F 23 . F_{1}=F_{2}~{};~{}~{}F_{4}=F_{5}=F_{6}=0~{};~{}~{}F_{11}=F_{22}~{};~{}~{}F_{44% }=F_{55}~{};~{}~{}F_{13}=F_{23}~{}.
  18. F 2 ( σ 1 + σ 2 ) + F 3 σ 3 + F 22 ( σ 1 2 + σ 2 2 ) + F 33 σ 3 2 + F 44 ( σ 4 2 + σ 5 2 ) + F 66 σ 6 2 + 2 F 12 σ 1 σ 2 + 2 F 23 ( σ 1 + σ 2 ) σ 3 1 \begin{aligned}\displaystyle F_{2}(\sigma_{1}+\sigma_{2})&\displaystyle+F_{3}% \sigma_{3}+F_{22}(\sigma_{1}^{2}+\sigma_{2}^{2})+F_{33}\sigma_{3}^{2}+F_{44}(% \sigma_{4}^{2}+\sigma_{5}^{2})+F_{66}\sigma_{6}^{2}\\ &\displaystyle\qquad+2F_{12}\sigma_{1}\sigma_{2}+2F_{23}(\sigma_{1}+\sigma_{2}% )\sigma_{3}\leq 1\end{aligned}
  19. F 66 = 2 ( F 11 - F 12 ) F_{66}=2(F_{11}-F_{12})
  20. F 22 F 33 - F 23 2 0 ; F 11 2 - F 12 2 0 . F_{22}~{}F_{33}-F_{23}^{2}\geq 0~{};~{}~{}F_{11}^{2}-F_{12}^{2}\geq 0~{}.
  21. σ 1 = σ 5 = σ 6 = 0 \sigma_{1}=\sigma_{5}=\sigma_{6}=0
  22. F 2 σ 2 + F 3 σ 3 + F 22 σ 2 2 + F 33 σ 3 2 + F 44 σ 4 2 + 2 F 23 σ 2 σ 3 1 F_{2}\sigma_{2}+F_{3}\sigma_{3}+F_{22}\sigma_{2}^{2}+F_{33}\sigma_{3}^{2}+F_{4% 4}\sigma_{4}^{2}+2F_{23}\sigma_{2}\sigma_{3}\leq 1
  23. F i , F i j F_{i},F_{ij}
  24. σ 1 c \sigma_{1c}
  25. σ 1 t \sigma_{1t}
  26. σ 3 c \sigma_{3c}
  27. σ 3 t \sigma_{3t}
  28. τ 23 \tau_{23}
  29. τ 12 \tau_{12}
  30. F 2 σ 2 + F 3 σ 3 + F 22 σ 2 2 + F 33 σ 3 2 + 2 F 23 σ 2 σ 3 = 1 - k 2 F_{2}\sigma_{2}+F_{3}\sigma_{3}+F_{22}\sigma_{2}^{2}+F_{33}\sigma_{3}^{2}+2F_{% 23}\sigma_{2}\sigma_{3}=1-k^{2}
  31. F 23 = - 1 2 F 22 F 33 ; k = σ 4 τ 23 . F_{23}=-\cfrac{1}{2}\sqrt{F_{22}F_{33}}~{};~{}~{}k=\cfrac{\sigma_{4}}{\tau_{23% }}~{}.
  32. σ 2 c = 4.6 \sigma_{2c}=4.6
  33. σ 2 t = 7.3 \sigma_{2t}=7.3
  34. σ 3 c = 6.3 \sigma_{3c}=6.3
  35. σ 3 t = 10 \sigma_{3t}=10
  36. 3 J ~ 2 + ( η 2 - 1 ) I ~ 1 2 = η 2 3~{}\tilde{J}_{2}+(\eta^{2}-1)~{}\tilde{I}_{1}^{2}=\eta^{2}
  37. J ~ 2 := 1 3 ( σ 1 2 σ 1 c 2 - σ 1 σ 2 σ 1 c σ 2 c + σ 2 2 σ 2 c 2 ) ; I ~ 1 := σ 1 σ 1 c + σ 2 σ 2 c \tilde{J}_{2}:=\tfrac{1}{3}\left(\cfrac{\sigma_{1}^{2}}{\sigma_{1c}^{2}}-% \cfrac{\sigma_{1}\sigma_{2}}{\sigma_{1c}\sigma_{2c}}+\cfrac{\sigma_{2}^{2}}{% \sigma_{2c}^{2}}\right)~{};~{}~{}\tilde{I}_{1}:=\cfrac{\sigma_{1}}{\sigma_{1c}% }+\cfrac{\sigma_{2}}{\sigma_{2c}}
  38. F 12 F_{12}

Tucker's_lemma.html

  1. B n B_{n}
  2. S n - 1 S_{n-1}
  3. S n - 1 S_{n-1}
  4. S n - 1 S_{n-1}
  5. L : V ( T ) { + 1 , - 1 , + 2 , - 2 , , + n , - n } L:V(T)\to\{+1,-1,+2,-2,...,+n,-n\}
  6. S n - 1 S_{n-1}
  7. L ( - v ) = - L ( v ) L(-v)=-L(v)
  8. v S n - 1 v\in S_{n-1}
  9. n = 2 n=2
  10. B n B_{n}
  11. ± 1 \pm 1
  12. ± 2 \pm 2
  13. n = 2 n=2

Tukey_lambda_distribution.html

  1. Q ( p ; λ ) = { 1 λ [ p λ - ( 1 - p ) λ ] , if λ 0 log ( p 1 - p ) , if λ = 0 , Q\left(p;\lambda\right)=\begin{cases}\frac{1}{\lambda}\left[p^{\lambda}-(1-p)^% {\lambda}\right],&\mbox{if }~{}\lambda\neq 0\\ \log(\frac{p}{1-p}),&\mbox{if }~{}\lambda=0,\end{cases}
  2. Q ( p ; λ ) = p ( λ - 1 ) + ( 1 - p ) ( λ - 1 ) . Q^{\prime}\left(p;\lambda\right)=p^{(\lambda-1)}+\left(1-p\right)^{(\lambda-1)}.
  3. Var [ X ] = 2 λ 2 ( 1 1 + 2 λ - Γ ( λ + 1 ) 2 Γ ( 2 λ + 2 ) ) . \operatorname{Var}[X]=\frac{2}{\lambda^{2}}\bigg(\frac{1}{1+2\lambda}-\frac{% \Gamma(\lambda+1)^{2}}{\Gamma(2\lambda+2)}\bigg).
  4. μ n = E [ X n ] = 1 λ n k = 0 n ( - 1 ) k ( n k ) B ( λ k + 1 , λ ( n - k ) + 1 ) . \mu_{n}=\operatorname{E}[X^{n}]=\frac{1}{\lambda^{n}}\sum_{k=0}^{n}(-1)^{k}{n% \choose k}\,B(\lambda k+1,\,\lambda(n-k)+1).
  5. L r λ = k = 0 r - 1 ( - 1 ) r - k - 1 ( r - 1 k ) ( r + k - 1 k ) ( 1 k + 1 + λ + ( - 1 ) r k + 1 + λ ) . L_{r}\lambda=\sum_{k=0}^{r-1}(-1)^{r-k-1}{\left({{r-1}\atop{k}}\right)}{\left(% {{r+k-1}\atop{k}}\right)}\left(\frac{1}{k+1+\lambda}+\frac{(-1)^{r}}{k+1+% \lambda}\right).
  6. L 1 = 0 L_{1}=0
  7. L 2 λ = - 2 1 + λ + 4 2 + λ L_{2}\lambda=-\frac{2}{1+\lambda}+\frac{4}{2+\lambda}
  8. L 3 = 0 L_{3}=0
  9. L 4 λ = - 2 1 + λ + 24 2 + λ - 60 3 + λ + 60 4 + λ L_{4}\lambda=-\frac{2}{1+\lambda}+\frac{24}{2+\lambda}-\frac{60}{3+\lambda}+% \frac{60}{4+\lambda}
  10. L 5 = 0 L_{5}=0
  11. L 6 λ = - 2 1 + λ + 60 2 + λ - 420 3 + λ + 1120 4 + λ - 1260 5 + λ + 504 6 + λ . L_{6}\lambda=-\frac{2}{1+\lambda}+\frac{60}{2+\lambda}-\frac{420}{3+\lambda}+% \frac{1120}{4+\lambda}-\frac{1260}{5+\lambda}+\frac{504}{6+\lambda}.
  12. \cap

Turán_sieve.html

  1. S ( A , P , z ) = | A p P ( z ) A p | . S(A,P,z)=\left|A\setminus\bigcup_{p\in P(z)}A_{p}\right|.
  2. | A p | = 1 f ( p ) X + R p \left|A_{p}\right|=\frac{1}{f(p)}X+R_{p}
  3. | A p q | = 1 f ( p ) f ( q ) X + R p , q \left|A_{pq}\right|=\frac{1}{f(p)f(q)}X+R_{p,q}
  4. U ( z ) = p P ( z ) f ( p ) . U(z)=\sum_{p\mid P(z)}f(p).
  5. S ( A , P , z ) X U ( z ) + 2 U ( z ) p P ( z ) | R p | + 1 U ( z ) 2 p , q P ( z ) | R p , q | . S(A,P,z)\leq\frac{X}{U(z)}+\frac{2}{U(z)}\sum_{p\mid P(z)}\left|R_{p}\right|+% \frac{1}{U(z)^{2}}\sum_{p,q\mid P(z)}\left|R_{p,q}\right|.

Tutte_matrix.html

  1. A i j = { x i j if ( i , j ) E and i < j - x j i if ( i , j ) E and i > j 0 otherwise A_{ij}=\begin{cases}x_{ij}\;\;\mbox{if}~{}\;(i,j)\in E\mbox{ and }~{}i<j\\ -x_{ji}\;\;\mbox{if}~{}\;(i,j)\in E\mbox{ and }~{}i>j\\ 0\;\;\;\;\mbox{otherwise}\end{cases}

Twisting_properties.html

  1. { x 1 , , x m } \{x_{1},\ldots,x_{m}\}
  2. M X = ( g θ , Z ) M_{X}=(g_{\theta},Z)
  3. F Θ ( θ ) F_{\Theta}(\theta)
  4. M X = ( g θ , Z ) M_{X}=(g_{\theta},Z)
  5. s y m b o l X = { X 1 , , X m } symbolX=\{X_{1},\ldots,X_{m}\}
  6. { g θ ( Z 1 ) , , g θ ( Z m ) } \{g_{\theta}(Z_{1}),\ldots,g_{\theta}(Z_{m})\}
  7. S = h 1 ( X 1 , , X m ) S=h_{1}(X_{1},\ldots,X_{m})
  8. s = h ( g θ ( z 1 ) , , g θ ( z m ) ) = ρ ( θ ; z 1 , , z m ) s=h(g_{\theta}(z_{1}),\ldots,g_{\theta}(z_{m}))=\rho(\theta;z_{1},\ldots,z_{m})
  9. s y m b o l z = { z 1 , , z m } symbolz=\{z_{1},\ldots,z_{m}\}
  10. s y m b o l Z symbolZ
  11. s y m b o l z symbolz
  12. s s θ θ s\geq s^{\prime}\leftrightarrow\theta\geq\theta^{\prime}
  13. s s θ θ s\geq s^{\prime}\leftrightarrow\theta\leq\theta^{\prime}
  14. s s^{\prime}
  15. θ \theta^{\prime}
  16. s s θ θ s s + s\geq s^{\prime}\rightarrow\theta\geq\theta^{\prime}\rightarrow s\geq s^{% \prime}+\ell
  17. > 0 \ell>0
  18. F Θ | S = s ( θ ) = F S | Θ = θ ( s ) F_{\Theta|S=s}(\theta)=F_{S|\Theta=\theta}(s)
  19. F Θ | S = s ( θ ) = 1 - F S | Θ = θ ( s ) F_{\Theta|S=s}(\theta)=1-F_{S|\Theta=\theta}(s)
  20. F Θ | S = s ( θ ) F_{\Theta|S=s}(\theta)
  21. > 0 \ell>0
  22. { x 1 , , x m } \{x_{1},\ldots,x_{m}\}
  23. \ell
  24. F Θ ( θ ) ( q 1 ( F S | Θ = θ ( s ) ) , q 2 ( F S | Θ = θ ( s ) ) ) F_{\Theta}(\theta)\in\left(q_{1}(F_{S|\Theta=\theta}(s)),q_{2}(F_{S|\Theta=% \theta}(s))\right)
  25. q 1 = q 2 q_{1}=q_{2}
  26. q 2 ( F S ( s ) ) = q 1 ( F S ( s - ) q_{2}(F_{S}(s))=q_{1}(F_{S}(s-\ell)
  27. q 1 ( F S ( s ) ) = q 2 ( F S ( s - ) q_{1}(F_{S}(s))=q_{2}(F_{S}(s-\ell)
  28. q i ( F S ) = 1 - F S q_{i}(F_{S})=1-F_{S}
  29. q i ( F S ) = F S q_{i}(F_{S})=F_{S}
  30. i = 1 , 2 i=1,2
  31. s y m b o l x symbolx
  32. ( k k ) ( s k s k ) (k\leq k^{\prime})\leftrightarrow(s_{k}\leq s_{k^{\prime}})
  33. ( λ λ ) ( s λ s λ ) (\lambda\leq\lambda^{\prime})\leftrightarrow(s_{\lambda^{\prime}}\leq s_{% \lambda})
  34. s k = i = 1 m x i s_{k}=\prod_{i=1}^{m}x_{i}
  35. s λ = i = 1 m x i s_{\lambda}=\sum_{i=1}^{m}x_{i}
  36. F Λ , K ( λ , k ) = F Λ | k ( λ | k ) F K ( k ) = F K | λ ( k | λ ) F Λ ( λ ) F_{\Lambda,K}(\lambda,k)=F_{\Lambda|k}(\lambda|k)F_{K}(k)=F_{K|\lambda}(k|% \lambda)F_{\Lambda}(\lambda)
  37. s k s_{k}
  38. r k = s k s λ m r_{k}=\frac{s_{k}}{s_{\lambda}^{m}}
  39. K K
  40. Λ \Lambda
  41. F Λ | k ( λ | k ) = 1 - Γ ( k m , λ s Λ ) Γ ( k m ) F_{\Lambda|k}(\lambda|k)=1-\frac{\Gamma(km,\lambda s_{\Lambda})}{\Gamma(km)}
  42. F K ( k ) = 1 - F R k ( r K ) F_{K}(k)=1-F_{R_{k}}(r_{K})
  43. s Λ s_{\Lambda}
  44. r K r_{K}
  45. Γ ( a , b ) \Gamma(a,b)
  46. F R k ( r K ) F_{R_{k}}(r_{K})
  47. m = 30 , s Λ = 72.82 m=30,s_{\Lambda}=72.82
  48. r K = r_{K}=
  49. 4.5 × 10 - 46 4.5\times 10^{-46}
  50. Λ \Lambda

Two-alternative_forced_choice.html

  1. d x = A d t + c d W , x ( 0 ) = 0 dx=Adt+cdW\ ,\ x(0)=0
  2. A A
  3. x ( 0 ) x(0)
  4. λ \lambda
  5. d x = ( λ x + A ) d t + c d W dx\ =\ (\lambda x+A)dt\ +\ cdW
  6. d y 1 = I 1 d t + c d W 1 d y 2 = I 2 d t + c d W 2 , y 1 ( 0 ) = y 2 ( 0 ) = 0 \begin{aligned}\displaystyle dy_{\,\text{1}}\ =\ I_{\,\text{1}}dt\ +\ cdW_{\,% \text{1}}\\ \displaystyle dy_{\,\text{2}}\ =\ I_{\,\text{2}}dt\ +\ cdW_{\,\text{2}}\end{% aligned},\quad y_{\,\text{1}}(0)\ =\ y_{\,\text{2}}(0)=0
  7. d y 1 = ( - k y 1 - w y 2 + I 1 ) d t + c d W 1 d y 2 = ( - k y 2 - w y 1 + I 2 ) d t + c d W 2 , y 1 ( 0 ) = y 2 ( 0 ) = 0 \begin{aligned}\displaystyle dy_{\,\text{1}}\ =\ (-ky_{\,\text{1}}-wy_{\,\text% {2}}+I_{\,\text{1}})dt\ +\ cdW_{\,\text{1}}\\ \displaystyle dy_{\,\text{2}}\ =\ (-ky_{\,\text{2}}-wy_{\,\text{1}}+I_{\,\text% {2}})dt\ +\ cdW_{\,\text{2}}\end{aligned},\quad y_{\,\text{1}}(0)\ =\ y_{\,% \text{2}}(0)=0
  8. k k
  9. w w
  10. d y 1 = I 1 d t + c d W 1 - u ( I 2 d t + c d W 2 ) d y 2 = I 2 d t + c d W 2 - u ( I 1 d t + c d W 1 ) , y 1 ( 0 ) = y 2 ( 0 ) = 0 \begin{aligned}\displaystyle dy_{\,\text{1}}\ =\ I_{\,\text{1}}dt\ +\ cdW_{\,% \text{1}}\ -\ u(I_{\,\text{2}}dt\ +\ cdW_{\,\text{2}})\\ \displaystyle dy_{\,\text{2}}\ =\ I_{\,\text{2}}dt\ +\ cdW_{\,\text{2}}\ -\ u(% I_{\,\text{1}}dt\ +\ cdW_{\,\text{1}})\end{aligned},\quad y_{\,\text{1}}(0)\ =% \ y_{\,\text{2}}(0)=0
  11. u u
  12. d y 1 = ( - k y 1 - w y 3 + v y 1 + I 1 ) d t + c d W 1 \displaystyle dy_{\,\text{1}}\ =\ (-ky_{\,\text{1}}-wy_{\,\text{3}}+vy_{\,% \text{1}}+I_{\,\text{1}})dt\ +\ cdW_{\,\text{1}}
  13. k inh k_{\,\text{inh}}
  14. w w^{\prime}

Two-dimensional_infrared_spectroscopy.html

  1. ω 3 \omega_{3}
  2. ω 1 \omega_{1}
  3. ω 1 \omega_{1}
  4. ω 3 \omega_{3}

Typical_price.html

  1. Typical Price = H + L + C 3 \,\text{Typical Price}=\frac{H+L+C}{3}

U-chart.html

  1. u ¯ ± 3 u ¯ n \bar{u}\pm 3\sqrt{\frac{\bar{u}}{n}}
  2. u ¯ i = j = 1 n no. of defects for x i j n \bar{u}_{i}=\frac{\sum_{j=1}^{n}\mbox{no. of defects for }~{}x_{ij}}{n}
  3. u ¯ ± 3 u ¯ n \bar{u}\pm 3\sqrt{\frac{\bar{u}}{n}}
  4. u ¯ \bar{u}
  5. u i = x i n i u_{i}=\frac{x_{i}}{n_{i}}

U-statistic.html

  1. f : R r R f\colon R^{r}\to R
  2. r r
  3. n r n\geq r
  4. f n : R n R f_{n}\colon R^{n}\to R
  5. φ ( 1 ) , , φ ( r ) \varphi(1),\ldots,\varphi(r)
  6. r r
  7. f ( x φ ) f(x_{\varphi})
  8. f n ( x 1 , , x n ) = ave f ( x φ ( 1 ) , , x φ ( r ) ) f_{n}(x_{1},\ldots,x_{n})=\operatorname{ave}f(x_{\varphi(1)},\ldots,x_{\varphi% (r)})
  9. r r
  10. { 1 , , n } \{1,\ldots,n\}
  11. f n ( x 1 , , x n ) f_{n}(x_{1},\ldots,x_{n})
  12. f ( x ) = x f(x)=x
  13. f n ( x ) = x ¯ n = ( x 1 + + x n ) / n f_{n}(x)=\bar{x}_{n}=(x_{1}+\cdots+x_{n})/n
  14. f ( x 1 , x 2 ) = | x 1 - x 2 | f(x_{1},x_{2})=|x_{1}-x_{2}|
  15. f n ( x 1 , , x n ) = i j | x i - x j | / ( n ( n - 1 ) ) f_{n}(x_{1},\ldots,x_{n})=\sum_{i\neq j}|x_{i}-x_{j}|/(n(n-1))
  16. n 2 n\geq 2
  17. f ( x 1 , x 2 ) = ( x 1 - x 2 ) 2 / 2 f(x_{1},x_{2})=(x_{1}-x_{2})^{2}/2
  18. f n ( x ) = ( x i - x ¯ n ) 2 / ( n - 1 ) f_{n}(x)=\sum(x_{i}-\bar{x}_{n})^{2}/(n-1)
  19. n - 1 n-1
  20. n 2 n\geq 2
  21. k k
  22. k 3 , n ( x ) = ( x i - x ¯ n ) 3 n / ( ( n - 1 ) ( n - 2 ) ) k_{3,n}(x)=\sum(x_{i}-\bar{x}_{n})^{3}n/((n-1)(n-2))
  23. n 3 n\geq 3
  24. f ( x 1 , x 2 , x 3 ) f(x_{1},x_{2},x_{3})
  25. f n ( x 1 , , x n ) f_{n}(x_{1},\ldots,x_{n})
  26. n n

Ultracold_neutrons.html

  1. V F ( p o l . ) = V F ( u n p o l . ) ± μ N B V_{F}(pol.)=V_{F}(unpol.)\pm\mu_{N}\cdot B
  2. μ N \mu_{N}
  3. B = μ 0 M B=\mu_{0}\cdot M
  4. μ ( E , θ ) = 2 η E cos 2 θ V F - E cos 2 θ \mu(E,\theta)=2\eta\sqrt{\frac{E\cos^{2}\theta}{V_{F}-E\cos^{2}\theta}}
  5. 885.7 ± 0.8 s 885.7\pm 0.8~{}s
  6. τ n = 885.4 ± 0.9 stat ± 0.4 syst s \tau_{n}=885.4\pm 0.9_{\mathrm{stat}}\pm 0.4_{\mathrm{syst}}~{}s
  7. 878.5 ± 0.7 stat ± 0.4 syst s 878.5~{}\pm 0.7_{\mathrm{stat}}\pm 0.4_{\mathrm{syst}}~{}s
  8. A 0 = - 0.1184 ± 0.0010 A_{0}=-0.1184\pm 0.0010

Ultralimit.html

  1. \mathbb{N}
  2. ω : 2 { 0 , 1 } \omega:2^{\mathbb{N}}\to\{0,1\}
  3. 2 2^{\mathbb{N}}
  4. \mathbb{N}
  5. ω ( ) = 1 \omega(\mathbb{N})=1
  6. \mathbb{N}
  7. F F\subseteq\mathbb{N}
  8. \mathbb{N}
  9. ( x n ) n (x_{n})_{n\in\mathbb{N}}
  10. x = lim ω x n x=\lim_{\omega}x_{n}
  11. ϵ > 0 \epsilon>0
  12. ω { n : d ( x n , x ) ϵ } = 1. \omega\{n:d(x_{n},x)\leq\epsilon\}=1.
  13. x = lim n x n x=\lim_{n\to\infty}x_{n}
  14. x = lim ω x n x=\lim_{\omega}x_{n}
  15. \mathbb{N}
  16. \mathbb{R}
  17. \mathbb{N}
  18. ( x n ) n (x_{n})_{n\in\mathbb{N}}
  19. d n ( x n , p n ) C d_{n}(x_{n},p_{n})\leq C
  20. 𝒜 \mathcal{A}
  21. 𝐱 = ( x n ) n \mathbf{x}=(x_{n})_{n\in\mathbb{N}}
  22. 𝐲 = ( y n ) n \mathbf{y}=(y_{n})_{n\in\mathbb{N}}
  23. d ^ ( 𝐱 , 𝐲 ) := lim ω d n ( x n , y n ) \hat{d}_{\infty}(\mathbf{x},\mathbf{y}):=\lim_{\omega}d_{n}(x_{n},y_{n})
  24. \sim
  25. 𝒜 \mathcal{A}
  26. 𝐱 , 𝐲 𝒜 \mathbf{x},\mathbf{y}\in\mathcal{A}
  27. 𝐱 𝐲 \mathbf{x}\sim\mathbf{y}
  28. d ^ ( 𝐱 , 𝐲 ) = 0. \hat{d}_{\infty}(\mathbf{x},\mathbf{y})=0.
  29. \sim
  30. 𝒜 . \mathcal{A}.
  31. ( X , d ) (X_{\infty},d_{\infty})
  32. X = 𝒜 / X_{\infty}=\mathcal{A}/{\sim}
  33. \sim
  34. [ 𝐱 ] , [ 𝐲 ] [\mathbf{x}],[\mathbf{y}]
  35. 𝐱 = ( x n ) n \mathbf{x}=(x_{n})_{n\in\mathbb{N}}
  36. 𝐲 = ( y n ) n \mathbf{y}=(y_{n})_{n\in\mathbb{N}}
  37. d ( [ 𝐱 ] , [ 𝐲 ] ) := d ^ ( 𝐱 , 𝐲 ) = lim ω d n ( x n , y n ) . d_{\infty}([\mathbf{x}],[\mathbf{y}]):=\hat{d}_{\infty}(\mathbf{x},\mathbf{y})% =\lim_{\omega}d_{n}(x_{n},y_{n}).
  38. d d_{\infty}
  39. X X_{\infty}
  40. ( X , d ) = lim ω ( X n , d n , p n ) (X_{\infty},d_{\infty})=\lim_{\omega}(X_{n},d_{n},p_{n})
  41. n n\in\mathbb{N}
  42. ( x n ) n , x n X n (x_{n})_{n},x_{n}\in X_{n}
  43. ( X , d ) (X_{\infty},d_{\infty})
  44. p n X n . p_{n}\in X_{n}.
  45. ( X , d ) = lim ω ( X n , d n ) (X_{\infty},d_{\infty})=\lim_{\omega}(X_{n},d_{n})
  46. ( X , d ) = lim ω ( X n , d n , p n ) (X_{\infty},d_{\infty})=\lim_{\omega}(X_{n},d_{n},p_{n})
  47. ( X , d ) = lim ω ( X n , d n , p n ) (X_{\infty},d_{\infty})=\lim_{\omega}(X_{n},d_{n},p_{n})
  48. ( X , d ) = lim ω ( X n , d n ) (X_{\infty},d_{\infty})=\lim_{\omega}(X_{n},d_{n})
  49. p n X n p_{n}\in X_{n}
  50. ( X , d ) = lim ω ( X n , d n , p n ) (X_{\infty},d_{\infty})=\lim_{\omega}(X_{n},d_{n},p_{n})
  51. ( X , d ) = lim ω ( X n , d n , p n ) (X_{\infty},d_{\infty})=\lim_{\omega}(X_{n},d_{n},p_{n})
  52. lim n κ n = - . \lim_{n\to\infty}\kappa_{n}=-\infty.
  53. ( X , d ) = lim ω ( X n , d n , p n ) (X_{\infty},d_{\infty})=\lim_{\omega}(X_{n},d_{n},p_{n})
  54. \mathbb{N}
  55. ( X , d n , p n ) (X,\frac{d}{n},p_{n})
  56. ( p n ) n (p_{n})_{n}\,
  57. C o n e ω ( X , d , ( p n ) n ) Cone_{\omega}(X,d,(p_{n})_{n})\,
  58. C o n e ω ( X , d ) Cone_{\omega}(X,d)\,
  59. C o n e ω ( X ) Cone_{\omega}(X)\,
  60. n n\in\mathbb{N}
  61. ( X , d ) = lim ω ( X n , d n ) (X_{\infty},d_{\infty})=\lim_{\omega}(X_{n},d_{n})
  62. A 1 = { n | ( X n , d n ) = ( X , d X ) } A_{1}=\{n|(X_{n},d_{n})=(X,d_{X})\}\,
  63. A 2 = { n | ( X n , d n ) = ( Y , d Y ) } A_{2}=\{n|(X_{n},d_{n})=(Y,d_{Y})\}\,
  64. A 1 A 2 = . A_{1}\cup A_{2}=\mathbb{N}.
  65. lim ω ( X n , d n ) \lim_{\omega}(X_{n},d_{n})
  66. lim ω ( X n , d n ) \lim_{\omega}(X_{n},d_{n})
  67. lim ω ( M , n d , p ) \lim_{\omega}(M,nd,p)
  68. lim ω ( M , n d , p ) \lim_{\omega}(M,nd,p)
  69. m \mathbb{R}^{m}
  70. ( m , d ) (\mathbb{R}^{m},d)
  71. C o n e ω ( m , d ) Cone_{\omega}(\mathbb{R}^{m},d)
  72. ( m , d ) (\mathbb{R}^{m},d)
  73. ( 2 , d ) (\mathbb{Z}^{2},d)
  74. C o n e ω ( 2 , d ) Cone_{\omega}(\mathbb{Z}^{2},d)
  75. ( 2 , d 1 ) (\mathbb{R}^{2},d_{1})
  76. d 1 d_{1}\,
  77. 2 \mathbb{R}^{2}
  78. C o n e ω ( X ) Cone_{\omega}(X)\,
  79. C o n e ω ( X ) Cone_{\omega}(X)\,
  80. C o n e ω ( X ) Cone_{\omega}(X)\,

Ultrasonic_grating.html

  1. d d
  2. λ \lambda
  3. θ \theta
  4. d sin θ = n λ d\sin\theta=n\lambda
  5. d = n λ / sin θ d=n\lambda/\sin\theta
  6. v / ν = n λ / sin θ v/\nu=n\lambda/\sin\theta
  7. v = ν n λ / sin θ v=\nu n\lambda/\sin\theta
  8. ν \nu
  9. λ \lambda\,\!
  10. λ c \lambda_{c}\,\!
  11. λ c sin θ = n λ \lambda_{c}\sin\theta=n\lambda\,\!
  12. θ \theta\,\!
  13. λ \lambda\,\!
  14. λ c \lambda_{c}\,\!
  15. v c v_{c}\,\!
  16. v c = η λ c v_{c}=\eta\lambda_{c}\,\!
  17. η \eta\,\!

Unbalanced_Oil_and_Vinegar.html

  1. m m
  2. n n
  3. m m
  4. n n
  5. n n
  6. T T
  7. S S
  8. P ´ \acute{P}
  9. T T
  10. y y
  11. y 1 , y 2 , , y n y_{1},y_{2},...,y_{n}
  12. S S
  13. P ´ \acute{P}
  14. y 1 \displaystyle y_{1}
  15. y = ( y 1 , y 2 , , y m ) y=(y_{1},y_{2},\ldots,y_{m})
  16. x = ( x 1 , x 2 , , x n ) x=(x_{1},x_{2},\ldots,x_{n})
  17. y y
  18. T T
  19. y 1 , y 2 , , y m y_{1},y_{2},...,y_{m}
  20. y i = γ i j k a j a ´ k + λ i j k a ´ j a ´ k + ξ i j a j + ξ ´ i j a ´ j + δ i y_{i}=\sum{\gamma_{ijk}a_{j}\acute{a}_{k}}+\sum{\lambda_{ijk}\acute{a}_{j}% \acute{a}_{k}}+\sum{\xi_{ij}a_{j}}+\sum{\acute{\xi}_{ij}\acute{a}_{j}}+\delta_% {i}
  21. y y
  22. x x
  23. γ i j k , λ i j k , ξ i j , ξ ´ i j , δ i \gamma_{ijk},\lambda_{ijk},\xi_{ij},\acute{\xi}_{ij},\delta_{i}
  24. a ´ j \acute{a}_{j}
  25. a i a_{i}
  26. A = ( a 1 , , a n , a ´ 1 , , a ´ v ) A=(a_{1},...,a_{n},\acute{a}_{1},...,\acute{a}_{v})
  27. A A
  28. S S
  29. x = S - 1 ( A ) x=S^{-1}(A)
  30. y 1 \displaystyle y_{1}
  31. y i y_{i}
  32. m m

Uncial_0243.html

  1. 𝔓 \mathfrak{P}

Uncial_0308.html

  1. 𝔓 \mathfrak{P}
  2. 𝔐 \mathfrak{M}
  3. 𝔓 \mathfrak{P}
  4. 𝔐 \mathfrak{M}
  5. 𝔓 \mathfrak{P}
  6. 𝔓 \mathfrak{P}

Underdetermined_system.html

  1. x + y + z = 1 x + y + z = 0 \begin{aligned}\displaystyle x+y+z&\displaystyle=1\\ \displaystyle x+y+z&\displaystyle=0\end{aligned}
  2. x + y + z = 1 x + y + 2 z = 3 \begin{aligned}\displaystyle x+y+z&\displaystyle=1\\ \displaystyle x+y+2z&\displaystyle=3\end{aligned}

Unfolding_(functions).html

  1. M M
  2. f : M . f:M\to\mathbb{R}.
  3. x 0 M x_{0}\in M
  4. y 0 y_{0}\in\mathbb{R}
  5. f ( x 0 ) = y 0 f(x_{0})=y_{0}
  6. N N
  7. k k
  8. N N
  9. F : M × N . F:M\times N\to\mathbb{R}.
  10. F F
  11. k k
  12. f f
  13. F ( x , 0 ) = f ( x ) F(x,0)=f(x)
  14. x . x.
  15. f : M f:M\to\mathbb{R}
  16. F : M × { 0 } F:M\times\{0\}\to\mathbb{R}
  17. f f
  18. F . F.
  19. f : 2 f:\mathbb{R}^{2}\to\mathbb{R}
  20. f ( x , y ) = x 2 + y 5 . f(x,y)=x^{2}+y^{5}.
  21. f f
  22. F : 2 × 3 F:\mathbb{R}^{2}\times\mathbb{R}^{3}\to\mathbb{R}
  23. F ( ( x , y ) , ( a , b , c ) ) = x 2 + y 5 + a y + b y 2 + c y 3 . F((x,y),(a,b,c))=x^{2}+y^{5}+ay+by^{2}+cy^{3}.
  24. x x
  25. y y
  26. a , a,
  27. b , b,
  28. c c
  29. \mathbb{R}
  30. f f
  31. M M
  32. \mathbb{R}
  33. C ( M , ) . C^{\infty}(M,\mathbb{R}).
  34. Φ : N C ( M , ) . \Phi:N\to C^{\infty}(M,\mathbb{R}).
  35. diff ( M ) × diff ( ) , \mbox{diff}~{}(M)\times\mbox{diff}~{}(\mathbb{R}),
  36. diff ( M ) \mbox{diff}~{}(M)
  37. M M
  38. C ( M , ) . C^{\infty}(M,\mathbb{R}).
  39. ( ϕ , ψ ) f = ψ f ϕ - 1 . (\phi,\psi)\cdot f=\psi\circ f\circ\phi^{-1}.
  40. g g
  41. f f
  42. M M
  43. \mathbb{R}
  44. g g
  45. f f
  46. Im ( Φ ) orb ( f ) \mbox{Im}~{}(\Phi)\pitchfork\mbox{orb}~{}(f)
  47. \pitchfork
  48. C ( M , ) C^{\infty}(M,\mathbb{R})
  49. Im ( Φ ) orb ( f ) \mbox{Im}~{}(\Phi)\pitchfork\mbox{orb}~{}(f)
  50. x 1 , , x n x_{1},\ldots,x_{n}
  51. M M
  52. 𝒪 ( x 1 , , x n ) \mathcal{O}(x_{1},\ldots,x_{n})
  53. f , f,
  54. J f J_{f}
  55. J f := f x 1 , , f x n . J_{f}:=\left\langle\frac{\partial f}{\partial x_{1}},\ldots,\frac{\partial f}{% \partial x_{n}}\right\rangle.
  56. f f
  57. 𝒪 ( x 1 , , x n ) J f \frac{\mathcal{O}(x_{1},\ldots,x_{n})}{J_{f}}
  58. f . f.
  59. f f
  60. f ( x , y ) = x 2 + y 5 . f(x,y)=x^{2}+y^{5}.
  61. 𝒪 ( x , y ) 2 x , 5 y 4 = { y , y 2 , y 3 } . \frac{\mathcal{O}(x,y)}{\langle 2x,5y^{4}\rangle}=\{y,y^{2},y^{3}\}\ .
  62. { y , y 2 , y 3 } \{y,y^{2},y^{3}\}
  63. F ( ( x , y ) , ( a , b , c ) ) = x 2 + y 5 + a y + b y 2 + c y 3 F((x,y),(a,b,c))=x^{2}+y^{5}+ay+by^{2}+cy^{3}

Uniform_10-polytope.html

  1. A ~ 9 {\tilde{A}}_{9}
  2. B ~ 9 {\tilde{B}}_{9}
  3. C ~ 9 {\tilde{C}}_{9}
  4. D ~ 9 {\tilde{D}}_{9}
  5. Q ¯ 9 {\bar{Q}}_{9}
  6. S ¯ 9 {\bar{S}}_{9}
  7. E 10 E_{10}
  8. T ¯ 9 {\bar{T}}_{9}
  9. E 10 E_{10}

Uniform_absolute-convergence.html

  1. f n : X f_{n}:X\to\mathbb{C}
  2. n = 0 f n ( x ) \sum_{n=0}^{\infty}f_{n}(x)
  3. n = 0 | f n ( x ) | \sum_{n=0}^{\infty}|f_{n}(x)|

Uniformly_hyperfinite_algebra.html

  1. A = n A n ¯ . A=\overline{\cup_{n}A_{n}}.
  2. A n M k n ( ) , A_{n}\simeq M_{k_{n}}(\mathbb{C}),
  3. ϕ n ( a ) = a I r , \phi_{n}(a)=a\otimes I_{r},
  4. δ ( A ) = p p t p \delta(A)=\prod_{p}p^{t_{p}}
  5. δ ( A ) = p p t p \delta(A)=\prod_{p}p^{t_{p}}
  6. α : H L ( H ) \alpha:H\rightarrow L(H)
  7. { α ( f n ) , α ( f m ) } = 0 and α ( f n ) * α ( f m ) + α ( f m ) α ( f n ) * = f m , f n I . \{\alpha(f_{n}),\alpha(f_{m})\}=0\quad\mbox{and}~{}\quad\alpha(f_{n})^{*}% \alpha(f_{m})+\alpha(f_{m})\alpha(f_{n})^{*}=\langle f_{m},f_{n}\rangle I.
  8. { α ( f n ) } . \{\alpha(f_{n})\}\;.
  9. C * ( α ( f 1 ) , , α ( f n ) ) C * ( α ( f 1 ) , , α ( f n + 1 ) ) C^{*}(\alpha(f_{1}),\cdots,\alpha(f_{n}))\hookrightarrow C^{*}(\alpha(f_{1}),% \cdots,\alpha(f_{n+1}))
  10. M 2 n M 2 n + 1 . M_{2^{n}}\hookrightarrow M_{2^{n+1}}.

UNIQUAC.html

  1. ln γ i = ln γ i C + ln γ i R \ln\gamma_{i}=\ln\gamma^{C}_{i}+\ln\gamma^{R}_{i}
  2. ln γ i C = ( 1 - V i + ln V i ) - z 2 q i ( 1 - V i F i + ln V i F i ) \ln\gamma_{i}^{C}=(1-V_{i}+\ln V_{i})-\frac{z}{2}q_{i}\left(1-\frac{V_{i}}{F_{% i}}+\ln\frac{V_{i}}{F_{i}}\right)
  3. V i = r i j r j x j V_{i}=\frac{r_{i}}{\sum_{j}r_{j}x_{j}}
  4. F i = q i j q j x j F_{i}=\frac{q_{i}}{\sum_{j}q_{j}x_{j}}
  5. { ln γ 1 C , = 1 - r 1 r 2 + ln r 1 r 2 - z 2 q 1 ( 1 - r 1 q 2 r 2 q 1 + ln r 1 q 2 r 2 q 1 ) ln γ 2 C , = 1 - r 2 r 1 + ln r 2 r 1 - z 2 q 2 ( 1 - r 2 q 1 r 1 q 2 + ln r 2 q 1 r 1 q 2 ) \begin{cases}\ln\gamma_{1}^{C,\infty}=1-\dfrac{r_{1}}{r_{2}}+\ln\dfrac{r_{1}}{% r_{2}}-\dfrac{z}{2}q_{1}\left(1-\dfrac{r_{1}q_{2}}{r_{2}q_{1}}+\ln\dfrac{r_{1}% q_{2}}{r_{2}q_{1}}\right)\\ \ln\gamma_{2}^{C,\infty}=1-\dfrac{r_{2}}{r_{1}}+\ln\dfrac{r_{2}}{r_{1}}-\dfrac% {z}{2}q_{2}\left(1-\dfrac{r_{2}q_{1}}{r_{1}q_{2}}+\ln\dfrac{r_{2}q_{1}}{r_{1}q% _{2}}\right)\end{cases}
  6. γ 1 C , = γ 2 C , = 1 \gamma_{1}^{C,\infty}=\gamma_{2}^{C,\infty}=1
  7. τ i j \tau_{ij}
  8. ln γ i R = q i ( 1 - ln j q j x j τ j i j q j x j - j q j x j τ i j k q k x k τ k j ) \ln\gamma_{i}^{R}=q_{i}\left(1-\ln\frac{\sum_{j}q_{j}x_{j}\tau_{ji}}{\sum_{j}q% _{j}x_{j}}-\sum_{j}{\frac{q_{j}x_{j}\tau_{ij}}{\sum_{k}q_{k}x_{k}\tau_{kj}}}\right)
  9. τ i j = e - Δ u i j / R T \tau_{ij}=e^{-\Delta u_{ij}/{RT}}
  10. γ i R = 1 \gamma_{i}^{R}=1
  11. τ i j \tau_{ij}
  12. ln τ i j = A i j + B i j / T + C i j ln ( T ) + D i j T + E i j / T 2 \ln\tau_{ij}=A_{ij}+B_{ij}/T+C_{ij}\ln(T)+D_{ij}T+E_{ij}/T^{2}

Unisolvent_functions.html

  1. [ f 1 ( x 1 ) f 1 ( x 2 ) f 1 ( x n ) ] , [ f 2 ( x 1 ) f 2 ( x 2 ) f 2 ( x n ) ] , , [ f n ( x 1 ) f n ( x 2 ) f n ( x n ) ] \begin{bmatrix}f_{1}(x_{1})\\ f_{1}(x_{2})\\ \vdots\\ f_{1}(x_{n})\end{bmatrix},\begin{bmatrix}f_{2}(x_{1})\\ f_{2}(x_{2})\\ \vdots\\ f_{2}(x_{n})\end{bmatrix},\dots,\begin{bmatrix}f_{n}(x_{1})\\ f_{n}(x_{2})\\ \vdots\\ f_{n}(x_{n})\end{bmatrix}

United_States_Standard_thread.html

  1. 1 / 4 {1}/{4}
  2. P = 0.24 D + 0.625 - 0.175 P=0.24\sqrt{D+0.625}-0.175

Units_of_measurement.html

  1. Z = n × [ Z ] = n [ Z ] . Z=n\times[Z]=n[Z].
  2. Z = n i × [ Z ] i Z=n_{i}\times[Z]_{i}
  3. [ Z ] i [Z]_{i}
  4. [ Z ] j [Z]_{j}
  5. [ Z ] i = c i j × [ Z ] j [Z]_{i}=c_{ij}\times[Z]_{j}
  6. Z = n i × ( c i j × [ Z ] j ) = ( n i × c i j ) × [ Z ] j Z=n_{i}\times(c_{ij}\times[Z]_{j})=(n_{i}\times c_{ij})\times[Z]_{j}
  7. n i n_{i}
  8. c i j c_{ij}
  9. Z = n i × [ Z ] i × ( c i j × [ Z ] j / [ Z ] i ) Z=n_{i}\times[Z]_{i}\times(c_{ij}\times[Z]_{j}/[Z]_{i})
  10. [ Z ] i [Z]_{i}
  11. [ Z ] j [Z]_{j}
  12. 52.8 ft s = 52.8 ft s 1 mi 5280 ft 3600 s 1 h = 52.8 × 3600 5280 mi / h = 36 mi / h 52.8\,\frac{\mathrm{ft}}{\mathrm{s}}=52.8\,\frac{\mathrm{ft}}{\mathrm{s}}\frac% {1\,\mathrm{mi}}{5280\,\mathrm{ft}}\frac{3600\,\mathrm{s}}{1\,\mathrm{h}}=% \frac{52.8\times 3600}{5280}\,\mathrm{mi/h}=36\,\mathrm{mi/h}
  13. 9 L 100 km = 9 L 100 km 1000000 μ L 1 L 1 km 1000 m = 9 × 1000000 100 × 1000 μ L / m = 90 μ L / m \mathrm{\frac{9\,\rm{L}}{100\,\rm{km}}}=\mathrm{\frac{9\,\rm{L}}{100\,\rm{km}}% }\mathrm{\frac{1000000\,\rm{\mu L}}{1\,\rm{L}}}\mathrm{\frac{1\,\rm{km}}{1000% \,\rm{m}}}=\frac{9\times 1000000}{100\times 1000}\,\mathrm{\mu L/m}=90\,% \mathrm{\mu L/m}

Universal_approximation_theorem.html

  1. φ ( ) \varphi(\cdot)
  2. I m I_{m}
  3. [ 0 , 1 ] m [0,1]^{m}
  4. I m I_{m}
  5. C ( I m ) C(I_{m})
  6. f C ( I m ) f\in C(I_{m})
  7. ε > 0 \varepsilon>0
  8. N N
  9. v i , b i v_{i},b_{i}\in\mathbb{R}
  10. i = 1 , , N i=1,\cdots,N
  11. F ( x ) = i = 1 N v i φ ( w i T x + b i ) F(x)=\sum_{i=1}^{N}v_{i}\varphi\left(w_{i}^{T}x+b_{i}\right)
  12. f f
  13. f f
  14. φ \varphi
  15. | F ( x ) - f ( x ) | < ε |F(x)-f(x)|<\varepsilon
  16. x I m x\in I_{m}
  17. F ( x ) F(x)
  18. C ( I m ) C(I_{m})
  19. I m I_{m}
  20. m \mathbb{R}^{m}

Universal_Century_technology.html

  1. He 2 3 + H 1 2 He 2 4 + p {}^{3}_{2}\mathrm{He}+{}^{2}_{1}\mathrm{H}\to{}^{4}_{2}\mathrm{He}+\mathrm{p}

Universal_variable_formulation.html

  1. d s d t = 1 r \frac{ds}{dt}=\frac{1}{r}
  2. r = r ( t ) r=r(t)
  3. d 2 𝐫 d t 2 + μ 𝐫 r 3 = 𝟎 \frac{d^{2}\mathbf{r}}{dt^{2}}+\mu\frac{\mathbf{r}}{r^{3}}=\mathbf{0}
  4. d 2 𝐫 d s 2 + α 𝐫 = - 𝐏 \frac{d^{2}\mathbf{r}}{ds^{2}}+\alpha\ \mathbf{r}=-\mathbf{P}
  5. α \alpha
  6. α = μ a \alpha=\frac{\mu}{a}
  7. d 3 𝐫 d s 3 + α d 𝐫 d s = 𝟎 \frac{d^{3}\mathbf{r}}{ds^{3}}+\alpha\frac{d\mathbf{r}}{ds}=\mathbf{0}
  8. 1 , s c 1 ( α s 2 ) , s 2 c 2 ( α s 2 ) , 1,\ s\ c_{1}(\alpha s^{2}),\ s^{2}\ c_{2}(\alpha s^{2}),
  9. c k ( x ) \ c_{k}(x)
  10. t - t 0 = r 0 s c 1 ( α s 2 ) + r 0 d r 0 d t s 2 c 2 ( α s 2 ) + μ s 3 c 3 ( α s 2 ) t-t_{0}=r_{0}\ s\ c_{1}(\alpha s^{2})+r_{0}\frac{dr_{0}}{dt}\ s^{2}\ c_{2}(% \alpha s^{2})+\mu\ s^{3}\ c_{3}(\alpha s^{2})
  11. t t
  12. s s
  13. f ( s ) \displaystyle f(s)
  14. t t
  15. 𝐫 = 𝐫 0 f ( s ) + 𝐯 0 g ( s ) \mathbf{r}=\mathbf{r}_{0}\ f(s)+\mathbf{v}_{0}\ g(s)
  16. t t
  17. f ˙ ( s ) \dot{f}(s)
  18. g ˙ ( s ) \dot{g}(s)
  19. 𝐯 = 𝐫 0 f ˙ ( s ) + 𝐯 0 g ˙ ( s ) \mathbf{v}=\mathbf{r}_{0}\ \dot{f}(s)+\mathbf{v}_{0}\ \dot{g}(s)
  20. 𝐫 \mathbf{r}
  21. 𝐯 \mathbf{v}
  22. t t
  23. 𝐫 0 \mathbf{r}_{0}
  24. 𝐯 0 \mathbf{v}_{0}
  25. t 0 t_{0}

Ursell_function.html

  1. E ( exp ( z X ) ) = n s n z n n ! = exp ( n u n z n n ! ) \operatorname{E}(\exp(zX))=\sum_{n}s_{n}\frac{z^{n}}{n!}=\exp\left(\sum_{n}u_{% n}\frac{z^{n}}{n!}\right)
  2. u n ( X 1 , , X n ) = z 1 z n log E ( exp z i X i ) | z i = 0 u_{n}(X_{1},\ldots,X_{n})=\frac{\partial}{\partial z_{1}}\cdots\frac{\partial}% {\partial z_{n}}\log E(\exp\sum z_{i}X_{i})\big|_{z_{i}=0}
  3. u 1 ( X 1 ) = E ( X 1 ) u_{1}(X_{1})=E(X_{1})
  4. u 2 ( X 1 , X 2 ) = E ( X 1 X 2 ) - E ( X 1 ) E ( X 2 ) u_{2}(X_{1},X_{2})=E(X_{1}X_{2})-E(X_{1})E(X_{2})
  5. u 3 ( X 1 , X 2 , X 3 ) = E ( X 1 X 2 X 3 ) - E ( X 1 ) E ( X 2 X 3 ) - E ( X 2 ) E ( X 3 X 1 ) - E ( X 3 ) E ( X 1 X 2 ) + 2 E ( X 1 ) E ( X 2 ) E ( X 3 ) u_{3}(X_{1},X_{2},X_{3})=E(X_{1}X_{2}X_{3})-E(X_{1})E(X_{2}X_{3})-E(X_{2})E(X_% {3}X_{1})-E(X_{3})E(X_{1}X_{2})+2E(X_{1})E(X_{2})E(X_{3})

Ursell_number.html

  1. U = H h ( λ h ) 2 = H λ 2 h 3 , U\,=\,\frac{H}{h}\left(\frac{\lambda}{h}\right)^{2}\,=\,\frac{H\,\lambda^{2}}{% h^{3}},

Urysohn_universal_space.html

  1. ( X , d ) , ( X , d ) (X,d),(X^{\prime},d^{\prime})
  2. ( x n ) n , ( x n ) n (x_{n})_{n},(x^{\prime}_{n})_{n}
  3. ϕ n : X X \phi_{n}:X\to X^{\prime}
  4. { x k : k < n } \{x_{k}:k<n\}
  5. { x k : k < n } \{x^{\prime}_{k}:k<n\}
  6. ϕ : X X \phi:X\to X^{\prime}

V-Cube_6.html

  1. 8 ! × 3 7 × 24 ! 6 4 ! 24 × 24 1.57 × 10 116 \frac{8!\times 3^{7}\times 24!^{6}}{4!^{24}\times 24}\approx 1.57\times 10^{116}

V-Cube_7.html

  1. 8 ! × 3 7 × 12 ! × 2 10 × 24 ! 8 4 ! 36 1.95 × 10 160 \frac{8!\times 3^{7}\times 12!\times 2^{10}\times 24!^{8}}{4!^{36}}\approx 1.9% 5\times 10^{160}

Vacuum_airship.html

  1. R R
  2. P P
  3. π R 2 P \pi R^{2}P
  4. σ = π R 2 P / 2 π R h = R P / 2 h \sigma=\pi R^{2}P/2\pi Rh=RP/2h
  5. h h
  6. h / R = ρ a / ( 3 ρ s ) h/R=\rho_{a}/(3\rho_{s})
  7. ρ a \rho_{a}
  8. ρ s \rho_{s}
  9. σ = ( 3 / 2 ) ( ρ s / ρ a ) P \sigma=(3/2)(\rho_{s}/\rho_{a})P
  10. 3.2 10 8 3.2\cdot 10^{8}
  11. P c r = 2 E h 2 3 ( 1 - μ 2 ) 1 R 2 P_{cr}=\frac{2Eh^{2}}{\sqrt{3(1-\mu^{2})}}\frac{1}{R^{2}}
  12. E E
  13. μ \mu
  14. E / ρ s 2 = 9 P c r 3 ( 1 - μ 2 ) 2 ρ a 2 E/\rho_{s}^{2}=\frac{9P_{cr}\sqrt{3(1-\mu^{2})}}{2\rho_{a}^{2}}
  15. 4.5 10 5 k g - 1 m 5 s - 2 4.5\cdot 10^{5}kg^{-1}m^{5}s^{-2}
  16. E / ρ s 2 1 10 5 E/\rho_{s}^{2}\approx 1\cdot 10^{5}

Vacuum_Rabi_oscillation.html

  1. ω = 2 f ( 𝐑 ) 𝐩 ϵ . \omega=\frac{2}{\hbar}\mathcal{E}f(\mathbf{R})\langle\mathbf{p\cdot\epsilon}% \rangle\ .
  2. \mathcal{E}

Validity.html

  1. and \and
  2. $\or$
  3. \top
  4. \leftrightarrow
  5. \rightarrow
  6. \leftarrow
  7. \bot
  8. \oplus
  9. \nrightarrow
  10. \nleftarrow
  11. ¬ \neg
  12. \uparrow
  13. \downarrow
  14. 𝒬 \mathcal{Q}
  15. 𝒬 \mathcal{Q}

Valuation_(logic).html

  1. v v
  2. { t , f } \{t,f\}
  3. v ( ϕ ) = [ [ ϕ ] ] v v(\phi)=[[\phi]]_{v}
  4. ϕ \phi

Valuation_(measure_theory).html

  1. ( X , 𝒯 ) \scriptstyle(X,\mathcal{T})
  2. v : 𝒯 + { + } v:\mathcal{T}\rightarrow\mathbb{R}^{+}\cup\{+\infty\}
  3. v ( ) = 0 Strictness property v ( U ) v ( V ) if U V U , V 𝒯 Monotonicity property v ( U V ) + v ( U V ) = v ( U ) + v ( V ) U , V 𝒯 Modularity property \begin{array}[]{lll}v(\varnothing)=0&&\scriptstyle{\,\text{Strictness property% }}\\ v(U)\leq v(V)&\mbox{if}~{}~{}U\subseteq V\quad U,V\in\mathcal{T}&\scriptstyle{% \,\text{Monotonicity property}}\\ v(U\cup V)+v(U\cap V)=v(U)+v(V)&\forall U,V\in\mathcal{T}&\scriptstyle{\,\text% {Modularity property}}\end{array}
  4. { U i } i I \scriptstyle\{U_{i}\}_{i\in I}
  5. i i
  6. j j
  7. I I
  8. k k
  9. U i U k \scriptstyle U_{i}\subseteq U_{k}
  10. U j U k \scriptstyle U_{j}\subseteq U_{k}
  11. v ( i I U i ) = sup i I v ( U i ) . v\left(\bigcup_{i\in I}U_{i}\right)=\sup_{i\in I}v(U_{i}).
  12. v ( U ) = i = 1 n a i δ x i ( U ) U 𝒯 v(U)=\sum_{i=1}^{n}a_{i}\delta_{x_{i}}(U)\quad\forall U\in\mathcal{T}
  13. a i a_{i}
  14. i i
  15. i i
  16. j j
  17. I I
  18. k k
  19. v i ( U ) v k ( U ) \scriptstyle v_{i}(U)\leq v_{k}(U)\!
  20. v j ( U ) v k ( U ) \scriptstyle v_{j}(U)\subseteq v_{k}(U)\!
  21. v ¯ ( U ) = sup i I v i ( U ) U 𝒯 . \bar{v}(U)=\sup_{i\in I}v_{i}(U)\quad\forall U\in\mathcal{T}.\,
  22. ( X , 𝒯 ) \scriptstyle(X,\mathcal{T})
  23. x x
  24. X X
  25. δ x ( U ) = { 0 if x U 1 if x U U 𝒯 \delta_{x}(U)=\begin{cases}0&\mbox{if}~{}~{}x\notin U\\ 1&\mbox{if}~{}~{}x\in U\end{cases}\quad\forall U\in\mathcal{T}

Valuative_criterion.html

  1. Hom Y ( Y , X ) Hom Y ( Spec K , X ) \,\text{Hom}_{Y}(Y^{\prime},X)\to\,\text{Hom}_{Y}(\,\text{Spec}K,X)

Van_der_Waerden_test.html

  1. A i j = Φ - 1 ( R ( X i j ) N + 1 ) A_{ij}=\Phi^{-1}\left(\frac{R(X_{ij})}{N+1}\right)
  2. A ¯ j = 1 n j i = 1 n j A i j j = 1 , 2 , , k \bar{A}_{j}=\frac{1}{n_{j}}\sum_{i=1}^{n_{j}}A_{ij}\quad j=1,2,\ldots,k
  3. s 2 = 1 N - 1 j = 1 k i = 1 n j A i j 2 s^{2}=\frac{1}{N-1}\sum_{j=1}^{k}\sum_{i=1}^{n_{j}}A_{ij}^{2}
  4. T 1 = 1 s 2 j = 1 k n j A ¯ j 2 T_{1}=\frac{1}{s^{2}}\sum_{j=1}^{k}n_{j}\bar{A}_{j}^{2}
  5. T 1 > χ α , k - 1 2 T_{1}>\chi_{\alpha,k-1}^{2}
  6. | A ¯ j 1 - A ¯ j 2 | > s t 1 - α / 2 N - 1 - T 1 N - k 1 n j 1 + 1 n j 2 \left|\bar{A}_{j_{1}}-\bar{A}_{j_{2}}\right|>s\,t_{1-\alpha/2}\sqrt{\frac{N-1-% T_{1}}{N-k}}\sqrt{\frac{1}{n_{j_{1}}}+\frac{1}{n_{j_{2}}}}

Van_Kampen_diagram.html

  1. G = A | R G=\langle A|R\,\rangle
  2. 𝒟 \mathcal{D}\,
  3. 𝒟 2 \mathcal{D}\subseteq\mathbb{R}^{2}\,
  4. 𝒟 \mathcal{D}\,
  5. 𝒟 \mathcal{D}\,
  6. 𝒟 2 \mathcal{D}\subseteq\mathbb{R}^{2}\,
  7. 𝒟 \mathcal{D}\,
  8. 𝒟 \mathcal{D}\,
  9. 2 \mathbb{R}^{2}\,
  10. 𝒟 \mathcal{D}\,
  11. 𝒟 \mathcal{D}\,
  12. 𝒟 \mathcal{D}\,
  13. 𝒟 \partial\mathcal{D}\,
  14. 𝒟 \mathcal{D}\,
  15. 𝒟 \mathcal{D}\,
  16. 𝒟 \mathcal{D}\,
  17. 𝒟 \mathcal{D}\,
  18. 𝒟 \mathcal{D}\,
  19. 𝒟 \mathcal{D}\,
  20. 𝒟 \mathcal{D}\,
  21. 𝒟 \mathcal{D}\,
  22. 𝒟 \mathcal{D}\,
  23. 𝒟 \mathcal{D}\,
  24. 𝒟 \mathcal{D}\,
  25. 𝒟 \mathcal{D}\,
  26. 𝒟 \mathcal{D}\,
  27. 𝒟 \mathcal{D}\,
  28. Area ( 𝒟 ) {\rm Area}(\mathcal{D})\,
  29. G = a , b | a b a - 1 b - 1 . G=\langle a,b|aba^{-1}b^{-1}\rangle.
  30. w = b - 1 b 3 a - 1 b - 2 a b - 1 b a - 1 a b - 1 b a - 1 a . w=b^{-1}b^{3}a^{-1}b^{-2}ab^{-1}ba^{-1}ab^{-1}ba^{-1}a.
  31. 𝒟 \mathcal{D}\,
  32. 𝒟 \mathcal{D}\,
  33. w = u 1 s 1 u 1 - 1 u n s n u n - 1 in F ( A ) , w=u_{1}s_{1}u_{1}^{-1}\cdots u_{n}s_{n}u_{n}^{-1}\,\text{ in }F(A),
  34. 𝒟 \mathcal{D}\,
  35. 𝒟 \mathcal{D}\,
  36. 𝒟 \mathcal{D}^{\prime}\,
  37. w = u s u - 1 w , w=usu^{-1}w^{\prime},\,
  38. 𝒟 \mathcal{D}^{\prime}\,
  39. 𝒟 \mathcal{D}\,
  40. 𝒟 0 \mathcal{D}_{0}\,
  41. 𝒟 0 \mathcal{D}_{0}\,
  42. 𝒟 0 \mathcal{D}_{0}\,
  43. 𝒟 0 , 𝒟 1 , 𝒟 2 , \mathcal{D}_{0},\mathcal{D}_{1},\mathcal{D}_{2},\dots\,
  44. 𝒟 k \mathcal{D}_{k}\,
  45. 𝒟 k \mathcal{D}_{k}\,
  46. 𝒟 \mathcal{D}\,
  47. 𝒟 \mathcal{D}\,
  48. Area ( w ) f ( | w | ) , {\rm Area}(w)\leq f(|w|),
  49. Dehn ( n ) = max { Area ( w ) : w = 1 in G , | w | n , w freely reduced . } {\rm Dehn}(n)=\max\{{\rm Area}(w):w=1\,\text{ in }G,|w|\leq n,w\,\text{ freely% reduced}.\}
  50. 𝒟 \mathcal{D}\,
  51. Area ( 𝒟 ) = Area ( w ) . {\rm Area}(\mathcal{D})={\rm Area}(w).

Vandermonde_polynomial.html

  1. X 1 , , X n X_{1},\dots,X_{n}
  2. V n = 1 i < j n ( X j - X i ) . V_{n}=\prod_{1\leq i<j\leq n}(X_{j}-X_{i}).
  3. ( X i - X j ) (X_{i}-X_{j})
  4. ( n 2 ) {\left({{n}\atop{2}}\right)}
  5. X i X_{i}
  6. V n = - V n , V_{n}=-V_{n},
  7. V n = 0 V_{n}=0
  8. ( X i - X j ) (X_{i}-X_{j})
  9. i j i\neq j
  10. Δ = V n 2 \Delta=V_{n}^{2}
  11. ( - 1 ) 2 = 1 (-1)^{2}=1
  12. Λ n \Lambda_{n}
  13. Λ n [ V n ] / V n 2 - Δ \Lambda_{n}[V_{n}]/\langle V_{n}^{2}-\Delta\rangle
  14. V n = a n - 1 1 i < j n ( X j - X i ) , V_{n}=a^{n-1}\prod_{1\leq i<j\leq n}(X_{j}-X_{i}),
  15. S U ( n ) SU(n)

Vapor–liquid–solid_method.html

  1. R = r o sin ( β o ) , R=\frac{r_{\mathrm{o}}}{\sin(\beta_{\mathrm{o}})},
  2. σ 1 cos ( β o ) = σ s - σ ls - τ r o \sigma_{\mathrm{1}}\cos(\beta_{\mathrm{o}})=\sigma_{\mathrm{s}}-\sigma_{% \mathrm{ls}}-\frac{\tau}{r_{\mathrm{o}}}
  3. σ 1 cos ( β o ) = σ s cos ( α ) - σ ls - τ r o \sigma_{\mathrm{1}}\cos(\beta_{\mathrm{o}})=\sigma_{\mathrm{s}}\cos(\alpha)-% \sigma_{\mathrm{ls}}-\frac{\tau}{r_{\mathrm{o}}}
  4. R min = 2 V l R T l n ( s ) σ l v R_{\mathrm{min}}=\frac{2V_{l}}{RTln(s)}\sigma_{lv}\,
  5. Δ μ \Delta\mu
  6. Δ μ = Δ μ o - 4 α Ω d \Delta\mu=\Delta\mu_{\mathrm{o}}-\frac{4\alpha\Omega}{d}
  7. d d\rightarrow\infty
  8. Ω \Omega
  9. α \alpha

Variable_structure_system.html

  1. 𝐱 ˙ = φ ( 𝐱 , t ) \dot{\mathbf{x}}=\varphi(\mathbf{x},t)
  2. 𝐱 [ x 1 , x 2 , , x n ] T n \mathbf{x}\triangleq[x_{1},x_{2},\ldots,x_{n}]^{\operatorname{T}}\in\mathbb{R}% ^{n}
  3. t t\in\mathbb{R}
  4. φ ( 𝐱 , t ) [ φ 1 ( 𝐱 , t ) , φ 2 ( 𝐱 , t ) , , φ n ( 𝐱 , t ) ] T : n + 1 n \varphi(\mathbf{x},t)\triangleq[\varphi_{1}(\mathbf{x},t),\varphi_{2}(\mathbf{% x},t),\ldots,\varphi_{n}(\mathbf{x},t)]^{\operatorname{T}}:\mathbb{R}^{n+1}% \mapsto\mathbb{R}^{n}

Variance_gamma_process.html

  1. W ( t ) W(t)
  2. θ t \theta t
  3. Γ ( t ; 1 , ν ) \Gamma(t;1,\nu)
  4. Γ ( t ; γ = 1 / ν , λ = 1 / ν ) \Gamma(t;\gamma=1/\nu,\lambda=1/\nu)
  5. X V G ( t ; σ , ν , θ ) := θ Γ ( t ; 1 , ν ) + σ W ( Γ ( t ; 1 , ν ) ) . X^{VG}(t;\sigma,\nu,\theta)\;:=\;\theta\,\Gamma(t;1,\nu)+\sigma\,W(\Gamma(t;1,% \nu))\quad.
  6. X V G ( t ; σ , ν , θ ) := Γ ( t ; μ p , μ p 2 ν ) - Γ ( t ; μ q , μ q 2 ν ) X^{VG}(t;\sigma,\nu,\theta)\;:=\;\Gamma(t;\mu_{p},\mu_{p}^{2}\,\nu)-\Gamma(t;% \mu_{q},\mu_{q}^{2}\,\nu)
  7. μ p := 1 2 θ 2 + 2 σ 2 ν + θ 2 and μ q := 1 2 θ 2 + 2 σ 2 ν - θ 2 . \mu_{p}:=\frac{1}{2}\sqrt{\theta^{2}+\frac{2\sigma^{2}}{\nu}}+\frac{\theta}{2}% \quad\quad\,\text{and}\quad\quad\mu_{q}:=\frac{1}{2}\sqrt{\theta^{2}+\frac{2% \sigma^{2}}{\nu}}-\frac{\theta}{2}\quad.
  8. σ \sigma
  9. ν \nu
  10. E [ X ( t ) ] = θ t E[X(t)]=\theta t
  11. V a r [ X ( t ) ] = ( θ 2 ν + σ 2 ) t Var[X(t)]=(\theta^{2}\nu+\sigma^{2})t
  12. E [ ( X ( t ) - E [ X ( t ) ] ) 3 ] = ( 2 θ 3 ν 2 + 3 σ 2 θ ν ) t E[(X(t)-E[X(t)])^{3}]=(2\theta^{3}\nu^{2}+3\sigma^{2}\theta\nu)t
  13. E [ ( X ( t ) - E [ X ( t ) ] ) 4 ] = ( 3 σ 4 ν + 12 σ 2 θ 2 ν 2 + 6 θ 4 ν 3 ) t + ( 3 σ 4 + 6 σ 2 θ 2 ν + 3 θ 4 ν 2 ) t 2 E[(X(t)-E[X(t)])^{4}]=(3\sigma^{4}\nu+12\sigma^{2}\theta^{2}\nu^{2}+6\theta^{4% }\nu^{3})t+(3\sigma^{4}+6\sigma^{2}\theta^{2}\nu+3\theta^{4}\nu^{2})t^{2}
  14. θ , σ , ν \theta,\sigma,\nu
  15. Δ t 1 , Δ t N \Delta t_{1},\dots\Delta t_{N}
  16. i = 1 N Δ t i = T . \sum_{i=1}^{N}\Delta t_{i}=T.
  17. Δ G i Γ ( Δ t i / ν , ν ) \Delta\,G_{i}\,\sim\Gamma(\Delta t_{i}/\nu,\nu)
  18. Z i 𝒩 ( 0 , 1 ) Z_{i}\sim\mathcal{N}(0,1)
  19. X ( t i ) = X ( t i - 1 ) + θ Δ G i + σ Δ G i Z i . X(t_{i})=X(t_{i-1})+\theta\Delta G_{i}+\sigma\sqrt{\Delta G_{i}}Z_{i}.
  20. X V G ( t ; σ , ν , θ ) = Γ ( t ; μ p , μ p 2 ν ) - Γ ( t ; μ q , μ q 2 ν ) X^{VG}(t;\sigma,\nu,\theta)\;=\;\Gamma(t;\mu_{p},\mu_{p}^{2}\,\nu)-\Gamma(t;% \mu_{q},\mu_{q}^{2}\,\nu)
  21. μ p , μ q , ν \mu_{p},\mu_{q},\nu
  22. θ , σ , ν , μ p , μ q \theta,\sigma,\nu,\mu_{p},\mu_{q}
  23. Δ t 1 , Δ t N \Delta t_{1},\dots\Delta t_{N}
  24. i = 1 N Δ t i = T . \sum_{i=1}^{N}\Delta t_{i}=T.
  25. γ i - Γ ( Δ t i / ν , ν μ q ) , γ i + Γ ( Δ t i / ν , ν μ p ) , \gamma_{i}^{-}\,\sim\,\Gamma(\Delta t_{i}/\nu,\nu\mu_{q}),\quad\gamma_{i}^{+}% \,\sim\,\Gamma(\Delta t_{i}/\nu,\nu\mu_{p}),
  26. X ( t i ) = X ( t i - 1 ) + Γ i + ( t ) - Γ i - ( t ) . X(t_{i})=X(t_{i-1})+\Gamma^{+}_{i}(t)-\Gamma^{-}_{i}(t).
  27. 1 ν \frac{1}{\nu}

Variational_integrator.html

  1. L ( t , q , v ) = 1 2 m v 2 - V ( q ) L(t,q,v)=\frac{1}{2}mv^{2}-V(q)
  2. m m
  3. V V
  4. L d ( t 0 , t 1 , q 0 , q 1 ) = t 1 - t 0 2 [ L ( t 0 , q 0 , q 1 - q 0 t 1 - t 0 ) + L ( t 1 , q 1 , q 1 - q 0 t 1 - t 0 ) ] t 0 t 1 d t L ( t , q ( t ) , v ( t ) ) L_{d}\left(t_{0},t_{1},q_{0},q_{1}\right)=\frac{t_{1}-t_{0}}{2}\left[L\left(t_% {0},q_{0},\frac{q_{1}-q_{0}}{t_{1}-t_{0}}\right)+L\left(t_{1},q_{1},\frac{q_{1% }-q_{0}}{t_{1}-t_{0}}\right)\right]\approx\int_{t_{0}}^{t_{1}}dt\,L(t,q(t),v(t))
  5. q ( t ) q 1 - q 0 t 1 - t 0 ( t - t 0 ) + q 0 q(t)\approx\frac{q_{1}-q_{0}}{t_{1}-t_{0}}\left(t-t_{0}\right)+q_{0}
  6. t 0 t_{0}
  7. t 1 t_{1}
  8. v ( q 1 - q 0 ) / ( t 1 - t 0 ) v\approx\left(q_{1}-q_{0}\right)/\left(t_{1}-t_{0}\right)
  9. L d ( t 0 , t 1 , q 0 , q 1 ) = t 0 t 1 d t L ( t , q ( t ) , v ( t ) ) + 𝒪 ( t 1 - t 0 ) 3 L_{d}\left(t_{0},t_{1},q_{0},q_{1}\right)=\int_{t_{0}}^{t_{1}}dt\,L(t,q(t),v(t% ))+\mathcal{O}\left(t_{1}-t_{0}\right)^{3}
  10. S d = L d ( t 0 , t 1 , q 0 , q 1 ) + L d ( t 1 , t 2 , q 1 , q 2 ) + S_{d}=L_{d}\left(t_{0},t_{1},q_{0},q_{1}\right)+L_{d}\left(t_{1},t_{2},q_{1},q% _{2}\right)+\ldots
  11. q 1 q_{1}
  12. S d q 1 = 0 = q 1 L d ( t 0 , t 1 , q 0 , q 1 ) + q 1 L d ( t 1 , t 2 , q 1 , q 2 ) \frac{\partial S_{d}}{\partial q_{1}}=0=\frac{\partial}{\partial q_{1}}L_{d}% \left(t_{0},t_{1},q_{0},q_{1}\right)+\frac{\partial}{\partial q_{1}}L_{d}\left% (t_{1},t_{2},q_{1},q_{2}\right)
  13. ( q 0 , q 1 ) (q_{0},q_{1})
  14. ( t 0 , t 1 , t 2 ) (t_{0},t_{1},t_{2})
  15. q 2 q_{2}
  16. q 2 = q 1 + t 2 - t 1 t 1 - t 0 ( q 1 - q 0 ) - ( t 2 - t 0 ) ( t 2 - t 1 ) 2 m d d q 1 V ( q 1 ) q_{2}=q_{1}+\frac{t_{2}-t_{1}}{t_{1}-t_{0}}\left(q_{1}-q_{0}\right)-\frac{% \left(t_{2}-t_{0}\right)\left(t_{2}-t_{1}\right)}{2m}\frac{d}{dq_{1}}V\left(q_% {1}\right)
  17. p 0 - q 0 L d ( t 0 , t 1 , q 0 , q 1 ) p_{0}\equiv-\frac{\partial}{\partial q_{0}}L_{d}\left(t_{0},t_{1},q_{0},q_{1}\right)
  18. p 1 q 1 L d ( t 0 , t 1 , q 0 , q 1 ) p_{1}\equiv\frac{\partial}{\partial q_{1}}L_{d}\left(t_{0},t_{1},q_{0},q_{1}\right)
  19. ( q 0 , p 0 ) (q_{0},p_{0})
  20. q 1 q_{1}
  21. p 1 p_{1}
  22. q 1 = q 0 + t 1 - t 0 m p 0 - ( t 1 - t 0 ) 2 2 m d d q 0 V ( q 0 ) q_{1}=q_{0}+\frac{t_{1}-t_{0}}{m}p_{0}-\frac{\left(t_{1}-t_{0}\right)^{2}}{2m}% \frac{d}{dq_{0}}V\left(q_{0}\right)
  23. p 1 = m q 1 - q 0 t 1 - t 0 - t 1 - t 0 2 d d q 1 V ( q 1 ) p_{1}=m\frac{q_{1}-q_{0}}{t_{1}-t_{0}}-\frac{t_{1}-t_{0}}{2}\frac{d}{dq_{1}}V% \left(q_{1}\right)
  24. q 2 q_{2}

Veblen_function.html

  1. α < β , \alpha<\beta\,,
  2. φ α ( φ β ( γ ) ) = φ β ( γ ) . \varphi_{\alpha}(\varphi_{\beta}(\gamma))=\varphi_{\beta}(\gamma)\,.
  3. φ α ( β ) < φ γ ( δ ) \varphi_{\alpha}(\beta)<\varphi_{\gamma}(\delta)\,
  4. α = γ \alpha=\gamma\,
  5. β < δ \beta<\delta\,
  6. α < γ \alpha<\gamma\,
  7. β < φ γ ( δ ) \beta<\varphi_{\gamma}(\delta)\,
  8. α > γ \alpha>\gamma\,
  9. φ α ( β ) < δ \varphi_{\alpha}(\beta)<\delta\,
  10. α = φ β 1 ( γ 1 ) + φ β 2 ( γ 2 ) + + φ β k ( γ k ) \alpha=\varphi_{\beta_{1}}(\gamma_{1})+\varphi_{\beta_{2}}(\gamma_{2})+\cdots+% \varphi_{\beta_{k}}(\gamma_{k})
  11. φ β m ( γ m ) φ β m + 1 ( γ m + 1 ) , \varphi_{\beta_{m}}(\gamma_{m})\geq\varphi_{\beta_{m+1}}(\gamma_{m+1})\,,
  12. γ m < φ β m ( γ m ) . \gamma_{m}<\varphi_{\beta_{m}}(\gamma_{m})\,.
  13. α [ n ] = φ β 1 ( γ 1 ) + + φ β k - 1 ( γ k - 1 ) + ( φ β k ( γ k ) [ n ] ) . \alpha[n]=\varphi_{\beta_{1}}(\gamma_{1})+\cdots+\varphi_{\beta_{k-1}}(\gamma_% {k-1})+(\varphi_{\beta_{k}}(\gamma_{k})[n])\,.
  14. γ < φ β ( γ ) , \gamma<\varphi_{\beta}(\gamma)\,,
  15. φ β ( γ ) [ n ] = φ β ( γ [ n ] ) . \varphi_{\beta}(\gamma)[n]=\varphi_{\beta}(\gamma[n])\,.
  16. φ 0 ( 0 ) \varphi_{0}(0)
  17. φ 0 ( γ + 1 ) = ω γ + 1 = ω γ ω , \varphi_{0}(\gamma+1)=\omega^{\gamma+1}=\omega^{\gamma}\cdot\omega\,,
  18. φ 0 ( γ + 1 ) [ n ] = φ 0 ( γ ) n = ω γ n . \varphi_{0}(\gamma+1)[n]=\varphi_{0}(\gamma)\cdot n=\omega^{\gamma}\cdot n\,.
  19. φ β + 1 ( 0 ) , \varphi_{\beta+1}(0)\,,
  20. φ β + 1 ( 0 ) [ 0 ] = 0 \varphi_{\beta+1}(0)[0]=0\,
  21. φ β + 1 ( 0 ) [ n + 1 ] = φ β ( φ β + 1 ( 0 ) [ n ] ) , \varphi_{\beta+1}(0)[n+1]=\varphi_{\beta}(\varphi_{\beta+1}(0)[n])\,,
  22. φ β ( 0 ) \varphi_{\beta}(0)
  23. φ β ( φ β ( 0 ) ) \varphi_{\beta}(\varphi_{\beta}(0))
  24. φ β + 1 ( γ + 1 ) \varphi_{\beta+1}(\gamma+1)
  25. φ β + 1 ( γ + 1 ) [ 0 ] = φ β + 1 ( γ ) + 1 \varphi_{\beta+1}(\gamma+1)[0]=\varphi_{\beta+1}(\gamma)+1\,
  26. φ β + 1 ( γ + 1 ) [ n + 1 ] = φ β ( φ β + 1 ( γ + 1 ) [ n ] ) . \varphi_{\beta+1}(\gamma+1)[n+1]=\varphi_{\beta}(\varphi_{\beta+1}(\gamma+1)[n% ])\,.
  27. β < φ β ( 0 ) \beta<\varphi_{\beta}(0)
  28. φ β ( 0 ) [ n ] = φ β [ n ] ( 0 ) . \varphi_{\beta}(0)[n]=\varphi_{\beta[n]}(0)\,.
  29. φ β ( γ + 1 ) \varphi_{\beta}(\gamma+1)
  30. φ β ( γ + 1 ) [ n ] = φ β [ n ] ( φ β ( γ ) + 1 ) . \varphi_{\beta}(\gamma+1)[n]=\varphi_{\beta[n]}(\varphi_{\beta}(\gamma)+1)\,.
  31. φ \varphi
  32. Γ 0 [ 0 ] = 0 \Gamma_{0}[0]=0\,
  33. Γ 0 [ n + 1 ] = φ Γ 0 [ n ] ( 0 ) . \Gamma_{0}[n+1]=\varphi_{\Gamma_{0}[n]}(0)\,.
  34. Γ β + 1 [ 0 ] = Γ β + 1 \Gamma_{\beta+1}[0]=\Gamma_{\beta}+1\,
  35. Γ β + 1 [ n + 1 ] = φ Γ β + 1 [ n ] ( 0 ) . \Gamma_{\beta+1}[n+1]=\varphi_{\Gamma_{\beta+1}[n]}(0)\,.
  36. β < Γ β \beta<\Gamma_{\beta}\,
  37. Γ β [ n ] = Γ β [ n ] . \Gamma_{\beta}[n]=\Gamma_{\beta[n]}\,.

Vector_control_(motor).html

  1. α \alpha
  2. β \beta
  3. α \alpha
  4. β \beta
  5. α \alpha
  6. β \beta
  7. α \alpha
  8. β \beta
  9. α \alpha
  10. β \beta
  11. α \alpha
  12. β \beta
  13. α \alpha
  14. β \beta
  15. α β γ \alpha\beta\gamma

Vector_meson_dominance.html

  1. ρ , ω \rho,\omega
  2. ϕ \phi
  3. ω \omega

Vector_space_model.html

  1. d j = ( w 1 , j , w 2 , j , , w t , j ) d_{j}=(w_{1,j},w_{2,j},\ldots,w_{t,j})
  2. q = ( w 1 , q , w 2 , q , , w n , q ) q=(w_{1,q},w_{2,q},\ldots,w_{n,q})
  3. cos θ = 𝐝 𝟐 𝐪 𝐝 𝟐 𝐪 \cos{\theta}=\frac{\mathbf{d_{2}}\cdot\mathbf{q}}{\left\|\mathbf{d_{2}}\right% \|\left\|\mathbf{q}\right\|}
  4. 𝐝 𝟐 𝐪 \mathbf{d_{2}}\cdot\mathbf{q}
  5. 𝐝 𝟐 \left\|\mathbf{d_{2}}\right\|
  6. 𝐪 \left\|\mathbf{q}\right\|
  7. 𝐪 = i = 1 n q i 2 \left\|\mathbf{q}\right\|=\sqrt{\sum_{i=1}^{n}q_{i}^{2}}
  8. 𝐯 d = [ w 1 , d , w 2 , d , , w N , d ] T \mathbf{v}_{d}=[w_{1,d},w_{2,d},\ldots,w_{N,d}]^{T}
  9. w t , d = tf t , d log | D | | { d D | t d } | w_{t,d}=\mathrm{tf}_{t,d}\cdot\log{\frac{|D|}{|\{d^{\prime}\in D\,|\,t\in d^{% \prime}\}|}}
  10. tf t , d \mathrm{tf}_{t,d}
  11. log | D | | { d D | t d } | \log{\frac{|D|}{|\{d^{\prime}\in D\,|\,t\in d^{\prime}\}|}}
  12. | D | |D|
  13. | { d D | t d } | |\{d^{\prime}\in D\,|\,t\in d^{\prime}\}|
  14. sim ( d j , q ) = 𝐝 𝐣 𝐪 𝐝 𝐣 𝐪 = i = 1 N w i , j w i , q i = 1 N w i , j 2 i = 1 N w i , q 2 \mathrm{sim}(d_{j},q)=\frac{\mathbf{d_{j}}\cdot\mathbf{q}}{\left\|\mathbf{d_{j% }}\right\|\left\|\mathbf{q}\right\|}=\frac{\sum_{i=1}^{N}w_{i,j}w_{i,q}}{\sqrt% {\sum_{i=1}^{N}w_{i,j}^{2}}\sqrt{\sum_{i=1}^{N}w_{i,q}^{2}}}

Vector_spaces_without_fields.html

  1. δ δ
  2. Δ Δ
  3. V V
  4. V V
  5. B V B\subseteq V
  6. δ δ
  7. δ ( b ) \delta(b)
  8. b b
  9. B B
  10. Δ Δ
  11. δ δ
  12. δ ( v + w ) = δ ( v ) + δ ( w ) \delta(v+w)=\delta(v)+\delta(w)
  13. δ ( γ ( v ) ) = γ ( δ ( v ) ) \delta(\gamma(v))=\gamma(\delta(v))
  14. γ Δ \gamma\in\Delta
  15. v , w V v,w\in V
  16. a F a\in F
  17. δ Δ \delta\in\Delta
  18. δ ( v ) = a v \delta(v)=av

Vector_spherical_harmonics.html

  1. Y l m ( θ , φ ) Y_{lm}(\theta,\varphi)
  2. 𝐘 l m = Y l m 𝐫 ^ \mathbf{Y}_{lm}=Y_{lm}\hat{\mathbf{r}}
  3. 𝚿 l m = r Y l m \mathbf{\Psi}_{lm}=r\nabla Y_{lm}
  4. 𝚽 l m = 𝐫 × Y l m \mathbf{\Phi}_{lm}=\vec{\mathbf{r}}\times\nabla Y_{lm}
  5. 𝐫 ^ \hat{\mathbf{r}}
  6. 𝐫 \vec{\mathbf{r}}
  7. r r
  8. θ \theta
  9. ϕ \phi
  10. 𝐄 = l = 0 m = - l l ( E l m r ( r ) 𝐘 l m + E l m ( 1 ) ( r ) 𝚿 l m + E l m ( 2 ) ( r ) 𝚽 l m ) \mathbf{E}=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\left(E^{r}_{lm}(r)\mathbf{Y}_{lm% }+E^{(1)}_{lm}(r)\mathbf{\Psi}_{lm}+E^{(2)}_{lm}(r)\mathbf{\Phi}_{lm}\right)
  11. E l m r E^{r}_{lm}
  12. E l m ( 1 ) E^{(1)}_{lm}
  13. E l m ( 2 ) E^{(2)}_{lm}
  14. 𝐘 l , - m = ( - 1 ) m 𝐘 l m * 𝚿 l , - m = ( - 1 ) m 𝚿 l m * 𝚽 l , - m = ( - 1 ) m 𝚽 l m * \mathbf{Y}_{l,-m}=(-1)^{m}\mathbf{Y}^{*}_{lm}\qquad\mathbf{\Psi}_{l,-m}=(-1)^{% m}\mathbf{\Psi}^{*}_{lm}\qquad\mathbf{\Phi}_{l,-m}=(-1)^{m}\mathbf{\Phi}^{*}_{lm}
  15. 𝐘 l m 𝚿 l m = 0 𝐘 l m 𝚽 l m = 0 𝚿 l m 𝚽 l m = 0 \mathbf{Y}_{lm}\cdot\mathbf{\Psi}_{lm}=0\qquad\mathbf{Y}_{lm}\cdot\mathbf{\Phi% }_{lm}=0\qquad\mathbf{\Psi}_{lm}\cdot\mathbf{\Phi}_{lm}=0
  16. 𝐘 l m 𝐘 l m * d Ω = δ l l δ m m \int\mathbf{Y}_{lm}\cdot\mathbf{Y}^{*}_{l^{\prime}m^{\prime}}\,\mathrm{d}% \Omega=\delta_{ll^{\prime}}\delta_{mm^{\prime}}
  17. 𝚿 l m 𝚿 l m * d Ω = l ( l + 1 ) δ l l δ m m \int\mathbf{\Psi}_{lm}\cdot\mathbf{\Psi}^{*}_{l^{\prime}m^{\prime}}\,\mathrm{d% }\Omega=l(l+1)\delta_{ll^{\prime}}\delta_{mm^{\prime}}
  18. 𝚽 l m 𝚽 l m * d Ω = l ( l + 1 ) δ l l δ m m \int\mathbf{\Phi}_{lm}\cdot\mathbf{\Phi}^{*}_{l^{\prime}m^{\prime}}\,\mathrm{d% }\Omega=l(l+1)\delta_{ll^{\prime}}\delta_{mm^{\prime}}
  19. 𝐘 l m 𝚿 l m * d Ω = 0 \int\mathbf{Y}_{lm}\cdot\mathbf{\Psi}^{*}_{l^{\prime}m^{\prime}}\,\mathrm{d}% \Omega=0
  20. 𝐘 l m 𝚽 l m * d Ω = 0 \int\mathbf{Y}_{lm}\cdot\mathbf{\Phi}^{*}_{l^{\prime}m^{\prime}}\,\mathrm{d}% \Omega=0
  21. 𝚿 l m 𝚽 l m * d Ω = 0 \int\mathbf{\Psi}_{lm}\cdot\mathbf{\Phi}^{*}_{l^{\prime}m^{\prime}}\,\mathrm{d% }\Omega=0
  22. E l m r = 𝐄 𝐘 l m * d Ω E^{r}_{lm}=\int\mathbf{E}\cdot\mathbf{Y}^{*}_{lm}\,\mathrm{d}\Omega
  23. E l m ( 1 ) = 1 l ( l + 1 ) 𝐄 𝚿 l m * d Ω E^{(1)}_{lm}=\frac{1}{l(l+1)}\int\mathbf{E}\cdot\mathbf{\Psi}^{*}_{lm}\,% \mathrm{d}\Omega
  24. E l m ( 2 ) = 1 l ( l + 1 ) 𝐄 𝚽 l m * d Ω E^{(2)}_{lm}=\frac{1}{l(l+1)}\int\mathbf{E}\cdot\mathbf{\Phi}^{*}_{lm}\,% \mathrm{d}\Omega
  25. ϕ = l = 0 m = - l l ϕ l m ( r ) Y l m ( θ , ϕ ) \phi=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\phi_{lm}(r)Y_{lm}(\theta,\phi)
  26. ϕ = l = 0 m = - l l ( d ϕ l m d r 𝐘 l m + ϕ l m r 𝚿 l m ) \nabla\phi=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\left(\frac{\mathrm{d}\phi_{lm}}{% \mathrm{d}r}\mathbf{Y}_{lm}+\frac{\phi_{lm}}{r}\mathbf{\Psi}_{lm}\right)
  27. ( f ( r ) 𝐘 l m ) = ( d f d r + 2 r f ) Y l m \nabla\cdot\left(f(r)\mathbf{Y}_{lm}\right)=\left(\frac{\mathrm{d}f}{\mathrm{d% }r}+\frac{2}{r}f\right)Y_{lm}
  28. ( f ( r ) 𝚿 l m ) = - l ( l + 1 ) r f Y l m \nabla\cdot\left(f(r)\mathbf{\Psi}_{lm}\right)=-\frac{l(l+1)}{r}fY_{lm}
  29. ( f ( r ) 𝚽 l m ) = 0 \nabla\cdot\left(f(r)\mathbf{\Phi}_{lm}\right)=0
  30. 𝐄 = l = 0 m = - l l ( d E l m r d r + 2 r E l m r - l ( l + 1 ) r E l m ( 1 ) ) Y l m \nabla\cdot\mathbf{E}=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\left(\frac{\mathrm{d}% E^{r}_{lm}}{\mathrm{d}r}+\frac{2}{r}E^{r}_{lm}-\frac{l(l+1)}{r}E^{(1)}_{lm}% \right)Y_{lm}
  31. 𝚽 l m \mathbf{\Phi}_{lm}
  32. × ( f ( r ) 𝐘 l m ) = - 1 r f 𝚽 l m \nabla\times\left(f(r)\mathbf{Y}_{lm}\right)=-\frac{1}{r}f\mathbf{\Phi}_{lm}
  33. × ( f ( r ) 𝚿 l m ) = ( d f d r + 1 r f ) 𝚽 l m \nabla\times\left(f(r)\mathbf{\Psi}_{lm}\right)=\left(\frac{\mathrm{d}f}{% \mathrm{d}r}+\frac{1}{r}f\right)\mathbf{\Phi}_{lm}
  34. × ( f ( r ) 𝚽 l m ) = - l ( l + 1 ) r f 𝐘 l m - ( d f d r + 1 r f ) 𝚿 l m \nabla\times\left(f(r)\mathbf{\Phi}_{lm}\right)=-\frac{l(l+1)}{r}f\mathbf{Y}_{% lm}-\left(\frac{\mathrm{d}f}{\mathrm{d}r}+\frac{1}{r}f\right)\mathbf{\Psi}_{lm}
  35. × 𝐄 = l = 0 m = - l l ( - l ( l + 1 ) r E l m ( 2 ) 𝐘 l m - ( d E l m ( 2 ) d r + 1 r E l m ( 2 ) ) 𝚿 l m + ( - 1 r E l m r + d E l m ( 1 ) d r + 1 r E l m ( 1 ) ) 𝚽 l m ) \nabla\times\mathbf{E}=\sum_{l=0}^{\infty}\sum_{m=-l}^{l}\left(-\frac{l(l+1)}{% r}E^{(2)}_{lm}\mathbf{Y}_{lm}-\left(\frac{\mathrm{d}E^{(2)}_{lm}}{\mathrm{d}r}% +\frac{1}{r}E^{(2)}_{lm}\right)\mathbf{\Psi}_{lm}+\left(-\frac{1}{r}E^{r}_{lm}% +\frac{\mathrm{d}E^{(1)}_{lm}}{\mathrm{d}r}+\frac{1}{r}E^{(1)}_{lm}\right)% \mathbf{\Phi}_{lm}\right)
  36. l = 0 l=0\,
  37. 𝐘 00 = 1 4 π 𝐫 ^ \mathbf{Y}_{00}=\sqrt{\frac{1}{4\pi}}\hat{\mathbf{r}}
  38. 𝚿 00 = 𝟎 \mathbf{\Psi}_{00}=\mathbf{0}
  39. 𝚽 00 = 𝟎 \mathbf{\Phi}_{00}=\mathbf{0}
  40. l = 1 l=1\,
  41. 𝐘 10 = 3 4 π cos θ 𝐫 ^ \mathbf{Y}_{10}=\sqrt{\frac{3}{4\pi}}\cos\theta\,\hat{\mathbf{r}}
  42. 𝐘 11 = - 3 8 π e i φ sin θ 𝐫 ^ \mathbf{Y}_{11}=-\sqrt{\frac{3}{8\pi}}\mathrm{e}^{\mathrm{i}\varphi}\sin\theta% \,\hat{\mathbf{r}}
  43. 𝚿 10 = - 3 4 π sin θ θ ^ \mathbf{\Psi}_{10}=-\sqrt{\frac{3}{4\pi}}\sin\theta\,\hat{\mathbf{\theta}}
  44. 𝚿 11 = - 3 8 π e i φ ( cos θ θ ^ + i φ ^ ) \mathbf{\Psi}_{11}=-\sqrt{\frac{3}{8\pi}}\mathrm{e}^{\mathrm{i}\varphi}\left(% \cos\theta\,\hat{\mathbf{\theta}}+\mathrm{i}\,\hat{\mathbf{\varphi}}\right)
  45. 𝚽 10 = - 3 4 π sin θ φ ^ \mathbf{\Phi}_{10}=-\sqrt{\frac{3}{4\pi}}\sin\theta\,\hat{\mathbf{\varphi}}
  46. 𝚽 11 = 3 8 π e i φ ( i θ ^ - cos θ φ ^ ) \mathbf{\Phi}_{11}=\sqrt{\frac{3}{8\pi}}\mathrm{e}^{\mathrm{i}\varphi}\left(% \mathrm{i}\,\hat{\mathbf{\theta}}-\cos\theta\,\hat{\mathbf{\varphi}}\right)
  47. ω \omega\,
  48. 𝐉 ^ = J ( r ) 𝚽 l m \hat{\mathbf{J}}=J(r)\mathbf{\Phi}_{lm}
  49. 𝐄 ^ = E ( r ) 𝚽 l m \hat{\mathbf{E}}=E(r)\mathbf{\Phi}_{lm}
  50. 𝐁 ^ = B r ( r ) 𝐘 l m + B ( 1 ) ( r ) 𝚿 l m \hat{\mathbf{B}}=B^{r}(r)\mathbf{Y}_{lm}+B^{(1)}(r)\mathbf{\Psi}_{lm}
  51. 𝐄 ^ = 0 \nabla\cdot\hat{\mathbf{E}}=0
  52. × 𝐄 ^ = - i ω 𝐁 ^ { l ( l + 1 ) r E = i ω B r d E d r + E r = i ω B ( 1 ) \nabla\times\hat{\mathbf{E}}=-\mathrm{i}\omega\hat{\mathbf{B}}\qquad% \Rightarrow\qquad\left\{\begin{array}[]{l}\displaystyle\frac{l(l+1)}{r}E=% \mathrm{i}\omega B^{r}\\ \\ \displaystyle\frac{\mathrm{d}E}{\mathrm{d}r}+\frac{E}{r}=\mathrm{i}\omega B^{(% 1)}\end{array}\right.
  53. 𝐁 ^ = 0 d B r d r + 2 r B r - l ( l + 1 ) r B ( 1 ) = 0 \nabla\cdot\hat{\mathbf{B}}=0\quad\Rightarrow\quad\frac{\mathrm{d}B^{r}}{% \mathrm{d}r}+\frac{2}{r}B^{r}-\frac{l(l+1)}{r}B^{(1)}=0
  54. × 𝐁 ^ = μ 0 𝐉 ^ + i μ 0 ε 0 ω 𝐄 ^ - B r r + d B ( 1 ) d r + B ( 1 ) r = μ 0 J + i ω μ 0 ε 0 E \nabla\times\hat{\mathbf{B}}=\mu_{0}\hat{\mathbf{J}}+\mathrm{i}\mu_{0}% \varepsilon_{0}\omega\hat{\mathbf{E}}\quad\Rightarrow\quad-\frac{B^{r}}{r}+% \frac{\mathrm{d}B^{(1)}}{\mathrm{d}r}+\frac{B^{(1)}}{r}=\mu_{0}J+\mathrm{i}% \omega\mu_{0}\varepsilon_{0}E
  55. 𝐯 = 0 \nabla\cdot\mathbf{v}=0
  56. 𝟎 = - p + η 2 𝐯 \mathbf{0}=-\nabla p+\eta\nabla^{2}\mathbf{v}
  57. 𝐯 = 𝟎 ( r = a ) \mathbf{v}=\mathbf{0}\quad(r=a)
  58. 𝐯 = - 𝐔 0 ( r ) \mathbf{v}=-\mathbf{U}_{0}\quad(r\to\infty)
  59. 𝐔 \mathbf{U}\,
  60. 𝐔 0 = U 0 ( cos θ 𝐫 ^ - sin θ θ ^ ) = U 0 ( 𝐘 10 + 𝚿 10 ) \mathbf{U}_{0}=U_{0}\left(\cos\theta\,\hat{\mathbf{r}}-\sin\theta\,\hat{% \mathbf{\theta}}\right)=U_{0}\left(\mathbf{Y}_{10}+\mathbf{\Psi}_{10}\right)
  61. p = p ( r ) Y 10 p=p(r)Y_{10}\,
  62. 𝐯 = v r ( r ) 𝐘 10 + v ( 1 ) ( r ) 𝚿 10 \mathbf{v}=v^{r}(r)\mathbf{Y}_{10}+v^{(1)}(r)\mathbf{\Psi}_{10}

Vectorial_addition_chain.html

  1. \leftarrow
  2. \leftarrow

Velocity.html

  1. 𝐯 ( t ) \mathbf{v}(t)
  2. Δ t Δt
  3. s y m b o l v ¯ = \Deltasymbol x Δ t . symbol{\bar{v}}=\frac{\Deltasymbol{x}}{\Delta\mathit{t}}.
  4. s y m b o l v ¯ = 1 t 1 - t 0 t 0 t 1 s y m b o l v ( t ) d t , symbol{\bar{v}}={1\over t_{1}-t_{0}}\int_{t_{0}}^{t_{1}}symbol{v}(t)\ dt,
  5. Δ s y m b o l x = t 0 t 1 s y m b o l v ( t ) d t \Delta symbol{x}=\int_{t_{0}}^{t_{1}}symbol{v}(t)\ dt
  6. Δ t = t 1 - t 0 . \Delta t=t_{1}-t_{0}.
  7. 𝐯 \mathbf{v}
  8. 𝐱 \mathbf{x}
  9. t t
  10. s y m b o l v = lim Δ t 0 Δ s y m b o l x Δ t = d s y m b o l x d t . symbol{v}=\lim_{{\Delta t}\to 0}\frac{\Delta symbol{x}}{\Delta t}=\frac{% dsymbol{x}}{d\mathit{t}}.
  11. 𝐯 \mathbf{v}
  12. t t
  13. 𝐱 \mathbf{x}
  14. 𝐯 ( t ) \mathbf{v}(t)
  15. 𝐱 ( t ) \mathbf{x}(t)
  16. 𝐬 \mathbf{s}
  17. 𝐬 \mathbf{s}
  18. s y m b o l x = s y m b o l v d t . symbol{x}=\int symbol{v}\ d\mathit{t}.
  19. 𝐯 \mathbf{v}
  20. t t
  21. s y m b o l a = d s y m b o l v d t . symbol{a}=\frac{dsymbol{v}}{d\mathit{t}}.
  22. a a
  23. t t
  24. s y m b o l v = s y m b o l a d t . symbol{v}=\int symbol{a}\ d\mathit{t}.
  25. s y m b o l v = s y m b o l u + s y m b o l a t symbol{v}=symbol{u}+symbol{a}t
  26. 𝐯 \mathbf{v}
  27. t t
  28. 𝐮 \mathbf{u}
  29. t = 0 t=0
  30. s y m b o l x = ( s y m b o l u + s y m b o l v ) 2 t = s y m b o l v ¯ t symbol{x}=\frac{(symbol{u}+symbol{v})}{2}\mathit{t}=symbol{\bar{v}}\mathit{t}
  31. v 2 = s y m b o l v \cdotsymbol v = ( s y m b o l u + s y m b o l a t ) ( s y m b o l u + s y m b o l a t ) = u 2 + 2 t ( s y m b o l a \cdotsymbol u ) + a 2 t 2 v^{2}=symbol{v}\cdotsymbol{v}=(symbol{u}+symbol{a}t)\cdot(symbol{u}+symbol{a}t% )=u^{2}+2t(symbol{a}\cdotsymbol{u})+a^{2}t^{2}
  32. ( 2 s y m b o l a ) \cdotsymbol x = ( 2 s y m b o l a ) ( s y m b o l u t + 1 2 s y m b o l a t 2 ) = 2 t ( s y m b o l a \cdotsymbol u ) + a 2 t 2 = v 2 - u 2 (2symbol{a})\cdotsymbol{x}=(2symbol{a})\cdot(symbol{u}t+\frac{1}{2}symbol{a}t^% {2})=2t(symbol{a}\cdotsymbol{u})+a^{2}t^{2}=v^{2}-u^{2}
  33. v 2 = u 2 + 2 ( s y m b o l a \cdotsymbol x ) \therefore v^{2}=u^{2}+2(symbol{a}\cdotsymbol{x})
  34. v = | 𝐯 | v=|\mathbf{v}|
  35. E k = 1 2 m v 2 E_{\,\text{k}}=\tfrac{1}{2}mv^{2}
  36. s y m b o l p = m s y m b o l v symbol{p}=msymbol{v}
  37. γ = 1 1 - v 2 c 2 \gamma=\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}
  38. v e = 2 G M r , v_{\,\text{e}}=\sqrt{\frac{2GM}{r}},
  39. s y m b o l v A relative to B = s y m b o l v - s y m b o l w symbol{v}_{A\,\text{ relative to }B}=symbol{v}-symbol{w}
  40. s y m b o l v B relative to A = s y m b o l w - s y m b o l v symbol{v}_{B\,\text{ relative to }A}=symbol{w}-symbol{v}
  41. v r e l = v - ( - w ) \,v_{rel}=v-(-w)
  42. v r e l = v - ( + w ) \,v_{rel}=v-(+w)
  43. s y m b o l v = s y m b o l v T + s y m b o l v R symbol{v}=symbol{v}_{T}+symbol{v}_{R}
  44. s y m b o l v T symbol{v}_{T}
  45. s y m b o l v R symbol{v}_{R}
  46. v R = s y m b o l v s y m b o l r | s y m b o l r | v_{R}=\frac{symbol{v}\cdot symbol{r}}{\left|symbol{r}\right|}
  47. s y m b o l r symbol{r}
  48. ω \omega
  49. v T = | s y m b o l r \timessymbol v | | s y m b o l r | = ω | s y m b o l r | v_{T}=\frac{|symbol{r}\timessymbol{v}|}{|symbol{r}|}=\omega|symbol{r}|
  50. ω = | s y m b o l r \timessymbol v | | s y m b o l r | 2 . \omega=\frac{|symbol{r}\timessymbol{v}|}{|symbol{r}|^{2}}.
  51. L = m r v T = m r 2 ω L=mrv_{T}=mr^{2}\omega\,
  52. m m\,
  53. r = s y m b o l r . r=\|symbol{r}\|.
  54. m r 2 mr^{2}

Velocity_made_good.html

  1. V M G , y ^ = V o c o s θ VMG,\hat{y}=V_{o}cos\theta
  2. V M G D , y ^ = V w c o s θ s ( 1 + β 100 ) | θ o - θ s | i c o s θ γ VMG_{D},\hat{y}={V_{w}\over cos\theta_{s}}(1+{\beta\over 100})^{|\theta_{o}-% \theta_{s}|\over i}cos\theta_{\gamma}
  3. V M G D , y ^ = V w c o s θ s ( 1 + β 100 ) | 180 - θ s - θ o | i c o s θ γ VMG_{D},\hat{y}={V_{w}\over cos\theta_{s}}(1+{\beta\over 100})^{|180^{\circ}-% \theta_{s}-\theta_{o}|\over i}cos\theta_{\gamma}
  4. V w V_{w}
  5. θ o \theta_{o}
  6. θ γ \theta_{\gamma}
  7. β \beta
  8. β \beta
  9. θ s \theta_{s}
  10. i i
  11. 5 5^{\circ}
  12. i i
  13. V w V_{w}
  14. θ o \theta_{o}
  15. θ γ \theta_{\gamma}
  16. V M G D , y ^ = V w c o s θ s ( 1 + β 100 ) | θ o - θ s | i c o s θ γ VMG_{D},\hat{y}={V_{w}\over cos\theta_{s}}(1+{\beta\over 100})^{|\theta_{o}-% \theta_{s}|\over i}cos\theta_{\gamma}
  17. f ( θ γ ) = d V M G U d θ γ = - V w c o s θ s ( 1 + β 100 ) | θ o - θ s | i s i n θ γ f^{\prime}(\theta_{\gamma})={dVMG_{U}\over d\theta_{\gamma}}={-V_{w}\over cos% \theta_{s}}(1+{\beta\over 100})^{|\theta_{o}-\theta_{s}|\over i}sin\theta_{\gamma}
  18. V w V_{w}
  19. θ o \theta_{o}
  20. θ γ \theta_{\gamma}
  21. β \beta
  22. θ s \theta_{s}
  23. i i
  24. V M G D , y ^ VMG_{D},\hat{y}
  25. f ( θ γ ) = d V M G U d θ γ f^{\prime}(\theta_{\gamma})={dVMG_{U}\over d\theta_{\gamma}}

Vertex_enumeration_problem.html

  1. A x b Ax\leq b

Vibration.html

  1. F s = - k x . F_{s}=-kx.\!
  2. Σ F = m a = m x ¨ = m d 2 x d t 2 . \Sigma\ F=ma=m\ddot{x}=m\frac{d^{2}x}{dt^{2}}.
  3. m x ¨ + k x = 0. \ m\ddot{x}+kx=0.
  4. x ( t ) = A cos ( 2 π f n t ) . x(t)=A\cos(2\pi f_{n}t).\!
  5. f n = 1 2 π k m . f_{n}={1\over{2\pi}}\sqrt{k\over m}.\!
  6. 1 2 k x 2 \tfrac{1}{2}kx^{2}
  7. 1 2 m v 2 \tfrac{1}{2}mv^{2}
  8. F d = - c v = - c x ˙ = - c d x d t . F_{d}=-cv=-c\dot{x}=-c\frac{dx}{dt}.\!
  9. m x ¨ + c x ˙ + k x = 0. m\ddot{x}+{c}\dot{x}+{k}x=0.
  10. c c = 2 k m . c_{c}=2\sqrt{km}.
  11. ζ \zeta
  12. ζ = c 2 k m . \zeta={c\over 2\sqrt{km}}.
  13. x ( t ) = X e - ζ ω n t cos ( 1 - ζ 2 ω n t - ϕ ) , ω n = 2 π f n . x(t)=Xe^{-\zeta\omega_{n}t}\cos({\sqrt{1-\zeta^{2}}\omega_{n}t-\phi}),\qquad% \omega_{n}=2\pi f_{n}.
  14. ϕ , \phi,
  15. f d , f_{d},
  16. f d = f n 1 - ζ 2 . f_{d}=f_{n}\sqrt{1-\zeta^{2}}.\,
  17. F = F 0 sin ( 2 π f t ) . F=F_{0}\sin{(2\pi ft)}.\!
  18. m x ¨ + c x ˙ + k x = F 0 sin ( 2 π f t ) . m\ddot{x}+{c}\dot{x}+{k}x=F_{0}\sin{(2\pi ft)}.
  19. x ( t ) = X sin ( 2 π f t + ϕ ) . x(t)=X\sin{(2\pi ft+\phi)}.\!
  20. ϕ . \phi.
  21. X = F 0 k 1 ( 1 - r 2 ) 2 + ( 2 ζ r ) 2 . X={F_{0}\over k}{1\over\sqrt{(1-r^{2})^{2}+(2\zeta r)^{2}}}.
  22. r = f f n . r=\frac{f}{f_{n}}.
  23. ϕ , \phi,
  24. ϕ = arctan ( 2 ζ r 1 - r 2 ) . \phi=\arctan{\left(\frac{2\zeta r}{1-r^{2}}\right)}.
  25. r 1 r\approx 1
  26. F 0 F_{0}
  27. F 0 . F_{0}.
  28. δ s t . \delta_{st}.
  29. δ s t . \delta_{st}.
  30. X ( i ω ) = H ( i ω ) F ( i ω ) o r H ( i ω ) = X ( i ω ) F ( i ω ) . X(i\omega)=H(i\omega)\cdot F(i\omega)\ \ or\ \ H(i\omega)={X(i\omega)\over F(i% \omega)}.
  31. H ( i ω ) H(i\omega)
  32. | H ( i ω ) | = | X ( i ω ) F ( i ω ) | = 1 k 1 ( 1 - r 2 ) 2 + ( 2 ζ r ) 2 , |H(i\omega)|=\left|{X(i\omega)\over F(i\omega)}\right|={1\over k}{1\over\sqrt{% (1-r^{2})^{2}+(2\zeta r)^{2}}},
  33. r = f f n = ω ω n . r=\frac{f}{f_{n}}=\frac{\omega}{\omega_{n}}.
  34. H ( i ω ) = - arctan ( 2 ζ r 1 - r 2 ) . \angle H(i\omega)=-\arctan{\left(\frac{2\zeta r}{1-r^{2}}\right)}.
  35. m 1 x 1 ¨ + ( c 1 + c 2 ) x 1 ˙ - c 2 x 2 ˙ + ( k 1 + k 2 ) x 1 - k 2 x 2 = f 1 , m_{1}\ddot{x_{1}}+{(c_{1}+c_{2})}\dot{x_{1}}-{c_{2}}\dot{x_{2}}+{(k_{1}+k_{2})% }x_{1}-{k_{2}}x_{2}=f_{1},
  36. m 2 x 2 ¨ - c 2 x 1 ˙ + ( c 2 + c 3 ) x 2 ˙ - k 2 x 1 + ( k 2 + k 3 ) x 2 = f 2 . m_{2}\ddot{x_{2}}-{c_{2}}\dot{x_{1}}+{(c_{2}+c_{3})}\dot{x_{2}}-{k_{2}}x_{1}+{% (k_{2}+k_{3})}x_{2}=f_{2}.\!
  37. [ m 1 0 0 m 2 ] { x 1 ¨ x 2 ¨ } + [ c 1 + c 2 - c 2 - c 2 c 2 + c 3 ] { x 1 ˙ x 2 ˙ } + [ k 1 + k 2 - k 2 - k 2 k 2 + k 3 ] { x 1 x 2 } = { f 1 f 2 } . \begin{bmatrix}m_{1}&0\\ 0&m_{2}\end{bmatrix}\begin{Bmatrix}\ddot{x_{1}}\\ \ddot{x_{2}}\end{Bmatrix}+\begin{bmatrix}c_{1}+c_{2}&-c_{2}\\ -c_{2}&c_{2}+c_{3}\end{bmatrix}\begin{Bmatrix}\dot{x_{1}}\\ \dot{x_{2}}\end{Bmatrix}+\begin{bmatrix}k_{1}+k_{2}&-k_{2}\\ -k_{2}&k_{2}+k_{3}\end{bmatrix}\begin{Bmatrix}x_{1}\\ x_{2}\end{Bmatrix}=\begin{Bmatrix}f_{1}\\ f_{2}\end{Bmatrix}.
  38. [ M ] { x ¨ } + [ C ] { x ˙ } + [ K ] { x } = { f } \begin{bmatrix}M\end{bmatrix}\begin{Bmatrix}\ddot{x}\end{Bmatrix}+\begin{% bmatrix}C\end{bmatrix}\begin{Bmatrix}\dot{x}\end{Bmatrix}+\begin{bmatrix}K\end% {bmatrix}\begin{Bmatrix}x\end{Bmatrix}=\begin{Bmatrix}f\end{Bmatrix}
  39. [ M ] , \begin{bmatrix}M\end{bmatrix},
  40. [ C ] , \begin{bmatrix}C\end{bmatrix},
  41. [ K ] \begin{bmatrix}K\end{bmatrix}
  42. [ M ] { x ¨ } + [ K ] { x } = 0. \begin{bmatrix}M\end{bmatrix}\begin{Bmatrix}\ddot{x}\end{Bmatrix}+\begin{% bmatrix}K\end{bmatrix}\begin{Bmatrix}x\end{Bmatrix}=0.
  43. { x } = { X } e i ω t . \begin{Bmatrix}x\end{Bmatrix}=\begin{Bmatrix}X\end{Bmatrix}e^{i\omega t}.
  44. { X } e i ω t \begin{Bmatrix}X\end{Bmatrix}e^{i\omega t}
  45. [ - ω 2 [ M ] + [ K ] ] { X } e i ω t = 0. \begin{bmatrix}-\omega^{2}\begin{bmatrix}M\end{bmatrix}+\begin{bmatrix}K\end{% bmatrix}\end{bmatrix}\begin{Bmatrix}X\end{Bmatrix}e^{i\omega t}=0.
  46. e i ω t e^{i\omega t}
  47. [ [ K ] - ω 2 [ M ] ] { X } = 0. \begin{bmatrix}\begin{bmatrix}K\end{bmatrix}-\omega^{2}\begin{bmatrix}M\end{% bmatrix}\end{bmatrix}\begin{Bmatrix}X\end{Bmatrix}=0.
  48. [ M ] - 1 \begin{bmatrix}M\end{bmatrix}^{-1}
  49. [ [ M ] - 1 [ K ] - ω 2 [ M ] - 1 [ M ] ] { X } = 0 \begin{bmatrix}\begin{bmatrix}M\end{bmatrix}^{-1}\begin{bmatrix}K\end{bmatrix}% -\omega^{2}\begin{bmatrix}M\end{bmatrix}^{-1}\begin{bmatrix}M\end{bmatrix}\end% {bmatrix}\begin{Bmatrix}X\end{Bmatrix}=0
  50. [ M ] - 1 [ K ] = [ A ] \begin{bmatrix}M\end{bmatrix}^{-1}\begin{bmatrix}K\end{bmatrix}=\begin{bmatrix% }A\end{bmatrix}
  51. λ = ω 2 \lambda=\omega^{2}\,
  52. [ [ A ] - λ [ I ] ] { X } = 0. \begin{bmatrix}\begin{bmatrix}A\end{bmatrix}-\lambda\begin{bmatrix}I\end{% bmatrix}\end{bmatrix}\begin{Bmatrix}X\end{Bmatrix}=0.
  53. ω 1 2 , ω 2 2 , ω N 2 \omega_{1}^{2},\omega_{2}^{2},\cdots\omega_{N}^{2}
  54. { X } \begin{Bmatrix}X\end{Bmatrix}
  55. [ ω r 2 ] = [ ω 1 2 0 0 ω N 2 ] \begin{bmatrix}^{\diagdown}\omega_{r\diagdown}^{2}\end{bmatrix}=\begin{bmatrix% }\omega_{1}^{2}&\cdots&0\\ \vdots&\ddots&\vdots\\ 0&\cdots&\omega_{N}^{2}\end{bmatrix}
  56. [ Ψ ] = [ { ψ 1 } { ψ 2 } { ψ N } ] . \begin{bmatrix}\Psi\end{bmatrix}=\begin{bmatrix}\begin{Bmatrix}\psi_{1}\end{% Bmatrix}\begin{Bmatrix}\psi_{2}\end{Bmatrix}\cdots\begin{Bmatrix}\psi_{N}\end{% Bmatrix}\end{bmatrix}.
  57. [ M ] = [ 1 0 0 1 ] \begin{bmatrix}M\end{bmatrix}=\begin{bmatrix}1&0\\ 0&1\end{bmatrix}
  58. [ K ] = [ 2000 - 1000 - 1000 2000 ] . \begin{bmatrix}K\end{bmatrix}=\begin{bmatrix}2000&-1000\\ -1000&2000\end{bmatrix}.
  59. [ A ] = [ 2000 - 1000 - 1000 2000 ] . \begin{bmatrix}A\end{bmatrix}=\begin{bmatrix}2000&-1000\\ -1000&2000\end{bmatrix}.
  60. [ ω r 2 ] = [ 1000 0 0 3000 ] . \begin{bmatrix}^{\diagdown}\omega_{r\diagdown}^{2}\end{bmatrix}=\begin{bmatrix% }1000&0\\ 0&3000\end{bmatrix}.
  61. ω = 2 π f \scriptstyle\omega=2\pi f
  62. f 1 = 5.033 Hz \scriptstyle f_{1}=5.033\mathrm{\ Hz}
  63. f 2 = 8.717 Hz \scriptstyle f_{2}=8.717\mathrm{\ Hz}
  64. [ Ψ ] = [ { ψ 1 } { ψ 2 } ] = [ { - 0.707 - 0.707 } 1 { 0.707 - 0.707 } 2 ] . \begin{bmatrix}\Psi\end{bmatrix}=\begin{bmatrix}\begin{Bmatrix}\psi_{1}\end{% Bmatrix}\begin{Bmatrix}\psi_{2}\end{Bmatrix}\end{bmatrix}=\begin{bmatrix}% \begin{Bmatrix}-0.707\\ -0.707\end{Bmatrix}_{1}\begin{Bmatrix}0.707\\ -0.707\end{Bmatrix}_{2}\end{bmatrix}.
  65. [ Ψ ] T [ M ] [ Ψ ] = [ m r ] , \begin{bmatrix}\Psi\end{bmatrix}^{T}\begin{bmatrix}M\end{bmatrix}\begin{% bmatrix}\Psi\end{bmatrix}=\begin{bmatrix}^{\diagdown}m_{r\diagdown}\end{% bmatrix},
  66. [ Ψ ] T [ K ] [ Ψ ] = [ k r ] . \begin{bmatrix}\Psi\end{bmatrix}^{T}\begin{bmatrix}K\end{bmatrix}\begin{% bmatrix}\Psi\end{bmatrix}=\begin{bmatrix}^{\diagdown}k_{r\diagdown}\end{% bmatrix}.
  67. [ m r ] \begin{bmatrix}^{\diagdown}m_{r\diagdown}\end{bmatrix}
  68. [ k r ] \begin{bmatrix}^{\diagdown}k_{r\diagdown}\end{bmatrix}
  69. { x } = [ Ψ ] { q } . \begin{Bmatrix}x\end{Bmatrix}=\begin{bmatrix}\Psi\end{bmatrix}\begin{Bmatrix}q% \end{Bmatrix}.
  70. [ M ] [ Ψ ] { q ¨ } + [ K ] [ Ψ ] { q } = 0. \begin{bmatrix}M\end{bmatrix}\begin{bmatrix}\Psi\end{bmatrix}\begin{Bmatrix}% \ddot{q}\end{Bmatrix}+\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}\Psi\end{% bmatrix}\begin{Bmatrix}q\end{Bmatrix}=0.
  71. [ Ψ ] T \begin{bmatrix}\Psi\end{bmatrix}^{T}
  72. [ Ψ ] T [ M ] [ Ψ ] { q ¨ } + [ Ψ ] T [ K ] [ Ψ ] { q } = 0. \begin{bmatrix}\Psi\end{bmatrix}^{T}\begin{bmatrix}M\end{bmatrix}\begin{% bmatrix}\Psi\end{bmatrix}\begin{Bmatrix}\ddot{q}\end{Bmatrix}+\begin{bmatrix}% \Psi\end{bmatrix}^{T}\begin{bmatrix}K\end{bmatrix}\begin{bmatrix}\Psi\end{% bmatrix}\begin{Bmatrix}q\end{Bmatrix}=0.
  73. [ m r ] { q ¨ } + [ k r ] { q } = 0. \begin{bmatrix}^{\diagdown}m_{r\diagdown}\end{bmatrix}\begin{Bmatrix}\ddot{q}% \end{Bmatrix}+\begin{bmatrix}^{\diagdown}k_{r\diagdown}\end{bmatrix}\begin{% Bmatrix}q\end{Bmatrix}=0.
  74. { x } = [ Ψ ] { q } \begin{Bmatrix}x\end{Bmatrix}=\begin{bmatrix}\Psi\end{bmatrix}\begin{Bmatrix}q% \end{Bmatrix}
  75. { x n } = q 1 { ψ } 1 + q 2 { ψ } 2 + q 3 { ψ } 3 + + q N { ψ } N . \begin{Bmatrix}x_{n}\end{Bmatrix}=q_{1}\begin{Bmatrix}\psi\end{Bmatrix}_{1}+q_% {2}\begin{Bmatrix}\psi\end{Bmatrix}_{2}+q_{3}\begin{Bmatrix}\psi\end{Bmatrix}_% {3}+\cdots+q_{N}\begin{Bmatrix}\psi\end{Bmatrix}_{N}.

Vibration_of_rotating_structures.html

  1. M * r ¨ E + ( D + G ) * r ˙ E + ( K + N ) * r E = M*\ddot{r}E+(D+G)*\dot{r}E+(K+N)*rE=
  2. = M * V * s ¨ E + B * V * s ˙ E + p E =M*V*\ddot{s}E+B*V*\dot{s}E+pE
  3. p U = A * V * s ˙ U pU=A*V*\dot{s}U
  4. s ˙ U \dot{s}U
  5. s ˙ U \dot{s}U
  6. s ˙ U \dot{s}U
  7. Ω \Omega
  8. Ω * ( I m a x - I m i n ) / I m i n \Omega*(Imax-Imin)/Imin
  9. Ω \Omega

Vibrations_of_a_circular_membrane.html

  1. u 12 u_{12}
  2. Ω \Omega
  3. a a
  4. t , t,
  5. ( x , y ) (x,y)
  6. Ω \Omega
  7. u ( x , y , t ) , u(x,y,t),
  8. Ω \partial\Omega
  9. Ω , \Omega,
  10. a a
  11. 2 u t 2 = c 2 ( 2 u x 2 + 2 u y 2 ) for ( x , y ) Ω \frac{\partial^{2}u}{\partial t^{2}}=c^{2}\left(\frac{\partial^{2}u}{\partial x% ^{2}}+\frac{\partial^{2}u}{\partial y^{2}}\right)\,\text{ for }(x,y)\in\Omega\,
  12. u = 0 on Ω . u=0\,\text{ on }\partial\Omega.\,
  13. Ω \Omega
  14. ( r , θ , z ) . (r,\theta,z).
  15. 2 u t 2 = c 2 ( 2 u r 2 + 1 r u r + 1 r 2 2 u θ 2 ) for 0 r < a , 0 θ 2 π \frac{\partial^{2}u}{\partial t^{2}}=c^{2}\left(\frac{\partial^{2}u}{\partial r% ^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}+\frac{1}{r^{2}}\frac{\partial^{% 2}u}{\partial\theta^{2}}\right)\,\text{ for }0\leq r<a,0\leq\theta\leq 2\pi\,
  16. u = 0 for r = a . u=0\,\text{ for }r=a.\,
  17. c c
  18. c = N r r * ρ h c=\sqrt{\frac{N_{rr}^{*}}{\rho h}}
  19. N r r * N_{rr}^{*}
  20. r = a r=a
  21. h h
  22. ρ \rho
  23. r r
  24. F = r N r r r = r N θ θ r F=rN^{r}_{rr}=rN^{r}_{\theta\theta}
  25. N θ θ r = N r r r N^{r}_{\theta\theta}=N^{r}_{rr}
  26. u u
  27. θ , \theta,
  28. 2 u t 2 = c 2 ( 2 u r 2 + 1 r u r ) . \frac{\partial^{2}u}{\partial t^{2}}=c^{2}\left(\frac{\partial^{2}u}{\partial r% ^{2}}+\frac{1}{r}\frac{\partial u}{\partial r}\right).
  29. u ( r , t ) = R ( r ) T ( t ) . u(r,t)=R(r)T(t).
  30. c 2 R ( r ) T ( t ) c^{2}R(r)T(t)
  31. T ′′ ( t ) c 2 T ( t ) = 1 R ( r ) ( R ′′ ( r ) + 1 r R ( r ) ) . \frac{T^{\prime\prime}(t)}{c^{2}T(t)}=\frac{1}{R(r)}\left(R^{\prime\prime}(r)+% \frac{1}{r}R^{\prime}(r)\right).
  32. r , r,
  33. t , t,
  34. K . K.
  35. T ( t ) T(t)
  36. R ( r ) R(r)
  37. T ′′ ( t ) = K c 2 T ( t ) T^{\prime\prime}(t)=Kc^{2}T(t)\,
  38. r R ′′ ( r ) + R ( r ) - K r R ( r ) = 0. rR^{\prime\prime}(r)+R^{\prime}(r)-KrR(r)=0.\,
  39. T ( t ) T(t)
  40. K > 0 , K>0,
  41. K = 0 , K=0,
  42. K < 0. K<0.
  43. K < 0 , K<0,
  44. K = - λ 2 . K=-\lambda^{2}.
  45. T ( t ) T(t)
  46. T ( t ) = A cos c λ t + B sin c λ t . T(t)=A\cos c\lambda t+B\sin c\lambda t.\,
  47. R ( r ) , R(r),
  48. K = - λ 2 , K=-\lambda^{2},
  49. R ( r ) = c 1 J 0 ( λ r ) + c 2 Y 0 ( λ r ) . R(r)=c_{1}J_{0}(\lambda r)+c_{2}Y_{0}(\lambda r).\,
  50. Y 0 Y_{0}
  51. r 0 , r\to 0,
  52. c 2 c_{2}
  53. c 1 = 1 , c_{1}=1,
  54. A A
  55. B B
  56. T ( t ) . T(t).
  57. R ( r ) = J 0 ( λ r ) . R(r)=J_{0}(\lambda r).
  58. u u
  59. R ( a ) = J 0 ( λ a ) = 0. R(a)=J_{0}(\lambda a)=0.
  60. J 0 J_{0}
  61. 0 < α 01 < α 02 < 0<\alpha_{01}<\alpha_{02}<\cdots
  62. λ a = α 0 n , \lambda a=\alpha_{0n},
  63. n = 1 , 2 , , n=1,2,\dots,
  64. R ( r ) = J 0 ( α 0 n a r ) . R(r)=J_{0}\left(\frac{\alpha_{0n}}{a}r\right).
  65. u u
  66. u 0 n ( r , t ) = ( A cos c λ 0 n t + B sin c λ 0 n t ) J 0 ( λ 0 n r ) for n = 1 , 2 , , u_{0n}(r,t)=\left(A\cos c\lambda_{0n}t+B\sin c\lambda_{0n}t\right)J_{0}\left(% \lambda_{0n}r\right)\,\text{ for }n=1,2,\dots,\,
  67. λ 0 n = α 0 n / a . \lambda_{0n}=\alpha_{0n}/a.
  68. u u
  69. θ , \theta,
  70. u ( r , θ , t ) = R ( r ) Θ ( θ ) T ( t ) . u(r,\theta,t)=R(r)\Theta(\theta)T(t).\,
  71. T ′′ ( t ) c 2 T ( t ) = R ′′ ( r ) R ( r ) + R ( r ) r R ( r ) + Θ ′′ ( θ ) r 2 Θ ( θ ) = K \frac{T^{\prime\prime}(t)}{c^{2}T(t)}=\frac{R^{\prime\prime}(r)}{R(r)}+\frac{R% ^{\prime}(r)}{rR(r)}+\frac{\Theta^{\prime\prime}(\theta)}{r^{2}\Theta(\theta)}=K
  72. K K
  73. T ( t ) T(t)
  74. K = - λ 2 K=-\lambda^{2}
  75. λ > 0 \lambda>0
  76. T ( t ) = A cos c λ t + B sin c λ t . T(t)=A\cos c\lambda t+B\sin c\lambda t.\,
  77. R ′′ ( r ) R ( r ) + R ( r ) r R ( r ) + Θ ′′ ( θ ) r 2 Θ ( θ ) = - λ 2 \frac{R^{\prime\prime}(r)}{R(r)}+\frac{R^{\prime}(r)}{rR(r)}+\frac{\Theta^{% \prime\prime}(\theta)}{r^{2}\Theta(\theta)}=-\lambda^{2}
  78. r 2 r^{2}
  79. λ 2 r 2 + r 2 R ′′ ( r ) R ( r ) + r R ( r ) R ( r ) = L \lambda^{2}r^{2}+\frac{r^{2}R^{\prime\prime}(r)}{R(r)}+\frac{rR^{\prime}(r)}{R% (r)}=L
  80. - Θ ′′ ( θ ) Θ ( θ ) = L , -\frac{\Theta^{\prime\prime}(\theta)}{\Theta(\theta)}=L,
  81. L . L.
  82. Θ ( θ ) \Theta(\theta)
  83. 2 π , 2\pi,
  84. θ \theta
  85. Θ ( θ ) = C cos m θ + D sin m θ , \Theta(\theta)=C\cos m\theta+D\sin m\theta,\,
  86. m = 0 , 1 , m=0,1,\dots
  87. C C
  88. D D
  89. L = m 2 . L=m^{2}.
  90. R ( r ) , R(r),
  91. J m J_{m}
  92. Y m . Y_{m}.
  93. R ( r ) = J m ( λ m n r ) , R(r)=J_{m}(\lambda_{mn}r),\,
  94. m = 0 , 1 , , m=0,1,\dots,
  95. n = 1 , 2 , , n=1,2,\dots,
  96. λ m n = α m n / a , \lambda_{mn}=\alpha_{mn}/a,
  97. α m n \alpha_{mn}
  98. n n
  99. J m . J_{m}.
  100. u m n ( r , θ , t ) = ( A cos c λ m n t + B sin c λ m n t ) J m ( λ m n r ) ( C cos m θ + D sin m θ ) u_{mn}(r,\theta,t)=\left(A\cos c\lambda_{mn}t+B\sin c\lambda_{mn}t\right)J_{m}% \left(\lambda_{mn}r\right)(C\cos m\theta+D\sin m\theta)
  101. m = 0 , 1 , , n = 1 , 2 , m=0,1,\dots,n=1,2,\dots
  102. ω = λ m n c \omega=\lambda_{mn}c
  103. u 01 u_{01}
  104. α 01 = 2.40483 \alpha_{01}=2.40483
  105. u 02 u_{02}
  106. α 02 = 5.52008 \alpha_{02}=5.52008
  107. u 03 u_{03}
  108. α 03 = 8.65373 \alpha_{03}=8.65373
  109. u 11 u_{11}
  110. α 11 = 3.83171 \alpha_{11}=3.83171
  111. u 12 u_{12}
  112. α 12 = 7.01559 \alpha_{12}=7.01559
  113. u 13 u_{13}
  114. α 13 = 10.1735 \alpha_{13}=10.1735
  115. u 21 u_{21}
  116. α 21 = 5.13562 \alpha_{21}=5.13562
  117. u 22 u_{22}
  118. α 22 = 8.41724 \alpha_{22}=8.41724
  119. u 23 u_{23}
  120. α 23 = 11.6198 \alpha_{23}=11.6198

Vickrey–Clarke–Groves_auction.html

  1. M = { t 1 , , t m } M=\{t_{1},\ldots,t_{m}\}
  2. N = { b 1 , , b n } N=\{b_{1},\ldots,b_{n}\}
  3. V N M V^{M}_{N}
  4. b i b_{i}
  5. t j t_{j}
  6. v i ( t j ) v_{i}(t_{j})
  7. b i b_{i}
  8. t j t_{j}
  9. v i ( t j ) v_{i}(t_{j})
  10. V N { b i } M - V N { b i } M { t j } V^{M}_{N\setminus\{b_{i}\}}-V^{M\setminus\{t_{j}\}}_{N\setminus\{b_{i}\}}
  11. b i b_{i}
  12. N { b i } N\setminus\{b_{i}\}
  13. t j t_{j}
  14. V N { b i } M . V^{M}_{N\setminus\{b_{i}\}}.
  15. b i b_{i}
  16. M { t j } M\setminus\{t_{j}\}
  17. V N { b i } M { t j } V^{M\setminus\{t_{j}\}}_{N\setminus\{b_{i}\}}
  18. b i b_{i}
  19. t j t_{j}
  20. b i b_{i}
  21. A A
  22. t j t_{j}
  23. v i ( t j ) = A , v_{i}(t_{j})=A,
  24. A - ( V N { b i } M - V N { b i } M { t j } ) . A-\left(V^{M}_{N\setminus\{b_{i}\}}-V^{M\setminus\{t_{j}\}}_{N\setminus\{b_{i}% \}}\right).
  25. b 1 b_{1}
  26. b 2 b_{2}
  27. t 1 t_{1}
  28. t 2 t_{2}
  29. v i , j v_{i,j}
  30. b i b_{i}
  31. t j t_{j}
  32. v 1 , 1 = 10 v_{1,1}=10
  33. v 1 , 2 = 5 v_{1,2}=5
  34. v 2 , 1 = 5 v_{2,1}=5
  35. v 2 , 2 = 3 v_{2,2}=3
  36. b 1 b_{1}
  37. b 2 b_{2}
  38. t 1 t_{1}
  39. t 1 t_{1}
  40. b 1 b_{1}
  41. 10 10
  42. t 2 t_{2}
  43. b 2 b_{2}
  44. 3 3
  45. 13 13
  46. b 2 b_{2}
  47. b 1 b_{1}
  48. t 1 t_{1}
  49. b 1 b_{1}
  50. 10 10
  51. b 2 b_{2}
  52. 10 - 10 = 0 10-10=0
  53. b 1 b_{1}
  54. t 1 t_{1}
  55. b 2 b_{2}
  56. 5 5
  57. b 1 b_{1}
  58. 5 - 3 = 2 5-3=2
  59. n n
  60. n n
  61. v ~ i j \tilde{v}_{ij}
  62. i i
  63. j j
  64. b ~ i j \tilde{b}_{ij}
  65. i i
  66. j j
  67. b i ( a ) = b ~ i k b_{i}(a)=\tilde{b}_{ik}
  68. i i
  69. k k
  70. a a
  71. a * a^{*}
  72. p i = [ max a A j i b j ( a ) ] - j i b j ( a * ) p_{i}=\left[\max_{a\in A}\sum_{j\neq i}b_{j}(a)\right]-\sum_{j\neq i}b_{j}(a^{% *})
  73. a * a^{*}
  74. b i b_{i}
  75. v i v_{i}
  76. t i t_{i}
  77. b i b_{i}
  78. t i t_{i}
  79. U i U_{i}
  80. b i b_{i}
  81. U i = v i - ( V N { b i } M - V N { b i } M { t i } ) = i v i - j i v j + V N { b i } M { t i } - V N { b i } M = i v i - V N { b i } M { t i } + V N { b i } M { t i } - V N { b i } M U_{i}=v_{i}-\left(V^{M}_{N\setminus\{b_{i}\}}-V^{M\setminus\{t_{i}\}}_{N% \setminus\{b_{i}\}}\right)=\sum_{i}v_{i}-\sum_{j\neq i}v_{j}+V^{M\setminus\{t_% {i}\}}_{N\setminus\{b_{i}\}}-V^{M}_{N\setminus\{b_{i}\}}=\sum_{i}v_{i}-V^{M% \setminus\{t_{i}\}}_{N\setminus\{b_{i}\}}+V^{M\setminus\{t_{i}\}}_{N\setminus% \{b_{i}\}}-V^{M}_{N\setminus\{b_{i}\}}
  82. = i v i - V N { b i } M =\sum_{i}v_{i}-V^{M}_{N\setminus\{b_{i}\}}
  83. V N { b i } M V^{M}_{N\setminus\{b_{i}\}}
  84. v i v_{i}
  85. i v i \sum_{i}v_{i}
  86. v i v_{i}
  87. U i - U j = [ v i + V N { b i } M { t i } ] - [ v j + V N { b i } M { t j } ] U_{i}-U_{j}=\left[v_{i}+V^{M\setminus\{t_{i}\}}_{N\setminus\{b_{i}\}}\right]-% \left[v_{j}+V^{M\setminus\{t_{j}\}}_{N\setminus\{b_{i}\}}\right]
  88. U i U_{i}
  89. b i b_{i}
  90. v i v_{i}
  91. t i t_{i}
  92. U j U_{j}
  93. b i b_{i}
  94. v i v^{\prime}_{i}
  95. t i t_{i}
  96. t j t_{j}
  97. v j v_{j}
  98. [ v j + V N { b i } M { t j } ] \left[v_{j}+V^{M\setminus\{t_{j}\}}_{N\setminus\{b_{i}\}}\right]
  99. t j t_{j}
  100. b i b_{i}
  101. t i t_{i}
  102. b i b_{i}
  103. [ v i + V N { b i } M { t i } ] - [ v j + V N { b i } M { t j } ] 0 \left[v_{i}+V^{M\setminus\{t_{i}\}}_{N\setminus\{b_{i}\}}\right]-\left[v_{j}+V% ^{M\setminus\{t_{j}\}}_{N\setminus\{b_{i}\}}\right]\geq 0
  104. U i U j U_{i}\geq U_{j}
  105. A A
  106. n n
  107. v i : A R + v_{i}:A\longrightarrow R_{+}
  108. i i
  109. a A a\in A
  110. i v i ( a ) \sum_{i}v_{i}(a)
  111. p i p_{i}
  112. p i = h i ( v - i ) - j i v j ( a ) p_{i}=h_{i}(v_{-i})-\sum_{j\neq i}v_{j}(a)
  113. v - i = ( v 1 , , v i - 1 , v i + 1 , , v n ) v_{-i}=(v_{1},\dots,v_{i-1},v_{i+1},\dots,v_{n})
  114. h i h_{i}
  115. h i ( v - i ) = 0 h_{i}(v_{-i})=0
  116. h i ( v - i ) = max b A j i v j ( b ) h_{i}(v_{-i})=\max_{b\in A}\sum_{j\neq i}v_{j}(b)
  117. v i v_{i}
  118. b i : A R + b_{i}:A\longrightarrow R_{+}
  119. a * a^{*}
  120. i b i ( a * ) \sum_{i}b_{i}(a^{*})
  121. p i = [ max a A j i b j ( a ) ] - j i b j ( a * ) . p_{i}=\left[\max_{a\in A}\sum_{j\neq i}b_{j}(a)\right]-\sum_{j\neq i}b_{j}(a^{% *}).
  122. i i
  123. v i ( a ) - p i 0 v_{i}(a)-p_{i}\geq 0
  124. p i 0 p_{i}\geq 0