wpmath0000005_2

Color_balance.html

  1. R = 240 R=240
  2. [ R G B ] = [ 255 / R w 0 0 0 255 / G w 0 0 0 255 / B w ] [ R G B ] \left[\begin{array}[]{c}R\\ G\\ B\end{array}\right]=\left[\begin{array}[]{ccc}255/R^{\prime}_{w}&0&0\\ 0&255/G^{\prime}_{w}&0\\ 0&0&255/B^{\prime}_{w}\end{array}\right]\left[\begin{array}[]{c}R^{\prime}\\ G^{\prime}\\ B^{\prime}\end{array}\right]
  3. R R
  4. G G
  5. B B
  6. R R^{\prime}
  7. G G^{\prime}
  8. B B^{\prime}
  9. R w R^{\prime}_{w}
  10. G w G^{\prime}_{w}
  11. B w B^{\prime}_{w}
  12. [ X Y Z ] = [ X w / X w 0 0 0 Y w / Y w 0 0 0 Z w / Z w ] [ X Y Z ] \left[\begin{array}[]{c}X\\ Y\\ Z\end{array}\right]=\left[\begin{array}[]{ccc}X_{w}/X^{\prime}_{w}&0&0\\ 0&Y_{w}/Y^{\prime}_{w}&0\\ 0&0&Z_{w}/Z^{\prime}_{w}\end{array}\right]\left[\begin{array}[]{c}X^{\prime}\\ Y^{\prime}\\ Z^{\prime}\end{array}\right]
  13. X X
  14. Y Y
  15. Z Z
  16. X w X_{w}
  17. Y w Y_{w}
  18. Z w Z_{w}
  19. X w X^{\prime}_{w}
  20. Y w Y^{\prime}_{w}
  21. Z w Z^{\prime}_{w}
  22. X X^{\prime}
  23. Y Y^{\prime}
  24. Z Z^{\prime}
  25. 𝐏 \mathbf{P}
  26. [ X Y Z ] = 𝐏 [ L R L G L B ] \left[\begin{array}[]{c}X\\ Y\\ Z\end{array}\right]=\mathbf{P}\left[\begin{array}[]{c}L_{R}\\ L_{G}\\ L_{B}\end{array}\right]
  27. L R L_{R}
  28. L G L_{G}
  29. L B L_{B}
  30. [ L R L G L B ] = 𝐏 - 𝟏 [ X w / X w 0 0 0 Y w / Y w 0 0 0 Z w / Z w ] 𝐏 [ L R L G L B ] \left[\begin{array}[]{c}L_{R}\\ L_{G}\\ L_{B}\end{array}\right]=\mathbf{P^{-1}}\left[\begin{array}[]{ccc}X_{w}/X^{% \prime}_{w}&0&0\\ 0&Y_{w}/Y^{\prime}_{w}&0\\ 0&0&Z_{w}/Z^{\prime}_{w}\end{array}\right]\mathbf{P}\left[\begin{array}[]{c}L_% {R^{\prime}}\\ L_{G^{\prime}}\\ L_{B^{\prime}}\end{array}\right]
  31. [ L M S ] = [ 1 / L w 0 0 0 1 / M w 0 0 0 1 / S w ] [ L M S ] \left[\begin{array}[]{c}L\\ M\\ S\end{array}\right]=\left[\begin{array}[]{ccc}1/L^{\prime}_{w}&0&0\\ 0&1/M^{\prime}_{w}&0\\ 0&0&1/S^{\prime}_{w}\end{array}\right]\left[\begin{array}[]{c}L^{\prime}\\ M^{\prime}\\ S^{\prime}\end{array}\right]
  32. L L
  33. M M
  34. S S
  35. L w L^{\prime}_{w}
  36. M w M^{\prime}_{w}
  37. S w S^{\prime}_{w}
  38. L L^{\prime}
  39. M M^{\prime}
  40. S S^{\prime}
  41. [ R G B ] = [ 255 / R w 0 0 0 255 / G w 0 0 0 255 / B w ] [ R G B ] \left[\begin{array}[]{c}R\\ G\\ B\end{array}\right]=\left[\begin{array}[]{ccc}255/R^{\prime}_{w}&0&0\\ 0&255/G^{\prime}_{w}&0\\ 0&0&255/B^{\prime}_{w}\end{array}\right]\left[\begin{array}[]{c}R^{\prime}\\ G^{\prime}\\ B^{\prime}\end{array}\right]

Color_model.html

  1. x = X X + Y + Z x=\frac{X}{X+Y+Z}
  2. y = Y X + Y + Z y=\frac{Y}{X+Y+Z}

Color_rendering_index.html

  1. D C = Δ u v = ( u r - u t ) 2 + ( v r - v t ) 2 DC={\Delta}_{uv}=\sqrt{(u_{r}-u_{t})^{2}+(v_{r}-v_{t})^{2}}
  2. Δ E i \Delta E_{i}
  3. R i = 100 - 4.6 Δ E i R_{i}=100-4.6\Delta E_{i}
  4. Δ E ¯ U V W \Delta\bar{E}_{UVW}
  5. R a R_{a}
  6. R a = 100 - 4.6 Δ E ¯ U V W R_{a}=100-4.6\Delta\bar{E}_{UVW}
  7. u c , i = 10.872 + 0.404 ( c r / c t ) c t , i - 4 ( d r / d t ) d t , i 16.518 + 1.481 ( c r / c t ) c t , i - ( d r / d t ) d t , i u_{c,i}=\frac{10.872+0.404(c_{r}/c_{t})c_{t,i}-4(d_{r}/d_{t})d_{t,i}}{16.518+1% .481(c_{r}/c_{t})c_{t,i}-(d_{r}/d_{t})d_{t,i}}
  8. v c , i = 5.520 16.518 + 1.481 ( c r / c t ) c t , i - ( d r / d t ) d t , i v_{c,i}=\frac{5.520}{16.518+1.481(c_{r}/c_{t})c_{t,i}-(d_{r}/d_{t})d_{t,i}}
  9. c = ( 4.0 - u - 10.0 v ) / v c=\left(4.0-u-10.0v\right)/v
  10. d = ( 1.708 v - 1.481 u + 0.404 ) / v d=\left(1.708v-1.481u+0.404\right)/v
  11. R a R_{a}
  12. x = 109.2 109.2 + 100.0 + 38.9 = 0.4402 x=\frac{109.2}{109.2+100.0+38.9}=0.4402
  13. y = 100 109.2 + 100.0 + 38.9 = 0.4031 y=\frac{100}{109.2+100.0+38.9}=0.4031
  14. u = 4 × 0.4402 - 2 × 0.4402 + 12 × 0.4031 + 3 = 0.2531 u=\frac{4\times 0.4402}{-2\times 0.4402+12\times 0.4031+3}=0.2531
  15. v = 6 × 0.4031 - 2 × 0.4402 + 12 × 0.4031 + 3 = 0.3477 v=\frac{6\times 0.4031}{-2\times 0.4402+12\times 0.4031+3}=0.3477
  16. D C \displaystyle DC
  17. C C T e s t . = - 449 n 3 + 3525 n 2 - 6823.3 n + 5520.33 CCT_{est.}=-449n^{3}+3525n^{2}-6823.3n+5520.33
  18. n = x - 0.3320 y - 0.1858 n=\frac{x-0.3320}{y-0.1858}
  19. ( x , y ) = ( 0.4402 , 0.4031 ) (x,y)=(0.4402,0.4031)
  20. R i R_{i}
  21. Δ E i \Delta E_{i}
  22. R i = 100 - 4.6 Δ E U V W R_{i}=100-4.6\Delta E_{UVW}
  23. Δ E U V W \Delta E_{UVW}
  24. R o u t = 10 ln [ exp ( R i n / 10 ) + 1 ] R_{out}=10\ln\left[\exp(R_{in}/10)+1\right]

Colors_of_noise.html

  1. f max 3627 × M M f_{\,\text{max}}\approx 3627\times{\,\text{M}_{\odot}\over\,\text{M}}

Colossally_abundant_number.html

  1. σ ( n ) n 1 + ε σ ( k ) k 1 + ε \frac{\sigma(n)}{n^{1+\varepsilon}}\geq\frac{\sigma(k)}{k^{1+\varepsilon}}
  2. σ ( n ) n 1 + ε \frac{\sigma(n)}{n^{1+\varepsilon}}
  3. c n = i = 1 n p i c_{n}=\prod_{i=1}^{n}p_{i}
  4. σ ( n ) < e γ n log log n 1.781072418 n log log n \sigma(n)<e^{\gamma}n\log\log n\approx 1.781072418\cdot n\log\log n\,
  5. σ ( n ) < H n + e x p ( H n ) l o g ( H n ) \sigma(n)<H_{n}+exp(H_{n})log(H_{n})
  6. σ ( n ) < e x p ( H n ) l o g ( H n ) \sigma(n)<exp(H_{n})log(H_{n})

Combinatorial_principles.html

  1. | i = 1 n A i | = i = 1 n | A i | - i , j : 1 i < j n | A i A j | + i , j , k : 1 i < j < k n | A i A j A k | - + ( - 1 ) n - 1 | A 1 A n | . \begin{aligned}\displaystyle\biggl|\bigcup_{i=1}^{n}A_{i}\biggr|&\displaystyle% {}=\sum_{i=1}^{n}\left|A_{i}\right|-\sum_{i,j\,:\,1\leq i<j\leq n}\left|A_{i}% \cap A_{j}\right|\\ &\displaystyle{}\qquad+\sum_{i,j,k\,:\,1\leq i<j<k\leq n}\left|A_{i}\cap A_{j}% \cap A_{k}\right|-\ \cdots\ +\left(-1\right)^{n-1}\left|A_{1}\cap\cdots\cap A_% {n}\right|.\end{aligned}
  2. G ( a n ; x ) = n = 0 a n x n . G(a_{n};x)=\sum_{n=0}^{\infty}a_{n}x^{n}.

Comic_book_price_guide.html

  1. + +
  2. - -

Comma_(music).html

  1. 81 64 80 81 = 1 5 4 1 = 5 4 {81\over 64}\cdot{80\over 81}={{1\cdot 5}\over{4\cdot 1}}={5\over 4}
  2. 32 27 81 80 = 2 3 1 5 = 6 5 {32\over 27}\cdot{81\over 80}={{2\cdot 3}\over{1\cdot 5}}={6\over 5}

Common-mode_rejection_ratio.html

  1. V + V_{+}
  2. V - V_{-}
  3. V o = A d ( V + - V - ) V_{\mathrm{o}}=A_{\mathrm{d}}(V_{+}-V_{-})
  4. A d A_{\mathrm{d}}
  5. V o = A d ( V + - V - ) + 1 2 A cm ( V + + V - ) , V_{\mathrm{o}}=A_{\mathrm{d}}(V_{+}-V_{-})+\tfrac{1}{2}A_{\mathrm{cm}}(V_{+}+V% _{-}),
  6. A cm A_{\mathrm{cm}}
  7. CMRR = ( A d | A cm | ) = 10 log 10 ( A d A cm ) 2 d B = 20 log 10 ( A d | A cm | ) d B \mathrm{CMRR}=\left(\frac{A_{\mathrm{d}}}{|A_{\mathrm{cm}}|}\right)=10\log_{10% }\left(\frac{A_{\mathrm{d}}}{A_{\mathrm{cm}}}\right)^{2}dB=20\log_{10}\left(% \frac{A_{\mathrm{d}}}{|A_{\mathrm{cm}}|}\right)dB

Common_drain.html

  1. A i = i out i in {A_{\mathrm{i}}}={i_{\mathrm{out}}\over i_{\mathrm{in}}}
  2. \infty
  3. \infty
  4. A v = v out v in {A_{\mathrm{v}}}={v_{\mathrm{out}}\over v_{\mathrm{in}}}
  5. g m R S g m R S + 1 \frac{g_{m}R_{\,\text{S}}}{g_{m}R_{\,\text{S}}+1}
  6. 1 \approx 1
  7. ( g m R S 1 ) (g_{m}R_{\,\text{S}}\gg 1)
  8. r in = v i n i i n r_{\mathrm{in}}=\frac{v_{in}}{i_{in}}
  9. \infty
  10. \infty
  11. r out = v o u t i o u t r_{\mathrm{out}}=\frac{v_{out}}{i_{out}}
  12. R S 1 g m = R S g m R S + 1 R_{\,\text{S}}\|\frac{1}{g_{m}}=\frac{R_{\,\text{S}}}{g_{m}R_{\,\text{S}}+1}
  13. 1 g m \approx\frac{1}{g_{m}}
  14. ( g m R S 1 ) (g_{m}R_{S}\gg 1)

Common_gate.html

  1. A i = i out i S | R L = 0 {A_{i}}={i_{\mathrm{out}}\over i_{\mathrm{S}}}\Big|_{R_{L}=0}
  2. 1 \ 1
  3. 1 \ 1
  4. A v = v out v S | R L = {A_{v}}={v_{\mathrm{out}}\over v_{\mathrm{S}}}\Big|_{R_{L}=\infty}
  5. ( ( g m + g m b ) r O + 1 ) R L r O + R L \begin{matrix}((g_{m}+g_{mb})r_{\mathrm{O}}+1)\frac{R_{L}}{r_{O}+R_{L}}\end{matrix}
  6. g m R L \ g_{m}R_{L}
  7. R in = v S i S R_{\mathrm{in}}=\frac{v_{S}}{i_{S}}
  8. R L + r O ( g m + g m b ) r O + 1 {{R_{L}+r_{O}}\over{(g_{m}+g_{mb})r_{O}+1}}
  9. 1 g m \begin{matrix}\frac{1}{g_{m}}\end{matrix}
  10. R out = v x i x R_{\mathrm{out}}=\frac{v_{x}}{i_{x}}
  11. ( 1 + ( g m + g m b ) r O ) R S + r O \ (1+(g_{m}+g_{mb})r_{O})R_{S}+r_{O}
  12. r O r_{O}
  13. A v g m R L 1 + g m R S {A_{\mathrm{v}}}\approx\begin{matrix}\frac{g_{m}R_{\mathrm{L}}}{1+g_{m}R_{S}}% \end{matrix}
  14. A v = R L R S or A v = g m R L A_{\mathrm{v}}=\begin{matrix}\frac{R_{L}}{R_{S}}\end{matrix}\ \ \mathrm{or}\ % \ A_{\mathrm{v}}=g_{m}R_{L}

Common_knowledge_(logic).html

  1. φ \varphi
  2. φ \varphi
  3. φ \varphi
  4. E G φ i G K i φ , E_{G}\varphi\Leftrightarrow\bigwedge_{i\in G}K_{i}\varphi,
  5. E G E G n - 1 φ E_{G}E_{G}^{n-1}\varphi
  6. E G n φ E_{G}^{n}\varphi
  7. E G 0 φ = φ E_{G}^{0}\varphi=\varphi
  8. C φ i = 0 E i φ C\varphi\Leftrightarrow\bigwedge_{i=0}^{\infty}E^{i}\varphi
  9. C G φ = ψ E G ( C G φ ) C_{G}\varphi=\psi\wedge E_{G}(C_{G}\varphi)
  10. ψ \psi
  11. E G ( φ C G φ ) E_{G}(\varphi\wedge C_{G}\varphi)
  12. φ \varphi
  13. R 1 , , R n R_{1},\dots,R_{n}
  14. S × S S\times S
  15. π \pi
  16. K i φ K_{i}\varphi
  17. φ \varphi
  18. ( s , t ) R i (s,t)\in R_{i}
  19. R i R_{i}
  20. R G R_{G}
  21. C G φ C_{G}\varphi
  22. φ \varphi
  23. ( s , t ) R G (s,t)\in R_{G}
  24. K i ( e ) = { s S | P i ( s ) e } K_{i}(e)=\{s\in S|P_{i}(s)\subset e\}
  25. E ( e ) = i K i ( e ) E(e)=\bigcap_{i}K_{i}(e)
  26. E 1 ( e ) = E ( e ) E^{1}(e)=E(e)
  27. E n + 1 ( e ) = E ( E n ( e ) ) E^{n+1}(e)=E(E^{n}(e))
  28. C ( e ) = n = 1 E n ( e ) . C(e)=\bigcap_{n=1}^{\infty}E^{n}(e).
  29. R i R_{i}
  30. P i P_{i}
  31. s E p s\in E^{p}
  32. E p E^{p}
  33. R G R_{G}
  34. P i P_{i}
  35. i G i\in G

Common_source.html

  1. A i i out i in A\text{i}\triangleq\frac{i\text{out}}{i\text{in}}\,
  2. \infty\,
  3. A v v out v in A\text{v}\triangleq\frac{v\text{out}}{v\text{in}}\,
  4. - g m R D 1 + g m R S \begin{matrix}-\frac{g_{\mathrm{m}}R\text{D}}{1+g_{\mathrm{m}}R\text{S}}\end{% matrix}\,
  5. r in v in i in r\text{in}\triangleq\frac{v\text{in}}{i\text{in}}\,
  6. \infty\,
  7. r out v out i out r\text{out}\triangleq\frac{v\text{out}}{i\text{out}}
  8. R D R\text{D}\,
  9. 1 + | A v | 1+|A\text{v}|\,
  10. i M = j ω C M v GS = j ω C M v G \ i_{\mathrm{M}}=j\omega C_{\mathrm{M}}v_{\mathrm{GS}}=j\omega C_{\mathrm{M}}v% _{\mathrm{G}}
  11. C M = C gd v GD v GS = C gd ( 1 - v D v G ) C_{\mathrm{M}}=C_{\mathrm{gd}}\frac{v_{\mathrm{GD}}}{v_{\mathrm{GS}}}=C_{% \mathrm{gd}}\left(1-\frac{v_{\mathrm{D}}}{v_{\mathrm{G}}}\right)
  12. v D v G - g m ( r O R L ) \frac{v_{\mathrm{D}}}{v_{\mathrm{G}}}\approx-g_{\mathrm{m}}(r_{\mathrm{O}}\|R_% {\mathrm{L}})
  13. C M = C gd ( 1 + g m ( r O R L ) ) C_{\mathrm{M}}=C_{\mathrm{gd}}\left(1+g_{\mathrm{m}}(r_{\mathrm{O}}\|R_{% \mathrm{L}})\right)
  14. v G = V A 1 / ( j ω C M ) 1 / ( j ω C M ) + R A = V A 1 1 + j ω C M R A v_{\mathrm{G}}=V_{\mathrm{A}}\frac{1/(j\omega C_{\mathrm{M}})}{1/(j\omega C_{% \mathrm{M}})+R_{\mathrm{A}}}=V_{\mathrm{A}}\frac{1}{1+j\omega C_{\mathrm{M}}R_% {\mathrm{A}}}
  15. f 3 d B = 1 2 π R A C M = 1 2 π R A [ C gd ( 1 + g m ( r O R L ) ] f_{\mathrm{3dB}}=\frac{1}{2\pi R_{\mathrm{A}}C_{\mathrm{M}}}=\frac{1}{2\pi R_{% \mathrm{A}}[C_{\mathrm{gd}}(1+g_{\mathrm{m}}(r_{\mathrm{O}}\|R_{\mathrm{L}})]}
  16. f 3 d B = 1 2 π R A ( C M + C gs ) = 1 2 π R A [ C gs + C gd ( 1 + g m ( r O R L ) ) ] f_{\mathrm{3dB}}=\frac{1}{2\pi R_{\mathrm{A}}(C_{\mathrm{M}}+C_{\mathrm{gs}})}% =\frac{1}{2\pi R_{\mathrm{A}}[C_{\mathrm{gs}}+C_{\mathrm{gd}}(1+g_{\mathrm{m}}% (r_{\mathrm{O}}\|R_{\mathrm{L}}))]}

Comodule.html

  1. ρ : M M C \rho:M\to M\otimes C
  2. ( i d Δ ) ρ = ( ρ i d ) ρ (id\otimes\Delta)\circ\rho=(\rho\otimes id)\circ\rho
  3. ( i d ε ) ρ = i d (id\otimes\varepsilon)\circ\rho=id
  4. M K M\otimes K
  5. M M\,
  6. C I C_{I}
  7. e i e_{i}
  8. i I i\in I
  9. C I C_{I}
  10. C I C_{I}
  11. C I C_{I}
  12. Δ ( e i ) = e i e i \Delta(e_{i})=e_{i}\otimes e_{i}
  13. C I C_{I}
  14. ε ( e i ) = 1 \varepsilon(e_{i})=1
  15. ρ \rho
  16. ρ ( v ) = v i e i \rho(v)=\sum v_{i}\otimes e_{i}
  17. v i v_{i}
  18. v v

Compactification_(physics).html

  1. M × C M\times C
  2. C C

Compensating_variation.html

  1. C V = e ( p 1 , u 1 ) - e ( p 1 , u 0 ) CV=e(p_{1},u_{1})-e(p_{1},u_{0})
  2. = w - e ( p 1 , u 0 ) =w-e(p_{1},u_{0})
  3. = e ( p 0 , u 0 ) - e ( p 1 , u 0 ) =e(p_{0},u_{0})-e(p_{1},u_{0})
  4. w w
  5. p 0 p_{0}
  6. p 1 p_{1}
  7. u 0 u_{0}
  8. u 1 u_{1}
  9. v ( p 1 , w - C V ) = u 0 v(p_{1},w-CV)=u_{0}
  10. e ( p 1 , v ( p 1 , w - C V ) ) = e ( p 1 , u 0 ) e(p_{1},v(p_{1},w-CV))=e(p_{1},u_{0})
  11. w - C V = e ( p 1 , u 0 ) w-CV=e(p_{1},u_{0})
  12. C V = w - e ( p 1 , u 0 ) CV=w-e(p_{1},u_{0})

Competitive_Lotka–Volterra_equations.html

  1. d x d t = r x ( 1 - x K ) . {dx\over dt}=rx\left(1-{x\over K}\right).
  2. d x 1 d t = r 1 x 1 ( 1 - ( x 1 + α 12 x 2 K 1 ) ) {dx_{1}\over dt}=r_{1}x_{1}\left(1-\left({x_{1}+\alpha_{12}x_{2}\over K_{1}}% \right)\right)
  3. d x 2 d t = r 2 x 2 ( 1 - ( x 2 + α 21 x 1 K 2 ) ) . {dx_{2}\over dt}=r_{2}x_{2}\left(1-\left({x_{2}+\alpha_{21}x_{1}\over K_{2}}% \right)\right).
  4. d x i d t = r i x i ( 1 - j = 1 N α i j x j K i ) \frac{dx_{i}}{dt}=r_{i}x_{i}\left(1-\frac{\sum_{j=1}^{N}\alpha_{ij}x_{j}}{K_{i% }}\right)
  5. d x i d t = r i x i ( 1 - j = 1 N α i j x j ) \frac{dx_{i}}{dt}=r_{i}x_{i}\left(1-\sum_{j=1}^{N}\alpha_{ij}x_{j}\right)
  6. Δ N - 1 = { x i : x i 0 , x i = 1 } \Delta_{N-1}=\left\{x_{i}:x_{i}\geq 0,\sum x_{i}=1\right\}
  7. r i = [ 1 0.72 1.53 1.27 ] α i j = [ 1 1.09 1.52 0 0 1 0.44 1.36 2.33 0 1 0.47 1.21 0.51 0.35 1 ] r_{i}=\begin{bmatrix}1\\ 0.72\\ 1.53\\ 1.27\end{bmatrix}\quad\alpha_{ij}=\begin{bmatrix}1&1.09&1.52&0\\ 0&1&0.44&1.36\\ 2.33&0&1&0.47\\ 1.21&0.51&0.35&1\end{bmatrix}
  8. K i = 1 K_{i}=1
  9. i i
  10. x ¯ = ( α i j ) - 1 [ 1 1 1 1 ] = [ 0.3013 0.4586 0.1307 0.3557 ] . \overline{x}=\left(\alpha_{ij}\right)^{-1}\begin{bmatrix}1\\ 1\\ 1\\ 1\end{bmatrix}=\begin{bmatrix}0.3013\\ 0.4586\\ 0.1307\\ 0.3557\end{bmatrix}.
  11. α i j = [ 1 α 1 0 0 α - 1 α - 1 1 α 1 0 0 0 α - 1 1 α 1 0 0 0 α - 1 1 α 1 α 1 0 0 α - 1 1 ] . \alpha_{ij}=\begin{bmatrix}1&\alpha_{1}&0&0&\alpha_{-1}\\ \alpha_{-1}&1&\alpha_{1}&0&0\\ 0&\alpha_{-1}&1&\alpha_{1}&0\\ 0&0&\alpha_{-1}&1&\alpha_{1}\\ \alpha_{1}&0&0&\alpha_{-1}&1\end{bmatrix}.
  12. x ¯ i = 1 j = 1 N α i j = 1 α - 1 + 1 + α 1 . \overline{x}_{i}=\frac{1}{\sum_{j=1}^{N}\alpha_{ij}}=\frac{1}{\alpha_{-1}+1+% \alpha_{1}}.
  13. λ k = j = 0 N - 1 c j γ k j \lambda_{k}=\sum_{j=0}^{N-1}c_{j}\gamma^{kj}
  14. γ = e i 2 π / N \gamma=e^{i2\pi/N}
  15. Re ( λ k ) = Re ( 1 + α - 2 e i 2 π k ( N - 2 ) / N + α - 1 e i 2 π k ( N - 1 ) / N + α 1 e i 2 π k / N + α 2 e i 4 π k / N ) \operatorname{Re}(\lambda_{k})=\operatorname{Re}\left(1+\alpha_{-2}e^{i2\pi k(% N-2)/N}+\alpha_{-1}e^{i2\pi k(N-1)/N}+\alpha_{1}e^{i2\pi k/N}+\alpha_{2}e^{i4% \pi k/N}\right)
  16. = 1 + ( α - 2 + α 2 ) cos ( i 4 π k N ) + ( α - 1 + α 1 ) cos ( i 2 π k N ) > 0 =1+(\alpha_{-2}+\alpha_{2})\cos\left(\frac{i4\pi k}{N}\right)+(\alpha_{-1}+% \alpha_{1})\cos\left(\frac{i2\pi k}{N}\right)>0
  17. α i j = [ 1 α 1 0 0 0 α - 1 1 α 1 0 0 0 α - 1 1 α 1 0 0 0 α - 1 1 α 1 0 0 0 α - 1 1 ] \alpha_{ij}=\begin{bmatrix}1&\alpha_{1}&0&0&0\\ \alpha_{-1}&1&\alpha_{1}&0&0\\ 0&\alpha_{-1}&1&\alpha_{1}&0\\ 0&0&\alpha_{-1}&1&\alpha_{1}\\ 0&0&0&\alpha_{-1}&1\end{bmatrix}

Complementary_event.html

  1. A ¯ \overline{A}
  2. P ( A c ) = 1 - P ( A ) . P(A^{c})=1-P(A).

Complete_Boolean_algebra.html

  1. A B A\land B
  2. a b a\cap b

Complete_set_of_commuting_observables.html

  1. A A
  2. B B
  3. A ^ \hat{A}
  4. B ^ \hat{B}
  5. A A
  6. B B
  7. A ^ \hat{A}
  8. B ^ \hat{B}
  9. A ^ \hat{A}
  10. B ^ \hat{B}
  11. [ A ^ , B ^ ] = 0 [\hat{A},\hat{B}]=0
  12. { | ψ n } \{|\psi_{n}\rangle\}
  13. A A
  14. B B
  15. { a n } \{a_{n}\}
  16. { b n } \{b_{n}\}
  17. A B | ψ n = A b n | ψ n = a n b n | ψ n = b n a n | ψ n = B A | ψ n AB|\psi_{n}\rangle=Ab_{n}|\psi_{n}\rangle=a_{n}b_{n}|\psi_{n}\rangle=b_{n}a_{n% }|\psi_{n}\rangle=BA|\psi_{n}\rangle
  18. | Ψ |\Psi\rangle
  19. { | ψ n } \{|\psi_{n}\rangle\}
  20. | Ψ = n c n | ψ n |\Psi\rangle=\sum_{n}c_{n}|\psi_{n}\rangle
  21. ( A B - B A ) | Ψ = n c n ( A B - B A ) | ψ n = 0 (AB-BA)|\Psi\rangle=\sum_{n}c_{n}(AB-BA)|\psi_{n}\rangle=0
  22. [ A , B ] = 0 [A,B]=0
  23. A A
  24. { | ψ n } \{|\psi_{n}\rangle\}
  25. A A
  26. { a n } \{a_{n}\}
  27. A A
  28. B B
  29. A ( B | ψ n ) = B A | ψ n = a n ( B | ψ n ) A(B|\psi_{n}\rangle)=BA|\psi_{n}\rangle=a_{n}(B|\psi_{n}\rangle)
  30. B | ψ n B|\psi_{n}\rangle
  31. A A
  32. a n a_{n}
  33. a n a_{n}
  34. B | ψ n B|\psi_{n}\rangle
  35. | ψ n |\psi_{n}\rangle
  36. b n b_{n}
  37. B | ψ n = b n | ψ n B|\psi_{n}\rangle=b_{n}|\psi_{n}\rangle
  38. | ψ n |\psi_{n}\rangle
  39. A A
  40. B B
  41. A A
  42. a n a_{n}
  43. g g
  44. | ψ n r , ( r = 1 , 2 , g ) |\psi_{nr}\rangle,(r=1,2,...g)
  45. [ A , B ] = 0 [A,B]=0
  46. B | ψ n r B|\psi_{nr}\rangle
  47. A A
  48. a n a_{n}
  49. B | ψ n r B|\psi_{nr}\rangle
  50. a n a_{n}
  51. B | ψ n r = s = 1 g c r s | ψ n s B|\psi_{nr}\rangle=\sum_{s=1}^{g}c_{rs}|\psi_{ns}\rangle
  52. c r s c_{rs}
  53. r r
  54. g g
  55. d r d_{r}
  56. B r = 1 g d r | ψ n r = r = 1 g s = 1 g d r c r s | ψ n s B\sum_{r=1}^{g}d_{r}|\psi_{nr}\rangle=\sum_{r=1}^{g}\sum_{s=1}^{g}d_{r}c_{rs}|% \psi_{ns}\rangle
  57. r = 1 g d r | ψ n r \sum_{r=1}^{g}d_{r}|\psi_{nr}\rangle
  58. B B
  59. b n b_{n}
  60. r = 1 g d r c r s = b n d s , s = 1 , 2 , g \sum_{r=1}^{g}d_{r}c_{rs}=b_{n}d_{s},s=1,2,...g
  61. g g
  62. d r d_{r}
  63. det [ c r s - b n δ r s ] = 0 \det[c_{rs}-b_{n}\delta_{rs}]=0
  64. g g
  65. b n b_{n}
  66. g g
  67. b n = b n ( k ) , k = 1 , 2 , g b_{n}=b_{n}^{(k)},k=1,2,...g
  68. d r d_{r}
  69. d r ( k ) d_{r}^{(k)}
  70. | ϕ n ( k ) = r = 1 g d r ( k ) | ψ n r |\phi_{n}^{(k)}\rangle=\sum_{r=1}^{g}d_{r}^{(k)}|\psi_{nr}\rangle
  71. A A
  72. B B
  73. a n a_{n}
  74. b n ( k ) b_{n}^{(k)}
  75. A A
  76. B B
  77. { | ψ n } \{|\psi_{n}\rangle\}
  78. A A
  79. B B
  80. A A
  81. B B
  82. A A
  83. B B
  84. | ψ n |\psi_{n}\rangle
  85. A A
  86. B B
  87. A | ψ n = a n | ψ n A|\psi_{n}\rangle=a_{n}|\psi_{n}\rangle
  88. B | ψ n = b n | ψ n B|\psi_{n}\rangle=b_{n}|\psi_{n}\rangle
  89. | ψ n |\psi_{n}\rangle
  90. A A
  91. B B
  92. a n a_{n}
  93. b n b_{n}
  94. 𝐫 \mathbf{r}
  95. x x
  96. y y
  97. z z
  98. 𝐩 \mathbf{p}
  99. p x p_{x}
  100. p y p_{y}
  101. p z p_{z}
  102. A , B , C A,B,C...
  103. A ^ \hat{A}
  104. A A
  105. { a n } \{a_{n}\}
  106. A ^ \hat{A}
  107. a n a_{n}
  108. | a n |a_{n}\rangle
  109. a n a_{n}
  110. B B
  111. A A
  112. A ^ \hat{A}
  113. B ^ \hat{B}
  114. ( a n , b n ) (a_{n},b_{n})
  115. { A , B } \{A,B\}
  116. A ^ \hat{A}
  117. ( a n , b n ) (a_{n},b_{n})
  118. C C
  119. A A
  120. B B
  121. A ^ \hat{A}
  122. B ^ \hat{B}
  123. C ^ \hat{C}
  124. ( a n , b n , c n ) (a_{n},b_{n},c_{n})
  125. { A , B , C } \{A,B,C\}
  126. { A , B , C , , } \{A,B,C,...,\}
  127. | ψ = i , j , k , c i , j , k , | a i , b j , c k , |\psi\rangle=\sum_{i,j,k,...}c_{i,j,k,...}|a_{i},b_{j},c_{k},...\rangle
  128. | a i , b j , c k , |a_{i},b_{j},c_{k},...\rangle
  129. A ^ , B ^ , C ^ \hat{A},\hat{B},\hat{C}
  130. A ^ | a i , b j , c k , = a i | a i , b j , c k , \hat{A}|a_{i},b_{j},c_{k},...\rangle=a_{i}|a_{i},b_{j},c_{k},...\rangle
  131. A , B , C , A,B,C,...
  132. | ψ |\psi\rangle
  133. a i , b j , c k , a_{i},b_{j},c_{k},...
  134. | c i , j , k , | 2 |c_{i,j,k,...}|^{2}
  135. 𝐋 \mathbf{L}
  136. [ L i , L j ] = i ϵ i j k L k [L_{i},L_{j}]=i\hbar\epsilon_{ijk}L_{k}
  137. 𝐋 \mathbf{L}
  138. L 2 L^{2}
  139. 𝐋 \mathbf{L}
  140. [ L x , L 2 ] = 0 , [ L y , L 2 ] = 0 , [ L z , L 2 ] = 0 [L_{x},L^{2}]=0,[L_{y},L^{2}]=0,[L_{z},L^{2}]=0
  141. H ^ = - 2 2 μ 2 - Z e 2 r \hat{H}=-\frac{\hbar^{2}}{2\mu}\nabla^{2}-\frac{Ze^{2}}{r}
  142. r r
  143. μ \mu
  144. 𝐋 \mathbf{L}
  145. [ 𝐋 , H ] = 0 , [ L 2 , H ] = 0 [\mathbf{L},H]=0,[L^{2},H]=0
  146. L 2 L^{2}
  147. 𝐋 \mathbf{L}
  148. L z L_{z}
  149. H H
  150. { H , L 2 , L z } \{H,L^{2},L_{z}\}
  151. | E n , l , m |E_{n},l,m\rangle
  152. H | E n , l , m = E n | E n , l , m H|E_{n},l,m\rangle=E_{n}|E_{n},l,m\rangle
  153. L 2 | E n , l , m = l ( l + 1 ) 2 | E n , l , m L^{2}|E_{n},l,m\rangle=l(l+1)\hbar^{2}|E_{n},l,m\rangle
  154. L z | E n , l , m = m | E n , l , m L_{z}|E_{n},l,m\rangle=m\hbar|E_{n},l,m\rangle
  155. { E n , l , m } \{E_{n},l,m\}
  156. { n , l , m } \{n,l,m\}
  157. H = - 2 2 m 2 H=-\frac{\hbar^{2}}{2m}\nabla^{2}
  158. [ H , 𝐓 ^ ] = 0 [H,\mathbf{\hat{T}}]=0
  159. H H
  160. Π \Pi
  161. [ H , Π ] = 0 [H,\Pi]=0
  162. { H , Π } \{H,\Pi\}
  163. | k |k\rangle
  164. | - k |-k\rangle
  165. H H
  166. H k = 2 k 2 2 m H_{k}=\frac{{\hbar^{2}}{k^{2}}}{2m}
  167. H | k = 2 k 2 2 m | k H|k\rangle=\frac{{\hbar^{2}}{k^{2}}}{2m}|k\rangle
  168. H | - k = 2 k 2 2 m | - k H|-k\rangle=\frac{{\hbar^{2}}{k^{2}}}{2m}|-k\rangle
  169. H H
  170. 𝐩 ^ \mathbf{\hat{p}}
  171. 𝐩 ^ | k = k | k \mathbf{\hat{p}}|k\rangle=k|k\rangle
  172. 𝐩 ^ | - k = - k | - k \mathbf{\hat{p}}|-k\rangle=-k|-k\rangle
  173. { 𝐩 ^ , H } \{\mathbf{\hat{p}},H\}
  174. 𝐉 𝟏 \mathbf{J_{1}}
  175. 𝐉 𝟐 \mathbf{J_{2}}
  176. J 1 2 J_{1}^{2}
  177. J 1 z J_{1z}
  178. | j 1 m 1 |j_{1}m_{1}\rangle
  179. J 2 2 J_{2}^{2}
  180. J 2 z J_{2z}
  181. | j 2 m 2 |j_{2}m_{2}\rangle
  182. J 1 2 | j 1 m 1 = j 1 ( j 1 + 1 ) 2 | j 1 m 1 J_{1}^{2}|j_{1}m_{1}\rangle=j_{1}(j_{1}+1)\hbar^{2}|j_{1}m_{1}\rangle
  183. J 1 z | j 1 m 1 = m 1 | j 1 m 1 J_{1z}|j_{1}m_{1}\rangle=m_{1}\hbar|j_{1}m_{1}\rangle
  184. J 2 2 | j 2 m 2 = j 2 ( j 2 + 1 ) 2 | j 2 m 2 J_{2}^{2}|j_{2}m_{2}\rangle=j_{2}(j_{2}+1)\hbar^{2}|j_{2}m_{2}\rangle
  185. J 2 z | j 2 m 2 = m 2 | j 2 m 2 J_{2z}|j_{2}m_{2}\rangle=m_{2}\hbar|j_{2}m_{2}\rangle
  186. | j 1 m 1 ; j 2 m 2 |j_{1}m_{1};j_{2}m_{2}\rangle
  187. | j 1 m 1 ; j 2 m 2 = | j 1 m 1 | j 2 m 2 |j_{1}m_{1};j_{2}m_{2}\rangle=|j_{1}m_{1}\rangle\otimes|j_{2}m_{2}\rangle
  188. { j 1 , m 1 , j 2 , m 2 } \{j_{1},m_{1},j_{2},m_{2}\}
  189. { J 1 2 , J 1 z , J 2 2 , J 2 z } \{J_{1}^{2},J_{1z},J_{2}^{2},J_{2z}\}
  190. 𝐉 = 𝐉 𝟏 + 𝐉 𝟐 \mathbf{J}=\mathbf{J_{1}}+\mathbf{J_{2}}
  191. J 2 J^{2}
  192. j ( j + 1 ) 2 j(j+1)\hbar^{2}
  193. j j
  194. j 1 + j 2 , j 1 + j 2 - 1 , , | j 1 - j 2 | j_{1}+j_{2},j_{1}+j_{2}-1,...,|j_{1}-j_{2}|
  195. J z J_{z}
  196. m m
  197. m = - j , - j + 1 , j - 1 , j m=-j,-j+1,...j-1,j
  198. J 2 J^{2}
  199. J z J_{z}
  200. | j 1 j 2 ; j m |j_{1}j_{2};jm\rangle
  201. { j 1 , j 2 , j , m } \{j_{1},j_{2},j,m\}
  202. { J 1 2 , J 2 2 , J 2 , J z } \{J_{1}^{2},J_{2}^{2},J^{2},J_{z}\}

Complex_conjugate_representation.html

  1. G G
  2. Π Π
  3. V V
  4. Π ¯ \overline{Π}
  5. V ¯ \overline{V}
  6. Π ¯ ( g ) \overline{Π}(g)
  7. Π ( g ) Π(g)
  8. g g
  9. G G
  10. Π ¯ \overline{Π}
  11. 𝐠 \mathbf{g}
  12. π π
  13. V V
  14. π ¯ \overline{π}
  15. V ¯ \overline{V}
  16. π ¯ ( X ) \overline{π}(X)
  17. π ( X ) π(X)
  18. X X
  19. 𝐠 \mathbf{g}
  20. π ¯ \overline{π}
  21. S p i n ( p + q ) Spin(p+q)
  22. S p i n ( p , q ) Spin(p,q)
  23. 𝔤 \mathfrak{g}
  24. π ¯ ( X ) \overline{π}(X)
  25. π ( X ¯ ) −π(\overline{X})
  26. X X
  27. 𝐠 \mathbf{g}

Complex_conjugate_vector_space.html

  1. V V\,
  2. V ¯ \overline{V}
  3. V V\,
  4. V ¯ \overline{V}
  5. V V\,
  6. V ¯ = { v ¯ v V } , \overline{V}=\{\overline{v}\mid v\in V\},
  7. v ¯ + w ¯ = v + w ¯ and α v ¯ = α ¯ v ¯ . \overline{v}+\overline{w}=\overline{\,v+w\,}\quad\,\text{and}\quad\alpha\,% \overline{v}=\overline{\,\overline{\alpha}\,v\,}.
  8. v v\,
  9. w w\,
  10. V V\,
  11. α \alpha\,
  12. α ¯ \overline{\alpha}
  13. α \alpha\,
  14. V V\,
  15. W W\,
  16. f : V W f\colon V\to W\,
  17. f ( v + v ) = f ( v ) + f ( v ) and f ( α v ) = α ¯ f ( v ) f(v+v^{\prime})=f(v)+f(v^{\prime})\quad\,\text{and}\quad f(\alpha v)=\overline% {\alpha}\,f(v)
  18. v , v V v,v^{\prime}\in V\,
  19. α \alpha\in\mathbb{C}
  20. V ¯ \overline{V}
  21. f : V W f\colon V\to W\,
  22. V ¯ W \overline{V}\to W
  23. v ¯ f ( v ) \overline{v}\mapsto f(v)
  24. V ¯ \overline{V}
  25. V V\,
  26. C : V V ¯ C\colon V\to\overline{V}
  27. C ( v ) = v ¯ C(v)=\overline{v}
  28. f : V ¯ W f\colon\overline{V}\to W
  29. f C : V W f\circ C\colon V\to W\,
  30. f : V W f\colon V\to W\,
  31. f ¯ : V ¯ W ¯ \overline{f}\colon\overline{V}\to\overline{W}
  32. f ¯ ( v ¯ ) = f ( v ) ¯ . \overline{f}(\overline{v})=\overline{\,f(v)\,}.
  33. f ¯ \overline{f}
  34. V V\,
  35. V ¯ \overline{V}
  36. f ¯ g ¯ = f g ¯ \overline{f}\circ\overline{g}=\overline{\,f\circ g\,}
  37. f f\,
  38. g g\,
  39. V V ¯ V\mapsto\overline{V}
  40. f f ¯ f\mapsto\overline{f}
  41. V V\,
  42. W W\,
  43. f f\,
  44. A A\,
  45. \mathcal{B}
  46. V V\,
  47. 𝒞 \mathcal{C}
  48. W W\,
  49. f ¯ \overline{f}
  50. A A\,
  51. ¯ \overline{\mathcal{B}}
  52. V ¯ \overline{V}
  53. 𝒞 ¯ \overline{\mathcal{C}}
  54. W ¯ \overline{W}
  55. V V\,
  56. V ¯ \overline{V}
  57. V V\,
  58. V ¯ \overline{V}
  59. C C\,
  60. V ¯ ¯ \overline{\overline{V}}
  61. V V\,
  62. V ¯ ¯ V \overline{\overline{V}}\to V
  63. v ¯ ¯ v . \overline{\overline{v}}\mapsto v.
  64. V V\,
  65. V V\,
  66. \mathcal{H}
  67. ¯ \overline{\mathcal{H}}
  68. \mathcal{H}^{\prime}
  69. \mathcal{H}
  70. v v
  71. v * v^{*}
  72. v v
  73. | ψ |\psi\rangle
  74. ψ | \langle\psi|

Complex_convexity.html

  1. Ω \Omega
  2. n \mathbb{C}^{n}
  3. \mathbb{C}

Complex_vector_bundle.html

  1. E E\otimes\mathbb{C}
  2. J : E E J:E\to E
  3. J x : E x E x J_{x}:E_{x}\to E_{x}
  4. J x 2 = - 1 J_{x}^{2}=-1
  5. J x J_{x}
  6. i i
  7. ( a + i b ) v = a v + J ( b v ) . (a+ib)v=av+J(bv).
  8. E ¯ \overline{E}
  9. E E ¯ = E E_{\mathbb{R}}\to\overline{E}_{\mathbb{R}}=E_{\mathbb{R}}
  10. E ¯ \overline{E}
  11. E ¯ \overline{E}
  12. c k ( E ¯ ) = ( - 1 ) k c k ( E ) c_{k}(\overline{E})=(-1)^{k}c_{k}(E)
  13. E ¯ \overline{E}
  14. E ¯ \overline{E}
  15. E * = Hom ( E , 𝒪 ) E^{*}=\operatorname{Hom}(E,\mathcal{O})
  16. 𝒪 \mathcal{O}
  17. ( E ) = E E (E\otimes\mathbb{C})_{\mathbb{R}}=E\oplus E

Compound_annual_growth_rate.html

  1. CAGR ( t 0 , t n ) = ( V ( t n ) / V ( t 0 ) ) 1 t n - t 0 - 1 \mathrm{CAGR}(t_{0},t_{n})=\left({V(t_{n})/V(t_{0})}\right)^{\frac{1}{t_{n}-t_% {0}}}-1
  2. V ( t 0 ) V(t_{0})
  3. V ( t n ) V(t_{n})
  4. t n - t 0 t_{n}-t_{0}
  5. t n - t 0 {t_{n}-t_{0}}
  6. CAGR ( 0 , 3 ) = ( 13000 9000 ) 1 3 - 1 = 0.13 = 13 % {\rm CAGR}(0,3)=\left(\frac{13000}{9000}\right)^{\frac{1}{3}}-1=0.13=13\%
  7. V ( t n ) = V ( t 0 ) × ( 1 + CAGR ) n V(t_{n})=V(t_{0})\times(1+{\rm CAGR})^{n}
  8. = V ( t 0 ) × ( 1 + CAGR ) × ( 1 + CAGR ) × ( 1 + CAGR ) =V(t_{0})\times(1+{\rm CAGR})\times(1+{\rm CAGR})\times(1+{\rm CAGR})
  9. = 9000 × 1.1304 × 1.1304 × 1.1304 = 13000 =9000\times 1.1304\times 1.1304\times 1.1304=13000
  10. AMR = x ¯ = 1 n i = 1 n x i = 1 n ( x 1 + + x n ) = 11.11 % + 10 % + 8.33 % 3 = 9.81 % . \,\text{AMR}=\bar{x}=\frac{1}{n}\sum_{i=1}^{n}x_{i}=\frac{1}{n}(x_{1}+\cdots+x% _{n})=\frac{11.11\%+10\%+8.33\%}{3}=9.81\%.
  11. V ( t n ) V(t_{n})
  12. V ( t 0 ) V(t_{0})
  13. AR = V f - V i V i = 13000 - 9000 9000 = 44.44 % . \,\text{AR}=\frac{V_{f}-V_{i}}{V_{i}}=\frac{13000-9000}{9000}=44.44\%.

Compression_body.html

  1. S S
  2. S × [ 0 , 1 ] S\times[0,1]
  3. S × { 1 } S\times\{1\}
  4. C C
  5. - C \partial_{-}C
  6. S × { 0 } S\times\{0\}
  7. C C
  8. - C = \partial_{-}C=\emptyset
  9. + C \partial_{+}C
  10. C \partial C
  11. S S
  12. S × { 0 } S\times\{0\}
  13. + C \partial_{+}C
  14. S × { 1 } S\times\{1\}
  15. - C \partial_{-}C
  16. C \partial C

Compton_wavelength.html

  1. λ = h m c \lambda=\frac{h}{mc}
  2. 2 π {2\pi}
  3. λ 2 π = m c \frac{\lambda}{2\pi}=\frac{\hbar}{mc}
  4. 2 ψ - 1 c 2 2 t 2 ψ = ( m c ) 2 ψ \mathbf{\nabla}^{2}\psi-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}\psi% =\left(\frac{mc}{\hbar}\right)^{2}\psi
  5. - i γ μ μ ψ + ( m c ) ψ = 0 -i\gamma^{\mu}\partial_{\mu}\psi+\left(\frac{mc}{\hbar}\right)\psi=0\,
  6. i t ψ = - 2 2 m 2 ψ - 1 4 π ϵ 0 Z e 2 r ψ i\hbar\frac{\partial}{\partial t}\psi=-\frac{\hbar^{2}}{2m}\nabla^{2}\psi-% \frac{1}{4\pi\epsilon_{0}}\frac{Ze^{2}}{r}\psi
  7. c \hbar c
  8. i c t ψ = - 1 2 ( m c ) 2 ψ - α Z r ψ \frac{i}{c}\frac{\partial}{\partial t}\psi=-\frac{1}{2}\left(\frac{\hbar}{mc}% \right)\nabla^{2}\psi-\frac{\alpha Z}{r}\psi
  9. E = h f = h c λ = m c 2 E=hf=\frac{hc}{\lambda}=mc^{2}
  10. Δ x Δ p 2 , \Delta x\,\Delta p\geq\frac{\hbar}{2},
  11. Δ p 2 Δ x . \Delta p\geq\frac{\hbar}{2\Delta x}.
  12. Δ x 1 2 ( m c ) . \Delta x\geq\frac{1}{2}\left(\frac{\hbar}{mc}\right).
  13. λ ¯ e λ e 2 π 386 fm \bar{\lambda}_{e}\equiv\tfrac{\lambda_{e}}{2\pi}\simeq 386~{}\textrm{fm}
  14. α 1 137 \alpha\simeq\tfrac{1}{137}
  15. a 0 = 1 α ( λ e 2 π ) 137 × λ ¯ e 5.29 × 10 4 fm a_{0}=\frac{1}{\alpha}\left(\frac{\lambda_{e}}{2\pi}\right)\simeq 137\times% \bar{\lambda}_{e}\simeq 5.29\times 10^{4}~{}\textrm{fm}
  16. r e = α ( λ e 2 π ) λ ¯ e 137 2.82 fm r_{e}=\alpha\left(\frac{\lambda_{e}}{2\pi}\right)\simeq\frac{\bar{\lambda}_{e}% }{137}\simeq 2.82~{}\textrm{fm}
  17. R = α 2 2 λ e R_{\infty}=\frac{\alpha^{2}}{2\lambda_{e}}
  18. σ T = 8 π 3 α 2 λ ¯ e 2 66.5 fm 2 \sigma_{T}=\frac{8\pi}{3}\alpha^{2}\bar{\lambda}_{e}^{2}\simeq 66.5~{}\textrm{% fm}^{2}
  19. α G \alpha_{G}
  20. P \ell_{P}
  21. P = λ e α G 2 π \ell_{P}=\lambda_{e}\,\frac{\sqrt{\alpha_{G}}}{2\pi}

Computational_Diffie–Hellman_assumption.html

  1. ( g , g a , g b ) (g,g^{a},g^{b})\,
  2. a , b { 0 , , q - 1 } , a,b\in\{0,\ldots,q-1\},\,
  3. g a b . g^{ab}.\,
  4. g g
  5. 𝔾 {\mathbb{G}}
  6. ( g , g a , g b ) , (g,g^{a},g^{b}),\,
  7. g a b g^{ab}
  8. a a
  9. g a g^{a}
  10. g g
  11. g a b g^{ab}
  12. g a b = ( g b ) a g^{ab}=(g^{b})^{a}
  13. ( g , g a , g b , g a b ) (g,g^{a},g^{b},g^{ab})
  14. g a b g^{ab}
  15. ( g , g a , g b ) (g,g^{a},g^{b})

Computational_indistinguishability.html

  1. { D n } n \scriptstyle\{D_{n}\}_{n\in\mathbb{N}}
  2. { E n } n \scriptstyle\{E_{n}\}_{n\in\mathbb{N}}
  3. δ ( n ) = | Pr x D n [ A ( x ) = 1 ] - Pr x E n [ A ( x ) = 1 ] | . \delta(n)=\left|\Pr_{x\leftarrow D_{n}}[A(x)=1]-\Pr_{x\leftarrow E_{n}}[A(x)=1% ]\right|.
  4. D n E n D_{n}\approx E_{n}\!\,
  5. n n\to\infty
  6. A A

Condensation_algorithm.html

  1. 𝐱 𝐭 \mathbf{x_{t}}
  2. t t
  3. 𝐳 𝟏 , , 𝐳 𝐭 \mathbf{z_{1},...,z_{t}}
  4. p ( 𝐱 𝐭 | 𝐳 𝟏 , , 𝐳 𝐭 ) p(\mathbf{x_{t}}|\mathbf{z_{1},...,z_{t}})
  5. p ( 𝐱 𝐭 | 𝐳 𝟏 , , 𝐳 𝐭 ) p(\mathbf{x_{t}}|\mathbf{z_{1},...,z_{t}})
  6. p ( 𝐱 𝐭 | 𝐳 𝟏 , , 𝐳 𝐭 ) p(\mathbf{x_{t}}|\mathbf{z_{1},...,z_{t}})
  7. { s t ( n ) , n = 1 , , N } \{s_{t}^{(n)},n=1,...,N\}
  8. π t ( n ) \pi_{t}^{(n)}
  9. p ( 𝐱 𝐭 | 𝐳 𝟏 , , 𝐳 𝐭 ) p(\mathbf{x_{t}}|\mathbf{z_{1},...,z_{t}})
  10. s t s_{t}
  11. π t \pi_{t}
  12. p ( 𝐱 𝐭 | 𝐱 𝐭 - 𝟏 ) p(\mathbf{x_{t}}|\mathbf{x_{t-1}})
  13. p ( 𝐱 𝐭 | 𝐱 𝐭 - 𝟏 ) p(\mathbf{x_{t}}|\mathbf{x_{t-1}})
  14. t = 0 t=0
  15. { s 0 ( n ) , n = 1 , , N } \{s_{0}^{(n)},n=1,...,N\}
  16. { π 0 ( n ) , n = 1 , , N } \{\pi_{0}^{(n)},n=1,...,N\}
  17. p ( 𝐱 𝐭 | 𝐳 𝟏 , , 𝐳 𝐭 ) p(\mathbf{x_{t}}|\mathbf{z_{1},...,z_{t}})
  18. p ( 𝐱 𝐭 | 𝐱 𝐭 - 𝟏 ) p(\mathbf{x_{t}}|\mathbf{x_{t-1}})
  19. { s t ( n ) } \{s_{t}^{(n)}\}
  20. 𝐳 𝐭 \mathbf{z_{t}}
  21. π t ( n ) = p ( 𝐳 𝐭 | s ( n ) ) j = 1 N p ( 𝐳 𝐭 | s ( j ) ) \pi_{t}^{(n)}=\frac{p(\mathbf{z_{t}}|s^{(n)})}{\sum_{j=1}^{N}p(\mathbf{z_{t}}|% s^{(j)})}
  22. { s t ( n ) } \{s_{t}^{(n)}\}
  23. p ( 𝐱 𝐭 | 𝐳 𝟏 , , 𝐳 𝐭 ) p(\mathbf{x_{t}}|\mathbf{z_{1},...,z_{t}})
  24. N N
  25. p ( 𝐱 𝐭 | 𝐱 𝐭 - 𝟏 ) p(\mathbf{x_{t}}|\mathbf{x_{t-1}})
  26. p ( 𝐱 𝐭 | 𝐱 𝐭 - 𝟏 ) e - 1 2 | | B - 1 ( ( 𝐱 𝐭 - 𝐱 ¯ ) - A ( 𝐱 𝐭 - 𝟏 - 𝐱 ¯ ) ) | | 2 ) p(\mathbf{x_{t}}|\mathbf{x_{t-1}})\propto e^{-\frac{1}{2}||B^{-1}((\mathbf{x_{% t}}-\mathbf{\bar{x}})-A(\mathbf{x_{t-1}}-\mathbf{\bar{x}}))||^{2})}
  27. 𝐱 ¯ \mathbf{\bar{x}}
  28. A A
  29. B B
  30. A A
  31. B B
  32. 𝐱 ¯ \mathbf{\bar{x}}
  33. p ( 𝐳 | 𝐱 ) p(\mathbf{z}|\mathbf{x})
  34. λ \lambda
  35. σ \sigma

Conditional_convergence.html

  1. n = 0 a n \scriptstyle\sum\limits_{n=0}^{\infty}a_{n}
  2. lim m n = 0 m a n \scriptstyle\lim\limits_{m\rightarrow\infty}\,\sum\limits_{n=0}^{m}\,a_{n}
  3. n = 0 | a n | = . \scriptstyle\sum\limits_{n=0}^{\infty}\left|a_{n}\right|=\infty.
  4. 1 - 1 2 + 1 3 - 1 4 + 1 5 - = n = 1 ( - 1 ) n + 1 n 1-{1\over 2}+{1\over 3}-{1\over 4}+{1\over 5}-\cdots=\sum\limits_{n=1}^{\infty% }{(-1)^{n+1}\over n}
  5. ln ( 2 ) \ln(2)\,\!
  6. sin ( x 2 ) \sin(x^{2})

Cone_(linear_algebra).html

  1. V { 0 } 𝐏 V . V\setminus\{0\}\to\mathbf{P}V.
  2. S = { x V : | x | = 1 } S=\{\,x\in V\;:\;|x|=1\,\}
  3. C = { x | x | : x C x 𝟎 } C^{\prime}=\bigg\{\,\frac{x}{|x|}\;:\;x\in C\wedge x\neq\mathbf{0}\,\bigg\}
  4. \cap
  5. \cap

Configuration_interaction.html

  1. Ψ = I = 0 c I Φ I S O = c 0 Φ 0 S O + c 1 Φ 1 S O + \Psi=\sum_{I=0}c_{I}\Phi_{I}^{SO}=c_{0}\Phi_{0}^{SO}+c_{1}\Phi_{1}^{SO}+{...}
  2. 𝐜 = 𝐞 𝕊 𝐜 , \mathbb{H}\mathbf{c}=\mathbf{e}\mathbb{S}\mathbf{c},
  3. i j = Φ i S O | 𝐇 e l | Φ j S O \mathbb{H}_{ij}=\left\langle\Phi_{i}^{SO}|\mathbf{H}^{el}|\Phi_{j}^{SO}\right\rangle
  4. 𝕊 i j = Φ i S O | Φ j S O \mathbb{S}_{ij}=\left\langle\Phi_{i}^{SO}|\Phi_{j}^{SO}\right\rangle
  5. Φ i S O | Φ j S O = δ i j \left\langle\Phi_{i}^{SO}|\Phi_{j}^{SO}\right\rangle=\delta_{ij}
  6. 𝕊 \mathbb{S}
  7. 𝐄 j \mathbf{E}^{j}
  8. 𝐜 I j \mathbf{c}_{I}^{j}

Confluent_hypergeometric_function.html

  1. M ( a , b , z ) M(a,b,z)
  2. U ( a , b , z ) U(a,b,z)
  3. Ψ ( a ; b ; z ) Ψ(a;b;z)
  4. z d 2 w d z 2 + ( b - z ) d w d z - a w = 0 , z\frac{d^{2}w}{dz^{2}}+(b-z)\frac{dw}{dz}-aw=0,
  5. z = 0 z=0
  6. z = z=\infty
  7. M ( a , b , z ) M(a,b,z)
  8. U ( a , b , z ) U(a,b,z)
  9. M ( a , b , z ) = n = 0 a ( n ) z n b ( n ) n ! = F 1 1 ( a ; b ; z ) , M(a,b,z)=\sum_{n=0}^{\infty}\frac{a^{(n)}z^{n}}{b^{(n)}n!}={}_{1}F_{1}(a;b;z),
  10. a ( 0 ) = 1 , a^{(0)}=1,
  11. a ( n ) = a ( a + 1 ) ( a + 2 ) ( a + n - 1 ) , a^{(n)}=a(a+1)(a+2)\cdots(a+n-1)\,,
  12. Φ ( a , b , z ) Φ(a,b,z)
  13. z z
  14. b = 0 , 1 , 2 , b=0,−1,−2,...
  15. b b
  16. b b
  17. M ( a , c , z ) = lim b F 1 2 ( a , b ; c ; z / b ) M(a,c,z)=\lim_{b\to\infty}{}_{2}F_{1}(a,b;c;z/b)
  18. U ( a , b , z ) U(a,b,z)
  19. Ψ ( a ; b ; z ) Ψ(a;b;z)
  20. U U
  21. U ( a , b , z ) = Γ ( 1 - b ) Γ ( a - b + 1 ) M ( a , b , z ) + Γ ( b - 1 ) Γ ( a ) z 1 - b M ( a - b + 1 , 2 - b , z ) . U(a,b,z)=\frac{\Gamma(1-b)}{\Gamma(a-b+1)}M(a,b,z)+\frac{\Gamma(b-1)}{\Gamma(a% )}z^{1-b}M(a-b+1,2-b,z).
  22. b b
  23. b b
  24. Γ ( b - 1 ) Γ ( a ) = 0 , \frac{\Gamma(b-1)}{\Gamma(a)}=0,
  25. a a
  26. U ( a , b , z ) U(a,b,z)
  27. M ( a , b , z ) M(a,b,z)
  28. b b
  29. M ( a , b , z ) M(a,b,z)
  30. a = b = 0 a=b=0
  31. w ( z ) = U ( 0 , 0 , z ) = 1 w(z)=U(0,0,z)=1
  32. w ( z ) = exp ( z ) . w(z)=\exp(z).
  33. U ( 0 , b , z ) = 1 U(0,b,z)=1
  34. w ( z ) = - z u - b e u d u w(z)=\int_{-\infty}^{z}u^{-b}e^{u}\mathrm{d}u
  35. z d 2 w d z 2 + ( b - z ) d w d z - ( m = 0 M a m z m ) w z\frac{d^{2}w}{dz^{2}}+(b-z)\frac{dw}{dz}-(\sum_{m=0}^{M}a_{m}z^{m})w
  36. z z
  37. ( A + B z ) d 2 w d z 2 + ( C + D z ) d w d z + ( E + F z ) w = 0 (A+Bz)\frac{d^{2}w}{dz^{2}}+(C+Dz)\frac{dw}{dz}+(E+Fz)w=0
  38. 0
  39. A + B z z A+Bz↦z
  40. z d 2 w d z 2 + ( C + D z ) d w d z + ( E + F z ) w = 0 z\frac{d^{2}w}{dz^{2}}+(C+Dz)\frac{dw}{dz}+(E+Fz)w=0
  41. C , D , E C,D,E
  42. F F
  43. z 1 D 2 - 4 F z z\mapsto\frac{1}{\sqrt{D^{2}-4F}}z
  44. z d 2 w d z 2 + ( C + D D 2 - 4 F z ) d w d z + ( E D 2 - 4 F + F D 2 - 4 F z ) w = 0 z\frac{d^{2}w}{dz^{2}}+\left(C+\frac{D}{\sqrt{D^{2}-4F}}z\right)\frac{dw}{dz}+% \left(\frac{E}{\sqrt{D^{2}-4F}}+\frac{F}{D^{2}-4F}z\right)w=0
  45. exp ( - ( 1 + D D 2 - 4 F ) z 2 ) w ( z ) , \exp\left(-\left(1+\frac{D}{\sqrt{D^{2}-4F}}\right)\frac{z}{2}\right)w(z),
  46. w ( z ) w(z)
  47. a = ( 1 + D D 2 - 4 F ) C 2 - E D 2 - 4 F , b = C . a=\left(1+\frac{D}{\sqrt{D^{2}-4F}}\right)\frac{C}{2}-\frac{E}{\sqrt{D^{2}-4F}% },\qquad b=C.
  48. exp ( - 1 2 D z ) w ( z ) , \exp\left(-\tfrac{1}{2}Dz\right)w(z),
  49. w ( z ) w(z)
  50. z w ′′ ( z ) + C w ( z ) + ( E - 1 2 C D ) w ( z ) = 0. zw^{\prime\prime}(z)+Cw^{\prime}(z)+\left(E-\tfrac{1}{2}CD\right)w(z)=0.
  51. M ( a , b , z ) M(a,b,z)
  52. M ( a , b , z ) = Γ ( b ) Γ ( a ) Γ ( b - a ) 0 1 e z u u a - 1 ( 1 - u ) b - a - 1 d u . M(a,b,z)=\frac{\Gamma(b)}{\Gamma(a)\Gamma(b-a)}\int_{0}^{1}e^{zu}u^{a-1}(1-u)^% {b-a-1}\,du.
  53. M ( a , a + b , i t ) M(a,a+b,it)
  54. U U
  55. U ( a , b , z ) = 1 Γ ( a ) 0 e - z t t a - 1 ( 1 + t ) b - a - 1 d t , ( Re a > 0 ) U(a,b,z)=\frac{1}{\Gamma(a)}\int_{0}^{\infty}e^{-zt}t^{a-1}(1+t)^{b-a-1}\,dt,% \quad(\operatorname{Re}\ a>0)
  56. R e z > 0 Rez>0
  57. M ( a , b , z ) = 1 2 π i Γ ( b ) Γ ( a ) - i i Γ ( - s ) Γ ( a + s ) Γ ( b + s ) ( - z ) s d s M(a,b,z)=\frac{1}{2\pi i}\frac{\Gamma(b)}{\Gamma(a)}\int_{-i\infty}^{i\infty}% \frac{\Gamma(-s)\Gamma(a+s)}{\Gamma(b+s)}(-z)^{s}ds
  58. Γ ( s ) Γ(−s)
  59. Γ ( a + s ) Γ(a+s)
  60. z z
  61. z z→∞
  62. a −a
  63. U ( a , b , z ) U(a,b,z)
  64. z z→∞
  65. z = x 𝐑 z=x∈\mathbf{R}
  66. x x→∞
  67. U ( a , b , x ) x 2 - a F 0 ( a , a - b + 1 ; ; - 1 x ) , U(a,b,x)\sim x^{-a}\,_{2}F_{0}\left(a,a-b+1;\,;-\frac{1}{x}\right),
  68. F 0 2 ( , ; ; - 1 / x ) {}_{2}F_{0}(\cdot,\cdot;;-1/x)
  69. z z
  70. | arg z | < 3 2 π . |\arg z|<\tfrac{3}{2}\pi.
  71. M ( a , b , z ) Γ ( b ) ( e z z a - b Γ ( a ) + ( - z ) - a Γ ( b - a ) ) M(a,b,z)\sim\Gamma(b)\left(\frac{e^{z}z^{a-b}}{\Gamma(a)}+\frac{(-z)^{-a}}{% \Gamma(b-a)}\right)
  72. z z
  73. - 3 2 π < arg z 1 2 π -\tfrac{3}{2}\pi<\arg z\leq\tfrac{1}{2}\pi
  74. Γ ( b a ) Γ(b−a)
  75. b a b−a
  76. z z
  77. Γ ( a ) Γ(a)
  78. z z
  79. e z z a - b e^{z}z^{a-b}
  80. z z→−∞
  81. M ( a , b , z ) M(a,b,z)
  82. U ( a , b , z ) U(a,b,z)
  83. e z ( - 1 ) a - b U ( b - a , b , - z ) e^{z}(-1)^{a-b}U(b-a,b,-z)
  84. M ( a , b , z ) M(a,b,z)
  85. M ( a ± 1 , b , z ) , M ( a , b ± 1 , z ) M(a±1,b,z),M(a,b±1,z)
  86. M ( a , b , z ) M(a,b,z)
  87. M ( a , b , z ) M(a,b,z)
  88. a , b a,b
  89. z z
  90. z d M d z = z a b M ( a + , b + ) = a ( M ( a + ) - M ) = ( b - 1 ) ( M ( b - ) - M ) = ( b - a ) M ( a - ) + ( a - b + z ) M = z ( a - b ) M ( b + ) / b + z M \begin{aligned}\displaystyle z\frac{dM}{dz}=z\frac{a}{b}M(a+,b+)&\displaystyle% =a(M(a+)-M)\\ &\displaystyle=(b-1)(M(b-)-M)\\ &\displaystyle=(b-a)M(a-)+(a-b+z)M\\ &\displaystyle=z(a-b)M(b+)/b+zM\\ \end{aligned}
  91. M ( a + m , b + n , z ) M(a+m,b+n,z)
  92. M ( a , b , z ) = e z M ( b - a , b , - z ) M(a,b,z)=e^{z}\,M(b-a,b,-z)
  93. U ( a , b , z ) = z 1 - b U ( 1 + a - b , 2 - b , z ) U(a,b,z)=z^{1-b}U\left(1+a-b,2-b,z\right)
  94. U ( a , b , z ) \displaystyle U(a,b,z)
  95. M ( a , b , x y x - 1 ) = ( 1 - x ) a n a ( n ) b ( n ) L n ( b - 1 ) ( y ) x n M\left(a,b,\frac{xy}{x-1}\right)=(1-x)^{a}\cdot\sum_{n}\frac{a^{(n)}}{b^{(n)}}% L_{n}^{(b-1)}(y)x^{n}
  96. b b
  97. M ( 0 , b , z ) = 1 M(0,b,z)=1
  98. U ( 0 , c , z ) = 1 U(0,c,z)=1
  99. M ( b , b , z ) = e z M(b,b,z)=e^{z}
  100. U ( a , a , z ) = e z z u - a e - u d u U(a,a,z)=e^{z}\int_{z}^{\infty}u^{-a}e^{-u}du
  101. U ( 1 , b , z ) Γ ( b - 1 ) + M ( 1 , b , z ) Γ ( b ) = z 1 - b e z \frac{U(1,b,z)}{\Gamma(b-1)}+\frac{M(1,b,z)}{\Gamma(b)}=z^{1-b}e^{z}
  102. U ( a , a + 1 , z ) = z - a U(a,a+1,z)=z^{-a}
  103. U ( - n , - 2 n , z ) U(-n,-2n,z)
  104. M ( n , b , z ) M(n,b,z)
  105. b = 2 a b=2a
  106. F 1 1 ( a , 2 a , x ) = e x 2 F 1 0 ( ; a + 1 2 ; x 2 16 ) = e x 2 ( x 4 ) 1 2 - a Γ ( a + 1 2 ) I a - 1 2 ( x 2 ) . {}_{1}F_{1}(a,2a,x)=e^{\frac{x}{2}}\,{}_{0}F_{1}\left(;a+\tfrac{1}{2};\tfrac{x% ^{2}}{16}\right)=e^{\frac{x}{2}}\left(\tfrac{x}{4}\right)^{\tfrac{1}{2}-a}% \Gamma\left(a+\tfrac{1}{2}\right)I_{a-\frac{1}{2}}\left(\tfrac{x}{2}\right).
  107. U ( a , 2 a , x ) = e x 2 π x 1 2 - a K a - 1 2 ( x 2 ) , U(a,2a,x)=\frac{e^{\frac{x}{2}}}{\sqrt{\pi}}x^{\tfrac{1}{2}-a}K_{a-\tfrac{1}{2% }}\left(\tfrac{x}{2}\right),
  108. a a
  109. 2 - a θ - a ( x 2 ) 2^{-a}\theta_{-a}\left(\tfrac{x}{2}\right)
  110. θ θ
  111. erf ( x ) = 2 π 0 x e - t 2 d t = 2 x π F 1 1 ( 1 2 , 3 2 , - x 2 ) . \mathrm{erf}(x)=\frac{2}{\sqrt{\pi}}\int_{0}^{x}e^{-t^{2}}dt=\frac{2x}{\sqrt{% \pi}}\ {}_{1}F_{1}\left(\tfrac{1}{2},\tfrac{3}{2},-x^{2}\right).
  112. M κ , μ ( z ) = e - z 2 z μ + 1 2 M ( μ - κ + 1 2 , 1 + 2 μ ; z ) M_{\kappa,\mu}(z)=e^{-\tfrac{z}{2}}z^{\mu+\tfrac{1}{2}}M\left(\mu-\kappa+% \tfrac{1}{2},1+2\mu;z\right)
  113. W κ , μ ( z ) = e - z 2 z μ + 1 2 U ( μ - κ + 1 2 , 1 + 2 μ ; z ) W_{\kappa,\mu}(z)=e^{-\tfrac{z}{2}}z^{\mu+\tfrac{1}{2}}U\left(\mu-\kappa+% \tfrac{1}{2},1+2\mu;z\right)
  114. p p
  115. p p
  116. E [ | N ( μ , σ 2 ) | p ] \displaystyle\operatorname{E}\left[\left|N\left(\mu,\sigma^{2}\right)\right|^{% p}\right]
  117. ( - 1 ) p (-1)^{p}
  118. M ( a + 1 , b + 1 , z ) M ( a , b , z ) = 1 1 - b - a b ( b + 1 ) z 1 + a + 1 ( b + 1 ) ( b + 2 ) z 1 - b - a + 1 ( b + 2 ) ( b + 3 ) z 1 + a + 2 ( b + 3 ) ( b + 4 ) z 1 - \frac{M(a+1,b+1,z)}{M(a,b,z)}=\cfrac{1}{1-\cfrac{{\displaystyle\frac{b-a}{b(b+% 1)}z}}{1+\cfrac{{\displaystyle\frac{a+1}{(b+1)(b+2)}z}}{1-\cfrac{{% \displaystyle\frac{b-a+1}{(b+2)(b+3)}z}}{1+\cfrac{{\displaystyle\frac{a+2}{(b+% 3)(b+4)}z}}{1-\ddots}}}}}
  119. z z

Conformational_isomerism.html

  1. K = e - Δ G / R T K={e^{-\Delta G/RT}}
  2. K K
  3. Δ G \Delta G
  4. R R
  5. T T
  6. N i N total = e - E rel / R T k = 1 N total e - E k / R T \frac{N_{i}}{N\text{total}}=\frac{e^{-E\text{rel}/RT}}{\sum_{k=1}^{N\text{% total}}e^{-E_{k}/RT}}

Conic_constant.html

  1. K = - e 2 , K=-e^{2},\,
  2. y 2 - 2 R x + ( K + 1 ) x 2 = 0 y^{2}-2Rx+(K+1)x^{2}=0

Conjugacy_class_sum.html

  1. C i ¯ = g C i g . \overline{C_{i}}=\sum_{g\in C_{i}}g.
  2. C 1 ¯ , , C k ¯ \overline{C_{1}},\ldots,\overline{C_{k}}
  3. Z ( 𝐂 G ) = { f 𝐂 G g 𝐂 G , f g = g f } Z(\mathbf{C}G)=\{f\in\mathbf{C}G\mid\forall g\in\mathbf{C}G,fg=gf\}
  4. f ( x y x - 1 ) = f ( y ) for x , y G f(xyx^{-1})=f(y)\,\text{ for }x,y\in G

Conjugate_gradient_method.html

  1. 𝐱 * \scriptstyle\mathbf{x}_{*}
  2. 𝐫 k = 𝐛 - 𝐀𝐱 k . \mathbf{r}_{k}=\mathbf{b}-\mathbf{Ax}_{k}.\,
  3. 𝐩 k = 𝐫 k - i < k 𝐩 i T 𝐀𝐫 k 𝐩 i T 𝐀𝐩 i 𝐩 i \mathbf{p}_{k}=\mathbf{r}_{k}-\sum_{i<k}\frac{\mathbf{p}_{i}^{\mathrm{T}}% \mathbf{A}\mathbf{r}_{k}}{\mathbf{p}_{i}^{\mathrm{T}}\mathbf{A}\mathbf{p}_{i}}% \mathbf{p}_{i}
  4. 𝐱 k + 1 = 𝐱 k + α k 𝐩 k \mathbf{x}_{k+1}=\mathbf{x}_{k}+\alpha_{k}\mathbf{p}_{k}
  5. α k = 𝐩 k T 𝐛 𝐩 k T 𝐀𝐩 k = 𝐩 k T ( 𝐫 k - 1 + 𝐀𝐱 k - 1 ) 𝐩 k T 𝐀𝐩 k = 𝐩 k T 𝐫 k - 1 𝐩 k T 𝐀𝐩 k , \alpha_{k}=\frac{\mathbf{p}_{k}^{\mathrm{T}}\mathbf{b}}{\mathbf{p}_{k}^{% \mathrm{T}}\mathbf{A}\mathbf{p}_{k}}=\frac{\mathbf{p}_{k}^{\mathrm{T}}(\mathbf% {r}_{k-1}+\mathbf{Ax}_{k-1})}{\mathbf{p}_{k}^{\mathrm{T}}\mathbf{A}\mathbf{p}_% {k}}=\frac{\mathbf{p}_{k}^{\mathrm{T}}\mathbf{r}_{k-1}}{\mathbf{p}_{k}^{% \mathrm{T}}\mathbf{A}\mathbf{p}_{k}},
  6. 𝐫 0 := 𝐛 - 𝐀𝐱 0 𝐩 0 := 𝐫 0 k := 0 repeat α k := 𝐫 k T 𝐫 k 𝐩 k T 𝐀𝐩 k 𝐱 k + 1 := 𝐱 k + α k 𝐩 k 𝐫 k + 1 := 𝐫 k - α k 𝐀𝐩 k if r k + 1 is sufficiently small then exit loop β k := 𝐫 k + 1 T 𝐫 k + 1 𝐫 k T 𝐫 k 𝐩 k + 1 := 𝐫 k + 1 + β k 𝐩 k k := k + 1 end repeat The result is 𝐱 k + 1 \begin{aligned}&\displaystyle\mathbf{r}_{0}:=\mathbf{b}-\mathbf{Ax}_{0}\\ &\displaystyle\mathbf{p}_{0}:=\mathbf{r}_{0}\\ &\displaystyle k:=0\\ &\displaystyle\hbox{repeat}\\ &\displaystyle\qquad\alpha_{k}:=\frac{\mathbf{r}_{k}^{\mathrm{T}}\mathbf{r}_{k% }}{\mathbf{p}_{k}^{\mathrm{T}}\mathbf{Ap}_{k}}\\ &\displaystyle\qquad\mathbf{x}_{k+1}:=\mathbf{x}_{k}+\alpha_{k}\mathbf{p}_{k}% \\ &\displaystyle\qquad\mathbf{r}_{k+1}:=\mathbf{r}_{k}-\alpha_{k}\mathbf{Ap}_{k}% \\ &\displaystyle\qquad\hbox{if }r_{k+1}\hbox{ is sufficiently small then exit % loop}\\ &\displaystyle\qquad\beta_{k}:=\frac{\mathbf{r}_{k+1}^{\mathrm{T}}\mathbf{r}_{% k+1}}{\mathbf{r}_{k}^{\mathrm{T}}\mathbf{r}_{k}}\\ &\displaystyle\qquad\mathbf{p}_{k+1}:=\mathbf{r}_{k+1}+\beta_{k}\mathbf{p}_{k}% \\ &\displaystyle\qquad k:=k+1\\ &\displaystyle\hbox{end repeat}\\ &\displaystyle\hbox{The result is }\mathbf{x}_{k+1}\end{aligned}
  7. β k \beta_{k}
  8. α k \alpha_{k}
  9. 𝐫 k + 1 \mathbf{r}_{k+1}
  10. 𝐫 k \mathbf{r}_{k}
  11. α k = 𝐫 k T 𝐫 k 𝐫 k T 𝐀𝐩 k = 𝐫 k T 𝐫 k 𝐩 k T 𝐀𝐩 k \alpha_{k}=\frac{\mathbf{r}_{k}^{\mathrm{T}}\mathbf{r}_{k}}{\mathbf{r}_{k}^{% \mathrm{T}}\mathbf{Ap}_{k}}=\frac{\mathbf{r}_{k}^{\mathrm{T}}\mathbf{r}_{k}}{% \mathbf{p}_{k}^{\mathrm{T}}\mathbf{Ap}_{k}}
  12. 𝐫 k = 𝐩 k - β k - 1 𝐩 k - 1 \mathbf{r}_{k}=\mathbf{p}_{k}-\mathbf{\beta}_{k-1}\mathbf{p}_{k-1}
  13. β k \beta_{k}
  14. 𝐩 k + 1 \mathbf{p}_{k+1}
  15. 𝐩 k \mathbf{p}_{k}
  16. β k \beta_{k}
  17. β k = - 𝐫 k + 1 T A 𝐩 k 𝐩 k T A 𝐩 k \beta_{k}=-\frac{\mathbf{r}_{k+1}^{\mathrm{T}}A\mathbf{p}_{k}}{\mathbf{p}_{k}^% {\mathrm{T}}A\mathbf{p}_{k}}
  18. 𝐫 k = 𝐫 k - 1 - α k - 1 A 𝐩 k - 1 \mathbf{r}_{k}=\mathbf{r}_{k-1}-\alpha_{k-1}A\mathbf{p}_{k-1}
  19. A 𝐩 k - 1 = 1 α k - 1 ( 𝐫 k - 1 - 𝐫 k ) A\mathbf{p}_{k-1}=\frac{1}{\alpha_{k-1}}(\mathbf{r}_{k-1}-\mathbf{r}_{k})
  20. β k \beta_{k}
  21. 𝐫 k + 1 T A 𝐩 k = 1 α k 𝐫 k + 1 T ( 𝐫 k - 𝐫 k + 1 ) = - 1 α k 𝐫 k + 1 T 𝐫 k + 1 \mathbf{r}_{k+1}^{\mathrm{T}}A\mathbf{p}_{k}=\frac{1}{\alpha_{k}}\mathbf{r}_{k% +1}^{\mathrm{T}}(\mathbf{r}_{k}-\mathbf{r}_{k+1})=-\frac{1}{\alpha_{k}}\mathbf% {r}_{k+1}^{\mathrm{T}}\mathbf{r}_{k+1}
  22. 𝐫 k + 1 \mathbf{r}_{k+1}
  23. 𝐫 k \mathbf{r}_{k}
  24. 𝐩 k T A 𝐩 k = ( 𝐫 k + β k - 1 𝐩 k - 1 ) T A 𝐩 k = 1 α k 𝐫 k T ( 𝐫 k - 𝐫 k + 1 ) = 1 α k 𝐫 k T 𝐫 k \mathbf{p}_{k}^{\mathrm{T}}A\mathbf{p}_{k}=(\mathbf{r}_{k}+\beta_{k-1}\mathbf{% p}_{k-1})^{\mathrm{T}}A\mathbf{p}_{k}=\frac{1}{\alpha_{k}}\mathbf{r}_{k}^{% \mathrm{T}}(\mathbf{r}_{k}-\mathbf{r}_{k+1})=\frac{1}{\alpha_{k}}\mathbf{r}_{k% }^{\mathrm{T}}\mathbf{r}_{k}
  25. 𝐩 k \mathbf{p}_{k}
  26. β \beta
  27. α k \alpha_{k}

Connected_category.html

  1. X = X 0 , X 1 , , X n - 1 , X n = Y X=X_{0},X_{1},\ldots,X_{n-1},X_{n}=Y
  2. f i : X i X i + 1 f_{i}:X_{i}\to X_{i+1}
  3. f i : X i + 1 X i f_{i}:X_{i+1}\to X_{i}

Consecutive_fifths.html

  1. D G {}_{G}^{D}
  2. E C {}_{C}^{E}
  3. D G {}_{G}^{D}

Conservative_extension.html

  1. T 2 T_{2}
  2. T 1 T_{1}
  3. T 2 T_{2}
  4. T 1 T_{1}
  5. T 1 T_{1}
  6. T 2 T_{2}
  7. T 2 T_{2}
  8. T 1 T_{1}
  9. T 1 T_{1}
  10. T 1 T_{1}
  11. T 2 T_{2}
  12. T 2 T_{2}
  13. T 1 T_{1}
  14. T 2 T_{2}
  15. T 1 T_{1}
  16. T 0 T_{0}
  17. T 1 T_{1}
  18. T 2 T_{2}
  19. T 2 T_{2}
  20. T 1 T_{1}
  21. T 1 T_{1}
  22. T 2 T_{2}

Consistency_criterion.html

  1. \mathcal{R}
  2. \mathcal{R}
  3. s V ( ) s_{V}(\mathcal{R})
  4. V V
  5. V = V 1 V 2 V=V_{1}\cup V_{2}
  6. V 1 V 2 = V_{1}\cap V_{2}=\emptyset
  7. ( I ) s V ( ) = s V 1 ( ) + s V 2 ( ) (I)\quad s_{V}(\mathcal{R})=s_{V_{1}}(\mathcal{R})+s_{V_{2}}(\mathcal{R})
  8. V V
  9. V = V 1 V 2 V=V_{1}\cup V_{2}
  10. \mathcal{R}
  11. \mathcal{R}
  12. \mathcal{R}^{\prime}
  13. ( I I ) : s V 1 ( ) > s V 1 ( ) (II)\quad\forall\mathcal{R}^{\prime}:s_{V_{1}}(\mathcal{R})>s_{V_{1}}(\mathcal% {R}^{\prime})
  14. ( I I I ) : s V 2 ( ) > s V 2 ( ) (III)\quad\forall\mathcal{R}^{\prime}:s_{V_{2}}(\mathcal{R})>s_{V_{2}}(% \mathcal{R}^{\prime})
  15. \mathcal{R}
  16. \mathcal{R}^{\prime}
  17. s V ( ) = ( I ) s V 1 ( ) + s V 2 ( ) > ( I I ) s V 1 ( ) + s V 2 ( ) > ( I I I ) s V 1 ( ) + s V 2 ( ) = ( I ) s V ( ) q . e . d . s_{V}(\mathcal{R})\stackrel{(I)}{=}s_{V_{1}}(\mathcal{R})+s_{V_{2}}(\mathcal{R% })\stackrel{(II)}{>}s_{V_{1}}(\mathcal{R}^{\prime})+s_{V_{2}}(\mathcal{R})% \stackrel{(III)}{>}s_{V_{1}}(\mathcal{R}^{\prime})+s_{V_{2}}(\mathcal{R}^{% \prime})\stackrel{(I)}{=}s_{V}(\mathcal{R}^{\prime})\quad q.e.d.

Consistent_estimator.html

  1. plim n T n = θ . \underset{n\to\infty}{\operatorname{plim}}\;T_{n}=\theta.
  2. plim n T n ( X θ ) = g ( θ ) , for all θ Θ . \underset{n\to\infty}{\operatorname{plim}}\;T_{n}(X^{\theta})=g(\theta),\ \ \,% \text{for all}\ \theta\in\Theta.
  3. ( T n - μ ) / ( σ / n ) \scriptstyle(T_{n}-\mu)/(\sigma/\sqrt{n})
  4. Pr [ | T n - μ | ε ] = Pr [ n | T n - μ | σ n ε / σ ] = 2 ( 1 - Φ ( n ε σ ) ) 0 \Pr\!\left[\,|T_{n}-\mu|\geq\varepsilon\,\right]=\Pr\!\left[\frac{\sqrt{n}\,% \big|T_{n}-\mu\big|}{\sigma}\geq\sqrt{n}\varepsilon/\sigma\right]=2\left(1-% \Phi\left(\frac{\sqrt{n}\,\varepsilon}{\sigma}\right)\right)\to 0
  5. Pr [ h ( T n - θ ) ε ] E [ h ( T n - θ ) ] ε , \Pr\!\big[h(T_{n}-\theta)\geq\varepsilon\big]\leq\frac{\operatorname{E}\big[h(% T_{n}-\theta)\big]}{\varepsilon},
  6. T n 𝑝 θ g ( T n ) 𝑝 g ( θ ) T_{n}\ \xrightarrow{p}\ \theta\ \quad\Rightarrow\quad g(T_{n})\ \xrightarrow{p% }\ g(\theta)
  7. T n + S n 𝑝 α + β , \displaystyle T_{n}+S_{n}\ \xrightarrow{p}\ \alpha+\beta,
  8. 1 n i = 1 n g ( X i ) 𝑝 E [ g ( X ) ] \frac{1}{n}\sum_{i=1}^{n}g(X_{i})\ \xrightarrow{p}\ \operatorname{E}[\,g(X)\,]
  9. 1 n x i + 1 n {1\over n}\sum x_{i}+{1\over n}
  10. n n\rightarrow\infty

Constraint_Handling_Rules.html

  1. h 1 , , h n | g 1 , , g m b 1 , , b o h_{1},\dots,h_{n}\,|\,g_{1},\dots,g_{m}\Longleftrightarrow b_{1},\dots,b_{o}
  2. h 1 , , h n h_{1},\dots,h_{n}
  3. g 1 , , g m g_{1},\dots,g_{m}
  4. b 1 , , b o b_{1},\dots,b_{o}
  5. h 1 , , h n | g 1 , , g m b 1 , , b o h_{1},\dots,h_{n}\,|\,g_{1},\dots,g_{m}\Longrightarrow b_{1},\dots,b_{o}
  6. h 1 , , h \ h + 1 , , h n | g 1 , , g m b 1 , , b o h_{1},\dots,h_{\ell}\,\backslash\,h_{\ell+1},\dots,h_{n}\,|\,g_{1},\dots,g_{m}% \Longleftrightarrow b_{1},\dots,b_{o}
  7. \ell
  8. \ \backslash
  9. n - n-\ell
  10. H k \ H r | G B H_{k}\,\backslash\,H_{r}\,|\,G\Longleftrightarrow B
  11. H k , H r , G , B H_{k},H_{r},G,B
  12. H k , H r H_{k},H_{r}
  13. B B
  14. G G
  15. H k , H r H_{k},H_{r}

Constructive_dilemma.html

  1. P Q , R S , P R Q S \frac{P\to Q,R\to S,PR}{\therefore QS}
  2. P Q P\to Q
  3. R S R\to S
  4. P R PR
  5. Q S QS
  6. ( P Q ) , ( R S ) , ( P R ) ( Q S ) (P\to Q),(R\to S),(PR)\vdash(QS)
  7. \vdash
  8. Q S QS
  9. P Q P\to Q
  10. R S R\to S
  11. Q S QS
  12. ( ( ( P Q ) and ( R S ) ) and ( P R ) ) ( Q S ) (((P\to Q)\and(R\to S))\and(PR))\to(QS)
  13. P P
  14. Q Q
  15. R R
  16. S S

Contact_angle.html

  1. γ S G \gamma_{SG}
  2. γ S L \gamma_{SL}
  3. γ L G \gamma_{LG}
  4. θ C \theta_{\mathrm{C}}
  5. γ SG - γ SL - γ LG cos θ C = 0 \gamma_{\mathrm{SG}}-\gamma_{\mathrm{SL}}-\gamma_{\mathrm{LG}}\cos\theta_{% \mathrm{C}}=0\,
  6. γ ( 1 + cos θ C ) = Δ W SLV \gamma(1+\cos\theta_{\mathrm{C}})=\Delta W_{\mathrm{SLV}}\,
  7. Δ W SLV \Delta W_{\mathrm{SLV}}
  8. f ( x , y ) f\left(x,y\right)
  9. κ m = 1 2 ( 1 + f x 2 ) f y y - 2 f x f y f x y + ( 1 + f y 2 ) f x x ( 1 + f x 2 + f y 2 ) 3 / 2 \kappa_{m}=\frac{1}{2}\frac{\left(1+{f_{x}}^{2}\right)f_{yy}-2f_{x}f_{y}f_{xy}% +\left(1+{f_{y}}^{2}\right)f_{xx}}{\left(1+{f_{x}}^{2}+{f_{y}}^{2}\right)^{3/2}}
  10. θ A \theta_{\mathrm{A}}
  11. θ R \theta_{\mathrm{R}}
  12. θ A - θ R \theta_{\mathrm{A}}-\theta_{\mathrm{R}}
  13. θ c \theta_{\mathrm{c}}
  14. θ A \theta_{\mathrm{A}}
  15. θ R \theta_{\mathrm{R}}
  16. θ c = arccos ( r A cos θ A + r R cos θ R r A + r R ) \theta_{\mathrm{c}}=\arccos\left(\frac{r_{\mathrm{A}}\cos{\theta_{\mathrm{A}}}% +r_{\mathrm{R}}\cos{\theta_{\mathrm{R}}}}{r_{\mathrm{A}}+r_{\mathrm{R}}}\right)
  17. r A = ( sin 3 θ A 2 - 3 cos θ A + cos 3 θ A ) 1 / 3 ; r R = ( sin 3 θ R 2 - 3 cos θ R + cos 3 θ R ) 1 / 3 r_{\mathrm{A}}=\left(\frac{\sin^{3}{\theta_{\mathrm{A}}}}{2-3\cos{\theta_{% \mathrm{A}}}+\cos^{3}{\theta_{\mathrm{A}}}}\right)^{1/3}~{};~{}~{}r_{\mathrm{R% }}=\left(\frac{\sin^{3}{\theta_{\mathrm{R}}}}{2-3\cos{\theta_{\mathrm{R}}}+% \cos^{3}{\theta_{\mathrm{R}}}}\right)^{1/3}

Continued_fraction_factorization.html

  1. k n , k + \sqrt{kn},\qquad k\in\mathbb{Z^{+}}
  2. O ( e 2 log n log log n ) = L n [ 1 / 2 , 2 ] O\left(e^{\sqrt{2\log n\log\log n}}\right)=L_{n}\left[1/2,\sqrt{2}\right]

Continuous-wave_radar.html

  1. f r = f t ( 1 + v / c 1 - v / c ) f_{r}=f_{t}\left(\frac{1+v/{c^{\prime}}}{1-v/c^{\prime}}\right)
  2. f d = f r - f t = 2 v f t ( c - v ) f_{d}=f_{r}-f_{t}=2v\frac{f_{t}}{(c^{\prime}-v)}
  3. c , ( v c ) \scriptstyle c^{\prime},(v\ll c^{\prime})
  4. ( c - v ) c \scriptstyle\left(c^{\prime}-v\right)\rightarrow c^{\prime}
  5. f d 2 v f t c f_{d}\approx 2v\frac{f_{t}}{c^{\prime}}
  6. I n s t r u m e n t e d R a n g e = F r - F t = S p e e d o f L i g h t ( 4 × M o d u l a t i o n F r e q u e n c y ) Instrumented\ Range=F_{r}-F_{t}=\frac{Speed\ of\ Light}{(4\times Modulation\ % Frequency)}
  7. f D f_{D}
  8. Δ t \Delta t
  9. Δ f \Delta f
  10. k = Δ f r a d a r Δ t r a d a r k=\frac{\Delta{f_{radar}}}{\Delta{t_{radar}}}
  11. f r a d a r f_{radar}
  12. t r a d a r t_{radar}
  13. Δ f e c h o = t r k \Delta{f_{echo}}=t_{r}k
  14. t r = Δ f e c h o k t_{r}=\frac{\Delta{f_{echo}}}{k}
  15. t r t_{r}
  16. d i s t o n e w a y = c t r 2 dist_{oneway}=\frac{c^{\prime}t_{r}}{2}
  17. c = c / n c^{\prime}=c/n
  18. R a n g e L i m i t = 0.5 c t r a d a r Range\ Limit=0.5\ c^{\prime}\ t_{radar}
  19. y ( t ) = cos ( 2 π ( f c + B cos ( 2 π f m t ) ) t ) y(t)=\cos\left(2\pi(f_{c}+B\cos\left(2\pi f_{m}t\right))t\right)\,
  20. B = f Δ f m B=\frac{f_{\Delta}}{f_{m}}
  21. y ( t ) = cos ( 2 π ( f c + B cos ( 2 π f m ( t + δ t ) ) ) ( t + δ t ) ) y(t)=\cos\left(2\pi(f_{c}+B\cos\left(2\pi f_{m}(t+\delta t)\right))(t+\delta t% )\right)\,
  22. δ t = \delta t=
  23. y ( t ) = cos ( 2 π ( f c + B cos ( 2 π f m ( t + δ t ) ) ) ( t + δ t ) ) cos ( 2 π ( f c + B cos ( 2 π f m t ) ) t ) y(t)=\cos\left(2\pi(f_{c}+B\cos\left(2\pi f_{m}(t+\delta t)\right))(t+\delta t% )\right)\ \cos\left(2\pi(f_{c}+B\cos\left(2\pi f_{m}t\right))t\right)\,
  24. y ( t ) cos ( - 4 t π B sin ( 2 π f m ( 2 t + δ t ) sin ( π f m δ t ) + 2 δ t π B cos ( 2 π f m ( t + δ t ) ) ) y(t)\approx\cos\left(-4t\pi B\sin(2\pi f_{m}(2t+\delta t)\sin(\pi f_{m}\delta t% )+2\delta t\pi B\cos(2\pi f_{m}(t+\delta t))\right)\,
  25. M o d u l a t i o n S p e c t r u m S p r e a d 2 ( B + 1 ) f m sin ( δ t ) Modulation\ Spectrum\ Spread\approx 2(B+1)f_{m}\sin(\delta t)
  26. R a n g e = 0.5 C / δ t Range=0.5C/\delta t

Continuous_linear_operator.html

  1. x 0 x_{0}
  2. A - 1 ( D ) + x 0 = A - 1 ( D + A x 0 ) A^{-1}(D)+x_{0}=A^{-1}(D+Ax_{0})\,\!

Continuous_signal.html

  1. f ( t ) = sin ( t ) , t f(t)=\sin(t),\quad t\in\mathbb{R}
  2. f ( t ) = sin ( t ) , t [ - π , π ] f(t)=\sin(t),\quad t\in[-\pi,\pi]
  3. f ( t ) = 0 f(t)=0
  4. f ( t ) = 1 t , t [ 0 , 1 ] f(t)=\frac{1}{t},\quad t\in[0,1]
  5. f ( t ) = 0 f(t)=0
  6. t = 0 t=0\,
  7. t - 1 t^{-1}
  8. t - 2 t^{-2}

Contorsion_tensor.html

  1. Γ i j k \Gamma_{ij}{}^{k}
  2. K a b c {K_{ab}}^{c}
  3. T i j k = Γ i j k - Γ j i k {T_{ij}}^{k}={\Gamma_{ij}}^{k}-{\Gamma_{ji}}^{k}
  4. K i j k = 1 2 ( T i j k - T j k i + T k i j ) , K_{ijk}=\frac{1}{2}(T_{ijk}-T_{jki}+T_{kij}),
  5. T i j k g k l T i j l T_{ijk}\equiv g_{kl}{T_{ij}}^{l}
  6. Γ k j i = Γ ¯ k j + i K k j i , {\Gamma_{kj}}^{i}=\bar{\Gamma}_{kj}{}^{i}+{K_{kj}}^{i},
  7. Γ ¯ k j i \bar{\Gamma}_{kj}{}^{i}

Contribution_margin.html

  1. c = P - V \,\text{c}=\,\text{P}-\,\text{V}
  2. C P = P - V P = Unit Contribution Margin Price = Total Contribution Margin Total Revenue \frac{\,\text{C}}{\,\text{P}}=\frac{\,\text{P}-\,\text{V}}{\,\text{P}}=\frac{% \,\text{Unit Contribution Margin}}{\,\text{Price}}=\frac{\,\text{Total % Contribution Margin}}{\,\text{Total Revenue}}
  3. PL \displaystyle\,\text{PL}
  4. TC = TFC + V × X \,\text{TC}=\,\text{TFC}+\,\text{V}\times\,\text{X}
  5. TR \displaystyle\,\text{TR}
  6. TVC = V × X \,\text{TVC}=\,\text{V}\times\,\text{X}
  7. PL \displaystyle\,\text{PL}

Control_volume.html

  1. d / d t d/dt\;
  2. D / D t D/Dt
  3. p = p ( t , x , y , z ) p=p(t,x,y,z)\;
  4. t t\;
  5. t + d t t+dt\;
  6. ( x , y , z ) (x,y,z)\;
  7. ( x + d x , y + d y , z + d z ) , (x+dx,y+dy,z+dz),\;
  8. d p dp\;
  9. d p = p t d t + p x d x + p y d y + p z d z dp=\frac{\partial p}{\partial t}dt+\frac{\partial p}{\partial x}dx+\frac{% \partial p}{\partial y}dy+\frac{\partial p}{\partial z}dz
  10. 𝐯 = ( v x , v y , v z ) , \mathbf{v}=(v_{x},v_{y},v_{z}),
  11. 𝐯 d t = ( v x d t , v y d t , v z d t ) , \mathbf{v}dt=(v_{x}dt,v_{y}dt,v_{z}dt),
  12. d p = p t d t + p x v x d t + p y v y d t + p z v z d t = ( p t + p x v x + p y v y + p z v z ) d t = ( p t + 𝐯 p ) d t . \begin{aligned}\displaystyle dp&\displaystyle=\frac{\partial p}{\partial t}dt+% \frac{\partial p}{\partial x}v_{x}dt+\frac{\partial p}{\partial y}v_{y}dt+% \frac{\partial p}{\partial z}v_{z}dt\\ &\displaystyle=\left(\frac{\partial p}{\partial t}+\frac{\partial p}{\partial x% }v_{x}+\frac{\partial p}{\partial y}v_{y}+\frac{\partial p}{\partial z}v_{z}% \right)dt\\ &\displaystyle=\left(\frac{\partial p}{\partial t}+\mathbf{v}\cdot\nabla p% \right)dt.\\ \end{aligned}
  13. p \nabla p
  14. d d t = t + 𝐯 . \frac{d}{dt}=\frac{\partial}{\partial t}+\mathbf{v}\cdot\nabla.
  15. D D t = t + 𝐮 . \frac{D}{Dt}=\frac{\partial}{\partial t}+\mathbf{u}\cdot\nabla.

Controller_(control_theory).html

  1. y s y_{s}
  2. y y
  3. u ( t ) = K c * e ( t ) + u 0 u(t)=K_{c}*e(t)+u_{0}
  4. u ( t ) u(t)
  5. e ( t ) = y s ( t ) - y ( t ) e(t)=y_{s}(t)-y(t)
  6. K c K_{c}
  7. u o u_{o}
  8. u u
  9. y y
  10. y y
  11. u u
  12. K c K_{c}
  13. y y
  14. y y
  15. u u
  16. K c K_{c}
  17. y y
  18. u u

Convective_available_potential_energy.html

  1. CAPE = z f z n g ( T v , parcel - T v , env T v , env ) d z \mathrm{CAPE}=\int_{z_{\mathrm{f}}}^{z_{\mathrm{n}}}g\left(\frac{T_{\mathrm{v,% parcel}}-T_{\mathrm{v,env}}}{T_{\mathrm{v,env}}}\right)\,dz
  2. z f z_{\mathrm{f}}
  3. z n z_{\mathrm{n}}
  4. T v , parcel T_{\mathrm{v,parcel}}
  5. T v , env T_{\mathrm{v,env}}
  6. g g

Convergent_series.html

  1. { a 1 , a 2 , a 3 , } \left\{a_{1},\ a_{2},\ a_{3},\dots\right\}
  2. S n S_{n}
  3. S n = k = 1 n a k . S_{n}=\sum_{k=1}^{n}a_{k}.
  4. { S 1 , S 2 , S 3 , } \left\{S_{1},\ S_{2},\ S_{3},\dots\right\}
  5. \ell
  6. ε > 0 \varepsilon>0
  7. N N
  8. n N n\geq\ N
  9. | S n - | ε . \left|S_{n}-\ell\right|\leq\ \varepsilon.
  10. 1 1 + 1 2 + 1 3 + 1 4 + 1 5 + 1 6 + . {1\over 1}+{1\over 2}+{1\over 3}+{1\over 4}+{1\over 5}+{1\over 6}+\cdots% \rightarrow\infty.
  11. 1 1 - 1 2 + 1 3 - 1 4 + 1 5 = ln ( 2 ) {1\over 1}-{1\over 2}+{1\over 3}-{1\over 4}+{1\over 5}\cdots=\ln(2)
  12. 1 2 + 1 3 + 1 5 + 1 7 + 1 11 + 1 13 + . {1\over 2}+{1\over 3}+{1\over 5}+{1\over 7}+{1\over 11}+{1\over 13}+\cdots% \rightarrow\infty.
  13. 1 1 + 1 3 + 1 6 + 1 10 + 1 15 + 1 21 + = 2. {1\over 1}+{1\over 3}+{1\over 6}+{1\over 10}+{1\over 15}+{1\over 21}+\cdots=2.
  14. 1 1 + 1 1 + 1 2 + 1 6 + 1 24 + 1 120 + = e . \frac{1}{1}+\frac{1}{1}+\frac{1}{2}+\frac{1}{6}+\frac{1}{24}+\frac{1}{120}+% \cdots=e.
  15. 1 1 + 1 4 + 1 9 + 1 16 + 1 25 + 1 36 + = π 2 6 . {1\over 1}+{1\over 4}+{1\over 9}+{1\over 16}+{1\over 25}+{1\over 36}+\cdots={% \pi^{2}\over 6}.
  16. 1 1 + 1 2 + 1 4 + 1 8 + 1 16 + 1 32 + = 2. {1\over 1}+{1\over 2}+{1\over 4}+{1\over 8}+{1\over 16}+{1\over 32}+\cdots=2.
  17. 1 1 - 1 2 + 1 4 - 1 8 + 1 16 - 1 32 + = 2 3 . {1\over 1}-{1\over 2}+{1\over 4}-{1\over 8}+{1\over 16}-{1\over 32}+\cdots={2% \over 3}.
  18. 1 1 + 1 1 + 1 2 + 1 3 + 1 5 + 1 8 + = ψ . \frac{1}{1}+\frac{1}{1}+\frac{1}{2}+\frac{1}{3}+\frac{1}{5}+\frac{1}{8}+\cdots% =\psi.
  19. { a n } \left\{a_{n}\right\}
  20. { b n } \left\{b_{n}\right\}
  21. 0 a n b n 0\leq\ a_{n}\leq\ b_{n}
  22. n = 1 b n \sum_{n=1}^{\infty}b_{n}
  23. n = 1 a n . \sum_{n=1}^{\infty}a_{n}.
  24. 0 b n a n 0\leq\ b_{n}\leq\ a_{n}
  25. n = 1 b n \sum_{n=1}^{\infty}b_{n}
  26. n = 1 a n . \sum_{n=1}^{\infty}a_{n}.
  27. a n > 0 a_{n}>0
  28. r r
  29. lim n | a n + 1 a n | = r . \lim_{n\to\infty}\left|{\frac{a_{n+1}}{a_{n}}}\right|=r.
  30. r = lim sup n | a n | n , r=\limsup_{n\rightarrow\infty}\sqrt[n]{|a_{n}|},
  31. f ( n ) = a n f(n)=a_{n}
  32. 1 f ( x ) d x = lim t 1 t f ( x ) d x < , \int_{1}^{\infty}f(x)\,dx=\lim_{t\to\infty}\int_{1}^{t}f(x)\,dx<\infty,
  33. { a n } , { b n } > 0 \left\{a_{n}\right\},\left\{b_{n}\right\}>0
  34. lim n a n b n \lim_{n\to\infty}\frac{a_{n}}{b_{n}}
  35. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  36. n = 1 b n \sum_{n=1}^{\infty}b_{n}
  37. n = 1 a n ( - 1 ) n \sum_{n=1}^{\infty}a_{n}(-1)^{n}
  38. { a n } \left\{a_{n}\right\}
  39. { a n } \left\{a_{n}\right\}
  40. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  41. k = 1 2 k a 2 k \sum_{k=1}^{\infty}2^{k}a_{2^{k}}
  42. { a 1 , a 2 , a 3 , } \left\{a_{1},\ a_{2},\ a_{3},\dots\right\}
  43. a n | a n | a_{n}\leq\ \left|a_{n}\right|
  44. n = 1 a n n = 1 | a n | . \sum_{n=1}^{\infty}a_{n}\leq\ \sum_{n=1}^{\infty}\left|a_{n}\right|.
  45. n = 1 | a n | \sum_{n=1}^{\infty}\left|a_{n}\right|
  46. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  47. n = 1 | a n | \sum_{n=1}^{\infty}\left|a_{n}\right|
  48. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  49. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  50. n = 1 | a n | \sum_{n=1}^{\infty}\left|a_{n}\right|
  51. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  52. { f 1 , f 2 , f 3 , } \left\{f_{1},\ f_{2},\ f_{3},\dots\right\}
  53. n = 1 f n \sum_{n=1}^{\infty}f_{n}
  54. { s n } \{s_{n}\}
  55. s n ( x ) = k = 1 n f k ( x ) s_{n}(x)=\sum_{k=1}^{n}f_{k}(x)
  56. n = 1 a n \sum_{n=1}^{\infty}a_{n}
  57. ε > 0 , \varepsilon>0,
  58. N N
  59. n m N n\geq m\geq N
  60. | k = m n a k | < ε , \left|\sum_{k=m}^{n}a_{k}\right|<\varepsilon,
  61. lim n m k = n n + m a k = 0. \lim_{n\to\infty\atop m\to\infty}\sum_{k=n}^{n+m}a_{k}=0.

Conversion_(chemistry).html

  1. i = 1 n ν i A i = j = 1 m μ j B j \sum_{i=1}^{n}\nu_{i}A_{i}=\sum_{j=1}^{m}\mu_{j}B_{j}
  2. ν i \nu_{i}
  3. μ j \mu_{j}
  4. X i , inst = n ˙ i , react n ˙ i , in X_{i,\,\text{inst}}=\frac{\dot{n}_{i,\,\text{react}}}{\dot{n}_{i,\,\text{in}}}
  5. X poly = m Pol i 0 t m ˙ i , in ( τ ) d τ X_{\,\text{poly}}=\frac{m_{\,\text{Pol}}}{\sum_{i}\int_{0}^{t}\dot{m}_{i,\,% \text{in}}(\tau)d\tau}
  6. X i = n i ( t = 0 ) - n i ( t ) n i ( t = 0 ) = 1 - n i ( t ) n i ( t = 0 ) X_{i}=\frac{n_{i}(t=0)-n_{i}(t)}{n_{i}(t=0)}=1-\frac{n_{i}(t)}{n_{i}(t=0)}
  7. X i = n i ( t = 0 ) + 0 t n ˙ i , in ( τ ) d τ - n i ( t ) n A ( t = 0 ) + 0 t end n ˙ i , in ( τ ) d τ X_{i}=\frac{n_{i}(t=0)+\int_{0}^{t}\dot{n}_{i,\,\text{in}}(\tau)d\tau-n_{i}(t)% }{n_{A}(t=0)+\int_{0}^{t_{\,\text{end}}}\dot{n}_{i,\,\text{in}}(\tau)d\tau}
  8. X i = n i ( t = 0 ) + 0 t n ˙ i , in ( τ ) d τ - n i ( t ) n A ( t = 0 ) + 0 t n ˙ i , in ( τ ) d τ X_{i}=\frac{n_{i}(t=0)+\int_{0}^{t}\dot{n}_{i,\,\text{in}}(\tau)d\tau-n_{i}(t)% }{n_{A}(t=0)+\int_{0}^{t}\dot{n}_{i,\,\text{in}}(\tau)d\tau}
  9. X i = n ˙ i , i n - n ˙ i , o u t n ˙ i , i n = 1 - n i , o u t n i , i n X_{i}=\frac{\dot{n}_{i,in}-\dot{n}_{i,out}}{\dot{n}_{i,in}}=1-\frac{n_{i,out}}% {n_{i,in}}
  10. Y p = n ˙ p , out - n ˙ p , in n ˙ k , in - n k , out only for Definition 1 | μ k ν p | Y_{p}=\frac{\dot{n}_{p,\,\text{out}}-\dot{n}_{p,\,\text{in}}}{\dot{n}_{k,\,% \text{in}}\underbrace{-n_{k,\,\text{out}}}_{\,\text{only for Definition 1}}}% \left|\frac{\mu_{k}}{\nu_{p}}\right|
  11. Y p = n ˙ p , out - n ˙ p , in n ˙ k , in | μ k ν p | Y_{p}=\frac{\dot{n}_{p,\,\text{out}}-\dot{n}_{p,\,\text{in}}}{\dot{n}_{k,\,% \text{in}}}\left|\frac{\mu_{k}}{\nu_{p}}\right|
  12. S p = n p ( t = 0 ) - n p ( t ) n k ( t = 0 ) + 0 t n ˙ k , in ( τ ) d τ - n k ( t ) | μ k ν p | S_{p}=\frac{n_{p}(t=0)-n_{p}(t)}{n_{k}(t=0)+\int_{0}^{t}\dot{n}_{k,\,\text{in}% }(\tau)d\tau-n_{k}(t)}\left|\frac{\mu_{k}}{\nu_{p}}\right|
  13. S p = n ˙ p , out - n ˙ p , in n ˙ k , in - n k , out | μ k ν p | S_{p}=\frac{\dot{n}_{p,\,\text{out}}-\dot{n}_{p,\,\text{in}}}{\dot{n}_{k,\,% \text{in}}-n_{k,\,\text{out}}}\left|\frac{\mu_{k}}{\nu_{p}}\right|
  14. Y p = X i S p Y_{p}=X_{i}\cdot S_{p}
  15. A B A\longrightarrow B
  16. A B A\longrightarrow B
  17. A C A\longrightarrow C
  18. X A = n A ( t = 0 ) - n A ( t ) n A ( t = 0 ) = 1 - n A ( t ) n A ( t = 0 ) = 100 - 10 100 = 0.9 = 90 % X_{A}=\frac{n_{A}(t=0)-n_{A}(t)}{n_{A}(t=0)}=1-\frac{n_{A}(t)}{n_{A}(t=0)}=% \frac{100-10}{100}=0.9=90\%
  19. Y B = n B ( t ) - n B ( t = 0 ) n A ( t = 0 ) + 0 t n ˙ A , in ( τ ) d τ | μ k ν p | = 72 - 0 100 + 0 1 1 = 0.72 = 72 % Y_{B}=\frac{n_{B}(t)-n_{B}(t=0)}{n_{A}(t=0)+\int_{0}^{t}\dot{n}_{A,\,\text{in}% }(\tau)d\tau}\left|\frac{\mu_{k}}{\nu_{p}}\right|=\frac{72-0}{100+0}\cdot\frac% {1}{1}=0.72=72\%
  20. S B = n B ( t = 0 ) - n ˙ B ( t ) n ˙ A ( t = 0 ) - n A ( t ) | μ k ν p | = 0 - 72 100 - 10 1 1 = 0.8 = 80 % S_{B}=\frac{{n}_{B}(t=0)-\dot{n}_{B}(t)}{\dot{n}_{A}(t=0)-n_{A}(t)}\left|\frac% {\mu_{k}}{\nu_{p}}\right|=\frac{0-72}{100-10}\cdot\frac{1}{1}=0.8=80\%
  21. Y p = X i S p Y_{p}=X_{i}\cdot S_{p}

Conversion_between_quaternions_and_Euler_angles.html

  1. 𝐪 = [ q 0 q 1 q 2 q 3 ] T \mathbf{q}=\begin{bmatrix}q_{0}&q_{1}&q_{2}&q_{3}\end{bmatrix}^{T}
  2. | 𝐪 | 2 = q 0 2 + q 1 2 + q 2 2 + q 3 2 = 1 |\mathbf{q}|^{2}=q_{0}^{2}+q_{1}^{2}+q_{2}^{2}+q_{3}^{2}=1
  3. 𝐪 0 = cos ( α / 2 ) \mathbf{q}_{0}=\cos(\alpha/2)
  4. 𝐪 1 = sin ( α / 2 ) cos ( β x ) \mathbf{q}_{1}=\sin(\alpha/2)\cos(\beta_{x})
  5. 𝐪 2 = sin ( α / 2 ) cos ( β y ) \mathbf{q}_{2}=\sin(\alpha/2)\cos(\beta_{y})
  6. 𝐪 3 = sin ( α / 2 ) cos ( β z ) \mathbf{q}_{3}=\sin(\alpha/2)\cos(\beta_{z})
  7. q = q 0 + i q 1 + j q 2 + k q 3 q=q_{0}+iq_{1}+jq_{2}+kq_{3}
  8. [ 1 - 2 ( q 2 2 + q 3 2 ) 2 ( q 1 q 2 - q 0 q 3 ) 2 ( q 0 q 2 + q 1 q 3 ) 2 ( q 1 q 2 + q 0 q 3 ) 1 - 2 ( q 1 2 + q 3 2 ) 2 ( q 2 q 3 - q 0 q 1 ) 2 ( q 1 q 3 - q 0 q 2 ) 2 ( q 0 q 1 + q 2 q 3 ) 1 - 2 ( q 1 2 + q 2 2 ) ] \begin{bmatrix}1-2(q_{2}^{2}+q_{3}^{2})&2(q_{1}q_{2}-q_{0}q_{3})&2(q_{0}q_{2}+% q_{1}q_{3})\\ 2(q_{1}q_{2}+q_{0}q_{3})&1-2(q_{1}^{2}+q_{3}^{2})&2(q_{2}q_{3}-q_{0}q_{1})\\ 2(q_{1}q_{3}-q_{0}q_{2})&2(q_{0}q_{1}+q_{2}q_{3})&1-2(q_{1}^{2}+q_{2}^{2})\end% {bmatrix}
  9. [ q 0 2 + q 1 2 - q 2 2 - q 3 2 2 ( q 1 q 2 - q 0 q 3 ) 2 ( q 0 q 2 + q 1 q 3 ) 2 ( q 1 q 2 + q 0 q 3 ) q 0 2 - q 1 2 + q 2 2 - q 3 2 2 ( q 2 q 3 - q 0 q 1 ) 2 ( q 1 q 3 - q 0 q 2 ) 2 ( q 0 q 1 + q 2 q 3 ) q 0 2 - q 1 2 - q 2 2 + q 3 2 ] \begin{bmatrix}q_{0}^{2}+q_{1}^{2}-q_{2}^{2}-q_{3}^{2}&2(q_{1}q_{2}-q_{0}q_{3}% )&2(q_{0}q_{2}+q_{1}q_{3})\\ 2(q_{1}q_{2}+q_{0}q_{3})&q_{0}^{2}-q_{1}^{2}+q_{2}^{2}-q_{3}^{2}&2(q_{2}q_{3}-% q_{0}q_{1})\\ 2(q_{1}q_{3}-q_{0}q_{2})&2(q_{0}q_{1}+q_{2}q_{3})&q_{0}^{2}-q_{1}^{2}-q_{2}^{2% }+q_{3}^{2}\end{bmatrix}
  10. q 0 + i q 1 + j q 2 + k q 3 q_{0}+iq_{1}+jq_{2}+kq_{3}
  11. [ cos θ cos ψ - cos ϕ sin ψ + sin ϕ sin θ cos ψ sin ϕ sin ψ + cos ϕ sin θ cos ψ cos θ sin ψ cos ϕ cos ψ + sin ϕ sin θ sin ψ - sin ϕ cos ψ + cos ϕ sin θ sin ψ - sin θ sin ϕ cos θ cos ϕ cos θ ] \begin{bmatrix}\cos\theta\cos\psi&-\cos\phi\sin\psi+\sin\phi\sin\theta\cos\psi% &\sin\phi\sin\psi+\cos\phi\sin\theta\cos\psi\\ \cos\theta\sin\psi&\cos\phi\cos\psi+\sin\phi\sin\theta\sin\psi&-\sin\phi\cos% \psi+\cos\phi\sin\theta\sin\psi\\ -\sin\theta&\sin\phi\cos\theta&\cos\phi\cos\theta\\ \end{bmatrix}
  12. 𝐪 𝐥𝐁 = [ cos ( ψ / 2 ) 0 0 sin ( ψ / 2 ) ] [ cos ( θ / 2 ) 0 sin ( θ / 2 ) 0 ] [ cos ( ϕ / 2 ) sin ( ϕ / 2 ) 0 0 ] = [ cos ( ϕ / 2 ) cos ( θ / 2 ) cos ( ψ / 2 ) + sin ( ϕ / 2 ) sin ( θ / 2 ) sin ( ψ / 2 ) sin ( ϕ / 2 ) cos ( θ / 2 ) cos ( ψ / 2 ) - cos ( ϕ / 2 ) sin ( θ / 2 ) sin ( ψ / 2 ) cos ( ϕ / 2 ) sin ( θ / 2 ) cos ( ψ / 2 ) + sin ( ϕ / 2 ) cos ( θ / 2 ) sin ( ψ / 2 ) cos ( ϕ / 2 ) cos ( θ / 2 ) sin ( ψ / 2 ) - sin ( ϕ / 2 ) sin ( θ / 2 ) cos ( ψ / 2 ) ] \mathbf{q_{lB}}=\begin{bmatrix}\cos(\psi/2)\\ 0\\ 0\\ \sin(\psi/2)\\ \end{bmatrix}\begin{bmatrix}\cos(\theta/2)\\ 0\\ \sin(\theta/2)\\ 0\\ \end{bmatrix}\begin{bmatrix}\cos(\phi/2)\\ \sin(\phi/2)\\ 0\\ 0\\ \end{bmatrix}=\begin{bmatrix}\cos(\phi/2)\cos(\theta/2)\cos(\psi/2)+\sin(\phi/% 2)\sin(\theta/2)\sin(\psi/2)\\ \sin(\phi/2)\cos(\theta/2)\cos(\psi/2)-\cos(\phi/2)\sin(\theta/2)\sin(\psi/2)% \\ \cos(\phi/2)\sin(\theta/2)\cos(\psi/2)+\sin(\phi/2)\cos(\theta/2)\sin(\psi/2)% \\ \cos(\phi/2)\cos(\theta/2)\sin(\psi/2)-\sin(\phi/2)\sin(\theta/2)\cos(\psi/2)% \\ \end{bmatrix}
  13. [ ϕ θ ψ ] = [ arctan 2 ( q 0 q 1 + q 2 q 3 ) 1 - 2 ( q 1 2 + q 2 2 ) arcsin ( 2 ( q 0 q 2 - q 3 q 1 ) ) arctan 2 ( q 0 q 3 + q 1 q 2 ) 1 - 2 ( q 2 2 + q 3 2 ) ] \begin{bmatrix}\phi\\ \theta\\ \psi\end{bmatrix}=\begin{bmatrix}\mbox{arctan}~{}\frac{2(q_{0}q_{1}+q_{2}q_{3}% )}{1-2(q_{1}^{2}+q_{2}^{2})}\\ \mbox{arcsin}~{}(2(q_{0}q_{2}-q_{3}q_{1}))\\ \mbox{arctan}~{}\frac{2(q_{0}q_{3}+q_{1}q_{2})}{1-2(q_{2}^{2}+q_{3}^{2})}\end{bmatrix}
  14. [ ϕ θ ψ ] = [ atan2 ( 2 ( q 0 q 1 + q 2 q 3 ) , 1 - 2 ( q 1 2 + q 2 2 ) ) arcsin ( 2 ( q 0 q 2 - q 3 q 1 ) ) atan2 ( 2 ( q 0 q 3 + q 1 q 2 ) , 1 - 2 ( q 2 2 + q 3 2 ) ) ] \begin{bmatrix}\phi\\ \theta\\ \psi\end{bmatrix}=\begin{bmatrix}\mbox{atan2}~{}(2(q_{0}q_{1}+q_{2}q_{3}),1-2(% q_{1}^{2}+q_{2}^{2}))\\ \mbox{arcsin}~{}(2(q_{0}q_{2}-q_{3}q_{1}))\\ \mbox{atan2}~{}(2(q_{0}q_{3}+q_{1}q_{2}),1-2(q_{2}^{2}+q_{3}^{2}))\end{bmatrix}
  15. ϕ \phi
  16. θ \theta
  17. ψ \psi

Convex_optimization.html

  1. X X
  2. f : 𝒳 f:\mathcal{X}\to\mathbb{R}
  3. 𝒳 \mathcal{X}
  4. X X
  5. x x^{\ast}
  6. 𝒳 \mathcal{X}
  7. f ( x ) f(x)
  8. x x^{\ast}
  9. f ( x ) f ( x ) f(x^{\ast})\leq f(x)
  10. x 𝒳 x\in\mathcal{X}
  11. f f
  12. x 𝒳 x^{\ast}\in\mathcal{X}
  13. f ( x ) = min { f ( x ) : x 𝒳 } , f(x^{\ast})=\min\{f(x):x\in\mathcal{X}\},
  14. 𝒳 n \mathcal{X}\subset\mathbb{R}^{n}
  15. f ( x ) : n f(x):\mathbb{R}^{n}\rightarrow\mathbb{R}
  16. 𝒳 \mathcal{X}
  17. f ( x ) f(x)
  18. n \mathbb{R}^{n}
  19. minimize \displaystyle\operatorname{minimize}
  20. f , g 1 g m : n f,g_{1}\ldots g_{m}:\mathbb{R}^{n}\rightarrow\mathbb{R}
  21. f ( x ) : n f(x):\mathbb{R}^{n}\to\mathbb{R}
  22. x x
  23. g i ( x ) 0 g_{i}(x)\leq 0
  24. g i g_{i}
  25. h i ( x ) = 0 h_{i}(x)=0
  26. h i h_{i}
  27. h i ( x ) = a i T x + b i h_{i}(x)=a_{i}^{T}x+b_{i}
  28. a i a_{i}
  29. b i b_{i}
  30. minimize 𝑥 \displaystyle\underset{x}{\operatorname{minimize}}
  31. h ( x ) = 0 h(x)=0
  32. h ( x ) 0 h(x)\leq 0
  33. - h ( x ) 0 -h(x)\leq 0
  34. h i ( x ) = 0 h_{i}(x)=0
  35. h i ( x ) h_{i}(x)
  36. h i ( x ) 0 h_{i}(x)\leq 0
  37. - h i ( x ) 0 -h_{i}(x)\leq 0
  38. h i ( x ) = 0 h_{i}(x)=0
  39. h i ( x ) h_{i}(x)
  40. f ( x ) f(x)
  41. g i ( x ) 0 g_{i}(x)\leq 0
  42. i = 1 m i=1\ldots m
  43. 𝒳 \mathcal{X}
  44. 𝒳 = { x X | g 1 ( x ) 0 , , g m ( x ) 0 } . \mathcal{X}=\left\{{x\in X|g_{1}(x)\leq 0,\ldots,g_{m}(x)\leq 0}\right\}.
  45. x 𝒳 x\in\mathcal{X}

Convex_polytope.html

  1. a 1 x 1 + a 2 x 2 + + a n x n b a_{1}x_{1}+a_{2}x_{2}+\cdots+a_{n}x_{n}\leq b
  2. a 11 x 1 \displaystyle a_{11}x_{1}
  3. A x b Ax\leq b
  4. A x b Ax\leq b
  5. v 1 , , v m v_{1},\dots,v_{m}

Convolution_(computer_science).html

  1. \ell
  2. = 4 \ell=4
  3. ( a , f , b ) ( n , i , e ) ( d , s , # ) ( # , h , # ) (a,f,b)(n,i,e)(d,s,\#)(\#,h,\#)
  4. ¯ \underline{\ell}
  5. ¯ = 2 \underline{\ell}=2
  6. \ell
  7. ( ( Σ { # } ) n ) * ((\Sigma\cup\{\#\})^{n})^{*}
  8. ( x 1 , y 1 , ) ( x 2 , y 2 , ) ( x , y , ) (x_{1},y_{1},\ldots)(x_{2},y_{2},\ldots)\ldots(x_{\ell},y_{\ell},\ldots)
  9. ( x 1 , y 1 , ) ( x 2 , y 2 , ) ( x ¯ , y ¯ , ) (x_{1},y_{1},\ldots)(x_{2},y_{2},\ldots)\ldots(x_{\underline{\ell}},y_{% \underline{\ell}},\ldots)
  10. ¯ \underline{\ell}
  11. # \#
  12. ¯ \underline{\ell}

Conway's_Soldiers.html

  1. x 0 = 1 {x^{0}=1}
  2. x n {x^{n}}
  3. n {n}
  4. x 2 + 2 x 3 {x^{2}+2x^{3}}
  5. x n {x^{n}}
  6. x n - 2 - x n - 1 - x n = x n - 2 ( 1 - x - x 2 ) {x^{n-2}-x^{n-1}-x^{n}=x^{n-2}(1-x-x^{2})}
  7. - x n - 1 {-x^{n-1}}
  8. x n + 2 - x n + 1 - x n = x n ( x 2 - x - 1 ) {x^{n+2}-x^{n+1}-x^{n}=x^{n}(x^{2}-x-1)}
  9. x x
  10. x x
  11. x 2 + x - 1 = 0 {x^{2}+x-1=0}
  12. x = 5 - 1 2 , - 5 - 1 2 {x=\frac{\sqrt{5}-1}{2},\frac{-\sqrt{5}-1}{2}}
  13. 5 - 1 2 = φ 0.61803 39887 {\frac{\sqrt{5}-1}{2}=\varphi\approx 0.61803\,39887\dots\,}
  14. φ 2 + φ - 1 = 0 {\varphi^{2}+\varphi-1=0}
  15. φ 2 = 1 - φ {\varphi^{2}=1-\varphi}
  16. φ {\varphi}
  17. φ 3 = φ - φ 2 {\varphi^{3}=\varphi-\varphi^{2}}
  18. φ 4 = φ 2 - φ 3 {\varphi^{4}=\varphi^{2}-\varphi^{3}}
  19. n = 2 φ n = 1 {\sum_{n=2}^{\infty}\varphi^{n}=1}
  20. k = 0 r k = 1 1 - r \sum_{k=0}^{\infty}r^{k}=\frac{1}{1-r}
  21. ( r < 1 ) (r<1)
  22. φ : \varphi:
  23. k = 2 φ k = 1 1 - φ - φ - 1 = 1 \sum_{k=2}^{\infty}\varphi^{k}=\frac{1}{1-\varphi}-\varphi-1=1
  24. φ + 2 φ 2 + 2 φ 3 + = φ + 2 ( φ 2 + φ 3 + φ 4 + ) {\varphi+2\varphi^{2}+2\varphi^{3}+\ldots=\varphi+2(\varphi^{2}+\varphi^{3}+% \varphi^{4}+\ldots)}
  25. φ \varphi
  26. S 1 = ( φ + 2 ( φ 2 + φ 3 + φ 4 ) ) ( 1 + φ + φ 2 + φ 3 + ) {{S}_{1}=(\varphi+2(\varphi^{2}+\varphi^{3}+\varphi^{4}\ldots))(1+\varphi+% \varphi^{2}+\varphi^{3}+\ldots)}
  27. n = 2 φ n = 1 \sum_{n=2}^{\infty}\varphi^{n}=1
  28. S 1 = ( φ + 2 ) ( 1 + φ + 1 ) = ( φ + 2 ) 2 = 4 + 4 φ + φ 2 = 5 + 3 φ {{S}_{1}=(\varphi+2)(1+\varphi+1)=(\varphi+2)^{2}=4+4\varphi+\varphi^{2}=5+3\varphi}
  29. φ 2 = 1 - φ {\varphi^{2}=1-\varphi}
  30. 5 + 3 φ {5+3\varphi}
  31. φ {\varphi}
  32. S 2 = φ ( 5 + 3 φ ) = 5 φ + 3 φ 2 = 5 φ + 3 ( 1 - φ ) = 3 + 2 φ {{S}_{2}=\varphi(5+3\varphi)=5\varphi+3\varphi^{2}=5\varphi+3(1-\varphi)=3+2\varphi}
  33. S 3 = 2 + φ {{S}_{3}=2+\varphi}
  34. S 4 = 1 + φ {{S}_{4}=1+\varphi}
  35. S 5 = 1 {{S}_{5}=1}
  36. φ 0 = 1 {\varphi^{0}=1}
  37. < m t p l > Q E D <mtpl>{{QED}}

Coplanarity.html

  1. ( x 3 - x 1 ) [ ( x 2 - x 1 ) × ( x 4 - x 3 ) ] = 0. (x_{3}-x_{1})\cdot[(x_{2}-x_{1})\times(x_{4}-x_{3})]=0.
  2. 𝐚 , 𝐛 \mathbf{a},\mathbf{b}
  3. 𝐜 \mathbf{c}
  4. ( 𝐜 𝐚 ^ ) 𝐚 ^ + ( 𝐜 𝐛 ^ ) 𝐛 ^ = 𝐜 , (\mathbf{c}\cdot\mathbf{\hat{a}})\mathbf{\hat{a}}+(\mathbf{c}\cdot\mathbf{\hat% {b}})\mathbf{\hat{b}}=\mathbf{c},
  5. 𝐚 ^ \mathbf{\hat{a}}
  6. 𝐚 \mathbf{a}
  7. 𝐜 \mathbf{c}
  8. 𝐚 \mathbf{a}
  9. 𝐜 \mathbf{c}
  10. 𝐛 \mathbf{b}
  11. 𝐜 \mathbf{c}
  12. [ w 1 w 2 w n x 1 x 2 x n y 1 y 2 y n z 1 z 2 z n ] \begin{bmatrix}w_{1}&w_{2}&\dots&w_{n}\\ x_{1}&x_{2}&\dots&x_{n}\\ y_{1}&y_{2}&\dots&y_{n}\\ z_{1}&z_{2}&\dots&z_{n}\end{bmatrix}

Copper_coulometer.html

  1. Q = 2 Δ m F 63.546 Q=\frac{2\Delta mF}{63.546}
  2. Δ m \Delta m

Copper–copper(II)_sulfate_electrode.html

  1. E = 0.337 + R T 2 F ln a Cu 2 + E=0.337+\frac{RT}{2F}\ln a_{\rm Cu^{2+}}

Copula_(probability_theory).html

  1. ( X 1 , X 2 , , X d ) (X_{1},X_{2},\dots,X_{d})
  2. F i ( x ) = [ X i x ] F_{i}(x)=\mathbb{P}[X_{i}\leq x]
  3. ( U 1 , U 2 , , U d ) = ( F 1 ( X 1 ) , F 2 ( X 2 ) , , F d ( X d ) ) (U_{1},U_{2},\dots,U_{d})=\left(F_{1}(X_{1}),F_{2}(X_{2}),\dots,F_{d}(X_{d})\right)
  4. ( X 1 , X 2 , , X d ) (X_{1},X_{2},\dots,X_{d})
  5. ( U 1 , U 2 , , U d ) (U_{1},U_{2},\dots,U_{d})
  6. C ( u 1 , u 2 , , u d ) = [ U 1 u 1 , U 2 u 2 , , U d u d ] . C(u_{1},u_{2},\dots,u_{d})=\mathbb{P}[U_{1}\leq u_{1},U_{2}\leq u_{2},\dots,U_% {d}\leq u_{d}].
  7. ( X 1 , X 2 , , X d ) (X_{1},X_{2},\dots,X_{d})
  8. F i F_{i}
  9. ( U 1 , U 2 , , U d ) (U_{1},U_{2},\dots,U_{d})
  10. ( X 1 , X 2 , , X d ) = ( F 1 - 1 ( U 1 ) , F 2 - 1 ( U 2 ) , , F d - 1 ( U d ) ) . (X_{1},X_{2},\dots,X_{d})=\left(F_{1}^{-1}(U_{1}),F_{2}^{-1}(U_{2}),\dots,F_{d% }^{-1}(U_{d})\right).
  11. F i - 1 F_{i}^{-1}
  12. F i F_{i}
  13. C ( u 1 , u 2 , , u d ) = [ X 1 F 1 - 1 ( u 1 ) , X 2 F 2 - 1 ( u 2 ) , , X d F d - 1 ( u d ) ] . C(u_{1},u_{2},\dots,u_{d})=\mathbb{P}[X_{1}\leq F_{1}^{-1}(u_{1}),X_{2}\leq F_% {2}^{-1}(u_{2}),\dots,X_{d}\leq F_{d}^{-1}(u_{d})].
  14. C : [ 0 , 1 ] d [ 0 , 1 ] C:[0,1]^{d}\rightarrow[0,1]
  15. [ 0 , 1 ] d [0,1]^{d}
  16. C : [ 0 , 1 ] d [ 0 , 1 ] C:[0,1]^{d}\rightarrow[0,1]
  17. C ( u 1 , , u i - 1 , 0 , u i + 1 , , u d ) = 0 C(u_{1},\dots,u_{i-1},0,u_{i+1},\dots,u_{d})=0
  18. C ( 1 , , 1 , u , 1 , , 1 ) = u C(1,\dots,1,u,1,\dots,1)=u
  19. B = i = 1 d [ x i , y i ] [ 0 , 1 ] d B=\prod_{i=1}^{d}[x_{i},y_{i}]\subseteq[0,1]^{d}
  20. B d C ( u ) = 𝐳 × i = 1 d { x i , y i } ( - 1 ) N ( 𝐳 ) C ( 𝐳 ) 0 , \int_{B}dC(u)=\sum_{\mathbf{z}\in\times_{i=1}^{d}\{x_{i},y_{i}\}}(-1)^{N(% \mathbf{z})}C(\mathbf{z})\geq 0,
  21. N ( 𝐳 ) = # { k : z k = x k } N(\mathbf{z})=\#\{k:z_{k}=x_{k}\}
  22. C : [ 0 , 1 ] × [ 0 , 1 ] [ 0 , 1 ] C:[0,1]\times[0,1]\rightarrow[0,1]
  23. C ( 0 , u ) = C ( u , 0 ) = 0 C(0,u)=C(u,0)=0
  24. C ( 1 , u ) = C ( u , 1 ) = u C(1,u)=C(u,1)=u
  25. C ( u 2 , v 2 ) - C ( u 2 , v 1 ) - C ( u 1 , v 2 ) + C ( u 1 , v 1 ) 0 C(u_{2},v_{2})-C(u_{2},v_{1})-C(u_{1},v_{2})+C(u_{1},v_{1})\geq 0
  26. 0 u 1 u 2 1 0\leq u_{1}\leq u_{2}\leq 1
  27. 0 v 1 v 2 1 0\leq v_{1}\leq v_{2}\leq 1
  28. H ( x 1 , , x d ) = [ X 1 x 1 , , X d x d ] H(x_{1},\dots,x_{d})=\mathbb{P}[X_{1}\leq x_{1},\dots,X_{d}\leq x_{d}]
  29. ( X 1 , X 2 , , X d ) (X_{1},X_{2},\dots,X_{d})
  30. F i ( x ) = [ X i x ] F_{i}(x)=\mathbb{P}[X_{i}\leq x]
  31. H ( x 1 , , x d ) = C ( F 1 ( x 1 ) , , F d ( x d ) ) , H(x_{1},\dots,x_{d})=C\left(F_{1}(x_{1}),\dots,F_{d}(x_{d})\right),
  32. C C
  33. H H
  34. Ran ( F 1 ) × × Ran ( F d ) \operatorname{Ran}(F_{1})\times\cdots\times\operatorname{Ran}(F_{d})
  35. F i F_{i}
  36. C : [ 0 , 1 ] d [ 0 , 1 ] C:[0,1]^{d}\rightarrow[0,1]
  37. F i ( x ) F_{i}(x)
  38. C ( F 1 ( x 1 ) , , F d ( x d ) ) C\left(F_{1}(x_{1}),\dots,F_{d}(x_{d})\right)
  39. C : [ 0 , 1 ] d [ 0 , 1 ] C:[0,1]^{d}\rightarrow[0,1]
  40. ( u 1 , , u d ) [ 0 , 1 ] d (u_{1},\dots,u_{d})\in[0,1]^{d}
  41. W ( u 1 , , u d ) C ( u 1 , , u d ) M ( u 1 , , u d ) . W(u_{1},\dots,u_{d})\leq C(u_{1},\dots,u_{d})\leq M(u_{1},\dots,u_{d}).
  42. W ( u 1 , , u d ) = max { 1 - d + i = 1 d u i , 0 } . W(u_{1},\ldots,u_{d})=\max\left\{1-d+\sum\limits_{i=1}^{d}{u_{i}},0\right\}.
  43. M ( u 1 , , u d ) = min { u 1 , , u d } . M(u_{1},\ldots,u_{d})=\min\{u_{1},\dots,u_{d}\}.
  44. C ~ \tilde{C}
  45. C ~ ( u ) = W ( u ) \tilde{C}(u)=W(u)
  46. max ( u + v - 1 , 0 ) C ( u , v ) min { u , v } \max(u+v-1,0)\leq C(u,v)\leq\min\{u,v\}
  47. [ 0 , 1 ] d [0,1]^{d}
  48. d \mathbb{R}^{d}
  49. R d × d R\in\mathbb{R}^{d\times d}
  50. R R
  51. C R Gauss ( u ) = Φ R ( Φ - 1 ( u 1 ) , , Φ - 1 ( u d ) ) , C_{R}^{\,\text{Gauss}}(u)=\Phi_{R}\left(\Phi^{-1}(u_{1}),\dots,\Phi^{-1}(u_{d}% )\right),
  52. Φ - 1 \Phi^{-1}
  53. Φ R \Phi_{R}
  54. R R
  55. c R Gauss ( u ) = 1 det R exp ( - 1 2 ( Φ - 1 ( u 1 ) Φ - 1 ( u d ) ) T ( R - 1 - 𝐈 ) ( Φ - 1 ( u 1 ) Φ - 1 ( u d ) ) ) , c_{R}^{\,\text{Gauss}}(u)=\frac{1}{\sqrt{\det{R}}}\exp\left(-\frac{1}{2}\begin% {pmatrix}\Phi^{-1}(u_{1})\\ \vdots\\ \Phi^{-1}(u_{d})\end{pmatrix}^{T}\cdot\left(R^{-1}-\mathbf{I}\right)\cdot% \begin{pmatrix}\Phi^{-1}(u_{1})\\ \vdots\\ \Phi^{-1}(u_{d})\end{pmatrix}\right),
  56. 𝐈 \mathbf{I}
  57. C ( u 1 , , u d ; θ ) = ψ [ - 1 ] ( ψ ( u 1 ; θ ) + + ψ ( u d ; θ ) ; θ ) C(u_{1},\dots,u_{d};\theta)=\psi^{[-1]}\left(\psi(u_{1};\theta)+\cdots+\psi(u_% {d};\theta);\theta\right)\,
  58. ψ : [ 0 , 1 ] × Θ [ 0 , ) \psi\!:[0,1]\times\Theta\rightarrow[0,\infty)
  59. ψ ( 1 ; θ ) = 0 \psi(1;\theta)=0
  60. θ \theta
  61. Θ \Theta
  62. ψ \psi
  63. ψ [ - 1 ] \psi^{[-1]}
  64. ψ [ - 1 ] ( t ; θ ) = { ψ - 1 ( t ; θ ) if 0 t ψ ( 0 ; θ ) 0 if ψ ( 0 ; θ ) t . \psi^{[-1]}(t;\theta)=\left\{\begin{array}[]{ll}\psi^{-1}(t;\theta)&\mbox{if }% ~{}0\leq t\leq\psi(0;\theta)\\ 0&\mbox{if }~{}\psi(0;\theta)\leq t\leq\infty.\end{array}\right.\,
  65. ψ - 1 \psi^{-1}\,
  66. ψ - 1 \psi^{-1}\,
  67. [ 0 , ) [0,\infty)
  68. d - 2 d-2
  69. ( - 1 ) k ψ - 1 , ( k ) ( t ; θ ) 0 (-1)^{k}\psi^{-1,(k)}(t;\theta)\geq 0\,
  70. t 0 t\geq 0
  71. k = 0 , 1 , , d - 2 k=0,1,\dots,d-2
  72. ( - 1 ) d - 2 ψ - 1 , ( d - 2 ) ( t ; θ ) (-1)^{d-2}\psi^{-1,(d-2)}(t;\theta)
  73. d d\in\mathbb{N}
  74. θ Θ \theta\in\Theta
  75. C θ ( u , v ) \,C_{\theta}(u,v)
  76. ψ θ ( t ) \,\psi_{\theta}(t)
  77. ψ θ - 1 ( t ) \,\psi_{\theta}^{-1}(t)
  78. θ \,\theta
  79. ( max { u - θ + v - θ - 1 ; 0 } ) - 1 / θ \left(\max\left\{u^{-\theta}+v^{-\theta}-1;0\right\}\right)^{-1/\theta}
  80. 1 θ ( t - θ - 1 ) \frac{1}{\theta}\,(t^{-\theta}-1)\,
  81. ( 1 + θ t ) - 1 / θ \left(1+\theta t\right)^{-1/\theta}
  82. θ [ - 1 , ) \ { 0 } \theta\in[-1,\infty)\backslash\{0\}
  83. u v 1 - θ ( 1 - u ) ( 1 - v ) \frac{uv}{1-\theta(1-u)(1-v)}
  84. log ( 1 - θ ( 1 - t ) t ) \log\!\left(\frac{1-\theta(1-t)}{t}\right)
  85. 1 - θ exp ( t ) - θ \frac{1-\theta}{\exp(t)-\theta}
  86. θ [ - 1 , 1 ) \theta\in[-1,1)
  87. exp ( - ( ( - log ( u ) ) θ + ( - log ( v ) ) θ ) 1 / θ ) \exp\!\left(-\left((-\log(u))^{\theta}+(-\log(v))^{\theta}\right)^{1/\theta}\right)
  88. ( - log ( t ) ) θ \left(-\log(t)\right)^{\theta}
  89. exp ( - t 1 / θ ) \exp\!\left(-t^{1/\theta}\right)
  90. θ [ 1 , ) \theta\in[1,\infty)
  91. - 1 θ log ( 1 + ( exp ( - θ u ) - 1 ) ( exp ( - θ v ) - 1 ) exp ( - θ ) - 1 ) -\frac{1}{\theta}\log\!\left(1+\frac{(\exp(-\theta u)-1)(\exp(-\theta v)-1)}{% \exp(-\theta)-1}\right)
  92. - log ( exp ( - θ t ) - 1 exp ( - θ ) - 1 ) -\log\!\left(\frac{\exp(-\theta t)-1}{\exp(-\theta)-1}\right)
  93. - 1 θ log ( 1 + exp ( - t ) ( exp ( - θ ) - 1 ) ) -\frac{1}{\theta}\,\log(1+\exp(-t)(\exp(-\theta)-1))
  94. θ \ { 0 } \theta\in\mathbb{R}\backslash\{0\}
  95. 1 - ( ( 1 - u ) θ + ( 1 - v ) θ - ( 1 - u ) θ ( 1 - v ) θ ) 1 / θ 1-\left((1-u)^{\theta}+(1-v)^{\theta}-(1-u)^{\theta}(1-v)^{\theta}\right)^{1/\theta}
  96. - log ( 1 - ( 1 - t ) θ ) -\log\!\left(1-(1-t)^{\theta}\right)
  97. 1 - ( 1 - exp ( - t ) ) 1 / θ 1-\left(1-\exp(-t)\right)^{1/\theta}
  98. θ [ 1 , ) \theta\in[1,\infty)
  99. u v uv
  100. - log ( t ) -\log(t)\,
  101. exp ( - t ) \exp(-t)\,
  102. ( X 1 i , X 2 i , , X d i ) , i = 1 , , n (X_{1}^{i},X_{2}^{i},\dots,X_{d}^{i}),\,i=1,\dots,n
  103. ( X 1 , X 2 , , X d ) (X_{1},X_{2},\dots,X_{d})
  104. ( U 1 i , U 2 i , , U d i ) = ( F 1 ( X 1 i ) , F 2 ( X 2 i ) , , F d ( X d i ) ) , i = 1 , , n . (U_{1}^{i},U_{2}^{i},\dots,U_{d}^{i})=\left(F_{1}(X_{1}^{i}),F_{2}(X_{2}^{i}),% \dots,F_{d}(X_{d}^{i})\right),\,i=1,\dots,n.
  105. F i F_{i}
  106. F k n ( x ) = 1 n i = 1 n 𝟏 ( X k i x ) F_{k}^{n}(x)=\frac{1}{n}\sum_{i=1}^{n}\mathbf{1}(X_{k}^{i}\leq x)
  107. ( U ~ 1 i , U ~ 2 i , , U ~ d i ) = ( F 1 n ( X 1 i ) , F 2 n ( X 2 i ) , , F d n ( X d i ) ) , i = 1 , , n . (\tilde{U}_{1}^{i},\tilde{U}_{2}^{i},\dots,\tilde{U}_{d}^{i})=\left(F_{1}^{n}(% X_{1}^{i}),F_{2}^{n}(X_{2}^{i}),\dots,F_{d}^{n}(X_{d}^{i})\right),\,i=1,\dots,n.
  108. C n ( u 1 , , u d ) = 1 n i = 1 n 𝟏 ( U ~ 1 i u 1 , , U ~ d i u d ) . C^{n}(u_{1},\dots,u_{d})=\frac{1}{n}\sum_{i=1}^{n}\mathbf{1}\left(\tilde{U}_{1% }^{i}\leq u_{1},\dots,\tilde{U}_{d}^{i}\leq u_{d}\right).
  109. U ~ k i = R k i / n \tilde{U}_{k}^{i}=R_{k}^{i}/n
  110. R k i R_{k}^{i}
  111. X k i X_{k}^{i}
  112. R k i = j = 1 n 𝟏 ( X k j X k i ) R_{k}^{i}=\sum_{j=1}^{n}\mathbf{1}(X_{k}^{j}\leq X_{k}^{i})
  113. g : d g:\mathbb{R}^{d}\rightarrow\mathbb{R}
  114. ( X 1 , , X d ) (X_{1},\dots,X_{d})
  115. H H
  116. 𝔼 [ g ( X 1 , , X d ) ] = d g ( x 1 , , x d ) d H ( x 1 , , x d ) . \mathbb{E}\left[g(X_{1},\dots,X_{d})\right]=\int_{\mathbb{R}^{d}}g(x_{1},\dots% ,x_{d})\,dH(x_{1},\dots,x_{d}).
  117. H H
  118. H ( x 1 , , x d ) = C ( F 1 ( x 1 ) , , F d ( x d ) ) H(x_{1},\dots,x_{d})=C(F_{1}(x_{1}),\dots,F_{d}(x_{d}))
  119. 𝔼 [ g ( X 1 , , X d ) ] = [ 0 , 1 ] d g ( F 1 - 1 ( u 1 ) , , F d - 1 ( u d ) ) d C ( u 1 , , u d ) . \mathbb{E}\left[g(X_{1},\dots,X_{d})\right]=\int_{[0,1]^{d}}g(F_{1}^{-1}(u_{1}% ),\dots,F_{d}^{-1}(u_{d}))\,dC(u_{1},\dots,u_{d}).
  120. 𝔼 [ g ( X 1 , , X d ) ] = [ 0 , 1 ] d g ( F 1 - 1 ( u 1 ) , , F d - 1 ( u d ) ) c ( u 1 , , u d ) d u 1 d u d . \mathbb{E}\left[g(X_{1},\dots,X_{d})\right]=\int_{[0,1]^{d}}g(F_{1}^{-1}(u_{1}% ),\dots,F_{d}^{-1}(u_{d}))c(u_{1},\dots,u_{d})\,du_{1}\cdots du_{d}.
  121. ( U 1 k , , U d k ) C ( k = 1 , , n ) (U_{1}^{k},\dots,U_{d}^{k})\sim C\;\;(k=1,\dots,n)
  122. ( X 1 , , X d ) (X_{1},\dots,X_{d})
  123. ( X 1 k , , X d k ) = ( F 1 - 1 ( U 1 k ) , , F d - 1 ( U d k ) ) H ( k = 1 , , n ) (X_{1}^{k},\dots,X_{d}^{k})=(F_{1}^{-1}(U_{1}^{k}),\dots,F_{d}^{-1}(U_{d}^{k})% )\sim H\;\;(k=1,\dots,n)
  124. 𝔼 [ g ( X 1 , , X d ) ] \mathbb{E}\left[g(X_{1},\dots,X_{d})\right]
  125. 𝔼 [ g ( X 1 , , X d ) ] 1 n k = 1 n g ( X 1 k , , X d k ) \mathbb{E}\left[g(X_{1},\dots,X_{d})\right]\approx\frac{1}{n}\sum_{k=1}^{n}g(X% _{1}^{k},\dots,X_{d}^{k})

Core_(game_theory).html

  1. ( N , v ) (N,v)
  2. N N
  3. v v
  4. x N x\in\mathbb{R}^{N}
  5. y y
  6. C C
  7. C C
  8. y y
  9. x i y i x_{i}\leq y_{i}
  10. i C i\in C
  11. i C i\in C
  12. x i < y i x_{i}<y_{i}
  13. C C
  14. y y
  15. C C
  16. i C y i v ( C ) \sum_{i\in C}y_{i}\geq v(C)
  17. x x
  18. y y
  19. x N x\in\mathbb{R}^{N}
  20. i N x i = v ( N ) \sum_{i\in N}x_{i}=v(N)
  21. i C x i v ( C ) \sum_{i\in C}x_{i}\geq v(C)
  22. C N C\subseteq N
  23. v ( S ) = { | S | / 2 , if | S | is even ; ( | S | - 1 ) / 2 , if | S | is odd . v(S)=\begin{cases}|S|/2,&\,\text{if }|S|\,\text{ is even};\\ (|S|-1)/2,&\,\text{if }|S|\,\text{ is odd}.\end{cases}
  24. x x
  25. y y

Corecursion.html

  1. 0 ! := 1 0!:=1
  2. n ! := n × ( n - 1 ) ! n!:=n\times(n-1)!
  3. 0 ! := 1 0!:=1
  4. n ! := n × ( n - 1 ) ! n!:=n\times(n-1)!
  5. n ! × ( n + 1 ) = : ( n + 1 ) ! n!\times(n+1)=:(n+1)!
  6. n , f = ( 0 , 1 ) : ( n + 1 , f × ( n + 1 ) ) n,f=(0,1):(n+1,f\times(n+1))
  7. n , f = 0 , 1 n,f=0,1
  8. n + 1 , f × ( n + 1 ) n+1,f\times(n+1)
  9. + 1 +1
  10. - 1 -1
  11. n ! n!

Correction_for_attenuation.html

  1. ρ = corr ( β ^ , θ ^ ) R β R θ . \rho=\frac{\mbox{corr}~{}(\hat{\beta},\hat{\theta})}{\sqrt{R_{\beta}R_{\theta}% }}.
  2. X X
  3. Y Y
  4. r x y r_{xy}
  5. r x x r_{xx}
  6. r y y r_{yy}
  7. X X
  8. Y Y
  9. r x y = r x y r x x r y y r_{x^{\prime}y^{\prime}}=\frac{r_{xy}}{\sqrt{r_{xx}r_{yy}}}
  10. X X
  11. Y Y
  12. X X^{\prime}
  13. Y Y^{\prime}
  14. r x y r_{x^{\prime}y^{\prime}}
  15. X X^{\prime}
  16. Y Y^{\prime}
  17. β \beta
  18. θ \theta
  19. β ^ \hat{\beta}
  20. θ ^ \hat{\theta}
  21. β \beta
  22. θ \theta
  23. β ^ = β + ϵ β , θ ^ = θ + ϵ θ , \hat{\beta}=\beta+\epsilon_{\beta},\quad\quad\hat{\theta}=\theta+\epsilon_{% \theta},
  24. ϵ β \epsilon_{\beta}
  25. ϵ θ \epsilon_{\theta}
  26. β ^ \hat{\beta}
  27. θ ^ \hat{\theta}
  28. corr ( β ^ , θ ^ ) = cov ( β ^ , θ ^ ) var [ β ^ ] var [ θ ^ ] \operatorname{corr}(\hat{\beta},\hat{\theta})=\frac{\operatorname{cov}(\hat{% \beta},\hat{\theta})}{\sqrt{\operatorname{var}[\hat{\beta}]\operatorname{var}[% \hat{\theta}}]}
  29. = cov ( β + ϵ β , θ + ϵ θ ) var [ β + ϵ β ] var [ θ + ϵ θ ] , =\frac{\operatorname{cov}(\beta+\epsilon_{\beta},\theta+\epsilon_{\theta})}{% \sqrt{\operatorname{var}[\beta+\epsilon_{\beta}]\operatorname{var}[\theta+% \epsilon_{\theta}]}},
  30. corr ( β ^ , θ ^ ) = cov ( β , θ ) ( var [ β ] + var [ ϵ β ] ) ( var [ θ ] + var [ ϵ θ ] ) \operatorname{corr}(\hat{\beta},\hat{\theta})=\frac{\operatorname{cov}(\beta,% \theta)}{\sqrt{(\operatorname{var}[\beta]+\operatorname{var}[\epsilon_{\beta}]% )(\operatorname{var}[\theta]+\operatorname{var}[\epsilon_{\theta}])}}
  31. = cov ( β , θ ) ( var [ β ] var [ θ ] ) . var [ β ] var [ θ ] ( var [ β ] + var [ ϵ β ] ) ( var [ θ ] + var [ ϵ θ ] ) =\frac{\operatorname{cov}(\beta,\theta)}{\sqrt{(\operatorname{var}[\beta]% \operatorname{var}[\theta])}}.\frac{\sqrt{\operatorname{var}[\beta]% \operatorname{var}[\theta]}}{\sqrt{(\operatorname{var}[\beta]+\operatorname{% var}[\epsilon_{\beta}])(\operatorname{var}[\theta]+\operatorname{var}[\epsilon% _{\theta}])}}
  32. = ρ R β R θ , =\rho\sqrt{R_{\beta}R_{\theta}},
  33. R β R_{\beta}
  34. β \beta
  35. R β R_{\beta}
  36. R β = var [ β ] var [ β ] + var [ ϵ β ] = var [ β ^ ] - var [ ϵ β ] var [ β ^ ] , R_{\beta}=\frac{\operatorname{var}[\beta]}{\operatorname{var}[\beta]+% \operatorname{var}[\epsilon_{\beta}]}=\frac{\operatorname{var}[\hat{\beta}]-% \operatorname{var}[\epsilon_{\beta}]}{\operatorname{var}[\hat{\beta}]},
  37. ϵ β \epsilon_{\beta}

Correlation_function_(statistical_mechanics).html

  1. s 1 s_{1}
  2. s 2 s_{2}
  3. R R
  4. R + r R+r
  5. t t
  6. t + τ t+\tau
  7. C ( r , τ ) = 𝐬 𝟏 ( R , t ) 𝐬 𝟐 ( R + r , t + τ ) - 𝐬 𝟏 ( R , t ) 𝐬 𝟐 ( R + r , t + τ ) . C(r,\tau)=\langle\mathbf{s_{1}}(R,t)\cdot\mathbf{s_{2}}(R+r,t+\tau)\rangle\ -% \langle\mathbf{s_{1}}(R,t)\rangle\langle\mathbf{s_{2}}(R+r,t+\tau)\rangle\,.
  8. \langle...\rangle
  9. s 1 s_{1}
  10. s 2 s_{2}
  11. 𝐬 𝟏 ( R , t ) 𝐬 𝟐 ( R + r , t + τ ) \langle\mathbf{s_{1}}(R,t)\rangle\langle\mathbf{s_{2}}(R+r,t+\tau)\rangle
  12. 𝐬 𝟏 ( R , t ) 𝐬 𝟐 ( R + r , t + τ ) \langle\mathbf{s_{1}}(R,t)\cdot\mathbf{s_{2}}(R+r,t+\tau)\rangle
  13. s 1 s_{1}
  14. s 2 s_{2}
  15. s 1 s_{1}
  16. s 2 s_{2}
  17. τ \tau
  18. τ = 0 \tau=0
  19. C ( r , 0 ) C(r,0)
  20. C ( r , 0 ) = 𝐬 𝟏 ( R , t ) 𝐬 𝟐 ( R + r , t ) - 𝐬 𝟏 ( R , t ) 𝐬 𝟐 ( R + r , t ) . C(r,0)=\langle\mathbf{s_{1}}(R,t)\cdot\mathbf{s_{2}}(R+r,t)\rangle\ -\langle% \mathbf{s_{1}}(R,t)\rangle\langle\mathbf{s_{2}}(R+r,t)\rangle\,.
  21. t t
  22. R R
  23. C ( r ) = 𝐬 𝟏 ( 0 ) 𝐬 𝟐 ( r ) - 𝐬 𝟏 ( 0 ) 𝐬 𝟐 ( r ) . C(r)=\langle\mathbf{s_{1}}(0)\cdot\mathbf{s_{2}}(r)\rangle\ -\langle\mathbf{s_% {1}}(0)\rangle\langle\mathbf{s_{2}}(r)\rangle\,.
  24. R R
  25. t t
  26. t + τ t+\tau
  27. C ( 0 , τ ) C(0,\tau)
  28. r = 0 r=0
  29. C ( 0 , τ ) = 𝐬 𝟏 ( R , t ) 𝐬 𝟐 ( R , t + τ ) - 𝐬 𝟏 ( R , t ) 𝐬 𝟐 ( R , t + τ ) . C(0,\tau)=\langle\mathbf{s_{1}}(R,t)\cdot\mathbf{s_{2}}(R,t+\tau)\rangle\ -% \langle\mathbf{s_{1}}(R,t)\rangle\langle\mathbf{s_{2}}(R,t+\tau)\rangle\,.
  30. C ( τ ) = 𝐬 𝟏 ( 0 ) 𝐬 𝟐 ( τ ) - 𝐬 𝟏 ( 0 ) 𝐬 𝟐 ( τ ) . C(\tau)=\langle\mathbf{s_{1}}(0)\cdot\mathbf{s_{2}}(\tau)\rangle\ -\langle% \mathbf{s_{1}}(0)\rangle\langle\mathbf{s_{2}}(\tau)\rangle\,.
  31. C ( r , τ ) C(r,\tau)
  32. τ \tau
  33. ξ \xi
  34. G ( r ) = 𝐬 ( R ) 𝐬 ( R + r ) - 𝐬 ( R ) 𝐬 ( R + r ) . G(r)=\langle\mathbf{s}(R)\cdot\mathbf{s}(R+r)\rangle\ -\langle\mathbf{s}(R)% \rangle\langle\mathbf{s}(R+r)\rangle\,.
  35. ξ \xi
  36. G ( r ) 1 r d - 2 + η exp ( - r ξ ) , G(r)\approx\frac{1}{r^{d-2+\eta}}\exp{\left(\frac{-r}{\xi}\right)}\,,
  37. η \eta
  38. T c T_{c}
  39. M 2 \langle M^{2}\rangle
  40. ξ | T - T c | - ν , \xi\propto|T-T_{c}|^{-\nu}\,,
  41. ν \nu
  42. C i 1 i 2 i n ( s 1 , s 2 , , s n ) = X i 1 ( s 1 ) X i 2 ( s 2 ) X i n ( s n ) . C_{i_{1}i_{2}\cdots i_{n}}(s_{1},s_{2},\cdots,s_{n})=\langle X_{i_{1}}(s_{1})X% _{i_{2}}(s_{2})\cdots X_{i_{n}}(s_{n})\rangle.

Correspondence_theorem_(group_theory).html

  1. N N
  2. G G
  3. A A
  4. G G
  5. N N
  6. G / N G/N
  7. G / N G/N
  8. G G
  9. N , N,
  10. N N
  11. G G
  12. G / N G/N
  13. G G
  14. H ¯ = H N . \bar{H}=HN.
  15. 𝒢 \mathcal{G}
  16. N A G N\subseteq A\subseteq G
  17. 𝒩 \mathcal{N}
  18. ϕ : 𝒢 𝒩 \phi:\mathcal{G}\to\mathcal{N}
  19. ϕ ( A ) = A / N \phi(A)=A/N
  20. A 𝒢 . A\in\mathcal{G}.
  21. 𝒢 \mathcal{G}
  22. A B A\subseteq B
  23. A B A^{\prime}\subseteq B^{\prime}
  24. A B A\subseteq B
  25. | B : A | = | B : A | |B:A|=|B^{\prime}:A^{\prime}|
  26. | B : A | |B:A|
  27. A , B / N = A , B , \langle A,B\rangle/N=\langle A^{\prime},B^{\prime}\rangle,
  28. A , B \langle A,B\rangle
  29. G G
  30. A B ; A\cup B;
  31. ( A B ) / N = A B (A\cap B)/N=A^{\prime}\cap B^{\prime}
  32. A A
  33. G G
  34. A A^{\prime}
  35. G / N G/N

Corwin_Hansch.html

  1. π \pi
  2. π \pi
  3. π \pi

Cosmological_decade.html

  1. - -\infty

Cost_curve.html

  1. S T C = P K . K + P L . L STC=P_{K}.K+P_{L}.L

Cost_of_carry.html

  1. F = S e ( r + s - c ) t F=Se^{(r+s-c)t}\,
  2. F F
  3. S S
  4. e e
  5. r r
  6. s s
  7. c c
  8. t t

Coulomb_collision.html

  1. Δ m e v Z e 2 4 π ϵ 0 1 v b \Delta m_{e}v_{\perp}\approx\frac{Ze^{2}}{4\pi\epsilon_{0}}\,\frac{1}{vb}
  2. 1 / v 2 1/v^{2}
  3. D v = ( Z e 2 4 π ϵ 0 ) 2 1 v 2 b 2 n v ( 2 π b d b ) = ( Z e 2 4 π ϵ 0 ) 2 2 π n v d b b D_{v\perp}=\int\left(\frac{Ze^{2}}{4\pi\epsilon_{0}}\right)^{2}\,\frac{1}{v^{2% }b^{2}}\,nv(2\pi b\,{\rm d}b)=\left(\frac{Ze^{2}}{4\pi\epsilon_{0}}\right)^{2}% \,\frac{2\pi n}{v}\,\int\frac{{\rm d}b}{b}
  4. Δ m e v \Delta m_{e}v_{\perp}
  5. b 0 = Z e 2 4 π ϵ 0 1 m e v 2 b_{0}=\frac{Ze^{2}}{4\pi\epsilon_{0}}\,\frac{1}{m_{e}v^{2}}
  6. λ D = ϵ 0 k T e n e e 2 \lambda_{D}=\sqrt{\frac{\epsilon_{0}kT_{e}}{n_{e}e^{2}}}

Coulometry.html

  1. [ C 5 H 5 NH ] SO 3 CH 3 + I 2 + H 2 O + 2 C 5 H 5 N [ C 5 H 5 NH ] SO 4 CH 3 + 2 [ C 5 H 5 NH ] I \mathrm{[C_{5}H_{5}NH]SO_{3}CH_{3}+I_{2}+H_{2}O+2C_{5}H_{5}N}\longrightarrow% \mathrm{[C_{5}H_{5}NH]SO_{4}CH_{3}+2[C_{5}H_{5}NH]I}
  2. Δ \Delta
  3. i i
  4. M M
  5. ρ \rho
  6. A A
  7. i M A ρ \triangle\propto\frac{iM}{A\rho}

Countable_chain_condition.html

  1. { 0 , 1 } 2 2 0 \{0,1\}^{2^{2^{\aleph_{0}}\ }}

Countersteering.html

  1. θ = arctan ( v 2 g r ) \theta=\arctan\left(\frac{v^{2}}{gr}\right)
  2. v v
  3. r r
  4. g g

Counting_problem_(complexity).html

  1. c R ( x ) = | { y R ( x , y ) } | c_{R}(x)=|\{y\mid R(x,y)\}|\,
  2. # R = { ( x , y ) y c R ( x ) } \#R=\{(x,y)\mid y\leq c_{R}(x)\}

Counting_single_transferable_votes.html

  1. (votes cast) (available seats) \frac{\,\text{(votes cast)}}{\,\text{(available seats)}}
  2. ( votes seats + 1 ) + 1 \left(\frac{\,\text{votes}}{\,\text{seats}+1}\right)+1
  3. 57 2 + 1 + 1 = 20 {57\over 2+1}+1=20
  4. n n
  5. 1 n \begin{matrix}\frac{1}{n}\end{matrix}
  6. 75 272 - 92 = 4 10 \textstyle\frac{75}{272-92}=\frac{4}{10}
  7. 1 5 \textstyle\frac{1}{5}
  8. 1 5 × 4 10 = 4 50 \textstyle\frac{1}{5}\times\frac{4}{10}=\frac{4}{50}
  9. Surplus Transfer Value = ( Total value of Candidate’s votes - Quota Total value of Candidate’s votes ) × Value of each vote \,\text{Surplus Transfer Value}=\left({{\,\text{Total value of Candidate's % votes}-\,\text{Quota}}\over\,\text{Total value of Candidate's votes}}\right)% \times\,\text{Value of each vote}
  10. 1 1
  11. 0
  12. w new = w old × Quota Candidate’s votes w\text{new}=w\text{old}\times\frac{\,\text{Quota}}{\,\text{Candidate's votes}}
  13. Candidate’s votes = Quota \,\text{Candidate's votes}=\,\text{Quota}
  14. 1 - nth Weighting 1-\,\text{nth Weighting}
  15. a a
  16. b b
  17. c c
  18. a a
  19. ( 1 - a ) b (1-a)b
  20. ( 1 - a ) ( 1 - b ) c (1-a)(1-b)c
  21. votes - excess seats + 1 , {{\,\text{votes}-\,\text{excess}}\over\,\text{seats}+1},

Coupling_(computer_programming).html

  1. Coupling ( C ) = 1 - 1 d i + 2 × c i + d o + 2 × c o + g d + 2 × g c + w + r \mathrm{Coupling}(C)=1-\frac{1}{d_{i}+2\times c_{i}+d_{o}+2\times c_{o}+g_{d}+% 2\times g_{c}+w+r}
  2. C = 1 - 1 1 + 0 + 1 + 0 + 0 + 0 + 1 + 0 = 1 - 1 3 = 0.67 C=1-\frac{1}{1+0+1+0+0+0+1+0}=1-\frac{1}{3}=0.67
  3. C = 1 - 1 5 + 2 × 5 + 5 + 2 × 5 + 10 + 0 + 3 + 4 = 0.98 C=1-\frac{1}{5+2\times 5+5+2\times 5+10+0+3+4}=0.98

Courant–Friedrichs–Lewy_condition.html

  1. n n
  2. n = 1 n=1
  3. n = 2 n=2
  4. n = 3 n=3
  5. C = u Δ t Δ x C max C=\frac{u\,\Delta t}{\Delta x}\leq C_{\max}
  6. u u
  7. Δ t \Delta t
  8. Δ x \Delta x
  9. C max C_{\max}
  10. C max = 1 C_{\max}=1
  11. C max C_{\max}
  12. C = u x Δ t Δ x + u y Δ t Δ y C max C=\frac{u_{x}\,\Delta t}{\Delta x}+\frac{u_{y}\,\Delta t}{\Delta y}\leq C_{\max}
  13. n n
  14. C = Δ t i = 1 n u x i Δ x i C max . C=\Delta t\sum_{i=1}^{n}\frac{u_{x_{i}}}{\Delta x_{i}}\leq C_{\max}.
  15. Δ x i , i = 1 , , n \Delta x_{i},i=1,\ldots,n

Cournot_competition.html

  1. N N
  2. c i ( q i ) c_{i}(q_{i})
  3. p 1 p_{1}
  4. p 2 p_{2}
  5. q 1 q_{1}
  6. q 2 q_{2}
  7. c c
  8. p 1 = p 2 = P ( q 1 + q 2 ) p_{1}=p_{2}=P(q_{1}+q_{2})
  9. Π 1 = q 1 ( P ( q 1 + q 2 ) - c ) \Pi_{1}=q_{1}(P(q_{1}+q_{2})-c)
  10. q 2 q_{2}
  11. P ( 0 + q 2 ) = P ( q 2 ) P(0+q_{2})=P(q_{2})
  12. q 1 q_{1}^{\prime}
  13. P ( q 1 + q 2 ) P(q_{1}^{\prime}+q_{2})
  14. d 1 ( q 2 ) d_{1}(q_{2})
  15. d 1 ( q 2 ) d_{1}(q_{2})
  16. q 2 q_{2}
  17. r 1 ( q 2 ) r_{1}(q_{2})
  18. d 1 ( q 2 ) d_{1}(q_{2})
  19. c c
  20. r 1 ( q 2 ) r_{1}(q_{2})
  21. q 1 ′′ ( q 2 ) q_{1}^{\prime\prime}(q_{2})
  22. q 1 ′′ ( q 2 ) q_{1}^{\prime\prime}(q_{2})
  23. q 2 q_{2}
  24. q 2 q_{2}
  25. q 2 = 0 q_{2}=0
  26. d 1 ( 0 ) = D d_{1}(0)=D
  27. q 1 ′′ ( 0 ) = q m q_{1}^{\prime\prime}(0)=q^{m}
  28. q m q^{m}
  29. q 2 = q c q_{2}=q^{c}
  30. P ( q c ) = c P(q^{c})=c
  31. q 1 ′′ ( q c ) = 0 q_{1}^{\prime\prime}(q^{c})=0
  32. d 1 ( q c ) d_{1}(q^{c})
  33. q 1 ′′ ( q 2 ) q_{1}^{\prime\prime}(q_{2})
  34. q 1 ′′ ( q 2 ) q_{1}^{\prime\prime}(q_{2})
  35. q 1 ′′ ( q 2 ) q_{1}^{\prime\prime}(q_{2})
  36. P ( q 1 + q 2 ) P(q_{1}+q_{2})
  37. C i ( q i ) C_{i}(q_{i})
  38. Π i = P ( q 1 + q 2 ) q i - C i ( q i ) \Pi_{i}=P(q_{1}+q_{2})\cdot q_{i}-C_{i}(q_{i})
  39. q i q_{i}
  40. Π i \Pi_{i}
  41. q j q_{j}
  42. i j i\neq j
  43. Π i \Pi_{i}
  44. q i q_{i}
  45. Π i \Pi_{i}
  46. q i q_{i}
  47. Π i q i = P ( q 1 + q 2 ) q i q i + P ( q 1 + q 2 ) - C i ( q i ) q i \frac{\partial\Pi_{i}}{\partial q_{i}}=\frac{\partial P(q_{1}+q_{2})}{\partial q% _{i}}\cdot q_{i}+P(q_{1}+q_{2})-\frac{\partial C_{i}(q_{i})}{\partial q_{i}}
  48. Π i q i = P ( q 1 + q 2 ) q i q i + P ( q 1 + q 2 ) - C i ( q i ) q i = 0 \frac{\partial\Pi_{i}}{\partial q_{i}}=\frac{\partial P(q_{1}+q_{2})}{\partial q% _{i}}\cdot q_{i}+P(q_{1}+q_{2})-\frac{\partial C_{i}(q_{i})}{\partial q_{i}}=0
  49. q i q_{i}
  50. q 1 q_{1}
  51. q 2 q_{2}
  52. q 1 q_{1}
  53. q 2 q_{2}
  54. P ( q 1 + q 2 ) = a - ( q 1 + q 2 ) P(q_{1}+q_{2})=a-(q_{1}+q_{2})
  55. C i ( q i ) C_{i}(q_{i})
  56. 2 C i ( q i ) q i 2 = 0 \frac{\partial^{2}C_{i}(q_{i})}{\partial q_{i}^{2}}=0
  57. C i ( q i ) q j = 0 , j i \frac{\partial C_{i}(q_{i})}{\partial q_{j}}=0,j\neq i
  58. Π i = ( a - ( q 1 + q 2 ) ) q i - C i ( q i ) \Pi_{i}=\bigg(a-(q_{1}+q_{2})\bigg)\cdot q_{i}-C_{i}(q_{i})
  59. ( a - ( q 1 + q 2 ) ) q i q i + a - ( q 1 + q 2 ) - C i ( q i ) q i = 0 \frac{\partial\bigg(a-(q_{1}+q_{2})\bigg)}{\partial q_{i}}\cdot q_{i}+a-(q_{1}% +q_{2})-\frac{\partial C_{i}(q_{i})}{\partial q_{i}}=0
  60. ( a - ( q 1 + q 2 ) ) q 1 q 1 + a - ( q 1 + q 2 ) - C 1 ( q 1 ) q 1 = 0 \frac{\partial\bigg(a-(q_{1}+q_{2})\bigg)}{\partial q_{1}}\cdot q_{1}+a-(q_{1}% +q_{2})-\frac{\partial C_{1}(q_{1})}{\partial q_{1}}=0
  61. - q 1 + a - ( q 1 + q 2 ) - C 1 ( q 1 ) q 1 = 0 \Rightarrow\ -q_{1}+a-(q_{1}+q_{2})-\frac{\partial C_{1}(q_{1})}{\partial q_{1% }}=0
  62. q 1 = a - q 2 - C 1 ( q 1 ) q 1 2 \Rightarrow\ q_{1}=\frac{a-q_{2}-\frac{\partial C_{1}(q_{1})}{\partial q_{1}}}% {2}
  63. q 2 = a - q 1 - C 2 ( q 2 ) q 2 2 \Rightarrow\ q_{2}=\frac{a-q_{1}-\frac{\partial C_{2}(q_{2})}{\partial q_{2}}}% {2}
  64. q 2 q_{2}
  65. q 1 q_{1}
  66. q 2 q_{2}
  67. q 1 = a - ( a - q 1 - C 2 ( q 2 ) q 2 2 ) - C 1 ( q 1 ) q 1 2 \ q_{1}=\frac{a-\left(\frac{a-q_{1}-\frac{\partial C_{2}(q_{2})}{\partial q_{2% }}}{2}\right)-\frac{\partial C_{1}(q_{1})}{\partial q_{1}}}{2}
  68. q 1 * = a + C 2 ( q 2 ) q 2 - 2 * C 1 ( q 1 ) q 1 3 \Rightarrow\ q_{1}^{*}=\frac{a+\frac{\partial C_{2}(q_{2})}{\partial q_{2}}-2*% \frac{\partial C_{1}(q_{1})}{\partial q_{1}}}{3}
  69. q 2 * = a + C 1 ( q 1 ) q 1 - 2 * C 2 ( q 2 ) q 2 3 \Rightarrow\ q_{2}^{*}=\frac{a+\frac{\partial C_{1}(q_{1})}{\partial q_{1}}-2^% {*}\frac{\partial C_{2}(q_{2})}{\partial q_{2}}}{3}
  70. ( q 1 * , q 2 * ) (q_{1}^{*},q_{2}^{*})
  71. P ( q 1 + q 2 ) = a - ( q 1 + q 2 ) P(q_{1}+q_{2})=a-(q_{1}+q_{2})
  72. p ( q ) = a - b q = a - b Q = p ( Q ) \ p(q)=a-bq=a-bQ=p(Q)
  73. c i ( q i ) = c q i \ c_{i}(q_{i})=cq_{i}
  74. q i = Q / N = a - c b ( N + 1 ) , \ q_{i}=Q/N=\frac{a-c}{b(N+1)},
  75. q i = N q = N ( a - c ) b ( N + 1 ) , \sum q_{i}=Nq=\frac{N(a-c)}{b(N+1)},
  76. p = a - b ( N q ) = a + N c N + 1 , \ p=a-b(Nq)=\frac{a+Nc}{N+1},
  77. Π i = ( a - c N + 1 ) 2 ( 1 b ) \Pi_{i}=\left(\frac{a-c}{N+1}\right)^{2}\left(\frac{1}{b}\right)
  78. lim N p = c \lim_{N\rightarrow\infty}p=c
  79. N N
  80. F F
  81. N = a - c F b - 1 N=\frac{a-c}{\sqrt{Fb}}-1
  82. q = F b b q=\frac{\sqrt{Fb}}{b}

Covering_group.html

  1. Z ( H ) Z ( G ) / K . Z(H)\cong Z(G)/K.
  2. π 1 ( H ) / π 1 ( G ) \pi_{1}(H)/\pi_{1}(G)
  3. 1 π 1 ( H ) H ~ H 1 1\to\pi_{1}(H)\to\tilde{H}\to H\to 1
  4. H ~ \tilde{H}
  5. H ~ , \tilde{H},
  6. H ~ / Z ( H ~ ) \tilde{H}/Z(\tilde{H})
  7. S L 2 ( ~ 𝐑 ) {\mathrm{S}\widetilde{\mathrm{L}_{2}(}\mathbf{R})}
  8. ϕ * : 𝔤 𝔥 \phi_{*}:\mathfrak{g}\to\mathfrak{h}
  9. 𝔤 \mathfrak{g}
  10. 𝔤 \mathfrak{g}

CPCTC.html

  1. A B C D E F \triangle ABC\cong\triangle DEF\,
  2. A B ¯ D E ¯ \overline{AB}\cong\overline{DE}\,
  3. B C ¯ E F ¯ \overline{BC}\cong\overline{EF}\,
  4. A C ¯ D F ¯ \overline{AC}\cong\overline{DF}\,
  5. B A C E D F \angle BAC\cong\angle EDF\,
  6. A B C D E F \angle ABC\cong\angle DEF\,
  7. B C A E F D \angle BCA\cong\angle EFD\,

Cramer–Shoup_cryptosystem.html

  1. G G
  2. q q
  3. g 1 , g 2 g_{1},g_{2}
  4. ( x 1 , x 2 , y 1 , y 2 , z ) ({x}_{1},{x}_{2},{y}_{1},{y}_{2},z)
  5. { 0 , , q - 1 } \{0,\ldots,q-1\}
  6. c = g 1 x 1 g 2 x 2 , d = g 1 y 1 g 2 y 2 , h = g 1 z c={g}_{1}^{x_{1}}g_{2}^{x_{2}},d={g}_{1}^{y_{1}}g_{2}^{y_{2}},h={g}_{1}^{z}
  7. ( c , d , h ) (c,d,h)
  8. G , q , g 1 , g 2 G,q,g_{1},g_{2}
  9. ( x 1 , x 2 , y 1 , y 2 , z ) (x_{1},x_{2},y_{1},y_{2},z)
  10. m m
  11. ( G , q , g 1 , g 2 , c , d , h ) (G,q,g_{1},g_{2},c,d,h)
  12. m m
  13. G G
  14. k k
  15. { 0 , , q - 1 } \{0,\ldots,q-1\}
  16. u 1 = g 1 k , u 2 = g 2 k u_{1}={g}_{1}^{k},u_{2}={g}_{2}^{k}
  17. e = h k m e=h^{k}m\,
  18. α = H ( u 1 , u 2 , e ) \alpha=H(u_{1},u_{2},e)\,
  19. v = c k d k α v=c^{k}d^{k\alpha}\,
  20. ( u 1 , u 2 , e , v ) (u_{1},u_{2},e,v)
  21. ( u 1 , u 2 , e , v ) (u_{1},u_{2},e,v)
  22. ( x 1 , x 2 , y 1 , y 2 , z ) (x_{1},x_{2},y_{1},y_{2},z)
  23. α = H ( u 1 , u 2 , e ) \alpha=H(u_{1},u_{2},e)\,
  24. u 1 x 1 u 2 x 2 ( u 1 y 1 u 2 y 2 ) α = v {u}_{1}^{x_{1}}u_{2}^{x_{2}}({u}_{1}^{y_{1}}u_{2}^{y_{2}})^{\alpha}=v\,
  25. m = e / ( u 1 z ) m=e/({u}_{1}^{z})\,
  26. u 1 z = g 1 k z = h k {u}_{1}^{z}={g}_{1}^{kz}=h^{k}\,
  27. m = e / h k . m=e/h^{k}.\,
  28. G G

Crank–Nicolson_method.html

  1. t t
  2. x x
  3. 2 {}^{2}
  4. u t = F ( u , x , t , u x , 2 u x 2 ) \frac{\partial u}{\partial t}=F\left(u,\,x,\,t,\,\frac{\partial u}{\partial x}% ,\,\frac{\partial^{2}u}{\partial x^{2}}\right)
  5. u ( i Δ x , n Δ t ) = u i n u(i\Delta x,\,n\Delta t)=u_{i}^{n}\,
  6. n n
  7. u i n + 1 - u i n Δ t = F i n ( u , x , t , u x , 2 u x 2 ) (forward Euler) \frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}=F_{i}^{n}\left(u,\,x,\,t,\,\frac{% \partial u}{\partial x},\,\frac{\partial^{2}u}{\partial x^{2}}\right)\qquad% \mbox{(forward Euler)}~{}
  8. u i n + 1 - u i n Δ t = F i n + 1 ( u , x , t , u x , 2 u x 2 ) (backward Euler) \frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}=F_{i}^{n+1}\left(u,\,x,\,t,\,\frac{% \partial u}{\partial x},\,\frac{\partial^{2}u}{\partial x^{2}}\right)\qquad% \mbox{(backward Euler)}~{}
  9. u i n + 1 - u i n Δ t = 1 2 [ F i n + 1 ( u , x , t , u x , 2 u x 2 ) + F i n ( u , x , t , u x , 2 u x 2 ) ] (Crank–Nicolson) . \frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}=\frac{1}{2}\left[F_{i}^{n+1}\left(u,\,x% ,\,t,\,\frac{\partial u}{\partial x},\,\frac{\partial^{2}u}{\partial x^{2}}% \right)+F_{i}^{n}\left(u,\,x,\,t,\,\frac{\partial u}{\partial x},\,\frac{% \partial^{2}u}{\partial x^{2}}\right)\right]\qquad\mbox{(Crank--Nicolson)}~{}.
  10. 𝒪 ( n ) \mathcal{O}(n)
  11. 𝒪 ( n 3 ) \mathcal{O}(n^{3})
  12. u t = a 2 u x 2 {\partial u\over\partial t}=a\frac{\partial^{2}u}{\partial x^{2}}
  13. u i n + 1 - u i n Δ t = a 2 ( Δ x ) 2 ( ( u i + 1 n + 1 - 2 u i n + 1 + u i - 1 n + 1 ) + ( u i + 1 n - 2 u i n + u i - 1 n ) ) \frac{u_{i}^{n+1}-u_{i}^{n}}{\Delta t}=\frac{a}{2(\Delta x)^{2}}\left((u_{i+1}% ^{n+1}-2u_{i}^{n+1}+u_{i-1}^{n+1})+(u_{i+1}^{n}-2u_{i}^{n}+u_{i-1}^{n})\right)
  14. r = a Δ t ( Δ x ) 2 r=\frac{a\Delta t}{(\Delta x)^{2}}
  15. - r u i + 1 n + 1 + 2 ( 1 + r ) u i n + 1 - r u i - 1 n + 1 = r u i + 1 n + 2 ( 1 - r ) u i n + r u i - 1 n -ru_{i+1}^{n+1}+2(1+r)u_{i}^{n+1}-ru_{i-1}^{n+1}=ru_{i+1}^{n}+2(1-r)u_{i}^{n}+% ru_{i-1}^{n}\,
  16. u i n + 1 u_{i}^{n+1}\,
  17. u t = a ( u ) 2 u x 2 \frac{\partial u}{\partial t}=a(u)\frac{\partial^{2}u}{\partial x^{2}}
  18. a a
  19. a i n ( u ) a_{i}^{n}(u)\,
  20. a i n + 1 ( u ) a_{i}^{n+1}(u)\,
  21. a i n + 1 ( u ) a_{i}^{n+1}(u)\,
  22. D x D_{x}
  23. C t = D x 2 C x 2 - U x C x - k ( C - C N ) - k ( C - C M ) \frac{\partial C}{\partial t}=D_{x}\frac{\partial^{2}C}{\partial x^{2}}-U_{x}% \frac{\partial C}{\partial x}-k(C-C_{N})-k(C-C_{M})
  24. C t C i j + 1 - C i j Δ t \frac{\partial C}{\partial t}\Rightarrow\frac{C_{i}^{j+1}-C_{i}^{j}}{\Delta t}
  25. 2 C x 2 1 2 ( Δ x ) 2 ( ( C i + 1 j + 1 - 2 C i j + 1 + C i - 1 j + 1 ) + ( C i + 1 j - 2 C i j + C i - 1 j ) ) \frac{\partial^{2}C}{\partial x^{2}}\Rightarrow\frac{1}{2(\Delta x)^{2}}\left(% (C_{i+1}^{j+1}-2C_{i}^{j+1}+C_{i-1}^{j+1})+(C_{i+1}^{j}-2C_{i}^{j}+C_{i-1}^{j}% )\right)
  26. C x 1 2 ( ( C i + 1 j + 1 - C i - 1 j + 1 ) 2 ( Δ x ) + ( C i + 1 j - C i - 1 j ) 2 ( Δ x ) ) \frac{\partial C}{\partial x}\Rightarrow\frac{1}{2}\left(\frac{(C_{i+1}^{j+1}-% C_{i-1}^{j+1})}{2(\Delta x)}+\frac{(C_{i+1}^{j}-C_{i-1}^{j})}{2(\Delta x)}\right)
  27. C N 1 2 ( C N i j + 1 + C N i j ) C_{N}\Rightarrow\frac{1}{2}(C_{Ni}^{j+1}+C_{Ni}^{j})
  28. λ = D x Δ t 2 Δ x 2 \lambda=\frac{D_{x}\Delta t}{2\Delta x^{2}}
  29. α = U x Δ t 4 Δ x \alpha=\frac{U_{x}\Delta t}{4\Delta x}
  30. β = k Δ t 2 \beta=\frac{k\Delta t}{2}
  31. - β C N i j + 1 - ( λ + α ) C i - 1 j + 1 + ( 1 + 2 λ + 2 β ) C i j + 1 - ( λ - α ) C i + 1 j + 1 - β C M i j + 1 = β C N i j + ( λ + α ) C i - 1 j + ( 1 - 2 λ - 2 β ) C i j + ( λ - α ) C i + 1 j + β C M i j . -\beta C_{Ni}^{j+1}-(\lambda+\alpha)C_{i-1}^{j+1}+(1+2\lambda+2\beta)C_{i}^{j+% 1}-(\lambda-\alpha)C_{i+1}^{j+1}-\beta C_{Mi}^{j+1}=\beta C_{Ni}^{j}+(\lambda+% \alpha)C_{i-1}^{j}+(1-2\lambda-2\beta)C_{i}^{j}+(\lambda-\alpha)C_{i+1}^{j}+% \beta C_{Mi}^{j}.
  32. - ( λ + α ) C i - 1 j + 1 + ( 1 + 2 λ + β ) C i j + 1 - ( λ - α ) C i + 1 j + 1 - β C M i j + 1 = + ( λ + α ) C i - 1 j + ( 1 - 2 λ - β ) C i j + ( λ - α ) C i + 1 j + β C M i j . -(\lambda+\alpha)C_{i-1}^{j+1}+(1+2\lambda+\beta)C_{i}^{j+1}-(\lambda-\alpha)C% _{i+1}^{j+1}-\beta C_{Mi}^{j+1}=+(\lambda+\alpha)C_{i-1}^{j}+(1-2\lambda-\beta% )C_{i}^{j}+(\lambda-\alpha)C_{i+1}^{j}+\beta C_{Mi}^{j}.
  33. - β C N i j + 1 - ( λ + α ) C i - 1 j + 1 + ( 1 + 2 λ + β ) C i j + 1 - ( λ - α ) C i + 1 j + 1 = β C N i j + ( λ + α ) C i - 1 j + ( 1 - 2 λ - β ) C i j + ( λ - α ) C i + 1 j . -\beta C_{Ni}^{j+1}-(\lambda+\alpha)C_{i-1}^{j+1}+(1+2\lambda+\beta)C_{i}^{j+1% }-(\lambda-\alpha)C_{i+1}^{j+1}=\beta C_{Ni}^{j}+(\lambda+\alpha)C_{i-1}^{j}+(% 1-2\lambda-\beta)C_{i}^{j}+(\lambda-\alpha)C_{i+1}^{j}.
  34. C 0 j C_{0}^{j}
  35. C 0 j + 1 C_{0}^{j+1}
  36. C N 0 j C_{N0}^{j}
  37. C M 0 j C_{M0}^{j}
  38. C x x = z = ( C i + 1 - C i - 1 ) 2 Δ x = 0. \frac{\partial C}{\partial x}_{x=z}=\frac{(C_{i+1}-C_{i-1})}{2\Delta x}=0.
  39. C i + 1 j + 1 = C i - 1 j + 1 . C_{i+1}^{j+1}=C_{i-1}^{j+1}.\,
  40. [ A A ] [ C j + 1 ] = [ B B ] [ C j ] + [ d ] \begin{bmatrix}AA\end{bmatrix}\begin{bmatrix}C^{j+1}\end{bmatrix}=[BB][C^{j}]+% [d]
  41. 𝐂 𝐣 + 𝟏 = [ C 11 j + 1 C 12 j + 1 C 13 j + 1 C 14 j + 1 C 21 j + 1 C 22 j + 1 C 23 j + 1 C 24 j + 1 C 31 j + 1 C 32 j + 1 C 33 j + 1 C 34 j + 1 ] \mathbf{C^{j+1}}=\begin{bmatrix}C_{11}^{j+1}\\ C_{12}^{j+1}\\ C_{13}^{j+1}\\ C_{14}^{j+1}\\ C_{21}^{j+1}\\ C_{22}^{j+1}\\ C_{23}^{j+1}\\ C_{24}^{j+1}\\ C_{31}^{j+1}\\ C_{32}^{j+1}\\ C_{33}^{j+1}\\ C_{34}^{j+1}\end{bmatrix}
  42. 𝐂 𝐣 = [ C 11 j C 12 j C 13 j C 14 j C 21 j C 22 j C 23 j C 24 j C 31 j C 32 j C 33 j C 34 j ] . \mathbf{C^{j}}=\begin{bmatrix}C_{11}^{j}\\ C_{12}^{j}\\ C_{13}^{j}\\ C_{14}^{j}\\ C_{21}^{j}\\ C_{22}^{j}\\ C_{23}^{j}\\ C_{24}^{j}\\ C_{31}^{j}\\ C_{32}^{j}\\ C_{33}^{j}\\ C_{34}^{j}\end{bmatrix}.
  43. 𝐀𝐀 = [ A A 1 A A 3 0 A A 3 A A 2 A A 3 0 A A 3 A A 1 ] \mathbf{AA}=\begin{bmatrix}AA1&AA3&0\\ AA3&AA2&AA3\\ 0&AA3&AA1\end{bmatrix}
  44. 𝐁𝐁 = [ B B 1 - A A 3 0 - A A 3 B B 2 - A A 3 0 - A A 3 B B 1 ] \mathbf{BB}=\begin{bmatrix}BB1&-AA3&0\\ -AA3&BB2&-AA3\\ 0&-AA3&BB1\end{bmatrix}
  45. 𝐀𝐀𝟏 = [ ( 1 + 2 λ + β ) - ( λ - α ) 0 0 - ( λ + α ) ( 1 + 2 λ + β ) - ( λ - α ) 0 0 - ( λ + α ) ( 1 + 2 λ + β ) - ( λ - α ) 0 0 - 2 λ ( 1 + 2 λ + β ) ] \mathbf{AA1}=\begin{bmatrix}(1+2\lambda+\beta)&-(\lambda-\alpha)&0&0\\ -(\lambda+\alpha)&(1+2\lambda+\beta)&-(\lambda-\alpha)&0\\ 0&-(\lambda+\alpha)&(1+2\lambda+\beta)&-(\lambda-\alpha)\\ 0&0&-2\lambda&(1+2\lambda+\beta)\end{bmatrix}
  46. 𝐀𝐀𝟐 = [ ( 1 + 2 λ + 2 β ) - ( λ - α ) 0 0 - ( λ + α ) ( 1 + 2 λ + 2 β ) - ( λ - α ) 0 0 - ( λ + α ) ( 1 + 2 λ + 2 β ) - ( λ - α ) 0 0 - 2 λ ( 1 + 2 λ + 2 β ) ] \mathbf{AA2}=\begin{bmatrix}(1+2\lambda+2\beta)&-(\lambda-\alpha)&0&0\\ -(\lambda+\alpha)&(1+2\lambda+2\beta)&-(\lambda-\alpha)&0\\ 0&-(\lambda+\alpha)&(1+2\lambda+2\beta)&-(\lambda-\alpha)\\ 0&0&-2\lambda&(1+2\lambda+2\beta)\end{bmatrix}
  47. 𝐀𝐀𝟑 = [ - β 0 0 0 0 - β 0 0 0 0 - β 0 0 0 0 - β ] \mathbf{AA3}=\begin{bmatrix}-\beta&0&0&0\\ 0&-\beta&0&0\\ 0&0&-\beta&0\\ 0&0&0&-\beta\end{bmatrix}
  48. 𝐁𝐁𝟏 = [ ( 1 - 2 λ - β ) ( λ - α ) 0 0 ( λ + α ) ( 1 - 2 λ - β ) ( λ - α ) 0 0 ( λ + α ) ( 1 - 2 λ - β ) ( λ - α ) 0 0 2 λ ( 1 - 2 λ - β ) ] \mathbf{BB1}=\begin{bmatrix}(1-2\lambda-\beta)&(\lambda-\alpha)&0&0\\ (\lambda+\alpha)&(1-2\lambda-\beta)&(\lambda-\alpha)&0\\ 0&(\lambda+\alpha)&(1-2\lambda-\beta)&(\lambda-\alpha)\\ 0&0&2\lambda&(1-2\lambda-\beta)\end{bmatrix}
  49. 𝐁𝐁𝟐 = [ ( 1 - 2 λ - 2 β ) ( λ - α ) 0 0 ( λ + α ) ( 1 - 2 λ - 2 β ) ( λ - α ) 0 0 ( λ + α ) ( 1 - 2 λ - 2 β ) ( λ - α ) 0 0 2 λ ( 1 - 2 λ - 2 β ) ] . \mathbf{BB2}=\begin{bmatrix}(1-2\lambda-2\beta)&(\lambda-\alpha)&0&0\\ (\lambda+\alpha)&(1-2\lambda-2\beta)&(\lambda-\alpha)&0\\ 0&(\lambda+\alpha)&(1-2\lambda-2\beta)&(\lambda-\alpha)\\ 0&0&2\lambda&(1-2\lambda-2\beta)\end{bmatrix}.
  50. 𝐝 = [ ( λ + α ) ( C 10 j + 1 + C 10 j ) 0 0 0 ( λ + α ) ( C 20 j + 1 + C 20 j ) 0 0 0 ( λ + α ) ( C 30 j + 1 + C 30 j ) 0 0 0 ] . \mathbf{d}=\begin{bmatrix}(\lambda+\alpha)(C_{10}^{j+1}+C_{10}^{j})\\ 0\\ 0\\ 0\\ (\lambda+\alpha)(C_{20}^{j+1}+C_{20}^{j})\\ 0\\ 0\\ 0\\ (\lambda+\alpha)(C_{30}^{j+1}+C_{30}^{j})\\ 0\\ 0\\ 0\end{bmatrix}.
  51. [ C j + 1 ] = [ A A - 1 ] ( [ B B ] [ C j ] + [ d ] ) . \begin{bmatrix}C^{j+1}\end{bmatrix}=\begin{bmatrix}AA^{-1}\end{bmatrix}([BB][C% ^{j}]+[d]).
  52. u t = a ( 2 u x 2 + 2 u y 2 ) \frac{\partial u}{\partial t}=a\left(\frac{\partial^{2}u}{\partial x^{2}}+% \frac{\partial^{2}u}{\partial y^{2}}\right)
  53. u i , j n + 1 = u i , j n + 1 2 a Δ t ( Δ x ) 2 [ ( u i + 1 , j n + 1 + u i - 1 , j n + 1 + u i , j + 1 n + 1 + u i , j - 1 n + 1 - 4 u i , j n + 1 ) + ( u i + 1 , j n + u i - 1 , j n + u i , j + 1 n + u i , j - 1 n - 4 u i , j n ) ] \begin{aligned}\displaystyle u_{i,j}^{n+1}&\displaystyle=u_{i,j}^{n}+\frac{1}{% 2}\frac{a\Delta t}{(\Delta x)^{2}}\big[(u_{i+1,j}^{n+1}+u_{i-1,j}^{n+1}+u_{i,j% +1}^{n+1}+u_{i,j-1}^{n+1}-4u_{i,j}^{n+1})\\ &\displaystyle\qquad{}+(u_{i+1,j}^{n}+u_{i-1,j}^{n}+u_{i,j+1}^{n}+u_{i,j-1}^{n% }-4u_{i,j}^{n})\big]\end{aligned}
  54. Δ x = Δ y \Delta x=\Delta y
  55. μ = a Δ t ( Δ x ) 2 . \mu=\frac{a\Delta t}{(\Delta x)^{2}}.
  56. ( 1 + 2 μ ) u i , j n + 1 - μ 2 ( u i + 1 , j n + 1 + u i - 1 , j n + 1 + u i , j + 1 n + 1 + u i , j - 1 n + 1 ) \displaystyle(1+2\mu)u_{i,j}^{n+1}-\frac{\mu}{2}\left(u_{i+1,j}^{n+1}+u_{i-1,j% }^{n+1}+u_{i,j+1}^{n+1}+u_{i,j-1}^{n+1}\right)
  57. u t = a u x x u_{t}=au_{xx}

CRC-based_framing.html

  1. x 8 + x 2 + x + 1 x^{8}+x^{2}+x+1

Creep_(Radiohead_song).html

  1. 5 ^ \hat{5}
  2. 5 ^ \hat{5}
  3. 6 ^ \hat{6}
  4. 6 ^ \hat{6}

Crest_factor.html

  1. C = | x | peak x rms . C={|x|_{\mathrm{peak}}\over x_{\mathrm{rms}}}.
  2. 𝑃𝐴𝑃𝑅 = | x | peak 2 x rms 2 = C 2 . \mathit{PAPR}={{|x|_{\mathrm{peak}}}^{2}\over{x_{\mathrm{rms}}}^{2}}=C^{2}.
  3. 1 2 0.707 {1\over\sqrt{2}}\approx 0.707
  4. 2 1.414 \sqrt{2}\approx 1.414
  5. 1 2 N \frac{1}{\sqrt{}}{2N}
  6. 2 N \sqrt{2N}
  7. 10 log 2 N 10\log 2N
  8. 1 2 0.707 {1\over\sqrt{2}}\approx 0.707
  9. 2 1.414 \sqrt{2}\approx 1.414
  10. 1 2 2 0.354 {1\over 2\sqrt{2}}\approx 0.354
  11. 2 2 2.83 2\sqrt{2}\approx 2.83\,
  12. 1 3 0.577 {1\over\sqrt{3}}\approx 0.577
  13. 3 1.732 \sqrt{3}\approx 1.732
  14. \geq
  15. t 1 T \sqrt{\frac{t_{1}}{T}}
  16. T t 1 \sqrt{\frac{T}{t_{1}}}
  17. 10 log T t 1 10\log\frac{T}{t_{1}}
  18. 3 7 \sqrt{\frac{3}{7}}
  19. 7 3 1.542 \sqrt{\frac{7}{3}}\approx 1.542
  20. \infty
  21. 1 3 0.577 {1\over\sqrt{3}}\approx 0.577
  22. 3 1.732 \sqrt{3}\approx 1.732
  23. σ \sigma
  24. \infty
  25. \infty

Critical_dimension.html

  1. S = d d x { 1 2 ( ϕ ) 2 + u ϕ 4 } , \displaystyle S=\int d^{d}x\left\{\frac{1}{2}\left(\nabla\phi\right)^{2}+u\phi% ^{4}\right\},
  2. S L . T . P = d d x { 1 2 ( 2 ϕ ) 2 + u ϕ 3 2 ϕ + w ϕ 6 } , \displaystyle S_{L.T.P}=\int d^{d}x\left\{\frac{1}{2}\left(\nabla^{2}\phi% \right)^{2}+u\phi^{3}\nabla^{2}\phi+w\phi^{6}\right\},
  3. x i x i b [ x i ] , ϕ i ϕ i b [ ϕ i ] . \displaystyle x_{i}\rightarrow x_{i}b^{\left[x_{i}\right]},\phi_{i}\rightarrow% \phi_{i}b^{\left[\phi_{i}\right]}.

Critical_exponent.html

  1. T c T_{c}
  2. f f
  3. τ := ( T - T c ) / T c \tau:=(T-T_{c})/T_{c}
  4. k k
  5. k = def lim τ 0 log | f ( τ ) | log | τ | . k\,\stackrel{\,\text{def}}{=}\,\lim_{\tau\to 0}{\log|f(\tau)|\over\log|\tau|}% \,\text{.}
  6. f ( τ ) τ k , τ 0 . f(\tau)\propto\tau^{k},\quad\tau\approx 0\,\text{.}
  7. f ( τ ) f(\tau)
  8. τ 0 \tau\to 0
  9. f ( τ ) = A τ k ( 1 + b τ k 1 + ) . f(\tau)=A\tau^{k}(1+b\tau^{k_{1}}+\cdots)\,\text{.}
  10. T c T_{c}
  11. Ψ \Psi
  12. T c T_{c}
  13. τ \tau
  14. τ \tau
  15. Ψ \Psi
  16. ρ - ρ c ρ c \frac{\rho-\rho_{c}}{\rho_{c}}
  17. τ \tau
  18. T - T c T c \frac{T-T_{c}}{T_{c}}
  19. f f
  20. C C
  21. - T 2 f T 2 -T\frac{\partial^{2}f}{\partial T^{2}}
  22. J J
  23. P - P c P c \frac{P-P_{c}}{P_{c}}
  24. χ \chi
  25. Ψ J \frac{\partial\Psi}{\partial J}
  26. ξ \xi
  27. d d
  28. ψ ( x ) ψ ( y ) \left\langle\psi(\vec{x})\psi(\vec{y})\right\rangle
  29. r r
  30. J = 0 J=0
  31. δ \delta
  32. τ \tau
  33. α \alpha
  34. C τ - α C\propto\tau^{-\alpha}
  35. γ \gamma
  36. χ τ - γ \chi\propto\tau^{-\gamma}
  37. ν \nu
  38. ξ τ - ν \xi\propto\tau^{-\nu}
  39. τ \tau
  40. α \alpha^{\prime}
  41. C ( - τ ) - α C\propto(-\tau)^{-\alpha^{\prime}}
  42. β \beta
  43. Ψ ( - τ ) β \Psi\propto(-\tau)^{\beta}
  44. γ \gamma^{\prime}
  45. χ ( - τ ) - γ \chi\propto(-\tau)^{-\gamma^{\prime}}
  46. ν \nu^{\prime}
  47. ξ ( - τ ) - ν \xi\propto(-\tau)^{-\nu^{\prime}}
  48. τ \tau
  49. δ \delta
  50. J Ψ δ J\propto\Psi^{\delta}
  51. η \eta
  52. ψ ( 0 ) ψ ( r ) r - d + 2 - η \left\langle\psi(0)\psi(r)\right\rangle\propto r^{-d+2-\eta}
  53. f ( J , T ) f(J,T)
  54. F [ J ; T ] F[J;T]
  55. α = α = 0 \alpha=\alpha^{\prime}=0\,
  56. β = 1 2 \beta=\frac{1}{2}\,
  57. γ = γ = 1 \gamma=\gamma^{\prime}=1\,
  58. δ = 3 \delta=3\,
  59. η = 0 \eta=0\,
  60. ν = 1 2 \nu=\frac{1}{2}\,
  61. α \alpha
  62. a Δ a^{\Delta}
  63. α α \alpha\equiv\alpha^{\prime}
  64. γ γ \gamma\equiv\gamma^{\prime}
  65. ν ν \nu\equiv\nu^{\prime}
  66. f ± ( k ξ , ) f_{\pm}(k\xi,\dots)
  67. k ξ 1 k\xi\ll 1
  68. k ξ 1 . k\xi\gg 1\,.
  69. ν d = 2 - α = 2 β + γ = β ( δ + 1 ) = γ δ + 1 δ - 1 \nu d=2-\alpha=2\beta+\gamma=\beta(\delta+1)=\gamma\frac{\delta+1}{\delta-1}\,
  70. 2 - η = γ ν = d δ - 1 δ + 1 2-\eta=\frac{\gamma}{\nu}=d\frac{\delta-1}{\delta+1}
  71. ν \,\nu
  72. τ char \tau_{\mathrm{char}}
  73. τ char ξ z \tau_{\mathrm{char}}\propto\xi^{z}
  74. z z

Critical_point_(mathematics).html

  1. f y ( x , y ) = 0. \frac{\partial f}{\partial y}(x,y)=0.
  2. g y ( x , y ) = 0. \frac{\partial g}{\partial y}(x,y)=0.
  3. f ( x ) = 1 - x 2 , f(x)=\sqrt{1-x^{2}},
  4. C C
  5. f ( x , y ) = 0 , f(x,y)=0,
  6. f f
  7. π y \pi_{y}
  8. π x \pi_{x}
  9. π y ( ( x , y ) ) = x \pi_{y}((x,y))=x
  10. π x ( ( x , y ) ) = y , \pi_{x}((x,y))=y,
  11. C C
  12. π y \pi_{y}
  13. C C
  14. π y \pi_{y}
  15. π y \pi_{y}
  16. f ( x , y ) = f y ( x , y ) = 0. f(x,y)=\frac{\partial f}{\partial y}(x,y)=0.
  17. π x \pi_{x}
  18. C C
  19. y = f ( x ) y=f(x)
  20. ( x , y ) (x,y)
  21. π x \pi_{x}
  22. x x
  23. f f
  24. C C
  25. π x \pi_{x}
  26. π y \pi_{y}
  27. C C
  28. f ( x , y ) = f x ( x , y ) = f y ( x , y ) = 0 f(x,y)=\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y)=0
  29. π y \pi_{y}
  30. C C
  31. f f
  32. π y ; \pi_{y};
  33. π x \pi_{x}
  34. x x
  35. y y
  36. Disc y ( f ) \operatorname{Disc}_{y}(f)
  37. f f
  38. y y
  39. x x
  40. x x
  41. π y \pi_{y}
  42. Disc y ( f ) \operatorname{Disc}_{y}(f)
  43. π y \pi_{y}
  44. x x
  45. y y
  46. f : V W f:V\rightarrow W
  47. V V
  48. W W
  49. p p
  50. V V
  51. f ( p ) f(p)
  52. φ : V R m \varphi:V\rightarrow{R}^{m}
  53. ψ : W R n . \psi:W\rightarrow{R}^{n}.
  54. p p
  55. f f
  56. φ ( p ) \varphi(p)
  57. ψ f φ - 1 . \psi\circ f\circ\varphi^{-1}.
  58. ψ f φ - 1 . \psi\circ f\circ\varphi^{-1}.
  59. V V
  60. n , \mathbb{R}^{n},
  61. P P
  62. V . V.
  63. P P
  64. V V
  65. V V
  66. V V

Critical_point_(thermodynamics).html

  1. ( p V ) T = ( 2 p V 2 ) T = 0 \left(\frac{\partial p}{\partial V}\right)_{T}=\left(\frac{\partial^{2}p}{% \partial V^{2}}\right)_{T}=0
  2. ( p / V ) T = 0 (\partial p/\partial V)_{T}=0
  3. T c = 8 a / 27 R b T_{c}=8a/27Rb
  4. V c = 3 n b V_{c}=3nb
  5. p c = a / 27 b 2 p_{c}=a/27b^{2}
  6. T r = T / T c T_{r}=T/T_{c}
  7. p r = p / p c p_{r}=p/p_{c}
  8. V r = V R T c / p c V_{r}=\frac{V}{RT_{c}/p_{c}}

Critical_variable.html

  1. ( P V ) C = 0 \left(\frac{\partial P}{\partial V}\right)_{C}=0
  2. ( 2 P V 2 ) C = 0 \left(\frac{\partial^{2}P}{\partial V^{2}}\right)_{C}=0
  3. P C = a 27 b 2 P_{C}=\frac{a}{27b^{2}}
  4. V C = 3 n b \displaystyle{V_{C}=3nb}
  5. T C = 8 a 27 b R T_{C}=\frac{8a}{27bR}

Cross_entropy.html

  1. q q
  2. p p
  3. p p
  4. q q
  5. H ( p , q ) = E p [ - log q ] = H ( p ) + D KL ( p q ) , H(p,q)=\operatorname{E}_{p}[-\log q]=H(p)+D_{\mathrm{KL}}(p\|q),\!
  6. H ( p ) H(p)
  7. p p
  8. D KL ( p | | q ) D_{\mathrm{KL}}(p||q)
  9. q q
  10. p p
  11. p p
  12. q q
  13. H ( p , q ) = - x p ( x ) log q ( x ) . H(p,q)=-\sum_{x}p(x)\,\log q(x).\!
  14. - X p ( x ) log q ( x ) d x . -\int_{X}p(x)\,\log q(x)\,dx.\!
  15. H ( p , q ) H(p,q)
  16. p p
  17. q q
  18. x i x_{i}
  19. X X
  20. q ( x i ) = 2 - l i q(x_{i})=2^{-l_{i}}
  21. X X
  22. l i l_{i}
  23. x i x_{i}
  24. Q Q
  25. P P
  26. P P
  27. Q Q
  28. H ( p , q ) = E p [ l i ] = E p [ log 1 q ( x i ) ] H(p,q)=\operatorname{E}_{p}[l_{i}]=\operatorname{E}_{p}\left[\log\frac{1}{q(x_% {i})}\right]
  29. H ( p , q ) = x i p ( x i ) log 1 q ( x i ) H(p,q)=\sum_{x_{i}}p(x_{i})\,\log\frac{1}{q(x_{i})}\!
  30. H ( p , q ) = - x p ( x ) log q ( x ) . H(p,q)=-\sum_{x}p(x)\,\log q(x).\!
  31. p p
  32. T T
  33. p p
  34. q q
  35. H ( T , q ) = - i = 1 N 1 N log 2 q ( x i ) H(T,q)=-\sum_{i=1}^{N}\frac{1}{N}\log_{2}q(x_{i})
  36. N N
  37. q ( x ) q(x)
  38. x x
  39. N N
  40. p ( x ) p(x)
  41. q q
  42. p p
  43. p p
  44. p = q p=q
  45. 0
  46. H ( p ) \mathrm{H}(p)
  47. q q
  48. p p
  49. q q
  50. D KL ( p q ) D_{\mathrm{KL}}(p\|q)
  51. H ( p , q ) H(p,q)
  52. p i p_{i}
  53. q i q_{i}
  54. 0
  55. 1 1
  56. y { 0 , 1 } y\in\{0,1\}
  57. 𝐱 \mathbf{x}
  58. g ( z ) = 1 / ( 1 + e - z ) g(z)=1/(1+e^{-z})
  59. y = 1 y=1
  60. q y = 1 = y ^ g ( 𝐰 𝐱 ) , q_{y=1}\ =\ \hat{y}\ \equiv\ g(\mathbf{w}\cdot\mathbf{x})\,,
  61. 𝐰 \mathbf{w}
  62. y = 0 y=0
  63. q y = 0 = 1 - y ^ q_{y=0}\ =\ 1-\hat{y}
  64. p y = 1 = y p_{y=1}=y
  65. p y = 0 = 1 - y p_{y=0}=1-y
  66. p { y , 1 - y } p\in\{y,1-y\}
  67. q { y ^ , 1 - y ^ } q\in\{\hat{y},1-\hat{y}\}
  68. p p
  69. q q
  70. H ( p , q ) = - i p i log q i = - y log y ^ - ( 1 - y ) log ( 1 - y ^ ) H(p,q)\ =\ -\sum_{i}p_{i}\log q_{i}\ =\ -y\log\hat{y}-(1-y)\log(1-\hat{y})
  71. N N
  72. n = 1 , , N n=1,\dots,N
  73. L ( 𝐰 ) \displaystyle L(\mathbf{w})
  74. y ^ n g ( 𝐰 𝐱 n ) \hat{y}_{n}\equiv g(\mathbf{w}\cdot\mathbf{x}_{n})
  75. g ( z ) g(z)

Cross_section_(geometry).html

  1. A A^{\prime}
  2. A = π r 2 A^{\prime}=\pi r^{2}
  3. A = 2 r h A^{\prime}=2rh
  4. A = π r 2 A^{\prime}=\pi r^{2}
  5. A A^{\prime}
  6. A = top d 𝐀 𝐫 ^ , A^{\prime}=\iint\limits_{\mathrm{top}}d\mathbf{A}\cdot\mathbf{\hat{r}},
  7. 𝐫 ^ \mathbf{\hat{r}}
  8. d 𝐀 d\mathbf{A}
  9. A A
  10. 𝐫 ^ \mathbf{\hat{r}}
  11. A = 1 2 A | d 𝐀 𝐫 ^ | A^{\prime}=\frac{1}{2}\iint\limits_{A}|d\mathbf{A}\cdot\mathbf{\hat{r}}|

Crossed_module.html

  1. ( g , h ) g h (g,h)\mapsto g\cdot h
  2. d : H G , d\colon H\longrightarrow G,\!
  3. d ( g h ) = g d ( h ) g - 1 d(g\cdot h)=gd(h)g^{-1}\!
  4. d ( h 1 ) h 2 = h 1 h 2 h 1 - 1 d(h_{1})\cdot h_{2}=h_{1}h_{2}h_{1}^{-1}\!
  5. d : N G d\colon N\longrightarrow G\!
  6. 1 A H G 1 1\to A\to H\to G\to 1\!
  7. d : H G d\colon H\to G\!
  8. d : π 2 ( X , A , x ) π 1 ( A , x ) d\colon\pi_{2}(X,A,x)\rightarrow\pi_{1}(A,x)\!
  9. Π : ( pairs of pointed spaces ) ( crossed modules ) \Pi\colon(\,\text{pairs of pointed spaces})\rightarrow(\,\text{crossed modules})
  10. F E B F\rightarrow E\rightarrow B\!
  11. d : π 1 ( F ) π 1 ( E ) d\colon\pi_{1}(F)\rightarrow\pi_{1}(E)\!
  12. M = ( d : H G ) M=(d\colon H\longrightarrow G)\!

Crossing_(physics).html

  1. A \mathrm{A}
  2. X \mathrm{X}
  3. B \mathrm{B}
  4. Y \mathrm{Y}
  5. A + B ¯ + X \scriptstyle\mathrm{A}+\bar{\mathrm{B}}+\mathrm{X}
  6. Y \mathrm{Y}
  7. B ¯ \scriptstyle\bar{\mathrm{B}}
  8. X \scriptstyle\mathrm{X}
  9. Y + A ¯ \scriptstyle\mathrm{Y}+\bar{\mathrm{A}}
  10. ( ϕ ( p ) + ) \mathcal{M}(\phi(p)+...\ \rightarrow...)
  11. ϕ ( p ) \phi(p)
  12. ϕ ¯ ( - p ) \bar{\phi}(-p)
  13. ϕ ( p ) \phi(p)
  14. ( ϕ ( p ) + ) = ( + ϕ ¯ ( - p ) ) \mathcal{M}(\phi(p)+...\rightarrow...)=\mathcal{M}(...\rightarrow...+\bar{\phi% }(-p))
  15. k 1 , k 2 , , k n k_{1},k_{2},...,k_{n}
  16. k = 1 n q k = p \sum_{k=1}^{n}q_{k}=p
  17. k = 1 n q k = - p \sum_{k=1}^{n}q_{k}=-p

Crown_molding.html

  1. = arctan ( sin ( spring angle ) tan ( wall angle/2 ) ) =\arctan\left(\frac{\sin{(\,\text{spring angle})}}{\tan{(\,\text{wall angle/2}% )}}\right)
  2. = arcsin ( cos ( spring angle ) * cos ( wall angle/2 ) ) =\arcsin\left(\cos{(\,\text{spring angle})}*\cos{(\,\text{wall angle/2})}\right)

Cryoscopic_constant.html

  1. T f = K f m i \triangle T_{f}=K_{f}\cdot m\cdot i
  2. Δ T \Delta T

Cryptographically_Generated_Address.html

  1. O ( 2 59 ) O(2^{59})
  2. 2 16 × S e c 2^{16\times Sec}
  3. O ( 2 59 + 16 × S e c ) O(2^{59+16\times Sec})

Crystal_momentum.html

  1. 𝐤 \mathbf{k}
  2. 𝐩 crystal 𝐤 {\mathbf{p}}_{\,\text{crystal}}\equiv\hbar{\mathbf{k}}
  3. \hbar
  4. V ( x ) V(x)
  5. V ( 𝐱 + 𝐚 ) = V ( 𝐱 ) , V({\mathbf{x}}+{\mathbf{a}})=V({\mathbf{x}}),
  6. a {a}
  7. a a
  8. T ( a ) T(a)
  9. ψ n ( 𝐱 ) = e i 𝐤 𝐱 u n 𝐤 ( 𝐱 ) , u n 𝐤 ( 𝐱 + 𝐚 ) = u n 𝐤 ( 𝐱 ) \psi_{n}({\mathbf{x}})=e^{i{\mathbf{k}{\mathbf{\cdot x}}}}u_{n{\mathbf{k}}}({% \mathbf{x}}),\qquad u_{n{\mathbf{k}}}({\mathbf{x}}+{\mathbf{a}})=u_{n{\mathbf{% k}}}({\mathbf{x}})
  10. ψ ( x ) \psi(x)
  11. u ( x ) u(x)
  12. k k
  13. 𝐩 crystal = 𝐤 . {\mathbf{p}}_{\,\text{crystal}}=\hbar{\mathbf{k}}.
  14. k k
  15. k = k + K , k^{\prime}=k+K,
  16. K K
  17. ψ n ( 𝐱 ) = e i 𝐤 𝐱 u n 𝐤 ( 𝐱 ) \psi_{n}({\mathbf{x}})=e^{i{\mathbf{k}{\mathbf{\cdot x}}}}u_{n{\mathbf{k}}}({% \mathbf{x}})
  18. k \hbar k
  19. k k
  20. k \hbar k
  21. 𝐯 n ( 𝐤 ) = 1 𝐤 E n ( 𝐤 ) . {\mathbf{v}}_{n}({\mathbf{k}})=\frac{1}{\hbar}\nabla_{\mathbf{k}}E_{n}({% \mathbf{k}}).
  22. 𝐯 n ( 𝐤 ) = 1 𝐤 E n ( 𝐤 ) , {\mathbf{v}}_{n}({\mathbf{k}})=\frac{1}{\hbar}\nabla_{\mathbf{k}}E_{n}({% \mathbf{k}}),
  23. 𝐩 ˙ crystal = - e ( 𝐄 - 1 c 𝐯 × 𝐇 ) {\mathbf{\dot{p}}}_{\,\text{crystal}}=-e\left({\mathbf{E}}-\frac{1}{c}{\mathbf% {v}}\times{\mathbf{H}}\right)
  24. 𝐩 = 2 m E kin sin θ {\mathbf{p_{\parallel}}}=\sqrt{2mE_{\,\text{kin}}}\sin\theta
  25. m m

Crystallization.html

  1. T ( S l i q u i d - S s o l i d ) > H l i q u i d - H s o l i d T(S_{liquid}-S_{solid})>H_{liquid}-H_{solid}
  2. G l i q u i d < G s o l i d G_{liquid}<G_{solid}
  3. B = d N d t = k n ( c - c * ) n B=\dfrac{dN}{dt}=k_{n}(c-c^{*})^{n}
  4. B = d N d t = k 1 M T j ( c - c * ) b B=\dfrac{dN}{dt}=k_{1}M_{T}^{j}(c-c^{*})^{b}

Cumulativity.html

  1. p \oplus_{p}
  2. ( X U p ) ( C U M ( X ) x , y ( X ( x ) X ( y ) x y ) x , y ( X ( x ) X ( y ) X ( x p y ) ) ) (\forall X\subseteq U_{p})(CUM(X)\iff\exists x,y(X(x)\wedge X(y)\wedge x\neq y% )\wedge\forall x,y(X(x)\wedge X(y)\Rightarrow X(x\oplus_{p}y)))
  3. ( x 1 , , x n , y 1 , , y n ) ( R ( x 1 , , x n ) R ( y 1 , , y n ) ) R ( x 1 y 1 , , x n y n ) (\forall x_{1},\ldots,x_{n},y_{1},\ldots,y_{n})(R(x_{1},\ldots,x_{n})\wedge R(% y_{1},\ldots,y_{n}))\rightarrow R(x_{1}\oplus y_{1},\ldots,x_{n}\oplus y_{n})

Cunningham_project.html

  1. ( b k n - 1 ) = ( b n - 1 ) r = 0 k - 1 b r n (b^{kn}-1)=(b^{n}-1)\sum_{r=0}^{k-1}b^{rn}
  2. ( b k n + 1 ) = ( b n + 1 ) r = 0 k - 1 ( - 1 ) r b r n (b^{kn}+1)=(b^{n}+1)\sum_{r=0}^{k-1}(-1)^{r}\cdot b^{rn}
  3. b n - 1 = d | n Φ d ( b ) b^{n}-1=\prod_{d|n}\Phi_{d}(b)
  4. b n + 1 = d | 2 n , d ! | n Φ d ( b ) b^{n}+1=\prod_{d|2n,d!|n}\Phi_{d}(b)
  5. Φ n ( b ) \Phi_{n}(b)
  6. k 1 mod 4 k\equiv 1\mod 4
  7. n k ( mod 2 k ) ; n\equiv k\;\;(\mathop{{\rm mod}}2k);
  8. k 2 , 3 ( mod 4 ) k\equiv 2,3\;\;(\mathop{{\rm mod}}4)
  9. n 2 k ( mod 4 k ) . n\equiv 2k\;\;(\mathop{{\rm mod}}4k).

Curie_constant.html

  1. C = μ 0 μ B 2 3 k B N g 2 J ( J + 1 ) C=\frac{\mu_{0}\mu_{B}^{2}}{3k_{B}}Ng^{2}J(J+1)
  2. N N
  3. g g
  4. μ B \mu_{B}
  5. J J
  6. k B k_{B}
  7. μ \mu
  8. C = 1 k B N μ 0 μ 2 C=\frac{1}{k_{B}}N\mu_{0}\mu^{2}
  9. 𝐌 = C T 𝐁 \mathbf{M}=\frac{C}{T}\mathbf{B}
  10. χ \chi
  11. 𝐌 \scriptstyle\mathbf{M}
  12. 𝐇 \scriptstyle\mathbf{H}
  13. χ = 𝐌 𝐇 \chi=\frac{\mathbf{M}}{\mathbf{H}}
  14. 𝐌 \scriptstyle\mathbf{M}
  15. T T

Curie–Weiss_law.html

  1. χ = C T - T c \chi=\frac{C}{T-T_{c}}
  2. χ 1 ( T - T c ) γ \chi\sim\frac{1}{(T-T_{c})^{\gamma}}
  3. ρ \rho
  4. A A
  5. A = T r ( A ρ ) \langle A\rangle=Tr(A\rho)
  6. | i |i\rangle
  7. ρ = i j ρ i j | i j | . \rho=\sum_{ij}\rho_{ij}|i\rangle\langle j|.
  8. i d d t ρ ( t ) = [ H , ρ ( t ) ] i\hbar\frac{d}{dt}\rho(t)=[H,\rho(t)]
  9. [ H , ρ ] = 0 [H,\rho]=0
  10. f ( H ) f(H)
  11. ρ = exp ( - H / T ) / Z \rho=\exp(-H/T)/Z
  12. Z = T r exp ( - H / T ) Z=Tr\exp(-H/T)
  13. H = - γ B σ 3 H=-\gamma\hbar B\sigma_{3}
  14. γ \gamma
  15. Z = 2 cosh ( γ B / ( 2 T ) ) Z=2\cosh(\gamma\hbar B/(2T))
  16. ρ ( B , T ) = 1 2 cosh ( γ B / ( 2 T ) ) ( exp ( - γ B / ( 2 T ) ) 0 0 exp ( γ B / ( 2 T ) ) ) . \rho(B,T)=\frac{1}{2\cosh(\gamma\hbar B/(2T))}\begin{pmatrix}\exp(-\gamma\hbar B% /(2T))&0\\ 0&\exp(\gamma\hbar B/(2T))\end{pmatrix}.
  17. J x = J y = 0 , J z = - 2 tanh ( γ B / ( 2 T ) ) . \langle J_{x}\rangle=\langle J_{y}\rangle=0,\langle J_{z}\rangle=-\frac{\hbar}% {2}\tanh(\gamma\hbar B/(2T)).
  18. B B
  19. Δ H = α J z B + β B 2 i ( x i 2 + y i 2 ) , \Delta H=\alpha J_{z}B+\beta B^{2}\sum_{i}(x_{i}^{2}+y_{i}^{2}),
  20. α , β \alpha,\beta
  21. i i
  22. Δ H \Delta H
  23. Δ H \Delta H
  24. | n |n\rangle
  25. Δ E n \Delta E_{n}
  26. | n |n\rangle
  27. Δ E n = n | Δ H | n + m , E m E n < t a b l e > E n - E m . I n o u r c a s e w e c a n i g n o r e < m a t h > B 3 \Delta E_{n}=\langle n|\Delta H|n\rangle+\sum_{m,E_{m}\neq E_{n}}\frac{<}{t}% able>{E_{n}-E_{m}}.Inourcasewecanignore<math>B^{3}
  28. Δ E n = α B n | J z | n + α 2 B 2 m , E m E n < t a b l e > E n - E m + β B 2 i n | x i 2 + y i 2 | n . I n c a s e o f d i a m a g n e t i c m a t e r i a l , t h e f i r s t t w o t e r m s a r e a b s e n t a s t h e y d o n t h a v e a n y a n g u l a r m o m e n t u m i n t h e i r g r o u n d s t a t e . I n c a s e o f p a r a m a g n e t i c m a t e r i a l a l l t h e t h r e e t e r m s c o n t r i b u t e . = = A d d i n g s p i n - s p i n i n t e r a c t i o n i n t h e H a m i l t o n i a n : I s i n g m o d e l = = S o f a r w e h a v e a s s u m e d t h a t t h e a t o m s d o n o t i n t e r a c t w i t h e a c h o t h e r . E v e n t h o u g h t h i s i s a r e a s o n a b l e a s s u m p t i o n i n c a s e o f d i a m a g n e t i c a n d p a r a m a g n e t i c s u b s t a n c e s , t h i s a s s u m p t i o n f a i l s i n c a s e o f f e r r o m a g n e t i s m w h e r e t h e s p i n s o f t h e a t o m t r y t o a l i g n w i t h e a c h o t h e r t o t h e e x t e n t p e r m i t t e d b y t h e t h e r m a l a g i t a t i o n . I n t h i s c a s e w e h a v e t o c o n s i d e r t h e H a m i l t o n i a n o f t h e e n s e m b l e o f t h e a t o m . S u c h a H a m i l t o n i a n w i l l c o n t a i n a l l t h e t e r m s d e s c r i b e d a b o v e f o r i n d i v i d u a l a t o m s a n d t e r m s c o r r e s p o n d i n g t o t h e i n t e r a c t i o n a m o n g t h e p a i r s o f a t o m . [ [ I s i n g m o d e l | I s i n g m o d e l ] ] i s o n e o f t h e s i m p l e s t a p p r o x i m a t i o n o f s u c h p a i r w i s e i n t e r a c t i o n . < m a t h > H p a i r s = - 1 2 R , R S ( R ) S ( R ) J ( R - R ) \Delta E_{n}=\alpha B\langle n|J_{z}|n\rangle+\alpha^{2}B^{2}\sum_{m,E_{m}\neq E% _{n}}\frac{<}{t}able>{E_{n}-E_{m}}+\beta B^{2}\sum_{i}\langle n|x_{i}^{2}+y_{i% }^{2}|n\rangle.Incaseofdiamagneticmaterial,thefirsttwotermsareabsentastheydon^% {\prime}thaveanyangularmomentumintheirgroundstate.% Incaseofparamagneticmaterialallthethreetermscontribute.\par ==Addingspin-% spininteractionintheHamiltonian:Isingmodel==\par Sofarwehaveassumedthattheatomsdonotinteractwitheachother% .% Eventhoughthisisareasonableassumptionincaseofdiamagneticandparamagneticsubstances% ,% thisassumptionfailsincaseofferromagnetismwherethespinsoftheatomtrytoalignwitheachothertotheextentpermittedbythethermalagitation% .InthiscasewehavetoconsidertheHamiltonianoftheensembleoftheatom.% SuchaHamiltonianwillcontainallthetermsdescribedaboveforindividualatomsandtermscorrespondingtotheinteractionamongthepairsofatom% .[[Ising_{m}odel|Isingmodel]]% isoneofthesimplestapproximationofsuchpairwiseinteraction.\par <math>H_{pairs}=% -\frac{1}{2}\sum_{R,R^{\prime}}S(R)\cdot S(R^{\prime})J(R-R^{\prime})
  29. R , R R,R^{\prime}
  30. J J
  31. R - R R-R^{\prime}
  32. J J
  33. χ = M H = M μ 0 B = C T . \chi=\frac{M}{H}=\frac{M\mu_{0}}{B}=\frac{C}{T}.
  34. C = μ B 2 3 k B N g 2 J ( J + 1 ) , C=\frac{\mu_{B}^{2}}{3k_{B}}Ng^{2}J(J+1),
  35. N N
  36. g g
  37. J J
  38. B + λ M B+λM
  39. λ λ
  40. χ = M μ 0 B \chi=\frac{M\mu_{0}}{B}
  41. M μ 0 B + λ M = C T \frac{M\mu_{0}}{B+\lambda M}=\frac{C}{T}
  42. χ = C T - C λ μ 0 \chi=\frac{C}{T-\frac{C\lambda}{\mu_{0}}}
  43. χ = C T - T c \chi=\frac{C}{T-T_{c}}
  44. T C = C λ μ 0 T_{C}=\frac{C\lambda}{\mu_{0}}

Current_quark.html

  1. m u m_{u}
  2. m c m_{c}
  3. m t m_{t}
  4. m d m_{d}
  5. m s m_{s}
  6. m b m_{b}

Cusp_neighborhood.html

  1. t ( z ) = ( 1 1 0 1 ) : z = 1 z + 1 0 z + 1 = z + 1 t(z)=\begin{pmatrix}1&1\\ 0&1\end{pmatrix}:z=\frac{1\cdot z+1}{0\cdot z+1}=z+1
  2. g = h - 1 t h g=h^{-1}th
  3. U = { z 𝐇 : z > 1 } U=\{z\in\mathbf{H}:\Im z>1\}
  4. γ ( U ) U = \gamma(U)\cap U=\emptyset
  5. γ G - g \gamma\in G-\langle g\rangle
  6. g \langle g\rangle
  7. E = U / g E=U/\langle g\rangle
  8. { z H : | z | > 1 , | Re ( z ) | < 1 2 } \left\{z\in H:\left|z\right|>1,\,\left|\,\mbox{Re}~{}(z)\,\right|<\frac{1}{2}\right\}
  9. d μ = d x d y y 2 d\mu=\frac{dxdy}{y^{2}}

Cutting-plane_method.html

  1. c T x \displaystyle c^{T}x
  2. x i + a ¯ i , j x j = b ¯ i x_{i}+\sum\bar{a}_{i,j}x_{j}=\bar{b}_{i}
  3. x i + a ¯ i , j x j - b ¯ i = b ¯ i - b ¯ i - ( a ¯ i , j - a ¯ i , j ) x j . x_{i}+\sum\lfloor\bar{a}_{i,j}\rfloor x_{j}-\lfloor\bar{b}_{i}\rfloor=\bar{b}_% {i}-\lfloor\bar{b}_{i}\rfloor-\sum(\bar{a}_{i,j}-\lfloor\bar{a}_{i,j}\rfloor)x% _{j}.
  4. b ¯ i - b ¯ i - ( a ¯ i , j - a ¯ i , j ) x j 0 \bar{b}_{i}-\lfloor\bar{b}_{i}\rfloor-\sum(\bar{a}_{i,j}-\lfloor\bar{a}_{i,j}% \rfloor)x_{j}\leq 0
  5. b ¯ i - b ¯ i - ( a ¯ i , j - a ¯ i , j ) x j = b ¯ i - b ¯ i > 0. \bar{b}_{i}-\lfloor\bar{b}_{i}\rfloor-\sum(\bar{a}_{i,j}-\lfloor\bar{a}_{i,j}% \rfloor)x_{j}=\bar{b}_{i}-\lfloor\bar{b}_{i}\rfloor>0.
  6. x k + ( a ¯ i , j - a ¯ i , j ) x j = b ¯ i - b ¯ i , x k 0 , x k an integer . x_{k}+\sum(\lfloor\bar{a}_{i,j}\rfloor-\bar{a}_{i,j})x_{j}=\lfloor\bar{b}_{i}% \rfloor-\bar{b}_{i},\,x_{k}\geq 0,\,x_{k}\mbox{ an integer}~{}.

Cyclic_homology.html

  1. H C n ( A ) Ω n A / d Ω n - 1 A i 1 H D R n - 2 i ( V ) . HC_{n}(A)\simeq\Omega^{n}\!A/d\Omega^{n-1}\!A\oplus\bigoplus_{i\geq 1}H^{n-2i}% _{DR}(V).
  2. C * C^{*}

Cyclic_number.html

  1. b p - 1 - 1 p \frac{b^{p-1}-1}{p}

Cylinder_stress.html

  1. σ θ = F t l \sigma_{\theta}=\dfrac{F}{tl}
  2. T = F l T=\dfrac{F}{l}
  3. σ θ = P r t \sigma_{\theta}=\dfrac{Pr}{t}
  4. σ θ = P r 2 t \sigma_{\theta}=\dfrac{Pr}{2t}
  5. σ θ \sigma_{\theta}\!
  6. σ z = F A = P d 2 ( d + 2 t ) 2 - d 2 \sigma_{z}=\dfrac{F}{A}=\dfrac{Pd^{2}}{(d+2t)^{2}-d^{2}}
  7. σ z = P r 2 t \sigma_{z}=\dfrac{Pr}{2t}
  8. σ r \sigma_{r}
  9. σ r = - P 2 \sigma_{r}=\dfrac{-P}{2}
  10. σ r = A - B r 2 \sigma_{r}=A-\dfrac{B}{r^{2}}
  11. σ θ = A + B r 2 \sigma_{\theta}=A+\dfrac{B}{r^{2}}
  12. R i = 0 R_{i}=0
  13. B = 0 B=0
  14. A = P o A=P_{o}

Damping_ratio.html

  1. ζ = c c c , \zeta=\frac{c}{c_{c}},
  2. ζ = actual damping critical damping , \zeta=\frac{\,\text{actual damping}}{\,\text{critical damping}},
  3. m d 2 x d t 2 + c d x d t + k x = 0 m\frac{d^{2}x}{dt^{2}}+c\frac{dx}{dt}+kx=0
  4. c c = 2 k m c_{c}=2\sqrt{km}
  5. c c = 2 m ω n c_{c}=2m\omega_{n}
  6. ω n = k / m \omega_{n}=\sqrt{k/m}
  7. d 2 x d t 2 + 2 ζ ω n d x d t + ω n 2 x = 0. \frac{d^{2}x}{dt^{2}}+2\zeta\omega_{n}\frac{dx}{dt}+\omega_{n}^{2}x=0.
  8. x ( t ) = C e s t , x(t)=Ce^{st},\,
  9. s = - ω n ( ζ ± ζ 2 - 1 ) . s=-\omega_{n}(\zeta\pm\sqrt{\zeta^{2}-1}).
  10. ζ 0 \zeta\to 0
  11. exp ( i ω n t ) \exp(i\omega_{n}t)
  12. exp ( i ω n 1 - ζ 2 t ) \exp(i\omega_{n}\sqrt{1-\zeta^{2}}t)
  13. ζ < 1 \zeta<1
  14. ζ > 1 \zeta>1
  15. ζ = 1 \zeta=1
  16. ζ = 1 2 Q = α ω 0 . \zeta=\frac{1}{2Q}={\alpha\over\omega_{0}}.
  17. ζ < 1 \zeta<1
  18. α \alpha
  19. α \alpha
  20. δ \delta
  21. ζ = δ ( 2 π ) 2 + δ 2 where δ ln x 1 x 2 . \zeta=\frac{\delta}{\sqrt{(2\pi)^{2}+\delta^{2}}}\qquad\,\text{where}\qquad% \delta\triangleq\ln\frac{x_{1}}{x_{2}}.

Darboux's_theorem.html

  1. ω = ϕ * ω 0 . \omega=\phi^{*}\omega_{0}.\,

Darcy_(unit).html

  1. Q = A κ Δ P μ Δ x Q=\frac{A\kappa\,\Delta P}{\mu\,\Delta x}
  2. Q Q\,
  3. A A\,
  4. κ \kappa\,
  5. μ \mu\,
  6. Δ P \Delta P\,
  7. Δ x \Delta x\,

Dark_matter_halo.html

  1. ρ ( r ) = constant ( r / a ) ( 1 + r / a ) 2 \rho(r)=\frac{\rm constant}{(r/a)(1+r/a)^{2}}
  2. ρ ( r ) = ρ 0 e - α r n . \rho(r)=\rho_{0}e^{-\alpha r^{n}}.

Davisson–Germer_experiment.html

  1. E E
  2. ν \nu
  3. E = h ν E=h\nu\,
  4. p p
  5. p = h λ , p=\frac{h}{\lambda},
  6. n λ = 2 d sin ( 90 - θ 2 ) , n\lambda=2d\sin\left(90^{\circ}-\frac{\theta}{2}\right),

De_Bruijn_sequence.html

  1. ( k ! ) k n - 1 k n \dfrac{\left(k!\right)^{k^{n-1}}}{k^{n}}
  2. 2 n 2^{n}
  3. 2 n 2^{n}
  4. n n
  5. 2 2 n - 1 - n 2^{2^{n-1}-n}
  6. 2 2 n - 1 - n 2^{2^{n-1}-n}

Dead_time.html

  1. P ( t ) d t = f e - f t d t P(t)dt=fe^{-ft}dt\,
  2. t = 0 t P ( t ) d t = 1 / f \langle t\rangle=\int_{0}^{\infty}tP(t)dt=1/f
  3. τ \tau
  4. t = 0 t=0
  5. t = τ t=\tau
  6. τ \tau
  7. P m ( t ) d t = 0 P_{m}(t)dt=0\,
  8. t τ t\leq\tau\,
  9. P m ( t ) d t = f e - f t d t τ f e - f t d t = f e - f ( t - τ ) d t P_{m}(t)dt=\frac{fe^{-ft}dt}{\int_{\tau}^{\infty}fe^{-ft}dt}=fe^{-f(t-\tau)}dt
  10. t > τ t>\tau\,
  11. t m = τ t P m ( t ) d t = t + τ \langle t_{m}\rangle=\int_{\tau}^{\infty}tP_{m}(t)dt=\langle t\rangle+\tau
  12. N m N_{m}
  13. T T
  14. N N m 1 - N m τ / T N\approx\frac{N_{m}}{1-N_{m}\tau/T}
  15. t i t_{i}
  16. t i t_{i}
  17. τ \tau
  18. t n t n = n ! \frac{\langle t^{n}\rangle}{\langle t\rangle^{n}}=n!

Debye_sheath.html

  1. m i / m e \sqrt{m_{\mathrm{i}}/m_{\mathrm{e}}}
  2. λ D \lambda_{\mathrm{D}}
  3. m i m_{\mathrm{i}}
  4. u 0 u_{0}
  5. 1 2 m i u ( x ) 2 = 1 2 m i u 0 2 - e φ ( x ) \frac{1}{2}m_{\mathrm{i}}\,u(x)^{2}=\frac{1}{2}m_{\mathrm{i}}\,u_{0}^{2}-e\,% \varphi(x)
  6. e e
  7. e = 1.602 e=1.602
  8. 10 - 19 10^{-19}
  9. C \mathrm{C}
  10. n 0 u 0 = n i ( x ) u ( x ) n_{0}\,u_{0}=n_{\mathrm{i}}(x)\,u(x)
  11. n e ( x ) = n 0 exp ( e φ ( x ) k B T e ) n_{\mathrm{e}}(x)=n_{0}\exp\Big(\frac{e\,\varphi(x)}{k_{\mathrm{B}}T_{\mathrm{% e}}}\Big)
  12. d 2 φ ( x ) d x 2 = e ( n e ( x ) - n i ( x ) ) ϵ 0 \frac{d^{2}\varphi(x)}{dx^{2}}=\frac{e(n_{\mathrm{e}}(x)-n_{\mathrm{i}}(x))}{% \epsilon_{0}}
  13. χ ( ξ ) = - e φ ( ξ ) k B T e \chi(\xi)=-\frac{e\varphi(\xi)}{k_{\mathrm{B}}T_{\mathrm{e}}}
  14. ξ = x λ D \xi=\frac{x}{\lambda_{\mathrm{D}}}
  15. 𝔐 = u o ( k B T e / m i ) 1 / 2 \mathfrak{M}=\frac{u_{\mathrm{o}}}{(k_{\mathrm{B}}T_{\mathrm{e}}/m_{\mathrm{i}% })^{1/2}}
  16. χ ′′ = ( 1 + 2 χ 𝔐 2 ) - 1 / 2 - e - χ \chi^{\prime\prime}=\left(1+\frac{2\chi}{\mathfrak{M}^{2}}\right)^{-1/2}-e^{-\chi}
  17. χ \chi^{\prime}
  18. 0 ξ χ χ ′′ d ξ 1 = 0 ξ ( 1 + 2 χ 𝔐 2 ) - 1 / 2 χ d ξ 1 - 0 ξ e - χ χ d ξ 1 \int_{0}^{\xi}\chi^{\prime}\chi^{\prime\prime}\,d\xi_{1}=\int_{0}^{\xi}\left(1% +\frac{2\chi}{\mathfrak{M}^{2}}\right)^{-1/2}\chi^{\prime}\,d\xi_{1}-\int_{0}^% {\xi}e^{-\chi}\chi^{\prime}\,d\xi_{1}
  19. ξ = 0 \xi=0
  20. χ = 0 \chi=0
  21. χ = 0 \chi^{\prime}=0
  22. 1 2 χ 2 = 𝔐 2 [ ( 1 + 2 χ 𝔐 2 ) 1 / 2 - 1 ] + e - χ - 1 \frac{1}{2}\chi^{\prime 2}=\mathfrak{M}^{2}\left[\left(1+\frac{2\chi}{% \mathfrak{M}^{2}}\right)^{1/2}-1\right]+e^{-\chi}-1
  23. χ \chi
  24. χ = 0 \chi=0
  25. 1 2 χ 2 ( - 1 𝔐 2 + 1 ) 0 \frac{1}{2}\chi^{2}\left(-\frac{1}{\mathfrak{M}^{2}}+1\right)\geq 0
  26. 𝔐 2 1 \mathfrak{M}^{2}\geq 1
  27. u 0 ( k B T e / m i ) 1 / 2 u_{0}\geq(k_{\mathrm{B}}T_{\mathrm{e}}/m_{\mathrm{i}})^{1/2}
  28. ( k B T e / 2 e ) (k_{\mathrm{B}}T_{\mathrm{e}}/2e)
  29. e - χ e^{-\chi}
  30. 2 χ / 𝔐 2 2\chi/\mathfrak{M}^{2}
  31. χ ′′ = 𝔐 ( 2 χ ) 1 / 2 \chi^{\prime\prime}=\frac{\mathfrak{M}}{(2\chi)^{1/2}}
  32. χ \chi^{\prime}
  33. 1 2 χ 2 = 𝔐 ( 2 χ ) 1 / 2 \frac{1}{2}\chi^{\prime 2}=\mathfrak{M}(2\chi)^{1/2}
  34. χ - 1 / 4 χ = 2 3 / 4 𝔐 1 / 2 \chi^{-1/4}\chi^{\prime}=2^{3/4}\mathfrak{M}^{1/2}
  35. 4 3 χ w 3 / 4 = 2 3 / 4 𝔐 1 / 2 d \frac{4}{3}\chi_{\mathrm{w}}^{3/4}=2^{3/4}\mathfrak{M}^{1/2}d
  36. χ w \chi_{\mathrm{w}}
  37. u 0 u_{0}
  38. ϕ \phi
  39. J = e n 0 u 0 E J=e\,n_{0}\,u_{0}\,E
  40. E = - d ϕ d x E=-\frac{d\phi}{dx}
  41. J = 4 9 ( 2 e m i ) 1 / 2 | φ w | 3 / 2 4 π d 2 J=\frac{4}{9}\left(\frac{2e}{m_{i}}\right)^{1/2}\frac{|\varphi_{w}|^{3/2}}{4% \pi d^{2}}
  42. J = j ion sat J=j_{\mathrm{ion}}^{\mathrm{sat}}
  43. d = 2 3 ( 2 e m i ) 1 / 4 | φ w | 3 / 4 2 π j ion sat d=\frac{2}{3}\left(\frac{2e}{m_{\mathrm{i}}}\right)^{1/4}\frac{|\varphi_{% \mathrm{w}}|^{3/4}}{2\sqrt{\pi j_{\mathrm{ion}}^{\mathrm{sat}}}}

Decay_correct.html

  1. A 0 = ( A t * e k t ) A_{0}=\left({A_{t}}*{e^{kt}}\right)
  2. A t A_{t}
  3. A 0 A_{0}
  4. ln ( 2 ) t 1 / 2 \frac{\ln(2)}{t_{1/2}}
  5. t 1 / 2 t_{1/2}
  6. t 1 / 2 t_{1/2}
  7. 0.693 12.7 {0.693\over 12.7}
  8. 1 0.3546 {1\over 0.3546}

Decay_heat.html

  1. P P 0 = 0.066 ( ( τ - τ s ) - 0.2 - τ - 0.2 ) \frac{P}{P_{0}}=0.066\left(\left(\tau-\tau_{s}\right)^{-0.2}-\tau^{-0.2}\right)
  2. P P
  3. P 0 P_{0}
  4. τ \tau
  5. τ s \tau_{s}
  6. P P 0 = i = 1 11 P i e - λ t . \frac{P}{P_{0}}=\sum_{i=1}^{11}P_{i}e^{-\lambda t}.

Decimation_(signal_processing).html

  1. y [ n ] = k = 0 K - 1 x [ n M - k ] h [ k ] , y[n]=\sum_{k=0}^{K-1}x[nM-k]\cdot h[k],
  2. B < 1 M 1 2 T , B<\tfrac{1}{M}\cdot\tfrac{1}{2T},
  3. 1 M \tfrac{1}{M}
  4. n = - x ( n T ) x [ n ] e - i 2 π f n T DTFT = 1 T k = - X ( f - k / T ) . \underbrace{\sum_{n=-\infty}^{\infty}\overbrace{x(nT)}^{x[n]}\ e^{-i2\pi fnT}}% _{\,\text{DTFT}}=\frac{1}{T}\sum_{k=-\infty}^{\infty}X(f-k/T).
  5. f \scriptstyle f
  6. n = - x ( n M T ) e - i 2 π f n ( M T ) = 1 M T k = - X ( f - k ( M T ) ) . \sum_{n=-\infty}^{\infty}x(n\cdot MT)\ e^{-i2\pi fn(MT)}=\frac{1}{MT}\sum_{k=-% \infty}^{\infty}X\left(f-\tfrac{k}{(MT)}\right).
  7. f , \scriptstyle f,
  8. B m a x = 1 M 1 2 T , B_{max}=\tfrac{1}{M}\cdot\tfrac{1}{2T},
  9. T B m a x = 1 M 1 2 = 0.5 M . TB_{max}=\tfrac{1}{M}\cdot\tfrac{1}{2}=\tfrac{0.5}{M}.
  10. z = e i ω . z=e^{i\omega}.
  11. n = - x [ n ] z - n = n = - x ( n T ) e - i ω n = 1 T k = - X ( ω 2 π T - k T ) X ( ω - 2 π k 2 π T ) , \sum_{n=-\infty}^{\infty}x[n]\ z^{-n}=\sum_{n=-\infty}^{\infty}x(nT)\ e^{-i% \omega n}=\frac{1}{T}\sum_{k=-\infty}^{\infty}\underbrace{X\left(\tfrac{\omega% }{2\pi T}-\tfrac{k}{T}\right)}_{X\left(\frac{\omega-2\pi k}{2\pi T}\right)},
  12. n = - x [ n M ] z - n = n = - x ( n M T ) e - i ω n = 1 M T k = - X ( ω 2 π M T - k M T ) X ( ω - 2 π k 2 π M T ) . \sum_{n=-\infty}^{\infty}x[nM]\ z^{-n}=\sum_{n=-\infty}^{\infty}x(nMT)\ e^{-i% \omega n}=\frac{1}{MT}\sum_{k=-\infty}^{\infty}\underbrace{X\left(\tfrac{% \omega}{2\pi MT}-\tfrac{k}{MT}\right)}_{X\left(\frac{\omega-2\pi k}{2\pi MT}% \right)}.
  13. 0.5 M \tfrac{0.5}{M}

Decoding_methods.html

  1. C 𝔽 2 n C\subset\mathbb{F}_{2}^{n}
  2. n n
  3. x , y x,y
  4. 𝔽 2 n \mathbb{F}_{2}^{n}
  5. d ( x , y ) d(x,y)
  6. x 𝔽 2 n x\in\mathbb{F}_{2}^{n}
  7. y C y\in C
  8. ( y sent x received ) \mathbb{P}(y\mbox{ sent}~{}\mid x\mbox{ received}~{})
  9. y y
  10. x x
  11. x 𝔽 2 n x\in\mathbb{F}_{2}^{n}
  12. y C y\in C
  13. ( x received y sent ) \mathbb{P}(x\mbox{ received}~{}\mid y\mbox{ sent}~{})
  14. y y
  15. x x
  16. y y
  17. ( x received y sent ) = ( x received , y sent ) ( y sent ) = ( y sent x received ) ( x received ) ( y sent ) . \begin{aligned}\displaystyle\mathbb{P}(x\mbox{ received}~{}\mid y\mbox{ sent}~% {})&\displaystyle{}=\frac{\mathbb{P}(x\mbox{ received}~{},y\mbox{ sent}~{})}{% \mathbb{P}(y\mbox{ sent}~{})}\\ &\displaystyle{}=\mathbb{P}(y\mbox{ sent}~{}\mid x\mbox{ received}~{})\cdot% \frac{\mathbb{P}(x\mbox{ received}~{})}{\mathbb{P}(y\mbox{ sent}~{})}.\end{aligned}
  18. ( x received ) \mathbb{P}(x\mbox{ received}~{})
  19. x x
  20. ( y sent ) \mathbb{P}(y\mbox{ sent}~{})
  21. ( x received y sent ) \mathbb{P}(x\mbox{ received}~{}\mid y\mbox{ sent}~{})
  22. y y
  23. ( y sent x received ) \mathbb{P}(y\mbox{ sent}~{}\mid x\mbox{ received}~{})
  24. x 𝔽 2 n x\in\mathbb{F}_{2}^{n}
  25. y C y\in C
  26. d ( x , y ) = # { i : x i y i } d(x,y)=\#\{i:x_{i}\not=y_{i}\}
  27. y y
  28. x x
  29. p p
  30. d ( x , y ) = d , d(x,y)=d,\,
  31. ( y received x sent ) \displaystyle\mathbb{P}(y\mbox{ received}~{}\mid x\mbox{ sent}~{})
  32. p p
  33. C 𝔽 2 n C\subset\mathbb{F}_{2}^{n}
  34. n n
  35. d d
  36. H H
  37. C C
  38. t = d - 1 2 t=\left\lfloor\frac{d-1}{2}\right\rfloor
  39. t t
  40. x 𝔽 2 n x\in\mathbb{F}_{2}^{n}
  41. e 𝔽 2 n e\in\mathbb{F}_{2}^{n}
  42. z = x + e z=x+e
  43. z z
  44. | C | |C|
  45. c C c\in C
  46. d ( c , z ) d ( y , z ) d(c,z)\leq d(y,z)
  47. y C y\in C
  48. H x = 0 Hx=0
  49. x C x\in C
  50. z = x + e z=x+e
  51. H z = H ( x + e ) = H x + H e = 0 + H e = H e Hz=H(x+e)=Hx+He=0+He=He
  52. t t
  53. H e He
  54. i = 0 t ( n i ) < | C | \begin{matrix}\sum_{i=0}^{t}{\left({{n}\atop{i}}\right)}<|C|\\ \end{matrix}
  55. H e He
  56. e 𝔽 2 n e\in\mathbb{F}_{2}^{n}
  57. e e
  58. x x
  59. x = z - e x=z-e

Defense_independent_pitching_statistics.html

  1. D I C E = 3.00 + 13 H R + 3 ( B B + H B P ) - 2 K I P DICE=3.00+\frac{13HR+3(BB+HBP)-2K}{IP}
  2. F I P = 13 H R + 3 B B - 2 K I P FIP=\frac{13HR+3BB-2K}{IP}
  3. F I P = 13 H R + 3 B B - 2 K I P + C FIP=\frac{13HR+3BB-2K}{IP}+C
  4. x F I P = 13 ( x H R ) + 3 B B - 2 K I P + C xFIP=\frac{13(xHR)+3BB-2K}{IP}+C

Degree_(graph_theory).html

  1. v v
  2. deg ( v ) \deg(v)
  3. deg v \deg v
  4. G = ( V , E ) G=(V,E)
  5. v V deg ( v ) = 2 | E | . \sum_{v\in V}\deg(v)=2|E|\,.

Degree_matrix.html

  1. G = ( V , E ) G=(V,E)
  2. | V | = n |V|=n
  3. D D
  4. G G
  5. n × n n\times n
  6. d i , j := { deg ( v i ) if i = j 0 otherwise d_{i,j}:=\left\{\begin{matrix}\deg(v_{i})&\mbox{if}~{}\ i=j\\ 0&\mbox{otherwise}\end{matrix}\right.
  7. deg ( v i ) \deg(v_{i})
  8. ( 4 0 0 0 0 0 0 3 0 0 0 0 0 0 2 0 0 0 0 0 0 3 0 0 0 0 0 0 3 0 0 0 0 0 0 1 ) \begin{pmatrix}4&0&0&0&0&0\\ 0&3&0&0&0&0\\ 0&0&2&0&0&0\\ 0&0&0&3&0&0\\ 0&0&0&0&3&0\\ 0&0&0&0&0&1\\ \end{pmatrix}
  9. k k

Degree_of_coherence.html

  1. g ( 1 ) g^{(1)}
  2. g ( 2 ) g^{(2)}
  3. g ( 1 ) ( 𝐫 1 , t 1 ; 𝐫 2 , t 2 ) = E * ( 𝐫 1 , t 1 ) E ( 𝐫 2 , t 2 ) [ | E ( 𝐫 1 , t 1 ) | 2 | E ( 𝐫 2 , t 2 ) | 2 ] 1 / 2 g^{(1)}(\mathbf{r}_{1},t_{1};\mathbf{r}_{2},t_{2})=\frac{\left\langle E^{*}(% \mathbf{r}_{1},t_{1})E(\mathbf{r}_{2},t_{2})\right\rangle}{\left[\left\langle% \left|E(\mathbf{r}_{1},t_{1})\right|^{2}\right\rangle\left\langle\left|E(% \mathbf{r}_{2},t_{2})\right|^{2}\right\rangle\right]^{1/2}}
  4. 𝐫 = z \mathbf{r}=z
  5. t 1 t_{1}
  6. τ = t 1 - t 2 \tau=t_{1}-t_{2}
  7. τ = t 1 - t 2 - z 1 - z 2 c \tau=t_{1}-t_{2}-\frac{z_{1}-z_{2}}{c}
  8. z 1 z 2 z_{1}\neq z_{2}
  9. g ( 1 ) ( τ ) = E * ( t ) E ( t + τ ) | E ( t ) | 2 g^{(1)}(\tau)=\frac{\left\langle E^{*}(t)E(t+\tau)\right\rangle}{\left\langle% \left|E(t)\right|^{2}\right\rangle}
  10. | g ( 1 ) ( τ ) | |g^{(1)}(\tau)|
  11. | g ( 1 ) ( 𝐫 1 , t 1 ; 𝐫 2 , t 2 ) | . \left|g^{(1)}(\mathbf{r}_{1},t_{1};\mathbf{r}_{2},t_{2})\right|.
  12. g ( 1 ) ( 0 ) = 1 g^{(1)}(0)=1
  13. g ( 1 ) ( τ ) = g ( 1 ) ( - τ ) * g^{(1)}(\tau)=g^{(1)}(-\tau)^{*}
  14. g ( 1 ) ( τ ) = e - i ω 0 τ g^{(1)}(\tau)=e^{-i\omega_{0}\tau}
  15. g ( 1 ) ( τ ) = e - i ω 0 τ - ( | τ | / τ c ) g^{(1)}(\tau)=e^{-i\omega_{0}\tau-(|\tau|/\tau_{c})}
  16. g ( 1 ) ( τ ) = e - i ω 0 τ - π 2 ( τ / τ c ) 2 g^{(1)}(\tau)=e^{-i\omega_{0}\tau-\frac{\pi}{2}(\tau/\tau_{c})^{2}}
  17. ω 0 \omega_{0}
  18. τ c \tau_{c}
  19. g ( 2 ) ( 𝐫 1 , t 1 ; 𝐫 2 , t 2 ) = E * ( 𝐫 1 , t 1 ) E * ( 𝐫 2 , t 2 ) E ( 𝐫 1 , t 1 ) E ( 𝐫 2 , t 2 ) | E ( 𝐫 1 , t 1 ) | 2 | E ( 𝐫 2 , t 2 ) | 2 g^{(2)}(\mathbf{r}_{1},t_{1};\mathbf{r}_{2},t_{2})=\frac{\left\langle E^{*}(% \mathbf{r}_{1},t_{1})E^{*}(\mathbf{r}_{2},t_{2})E(\mathbf{r}_{1},t_{1})E(% \mathbf{r}_{2},t_{2})\right\rangle}{\left\langle\left|E(\mathbf{r}_{1},t_{1})% \right|^{2}\right\rangle\left\langle\left|E(\mathbf{r}_{2},t_{2})\right|^{2}% \right\rangle}
  20. g ( 2 ) g^{(2)}
  21. g ( 2 ) ( τ ) = I ( t ) I ( t + τ ) I ( t ) 2 g^{(2)}(\tau)=\frac{\left\langle I(t)I(t+\tau)\right\rangle}{\left\langle I(t)% \right\rangle^{2}}
  22. g ( 2 ) ( τ ) = g ( 2 ) ( - τ ) g^{(2)}(\tau)=g^{(2)}(-\tau)
  23. 1 g ( 2 ) ( 0 ) 1\leq g^{(2)}(0)\leq\infty
  24. g ( 2 ) ( τ ) g ( 2 ) ( 0 ) g^{(2)}(\tau)\leq g^{(2)}(0)
  25. g ( 2 ) = 1 g^{(2)}=1
  26. g ( 2 ) g^{(2)}
  27. g ( 2 ) ( τ ) = 1 + | g ( 1 ) ( τ ) | 2 g^{(2)}(\tau)=1+|g^{(1)}(\tau)|^{2}
  28. | g ( 1 ) ( τ ) | |g^{(1)}(\tau)|
  29. g ( 2 ) ( τ ) g^{(2)}(\tau)
  30. g ( 2 ) ( τ ) = 1 g^{(2)}(\tau)=1
  31. g ( 2 ) g^{(2)}
  32. g ( 2 ) ( 0 ) = 0 g^{(2)}(0)=0
  33. g ( 2 ) ( 0 ) = n ( n - 1 ) n 2 g^{(2)}(0)=\frac{\left\langle n(n-1)\right\rangle}{\left\langle n\right\rangle% ^{2}}
  34. n n
  35. g ( n ) ( 𝐫 1 , t 1 ; 𝐫 2 , t 2 ; ; 𝐫 2 n , t 2 n ) = E * ( 𝐫 1 , t 1 ) E * ( 𝐫 2 , t 2 ) E * ( 𝐫 n , t n ) E ( 𝐫 n + 1 , t n + 1 ) E ( 𝐫 n + 2 , t n + 2 ) E ( 𝐫 2 n , t 2 n ) [ | E ( 𝐫 1 , t 1 ) | 2 | E ( 𝐫 2 , t 2 ) | 2 | E ( 𝐫 2 n , t 2 n ) | 2 ] 1 / 2 g^{(n)}(\mathbf{r}_{1},t_{1};\mathbf{r}_{2},t_{2};\dots;\mathbf{r}_{2n},t_{2n}% )=\frac{\left\langle E^{*}(\mathbf{r}_{1},t_{1})E^{*}(\mathbf{r}_{2},t_{2})% \cdots E^{*}(\mathbf{r}_{n},t_{n})E(\mathbf{r}_{n+1},t_{n+1})E(\mathbf{r}_{n+2% },t_{n+2})\dots E(\mathbf{r}_{2n},t_{2n})\right\rangle}{\left[\left\langle% \left|E(\mathbf{r}_{1},t_{1})\right|^{2}\right\rangle\left\langle\left|E(% \mathbf{r}_{2},t_{2})\right|^{2}\right\rangle\cdots\left\langle\left|E(\mathbf% {r}_{2n},t_{2n})\right|^{2}\right\rangle\right]^{1/2}}
  36. g ( n ) ( 𝐫 1 , t 1 ; 𝐫 2 , t 2 ; ; 𝐫 n , t n ) = E * ( 𝐫 1 , t 1 ) E * ( 𝐫 2 , t 2 ) E * ( 𝐫 n , t n ) E ( 𝐫 1 , t 1 ) E ( 𝐫 2 , t 2 ) E ( 𝐫 n , t n ) | E ( 𝐫 1 , t 1 ) | 2 | E ( 𝐫 2 , t 2 ) | 2 | E ( 𝐫 n , t n ) | 2 g^{(n)}(\mathbf{r}_{1},t_{1};\mathbf{r}_{2},t_{2};\dots;\mathbf{r}_{n},t_{n})=% \frac{\left\langle E^{*}(\mathbf{r}_{1},t_{1})E^{*}(\mathbf{r}_{2},t_{2})% \cdots E^{*}(\mathbf{r}_{n},t_{n})E(\mathbf{r}_{1},t_{1})E(\mathbf{r}_{2},t_{2% })\cdots E(\mathbf{r}_{n},t_{n})\right\rangle}{\left\langle\left|E(\mathbf{r}_% {1},t_{1})\right|^{2}\right\rangle\left\langle\left|E(\mathbf{r}_{2},t_{2})% \right|^{2}\right\rangle\cdots\left\langle\left|E(\mathbf{r}_{n},t_{n})\right|% ^{2}\right\rangle}
  37. g ( n ) ( 𝐫 1 , t 1 ; 𝐫 2 , t 2 ; ; 𝐫 n , t n ) = I ( 𝐫 1 , t 1 ) I ( 𝐫 2 , t 2 ) I ( 𝐫 n , t n ) I ( 𝐫 1 , t 1 ) I ( 𝐫 2 , t 2 ) I ( 𝐫 n , t n ) g^{(n)}(\mathbf{r}_{1},t_{1};\mathbf{r}_{2},t_{2};\dots;\mathbf{r}_{n},t_{n})=% \frac{\langle I(\mathbf{r}_{1},t_{1})I(\mathbf{r}_{2},t_{2})\cdots I(\mathbf{r% }_{n},t_{n})\rangle}{\langle I(\mathbf{r}_{1},t_{1})\rangle\langle I(\mathbf{r% }_{2},t_{2})\rangle\cdots\langle I(\mathbf{r}_{n},t_{n})\rangle}
  38. g ( n ) ( 𝐫 1 , t 1 ; 𝐫 2 , t 2 ; ; 𝐫 n , t n ) = 1 g^{(n)}(\mathbf{r}_{1},t_{1};\mathbf{r}_{2},t_{2};\dots;\mathbf{r}_{n},t_{n})=1
  39. g ( n ) ( ) = 0 g^{(n)}(\infty)=0
  40. g ( n ) ( ) = 1 g^{(n)}(\infty)=1
  41. g ( n ) ( 0 ) = n ! g^{(n)}(0)=n!
  42. g ( n ) g^{(n)}
  43. E * E ^ - E^{*}\rightarrow\hat{E}^{-}
  44. E E ^ + . E\rightarrow\hat{E}^{+}.
  45. E ^ - = - i ( ω 2 ϵ 0 V ) 1 / 2 a ^ e - i ( k z - w t ) \hat{E}^{-}=-i\left(\frac{\hbar\omega}{2\epsilon_{0}V}\right)^{1/2}\hat{a}^{% \dagger}e^{-i(kz-wt)}
  46. g ( 2 ) = a ^ ( t ) a ^ ( t + τ ) a ^ ( t + τ ) a ^ ( t ) a ^ ( t ) a ^ ( t ) 2 g^{(2)}=\frac{\left\langle\hat{a}^{\dagger}(t)\hat{a}^{\dagger}(t+\tau)\hat{a}% (t+\tau)\hat{a}(t)\right\rangle}{\left\langle\hat{a}^{\dagger}(t)\hat{a}(t)% \right\rangle^{2}}
  47. g ( 2 ) ( τ ) < g ( 2 ) ( 0 ) g^{(2)}(\tau)<g^{(2)}(0)
  48. g ( 2 ) ( τ ) > g ( 2 ) ( 0 ) g^{(2)}(\tau)>g^{(2)}(0)

Degree_of_curvature.html

  1. R = A * 180 D A * π R=\frac{A*180}{D_{A}*\pi}
  2. A A
  3. R R
  4. D A D_{A}
  5. R = C 2 sin ( D C 2 ) R=\frac{\frac{C}{2}}{\sin(\frac{D_{C}}{2})}
  6. C C
  7. R R
  8. D C D_{C}

Dehn_surgery.html

  1. M M
  2. L M L\subset M
  3. M M
  4. L L
  5. L L
  6. M M
  7. M M
  8. L L
  9. M M
  10. M L M\setminus L
  11. T T
  12. γ \gamma
  13. γ \gamma
  14. γ \gamma
  15. [ γ ] [\gamma]

Delayed_neutron.html

  1. β = precursor atoms prompt neutrons + precursor atoms . \beta=\frac{\mbox{precursor atoms}~{}}{\mbox{prompt neutrons}~{}+\mbox{% precursor atoms}~{}}.
  2. D N F = delayed neutrons prompt neutrons + delayed neutrons . DNF=\frac{\mbox{delayed neutrons}~{}}{\mbox{prompt neutrons}~{}+\mbox{delayed % neutrons}~{}}.

Delta-sigma_modulation.html

  1. f f
  2. v v
  3. f = k v f=k\cdot v
  4. P P
  5. Σ \Sigma
  6. P f = k P v P\cdot f=k\cdot P\cdot v
  7. k P k\cdot P
  8. Σ \Sigma
  9. v v
  10. P P
  11. V V
  12. d t \operatorname{d}t
  13. V d t V\operatorname{d}t
  14. V d t V\operatorname{d}t
  15. Δ = V d t \Delta=\int V\operatorname{d}t
  16. p = 1 f = 1 k v p=\frac{1}{f}=\frac{1}{k\cdot v}
  17. Δ \Delta
  18. Δ \Delta
  19. Δ \Delta
  20. v v
  21. A A
  22. A v = Δ p = Δ f = Δ k v A\cdot v=\frac{\Delta}{p}=\Delta\cdot f=\Delta\cdot k\cdot v
  23. A = Δ k A=\Delta\cdot k
  24. a + b = ( a + b ) \scriptstyle\int a\,+\,\int b\,=\,\int(a\,+\,b)
  25. u \scriptstyle u
  26. y DM = Quantize ( u - y DM ) . y\text{DM}=\int\operatorname{Quantize}\left(u-y\text{DM}\right).\,
  27. y SDM = Quantize ( ( u - y SDM ) ) . y\text{SDM}=\operatorname{Quantize}\left(\int\left(u-y\text{SDM}\right)\right).\,
  28. e rms = 1 Δ - Δ / 2 + Δ / 2 e 2 d e = Δ 2 3 e_{\mathrm{rms}}\,=\sqrt{\,\frac{1}{\Delta}\int_{-\Delta/2}^{+\Delta/2}e^{2}\,% de\,}=\,\frac{\Delta}{2\sqrt{3}}
  29. f s f_{\mathrm{s}}
  30. 2 f 0 2f_{0}
  31. OSR = f s 2 f 0 = 1 2 f 0 τ \mathrm{OSR}\,=\,\frac{f_{s}}{2f_{0}}\,=\,\frac{1}{2f_{0}\tau}
  32. n 0 = e r m s O S R \mathrm{n_{0}}\,=\,\frac{e_{rms}}{\sqrt{OSR}}
  33. f 0 \scriptstyle f_{0}
  34. f s \scriptstyle f_{\mathrm{s}}
  35. H n ( z ) = [ 1 - z - 1 ] \scriptstyle H_{n}(z)\,=\,\left[1-z^{-1}\right]
  36. f s f 0 \scriptstyle f_{s}\,\gg\,f_{0}
  37. n 0 = e r m s π 3 ( 2 f 0 τ ) 3 2 \mathrm{n_{0}}\,=\,e_{rms}\frac{\pi}{\sqrt{3}}\,(2f_{0}\tau)^{\frac{3}{2}}
  38. H n ( z ) = [ 1 - z - 1 ] 2 \scriptstyle H_{n}(z)\,=\,\left[1-z^{-1}\right]^{2}
  39. n 0 = e r m s π 2 5 ( 2 f 0 τ ) 5 2 \mathrm{n_{0}}\,=\,e_{rms}\frac{\pi^{2}}{\sqrt{5}}\,(2f_{0}\tau)^{\frac{5}{2}}
  40. N \scriptstyle\mathrm{N}
  41. n 0 = e r m s π n 2 n + 1 ( 2 f 0 τ ) 2 n + 1 2 \mathrm{n_{0}}\,=\,e_{rms}\frac{\pi^{n}}{\sqrt{2n+1}}\,(2f_{0}\tau)^{\frac{2n+% 1}{2}}
  42. 10 log ( 2 2 N + 1 ) d B \scriptstyle 10\log(2^{2N+1})\,dB
  43. N \scriptstyle\mathrm{N}