wpmath0000006_2

Cramér–Wold_theorem.html

  1. R k R^{k}
  2. X ¯ n = ( X n 1 , , X n k ) \overline{X}_{n}=(X_{n1},\dots,X_{nk})\;
  3. X ¯ = ( X 1 , , X k ) \;\overline{X}=(X_{1},\dots,X_{k})
  4. X ¯ n \overline{X}_{n}
  5. X ¯ \overline{X}
  6. i = 1 k t i X n i n 𝐷 i = 1 k t i X i . \sum_{i=1}^{k}t_{i}X_{ni}\overset{D}{\underset{n\rightarrow\infty}{\rightarrow% }}\sum_{i=1}^{k}t_{i}X_{i}.
  7. ( t 1 , , t k ) k (t_{1},\dots,t_{k})\in\mathbb{R}^{k}
  8. X ¯ n \overline{X}_{n}
  9. X ¯ \overline{X}

Creative_and_productive_sets.html

  1. φ i \varphi_{i}
  2. f f
  3. i i\in\mathbb{N}
  4. W i A W_{i}\subseteq A
  5. f ( i ) A W i . f(i)\in A\setminus W_{i}.
  6. f f
  7. A . A.
  8. A \mathbb{N}\setminus A
  9. K = { i i W i } K=\{i\mid i\in W_{i}\}
  10. K ¯ = { i i W i } \bar{K}=\{i\mid i\not\in W_{i}\}
  11. i K ¯ i\in\bar{K}
  12. i W i i\not\in W_{i}
  13. i K ¯ i\in\bar{K}
  14. i K i\in K
  15. i W i i\in W_{i}
  16. W i K ¯ W_{i}\subseteq\bar{K}
  17. i K ¯ i\in\bar{K}
  18. i K ¯ i\in\bar{K}
  19. i W i i\not\in W_{i}
  20. i W i i\in W_{i}
  21. i K i\in K
  22. i W i i\not\in W_{i}
  23. K K
  24. K ¯ \bar{K}
  25. K ¯ \bar{K}
  26. K ¯ \bar{K}
  27. K K
  28. d ( x ) = [ [ x ] ] ( x ) + 1 d(x)=[[x]](x)+1
  29. K K
  30. d ( x ) d(x)
  31. Φ \Phi
  32. Φ \Phi
  33. Φ ( w , x ) = \Phi(w,x)=
  34. w w
  35. x x
  36. f f
  37. f ( x ) = Φ ( e , x ) f(x)=\Phi(e,x)
  38. x x
  39. e e
  40. f f
  41. Φ ( e , x ) = [ [ e ] ] ( x ) \Phi(e,x)=[[e]](x)
  42. d ( x ) = [ [ x ] ] ( x ) + 1 d(x)=[[x]](x)+1

Credit_card_interest.html

  1. E A R = ( 1 + A P R n ) n - 1 EAR=(1+{APR\over n})^{n}-1
  2. n n
  3. ( 1 + 0.1299 365 ) 365 - 1 (1+{0.1299\over 365})^{365}-1
  4. ( 1 + 0.1299 12 ) 12 - 1 (1+{0.1299\over 12})^{12}-1
  5. 0.1299 12 {0.1299\over 12}
  6. ( 1 + 0.1299 365 ) 28 - 1 (1+{0.1299\over 365})^{28}-1
  7. 0.1299 12 {0.1299\over 12}
  8. ( 1 + 0.1299 365 ) 31 - 1 (1+{0.1299\over 365})^{31}-1

Critical_distance.html

  1. d c d_{c}
  2. d c = 1 4 γ A π 0.057 γ V R T 60 , d_{c}=\frac{1}{4}\sqrt{\frac{\gamma A}{\pi}}\approx 0.057\sqrt{\frac{\gamma V}% {RT_{60}}},
  3. γ \gamma
  4. γ = 1 \gamma=1
  5. A A
  6. V V
  7. R T 60 RT_{60}
  8. R T 60 = V / 6 A RT_{60}=V/6A

Critical_ionization_velocity.html

  1. 1 2 m v 2 = e V i o n \frac{1}{2}mv^{2}=eV_{ion}
  2. e V i o n = E g = k M m r eV_{ion}=E_{g}=\frac{kMm}{r}
  3. r i = E g = k M m e V i o n = 6.9 10 - 20 M m V i o n = 13.5 10 13 m V i o n c m r_{i}=E_{g}=\frac{kMm}{eV_{ion}}=6.9\cdot 10^{-20}M\frac{m^{\prime}}{V_{ion}}=% 13.5\cdot 10^{13}\frac{m^{\prime}}{V_{ion}}cm

Critical_radius.html

  1. G = 4 π 3 r 3 G v + 4 π r 2 γ G=\frac{4\pi}{3}r^{3}G_{v}+4\pi r^{2}\gamma
  2. G v G_{v}
  3. γ \gamma
  4. r c r_{c}
  5. G G
  6. d G d r = 4 π r c 2 G v + 8 π r c γ = 0 \frac{dG}{dr}=4\pi r_{c}^{2}G_{v}+8\pi r_{c}\gamma=0
  7. r c = - 2 γ G v r_{c}=-\frac{2\gamma}{G_{v}}
  8. γ \gamma
  9. G v G_{v}

Cross-covariance.html

  1. X = ( X t ) X=(X_{t})
  2. Y = ( Y t ) Y=(Y_{t})
  3. μ t = E [ X t ] \mu_{t}=E[X_{t}]
  4. ν t = E [ Y t ] \nu_{t}=E[Y_{t}]
  5. C X Y ( t , s ) = c o v ( X t , Y s ) = E [ ( X t - μ t ) ( Y s - ν s ) ] = E [ X t Y s ] - μ t ν s . C_{XY}(t,s)=cov(X_{t},Y_{s})=E[(X_{t}-\mu_{t})(Y_{s}-\nu_{s})]=E[X_{t}Y_{s}]-% \mu_{t}\nu_{s}.\,
  6. X = ( X 1 , X 2 , , X n ) X=(X_{1},X_{2},...,X_{n})
  7. Y = ( Y 1 , Y 2 , , Y n ) Y=(Y_{1},Y_{2},...,Y_{n})
  8. C X Y C_{XY}
  9. C X Y ( j , k ) = c o v ( X j , Y k ) . C_{XY}(j,k)=cov(X_{j},Y_{k}).\,
  10. cov ( X , Y ) = E [ ( X - μ X ) ( Y - μ Y ) ] , \operatorname{cov}(X,Y)=\operatorname{E}[(X-\mu_{X})(Y-\mu_{Y})^{\prime}],
  11. ( f g ) i = def j f j * g i + j (f\star g)_{i}\ \stackrel{\mathrm{def}}{=}\ \sum_{j}f^{*}_{j}\,g_{i+j}
  12. ( f g ) ( x ) = def f * ( t ) g ( x + t ) d t (f\star g)(x)\ \stackrel{\mathrm{def}}{=}\ \int f^{*}(t)g(x+t)\,dt
  13. f ( t ) g ( t ) = f * ( - t ) * g ( t ) , f(t)\star g(t)=f^{*}(-t)*g(t),

Cross_fluid.html

  1. μ eff ( γ ˙ ) = μ 0 1 + ( μ 0 γ ˙ τ * ) 1 - n \mu_{\operatorname{eff}}(\dot{\gamma})=\frac{\mu_{0}}{1+({\frac{\mu_{0}\dot{% \gamma}}{\tau^{*}}})^{1-n}}
  2. μ eff ( γ ˙ ) \mu_{\operatorname{eff}}(\dot{\gamma})
  3. μ 0 \mu_{0}
  4. τ * \tau^{*}
  5. μ 0 γ ˙ τ * \mu_{0}\dot{\gamma}\ll\tau^{*}
  6. μ 0 γ ˙ τ * \mu_{0}\dot{\gamma}\gg\tau^{*}

Crossed_product.html

  1. G G
  2. N G N\rtimes G
  3. C [ N ] G C[N]\rtimes G
  4. N G N\rtimes G
  5. A G A\rtimes G
  6. A G A\rtimes G
  7. A A
  8. L ( X ) L^{\infty}(X)
  9. A G A\rtimes G
  10. A A
  11. G G
  12. Γ \Gamma
  13. χ \chi
  14. G G
  15. K K
  16. ( χ k ) ( h ) = χ ( h ) k ( h ) (\chi\cdot k)(h)=\chi(h)k(h)
  17. Γ \Gamma
  18. A B ( L 2 ( G ) ) A\otimes B(L^{2}(G))
  19. ( A G ) Γ (A\rtimes G)\rtimes\Gamma
  20. G G
  21. Γ \Gamma
  22. A A
  23. L 2 ( G ) L^{2}(G)
  24. ( g f ) ( h ) = f ( h g ) (g\cdot f)(h)=f(hg)
  25. A A
  26. Γ \Gamma
  27. G G
  28. I I II_{\infty}
  29. R R
  30. I I I 1 III_{1}
  31. ( log λ ) Z (\log\lambda)Z
  32. λ \lambda
  33. I I I λ III_{\lambda}
  34. I I I 0 III_{0}
  35. L ( X ) Z L^{\infty}(X)\rtimes Z
  36. Z Z
  37. A G A\rtimes G

Crunode.html

  1. f x \partial f\over\partial x
  2. f y \partial f\over\partial y

Crystal_Ball_function.html

  1. f ( x ; α , n , x ¯ , σ ) = N { exp ( - ( x - x ¯ ) 2 2 σ 2 ) , for x - x ¯ σ > - α A ( B - x - x ¯ σ ) - n , for x - x ¯ σ - α f(x;\alpha,n,\bar{x},\sigma)=N\cdot\begin{cases}\exp(-\frac{(x-\bar{x})^{2}}{2% \sigma^{2}}),&\mbox{for }~{}\frac{x-\bar{x}}{\sigma}>-\alpha\\ A\cdot(B-\frac{x-\bar{x}}{\sigma})^{-n},&\mbox{for }~{}\frac{x-\bar{x}}{\sigma% }\leqslant-\alpha\end{cases}
  2. A = ( n | α | ) n exp ( - | α | 2 2 ) A=\left(\frac{n}{\left|\alpha\right|}\right)^{n}\cdot\exp\left(-\frac{\left|% \alpha\right|^{2}}{2}\right)
  3. B = n | α | - | α | B=\frac{n}{\left|\alpha\right|}-\left|\alpha\right|
  4. N = 1 σ ( C + D ) N=\frac{1}{\sigma(C+D)}
  5. C = n | α | 1 n - 1 exp ( - | α | 2 2 ) C=\frac{n}{\left|\alpha\right|}\cdot\frac{1}{n-1}\cdot\exp\left(-\frac{\left|% \alpha\right|^{2}}{2}\right)
  6. D = π 2 ( 1 + erf ( | α | 2 ) ) D=\sqrt{\frac{\pi}{2}}\left(1+\operatorname{erf}\left(\frac{\left|\alpha\right% |}{\sqrt{2}}\right)\right)
  7. N N
  8. α \alpha
  9. n n
  10. x ¯ \bar{x}
  11. σ \sigma

Crystallographic_restriction_theorem.html

  1. 𝐫 = m 𝐫 \mathbf{r}^{\prime}=m\mathbf{r}\,
  2. m m
  3. r = | 𝐫 | r=|\mathbf{r}|
  4. r = | 𝐫 | r^{\prime}=|\mathbf{r}^{\prime}|
  5. r = 2 r cos α - r . r^{\prime}=2r\cos\alpha-r.
  6. cos α = m + 1 2 = M 2 \cos\alpha=\frac{m+1}{2}=\frac{M}{2}
  7. M = m + 1 M=m+1
  8. | cos α | 1 |\cos\alpha|\leq 1
  9. M { - 2 , - 1 , 0 , 1 , 2 } M\in\{-2,-1,0,1,2\}
  10. α \alpha
  11. 2 a cos θ = 2 a cos 2 π n 2a\cos{\theta}=2a\cos{\frac{2\pi}{n}}
  12. 2 cos 2 π n = m 2\cos{\frac{2\pi}{n}}=m
  13. [ 1 / 2 - 3 / 2 3 / 2 1 / 2 ] \begin{bmatrix}{1/2}&-{\sqrt{3}/2}\\ {\sqrt{3}/2}&{1/2}\end{bmatrix}
  14. [ 1 / 2 - 1 / 2 1 / 2 1 / 2 ] \begin{bmatrix}{1/\sqrt{2}}&-{1/\sqrt{2}}\\ {1/\sqrt{2}}&{1/\sqrt{2}}\end{bmatrix}
  15. [ 0 - 1 1 1 ] \begin{bmatrix}0&-1\\ 1&1\end{bmatrix}
  16. A = [ 0 0 0 - 1 1 0 0 0 0 - 1 0 0 0 0 - 1 0 ] . A=\begin{bmatrix}0&0&0&-1\\ 1&0&0&0\\ 0&-1&0&0\\ 0&0&-1&0\end{bmatrix}.
  17. B = [ - 1 / 2 0 - 1 / 2 2 / 2 1 / 2 2 / 2 - 1 / 2 0 - 1 / 2 0 - 1 / 2 - 2 / 2 - 1 / 2 2 / 2 1 / 2 0 ] B=\begin{bmatrix}-1/2&0&-1/2&\sqrt{2}/2\\ 1/2&\sqrt{2}/2&-1/2&0\\ -1/2&0&-1/2&-\sqrt{2}/2\\ -1/2&\sqrt{2}/2&1/2&0\end{bmatrix}
  18. B A B - 1 = [ 2 / 2 2 / 2 0 0 - 2 / 2 2 / 2 0 0 0 0 - 2 / 2 2 / 2 0 0 - 2 / 2 - 2 / 2 ] . BAB^{-1}=\begin{bmatrix}\sqrt{2}/2&\sqrt{2}/2&0&0\\ -\sqrt{2}/2&\sqrt{2}/2&0&0\\ 0&0&-\sqrt{2}/2&\sqrt{2}/2\\ 0&0&-\sqrt{2}/2&-\sqrt{2}/2\end{bmatrix}.

Cue_validity.html

  1. f i f_{i}
  2. c j c_{j}
  3. p ( c j | f i ) p(c_{j}|f_{i})
  4. p ( c j | f i ) - p ( c j ) p(c_{j}|f_{i})-p(c_{j})
  5. f p - i n t f_{p\mbox{-}~{}int}
  6. p ( c r a t i o n a l | f p - i n t ) = 1 p(c_{rational}|f_{p\mbox{-}~{}int})=1
  7. p ( c i r r a t i o n a l | f p - i n t ) = 0 p(c_{irrational}|f_{p\mbox{-}~{}int})=0
  8. p ( c e v e n | f p - i n t ) = 0.5 p(c_{even}|f_{p\mbox{-}~{}int})=0.5

Current_(mathematics).html

  1. Ω c m ( M ) \Omega_{c}^{m}(M)
  2. M M
  3. Ω c m ( M ) \Omega_{c}^{m}(M)
  4. T : Ω c m ( M ) T\colon\Omega_{c}^{m}(M)\to\mathbb{R}
  5. ω k \omega_{k}
  6. k k
  7. T ( ω k ) T(\omega_{k})
  8. 𝒟 m ( M ) \mathcal{D}_{m}(M)
  9. M M
  10. ( T + S ) ( ω ) := T ( ω ) + S ( ω ) , ( λ T ) ( ω ) := λ T ( ω ) . (T+S)(\omega):=T(\omega)+S(\omega),\qquad(\lambda T)(\omega):=\lambda T(\omega).
  11. T 𝒟 m ( M ) T\in\mathcal{D}_{m}(M)
  12. U M U\subset M
  13. T ( ω ) = 0 T(\omega)=0
  14. ω Ω c m ( U ) \omega\in\Omega_{c}^{m}(U)
  15. 𝒟 m ( M ) \mathcal{D}_{m}(M)
  16. M M
  17. m ( M ) \mathcal{E}_{m}(M)
  18. [ [ M ] ] [[M]]
  19. [ [ M ] ] ( ω ) = M ω . [[M]](\omega)=\int_{M}\omega.\,
  20. [ [ M ] ] ( ω ) = M ω = M d ω = [ [ M ] ] ( d ω ) . [[\partial M]](\omega)=\int_{\partial M}\omega=\int_{M}d\omega=[[M]](d\omega).
  21. : 𝒟 m + 1 𝒟 m \partial\colon\mathcal{D}_{m+1}\to\mathcal{D}_{m}
  22. ( T ) ( ω ) := T ( d ω ) (\partial T)(\omega):=T(d\omega)\,
  23. \partial
  24. T k ( ω ) T ( ω ) , ω . T_{k}(\omega)\to T(\omega),\qquad\forall\omega.\,
  25. ω := sup { | ω , ξ | : ξ is a unit, simple, m -vector } . \|\omega\|:=\sup\{|\langle\omega,\xi\rangle|\colon\xi\mbox{ is a unit, simple,% }~{}m\mbox{-vector}~{}\}.
  26. 𝐌 ( T ) := sup { T ( ω ) : sup x || ω ( x ) || 1 } . \mathbf{M}(T):=\sup\{T(\omega)\colon\sup_{x}||\omega(x)||\leq 1\}.
  27. Ω c 0 ( n ) C c ( n ) \Omega_{c}^{0}(\mathbb{R}^{n})\equiv C^{\infty}_{c}(\mathbb{R}^{n})\,
  28. T ( f ) = f ( 0 ) . T(f)=f(0).\,
  29. μ \mu
  30. T ( f ) = f ( x ) d μ ( x ) . T(f)=\int f(x)\,d\mu(x).
  31. T ( a d x d y + b d y d z + c d x d z ) = 0 1 0 1 b ( x , y , 0 ) d x d y . T(a\,dx\wedge dy+b\,dy\wedge dz+c\,dx\wedge dz)=\int_{0}^{1}\int_{0}^{1}b(x,y,% 0)\,dx\,dy.

Current_sheet.html

  1. R 𝐁 𝐝𝐬 = μ 0 I e n c \oint_{R}\mathbf{B}\cdot\mathbf{ds}=\mu_{0}I_{enc}
  2. R B d s cos θ = μ 0 I e n c \oint_{R}Bds\cos{\theta}=\mu_{0}I_{enc}
  3. 𝐁 𝐝𝐬 = 0 \mathbf{B}\cdot\mathbf{ds}=0
  4. cos ( 90 ) = 0 \cos(90)=0
  5. cos ( 0 ) = 1 \cos(0)=1
  6. 2 S B d s = μ 0 I e n c 2\int_{S}Bds=\mu_{0}I_{enc}
  7. 2 B S d s = μ 0 I e n c 2B\int_{S}ds=\mu_{0}I_{enc}
  8. 2 B L = μ 0 I e n c 2BL=\mu_{0}I_{enc}
  9. B = μ 0 I e n c 2 L B=\frac{\mu_{0}I_{enc}}{2L}
  10. B = μ 0 I N L 2 L B=\frac{\mu_{0}INL}{2L}
  11. B = μ 0 I N 2 B=\frac{\mu_{0}IN}{2}
  12. 𝐁 = B 0 tanh ( x / L ) 𝐞 z \mathbf{B}=B_{0}\tanh(x/L)\mathbf{e}_{z}

Current_sources_and_sinks.html

  1. Φ ( r ) = I 4 π r σ \Phi(r)={I\over 4\pi r\sigma}
  2. Φ \Phi
  3. r r
  4. I I
  5. σ \sigma

Current–voltage_characteristic.html

  1. V G S - V t h V_{GS}-V_{th}

Curtal_sonnet.html

  1. 12 2 + 9 2 = 21 2 = 10 1 2 {12\over 2}+{9\over 2}={21\over 2}=10{1\over 2}
  2. 12 2 = 6 {12\over 2}=6
  3. 9 2 = 4 1 2 {9\over 2}=4{1\over 2}

Curvature_collineation.html

  1. X R a = b c d 0 \mathcal{L}_{X}R^{a}{}_{bcd}=0
  2. R a b c d R^{a}{}_{bcd}
  3. C C ( M ) CC(M)

Curvature_invariant.html

  1. R a b c d R a b c d R_{abcd}\,R^{abcd}
  2. R a b c d R a b c d R_{abcd}\,{{}^{\star}\!R}^{abcd}

Curvature_invariant_(general_relativity).html

  1. K 1 = R a b c d R a b c d K_{1}=R_{abcd}\,R^{abcd}
  2. K 2 = R a b c d R a b c d K_{2}={{}^{\star}R}_{abcd}\,R^{abcd}
  3. K 3 = R a b c d R a b c d K_{3}={{}^{\star}R^{\star}}_{abcd}\,R^{abcd}
  4. I 1 = C a b c d C a b c d I_{1}=C_{abcd}\,C^{abcd}
  5. I 2 = C a b c d C a b c d I_{2}={{}^{\star}C}_{abcd}\,C^{abcd}
  6. C a b c d = - C a b c d {{}^{\star}C^{\star}}_{abcd}=-C_{abcd}
  7. K 1 = I 1 + 2 R a b R a b - 1 3 R 2 K_{1}=I_{1}+2\,R_{ab}\,R^{ab}-\frac{1}{3}\,R^{2}
  8. K 3 = - I 1 + 2 R a b R a b - 2 3 R 2 K_{3}=-I_{1}+2\,R_{ab}\,R^{ab}-\frac{2}{3}\,R^{2}
  9. X \vec{X}
  10. E [ X ] a b = R a m b n X m X n E[\vec{X}]_{ab}=R_{ambn}\,X^{m}\,X^{n}
  11. B [ X ] a b = R a m b n X m X n B[\vec{X}]_{ab}={{}^{\star}R}_{ambn}\,X^{m}\,X^{n}
  12. L [ X ] a b = R a m b n X m X n L[\vec{X}]_{ab}={{}^{\star}R^{\star}}_{ambn}\,X^{m}\,X^{n}
  13. K 1 / 4 K_{1}/4
  14. - K 2 / 8 -K_{2}/8
  15. K 3 / 8 K_{3}/8
  16. I 1 - i I 2 = 16 ( 3 Ψ 2 2 + Ψ 0 Ψ 4 - 4 Ψ 1 Ψ 3 ) I_{1}-i\,I_{2}=16\,\left(3\Psi_{2}^{2}+\Psi_{0}\,\Psi_{4}-4\,\Psi_{1}\Psi_{3}\right)
  17. R a b R a b R_{ab}\,R^{ab}
  18. I 2 I_{2}

Curved_space.html

  1. d x 2 + d y 2 d l 2 dx^{2}+dy^{2}\neq dl^{2}
  2. ( x , y , z ) \left(x^{\prime},y^{\prime},z^{\prime}\right)
  3. d x 2 + d y 2 + d z 2 d l 2 dx^{\prime 2}+dy^{\prime 2}+dz^{\prime 2}\neq dl^{\prime 2}\,
  4. x , y , z , w x,y,z,w
  5. d x 2 + d y 2 + d z 2 + d w 2 = d l 2 dx^{2}+dy^{2}+dz^{2}+dw^{2}=dl^{2}\,
  6. x x
  7. x x^{\prime}
  8. x 2 + y 2 + z 2 + w 2 = constant x^{2}+y^{2}+z^{2}+w^{2}=\textrm{constant}\,
  9. κ - 1 R 2 \kappa^{-1}R^{2}
  10. R 2 R^{2}\,
  11. κ \plusmn 1 \kappa\equiv\plusmn 1
  12. w w
  13. x d x + y d y + z d z + w d w = 0 xdx+ydy+zdz+wdw=0\,
  14. d w = - w - 1 ( x d x + y d y + z d z ) dw=-w^{-1}(xdx+ydy+zdz)\,
  15. d w dw
  16. d l 2 = d x 2 + d y 2 + d z 2 + ( x d x + y d y + z d z ) 2 κ - 1 R 2 - x 2 - y 2 - z 2 dl^{2}=dx^{2}+dy^{2}+dz^{2}+\frac{(xdx+ydy+zdz)^{2}}{\kappa^{-1}R^{2}-x^{2}-y^% {2}-z^{2}}
  17. x = r sin θ cos ϕ x=r\sin\theta\cos\phi
  18. y = r sin θ sin ϕ y=r\sin\theta\sin\phi
  19. z = r cos θ z=r\cos\theta
  20. d l 2 = d r 2 1 - κ r 2 R 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 dl^{2}=\frac{dr^{2}}{1-\kappa\frac{r^{2}}{R^{2}}}+r^{2}d\theta^{2}+r^{2}\sin^{% 2}\theta d\phi^{2}
  21. d l 2 = e - λ ( r ) d r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 dl^{2}=e^{-\lambda(r)}{dr^{2}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\phi^{2}\,
  22. λ = 0 \lambda=0
  23. R a b R_{ab}
  24. R R
  25. R a b = g a b R R_{ab}=g_{ab}R
  26. λ = - 1 2 ln ( 1 - k r 2 ) \lambda=-\frac{1}{2}\ln\left(1-kr^{2}\right)
  27. k R 2 k\equiv\frac{R}{2}
  28. d l 2 = d r 2 1 - k r 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 dl^{2}=\frac{dr^{2}}{1-k{r^{2}}}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\phi^{2}
  29. k k
  30. d l 2 = d r 2 1 - κ r 2 R 2 + r 2 d θ 2 + r 2 sin 2 θ d ϕ 2 dl^{2}=\frac{dr^{2}}{1-\kappa\frac{r^{2}}{R^{2}}}+r^{2}d\theta^{2}+r^{2}\sin^{% 2}\theta d\phi^{2}
  31. R R
  32. κ \kappa
  33. κ \kappa
  34. κ = + 1 \kappa=+1
  35. κ = - 1 \kappa=-1
  36. ( 4 / 3 ) π r 3 (4/3)\pi r^{3}

Cut_(graph_theory).html

  1. C = ( S , T ) C=(S,T)
  2. V V
  3. G = ( V , E ) G=(V,E)
  4. C = ( S , T ) C=(S,T)
  5. { ( u , v ) E | u S , v T } \{(u,v)\in E|u\in S,v\in T\}
  6. O ( log n ) O(\sqrt{\log n})

Cycles_and_fixed_points.html

  1. π = ( 1 6 7 2 5 4 8 3 2 8 7 4 5 3 6 1 ) = ( 1 2 4 3 5 6 7 8 2 4 3 1 5 8 7 6 ) \pi=\begin{pmatrix}1&6&7&2&5&4&8&3\\ 2&8&7&4&5&3&6&1\end{pmatrix}=\begin{pmatrix}1&2&4&3&5&6&7&8\\ 2&4&3&1&5&8&7&6\end{pmatrix}
  2. f ( k , 1 ) = i = 1 k ( - 1 ) i + 1 ( k i ) i ( k - i ) ! f(k,1)=\sum_{i=1}^{k}(-1)^{i+1}{k\choose i}i(k-i)!
  3. f ( k , 0 ) = k ! - i = 1 k ( - 1 ) i + 1 ( k i ) ( k - i ) ! f(k,0)=k!-\sum_{i=1}^{k}(-1)^{i+1}{k\choose i}(k-i)!
  4. f ( k , 0 ) = ( k - 1 ) ( f ( k - 1 , 0 ) + f ( k - 2 , 0 ) ) f(k,0)=(k-1)(f(k-1,0)+f(k-2,0))
  5. f ( k , 0 ) = k ! i = 2 k ( - 1 ) i / i ! f(k,0)=k!\sum_{i=2}^{k}(-1)^{i}/i!
  6. f ( k , 0 ) k ! / e f(k,0)\approx k!/e

Cycles_per_instruction.html

  1. C P I = Σ ( I I C ) ( C C I ) I C CPI=\frac{\Sigma(IIC)(CCI)}{IC}
  2. CPI = 5 × 50 + 4 × 15 + 4 × 25 + 3 × 8 + 3 × 2 100 = 4.05 \,\text{CPI}=\frac{5\times 50+4\times 15+4\times 25+3\times 8+3\times 2}{100}=% 4.05
  3. CPI = 45000 × 1 + 32000 × 2 + 15000 × 2 + 8000 × 2 100000 = 155000 100000 = 1.55 \,\text{CPI}=\frac{45000\times 1+32000\times 2+15000\times 2+8000\times 2}{100% 000}=\frac{155000}{100000}=1.55
  4. Effective processor performance = MIPS = clock frequency CPI × 1000000 = 400 × 1000000 1.55 × 1000000 = 258 MIPS \,\text{Effective processor performance}=\,\text{MIPS}=\frac{\,\text{clock % frequency}}{\,\text{CPI}\times 1000000}=\frac{400\times 1000000}{1.55\times 10% 00000}=258\,\,\text{MIPS}
  5. Execution time ( T ) = CPI × Instruction count × clock time = CPI × Instruction Count frequency = 1.55 × 100000 400 × 1000000 = 1.55 4000 = 0.387 ms \,\text{Execution time}(T)=\,\text{CPI}\times\,\text{Instruction count}\times% \,\text{clock time}=\frac{\,\text{CPI}\times\,\text{Instruction Count}}{\,% \text{frequency}}=\frac{1.55\times 100000}{400\times 1000000}=\frac{1.55}{4000% }=0.387\,\,\text{ms}

Cyclometer.html

  1. A D \displaystyle AD
  2. A D = (pjxroquctwzsy)(kvgledmanhfib) B E = (kxtcoigweh)(zvfbsylrnp)(ujd)(mqa) C F = (yvxqtdhpim)(skgrjbcolw)(un)(fa)(e)(z) \begin{aligned}\displaystyle AD&\displaystyle=\texttt{(pjxroquctwzsy)(% kvgledmanhfib)}\\ \displaystyle BE&\displaystyle=\texttt{(kxtcoigweh)(zvfbsylrnp)(ujd)(mqa)}\\ \displaystyle CF&\displaystyle=\texttt{(yvxqtdhpim)(skgrjbcolw)(un)(fa)(e)(z)}% \\ \end{aligned}
  3. A B C D E F ABCDEF
  4. A * B * C * D * E * F * A*B*C*D*E*F*
  5. A A
  6. B B
  7. A * D * \displaystyle A^{*}D^{*}
  8. S S
  9. C F CF
  10. C F CF
  11. C * F * C*F*
  12. A * D * A*D*
  13. B * E * B*E*
  14. C * F * C*F*
  15. A D = (pjxroquctwzsy)(kvgledmanhfib) A * D * = (sjxroqtcuzwpy)(kngledamvhifb) B E = (kxtcoigweh)(zvfbsylrnp)(ujd)(mqa) B * E * = (kxucofgzeh)(wnibpylrvs)(tjd)(aqm) C F = (yvxqtdhpim)(skgrjbcolw)(un)(fa)(e)(z) C * F * = (ynxqudhsfa)(pkgrjbcolz)(tv)(im)(e)(w) \begin{aligned}\displaystyle AD&\displaystyle=\texttt{(pjxroquctwzsy)(% kvgledmanhfib)}\\ \displaystyle A^{*}D^{*}&\displaystyle=\texttt{(sjxroqtcuzwpy)(kngledamvhifb)}% \\ \displaystyle BE&\displaystyle=\texttt{(kxtcoigweh)(zvfbsylrnp)(ujd)(mqa)}\\ \displaystyle B^{*}E^{*}&\displaystyle=\texttt{(kxucofgzeh)(wnibpylrvs)(tjd)(% aqm)}\\ \displaystyle CF&\displaystyle=\texttt{(yvxqtdhpim)(skgrjbcolw)(un)(fa)(e)(z)}% \\ \displaystyle C^{*}F^{*}&\displaystyle=\texttt{(ynxqudhsfa)(pkgrjbcolz)(tv)(im% )(e)(w)}\\ \end{aligned}
  16. n n
  17. S S
  18. A D AD
  19. B E BE
  20. C F CF

Cylinder_set.html

  1. X = α X α \textstyle X=\prod_{\alpha}X_{\alpha}\,
  2. X α X_{\alpha}
  3. α \alpha
  4. p α : X X α p_{\alpha}:X\to X_{\alpha}
  5. α \alpha
  6. U X α U\subset X_{\alpha}
  7. p α - 1 ( U ) p_{\alpha}^{-1}(U)
  8. X X
  9. S = { 1 , 2 , , n } S=\{1,2,\ldots,n\}
  10. S = { x = ( , x - 1 , x 0 , x 1 , ) : x k S k } S^{\mathbb{Z}}=\{x=(\ldots,x_{-1},x_{0},x_{1},\ldots):x_{k}\in S\;\forall k\in% \mathbb{Z}\}
  11. \mathbb{Z}
  12. S S
  13. S S^{\mathbb{Z}}
  14. C t [ a ] = { x S : x t = a } . C_{t}[a]=\{x\in S^{\mathbb{Z}}:x_{t}=a\}.
  15. C t [ a 0 , , a m ] = C t [ a 0 ] C t + 1 [ a 1 ] C t + m [ a m ] = { x S : x t = a 0 , , x t + m = a m } . C_{t}[a_{0},\cdots,a_{m}]=C_{t}[a_{0}]\,\cap\,C_{t+1}[a_{1}]\,\cap\cdots\cap\,% C_{t+m}[a_{m}]=\{x\in S^{\mathbb{Z}}:x_{t}=a_{0},\ldots,x_{t+m}=a_{m}\}.
  16. V V
  17. C A [ f 0 , , f m ] = { x V : ( f 1 ( x ) , f 2 ( x ) , , f m ( x ) ) A } C_{A}[f_{0},\cdots,f_{m}]=\{x\in V:(f_{1}(x),f_{2}(x),\cdots,f_{m}(x))\in A\}
  18. A K n A\subset K^{n}
  19. K n K^{n}
  20. f j f_{j}
  21. V V
  22. f j ( V * ) n f_{j}\in(V^{*})^{\otimes n}
  23. V V
  24. f j ( V ) n f_{j}\in(V^{\prime})^{\otimes n}
  25. f j f_{j}
  26. S S^{\mathbb{Z}}
  27. 1 / 2 m 1/2^{m}
  28. S S^{\mathbb{Z}}

Cylindric_numbering.html

  1. ν \nu
  2. μ \mu
  3. f f
  4. ν = μ f \nu=\mu\circ f
  5. f f
  6. μ \mu
  7. f f
  8. ν \nu
  9. ν 1 c ( ν ) . \nu\equiv_{1}c(\nu).
  10. S S
  11. 1 S : { 0 , 1 } 1_{S}:\mathbb{N}\to\{0,1\}
  12. ν ν = ν \nu\circ\nu=\nu

Cylindrification.html

  1. ν \nu
  2. c ( ν ) c(\nu)
  3. Domain ( c ( ν ) ) := { n , k | n Domain ( ν ) } \mathrm{Domain}(c(\nu)):=\{\langle n,k\rangle|n\in\mathrm{Domain}(\nu)\}
  4. c ( ν ) n , k := ν ( i ) c(\nu)\langle n,k\rangle:=\nu(i)
  5. n , k \langle n,k\rangle
  6. ν \nu
  7. μ \mu
  8. ν μ c ( ν ) 1 c ( μ ) \nu\leq\mu\Leftrightarrow c(\nu)\leq_{1}c(\mu)
  9. ν 1 c ( ν ) \nu\leq_{1}c(\nu)

Cynoscion_nebulosus.html

  1. W = c L b W=cL^{b}\!\,

D-term.html

  1. θ 1 , θ 2 , θ ¯ 1 , θ ¯ 2 \theta^{1},\theta^{2},\bar{\theta}^{1},\bar{\theta}^{2}
  2. D θ 1 θ 2 θ ¯ 1 θ ¯ 2 D\theta^{1}\theta^{2}\bar{\theta}^{1}\bar{\theta}^{2}
  3. θ \theta

D54_(protocol).html

  1. 10 - 6 / R t 10^{-}6/Rt

D_(disambiguation).html

  1. 𝔻 \mathbb{D}
  2. 𝐃 \mathbf{D}

Daniel_Goldston.html

  1. lim inf n p n + 1 - p n log p n = 0 \liminf_{n\to\infty}\frac{p_{n+1}-p_{n}}{\log p_{n}}=0
  2. p n p_{n}
  3. c > 0 c>0
  4. p n p_{n}
  5. p n + 1 p_{n+1}
  6. c c
  7. p n + 1 - p n < c log p n p_{n+1}-p_{n}<c\log p_{n}

Daniell_integral.html

  1. H H
  2. X X
  3. H H
  4. h ( x ) h(x)
  5. H H
  6. | h ( x ) | |h(x)|
  7. I h Ih
  8. α \alpha
  9. β \beta
  10. I ( α h + β k ) = α I h + β I k I(\alpha h+\beta k)=\alpha Ih+\beta Ik
  11. h ( x ) 0 h(x)\geq 0
  12. I h 0 Ih\geq 0
  13. h n ( x ) h_{n}(x)
  14. h 1 h k h_{1}\geq\cdots\geq h_{k}\geq\cdots
  15. H H
  16. x x
  17. X X
  18. I h n 0 Ih_{n}\to 0
  19. I I
  20. Z Z
  21. X X
  22. ϵ > 0 \epsilon>0
  23. h p ( x ) h_{p}(x)
  24. I h p < ϵ Ih_{p}<\epsilon
  25. sup p h p ( x ) 1 \sup_{p}h_{p}(x)\geq 1
  26. Z Z
  27. X X
  28. L + L^{+}
  29. h n h_{n}
  30. I h n Ih_{n}
  31. f f
  32. L + L^{+}
  33. I f = lim n I h n If=\lim_{n\to\infty}Ih_{n}
  34. h n h_{n}
  35. L + L^{+}
  36. L L
  37. ϕ \phi
  38. I + ϕ = inf f I f I^{+}\phi=\inf_{f}If
  39. f f
  40. L + L^{+}
  41. f ϕ f\geq\phi
  42. I - ϕ = - I + ( - ϕ ) I^{-}\phi=-I^{+}(-\phi)
  43. L L
  44. X ϕ ( x ) d x = I + ϕ = I - ϕ . \int_{X}\phi(x)dx=I^{+}\phi=I^{-}\phi.
  45. L L
  46. ϕ ( x ) \phi(x)
  47. ϕ = f - g \phi=f-g
  48. f f
  49. g g
  50. L + L^{+}
  51. ϕ ( x ) \phi(x)
  52. X ϕ ( x ) d x = I f - I g \int_{X}\phi(x)dx=If-Ig\,
  53. ϕ \phi
  54. f f
  55. g g
  56. χ ( x ) \chi(x)

Data_assimilation.html

  1. J ( 𝐱 ) = ( 𝐱 - 𝐱 b ) T 𝐁 - 1 ( 𝐱 - 𝐱 b ) + ( 𝐲 - H [ 𝐱 ] ) T 𝐑 - 1 ( 𝐲 - H [ 𝐱 ] ) , J(\mathbf{x})=(\mathbf{x}-\mathbf{x}_{b})^{\mathrm{T}}\mathbf{B}^{-1}(\mathbf{% x}-\mathbf{x}_{b})+(\mathbf{y}-\mathit{H}[\mathbf{x}])^{\mathrm{T}}\mathbf{R}^% {-1}(\mathbf{y}-\mathit{H}[\mathbf{x}]),
  2. 𝐁 \mathbf{B}
  3. 𝐑 \mathbf{R}
  4. J ( 𝐱 ) = 2 𝐁 - 1 ( 𝐱 - 𝐱 b ) - 2 H T 𝐑 - 1 ( 𝐲 - H [ 𝐱 ] ) \nabla J(\mathbf{x})=2\mathbf{B}^{-1}(\mathbf{x}-\mathbf{x}_{b})-2\mathit{H}^{% T}\mathbf{R}^{-1}(\mathbf{y}-\mathit{H}[\mathbf{x}])
  5. J ( 𝐱 ) = ( 𝐱 - 𝐱 b ) T 𝐁 - 1 ( 𝐱 - 𝐱 b ) + i = 0 n ( 𝐲 i - H i [ 𝐱 i ] ) T 𝐑 i - 1 ( 𝐲 i - H i [ 𝐱 i ] ) J(\mathbf{x})=(\mathbf{x}-\mathbf{x}_{b})^{\mathrm{T}}\mathbf{B}^{-1}(\mathbf{% x}-\mathbf{x}_{b})+\sum_{i=0}^{n}(\mathbf{y}_{i}-\mathit{H}_{i}[\mathbf{x}_{i}% ])^{\mathrm{T}}\mathbf{R}_{i}^{-1}(\mathbf{y}_{i}-\mathit{H}_{i}[\mathbf{x}_{i% }])
  6. H \mathit{H}

Davis–Putnam_algorithm.html

  1. c c
  2. n n

Day_count_convention.html

  1. Interest = Principal × CouponRate × Factor \mathrm{Interest}=\mathrm{Principal}\times\mathrm{CouponRate}\times\mathrm{Factor}
  2. Factor = 360 × ( Y 2 - Y 1 ) + 30 × ( M 2 - M 1 ) + ( D 2 - D 1 ) 360 \mathrm{Factor}=\frac{360\times(Y_{2}-Y_{1})+30\times(M_{2}-M_{1})+(D_{2}-D_{1% })}{360}
  3. CouponFactor = 360 × ( Y 3 - Y 1 ) + 30 × ( M 3 - M 1 ) + ( D 3 - D 1 ) 360 \mathrm{CouponFactor}=\frac{360\times(Y_{3}-Y_{1})+30\times(M_{3}-M_{1})+(D_{3% }-D_{1})}{360}
  4. CouponFactor = 1 Freq \mathrm{CouponFactor}=\frac{1}{\mathrm{Freq}}
  5. Factor = Days ( Date1 , Date2 ) Freq × Days ( Date1 , Date3 ) \mathrm{Factor}=\frac{\mathrm{Days}(\mathrm{Date1},\mathrm{Date2})}{\mathrm{% Freq}\times\mathrm{Days}(\mathrm{Date1},\mathrm{Date3})}
  6. CouponFactor = 1 Freq \mathrm{CouponFactor}=\frac{1}{\mathrm{Freq}}
  7. Factor = Days not in leap year 365 + Days in leap year 366 \mathrm{Factor}=\frac{\mbox{Days not in leap year}~{}}{365}+\frac{\mbox{Days % in leap year}~{}}{366}
  8. Factor = Days ( Date1 , Date2 ) 365 \mathrm{Factor}=\frac{\mathrm{Days}(\mathrm{Date1},\mathrm{Date2})}{365}
  9. Factor = Days ( Date1 , Date2 ) 360 \mathrm{Factor}=\frac{\mathrm{Days}(\mathrm{Date1},\mathrm{Date2})}{360}
  10. Factor = Days ( Date1 , Date2 ) 364 \mathrm{Factor}=\frac{\mathrm{Days}(\mathrm{Date1},\mathrm{Date2})}{364}
  11. Factor = Days ( Date1 , Date2 ) DiY \mathrm{Factor}=\frac{\mathrm{Days}(\mathrm{Date1},\mathrm{Date2})}{\mathrm{% DiY}}
  12. Factor = Days ( Date1 , Date2 ) DiY \mathrm{Factor}=\frac{\mathrm{Days}(\mathrm{Date1},\mathrm{Date2})}{\mathrm{% DiY}}

DBFS.html

  1. DR = SNR = 20 log 10 ( 2 n 3 2 ) 6.0206 n + 1.761 \mathrm{DR}=\mathrm{SNR}=20\log_{10}{\left(2^{n}\sqrt{\tfrac{3}{2}}\right)}% \approx 6.0206\cdot n+1.761
  2. DR = SNR = 20 log 10 ( 2 16 3 2 ) 6.0206 16 + 1.761 98.09 \mathrm{DR}=\mathrm{SNR}=20\log_{10}{\left(2^{16}\sqrt{\tfrac{3}{2}}\right)}% \approx 6.0206\cdot 16+1.761\approx 98.09\,

DBZ_(meteorology).html

  1. Z = 0 D m a x N 0 e - Λ D D 6 d D Z=\int_{0}^{Dmax}N_{0}e^{-\Lambda D}D^{6}dD
  2. d B Z 10 log 10 Z Z 0 dBZ\propto 10\,\log_{10}\frac{Z}{Z_{0}}
  3. mm hr = ( 10 ( d B Z / 10 ) 200 ) 5 8 \frac{\mathrm{mm}}{\mathrm{hr}}=\left(\frac{10^{(dBZ/10)}}{200}\right)^{5\over 8}

De_Arte_Combinatoria.html

  1. ( n r ) = ( n - 1 r ) + ( n - 1 r - 1 ) {n\choose r}={n-1\choose r}+{n-1\choose r-1}

De_Bruijn_graph.html

  1. S := { s 1 , , s m } S:=\{s_{1},\dots,s_{m}\}
  2. V = S n = { ( s 1 , , s 1 , s 1 ) , ( s 1 , , s 1 , s 2 ) , , ( s 1 , , s 1 , s m ) , ( s 1 , , s 2 , s 1 ) , , ( s m , , s m , s m ) } . V=S^{n}=\{(s_{1},\dots,s_{1},s_{1}),(s_{1},\dots,s_{1},s_{2}),\dots,(s_{1},% \dots,s_{1},s_{m}),(s_{1},\dots,s_{2},s_{1}),\dots,(s_{m},\dots,s_{m},s_{m})\}.
  3. E = { ( ( v 1 , v 2 , , v n ) , ( v 2 , , v n , s i ) ) : i = 1 , , m } . E=\{((v_{1},v_{2},\dots,v_{n}),(v_{2},\dots,v_{n},s_{i})):i=1,\dots,m\}.
  4. n = 1 n=1
  5. m 2 m^{2}
  6. m m
  7. m m
  8. n n
  9. ( n - 1 ) (n-1)
  10. n n
  11. ( n - 1 ) (n-1)
  12. n n
  13. ( n - 1 ) (n-1)
  14. x m x mod 1 x\mapsto mx\ \bmod\ 1

De_Polignac's_formula.html

  1. n ! = prime p n p s p ( n ) , n!=\prod_{\,\text{prime }p\leq n}p^{s_{p}(n)},
  2. s p ( n ) = j = 1 n p j , s_{p}(n)=\sum_{j=1}^{\infty}\left\lfloor\frac{n}{p^{j}}\right\rfloor,
  3. s p ( n ) = j = 1 log p ( n ) n p j s_{p}(n)=\sum_{j=1}^{\lfloor\log_{p}(n)\rfloor}\left\lfloor\frac{n}{p^{j}}\right\rfloor
  4. x n = x n \left\lfloor\frac{x}{n}\right\rfloor=\left\lfloor\frac{\lfloor x\rfloor}{n}\right\rfloor
  5. n ! = i = 1 π ( n ) p i s p i ( n ) = i = 1 π ( n ) p i j = 1 log p i ( n ) n p i j \displaystyle n!=\prod_{i=1}^{\pi(n)}p_{i}^{s_{p_{i}}(n)}=\prod_{i=1}^{\pi(n)}% p_{i}^{\sum_{j=1}^{\lfloor\log_{p_{i}}(n)\rfloor}\left\lfloor\frac{n}{{p_{i}}^% {j}}\right\rfloor}
  6. π ( n ) \pi(n)

De_Rham–Weil_theorem.html

  1. \mathcal{F}
  2. X X
  3. \mathcal{F}^{\bullet}
  4. \mathcal{F}
  5. H q ( X , ) H q ( ( X ) ) , H^{q}(X,\mathcal{F})\cong H^{q}(\mathcal{F}^{\bullet}(X)),
  6. H q ( X , ) H^{q}(X,\mathcal{F})
  7. q q
  8. X X
  9. . \mathcal{F}.

De_Sitter_double_star_experiment.html

  1. c c
  2. v v
  3. ( c + v ) (c+v)
  4. ( c - v ) (c-v)
  5. k < 2 × 10 - 9 k<2\times 10^{-9}

Debye–Hückel_equation.html

  1. γ \ \gamma\,
  2. a C \ a_{C}\,
  3. a C = γ [ C ] [ C ] \ a_{C}=\gamma\frac{[C]}{[C^{\ominus}]}\,
  4. γ \ \gamma\,
  5. [ C ] \ [C^{\ominus}]\,
  6. [ C ] [C]
  7. [ C ] [C]
  8. [ C ] [C^{\ominus}]
  9. ln ( γ i ) = - z i 2 q 2 κ 8 π ε r ε 0 k B T = - z i 2 q 3 N A 1 / 2 4 π ( ε r ε 0 k B T ) 3 / 2 I 2 = - A z i 2 I \ln(\gamma_{i})=-\frac{z_{i}^{2}q^{2}\kappa}{8\pi\varepsilon_{r}\varepsilon_{0% }k_{B}T}=-\frac{z_{i}^{2}q^{3}N^{1/2}_{\mathrm{A}}}{4\pi(\varepsilon_{r}% \varepsilon_{0}k_{B}T)^{3/2}}\sqrt{\frac{I}{2}}=-Az_{i}^{2}\sqrt{I}
  10. z i z_{i}
  11. q q
  12. κ \kappa
  13. ε r \varepsilon_{r}
  14. ε 0 \varepsilon_{0}
  15. k B k_{B}
  16. T T
  17. N A N_{\mathrm{A}}
  18. I I
  19. A A
  20. I I
  21. A A
  22. 1.172 mol - 1 / 2 kg 1 / 2 1.172\,\text{ mol}^{-1/2}\,\text{ kg}^{1/2}
  23. log 10 \log 10
  24. 0.509 mol - 1 / 2 kg 1 / 2 0.509\,\text{ mol}^{-1/2}\,\text{ kg}^{1/2}
  25. i = 1 s x i ν i = K \prod_{i=1}^{s}x_{i}^{\nu_{i}}=K
  26. \textstyle\prod
  27. i i
  28. s s
  29. x i x_{i}
  30. i i
  31. ν i \nu_{i}
  32. i i
  33. K K
  34. γ K \gamma K
  35. γ \gamma
  36. γ \gamma
  37. γ i \gamma_{i}
  38. log ( γ ) = i = 1 s ν i log ( γ i ) . \log(\gamma)=\sum_{i=1}^{s}\nu_{i}\log(\gamma_{i}).
  39. Φ \Phi
  40. Ξ \Xi
  41. Φ = S - U T = - A T \Phi=S-\frac{U}{T}=-\frac{A}{T}
  42. Φ \Phi
  43. S S
  44. U U
  45. T T
  46. A A
  47. Φ \Phi
  48. d Φ = P T d V + U T 2 d T d\Phi=\frac{P}{T}dV+\frac{U}{T^{2}}dT
  49. P P
  50. V V
  51. P T = Φ V \frac{P}{T}=\frac{\partial\Phi}{\partial V}
  52. U T 2 = Φ T , \frac{U}{T^{2}}=\frac{\partial\Phi}{\partial T},
  53. U = U k + U e U=U_{k}+U_{e}
  54. k k
  55. e e
  56. Φ = Φ k + Φ e \Phi=\Phi_{k}+\Phi_{e}
  57. Φ e = U e T 2 d T \Phi_{e}=\int\frac{U_{e}}{T^{2}}dT
  58. Φ e = P e T d V + U e T 2 d T . \Phi_{e}=\int\frac{P_{e}}{T}dV+\int\frac{U_{e}}{T^{2}}\,dT.
  59. Φ e = Ξ e \Phi_{e}=\Xi_{e}
  60. Ξ \Xi
  61. Ξ = S - U + P V T = Φ - P V T = - G T . \Xi=S-\frac{U+PV}{T}=\Phi-\frac{PV}{T}=-\frac{G}{T}.
  62. G G
  63. Ξ \Xi
  64. d Ξ = - V T d P + U + P V T 2 d T . d\Xi=-\frac{V}{T}dP+\frac{U+PV}{T^{2}}dT.
  65. P e P_{e}
  66. Ξ = Ξ k + Ξ e \Xi=\Xi_{k}+\Xi_{e}
  67. Ξ e = Φ e = U e T 2 d T . \Xi_{e}=\Phi_{e}=\int\frac{U_{e}}{T^{2}}dT.
  68. Ξ k = i = 0 s N i ( ξ i - k B l n ( x i ) ) . \Xi_{k}=\sum_{i=0}^{s}N_{i}(\xi_{i}-k_{B}ln(x_{i})).
  69. i i
  70. s s
  71. N i N_{i}
  72. ξ i \xi_{i}
  73. k B k_{B}
  74. x i x_{i}
  75. ξ i \xi_{i}
  76. ξ i = s i - u i + P v i T . \xi_{i}=s_{i}-\frac{u_{i}+Pv_{i}}{T}.
  77. Ξ k \Xi_{k}
  78. U U
  79. i = 1 s N i z i = 0. \sum_{i=1}^{s}N_{i}z_{i}=0.
  80. N i N_{i}
  81. z i z_{i}
  82. P P
  83. z i q φ z_{i}q\varphi
  84. q q
  85. φ \varphi
  86. P P
  87. n i n_{i}
  88. n i 0 n^{0}_{i}
  89. e - z i q φ k B T e^{-\frac{z_{i}q\varphi}{k_{B}T}}
  90. k B k_{B}
  91. T T
  92. n i = N i V e - z i q φ k B T = n i 0 e - z i q φ k B T n_{i}=\frac{N_{i}}{V}e^{-\dfrac{z_{i}q\varphi}{k_{B}T}}=n^{0}_{i}e^{-\dfrac{z_% {i}q\varphi}{k_{B}T}}
  93. ρ = i z i q n i = i z i q n i 0 e - z i q φ k B T . \rho=\sum_{i}z_{i}qn_{i}=\sum_{i}z_{i}qn^{0}_{i}e^{-\frac{z_{i}q\varphi}{k_{B}% T}}.
  94. 2 φ = - ρ ε r ε 0 = - i z i q n i 0 ε r ε 0 e - z i q φ k B T . {\nabla}^{2}\varphi=-\frac{\rho}{\varepsilon_{r}\varepsilon_{0}}=-\sum_{i}% \frac{z_{i}qn^{0}_{i}}{\varepsilon_{r}\varepsilon_{0}}e^{-\frac{z_{i}q\varphi}% {k_{B}T}}.
  95. e x = 1 + x e^{x}=1+x
  96. 0 < x 1 0<x\ll 1
  97. - i z i q n i 0 ε r ε 0 e - z i q φ k B T - i z i q n i 0 ε r ε 0 ( 1 - z i q φ k B T ) = - ( i z i q n i 0 ε r ε 0 - i z i 2 q 2 n i 0 φ ε r ε 0 k B T ) . -\sum_{i}\frac{z_{i}qn^{0}_{i}}{\varepsilon_{r}\varepsilon_{0}}e^{-\frac{z_{i}% q\varphi}{k_{B}T}}\approx-\sum_{i}\frac{z_{i}qn^{0}_{i}}{\varepsilon_{r}% \varepsilon_{0}}\left(1-\frac{z_{i}q\varphi}{k_{B}T}\right)=-\left(\sum_{i}% \frac{z_{i}qn^{0}_{i}}{\varepsilon_{r}\varepsilon_{0}}-\sum_{i}\frac{z_{i}^{2}% q^{2}n^{0}_{i}\varphi}{\varepsilon_{r}\varepsilon_{0}k_{B}T}\right).
  98. 2 φ = i z i 2 q 2 n i 0 φ ε r ε 0 k B T , {\nabla}^{2}\varphi=\sum_{i}\frac{z_{i}^{2}q^{2}n^{0}_{i}\varphi}{\varepsilon_% {r}\varepsilon_{0}k_{B}T},
  99. κ 2 \kappa^{2}
  100. I I
  101. κ 2 = i z i 2 q 2 n i 0 ε r ε 0 k B T = 2 I q 2 ε r ε 0 k B T , \kappa^{2}=\sum_{i}\frac{z_{i}^{2}q^{2}n^{0}_{i}}{\varepsilon_{r}\varepsilon_{% 0}k_{B}T}=\frac{2Iq^{2}}{\varepsilon_{r}\varepsilon_{0}k_{B}T},
  102. I = 1 2 i z i 2 n i 0 . I=\frac{1}{2}\sum_{i}z_{i}^{2}n^{0}_{i}.
  103. 2 φ = κ 2 φ . {\nabla}^{2}\varphi=\kappa^{2}\varphi.
  104. κ - 1 \kappa^{-1}
  105. r = 0 r=0
  106. 2 φ = 1 r 2 r ( r 2 φ ( r ) r ) = 2 φ ( r ) r 2 + 2 r φ ( r ) r = κ 2 φ ( r ) . {\nabla}^{2}\varphi=\frac{1}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac% {\partial\varphi(r)}{\partial r}\right)=\frac{\partial^{2}\varphi(r)}{\partial r% ^{2}}+\frac{2}{r}\frac{\partial\varphi(r)}{\partial r}=\kappa^{2}\varphi(r).
  107. κ \kappa
  108. φ ( r ) = A e - κ 2 r r + A e κ 2 r 2 r κ 2 = A e - κ r r + A ′′ e κ r r = A e - κ r r . \varphi(r)=A\frac{e^{-\sqrt{\kappa^{2}}r}}{r}+A^{\prime}\frac{e^{\sqrt{\kappa^% {2}}r}}{2r\sqrt{\kappa^{2}}}=A\frac{e^{-\kappa r}}{r}+A^{\prime\prime}\frac{e^% {\kappa r}}{r}=A\frac{e^{-\kappa r}}{r}.
  109. A A
  110. A A^{\prime}
  111. A ′′ A^{\prime\prime}
  112. A ′′ A^{\prime\prime}
  113. a i a_{i}
  114. a i a_{i}
  115. φ p c ( r ) = 1 4 π ε r ε 0 z i q r . \varphi_{pc}(r)={1\over 4\pi\varepsilon_{r}\varepsilon_{0}}{z_{i}q\over r}.
  116. φ s p ( r ) = φ p c ( r ) + B i = 1 4 π ε r ε 0 z i q r + B i , \varphi_{sp}(r)=\varphi_{pc}(r)+B_{i}={1\over 4\pi\varepsilon_{r}\varepsilon_{% 0}}{z_{i}q\over r}+B_{i},
  117. B i B_{i}
  118. B i B_{i}
  119. r r
  120. a i a_{i}
  121. a i a_{i}
  122. φ ( r ) \varphi(r)
  123. φ ( a i ) = A i e - κ a i a i = 1 4 π ε r ε 0 z i q a i + B i = φ s p ( a i ) , \varphi(a_{i})=A_{i}\frac{e^{-\kappa a_{i}}}{a_{i}}={1\over 4\pi\varepsilon_{r% }\varepsilon_{0}}{z_{i}q\over a_{i}}+B_{i}=\varphi_{sp}(a_{i}),
  124. φ ( a i ) = - A i e - κ a i ( 1 + κ a i ) a i 2 = - 1 4 π ε r ε 0 z i q a i 2 = φ s p ( a i ) , \varphi^{\prime}(a_{i})=-\frac{A_{i}e^{-\kappa a_{i}}(1+\kappa a_{i})}{a_{i}^{% 2}}=-{1\over 4\pi\varepsilon_{r}\varepsilon_{0}}{z_{i}q\over a_{i}^{2}}=% \varphi_{sp}^{\prime}(a_{i}),
  125. A i = z i q 4 π ε r ε 0 e κ a i 1 + κ a i , A_{i}=\frac{z_{i}q}{4\pi\varepsilon_{r}\varepsilon_{0}}\frac{e^{\kappa a_{i}}}% {1+\kappa a_{i}},
  126. B i = - z i q κ 4 π ε r ε 0 1 1 + κ a i . B_{i}=-\frac{z_{i}q\kappa}{4\pi\varepsilon_{r}\varepsilon_{0}}\frac{1}{1+% \kappa a_{i}}.
  127. u i = z i q B i = - z i 2 q 2 κ 4 π ε r ε 0 1 1 + κ a i . u_{i}=z_{i}qB_{i}=-\frac{z_{i}^{2}q^{2}\kappa}{4\pi\varepsilon_{r}\varepsilon_% {0}}\frac{1}{1+\kappa a_{i}}.
  128. U e = 1 2 i = 1 s N i u i = - i = 1 s N i z i 2 2 q 2 κ 4 π ε r ε 0 1 1 + κ a i U_{e}=\frac{1}{2}\sum_{i=1}^{s}N_{i}u_{i}=-\sum_{i=1}^{s}\frac{N_{i}z_{i}^{2}}% {2}\frac{q^{2}\kappa}{4\pi\varepsilon_{r}\varepsilon_{0}}\frac{1}{1+\kappa a_{% i}}
  129. 2 φ ( r ) r 2 + 2 r φ ( r ) r = I q φ ( r ) ε r ε 0 k b T = κ 2 φ ( r ) . \frac{\partial^{2}\varphi(r)}{\partial r^{2}}+\frac{2}{r}\frac{\partial\varphi% (r)}{\partial r}=\frac{Iq\varphi(r)}{\varepsilon_{r}\varepsilon_{0}k_{b}T}=% \kappa^{2}\varphi(r).
  130. π 1 = q φ ( r ) k b T = Φ ( R ( r ) ) \pi_{1}=\frac{q\varphi(r)}{k_{b}T}=\Phi(R(r))
  131. π 2 = ε r \pi_{2}=\varepsilon_{r}
  132. π 3 = a k b T ε 0 q 2 \pi_{3}=\frac{ak_{b}T\varepsilon_{0}}{q^{2}}
  133. π 4 = a 3 I \pi_{4}=a^{3}I
  134. π 5 = z 0 \pi_{5}=z_{0}
  135. π 6 = r a = R ( r ) . \pi_{6}=\frac{r}{a}=R(r).
  136. Φ \Phi
  137. R R
  138. ( κ a ) 2 (\kappa a)^{2}
  139. Z 0 Z_{0}
  140. z 0 z_{0}
  141. π 4 π 2 π 3 = a 2 q 2 I ε r ε 0 k b T = ( κ a ) 2 \frac{\pi_{4}}{\pi_{2}\pi_{3}}=\frac{a^{2}q^{2}I}{\varepsilon_{r}\varepsilon_{% 0}k_{b}T}=(\kappa a)^{2}
  142. π 5 π 2 π 3 = z 0 q 2 4 π a ε r ε 0 k b T = Z 0 \frac{\pi_{5}}{\pi_{2}\pi_{3}}=\frac{z_{0}q^{2}}{4\pi a\varepsilon_{r}% \varepsilon_{0}k_{b}T}=Z_{0}
  143. π \pi
  144. φ ( r ) \varphi(r)
  145. Φ ( R ( r ) ) \Phi(R(r))
  146. R ( r ) R(r)
  147. r r
  148. R ( r ) = a {R^{\prime}}(r)=a
  149. r r
  150. R R
  151. I I
  152. ( κ a ) 2 (\kappa a)^{2}
  153. z 0 z_{0}
  154. Z 0 Z_{0}
  155. Φ ( R ) R | R = 1 = - Z 0 \frac{\partial\Phi(R)}{\partial R}\bigg|_{R=1}=-Z_{0}
  156. Φ ( ) = 0 \Phi(\infty)=0
  157. 2 Φ ( R ) R 2 + 2 R Φ ( R ) R = ( κ a ) 2 Φ ( R ) . \frac{\partial^{2}\Phi(R)}{\partial R^{2}}+\frac{2}{R}\frac{\partial\Phi(R)}{% \partial R}=(\kappa a)^{2}\Phi(R).
  158. ( κ a ) 2 (\kappa a)^{2}
  159. Z 0 Z_{0}
  160. ( κ a ) 2 (\kappa a)^{2}
  161. - log 10 ( γ ) = A | z + z - | μ 1 + B a μ \ -\log_{10}(\gamma)=\frac{A|z_{+}z_{-}|\sqrt{\mu}}{1+Ba\sqrt{\mu}}\,
  162. γ \ \gamma\,
  163. z \ z\,
  164. μ \ \mu\,
  165. a \ a\,
  166. A \ A\,
  167. B \ B\,

Decade_(log_scale).html

  1. f 1 f_{1}
  2. f 2 f_{2}
  3. log 10 ( f 2 / f 1 ) log 10 ( 10 ) \log_{10}(f_{2}/f_{1})\over\log_{10}(10)
  4. ln f 2 - ln f 1 ln 10 \ln f_{2}-\ln f_{1}\over\ln 10
  5. log 10 ( 150000 / 15 ) = 4 \log_{10}(150000/15)=4
  6. log 10 ( 4.7 × 10 6 / 3.2 × 10 9 ) = - 2.83 \log_{10}(4.7\times 10^{6}/3.2\times 10^{9})=-2.83
  7. log 10 ( 2 ) = 0.301 \log_{10}(2)=0.301
  8. 220 × 10 - 3 = 0.22 220\times 10^{-3}=0.22
  9. 10 × 10 1.5 = 316.23 10\times 10^{1.5}=316.23
  10. 10 1 / 30 = 1.079775 10^{1/30}=1.079775

Decarburization.html

  1. C + CO 2 2 CO \mathrm{C+CO_{2}\ \rightleftharpoons\ 2\ CO\quad}
  2. C + H 2 O CO + H 2 \mathrm{C+H_{2}O\ \rightleftharpoons\ CO+H_{2}}
  3. C + 2 H 2 CH 4 \mathrm{C+2\ H_{2}\ \rightleftharpoons\ CH_{4}}
  4. C + 1 2 O 2 CO \mathrm{C+\tfrac{1}{2}O_{2}\ \Rightarrow\ CO}
  5. C + O 2 CO 2 \mathrm{C+O_{2}\ \Rightarrow\ CO_{2}}
  6. C + FeO CO + Fe \mathrm{C+FeO\ \Rightarrow\ CO+Fe}

Deformation_retract.html

  1. r : X A r:X\to A
  2. ι : A X \iota:A\hookrightarrow X
  3. r ι = i d A , r\circ\iota=id_{A},
  4. r : X A r:X\to A
  5. ι r \iota\circ r
  6. s : X X s:X\to X
  7. A U X A\subset U\subset X
  8. F : X × [ 0 , 1 ] X F:X\times[0,1]\to X\,
  9. F ( x , 0 ) = x , F ( x , 1 ) A , and F ( a , 1 ) = a . F(x,0)=x,\;F(x,1)\in A,\quad\mbox{and}~{}\quad F(a,1)=a.
  10. F ( a , t ) = a F(a,t)=a\,
  11. F ( x , t ) = ( ( 1 - t ) + t x ) x . F(x,t)=\left((1-t)+{t\over\|x\|}\right)x.
  12. u : X I u:X\rightarrow I
  13. I = [ 0 , 1 ] I=[0,1]
  14. A = u - 1 ( 0 ) A=u^{-1}(0)
  15. H : X × I X H:X\times I\rightarrow X
  16. H ( x , 0 ) = x H(x,0)=x
  17. x X x\in X
  18. H ( a , t ) = a H(a,t)=a
  19. ( a , t ) A × I (a,t)\in A\times I
  20. h ( x , 1 ) A h(x,1)\in A
  21. x u - 1 [ 0 , 1 ) x\in u^{-1}[0,1)
  22. f : A Y f:A\rightarrow Y
  23. g : X Y g:X\rightarrow Y
  24. g = f r g=f\circ r\,

Degenerate_energy_levels.html

  1. N N
  2. A A
  3. N × N N×N
  4. X X
  5. λ λ
  6. A X = λ X AX=\lambda X
  7. λ λ
  8. A A
  9. X X
  10. λ λ
  11. λ λ
  12. λ λ
  13. λ λ
  14. A X 1 = λ X 1 AX_{1}=\lambda X_{1}
  15. A X 2 = λ X 2 AX_{2}=\lambda X_{2}
  16. X 1 X_{1}
  17. X 2 X_{2}
  18. H ^ \hat{H}
  19. | ψ 1 |\psi_{1}\rangle
  20. | ψ 2 |\psi_{2}\rangle
  21. E E
  22. H ^ | ψ 1 = E | ψ 1 \hat{H}|\psi_{1}\rangle=E|\psi_{1}\rangle
  23. H ^ | ψ 2 = E | ψ 2 \hat{H}|\psi_{2}\rangle=E|\psi_{2}\rangle
  24. | ψ = c 1 | ψ 1 + c 2 | ψ 2 |\psi\rangle=c_{1}|\psi_{1}\rangle+c_{2}|\psi_{2}\rangle
  25. c 1 c_{1}
  26. c 2 c_{2}
  27. | ψ 1 |\psi_{1}\rangle
  28. | ψ 2 |\psi_{2}\rangle
  29. H ^ | ψ = H ^ ( c 1 | ψ 1 + c 2 | ψ 2 ) \hat{H}|\psi\rangle=\hat{H}(c_{1}|\psi_{1}\rangle+c_{2}|\psi_{2}\rangle)
  30. = ( c 1 H ^ | ψ 1 + c 2 H ^ | ψ 2 ) =(c_{1}\hat{H}|\psi_{1}\rangle+c_{2}\hat{H}|\psi_{2}\rangle)
  31. = E ( c 1 | ψ 1 + c 2 | ψ 2 ) =E(c_{1}|\psi_{1}\rangle+c_{2}|\psi_{2}\rangle)
  32. = E | ψ =E|\psi\rangle
  33. | ψ |\psi\rangle
  34. H ^ \hat{H}
  35. E E
  36. H ^ \hat{H}
  37. E n E_{n}
  38. E n E_{n}
  39. | E n , i |E_{n,i}\rangle
  40. | ψ |\psi\rangle
  41. E n E_{n}
  42. P ( E n ) = i = 1 g n | E n , i | ψ | 2 P(E_{n})=\sum_{i=1}^{g_{n}}|\langle E_{n,i}|\psi\rangle|^{2}
  43. | ψ |\psi\rangle
  44. V ( x ) V(x)
  45. - 2 2 m 2 ψ x 2 + V ψ = E ψ -\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi}{\partial x^{2}}+V\psi=E\psi
  46. E E
  47. V ( x ) V(x)
  48. | ψ 1 |\psi_{1}\rangle
  49. | ψ 2 |\psi_{2}\rangle
  50. E E
  51. - 2 2 m 2 ψ 1 x 2 + V ψ 1 = E ψ 1 -\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi_{1}}{\partial x^{2}}+V\psi_{1}=E% \psi_{1}
  52. - 2 2 m 2 ψ 2 x 2 + V ψ 2 = E ψ 2 -\frac{\hbar^{2}}{2m}\frac{\partial^{2}\psi_{2}}{\partial x^{2}}+V\psi_{2}=E% \psi_{2}
  53. ψ 2 \psi_{2}
  54. ψ 1 \psi_{1}
  55. ψ 1 d 2 d x 2 ψ 2 - ψ 2 d 2 d x 2 ψ 1 = 0 \psi_{1}\frac{d^{2}}{dx^{2}}\psi_{2}-\psi_{2}\frac{d^{2}}{dx^{2}}\psi_{1}=0
  56. ψ 1 ψ 2 x - ψ 2 ψ 1 x = c o n s t a n t \psi_{1}\frac{\partial\psi_{2}}{\partial x}-\psi_{2}\frac{\partial\psi_{1}}{% \partial x}=constant
  57. ψ 1 ( x ) = c ψ 2 ( x ) \psi_{1}(x)=c\psi_{2}(x)
  58. c c
  59. x x\to\infty
  60. x - x\to-\infty
  61. L x L_{x}
  62. L y L_{y}
  63. | ψ |\psi\rangle
  64. - 2 2 m ( 2 ψ x 2 + 2 ψ y 2 ) = E ψ -\frac{\hbar^{2}}{2m}\left(\frac{\partial^{2}\psi}{{\partial x}^{2}}+\frac{% \partial^{2}\psi}{{\partial y}^{2}}\right)=E\psi
  65. E n x , n y = π 2 2 2 m ( n x 2 / L x 2 + n y 2 / L y 2 ) E_{n_{x},n_{y}}=\frac{\pi^{2}\hbar^{2}}{2m}(n_{x}^{2}/L_{x}^{2}+n_{y}^{2}/L_{y% }^{2})
  66. ψ n x , n y ( x , y ) = 2 / L x L y sin ( n x π x / L x ) sin ( n y π y / L y ) \psi_{n_{x},n_{y}}(x,y)=2/\sqrt{L_{x}L_{y}}\sin(n_{x}\pi x/L_{x})\sin(n_{y}\pi y% /L_{y})
  67. n x , n y = 1 , 2 , 3... n_{x},n_{y}=1,2,3...
  68. n x n_{x}
  69. n y n_{y}
  70. E 1 , 1 = π 2 2 2 m ( 1 / L x 2 + 1 / L y 2 ) E_{1,1}=\pi^{2}\frac{\hbar^{2}}{2m}(1/L_{x}^{2}+1/L_{y}^{2})
  71. L x L_{x}
  72. L y L_{y}
  73. L x / L y = p / q L_{x}/L_{y}=p/q
  74. ( n x , n y ) (n_{x},n_{y})
  75. ( p n y / q , q n x / p ) (pn_{y}/q,qn_{x}/p)
  76. L x = L y = L L_{x}=L_{y}=L
  77. E n x , n y = π 2 2 2 m L 2 ( n x 2 + n y 2 ) E_{n_{x},n_{y}}=\frac{\pi^{2}\hbar^{2}}{2mL^{2}}(n_{x}^{2}+n_{y}^{2})
  78. n x n_{x}
  79. n y n_{y}
  80. n x n_{x}
  81. n y n_{y}
  82. A ^ \hat{A}
  83. B ^ \hat{B}
  84. [ A ^ , B ^ ] = 0 [\hat{A},\hat{B}]=0
  85. | ψ |\psi\rangle
  86. A ^ \hat{A}
  87. B ^ | ψ \hat{B}|\psi\rangle
  88. A ^ \hat{A}
  89. λ \lambda
  90. B ^ | ψ \hat{B}|\psi\rangle
  91. E λ E_{\lambda}
  92. A ^ \hat{A}
  93. B ^ \hat{B}
  94. λ \lambda
  95. A ^ \hat{A}
  96. A ^ \hat{A}
  97. B ^ \hat{B}
  98. B ^ \hat{B}
  99. A ^ \hat{A}
  100. A ^ \hat{A}
  101. B ^ \hat{B}
  102. A ^ \hat{A}
  103. A ^ \hat{A}
  104. B ^ \hat{B}
  105. A ^ \hat{A}
  106. B ^ \hat{B}
  107. A ^ \hat{A}
  108. A ^ \hat{A}
  109. B ^ \hat{B}
  110. A ^ \hat{A}
  111. B ^ \hat{B}
  112. C ^ \hat{C}
  113. A ^ \hat{A}
  114. B ^ \hat{B}
  115. | r |r\rangle
  116. r | P | ψ = ψ ( - r ) \langle r|P|\psi\rangle=\psi(-r)
  117. ± 1 \pm 1
  118. A ^ \hat{A}
  119. A ~ = P A ^ P \tilde{A}=P\hat{A}P
  120. [ P , A ^ ] = 0 [P,\hat{A}]=0
  121. B ^ \hat{B}
  122. P B ^ + B ^ P = 0 P\hat{B}+\hat{B}P=0
  123. P ^ 2 \hat{P}^{2}
  124. H ^ \hat{H}
  125. H ^ \hat{H}
  126. P | ψ P|\psi\rangle
  127. H ^ \hat{H}
  128. | ψ |\psi\rangle
  129. S S
  130. S S
  131. H = S H S - 1 = S H S H^{\prime}=SHS^{-1}=SHS^{\dagger}
  132. S S
  133. S S
  134. S H S = H SHS^{\dagger}=H
  135. S H S - 1 = H SHS^{-1}=H
  136. H S = S H HS=SH
  137. [ S , H ] = 0 [S,H]=0
  138. | α |\alpha\rangle
  139. H | α = E | α H|\alpha\rangle=E|\alpha\rangle
  140. H S | α = S H | α = S E | α = E S | α HS|\alpha\rangle=SH|\alpha\rangle=SE|\alpha\rangle=ES|\alpha\rangle
  141. S | α S|\alpha\rangle
  142. E E
  143. | α |\alpha\rangle
  144. S | α S|\alpha\rangle
  145. S S
  146. ϵ \epsilon
  147. S ( ϵ ) | α S(\epsilon)|\alpha\rangle
  148. L 2 ^ \hat{L^{2}}
  149. L z ^ \hat{L_{z}}
  150. S 2 ^ \hat{S^{2}}
  151. S z ^ \hat{S_{z}}
  152. I 2 ^ \hat{I^{2}}
  153. I z ^ \hat{I_{z}}
  154. n n
  155. n n
  156. l = 0 l=0
  157. n - 1 n-1
  158. ( 2 l + 1 ) (2l+1)
  159. l z = - l l_{z}=-l
  160. l l
  161. l z l_{z}
  162. F = - k r F=-kr
  163. V ( r ) V(r)
  164. V ( r ) = 1 / 2 ( m ω 2 r 2 ) V(r)=1/2(m\omega^{2}r^{2})
  165. ω \omega
  166. k / m \sqrt{k/m}
  167. - 2 2 m ( 2 ψ x 2 + 2 ψ y 2 + 2 ψ z 2 ) + ( 1 / 2 ) m ω 2 ( x 2 + y 2 + z 2 ) ψ = E ψ -\frac{\hbar^{2}}{2m}\left(\frac{\partial^{2}\psi}{\partial x^{2}}+\frac{% \partial^{2}\psi}{\partial y^{2}}+\frac{\partial^{2}\psi}{\partial z^{2}}% \right)+(1/2){m\omega^{2}(x^{2}+y^{2}+z^{2})\psi}=E\psi
  168. E n x , n y , n z = ( n x + n y + n z + 3 / 2 ) ω E_{n_{x},n_{y},n_{z}}=(n_{x}+n_{y}+n_{z}+3/2)\hbar\omega
  169. E n = ( n + 3 / 2 ) ω E_{n}=(n+3/2)\hbar\omega
  170. n x , n y , n z {n_{x},n_{y},n_{z}}
  171. n x + n y + n z = n n_{x}+n_{y}+n_{z}=n
  172. n x = 0 n ( n - n x + 1 ) = ( n + 1 ) ( n + 2 ) / 2 \sum_{n_{x}=0}^{n}(n-n_{x}+1)=(n+1)(n+2)/2
  173. H 0 ^ \hat{H_{0}}
  174. V ^ \hat{V}
  175. H ^ = H 0 ^ + V ^ \hat{H}=\hat{H_{0}}+\hat{V}
  176. | ψ 0 = | n 0 + k 0 V k 0 / ( E 0 - E k ) | n k |\psi_{0}\rangle=|n_{0}\rangle+\sum_{k\neq 0}V_{k0}/(E_{0}-E_{k})|n_{k}\rangle
  177. E 0 = E k E_{0}=E_{k}
  178. H 0 ^ \hat{H_{0}}
  179. | m |m\rangle
  180. | ψ j |\psi_{j}\rangle
  181. | ψ j = i | m i m i | ψ j = i c j i | m i |\psi_{j}\rangle=\sum_{i}|m_{i}\rangle\langle m_{i}|\psi_{j}\rangle=\sum_{i}c_% {ji}|m_{i}\rangle
  182. [ H 0 ^ + V ^ ] ψ j = [ H 0 ^ + V ^ ] i c j i | m i = E j i c j i | m i [\hat{H_{0}}+\hat{V}]\psi_{j}\rangle=[\hat{H_{0}}+\hat{V}]\sum_{i}c_{ji}|m_{i}% \rangle=E_{j}\sum_{i}c_{ji}|m_{i}\rangle
  183. E j E_{j}
  184. E E
  185. H 0 ^ \hat{H_{0}}
  186. i c j i V ^ | m i = ( E j - E ) i c j i | m i = Δ E j i c j i | m i \sum_{i}c_{ji}\hat{V}|m_{i}\rangle=(E_{j}-E)\sum_{i}c_{ji}|m_{i}\rangle=\Delta E% _{j}\sum_{i}c_{ji}|m_{i}\rangle
  187. m k | \langle m_{k}|
  188. i c j i [ m k | V ^ | m i - δ i k ( E j - E ) ] = 0 \sum_{i}c_{ji}[\langle m_{k}|\hat{V}|m_{i}\rangle-\delta_{ik}(E_{j}-E)]=0
  189. V i k = m i | V ^ | m k V_{ik}=\langle m_{i}|\hat{V}|m_{k}\rangle
  190. | V 11 - Δ E j V 12 V 1 N V 21 V 22 - Δ E j V 2 N V N 1 V N 2 V N N - Δ E j | . \begin{vmatrix}V_{11}-\Delta E_{j}&V_{12}&\dots&V_{1N}\\ V_{21}&V_{22}-\Delta E_{j}&\dots&V_{2N}\\ \vdots&\vdots&\ddots&\vdots\\ V_{N1}&V_{N2}&\dots&V_{NN}-\Delta E_{j}\end{vmatrix}.\,
  191. | m |m\rangle
  192. V ^ \hat{V}
  193. H 0 ^ \hat{H_{0}}
  194. E 1 E_{1}
  195. E 2 E_{2}
  196. E 1 = E 2 = E E_{1}=E_{2}=E
  197. W W
  198. 𝐖 = [ 0 W 12 W 12 * 0 ] . \mathbf{W}=\begin{bmatrix}0&W_{12}\\ W_{12}^{*}&0\end{bmatrix}.
  199. E + = E + | W 12 | E_{+}=E+|W_{12}|
  200. E - = E - | W 12 | E_{-}=E-|W_{12}|
  201. H r = - p 4 / 8 m 3 c 2 H_{r}=-p^{4}/8m^{3}c^{2}
  202. p p
  203. m m
  204. | n l m |nlm\rangle
  205. E r = ( - 1 / 8 m 3 c 2 ) n l m | p 4 | n l m E_{r}=(-1/8m^{3}c^{2})\langle nlm|p^{4}|nlm\rangle
  206. p 4 = 4 m 2 ( H 0 + e 2 / r ) 2 p^{4}=4m^{2}(H^{0}+e^{2}/r)^{2}
  207. E r = ( - 1 / 2 m c 2 ) [ E n 2 + 2 E n e 2 1 / r + e 4 1 / r 2 ] E_{r}=(-1/2mc^{2})[E_{n}^{2}+2E_{n}e^{2}\langle 1/r\rangle+e^{4}\langle 1/r^{2% }\rangle]
  208. = ( - 1 / 2 ) m c 2 α 4 [ - 3 / ( 4 n 4 ) + 1 / n 3 ( l + 1 / 2 ) ] =(-1/2)mc^{2}\alpha^{4}[-3/(4n^{4})+1/{n^{3}(l+1/2)}]
  209. α \alpha
  210. H s o = - ( e / m c ) m L / r 3 = [ ( e 2 / ( m 2 c 2 r 3 ) ) S L ] H_{so}=-(e/mc){\vec{m}\cdot\vec{L}/r^{3}}=[(e^{2}/(m^{2}c^{2}r^{3}))\vec{S}% \cdot\vec{L}]
  211. H s o = ( e 2 / ( 4 m 2 c 2 r 3 ) ) [ J 2 - L 2 - S 2 ] H_{so}=(e^{2}/(4m^{2}c^{2}r^{3}))[\vec{J}^{2}-\vec{L}^{2}-\vec{S}^{2}]
  212. | j , m , l , 1 / 2 |j,m,l,1/2\rangle
  213. E s o = ( 2 e 2 ) / ( 4 m 2 c 2 ) [ j ( j + 1 ) - l ( l + 1 ) - 3 / 4 ] / ( ( a 0 ) 3 n 3 ( l ( l + 1 / 2 ) ( l + 1 ) ) ] E_{so}=(\hbar^{2}e^{2})/(4m^{2}c^{2})[j(j+1)-l(l+1)-3/4]/((a_{0})^{3}n^{3}(l(l% +1/2)(l+1))]
  214. a 0 a_{0}
  215. E f s = - ( m c 2 α 4 / ( 2 n 3 ) ) [ 1 / ( j + 1 / 2 ) - 3 / 4 n ] E_{fs}=-(mc^{2}\alpha^{4}/(2n^{3}))[1/(j+1/2)-3/4n]
  216. j = l ± 1 / 2 j=l\pm 1/2
  217. m \vec{m}
  218. L \vec{L}
  219. S \vec{S}
  220. V ^ = - ( m l + m s ) B \hat{V}=-(\vec{m_{l}}+\vec{m_{s}})\cdot\vec{B}
  221. m l = - e L / 2 m m_{l}=-e\vec{L}/2m
  222. m s = - e S / m m_{s}=-e\vec{S}/m
  223. V ^ = e ( L + 2 S ) B / 2 m \hat{V}=e(\vec{L}+2\vec{S})\cdot\vec{B}/2m
  224. L \vec{L}
  225. S \vec{S}
  226. E z = - μ B g j B m j E_{z}=-\mu_{B}g_{j}Bm_{j}
  227. μ B = e / 2 m \mu_{B}={e\hbar}/2m
  228. m j m_{j}
  229. V ^ = e B ( L z + 2 S z ) / 2 m \hat{V}=eB(L_{z}+2S_{z})/2m
  230. V ^ = e B ( m l + 2 m s ) / 2 m \hat{V}=eB(m_{l}+2m_{s})/2m
  231. ± 1 / 2 \pm 1/2
  232. H ^ s = - | e | E z \hat{H}_{s}=-|e|Ez
  233. H ^ s \hat{H}_{s}
  234. | n l m |nlm\rangle
  235. n l m l | z | n 1 l 1 m l 1 0 \langle nlm_{l}|z|n_{1}l_{1}m_{l1}\rangle\neq 0
  236. l = l 1 ± 1 l=l_{1}\pm 1
  237. m l = m l 1 m_{l}=m_{l1}
  238. | 2 , 0 , 0 |2,0,0\rangle
  239. | 2 , 1 , 0 |2,1,0\rangle
  240. Δ E 2 , 1 , m l = ± | e | ( 2 ) / ( m e e 2 ) E \Delta E_{2,1,m_{l}}=\pm|e|(\hbar^{2})/(m_{e}e^{2})E

Degree_distribution.html

  1. P ( k ) = ( n - 1 k ) p k ( 1 - p ) n - 1 - k , P(k)={n-1\choose k}p^{k}(1-p)^{n-1-k},

Degree_of_a_continuous_mapping.html

  1. n n
  2. S n S^{n}
  3. n = 1 n=1
  4. f : S n S n f\colon S^{n}\to S^{n}
  5. f f
  6. f * : H n ( S n ) H n ( S n ) f_{*}\colon H_{n}\left(S^{n}\right)\to H_{n}\left(S^{n}\right)
  7. H n ( ) H_{n}\left(\cdot\right)
  8. n n
  9. H n ( S n ) H_{n}\left(S^{n}\right)\cong\mathbb{Z}
  10. f * f_{*}
  11. f * : x α x f_{*}\colon x\mapsto\alpha x
  12. α \alpha\in\mathbb{Z}
  13. α \alpha
  14. f f
  15. f * ( [ X ] ) = deg ( f ) [ Y ] . f_{*}([X])=\deg(f)[Y]\,.
  16. f - 1 ( p ) = { x 1 , x 2 , , x n } . f^{-1}(p)=\{x_{1},x_{2},\ldots,x_{n}\}\,.
  17. f * [ c ] , [ ω ] = [ c ] , f * [ ω ] , \langle f_{*}[c],[\omega]\rangle=\langle[c],f^{*}[\omega]\rangle,
  18. deg f Y ω = X f * ω \deg f\int_{Y}\omega=\int_{X}f^{*}\omega\,
  19. Ω \R n \Omega\subset\R^{n}
  20. f : Ω ¯ \R n f:\bar{\Omega}\to\R^{n}
  21. p p
  22. f f
  23. p f ( Ω ) p\notin f(\partial\Omega)
  24. deg ( f , Ω , p ) \deg(f,\Omega,p)
  25. deg ( f , Ω , p ) := y f - 1 ( p ) sgn det D f ( y ) \deg(f,\Omega,p):=\sum_{y\in f^{-1}(p)}\operatorname{sgn}\det Df(y)
  26. D f ( y ) Df(y)
  27. f f
  28. y y
  29. p p
  30. deg ( f , Ω , p ) = deg ( f , Ω , p ) \deg(f,\Omega,p)=\deg(f,\Omega,p^{\prime})
  31. p p^{\prime}
  32. p p
  33. deg ( f , Ω ¯ , p ) 0 \deg(f,\bar{\Omega},p)\neq 0
  34. x Ω x\in\Omega
  35. f ( x ) = p f(x)=p
  36. deg ( id , Ω , y ) = 1 \deg(\operatorname{id},\Omega,y)=1
  37. y Ω y\in\Omega
  38. deg ( f , Ω , y ) = deg ( f , Ω 1 , y ) + deg ( f , Ω 2 , y ) \deg(f,\Omega,y)=\deg(f,\Omega_{1},y)+\deg(f,\Omega_{2},y)
  39. Ω 1 , Ω 2 \Omega_{1},\Omega_{2}
  40. Ω = Ω 1 Ω 2 \Omega=\Omega_{1}\cup\Omega_{2}
  41. y f ( Ω ¯ ( Ω 1 Ω 2 ) ) y\not\in f(\overline{\Omega}\setminus(\Omega_{1}\cup\Omega_{2}))
  42. f f
  43. g g
  44. F ( t ) F(t)
  45. F ( 0 ) = f , F ( 1 ) = g F(0)=f,\,F(1)=g
  46. p F ( t ) ( Ω ) p\notin F(t)(\partial\Omega)
  47. deg ( f , Ω , p ) = deg ( g , Ω , p ) \deg(f,\Omega,p)=\deg(g,\Omega,p)
  48. p deg ( f , Ω , p ) p\mapsto\deg(f,\Omega,p)
  49. \R n - f ( Ω ) \R^{n}-f(\partial\Omega)
  50. f , g : S n S n f,g:S^{n}\to S^{n}\,
  51. deg ( f ) = deg ( g ) \deg(f)=\deg(g)
  52. [ S n , S n ] = π n S n 𝐙 [S^{n},S^{n}]=\pi_{n}S^{n}\to\mathbf{Z}
  53. n n
  54. f , g : M S n f,g:M\to S^{n}
  55. deg ( f ) = deg ( g ) . \deg(f)=\deg(g).
  56. f : S n S n f:S^{n}\to S^{n}
  57. F : B n S n F:B_{n}\to S^{n}
  58. deg ( f ) = 0 \deg(f)=0
  59. S n S^{n}

Degree_of_an_algebraic_variety.html

  1. d i m ( V ) + d i m ( L ) = n . dim(V)+dim(L)=n.

Degree_of_anonymity.html

  1. d d
  2. d d
  3. A , B , C A,B,C
  4. D D
  5. Q , R , S Q,R,S
  6. T T
  7. { A , B } T \{A,B\}\in T
  8. { A , B , C } S \{A,B,C\}\in S
  9. { A , B , C , D } Q , R \{A,B,C,D\}\in Q,R
  10. \in
  11. 2 2
  12. A , B A,B
  13. C C
  14. R R
  15. S S
  16. S S
  17. E E
  18. 1 1
  19. S S
  20. H ( X ) := i = 1 N [ p i lg ( 1 p i ) ] H(X):=\sum_{i=1}^{N}\left[p_{i}\cdot\lg\left(\frac{1}{p_{i}}\right)\right]
  21. H ( X ) H(X)
  22. N N
  23. p i p_{i}
  24. i i
  25. ( 1 N ) \left(\frac{1}{N}\right)
  26. H M := H ( X ) lg ( N ) H_{M}:=H(X)\leftarrow\lg(N)
  27. H M H_{M}
  28. d := 1 - H M - H ( X ) H M = H ( X ) H M d:=1-\frac{H_{M}-H(X)}{H_{M}}=\frac{H(X)}{H_{M}}
  29. d d
  30. d d
  31. p f p_{f}
  32. C C
  33. N N
  34. H M lg ( N - C ) H_{M}\leftarrow\lg(N-C)
  35. H ( x ) N - p f ( N - C - 1 ) N lg [ N N - p f ( N - C - 1 ) ] + p f N - C - 1 N lg [ N / p f ] H(x)\leftarrow\frac{N-p_{f}\cdot(N-C-1)}{N}\cdot\lg\left[\frac{N}{N-p_{f}\cdot% (N-C-1)}\right]+p_{f}\cdot\frac{N-C-1}{N}\cdot\lg\left[N/p_{f}\right]
  36. d d
  37. H M H_{M}
  38. H ( X ) lg ( S ) H(X)\leftarrow\lg(S)
  39. S S
  40. H ( X ) lg ( L ) H(X)\leftarrow\lg(L)
  41. L L
  42. S S

Degree_of_polymerization.html

  1. D P n X n = M n M 0 DP_{n}\equiv X_{n}=\frac{M_{n}}{M_{0}}

Degrees_of_freedom_(mechanics).html

  1. M = 6 n = 6 ( N - 1 ) , M=6n=6(N-1),\!
  2. M = 6 n - i = 1 j ( 6 - f i ) = 6 ( N - 1 - j ) + i = 1 j f i M=6n-\sum_{i=1}^{j}\ (6-f_{i})=6(N-1-j)+\sum_{i=1}^{j}\ f_{i}
  3. M = i = 1 j f i M=\sum_{i=1}^{j}\ f_{i}
  4. M = i = 1 j f i - 6 M=\sum_{i=1}^{j}\ f_{i}-6
  5. M = 3 ( N - 1 - j ) + i = 1 j f i , M=3(N-1-j)+\sum_{i=1}^{j}\ f_{i},
  6. M = i = 1 j f i , M=\sum_{i=1}^{j}\ f_{i},
  7. M = i = 1 j f i - 3. M=\sum_{i=1}^{j}\ f_{i}-3.

Degrees_of_freedom_(statistics).html

  1. X 1 , , X n X_{1},\dots,X_{n}
  2. X ¯ n = X 1 + + X n n \overline{X}_{n}={X_{1}+\cdots+X_{n}\over n}
  3. X i - X ¯ n X_{i}-\overline{X}_{n}\,
  4. Y i = a + b x i + e i for i = 1 , , n Y_{i}=a+bx_{i}+e_{i}\,\text{ for }i=1,\dots,n
  5. a ^ \widehat{a}
  6. b ^ \widehat{b}
  7. e i ^ = y i - ( a ^ + b ^ x i ) \widehat{e_{i}}=y_{i}-(\widehat{a}+\widehat{b}x_{i})\,
  8. e 1 ^ + + e n ^ = 0 , \widehat{e_{1}}+\cdots+\widehat{e_{n}}=0,\,
  9. x 1 e 1 ^ + + x n e n ^ = 0. x_{1}\widehat{e_{1}}+\cdots+x_{n}\widehat{e_{n}}=0.\,
  10. X 1 , , X n . X_{1},\dots,X_{n}.\,
  11. ( X 1 X n ) . \begin{pmatrix}X_{1}\\ \vdots\\ X_{n}\end{pmatrix}.
  12. X ¯ \bar{X}
  13. ( X 1 X n ) = X ¯ ( 1 1 ) + ( X 1 - X ¯ X n - X ¯ ) . \begin{pmatrix}X_{1}\\ \vdots\\ X_{n}\end{pmatrix}=\bar{X}\begin{pmatrix}1\\ \vdots\\ 1\end{pmatrix}+\begin{pmatrix}X_{1}-\bar{X}\\ \vdots\\ X_{n}-\bar{X}\end{pmatrix}.
  14. X ¯ \bar{X}
  15. i = 1 n ( X i - X ¯ ) = 0 \sum_{i=1}^{n}(X_{i}-\bar{X})=0
  16. i = 1 n ( X i - X ¯ ) 2 = X 1 - X ¯ X n - X ¯ 2 . \sum_{i=1}^{n}(X_{i}-\bar{X})^{2}=\begin{Vmatrix}X_{1}-\bar{X}\\ \vdots\\ X_{n}-\bar{X}\end{Vmatrix}^{2}.
  17. X i X_{i}
  18. σ 2 \sigma^{2}
  19. σ 2 \sigma^{2}
  20. n ( X ¯ - μ 0 ) i = 1 n ( X i - X ¯ ) 2 / ( n - 1 ) \frac{\sqrt{n}(\bar{X}-\mu_{0})}{\sqrt{\sum\limits_{i=1}^{n}(X_{i}-\bar{X})^{2% }/(n-1)}}
  21. μ 0 \mu_{0}
  22. X 1 , , X n X_{1},\ldots,X_{n}
  23. Y 1 , , Y n Y_{1},\ldots,Y_{n}
  24. Z 1 , , Z n Z_{1},\ldots,Z_{n}
  25. X i = M ¯ + ( X ¯ - M ¯ ) + ( X i - X ¯ ) Y i = M ¯ + ( Y ¯ - M ¯ ) + ( Y i - Y ¯ ) Z i = M ¯ + ( Z ¯ - M ¯ ) + ( Z i - Z ¯ ) \begin{aligned}\displaystyle X_{i}&\displaystyle=\bar{M}+(\bar{X}-\bar{M})+(X_% {i}-\bar{X})\\ \displaystyle Y_{i}&\displaystyle=\bar{M}+(\bar{Y}-\bar{M})+(Y_{i}-\bar{Y})\\ \displaystyle Z_{i}&\displaystyle=\bar{M}+(\bar{Z}-\bar{M})+(Z_{i}-\bar{Z})% \end{aligned}
  26. X ¯ , Y ¯ , Z ¯ \bar{X},\bar{Y},\bar{Z}
  27. M ¯ = ( X ¯ + Y ¯ + Z ¯ ) / 3 \bar{M}=(\bar{X}+\bar{Y}+\bar{Z})/3
  28. ( X 1 X n Y 1 Y n Z 1 Z n ) = M ¯ ( 1 1 1 1 1 1 ) + ( X ¯ - M ¯ X ¯ - M ¯ Y ¯ - M ¯ Y ¯ - M ¯ Z ¯ - M ¯ Z ¯ - M ¯ ) + ( X 1 - X ¯ X n - X ¯ Y 1 - Y ¯ Y n - Y ¯ Z 1 - Z ¯ Z n - Z ¯ ) . \begin{pmatrix}X_{1}\\ \vdots\\ X_{n}\\ Y_{1}\\ \vdots\\ Y_{n}\\ Z_{1}\\ \vdots\\ Z_{n}\end{pmatrix}=\bar{M}\begin{pmatrix}1\\ \vdots\\ 1\\ 1\\ \vdots\\ 1\\ 1\\ \vdots\\ 1\end{pmatrix}+\begin{pmatrix}\bar{X}-\bar{M}\\ \vdots\\ \bar{X}-\bar{M}\\ \bar{Y}-\bar{M}\\ \vdots\\ \bar{Y}-\bar{M}\\ \bar{Z}-\bar{M}\\ \vdots\\ \bar{Z}-\bar{M}\end{pmatrix}+\begin{pmatrix}X_{1}-\bar{X}\\ \vdots\\ X_{n}-\bar{X}\\ Y_{1}-\bar{Y}\\ \vdots\\ Y_{n}-\bar{Y}\\ Z_{1}-\bar{Z}\\ \vdots\\ Z_{n}-\bar{Z}\end{pmatrix}.
  29. X ¯ - M ¯ \bar{X}-\bar{M}
  30. Y ¯ - M ¯ \bar{Y}-\bar{M}
  31. Z ¯ - M ¯ \overline{Z}-\overline{M}
  32. SSTr = n ( X ¯ - M ¯ ) 2 + n ( Y ¯ - M ¯ ) 2 + n ( Z ¯ - M ¯ ) 2 \,\text{SSTr}=n(\bar{X}-\bar{M})^{2}+n(\bar{Y}-\bar{M})^{2}+n(\bar{Z}-\bar{M})% ^{2}
  33. SSE = i = 1 n ( X i - X ¯ ) 2 + i = 1 n ( Y i - Y ¯ ) 2 + i = 1 n ( Z i - Z ¯ ) 2 \,\text{SSE}=\sum_{i=1}^{n}(X_{i}-\bar{X})^{2}+\sum_{i=1}^{n}(Y_{i}-\bar{Y})^{% 2}+\sum_{i=1}^{n}(Z_{i}-\bar{Z})^{2}
  34. X i ; i = 1 , , n X_{i};i=1,\ldots,n
  35. ( μ , σ 2 ) (\mu,\sigma^{2})
  36. i = 1 n ( X i - X ¯ ) 2 σ 2 \frac{\sum\limits_{i=1}^{n}(X_{i}-\bar{X})^{2}}{\sigma^{2}}
  37. { X i - X ¯ } \{X_{i}-\bar{X}\}
  38. y ^ = H y , \hat{y}=Hy,\,
  39. y ^ \hat{y}
  40. || H y || 2 ||Hy||^{2}
  41. || y - H y || 2 ||y-Hy||^{2}
  42. tr ( H ) = i h i i = i y ^ i y i , \mathrm{tr}(H)=\sum_{i}h_{ii}=\sum_{i}\frac{\partial\hat{y}_{i}}{\partial y_{i% }},
  43. r ^ = y - H y \hat{r}=y-Hy
  44. σ ^ 2 = r ^ 2 tr ( ( I - H ) ( I - H ) ) , \hat{\sigma}^{2}=\frac{\|\hat{r}\|^{2}}{\hbox{tr}\left((I-H)^{\prime}(I-H)% \right)},
  45. σ ^ 2 = r ^ 2 n - tr ( 2 H - H H ) = r ^ 2 n - 2 tr ( H ) + tr ( H H ) \hat{\sigma}^{2}=\frac{\|\hat{r}\|^{2}}{n-\mathrm{tr}(2H-HH^{\prime})}=\frac{% \|\hat{r}\|^{2}}{n-2\,\mathrm{tr}(H)+\mathrm{tr}(HH^{\prime})}
  46. σ ^ 2 r ^ 2 n - 1.25 tr ( H ) + 0.5 . \hat{\sigma}^{2}\approx\frac{\|\hat{r}\|^{2}}{n-1.25\,\mathrm{tr}(H)+0.5}.
  47. r ^ 2 \|\hat{r}\|^{2}
  48. r ^ Σ - 1 r ^ \hat{r}^{\prime}\Sigma^{-1}\hat{r}
  49. || y - H y || 2 ||y-Hy||^{2}

Delta_method.html

  1. n [ X n - θ ] 𝐷 𝒩 ( 0 , σ 2 ) , {\sqrt{n}[X_{n}-\theta]\,\xrightarrow{D}\,\mathcal{N}(0,\sigma^{2})},
  2. 𝐷 \xrightarrow{D}
  3. n [ g ( X n ) - g ( θ ) ] 𝐷 𝒩 ( 0 , σ 2 [ g ( θ ) ] 2 ) {\sqrt{n}[g(X_{n})-g(\theta)]\,\xrightarrow{D}\,\mathcal{N}(0,\sigma^{2}[g^{% \prime}(\theta)]^{2})}
  4. g ( θ ) g′(θ)
  5. g ( θ ) g′(θ)
  6. g ( X n ) = g ( θ ) + g ( θ ~ ) ( X n - θ ) , g(X_{n})=g(\theta)+g^{\prime}(\tilde{\theta})(X_{n}-\theta),
  7. θ ~ \tilde{\theta}
  8. X n 𝑃 θ X_{n}\,\xrightarrow{P}\,\theta
  9. X n < θ ~ < θ X_{n}<\tilde{\theta}<\theta
  10. θ ~ 𝑃 θ \tilde{\theta}\,\xrightarrow{P}\,\theta
  11. g ( θ ) g′(θ)
  12. g ( θ ~ ) 𝑃 g ( θ ) , g^{\prime}(\tilde{\theta})\,\xrightarrow{P}\,g^{\prime}(\theta),
  13. 𝑃 \xrightarrow{P}
  14. n \sqrt{n}
  15. n [ g ( X n ) - g ( θ ) ] = g ( θ ~ ) n [ X n - θ ] . \sqrt{n}[g(X_{n})-g(\theta)]=g^{\prime}\left(\tilde{\theta}\right)\sqrt{n}[X_{% n}-\theta].
  16. n [ X n - θ ] 𝐷 𝒩 ( 0 , σ 2 ) {\sqrt{n}[X_{n}-\theta]\xrightarrow{D}\mathcal{N}(0,\sigma^{2})}
  17. n [ g ( X n ) - g ( θ ) ] 𝐷 𝒩 ( 0 , σ 2 [ g ( θ ) ] 2 ) . {\sqrt{n}[g(X_{n})-g(\theta)]\xrightarrow{D}\mathcal{N}(0,\sigma^{2}[g^{\prime% }(\theta)]^{2})}.
  18. n [ g ( X n ) - g ( θ ) ] \displaystyle\sqrt{n}[g(X_{n})-g(\theta)]
  19. n ( B - β ) 𝐷 N ( 0 , Σ ) , \sqrt{n}\left(B-\beta\right)\,\xrightarrow{D}\,N\left(0,\Sigma\right),
  20. h ( B ) h ( β ) + h ( β ) T ( B - β ) h(B)\approx h(\beta)+\nabla h(\beta)^{T}\cdot(B-\beta)
  21. Var ( h ( B ) ) Var ( h ( β ) + h ( β ) T ( B - β ) ) = Var ( h ( β ) + h ( β ) T B - h ( β ) T β ) = Var ( h ( β ) T B ) = h ( β ) T Cov ( B ) h ( β ) = h ( β ) T ( Σ / n ) h ( β ) \begin{aligned}\displaystyle\operatorname{Var}\left(h(B)\right)&\displaystyle% \approx\operatorname{Var}\left(h(\beta)+\nabla h(\beta)^{T}\cdot(B-\beta)% \right)\\ &\displaystyle=\operatorname{Var}\left(h(\beta)+\nabla h(\beta)^{T}\cdot B-% \nabla h(\beta)^{T}\cdot\beta\right)\\ &\displaystyle=\operatorname{Var}\left(\nabla h(\beta)^{T}\cdot B\right)\\ &\displaystyle=\nabla h(\beta)^{T}\cdot\operatorname{Cov}(B)\cdot\nabla h(% \beta)\\ &\displaystyle=\nabla h(\beta)^{T}\cdot(\Sigma/n)\cdot\nabla h(\beta)\end{aligned}
  22. n ( h ( B ) - h ( β ) ) 𝐷 N ( 0 , h ( β ) T Σ h ( β ) ) \sqrt{n}\left(h(B)-h(\beta)\right)\,\xrightarrow{D}\,N\left(0,\nabla h(\beta)^% {T}\cdot\Sigma\cdot\nabla h(\beta)\right)
  23. n ( h ( B ) - h ( β ) ) 𝐷 N ( 0 , σ 2 ( h ( β ) ) 2 ) . \sqrt{n}\left(h(B)-h(\beta)\right)\,\xrightarrow{D}\,N\left(0,\sigma^{2}\cdot% \left(h^{\prime}(\beta)\right)^{2}\right).
  24. n [ X n n - p ] 𝐷 N ( 0 , p ( 1 - p ) ) , {\sqrt{n}\left[\frac{X_{n}}{n}-p\right]\,\xrightarrow{D}\,N(0,p(1-p))},
  25. g ( θ ) = l o g ( θ ) g(θ)=log(θ)
  26. n [ log ( X n n ) - log ( p ) ] 𝐷 N ( 0 , p ( 1 - p ) [ 1 / p ] 2 ) {\sqrt{n}\left[\log\left(\frac{X_{n}}{n}\right)-\log(p)\right]\,\xrightarrow{D% }\,N(0,p(1-p)[1/p]^{2})}
  27. log ( X n n ) \log\left(\frac{X_{n}}{n}\right)
  28. 1 - p p n . \frac{1-p}{p\,n}.
  29. p ^ \hat{p}
  30. q ^ \hat{q}
  31. p ^ q ^ \frac{\hat{p}}{\hat{q}}
  32. 1 - p ^ p ^ n + 1 - q ^ q ^ m . \frac{1-\hat{p}}{\hat{p}\,n}+\frac{1-\hat{q}}{\hat{q}\,m}.
  33. Var ( h r ) = \displaystyle\operatorname{Var}\left(h_{r}\right)=
  34. h < s u b > r h<sub>r

Delta_neutral.html

  1. Δ \Delta
  2. V 0 V_{0}
  3. V V
  4. S 0 S_{0}
  5. V S \tfrac{\partial V}{\partial S}
  6. C ( s ) C(s)\,
  7. ( ϵ ) (\epsilon\,)
  8. C ( s + ϵ ) = C ( s ) + ϵ C ( s ) + 1 / 2 ϵ 2 C ′′ ( s ) + C(s+\epsilon\,)=C(s)+\epsilon\,C^{\prime}(s)+{1/2}\,\epsilon^{2}\,C^{\prime% \prime}(s)+...
  9. C ( s ) = Δ C^{\prime}(s)=\Delta\,
  10. C ′′ ( s ) = Γ C^{\prime\prime}(s)=\Gamma\,
  11. Δ \Delta\,
  12. Γ \Gamma\,

Dense_graph.html

  1. D = 2 | E | | V | ( | V | - 1 ) D=\frac{2|E|}{|V|\,(|V|-1)}
  2. D = | E | | V | ( | V | - 1 ) D=\frac{|E|}{|V|\,(|V|-1)}

Dense_plasma_focus.html

  1. I a p \frac{I}{a\sqrt{p}}
  2. I I
  3. a a
  4. p p

Dependence_relation.html

  1. X X
  2. \triangleleft
  3. a a
  4. X X
  5. S S
  6. X X
  7. a S a\triangleleft S
  8. a S a\in S
  9. a S a\triangleleft S
  10. a S a\triangleleft S
  11. S 0 S_{0}
  12. S S
  13. a S 0 a\triangleleft S_{0}
  14. T T
  15. X X
  16. b S b\in S
  17. b T b\triangleleft T
  18. a S a\triangleleft S
  19. a T a\triangleleft T
  20. a S a\triangleleft S
  21. a S - { b } a\not\!\triangleleft S-\{b\}
  22. b S b\in S
  23. b ( S - { b } ) { a } b\triangleleft(S-\{b\})\cup\{a\}
  24. \triangleleft
  25. X X
  26. S S
  27. X X
  28. a S - { a } a\not\!\triangleleft S-\{a\}
  29. a S . a\in S.
  30. S T S\subseteq T
  31. S S
  32. T T
  33. t S t\triangleleft S
  34. t T . t\in T.
  35. S S
  36. X X
  37. S S
  38. S S
  39. X . X.
  40. X X
  41. \triangleleft
  42. X X
  43. . \triangleleft.
  44. X X
  45. V V
  46. F . F.
  47. \triangleleft
  48. υ S \upsilon\triangleleft S
  49. υ \upsilon
  50. S S
  51. K K
  52. F . F.
  53. \triangleleft
  54. α S \alpha\triangleleft S
  55. α \alpha
  56. F ( S ) . F(S).
  57. \triangleleft

Depth_of_focus.html

  1. t = 2 N c v f , t=2Nc\frac{v}{f}\,,
  2. t = 2 N c ( 1 + m ) . t=2Nc\left(1+m\right)\,.
  3. t 2 N c . t\approx 2Nc\,.
  4. Δ f \Delta f
  5. ± λ / 4 \pm\lambda/4
  6. Δ f = ± 2 λ N 2 \Delta f=\pm 2\lambda N^{2}

Descartes'_rule_of_signs.html

  1. f ( x ) = a x 3 + b x 2 + c x + d f(x)=ax^{3}+bx^{2}+cx+d
  2. - x -x
  3. f ( - x ) = a ( - x ) 3 + b ( - x ) 2 + c ( - x ) + d = - a x 3 + b x 2 - c x + d g ( x ) . f(-x)=a(-x)^{3}+b(-x)^{2}+c(-x)+d=-ax^{3}+bx^{2}-cx+d\equiv g(x).
  4. g ( x ) g(x)
  5. x i x_{i}
  6. g ( x ) = f ( - x ) g(x)=f(-x)
  7. ( - x i ) (-x_{i})
  8. x i x_{i}
  9. f ( x ) = + x 3 + x 2 - x - 1 f(x)=+x^{3}+x^{2}-x-1\,
  10. f ( - x ) f(-x)
  11. f ( - x ) = - x 3 + x 2 + x - 1 f(-x)=-x^{3}+x^{2}+x-1\,
  12. f ( x ) = ( x + 1 ) 2 ( x - 1 ) , f(x)=(x+1)^{2}(x-1),\,
  13. f ( - x ) = - ( x - 1 ) 2 ( x + 1 ) , f(-x)=-(x-1)^{2}(x+1),\,
  14. n - ( p + q ) , n-(p+q),\,
  15. f ( x ) = x 3 - 1 , f(x)=x^{3}-1\,,
  16. f ( - x ) = - x 3 - 1 , f(-x)=-x^{3}-1\,,
  17. 3 - ( 1 + 0 ) = 2 . 3-(1+0)=2\,.

Detailed_balance.html

  1. A 1 A 2 A n A 1 A_{1}\to A_{2}\to\cdots\to A_{n}\to A_{1}
  2. i α i A i j β j B j \sum_{i}\alpha_{i}A_{i}\to\sum_{j}\beta_{j}B_{j}
  3. j β j B j i α i A i \sum_{j}\beta_{j}B_{j}\to\sum_{i}\alpha_{i}A_{i}
  4. A i , B j A_{i},B_{j}
  5. α i , β j 0 \alpha_{i},\beta_{j}\geq 0
  6. A i A_{i}
  7. A v + A w A v + A w , A_{v}+A_{w}\to A_{v^{\prime}}+A_{w^{\prime}},
  8. A v A_{v}
  9. A v A_{v}
  10. A - v A_{-v}
  11. π i P i j = π j P j i , \pi_{i}P_{ij}=\pi_{j}P_{ji}\,,
  12. π ( s ) P ( s , s ) = π ( s ) P ( s , s ) . \pi(s^{\prime})P(s^{\prime},s)=\pi(s)P(s,s^{\prime})\,.
  13. P ( a , b ) P ( b , c ) P ( c , a ) = P ( a , c ) P ( c , b ) P ( b , a ) . P(a,b)P(b,c)P(c,a)=P(a,c)P(c,b)P(b,a)\,.
  14. A 1 A 2 A 3 A 1 A_{1}\to A_{2}\to A_{3}\to A_{1}
  15. i α r i A i j β r j A j ( r = 1 , , m ) , \sum_{i}\alpha_{ri}A_{i}\to\sum_{j}\beta_{rj}A_{j}\;\;(r=1,\ldots,m)\,,
  16. A i A_{i}
  17. α r i , β r j 0 \alpha_{ri},\beta_{rj}\geq 0
  18. s y m b o l Γ = ( γ r i ) symbol{\Gamma}=(\gamma_{ri})
  19. γ r i = β r i - α r i \gamma_{ri}=\beta_{ri}-\alpha_{ri}
  20. γ r \gamma_{r}
  21. s y m b o l Γ symbol{\Gamma}
  22. γ r i = β r i - α r i \gamma_{ri}=\beta_{ri}-\alpha_{ri}
  23. w r = k r i = 1 n a i α r i , w_{r}=k_{r}\prod_{i=1}^{n}a_{i}^{\alpha_{ri}}\,,
  24. a i 0 a_{i}\geq 0
  25. A i A_{i}
  26. k r > 0 k_{r}>0
  27. k r + = k r k_{r}^{+}=k_{r}
  28. w r + = w r w_{r}^{+}=w_{r}
  29. k r - k_{r}^{-}
  30. w r - w_{r}^{-}
  31. K r = k r + / k r - K_{r}=k_{r}^{+}/k_{r}^{-}
  32. k r k_{r}
  33. a i eq > 0 a_{i}^{\rm eq}>0
  34. w r + = w r - w_{r}^{+}=w_{r}^{-}
  35. i γ r i x i = ln k r + - ln k r - = ln K r \sum_{i}\gamma_{ri}x_{i}=\ln k_{r}^{+}-\ln k_{r}^{-}=\ln K_{r}
  36. x i = ln a i eq x_{i}=\ln a_{i}^{\rm eq}
  37. a i eq > 0 a_{i}^{\rm eq}>0
  38. k r + > 0 k_{r}^{+}>0
  39. k r - > 0 k_{r}^{-}>0
  40. k r - > 0 k_{r}^{-}>0
  41. k r + > 0 k_{r}^{+}>0
  42. s y m b o l λ = ( λ r ) symbol{\lambda}=(\lambda_{r})
  43. s y m b o l λ Γ = 0 ( i.e. r λ r γ r i = 0 for all i ) symbol{\lambda\Gamma}=0\;\;\left(\mbox{i.e.}~{}\;\;\sum_{r}\lambda_{r}\gamma_{% ri}=0\;\;\mbox{for all}~{}\;\;i\right)
  44. r = 1 m ( k r + ) λ r = r = 1 m ( k r - ) λ r . \prod_{r=1}^{m}(k_{r}^{+})^{\lambda_{r}}=\prod_{r=1}^{m}(k_{r}^{-})^{\lambda_{% r}}\,.
  45. s y m b o l λ Γ = 0 symbol{\lambda\Gamma}=0
  46. A 1 A 2 A_{1}\rightleftharpoons A_{2}
  47. A 2 A 3 A_{2}\rightleftharpoons A_{3}
  48. A 3 A 1 A_{3}\rightleftharpoons A_{1}
  49. A 1 + A 2 2 A 3 A_{1}+A_{2}\rightleftharpoons 2A_{3}
  50. k 1 + k 2 + k 3 + = k 1 - k 2 - k 3 - k_{1}^{+}k_{2}^{+}k_{3}^{+}=k_{1}^{-}k_{2}^{-}k_{3}^{-}
  51. k 3 + k 4 + / k 2 + = k 3 - k 4 - / k 2 - k_{3}^{+}k_{4}^{+}/k_{2}^{+}=k_{3}^{-}k_{4}^{-}/k_{2}^{-}
  52. γ 1 + γ 2 + γ 3 = 0 \gamma_{1}+\gamma_{2}+\gamma_{3}=0
  53. γ 3 + γ 4 - γ 2 = 0 \gamma_{3}+\gamma_{4}-\gamma_{2}=0
  54. a i = exp ( μ i - μ i R T ) a_{i}=\exp\left(\frac{\mu_{i}-\mu^{\ominus}_{i}}{RT}\right)
  55. μ i = R T ln c i + μ i \mu_{i}=RT\ln c_{i}+\mu^{\ominus}_{i}
  56. a j = c j a_{j}=c_{j}
  57. F = R T i N i ( ln ( N i V ) - 1 + μ i ( T ) R T ) F=RT\sum_{i}N_{i}\left(\ln\left(\frac{N_{i}}{V}\right)-1+\frac{\mu^{\ominus}_{% i}(T)}{RT}\right)
  58. μ i = F ( T , V , N ) / N j \mu_{i}=\partial F(T,V,N)/\partial N_{j}
  59. d N i d t = V r γ r i ( w r + - w r - ) . \frac{dN_{i}}{dt}=V\sum_{r}\gamma_{ri}(w^{+}_{r}-w^{-}_{r}).
  60. w r + ( c eq , T ) = w r - ( c eq , T ) = w r eq w^{+}_{r}(c^{\rm eq},T)=w^{-}_{r}(c^{\rm eq},T)=w^{\rm eq}_{r}
  61. w r + = w r eq exp ( i α r i ( μ i - μ i eq ) R T ) ; w r - = w r eq exp ( i β r i ( μ i - μ i eq ) R T ) ; w^{+}_{r}=w^{\rm eq}_{r}\exp\left(\sum_{i}\frac{\alpha_{ri}(\mu_{i}-\mu^{\rm eq% }_{i})}{RT}\right);\;\;w^{-}_{r}=w^{\rm eq}_{r}\exp\left(\sum_{i}\frac{\beta_{% ri}(\mu_{i}-\mu^{\rm eq}_{i})}{RT}\right);
  62. μ i eq = μ i ( c eq , T ) \mu^{\rm eq}_{i}=\mu_{i}(c^{\rm eq},T)
  63. d F d t = i F ( T , V , N ) N i d N i d t = i μ i d N i d t = - V R T r ( ln w r + - ln w r - ) ( w r + - w r - ) 0 \frac{dF}{dt}=\sum_{i}\frac{\partial F(T,V,N)}{\partial N_{i}}\frac{dN_{i}}{dt% }=\sum_{i}\mu_{i}\frac{dN_{i}}{dt}=-VRT\sum_{r}(\ln w_{r}^{+}-\ln w_{r}^{-})(w% _{r}^{+}-w_{r}^{-})\leq 0
  64. ln w r + - ln w r - \ln w_{r}^{+}-\ln w_{r}^{-}
  65. w r + - w r - w_{r}^{+}-w_{r}^{-}
  66. w r + = w r eq ( 1 + i α r i ( μ i - μ i eq ) R T ) ; w r - = w r eq ( 1 + i β r i ( μ i - μ i eq ) R T ) ; w^{+}_{r}=w^{\rm eq}_{r}\left(1+\sum_{i}\frac{\alpha_{ri}(\mu_{i}-\mu^{\rm eq}% _{i})}{RT}\right);\;\;w^{-}_{r}=w^{\rm eq}_{r}\left(1+\sum_{i}\frac{\beta_{ri}% (\mu_{i}-\mu^{\rm eq}_{i})}{RT}\right);
  67. γ r i = β r i - α r i \gamma_{ri}=\beta_{ri}-\alpha_{ri}
  68. d N i d t = - V j [ r w r eq γ r i γ r j ] μ j - μ j eq R T . \frac{dN_{i}}{dt}=-V\sum_{j}\left[\sum_{r}w^{\rm eq}_{r}\gamma_{ri}\gamma_{rj}% \right]\frac{\mu_{j}-\mu^{\rm eq}_{j}}{RT}.
  69. X j X_{j}
  70. L i j L_{ij}
  71. X j = μ j - μ j eq T ; d N i d t = j L i j X j X_{j}=\frac{\mu_{j}-\mu^{\rm eq}_{j}}{T};\;\;\frac{dN_{i}}{dt}=\sum_{j}L_{ij}X% _{j}
  72. L i j L_{ij}
  73. L i j = - V R r w r eq γ r i γ r j L_{ij}=-\frac{V}{R}\sum_{r}w^{\rm eq}_{r}\gamma_{ri}\gamma_{rj}
  74. L i j = L j i L_{ij}=L_{ji}
  75. L L
  76. γ r \gamma_{r}
  77. d N i d t = V r γ r i w r = V r ( β r i - α r i ) w r \frac{dN_{i}}{dt}=V\sum_{r}\gamma_{ri}w_{r}=V\sum_{r}(\beta_{ri}-\alpha_{ri})w% _{r}
  78. α r = α r i \alpha_{r}=\alpha_{ri}
  79. β r = β r i \beta_{r}=\beta_{ri}
  80. Y Y
  81. α r , β r \alpha_{r},\beta_{r}
  82. ν Y \nu\in Y
  83. R ν + = { r | α r = ν } ; R ν - = { r | β r = ν } R_{\nu}^{+}=\{r|\alpha_{r}=\nu\};\;\;\;R_{\nu}^{-}=\{r|\beta_{r}=\nu\}
  84. r R ν + r\in R_{\nu}^{+}
  85. ν \nu
  86. α r \alpha_{r}
  87. r R ν - r\in R_{\nu}^{-}
  88. ν \nu
  89. β r \beta_{r}
  90. ν Y \nu\in Y
  91. r R ν - w r = r R ν + w r \sum_{r\in R_{\nu}^{-}}w_{r}=\sum_{r\in R_{\nu}^{+}}w_{r}
  92. d N d t = V r γ r w r = 0 \frac{dN}{dt}=V\sum_{r}\gamma_{r}w_{r}=0
  93. d F / d t 0 dF/dt\geq 0
  94. i α r i A i i β r i A i \sum_{i}\alpha_{ri}A_{i}\to\sum_{i}\beta_{ri}A_{i}
  95. w r = φ r exp ( i α r i μ i R T ) w_{r}=\varphi_{r}\exp\left(\sum_{i}\frac{\alpha_{ri}\mu_{i}}{RT}\right)
  96. μ i = F ( T , V , N ) / N i \mu_{i}=\partial F(T,V,N)/\partial N_{i}
  97. F ( T , V , N ) F(T,V,N)
  98. φ r 0 \varphi_{r}\geq 0
  99. d N i d t = V r γ r i w r \frac{dN_{i}}{dt}=V\sum_{r}\gamma_{ri}w_{r}
  100. θ ( λ ) \theta(\lambda)
  101. λ [ 0 , 1 ] \lambda\in[0,1]
  102. θ ( λ ) = r φ r exp ( i ( λ α r i + ( 1 - λ ) β r i ) ) μ i R T ) \theta(\lambda)=\sum_{r}\varphi_{r}\exp\left(\sum_{i}\frac{(\lambda\alpha_{ri}% +(1-\lambda)\beta_{ri}))\mu_{i}}{RT}\right)
  103. θ ( λ ) \theta(\lambda)
  104. α ~ ρ ( λ ) = λ α ρ + ( 1 - λ ) β ρ \tilde{\alpha}_{\rho}(\lambda)=\lambda\alpha_{\rho}+(1-\lambda)\beta_{\rho}
  105. λ = 1 \lambda=1
  106. θ ( λ ) \theta(\lambda)
  107. θ ′′ ( λ ) 0 \theta^{\prime\prime}(\lambda)\geq 0
  108. d F d t = - V R T d θ ( λ ) d λ | λ = 1 \frac{dF}{dt}=-VRT\left.\frac{d\theta(\lambda)}{d\lambda}\right|_{\lambda=1}
  109. θ ( λ ) \theta(\lambda)
  110. d F d t < 0 if and only if θ ( λ ) < θ ( 1 ) for some λ < 1 ; d F d t 0 if and only if θ ( λ ) θ ( 1 ) for some λ < 1 \frac{dF}{dt}<0\mbox{ if and only if }~{}\theta(\lambda)<\theta(1)\mbox{ for % some }~{}\lambda<1;\;\;\;\frac{dF}{dt}\leq 0\mbox{ if and only if }~{}\theta(% \lambda)\leq\theta(1)\mbox{ for some }~{}\lambda<1
  111. θ ( 0 ) θ ( 1 ) \theta(0)\equiv\theta(1)
  112. d F / d t 0 {dF}/{dt}\leq 0
  113. A 1 A 2 A 3 A 1 A_{1}\to A_{2}\to A_{3}\to A_{1}
  114. A 1 A 2 A 3 A 1 A_{1}\to A_{2}\to A_{3}\leftarrow A_{1}

Determinacy.html

  1. a 0 , a 1 , a 2 , A \langle a_{0},a_{1},a_{2},\ldots\rangle\in A
  2. σ ( ) , a 1 , σ ( σ ( ) , a 1 ) , a 3 , \langle\sigma(\langle\rangle),a_{1},\sigma(\langle\sigma(\langle\rangle),a_{1}% \rangle),a_{3},\ldots\rangle
  3. 𝚷 3 0 \mathbf{\Pi}^{0}_{3}
  4. Π 3 0 \Pi^{0}_{3}
  5. n n
  6. Σ 1 1 ( r ) \Sigma^{1}_{1}(r)
  7. Σ 1 1 ( r ) \Sigma^{1}_{1}(r)
  8. Σ 1 1 ( r ) \Sigma^{1}_{1}(r)
  9. ( ( x 0 , y 0 ) , ( x 1 , y 1 ) , ) ((x_{0},y_{0}),(x_{1},y_{1}),...)
  10. T s T_{s}
  11. T s T_{s}
  12. T s T_{s}
  13. T s T_{s}
  14. T s T_{s}
  15. Δ 2 1 \Delta^{1}_{2}
  16. Δ 2 1 \Delta^{1}_{2}
  17. ω 2 L [ x ] \omega_{2}^{L[x]}

Developmental_disability.html

  1. D Q = D e v e l o p m e n t a l a g e C h r o n o l o g i c a l a g e * 100 DQ=\frac{Developmental\ age}{Chronological\ age}*100

Deviance_information_criterion.html

  1. D ( θ ) = - 2 log ( p ( y | θ ) ) + C D(\theta)=-2\log(p(y|\theta))+C\,
  2. y y\,
  3. θ \theta\,
  4. p ( y | θ ) p(y|\theta)\,
  5. C C\,
  6. D ¯ = 𝐄 θ [ D ( θ ) ] \bar{D}=\mathbf{E}^{\theta}[D(\theta)]
  7. p D = D ¯ - D ( θ ¯ ) p_{D}=\bar{D}-D(\bar{\theta})
  8. θ ¯ \bar{\theta}
  9. θ \theta\,
  10. p D = p V = 1 2 var ^ ( D ( θ ) ) p_{D}=p_{V}=\frac{1}{2}\widehat{\operatorname{var}}\left(D(\theta)\right)
  11. 𝐷𝐼𝐶 = p D + D ¯ , \mathit{DIC}=p_{D}+\bar{D},
  12. 𝐷𝐼𝐶 = D ( θ ¯ ) + 2 p D . \mathit{DIC}=D(\bar{\theta})+2p_{D}.
  13. D ¯ \bar{D}
  14. p D p_{D}\,
  15. D ¯ \bar{D}
  16. p D p_{D}\,
  17. θ \theta\,
  18. D ¯ \bar{D}
  19. D ( θ ) D(\theta)\,
  20. θ \theta\,
  21. D ( θ ¯ ) D(\bar{\theta})
  22. D D\,
  23. θ \theta\,
  24. 𝐼𝐶 = D ¯ + 2 p D = - 2 𝐄 θ [ log ( p ( y | θ ) ) ] + 2 p D . \mathit{IC}=\bar{D}+2p_{D}=-2\mathbf{E}^{\theta}[\log(p(y|\theta))]+2p_{D}.

Diabatic.html

  1. 𝐫 \mathbf{r}
  2. 𝐑 \mathbf{R}
  3. E 1 ( 𝐑 ) E 2 ( 𝐑 ) E_{1}(\mathbf{R})\approx E_{2}(\mathbf{R})
  4. χ 1 ( 𝐫 ; 𝐑 ) \chi_{1}(\mathbf{r};\mathbf{R})\,
  5. χ 2 ( 𝐫 ; 𝐑 ) \chi_{2}(\mathbf{r};\mathbf{R})\,
  6. T n = A α = x , y , z P A α P A α 2 M A with P A α = - i A α - i R A α . T_{\mathrm{n}}=\sum_{A}\sum_{\alpha=x,y,z}\frac{P_{A\alpha}P_{A\alpha}}{2M_{A}% }\quad\mathrm{with}\quad P_{A\alpha}=-i\nabla_{A\alpha}\equiv-i\frac{\partial% \quad}{\partial R_{A\alpha}}.
  7. T n T_{\textrm{n}}
  8. T n ( 𝐑 ) k k χ k | T n | χ k ( 𝐫 ) = δ k k T n + A , α 1 M A χ k | ( P A α χ k ) ( 𝐫 ) P A α + χ k | ( T n χ k ) ( 𝐫 ) . \mathrm{T_{n}}(\mathbf{R})_{k^{\prime}k}\equiv\langle\chi_{k^{\prime}}|T_{n}|% \chi_{k}\rangle_{(\mathbf{r})}=\delta_{k^{\prime}k}T_{\textrm{n}}+\sum_{A,% \alpha}\frac{1}{M_{A}}\langle\chi_{k^{\prime}}|\big(P_{A\alpha}\chi_{k}\big)% \rangle_{(\mathbf{r})}P_{A\alpha}+\langle\chi_{k^{\prime}}|\big(T_{\mathrm{n}}% \chi_{k}\big)\rangle_{(\mathbf{r})}.
  9. ( 𝐫 ) {(\mathbf{r})}
  10. T n ( 𝐑 ) k p = T n ( 𝐑 ) p k \mathrm{T_{n}}(\mathbf{R})_{kp}=\mathrm{T_{n}}(\mathbf{R})_{pk}
  11. Ψ ( 𝐫 , 𝐑 ) = χ 1 ( 𝐫 ; 𝐑 ) Φ 1 ( 𝐑 ) + χ 2 ( 𝐫 ; 𝐑 ) Φ 2 ( 𝐑 ) , \Psi(\mathbf{r},\mathbf{R})=\chi_{1}(\mathbf{r};\mathbf{R})\Phi_{1}(\mathbf{R}% )+\chi_{2}(\mathbf{r};\mathbf{R})\Phi_{2}(\mathbf{R}),
  12. ( E 1 ( 𝐑 ) + T n ( 𝐑 ) 11 T n ( 𝐑 ) 12 T n ( 𝐑 ) 21 E 2 ( 𝐑 ) + T n ( 𝐑 ) 22 ) s y m b o l Φ ( 𝐑 ) = E s y m b o l Φ ( 𝐑 ) with s y m b o l Φ ( 𝐑 ) ( Φ 1 ( 𝐑 ) Φ 2 ( 𝐑 ) ) . \begin{pmatrix}E_{1}(\mathbf{R})+\mathrm{T_{n}}(\mathbf{R})_{11}&\mathrm{T_{n}% }(\mathbf{R})_{12}\\ \mathrm{T_{n}}(\mathbf{R})_{21}&E_{2}(\mathbf{R})+\mathrm{T_{n}}(\mathbf{R})_{% 22}\\ \end{pmatrix}symbol{\Phi}(\mathbf{R})=E\,symbol{\Phi}(\mathbf{R})\quad\mathrm{% with}\quad symbol{\Phi}(\mathbf{R})\equiv\begin{pmatrix}\Phi_{1}(\mathbf{R})\\ \Phi_{2}(\mathbf{R})\\ \end{pmatrix}.
  13. χ 1 \chi_{1}\,
  14. χ 2 \chi_{2}\,
  15. ( φ 1 ( 𝐫 ; 𝐑 ) φ 2 ( 𝐫 ; 𝐑 ) ) = ( cos γ ( 𝐑 ) sin γ ( 𝐑 ) - sin γ ( 𝐑 ) cos γ ( 𝐑 ) ) ( χ 1 ( 𝐫 ; 𝐑 ) χ 2 ( 𝐫 ; 𝐑 ) ) \begin{pmatrix}\varphi_{1}(\mathbf{r};\mathbf{R})\\ \varphi_{2}(\mathbf{r};\mathbf{R})\\ \end{pmatrix}=\begin{pmatrix}\cos\gamma(\mathbf{R})&\sin\gamma(\mathbf{R})\\ -\sin\gamma(\mathbf{R})&\cos\gamma(\mathbf{R})\\ \end{pmatrix}\begin{pmatrix}\chi_{1}(\mathbf{r};\mathbf{R})\\ \chi_{2}(\mathbf{r};\mathbf{R})\\ \end{pmatrix}
  16. γ ( 𝐑 ) \gamma(\mathbf{R})
  17. χ k | ( P A α χ k ) ( 𝐫 ) \langle\chi_{k^{\prime}}|\big(P_{A\alpha}\chi_{k}\big)\rangle_{(\mathbf{r})}
  18. k , k = 1 , 2 k^{\prime},k=1,2
  19. φ k | ( P A α φ k ) ( 𝐫 ) = 0 for k = 1 , 2. \langle{\varphi_{k}}|\big(P_{A\alpha}\varphi_{k}\big)\rangle_{(\mathbf{r})}=0% \quad\textrm{for}\quad k=1,\,2.
  20. φ k \varphi_{k}
  21. P A α P_{A\alpha}\,
  22. φ 2 | ( P A α φ 1 ) ( 𝐫 ) = ( P A α γ ( 𝐑 ) ) + χ 2 | ( P A α χ 1 ) ( 𝐫 ) . \langle{\varphi_{2}}|\big(P_{A\alpha}\varphi_{1}\big)\rangle_{(\mathbf{r})}=% \big(P_{A\alpha}\gamma(\mathbf{R})\big)+\langle\chi_{2}|\big(P_{A\alpha}\chi_{% 1}\big)\rangle_{(\mathbf{r})}.
  23. γ ( 𝐑 ) \gamma(\mathbf{R})
  24. ( P A α γ ( 𝐑 ) ) + χ 2 | ( P A α χ 1 ) ( 𝐫 ) = 0 \big(P_{A\alpha}\gamma(\mathbf{R})\big)+\langle\chi_{2}|\big(P_{A\alpha}\chi_{% 1}\big)\rangle_{(\mathbf{r})}=0
  25. φ 1 \varphi_{1}
  26. φ 2 \varphi_{2}
  27. φ 1 \varphi_{1}
  28. φ 2 \varphi_{2}
  29. γ ( 𝐑 ) \gamma(\mathbf{R})
  30. F A α ( 𝐑 ) = - A α V ( 𝐑 ) with V ( 𝐑 ) γ ( 𝐑 ) and F A α ( 𝐑 ) χ 2 | ( i P A α χ 1 ) ( 𝐫 ) . F_{A\alpha}(\mathbf{R})=-\nabla_{A\alpha}V(\mathbf{R})\qquad\mathrm{with}\;\;V% (\mathbf{R})\equiv\gamma(\mathbf{R})\;\;\mathrm{and}\;\;F_{A\alpha}(\mathbf{R}% )\equiv\langle\chi_{2}|\big(iP_{A\alpha}\chi_{1}\big)\rangle_{(\mathbf{r})}.
  31. F A α ( 𝐑 ) F_{A\alpha}(\mathbf{R})
  32. A α F B β ( 𝐑 ) - B β F A α ( 𝐑 ) = 0. \nabla_{A\alpha}F_{B\beta}(\mathbf{R})-\nabla_{B\beta}F_{A\alpha}(\mathbf{R})=0.
  33. γ ( 𝐑 ) \gamma(\mathbf{R})
  34. φ k | T n | φ k ( 𝐫 ) = δ k k T n . \langle\varphi_{k^{\prime}}|T_{n}|\varphi_{k}\rangle_{(\mathbf{r})}=\delta_{k^% {\prime}k}T_{n}.
  35. ( T n + E 1 ( 𝐑 ) + E 2 ( 𝐑 ) 2 0 0 T n + E 1 ( 𝐑 ) + E 2 ( 𝐑 ) 2 ) s y m b o l Φ ~ ( 𝐑 ) + E 2 ( 𝐑 ) - E 1 ( 𝐑 ) 2 ( cos 2 γ sin 2 γ sin 2 γ - cos 2 γ ) s y m b o l Φ ~ ( 𝐑 ) = E s y m b o l Φ ~ ( 𝐑 ) . \begin{pmatrix}T_{\mathrm{n}}+\frac{E_{1}(\mathbf{R})+E_{2}(\mathbf{R})}{2}&0% \\ 0&T_{\mathrm{n}}+\frac{E_{1}(\mathbf{R})+E_{2}(\mathbf{R})}{2}\end{pmatrix}% \tilde{symbol{\Phi}}(\mathbf{R})+\tfrac{E_{2}(\mathbf{R})-E_{1}(\mathbf{R})}{2% }\begin{pmatrix}\cos 2\gamma&\sin 2\gamma\\ \sin 2\gamma&-\cos 2\gamma\end{pmatrix}\tilde{symbol{\Phi}}(\mathbf{R})=E% \tilde{symbol{\Phi}}(\mathbf{R}).
  36. E 1 ( 𝐑 ) E_{1}(\mathbf{R})
  37. E 2 ( 𝐑 ) E_{2}(\mathbf{R})
  38. T n T_{\mathrm{n}}\,
  39. γ ( 𝐑 ) \gamma(\mathbf{R})
  40. γ ( 𝐑 ) \gamma(\mathbf{R})
  41. Ψ ( 𝐫 , 𝐑 ) = φ 1 ( 𝐫 ; 𝐑 ) Φ ~ 1 ( 𝐑 ) + φ 2 ( 𝐫 ; 𝐑 ) Φ ~ 2 ( 𝐑 ) . \Psi(\mathbf{r},\mathbf{R})=\varphi_{1}(\mathbf{r};\mathbf{R})\tilde{\Phi}_{1}% (\mathbf{R})+\varphi_{2}(\mathbf{r};\mathbf{R})\tilde{\Phi}_{2}(\mathbf{R}).
  42. R A α \mathrm{R_{A\alpha}}
  43. R A α \mathrm{R_{A\alpha}}
  44. R B β \mathrm{R_{B\beta}}
  45. γ \gamma
  46. γ \gamma
  47. 𝐀 \mathbf{A}
  48. 𝐀 + 𝐅𝐀 = 𝟎 \nabla\mathbf{A+FA=0}
  49. 𝐅 \mathbf{F}
  50. 𝐅 j k = χ j χ k ; j , k = 1 , 2 , , M \ \mathbf{F}_{jk}=\langle\chi_{j}\mid\nabla\chi_{k}\rangle;\qquad j,k=1,2,% \ldots,M
  51. \nabla
  52. = { q 1 , q 2 , , q N } \nabla=\left\{\frac{\partial\quad}{\partial q_{1}},\quad\frac{\partial\quad}{% \partial q_{2}},\ldots,\frac{\partial\quad}{\partial q_{N}}\right\}
  53. | χ k ( 𝐫 𝐪 ) ; k = 1 , M |\chi_{k}(\mathbf{r\mid q})\rangle;\ k=1,M
  54. 𝐫 \mathbf{r}
  55. 𝐪 \mathbf{q}
  56. 𝐀 \mathbf{A}
  57. Γ \Gamma
  58. 𝐖 \mathbf{W}
  59. 𝐖 = 𝐀 * 𝐮𝐀 \mathbf{W}=\mathbf{A}^{*}\mathbf{uA}
  60. 𝐮 j \mathbf{u}_{j}
  61. 𝐖 \mathbf{W}
  62. 𝐀 \mathbf{A}
  63. 𝐀 \mathbf{A}
  64. 𝐅 \mathbf{F}
  65. G q i q j = 𝐅 q i q j - 𝐅 q j q i - [ 𝐅 q i , 𝐅 q j ] = 0. G_{{q_{i}}{q_{j}}}=\frac{{\partial}\mathbf{F}_{q_{i}}}{\partial q_{j}}-\frac{{% \partial}\mathbf{F}_{q_{j}}}{\partial q_{i}}-\left[\mathbf{F}_{q_{i}},\mathbf{% F}_{q_{j}}\right]=0.
  66. 𝐆 \mathbf{G}
  67. 𝐀 \mathbf{A}
  68. Γ \Gamma
  69. 𝐀 ( 𝐪 | Γ ) = P ^ exp \mathbf{A}\left(\mathbf{q}|\Gamma\right)=\hat{P}\exp
  70. ( - 𝐪 𝟎 𝐪 𝐅 ( 𝐪 Γ ) d 𝐪 ) \left(-\int_{\mathbf{q_{0}}}^{\mathbf{q}}\mathbf{F}\left(\mathbf{q^{\prime}}% \mid\Gamma\right)\cdot d\mathbf{q^{\prime}}\right)
  71. P ^ \hat{P}
  72. 𝐪 \mathbf{q}
  73. 𝐪 𝟎 \mathbf{q_{0}}
  74. Γ \Gamma
  75. 𝐀 \mathbf{A}
  76. 𝐐 j ( γ i j ) \mathbf{Q}_{j}(\gamma_{ij})
  77. 𝐐 13 = ( cos γ 13 0 sin γ 13 0 1 0 - sin γ 13 0 cos γ 13 ) \mathbf{Q}_{13}=\begin{pmatrix}\cos\gamma_{13}&0&\sin\gamma_{13}\\ 0&1&0\\ -\sin\gamma_{13}&0&\cos\gamma_{13}\end{pmatrix}
  78. 𝐀 = 𝐐 k l 𝐐 m n 𝐐 p q \mathbf{A}=\mathbf{Q}_{kl}\mathbf{Q}_{mn}\mathbf{Q}_{pq}
  79. γ i j {\gamma}_{ij}
  80. 𝐀 = 𝐐 12 𝐐 23 𝐐 13 \mathbf{A}=\mathbf{Q}_{12}\mathbf{Q}_{23}\mathbf{Q}_{13}
  81. γ 12 {\gamma}_{12}
  82. γ 23 {\gamma}_{23}
  83. γ 12 = - F 12 - tan γ 23 ( - F 13 cos γ 12 + F 23 sin γ 12 ) \nabla\gamma_{12}=-F_{12}-\tan{\gamma}_{23}(-F_{13}\cos\gamma_{12}+F_{23}\sin% \gamma_{12})
  84. γ 23 = - ( F 23 cos γ 12 + F 13 sin γ 12 ) \nabla\gamma_{23}=-(F_{23}\cos\gamma_{12}+F_{13}\sin\gamma_{12})
  85. γ 13 \gamma_{13}
  86. γ 13 = ( cos γ 23 ) - 1 ( - F 13 cos γ 12 + F 23 sin γ 12 ) \nabla\gamma_{13}=(\cos\gamma_{23})^{-1}(-F_{13}\cos\gamma_{12}+F_{23}\sin% \gamma_{12})
  87. γ 12 \gamma_{12}
  88. γ 23 \gamma_{23}
  89. 𝐀 \mathbf{A}
  90. A \mathrm{A}
  91. A = ( cos γ - sin γ sin γ cos γ ) \mathrm{A}=\begin{pmatrix}\cos\gamma&-\sin\gamma\\ \sin\gamma&\cos\gamma\end{pmatrix}
  92. A \mathrm{A}
  93. γ \gamma
  94. γ + 𝐅 𝟏𝟐 = 𝟎 γ ( 𝐪 Γ ) = - 𝐪 𝟎 𝐪 𝐅 12 ( 𝐪 Γ ) d 𝐪 \nabla\mathbf{\gamma+F_{12}=0}\cdot\Longrightarrow\cdot\gamma\left(\mathbf{q}% \mid\Gamma\right)=-\int_{\mathbf{q_{0}}}^{\mathbf{q}}\mathbf{F}_{12}\left(% \mathbf{q^{\prime}}\mid\Gamma\right)\cdot d\mathbf{q^{\prime}}
  95. F 12 \mathrm{F}_{12}
  96. Γ \Gamma

Diagonal_form.html

  1. Σ a i x i m \Sigma a_{i}{x_{i}}^{m}
  2. X 2 + Y 2 - Z 2 = 0 X^{2}+Y^{2}-Z^{2}=0
  3. X 2 - Y 2 - Z 2 = 0 X^{2}-Y^{2}-Z^{2}=0
  4. x 0 3 + x 1 3 + x 2 3 + x 3 3 = 0 x_{0}^{3}+x_{1}^{3}+x_{2}^{3}+x_{3}^{3}=0
  5. x 0 4 + x 1 4 + x 2 4 + x 3 4 = 0 x_{0}^{4}+x_{1}^{4}+x_{2}^{4}+x_{3}^{4}=0

Diamond_v._Diehr.html

  1. k = A e - E a / R T k=Ae^{{-E_{a}}/{RT}}

Dickey–Fuller_test.html

  1. y t = ρ y t - 1 + u t y_{t}=\rho y_{t-1}+u_{t}\,
  2. y t y_{t}
  3. t t
  4. ρ \rho
  5. u t u_{t}
  6. ρ = 1 \rho=1
  7. y t = ( ρ - 1 ) y t - 1 + u t = δ y t - 1 + u t \nabla y_{t}=(\rho-1)y_{t-1}+u_{t}=\delta y_{t-1}+u_{t}\,
  8. \nabla
  9. δ = 0 \delta=0
  10. δ ρ - 1 \delta\equiv\rho-1
  11. t t
  12. y t = δ y t - 1 + u t \nabla y_{t}=\delta y_{t-1}+u_{t}\,
  13. y t = a 0 + δ y t - 1 + u t \nabla y_{t}=a_{0}+\delta y_{t-1}+u_{t}\,
  14. y t = a 0 + a 1 t + δ y t - 1 + u t \nabla y_{t}=a_{0}+a_{1}t+\delta y_{t-1}+u_{t}\,
  15. δ = 0 \delta=0
  16. δ = 0 \delta=0
  17. δ \delta
  18. y y
  19. y t = a 0 + u t \nabla y_{t}=a_{0}+u_{t}\,
  20. y t = y 0 + i = 1 t u i + a 0 t y_{t}=y_{0}+\sum_{i=1}^{t}u_{i}+a_{0}t
  21. a 0 t a_{0}t
  22. y 0 + i = 1 t u i y_{0}+\sum_{i=1}^{t}u_{i}
  23. a 0 a_{0}

Dielectric_complex_reluctance.html

  1. F - 1 F^{-1}
  2. Z ϵ = U ˙ Q ˙ = U ˙ m Q ˙ m = z ϵ e j ϕ Z_{\epsilon}=\frac{\dot{U}}{\dot{Q}}=\frac{\dot{U}_{m}}{\dot{Q}_{m}}=z_{% \epsilon}e^{j\phi}
  3. U ˙ \dot{U}
  4. U ˙ m \dot{U}_{m}
  5. Q ˙ \dot{Q}
  6. Q ˙ m \dot{Q}_{m}
  7. z ϵ z_{\epsilon}
  8. e e
  9. j ϕ = j ( β - α ) j\phi=j\left(\beta-\alpha\right)
  10. j j
  11. β \beta
  12. α \alpha
  13. ϕ \phi
  14. Z ϵ = 1 ϵ ˙ ϵ 0 l S Z_{\epsilon}=\frac{1}{\dot{\epsilon}\epsilon_{0}}\frac{l}{S}
  15. l l
  16. S S
  17. ϵ ˙ ϵ 0 \dot{\epsilon}\epsilon_{0}

Dielectric_reluctance.html

  1. z ϵ = U Q = U m Q m z_{\epsilon}=\frac{U}{Q}=\frac{U_{m}}{Q_{m}}
  2. z ϵ = 1 ϵ ϵ 0 l S z_{\epsilon}=\frac{1}{\epsilon\epsilon_{0}}\frac{l}{S}
  3. l l
  4. S S
  5. ϵ ϵ 0 \epsilon\epsilon_{0}

Difference_set.html

  1. ( v , k , λ ) (v,k,\lambda)
  2. D D
  3. k k
  4. G G
  5. v v
  6. G G
  7. d 1 d 2 - 1 d_{1}d_{2}^{-1}
  8. D D
  9. λ \lambda
  10. D D
  11. G G
  12. λ = 1 \lambda=1
  13. G G
  14. G G
  15. D D
  16. λ \lambda
  17. k 2 - k k^{2}-k
  18. D D
  19. k 2 - k = ( v - 1 ) λ k^{2}-k=(v-1)\lambda
  20. D D
  21. g G g\in G
  22. g D = { g d : d D } gD=\{gd:d\in D\}
  23. D D
  24. D + g D+g
  25. ( v , k , λ ) (v,k,\lambda)
  26. ( v , v - k , v - 2 k + λ ) (v,v-k,v-2k+\lambda)
  27. D D
  28. D D
  29. d e v ( D ) dev(D)
  30. v v
  31. v v
  32. k k
  33. k k
  34. λ \lambda
  35. λ \lambda
  36. G G
  37. λ = 1 \lambda=1
  38. / 7 \mathbb{Z}/7\mathbb{Z}
  39. { 1 , 2 , 4 } \{1,2,4\}
  40. D 1 D_{1}
  41. G 1 G_{1}
  42. D 2 D_{2}
  43. G 2 G_{2}
  44. ψ \psi
  45. G 1 G_{1}
  46. G 2 G_{2}
  47. D 1 ψ = { d ψ : d D 1 } = g D 2 D_{1}^{\psi}=\{d^{\psi}\colon d\in D_{1}\}=gD_{2}
  48. g G 2 g\in G_{2}
  49. d e v ( D 1 ) dev(D_{1})
  50. d e v ( D 2 ) dev(D_{2})
  51. D D
  52. G G
  53. ϕ \phi
  54. G G
  55. D ϕ = g D D^{\phi}=gD
  56. g G g\in G
  57. G G
  58. ϕ \phi
  59. h h t h\mapsto h^{t}
  60. t t
  61. k - λ k-\lambda
  62. g g p g\mapsto g^{p}
  63. p > λ p>\lambda
  64. G G
  65. D D
  66. ( v , k , λ ) (v,k,\lambda)
  67. G G
  68. v * v^{*}
  69. t t
  70. v v
  71. m > λ m>\lambda
  72. k - λ k-\lambda
  73. t p i ( mod v * ) t\equiv p^{i}\ \;\;(\mathop{{\rm mod}}v^{*})
  74. D D
  75. G G
  76. D D
  77. D D
  78. D D
  79. ( ( q n + 2 - 1 ) / ( q - 1 ) , ( q n + 1 - 1 ) / ( q - 1 ) , ( q n - 1 ) / ( q - 1 ) ) ((q^{n+2}-1)/(q-1),(q^{n+1}-1)/(q-1),(q^{n}-1)/(q-1))
  80. q q
  81. n n
  82. ( 4 n - 1 , 2 n - 1 , n - 1 ) (4n-1,2n-1,n-1)
  83. n n
  84. ( 4 n 2 , 2 n 2 - n , n 2 - n ) (4n^{2},2n^{2}-n,n^{2}-n)
  85. n n
  86. ( q n + 1 ( 1 + ( q n + 1 - 1 ) / ( q - 1 ) ) , q n ( q n + 1 - 1 ) / ( q - 1 ) , q n ( q n - 1 ) ( q - 1 ) ) (q^{n+1}(1+(q^{n+1}-1)/(q-1)),q^{n}(q^{n+1}-1)/(q-1),q^{n}(q^{n}-1)(q-1))
  87. q q
  88. n n
  89. ( 3 n + 1 ( 3 n + 1 - 1 ) / 2 , 3 n ( 3 n + 1 + 1 ) / 2 , 3 n ( 3 n + 1 ) / 2 ) (3^{n+1}(3^{n+1}-1)/2,3^{n}(3^{n+1}+1)/2,3^{n}(3^{n}+1)/2)
  90. n n
  91. ( 4 q 2 n ( q 2 n - 1 ) / ( q - 1 ) , q 2 n - 1 ( 1 + 2 ( q 2 n - 1 ) / ( q + 1 ) ) , q 2 n - 1 ( q 2 n - 1 + 1 ) ( q - 1 ) / ( q + 1 ) ) (4q^{2n}(q^{2n}-1)/(q-1),q^{2n-1}(1+2(q^{2n}-1)/(q+1)),q^{2n-1}(q^{2n-1}+1)(q-% 1)/(q+1))
  92. q q
  93. n n
  94. GF ( q ) {\rm GF}(q)
  95. q q
  96. q q
  97. G = ( GF ( q ) , + ) G=({\rm GF}(q),+)
  98. GF ( q ) * {\rm GF}(q)^{*}
  99. ( 4 n - 1 , 2 n - 1 , n - 1 ) (4n-1,2n-1,n-1)
  100. q = 4 n - 1 q=4n-1
  101. G = ( GF ( q ) , + ) G=({\rm GF}(q),+)
  102. D D
  103. ( ( q n + 2 - 1 ) / ( q - 1 ) , ( q n + 1 - 1 ) / ( q - 1 ) , ( q n - 1 ) / ( q - 1 ) ) ((q^{n+2}-1)/(q-1),(q^{n+1}-1)/(q-1),(q^{n}-1)/(q-1))
  104. G = GF ( q n + 2 ) * / GF ( q ) * G={\rm GF}(q^{n+2})^{*}/{\rm GF}(q)^{*}
  105. D = { x G | Tr q n + 2 / q ( x ) = 0 } D=\{x\in G~{}|~{}{\rm Tr}_{q^{n+2}/q}(x)=0\}
  106. ( ( q n + 2 - 1 ) / ( q - 1 ) , ( q n + 1 - 1 ) / ( q - 1 ) , ( q n - 1 ) / ( q - 1 ) ) ((q^{n+2}-1)/(q-1),(q^{n+1}-1)/(q-1),(q^{n}-1)/(q-1))
  107. Tr q n + 2 / q : GF ( q n + 2 ) GF ( q ) {\rm Tr}_{q^{n+2}/q}:{\rm GF}(q^{n+2})\rightarrow{\rm GF}(q)
  108. Tr q n + 2 / q ( x ) = x + x q + + x q n + 1 {\rm Tr}_{q^{n+2}/q}(x)=x+x^{q}+\cdots+x^{q^{n+1}}
  109. ( q 2 + 2 q , q 2 + 2 q - 1 2 , q 2 + 2 q - 3 4 ) \left(q^{2}+2q,\frac{q^{2}+2q-1}{2},\frac{q^{2}+2q-3}{4}\right)
  110. q q
  111. q + 2 q+2
  112. G = ( GF ( q ) , + ) ( GF ( q + 2 ) , + ) G=({\rm GF}(q),+)\oplus({\rm GF}(q+2),+)
  113. D = { ( x , y ) : y = 0 or x and y are non-zero and both are squares or both are non-squares } . D=\{(x,y)\colon y=0\,\text{ or }x\,\text{ and }y\,\text{ are non-zero and both% are squares or both are non-squares}\}.
  114. ( v , k , λ , s ) (v,k,\lambda,s)
  115. B = { B 1 , B s } B=\{B_{1},...B_{s}\}
  116. G G
  117. G G
  118. v v
  119. B i B_{i}
  120. k k
  121. i i
  122. G G
  123. d 1 d 2 - 1 d_{1}d_{2}^{-1}
  124. B i B_{i}
  125. i i
  126. d 1 , d 2 d_{1},d_{2}
  127. B i B_{i}
  128. λ \lambda
  129. s = 1 s=1
  130. s ( k 2 - k ) = ( v - 1 ) λ s(k^{2}-k)=(v-1)\lambda
  131. d e v ( B ) = { B i + g : i = 1 , , s , g G } dev(B)=\{B_{i}+g:i=1,...,s,g\in G\}
  132. d e v ( B ) dev(B)
  133. B B

Different_ideal.html

  1. x tr x 2 x\mapsto\operatorname{tr}~{}x^{2}
  2. δ ( α ) = ( α - α ( i ) ) \delta(\alpha)=\prod\left({\alpha-\alpha^{(i)}}\right)
  3. Ω O L / O K 1 \Omega^{1}_{O_{L}/O_{K}}
  4. δ L / K = { x O L : x d y = 0 for all y O L } . \delta_{L/K}=\{x\in O_{L}:x\mathrm{d}y=0\,\text{ for all }y\in O_{L}\}.
  5. i = 0 ( | G i | - 1 ) . \sum_{i=0}^{\infty}(|G_{i}|-1).

Differentiable_manifold.html

  1. φ α : U α 𝐑 n \varphi_{\alpha}\colon U_{\alpha}\to{\mathbf{R}}^{n}
  2. φ α β = φ β φ α - 1 | φ α ( U α U β ) : φ α ( U α U β ) φ β ( U α U β ) . \varphi_{\alpha\beta}=\varphi_{\beta}\circ\varphi_{\alpha}^{-1}|_{\varphi_{% \alpha}(U_{\alpha}\cap U_{\beta})}\colon\varphi_{\alpha}(U_{\alpha}\cap U_{% \beta})\to\varphi_{\beta}(U_{\alpha}\cap U_{\beta}).
  3. i U i \cup_{i}\,U_{i}
  4. f | U i Γ f|_{U_{i}}\in\Gamma
  5. f ϕ - 1 : ϕ ( U ) 𝐑 n 𝐑 f\circ\phi^{-1}\colon\phi(U)\subset{\mathbf{R}}^{n}\to{\mathbf{R}}
  6. d d t f ( γ ( t ) ) | t = 0 . \left.\frac{d}{dt}f(\gamma(t))\right|_{t=0}.
  7. d d t ϕ γ 1 ( t ) | t = 0 = d d t ϕ γ 2 ( t ) | t = 0 \left.\frac{d}{dt}\phi\circ\gamma_{1}(t)\right|_{t=0}=\left.\frac{d}{dt}\phi% \circ\gamma_{2}(t)\right|_{t=0}
  8. γ 1 γ 2 d d t ϕ γ 1 ( t ) | t = 0 = d d t ϕ γ 2 ( t ) | t = 0 \gamma_{1}\equiv\gamma_{2}\iff\left.\frac{d}{dt}\phi\circ\gamma_{1}(t)\right|_% {t=0}=\left.\frac{d}{dt}\phi\circ\gamma_{2}(t)\right|_{t=0}
  9. X f ( p ) := d d t f ( γ ( t ) ) | t = 0 . Xf(p):=\left.\frac{d}{dt}f(\gamma(t))\right|_{t=0}.
  10. X X f ( p ) X\mapsto Xf(p)
  11. d f ( p ) : T p M 𝐑 . df(p)\colon T_{p}M\to{\mathbf{R}}.
  12. i ϕ i ( x ) = 1. \sum_{i}\phi_{i}(x)=1.\,
  13. k = x k \partial_{k}=\frac{\partial}{\partial x_{k}}
  14. d f ( p ) : T p M T f ( p ) N . df(p)\colon T_{p}M\to T_{f(p)}N.
  15. ( y 1 f , , y n f , x 1 , , x m - n ) (y_{1}\circ f,\ldots,y_{n}\circ f,x_{1},\ldots,x_{m-n})
  16. [ A , B ] := A B = - B A . [A,B]:=\mathcal{L}_{A}B=-\mathcal{L}_{B}A.
  17. d : 𝒞 ( M ) T * ( M ) : f d f \mathrm{d}\colon\mathcal{C}(M)\to\mathrm{T}^{*}(M):f\mapsto\mathrm{d}f
  18. d f : T ( M ) 𝒞 ( M ) : V V ( f ) . \mathrm{d}f\colon\mathrm{T}(M)\to\mathcal{C}(M):V\mapsto V(f).
  19. d ( ω η ) = d ω η + ( - 1 ) deg ω ( ω d η ) . \mathrm{d}(\omega\wedge\eta)=\mathrm{d}\omega\wedge\eta+(-1)^{{\rm deg\,}% \omega}(\omega\wedge\mathrm{d}\eta).

Differential_evolution.html

  1. f : n f:\mathbb{R}^{n}\to\mathbb{R}
  2. f f
  3. m m
  4. f ( m ) f ( p ) f(m)\leq f(p)
  5. p p
  6. m m
  7. h := - f h:=-f
  8. 𝐱 n \mathbf{x}\in\mathbb{R}^{n}
  9. 𝐱 \mathbf{x}
  10. 𝐱 \mathbf{x}
  11. 𝐚 , 𝐛 \mathbf{a},\mathbf{b}
  12. 𝐜 \mathbf{c}
  13. 𝐱 \mathbf{x}
  14. R { 1 , , n } R\in\{1,\ldots,n\}
  15. n n
  16. 𝐲 = [ y 1 , , y n ] \mathbf{y}=[y_{1},\ldots,y_{n}]
  17. i i
  18. r i U ( 0 , 1 ) r_{i}\equiv U(0,1)
  19. r i < CR r_{i}<\,\text{CR}
  20. i = R i=R
  21. y i = a i + F × ( b i - c i ) y_{i}=a_{i}+F\times(b_{i}-c_{i})
  22. y i = x i y_{i}=x_{i}
  23. 𝐱 \mathbf{x}
  24. 𝐳 = 𝐚 + F × ( 𝐛 - 𝐜 ) \mathbf{z}=\mathbf{a}+F\times(\mathbf{b}-\mathbf{c})
  25. f ( 𝐲 ) < f ( 𝐱 ) f(\mathbf{y})<f(\mathbf{x})
  26. 𝐱 \mathbf{x}
  27. 𝐲 \mathbf{y}
  28. F [ 0 , 2 ] F\in[0,2]
  29. CR [ 0 , 1 ] \,\text{CR}\in[0,1]
  30. NP 4 \,\text{NP}\geq 4
  31. NP \,\text{NP}
  32. F \,\text{F}
  33. CR \,\text{CR}
  34. F , CR F,\,\text{CR}
  35. NP \,\text{NP}

Differential_graded_algebra.html

  1. d : A A d\colon A\to A
  2. - 1 -1
  3. d d = 0 d\circ d=0
  4. d ( a b ) = ( d a ) b + ( - 1 ) deg ( a ) a ( d b ) d(a\cdot b)=(da)\cdot b+(-1)^{\operatorname{deg}(a)}a\cdot(db)
  5. H * ( A ) = ker ( d ) / im ( d ) H_{*}(A)=\ker(d)/\operatorname{im}(d)
  6. ( A , d ) (A,d)

Diffusion_MRI.html

  1. ρ \rho
  2. J J
  3. J ( x , t ) = - D ρ ( x , t ) J(x,t)=-D\nabla\rho(x,t)
  4. ρ ( x , t ) t = - J ( x , t ) \frac{\partial\rho(x,t)}{\partial t}=-\nabla\cdot J(x,t)
  5. ρ ( x , t ) t = D 2 ρ ( x , t ) . \frac{\partial\rho(x,t)}{\partial t}=D\nabla^{2}\rho(x,t).
  6. d M d t = γ M × B - M x i + M y j T 2 - ( M z - M 0 ) k T 1 \frac{d{M}}{dt}=\gamma{M}\times{B}-\frac{M_{x}\vec{i}+M_{y}\vec{j}}{T_{2}}-% \frac{(M_{z}-M_{0})\vec{k}}{T_{1}}
  7. M M
  8. d M d t = γ M × B - M x i + M y j T 2 - ( M z - M 0 ) k T 1 + D M \frac{d{M}}{dt}=\gamma{M}\times{B}-\frac{M_{x}\vec{i}+M_{y}\vec{j}}{T_{2}}-% \frac{(M_{z}-M_{0})\vec{k}}{T_{1}}+\nabla\cdot D\nabla{M}
  9. D D
  10. D = D I = D [ 1 0 0 0 1 0 0 0 1 ] , {D}=D\cdot\vec{I}=D\cdot\begin{bmatrix}1&0&0\\ 0&1&0\\ 0&0&1\end{bmatrix},
  11. M = M bloch e - 1 3 γ 2 G 2 t 3 e - b D 0 {M}={M}_{\,\text{bloch}}e^{-\frac{1}{3}\gamma^{2}G^{2}t^{3}}\sim e^{-bD_{0}}
  12. A A
  13. A = e - i , j b i j D i j A=e^{-\sum_{i,j}b_{ij}D_{ij}}
  14. b i j b_{ij}
  15. G x , G_{x},
  16. G y G_{y}
  17. G z G_{z}
  18. T 1 T_{1}
  19. T 2 T_{2}
  20. S ( T E ) S 0 = exp [ - γ 2 G 2 δ 2 ( Δ - δ 3 ) D ] \frac{S(TE)}{S_{0}}=\exp\left[-\gamma^{2}G^{2}\delta^{2}\left(\Delta-\frac{% \delta}{3}\right)D\right]
  21. S 0 S_{0}
  22. S S
  23. γ \gamma
  24. G G
  25. δ \delta
  26. Δ \Delta
  27. D D
  28. S ( T E ) S 0 = exp ( - b A D C ) \frac{S(TE)}{S_{0}}=\exp(-b\cdot ADC)
  29. D D
  30. A D C ADC
  31. b 1 b_{1}
  32. b 2 b_{2}
  33. ADC ( x , y , z ) = ln [ S 2 ( x , y , z ) / S 1 ( x , y , z ) ] / ( b 1 - b 2 ) \mathrm{ADC}(x,y,z)=\ln[S_{2}(x,y,z)/S_{1}(x,y,z)]/(b_{1}-b_{2})
  34. ( vector length ) 2 = B X 2 + B Y 2 + B Z 2 (\,\text{vector length})^{2}=B_{X}^{2}+B_{Y}^{2}+B_{Z}^{2}\,
  35. diffusion vector angle between B X and B Y = arctan B Y B X \,\text{diffusion vector angle between }B_{X}\,\text{ and }B_{Y}=\arctan\frac{% B_{Y}}{B_{X}}
  36. diffusion vector angle between B X and B Z = arctan B Z B X \,\text{diffusion vector angle between }B_{X}\,\text{ and }B_{Z}=\arctan\frac{% B_{Z}}{B_{X}}
  37. diffusion vector angle between B Y and B Z = arctan B Y B Z \,\text{diffusion vector angle between }B_{Y}\,\text{ and }B_{Z}=\arctan\frac{% B_{Y}}{B_{Z}}
  38. ( A D C x + A D C y + A D C z ) / 3 = A D C i (ADC_{x}+ADC_{y}+ADC_{z})/3=ADC_{i}
  39. λ = ( λ 2 + λ 3 ) / 2. \lambda_{\perp}=(\lambda_{2}+\lambda_{3})/2.
  40. λ \lambda_{\perp}
  41. tr ( Λ ) = λ 1 + λ 2 + λ 3 \mathrm{tr}(\Lambda)=\lambda_{1}+\lambda_{2}+\lambda_{3}
  42. Λ \Lambda
  43. λ 1 \lambda_{1}
  44. λ 2 \lambda_{2}
  45. λ 3 \lambda_{3}
  46. M D = ( λ 1 + λ 2 + λ 3 ) / 3 MD=(\lambda_{1}+\lambda_{2}+\lambda_{3})/3
  47. tr ( Λ ) / 3 \displaystyle\mathrm{tr}(\Lambda)/3
  48. V V
  49. D D
  50. 1 / 2 1/\sqrt{2}
  51. F A = 3 ( ( λ 1 - 𝔼 [ λ ] ) 2 + ( λ 2 - 𝔼 [ λ ] ) 2 + ( λ 3 - 𝔼 [ λ ] ) 2 ) 2 ( λ 1 2 + λ 2 2 + λ 3 2 ) FA=\frac{\sqrt{3((\lambda_{1}-\mathbb{E}[\lambda])^{2}+(\lambda_{2}-\mathbb{E}% [\lambda])^{2}+(\lambda_{3}-\mathbb{E}[\lambda])^{2})}}{\sqrt{2(\lambda_{1}^{2% }+\lambda_{2}^{2}+\lambda_{3}^{2})}}
  52. λ 1 λ 2 λ 3 0 \lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\geq 0
  53. λ 1 λ 2 λ 3 \lambda_{1}\gg\lambda_{2}\simeq\lambda_{3}
  54. C l = λ 1 - λ 2 λ 1 + λ 2 + λ 3 C_{l}=\frac{\lambda_{1}-\lambda_{2}}{\lambda_{1}+\lambda_{2}+\lambda_{3}}
  55. λ 1 λ 2 λ 3 \lambda_{1}\simeq\lambda_{2}\gg\lambda_{3}
  56. C p = 2 ( λ 2 - λ 3 ) λ 1 + λ 2 + λ 3 C_{p}=\frac{2(\lambda_{2}-\lambda_{3})}{\lambda_{1}+\lambda_{2}+\lambda_{3}}
  57. λ 1 λ 2 λ 3 \lambda_{1}\simeq\lambda_{2}\simeq\lambda_{3}
  58. C s = 3 λ 3 λ 1 + λ 2 + λ 3 C_{s}=\frac{3\lambda_{3}}{\lambda_{1}+\lambda_{2}+\lambda_{3}}
  59. C a = C l + C p = 1 - C s = λ 1 + λ 2 - 2 λ 3 λ 1 + λ 2 + λ 3 C_{a}=C_{l}+C_{p}=1-C_{s}=\frac{\lambda_{1}+\lambda_{2}-2\lambda_{3}}{\lambda_% {1}+\lambda_{2}+\lambda_{3}}
  60. R A = ( λ 1 - 𝔼 [ λ ] ) 2 + ( λ 2 - 𝔼 [ λ ] ) 2 + ( λ 3 - 𝔼 [ λ ] ) 2 3 𝔼 [ λ ] RA=\frac{\sqrt{(\lambda_{1}-\mathbb{E}[\lambda])^{2}+(\lambda_{2}-\mathbb{E}[% \lambda])^{2}+(\lambda_{3}-\mathbb{E}[\lambda])^{2}}}{\sqrt{3\mathbb{E}[% \lambda]}}
  61. V R = λ 1 λ 2 λ 3 𝔼 [ λ ] 3 VR=\frac{\lambda_{1}\lambda_{2}\lambda_{3}}{\mathbb{E}[\lambda]^{3}}
  62. D ¯ = | D \color r e d x x D x y D x z D x y D \color r e d y y D y z D x z D y z D \color r e d z z | \bar{D}=\begin{vmatrix}D_{\color{red}xx}&D_{xy}&D_{xz}\\ D_{xy}&D_{\color{red}yy}&D_{yz}\\ D_{xz}&D_{yz}&D_{\color{red}zz}\end{vmatrix}

Digital_cinematography.html

  1. 1024 n 1024n

Dihedral_group_of_order_6.html

  1. r , a r 3 = 1 , a 2 = 1 , a r a = r - 1 \langle r,a\mid r^{3}=1,a^{2}=1,ara=r^{-1}\rangle
  2. r , a r 3 , a 2 , a r a r \langle r,a\mid r^{3},a^{2},arar\rangle
  3. a , b a 2 = b 2 = ( a b ) 3 = 1 \langle a,b\mid a^{2}=b^{2}=(ab)^{3}=1\rangle
  4. a , b a 2 , b 2 , ( a b ) 3 \langle a,b\mid a^{2},b^{2},(ab)^{3}\rangle
  5. G = A 3 H G=\mathrm{A}_{3}\rtimes H
  6. C 3 φ C 2 \mathrm{C}_{3}\rtimes_{\varphi}\mathrm{C}_{2}
  7. C 3 × C 2 \mathrm{C}_{3}\times\mathrm{C}_{2}
  8. G x = { g x g G } Gx=\left\{g\cdot x\mid g\in G\right\}
  9. G x = { g G g x = x } G_{x}=\{g\in G\mid g\cdot x=x\}
  10. | X / G | = 1 | G | g G | X g | \left|X/G\right|=\frac{1}{\left|G\right|}\sum_{g\in G}\left|X^{g}\right|
  11. I I
  12. ρ 1 \rho_{1}
  13. ρ 2 \rho_{2}
  14. 3 \mathbb{C}^{3}
  15. I I
  16. ρ 2 \rho_{2}
  17. I I
  18. ( λ , λ , λ ) , λ (\lambda,\lambda,\lambda),\lambda\in\mathbb{C}
  19. ρ 2 \rho_{2}
  20. ( λ 1 , λ 2 , - λ 1 - λ 2 ) (\lambda_{1},\lambda_{2},-\lambda_{1}-\lambda_{2})
  21. ρ 1 \rho_{1}
  22. 2 \mathbb{Z}_{2}
  23. 1 2 + 1 2 + 2 2 = 6 1^{2}+1^{2}+2^{2}=6
  24. z z
  25. 1 - z , 1 / z , z / ( z - 1 ) 1-z,1/z,z/(z-1)
  26. ( z - 1 ) / z , 1 / ( 1 - z ) (z-1)/z,1/(1-z)
  27. S 3 PGL ( 2 , 2 ) . S_{3}\approx\mathrm{PGL}(2,2).
  28. - 1 = [ - 1 : 1 ] , -1=[-1:1],
  29. 2 = 1 / 2 = - 1 2=1/2=-1
  30. S 3 S 4 \mathrm{S}_{3}\hookrightarrow\mathrm{S}_{4}
  31. - 1 -1

Dini_derivative.html

  1. f : , f:{\mathbb{R}}\rightarrow{\mathbb{R}},
  2. f + , f^{\prime}_{+},\,
  3. f + ( t ) lim sup h 0 + f ( t + h ) - f ( t ) h f^{\prime}_{+}(t)\triangleq\limsup_{h\to{0+}}\frac{f(t+h)-f(t)}{h}
  4. lim sup \limsup
  5. f - , f^{\prime}_{-},\,
  6. f - ( t ) lim inf h 0 + f ( t + h ) - f ( t ) h f^{\prime}_{-}(t)\triangleq\liminf_{h\to{0+}}\frac{f(t+h)-f(t)}{h}
  7. lim inf \liminf
  8. f f
  9. t t
  10. d d
  11. f + ( t , d ) lim sup h 0 + f ( t + h d ) - f ( t ) h . f^{\prime}_{+}(t,d)\triangleq\limsup_{h\to{0+}}\frac{f(t+hd)-f(t)}{h}.
  12. f f
  13. f + f^{\prime}_{+}\,
  14. f f
  15. t t
  16. t t
  17. t t
  18. D + f ( t ) D^{+}f(t)\,
  19. f + ( t ) , f^{\prime}_{+}(t),\,
  20. D + f ( t ) D_{+}f(t)\,
  21. f - ( t ) . f^{\prime}_{-}(t).\,
  22. D - f ( t ) lim sup h 0 - f ( t + h ) - f ( t ) h D^{-}f(t)\triangleq\limsup_{h\to{0-}}\frac{f(t+h)-f(t)}{h}
  23. D - f ( t ) lim inf h 0 - f ( t + h ) - f ( t ) h . D_{-}f(t)\triangleq\liminf_{h\to{0-}}\frac{f(t+h)-f(t)}{h}.
  24. D D
  25. + +\infty
  26. - -\infty

Dirac_equation_in_the_algebra_of_physical_space.html

  1. i ¯ Ψ 𝐞 3 + e A ¯ Ψ = m Ψ ¯ i\bar{\partial}\Psi\mathbf{e}_{3}+e\bar{A}\Psi=m\bar{\Psi}^{\dagger}
  2. Ψ = ψ 11 P 3 - ψ 12 P 3 𝐞 1 + ψ 21 𝐞 1 P 3 + ψ 22 P ¯ 3 , \Psi=\psi_{11}P_{3}-\psi_{12}P_{3}\mathbf{e}_{1}+\psi_{21}\mathbf{e}_{1}P_{3}+% \psi_{22}\bar{P}_{3},
  3. Ψ ( ψ 11 ψ 12 ψ 21 ψ 22 ) \Psi\rightarrow\begin{pmatrix}\psi_{11}&\psi_{12}\\ \psi_{21}&\psi_{22}\end{pmatrix}
  4. Ψ ¯ ( ψ 22 * - ψ 21 * - ψ 12 * ψ 11 * ) \bar{\Psi}^{\dagger}\rightarrow\begin{pmatrix}\psi_{22}^{*}&-\psi_{21}^{*}\\ -\psi_{12}^{*}&\psi_{11}^{*}\end{pmatrix}
  5. P 3 = 1 2 ( 1 + 𝐞 3 ) , P_{3}=\frac{1}{2}(1+\mathbf{e}_{3}),
  6. Ψ L = Ψ ¯ P 3 \Psi_{L}=\bar{\Psi}^{\dagger}P_{3}
  7. Ψ R = Ψ P 3 \Psi_{R}=\Psi P_{3}
  8. Ψ L ( ψ 22 * 0 - ψ 12 * 0 ) \Psi_{L}\rightarrow\begin{pmatrix}\psi_{22}^{*}&0\\ -\psi_{12}^{*}&0\end{pmatrix}
  9. Ψ R ( ψ 11 0 ψ 21 0 ) \Psi_{R}\rightarrow\begin{pmatrix}\psi_{11}&0\\ \psi_{21}&0\end{pmatrix}
  10. i Ψ ¯ 𝐞 3 + e A Ψ ¯ = m Ψ i\partial\bar{\Psi}^{\dagger}\mathbf{e}_{3}+eA\bar{\Psi}^{\dagger}=m\Psi
  11. ( 0 i ¯ i 0 ) ( Ψ ¯ P 3 Ψ P 3 ) = m ( Ψ ¯ P 3 Ψ P 3 ) \begin{pmatrix}0&i\bar{\partial}\\ i\partial&0\end{pmatrix}\begin{pmatrix}\bar{\Psi}^{\dagger}P_{3}\\ \Psi P_{3}\end{pmatrix}=m\begin{pmatrix}\bar{\Psi}^{\dagger}P_{3}\\ \Psi P_{3}\end{pmatrix}
  12. ( 0 i 0 + i i 0 - i 0 ) ( Ψ L Ψ R ) = m ( Ψ L Ψ R ) \begin{pmatrix}0&i\partial_{0}+i\nabla\\ i\partial_{0}-i\nabla&0\end{pmatrix}\begin{pmatrix}\Psi_{L}\\ \Psi_{R}\end{pmatrix}=m\begin{pmatrix}\Psi_{L}\\ \Psi_{R}\end{pmatrix}
  13. i ( ( 0 1 1 0 ) 0 + ( 0 σ - σ 0 ) ) ( ψ L ψ R ) = m ( ψ L ψ R ) , i\left(\begin{pmatrix}0&1\\ 1&0\end{pmatrix}\partial_{0}+\begin{pmatrix}0&\sigma\\ -\sigma&0\end{pmatrix}\cdot\nabla\right)\begin{pmatrix}\psi_{L}\\ \psi_{R}\end{pmatrix}=m\begin{pmatrix}\psi_{L}\\ \psi_{R}\end{pmatrix},
  14. ψ L ( ψ 22 * - ψ 12 * ) \psi_{L}\rightarrow\begin{pmatrix}\psi_{22}^{*}\\ -\psi_{12}^{*}\end{pmatrix}
  15. ψ R ( ψ 11 ψ 21 ) \psi_{R}\rightarrow\begin{pmatrix}\psi_{11}\\ \psi_{21}\end{pmatrix}
  16. i γ μ μ ψ = m ψ , i\gamma^{\mu}\partial_{\mu}\psi=m\psi,
  17. ψ = ( ψ 22 * - ψ 12 * ψ 11 ψ 21 ) \psi_{=}\begin{pmatrix}\psi_{22}^{*}\\ -\psi_{12}^{*}\\ \psi_{11}\\ \psi_{21}\end{pmatrix}
  18. Ψ \Psi
  19. Φ \Phi
  20. ψ \psi
  21. ϕ \phi
  22. ϕ γ 0 ψ = Φ ¯ Ψ + ( Ψ ¯ Φ ) S \phi^{\dagger}\gamma^{0}\psi=\langle\bar{\Phi}\Psi+(\bar{\Psi}\Phi)^{\dagger}% \rangle_{S}
  23. ψ γ 0 ψ = 2 Ψ ¯ Ψ S R \psi^{\dagger}\gamma^{0}\psi=2\langle\bar{\Psi}\Psi\rangle_{SR}
  24. Ψ Ψ = Ψ R 0 \Psi\rightarrow\Psi^{\prime}=\Psi R_{0}
  25. i ¯ Ψ 𝐞 3 i ¯ Ψ R 0 𝐞 3 R 0 R 0 = ( i ¯ Ψ 𝐞 3 ) R 0 , i\bar{\partial}\Psi\mathbf{e}_{3}\rightarrow i\bar{\partial}\Psi R_{0}\mathbf{% e}_{3}R_{0}^{\dagger}R_{0}=(i\bar{\partial}\Psi\mathbf{e}_{3}^{\prime})R_{0},
  26. 𝐞 3 𝐞 3 = R 0 𝐞 3 R 0 \mathbf{e}_{3}\rightarrow\mathbf{e}_{3}^{\prime}=R_{0}\mathbf{e}_{3}R_{0}^{\dagger}
  27. m Ψ ¯ m ( Ψ R 0 ) ¯ = m Ψ ¯ R 0 , m\overline{\Psi^{\dagger}}\rightarrow m\overline{(\Psi R_{0})^{\dagger}}=m% \overline{\Psi^{\dagger}}R_{0},
  28. R = exp ( - i e χ 𝐞 3 ) R=\exp(-ie\chi\mathbf{e}_{3})
  29. i ¯ Ψ 𝐞 3 ( i ¯ Ψ ) R 𝐞 3 + ( e ¯ χ ) Ψ R i\bar{\partial}\Psi\mathbf{e}_{3}\rightarrow(i\bar{\partial}\Psi)R\mathbf{e}_{% 3}+(e\bar{\partial}\chi)\Psi R
  30. i ¯ Ψ 𝐞 3 - e A ¯ Ψ ( i ¯ Ψ R 𝐞 3 R - e ( A + χ ) ¯ Ψ ) R , i\bar{\partial}\Psi\mathbf{e}_{3}-e\bar{A}\Psi\rightarrow(i\bar{\partial}\Psi R% \mathbf{e}_{3}R^{\dagger}-e\overline{(A+\partial\chi)}\Psi)R,
  31. J = Ψ Ψ , J=\Psi\Psi^{\dagger},
  32. ¯ J S = 0 \left\langle\bar{\partial}J\right\rangle_{S}=0
  33. ( - ¯ + A A ¯ ) Ψ - i ( 2 e A ¯ S + e F ) Ψ 𝐞 3 = m 2 Ψ (-\partial\bar{\partial}+A\bar{A})\Psi-i(2e\left\langle A\bar{\partial}\right% \rangle_{S}+eF)\Psi\mathbf{e}_{3}=m^{2}\Psi
  34. p = p 0 + 𝐩 p=p^{0}+\mathbf{p}
  35. p 0 > 0 p^{0}>0
  36. Ψ = p m R ( 0 ) exp ( - i p x ¯ S 𝐞 3 ) . \Psi=\sqrt{\frac{p}{m}}R(0)\exp(-i\left\langle p\bar{x}\right\rangle_{S}% \mathbf{e}_{3}).
  37. Ψ Ψ ¯ = 1 \Psi\bar{\Psi}=1
  38. u = p m u=\frac{p}{m}
  39. J = Ψ Ψ = p m J=\Psi{\Psi}^{\dagger}=\frac{p}{m}
  40. p = - | p 0 | - 𝐩 = - p p=-|p^{0}|-\mathbf{p}=-p^{\prime}
  41. Ψ = i p m R ( 0 ) exp ( i p x ¯ S 𝐞 3 ) , \Psi=i\sqrt{\frac{p^{\prime}}{m}}R(0)\exp(i\left\langle p^{\prime}\bar{x}% \right\rangle_{S}\mathbf{e}_{3}),
  42. Ψ Ψ ¯ = - 1 \Psi\bar{\Psi}=-1
  43. u = p m u=\frac{p}{m}
  44. J = Ψ Ψ = - p m , J=\Psi{\Psi}^{\dagger}=-\frac{p}{m},
  45. d t d τ = p m S < 0 \frac{dt}{d\tau}=\left\langle\frac{p}{m}\right\rangle_{S}<0
  46. L = i Ψ ¯ 𝐞 3 Ψ ¯ - e A Ψ ¯ Ψ ¯ - m Ψ Ψ ¯ S L=\langle i\partial\bar{\Psi}^{\dagger}\mathbf{e}_{3}\bar{\Psi}-eA\bar{\Psi}^{% \dagger}\bar{\Psi}-m\Psi\bar{\Psi}\rangle_{S}

Direct_image_functor.html

  1. f * : S h ( X ) S h ( Y ) f_{*}:Sh(X)\to Sh(Y)
  2. f * F ( U ) := F ( f - 1 ( U ) ) , f_{*}F(U):=F(f^{-1}(U)),
  3. U H q ( f - 1 ( U ) , F ) . U\mapsto H^{q}(f^{-1}(U),F).
  4. f : X Y f:X\to Y
  5. , 𝒢 \mathcal{F},\mathcal{G}
  6. Hom 𝐒𝐡 ( X ) ( f - 1 𝒢 , ) = Hom 𝐒𝐡 ( Y ) ( 𝒢 , f * ) \mathrm{Hom}_{\mathbf{Sh}(X)}(f^{-1}\mathcal{G},\mathcal{F})=\mathrm{Hom}_{% \mathbf{Sh}(Y)}(\mathcal{G},f_{*}\mathcal{F})
  7. ( f * ) y (f_{*}\mathcal{F})_{y}
  8. y \mathcal{F}_{y}
  9. y X y\in X

Direct_simulation_Monte_Carlo.html

  1. 2 {}^{2}

Direct_stiffness_method.html

  1. 𝐐 m = 𝐤 m 𝐪 m + 𝐐 o m ( 1 ) \mathbf{Q}^{m}=\mathbf{k}^{m}\mathbf{q}^{m}+\mathbf{Q}^{om}\qquad\qquad\qquad% \mathrm{(1)}
  2. 𝐐 m \mathbf{Q}^{m}
  3. 𝐤 m \mathbf{k}^{m}
  4. 𝐪 m \mathbf{q}^{m}
  5. 𝐐 o m \mathbf{Q}^{om}
  6. 𝐪 m = 0 \mathbf{q}^{m}=0
  7. 𝐪 m \mathbf{q}^{m}
  8. 𝐐 m \mathbf{Q}^{m}
  9. 𝐪 m \mathbf{q}^{m}
  10. 𝐐 m \mathbf{Q}^{m}
  11. 𝐑 = 𝐊𝐫 + 𝐑 o ( 2 ) \mathbf{R}=\mathbf{Kr}+\mathbf{R}^{o}\qquad\qquad\qquad\mathrm{(2)}
  12. 𝐑 \mathbf{R}
  13. 𝐊 \mathbf{K}
  14. 𝐤 m \mathbf{k}^{m}
  15. 𝐫 \mathbf{r}
  16. 𝐑 o \mathbf{R}^{o}
  17. 𝐐 o m \mathbf{Q}^{om}
  18. 𝐤 m \mathbf{k}^{m}
  19. 𝐫 = 𝐊 - 1 ( 𝐑 - 𝐑 o ) ( 3 ) \mathbf{r}=\mathbf{K}^{-1}(\mathbf{R}-\mathbf{R}^{o})\qquad\qquad\qquad\mathrm% {(3)}
  20. 𝐪 m \mathbf{q}^{m}
  21. 𝐪 m \mathbf{q}^{m}
  22. 𝐐 o m \mathbf{Q}^{om}
  23. 𝐊 \mathbf{K}
  24. 𝐑 o \mathbf{R}^{o}
  25. 𝐤 m \mathbf{k}^{m}
  26. 𝐐 o m \mathbf{Q}^{om}
  27. [ f x 1 f y 1 f x 2 f y 2 ] = [ k 11 k 12 k 13 k 14 k 21 k 22 k 23 k 24 k 31 k 32 k 33 k 34 k 41 k 42 k 43 k 44 ] [ u x 1 u y 1 u x 2 u y 2 ] \begin{bmatrix}f_{x1}\\ f_{y1}\\ f_{x2}\\ f_{y2}\\ \end{bmatrix}=\begin{bmatrix}k_{11}&k_{12}&k_{13}&k_{14}\\ k_{21}&k_{22}&k_{23}&k_{24}\\ k_{31}&k_{32}&k_{33}&k_{34}\\ k_{41}&k_{42}&k_{43}&k_{44}\\ \end{bmatrix}\begin{bmatrix}u_{x1}\\ u_{y1}\\ u_{x2}\\ u_{y2}\\ \end{bmatrix}
  28. [ f x 1 f y 1 m z 1 f x 2 f y 2 m z 2 ] = [ k 11 k 12 k 13 k 14 k 15 k 16 k 21 k 22 k 23 k 24 k 25 k 26 k 31 k 32 k 33 k 34 k 35 k 36 k 41 k 42 k 43 k 44 k 45 k 46 k 51 k 52 k 53 k 54 k 55 k 56 k 61 k 62 k 63 k 64 k 65 k 66 ] [ u x 1 u y 1 θ z 1 u x 2 u y 2 θ z 2 ] \begin{bmatrix}f_{x1}\\ f_{y1}\\ m_{z1}\\ f_{x2}\\ f_{y2}\\ m_{z2}\\ \end{bmatrix}=\begin{bmatrix}k_{11}&k_{12}&k_{13}&k_{14}&k_{15}&k_{16}\\ k_{21}&k_{22}&k_{23}&k_{24}&k_{25}&k_{26}\\ k_{31}&k_{32}&k_{33}&k_{34}&k_{35}&k_{36}\\ k_{41}&k_{42}&k_{43}&k_{44}&k_{45}&k_{46}\\ k_{51}&k_{52}&k_{53}&k_{54}&k_{55}&k_{56}\\ k_{61}&k_{62}&k_{63}&k_{64}&k_{65}&k_{66}\\ \end{bmatrix}\begin{bmatrix}u_{x1}\\ u_{y1}\\ \theta_{z1}\\ u_{x2}\\ u_{y2}\\ \theta_{z2}\\ \end{bmatrix}
  29. [ f x 1 f y 1 f x 2 f y 2 ] = E A L [ c 2 s c - c 2 - s c s c s 2 - s c - s 2 - c 2 - s c c 2 s c - s c - s 2 s c s 2 ] [ u x 1 u y 1 u x 2 u y 2 ] s = sin β c = cos β \begin{bmatrix}f_{x1}\\ f_{y1}\\ f_{x2}\\ f_{y2}\\ \end{bmatrix}=\frac{EA}{L}\begin{bmatrix}c^{2}&sc&-c^{2}&-sc\\ sc&s^{2}&-sc&-s^{2}\\ -c^{2}&-sc&c^{2}&sc\\ -sc&-s^{2}&sc&s^{2}\\ \end{bmatrix}\begin{bmatrix}u_{x1}\\ u_{y1}\\ u_{x2}\\ u_{y2}\\ \end{bmatrix}\begin{array}[]{ r }s=\sin\beta\\ c=\cos\beta\\ \end{array}
  30. [ f x 1 f y 1 f x 2 f y 2 ] = E A L [ c x c x c x c y - c x c x - c x c y c y c x c y c y - c y c x - c y c y - c x c x - c x c y c x c x c x c y - c y c x - c y c y c y c x c y c y ] [ u x 1 u y 1 u x 2 u y 2 ] \left[\begin{array}[]{c}f_{x1}\\ f_{y1}\\ \hline f_{x2}\\ f_{y2}\\ \end{array}\right]=\frac{EA}{L}\left[\begin{array}[]{c c|c c}c_{x}c_{x}&c_{x}c% _{y}&-c_{x}c_{x}&-c_{x}c_{y}\\ c_{y}c_{x}&c_{y}c_{y}&-c_{y}c_{x}&-c_{y}c_{y}\\ \hline-c_{x}c_{x}&-c_{x}c_{y}&c_{x}c_{x}&c_{x}c_{y}\\ -c_{y}c_{x}&-c_{y}c_{y}&c_{y}c_{x}&c_{y}c_{y}\\ \end{array}\right]\left[\begin{array}[]{c}u_{x1}\\ u_{y1}\\ \hline u_{x2}\\ u_{y2}\\ \end{array}\right]
  31. c x c_{x}
  32. c y c_{y}
  33. k ( 1 ) = E A L [ 1 0 - 1 0 0 0 0 0 - 1 0 1 0 0 0 0 0 ] K ( 1 ) = E A L [ 1 0 - 1 0 0 0 0 0 0 0 0 0 - 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ] k^{(1)}=\frac{EA}{L}\begin{bmatrix}1&0&-1&0\\ 0&0&0&0\\ -1&0&1&0\\ 0&0&0&0\\ \end{bmatrix}\rightarrow K^{(1)}=\frac{EA}{L}\begin{bmatrix}1&0&-1&0&0&0\\ 0&0&0&0&0&0\\ -1&0&1&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ 0&0&0&0&0&0\\ \end{bmatrix}

Direct_sum.html

  1. 𝐑 𝐑 \mathbf{R}\oplus\mathbf{R}
  2. 𝐑 \mathbf{R}
  3. 𝐑 𝐑 \mathbf{R}\oplus\mathbf{R}
  4. A A
  5. B B
  6. A B A\oplus B
  7. ( a , b ) (a,b)
  8. a A a\in A
  9. b B b\in B
  10. ( a , b ) + ( c , d ) (a,b)+(c,d)
  11. ( a + c , b + d ) (a+c,b+d)
  12. A B C A\oplus B\oplus C
  13. A , B , A,B,
  14. C C
  15. x y xy
  16. ( A i ) i I (A_{i})_{i\in I}
  17. i I A i \bigoplus_{i\in I}A_{i}
  18. ( a i ) i I (a_{i})_{i\in I}
  19. a i A i a_{i}\in A_{i}
  20. a i = 0 a_{i}=0
  21. i I A i \bigoplus_{i\in I}A_{i}
  22. i I A i \prod_{i\in I}A_{i}
  23. I I
  24. ( x 1 , y 1 ) + ( x 2 , y 2 ) = ( x 1 + x 2 , y 1 + y 2 ) (x_{1},y_{1})+(x_{2},y_{2})=(x_{1}+x_{2},y_{1}+y_{2})
  25. A A
  26. B B
  27. A B A\oplus B
  28. A i A_{i}
  29. i I i\in I
  30. A = i I A i \textstyle A=\bigoplus_{i\in I}A_{i}
  31. + +
  32. * *
  33. R R
  34. R R R\oplus R
  35. Z 6 Z_{6}
  36. { 0 , 1 , 2 , 3 , 4 , 5 } \{0,1,2,3,4,5\}
  37. Z 6 = { 0 , 3 } { 0 , 2 , 4 } Z_{6}=\{0,3\}\oplus\{0,2,4\}
  38. ( A , ) (A,\ast)
  39. ( B , ) (B,\cdot)
  40. A B A\oplus B
  41. A × B A\times B
  42. \circ
  43. ( a 1 , b 1 ) ( a 2 , b 2 ) = ( a 1 a 2 , b 1 b 2 ) (a_{1},b_{1})\circ(a_{2},b_{2})=(a_{1}\ast a_{2},b_{1}\cdot b_{2})
  44. i I A i \bigoplus_{i\in I}A_{i}
  45. ( a i ) j I A j \textstyle(a_{i})\in\prod_{j\in I}A_{j}
  46. R S R\oplus S
  47. R × S R\times S
  48. R × S R\times S
  49. R R × S R\to R\times S
  50. R × S R\times S
  51. ( R i ) i I (R_{i})_{i\in I}
  52. i I A i \bigoplus_{i\in I}A_{i}
  53. π j : i I A i A j \pi_{j}\colon\bigoplus_{i\in I}A_{i}\to A_{j}
  54. α j : A j i I A i \alpha_{j}\colon A_{j}\to\bigoplus_{i\in I}A_{i}
  55. g j : A j B g_{j}\colon A_{j}\to B
  56. g : i I A i B g\colon\bigoplus_{i\in I}A_{i}\to B
  57. g α j = g j g\alpha_{j}=g_{j}

Direction_cosine.html

  1. 𝐯 = v x 𝐞 x + v y 𝐞 y + v z 𝐞 z {\mathbf{v}}=v\text{x}\mathbf{e}\text{x}+v\text{y}\mathbf{e}\text{y}+v\text{z}% \mathbf{e}\text{z}
  2. α = cos a = 𝐯 𝐞 x | 𝐯 | = v x v x 2 + v y 2 + v z 2 , β = cos b = 𝐯 𝐞 y | 𝐯 | = v y v x 2 + v y 2 + v z 2 , γ = cos c = 𝐯 𝐞 z | 𝐯 | = v z v x 2 + v y 2 + v z 2 . \begin{aligned}\displaystyle\alpha&\displaystyle=\cos a=\frac{{\mathbf{v}}% \cdot\mathbf{e}\text{x}}{\left|{\mathbf{v}}\right|}&\displaystyle=\frac{v\text% {x}}{\sqrt{v\text{x}^{2}+v\text{y}^{2}+v\text{z}^{2}}},\\ \displaystyle\beta&\displaystyle=\cos b=\frac{{\mathbf{v}}\cdot\mathbf{e}\text% {y}}{\left|{\mathbf{v}}\right|}&\displaystyle=\frac{v\text{y}}{\sqrt{v\text{x}% ^{2}+v\text{y}^{2}+v\text{z}^{2}}},\\ \displaystyle\gamma&\displaystyle=\cos c=\frac{{\mathbf{v}}\cdot\mathbf{e}% \text{z}}{\left|{\mathbf{v}}\right|}&\displaystyle=\frac{v\text{z}}{\sqrt{v% \text{x}^{2}+v\text{y}^{2}+v\text{z}^{2}}}.\end{aligned}
  3. cos 2 a + cos 2 b + cos 2 c = 1 . \cos^{2}a+\cos^{2}b+\cos^{2}c=1\,.

Direction_vector.html

  1. A B \overrightarrow{AB}
  2. A A
  3. B B

Dirichlet's_approximation_theorem.html

  1. | q α - p | 1 N + 1 \left|q\alpha-p\right|\leq\frac{1}{N+1}
  2. | α - p q | < 1 q 2 \left|\alpha-\frac{p}{q}\right|<\frac{1}{q^{2}}
  3. α 1 , , α d \alpha_{1},...,\alpha_{d}
  4. N N
  5. p 1 , , p d , q \Z , 1 q N p_{1},...,p_{d},q\in\Z,1\leq q\leq N
  6. | α i - p i q | 1 q N 1 / d . \left|\alpha_{i}-\frac{p_{i}}{q}\right|\leq\frac{1}{qN^{1/d}}.
  7. S = { ( x , y ) \R 2 ; - N - 1 2 x N + 1 2 , | α x - y | 1 N } S=\{(x,y)\in\R^{2};-N-\frac{1}{2}\leq x\leq N+\frac{1}{2},|\alpha x-y|\leq% \frac{1}{N}\}
  8. S S
  9. 4 4
  10. S = { ( x , y 1 , , y d ) \R 1 + d ; - N - 1 2 x N + 1 2 , | α i x - y i | 1 N 1 / d } S=\{(x,y_{1},\dots,y_{d})\in\R^{1+d};-N-\frac{1}{2}\leq x\leq N+\frac{1}{2},|% \alpha_{i}x-y_{i}|\leq\frac{1}{N^{1/d}}\}

Dirichlet's_test.html

  1. { a n } \{a_{n}\}
  2. { b n } \{b_{n}\}
  3. a n a n + 1 a_{n}\geq a_{n+1}
  4. lim n a n = 0 \lim_{n\rightarrow\infty}a_{n}=0
  5. | n = 1 N b n | M \left|\sum^{N}_{n=1}b_{n}\right|\leq M
  6. n = 1 a n b n \sum^{\infty}_{n=1}a_{n}b_{n}
  7. S n = k = 0 n a k b k S_{n}=\sum_{k=0}^{n}a_{k}b_{k}
  8. B n = k = 0 n b k B_{n}=\sum_{k=0}^{n}b_{k}
  9. S n = a n + 1 B n + k = 0 n B k ( a k - a k + 1 ) S_{n}=a_{n+1}B_{n}+\sum_{k=0}^{n}B_{k}(a_{k}-a_{k+1})
  10. B n B_{n}
  11. a n 0 a_{n}\rightarrow 0
  12. a n + 1 B n 0 a_{n+1}B_{n}\to 0
  13. a n a_{n}
  14. a k - a k + 1 a_{k}-a_{k+1}
  15. | B k ( a k - a k + 1 ) | M ( a k - a k + 1 ) |B_{k}(a_{k}-a_{k+1})|\leq M(a_{k}-a_{k+1})
  16. k = 0 n M ( a k - a k + 1 ) = M k = 0 n ( a k - a k + 1 ) \sum_{k=0}^{n}M(a_{k}-a_{k+1})=M\sum_{k=0}^{n}(a_{k}-a_{k+1})
  17. M ( a 0 - a n + 1 ) M(a_{0}-a_{n+1})
  18. M a 0 Ma_{0}
  19. k = 0 M ( a k - a k + 1 ) \sum_{k=0}^{\infty}M(a_{k}-a_{k+1})
  20. k = 0 | B k ( a k - a k + 1 ) | \sum_{k=0}^{\infty}|B_{k}(a_{k}-a_{k+1})|
  21. k = 0 B k ( a k - a k + 1 ) \sum_{k=0}^{\infty}B_{k}(a_{k}-a_{k+1})
  22. S n S_{n}
  23. b n = ( - 1 ) n | n = 1 N b n | 1 b_{n}=(-1)^{n}\Rightarrow\left|\sum_{n=1}^{N}b_{n}\right|\leq 1
  24. n = 1 a n sin n \sum_{n=1}^{\infty}a_{n}\sin n
  25. { a n } \{a_{n}\}

Dirichlet_beta_function.html

  1. β ( s ) = n = 0 ( - 1 ) n ( 2 n + 1 ) s , \beta(s)=\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(2n+1)^{s}},
  2. β ( s ) = 1 Γ ( s ) 0 x s - 1 e - x 1 + e - 2 x d x . \beta(s)=\frac{1}{\Gamma(s)}\int_{0}^{\infty}\frac{x^{s-1}e^{-x}}{1+e^{-2x}}\,dx.
  3. β ( s ) = 4 - s ( ζ ( s , 1 4 ) - ζ ( s , 3 4 ) ) . \beta(s)=4^{-s}\left(\zeta\left(s,{1\over 4}\right)-\zeta\left(s,{3\over 4}% \right)\right).
  4. β ( s ) = 2 - s Φ ( - 1 , s , 1 2 ) , \beta(s)=2^{-s}\Phi\left(-1,s,{{1}\over{2}}\right),
  5. β ( s ) = 1 2 s n = 0 ( - 1 ) n ( n + 1 2 ) s = 1 ( - 2 ) 2 s ( s - 1 ) ! [ ψ ( s - 1 ) ( 1 4 ) - ψ ( s - 1 ) ( 3 4 ) ] . \beta(s)=\frac{1}{2^{s}}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{\left(n+\frac{1}{2}% \right)^{s}}=\frac{1}{(-2)^{2s}(s-1)!}\left[\psi^{(s-1)}\left(\frac{1}{4}% \right)-\psi^{(s-1)}\left(\frac{3}{4}\right)\right].
  6. β ( 0 ) = 1 2 , \beta(0)=\frac{1}{2},
  7. β ( 1 ) = tan - 1 ( 1 ) = π 4 , \beta(1)\;=\;\tan^{-1}(1)\;=\;\frac{\pi}{4},
  8. β ( 2 ) = G , \beta(2)\;=\;G,
  9. β ( 3 ) = π 3 32 , \beta(3)\;=\;\frac{\pi^{3}}{32},
  10. β ( 4 ) = 1 768 ( ψ 3 ( 1 4 ) - 8 π 4 ) , \beta(4)\;=\;\frac{1}{768}(\psi_{3}(\frac{1}{4})-8\pi^{4}),
  11. β ( 5 ) = 5 π 5 1536 , \beta(5)\;=\;\frac{5\pi^{5}}{1536},
  12. β ( 7 ) = 61 π 7 184320 , \beta(7)\;=\;\frac{61\pi^{7}}{184320},
  13. ψ 3 ( 1 / 4 ) \psi_{3}(1/4)
  14. β ( 2 k + 1 ) = ( - 1 ) k E 2 k π 2 k + 1 4 k + 1 ( 2 k ) ! , \beta(2k+1)={{{({-1})^{k}}{E_{2k}}{\pi^{2k+1}}\over{4^{k+1}}(2k)!}},
  15. E n \!\ E_{n}
  16. β ( - k ) = E k 2 . \beta(-k)={{E_{k}}\over{2}}.

Discriminant_of_an_algebraic_number_field.html

  1. Δ K = ( det ( σ 1 ( b 1 ) σ 1 ( b 2 ) σ 1 ( b n ) σ 2 ( b 1 ) σ n ( b 1 ) σ n ( b n ) ) ) 2 . \Delta_{K}=\left(\operatorname{det}\left(\begin{array}[]{cccc}\sigma_{1}(b_{1}% )&\sigma_{1}(b_{2})&\cdots&\sigma_{1}(b_{n})\\ \sigma_{2}(b_{1})&\ddots&&\vdots\\ \vdots&&\ddots&\vdots\\ \sigma_{n}(b_{1})&\cdots&\cdots&\sigma_{n}(b_{n})\end{array}\right)\right)^{2}.
  2. K = 𝐐 ( d ) K=\mathbf{Q}(\sqrt{d})
  3. Δ K = { d if d 1 ( mod 4 ) 4 d if d 2 , 3 ( mod 4 ) . \Delta_{K}=\left\{\begin{array}[]{ll}d&\,\text{if }d\equiv 1\;\;(\mathop{{\rm mod% }}4)\\ 4d&\,\text{if }d\equiv 2,3\;\;(\mathop{{\rm mod}}4).\\ \end{array}\right.
  4. Δ K n = ( - 1 ) φ ( n ) / 2 n φ ( n ) p | n p φ ( n ) / ( p - 1 ) \Delta_{K_{n}}=(-1)^{\varphi(n)/2}\frac{n^{\varphi(n)}}{\displaystyle\prod_{p|% n}p^{\varphi(n)/(p-1)}}
  5. φ ( n ) \varphi(n)
  6. 1 i < j n ( α i - α j ) 2 \prod_{1\leq i<j\leq n}(\alpha_{i}-\alpha_{j})^{2}
  7. Δ K 0 or 1 ( mod 4 ) . \Delta_{K}\equiv 0\,\text{ or }1\;\;(\mathop{{\rm mod}}4).
  8. | Δ K | 1 / 2 n n n ! ( π 4 ) r 2 n n n ! ( π 4 ) n / 2 . |\Delta_{K}|^{1/2}\geq\frac{n^{n}}{n!}\left(\frac{\pi}{4}\right)^{r_{2}}\geq% \frac{n^{n}}{n!}\left(\frac{\pi}{4}\right)^{n/2}.
  9. Δ K / F = 𝒩 L / F ( Δ K / L ) Δ L / F [ K : L ] \Delta_{K/F}=\mathcal{N}_{L/F}\left({\Delta_{K/L}}\right)\Delta_{L/F}^{[K:L]}
  10. 𝒩 \mathcal{N}
  11. α ( ρ , σ ) 60.8 ρ 22.3 σ \alpha(\rho,\sigma)\geq 60.8^{\rho}22.3^{\sigma}
  12. α ( ρ , σ ) 215.3 ρ 44.7 σ . \alpha(\rho,\sigma)\geq 215.3^{\rho}44.7^{\sigma}.
  13. K 𝐐 𝐑 K\otimes_{\mathbf{Q}}\mathbf{R}
  14. | Δ K | \sqrt{|\Delta_{K}|}
  15. 2 - r 2 | Δ K | 2^{-r_{2}}\sqrt{|\Delta_{K}|}

Dissolution_(chemistry).html

  1. \rightleftharpoons
  2. a N a + a C l - = K s p a_{Na^{+}}\cdot a_{Cl^{-}}=K_{sp}
  3. d m d t = A D d ( C s - C b ) \frac{dm}{dt}=A\frac{D}{d}(C_{s}-C_{b})

Distribution_(differential_geometry).html

  1. M M
  2. C C^{\infty}
  3. m m
  4. n m n\leq m
  5. x M x\in M
  6. n n
  7. Δ x T x ( M ) \Delta_{x}\subset T_{x}(M)
  8. N x M N_{x}\subset M
  9. x x
  10. n n
  11. X 1 , , X n X_{1},\ldots,X_{n}
  12. y N x y\in N_{x}
  13. X 1 ( y ) , , X n ( y ) X_{1}(y),\ldots,X_{n}(y)
  14. Δ y . \Delta_{y}.
  15. Δ \Delta
  16. Δ x \Delta_{x}
  17. x M x\in M
  18. Δ \Delta
  19. n n
  20. M M
  21. C C^{\infty}
  22. n n
  23. M . M.
  24. { X 1 , , X n } \{X_{1},\ldots,X_{n}\}
  25. Δ . \Delta.
  26. Δ \Delta
  27. M M
  28. x M x\in M
  29. { X 1 , , X n } \{X_{1},\ldots,X_{n}\}
  30. x x
  31. 1 i , j n 1\leq i,j\leq n
  32. [ X i , X j ] [X_{i},X_{j}]
  33. { X 1 , , X n } . \{X_{1},\ldots,X_{n}\}.
  34. [ X i , X j ] [X_{i},X_{j}]
  35. { X 1 , , X n } . \{X_{1},\ldots,X_{n}\}.
  36. [ Δ , Δ ] Δ . [\Delta,\Delta]\subset\Delta.
  37. Δ x T x M \Delta_{x}\subset T_{x}M
  38. Δ x \Delta_{x}
  39. Δ x \Delta_{x}

Distribution_ensemble.html

  1. X = { X i } i I X=\{X_{i}\}_{i\in I}
  2. I I
  3. X i X_{i}
  4. I = 𝒩 I=\mathcal{N}
  5. X n X_{n}
  6. U = { U n } n U=\{U_{n}\}_{n\in\mathbb{N}}

Diversity_combining.html

  1. k = 1 N S N R k \sum_{k=1}^{N}SNR_{k}
  2. S N R k SNR_{k}
  3. k k
  4. k = 1 N 1 k \sum_{k=1}^{N}\frac{1}{k}

Diversity_index.html

  1. D q = 1 M q - 1 = 1 i = 1 R p i p i q - 1 q - 1 = ( i = 1 R p i q ) 1 / ( 1 - q ) {}^{q}\!D={1\over M_{q-1}}={1\over\sqrt[q-1]{{\sum_{i=1}^{R}p_{i}p_{i}^{q-1}}}% }=\left({\sum_{i=1}^{R}p_{i}^{q}}\right)^{1/(1-q)}
  2. D 1 = 1 i = 1 R p i p i = exp ( - i = 1 R p i ln ( p i ) ) {}^{1}\!D={1\over{\prod_{i=1}^{R}p_{i}^{p_{i}}}}=\exp\left(-\sum_{i=1}^{R}p_{i% }\ln(p_{i})\right)
  3. p i p_{i}
  4. p i p_{i}
  5. D 0 {}^{0}\!D
  6. D q {}^{q}\!D
  7. D q = ( i = 1 R p i q ) 1 / ( 1 - q ) {}^{q}\!D=\left({\sum_{i=1}^{R}p_{i}^{q}}\right)^{1/(1-q)}
  8. H = - i = 1 R p i ln p i H^{\prime}=-\sum_{i=1}^{R}p_{i}\ln p_{i}
  9. p i p_{i}
  10. p i p_{i}
  11. H = - i = 1 R p i ln p i = - i = 1 R ln p i p i H^{\prime}=-\sum_{i=1}^{R}p_{i}\ln p_{i}=-\sum_{i=1}^{R}\ln p_{i}^{p_{i}}
  12. H = - ( ln p 1 p 1 + ln p 2 p 2 + ln p 3 p 3 + + ln p R p R ) H^{\prime}=-(\ln p_{1}^{p_{1}}+\ln p_{2}^{p_{2}}+\ln p_{3}^{p_{3}}+\cdots+\ln p% _{R}^{p_{R}})
  13. H = - ln p 1 p 1 p 2 p 2 p 3 p 3 p R p R = ln ( 1 p 1 p 1 p 2 p 2 p 3 p 3 p R p R ) = ln ( 1 i = 1 R p i p i ) H^{\prime}=-\ln p_{1}^{p_{1}}p_{2}^{p_{2}}p_{3}^{p_{3}}\cdots p_{R}^{p_{R}}=% \ln\left({1\over p_{1}^{p_{1}}p_{2}^{p_{2}}p_{3}^{p_{3}}\cdots p_{R}^{p_{R}}}% \right)=\ln\left({1\over{\prod_{i=1}^{R}p_{i}^{p_{i}}}}\right)
  14. p i p_{i}
  15. p i p_{i}
  16. p i p_{i}
  17. p i p_{i}
  18. p i p_{i}
  19. H q = 1 1 - q ln ( i = 1 R p i q ) {}^{q}H=\frac{1}{1-q}\;\ln\left(\sum_{i=1}^{R}p_{i}^{q}\right)
  20. H q = ln ( 1 i = 1 R p i p i q - 1 q - 1 ) = ln ( D q ) {}^{q}H=\ln\left({1\over\sqrt[q-1]{{\sum_{i=1}^{R}p_{i}p_{i}^{q-1}}}}\right)=% \ln({}^{q}\!D)
  21. λ = i = 1 R p i 2 \lambda=\sum_{i=1}^{R}p_{i}^{2}
  22. p i p_{i}
  23. λ 1 / R \lambda\geq 1/R
  24. l = i = 1 R n i ( n i - 1 ) N ( N - 1 ) l=\frac{\sum_{i=1}^{R}n_{i}(n_{i}-1)}{N(N-1)}
  25. n i n_{i}
  26. 1 / λ = 1 i = 1 R p i 2 = D 2 1/\lambda={1\over\sum_{i=1}^{R}p_{i}^{2}}={}^{2}D
  27. 1 - λ = 1 - i = 1 R p i 2 = 1 - 1 / D 2 1-\lambda=1-\sum_{i=1}^{R}p_{i}^{2}=1-1/{}^{2}D
  28. p i p_{i}
  29. p i p_{i}
  30. 1 / D 1/{}^{\infty}\!D

Dogic.html

  1. 59 ! × 20 ! × 3 19 2 × 5 ! 12 2.20 × 10 82 \frac{59!\times 20!\times 3^{19}}{2\times 5!^{12}}\approx 2.20\times 10^{82}
  2. 59 ! × 20 ! 2 11 × 6 ! 10 4.40 × 10 66 \frac{59!\times 20!}{2^{11}\times 6!^{10}}\approx 4.40\times 10^{66}

Dolbeault_cohomology.html

  1. ¯ : Γ ( Ω p , q ) Γ ( Ω p , q + 1 ) \bar{\partial}:\Gamma(\Omega^{p,q})\rightarrow\Gamma(\Omega^{p,q+1})
  2. ¯ 2 = 0 \bar{\partial}^{2}=0
  3. H p , q ( M , ) = ker ( ¯ : Γ ( Ω p , q , M ) Γ ( Ω p , q + 1 , M ) ) ¯ Γ ( Ω p , q - 1 ) . H^{p,q}(M,\mathbb{C})=\frac{\hbox{ker}\left(\bar{\partial}:\Gamma(\Omega^{p,q}% ,M)\rightarrow\Gamma(\Omega^{p,q+1},M)\right)}{\bar{\partial}\Gamma(\Omega^{p,% q-1})}.
  4. 𝒪 ( E ) \mathcal{O}(E)
  5. 𝒪 ( E ) \mathcal{O}(E)
  6. H p , q ( M ) H q ( M , Ω p ) H^{p,q}(M)\cong H^{q}(M,\Omega^{p})
  7. p , q \mathcal{F}^{p,q}
  8. C C^{\infty}
  9. ( p , q ) (p,q)
  10. ¯ \overline{\partial}
  11. Ω p , q ¯ p , q + 1 ¯ p , q + 2 ¯ \Omega^{p,q}\xrightarrow{\overline{\partial}}\mathcal{F}^{p,q+1}\xrightarrow{% \overline{\partial}}\mathcal{F}^{p,q+2}\xrightarrow{\overline{\partial}}\cdots\,

Doppler_broadening.html

  1. f = f 0 ( 1 + v c ) f=f_{0}\left(1+\frac{v}{c}\right)
  2. f \ f
  3. f 0 \ f_{0}
  4. v \ v
  5. c c
  6. P v ( v ) d v \,P_{v}(v)dv
  7. v \,v
  8. v + d v \,v+dv
  9. P f ( f ) d f = P v ( v f ) d v d f d f P_{f}(f)df=P_{v}(v_{f})\frac{dv}{df}df
  10. v f = c ( f f 0 - 1 ) \,v_{f}=c\left(\frac{f}{f_{0}}-1\right)
  11. f 0 \,f_{0}
  12. f \,f
  13. P f ( f ) d f = c f 0 P v ( c ( f f 0 - 1 ) ) d f P_{f}(f)df=\frac{c}{f_{0}}P_{v}\left(c\left(\frac{f}{f_{0}}-1\right)\right)df
  14. λ \,\lambda
  15. λ - λ 0 λ 0 - f - f 0 f 0 \frac{\lambda-\lambda_{0}}{\lambda_{0}}\approx-\frac{f-f_{0}}{f_{0}}
  16. P λ ( λ ) d λ = c λ 0 P v ( c ( 1 - λ λ 0 ) ) d λ P_{\lambda}(\lambda)d\lambda=\frac{c}{\lambda_{0}}P_{v}\left(c\left(1-\frac{% \lambda}{\lambda_{0}}\right)\right)d\lambda
  17. P v ( v ) d v = m 2 π k T exp ( - m v 2 2 k T ) d v P_{v}(v)dv=\sqrt{\frac{m}{2\pi kT}}\,\exp\left(-\frac{mv^{2}}{2kT}\right)dv
  18. m \,m
  19. T \,T
  20. k \,k
  21. P f ( f ) d f = ( c f 0 ) m 2 π k T exp ( - m [ c ( f f 0 - 1 ) ] 2 2 k T ) d f P_{f}(f)df=\left(\frac{c}{f_{0}}\right)\sqrt{\frac{m}{2\pi kT}}\,\exp\left(-% \frac{m\left[c\left(\frac{f}{f_{0}}-1\right)\right]^{2}}{2kT}\right)df
  22. P f ( f ) d f = m c 2 2 π k T f 0 2 exp ( - m c 2 ( f - f 0 ) 2 2 k T f 0 2 ) d f P_{f}(f)df=\sqrt{\frac{mc^{2}}{2\pi kT{f_{0}}^{2}}}\,\exp\left(-\frac{mc^{2}% \left(f-f_{0}\right)^{2}}{2kT{f_{0}}^{2}}\right)df
  23. σ f = k T m c 2 f 0 \sigma_{f}=\sqrt{\frac{kT}{mc^{2}}}f_{0}
  24. Δ f FWHM = 8 k T ln 2 m c 2 f 0 \Delta f_{\,\text{FWHM}}=\sqrt{\frac{8kT\ln 2}{mc^{2}}}f_{0}

Doubly_stochastic_matrix.html

  1. A = ( a i j ) A=(a_{ij})
  2. i a i j = j a i j = 1 \sum_{i}a_{ij}=\sum_{j}a_{ij}=1
  3. n × n n\times n
  4. B n B_{n}
  5. ( n - 1 ) 2 (n-1)^{2}
  6. n 2 n^{2}
  7. 2 n - 1 2n-1
  8. 2 n - 1 2n-1
  9. 2 n 2n
  10. B n B_{n}
  11. n × n n\times n
  12. B n B_{n}
  13. n = 2 n=2
  14. n n
  15. n ! / n n n!/n^{n}
  16. 1 / n 1/n

Draft:Brent–Kung_adder.html

  1. O ( l o g 2 ( n ) ) O(log_{2}(n))
  2. O ( l o g 2 ( n ) ) O(log_{2}(n))
  3. O ( n * l o g 2 ( n ) ) O(n*log_{2}(n))
  4. O ( n ) O(n)
  5. O ( l o g 2 ( n ) ) O(log_{2}(n))
  6. O ( l o g 2 ( n ) ) O(log_{2}(n))
  7. O ( n ) O(n)

Drag_(physics).html

  1. F D = 1 2 ρ v 2 C D A F_{D}\,=\,\tfrac{1}{2}\,\rho\,v^{2}\,C_{D}\,A
  2. F D F_{D}
  3. ρ \rho
  4. v v
  5. A A
  6. C D C_{D}
  7. R e = v D ν R_{e}=\frac{vD}{\nu}
  8. D D
  9. ν {\nu}
  10. μ {\mu}
  11. F D = 1 2 ρ v 2 C d A , F_{D}\,=\,\tfrac{1}{2}\,\rho\,v^{2}\,C_{d}\,A,
  12. P d = 𝐅 d 𝐯 = 1 2 ρ v 3 A C d P_{d}=\mathbf{F}_{d}\cdot\mathbf{v}=\tfrac{1}{2}\rho v^{3}AC_{d}
  13. v ( t ) = 2 m g ρ A C d tanh ( t g ρ C d A 2 m ) . v(t)=\sqrt{\frac{2mg}{\rho AC_{d}}}\tanh\left(t\sqrt{\frac{g\rho C_{d}A}{2m}}% \right).\,
  14. v t = 2 m g ρ A C d . v_{t}=\sqrt{\frac{2mg}{\rho AC_{d}}}.\,
  15. v t = g d ρ o b j ρ . v_{t}=\sqrt{gd\frac{\rho_{obj}}{\rho}}.\,
  16. v t = 90 d , v_{t}=90\sqrt{d},\,
  17. d \mathbf{}d
  18. v t \mathbf{}v_{t}
  19. d \mathbf{}d
  20. v t \mathbf{}v_{t}
  21. d \mathbf{}d
  22. v t \mathbf{}v_{t}
  23. d \mathbf{}d
  24. v t \mathbf{}v_{t}
  25. R e < 1 R_{e}<1
  26. 𝐅 d = - b 𝐯 \mathbf{F}_{d}=-b\mathbf{v}\,
  27. b \mathbf{}b
  28. 𝐯 \mathbf{v}
  29. v ( t ) = ( ρ - ρ 0 ) V g b ( 1 - e - b t / m ) v(t)=\frac{(\rho-\rho_{0})Vg}{b}\left(1-e^{-bt/m}\right)
  30. v t = ( ρ - ρ 0 ) V g b \mathbf{}v_{t}=\frac{(\rho-\rho_{0})Vg}{b}
  31. b \mathbf{}b
  32. b = 6 π η r b=6\pi\eta r\,
  33. r \mathbf{}r
  34. η \mathbf{}\eta
  35. 𝐅 d = - 6 π η r 𝐯 . \mathbf{F}_{d}=-6\pi\eta r\,\mathbf{v}.
  36. r \mathbf{}r
  37. v \mathbf{}v

Drag_count.html

  1. \Rightarrow
  2. C d C_{d}
  3. Δ C d \Delta C_{\mathrm{d}}\,
  4. Δ C d = 10 4 2 F d ρ v 2 A , \Delta C_{\mathrm{d}}=10^{4}\dfrac{2F_{\mathrm{d}}}{\rho v^{2}A}\,,
  5. F d F_{\mathrm{d}}\,
  6. ρ \rho\,
  7. v v\,
  8. A A\,

Drell–Yan_process.html

  1. σ p d 2 σ p p = 1 2 [ 1 + d ¯ ( x ) u ¯ ( x ) ] \frac{\sigma^{pd}}{2\sigma^{pp}}=\frac{1}{2}\left[1+\frac{\bar{d}(x)}{\bar{u}(% x)}\right]

Drug_action.html

  1. [ d r u g ] [ r e c e p t o r ] [ c o m p l e x ] \frac{[drug][receptor]}{[complex]}

Dual_curve.html

  1. x f x ( p , q , r ) + y f y ( p , q , r ) + z f z ( p , q , r ) = 0. x\frac{\partial f}{\partial x}(p,q,r)+y\frac{\partial f}{\partial y}(p,q,r)+z% \frac{\partial f}{\partial z}(p,q,r)=0.
  2. X = λ f x ( p , q , r ) , Y = λ f y ( p , q , r ) , Z = λ f z ( p , q , r ) . X=\lambda\frac{\partial f}{\partial x}(p,q,r),\,Y=\lambda\frac{\partial f}{% \partial y}(p,q,r),\,Z=\lambda\frac{\partial f}{\partial z}(p,q,r).
  3. X = 2 λ a p , Y = 2 λ b q , Z = 2 λ c r , X p + Y q + Z r = 0. X=2\lambda ap,\,Y=2\lambda bq,\,Z=2\lambda cr,\,Xp+Yq+Zr=0.
  4. X 2 2 λ a + Y 2 2 λ b + Z 2 2 λ c = 0. \frac{X^{2}}{2\lambda a}+\frac{Y^{2}}{2\lambda b}+\frac{Z^{2}}{2\lambda c}=0.
  5. X 2 a + Y 2 b + Z 2 c = 0. \frac{X^{2}}{a}+\frac{Y^{2}}{b}+\frac{Z^{2}}{c}=0.
  6. X = y y x - x y X=\frac{y^{\prime}}{yx^{\prime}-xy^{\prime}}
  7. Y = x x y - y x Y=\frac{x^{\prime}}{xy^{\prime}-yx^{\prime}}
  8. F ( x 0 , , x n ) F(x_{0},\ldots,x_{n})
  9. x = ( x 0 , , x n ) ( F / x 0 ( x ) , , F / x n ( x ) ) x=(x_{0},\ldots,x_{n})\mapsto(\partial F/\partial x_{0}(x),\ldots,\partial F/% \partial x_{n}(x))
  10. ( a 0 : , a n ) (a_{0}:\ldots,a_{n})
  11. a 0 x 0 + + a n x n = 0 a_{0}x_{0}+\ldots+a_{n}x_{n}=0

Ductal.html

  1. t / m 3 t/m^{3}

Duffing_equation.html

  1. x ¨ + δ x ˙ + α x + β x 3 = γ cos ( ω t ) \ddot{x}+\delta\dot{x}+\alpha x+\beta x^{3}=\gamma\cos(\omega t)\,
  2. x ˙ \dot{x}
  3. x ¨ \ddot{x}
  4. δ \delta
  5. α \alpha
  6. β \beta
  7. γ \gamma
  8. ω \omega
  9. δ \delta
  10. α \alpha
  11. β \beta
  12. β = 0 \beta=0
  13. γ \gamma
  14. γ = 0 \gamma=0
  15. ω \omega
  16. x 3 x^{3}
  17. δ = 0 \delta=0
  18. γ = 0 \gamma=0
  19. γ = δ = 0 , \gamma=\delta=0,
  20. x ˙ \dot{x}
  21. x ˙ ( x ¨ + α x + β x 3 ) = 0 d d t [ 1 2 ( x ˙ ) 2 + 1 2 α x 2 + 1 4 β x 4 ] = 0 1 2 ( x ˙ ) 2 + 1 2 α x 2 + 1 4 β x 4 = H , \begin{aligned}&\displaystyle\dot{x}\left(\ddot{x}+\alpha x+\beta x^{3}\right)% =0\\ &\displaystyle\Rightarrow\frac{\,\text{d}}{\,\text{d}t}\left[\tfrac{1}{2}\left% (\dot{x}\right)^{2}+\tfrac{1}{2}\alpha x^{2}+\tfrac{1}{4}\beta x^{4}\right]=0% \\ &\displaystyle\Rightarrow\tfrac{1}{2}\left(\dot{x}\right)^{2}+\tfrac{1}{2}% \alpha x^{2}+\tfrac{1}{4}\beta x^{4}=H,\end{aligned}
  22. x ( 0 ) x(0)
  23. x ˙ ( 0 ) . \dot{x}(0).
  24. y = x ˙ y=\dot{x}
  25. x ˙ = + H y , \dot{x}=+\frac{\partial H}{\partial y},
  26. y ˙ = - H x \dot{y}=-\frac{\partial H}{\partial x}
  27. H = 1 2 y 2 + 1 2 α x 2 + 1 4 β x 4 . \quad H=\tfrac{1}{2}y^{2}+\tfrac{1}{2}\alpha x^{2}+\tfrac{1}{4}\beta x^{4}.
  28. α \alpha
  29. β \beta
  30. | x | 2 H / α |x|\leq\sqrt{2H/\alpha}
  31. | x ˙ | 2 H , |\dot{x}|\leq\sqrt{2H},

Duffing_map.html

  1. x n + 1 = y n x_{n+1}=y_{n}\,
  2. y n + 1 = - b x n + a y n - y n 3 . y_{n+1}=-bx_{n}+ay_{n}-y_{n}^{3}.\,

Duncan's_new_multiple_range_test.html

  1. m 1 , m 2 , , m n m_{1},m_{2},...,m_{n}
  2. μ 1 , μ 2 , , μ 2 \mu_{1},\mu_{2},...,\mu_{2}
  3. σ \sigma
  4. s m s_{m}
  5. n 2 n_{2}
  6. S m S_{m}
  7. n 2 S m 2 σ m 2 \frac{n_{2}\cdot S_{m}^{2}}{\sigma^{2}_{m}}
  8. χ 2 \chi^{2}
  9. n 2 n_{2}
  10. α p \alpha_{p}
  11. α p = 1 - γ p \alpha_{p}=1-\gamma_{p}
  12. γ p = ( 1 - α ) ( p - 1 ) \gamma_{p}=(1-\alpha)^{(p-1)}
  13. p p
  14. α p \alpha_{p}
  15. α p \alpha_{p}
  16. p - 2 p-2
  17. R ( p , α ) R_{(p,\alpha)}
  18. p p
  19. m i m_{i}
  20. m j m_{j}
  21. m ( i - 1 ) m_{(i-1)}
  22. m i - m j m_{i}-m_{j}
  23. σ m R ( p , α ) \sigma_{m}\cdot R_{(p,\alpha)}
  24. P = i - j , α = α p P=i-j,\alpha=\alpha_{p}
  25. m i - m j m_{i}-m_{j}
  26. ( m j , m j + 1 , , m I ) (m_{j},m_{j+1},...,m_{I})
  27. Q ( p , ν , γ ( p , α ) ) Q_{(p,\nu,\gamma_{(p,\alpha)})}
  28. γ α \gamma_{\alpha}
  29. ν \nu
  30. r ( p , ν , α ) r_{(p,\nu,\alpha)}
  31. r ( p , ν , α ) = Q ( p , ν , γ ( p , α ) ) r_{(p,\nu,\alpha)}=Q_{(p,\nu,\gamma_{(p,\alpha)})}
  32. r ( p , ν , α ) = m a x ( Q ( p , ν , γ ( p , α ) ) , r ( p - 1 , ν , α ) ) r_{(p,\nu,\alpha)}=max(Q_{(p,\nu,\gamma_{(p,\alpha)})},r_{(p-1,\nu,\alpha)})
  33. R ( p , ν , α ) = σ m r ( p , ν , α ) R_{(}p,\nu,\alpha)=\sigma_{m}\cdot r_{(p,\nu,\alpha)}
  34. ν \nu
  35. s m = 1.796 s_{m}=1.796
  36. ν = 20 \nu=20
  37. r ( p , ν , α ) r_{(p,\nu,\alpha)}
  38. r ( 2 , 20 , 0.05 ) = 2.95 r_{(2,20,0.05)}=2.95
  39. r ( 3 , 20 , 0.05 ) = 3.10 r_{(3,20,0.05)}=3.10
  40. r ( 4 , 20 , 0.05 ) = 3.18 r_{(4,20,0.05)}=3.18
  41. r ( 5 , 20 , 0.05 ) = 3.25 r_{(5,20,0.05)}=3.25
  42. R ( p , ν , α ) = σ m * r ( p , ν , α ) R_{(p,\nu,\alpha)}=\sigma_{m}*r_{(p,\nu,\alpha)}
  43. R ( 2 , 20 , 0.05 ) = 3.75 R_{(2,20,0.05)}=3.75
  44. R ( 3 , 20 , 0.05 ) = 3.94 R_{(3,20,0.05)}=3.94
  45. R ( 4 , 20 , 0.05 ) = 4.04 R_{(4,20,0.05)}=4.04
  46. R ( 5 , 20 , 0.05 ) = 4.13 R_{(5,20,0.05)}=4.13
  47. R ( 5 , 20 , 0.05 ) = 4.13. R_{(5,20,0.05)}=4.13.
  48. R ( 4 , 20 , 0.05 ) = 4.04 R_{(4,20,0.05)}=4.04
  49. 4 v s .1 : 21.6 - 9.8 = 11.8 > 4.13 ( R 5 ) 4vs.1:21.6-9.8=11.8>4.13(R_{5})
  50. 4 v s .5 : 21.6 - 10.8 = 10.8 > 4.04 ( R 4 ) 4vs.5:21.6-10.8=10.8>4.04(R_{4})
  51. 4 v s .2 : 21.6 - 15.4 = 6.2 > 3.94 ( R 3 ) 4vs.2:21.6-15.4=6.2>3.94(R_{3})
  52. 4 v s .3 : 21.6 - 17.6 = 4.0 > 3.75 ( R 2 ) 4vs.3:21.6-17.6=4.0>3.75(R_{2})
  53. 3 v s .1 : 17.6 - 9.8 = 7.8 > 4.04 ( R 4 ) 3vs.1:17.6-9.8=7.8>4.04(R_{4})
  54. 3 v s .5 : 17.6 - 10.8 = 6.8 > 3.94 ( R 3 ) 3vs.5:17.6-10.8=6.8>3.94(R_{3})
  55. 3 v s .2 : 17.6 - 15.4 = 2.2 < 3.75 ( R 2 ) 3vs.2:17.6-15.4=2.2<3.75(R_{2})
  56. 2 v s .1 : 15.4 - 9.8 = 5.6 > 3.94 ( R 3 ) 2vs.1:15.4-9.8=5.6>3.94(R_{3})
  57. 2 v s .5 : 15.4 - 10.8 = 4.6 > 3.75 ( R 2 ) 2vs.5:15.4-10.8=4.6>3.75(R_{2})
  58. 5 v s .1 : 10.8 - 9.8 = 1.0 < 3.75 ( R 2 ) 5vs.1:10.8-9.8=1.0<3.75(R_{2})
  59. γ 2 , α = 1 - α \gamma_{2,\alpha}={1-\alpha}
  60. γ 2 , α = 1 - α \gamma_{2,\alpha}={1-\alpha}
  61. γ p , α = γ 2 , α p - 1 = ( 1 - α ) p - 1 \gamma_{p,\alpha}=\gamma_{2,\alpha}^{p-1}=(1-\alpha)^{p-1}
  62. α p = 1 - γ p \alpha_{p}=1-\gamma_{p}
  63. γ 2 , α = 1 - α \gamma_{2,\alpha}={1-\alpha}
  64. γ 2 , α p - 1 \gamma_{2,\alpha}^{p-1}
  65. α p \alpha_{p}
  66. α = 0.05 \alpha=0.05
  67. : γ p , α :\gamma_{p,\alpha}
  68. H 0 : α p H_{0}:\alpha_{p}
  69. γ 2 = 0.95 \gamma_{2}=0.95
  70. ( 100 2 ) 100\choose 2
  71. 1 - 0.95 = 0.05 1-0.95=0.05
  72. ( 100 3 ) 100\choose 3
  73. 1 - ( 0.95 ) 2 = 0.097 1-(0.95)^{2}=0.097
  74. ( 100 4 ) 100\choose 4
  75. 1 - ( 0.95 ) 3 = 0.143 1-(0.95)^{3}=0.143
  76. α p α \alpha_{p}\geq\alpha

Duocylinder.html

  1. D = { ( x , y , z , w ) | x 2 + y 2 r 1 2 , z 2 + w 2 r 2 2 } D=\{(x,y,z,w)|x^{2}+y^{2}\leq r_{1}^{2},\ z^{2}+w^{2}\leq r_{2}^{2}\}
  2. x 2 + y 2 = r 1 2 , z 2 + w 2 r 2 2 x^{2}+y^{2}=r_{1}^{2},z^{2}+w^{2}\leq r_{2}^{2}
  3. z 2 + w 2 = r 2 2 , x 2 + y 2 r 1 2 z^{2}+w^{2}=r_{2}^{2},x^{2}+y^{2}\leq r_{1}^{2}

Duoprism.html

  1. P 1 × P 2 = { ( x , y , z , w ) | ( x , y ) P 1 , ( z , w ) P 2 } P_{1}\times P_{2}=\{(x,y,z,w)|(x,y)\in P_{1},(z,w)\in P_{2}\}
  2. T ¯ 7 {\bar{T}}_{7}

Duru–Kleinert_transformation.html

  1. 1 / r 1/r

Dust_solution.html

  1. T μ ν = ρ U μ U ν T^{\mu\nu}=\rho U^{\mu}U^{\nu}
  2. U μ U^{\mu}
  3. ρ \rho
  4. χ ( λ ) = λ 4 + a 3 λ 3 + a 2 λ 2 + a 1 λ + a 0 \chi(\lambda)=\lambda^{4}+a_{3}\,\lambda^{3}+a_{2}\,\lambda^{2}+a_{1}\,\lambda% +a_{0}
  5. χ ( λ ) = ( λ - 8 π μ ) λ 3 \chi(\lambda)=\left(\lambda-8\pi\mu\right)\,\lambda^{3}
  6. a 0 = a 1 = a 2 = 0 a_{0}\,=a_{1}=a_{2}=0
  7. t 2 = t 1 2 , t 3 = t 1 3 , t 4 = t 1 4 t_{2}=t_{1}^{2},\;\;t_{3}=t_{1}^{3},\;\;t_{4}=t_{1}^{4}
  8. G a a = - R {G^{a}}_{a}=-R
  9. G a b G b a = R 2 {G^{a}}_{b}\,{G^{b}}_{a}=R^{2}
  10. G a b G b c G c a = - R 3 {G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{a}=-R^{3}
  11. G a b G b c G c d G d a = R 4 {G^{a}}_{b}\,{G^{b}}_{c}\,{G^{c}}_{d}\,{G^{d}}_{a}=R^{4}

Dym_equation.html

  1. u t = u 3 u x x x . u_{t}=u^{3}u_{xxx}.\,
  2. v t = ( v - 1 / 2 ) x x x . v_{t}=(v^{-1/2})_{xxx}.\,
  3. u ( t , x ) = [ - 3 α ( x + 4 α 2 t ) ] 2 / 3 . u(t,x)=\left[-3\alpha\left(x+4\alpha^{2}t\right)\right]^{2/3}.

Dynamical_billiards.html

  1. H ( p , q ) = p 2 2 m + V ( q ) H(p,q)=\frac{p^{2}}{2m}+V(q)
  2. V ( q ) \scriptstyle V(q)
  3. Ω \scriptstyle\Omega
  4. V ( q ) = { 0 q Ω q Ω V(q)=\begin{cases}0&q\in\Omega\\ \infty&q\notin\Omega\end{cases}
  5. H ( p , q ) = 1 2 m p i p j g i j ( q ) + V ( q ) H(p,q)=\frac{1}{2m}p^{i}p^{j}g_{ij}(q)+V(q)
  6. g i j ( q ) \scriptstyle g_{ij}(q)
  7. q Ω \scriptstyle q\;\in\;\Omega
  8. ρ > 0 \rho>0
  9. B i M B_{i}\subset M
  10. i = 1 , , n i=1,\ldots,n
  11. B = M ( i = 1 n I n t ( B i ) ) B=M\ (\bigcup_{i=1}^{n}Int(B_{i}))
  12. I n t ( B i ) Int(B_{i})
  13. B i B_{i}
  14. B M B\subset M
  15. B i B j B_{i}\cap B_{j}
  16. i j i\neq j
  17. Π n \scriptstyle\Pi\,\subset\,\mathbb{R}^{n}
  18. Γ \scriptstyle\Gamma
  19. Γ \scriptstyle\Gamma
  20. Π \scriptstyle\Pi
  21. Γ \scriptstyle\Gamma
  22. f ( γ , t ) \scriptstyle f(\gamma,\,t)
  23. Γ × 1 \scriptstyle\Gamma\,\times\,\mathbb{R}^{1}
  24. 1 \scriptstyle\mathbb{R}^{1}
  25. γ Γ \scriptstyle\gamma\,\in\,\Gamma
  26. t 1 \scriptstyle t\,\in\,\mathbb{R}^{1}
  27. v \scriptstyle v
  28. Γ \scriptstyle\Gamma
  29. γ Γ \scriptstyle\gamma\,\in\,\Gamma
  30. t * \scriptstyle t^{*}
  31. t * \scriptstyle t^{*}
  32. v * \scriptstyle v^{*}
  33. Γ * \scriptstyle\Gamma^{*}
  34. Γ \scriptstyle\Gamma
  35. γ \scriptstyle\gamma
  36. t * \scriptstyle t^{*}
  37. Γ \scriptstyle\Gamma
  38. γ \scriptstyle\gamma
  39. f t ( γ , t * ) \scriptstyle\frac{\partial f}{\partial t}(\gamma,\,t^{*})
  40. f \scriptstyle f
  41. Γ * \scriptstyle\Gamma^{*}
  42. Π \scriptstyle\Pi
  43. f t ( γ , t ) > 0 \scriptstyle\frac{\partial f}{\partial t}(\gamma,\,t)\;>\;0
  44. v * \scriptstyle v^{*}
  45. Π \scriptstyle\Pi
  46. Π \scriptstyle\Pi
  47. Γ \scriptstyle\Gamma
  48. v * \scriptstyle v^{*}
  49. Π \scriptstyle\Pi
  50. Γ \scriptstyle\Gamma
  51. γ \scriptstyle\gamma
  52. t ~ > t * \scriptstyle\tilde{t}\;>\;t^{*}
  53. f ( γ , t ) \scriptstyle f(\gamma,\,t)
  54. t \scriptstyle t
  55. f t = 0 \scriptstyle\frac{\partial f}{\partial t}\;=\;0
  56. Γ \scriptstyle\Gamma
  57. H ψ = E ψ \scriptstyle H\psi\;=\;E\psi
  58. - 2 2 m 2 ψ n ( q ) = E n ψ n ( q ) -\frac{\hbar^{2}}{2m}\nabla^{2}\psi_{n}(q)=E_{n}\psi_{n}(q)
  59. 2 \scriptstyle\nabla^{2}
  60. Ω \scriptstyle\Omega
  61. ψ n ( q ) = 0 for q Ω \psi_{n}(q)=0\quad\mbox{for}~{}\quad q\notin\Omega
  62. Ω ψ m ¯ ( q ) ψ n ( q ) d q = δ m n \int_{\Omega}\overline{\psi_{m}}(q)\psi_{n}(q)\,dq=\delta_{mn}
  63. ( 2 + k 2 ) ψ = 0 \left(\nabla^{2}+k^{2}\right)\psi=0
  64. k 2 = 1 2 2 m E n k^{2}=\frac{1}{\hbar^{2}}2mE_{n}
  65. 0 \scriptstyle\hbar\;\to\;0
  66. m \scriptstyle m\;\to\;\infty

Dynkin_system.html

  1. Ω \Omega
  2. D D
  3. D D
  4. D D
  5. D D
  6. D D
  7. D D
  8. D D
  9. n = 1 A n D \bigcup_{n=1}^{\infty}A_{n}\in D
  10. D D
  11. D D
  12. D D
  13. n = 1 A n D \bigcup_{n=1}^{\infty}A_{n}\in D
  14. 𝒥 \mathcal{J}
  15. Ω \Omega
  16. D { 𝒥 } D\{\mathcal{J}\}
  17. 𝒥 \mathcal{J}
  18. D ~ \tilde{D}
  19. 𝒥 \mathcal{J}
  20. D { 𝒥 } D ~ D\{\mathcal{J}\}\subseteq\tilde{D}
  21. D { 𝒥 } D\{\mathcal{J}\}
  22. 𝒥 \mathcal{J}
  23. D { } = { , Ω } D\{\emptyset\}=\{\emptyset,\Omega\}
  24. Ω = { 1 , 2 , 3 , 4 } \Omega=\{1,2,3,4\}
  25. 𝒥 = { 1 } \mathcal{J}=\{1\}
  26. D { 𝒥 } = { , { 1 } , { 2 , 3 , 4 } , Ω } D\{\mathcal{J}\}=\{\emptyset,\{1\},\{2,3,4\},\Omega\}
  27. P P
  28. D D
  29. P D P\subseteq D
  30. σ { P } D \sigma\{P\}\subseteq D
  31. P P
  32. D D