wpmath0000012_1

Bankoff_circle.html

  1. R = 1 2 r ( 1 - r ) . R=\frac{1}{2}r\left(1-r\right).

Bargaining_problem.html

  1. v = ( v 1 , v 2 ) v=(v_{1},v_{2})
  2. v 1 v_{1}
  3. v 2 v_{2}
  4. F F
  5. 𝐑 2 \,\textbf{R}^{2}
  6. F F
  7. F F
  8. ϕ \phi
  9. F F
  10. v v
  11. d = 0 d=0
  12. ( u ( x ) - u ( d ) ) ( v ( y ) - v ( d ) ) (u(x)-u(d))(v(y)-v(d))
  13. u ( d ) u(d)
  14. v ( d ) v(d)
  15. g 1 g_{1}
  16. g 2 g_{2}
  17. ϕ \phi
  18. ϕ 1 / ϕ 2 = g 1 / g 2 \phi_{1}/\phi_{2}=g_{1}/g_{2}

Barkhausen_stability_criterion.html

  1. | β A | = 1 |\beta A|=1\,
  2. β A = 2 π n , n 0 , 1 , 2 , . \angle\beta A=2\pi n,n\in 0,1,2,\dots\,.

Barnes_integral.html

  1. F 1 2 ( a , b ; c ; z ) = Γ ( c ) Γ ( a ) Γ ( b ) 1 2 π i - i i Γ ( a + s ) Γ ( b + s ) Γ ( - s ) Γ ( c + s ) ( - z ) s d s . {}_{2}F_{1}(a,b;c;z)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(b)}\frac{1}{2\pi i}\int_% {-i\infty}^{i\infty}\frac{\Gamma(a+s)\Gamma(b+s)\Gamma(-s)}{\Gamma(c+s)}(-z)^{% s}\,ds.
  2. 1 2 π i - i i Γ ( a + s ) Γ ( b + s ) Γ ( c - s ) Γ ( d - s ) d s = Γ ( a + c ) Γ ( a + d ) Γ ( b + c ) Γ ( b + d ) Γ ( a + b + c + d ) . \frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\Gamma(a+s)\Gamma(b+s)\Gamma(c-s)% \Gamma(d-s)ds=\frac{\Gamma(a+c)\Gamma(a+d)\Gamma(b+c)\Gamma(b+d)}{\Gamma(a+b+c% +d)}.
  3. 1 2 π i - i i Γ ( a + s ) Γ ( b + s ) Γ ( c + s ) Γ ( 1 - d - s ) Γ ( - s ) Γ ( e + s ) d s \frac{1}{2\pi i}\int_{-i\infty}^{i\infty}\frac{\Gamma(a+s)\Gamma(b+s)\Gamma(c+% s)\Gamma(1-d-s)\Gamma(-s)}{\Gamma(e+s)}ds
  4. = Γ ( a ) Γ ( b ) Γ ( c ) Γ ( 1 - d + a ) Γ ( 1 - d + b ) Γ ( 1 - d + c ) Γ ( e - a ) Γ ( e - b ) Γ ( e - c ) =\frac{\Gamma(a)\Gamma(b)\Gamma(c)\Gamma(1-d+a)\Gamma(1-d+b)\Gamma(1-d+c)}{% \Gamma(e-a)\Gamma(e-b)\Gamma(e-c)}

Bartlett's_bisection_theorem.html

  1. Z o c Z_{oc}
  2. Z s c Z_{sc}
  3. Z s c Z_{sc}
  4. Z o c Z_{oc}
  5. Z s c Z_{sc}
  6. 2 Z o c 2Z_{oc}
  7. I = 2 E 2 Z o c I=\frac{2E}{2Z_{oc}}
  8. E I = Z o c \frac{E}{I}=Z_{oc}
  9. 2 Z s c 2Z_{sc}
  10. E I = Z s c \frac{E}{I}=Z_{sc}
  11. Z o c Z_{oc}
  12. Z s c Z_{sc}
  13. A = V 2 E = 2 R 2 R 1 + R 2 A=\frac{V_{2}}{E}=\frac{2R_{2}}{R_{1}+R_{2}}

Barwise_compactness_theorem.html

  1. A A
  2. L L
  3. A A
  4. Γ \Gamma
  5. L A L_{A}
  6. Γ \Gamma
  7. Σ 1 \Sigma_{1}
  8. A A
  9. A A
  10. Γ \Gamma
  11. Γ \Gamma

Barycentric-sum_problem.html

  1. Z n Z_{n}
  2. Z n Z_{n}
  3. | S | |S|
  4. S Z n S\subseteq Z_{n}
  5. | S | 2 |S|\geq 2
  6. a j a_{j}
  7. a i S a i = | S | a j \sum\limits_{a_{i}\in S}a_{i}=|S|a_{j}
  8. S S
  9. a j a_{j}
  10. t t
  11. Z 8 \subseteq Z_{8}
  12. Z 8 \subseteq Z_{8}
  13. n + k - 1 n+k-1
  14. Z n Z_{n}
  15. Z n Z_{n}
  16. 2 n - 1 2n-1
  17. Z n Z_{n}

Basis_(universal_algebra).html

  1. b ( i ) b(i)
  2. b ( i ) b(i)
  3. I I
  4. I I
  5. I I
  6. \ell
  7. ( m ) \ell(m)
  8. p i p_{i}
  9. I I
  10. p i ( m ) = m ( i ) p_{i}(m)=m(i)
  11. I I
  12. I I
  13. I I
  14. \ell
  15. ( b ) \ell(b)
  16. I I
  17. ( b ) = ′′ ( b ) \ell^{\prime}(b)=\ell^{\prime\prime}(b)
  18. = ′′ \ell^{\prime}=\ell^{\prime\prime}
  19. χ \chi
  20. I I
  21. χ ( ( b ) ) = \chi({\ell(b)})=\ell
  22. \ell
  23. ϱ \varrho
  24. ϱ ( e ) = m \varrho(e)=m
  25. m ( i ) = [ ϱ ( e ) ] i = e ( b ( i ) ) m(i)=[\varrho(e)]_{i}=e(b(i))
  26. ϱ \varrho
  27. m = ϱ ( e ) m=\varrho(e)
  28. η \eta
  29. η ( m ) \eta(m)
  30. ϱ ( η ( m ) ) = m \varrho(\eta(m))=m
  31. [ χ ( a ) ] m = [ η ( m ) ] a [\chi(a)]_{m}=[\eta(m)]_{a}
  32. I = { 0 , 1 , n - 1 } I=\{0,1,\ldots n-1\}
  33. n > 0 n>0
  34. \ell
  35. ( b ) = c 0 b 0 + c 1 b 1 + c n - 1 b n - 1 \ell(b)=c_{0}b_{0}+c_{1}b_{1}+\ldots c_{n-1}b_{n-1}
  36. m ( i ) m(i)
  37. ϱ \varrho
  38. η \eta
  39. ( 0 - 1 2 - 2 3 1 1 0 2 ) η 𝗅 ϱ { x 0 = - x 1 + 2 x 2 x 1 = - 2 x 0 + 3 x 1 + x 2 x 2 = x 0 + 2 x 2 {}\qquad\left(\begin{array}[]{rrc}0&-1&2\\ -2&3&1\\ 1&0&2\end{array}\right)\quad\begin{array}[]{c}\stackrel{\eta}{\longmapsto}\\ \stackrel{\varrho}{\longleftarrow\!\!{}^{{}_{\!{}_{\mathsf{l}}}}}\end{array}% \quad\left\{\begin{array}[]{rcrccr}x^{\prime}_{0}&=&&-x_{1}&+&2x_{2}\\ x^{\prime}_{1}&=&-2x_{0}&+3x_{1}&+&x_{2}\\ x^{\prime}_{2}&=&x_{0}&&+&2x_{2}\end{array}\right.
  40. \ell
  41. c i m ( i ) c_{i}m(i)
  42. \ell
  43. I I
  44. ( m ) \ell(m)
  45. ( m ) \ell^{\prime}(m^{\prime})
  46. m m^{\prime}
  47. \ell^{\prime}
  48. \ell
  49. \ell^{\prime}
  50. \ell
  51. I I
  52. I I
  53. I = { 𝖺 , 𝖻 , 𝖼 , } I=\{\mathsf{a,b,c,}\ldots\}
  54. ϵ \epsilon
  55. v w vw
  56. ϵ \epsilon
  57. I I
  58. i {i}
  59. 𝗂 \mathsf{i}
  60. b ( 𝗂 ) = i b(\mathsf{i})=i
  61. i i
  62. 𝗂 \mathsf{i}
  63. { ( , 𝗂 ) } \{(\emptyset,\mathsf{i})\}
  64. ( , 𝗂 ) (\emptyset,\mathsf{i})
  65. ( 𝗂 , ) (\mathsf{i},\emptyset)
  66. m = ϱ ( e ) m=\varrho(e)
  67. i i
  68. w = m ( 𝗂 ) w=m(\mathsf{i})
  69. u = i 0 i 1 i k u={i}_{0}{i}_{1}\cdots{i}_{k}
  70. e ( u ) = m ( 𝗂 0 ) m ( 𝗂 1 ) m ( 𝗂 k ) e(u)=m(\mathsf{i}_{0})m(\mathsf{i}_{1})\cdots m(\mathsf{i}_{k})
  71. ϱ \varrho

Basset_force.html

  1. u t = ν c 2 u y 2 , \frac{\partial u}{\partial t}=\nu_{c}\,\frac{\partial^{2}u}{\partial y^{2}},
  2. u = u 0 - u 0 erf ( y 2 ν c t ) = u 0 erfc ( y 2 ν c t ) , u=u_{0}-u_{0}\,\operatorname{erf}\left(\frac{y}{2\sqrt{\nu_{c}t}}\right)=u_{0}% \operatorname{erfc}\left(\frac{y}{2\sqrt{\nu_{c}t}}\right),
  3. τ = ρ c μ c π 0 t u p t t - t d t , \tau=\sqrt{\frac{\rho_{c}\mu_{c}}{\pi}}\int\limits_{0}^{t}\frac{\frac{\partial u% _{p}}{\partial t^{\prime}}}{\sqrt{t-t^{\prime}}}\,dt^{\prime},
  4. 𝐅 = 3 2 D 2 π ρ c μ c 0 t D 𝐮 D t - 𝐯 t t - t d t , \mathbf{F}=\frac{3}{2}D^{2}\sqrt{\pi\rho_{c}\mu_{c}}\int\limits_{0}^{t}\frac{% \frac{D\mathbf{u}}{Dt^{\prime}}-\frac{\partial\mathbf{v}}{\partial t^{\prime}}% }{\sqrt{t-t^{\prime}}}dt^{\prime},
  5. D / D t {D}/{Dt}

Bass–Serre_theory.html

  1. E E , e e ¯ E\to E,\ e\mapsto\overline{e}
  2. e ¯ \overline{e}
  3. e ¯ ¯ = e \overline{\overline{e}}=e
  4. e ¯ \overline{e}
  5. e ¯ \overline{e}
  6. e ¯ \overline{e}
  7. A e = A e ¯ A_{e}=A_{\overline{e}}
  8. α e : A e A o ( e ) \alpha_{e}:A_{e}\to A_{o(e)}
  9. α e ¯ : A e A t ( e ) \alpha_{\overline{e}}:A_{e}\to A_{t(e)}
  10. ( v V A v ) F ( E ) (\ast_{v\in V}A_{v})\ast F(E)
  11. e ¯ α e ( g ) e = α e ¯ ( g ) \overline{e}\alpha_{e}(g)e=\alpha_{\overline{e}}(g)
  12. g A e g\in A_{e}
  13. e ¯ \overline{e}
  14. ( v V A v ) * F ( E + T ) (\ast_{v\in V}A_{v})*F(E^{+}T)
  15. g A e g\in A_{e}
  16. B = v V A v / ncl { α e ( g ) = ω e ( g ) , where e E + T , g G e } B=\ast_{v\in V}A_{v}/{\rm ncl}\{\alpha_{e}(g)=\omega_{e}(g),\,\text{ where }e% \in E^{+}T,g\in G_{e}\}
  17. B , E + ( A - T ) | e - 1 α e ( g ) e = ω e ( g ) where e E + ( A - T ) , g G e , \langle B,E^{+}(A-T)|e^{-1}\alpha_{e}(g)e=\omega_{e}(g)\,\text{ where }e\in E^% {+}(A-T),g\in G_{e}\rangle,
  18. { e | e E + ( A - T ) } \{e|e\in E^{+}(A-T)\}
  19. G = ( A v ) F ( X ) G=(\ast A_{v})\ast F(X)
  20. g = a 0 e 1 a 1 e n a n , g=a_{0}e_{1}a_{1}\dots e_{n}a_{n},
  21. A v 0 A_{v_{0}}
  22. a i ω e i ( A e i ) a_{i}\in\omega_{e_{i}}(A_{e_{i}})
  23. 𝐀 ~ \tilde{\mathbf{A}}
  24. 𝐀 ~ / π 1 ( 𝐀 , v ) \tilde{\mathbf{A}}/\pi_{1}(\mathbf{A},v)
  25. e ¯ \overline{e}
  26. 𝐀 ~ \tilde{\mathbf{A}}
  27. j : X 𝐀 ~ j:X\to\tilde{\mathbf{A}}
  28. e ¯ \overline{e}
  29. α = α e : C H , ω = ω e : C K \alpha=\alpha_{e}:C\to H,\omega=\omega_{e}:C\to K
  30. G = H C K = H K / ncl { α ( c ) = ω ( c ) , c C } . G=H\ast_{C}K=H\ast K/{\rm ncl}\{\alpha(c)=\omega(c),c\in C\}.
  31. X = 𝐀 ~ X=\tilde{\mathbf{A}}
  32. V X = { g K : g G } { g H : g G } . VX=\{gK:g\in G\}\sqcup\{gH:g\in G\}.
  33. e ¯ \overline{e}
  34. α = α e : C B , ω = ω e : C B \alpha=\alpha_{e}:C\to B,\omega=\omega_{e}:C\to B
  35. G = B , e | e - 1 α ( c ) e = ω ( c ) , c C . G=\langle B,e|e^{-1}\alpha(c)e=\omega(c),c\in C\rangle.
  36. ϕ = ω α - 1 : H K \phi=\omega\circ\alpha^{-1}:H\to K
  37. G = B , e | e - 1 h e = ϕ ( h ) , h H . G=\langle B,e|e^{-1}he=\phi(h),h\in H\rangle.\,
  38. X = 𝐀 ~ X=\tilde{\mathbf{A}}
  39. 𝐀 ~ \tilde{\mathbf{A}}
  40. A ~ \tilde{A}
  41. 𝐀 ~ \tilde{\mathbf{A}}
  42. A ~ \tilde{A}
  43. X ( g ) = min { d ( x , g x ) | x V X } . \ell_{X}(g)=\min\{d(x,gx)|x\in VX\}.
  44. X : G 𝐙 , g G X ( g ) \ell_{X}:G\to\mathbf{Z},\quad g\in G\mapsto\ell_{X}(g)
  45. v o l ( 𝐀 ) = v V 1 | A v | . vol(\mathbf{A})=\sum_{v\in V}\frac{1}{|A_{v}|}.

Bateman_transform.html

  1. ϕ ( w , x , y , z ) = γ f ( ( w + i x ) + ( i y + z ) ζ , ( i y - z ) + ( w - i x ) ζ , ζ ) d ζ \phi(w,x,y,z)=\oint_{\gamma}f\left((w+ix)+(iy+z)\zeta,(iy-z)+(w-ix)\zeta,\zeta% \right)\,d\zeta

Battery_(electricity).html

  1. 1 \mathcal{E}_{1}
  2. 2 \mathcal{E}_{2}
  3. 2 - 1 \mathcal{E}_{2}-\mathcal{E}_{1}
  4. Δ V b a t \displaystyle{\Delta V_{bat}}
  5. \mathcal{E}
  6. t = Q P I k t=\frac{Q_{P}}{I^{k}}
  7. Q P Q_{P}
  8. I I
  9. t t
  10. k k

Baumol–Tobin_model.html

  1. Y Y
  2. Y Y
  3. C C
  4. N N
  5. i i
  6. N C NC
  7. i M iM
  8. M M
  9. Y / 2 Y/2
  10. Y / 4 Y/4
  11. Y / 2 N Y/2N
  12. N C + Y i 2 N NC+\frac{Yi}{2N}
  13. N N
  14. C - Y i 2 N 2 = 0 C-\frac{Yi}{2{N^{2}}}=0
  15. N * = ( Y i 2 C ) 1 2 N^{*}=\left(\frac{Yi}{2C}\right)^{\frac{1}{2}}
  16. M = ( C Y 2 i ) 1 2 M=\left(\frac{CY}{2i}\right)^{\frac{1}{2}}
  17. L ( Y , i ) = M P = ( C Y 2 i ) 1 2 L(Y,i)=\frac{M}{P}=\left(\frac{CY}{2i}\right)^{\frac{1}{2}}

Baum–Sweet_sequence.html

  1. b n = { 0 if m is even b ( m - 1 ) / 2 if m is odd . b_{n}=\begin{cases}0&\,\text{if }m\,\text{ is even}\\ b_{(m-1)/2}&\,\text{if }m\,\text{ is odd}.\end{cases}

BCK_algebra.html

  1. ( X ; , 0 ) \left(X;\ast,0\right)
  2. ( 2 , 0 ) \left(2,0\right)
  3. x , y , z X x,y,z\in X
  4. 0
  5. x y x\ast y
  6. y y
  7. x x
  8. ( ( x y ) ( x z ) ) ( z y ) = 0 \left(\left(x\ast y\right)\ast\left(x\ast z\right)\right)\ast\left(z\ast y% \right)=0
  9. ( x ( x y ) ) y = 0 \left(x\ast\left(x\ast y\right)\right)\ast y=0
  10. x x = 0 x\ast x=0
  11. x y = 0 and y x = 0 x = y x\ast y=0\and y\ast x=0\implies x=y
  12. x 0 = 0 x = 0 x\ast 0=0\implies x=0
  13. ( X ; , 0 ) \left(X;\ast,0\right)
  14. x X : 0 x = 0 \forall x\in X:0\ast x=0
  15. x ( x y ) = y ( y x ) x\ast(x\ast y)=y\ast(y\ast x)

Beauville–Laszlo_theorem.html

  1. F = M A A f = M f G = M A A ^ . F=M\otimes_{A}A_{f}=M_{f}\qquad G=M\otimes_{A}\hat{A}.
  2. G = M ^ G=\widehat{M}
  3. ϕ : G f F A f A ^ f = F A A ^ . \phi\colon G_{f}\xrightarrow{\sim}F\otimes_{A_{f}}\hat{A}_{f}=F\otimes_{A}\hat% {A}.
  4. α : M f F β : M A A ^ G \alpha\colon M_{f}\xrightarrow{\sim}F\qquad\beta\colon M\otimes_{A}\hat{A}% \xrightarrow{\sim}G
  5. G f = G A A f β - 1 1 M A A ^ A A f = M f A A ^ α 1 F A A ^ . G_{f}=G\otimes_{A}A_{f}\xrightarrow{\beta^{-1}\otimes 1}M\otimes_{A}\hat{A}% \otimes_{A}A_{f}=M_{f}\otimes_{A}\hat{A}\xrightarrow{\alpha\otimes 1}F\otimes_% {A}\hat{A}.
  6. 𝐌 f ( A ) 𝐌 f ( A ^ ) 𝐌 ( A f ) 𝐌 ( A ^ f ) \begin{array}[]{ccc}\mathbf{M}_{f}(A)&\longrightarrow&\mathbf{M}_{f}(\hat{A})% \\ \downarrow&&\downarrow\\ \mathbf{M}(A_{f})&\longrightarrow&\mathbf{M}(\hat{A}_{f})\end{array}
  7. 𝐕𝐞𝐜𝐭 r ( X R ) 𝐕𝐞𝐜𝐭 r ( D R ) 𝐕𝐞𝐜𝐭 r ( ( X x ) R ) 𝐕𝐞𝐜𝐭 r ( D R 0 ) \begin{array}[]{ccc}\mathbf{Vect}_{r}(X_{R})&\longrightarrow&\mathbf{Vect}_{r}% (D_{R})\\ \downarrow&&\downarrow\\ \mathbf{Vect}_{r}((X\setminus x)_{R})&\longrightarrow&\mathbf{Vect}_{r}(D_{R}^% {0})\end{array}

Behavior_of_DEVS.html

  1. = < X , Y , S , s 0 , t a , δ e x t , δ i n t , λ Align g t ; \mathcal{M}=<X,Y,S,s_{0},ta,\delta_{ext},\delta_{int},\lambda&gt;
  2. δ e x t : Q × X S \delta_{ext}:Q\times X\rightarrow S
  3. Q = { ( s , t e ) | s S , t e ( 𝕋 [ 0 , t a ( s ) ] ) } Q=\{(s,t_{e})|s\in S,t_{e}\in(\mathbb{T}\cap[0,ta(s)])\}
  4. t e t_{e}
  5. 𝕋 = [ 0 , ) \mathbb{T}=[0,\infty)
  6. \mathcal{M}
  7. 𝒢 = < Z , Q , Q 0 , Q A , Δ Align g t ; \mathcal{G}=<Z,Q,Q_{0},Q_{A},\Delta&gt;
  8. Z = X Y ϕ Z=X\cup Y^{\phi}
  9. Q = Q A Q N Q=Q_{A}\cup Q_{N}
  10. Q N = { s ¯ S } Q_{N}=\{\bar{s}\not\in S\}
  11. Q 0 = { ( s 0 , 0 ) } \,Q_{0}=\{(s_{0},0)\}
  12. Q A = . Q . Q_{A}=\mathcal{M}.Q.
  13. Δ Q × Ω Z , [ t l , t u ] × Q \Delta\subseteq Q\times\Omega_{Z,[t_{l},t_{u}]}\times Q
  14. q Q N q\in Q_{N}
  15. q Q A q\in Q_{A}
  16. q Q N q\in Q_{N}
  17. ω Ω Z , [ t l , t u ] \omega\in\Omega_{Z,[t_{l},t_{u}]}
  18. ( q , ω , q ) Δ . (q,\omega,q)\in\Delta.
  19. q = ( s , t e ) Q A q=(s,t_{e})\in Q_{A}
  20. t 𝕋 t\in\mathbb{T}
  21. ω Ω Z , [ t l , t u ] \omega\in\Omega_{Z,[t_{l},t_{u}]}
  22. ω \omega
  23. ω = ϵ [ t , t + d t ] \omega=\epsilon_{[t,t+dt]}
  24. ( q , ω , ( s , t e + d t ) ) Δ . \,(q,\omega,(s,t_{e}+dt))\in\Delta.\,
  25. ω \omega
  26. ω = ( x , t ) \omega=(x,t)
  27. x X x\in X
  28. ( q , ω , ( δ e x t ( q , x ) , 0 ) ) Δ . (q,\omega,(\delta_{ext}(q,x),0))\in\Delta.
  29. ω \omega
  30. ω = ( y , t ) \omega=(y,t)
  31. y Y ϕ y\in Y^{\phi}
  32. { ( q , ω , ( δ i n t ( s ) , 0 ) ) Δ if t e = t a ( s ) , y = λ ( s ) ( q , ω , s ¯ ) otherwise . \begin{cases}(q,\omega,(\delta_{int}(s),0))\in\Delta&\textrm{if }~{}t_{e}=ta(s% ),y=\lambda(s)\\ (q,\omega,\bar{s})&\textrm{otherwise}.\end{cases}
  33. = < X , Y , S , s 0 , t a , δ e x t , δ i n t , λ Align g t ; \mathcal{M}=<X,Y,S,s_{0},ta,\delta_{ext},\delta_{int},\lambda&gt;
  34. Q = { ( s , t s , t e ) | s S , t s 𝕋 , t e ( 𝕋 [ 0 , t s ] ) } Q=\{(s,t_{s},t_{e})|s\in S,t_{s}\in\mathbb{T}^{\infty},t_{e}\in(\mathbb{T}\cap% [0,t_{s}])\}
  35. t s t_{s}
  36. s s
  37. t e t_{e}
  38. t s t_{s}
  39. 𝕋 = [ 0 , ) { } \mathbb{T}^{\infty}=[0,\infty)\cup\{\infty\}
  40. δ e x t : Q × X S × { 0 , 1 } \delta_{ext}:Q\times X\rightarrow S\times\{0,1\}
  41. Q = 𝒟 Q=\mathcal{D}
  42. 𝒢 = < Z , Q , Q 0 , Q A , Δ Align g t ; \mathcal{G}=<Z,Q,Q_{0},Q_{A},\Delta&gt;
  43. Z = X Y ϕ Z=X\cup Y^{\phi}
  44. Q = Q A Q N Q=Q_{A}\cup Q_{N}
  45. Q N = { s ¯ S } Q_{N}=\{\bar{s}\not\in S\}
  46. Q 0 = { ( s 0 , t a ( s 0 ) , 0 ) } \,Q_{0}=\{(s_{0},ta(s_{0}),0)\}
  47. Q A = . Q Q_{A}=\mathcal{M}.Q
  48. Δ Q × Ω Z , [ t l , t u ] × Q \Delta\subseteq Q\times\Omega_{Z,[t_{l},t_{u}]}\times Q
  49. q Q N q\in Q_{N}
  50. q Q A q\in Q_{A}
  51. q Q N q\in Q_{N}
  52. ω Ω Z , [ t l , t u ] \omega\in\Omega_{Z,[t_{l},t_{u}]}
  53. ( q , ω , q ) Δ . (q,\omega,q)\in\Delta.
  54. q = ( s , t s , t e ) Q A q=(s,t_{s},t_{e})\in Q_{A}
  55. t 𝕋 t\in\mathbb{T}
  56. ω Ω Z , [ t l , t u ] \omega\in\Omega_{Z,[t_{l},t_{u}]}
  57. ω \omega
  58. ω = ϵ [ t , t + d t ] \omega=\epsilon_{[t,t+dt]}
  59. ( q , ω , ( s , t s , t e + d t ) ) Δ . (q,\omega,(s,t_{s},t_{e}+dt))\in\Delta.
  60. ω \omega
  61. ω = ( x , t ) \omega=(x,t)
  62. x X x\in X
  63. { ( q , ω , ( s , t a ( s ) , 0 ) ) Δ if δ e x t ( s , t s , t e , x ) = ( s , 1 ) , ( q , ω , ( s , t s , t e ) ) Δ otherwise, i.e. δ e x t ( s , t s , t e , x ) = ( s , 0 ) . \begin{cases}(q,\omega,(s^{\prime},ta(s^{\prime}),0))\in\Delta&\textrm{if }~{}% \delta_{ext}(s,t_{s},t_{e},x)=(s^{\prime},1),\\ (q,\omega,(s^{\prime},t_{s},t_{e}))\in\Delta&\textrm{otherwise, i.e. }~{}% \delta_{ext}(s,t_{s},t_{e},x)=(s^{\prime},0).\end{cases}
  64. ω \omega
  65. ω = ( y , t ) \omega=(y,t)
  66. y Y ϕ y\in Y^{\phi}
  67. { ( q , ω , ( s , t a ( s ) , 0 ) ) Δ if t e = t s , y = λ ( s ) , δ i n t ( s ) = s , ( q , ω , s ¯ ) Δ otherwise . \begin{cases}(q,\omega,(s^{\prime},ta(s^{\prime}),0))\in\Delta&\textrm{if }~{}% t_{e}=t_{s},y=\lambda(s),\delta_{int}(s)=s^{\prime},\\ (q,\omega,\bar{s})\in\Delta&\textrm{otherwise}.\end{cases}
  68. q = ( s , t e ) Q q=(s,t_{e})\in Q
  69. t a ( s ) = σ \,ta(s)=\sigma
  70. σ \sigma
  71. S = { ( d , σ ) | d S , σ 𝕋 } S=\{(d,\sigma)|d\in S^{\prime},\sigma\in\mathbb{T}^{\infty}\}
  72. S S^{\prime}
  73. x X x\in X
  74. t e t_{e}
  75. x x
  76. σ = σ - t e \,\sigma=\sigma-t_{e}
  77. δ e x t \delta_{ext}
  78. σ \sigma
  79. s = ( d , σ ) S s=(d,\sigma)\in S
  80. t e = 0 t_{e}=0
  81. σ \sigma
  82. δ e x t \delta_{ext}
  83. Δ \Delta
  84. q = ( s , t s , t e ) Q q=(s,t_{s},t_{e})\in Q
  85. σ \sigma
  86. σ = t s - t e . \,\sigma=t_{s}-t_{e}.
  87. x X x\in X
  88. t e t_{e}
  89. δ e x t ( q , x ) = ( s , 1 ) \delta_{ext}(q,x)=(s^{\prime},1)
  90. x x
  91. δ e x t ( q , x ) = ( s , 0 ) \delta_{ext}(q,x)=(s^{\prime},0)
  92. σ \sigma
  93. S S
  94. | S | |S|

Beltrami_identity.html

  1. I [ u ] = a b L [ x , u ( x ) , u ( x ) ] d x , I[u]=\int_{a}^{b}L[x,u(x),u^{\prime}(x)]\,dx\,,
  2. a , b a,b
  3. u ( x ) = d u / d x u′(x)=du/dx
  4. L / x = 0 ∂L/∂x=0
  5. C C
  6. L u = d d x L u . \frac{\partial L}{\partial u}=\frac{d}{dx}\frac{\partial L}{\partial u^{\prime% }}\,.
  7. u u′
  8. u L u = u d d x L u . u^{\prime}\frac{\partial L}{\partial u}=u^{\prime}\frac{d}{dx}\frac{\partial L% }{\partial u^{\prime}}\,.
  9. d L d x = L u u + L u u ′′ + L x , {dL\over dx}={\partial L\over\partial u}u^{\prime}+{\partial L\over\partial u^% {\prime}}u^{\prime\prime}+{\partial L\over\partial x}\,,
  10. u L u = d L d x - L u u ′′ - L x . u^{\prime}{\partial L\over\partial u}={dL\over dx}-{\partial L\over\partial u^% {\prime}}u^{\prime\prime}-{\partial L\over\partial x}\,.
  11. u L / u u′∂L/∂u
  12. d L d x - L u u ′′ - L x - u d d x L u = 0 . {dL\over dx}-{\partial L\over\partial u^{\prime}}u^{\prime\prime}-{\partial L% \over\partial x}-u^{\prime}\frac{d}{dx}\frac{\partial L}{\partial u^{\prime}}=% 0\,.
  13. u d d x L u = d d x ( L u u ) - L u u ′′ , u^{\prime}\frac{d}{dx}\frac{\partial L}{\partial u^{\prime}}=\frac{d}{dx}\left% (\frac{\partial L}{\partial u^{\prime}}u^{\prime}\right)-\frac{\partial L}{% \partial u^{\prime}}u^{\prime\prime}\,,
  14. d d x ( L - u L u ) = L x . {d\over dx}\left({L-u^{\prime}\frac{\partial L}{\partial u^{\prime}}}\right)={% \partial L\over\partial x}\,.
  15. L / x = 0 ∂L/∂x=0
  16. d d x ( L - u L u ) = 0 , {d\over dx}\left({L-u^{\prime}\frac{\partial L}{\partial u^{\prime}}}\right)=0\,,
  17. L - u L u = C , L-u^{\prime}\frac{\partial L}{\partial u^{\prime}}=C\,,
  18. C C
  19. y = y ( x ) y=y(x)
  20. I [ y ] = 0 a 1 + y 2 y d x . I[y]=\int_{0}^{a}\sqrt{{1+y^{\prime\,2}}\over y}dx\,.
  21. L ( y , y ) = 1 + y 2 y L(y,y^{\prime})=\sqrt{{1+y^{\prime\,2}}\over y}
  22. x x
  23. L - y L y = C . L-y^{\prime}\frac{\partial L}{\partial y^{\prime}}=C\,.
  24. L L
  25. y ( 1 + y 2 ) = 1 / C 2 (constant) , y(1+y^{\prime\,2})=1/C^{2}~{}~{}\text{(constant)}\,,
  26. x = A ( ϕ - sin ϕ ) x=A(\phi-\sin\phi)
  27. y = A ( 1 - cos ϕ ) y=A(1-\cos\phi)
  28. A A
  29. φ φ

Benjamin–Bona–Mahony_equation.html

  1. u t + u x + u u x - u x x t = 0. u_{t}+u_{x}+uu_{x}-u_{xxt}=0.\,
  2. u t - 2 u t + div φ ( u ) = 0. u_{t}-\nabla^{2}u_{t}+\operatorname{div}\,\varphi(u)=0.\,
  3. φ \varphi
  4. \mathbb{R}
  5. n \mathbb{R}^{n}
  6. u = 3 c 2 1 - c 2 sech 2 1 2 ( c x - c t 1 - c 2 + δ ) , u=3\frac{c^{2}}{1-c^{2}}\operatorname{sech}^{2}\frac{1}{2}\left(cx-\frac{ct}{1% -c^{2}}+\delta\right),
  7. δ \delta
  8. | c | < 1 |c|<1
  9. x x
  10. 1 / ( 1 - c 2 ) . 1/(1-c^{2}).
  11. u t = - 𝒟 δ H δ u , u_{t}=-\mathcal{D}\frac{\delta H}{\delta u},\,
  12. H = - + ( 1 2 u 2 + 1 6 u 3 ) d x H=\int_{-\infty}^{+\infty}\left(\tfrac{1}{2}u^{2}+\tfrac{1}{6}u^{3}\right)\,\,% \text{d}x\,
  13. 𝒟 = ( 1 - x 2 ) - 1 x . \mathcal{D}=\left(1-\partial_{x}^{2}\right)^{-1}\,\partial_{x}.
  14. δ H / δ u \delta H/\delta u
  15. H ( u ) H(u)
  16. u ( x ) , u(x),
  17. x \partial_{x}
  18. x . x.
  19. u u
  20. u = - v - 1 u=-v-1
  21. v t - v x x t = v v x . v_{t}-v_{xxt}=v\,v_{x}.
  22. v t - ( v x t + 1 2 v 2 ) x = 0 , ( 1 2 v 2 + 1 2 v x 2 ) t - ( v v x t + 1 3 v 3 ) x = 0 and ( 1 3 v 3 ) t + ( v t 2 - v x t 2 - v 2 v x t - 1 4 v 4 ) x = 0. \begin{aligned}\displaystyle v_{t}&\displaystyle-\left(v_{xt}+\tfrac{1}{2}v^{2% }\right)_{x}=0,\\ \displaystyle\left(\tfrac{1}{2}v^{2}+\tfrac{1}{2}v_{x}^{2}\right)_{t}&% \displaystyle-\left(v\,v_{xt}+\tfrac{1}{3}v^{3}\right)_{x}=0\qquad\,\text{and}% \\ \displaystyle\left(\tfrac{1}{3}v^{3}\right)_{t}&\displaystyle+\left(v_{t}^{2}-% v_{xt}^{2}-v^{2}\,v_{xt}-\tfrac{1}{4}v^{4}\right)_{x}=0.\end{aligned}
  23. u u
  24. v = - u - 1. v=-u-1.

Benjamin–Ono_equation.html

  1. u t + u u x + H u x x = 0 u_{t}+uu_{x}+Hu_{xx}=0

Benz_plane.html

  1. \infty
  2. \R 2 \textstyle\R^{2}
  3. y = a x 2 + b x + c y=ax^{2}+bx+c
  4. y = a x 2 + b x + c y=ax^{2}+bx+c
  5. ( , a ) (\infty,a)
  6. ( \R ) × \R (\R\cup{\infty})\times\R
  7. \R 3 \R^{3}
  8. \R 2 \R^{2}
  9. y = m x + d , m 0 y=mx+d,m\neq 0
  10. y = a x - b + c , a 0 y=\tfrac{a}{x-b}+c,a\neq 0
  11. ( , ) (\infty,\infty)
  12. y = a x - b + c , a 0 y=\tfrac{a}{x-b}+c,a\neq 0
  13. ( b , ) , ( , c ) (b,\infty),(\infty,c)
  14. ( \R ) 2 (\R\cup{\infty})^{2}

Berezin_integral.html

  1. Λ n \Lambda^{n}
  2. θ 1 , , θ n \theta_{1},\dots,\theta_{n}
  3. θ 1 , , θ n \theta_{1},\dots,\theta_{n}
  4. Λ n \Lambda^{n}
  5. Λ n d θ \int_{\Lambda^{n}}\cdot\textrm{d}\theta
  6. Λ n θ n θ 1 d θ = 1 , \int_{\Lambda^{n}}\theta_{n}\cdots\theta_{1}\,\mathrm{d}\theta=1,
  7. Λ n f θ i d θ = 0 , i = 1 , , n \int_{\Lambda^{n}}\frac{\partial f}{\partial\theta_{i}}\,\mathrm{d}\theta=0,\ % i=1,\dots,n
  8. f Λ n , f\in\Lambda^{n},
  9. / θ i \partial/\partial\theta_{i}
  10. Λ n f ( θ ) d θ = Λ 1 ( Λ 1 ( Λ 1 f ( θ ) d θ 1 ) d θ 2 ) d θ n \int_{\Lambda^{n}}f\left(\theta\right)\,\mathrm{d}\theta=\int_{\Lambda^{1}}% \left(\cdots\int_{\Lambda^{1}}\left(\int_{\Lambda^{1}}f\left(\theta\right)\,% \mathrm{d}\theta_{1}\right)\,\mathrm{d}\theta_{2}\cdots\right)\mathrm{d}\theta% _{n}
  11. f = g ( θ ) θ 1 f=g\left(\theta^{\prime}\right)\theta_{1}
  12. g ( θ ) , g\left(\theta^{\prime}\right),
  13. θ = ( θ 2 , , θ n ) \theta^{\prime}=\left(\theta_{2},...,\theta_{n}\right)
  14. f = g ( θ ) f=g\left(\theta^{\prime}\right)
  15. θ 2 \theta_{2}
  16. θ i = θ i ( ξ 1 , , ξ n ) , i = 1 , , n , \theta_{i}=\theta_{i}\left(\xi_{1},...,\xi_{n}\right),\ i=1,...,n,
  17. ξ 1 , , ξ n \xi_{1},...,\xi_{n}
  18. D = { θ i ξ j , i , j = 1 , , n } , D=\left\{\frac{\partial\theta_{i}}{\partial\xi_{j}},\ i,j=1,...,n\right\},
  19. f ( θ ) d θ = f ( θ ( ξ ) ) ( det D ) - 1 d ξ . \int f\left(\theta\right)\mathrm{d}\theta=\int f\left(\theta\left(\xi\right)% \right)\left(\det D\right)^{-1}\mathrm{d}\xi.
  20. Λ m n \Lambda^{m\mid n}
  21. x = x 1 , , x m x=x_{1},...,x_{m}
  22. θ 1 , , θ n \theta_{1},...,\theta_{n}
  23. ( m n ) \left(m\mid n\right)
  24. f = f ( x , θ ) Λ m n f=f\left(x,\theta\right)\in\Lambda^{m\mid n}
  25. x x
  26. X m X\subset\mathbb{R}^{m}
  27. Λ n . \Lambda^{n}.
  28. [ - F o r m u l a E r r o r - ] [-FormulaError-]
  29. K m . K\subset\mathbb{R}^{m}.
  30. Λ m n f ( x , θ ) d θ d x = m d x Λ n f ( x , θ ) d θ . \int_{\Lambda^{m\mid n}}f\left(x,\theta\right)\mathrm{d}\theta\mathrm{d}x=\int% _{\mathbb{R}^{m}}\mathrm{d}x\int_{\Lambda^{n}}f\left(x,\theta\right)\mathrm{d}\theta.
  31. x i = x i ( y , ξ ) , i = 1 , , m ; θ j = θ j ( y , ξ ) , j = 1 , , n x_{i}=x_{i}\left(y,\xi\right),\ i=1,...,m;\ \theta_{j}=\theta_{j}\left(y,\xi% \right),j=1,...,n
  32. x i , y i x_{i},y_{i}
  33. θ j , ξ j \theta_{j},\xi_{j}
  34. ξ \xi
  35. y . y.
  36. J = ( x , θ ) ( y , ξ ) = ( A B C D ) , \mathrm{J}=\frac{\partial\left(x,\theta\right)}{\partial\left(y,\xi\right)}=% \left(\begin{array}[c]{cc}A&B\\ C&D\end{array}\right),
  37. / y j \partial/\partial y_{j}
  38. Λ m n \Lambda^{m\mid n}
  39. A = x / y A=\partial x/\partial y
  40. D = θ / ξ D=\partial\theta/\partial\xi
  41. B = x / ξ , C = θ / y B=\partial x/\partial\xi,\ C=\partial\theta/\partial y
  42. / ξ j \partial/\partial\xi_{j}
  43. J \mathrm{J}
  44. Ber J = det ( A - B D - 1 C ) det D - 1 \mathrm{Ber~{}J}=\det\left(A-BD^{-1}C\right)\det D^{-1}
  45. det D \det D
  46. Λ m n . \Lambda^{m\mid n}.
  47. x i = x i ( y , 0 ) x_{i}=x_{i}\left(y,0\right)
  48. F : Y X F:Y\rightarrow X
  49. X , Y X,\ Y
  50. m \mathbb{R}^{m}
  51. ξ θ = θ ( y , ξ ) \xi\mapsto\theta=\theta\left(y,\xi\right)
  52. y Y . y\in Y.
  53. Λ m n f ( x , θ ) d θ d x = Λ m n f ( x ( y , ξ ) , θ ( y , ξ ) ) ε Ber J d ξ d y \int_{\Lambda^{m\mid n}}f\left(x,\theta\right)\mathrm{d}\theta\mathrm{d}x=\int% _{\Lambda^{m\mid n}}f\left(x\left(y,\xi\right),\theta\left(y,\xi\right)\right)% \varepsilon\mathrm{Ber~{}J~{}d}\xi\mathrm{d}y
  54. = Λ m n f ( x ( y , ξ ) , θ ( y , ξ ) ) ε det ( A - B D - 1 C ) det D d ξ d y , =\int_{\Lambda^{m\mid n}}f\left(x\left(y,\xi\right),\theta\left(y,\xi\right)% \right)\varepsilon\frac{\det\left(A-BD^{-1}C\right)}{\det D}\mathrm{d}\xi% \mathrm{d}y,
  55. ε = sgn det x ( y , 0 ) / y \varepsilon=\mathrm{sgn~{}\det}\partial x\left(y,0\right)/\partial y
  56. F . F.
  57. f ( x ( y , ξ ) , θ ( y , ξ ) ) f\left(x\left(y,\xi\right),\theta\left(y,\xi\right)\right)
  58. x i ( y , ξ ) x_{i}\left(y,\xi\right)
  59. ξ . \xi.
  60. x i ( y , ξ ) = x i ( y , 0 ) + δ i , x_{i}\left(y,\xi\right)=x_{i}\left(y,0\right)+\delta_{i},
  61. δ i , i = 1 , , m \delta_{i},\ i=1,...,m
  62. Λ m n \Lambda^{m\mid n}
  63. f ( x ( y , ξ ) , θ ) = f ( x ( y , 0 ) , θ ) + i f x i ( x ( y , 0 ) , θ ) δ i + 1 2 i , j 2 f x i x j ( x ( y , 0 ) , θ ) δ i δ j + , f\left(x\left(y,\xi\right),\theta\right)=f\left(x\left(y,0\right),\theta\right% )+\sum_{i}\frac{\partial f}{\partial x_{i}}\left(x\left(y,0\right),\theta% \right)\delta_{i}+\frac{1}{2}\sum_{i,j}\frac{\partial^{2}f}{\partial x_{i}% \partial x_{j}}\left(x\left(y,0\right),\theta\right)\delta_{i}\delta_{j}+...,

Bernstein's_constant.html

  1. 1 3 + 1 1 + 1 1 + 1 3 + 1 9 + \cfrac{1}{3+\cfrac{1}{1+\cfrac{1}{1+\cfrac{1}{3+\cfrac{1}{9+\ddots}}}}}
  2. β = lim n 2 n E 2 n ( f ) , \beta=\lim_{n\to\infty}2nE_{2n}(f),\,
  3. 1 2 π = 0.28209 . \frac{1}{2\sqrt{\pi}}=0.28209\dots\,.
  4. β = 0.280169499023 . \beta=0.280169499023\dots\,.

Bertrand–Diquet–Puiseux_theorem.html

  1. K ( p ) = lim r 0 + 3 2 π r - C ( r ) π r 3 = lim r 0 + 12 π r 2 - A ( r ) π r 4 . K(p)=\lim_{r\to 0^{+}}3\frac{2\pi r-C(r)}{\pi r^{3}}=\lim_{r\to 0^{+}}12\frac{% \pi r^{2}-A(r)}{\pi r^{4}}.

Besicovitch_covering_theorem.html

  1. E i = 1 c N B A i B . E\subseteq\bigcup_{i=1}^{c_{N}}\bigcup_{B\in A_{i}}B.
  2. 𝟏 E S 𝐆 := B 𝐆 𝟏 B b N . \mathbf{1}_{E}\leq S_{\mathbf{G}}:=\sum_{B\in\mathbf{G}}\mathbf{1}_{B}\leq b_{% N}.
  3. f * f^{*}
  4. × 0 = 0 \infty\times 0=0
  5. f * ( x ) = sup r > 0 ( μ ( B ( x , r ) ) - 1 B ( x , r ) | f ( y ) | d μ ( y ) ) . f^{*}(x)=\sup_{r>0}\Bigl(\mu(B(x,r))^{-1}\int_{B(x,r)}|f(y)|\,d\mu(y)\Bigr).
  6. λ μ ( { x : f * ( x ) > λ } ) b N | f | d μ . \lambda\,\mu\bigl(\{x:f^{*}(x)>\lambda\}\bigr)\leq b_{N}\,\int|f|\,d\mu.
  7. f * ( x ) > λ f^{*}(x)>\lambda
  8. 𝟏 B | f | d μ = B | f ( y ) | d μ ( y ) > λ μ ( B ) . \int\mathbf{1}_{B}\,|f|\ d\mu=\int_{B}|f(y)|\,d\mu(y)>\lambda\,\mu(B).
  9. λ μ ( E ) λ B 𝐆 μ ( B ) B 𝐆 𝟏 B | f | d μ = S 𝐆 | f | d μ b N | f | d μ , \begin{aligned}\displaystyle\lambda\,\mu(E^{\prime})&\displaystyle\leq\lambda% \,\sum_{B\in\mathbf{G}}\mu(B)\\ &\displaystyle\leq\sum_{B\in\mathbf{G}}\int\mathbf{1}_{B}\,|f|\,d\mu=\int S_{% \mathbf{G}}\,|f|\,d\mu\leq b_{N}\,\int|f|\,d\mu,\end{aligned}

Besov_space.html

  1. B p , q s ( 𝐑 ) B^{s}_{p,q}(\mathbf{R})
  2. 1 p , q 1≤p,q≤∞
  3. Δ h f ( x ) = f ( x - h ) - f ( x ) \Delta_{h}f(x)=f(x-h)-f(x)
  4. ω p 2 ( f , t ) = sup | h | t Δ h 2 f p \omega^{2}_{p}(f,t)=\sup_{|h|\leq t}\left\|\Delta^{2}_{h}f\right\|_{p}
  5. n n
  6. s = n + α s=n+α
  7. B p , q s ( 𝐑 ) B^{s}_{p,q}(\mathbf{R})
  8. f f
  9. f W s , p ( 𝐑 ) , 0 | ω p 2 ( f ( n ) , t ) t α | q d t t < . f\in W^{s,p}(\mathbf{R}),\qquad\int_{0}^{\infty}\left|\frac{\omega^{2}_{p}% \left(f^{(n)},t\right)}{t^{\alpha}}\right|^{q}\frac{dt}{t}<\infty.
  10. B p , q s ( 𝐑 ) B^{s}_{p,q}(\mathbf{R})
  11. f B p , q s ( 𝐑 ) = ( f W n , p ( 𝐑 ) q + 0 | ω p 2 ( f ( n ) , t ) t α | q d t t ) 1 q \left\|f\right\|_{B^{s}_{p,q}(\mathbf{R})}=\left(\|f\|_{W^{n,p}(\mathbf{R})}^{% q}+\int_{0}^{\infty}\left|\frac{\omega^{2}_{p}\left(f^{(n)},t\right)}{t^{% \alpha}}\right|^{q}\frac{dt}{t}\right)^{\frac{1}{q}}
  12. B 2 , 2 s ( 𝐑 ) B^{s}_{2,2}(\mathbf{R})
  13. H s ( 𝐑 ) H^{s}(\mathbf{R})
  14. p = q p=q
  15. B p , p s ( 𝐑 ) = W s , p ( 𝐑 ) B^{s}_{p,p}(\mathbf{R})=W^{s,p}(\mathbf{R})

Best_linear_unbiased_prediction.html

  1. Y j = μ + x j T β + ξ j + ε j , Y_{j}=\mu+x_{j}^{T}\beta+\xi_{j}+\varepsilon_{j},\,
  2. Y k ~ = μ + x k T β + ξ k , \tilde{Y_{k}}=\mu+x_{k}^{T}\beta+\xi_{k},
  3. Y k ^ = j = 1 n c j , k Y j , \widehat{Y_{k}}=\sum_{j=1}^{n}c_{j,k}Y_{j},
  4. V = Var ( Y k ~ - Y k ^ ) , V=\operatorname{Var}(\tilde{Y_{k}}-\widehat{Y_{k}}),
  5. E ( Y k ~ - Y k ^ ) = 0. \operatorname{E}(\tilde{Y_{k}}-\widehat{Y_{k}})=0.
  6. Y k ~ \tilde{Y_{k}}
  7. Y k Y_{k}
  8. Y k ^ \widehat{Y_{k}}

Bi-isotropic_material.html

  1. D = ε E + ξ H D=\varepsilon E+\xi H\,
  2. B = μ H + ζ E B=\mu H+\zeta E\,
  3. χ - i κ = ξ ε μ \chi-i\kappa=\frac{\xi}{\sqrt{\varepsilon\mu}}
  4. χ + i κ = ζ ε μ \chi+i\kappa=\frac{\zeta}{\sqrt{\varepsilon\mu}}
  5. D = ε E + ( χ - i κ ) ε μ H D=\varepsilon E+(\chi-i\kappa)\sqrt{\varepsilon\mu}H
  6. B = μ H + ( χ + i κ ) ε μ E B=\mu H+(\chi+i\kappa)\sqrt{\varepsilon\mu}E
  7. κ = 0 \kappa=0\,
  8. κ 0 \kappa\neq 0
  9. χ = 0 \chi=0\,
  10. χ 0 \chi\neq 0
  11. D = ε E - i κ ε μ H D=\varepsilon E-i\kappa\sqrt{\varepsilon\mu}H
  12. B = μ H + χ ε μ E B=\mu H+\chi\sqrt{\varepsilon\mu}E

Bias_ratio.html

  1. r i = r_{i}=
  2. Bias Ratio = BR = Count ( r i | r i [ 0 , σ ] ) 1 + Count ( r i | r i [ - σ , 0 ) ) \mathrm{Bias\ Ratio}=\mathrm{BR}=\frac{\mathrm{Count}(r_{i}|r_{i}\in[0,\sigma]% )}{1+\mathrm{Count}(r_{i}|r_{i}\in[-\sigma,0))}
  3. 0 BR n 0\leq\mathrm{BR}\leq n
  4. r i 0 , i , r_{i}\leq 0,\forall{i},
  5. r i \forall{r_{i}}
  6. r i > 0 , r i > 1.0 σ r_{i}>0,\ r_{i}>1.0\sigma
  7. r i r_{i}

Bidirectional_associative_memory.html

  1. M = X i t Y i M=\sum{{}^{t}\!X_{i}Y_{i}}
  2. X i t {}^{t}\!X_{i}
  3. M = [ 2 0 0 - 2 0 - 2 2 0 2 0 0 - 2 - 2 0 0 2 0 2 - 2 0 - 2 0 0 2 ] M=\left[{\begin{array}[]{*{10}c}2&0&0&-2\\ 0&-2&2&0\\ 2&0&0&-2\\ -2&0&0&2\\ 0&2&-2&0\\ -2&0&0&2\\ \end{array}}\right]

Bilateral_hypergeometric_series.html

  1. H p p ( a 1 , , a p ; b 1 , , b p ; z ) = H p p ( a 1 a p b 1 b p ; z ) = n = - ( a 1 ) n ( a 2 ) n ( a p ) n ( b 1 ) n ( b 2 ) n ( b p ) n z n {}_{p}H_{p}(a_{1},\ldots,a_{p};b_{1},\ldots,b_{p};z)={}_{p}H_{p}\left(\begin{% matrix}a_{1}&\ldots&a_{p}\\ b_{1}&\ldots&b_{p}\\ \end{matrix};z\right)=\sum_{n=-\infty}^{\infty}\frac{(a_{1})_{n}(a_{2})_{n}% \ldots(a_{p})_{n}}{(b_{1})_{n}(b_{2})_{n}\ldots(b_{p})_{n}}z^{n}
  2. ( a ) n = a ( a + 1 ) ( a + 2 ) ( a + n - 1 ) (a)_{n}=a(a+1)(a+2)\cdots(a+n-1)\,
  3. ( b 1 + b n - a 1 - - a n ) > 1. \Re(b_{1}+\cdots b_{n}-a_{1}-\cdots-a_{n})>1.
  4. H 2 2 ( a , b ; c , d ; 1 ) = - ( a ) n ( b ) n ( c ) n ( d ) n = Γ ( d ) Γ ( c ) Γ ( 1 - a ) Γ ( 1 - b ) Γ ( c + d - a - b - 1 ) Γ ( c - a ) Γ ( c - b ) Γ ( d - a ) Γ ( d - b ) {}_{2}H_{2}(a,b;c,d;1)=\sum_{-\infty}^{\infty}\frac{(a)_{n}(b)_{n}}{(c)_{n}(d)% _{n}}=\frac{\Gamma(d)\Gamma(c)\Gamma(1-a)\Gamma(1-b)\Gamma(c+d-a-b-1)}{\Gamma(% c-a)\Gamma(c-b)\Gamma(d-a)\Gamma(d-b)}
  5. n = - Γ ( a + n ) Γ ( b + n ) Γ ( c + n ) Γ ( d + n ) = π 2 sin ( π a ) sin ( π b ) Γ ( c + d - a - b - 1 ) Γ ( c - a ) Γ ( d - a ) Γ ( c - b ) Γ ( d - b ) . \sum_{n=-\infty}^{\infty}\frac{\Gamma(a+n)\Gamma(b+n)}{\Gamma(c+n)\Gamma(d+n)}% =\frac{\pi^{2}}{\sin(\pi a)\sin(\pi b)}\frac{\Gamma(c+d-a-b-1)}{\Gamma(c-a)% \Gamma(d-a)\Gamma(c-b)\Gamma(d-b)}.
  6. H 3 3 ( a , b , f + 1 ; d , e , f ; 1 ) = - ( a ) n ( b ) n ( f + 1 ) n ( d ) n ( e ) n ( f ) n = λ Γ ( d ) Γ ( e ) Γ ( 1 - a ) Γ ( 1 - b ) Γ ( d + e - a - b - 2 ) Γ ( d - a ) Γ ( d - b ) Γ ( e - a ) Γ ( e - b ) {}_{3}H_{3}(a,b,f+1;d,e,f;1)=\sum_{-\infty}^{\infty}\frac{(a)_{n}(b)_{n}(f+1)_% {n}}{(d)_{n}(e)_{n}(f)_{n}}=\lambda\frac{\Gamma(d)\Gamma(e)\Gamma(1-a)\Gamma(1% -b)\Gamma(d+e-a-b-2)}{\Gamma(d-a)\Gamma(d-b)\Gamma(e-a)\Gamma(e-b)}
  7. λ = f - 1 [ ( f - a ) ( f - b ) - ( 1 + f - d ) ( 1 + f - e ) ] . \lambda=f^{-1}\left[(f-a)(f-b)-(1+f-d)(1+f-e)\right].

Bilevel_optimization.html

  1. min x X , y Y F ( x , y ) \min\limits_{x\in X,y\in Y}\;\;F(x,y)
  2. G i ( x , y ) 0 G_{i}(x,y)\leq 0
  3. i { 1 , 2 , , I } i\in\{1,2,\ldots,I\}
  4. y arg min z Y { f ( x , z ) : g j ( x , z ) 0 , j { 1 , 2 , , J } } y\in\arg\min\limits_{z\in Y}\{f(x,z):g_{j}(x,z)\leq 0,j\in\{1,2,\ldots,J\}\}
  5. F , f : R n x × R n y R F,f:R^{n_{x}}\times R^{n_{y}}\to R
  6. G i , g j : R n x × R n y R G_{i},g_{j}:R^{n_{x}}\times R^{n_{y}}\to R
  7. X R n x X\subseteq R^{n_{x}}
  8. Y R n y . Y\subseteq R^{n_{y}}.
  9. F F
  10. f f
  11. x x
  12. y y
  13. G i G_{i}
  14. g j g_{j}
  15. min x X , y Y F ( x , y ) = ( F 1 ( x , y ) , F 2 ( x , y ) , , F p ( x , y ) ) \min\limits_{x\in X,y\in Y}\;\;F(x,y)=(F_{1}(x,y),F_{2}(x,y),\ldots,F_{p}(x,y))
  16. G i ( x , y ) 0 G_{i}(x,y)\leq 0
  17. i { 1 , 2 , , I } i\in\{1,2,\ldots,I\}
  18. y arg min z Y { f ( x , z ) = ( f 1 ( x , z ) , f 2 ( x , z ) , , f q ( x , z ) ) : g j ( x , z ) 0 , j { 1 , 2 , , J } } y\in\arg\min\limits_{z\in Y}\{f(x,z)=(f_{1}(x,z),f_{2}(x,z),\ldots,f_{q}(x,z))% :g_{j}(x,z)\leq 0,j\in\{1,2,\ldots,J\}\}
  19. F : R n x × R n y R p F:R^{n_{x}}\times R^{n_{y}}\to R^{p}
  20. f : R n x × R n y R q f:R^{n_{x}}\times R^{n_{y}}\to R^{q}
  21. G i , g j : R n x × R n y R G_{i},g_{j}:R^{n_{x}}\times R^{n_{y}}\to R
  22. X R n x X\subseteq R^{n_{x}}
  23. Y R n y . Y\subseteq R^{n_{y}}.
  24. F F
  25. p p
  26. f f
  27. q q
  28. x x
  29. y y
  30. G i G_{i}
  31. g j g_{j}

Bingham_distribution.html

  1. f ( 𝐱 ; M , Z ) d S n - 1 = F 1 1 ( 1 2 ; n 2 ; Z ) - 1 exp ( tr Z M T 𝐱𝐱 T M ) d S n - 1 f(\mathbf{x}\,;\,M,Z)\;dS^{n-1}\;=\;{}_{1}F_{1}({\textstyle\frac{1}{2}};{% \textstyle\frac{n}{2}};Z)^{-1}\;\cdot\;\exp\left({\textrm{tr}\;ZM^{T}\mathbf{x% }\mathbf{x}^{T}M}\right)\;dS^{n-1}
  2. f ( 𝐱 ; M , Z ) d S n - 1 = F 1 1 ( 1 2 ; n 2 ; Z ) - 1 exp ( 𝐱 T M Z M T 𝐱 ) d S n - 1 f(\mathbf{x}\,;\,M,Z)\;dS^{n-1}\;=\;{}_{1}F_{1}({\textstyle\frac{1}{2}};{% \textstyle\frac{n}{2}};Z)^{-1}\;\cdot\;\exp\left({\mathbf{x}^{T}MZM^{T}\mathbf% {x}}\right)\;dS^{n-1}
  3. F 1 1 ( ; , ) {}_{1}F_{1}(\cdot;\cdot,\cdot)

Binomial_differential_equation.html

  1. ( y ) m = f ( x , y ) , \left(y^{\prime}\right)^{m}=f(x,y),\,
  2. m m
  3. f ( x , y ) f(x,y)

Biogas_upgrader.html

  1. \Leftrightarrow

Bioheat_transfer.html

  1. d 2 T d x 2 + q ˙ m + q ˙ p k = 0 [ 1 ] \frac{d^{2}T}{dx^{2}}+\frac{\dot{q}_{m}+\dot{q}_{p}}{k}=0\quad[1]
  2. q ˙ m \dot{q}_{m}
  3. q ˙ p \dot{q}_{p}
  4. k k
  5. ω \omega
  6. q ˙ p = ω ρ b c b ( T a - T ) [ 2 ] \dot{q}_{p}=\omega\rho_{b}c_{b}(T_{a}-T)\quad[2]
  7. ρ b \rho_{b}
  8. c b c_{b}
  9. d 2 T d x 2 + q ˙ m + ω ρ b c b ( T a - T ) k = 0 [ 3 ] \frac{d^{2}T}{dx^{2}}+\frac{\dot{q}_{m}+\omega\rho_{b}c_{b}(T_{a}-T)}{k}=0% \quad[3]

Bipartite_dimension.html

  1. K n K_{n}
  2. log 2 n \lceil\log_{2}n\rceil
  3. σ ( n ) \sigma(n)
  4. σ ( n ) = min { k n ( k k / 2 ) } \sigma(n)=\min\left\{\,k\mid n\leq{\left({{k}\atop{\lfloor k/2\rfloor}}\right)% }\,\right\}
  5. P n P_{n}
  6. d ( P n ) = n / 2 d(P_{n})=\lfloor n/2\rfloor
  7. C n C_{n}
  8. d ( C n ) = n / 2 d(C_{n})=\lceil n/2\rceil
  9. G = ( V , E ) G=(V,E)
  10. k k
  11. k k
  12. 𝒮 = { S 1 , , S n } \mathcal{S}=\{S_{1},\ldots,S_{n}\}
  13. 𝒰 \mathcal{U}
  14. 𝒮 \mathcal{S}
  15. = { B 1 , , B } \mathcal{B}=\{B_{1},\ldots,B_{\ell}\}
  16. 𝒰 \mathcal{U}
  17. S i S_{i}
  18. \mathcal{B}
  19. 𝒰 \mathcal{U}
  20. 𝒮 = { S 1 , , S n } \mathcal{S}=\{S_{1},\ldots,S_{n}\}
  21. 𝒰 \mathcal{U}
  22. k k
  23. 𝒮 \mathcal{S}
  24. O ( log n ) O(\log\,\!n)
  25. | V | 1 / 3 - ϵ |V|^{1/3-\epsilon}
  26. ϵ > 0 \epsilon>0
  27. O ( f ( k ) ) + n 3 O(f(k))+n^{3}
  28. f ( k ) = 2 k 2 k - 1 + 3 k f(k)=2^{k2^{k-1}+3k}

Bipartite_double_cover.html

  1. [ 0 A A T 0 ] , \left[\begin{matrix}0&A\\ A^{T}&0\end{matrix}\right],

Birch's_law.html

  1. v p = a ( M avg ) + b ρ , v_{p}=a(M_{\mathrm{avg}})+b\rho,
  2. a ( x ) a(x)
  3. < m a t h > ρ <math>\rho

Birkhoff's_representation_theorem.html

  1. m ( x , y , z ) = ( x y ) ( x z ) ( y z ) = ( x y ) ( x z ) ( y z ) m(x,y,z)=(x\vee y)\wedge(x\vee z)\wedge(y\vee z)=(x\wedge y)\vee(x\wedge z)% \vee(y\wedge z)

Birnbaum–Orlicz_space.html

  1. n | f ( x ) | log + | f ( x ) | d x < . \int_{\mathbb{R}^{n}}|f(x)|\log^{+}|f(x)|\,dx<\infty.
  2. Φ ( x ) x , as x , \frac{\Phi(x)}{x}\to\infty,\quad\mathrm{as\ \ }x\to\infty,
  3. Φ ( x ) x 0 , as x 0. \frac{\Phi(x)}{x}\to 0,\quad\mathrm{as\ \ }x\to 0.
  4. L Φ L^{\dagger}_{\Phi}
  5. X Φ ( | f | ) d μ \int_{X}\Phi(|f|)\,d\mu
  6. L Φ L^{\dagger}_{\Phi}
  7. L Φ L_{\Phi}
  8. L Φ L_{\Phi}
  9. Ψ ( x ) = 0 x ( Φ ) - 1 ( t ) d t . \Psi(x)=\int_{0}^{x}(\Phi^{\prime})^{-1}(t)\,dt.
  10. a b Φ ( a ) + Ψ ( b ) . ab\leq\Phi(a)+\Psi(b).
  11. f Φ = sup { f g 1 Ψ | g | d μ 1 } . \|f\|_{\Phi}=\sup\left\{\|fg\|_{1}\mid\int\Psi\circ|g|\,d\mu\leq 1\right\}.
  12. L Φ L_{\Phi}
  13. f Φ = inf { k ( 0 , ) X Φ ( | f | / k ) d μ 1 } , \|f\|^{\prime}_{\Phi}=\inf\left\{k\in(0,\infty)\mid\int_{X}\Phi(|f|/k)\,d\mu% \leq 1\right\},
  14. L Φ L^{\dagger}_{\Phi}
  15. L Φ L_{\Phi}
  16. L Φ L_{\Phi}
  17. L Φ L^{\dagger}_{\Phi}
  18. 1 < p < 1<p<\infty
  19. φ ( t ) = | t | p \varphi(t)=|t|^{p}
  20. u L φ ( X ) = u L p ( X ) \|u\|_{L^{\varphi}(X)}=\|u\|_{L^{p}(X)}
  21. L φ ( X ) = L p ( X ) L^{\varphi}(X)=L^{p}(X)
  22. L φ ( X ) L^{\varphi}(X)
  23. X n X\subseteq\mathbb{R}^{n}
  24. X \partial X
  25. W 0 1 , p ( X ) L φ ( X ) W_{0}^{1,p}(X)\subseteq L^{\varphi}(X)
  26. φ ( t ) := exp ( | t | p / ( p - 1 ) ) - 1. \varphi(t):=\exp\left(|t|^{p/(p-1)}\right)-1.
  27. X n X\subseteq\mathbb{R}^{n}
  28. X \partial X
  29. W 0 k , p ( X ) W_{0}^{k,p}(X)
  30. k p = n kp=n
  31. C 1 , C 2 > 0 C_{1},C_{2}>0
  32. X exp ( ( | u ( x ) | C 1 D k u L p ( X ) ) p / ( p - 1 ) ) d x C 2 | X | . \int_{X}\exp\left(\left(\frac{|u(x)|}{C_{1}\|\mathrm{D}^{k}u\|_{L^{p}(X)}}% \right)^{p/(p-1)}\right)\,\mathrm{d}x\leq C_{2}|X|.
  33. X Ψ inf { k ( 0 , ) E [ Ψ ( | X | / k ) ] 1 } . \|X\|_{\Psi}\triangleq\inf\left\{k\in(0,\infty)\mid E[\Psi(|X|/k)]\leq 1\right\}.
  34. Ψ ( x ) = x p \Psi(x)=x^{p}
  35. Ψ q ( x ) = exp ( x q ) - 1 \Psi_{q}(x)=\exp(x^{q})-1
  36. q 1 q\geq 1
  37. Ψ 2 \Psi_{2}
  38. Ψ 1 \Psi_{1}
  39. Ψ p \Psi_{p}
  40. X Ψ p = c lim x f X ( x ) exp ( | x / c | p ) = 0 , \|X\|_{\Psi_{p}}=c\rightarrow\lim_{x\rightarrow\infty}f_{X}(x)\exp(|x/c|^{p})=0,
  41. exp ( - | x / c | p ) \exp(-|x/c|^{p})
  42. Ψ 1 \Psi_{1}
  43. M X ( t ) = ( 1 - 2 t ) - K / 2 M_{X}(t)=(1-2t)^{-K/2}
  44. Ψ 1 \Psi_{1}
  45. X Ψ 1 - 1 = M X - 1 ( 2 ) = ( 1 - 4 - 1 / K ) / 2. \|X\|_{\Psi_{1}}^{-1}=M_{X}^{-1}(2)=(1-4^{-1/K})/2.

Birnbaum–Saunders_distribution.html

  1. P ( X ω ) = Φ ( ω - n μ σ n ) P(X\leq\omega)=\Phi\left(\frac{\omega-n\mu}{\sigma\sqrt{n}}\right)
  2. P ( T t ) = 1 - Φ ( ω - t μ σ t ) = Φ ( t μ - ω σ t ) = Φ ( μ t σ - ω σ t ) = Φ ( μ ω σ [ ( t ω / μ ) 0.5 - ( ω / μ t ) 0.5 ] ) P(T\leq t)=1-\Phi\left(\frac{\omega-t\mu}{\sigma\sqrt{t}}\right)=\Phi\left(% \frac{t\mu-\omega}{\sigma\sqrt{t}}\right)=\Phi\left(\frac{\mu\sqrt{t}}{\sigma}% -\frac{\omega}{\sigma\sqrt{t}}\right)=\Phi\left(\frac{\sqrt{\mu\omega}}{\sigma% }\left[\left(\frac{t}{\omega/\mu}\right)^{0.5}-\left(\frac{\omega/\mu}{t}% \right)^{0.5}\right]\right)
  3. F ( x ; α , β ) = Φ ( 1 α [ ( x β ) 0.5 - ( β x ) 0.5 ] ) F(x;\alpha,\beta)=\Phi\left(\frac{1}{\alpha}\left[\left(\frac{x}{\beta}\right)% ^{0.5}-\left(\frac{\beta}{x}\right)^{0.5}\right]\right)
  4. μ = β ( 1 + α 2 2 ) \mu=\beta(1+\frac{\alpha^{2}}{2})
  5. σ 2 = ( α β ) 2 ( 1 + 5 α 2 4 ) \sigma^{2}=(\alpha\beta)^{2}(1+\frac{5\alpha^{2}}{4})
  6. γ = 16 α 2 ( 11 α 2 + 6 ) ( 5 α 2 + 4 ) 3 \gamma=\frac{16\alpha^{2}(11\alpha^{2}+6)}{(5\alpha^{2}+4)^{3}}
  7. κ = 3 + 6 α 2 ( 93 α 2 + 41 ) ( 5 α 2 + 4 ) 2 \kappa=3+\frac{6\alpha^{2}(93\alpha^{2}+41)}{(5\alpha^{2}+4)^{2}}
  8. { 2 α 2 β x 2 ( β + x ) f ( x ) + f ( x ) ( - β 3 + x 3 + ( α 2 + 1 ) β x 2 + ( 3 α 2 - 1 ) β 2 x ) = 0 , f ( 1 ) = ( β + 1 ) e - ( β - 1 ) 2 2 α 2 β 2 2 π α β } \left\{\begin{array}[]{l}2\alpha^{2}\beta x^{2}(\beta+x)f^{\prime}(x)+f(x)% \left(-\beta^{3}+x^{3}+\left(\alpha^{2}+1\right)\beta x^{2}+\left(3\alpha^{2}-% 1\right)\beta^{2}x\right)=0,\\ f(1)=\frac{(\beta+1)e^{-\frac{(\beta-1)^{2}}{2\alpha^{2}\beta}}}{2\sqrt{2\pi}% \alpha\sqrt{\beta}}\end{array}\right\}
  9. X = 1 2 [ ( T β ) 0.5 - ( T β ) - 0.5 ] X=\frac{1}{2}\left[\left(\frac{T}{\beta}\right)^{0.5}-\left(\frac{T}{\beta}% \right)^{-0.5}\right]
  10. T = β ( 1 + 2 X 2 + 2 X ( 1 + X 2 ) 0.5 ) T=\beta\left(1+2X^{2}+2X(1+X^{2})^{0.5}\right)
  11. f ( x ) = x - μ β + β x - μ 2 γ ( x - μ ) ϕ ( x - μ β - β x - μ γ ) x > μ ; γ , β > 0 f(x)=\frac{\sqrt{\frac{x-\mu}{\beta}}+\sqrt{\frac{\beta}{x-\mu}}}{2\gamma\left% (x-\mu\right)}\phi\left(\frac{\sqrt{\frac{x-\mu}{\beta}}-\sqrt{\frac{\beta}{x-% \mu}}}{\gamma}\right)\quad x>\mu;\gamma,\beta>0
  12. ϕ \phi
  13. f ( x ) = x + 1 x 2 γ x ϕ ( x - 1 x γ ) x > 0 ; γ > 0 f(x)=\frac{\sqrt{x}+\sqrt{\frac{1}{x}}}{2\gamma x}\phi\left(\frac{\sqrt{x}-% \sqrt{\frac{1}{x}}}{\gamma}\right)\quad x>0;\gamma>0
  14. F ( x ) = Φ ( x - 1 x γ ) x > 0 ; γ > 0 F(x)=\Phi\left(\frac{\sqrt{x}-\sqrt{\frac{1}{x}}}{\gamma}\right)\quad x>0;% \gamma>0
  15. G ( p ) = 1 4 [ γ Φ - 1 ( p ) + 4 + ( γ Φ - 1 ( p ) ) 2 ] 2 G(p)=\frac{1}{4}\left[\gamma\Phi^{-1}(p)+\sqrt{4+\left(\gamma\Phi^{-1}(p)% \right)^{2}}\right]^{2}

Bismut_connection.html

  1. \nabla
  2. g = 0 \nabla g=0
  3. J = 0 \nabla J=0
  4. T ( X , Y ) T(X,Y)
  5. T ( X , Y , Z ) = g ( T ( X , Y ) , Z ) T(X,Y,Z)=g(T(X,Y),Z)
  6. - , - \langle-,-\rangle
  7. X , J Y = - J X , Y \langle X,JY\rangle=-\langle JX,Y\rangle
  8. \nabla
  9. T T
  10. T ( Z , X , Y ) = - 1 2 Z , ( X J ) Y T(Z,X,Y)=-\frac{1}{2}\langle Z,(\nabla_{X}J)Y\rangle
  11. + T \nabla+T
  12. Γ β γ α - 1 2 J δ α β J γ δ \Gamma^{\alpha}_{\beta\gamma}-\frac{1}{2}J^{\alpha}_{~{}\delta}\nabla_{\beta}J% ^{\delta}_{~{}\gamma}
  13. Γ β γ α \Gamma^{\alpha}_{\beta\gamma}
  14. T T
  15. T ( Z , X , Y ) + cyc in X , Y , Z = T ( Z , X , Y ) + S ( Z , X , Y ) T(Z,X,Y)+\textrm{cyc~{}in~{}}X,Y,Z=T(Z,X,Y)+S(Z,X,Y)
  16. S S
  17. S ( Z , X , Y ) = - 1 2 X , J ( Y J ) Z - 1 2 Y , J ( Z J ) X . S(Z,X,Y)=-\frac{1}{2}\langle X,J(\nabla_{Y}J)Z\rangle-\frac{1}{2}\langle Y,J(% \nabla_{Z}J)X\rangle.
  18. S S
  19. S ( Z , X , J Y ) = - S ( J Z , X , Y ) S(Z,X,JY)=-S(JZ,X,Y)
  20. S ( Z , X , J Y ) + S ( J Z , X , Y ) = - 1 2 J X , ( - ( J Y J ) Z - ( J Z J ) Y + ( J Y J ) Z + ( J Z J ) Y ) = - 1 2 J X , R e ( ( 1 - i J ) [ ( 1 + i J ) Y , ( 1 + i J ) Z ] ) . \begin{aligned}\displaystyle S(Z,X,JY)+S(JZ,X,Y)&\displaystyle=-\frac{1}{2}% \langle JX,\big(-(\nabla_{JY}J)Z-(J\nabla_{Z}J)Y+(J\nabla_{Y}J)Z+(\nabla_{JZ}J% )Y\big)\rangle\\ &\displaystyle=-\frac{1}{2}\langle JX,Re\big((1-iJ)[(1+iJ)Y,(1+iJ)Z]\big)% \rangle.\end{aligned}
  21. J J
  22. Γ β γ α + T β γ α + S β γ α . \Gamma^{\alpha}_{\beta\gamma}+T^{\alpha}_{~{}\beta\gamma}+S^{\alpha}_{~{}\beta% \gamma}.

BL_(logic).html

  1. \rightarrow
  2. \otimes
  3. \otimes
  4. \bot
  5. 0
  6. 0 ¯ \overline{0}
  7. \wedge
  8. A B A ( A B ) A\wedge B\equiv A\otimes(A\rightarrow B)
  9. ¬ \neg
  10. ¬ A A \neg A\equiv A\rightarrow\bot
  11. \leftrightarrow
  12. A B ( A B ) ( B A ) A\leftrightarrow B\equiv(A\rightarrow B)\wedge(B\rightarrow A)
  13. ( A B ) ( B A ) . (A\rightarrow B)\otimes(B\rightarrow A).
  14. \vee
  15. A B ( ( A B ) B ) ( ( B A ) A ) A\vee B\equiv((A\rightarrow B)\rightarrow B)\wedge((B\rightarrow A)\rightarrow A)
  16. \top
  17. 1 1
  18. 1 ¯ \overline{1}
  19. \top\equiv\bot\rightarrow\bot
  20. A A
  21. A B A\rightarrow B
  22. B . B.
  23. ( BL1 ) : ( A B ) ( ( B C ) ( A C ) ) ( BL2 ) : A B A ( BL3 ) : A B B A ( BL4 ) : A ( A B ) B ( B A ) ( BL5a ) : ( A ( B C ) ) ( A B C ) ( BL5b ) : ( A B C ) ( A ( B C ) ) ( BL6 ) : ( ( A B ) C ) ( ( ( B A ) C ) C ) ( BL7 ) : A \begin{array}[]{ll}{\rm(BL1)}\colon&(A\rightarrow B)\rightarrow((B\rightarrow C% )\rightarrow(A\rightarrow C))\\ {\rm(BL2)}\colon&A\otimes B\rightarrow A\\ {\rm(BL3)}\colon&A\otimes B\rightarrow B\otimes A\\ {\rm(BL4)}\colon&A\otimes(A\rightarrow B)\rightarrow B\otimes(B\rightarrow A)% \\ {\rm(BL5a)}\colon&(A\rightarrow(B\rightarrow C))\rightarrow(A\otimes B% \rightarrow C)\\ {\rm(BL5b)}\colon&(A\otimes B\rightarrow C)\rightarrow(A\rightarrow(B% \rightarrow C))\\ {\rm(BL6)}\colon&((A\rightarrow B)\rightarrow C)\rightarrow(((B\rightarrow A)% \rightarrow C)\rightarrow C)\\ {\rm(BL7)}\colon&\bot\rightarrow A\end{array}

Black–Scholes_equation.html

  1. V t + 1 2 σ 2 S 2 2 V S 2 + r S V S - r V = 0 \frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{% \partial S^{2}}+rS\frac{\partial V}{\partial S}-rV=0
  2. σ \sigma
  3. V t + 1 2 σ 2 S 2 2 V S 2 = r V - r S V S \frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{% \partial S^{2}}=rV-rS\frac{\partial V}{\partial S}
  4. V S \frac{\partial V}{\partial S}
  5. d S S = μ d t + σ d W \frac{dS}{S}=\mu\,dt+\sigma\,dW\,
  6. σ 2 d t \sigma^{2}dt
  7. V ( S , T ) V(S,T)
  8. V V
  9. S S
  10. t t
  11. d V = ( μ S V S + V t + 1 2 σ 2 S 2 2 V S 2 ) d t + σ S V S d W dV=\left(\mu S\frac{\partial V}{\partial S}+\frac{\partial V}{\partial t}+% \frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}}\right)dt+\sigma S% \frac{\partial V}{\partial S}\,dW
  12. V S \frac{\partial V}{\partial S}
  13. t t
  14. Π = - V + V S S \Pi=-V+\frac{\partial V}{\partial S}S
  15. [ t , t + Δ t ] [t,t+\Delta t]
  16. Δ Π = - Δ V + V S Δ S \Delta\Pi=-\Delta V+\frac{\partial V}{\partial S}\,\Delta S
  17. Δ S = μ S Δ t + σ S Δ W \Delta S=\mu S\,\Delta t+\sigma S\,\Delta W\,
  18. Δ V = ( μ S V S + V t + 1 2 σ 2 S 2 2 V S 2 ) Δ t + σ S V S Δ W \Delta V=\left(\mu S\frac{\partial V}{\partial S}+\frac{\partial V}{\partial t% }+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{\partial S^{2}}\right)\Delta t% +\sigma S\frac{\partial V}{\partial S}\,\Delta W
  19. Δ Π \Delta\Pi
  20. Δ Π = ( - V t - 1 2 σ 2 S 2 2 V S 2 ) Δ t \Delta\Pi=\left(-\frac{\partial V}{\partial t}-\frac{1}{2}\sigma^{2}S^{2}\frac% {\partial^{2}V}{\partial S^{2}}\right)\Delta t
  21. Δ W \Delta W
  22. r r
  23. [ t , t + Δ t ] [t,t+\Delta t]
  24. r Π Δ t = Δ Π r\Pi\,\Delta t=\Delta\Pi
  25. Δ Π \Delta\Pi
  26. ( - V t - 1 2 σ 2 S 2 2 V S 2 ) Δ t = r ( - V + S V S ) Δ t \left(-\frac{\partial V}{\partial t}-\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^% {2}V}{\partial S^{2}}\right)\Delta t=r\left(-V+S\frac{\partial V}{\partial S}% \right)\Delta t
  27. V t + 1 2 σ 2 S 2 2 V S 2 + r S V S - r V = 0 \frac{\partial V}{\partial t}+\frac{1}{2}\sigma^{2}S^{2}\frac{\partial^{2}V}{% \partial S^{2}}+rS\frac{\partial V}{\partial S}-rV=0
  28. V V
  29. S S
  30. t t
  31. d S ( t ) S ( t ) = r d t + σ d W ( t ) \frac{dS(t)}{S(t)}=r\ dt+\sigma dW(t)
  32. exp ( - r t ) V ( t , S ( t ) ) \exp(-rt)V(t,S(t))
  33. C ( 0 , t ) = 0 for all t C ( S , t ) S as S C ( S , T ) = max { S - K , 0 } \begin{aligned}\displaystyle C(0,t)&\displaystyle=0\,\text{ for all }t\\ \displaystyle C(S,t)&\displaystyle\rightarrow S\,\text{ as }S\rightarrow\infty% \\ \displaystyle C(S,T)&\displaystyle=\max\{S-K,0\}\end{aligned}
  34. 𝔼 [ max { S - K , 0 } ] \mathbb{E}\left[\max\{S-K,0\}\right]
  35. τ = T - t u = C e r τ x = ln ( S K ) + ( r - 1 2 σ 2 ) τ \begin{aligned}\displaystyle\tau&\displaystyle=T-t\\ \displaystyle u&\displaystyle=Ce^{r\tau}\\ \displaystyle x&\displaystyle=\ln\left(\frac{S}{K}\right)+\left(r-\frac{1}{2}% \sigma^{2}\right)\tau\end{aligned}
  36. u τ = 1 2 σ 2 2 u x 2 \frac{\partial u}{\partial\tau}=\frac{1}{2}\sigma^{2}\frac{\partial^{2}u}{% \partial x^{2}}
  37. C ( S , T ) = max { S - K , 0 } C(S,T)=\max\{S-K,0\}
  38. u ( x , 0 ) = u 0 ( x ) K ( e max { x , 0 } - 1 ) u(x,0)=u_{0}(x)\equiv K(e^{\max\{x,0\}}-1)
  39. u ( x , τ ) = 1 σ 2 π τ - u 0 [ y ] exp [ - ( x - y ) 2 2 σ 2 τ ] d y u(x,\tau)=\frac{1}{\sigma\sqrt{2\pi\tau}}\int_{-\infty}^{\infty}{u_{0}[y]\exp{% \left[-\frac{(x-y)^{2}}{2\sigma^{2}\tau}\right]}}\,dy
  40. u ( x , τ ) = K e x + 1 2 σ 2 τ N ( d 1 ) - K N ( d 2 ) u(x,\tau)=Ke^{x+\frac{1}{2}\sigma^{2}\tau}N(d_{1})-KN(d_{2})
  41. d 1 = 1 σ τ [ ( x + 1 2 σ 2 τ ) + 1 2 σ 2 τ ] d 2 = 1 σ τ [ ( x + 1 2 σ 2 τ ) - 1 2 σ 2 τ ] \begin{aligned}\displaystyle d_{1}&\displaystyle=\frac{1}{\sigma\sqrt{\tau}}% \left[\left(x+\frac{1}{2}\sigma^{2}\tau\right)+\frac{1}{2}\sigma^{2}\tau\right% ]\\ \displaystyle d_{2}&\displaystyle=\frac{1}{\sigma\sqrt{\tau}}\left[\left(x+% \frac{1}{2}\sigma^{2}\tau\right)-\frac{1}{2}\sigma^{2}\tau\right]\end{aligned}
  42. u , x , τ u,x,\tau

Blade_(geometry).html

  1. k k
  2. k k
  3. k k
  4. 𝐚 \mathbf{a}
  5. 𝐛 \mathbf{b}
  6. 𝐚 𝐛 . \mathbf{a}\wedge\mathbf{b}.
  7. 𝐚 \mathbf{a}
  8. 𝐛 \mathbf{b}
  9. 𝐜 \mathbf{c}
  10. 𝐚 𝐛 𝐜 . \mathbf{a}\wedge\mathbf{b}\wedge\mathbf{c}.
  11. n n
  12. n 1 n−1
  13. n n
  14. n n
  15. n n
  16. k ( n k ) + 1 k(n−k)+1
  17. k k
  18. n n
  19. n n
  20. k k
  21. k k
  22. k k
  23. v 1 v k v_{1}\wedge\cdots\wedge v_{k}
  24. k × k k×k
  25. 𝔽 k \mathbb{F}^{k}
  26. k × ( n k ) k× (n−k)

Blazed_grating.html

  1. d d
  2. θ B \theta_{B}
  3. θ B \theta_{B}
  4. θ B \theta_{B}
  5. α = β = θ B \alpha=\beta=\theta_{B}
  6. d ( sin α + sin β ) = m λ d\left(\sin{\alpha}+\sin{\beta}\right)=m\lambda
  7. d d
  8. α \alpha
  9. β \beta
  10. m m
  11. λ \lambda
  12. 2 d sin θ B = m λ 2d\sin{\theta_{B}}=m\lambda
  13. θ B \theta_{B}
  14. θ B = arcsin m λ 2 d . \theta_{B}=\arcsin{\frac{m\lambda}{2d}}\ .

Blend_modes.html

  1. f ( a , b ) = b f(a,b)=b
  2. f ( a , b ) = a b f(a,b)=ab
  3. f ( a , b ) = 1 - ( 1 - a ) ( 1 - b ) f(a,b)=1-(1-a)(1-b)
  4. f ( a , b ) = { 2 a b , if a < 0.5 1 - 2 ( 1 - a ) ( 1 - b ) , otherwise f(a,b)=\begin{cases}2ab,&\mbox{if }~{}a<0.5\\ 1-2(1-a)(1-b),&\mbox{otherwise}\end{cases}
  5. f p h o t o s h o p ( a , b ) = { 2 a b + a 2 ( 1 - 2 b ) , if b < 0.5 2 a ( 1 - b ) + a ( 2 b - 1 ) , otherwise f_{photoshop}(a,b)=\begin{cases}2ab+a^{2}(1-2b),&\mbox{if }~{}b<0.5\\ 2a(1-b)+\sqrt{a}(2b-1),&\mbox{otherwise}\end{cases}
  6. f p e g t o p ( a , b ) = ( 1 - 2 b ) a 2 + 2 b a f_{pegtop}(a,b)=(1-2b)a^{2}+2ba
  7. f i l l u s i o n s . h u ( a , b ) = a ( 2 2 ( 0.5 - b ) ) f_{illusions.hu}(a,b)=a^{(2^{2(0.5-b)})}
  8. f w 3 c ( a , b ) = { a - ( 1 - 2 b ) a ( 1 - a ) if b 0.5 a + ( 2 b - 1 ) ( g w 3 c ( a ) - a ) otherwise f_{w3c}(a,b)=\begin{cases}a-(1-2b)\cdot a\cdot(1-a)&\,\text{if }b\leq 0.5\\ a+(2b-1)\cdot(g_{w3c}(a)-a)&\,\text{otherwise}\end{cases}
  9. g w 3 c ( a ) = { ( ( 16 a - 12 ) a + 4 ) a if a 0.25 a otherwise g_{w3c}(a)=\begin{cases}((16a-12)\cdot a+4)\cdot a&\,\text{if }a\leq 0.25\\ \sqrt{a}&\,\text{otherwise}\end{cases}
  10. [ m i n ( r 1 , r 2 ) , m i n ( g 1 , g 2 ) , m i n ( b 1 , b 2 ) ] [min(r_{1},r_{2}),min(g_{1},g_{2}),min(b_{1},b_{2})]
  11. [ m a x ( r 1 , r 2 ) , m a x ( g 1 , g 2 ) , m a x ( b 1 , b 2 ) ] [max(r_{1},r_{2}),max(g_{1},g_{2}),max(b_{1},b_{2})]

Bloch_equations.html

  1. d M x ( t ) d t = γ ( M ( t ) × B ( t ) ) x - M x ( t ) T 2 \frac{dM_{x}(t)}{dt}=\gamma({M}(t)\times{B}(t))_{x}-\frac{M_{x}(t)}{T_{2}}
  2. d M y ( t ) d t = γ ( M ( t ) × B ( t ) ) y - M y ( t ) T 2 \frac{dM_{y}(t)}{dt}=\gamma({M}(t)\times{B}(t))_{y}-\frac{M_{y}(t)}{T_{2}}
  3. d M z ( t ) d t = γ ( M ( t ) × B ( t ) ) z - M z ( t ) - M 0 T 1 \frac{dM_{z}(t)}{dt}=\gamma({M}(t)\times{B}(t))_{z}-\frac{M_{z}(t)-M_{0}}{T_{1}}
  4. d M x ( t ) d t = γ ( M ( t ) × B ( t ) ) x \frac{dM_{x}(t)}{dt}=\gamma({M}(t)\times{B}(t))_{x}
  5. d M y ( t ) d t = γ ( M ( t ) × B ( t ) ) y \frac{dM_{y}(t)}{dt}=\gamma({M}(t)\times{B}(t))_{y}
  6. d M z ( t ) d t = γ ( M ( t ) × B ( t ) ) z \frac{dM_{z}(t)}{dt}=\gamma({M}(t)\times{B}(t))_{z}
  7. d M ( t ) d t = γ M ( t ) × B ( t ) \frac{d{M}(t)}{dt}=\gamma{M}(t)\times{B}(t)
  8. ( - M x T 2 , - M y T 2 , - M z - M 0 T 1 ) \left(-\frac{M_{x}}{T_{2}},-\frac{M_{y}}{T_{2}},-\frac{M_{z}-M_{0}}{T_{1}}\right)
  9. d M x ( t ) d t = γ ( M y ( t ) B z ( t ) - M z ( t ) B y ( t ) ) - M x ( t ) T 2 \frac{dM_{x}(t)}{dt}=\gamma\left(M_{y}(t)B_{z}(t)-M_{z}(t)B_{y}(t)\right)-% \frac{M_{x}(t)}{T_{2}}
  10. d M y ( t ) d t = γ ( M z ( t ) B x ( t ) - M x ( t ) B z ( t ) ) - M y ( t ) T 2 \frac{dM_{y}(t)}{dt}=\gamma\left(M_{z}(t)B_{x}(t)-M_{x}(t)B_{z}(t)\right)-% \frac{M_{y}(t)}{T_{2}}
  11. d M z ( t ) d t = γ ( M x ( t ) B y ( t ) - M y ( t ) B x ( t ) ) - M z ( t ) - M 0 T 1 \frac{dM_{z}(t)}{dt}=\gamma\left(M_{x}(t)B_{y}(t)-M_{y}(t)B_{x}(t)\right)-% \frac{M_{z}(t)-M_{0}}{T_{1}}
  12. M x y = M x + i M y and B x y = B x + i B y M_{xy}=M_{x}+iM_{y}\,\text{ and }B_{xy}=B_{x}+iB_{y}\,
  13. d M x y ( t ) d t = - i γ ( M x y ( t ) B z ( t ) - M z ( t ) B x y ( t ) ) - M x y T 2 \frac{dM_{xy}(t)}{dt}=-i\gamma\left(M_{xy}(t)B_{z}(t)-M_{z}(t)B_{xy}(t)\right)% -\frac{M_{xy}}{T_{2}}
  14. d M z ( t ) d t = i γ 2 ( M x y ( t ) B x y ( t ) ¯ - M x y ¯ ( t ) B x y ( t ) ) - M z - M 0 T 1 \frac{dM_{z}(t)}{dt}=i\frac{\gamma}{2}\left(M_{xy}(t)\overline{B_{xy}(t)}-% \overline{M_{xy}}(t)B_{xy}(t)\right)-\frac{M_{z}-M_{0}}{T_{1}}
  15. M x y ¯ = M x - i M y \overline{M_{xy}}=M_{x}-iM_{y}
  16. d d t ( M x M y M z ) = ( - 1 T 2 γ B z - γ B y - γ B z - 1 T 2 γ B x γ B y - γ B x - 1 T 1 ) ( M x M y M z ) + ( 0 0 M 0 T 1 ) \frac{d}{dt}\left(\begin{array}[]{c}M_{x}\\ M_{y}\\ M_{z}\end{array}\right)=\left(\begin{array}[]{ccc}-\frac{1}{T_{2}}&\gamma B_{z% }&-\gamma B_{y}\\ -\gamma B_{z}&-\frac{1}{T_{2}}&\gamma B_{x}\\ \gamma B_{y}&-\gamma B_{x}&-\frac{1}{T_{1}}\end{array}\right)\left(\begin{% array}[]{c}M_{x}\\ M_{y}\\ M_{z}\end{array}\right)+\left(\begin{array}[]{c}0\\ 0\\ \frac{M_{0}}{T_{1}}\end{array}\right)
  17. d M x y ( t ) d t = - i γ M x y ( t ) B 0 \frac{dM_{xy}(t)}{dt}=-i\gamma M_{xy}(t)B_{0}
  18. d M z ( t ) d t = 0 \frac{dM_{z}(t)}{dt}=0
  19. M x y ( t ) = M x y ( 0 ) e - i γ B 0 t M_{xy}(t)=M_{xy}(0)e^{-i\gamma B_{0}t}
  20. M z ( t ) = M 0 = const M_{z}(t)=M_{0}=\,\text{const}\,
  21. M x y ( t ) = M x y ( 0 ) e - i γ B z 0 t = M x y ( 0 ) [ cos ( ω 0 t ) - i sin ( ω 0 t ) ] M_{xy}(t)=M_{xy}(0)e^{-i\gamma B_{z0}t}=M_{xy}(0)\left[\cos(\omega_{0}t)-i\sin% (\omega_{0}t)\right]
  22. M x ( t ) = Re ( M x y ( t ) ) = M x y ( 0 ) cos ( ω 0 t ) M_{x}(t)=\text{Re}\left(M_{xy}(t)\right)=M_{xy}(0)\cos(\omega_{0}t)
  23. M y ( t ) = Im ( M x y ( t ) ) = - M x y ( 0 ) sin ( ω 0 t ) M_{y}(t)=\text{Im}\left(M_{xy}(t)\right)=-M_{xy}(0)\sin(\omega_{0}t)
  24. M z ( t ) = M z ( t ) M_{z}^{\prime}(t)=M_{z}(t)\,
  25. M x y ( t ) = e + i Ω t M x y ( t ) M_{xy}^{\prime}(t)=e^{+i\Omega t}M_{xy}(t)\,
  26. d M x y ( t ) d t = d ( M x y ( t ) e + i Ω t ) d t = e + i Ω t d M x y ( t ) d t + i Ω e + i Ω t M x y = e + i Ω t d M x y ( t ) d t + i Ω M x y \frac{dM_{xy}^{\prime}(t)}{dt}=\frac{d\left(M_{xy}(t)e^{+i\Omega t}\right)}{dt% }=e^{+i\Omega t}\frac{dM_{xy}(t)}{dt}+i\Omega e^{+i\Omega t}M_{xy}=e^{+i\Omega t% }\frac{dM_{xy}(t)}{dt}+i\Omega M_{xy}^{\prime}
  27. d M x y ( t ) d t \displaystyle\frac{dM_{xy}^{\prime}(t)}{dt}
  28. d M x y ( t ) d t \displaystyle\frac{dM_{xy}^{\prime}(t)}{dt}
  29. d M z ( t ) d t = i γ 2 ( M x y ( t ) B x y ( t ) ¯ - M x y ¯ ( t ) B x y ( t ) ) - M z - M 0 T 1 \frac{dM_{z}(t)}{dt}=i\frac{\gamma}{2}\left(M^{\prime}_{xy}(t)\overline{B^{% \prime}_{xy}(t)}-\overline{M^{\prime}_{xy}}(t)B^{\prime}_{xy}(t)\right)-\frac{% M_{z}-M_{0}}{T_{1}}
  30. B x ( t ) = B 1 cos ω t B_{x}(t)=B_{1}\cos\omega t
  31. B y ( t ) = - B 1 sin ω t B_{y}(t)=-B_{1}\sin\omega t
  32. B z ( t ) = B 0 B_{z}(t)=B_{0}
  33. ϵ = γ B 1 \epsilon=\gamma B_{1}
  34. Δ = γ B 0 - ω \Delta=\gamma B_{0}-\omega
  35. d d t ( M x M y M z ) = ( - 1 T 2 Δ 0 - Δ - 1 T 2 ϵ 0 - ϵ - 1 T 1 ) ( M x M y M z ) + ( 0 0 M 0 T 1 ) \frac{d}{dt}\left(\begin{array}[]{c}M^{\prime}_{x}\\ M^{\prime}_{y}\\ M^{\prime}_{z}\end{array}\right)=\left(\begin{array}[]{ccc}-\frac{1}{T_{2}}&% \Delta&0\\ -\Delta&-\frac{1}{T_{2}}&\epsilon\\ 0&-\epsilon&-\frac{1}{T_{1}}\end{array}\right)\left(\begin{array}[]{c}M^{% \prime}_{x}\\ M^{\prime}_{y}\\ M^{\prime}_{z}\end{array}\right)+\left(\begin{array}[]{c}0\\ 0\\ \frac{M_{0}}{T_{1}}\end{array}\right)
  36. d M x y ( t ) d t = - M x y T 2 \frac{dM_{xy}^{\prime}(t)}{dt}=-\frac{M_{xy}^{\prime}}{T_{2}}
  37. M x y ( t ) = M x y ( 0 ) e - t / T 2 M_{xy}^{\prime}(t)=M_{xy}^{\prime}(0)e^{-t/T_{2}}
  38. d M z ( t ) d t = - M z ( t ) - M z , eq T 1 \frac{dM_{z}(t)}{dt}=-\frac{M_{z}(t)-M_{z,\mathrm{eq}}}{T_{1}}
  39. M z ( t ) = M z , eq - [ M z , eq - M z ( 0 ) ] e - t / T 1 M_{z}(t)=M_{z,\mathrm{eq}}-[M_{z,\mathrm{eq}}-M_{z}(0)]e^{-t/T_{1}}
  40. d M x y ( t ) d t = i γ B x y M z ( t ) \displaystyle\frac{dM_{xy}^{\prime}(t)}{dt}=i\gamma B_{xy}^{\prime}M_{z}(t)
  41. d M z ( t ) d t = i γ 2 ( M x y ( t ) B x y ¯ - M x y ¯ ( t ) B x y ) \frac{dM_{z}(t)}{dt}=i\frac{\gamma}{2}\left(M^{\prime}_{xy}(t)\overline{B^{% \prime}_{xy}}-\overline{M^{\prime}_{xy}}(t)B^{\prime}_{xy}\right)

Block_Wiedemann_algorithm.html

  1. n × n n\times n
  2. x base x_{\mathrm{base}}
  3. x = M x base x=Mx_{\mathrm{base}}
  4. S = [ x , M x , M 2 x , ] S=\left[x,Mx,M^{2}x,\ldots\right]
  5. S y = [ y x , y M x , y M 2 x ] S_{y}=\left[y\cdot x,y\cdot Mx,y\cdot M^{2}x\ldots\right]
  6. n 0 n_{0}
  7. r = 0 n 0 p r M r = 0 \sum_{r=0}^{n_{0}}p_{r}M^{r}=0
  8. r = 0 n 0 y ( p r ( M r x ) ) = 0 \sum_{r=0}^{n_{0}}y\cdot(p_{r}(M^{r}x))=0
  9. S S
  10. S y S_{y}
  11. q 0 q L q_{0}\ldots q_{L}
  12. i = 0 L q i S y [ i + r ] = 0 r \sum_{i=0}^{L}q_{i}S_{y}[{i+r}]=0\forall r
  13. y S y\cdot S
  14. S S
  15. i = 0 L q i M i x = 0 \sum_{i=0}^{L}q_{i}M^{i}x=0
  16. x x
  17. M i = 0 L q i M i x base = 0 M\sum_{i=0}^{L}q_{i}M^{i}x_{\mathrm{base}}=0
  18. i = 0 L q i M i x base \sum_{i=0}^{L}q_{i}M^{i}x_{\mathrm{base}}
  19. M M
  20. y i M t x j y_{i}\cdot M^{t}x_{j}
  21. i = 0 i max , j = 0 j max , t = 0 t max i=0\ldots i_{\max},j=0\ldots j_{\max},t=0\ldots t_{\max}
  22. i max , j max , t max i_{\max},j_{\max},t_{\max}
  23. t max > d i max + d j max + O ( 1 ) t_{\max}>\frac{d}{i_{\max}}+\frac{d}{j_{\max}}+O(1)
  24. y i y_{i}
  25. y i y_{i}
  26. i max i_{\max}

Blossom_(functional).html

  1. [ f ] , \mathcal{B}[f],
  2. [ f ] ( u 1 , , u d ) = [ f ] ( π ( u 1 , , u d ) ) , \mathcal{B}[f](u_{1},\dots,u_{d})=\mathcal{B}[f]\big(\pi(u_{1},\dots,u_{d})% \big),\,
  3. [ f ] ( α u + β v , ) = α [ f ] ( u , ) + β [ f ] ( v , ) , when α + β = 1. \mathcal{B}[f](\alpha u+\beta v,\dots)=\alpha\mathcal{B}[f](u,\dots)+\beta% \mathcal{B}[f](v,\dots),\,\text{ when }\alpha+\beta=1.\,
  4. [ f ] ( u , , u ) = f ( u ) . \mathcal{B}[f](u,\dots,u)=f(u).\,

Blum–Micali_algorithm.html

  1. p p
  2. g g
  3. p p
  4. x 0 x_{0}
  5. x i + 1 = g x i mod p x_{i+1}=g^{x_{i}}\ \bmod{\ p}
  6. i i
  7. x i < p - 1 2 x_{i}<\frac{p-1}{2}
  8. x i x_{i}
  9. n - c - 1 n-c-1
  10. x i x_{i}
  11. c c
  12. p p
  13. p p

Bochner–Riesz_mean.html

  1. ( ξ ) + = { ξ , if ξ > 0 0 , otherwise . (\xi)_{+}=\begin{cases}\xi,&\mbox{if }~{}\xi>0\\ 0,&\mbox{otherwise}~{}.\end{cases}
  2. f f
  3. 𝕋 n \mathbb{T}^{n}
  4. f ^ ( k ) \hat{f}(k)
  5. k n k\in\mathbb{Z}^{n}
  6. δ \delta
  7. B R δ f B_{R}^{\delta}f
  8. R > 0 R>0
  9. Re ( δ ) > 0 \mbox{Re}~{}(\delta)>0
  10. B R δ f ( θ ) = k n | k | R ( 1 - | k | 2 R 2 ) + δ f ^ ( k ) e 2 π i k θ . B_{R}^{\delta}f(\theta)=\underset{|k|\leq R}{\sum_{k\in\mathbb{Z}^{n}}}\left(1% -\frac{|k|^{2}}{R^{2}}\right)_{+}^{\delta}\hat{f}(k)e^{2\pi ik\cdot\theta}.
  11. f f
  12. n \mathbb{R}^{n}
  13. f ^ ( ξ ) \hat{f}(\xi)
  14. δ \delta
  15. S R δ f S_{R}^{\delta}f
  16. R > 0 R>0
  17. Re ( δ ) > 0 \mbox{Re}~{}(\delta)>0
  18. S R δ f ( x ) = | ξ | R ( 1 - | ξ | 2 R 2 ) + δ f ^ ( ξ ) e 2 π i x ξ d ξ . S_{R}^{\delta}f(x)=\int_{|\xi|\leq R}\left(1-\frac{|\xi|^{2}}{R^{2}}\right)_{+% }^{\delta}\hat{f}(\xi)e^{2\pi ix\cdot\xi}\,d\xi.
  19. δ > 0 \delta>0
  20. n = 1 n=1
  21. S R δ S_{R}^{\delta}
  22. B R δ B_{R}^{\delta}
  23. L p L^{p}
  24. δ = 0 \delta=0
  25. δ n - 1 2 \delta\leq\tfrac{n-1}{2}
  26. δ \delta
  27. p p
  28. L p L^{p}
  29. n 2 n\geq 2
  30. δ = 0 \delta=0
  31. L p L^{p}
  32. p 2 p\neq 2
  33. n \mathbb{R}^{n}
  34. 𝕋 n \mathbb{T}^{n}
  35. p ( 1 , ) p\in(1,\infty)
  36. L p L^{p}
  37. δ \delta
  38. ( 1 - | ξ | 2 ) + δ (1-|\xi|^{2})^{\delta}_{+}
  39. L p L^{p}
  40. n = 2 n=2
  41. n 3 n\geq 3
  42. n = 1 n=1
  43. p ( 1 , ) p\in(1,\infty)
  44. δ = 0 \delta=0
  45. L p L^{p}

Bode's_sensitivity_integral.html

  1. 0 ln | S ( i ω ) | d ω = 0 ln | 1 1 + L ( i ω ) | d ω = π R e ( p k ) - π 2 lim s s L ( s ) \int_{0}^{\infty}\ln|S(i\omega)|d\omega=\int_{0}^{\infty}\ln\left|\frac{1}{1+L% (i\omega)}\right|d\omega=\pi\sum Re(p_{k})-\frac{\pi}{2}\lim_{s\rightarrow% \infty}sL(s)
  2. p k p_{k}
  3. 0 ln | S ( i ω ) | d ω = 0 \int_{0}^{\infty}\ln|S(i\omega)|d\omega=0

Body_wave_magnitude.html

  1. m b m_{b}
  2. m b m_{b}

Bogacki–Shampine_method.html

  1. y = f ( t , y ) y^{\prime}=f(t,y)
  2. y n y_{n}
  3. t n t_{n}
  4. h n h_{n}
  5. h n = t n + 1 - t n h_{n}=t_{n+1}-t_{n}
  6. k 1 \displaystyle k_{1}
  7. z n + 1 z_{n+1}
  8. y n + 1 y_{n+1}
  9. y n + 1 y_{n+1}
  10. y n + 1 y_{n+1}
  11. z n + 1 z_{n+1}
  12. k 4 k_{4}
  13. k 1 k_{1}

Bogoliubov_causality_condition.html

  1. g : M [ 0 , 1 ] g:M\to[0,1]
  2. M M
  3. g ( x ) = 0 g(x)=0
  4. x x
  5. x x
  6. g ( x ) = 1 g(x)=1
  7. x x
  8. x , y M x,y\in M
  9. x y x\leq y
  10. x x
  11. y y
  12. S ( g ) S(g)
  13. g g
  14. δ δ g ( x ) ( δ S ( g ) δ g ( y ) S ( g ) ) = 0 for x y . \frac{\delta}{\delta g(x)}\left(\frac{\delta S(g)}{\delta g(y)}S^{\dagger}(g)% \right)=0\mbox{ for }~{}x\leq y.

Bogoliubov_inner_product.html

  1. A A
  2. X , Y A = 0 1 Tr [ e x A X e ( 1 - x ) A Y ] d x \langle X,Y\rangle_{A}=\int\limits_{0}^{1}{\rm Tr}[{\rm e}^{xA}X^{\dagger}{\rm e% }^{(1-x)A}Y]dx
  3. X , X A 0 \langle X,X\rangle_{A}\geq 0
  4. X , Y A = Y , X A \langle X,Y\rangle_{A}=\langle Y,X\rangle_{A}
  5. A A
  6. A = β H A=\beta H
  7. H H
  8. β \beta
  9. X , Y β H = 0 1 e x β H X e - x β H Y d x \langle X,Y\rangle_{\beta H}=\int\limits_{0}^{1}\langle{\rm e}^{x\beta H}X^{% \dagger}{\rm e}^{-x\beta H}Y\rangle dx
  10. \langle\dots\rangle
  11. H H
  12. β \beta
  13. X , Y β H = 2 t s Tr e β H + t X + s Y | t = s = 0 \langle X,Y\rangle_{\beta H}=\frac{\partial^{2}}{\partial t\partial s}{\rm Tr}% \,{\rm e}^{\beta H+tX+sY}\bigg|_{t=s=0}

Bogomolov_conjecture.html

  1. K ¯ \overline{K}
  2. h ^ \hat{h}
  3. ϵ > 0 \epsilon>0
  4. { P C ( K ¯ ) : h ^ ( P ) < ϵ } \{P\in C(\overline{K}):\hat{h}(P)<\epsilon\}
  5. h ^ ( P ) = 0 \hat{h}(P)=0
  6. h ^ \hat{h}
  7. X A X\subset A
  8. ϵ > 0 \epsilon>0
  9. { P X ( K ¯ ) : h ^ ( P ) < ϵ } \{P\in X(\overline{K}):\hat{h}(P)<\epsilon\}

Bohr_equation.html

  1. 𝐕 d 𝐕 t = P a CO 2 - P e CO 2 P a CO 2 \frac{\mathbf{V}_{\mathrm{d}}}{\mathbf{V}_{\mathrm{t}}}=\frac{{\mathrm{P}_{% \mathrm{a}}{\mathrm{CO}}_{\mathrm{2}}}-{\mathrm{P}_{\mathrm{e}}{\mathrm{CO}}_{% \mathrm{2}}}}{\mathrm{P}_{\mathrm{a}}{\mathrm{CO}}_{\mathrm{2}}}
  2. V a V_{a}
  3. V t V_{t}
  4. V a + V d V_{a}+V_{d}
  5. V a V_{a}
  6. V t - V d V_{t}-V_{d}
  7. V t × F e = V a × F a V_{t}\times F_{e}=V_{a}\times F_{a}
  8. V t × F e = ( V t - V d ) × F a V_{t}\times F_{e}=(V_{t}-V_{d})\times F_{a}
  9. V t × F e = V t × F a - V d × F a V_{t}\times F_{e}=V_{t}\times F_{a}-V_{d}\times F_{a}
  10. V d × F a = V t × F a - V t × F e V_{d}\times F_{a}=V_{t}\times F_{a}-V_{t}\times F_{e}
  11. V d × F a = V t × ( F a - F e ) V_{d}\times F_{a}=V_{t}\times(F_{a}-F_{e})
  12. V d / V t = F a - F e F a V_{d}/V_{t}=\frac{F_{a}-F_{e}}{F_{a}}
  13. F a × P t o t = P a C O 2 F_{a}\times Ptot=PaCO_{2}
  14. F e × P t o t = P e C O 2 F_{e}\times Ptot=PeCO_{2}
  15. V d / V t = F a C O 2 - F e C O 2 F a C O 2 × P t o t P t o t V_{d}/V_{t}=\frac{FaCO_{2}-FeCO_{2}}{FaCO_{2}}\times\frac{Ptot}{Ptot}
  16. V d / V t = P a C O 2 - P e C O 2 P a C O 2 V_{d}/V_{t}=\frac{PaCO_{2}-PeCO_{2}}{PaCO_{2}}

Bohr–van_Leeuwen_theorem.html

  1. exp ( - U / k B T ) \exp(-U/k\text{B}T)
  2. U U
  3. k B k\text{B}
  4. T T
  5. ( m v 2 / 2 ) (mv^{2}/2)
  6. m m
  7. v v
  8. q q
  9. 𝐯 \mathbf{v}
  10. 𝐅 = q ( 𝐄 + 𝐯 × 𝐁 ) , \mathbf{F}=q\left(\mathbf{E}+\mathbf{v}\times\mathbf{B}\right),
  11. 𝐄 \mathbf{E}
  12. 𝐁 \mathbf{B}
  13. 𝐅 𝐯 = q 𝐄 𝐯 \mathbf{F}\cdot\mathbf{v}=q\mathbf{E}\cdot\mathbf{v}
  14. 𝐁 \mathbf{B}
  15. N N
  16. e e
  17. m e m\text{e}
  18. 𝐫 \mathbf{r}
  19. 𝐯 \mathbf{v}
  20. 𝐣 = e 𝐯 \mathbf{j}=e\mathbf{v}
  21. μ = 1 2 c 𝐫 × 𝐣 = e 2 c 𝐫 × 𝐯 . \mathbf{\mu}=\frac{1}{2c}\mathbf{r}\times\mathbf{j}=\frac{e}{2c}\mathbf{r}% \times\mathbf{v}.
  22. μ = i = 1 N 𝐚 i 𝐫 ˙ i , \mu=\sum_{i=1}^{N}\mathbf{a}_{i}\cdot\dot{\mathbf{r}}_{i},
  23. 𝐚 i \mathbf{a}_{i}
  24. { 𝐫 i , i = 1 N } \{\mathbf{r}_{i},i=1\ldots N\}
  25. 𝐩 n \mathbf{p}_{n}
  26. 𝐫 n \mathbf{r}_{n}
  27. d P exp [ - ( 𝐩 1 , , 𝐩 N ; 𝐫 1 , , 𝐫 N ) k B T ] d 𝐩 1 , , d 𝐩 N d 𝐫 1 , , d 𝐫 N , dP\propto\exp{\left[-\frac{\mathcal{H}(\mathbf{p}_{1},\ldots,\mathbf{p}_{N};% \mathbf{r}_{1},\ldots,\mathbf{r}_{N})}{k\text{B}T}\right]}d\mathbf{p}_{1},% \ldots,d\mathbf{p}_{N}d\mathbf{r}_{1},\ldots,d\mathbf{r}_{N},
  28. \mathcal{H}
  29. f ( 𝐩 1 , , 𝐩 N ; 𝐫 1 , , 𝐫 N ) f(\mathbf{p}_{1},\ldots,\mathbf{p}_{N};\mathbf{r}_{1},\ldots,\mathbf{r}_{N})
  30. f = f d P d P . \langle f\rangle=\frac{\int fdP}{\int dP}.
  31. = 1 2 m e i = 1 N ( 𝐩 i - e c 𝐀 i ) 2 + e ϕ ( 𝐪 ) , \mathcal{H}=\frac{1}{2m\text{e}}\sum_{i=1}^{N}\left(\mathbf{p}_{i}-\frac{e}{c}% \mathbf{A}_{i}\right)^{2}+e\phi(\mathbf{q}),
  32. 𝐀 i \mathbf{A}_{i}
  33. ϕ ( 𝐪 ) \phi(\mathbf{q})
  34. 𝐩 i \mathbf{p}_{i}
  35. 𝐫 i \mathbf{r}_{i}
  36. 𝐩 ˙ i = / 𝐫 i 𝐫 ˙ i = - / 𝐩 i . \begin{aligned}\displaystyle\dot{\mathbf{p}}_{i}&\displaystyle=\partial% \mathcal{H}/\partial\mathbf{r}_{i}\\ \displaystyle\dot{\mathbf{r}}_{i}&\displaystyle=-\partial\mathcal{H}/\partial% \mathbf{p}_{i}.\end{aligned}
  37. 𝐫 ˙ i 𝐩 i , \dot{\mathbf{r}}_{i}\propto\mathbf{p}_{i},
  38. μ \mu
  39. 𝐩 i \mathbf{p}_{i}
  40. μ = μ d P d P , \langle\mu\rangle=\frac{\int\mu dP}{\int dP},
  41. - p d p , \int_{-\infty}^{\infty}pdp,
  42. p p
  43. p p
  44. μ = 0 \langle\mu\rangle=0

Bolometric_correction.html

  1. B C = M b - M v BC=M_{b}-M_{v}\!\,
  2. M b o l = 0 M_{bol}=0
  3. M b o l Sun = 4.74 M_{bol_{\rm Sun}}=4.74
  4. m b o l = 0 m_{bol}=0
  5. f o = 2.518021002... e - 8 W / m 2 f_{o}=2.518021002...e-8W/m2
  6. m b o l Sun = - 26.832 m_{bol_{\rm Sun}}=-26.832

Boltzmann-Matano_analysis.html

  1. c t = x [ D ( c ) c x ] Flux \frac{\partial c}{\partial t}=\frac{\partial}{\partial x}\overbrace{\left[D(c)% \frac{\partial c}{\partial x}\right]}\text{Flux}
  2. ξ = x 2 t \xi=\frac{x}{2\sqrt{t}}
  3. ξ t = - x 4 t 3 / 2 = - ξ 2 t \frac{\partial\xi}{\partial t}=-\frac{x}{4t^{3/2}}=-\frac{\xi}{2t}
  4. ξ x = 1 2 t \frac{\partial\xi}{\partial x}=\frac{1}{2\sqrt{t}}
  5. c t = c ξ ξ t = - ξ 2 t c ξ \frac{\partial c}{\partial t}=\frac{\partial c}{\partial\xi}\frac{\partial\xi}% {\partial t}=-\frac{\xi}{2t}\frac{\partial c}{\partial\xi}
  6. c x = c ξ ξ x = 1 2 t c ξ \frac{\partial c}{\partial x}=\frac{\partial c}{\partial\xi}\frac{\partial\xi}% {\partial x}=\frac{1}{2\sqrt{t}}\frac{\partial c}{\partial\xi}
  7. - ξ 2 t c ξ = 1 2 t x [ D ( c ) c ξ ] -\frac{\xi}{2t}\frac{\partial c}{\partial\xi}=\frac{1}{2\sqrt{t}}\frac{% \partial}{\partial x}\left[D(c)\frac{\partial c}{\partial\xi}\right]
  8. - ξ 2 t c ξ = 1 4 t ξ [ D ( c ) c ξ ] -\frac{\xi}{2t}\frac{\partial c}{\partial\xi}=\frac{1}{4t}\frac{\partial}{% \partial\xi}\left[D(c)\frac{\partial c}{\partial\xi}\right]
  9. - 2 ξ d c d ξ = d d ξ [ D ( c ) d c d ξ ] -2\xi\frac{\mathrm{d}c}{\mathrm{d}\xi}=\frac{\mathrm{d}}{\mathrm{d}\xi}\left[D% (c)\frac{\mathrm{d}c}{\mathrm{d}\xi}\right]
  10. x t x\propto\sqrt{t}
  11. t x 2 t\propto x^{2}
  12. c = c L x < 0 c=c_{L}\qquad\forall x<0
  13. c = c R x > 0 c=c_{R}\qquad\forall x>0
  14. - 2 c R c * ξ d c = c = c R c = c * d [ D ( c ) ( d c d ξ ) ] -2\int_{c_{R}}^{c^{*}}\xi\mathrm{d}c=\int_{c=c_{R}}^{c=c^{*}}\mathrm{d}\left[D% (c)\left(\frac{\mathrm{d}c}{\mathrm{d}\xi}\right)\right]
  15. - 1 2 t c R c * x d c = [ D ( c ) ( d c d x ) ] c = c R c = c * -\frac{1}{2t}\int_{c_{R}}^{c^{*}}x\mathrm{d}c=\left[D(c)\left(\frac{\mathrm{d}% c}{\mathrm{d}x}\right)\right]_{c=c_{R}}^{c=c^{*}}
  16. D ( c * ) = - 1 2 t c R c * x d c ( d c / d x ) c = c * D(c^{*})=-\frac{1}{2t}\frac{\int^{c^{*}}_{c_{R}}x\mathrm{d}c}{(\mathrm{d}c/% \mathrm{d}x)_{c=c^{*}}}
  17. 0 = c R c L ( x - X M ) d c 0=\int^{c_{L}}_{c_{R}}(x-X_{M})\mathrm{d}c
  18. X M = 1 c L - c R c R c L x d c X_{M}=\frac{1}{c_{L}-c_{R}}\int^{c_{L}}_{c_{R}}x\mathrm{d}c

Bondi_accretion.html

  1. M ˙ π R 2 ρ v , \dot{M}\simeq\pi R^{2}\rho v,
  2. ρ \rho
  3. v v
  4. c s c_{s}
  5. R R
  6. 2 G M R c s , \sqrt{\frac{2GM}{R}}\simeq c_{s},
  7. R 2 G M c s 2 R\simeq\frac{2GM}{c_{s}^{2}}
  8. M ˙ 4 π ρ G 2 M 2 c s 3 \dot{M}\simeq\frac{4\pi\rho G^{2}M^{2}}{c_{s}^{3}}

Boneh–Franklin_scheme.html

  1. G 1 \textstyle G_{1}
  2. G 2 \textstyle G_{2}
  3. G 1 \textstyle G_{1}
  4. p \textstyle p
  5. p 2 mod 3 \textstyle p\equiv 2\mod 3
  6. E : y 2 = x 3 + 1 \textstyle E:y^{2}=x^{3}+1
  7. / p \textstyle\mathbb{Z}/p\mathbb{Z}
  8. 4 a 3 + 27 b 2 = 27 = 3 3 \textstyle 4a^{3}+27b^{2}=27=3^{3}
  9. 0 \textstyle 0
  10. p = 3 \textstyle p=3
  11. q > 3 \textstyle q>3
  12. p + 1 \textstyle p+1
  13. E \textstyle E
  14. P E \textstyle P\in E
  15. q \textstyle q
  16. G 1 \textstyle G_{1}
  17. P \textstyle P
  18. { n P n { 0 , , q - 1 } } \textstyle\left\{nP\|n\in\left\{0,\ldots,q-1\right\}\right\}
  19. G 2 \textstyle G_{2}
  20. q \textstyle q
  21. G F ( p 2 ) * \textstyle GF\left(p^{2}\right)^{*}
  22. G 1 \textstyle G_{1}
  23. P \textstyle P
  24. G 2 \textstyle G_{2}
  25. q \textstyle q
  26. k \textstyle k
  27. e \textstyle e
  28. K m = s q * \textstyle K_{m}=s\in\mathbb{Z}_{q}^{*}
  29. K p u b = s P \textstyle K_{pub}=sP
  30. H 1 : { 0 , 1 } * G 1 * \textstyle H_{1}:\left\{0,1\right\}^{*}\rightarrow G_{1}^{*}
  31. H 2 : G 2 { 0 , 1 } n \textstyle H_{2}:G_{2}\rightarrow\left\{0,1\right\}^{n}
  32. n \textstyle n
  33. = { 0 , 1 } n , 𝒞 = G 1 * × { 0 , 1 } n \textstyle\mathcal{M}=\left\{0,1\right\}^{n},\mathcal{C}=G_{1}^{*}\times\left% \{0,1\right\}^{n}
  34. I D { 0 , 1 } * \textstyle ID\in\left\{0,1\right\}^{*}
  35. Q I D = H 1 ( I D ) \textstyle Q_{ID}=H_{1}\left(ID\right)
  36. d I D = s Q I D \textstyle d_{ID}=sQ_{ID}
  37. m \textstyle m\in\mathcal{M}
  38. c \textstyle c
  39. Q I D = H 1 ( I D ) G 1 * \textstyle Q_{ID}=H_{1}\left(ID\right)\in G_{1}^{*}
  40. r q * \textstyle r\in\mathbb{Z}_{q}^{*}
  41. g I D = e ( Q I D , K p u b ) G 2 \textstyle g_{ID}=e\left(Q_{ID},K_{pub}\right)\in G_{2}
  42. c = ( r P , m H 2 ( g I D r ) ) \textstyle c=\left(rP,m\oplus H_{2}\left(g_{ID}^{r}\right)\right)
  43. K p u b \textstyle K_{pub}
  44. c = ( u , v ) 𝒞 \textstyle c=\left(u,v\right)\in\mathcal{C}
  45. m = v H 2 ( e ( d I D , u ) ) \textstyle m=v\oplus H_{2}\left(e\left(d_{ID},u\right)\right)
  46. H 2 \textstyle H_{2}
  47. H 2 ( g I D r ) \textstyle H_{2}\left(g_{ID}^{r}\right)
  48. H 2 ( e ( d I D , u ) ) \textstyle H_{2}\left(e\left(d_{ID},u\right)\right)
  49. H 2 ( e ( d I D , u ) ) \displaystyle H_{2}\left(e\left(d_{ID},u\right)\right)

Boneh–Lynn–Shacham.html

  1. e : G × G G T e\colon G\times G\rightarrow G_{T}
  2. G G
  3. G T G_{T}
  4. r r
  5. g g
  6. G G
  7. g g
  8. g x g^{x}
  9. g y g^{y}
  10. e e
  11. g x y g^{xy}
  12. g z g^{z}
  13. g z = g x y g^{z}=g^{xy}
  14. x x
  15. y y
  16. z z
  17. e ( g x , g y ) = e ( g , g z ) e(g^{x},g^{y})=e(g,g^{z})
  18. x + y + z x+y+z
  19. e ( g x , g y ) = e ( g , g ) x y = e ( g , g ) z = e ( g , g z ) e(g^{x},g^{y})=e(g,g)^{xy}=e(g,g)^{z}=e(g,g^{z})
  20. G T G_{T}
  21. x y = z xy=z
  22. x x
  23. x x
  24. g x g^{x}
  25. x x
  26. m m
  27. m m
  28. h = H ( m ) h=H(m)
  29. σ = h x \sigma=h^{x}
  30. σ \sigma
  31. g x g^{x}
  32. e ( σ , g ) = e ( H ( m ) , g x ) e(\sigma,g)=e(H(m),g^{x})

Bootstrapping_populations.html

  1. { x 1 , , x m } \{x_{1},\ldots,x_{m}\}
  2. s y m b o l θ symbol\theta
  3. s y m b o l Θ symbol\Theta
  4. { s 1 , , s k } \{s_{1},\ldots,s_{k}\}
  5. { x 1 , , x m } \{x_{1},\ldots,x_{m}\}
  6. x i x_{i}
  7. z i z_{i}
  8. ( g s y m b o l θ , Z ) (g_{symbol\theta},Z)
  9. s j s_{j}
  10. s y m b o l x = { x 1 , , x m } symbolx=\{x_{1},\ldots,x_{m}\}
  11. ( g s y m b o l θ , Z ) (g_{symbol\theta},Z)
  12. s y m b o l x = { g s y m b o l θ ( z 1 ) , , g s y m b o l θ ( z m ) } symbolx=\{g_{symbol\theta}(z_{1}),\ldots,g_{symbol\theta}(z_{m})\}
  13. s y m b o l θ = ( θ 1 , , θ k ) symbol\theta=(\theta_{1},\ldots,\theta_{k})
  14. s 1 = h 1 ( x 1 , , x m ) , s_{1}=h_{1}(x_{1},\ldots,x_{m}),
  15. \vdots\ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots
  16. s k = h k ( x 1 , , x m ) , s_{k}=h_{k}(x_{1},\ldots,x_{m}),
  17. s 1 = h 1 ( g s y m b o l θ ( z 1 ) , , g s y m b o l θ ( z m ) ) = ρ 1 ( s y m b o l θ ; z 1 , , z m ) s_{1}=h_{1}(g_{symbol\theta}(z_{1}),\ldots,g_{symbol\theta}(z_{m}))=\rho_{1}(% symbol\theta;z_{1},\ldots,z_{m})
  18. \vdots\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vdots\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ % \ \ \ \ \ \ \ \ \ \ \ \ \vdots
  19. s k = h k ( g s y m b o l θ ( z 1 ) , , g s y m b o l θ ( z m ) ) = ρ k ( s y m b o l θ ; z 1 , , z m ) . s_{k}=h_{k}(g_{symbol\theta}(z_{1}),\ldots,g_{symbol\theta}(z_{m}))=\rho_{k}(% symbol\theta;z_{1},\ldots,z_{m}).
  20. { z 1 , , z m } \{z_{1},\ldots,z_{m}\}
  21. s y m b o l θ symbol\theta
  22. s i s_{i}
  23. Θ j \Theta_{j}
  24. F ^ Θ j ( θ ) = i = 1 N 1 N I ( - , θ ] ( θ ˘ j , i ) \widehat{F}_{\Theta_{j}}(\theta)=\sum_{i=1}^{N}\frac{1}{N}I_{(-\infty,\theta]}% (\breve{\theta}_{j,i})
  25. θ ˘ j , i \breve{\theta}_{j,i}
  26. I ( - , θ ] ( θ ˘ j , i ) I_{(-\infty,\theta]}(\breve{\theta}_{j,i})
  27. θ ˘ j , i \breve{\theta}_{j,i}
  28. ( - , θ ] . (-\infty,\theta].
  29. s y m b o l Θ symbol\Theta
  30. s y m b o l s s y m b o l Θ symbols_{symbol\Theta}
  31. { x 1 , , x m } \{x_{1},\ldots,x_{m}\}
  32. s y m b o l θ symbol\theta
  33. s y m b o l S symbolS
  34. s y m b o l Θ symbol\Theta
  35. s y m b o l s s y m b o l Θ symbols_{symbol\Theta}
  36. s y m b o l S symbolS
  37. s y m b o l z ˘ i \breve{symbolz}_{i}
  38. s y m b o l θ ˘ i = Inv ( s y m b o l s , s y m b o l z i ) \breve{symbol\theta}_{i}=\mathrm{Inv}(symbols,symbolz_{i})
  39. s y m b o l s = s y m b o l s s y m b o l Θ symbols=symbols_{symbol\Theta}
  40. s y m b o l z i = { z ˘ 1 , , z ˘ m } symbolz_{i}=\{\breve{z}_{1},\ldots,\breve{z}_{m}\}
  41. s y m b o l θ ˘ i \breve{symbol\theta}_{i}
  42. s y m b o l Θ symbol\Theta
  43. s Λ = j = 1 m x j s_{\Lambda}=\sum_{j=1}^{m}x_{j}
  44. iii) Inv ( s Λ , s y m b o l u i ) = j = 1 m ( - log u i j ) / s Λ \,\text{ iii) Inv}(s_{\Lambda},symbolu_{i})=\sum_{j=1}^{m}(-\log u_{ij})/s_{\Lambda}
  45. [ 0 , a ] [0,a]
  46. s A = max j = 1 , , m x j s_{A}=\max_{j=1,\ldots,m}x_{j}
  47. iii) Inv ( s A , s y m b o l u i ) = s A / max j = 1 , , m { u i j } \,\text{iii) Inv}(s_{A},symbolu_{i})=s_{A}/\max_{j=1,\ldots,m}\{u_{ij}\}
  48. s y m b o l x symbolx
  49. a a
  50. F X ( x ) = 1 - ( k x ) a F_{X}(x)=1-\left(\frac{k}{x}\right)^{a}
  51. ( g ( a , k ) , U ) (g_{(a,k)},U)
  52. [ 0 , 1 ] [0,1]
  53. g ( a , k ) g_{(a,k)}
  54. x = g ( a , k ) = ( 1 - u ) - 1 a k x=g_{(a,k)}=(1-u)^{-\frac{1}{a}}k
  55. s y m b o l s s y m b o l Θ symbols_{s}ymbol\Theta
  56. A A
  57. s 1 = i = 1 m log x i , s 2 = min { x i } s_{1}=\sum_{i=1}^{m}\log x_{i},s_{2}=\min\{x_{i}\}
  58. s 1 = i = 1 m - 1 a log ( 1 - u i ) + m log k s_{1}=\sum_{i=1}^{m}-\frac{1}{a}\log(1-u_{i})+m\log k
  59. s 2 = ( 1 - u min ) - 1 a k s_{2}=(1-u_{\min})^{-\frac{1}{a}}k
  60. u min = min { u i } u_{\min}=\min\{u_{i}\}
  61. ( A , K ) (A,K)
  62. x min x_{\mathrm{min}}

Borda–Carnot_equation.html

  1. Δ E = ξ 1 2 ρ ( v 1 - v 2 ) 2 , \Delta E\,=\,\xi\,{\scriptstyle\frac{1}{2}}\,\rho\,\left(v_{1}\,-\,v_{2}\right% )^{2},
  2. p 1 + 1 2 ρ v 1 2 + ρ g z 1 = p 2 + 1 2 ρ v 2 2 + ρ g z 2 + Δ E , p_{1}\,+\,{\scriptstyle\frac{1}{2}}\,\rho\,v_{1}^{2}\,+\,\rho\,g\,z_{1}\,=\,p_% {2}\,+\,{\scriptstyle\frac{1}{2}}\,\rho\,v_{2}^{2}\,+\,\rho\,g\,z_{2}\,+\,% \Delta E,
  3. Δ E = Δ ( p + 1 2 ρ v 2 ) . \Delta E\,=\,\Delta\left(p\,+\,{\scriptstyle\frac{1}{2}}\,\rho\,v^{2}\right).
  4. Δ E = ρ g Δ H , \Delta E\,=\,\rho\,g\,\Delta H,
  5. H = h + v 2 2 g , H\,=\,h\,+\,\frac{v^{2}}{2g},
  6. A 1 v 1 = A 2 v 2 A_{1}\,v_{1}\,=A_{2}\,v_{2}
  7. v 2 = A 1 A 2 v 1 . v_{2}\,=\,\frac{A_{1}}{A_{2}}\,v_{1}.
  8. Δ E = 1 2 ρ ( 1 - A 1 A 2 ) 2 v 1 2 . \Delta E\,=\,\frac{1}{2}\,\rho\,\left(1\,-\,\frac{A_{1}}{A_{2}}\right)^{2}\,v_% {1}^{2}.
  9. Δ H = Δ E ρ g = 1 2 g ( 1 - A 1 A 2 ) 2 v 1 2 . \Delta H\,=\,\frac{\Delta E}{\rho\,g}\,=\,\frac{1}{2\,g}\,\left(1\,-\,\frac{A_% {1}}{A_{2}}\right)^{2}\,v_{1}^{2}.
  10. Δ p = p 1 - p 2 = - ρ A 1 A 2 ( 1 - A 1 A 2 ) v 1 2 , \Delta p\,=\,p_{1}\,-\,p_{2}\,=\,-\,\rho\,\frac{A_{1}}{A_{2}}\left(1\,-\,\frac% {A_{1}}{A_{2}}\right)\,v_{1}^{2},
  11. Δ h = h 1 - h 2 = - 1 g A 1 A 2 ( 1 - A 1 A 2 ) v 1 2 . \Delta h\,=\,h_{1}\,-\,h_{2}\,=\,-\,\frac{1}{g}\,\frac{A_{1}}{A_{2}}\left(1\,-% \,\frac{A_{1}}{A_{2}}\right)\,v_{1}^{2}.
  12. μ = A 3 A 2 , \mu\,=\,\frac{A_{3}}{A_{2}},
  13. A 1 v 1 = A 2 v 2 = A 3 v 3 , A_{1}\,v_{1}\,=\,A_{2}\,v_{2}\,=\,A_{3}\,v_{3},
  14. Δ E = 1 2 ρ ( v 3 - v 2 ) 2 = 1 2 ρ ( 1 μ - 1 ) 2 v 2 2 = 1 2 ρ ( 1 μ - 1 ) 2 ( A 1 A 2 ) 2 v 1 2 . \Delta E\,=\,\frac{1}{2}\,\rho\,\left(v_{3}\,-\,v_{2}\right)^{2}\,=\,\frac{1}{% 2}\,\rho\,\left(\frac{1}{\mu}\,-\,1\right)^{2}\,v_{2}^{2}\,=\,\frac{1}{2}\,% \rho\,\left(\frac{1}{\mu}\,-\,1\right)^{2}\,\left(\frac{A_{1}}{A_{2}}\right)^{% 2}\,v_{1}^{2}.
  15. μ = 0.63 + 0.37 ( A 2 A 1 ) 3 . \mu\,=\,0.63\,+\,0.37\,\left(\frac{A_{2}}{A_{1}}\right)^{3}.
  16. S = A ( 1 2 ρ v 2 + p ) . S=A\,\left(\frac{1}{2}\,\rho\,v^{2}+p\right).
  17. F = ( A 2 - A 1 ) p 1 , F=(A_{2}-A_{1})\,p_{1},
  18. S 1 - S 2 + F = A 1 ( ρ v 1 2 + p 1 ) - A 2 ( ρ v 2 2 + p 2 ) + ( A 2 - A 1 ) p 1 = 0. S_{1}-S_{2}+F=A_{1}\,\left(\rho\,v_{1}^{2}+p_{1}\right)-A_{2}\,\left(\rho\,v_{% 2}^{2}+p_{2}\right)+(A_{2}-A_{1})\,p_{1}=0.
  19. Δ p = p 1 - p 2 = - ρ ( A 1 A 2 v 1 2 - v 2 2 ) = - ρ A 1 A 2 ( 1 - A 1 A 2 ) v 1 2 , \Delta p=p_{1}-p_{2}=-\rho\,\left(\frac{A_{1}}{A_{2}}\,v_{1}^{2}-v_{2}^{2}% \right)=-\rho\,\frac{A_{1}}{A_{2}}\,\left(1-\frac{A_{1}}{A_{2}}\right)\,v_{1}^% {2},
  20. Δ E \displaystyle\Delta E

Borderline_tree.html

  1. 75.625 B A F \sqrt{\frac{75.625}{BAF}}
  2. 0.25 B A F \sqrt{\frac{0.25}{BAF}}

Born–Landé_equation.html

  1. E = - N A M z + z - e 2 4 π ϵ 0 r 0 ( 1 - 1 n ) E=-\frac{N_{A}Mz^{+}z^{-}e^{2}}{4\pi\epsilon_{0}r_{0}}\left(1-\frac{1}{n}\right)
  2. × 10 19 \times 10^{−}19
  3. × 10 10 \times 10^{−}10
  4. E pair E\text{pair}
  5. E pair = - z 2 e 2 4 π ϵ 0 r E\text{pair}=-\frac{z^{2}e^{2}}{4\pi\epsilon_{0}r}
  6. z z
  7. e e
  8. × 10 19 \times 10^{−}19
  9. ϵ 0 \epsilon_{0}
  10. 4 π ϵ 0 4\pi\epsilon_{0}
  11. × 10 10 \times 10^{−}10
  12. r r
  13. E M E_{M}
  14. E M = - z 2 e 2 M 4 π ϵ 0 r E_{M}=-\frac{z^{2}e^{2}M}{4\pi\epsilon_{0}r}
  15. M M
  16. r r
  17. 1 / r n 1/r^{n}
  18. E R E_{R}
  19. E R = B r n \,E_{R}=\frac{B}{r^{n}}
  20. B B
  21. r r
  22. n n
  23. E ( r ) = - z 2 e 2 M 4 π ϵ 0 r + B r n E(r)=-\frac{z^{2}e^{2}M}{4\pi\epsilon_{0}r}+\frac{B}{r^{n}}
  24. r r
  25. r 0 r_{0}
  26. B B
  27. d E d r \displaystyle\frac{\mathrm{d}E}{\mathrm{d}r}
  28. B B
  29. r 0 r_{0}
  30. E ( r 0 ) = - M z 2 e 2 4 π ϵ 0 r 0 ( 1 - 1 n ) E(r_{0})=-\frac{Mz^{2}e^{2}}{4\pi\epsilon_{0}r_{0}}\left(1-\frac{1}{n}\right)

Boundary_current.html

  1. D + h D+h
  2. f ( D + h ) v - F cos ( π y b ) - R u - g ( D + h ) h x = 0 ( 1 ) f(D+h)v-F\cos\left(\frac{\pi y}{b}\right)-Ru-g(D+h)\frac{\partial h}{\partial x% }=0\qquad(1)
  3. - f ( D + h ) u - R v - g ( D + h ) h y = 0 ( 2 ) \quad-f(D+h)u-Rv-g(D+h)\frac{\partial h}{\partial y}=0\qquad\qquad(2)
  4. [ ( D + h ) u ] x + [ ( D + h ) v ] y = 0 ( 3 ) \qquad\qquad\frac{\partial[(D+h)u]}{\partial x}+\frac{\partial[(D+h)v]}{% \partial y}=0\qquad\qquad\qquad(3)
  5. f f
  6. R R
  7. g g\,\,
  8. - F cos ( π y b ) -F\cos\left(\frac{\pi y}{b}\right)
  9. y = 0 y=0
  10. y = b y=b
  11. y \frac{\partial}{\partial y}
  12. x \frac{\partial}{\partial x}
  13. v ( D + h ) ( f y ) + π F b sin ( π y b ) + R ( v x - u y ) = 0 ( 4 ) v(D+h)\left(\frac{\partial f}{\partial y}\right)+\frac{\pi F}{b}\sin\left(% \frac{\pi y}{b}\right)+R\left(\frac{\partial v}{\partial x}-\frac{\partial u}{% \partial y}\right)=0\quad(4)
  14. ψ \psi
  15. D h D>>h
  16. 2 ψ + α ( ψ x ) = γ sin ( π y b ) ( 5 ) \nabla^{2}\psi+\alpha\left(\frac{\partial\psi}{\partial x}\right)=\gamma\sin% \left(\frac{\pi y}{b}\right)\qquad(5)
  17. α = ( D R ) ( f y ) \alpha=\left(\frac{D}{R}\right)\left(\frac{\partial f}{\partial y}\right)
  18. γ = π F R b \gamma=\frac{\pi F}{Rb}
  19. ψ \psi
  20. α \alpha

Box–Cox_distribution.html

  1. f ( y ) = 1 ( 1 - I ( f < 0 ) - sgn ( f ) Φ ( 0 , m , s ) ) 2 π s 2 exp { - 1 2 s 2 ( y f f - m ) 2 } f(y)=\frac{1}{\left(1-I(f<0)-\operatorname{sgn}(f)\Phi(0,m,\sqrt{s})\right)% \sqrt{2\pi s^{2}}}\exp\left\{-\frac{1}{2s^{2}}\left(\frac{y^{f}}{f}-m\right)^{% 2}\right\}

Branch-decomposition.html

  1. b - 1 k 3 2 b - 1. b-1\leq k\leq\left\lfloor\frac{3}{2}b\right\rfloor-1.

Brazilian_cruzado_novo.html

  1. NCzS \mathrm{NCzS}\!\!\!\|

Bresler_Pister_yield_criterion.html

  1. J 2 = A + B I 1 + C I 1 2 \sqrt{J_{2}}=A+B~{}I_{1}+C~{}I_{1}^{2}
  2. I 1 I_{1}
  3. J 2 J_{2}
  4. A , B , C A,B,C
  5. A , B , C A,B,C
  6. σ c \sigma_{c}
  7. σ t \sigma_{t}
  8. σ b \sigma_{b}
  9. B = ( σ t - σ c 3 ( σ t + σ c ) ) ( 4 σ b 2 - σ b ( σ c + σ t ) + σ c σ t 4 σ b 2 + 2 σ b ( σ t - σ c ) - σ c σ t ) C = ( 1 3 ( σ t + σ c ) ) ( σ b ( 3 σ t - σ c ) - 2 σ c σ t 4 σ b 2 + 2 σ b ( σ t - σ c ) - σ c σ t ) A = σ c 3 + c 1 σ c - c 2 σ c 2 \begin{aligned}\displaystyle B=&\displaystyle\left(\cfrac{\sigma_{t}-\sigma_{c% }}{\sqrt{3}(\sigma_{t}+\sigma_{c})}\right)\left(\cfrac{4\sigma_{b}^{2}-\sigma_% {b}(\sigma_{c}+\sigma_{t})+\sigma_{c}\sigma_{t}}{4\sigma_{b}^{2}+2\sigma_{b}(% \sigma_{t}-\sigma_{c})-\sigma_{c}\sigma_{t}}\right)\\ \displaystyle C=&\displaystyle\left(\cfrac{1}{\sqrt{3}(\sigma_{t}+\sigma_{c})}% \right)\left(\cfrac{\sigma_{b}(3\sigma_{t}-\sigma_{c})-2\sigma_{c}\sigma_{t}}{% 4\sigma_{b}^{2}+2\sigma_{b}(\sigma_{t}-\sigma_{c})-\sigma_{c}\sigma_{t}}\right% )\\ \displaystyle A=&\displaystyle\cfrac{\sigma_{c}}{\sqrt{3}}+c_{1}\sigma_{c}-c_{% 2}\sigma_{c}^{2}\end{aligned}
  10. σ 1 , σ 2 , σ 3 \sigma_{1},\sigma_{2},\sigma_{3}
  11. 1 6 [ ( σ 1 - σ 2 ) 2 + ( σ 2 - σ 3 ) 2 + ( σ 3 - σ 1 ) 2 ] 1 / 2 - A - B ( σ 1 + σ 2 + σ 3 ) - C ( σ 1 + σ 2 + σ 3 ) 2 = 0 . \cfrac{1}{\sqrt{6}}\left[(\sigma_{1}-\sigma_{2})^{2}+(\sigma_{2}-\sigma_{3})^{% 2}+(\sigma_{3}-\sigma_{1})^{2}\right]^{1/2}-A-B~{}(\sigma_{1}+\sigma_{2}+% \sigma_{3})-C~{}(\sigma_{1}+\sigma_{2}+\sigma_{3})^{2}=0~{}.
  12. σ t = σ 1 \sigma_{t}=\sigma_{1}
  13. 1 3 σ t - A - B σ t - C σ t 2 = 0 . \cfrac{1}{\sqrt{3}}~{}\sigma_{t}-A-B\sigma_{t}-C\sigma_{t}^{2}=0~{}.
  14. - σ c = σ 1 -\sigma_{c}=\sigma_{1}
  15. 1 3 σ c - A + B σ c - C σ c 2 = 0 . \cfrac{1}{\sqrt{3}}~{}\sigma_{c}-A+B\sigma_{c}-C\sigma_{c}^{2}=0~{}.
  16. - σ b = σ 1 = σ 2 -\sigma_{b}=\sigma_{1}=\sigma_{2}
  17. 1 3 σ b - A + 2 B σ b - 4 C σ b 2 = 0 . \cfrac{1}{\sqrt{3}}~{}\sigma_{b}-A+2B\sigma_{b}-4C\sigma_{b}^{2}=0~{}.
  18. A , B , C A,B,C
  19. A := 1 3 σ c σ t σ b ( σ t + 8 σ b - 3 σ c ) ( σ c + σ t ) ( 2 σ b - σ c ) ( 2 σ b + σ t ) B := 1 3 ( σ c - σ t ) ( σ b σ c + σ b σ t - σ c σ t - 4 σ b 2 ) ( σ c + σ t ) ( 2 σ b - σ c ) ( 2 σ b + σ t ) C := 1 3 3 σ b σ t - σ b σ c - 2 σ c σ t ( σ c + σ t ) ( 2 σ b - σ c ) ( 2 σ b + σ t ) \begin{aligned}\displaystyle A:=&\displaystyle\cfrac{1}{\sqrt{3}}~{}\cfrac{% \sigma_{c}\sigma_{t}\sigma_{b}(\sigma_{t}+8\sigma_{b}-3\sigma_{c})}{(\sigma_{c% }+\sigma_{t})(2\sigma_{b}-\sigma_{c})(2\sigma_{b}+\sigma_{t})}\\ \displaystyle B:=&\displaystyle\cfrac{1}{\sqrt{3}}~{}\cfrac{(\sigma_{c}-\sigma% _{t})(\sigma_{b}\sigma_{c}+\sigma_{b}\sigma_{t}-\sigma_{c}\sigma_{t}-4\sigma_{% b}^{2})}{(\sigma_{c}+\sigma_{t})(2\sigma_{b}-\sigma_{c})(2\sigma_{b}+\sigma_{t% })}\\ \displaystyle C:=&\displaystyle\cfrac{1}{\sqrt{3}}~{}\cfrac{3\sigma_{b}\sigma_% {t}-\sigma_{b}\sigma_{c}-2\sigma_{c}\sigma_{t}}{(\sigma_{c}+\sigma_{t})(2% \sigma_{b}-\sigma_{c})(2\sigma_{b}+\sigma_{t})}\end{aligned}
  20. σ c = 1 , σ t = 0.3 , σ b = 1.7 \sigma_{c}=1,\sigma_{t}=0.3,\sigma_{b}=1.7
  21. π \pi
  22. σ c = 1 , σ t = 0.3 , σ b = 1.7 \sigma_{c}=1,\sigma_{t}=0.3,\sigma_{b}=1.7
  23. σ 1 - σ 2 \sigma_{1}-\sigma_{2}
  24. σ c = 1 , σ t = 0.3 , σ b = 1.7 \sigma_{c}=1,\sigma_{t}=0.3,\sigma_{b}=1.7
  25. σ e \sigma_{e}
  26. σ m \sigma_{m}
  27. σ e = a + b σ m + c σ m 2 ; σ e = 3 J 2 , σ m = I 1 / 3 . \sigma_{e}=a+b~{}\sigma_{m}+c~{}\sigma_{m}^{2}~{};~{}~{}\sigma_{e}=\sqrt{3J_{2% }}~{},~{}~{}\sigma_{m}=I_{1}/3~{}.
  28. J 2 = 1 3 I 1 - 1 2 3 ( σ t σ c 2 - σ t 2 ) I 1 2 \sqrt{J_{2}}=\cfrac{1}{\sqrt{3}}~{}I_{1}-\cfrac{1}{2\sqrt{3}}~{}\left(\cfrac{% \sigma_{t}}{\sigma_{c}^{2}-\sigma_{t}^{2}}\right)~{}I_{1}^{2}
  29. σ c \sigma_{c}
  30. σ t \sigma_{t}
  31. J 2 = { 1 3 σ t - 0.03 3 ρ ρ m σ t I 1 2 - 1 3 σ c + 0.03 3 ρ ρ m σ c I 1 2 \sqrt{J_{2}}=\begin{cases}\cfrac{1}{\sqrt{3}}~{}\sigma_{t}-0.03\sqrt{3}\cfrac{% \rho}{\rho_{m}~{}\sigma_{t}}~{}I_{1}^{2}\\ -\cfrac{1}{\sqrt{3}}~{}\sigma_{c}+0.03\sqrt{3}\cfrac{\rho}{\rho_{m}~{}\sigma_{% c}}~{}I_{1}^{2}\end{cases}
  32. ρ \rho
  33. ρ m \rho_{m}

Breusch–Godfrey_test.html

  1. Y t = α 0 + α 1 X t , 1 + α 2 X t , 2 + u t Y_{t}=\alpha_{0}+\alpha_{1}X_{t,1}+\alpha_{2}X_{t,2}+u_{t}\,
  2. u t = ρ 1 u t - 1 + ρ 2 u t - 2 + + ρ p u t - p + ε t . u_{t}=\rho_{1}u_{t-1}+\rho_{2}u_{t-2}+\cdots+\rho_{p}u_{t-p}+\varepsilon_{t}.\,
  3. u ^ t \hat{u}_{t}
  4. u ^ t = α 0 + α 1 X t , 1 + α 2 X t , 2 + ρ 1 u ^ t - 1 + ρ 2 u ^ t - 2 + + ρ p u ^ t - p + ε t \hat{u}_{t}=\alpha_{0}+\alpha_{1}X_{t,1}+\alpha_{2}X_{t,2}+\rho_{1}\hat{u}_{t-% 1}+\rho_{2}\hat{u}_{t-2}+\cdots+\rho_{p}\hat{u}_{t-p}+\varepsilon_{t}\,
  5. R 2 R^{2}
  6. n R 2 χ p 2 , nR^{2}\,\sim\,\chi^{2}_{p},\,
  7. H 0 : { ρ i = 0 for all i } {H_{0}:\{\rho_{i}=0\,\text{ for all }i\}}
  8. u ^ t \hat{u}_{t}
  9. n = T - p , n=T-p,\,

Bridge_(interpersonal).html

  1. n 1 n_{1}
  2. n 2 n_{2}
  3. e e
  4. n 1 n_{1}
  5. n 2 n_{2}
  6. e e
  7. e e
  8. n 1 n_{1}
  9. n 2 n_{2}
  10. e e

Bridged_T_delay_equaliser.html

  1. ϕ = ω T D \phi=\omega T_{D}\,
  2. ω 0 = 1 L C = 1 L C \omega_{0}=\frac{1}{\sqrt{LC}}=\frac{1}{\sqrt{L^{\prime}C^{\prime}}}
  3. T D ( m a x ) = 1 2 f m T_{D(max)}=\frac{1}{2f_{m}}

Brillouin's_theorem.html

  1. | ψ 0 |\psi_{0}\rangle
  2. ψ 0 | H ^ | ψ a r = 0 \langle\psi_{0}|\hat{H}|\psi_{a}^{r}\rangle=0
  3. h ( 1 ) = - 1 2 1 2 - α Z α r 1 α h(1)=-\frac{1}{2}\nabla^{2}_{1}-\sum_{\alpha}\frac{Z_{\alpha}}{r_{1\alpha}}
  4. j | r 1 - r j | - 1 \sum_{j}|r_{1}-r_{j}|^{-1}
  5. ψ 0 | H ^ | ψ a r = a | h | r + b a b | | r b \langle\psi_{0}|\hat{H}|\psi_{a}^{r}\rangle=\langle a|h|r\rangle+\sum_{b}% \langle ab||rb\rangle
  6. χ a | f | χ r \langle\chi_{a}|f|\chi_{r}\rangle

BRP-PACU.html

  1. 1 / p L ( i k ) 1/p_{L}(ik)

Brumer–Stark_conjecture.html

  1. K / k K/k
  2. S S
  3. k k
  4. K / k K/k
  5. S S
  6. θ ( s ) θ(s)
  7. S S
  8. 𝐂 G G \mathbf{C}GG
  9. G G
  10. K / k K/k
  11. s = 1 s=1
  12. K K
  13. G G
  14. A A
  15. 𝐙 G G \mathbf{Z}GG
  16. θ ( 0 ) θ(0)
  17. 𝐐 G G \mathbf{Q}GG
  18. A θ ( 0 ) Aθ(0)
  19. 𝐙 G G \mathbf{Z}GG
  20. W θ ( 0 ) Wθ(0)
  21. 𝐙 G G \mathbf{Z}GG
  22. W W
  23. K K
  24. G G
  25. W θ ( 0 ) Wθ(0)
  26. 𝔞 \mathfrak{a}
  27. K K
  28. ε ε
  29. 𝔞 W θ ( 0 ) = ( ε ) . \mathfrak{a}^{W\theta(0)}=(\varepsilon).
  30. K ( ε 1 W ) / k K\left(\varepsilon^{\frac{1}{W}}\right)/k
  31. | ε | < s u b > ν |ε|<sub>ν

Brun_sieve.html

  1. S ( A , P , z ) = | A p P ( z ) A p | . S(A,P,z)=\left|A\setminus\bigcup_{p\in P(z)}A_{p}\right|.
  2. | A d | = w ( d ) d X + R d \left|A_{d}\right|=\frac{w(d)}{d}X+R_{d}
  3. W ( z ) = p P ( z ) ( 1 - w ( p ) p ) . W(z)=\prod_{p\in P(z)}\left(1-\frac{w(p)}{p}\right).
  4. p P z w ( p ) p < D log log z + E ; \sum_{p\in P_{z}}\frac{w(p)}{p}<D\log\log z+E;
  5. S ( A , P , z ) = X W ( z ) ( 1 + O ( ( log z ) - b log b ) ) + O ( z b log log z ) S(A,P,z)=X\cdot W(z)\cdot\left({1+O\left((\log z)^{-b\log b}\right)}\right)+O% \left(z^{b\log\log z}\right)
  6. S ( A , P , z ) = X W ( z ) ( 1 + o ( 1 ) ) . S(A,P,z)=X\cdot W(z)(1+o(1)).\,

Brun–Titchmarsh_theorem.html

  1. π ( x ; q , a ) \pi(x;q,a)
  2. π ( x ; q , a ) 2 x φ ( q ) log ( x / q ) \pi(x;q,a)\leq{2x\over\varphi(q)\log(x/q)}
  3. q x 9 / 20 q\leq x^{9/20}
  4. π ( x ; q , a ) ( 2 + o ( 1 ) ) x φ ( q ) ln ( x / q 3 / 8 ) \pi(x;q,a)\leq{(2+o(1))x\over\varphi(q)\ln(x/q^{3/8})}
  5. π ( x ; q , a ) = x φ ( q ) log ( x ) ( 1 + O ( 1 log x ) ) \pi(x;q,a)=\frac{x}{\varphi(q)\log(x)}\left({1+O\left(\frac{1}{\log x}\right)}\right)

Bs_space.html

  1. sup n | i = 1 n x i | \sup_{n}\left|\sum_{i=1}^{n}x_{i}\right|
  2. x b s = sup n | i = 1 n x i | . \left\|x\right\|_{bs}=\sup_{n}\left|\sum_{i=1}^{n}x_{i}\right|.
  3. i = 1 x i \sum_{i=1}^{\infty}x_{i}
  4. T ( x 1 , x 2 , ) = ( x 1 , x 1 + x 2 , x 1 + x 2 + x 3 , ) . T(x_{1},x_{2},\dots)=(x_{1},x_{1}+x_{2},x_{1}+x_{2}+x_{3},\dots).

BSWW_Beach_Soccer_rankings.html

  1. Points Earned = ( 3 × Number of games won in normal time ) + ( 2 × Number of games won in extra time or on penalties ) \,\text{Points Earned}=(3\times\,\text{Number of games won in normal time})+(2% \times\,\text{Number of games won in extra time or on penalties})
  2. Points Earned = Points earned in 2008 100 × 40 + Points earned in 2009 100 × 70 + Points earned in 2010 \,\text{Points Earned}=\frac{\,\text{Points earned in 2008}}{100}\times 40+% \frac{\,\text{Points earned in 2009}}{100}\times 70+\,\text{Points earned in 2% 010}

BTZ_black_hole.html

  1. d s 2 = - ( r 2 - r + 2 ) ( r 2 - r - 2 ) l 2 r 2 d t 2 + l 2 r 2 d r 2 ( r 2 - r + 2 ) ( r 2 - r - 2 ) + r 2 ( d ϕ - r + r - l r 2 d t ) 2 ds^{2}=-\frac{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}{l^{2}r^{2}}dt^{2}+\frac{l^{2% }r^{2}dr^{2}}{(r^{2}-r_{+}^{2})(r^{2}-r_{-}^{2})}+r^{2}\left(d\phi-\frac{r_{+}% r_{-}}{lr^{2}}dt\right)^{2}
  2. r + , r - r_{+},~{}r_{-}
  3. l l
  4. M = r + 2 + r - 2 l 2 , J = 2 r + r - l M=\frac{r_{+}^{2}+r_{-}^{2}}{l^{2}},~{}~{}~{}~{}~{}J=\frac{2r_{+}r_{-}}{l}

Bulging_factor.html

  1. b u l g i n g f a c t o r = S I F ( c u r v e d ) / S I F ( f l a t ) bulgingfactor=SIF(curved)/SIF(flat)

Burau_representation.html

  1. n n
  2. n n
  3. 𝐙 Z 𝐙 \mathbf{Z}Z\mathbf{Z}≅≅
  4. n 1 n− 1
  5. Γ Γ
  6. γ γ
  7. n n
  8. D = { ( x 1 , , x n ) 𝐙 n : x 1 + + x n = 0 } , D=\left\{\left(x_{1},\cdots,x_{n}\right)\in\mathbf{Z}^{n}:x_{1}+\cdots+x_{n}=0% \right\},
  9. σ i ( I i - 1 0 0 0 0 1 - t t 0 0 1 0 0 0 0 0 I n - i - 1 ) , \sigma_{i}\mapsto\left(\begin{array}[]{c|cc|c}I_{i-1}&0&0&0\\ \hline 0&1-t&t&0\\ 0&1&0&0\\ \hline 0&0&0&I_{n-i-1}\end{array}\right),
  10. 1 i n 1 1≤i≤n−1
  11. k × k k×k
  12. n 3 n≥3
  13. σ 1 ( - t 1 0 0 1 0 0 0 I n - 3 ) , \sigma_{1}\mapsto\left(\begin{array}[]{cc|c}-t&1&0\\ 0&1&0\\ \hline 0&0&I_{n-3}\end{array}\right),
  14. σ i ( I i - 2 0 0 0 0 0 1 0 0 0 0 t - t 1 0 0 0 0 1 0 0 0 0 0 I n - i - 2 ) , 2 i n - 2 , \sigma_{i}\mapsto\left(\begin{array}[]{c|ccc|c}I_{i-2}&0&0&0&0\\ \hline 0&1&0&0&0\\ 0&t&-t&1&0\\ 0&0&0&1&0\\ \hline 0&0&0&0&I_{n-i-2}\end{array}\right),\quad 2\leq i\leq n-2,
  15. σ n - 1 ( I n - 3 0 0 0 1 0 0 t - t ) , \sigma_{n-1}\mapsto\left(\begin{array}[]{c|cc}I_{n-3}&0&0\\ \hline 0&1&0\\ 0&t&-t\end{array}\right),
  16. n = 2 n=2
  17. σ 1 ( - t ) . \sigma_{1}\mapsto\left(-t\right).
  18. t t
  19. 0 , 11 0,11
  20. σ σ
  21. n n
  22. n n
  23. t t
  24. ( i , j ) (i,j)
  25. σ σ
  26. i i
  27. j j
  28. K K
  29. f f
  30. K K
  31. 1 - t 1 - t n det ( I - f * ) , \frac{1-t}{1-t^{n}}\det(I-f_{*}),
  32. f f
  33. 1 - t 1 - t n det ( I - f * ) = t , \frac{1-t}{1-t^{n}}\det(I-f_{*})=t,
  34. 1 1
  35. n 5 n≥5
  36. ψ 1 = σ 3 - 1 σ 2 σ 1 2 σ 2 σ 4 3 σ 3 σ 2 , ψ 2 = σ 4 - 1 σ 3 σ 2 σ 1 - 2 σ 2 σ 1 2 σ 2 2 σ 1 σ 4 5 . \psi_{1}=\sigma_{3}^{-1}\sigma_{2}\sigma_{1}^{2}\sigma_{2}\sigma_{4}^{3}\sigma% _{3}\sigma_{2},\quad\psi_{2}=\sigma_{4}^{-1}\sigma_{3}\sigma_{2}\sigma_{1}^{-2% }\sigma_{2}\sigma_{1}^{2}\sigma_{2}^{2}\sigma_{1}\sigma_{4}^{5}.
  37. [ ψ 1 - 1 σ 4 ψ 1 , ψ 2 - 1 σ 4 σ 3 σ 2 σ 1 2 σ 2 σ 3 σ 4 ψ 2 ] . [\psi_{1}^{-1}\sigma_{4}\psi_{1},\psi_{2}^{-1}\sigma_{4}\sigma_{3}\sigma_{2}% \sigma_{1}^{2}\sigma_{2}\sigma_{3}\sigma_{4}\psi_{2}].
  38. n = 2 , 3 n=2, 3
  39. n = 4 n=4
  40. n = 4 n=4
  41. t t
  42. 1 1
  43. n 6 n≥6

Bures_metric.html

  1. [ d ( ρ , ρ + d ρ ) ] 2 = 1 2 tr ( d ρ G ) , [d(\rho,\rho+d\rho)]^{2}=\frac{1}{2}\mbox{tr}~{}(d\rho G),
  2. G G
  3. ρ G + G ρ = d ρ \rho G+G\rho=d\rho
  4. D B ( ρ 1 , ρ 2 ) 2 = 2 ( 1 - F ( ρ 1 , ρ 2 ) ) , D_{B}(\rho_{1},\rho_{2})^{2}=2(1-\sqrt{F(\rho_{1},\rho_{2})}),
  5. F ( ρ 1 , ρ 2 ) = [ tr ( ρ 1 ρ 2 ρ 1 ) ] 2 F(\rho_{1},\rho_{2})=\left[\mbox{tr}~{}(\sqrt{\sqrt{\rho_{1}}\rho_{2}\sqrt{% \rho_{1}}})\right]^{2}
  6. D A ( ρ 1 , ρ 2 ) = arccos F ( ρ 1 , ρ 2 ) , D_{A}(\rho_{1},\rho_{2})=\arccos\sqrt{F(\rho_{1},\rho_{2})},
  7. [ d ( ρ , ρ + d ρ ) ] 2 = 1 2 tr ( d ρ d θ μ L ν ) d θ μ d θ ν , [d(\rho,\rho+d\rho)]^{2}=\frac{1}{2}\mbox{tr}~{}\left(\frac{d\rho}{d\theta^{% \mu}}L_{\nu}\right)d\theta^{\mu}d\theta^{\nu},
  8. L μ L_{\mu}
  9. ρ L μ + L μ ρ 2 = d ρ d θ μ . \frac{\rho L_{\mu}+L_{\mu}\rho}{2}=\frac{d\rho}{d\theta^{\mu}}.
  10. [ d ( ρ , ρ + d ρ ) ] 2 = 1 2 tr [ ρ L μ L ν + L ν L μ 2 ] d θ μ d θ ν [d(\rho,\rho+d\rho)]^{2}=\frac{1}{2}\mbox{tr}~{}\left[\rho\frac{L_{\mu}L_{\nu}% +L_{\nu}L_{\mu}}{2}\right]d\theta^{\mu}d\theta^{\nu}
  11. J μ ν = tr [ ρ L μ L ν + L ν L μ 2 ] . J_{\mu\nu}=\mbox{tr}~{}\left[\rho\frac{L_{\mu}L_{\nu}+L_{\nu}L_{\mu}}{2}\right].
  12. g μ ν g_{\mu\nu}
  13. J μ ν = 4 g μ ν J_{\mu\nu}=4g_{\mu\nu}
  14. [ d ( ρ , ρ + d ρ ) ] 2 = 1 4 tr [ d ρ d ρ + 1 det ( ρ ) ( 𝟏 - ρ ) d ρ ( 𝟏 - ρ ) d ρ ] [d(\rho,\rho+d\rho)]^{2}=\frac{1}{4}\mbox{tr}~{}\left[d\rho d\rho+\frac{1}{% \det(\rho)}(\mathbf{1}-\rho)d\rho(\mathbf{1}-\rho)d\rho\right]
  15. [ d ( ρ , ρ + d ρ ) ] 2 = 1 4 tr [ d ρ d ρ + 3 1 - tr ρ 3 ( 𝟏 - ρ ) d ρ ( 𝟏 - ρ ) d ρ + 3 det ρ 1 - tr ρ 3 ( 𝟏 - ρ - 1 ) d ρ ( 𝟏 - ρ - 1 ) d ρ ] [d(\rho,\rho+d\rho)]^{2}=\frac{1}{4}\mbox{tr}~{}\left[d\rho d\rho+\frac{3}{1-% \mbox{tr}~{}\rho^{3}}(\mathbf{1}-\rho)d\rho(\mathbf{1}-\rho)d\rho+\frac{3\det{% \rho}}{1-\mbox{tr}~{}\rho^{3}}(\mathbf{1}-\rho^{-1})d\rho(\mathbf{1}-\rho^{-1}% )d\rho\right]
  16. [ d ( ρ , ρ + d ρ ) ] 2 = 1 2 j , k = 1 n | j | d ρ | k | 2 λ j + λ k . [d(\rho,\rho+d\rho)]^{2}=\frac{1}{2}\sum_{j,k=1}^{n}\frac{|\langle j|d\rho|k% \rangle|^{2}}{\lambda_{j}+\lambda_{k}}.
  17. ρ = 1 2 ( I + s y m b o l r σ ) \rho=\frac{1}{2}(I+symbol{r\cdot\sigma})
  18. r 2 = def s y m b o l r r 1 r^{2}\stackrel{\mathrm{def}}{=}symbol{r\cdot r}\leq 1
  19. 𝗀 = 𝖨 4 + s y m b o l r r 4 ( 1 - r 2 ) \mathsf{g}=\frac{\mathsf{I}}{4}+\frac{symbol{r\otimes r}}{4(1-r^{2})}
  20. d V B = d 3 s y m b o l r 8 1 - r 2 , dV_{B}=\frac{d^{3}symbol{r}}{8\sqrt{1-r^{2}}},
  21. V B = r 2 1 d 3 s y m b o l r 8 1 - r 2 = π 2 8 V_{B}=\iiint_{r^{2}\leq 1}\frac{d^{3}symbol{r}}{8\sqrt{1-r^{2}}}=\frac{\pi^{2}% }{8}
  22. F ( ρ 1 , ρ 2 ) = tr ( ρ 1 ρ 2 ρ 1 ) F(\rho_{1},\rho_{2})=\mbox{tr}~{}(\sqrt{\sqrt{\rho_{1}}\rho_{2}\sqrt{\rho_{1}}})

Burning_Index.html

  1. F L = j [ ( S C 60 ) ( 25 ( E R C ) ) ] 0.46 F_{L}=j\left[\left(\frac{SC}{60}\right)(25(ERC))\right]^{0.46}
  2. B I = j 1 F L BI=j_{1}\ F_{L}
  3. j 1 j_{1}

Butler–Volmer_equation.html

  1. I = A j 0 { exp [ α a n F R T ( E - E e q ) ] - exp [ - α c n F R T ( E - E e q ) ] } I=A\cdot j_{0}\cdot\left\{\exp\left[\frac{\alpha_{a}nF}{RT}(E-E_{eq})\right]-% \exp\left[-{\frac{\alpha_{c}nF}{RT}}(E-E_{eq})\right]\right\}
  2. j = j 0 { exp [ α a n F η R T ] - exp [ - α c n F η R T ] } j=j_{0}\cdot\left\{\exp\left[\frac{\alpha_{a}nF\eta}{RT}\right]-\exp\left[-{% \frac{\alpha_{c}nF\eta}{RT}}\right]\right\}
  3. I I
  4. A A
  5. j j
  6. j o j_{o}
  7. E E
  8. E e q E_{eq}
  9. T T
  10. n n
  11. F F
  12. R R
  13. α c \alpha_{c}
  14. α a \alpha_{a}
  15. η \eta
  16. η = ( E - E e q ) \eta=(E-E_{eq})
  17. i l i m i t i n g = n F D δ C * i_{limiting}=\frac{nFD}{\delta}C^{*}
  18. i = i 0 n F R T ( E - E e q ) i=i_{0}\frac{nF}{RT}(E-E_{eq})
  19. E - E e q = a - b log ( i ) E-E_{eq}=a-b\log(i)
  20. E - E e q = a + b log ( i ) E-E_{eq}=a+b\log(i)

Bx-tree.html

  1. B x v a l u e ( O , t ) = [ i n d e x p a r t i t i o n ] 2 + [ x r e p ] 2 B^{x}value\left(O,t\right)=\left[indexpartition\right]_{2}+\left[xrep\right]_{2}
  2. [ X ] 2 \left[X\right]_{2}
  3. q = ( [ q x 1 , q y 1 ] ; [ q x 2 ; q y 2 ] ) q=\left(\left[qx1,qy1\right];\left[qx2;qy2\right]\right)
  4. t q tq

Byron_Lavoy_Cockrell.html

  1. V e V_{e}

C-chart.html

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

C_space.html

  1. x = sup n | x n | \|x\|_{\infty}=\sup_{n}|x_{n}|
  2. x 0 lim n y n + i = 1 x i y i . x_{0}\lim_{n\to\infty}y_{n}+\sum_{i=1}^{\infty}x_{i}y_{i}.
  3. i = 0 x i y i . \sum_{i=0}^{\infty}x_{i}y_{i}.

CAIFI.html

  1. CAIFI = Total Number of Customer Interruptions Number of Distinct Customers Interrupted \mbox{CAIFI}~{}=\frac{\mbox{Total Number of Customer Interruptions}~{}}{\mbox{% Number of Distinct Customers Interrupted}~{}}

Calculated_Match_Average.html

  1. ( Total points Total rides ) × 4 \left(\frac{\hbox{Total points}}{\hbox{Total rides}}\right)\times 4

Calibrated_geometry.html

  1. Σ vol Σ = Σ ϕ = Σ ϕ Σ vol Σ \int_{\Sigma}\mathrm{vol}_{\Sigma}=\int_{\Sigma}\phi=\int_{\Sigma^{\prime}}% \phi\leq\int_{\Sigma^{\prime}}\mathrm{vol}_{\Sigma^{\prime}}

Calogero–Degasperis–Fokas_equation.html

  1. u x x x - 1 8 u x 3 + u x ( A e u + B e - u ) = 0. \displaystyle u_{xxx}-\frac{1}{8}u_{x}^{3}+u_{x}\left(Ae^{u}+Be^{-u}\right)=0.

Camassa–Holm_equation.html

  1. u = m 1 e - | x - x 1 | + m 2 e - | x - x 2 | . u=m_{1}\,e^{-|x-x_{1}|}+m_{2}\,e^{-|x-x_{2}|}.
  2. x 1 ( t ) x_{1}(t)
  3. x 2 ( t ) x_{2}(t)
  4. m 1 ( t ) m_{1}(t)
  5. m 2 ( t ) , m_{2}(t),
  6. u t + 2 κ u x - u x x t + 3 u u x = 2 u x u x x + u u x x x . u_{t}+2\kappa u_{x}-u_{xxt}+3uu_{x}=2u_{x}u_{xx}+uu_{xxx}.\,
  7. u t + u u x + p x = 0 , p - p x x = 2 κ u + u 2 + 1 2 ( u x ) 2 , \begin{aligned}\displaystyle u_{t}+uu_{x}+p_{x}&\displaystyle=0,\\ \displaystyle p-p_{xx}&\displaystyle=2\kappa u+u^{2}+\frac{1}{2}\left(u_{x}% \right)^{2},\end{aligned}
  8. ω = 2 κ k 1 + k 2 , \omega=2\kappa\frac{k}{1+k^{2}},
  9. m = u - u x x + κ , m=u-u_{xx}+\kappa,\,
  10. m t = - 𝒟 1 δ 1 δ m with 𝒟 1 = m x + x m and 1 = 1 2 u 2 + ( u x ) 2 d x , m t = - 𝒟 2 δ 2 δ m with 𝒟 2 = x + 3 x 3 and 2 = 1 2 u 3 + u ( u x x ) 2 - κ u 2 d x . \begin{aligned}\displaystyle m_{t}&\displaystyle=-\mathcal{D}_{1}\frac{\delta% \mathcal{H}_{1}}{\delta m}&&\displaystyle\,\text{ with }&\displaystyle\mathcal% {D}_{1}&\displaystyle=m\frac{\partial}{\partial x}+\frac{\partial}{\partial x}% m&\displaystyle\,\text{ and }\mathcal{H}_{1}&\displaystyle=\frac{1}{2}\int u^{% 2}+\left(u_{x}\right)^{2}\;\,\text{d}x,\\ \displaystyle m_{t}&\displaystyle=-\mathcal{D}_{2}\frac{\delta\mathcal{H}_{2}}% {\delta m}&&\displaystyle\,\text{ with }&\displaystyle\mathcal{D}_{2}&% \displaystyle=\frac{\partial}{\partial x}+\frac{\partial^{3}}{\partial x^{3}}&% \displaystyle\,\text{ and }\mathcal{H}_{2}&\displaystyle=\frac{1}{2}\int u^{3}% +u\left(u_{xx}\right)^{2}-\kappa u^{2}\;\,\text{d}x.\end{aligned}
  11. m = u - u x x + κ m=u-u_{xx}+\kappa\,
  12. u ( t , x ) = f ( x - c t ) u(t,x)=f(x-ct)\,
  13. κ > 0 \kappa>0
  14. c = x / ( κ t ) c=x/(\kappa t)
  15. c > 2 c>2
  16. 0 < c < 2 0<c<2
  17. t - 1 / 2 t^{-1/2}
  18. - 1 / 4 < c < 0 -1/4<c<0
  19. c < - 1 / 4 c<-1/4
  20. κ = 0 \kappa=0

Canadian_traveller_problem.html

  1. 𝒢 ( V , E , F ) = { ( V , E + F ) | F F } , E F = \mathcal{G}(V,E,F)=\{(V,E+F^{\prime})|F^{\prime}\subseteq F\},E\cap F=\emptyset
  2. G 𝒢 ( V , E , F ) G\in\mathcal{G}(V,E,F)
  3. w i j w_{ij}
  4. E B ( v , V ) E_{B}(v,V)
  5. G 𝒢 ( V , E , F ) G\in\mathcal{G}(V,E,F)
  6. d B ( s , t ) d_{B}(s,t)
  7. π \pi
  8. ( 𝒫 ( E ) , 𝒫 ( F ) , V ) (\mathcal{P}(E),\mathcal{P}(F),V)
  9. V V
  10. 𝒫 ( X ) \mathcal{P}(X)
  11. c ( π , B ) c(\pi,B)
  12. π \pi
  13. G = ( V , B ) G=(V,B)
  14. v 0 = s , E 0 = E v_{0}=s,E_{0}=E
  15. F 0 = F F_{0}=F
  16. i = 0 , 1 , 2 , i=0,1,2,...
  17. E i + 1 = E i E B ( v i , V ) E_{i+1}=E_{i}\cup E_{B}(v_{i},V)
  18. F i + 1 = F i - E F ( v i , V ) F_{i+1}=F_{i}-E_{F}(v_{i},V)
  19. v i + 1 = π ( E i + 1 , F i + 1 , v i ) v_{i+1}=\pi(E_{i+1},F_{i+1},v_{i})
  20. v T = t v_{T}=t
  21. c ( π , B ) = i = 0 T - 1 w v i , v i + 1 c(\pi,B)=\sum_{i=0}^{T-1}w_{v_{i},v_{i+1}}
  22. c ( π , B ) = c(\pi,B)=\infty
  23. E i E_{i}
  24. F i F_{i}
  25. E i E_{i}
  26. F i F_{i}
  27. ( V , E , F , s , t , r ) (V,E,F,s,t,r)
  28. π \pi
  29. ( V , B ) 𝒢 ( V , E , F ) (V,B)\in\mathcal{G}(V,E,F)
  30. c ( π , B ) c(\pi,B)
  31. d B ( s , t ) d_{B}(s,t)

Capillary_condensation.html

  1. ln P v P s a t = - 2 H γ V l R T \ln\frac{P_{v}}{P_{sat}}=-\frac{2H\gamma V_{l}}{RT}
  2. P v \ P_{v}
  3. P s a t \ P_{sat}
  4. H \ H
  5. γ \ \gamma
  6. V l \ V_{l}
  7. R \ R
  8. T \ T
  9. Δ P \scriptstyle\Delta P
  10. Δ P = 2 H γ \ \Delta P=2H\gamma
  11. H = 1 2 ( 1 R 1 + 1 R 2 ) \ H=\frac{1}{2}(\frac{1}{R_{1}}+\frac{1}{R_{2}})
  12. R 1 = r \ R_{1}=r
  13. R 2 = \ R_{2}=\infty
  14. H = 1 2 r \ H=\frac{1}{2r}
  15. R 1 = r \ R_{1}=r
  16. R 2 = r \ R_{2}=r
  17. H = 1 r \ H=\frac{1}{r}
  18. θ \ \theta
  19. θ \ \theta
  20. θ \ \theta
  21. θ \ \theta

Capstan_equation.html

  1. T load = T hold e μ ϕ , T\text{load}=T\text{hold}\ e^{\mu\phi}~{},
  2. T load T\text{load}
  3. T hold T\text{hold}
  4. μ \mu
  5. ϕ \phi
  6. ϕ = 2 π \phi=2\pi\,
  7. T load T\text{load}
  8. ϕ \phi
  9. e μ ϕ e^{\mu\phi}\,
  10. F F
  11. T T
  12. F φ F\varphi
  13. T hold sin ( φ ) T\text{hold}\sin(\varphi)
  14. F φ = T hold sin ( φ ) F\varphi=T\text{hold}\sin(\varphi)
  15. φ \varphi
  16. sin ( φ ) = φ \sin(\varphi)=\varphi
  17. T hold = T load = T T\text{hold}=T\text{load}=T
  18. φ \varphi
  19. F = T F=T
  20. d φ d\varphi
  21. μ F d φ = μ T d φ \mu{F}d\varphi=\mu{T}d\varphi
  22. μ \mu
  23. d T dT
  24. d φ d\varphi
  25. d T = μ T d φ dT=\mu{T}d\varphi
  26. 1 T d T = μ d φ \frac{1}{T}dT=\mu d\varphi
  27. T hold T load 1 T d T = 0 ϕ μ d φ \int_{T\text{hold}}^{T\text{load}}\frac{1}{T}\;{dT}=\int_{0}^{\phi}\mu\;{d}\varphi
  28. ln T load - ln T hold = ln T load T hold = μ ϕ \ln T\text{load}-\ln T\text{hold}=\ln\frac{T\text{load}}{T\text{hold}}=\mu\phi
  29. T load T hold = e μ ϕ \frac{T\text{load}}{T\text{hold}}={e}^{\mu\phi}
  30. T load = T hold e μ ϕ T\text{load}=T\text{hold}{e}^{\mu\phi}

Carathéodory's_criterion.html

  1. λ * \lambda^{*}
  2. n \mathbb{R}^{n}
  3. E n E\subseteq\mathbb{R}^{n}
  4. E E
  5. λ * ( A ) = λ * ( A E ) + λ * ( A E c ) \lambda^{*}(A)=\lambda^{*}(A\cap E)+\lambda^{*}(A\cap E^{c})
  6. A n A\subseteq\mathbb{R}^{n}
  7. A A

Carathéodory's_existence_theorem.html

  1. y ( t ) = f ( t , y ( t ) ) y^{\prime}(t)=f(t,y(t))\,
  2. y ( t 0 ) = y 0 , y(t_{0})=y_{0},\,
  3. R = { ( t , y ) 𝐑 × 𝐑 n : | t - t 0 | a , | y - y 0 | b } . R=\{(t,y)\in\mathbf{R}\times\mathbf{R}^{n}\,:\,|t-t_{0}|\leq a,|y-y_{0}|\leq b\}.
  4. y ( t ) = H ( t ) , y ( 0 ) = 0 , y^{\prime}(t)=H(t),\quad y(0)=0,
  5. H ( t ) = { 0 , if t 0 ; 1 , if t > 0. H(t)=\begin{cases}0,&\,\text{if }t\leq 0;\\ 1,&\,\text{if }t>0.\end{cases}
  6. y ( t ) = 0 t H ( s ) d s = { 0 , if t 0 ; t , if t > 0 y(t)=\int_{0}^{t}H(s)\,\mathrm{d}s=\begin{cases}0,&\,\text{if }t\leq 0;\\ t,&\,\text{if }t>0\end{cases}
  7. t = 0 t=0
  8. y = f ( t , y ) y^{\prime}=f(t,y)
  9. y ( t 0 ) = y 0 y(t_{0})=y_{0}
  10. y ( t ) = f ( t , y ( t ) ) , y ( t 0 ) = y 0 , y^{\prime}(t)=f(t,y(t)),\quad y(t_{0})=y_{0},\,
  11. f f
  12. R = { ( t , y ) | | t - t 0 | a , | y - y 0 | b } R=\{(t,y)\,|\,|t-t_{0}|\leq a,|y-y_{0}|\leq b\}
  13. f f
  14. f ( t , y ) f(t,y)
  15. y y
  16. t t
  17. f ( t , y ) f(t,y)
  18. t t
  19. y y
  20. m ( t ) m(t)
  21. | t - t 0 | a |t-t_{0}|\leq a
  22. | f ( t , y ) | m ( t ) |f(t,y)|\leq m(t)
  23. ( t , y ) R (t,y)\in R

Carmichael's_totient_function_conjecture.html

  1. 10 10 7 10^{10^{7}}
  2. 10 10 10 10^{10^{10}}

Carrier_scattering.html

  1. E c - E d b = U + K E = 1 2 U ( 1 ) E_{c}-E_{db}=U+KE=\frac{1}{2}U\;\;(1)
  2. n λ = 2 π r ( 2 ) n\lambda=2\pi r\;\;(2)
  3. K E = p 2 2 m * = h 2 n 2 8 m * π 2 r 2 = - U 2 = q 2 8 π ε ε r r ( 3 ) KE=\frac{p^{2}}{2m^{*}}=\frac{h^{2}n^{2}}{8m^{*}\pi^{2}r^{2}}=-\frac{U}{2}=% \frac{q^{2}}{8\pi\varepsilon\varepsilon_{r}r}\;\;(3)
  4. r = 4 π 2 n 2 ε ε r q 2 m * ( 4 ) r=\frac{4\pi\hbar^{2}n^{2}\varepsilon\varepsilon_{r}}{q^{2}m^{*}}\;\;(4)
  5. E c - E d b = U 2 = m * q 4 8 h 2 ( ε ε r ) 2 ( 5 ) E_{c}-E_{db}=\frac{U}{2}=\frac{m^{*}q^{4}}{8h^{2}(\varepsilon\varepsilon_{r})^% {2}}\;\;(5)
  6. S k k = 2 π | < f | H | i > | 2 δ ( E f - E i ) ( 6 ) S_{k^{\prime}k}=\frac{2\pi}{\hbar}|<f|H^{\prime}|i>|^{2}\delta(E_{f}-E_{i})\;% \;(6)
  7. < f | H | i > M k k = 1 V d r ¯ H e i r ¯ ( k ¯ - k ¯ ) = 1 V q ¯ d r ¯ H q ¯ e i r ¯ ( k ¯ - k ¯ + q ¯ ) = 1 V q ¯ H q ¯ δ k ¯ - k ¯ , q ¯ = 1 V H q ¯ ( 7 ) <f|H^{\prime}|i>\equiv M_{k^{\prime}k}=\frac{1}{V}\int d\bar{r}H^{\prime}e^{i% \bar{r}(\bar{k}-\bar{k}^{\prime})}=\frac{1}{V}\sum_{\bar{q}}\int d\bar{r}H_{% \bar{q}}e^{i\bar{r}(\bar{k}-\bar{k}^{\prime}+\bar{q})}=\frac{1}{V}\sum_{\bar{q% }}H_{\bar{q}}\delta_{\bar{k}-\bar{k}^{\prime}},_{\bar{q}}=\frac{1}{V}H_{\bar{q% }}\;\;(7)
  8. 2 V ( r ¯ ) = - e δ ( r ¯ ) ε ε r = - q ¯ q ¯ 2 V q ¯ e i q ¯ r ¯ = - e V ε ε r q ¯ e i q ¯ r ¯ ( 8 ) \nabla^{2}V(\bar{r})=\frac{-e\delta(\bar{r})}{\varepsilon\varepsilon_{r}}=-% \sum_{\bar{q}}\bar{q}^{2}V_{\bar{q}}e^{i\bar{q}\bar{r}}=\frac{-e}{V\varepsilon% \varepsilon_{r}}\ \sum_{\bar{q}}e^{i\bar{q}\bar{r}}\;\;(8)
  9. H q ¯ = - e V q ¯ = - e 2 q ¯ 2 ε ε r V = - e 2 ( q ¯ 2 + q s 2 ) ε ε r V ( 9 ) H_{\bar{q}}=-eV_{\bar{q}}=\frac{-e^{2}}{\bar{q}^{2}\varepsilon\varepsilon_{r}V% }=\frac{-e^{2}}{(\bar{q}^{2}+q_{s}^{2})\varepsilon\varepsilon_{r}V}\;\;(9)
  10. 1 τ = k ¯ , k ¯ S k ¯ k ¯ = n k ¯ 2 π e 4 δ ( E k ¯ - E k ¯ ) ε ε r V [ q ¯ 2 - q s 2 ] 2 = n e 4 4 π 2 ε ε r d k d θ d ϕ k 2 s i n θ δ ( E k ¯ - E k ¯ ) [ q ¯ 2 - q s 2 ] 2 ( 10 ) \frac{1}{\tau}=\sum_{\bar{k}^{\prime},\bar{k}}S_{\bar{k}^{\prime}\bar{k}}=n% \sum_{\bar{k}}\frac{2\pi}{\hbar}\frac{e^{4}\delta(E_{\bar{k}}-E_{\bar{k}^{% \prime}})}{\varepsilon\varepsilon_{r}V[\bar{q}^{2}-q_{s}^{2}]^{2}}=\frac{ne^{4% }}{4\pi^{2}\hbar\varepsilon\varepsilon_{r}}\int\int\int dkd\theta d\phi\frac{k% ^{2}sin\theta\;\delta(E_{\bar{k}}-E_{\bar{k}^{\prime}})}{[\bar{q}^{2}-q_{s}^{2% }]^{2}}\;\;(10)
  11. 1 τ = n e 4 2 π 2 m * ( E c - E d b ) 2 ε ε r ( 1 q s 2 - 1 q s 2 + 8 m * ( E c - E d b ) 2 ) ( 11 ) \frac{1}{\tau}=\frac{ne^{4}}{2\pi\sqrt{2m^{*}(E_{c}-E_{db})}\hbar^{2}% \varepsilon\varepsilon_{r}}(\frac{1}{q_{s}^{2}}-\frac{1}{q_{s}^{2}+\frac{8m^{*% }(E_{c}-E_{db})}{\hbar^{2}}})\;\;(11)
  12. Δ V 0 \Delta V_{0}
  13. Δ V 0 / V 0 = u ( r , t ) \Delta V_{0}/V_{0}=\bigtriangledown u(r,t)
  14. u ( r , t ) e x p ± ( i q r - i ω t ) u(r,t)\propto exp\pm(iqr-i\omega t)
  15. Δ E C B = d E C B d V 0 Δ V 0 = V 0 d E C B d V 0 Δ V 0 V 0 = Z D P u ( r , t ) ( 12 ) \Delta E_{CB}=\frac{\mathrm{d}E_{CB}}{\mathrm{d}V_{0}}\Delta V_{0}=V_{0}\frac{% \mathrm{d}E_{CB}}{\mathrm{d}V_{0}}\frac{\Delta V_{0}}{V_{0}}=Z_{DP}\cdot% \bigtriangledown u(r,t)\;\;(12)
  16. Δ E C B T o t = H ^ i n t = Z D P u ( r , t ) N q + 1 2 ± 1 2 = ± i q Z D P u ( r , t ) N q + 1 2 ± 1 2 ( 13 ) \Delta E_{CB}^{Tot}=\widehat{H}_{int}=Z_{DP}\cdot\bigtriangledown u(r,t)\sqrt{% N_{q}+\frac{1}{2}\pm\frac{1}{2}}=\pm iqZ_{DP}\cdot\bigtriangledown u(r,t)\sqrt% {N_{q}+\frac{1}{2}\pm\frac{1}{2}}\;\;(13)
  17. q u ( r , t ) q\perp u(r,t)
  18. < k | H ^ i n t | k ± i q Z D P u ( r , t ) N q + 1 2 ± 1 2 δ k , k ± q ( 14 ) <k^{\prime}|\widehat{H}_{int}|k>=\pm iqZ_{DP}\cdot\bigtriangledown u(r,t)\sqrt% {N_{q}+\frac{1}{2}\pm\frac{1}{2}}\delta_{k^{\prime},k\pm q}\;\;(14)
  19. | < k | H ^ i n t | k > | 2 = Z D P 2 ω q 2 V ρ c 2 ( N q + 1 2 ± 1 2 ) δ k , k ± q ( 15 ) |<k^{\prime}|\widehat{H}_{int}|k>|^{2}=Z_{DP}^{2}\frac{\hbar\omega_{q}}{2V\rho c% ^{2}}(N_{q}+\frac{1}{2}\pm\frac{1}{2})\delta_{k^{\prime},k\pm q}\;\;(15)
  20. S k k A c = 2 π Z D P 2 ω q 2 V ρ c 2 ( N q + 1 2 ± 1 2 ) δ k , k ± q δ [ E ( k ) - E ( k ) ± ω q ] ( 16 ) S_{k^{\prime}k}^{Ac}=\frac{2\pi}{\hbar}Z_{DP}^{2}\frac{\hbar\omega_{q}}{2V\rho c% ^{2}}(N_{q}+\frac{1}{2}\pm\frac{1}{2})\delta_{k^{\prime},k\pm q}\delta[E(k^{% \prime})-E(k)\pm\hbar\omega_{q}]\;\;(16)
  21. 1 τ = k S k k A c = k S k ± q , k A c \frac{1}{\tau}=\sum_{k^{\prime}}S_{k^{\prime}k}^{Ac}=\sum_{k}S_{k\pm q,k}^{Ac}
  22. = 2 π Z D P 2 ω q 2 V ρ c 2 ( k T ω q ) k δ k , k ± q δ [ E ( k ) - E ( k ) ± ω q ] =\frac{2\pi}{\hbar}Z_{DP}^{2}\frac{\hbar\omega_{q}}{2V\rho c^{2}}(\frac{kT}{% \hbar\omega_{q}})\sum_{k}\delta_{k^{\prime},k\pm q}\delta[E(k^{\prime})-E(k)% \pm\hbar\omega_{q}]
  23. = 2 π Z D P 2 k T 2 V ρ c 2 V × g ( E ) =\frac{2\pi}{\hbar}Z_{DP}^{2}\frac{kT}{2V\rho c^{2}}V\times g(E)
  24. = 2 π Z D P 2 m * 3 2 k T ρ 4 c 2 E - E C B ( 17 ) =\frac{\sqrt{2}}{\pi}\frac{Z_{DP}^{2}m^{*\frac{3}{2}}kT}{\rho\hbar^{4}c^{2}}% \sqrt{E-E_{CB}}\;\;(17)
  25. 1 τ = k S k k O p = 2 π Z D P 2 ω 2 V ρ c 2 ( N q + 1 2 ± 1 2 ) k δ k , k ± q δ [ E ( k ) - E ( k ) ± ω ] \frac{1}{\tau}=\sum_{k^{\prime}}S_{k^{\prime}k}^{Op}=\frac{2\pi}{\hbar}Z_{DP}^% {2}\frac{\hbar\omega}{2V\rho c^{2}}(N_{q}+\frac{1}{2}\pm\frac{1}{2})\sum_{k^{% \prime}}\delta_{k^{\prime},k\pm q}\delta[E(k^{\prime})-E(k)\pm\hbar\omega]
  26. = Z D P 2 ω 8 π 2 ρ c 2 ( N q + 1 2 ± 1 2 ) g ( E ± ω ) ( 18 ) =Z_{DP}^{2}\frac{\hbar\omega}{8\pi^{2}\hbar\rho c^{2}}(N_{q}+\frac{1}{2}\pm% \frac{1}{2})g(E\pm\hbar\omega)\;\;(18)

Cartan–Eilenberg_resolution.html

  1. 𝒜 \mathcal{A}
  2. 𝒜 \mathcal{A}
  3. F : 𝒜 F\colon\mathcal{A}\to\mathcal{B}

Casio_fx-3650P.html

  1. x ¯ \bar{x}
  2. y ¯ \bar{y}
  3. x ^ \hat{x}
  4. y ^ \hat{y}
  5. x ^ \hat{x}
  6. x ^ \hat{x}
  7. \Rightarrow
  8. \geqq
  9. \Rightarrow
  10. \Rightarrow
  11. \textstyle\int

Castability.html

  1. V c V b \frac{V_{c}}{V_{b}}
  2. 6 ( V c ) 2 / 3 A c \frac{6(V_{c})^{2/3}}{A_{c}}
  3. 1 ( 1 + n f ) 0.5 \frac{1}{(1+n_{f})^{0.5}}

Category:Singular_integrals.html

  1. T ( f ) ( x ) = K ( x , y ) f ( y ) d y , T(f)(x)=\int K(x,y)f(y)\,dy,

Category:Surgery_theory.html

  1. > 4 >4

Cation-anion_radius_ratio.html

  1. r C / r A r_{C}/r_{A}

Cauchy_elastic_material.html

  1. s y m b o l σ symbol{\sigma}
  2. s y m b o l F symbol{F}
  3. s y m b o l σ = 𝒢 ( s y m b o l F ) \ symbol{\sigma}=\mathcal{G}(symbol{F})
  4. 𝒢 \mathcal{G}
  5. 𝒢 \mathcal{G}
  6. s y m b o l σ * = 𝒢 ( s y m b o l F * ) symbol{\sigma}^{*}=\mathcal{G}(symbol{F}^{*})
  7. σ \sigma
  8. F F
  9. s y m b o l σ * \displaystyle symbol{\sigma}^{*}
  10. s y m b o l R symbol{R}
  11. 𝒢 \mathcal{G}
  12. s y m b o l σ symbol{\sigma}
  13. s y m b o l B = s y m b o l F \cdotsymbol F T symbol{B}=symbol{F}\cdotsymbol{F}^{T}
  14. s y m b o l σ = ( s y m b o l B ) . \ symbol{\sigma}=\mathcal{H}(symbol{B}).
  15. h h
  16. s y m b o l σ * = ( s y m b o l B * ) s y m b o l R s y m b o l σ s y m b o l R T = ( s y m b o l F * ( s y m b o l F * ) T ) s y m b o l R ( s y m b o l B ) \cdotsymbol R T = ( s y m b o l R \cdotsymbol F \cdotsymbol F T \cdotsymbol R T ) s y m b o l R ( s y m b o l B ) s y m b o l R T = ( s y m b o l R \cdotsymbol B \cdotsymbol R T ) . \ \begin{array}[]{rrcl}&symbol{\sigma}^{*}&=&\mathcal{H}(symbol{B}^{*})\\ \Rightarrow&symbol{R}\cdot symbol{\sigma}\cdot symbol{R}^{T}&=&\mathcal{H}(% symbol{F}^{*}\cdot(symbol{F}^{*})^{T})\\ \Rightarrow&symbol{R}\cdot\mathcal{H}(symbol{B})\cdotsymbol{R}^{T}&=&\mathcal{% H}(symbol{R}\cdotsymbol{F}\cdotsymbol{F}^{T}\cdotsymbol{R}^{T})\\ \Rightarrow&symbol{R}\cdot\mathcal{H}(symbol{B})\cdot symbol{R}^{T}&=&\mathcal% {H}(symbol{R}\cdotsymbol{B}\cdotsymbol{R}^{T}).\end{array}

Cauchy_momentum_equation.html

  1. D 𝐮 D t = 1 ρ s y m b o l σ + 𝐠 \frac{D\mathbf{u}}{Dt}=\frac{1}{\rho}\nabla\cdot symbol{\sigma}+\mathbf{g}
  2. ρ \rho
  3. s y m b o l σ symbol{\sigma}
  4. 𝐠 \mathbf{g}
  5. 𝐮 \mathbf{u}
  6. j t + F = s \frac{\partial j}{\partial t}+\nabla\cdot F=s
  7. j j
  8. F F
  9. 𝐬 \mathbf{s}
  10. i t h i^{th}
  11. m a i = F i ma_{i}=F_{i}\,
  12. Ω ρ D u i D t d V = Ω j σ i j d V + Ω ρ g i d V \int_{\Omega}\rho\frac{Du_{i}}{Dt}\,dV=\int_{\Omega}\nabla_{j}\sigma_{i}^{j}\,% dV+\int_{\Omega}\rho g_{i}\,dV
  13. Ω ( ρ D u i D t - j σ i j - ρ g i ) d V = 0 \int_{\Omega}(\rho\frac{Du_{i}}{Dt}-\nabla_{j}\sigma_{i}^{j}-\rho g_{i})\,dV=0
  14. ρ D u i D t - j σ i j - ρ g i = 0 \rho\frac{Du_{i}}{Dt}-\nabla_{j}\sigma_{i}^{j}-\rho g_{i}=0
  15. D u i D t - j σ i j ρ - g i = 0 \frac{Du_{i}}{Dt}-\frac{\nabla_{j}\sigma_{i}^{j}}{\rho}-g_{i}=0
  16. Ω \Omega
  17. F i F_{i}
  18. j = ρ u \displaystyle{j}=\rho u
  19. j j
  20. F F
  21. 𝐬 \mathbf{s}
  22. 𝐮 𝐮 \mathbf{u}\cdot\nabla\mathbf{u}
  23. ( 𝐮 ) 𝐮 (\mathbf{u}\cdot\nabla)\,\mathbf{u}
  24. 𝐮 ( 𝐮 ) , \mathbf{u}\cdot(\nabla\mathbf{u}),
  25. 𝐮 \nabla\mathbf{u}
  26. 𝐮 . \mathbf{u}.
  27. \nabla
  28. ( 𝐮 ) 𝐮 (\mathbf{u}\cdot\nabla)\mathbf{u}
  29. 𝐮 \mathbf{u}\cdot\nabla
  30. 𝐮 . \nabla\mathbf{u}.
  31. 𝐮 \nabla\mathbf{u}
  32. [ 𝐮 ] m i = m v i \left[\nabla\mathbf{u}\right]_{mi}=\partial_{m}v_{i}
  33. [ 𝐮 ( 𝐮 ) ] i = v m m v i = [ ( 𝐮 ) 𝐮 ] i \left[\mathbf{u}\cdot\left(\nabla\mathbf{u}\right)\right]_{i}=v_{m}\partial_{m% }v_{i}=\left[(\mathbf{u}\cdot\nabla)\mathbf{u}\right]_{i}
  34. 𝐯 × ( × 𝐚 ) = a ( 𝐯 𝐚 ) - 𝐯 𝐚 , \mathbf{v\ \times}\left(\mathbf{\nabla\times a}\right)=\nabla_{a}\left(\mathbf% {v\cdot a}\right)-\mathbf{v\cdot\nabla}\mathbf{a}\ ,
  35. 𝐮 𝐮 = ( 𝐮 2 2 ) + ( × 𝐮 ) × 𝐮 . \mathbf{u}\cdot\nabla\mathbf{u}=\nabla\left(\frac{\|\mathbf{u}\|^{2}}{2}\right% )+\left(\nabla\times\mathbf{u}\right)\times\mathbf{u}.
  36. 𝐮 t + 1 2 ( u 2 ) + ( × 𝐮 ) × 𝐮 = 1 ρ s y m b o l σ + 𝐠 \frac{\partial\mathbf{u}}{\partial t}+\frac{1}{2}\nabla(u^{2})+(\nabla\times% \mathbf{u})\times\mathbf{u}=\frac{1}{\rho}\nabla\cdot symbol\sigma+\mathbf{g}
  37. ( s y m b o l σ ρ ) = 1 ρ s y m b o l σ - 1 ρ 2 s y m b o l σ ρ \nabla\cdot\left(\frac{symbol\sigma}{\rho}\right)=\frac{1}{\rho}\nabla\cdot symbol% \sigma-\frac{1}{\rho^{2}}symbol\sigma\cdot\nabla\rho
  38. ( 1 2 u 2 + s y m b o l σ ρ ) - 𝐠 = 1 ρ 2 s y m b o l σ ρ + 𝐮 × ( × 𝐮 ) - 𝐮 t \nabla\cdot\left(\frac{1}{2}u^{2}+\frac{symbol\sigma}{\rho}\right)-\mathbf{g}=% \frac{1}{\rho^{2}}symbol\sigma\cdot\nabla\rho+\mathbf{u}\times(\nabla\times% \mathbf{u})-\frac{\partial\mathbf{u}}{\partial t}
  39. ( 1 2 u 2 + ϕ + s y m b o l σ ρ ) = 1 ρ 2 s y m b o l σ ρ + 𝐮 × ( × 𝐮 ) - 𝐮 t \nabla\cdot\left(\frac{1}{2}u^{2}+\phi+\frac{symbol\sigma}{\rho}\right)=\frac{% 1}{\rho^{2}}symbol\sigma\cdot\nabla\rho+\mathbf{u}\times(\nabla\times\mathbf{u% })-\frac{\partial\mathbf{u}}{\partial t}
  40. ( 1 2 u 2 + ϕ + s y m b o l σ ρ ) = 1 ρ 2 s y m b o l σ ρ + 𝐮 × ( × 𝐮 ) \nabla\cdot\left(\frac{1}{2}u^{2}+\phi+\frac{symbol\sigma}{\rho}\right)=\frac{% 1}{\rho^{2}}symbol\sigma\cdot\nabla\rho+\mathbf{u}\times(\nabla\times\mathbf{u})
  41. 𝐮 ( 1 2 u 2 + ϕ + s y m b o l σ ρ ) = 1 ρ 2 𝐮 ( s y m b o l σ ρ ) \mathbf{u}\cdot\nabla\cdot\left(\frac{1}{2}u^{2}+\phi+\frac{symbol\sigma}{\rho% }\right)=\frac{1}{\rho^{2}}\mathbf{u}\cdot(symbol\sigma\cdot\nabla\rho)
  42. 𝐮 ρ = 0 \mathbf{u}\cdot\nabla\rho=0
  43. 𝐮 ( 1 2 u 2 + ϕ + p ρ ) = p ρ 2 𝐮 ρ \mathbf{u}\cdot\nabla\cdot\left(\frac{1}{2}u^{2}+\phi+\frac{p}{\rho}\right)=% \frac{p}{\rho^{2}}\mathbf{u}\cdot\nabla\rho
  44. 𝐮 ( 1 2 u 2 + ϕ + p ρ ) = 0 \mathbf{u}\cdot\nabla\left(\frac{1}{2}u^{2}+\phi+\frac{p}{\rho}\right)=0
  45. b l 1 2 u 2 + ϕ + p ρ b_{l}\equiv\frac{1}{2}u^{2}+\phi+\frac{p}{\rho}
  46. 𝐮 b l = 0 \mathbf{u}\cdot\nabla b_{l}=0
  47. ω = × 𝐮 \omega=\nabla\times\mathbf{u}
  48. 𝐮 𝐮 = ( 𝐮 2 2 ) . \mathbf{u}\cdot\nabla\mathbf{u}=\nabla\left(\frac{\|\mathbf{u}\|^{2}}{2}\right).
  49. p \nabla p
  50. \cdotsymbol τ \nabla\cdotsymbol\tau
  51. p \nabla p
  52. \cdotsymbol τ \nabla\cdotsymbol\tau
  53. s y m b o l τ symbol\tau
  54. s y m b o l σ = - p 𝟏 + s y m b o l τ symbol\sigma=-p\mathbf{1}+symbol\tau
  55. 𝟏 \mathbf{1}
  56. s y m b o l τ symbol{\tau}
  57. s y m b o l σ = - p + s y m b o l τ . \nabla\cdot symbol{\sigma}=-\nabla p+\nabla\cdot symbol{\tau}.
  58. s y m b o l τ symbol\tau
  59. 𝐠 \mathbf{g}
  60. χ \chi\,
  61. 𝐠 = χ , \mathbf{g}=\nabla\chi,
  62. - ρ g z -\rho gz
  63. h = p - χ . h=p-\chi.
  64. - p + 𝐠 = - p + χ = - ( p - χ ) = - h . -\nabla p+\mathbf{g}=-\nabla p+\nabla\chi=-\nabla\left(p-\chi\right)=-\nabla h.
  65. ρ * ρ ρ 0 , \rho^{*}\equiv\frac{\rho}{\rho_{0}},
  66. u * u u 0 , u^{*}\equiv\frac{u}{u_{0}},
  67. r * r r 0 , r^{*}\equiv\frac{r}{r_{0}},
  68. t * u 0 r 0 t , t^{*}\equiv\frac{u_{0}}{r_{0}}t,
  69. * r 0 , \nabla^{*}\equiv r_{0}\nabla,
  70. 𝐠 * 𝐠 g 0 , \mathbf{g}^{*}\equiv\frac{\mathbf{g}}{g_{0}},
  71. p * p p 0 , p^{*}\equiv\frac{p}{p_{0}},
  72. s y m b o l τ * s y m b o l τ τ 0 , symbol\tau^{*}\equiv\frac{symbol\tau}{\tau_{0}},
  73. ρ 0 u 0 2 r 0 ρ * u * t * + * r 0 ( ρ 0 u 0 2 ρ * u * u * + p 0 p * ) = - τ 0 r 0 * s y m b o l τ * + g 0 𝐠 * \frac{\rho_{0}u_{0}^{2}}{r_{0}}{\partial\rho^{*}u^{*}\over\partial t^{*}}+% \frac{\nabla^{*}}{r_{0}}\cdot\left(\rho_{0}u_{0}^{2}\rho^{*}u^{*}\otimes u^{*}% +p_{0}p^{*}\right)=-\frac{\tau_{0}}{r_{0}}\nabla^{*}\cdot symbol\tau^{*}+g_{0}% \mathbf{g}^{*}
  74. ρ * u * t * + * ( ρ * u * u * + p 0 ρ 0 u 0 2 p * ) = - τ 0 ρ 0 u 0 2 * s y m b o l τ * + g 0 r 0 u 0 2 𝐠 * {\partial\rho^{*}u^{*}\over\partial t^{*}}+\nabla^{*}\cdot\left(\rho^{*}u^{*}% \otimes u^{*}+\frac{p_{0}}{\rho_{0}u_{0}^{2}}p^{*}\right)=-\frac{\tau_{0}}{% \rho_{0}u_{0}^{2}}\nabla^{*}\cdot symbol\tau^{*}+\frac{g_{0}r_{0}}{u_{0}^{2}}% \mathbf{g}^{*}
  75. Fr = u 0 2 g 0 r 0 , \mathrm{Fr}=\frac{u_{0}^{2}}{g_{0}r_{0}},
  76. Eu = p 0 ρ 0 u 0 2 , \mathrm{Eu}=\frac{p_{0}}{\rho_{0}u_{0}^{2}},
  77. C f = 2 τ 0 ρ 0 u 0 2 , \mathrm{C_{f}}=\frac{2\tau_{0}}{\rho_{0}u_{0}^{2}},
  78. j = ρ u j=\rho u
  79. f = ρ g f=\rho g
  80. x : u x t + u x u x x + u y u x y + u z u x z = 1 ρ ( - P x + τ x x x + τ x y y + τ x z z ) + g x y : u y t + u x u y x + u y u y y + u z u y z = 1 ρ ( - P y + τ y x x + τ y y y + τ y z z ) + g y z : u z t + u x u z x + u y u z y + u z u z z = 1 ρ ( - P z + τ z x x + τ z y y + τ z z z ) + g z . \begin{aligned}\displaystyle x:\;\;\frac{\partial u_{x}}{\partial t}+u_{x}% \frac{\partial u_{x}}{\partial x}+u_{y}\frac{\partial u_{x}}{\partial y}+u_{z}% \frac{\partial u_{x}}{\partial z}&\displaystyle=\frac{1}{\rho}\left(-\frac{% \partial P}{\partial x}+\frac{\partial\tau_{xx}}{\partial x}+\frac{\partial% \tau_{xy}}{\partial y}+\frac{\partial\tau_{xz}}{\partial z}\right)+g_{x}\\ \displaystyle y:\;\;\frac{\partial u_{y}}{\partial t}+u_{x}\frac{\partial u_{y% }}{\partial x}+u_{y}\frac{\partial u_{y}}{\partial y}+u_{z}\frac{\partial u_{y% }}{\partial z}&\displaystyle=\frac{1}{\rho}\left(-\frac{\partial P}{\partial y% }+\frac{\partial\tau_{yx}}{\partial x}+\frac{\partial\tau_{yy}}{\partial y}+% \frac{\partial\tau_{yz}}{\partial z}\right)+g_{y}\\ \displaystyle z:\;\;\frac{\partial u_{z}}{\partial t}+u_{x}\frac{\partial u_{z% }}{\partial x}+u_{y}\frac{\partial u_{z}}{\partial y}+u_{z}\frac{\partial u_{z% }}{\partial z}&\displaystyle=\frac{1}{\rho}\left(-\frac{\partial P}{\partial z% }+\frac{\partial\tau_{zx}}{\partial x}+\frac{\partial\tau_{zy}}{\partial y}+% \frac{\partial\tau_{zz}}{\partial z}\right)+g_{z}.\end{aligned}
  81. r : u r t + u r u r r + u ϕ r u r ϕ + u z u r z - u ϕ 2 r = - 1 ρ P r + 1 r ρ ( r τ r r ) r + 1 r ρ τ ϕ r ϕ + 1 ρ τ z r z - τ ϕ ϕ r ρ + g r r:\;\;\frac{\partial u_{r}}{\partial t}+u_{r}\frac{\partial u_{r}}{\partial r}% +\frac{u_{\phi}}{r}\frac{\partial u_{r}}{\partial\phi}+u_{z}\frac{\partial u_{% r}}{\partial z}-\frac{u_{\phi}^{2}}{r}=-\frac{1}{\rho}\frac{\partial P}{% \partial r}+\frac{1}{r\,\rho}\frac{\partial{(r{\tau_{rr})}}}{\partial r}+\frac% {1}{r\,\rho}\frac{\partial{\tau_{\phi r}}}{\partial\phi}+\frac{1}{\rho}\frac{% \partial{\tau_{zr}}}{\partial z}-\frac{\tau_{\phi\phi}}{r\,\rho}+g_{r}
  82. ϕ : u ϕ t + u r u ϕ r + u ϕ r u ϕ ϕ + u z u ϕ z + u r u ϕ r = - 1 r ρ P ϕ + 1 r ρ τ ϕ ϕ ϕ + 1 r 2 ρ ( r 2 τ r ϕ ) r + 1 ρ τ z ϕ z + g ϕ \phi:\;\;\frac{\partial u_{\phi}}{\partial t}+u_{r}\frac{\partial u_{\phi}}{% \partial r}+\frac{u_{\phi}}{r}\frac{\partial u_{\phi}}{\partial\phi}+u_{z}% \frac{\partial u_{\phi}}{\partial z}+\frac{u_{r}u_{\phi}}{r}=-\frac{1}{r\,\rho% }\frac{\partial P}{\partial\phi}+\frac{1}{r\,\rho}\frac{\partial{\tau_{\phi% \phi}}}{\partial\phi}+\frac{1}{r^{2}\,\rho}\frac{\partial{(r^{2}{\tau_{r\phi})% }}}{\partial r}+\frac{1}{\rho}\frac{\partial{\tau_{z\phi}}}{\partial z}+g_{\phi}
  83. z : u z t + u r u z r + u ϕ r u z ϕ + u z u z z = - 1 ρ P z + 1 ρ τ z z z + 1 r ρ τ ϕ z ϕ + 1 r ρ ( r τ r z ) r + g z . z:\;\;\frac{\partial u_{z}}{\partial t}+u_{r}\frac{\partial u_{z}}{\partial r}% +\frac{u_{\phi}}{r}\frac{\partial u_{z}}{\partial\phi}+u_{z}\frac{\partial u_{% z}}{\partial z}=-\frac{1}{\rho}\frac{\partial P}{\partial z}+\frac{1}{\rho}% \frac{\partial{\tau_{zz}}}{\partial z}+\frac{1}{r\,\rho}\frac{\partial{\tau_{% \phi z}}}{\partial\phi}+\frac{1}{r\,\rho}\frac{\partial{(r{\tau_{rz})}}}{% \partial r}+g_{z}.
  84. j = ( ρ u 1 ρ u 2 ρ u 3 ) ; s = ( ρ g 1 ρ g 2 ρ g 3 ) ; F = ( ρ u 1 2 + σ 11 ρ u 1 u 2 + σ 12 ρ u 1 u 3 + σ 13 ρ u 2 u 1 + σ 12 ρ u 2 2 + σ 22 ρ u 2 u 3 + σ 23 ρ u 3 u 1 + σ 13 ρ u 3 u 2 + σ 23 ρ u 3 2 + σ 33 ) . {j}=\begin{pmatrix}\rho u_{1}\\ \rho u_{2}\\ \rho u_{3}\end{pmatrix};\quad{s}=\begin{pmatrix}\rho g_{1}\\ \rho g_{2}\\ \rho g_{3}\end{pmatrix};\quad{F}=\begin{pmatrix}\rho u_{1}^{2}+\sigma_{11}&% \rho u_{1}u_{2}+\sigma_{12}&\rho u_{1}u_{3}+\sigma_{13}\\ \rho u_{2}u_{1}+\sigma_{12}&\rho u_{2}^{2}+\sigma_{22}&\rho u_{2}u_{3}+\sigma_% {23}\\ \rho u_{3}u_{1}+\sigma_{13}&\rho u_{3}u_{2}+\sigma_{23}&\rho u_{3}^{2}+\sigma_% {33}\end{pmatrix}.

Causes_of_the_United_States_housing_bubble.html

  1. r IT r_{\rm IT}
  2. r PT r_{\rm PT}
  3. r M r_{\rm M}
  4. cost ( ( r 1 - ( 1 + r ) - N + r PT ) \scriptstyle{\scriptstyle{\rm cost}}\approx\Big(\big(\frac{r}{1-(1+r)^{-N}}+r_% {\rm PT}\big)
  5. × ( 1 - r IT ) + r M ) × P / 12. \scriptstyle{}\qquad{}\times(1-r_{\rm IT})+r_{\rm M}\Big)\times P/12.
  6. P / E ratio = < m t p l > PriceRent - Expenses . \textstyle{\scriptstyle{\rm P/E\ ratio}}=\frac{<}{m}tpl>{{\rm Price}}{{\rm Rent% }-{\rm Expenses}}.
  7. monthly payment = r 1 - ( 1 + r ) - N W h o × Principal \textstyle{{\rm monthly}\atop{\rm payment}}=\frac{r}{1-(1+r)^{-N}}{% \scriptstyle{{Who}\times\rm Principal}}
  8. ( 1 + r / K ) N K e N r \scriptstyle(1+r/K)^{NK}\approx e^{Nr}
  9. Δ Principal < m t p l > Principal - ( 1 - N r e - N r 1 - e - N r ) Δ r r \textstyle\frac{\Delta{\rm Principal}}{<}mtpl>{{\rm Principal}}\approx-\left(1% -\frac{Nre^{-Nr}}{1-e^{-Nr}}\right)\frac{\Delta r}{r}
  10. Δ Principal < m t p l > Principal - Δ r r \textstyle\frac{\Delta{\rm Principal}}{<}mtpl>{{\rm Principal}}\approx-\frac{% \Delta r}{r}

Cavity_quantum_electrodynamics.html

  1. | e | n - 1 | g | n |e\rangle|n-1\rangle\leftrightarrow|g\rangle|n\rangle
  2. ( α | g + β | e ) | 0 | g ( α | 0 + β | 1 ) (\alpha|g\rangle+\beta|e\rangle)|0\rangle\leftrightarrow|g\rangle(\alpha|0% \rangle+\beta|1\rangle)
  3. | e | 0 |e\rangle|0\rangle
  4. ( | e | 0 + | g | 1 ) / 2 (|e\rangle|0\rangle+|g\rangle|1\rangle)/\sqrt{2}

Centered_trochoid.html

  1. z = r 1 e i ω 1 t + r 2 e i ω 2 t , z=r_{1}e^{i\omega_{1}t}+r_{2}e^{i\omega_{2}t},\,
  2. x = r 1 cos ( ω 1 t ) + r 2 cos ( ω 2 t ) , y = r 1 sin ( ω 1 t ) + r 2 sin ( ω 2 t ) , x=r_{1}\cos(\omega_{1}t)+r_{2}\cos(\omega_{2}t),y=r_{1}\sin(\omega_{1}t)+r_{2}% \sin(\omega_{2}t),\,
  3. r 1 , r 2 , ω 1 , ω 2 0 , ω 1 ω 2 . r_{1},r_{2},\omega_{1},\omega_{2}\neq 0,\quad\omega_{1}\neq\omega_{2}.\,
  4. ω 1 / ω 2 \omega_{1}/\omega_{2}
  5. | r 1 | + | r 2 | |r_{1}|+|r_{2}|
  6. | | r 1 | - | r 2 | | ||r_{1}|-|r_{2}||
  7. ω 1 \omega_{1}
  8. ω 2 \omega_{2}
  9. b b
  10. a a
  11. p p
  12. f ( t ) = a e i t f(t)=ae^{it}
  13. r ( t ) = b e i ( a / b ) t r(t)=be^{i(a/b)t}
  14. r ( t ) = - b e - i ( a / b ) t r(t)=-be^{-i(a/b)t}
  15. r ( t ) = c e i ( a / c ) t r(t)=ce^{i(a/c)t}
  16. | c | = b |c|=b
  17. p p
  18. d d
  19. f ( t ) + ( d - r ( t ) ) f ( t ) r ( t ) \displaystyle f(t)+(d-r(t)){f^{\prime}(t)\over r^{\prime}(t)}
  20. r 1 = a - c r_{1}=a-c
  21. r 2 = d r_{2}=d
  22. ω 1 = 1 \omega_{1}=1
  23. ω 2 = 1 - a / c \omega_{2}=1-a/c
  24. r 1 r_{1}
  25. r 2 r_{2}
  26. ω 1 \omega_{1}
  27. ω 2 \omega_{2}
  28. r 1 e i ω 1 t + r 2 e i ω 2 t r_{1}e^{i\omega_{1}t}+r_{2}e^{i\omega_{2}t}
  29. r 1 e i t + r 2 e i ( ω 2 / ω 1 ) t r_{1}e^{it}+r_{2}e^{i(\omega_{2}/\omega_{1})t}
  30. r 1 = a - c r_{1}=a-c
  31. r 2 = d r_{2}=d
  32. ω 2 / ω 1 = 1 - a / c \omega_{2}/\omega_{1}=1-a/c
  33. a a
  34. c c
  35. d d
  36. a = r 1 ( 1 - ω 1 / ω 2 ) , c = - r 1 ω 1 / ω 2 , d = r 2 . a=r_{1}(1-\omega_{1}/\omega_{2}),\ c=-r_{1}{\omega_{1}/\omega_{2}},\ d=r_{2}.
  37. r 1 e i ω 1 t + r 2 e i ω 2 t r_{1}e^{i\omega_{1}t}+r_{2}e^{i\omega_{2}t}
  38. a a
  39. c c
  40. d d
  41. 2 e i t - e 2 i t 2e^{it}-e^{2it}
  42. r 1 = 2 , r 2 = - 1 , ω 1 = 1 , ω 2 = 2 r_{1}=2,r_{2}=-1,\omega_{1}=1,\omega_{2}=2
  43. a = 2 ( 1 - 1 / 2 ) = 1 , c = - 2 ( 1 / 2 ) = - 1 , d = - 1 a=2(1-1/2)=1,c=-2(1/2)=-1,d=-1
  44. r 1 = - 1 , r 2 = 2 , ω 1 = 2 , ω 2 = 1 r_{1}=-1,r_{2}=2,\omega_{1}=2,\omega_{2}=1
  45. a = - 1 ( 1 - 2 ) = 1 , b = - ( - 1 ) ( 2 ) = 2 , d = 2. a=-1(1-2)=1,b=-(-1)(2)=2,d=2.
  46. ω 1 = - ω 2 \omega_{1}=-\omega_{2}
  47. r 1 e i t + r 2 e - i t r_{1}e^{it}+r_{2}e^{-it}
  48. x = ( r 1 + r 2 ) cos t , y = ( r 1 - r 2 ) sin t x=(r_{1}+r_{2})\cos t,y=(r_{1}-r_{2})\sin t\,\!
  49. | r 1 | | r 2 | |r_{1}|\neq|r_{2}|
  50. 2 | r 1 + r 2 | 2|r_{1}+r_{2}|
  51. 2 | r 1 - r 2 | 2|r_{1}-r_{2}|
  52. a a
  53. c c
  54. d d
  55. a = 2 r 1 , c = r 1 , d = r 2 a=2r_{1},c=r_{1},d=r_{2}\,\!
  56. a = 2 r 2 , c = r 2 , d = r 1 a=2r_{2},c=r_{2},d=r_{1}\,\!
  57. ω 1 = - ω 2 \omega_{1}=-\omega_{2}
  58. r 1 = r 2 = r r_{1}=r_{2}=r
  59. a = 2 r , b = r , c = r a=2r,b=r,c=r\,\!
  60. x = 2 r cos t , y = 0 x=2r\cos t,y=0\,\!
  61. r 1 = r , r 2 = - r r_{1}=r,r_{2}=-r\,\!
  62. a = 2 r , c = r , d = - r a=2r,c=r,d=-r\,\!
  63. a = - 2 r , c = - r , d = r a=-2r,c=-r,d=r\,\!
  64. x = 0 , y = 2 r sin t x=0,y=2r\sin t\,\!
  65. ω 1 = - ω 2 , | r 1 | = | r 2 | \omega_{1}=-\omega_{2},\ |r_{1}|=|r_{2}|

Centrifugal_force.html

  1. 𝐫 \mathbf{r}
  2. d d 𝐫 / d t dd\mathbf{r}/dt
  3. r = 0 r=0
  4. ω \mathbf{ω}
  5. d s y m b o l r d t = [ d s y m b o l r d t ] + s y m b o l ω × s y m b o l r , \frac{\operatorname{d}symbol{r}}{\operatorname{d}t}=\left[\frac{\operatorname{% d}symbol{r}}{\operatorname{d}t}\right]+symbol{\omega}\times symbol{r}\ ,
  6. × \times
  7. . \mathbf{....}
  8. s y m b o l ω × s y m b o l r symbol{\omega}\times symbol{r}
  9. 𝐫 \mathbf{r}
  10. ω \mathbf{ω}
  11. ω \mathbf{ω}
  12. ω ω
  13. s y m b o l F = m s y m b o l a , symbol{F}=msymbol{a}\ ,
  14. 𝐅 \mathbf{F}
  15. 𝐚 \mathbf{a}
  16. s y m b o l a = d 2 s y m b o l r d t 2 , symbol{a}=\frac{\operatorname{d}^{2}symbol{r}}{\operatorname{d}t^{2}}\ ,
  17. 𝐫 \mathbf{r}
  18. s y m b o l a \displaystyle symbol{a}
  19. s y m b o l F - m d s y m b o l ω d t \timessymbol r - 2 m s y m b o l ω × [ d s y m b o l r d t ] - m s y m b o l ω × ( s y m b o l ω × s y m b o l r ) symbol{F}-m\frac{\operatorname{d}symbol{\omega}}{\operatorname{d}t}% \timessymbol{r}-2msymbol{\omega}\times\left[\frac{\operatorname{d}symbol{r}}{% \operatorname{d}t}\right]-msymbol{\omega}\times(symbol{\omega}\times symbol{r})
  20. = m [ d 2 s y m b o l r d t 2 ] . =m\left[\frac{\operatorname{d}^{2}symbol{r}}{\operatorname{d}t^{2}}\right]\ .
  21. m d s y m b o l ω / d t \timessymbol r m\operatorname{d}symbol{\omega}/\operatorname{d}t\timessymbol{r}
  22. 2 m s y m b o l ω × [ d s y m b o l r / d t ] 2msymbol{\omega}\times\left[\operatorname{d}symbol{r}/\operatorname{d}t\right]
  23. m s y m b o l ω × ( s y m b o l ω × s y m b o l r ) msymbol{\omega}\times(symbol{\omega}\times symbol{r})
  24. ( s y m b o l ω = 0 ) (symbol\omega=0)
  25. ( r , θ ) (r,\ \theta)

Chameleon_particle.html

  1. m eff ρ α m_{\textrm{eff}}\sim\rho^{\alpha}
  2. α 1 \alpha\simeq 1
  3. β > 5.8 * 10 8 \beta>5.8*10^{8}
  4. V eff = V ( Φ ) + e β Φ / M P ρ V_{\textrm{eff}}=V(\Phi)+e^{\beta\Phi/M^{\prime}_{P}}\rho
  5. ρ \rho
  6. V ( Φ ) V(\Phi)
  7. M P M^{\prime}_{P}

Change-making_problem.html

  1. n n
  2. W W
  3. j = 1 n x j \sum_{j=1}^{n}x_{j}
  4. j = 1 n w j x j = W . \sum_{j=1}^{n}w_{j}x_{j}=W.

Change_of_variables_(PDE).html

  1. V t + 1 2 S 2 2 V S 2 + S V S - V = 0. \frac{\partial V}{\partial t}+\frac{1}{2}S^{2}\frac{\partial^{2}V}{\partial S^% {2}}+S\frac{\partial V}{\partial S}-V=0.
  2. u τ = 2 u x 2 \frac{\partial u}{\partial\tau}=\frac{\partial^{2}u}{\partial x^{2}}
  3. V ( S , t ) = v ( x ( S ) , τ ( t ) ) V(S,t)=v(x(S),\tau(t))
  4. x ( S ) = ln ( S ) x(S)=\ln(S)
  5. τ ( t ) = 1 2 ( T - t ) \tau(t)=\frac{1}{2}(T-t)
  6. v ( x , τ ) = exp ( - ( 1 / 2 ) x - ( 9 / 4 ) τ ) u ( x , τ ) v(x,\tau)=\exp(-(1/2)x-(9/4)\tau)u(x,\tau)
  7. V ( S , t ) V(S,t)
  8. v ( x ( S ) , τ ( t ) ) v(x(S),\tau(t))
  9. 1 2 ( - 2 v ( x ( S ) , τ ) + 2 τ t v τ + S ( ( 2 x S + S 2 x S 2 ) v x + S ( x S ) 2 2 v x 2 ) ) = 0. \frac{1}{2}\left(-2v(x(S),\tau)+2\frac{\partial\tau}{\partial t}\frac{\partial v% }{\partial\tau}+S\left(\left(2\frac{\partial x}{\partial S}+S\frac{\partial^{2% }x}{\partial S^{2}}\right)\frac{\partial v}{\partial x}+S\left(\frac{\partial x% }{\partial S}\right)^{2}\frac{\partial^{2}v}{\partial x^{2}}\right)\right)=0.
  10. x ( S ) x(S)
  11. τ ( t ) \tau(t)
  12. ln ( S ) \ln(S)
  13. 1 2 ( T - t ) \frac{1}{2}(T-t)
  14. 1 2 ( - 2 v ( ln ( S ) , 1 2 ( T - t ) ) - v ( ln ( S ) , 1 2 ( T - t ) ) τ + v ( ln ( S ) , 1 2 ( T - t ) ) x + 2 v ( ln ( S ) , 1 2 ( T - t ) ) x 2 ) = 0. \frac{1}{2}\left(-2v(\ln(S),\frac{1}{2}(T-t))-\frac{\partial v(\ln(S),\frac{1}% {2}(T-t))}{\partial\tau}+\frac{\partial v(\ln(S),\frac{1}{2}(T-t))}{\partial x% }+\frac{\partial^{2}v(\ln(S),\frac{1}{2}(T-t))}{\partial x^{2}}\right)=0.
  15. ln ( S ) \ln(S)
  16. 1 2 ( T - t ) \frac{1}{2}(T-t)
  17. x ( S ) x(S)
  18. τ ( t ) \tau(t)
  19. 1 2 \frac{1}{2}
  20. - 2 v - v τ + v x + 2 v x 2 = 0. -2v-\frac{\partial v}{\partial\tau}+\frac{\partial v}{\partial x}+\frac{% \partial^{2}v}{\partial x^{2}}=0.
  21. v ( x , τ ) v(x,\tau)
  22. exp ( - ( 1 / 2 ) x - ( 9 / 4 ) τ ) u ( x , τ ) \exp(-(1/2)x-(9/4)\tau)u(x,\tau)
  23. - exp ( - ( 1 / 2 ) x - ( 9 / 4 ) τ ) u ( x , τ ) -\exp(-(1/2)x-(9/4)\tau)u(x,\tau)
  24. u ( x , t ) u(x,t)
  25. x 1 , x 2 x_{1},x_{2}
  26. a ( x , t ) , b ( x , t ) a(x,t),b(x,t)
  27. x 1 = a ( x , t ) x_{1}=a(x,t)
  28. x 2 = b ( x , t ) x_{2}=b(x,t)
  29. e ( x 1 , x 2 ) , f ( x 1 , x 2 ) e(x_{1},x_{2}),f(x_{1},x_{2})
  30. x = e ( x 1 , x 2 ) x=e(x_{1},x_{2})
  31. t = f ( x 1 , x 2 ) t=f(x_{1},x_{2})
  32. x 1 = a ( e ( x 1 , x 2 ) , f ( x 1 , x 2 ) ) x_{1}=a(e(x_{1},x_{2}),f(x_{1},x_{2}))
  33. x 2 = b ( e ( x 1 , x 2 ) , f ( x 1 , x 2 ) ) x_{2}=b(e(x_{1},x_{2}),f(x_{1},x_{2}))
  34. x = e ( a ( x , t ) , b ( x , t ) ) x=e(a(x,t),b(x,t))
  35. t = f ( a ( x , t ) , b ( x , t ) ) t=f(a(x,t),b(x,t))
  36. \mathcal{L}
  37. u ( x , t ) = 0 \mathcal{L}u(x,t)=0
  38. v ( x 1 , x 2 ) = 0 \mathcal{L}v(x_{1},x_{2})=0
  39. v ( x 1 , x 2 ) = u ( e ( x 1 , x 2 ) , f ( x 1 , x 2 ) ) v(x_{1},x_{2})=u(e(x_{1},x_{2}),f(x_{1},x_{2}))
  40. u ( x , t ) = 0 \mathcal{L}u(x,t)=0
  41. v ( x 1 , x 2 ) = 0 : \mathcal{L}v(x_{1},x_{2})=0:
  42. v ( x 1 ( x , t ) , x 2 ( x , t ) ) = 0 \mathcal{L}v(x_{1}(x,t),x_{2}(x,t))=0
  43. e 1 e_{1}
  44. a ( x , t ) a(x,t)
  45. x 1 ( x , t ) x_{1}(x,t)
  46. b ( x , t ) b(x,t)
  47. x 2 ( x , t ) x_{2}(x,t)
  48. e 1 e_{1}
  49. e 2 e_{2}
  50. x x
  51. e ( x 1 , x 2 ) e(x_{1},x_{2})
  52. t t
  53. f ( x 1 , x 2 ) f(x_{1},x_{2})
  54. v ( x 1 , x 2 ) = 0 \mathcal{L}v(x_{1},x_{2})=0
  55. x x
  56. t t
  57. n n
  58. x ˙ i = H / p j \dot{x}_{i}=\partial H/\partial p_{j}
  59. p ˙ j = - H / x j \dot{p}_{j}=-\partial H/\partial x_{j}
  60. n n
  61. I i I_{i}
  62. { x 1 , , x n , p 1 , , p n } \{x_{1},\dots,x_{n},p_{1},\dots,p_{n}\}
  63. { I 1 , I n , φ 1 , , φ n } \{I_{1},\dots I_{n},\varphi_{1},\dots,\varphi_{n}\}
  64. I ˙ i = 0 \dot{I}_{i}=0
  65. φ ˙ i = ω i ( I 1 , , I n ) \dot{\varphi}_{i}=\omega_{i}(I_{1},\dots,I_{n})
  66. ω 1 , , ω n \omega_{1},\dots,\omega_{n}
  67. I 1 , , I n I_{1},\dots,I_{n}
  68. I 1 , , I n I_{1},\dots,I_{n}
  69. φ 1 , , φ n \varphi_{1},\dots,\varphi_{n}
  70. x ˙ = 2 p \dot{x}=2p
  71. p ˙ = - 2 x \dot{p}=-2x
  72. H ( x , p ) = x 2 + p 2 H(x,p)=x^{2}+p^{2}
  73. I ˙ = 0 \dot{I}=0
  74. φ ˙ = 1 \dot{\varphi}=1
  75. I I
  76. φ \varphi
  77. I = p 2 + q 2 I=p^{2}+q^{2}
  78. tan ( φ ) = p / x \tan(\varphi)=p/x

Characteristic_velocity.html

  1. c * c^{*}
  2. c * = p 1 × A t m ˙ c^{*}=\frac{p_{1}\times A_{t}}{\dot{m}}
  3. c * c^{*}
  4. p 1 p_{1}
  5. A t A_{t}
  6. m ˙ \dot{m}

Charlier_polynomials.html

  1. C n ( x ; μ ) = F 0 2 ( - n , - x , - 1 / μ ) = ( - 1 ) n n ! L n ( - 1 - x ) ( - 1 μ ) , C_{n}(x;\mu)={}_{2}F_{0}(-n,-x,-1/\mu)=(-1)^{n}n!L_{n}^{(-1-x)}\left(-\frac{1}% {\mu}\right),\,
  2. L L
  3. x = 0 μ x x ! C n ( x ; μ ) C m ( x ; μ ) = μ - n e μ n ! δ n m , μ > 0. \sum_{x=0}^{\infty}\frac{\mu^{x}}{x!}C_{n}(x;\mu)C_{m}(x;\mu)=\mu^{-n}e^{\mu}n% !\delta_{nm},\quad\mu>0.

Charlot_equation.html

  1. [ H + ] = K a C a - Δ C b + Δ \mathrm{[H^{+}]}=K_{a}\frac{C_{a}-\Delta}{C_{b}+\Delta}
  2. [ H + ] 3 + ( K a + C b ) [ H + ] 2 - ( K w + K a C a ) [ H + ] - K a K w = 0 \mathrm{[H^{+}]^{3}}+(K_{a}+C_{b})\mathrm{[H^{+}]^{2}}-(K_{w}+K_{a}C_{a})% \mathrm{[H^{+}]}-K_{a}K_{w}=0
  3. K a = [ H + ] [ A - ] [ HA ] K_{a}=\mathrm{\frac{[H^{+}][A^{-}]}{[HA]}}
  4. [ H + ] = K a [ HA ] [ A - ] \mathrm{[H^{+}]=\mathit{K_{a}}\frac{[HA]}{[A^{-}]}}
  5. [ M + ] + [ H + ] = [ A - ] + [ OH - ] \mathrm{[M^{+}]+[H^{+}]=[A^{-}]+[OH^{-}]}
  6. [ A - ] = C b + [ H + ] - [ OH - ] = C b + Δ \mathrm{[A^{-}]=\mathit{C_{b}}+[H^{+}]-[OH^{-}]}=C_{b}+\Delta
  7. [ HA ] + [ A - ] = C a + C b \mathrm{[HA]+[A^{-}]}=C_{a}+C_{b}
  8. [ HA ] = C a - Δ \mathrm{[HA]}=C_{a}-\Delta

Chase_(algorithm).html

  1. π S 1 ( R ) π S 2 ( R ) π S k ( R ) \pi_{S_{1}}(R)\bowtie\pi_{S_{2}}(R)\bowtie...\bowtie\pi_{S_{k}}(R)
  2. π S i ( R ) \pi_{S_{i}}(R)
  3. π S i ( R ) \pi_{S_{i}}(R)

Chebychev–Grübler–Kutzbach_criterion.html

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

Chebyshev_center.html

  1. Q Q
  2. Q Q
  3. Q Q
  4. x ^ \hat{x}
  5. x x
  6. Q Q
  7. x ^ \hat{x}
  8. Q Q
  9. x ^ \hat{x}
  10. x ^ \hat{x}
  11. min x ^ , r { r : x ^ - x 2 r , x Q } \min_{{\hat{x}},r}\left\{r:\left\|{\hat{x}}-x\right\|^{2}\leq r,\forall x\in Q\right\}
  12. arg min x ^ max x Q x - x ^ 2 . \operatorname*{\arg\min}_{\hat{x}}\max_{x\in Q}\left\|x-\hat{x}\right\|^{2}.
  13. Q Q
  14. k k
  15. min x ^ max x { x ^ - x 2 : f i ( x ) 0 , 0 i k } \min_{\hat{x}}\max_{x}\left\{\left\|{\hat{x}}-x\right\|^{2}:f_{i}(x)\leq 0,0% \leq i\leq k\right\}
  16. f i ( x ) = x T Q i x + 2 g i T x + d i 0 , 0 i k . f_{i}(x)=x^{T}Q_{i}x+2g_{i}^{T}x+d_{i}\leq 0,0\leq i\leq k.\,
  17. Δ = x x T \Delta=xx^{T}
  18. min x ^ max ( Δ , x ) G { x ^ 2 - 2 x ^ T x + Tr ( Δ ) } \min_{\hat{x}}\max_{(\Delta,x)\in G}\left\{\left\|{\hat{x}}\right\|^{2}-2{\hat% {x}}^{T}x+\operatorname{Tr}(\Delta)\right\}
  19. Tr ( ) \operatorname{Tr}(\cdot)
  20. G = { ( Δ , x ) : f i ( Δ , x ) 0 , 0 i k , Δ = x x T } G=\left\{(\Delta,x):{\rm{f}}_{i}(\Delta,x)\leq 0,0\leq i\leq k,\Delta=xx^{T}\right\}
  21. f i ( Δ , x ) = Tr ( Q i Δ ) + 2 g i T x + d i . f_{i}(\Delta,x)=\operatorname{Tr}(Q_{i}\Delta)+2g_{i}^{T}x+d_{i}.
  22. Δ \Delta
  23. Δ x x T \Delta\geq xx^{T}
  24. Δ - x x T S + \Delta-xx^{T}\in S_{+}
  25. S + S_{+}
  26. R C C = max ( Δ , x ) T { - x 2 + Tr ( Δ ) } RCC=\max_{(\Delta,x)\in{T}}\left\{-\left\|x\right\|^{2}+\operatorname{Tr}(% \Delta)\right\}
  27. T = { ( Δ , x ) : f i ( Δ , x ) 0 , 0 i k , Δ xx T } . {T}=\left\{(\Delta,x):\rm{f}_{i}(\Delta,x)\leq 0,0\leq i\leq k,\Delta\geq xx^{% T}\right\}.
  28. x ^ C L S = arg min x C y - A x 2 {\hat{x}}_{CLS}=\operatorname*{\arg\min}_{x\in C}\left\|y-Ax\right\|^{2}
  29. C = { x : f i ( x ) = x T Q i x + 2 g i T x + d i 0 , 1 i k } {C}=\left\{x:f_{i}(x)=x^{T}Q_{i}x+2g_{i}^{T}x+d_{i}\leq 0,1\leq i\leq k\right\}
  30. Q i 0 , g i R m , d i R . Q_{i}\geq 0,g_{i}\in R^{m},d_{i}\in R.
  31. max ( Δ , < m t p l > x ) V { - x 2 + Tr ( Δ ) } \max_{(\Delta,<mtpl>{{x}})\in{V}}\left\{{-\left\|{{x}}\right\|^{2}+% \operatorname{Tr}(\Delta)}\right\}
  32. V = { ( Δ , x ) : x C Tr ( A T A Δ ) - 2 y T A T x + y 2 - ρ 0 , Δ xx T } . V=\left\{\begin{array}[]{c}(\Delta,x):x\in C\\ \operatorname{Tr}(A^{T}A\Delta)-2y^{T}A^{T}x+\left\|y\right\|^{2}-\rho\leq 0,% \rm{}\Delta\geq xx^{T}\\ \end{array}\right\}.
  33. ( x , Δ ) (x,\Delta)
  34. T V T\in V
  35. Q Q
  36. Q Q
  37. l a T x u l\leq a^{T}x\leq u
  38. ( a T x - l ) ( a T x - u ) 0. (a^{T}x-l)(a^{T}x-u)\leq 0.

Chebyshev–Gauss_quadrature.html

  1. - 1 + 1 f ( x ) 1 - x 2 d x \int_{-1}^{+1}\frac{f(x)}{\sqrt{1-x^{2}}}\,dx
  2. - 1 + 1 1 - x 2 g ( x ) d x . \int_{-1}^{+1}\sqrt{1-x^{2}}g(x)\,dx.
  3. - 1 + 1 f ( x ) 1 - x 2 d x i = 1 n w i f ( x i ) \int_{-1}^{+1}\frac{f(x)}{\sqrt{1-x^{2}}}\,dx\approx\sum_{i=1}^{n}w_{i}f(x_{i})
  4. x i = cos ( 2 i - 1 2 n π ) x_{i}=\cos\left(\frac{2i-1}{2n}\pi\right)
  5. w i = π n . w_{i}=\frac{\pi}{n}.
  6. - 1 + 1 1 - x 2 g ( x ) d x i = 1 n w i g ( x i ) \int_{-1}^{+1}\sqrt{1-x^{2}}g(x)\,dx\approx\sum_{i=1}^{n}w_{i}g(x_{i})
  7. x i = cos ( i n + 1 π ) x_{i}=\cos\left(\frac{i}{n+1}\pi\right)
  8. w i = π n + 1 sin 2 ( i n + 1 π ) . w_{i}=\frac{\pi}{n+1}\sin^{2}\left(\frac{i}{n+1}\pi\right).\,

Cheeger_constant.html

  1. h ( M ) = inf E S ( E ) min ( V ( A ) , V ( B ) ) , h(M)=\inf_{E}\frac{S(E)}{\min(V(A),V(B))},
  2. λ 1 ( M ) , \scriptstyle{\lambda_{1}(M)},
  3. λ 1 ( M ) h 2 ( M ) 4 . \lambda_{1}(M)\geq\frac{h^{2}(M)}{4}.
  4. λ 1 ( M ) \scriptstyle{\lambda_{1}(M)}
  5. λ 1 ( M ) 2 a ( n - 1 ) h ( M ) + 10 h 2 ( M ) . \lambda_{1}(M)\leq 2a(n-1)h(M)+10h^{2}(M).

Chemical_force_microscopy.html

  1. F a d = 3 2 π R W S T M F_{ad}=\frac{3}{2}\pi RW_{STM}
  2. r = ( 3 π γ R 2 K ) 1 3 r=(\frac{3\pi\gamma R^{2}}{K})^{\frac{1}{3}}
  3. π r 2 / A F G \scriptstyle\pi r^{2}/A_{FG}
  4. W = F d x 1 2 F m a x Δ x W=\int Fdx\approx\frac{1}{2}F_{max}\Delta x

Chézy_formula.html

  1. v = C R i , v=C\sqrt{R\,i},\,
  2. v v
  3. C C
  4. R R
  5. i i
  6. C = 1 n R 1 / 6 C=\frac{1}{n}R^{1/6}
  7. C C
  8. R R
  9. n n

Choice_sequence.html

  1. \mathbb{N}
  2. i i
  3. i - 1 i-1
  4. f : f:\mathbb{N}\mapsto\mathbb{N}
  5. α \alpha
  6. α 0 \alpha_{0}
  7. α 1 \alpha_{1}
  8. α \alpha
  9. α \alpha
  10. α \alpha
  11. α 0 , α 1 , , α k \langle\alpha_{0},\alpha_{1},\ldots,\alpha_{k}\rangle
  12. α \alpha
  13. k k
  14. k k\in\mathbb{N}
  15. { 1 , 2 , 3 , 4 , 5 , 6 } \{1,2,3,4,5,6\}
  16. α n \alpha\in n
  17. α \alpha
  18. n n
  19. α \alpha
  20. n n
  21. n n
  22. A ( α ) n [ α n β n [ A ( β ) ] ] A(\alpha)\rightarrow\exists n[\alpha\in n\,\land\,\forall\beta\in n[A(\beta)]]
  23. A A
  24. A A
  25. α \alpha
  26. A A
  27. α \alpha
  28. A A
  29. α \alpha
  30. A A
  31. β \beta
  32. A ( α ) A(\alpha)
  33. β \beta
  34. α \alpha
  35. n n
  36. β \beta
  37. n α [ α n ] \forall n\,\exists\alpha[\alpha\in n]
  38. n n
  39. α \alpha

Christoph_Cremer.html

  1. 4 π 4\pi

Chudnovsky_algorithm.html

  1. d = - 163 d=-163
  2. j ( 1 + - 163 2 ) = - 640320 3 j\big(\tfrac{1+\sqrt{-163}}{2}\big)=-640320^{3}
  3. 1 π = 12 k = 0 ( - 1 ) k ( 6 k ) ! ( 545140134 k + 13591409 ) ( 3 k ) ! ( k ! ) 3 ( 640320 3 ) k + 1 / 2 . \frac{1}{\pi}=12\sum^{\infty}_{k=0}\frac{(-1)^{k}(6k)!(545140134k+13591409)}{(% 3k)!(k!)^{3}(640320^{3})^{k+1/2}}.\!
  4. e π 163 640320 3 + 743.99999999999925 e^{\pi\sqrt{163}}\approx 640320^{3}+743.99999999999925\dots

Church–Kleene_ordinal.html

  1. ω 1 CK \omega^{\mathrm{CK}}_{1}

Cinquefoil_knot.html

  1. Δ ( t ) = t 2 - t + 1 - t - 1 + t - 2 \Delta(t)=t^{2}-t+1-t^{-1}+t^{-2}
  2. ( z ) = z 4 + 3 z 2 + 1 \nabla(z)=z^{4}+3z^{2}+1
  3. V ( q ) = q - 2 + q - 4 - q - 5 + q - 6 - q - 7 . V(q)=q^{-2}+q^{-4}-q^{-5}+q^{-6}-q^{-7}.

Circle_criterion.html

  1. φ ( v , t ) \varphi(v,t)
  2. [ μ 1 , μ 2 ] [\mu_{1},\mu_{2}]
  3. [ - 1 / μ 1 , - 1 / μ 2 ] [-1/\mu_{1},-1/\mu_{2}]
  4. 𝐱 ˙ = 𝐀𝐱 + 𝐁𝐰 , \dot{\mathbf{x}}=\mathbf{Ax}+\mathbf{Bw},
  5. 𝐯 = 𝐂𝐱 , \mathbf{v}=\mathbf{Cx},
  6. 𝐰 = φ ( v , t ) . \mathbf{w}=\varphi(v,t).
  7. μ 1 v φ ( v , t ) μ 2 v , v , t \mu_{1}v\leq\varphi(v,t)\leq\mu_{2}v,\ \forall v,t
  8. det ( i ω I n - A ) 0 , ω R - 1 and μ 0 [ μ 1 , μ 2 ] : A + μ 0 B C \det(i\omega I_{n}-A)\neq 0,\ \forall\omega\in R^{-1}\,\text{ and }\exists\mu_% {0}\in[\mu_{1},\mu_{2}]\,:\,A+\mu_{0}BC
  9. [ ( μ 2 C ( i ω I n - A ) - 1 B - 1 ) ( 1 - μ 1 C ( i ω I n - A ) - 1 B ) ] < 0 ω R - 1 . \Re\left[(\mu_{2}C(i\omega I_{n}-A)^{-1}B-1)(1-\mu_{1}C(i\omega I_{n}-A)^{-1}B% )\right]<0\ \forall\omega\in R^{-1}.
  10. c > 0 , δ > 0 \exists c>0,\delta>0
  11. | x ( t ) | c e - δ t | x ( 0 ) | , t 0. |x(t)|\leq ce^{-\delta t}|x(0)|,\ \forall t\geq 0.

Circle_packing.html

  1. η h = π 2 3 0.9069. \eta_{h}=\frac{\pi}{2\sqrt{3}}\approx 0.9069.

Circle_packing_theorem.html

  1. K v 2 K_{v}\subset\mathbb{R}^{2}
  2. P = ( K v : v V ) P=(K^{\prime}_{v}:v\in V)
  3. K v K u K^{\prime}_{v}\cap K^{\prime}_{u}\neq\varnothing
  4. [ v , u ] E [v,u]\in E
  5. v V v\in V
  6. K v K^{\prime}_{v}
  7. K v K_{v}

Circlotron.html

  1. Z = R p / ( 2 + μ ) Z=R_{p}/(2+\mu)

Circular-arc_graph.html

  1. I 1 , I 2 , , I n C 1 I_{1},I_{2},\ldots,I_{n}\subset C_{1}
  2. V = { I 1 , I 2 , , I n } V=\{I_{1},I_{2},\ldots,I_{n}\}
  3. { I α , I β } E I α I β . \{I_{\alpha},I_{\beta}\}\in E\iff I_{\alpha}\cap I_{\beta}\neq\varnothing.
  4. 𝒪 ( n 3 ) {\mathcal{O}}(n^{3})
  5. ( 𝒪 ( n + m ) ) ({\mathcal{O}}(n+m))
  6. G = ( V , E ) G=(V,E)
  7. G G
  8. G G
  9. ( 𝒪 ( n + m ) ) ({\mathcal{O}}(n+m))
  10. G G
  11. 𝒪 ( n 3 ) {\mathcal{O}(n^{3})}

Circular_coloring.html

  1. χ c ( G ) \chi_{c}(G)
  2. χ c ( G ) \chi_{c}(G)
  3. r r
  4. V ( G ) V(G)
  5. 1 r \geq\frac{1}{r}
  6. χ c ( G ) \chi_{c}(G)
  7. n k \frac{n}{k}
  8. V ( G ) V(G)
  9. / n {\mathbb{Z}}/n{\mathbb{Z}}
  10. k \geq k
  11. C C
  12. | E ( C ) | |E(C)|
  13. χ c ( G ) \chi_{c}(G)
  14. G G
  15. χ c ( G ) χ ( G ) \chi_{c}(G)\leq\chi(G)
  16. χ c ( G ) = χ ( G ) \lceil\chi_{c}(G)\rceil=\chi(G)

Clairaut's_relation.html

  1. r ( t ) cos θ ( t ) = constant . r(t)\cos\theta(t)=\,\text{constant}.\,

Clary_Ranch.html

  1. ± \pm

Classical_electromagnetism_and_special_relativity.html

  1. 𝐄 \stackrel{\mathbf{E}_{\parallel}}{}
  2. 𝐄 \stackrel{\mathbf{E}_{\bot}}{}
  3. 𝐄 = 𝐄 𝐁 = 𝐁 𝐄 = γ ( 𝐄 + 𝐯 × 𝐁 ) 𝐁 = γ ( 𝐁 - 1 c 2 𝐯 × 𝐄 ) \begin{aligned}&\displaystyle\mathbf{{E}_{\parallel}}^{\prime}=\mathbf{{E}_{% \parallel}}\\ &\displaystyle\mathbf{{B}_{\parallel}}^{\prime}=\mathbf{{B}_{\parallel}}\\ &\displaystyle\mathbf{{E}_{\bot}}^{\prime}=\gamma\left(\mathbf{E}_{\bot}+% \mathbf{v}\times\mathbf{B}\right)\\ &\displaystyle\mathbf{{B}_{\bot}}^{\prime}=\gamma\left(\mathbf{B}_{\bot}-\frac% {1}{c^{2}}\mathbf{v}\times\mathbf{E}\right)\end{aligned}
  4. γ = def 1 1 - v 2 / c 2 \gamma\ \overset{\underset{\mathrm{def}}{}}{=}\ \frac{1}{\sqrt{1-v^{2}/{c}^{2}}}
  5. 𝐄 = γ ( 𝐄 + 𝐯 × 𝐁 ) - ( γ - 1 ) ( 𝐄 𝐯 ^ ) 𝐯 ^ 𝐁 = γ ( 𝐁 - 𝐯 × 𝐄 c 2 ) - ( γ - 1 ) ( 𝐁 𝐯 ^ ) 𝐯 ^ \begin{aligned}&\displaystyle\mathbf{E}^{\prime}=\gamma\left(\mathbf{E}+% \mathbf{v}\times\mathbf{B}\right)-\left({\gamma-1}\right)(\mathbf{E}\cdot% \mathbf{\hat{v}})\mathbf{\hat{v}}\\ &\displaystyle\mathbf{B}^{\prime}=\gamma\left(\mathbf{B}-\frac{\mathbf{v}% \times\mathbf{E}}{c^{2}}\right)-\left({\gamma-1}\right)(\mathbf{B}\cdot\mathbf% {\hat{v}})\mathbf{\hat{v}}\\ \end{aligned}
  6. 𝐅 = q 𝐄 + q 𝐮 × 𝐁 \mathbf{F}=q\mathbf{E}+q\mathbf{u}\times\mathbf{B}
  7. 𝐅 = q 𝐄 + q 𝐮 × 𝐁 \mathbf{F^{\prime}}=q\mathbf{E^{\prime}}+q\mathbf{u^{\prime}}\times\mathbf{B^{% \prime}}
  8. u x = u x + v 1 + ( v u x ) / c 2 u y = u y / γ 1 + ( v u x ) / c 2 u z = u z / γ 1 + ( v u x ) / c 2 \begin{aligned}&\displaystyle u_{x}^{\prime}=\frac{u_{x}+v}{1+(v\ u_{x})/c^{2}% }\\ &\displaystyle u_{y}^{\prime}=\frac{u_{y}/\gamma}{1+(v\ u_{x})/c^{2}}\\ &\displaystyle u_{z}^{\prime}=\frac{u_{z}/\gamma}{1+(v\ u_{x})/c^{2}}\end{aligned}
  9. E x = E x \displaystyle E^{\prime}_{x}=E_{x}
  10. 𝐃 = ϵ 0 𝐄 , 𝐁 = μ 0 𝐇 , c 2 = 1 ϵ 0 μ 0 , \mathbf{D}=\epsilon_{0}\mathbf{E}\,,\quad\mathbf{B}=\mu_{0}\mathbf{H}\,,\quad c% ^{2}=\frac{1}{\epsilon_{0}\mu_{0}}\,,
  11. 𝐃 = γ ( 𝐃 + 1 c 2 𝐯 × 𝐇 ) + ( 1 - γ ) ( 𝐃 𝐯 ^ ) 𝐯 ^ 𝐇 = γ ( 𝐇 - 𝐯 × 𝐃 ) + ( 1 - γ ) ( 𝐇 𝐯 ^ ) 𝐯 ^ \begin{aligned}\displaystyle\mathbf{D}^{\prime}&\displaystyle=\gamma\left(% \mathbf{D}+\frac{1}{c^{2}}\mathbf{v}\times\mathbf{H}\right)+(1-\gamma)(\mathbf% {D}\cdot\mathbf{\hat{v}})\mathbf{\hat{v}}\\ \displaystyle\mathbf{H}^{\prime}&\displaystyle=\gamma\left(\mathbf{H}-\mathbf{% v}\times\mathbf{D}\right)+(1-\gamma)(\mathbf{H}\cdot\mathbf{\hat{v}})\mathbf{% \hat{v}}\\ \end{aligned}
  12. φ = γ ( φ - v A ) A = γ ( A - v φ / c 2 ) A = A \begin{aligned}&\displaystyle\varphi^{\prime}=\gamma(\varphi-vA_{\parallel})\\ &\displaystyle A_{\parallel}^{\prime}=\gamma(A_{\parallel}-v\varphi/c^{2})\\ &\displaystyle A_{\bot}^{\prime}=A_{\bot}\end{aligned}
  13. A \scriptstyle A_{\parallel}
  14. A \scriptstyle A_{\bot}
  15. 𝐀 \displaystyle\mathbf{A}^{\prime}
  16. J = γ ( J - v ρ ) ρ = γ ( ρ - v J / c 2 ) J = J \begin{aligned}&\displaystyle J_{\parallel}^{\prime}=\gamma(J_{\parallel}-v% \rho)\\ &\displaystyle\rho^{\prime}=\gamma(\rho-vJ_{\parallel}/c^{2})\\ &\displaystyle J_{\bot}^{\prime}=J_{\bot}\end{aligned}
  17. 𝐉 \displaystyle\mathbf{J}^{\prime}
  18. 𝐄 \displaystyle\mathbf{E}^{\prime}
  19. F μ ν = ( 0 - E x / c - E y / c - E z / c E x / c 0 - B z B y E y / c B z 0 - B x E z / c - B y B x 0 ) F^{\mu\nu}=\begin{pmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\ E_{x}/c&0&-B_{z}&B_{y}\\ E_{y}/c&B_{z}&0&-B_{x}\\ E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}
  20. G μ ν = ( 0 - B x - B y - B z B x 0 E z / c - E y / c B y - E z / c 0 E x / c B z E y / c - E x / c 0 ) G^{\mu\nu}=\begin{pmatrix}0&-B_{x}&-B_{y}&-B_{z}\\ B_{x}&0&E_{z}/c&-E_{y}/c\\ B_{y}&-E_{z}/c&0&E_{x}/c\\ B_{z}&E_{y}/c&-E_{x}/c&0\end{pmatrix}
  21. F α β = Λ α Λ β μ F μ ν ν F^{\prime\alpha\beta}=\Lambda^{\alpha}{}_{\mu}\Lambda^{\beta}{}_{\nu}F^{\mu\nu}
  22. J α = ( c ρ J x J y J z ) J^{\alpha}=\begin{pmatrix}c\rho&J_{x}&J_{y}&J_{z}\end{pmatrix}
  23. F α β = η α λ η β μ F λ μ F_{\alpha\beta}=\eta_{\alpha\lambda}\eta_{\beta\mu}F^{\lambda\mu}
  24. ϵ δ α β γ F β γ x α = F α β x γ + F γ α x β + F β γ x α = 0 \epsilon^{\delta\alpha\beta\gamma}\dfrac{\partial F_{\beta\gamma}}{\partial x^% {\alpha}}=\dfrac{\partial F_{\alpha\beta}}{\partial x^{\gamma}}+\dfrac{% \partial F_{\gamma\alpha}}{\partial x^{\beta}}+\dfrac{\partial F_{\beta\gamma}% }{\partial x^{\alpha}}=0
  25. ϵ α β γ δ \epsilon^{\alpha\beta\gamma\delta}
  26. α β γ \begin{array}[]{rc}&\scriptstyle{\alpha\,\,\longrightarrow\,\,\beta}\\ &\nwarrow_{\gamma}\swarrow\end{array}
  27. F α β = A β x α - A α x β , F^{\alpha\beta}=\frac{\partial A^{\beta}}{\partial x_{\alpha}}-\frac{\partial A% ^{\alpha}}{\partial x_{\beta}}\,,
  28. A α = ( φ / c , A x , A y , A z ) , A^{\alpha}=(\varphi/c,A_{x},A_{y},A_{z})\,,
  29. x α = ( c t , - x , - y , - z ) x_{\alpha}=(ct,-x,-y,-z)
  30. \Box

Classical_mechanics.html

  1. 𝐯 = d 𝐫 d t \mathbf{v}={\mathrm{d}\mathbf{r}\over\mathrm{d}t}\,\!
  2. 𝐮 = 𝐮 - 𝐯 . \mathbf{u}^{\prime}=\mathbf{u}-\mathbf{v}\,.
  3. 𝐯 = 𝐯 - 𝐮 . \mathbf{v^{\prime}}=\mathbf{v}-\mathbf{u}\,.
  4. 𝐮 = ( u - v ) 𝐝 . \mathbf{u}^{\prime}=(u-v)\mathbf{d}\,.
  5. u = u - v . u^{\prime}=u-v\,.
  6. 𝐚 = d 𝐯 d t = d 2 𝐫 d t 2 . \mathbf{a}={\mathrm{d}\mathbf{v}\over\mathrm{d}t}={\mathrm{d^{2}}\mathbf{r}% \over\mathrm{d}t^{2}}.
  7. 𝐅 = d 𝐩 d t = d ( m 𝐯 ) d t . \mathbf{F}={\mathrm{d}\mathbf{p}\over\mathrm{d}t}={\mathrm{d}(m\mathbf{v})% \over\mathrm{d}t}.
  8. 𝐅 = m 𝐚 . \mathbf{F}=m\mathbf{a}\,.
  9. 𝐅 R = - λ 𝐯 , \mathbf{F}_{\rm R}=-\lambda\mathbf{v}\,,
  10. - λ 𝐯 = m 𝐚 = m d 𝐯 d t . -\lambda\mathbf{v}=m\mathbf{a}=m{\mathrm{d}\mathbf{v}\over\mathrm{d}t}\,.
  11. 𝐯 = 𝐯 0 e - λ t / m \mathbf{v}=\mathbf{v}_{0}e^{-\lambda t/m}
  12. W = 𝐅 Δ 𝐫 . W=\mathbf{F}\cdot\Delta\mathbf{r}\,.
  13. W = C 𝐅 ( 𝐫 ) d 𝐫 . W=\int_{C}\mathbf{F}(\mathbf{r})\cdot\mathrm{d}\mathbf{r}\,.
  14. E k = 1 2 m v 2 . E_{\mathrm{k}}=\tfrac{1}{2}mv^{2}\,.
  15. W = Δ E k = E k , 2 - E k , 1 = 1 2 m ( v 2 2 - v 1 2 ) . W=\Delta E_{\mathrm{k}}=E_{\mathrm{k,2}}-E_{\mathrm{k,1}}=\tfrac{1}{2}m\left(v% _{2}^{\,2}-v_{1}^{\,2}\right)\,.
  16. 𝐅 = - E p . \mathbf{F}=-\mathbf{\nabla}E_{\mathrm{p}}\,.
  17. 𝐅 Δ 𝐫 = - E p Δ 𝐫 = - Δ E p - Δ E p = Δ E k Δ ( E k + E p ) = 0 . \mathbf{F}\cdot\Delta\mathbf{r}=-\mathbf{\nabla}E_{\mathrm{p}}\cdot\Delta% \mathbf{r}=-\Delta E_{\mathrm{p}}\Rightarrow-\Delta E_{\mathrm{p}}=\Delta E_{% \mathrm{k}}\Rightarrow\Delta(E_{\mathrm{k}}+E_{\mathrm{p}})=0\,.
  18. E = E k + E p , \sum E=E_{\mathrm{k}}+E_{\mathrm{p}}\,,
  19. 𝐩 = m 𝐯 1 - ( v 2 / c 2 ) , \mathbf{p}=\frac{m\mathbf{v}}{\sqrt{1-(v^{2}/c^{2})}}\,,
  20. 𝐩 m 𝐯 . \mathbf{p}\approx m\mathbf{v}\,.
  21. f = f c m 0 m 0 + T / c 2 , f=f_{\mathrm{c}}\frac{m_{0}}{m_{0}+T/c^{2}}\,,
  22. λ = h p \lambda=\frac{h}{p}

Clausen's_formula.html

  1. F 1 2 [ a b a + b + 1 / 2 ; x ] 2 = 3 F 2 [ 2 a 2 b a + b a + b + 1 / 2 2 a + 2 b ; x ] \;{}_{2}F_{1}\left[\begin{matrix}a&b\\ a+b+1/2\end{matrix};x\right]^{2}=\;_{3}F_{2}\left[\begin{matrix}2a&2b&a+b\\ a+b+1/2&2a+2b\end{matrix};x\right]

Clearing_factor.html

  1. t t
  2. s s
  3. t = k s t=\frac{k}{s}
  4. ω \omega
  5. r r
  6. k = ln ( r max / r min ) ω 2 × 10 - 13 3600 k=\frac{\ln(r_{\rm{max}}/r_{\rm{min}})}{\omega^{2}}\times\frac{10^{-13}}{3600}
  7. k = 2.53 10 5 × ln ( r max / r min ) ( RPM / 1000 ) 2 k=\frac{2.53\cdot 10^{5}\times\ln(r_{\rm{max}}/r_{\rm{min}})}{(\rm{RPM}/1000)^% {2}}
  8. k k
  9. k k
  10. k adj = k ( maximum rotor-speed actual rotor-speed ) k_{\rm{adj}}=k\left(\frac{\mbox{maximum rotor-speed}~{}}{\mbox{actual rotor-% speed}~{}}\right)
  11. S S
  12. T = K S T=\frac{K}{S}
  13. T T
  14. S S
  15. T 1 K 1 = T 2 K 2 \frac{T_{1}}{K_{1}}=\frac{T_{2}}{K_{2}}
  16. T 1 T_{1}
  17. K 1 K_{1}
  18. K 2 K_{2}
  19. T 2 T_{2}

Cleavable_detergent.html

  1. \longrightarrow
  2. + +

Clique-width.html

  1. G G
  2. G G
  3. G H G\oplus H
  4. i j i\neq j

Closed_geodesic.html

  1. γ : M \gamma:\mathbb{R}\rightarrow M
  2. Λ M \Lambda M
  3. E : Λ M E:\Lambda M\rightarrow\mathbb{R}
  4. E ( γ ) = 0 1 g γ ( t ) ( γ ˙ ( t ) , γ ˙ ( t ) ) d t . E(\gamma)=\int_{0}^{1}g_{\gamma(t)}(\dot{\gamma}(t),\dot{\gamma}(t))\,\mathrm{% d}t.
  5. γ \gamma
  6. t γ ( p t ) t\mapsto\gamma(pt)
  7. γ \gamma
  8. γ m \gamma^{m}
  9. m m\in\mathbb{N}
  10. γ m ( t ) := γ ( m t ) \gamma^{m}(t):=\gamma(mt)
  11. S n n + 1 S^{n}\subset\mathbb{R}^{n+1}

Closest_string.html

  1. O ( k L + k d d d ) O(kL+kd\cdot d^{d})

Cluster_expansion.html

  1. . H 0 = i N p i 2 2 m \big.H_{0}=\sum_{i}^{N}\frac{p_{i}^{2}}{2m}
  2. . Z 0 = 1 N ! h 3 N i d p i d r i exp { - β H 0 ( { r i , p i } ) } = V N N ! h 3 N ( 2 π m β ) 3 N 2 . \big.Z_{0}=\frac{1}{N!h^{3N}}\int\prod_{i}d\vec{p}_{i}\;d\vec{r}_{i}\exp\left% \{-\beta H_{0}(\{r_{i},p_{i}\})\right\}=\frac{V^{N}}{N!h^{3N}}\left(\frac{2\pi m% }{\beta}\right)^{\frac{3N}{2}}.
  3. . F 0 = - k B T ln Z 0 \big.F_{0}=-k_{B}T\ln Z_{0}
  4. . U ( { r i } ) = i = 1 , i < j N u 2 ( | r i - r j | ) = i = 1 , i < j N u 2 ( r i j ) . \big.U\left(\{r_{i}\}\right)=\sum_{i=1,i<j}^{N}u_{2}(|\vec{r}_{i}-\vec{r}_{j}|% )=\sum_{i=1,i<j}^{N}u_{2}(r_{ij}).
  5. . Z = Z 0 Q \big.Z=Z_{0}\ Q
  6. F = F 0 - k B T ln ( Q ) F=F_{0}-k_{B}T\!\ln\left(Q\right)
  7. Q = 1 V N i d r i exp { - β i = 1 , i < j N u 2 ( r i j ) } . Q=\frac{1}{V^{N}}\int\prod_{i}d\vec{r}_{i}\exp\left\{-\beta\sum_{i=1,i<j}^{N}u% _{2}(r_{ij})\right\}.
  8. Q Q
  9. u 2 ( r ) u_{2}(r)
  10. Q Q
  11. exp { - β i = 1 , i < j N u 2 ( r i j ) } = i = 1 , i < j N exp { - β u 2 ( r i j ) } \exp\left\{-\beta\sum_{i=1,i<j}^{N}u_{2}(r_{ij})\right\}=\prod_{i=1,i<j}^{N}% \exp\left\{-\beta u_{2}(r_{ij})\right\}
  12. f i j f_{ij}
  13. exp { - β u 2 ( r i j ) } = 1 + f i j \exp\left\{-\beta u_{2}(r_{ij})\right\}=1+f_{ij}
  14. . Q = 1 V N i d r i i = 1 , i < j N ( 1 + f i j ) \big.Q=\frac{1}{V^{N}}\int\prod_{i}d\vec{r}_{i}\prod_{i=1,i<j}^{N}\left(1+f_{% ij}\right)
  15. f i j f_{ij}
  16. i = 1 , i < j N ( 1 + f i j ) = 1 + i = 1 , i < j N f i j + i = 1 , i < j , k = 1 , k < l N f i j f k l + \prod_{i=1,i<j}^{N}\left(1+f_{ij}\right)=1+\sum_{i=1,i<j}^{N}\;f_{ij}+\sum_{i=% 1,i<j,\atop k=1,k<l}^{N}\;f_{ij}\;f_{kl}+\cdots
  17. Q Q
  18. . Q = 1 + N V α 1 + N ( N - 1 ) 2 V 2 α 2 + . \big.Q=1+\frac{N}{V}\alpha_{1}+\frac{N\;(N-1)}{2\;V^{2}}\alpha_{2}+\cdots.
  19. P V = N k B T ( 1 + N V B 2 ( T ) + N 2 V 2 B 3 ( T ) + N 3 V 3 B 4 ( T ) + ) PV=Nk_{B}T\left(1+\frac{N}{V}B_{2}(T)+\frac{N^{2}}{V^{2}}B_{3}(T)+\frac{N^{3}}% {V^{3}}B_{4}(T)+\cdots\right)
  20. B i ( T ) B_{i}(T)
  21. B 2 ( T ) B_{2}(T)
  22. B 3 ( T ) B_{3}(T)

Coarea_formula.html

  1. B V BV
  2. Ω g ( x ) | u ( x ) | d x = - ( u - 1 ( t ) g ( x ) d H n - 1 ( x ) ) d t \int_{\Omega}g(x)|\nabla u(x)|\,dx=\int_{-\infty}^{\infty}\left(\int_{u^{-1}(t% )}g(x)\,dH_{n-1}(x)\right)\,dt
  3. Ω | u | = - H n - 1 ( u - 1 ( t ) ) d t , \int_{\Omega}|\nabla u|=\int_{-\infty}^{\infty}H_{n-1}(u^{-1}(t))\,dt,
  4. n f d x = 0 { B ( x 0 ; r ) f d S } d r . \int_{\mathbb{R}^{n}}f\,dx=\int_{0}^{\infty}\left\{\int_{\partial B(x_{0};r)}f% \,dS\right\}\,dr.
  5. ( n | u | n / ( n - 1 ) ) n - 1 n n - 1 ω n - 1 / n n | u | \left(\int_{\mathbb{R}^{n}}|u|^{n/(n-1)}\right)^{\frac{n-1}{n}}\leq n^{-1}% \omega_{n}^{-1/n}\int_{\mathbb{R}^{n}}|\nabla u|

Cobalt(II,III)_oxide.html

  1. \overrightarrow{\leftarrow}

Cochran's_Q_test.html

  1. \cdots
  2. \cdots
  3. \cdots
  4. \cdots
  5. \vdots
  6. \vdots
  7. \vdots
  8. \ddots
  9. \vdots
  10. \cdots
  11. T = k ( k - 1 ) j = 1 k ( X j - N k ) 2 i = 1 b X i ( k - X i ) T=k\left(k-1\right)\frac{\sum\limits_{j=1}^{k}\left(X_{\bullet j}-\frac{N}{k}% \right)^{2}}{\sum\limits_{i=1}^{b}X_{i\bullet}\left(k-X_{i\bullet}\right)}
  12. T > χ 1 - α , k - 1 2 T>\chi^{2}_{1-\alpha,k-1}

Cocks_IBE_scheme.html

  1. n = p q \textstyle n=pq
  2. p , q , p q 3 mod 4 \textstyle p,q,\,p\equiv q\equiv 3\mod 4
  3. = { - 1 , 1 } , 𝒞 = n \textstyle\mathcal{M}=\left\{-1,1\right\},\mathcal{C}=\mathbb{Z}_{n}
  4. f : { 0 , 1 } * n \textstyle f:\left\{0,1\right\}^{*}\rightarrow\mathbb{Z}_{n}
  5. I D \textstyle ID
  6. a \textstyle a
  7. ( a n ) = 1 \textstyle\left(\frac{a}{n}\right)=1
  8. I D \textstyle ID
  9. f \textstyle f
  10. r = a ( n + 5 - p - q ) / 8 mod n \textstyle r=a^{(n+5-p-q)/8}\bmod n
  11. r 2 = a mod n \textstyle r^{2}=a\bmod n
  12. r 2 = - a mod n \textstyle r^{2}=-a\bmod n
  13. r \textstyle r
  14. 1 \textstyle 1
  15. - 1 \textstyle-1
  16. m \textstyle m\in\mathcal{M}
  17. I D \textstyle ID
  18. t 1 \textstyle t_{1}
  19. m = ( t 1 n ) \textstyle m=\left(\frac{t_{1}}{n}\right)
  20. t 2 \textstyle t_{2}
  21. m = ( t 2 n ) \textstyle m=\left(\frac{t_{2}}{n}\right)
  22. t 1 \textstyle t_{1}
  23. c 1 = t 1 + a t 1 - 1 mod n \textstyle c_{1}=t_{1}+at_{1}^{-1}\bmod n
  24. c 2 = t 2 - a t 2 - 1 c_{2}=t_{2}-at_{2}^{-1}
  25. s = ( c 1 , c 2 ) \textstyle s=(c_{1},c_{2})
  26. s = ( c 1 , c 2 ) s=(c_{1},c_{2})
  27. I D ID
  28. α = c 1 + 2 r \alpha=c_{1}+2r
  29. r 2 = a r^{2}=a
  30. α = c 2 + 2 r \alpha=c_{2}+2r
  31. m = ( α n ) m=\left(\frac{\alpha}{n}\right)
  32. I D ID
  33. r r
  34. a a
  35. - a -a
  36. p q 3 mod 4 \textstyle p\equiv q\equiv 3\bmod 4
  37. ( - 1 p ) = ( - 1 q ) = - 1 \left(\frac{-1}{p}\right)=\left(\frac{-1}{q}\right)=-1
  38. ( a n ) ( a p ) = ( a q ) \textstyle\left(\frac{a}{n}\right)\Rightarrow\left(\frac{a}{p}\right)=\left(% \frac{a}{q}\right)
  39. a \textstyle a
  40. - a \textstyle-a
  41. n \textstyle n
  42. r \textstyle r
  43. a \textstyle a
  44. - a \textstyle-a
  45. r 2 \displaystyle r^{2}
  46. a \textstyle a
  47. - a \textstyle-a
  48. ( s + 2 r n ) \displaystyle\left(\frac{s+2r}{n}\right)
  49. n \textstyle n
  50. t \textstyle t
  51. t \textstyle t
  52. r r

Codex_Athous_Lavrensis.html

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

Codex_Augiensis.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Codex_Nitriensis.html

  1. 𝔓 \mathfrak{P}

Codex_Porphyrianus.html

  1. 𝔓 \mathfrak{P}
  2. 𝔓 \mathfrak{P}

Codex_Regius_(New_Testament).html

  1. 𝔓 \mathfrak{P}

Cognate_linkage.html

  1. K L = sin ( α ) sin ( β ) K_{L}=\frac{\sin(\alpha)}{\sin(\beta)}
  2. K R = sin ( γ ) sin ( β ) K_{R}=\frac{\sin(\gamma)}{\sin(\beta)}
  3. ( + s ) < ( P + q ) (\ell+s)<(P+q)

Coherence_(signal_processing).html

  1. C x y ( f ) = | G x y ( f ) | 2 G x x ( f ) G y y ( f ) C_{xy}(f)=\frac{|G_{xy}(f)|^{2}}{G_{xx}(f)G_{yy}(f)}
  2. 0 C x y ( f ) 1 0\leq C_{xy}(f)\leq 1
  3. y ( t ) = h ( t ) * x ( t ) y(t)=h(t)*x(t)
  4. Y ( f ) = H ( f ) X ( f ) Y(f)=H(f)X(f)
  5. G y y = | H ( f ) | 2 G x x ( f ) G_{yy}=|H(f)|^{2}G_{xx}(f)
  6. G x y = H ( f ) G x x ( f ) G_{xy}=H(f)G_{xx}(f)
  7. G x x ( f ) G_{xx}(f)
  8. C x y ( f ) = | H ( f ) G x x ( f ) | 2 G x x ( f ) G y y ( f ) = | H ( f ) G x x ( f ) | 2 G x x 2 ( f ) | H ( f ) | 2 = | G x x ( f ) | 2 G x x 2 ( f ) = 1 C_{xy}(f)=\frac{|H(f)G_{xx}(f)|^{2}}{G_{xx}(f)G_{yy}(f)}=\frac{|H(f)G_{xx}(f)|% ^{2}}{G_{xx}^{2}(f)|H(f)|^{2}}=\frac{|G_{xx}(f)|^{2}}{G_{xx}^{2}(f)}=1
  9. C x y 1 C_{xy}\leq 1
  10. 1 - C x y 1-C_{xy}
  11. G v v = C x y G y y G_{vv}=C_{xy}G_{yy}
  12. G v v G_{vv}
  13. γ 2 \gamma^{2}

Cohomology_with_compact_support.html

  1. Ω c k ( X ) \Omega^{k}_{\mathrm{c}}(X)
  2. H c q ( X ) H^{q}_{\mathrm{c}}(X)
  3. ( Ω c ( X ) , d ) (\Omega^{\bullet}_{\mathrm{c}}(X),d)
  4. 0 Ω c 0 ( X ) Ω c 1 ( X ) Ω c 2 ( X ) 0\to\Omega^{0}_{\mathrm{c}}(X)\to\Omega^{1}_{\mathrm{c}}(X)\to\Omega^{2}_{% \mathrm{c}}(X)\to\cdots
  5. H c q ( X ) H^{q}_{\mathrm{c}}(X)
  6. j * : Ω c ( U ) Ω c ( X ) j_{*}:\Omega^{\bullet}_{\mathrm{c}}(U)\to\Omega^{\bullet}_{\mathrm{c}}(X)
  7. j * : H c q ( U ) H c q ( X ) j_{*}:H^{q}_{\mathrm{c}}(U)\to H^{q}_{\mathrm{c}}(X)
  8. f * : Ω c q ( X ) Ω c q ( Y ) I g I d x i 1 d x i q I ( g I f ) d ( x i 1 f ) d ( x i q f ) f^{*}:\Omega^{q}_{\mathrm{c}}(X)\to\Omega^{q}_{\mathrm{c}}(Y)\sum_{I}g_{I}\,dx% _{i_{1}}\wedge\ldots\wedge dx_{i_{q}}\mapsto\sum_{I}(g_{I}\circ f)\,d(x_{i_{1}% }\circ f)\wedge\ldots\wedge d(x_{i_{q}}\circ f)
  9. H c q ( X ) H c q ( Y ) H^{q}_{\mathrm{c}}(X)\to H^{q}_{\mathrm{c}}(Y)
  10. H c q ( U ) j * H c q ( X ) i * H c q ( Z ) 𝛿 H c q + 1 ( U ) \cdots\to H^{q}_{\mathrm{c}}(U)\overset{j_{*}}{\longrightarrow}H^{q}_{\mathrm{% c}}(X)\overset{i^{*}}{\longrightarrow}H^{q}_{\mathrm{c}}(Z)\overset{\delta}{% \longrightarrow}H^{q+1}_{\mathrm{c}}(U)\to\cdots
  11. H c q ( U V ) H c q ( U ) H c q ( V ) H c q ( X ) 𝛿 H c q + 1 ( U V ) \cdots\to H^{q}_{\mathrm{c}}(U\cap V)\to H^{q}_{\mathrm{c}}(U)\oplus H^{q}_{% \mathrm{c}}(V)\to H^{q}_{\mathrm{c}}(X)\overset{\delta}{\longrightarrow}H^{q+1% }_{\mathrm{c}}(U\cap V)\to\cdots

Collaboration_graph.html

  1. k k
  2. k k
  3. 1 k \tfrac{1}{k}
  4. k k

Collage_theorem.html

  1. 𝕏 \mathbb{X}
  2. L L
  3. 𝕏 \mathbb{X}
  4. ϵ 0 \epsilon\geq 0
  5. { 𝕏 ; w 1 , w 2 , , w N } \{\mathbb{X};w_{1},w_{2},\dots,w_{N}\}
  6. 0 s < 1 0\leq s<1
  7. w i w_{i}
  8. h ( L , n = 1 N w n ( L ) ) ε , h\left(L,\bigcup_{n=1}^{N}w_{n}(L)\right)\leq\varepsilon,
  9. h ( d ) h(d)
  10. h ( L , A ) ε 1 - s h(L,A)\leq\frac{\varepsilon}{1-s}
  11. h ( L , A ) ( 1 - s ) - 1 h ( L , n = 1 N w n ( L ) ) h(L,A)\leq(1-s)^{-1}h\left(L,\cup_{n=1}^{N}w_{n}(L)\right)\quad
  12. 𝕏 \mathbb{X}
  13. L L
  14. L L

Collapse_(topology).html

  1. K K
  2. τ , σ K \tau,\sigma\in K
  3. τ σ \tau\subset\sigma
  4. dim τ < dim σ \dim\tau<\dim\sigma
  5. σ \sigma
  6. τ \tau
  7. τ \tau
  8. γ \gamma
  9. τ γ σ \tau\subseteq\gamma\subseteq\sigma
  10. τ \tau

Collapsing_manifold.html

  1. | sec ( M i ) | 1 |\sec(M_{i})|\leq 1
  2. M i M_{i}
  3. n n
  4. sec ( M i ) \sec(M_{i})
  5. ε ( n ) \varepsilon(n)
  6. | sec ( M i ) | 1 |\sec(M_{i})|\leq 1
  7. Inj ( M i ) < ε ( n ) {\rm Inj}(M_{i})<\varepsilon(n)
  8. M i M_{i}
  9. Inj ( M ) {\rm Inj}(M)
  10. sec ( M i ) - 1 \sec(M_{i})\geq-1
  11. g i g_{i}
  12. sec ( M , g i ) - 1 / n \sec(M,g_{i})\geq-1/n
  13. diam ( M , g i ) 1 / n {\rm diam}(M,g_{i})\leq 1/n
  14. X X
  15. M i n M^{n}_{i}
  16. X X
  17. i i
  18. ( δ , n ) (\delta,n)
  19. X X
  20. ε ( n ) \varepsilon(n)
  21. δ ( n ) \delta(n)
  22. M M
  23. Vol ( M ) < ε ( n ) {\rm Vol}(M)<\varepsilon(n)
  24. M M
  25. δ ( n ) \delta(n)

Comb_space.html

  1. \R 2 \R^{2}
  2. \R 2 \R^{2}
  3. { 1 / n | n { 0 } } \{1/n|n\in\mathbb{N}\setminus\{0\}\}
  4. ( { 0 } × [ 0 , 1 ] ) ( K × [ 0 , 1 ] ) ( [ 0 , 1 ] × { 0 } ) (\{0\}\times[0,1])\cup(K\times[0,1])\cup([0,1]\times\{0\})
  5. \R 2 \R^{2}
  6. ( { 0 } × { 0 , 1 } ) ( K × [ 0 , 1 ] ) ( [ 0 , 1 ] × { 0 } ) (\{0\}\times\{0,1\})\cup(K\times[0,1])\cup([0,1]\times\{0\})
  7. { 0 } × ( 0 , 1 ) \{0\}\times(0,1)
  8. { 0 } × ( 0 , 1 ] \{0\}\times(0,1]
  9. \subset
  10. \subset

Combinatory_categorial_grammar.html

  1. β : Y α : X \ Y β α : X = = = C o m p o s i t i o n C o m b i n a t o r s = = = T h e c o m p o s i t i o n c o m b i n a t o r s , o f t e n d e n o t e d b y < m a t h > B > \dfrac{\beta:Y\qquad\alpha:X\backslash Y}{\beta\alpha:X}\par ===% CompositionCombinators===\par Thecompositioncombinators,oftendenotedby<math>B% _{>}
  2. B f o r b a c k w a r d c o m p o s i t i o n , a r e s i m i l a r t o f u n c t i o n c o m p o s i t i o n f r o m m a t h e m a t i c s , a n d c a n b e d e f i n e d a s f o l l o w s : < m a t h > α : X / Y β : Y / Z α β : X / Z B Align g t ; B_{f}orbackwardcomposition,aresimilartofunctioncompositionfrommathematics,% andcanbedefinedasfollows:<math>\dfrac{\alpha:X/Y\qquad\beta:Y/Z}{\alpha\beta:X% /Z}B_{&}gt;
  3. β : Y \ Z α : X \ Y β α : X \ Z B = = = T y p e - r a i s i n g C o m b i n a t o r s = = = T h e t y p e - r a i s i n g c o m b i n a t o r s , o f t e n d e n o t e d a s < m a t h > T Align g t ; \dfrac{\beta:Y\backslash Z\qquad\alpha:X\backslash Y}{\beta\alpha:X\backslash Z% }B_{\par }===Type-raisingCombinators===\par Thetype-raisingcombinators,% oftendenotedas<math>T_{&}gt;
  4. T f o r b a c k w a r d t y p e - r a i s i n g , t a k e a r g u m e n t t y p e s ( u s u a l l y p r i m i t i v e t y p e s ) t o f u n c t o r t y p e s , w h i c h t a k e a s t h e i r a r g u m e n t t h e f u n c t o r s t h a t , b e f o r e t y p e - r a i s i n g , w o u l d h a v e t a k e n t h e m a s a r g u m e n t s . < m a t h > α : X α : T / ( T \ X ) T Align g t ; T_{f}orbackwardtype-raising,takeargumenttypes(usuallyprimitivetypes)% tofunctortypes,whichtakeastheirargumentthefunctorsthat,beforetype-raising,% wouldhavetakenthemasarguments.\par <math>\dfrac{\alpha:X}{\alpha:T/(T% \backslash X)}T_{&}gt;
  5. α : X α : T \ ( T / X ) T = = E x a m p l e = = T h e s e n t e n c e " t h e d o g b i t J o h n " h a s a n u m b e r o f d i f f e r e n t p o s s i b l e p r o o f s . B e l o w a r e a f e w o f t h e m . T h e v a r i e t y o f p r o o f s d e m o n s t r a t e s t h e f a c t t h a t i n C C G , s e n t e n c e s d o n t h a v e a s i n g l e s t r u c t u r e , a s i n o t h e r m o d e l s o f g r a m m a r . L e t t h e t y p e s o f t h e s e l e x i c a l i t e m s b e < m a t h > t h e : N P / N d o g : N J o h n : N P b i t : ( S \ N P ) / N P \dfrac{\alpha:X}{\alpha:T\backslash(T/X)}T_{\par }==Example==\par Thesentence% "thedogbitJohn"hasanumberofdifferentpossibleproofs.Belowareafewofthem.% ThevarietyofproofsdemonstratesthefactthatinCCG,sentencesdon^{\prime}% thaveasinglestructure,asinothermodelsofgrammar.\par Letthetypesoftheselexicalitemsbe% \par <math>the:NP/N\qquad dog:N\qquad John:NP\qquad bit:(S\backslash NP)/NP
  6. t h e N P / N d o g N N P > b i t ( S \ N P ) / N P J o h n N P S \ N P > S < \dfrac{\dfrac{\dfrac{the}{NP/N}\qquad\dfrac{dog}{N}}{NP}>\qquad\dfrac{\dfrac{% bit}{(S\backslash NP)/NP}\qquad\dfrac{John}{NP}}{S\backslash NP}>}{S}<
  7. t h e N P / N d o g N N P > S / ( S \ N P ) T > b i t ( S \ N P ) / N P S / N P B > J o h n N P S Align g t ; \dfrac{\dfrac{\dfrac{\dfrac{\dfrac{the}{NP/N}\dfrac{dog}{N}\qquad}{NP}>}{S/(S% \backslash NP)}T_{>}\qquad\dfrac{bit}{(S\backslash NP)/NP}}{S/NP}B_{>}\qquad% \dfrac{John}{NP}}{S}&gt;
  8. a n b n c n d n : n 0 {a^{n}b^{n}c^{n}d^{n}:n\geq 0}

Common_integrals_in_quantum_field_theory.html

  1. G - e - 1 2 x 2 d x G\equiv\int_{-\infty}^{\infty}e^{-{1\over 2}x^{2}}\,dx
  2. G 2 = ( - e - 1 2 x 2 d x ) ( - e - 1 2 y 2 d y ) = 2 π 0 r e - 1 2 r 2 d r = 2 π 0 e - w d w = 2 π . G^{2}=\left(\int_{-\infty}^{\infty}e^{-{1\over 2}x^{2}}\,dx\right)\cdot\left(% \int_{-\infty}^{\infty}e^{-{1\over 2}y^{2}}\,dy\right)=2\pi\int_{0}^{\infty}re% ^{-{1\over 2}r^{2}}\,dr=2\pi\int_{0}^{\infty}e^{-w}\,dw=2\pi.
  3. - e - 1 2 x 2 d x = 2 π . \int_{-\infty}^{\infty}e^{-{1\over 2}x^{2}}\,dx=\sqrt{2\pi}.
  4. - e - 1 2 a x 2 d x = 2 π a \int_{-\infty}^{\infty}e^{-{1\over 2}ax^{2}}\,dx=\sqrt{2\pi\over a}
  5. x x a x\to{x\over\sqrt{a}}
  6. - x 2 e - 1 2 a x 2 d x = - 2 d d a - e - 1 2 a x 2 d x = - 2 d d a ( 2 π a ) 1 2 = ( 2 π a ) 1 2 1 a \int_{-\infty}^{\infty}x^{2}e^{-{1\over 2}ax^{2}}\,dx=-2{d\over da}\int_{-% \infty}^{\infty}e^{-{1\over 2}ax^{2}}\,dx=-2{d\over da}\left({2\pi\over a}% \right)^{1\over 2}=\left({2\pi\over a}\right)^{1\over 2}{1\over a}
  7. - x 4 e - 1 2 a x 2 d x = ( - 2 d d a ) ( - 2 d d a ) - e - 1 2 a x 2 d x = ( - 2 d d a ) ( - 2 d d a ) ( 2 π a ) 1 2 = ( 2 π a ) 1 2 3 a 2 \int_{-\infty}^{\infty}x^{4}e^{-{1\over 2}ax^{2}}\,dx=\left(-2{d\over da}% \right)\left(-2{d\over da}\right)\int_{-\infty}^{\infty}e^{-{1\over 2}ax^{2}}% \,dx=\left(-2{d\over da}\right)\left(-2{d\over da}\right)\left({2\pi\over a}% \right)^{1\over 2}=\left({2\pi\over a}\right)^{1\over 2}{3\over a^{2}}
  8. - x 2 n e - 1 2 a x 2 d x = ( 2 π a ) 1 2 1 a n ( 2 n - 1 ) ( 2 n - 3 ) 5 3 1 = ( 2 π a ) 1 2 1 a n ( 2 n - 1 ) ! ! \int_{-\infty}^{\infty}x^{2n}e^{-{1\over 2}ax^{2}}\,dx=\left({2\pi\over a}% \right)^{1\over{2}}{1\over a^{n}}\left(2n-1\right)\left(2n-3\right)\cdots 5% \cdot 3\cdot 1=\left({2\pi\over a}\right)^{1\over{2}}{1\over a^{n}}\left(2n-1% \right)!!
  9. - exp ( - 1 2 a x 2 + J x ) d x \int_{-\infty}^{\infty}\exp\left(-{1\over 2}ax^{2}+Jx\right)dx
  10. ( - 1 2 a x 2 + J x ) = - 1 2 a ( x 2 - 2 J x a + J 2 a 2 - J 2 a 2 ) = - 1 2 a ( x - J a ) 2 + J 2 2 a \left(-{1\over 2}ax^{2}+Jx\right)=-{1\over 2}a\left(x^{2}-{2Jx\over a}+{J^{2}% \over a^{2}}-{J^{2}\over a^{2}}\right)=-{1\over 2}a\left(x-{J\over a}\right)^{% 2}+{J^{2}\over 2a}
  11. - exp ( - 1 2 a x 2 + J x ) d x \displaystyle\int_{-\infty}^{\infty}\exp\left(-{1\over 2}ax^{2}+Jx\right)\,dx
  12. - exp ( - 1 2 a x 2 + i J x ) d x = ( 2 π a ) 1 2 exp ( - J 2 2 a ) \int_{-\infty}^{\infty}\exp\left(-{1\over 2}ax^{2}+iJx\right)dx=\left({2\pi% \over a}\right)^{1\over 2}\exp\left(-{J^{2}\over 2a}\right)
  13. J J
  14. x x
  15. a a
  16. x x
  17. J J
  18. - exp ( 1 2 i a x 2 + i J x ) d x . \int_{-\infty}^{\infty}\exp\left({1\over 2}iax^{2}+iJx\right)dx.
  19. a a
  20. J J
  21. ( 1 2 i a x 2 + i J x ) = 1 2 i a ( x 2 + 2 J x a + ( J a ) 2 - ( J a ) 2 ) = - 1 2 a i ( x + J a ) 2 - i J 2 2 a . \left({1\over 2}iax^{2}+iJx\right)={1\over 2}ia\left(x^{2}+{2Jx\over a}+\left(% {J\over a}\right)^{2}-\left({J\over a}\right)^{2}\right)=-{1\over 2}{a\over i}% \left(x+{J\over a}\right)^{2}-{iJ^{2}\over 2a}.
  22. - exp ( 1 2 i a x 2 + i J x ) d x = ( 2 π i a ) 1 2 exp ( - i J 2 2 a ) . \int_{-\infty}^{\infty}\exp\left({1\over 2}iax^{2}+iJx\right)dx=\left({2\pi i% \over a}\right)^{1\over 2}\exp\left({-iJ^{2}\over 2a}\right).
  23. a a
  24. exp ( - 1 2 x A x + J x ) d n x = ( 2 π ) n det A exp ( 1 2 J A - 1 J ) \int\exp\left(-\frac{1}{2}x\cdot A\cdot x+J\cdot x\right)d^{n}x=\sqrt{\frac{(2% \pi)^{n}}{\det A}}\exp\left({1\over 2}J\cdot A^{-1}\cdot J\right)
  25. A A
  26. A A
  27. D = O - 1 A O = O T A O D=O^{-1}AO=O^{T}AO
  28. D D
  29. O O
  30. n n
  31. exp ( - 1 2 A i j x i x j ) d 2 x = ( 2 π ) 2 det A \int\exp\left(-\frac{1}{2}A_{ij}x^{i}x^{j}\right)d^{2}x=\sqrt{\frac{(2\pi)^{2}% }{\det A}}
  32. A A
  33. A = [ a c c b ] A=\begin{bmatrix}a&c\\ c&b\end{bmatrix}
  34. A i j x i x j x T A x = x T ( O O T ) A ( O O T ) x = ( x T O ) ( O T A O ) ( O T x ) A_{ij}x^{i}x^{j}\equiv x^{T}Ax=x^{T}\left(OO^{T}\right)A\left(OO^{T}\right)x=% \left(x^{T}O\right)\left(O^{T}AO\right)\left(O^{T}x\right)
  35. A A
  36. O O
  37. O O
  38. A A
  39. O O
  40. A A
  41. λ λ
  42. A A
  43. [ a c c b ] [ u v ] = λ [ u v ] . \begin{bmatrix}a&c\\ c&b\end{bmatrix}\begin{bmatrix}u\\ v\end{bmatrix}=\lambda\begin{bmatrix}u\\ v\end{bmatrix}.
  44. ( a - λ ) ( b - λ ) - c 2 = 0 (a-\lambda)(b-\lambda)-c^{2}=0
  45. λ ± = 1 2 ( a + b ) ± 1 2 ( a - b ) 2 + 4 c 2 . \lambda_{\pm}={1\over 2}(a+b)\pm{1\over 2}\sqrt{(a-b)^{2}+4c^{2}}.
  46. v = - ( a - λ ± ) u c , v = - c u ( b - λ ± ) . v=-{\left(a-\lambda_{\pm}\right)u\over c},\qquad v=-{cu\over\left(b-\lambda_{% \pm}\right)}.
  47. a - λ ± c = c b - λ ± . {a-\lambda_{\pm}\over c}={c\over b-\lambda_{\pm}}.
  48. a - λ ± c = - b - λ c . {a-\lambda_{\pm}\over c}=-{b-\lambda_{\mp}\over c}.
  49. [ 1 η - a - λ - c η ] , [ - b - λ + c η 1 η ] \begin{bmatrix}\frac{1}{\eta}\\ -\frac{a-\lambda_{-}}{c\eta}\end{bmatrix},\qquad\begin{bmatrix}-\frac{b-% \lambda_{+}}{c\eta}\\ \frac{1}{\eta}\end{bmatrix}
  50. η η
  51. η = 1 + ( a - λ - c ) 2 = 1 + ( b - λ + c ) 2 . \eta=\sqrt{1+\left(\frac{a-\lambda_{-}}{c}\right)^{2}}=\sqrt{1+\left(\frac{b-% \lambda_{+}}{c}\right)^{2}}.
  52. O = [ 1 η - b - λ + c η - a - λ - c η 1 η ] . O=\begin{bmatrix}\frac{1}{\eta}&-\frac{b-\lambda_{+}}{c\eta}\\ -\frac{a-\lambda_{-}}{c\eta}&\frac{1}{\eta}\end{bmatrix}.
  53. d e t ( O ) = 1 det(O)=1
  54. sin ( θ ) = - a - λ - c η \sin(\theta)=-\frac{a-\lambda_{-}}{c\eta}
  55. O = [ cos ( θ ) - sin ( θ ) sin ( θ ) cos ( θ ) ] O=\begin{bmatrix}\cos(\theta)&-\sin(\theta)\\ \sin(\theta)&\cos(\theta)\end{bmatrix}
  56. O - 1 = O T = [ cos ( θ ) sin ( θ ) - sin ( θ ) cos ( θ ) ] . O^{-1}=O^{T}=\begin{bmatrix}\cos(\theta)&\sin(\theta)\\ -\sin(\theta)&\cos(\theta)\end{bmatrix}.
  57. D = O T A O = [ λ - 0 0 λ + ] D=O^{T}AO=\begin{bmatrix}\lambda_{-}&0\\ 0&\lambda_{+}\end{bmatrix}
  58. [ 1 0 ] , [ 0 1 ] \begin{bmatrix}1\\ 0\end{bmatrix},\qquad\begin{bmatrix}0\\ 1\end{bmatrix}
  59. A = [ 2 1 1 1 ] A=\begin{bmatrix}2&1\\ 1&1\end{bmatrix}
  60. λ ± = 3 2 ± 5 2 . \lambda_{\pm}={3\over 2}\pm{\sqrt{5}\over 2}.
  61. 1 η [ 1 - 1 2 - 5 2 ] , 1 η [ 1 2 + 5 2 1 ] {1\over\eta}\begin{bmatrix}1\\ -{1\over 2}-{\sqrt{5}\over 2}\end{bmatrix},\qquad{1\over\eta}\begin{bmatrix}{1% \over 2}+{\sqrt{5}\over 2}\\ 1\end{bmatrix}
  62. η = 5 2 + 5 2 . \eta=\sqrt{{5\over 2}+{\sqrt{5}\over 2}}.
  63. O = [ 1 η 1 η ( 1 2 + 5 2 ) 1 η ( - 1 2 - 5 2 ) 1 η ] O - 1 = [ 1 η 1 η ( - 1 2 - 5 2 ) 1 η ( 1 2 + 5 2 ) 1 η ] \begin{aligned}\displaystyle O&\displaystyle=\begin{bmatrix}\frac{1}{\eta}&% \frac{1}{\eta}\left({1\over 2}+{\sqrt{5}\over 2}\right)\\ \frac{1}{\eta}\left(-{1\over 2}-{\sqrt{5}\over 2}\right)&{1\over\eta}\end{% bmatrix}\\ \displaystyle O^{-1}&\displaystyle=\begin{bmatrix}\frac{1}{\eta}&\frac{1}{\eta% }\left(-{1\over 2}-{\sqrt{5}\over 2}\right)\\ \frac{1}{\eta}\left({1\over 2}+{\sqrt{5}\over 2}\right)&\frac{1}{\eta}\end{% bmatrix}\end{aligned}
  64. D = O T A O = [ λ - 0 0 λ + ] = [ 3 2 - 5 2 0 0 3 2 + 5 2 ] D=O^{T}AO=\begin{bmatrix}\lambda_{-}&0\\ 0&\lambda_{+}\end{bmatrix}=\begin{bmatrix}{3\over 2}-{\sqrt{5}\over 2}&0\\ 0&{3\over 2}+{\sqrt{5}\over 2}\end{bmatrix}
  65. [ 1 0 ] , [ 0 1 ] \begin{bmatrix}1\\ 0\end{bmatrix},\qquad\begin{bmatrix}0\\ 1\end{bmatrix}
  66. exp ( - 1 2 x T A x ) d 2 x = exp ( - 1 2 j = 1 2 λ j y j 2 ) d 2 y \int\exp\left(-\frac{1}{2}x^{T}Ax\right)d^{2}x=\int\exp\left(-\frac{1}{2}\sum_% {j=1}^{2}\lambda_{j}y_{j}^{2}\right)\,d^{2}y
  67. y = O T x . y=O^{T}x.
  68. d y 2 = d x 2 dy^{2}=dx^{2}
  69. exp ( - 1 2 x T A x ) d 2 x = exp ( - 1 2 j = 1 2 λ j y j 2 ) d 2 y = j = 1 2 ( 2 π λ j ) 1 2 = ( ( 2 π ) 2 j = 1 2 λ j ) 1 2 = ( ( 2 π ) 2 det ( O - 1 A O ) ) 1 2 = ( ( 2 π ) 2 det ( A ) ) 1 2 \begin{aligned}\displaystyle\int\exp\left(-\frac{1}{2}x^{T}Ax\right)d^{2}x&% \displaystyle=\int\exp\left(-\frac{1}{2}\sum_{j=1}^{2}\lambda_{j}y_{j}^{2}% \right)d^{2}y\\ &\displaystyle=\prod_{j=1}^{2}\left({2\pi\over\lambda_{j}}\right)^{1\over 2}\\ &\displaystyle=\left({(2\pi)^{2}\over\prod_{j=1}^{2}\lambda_{j}}\right)^{1% \over 2}\\ &\displaystyle=\left({(2\pi)^{2}\over\det{\left(O^{-1}AO\right)}}\right)^{1% \over 2}\\ &\displaystyle=\left({(2\pi)^{2}\over\det{\left(A\right)}}\right)^{1\over 2}% \end{aligned}
  70. exp ( - 1 2 x A x + J x ) d n x = ( 2 π ) n det A exp ( 1 2 J A - 1 J ) \int\exp\left(-\frac{1}{2}x\cdot A\cdot x+J\cdot x\right)d^{n}x=\sqrt{\frac{(2% \pi)^{n}}{\det A}}\exp\left({1\over 2}J\cdot A^{-1}\cdot J\right)
  71. exp ( - 1 2 x A x + i J x ) d n x = ( 2 π ) n det A exp ( - 1 2 J A - 1 J ) \int\exp\left(-\frac{1}{2}x\cdot A\cdot x+iJ\cdot x\right)d^{n}x=\sqrt{\frac{(% 2\pi)^{n}}{\det A}}\exp\left(-{1\over 2}J\cdot A^{-1}\cdot J\right)
  72. exp ( - i 2 x A x + i J x ) d n x = ( 2 π i ) n det A exp ( - i 2 J A - 1 J ) \int\exp\left(-\frac{i}{2}x\cdot A\cdot x+iJ\cdot x\right)d^{n}x=\sqrt{\frac{(% 2\pi i)^{n}}{\det A}}\exp\left(-{i\over 2}J\cdot A^{-1}\cdot J\right)
  73. exp [ d 4 x ( - 1 2 φ A ^ φ + J φ ) ] D φ \int\exp\left[\int d^{4}x\left(-\frac{1}{2}\varphi\hat{A}\varphi+J\varphi% \right)\right]D\varphi
  74. A ^ \hat{A}
  75. φ \varphi
  76. J J
  77. D φ D\varphi
  78. exp ( - 1 2 φ A ^ φ + J φ ) D φ exp ( 1 2 d 4 x d 4 y J ( x ) D ( x - y ) J ( y ) ) \int\exp\left(-\frac{1}{2}\varphi\hat{A}\varphi+J\varphi\right)D\varphi\;% \propto\;\exp\left({1\over 2}\int d^{4}x\;d^{4}yJ(x)D(x-y)J(y)\right)
  79. A ^ D ( x - y ) = δ 4 ( x - y ) \hat{A}D(x-y)=\delta^{4}(x-y)
  80. D ( x y ) D(x−y)
  81. A ^ \hat{A}
  82. δ 4 ( x - y ) \delta^{4}(x-y)
  83. exp [ d 4 x ( - 1 2 φ A ^ φ + i J φ ) ] D φ exp ( - 1 2 d 4 x d 4 y J ( x ) D ( x - y ) J ( y ) ) , \int\exp\left[\int d^{4}x\left(-\frac{1}{2}\varphi\hat{A}\varphi+iJ\varphi% \right)\right]D\varphi\;\propto\;\exp\left(-{1\over 2}\int d^{4}x\;d^{4}yJ(x)D% (x-y)J(y)\right),
  84. exp [ i d 4 x ( 1 2 φ A ^ φ + J φ ) ] D φ exp ( i 2 d 4 x d 4 y J ( x ) D ( x - y ) J ( y ) ) . \int\exp\left[i\int d^{4}x\left(\frac{1}{2}\varphi\hat{A}\varphi+J\varphi% \right)\right]D\varphi\;\propto\;\exp\left({i\over 2}\int d^{4}x\;d^{4}yJ(x)D(% x-y)J(y)\right).
  85. - exp ( - 1 f ( q ) ) d n q \int_{-\infty}^{\infty}\exp\left(-{1\over\hbar}f(q)\right)d^{n}q
  86. \hbar
  87. q = q 0 q=q_{0}
  88. - exp [ - 1 ( f ( q 0 ) + 1 2 ( q - q 0 ) 2 f ′′ ( q - q 0 ) + ) ] d n q \int_{-\infty}^{\infty}\exp\left[-{1\over\hbar}\left(f\left(q_{0}\right)+{1% \over 2}\left(q-q_{0}\right)^{2}f^{\prime\prime}\left(q-q_{0}\right)+\cdots% \right)\right]d^{n}q
  89. f ′′ f^{\prime\prime}
  90. - exp [ - 1 ( f ( q ) ) ] d n q exp [ - 1 ( f ( q 0 ) ) ] ( 2 π ) n det f ′′ . \int_{-\infty}^{\infty}\exp\left[-{1\over\hbar}(f(q))\right]d^{n}q\approx\exp% \left[-{1\over\hbar}\left(f\left(q_{0}\right)\right)\right]\sqrt{(2\pi\hbar)^{% n}\over\det f^{\prime\prime}}.
  91. exp ( i S ( q , q ˙ ) ) D q \int\exp\left({i\over\hbar}S\left(q,\dot{q}\right)\right)Dq
  92. S ( q , q ˙ ) S\left(q,\dot{q}\right)
  93. \hbar
  94. d 4 k ( 2 π ) 4 exp ( i k ( x - y ) ) = δ 4 ( x - y ) . \int\frac{d^{4}k}{(2\pi)^{4}}\exp(ik(x-y))=\delta^{4}(x-y).
  95. N N
  96. d N k ( 2 π ) N exp ( i k ( x - y ) ) = δ N ( x - y ) . \int\frac{d^{N}k}{(2\pi)^{N}}\exp(ik(x-y))=\delta^{N}(x-y).
  97. - 1 4 π 2 ( 1 r ) = δ ( 𝐫 ) -{1\over 4\pi}\nabla^{2}\left({1\over r}\right)=\delta\left(\mathbf{r}\right)
  98. r 2 = 𝐫 𝐫 r^{2}=\mathbf{r}\cdot\mathbf{r}
  99. d 3 k ( 2 π ) 3 exp ( i 𝐤 𝐫 ) k 2 = 1 4 π r . \int\frac{d^{3}k}{(2\pi)^{3}}{\exp\left(i\mathbf{k}\cdot\mathbf{r}\right)\over k% ^{2}}={1\over 4\pi r}.
  100. d 3 k ( 2 π ) 3 exp ( i 𝐤 𝐫 ) k 2 + m 2 = e - m r 4 π r \int\frac{d^{3}k}{(2\pi)^{3}}{\exp\left(i\mathbf{k}\cdot\mathbf{r}\right)\over k% ^{2}+m^{2}}={e^{-mr}\over 4\pi r}
  101. r 2 = 𝐫 𝐫 , k 2 = 𝐤 𝐤 . r^{2}=\mathbf{r}\cdot\mathbf{r},\qquad k^{2}=\mathbf{k}\cdot\mathbf{k}.
  102. 1 4 π r 1\frac{4}{πr}
  103. d 3 k ( 2 π ) 3 exp ( i 𝐤 𝐫 ) k 2 + m 2 \displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\exp\left(i\mathbf{k}\cdot% \mathbf{r}\right)}{k^{2}+m^{2}}
  104. d 3 k ( 2 π ) 3 ( 𝐤 ^ 𝐫 ^ ) 2 exp ( i 𝐤 𝐫 ) k 2 + m 2 = e - m r 4 π r { 1 + 2 m r - 2 ( m r ) 2 ( e m r - 1 ) } \int\frac{d^{3}k}{(2\pi)^{3}}\left(\mathbf{\hat{k}}\cdot\mathbf{\hat{r}}\right% )^{2}\frac{\exp\left(i\mathbf{k}\cdot\mathbf{r}\right)}{k^{2}+m^{2}}=\frac{e^{% -mr}}{4\pi r}\left\{1+\frac{2}{mr}-\frac{2}{(mr)^{2}}\left(e^{mr}-1\right)\right\}
  105. d 3 k ( 2 π ) 3 ( 𝐤 ^ 𝐫 ^ ) 2 exp ( i 𝐤 𝐫 ) k 2 + m 2 \displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\left(\mathbf{\hat{k}}\cdot\mathbf{% \hat{r}}\right)^{2}\frac{\exp\left(i\mathbf{k}\cdot\mathbf{r}\right)}{k^{2}+m^% {2}}
  106. m m
  107. 1 1
  108. d 3 k ( 2 π ) 3 𝐤 ^ 𝐤 ^ exp ( i 𝐤 𝐫 ) k 2 + m 2 = 1 2 e - m r 4 π r ( [ 𝟏 - 𝐫 ^ 𝐫 ^ ] + { 1 + 2 m r - 2 ( m r ) 2 ( e m r - 1 ) } [ 𝟏 + 𝐫 ^ 𝐫 ^ ] ) \int\frac{d^{3}k}{(2\pi)^{3}}\mathbf{\hat{k}}\mathbf{\hat{k}}\frac{\exp\left(i% \mathbf{k}\cdot\mathbf{r}\right)}{k^{2}+m^{2}}={1\over 2}\frac{e^{-mr}}{4\pi r% }\left(\left[\mathbf{1}-\mathbf{\hat{r}}\mathbf{\hat{r}}\right]+\left\{1+\frac% {2}{mr}-{2\over(mr)^{2}}\left(e^{mr}-1\right)\right\}\left[\mathbf{1}+\mathbf{% \hat{r}}\mathbf{\hat{r}}\right]\right)
  109. d 3 k ( 2 π ) 3 𝐤 ^ 𝐤 ^ exp ( i 𝐤 𝐫 ) k 2 + m 2 \displaystyle\int\frac{d^{3}k}{(2\pi)^{3}}\mathbf{\hat{k}}\mathbf{\hat{k}}% \frac{\exp\left(i\mathbf{k}\cdot\mathbf{r}\right)}{k^{2}+m^{2}}
  110. m m
  111. 1 2 1 4 π r [ 𝟏 - 𝐫 ^ 𝐫 ^ ] . {1\over 2}{1\over 4\pi r}\left[\mathbf{1}-\mathbf{\hat{r}}\mathbf{\hat{r}}% \right].
  112. d 3 k ( 2 π ) 3 [ 𝟏 - 𝐤 ^ 𝐤 ^ ] exp ( i 𝐤 𝐫 ) k 2 + m 2 = 1 2 e - m r 4 π r { 2 ( m r ) 2 ( e m r - 1 ) - 2 m r } [ 𝟏 + 𝐫 ^ 𝐫 ^ ] \int\frac{d^{3}k}{(2\pi)^{3}}\left[\mathbf{1}-\mathbf{\hat{k}}\mathbf{\hat{k}}% \right]{\exp\left(i\mathbf{k}\cdot\mathbf{r}\right)\over k^{2}+m^{2}}={1\over 2% }{e^{-mr}\over 4\pi r}\left\{{2\over(mr)^{2}}\left(e^{mr}-1\right)-{2\over mr}% \right\}\left[\mathbf{1}+\mathbf{\hat{r}}\mathbf{\hat{r}}\right]
  113. 1 2 1 4 π r [ 𝟏 + 𝐫 ^ 𝐫 ^ ] . {1\over 2}{1\over 4\pi r}\left[\mathbf{1}+\mathbf{\hat{r}}\mathbf{\hat{r}}% \right].
  114. 1 4 π m 2 r 3 [ 𝟏 + 𝐫 ^ 𝐫 ^ ] . \frac{1}{4\pi m^{2}r^{3}}\left[\mathbf{1}+\mathbf{\hat{r}}\mathbf{\hat{r}}% \right].
  115. 0 2 π d φ 2 π exp ( i p cos ( φ ) ) = J 0 ( p ) \int_{0}^{2\pi}{d\varphi\over 2\pi}\exp\left(ip\cos(\varphi)\right)=J_{0}(p)
  116. 0 2 π d φ 2 π cos ( φ ) exp ( i p cos ( φ ) ) = i J 1 ( p ) . \int_{0}^{2\pi}{d\varphi\over 2\pi}\cos(\varphi)\exp\left(ip\cos(\varphi)% \right)=iJ_{1}(p).
  117. 0 k d k k 2 + m 2 J 0 ( k r ) = K 0 ( m r ) . \int_{0}^{\infty}{k\;dk\over k^{2}+m^{2}}J_{0}\left(kr\right)=K_{0}(mr).
  118. m r 1 mr<<1
  119. K 0 ( m r ) - ln ( m r 2 ) + 0.5772. K_{0}(mr)\to-\ln\left({mr\over 2}\right)+0.5772.
  120. 0 k d k k 2 + m 2 J 1 2 ( k r ) = I 1 ( m r ) K 1 ( m r ) . \int_{0}^{\infty}{k\;dk\over k^{2}+m^{2}}J_{1}^{2}(kr)=I_{1}(mr)K_{1}(mr).
  121. o k d k k 2 + m 2 J 1 2 ( k r ) 1 2 [ 1 - 1 8 ( m r ) 2 ] . \int_{o}^{\infty}{k\;dk\over k^{2}+m^{2}}J_{1}^{2}(kr)\to{1\over 2}\left[1-{1% \over 8}(mr)^{2}\right].
  122. o k d k k 2 + m 2 J 1 2 ( k r ) 1 2 ( 1 m r ) . \int_{o}^{\infty}{k\;dk\over k^{2}+m^{2}}J_{1}^{2}(kr)\to{1\over 2}\left({1% \over mr}\right).
  123. 0 k d k k 2 + m 2 J ν 2 ( k r ) = I ν ( m r ) K ν ( m r ) ( ν ) > - 1. \int_{0}^{\infty}{k\;dk\over k^{2}+m^{2}}J_{\nu}^{2}(kr)=I_{\nu}(mr)K_{\nu}(mr% )\qquad\Re(\nu)>-1.
  124. 2 a 2 n + 2 n ! 0 d r r 2 n + 1 exp ( - a 2 r 2 ) J 0 ( k r ) = M ( n + 1 , 1 , - k 2 4 a 2 ) . {2a^{2n+2}\over n!}\int_{0}^{\infty}{dr}\;r^{2n+1}\exp\left(-a^{2}r^{2}\right)% J_{0}(kr)=M\left(n+1,1,-{k^{2}\over 4a^{2}}\right).