wpmath0000009_13

Tricorn_(mathematics).html

  1. z z ¯ 2 + c z\mapsto\bar{z}^{2}+c
  2. z z 2 + c z\mapsto z^{2}+c

Trigonometric_moment_problem.html

  1. α k = 1 2 π 0 2 π e - i k t d μ ( t ) . \alpha_{k}=\frac{1}{2\pi}\int_{0}^{2\pi}e^{-ikt}\,d\mu(t).
  2. A = ( α 0 α 1 α n α 1 ¯ α 0 α n - 1 α n ¯ α n - 1 ¯ α 0 ) A=\left(\begin{matrix}\alpha_{0}&\alpha_{1}&\cdots&\alpha_{n}\\ \bar{\alpha_{1}}&\alpha_{0}&\cdots&\alpha_{n-1}\\ \vdots&\vdots&\ddots&\vdots\\ \bar{\alpha_{n}}&\bar{\alpha_{n-1}}&\cdots&\alpha_{0}\\ \end{matrix}\right)
  3. ( , , ) (\mathcal{H},\langle\;,\;\rangle)
  4. \mathcal{H}
  5. \mathcal{E}
  6. \mathcal{F}
  7. V : V:\mathcal{E}\rightarrow\mathcal{F}
  8. V [ e k ] = [ e k + 1 ] for k = 0 n - 1. V[e_{k}]=[e_{k+1}]\quad\mbox{for}~{}\quad k=0\ldots n-1.
  9. V [ e j ] , V [ e k ] = [ e j + 1 ] , [ e k + 1 ] = A j + 1 , k + 1 = A j , k = [ e j + 1 ] , [ e k + 1 ] , \langle V[e_{j}],V[e_{k}]\rangle=\langle[e_{j+1}],[e_{k+1}]\rangle=A_{j+1,k+1}% =A_{j,k}=\langle[e_{j+1}],[e_{k+1}]\rangle,
  10. \mathcal{H}
  11. ( U * ) k [ e n + 1 ] , [ e n + 1 ] = 𝐓 z k d m . \langle(U^{*})^{k}[e_{n+1}],[e_{n+1}]\rangle=\int_{\mathbf{T}}z^{k}dm.
  12. ( U * ) k [ e n + 1 ] , [ e n + 1 ] = ( V * ) k [ e n + 1 ] , [ e n + 1 ] = [ e n + 1 - k ] , [ e n + 1 ] = A n + 1 , n + 1 - k = α k ¯ . \langle(U^{*})^{k}[e_{n+1}],[e_{n+1}]\rangle=\langle(V^{*})^{k}[e_{n+1}],[e_{n% +1}]\rangle=\langle[e_{n+1-k}],[e_{n+1}]\rangle=A_{n+1,n+1-k}=\bar{\alpha_{k}}.
  13. 𝐓 z - k d m = 𝐓 z ¯ k d m = α k . \int_{\mathbf{T}}z^{-k}dm=\int_{\mathbf{T}}\bar{z}^{k}dm=\alpha_{k}.
  14. 1 2 π 0 2 π e - i k t d μ ( t ) = α k \frac{1}{2\pi}\int_{0}^{2\pi}e^{-ikt}d\mu(t)=\alpha_{k}

Trimaximal_mixing.html

  1. 3 × 3 3\times 3
  2. U U
  3. | U a i | = 1 / 3 |U_{ai}|=1/\sqrt{3}
  4. a , i = 1 , 2 , 3 a,i=1,2,3
  5. U = [ 1 3 1 3 1 3 ω 3 1 3 ω ¯ 3 ω ¯ 3 1 3 ω 3 ] ( | U i α | 2 ) = [ 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 1 3 ] U=\begin{bmatrix}\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{1}{\sqrt{3}}\\ \frac{\omega}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{\bar{\omega}}{\sqrt{3}}\\ \frac{\bar{\omega}}{\sqrt{3}}&\frac{1}{\sqrt{3}}&\frac{\omega}{\sqrt{3}}\end{% bmatrix}\Rightarrow(|U_{i\alpha}|^{2})=\begin{bmatrix}\frac{1}{3}&\frac{1}{3}&% \frac{1}{3}\\ \frac{1}{3}&\frac{1}{3}&\frac{1}{3}\\ \frac{1}{3}&\frac{1}{3}&\frac{1}{3}\end{bmatrix}
  6. ω = exp ( i 2 π / 3 ) \omega=\exp(i2\pi/3)
  7. ω ¯ = exp ( - i 2 π / 3 ) \bar{\omega}=\exp(-i2\pi/3)
  8. θ 12 = θ 23 = π / 4 \theta_{12}=\theta_{23}=\pi/4
  9. θ 13 = sin - 1 ( 1 / 3 ) \theta_{13}=\sin^{-1}(1/\sqrt{3})
  10. δ = π / 2 \delta=\pi/2
  11. C P CP
  12. J J
  13. | J | = 1 / ( 6 3 ) |J|=1/(6\sqrt{3})

Trimean.html

  1. T M = Q 1 + 2 Q 2 + Q 3 4 TM=\frac{Q_{1}+2Q_{2}+Q_{3}}{4}
  2. T M = 1 2 ( Q 2 + Q 1 + Q 3 2 ) TM=\frac{1}{2}\left(Q_{2}+\frac{Q_{1}+Q_{3}}{2}\right)

Triple_correlation.html

  1. - f * ( x ) f ( x + s 1 ) f ( x + s 2 ) d x \int_{-\infty}^{\infty}f^{*}(x)f(x+s_{1})f(x+s_{2})dx
  2. L 1 ( G ) L_{1}(G)

Triple_modular_redundancy.html

  1. x y y z x z xy\lor yz\lor xz

Triple_product_property.html

  1. G G
  2. S , T , U G S,T,U\subset G
  3. G G
  4. s , s S s,s^{\prime}\in S
  5. t , t T t,t^{\prime}\in T
  6. u , u U u,u^{\prime}\in U
  7. s s - 1 t t - 1 u u - 1 = 1 s = s , t = t , u = u s^{\prime}s^{-1}t^{\prime}t^{-1}u^{\prime}u^{-1}=1\Rightarrow s^{\prime}=s,t^{% \prime}=t,u^{\prime}=u
  8. 1 1
  9. G G

Trisops.html

  1. × B = α B v = ± β B \begin{aligned}\displaystyle\vec{\nabla}\times\vec{B}=\alpha\vec{B}\\ \displaystyle\vec{v}=\pm\beta\vec{B}\end{aligned}
  2. j × B \vec{j}\times\vec{B}
  3. ρ × v \rho\vec{\nabla}\times\vec{v}
  4. ρ \rho

Trivialism.html

  1. p T p \forall pTp
  2. p T p p\leftrightarrow Tp

Trophic_level.html

  1. T L i = 1 + j ( T L j D C i j ) TL_{i}=1+\sum_{j}(TL_{j}\cdot DC_{ij})\!
  2. T L j TL_{j}
  3. D C i j DC_{ij}
  4. T L y = i ( T L i Y i y ) i Y i y TL_{y}=\frac{\sum_{i}(TL_{i}\cdot Y_{iy})}{\sum_{i}Y_{iy}}
  5. Y i y Y_{iy}
  6. T L i TL_{i}
  7. F i B y = log Y y / ( T E ) T L y Y 0 / ( T E ) T L 0 FiB_{y}=\log\frac{Y_{y}/(TE)^{TL_{y}}}{Y_{0}/(TE)^{TL_{0}}}
  8. Y y Y_{y}
  9. T L y {TL}_{y}
  10. Y 0 Y_{0}
  11. T L 0 {TL}_{0}
  12. T E TE

Tropical_cyclone.html

  1. M M
  2. M = 1 2 f r 2 + v r M=\frac{1}{2}fr^{2}+vr
  3. f f
  4. v v
  5. r r
  6. f = 0 f=0
  7. v p v_{p}
  8. v p 2 = C k C d T s - T o T o Δ k v_{p}^{2}=\frac{C_{k}}{C_{d}}\frac{T_{s}-T_{o}}{T_{o}}\Delta k
  9. T s T_{s}
  10. T o T_{o}
  11. Δ k \Delta k
  12. C k C_{k}
  13. C d C_{d}
  14. Δ k = k s * - k \Delta k=k^{*}_{s}-k
  15. k s * k^{*}_{s}
  16. k k
  17. Δ k \Delta k
  18. W i n = W o u t W_{in}=W_{out}
  19. W o u t W_{out}
  20. W o u t = C d ρ | 𝐮 | 3 W_{out}=C_{d}\rho|\mathbf{u}|^{3}
  21. ρ \rho
  22. | 𝐮 | |\mathbf{u}|
  23. W i n W_{in}
  24. W i n = ϵ Q i n W_{in}=\epsilon Q_{in}
  25. ϵ \epsilon
  26. Q i n Q_{in}
  27. ϵ = T s - T o T s \epsilon=\frac{T_{s}-T_{o}}{T_{s}}
  28. k = C p T + L v q k=C_{p}T+L_{v}q
  29. C p C_{p}
  30. T T
  31. L v L_{v}
  32. q q
  33. Q i n : k Q_{in:k}
  34. Q i n : k = C k ρ | 𝐮 | Δ k Q_{in:k}=C_{k}\rho|\mathbf{u}|\Delta k
  35. Δ k = k s * - k \Delta k=k^{*}_{s}-k
  36. W o u t W_{out}
  37. Q i n : f r i c t i o n = C d ρ | 𝐮 | 3 Q_{in:friction}=C_{d}\rho|\mathbf{u}|^{3}
  38. W i n = T s - T o T s ( C k ρ | 𝐮 | Δ k + C d ρ | 𝐮 | 3 ) W_{in}=\frac{T_{s}-T_{o}}{T_{s}}\left(C_{k}\rho|\mathbf{u}|\Delta k+C_{d}\rho|% \mathbf{u}|^{3}\right)
  39. W i n = W o u t W_{in}=W_{out}
  40. | 𝐮 | v |\mathbf{u}|\approx v
  41. v p v_{p}
  42. Q i n : f r i c t i o n Q_{in:friction}
  43. T s T o \frac{T_{s}}{T_{o}}
  44. T s T_{s}
  45. T o T_{o}
  46. v p = T s T o C k C d ( C A P E s * - C A P E b ) | m v_{p}=\sqrt{\frac{T_{s}}{T_{o}}\frac{C_{k}}{C_{d}}(CAPE^{*}_{s}-CAPE_{b})|_{m}}
  47. C A P E s * CAPE^{*}_{s}
  48. C A P E b CAPE_{b}
  49. T s T_{s}
  50. T o T_{o}
  51. ϵ = 1 / 3 \epsilon=1/3
  52. C k / C d C_{k}/C_{d}
  53. C d C_{d}
  54. C k C_{k}
  55. v p v_{p}
  56. Δ k \Delta k
  57. v p v_{p}

Tropical_cyclone_forecasting.html

  1. V = A + B e C ( T - T 0 ) V=A+B\cdot e^{C(T-T_{0})}
  2. V V
  3. T T
  4. T 0 T_{0}
  5. A A
  6. B B
  7. C C
  8. A = 28.2 A=28.2
  9. B = 55.8 B=55.8
  10. C = 0.1813 C=0.1813

TrueSkill.html

  1. 𝒩 \mathcal{N}
  2. μ \mu
  3. σ \sigma
  4. μ \mu
  5. 𝒩 ( x ) \mathcal{N}(x)
  6. x x
  7. μ = 25 \mu=25
  8. σ = 25 / 3 \sigma=25/3
  9. μ \mu
  10. σ \sigma
  11. ( μ , σ ) (\mu,\sigma)
  12. R = μ - 3 × σ R=\mu-3\times\sigma
  13. R = 25 - 3 25 3 = 0 R=25-3\cdot\frac{25}{3}=0

Truncated_normal_distribution.html

  1. X N ( μ , σ 2 ) X\sim N(\mu,\sigma^{2})\!
  2. X ( a , b ) , - a < b X\in(a,b),\;-\infty\leq a<b\leq\infty
  3. X X
  4. a < X < b a<X<b
  5. a x b a\leq x\leq b
  6. f ( x ; μ , σ , a , b ) = 1 σ ϕ ( x - μ σ ) Φ ( b - μ σ ) - Φ ( a - μ σ ) f(x;\mu,\sigma,a,b)=\frac{\frac{1}{\sigma}\phi(\frac{x-\mu}{\sigma})}{\Phi(% \frac{b-\mu}{\sigma})-\Phi(\frac{a-\mu}{\sigma})}
  7. ϕ ( ξ ) = 1 2 π exp ( - 1 2 ξ 2 ) \scriptstyle{\phi(\xi)=\frac{1}{\sqrt{2\pi}}\exp{(-\frac{1}{2}\xi^{2}})}
  8. Φ ( ) \scriptstyle{\Phi(\cdot)}
  9. b = \scriptstyle{b=\infty}
  10. Φ ( b - μ σ ) = 1 \scriptstyle{\Phi(\frac{b-\mu}{\sigma})=1}
  11. a = - \scriptstyle{a=-\infty}
  12. Φ ( a - μ σ ) = 0 \scriptstyle{\Phi(\frac{a-\mu}{\sigma})=0}
  13. E ( X a < X < b ) = μ + ϕ ( a - μ σ ) - ϕ ( b - μ σ ) Φ ( b - μ σ ) - Φ ( a - μ σ ) σ \operatorname{E}(X\mid a<X<b)=\mu+\frac{\phi(\frac{a-\mu}{\sigma})-\phi(\frac{% b-\mu}{\sigma})}{\Phi(\frac{b-\mu}{\sigma})-\Phi(\frac{a-\mu}{\sigma})}\sigma\!
  14. Var ( X a < X < b ) = σ 2 [ 1 + a - μ σ ϕ ( a - μ σ ) - b - μ σ ϕ ( b - μ σ ) Φ ( b - μ σ ) - Φ ( a - μ σ ) - ( ϕ ( a - μ σ ) - ϕ ( b - μ σ ) Φ ( b - μ σ ) - Φ ( a - μ σ ) ) 2 ] \operatorname{Var}(X\mid a<X<b)=\sigma^{2}\left[1+\frac{\frac{a-\mu}{\sigma}% \phi(\frac{a-\mu}{\sigma})-\frac{b-\mu}{\sigma}\phi(\frac{b-\mu}{\sigma})}{% \Phi(\frac{b-\mu}{\sigma})-\Phi(\frac{a-\mu}{\sigma})}-\left(\frac{\phi(\frac{% a-\mu}{\sigma})-\phi(\frac{b-\mu}{\sigma})}{\Phi(\frac{b-\mu}{\sigma})-\Phi(% \frac{a-\mu}{\sigma})}\right)^{2}\right]\!
  15. E ( X X > a ) = μ + σ λ ( α ) \operatorname{E}(X\mid X>a)=\mu+\sigma\lambda(\alpha)\!
  16. Var ( X X > a ) = σ 2 [ 1 - δ ( α ) ] , \operatorname{Var}(X\mid X>a)=\sigma^{2}[1-\delta(\alpha)],\!
  17. α = ( a - μ ) / σ , λ ( α ) = ϕ ( α ) / [ 1 - Φ ( α ) ] \alpha=(a-\mu)/\sigma,\;\lambda(\alpha)=\phi(\alpha)/[1-\Phi(\alpha)]\;
  18. δ ( α ) = λ ( α ) [ λ ( α ) - α ] \;\delta(\alpha)=\lambda(\alpha)[\lambda(\alpha)-\alpha]
  19. E ( X X < b ) = μ - σ ϕ ( β ) Φ ( β ) \operatorname{E}(X\mid X<b)=\mu-\sigma\frac{\phi(\beta)}{\Phi(\beta)}\!
  20. Var ( X X < b ) = σ 2 [ 1 - β ϕ ( β ) Φ ( β ) - ( ϕ ( β ) Φ ( β ) ) 2 ] , \operatorname{Var}(X\mid X<b)=\sigma^{2}\left[1-\beta\frac{\phi(\beta)}{\Phi(% \beta)}-\left(\frac{\phi(\beta)}{\Phi(\beta)}\right)^{2}\right],\!
  21. β = ( b - μ ) / σ . \beta=(b-\mu)/\sigma.
  22. { σ 2 f ( x ) + f ( x ) ( x - μ ) = 0 , f ( 0 ) = 2 π e - μ 2 2 σ 2 σ ( erf ( μ - a 2 σ ) - erf ( μ - b 2 σ ) ) } \left\{\sigma^{2}f^{\prime}(x)+f(x)(x-\mu)=0,f(0)=\frac{\sqrt{\frac{2}{\pi}}e^% {-\frac{\mu^{2}}{2\sigma^{2}}}}{\sigma\left(\,\text{erf}\left(\frac{\mu-a}{% \sqrt{2}\sigma}\right)-\,\text{erf}\left(\frac{\mu-b}{\sqrt{2}\sigma}\right)% \right)}\right\}
  23. x = Φ - 1 ( Φ ( α ) + U ( Φ ( β ) - Φ ( α ) ) ) σ + μ x=\Phi^{-1}(\Phi(\alpha)+U\cdot(\Phi(\beta)-\Phi(\alpha)))\sigma+\mu
  24. Φ \Phi
  25. Φ - 1 \Phi^{-1}
  26. U U
  27. ( 0 , 1 ) (0,1)
  28. ( a , b ) (a,b)
  29. Φ \Phi
  30. Φ - 1 \Phi^{-1}

Truncation_error.html

  1. sin ( x ) x - 1 6 x 3 \sin(x)\approx x-\tfrac{1}{6}x^{3}
  2. x x

Tschirnhausen_cubic.html

  1. r = a sec 3 ( θ / 3 ) . r=a\sec^{3}(\theta/3).
  2. t = tan ( θ / 3 ) t=\tan(\theta/3)
  3. x = a cos θ sec 3 θ 3 = a ( cos 3 θ 3 - 3 cos θ 3 sin 2 θ 3 ) sec 3 θ 3 x=a\cos\theta\sec^{3}\frac{\theta}{3}=a(\cos^{3}\frac{\theta}{3}-3\cos\frac{% \theta}{3}\sin^{2}\frac{\theta}{3})\sec^{3}\frac{\theta}{3}
  4. = a ( 1 - 3 tan 2 θ 3 ) = a ( 1 - 3 t 2 ) =a\left(1-3\tan^{2}\frac{\theta}{3}\right)=a(1-3t^{2})
  5. y = a sin θ sec 3 θ 3 = a ( 3 cos 2 θ 3 sin θ 3 - sin 3 θ 3 ) sec 3 θ 3 y=a\sin\theta\sec^{3}\frac{\theta}{3}=a\left(3\cos^{2}\frac{\theta}{3}\sin% \frac{\theta}{3}-\sin^{3}\frac{\theta}{3}\right)\sec^{3}\frac{\theta}{3}
  6. = a ( 3 tan θ 3 - tan 3 θ 3 ) = a t ( 3 - t 2 ) =a\left(3\tan\frac{\theta}{3}-\tan^{3}\frac{\theta}{3}\right)=at(3-t^{2})
  7. 27 a y 2 = ( a - x ) ( 8 a + x ) 2 27ay^{2}=(a-x)(8a+x)^{2}
  8. x = 3 a ( 3 - t 2 ) x=3a(3-t^{2})
  9. y = a t ( 3 - t 2 ) y=at(3-t^{2})
  10. x 3 = 9 a ( x 2 - 3 y 2 ) x^{3}=9a\left(x^{2}-3y^{2}\right)
  11. r = 9 a ( sec θ - 3 sec θ tan 2 θ ) r=9a\left(\sec\theta-3\sec\theta\tan^{2}\theta\right)

Tubulin_GTPase.html

  1. \rightleftharpoons

Tupper's_self-referential_formula.html

  1. 1 2 < mod ( y 17 2 - 17 x - mod ( y , 17 ) , 2 ) , {1\over 2}<\left\lfloor\mathrm{mod}\left(\left\lfloor{y\over 17}\right\rfloor 2% ^{-17\lfloor x\rfloor-\mathrm{mod}(\lfloor y\rfloor,17)},2\right)\right\rfloor,
  2. \lfloor\cdot\rfloor
  3. mod \mathrm{mod}

Turbulent_Prandtl_number.html

  1. ϵ M \epsilon_{M}
  2. ϵ H \epsilon_{H}
  3. - u v ¯ = ϵ M u ¯ y -\overline{u^{\prime}v^{\prime}}=\epsilon_{M}\frac{\partial\bar{u}}{\partial y}
  4. - v T ¯ = ϵ H T ¯ y -\overline{v^{\prime}T^{\prime}}=\epsilon_{H}\frac{\partial\bar{T}}{\partial y}
  5. - u v ¯ -\overline{u^{\prime}v^{\prime}}
  6. - v T ¯ -\overline{v^{\prime}T^{\prime}}
  7. Pr t = ϵ M ϵ H . \mathrm{Pr}_{\mathrm{t}}=\frac{\epsilon_{M}}{\epsilon_{H}}.
  8. u ¯ u ¯ x + v ¯ u ¯ y = - 1 ρ d P ¯ d x + y [ ( ν u ¯ y - u v ¯ ) ] . \bar{u}\frac{\partial\bar{u}}{\partial x}+\bar{v}\frac{\partial\bar{u}}{% \partial y}=-\frac{1}{\rho}\frac{d\bar{P}}{dx}+\frac{\partial}{\partial y}% \left[(\nu\frac{\partial\bar{u}}{\partial y}-\overline{u^{\prime}v^{\prime}})% \right].
  9. u ¯ T ¯ x + v ¯ T ¯ y = y ( α T ¯ y - v T ¯ ) . \bar{u}\frac{\partial\bar{T}}{\partial x}+\bar{v}\frac{\partial\bar{T}}{% \partial y}=\frac{\partial}{\partial y}\left(\alpha\frac{\partial\bar{T}}{% \partial y}-\overline{v^{\prime}T^{\prime}}\right).
  10. u ¯ u ¯ x + v ¯ u ¯ y = - 1 ρ d P ¯ d x + y [ ( ν + ϵ M ) u ¯ y ] \bar{u}\frac{\partial\bar{u}}{\partial x}+\bar{v}\frac{\partial\bar{u}}{% \partial y}=-\frac{1}{\rho}\frac{d\bar{P}}{dx}+\frac{\partial}{\partial y}% \left[(\nu+\epsilon_{M})\frac{\partial\bar{u}}{\partial y}\right]
  11. u ¯ T ¯ x + v ¯ T ¯ y = y [ ( α + ϵ H ) T ¯ y ] . \bar{u}\frac{\partial\bar{T}}{\partial x}+\bar{v}\frac{\partial\bar{T}}{% \partial y}=\frac{\partial}{\partial y}\left[(\alpha+\epsilon_{H})\frac{% \partial\bar{T}}{\partial y}\right].
  12. u ¯ T ¯ x + v ¯ T ¯ y = y [ ( α + ϵ M Pr t ) T ¯ y ] . \bar{u}\frac{\partial\bar{T}}{\partial x}+\bar{v}\frac{\partial\bar{T}}{% \partial y}=\frac{\partial}{\partial y}\left[(\alpha+\frac{\epsilon_{M}}{% \mathrm{Pr}_{\mathrm{t}}})\frac{\partial\bar{T}}{\partial y}\right].

Two-body_problem_in_general_relativity.html

  1. F = G M m r 2 F=G\frac{Mm}{r^{2}}
  2. d s 2 = d x 2 + d y 2 + d z 2 ds^{2}=dx^{2}+dy^{2}+dz^{2}\,\!
  3. d s 2 = F ( x , y , z ) d x 2 + G ( x , y , z ) d y 2 + H ( x , y , z ) d z 2 ds^{2}=F(x,y,z)dx^{2}+G(x,y,z)dy^{2}+H(x,y,z)dz^{2}\,\!
  4. d s 2 = d r 2 + r 2 d θ 2 + r 2 sin 2 θ d φ 2 ds^{2}=dr^{2}+r^{2}d\theta^{2}+r^{2}\sin^{2}\theta d\varphi^{2}\,\!
  5. d s 2 = g x x d x 2 + g x y d x d y + g x z d x d z + + g z y d z d y + g z z d z 2 ds^{2}=g_{xx}dx^{2}+g_{xy}dxdy+g_{xz}dxdz+\cdots+g_{zy}dzdy+g_{zz}dz^{2}\,\!
  6. c 2 d τ 2 = c 2 d t 2 - d x 2 - d y 2 - d z 2 c^{2}d\tau^{2}=c^{2}dt^{2}-dx^{2}-dy^{2}-dz^{2}\,\!
  7. c 2 d τ 2 = c 2 d t 2 - d r 2 - r 2 d θ 2 - r 2 sin 2 θ d φ 2 c^{2}d\tau^{2}=c^{2}dt^{2}-dr^{2}-r^{2}d\theta^{2}-r^{2}\sin^{2}\theta d% \varphi^{2}\,\!
  8. c 2 d τ 2 = g μ ν d x μ d x ν c^{2}d\tau^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}\,\!
  9. d 2 x μ d q 2 + Γ ν λ μ d x ν d q d x λ d q = 0 \frac{d^{2}x^{\mu}}{dq^{2}}+\Gamma^{\mu}_{\nu\lambda}\frac{dx^{\nu}}{dq}\frac{% dx^{\lambda}}{dq}=0
  10. r s = 2 G M c 2 r_{s}=\frac{2GM}{c^{2}}
  11. 3 / 8 {3}/{8}
  12. ( d r d τ ) 2 = E 2 m 2 c 2 - ( 1 - r s r ) ( c 2 + h 2 r 2 ) . \left(\frac{dr}{d\tau}\right)^{2}=\frac{E^{2}}{m^{2}c^{2}}-\left(1-\frac{r_{s}% }{r}\right)\left(c^{2}+\frac{h^{2}}{r^{2}}\right).
  13. 𝐡 = 𝐫 × 𝐯 = 𝐋 μ \mathbf{h}=\mathbf{r}\times\mathbf{v}={\mathbf{L}\over\mu}
  14. ( d r d φ ) 2 = r 4 b 2 - ( 1 - r s r ) ( r 4 a 2 + r 2 ) \left(\frac{dr}{d\varphi}\right)^{2}=\frac{r^{4}}{b^{2}}-\left(1-\frac{r_{s}}{% r}\right)\left(\frac{r^{4}}{a^{2}}+r^{2}\right)
  15. φ = 1 r 2 [ 1 b 2 - ( 1 - r s r ) ( 1 a 2 + 1 r 2 ) ] - 1 / 2 d r . \varphi=\int\frac{1}{r^{2}}\left[\frac{1}{b^{2}}-\left(1-\frac{r_{\mathrm{s}}}% {r}\right)\left(\frac{1}{a^{2}}+\frac{1}{r^{2}}\right)\right]^{-1/2}dr.
  16. φ = d r r 2 1 b 2 - ( 1 - r s r ) 1 r 2 \varphi=\int\frac{dr}{r^{2}\sqrt{\frac{1}{b^{2}}-\left(1-\frac{r_{s}}{r}\right% )\frac{1}{r^{2}}}}
  17. δ φ 2 r s b = 4 G M c 2 b . \delta\varphi\approx\frac{2r_{s}}{b}=\frac{4GM}{c^{2}b}.
  18. ( d r d τ ) 2 = E 2 m 2 c 2 - c 2 + r s c 2 r - h 2 r 2 + r s h 2 r 3 \left(\frac{dr}{d\tau}\right)^{2}=\frac{E^{2}}{m^{2}c^{2}}-c^{2}+\frac{r_{s}c^% {2}}{r}-\frac{h^{2}}{r^{2}}+\frac{r_{s}h^{2}}{r^{3}}
  19. 1 2 m ( d r d τ ) 2 = [ E 2 2 m c 2 - 1 2 m c 2 ] + G M m r - L 2 2 μ r 2 + G ( M + m ) L 2 c 2 μ r 3 \frac{1}{2}m\left(\frac{dr}{d\tau}\right)^{2}=\left[\frac{E^{2}}{2mc^{2}}-% \frac{1}{2}mc^{2}\right]+\frac{GMm}{r}-\frac{L^{2}}{2\mu r^{2}}+\frac{G(M+m)L^% {2}}{c^{2}\mu r^{3}}
  20. V ( r ) = - G M m r + L 2 2 μ r 2 - G ( M + m ) L 2 c 2 μ r 3 V(r)=-\frac{GMm}{r}+\frac{L^{2}}{2\mu r^{2}}-\frac{G(M+m)L^{2}}{c^{2}\mu r^{3}}
  21. δ φ 6 π G ( M + m ) c 2 A ( 1 - e 2 ) \delta\varphi\approx\frac{6\pi G(M+m)}{c^{2}A\left(1-e^{2}\right)}
  22. V ( r ) = m c 2 2 [ - r s r + a 2 r 2 - r s a 2 r 3 ] . V(r)=\frac{mc^{2}}{2}\left[-\frac{r_{s}}{r}+\frac{a^{2}}{r^{2}}-\frac{r_{s}a^{% 2}}{r^{3}}\right].
  23. F = - d V d r = - m c 2 2 r 4 [ r s r 2 - 2 a 2 r + 3 r s a 2 ] = 0 ; F=-\frac{dV}{dr}=-\frac{mc^{2}}{2r^{4}}\left[r_{s}r^{2}-2a^{2}r+3r_{s}a^{2}% \right]=0;
  24. r outer = a 2 r s ( 1 + 1 - 3 r s 2 a 2 ) r_{\mathrm{outer}}=\frac{a^{2}}{r_{s}}\left(1+\sqrt{1-\frac{3r_{s}^{2}}{a^{2}}% }\right)
  25. r inner = a 2 r s ( 1 - 1 - 3 r s 2 a 2 ) = 3 a 2 r outer , r_{\mathrm{inner}}=\frac{a^{2}}{r_{s}}\left(1-\sqrt{1-\frac{3r_{s}^{2}}{a^{2}}% }\right)=\frac{3a^{2}}{r_{\mathrm{outer}}},
  26. r outer 2 a 2 r s r_{\mathrm{outer}}\approx\frac{2a^{2}}{r_{s}}
  27. r inner 3 2 r s r_{\mathrm{inner}}\approx\frac{3}{2}r_{s}
  28. r outer 3 = G ( M + m ) ω φ 2 r_{\mathrm{outer}}^{3}=\frac{G(M+m)}{\omega_{\varphi}^{2}}
  29. G M m r 2 = μ ω φ 2 r \frac{GMm}{r^{2}}=\mu\omega_{\varphi}^{2}r
  30. μ \mu
  31. ω φ 2 G M r outer 3 = ( r s c 2 2 r outer 3 ) = ( r s c 2 2 ) ( r s 3 8 a 6 ) = c 2 r s 4 16 a 6 \omega_{\varphi}^{2}\approx\frac{GM}{r_{\mathrm{outer}}^{3}}=\left(\frac{r_{s}% c^{2}}{2r_{\mathrm{outer}}^{3}}\right)=\left(\frac{r_{s}c^{2}}{2}\right)\left(% \frac{r_{s}^{3}}{8a^{6}}\right)=\frac{c^{2}r_{s}^{4}}{16a^{6}}
  32. r outer r inner 3 r s r_{\mathrm{outer}}\approx r_{\mathrm{inner}}\approx 3r_{s}
  33. 3 / 2 {3}/{2}
  34. 3 / 2 {3}/{2}
  35. 3 / 2 {3}/{2}
  36. ω r 2 = 1 m [ d 2 V d r 2 ] r = r outer \omega_{r}^{2}=\frac{1}{m}\left[\frac{d^{2}V}{dr^{2}}\right]_{r=r_{\mathrm{% outer}}}
  37. ω r 2 = ( c 2 r s 2 r outer 4 ) ( r outer - r inner ) = ω φ 2 1 - 3 r s 2 a 2 \omega_{r}^{2}=\left(\frac{c^{2}r_{s}}{2r_{\mathrm{outer}}^{4}}\right)\left(r_% {\mathrm{outer}}-r_{\mathrm{inner}}\right)=\omega_{\varphi}^{2}\sqrt{1-\frac{3% r_{s}^{2}}{a^{2}}}
  38. ω r = ω φ ( 1 - 3 r s 2 4 a 2 + ) \omega_{r}=\omega_{\varphi}\left(1-\frac{3r_{s}^{2}}{4a^{2}}+\cdots\right)
  39. δ φ = T ( ω φ - ω r ) 2 π ( 3 r s 2 4 a 2 ) = 3 π m 2 c 2 2 L 2 r s 2 \delta\varphi=T\left(\omega_{\varphi}-\omega_{r}\right)\approx 2\pi\left(\frac% {3r_{s}^{2}}{4a^{2}}\right)=\frac{3\pi m^{2}c^{2}}{2L^{2}}r_{s}^{2}
  40. δ φ 3 π m 2 c 2 2 L 2 ( 4 G 2 M 2 c 4 ) = 6 π G 2 M 2 m 2 c 2 L 2 \delta\varphi\approx\frac{3\pi m^{2}c^{2}}{2L^{2}}\left(\frac{4G^{2}M^{2}}{c^{% 4}}\right)=\frac{6\pi G^{2}M^{2}m^{2}}{c^{2}L^{2}}
  41. h 2 G ( M + m ) = A ( 1 - e 2 ) \frac{h^{2}}{G(M+m)}=A\left(1-e^{2}\right)
  42. δ φ 6 π G ( M + m ) c 2 A ( 1 - e 2 ) \delta\varphi\approx\frac{6\pi G(M+m)}{c^{2}A\left(1-e^{2}\right)}
  43. - d E d t = 32 G 4 m 1 2 m 2 2 ( m 1 + m 2 ) 5 c 5 a 5 ( 1 - e 2 ) 7 / 2 ( 1 + 73 24 e 2 + 37 96 e 4 ) -\Bigl\langle\frac{dE}{dt}\Bigr\rangle=\frac{32G^{4}m_{1}^{2}m_{2}^{2}\left(m_% {1}+m_{2}\right)}{5c^{5}a^{5}\left(1-e^{2}\right)^{7/2}}\left(1+\frac{73}{24}e% ^{2}+\frac{37}{96}e^{4}\right)
  44. - d L z d t = 32 G 7 / 2 m 1 2 m 2 2 m 1 + m 2 5 c 5 a 7 / 2 ( 1 - e 2 ) 2 ( 1 + 7 8 e 2 ) -\Bigl\langle\frac{dL_{z}}{dt}\Bigr\rangle=\frac{32G^{7/2}m_{1}^{2}m_{2}^{2}% \sqrt{m_{1}+m_{2}}}{5c^{5}a^{7/2}\left(1-e^{2}\right)^{2}}\left(1+\frac{7}{8}e% ^{2}\right)
  45. - d P b d t = 192 G 5 / 3 m 1 m 2 ( m 1 + m 2 ) - 1 / 3 5 c 5 ( 1 - e 2 ) 7 / 2 ( 1 + 73 24 e 2 + 37 96 e 4 ) ( P b 2 π ) - 5 / 3 -\Bigl\langle\frac{dP_{b}}{dt}\Bigr\rangle=\frac{192G^{5/3}m_{1}m_{2}\left(m_{% 1}+m_{2}\right)^{-1/3}}{5c^{5}\left(1-e^{2}\right)^{7/2}}\left(1+\frac{73}{24}% e^{2}+\frac{37}{96}e^{4}\right)\left(\frac{P_{b}}{2\pi}\right)^{-5/3}

Two-center_bipolar_coordinates.html

  1. c 1 c_{1}
  2. c 2 c_{2}
  3. ( x , y ) (x,\ y)
  4. ( r 1 , r 2 ) (r_{1},\ r_{2})
  5. x = r 2 2 - r 1 2 4 a x=\frac{r_{2}^{2}-r_{1}^{2}}{4a}
  6. y = ± 1 4 a 16 a 2 r 2 2 - ( r 2 2 - r 1 2 + 4 a 2 ) 2 y=\pm\frac{1}{4a}\sqrt{16a^{2}r_{2}^{2}-(r_{2}^{2}-r_{1}^{2}+4a^{2})^{2}}
  7. ( + a , 0 ) (+a,\ 0)
  8. ( - a , 0 ) (-a,\ 0)
  9. r = r 1 2 + r 2 2 - 2 a 2 2 r=\sqrt{\frac{r_{1}^{2}+r_{2}^{2}-2a^{2}}{2}}
  10. θ = arctan ( r 1 4 - 8 a 2 r 1 2 - 2 r 1 2 r 2 2 - ( 4 a 2 - r 2 2 ) 2 r 2 2 - r 1 2 ) \theta=\arctan\left(\frac{\sqrt{r_{1}^{4}-8a^{2}r_{1}^{2}-2r_{1}^{2}r_{2}^{2}-% (4a^{2}-r_{2}^{2})^{2}}}{r_{2}^{2}-r_{1}^{2}}\right)\,\!
  11. 2 a 2a

Two-dimensional_point_vortex_gas.html

  1. k i d x i d t = H y i , k i d y i d t = - H x i , k_{i}\frac{dx_{i}}{dt}=\frac{\partial H}{\partial y_{i}},\qquad k_{i}\frac{dy_% {i}}{dt}=-\frac{\partial H}{\partial x_{i}},

Two-dimensional_singular_value_decomposition.html

  1. X = ( x 1 , , x n ) X=(x_{1},...,x_{n})
  2. F F
  3. G G
  4. F = X X T F=XX^{T}
  5. G = X T X G=X^{T}X
  6. U = ( u 1 , , u n ) U=(u_{1},...,u_{n})
  7. V = ( v 1 , , v n ) V=(v_{1},...,v_{n})
  8. V V T = I , U U T = I VV^{T}=I,UU^{T}=I
  9. X = U U T X V V T = U ( U T X V ) V T = U Σ V T X=UU^{T}XVV^{T}=U(U^{T}XV)V^{T}=U\Sigma V^{T}
  10. K K
  11. U , V U,V
  12. X X
  13. ( X 1 , , X n ) (X_{1},...,X_{n})
  14. i X i = 0 \sum_{i}X_{i}=0
  15. F = i X i X i T F=\sum_{i}X_{i}X_{i}^{T}
  16. G = i X i T X i G=\sum_{i}X_{i}^{T}X_{i}
  17. U U
  18. V V
  19. X i X_{i}
  20. X i = U U T X i V V T = U ( U T X i V ) V T = U M i V T X_{i}=UU^{T}X_{i}VV^{T}=U(U^{T}X_{i}V)V^{T}=UM_{i}V^{T}
  21. ( X 1 , , X n ) (X_{1},...,X_{n})
  22. J = i | X i - L M i R T | 2 J=\sum_{i}|X_{i}-LM_{i}R^{T}|^{2}

Tychonoff_plank.html

  1. [ 0 , ω 1 ] [0,\omega_{1}]
  2. [ 0 , ω ] [0,\omega]
  3. ω \omega
  4. ω 1 \omega_{1}
  5. = ( ω 1 , ω ) \infty=(\omega_{1},\omega)
  6. { } \{\infty\}

Ubbelohde_viscometer.html

  1. d V d t = v π R 2 = π R 4 8 η ( - Δ P Δ x ) = π R 4 8 η | Δ P | L , \frac{dV}{dt}=v\pi R^{2}=\frac{\pi R^{4}}{8\eta}\left(\frac{-\Delta P}{\Delta x% }\right)=\frac{\pi R^{4}}{8\eta}\frac{|\Delta P|}{L},
  2. d v d t \frac{dv}{dt}
  3. Δ P = ρ g Δ H \Delta P=\rho g\Delta H\,
  4. η r = η η 0 = t ρ t 0 ρ 0 , \eta_{r}=\frac{\eta}{\eta_{0}}=\frac{t\rho}{t_{0}\rho_{0}},
  5. ρ ρ 0 \rho\simeq\rho_{0}\,
  6. η s p = η r - 1 = t - t 0 t 0 . \eta_{sp}=\eta_{r}-1=\frac{t-t_{0}}{t_{0}}.\,
  7. η s p = [ η ] c + k [ η ] 2 c 2 + \eta_{sp}=[\eta]c+k[\eta]^{2}c^{2}+\cdots\,
  8. η s p c = [ η ] + k [ η ] 2 c + , \frac{\eta_{sp}}{c}=[\eta]+k[\eta]^{2}c+\cdots,\,
  9. η s p c \frac{\eta_{sp}}{c}

Ultraviolet_photoelectron_spectroscopy.html

  1. E K E_{K}
  2. E K = h ν - I E_{K}=h\nu-I\,

Umbilical_point.html

  1. a x 3 + 3 b x 2 y + 3 c x y 2 + d y 3 ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3}
  2. λ ( x , y ) \lambda(x,y)
  3. λ \lambda
  4. x 2 y - y 3 x^{2}y-y^{3}
  5. x 2 y x^{2}y
  6. x 2 y + y 3 x^{2}y+y^{3}
  7. x 3 x^{3}
  8. z 3 + 3 β ¯ z 2 z ¯ + 3 β z z ¯ 2 + z ¯ 3 z^{3}+3\overline{\beta}z^{2}\overline{z}+3\beta z\overline{z}^{2}+\overline{z}% ^{3}
  9. β \beta
  10. β = 1 3 ( 2 e i θ + e - 2 i θ ) \beta=\tfrac{1}{3}(2e^{i\theta}+e^{-2i\theta})
  11. | β | = 1 \left|\beta\right|=1
  12. β \beta
  13. | β | = 1 3 \left|\beta\right|=\tfrac{1}{3}
  14. F : 2 2 F:\mathbb{R}^{2}\rightarrow\mathbb{R}^{2}
  15. F ( x , y ) = ( x 2 + y 2 , a x 3 + 3 b x 2 y + 3 c x y 2 + d y 3 ) F(x,y)=(x^{2}+y^{2},ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3})
  16. b x 3 + ( 2 c - a ) x 2 y + ( d - 2 b ) x y 2 - c y 3 bx^{3}+(2c-a)x^{2}y+(d-2b)xy^{2}-cy^{3}
  17. β = - 2 e i θ - e - 2 i θ \beta=-2e^{i\theta}-e^{-2i\theta}
  18. β \beta
  19. z = 1 2 κ ( x 2 + y 2 ) + 1 3 ( a x 3 + 3 b x 2 y + 3 c x y 2 + d y 3 ) + z=\tfrac{1}{2}\kappa(x^{2}+y^{2})+\tfrac{1}{3}(ax^{3}+3bx^{2}y+3cxy^{2}+dy^{3}% )+\ldots
  20. κ \kappa
  21. ν \nu
  22. ν \nu

Umbrella_sampling.html

  1. π ( 𝐫 N ) = w ( 𝐫 N ) exp ( - U ( 𝐫 N ) / k B T ) w ( 𝐫 N ) exp ( - U ( 𝐫 N ) / k B T ) d 𝐫 N , \pi(\mathbf{r}^{N})=\frac{w(\,\textbf{r}^{N})\exp{(-U(\mathbf{r}^{N}})/k_{B}T)% }{\int{w(\mathbf{r^{\prime}}^{N})\exp{(-U(\mathbf{r^{\prime}}^{N}})/k_{B}T)}d% \mathbf{r^{\prime}}^{N}},
  2. A = A / w π 1 / w π , \langle A\rangle=\frac{\langle A/w\rangle_{\pi}}{\langle 1/w\rangle_{\pi}},
  3. π \pi
  4. V ( 𝐫 N ) = - k B T ln w ( 𝐫 N ) V(\mathbf{r}^{N})=-k_{B}T\ln w(\mathbf{r}^{N})
  5. Q Q
  6. F 0 ( Q ) = F π ( Q ) - V ( Q ) F_{0}(Q)=F_{\pi}(Q)-V(Q)
  7. F 0 ( Q ) F_{0}(Q)
  8. F π ( Q ) F_{\pi}(Q)

Unate_function.html

  1. f ( x 1 , x 2 , , x n ) f(x_{1},x_{2},\ldots,x_{n})
  2. x i x_{i}
  3. x j x_{j}
  4. j i j\neq i
  5. f ( x 1 , x 2 , , x i - 1 , 1 , x i + 1 , , x n ) f ( x 1 , x 2 , , x i - 1 , 0 , x i + 1 , , x n ) . f(x_{1},x_{2},\ldots,x_{i-1},1,x_{i+1},\ldots,x_{n})\geq f(x_{1},x_{2},\ldots,% x_{i-1},0,x_{i+1},\ldots,x_{n}).\,
  6. x i x_{i}
  7. f ( x 1 , x 2 , , x i - 1 , 0 , x i + 1 , , x n ) f ( x 1 , x 2 , , x i - 1 , 1 , x i + 1 , , x n ) . f(x_{1},x_{2},\ldots,x_{i-1},0,x_{i+1},\ldots,x_{n})\geq f(x_{1},x_{2},\ldots,% x_{i-1},1,x_{i+1},\ldots,x_{n}).\,
  8. x i x_{i}
  9. x i x_{i}
  10. x i x_{i}

Unbiased_estimation_of_standard_deviation.html

  1. s = 1 n - 1 i = 1 n ( x i - x ¯ ) 2 , s=\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}\,,
  2. { x 1 , x 2 , , x n } \{x_{1},x_{2},\ldots,x_{n}\}
  3. x ¯ \overline{x}
  4. n - 1 s / σ \sqrt{n-1}\,s/\sigma
  5. E [ s ] = c 4 ( n ) σ \operatorname{E}[s]=c_{4}(n)\sigma\,
  6. μ 1 ( n - 1 ) / n - 1 . \mu_{1}(n-1)/\sqrt{n-1}.
  7. c 4 ( n ) = 2 n - 1 Γ ( n 2 ) Γ ( n - 1 2 ) = 1 - 1 4 n - 7 32 n 2 - 19 128 n 3 + O ( n - 4 ) c_{4}(n)\,=\,\sqrt{\frac{2}{n-1}}\,\,\,\frac{\Gamma\left(\frac{n}{2}\right)}{% \Gamma\left(\frac{n-1}{2}\right)}\,=\,1-\frac{1}{4n}-\frac{7}{32n^{2}}-\frac{1% 9}{128n^{3}}+O(n^{-4})
  8. 2 π \sqrt{\frac{2}{\pi}}
  9. π 2 \frac{\sqrt{\pi}}{2}
  10. 2 2 3 π 2\,\sqrt{\frac{2}{3\pi}}
  11. 3 4 π 2 \frac{3}{4}\,\sqrt{\frac{\pi}{2}}
  12. 8 3 2 5 π \frac{8}{3}\,\sqrt{\frac{2}{5\pi}}
  13. 5 3 π 16 \frac{5\sqrt{3\pi}}{16}
  14. 16 5 2 7 π \frac{16}{5}\,\sqrt{\frac{2}{7\pi}}
  15. 35 π 64 \frac{35\sqrt{\pi}}{64}
  16. 128 105 2 π \frac{128}{105}\,\sqrt{\frac{2}{\pi}}
  17. 2 π ( 2 k - 1 ) 2 2 k - 2 ( k - 1 ) ! 2 ( 2 k - 2 ) ! \sqrt{\frac{2}{\pi\left(2k-1\right)}}\,\frac{2^{2k-2}\left(k-1\right)!^{2}}{% \left(2k-2\right)!}
  18. π k ( 2 k - 1 ) ! 2 2 k - 1 ( k - 1 ) ! 2 \sqrt{\frac{\pi}{k}}\,\frac{\left(2k-1\right)!}{2^{2k-1}\left(k-1\right)!^{2}}
  19. σ 1 - c 4 2 \sigma\sqrt{1-c_{4}^{2}}
  20. σ c 4 - 2 - 1 . \sigma\sqrt{c_{4}^{-2}-1}.
  21. σ ^ = 1 n - 1.5 i = 1 n ( x i - x ¯ ) 2 \hat{\sigma}=\sqrt{\frac{1}{n-1.5}\sum_{i=1}^{n}(x_{i}-\bar{x})^{2}}
  22. E [ σ ^ ] = σ ( 1 + 1 16 n 2 + 3 16 n 3 + O ( n - 4 ) ) . \operatorname{E}[\hat{\sigma}]=\sigma\cdot\Big(1+\frac{1}{16n^{2}}+\frac{3}{16% n^{3}}+O(n^{-4})\Big).
  23. σ ^ = 1 n - 1.5 - 1 4 γ 2 i = 1 n ( x i - x ¯ ) 2 , \hat{\sigma}=\sqrt{\frac{1}{n-1.5-\tfrac{1}{4}\gamma_{2}}\sum_{i=1}^{n}(x_{i}-% \bar{x})^{2}},
  24. E [ s 2 ] = σ 2 [ 1 - 2 n - 1 k = 1 n - 1 ( 1 - k n ) ρ k ] {\rm E}\left[{s^{2}}\right]\,\,=\,\,\sigma^{2}\,\left[{1\,\,\,-\,\,\,{2\over{n% -\,\,1}}\,\,\sum\limits_{k\,=\,1}^{n\,-1}{\,\left({1\,\,-\,\,{k\over n}}\right% )\rho_{k}}}\right]
  25. ρ k \rho_{k}
  26. s 2 s^{2}
  27. ρ k = ( 1 - α ) k \rho_{k}=\,\,\left({\,1\,\,-\,\,\alpha\,}\right)^{k}
  28. VRR = α 2 - α {\rm VRR}\,\,\,=\,\,{\alpha\over{2\,\,-\,\,\alpha}}
  29. Var [ x ¯ ] = σ 2 n [ 1 + 2 k = 1 n - 1 ( 1 - k n ) ρ k ] . {\rm Var}\left[\bar{x}\right]\,\,\,=\,\,{{\sigma^{2}}\over n}\,\left[{1\,\,\,+% \,\,\,2\,\sum\limits_{k\,=\,1}^{n-1}{\left({1\,\,-\,\,{k\over n}}\right)\rho_{% k}}}\right].
  30. γ 1 1 - 2 n - 1 k = 1 n - 1 ( 1 - k n ) ρ k γ 2 1 + 2 k = 1 n - 1 ( 1 - k n ) ρ k \gamma_{1}\,\,\equiv\,\,1\,\,\,-\,\,{2\over{n\,\,-\,\,1}}\,\,\sum\limits_{k\,=% \,1}^{n\,-\,1}{\,\left({1\,\,-\,\,{k\over n}}\right)}\,\rho_{k}\,\,\,\,\,\,\,% \,\,\,\,\,\gamma_{2}\,\,\equiv\,\,1\,\,\,+\,\,2\,\sum\limits_{k\,=\,1}^{n\,-\,% 1}{\,\left({1\,\,-\,\,{k\over n}}\right)}\,\rho_{k}
  31. E [ s 2 ] = σ 2 γ 1 E [ s 2 γ 1 ] = σ 2 {\rm E}\left[{s^{2}}\right]\,\,=\,\,\sigma^{2}\,\gamma_{1}\,\,\,\,\,\,% \Rightarrow\,\,\,\,\,\,{\rm E}\left[{{{s^{2}}\over{\gamma_{1}}}}\right]\,\,\,=% \,\,\,\sigma^{2}
  32. γ 1 \gamma_{1}
  33. Var [ x ¯ ] = σ 2 n γ 2 {\rm Var}\left[{\bar{x}}\right]\,\,\,=\,\,\,{{\sigma^{2}}\over n}\,\,\gamma_{2}
  34. σ 2 \sigma^{2}
  35. Var [ x ¯ ] = E [ s 2 γ 1 ( γ 2 n ) ] = E [ s 2 n { n - 1 n γ 2 - 1 } ] {\rm Var}\left[{\bar{x}}\right]\,\,\,=\,\,\,{\rm E}\left[{{{s^{2}}\over{\gamma% _{1}}}\left({{{\gamma_{2}}\over n}}\right)}\right]\,\,\,\,=\,\,\,{\rm E}\left[% {{{s^{2}}\over n}\left\{{{{n\,\,-\,\,1}\over{{n\over{\gamma_{2}}}-\,\,1}}}% \right\}}\right]
  36. ρ k \rho_{k}
  37. E [ s ] E [ s 2 ] σ γ 1 {\rm E}[s]\,\,\,\neq\,\,\sqrt{\,{\rm E}\left[{s^{2}}\right]}\,\,\,\neq\,\,\,% \sigma\,\sqrt{\,\gamma_{1}}
  38. E [ s ] = σ θ γ 1 σ ^ = s θ γ 1 {\rm E}\left[s\right]\,\,\,=\,\,\,\sigma\,\,\theta\sqrt{\,\gamma_{1}}\,\,\,\,% \,\,\,\,\Rightarrow\,\,\,\,\,\,\,\hat{\sigma}\,\,=\,\,{s\over{\theta\,\sqrt{\,% \gamma_{1}}}}
  39. E [ s ] σ γ 1 σ ^ s γ 1 {\rm E}[s]\,\,\approx\,\,\sigma\,\sqrt{\,\gamma_{1}}\,\,\,\,\,\,\,\,\,\,% \Rightarrow\,\,\,\,\,\,\,\,\,\hat{\sigma}\,\,\,\approx\,\,\,{s\over{\sqrt{\,% \gamma_{1}}}}
  40. Var [ x ¯ ] = σ 2 n γ 2 {\rm Var}\left[{\bar{x}}\right]\,\,\,=\,\,\,{{\sigma^{2}}\over n}\,\,\gamma_{2}\,
  41. σ x ¯ = σ n γ 2 \sigma_{\bar{x}}\,\,\,\,=\,\,\,{\sigma\over{\sqrt{\,n}}}\,\,\sqrt{\,\gamma_{2}}
  42. σ ^ x ¯ = s θ n γ 2 γ 1 \hat{\sigma}_{\bar{x}}\,\,=\,\,\,{{s\,}\over{\theta\,\sqrt{\,n}}}{{\sqrt{\,% \gamma_{2}}}\over{\sqrt{\,\gamma_{1}}}}
  43. σ ^ x ¯ = s c 4 n \hat{\sigma}_{\bar{x}}\,\,\,=\,\,\,{s\over{c_{4}\sqrt{\,n}}}
  44. σ ^ x ¯ s n γ 2 γ 1 = s n n - 1 n γ 2 - 1 \hat{\sigma}_{\bar{x}}\,\,\,\approx\,\,\,\,{{s\,}\over{\sqrt{\,n}}}{{\sqrt{\,% \gamma_{2}}}\over{\sqrt{\,\gamma_{1}}}}\,\,\,\,=\,\,\,\,{{s\,}\over{\sqrt{\,n}% }}\sqrt{{{n\,\,-\,\,1}\over{{n\over{\gamma_{2}}}\,\,-\,\,1}}}

Undefined_(mathematics).html

  1. f ( x ) = x f(x)=\sqrt{x}
  2. x x
  3. f f
  4. f ( x ) = 1 x f(x)=\frac{1}{x}
  5. x = 0 x=0
  6. f ( x ) = x f(x)=\sqrt{x}
  7. x x
  8. \infty
  9. - -\infty
  10. { a n } \left\{a_{n}\right\}\rightarrow\infty
  11. ± \pm\infty
  12. x + = x+\infty=\infty
  13. x { } \forall x\in\mathbb{R}\cup\{\infty\}
  14. - + x = - -\infty+x=-\infty
  15. x { - } \forall x\in\mathbb{R}\cup\{-\infty\}
  16. x = x\cdot\infty=\infty
  17. x + \forall x\in\mathbb{R}^{+}
  18. \infty
  19. - \infty-\infty
  20. 0 0\cdot\infty
  21. 0
  22. \frac{\infty}{\infty}
  23. z z\in\mathbb{C}
  24. z z
  25. z z
  26. z z

Underwater_acoustics.html

  1. c c\,
  2. f f\,
  3. λ \lambda\,
  4. c = f λ c=f\cdot\lambda
  5. u u\,
  6. p p\,
  7. ρ \rho\,
  8. c c\,
  9. p = c u ρ p=c\cdot u\cdot\rho
  10. c c
  11. ρ \rho\,
  12. I = q 2 / ( ρ c ) I=q^{2}/(\rho c)\,
  13. q q\,
  14. R = - e - 2 k 2 h 2 s i n 2 A R=-e^{-2k^{2}h^{2}sin^{2}A}
  15. I s I_{s}
  16. I r I_{r}
  17. P L = 10 l o g ( I s / I r ) PL=10log(I_{s}/I_{r})
  18. I r I_{r}
  19. P L = 20 l o g ( p s / p r ) PL=20log(p_{s}/p_{r})
  20. p s p_{s}
  21. p r p_{r}

Unduloid.html

  1. sn ( u , k ) \operatorname{sn}(u,k)
  2. dn ( u , k ) \operatorname{dn}(u,k)
  3. F ( z , k ) \operatorname{F}(z,k)
  4. E ( z , k ) \operatorname{E}(z,k)
  5. x ( u ) = - a ( 1 - e ) ( F ( sn ( u , k ) , k ) + F ( 1 , k ) ) - a ( 1 + e ) ( E ( sn ( u , k ) , k ) + E ( 1 , k ) ) \operatorname{x}(u)=-a(1-e)(\operatorname{F}(\operatorname{sn}(u,k),k)+% \operatorname{F}(1,k))-a(1+e)(\operatorname{E}(\operatorname{sn}(u,k),k)+% \operatorname{E}(1,k))\,
  6. y ( u ) = a ( 1 + e ) dn ( u , k ) \operatorname{y}(u)=a(1+e)\operatorname{dn}(u,k)\,
  7. X ( u , v ) = x ( u ) , y ( u ) cos ( v ) , y ( u ) sin ( v ) \operatorname{X}(u,v)=\langle\operatorname{x}(u),\operatorname{y}(u)\cos(v),% \operatorname{y}(u)\sin(v)\rangle\,

Uniform_integrability.html

  1. ( X , 𝔐 , μ ) (X,\mathfrak{M},\mu)
  2. Φ L 1 ( μ ) \Phi\subset L^{1}(\mu)
  3. ϵ > 0 \epsilon>0
  4. δ > 0 \delta>0
  5. | E f d μ | < ϵ \left|\int_{E}fd\mu\right|<\epsilon
  6. f Φ f\in\Phi
  7. μ ( E ) < δ . \mu(E)<\delta.
  8. 𝒞 \mathcal{C}
  9. ϵ > 0 \epsilon>0
  10. K [ 0 , ) K\in[0,\infty)
  11. E ( | X | I | X | K ) ϵ for all X 𝒞 E(|X|I_{|X|\geq K})\leq\epsilon\ \,\text{ for all X}\in\mathcal{C}
  12. I | X | K I_{|X|\geq K}
  13. I | X | K = { 1 if | X | K , 0 if | X | < K . I_{|X|\geq K}=\begin{cases}1&\,\text{if }|X|\geq K,\\ 0&\,\text{if }|X|<K.\end{cases}
  14. 𝒞 \mathcal{C}
  15. K K
  16. X X
  17. 𝒞 \mathcal{C}
  18. E ( | X | ) K \mathrm{E}(|X|)\leqslant K
  19. ϵ > 0 \epsilon>0
  20. δ > 0 \delta>0
  21. A A
  22. P ( A ) δ \mathrm{P}(A)\leqslant\delta
  23. X X
  24. 𝒞 \mathcal{C}
  25. E ( | X | : A ) ϵ \mathrm{E}(|X|:A)\leqslant\epsilon
  26. lim K sup X 𝒞 E ( | X | I | X | K ) = 0. \lim_{K\to\infty}\sup_{X\in\mathcal{C}}E(|X|I_{|X|\geq K})=0.
  27. Ω = [ 0 , 1 ] \Omega=[0,1]\subset\mathbb{R}
  28. X n ( ω ) = { n , ω ( 0 , 1 / n ) , 0 , otherwise. X_{n}(\omega)=\begin{cases}n,&\omega\in(0,1/n),\\ 0,&\,\text{otherwise.}\end{cases}
  29. X n L 1 X_{n}\in L^{1}
  30. E ( | X n | ) = 1 , E(|X_{n}|)=1\ ,
  31. E ( | X n | , | X n | K ) = 1 for all n K , E(|X_{n}|,|X_{n}|\geq K)=1\ \,\text{ for all }n\geq K,
  32. X n 0 X_{n}\to 0
  33. L 1 L^{1}
  34. X n X_{n}
  35. δ \delta
  36. ( 0 , 1 / n ) (0,1/n)
  37. δ \delta
  38. E [ | X m | : ( 0 , 1 / n ) ] = 1 E[|X_{m}|:(0,1/n)]=1
  39. m n m\geq n
  40. X X
  41. E ( | X | ) = E ( | X | , | X | > K ) + E ( | X | , | X | < K ) E(|X|)=E(|X|,|X|>K)+E(|X|,|X|<K)
  42. L 1 L^{1}
  43. X n X_{n}
  44. Y Y
  45. | X n ( ω ) | | Y ( ω ) | , Y ( ω ) 0 , E ( Y ) < , \ |X_{n}(\omega)|\leq|Y(\omega)|,\ Y(\omega)\geq 0,\ E(Y)<\infty,
  46. 𝒞 \mathcal{C}
  47. { X n } \{X_{n}\}
  48. L p L^{p}
  49. p > 1 p>1
  50. X n L 1 ( μ ) X_{n}\subset L^{1}(\mu)
  51. σ ( L 1 , L ) \sigma(L^{1},L^{\infty})
  52. { X α } α \Alpha L 1 ( μ ) \{X_{\alpha}\}_{\alpha\in\Alpha}\subset L^{1}(\mu)
  53. G ( t ) G(t)
  54. lim t G ( t ) t = \lim_{t\to\infty}\frac{G(t)}{t}=\infty
  55. sup α E ( G ( | X α | ) ) < . \sup_{\alpha}E(G(|X_{\alpha}|))<\infty.
  56. { X n } \{X_{n}\}
  57. X X
  58. L 1 L_{1}
  59. X X

Uniform_tiling.html

  1. A ~ 2 {\tilde{A}}_{2}
  2. B ~ 2 {\tilde{B}}_{2}
  3. G ~ 2 {\tilde{G}}_{2}
  4. I ~ 1 {\tilde{I}}_{1}
  5. I ~ 1 {\tilde{I}}_{1}
  6. I ~ 1 {\tilde{I}}_{1}
  7. I ~ 1 {\tilde{I}}_{1}
  8. A ~ 2 {\tilde{A}}_{2}

Uniformly_convex_space.html

  1. 0 < ϵ 2 0<\epsilon\leq 2
  2. δ > 0 \delta>0
  3. x = 1 \|x\|=1
  4. y = 1 , \|y\|=1,
  5. x + y 2 1 - δ . \left\|\frac{x+y}{2}\right\|\geq 1-\delta.
  6. x - y ϵ \|x-y\|\leq\epsilon
  7. { f n } n = 1 \{f_{n}\}_{n=1}^{\infty}
  8. f f
  9. f n f , \|f_{n}\|\to\|f\|,
  10. f n f_{n}
  11. f f
  12. f n - f 0 \|f_{n}-f\|\to 0
  13. X X
  14. X * X^{*}
  15. ( 1 < p < ) (1<p<\infty)
  16. L L^{\infty}
  17. 2 \mathbb{R}^{2}
  18. x = ( 1 , 1 ) x=(1,1)
  19. y = ( 0 , 1 ) y=(0,1)
  20. x = y = 1 \|x\|_{\infty}=\|y\|_{\infty}=1
  21. x + y = ( 1 , 2 ) = 2 \|x+y\|_{\infty}=\|(1,2)\|_{\infty}=2
  22. x - y = ( 1 , 0 ) = 1 \|x-y\|_{\infty}=\|(1,0)\|_{\infty}=1

Union_of_two_regular_languages.html

  1. L 1 L_{1}
  2. L 2 L_{2}
  3. L 1 L 2 L_{1}\cup L_{2}
  4. L 1 L_{1}
  5. L 2 L_{2}
  6. N 1 , N 2 N_{1},\ N_{2}
  7. L 1 L_{1}
  8. L 2 L_{2}
  9. N 1 = ( Q 1 , Σ , T 1 , q 1 , A 1 ) N_{1}=(Q_{1},\ \Sigma,\ T_{1},\ q_{1},\ A_{1})
  10. N 2 = ( Q 2 , Σ , T 2 , q 2 , A 2 ) N_{2}=(Q_{2},\ \Sigma,\ T_{2},\ q_{2},\ A_{2})
  11. N = ( Q , Σ , T , q 0 , A 1 A 2 ) N=(Q,\ \Sigma,\ T,\ q_{0},\ A_{1}\cup A_{2})
  12. Q = Q 1 Q 2 { q 0 } Q=Q_{1}\cup Q_{2}\cup\{q_{0}\}
  13. T ( q , x ) = { T 1 ( q , x ) if q Q 1 T 2 ( q , x ) if q Q 2 { q 1 , q 2 } if q = q 0 a n d x = ϵ if q = q 0 a n d x ϵ T(q,x)=\left\{\begin{array}[]{lll}T_{1}(q,x)&\mbox{if}&q\in Q_{1}\\ T_{2}(q,x)&\mbox{if}&q\in Q_{2}\\ \{q_{1},q_{2}\}&\mbox{if}&q=q_{0}\ and\ x=\epsilon\\ \emptyset&\mbox{if}&q=q_{0}\ and\ x\neq\epsilon\end{array}\right.
  14. p x , T q p\stackrel{x,T}{\rightarrow}q
  15. q E ( T ( p , x ) ) q\in E(T(p,x))
  16. w w
  17. L 1 L 2 L_{1}\cup L_{2}
  18. w L 1 w\in L_{1}
  19. w = x 1 x 2 x m w=x_{1}x_{2}\cdots x_{m}
  20. m 0 , x i Σ m\geq 0,x_{i}\in\Sigma
  21. N 1 N_{1}
  22. x 1 x 2 x m x_{1}x_{2}\cdots x_{m}
  23. r 0 , r 1 , r m Q 1 r_{0},r_{1},\cdots r_{m}\in Q_{1}
  24. q 1 ϵ , T 1 r 0 x 1 , T 1 r 1 x 2 , T 1 r 2 r m - 1 x m , T 1 r m , r m A 1 q_{1}\stackrel{\epsilon,T_{1}}{\rightarrow}r_{0}\stackrel{x_{1},T_{1}}{% \rightarrow}r_{1}\stackrel{x_{2},T_{1}}{\rightarrow}r_{2}\cdots r_{m-1}% \stackrel{x_{m},T_{1}}{\rightarrow}r_{m},r_{m}\in A_{1}
  25. T 1 ( q , x ) = T ( q , x ) q Q 1 x Σ T_{1}(q,x)=T(q,x)\ \forall q\in Q_{1}\forall x\in\Sigma
  26. r 0 E ( T 1 ( q 1 , ϵ ) ) r 0 E ( T ( q 1 , ϵ ) ) r_{0}\in E(T_{1}(q_{1},\epsilon))\Rightarrow r_{0}\in E(T(q_{1},\epsilon))
  27. r 1 E ( T 1 ( r 0 , x 1 ) ) r 1 E ( T ( r 0 , x 1 ) ) r_{1}\in E(T_{1}(r_{0},x_{1}))\Rightarrow r_{1}\in E(T(r_{0},x_{1}))
  28. \vdots
  29. r m E ( T 1 ( r m - 1 , x m ) ) r m E ( T ( r m - 1 , x m ) ) r_{m}\in E(T_{1}(r_{m-1},x_{m}))\Rightarrow r_{m}\in E(T(r_{m-1},x_{m}))
  30. T T
  31. T 1 T_{1}
  32. q 1 ϵ , T r 0 x 1 , T r 1 x 2 , T r 2 r m - 1 x m , T r m , r m A 1 A 2 , r 0 , r 1 , r m Q q_{1}\stackrel{\epsilon,T}{\rightarrow}r_{0}\stackrel{x_{1},T}{\rightarrow}r_{% 1}\stackrel{x_{2},T}{\rightarrow}r_{2}\cdots r_{m-1}\stackrel{x_{m},T}{% \rightarrow}r_{m},r_{m}\in A_{1}\cup A_{2},r_{0},r_{1},\cdots r_{m}\in Q
  33. T ( q 0 , ϵ ) = { q 1 , q 2 } q 1 T ( q 0 , ϵ ) q 1 E ( T ( q 0 , ϵ ) ) q 0 ϵ , T q 1 \begin{array}[]{lcl}T(q_{0},\epsilon)=\{q_{1},q_{2}\}&\Rightarrow&q_{1}\in T(q% _{0},\epsilon)\\ \\ &\Rightarrow&q_{1}\in E(T(q_{0},\epsilon))\\ \\ &\Rightarrow&q_{0}\stackrel{\epsilon,T}{\rightarrow}q_{1}\end{array}
  34. q 0 ϵ , T q 1 ϵ , T r 0 q 0 ϵ , T r 0 q_{0}\stackrel{\epsilon,T}{\rightarrow}q_{1}\stackrel{\epsilon,T}{\rightarrow}% r_{0}\Rightarrow q_{0}\stackrel{\epsilon,T}{\rightarrow}r_{0}
  35. q 0 ϵ , T r 0 x 1 , T r 1 x 2 , T r 2 r m - 1 x m , T r m , r m A 1 A 2 , r 0 , r 1 , r m Q q_{0}\stackrel{\epsilon,T}{\rightarrow}r_{0}\stackrel{x_{1},T}{\rightarrow}r_{% 1}\stackrel{x_{2},T}{\rightarrow}r_{2}\cdots r_{m-1}\stackrel{x_{m},T}{% \rightarrow}r_{m},r_{m}\in A_{1}\cup A_{2},r_{0},r_{1},\cdots r_{m}\in Q
  36. N N
  37. x 1 x 2 x m x_{1}x_{2}\cdots x_{m}
  38. L 1 L 2 L_{1}\cup L_{2}
  39. L 1 L_{1}
  40. L 2 L_{2}
  41. ϵ \epsilon

Unit_valuation_system.html

  1. 𝐔𝐧𝐢𝐭 𝐕𝐚𝐥𝐮𝐞 = 𝐌𝐚𝐫𝐤𝐞𝐭 𝐕𝐚𝐥𝐮𝐞 𝐨𝐟 𝐈𝐧𝐯𝐞𝐬𝐭𝐦𝐞𝐧𝐭𝐬 + 𝐎𝐭𝐡𝐞𝐫 𝐀𝐬𝐬𝐞𝐭𝐬 + 𝐈𝐧𝐜𝐨𝐦𝐞 - 𝐋𝐢𝐚𝐛𝐢𝐥𝐢𝐭𝐢𝐞𝐬 - 𝐄𝐱𝐩𝐞𝐧𝐬𝐞𝐬 𝐍𝐮𝐦𝐛𝐞𝐫 𝐨𝐟 𝐀𝐥𝐥𝐨𝐜𝐚𝐭𝐞𝐝 𝐔𝐧𝐢𝐭𝐬 \mathbf{Unit\;Value}={\mathbf{Market\;Value\;of\;Investments+Other\;Assets+% Income-Liabilities-Expenses}\over\mathbf{Number\;of\;Allocated\;Units}}

Unitarity_(physics).html

  1. e - i H ^ t e^{-i\hat{H}t}
  2. | M | 2 |M|^{2}
  3. | M | 2 Im ( M ) |M|^{2}\leq\mbox{Im}~{}(M)

Urocanase.html

  1. \rightleftharpoons

UTP—glucose-1-phosphate_uridylyltransferase.html

  1. \rightleftharpoons

V-optimal_histograms.html

  1. W = j = 1 J n j V j , W=\sum_{j=1}^{J}n_{j}V_{j}\,,

VALBOND.html

  1. E ( α ) = k ( S m a x - S ( α ) ) E(\alpha)=k(S^{max}-S(\alpha))
  2. S m a x = 1 1 + m + n ( 1 + 3 m + 5 n ) S^{max}=\sqrt{\frac{1}{1+m+n}}(1+\sqrt{3m}+\sqrt{5n})
  3. S ( α ) = S m a x 1 - 1 - 1 - Δ 2 2 S(\alpha)=S^{max}\sqrt{1-\frac{1-\sqrt{1-\Delta^{2}}}{2}}
  4. Δ = 1 1 + m + n [ 1 + m cos α + n 2 ( 3 cos 2 α - 1 ) ] \Delta=\frac{1}{1+m+n}\left[1+m\cos\alpha+\frac{n}{2}(3\cos^{2}\alpha-1)\right]
  5. % p i = n p w t i j w t j \%p_{i}=\frac{n_{p}wt_{i}}{\sum_{j}wt_{j}}
  6. E t o t = j c j E j E_{tot}=\sum_{j}c_{j}E_{j}
  7. c j = i = 1 h y p e cos 2 α i j = 1 c o n f i g i = 1 h y p e cos 2 α i c_{j}=\frac{\displaystyle\prod_{i=1}^{hype}\cos^{2}\alpha_{i}}{\displaystyle% \sum_{j=1}^{config}\prod_{i=1}^{hype}\cos^{2}\alpha_{i}}
  8. E ( α ) = B O F × k α [ 1 - Δ ( α + π ) 2 ] E(\alpha)=BOF\times k_{\alpha}[1-\Delta(\alpha+\pi)^{2}]
  9. E o f f s e t = i = 1 c o n f i g c i j = 1 h y p e E N i j a + E N i j b 2 E_{offset}=\sum_{i=1}^{config}c_{i}\sum_{j=1}^{hype}\frac{EN_{ija}+EN_{ijb}}{2}
  10. E N i j a = 30 × ( e n l i g - e n c . a . ) × s s EN_{ija}=30\times(en_{lig}-en_{c.a.})\times ss

Valve_audio_amplifier_technical_specification.html

  1. k B T B k_{B}TB
  2. k B k_{B}
  3. 4 k B T B R \sqrt{4k_{B}\cdot T\cdot B\cdot R}
  4. 4 k B T B / R \sqrt{4k_{B}\cdot T\cdot B/R}

Valve_RF_amplifier.html

  1. k B T B k_{B}TB
  2. 4 * k B * T * B * R ) 1 / 2 4*k_{B}*T*B*R)^{1/2}
  3. 4 * k B * T * B / R ) 1 / 2 4*k_{B}*T*B/R)^{1/2}
  4. 1 / 10 1 / 2 1/10^{1/2}

Van_der_Waerden_number.html

  1. W ( r , k ) 2 2 r 2 2 k + 9 W(r,k)\leq 2^{2^{r^{2^{2^{k+9}}}}}
  2. p 2 p W ( 2 , p + 1 ) p\cdot 2^{p}\leq W(2,p+1)
  3. 5 \mathcal{E}^{5}

Van_Wijngaarden_transformation.html

  1. s 0 , k = n = 0 k ( - 1 ) n a n s_{0,k}=\sum_{n=0}^{k}(-1)^{n}a_{n}
  2. s j + 1 , k = s j , k + s j , k + 1 2 \,s_{j+1,k}=\frac{s_{j,k}+s_{j,k+1}}{2}
  3. s j , 0 \scriptstyle s_{j,0}
  4. a 0 , a 1 , , a 12 \scriptstyle a_{0},a_{1},\ldots,a_{12}
  5. s 8 , 4 \scriptstyle s_{8,4}
  6. s 12 , 0 . \scriptstyle s\,_{12,0}.
  7. 1 - 1 3 + 1 5 - 1 7 + = π 4 = 0.7853981 \scriptstyle 1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots=\frac{\pi}{4}=0.7853% 981\ldots
  8. s 0 , 12 = 0.8046006... ( + 2.4 % ) \scriptstyle\,s_{0,12}=0.8046006...(+2.4\%)
  9. s 12 , 0 = 0.7854002... ( + 2.6 × 10 - 6 ) \scriptstyle\,s_{12,0}=0.7854002...(+2.6\times 10^{-6})
  10. s 8 , 4 = 0.7853982... ( + 4.7 × 10 - 8 ) \scriptstyle\,s_{8,4}=0.7853982...(+4.7\times 10^{-8})

Vapor_quality.html

  1. χ \chi
  2. χ = m v a p o r m t o t a l \chi=\frac{m_{vapor}}{m_{total}}
  3. m m
  4. χ = y - y f y g - y f \chi=\frac{y-y_{f}}{y_{g}-y_{f}}
  5. y y
  6. y f y_{f}
  7. y g - y f y_{g}-y_{f}
  8. y f y_{f}
  9. y g y_{g}
  10. χ = m v m l + m v \chi=\frac{m_{v}}{m_{l}+m_{v}}
  11. m v m_{v}
  12. m l m_{l}

Variable-order_Markov_model.html

  1. A A
  2. x 1 n = x 1 x 2 x n x_{1}^{n}=x_{1}x_{2}\dots x_{n}
  3. n n
  4. x i A x_{i}\in A
  5. i i
  6. i i
  7. n n
  8. x i x_{i}
  9. x i + 1 x_{i+1}
  10. x i x i + 1 x_{i}x_{i+1}
  11. x 1 n x_{1}^{n}
  12. P P
  13. P ( x i | s ) P(x_{i}|s)
  14. x i A x_{i}\in A
  15. s A * s\in A^{*}
  16. P ( x i | s ) P(x_{i}|s)
  17. s s
  18. D D
  19. s s
  20. D D

Variable-range_hopping.html

  1. σ = σ 0 e - ( T 0 / T ) 1 / 4 \sigma=\sigma_{0}e^{-(T_{0}/T)^{1/4}}
  2. σ = σ 0 e - ( T 0 / T ) 1 / ( d + 1 ) \sigma=\sigma_{0}e^{-(T_{0}/T)^{1/(d+1)}}
  3. \textstyle\mathcal{R}
  4. R \textstyle R
  5. P exp [ - 2 α R - W k T ] P\sim\exp\left[-2\alpha R-\frac{W}{kT}\right]
  6. = 2 α R + W / k T \textstyle\mathcal{R}=2\alpha R+W/kT
  7. P exp ( - ) \textstyle P\sim\exp(-\mathcal{R})
  8. \textstyle\mathcal{R}
  9. σ exp ( - ¯ n n ) \sigma\sim\exp(-\overline{\mathcal{R}}_{nn})
  10. ¯ n n \textstyle\overline{\mathcal{R}}_{nn}
  11. 𝒩 ( ) \textstyle\mathcal{N}(\mathcal{R})
  12. \textstyle\mathcal{R}
  13. 𝒩 ( ) = K d + 1 \mathcal{N}(\mathcal{R})=K\mathcal{R}^{d+1}
  14. K = N π k T 3 × 2 d α d \textstyle K=\frac{N\pi kT}{3\times 2^{d}\alpha^{d}}
  15. ¯ n n \textstyle\overline{\mathcal{R}}_{nn}
  16. \textstyle\mathcal{R}
  17. P n n ( ) = 𝒩 ( ) exp [ - 𝒩 ( ) ] P_{nn}(\mathcal{R})=\frac{\partial\mathcal{N}(\mathcal{R})}{\partial\mathcal{R% }}\exp[-\mathcal{N}(\mathcal{R})]
  18. ¯ n n = 0 ( d + 1 ) K d + 1 exp ( - K d + 1 ) d \overline{\mathcal{R}}_{nn}=\int_{0}^{\infty}(d+1)K\mathcal{R}^{d+1}\exp(-K% \mathcal{R}^{d+1})d\mathcal{R}
  19. t = K d + 1 \textstyle t=K\mathcal{R}^{d+1}
  20. Γ ( z ) = 0 t z - 1 e - t d t \textstyle\Gamma(z)=\int_{0}^{\infty}t^{z-1}e^{-t}\,\mathrm{d}t
  21. ¯ n n = Γ ( d + 2 d + 1 ) K 1 d + 1 \overline{\mathcal{R}}_{nn}=\frac{\Gamma(\frac{d+2}{d+1})}{K^{\frac{1}{d+1}}}
  22. σ exp ( - T - 1 d + 1 ) \sigma\propto\exp\left(-T^{-\frac{1}{d+1}}\right)

Variational_Monte_Carlo.html

  1. | Ψ ( a ) |\Psi(a)\rangle
  2. a a
  3. a a
  4. \mathcal{H}
  5. X X
  6. E ( a ) = Ψ ( a ) | | Ψ ( a ) Ψ ( a ) | Ψ ( a ) = | Ψ ( X , a ) | 2 Ψ ( X , a ) Ψ ( X , a ) d X | Ψ ( X , a ) | 2 d X . E(a)=\frac{\langle\Psi(a)|\mathcal{H}|\Psi(a)\rangle}{\langle\Psi(a)|\Psi(a)% \rangle}=\frac{\int|\Psi(X,a)|^{2}\frac{\mathcal{H}\Psi(X,a)}{\Psi(X,a)}\,dX}{% \int|\Psi(X,a)|^{2}\,dX}.
  7. | Ψ ( X , a ) | 2 | Ψ ( X , a ) | 2 d X \frac{|\Psi(X,a)|^{2}}{\int|\Psi(X,a)|^{2}\,dX}
  8. E ( a ) E(a)
  9. E loc ( X ) = Ψ ( X , a ) Ψ ( X , a ) E_{\textrm{loc}}(X)=\frac{\mathcal{H}\Psi(X,a)}{\Psi(X,a)}
  10. E ( a ) E(a)
  11. a a
  12. X X
  13. Ψ \Psi
  14. Ψ ( X ) = exp ( u ( r i j ) ) \Psi(X)=\exp(\sum{u(r_{ij})})
  15. r i j r_{ij}
  16. u ( r ) u(r)

Vector_measure.html

  1. ( Ω , ) (\Omega,\mathcal{F})
  2. X X
  3. μ : X \mu:\mathcal{F}\to X
  4. A A
  5. B B
  6. \mathcal{F}
  7. μ ( A B ) = μ ( A ) + μ ( B ) . \mu(A\cup B)=\mu(A)+\mu(B).
  8. μ \mu
  9. ( A i ) i = 1 (A_{i})_{i=1}^{\infty}
  10. \mathcal{F}
  11. \mathcal{F}
  12. μ ( i = 1 A i ) = i = 1 μ ( A i ) \mu\left(\bigcup_{i=1}^{\infty}A_{i}\right)=\sum_{i=1}^{\infty}\mu(A_{i})
  13. X . X.
  14. μ \mu
  15. ( A i ) i = 1 (A_{i})_{i=1}^{\infty}
  16. lim n μ ( i = n A i ) = 0 , ( * ) \lim_{n\to\infty}\left\|\mu\left(\displaystyle\bigcup_{i=n}^{\infty}A_{i}% \right)\right\|=0,\quad\quad\quad(*)
  17. \|\cdot\|
  18. X . X.
  19. [ 0 , ) , [0,\infty),
  20. [ 0 , 1 ] [0,1]
  21. \mathcal{F}
  22. A A
  23. μ ( A ) = χ A \mu(A)=\chi_{A}\,
  24. χ \chi
  25. A . A.
  26. μ \mu
  27. μ , \mu,
  28. \mathcal{F}
  29. L ( [ 0 , 1 ] ) , L^{\infty}([0,1]),
  30. μ , \mu,
  31. \mathcal{F}
  32. L 1 ( [ 0 , 1 ] ) , L^{1}([0,1]),
  33. μ : X , \mu:\mathcal{F}\to X,
  34. | μ | |\mu|
  35. μ \mu
  36. | μ | ( A ) = sup i = 1 n μ ( A i ) |\mu|(A)=\sup\sum_{i=1}^{n}\|\mu(A_{i})\|
  37. A = i = 1 n A i A=\bigcup_{i=1}^{n}A_{i}
  38. A A
  39. A A
  40. \mathcal{F}
  41. \|\cdot\|
  42. X . X.
  43. μ \mu
  44. [ 0 , ] . [0,\infty].
  45. || μ ( A ) || | μ | ( A ) ||\mu(A)||\leq|\mu|(A)
  46. A A
  47. . \mathcal{F}.
  48. | μ | ( Ω ) |\mu|(\Omega)
  49. μ \mu
  50. μ \mu
  51. μ \mu
  52. | μ | |\mu|

Vegard's_law.html

  1. a A ( 1 - x ) B x = ( 1 - x ) a A + 𝑥𝑎 B \mathit{a}_{\mathrm{A}_{(1-x)}\mathrm{B}_{x}}=\mathit{(1-x)}\mathit{a}_{% \mathrm{A}}+\mathit{x}\mathit{a}_{\mathrm{B}}
  2. a a
  3. a InP x As ( 1 - x ) = 𝑥𝑎 InP + ( 1 - x ) a InAs \mathit{a}_{\mathrm{InP}_{x}\mathrm{As}_{(1-x)}}=\mathit{x}\mathit{a}_{\mathrm% {InP}}+(1-\mathit{x})\mathit{a}_{\mathrm{InAs}}
  4. E g \mathit{E_{g}}
  5. E g , InPAs = 𝑥𝐸 g , InP + ( 1 - x ) E g , InAs \mathit{E_{g,\mathrm{InPAs}}}=\mathit{x}\mathit{E_{g,\mathrm{InP}}}+(1-\mathit% {x})\mathit{E_{g,\mathrm{InAs}}}
  6. b b
  7. E g , InPAs = 𝑥𝐸 g , InP + ( 1 - x ) E g , InAs - 𝑏𝑥 ( 1 - x ) \mathit{E_{g,\mathrm{InPAs}}}=\mathit{x}\mathit{E_{g,\mathrm{InP}}}+(1-\mathit% {x})\mathit{E_{g,\mathrm{InAs}}}-\mathit{bx}(1-\mathit{x})

Verlet_list.html

  1. R c + 2 n d R_{c}+2nd
  2. R c R_{c}
  3. d d
  4. N 2 N^{2}
  5. N N
  6. n n
  7. N n 2 Nn^{2}
  8. N N NN
  9. n n
  10. O ( N 2 ) O(N^{2})
  11. O ( N 5 / 3 ) O(N^{5/3})
  12. O ( N ) O(N)

Vertex_(geometry).html

  1. V - E + F = 2 , V-E+F=2,

Vertical_axis_wind_turbine.html

  1. W \vec{W}
  2. U \vec{U}
  3. - ω × R -\vec{\omega}\times\vec{R}
  4. W = U + ( - ω × R ) \vec{W}=\vec{U}+\left(-\vec{\omega}\times\vec{R}\right)
  5. θ = 0 \theta=0{}^{\circ}
  6. θ = 180 \theta=180{}^{\circ}
  7. θ \theta
  8. α \alpha
  9. W = U 1 + 2 λ cos θ + λ 2 W=U\sqrt{1+2\lambda\cos\theta+\lambda^{2}}
  10. α = tan - 1 ( sin θ cos θ + λ ) \alpha=\tan^{-1}\left(\frac{\sin\theta}{\cos\theta+\lambda}\right)
  11. λ = ω R U \lambda=\frac{\omega R}{U}
  12. C L = F L 1 / 2 ρ A W 2 ; C D = D 1 / 2 ρ A W 2 ; C T = T 1 / 2 ρ A U 2 R ; C N = N 1 / 2 ρ A U 2 C_{L}=\frac{F_{L}}{{1}/{2}\;\rho AW^{2}}\,\text{ };\,\text{ }C_{D}=\frac{D}{{1% }/{2}\;\rho AW^{2}}\,\text{ };\,\text{ }C_{T}=\frac{T}{{1}/{2}\;\rho AU^{2}R}% \,\text{ };\,\text{ }C_{N}=\frac{N}{{1}/{2}\;\rho AU^{2}}
  13. P = 1 2 C p ρ A ν 3 P=\frac{1}{2}C_{p}\rho A\nu^{3}
  14. C p C_{p}
  15. ρ \rho
  16. A A
  17. ν \nu

Virtual_fixture.html

  1. 𝐩 = [ x , y , z ] \mathbf{p}=\left[x,y,z\right]
  2. 𝐫 = [ r x , r y , r z ] \mathbf{r}=\left[r_{\textrm{x}},r_{\textrm{y}},r_{\textrm{z}}\right]
  3. F r F_{\textrm{r}}
  4. 𝐮 \mathbf{u}
  5. 𝐯 = 𝐱 ˙ = [ 𝐩 ˙ , 𝐫 ˙ ] \mathbf{v}=\dot{\mathbf{x}}=\left[\dot{\mathbf{p}},\dot{\mathbf{r}}\right]
  6. 𝐯 op \mathbf{v}_{\textrm{op}}
  7. 𝐮 \mathbf{u}
  8. 𝐯 op \mathbf{v}_{\textrm{op}}
  9. 𝐮 = c 𝐯 op \mathbf{u}=c\cdot\mathbf{v}_{\textrm{op}}
  10. c = 1 c=1
  11. c c
  12. 𝐂 \mathbf{C}
  13. 𝐱 ˙ \dot{\mathbf{x}}
  14. 𝐂 \mathbf{C}
  15. c c
  16. 𝐱 6 \mathbf{x}\in\mathbb{R}^{6}
  17. 𝐩 3 \mathbf{p}\in\mathbb{R}^{3}
  18. 𝐃 ( t ) 6 × n , n [ 1..6 ] \mathbf{D}(t)\in\mathbb{R}^{6\times n},~{}n\in[1..6]
  19. t t
  20. n = 1 n=1
  21. 6 \mathbb{R}^{6}
  22. n = 2 n=2
  23. 𝐃 \mathbf{D}
  24. Span ( 𝐃 ) \displaystyle\textrm{Span}(\mathbf{D})
  25. 𝐃 \mathbf{D}
  26. Span ( 𝐃 ) [ 𝐃 ] = 𝐃 ( 𝐃 T 𝐃 ) 𝐃 T \textrm{Span}(\mathbf{D})\equiv\left[\mathbf{D}\right]=\mathbf{D}(\mathbf{D}^{% T}\mathbf{D})^{\dagger}\mathbf{D}^{T}
  27. 𝐃 \mathbf{D}^{\dagger}
  28. 𝐃 \mathbf{D}
  29. 𝐯 D [ 𝐃 ] 𝐯 op and 𝐯 τ 𝐯 op - 𝐯 D = 𝐃 𝐯 op \mathbf{v}_{\textrm{D}}\equiv\left[\mathbf{D}\right]\mathbf{v}_{\textrm{op}}% \textrm{~{}and~{}}\mathbf{v}_{\tau}\equiv\mathbf{v}_{\textrm{op}}-\mathbf{v}_{% \textrm{D}}=\langle\mathbf{D}\rangle\mathbf{v}_{\textrm{op}}
  30. 𝐯 = c 𝐯 op = c ( 𝐯 D + 𝐯 τ ) \mathbf{v}=c\cdot\mathbf{v}_{\textrm{op}}=c\left(\mathbf{v}_{\textrm{D}}+% \mathbf{v}_{\tau}\right)
  31. 𝐯 = c ( 𝐯 D + c τ 𝐯 τ ) = c ( [ 𝐃 ] + c τ 𝐃 ) 𝐯 op \mathbf{v}=c\left(\mathbf{v}_{\textrm{D}}+c_{\tau}\cdot\mathbf{v}_{\tau}\right% )=c\left(\left[\mathbf{D}\right]+c_{\tau}\langle\mathbf{D}\rangle\right)% \mathbf{v}_{\textrm{op}}

Virtual_temperature.html

  1. T v T_{v}
  2. R x = 1000 R * M x , {R_{x}}=1000\frac{R^{*}}{M_{x}}\,,
  3. R * R^{*}
  4. M x M_{x}
  5. x x
  6. M a i r = e p M v + p d p M d , {M_{air}}=\frac{e}{p}M_{v}+\frac{p_{d}}{p}M_{d}\,,
  7. e e
  8. p d p_{d}
  9. M v M_{v}
  10. M d M_{d}
  11. p p
  12. p = p d + e . {p}={p_{d}}+{e}\,.
  13. m d m_{d}
  14. m v m_{v}
  15. V V
  16. ρ = m d + m v V = ρ d + ρ v , {\rho}=\frac{m_{d}+m_{v}}{V}=\rho_{d}+\rho_{v}\,,
  17. ρ d \rho_{d}
  18. ρ v \rho_{v}
  19. e = ρ v R v T {e}=\rho_{v}R_{v}T\,
  20. p d = ρ d R d T . {p_{d}}=\rho_{d}R_{d}T\,.
  21. ρ = p - e R d T + e R v T . {\rho}=\frac{p-e}{R_{d}T}+\frac{e}{R_{v}T}\,.
  22. p p
  23. ϵ = R d R v = M v M d \textstyle\epsilon=\frac{R_{d}}{R_{v}}=\frac{M_{v}}{M_{d}}
  24. p = ρ R d T v , {p}={\rho}R_{d}T_{v}\,,
  25. T v T_{v}
  26. T v = T 1 - e p ( 1 - ϵ ) . {T_{v}}=\frac{T}{1-\frac{e}{p}(1-{\epsilon})}\,.
  27. e p , \scriptstyle\frac{e}{p}\,,
  28. T v T_{v}
  29. w w
  30. e p = w w + ϵ , \frac{e}{p}=\frac{w}{w+{\epsilon}}\,,
  31. T v = T w + ϵ ϵ ( 1 + w ) . {T_{v}}=T\frac{w+\epsilon}{\epsilon(1+w)}\,.
  32. w w
  33. 10 - 3 10^{-3}
  34. ϵ \epsilon
  35. T v T ( 1 + 0.61 w ) . {T_{v}}\approx T(1+0.61w)\,.
  36. T T
  37. w w
  38. T v T + w 6 . {T_{v}}\approx T+\frac{w}{6}\,.

Virtually.html

  1. N H N\rtimes H
  2. N H N\rtimes H
  3. N H N\rtimes H
  4. N H N\rtimes H
  5. N H N\rtimes H
  6. N H N\rtimes H

Visibility_polygon.html

  1. S S
  2. 2 \mathbb{R}^{2}
  3. p p
  4. 2 \mathbb{R}^{2}
  5. V V
  6. 2 \mathbb{R}^{2}
  7. q q
  8. V V
  9. p q pq
  10. S S
  11. p p
  12. S S
  13. p p
  14. S S
  15. V V
  16. \emptyset
  17. θ = 0 , , 2 π \theta=0,\cdots,2\pi
  18. θ \theta
  19. r r
  20. \infty
  21. S S
  22. r r
  23. r r
  24. p p
  25. ( θ , r ) (\theta,r)
  26. V V
  27. V V
  28. p p
  29. S S
  30. V V
  31. \emptyset
  32. b b
  33. S S
  34. v v
  35. b b
  36. p p
  37. v v
  38. r r
  39. p p
  40. v v
  41. θ \theta
  42. v v
  43. p p
  44. b b^{\prime}
  45. S S
  46. r r
  47. r r
  48. p p
  49. b b^{\prime}
  50. ( θ , r ) (\theta,r)
  51. V V
  52. V V
  53. O ( n 2 ) O(n^{2})
  54. n n
  55. O ( n ) O(n)
  56. O ( n log n ) O(n\log n)
  57. 𝒫 \mathcal{P}
  58. p p
  59. 𝒫 \mathcal{P}
  60. p p
  61. 𝒫 \mathcal{P}
  62. 𝒮 = s 0 , s 1 , , s t \mathcal{S}=s_{0},s_{1},\cdots,s_{t}
  63. s t s_{t}
  64. p p
  65. 𝒮 \mathcal{S}
  66. 𝒮 \mathcal{S}
  67. 𝒮 \mathcal{S}
  68. h h
  69. n n
  70. Θ ( n + h log h ) \Theta(n+h\log h)
  71. n n
  72. Θ ( n log n ) \Theta(n\log n)
  73. Θ ( n α ( n ) ) \Theta(n\alpha(n))
  74. α ( n ) \alpha(n)
  75. Θ ( n log n ) \Theta(n\log n)

Vitale's_random_Brunn–Minkowski_inequality.html

  1. K = max { v n | v K } \|K\|=\max\left\{\left.\|v\|_{\mathbb{R}^{n}}\right|v\in K\right\}
  2. E [ X ] = { E [ V ] | V is a selection of X and E V < + } . \mathrm{E}[X]=\{\mathrm{E}[V]|V\mbox{ is a selection of }~{}X\mbox{ and }~{}% \mathrm{E}\|V\|<+\infty\}.
  3. ( vol ( E [ X ] ) ) 1 / n E [ vol ( X ) 1 / n ] , \left(\mathrm{vol}\left(\mathrm{E}[X]\right)\right)^{1/n}\geq\mathrm{E}\left[% \mathrm{vol}(X)^{1/n}\right],

Vitali_covering_lemma.html

  1. B 1 , , B n B_{1},\ldots,B_{n}
  2. B j 1 , B j 2 , , B j m B_{j_{1}},B_{j_{2}},\dots,B_{j_{m}}
  3. B 1 B 2 B n 3 B j 1 3 B j 2 3 B j m B_{1}\cup B_{2}\cup\ldots\cup B_{n}\subseteq 3B_{j_{1}}\cup 3B_{j_{2}}\cup% \ldots\cup 3B_{j_{m}}
  4. 3 B j k 3B_{j_{k}}
  5. B j k B_{j_{k}}
  6. { B j : j J } \{B_{j}:j\in J\}
  7. sup { rad ( B j ) : j J } < \sup\,\{\mathrm{rad}(B_{j}):j\in J\}<\infty
  8. rad ( B j ) \mathrm{rad}(B_{j})
  9. { B j : j J } , J J \{B_{j}:j\in J^{\prime}\},\quad J^{\prime}\subset J
  10. j J B j j J 5 B j . \bigcup_{j\in J}B_{j}\subseteq\bigcup_{j\in J^{\prime}}5\,B_{j}.
  11. B j 1 B_{j_{1}}
  12. B j 1 , , B j k B_{j_{1}},\dots,B_{j_{k}}
  13. B 1 , , B n B_{1},\dots,B_{n}
  14. B j 1 B j 2 B j k B_{j_{1}}\cup B_{j_{2}}\cup\cdots\cup B_{j_{k}}
  15. B j k + 1 B_{j_{k+1}}
  16. X := k = 1 m 3 B j k X:=\bigcup_{k=1}^{m}3\,B_{j_{k}}
  17. B i X B_{i}\subset X
  18. i = 1 , 2 , , n i=1,2,\dots,n
  19. i { j 1 , , j m } i\in\{j_{1},\dots,j_{m}\}
  20. k { 1 , , m } k\in\{1,\dots,m\}
  21. B j k B_{j_{k}}
  22. B j k B_{j_{k}}
  23. B i 3 B j k X B_{i}\subset 3\,B_{j_{k}}\subset X
  24. { B j , j J } \{B_{j},j\in J^{\prime}\}
  25. j J B j j J 5 B j . \bigcup_{j\in J}B_{j}\subseteq\bigcup_{j\in J^{\prime}}5\,B_{j}.
  26. 𝐇 n + 1 = { B 𝐅 n + 1 : B C = , C 𝐆 0 𝐆 1 𝐆 n } , \mathbf{H}_{n+1}=\{B\in\mathbf{F}_{n+1}:\ B\cap C=\emptyset,\ \ \forall C\in% \mathbf{G}_{0}\cup\mathbf{G}_{1}\cup\ldots\cup\mathbf{G}_{n}\},
  27. 𝐆 := n = 0 𝐆 n \mathbf{G}:=\bigcup_{n=0}^{\infty}\mathbf{G}_{n}
  28. λ d \lambda_{d}
  29. { B j : j J } \{B_{j}:j\in J\}
  30. { B j : j J } \{B_{j}:j\in J^{\prime}\}
  31. j J 5 B j j J B j E \bigcup_{j\in J^{\prime}}5B_{j}\supset\bigcup_{j\in J}B_{j}\supset E
  32. λ d ( E ) λ d ( j J B j ) λ d ( j J 5 B j ) j J λ d ( 5 B j ) . \lambda_{d}(E)\leq\lambda_{d}\Bigl(\bigcup_{j\in J}B_{j}\Bigr)\leq\lambda_{d}% \Bigl(\bigcup_{j\in J^{\prime}}5B_{j}\Bigr)\leq\sum_{j\in J^{\prime}}\lambda_{% d}(5B_{j}).
  33. j J λ d ( 5 B j ) = 5 d j J λ d ( B j ) \sum_{j\in J^{\prime}}\lambda_{d}(5B_{j})=5^{d}\sum_{j\in J^{\prime}}\lambda_{% d}(B_{j})
  34. λ d ( E ) 5 d j J λ d ( B j ) . \lambda_{d}(E)\leq 5^{d}\sum_{j\in J^{\prime}}\lambda_{d}(B_{j}).
  35. 𝒱 \mathcal{V}
  36. 𝒱 \mathcal{V}
  37. 𝒱 \mathcal{V}
  38. 𝒱 \mathcal{V}
  39. 𝒱 \mathcal{V}
  40. diam ( V ) d C λ d ( V ) \mathrm{diam}(V)^{d}\leq C\,\lambda_{d}(V)
  41. 𝒱 \mathcal{V}
  42. 𝒱 \mathcal{V}
  43. 𝒱 \mathcal{V}
  44. 𝒱 \mathcal{V}
  45. { U j } 𝒱 \{U_{j}\}\subseteq\mathcal{V}
  46. λ d ( E j U j ) = 0. \lambda_{d}\Bigl(E\setminus\bigcup_{j}U_{j}\Bigr)=0.
  47. 𝒱 \mathcal{V}
  48. 𝒱 \mathcal{V}
  49. { U j } 𝒱 \{U_{j}\}\subseteq\mathcal{V}
  50. H s ( E \ j U j ) = 0 or j diam ( U j ) s = . H^{s}\left(E\backslash\bigcup_{j}U_{j}\right)=0\ \mbox{ or }~{}\sum_{j}\mathrm% {diam}(U_{j})^{s}=\infty.
  51. H s ( E ) j diam ( U j ) s + ε . H^{s}(E)\leq\sum_{j}\mathrm{diam}(U_{j})^{s}+\varepsilon.
  52. { U j } \{U_{j}\}
  53. j diam ( U j ) d C j λ d ( U j ) C λ d ( B ) < + \sum_{j}\mathrm{diam}(U_{j})^{d}\leq C\sum_{j}\lambda_{d}(U_{j})\leq C\,% \lambda_{d}(B)<+\infty
  54. { λ d ( C ) : C G } = n = 0 ( { λ d ( C ) : C G n } ) λ d ( B ( r + 2 ) ) < + . \sum\{\lambda_{d}(C):C\in G\}=\sum_{n=0}^{\infty}\Bigl(\sum\{\lambda_{d}(C):C% \in G_{n}\}\Bigr)\leq\lambda_{d}(B(r+2))<+\infty.
  55. { λ d ( C ) : C G n , n > N } < ε . \sum\{\lambda_{d}(C):C\in G_{n},\,n>N\}<\varepsilon.
  56. Z U N := { 5 C : C G n , n > N } Z\subset U_{N}:=\bigcup\,\{5\,C:C\in G_{n},\,n>N\}
  57. λ d ( U N ) { λ d ( 5 C ) : C G n , n > N } = 5 d { λ d ( C ) : C G n , n > N } < 5 d ε . \lambda_{d}(U_{N})\leq\sum\{\lambda_{d}(5\,C):C\in G_{n},\,n>N\}=5^{d}\sum\{% \lambda_{d}(C):C\in G_{n},\,n>N\}<5^{d}\varepsilon.

Viviani's_theorem.html

  1. u a 2 \frac{u\cdot a}{2}
  2. s a 2 \frac{s\cdot a}{2}
  3. t a 2 \frac{t\cdot a}{2}
  4. u a 2 + s a 2 + t a 2 = h a 2 \frac{u\cdot a}{2}+\frac{s\cdot a}{2}+\frac{t\cdot a}{2}=\frac{h\cdot a}{2}

Volatility_(finance).html

  1. σ T = σ T . \sigma_{T}=\sigma\sqrt{T}.\,
  2. σ = σ S D P . \sigma=\frac{\sigma_{SD}}{\sqrt{P}}.\,
  3. σ annual = 0.01 1 252 = 0.01 252 = 0.1587. \sigma\text{annual}={0.01\over\sqrt{\tfrac{1}{252}}}=0.01\sqrt{252}=0.1587.
  4. σ monthly = 0.1587 1 12 = 0.0458. \sigma\text{monthly}=0.1587\sqrt{\tfrac{1}{12}}=0.0458.
  5. σ T = T 1 / α σ . \sigma_{T}=T^{1/\alpha}\sigma.\,
  6. log ( 1 + y ) = y - 1 2 y 2 + 1 3 y 3 - 1 4 y 4 + \log(1+y)=y-\tfrac{1}{2}y^{2}+\tfrac{1}{3}y^{3}-\tfrac{1}{4}y^{4}+...
  7. CAGR AR - 1 2 σ 2 \mathrm{CAGR}\approx\mathrm{AR}-\tfrac{1}{2}\sigma^{2}
  8. CAGR AR - 1 2 k σ 2 \mathrm{CAGR}\approx\mathrm{AR}-\tfrac{1}{2}k\sigma^{2}

Voltage_graph.html

  1. α : E ( G ) Π \alpha:E(G)\rightarrow\Pi
  2. ( G , α : E ( G ) Π ) (G,\alpha:E(G)\rightarrow\Pi)
  3. ( G , α : E ( G ) Π ) (G,\alpha:E(G)\rightarrow\Pi)
  4. ( G , α : E ( G ) n ) (G,\alpha:E(G)\rightarrow\mathbb{Z}_{n})
  5. G ~ \tilde{G}
  6. V ~ = V × n \tilde{V}=V\times\mathbb{Z}_{n}
  7. E ~ = E × n \tilde{E}=E\times\mathbb{Z}_{n}
  8. ( v , k ) (v,\ k)
  9. ( w , k + α ( e ) ) (w,\ k+\alpha(e))
  10. d \mathbb{Z}^{d}

Volume_operator.html

  1. V ( R ) V(R)
  2. R R
  3. R R
  4. ψ \psi
  5. ψ , V ( R ) ψ \langle\psi,V(R)\psi\rangle
  6. R R

Von_Kármán_constant.html

  1. u = u κ ln z z 0 , u=\frac{u_{\star}}{\kappa}\ln\frac{z}{z_{0}},
  2. u u
  3. u u_{\star}
  4. u = τ w ρ , u_{\star}=\sqrt{\frac{\tau_{w}}{\rho}},

Von_Neumann's_theorem.html

  1. ( T * T ) * = T * T (T^{*}T)^{*}=T^{*}T

Wagner_VI_projection.html

  1. 2 / 3 {2}/\sqrt{3}
  2. x = λ 1 - 3 ( ϕ π ) 2 x=\lambda\sqrt{1-3\left(\frac{\phi}{\pi}\right)^{2}}
  3. y = ϕ y=\phi\,

Wait_Calculation.html

  1. T n o w T t = ( 1 + r ) t 2 \frac{T_{now}}{T_{t}}={(1+r)}^{\tfrac{t}{2}}

Wallenius'_noncentral_hypergeometric_distribution.html

  1. wnchypg ( x ; n , m 1 , m 2 , ω ) = wnchypg ( n - x ; n , m 2 , m 1 , 1 / ω ) . \operatorname{wnchypg}(x;n,m_{1},m_{2},\omega)=\operatorname{wnchypg}(n-x;n,m_% {2},m_{1},1/\omega)\,.
  2. wnchypg ( x ; n , m 1 , m 2 , ω ) = \operatorname{wnchypg}(x;n,m_{1},m_{2},\omega)=
  3. wnchypg ( x - 1 ; n - 1 , m 1 , m 2 , ω ) ( m 1 - x + 1 ) ω ( m 1 - x + 1 ) ω + m 2 + x - n + \operatorname{wnchypg}(x-1;n-1,m_{1},m_{2},\omega)\frac{(m_{1}-x+1)\omega}{(m_% {1}-x+1)\omega+m_{2}+x-n}+
  4. wnchypg ( x ; n - 1 , m 1 , m 2 , ω ) m 2 + x - n + 1 ( m 1 - x ) ω + m 2 + x - n + 1 \operatorname{wnchypg}(x;n-1,m_{1},m_{2},\omega)\frac{m_{2}+x-n+1}{(m_{1}-x)% \omega+m_{2}+x-n+1}
  5. wnchypg ( x ; n , m 1 , m 2 , ω ) = \operatorname{wnchypg}(x;n,m_{1},m_{2},\omega)=
  6. wnchypg ( x - 1 ; n - 1 , m 1 - 1 , m 2 , ω ) m 1 ω m 1 ω + m 2 + \operatorname{wnchypg}(x-1;n-1,m_{1}-1,m_{2},\omega)\frac{m_{1}\omega}{m_{1}% \omega+m_{2}}+
  7. wnchypg ( x ; n - 1 , m 1 , m 2 - 1 , ω ) m 2 m 1 ω + m 2 . \operatorname{wnchypg}(x;n-1,m_{1},m_{2}-1,\omega)\frac{m_{2}}{m_{1}\omega+m_{% 2}}\,.
  8. f 1 ( x ) wnchypg ( x ; n , m 1 , m 2 , ω ) f 2 ( x ) , for ω < 1 , \operatorname{f}_{1}(x)\leq\operatorname{wnchypg}(x;n,m_{1},m_{2},\omega)\leq% \operatorname{f}_{2}(x)\,,\,\,\,\text{for}\,\,\omega<1\,,
  9. f 1 ( x ) wnchypg ( x ; n , m 1 , m 2 , ω ) f 2 ( x ) , for ω > 1 , where \operatorname{f}_{1}(x)\geq\operatorname{wnchypg}(x;n,m_{1},m_{2},\omega)\geq% \operatorname{f}_{2}(x)\,,\,\,\,\text{for}\,\,\omega>1\,,\,\text{where}
  10. f 1 ( x ) = ( m 1 x ) ( m 2 n - x ) n ! ( m 1 + m 2 / ω ) x ¯ ( m 2 + ω ( m 1 - x ) ) n - x ¯ \operatorname{f}_{1}(x)={\left({{m_{1}}\atop{x}}\right)}{\left({{m_{2}}\atop{n% -x}}\right)}\frac{n!}{(m_{1}+m_{2}/\omega)^{\underline{x}}\,(m_{2}+\omega(m_{1% }-x))^{\underline{n-x}}}
  11. f 2 ( x ) = ( m 1 x ) ( m 2 n - x ) n ! ( m 1 + ( m 2 - x 2 ) / ω ) x ¯ ( m 2 + ω m 1 ) n - x ¯ , \operatorname{f}_{2}(x)={\left({{m_{1}}\atop{x}}\right)}{\left({{m_{2}}\atop{n% -x}}\right)}\frac{n!}{(m_{1}+(m_{2}-x_{2})/\omega)^{\underline{x}}\,(m_{2}+% \omega m_{1})^{\underline{n-x}}}\,,
  12. a b ¯ = a ( a - 1 ) ( a - b + 1 ) a^{\underline{b}}=a(a-1)\ldots(a-b+1)
  13. mwnchypg ( ( 0 , , 0 , x j , 0 , ) ; n , 𝐦 , s y m b o l ω ) = m j n ¯ ( 1 ω j i = 1 c m i ω i ) n ¯ \operatorname{mwnchypg}((0,\ldots,0,x_{j},0,\ldots);n,\mathbf{m},symbol{\omega% })=\frac{m_{j}^{\,\,\underline{n}}}{\left(\frac{1}{\omega_{j}}\sum_{i=1}^{c}m_% {i}\omega_{i}\right)^{\underline{n}}}
  14. μ i = m i ( 1 - e ω i θ ) \mu_{i}=m_{i}(1-e^{\omega_{i}\theta})
  15. i = 1 c μ i = n \sum_{i=1}^{c}\mu_{i}=n
  16. mwnchypg ( 𝐱 ; n , 𝐦 , s y m b o l ω ) = mwnchypg ( 𝐱 ; n , 𝐦 , r s y m b o l ω ) \operatorname{mwnchypg}(\mathbf{x};n,\mathbf{m},symbol{\omega})=\operatorname{% mwnchypg}(\mathbf{x};n,\mathbf{m},rsymbol{\omega})\,\,
  17. r + r\in\mathbb{R}_{+}
  18. mwnchypg ( 𝐱 ; n , 𝐦 , ( ω 1 , , ω c - 1 , ω c - 1 ) ) = \operatorname{mwnchypg}\left(\mathbf{x};n,\mathbf{m},(\omega_{1},\ldots,\omega% _{c-1},\omega_{c-1})\right)\,=
  19. mwnchypg ( ( x 1 , , x c - 1 + x c ) ; n , ( m 1 , , m c - 1 + m c ) , ( ω 1 , , ω c - 1 ) ) \operatorname{mwnchypg}\left((x_{1},\ldots,x_{c-1}+x_{c});n,(m_{1},\ldots,m_{c% -1}+m_{c}),(\omega_{1},\ldots,\omega_{c-1})\right)\,\cdot
  20. hypg ( x c ; x c - 1 + x c , m c , m c - 1 + m c ) , \operatorname{hypg}(x_{c};x_{c-1}+x_{c},m_{c},m_{c-1}+m_{c})\,,
  21. hypg ( x ; n , m , N ) \operatorname{hypg}(x;n,m,N)

Walter_Weldon.html

  1. MnO 2 + 4 HCl MnCl 2 + Cl 2 + 2 H 2 O \mathrm{MnO_{2}+4\ HCl\longrightarrow MnCl_{2}+Cl_{2}+2\ H_{2}O}

Walter_Zinn.html

  1. R c r i t = - π M k - 1 R_{crit}=-\frac{\pi M}{\sqrt{k-1}}

Wang_Ganchang.html

  1. π - + C Σ ¯ - + K 0 + K ¯ 0 + K - + p + + π + + π - + n u c l e u s r e c o i l \pi^{-}+C\to\bar{\Sigma}^{-}+K^{0}+\bar{K}^{0}+K^{-}+p^{+}+\pi^{+}+\pi^{-}+% nucleus~{}~{}~{}recoil
  2. Σ ¯ - n ¯ 0 + π - \bar{\Sigma}^{-}\to\bar{n}^{0}+\pi^{-}

Warnock_algorithm.html

  1. O ( n p ) O(np)

Water_model.html

  1. E a b = i on a j on b k C q i q j r i j + A r OO 12 - B r OO 6 E_{ab}=\sum_{i}^{\,\text{on }a}\sum_{j}^{\,\text{on }b}\frac{k_{C}q_{i}q_{j}}{% r_{ij}}+\frac{A}{{r_{\,\text{OO}}}^{12}}-\frac{B}{{r_{\,\text{OO}}}^{6}}
  2. E p o l = 1 2 i ( μ - μ 0 ) 2 α i E_{pol}=\frac{1}{2}\sum_{i}\frac{(\mu-\mu^{0})^{2}}{\alpha_{i}}
  3. S ( r i j ) = { 0 , if r i j R L ( r i j - R L ) 2 ( 3 R U - R L - 2 r i j ) ( R U - R L ) 2 , if R L r i j R U 1 , if R U r i j S(r_{ij})=\begin{cases}0,&\mbox{if }~{}r_{ij}\leq R_{L}\\ \frac{(r_{ij}-R_{L})^{2}(3R_{U}-R_{L}-2r_{ij})}{(R_{U}-R_{L})^{2}},&\mbox{if }% ~{}R_{L}\leq r_{ij}\leq R_{U}\\ 1,&\mbox{if }~{}R_{U}\leq r_{ij}\end{cases}

Watterson_estimator.html

  1. θ = 4 N e μ \theta=4N_{e}\mu
  2. N e N_{e}
  3. μ \mu
  4. n N e n\ll N_{e}
  5. θ \theta
  6. θ ^ w {\hat{\theta}_{w}}
  7. θ ^ w = K a n , {\hat{\theta}_{w}}={K\over a_{n}},
  8. a n = i = 1 n - 1 1 i a_{n}=\sum^{n-1}_{i=1}{1\over i}
  9. θ ^ w {\hat{\theta}_{w}}

Watts_and_Strogatz_model.html

  1. β \beta
  2. N N
  3. K K
  4. β \beta
  5. 0 β 1 0\leq\beta\leq 1
  6. N K ln ( N ) 1 N\gg K\gg\ln(N)\gg 1
  7. N N
  8. N K 2 \frac{NK}{2}
  9. N N
  10. K K
  11. K / 2 K/2
  12. n 0 n N - 1 n_{0}\ldots n_{N-1}
  13. ( n i , n j ) (n_{i},n_{j})
  14. 0 < | i - j | mod ( N - 1 - K 2 ) K 2 0<|i-j|\mod{(}N-1-{\frac{K}{2}}{)}\leq\frac{K}{2}
  15. n i = n 0 , , n N - 1 n_{i}=n_{0},\dots,n_{N-1}
  16. ( n i , n j ) (n_{i},n_{j})
  17. i < j i<j
  18. β \beta
  19. ( n i , n j ) (n_{i},n_{j})
  20. ( n i , n k ) (n_{i},n_{k})
  21. k k
  22. k i k\neq i
  23. ( n i , n k ) (n_{i},n_{k^{\prime}})
  24. k = k k^{\prime}=k
  25. β N K 2 \beta\frac{NK}{2}
  26. β \beta
  27. β = 0 \beta=0
  28. β = 1 \beta=1
  29. G ( n , p ) G(n,p)
  30. n = N n=N
  31. p = N K 2 ( N 2 ) p=\frac{NK}{2{N\choose 2}}
  32. l ( 0 ) = N / 2 K 1 l(0)=N/2K\gg 1
  33. β 1 \beta\rightarrow 1
  34. l ( 1 ) = ln N ln K l(1)=\frac{\ln{N}}{\ln{K}}
  35. 0 < β < 1 0<\beta<1
  36. β \beta
  37. C ( 0 ) = 3 ( K - 2 ) 4 ( K - 1 ) C(0)=\frac{3(K-2)}{4(K-1)}
  38. 3 / 4 3/4
  39. K K
  40. β 1 \beta\rightarrow 1
  41. C ( 1 ) = K / N C(1)=K/N
  42. β \beta
  43. C ( β ) C^{\prime}(\beta)
  44. C ( β ) 3 × number of triangles number of connected triples C^{\prime}(\beta)\equiv\frac{3\times\mbox{number of triangles}~{}}{\mbox{% number of connected triples}~{}}
  45. C ( β ) C ( 0 ) ( 1 - β ) 3 . C^{\prime}(\beta)\sim C(0)\left(1-\beta\right)^{3}.
  46. K K
  47. β 1 \beta\rightarrow 1
  48. 0 < β < 1 0<\beta<1
  49. P ( k ) = n = 0 f ( k , K ) C K / 2 n ( 1 - β ) n β K / 2 - n ( β K / 2 ) k - K / 2 - n ( k - K / 2 - n ) ! e - β K / 2 P(k)=\sum_{n=0}^{f\left(k,K\right)}C^{n}_{K/2}\left(1-\beta\right)^{n}\beta^{K% /2-n}\frac{(\beta K/2)^{k-K/2-n}}{\left(k-K/2-n\right)!}e^{-\beta K/2}
  50. k i k_{i}
  51. i t h i^{th}
  52. k K / 2 k\geq K/2
  53. f ( k , K ) = min ( k - K / 2 , K / 2 ) f(k,K)=\min(k-K/2,K/2)
  54. k = K k=K
  55. | k - K | |k-K|

Weak_measurement.html

  1. | ϕ i |\phi_{i}\rangle
  2. | ϕ f |\phi_{f}\rangle
  3. A w = ϕ f | 𝐀 ^ | ϕ i ϕ f | ϕ i . A_{w}=\frac{\langle\phi_{f}|\hat{\mathbf{A}}|\phi_{i}\rangle}{\langle\phi_{f}|% \phi_{i}\rangle}.
  4. | ϕ f |\phi_{f}\rangle
  5. | ϕ i |\phi_{i}\rangle

Weakly_contractible.html

  1. S S^{\infty}
  2. S n , n 1 S^{n},n\geq 1
  3. S S^{\infty}

Weibull_modulus.html

  1. f ( x ; x 0 , λ , k ) = { k λ ( x - x 0 λ ) k - 1 e - ( ( x - x 0 ) / λ ) k x x 0 , 0 x < x 0 , f(x;x_{0},\lambda,k)=\begin{cases}\frac{k}{\lambda}\left(\frac{x-x_{0}}{% \lambda}\right)^{k-1}e^{-((x-x_{0})/\lambda)^{k}}&x\geq x_{0},\\ 0&x<x_{0},\end{cases}

Weighted_matroid.html

  1. A = { e 1 , e 2 , , e r } A=\{e_{1},e_{2},\ldots,e_{r}\}
  2. B = { f 1 , f 2 , , f r } B=\{f_{1},f_{2},\ldots,f_{r}\}
  3. k k
  4. 1 k r 1\leq k\leq r
  5. O 1 = { e 1 , , e k - 1 } O_{1}=\{e_{1},\ldots,e_{k-1}\}
  6. O 2 = { f 1 , , f k } O_{2}=\{f_{1},\ldots,f_{k}\}
  7. O 1 O_{1}
  8. O 2 O_{2}
  9. O 2 O_{2}
  10. O 1 O_{1}
  11. e k e_{k}
  12. O 1 O_{1}
  13. e k e_{k}
  14. O 2 O_{2}
  15. e k e_{k}
  16. f k f_{k}
  17. k k
  18. A A
  19. B B
  20. I I
  21. I 1 I_{1}
  22. I 2 I_{2}
  23. | I 1 | < | I 2 | |I_{1}|<|I_{2}|
  24. e I 2 I 1 e\in I_{2}\setminus I_{1}
  25. I 1 e I_{1}\cup e
  26. ϵ > 0 \epsilon>0
  27. τ > 0 \tau>0
  28. ( 1 + 2 ϵ ) | I 1 | + τ | E | < | I 2 | (1+2\epsilon)|I_{1}|+\tau|E|<|I_{2}|
  29. I 1 I 2 I_{1}\cup I_{2}
  30. 2 2
  31. 2 + 2 ϵ 2+2\epsilon
  32. I 1 I 2 I_{1}\setminus I_{2}
  33. 1 + ϵ 1+\epsilon
  34. 1 + 2 ϵ 1+2\epsilon
  35. I 2 I 1 I_{2}\setminus I_{1}
  36. 1 1
  37. 1 + ϵ 1+\epsilon
  38. 0
  39. τ \tau
  40. I 1 I_{1}
  41. I 2 I 1 I_{2}\setminus I_{1}
  42. ( 1 + 2 ϵ ) | I 1 | + τ | E | + | I 1 I 2 | (1+2\epsilon)|I_{1}|+\tau|E|+|I_{1}\cup I_{2}|
  43. I 2 I_{2}

Weisz-Prater_Criterion.html

  1. N W - P = R p 2 C s D e f f 3 β N_{W-P}=\dfrac{\mathfrak{R}R^{2}_{p}}{C_{s}D_{eff}}\leq 3\beta
  2. \mathfrak{R}
  3. R p R_{p}
  4. C s C_{s}
  5. D e f f D_{eff}
  6. η \eta
  7. β \beta
  8. η = 3 R p 3 0 R p [ 1 - β ( 1 - r / R p ) n ] r 2 d r \eta=\dfrac{3}{R^{3}_{p}}\int_{0}^{R_{p}}[1-\beta(1-r/R_{p})^{n}]r^{2}\ dr
  9. η = 1 - n β 4 \eta=1-\dfrac{n\beta}{4}
  10. η 0.95 \eta\geq 0.95
  11. β \beta
  12. N W - P 0.3 N_{W-P}\leq 0.3

Weitzenböck's_inequality.html

  1. a a
  2. b b
  3. c c
  4. Δ \Delta
  5. a 2 + b 2 + c 2 4 3 Δ . a^{2}+b^{2}+c^{2}\geq 4\sqrt{3}\,\Delta.
  6. Δ \displaystyle\Delta
  7. ( a 2 - b 2 ) 2 + ( b 2 - c 2 ) 2 + ( c 2 - a 2 ) 2 0 \displaystyle(a^{2}-b^{2})^{2}+(b^{2}-c^{2})^{2}+(c^{2}-a^{2})^{2}\geq 0
  8. a = b = c a=b=c
  9. a 2 + b 2 + c 2 \displaystyle a^{2}+b^{2}+c^{2}
  10. a = b = c a=b=c
  11. 3 24 ( a 2 + b 2 + c 2 - 4 3 Δ ) , \frac{\sqrt{3}}{24}(a^{2}+b^{2}+c^{2}-4\sqrt{3}\Delta),

Well-pointed_category.html

  1. 1 1
  2. f , g : A B f,g:A\to B
  3. f g f\neq g
  4. p : 1 A p:1\to A
  5. f p g p f\circ p\neq g\circ p
  6. p p

Well_drainage.html

  1. Q = 2 π K ( D b - D m ) ( D w - D m ) ln R i R w Q=2\pi K\frac{\left(D_{b}-D_{m}\right)\left(D_{w}-D_{m}\right)}{\ln\frac{R_{i}% }{R_{w}}}
  2. D b D_{b}
  3. D m D_{m}
  4. D w D_{w}
  5. R i R_{i}
  6. R w R_{w}
  7. R i = ( A t π N ) R_{i}=\sqrt{\left(\frac{A_{t}}{\pi N}\right)}
  8. A t A_{t}
  9. Q = q A t N F w Q=q\frac{A_{t}}{NF_{w}}
  10. F w F_{w}
  11. D w - D m = q A t 2 π K ( D b - D m ) N F w ln ( R i R w ) D_{w}-D_{m}=\frac{qA_{t}}{2\pi K(D_{b}-D_{m})NF_{w}}\ln\left(\frac{R_{i}}{R_{w% }}\right)

Well_kill.html

  1. P = h g ρ P=hg\rho
  2. P = h γ P=h\gamma
  3. h = P γ h=\frac{P}{\gamma}
  4. h = 38 M P a 16 k N m - 3 h=\frac{38\,MPa}{16\,kNm^{-3}}
  5. h = 2375 m h=2375\,m

Weyl_integral.html

  1. n = - a n e i n θ \sum_{n=-\infty}^{\infty}a_{n}e^{in\theta}
  2. n = - ( i n ) s a n e i n θ \sum_{n=-\infty}^{\infty}(in)^{s}a_{n}e^{in\theta}

Weyl–Brauer_matrices.html

  1. n n
  2. n n
  3. n n
  4. k k
  5. k k
  6. V V
  7. q 1 2 + + q k 2 + p 1 2 + + p k 2 ( + p n 2 ) , q_{1}^{2}+\dots+q_{k}^{2}+p_{1}^{2}+\dots+p_{k}^{2}~{}~{}(+p_{n}^{2})~{},
  8. 1 = ( 1 0 0 1 ) , 1 = ( 1 0 0 - 1 ) , P = ( 0 1 1 0 ) , Q = ( 0 i - i 0 ) \begin{matrix}{1}=\left(\begin{matrix}1&0\\ 0&1\end{matrix}\right),&{1}^{\prime}=\left(\begin{matrix}1&0\\ 0&-1\end{matrix}\right),\\ P=\left(\begin{matrix}0&1\\ 1&0\end{matrix}\right),&Q=\left(\begin{matrix}0&i\\ -i&0\end{matrix}\right)\end{matrix}
  9. P i = 1 1 P 1 1 P_{i}={1}^{\prime}\otimes\dots\otimes{1}^{\prime}\otimes P\otimes{1}\otimes% \dots\otimes{1}
  10. Q i = 1 1 Q 1 1 Q_{i}={1}^{\prime}\otimes\dots\otimes{1}^{\prime}\otimes Q\otimes{1}\otimes% \dots\otimes{1}
  11. \otimes
  12. P i 2 = 1 , i = 1 , 2 , , 2 k P_{i}^{2}=1,i=1,2,...,2k
  13. P i P j = - P j P i P_{i}P_{j}=-P_{j}P_{i}
  14. P i R ( P ) i = j R i j P j P_{i}\mapsto R(P)_{i}=\sum_{j}R_{ij}P_{j}
  15. R ( P ) i = S ( R ) P i S ( R ) - 1 R(P)_{i}=S(R)P_{i}S(R)^{-1}
  16. C = P Q P Q . C=P\otimes Q\otimes P\otimes\dots\otimes Q.
  17. t P i t S ( R ) - 1 P i t t S ( R ) = ( C S ( R ) C - 1 ) t P i ( C S ( R ) C - 1 ) - 1 \hbox{ }^{t}P_{i}\rightarrow\,^{t}S(R)^{-1}\,{}^{t}P_{i}\,^{t}S(R)=(CS(R)C^{-1% })\,^{t}P_{i}(CS(R)C^{-1})^{-1}
  18. U = 1 1 U={1}^{\prime}\otimes\dots\otimes{1}^{\prime}
  19. P n = 1 1 P_{n}={1}^{\prime}\otimes\dots\otimes{1}^{\prime}
  20. R ( P ) i = j R i j P j R(P)_{i}=\sum_{j}R_{ij}P_{j}
  21. R ( P ) i = S ( R ) P i S ( R ) - 1 R(P)_{i}=S(R)P_{i}S(R)^{-1}

Wheel_chock.html

  1. c h o c k h e i g h t r a d i u s o f t h e w h e e l ÷ 0.032 \frac{chock\ height}{radius\ of\ the\ wheel}\div 0.032

Whewell_equation.html

  1. φ \varphi
  2. s s
  3. s s
  4. φ \varphi
  5. d r d s = ( d x / d s d y / d s ) = ( cos φ sin φ ) since | d r d s | = 1 , \frac{d\vec{r}}{ds}=\begin{pmatrix}dx/ds\\ dy/ds\end{pmatrix}=\begin{pmatrix}\cos\varphi\\ \sin\varphi\end{pmatrix}\quad\text{since}\quad\left|\frac{d\vec{r}}{ds}\right|% =1,
  6. d y d x = tan φ . \frac{dy}{dx}=\tan\varphi.
  7. x = cos φ d s x=\int\cos\varphi\,ds
  8. y = sin φ d s y=\int\sin\varphi\,ds
  9. κ = d φ d s , \kappa=\frac{d\varphi}{ds},
  10. φ = c \varphi=c
  11. s = a φ s=a\varphi
  12. s = a tan φ s=a\tan\varphi

Wien_bridge.html

  1. ω 2 = 1 R x R 2 C x C 2 \omega^{2}={1\over R_{x}R_{2}C_{x}C_{2}}
  2. C x C 2 = R 4 R 3 - R 2 R x . {C_{x}\over C_{2}}={R_{4}\over R_{3}}-{R_{2}\over R_{x}}\,.

Wilhelmy_plate.html

  1. γ \gamma
  2. γ = F l cos ( θ ) \gamma=\frac{F}{l\cos(\theta)}
  3. l l
  4. 2 w + 2 d 2w+2d
  5. w w
  6. d d
  7. θ \theta
  8. θ = 0 \theta=0

William_Rutherford_(mathematician).html

  1. π 4 = 4 arctan ( 1 5 ) - arctan ( 1 70 ) + arctan ( 1 99 ) {\pi\over 4}=4\arctan\left({1\over 5}\right)-\arctan\left({1\over 70}\right)+% \arctan\left({1\over 99}\right)

Wilson–Bappu_effect.html

  1. M V = 33.2 - 18.0 l o g ( W 0 ) M_{V}=33.2-18.0log(W_{0})

Wind_engineering.html

  1. v z = v g ( z z g ) 1 α , 0 < z < z g \ v_{z}=v_{g}\cdot\left(\frac{z}{z_{g}}\right)^{\frac{1}{\alpha}},0<z<z_{g}
  2. v z \ v_{z}
  3. z \ z
  4. v g \ v_{g}
  5. z g \ z_{g}
  6. α \ \alpha
  7. v w ( h ) = v r e f ( h h r e f ) a \ v_{w}(h)=v_{ref}\cdot\left(\frac{h}{h_{ref}}\right)^{a}
  8. v w ( h ) \ v_{w}(h)
  9. h h
  10. v r e f \ v_{ref}
  11. h r e f h_{ref}
  12. a \ a

Wirtinger_inequality_(2-forms).html

  1. M M
  2. k k
  3. ( 2 k ) (2k)
  4. k ! k!
  5. ω k ( ζ ) k ! . \omega^{k}(\zeta)\leq k!\,.
  6. ω k k ! \textstyle{\frac{\omega^{k}}{k!}}
  7. M M

Woods–Saxon_potential.html

  1. = =
  2. V ( r ) = - V 0 1 + exp ( r - R a ) V(r)=-\frac{V_{0}}{1+\exp({r-R\over a})}
  3. R = r 0 A 1 / 3 R=r_{0}A^{1/3}

Wythoff's_game.html

  1. n k = k ϕ = m k ϕ - m k n_{k}=\lfloor k\phi\rfloor=\lfloor m_{k}\phi\rfloor-m_{k}\,
  2. m k = k ϕ 2 = n k ϕ = n k + k m_{k}=\lfloor k\phi^{2}\rfloor=\lceil n_{k}\phi\rceil=n_{k}+k\,
  3. 1 ϕ + 1 ϕ 2 = 1 . \frac{1}{\phi}+\frac{1}{\phi^{2}}=1\,.

X-bar_chart.html

  1. x ¯ \bar{x}
  2. x ¯ \bar{x}

Yenice,_Çanakkale.html

  1. M w M_{\mathrm{w}}

Yetter–Drinfeld_category.html

  1. Δ \Delta
  2. ( V , s y m b o l . ) (V,symbol{.})
  3. s y m b o l . : H V V symbol{.}:H\otimes V\to V
  4. ( V , δ ) (V,\delta\;)
  5. δ : V H V \delta:V\to H\otimes V
  6. s y m b o l . symbol{.}
  7. δ \delta
  8. δ ( h s y m b o l . v ) = h ( 1 ) v ( - 1 ) S ( h ( 3 ) ) h ( 2 ) s y m b o l . v ( 0 ) \delta(hsymbol{.}v)=h_{(1)}v_{(-1)}S(h_{(3)})\otimes h_{(2)}symbol{.}v_{(0)}
  9. h H , v V h\in H,v\in V
  10. ( Δ id ) Δ ( h ) = h ( 1 ) h ( 2 ) h ( 3 ) H H H (\Delta\otimes\mathrm{id})\Delta(h)=h_{(1)}\otimes h_{(2)}\otimes h_{(3)}\in H% \otimes H\otimes H
  11. h H h\in H
  12. δ ( v ) = v ( - 1 ) v ( 0 ) \delta(v)=v_{(-1)}\otimes v_{(0)}
  13. δ ( v ) = 1 v \delta(v)=1\otimes v
  14. V = k { v } V=k\{v\}
  15. h s y m b o l . v = ϵ ( h ) v hsymbol{.}v=\epsilon(h)v
  16. δ ( v ) = 1 v \delta(v)=1\otimes v
  17. V = g G V g V=\bigoplus_{g\in G}V_{g}
  18. V g V_{g}
  19. V = g G V g V=\bigoplus_{g\in G}V_{g}
  20. g . V h V g h g - 1 g.V_{h}\subset V_{ghg^{-1}}
  21. k = k=\mathbb{C}\;
  22. [ g ] G [g]\subset G\;
  23. χ , X \chi,X\;
  24. C e n t ( g ) Cent(g)\;
  25. g [ g ] g\in[g]
  26. V = 𝒪 [ g ] χ = 𝒪 [ g ] X V = h [ g ] V h = h [ g ] X V=\mathcal{O}_{[g]}^{\chi}=\mathcal{O}_{[g]}^{X}\qquad V=\bigoplus_{h\in[g]}V_% {h}=\bigoplus_{h\in[g]}X
  27. 𝒪 [ g ] χ \mathcal{O}_{[g]}^{\chi}
  28. χ , X \chi,X\;
  29. I n d C e n t ( g ) G ( χ ) = k G k C e n t ( g ) X Ind_{Cent(g)}^{G}(\chi)=kG\otimes_{kCent(g)}X
  30. t v k G k C e n t ( g ) X = V t\otimes v\in kG\otimes_{kCent(g)}X=V
  31. t v V t g t - 1 t\otimes v\in V_{tgt^{-1}}
  32. V V\;
  33. t i t_{i}\;
  34. C e n t ( g ) Cent(g)\;
  35. h v [ g ] × X t i v k G k C e n t ( g ) X with uniquely h = t i g t i - 1 h\otimes v\subset[g]\times X\;\;\leftrightarrow\;\;t_{i}\otimes v\in kG\otimes% _{kCent(g)}X\qquad\,\text{with uniquely}\;\;h=t_{i}gt_{i}^{-1}
  36. h v V h h\otimes v\in V_{h}
  37. c V , W : V W W V c_{V,W}:V\otimes W\to W\otimes V
  38. c ( v w ) := v ( - 1 ) s y m b o l . w v ( 0 ) , c(v\otimes w):=v_{(-1)}symbol{.}w\otimes v_{(0)},
  39. c V , W - 1 ( w v ) := v ( 0 ) S ( v ( - 1 ) ) s y m b o l . w . c_{V,W}^{-1}(w\otimes v):=v_{(0)}\otimes S(v_{(-1)})symbol{.}w.
  40. ( c V , W id U ) ( id V c U , W ) ( c U , V id W ) = ( id W c U , V ) ( c U , W id V ) ( id U c V , W ) : U V W W V U . (c_{V,W}\otimes\mathrm{id}_{U})(\mathrm{id}_{V}\otimes c_{U,W})(c_{U,V}\otimes% \mathrm{id}_{W})=(\mathrm{id}_{W}\otimes c_{U,V})(c_{U,W}\otimes\mathrm{id}_{V% })(\mathrm{id}_{U}\otimes c_{V,W}):U\otimes V\otimes W\to W\otimes V\otimes U.
  41. 𝒞 \mathcal{C}
  42. 𝒴 H H 𝒟 {}^{H}_{H}\mathcal{YD}

Young_measure.html

  1. { f k } k = 1 \{f_{k}\}_{k=1}^{\infty}
  2. L ( U , m ) L^{\infty}(U,\mathbb{R}^{m})
  3. U U
  4. n \mathbb{R}^{n}
  5. { f k j } j = 1 { f k } k = 1 \{f_{k_{j}}\}_{j=1}^{\infty}\subset\{f_{k}\}_{k=1}^{\infty}
  6. x U x\in U
  7. ν x \nu_{x}
  8. m \mathbb{R}^{m}
  9. F C ( m ) F\in C(\mathbb{R}^{m})
  10. F ( f k j ) m F ( y ) d ν ( y ) F(f_{k_{j}})\overset{\ast}{\rightharpoonup}\int_{\mathbb{R}^{m}}F(y)d\nu_{% \cdot}(y)
  11. L ( U ) L^{\infty}(U)
  12. ν x \nu_{x}
  13. { f k } k = 1 \{f_{k}\}_{k=1}^{\infty}
  14. I ( u ) = 0 1 ( u x 2 - 1 ) 2 + u 2 d x I(u)=\int_{0}^{1}(u_{x}^{2}-1)^{2}+u^{2}dx
  15. u ( 0 ) = u ( 1 ) = 0 u(0)=u(1)=0
  16. ν x = 1 2 δ - 1 + 1 2 δ 1 \nu_{x}=\frac{1}{2}\delta_{-1}+\frac{1}{2}\delta_{1}
  17. ± 1 \pm 1
  18. ± 1 \pm 1

Young–Laplace_equation.html

  1. Δ p \displaystyle\Delta p
  2. Δ p \Delta p
  3. n ^ \hat{n}
  4. H H
  5. R 1 R_{1}
  6. R 2 R_{2}
  7. 2 H 2H
  8. Δ p = 2 γ R . \Delta p=\frac{2\gamma}{R}.
  9. R = a cos θ R=\frac{a}{\cos\theta}
  10. Δ p = 2 γ cos θ a . \Delta p=\frac{2\gamma\cos\theta}{a}.
  11. h = 2 γ cos θ ρ g a . h=\frac{2\gamma\cos\theta}{\rho ga}.
  12. h 1.4 × 10 - 5 a h\approx{{1.4\times 10^{-5}}\over a}
  13. Δ p = ρ g h - γ ( 1 R 1 + 1 R 2 ) \Delta p=\rho gh-\gamma\left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right)
  14. L c = γ ρ g , L_{c}=\sqrt{\frac{\gamma}{\rho g}},
  15. p c = γ L c = γ ρ g . p_{c}=\frac{\gamma}{L_{c}}=\sqrt{\gamma\rho g}.
  16. h * - Δ p * = ( 1 R 1 * + 1 R 2 * ) . h^{*}-\Delta p^{*}=\left(\frac{1}{{R_{1}}^{*}}+\frac{1}{{R_{2}}^{*}}\right).
  17. r ′′ ( 1 + r 2 ) 3 2 - 1 r ( z ) 1 + r 2 = z - Δ p * \frac{r^{\prime\prime}}{(1+r^{\prime 2})^{\frac{3}{2}}}-\frac{1}{r(z)\sqrt{1+r% ^{\prime 2}}}=z-\Delta p^{*}
  18. z ′′ ( 1 + z 2 ) 3 2 + z r ( 1 + z 2 ) 1 2 = Δ p * - z ( r ) . \frac{z^{\prime\prime}}{(1+z^{\prime 2})^{\frac{3}{2}}}+\frac{z^{\prime}}{r(1+% z^{\prime 2})^{\frac{1}{2}}}=\Delta p^{*}-z(r).

Z-channel_(information_theory).html

  1. 𝖼𝖺𝗉 ( ) \mathsf{cap}(\mathbb{Z})
  2. \mathbb{Z}
  3. 𝖼𝖺𝗉 ( ) = max α { 𝖧 ( Y ) - 𝖧 ( Y X ) } = max α { 𝖧 ( Y ) - x { 0 , 1 } 𝖧 ( Y X = x ) 𝖯𝗋𝗈𝖻 { X = x } } \mathsf{cap}(\mathbb{Z})=\max_{\alpha}\{\mathsf{H}(Y)-\mathsf{H}(Y\mid X)\}=% \max_{\alpha}\Bigl\{\mathsf{H}(Y)-\sum_{x\in\{0,1\}}\mathsf{H}(Y\mid X=x)% \mathsf{Prob}\{X=x\}\Bigr\}
  4. = max α { 𝖧 ( ( 1 - α ) ( 1 - p ) ) - 𝖧 ( Y X = 1 ) 𝖯𝗋𝗈𝖻 { X = 1 } } =\max_{\alpha}\{\mathsf{H}((1-\alpha)(1-p))-\mathsf{H}(Y\mid X=1)\mathsf{Prob}% \{X=1\}\}
  5. = max α { 𝖧 ( ( 1 - α ) ( 1 - p ) ) - ( 1 - α ) 𝖧 ( p ) } , =\max_{\alpha}\{\mathsf{H}((1-\alpha)(1-p))-(1-\alpha)\mathsf{H}(p)\},
  6. 𝖧 ( ) \mathsf{H}(\cdot)
  7. α = 1 - 1 ( 1 - p ) ( 1 + 2 𝖧 ( p ) / ( 1 - p ) ) , \alpha=1-\frac{1}{(1-p)(1+2^{\mathsf{H}(p)/(1-p)})},
  8. 𝖼𝖺𝗉 ( ) \mathsf{cap}(\mathbb{Z})
  9. 𝖼𝖺𝗉 ( ) = 𝖧 ( 1 1 + 2 𝗌 ( p ) ) - 𝗌 ( p ) 1 + 2 𝗌 ( p ) = log 2 ( 1 + 2 - 𝗌 ( p ) ) = log 2 ( 1 + ( 1 - p ) p p / ( 1 - p ) ) where 𝗌 ( p ) = 𝖧 ( p ) 1 - p . \mathsf{cap}(\mathbb{Z})=\mathsf{H}\left(\frac{1}{1+2^{\mathsf{s}(p)}}\right)-% \frac{\mathsf{s}(p)}{1+2^{\mathsf{s}(p)}}=\log_{2}(1{+}2^{-\mathsf{s}(p)})=% \log_{2}\left(1+(1-p)p^{p/(1-p)}\right)\;\textrm{ where }\;\mathsf{s}(p)=\frac% {\mathsf{H}(p)}{1-p}.
  10. 𝖼𝖺𝗉 ( ) 1 - 0.5 𝖧 ( p ) \mathsf{cap}(\mathbb{Z})\approx 1-0.5\mathsf{H}(p)\,
  11. 1 - 𝖧 ( p ) 1{-}\mathsf{H}(p)
  12. 𝖽 A ( 𝐱 , 𝐲 ) \mathsf{d}_{A}(\mathbf{x},\mathbf{y})
  13. 𝐱 , 𝐲 { 0 , 1 } n \mathbf{x},\mathbf{y}\in\{0,1\}^{n}
  14. 𝖽 A ( 𝐱 , 𝐲 ) = max { | { i x i = 0 , y i = 1 } | , | { i x i = 1 , y i = 0 } | } . \mathsf{d}_{A}(\mathbf{x},\mathbf{y})\stackrel{\vartriangle}{=}\max\left\{\big% |\{i\mid x_{i}=0,y_{i}=1\}\big|,\big|\{i\mid x_{i}=1,y_{i}=0\}\big|\right\}.
  15. V t ( 𝐱 ) V_{t}(\mathbf{x})
  16. 𝐱 { 0 , 1 } n \mathbf{x}\in\{0,1\}^{n}
  17. 𝐱 \mathbf{x}
  18. V t ( 𝐱 ) = { 𝐲 { 0 , 1 } n 𝖽 A ( 𝐱 , 𝐲 ) t } . V_{t}(\mathbf{x})=\{\mathbf{y}\in\{0,1\}^{n}\mid\mathsf{d}_{A}(\mathbf{x},% \mathbf{y})\leq t\}.
  19. 𝒞 \mathcal{C}
  20. 𝐜 𝐜 { 0 , 1 } n \mathbf{c}\neq\mathbf{c}^{\prime}\in\{0,1\}^{n}
  21. V t ( 𝐜 ) V t ( 𝐜 ) = V_{t}(\mathbf{c})\cap V_{t}(\mathbf{c}^{\prime})=\emptyset
  22. M ( n , t ) M(n,t)
  23. M ( n , t ) 2 n + 1 j = 0 t ( ( n / 2 j ) + ( n / 2 j ) ) . M(n,t)\leq\frac{2^{n+1}}{\sum_{j=0}^{t}{\left({\left({{\lfloor n/2\rfloor}% \atop{j}}\right)}+{\left({{\lceil n/2\rceil}\atop{j}}\right)}\right)}}.
  24. B 0 = 2 , B i = min 0 j < i { B j + A ( n + t + i - j - 1 , 2 t + 2 , t + i ) } B_{0}=2,\quad B_{i}=\min_{0\leq j<i}\{B_{j}+A(n{+}t{+}i{-}j{-}1,2t{+}2,t{+}i)\}
  25. i > 0 i>0
  26. M ( n , t ) B n - 2 t - 1 . M(n,t)\leq B_{n-2t-1}.

Zakai_equation.html

  1. d x = f ( x , t ) d t + d w dx=f(x,t)dt+dw
  2. d z = h ( x , t ) d t + d v dz=h(x,t)dt+dv
  3. d w , d v dw,dv
  4. p ( x , t ) p(x,t)
  5. d p = L ( p ) d t + p h T d z dp=L(p)dt+ph^{T}dz
  6. L = - ( f i p ) x i + 1 2 2 p x i x j L=-\sum\frac{\partial(f_{i}p)}{\partial x_{i}}+\frac{1}{2}\sum\frac{\partial^{% 2}p}{\partial x_{i}\partial x_{j}}

Zero-fuel_weight.html

  1. Z F W + F O B = T O W ZFW+FOB=TOW
  2. M a x P a y l o a d = M Z F W - O E W MaxPayload=MZFW-OEW

Zero_dagger.html

  1. ( L , , U ) (L,\in,U)

Zero_Forcing_Equalizer.html

  1. F ( f ) F(f)
  2. C ( f ) C(f)
  3. C ( f ) = 1 / F ( f ) C(f)=1/F(f)
  4. F ( f ) C ( f ) = 1 F(f)C(f)=1
  5. C ( f ) C(f)
  6. C ( f ) = 1 / ( F ( f ) + k ) C(f)=1/(F(f)+k)

Zerodur.html

  1. CTE ± 0.007 × 10 - 6 / K \mathrm{CTE}\pm 0.007\times 10^{-6}/K

Zeuthen_strategy.html

  1. Risk ( i , t ) = { 1 U i ( δ ( i , t ) ) = 0 U i ( δ ( i , t ) ) - U i ( δ ( j , t ) ) U i ( δ ( i , t ) ) otherwise \,\text{Risk}(i,t)=\begin{cases}1&U_{i}(\delta(i,t))=0\\ \frac{U_{i}(\delta(i,t))-U_{i}(\delta(j,t))}{U_{i}(\delta(i,t))}&\,\text{% otherwise}\end{cases}
  2. δ = arg max δ S C ( A , t ) { U A ( δ ) } \delta^{\prime}=\arg\max_{\delta\in{SC(A,t)}}\{U_{A}(\delta)\}
  3. δ ( A , 0 ) = arg max δ N S U A ( δ ) \delta(A,0)=\arg\max_{\delta\in{NS}}U_{A}(\delta)
  4. δ = arg max δ N S { π ( δ ) } , \delta=\arg\max_{\delta^{\prime}\in{NS}}\{\pi(\delta^{\prime})\},
  5. t t
  6. R i s k ( A , t ) R i s k ( B , t ) . Risk(A,t)\leq Risk(B,t).
  7. U A ( δ A ) - U A ( δ B ) U A ( δ A ) U B ( δ B ) - U B ( δ A ) U B ( δ B ) \frac{U_{A}(\delta_{A})-U_{A}(\delta_{B})}{U_{A}(\delta_{A})}\leq\frac{U_{B}(% \delta_{B})-U_{B}(\delta_{A})}{U_{B}(\delta_{B})}
  8. U B ( δ B ) ( U A ( δ A ) - U A ( δ B ) ) U A ( δ A ) ( U B ( δ B ) - U B ( δ A ) ) U_{B}(\delta_{B})(U_{A}(\delta_{A})-U_{A}(\delta_{B}))\leq U_{A}(\delta_{A})(U% _{B}(\delta_{B})-U_{B}(\delta_{A}))
  9. U A ( δ A ) U B ( δ B ) - U A ( δ B ) U B ( δ B ) U A ( δ A ) U B ( δ B ) - U A ( δ A ) U B ( δ A ) U_{A}(\delta_{A})U_{B}(\delta_{B})-U_{A}(\delta_{B})U_{B}(\delta_{B})\leq U_{A% }(\delta_{A})U_{B}(\delta_{B})-U_{A}(\delta_{A})U_{B}(\delta_{A})
  10. - U A ( δ B ) U B ( δ B ) - U A ( δ A ) U B ( δ A ) -U_{A}(\delta_{B})U_{B}(\delta_{B})\leq-U_{A}(\delta_{A})U_{B}(\delta_{A})
  11. U A ( δ A ) U B ( δ A ) U A ( δ B ) U B ( δ B ) U_{A}(\delta_{A})U_{B}(\delta_{A})\leq U_{A}(\delta_{B})U_{B}(\delta_{B})
  12. π ( δ A ) π ( δ B ) \pi(\delta_{A})\leq\pi(\delta_{B})
  13. δ A \delta_{A}

Zobel_network.html

  1. Z Z 0 = Z 0 Z \frac{Z}{Z_{0}}=\frac{Z_{0}}{Z^{\prime}}
  2. Z = 1 Z Z=\frac{1}{Z^{\prime}}
  3. Z \scriptstyle Z^{\prime}
  4. Z \scriptstyle Z
  5. 1 Z in = 1 Z 0 + Z + 1 Z + Z 0 \frac{1}{Z\text{in}}=\frac{1}{Z_{0}+Z^{\prime}}+\frac{1}{Z+Z_{0}}
  6. Z = Z 0 2 Z Z^{\prime}=\frac{Z_{0}^{2}}{Z}
  7. Z in = Z 0 Z\text{in}=Z_{0}
  8. Z 0 = R 0 Z_{0}=R_{0}\!\,
  9. A ( ω ) = Z 0 Z + Z 0 A(\omega)=\frac{Z_{0}}{Z+Z_{0}}
  10. Z B = Z 0 Z_{B}=Z_{0}\,\!
  11. Z = R Z=R\,\!
  12. Z = R = R 0 2 R Z^{\prime}=R^{\prime}=\frac{R_{0}^{2}}{R}
  13. L = 20 log ( R R 0 + 1 ) dB L=20\log\left(\frac{R}{R_{0}}+1\right)\,\,\text{dB}
  14. Z = i ω L Z=i\omega L\!\,
  15. Z = 1 i ω C Z^{\prime}=\frac{1}{i\omega C^{\prime}}
  16. C = L R 0 2 C^{\prime}=\frac{L}{R_{0}^{2}}
  17. A ( ω ) = R 0 i ω L + R 0 A(\omega)=\frac{R_{0}}{i\omega L+R_{0}}
  18. ω c = R 0 L \omega_{c}=\frac{R_{0}}{L}
  19. A ( ω ) R 0 i ω L A(\omega)\approx\frac{R_{0}}{i\omega L}
  20. Z = 1 i ω C Z=\frac{1}{i\omega C}
  21. Z = i ω L Z^{\prime}=i\omega L^{\prime}\!\,
  22. L = C R 0 2 L^{\prime}=CR_{0}^{2}\!\,
  23. A ( ω ) = i ω C R 0 1 + i ω C R 0 A(\omega)=\frac{i\omega CR_{0}}{1+i\omega CR_{0}}
  24. ω c = 1 C R 0 \omega_{c}=\frac{1}{CR_{0}}
  25. A ( ω ) i ω C R 0 A(\omega)\approx i\omega CR_{0}
  26. Z = i ω L + 1 i ω C Z=i\omega L+\frac{1}{i\omega C}
  27. Y = 1 Z = i ω C + 1 i ω L Y^{\prime}=\frac{1}{Z^{\prime}}=i\omega C^{\prime}+\frac{1}{i\omega L^{\prime}}
  28. A ( ω ) = i ω C R 0 1 + i ω C R 0 - ω 2 L C A(\omega)=\frac{i\omega CR_{0}}{1+i\omega CR_{0}-\omega^{2}LC}
  29. ω c = 1 2 L C ( ± R 0 C + R 0 2 C 2 + 4 L C ) \omega_{c}=\frac{1}{2LC}\left(\pm R_{0}C+\sqrt{R_{0}^{2}C^{2}+4LC}\right)
  30. Δ ω \displaystyle\Delta\omega
  31. ω 0 = 1 L C \omega_{0}=\sqrt{\frac{1}{LC}}
  32. ω m 2 = ( Δ ω 2 ) 2 + ω 0 2 \omega_{m}^{2}=\left(\frac{\Delta\omega}{2}\right)^{2}+\omega_{0}^{2}
  33. Y = 1 Z = i ω C + 1 i ω L Y=\frac{1}{Z}=i\omega C+\frac{1}{i\omega L}
  34. Z = i ω L + 1 i ω C Z^{\prime}=i\omega L^{\prime}+\frac{1}{i\omega C^{\prime}}
  35. Δ ω = 1 C R 0 \Delta\omega=\frac{1}{CR_{0}}
  36. ω m = ( 1 2 R 0 C ) 2 + 1 L C \omega_{m}=\sqrt{\left(\frac{1}{2R_{0}C}\right)^{2}+\frac{1}{LC}}
  37. R 0 = 1 \scriptstyle R_{0}\;=\;1
  38. ω c = 1 \scriptstyle\omega_{c}\;=\;1
  39. f c 1 \scriptstyle f_{c1}
  40. f c 2 \scriptstyle f_{c2}
  41. f c \scriptstyle f_{c}
  42. Z B \scriptstyle Z_{B}
  43. R 0 \scriptstyle R_{0}
  44. R 0 \scriptstyle R_{0}
  45. R 0 \scriptstyle R_{0}
  46. R 0 \scriptstyle R_{0}
  47. V O V i n = R 0 Z + R 0 \frac{V_{O}}{V_{in}}=\frac{R_{0}}{Z+R_{0}}
  48. 1 / 2 {1}/{2}
  49. R 0 \scriptstyle\approx R_{0}
  50. R B = \scriptstyle R_{B}\;=\;\infty
  51. f \scriptstyle\sqrt{f}
  52. A ( ω ) = R 0 Z i n + R 0 A(\omega)=\frac{R_{0}}{Z_{in}+R_{0}}

Zoeppritz_equations.html

  1. R ( θ ) = R ( 0 ) + G sin 2 θ + F ( tan 2 θ - sin 2 θ ) R(\theta)=R(0)+G\sin^{2}\theta+F(\tan^{2}\theta-\sin^{2}\theta)
  2. R ( 0 ) = 1 2 ( Δ V P V P + Δ ρ ρ ) R(0)=\frac{1}{2}\left(\frac{\Delta V_{\mathrm{P}}}{V_{\mathrm{P}}}+\frac{% \Delta\rho}{\rho}\right)
  3. G = 1 2 Δ V P V P - 2 V S 2 V P 2 ( Δ ρ ρ + 2 Δ V S V S ) G=\frac{1}{2}\frac{\Delta V_{\mathrm{P}}}{V_{\mathrm{P}}}-2\frac{V^{2}_{% \mathrm{S}}}{V^{2}_{\mathrm{P}}}\left(\frac{\Delta\rho}{\rho}+2\frac{\Delta V_% {\mathrm{S}}}{V_{\mathrm{S}}}\right)
  4. F = 1 2 Δ V P V P F=\frac{1}{2}\frac{\Delta V_{\mathrm{P}}}{V_{\mathrm{P}}}
  5. θ {\theta}
  6. V p {V_{p}}
  7. Δ V p {{\Delta}V_{p}}
  8. V s {V_{s}}
  9. Δ V s {{\Delta}V_{s}}
  10. < m t p l > ρ <mtpl>{{\rho}}
  11. Δ ρ {{\Delta}{\rho}}
  12. R ( θ ) = R ( 0 ) - A sin 2 θ R(\theta)=R(0)-A\sin^{2}\theta
  13. A = 2 V S 2 V P 2 ( Δ ρ ρ + 2 Δ V S V S ) A=2\frac{V^{2}_{\mathrm{S}}}{V^{2}_{\mathrm{P}}}\left(\frac{\Delta\rho}{\rho}+% 2\frac{\Delta V_{\mathrm{S}}}{V_{\mathrm{S}}}\right)
  14. R ( θ ) = R ( 0 ) + G sin 2 θ R(\theta)=R(0)+G\sin^{2}\theta

Zonal_polynomial.html

  1. ( S 2 n , H n ) (S_{2n},H_{n})
  2. H n H_{n}
  3. ( G l n ( ) , O n ) (Gl_{n}(\mathbb{R}),O_{n})
  4. [ H n \ S 2 n / H n ] \mathbb{C}[H_{n}\backslash S_{2n}/H_{n}]
  5. [ O d ( ) \ M d ( ) / O d ( ) ] \mathbb{C}[O_{d}(\mathbb{R})\backslash M_{d}(\mathbb{R})/O_{d}(\mathbb{R})]
  6. α = 2 \alpha=2

Zuckerman_functor.html

  1. Γ g , L K g , K ( W ) = hom R ( g , L K ) ( R ( g , K ) , W ) K \Gamma^{g,K}_{g,L\cap K}(W)=\hom_{R(g,L\cap K)}(R(g,K),W)_{K}
  2. Π g , L K g , K ( W ) = R ( g , K ) R ( g , L K ) W . \Pi^{g,K}_{g,L\cap K}(W)=R(g,K)\otimes_{R(g,L\cap K)}W.

Łukasiewicz_logic.html

  1. \rightarrow
  2. ¬ \neg
  3. \leftrightarrow
  4. \wedge
  5. \otimes
  6. \vee
  7. \oplus
  8. 0 ¯ \overline{0}
  9. 1 ¯ \overline{1}
  10. A ( B A ) A\rightarrow(B\rightarrow A)
  11. ( A B ) ( ( B C ) ( A C ) ) (A\rightarrow B)\rightarrow((B\rightarrow C)\rightarrow(A\rightarrow C))
  12. ( ( A B ) B ) ( ( B A ) A ) ((A\rightarrow B)\rightarrow B)\rightarrow((B\rightarrow A)\rightarrow A)
  13. ( ¬ B ¬ A ) ( A B ) . (\neg B\rightarrow\neg A)\rightarrow(A\rightarrow B).
  14. ( A B ) ( A ( A B ) ) (A\wedge B)\rightarrow(A\otimes(A\rightarrow B))
  15. ¬ ¬ A A . \neg\neg A\rightarrow A.
  16. w ( θ ϕ ) = F ( w ( θ ) , w ( ϕ ) ) w(\theta\circ\phi)=F_{\circ}(w(\theta),w(\phi))
  17. , \circ,
  18. w ( ¬ θ ) = F ¬ ( w ( θ ) ) , w(\neg\theta)=F_{\neg}(w(\theta)),
  19. w ( 0 ¯ ) = 0 w(\overline{0})=0
  20. w ( 1 ¯ ) = 1 , w(\overline{1})=1,
  21. F ( x , y ) = min { 1 , 1 - x + y } F_{\rightarrow}(x,y)=\min\{1,1-x+y\}
  22. F ( x , y ) = 1 - | x - y | F_{\leftrightarrow}(x,y)=1-|x-y|
  23. F ¬ ( x ) = 1 - x F_{\neg}(x)=1-x
  24. F ( x , y ) = min { x , y } F_{\wedge}(x,y)=\min\{x,y\}
  25. F ( x , y ) = max { x , y } F_{\vee}(x,y)=\max\{x,y\}
  26. F ( x , y ) = max { 0 , x + y - 1 } F_{\otimes}(x,y)=\max\{0,x+y-1\}
  27. F ( x , y ) = min { 1 , x + y } . F_{\oplus}(x,y)=\min\{1,x+y\}.
  28. F F_{\otimes}
  29. F F_{\oplus}
  30. F F_{\rightarrow}
  31. A A
  32. A A
  33. A A
  34. A A

Π_pad.html

  1. Z 1 Z_{1}\,\!
  2. Z 2 Z_{2}\,\!
  3. A = V out V in A=\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}
  4. L = V in V out L=\frac{V_{\mathrm{in}}}{V_{\mathrm{out}}}
  5. L dB = 20 log L L_{\mathrm{dB}}=20\log L\,\!
  6. L = e γ L=e^{\gamma}\,
  7. γ \gamma\,
  8. Y i Π = Y 2 + Y Z Y_{\mathrm{i\Pi}}=\sqrt{Y^{2}+\frac{Y}{Z}}
  9. Z iT = Z 2 + Z Y Z_{\mathrm{iT}}=\sqrt{Z^{2}+\frac{Z}{Y}}
  10. L L1 = Z i Π Y iT e γ L L_{\mathrm{L1}}=\sqrt{Z_{\mathrm{i\Pi}}Y_{\mathrm{iT}}}\ e^{\gamma_{\mathrm{L}}}
  11. γ L = sinh - 1 Z Y \gamma_{\mathrm{L}}=\sinh^{-1}{\sqrt{ZY}}
  12. L L2 = Z iT Y i Π e γ L L_{\mathrm{L2}}=\sqrt{Z_{\mathrm{iT}}Y_{\mathrm{i\Pi}}}\ e^{\gamma_{\mathrm{L}}}
  13. L Π = L L1 L L2 = e 2 γ L = e γ Π L_{\mathrm{\Pi}}=L_{\mathrm{L1}}L_{\mathrm{L2}}=e^{2\gamma_{\mathrm{L}}}=e^{% \gamma_{\mathrm{\Pi}}}\,
  14. γ Π = 2 γ L = 2 sinh - 1 R 2 2 R 1 \gamma_{\mathrm{\Pi}}=2\gamma_{\mathrm{L}}=2\sinh^{-1}{\sqrt{\frac{R_{2}}{2R_{% 1}}}}\,
  15. 1 Z 0 = 1 R 1 2 + 2 R 1 R 2 \frac{1}{Z_{0}}=\sqrt{\frac{1}{{R_{1}}^{2}}+\frac{2}{R_{1}R_{2}}}
  16. Z 0 = Z 1 = Z 2 Z_{0}=Z_{1}=Z_{2}\,
  17. R 1 = Z 0 coth ( γ Π 2 ) = Z 0 1 + A 1 - A R_{1}=Z_{0}\coth\left(\frac{\gamma_{\mathrm{\Pi}}}{2}\right)=Z_{0}\frac{1+A}{1% -A}
  18. A = V out V in A=\frac{V_{\mathrm{out}}}{V_{\mathrm{in}}}
  19. R 2 = 2 R 1 ( R 1 Z 0 ) 2 - 1 R_{2}=\frac{2R_{1}}{\left(\frac{R_{1}}{Z_{0}}\right)^{2}-1}