wpmath0000007_0

't_Hooft_symbol.html

  1. η μ ν a = { ϵ a μ ν μ , ν = 1 , 2 , 3 - δ a ν μ = 4 δ a μ ν = 4 0 μ = ν = 4 . \eta^{a}_{\mu\nu}=\begin{cases}\epsilon^{a\mu\nu}&\mu,\nu=1,2,3\\ -\delta^{a\nu}&\mu=4\\ \delta^{a\mu}&\nu=4\\ 0&\mu=\nu=4\end{cases}.
  2. a = 1 , 2 , 3 ; μ , ν = 1 , 2 , 3 , 4 ; ϵ 1234 = + 1 a=1,2,3;~{}\mu,\nu=1,2,3,4;~{}\epsilon_{1234}=+1
  3. η a μ ν = ϵ a μ ν 4 + δ a μ δ ν 4 - δ a ν δ μ 4 \eta_{a\mu\nu}=\epsilon_{a\mu\nu 4}+\delta_{a\mu}\delta_{\nu 4}-\delta_{a\nu}% \delta_{\mu 4}
  4. η ¯ a μ ν = ϵ a μ ν 4 - δ a μ δ ν 4 + δ a ν δ μ 4 \bar{\eta}_{a\mu\nu}=\epsilon_{a\mu\nu 4}-\delta_{a\mu}\delta_{\nu 4}+\delta_{% a\nu}\delta_{\mu 4}
  5. η a μ ν = 1 2 ϵ μ ν ρ σ η a ρ σ , η ¯ a μ ν = - 1 2 ϵ μ ν ρ σ η ¯ a ρ σ \eta_{a\mu\nu}=\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\eta_{a\rho\sigma}\ ,% \qquad\bar{\eta}_{a\mu\nu}=-\frac{1}{2}\epsilon_{\mu\nu\rho\sigma}\bar{\eta}_{% a\rho\sigma}
  6. ϵ a b c η b μ ν η c ρ σ = δ μ ρ η a ν σ + δ ν σ η a μ ρ - δ μ σ η a ν ρ - δ ν ρ η a μ σ \epsilon_{abc}\eta_{b\mu\nu}\eta_{c\rho\sigma}=\delta_{\mu\rho}\eta_{a\nu% \sigma}+\delta_{\nu\sigma}\eta_{a\mu\rho}-\delta_{\mu\sigma}\eta_{a\nu\rho}-% \delta_{\nu\rho}\eta_{a\mu\sigma}
  7. η a μ ν η a ρ σ = δ μ ρ δ ν σ - δ μ σ δ ν ρ + ϵ μ ν ρ σ , \eta_{a\mu\nu}\eta_{a\rho\sigma}=\delta_{\mu\rho}\delta_{\nu\sigma}-\delta_{% \mu\sigma}\delta_{\nu\rho}+\epsilon_{\mu\nu\rho\sigma}\ ,
  8. η a μ ρ η b μ σ = δ a b δ ρ σ + ϵ a b c η c ρ σ , \eta_{a\mu\rho}\eta_{b\mu\sigma}=\delta_{ab}\delta_{\rho\sigma}+\epsilon_{abc}% \eta_{c\rho\sigma}\ ,
  9. ϵ μ ν ρ θ η a σ θ = δ σ μ η a ν ρ + δ σ ρ η a μ ν - δ σ ν η a μ ρ , \epsilon_{\mu\nu\rho\theta}\eta_{a\sigma\theta}=\delta_{\sigma\mu}\eta_{a\nu% \rho}+\delta_{\sigma\rho}\eta_{a\mu\nu}-\delta_{\sigma\nu}\eta_{a\mu\rho}\ ,
  10. η a μ ν η a μ ν = 12 , η a μ ν η b μ ν = 4 δ a b , η a μ ρ η a μ σ = 3 δ ρ σ . \eta_{a\mu\nu}\eta_{a\mu\nu}=12\ ,\quad\eta_{a\mu\nu}\eta_{b\mu\nu}=4\delta_{% ab}\ ,\quad\eta_{a\mu\rho}\eta_{a\mu\sigma}=3\delta_{\rho\sigma}\ .
  11. η ¯ \bar{\eta}
  12. η ¯ a μ ν η ¯ a ρ σ = δ μ ρ δ ν σ - δ μ σ δ ν ρ - ϵ μ ν ρ σ . \bar{\eta}_{a\mu\nu}\bar{\eta}_{a\rho\sigma}=\delta_{\mu\rho}\delta_{\nu\sigma% }-\delta_{\mu\sigma}\delta_{\nu\rho}-\epsilon_{\mu\nu\rho\sigma}\ .
  13. ϵ μ ν ρ θ η ¯ a σ θ = - δ σ μ η ¯ a ν ρ - δ σ ρ η ¯ a μ ν + δ σ ν η ¯ a μ ρ , \epsilon_{\mu\nu\rho\theta}\bar{\eta}_{a\sigma\theta}=-\delta_{\sigma\mu}\bar{% \eta}_{a\nu\rho}-\delta_{\sigma\rho}\bar{\eta}_{a\mu\nu}+\delta_{\sigma\nu}% \bar{\eta}_{a\mu\rho}\ ,
  14. η a μ ν η ¯ b μ ν = 0 \eta_{a\mu\nu}\bar{\eta}_{b\mu\nu}=0

(a,b)-tree.html

  1. a a
  2. b b
  3. a a
  4. b b
  5. 2 a ( b + 1 ) / 2 2≤a≤(b+1)/2
  6. b b
  7. a a
  8. b b
  9. 2 a ( b + 1 ) / 2 2≤a≤(b+1)/2
  10. T T
  11. a a
  12. b b
  13. b b
  14. v v
  15. ρ v \rho_{v}
  16. S v [ 1 ρ v ] S_{v}[1\dots\rho_{v}]
  17. H v [ 1 ρ v - 1 ] H_{v}[1\dots\rho_{v}-1]
  18. H v [ i ] H_{v}[i]
  19. S v [ i ] S_{v}[i]

(SAT,_ε-UNSAT).html

  1. L ( SAT , ϵ - UNSAT ) L\leq(\mbox{SAT}~{},\epsilon-\mbox{UNSAT}~{})
  2. y 1 , y 2 , , y m y_{1},y_{2},\ldots,y_{m}
  3. ϕ = r ϕ r \phi=\bigwedge_{r}\phi_{r}
  4. ϕ r \phi_{r}
  5. ϕ r = f r ( y i 1 , y i 2 , , y i q ) \phi_{r}=f_{r}(y_{i_{1}},y_{i_{2}},\ldots,y_{i_{q}})
  6. f r f_{r}
  7. 2 q 2^{q}
  8. q 2 q q2^{q}
  9. x L x\in L
  10. ϕ ( y 1 , y 2 , , y m ) \phi(y_{1},y_{2},\ldots,y_{m})
  11. x L x\notin L
  12. y 1 , y 2 , , y m y_{1},y_{2},\ldots,y_{m}
  13. f r f_{r}
  14. ϵ = 1 2 q 2 q \epsilon=\frac{1}{2q2^{q}}
  15. L ( SAT , ϵ - UNSAT ) L\leq(\mbox{SAT}~{},\epsilon-\mbox{UNSAT}~{})
  16. ( S A T , ϵ - U N S A T ) P C P ( O ( log n ) , O ( 1 ) ) (SAT,\epsilon-UNSAT)\in PCP(O(\log n),O(1))
  17. O ( 1 / ϵ ) O(1/\epsilon)
  18. log n \log n
  19. n n
  20. O ( log n / ϵ ) = O ( log n ) O(\log n/\epsilon)=O(\log n)
  21. O ( 3 / ϵ ) = O ( 1 ) O(3/\epsilon)=O(1)
  22. ( 1 - ϵ ) 1 / ϵ 1 / e < 1 / 2 (1-\epsilon)^{1/\epsilon}\leq 1/e<1/2
  23. ( S A T , ϵ - U N S A T ) P C P ( O ( log n ) , O ( 1 ) ) (SAT,\epsilon-UNSAT)\in PCP(O(\log n),O(1))

*-autonomous_category.html

  1. \bot
  2. \bot
  3. A , : A ( A ) \partial_{A,\bot}:A\to(A\Rightarrow\bot)\Rightarrow\bot
  4. eval A , A γ A , A : ( A ) A \mathrm{eval}_{A,A\Rightarrow\bot}\circ\gamma_{A\Rightarrow\bot,A}:(A% \Rightarrow\bot)\otimes A\to\bot
  5. \bot
  6. A , \partial_{A,\bot}
  7. ( - ) * : C op C (-)^{*}:C^{\mathrm{op}}\to C
  8. A A * * A\cong{A^{*}}^{*}
  9. Hom ( A B , C * ) Hom ( A , ( B C ) * ) \mathrm{Hom}(A\otimes B,C^{*})\cong\mathrm{Hom}(A,(B\otimes C)^{*})
  10. = I * \bot=I^{*}
  11. A * B * ( B A ) * A^{*}\otimes B^{*}\to(B\otimes A)^{*}

0_(year).html

  1. n n

177_(number).html

  1. [ 17 89 71 113 59 5 47 29 101 ] . \begin{bmatrix}17&89&71\\ 113&59&5\\ 47&29&101\end{bmatrix}.

183_(number).html

  1. n 2 - n + 1 n^{2}-n+1
  2. n = 14 n=14

2-choice_hashing.html

  1. log 2 ln n + θ ( m / n ) \log_{2}\ln n+\theta(m/n)
  2. θ ( l o g ( l o g ( n ) ) ) θ(log(log(n)))

2-sided.html

  1. F F
  2. M M
  3. M M
  4. h : F × [ - 1 , 1 ] M h\colon F\times[-1,1]\to M
  5. h ( x , 0 ) = x h(x,0)=x
  6. x F x\in F
  7. h ( F × [ - 1 , 1 ] ) M = h ( F × [ - 1 , 1 ] ) h(F\times[-1,1])\cap\partial M=h(\partial F\times[-1,1])

257-gon.html

  1. A = 257 4 t 2 cot π 257 A=\frac{257}{4}t^{2}\cot\frac{\pi}{257}
  2. cos π 257 \cos\frac{\pi}{257}
  3. cos 2 π 257 \cos\frac{2\pi}{257}
  4. 257 2 = 128 \left\lfloor\frac{257}{2}\right\rfloor=128
  5. \scriptstyle\angle{}
  6. \scriptstyle\angle{}

2DEG.html

  1. μ \mu
  2. μ \mu
  3. \ell
  4. = v F τ = 2 π n μ e 5.2 μ m × μ [ 10 6 cm 2 / Vs ] n [ 10 11 cm - 2 ] \ell=v_{F}\tau=\sqrt{2\pi n}\frac{\hbar\mu}{e}\approx 5.2\ \mu\mathrm{m}\times% \mu\ [10^{6}\ \mathrm{cm^{2}/Vs}]\sqrt{n\ [10^{11}\ \mathrm{cm^{-2}}]}
  5. n n
  6. μ \mu
  7. n n

3char.html

  1. 26 * 36 * 36 26*36*36
  2. 36 3 36^{3}

4-Hydroxybenzoic_acid.html

  1. \overrightarrow{\leftarrow}

4Pi_microscope.html

  1. Ω \Omega
  2. Ω = 2 π \Omega=2\pi
  3. Ω = 4 π \Omega=4\pi
  4. Ω \Omega
  5. 1.3 π 1.3\pi
  6. 4 π 4\pi
  7. 4 π 4\pi

53_equal_temperament.html

  1. ( 3 / 2 ) 53 (3/2)^{53}
  2. ( 2 / 1 ) 31 (2/1)^{31}
  3. 177147 / 176776 177147/176776
  4. ( 3 53 / 2 84 = 19383245667680019896796723 / 19342813113834066795298816 ) (3^{53}/2^{84}=19383245667680019896796723/19342813113834066795298816)

65537-gon.html

  1. A = 65537 4 t 2 cot π 65537 A=\frac{65537}{4}t^{2}\cot\frac{\pi}{65537}
  2. cos π 65537 \cos\frac{\pi}{65537}
  3. cos 2 π 65537 \cos\frac{2\pi}{65537}
  4. 65537 2 = 32768 \left\lfloor\frac{65537}{2}\right\rfloor=32768

Abel's_identity.html

  1. y ′′ + p ( x ) y + q ( x ) y = 0 y^{\prime\prime}+p(x)y^{\prime}+q(x)\,y=0
  2. W ( y 1 , y 2 ) ( x ) = | y 1 ( x ) y 2 ( x ) y 1 ( x ) y 2 ( x ) | = y 1 ( x ) y 2 ( x ) - y 1 ( x ) y 2 ( x ) , x I , W(y_{1},y_{2})(x)=\begin{vmatrix}y_{1}(x)&y_{2}(x)\\ y^{\prime}_{1}(x)&y^{\prime}_{2}(x)\end{vmatrix}=y_{1}(x)\,y^{\prime}_{2}(x)-y% ^{\prime}_{1}(x)\,y_{2}(x),\qquad x\in I,
  3. W ( y 1 , y 2 ) ( x ) = W ( y 1 , y 2 ) ( x 0 ) exp ( - x 0 x p ( ξ ) d ξ ) , x I , W(y_{1},y_{2})(x)=W(y_{1},y_{2})(x_{0})\exp\biggl(-\int_{x_{0}}^{x}p(\xi)\,% \textrm{d}\xi\biggr),\qquad x\in I,
  4. W \displaystyle W^{\prime}
  5. y ′′ y^{\prime\prime}
  6. y ′′ = - ( p y + q y ) . y^{\prime\prime}=-(py^{\prime}+qy).\,
  7. W \displaystyle W^{\prime}
  8. V ( x ) = W ( x ) exp ( x 0 x p ( ξ ) d ξ ) , x I , V(x)=W(x)\exp\left(\int_{x_{0}}^{x}p(\xi)\,\textrm{d}\xi\right),\qquad x\in I,
  9. V ( x ) = ( W ( x ) + W ( x ) p ( x ) ) exp ( x 0 x p ( ξ ) d ξ ) = 0 , x I , V^{\prime}(x)=\bigl(W^{\prime}(x)+W(x)p(x)\bigr)\exp\biggl(\int_{x_{0}}^{x}p(% \xi)\,\textrm{d}\xi\biggr)=0,\qquad x\in I,
  10. y ( n ) + p n - 1 ( x ) y ( n - 1 ) + + p 1 ( x ) y + p 0 ( x ) y = 0 , y^{(n)}+p_{n-1}(x)\,y^{(n-1)}+\cdots+p_{1}(x)\,y^{\prime}+p_{0}(x)\,y=0,
  11. W ( y 1 , , y n ) ( x ) = | y 1 ( x ) y 2 ( x ) y n ( x ) y 1 ( x ) y 2 ( x ) y n ( x ) y 1 ( n - 1 ) ( x ) y 2 ( n - 1 ) ( x ) y n ( n - 1 ) ( x ) | , x I , W(y_{1},\ldots,y_{n})(x)=\begin{vmatrix}y_{1}(x)&y_{2}(x)&\cdots&y_{n}(x)\\ y^{\prime}_{1}(x)&y^{\prime}_{2}(x)&\cdots&y^{\prime}_{n}(x)\\ \vdots&\vdots&\ddots&\vdots\\ y_{1}^{(n-1)}(x)&y_{2}^{(n-1)}(x)&\cdots&y_{n}^{(n-1)}(x)\end{vmatrix},\qquad x% \in I,
  12. W ( y 1 , , y n ) ( x ) = W ( y 1 , , y n ) ( x 0 ) exp ( - x 0 x p n - 1 ( ξ ) d ξ ) , x I , W(y_{1},\ldots,y_{n})(x)=W(y_{1},\ldots,y_{n})(x_{0})\exp\biggl(-\int_{x_{0}}^% {x}p_{n-1}(\xi)\,\textrm{d}\xi\biggr),\qquad x\in I,
  13. W = - p n - 1 W , W^{\prime}=-p_{n-1}\,W,
  14. W \displaystyle W^{\prime}
  15. W = | y 1 y 2 y n y 1 y 2 y n y 1 ( n - 2 ) y 2 ( n - 2 ) y n ( n - 2 ) y 1 ( n ) y 2 ( n ) y n ( n ) | . W^{\prime}=\begin{vmatrix}y_{1}&y_{2}&\cdots&y_{n}\\ y^{\prime}_{1}&y^{\prime}_{2}&\cdots&y^{\prime}_{n}\\ \vdots&\vdots&\ddots&\vdots\\ y_{1}^{(n-2)}&y_{2}^{(n-2)}&\cdots&y_{n}^{(n-2)}\\ y_{1}^{(n)}&y_{2}^{(n)}&\cdots&y_{n}^{(n)}\end{vmatrix}.
  16. y i ( n ) + p n - 2 y i ( n - 2 ) + + p 1 y i + p 0 y i = - p n - 1 y i ( n - 1 ) y_{i}^{(n)}+p_{n-2}\,y_{i}^{(n-2)}+\cdots+p_{1}\,y^{\prime}_{i}+p_{0}\,y_{i}=-% p_{n-1}\,y_{i}^{(n-1)}
  17. W = | y 1 y 2 y n y 1 y 2 y n y 1 ( n - 2 ) y 2 ( n - 2 ) y n ( n - 2 ) - p n - 1 y 1 ( n - 1 ) - p n - 1 y 2 ( n - 1 ) - p n - 1 y n ( n - 1 ) | = - p n - 1 W . W^{\prime}=\begin{vmatrix}y_{1}&y_{2}&\cdots&y_{n}\\ y^{\prime}_{1}&y^{\prime}_{2}&\cdots&y^{\prime}_{n}\\ \vdots&\vdots&\ddots&\vdots\\ y_{1}^{(n-2)}&y_{2}^{(n-2)}&\cdots&y_{n}^{(n-2)}\\ -p_{n-1}\,y_{1}^{(n-1)}&-p_{n-1}\,y_{2}^{(n-1)}&\cdots&-p_{n-1}\,y_{n}^{(n-1)}% \end{vmatrix}=-p_{n-1}W.
  18. Φ ( x ) = ( y 1 ( x ) y 2 ( x ) y n ( x ) y 1 ( x ) y 2 ( x ) y n ( x ) y 1 ( n - 2 ) ( x ) y 2 ( n - 2 ) ( x ) y n ( n - 2 ) ( x ) y 1 ( n - 1 ) ( x ) y 2 ( n - 1 ) ( x ) y n ( n - 1 ) ( x ) ) , x I , \Phi(x)=\begin{pmatrix}y_{1}(x)&y_{2}(x)&\cdots&y_{n}(x)\\ y^{\prime}_{1}(x)&y^{\prime}_{2}(x)&\cdots&y^{\prime}_{n}(x)\\ \vdots&\vdots&\ddots&\vdots\\ y_{1}^{(n-2)}(x)&y_{2}^{(n-2)}(x)&\cdots&y_{n}^{(n-2)}(x)\\ y_{1}^{(n-1)}(x)&y_{2}^{(n-1)}(x)&\cdots&y_{n}^{(n-1)}(x)\end{pmatrix},\qquad x% \in I,
  19. ( y y ′′ y ( n - 1 ) y ( n ) ) = ( 0 1 0 0 0 0 1 0 0 0 0 1 - p 0 ( x ) - p 1 ( x ) - p 2 ( x ) - p n - 1 ( x ) ) ( y y y ( n - 2 ) y ( n - 1 ) ) . \begin{pmatrix}y^{\prime}\\ y^{\prime\prime}\\ \vdots\\ y^{(n-1)}\\ y^{(n)}\end{pmatrix}=\begin{pmatrix}0&1&0&\cdots&0\\ 0&0&1&\cdots&0\\ \vdots&\vdots&\vdots&\ddots&\vdots\\ 0&0&0&\cdots&1\\ -p_{0}(x)&-p_{1}(x)&-p_{2}(x)&\cdots&-p_{n-1}(x)\end{pmatrix}\begin{pmatrix}y% \\ y^{\prime}\\ \vdots\\ y^{(n-2)}\\ y^{(n-1)}\end{pmatrix}.

Abraham–Lorentz_force.html

  1. 𝐅 rad = μ 0 q 2 6 π c 𝐚 ˙ = q 2 6 π ϵ 0 c 3 𝐚 ˙ \mathbf{F}_{\mathrm{rad}}=\frac{\mu_{0}q^{2}}{6\pi c}\mathbf{\dot{a}}=\frac{q^% {2}}{6\pi\epsilon_{0}c^{3}}\mathbf{\dot{a}}
  2. 𝐅 rad = 2 3 q 2 c 3 𝐚 ˙ . \mathbf{F}_{\mathrm{rad}}={2\over 3}\frac{q^{2}}{c^{3}}\mathbf{\dot{a}}.
  3. 𝐚 ˙ \mathbf{\dot{a}}
  4. P = μ 0 q 2 6 π c 𝐚 2 P=\frac{\mu_{0}q^{2}}{6\pi c}\mathbf{a}^{2}
  5. τ 1 \tau_{1}
  6. τ 2 \tau_{2}
  7. τ 1 τ 2 𝐅 rad 𝐯 d t = τ 1 τ 2 - P d t = - τ 1 τ 2 μ 0 q 2 6 π c 𝐚 2 d t = - τ 1 τ 2 μ 0 q 2 6 π c d 𝐯 d t d 𝐯 d t d t \int_{\tau_{1}}^{\tau_{2}}\mathbf{F}_{\mathrm{rad}}\cdot\mathbf{v}dt=\int_{% \tau_{1}}^{\tau_{2}}-Pdt=-\int_{\tau_{1}}^{\tau_{2}}\frac{\mu_{0}q^{2}}{6\pi c% }\mathbf{a}^{2}dt=-\int_{\tau_{1}}^{\tau_{2}}\frac{\mu_{0}q^{2}}{6\pi c}\frac{% d\mathbf{v}}{dt}\cdot\frac{d\mathbf{v}}{dt}dt
  8. τ 1 τ 2 𝐅 rad 𝐯 d t = - μ 0 q 2 6 π c d 𝐯 d t 𝐯 | τ 1 τ 2 + τ 1 τ 2 μ 0 q 2 6 π c d 2 𝐯 d t 2 𝐯 d t = - 0 + τ 1 τ 2 μ 0 q 2 6 π c 𝐚 ˙ 𝐯 d t \int_{\tau_{1}}^{\tau_{2}}\mathbf{F}_{\mathrm{rad}}\cdot\mathbf{v}dt=-\frac{% \mu_{0}q^{2}}{6\pi c}\frac{d\mathbf{v}}{dt}\cdot\mathbf{v}\bigg|_{\tau_{1}}^{% \tau_{2}}+\int_{\tau_{1}}^{\tau_{2}}\frac{\mu_{0}q^{2}}{6\pi c}\frac{d^{2}% \mathbf{v}}{dt^{2}}\cdot\mathbf{v}dt=-0+\int_{\tau_{1}}^{\tau_{2}}\frac{\mu_{0% }q^{2}}{6\pi c}\mathbf{\dot{a}}\cdot\mathbf{v}dt
  9. 𝐅 rad = μ 0 q 2 6 π c 𝐚 ˙ \mathbf{F}_{\mathrm{rad}}=\frac{\mu_{0}q^{2}}{6\pi c}\mathbf{\dot{a}}
  10. 𝐅 ext \mathbf{F}_{\mathrm{ext}}
  11. m 𝐯 ˙ = 𝐅 rad + 𝐅 ext = m t 0 < m t p l > v ¨ + 𝐅 ext . m\dot{\mathbf{v}}=\mathbf{F}_{\mathrm{rad}}+\mathbf{F}_{\mathrm{ext}}=mt_{0}% \ddot{\mathbf{<}mtpl>{{v}}}+\mathbf{F}_{\mathrm{ext}}.
  12. t 0 = μ 0 q 2 6 π m c . t_{0}=\frac{\mu_{0}q^{2}}{6\pi mc}.
  13. m 𝐯 ˙ = 1 t 0 t exp ( - t - t t 0 ) 𝐅 ext ( t ) d t . m\dot{\mathbf{v}}={1\over t_{0}}\int_{t}^{\infty}\exp\left(-{t^{\prime}-t\over t% _{0}}\right)\,\mathbf{F}_{\mathrm{ext}}(t^{\prime})\,dt^{\prime}.
  14. exp ( - t - t t 0 ) \exp\left(-{t^{\prime}-t\over t_{0}}\right)
  15. t 0 t_{0}
  16. t 0 t_{0}
  17. 10 - 24 10^{-24}
  18. F μ rad = μ o q 2 6 π m c [ d 2 p μ d τ 2 - p μ m 2 c 2 ( d p ν d τ d p ν d τ ) ] . F^{\mathrm{rad}}_{\mu}=\frac{\mu_{o}q^{2}}{6\pi mc}\left[\frac{d^{2}p_{\mu}}{d% \tau^{2}}-\frac{p_{\mu}}{m^{2}c^{2}}\left(\frac{dp_{\nu}}{d\tau}\frac{dp^{\nu}% }{d\tau}\right)\right].
  19. P = μ o q 2 a 2 γ 6 6 π c , P=\frac{\mu_{o}q^{2}a^{2}\gamma^{6}}{6\pi c},
  20. 1 Δ t 0 t P d t = 1 Δ t 0 t 𝐅 𝐯 d t . \frac{1}{\Delta t}\int_{0}^{t}Pdt=\frac{1}{\Delta t}\int_{0}^{t}\,\textbf{F}% \cdot\,\textbf{v}\,dt.

Abrikosov_vortex.html

  1. ξ \sim\xi
  2. λ \lambda
  3. λ > ξ \lambda>\xi
  4. Φ 0 \Phi_{0}
  5. B ( r ) = Φ 0 2 π λ 2 K 0 ( r λ ) λ r exp ( - r λ ) , B(r)=\frac{\Phi_{0}}{2\pi\lambda^{2}}K_{0}\left(\frac{r}{\lambda}\right)% \approx\sqrt{\frac{\lambda}{r}}\exp\left(-\frac{r}{\lambda}\right),
  6. K 0 ( z ) K_{0}(z)
  7. r 0 r\to 0
  8. B ( r ) ln ( λ / r ) B(r)\propto\ln(\lambda/r)
  9. r ξ r\lesssim\xi
  10. B ( 0 ) Φ 0 2 π λ 2 ln κ , B(0)\approx\frac{\Phi_{0}}{2\pi\lambda^{2}}\ln\kappa,
  11. κ > 1 / 2 \kappa>1/\sqrt{2}
  12. H H
  13. H c 1 H_{c1}
  14. H c 2 H_{c2}
  15. Φ 0 \Phi_{0}

Absolute_return.html

  1. 1000 × 10 = 10 , 000 1000\times 10=10,000
  2. 1000 × 9.5 = 9 , 500 1000\times 9.5=9,500
  3. ( 10 , 000 - 9 , 500 ) / 10 , 000 = 5 % (10,000-9,500)/10,000=5\%
  4. 5 % - 2 % = 3 % 5\%-2\%=3\%
  5. ( 200 M + 200 M ) / 100 M = 4 (200M+200M)/100M=4
  6. 4 % - 2 % = 2 % 4\%-2\%=2\%
  7. 5 × 2 % = 10 % 5\times 2\%=10\%
  8. 4 × 3 % = 12 % 4\times 3\%=12\%

Absolute_value_(algebra).html

  1. | p n a b | p = p - n . \left|p^{n}\frac{a}{b}\right|_{p}=p^{-n}.
  2. | x / y | = | x | / | y | . |x/y|=|x|/|y|.\,

Absolutely_simple_group.html

  1. G G
  2. G G
  3. { e } \{e\}
  4. G G

Absorption_wavemeter.html

  1. f = 1 2 π L C f={1\over 2\pi\sqrt{LC}}

Access_Control_Matrix.html

  1. O O
  2. S S
  3. R R
  4. r ( s , o ) r(s,o)
  5. s S s\in S
  6. o O o\in O
  7. r ( s , o ) R r(s,o)\subseteq R

Accessibility_relation.html

  1. R R\,\!
  2. p p\,\!
  3. p p\,\!
  4. w w\,\!
  5. R ( w * , w ) . R(w^{*},w).\,\!
  6. p p\,\!
  7. p p\,\!
  8. w w\,\!
  9. R ( w * , w ) R(w^{*},w)\,\!
  10. ( p q ) ( p q ) (\Box p\lor\Box q)\rightarrow\Box(p\lor q)
  11. R R\,\!
  12. R R\,\!
  13. R R\,\!
  14. \Box
  15. \Diamond
  16. \Box
  17. \Box
  18. A A\,\!
  19. A A\,\!
  20. A A\,\!
  21. A \Box A
  22. A \Diamond A
  23. \Box
  24. A A\,\!
  25. A \Diamond A
  26. p p\,\!
  27. q q\,\!
  28. p p\,\!
  29. \Box
  30. q q\,\!
  31. \Diamond
  32. q q\,\!
  33. p ¬ ¬ p \Box p\leftrightarrow\lnot\Diamond\lnot p
  34. p p\,\!
  35. p ¬ ¬ p \Box p\leftrightarrow\lnot\Diamond\lnot p
  36. p p\,\!
  37. p ¬ ¬ p \Box p\leftrightarrow\lnot\Diamond\lnot p
  38. p p\,\!
  39. p ¬ ¬ p \Box p\leftrightarrow\lnot\Diamond\lnot p
  40. p ¬ ¬ p \Diamond p\leftrightarrow\lnot\Box\lnot p
  41. p p\,\!
  42. p ¬ ¬ p \Diamond p\leftrightarrow\lnot\Box\lnot p
  43. p p\,\!
  44. ( p q ) ( p q ) \Box(p\land q)\leftrightarrow(\Box p\land\Box q)
  45. ( p q ) ( p q ) (\Box p\lor\Box q)\rightarrow\Box(p\lor q)
  46. ( p q ) ( p q ) \Box(p\to q)\to(\Box p\to\Box q)
  47. p p \Box p\rightarrow p
  48. p p \Box p\rightarrow\Box\Box p
  49. p p \Diamond p\rightarrow\Box\Diamond p
  50. p p p\rightarrow\Box\Diamond p
  51. ( p q ) ( p q ) (\Box p\lor\Box q)\rightarrow\Box(p\lor q)
  52. R ( w 1 , w 2 ) R(w_{1},w_{2})\,\!
  53. w 1 w_{1}\,\!
  54. w 2 w_{2}\,\!
  55. w 2 w_{2}\,\!
  56. w 1 w_{1}\,\!
  57. ( p q ) ( p q ) (\Box p\lor\Box q)\rightarrow\Box(p\lor q)
  58. ( p q ) ( p q ) (\Box p\lor\Box q)\rightarrow\Box(p\lor q)
  59. R R\,\!
  60. R R\,\!
  61. R R\,\!
  62. p p\,\!
  63. p p\,\!
  64. w w\,\!
  65. R ( w * , w ) R(w^{*},w)\,\!
  66. p p\,\!
  67. p p\,\!
  68. w w\,\!
  69. R ( w * , w ) R(w^{*},w)\,\!
  70. R R\,\!
  71. R R\,\!
  72. R R\,\!
  73. A A\,\!
  74. A A\,\!
  75. A A\,\!
  76. A A\,\!
  77. A \Box A
  78. A A\,\!
  79. R R\,\!
  80. w w^{\prime}\,\!
  81. w w\,\!
  82. w w\,\!
  83. A \Box A
  84. w w\,\!
  85. A A\,\!
  86. w w^{\prime}\,\!
  87. w w\,\!
  88. w ′′ w^{\prime\prime}\,\!
  89. w w^{\prime}\,\!
  90. w ′′ w^{\prime\prime}\,\!
  91. A \Box A
  92. w w\,\!
  93. w w^{\prime}\,\!
  94. w ′′ w^{\prime\prime}\,\!
  95. A \Box\Box A
  96. w w\,\!
  97. R R\,\!
  98. A \Box\Diamond A
  99. w w\,\!
  100. w w^{\prime}\,\!
  101. w w\,\!
  102. w ′′ w^{\prime\prime}\,\!
  103. w w^{\prime}\,\!
  104. A A\,\!
  105. A A\,\!
  106. w w^{\prime}\,\!
  107. w w\,\!
  108. w w\,\!
  109. w w\,\!
  110. w w^{\prime}\,\!
  111. A A\,\!
  112. A \Diamond A
  113. w w\,\!
  114. A \Diamond A
  115. A \Box\Diamond A
  116. R R\,\!
  117. w w^{\prime}\,\!
  118. w w\,\!
  119. w w\,\!
  120. w w^{\prime}\,\!
  121. A A\,\!
  122. w w\,\!
  123. w w^{\prime}\,\!
  124. w w\,\!
  125. w w\,\!
  126. w w^{\prime}\,\!
  127. A A\,\!
  128. A A\,\!
  129. w w^{\prime}\,\!
  130. w w\,\!
  131. A A\,\!
  132. P P\,\!
  133. P P\,\!
  134. P P\,\!
  135. x x\,\!
  136. P P\,\!
  137. x x\,\!
  138. P P\,\!
  139. P P\,\!
  140. w ′′ w^{\prime\prime}\,\!
  141. w w\,\!
  142. P P\,\!
  143. w w^{\prime}\,\!
  144. w w\,\!
  145. I I\,\!
  146. w w\,\!
  147. I I\,\!
  148. w = w w^{\prime}=w\,\!
  149. I I\,\!
  150. w = w w^{\prime}=w\,\!
  151. w ′′ = w w^{\prime\prime}=w^{\prime}\,\!
  152. I I\,\!
  153. w ′′ = w w^{\prime\prime}=w\,\!

Accidental_release_source_terms.html

  1. γ \gamma
  2. Q = C A k ρ P ( 2 k + 1 ) ( k + 1 ) / ( k - 1 ) Q\;=\;C\;A\;\sqrt{\;k\;\rho\;P\;\bigg(\frac{2}{k+1}\bigg)^{(k+1)/(k-1)}}
  3. Q = C A P ( k M Z R T ) ( 2 k + 1 ) ( k + 1 ) / ( k - 1 ) Q\;=\;C\;A\;P\;\sqrt{\bigg(\frac{\;\,k\;M}{Z\;R\;T}\bigg)\bigg(\frac{2}{k+1}% \bigg)^{(k+1)/(k-1)}}
  4. Q = C A 2 ρ P ( k k - 1 ) [ ( P A P ) 2 / k - ( P A P ) ( k + 1 ) / k ] Q\;=\;C\;A\;\sqrt{\;2\;\rho\;P\;\bigg(\frac{k}{k-1}\bigg)\Bigg[\,\bigg(\frac{% \;P_{A}}{P}\bigg)^{2/k}-\;\,\bigg(\frac{\;P_{A}}{P}\bigg)^{(k+1)/k}\;\Bigg]}
  5. Q = C A P ( 2 M Z R T ) ( k k - 1 ) [ ( P A P ) 2 / k - ( P A P ) ( k + 1 ) / k ] Q\;=\;C\;A\;P\;\sqrt{\bigg(\frac{2\;M}{Z\;R\;T}\bigg)\bigg(\frac{k}{k-1}\bigg)% \Bigg[\,\bigg(\frac{\;P_{A}}{P}\bigg)^{2/k}-\;\,\bigg(\frac{\;P_{A}}{P}\bigg)^% {(k+1)/k}\;\Bigg]}
  6. ρ \rho
  7. Q = C ρ A A 2 P ρ k k - 1 [ 1 - ( P A P ) ( k - 1 ) / k ) ] Q=C\;\rho_{A}\;A\;\sqrt{\frac{\;\,2\;P}{\rho}\cdot\frac{k}{k-1}\cdot{\Bigg[\;1% -{\bigg(\frac{P_{A}}{P}\bigg)^{(k-1)/k)}}\Bigg]}}
  8. ρ \rho
  9. ρ A = M P A R T A \rho_{A}=\frac{M\;P_{A}}{R\;T_{A}}
  10. T A T = ( P A P ) ( k - 1 ) / k \frac{T_{A}}{T}=\bigg(\frac{P_{A}}{P}\bigg)^{(k-1)/k}
  11. ρ \rho
  12. ρ A = M P ( k - 1 ) / k R T P A - 1 / k \rho_{A}=\frac{M\;P^{\;(k-1)/k}}{R\;T\;P_{A}^{\ -1/k}}
  13. E = ( 4.161 × 10 - 5 ) u 0.75 T F M ( P S / P H ) E=(4.161\times{10^{-5}})\cdot u^{0.75}\cdot T_{F}\cdot M\cdot(P_{S}/P_{H})
  14. P H = 760 e 65.3319 - ( 7245.2 / T A ) - ( 8.22 ln T a ) + ( 0.0061557 T A ) P_{H}=760\cdot e^{65.3319-(7245.2/T_{A})-(8.22\;\ln T_{a})+(0.0061557\;T_{A})}
  15. e e
  16. ln \ln
  17. E = 0.1288 A P M 0.667 u 0.78 T E=\frac{0.1288\cdot A\cdot P\cdot M^{0.667}\cdot u^{0.78}}{T}
  18. E = k P M R T A E=\frac{k\cdot P\cdot M}{R\cdot T_{A}}
  19. E = ( 0.0001 M ) ( 7.7026 - 0.0288 B ) e - ( 0.0077 B ) - 0.1376 E=(0.0001\;M)(7.7026-0.0288\;B)\;e^{-(0.0077\;B)-0.1376}
  20. X = 100 H s L - H a L H a V - H a L X=100\;\frac{H_{s}^{L}-H_{a}^{L}}{H_{a}^{V}-H_{a}^{L}}
  21. X = 100 c p ( T s - T b ) / H X=100\cdot c_{p}\cdot(T_{s}-T_{b})/H

Accuracy_paradox.html

  1. A ( M ) = T N + T P T N + F P + F N + T P \mathrm{A}(M)=\frac{TN+TP}{TN+FP+FN+TP}
  2. A ( M ) = 9 , 700 + 100 9 , 700 + 150 + 50 + 100 = 98.0 % \mathrm{A}(M)=\frac{9,700+100}{9,700+150+50+100}=98.0\%
  3. A ( M ) = 9 , 850 + 0 9 , 850 + 150 + 0 + 0 = 98.5 % \mathrm{A}(M)=\frac{9,850+0}{9,850+150+0+0}=98.5\%

Acoustic_resonance.html

  1. f = n v 2 L f={nv\over 2L}
  2. v = T ρ v=\sqrt{T\over\rho}
  3. f = n T ρ 2 L = n T m / L 2 L f={n\sqrt{T\over\rho}\over 2L}={n\sqrt{T\over m/L}\over 2L}
  4. f = n v 2 L f={nv\over 2L}
  5. f = n v 2 ( L + 0.8 d ) f={nv\over 2(L+0.8d)}
  6. f = v 2 ( L + 0.8 d ) f={v\over 2(L+0.8d)}
  7. f ( 2 ( L + 0.8 d ) ) = v {f(2(L+0.8d))}=v
  8. f λ = v {f\lambda}=v
  9. λ = 2 ( L + 0.8 d ) \lambda={2(L+0.8d)}
  10. f = n v 4 L f={nv\over 4L}
  11. f = n v 4 ( L + 0.4 d ) f={nv\over 4(L+0.4d)}
  12. f = v 4 ( L + 0.4 d ) f={v\over 4(L+0.4d)}
  13. f ( 4 ( L + 0.4 d ) ) = v {f(4(L+0.4d))}=v
  14. f λ = v {f\lambda}=v
  15. λ = 4 ( L + 0.4 d ) \lambda={4(L+0.4d)}
  16. k L = n π - tan - 1 k x kL=n\pi-\tan^{-1}kx
  17. k = 2 π f / v k=2\pi f/v
  18. k ( L + x ) n π k(L+x)\approx n\pi
  19. f = v 2 ( L x ) 2 + ( m L y ) 2 + ( n L z ) 2 f={v\over 2}\sqrt{\left({\ell\over L_{x}}\right)^{2}+\left({m\over L_{y}}% \right)^{2}+\left({n\over L_{z}}\right)^{2}}
  20. \ell
  21. n n
  22. m m
  23. f = v 2 π A V 0 L e q f=\frac{v}{2\pi}\sqrt{\frac{A}{V_{0}L_{eq}}}
  24. L e q L_{eq}
  25. L e q = L + 0.75 d L_{eq}=L+0.75d
  26. L e q = L + 0.85 d L_{eq}=L+0.85d
  27. f = v d π 3 8 L e q D 3 f=\frac{vd}{\pi}\sqrt{\frac{3}{8L_{eq}D^{3}}}
  28. f = v π 3 d 8 ( 0.85 ) D 3 f=\frac{v}{\pi}\sqrt{\frac{3d}{8(0.85)D^{3}}}
  29. f = 72.6 d D 3 f=72.6\sqrt{\frac{d}{D^{3}}}

Active_networking.html

  1. X X
  2. Y Y
  3. X X
  4. X d a t a X^{data}
  5. X c o d e X^{code}
  6. X c o d e X^{code}

Actuarial_present_value.html

  1. A x \,A_{x}\!
  2. A ¯ x \,\overline{A}_{x}\!
  3. Z = v T = ( 1 + i ) - T = e - δ T \,Z=v^{T}=(1+i)^{-T}=e^{-\delta T}
  4. E ( Z ) \,E(Z)
  5. A x \displaystyle A_{x}
  6. p x t {}_{t}p_{x}
  7. q x + t \,q_{x+t}
  8. A ¯ x = E [ v T ] = 0 v t f T ( t ) d t = 0 v t t p x μ x + t d t , \,\overline{A}_{x}\!=E[v^{T}]=\int_{0}^{\infty}v^{t}f_{T}(t)\,dt=\int_{0}^{% \infty}v^{t}\,_{t}p_{x}\mu_{x+t}\,dt,
  9. f T f_{T}
  10. p x t \,{}_{t}p_{x}\!
  11. x x
  12. x + t x+t
  13. μ x + t \mu_{x+t}
  14. x + t x+t
  15. x x
  16. E x n = P r [ G > x + n ] v n = n p x v n \,{}_{n}E_{x}=Pr[G>x+n]v^{n}=\,_{n}p_{x}v^{n}
  17. 100 , 000 A x 1 : 3 ¯ | = 100 , 000 t = 1 3 v t P r [ T ( G , x ) = t ] 100,000\,A_{\stackrel{1}{x}:{\overline{3}|}}=100,000\sum_{t=1}^{3}v^{t}Pr[T(G,% x)=t]
  18. P r [ T ( G , x ) = 1 ] = 0.1 , P r [ T ( G , x ) = 2 ] = 0.9 ( 0.1 ) = 0.09 , P r [ T ( G , x ) = 3 ] = 0.9 2 ( 0.1 ) = 0.081 , Pr[T(G,x)=1]=0.1,\quad Pr[T(G,x)=2]=0.9(0.1)=0.09,\quad Pr[T(G,x)=3]=0.9^{2}(0% .1)=0.081,
  19. A x 1 : 3 ¯ | = 0.1 ( 1.06 ) - 1 + 0.09 ( 1.06 ) - 2 + 0.081 ( 1.06 ) - 3 = 0.24244846 , \,A_{\stackrel{1}{x}:{\overline{3}|}}=0.1(1.06)^{-1}+0.09(1.06)^{-2}+0.081(1.0% 6)^{-3}=0.24244846,
  20. Y = a ¯ T ( x ) | ¯ = 1 - ( 1 + i ) - T δ = 1 - v T ( x ) δ , Y=\overline{a}_{\overline{T(x)|}}=\frac{1-(1+i)^{-T}}{\delta}=\frac{1-v^{T}(x)% }{\delta},
  21. a ¯ x = 0 a ¯ t | ¯ f T ( t ) d t = 0 a ¯ t | ¯ p x t μ x + t d t . \,\overline{a}_{x}=\int_{0}^{\infty}\overline{a}_{\overline{t|}}f_{T}(t)\,dt=% \int_{0}^{\infty}\overline{a}_{\overline{t|}}\,{}_{t}p_{x}\mu_{x+t}\,dt.
  22. a ¯ x = 0 v t [ 1 - F T ( t ) ] d t = 0 v t t p x d t \,\overline{a}_{x}=\int_{0}^{\infty}v^{t}[1-F_{T}(t)]\,dt=\int_{0}^{\infty}v^{% t}\,_{t}p_{x}\,dt\,
  23. a x = k = 1 v t [ 1 - F T ( t ) ] = t = 1 v t t p x . a_{x}=\sum_{k=1}^{\infty}v^{t}[1-F_{T}(t)]=\sum_{t=1}^{\infty}v^{t}\,_{t}p_{x}.
  24. A x = 1 - i v a ¨ x \,A_{x}=1-iv\ddot{a}_{x}
  25. A x = 1 - d a ¨ x \,A_{x}=1-d\ddot{a}_{x}
  26. A ¯ x = 1 - δ a ¯ x . \,\overline{A}_{x}=1-\delta\overline{a}_{x}.

Actuarial_reserves.html

  1. K ( x ) K(x)
  2. P P
  3. L L
  4. K ( x ) K(x)
  5. L = v K ( x ) + 1 - P a ¨ K ( x ) + 1 ¯ | L=v^{K(x)+1}-P\ddot{a}_{\overline{K(x)+1}|}
  6. L t = v K ( x ) + 1 - t - P a ¨ K ( x ) + 1 - t | ¯ {}_{t}L=v^{K(x)+1-t}-P\ddot{a}_{\overline{K(x)+1-t|}}
  7. L = v K ( x ) + 1 - P a ¨ K ( x ) + 1 | ¯ L=v^{K(x)+1}-P\ddot{a}_{\overline{K(x)+1|}}
  8. E [ L ] = E [ v K ( x ) + 1 - P a ¨ K ( x ) + 1 | ¯ ] \operatorname{E}[L]=\operatorname{E}[v^{K(x)+1}-P\ddot{a}_{\overline{K(x)+1|}}]
  9. E [ L ] = E [ v K ( x ) + 1 ] - P E [ a ¨ K ( x ) + 1 | ¯ ] \operatorname{E}[L]=\operatorname{E}[v^{K(x)+1}]-P\operatorname{E}[\ddot{a}_{% \overline{K(x)+1|}}]
  10. V x 0 = A x - P a ¨ x {}_{0}\!V_{x}=A_{x}-P\cdot\ddot{a}_{x}
  11. P = A x a ¨ x P=\frac{A_{x}}{\ddot{a}_{x}}
  12. P x P_{x}
  13. L t = v K ( x ) + 1 - t - P x a ¨ K ( x ) + 1 - t | ¯ {}_{t}L=v^{K(x)+1-t}-P_{x}\ddot{a}_{\overline{K(x)+1-t|}}
  14. E [ L t K ( x ) > t ] = E [ v K ( x ) + 1 - t K ( x ) > t ] - P x E [ a ¨ K ( x ) + 1 - t | ¯ K ( x ) > t ] \operatorname{E}[{}_{t}L\mid K(x)>t]=\operatorname{E}[v^{K(x)+1-t}\mid K(x)>t]% -P_{x}\operatorname{E}[\ddot{a}_{\overline{K(x)+1-t|}}\mid K(x)>t]
  15. V x t = A x + t - P x a ¨ x + t {}_{t}\!V_{x}=A_{x+t}-P_{x}\cdot\ddot{a}_{x+t}
  16. P x = A x a ¨ x {}P_{x}=\frac{A_{x}}{\ddot{a}_{x}}

Acutance.html

  1. A = ( D 1 - D 2 ) 1 N n = 1 N G n 2 A=\left(D_{1}-D_{2}\right)\frac{1}{N}\sum_{n=1}^{N}G_{n}^{2}

Adams_spectral_sequence.html

  1. n \mathbb{R}^{n}

Additive_identity.html

  1. 0 = ( 0 0 0 0 ) 0=\begin{pmatrix}0&0\\ 0&0\end{pmatrix}
  2. s 0 \displaystyle s\cdot 0

Additive_increase::multiplicative_decrease.html

  1. a > 0 a>0
  2. 0 < b < 1 0<b<1
  3. w ( t + 1 ) = { w ( t ) + a if congestion is not detected w ( t ) × b if congestion is detected w(t+1)=\begin{cases}w(t)+a&\,\text{ if congestion is not detected}\\ w(t)\times b&\,\text{ if congestion is detected}\end{cases}

Additive_number_theory.html

  1. A + B = { a + b : a A , b B } A+B=\{a+b:a\in A,b\in B\}
  2. h A = A + + A . hA=\underset{h}{\underbrace{A+\cdots+A}}.
  3. A k = { 0 k , 1 k , 2 k , 3 k , } A_{k}=\{0^{k},1^{k},2^{k},3^{k},\ldots\}

Adiabatic_flame_temperature.html

  1. T a d T_{ad}
  2. T a d T_{ad}
  3. Q P R - W P R = U P - U R {}_{R}Q_{P}-{}_{R}W_{P}=U_{P}-U_{R}
  4. Q P R {}_{R}Q_{P}
  5. W P R {}_{R}W_{P}
  6. U R U_{R}
  7. U P U_{P}
  8. W P R = R P p d V = 0 {}_{R}W_{P}=\int\limits_{R}^{P}{pdV}=0
  9. Q P R = 0 {}_{R}Q_{P}=0
  10. U P = U R U_{P}=U_{R}
  11. U P = U R m P u P = m R u R u P = u R U_{P}=U_{R}\Rightarrow m_{P}u_{P}=m_{R}u_{R}\Rightarrow u_{P}=u_{R}
  12. W P R = R P p d V = p ( V P - V R ) {}_{R}W_{P}=\int\limits_{R}^{P}{pdV}=p\left({V_{P}-V_{R}}\right)
  13. Q P R = 0 {}_{R}Q_{P}=0
  14. - p ( V P - V R ) = U P - U R U P + p V P = U R + p V R -p\left({V_{P}-V_{R}}\right)=U_{P}-U_{R}\Rightarrow U_{P}+pV_{P}=U_{R}+pV_{R}
  15. H P = H R H_{P}=H_{R}
  16. H P = H R m P h P = m R h R h P = h R H_{P}=H_{R}\Rightarrow m_{P}h_{P}=m_{R}h_{R}\Rightarrow h_{P}=h_{R}
  17. C O 2 CO_{2}
  18. H 2 O H_{2}O
  19. C α H β O γ N δ + ( a O 2 + b N 2 ) ν 1 CO 2 + ν 2 H 2 O + ν 3 N 2 + ν 4 O 2 {\rm{C}}_{\alpha}{\rm{H}}_{\beta}{\rm{O}}_{\gamma}{\rm{N}}_{\delta}+\left({a{% \rm{O}}_{\rm{2}}+b{\rm{N}}_{\rm{2}}}\right)\to\nu_{1}{\rm{CO}}_{\rm{2}}+\nu_{2% }{\rm{H}}_{\rm{2}}{\rm{O}}+\nu_{3}{\rm{N}}_{\rm{2}}+\nu_{4}{\rm{O}}_{\rm{2}}
  20. C O CO
  21. H 2 H_{2}
  22. C α H β O γ N δ + ( a O 2 + b N 2 ) ν 1 CO 2 + ν 2 H 2 O + ν 3 N 2 + ν 5 CO + ν 6 H 2 {\rm{C}}_{\alpha}{\rm{H}}_{\beta}{\rm{O}}_{\gamma}{\rm{N}}_{\delta}+\left({a{% \rm{O}}_{\rm{2}}+b{\rm{N}}_{\rm{2}}}\right)\to\nu_{1}{\rm{CO}}_{\rm{2}}+\nu_{2% }{\rm{H}}_{\rm{2}}{\rm{O}}+\nu_{3}{\rm{N}}_{\rm{2}}+\nu_{5}{\rm{CO}}+\nu_{6}{% \rm{H}}_{\rm{2}}
  23. CO 2 + H 2 CO + H 2 O {\rm{CO}}_{\rm{2}}+H_{2}\Leftrightarrow{\rm{CO}}+{\rm{H}}_{\rm{2}}{\rm{O}}

Adjunction_(field_theory).html

  1. 𝒯 \mathcal{T}
  2. F ( A ) = T 𝒯 F ( T ) F(A)=\bigcup_{T\in\mathcal{T}}F(T)

ADM_formalism.html

  1. Σ t \Sigma_{t}
  2. t t
  3. x i x^{i}
  4. γ i j ( t , x k ) \gamma_{ij}(t,x^{k})
  5. π i j ( t , x k ) \pi^{ij}(t,x^{k})
  6. γ i j \gamma_{ij}
  7. π i j \pi^{ij}
  8. N N
  9. N i N_{i}
  10. Σ t \Sigma_{t}
  11. g i j g_{ij}
  12. g μ ν ( 4 ) {{}^{(4)}}g_{\mu\nu}
  13. i \partial_{i}
  14. i \nabla_{i}
  15. g g
  16. π = g i j π i j \pi=g^{ij}\pi_{ij}
  17. = R ( 4 ) g ( 4 ) \mathcal{L}={{}^{(4)}R}\sqrt{{}^{(4)}g}
  18. g i j = g i j ( 4 ) g_{ij}={{}^{(4)}}g_{ij}\,\!
  19. π i j = g ( 4 ) ( Γ p q 0 ( 4 ) - g p q Γ r s 0 ( 4 ) g r s ) g i p g j q \pi^{ij}=\sqrt{{}^{(4)}g}\left({{}^{(4)}}\Gamma^{0}_{pq}-g_{pq}{{}^{(4)}}% \Gamma^{0}_{rs}g^{rs}\right)g^{ip}g^{jq}
  20. Γ i j 0 ( 4 ) {{}^{(4)}}\Gamma^{0}_{ij}
  21. N = ( - g 00 ( 4 ) ) - 1 / 2 N=\left(-{{}^{(4)}g^{00}}\right)^{-1/2}
  22. N i = g 0 i ( 4 ) N_{i}={{}^{(4)}g_{0i}}\,\!
  23. = - g i j t π i j - N H - N i P i - 2 i ( π i j N j - 1 2 π N i + i N g ) \mathcal{L}=-g_{ij}\partial_{t}\pi^{ij}-NH-N_{i}P^{i}-2\partial_{i}(\pi^{ij}N_% {j}-\frac{1}{2}\pi N^{i}+\nabla^{i}N\sqrt{g})
  24. H = - g [ R ( 3 ) + g - 1 ( 1 2 π 2 - π i j π i j ) ] H=-\sqrt{g}\left[{}^{(3)}R+g^{-1}\left(\frac{1}{2}\pi^{2}-\pi^{ij}\pi_{ij}% \right)\right]
  25. P i = - 2 π i j ; j P^{i}=-2\pi^{ij}{}_{;j}
  26. g i j g_{ij}
  27. π i j \pi^{ij}
  28. t g i j = 2 N g - 1 / 2 ( π i j - 1 2 π g i j ) + N i ; j + N j ; i \partial_{t}g_{ij}=2Ng^{-1/2}(\pi_{ij}-\frac{1}{2}\pi g_{ij})+N_{i;j}+N_{j;i}
  29. t π i j = - N g ( R i j - 1 2 R g i j ) + 1 2 N g - 1 / 2 g i j ( π m n π m n - 1 2 π 2 ) - 2 N g - 1 / 2 ( π i n π n - j 1 2 π π i j ) \partial_{t}\pi^{ij}=-N\sqrt{g}(R^{ij}-\frac{1}{2}Rg^{ij})+\frac{1}{2}Ng^{-1/2% }g^{ij}(\pi^{mn}\pi_{mn}-\frac{1}{2}\pi^{2})-2Ng^{-1/2}(\pi^{in}\pi_{n}{}^{j}-% \frac{1}{2}\pi\pi^{ij})
  30. - g ( i j N - g i j n n N ) + n ( π i j N n ) - N i π n j ; n - N j π n i ; n -\sqrt{g}(\nabla^{i}\nabla^{j}N-g^{ij}\nabla^{n}\nabla_{n}N)+\nabla_{n}(\pi^{% ij}N^{n})-N^{i}{}_{;n}\pi^{nj}-N^{j}{}_{;n}\pi^{ni}
  31. H = 0 H=0
  32. P i = 0 P^{i}=0
  33. π i j ( t , x k ) \pi^{ij}(t,x^{k})
  34. g ^ i j ( t , x k ) g i j ( t , x k ) \hat{g}_{ij}(t,x^{k})\to g_{ij}(t,x^{k})
  35. π ^ i j ( t , x k ) - i δ δ g i j ( t , x k ) \hat{\pi}^{ij}(t,x^{k})\to-i\frac{\delta}{\delta g_{ij}(t,x^{k})}

Admissible_ordinal.html

  1. ω 1 CK \omega_{1}^{\mathrm{CK}}
  2. ω α CK \omega_{\alpha}^{\mathrm{CK}}
  3. α \alpha

AdS::QCD_correspondence.html

  1. N N
  2. N N
  3. 10 - 11 10^{-11}
  4. η \eta
  5. s s
  6. η s 4 π k \frac{\eta}{s}\approx\frac{\hbar}{4\pi k}
  7. \hbar
  8. k k
  9. η / s \eta/s
  10. q ^ \widehat{q}
  11. q ^ \widehat{q}

Advantage_(cryptography).html

  1. A d v ( A ) = | Pr [ A ( F ) = 1 ] - Pr [ A ( G ) = 1 ] | Adv(A)=|\Pr[A(F)=1]-\Pr[A(G)=1]|
  2. 10 3.47 × 10 20 10^{3.47\times 10^{20}}

Aerodynamic_center.html

  1. d C m d C L = 0 {dC_{m}\over dC_{L}}=0
  2. C L C_{L}
  3. d C m d α < 0 {dC_{m}\over d\alpha}<0
  4. d C z d α > 0 {dC_{z}\over d\alpha}>0
  5. d C n d β > 0 {dC_{n}\over d\beta}>0
  6. d C y d β > 0 {dC_{y}\over d\beta}>0
  7. C z = C L c o s ( α ) + C d s i n ( α ) {C_{z}=C_{L}cos(\alpha)+C_{d}sin(\alpha)}
  8. C x = C L s i n ( α ) - C d c o s ( α ) {C_{x}=C_{L}sin(\alpha)-C_{d}cos(\alpha)}
  9. X A C = X r e f + c d C m d C z X_{AC}=X_{ref}+c{dC_{m}\over dC_{z}}
  10. c o s ( α ) = 1 cos({\alpha})=1
  11. s i n ( α ) sin({\alpha})
  12. α \alpha
  13. β = 0 {\beta}=0
  14. C z = C L - C d * α C_{z}=C_{L}-C_{d}*\alpha
  15. C z = C L C_{z}=C_{L}
  16. X A C = X r e f + c d C m d C L X_{AC}=X_{ref}+c{dC_{m}\over dC_{L}}
  17. Y A C = Y r e f Y_{AC}=Y_{ref}
  18. Z A C = Z r e f Z_{AC}=Z_{ref}
  19. X A C = X r e f + c d C m d C z + c d C n d C y X_{AC}=X_{ref}+c{dC_{m}\over dC_{z}}+c{dC_{n}\over dC_{y}}
  20. Y A C = Y r e f + c d C l d C z + c d C n d C x Y_{AC}=Y_{ref}+c{dC_{l}\over dC_{z}}+c{dC_{n}\over dC_{x}}
  21. Z A C = Z r e f + c d C l d C y + c d C m d C x Z_{AC}=Z_{ref}+c{dC_{l}\over dC_{y}}+c{dC_{m}\over dC_{x}}
  22. S M = X A C - X C G c SM={X_{AC}-X_{CG}\over c}
  23. C n C_{n}
  24. C m C_{m}
  25. C l C_{l}
  26. C x C_{x}
  27. C y C_{y}
  28. C z C_{z}
  29. α \alpha
  30. β \beta

Affine_curvature.html

  1. P 1 , P 2 , P 3 , P 4 P . P_{1},P_{2},P_{3},P_{4}\to P.
  2. β ( t ) \beta(t)
  3. a = ( a 1 , a 2 ) a=(a_{1},\;a_{2})
  4. b = ( b 1 , b 2 ) b=(b_{1},\;b_{2})
  5. det [ a b ] = a 1 b 2 - a 2 b 1 . \det\left[a\;b\right]=a_{1}b_{2}-a_{2}b_{1}.
  6. det [ d β d t d 2 β d t 2 ] \det\begin{bmatrix}\frac{d\beta}{dt}&\frac{d^{2}\beta}{dt^{2}}\end{bmatrix}
  7. det [ d β d t d 2 β d t 2 ] \displaystyle\det\begin{bmatrix}\frac{d\beta}{dt}&\frac{d^{2}\beta}{dt^{2}}% \end{bmatrix}
  8. det [ d β d s d 2 β d s 2 ] = 1 \det\begin{bmatrix}\frac{d\beta}{ds}&\frac{d^{2}\beta}{ds^{2}}\end{bmatrix}=1
  9. s ( t ) = a t det [ d β d t d 2 β d t 2 ] 3 d t . s(t)=\int_{a}^{t}\sqrt[3]{\det\begin{bmatrix}\frac{d\beta}{dt}&\frac{d^{2}% \beta}{dt^{2}}\end{bmatrix}}\,\,dt.
  10. k ( s ) = det [ β ′′ ( s ) β ′′′ ( s ) ] . k(s)=\det\begin{bmatrix}\beta^{\prime\prime}(s)&\beta^{\prime\prime\prime}(s)% \end{bmatrix}.
  11. t ( x ( t ) , y ( t ) ) , t\mapsto(x(t),y(t)),
  12. k ( t ) \displaystyle k(t)
  13. k = - 1 2 ( 1 ( y ′′ ) 2 / 3 ) ′′ = 1 3 y ′′′′ ( y ′′ ) 5 / 3 - 5 9 ( y ′′′ ) 2 ( y ′′ ) 8 / 3 k=-\frac{1}{2}\left(\frac{1}{(y^{\prime\prime})^{2/3}}\right)^{\prime\prime}=% \frac{1}{3}\frac{y^{\prime\prime\prime\prime}}{(y^{\prime\prime})^{5/3}}-\frac% {5}{9}\frac{(y^{\prime\prime\prime})^{2}}{(y^{\prime\prime})^{8/3}}
  14. σ = k ( s ) d s , \sigma=\int\sqrt{k(s)}\,ds,
  15. κ = 1 k 3 / 2 d k d s . \kappa=\frac{1}{k^{3/2}}\frac{dk}{ds}.
  16. C β ( s ) = [ β ( s ) β ′′ ( s ) ] . C_{\beta}(s)=\begin{bmatrix}\beta^{\prime}(s)&\beta^{\prime\prime}(s)\end{% bmatrix}.
  17. k = det ( C β ) . k=\det(C_{\beta}^{\prime}).\,
  18. C β = C β [ 0 - k 1 0 ] . C_{\beta}^{\prime}=C_{\beta}\begin{bmatrix}0&-k\\ 1&0\end{bmatrix}.
  19. C β ( s ) = exp { s [ 0 - k 1 0 ] } = [ cos k s k sin k s - 1 k sin k s cos k s ] . \begin{aligned}\displaystyle C_{\beta}(s)&\displaystyle=\exp\left\{s\cdot% \begin{bmatrix}0&-k\\ 1&0\end{bmatrix}\right\}\\ &\displaystyle=\begin{bmatrix}\cos\sqrt{k}s&\sqrt{k}\sin\sqrt{k}s\\ -\frac{1}{\sqrt{k}}\sin\sqrt{k}s&\cos\sqrt{k}s\end{bmatrix}.\end{aligned}
  20. C β ( s ) = [ 1 0 s 1 ] C_{\beta}(s)=\begin{bmatrix}1&0\\ s&1\end{bmatrix}
  21. β ( s ) = ( s , s 2 / 2 ) \beta(s)=(s,s^{2}/2)\,
  22. β ( s ) = ( cos k s , 1 k sin k s ) \beta^{\prime}(s)=\left(\cos\sqrt{k}s,\frac{1}{\sqrt{k}}\sin\sqrt{k}s\right)
  23. β ( s ) = ( 1 k sin k s , - 1 k cos k s ) \beta(s)=\left(\frac{1}{\sqrt{k}}\sin\sqrt{k}s,-\frac{1}{k}\cos\sqrt{k}s\right)
  24. C β ( s ) = [ cosh | k | s | k | sinh | k | s 1 | k | sinh | k | s cosh | k | s ] . C_{\beta}(s)=\begin{bmatrix}\cosh\sqrt{|k|}s&\sqrt{|k|}\sinh\sqrt{|k|}s\\ \frac{1}{\sqrt{|k|}}\sinh\sqrt{|k|}s&\cosh\sqrt{|k|}s\end{bmatrix}.
  25. β ( s ) = ( 1 | k | sinh | k | s , 1 | k | cosh | k | s ) \beta(s)=\left(\frac{1}{\sqrt{|k|}}\sinh\sqrt{|k|}s,\frac{1}{|k|}\cosh\sqrt{|k% |}s\right)
  26. - | k | x 2 + | k | 2 y 2 = 1. -|k|x^{2}+|k|^{2}y^{2}=1.
  27. C β = C β [ 0 - k 1 0 ] C_{\beta}^{\prime}=C_{\beta}\begin{bmatrix}0&-k\\ 1&0\end{bmatrix}
  28. x a x + b y + α y c x + d y + β , \begin{aligned}\displaystyle x&\displaystyle\mapsto ax+by+\alpha\\ \displaystyle y&\displaystyle\mapsto cx+dy+\beta,\end{aligned}
  29. T 1 = x , T 2 = y T_{1}=\partial_{x},\quad T_{2}=\partial y
  30. X 1 = x y , X 2 = y x , H = x x - y y . X_{1}=x\partial_{y},\quad X_{2}=y\partial_{x},\quad H=x\partial_{x}-y\partial_% {y}.
  31. ( x , y , y ) . (x,y,y^{\prime}).\,
  32. T 1 ( 1 ) , T 2 ( 1 ) , X 1 ( 1 ) , X 2 ( 1 ) , H ( 1 ) T_{1}^{(1)},T_{2}^{(1)},X_{1}^{(1)},X_{2}^{(1)},H^{(1)}
  33. T 1 , T 2 , X 1 , X 2 , H T_{1},T_{2},X_{1},X_{2},H
  34. θ 1 = d y - y d x . \theta_{1}=dy-y^{\prime}dx.
  35. L X ( 1 ) θ 1 0 ( mod θ 1 ) L_{X^{(1)}}\theta_{1}\equiv 0\;\;(\mathop{{\rm mod}}\theta_{1})
  36. ( x , y , y , y ′′ , , y ( k ) ) . (x,y,y^{\prime},y^{\prime\prime},\dots,y^{(k)}).
  37. L X ( k ) θ k 0 ( mod θ 1 , , θ k ) L_{X^{(k)}}\theta_{k}\equiv 0\;\;(\mathop{{\rm mod}}\theta_{1},\dots,\theta_{k})
  38. θ i = d y ( i - 1 ) - y ( i ) d x . \theta_{i}=dy^{(i-1)}-y^{(i)}dx.
  39. T 1 ( 4 ) = x , T 2 ( 4 ) = y T_{1}^{(4)}=\partial_{x},\quad T_{2}^{(4)}=\partial_{y}
  40. X 1 ( 4 ) = x y + y X_{1}^{(4)}=x\partial_{y}+\partial_{y^{\prime}}
  41. X 2 ( 4 ) = y x - y 2 y - 3 y y ′′ y ′′ - ( 3 y ′′ 2 + 4 y y ′′′ ) y ′′′ - ( 10 y ′′ y ′′′ + 5 y y ′′′′ ) y ′′′′ \begin{aligned}\displaystyle X_{2}^{(4)}=y\partial_{x}&\displaystyle-y^{\prime 2% }\partial_{y^{\prime}}-3y^{\prime}y^{\prime\prime}\partial_{y^{\prime\prime}}-% (3y^{\prime\prime 2}+4y^{\prime}y^{\prime\prime\prime})\partial_{y^{\prime% \prime\prime}}\\ &\displaystyle-(10y^{\prime\prime}y^{\prime\prime\prime}+5y^{\prime}y^{\prime% \prime\prime\prime})\partial_{y^{\prime\prime\prime\prime}}\end{aligned}
  42. H ( 4 ) = x x - y y - 2 y y - 3 y ′′ y ′′ - 4 y ′′′ y ′′′ - 5 y ′′′′ y ′′′′ . H^{(4)}=x\partial_{x}-y\partial_{y}-2y^{\prime}\partial_{y^{\prime}}-3y^{% \prime\prime}\partial_{y^{\prime\prime}}-4y^{\prime\prime\prime}\partial_{y^{% \prime\prime\prime}}-5y^{\prime\prime\prime\prime}\partial_{y^{\prime\prime% \prime\prime}}.
  43. k = 1 3 y ′′′′ ( y ′′ ) 5 / 3 - 5 9 ( y ′′′ ) 2 ( y ′′ ) 8 / 3 k=\frac{1}{3}\frac{y^{\prime\prime\prime\prime}}{(y^{\prime\prime})^{5/3}}-% \frac{5}{9}\frac{(y^{\prime\prime\prime})^{2}}{(y^{\prime\prime})^{8/3}}
  44. T 1 ( 4 ) k = T 2 ( 4 ) k = X 1 ( 4 ) k = 0. T_{1}^{(4)}k=T_{2}^{(4)}k=X_{1}^{(4)}k=0.
  45. X 2 ( 4 ) k = 1 2 [ H , X 1 ] ( 4 ) k = 1 2 [ H ( 4 ) , X 1 ( 4 ) ] k = 0. X_{2}^{(4)}k=\frac{1}{2}[H,X1]^{(4)}k=\frac{1}{2}[H^{(4)},X1^{(4)}]k=0.
  46. T 1 ( 4 ) , T 2 ( 4 ) , X 1 ( 4 ) , X 2 ( 4 ) , H ( 4 ) T_{1}^{(4)},T_{2}^{(4)},X_{1}^{(4)},X_{2}^{(4)},H^{(4)}
  47. v = γ κ - 1 3 , v=\gamma\kappa^{-\frac{1}{3}},
  48. v v
  49. κ \kappa
  50. γ \gamma

Affine_hull.html

  1. aff ( S ) = { i = 1 k α i x i | k > 0 , x i S , α i , i = 1 k α i = 1 } . \operatorname{aff}(S)=\left\{\sum_{i=1}^{k}\alpha_{i}x_{i}\Bigg|k>0,\,x_{i}\in S% ,\,\alpha_{i}\in\mathbb{R},\,\sum_{i=1}^{k}\alpha_{i}=1\right\}.
  2. aff ( aff ( S ) ) = aff ( S ) \mathrm{aff}(\mathrm{aff}(S))=\mathrm{aff}(S)
  3. aff ( S ) \mathrm{aff}(S)
  4. α i \alpha_{i}
  5. α i \alpha_{i}

Air-to-cloth_ratio.html

  1. v = V / Δ t A v={V/\Delta t\over{A}}

Air_flow_bench.html

  1. V = 1096.7 H / d V=1096.7\sqrt{H/d}
  2. V = [ 2 * H ] / d V=\sqrt{[2*H]/d}

Albers_projection.html

  1. x = ρ sin θ x=\rho\sin\theta
  2. y = ρ 0 - ρ cos θ y=\rho_{0}-\rho\cos\theta
  3. n = 1 2 ( sin φ 1 + sin φ 2 ) n={\tfrac{1}{2}}(\sin\varphi_{1}+\sin\varphi_{2})
  4. θ = n ( λ - λ 0 ) \theta=n(\lambda-\lambda_{0})
  5. C = cos 2 φ 1 + 2 n sin φ 1 C=\cos^{2}\varphi_{1}+2n\sin\varphi_{1}
  6. ρ = C - 2 n sin φ n \rho=\frac{\sqrt{C-2n\sin\varphi}}{n}
  7. ρ 0 = C - 2 n sin φ 0 n \rho_{0}=\frac{\sqrt{C-2n\sin\varphi_{0}}}{n}

Alcator_C-Mod.html

  1. η = d ln T / d ln n \eta=\,\text{d}\ln T/\,\text{d}\ln n
  2. η H = n H / ( n H + n D ) \eta_{H}=n_{H}/(n_{H}+n_{D})
  3. η H e 3 = n H e 3 / n e \eta_{He3}=n_{He3}/n_{e}

Alexander_Aitken.html

  1. π \pi

Alexander_Gelfond.html

  1. 2 2 2^{\sqrt{2}}
  2. e π e^{\pi}\,

Algebra_of_Communicating_Processes.html

  1. a , b , c , \mathit{a,b,c,...}
  2. δ \delta
  3. τ \tau
  4. + +
  5. \cdot
  6. ( a + b ) c (a+b)\cdot c
  7. a \mathit{a}
  8. b \mathit{b}
  9. c \mathit{c}
  10. a \mathit{a}
  11. b \mathit{b}
  12. | | ||
  13. | |\lfloor
  14. ( a b ) | | ( c d ) (a\cdot b)||(c\cdot d)
  15. a , b , c , d a,b,c,d
  16. a b c d , a c b d , a c d b , c a b d , c a d b , c d a b abcd,acbd,acdb,cabd,cadb,cdab
  17. ( a b ) | ( c d ) (a\cdot b)|\lfloor(c\cdot d)
  18. a b c d , a c b d , a c d b abcd,acbd,acdb
  19. a \mathit{a}
  20. | |
  21. r ( d ) r(d)
  22. w ( d ) w(d)
  23. d D = { 1 , 2 , 3 , } d\in D=\{1,2,3,\ldots\}
  24. ( d D r ( d ) y ) | ( w ( 1 ) z ) \left(\sum_{d\in D}r(d)\cdot y\right)|(w(1)\cdot z)
  25. 1 1
  26. d \mathit{d}
  27. 1 1
  28. d \mathit{d}
  29. y \mathit{y}
  30. y \mathit{y}
  31. z \mathit{z}
  32. τ I \tau_{I}
  33. τ \tau
  34. τ { c } ( ( a + b ) c ) = ( a + b ) τ \tau_{\{c\}}((a+b)\cdot c)=(a+b)\cdot\tau
  35. a + b a+b
  36. c \mathit{c}
  37. τ \tau
  38. x + y = y + x ( x + y ) + z = x + ( y + z ) x + x = x ( x + y ) z = ( x z ) + ( y z ) ( x y ) z = x ( y z ) \begin{matrix}x+y&=&y+x\\ (x+y)+z&=&x+(y+z)\\ x+x&=&x\\ (x+y)\cdot z&=&(x\cdot z)+(y\cdot z)\\ (x\cdot y)\cdot z&=&x\cdot(y\cdot z)\end{matrix}
  39. δ \delta
  40. δ + x = x δ x = δ \begin{matrix}\delta+x&=&x\\ \delta\cdot x&=&\delta\end{matrix}
  41. x | | y = x | y + y | x + x | y a x | y = a ( x | | y ) a | y = a y ( x + y ) | z = ( x | z ) + ( y | z ) a x | b = ( a | b ) x a | b x = ( a | b ) x a x | b y = ( a | b ) ( x | | y ) ( x + y ) | z = x | z + y | z x | ( y + z ) = x | y + x | z \begin{matrix}x||y&=&x|\lfloor y+y|\lfloor x+x|y\\ a\cdot x|\lfloor y&=&a\cdot(x||y)\\ a|\lfloor y&=&a\cdot y\\ (x+y)|\lfloor z&=&(x|\lfloor z)+(y|\lfloor z)\\ a\cdot x|b&=&(a|b)\cdot x\\ a|b\cdot x&=&(a|b)\cdot x\\ a\cdot x|b\cdot y&=&(a|b)\cdot(x||y)\\ (x+y)|z&=&x|z+y|z\\ x|(y+z)&=&x|y+x|z\end{matrix}
  42. | : A × A A |:A\times A\rightarrow A
  43. δ \delta
  44. a | a c a|a\rightarrow c
  45. a | a a|a
  46. c c
  47. H \partial_{H}
  48. H A H\subseteq A
  49. H H
  50. a | b = b | a ( a | b ) | c = a | ( b | c ) a | δ = δ H ( a ) = a if a H H ( a ) = δ if a H H ( x + y ) = H ( x ) + H ( y ) H ( x y ) = H ( x ) H ( y ) \begin{matrix}a|b&=&b|a\\ (a|b)|c&=&a|(b|c)\\ a|\delta&=&\delta\\ \partial_{H}(a)&=&a\mbox{ if }~{}a\notin H\\ \partial_{H}(a)&=&\delta\mbox{ if }~{}a\in H\\ \partial_{H}(x+y)&=&\partial_{H}(x)+\partial_{H}(y)\\ \partial_{H}(x\cdot y)&=&\partial_{H}(x)\cdot\partial_{H}(y)\\ \end{matrix}
  51. τ I ( τ ) = τ τ I ( a ) = a if a I τ I ( a ) = τ if a I τ I ( x + y ) = τ I ( x ) + τ I ( y ) τ I ( x y ) = τ I ( x ) τ I ( y ) H ( τ ) = τ x τ = x τ x = τ x + x a ( τ x + y ) = a ( τ x + y ) + a x τ x | y = τ ( x | | y ) τ | x = τ x τ | x = δ x | τ = δ τ x | y = x | y x | τ y = x | y \begin{matrix}\tau_{I}(\tau)&=&\tau\\ \tau_{I}(a)&=&a\mbox{ if }~{}a\notin I\\ \tau_{I}(a)&=&\tau\mbox{ if }~{}a\in I\\ \tau_{I}(x+y)&=&\tau_{I}(x)+\tau_{I}(y)\\ \tau_{I}(x\cdot y)&=&\tau_{I}(x)\cdot\tau_{I}(y)\\ \partial_{H}(\tau)&=&\tau\\ x\cdot\tau&=&x\\ \tau\cdot x&=&\tau\cdot x+x\\ a\cdot(\tau\cdot x+y)&=&a\cdot(\tau\cdot x+y)+a\cdot x\\ \tau\cdot x|\lfloor y&=&\tau\cdot(x||y)\\ \tau|\lfloor x&=&\tau\cdot x\\ \tau|x&=&\delta\\ x|\tau&=&\delta\\ \tau\cdot x|y&=&x|y\\ x|\tau\cdot y&=&x|y\end{matrix}

Algebraic_connectivity.html

  1. 1 n D \frac{1}{nD}
  2. 4 n D \frac{4}{nD}

Algebraic_expression.html

  1. 3 x 2 - 2 x y + c 3x^{2}-2xy+c
  2. 1 2 \tfrac{1}{2}
  3. 1 - x 2 1 + x 2 \sqrt{\frac{1-x^{2}}{1+x^{2}}}
  4. 3 x 2 - 2 x y + c y 3 - 1 \frac{3x^{2}-2xy+c}{y^{3}-1}
  5. 1 - x 2 1 + x 2 \sqrt{\frac{1-x^{2}}{1+x^{2}}}
  6. P ( x ) Q ( x ) \frac{P(x)}{Q(x)}
  7. x , y x,y
  8. a , b , c a,b,c
  9. x , y x,y
  10. z z
  11. x 2 x^{2}
  12. x x
  13. 1 x 2 1x^{2}
  14. x 2 x^{2}
  15. 3 x 1 3x^{1}
  16. 3 x 3x
  17. 3 x 0 3x^{0}
  18. 3 3
  19. x 0 x^{0}
  20. 1 1
  21. x < s u p > 2 + 4 x + 4 x<sup>2+4x+4

Algorithmically_random_sequence.html

  1. K ( w ) | w | - c K(w)\geq|w|-c
  2. U i U_{i}
  3. U i + 1 U i U_{i+1}\subseteq U_{i}
  4. μ ( U i ) 2 - i \mu(U_{i})\leq 2^{-i}
  5. G δ G_{\delta}
  6. U i U_{i}
  7. G δ G_{\delta}
  8. d : { 0 , 1 } * [ 0 , ) d:\{0,1\}^{*}\to[0,\infty)
  9. d ( w ) = ( d ( w 0 ) + d ( w 1 ) ) / 2 d(w)=(d(w^{\smallfrown}0)+d(w^{\smallfrown}1))/2
  10. a b a^{\smallfrown}b
  11. lim sup n d ( S n ) = , \limsup_{n\to\infty}d(S\upharpoonright n)=\infty,
  12. S n S\upharpoonright n
  13. d ^ : { 0 , 1 } * × 𝒩 \widehat{d}:\{0,1\}^{*}\times\mathcal{N}\to{\mathbb{Q}}
  14. d ^ ( w , t ) d ^ ( w , t + 1 ) < d ( w ) , \widehat{d}(w,t)\leq\widehat{d}(w,t+1)<d(w),
  15. lim t d ^ ( w , t ) = d ( w ) . \lim_{t\to\infty}\widehat{d}(w,t)=d(w).
  16. Σ 2 0 \Sigma^{0}_{2}
  17. Σ 2 0 \Sigma^{0}_{2}
  18. Σ 2 0 \Sigma^{0}_{2}
  19. Δ 2 0 \Delta^{0}_{2}
  20. Δ 1 0 \Delta_{1}^{0}
  21. Σ 1 0 \Sigma_{1}^{0}
  22. Π 1 0 \Pi_{1}^{0}
  23. Δ 2 0 \Delta_{2}^{0}
  24. \emptyset^{\prime}
  25. ( n ) \emptyset^{(n)}
  26. ( n - 1 ) \emptyset^{(n-1)}
  27. Δ 2 0 \Delta^{0}_{2}
  28. Δ 2 0 \Delta^{0}_{2}
  29. Δ 2 0 \Delta^{0}_{2}
  30. K ( w ) K ( | w | ) + b K(w)\leq K(|w|)+b

Alias_analysis.html

  1. A i A_{i}
  2. B j B_{j}
  3. i , j i,j
  4. i = j i=j
  5. A i A_{i}
  6. B j B_{j}
  7. i j i\neq j

Allais_paradox.html

  1. L 1 L_{1}
  2. L 2 L_{2}
  3. L 1 L_{1}
  4. L 3 L_{3}
  5. p p
  6. L 2 L_{2}
  7. L 3 L_{3}
  8. p p
  9. L 3 L_{3}
  10. L 1 L_{1}
  11. L 2 L_{2}
  12. L 3 L_{3}
  13. 1.00 U ( $ 1 M ) > 0.89 U ( $ 1 M ) + 0.01 U ( $ 0 M ) + 0.1 U ( $ 5 M ) 1.00U(\$1\,\text{ M})>0.89U(\$1\,\text{ M})+0.01U(\$0\,\text{ M})+0.1U(\$5\,% \text{ M})\,
  14. 0.89 U ( $ 0 M ) + 0.11 U ( $ 1 M ) < 0.9 U ( $ 0 M ) + 0.1 U ( $ 5 M ) 0.89U(\$0\,\text{ M})+0.11U(\$1\,\text{ M})<0.9U(\$0\,\text{ M})+0.1U(\$5\,% \text{ M})\,
  15. 0.11 U ( $ 1 M ) < 0.01 U ( $ 0 M ) + 0.1 U ( $ 5 M ) 0.11U(\$1\,\text{ M})<0.01U(\$0\,\text{ M})+0.1U(\$5\,\text{ M})\,
  16. 1.00 U ( $ 1 M ) - 0.89 U ( $ 1 M ) < 0.01 U ( $ 0 M ) + 0.1 U ( $ 5 M ) 1.00U(\$1\,\text{ M})-0.89U(\$1\,\text{ M})<0.01U(\$0\,\text{ M})+0.1U(\$5\,% \text{ M})\,
  17. 1.00 U ( $ 1 M ) < 0.89 U ( $ 1 M ) + 0.01 U ( $ 0 M ) + 0.1 U ( $ 5 M ) , 1.00U(\$1\,\text{ M})<0.89U(\$1\,\text{ M})+0.01U(\$0\,\text{ M})+0.1U(\$5\,% \text{ M}),\,

Alligation.html

  1. 1 2 × 120 + 1 4 × 100 + 1 4 × 150 = 122.5 {1\over 2}\times 120+{1\over 4}\times 100+{1\over 4}\times 150=122.5
  2. 2 + 1 2 = 5 2 2+{1\over 2}={5\over 2}
  3. 16 5 16\over 5
  4. 16 5 × 1 2 = 8 5 {16\over 5}\times{1\over 2}={8\over 5}
  5. 16 5 × 2 = 32 5 {16\over 5}\times 2={32\over 5}

ALOPEX.html

  1. Δ W i j ( n ) = γ Δ W i j ( n - 1 ) Δ R ( n ) + r i ( n ) \Delta\ W_{ij}(n)=\gamma\ \Delta\ W_{ij}(n-1)\Delta\ R(n)+r_{i}(n)
  2. n 0 n\geq 0
  3. Δ W i j ( n ) \Delta\ W_{ij}(n)
  4. W i j \ W_{ij}
  5. n n
  6. Δ R ( n ) \Delta\ R(n)
  7. R , \ R,
  8. n n
  9. γ \gamma
  10. ( γ < 0 (\gamma\ <0
  11. R , R,
  12. γ > 0 \gamma\ >0
  13. R ) R\ )
  14. r i ( n ) N ( 0 , σ 2 ) r_{i}(n)\sim\ N(0,\sigma\ ^{2})
  15. W i j ( n ) W_{ij}(n)
  16. Δ \Delta
  17. W i j ( n - 1 ) W_{ij}(n-1)
  18. Δ \Delta
  19. R ( n ) R(n)
  20. γ \gamma
  21. r i j ( n ) r_{ij}(n)

Alpha_Caeli.html

  1. [ M H ] = - 0.10 \begin{smallmatrix}\left[\frac{M}{H}\right]\ =\ -0.10\end{smallmatrix}

Alpha_diversity.html

  1. D α q = 1 j = 1 N i = 1 S p i j p i | j q - 1 q - 1 {}^{q}\!D_{\alpha}=\dfrac{1}{\sqrt[q-1]{\sum_{j=1}^{N}{\sum_{i=1}^{S}p_{ij}p_{% i|j}^{q-1}}}}
  2. p i | j p_{i|j}
  3. p i j p_{ij}
  4. p i | j p_{i|j}
  5. D α q = j = 1 N w j ( D α j q ) 1 - q 1 - q {}^{q}\!D_{\alpha}=\sqrt[1-q]{\sum_{j=1}^{N}w_{j}({}^{q}\!D_{\alpha j})^{1-q}}
  6. w j w_{j}

Alphabet_(formal_languages).html

  1. Σ \Sigma
  2. Σ * \Sigma^{*}
  3. Σ \Sigma
  4. * {}^{*}
  5. Σ * \Sigma^{*}
  6. Σ \Sigma
  7. Σ \Sigma^{\infty}
  8. Σ 𝒩 \Sigma^{\mathcal{N}}
  9. Σ ω \Sigma^{\omega}
  10. Σ \Sigma

Alternant_matrix.html

  1. M = [ f 1 ( α 1 ) f 2 ( α 1 ) f n ( α 1 ) f 1 ( α 2 ) f 2 ( α 2 ) f n ( α 2 ) f 1 ( α 3 ) f 2 ( α 3 ) f n ( α 3 ) f 1 ( α m ) f 2 ( α m ) f n ( α m ) ] M=\begin{bmatrix}f_{1}(\alpha_{1})&f_{2}(\alpha_{1})&\dots&f_{n}(\alpha_{1})\\ f_{1}(\alpha_{2})&f_{2}(\alpha_{2})&\dots&f_{n}(\alpha_{2})\\ f_{1}(\alpha_{3})&f_{2}(\alpha_{3})&\dots&f_{n}(\alpha_{3})\\ \vdots&\vdots&\ddots&\vdots\\ f_{1}(\alpha_{m})&f_{2}(\alpha_{m})&\dots&f_{n}(\alpha_{m})\\ \end{bmatrix}
  2. M i , j = f j ( α i ) M_{i,j}=f_{j}(\alpha_{i})
  3. f i ( α ) = α i - 1 f_{i}(\alpha)=\alpha^{i-1}
  4. f i ( α ) = α q i - 1 f_{i}(\alpha)=\alpha^{q^{i-1}}
  5. n = m n=m
  6. f j ( x ) f_{j}(x)
  7. α i = α j \alpha_{i}=\alpha_{j}
  8. i < j i<j
  9. ( α j - α i ) (\alpha_{j}-\alpha_{i})
  10. 1 i < j n 1\leq i<j\leq n
  11. V = [ 1 α 1 α 1 n - 1 1 α 2 α 2 n - 1 1 α 3 α 3 n - 1 1 α n α n n - 1 ] V=\begin{bmatrix}1&\alpha_{1}&\dots&\alpha_{1}^{n-1}\\ 1&\alpha_{2}&\dots&\alpha_{2}^{n-1}\\ 1&\alpha_{3}&\dots&\alpha_{3}^{n-1}\\ \vdots&\vdots&\ddots&\vdots\\ 1&\alpha_{n}&\dots&\alpha_{n}^{n-1}\\ \end{bmatrix}
  12. i < j ( α j - α i ) = det V \prod_{i<j}(\alpha_{j}-\alpha_{i})=\det V
  13. det M det V \frac{\det M}{\det V}
  14. f j ( x ) = x m j f_{j}(x)=x^{m_{j}}

Alternating_factorial.html

  1. af ( n ) = i = 1 n ( - 1 ) n - i i ! \mathrm{af}(n)=\sum_{i=1}^{n}(-1)^{n-i}i!
  2. af ( n ) = n ! - af ( n - 1 ) \mathrm{af}(n)=n!-\mathrm{af}(n-1)
  3. 1 2 i = 0 n - 1 i ! {1\over 2}\sum_{i=0}^{n-1}i!

Alternating_permutation.html

  1. E ( 0 , 0 ) = 1 E(0,0)=1
  2. E ( n , 0 ) = 0 for n > 0 E(n,0)=0\qquad\mbox{for }~{}n>0
  3. E ( n , k ) = E ( n , k - 1 ) + E ( n - 1 , n - k ) E(n,k)=E(n,k-1)+E(n-1,n-k)
  4. sec x = 1 + 1 2 ! x 2 + 5 4 ! x 4 + = n = 0 A 2 n x 2 n ( 2 n ) ! . \sec x=1+\frac{1}{2!}x^{2}+\frac{5}{4!}x^{4}+\cdots=\sum_{n=0}^{\infty}A_{2n}{% x^{2n}\over({2n})!}.
  5. tan x = 1 1 ! x + 2 3 ! x 3 + 16 5 ! x 5 + = n = 0 A 2 n + 1 x 2 n + 1 ( 2 n + 1 ) ! . \tan x=\frac{1}{1!}x+\frac{2}{3!}x^{3}+\frac{16}{5!}x^{5}+\cdots=\sum_{n=0}^{% \infty}A_{2n+1}{x^{2n+1}\over({2n+1})!}.
  6. B 2 n = ( - 1 ) n - 1 2 n 4 2 n - 2 2 n A 2 n - 1 B_{2n}=(-1)^{n-1}\frac{2n}{4^{2n}-2^{2n}}A_{2n-1}
  7. n = 0 A n x n n ! = sec x + tan x = tan ( x 2 + π 4 ) . \sum_{n=0}^{\infty}A_{n}{x^{n}\over n!}=\sec x+\tan x=\tan\left({x\over 2}+{% \pi\over 4}\right).
  8. A n = i n + 1 k = 1 n + 1 j = 0 k ( k j ) ( - 1 ) j ( k - 2 j ) n + 1 2 k i k k A_{n}=i^{n+1}\sum_{k=1}^{n+1}\sum_{j=0}^{k}{k\choose{j}}\frac{(-1)^{j}(k-2j)^{% n+1}}{2^{k}i^{k}k}
  9. a 1 > a 2 < a 3 > a 4 < a_{1}>a_{2}<a_{3}>a_{4}<\cdots\,
  10. a 1 < a 2 > a 3 < a 4 > . a_{1}<a_{2}>a_{3}<a_{4}>\cdots.\,
  11. a i n + 1 - a i . ) a_{i}\mapsto n+1-a_{i}.)\,
  12. E 0 = 1 , E 1 = 1 , E 2 = 1 , E 3 = 2 , E 4 = 5 , E 5 = 16 , E 6 = 61 , E 7 = 272 , E_{0}=1,\quad E_{1}=1,\quad E_{2}=1,\quad E_{3}=2,\quad E_{4}=5,\quad E_{5}=16% ,\quad E_{6}=61,\quad E_{7}=272,\quad\ldots
  13. f ( x ) = n = 0 E n x n n ! = sec x + tan x . f(x)=\sum_{n=0}^{\infty}E_{n}\frac{x^{n}}{n!}=\sec x+\tan x.
  14. π \pi
  15. 2 E n + 1 = k = 0 n ( n k ) E k E n - k if n 1. ( 1 ) 2E_{n+1}=\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}E_{k}E_{n-k}\,\text{ if }n% \geq 1.\qquad\qquad(1)
  16. a k , a k - 1 , a k - 2 , , a 1 , n + 1 , b 1 , b 2 , b 3 , b n - k a_{k},a_{k-1},a_{k-2},\ldots,a_{1},n+1,b_{1},b_{2},b_{3},\ldots b_{n-k}\,
  17. ( n k ) E k E n - k . {\left({{n}\atop{k}}\right)}E_{k}E_{n-k}.\,
  18. 2 f ( x ) = 2 E 0 + 2 E 1 x + 2 n = 1 E n + 1 x n + 1 ( n + 1 ) ! = 2 E 0 + 2 E 1 x + n = 1 k = 0 n ( n k ) E k E n - k x n + 1 ( n + 1 ) ! . 2f(x)=2E_{0}+2E_{1}x+2\sum_{n=1}^{\infty}E_{n+1}\frac{x^{n+1}}{(n+1)!}=2E_{0}+% 2E_{1}x+\sum_{n=1}^{\infty}\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}E_{k}E_{n% -k}\frac{x^{n+1}}{(n+1)!}.
  19. 2 f ( x ) = 2 E 1 + 2 n = 1 E n + 1 x n n ! = 2 E 1 + n = 1 k = 0 n ( n k ) E k E n - k x n n ! . 2f^{\prime}(x)=2E_{1}+2\sum_{n=1}^{\infty}E_{n+1}\frac{x^{n}}{n!}=2E_{1}+\sum_% {n=1}^{\infty}\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}E_{k}E_{n-k}\frac{x^{n% }}{n!}.
  20. E 1 + n = 0 k = 0 n ( n k ) E k E n - k x n n ! = 1 + n = 0 k = 0 n ( E k x k k ! ) ( E n - k x n - k ( n - k ) ! ) . E_{1}+\sum_{n=0}^{\infty}\sum_{k=0}^{n}{\left({{n}\atop{k}}\right)}E_{k}E_{n-k% }\frac{x^{n}}{n!}=1+\sum_{n=0}^{\infty}\sum_{k=0}^{n}\left(E_{k}\frac{x^{k}}{k% !}\right)\left(E_{n-k}\frac{x^{n-k}}{(n-k)!}\right).
  21. 1 + i = 0 j = 0 E i x i i ! E j x j j ! 1+\sum_{i=0}^{\infty}\sum_{j=0}^{\infty}E_{i}\frac{x^{i}}{i!}\cdot E_{j}\frac{% x^{j}}{j!}
  22. E i x i i ! E_{i}\frac{x^{i}}{i!}
  23. 1 + i = 0 ( E i x i i ! j = 0 E j x j j ! ) . 1+\sum_{i=0}^{\infty}\left(E_{i}\frac{x^{i}}{i!}\sum_{j=0}^{\infty}E_{j}\frac{% x^{j}}{j!}\right).
  24. 1 + ( j = 0 E j x j j ! ) ( i = 0 E i x i i ! ) . 1+\left(\sum_{j=0}^{\infty}E_{j}\frac{x^{j}}{j!}\right)\left(\sum_{i=0}^{% \infty}E_{i}\frac{x^{i}}{i!}\right).
  25. 1 + ( f ( x ) ) 2 . 1+(f(x))^{2}.\,
  26. 2 f ( x ) = 1 + ( f ( x ) ) 2 . 2f^{\prime}(x)=1+(f(x))^{2}.\,
  27. f ( x ) = tan ( x 2 + constant ) . f(x)=\tan\left(\frac{x}{2}+\,\text{constant}\right).
  28. f ( x ) = tan ( x 2 + π 4 ) . f(x)=\tan\left(\frac{x}{2}+\frac{\pi}{4}\right).
  29. f ( x ) = sec x + tan x . f(x)=\sec x+\tan x.\,

Amagat.html

  1. η \eta
  2. η = n n 0 \eta=\frac{n}{n_{0}}
  3. × 10 2 5 \times 10^{2}5
  4. η = ( p p 0 ) ( T 0 T ) amg \eta=\left(\frac{p}{p_{0}}\right)\left(\frac{T_{0}}{T}\right)\,{\rm amg}
  5. η = ( 1 atm p 0 ) ( 273.15 K ( 273.15 + 20 ) K ) amg = 0.932 amg \eta=\left(\frac{1\,{\rm atm}}{p_{0}}\right)\left(\frac{273.15\,{\rm K}}{(273.% 15+20)\,{\rm K}}\right){\rm amg}=0.932\,{\rm amg}

Amortization_calculator.html

  1. A = P i ( 1 + i ) n ( 1 + i ) n - 1 = P × i 1 - ( 1 + i ) - n = P ( i + i ( 1 + i ) n - 1 ) A=P\frac{i(1+i)^{n}}{(1+i)^{n}-1}=\frac{P\times i}{1-(1+i)^{-n}}=P\left(i+% \frac{i}{(1+i)^{n}-1}\right)
  2. n = 30 years × 12 months/year = 360 months n=30\,\text{ years}\times 12\,\text{ months/year}=360\,\text{ months}
  3. i i
  4. A A
  5. p ( t ) p(t)
  6. t t
  7. A A
  8. r = 1 + i r=1+i
  9. p ( 0 ) = P \;p(0)=P
  10. p ( 1 ) = p ( 0 ) r - A = P r - A \;p(1)=p(0)r-A=Pr-A
  11. p ( 2 ) = p ( 1 ) r - A = P r 2 - A r - A \;p(2)=p(1)r-A=Pr^{2}-Ar-A
  12. p ( 3 ) = p ( 2 ) r - A = P r 3 - A r 2 - A r - A \;p(3)=p(2)r-A=Pr^{3}-Ar^{2}-Ar-A
  13. p ( t ) = P r t - A k = 0 t - 1 r k \;p(t)=Pr^{t}-A\sum_{k=0}^{t-1}r^{k}
  14. k = 0 t - 1 r k = 1 + r + r 2 + + r t - 1 = r t - 1 r - 1 \;\sum_{k=0}^{t-1}r^{k}=1+r+r^{2}+...+r^{t-1}=\frac{r^{t}-1}{r-1}
  15. p ( t ) = P r t - A r t - 1 r - 1 \;p(t)=Pr^{t}-A\frac{r^{t}-1}{r-1}
  16. n n
  17. p ( n ) = P r n - A r n - 1 r - 1 = 0 \;p(n)=Pr^{n}-A\frac{r^{n}-1}{r-1}=0
  18. A = P r n ( r - 1 ) r n - 1 = P ( i + 1 ) n ( ( i + \cancel 1 ) - \cancel 1 ) ( i + 1 ) n - 1 = P i ( 1 + i ) n ( 1 + i ) n - 1 \;A=P\frac{r^{n}(r-1)}{r^{n}-1}=P\frac{(i+1)^{n}((i+\cancel{1})-\cancel{1})}{(% i+1)^{n}-1}=P\frac{i(1+i)^{n}}{(1+i)^{n}-1}
  19. A P = i 1 - ( 1 + i ) - n \frac{A}{P}=\frac{i}{1-(1+i)^{-n}}
  20. p ( t ) P = 1 - ( 1 + i ) t - 1 ( 1 + i ) n - 1 \frac{p(t)}{P}=1-\frac{(1+i)^{t}-1}{(1+i)^{n}-1}

Anchor_(climbing).html

  1. F L o a d F_{Load}
  2. θ V \theta_{V}
  3. F A n c h o r = F L o a d 2 cos ( θ V / 2 ) F_{Anchor}=\frac{F_{Load}}{2\cos(\theta_{V}/2)}

Ancient_Egyptian_multiplication.html

  1. 2 97 * 56 56 = 112 56 * 97 = 97 + 8 + 7 56 * 97 \frac{2}{97}*\frac{56}{56}=\frac{112}{56*97}=\frac{97+8+7}{56*97}
  2. 2 97 = 1 56 + 1 679 + 1 776 \frac{2}{97}=\frac{1}{56}+\frac{1}{679}+\frac{1}{776}
  3. 26 97 * 4 4 = 104 4 * 97 = 97 + 4 + 2 + 1 4 * 97 \frac{26}{97}*\frac{4}{4}=\frac{104}{4*97}=\frac{97+4+2+1}{4*97}
  4. 26 97 = 1 4 + 1 97 + 1 194 + 1 388 \frac{26}{97}=\frac{1}{4}+\frac{1}{97}+\frac{1}{194}+\frac{1}{388}
  5. 28 97 = 1 4 + 1 56 + 1 97 + 1 194 + 1 388 + 1 679 + 1 776 \frac{28}{97}=\frac{1}{4}+\frac{1}{56}+\frac{1}{97}+\frac{1}{194}+\frac{1}{388% }+\frac{1}{679}+\frac{1}{776}

Anderson's_rule.html

  1. χ \chi
  2. E g E_{\rm g}
  3. Δ E v \Delta E_{\rm v}
  4. Δ E c \Delta E_{\rm c}
  5. Δ E c = χ 2 - χ 1 \Delta E_{\rm c}=\chi_{2}-\chi_{1}\,
  6. Δ E v = ( χ 1 + E g1 ) - ( χ 2 + E g2 ) \Delta E_{\rm v}=(\chi_{\rm 1}+E_{\rm g1})-(\chi_{\rm 2}+E_{\rm g2})\,

Anderson–Darling_test.html

  1. F F
  2. F n F_{n}
  3. F F
  4. F n F_{n}
  5. n - ( F n ( x ) - F ( x ) ) 2 w ( x ) d F ( x ) , n\int_{-\infty}^{\infty}(F_{n}(x)-F(x))^{2}\,w(x)\,dF(x),
  6. w ( x ) w(x)
  7. w ( x ) = 1 w(x)=1
  8. A = n - ( F n ( x ) - F ( x ) ) 2 F ( x ) ( 1 - F ( x ) ) d F ( x ) , A=n\int_{-\infty}^{\infty}\frac{(F_{n}(x)-F(x))^{2}}{F(x)\;(1-F(x))}\,dF(x),
  9. w ( x ) = [ F ( x ) ( 1 - F ( x ) ) ] - 1 w(x)=[F(x)\;(1-F(x))]^{-1}
  10. A A
  11. { Y 1 < < Y n } \{Y_{1}<\cdots<Y_{n}\}
  12. Φ \Phi
  13. A 2 = - n - S , A^{2}=-n-S\,,
  14. S = i = 1 n 2 i - 1 n [ ln ( Φ ( Y i ) ) + ln ( 1 - Φ ( Y n + 1 - i ) ) ] . S=\sum_{i=1}^{n}\frac{2i-1}{n}\left[\ln(\Phi(Y_{i}))+\ln\left(1-\Phi(Y_{n+1-i}% )\right)\right].
  15. Φ \Phi
  16. A 2 A^{2}
  17. μ \mu
  18. σ 2 \sigma^{2}
  19. σ 2 \sigma^{2}
  20. μ \mu
  21. μ \mu
  22. σ 2 \sigma^{2}
  23. μ \mu
  24. σ 2 \sigma^{2}
  25. X i X_{i}
  26. i = 1 , n i=1,\ldots n
  27. X X
  28. μ ^ = { μ , if the mean is known. X ¯ , = 1 n i = 1 n X i otherwise. \hat{\mu}=\begin{cases}\mu,&\,\text{if the mean is known.}\\ \bar{X},=\frac{1}{n}\sum_{i=1}^{n}X_{i}&\,\text{otherwise.}\end{cases}
  29. σ ^ 2 = { σ 2 , if the variance is known. 1 n i = 1 n ( X i - μ ) 2 , if the variance is not known, but the mean is. 1 n - 1 i = 1 n ( X i - X ¯ ) 2 , otherwise. \hat{\sigma}^{2}=\begin{cases}\sigma^{2},&\,\text{if the variance is known.}\\ \frac{1}{n}\sum_{i=1}^{n}(X_{i}-\mu)^{2},&\,\text{if the variance is not known% , but the mean is.}\\ \frac{1}{n-1}\sum_{i=1}^{n}(X_{i}-\bar{X})^{2},&\,\text{otherwise.}\end{cases}
  30. X i X_{i}
  31. Y i Y_{i}
  32. Y i = X i - μ ^ σ ^ . Y_{i}=\frac{X_{i}-\hat{\mu}}{\hat{\sigma}}.
  33. Φ \Phi
  34. A 2 A^{2}
  35. A 2 = - n - 1 n i = 1 n ( 2 i - 1 ) ( ln Φ ( Y i ) + ln ( 1 - Φ ( Y n + 1 - i ) ) ) . A^{2}=-n-\frac{1}{n}\sum_{i=1}^{n}(2i-1)(\ln\Phi(Y_{i})+\ln(1-\Phi(Y_{n+1-i}))).
  36. A 2 = - n - 1 n i = 1 n [ ( 2 i - 1 ) ln Φ ( Y i ) + ( 2 ( n - i ) + 1 ) ln ( 1 - Φ ( Y i ) ) ] . A^{2}=-n-\frac{1}{n}\sum_{i=1}^{n}\left[(2i-1)\ln\Phi(Y_{i})+(2(n-i)+1)\ln(1-% \Phi(Y_{i}))\right].
  37. A * 2 = { A 2 ( 1 + 4 n - 25 n 2 ) , if the variance and the mean are both unknown. A 2 , otherwise. A^{*2}=\begin{cases}A^{2}\left(1+\frac{4}{n}-\frac{25}{n^{2}}\right),&\,\text{% if the variance and the mean are both unknown.}\\ A^{2},&\,\text{otherwise.}\end{cases}
  38. A * 2 A^{*2}
  39. n 5 n\geq 5
  40. σ ^ \hat{\sigma}
  41. Φ ( Y i ) = \Phi(Y_{i})=
  42. A 2 A^{2}
  43. A * 2 = A 2 ( 1 + 0.75 n + 2.25 n 2 ) . A^{*2}=A^{2}\left(1+\frac{0.75}{n}+\frac{2.25}{n^{2}}\right).
  44. A * 2 A^{*2}
  45. A * 2 A^{*2}
  46. X i X_{i}

Andreotti–Frankel_theorem.html

  1. V V
  2. n n
  3. V V
  4. n n
  5. V V
  6. V V
  7. V r V\subseteq\mathbb{C}^{r}
  8. n n
  9. V V
  10. C W CW
  11. n \leq n
  12. H i ( V ; Z ) = 0 , for i > n H^{i}(V;Z)=0,\,\text{ for }i>n\,
  13. H i ( V ; Z ) = 0 , for i > n . H_{i}(V;Z)=0,\,\text{ for }i>n.\,
  14. n n

Angle_notation.html

  1. 1 \ang θ 1\ang\theta
  2. ( cos θ , sin θ ) (\cos\theta,\sin\theta)\,
  3. cos θ + j sin θ = e j θ \cos\theta+j\sin\theta=e^{j\theta}
  4. j 2 = - 1 j^{2}=-1
  5. A A
  6. θ \theta
  7. A \ang θ . A\ang\theta.
  8. 1 \ang 90 1\ang 90
  9. 1 \ang 90 , 1\ang 90^{\circ},
  10. ( 0 , 1 ) (0,1)\,
  11. e j π / 2 . e^{j\pi/2}.\,

Angular_distance.html

  1. θ \theta
  2. a a
  3. D D
  4. tan ( a D ) \tan(\frac{a}{D})
  5. θ a D \theta\approx\dfrac{a}{D}

Angular_momentum_operator.html

  1. 𝐋 = 𝐫 × 𝐩 \mathbf{L}=\mathbf{r}\times\mathbf{p}
  2. 𝐋 = ( L x , L y , L z ) \mathbf{L}=(L_{x},L_{y},L_{z})
  3. 𝐉 = 𝐋 + 𝐒 . \mathbf{J}=\mathbf{L}+\mathbf{S}.
  4. 𝐋 = 𝐫 × 𝐩 \mathbf{L}=\mathbf{r}\times\mathbf{p}
  5. 𝐋 = - i ( 𝐫 × ) \mathbf{L}=-i\hbar(\mathbf{r}\times\nabla)
  6. 𝐋 = ( L x , L y , L z ) \mathbf{L}=(L_{x},L_{y},L_{z})
  7. [ L x , L y ] = i L z , [ L y , L z ] = i L x , [ L z , L x ] = i L y , [L_{x},L_{y}]=i\hbar L_{z},\;\;[L_{y},L_{z}]=i\hbar L_{x},\;\;[L_{z},L_{x}]=i% \hbar L_{y},
  8. [ X , Y ] X Y - Y X . [X,Y]\equiv XY-YX.
  9. [ L l , L m ] = i n = 1 3 ε l m n L n [L_{l},L_{m}]=i\hbar\sum_{n=1}^{3}\varepsilon_{lmn}L_{n}
  10. 𝐋 × 𝐋 = i 𝐋 \mathbf{L}\times\mathbf{L}=i\hbar\mathbf{L}
  11. [ x l , p m ] = i δ l m [x_{l},p_{m}]=i\hbar\delta_{lm}
  12. { L i , L j } = ε i j k L k \{L_{i},L_{j}\}=\varepsilon_{ijk}L_{k}\!
  13. { , } \{,\}
  14. [ S l , S m ] = i n = 1 3 ε l m n S n , [ J l , J m ] = i n = 1 3 ε l m n J n [S_{l},S_{m}]=i\hbar\sum_{n=1}^{3}\varepsilon_{lmn}S_{n},\quad[J_{l},J_{m}]=i% \hbar\sum_{n=1}^{3}\varepsilon_{lmn}J_{n}
  15. L 2 L x 2 + L y 2 + L z 2 L^{2}\equiv L_{x}^{2}+L_{y}^{2}+L_{z}^{2}
  16. [ L 2 , L x ] = [ L 2 , L y ] = [ L 2 , L z ] = 0 . [L^{2},L_{x}]=[L^{2},L_{y}]=[L^{2},L_{z}]=0~{}.\,
  17. [ L 2 , L x ] = [ L x 2 , L x ] + [ L y 2 , L x ] + [ L z 2 , L x ] [L^{2},L_{x}]=[L_{x}^{2},L_{x}]+[L_{y}^{2},L_{x}]+[L_{z}^{2},L_{x}]
  18. = L y [ L y , L x ] + [ L y , L x ] L y + L z [ L z , L x ] + [ L z , L x ] L z =L_{y}[L_{y},L_{x}]+[L_{y},L_{x}]L_{y}+L_{z}[L_{z},L_{x}]+[L_{z},L_{x}]L_{z}
  19. = L y ( - i L z ) + ( - i L z ) L y + L z ( i L y ) + ( i L y ) L z =L_{y}(-i\hbar L_{z})+(-i\hbar L_{z})L_{y}+L_{z}(i\hbar L_{y})+(i\hbar L_{y})L% _{z}
  20. = 0 =0
  21. [ 𝐋 𝐋 , 𝐋 𝐧 ] = [ L 2 , 𝐋 𝐧 ] = 0 , [\mathbf{L}\cdot\mathbf{L},\mathbf{L}\cdot\mathbf{n}]=[L^{2},\mathbf{L}\cdot% \mathbf{n}]=0~{},
  22. [ S 2 , S i ] = 0 , [ J 2 , J i ] = 0. \begin{aligned}\displaystyle{[}S^{2},S_{i}]&\displaystyle=0,\\ \displaystyle{[}J^{2},J_{i}]&\displaystyle=0.\end{aligned}
  23. σ L x σ L y 2 | L z | . \sigma_{L_{x}}\sigma_{L_{y}}\geq\frac{\hbar}{2}\left|\langle L_{z}\rangle% \right|.
  24. σ X \sigma_{X}
  25. X \langle X\rangle
  26. L x = L y = L z = 0 L_{x}=L_{y}=L_{z}=0
  27. \hbar
  28. ( m ) (\hbar m)
  29. m { , - 2 , - 1 , 0 , 1 , 2 , } m\in\{\ldots,-2,-1,0,1,2,\ldots\}
  30. ( m ) (\hbar m)
  31. m { , - 1 , - 0.5 , 0 , 0.5 , 1 , 1.5 , } m\in\{\ldots,-1,-0.5,0,0.5,1,1.5,\ldots\}
  32. L 2 L^{2}
  33. ( 2 ( + 1 ) ) (\hbar^{2}\ell(\ell+1))
  34. { 0 , 1 , 2 , } \ell\in\{0,1,2,\ldots\}
  35. L 2 L x 2 + L y 2 + L z 2 L^{2}\equiv L_{x}^{2}+L_{y}^{2}+L_{z}^{2}
  36. \ell
  37. S 2 S^{2}
  38. ( 2 s ( s + 1 ) ) (\hbar^{2}s(s+1))
  39. s { 0 , 0.5 , 1 , 1.5 , } s\in\{0,0.5,1,1.5,\ldots\}
  40. J 2 J^{2}
  41. ( 2 j ( j + 1 ) ) (\hbar^{2}j(j+1))
  42. j { 0 , 0.5 , 1 , 1.5 , } j\in\{0,0.5,1,1.5,\ldots\}
  43. L 2 L^{2}
  44. L z L_{z}
  45. ( 2 ( + 1 ) ) (\hbar^{2}\ell(\ell+1))
  46. L 2 L^{2}
  47. ( m ) (\hbar m_{\ell})
  48. L z L_{z}
  49. { 0 , 1 , 2 , } \ell\in\{0,1,2,\ldots\}
  50. m { - , ( - + 1 ) , , ( - 1 ) , } m_{\ell}\in\{-\ell,(-\ell+1),\ldots,(\ell-1),\ell\}
  51. S 2 S^{2}
  52. S z S_{z}
  53. ( 2 s ( s + 1 ) ) (\hbar^{2}s(s+1))
  54. S 2 S^{2}
  55. ( m s ) (\hbar m_{s})
  56. S z S_{z}
  57. s { 0 , 0.5 , 1 , 1.5 , } s\in\{0,0.5,1,1.5,\ldots\}
  58. m s { - s , ( - s + 1 ) , , ( s - 1 ) , s } m_{s}\in\{-s,(-s+1),\ldots,(s-1),s\}
  59. J 2 J^{2}
  60. J z J_{z}
  61. ( 2 j ( j + 1 ) ) (\hbar^{2}j(j+1))
  62. J 2 J^{2}
  63. ( m j ) (\hbar m_{j})
  64. J z J_{z}
  65. j { 0 , 0.5 , 1 , 1.5 , } j\in\{0,0.5,1,1.5,\ldots\}
  66. m j { - j , ( - j + 1 ) , , ( j - 1 ) , j } m_{j}\in\{-j,(-j+1),\ldots,(j-1),j\}
  67. J + \displaystyle J_{+}
  68. | ψ |\psi\rangle
  69. J 2 J^{2}
  70. J z J_{z}
  71. J 2 J^{2}
  72. J z J_{z}
  73. J + | ψ J_{+}|\psi\rangle
  74. J - | ψ J_{-}|\psi\rangle
  75. J 2 J^{2}
  76. J z | ψ J_{z}|\psi\rangle
  77. \hbar
  78. J x 2 , J y 2 , J z 2 J_{x}^{2},J_{y}^{2},J_{z}^{2}
  79. J 2 = J x 2 + J y 2 + J z 2 J^{2}=J_{x}^{2}+J_{y}^{2}+J_{z}^{2}
  80. ( J 2 - J z 2 ) = ( J x 2 + J y 2 ) (J^{2}-J_{z}^{2})=(J_{x}^{2}+J_{y}^{2})
  81. | ψ |\psi\rangle
  82. J 2 J^{2}
  83. J z J_{z}
  84. J 2 J^{2}
  85. 2 j ( j + 1 ) \hbar^{2}j(j+1)
  86. J 2 J^{2}
  87. J z J_{z}
  88. m \hbar m
  89. | ψ |\psi\rangle
  90. | ψ = | j , m |\psi\rangle=|j,m\rangle
  91. { , J - J - | j , m , J - | j , m , | j , m , J + | j , m , J + J + | j , m , } \{\ldots\;,\;J_{-}J_{-}|j,m\rangle\;,\;J_{-}|j,m\rangle\;,\;|j,m\rangle\;,\;J_% {+}|j,m\rangle\;,\;J_{+}J_{+}|j,m\rangle\;,\;\ldots\}
  92. J 2 J^{2}
  93. J z J_{z}
  94. \hbar
  95. ( J 2 - J z 2 ) (J^{2}-J_{z}^{2})
  96. J 2 J^{2}
  97. J z 2 J_{z}^{2}
  98. J 2 J^{2}
  99. J z 2 J_{z}^{2}
  100. | j , m m a x |j,m_{max}\rangle
  101. J + | j , m m a x = 0 J_{+}|j,m_{max}\rangle=0
  102. J + | j , m = j ( j + 1 ) - m ( m + 1 ) | j , m + 1 J_{+}|j,m\rangle=\hbar\sqrt{j(j+1)-m(m+1)}|j,m+1\rangle
  103. j ( j + 1 ) = m m a x ( m m a x + 1 ) j(j+1)=m_{max}(m_{max}+1)
  104. j = m j=m
  105. j = - m - 1 j=-m-1
  106. J 2 - J z 2 J^{2}-J_{z}^{2}
  107. 2 j ( j + 1 ) ( m ) 2 \hbar^{2}j(j+1)\geq(\hbar m)^{2}
  108. m m a x = j m_{max}=j
  109. | j , m m i n |j,m_{min}\rangle
  110. J - | j , m m i n = 0 J_{-}|j,m_{min}\rangle=0
  111. J - | j , m = j ( j + 1 ) - m ( m - 1 ) | j , m - 1 J_{-}|j,m\rangle=\hbar\sqrt{j(j+1)-m(m-1)}|j,m-1\rangle
  112. m m i n = - j m_{min}=-j
  113. ( j - ( - j ) ) (j-(-j))
  114. = 2 \ell=2
  115. m = - 2 , - 1 , 0 , 1 , 2 m_{\ell}=-2,-1,0,1,2
  116. | L | = L 2 = 6 |L|=\sqrt{L^{2}}=\hbar\sqrt{6}
  117. 6 \hbar\sqrt{6}
  118. L z L_{z}
  119. L x L_{x}
  120. L y L_{y}
  121. \ell
  122. m m_{\ell}
  123. L L
  124. L z / L_{z}/\hbar
  125. R ( n ^ , ϕ ) R(\hat{n},\phi)
  126. n ^ \hat{n}
  127. ϕ \phi
  128. ϕ 0 \phi\rightarrow 0
  129. R ( n ^ , ϕ ) R(\hat{n},\phi)
  130. J n ^ J_{\hat{n}}
  131. n ^ \hat{n}
  132. J n ^ i lim ϕ 0 R ( n ^ , ϕ ) - 1 ϕ J_{\hat{n}}\equiv i\hbar\lim_{\phi\rightarrow 0}\frac{R(\hat{n},\phi)-1}{\phi}
  133. R ( n ^ , ϕ 1 + ϕ 2 ) = R ( n ^ , ϕ 1 ) R ( n ^ , ϕ 2 ) R(\hat{n},\phi_{1}+\phi_{2})=R(\hat{n},\phi_{1})R(\hat{n},\phi_{2})
  134. R ( n ^ , ϕ ) = exp ( - i ϕ J n ^ / ) R(\hat{n},\phi)=\exp(-i\phi J_{\hat{n}}/\hbar)
  135. R spatial ( n ^ , ϕ ) = exp ( - i ϕ L n ^ / ) , R_{\mathrm{spatial}}(\hat{n},\phi)=\exp(-i\phi L_{\hat{n}}/\hbar),
  136. R internal ( n ^ , ϕ ) = exp ( - i ϕ S n ^ / ) , R_{\mathrm{internal}}(\hat{n},\phi)=\exp(-i\phi S_{\hat{n}}/\hbar),
  137. R ( n ^ , ϕ ) = R internal ( n ^ , ϕ ) R spatial ( n ^ , ϕ ) R(\hat{n},\phi)=R_{\mathrm{internal}}(\hat{n},\phi)R_{\mathrm{spatial}}(\hat{n% },\phi)
  138. R ( n ^ , 360 ) = 1 R(\hat{n},360^{\circ})=1
  139. R ( n ^ , 360 ) = - 1 R(\hat{n},360^{\circ})=-1
  140. R ( n ^ , 360 ) = + 1 R(\hat{n},360^{\circ})=+1
  141. R spatial ( n ^ , 360 ) = + 1 R_{\mathrm{spatial}}(\hat{n},360^{\circ})=+1
  142. R spatial R_{\mathrm{spatial}}
  143. R R
  144. R internal R_{\mathrm{internal}}
  145. + 1 = R spatial ( z ^ , 360 ) = exp ( - 2 π i L z / ) +1=R_{\mathrm{spatial}}(\hat{z},360^{\circ})=\exp(-2\pi iL_{z}/\hbar)
  146. L z | ψ = m | ψ L_{z}|\psi\rangle=m\hbar|\psi\rangle
  147. e - 2 π i m = 1 e^{-2\pi im}=1
  148. | ψ 0 |\psi_{0}\rangle
  149. R ( n ^ , ϕ ) | ψ 0 R(\hat{n},\phi)|\psi_{0}\rangle
  150. n ^ \hat{n}
  151. ϕ \phi
  152. R H R - 1 = H RHR^{-1}=H
  153. [ H , R ] = 0 [H,R]=0
  154. [ H , 𝐉 ] = 𝟎 [H,\mathbf{J}]=\mathbf{0}
  155. | 𝐫 | |\mathbf{r}|
  156. ( J 1 ) z , ( J 1 ) 2 , ( J 2 ) z , ( J 2 ) 2 (J_{1})_{z},(J_{1})^{2},(J_{2})_{z},(J_{2})^{2}
  157. ( J 1 ) 2 , ( J 2 ) 2 , J 2 , J z (J_{1})^{2},(J_{2})^{2},J^{2},J_{z}
  158. ( J 1 ) 2 , ( J 2 ) 2 , J 2 (J_{1})^{2},(J_{2})^{2},J^{2}
  159. j { | j 1 - j 2 | , ( | j 1 - j 2 | + 1 ) , , ( j 1 + j 2 ) } j\in\{|j_{1}-j_{2}|,(|j_{1}-j_{2}|+1),\ldots,(j_{1}+j_{2})\}
  160. L 2 , S 2 , J 2 L^{2},S^{2},J^{2}
  161. 𝐋 \displaystyle\mathbf{L}
  162. L 2 l , m \displaystyle L^{2}\mid l,m\rangle
  163. θ , ϕ | l , m = Y l , m ( θ , ϕ ) \langle\theta,\phi|l,m\rangle=Y_{l,m}(\theta,\phi)

Annihilator_method.html

  1. P ( D ) y = f ( x ) P(D)y=f(x)
  2. A ( D ) A(D)
  3. A ( D ) f ( x ) = 0 A(D)f(x)=0
  4. A ( D ) A(D)
  5. ( A ( D ) P ( D ) ) y = 0 \big(A(D)P(D)\big)y=0
  6. { y 1 , , y n } \{y_{1},\ldots,y_{n}\}
  7. y ′′ - 4 y + 5 y = sin ( k x ) y^{\prime\prime}-4y^{\prime}+5y=\sin(kx)
  8. P ( D ) = D 2 - 4 D + 5 P(D)=D^{2}-4D+5
  9. sin ( k x ) \sin(kx)
  10. A ( D ) = D 2 + k 2 A(D)=D^{2}+k^{2}
  11. A ( z ) P ( z ) A(z)P(z)
  12. { 2 + i , 2 - i , i k , - i k } \{2+i,2-i,ik,-ik\}
  13. A ( D ) P ( D ) A(D)P(D)
  14. { y 1 , y 2 , y 3 , y 4 } = { e ( 2 + i ) x , e ( 2 - i ) x , e i k x , e - i k x } . \{y_{1},y_{2},y_{3},y_{4}\}=\{e^{(2+i)x},e^{(2-i)x},e^{ikx},e^{-ikx}\}.
  15. y = c 1 y 1 + c 2 y 2 + c 3 y 3 + c 4 y 4 y=c_{1}y_{1}+c_{2}y_{2}+c_{3}y_{3}+c_{4}y_{4}
  16. sin ( k x ) \displaystyle\sin(kx)
  17. i = ( k 2 + 4 i k - 5 ) c 3 + ( - k 2 + 4 i k + 5 ) c 4 i=(k^{2}+4ik-5)c_{3}+(-k^{2}+4ik+5)c_{4}
  18. 0 = ( k 2 + 4 i k - 5 ) c 3 + ( k 2 - 4 i k - 5 ) c 4 0=(k^{2}+4ik-5)c_{3}+(k^{2}-4ik-5)c_{4}
  19. c 3 = i 2 ( k 2 + 4 i k - 5 ) c_{3}=\frac{i}{2(k^{2}+4ik-5)}
  20. c 4 = i 2 ( - k 2 + 4 i k + 5 ) c_{4}=\frac{i}{2(-k^{2}+4ik+5)}
  21. y \displaystyle y
  22. y p = 4 k cos ( k x ) + ( 5 - k 2 ) sin ( k x ) k 4 + 6 k 2 + 25 y_{p}=\frac{4k\cos(kx)+(5-k^{2})\sin(kx)}{k^{4}+6k^{2}+25}
  23. y c = c 1 y 1 + c 2 y 2 y_{c}=c_{1}y_{1}+c_{2}y_{2}
  24. c 1 c_{1}
  25. c 2 c_{2}
  26. y 1 = e ( 2 + i ) x y_{1}=e^{(2+i)x}
  27. y 2 = e ( 2 - i ) x y_{2}=e^{(2-i)x}
  28. e ( 2 + i ) x = e 2 x e i x = e 2 x ( cos x + i sin x ) e^{(2+i)x}=e^{2x}e^{ix}=e^{2x}(\cos x+i\sin x)
  29. e ( 2 - i ) x = e 2 x e - i x = e 2 x ( cos x - i sin x ) e^{(2-i)x}=e^{2x}e^{-ix}=e^{2x}(\cos x-i\sin x)
  30. c 1 y 1 + c 2 y 2 = c 1 e 2 x ( cos x + i sin x ) + c 2 e 2 x ( cos x - i sin x ) = ( c 1 + c 2 ) e 2 x cos x + i ( c 1 - c 2 ) e 2 x sin x c_{1}y_{1}+c_{2}y_{2}=c_{1}e^{2x}(\cos x+i\sin x)+c_{2}e^{2x}(\cos x-i\sin x)=% (c_{1}+c_{2})e^{2x}\cos x+i(c_{1}-c_{2})e^{2x}\sin x
  31. y c = e 2 x ( c 1 cos x + c 2 sin x ) y_{c}=e^{2x}(c_{1}\cos x+c_{2}\sin x)

Annual_percentage_yield.html

  1. A P Y = ( 1 + i nom N ) N - 1 APY=\left(1+\frac{i\text{nom}}{N}\right)^{N}-1
  2. i n o m i_{nom}
  3. N N
  4. A P Y e i nom - 1 , APY\approx e^{i\text{nom}}-1,
  5. A P Y = 100 [ ( 1 + I n t e r e s t P r i n c i p a l ) 365 / D a y s i n t e r m - 1 ] APY=100\left[\left(1+\frac{Interest}{Principal}\right)^{365/Days~{}in~{}term}-% 1\right]
  6. I n t e r e s t = P r i n c i p a l [ ( A P Y 100 + 1 ) D a y s i n t e r m / 365 - 1 ] Interest=Principal\left[\left(\frac{APY}{100}+1\right)^{Days~{}in~{}term/365}-% 1\right]

Anomalous_monism.html

  1. T f T_{f}
  2. T p T_{p}
  3. P P^{\prime}
  4. P ′′ P^{\prime\prime}
  5. T f T_{f}
  6. T p T_{p}
  7. T p T_{p}^{\prime}
  8. T f + T p + T p + T p ′′ T_{f}+T_{p}+T_{p}^{\prime}+T_{p}^{\prime\prime}

Antenna_measurement.html

  1. d = 2 D 2 λ d={{2D^{2}}\over{\lambda}}
  2. λ {\lambda}
  3. D 2 = P S 3 d B \displaystyle D^{2}=\frac{P}{S}\propto\!\,3dB
  4. n ~ = P r P r + P l \tilde{n}=\frac{P_{r}}{P_{r}+P_{l}}
  5. G = ( P S ) a n t ( P S ) i s o G={\left({P\over S}\right)_{ant}\over\left({P\over S}\right)_{iso}}\,\!
  6. E θ \scriptstyle{E_{\theta}}
  7. r \scriptstyle{r}
  8. E θ = A I r E_{\theta}={AI\over r}
  9. I \scriptstyle{I}
  10. A \scriptstyle{A}
  11. r \scriptstyle{r}
  12. P S = c ε 2 E θ 2 = 1 2 E θ 2 Z {P\over S}={c\varepsilon_{\circ}\over 2}{E_{\theta}}^{2}={1\over 2}{{E_{\theta% }}^{2}\over Z_{\circ}}\,\!
  13. Z = < m t p l > μ ε = 376.730313461 Ω \scriptstyle{Z_{\circ}=\sqrt{<}mtpl>{{\mu_{\circ}\over\varepsilon_{\circ}}}=37% 6.730313461\,\Omega}\,\!
  14. ( P S ) a n t = 1 2 Z A 2 I 2 r 2 \left({P\over S}\right)_{ant}={1\over 2Z_{\circ}}{A^{2}I^{2}\over r^{2}}\,\!
  15. R s \scriptstyle{R_{s}}
  16. 1 2 R s I 2 \scriptstyle{{1\over 2}R_{s}I^{2}}
  17. r \scriptstyle{r}
  18. ( P S ) i s o = 1 2 R s I 2 4 π r 2 \left({P\over S}\right)_{iso}={{1\over 2}R_{s}I^{2}\over 4\pi r^{2}}\,\!
  19. G = 1 2 Z A 2 I 2 r 2 1 2 R s I 2 4 π r 2 = A 2 30 R s G={{1\over 2Z_{\circ}}{A^{2}I^{2}\over r^{2}}\over{{1\over 2}R_{s}I^{2}\over 4% \pi r^{2}}}={A^{2}\over 30R_{s}}\,\!
  20. R λ 2 = 60 Cin ( 2 π ) = 60 [ ln ( 2 π γ ) - Ci ( 2 π ) ] = 120 0 π 2 cos ( π 2 cos θ ) 2 sin θ d θ , = 15 [ 2 π 2 - 1 3 π 4 + 4 135 π 6 - 1 630 π 8 + 4 70875 π 10 - ( - 1 ) n ( 2 π ) 2 n n ( 2 n ) ! ] , = 73.1296 Ω ; \begin{aligned}\displaystyle R_{\frac{\lambda}{2}}&\displaystyle=60% \operatorname{Cin}(2\pi)=60\left[\ln(2\pi\gamma)-\operatorname{Ci}(2\pi)\right% ]=120\int_{0}^{\frac{\pi}{2}}\frac{\cos\left(\frac{\pi}{2}\cos\theta\right)^{2% }}{\sin\theta}d\theta,\\ &\displaystyle=15\left[2\pi^{2}-\frac{1}{3}\pi^{4}+\frac{4}{135}\pi^{6}-\frac{% 1}{630}\pi^{8}+\frac{4}{70875}\pi^{10}\ldots-(-1)^{n}\frac{(2\pi)^{2n}}{n(2n)!% }\right],\\ &\displaystyle=73.1296\ldots\;\Omega;\end{aligned}\,\!
  21. G λ 2 = 60 2 30 R λ 2 = 3600 30 R λ 2 = 120 R λ 2 = 1 0 π 2 cos ( π 2 cos θ ) 2 sin θ d θ , 120 73.1296 1.6409224 2.15088 dBi ; \begin{aligned}\displaystyle G_{\frac{\lambda}{2}}&\displaystyle=\frac{60^{2}}% {30R_{\frac{\lambda}{2}}}=\frac{3600}{30R_{\frac{\lambda}{2}}}=\frac{120}{R_{% \frac{\lambda}{2}}}=\frac{1}{{}^{\int_{0}^{\frac{\pi}{2}}\frac{\cos\left(\frac% {\pi}{2}\cos\theta\right)^{2}}{\sin\theta}d\theta}},\\ &\displaystyle\approx\frac{120}{73.1296}\approx 1.6409224\approx 2.15088\ \,% \mathrm{dBi};\end{aligned}\,\!
  22. q \scriptstyle{q}
  23. E = - q 4 π ε [ e r r 2 + r c d d t ( e r r 2 ) + 1 c 2 d 2 d t 2 ( e r ) ] \vec{E}={-q\over 4\pi\varepsilon_{\circ}}\left[{\vec{e}_{r^{\prime}}\over r^{% \prime 2}}+{r^{\prime}\over c}{d\ \over dt}\left({\vec{e}_{r^{\prime}}\over r^% {\prime 2}}\right)+{1\over c^{2}}{d^{2}\ \over dt^{2}}\left(\vec{e}_{r^{\prime% }}\right)\right]\,
  24. c \scriptstyle{c}
  25. ε \scriptstyle{\varepsilon_{\circ}}
  26. r \scriptstyle{r^{\prime}}
  27. E \scriptstyle{\vec{E}}
  28. r c \scriptstyle{r^{\prime}\over c}
  29. e r \textstyle{\vec{e}_{r^{\prime}}}
  30. E \scriptstyle{\vec{E}}
  31. r c \scriptstyle{r^{\prime}\over c}
  32. 1 r 2 \textstyle{1\over r^{2}}
  33. 1 r \textstyle{1\over r}
  34. E = - q 4 π ε c 2 d 2 d t 2 ( e r ) = - q 10 - 7 d 2 d t 2 ( e r ) \vec{E}={-q\over 4\pi\varepsilon c^{2}_{\circ}}{d^{2}\ \over dt^{2}}\left(\vec% {e}_{r^{\prime}}\right)=-q10^{-7}{d^{2}\ \over dt^{2}}\left(\vec{e}_{r^{\prime% }}\right)\,
  35. E θ \scriptstyle{\vec{E}_{\theta}}
  36. r \textstyle{r}
  37. \scriptstyle{\ell_{\circ}}
  38. ω \scriptstyle{\omega}
  39. P = q 2 ω 4 2 12 π ε c 3 P={q^{2}\omega^{4}\ell_{\circ}^{2}\over 12\pi\varepsilon_{\circ}c^{3}}
  40. d \scriptstyle{d\ell}
  41. I \scriptstyle{I}
  42. d E θ ( t + r c ) = - d sin θ 4 π ε c 2 r d I d t dE_{\theta}(t+\textstyle{r\over c})=\displaystyle{-d\ell\sin\theta\over 4\pi% \varepsilon_{\circ}c^{2}r}{dI\over dt}\,
  43. θ \scriptstyle{\theta}
  44. t + r c \scriptstyle{t+{r\over c}}
  45. < m t p l > r c \scriptstyle<mtpl>{{r\over c}}
  46. θ \scriptstyle{\theta}
  47. I = I e j ω t I=I_{\circ}e^{j\omega t}
  48. I \scriptstyle{I_{\circ}}
  49. < m t p l > ω = 2 π f \scriptstyle<mtpl>{{\omega=2\pi f}}
  50. j = - 1 \scriptstyle{j=\sqrt{-1}}
  51. d E θ ( t + < m t p l > r c ) = - d j ω 4 π ε c 2 sin θ r e j ω t dE_{\theta}(t+\textstyle<mtpl>{{r\over c}})=\displaystyle{-d\ell j\omega\over 4% \pi\varepsilon_{\circ}c^{2}}{\sin\theta\over r}e^{j\omega t}\,
  52. t \textstyle{t}\,
  53. d E θ ( t ) = - d j ω 4 π ε c 2 sin θ r e j ( ω t - ω c r ) dE_{\theta}(t)={-d\ell j\omega\over 4\pi\varepsilon_{\circ}c^{2}}{\sin\theta% \over r}e^{j\left(\omega t-{\omega\over c}r\right)}\,
  54. V a = R a G a λ cos ψ π Z E b V_{a}={\sqrt{R_{a}G_{a}}\,\lambda\cos\psi\over\sqrt{\pi Z_{\circ}}}E_{b}
  55. V a \scriptstyle{V_{a}}
  56. Z a \scriptstyle{Z_{a}}
  57. R a \scriptstyle{R_{a}}
  58. Z a \scriptstyle{Z_{a}}\,
  59. G a \scriptstyle{G_{a}}
  60. λ \scriptstyle{\lambda}
  61. E b \scriptstyle{E_{b}}
  62. ψ \scriptstyle{\psi}
  63. ψ \scriptstyle{\psi}
  64. cos ψ \scriptstyle{\cos\psi}
  65. Z = < m t p l > μ ε = 376.730313461 Ω \scriptstyle{Z_{\circ}=\sqrt{<}mtpl>{{\mu_{\circ}\over\varepsilon_{\circ}}}=37% 6.730313461\ \Omega}
  66. = R a G a λ cos ψ π Z =\displaystyle{{{\sqrt{R_{a}G_{a}}\lambda\cos\psi\over\sqrt{\pi Z_{\circ}}}}}\,
  67. = G a λ 2 4 π Z E b 2 =\displaystyle{{G_{a}\lambda^{2}\over 4\pi Z_{\circ}}E_{b}^{2}}\,
  68. = G a 4 π λ 2 =\displaystyle{{G_{a}\over 4\pi}\lambda^{2}}\,
  69. λ \scriptstyle{\lambda}

Antiunitary_operator.html

  1. U : H 1 H 2 U:H_{1}\to H_{2}\,
  2. U x , U y = x , y ¯ \langle Ux,Uy\rangle=\overline{\langle x,y\rangle}
  3. x x
  4. y y
  5. H 1 H_{1}
  6. H 1 = H 2 H_{1}=H_{2}
  7. H H
  8. | T x , T y | = | x , y | |\langle Tx,Ty\rangle|=|\langle x,y\rangle|
  9. x x
  10. y y
  11. H H
  12. U x , U y = x , y ¯ = y , x \langle Ux,Uy\rangle=\overline{\langle x,y\rangle}=\langle y,x\rangle
  13. x , y x,y
  14. U U
  15. U U
  16. U 2 U^{2}
  17. U 2 x , U 2 y = U x , U y ¯ = x , y . \langle U^{2}x,U^{2}y\rangle=\overline{\langle Ux,Uy\rangle}=\langle x,y\rangle.
  18. V V
  19. V K VK
  20. K K
  21. U U
  22. U K UK
  23. U U
  24. U * U^{*}
  25. U * x , y = x , U y ¯ \langle U^{*}x,y\rangle=\overline{\langle x,Uy\rangle}
  26. U U
  27. U U * = U * U = 1. UU^{*}=U^{*}U=1.
  28. U U
  29. K , K z = z ¯ , K,Kz=\overline{z},
  30. U = σ y K = ( 0 - i i 0 ) K , U=\sigma_{y}K=\begin{pmatrix}0&-i\\ i&0\end{pmatrix}K,
  31. σ y \sigma_{y}
  32. K K
  33. U 2 = - 1 U^{2}=-1
  34. W θ W_{\theta}
  35. 0 θ π 0\leq\theta\leq\pi
  36. W 0 : C C W_{0}:C\rightarrow C
  37. W 0 ( z ) = z ¯ W_{0}(z)=\overline{z}\,
  38. 0 < θ π 0<\theta\leq\pi
  39. W θ W_{\theta}
  40. W θ ( ( z 1 , z 2 ) ) = ( e i θ / 2 z 2 ¯ , e - i θ / 2 z 1 ¯ ) . W_{\theta}((z_{1},z_{2}))=(e^{i\theta/2}\overline{z_{2}},e^{-i\theta/2}% \overline{z_{1}}).\,
  41. 0 < θ π 0<\theta\leq\pi
  42. W θ ( W θ ( ( z 1 , z 2 ) ) ) = ( e i θ z 1 , e - i θ z 2 ) , W_{\theta}(W_{\theta}((z_{1},z_{2})))=(e^{i\theta}z_{1},e^{-i\theta}z_{2}),\,
  43. W θ W_{\theta}
  44. W 0 W_{0}

APBRmetrics.html

  1. Possessions = .96 * ( FGA - ORb + TO + ( .44 * FTA ) ) \mathrm{Possessions}=\mathrm{.96}*(\mathrm{FGA}-\mathrm{ORb}+\mathrm{TO}+(.44*% \mathrm{FTA}))
  2. OffensiveRating = PointsScored * 100 Possessions \mathrm{OffensiveRating}=\frac{\mathrm{PointsScored*100}}{\mathrm{Possessions}}
  3. DefensiveRating = PointsAllowed * 100 Possessions \mathrm{DefensiveRating}=\frac{\mathrm{PointsAllowed*100}}{\mathrm{Possessions}}
  4. eFG % = ( FGM + .5 * 3 F G M ) / FGA \mathrm{eFG\%}=(\mathrm{FGM}+\mathrm{.5*3FGM})/\mathrm{FGA}

APEX_system.html

  1. A 2 T = B S x K , \frac{A^{2}}{T}=\frac{BS_{x}}{K}\,,
  2. A A
  3. T T
  4. B B
  5. S x S_{x}
  6. K K
  7. B B
  8. L L
  9. k k
  10. N N
  11. t t
  12. K K
  13. E v E_{v}
  14. E v = log 2 A 2 T = log 2 B S x K . E_{v}=\log_{2}{\frac{A^{2}}{T}}=\log_{2}{\frac{BS_{x}}{K}}\,.
  15. E v E_{v}
  16. E v E_{v}
  17. E v E_{v}
  18. E v E_{v}
  19. E v E_{v}
  20. A A
  21. T T
  22. A 2 T = L S . \frac{A^{2}}{T}=LS\,.
  23. K K
  24. K K
  25. E v = A v + T v = B v + S v , E_{v}=A_{v}+T_{v}=B_{v}+S_{v}\,,
  26. A v A_{v}
  27. A v = A_{v}=
  28. log 2 \log_{2}
  29. A 2 A^{2}
  30. T v T_{v}
  31. T v = log 2 T_{v}=\log_{2}
  32. ( 1 / T ) (1/T)
  33. E v E_{v}
  34. E v = A v + T v E_{v}=A_{v}+T_{v}
  35. S v S_{v}
  36. S v = log 2 S_{v}=\log_{2}
  37. ( N S x ) (NS_{x})
  38. B v B_{v}
  39. B v = log 2 B_{v}=\log_{2}
  40. ( B / N K ) (B/NK)
  41. N N
  42. S x S_{x}
  43. S v S_{v}
  44. N N
  45. 2 - 7 / 4 2^{-7/4}
  46. 2 - 7 / 4 2^{-7/4}
  47. N N
  48. K K
  49. A 2 T = I S x C , \frac{A^{2}}{T}=\frac{IS_{x}}{C}\,,
  50. I I
  51. C C
  52. I I
  53. E E
  54. E v = A v + T v = I v + S v , E_{v}=A_{v}+T_{v}=I_{v}+S_{v}\,,
  55. I v I_{v}
  56. I v = log 2 I_{v}=\log_{2}
  57. ( I / N C ) (I/NC)
  58. L W LW
  59. E v E_{v}
  60. A v A_{v}
  61. L W k LWk
  62. T v T_{v}
  63. L W t LWt
  64. S v S_{v}
  65. B v B_{v}
  66. v v
  67. E E
  68. B B
  69. I I
  70. K K
  71. f f
  72. N N
  73. 2 - 7 / 4 2^{-7/4}
  74. S v S_{\mathrm{v}}
  75. B v B_{\mathrm{v}}
  76. N N
  77. K K
  78. B v = log 2 B N K . B_{\mathrm{v}}=\log_{2}\frac{B}{NK}\,.
  79. N N
  80. B v B_{\mathrm{v}}
  81. K K
  82. B v B_{\mathrm{v}}
  83. B B
  84. K = 1 / N K=1/N
  85. B B
  86. B B
  87. B B
  88. S v S_{\mathrm{v}}

Apollonian_circles.html

  1. { X d ( X , C ) d ( X , D ) = r } . \left\{X\mid\frac{d(X,C)}{d(X,D)}=r\right\}.
  2. { X C X ^ D = θ } . \left\{X\mid\;C\hat{X}D\;=\theta\right\}.
  3. i s o p t ( θ ) = { X ( X C , X D ) = θ + 2 k π } . isopt(\theta)=\left\{X\mid\left(\overrightarrow{XC},\overrightarrow{XD}\right)% =\theta+2k\pi\right\}.
  4. i s o p t ( θ + π ) isopt(\theta+\pi)
  5. full red circle = { X ( ( X C ) , ( X D ) ) = θ + k π } {\rm full\;red\;circle}=\left\{X\mid\;\left((XC),(XD)\right)\;=\theta+k\pi\right\}
  6. ( x - p ) 2 + ( y - q ) 2 = r 2 , \displaystyle(x-p)^{2}+(y-q)^{2}=r^{2},
  7. α ( x 2 + y 2 ) - 2 β x - 2 γ y + δ = 0 , \displaystyle\alpha(x^{2}+y^{2})-2\beta x-2\gamma y+\delta=0,
  8. z ( α 1 , β 1 , γ 1 , δ 1 ) + ( 1 - z ) ( α 2 , β 2 , γ 2 , δ 2 ) \displaystyle z(\alpha_{1},\beta_{1},\gamma_{1},\delta_{1})+(1-z)(\alpha_{2},% \beta_{2},\gamma_{2},\delta_{2})

Apollonius'_theorem.html

  1. A B 2 + A C 2 = 2 ( A D 2 + B D 2 ) . AB^{2}+AC^{2}=2(AD^{2}+BD^{2}).\,
  2. b 2 = m 2 + d 2 - 2 d m cos θ c 2 = m 2 + d 2 - 2 d m cos θ = m 2 + d 2 + 2 d m cos θ . \begin{aligned}\displaystyle b^{2}&\displaystyle=m^{2}+d^{2}-2dm\cos\theta\\ \displaystyle c^{2}&\displaystyle=m^{2}+d^{2}-2dm\cos\theta^{\prime}\\ &\displaystyle=m^{2}+d^{2}+2dm\cos\theta.\end{aligned}
  3. b 2 + c 2 = 2 m 2 + 2 d 2 b^{2}+c^{2}=2m^{2}+2d^{2}\,

Append.html

  1. n n

Approximate_string_matching.html

  1. P = p 1 p 2 p m P=p_{1}p_{2}...p_{m}
  2. T = t 1 t 2 t n T=t_{1}t_{2}\dots t_{n}
  3. T j , j = t j t j T_{j^{\prime},j}=t_{j^{\prime}}\dots t_{j}
  4. P i P_{i}
  5. T j , j T_{j^{\prime},j}

Approximations_of_π.html

  1. π \pi
  2. π \pi
  3. π \pi
  4. π \pi
  5. π \pi
  6. π \pi
  7. π \pi
  8. π \pi
  9. 22 / 7 {22}/{7}
  10. π \pi
  11. π \pi
  12. 25 / 8 = 3.125 25/8=3.125
  13. π \pi
  14. 256 / 81 {256}/{81}
  15. 339 / 108 3.139 339/108≈3.139
  16. 223 / 71 {223}/{71}
  17. 377 / 120 {377}/{120}
  18. π \pi
  19. π 3927 / 1250 = 3.1416 π≈3927/1250=3.1416
  20. π \pi
  21. π \pi
  22. π 22 / 7 π≈22/7
  23. π 355 / 113 π≈355/113
  24. π \pi
  25. π \pi
  26. π 62832 / 20000 = 3.1416 π≈62832/20000=3.1416
  27. π 62832 / 20000 = 3.1416 π≈62832/20000=3.1416
  28. π \pi
  29. π \pi
  30. π \pi
  31. π \pi
  32. π = 12 k = 0 ( - 3 ) - k 2 k + 1 = 12 k = 0 ( - 1 3 ) k 2 k + 1 = 12 ( 1 - 1 3 3 + 1 5 3 2 - 1 7 3 3 + ) \pi=\sqrt{12}\sum^{\infty}_{k=0}\frac{(-3)^{-k}}{2k+1}=\sqrt{12}\sum^{\infty}_% {k=0}\frac{(-\frac{1}{3})^{k}}{2k+1}=\sqrt{12}\left(1-{1\over 3\cdot 3}+{1% \over 5\cdot 3^{2}}-{1\over 7\cdot 3^{3}}+\cdots\right)
  33. π \pi
  34. π \pi
  35. n 2 + 1 4 n 3 + 5 n \frac{n^{2}+1}{4n^{3}+5n}
  36. π / 4 {\pi}/{4}
  37. π \pi
  38. n n
  39. π \pi
  40. 2 π 6.28318530717958648 , 2\pi\approx 6.28318530717958648,\,
  41. π 3.14159265358979324. \pi\approx 3.14159265358979324.\,
  42. π \pi
  43. π \pi
  44. π \pi
  45. π \pi
  46. π \pi
  47. π \pi
  48. 1 π = 2 2 9801 k = 0 ( 4 k ) ! ( 1103 + 26390 k ) ( k ! ) 4 396 4 k \frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum^{\infty}_{k=0}\frac{(4k)!(1103+26390k% )}{(k!)^{4}396^{4k}}
  49. π \pi
  50. π \pi
  51. π \pi
  52. π \pi
  53. π \pi
  54. π \pi
  55. π \pi
  56. π \pi
  57. π \pi
  58. 1 π = 12 k = 0 ( - 1 ) k ( 6 k ) ! ( 13591409 + 545140134 k ) ( 3 k ) ! ( k ! ) 3 640320 3 k + 3 / 2 . \frac{1}{\pi}=12\sum^{\infty}_{k=0}\frac{(-1)^{k}(6k)!(13591409+545140134k)}{(% 3k)!(k!)^{3}640320^{3k+3/2}}.
  59. π \pi
  60. π \pi
  61. π \pi
  62. π \pi
  63. π \pi
  64. π \pi
  65. π \pi
  66. 22 / 7 {22}/{7}
  67. 355 / 113 {355}/{113}
  68. π \pi
  69. π = 3.2 π=3.2
  70. π \pi
  71. π \pi
  72. π \pi
  73. π \pi
  74. 111 / 106 {111}/{106}
  75. 333 / 106 {333}/{106}
  76. π \pi
  77. π \pi
  78. π = 16 / 5 = 3.2 π=16/5=3.2
  79. π \pi
  80. π \pi
  81. π \pi
  82. π 4 = 4 arctan 1 5 - arctan 1 239 \frac{\pi}{4}=4\arctan\frac{1}{5}-\arctan\frac{1}{239}
  83. ( 5 + i ) 4 ( 239 - i ) = 2 2 13 4 ( 1 + i ) . (5+i)^{4}\cdot(239-i)=2^{2}\cdot 13^{4}(1+i).\!
  84. x x
  85. y y
  86. x x
  87. y y
  88. π 4 = 12 arctan 1 49 + 32 arctan 1 57 - 5 arctan 1 239 + 12 arctan 1 110443 \frac{\pi}{4}=12\arctan\frac{1}{49}+32\arctan\frac{1}{57}-5\arctan\frac{1}{239% }+12\arctan\frac{1}{110443}\!
  89. π 4 = 44 arctan 1 57 + 7 arctan 1 239 - 12 arctan 1 682 + 24 arctan 1 12943 \frac{\pi}{4}=44\arctan\frac{1}{57}+7\arctan\frac{1}{239}-12\arctan\frac{1}{68% 2}+24\arctan\frac{1}{12943}\!
  90. π \pi
  91. π \displaystyle\pi
  92. π = 12 k = 0 ( - 3 ) - k 2 k + 1 = 12 k = 0 ( - 1 3 ) k 2 k + 1 = 12 ( 1 1 3 0 - 1 3 3 1 + 1 5 3 2 - 1 7 3 3 + ) \pi=\sqrt{12}\sum^{\infty}_{k=0}\frac{(-3)^{-k}}{2k+1}=\sqrt{12}\sum^{\infty}_% {k=0}\frac{(-\frac{1}{3})^{k}}{2k+1}=\sqrt{12}\left({1\over 1\cdot 3^{0}}-{1% \over 3\cdot 3^{1}}+{1\over 5\cdot 3^{2}}-{1\over 7\cdot 3^{3}}+\cdots\right)
  93. π = 20 arctan 1 7 + 8 arctan 3 79 {\pi}=20\arctan\frac{1}{7}+8\arctan\frac{3}{79}
  94. π 2 = k = 0 k ! ( 2 k + 1 ) ! ! = k = 0 2 k k ! 2 ( 2 k + 1 ) ! = 1 + 1 3 ( 1 + 2 5 ( 1 + 3 7 ( 1 + ) ) ) \frac{\pi}{2}=\sum_{k=0}^{\infty}\frac{k!}{(2k+1)!!}=\sum_{k=0}^{\infty}\cfrac% {2^{k}k!^{2}}{(2k+1)!}=1+\frac{1}{3}\left(1+\frac{2}{5}\left(1+\frac{3}{7}% \left(1+\cdots\right)\right)\right)
  95. 1 π = 2 2 9801 k = 0 ( 4 k ) ! ( 1103 + 26390 k ) ( k ! ) 4 396 4 k \frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum^{\infty}_{k=0}\frac{(4k)!(1103+26390k% )}{(k!)^{4}396^{4k}}
  96. 1 π = 12 k = 0 ( - 1 ) k ( 6 k ) ! ( 13591409 + 545140134 k ) ( 3 k ) ! ( k ! ) 3 640320 3 k + 3 / 2 \frac{1}{\pi}=12\sum^{\infty}_{k=0}\frac{(-1)^{k}(6k)!(13591409+545140134k)}{(% 3k)!(k!)^{3}640320^{3k+3/2}}
  97. π \pi
  98. π \pi
  99. y 0 = 2 - 1 , a 0 = 6 - 4 2 y_{0}=\sqrt{2}-1,\ a_{0}=6-4\sqrt{2}
  100. y k + 1 = ( 1 - f ( y k ) ) / ( 1 + f ( y k ) ) , a k + 1 = a k ( 1 + y k + 1 ) 4 - 2 2 k + 3 y k + 1 ( 1 + y k + 1 + y k + 1 2 ) y_{k+1}=(1-f(y_{k}))/(1+f(y_{k}))~{},~{}a_{k+1}=a_{k}(1+y_{k+1})^{4}-2^{2k+3}y% _{k+1}(1+y_{k+1}+y_{k+1}^{2})
  101. f ( y ) = ( 1 - y 4 ) 1 / 4 f(y)=(1-y^{4})^{1/4}
  102. 1 / a k 1/a_{k}
  103. π \pi
  104. π \pi
  105. 1 / π {1}/{\pi}
  106. π 4 = 12 arctan 1 49 + 32 arctan 1 57 - 5 arctan 1 239 + 12 arctan 1 110443 \frac{\pi}{4}=12\arctan\frac{1}{49}+32\arctan\frac{1}{57}-5\arctan\frac{1}{239% }+12\arctan\frac{1}{110443}
  107. π 4 = 44 arctan 1 57 + 7 arctan 1 239 - 12 arctan 1 682 + 24 arctan 1 12943 \frac{\pi}{4}=44\arctan\frac{1}{57}+7\arctan\frac{1}{239}-12\arctan\frac{1}{68% 2}+24\arctan\frac{1}{12943}
  108. π \pi
  109. π \pi
  110. 60 {}_{60}
  111. 3 + 8 60 + 29 60 2 + 44 60 3 = 3.14159 259 + 3+\frac{8}{60}+\frac{29}{60^{2}}+\frac{44}{60^{3}}=3.14159\ 259^{+}
  112. π \pi
  113. 2 + 3 = 3.146 + \sqrt{2}+\sqrt{3}=3.146^{+}
  114. π \pi
  115. 15 - 3 + 1 = 3.140 + \sqrt{15}-\sqrt{3}+1=3.140^{+}
  116. 31 3 = 3.1413 + \sqrt[3]{31}=3.1413^{+}
  117. 7 + 6 + 5 = 3.1416 + \sqrt{7+\sqrt{6+\sqrt{5}}}=3.1416^{+}
  118. 9 5 + 9 5 = 3.1416 + \frac{9}{5}+\sqrt{\frac{9}{5}}=3.1416^{+}
  119. 7 7 4 9 = 3.14156 + \frac{7^{7}}{4^{9}}=3.14156^{+}
  120. 355 113 = 3.14159 29 + \frac{355}{113}=3.14159\ 29^{+}
  121. 3 4 + 2 4 + 1 2 + ( 2 3 ) 2 4 = 2143 22 4 = 3.14159 2652 + \sqrt[4]{3^{4}+2^{4}+\frac{1}{2+(\frac{2}{3})^{2}}}=\sqrt[4]{\frac{2143}{22}}=% 3.14159\ 2652^{+}
  122. π \pi
  123. 63 25 × 17 + 15 5 7 + 15 5 = 3.14159 26538 + \frac{63}{25}\times\frac{17+15\sqrt{5}}{7+15\sqrt{5}}=3.14159\ 26538^{+}
  124. 10 100 11222.11122 193 = 3.14159 26536 + \sqrt[193]{\frac{10^{100}}{11222.11122}}=3.14159\ 26536^{+}
  125. π \pi
  126. π \pi
  127. 80 15 ( 5 4 + 53 89 ) 3 2 3308 ( 5 4 + 53 89 ) - 3 89 \frac{80\sqrt{15}(5^{4}+53\sqrt{89})^{\frac{3}{2}}}{3308(5^{4}+53\sqrt{89})-3% \sqrt{89}}
  128. d d
  129. h h
  130. d d
  131. 5 4 + 53 89 \scriptstyle 5^{4}+53\sqrt{89}
  132. U 89 = 500 + 53 89 \scriptstyle U_{89}=500+53\sqrt{89}
  133. x x
  134. y y
  135. x x
  136. y y
  137. ln ( 640320 3 + 744 ) 163 = 3.14159 26535 89793 23846 26433 83279 + \frac{\ln(640320^{3}+744)}{\sqrt{163}}=3.14159\ 26535\ 89793\ 23846\ 26433\ 83% 279^{+}
  138. ln ( 5280 3 ( 236674 + 30303 61 ) 3 + 744 ) 427 \frac{\ln(5280^{3}(236674+30303\sqrt{61})^{3}+744)}{\sqrt{427}}
  139. ln ( ( 2 u ) 6 + 24 ) 3502 \frac{\ln\big((2u)^{6}+24\big)}{\sqrt{3502}}
  140. u = ( a + a 2 - 1 ) 2 ( b + b 2 - 1 ) 2 ( c + c 2 - 1 ) ( d + d 2 - 1 ) u=(a+\sqrt{a^{2}-1})^{2}(b+\sqrt{b^{2}-1})^{2}(c+\sqrt{c^{2}-1})(d+\sqrt{d^{2}% -1})
  141. a = 1 2 ( 23 + 4 34 ) b = 1 2 ( 19 2 + 7 17 ) c = ( 429 + 304 2 ) d = 1 2 ( 627 + 442 2 ) \begin{aligned}\displaystyle a&\displaystyle=\tfrac{1}{2}(23+4\sqrt{34})\\ \displaystyle b&\displaystyle=\tfrac{1}{2}(19\sqrt{2}+7\sqrt{17})\\ \displaystyle c&\displaystyle=(429+304\sqrt{2})\\ \displaystyle d&\displaystyle=\tfrac{1}{2}(627+442\sqrt{2})\end{aligned}
  142. τ = - 3502 \tau=\sqrt{-3502}
  143. π \pi
  144. π \pi
  145. 3 1 , 22 7 , 333 106 , 355 113 , 103993 33102 , 104348 33215 , 208341 66317 , 312689 99532 , 833719 265381 , 1146408 364913 , 4272943 1360120 , 5419351 1725033 \frac{3}{1},\frac{22}{7},\frac{333}{106},\frac{355}{113},\frac{103993}{33102},% \frac{104348}{33215},\frac{208341}{66317},\frac{312689}{99532},\frac{833719}{2% 65381},\frac{1146408}{364913},\frac{4272943}{1360120},\frac{5419351}{1725033}
  146. 355 113 \frac{355}{113}
  147. π \pi
  148. π \pi
  149. π / 4 π/4
  150. A = π r 2 . A=\pi r^{2}.
  151. r r
  152. r r
  153. x x
  154. y y
  155. d = x 2 + y 2 . d=\sqrt{x^{2}+y^{2}}.\!
  156. x x
  157. y y
  158. x x
  159. y y
  160. r r
  161. r r
  162. x x
  163. y y
  164. x 2 + y 2 r . \sqrt{x^{2}+y^{2}}\leq r.\!
  165. π \pi
  166. r r
  167. π = lim r 1 r 2 x = - r r y = - r r { 1 if x 2 + y 2 r 0 if x 2 + y 2 > r . \pi=\lim_{r\to\infty}\frac{1}{r^{2}}\sum_{x=-r}^{r}\;\sum_{y=-r}^{r}\begin{% cases}1&\,\text{if }\sqrt{x^{2}+y^{2}}\leq r\\ 0&\,\text{if }\sqrt{x^{2}+y^{2}}>r.\end{cases}
  168. r r
  169. x x
  170. y y
  171. x x
  172. y y
  173. r r
  174. r r
  175. r r
  176. r r
  177. r r
  178. π \pi
  179. r r
  180. π \pi
  181. r r
  182. π \pi
  183. π \pi
  184. π \pi
  185. π = 3 + 1 2 6 + 3 2 6 + 5 2 6 + \pi={3+\cfrac{1^{2}}{6+\cfrac{3^{2}}{6+\cfrac{5^{2}}{6+\ddots\,}}}}\!
  186. π = 4 1 + 1 2 3 + 2 2 5 + 3 2 7 + \pi=\cfrac{4}{1+\cfrac{1^{2}}{3+\cfrac{2^{2}}{5+\cfrac{3^{2}}{7+\ddots}}}}\!
  187. π = 4 n = 0 ( - 1 ) n 2 n + 1 = 4 ( 1 1 - 1 3 + 1 5 - 1 7 + - ) = 4 1 + 1 2 2 + 3 2 2 + 5 2 2 + \pi=4\sum_{n=0}^{\infty}\cfrac{(-1)^{n}}{2n+1}=4\left(\frac{1}{1}-\frac{1}{3}+% \frac{1}{5}-\frac{1}{7}+-\cdots\right)\!=\cfrac{4}{1+\cfrac{1^{2}}{2+\cfrac{3^% {2}}{2+\cfrac{5^{2}}{2+\ddots}}}}\!
  188. x x
  189. x x
  190. π \pi
  191. 4 arctan ( 1 ) = π 4\arctan(1)=\pi\!
  192. π = 2 ( 1 + 1 3 + 1 2 3 5 + 1 2 3 3 5 7 + 1 2 3 4 3 5 7 9 + 1 2 3 4 5 3 5 7 9 11 + ) \pi=2\left(1+\cfrac{1}{3}+\cfrac{1\cdot 2}{3\cdot 5}+\cfrac{1\cdot 2\cdot 3}{3% \cdot 5\cdot 7}+\cfrac{1\cdot 2\cdot 3\cdot 4}{3\cdot 5\cdot 7\cdot 9}+\cfrac{% 1\cdot 2\cdot 3\cdot 4\cdot 5}{3\cdot 5\cdot 7\cdot 9\cdot 11}+\cdots\right)\!
  193. = 2 n = 0 n ! ( 2 n + 1 ) ! ! = n = 0 2 n + 1 n ! 2 ( 2 n + 1 ) ! = n = 0 2 n + 1 ( 2 n n ) ( 2 n + 1 ) =2\sum_{n=0}^{\infty}\cfrac{n!}{(2n+1)!!}=\sum_{n=0}^{\infty}\cfrac{2^{n+1}n!^% {2}}{(2n+1)!}=\sum_{n=0}^{\infty}\cfrac{2^{n+1}}{{\left({{2n}\atop{n}}\right)}% (2n+1)}\!
  194. = 2 + 2 3 + 4 15 + 4 35 + 16 315 + 16 693 + 32 3003 + 32 6435 + 256 109395 + 256 230945 + =2+\frac{2}{3}+\frac{4}{15}+\frac{4}{35}+\frac{16}{315}+\frac{16}{693}+\frac{3% 2}{3003}+\frac{32}{6435}+\frac{256}{109395}+\frac{256}{230945}+\cdots\!
  195. sin ( π 6 ) = 1 2 \sin\left(\frac{\pi}{6}\right)=\frac{1}{2}\!
  196. π = 6 sin - 1 ( 1 2 ) = 6 ( 1 2 + 1 2 3 2 3 + 1 3 2 4 5 2 5 + 1 3 5 2 4 6 7 2 7 + ) \pi=6\sin^{-1}\left(\frac{1}{2}\right)=6\left(\frac{1}{2}+\frac{1}{2\cdot 3% \cdot 2^{3}}+\frac{1\cdot 3}{2\cdot 4\cdot 5\cdot 2^{5}}+\frac{1\cdot 3\cdot 5% }{2\cdot 4\cdot 6\cdot 7\cdot 2^{7}}+\cdots\!\right)
  197. = 3 16 0 1 + 6 16 1 3 + 18 16 2 5 + 60 16 3 7 + = n = 0 3 ( 2 n n ) 16 n ( 2 n + 1 ) =\frac{3}{16^{0}\cdot 1}+\frac{6}{16^{1}\cdot 3}+\frac{18}{16^{2}\cdot 5}+% \frac{60}{16^{3}\cdot 7}+\cdots\!=\sum_{n=0}^{\infty}\frac{3\cdot{\left({{2n}% \atop{n}}\right)}}{16^{n}(2n+1)}
  198. = 3 + 1 8 + 9 640 + 15 7168 + 35 98304 + 189 2883584 + 693 54525952 + 429 167772160 + =3+\frac{1}{8}+\frac{9}{640}+\frac{15}{7168}+\frac{35}{98304}+\frac{189}{28835% 84}+\cfrac{693}{54525952}+\frac{429}{167772160}+\cdots\!
  199. π \pi
  200. N N
  201. N log ( N ) log ( log ( N ) ) N\,\log(N)\,\log(\log(N))
  202. π \pi
  203. π \pi
  204. π = n = 0 ( 4 8 n + 1 - 2 8 n + 4 - 1 8 n + 5 - 1 8 n + 6 ) ( 1 16 ) n \pi=\sum_{n=0}^{\infty}\left(\frac{4}{8n+1}-\frac{2}{8n+4}-\frac{1}{8n+5}-% \frac{1}{8n+6}\right)\left(\frac{1}{16}\right)^{n}\!
  205. n n
  206. π \pi
  207. n n
  208. n n
  209. π \pi
  210. π + 3 = n = 1 n 2 n n ! 2 ( 2 n ) ! \pi+3=\sum_{n=1}^{\infty}\frac{n2^{n}n!^{2}}{(2n)!}
  211. n n
  212. π \pi
  213. π = 1 2 6 n = 0 ( - 1 ) n 2 10 n ( - 2 5 4 n + 1 - 1 4 n + 3 + 2 8 10 n + 1 - 2 6 10 n + 3 - 2 2 10 n + 5 - 2 2 10 n + 7 + 1 10 n + 9 ) \pi=\frac{1}{2^{6}}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{2^{10n}}\left(-\frac{2^{% 5}}{4n+1}-\frac{1}{4n+3}+\frac{2^{8}}{10n+1}-\frac{2^{6}}{10n+3}-\frac{2^{2}}{% 10n+5}-\frac{2^{2}}{10n+7}+\frac{1}{10n+9}\right)\!
  214. π \pi
  215. π \pi
  216. π \pi
  217. π = k = 0 1 16 k ( 4 8 k + 1 - 2 8 k + 4 - 1 8 k + 5 - 1 8 k + 6 ) . \pi=\sum_{k=0}^{\infty}\frac{1}{16^{k}}\left(\frac{4}{8k+1}-\frac{2}{8k+4}-% \frac{1}{8k+5}-\frac{1}{8k+6}\right).\!
  218. π \pi
  219. π \pi
  220. π = 1 2 6 n = 0 ( - 1 ) n 2 10 n ( - 2 5 4 n + 1 - 1 4 n + 3 + 2 8 10 n + 1 - 2 6 10 n + 3 - 2 2 10 n + 5 - 2 2 10 n + 7 + 1 10 n + 9 ) \pi=\frac{1}{2^{6}}\sum_{n=0}^{\infty}\frac{{(-1)}^{n}}{2^{10n}}\left(-\frac{2% ^{5}}{4n+1}-\frac{1}{4n+3}+\frac{2^{8}}{10n+1}-\frac{2^{6}}{10n+3}-\frac{2^{2}% }{10n+5}-\frac{2^{2}}{10n+7}+\frac{1}{10n+9}\right)\!
  221. π \pi
  222. π 2 = k = 0 k ! ( 2 k + 1 ) ! ! = k = 0 2 k k ! 2 ( 2 k + 1 ) ! = 1 + 1 3 ( 1 + 2 5 ( 1 + 3 7 ( 1 + ) ) ) \frac{\pi}{2}=\sum_{k=0}^{\infty}\frac{k!}{(2k+1)!!}=\sum_{k=0}^{\infty}\frac{% 2^{k}k!^{2}}{(2k+1)!}=1+\frac{1}{3}\left(1+\frac{2}{5}\left(1+\frac{3}{7}\left% (1+\cdots\right)\right)\right)\!
  223. 1 π = 2 2 9801 k = 0 ( 4 k ) ! ( 1103 + 26390 k ) ( k ! ) 4 396 4 k \frac{1}{\pi}=\frac{2\sqrt{2}}{9801}\sum^{\infty}_{k=0}\frac{(4k)!(1103+26390k% )}{(k!)^{4}396^{4k}}\!
  224. π \pi
  225. 1 π = 12 k = 0 ( - 1 ) k ( 6 k ) ! ( 13591409 + 545140134 k ) ( 3 k ) ! ( k ! ) 3 640320 3 k + 3 / 2 \frac{1}{\pi}=12\sum^{\infty}_{k=0}\frac{(-1)^{k}(6k)!(13591409+545140134k)}{(% 3k)!(k!)^{3}640320^{3k+3/2}}\!
  226. π \pi
  227. π \pi
  228. π \pi
  229. π \pi
  230. π \pi
  231. π \pi
  232. y y
  233. y y
  234. e e
  235. 2 √2
  236. e e
  237. 2 √2
  238. 3 √3
  239. π \pi
  240. π \pi
  241. π \pi
  242. π \pi
  243. π \pi
  244. π \pi
  245. π \pi
  246. y y

Archimedes'_cattle_problem.html

  1. 7.76 × 10 206544 7.76\times 10^{206544}
  2. = ( 1 2 + 1 3 ) =\left(\frac{1}{2}+\frac{1}{3}\right)
  3. = ( 1 4 + 1 5 ) =\left(\frac{1}{4}+\frac{1}{5}\right)
  4. = ( 1 6 + 1 7 ) =\left(\frac{1}{6}+\frac{1}{7}\right)
  5. = ( 1 3 + 1 4 ) =\left(\frac{1}{3}+\frac{1}{4}\right)
  6. = ( 1 4 + 1 5 ) =\left(\frac{1}{4}+\frac{1}{5}\right)
  7. = ( 1 5 + 1 6 ) =\left(\frac{1}{5}+\frac{1}{6}\right)
  8. = ( 1 6 + 1 7 ) =\left(\frac{1}{6}+\frac{1}{7}\right)
  9. W , B , D , W,B,D,
  10. Y Y
  11. w , b , d , w,b,d,
  12. y y
  13. W \displaystyle W
  14. B \displaystyle B
  15. n = ( w 4658 j - w - 4658 j ) 2 ( 4657 ) ( 79072 ) n=\frac{(w^{4658j}-w^{-4658j})^{2}}{(4657)(79072)}\,
  16. w = 300426607914281713365 609 + 84129507677858393258 7766 w=300426607914281713365\sqrt{609}+84129507677858393258\sqrt{7766}
  17. w 2 = u + v ( 609 ) ( 7766 ) w^{2}=u+v\sqrt{(609)(7766)}\,
  18. u 2 - ( 609 ) ( 7766 ) v 2 = 1 u^{2}-(609)(7766)v^{2}=1\,
  19. 7.76 × 10 206544 7.76\times 10^{206544}
  20. B + W = 7 , 460 , 514 k + 10 , 366 , 482 k = ( 2 2 ) ( 3 ) ( 11 ) ( 29 ) ( 4657 ) k B+W=7,460,514k+10,366,482k=(2^{2})(3)(11)(29)(4657)k\,
  21. D + Y = t 2 + t 2 D+Y=\frac{t^{2}+t}{2}
  22. t = - 1 ± 1 + 8 ( D + Y ) 2 t=\frac{-1\pm\sqrt{1+8(D+Y)}}{2}
  23. p 2 - ( 4 ) ( 609 ) ( 7766 ) ( 4657 2 ) q 2 = 1 p^{2}-(4)(609)(7766)(4657^{2})q^{2}=1\,

Aridity_index.html

  1. R / 2 R/2
  2. R = 2 × T R=2\times T
  3. R = 2 × T + 14 R=2\times T+14
  4. R = 2 × T + 28 R=2\times T+28
  5. T T
  6. A I T = 100 × d n AI_{T}=100\times\frac{d}{n}
  7. d d
  8. n n
  9. A I B = 100 × R L P AI_{B}=100\times\frac{R}{LP}
  10. R R
  11. P P
  12. L L
  13. R R
  14. L L
  15. P P
  16. A I U = P P E T AI_{U}=\frac{P}{PET}
  17. P E T PET
  18. P P
  19. P E T PET
  20. P P

Armstrong's_axioms.html

  1. F + F^{+}
  2. F F
  3. F + F^{+}
  4. R ( U ) , F \langle R(U),F\rangle
  5. U U
  6. F F
  7. f f
  8. F F
  9. F f F\models f
  10. r r
  11. R R
  12. F F
  13. f f
  14. F + F^{+}
  15. F F
  16. A A
  17. f f
  18. F F
  19. A A
  20. F A f F\vdash_{A}f
  21. f f
  22. A A
  23. F F
  24. F A * F^{*}_{A}
  25. F F
  26. A A
  27. A A
  28. F A * F + F^{*}_{A}\subseteq F^{+}
  29. A A
  30. F F
  31. A A
  32. F + F A * F^{+}\subseteq F^{*}_{A}
  33. A A
  34. F F
  35. R ( U ) R(U)
  36. U U
  37. X X
  38. Y Y
  39. Z Z
  40. U U
  41. X X
  42. Y Y
  43. X Y XY
  44. X Y X\cup Y
  45. Y X Y\subseteq X
  46. X Y X\to Y
  47. X Y X\to Y
  48. X Z Y Z XZ\to YZ
  49. Z Z
  50. X Y X\to Y
  51. Y Z Y\to Z
  52. X Z X\to Z
  53. X Y X\to Y
  54. X Z X\to Z
  55. X Y Z X\to YZ
  56. X Y Z X\to YZ
  57. X Y X\to Y
  58. X Z X\to Z
  59. X Y X\to Y
  60. Y Z W YZ\to W
  61. X Z W XZ\to W
  62. F F
  63. F + F^{+}

Arnold_tongue.html

  1. θ n + 1 = θ n + Ω - K 2 π sin ( 2 π θ n ) . \theta_{n+1}=\theta_{n}+\Omega-\frac{K}{2\pi}\sin(2\pi\theta_{n}).
  2. θ \theta
  3. θ n \theta_{n}
  4. ω = lim n 1 n θ n n . \omega=\lim_{n\to\infty}\frac{\sum_{1}^{n}\theta_{n}}{n}.
  5. Ω = p / q \Omega=p/q
  6. K 0 K\to 0
  7. ω = p / q \omega=p/q
  8. ω = 1 / 2 \omega=1/2
  9. θ n \theta_{n}
  10. θ n \theta_{n}
  11. p / q p/q
  12. K 0 K\neq 0
  13. θ n + 1 = θ n + p n + K 2 π sin ( 2 π θ n ) \theta_{n+1}=\theta_{n}+p_{n}+{\frac{K}{2\pi}}\sin(2\pi\theta_{n})
  14. p n + 1 = θ n + 1 - θ n p_{n+1}=\theta_{n+1}-\theta_{n}
  15. p n p_{n}

Arnold_Walfisz.html

  1. σ \sigma
  2. ϕ \phi

Aromatic_ring_current.html

  1. HOMA = 1 - 257.7 / n i n ( d opt - d i ) 2 , \mathrm{HOMA}=1-257.7/n\sum^{n}_{i}(d_{\rm opt}-d_{i})^{2}\,,

Arsenic_pentoxide.html

  1. \overrightarrow{\leftarrow}

Assumption-based_planning.html

  1. e = ( P R ) C T e=\frac{\sum(PR)}{CT}

Asymptotically_optimal_algorithm.html

  1. lim n t ( n ) b ( n ) < . \lim_{n\rightarrow\infty}\frac{t(n)}{b(n)}<\infty.

Atan2.html

  1. x x
  2. y y
  3. a t a n 2 ( y , x ) atan2(y,x)
  4. x x
  5. ( x , y ) (x,y)
  6. y > 0 y>0
  7. y Align l t ; 0 y&lt;0
  8. a t a n 2 atan2
  9. x + i y x+iy
  10. a t a n 2 ( y , x ) = P r a r g ( x + i y ) = A r g ( x + i y ) atan2(y,x)=Prarg(x+iy)=Arg(x+iy)
  11. 2 π
  12. a t a n 2 atan2
  13. ( π , π ] (−π,π]
  14. ( x , y ) (x,y)
  15. x x
  16. ( 1 , 1 ) (1,1)
  17. a r c t a n ( 1 / 1 ) arctan(1/1)
  18. π / 4 π/4
  19. 45 ° 45°
  20. x x
  21. ( 1 , 1 ) (−1,−1)
  22. a r c t a n ( 1 / 1 ) arctan(−1/−1)
  23. π / 4 π/4
  24. 3 π / 4 −3π/4
  25. 135 ° −135°
  26. a t a n 2 atan2
  27. a t a n 2 ( 1 , 1 ) = π / 4 atan2(1,1)=π/4
  28. a t a n 2 ( 1 , 1 ) = 3 π / 4 atan2(−1,−1)=−3π/4
  29. ± π / 2 ±π/2
  30. ± 90 ° ±90°
  31. x x
  32. ( 0 , 1 ) (0,1)
  33. a r c t a n ( 1 / 0 ) arctan(1/0)
  34. a t a n 2 ( 1 , 0 ) atan2(1,0)
  35. π / 2 π/2
  36. a r c t a n arctan
  37. ( π / 2 , π / 2 ) (−π/2,π/2)
  38. atan2 ( y , x ) = { arctan ( y x ) if x > 0 , arctan ( y x ) + π if x < 0 and y 0 , arctan ( y x ) - π if x < 0 and y < 0 , + π 2 if x = 0 and y > 0 , - π 2 if x = 0 and y < 0 , undefined if x = 0 and y = 0. \operatorname{atan2}(y,\,x)=\begin{cases}\arctan(\frac{y}{x})&\,\text{if }x>0,% \\ \arctan(\frac{y}{x})+\pi&\,\text{if }x<0\,\text{ and }y\geq 0,\\ \arctan(\frac{y}{x})-\pi&\,\text{if }x<0\,\text{ and }y<0,\\ +\frac{\pi}{2}&\,\text{if }x=0\,\text{ and }y>0,\\ -\frac{\pi}{2}&\,\text{if }x=0\,\text{ and }y<0,\\ \,\text{undefined}&\,\text{if }x=0\,\text{ and }y=0.\end{cases}
  39. ( π , π ] (−π,π]
  40. [ 0 , 2 π ) [0,2π)
  41. 2 π
  42. a t a n 2 ( 0 , 0 ) atan2(0,0)
  43. [ π , π ] [−π,π]
  44. a t a n 2 ( 0 , 0 ) atan2(0,0)
  45. π π
  46. a t a n 2 ( 0 , x ) atan2(−0,x)
  47. x x
  48. a t a n 2 ( + 0 , x ) atan2(+0,x)
  49. x x
  50. a t a n 2 atan2
  51. a t a n ( y ) atan(y)
  52. a t a n 2 ( y , 1 ) atan2(y,1)
  53. a t a n 2 atan2
  54. atan2 ( y , x ) = { 2 arctan ( y x 2 + y 2 + x ) if x > 0 or y 0 , π if x < 0 and y = 0 , undefined if x = 0 and y = 0. \operatorname{atan2}(y,\,x)=\begin{cases}2\arctan\biggl(\frac{y}{\sqrt{x^{2}+y% ^{2}}+x}\biggr)&\,\text{if }x>0\,\text{ or }y\neq 0,\\ \pi&\,\text{if }x<0\,\text{ and }y=0,\\ \,\text{undefined}&\,\text{if }x=0\,\text{ and }y=0.\end{cases}
  55. a t a n 2 ( 0 , 0 ) atan2(0,0)
  56. a t a n 2 ( 0 , 0 ) atan2(0,0)
  57. atan2 ( y , x ) = { 2 arctan ( x 2 + y 2 - x y ) if y 0 , 0 if x > 0 and y = 0 , π if x < 0 and y = 0 , undefined if x = 0 and y = 0. \operatorname{atan2}(y,\,x)=\begin{cases}2\arctan\biggl(\frac{\sqrt{x^{2}+y^{2% }}-x}{y}\biggr)&\,\text{if }y\neq 0,\\ 0&\,\text{if }x>0\,\text{ and }y=0,\\ \pi&\,\text{if }x<0\,\text{ and }y=0,\\ \,\text{undefined}&\,\text{if }x=0\,\text{ and }y=0.\end{cases}
  58. [ π , π ] [−π,π]
  59. π π
  60. π π
  61. π π
  62. a r g arg
  63. x + y i = ( x , y ) x+yi=(x,y)
  64. A t a n ( y , x ) = A r g ( x + y i ) Atan(y,x)=Arg(x+yi)
  65. a t a n 2 ( y , x ) atan2(y,x)
  66. ( x , y ) (x,y)
  67. a t a n 2 atan2
  68. x x
  69. 0
  70. ( 1 , 0 ) (1,0)
  71. ( 0 , 1 ) (0,1)
  72. π / 2 π/2
  73. ( 1 , 0 ) (−1,0)
  74. π π
  75. ( 0 , 1 ) (0,−1)
  76. 3 π / 2 3π/2
  77. ( 1 , 0 ) (1,0)
  78. 0 = ( n 2 π m o d 2 π ) 0=(n2πmod2π)
  79. a t a n 2 atan2
  80. a t a n 2 ( y , x ) atan2(y,x)
  81. a r c t a n ( y / x ) arctan(y/x)
  82. a t a n 2 ( y , x ) atan2(y,x)
  83. a r c t a n ( y / x ) arctan(y/x)
  84. x > 0 x>0
  85. a t a n 2 atan2
  86. a t a n 2 atan2
  87. a r c t a n ( y / x ) arctan(y/x)
  88. x > 0 x>0
  89. y 0 y≠0
  90. atan2 x ( y , x ) = arctan x ( y / x ) = - y x 2 + y 2 , \frac{\partial\operatorname{atan2}}{\partial x}(y,\,x)=\frac{\partial\arctan}{% \partial x}(y/x)=-\frac{y}{x^{2}+y^{2}},
  91. atan2 y ( y , x ) = arctan y ( y / x ) = x x 2 + y 2 . \frac{\partial\operatorname{atan2}}{\partial y}(y,\,x)=\frac{\partial\arctan}{% \partial y}(y/x)=\frac{x}{x^{2}+y^{2}}.
  92. a t a n 2 atan2
  93. θ ( x , y ) = a t a n 2 ( y , x ) θ(x,y)=atan2(y,x)
  94. d θ \displaystyle\mathrm{d}\theta
  95. a t a n 2 atan2
  96. y y
  97. a t a n 2 atan2

Atkinson_friction_factor.html

  1. k k
  2. k = 1 2 ρ f , k=\frac{1}{2}\rho f,
  3. ρ \rho
  4. f f
  5. k = 1 2 ρ λ 4 , k=\frac{1}{2}\rho\frac{\lambda}{4},
  6. λ \lambda
  7. k k

Atkinson_index.html

  1. ε \varepsilon
  2. ε \varepsilon
  3. ε \varepsilon
  4. ε \varepsilon
  5. ε \varepsilon
  6. ε = 0 \varepsilon=0
  7. A ε A_{\varepsilon}
  8. ε = \varepsilon=\infty
  9. A ε = 1 A_{\varepsilon}=1
  10. A ε A_{\varepsilon}
  11. ε \varepsilon
  12. ε \varepsilon
  13. A ε A_{\varepsilon}
  14. A ε A_{\varepsilon}
  15. A ε ( y 1 , , y N ) = { 1 - 1 μ ( 1 N i = 1 N y i 1 - ε ) 1 / ( 1 - ε ) for 0 ϵ 1 1 - 1 μ ( i = 1 N y i ) 1 / N for ε = 1 , A_{\varepsilon}(y_{1},\ldots,y_{N})=\begin{cases}1-\frac{1}{\mu}\left(\frac{1}% {N}\sum_{i=1}^{N}y_{i}^{1-\varepsilon}\right)^{1/(1-\varepsilon)}&\mbox{for}~{% }\ 0\leq\epsilon\neq 1\\ 1-\frac{1}{\mu}\left(\prod_{i=1}^{N}y_{i}\right)^{1/N}&\mbox{for}~{}\ % \varepsilon=1,\end{cases}
  16. y i y_{i}
  17. μ \mu
  18. A ε ( y 1 , , y N ) = A ε ( y σ ( 1 ) , , y σ ( N ) ) A_{\varepsilon}(y_{1},\ldots,y_{N})=A_{\varepsilon}(y_{\sigma(1)},\ldots,y_{% \sigma(N)})
  19. σ \sigma
  20. A ε ( y 1 , , y N ) = 0 A_{\varepsilon}(y_{1},\ldots,y_{N})=0
  21. y i = μ y_{i}=\mu
  22. i i
  23. Δ > 0 \Delta>0
  24. y i y_{i}
  25. y j y_{j}
  26. y i - Δ > y j + Δ y_{i}-\Delta>y_{j}+\Delta
  27. A ε ( { y 1 , , y N } , , { y 1 , , y N } ) = A ε ( y 1 , , y N ) A_{\varepsilon}(\{y_{1},\ldots,y_{N}\},\ldots,\{y_{1},\ldots,y_{N}\})=A_{% \varepsilon}(y_{1},\ldots,y_{N})
  28. A ε ( y 1 , , y N ) = A ε ( k y 1 , , k y N ) A_{\varepsilon}(y_{1},\ldots,y_{N})=A_{\varepsilon}(ky_{1},\ldots,ky_{N})
  29. k > 0 k>0
  30. A ε ( y g i : g = 1 , , G , i = 1 , , N g ) = g = 1 G w g A ε ( y g 1 , , y g , N g ) + A ε ( μ 1 , , μ G ) A_{\varepsilon}(y_{gi}:g=1,\ldots,G,i=1,\ldots,N_{g})=\sum_{g=1}^{G}w_{g}A_{% \varepsilon}(y_{g1},\ldots,y_{g,N_{g}})+A_{\varepsilon}(\mu_{1},\ldots,\mu_{G})
  31. g g
  32. i i
  33. μ g \mu_{g}
  34. g g
  35. w g w_{g}
  36. μ g , μ , N \mu_{g},\mu,N
  37. N g N_{g}

Atkinson_resistance.html

  1. R R
  2. Δ P = ρ a c t u a l ρ r e f R Q 2 \Delta P=\frac{\rho_{actual}}{\rho_{ref}}RQ^{2}
  3. Δ P \Delta P
  4. ρ a c t u a l \rho_{actual}
  5. ρ r e f \rho_{ref}
  6. R R
  7. Q Q
  8. Δ P = 1 2 ρ f L S A v 2 \Delta P=\frac{1}{2}\rho fL\frac{S}{A}v^{2}
  9. Δ P \Delta P
  10. ρ \rho
  11. f f
  12. L L
  13. S S
  14. A A
  15. v v
  16. v v
  17. Q / A Q/A
  18. 1 / 2 ρ 1/2\rho
  19. R R
  20. R = 1 2 ρ f L S A 3 1 2 ρ f L R h A 2 1 2 ρ 4 f L D h A 2 1 2 ρ λ L D h A 2 R=\frac{1}{2}\rho\frac{fLS}{A^{3}}\equiv\frac{1}{2}\rho\frac{fL}{R_{h}A^{2}}% \equiv\frac{1}{2}\rho\frac{4fL}{D_{h}A^{2}}\equiv\frac{1}{2}\rho\frac{\lambda L% }{D_{h}A^{2}}
  21. R h R_{h}
  22. D h D_{h}
  23. λ \lambda
  24. 1 / 2 1/2
  25. R = k L S A 3 R=\frac{kLS}{A^{3}}
  26. k k
  27. Δ P = ρ a c t u a l ρ r e f R Q 2 \Delta P=\frac{\rho_{actual}}{\rho_{ref}}RQ^{2}
  28. Δ P \Delta P
  29. ρ a c t u a l \rho_{actual}
  30. ρ r e f \rho_{ref}
  31. R R
  32. Q Q
  33. 1 gaul = 1 atkinson × 10 6 × ( m e t r e s f e e t ) 8 k i l o g r a m s p o u n d s × g 1 × 10 6 × 0.3048 8 0.4536 × 9.80665 16.747 atkinsons 1\mbox{ gaul}~{}=1\mbox{ atkinson}~{}\times\frac{10^{6}\times\left(\frac{% metres}{feet}\right)^{8}}{\frac{kilograms}{pounds}\times g}\equiv 1\times\frac% {10^{6}\times 0.3048^{8}}{0.4536\times 9.80665}\equiv 16.747\mbox{ atkinsons}~{}
  34. g g

Atmospheric_dispersion_modeling.html

  1. C = Q u f σ y 2 π g 1 + g 2 + g 3 σ z 2 π C=\frac{\;Q}{u}\cdot\frac{\;f}{\sigma_{y}\sqrt{2\pi}}\;\cdot\frac{\;g_{1}+g_{2% }+g_{3}}{\sigma_{z}\sqrt{2\pi}}
  2. f f
  3. exp [ - y 2 / ( 2 σ y 2 ) ] \exp\;[-\,y^{2}/\,(2\;\sigma_{y}^{2}\;)\;]
  4. g g
  5. g 1 + g 2 + g 3 \,g_{1}+g_{2}+g_{3}
  6. g 1 g_{1}
  7. exp [ - ( z - H ) 2 / ( 2 σ z 2 ) ] \;\exp\;[-\,(z-H)^{2}/\,(2\;\sigma_{z}^{2}\;)\;]
  8. g 2 g_{2}
  9. exp [ - ( z + H ) 2 / ( 2 σ z 2 ) ] \;\exp\;[-\,(z+H)^{2}/\,(2\;\sigma_{z}^{2}\;)\;]
  10. g 3 g_{3}
  11. m = 1 { exp [ - ( z - H - 2 m L ) 2 / ( 2 σ z 2 ) ] \sum_{m=1}^{\infty}\;\big\{\exp\;[-\,(z-H-2mL)^{2}/\,(2\;\sigma_{z}^{2}\;)\;]
  12. + exp [ - ( z + H + 2 m L ) 2 / ( 2 σ z 2 ) ] +\,\exp\;[-\,(z+H+2mL)^{2}/\,(2\;\sigma_{z}^{2}\;)\;]
  13. + exp [ - ( z + H - 2 m L ) 2 / ( 2 σ z 2 ) ] +\,\exp\;[-\,(z+H-2mL)^{2}/\,(2\;\sigma_{z}^{2}\;)\;]
  14. + exp [ - ( z - H + 2 m L ) 2 / ( 2 σ z 2 ) ] } +\,\exp\;[-\,(z-H+2mL)^{2}/\,(2\;\sigma_{z}^{2}\;)\;]\big\}
  15. C C
  16. Q Q
  17. u u
  18. H H
  19. σ z \sigma_{z}
  20. σ y \sigma_{y}
  21. L L
  22. exp \exp
  23. g 3 g_{3}
  24. σ z \sigma_{z}
  25. σ y \sigma_{y}
  26. σ y \sigma_{y}
  27. σ z \sigma_{z}

Atomic_formula.html

  1. t c x f ( t 1 , , t n ) t\equiv c\mid x\mid f(t_{1},...,t_{n})
  2. A , B , P ( t 1 , , t n ) A B A B A B x . A x . A A,B,...\equiv P(t_{1},...,t_{n})\mid A\wedge B\mid\top\mid A\vee B\mid\bot\mid A% \supset B\mid\forall x.\ A\mid\exists x.\ A
  3. P ( x ) P(x)
  4. Q ( y , f ( x ) ) Q(y,f(x))
  5. R ( z ) R(z)

Atomic_layer_deposition.html

  1. R a b s = S * F {R_{abs}}=S*F
  2. 2 {}_{2}
  3. 2 {}_{2}
  4. 2 {}_{2}
  5. 2 {}_{2}
  6. 4 {}_{4}
  7. 2 {}_{2}
  8. 2 {}_{2}
  9. 3 {}_{3}
  10. 2 {}_{2}
  11. 3 {}_{3}
  12. 4 {}_{4}
  13. 2 {}_{2}
  14. 4 {}_{4}
  15. 2 {}_{2}
  16. 5 {}_{5}
  17. 5 {}_{5}
  18. 2 {}_{2}
  19. 3 {}_{3}
  20. 5 {}_{5}
  21. 4 {}_{4}
  22. 3 {}_{3}
  23. 3 {}_{3}
  24. 2 {}_{2}
  25. 2 {}_{2}
  26. 2 {}_{2}
  27. 2 {}_{2}
  28. 3 {}_{3}
  29. 3 {}_{3}
  30. 2 {}_{2}
  31. 3 {}_{3}
  32. 3 {}_{3}
  33. 2 {}_{2}
  34. 2 {}_{2}
  35. 2 {}_{2}
  36. 3 {}_{3}
  37. 2 {}_{2}
  38. 3 {}_{3}
  39. 2 {}_{2}
  40. 3 {}_{3}
  41. 2 {}_{2}
  42. 3 {}_{3}
  43. 3 {}_{3}
  44. 2 {}_{2}
  45. 3 {}_{3}
  46. 4 {}_{4}
  47. 2 {}_{2}
  48. 3 {}_{3}
  49. 2 {}_{2}
  50. 2 {}_{2}
  51. 2 {}_{2}
  52. 2 {}_{2}
  53. 3 {}_{3}
  54. 2 {}_{2}
  55. 2 {}_{2}
  56. 4 {}_{4}
  57. 2 {}_{2}
  58. 2 {}_{2}
  59. 2 {}_{2}
  60. 3 {}^{3}
  61. 2 {}_{2}
  62. 2 {}_{2}
  63. 2 {}_{2}
  64. 4 {}_{4}
  65. 3 {}_{3}
  66. 2 {}_{2}
  67. 4 {}_{4}
  68. 2 {}_{2}
  69. 2 {}_{2}
  70. 2 {}_{2}
  71. 6 {}_{6}
  72. 5 {}_{5}
  73. 3 {}_{3}
  74. 5 {}_{5}
  75. 2 {}_{2}
  76. 6 {}_{6}
  77. 2 {}_{2}
  78. 3 {}_{3}
  79. 2 {}_{2}
  80. 6 {}_{6}
  81. 2 {}_{2}
  82. 6 {}_{6}
  83. 3 {}_{3}
  84. 2 {}_{2}
  85. 2 {}_{2}
  86. 3 {}_{3}
  87. 2 {}_{2}

Atomic_line_filter.html

  1. β = 𝒱 B d \beta=\mathcal{V}Bd

Atomic_mirror.html

  1. K ~{}\vec{K}~{}
  2. L ~{}L~{}
  3. L ~{}L~{}
  4. p = K L θ ~{}p=\sqrt{KL~{}}~{}\theta~{}

Atomic_packing_factor.html

  1. APF = N particle V particle V unitcell \mathrm{APF}=\frac{N_{\mathrm{particle}}V_{\mathrm{particle}}}{V_{\mathrm{% unitcell}}}
  2. a = 4 r 3 . a=\frac{4r}{\sqrt{3}}.
  3. APF = N atoms V atom V crystal = 2 ( 4 / 3 ) π r 3 ( 4 r / 3 ) 3 \mathrm{APF}=\frac{N_{\mathrm{atoms}}V_{\mathrm{atom}}}{V_{\mathrm{crystal}}}=% \frac{2(4/3)\pi r^{3}}{(4r/\sqrt{3})^{3}}
  4. = π 3 8 0.68. =\frac{\pi\sqrt{3}}{8}\approx 0.68.\,\!
  5. a = 2 r a=2r
  6. c = 2 3 ( 4 r ) . c=\sqrt{\frac{2}{3}}(4r).
  7. APF = N atoms V atom V crystal = 6 ( 4 / 3 ) π r 3 [ ( 3 3 ) / 2 ] ( a 2 ) ( c ) \mathrm{APF}=\frac{N_{\mathrm{atoms}}\cdot V_{\mathrm{atom}}}{V_{\mathrm{% crystal}}}=\frac{6\cdot(4/3)\pi r^{3}}{[(3\sqrt{3})/2](a^{2})(c)}
  8. = 6 ( 4 / 3 ) π r 3 [ ( 3 3 ) / 2 ] ( 2 r ) 2 ( 2 3 ) ( 4 r ) = 6 ( 4 / 3 ) π r 3 [ ( 3 3 ) / 2 ] ( 2 3 ) ( 16 r 3 ) =\frac{6(4/3)\pi r^{3}}{[(3\sqrt{3})/2](2r)^{2}(\sqrt{\frac{2}{3}})(4r)}=\frac% {6(4/3)\pi r^{3}}{[(3\sqrt{3})/2](\sqrt{\frac{2}{3}})(16r^{3})}
  9. = π 18 0.74. =\frac{\pi}{\sqrt{18}}\approx 0.74.\,\!

Attenuator_(electronics).html

  1. A = 10 - L o s s / 20 R a = R b = Z S 1 - A 1 + A R c = Z s 2 - R b 2 2 R b A=10^{-Loss/20}\qquad R_{a}=R_{b}=Z_{S}\frac{1-A}{1+A}\qquad R_{c}=\frac{Z_{s}% ^{2}-R_{b}^{2}}{2R_{b}}\qquad\,
  2. A = 10 - L o s s / 20 R x = R y = Z S 1 + A 1 - A R z = 2 R x ( R x Z S ) 2 - 1 ] A=10^{-Loss/20}\qquad R_{x}=R_{y}=Z_{S}\frac{1+A}{1-A}\qquad R_{z}=\frac{2R_{x% }}{\left(\frac{R_{x}}{Z_{S}}\right)^{2}-1}]\qquad\,
  3. R q = Z m ρ - 1 R p = Z m ρ - 1 R_{q}=\frac{Z_{m}}{\sqrt{\rho-1}}\qquad R_{p}=Z_{m}\sqrt{\rho-1}
  4. Loss = 20 log 10 ( ρ - 1 + ρ ) where ρ = Z 1 Z 2 Z m = Z 1 Z 2 \,\text{Loss}=20\log_{10}\left(\sqrt{\rho-1}+\sqrt{\rho}\right)\quad\,\text{% where}\quad\rho=\frac{Z_{1}}{Z_{2}}\quad Z_{m}=\sqrt{Z_{1}Z_{2}}\,\text{ }\,
  5. R z = R a R b + R a R c + R b R c R c R x = R a R b + R a R c + R b R c R b R y = R a R b + R a R c + R b R c R a . R_{z}=\frac{R_{a}R_{b}+R_{a}R_{c}+R_{b}R_{c}}{R_{c}}\qquad R_{x}=\frac{R_{a}R_% {b}+R_{a}R_{c}+R_{b}R_{c}}{R_{b}}\qquad R_{y}=\frac{R_{a}R_{b}+R_{a}R_{c}+R_{b% }R_{c}}{R_{a}}.\qquad\,
  6. R c = R x R y R x + R y + R z R a = R z R x R x + R y + R z R b = R z R y R x + R y + R z R_{c}=\frac{R_{x}R_{y}}{R_{x}+R_{y}+R_{z}}\qquad R_{a}=\frac{R_{z}R_{x}}{R_{x}% +R_{y}+R_{z}}\qquad R_{b}=\frac{R_{z}R_{y}}{R_{x}+R_{y}+R_{z}}\qquad\,
  7. V 1 = Z 11 I 1 + Z 12 I 2 V 2 = Z 21 I 1 + Z 22 I 2 with Z 12 = Z 21 V_{1}=Z_{11}I_{1}+Z_{12}I_{2}\qquad V_{2}=Z_{21}I_{1}+Z_{22}I_{2}\qquad\,\text% {with}\qquad Z_{12}=Z_{21}\,
  8. Z 21 = R c Z 11 = R c + R a Z 22 = R c + R b Z_{21}=R_{c}\qquad Z_{11}=R_{c}+R_{a}\qquad Z_{22}=R_{c}+R_{b}\,
  9. R c = Z 21 R a = Z 11 - Z 21 R b = Z 22 - Z 21 R_{c}=Z_{21}\qquad R_{a}=Z_{11}-Z_{21}\qquad R_{b}=Z_{22}-Z_{21}\,
  10. R z = Z 11 Z 22 - Z 21 2 Z 21 R x = Z 11 Z 22 - Z 21 2 Z 22 - Z 21 R y = Z 11 Z 22 - Z 21 2 Z 11 - Z 21 R_{z}=\frac{Z_{11}Z_{22}-Z_{21}^{2}}{Z_{21}}\qquad R_{x}=\frac{Z_{11}Z_{22}-Z_% {21}^{2}}{Z_{22}-Z_{21}}\qquad R_{y}=\frac{Z_{11}Z_{22}-Z_{21}^{2}}{Z_{11}-Z_{% 21}}\qquad
  11. I 1 = Y 11 V 1 + Y 12 V 2 I 2 = Y 21 V 1 + Y 22 V 2 with Y 12 = Y 21 I_{1}=Y_{11}V_{1}+Y_{12}V_{2}\qquad I_{2}=Y_{21}V_{1}+Y_{22}V_{2}\qquad\,\text% {with}\qquad Y_{12}=Y_{21}\,
  12. Y 21 = 1 R z Y 11 = 1 R x + 1 R z Y 22 = 1 R y + 1 R z Y_{21}=\frac{1}{R_{z}}\qquad Y_{11}=\frac{1}{R_{x}}+\frac{1}{R_{z}}\qquad Y_{2% 2}=\frac{1}{R_{y}}+\frac{1}{R_{z}}\,
  13. R z = 1 Y 21 R x = 1 Y 11 - Y 21 R y = 1 Y 22 - Y 21 R_{z}=\frac{1}{Y_{21}}\qquad R_{x}=\frac{1}{Y_{11}-Y_{21}}\qquad R_{y}=\frac{1% }{Y_{22}-Y_{21}}\,
  14. L o s s m i n = 20 l o g 10 ( ρ - 1 + ρ ) where ρ = max [ Z S , Z L o a d ] min [ Z S , Z L o a d ] Loss_{min}=20\ log_{10}\left(\sqrt{\rho-1}+\sqrt{\rho}\quad\right)\,\quad\,% \text{where}\quad\rho=\frac{\max[Z_{S},Z_{Load}]}{\min[Z_{S},Z_{Load}]}\,
  15. A = 10 - L o s s / 20 Z 11 = Z S 1 + A 2 1 - A 2 Z 22 = Z L o a d 1 + A 2 1 - A 2 Z 21 = 2 A Z S Z L o a d 1 - A 2 A=10^{-Loss/20}\qquad Z_{11}=Z_{S}\frac{1+A^{2}}{1-A^{2}}\qquad Z_{22}=Z_{Load% }\frac{1+A^{2}}{1-A^{2}}\qquad Z_{21}=2\frac{A\sqrt{Z_{S}Z_{Load}}}{1-A^{2}}\,

Atwater_system.html

  1. Metabolisable Energy = ( Gross Energy in Food ) - ( Energy lost in Faeces, Urine, Secretions and Gases ) . {\,\text{Metabolisable Energy}}=\left(\,\text{Gross Energy in Food}\right)-% \left(\,\text{Energy lost in Faeces, Urine, Secretions and Gases}\right).
  2. G E = G E p + G E f + G E c h o {GE}={{GE}_{p}+{GE}_{f}+{GE}_{cho}}\,
  3. G E F = G E p F + G E f F + G E c h o F {GE}^{F}={{GE}_{p}^{F}+{GE}_{f}^{F}+{GE}_{cho}^{F}}\,
  4. Digestible energy = G E p ( D p ) + G E f ( D f ) + G E c h o ( D c h o ) \,\text{Digestible energy}={{GE}_{p}(D_{p})}+{{GE}_{f}(D_{f})}+{{GE}_{cho}(D_{% cho})}\,
  5. intake - faecal excretion intake \frac{\,\text{intake}-\,\text{faecal excretion}}{\,\text{intake}}
  6. M E = ( G E p - 7.9 6.25 ) D p + G E f D f + G E c h o D c h o {ME}=\left({GE}_{p}-\frac{7.9}{6.25}\right)D_{p}+{GE}_{f}D_{f}+{GE}_{cho}D_{% cho}\,
  7. intake - faecal excretion intake , \frac{\,\text{intake}-\,\text{faecal excretion}}{\,\text{intake}},

Augmented_Dickey–Fuller_test.html

  1. Δ y t = α + β t + γ y t - 1 + δ 1 Δ y t - 1 + + δ p - 1 Δ y t - p + 1 + ε t , \Delta y_{t}=\alpha+\beta t+\gamma y_{t-1}+\delta_{1}\Delta y_{t-1}+\cdots+% \delta_{p-1}\Delta y_{t-p+1}+\varepsilon_{t},
  2. α \alpha
  3. β \beta
  4. p p
  5. α = 0 \alpha=0
  6. β = 0 \beta=0
  7. β = 0 \beta=0
  8. γ = 0 \gamma=0
  9. γ < 0. \gamma<0.
  10. D F τ = γ ^ S E ( γ ^ ) DF_{\tau}=\frac{\hat{\gamma}}{SE(\hat{\gamma})}
  11. γ = 0 \gamma=0
  12. y t - 1 y_{t-1}
  13. y t y_{t}
  14. Δ y t - k \Delta y_{t-k}
  15. γ = 0 \gamma=0
  16. D F τ DF_{\tau}

Augmented_major_seventh_chord.html

  1. 7 M {}_{M}^{7}

Augmented_matrix.html

  1. A = [ 1 3 2 2 0 1 5 2 2 ] , B = [ 4 3 1 ] , A=\begin{bmatrix}1&3&2\\ 2&0&1\\ 5&2&2\end{bmatrix},\quad B=\begin{bmatrix}4\\ 3\\ 1\end{bmatrix},
  2. ( A | B ) = [ 1 3 2 4 2 0 1 3 5 2 2 1 ] . (A|B)=\left[\begin{array}[]{ccc|c}1&3&2&4\\ 2&0&1&3\\ 5&2&2&1\end{array}\right].
  3. C = [ 1 3 - 5 0 ] . C=\begin{bmatrix}1&3\\ -5&0\end{bmatrix}.
  4. ( C | I ) = [ 1 3 1 0 - 5 0 0 1 ] (C|I)=\left[\begin{array}[]{cc|cc}1&3&1&0\\ -5&0&0&1\end{array}\right]
  5. ( I | C - 1 ) = [ 1 0 0 - 1 5 0 1 1 3 1 15 ] (I|C^{-1})=\left[\begin{array}[]{cc|cc}1&0&0&-\frac{1}{5}\\ 0&1&\frac{1}{3}&\frac{1}{15}\end{array}\right]
  6. A = [ 1 1 2 1 1 1 2 2 2 ] , A=\begin{bmatrix}1&1&2\\ 1&1&1\\ 2&2&2\\ \end{bmatrix},
  7. ( A | B ) = [ 1 1 2 3 1 1 1 1 2 2 2 2 ] . (A|B)=\left[\begin{array}[]{ccc|c}1&1&2&3\\ 1&1&1&1\\ 2&2&2&2\end{array}\right].
  8. A = [ 1 1 2 1 1 1 2 2 2 ] , A=\begin{bmatrix}1&1&2\\ 1&1&1\\ 2&2&2\\ \end{bmatrix},
  9. ( A | B ) = [ 1 1 2 3 1 1 1 1 2 2 2 5 ] . (A|B)=\left[\begin{array}[]{ccc|c}1&1&2&3\\ 1&1&1&1\\ 2&2&2&5\end{array}\right].
  10. x + 2 y + 3 z = 0 3 x + 4 y + 7 z = 2 6 x + 5 y + 9 z = 11 \begin{aligned}\displaystyle x+2y+3z&\displaystyle=0\\ \displaystyle 3x+4y+7z&\displaystyle=2\\ \displaystyle 6x+5y+9z&\displaystyle=11\end{aligned}
  11. A = [ 1 2 3 3 4 7 6 5 9 ] , B = [ 0 2 11 ] , A=\begin{bmatrix}1&2&3\\ 3&4&7\\ 6&5&9\end{bmatrix},\quad B=\begin{bmatrix}0\\ 2\\ 11\end{bmatrix},
  12. ( A | B ) = [ 1 2 3 0 3 4 7 2 6 5 9 11 ] (A|B)=\left[\begin{array}[]{ccc|c}1&2&3&0\\ 3&4&7&2\\ 6&5&9&11\end{array}\right]
  13. [ 1 0 0 4 0 1 0 1 0 0 1 - 2 ] , \left[\begin{array}[]{ccc|c}1&0&0&4\\ 0&1&0&1\\ 0&0&1&-2\\ \end{array}\right],

Autocorrelation_technique.html

  1. S ( ω ) S(\omega)
  2. R ( 1 ) = 1 2 π - π π S ( ω ) e i ω 1 d ω . R(1)=\frac{1}{2\pi}\int_{-\pi}^{\pi}S(\omega)e^{i\,\omega\,1}d\omega.
  3. S ( ω ) = def δ ( ω - ω 0 ) S(\omega)\ \stackrel{\mathrm{def}}{=}\ \delta(\omega-\omega_{0})
  4. R ( 1 ) = 1 2 π - π π δ ( ω - ω 0 ) e i ω d ω R(1)=\frac{1}{2\pi}\int_{-\pi}^{\pi}\delta(\omega-\omega_{0})e^{i\,\omega}d\omega
  5. R ( 1 ) = 1 2 π e i ω 0 R(1)=\frac{1}{2\pi}e^{i\,\omega_{0}}
  6. R ( 1 ) R(1)
  7. ω = R N ( 1 ) = tan - 1 i m { R N ( 1 ) } r e { R N ( 1 ) } . \omega=\angle R_{N}(1)=\tan^{-1}\frac{im\{R_{N}(1)\}}{re\{R_{N}(1)\}}.
  8. v a r { ω } = 2 N ( 1 - | R N ( 1 ) | R N ( 0 ) ) . var\{\omega\}=\frac{2}{N}\left(1-\frac{|R_{N}(1)|}{R_{N}(0)}\right).

Automatic_group.html

  1. A A^{\ast}
  2. a A { 1 } a\in A\cup\{1\}
  3. w 1 a = w 2 w_{1}a=w_{2}

Autoxidation.html

  1. ROOH + RH energy RO + OH + RH RO + H 2 O + R \mathrm{ROOH+RH\ \xrightarrow{energy}\ RO{\cdot}+{\cdot}OH+RH\ \longrightarrow% {}\ RO{\cdot}+H_{2}O+R{\cdot}\quad}
  2. RO + RH H - abstraction R + ROH \mathrm{RO{\cdot}+RH\ \xrightarrow{H-abstraction}\ R{\cdot}+ROH\quad}
  3. R + O 2 fast ROO \mathrm{R{{}^{\cdot}}+O_{2}\ \xrightarrow{fast}\ ROO{{}^{\cdot}}}
  4. ROO + RH H - abstraction ROOH + R \mathrm{ROO{{}^{\cdot}}+RH\ \xrightarrow{H-abstraction}\ ROOH+{{}^{\cdot}}R}
  5. 2 R O O 2 RO + O 2 ROH + QO + O 2 \mathrm{2ROO{{}^{\cdot}}\ \xrightarrow{}\ 2RO{{}^{\cdot}}+O_{2}\ % \longrightarrow{}\ ROH+QO+O_{2}}
  6. ROOH + ROO ROOH + Q OOH ROOH + QO + OH \mathrm{ROOH+ROO{{}^{\cdot}}\ \longrightarrow{}\ ROOH+Q{{}^{\cdot}}OOH\ % \longrightarrow{}\ ROOH+QO+^{\cdot}OH}
  7. ROOH + QO + OH + RH ROOH + QO + H 2 O + R RO + ROH + QO + H 2 O \mathrm{ROOH+QO+^{\cdot}OH+RH\ \longrightarrow{}\ ROOH+QO+H_{2}O+R^{\cdot}\ % \longrightarrow{}\ RO^{\cdot}+ROH+QO+H_{2}O}
  8. r init = k init [ ROOH ] = k term [ ROO ] 2 \mathrm{r_{init}=k_{init}\cdot[ROOH]=k_{term}\cdot[ROO^{\cdot}]^{2}}
  9. r prop = k prop [ RH ] [ ROO ] = k prop [ RH ] k init k term [ ROOH ] \mathrm{r_{prop}=k_{prop}\cdot[RH]\cdot[ROO^{\cdot}]=k_{prop}\cdot[RH]\cdot% \sqrt[\,]{\frac{k_{init}}{k_{term}}}\cdot\sqrt[\,]{[ROOH]}}

Auxiliary_field.html

  1. A A
  2. a u x = 1 2 ( A , A ) + ( f ( φ ) , A ) \mathcal{L}_{aux}=\frac{1}{2}(A,A)+(f(\varphi),A)
  3. A A
  4. A ( φ ) = - f ( φ ) A(\varphi)=-f(\varphi)
  5. a u x = - 1 2 ( f ( φ ) , f ( φ ) ) \mathcal{L}_{aux}=-\frac{1}{2}(f(\varphi),f(\varphi))
  6. 0 \mathcal{L}_{0}
  7. φ \varphi
  8. = 0 ( φ ) + a u x = 0 ( φ ) - 1 2 ( f ( φ ) , f ( φ ) ) \mathcal{L}=\mathcal{L}_{0}(\varphi)+\mathcal{L}_{aux}=\mathcal{L}_{0}(\varphi% )-\frac{1}{2}(f(\varphi),f(\varphi))
  9. φ \varphi
  10. 0 \mathcal{L}_{0}
  11. 𝒮 = d n x . \mathcal{S}=\int{\mathcal{L}\,d^{n}x}.
  12. - d A e - 1 2 A 2 + A f = 2 π e - f 2 2 \int_{-\infty}^{\infty}\!dA\,e^{-\frac{1}{2}A^{2}+Af}=\sqrt{2\pi}e^{-\frac{f^{% 2}}{2}}

Average_fixed_cost.html

  1. A F C = F C Q . AFC=\frac{FC}{Q}.
  2. A T C = A V C + A F C ATC=AVC+AFC

Average_variable_cost.html

  1. AVC = VC Q \,\text{AVC}=\frac{\,\text{VC}}{\,\text{Q}}
  2. AVC + AFC = ATC . \,\text{AVC}+\,\text{AFC}=\,\text{ATC}.

Axial_multipole_moments.html

  1. 1 R \frac{1}{R}
  2. λ ( z ) \lambda(z)
  3. z = a z=a
  4. Φ ( 𝐫 ) = q 4 π ε 1 R = q 4 π ε 1 r 2 + a 2 - 2 a r cos θ . \Phi(\mathbf{r})=\frac{q}{4\pi\varepsilon}\frac{1}{R}=\frac{q}{4\pi\varepsilon% }\frac{1}{\sqrt{r^{2}+a^{2}-2ar\cos\theta}}.
  5. 1 r \frac{1}{r}
  6. ( a / r ) < 1 (a/r)<1
  7. Φ ( 𝐫 ) = q 4 π ε r k = 0 ( a r ) k P k ( cos θ ) 1 4 π ε k = 0 M k ( 1 r k + 1 ) P k ( cos θ ) \Phi(\mathbf{r})=\frac{q}{4\pi\varepsilon r}\sum_{k=0}^{\infty}\left(\frac{a}{% r}\right)^{k}P_{k}(\cos\theta)\equiv\frac{1}{4\pi\varepsilon}\sum_{k=0}^{% \infty}M_{k}\left(\frac{1}{r^{k+1}}\right)P_{k}(\cos\theta)
  8. M k q a k M_{k}\equiv qa^{k}
  9. M 0 = q M_{0}=q
  10. M 1 = q a M_{1}=qa
  11. M 2 q a 2 M_{2}\equiv qa^{2}
  12. 1 a \frac{1}{a}
  13. ( r / a ) < 1 (r/a)<1
  14. Φ ( 𝐫 ) = q 4 π ε a k = 0 ( r a ) k P k ( cos θ ) 1 4 π ε k = 0 I k r k P k ( cos θ ) \Phi(\mathbf{r})=\frac{q}{4\pi\varepsilon a}\sum_{k=0}^{\infty}\left(\frac{r}{% a}\right)^{k}P_{k}(\cos\theta)\equiv\frac{1}{4\pi\varepsilon}\sum_{k=0}^{% \infty}I_{k}r^{k}P_{k}(\cos\theta)
  15. I k q a k + 1 I_{k}\equiv\frac{q}{a^{k+1}}
  16. λ ( ζ ) d ζ \lambda(\zeta)\ d\zeta
  17. λ ( ζ ) \lambda(\zeta)
  18. z = ζ z=\zeta
  19. | ζ | \left|\zeta\right|
  20. λ ( ζ ) \lambda(\zeta)
  21. ζ max \zeta\text{max}
  22. Φ ( 𝐫 ) = 1 4 π ε k = 0 M k ( 1 r k + 1 ) P k ( cos θ ) \Phi(\mathbf{r})=\frac{1}{4\pi\varepsilon}\sum_{k=0}^{\infty}M_{k}\left(\frac{% 1}{r^{k+1}}\right)P_{k}(\cos\theta)
  23. M k M_{k}
  24. M k d ζ λ ( ζ ) ζ k M_{k}\equiv\int d\zeta\ \lambda(\zeta)\zeta^{k}
  25. M 0 d ζ λ ( ζ ) M_{0}\equiv\int d\zeta\ \lambda(\zeta)
  26. M 1 d ζ λ ( ζ ) ζ M_{1}\equiv\int d\zeta\ \lambda(\zeta)\ \zeta
  27. M 2 d ζ λ ( ζ ) ζ 2 M_{2}\equiv\int d\zeta\ \lambda(\zeta)\ \zeta^{2}
  28. r r
  29. 1 r \frac{1}{r}
  30. 1 r 2 \frac{1}{r^{2}}
  31. 1 r 3 \frac{1}{r^{3}}
  32. ζ max r 1 \frac{\zeta\text{max}}{r}\ll 1
  33. M k M_{k}^{\prime}
  34. M k d ζ λ ( ζ ) ( ζ + b ) k M_{k}^{\prime}\equiv\int d\zeta\ \lambda(\zeta)\ \left(\zeta+b\right)^{k}
  35. ( ζ + b ) l = ζ l + l b ζ l - 1 + + l ζ b l - 1 + b l \left(\zeta+b\right)^{l}=\zeta^{l}+lb\zeta^{l-1}+\ldots+l\zeta b^{l-1}+b^{l}
  36. M k = M k + l b M k - 1 + + l b l - 1 M 1 + b l M 0 M_{k}^{\prime}=M_{k}+lbM_{k-1}+\ldots+lb^{l-1}M_{1}+b^{l}M_{0}
  37. M k - 1 , M k - 2 , , M 1 , M 0 M_{k-1},M_{k-2},\ldots,M_{1},M_{0}
  38. M k = M k M_{k}^{\prime}=M_{k}
  39. | ζ | \left|\zeta\right|
  40. λ ( ζ ) \lambda(\zeta)
  41. ζ m i n \zeta_{min}
  42. Φ ( 𝐫 ) = 1 4 π ε k = 0 I k r k P k ( cos θ ) \Phi(\mathbf{r})=\frac{1}{4\pi\varepsilon}\sum_{k=0}^{\infty}I_{k}r^{k}P_{k}(% \cos\theta)
  43. I k I_{k}
  44. I k d ζ λ ( ζ ) ζ k + 1 I_{k}\equiv\int d\zeta\ \frac{\lambda(\zeta)}{\zeta^{k+1}}
  45. \neq
  46. M 0 d ζ λ ( ζ ) ζ M_{0}\equiv\int d\zeta\ \frac{\lambda(\zeta)}{\zeta}
  47. M 1 d ζ λ ( ζ ) ζ 2 M_{1}\equiv\int d\zeta\ \frac{\lambda(\zeta)}{\zeta^{2}}
  48. r r
  49. r r
  50. r 2 r^{2}
  51. r ζ m i n 1 \frac{r}{\zeta_{min}}\ll 1

Axiomatic_design.html

  1. [ F R 1 F R 2 ] = [ A 11 A 12 A 21 A 22 ] [ D P 1 D P 2 ] \begin{bmatrix}FR_{1}\\ FR_{2}\end{bmatrix}=\begin{bmatrix}A_{11}&A_{12}\\ A_{21}&A_{22}\end{bmatrix}\begin{bmatrix}DP_{1}\\ DP_{2}\end{bmatrix}

Axiomatic_quantum_field_theory.html

  1. t t
  2. τ = - - 1 t \tau=-\sqrt{-1}t
  3. - 1 \sqrt{-1}

Babylonian_mathematics.html

  1. 2 \sqrt{2}
  2. a b = ( a + b ) 2 - a 2 - b 2 2 ab=\frac{(a+b)^{2}-a^{2}-b^{2}}{2}
  3. a b = ( a + b ) 2 - ( a - b ) 2 4 ab=\frac{(a+b)^{2}-(a-b)^{2}}{4}
  4. a b = a × 1 b \frac{a}{b}=a\times\frac{1}{b}
  5. 1 13 = 7 91 = 7 × 1 91 7 × 1 90 = 7 × 40 3600 = 280 3600 = 4 60 + 40 3600 . \frac{1}{13}=\frac{7}{91}=7\times\frac{1}{91}\approx 7\times\frac{1}{90}=7% \times\frac{40}{3600}=\frac{280}{3600}=\frac{4}{60}+\frac{40}{3600}.
  6. x 2 + b x = c \ x^{2}+bx=c
  7. x = - b 2 + ( b 2 ) 2 + c x=-\frac{b}{2}+\sqrt{\left(\frac{b}{2}\right)^{2}+c}
  8. a x 3 + b x 2 = c . \ ax^{3}+bx^{2}=c.
  9. ( a x b ) 3 + ( a x b ) 2 = c a 2 b 3 . \left(\frac{ax}{b}\right)^{3}+\left(\frac{ax}{b}\right)^{2}=\frac{ca^{2}}{b^{3% }}.
  10. y 3 + y 2 = c a 2 b 3 y^{3}+y^{2}=\frac{ca^{2}}{b^{3}}
  11. ( a , b , c ) \scriptstyle(a,b,c)
  12. a 2 + b 2 = c 2 \scriptstyle a^{2}+b^{2}=c^{2}
  13. π \pi
  14. π \pi
  15. 25 / 8 = 3.125 25/8=3.125
  16. π \pi
  17. π \pi
  18. π \pi

Bach's_algorithm.html

  1. N 2 < x N \frac{N}{2}<x\leq N
  2. p a N p^{a}\leq N
  3. M = N p a M=\frac{N}{p^{a}}
  4. x = p a y x=p^{a}y
  5. p a p^{a}

Background_field_method.html

  1. ϕ ( x ) = B ( x ) + η ( x ) \phi(x)=B(x)+\eta(x)
  2. Z [ J ] = 𝒟 ϕ e i d d x ( [ ϕ ( x ) ] + J ( x ) ϕ ( x ) ) Z[J]=\int\mathcal{D}\phi e^{i\int d^{d}x(\mathcal{L}[\phi(x)]+J(x)\phi(x))}
  3. ( x ) \mathcal{L}(x)
  4. ϕ ( x ) = B ( x ) + η ( x ) . \phi(x)=B(x)+\eta(x)\,.
  5. δ S δ ϕ | ϕ = B = 0 \left.\frac{\delta S}{\delta\phi}\right|_{\phi=B}=0
  6. d d x ( [ ϕ ( x ) ] + J ( x ) ϕ ( x ) ) \displaystyle\int d^{d}x(\mathcal{L}[\phi(x)]+J(x)\phi(x))
  7. Z [ J ] = e i d d x ( [ B ( x ) ] + J ( x ) B ( x ) ) 𝒟 η e i 2 d d x d d y δ 2 δ ϕ ( x ) δ ϕ ( y ) [ B ] η ( x ) η ( y ) + . Z[J]=e^{i\int d^{d}x(\mathcal{L}[B(x)]+J(x)B(x))}\int\mathcal{D}\eta e^{\frac{% i}{2}\int d^{d}xd^{d}y\frac{\delta^{2}\mathcal{L}}{\delta\phi(x)\delta\phi(y)}% [B]\eta(x)\eta(y)+\cdots}.
  8. Z [ J ] = C e i d d x ( [ B ( x ) ] + J ( x ) B ( x ) ) ( det δ 2 δ ϕ ( x ) δ ϕ ( y ) [ B ] ) - 1 / 2 + Z[J]=Ce^{i\int d^{d}x(\mathcal{L}[B(x)]+J(x)B(x))}\left(\det\frac{\delta^{2}% \mathcal{L}}{\delta\phi(x)\delta\phi(y)}[B]\right)^{-1/2}+\cdots

Background_radiation_equivalent_time.html

  1. B R E T = S V d o s e S V b a c k g r o u n d 365 BRET=\frac{{SV}_{dose}}{{SV}_{background}}\cdot 365\,

Backjumping.html

  1. x 1 , , x n x_{1},\ldots,x_{n}
  2. x 1 = a 1 , , x k = a k x_{1}=a_{1},\ldots,x_{k}=a_{k}
  3. x k + 1 x_{k+1}
  4. x 1 = a 1 , , x k = a k x_{1}=a_{1},\ldots,x_{k}=a_{k}
  5. x k x_{k}
  6. x k + 1 x_{k+1}
  7. j < k j<k
  8. x 1 , , x j = a 1 , , a j x_{1},\ldots,x_{j}=a_{1},\ldots,a_{j}
  9. x k + 1 x_{k+1}
  10. x j x_{j}
  11. x k x_{k}
  12. x 1 x 2 x 3 x 4 x_{1}x_{2}x_{3}x_{4}
  13. x 5 x_{5}
  14. x 4 x_{4}
  15. x 1 x 2 x 5 = 211 x_{1}x_{2}x_{5}=211
  16. x 1 x 5 = 22 x_{1}x_{5}=22
  17. x 1 x 2 x 5 = 213 x_{1}x_{2}x_{5}=213
  18. x 1 x 2 x_{1}x_{2}
  19. x 2 x_{2}
  20. x k + 1 x_{k+1}
  21. x j x_{j}
  22. x 1 , , x j x_{1},\ldots,x_{j}
  23. x k + 1 x_{k+1}
  24. j j
  25. x k + 1 x_{k+1}
  26. x 1 , , x k = a 1 , , a k x_{1},\ldots,x_{k}=a_{1},\ldots,a_{k}
  27. x k + 1 x_{k+1}
  28. x k + 1 x_{k+1}
  29. a k + 1 a_{k+1}
  30. x 1 , , x k = a 1 , , a k x_{1},\ldots,x_{k}=a_{1},\ldots,a_{k}
  31. x k + 1 = a k + 1 x_{k+1}=a_{k+1}
  32. a k + 1 a_{k+1}
  33. x k + 1 x_{k+1}
  34. x 1 = a 1 x_{1}=a_{1}
  35. x k - 1 = a k - 1 x_{k-1}=a_{k-1}
  36. x k = a k x_{k}=a_{k}
  37. x k + 1 = a k + 1 x_{k+1}=a_{k+1}
  38. x 1 = a 1 x_{1}=a_{1}
  39. x k - 1 = a k - 1 x_{k-1}=a_{k-1}
  40. x k + 1 = a k + 1 x_{k+1}=a_{k+1}
  41. x 1 = a 1 x_{1}=a_{1}
  42. x k + 1 = a k + 1 x_{k+1}=a_{k+1}
  43. x k + 1 = a k + 1 x_{k+1}=a_{k+1}
  44. x k + 1 x_{k+1}
  45. x k + 1 = a k + 1 x_{k+1}=a_{k+1}
  46. x k + 1 x_{k+1}
  47. x k + 1 = a k + 1 x_{k+1}=a_{k+1}
  48. x 1 , , x k x_{1},\ldots,x_{k}
  49. x k + 1 x_{k+1}
  50. x k + 1 x_{k+1}
  51. x k x_{k}
  52. x k - 2 x_{k-2}
  53. x k - 1 x_{k-1}
  54. x k x_{k}
  55. x k - 2 x_{k-2}
  56. x k - 2 x_{k-2}
  57. x k + 1 x_{k+1}
  58. x i x_{i}
  59. x 1 , , x i x_{1},\ldots,x_{i}
  60. x k + 1 , x k + 2 , x_{k+1},x_{k+2},...
  61. x k + 1 x_{k+1}
  62. x k + 1 x_{k+1}
  63. x k + 1 x_{k+1}
  64. x l x_{l}
  65. x l x_{l}
  66. x m x_{m}
  67. x l x_{l}
  68. x m x_{m}
  69. x l x_{l}
  70. x m x_{m}
  71. x l x_{l}
  72. x m x_{m}
  73. x l x_{l}
  74. x m x_{m}
  75. x m x_{m}
  76. x l x_{l}
  77. x 1 , , x k x_{1},\ldots,x_{k}
  78. x k + 1 , x k + 2 , x_{k+1},x_{k+2},...
  79. x i x_{i}
  80. i > k i>k
  81. k k
  82. x i x_{i}
  83. x k + 1 x_{k+1}
  84. x k + 1 x_{k+1}
  85. C C
  86. D D
  87. C C
  88. D D
  89. D D
  90. C C
  91. i i
  92. x 1 , , x i x_{1},\ldots,x_{i}
  93. k + 1 k+1
  94. x k + 1 x_{k+1}
  95. k k

Backmarking.html

  1. x 1 , , x n x_{1},\ldots,x_{n}
  2. x i x_{i}
  3. x i x_{i}
  4. a a
  5. x i x_{i}
  6. a a
  7. j < i j<i
  8. x 1 , , x j , x i x_{1},\ldots,x_{j},x_{i}
  9. x i x_{i}
  10. x i x_{i}
  11. k < i k<i
  12. x i x_{i}
  13. a a
  14. x 1 , x i x_{1},x_{i}
  15. x 1 , x 2 , x i x_{1},x_{2},x_{i}
  16. x 1 , x 2 , x 3 , x i x_{1},x_{2},x_{3},x_{i}
  17. x i x_{i}
  18. x j x_{j}
  19. x j x_{j}
  20. x i x_{i}
  21. i > j i>j
  22. k k
  23. m i n ( k , j ) min(k,j)
  24. x i = a x_{i}=a
  25. x i x_{i}
  26. x i = a x_{i}=a
  27. k k
  28. x i x_{i}
  29. j j
  30. x 1 , , x j , x i x_{1},\ldots,x_{j},x_{i}
  31. x i x_{i}
  32. a a
  33. j < k j<k
  34. x 1 , , x j , x i x_{1},\ldots,x_{j},x_{i}
  35. j k j\geq k
  36. x 1 , , x k , x i x_{1},\ldots,x_{k},x_{i}
  37. x 1 , , x i x_{1},\ldots,x_{i}

Bagnold_number.html

  1. Ba = ρ d 2 λ 1 / 2 γ μ \mathrm{Ba}=\frac{\rho d^{2}\lambda^{1/2}\gamma}{\mu}
  2. ρ \rho
  3. d d
  4. γ ˙ \dot{\gamma}
  5. μ \mu
  6. λ \lambda
  7. λ = 1 ( ϕ 0 / ϕ ) 1 3 - 1 \lambda=\frac{1}{\left(\phi_{0}/\phi\right)^{\frac{1}{3}}-1}
  8. ϕ \phi
  9. ϕ 0 \phi_{0}

Baire_function.html

  1. x 2 sin ( 1 / x ) x^{2}\sin(1/x)
  2. g n ( x ) = max ( 0 , 1 - n d ( x , C ) ) g_{n}(x)=\max(0,{1-nd(x,C)})
  3. d ( x , C ) d(x,C)
  4. χ \chi_{\mathbb{Q}}

Bairstow's_method.html

  1. x 2 + u x + v x^{2}+ux+v
  2. P ( x ) = i = 0 n a i x i P(x)=\sum_{i=0}^{n}a_{i}x^{i}
  3. x 2 + u x + v x^{2}+ux+v
  4. Q ( x ) = i = 0 n - 2 b i x i Q(x)=\sum_{i=0}^{n-2}b_{i}x^{i}
  5. c x + d cx+d
  6. P ( x ) = ( x 2 + u x + v ) ( i = 0 n - 2 b i x i ) + ( c x + d ) . P(x)=(x^{2}+ux+v)\left(\sum_{i=0}^{n-2}b_{i}x^{i}\right)+(cx+d).
  7. Q ( x ) Q(x)
  8. x 2 + u x + v x^{2}+ux+v
  9. R ( x ) = i = 0 n - 4 f i x i R(x)=\sum_{i=0}^{n-4}f_{i}x^{i}
  10. g x + h gx+h
  11. Q ( x ) = ( x 2 + u x + v ) ( i = 0 n - 4 f i x i ) + ( g x + h ) . Q(x)=(x^{2}+ux+v)\left(\sum_{i=0}^{n-4}f_{i}x^{i}\right)+(gx+h).
  12. c , d , g , h c,\,d,\,g,\,h
  13. { b i } , { f i } \{b_{i}\},\;\{f_{i}\}
  14. u u
  15. v v
  16. b n = b n - 1 = 0 , f n = f n - 1 = 0 , b i = a i + 2 - u b i + 1 - v b i + 2 f i = b i + 2 - u f i + 1 - v f i + 2 ( i = n - 2 , , 0 ) , c = a 1 - u b 0 - v b 1 , g = b 1 - u f 0 - v f 1 , d = a 0 - v b 0 , h = b 0 - v f 0 . \begin{aligned}\displaystyle b_{n}&\displaystyle=b_{n-1}=0,&\displaystyle f_{n% }&\displaystyle=f_{n-1}=0,\\ \displaystyle b_{i}&\displaystyle=a_{i+2}-ub_{i+1}-vb_{i+2}&\displaystyle f_{i% }&\displaystyle=b_{i+2}-uf_{i+1}-vf_{i+2}\qquad(i=n-2,\ldots,0),\\ \displaystyle c&\displaystyle=a_{1}-ub_{0}-vb_{1},&\displaystyle g&% \displaystyle=b_{1}-uf_{0}-vf_{1},\\ \displaystyle d&\displaystyle=a_{0}-vb_{0},&\displaystyle h&\displaystyle=b_{0% }-vf_{0}.\end{aligned}
  17. c ( u , v ) = d ( u , v ) = 0. c(u,v)=d(u,v)=0.\,
  18. u u
  19. v v
  20. [ u v ] := [ u v ] - [ c u c v d u d v ] - 1 [ c d ] := [ u v ] - 1 v g 2 + h ( h - u g ) [ - h g - g v g u - h ] [ c d ] \begin{bmatrix}u\\ v\end{bmatrix}:=\begin{bmatrix}u\\ v\end{bmatrix}-\begin{bmatrix}\frac{\partial c}{\partial u}&\frac{\partial c}{% \partial v}\\ \frac{\partial d}{\partial u}&\frac{\partial d}{\partial v}\end{bmatrix}^{-1}% \begin{bmatrix}c\\ d\end{bmatrix}:=\begin{bmatrix}u\\ v\end{bmatrix}-\frac{1}{vg^{2}+h(h-ug)}\begin{bmatrix}-h&g\\ -gv&gu-h\end{bmatrix}\begin{bmatrix}c\\ d\end{bmatrix}
  21. f ( x ) = 6 x 5 + 11 x 4 - 33 x 3 - 33 x 2 + 11 x + 6. f(x)=6\,x^{5}+11\,x^{4}-33\,x^{3}-33\,x^{2}+11\,x+6.
  22. u = a n - 1 a n = 11 6 ; v = a n - 2 a n = - 33 6 . u=\frac{a_{n-1}}{a_{n}}=\frac{11}{6};\quad v=\frac{a_{n-2}}{a_{n}}=-\frac{33}{% 6}.\,
  23. f ( x ) = x 5 - 1 f(x)=x^{5}-1
  24. f ( x ) = x 6 - x f(x)=x^{6}-x
  25. f ( x ) = 6 x 5 + 11 x 4 - 33 x 3 - 33 x 2 + 11 x + 6 \begin{aligned}\displaystyle f(x)=&\displaystyle 6x^{5}+11x^{4}-33x^{3}\\ &\displaystyle-33x^{2}+11x+6\end{aligned}
  26. ( s , t ) [ - 3 , 3 ] 2 (s,t)\in[-3,3]^{2}
  27. s ± i t s\pm it
  28. x 2 + u x + v = ( x - s ) 2 + t 2 x^{2}+ux+v=(x-s)^{2}+t^{2}
  29. s ± t s\pm t
  30. x 2 + u x + v = ( x - s ) 2 - t 2 x^{2}+ux+v=(x-s)^{2}-t^{2}
  31. ( u , v ) = ( - 2 s , s 2 + t | t | ) (u,\,v)=(-2s,\,s^{2}+t\,|t|)

Baker_percentage.html

  1. b a k e r s p e r c e n t a g e i n g r e d i e n t = 100 % × m a s s i n g r e d i e n t m a s s f l o u r baker^{\prime}s\ percentage_{ingredient}=100\%\times\frac{mass_{ingredient}}{% mass_{flour}}
  2. {}^{†}
  3. {}^{††}
  4. w e i g h t i n g r e d i e n t = w e i g h t f l o u r × b a k e r s p e r c e n t a g e i n g r e d i e n t 100 % = w e i g h t f l o u r × b a k e r s p e r c e n t a g e i n g r e d i e n t \begin{array}[]{rcl}weight_{ingredient}&=&\frac{weight_{flour}\ \times\ baker^% {\prime}s\ percentage_{ingredient}}{100\%}\\ &=&{weight_{flour}\times baker^{\prime}s\ percentage_{ingredient}}\\ \end{array}
  5. i n g r e d i e n t m a s s / f l o u r m a s s {ingredientmass}/{flourmass}
  6. f o r m u l a m a s s = m a s s f l o u r × f o r m u l a p e r c e n t a g e f o r m u l a m a s s f o r m u l a p e r c e n t a g e = m a s s f l o u r \begin{array}[]{rcl}formula\ mass&=&mass_{flour}\times formula\ percentage\\ \frac{formula\ mass}{formula\ percentage}&=&mass_{flour}\\ \end{array}
  7. m a s s i n g r e d i e n t = f o r m u l a m a s s × t r u e p e r c e n t a g e i n g r e d i e n t t r u e p e r c e n t a g e i n g r e d i e n t = b a k e r s p e r c e n t a g e i n g r e d i e n t f o r m u l a p e r c e n t a g e × 100 % m a s s i n g r e d i e n t = f o r m u l a m a s s × b a k e r s p e r c e n t a g e i n g r e d i e n t f o r m u l a p e r c e n t a g e = f o r m u l a m a s s × b a k e r s p e r c e n t a g e i n g r e d i e n t f o r m u l a p e r c e n t a g e \begin{array}[]{rcl}mass_{ingredient}&=&formula\ mass\times true\ percentage_{% ingredient}\\ true\ percentage_{ingredient}&=&\frac{baker^{\prime}s\ percentage_{ingredient}% }{formula\ percentage}\times 100\%\\ mass_{ingredient}&=&formula\ mass\times\frac{baker^{\prime}s\ percentage_{% ingredient}}{formula\ percentage}\\ &=&\frac{formula\ mass\ \times\ baker^{\prime}s\ percentage_{ingredient}}{% formula\ percentage}\end{array}
  8. m a s s i n g r e d i e n t = f o r m u l a m a s s f o r m u l a p e r c e n t a g e × b a k e r s p e r c e n t a g e i n g r e d i e n t = m a s s f l o u r × b a k e r s p e r c e n t a g e i n g r e d i e n t \begin{array}[]{rcl}mass_{ingredient}&=&\frac{formula\ mass}{formula\ % percentage}\times baker^{\prime}s\ percentage_{ingredient}\\ &=&mass_{flour}\times baker^{\prime}s\ percentage_{ingredient}\end{array}
  9. f o r m u l a m a s s × t r u e p e r c e n t a g e i n g r e d i e n t = m a s s f l o u r × b a k e r s p e r c e n t a g e i n g r e d i e n t formula\ mass\ \times\ true\ percentage_{ingredient}\ =\ mass_{flour}\ \times% \ baker^{\prime}s\ percentage_{ingredient}
  10. {}^{†}
  11. {}^{††}

Balanced_prime.html

  1. p n p_{n}
  2. p n = p n - 1 + p n + 1 2 . p_{n}={{p_{n-1}+p_{n+1}}\over 2}.
  3. 2 = 1 + 3 2 . 2={1+3\over 2}.
  4. p n = 1213266377 × 2 35000 + 2429 , p n - 1 = p n - 2430 , p n + 1 = p n + 2430. p_{n}=1213266377\times 2^{35000}+2429,\quad p_{n-1}=p_{n}-2430,\quad p_{n+1}=p% _{n}+2430.
  5. p k p_{k}
  6. p k = i = 1 n ( p k - i + p k + i ) 2 n . p_{k}={\sum_{i=1}^{n}({p_{k-i}+p_{k+i})}\over 2n}.

Balian–Low_theorem.html

  1. g m , n ( x ) = e 2 π i m b x g ( x - n a ) , g_{m,n}(x)=e^{2\pi imbx}g(x-na),
  2. { g m , n : m , n } \{g_{m,n}:m,n\in\mathbb{Z}\}
  3. L 2 ( ) , L^{2}(\mathbb{R}),
  4. - x 2 | g ( x ) | 2 d x = or - ξ 2 | g ^ ( ξ ) | 2 d ξ = . \int_{-\infty}^{\infty}x^{2}|g(x)|^{2}\;dx=\infty\quad\textrm{or}\quad\int_{-% \infty}^{\infty}\xi^{2}|\hat{g}(\xi)|^{2}\;d\xi=\infty.

Ball_screw.html

  1. T = F l 2 π ν T=\frac{Fl}{2\pi\nu}
  2. T \mathit{T}
  3. F \mathit{F}
  4. l \mathit{l}
  5. ν \nu

Ballistic_conduction.html

  1. 10 - 9 10^{-9}
  2. L λ M F P L\leq\lambda_{MFP}
  3. L L
  4. λ M F P \lambda_{MFP}
  5. 1 λ MFP = 1 λ el - el + 1 λ ap + 1 λ op , ems + 1 λ op , abs + 1 λ impurity + 1 λ defect + 1 λ boundary \frac{1}{\lambda_{\mathrm{MFP}}}=\frac{1}{\lambda_{\mathrm{el-el}}}+\frac{1}{% \lambda_{\mathrm{ap}}}+\frac{1}{\lambda_{\mathrm{op,ems}}}+\frac{1}{\lambda_{% \mathrm{op,abs}}}+\frac{1}{\lambda_{\mathrm{impurity}}}+\frac{1}{\lambda_{% \mathrm{defect}}}+\frac{1}{\lambda_{\mathrm{boundary}}}
  6. λ el - el \lambda_{\mathrm{el-el}}
  7. λ ap \lambda_{\mathrm{ap}}
  8. λ op , ems \lambda_{\mathrm{op,ems}}
  9. λ op , abs \lambda_{\mathrm{op,abs}}
  10. λ impurity \lambda_{\mathrm{impurity}}
  11. λ defect \lambda_{\mathrm{defect}}
  12. λ boundary \lambda_{\mathrm{boundary}}
  13. λ MFP \lambda_{\mathrm{MFP}}
  14. I A B = g s e h E F B E F A M ( E ) f ( E ) T ( E ) d E I_{AB}=\frac{g_{s}e}{h}\int_{E_{F_{B}}}^{E_{F_{A}}}M(E)f^{^{\prime}}(E)T(E)dE
  15. g s = 2 g_{s}=2
  16. E F A E_{F_{A}}
  17. E F B E_{F_{B}}
  18. M ( E ) M(E)
  19. f ( E ) f^{^{\prime}}(E)
  20. T ( E ) T(E)
  21. G = I V G=\frac{I}{V}
  22. e V = E F A - E F B eV=E_{F_{A}}-E_{F_{B}}
  23. G = 2 e 2 h M T G=\frac{2e^{2}}{h}MT
  24. G G
  25. T M M c o n t a c t T\approx\frac{M}{M_{contact}}
  26. 12.9 k Ω M \frac{12.9k\Omega}{M}
  27. λ M F P 1 μ \lambda_{MFP}\approx 1{\mu}

Ballistic_pendulum.html

  1. g g
  2. h h
  3. v 1 v_{1}
  4. K initial = U final K_{\textrm{initial}}=U_{\textrm{final}}\,
  5. 1 2 ( m b + m p ) v 1 2 = ( m b + m p ) g h \begin{matrix}\frac{1}{2}\end{matrix}(m_{\textrm{b}}+m_{\textrm{p}})\cdot v_{1% }^{2}=(m_{\textrm{b}}+m_{\textrm{p}})\cdot g\cdot h
  6. v 1 = 2 g h . v_{1}=\sqrt{2\cdot g\cdot h}.
  7. p = m v p=mv
  8. v 0 v_{0}
  9. p [b] = p [p + b] p_{\textrm{[b]}}=p_{\textrm{[p + b]}}\,
  10. m b v 0 = ( m b + m p ) 2 g h m_{\textrm{b}}\cdot v_{0}=(m_{\textrm{b}}+m_{\textrm{p}})\cdot\sqrt{2\cdot g% \cdot h}
  11. v 0 = ( m b + m p ) 2 g h m b = ( 1 + m p m b ) 2 g h v_{0}=\frac{(m_{\textrm{b}}+m_{\textrm{p}})\cdot\sqrt{2\cdot g\cdot h}}{m_{% \textrm{b}}}=(1+\frac{m_{\textrm{p}}}{m_{\textrm{b}}})\cdot\sqrt{2\cdot g\cdot h}
  12. v = 614.58 g c p + b b i r n v=614.58gc\cdot\frac{p+b}{birn}
  13. v v
  14. b b
  15. p p
  16. g g
  17. i i
  18. c c
  19. r r
  20. n n
  21. m v c f = M b k g h mvcf=Mbk^{\prime}\sqrt{gh}
  22. m m
  23. v v
  24. c c
  25. f f
  26. M M
  27. b b
  28. k k^{\prime}
  29. g g
  30. h h
  31. k k^{\prime}
  32. T = π k 2 g h T=\pi\sqrt{\frac{k^{\prime 2}}{gh}}
  33. T T
  34. V = M p M b 0.2018 D V=\frac{Mp}{Mb}0.2018D
  35. V V
  36. M p Mp
  37. M b Mb
  38. D D
  39. C = p i T 12 C=\frac{pi}{T12}
  40. V = M p M b C D V=\frac{Mp}{Mb}CD