wpmath0000008_12

Subgame_perfect_equilibrium.html

  1. { L , R } \{L,R\}
  2. { K , U } 2 \{K,U\}^{2}
  3. L , ( U , U ) L,(U,U)
  4. 2 2 2_{2}
  5. K K
  6. U U
  7. 2 2 2_{2}
  8. 2 2 2_{2}
  9. ( U , K ) (U,K)
  10. R R
  11. R , ( U , K ) R,(U,K)
  12. 2 1 2_{1}
  13. 2 2 2_{2}

Subnormal_operator.html

  1. N = [ A B 0 C ] N=\begin{bmatrix}A&B\\ 0&C\end{bmatrix}
  2. B : H H , and C : H H . B:H^{\perp}\rightarrow H,\quad\mbox{and}~{}\quad C:H^{\perp}\rightarrow H^{% \perp}.
  3. H H H\oplus H
  4. U = [ A I - A A * 0 - A * ] . U=\begin{bmatrix}A&I-AA^{*}\\ 0&-A^{*}\end{bmatrix}.
  5. V = [ U I - U U * 0 - U * ] = [ U D U * 0 - U * ] . V=\begin{bmatrix}U&I-UU^{*}\\ 0&-U^{*}\end{bmatrix}=\begin{bmatrix}U&D_{U^{*}}\\ 0&-U^{*}\end{bmatrix}.
  6. Q = [ P 0 0 P ] . Q=\begin{bmatrix}P&0\\ 0&P\end{bmatrix}.
  7. N * N = Q V * V Q = Q 2 = [ P 2 0 0 P 2 ] . N^{*}N=QV^{*}VQ=Q^{2}=\begin{bmatrix}P^{2}&0\\ 0&P^{2}\end{bmatrix}.
  8. N N * = [ U P 2 U * + D U * P 2 D U * - D U * P 2 U - U * P 2 D U * U * P 2 U ] . NN^{*}=\begin{bmatrix}UP^{2}U^{*}+D_{U^{*}}P^{2}D_{U^{*}}&-D_{U^{*}}P^{2}U\\ -U^{*}P^{2}D_{U^{*}}&U^{*}P^{2}U\end{bmatrix}.
  9. B ( , a - 1 , a ^ 0 , a 1 , ) = ( , a ^ - 1 , a 0 , a 1 , ) , B(\cdots,a_{-1},{\hat{a}_{0}},a_{1},\cdots)=(\cdots,{\hat{a}_{-1}},a_{0},a_{1}% ,\cdots),
  10. B = [ A I - A A * 0 A * ] . B=\begin{bmatrix}A&I-AA^{*}\\ 0&A^{*}\end{bmatrix}.
  11. B = [ A I - A A * 0 - A * ] B^{\prime}=\begin{bmatrix}A&I-AA^{*}\\ 0&-A^{*}\end{bmatrix}
  12. B ( , a - 2 , a - 1 , a ^ 0 , a 1 , a 2 , ) = ( , - a - 2 , a ^ - 1 , a 0 , a 1 , a 2 , ) . B^{\prime}(\cdots,a_{-2},a_{-1},{\hat{a}_{0}},a_{1},a_{2},\cdots)=(\cdots,-a_{% -2},{\hat{a}_{-1}},a_{0},a_{1},a_{2},\cdots).
  13. U : K 1 K 2 . U:K_{1}\rightarrow K_{2}.
  14. U B 1 = B 2 U . UB_{1}=B_{2}U.\,
  15. i = 0 n ( B 1 * ) i h i = h 0 + B 1 * h 1 + ( B 1 * ) 2 h 2 + + ( B 1 * ) n h n where h i H . \sum_{i=0}^{n}(B_{1}^{*})^{i}h_{i}=h_{0}+B_{1}^{*}h_{1}+(B_{1}^{*})^{2}h_{2}+% \cdots+(B_{1}^{*})^{n}h_{n}\quad\mbox{where}~{}\quad h_{i}\in H.
  16. U i = 0 n ( B 1 * ) i h i = i = 0 n ( B 2 * ) i h i U\sum_{i=0}^{n}(B_{1}^{*})^{i}h_{i}=\sum_{i=0}^{n}(B_{2}^{*})^{i}h_{i}
  17. i = 0 n ( B 1 * ) i h i , j = 0 n ( B 1 * ) j h j = i j h i , ( B 1 ) i ( B 1 * ) j h j = i j ( B 2 ) j h i , ( B 2 ) i h j = i = 0 n ( B 2 * ) i h i , j = 0 n ( B 2 * ) j h j , \langle\sum_{i=0}^{n}(B_{1}^{*})^{i}h_{i},\sum_{j=0}^{n}(B_{1}^{*})^{j}h_{j}% \rangle=\sum_{ij}\langle h_{i},(B_{1})^{i}(B_{1}^{*})^{j}h_{j}\rangle=\sum_{ij% }\langle(B_{2})^{j}h_{i},(B_{2})^{i}h_{j}\rangle=\langle\sum_{i=0}^{n}(B_{2}^{% *})^{i}h_{i},\sum_{j=0}^{n}(B_{2}^{*})^{j}h_{j}\rangle,
  18. if g = i = 0 n ( B 1 * ) i h i , \mbox{if}~{}\quad g=\sum_{i=0}^{n}(B_{1}^{*})^{i}h_{i},
  19. then U B 1 g = B 2 U g = i = 0 n ( B 2 * ) i A h i . \mbox{then}~{}\quad UB_{1}g=B_{2}Ug=\sum_{i=0}^{n}(B_{2}^{*})^{i}Ah_{i}.

Substitution_tiling.html

  1. T 1 , T 2 , , T m T_{1},T_{2},\dots,T_{m}
  2. Q Q
  3. Q T i QT_{i}
  4. T j T_{j}
  5. d {\mathbb{R}}^{d}
  6. P = { T 1 , T 2 , , T m } {P}=\{T_{1},T_{2},\dots,T_{m}\}
  7. T i T_{i}
  8. ( T i , φ ) (T_{i},\varphi)
  9. φ \varphi
  10. d {\mathbb{R}}^{d}
  11. φ ( T i ) \varphi(T_{i})
  12. ( Q , σ ) (Q,\sigma)
  13. Q : d d Q:{\mathbb{R}}^{d}\to{\mathbb{R}}^{d}
  14. σ \sigma
  15. T i T_{i}
  16. Q T i QT_{i}
  17. σ \sigma
  18. σ ( T ) \sigma({T})
  19. Q σ ( W ) Q_{\sigma}({W})
  20. σ ( T ) = ( T i , φ ) T { ( T j , Q φ Q - 1 ρ ) : ( T j , ρ ) σ ( T i ) } . \sigma({T})=\bigcup_{(T_{i},\varphi)\in{T}}\{(T_{j},Q\circ\varphi\circ Q^{-1}% \circ\rho):(T_{j},\rho)\in\sigma(T_{i})\}.
  21. ( Q , σ ) (Q,\sigma)
  22. d {\mathbb{R}}^{d}
  23. σ k ( T i ) \sigma^{k}(T_{i})
  24. ( Q , σ ) (Q,\sigma)

Sulfite_reductase_(ferredoxin).html

  1. \rightleftharpoons

Sunflower_(mathematics).html

  1. ω 2 \omega_{2}
  2. W W
  3. ω 2 \omega_{2}
  4. ω 2 \omega_{2}
  5. ω 2 \omega_{2}
  6. A α : α < ω 2 \langle A_{\alpha}:\alpha<\omega_{2}\rangle
  7. W W
  8. cf ( α ) = ω 1 {\rm cf}(\alpha)=\omega_{1}
  9. f ( α ) = sup ( A α α ) f(\alpha)={\rm sup}(A_{\alpha}\cap\alpha)
  10. S S
  11. ω 2 \omega_{2}
  12. f f
  13. β \beta
  14. S S
  15. S S S^{\prime}\subseteq S
  16. ω 2 \omega_{2}
  17. i < j i<j
  18. S S^{\prime}
  19. A i j A_{i}\subseteq j
  20. ω 1 \omega_{1}
  21. β \beta

Super_star_cluster.html

  1. n e = 10 3 n_{e}=10^{3}
  2. 10 6 10^{6}
  3. - 3 {}^{-3}
  4. P / P/
  5. k b k_{b}
  6. = 10 7 =10^{7}
  7. 10 10 10^{10}
  8. - 3 {}^{-3}

Supercharger.html

  1. T 2 T 1 \frac{T_{2}}{T_{1}}
  2. = =\,\!
  3. ( p 2 p 1 ) γ - 1 γ \left(\frac{p_{2}}{p_{1}}\right)^{\frac{\gamma-1}{\gamma}}
  4. T 1 T_{1}\,\!
  5. T 2 T_{2}\,\!
  6. p 1 p_{1}\,\!
  7. p 2 p_{2}\,\!
  8. γ \gamma\,\!
  9. C p / C v C_{p}/C_{v}\,\!
  10. C p C_{p}\,\!
  11. C v C_{v}\,\!
  12. 1 / 3 {1}/{3}
  13. 1 / 3 {1}/{3}
  14. 1 / 3 {1}/{3}

Superoperator.html

  1. i t Ψ = H ^ Ψ i\hbar\frac{\partial}{\partial t}\Psi=\hat{H}\Psi
  2. ψ \psi
  3. H ^ \hat{H}
  4. \mathcal{H}
  5. i t ρ = [ ρ ] i\hbar\frac{\partial}{\partial t}\rho=\mathcal{H}[\rho]
  6. [ ρ ] = [ H ^ , ρ ] H ^ ρ - ρ H ^ \mathcal{H}[\rho]=[\hat{H},\rho]\equiv\hat{H}\rho-\rho\hat{H}
  7. H ^ = H ^ ( P ^ ) \hat{H}=\hat{H}(\hat{P})
  8. Δ H ^ Δ P ^ \frac{\Delta\hat{H}}{\Delta\hat{P}}
  9. H ( P ) = P 3 = P P P H(P)=P^{3}=PPP
  10. Δ H Δ P [ X ] = X P 2 + P X P + P 2 X \frac{\Delta H}{\Delta P}[X]=XP^{2}+PXP+P^{2}X

Superpattern.html

  1. ( n k ) k ! , {\textstyle\left({{n}\atop{k}}\right)}\geq k!,

Support_(measure_theory).html

  1. supp ( μ ) := { A Σ μ ( A ) > 0 } ¯ \mathrm{supp}(\mu):=\overline{\{A\in\Sigma\mid\mu(A)>0\}}
  2. { x X for some open N x x , μ ( N x ) > 0 } \{x\in X\mid\,\text{for some open }N_{x}\ni x,\mu(N_{x})>0\}
  3. supp ( μ ) := { x X x N x T μ ( N x ) > 0 } . \mathrm{supp}(\mu):=\{x\in X\mid x\in N_{x}\in T\implies\mu(N_{x})>0\}.
  4. U T and U C μ ( U C ) > 0. U\in T\,\text{ and }U\cap C\neq\varnothing\implies\mu(U\cap C)>0.
  5. A X supp ( μ ) μ ( A ) = 0. A\subseteq X\setminus\mathrm{supp}(\mu)\implies\mu(A)=0.
  6. X f ( x ) d μ ( x ) = supp ( μ ) f ( x ) d μ ( x ) . \int_{X}f(x)\,\mathrm{d}\mu(x)=\int_{\mathrm{supp}(\mu)}f(x)\,\mathrm{d}\mu(x).
  7. μ \mu
  8. \mathbb{R}
  9. ( A f ) ( x ) = x f ( x ) (Af)(x)=xf(x)
  10. D ( A ) = { f L 2 ( , d μ ) x f ( x ) L 2 ( , d μ ) } D(A)=\{f\in L^{2}(\mathbb{R},d\mu)\mid xf(x)\in L^{2}(\mathbb{R},d\mu)\}
  11. x x x\mapsto x
  12. μ \mu
  13. μ ( A ) := λ ( A ( 0 , 1 ) ) \mu(A):=\lambda(A\cap(0,1))
  14. μ = μ + - μ - , \mu=\mu^{+}-\mu^{-},
  15. supp ( μ ) := supp ( μ + ) supp ( μ - ) . \mathrm{supp}(\mu):=\mathrm{supp}(\mu^{+})\cup\mathrm{supp}(\mu^{-}).

Surface-area-to-volume_ratio.html

  1. SA:V = 6 cm 2 1 cm 3 = 6 cm - 1 \mbox{SA:V}~{}=\frac{6~{}\mbox{cm}~{}^{2}}{1~{}\mbox{cm}~{}^{3}}=6~{}\mbox{cm}% ~{}^{-1}
  2. a a
  3. 3 a 2 \sqrt{3}a^{2}
  4. 2 a 3 12 \frac{\sqrt{2}a^{3}}{12}
  5. 6 6 a 14.697 a \frac{6\sqrt{6}}{a}\approx\frac{14.697}{a}
  6. 6 a 2 6a^{2}
  7. a 3 a^{3}
  8. 6 a \frac{6}{a}
  9. 2 3 a 2 2\sqrt{3}a^{2}
  10. 1 3 2 a 3 \frac{1}{3}\sqrt{2}a^{3}
  11. 3 6 a 7.348 a \frac{3\sqrt{6}}{a}\approx\frac{7.348}{a}
  12. 3 25 + 10 5 a 2 3\sqrt{25+10\sqrt{5}}a^{2}
  13. 1 4 ( 15 + 7 5 ) a 3 \frac{1}{4}(15+7\sqrt{5})a^{3}
  14. 12 25 + 10 5 ( 15 + 7 5 ) a 2.694 a \frac{12\sqrt{25+10\sqrt{5}}}{(15+7\sqrt{5})a}\approx\frac{2.694}{a}
  15. 5 3 a 2 5\sqrt{3}a^{2}
  16. 5 12 ( 3 + 5 ) a 3 \frac{5}{12}(3+\sqrt{5})a^{3}
  17. 12 3 ( 3 + 5 ) a 3.970 a \frac{12\sqrt{3}}{(3+\sqrt{5})a}\approx\frac{3.970}{a}
  18. 4 π a 2 4\pi a^{2}
  19. 4 π a 3 3 \frac{4\pi a^{3}}{3}
  20. 3 a \frac{3}{a}

Surface_feet_per_minute.html

  1. Spindle speed (RPM) = S F M π × 1 12 × stock diameter (in) S F M 0.2618 × stock diameter (in) \,\text{Spindle speed (RPM)}=\frac{SFM}{\pi\times\frac{1}{12}\times\,\text{% stock diameter (in)}}\approx\frac{SFM}{0.2618\times\,\text{stock diameter (in)}}
  2. S F M = stock diameter (in) × π × 1 12 × RPM stock diameter (in) × 0.2618 × RPM SFM=\,\text{stock diameter (in)}\times\pi\times\frac{1}{12}\times\,\text{RPM}% \approx\,\text{stock diameter (in)}\times 0.2618\times\,\text{RPM}

Surface_photovoltage.html

  1. L L
  2. τ bulk \tau_{\mathrm{bulk}}
  3. L = D τ bulk L=\sqrt{D\tau_{\mathrm{bulk}}}
  4. D D
  5. L meas = D τ eff L_{\mathrm{meas}}=\sqrt{D\tau_{\mathrm{eff}}}
  6. 1 τ eff = 1 τ bulk + 2 s d \frac{1}{\tau_{\mathrm{eff}}}=\frac{1}{\tau_{\mathrm{bulk}}}+\frac{2s}{d}
  7. τ eff \tau_{\mathrm{eff}}
  8. τ bulk \tau_{\mathrm{bulk}}
  9. s s
  10. d d

Surface_reconstruction.html

  1. 3 × 3 \sqrt{3}\times\sqrt{3}
  2. 31 × 31 \sqrt{31}\times\sqrt{31}
  3. a a
  4. b b
  5. a s a_{s}
  6. b s b_{s}
  7. a s = G 11 a + G 12 b a_{s}=G_{11}a+G_{12}b
  8. b s = G 21 a + G 22 b b_{s}=G_{21}a+G_{22}b
  9. G = ( G 11 G 12 G 21 G 22 ) G=\begin{pmatrix}G_{11}&G_{12}\\ G_{21}&G_{22}\end{pmatrix}
  10. ϕ \phi

Surgery_theory.html

  1. ϕ \phi
  2. M := ( M - int im ϕ ) ϕ | S p × S q - 1 ( D p + 1 × S q - 1 ) . M^{\prime}:=(M-\operatorname{int~{}im}\phi)\;\cup_{\phi|_{S^{p}\times S^{q-1}}% }(D^{p+1}\!\times\!S^{q-1}).
  3. ϕ \phi
  4. L := L ϕ ( D p + 1 × D q ) . L^{\prime}:=L\;\cup_{\phi}(D^{p+1}\!\!\times\!D^{q}).
  5. L = ( L - int im ϕ ) ϕ | S p × S q - 1 ( D p + 1 × S q - 1 ) . \partial L^{\prime}=(\partial L-\operatorname{int~{}im}\phi)\;\cup_{\phi|_{S^{% p}\times S^{q-1}}}(D^{p+1}\!\times\!S^{q-1}).
  6. W := ( M × I ) S p × D q × { 1 } ( D p + 1 × D q ) W:=(M\times I)\;\cup_{S^{p}\times D^{q}\times\{1\}}(D^{p+1}\!\times\!D^{q})
  7. S n = D n + 1 ( D p + 1 × D q ) = S p × D q D p + 1 × S q - 1 S^{n}=\partial D^{n+1}\approx\partial(D^{p+1}\times D^{q})=S^{p}\times D^{q}\;% \cup\;D^{p+1}\times S^{q-1}
  8. D p + 1 × S q - 1 D p + 1 × S q - 1 = S p + 1 × S q - 1 D^{p+1}\times S^{q-1}\;\cup\;D^{p+1}\times S^{q-1}=S^{p+1}\times S^{q-1}
  9. M := f - 1 ( c + ϵ ) M^{\prime}:=f^{-1}(c+\epsilon)
  10. M := f - 1 ( c - ϵ ) M:=f^{-1}(c-\epsilon)
  11. W := f - 1 ( [ c - ϵ , c + ϵ ] ) W:=f^{-1}([c-\epsilon,c+\epsilon])
  12. - x 2 + y 2 -\|x\|^{2}+\|y\|^{2}
  13. x R p + 1 , y R q x\in R^{p+1},y\in R^{q}
  14. W M × I S p × D q D p + 1 × D q W\cong M\times I\cup_{S^{p}\times D^{q}}D^{p+1}\times D^{q}
  15. D p + 1 × D q D^{p+1}\times D^{q}
  16. α π p ( M ) \alpha\in\pi_{p}(M)
  17. α π p ( M ) \alpha\in\pi_{p}(M)
  18. 𝐙 [ π 1 ( X ) ] \mathbf{Z}[\pi_{1}(X)]
  19. σ ( f ) L n ( 𝐙 [ π 1 ( X ) ] ) \sigma(f)\in L_{n}(\mathbf{Z}[\pi_{1}(X)])
  20. π 1 ( X ) = 0 \pi_{1}(X)=0
  21. L 4 k ( 𝐙 ) L_{4k}(\mathbf{Z})
  22. 𝐙 \mathbf{Z}
  23. σ ( X ) - σ ( M ) \sigma(X)-\sigma(M)
  24. [ X , G / O ] L n ( 𝐙 [ π 1 ( X ) ] ) . [X,G/O]\to L_{n}(\mathbf{Z}[\pi_{1}(X)]).
  25. H * ( X ) H n - * ( X ) H^{*}(X)\cong H_{n-*}(X)

Swinging_Atwood's_machine.html

  1. m m
  2. M M
  3. M = m M=m
  4. M > m M>m
  5. m m
  6. M M
  7. T T
  8. T \displaystyle T
  9. r r
  10. θ \theta
  11. U U
  12. U \displaystyle U
  13. \mathcal{L}
  14. \mathcal{H}
  15. \displaystyle\mathcal{L}
  16. p r p_{r}
  17. p θ p_{\theta}
  18. p r \displaystyle p_{r}
  19. r r
  20. θ \theta
  21. θ \theta
  22. θ \displaystyle\frac{\partial{\mathcal{L}}}{\partial\theta}
  23. r r
  24. r \displaystyle\frac{\partial{\mathcal{L}}}{\partial r}
  25. μ = M m \mu=\frac{M}{m}
  26. ( μ + 1 ) r ¨ - r θ ˙ 2 + g ( μ - cos θ ) = 0 (\mu+1)\ddot{r}-r\dot{\theta}^{2}+g(\mu-\cos{\theta})=0
  27. r r
  28. θ \theta
  29. p r p_{r}
  30. p θ p_{\theta}
  31. r ˙ \displaystyle\dot{r}
  32. θ \theta
  33. θ ˙ \dot{\theta}
  34. r ¨ = g 1 - μ 1 + μ = g m - M m + M \ddot{r}=g\frac{1-\mu}{1+\mu}=g\frac{m-M}{m+M}
  35. r r
  36. θ \theta
  37. p r p_{r}
  38. p θ p_{\theta}
  39. I I
  40. R R
  41. ( r , θ , r ˙ , θ ˙ ) = 1 2 M t ( R θ ˙ - r ˙ ) 2 + 1 2 m r 2 θ ˙ 2 T + g r ( M - m cos θ ) + g R ( m sin θ - M θ ) U , \mathcal{H}\left(r,\theta,\dot{r},\dot{\theta}\right)=\underbrace{\frac{1}{2}M% _{t}\left(R\dot{\theta}-\dot{r}\right)^{2}+\frac{1}{2}mr^{2}\dot{\theta}^{2}}_% {T}+\underbrace{gr\left(M-m\cos{\theta}\right)+gR\left(m\sin{\theta}-M\theta% \right)}_{U},
  42. M M
  43. M t = M + m + I R 2 M_{t}=M+m+\frac{I}{R^{2}}
  44. R R
  45. I I
  46. μ t ( r ¨ - R θ ¨ ) = r θ ˙ 2 + g ( cos θ - μ ) r θ ¨ = - 2 r ˙ θ ˙ + R θ ˙ 2 - g sin θ \begin{aligned}\displaystyle\mu_{t}(\ddot{r}-R\ddot{\theta})&\displaystyle=r% \dot{\theta}^{2}+g(\cos{\theta}-\mu)\\ \displaystyle r\ddot{\theta}&\displaystyle=-2\dot{r}\dot{\theta}+R\dot{\theta}% ^{2}-g\sin{\theta}\\ \end{aligned}
  47. μ t = M t / m \mu_{t}=M_{t}/m
  48. μ = M / m = 3 \mu=M/m=3
  49. μ = 4 n 2 - 1 = 3 , 15 , 35 , \mu=4n^{2}-1=3,15,35,...
  50. μ = 3 \mu=3
  51. r = 0 , r ˙ = v , θ = θ 0 , θ ˙ = 0 r=0,\dot{r}=v,\theta=\theta_{0},\dot{\theta}=0
  52. θ 0 \theta_{0}
  53. θ 0 \theta_{0}
  54. μ = 2 \mu=2
  55. θ 0 = π 2 \theta_{0}=\frac{\pi}{2}
  56. μ = 3 \mu=3
  57. θ 0 = π 2 \theta_{0}=\frac{\pi}{2}
  58. μ = 5 \mu=5
  59. θ 0 = π 2 \theta_{0}=\frac{\pi}{2}
  60. μ = 6 \mu=6
  61. θ 0 = π 2 \theta_{0}=\frac{\pi}{2}
  62. μ = 16 \mu=16
  63. θ 0 = π 2 \theta_{0}=\frac{\pi}{2}
  64. μ = 19 \mu=19
  65. θ 0 = π 2 \theta_{0}=\frac{\pi}{2}
  66. μ = 21 \mu=21
  67. θ 0 = π 2 \theta_{0}=\frac{\pi}{2}
  68. μ = 24 \mu=24
  69. θ 0 = π 2 \theta_{0}=\frac{\pi}{2}
  70. r ( t + τ ) = r ( t ) , θ ( t + τ ) = θ ( t ) r(t+\tau)=r(t),\,\theta(t+\tau)=\theta(t)
  71. μ = 1.665 \mu=1.665
  72. θ 0 = π 2 \theta_{0}=\frac{\pi}{2}
  73. μ = 2.394 \mu=2.394
  74. θ 0 = π 2 \theta_{0}=\frac{\pi}{2}
  75. μ = 1.1727 \mu=1.1727
  76. θ 0 = π 2 \theta_{0}=\frac{\pi}{2}
  77. μ = 1.555 \mu=1.555
  78. θ 0 = π 2 \theta_{0}=\frac{\pi}{2}
  79. r ( 0 ) = 0 r(0)=0
  80. r r
  81. r r
  82. μ = 10 \mu=10
  83. θ 0 = π 2 \theta_{0}=\frac{\pi}{2}
  84. μ = 25 \mu=25
  85. θ 0 = π 2 \theta_{0}=\frac{\pi}{2}
  86. θ 0 \theta_{0}
  87. r ( τ ) = r ( 0 ) = 0 , τ > 0 r(\tau)=r(0)=0,\,\tau>0
  88. M M
  89. m m
  90. r = r 0 r=r_{0}
  91. θ = θ 0 \theta=\theta_{0}
  92. E = 1 2 M r ˙ 2 + 1 2 m ( r ˙ 2 + r 2 θ ˙ 2 ) + M g r - m g r cos θ = M g r 0 - m g r 0 cos θ 0 E=\frac{1}{2}M\dot{r}^{2}+\frac{1}{2}m\left(\dot{r}^{2}+r^{2}\dot{\theta}^{2}% \right)+Mgr-mgr\cos{\theta}=Mgr_{0}-mgr_{0}\cos{\theta_{0}}
  93. M g r - m g r cos θ = M g r 0 - m g r 0 cos θ 0 Mgr-mgr\cos{\theta}=Mgr_{0}-mgr_{0}\cos{\theta_{0}}
  94. r r
  95. r \displaystyle r
  96. h h
  97. 1 μ \frac{1}{\mu}
  98. μ > 1 \mu>1
  99. μ = 1 \mu=1
  100. μ < 1 \mu<1
  101. μ \mu

Swiss_Formula.html

  1. T new = A × T old A + T old T\text{new}=\frac{A\times T\text{old}}{A+T\text{old}}

Symmetric_hypergraph_theorem.html

  1. G G
  2. S S
  3. x x
  4. y y
  5. S S
  6. f f
  7. G G
  8. f ( x ) = y f(x)=y
  9. H = ( S , E ) H=(S,E)
  10. m = | S | m=|S|
  11. χ ( H ) \chi(H)
  12. H H
  13. α ( H ) \alpha(H)
  14. H H
  15. χ ( H ) < 1 + m α ( H ) ln m . \chi(H)<1+\frac{m}{\alpha(H)}\ln m.

Symmetric_level-index_arithmetic.html

  1. X = e e e e f X=e^{e^{e^{...^{e^{f}}}}}
  2. 0 f < 1 0\leq f<1
  3. l 0 l\geq 0
  4. X = 1234567 = e e e 0.9711308 X=1234567=e^{e^{e^{0.9711308}}}
  5. x = l + f = 3 + 0.9711308 = 3.9711308 x=l+f=3+0.9711308=3.9711308\,
  6. ψ ( X ) = { X if 0 X < 1 1 + ψ ( ln X ) if X 1 \psi(X)=\left\{\begin{matrix}X&\mathrm{if}\quad 0\leq X<1\\ 1+\psi(\ln X)&\mathrm{if}\quad X\geq 1\end{matrix}\right.
  7. [ 0 , ) [0,\infty)
  8. ϕ ( x ) = { x if 0 x < 1 e ϕ ( x - 1 ) if x 1 \phi(x)=\left\{\begin{matrix}x&\mathrm{if}\quad 0\leq x<1\\ e^{\phi(x-1)}&\mathrm{if}\quad x\geq 1\end{matrix}\right.
  9. d ϕ ( x ) d x | x = 1 = d ϕ ( e x ) d x | x = 0 \frac{d\phi(x)}{dx}|_{x=1}=\frac{d\phi(e^{x})}{dx}|_{x=0}
  10. X = s X ϕ ( x ) r X X=s_{X}\phi(x)^{r_{X}}
  11. s X = sgn ( X ) , r X = sgn ( | X | - | X | - 1 ) , x = ψ ( max ( | X | , | X | - 1 ) ) = ψ ( | X | r X ) s_{X}=\,\text{sgn}(X),r_{X}=\,\text{sgn}(|X|-|X|^{-1}),x=\psi(\max(|X|,|X|^{-1% }))=\psi(|X|^{r_{X}})
  12. s 0 = + 1 , r 0 = + 1 , x = 0.0 s_{0}=+1,r_{0}=+1,x=0.0\,
  13. s 1 = + 1 , r 1 = + 1 , x = 1.0 s_{1}=+1,r_{1}=+1,x=1.0\,
  14. X = - 1 1234567 = - e - e e 0.9711308 X=-\dfrac{1}{1234567}=-e^{-e^{e^{0.9711308}}}
  15. x = - ϕ ( 3.9711308 ) - 1 x=-\phi(3.9711308)^{-1}\,

Symmetrization.html

  1. S S
  2. A A
  3. α : S × S A \alpha:S\times S\to A
  4. α \alpha
  5. α ( s , t ) = α ( t , s ) \alpha(s,t)=\alpha(t,s)
  6. s , t S s,t\in S
  7. α : S × S A \alpha\colon S\times S\to A
  8. ( x , y ) α ( x , y ) + α ( y , x ) (x,y)\mapsto\alpha(x,y)+\alpha(y,x)
  9. α : S × S A \alpha\colon S\times S\to A
  10. ( x , y ) α ( x , y ) - α ( y , x ) (x,y)\mapsto\alpha(x,y)-\alpha(y,x)
  11. 2 α . 2\alpha.
  12. 𝐙 / 2 , \mathbf{Z}/2,
  13. 1 = - 1 1=-1
  14. S 2 = C 2 \mathrm{S}_{2}=\mathrm{C}_{2}
  15. n ! n!
  16. n ! / 2 n!/2
  17. n ! / 2 n!/2
  18. p > n , p>n,
  19. n > 2 n>2

Symplectic_filling.html

  1. \partial
  2. ω | ξ > 0 \omega|_{\xi}>0
  3. \partial
  4. \partial
  5. { x 2 : | x | = 1 } \{x\in\mathbb{C}^{2}:|x|=1\}\,
  6. 2 \mathbb{C}^{2}
  7. - 1 \sqrt{-1}

Symplectic_sum.html

  1. M 1 M_{1}
  2. M 2 M_{2}
  3. 2 n 2n
  4. V V
  5. ( 2 n - 2 ) (2n-2)
  6. M 1 M_{1}
  7. M 2 M_{2}
  8. j i : V M i , j_{i}:V\hookrightarrow M_{i},
  9. e ( N M 1 V ) = - e ( N M 2 V ) . e(N_{M_{1}}V)=-e(N_{M_{2}}V).
  10. ψ : N M 1 V N M 2 V \psi:N_{M_{1}}V\to N_{M_{2}}V
  11. ( M 1 , V ) # ( M 2 , V ) (M_{1},V)\#(M_{2},V)
  12. M i M_{i}
  13. ψ \psi
  14. V V
  15. M 1 M_{1}
  16. M 2 M_{2}
  17. V V
  18. M M
  19. V V
  20. X X
  21. V V
  22. X i M i X_{i}\subseteq M_{i}
  23. V V
  24. V V
  25. M i M_{i}
  26. 2 k 2k
  27. 2 k 2k
  28. 2 k 2k
  29. 2 k 2k
  30. ω \omega
  31. [ ω ] H 2 ( 𝕊 2 k , ) . [\omega]\in H^{2}(\mathbb{S}^{2k},\mathbb{R}).
  32. 2 k = 2 2k=2
  33. M M
  34. V V
  35. V V
  36. M M
  37. 1 \mathbb{CP}^{1}
  38. P := ( N M V ) . P:=\mathbb{P}(N_{M}V\oplus\mathbb{C}).
  39. P P
  40. V V
  41. V 0 V_{0}
  42. V V
  43. M M
  44. V V_{\infty}
  45. ( M , V ) (M,V)
  46. ( P , V ) (P,V_{\infty})
  47. M M
  48. V 0 V_{0}
  49. V V
  50. ( M , V ) = ( ( M , V ) # ( P , V ) , V 0 ) . (M,V)=((M,V)\#(P,V_{\infty}),V_{0}).
  51. ( M , V ) (M,V)
  52. P P
  53. M 1 M_{1}
  54. M 2 M_{2}
  55. V V
  56. j 1 j_{1}
  57. j 2 j_{2}
  58. ψ \psi
  59. ( 2 n + 2 ) (2n+2)
  60. Z Z
  61. Z D Z\to D\subseteq\mathbb{C}
  62. Z 0 = M 1 V M 2 Z_{0}=M_{1}\cup_{V}M_{2}
  63. M i M_{i}
  64. V V
  65. Z ϵ Z_{\epsilon}
  66. M i M_{i}
  67. η \eta
  68. N M 1 V N M 2 V . N_{M_{1}}V\otimes_{\mathbb{C}}N_{M_{2}}V.
  69. N M 1 V N M 2 V , N_{M_{1}}V\oplus N_{M_{2}}V,
  70. v i v_{i}
  71. V V
  72. M i M_{i}
  73. v 1 v 2 = ϵ η v_{1}\otimes v_{2}=\epsilon\eta
  74. ϵ \epsilon\in\mathbb{C}
  75. M i V M_{i}\setminus V
  76. V V
  77. Z ϵ Z_{\epsilon}
  78. ϵ \epsilon
  79. Z ϵ Z_{\epsilon}
  80. Z D Z\to D
  81. Z 0 Z_{0}
  82. P P
  83. M M
  84. M M
  85. V V
  86. 1 \mathbb{CP}^{1}
  87. P P
  88. V V
  89. P P
  90. M M
  91. ( M , V ) = ( M , V ) # ( P , V ) . (M,V)=(M,V)\#(P,V_{\infty}).
  92. M M

Syntactic_predicate.html

  1. L = { a n b n c n : n 1 } L=\{a^{n}b^{n}c^{n}:n\geq 1\}
  2. L 1 = { a m b n c n : m , n 1 } L_{1}=\{a^{m}b^{n}c^{n}:m,n\geq 1\}
  3. L 2 = { a n b n c m : m , n 1 } L_{2}=\{a^{n}b^{n}c^{m}:m,n\geq 1\}
  4. L 3 = L 1 L 2 L_{3}=L_{1}\cap L_{2}
  5. L = { a n b n c n | n 1 } L=\{a^{n}b^{n}c^{n}|n\geq 1\}
  6. L = { a n b n c n | n 0 } L=\{a^{n}b^{n}c^{n}|n\geq 0\}

Szilassi_polyhedron.html

  1. h = ( f - 4 ) ( f - 3 ) 12 . h=\frac{(f-4)(f-3)}{12}.

T-stage.html

  1. w 2 = ( w 2 T 3 / P 3 ) ) * ( P 3 / P 2 ) * ( P 2 / P 1 ) * ( P 1 / T 1 ) / ( T 3 / T 2 * T 2 / T 1 ) , w_{2}=(w_{2}\sqrt{T_{3}}/P_{3}))*(P_{3}/P_{2})*(P_{2}/P_{1})*(P_{1}/\sqrt{T_{1% }})/(\sqrt{T_{3}/T_{2}}*\sqrt{T_{2}/T_{1}}),
  2. w 2 w_{2}\,
  3. ( w 2 T 3 / P 3 ) (w_{2}\sqrt{T_{3}}/P_{3})\,
  4. P 3 / P 2 P_{3}/P_{2}\,
  5. P 2 / P 1 P_{2}/P_{1}\,
  6. P 1 P_{1}\,
  7. T 1 T_{1}\,
  8. T 3 / T 2 T_{3}/T_{2}\,
  9. T 2 / T 1 T_{2}/T_{1}\,
  10. P 2 / P 1 P_{2}/P_{1}\,
  11. P 2 / P 1 P_{2}/P_{1}\,
  12. w 2 w_{2}\,

T_(disambiguation).html

  1. τ \tau
  2. 𝕋 n \mathbb{T}^{n}
  3. n / n \mathbb{R}^{n}/\mathbb{Z}^{n}
  4. 𝕋 \mathbb{T}

Table_of_polyhedron_dihedral_angles.html

  1. π - arccos ( 1 3 ) \pi-\arccos{\left(\frac{1}{3}\right)}
  2. π - arccos ( 1 3 ) \pi-\arccos{\left(\frac{1}{\sqrt{3}}\right)}
  3. π - arccos ( ( 5 + 2 5 ) 15 ) \pi-\arccos{\left(\sqrt{\frac{(5+2\sqrt{5})}{15}}\right)}

Tach_timer.html

  1. t o t a l r e v o l u t i o n s / ( 2400 * 60 ) total\ revolutions/(2400*60)

Tacit_collusion.html

  1. δ \delta
  2. t = 1 δ t 35 = δ 1 - δ 35 \sum_{t=1}^{\infty}\delta^{t}35=\frac{\delta}{1-\delta}35
  3. 30 < δ 1 - δ 35 δ > 6 13 30<\frac{\delta}{1-\delta}35\Leftrightarrow\delta>\frac{6}{13}

Tacit_programming.html

  1. hom ( A × B , C ) hom ( B , C A ) \hom(A\times B,C)\cong\hom(B,C^{A})

Tacnode.html

  1. ( y - x 2 ) ( y + x 2 ) = 0. (y-x^{2})(y+x^{2})=0.
  2. ( x 2 + y 2 - 3 x ) 2 - 4 x 2 ( 2 - x ) = 0. (x^{2}+y^{2}-3x)^{2}-4x^{2}(2-x)=0.

Tafel_equation.html

  1. Δ V = A × ln ( i i 0 ) \Delta V=A\times\ln\left(\frac{i}{i_{0}}\right)
  2. Δ V \Delta V
  3. A A
  4. i i
  5. i 0 i_{0}
  6. k T e < A \frac{kT}{e}<A
  7. A = k T e α A=\frac{kT}{e\alpha}
  8. k k
  9. T T
  10. e e
  11. α \alpha
  12. i = n F k exp ( ± α F Δ V R T ) i=nFk\exp\left(\pm\alpha F\frac{\Delta V}{RT}\right)
  13. i = i 0 n F R T Δ E i=i_{0}\frac{nF}{RT}\Delta E

Tanh-sinh_quadrature.html

  1. x = tanh ( 1 2 π sinh t ) x=\tanh(\tfrac{1}{2}\pi\sinh t)\,
  2. - 1 1 f ( x ) d x k = - w k f ( x k ) , \int_{-1}^{1}f(x)\,dx\approx\sum_{k=-\infty}^{\infty}w_{k}f(x_{k}),
  3. x k = tanh ( 1 2 π sinh k h ) x_{k}=\tanh(\tfrac{1}{2}\pi\sinh kh)
  4. w k = 1 2 h π cosh k h cosh 2 ( 1 2 π sinh k h ) . w_{k}=\frac{\tfrac{1}{2}h\pi\cosh kh}{\cosh^{2}(\tfrac{1}{2}\pi\sinh kh)}.

Tantalum(IV)_sulfide.html

  1. \overrightarrow{\leftarrow}

Tapestry_(DHT).html

  1. log B N \log_{B}N
  2. c * B * log B N c*B*\log_{B}N

Tarjan's_strongly_connected_components_algorithm.html

  1. O ( | V | + | E | ) O(|V|+|E|)

Tarski–Grothendieck_set_theory.html

  1. A A
  2. { A } \{A\}
  3. F F
  4. A A
  5. F F
  6. F ( x ) F(x)
  7. x x
  8. A A
  9. x x
  10. y y
  11. x x
  12. y y
  13. y y
  14. y y
  15. y y
  16. x y [ x y z y ( 𝒫 ( z ) y 𝒫 ( z ) y ) z 𝒫 ( y ) ( ¬ z y z y ) ] \forall x\exists y[x\in y\wedge\forall z\in y(\mathcal{P}(z)\subseteq y\wedge% \mathcal{P}(z)\in y)\wedge\forall z\in\mathcal{P}(y)(\neg z\approx y\to z\in y)]
  17. 𝒫 ( x ) \mathcal{P}(x)
  18. \approx
  19. x x
  20. { a , b } = { b , a } \{a,b\}=\{b,a\}
  21. { { a , b } , { a } } = ( a , b ) ( b , a ) \{\{a,b\},\{a\}\}=(a,b)\neq(b,a)
  22. Y Y
  23. Y Y

Tate_cohomology_group.html

  1. H ^ n ( G , A ) \hat{H}^{n}(G,A)
  2. H ^ n ( G , A ) = H n ( G , A ) \hat{H}^{n}(G,A)=H^{n}(G,A)
  3. H ^ 0 ( G , A ) = \hat{H}^{0}(G,A)=
  4. H ^ - 1 ( G , A ) = \hat{H}^{-1}(G,A)=
  5. H ^ n ( G , A ) = H - ( n + 1 ) ( G , A ) \hat{H}^{n}(G,A)=H_{-(n+1)}(G,A)
  6. 0 A B C 0 0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow 0
  7. H ^ n ( G , A ) H ^ n ( G , B ) H ^ n ( G , C ) H ^ n + 1 ( G , A ) H ^ n + 1 ( G , B ) \cdots\longrightarrow\hat{H}^{n}(G,A)\longrightarrow\hat{H}^{n}(G,B)% \longrightarrow\hat{H}^{n}(G,C)\longrightarrow\hat{H}^{n+1}(G,A)% \longrightarrow\hat{H}^{n+1}(G,B)\cdots
  8. H ^ n ( G , ) H ^ n + 2 ( G , A ) \hat{H}^{n}(G,\mathbb{Z})\longrightarrow\hat{H}^{n+2}(G,A)
  9. H ^ n ( G , A ) \hat{H}^{n}(G,A)

Tau_Ceti_in_fiction.html

  1. L \begin{smallmatrix}L_{\odot}\end{smallmatrix}

TC0.html

  1. AC 0 AC [ p ] 0 TC 0 NC . 1 \mbox{AC}~{}^{0}\subsetneq\mbox{AC}~{}^{0}[p]\subsetneq\mbox{TC}~{}^{0}% \subseteq\mbox{NC}~{}^{1}.
  2. TC 0 PP \mbox{TC}~{}^{0}\subsetneq\mbox{PP}~{}

Telegrapher's_equations.html

  1. R R
  2. L L
  3. C C
  4. G G
  5. 1 / G 1/G
  6. R R^{\prime}
  7. L L^{\prime}
  8. C C^{\prime}
  9. G G^{\prime}
  10. G ( f ) = G 1 ( f f 1 ) g e G(f)=G_{1}\left(\frac{f}{f_{1}}\right)^{ge}
  11. f 1 = 1 MHz f_{1}=1\;\mathrm{MHz}
  12. G 1 = 29.11 μ S / km = 8.873 μ S / kft G_{1}=29.11\;\mathrm{\mu S/km}=8.873\;\mathrm{\mu S/kft}
  13. g e = 0.87 ge=0.87
  14. f 0.5 f^{0.5}\,
  15. f g e f^{ge}\,
  16. V = V ( x , t ) V=V(x,t)
  17. I = I ( x , t ) I=I(x,t)
  18. V x = - L I t \frac{\partial V}{\partial x}=-L\frac{\partial I}{\partial t}
  19. I x = - C V t \frac{\partial I}{\partial x}=-C\frac{\partial V}{\partial t}
  20. 2 V t 2 - u 2 2 V x 2 = 0 \frac{\partial^{2}V}{{\partial t}^{2}}-u^{2}\frac{\partial^{2}V}{{\partial x}^% {2}}=0
  21. 2 I t 2 - u 2 2 I x 2 = 0 \frac{\partial^{2}I}{{\partial t}^{2}}-u^{2}\frac{\partial^{2}I}{{\partial x}^% {2}}=0
  22. u = 1 L C u=\frac{1}{\sqrt{LC}}
  23. V ( x , t ) = Re { V ( x ) e j ω t } V(x,t)=\mathrm{Re}\{V(x)\cdot e^{j\omega t}\}
  24. I ( x , t ) = Re { I ( x ) e j ω t } I(x,t)=\mathrm{Re}\{I(x)\cdot e^{j\omega t}\}
  25. ω \omega
  26. d V d x = - j ω L I \frac{dV}{dx}=-j\omega LI
  27. d I d x = - j ω C V \frac{dI}{dx}=-j\omega CV
  28. d 2 V d x 2 + k 2 V = 0 \frac{d^{2}V}{dx^{2}}+k^{2}V=0
  29. d 2 I d x 2 + k 2 I = 0 \frac{d^{2}I}{dx^{2}}+k^{2}I=0
  30. k = ω L C = ω u . k=\omega\sqrt{LC}={\omega\over u}.
  31. V ( x ) = V 1 e - j k x + V 2 e + j k x V(x)=V_{1}e^{-jkx}+V_{2}e^{+jkx}
  32. I ( x ) = V 1 Z 0 e - j k x - V 2 Z 0 e + j k x I(x)={V_{1}\over Z_{0}}e^{-jkx}-{V_{2}\over Z_{0}}e^{+jkx}
  33. Z 0 Z_{0}
  34. Z 0 = L C Z_{0}=\sqrt{{L\over C}}
  35. V 1 V_{1}
  36. V 2 V_{2}
  37. V ( x , t ) = f 1 ( x - u t ) + f 2 ( x + u t ) V(x,t)\ =\ f_{1}(x-ut)+f_{2}(x+ut)
  38. f 1 f_{1}
  39. f 2 f_{2}
  40. u = 1 L C u=\frac{1}{\sqrt{LC}}
  41. I ( x , t ) = f 1 ( x - u t ) Z 0 - f 2 ( x + u t ) Z 0 I(x,t)\ =\ \frac{f_{1}(x-ut)}{Z_{0}}-\frac{f_{2}(x+ut)}{Z_{0}}
  42. x V ( x , t ) = - L t I ( x , t ) - R I ( x , t ) \frac{\partial}{\partial x}V(x,t)=-L\frac{\partial}{\partial t}I(x,t)-RI(x,t)
  43. x I ( x , t ) = - C t V ( x , t ) - G V ( x , t ) \frac{\partial}{\partial x}I(x,t)=-C\frac{\partial}{\partial t}V(x,t)-GV(x,t)
  44. 2 x 2 V = L C 2 t 2 V + ( R C + G L ) t V + G R V \frac{\partial^{2}}{{\partial x}^{2}}V=LC\frac{\partial^{2}}{{\partial t}^{2}}% V+(RC+GL)\frac{\partial}{\partial t}V+GRV
  45. 2 x 2 I = L C 2 t 2 I + ( R C + G L ) t I + G R I \frac{\partial^{2}}{{\partial x}^{2}}I=LC\frac{\partial^{2}}{{\partial t}^{2}}% I+(RC+GL)\frac{\partial}{\partial t}I+GRI
  46. V 1 = V 2 cosh ( γ x ) + I 2 Z sinh ( γ x ) V_{1}=V_{2}\cosh(\gamma x)+I_{2}Z\sinh(\gamma x)\,
  47. I 1 = V 2 1 Z sinh ( γ x ) + I 2 cosh ( γ x ) . I_{1}=V_{2}\frac{1}{Z}\sinh(\gamma x)+I_{2}\cosh(\gamma x).\,
  48. E s , E L , I s , I L , l E_{s},E_{L},I_{s},I_{L},l\,
  49. V 1 , V 2 , I 1 , I 2 , x V_{1},V_{2},I_{1},I_{2},x\,
  50. V 1 V_{1}\,
  51. I 1 I_{1}\,
  52. V 2 V_{2}\,
  53. I 2 I_{2}\,
  54. V 1 V_{1}\,
  55. I 1 I_{1}\,
  56. V 2 V_{2}\,
  57. I 2 I_{2}\,
  58. V 1 V_{1}\,
  59. V 2 V_{2}\,
  60. V 1 V_{1}\,
  61. V 2 V_{2}\,
  62. I 1 I_{1}\,
  63. I 2 I_{2}\,
  64. V 1 V_{1}\,
  65. V 2 V_{2}\,

Terminal_and_nonterminal_symbols.html

  1. N N
  2. Σ \Sigma
  3. N N
  4. P P
  5. ( Σ N ) * N ( Σ N ) * ( Σ N ) * (\Sigma\cup N)^{*}N(\Sigma\cup N)^{*}\rightarrow(\Sigma\cup N)^{*}
  6. * {}^{*}
  7. \cup
  8. ( Σ N ) * (\Sigma\cup N)^{*}
  9. N N
  10. Λ \Lambda
  11. e e
  12. ϵ \epsilon
  13. S N S\in N
  14. < N , Σ , P , S Align g t ; <N,\Sigma,P,S&gt;

Terminating_Reliable_Broadcast.html

  1. SF \mathrm{SF}
  2. SF \mathrm{SF}
  3. m m
  4. m m
  5. m SF m\neq\mathrm{SF}
  6. m m
  7. m m
  8. m m
  9. SF \mathrm{SF}

Tesla_Roadster.html

  1. W · h / g a l {W·h}/{gal}
  2. 33705 W h gal ge 135 W h km × 1.6 km mi × 77.6 % = charging eff . 120 mpg ge = 1.95 L ge 100 km \frac{33705\,\frac{\mathrm{W\cdot h}}{\mathrm{gal_{ge}}}}{135\,\frac{\mathrm{W% \cdot h}}{\mathrm{km}}\times\frac{1.6\,\mathrm{km}}{\mathrm{mi}}}\times 77.6\%% {\mathrm{{}_{charging\ eff.}}}=120\,\mathrm{mpg_{ge}}=1.95\frac{\mathrm{L_{ge}% }}{100\,\mathrm{km}}
  3. W · h / g a l {W·h}/{gal}
  4. W · h / g a l < s u b > U S {W·h}/{gal<sub>US}

Test_probe.html

  1. R s R_{s}
  2. R R
  3. N N
  4. R s = R / N 2 R_{s}=R/N^{2}

Tetradecagon.html

  1. A = 14 4 a 2 cot π 14 15.3345 a 2 A=\frac{14}{4}a^{2}\cot\frac{\pi}{14}\simeq 15.3345a^{2}

Tetralemma.html

  1. X X
  2. ¬ X \neg X
  3. X ¬ X X\land\neg X
  4. ¬ ( X ¬ X ) \neg(X\lor\neg X)

The_Coal_Question.html

  1. 0 110 82.17 ( 1.035 ) t d t \int_{0}^{110}82.17(1.035)^{t}\,dt
  2. 36 105 - 1 = 3.4 % \sqrt[105]{36}-1=3.4\%
  3. 2.1 34 - 1 = 2.2 % \sqrt[34]{2.1}-1=2.2\%

Theorem_of_corresponding_states.html

  1. Z c = p c V c n c k B T c Z_{c}=\frac{p_{c}V_{c}}{n_{c}k_{B}T_{c}}
  2. c c
  3. 3 / 8 = 0.375 3/8=0.375

Theoretical_and_experimental_justification_for_the_Schrödinger_equation.html

  1. 2 𝐄 - 1 c 2 2 𝐄 t 2 = 0 \nabla^{2}\mathbf{E}\ -\ {1\over c^{2}}{\partial^{2}\mathbf{E}\over\partial t^% {2}}\ \ =\ \ 0
  2. 2 𝐁 - 1 c 2 2 𝐁 t 2 = 0 \nabla^{2}\mathbf{B}\ -\ {1\over c^{2}}{\partial^{2}\mathbf{B}\over\partial t^% {2}}\ \ =\ \ 0
  3. × 𝐄 = - 1 c 𝐁 t \nabla\times\mathbf{E}=-{1\over c}\frac{\partial\mathbf{B}}{\partial t}
  4. 𝐄 ( 𝐫 , t ) = 𝐄 Re { | ζ exp [ i ( k z - ω t ) ] } 𝐄 Re { | ϕ } \mathbf{E}(\mathbf{r},t)=\mid\mathbf{E}\mid\mathrm{Re}\left\{|\zeta\rangle\exp% \left[i\left(kz-\omega t\right)\right]\right\}\equiv\mid\mathbf{E}\mid\mathrm{% Re}\left\{|\phi\rangle\right\}
  5. 𝐁 ( 𝐫 , t ) = 𝐳 ^ × 𝐄 ( 𝐫 , t ) \mathbf{B}(\mathbf{r},t)=\hat{\mathbf{z}}\times\mathbf{E}(\mathbf{r},t)
  6. ω = c k \omega=ck
  7. c c
  8. 𝐄 \mid\mathbf{E}\mid
  9. | ζ ( ζ x ζ y ) = ( cos θ exp ( i α x ) sin θ exp ( i α y ) ) |\zeta\rangle\equiv\begin{pmatrix}\zeta_{x}\\ \zeta_{y}\end{pmatrix}=\begin{pmatrix}\cos\theta\exp\left(i\alpha_{x}\right)\\ \sin\theta\exp\left(i\alpha_{y}\right)\end{pmatrix}
  10. θ , α x , and α y \theta,\;\alpha_{x},\;\mbox{and}~{}\;\alpha_{y}
  11. | ϕ = exp [ i ( k z - ω t ) ] | ζ |\phi\rangle=\exp\left[i\left(kz-\omega t\right)\right]|\zeta\rangle
  12. c = 1 8 π [ 𝐄 2 ( 𝐫 , t ) + 𝐁 2 ( 𝐫 , t ) ] \mathcal{E}_{c}=\frac{1}{8\pi}\left[\mathbf{E}^{2}(\mathbf{r},t)+\mathbf{B}^{2% }(\mathbf{r},t)\right]
  13. c = 𝐄 2 8 π \mathcal{E}_{c}=\frac{\mid\mathbf{E}\mid^{2}}{8\pi}
  14. f x = 𝐄 2 cos 2 θ 𝐄 2 = ϕ x * ϕ x f_{x}=\frac{\mid\mathbf{E}\mid^{2}\cos^{2}\theta}{\mid\mathbf{E}\mid^{2}}=\phi% _{x}^{*}\phi_{x}
  15. ϕ x * ϕ x + ϕ y * ϕ y = ϕ | ϕ = 1 \phi_{x}^{*}\phi_{x}+\phi_{y}^{*}\phi_{y}=\langle\phi|\phi\rangle=1
  16. s y m b o l 𝒫 = 1 4 π c 𝐄 ( 𝐫 , t ) × 𝐁 ( 𝐫 , t ) symbol{\mathcal{P}}={1\over 4\pi c}\mathbf{E}(\mathbf{r},t)\times\mathbf{B}(% \mathbf{r},t)
  17. 𝒫 c = c \mathcal{P}c=\mathcal{E}_{c}
  18. s y m b o l = 𝐫 × s y m b o l 𝒫 = 1 4 π c 𝐫 × [ 𝐄 ( 𝐫 , t ) × 𝐁 ( 𝐫 , t ) ] symbol{\mathcal{L}}=\mathbf{r}\times symbol{\mathcal{P}}={1\over 4\pi c}% \mathbf{r}\times\left[\mathbf{E}(\mathbf{r},t)\times\mathbf{B}(\mathbf{r},t)\right]
  19. = 𝐄 2 8 π ω ( R | ϕ 2 - L | ϕ 2 ) = 1 ω c ( ϕ R 2 - ϕ L 2 ) \mathcal{L}={{\mid\mathbf{E}\mid^{2}}\over{8\pi\omega}}\left(\mid\langle R|% \phi\rangle\mid^{2}-\mid\langle L|\phi\rangle\mid^{2}\right)={1\over\omega}% \mathcal{E}_{c}\left(\mid\phi_{R}\mid^{2}-\mid\phi_{L}\mid^{2}\right)
  20. | R 1 2 ( 1 i ) |R\rangle\equiv{1\over\sqrt{2}}\begin{pmatrix}1\\ i\end{pmatrix}
  21. | L 1 2 ( 1 - i ) |L\rangle\equiv{1\over\sqrt{2}}\begin{pmatrix}1\\ -i\end{pmatrix}
  22. ( t + τ ) (t+\tau)
  23. | ϕ ( t + τ ) = U ^ ( τ ) | ϕ ( t ) |\phi(t+\tau)\rangle=\hat{U}(\tau)|\phi(t)\rangle
  24. ϕ ( t + τ ) | ϕ ( t + τ ) = ϕ ( t ) | U ^ ( τ ) U ^ ( τ ) | ϕ ( t ) = ϕ ( t ) | ϕ ( t ) = 1 \langle\phi(t+\tau)|\phi(t+\tau)\rangle=\langle\phi(t)|\hat{U}^{\dagger}(\tau)% \hat{U}(\tau)|\phi(t)\rangle=\langle\phi(t)|\phi(t)\rangle=1
  25. U U^{\dagger}
  26. U ^ U ^ = I \hat{U}^{\dagger}\hat{U}=I
  27. τ \tau
  28. d t dt
  29. U ^ I - i H ^ τ \hat{U}\approx I-i\hat{H}\tau
  30. U ^ I + i H ^ τ \hat{U}^{\dagger}\approx I+i\hat{H}^{\dagger}\tau
  31. I = U ^ U ^ ( I + i H ^ τ ) ( I - i H ^ τ ) I + i H ^ τ - i H ^ τ + H ^ H ^ τ 2 I=\hat{U}^{\dagger}\hat{U}\approx\left(I+i\hat{H}^{\dagger}\tau\right)\left(I-% i\hat{H}\tau\right)\approx I+i\hat{H}^{\dagger}\tau-i\hat{H}\tau+\hat{H}^{% \dagger}\hat{H}\tau^{2}
  32. τ \tau
  33. τ 2 \tau^{2}
  34. τ \tau
  35. H ^ = H ^ \hat{H}=\hat{H}^{\dagger}
  36. τ = d t \tau=dt\,
  37. U ^ U ^ = I \hat{U}^{\dagger}\hat{U}=I
  38. | ϕ ( t + d t ) - | ϕ ( t ) = - i H ^ d t | ϕ ( t ) |\phi(t+dt)\rangle-|\phi(t)\rangle=-i\hat{H}dt|\phi(t)\rangle
  39. ϕ \mid\phi\rangle
  40. ϵ = ω \epsilon=\hbar\omega
  41. \hbar
  42. N N
  43. V V
  44. N ω N\hbar\omega
  45. N ω V {N\hbar\omega\over V}
  46. N N
  47. N ω V = c = 𝐄 2 8 π {N\hbar\omega\over V}=\mathcal{E}_{c}=\frac{\mid\mathbf{E}\mid^{2}}{8\pi}
  48. N = V 8 π ω 𝐄 2 N=\frac{V}{8\pi\hbar\omega}\mid\mathbf{E}\mid^{2}
  49. 𝒫 c = N ω c V = N k V \mathcal{P}_{c}={N\hbar\omega\over cV}={N\hbar k\over V}
  50. k \hbar k
  51. h λ h\over\lambda
  52. = 1 ω c ( ψ R 2 - ψ L 2 ) = V ( ψ R 2 - ψ L 2 ) \mathcal{L}={1\over\omega}\mathcal{E}_{c}\left(\mid\psi_{R}\mid^{2}-\mid\psi_{% L}\mid^{2}\right)={\hbar\over V}\left(\mid\psi_{R}\mid^{2}-\mid\psi_{L}\mid^{2% }\right)
  53. l z = ( ψ R 2 - ψ L 2 ) l_{z}=\hbar\left(\mid\psi_{R}\mid^{2}-\mid\psi_{L}\mid^{2}\right)
  54. ψ R 2 \mid\psi_{R}\mid^{2}
  55. \hbar
  56. ψ L 2 \mid\psi_{L}\mid^{2}
  57. - -\hbar
  58. ± \pm\hbar
  59. \hbar
  60. | R |R\rangle
  61. | L |L\rangle
  62. S ^ | R R | - | L L | = ( 0 - i i 0 ) \hat{S}\equiv|R\rangle\langle R|-|L\rangle\langle L|=\begin{pmatrix}0&-i\\ i&0\end{pmatrix}
  63. | R |R\rangle
  64. | L |L\rangle
  65. ψ | S ^ | ψ = ψ R 2 - ψ L 2 \langle\psi|\hat{S}|\psi\rangle=\mid\psi_{R}\mid^{2}-\mid\psi_{L}\mid^{2}
  66. ω \omega
  67. k k
  68. E = ω E=\hbar\omega
  69. p = k p=\hbar k
  70. E 0 = m c 2 = ω 0 E_{0}=mc^{2}=\hbar\omega_{0}
  71. p 0 = 0 = k 0 p_{0}=0=\hbar k_{0}
  72. v ϕ = ω 0 k 0 v_{\phi}={\omega_{0}\over k_{0}}
  73. cos ( ω 0 t ) \cos(\omega_{0}t)
  74. ω c = ω 0 γ \omega_{c}={\omega_{0}\over\gamma}
  75. γ = 1 1 - v 2 c 2 \gamma={1\over\sqrt{1-{v^{2}\over c^{2}}}}
  76. ω c t = ω 0 γ ( γ t 0 ) = ω 0 t 0 \omega_{c}t={\omega_{0}\over\gamma}(\gamma t_{0})=\omega_{0}t_{0}
  77. t 0 t_{0}
  78. E = γ m c 2 = ω = γ ω 0 E=\gamma mc^{2}=\hbar\omega=\gamma\hbar\omega_{0}
  79. p = γ m v = k p=\gamma mv=\hbar k
  80. v ϕ = ω k = E p = c 2 v v_{\phi}={\omega\over k}={E\over p}={c^{2}\over v}
  81. ω t - k z = ω t - ω v ϕ v t = ω t ( 1 - v 2 c 2 ) = ω t γ 2 = 1 γ 2 γ m c 2 ( γ t 0 ) = ω 0 t 0 = ω c t \omega t-kz=\omega t-{\omega\over v_{\phi}}vt=\omega t\left(1-{v^{2}\over c^{2% }}\right)={\omega t\over\gamma^{2}}={1\over\gamma^{2}}{\gamma mc^{2}\over\hbar% }(\gamma t_{0})=\omega_{0}t_{0}=\omega_{c}t
  82. ω n = R ( 1 2 2 - 1 n 2 ) n = 3 , 4 , 5 , \hbar\omega_{n}=R\left(\frac{1}{2^{2}}-\frac{1}{n^{2}}\right)\quad n=3,4,5,...
  83. L = n {L}=n\hbar
  84. m v 2 r = e M 2 r 2 {mv^{2}\over r}={{e_{M}}^{2}\over r^{2}}
  85. e M = e 4 π ϵ 0 {e_{M}}={e\over{4\pi\epsilon_{0}}}
  86. E = 1 2 m v 2 - e M 2 r = - 1 2 e M 2 r E={1\over 2}mv^{2}-{{e_{M}}^{2}\over r}=-{1\over 2}{{e_{M}}^{2}\over r}
  87. L = m v r = n L=mvr=n\hbar
  88. r = n 2 2 m e M 2 r={n^{2}\hbar^{2}\over m{e_{M}}^{2}}
  89. E n = - 1 2 e M 2 r = - 1 2 ( m e M 4 2 ) 1 n 2 E_{n}=-{1\over 2}{{e_{M}}^{2}\over r}=-{1\over 2}\left({m{e_{M}}^{4}\over\hbar% ^{2}}\right){1\over n^{2}}
  90. p = m v = k p=mv=\hbar k
  91. L = m v r = k r = ( 2 π λ ) r L=mvr=\hbar kr=\hbar\left({2\pi\over\lambda}\right)r
  92. λ \lambda
  93. λ = 2 π r n \lambda={2\pi r\over n}
  94. L = n L=n\hbar
  95. | ϕ ( t + d t ) - | ϕ ( t ) = - i H ^ d t | ϕ ( t ) |\phi(t+dt)\rangle-|\phi(t)\rangle=-i\hat{H}dt|\phi(t)\rangle
  96. | ψ ( t + d t ) - | ψ ( t ) = - i H ^ d t | ψ ( t ) |\psi(t+dt)\rangle-|\psi(t)\rangle=-i\hat{H}dt|\psi(t)\rangle
  97. | ψ ( t ) |\psi(t)\rangle
  98. λ \lambda
  99. | ψ ( t ) ( ψ j - 1 ( t ) ψ j ( t ) ψ j + 1 ( t ) ) |\psi(t)\rangle\equiv\begin{pmatrix}\vdots\\ \psi_{j-1}(t)\\ \psi_{j}(t)\\ \psi_{j+1}(t)\\ \vdots\end{pmatrix}
  100. ψ j ( t + d t ) - ψ j ( t ) = - i k H j k d t ψ k ( t ) \psi_{j}(t+dt)-\psi_{j}(t)=-i\sum_{k}H_{jk}\,dt\,\psi_{k}(t)
  101. H j k = H k j * H_{jk}=H^{*}_{kj}
  102. H j ± 1 , j 0 H_{j\pm 1,j}\neq 0
  103. H j , j 0 H_{j,j}\neq 0
  104. H j + 1 , j = H j , j - 1 H R H_{j+1,j}=H_{j,j-1}\equiv H_{R}
  105. H j - 1 , j = H j , j + 1 H L H_{j-1,j}=H_{j,j+1}\equiv H_{L}
  106. i ψ j ( t ) t = H L ψ j + 1 ( t ) - H R ψ j ( t ) + H R ψ j - 1 ( t ) - H L ψ j ( t ) + H j j ψ j ( t ) i{\partial\psi_{j}(t)\over\partial t}=H_{L}\psi_{j+1}(t)-H_{R}\psi_{j}(t)+H_{R% }\psi_{j-1}(t)-H_{L}\psi_{j}(t)+H_{jj}\psi_{j}(t)
  107. λ \lambda
  108. H R = H L H 0 H_{R}=H_{L}\equiv H_{0}
  109. i ψ j ( t ) t = H 0 λ 2 2 ψ j ( t ) x 2 + H j j ψ j ( t ) i{\partial\psi_{j}(t)\over\partial t}=H_{0}{\lambda^{2}}{\partial^{2}\psi_{j}(% t)\over\partial x^{2}}+H_{jj}\psi_{j}(t)
  110. exp [ i ( k x - ω t ) ] \exp\left[i\left(kx-\omega t\right)\right]
  111. E = ω = p 2 2 m = 2 k 2 2 m E=\hbar\omega={p^{2}\over 2m}={\hbar^{2}k^{2}\over 2m}
  112. H 0 λ 2 = - 2 m H_{0}\lambda^{2}=-{\hbar\over 2m}
  113. H j j = 0 H_{jj}=0
  114. H j j H_{jj}
  115. i ψ ( x , t ) t = - 2 2 m 2 ψ ( x , t ) x 2 + U ( x ) ψ ( x , t ) i\hbar{\partial\psi(x,t)\over\partial t}=-\frac{\hbar^{2}}{2m}\frac{\partial^{% 2}\psi(x,t)}{\partial x^{2}}+U(x)\psi(x,t)
  116. ψ ( x , t ) 1 λ ψ j ( t ) \psi(x,t)\equiv{1\over\sqrt{\lambda}}\psi_{j}(t)
  117. 1 = - ψ * ( x , t ) ψ ( x , t ) d x 1=\int_{-\infty}^{\infty}\psi^{*}(x,t)\psi(x,t)dx
  118. - 2 2 m 2 ψ + U ψ = i t ψ -\frac{\hbar^{2}}{2m}{\nabla^{2}\psi}+U\psi=i\hbar{\partial\over\partial t}\psi

Theoretical_plate.html

  1. N a = N t E N_{a}=\frac{N_{t}}{E}
  2. N a N_{a}
  3. N t N_{t}
  4. E E
  5. N t = H H E T P N_{t}=\frac{H}{HETP}
  6. N t N_{t}
  7. H H
  8. H E T P HETP
  9. N t = H H E T P N_{t}=\frac{H}{HETP}

Theory_of_everything_(philosophy).html

  1. t t ( t E t ) \forall t\,\exists t^{\prime}\,(t^{\prime}\ E\ t)
  2. t ( T * E t ) \forall t\,(T^{*}\ E\ t)
  3. t ( ( t E T * ) ( t = T * ) ) \forall t\,((t\ E\ T^{*})\to(t=T^{*}))
  4. t ( t E t ) \nexists t\,(t\ E\ t)

Thermal_contact_conductance.html

  1. h c h_{c}
  2. q = - k A d T d x q=-kA\frac{dT}{dx}
  3. q q
  4. k k
  5. A A
  6. d T / d x dT/dx
  7. q = T 1 - T 3 Δ x A / ( k A A ) + 1 / ( h c A ) + Δ x B / ( k B A ) q=\frac{T_{1}-T_{3}}{\Delta x_{A}/(k_{A}A)+1/(h_{c}A)+\Delta x_{B}/(k_{B}A)}
  8. k A k_{A}
  9. k B k_{B}
  10. A A
  11. 1 / h c 1/h_{c}
  12. h c h_{c}
  13. σ \sigma
  14. A A

Thermal_death_time.html

  1. L = 10 ( T - T Ref ) / z L=10^{(T-T_{\mathrm{Ref}})/z}
  2. j = j I I j={jI\over I}
  3. log g = log j I - B B f h \log g=\log jI-{B_{B}\over f_{h}}
  4. z = T 2 - T 1 log D 1 - log D 2 z=\frac{T_{2}-T_{1}}{\log D_{1}-\log D_{2}}

Thermodynamic_databases_for_pure_substances.html

  1. C P ( T ) = { lim Δ T 0 Δ H Δ T } = ( H T ) p C_{P}(T)=\left\{\lim_{\Delta T\to 0}\frac{\Delta H}{\Delta T}\right\}=\left(% \frac{\partial H}{\partial T}\right)_{p}
  2. γ \gamma
  3. P b ( c , l ) + 2 H C l ( g ) P b C l 2 + H 2 ( g ) Pb(c,l)+2HCl(g)\Rightarrow PbCl_{2}+H_{2}(g)
  4. H ( T ) = Δ H f o r m , 298 + [ H T - H 298 ] H(T)=\Delta H^{\circ}_{form,298}+[H_{T}-H_{298}]
  5. Δ H f o r m = H ( T ) c o m p o u n d - { H ( T ) e l e m e n t s } \Delta H^{\circ}_{form}=H(T)compound-\sum\left\{H(T)elements\right\}
  6. G ( T ) = H ( T ) - T × S ( T ) G(T)=H(T)-T\times S(T)
  7. Δ G f o r m = G ( T ) c o m p o u n d - { G ( T ) e l e m e n t s } \Delta G^{\circ}_{form}=G(T)compound-\sum\left\{G(T)elements\right\}
  8. ( H 298 - G T ) / T = S T - ( H T - H 298 ) / T (H^{\circ}_{298}-G^{\circ}_{T})/T=S^{\circ}_{T}-(H_{T}-H_{298})/T
  9. log 10 ( K e q ) = - Δ G f o r m / ( 19.1448 T ) \log_{10}\left(K_{eq}\right)=-\Delta G^{\circ}_{form}/(19.1448T)
  10. H T - H 298 = A ( T ) + B ( T 2 ) + C ( T - 1 ) + D ( T 0.5 ) + E ( T 3 ) + F H_{T}-H_{298}=A(T)+B(T^{2})+C(T^{-1})+D(T^{0.5})+E(T^{3})+F\,
  11. C P = A + 2 B ( T ) - C ( T - 2 ) + 1 2 D ( T - 0.5 ) + 3 E ( T 2 ) C_{P}=A+2B(T)-C(T^{-2})+\textstyle\frac{1}{2}D(T^{-0.5})+3E(T^{2})\,
  12. S T = A ( ln T ) + 2 B ( T ) + 1 2 C ( T - 2 ) - D ( T - 1 2 ) + 1 1 2 E ( T 2 ) + F S^{\circ}_{T}=A(\ln T)+2B(T)+\textstyle\frac{1}{2}C(T^{-2})-D(T^{\textstyle-% \frac{1}{2}})+1\textstyle\frac{1}{2}E(T^{2})+F^{\prime}
  13. Δ G f o r m = ( Δ A - Δ F ) T - Δ A ( T ln T ) - Δ B ( T 2 ) + 1 2 Δ C ( T - 1 ) + 2 Δ D ( T 1 2 ) \Delta G^{\circ}_{form}=(\Delta A-\Delta F^{\prime})T-\Delta A(T\ln T)-\Delta B% (T^{2})+\textstyle\frac{1}{2}\Delta C(T^{-1})+2\Delta D(T^{\textstyle\frac{1}{% 2}})
  14. - 1 2 Δ E ( T 3 ) + Δ F + Δ H f o r m 298 -\textstyle\frac{1}{2}\Delta E(T^{3})+\Delta F+\Delta H^{\circ}_{form298}
  15. Δ G f o r m = α T + β ( T ln T ) + χ \Delta G^{\circ}_{form}=\alpha T+\beta(T\ln T)+\chi

Thermophoresis.html

  1. χ t = ( D χ + D T χ ( 1 - χ ) T ) \frac{\partial\chi}{\partial t}=\nabla\cdot(D\,\nabla\chi+D_{T}\,\chi(1-\chi)% \,\nabla T)
  2. D D
  3. D T D_{T}
  4. S T = D T D S_{T}=\frac{D_{T}}{D}

Thin_plate_spline.html

  1. z z
  2. x x
  3. y y
  4. K K
  5. 2 ( K + 3 ) 2(K+3)
  6. 2 K 2K
  7. x x
  8. f ( x ) f(x)
  9. { y i } \{y_{i}\}
  10. { x i } \{x_{i}\}
  11. E t p s ( f ) = i = 1 K y i - f ( x i ) 2 E_{tps}(f)=\sum_{i=1}^{K}\|y_{i}-f(x_{i})\|^{2}
  12. λ \lambda
  13. E t p s , s m o o t h ( f ) = i = 1 K y i - f ( x i ) 2 + λ [ ( 2 f x 1 2 ) 2 + 2 ( 2 f x 1 x 2 ) 2 + ( 2 f x 2 2 ) 2 ] d x 1 d x 2 E_{tps,smooth}(f)=\sum_{i=1}^{K}\|y_{i}-f(x_{i})\|^{2}+\lambda\iint\left[\left% (\frac{\partial^{2}f}{\partial x_{1}^{2}}\right)^{2}+2\left(\frac{\partial^{2}% f}{\partial x_{1}\partial x_{2}}\right)^{2}+\left(\frac{\partial^{2}f}{% \partial x_{2}^{2}}\right)^{2}\right]\textrm{d}x_{1}\,\textrm{d}x_{2}
  14. f f
  15. { w i , i = 1 , 2 , , K } \{w_{i},i=1,2,\ldots,K\}
  16. x x
  17. f ( x ) f(x)
  18. f ( x ) = i = 1 K c i φ ( x - w i ) f(x)=\sum_{i=1}^{K}c_{i}\varphi(\left\|x-w_{i}\right\|)
  19. \left\|\cdot\right\|
  20. { c i } \{c_{i}\}
  21. φ ( r ) = r 2 log r \varphi(r)=r^{2}\log r
  22. D = 2 D=2
  23. y i y_{i}
  24. ( 1 , y i x , y i y ) (1,y_{ix},y_{iy})
  25. f f
  26. α \alpha
  27. d d
  28. c c
  29. α = { d , c } \alpha=\{d,c\}
  30. f t p s ( z , α ) = f t p s ( z , d , c ) = z d + i = 1 K ϕ ( z - x i ) c i f_{tps}(z,\alpha)=f_{tps}(z,d,c)=z\cdot d+\sum_{i=1}^{K}\phi(\|z-x_{i}\|)\cdot c% _{i}
  31. ( D + 1 ) × ( D + 1 ) (D+1)\times(D+1)
  32. z z
  33. 1 × ( D + 1 ) 1\times(D+1)
  34. K × ( D + 1 ) K\times(D+1)
  35. ϕ ( z ) \phi(z)
  36. 1 × K 1\times K
  37. z z
  38. ϕ i ( z ) = z - x i 2 log z - x i \phi_{i}(z)=\|z-x_{i}\|^{2}\log\|z-x_{i}\|
  39. D D
  40. { w i } \{w_{i}\}
  41. { x i } \{x_{i}\}
  42. { x i } \{x_{i}\}
  43. f f
  44. E t p s E_{tps}
  45. E t p s ( d , c ) = Y - X d - Φ c 2 + λ c T Φ c E_{tps}(d,c)=\|Y-Xd-\Phi c\|^{2}+\lambda c^{T}\Phi c
  46. Y Y
  47. X X
  48. y i y_{i}
  49. x i x_{i}
  50. Φ \Phi
  51. ( K × K ) (K\times K)
  52. ϕ ( x i - x j ) \phi(\|x_{i}-x_{j}\|)
  53. Φ \Phi
  54. c c
  55. X = [ Q 1 | Q 2 ] ( R 0 ) X=[Q_{1}|Q_{2}]\left(\begin{array}[]{cc}R\\ 0\end{array}\right)
  56. K × ( D + 1 ) K\times(D+1)
  57. K × ( K - D - 1 ) K\times(K-D-1)
  58. R R
  59. E t p s ( γ , d ) = Q 2 T Y - Q 2 T Φ Q 2 γ 2 + Q 1 T Y - R d - Q 1 T Φ Q 2 γ 2 + λ trace ( γ T Q 2 T Φ Q 2 γ ) E_{tps}(\gamma,d)=\|Q_{2}^{T}Y-Q_{2}^{T}\Phi Q_{2}\gamma\|^{2}+\|Q_{1}^{T}Y-Rd% -Q_{1}^{T}\Phi Q_{2}\gamma\|^{2}+\lambda\textrm{trace}(\gamma^{T}Q_{2}^{T}\Phi Q% _{2}\gamma)
  60. γ \gamma
  61. ( K - D - 1 ) × ( D + 1 ) (K-D-1)\times(D+1)
  62. c = Q 2 γ c=Q_{2}\gamma
  63. X T c = 0 X^{T}c=0
  64. γ \gamma
  65. d d
  66. c ^ = Q 2 ( Q 2 T Φ Q 2 + λ I ( k - D - 1 ) ) - 1 Q 2 T Y \hat{c}=Q_{2}(Q_{2}^{T}\Phi Q_{2}+\lambda I_{(k-D-1)})^{-1}Q_{2}^{T}Y
  67. d ^ = R - 1 Q 1 T ( Y - Φ c ^ ) \hat{d}=R^{-1}Q_{1}^{T}(Y-\Phi\hat{c})
  68. ( c ^ , d ^ ) (\hat{c},\hat{d})
  69. E b e n d i n g = λ trace [ Q 2 ( Q 2 T Φ Q 2 + λ I ( k - D - 1 ) ) - 1 Q 2 T Y Y T ] E_{bending}=\lambda\,\textrm{trace}[Q_{2}(Q_{2}^{T}\Phi Q_{2}+\lambda I_{(k-D-% 1)})^{-1}Q_{2}^{T}YY^{T}]

Three-dimensional_graph.html

  1. f ( x , y ) = sin x 2 cos y 2 f(x,y)=\sin{x^{2}}\cdot\cos{y^{2}}
  2. { ( x , y , sin x 2 cos y 2 ) : x , y } \{(x,y,\sin{x^{2}}\cdot\cos{y^{2}}):x,y\in\mathbb{R}\}
  3. x 2 + y 2 + z 2 = r 2 x^{2}+y^{2}+z^{2}=r^{2}

Three-phase_traffic_theory.html

  1. q q
  2. k k
  3. q m a x q_{max}
  4. k c r i t k_{crit}
  5. v f r e e m i n v^{min}_{free}
  6. v min free = q max k crit v^{\min}\text{free}=\frac{q_{\max}}{k\text{crit}}
  7. v g v_{g}
  8. G G
  9. g safe g_{\rm safe}
  10. g g
  11. G G
  12. g > G g>G
  13. g safe g_{\rm safe}
  14. g < g safe g<g_{\rm safe}
  15. g > G g>G
  16. g < g safe g<g_{\rm safe}
  17. g safe g G g_{\rm safe}\leq g\leq G
  18. g safe g_{\rm safe}
  19. g g
  20. g safe g G g_{\rm safe}\leq g\leq G
  21. C m i n C_{min}
  22. C m a x C_{max}
  23. C m a x C_{max}
  24. C m a x C_{max}
  25. C m i n C_{min}
  26. C m i n C_{min}
  27. q q
  28. C m i n q < C m a x . C_{min}\leq q<C_{max}.
  29. C m i n C_{min}
  30. C m a x C_{max}
  31. C m i n C_{min}
  32. C m a x C_{max}
  33. v g v_{g}
  34. v g v_{g}
  35. q o u t q_{out}
  36. v g v_{g}
  37. C m i n C_{min}
  38. q o u t q_{out}
  39. C m i n C_{min}
  40. q o u t q_{out}
  41. q i n q_{in}
  42. q o u t q_{out}
  43. C m i n C_{min}
  44. q o u t q_{out}
  45. B 1 B_{1}
  46. B 2 B_{2}
  47. B 3 B_{3}
  48. C m i n C_{min}

Three-two_pull_down.html

  1. 23.976 29.97 = 4 5 \frac{23.976}{29.97}=\frac{4}{5}

Threshold_cryptosystem.html

  1. n n

Threshold_energy.html

  1. p + p p + p + π 0 p+p\to p+p+\pi^{0}
  2. E = 2 γ m p c 2 = 2 m p c 2 + m π c 2 E=2\gamma m_{p}c^{2}=2m_{p}c^{2}+m_{\pi}c^{2}
  3. γ = 1 1 - β 2 = 2 m p c 2 + m π c 2 2 m p c 2 \gamma=\frac{1}{\sqrt{1-\beta^{2}}}=\frac{2m_{p}c^{2}+m_{\pi}c^{2}}{2m_{p}c^{2}}
  4. β 2 = 1 - ( 2 m p c 2 2 m p c 2 + m π c 2 ) 2 0.13 \beta^{2}=1-(\frac{2m_{p}c^{2}}{2m_{p}c^{2}+m_{\pi}c^{2}})^{2}\approx 0.13
  5. v lab = u cm + v cm 1 + u cm v cm / c 2 0.64 c v\text{lab}=\frac{u\text{cm}+v\text{cm}}{1+u\text{cm}v\text{cm}/c^{2}}\approx 0% .64c
  6. E = γ m p c 2 = m p c 2 1 - ( v lab / c ) 2 = 1221 E=\gamma m_{p}c^{2}=\frac{m_{p}c^{2}}{\sqrt{1-(v\text{lab}/c)^{2}}}=1221\,
  7. E 1 E_{1}
  8. p 1 p_{1}
  9. m 1 m_{1}
  10. E 2 = m 2 E_{2}=m_{2}
  11. E 1 , thr E_{1,\,\text{thr}}
  12. m a m_{a}
  13. m b m_{b}
  14. m c m_{c}
  15. 1 + 2 a + b + c , 1+2\to a+b+c,
  16. E cm = m a c 2 + m b c 2 + m c c 2 = E ^ 1 + E ^ 2 = γ ( E 1 - β p 1 c ) + γ m 2 c 2 E\text{cm}=m_{a}c^{2}+m_{b}c^{2}+m_{c}c^{2}=\hat{E}_{1}+\hat{E}_{2}=\gamma(E_{% 1}-\beta p_{1}c)+\gamma m_{2}c^{2}
  17. E cm E\text{cm}
  18. γ = E 1 + m 2 c 2 E cm \gamma=\frac{E_{1}+m_{2}c^{2}}{E\text{cm}}
  19. β = p 1 c E 1 + m 2 c 2 \beta=\frac{p_{1}c}{E_{1}+m_{2}c^{2}}
  20. p 1 2 c 2 = E 1 2 - m 1 2 c 4 p_{1}^{2}c^{2}=E_{1}^{2}-m_{1}^{2}c^{4}
  21. E 1 , thr = ( m a c 2 + m b c 2 + m c c 2 ) 2 - m 1 2 c 4 - m 2 2 c 4 2 m 2 c 2 E_{1,\,\text{thr}}=\frac{(m_{a}c^{2}+m_{b}c^{2}+m_{c}c^{2})^{2}-m_{1}^{2}c^{4}% -m_{2}^{2}c^{4}}{2m_{2}c^{2}}

Thue_number.html

  1. Σ 2 P \Sigma_{2}^{P}

Thyroxine_5-deiodinase.html

  1. \rightleftharpoons

TI-36.html

  1. π \pi
  2. π \pi

Tidal_heating.html

  1. q t i d q_{tid}
  2. q t i d = 63 q_{tid}=63
  3. ρ \rho
  4. n 5 n^{5}
  5. r 4 r^{4}
  6. e 2 / ( 38 e^{2}/(38
  7. μ \mu
  8. Q ) Q)
  9. ρ \rho
  10. n n
  11. r r
  12. e e
  13. μ \mu
  14. Q Q

Tight_closure.html

  1. R R
  2. p > 0 p>0
  3. p p
  4. I I
  5. R R
  6. I I
  7. I * I^{*}
  8. R R
  9. I I
  10. I * I^{*}
  11. z I * z\in I^{*}
  12. c R c\in R
  13. c c
  14. R R
  15. c z p e I [ p e ] cz^{p^{e}}\in I^{[p^{e}]}
  16. e 0 e\gg 0
  17. R R
  18. e > 0 e>0
  19. I [ p e ] I^{[p^{e}]}
  20. R R
  21. p e p^{e}
  22. I I
  23. e e
  24. I I
  25. I = I * I=I^{*}
  26. F F
  27. F F
  28. F F
  29. F F

Tightness_of_measures.html

  1. | μ | ( X K ε ) < ε . |\mu|(X\setminus K_{\varepsilon})<\varepsilon.
  2. | μ | |\mu|
  3. μ \mu
  4. μ ( K ε ) > 1 - ε . \mu(K_{\varepsilon})>1-\varepsilon.\,
  5. [ 0 , ω 1 ] [0,\omega_{1}]
  6. μ \mu
  7. { μ } \{\mu\}
  8. M 1 := { δ n | n } M_{1}:=\{\delta_{n}|n\in\mathbb{N}\}
  9. M 2 := { δ 1 / n | n } M_{2}:=\{\delta_{1/n}|n\in\mathbb{N}\}
  10. Γ = { γ i | i I } , \Gamma=\{\gamma_{i}|i\in I\},
  11. { μ i | i I } n \{\mu_{i}|i\in I\}\subseteq\mathbb{R}^{n}
  12. { σ i 2 | i I } \{\sigma_{i}^{2}|i\in I\}\subseteq\mathbb{R}
  13. lim sup δ 0 δ log μ δ ( X K η ) < - η . \limsup_{\delta\downarrow 0}\delta\log\mu_{\delta}(X\setminus K_{\eta})<-\eta.

Time-bin_encoding.html

  1. | 0 |0\rangle
  2. | 1 |1\rangle
  3. | ψ = α | 0 + β | 1 , |\psi\rangle=\alpha|0\rangle+\beta|1\rangle,\,
  4. | ψ = | 0 + e i ϕ | 1 2 , |\psi\rangle=\frac{|0\rangle+e^{i\phi}|1\rangle}{\sqrt{2}},
  5. | 0 |0\rangle
  6. | 1 |1\rangle

Time-domain_reflectometry.html

  1. ρ = R L - Z 0 R L + Z 0 \rho=\frac{R_{L}-Z_{0}}{R_{L}+Z_{0}}
  2. Z 0 Z_{0}

Time-evolving_block_decimation.html

  1. | Ψ H N |\Psi\rangle\in H^{{\otimes}N}
  2. | Ψ |\Psi\rangle
  3. M N M^{N}
  4. | i 1 , i 2 , . . , i N - 1 , i N |i_{1},i_{2},..,i_{N-1},i_{N}\rangle
  5. | Ψ = i = 1 M c i 1 i 2 . . i N | i 1 , i 2 , . . , i N - 1 , i N |\Psi\rangle=\sum\limits_{i=1}^{M}c_{i_{1}i_{2}..i_{N}}|{i_{1},i_{2},..,i_{N-1% },i_{N}}\rangle
  6. c i 1 i 2 . . i N c_{i_{1}i_{2}..i_{N}}
  7. c i 1 i 2 . . i N = α 1 , . . , α N - 1 = 0 χ Γ α 1 [ 1 ] i 1 λ α 1 [ 1 ] Γ α 1 α 2 [ 2 ] i 2 λ α 2 [ 2 ] Γ α 2 α 3 [ 3 ] i 3 λ α 3 [ 3 ] . . Γ α N - 2 α N - 1 [ N - 1 ] i N - 1 λ α N - 1 [ N - 1 ] Γ α N - 1 [ N ] i N c_{i_{1}i_{2}..i_{N}}=\sum\limits_{\alpha_{1},..,\alpha_{N-1}=0}^{\chi}\Gamma^% {[1]i_{1}}_{\alpha_{1}}\lambda^{[1]}_{\alpha_{1}}\Gamma^{[2]i_{2}}_{\alpha_{1}% \alpha_{2}}\lambda^{[2]}_{\alpha_{2}}\Gamma^{[3]i_{3}}_{\alpha_{2}\alpha_{3}}% \lambda^{[3]}_{\alpha_{3}}\cdot..\cdot\Gamma^{[{N-1}]i_{N-1}}_{\alpha_{N-2}% \alpha_{N-1}}\lambda^{[N-1]}_{\alpha_{N-1}}\Gamma^{[N]i_{N}}_{\alpha_{N-1}}
  8. | Ψ H A H B |\Psi\rangle\in{H_{A}\otimes H_{B}}
  9. | Ψ |{\Psi}\rangle
  10. | Ψ = i = 1 M a i | Φ i A Φ i B |{\Psi}\rangle=\sum\limits_{i=1}^{M}a_{i}|{\Phi^{A}_{i}\Phi^{B}_{i}}\rangle
  11. | Φ i A Φ i B = | Φ i A | Φ i B |{\Phi^{A}_{i}\Phi^{B}_{i}}\rangle=|{\Phi^{A}_{i}}\rangle\otimes|{\Phi^{B}_{i}}\rangle
  12. | Φ i A |{\Phi^{A}_{i}}\rangle
  13. H A H_{A}
  14. | Φ i B |{\Phi^{B}_{i}}\rangle
  15. H B {H_{B}}
  16. a i a_{i}
  17. i = 1 M a i 2 = 1 \sum\limits_{i=1}^{M}a^{2}_{i}=1
  18. M max = min ( dim ( < m t p l > H A ) , dim ( H B ) ) M_{\max}=\min(\dim(<mtpl>{{H_{A}}}),\dim({{H_{B}}}))
  19. | Ψ = 1 2 2 ( | 00 + 3 | 01 + 3 | 10 + | 11 ) |{\Psi}\rangle=\frac{1}{2\sqrt{2}}\left(|{00}\rangle+{\sqrt{3}}|{01}\rangle+{% \sqrt{3}}|{10}\rangle+|{11}\rangle\right)
  20. | Ψ = 3 + 1 2 2 | ϕ 1 A ϕ 1 B + 3 - 1 2 2 | ϕ 2 A ϕ 2 B |{\Psi}\rangle=\frac{\sqrt{3}+1}{2\sqrt{2}}|{\phi^{A}_{1}\phi^{B}_{1}}\rangle+% \frac{\sqrt{3}-1}{2\sqrt{2}}|{\phi^{A}_{2}\phi^{B}_{2}}\rangle
  21. | ϕ 1 A = 1 2 ( | 0 A + | 1 A ) , | ϕ 1 B = 1 2 ( | 0 B + | 1 B ) , | ϕ 2 A = 1 2 ( | 0 A - | 1 A ) , | ϕ 2 B = 1 2 ( | 1 B - | 0 B ) |{\phi^{A}_{1}}\rangle=\frac{1}{\sqrt{2}}(|{0_{A}}\rangle+|{1_{A}}\rangle),\ % \ |{\phi^{B}_{1}}\rangle=\frac{1}{\sqrt{2}}(|{0_{B}}\rangle+|{1_{B}}\rangle),% \ \ |{\phi^{A}_{2}}\rangle=\frac{1}{\sqrt{2}}(|{0_{A}}\rangle-|{1_{A}}\rangle)% ,\ \ |{\phi^{B}_{2}}\rangle=\frac{1}{\sqrt{2}}(|{1_{B}}\rangle-|{0_{B}}\rangle)
  22. | Φ = 1 3 | 00 + 1 6 | 01 - i 3 | 10 - i 6 | 11 |{\Phi}\rangle=\frac{1}{\sqrt{3}}|{00}\rangle+\frac{1}{\sqrt{6}}|{01}\rangle-% \frac{i}{\sqrt{3}}|{10}\rangle-\frac{i}{\sqrt{6}}|{11}\rangle
  23. | Φ = ( 1 3 | 0 A - i 3 | 1 A ) ( | 0 B + 1 2 | 1 B ) |{\Phi}\rangle={(}\frac{1}{\sqrt{3}}|{0_{A}}\rangle-\frac{i}{\sqrt{3}}|{1_{A}}% \rangle{)}\otimes{(}|{0_{B}}\rangle+\frac{1}{\sqrt{2}}|{1_{B}}\rangle{)}
  24. [ 1 ] : [ 2.. N ] [1]:[2..N]
  25. λ α 1 [ 1 ] \lambda^{[1]}_{{\alpha}_{1}}
  26. | Φ α 1 [ 1 ] | Φ α 1 [ 2.. N ] |{\Phi^{[1]}_{\alpha_{1}}}\rangle|{\Phi^{[2..N]}_{\alpha_{1}}}\rangle
  27. | Φ α 1 [ 1 ] |{\Phi^{[1]}_{\alpha_{1}}}\rangle
  28. | Ψ = i 1 , α 1 = 1 M , χ Γ α 1 [ 1 ] i 1 λ α 1 [ 1 ] | i 1 | Φ α 1 [ 2.. N ] |{\Psi}\rangle=\sum\limits_{i_{1},{\alpha_{1}=1}}^{M,\chi}\Gamma^{[1]i_{1}}_{% \alpha_{1}}\lambda^{[1]}_{\alpha_{1}}|{i_{1}}\rangle|{\Phi^{[2..N]}_{\alpha_{1% }}}\rangle
  29. | Φ α 1 [ 2.. N ] |{\Phi^{[2..N]}_{\alpha_{1}}}\rangle
  30. | Φ α 1 [ 2.. N ] = i 2 | i 2 | τ α 1 i 2 [ 3.. N ] |{\Phi^{[2..N]}_{\alpha_{1}}}\rangle=\sum_{i_{2}}|{i_{2}}\rangle|{\tau^{[3..N]% }_{\alpha_{1}i_{2}}}\rangle
  31. | τ α 1 i 2 [ 3.. N ] |{\tau^{[3..N]}_{\alpha_{1}i_{2}}}\rangle
  32. | τ α 1 i 2 [ 3.. N ] |{\tau^{[3..N]}_{\alpha_{1}i_{2}}}\rangle
  33. χ \chi
  34. | Φ α 2 [ 3.. N ] |{\Phi^{[3..N]}_{\alpha_{2}}}\rangle
  35. λ α 2 [ 2 ] \lambda^{[2]}_{{\alpha}_{2}}
  36. | τ α 1 i 2 [ 3.. N ] = α 2 Γ α 1 α 2 [ 2 ] i 2 λ α 2 [ 2 ] | Φ α 2 [ 3.. N ] |\tau^{[3..N]}_{\alpha_{1}i_{2}}\rangle=\sum_{\alpha_{2}}\Gamma^{[2]i_{2}}_{% \alpha_{1}\alpha_{2}}\lambda^{[2]}_{{\alpha}_{2}}|{\Phi^{[3..N]}_{\alpha_{2}}}\rangle
  37. | Ψ = i 1 , i 2 , α 1 , α 2 Γ α 1 [ 1 ] i 1 λ α 1 [ 1 ] Γ α 1 α 2 [ 2 ] i 2 λ α 2 [ 2 ] | i 1 i 2 | Φ α 2 [ 3.. N ] |{\Psi}\rangle=\sum_{i_{1},i_{2},\alpha_{1},\alpha_{2}}\Gamma^{[1]i_{1}}_{% \alpha_{1}}\lambda^{[1]}_{\alpha_{1}}\Gamma^{[2]i_{2}}_{\alpha_{1}\alpha_{2}}% \lambda^{[2]}_{{\alpha}_{2}}|{i_{1}i_{2}}\rangle|{\Phi^{[3..N]}_{\alpha_{2}}}\rangle
  38. Γ \Gamma
  39. ( N - 1 ) t h (N-1)^{th}
  40. N t h N^{th}
  41. k t h k^{th}
  42. [ 1.. k ] : [ k + 1.. N ] [1..k]:[k+1..N]
  43. | Ψ = α k λ α k [ k ] | Φ α k [ 1.. k ] | Φ α k [ k + 1.. N ] |{\Psi}\rangle=\sum_{\alpha_{k}}\lambda^{[k]}_{{\alpha}_{k}}|{\Phi^{[1..k]}_{% \alpha_{k}}}\rangle|{\Phi^{[k+1..N]}_{\alpha_{k}}}\rangle
  44. | Φ α k [ 1.. k ] = α 1 , α 2 . . α k - 1 Γ α 1 [ 1 ] i 1 λ α 1 [ 1 ] Γ α k - 1 α k [ k ] i k | i 1 i 2 . . i k |{\Phi^{[1..k]}_{\alpha_{k}}}\rangle=\sum_{\alpha_{1},\alpha_{2}..\alpha_{k-1}% }\Gamma^{[1]i_{1}}_{\alpha_{1}}\lambda^{[1]}_{\alpha_{1}}\cdot\cdot\Gamma^{[k]% i_{k}}_{\alpha_{k-1}\alpha_{k}}|{i_{1}i_{2}..i_{k}}\rangle
  45. | Φ α k [ k + 1.. N ] = α k + 1 , α k + 2 . . α N Γ α k α k + 1 [ k + 1 ] i k + 1 λ α k + 1 [ k + 1 ] λ α N - 1 N - 1 Γ α N - 1 [ N ] i N | i k + 1 i k + 2 . . i N |{\Phi^{[k+1..N]}_{\alpha_{k}}}\rangle=\sum_{\alpha_{k+1},\alpha_{k+2}..\alpha% _{N}}\Gamma^{[k+1]i_{k+1}}_{\alpha_{k}\alpha_{k+1}}\lambda^{[k+1]}_{\alpha_{k+% 1}}\cdot\cdot\lambda^{N-1}_{\alpha_{N-1}}\Gamma^{[N]i_{N}}_{\alpha_{N-1}}|{i_{% k+1}i_{k+2}..i_{N}}\rangle
  46. M N M^{N}
  47. χ 2 M ( N - 2 ) + 2 χ M + ( N - 1 ) χ {\chi}^{2}{\cdot}M(N-2)+2{\chi}M+(N-1)\chi
  48. c i 1 i 2 . . i N c_{i_{1}i_{2}..i_{N}}
  49. χ \chi
  50. M N / 2 M^{N/2}
  51. M N + 1 ( N - 2 ) M^{N+1}{\cdot}(N-2)
  52. χ 2 {\chi}^{2}
  53. M N M^{N}
  54. α \alpha
  55. λ α l [ l ] e - K α l , K > 0. \lambda^{[l]}_{{\alpha}_{l}}{\sim}e^{-K\alpha_{l}},\ K>0.
  56. | Ψ = 1 α l = 1 χ c | λ α l [ l ] | 2 α l = 1 χ c λ α l [ l ] | Φ α l [ 1.. l ] | Φ α l [ l + 1.. N ] , |{\Psi}\rangle=\frac{1}{\sqrt{\sum\limits_{{\alpha_{l}}=1}^{{\chi}_{c}}{|% \lambda^{[l]}_{{\alpha}_{l}}|}^{2}}}\cdot\sum\limits_{{{\alpha}_{l}}=1}^{{\chi% }_{c}}\lambda^{[l]}_{{\alpha}_{l}}|{\Phi^{[1..l]}_{\alpha_{l}}}\rangle|{\Phi^{% [l+1..N]}_{\alpha_{l}}}\rangle,
  57. χ c \chi_{c}
  58. χ c \chi_{c}
  59. 0.0001 0.0001
  60. 2 14 2^{14}
  61. 2 50 2^{50}
  62. ψ \psi
  63. ψ \psi
  64. M N M^{N}
  65. c i 1 i 2 . . i N c_{i_{1}i_{2}..i_{N}}
  66. χ \chi
  67. | ψ |{\psi}\rangle
  68. Γ [ k - 1 ] \Gamma^{[k-1]}
  69. Γ [ k + 1 ] \Gamma^{[k+1]}
  70. Γ \Gamma
  71. Γ [ k ] \Gamma^{[k]}
  72. < m t p l > O ( M 2 χ 2 ) <mtpl>{{O}}(M^{2}\cdot\chi^{2})
  73. Γ α k - 1 α k [ k ] i k = j U j k i k Γ α k - 1 α k [ k ] j k . \Gamma^{{}^{\prime}[k]i_{k}}_{\alpha_{k-1}\alpha_{k}}=\sum_{j}U^{i_{k}}_{j_{k}% }\Gamma^{[k]j_{k}}_{\alpha_{k-1}\alpha_{k}}.
  74. Γ \Gamma
  75. λ \lambda
  76. Γ [ k ] \Gamma^{[k]}
  77. Γ [ k + 1 ] \Gamma^{[k+1]}
  78. < m t p l > O ( M χ 3 ) <mtpl>{{O}}({M\cdot\chi}^{3})
  79. | ψ |{\psi}\rangle
  80. H = J < m t p l > H C H D K . {{H}=J{<mtpl>{{\otimes}}}H_{C}{\otimes}H_{D}{\otimes}K}.\,
  81. ρ J = T r C D K | ψ ψ | \rho^{J}=Tr_{CDK}|\psi\rangle\langle\psi|
  82. ρ [ 1.. k - 1 ] = α ( λ α [ k - 1 ] ) 2 | Φ α [ 1.. k - 1 ] Φ α [ 1.. k - 1 ] | = α ( λ α [ k - 1 ] ) 2 | α α | . \rho^{[1..{k-1}]}=\sum_{\alpha}{(\lambda^{[k-1]}_{\alpha})}^{2}|{\Phi^{[1..{k-% 1}]}_{\alpha}}\rangle\langle{\Phi^{[1..{k-1}]}_{\alpha}}|=\sum_{\alpha}{(% \lambda^{[k-1]}_{\alpha})^{2}}|{\alpha}\rangle\langle{\alpha}|.
  83. ρ [ k + 2.. N ] = γ ( λ γ [ k + 1 ] ) 2 | Φ γ [ k + 2.. N ] Φ γ [ k + 2.. N ] | = γ ( λ γ [ k + 1 ] ) 2 | γ γ | . \rho^{[{k+2}..{N}]}=\sum_{\gamma}{(\lambda^{[k+1]}_{\gamma})^{2}}|{\Phi^{[{k+2% }..N]}_{\gamma}}\rangle\langle{\Phi^{[{k+2}..N]}_{\gamma}}|=\sum_{\gamma}{(% \lambda^{[k+1]}_{\gamma})^{2}}|{\gamma}\rangle\langle{\gamma}|.
  84. H C H_{C}
  85. H D H_{D}
  86. | ψ |{\psi}\rangle
  87. | ψ = α , β , γ = 1 χ i , j = 1 M λ α [ C - 1 ] Γ α β [ C ] i λ β [ C ] Γ β γ [ D ] j λ γ [ D ] | α i j γ |{\psi}\rangle=\sum\limits_{\alpha,\beta,\gamma=1}^{\chi}\sum\limits_{i,j=1}^{% M}\lambda^{[C-1]}_{\alpha}\Gamma^{[C]i}_{\alpha\beta}\lambda^{[C]}_{\beta}% \Gamma^{[D]j}_{\beta\gamma}\lambda^{[D]}_{\gamma}|{{\alpha}ij{\gamma}}\rangle
  88. Γ [ C ] \Gamma^{[C]}
  89. λ \lambda
  90. Γ [ D ] . \Gamma^{[D]}.
  91. | ψ = V | ψ |{\psi^{\prime}}\rangle=V|{\psi}\rangle
  92. | ψ = α , γ = 1 χ i , j = 1 M λ α Θ α γ i j λ γ | α i j γ |{\psi^{\prime}}\rangle=\sum\limits_{\alpha,\gamma=1}^{\chi}\sum\limits_{i,j=1% }^{M}\lambda_{\alpha}\Theta^{ij}_{\alpha\gamma}\lambda_{\gamma}|{{\alpha}ij% \gamma}\rangle
  93. Θ α γ i j = β = 1 χ m , n = 1 M V m n i j Γ α β [ C ] m λ β Γ β γ [ D ] n . \Theta^{ij}_{\alpha\gamma}=\sum\limits_{\beta=1}^{\chi}\sum\limits_{m,n=1}^{M}% V^{ij}_{mn}\Gamma^{[C]m}_{\alpha\beta}\lambda_{\beta}\Gamma^{[D]n}_{\beta% \gamma}.
  94. λ \lambda
  95. < m t p l > Γ <mtpl>{{\Gamma}}
  96. ρ [ D K ] \rho^{{}^{\prime}[DK]}
  97. ρ [ D K ] = T r J C | ψ ψ | = j , j , γ , γ ρ γ γ j j | j γ j γ | . \rho^{{}^{\prime}[DK]}=Tr_{JC}|{\psi^{\prime}}\rangle\langle{\psi^{\prime}}|=% \sum_{j,j^{\prime},\gamma,\gamma^{\prime}}\rho^{jj^{\prime}}_{\gamma\gamma^{% \prime}}|{j\gamma}\rangle\langle{j^{\prime}\gamma^{\prime}}|.
  98. λ \lambda
  99. { | j γ } \{|{j\gamma}\rangle\}
  100. Γ [ < m t p l > D ] \Gamma^{[<mtpl>{{D]}}}
  101. | Φ [ < m t p l > D K ] = j , γ Γ β γ [ D ] j λ γ | j γ . |{\Phi^{{}^{\prime}[<mtpl>{{DK}}]}}\rangle=\sum_{j,\gamma}\Gamma^{{}^{\prime}[% {{D}}]j}_{\beta\gamma}\lambda_{\gamma}|{j\gamma}\rangle.
  102. λ β | Φ β [ < m t p l > J C ] = Φ β [ D K ] | ψ = i , j , α , γ ( Γ β γ [ D ] j ) * Θ α γ i j ( λ γ ) 2 λ α | α i \lambda^{{}^{\prime}}_{\beta}|{\Phi^{{}^{\prime}[<mtpl>{{JC}}]}_{\beta}}% \rangle=\langle{\Phi^{{}^{\prime}[{DK}]}_{\beta}}|{\psi^{\prime}}\rangle=\sum_% {i,j,\alpha,\gamma}(\Gamma^{{}^{\prime}[{D}]j}_{\beta\gamma})^{*}\Theta^{ij}_{% \alpha\gamma}(\lambda_{\gamma})^{2}\lambda_{\alpha}|{{\alpha}i}\rangle
  103. { | i α } \{|{i\alpha}\rangle\}
  104. Γ [ C ] \Gamma^{[{C}]}
  105. | Φ [ < m t p l > J C ] = i , α Γ α β [ C ] i λ α | α i . |{\Phi^{{}^{\prime}[<mtpl>{{JC}}]}}\rangle=\sum_{i,\alpha}\Gamma^{{}^{\prime}[% {{C}}]i}_{\alpha\beta}\lambda_{\alpha}|{{\alpha}i}\rangle.
  106. < m t p l > O ( M χ 2 ) <mtpl>{{O}}(M{\cdot}{\chi}^{2})
  107. Θ α γ i j \Theta^{ij}_{\alpha\gamma}
  108. β \beta
  109. m \it{m}
  110. n \it{n}
  111. γ , α , i , j \gamma,\alpha,{\it{i,j}}
  112. < m t p l > O ( M 4 χ 3 ) <mtpl>{{O}}(M^{4}{\cdot}{\chi}^{3})
  113. ρ γ γ < m t p l > j j \rho^{<mtpl>{{jj^{\prime}}}}_{\gamma\gamma^{\prime}}
  114. λ β | Φ β [ 𝐽𝐶 ] \lambda^{{}^{\prime}}_{\beta}|{\Phi^{{}^{\prime}[{\it{JC}}]}_{\beta}}\rangle
  115. O ( M 3 χ 3 ) {\it{O}}(M^{3}{\cdot}{\chi}^{3})
  116. O ( M 2 χ 3 ) {\it{O}}(M^{2}{\cdot}{\chi}^{3})
  117. M = 2 {\it{M}}=2
  118. N {\it{N}}
  119. H n = l = 1 N K 1 [ l ] + l = 1 N K 2 [ l , l + 1 ] . H_{n}=\sum\limits_{l=1}^{N}K^{[l]}_{1}+\sum\limits_{l=1}^{N}K^{[l,l+1]}_{2}.
  120. H n H_{n}
  121. H n = F + G H_{n}=F+G
  122. F e v e n l ( K 1 l + K 2 l , l + 1 ) = e v e n l F [ l ] , F\equiv\sum_{even\ \ l}(K^{l}_{1}+K^{l,l+1}_{2})=\sum_{even\ \ l}F^{[l]},
  123. G o d d l ( K 1 l + K 2 l , l + 1 ) = o d d l G [ l ] . G\equiv\sum_{odd\ \ l}(K^{l}_{1}+K^{l,l+1}_{2})=\sum_{odd\ \ l}G^{[l]}.
  124. [ F [ l ] , F [ l ] ] = 0 [F^{[l]},F^{[l^{\prime}]}]=0
  125. [ G [ l ] , G [ l ] ] = 0 [G^{[l]},G^{[l^{\prime}]}]=0
  126. e ( A + B ) = lim n ( e < m t p l > A n e B n ) n e^{(A+B)}=\lim_{n\rightarrow\infty}\Bigl(e^{\frac{<}{m}tpl>{{A}}{n}}e^{\frac{{% B}}{n}}\Bigr)^{n}
  127. e δ ( A + B ) = lim δ 0 [ e δ A e δ B ] + O ( δ 2 ) . e^{{\delta}(A+B)}=\lim_{\delta\rightarrow 0}[e^{{\delta}A}e^{{\delta}B}]+{{\it% {O}}}(\delta^{2}).
  128. δ 0 \delta\rightarrow 0
  129. e - i H t e^{-iHt}
  130. e - i H n T = [ e - i H n δ ] T / δ = [ e < m t p l > δ 2 F e δ G e δ 2 F ] n e^{-iH_{n}T}=[e^{-iH_{n}\delta}]^{T/{\delta}}=[e^{\frac{<}{m}tpl>{{\delta}}{2}% F}e^{{\delta}G}e^{\frac{{\delta}}{2}F}]^{n}
  131. n = T δ n=\frac{T}{\delta}
  132. n {\it{n}}
  133. e < m t p l > δ 2 F e^{\frac{<}{m}tpl>{{\delta}}{2}F}
  134. e δ G e^{{\delta}G}
  135. e < m t p l > δ 2 F = o d d l e δ 2 F [ l ] e^{\frac{<}{m}tpl>{{\delta}}{2}F}=\prod_{odd\ \ l}e^{\frac{{\delta}}{2}F^{[l]}}
  136. e δ G = e v e n l e δ G [ l ] e^{{\delta}G}=\prod_{even\ \ l}e^{{\delta}G^{[l]}}
  137. F [ l ] F^{[l]}
  138. F [ l ] F^{[l^{\prime}]}
  139. G [ l ] G^{[l]}
  140. G [ l ] G^{[l^{\prime}]}
  141. l l l{\neq}l^{\prime}
  142. | ψ ~ t + δ = e - i < m t p l > δ 2 F e - i δ G e - i δ 2 F | ψ ~ t . |{\tilde{\psi}_{t+\delta}}\rangle=e^{-i\frac{<}{m}tpl>{{\delta}}{2}F}e^{{-i% \delta}G}e^{\frac{{-i\delta}}{2}F}|{\tilde{\psi}_{t}}\rangle.
  143. δ \delta
  144. e - i < m t p l > δ 2 F [ l ] e^{-i\frac{<}{m}tpl>{{\delta}}{2}F^{[l]}}
  145. e - i δ G [ l ] e^{{-i\delta}G^{[l]}}
  146. e - i < m t p l > δ 2 F [ l ] e^{-i\frac{<}{m}tpl>{{\delta}}{2}F^{[l]}}
  147. D {\it{D}}
  148. | ψ 0 |{\psi_{0}}\rangle
  149. | ψ T |{\psi_{T}}\rangle
  150. H n H_{n}
  151. D {\it{D}}
  152. χ c \chi_{c}
  153. D {\it{D}}
  154. | ψ P |{\psi_{P}}\rangle
  155. | ψ 0 |{\psi_{0}}\rangle
  156. | ψ g r |{\psi_{gr}}\rangle
  157. H ~ \tilde{H}
  158. | ψ 0 = Q | ψ g r |{\psi_{0}}\rangle=Q|{\psi_{gr}}\rangle
  159. H ~ \tilde{H}
  160. | ψ g r |{\psi_{gr}}\rangle
  161. | ψ g r = lim τ e - H ~ τ | ψ P || e - H ~ τ | ψ P || , |{\psi_{gr}}\rangle=\lim_{\tau\rightarrow\infty}\frac{e^{-\tilde{H}\tau}|{\psi% _{P}}\rangle}{||e^{-\tilde{H}\tau}|{\psi_{P}}\rangle||},
  162. H 1 H_{1}
  163. | ψ P |{\psi_{P}}\rangle
  164. H ~ \tilde{H}
  165. | ψ 0 |{\psi_{0}}\rangle
  166. | ψ T |{\psi_{T}}\rangle
  167. H n H_{n}
  168. | ψ < m t p l > T = e - i H n T | ψ 0 |{\psi_{<}mtpl>{{T}}}\rangle=e^{-iH_{n}T}|{\psi_{0}}\rangle
  169. p 𝑡ℎ {\it{p^{th}}}
  170. δ p + 1 {\delta}^{p+1}
  171. n = T δ n=\frac{T}{\delta}
  172. ϵ = T δ δ p + 1 = T δ p \epsilon=\frac{T}{\delta}\delta^{p+1}=T\delta^{p}
  173. | ψ ~ T r |{\tilde{\psi}_{Tr}}\rangle
  174. | ψ ~ T r = 1 - ϵ 2 | ψ T r + ϵ | ψ T r |{\tilde{\psi}_{Tr}}\rangle=\sqrt{1-{\epsilon}^{2}}|{\psi_{Tr}}\rangle+{% \epsilon}|{\psi^{\bot}_{Tr}}\rangle
  175. | ψ T r |{\psi_{Tr}}\rangle
  176. | ψ T r |{\psi^{\bot}_{Tr}}\rangle
  177. T {\it{T}}
  178. ϵ ( T ) = 1 - | ψ T r ~ | ψ < m t p l > T r | 2 = 1 - 1 + ϵ 2 = ϵ 2 \epsilon({{{\it{T}}}})=1-|\langle{\tilde{\psi_{Tr}}}|{\psi_{<}mtpl>{{Tr}}}% \rangle|^{2}=1-1+\epsilon^{2}=\epsilon^{2}
  179. ϵ ( D ) = 1 - n = 1 N - 1 ( 1 - ϵ n ) \epsilon({{{\it{D}}}})=1-\prod\limits_{n=1}^{N-1}(1-\epsilon_{n})
  180. ϵ n = α = χ c χ ( λ α [ n ] ) 2 \epsilon_{n}=\sum\limits_{\alpha=\chi_{c}}^{\chi}(\lambda^{[n]}_{\alpha})^{2}
  181. n {\it{n}}
  182. | ψ |{\psi}\rangle
  183. n {\it{n}}
  184. | ψ = 1 - ϵ n | ψ D + ϵ n | ψ D |{\psi}\rangle=\sqrt{1-\epsilon_{n}}|{\psi_{D}}\rangle+\sqrt{\epsilon_{n}}|{% \psi^{\bot}_{D}}\rangle
  185. | ψ D = 1 1 - ϵ n α n = 1 χ c λ α n [ n ] | Φ α n [ 1.. n ] | Φ α n [ n + 1.. N ] |{\psi_{D}}\rangle=\frac{1}{\sqrt{1-\epsilon_{n}}}\sum\limits_{{{\alpha}_{n}}=% 1}^{{\chi}_{c}}\lambda^{[n]}_{{\alpha}_{n}}|{\Phi^{[1..n]}_{\alpha_{n}}}% \rangle|{\Phi^{[n+1..N]}_{\alpha_{n}}}\rangle
  186. | ψ D = 1 ϵ n α n = χ c < m t p l > χ λ α n [ n ] | Φ α n [ 1.. n ] | Φ α n [ n + 1.. N ] |{\psi^{\bot}_{D}}\rangle=\frac{1}{\sqrt{\epsilon_{n}}}\sum\limits_{{{\alpha}_% {n}}={\chi}_{c}}^{<}mtpl>{{\chi}}\lambda^{[n]}_{{\alpha}_{n}}|{\Phi^{[1..n]}_{% \alpha_{n}}}\rangle|{\Phi^{[n+1..N]}_{\alpha_{n}}}\rangle
  187. ψ D | ψ D = 0 \langle\psi^{\bot}_{D}|\psi_{D}\rangle=0
  188. ϵ n = 1 - | ψ | ψ D | 2 = α = χ c χ ( λ α [ n ] ) 2 \epsilon_{n}=1-|\langle{\psi}|\psi_{D}\rangle|^{2}=\sum\limits_{\alpha=\chi_{c% }}^{\chi}(\lambda^{[n]}_{\alpha})^{2}
  189. | ψ D = 1 - ϵ n + 1 | ψ D + ϵ n + 1 | ψ D |{\psi_{D}}\rangle=\sqrt{1-\epsilon_{n+1}}|{{{\psi}^{\prime}}_{D}}\rangle+% \sqrt{\epsilon_{n+1}}|{{\psi^{\prime}}^{\bot}_{D}}\rangle
  190. ϵ = 1 - | ψ | ψ D | 2 = 1 - ( 1 - ϵ n + 1 ) | ψ | ψ D | 2 = 1 - ( 1 - ϵ n + 1 ) ( 1 - ϵ n ) \epsilon=1-|\langle{\psi}|\psi^{\prime}_{D}\rangle|^{2}=1-(1-\epsilon_{n+1})|% \langle{\psi}|\psi_{D}\rangle|^{2}=1-(1-\epsilon_{n+1})(1-\epsilon_{n})
  191. D {\it{D}}
  192. l {\it{l}}
  193. ( [ 1.. l ] : [ l + 1.. N ] ) ([1..l]:[l+1..N])
  194. R = α l = 1 χ c | λ α l [ l ] | 2 = 1 - ϵ l R={\sum\limits_{{\alpha_{l}}=1}^{{\chi}_{c}}{|\lambda^{[l]}_{{\alpha}_{l}}|}^{% 2}}={1-\epsilon_{l}}
  195. l - 1 {\it{l}}-1
  196. || Φ α l - 1 [ l - 1.. N ] || ||{\Phi^{[l-1..N]}_{\alpha_{l-1}}}||
  197. n 1 = 1 = α l = 1 χ c ( c α l - 1 α l ) 2 ( λ α l [ l ] ) 2 + α l = χ c χ ( c α l - 1 α l ) 2 ( λ α l [ l ] ) 2 = S 1 + S 2 n_{1}=1=\sum\limits_{\alpha_{l}=1}^{\chi_{c}}(c_{\alpha_{l-1}\alpha_{l}})^{2}(% \lambda^{[l]}_{\alpha_{l}})^{2}+\sum\limits_{\alpha_{l}=\chi_{c}}^{\chi}(c_{% \alpha_{l-1}\alpha_{l}})^{2}(\lambda^{[l]}_{\alpha_{l}})^{2}=S_{1}+S_{2}
  198. ( c α l - 1 α l ) 2 = i l = 1 d ( Γ α l - 1 α l [ l ] i l ) * Γ α l - 1 α l [ l ] i l (c_{\alpha_{l-1}\alpha_{l}})^{2}=\sum\limits_{i_{l}=1}^{d}(\Gamma^{[l]i_{l}}_{% \alpha_{l-1}\alpha_{l}})^{*}\Gamma^{[l]i_{l}}_{\alpha_{l-1}\alpha_{l}}
  199. n 2 = α l = 1 χ c ( c α < m t p l > l - 1 α l ) 2 ( λ α l [ l ] ) 2 = α l = 1 χ c ( c α l - 1 α l ) 2 ( λ α l [ l ] ) 2 R = S 1 R n_{2}=\sum\limits_{\alpha_{l}=1}^{\chi_{c}}(c_{\alpha_{<mtpl>{{l-1}}}\alpha_{l% }})^{2}\cdot({\lambda^{\prime}}^{[l]}_{\alpha_{l}})^{2}=\sum\limits_{\alpha_{l% }=1}^{\chi_{c}}(c_{\alpha_{{{l-1}}}\alpha_{l}})^{2}\frac{(\lambda^{[l]}_{% \alpha_{l}})^{2}}{R}=\frac{S_{1}}{R}
  200. ϵ = n 2 - n 1 = n 2 - 1 \epsilon=n_{2}-n_{1}=n_{2}-1
  201. ϵ = S 1 R - 1 1 - R R = ϵ l 1 - ϵ l 0 a s ϵ l < m t p l > 0 \epsilon=\frac{S_{1}}{R}-1\leq\frac{1-R}{R}=\frac{\epsilon_{l}}{1-\epsilon_{l}% }{\rightarrow}0\ \ as\ \ {{\epsilon_{l}{\rightarrow}<mtpl>{{0}}}}
  202. | ψ D | ψ D | 2 = 1 - ϵ l 1 - ϵ l = 1 - 2 ϵ l 1 - ϵ l |\langle{\psi_{D}}|\psi_{D}\rangle|^{2}=1-\frac{\epsilon_{l}}{1-\epsilon_{l}}=% \frac{1-2\epsilon_{l}}{1-\epsilon_{l}}
  203. ϵ ( < m t p l > D ) = 1 - n = 1 N - 1 ( 1 - ϵ n ) n = 1 N - 1 1 - 2 ϵ n 1 - ϵ n = 1 - n = 1 N - 1 ( 1 - 2 ϵ n ) \epsilon(<mtpl>{{{{D}}}})=1-\prod\limits_{n=1}^{N-1}(1-\epsilon_{n})\prod% \limits_{n=1}^{N-1}\frac{1-2\epsilon_{n}}{1-\epsilon_{n}}=1-\prod\limits_{n=1}% ^{N-1}(1-2\epsilon_{n})

Tisserand's_parameter.html

  1. a a\,\!
  2. e e\,\!
  3. i i\,\!
  4. a P a_{P}
  5. T P = a P a + 2 a a P ( 1 - e 2 ) cos i T_{P}\ =\frac{a_{P}}{a}+2\cdot\sqrt{\frac{a}{a_{P}}(1-e^{2})}\cos i
  6. T J > 3 T_{J}>3
  7. 2 < T J < 3 2<T_{J}<3
  8. a ( 1 - e 2 ) cos i \sqrt{a(1-e^{2})}\cos i

Titchmarsh_convolution_theorem.html

  1. ϕ ( t ) \phi\,(t)
  2. ψ ( t ) \psi(t)\,
  3. 0 x ϕ ( t ) ψ ( x - t ) d t = 0 \int_{0}^{x}\phi(t)\psi(x-t)\,dt=0
  4. 0 < x < κ 0<x<\kappa\,
  5. λ 0 \lambda\geq 0
  6. μ 0 \mu\geq 0
  7. λ + μ κ \lambda+\mu\geq\kappa
  8. ϕ ( t ) = 0 \phi(t)=0\,
  9. ( 0 , λ ) (0,\lambda)\,
  10. ψ ( t ) = 0 \psi(t)=0\,
  11. ( 0 , μ ) (0,\mu)\,
  12. ϕ , ψ L 1 ( ) \phi,\,\psi\in L^{1}(\mathbb{R})
  13. inf supp ϕ ψ = inf supp ϕ + inf supp ψ \inf\mathop{\rm supp}\,\phi\ast\psi=\inf\mathop{\rm supp}\,\phi+\inf\mathop{% \rm supp}\,\psi
  14. sup supp ϕ ψ = sup supp ϕ + sup supp ψ \sup\mathop{\rm supp}\,\phi\ast\psi=\sup\mathop{\rm supp}\,\phi+\sup\mathop{% \rm supp}\,\psi
  15. supp ϕ ψ supp ϕ + supp ψ {\rm supp}\,\phi\ast\psi\subset\mathop{\rm supp}\,\phi+\mathop{\rm supp}\,\psi
  16. ϕ , ψ ( n ) \phi,\,\psi\in\mathcal{E}^{\prime}(\mathbb{R}^{n})
  17. c . h . supp ϕ ψ = c . h . supp ϕ + c . h . supp ψ . \mathop{c.h.}\mathop{\rm supp}\,\phi\ast\psi=\mathop{c.h.}\mathop{\rm supp}\,% \phi+\mathop{c.h.}\mathop{\rm supp}\,\psi.
  18. c . h . \mathop{c.h.}
  19. ( n ) \mathcal{E}^{\prime}(\mathbb{R}^{n})

Tonelli–Hobson_test.html

  1. ( | f ( x , y ) | d x ) d y \int_{\mathbb{R}}\left(\int_{\mathbb{R}}|f(x,y)|\,dx\right)\,dy
  2. ( | f ( x , y ) | d y ) d x \int_{\mathbb{R}}\left(\int_{\mathbb{R}}|f(x,y)|\,dy\right)\,dx

Tool_wear.html

  1. V c T n = C V_{c}T^{n}=C
  2. V c T n × D x S y = C V_{c}T^{n}\times D^{x}S^{y}=C
  3. V c V_{c}

Topological_conjugacy.html

  1. f f
  2. g g
  3. h h
  4. g = h - 1 f h , g=h^{-1}\circ f\circ h,
  5. f f
  6. g g
  7. g n = h - 1 f n h , g^{n}=h^{-1}\circ f^{n}\circ h,
  8. X X
  9. Y Y
  10. f : X X f\colon X\to X
  11. g : Y Y g\colon Y\to Y
  12. f f
  13. g g
  14. h : Y X h\colon Y\to X
  15. f h = h g f\circ h=h\circ g
  16. h h
  17. f f
  18. g g
  19. h h
  20. f f
  21. g g
  22. φ φ
  23. X X
  24. ψ ψ
  25. Y Y
  26. h : Y X h\colon Y\to X
  27. φ ( h ( y ) , t ) = h ψ ( y , t ) \varphi(h(y),t)=h\psi(y,t)
  28. y Y y\in Y
  29. t t\in\mathbb{R}
  30. h h
  31. ψ ψ
  32. φ φ
  33. f f
  34. g g
  35. g g
  36. f f
  37. g = h - 1 f h g=h^{-1}\circ f\circ h
  38. g n = h - 1 f n h g^{n}=h^{-1}\circ f^{n}\circ h
  39. φ ( , t ) \varphi(\cdot,t)
  40. ψ ( , t ) \psi(\cdot,t)
  41. t t
  42. φ φ
  43. ψ ψ
  44. X X
  45. ψ ψ
  46. φ φ
  47. h : Y X h:Y\to X
  48. ψ ψ
  49. φ φ
  50. 𝒪 \mathcal{O}
  51. h ( 𝒪 ( y , ψ ) ) = { h ( ψ ( y , t ) ) : t } = { φ ( h ( y ) , t ) : t } = 𝒪 ( h ( y ) , φ ) h(\mathcal{O}(y,\psi))=\{h(\psi(y,t)):t\in\mathbb{R}\}=\{\varphi(h(y),t):t\in% \mathbb{R}\}=\mathcal{O}(h(y),\varphi)
  52. y Y y\in Y
  53. y Y y\in Y
  54. δ > 0 \delta>0
  55. 0 < | s | < t < δ 0<|s|<t<\delta
  56. s s
  57. φ ( h ( y ) , s ) = h ( ψ ( y , t ) ) \varphi(h(y),s)=h(\psi(y,t))
  58. s > 0 s>0
  59. ψ ψ
  60. φ φ
  61. x = f ( x ) x^{\prime}=f(x)
  62. y = g ( y ) y^{\prime}=g(y)
  63. h : X Y h:X\to Y
  64. f ( x ) = M - 1 ( x ) g ( h ( x ) ) where M ( x ) = d h ( x ) d x . f(x)=M^{-1}(x)g(h(x))\quad\,\text{where}\quad M(x)=\frac{\mathrm{d}h(x)}{% \mathrm{d}x}.
  65. y = h ( x ) y=h(x)
  66. x = f ( x ) x^{\prime}=f(x)
  67. x = g ( x ) x^{\prime}=g(x)
  68. μ : X 𝐑 \mu:X\to\mathbf{R}
  69. g ( x ) = μ ( x ) f ( x ) g(x)=\mu(x)f(x)
  70. x = A x x^{\prime}=Ax
  71. A A

Topological_entropy.html

  1. C D C\vee D
  2. H ( C , f ) = lim n 1 n H ( C f - 1 C f - n + 1 C ) . H(C,f)=\lim_{n\to\infty}\frac{1}{n}H(C\vee f^{-1}C\vee\ldots\vee f^{-n+1}C).
  3. H ( C f - 1 C f - n + 1 C ) H(C\vee f^{-1}C\vee\ldots\vee f^{-n+1}C)
  4. d n ( x , y ) = max { d ( f i ( x ) , f i ( y ) ) : 0 i < n } . d_{n}(x,y)=\max\{d(f^{i}(x),f^{i}(y)):0\leq i<n\}.
  5. h ( f ) = lim ϵ 0 ( lim sup n 1 n log N ( n , ϵ ) ) . h(f)=\lim_{\epsilon\to 0}\left(\limsup_{n\to\infty}\frac{1}{n}\log N(n,% \epsilon)\right).
  6. f f
  7. X X
  8. C C
  9. f f
  10. C C
  11. f f
  12. h ( f ) = H ( f , C ) h(f)=H(f,C)
  13. f : X X f:X\rightarrow X
  14. X X
  15. h μ ( f ) h_{\mu}(f)
  16. f f
  17. μ \mu
  18. M ( X , f ) M(X,f)
  19. f f
  20. h ( f ) = sup μ M ( X , f ) h μ ( f ) h(f)=\sup_{\mu\in M(X,f)}h_{\mu}(f)
  21. h μ h_{\mu}
  22. μ h μ ( f ) : M ( X , f ) \R \mu\mapsto h_{\mu}(f):M(X,f)\rightarrow\R
  23. f f
  24. μ \mu
  25. f f
  26. μ \mu
  27. σ : Σ k Σ \sigma:\Sigma_{k}\rightarrow\Sigma
  28. x n x n - 1 x_{n}\mapsto x_{n-1}
  29. { 1 , , k } \{1,\dots,k\}
  30. C = { [ 1 ] , , [ k ] } C=\{[1],\dots,[k]\}
  31. Σ k \Sigma_{k}
  32. j = 0 n σ - 1 ( C ) \bigvee_{j=0}^{n}\sigma^{-1}(C)
  33. Σ k \Sigma_{k}
  34. n 𝒩 n\in\mathcal{N}
  35. k n k^{n}
  36. C C
  37. h ( σ ) = h ( σ , C ) = lim n 1 n log k n = log k h(\sigma)=h(\sigma,C)=\lim_{n\rightarrow\infty}\frac{1}{n}\log k^{n}=\log k
  38. ( 1 k , , 1 k ) (\frac{1}{k},\dots,\frac{1}{k})
  39. log k \log k
  40. A A
  41. k × k k\times k
  42. { 0 , 1 } \{0,1\}
  43. σ : Σ A Σ A \sigma:\Sigma_{A}\rightarrow\Sigma_{A}
  44. h ( σ ) = log λ h(\sigma)=\log\lambda
  45. λ \lambda
  46. A A

Toric_section.html

  1. ( x 2 + y 2 ) 2 + a x 2 + b y 2 + c x + d y + e = 0. \left(x^{2}+y^{2}\right)^{2}+ax^{2}+by^{2}+cx+dy+e=0.

Torsion_constant.html

  1. θ = T L J G \theta=\frac{TL}{JG}
  2. θ \theta
  3. J z z = J x x + J y y = π r 4 4 + π r 4 4 = π r 4 2 J_{zz}=J_{xx}+J_{yy}=\frac{\pi r^{4}}{4}+\frac{\pi r^{4}}{4}=\frac{\pi r^{4}}{2}
  4. J = π D 4 32 J=\frac{\pi D^{4}}{32}
  5. J π a 3 b 3 a 2 + b 2 J\approx\frac{\pi a^{3}b^{3}}{a^{2}+b^{2}}
  6. J 2.25 a 4 J\approx\,2.25a^{4}
  7. J β a b 3 J\approx\beta ab^{3}
  8. β \beta
  9. β \beta
  10. \infty
  11. J a b 3 ( 1 3 - 0.21 b a ( 1 - b 4 12 a 4 ) ) J\approx ab^{3}\left(\frac{1}{3}-0.21\frac{b}{a}\left(1-\frac{b^{4}}{12a^{4}}% \right)\right)
  12. J = 1 3 U t 3 J=\frac{1}{3}Ut^{3}
  13. J = 2 3 π r t 3 J=\frac{2}{3}\pi rt^{3}

Torsten_Carleman.html

  1. | K ( x , y ) | 2 d y < \int|K(x,y)|^{2}dy<\infty
  2. n = 1 ( a 1 a 2 a n ) 1 / n e n = 1 a n , \sum_{n=1}^{\infty}\left(a_{1}a_{2}\cdots a_{n}\right)^{1/n}\leq e\sum_{n=1}^{% \infty}a_{n},
  3. d 𝐮 / d d{\mathbf{u}}/{d}

Tortuosity.html

  1. τ = L C \tau=\frac{L}{C}
  2. τ = < m t p l > N - 1 L i = 1 N ( L i S i - 1 ) \tau=\frac{<}{m}tpl>{{N-1}}{L}\cdot\sum\limits_{i=1}^{N}{\left({\frac{{L_{i}}}% {{S_{i}}}-1}\right)}
  3. d < m t p l > d x log ( κ ) = κ κ \frac{d}{<}mtpl>{{dx}}\log\left(\kappa\right)=\frac{{\kappa^{\prime}}}{\kappa}
  4. τ = t 1 t 2 ( κ ( t ) ) 2 d t L \tau=\frac{{\int\limits_{t_{1}}^{t_{2}}{\left({\kappa^{\prime}\left(t\right)}% \right)^{2}}dt}}{L}

Total_correlation.html

  1. { X 1 , X 2 , , X n } \{X_{1},X_{2},\ldots,X_{n}\}
  2. C ( X 1 , X 2 , , X n ) C(X_{1},X_{2},\ldots,X_{n})
  3. p ( X 1 , , X n ) p(X_{1},\ldots,X_{n})
  4. p ( X 1 ) p ( X 2 ) p ( X n ) p(X_{1})p(X_{2})\cdots p(X_{n})
  5. C ( X 1 , X 2 , , X n ) D KL [ p ( X 1 , , X n ) p ( X 1 ) p ( X 2 ) p ( X n ) ] . C(X_{1},X_{2},\ldots,X_{n})\equiv\operatorname{D_{KL}}\left[p(X_{1},\ldots,X_{% n})\|p(X_{1})p(X_{2})\cdots p(X_{n})\right]\;.
  6. C ( X 1 , X 2 , , X n ) = [ i = 1 n H ( X i ) ] - H ( X 1 , X 2 , , X n ) C(X_{1},X_{2},\ldots,X_{n})=\left[\sum_{i=1}^{n}H(X_{i})\right]-H(X_{1},X_{2},% \ldots,X_{n})
  7. H ( X i ) H(X_{i})
  8. X i X_{i}\,
  9. H ( X 1 , X 2 , , X n ) H(X_{1},X_{2},\ldots,X_{n})
  10. { X 1 , X 2 , , X n } \{X_{1},X_{2},\ldots,X_{n}\}
  11. { X 1 , X 2 , , X n } \{X_{1},X_{2},\ldots,X_{n}\}
  12. C ( X 1 , X 2 , , X n ) = x 1 𝒳 1 x 2 𝒳 2 x n 𝒳 n p ( x 1 , x 2 , , x n ) log p ( x 1 , x 2 , , x n ) p ( x 1 ) p ( x 2 ) p ( x n ) . C(X_{1},X_{2},\ldots,X_{n})=\sum_{x_{1}\in\mathcal{X}_{1}}\sum_{x_{2}\in% \mathcal{X}_{2}}\ldots\sum_{x_{n}\in\mathcal{X}_{n}}p(x_{1},x_{2},\ldots,x_{n}% )\log\frac{p(x_{1},x_{2},\ldots,x_{n})}{p(x_{1})p(x_{2})\cdots p(x_{n})}.
  13. i = 1 n H ( X i ) \begin{matrix}\sum_{i=1}^{n}H(X_{i})\end{matrix}
  14. H ( X 1 , X 2 , , X n ) H(X_{1},X_{2},\ldots,X_{n})
  15. p ( X 1 , X 2 , , X n ) p(X_{1},X_{2},\ldots,X_{n})
  16. p ( X 1 ) p ( X 2 ) p ( X n ) p(X_{1})p(X_{2})\cdots p(X_{n})
  17. C max = i = 1 n H ( X i ) - max X i H ( X i ) , C_{\max}=\sum_{i=1}^{n}H(X_{i})-\max\limits_{X_{i}}H(X_{i}),
  18. C ( X 1 , X 2 , , X n | Y = y ) D KL [ p ( X 1 , , X n | Y = y ) p ( X 1 | Y = y ) p ( X 2 | Y = y ) p ( X n | Y = y ) ] . C(X_{1},X_{2},\ldots,X_{n}|Y=y)\equiv\operatorname{D_{KL}}\left[p(X_{1},\ldots% ,X_{n}|Y=y)\|p(X_{1}|Y=y)p(X_{2}|Y=y)\cdots p(X_{n}|Y=y)\right]\;.
  19. C ( X 1 , X 2 , , X n | Y = y ) = i = 1 n H ( X i | Y = y ) - H ( X 1 , X 2 , , X n | Y = y ) C(X_{1},X_{2},\ldots,X_{n}|Y=y)=\sum_{i=1}^{n}H(X_{i}|Y=y)-H(X_{1},X_{2},% \ldots,X_{n}|Y=y)

Total_maximum_daily_load.html

  1. T M D L = W L A + L A + M O S TMDL=WLA+LA+MOS

Total_variation_diminishing.html

  1. u t + a u x = 0 , \frac{\partial u}{\partial t}+a\frac{\partial u}{\partial x}=0,
  2. T V = | u x | d x , TV=\int\left|\frac{\partial u}{\partial x}\right|dx,
  3. T V = j | u j + 1 - u j | . TV=\sum_{j}\left|u_{j+1}-u_{j}\right|.
  4. T V ( u n + 1 ) T V ( u n ) . TV\left(u^{n+1}\right)\leq TV\left(u^{n}\right).
  5. u n u^{n}
  6. u n + 1 u^{n+1}
  7. ( ρ 𝐮 ϕ ) = ( Γ ϕ ) + S ϕ \nabla\cdot(\rho\mathbf{u}\phi)\,=\nabla\cdot(\Gamma\nabla\phi)+S_{\phi}\;
  8. ρ \rho
  9. 𝐮 \mathbf{u}
  10. ϕ \phi
  11. Γ \Gamma
  12. S ϕ S_{\phi}
  13. ϕ \phi
  14. A 𝐧 ( ρ 𝐮 ϕ ) d A = A 𝐧 ( Γ ϕ ) d A + C V S ϕ d V \int_{A}\mathbf{n}\cdot(\rho\mathbf{u}\phi)\,dA=\int_{A}\mathbf{n}\cdot(\Gamma% \nabla\phi)\,dA+\int_{C}VS_{\phi}\,dV
  15. \;
  16. 𝐧 \mathbf{n}
  17. ( ρ 𝐮 ϕ A ) r - ( ρ 𝐮 ϕ A ) l = ( Γ A ϕ x ) r - ( Γ A ϕ x ) l (\rho\mathbf{u}\phi A)_{r}-(\rho\mathbf{u}\phi A)_{l}=\left(\Gamma A\frac{% \partial\phi}{\partial x}\right)_{r}-\left(\Gamma A\frac{\partial\phi}{% \partial x}\right)_{l}
  18. ϕ x = δ ϕ δ x \frac{\partial\phi}{\partial x}=\frac{\delta\phi}{\delta x}
  19. A r = A l ; A_{r}=A_{l};\,
  20. ( ρ 𝐮 ϕ ) r - ( ρ 𝐮 ϕ ) l = ( Γ δ x δ ϕ ) r - ( Γ δ x δ ϕ ) l . (\rho\mathbf{u}\phi)_{r}-(\rho\mathbf{u}\phi)_{l}\,=\left(\frac{\Gamma}{\delta x% }\delta\phi\right)_{r}-\left(\frac{\Gamma}{\delta x}\delta\phi\right)_{l}.
  21. F r = ( ρ 𝐮 ) r ; and F l = ( ρ 𝐮 ) l ; F_{r}=(\rho\mathbf{u})_{r};\,\text{ and }F_{l}=(\rho\mathbf{u})_{l};
  22. D l = ( Γ δ x ) l ; D r = ( Γ δ x ) r ; D_{l}=\left(\frac{\Gamma}{\delta x}\right)_{l};\qquad D_{r}=\left(\frac{\Gamma% }{\delta x}\right)_{r};
  23. δ ϕ r = ϕ R - ϕ P ; and δ x r = x P R ; \delta\phi_{r}=\phi_{R}-\phi_{P};\,\text{ and }\delta x_{r}=x_{PR};
  24. δ ϕ l = ϕ P - ϕ L ; and δ x l = x L P ; \delta\phi_{l}=\phi_{P}-\phi_{L};\,\text{ and }\delta x_{l}=x_{LP};
  25. F r ϕ r - F l ϕ l = D r ( ϕ R - ϕ P ) - D l ( ϕ P - ϕ L ) ; F_{r}\phi_{r}-F_{l}\phi_{l}=D_{r}(\phi_{R}-\phi_{P})-D_{l}(\phi_{P}-\phi_{L});
  26. ( ρ 𝐮 ) r - ( ρ 𝐮 ) l = 0 ; ( O R ) F r - F l = 0 ; ( O R ) F r = F l = F ; (\rho\mathbf{u})_{r}-(\rho\mathbf{u})_{l}\,=0;(OR)F_{r}-F_{l}=0;(OR)F_{r}=F_{l% }=F;
  27. Γ l = Γ r ; δ x L P = δ x P R = δ x \Gamma_{l}=\Gamma_{r};\qquad\delta x_{LP}=\delta x_{PR}=\delta x
  28. D l = D r = D D_{l}=D_{r}=D
  29. F ( ϕ r - ϕ l ) = D ( ϕ R - 2 ϕ P + ϕ L ) F(\phi_{r}-\phi_{l})=D(\phi_{R}-2\phi_{P}+\phi_{L})
  30. P ( ϕ r - ϕ l ) = ( ϕ R - 2 ϕ P + ϕ L ) P(\phi_{r}-\phi_{l})=(\phi_{R}-2\phi_{P}+\phi_{L})
  31. P = F D = ρ 𝐮 δ x Γ . P=\frac{F}{D}=\frac{\rho\mathbf{u}\delta x}{\Gamma}.
  32. ϕ r \phi_{r}
  33. ϕ l \phi_{l}
  34. P ϕ r = 1 2 ( P + | P | ) [ f r + ϕ R + ( 1 - f r + ) ϕ L ] + 1 2 ( P - | P | ) [ f r - ϕ P + ( 1 - f r - ) ϕ R R ] P\phi_{r}=\frac{1}{2}(P+|P|)[f_{r}^{+}\phi_{R}+(1-f_{r}^{+})\phi_{L}]+\frac{1}% {2}(P-|P|)[f_{r}^{-}\phi_{P}+(1-f_{r}^{-})\phi_{RR}]
  35. P ϕ l = 1 2 ( P + | P | ) [ f l + ϕ P + ( 1 - f l + ) ϕ L L ] + 1 2 ( P - | P | ) [ f l - ϕ L + ( 1 - f l - ) ϕ R ] P\phi_{l}=\frac{1}{2}(P+|P|)[f_{l}^{+}\phi_{P}+(1-f_{l}^{+})\phi_{LL}]+\frac{1% }{2}(P-|P|)[f_{l}^{-}\phi_{L}+(1-f_{l}^{-})\phi_{R}]
  36. P P
  37. f f
  38. f = f ( ϵ ) = f ϕ U - ϕ U U ϕ D - ϕ U U f=f(\epsilon)=f\frac{\phi_{U}-\phi_{UU}}{\phi_{D}-\phi_{UU}}
  39. f + f^{+}
  40. f - f^{-}
  41. f r + is a function of ϕ P - ϕ L ϕ R - ϕ L . f r - is a function of ϕ R - ϕ R R ϕ P - ϕ R R , f l + is a function of ϕ L - ϕ L L ϕ P - ϕ L L , and f l - is a function of ϕ P - ϕ R ϕ L - ϕ R \begin{aligned}&\displaystyle f_{r}^{+}\,\text{ is a function of }\dfrac{\phi_% {P}-\phi_{L}}{\phi_{R}-\phi_{L}}.\\ &\displaystyle f_{r}^{-}\,\text{ is a function of }\dfrac{\phi_{R}-\phi_{RR}}{% \phi_{P}-\phi_{RR}},\\ &\displaystyle f_{l}^{+}\,\text{ is a function of }\dfrac{\phi_{L}-\phi_{LL}}{% \phi_{P}-\phi_{LL}},\,\text{ and}\\ &\displaystyle f_{l}^{-}\,\text{ is a function of }\dfrac{\phi_{P}-\phi_{R}}{% \phi_{L}-\phi_{R}}\end{aligned}
  42. P P
  43. ( P - | P | ) = 0 (P-|P|)=0
  44. f - f^{-}
  45. P P
  46. ( P + | P | ) = 0 (P+|P|)=0
  47. f + f^{+}
  48. ϕ r \phi_{r}
  49. ϕ r \phi_{r}

Townsend_discharge.html

  1. I I
  2. I I 0 = e α n d , \frac{I}{I_{0}}=e^{\alpha_{n}d},\,
  3. I I
  4. I 0 I_{0}
  5. e e
  6. α n \alpha_{n}
  7. d d
  8. I I
  9. d d
  10. α p \alpha_{p}
  11. I I 0 = ( α n - α p ) e ( α n - α p ) d α n - α p e ( α n - α p ) d I I 0 e α n d 1 - ( α p / α n ) e α n d \frac{I}{I_{0}}=\frac{(\alpha_{n}-\alpha_{p})e^{(\alpha_{n}-\alpha_{p})d}}{% \alpha_{n}-\alpha_{p}e^{(\alpha_{n}-\alpha_{p})d}}\qquad\Longrightarrow\qquad% \frac{I}{I_{0}}\cong\frac{e^{\alpha_{n}d}}{1-({\alpha_{p}/\alpha_{n}})e^{% \alpha_{n}d}}
  12. α p α n \alpha_{p}\ll\alpha_{n}
  13. ϵ i \epsilon_{i}
  14. I I 0 = e α n d 1 - ϵ i ( e α n d - 1 ) . \frac{I}{I_{0}}=\frac{e^{\alpha_{n}d}}{1-{\epsilon_{i}}\left(e^{\alpha_{n}d}-1% \right)}.
  15. f 1 R 1 C 1 ln V 1 - V GLOW V 1 - V TWN , f\cong\frac{1}{R_{1}C_{1}\ln\frac{V_{1}-V\text{GLOW}}{V_{1}-V\text{TWN}}},
  16. V GLOW V\text{GLOW}
  17. V TWN V\text{TWN}
  18. C 1 C_{1}
  19. R 1 R_{1}
  20. V 1 V_{1}

Traced_monoidal_category.html

  1. Tr X , Y U : 𝐂 ( X U , Y U ) 𝐂 ( X , Y ) \mathrm{Tr}^{U}_{X,Y}:\mathbf{C}(X\otimes U,Y\otimes U)\to\mathbf{C}(X,Y)
  2. id U \,\text{id}_{U}
  3. f : X U Y U f:X\otimes U\to Y\otimes U
  4. g : X X g:X^{\prime}\to X
  5. Tr X , Y U ( f ) g = Tr X , Y U ( f ( g U ) ) \mathrm{Tr}^{U}_{X,Y}(f)g=\mathrm{Tr}^{U}_{X^{\prime},Y}(f(g\otimes U))
  6. f : X U Y U f:X\otimes U\to Y\otimes U
  7. g : Y Y g:Y\to Y^{\prime}
  8. g Tr X , Y U ( f ) = Tr X , Y U ( ( g U ) f ) g\mathrm{Tr}^{U}_{X,Y}(f)=\mathrm{Tr}^{U}_{X,Y^{\prime}}((g\otimes U)f)
  9. f : X U Y U f:X\otimes U\to Y\otimes U^{\prime}
  10. g : U U g:U^{\prime}\to U
  11. Tr X , Y U ( ( Y g ) f ) = Tr X , Y U ( f ( X g ) ) \mathrm{Tr}^{U}_{X,Y}((Y\otimes g)f)=\mathrm{Tr}^{U^{\prime}}_{X,Y}(f(X\otimes g))
  12. f : X I Y I f:X\otimes I\to Y\otimes I
  13. Tr X , Y I ( f ) = f \mathrm{Tr}^{I}_{X,Y}(f)=f
  14. f : X U V Y U V f:X\otimes U\otimes V\to Y\otimes U\otimes V
  15. Tr X , Y U V ( f ) = Tr X , Y U ( Tr X U , Y U V ( f ) ) \mathrm{Tr}^{U\otimes V}_{X,Y}(f)=\mathrm{Tr}^{U}_{X,Y}(\mathrm{Tr}^{V}_{X% \otimes U,Y\otimes U}(f))
  16. f : X U Y U f:X\otimes U\to Y\otimes U
  17. g : W Z g:W\to Z
  18. g Tr X , Y U ( f ) = Tr W X , Z Y U ( g f ) g\otimes\mathrm{Tr}^{U}_{X,Y}(f)=\mathrm{Tr}^{U}_{W\otimes X,Z\otimes Y}(g% \otimes f)
  19. Tr X , X X ( γ X , X ) = X \mathrm{Tr}^{X}_{X,X}(\gamma_{X,X})=X
  20. γ \gamma

Trailing_zero.html

  1. f ( n ) = i = 1 k n 5 i = n 5 + n 5 2 + n 5 3 + + n 5 k , f(n)=\sum_{i=1}^{k}\left\lfloor\frac{n}{5^{i}}\right\rfloor=\left\lfloor\frac{% n}{5}\right\rfloor+\left\lfloor\frac{n}{5^{2}}\right\rfloor+\left\lfloor\frac{% n}{5^{3}}\right\rfloor+\cdots+\left\lfloor\frac{n}{5^{k}}\right\rfloor,\,
  2. 5 k + 1 > n , 5^{k+1}>n,\,
  3. a \lfloor a\rfloor
  4. 32 5 + 32 5 2 = 6 + 1 = 7 \left\lfloor\frac{32}{5}\right\rfloor+\left\lfloor\frac{32}{5^{2}}\right% \rfloor=6+1=7\,
  5. q i = n 5 i , q_{i}=\left\lfloor\frac{n}{5^{i}}\right\rfloor,\,
  6. q 0 \displaystyle q_{0}

Trajpar.html

  1. d 1 = s i n ( t r a j p a r 8 π ) d1=sin(trajpar\cdot 8\pi)
  2. d 1 = 1 + t r a j p a r 2 d1=1+trajpar^{2}

Transcendental_equation.html

  1. x = e - x x=e^{-x}
  2. x = cos x x=\cos x
  3. ln x = 3 \ln x=3
  4. x = e 3 x=e^{3}
  5. sin x = 0 \sin x=0
  6. x = π n x=\pi n
  7. n n
  8. cos x = sin 2 x \cos x=\sin{2x}
  9. sin 2 x = 2 sin x cos x \sin 2x=2\sin x\cos x
  10. cos x = 0 \cos x=0
  11. 2 sin x = 1 2\sin x=1
  12. x = 2 π n + π / 2 x=2\pi n+\pi/2
  13. x = 2 π m / 3 + π / 6 x={2\pi m}/3+\pi/6
  14. m , n m,n
  15. e x = x e^{x}=x
  16. e x > x e^{x}>x
  17. x x
  18. sin x = x \sin x=x
  19. x = 0 x=0
  20. 3 2 x - 2 = 4 x 3\;2^{x}-2=4^{x}
  21. 3 q - 2 = q 2 3\;q-2=q^{2}
  22. q = 1 q=1
  23. q = 2 q=2
  24. 2 x = q 2^{x}=q
  25. x = 0 x=0
  26. x = 1 x=1
  27. k 1 k\approx 1
  28. sin x = k x \sin x=kx
  29. ( 1 - k ) x - x 3 / 3 (1-k)x-x^{3}/3
  30. x = 0 x=0
  31. x = \plusmn 6 1 - k x=\plusmn\sqrt{6}\sqrt{1-k}
  32. x = e - x x=e^{-x}

Transcendental_perspectivism.html

  1. X p + X o N + E = X p o N X_{p}+X_{o^{N}}+E=X_{po^{N}}
  2. X X
  3. X p X_{p}
  4. X o N X_{o^{N}}
  5. E E
  6. X p o N X_{po^{N}}
  7. X p o N X_{po^{N}}

Transferable_belief_model.html

  1. Pr ( Head ) + Pr ( Tail ) = 1 \scriptstyle\Pr(\mathrm{Head})+\Pr(\mathrm{Tail})=1
  2. Pr ( ) + Pr ( Head ) + Pr ( Tail ) + Pr ( Head , Tail ) = 1. \Pr(\emptyset)+\Pr(\mathrm{Head})+\Pr(\mathrm{Tail})+\Pr(\mathrm{Head,Tail})=1.

Translational_partition_function.html

  1. q T q_{T}
  2. q T = V Λ 3 = V ( 2 π m k T ) 3 2 h 3 q_{T}=\frac{V}{\Lambda^{3}}=V\frac{(2\pi mkT)^{\frac{3}{2}}}{h^{3}}
  3. Λ = h β 2 π m \Lambda=h\sqrt{\frac{\beta}{2\pi m}}
  4. β = 1 k T \beta=\frac{1}{kT}
  5. V V
  6. Λ \Lambda
  7. h h
  8. m m
  9. k k
  10. T T

Transmission_coefficient.html

  1. T = J trans n ^ J inc n ^ , T=\frac{\vec{J}_{\mathrm{trans}}\cdot\hat{n}}{\vec{J}_{\mathrm{inc}}\cdot\hat{% n}},
  2. n ^ \hat{n}
  3. R = J refl - n ^ J inc n ^ = | J refl | | J inc | R=\frac{\vec{J}_{\mathrm{refl}}\cdot-\hat{n}}{\vec{J}_{\mathrm{inc}}\cdot\hat{% n}}=\frac{|J_{\mathrm{refl}}|}{|J_{\mathrm{inc}}|}
  4. T = exp ( - 2 x 1 x 2 d x 2 m 2 ( V ( x ) - E ) ) ( 1 + 1 4 exp ( - 2 x 1 x 2 d x 2 m 2 ( V ( x ) - E ) ) ) 2 , T=\frac{\displaystyle\exp\left(-2\int_{x_{1}}^{x_{2}}dx\sqrt{\frac{2m}{\hbar^{% 2}}\left(V(x)-E\right)}\,\right)}{\displaystyle\left(1+\frac{1}{4}\exp\left(-2% \int_{x_{1}}^{x_{2}}dx\sqrt{\frac{2m}{\hbar^{2}}\left(V(x)-E\right)}\,\right)% \right)^{2}}\ ,
  5. x 1 , x 2 x_{1},\,x_{2}
  6. 0 \hbar\rightarrow 0
  7. T 16 E U 0 ( 1 - E U 0 ) exp ( - 2 L 2 m 2 ( U 0 - E ) ) T\approx 16\frac{E}{U_{0}}\left(1-\frac{E}{U_{0}}\right)\exp\left(-2L\sqrt{% \frac{2m}{\hbar^{2}}(U_{0}-E)}\right)
  8. L = x 2 - x 1 L=x_{2}-x_{1}

Transmission_time.html

  1. 2 * 10 8 2*10^{8}
  2. 3 * 10 8 3*10^{8}

Transport_of_structure.html

  1. ϕ : V W \phi\colon V\to W
  2. ( , ) (\cdot,\cdot)
  3. W W
  4. [ , ] [\cdot,\cdot]
  5. [ v 1 , v 2 ] = ( ϕ ( v 1 ) , ϕ ( v 2 ) ) [v_{1},v_{2}]=(\phi(v_{1}),\phi(v_{2}))\;
  6. ϕ \phi
  7. ϕ \phi
  8. [ , ] [\cdot,\cdot]
  9. ϕ \phi
  10. ϕ : X M \phi\colon X\to M
  11. ϕ \phi
  12. M M
  13. c : U n c\colon U\to\mathbb{R}^{n}\;
  14. U = ϕ - 1 ( U ) U^{\prime}=\phi^{-1}(U)\;
  15. c = c ϕ c^{\prime}=c\circ\phi\;
  16. ϕ \phi
  17. c : U n c\colon U\to\mathbb{R}^{n}\;
  18. d : V n d\colon V\to\mathbb{R}^{n}\;
  19. d c - 1 : c ( U V ) n d\circ c^{-1}\colon c(U\cap V)\to\mathbb{R}^{n}\;
  20. n \mathbb{R}^{n}
  21. ϕ - 1 ( U ) ϕ - 1 ( V ) = ϕ - 1 ( U V ) \phi^{-1}(U)\cap\phi^{-1}(V)=\phi^{-1}(U\cap V)\;
  22. c ( U V ) = ( c ϕ ) ( ϕ - 1 ( U V ) ) = c ( U V ) c^{\prime}(U^{\prime}\cap V^{\prime})=(c\circ\phi)(\phi^{-1}(U\cap V))=c(U\cap V)\;
  23. d ( c ) - 1 = ( d ϕ ) ( c ϕ ) - 1 = d ( ϕ ϕ - 1 ) c - 1 = d c - 1 d^{\prime}\circ(c^{\prime})^{-1}=(d\circ\phi)\circ(c\circ\phi)^{-1}=d\circ(% \phi\circ\phi^{-1})\circ c^{-1}=d\circ c^{-1}\;
  24. U U^{\prime}
  25. V V^{\prime}
  26. S 7 S^{7}

Transportation_theory_(mathematics).html

  1. c ( T ) := m M c ( m , T ( m ) ) c(T):=\sum_{m\in M}c(m,T(m))
  2. inf { X c ( x , T ( x ) ) d μ ( x ) | T ( μ ) = ν } , \inf\left\{\left.\int_{X}c(x,T(x))\,\mathrm{d}\mu(x)\;\right|\;T(\mu)=\nu% \right\},
  3. inf { X × Y c ( x , y ) d γ ( x , y ) | γ Γ ( μ , ν ) } , \inf\left\{\left.\int_{X\times Y}c(x,y)\,\mathrm{d}\gamma(x,y)\right|\gamma\in% \Gamma(\mu,\nu)\right\},
  4. sup ( X φ ( x ) d μ ( x ) + Y ψ ( y ) d ν ( y ) ) , \sup\left(\int_{X}\varphi(x)\,\mathrm{d}\mu(x)+\int_{Y}\psi(y)\,\mathrm{d}\nu(% y)\right),
  5. φ ( x ) + ψ ( y ) c ( x , y ) . \varphi(x)+\psi(y)\leq c(x,y).
  6. μ , ν 𝒫 p ( 𝐑 ) \mu,\nu\in\mathcal{P}_{p}(\mathbf{R})
  7. F ν - 1 F μ : 𝐑 𝐑 F_{\nu}^{-1}\circ F_{\mu}:\mathbf{R}\to\mathbf{R}
  8. min γ Γ ( μ , ν ) 𝐑 2 c ( x , y ) d γ ( x , y ) = 0 1 c ( F μ - 1 ( s ) , F ν - 1 ( s ) ) d s . \min_{\gamma\in\Gamma(\mu,\nu)}\int_{\mathbf{R}^{2}}c(x,y)\,\mathrm{d}\gamma(x% ,y)=\int_{0}^{1}c\left(F_{\mu}^{-1}(s),F_{\nu}^{-1}(s)\right)\,\mathrm{d}s.
  9. 𝒫 p ( X ) \mathcal{P}_{p}(X)
  10. 𝒫 p r ( X ) \mathcal{P}_{p}^{r}(X)
  11. μ 𝒫 p ( X ) \mu\in\mathcal{P}_{p}(X)
  12. μ 𝒫 p r ( X ) \mu\in\mathcal{P}_{p}^{r}(X)
  13. ν 𝒫 p ( X ) \nu\in\mathcal{P}_{p}(X)
  14. c ( x , y ) = | x - y | p / p c(x,y)=|x-y|^{p}/p
  15. κ = ( id X × r ) * ( μ ) Γ ( μ , ν ) . \kappa=(\mathrm{id}_{X}\times r)_{*}(\mu)\in\Gamma(\mu,\nu).
  16. r ( x ) = x - | ϕ ( x ) | q - 2 ϕ ( x ) r(x)=x-|\nabla\phi(x)|^{q-2}\nabla\phi(x)

Transverse_mass.html

  1. m T 2 = m 2 + p x 2 + p y 2 m_{T}^{2}=m^{2}+p_{x}^{2}+p_{y}^{2}\,
  2. p x p_{x}
  3. p y p_{y}
  4. m m
  5. M T 2 = ( E T , 1 + E T , 2 ) 2 - ( p T , 1 + p T , 2 ) 2 M_{T}^{2}=(E_{T,1}+E_{T,2})^{2}-(\overrightarrow{p}_{T,1}+\overrightarrow{p}_{% T,2})^{2}
  6. E T E_{T}
  7. m m
  8. E T 2 = m 2 + ( p T ) 2 E_{T}^{2}=m^{2}+(\overrightarrow{p}_{T})^{2}
  9. M T 2 = m 1 2 + m 2 2 + 2 ( E T , 1 E T , 2 - p T , 1 p T , 2 ) M_{T}^{2}=m_{1}^{2}+m_{2}^{2}+2\left(E_{T,1}E_{T,2}-\overrightarrow{p}_{T,1}% \cdot\overrightarrow{p}_{T,2}\right)
  10. m 1 = m 2 = 0 m_{1}=m_{2}=0
  11. E T = | p T | E_{T}=|\overrightarrow{p}_{T}|
  12. M T 2 2 E T , 1 E T , 2 ( 1 - cos ϕ ) M_{T}^{2}\rightarrow 2E_{T,1}E_{T,2}\left(1-\cos\phi\right)
  13. ϕ \phi
  14. M T M_{T}
  15. M T M M_{T}\leq M
  16. W W
  17. m T m_{T}
  18. M T M_{T}
  19. m T m_{T}
  20. M T M_{T}

Tree-graded_space.html

  1. X X
  2. X X

Triangle-free_graph.html

  1. Ω ( n log n ) \Omega(\sqrt{n\log n})
  2. O ( n log n ) O(\sqrt{n\log n})
  3. O ( n log n ) O(\sqrt{n\log n})
  4. Θ ( t 2 log t ) \Theta(\tfrac{t^{2}}{\log t})
  5. Ω ( t 2 log t ) \Omega(\tfrac{t^{2}}{\log t})
  6. O ( m 1 / 3 ( log m ) 2 / 3 ) O\left(\frac{m^{1/3}}{(\log m)^{2/3}}\right)

Trigonometry_in_Galois_fields.html

  1. ζ \zeta
  2. \equiv
  3. \angle
  4. ζ i \zeta^{i}
  5. ζ \zeta
  6. cos k ( ζ i ) = ( 2 - 1 mod p ) ( ζ i k + ζ - i k ) , \cos_{k}(\angle\zeta^{i})=(2^{-1}\bmod{p})\cdot(\zeta^{ik}+\zeta^{-ik}),
  7. sin k ( ζ i ) = ( 1 / j ) ( 2 - 1 mod p ) ( ζ i k - ζ - i k ) , \sin_{k}(\angle\zeta^{i})=(1/j)(2^{-1}\bmod{p})\cdot(\zeta^{ik}-\zeta^{-ik}),
  8. ζ \angle\zeta
  9. ζ \angle\zeta
  10. sin k 2 ( i ) + cos k 2 ( i ) 1. \sin_{k}^{2}(i)+\cos_{k}^{2}(i)\equiv 1.\,
  11. cos k ( i ) cos k ( - i ) . \cos_{k}(i)\equiv\cos_{k}(-i).\,
  12. sin k ( i ) - sin k ( - i ) . \sin_{k}(i)\equiv-\sin_{k}(-i).\,
  13. ζ k i cos k ( i ) + j sin k ( i ) . \zeta^{ki}\equiv\cos_{k}(i)+j\sin_{k}(i).\,
  14. cos k ( i + t ) cos k ( i ) cos k ( t ) - sin k ( i ) sin k ( t ) , \cos_{k}(i+t)\equiv\cos_{k}(i)\cos_{k}(t)-\sin_{k}(i)\sin_{k}(t),
  15. sin k ( i + t ) sin k ( i ) cos k ( t ) + sin k ( t ) cos k ( i ) . \sin_{k}(i+t)\equiv\sin_{k}(i)\cos_{k}(t)+\sin_{k}(t)\cos_{k}(i).
  16. cos k 2 ( i ) ( 2 - 1 mod p ) ( 1 + cos k ( 2 i ) ) , \cos_{k}^{2}(i)\equiv(2^{-1}\bmod{p})\cdot(1+\cos_{k}(2i)),
  17. sin k 2 ( i ) ( 2 - 1 mod p ) ( 1 - cos k ( 2 i ) . \sin_{k}^{2}(i)\equiv(2^{-1}\bmod{p})\cdot(1-\cos_{k}(2i).
  18. cos k ( i ) cos i ( k ) , \cos_{k}(i)\equiv\cos_{i}(k),
  19. sin k ( i ) sin i ( k ) . \sin_{k}(i)\equiv\sin_{i}(k).
  20. ( k + t ) = ( i + r ) = N , (k+t)=(i+r)=N,\,
  21. cos k ( i ) cos r ( t ) , \cos_{k}(i)\equiv\cos_{r}(t),\,
  22. sin k ( i ) sin r ( t ) . \sin_{k}(i)\equiv\sin_{r}(t).\,
  23. cos k ( i + N ) cos k ( i ) , \cos_{k}(i+N)\equiv\cos_{k}(i),\,
  24. sin k ( i + N ) sin k ( i ) . \sin_{k}(i+N)\equiv\sin_{k}(i).\,
  25. ( i + t ) = N , (i+t)=N,\,
  26. cos k ( i ) cos k ( t ) , \cos_{k}(i)\equiv\cos_{k}(t),\,
  27. sin k ( i ) - sin k ( t ) . \sin_{k}(i)\equiv-\sin_{k}(t).\,
  28. cos k ( i ) \cos_{k}(i)
  29. k = 0 N - 1 cos k ( i ) . \sum_{k=0}^{N-1}\cos_{k}(i)\equiv.
  30. sin k ( i ) \sin_{k}(i)
  31. k = 0 N - 1 sin k ( i ) 0. \sum_{k=0}^{N-1}\sin_{k}(i)\equiv 0.
  32. k = 0 N - 1 [ cos k ( i ) sin k ( t ) ] 0. \sum_{k=0}^{N-1}[\cos_{k}(i)\sin_{k}(t)]\equiv 0.
  33. \equiv
  34. ζ \zeta\in
  35. ζ \zeta\in
  36. ζ \zeta
  37. G θ G_{\theta}
  38. ζ = r ϵ θ \zeta=r\cdot\epsilon^{\theta}
  39. \mathbb{R}
  40. a G F ( p ) a\in GF(p)
  41. | a | = { a , if a ( p - 1 ) / 2 1 mod p , - a , if a ( p - 1 ) / 2 - 1 mod p . \mathcal{|}a|=\begin{cases}a,&\textrm{if }a^{(p-1)/2}\equiv 1\bmod{p},\\ -a,&\textrm{if }a^{(p-1)/2}\equiv-1\bmod{p}.\end{cases}
  42. a + j b = | a 2 + b 2 | . \mid a+jb\mid=\left|\sqrt{\mid a^{2}+b^{2}\mid}\right|.
  43. θ \theta
  44. θ \theta
  45. ζ \zeta
  46. ζ / r = ϵ θ \zeta/r=\epsilon^{\theta}
  47. ζ \zeta
  48. ϵ \epsilon
  49. ϵ \epsilon
  50. θ \theta
  51. j θ j\theta
  52. \mathbb{C}
  53. \equiv
  54. \equiv
  55. G θ G_{\theta}
  56. \equiv
  57. r = | ( | 6 2 + 16 2 | ) | | | 13 | | r=|\sqrt{(|6^{2}+16^{2}|)}|\equiv|\sqrt{|13|}|\equiv
  58. ϵ = ζ \epsilon=\zeta
  59. \equiv
  60. β = ϵ \beta=\epsilon
  61. ϵ 64 / N \epsilon^{64/N}
  62. r = | ( | 6 2 + 4 2 | ) | | | 3 | | r=|\sqrt{(|6^{2}+4^{2}|)}|\equiv|\sqrt{|3|}|\equiv
  63. \equiv
  64. \equiv
  65. \equiv
  66. θ \theta
  67. ζ \zeta
  68. ζ \zeta
  69. \equiv
  70. cos ( i ) = ( 2 - 1 mod p ) ( ϵ i + ϵ - i ) , \cos(i)=(2^{-1}\bmod{p})\cdot(\epsilon^{i}+\epsilon^{-i}),
  71. sin ( i ) = ( 1 / j ) ( 2 - 1 mod p ) ( ϵ i - ϵ - i ) , \sin(i)=(1/j)(2^{-1}\bmod{p})\cdot(\epsilon^{i}-\epsilon^{-i}),
  72. ϵ \epsilon
  73. ζ \zeta
  74. ϵ \epsilon
  75. \in
  76. ζ \zeta
  77. ϵ \epsilon
  78. ϵ \epsilon
  79. ϵ i \epsilon^{i}
  80. ϵ - i \epsilon^{-i}
  81. ϵ i \epsilon^{i}
  82. ϵ - i \epsilon^{-i}
  83. ϵ \epsilon
  84. cos ( i ) = Re { ϵ i } \cos(i)=\mathrm{Re}\{\epsilon^{i}\}
  85. sin ( i ) = Im { ϵ i } \sin(i)=\mathrm{Im}\{\epsilon^{i}\}
  86. ϵ \epsilon
  87. ζ \zeta
  88. \equiv
  89. ζ \zeta

Triple_product_rule.html

  1. ( x y ) z ( y z ) x ( z x ) y = - 1. \left(\frac{\partial x}{\partial y}\right)_{z}\left(\frac{\partial y}{\partial z% }\right)_{x}\left(\frac{\partial z}{\partial x}\right)_{y}=-1.
  2. ( x y ) z = - ( z y ) x ( z x ) y \left(\frac{\partial x}{\partial y}\right)_{z}=-\frac{\left(\frac{\partial z}{% \partial y}\right)_{x}}{\left(\frac{\partial z}{\partial x}\right)_{y}}
  3. d z = ( z x ) y d x + ( z y ) x d y dz=\left(\frac{\partial z}{\partial x}\right)_{y}dx+\left(\frac{\partial z}{% \partial y}\right)_{x}dy
  4. d y = ( y x ) z d x dy=\left(\frac{\partial y}{\partial x}\right)_{z}dx
  5. 0 = ( z x ) y d x + ( z y ) x ( y x ) z d x 0=\left(\frac{\partial z}{\partial x}\right)_{y}\,dx+\left(\frac{\partial z}{% \partial y}\right)_{x}\left(\frac{\partial y}{\partial x}\right)_{z}\,dx
  6. ( z x ) y = - ( z y ) x ( y x ) z \left(\frac{\partial z}{\partial x}\right)_{y}=-\left(\frac{\partial z}{% \partial y}\right)_{x}\left(\frac{\partial y}{\partial x}\right)_{z}
  7. ( x y ) z ( y z ) x ( z x ) y = - 1 \left(\frac{\partial x}{\partial y}\right)_{z}\left(\frac{\partial y}{\partial z% }\right)_{x}\left(\frac{\partial z}{\partial x}\right)_{y}=-1
  8. d x = ( x y ) z d y + ( x z ) y d z dx=\left(\frac{\partial x}{\partial y}\right)_{z}dy+\left(\frac{\partial x}{% \partial z}\right)_{y}dz
  9. d y = ( y x ) z d x + ( y z ) x d z dy=\left(\frac{\partial y}{\partial x}\right)_{z}dx+\left(\frac{\partial y}{% \partial z}\right)_{x}dz
  10. d z = ( z x ) y d x + ( z y ) x d y dz=\left(\frac{\partial z}{\partial x}\right)_{y}dx+\left(\frac{\partial z}{% \partial y}\right)_{x}dy
  11. d x = ( x y ) z [ ( y x ) z d x + ( y z ) x d z ] + ( x z ) y d z dx=\left(\frac{\partial x}{\partial y}\right)_{z}\left[\left(\frac{\partial y}% {\partial x}\right)_{z}dx+\left(\frac{\partial y}{\partial z}\right)_{x}dz% \right]+\left(\frac{\partial x}{\partial z}\right)_{y}dz
  12. d x = ( x y ) z ( y x ) z d x + ( x y ) z ( y z ) x ( z x ) y d x + ( x y ) z ( y z ) x ( z y ) x d y + ( x z ) y ( z x ) y d x + ( x z ) y ( z y ) x d y dx=\left(\frac{\partial x}{\partial y}\right)_{z}\left(\frac{\partial y}{% \partial x}\right)_{z}dx+\left(\frac{\partial x}{\partial y}\right)_{z}\left(% \frac{\partial y}{\partial z}\right)_{x}\left(\frac{\partial z}{\partial x}% \right)_{y}dx+\left(\frac{\partial x}{\partial y}\right)_{z}\left(\frac{% \partial y}{\partial z}\right)_{x}\left(\frac{\partial z}{\partial y}\right)_{% x}dy+\left(\frac{\partial x}{\partial z}\right)_{y}\left(\frac{\partial z}{% \partial x}\right)_{y}dx+\left(\frac{\partial x}{\partial z}\right)_{y}\left(% \frac{\partial z}{\partial y}\right)_{x}dy
  13. 1 = ( x y ) z ( y x ) z + ( x y ) z ( y z ) x ( z x ) y + ( x z ) y ( z x ) y 1=\left(\frac{\partial x}{\partial y}\right)_{z}\left(\frac{\partial y}{% \partial x}\right)_{z}+\left(\frac{\partial x}{\partial y}\right)_{z}\left(% \frac{\partial y}{\partial z}\right)_{x}\left(\frac{\partial z}{\partial x}% \right)_{y}+\left(\frac{\partial x}{\partial z}\right)_{y}\left(\frac{\partial z% }{\partial x}\right)_{y}
  14. ( x y ) z ( y z ) x ( z x ) y = - 1 \left(\frac{\partial x}{\partial y}\right)_{z}\left(\frac{\partial y}{\partial z% }\right)_{x}\left(\frac{\partial z}{\partial x}\right)_{y}=-1

Triruthenium_dodecacarbonyl.html

  1. \overrightarrow{\leftarrow}

Truncated_distribution.html

  1. X X
  2. f ( x ) f(x)
  3. F ( x ) F(x)
  4. y = ( a , b ] y=(a,b]
  5. X X
  6. a < X b a<X\leq b
  7. f ( x | a < X b ) = g ( x ) F ( b ) - F ( a ) = T r ( x ) f(x|a<X\leq b)=\frac{g(x)}{F(b)-F(a)}=Tr(x)
  8. g ( x ) = f ( x ) g(x)=f(x)
  9. a < x b a<x\leq b
  10. g ( x ) = 0 g(x)=0
  11. T r ( x ) Tr(x)
  12. g ( x ) g(x)
  13. g ( x ) g(x)
  14. f ( x ) f(x)
  15. T r ( x ) Tr(x)
  16. g ( x ) g(x)
  17. T r ( x ) Tr(x)
  18. T r ( x ) Tr(x)
  19. f ( x | a < X b ) f(x|a<X\leq b)
  20. a b f ( x | a < X b ) d x = 1 F ( b ) - F ( a ) a b g ( x ) d x = 1 \int_{a}^{b}f(x|a<X\leq b)dx=\frac{1}{F(b)-F(a)}\int_{a}^{b}g(x)dx=1
  21. f ( x | X > y ) = g ( x ) 1 - F ( y ) f(x|X>y)=\frac{g(x)}{1-F(y)}
  22. g ( x ) = f ( x ) g(x)=f(x)
  23. y < x y<x
  24. g ( x ) = 0 g(x)=0
  25. F ( x ) F(x)
  26. f ( x | X y ) = g ( x ) F ( y ) f(x|X\leq y)=\frac{g(x)}{F(y)}
  27. g ( x ) = f ( x ) g(x)=f(x)
  28. x y x\leq y
  29. g ( x ) = 0 g(x)=0
  30. F ( x ) F(x)
  31. f ( x ) f(x)
  32. F ( x ) F(x)
  33. X X
  34. y y
  35. E ( X | X > y ) = y x g ( x ) d x 1 - F ( y ) E(X|X>y)=\frac{\int_{y}^{\infty}xg(x)dx}{1-F(y)}
  36. g ( x ) g(x)
  37. g ( x ) = f ( x ) g(x)=f(x)
  38. y < x y<x
  39. g ( x ) = 0 g(x)=0
  40. a a
  41. b b
  42. f ( x ) f(x)
  43. E ( u ( X ) | X > y ) E(u(X)|X>y)
  44. u ( X ) u(X)
  45. X X
  46. f ( x ) f(x)
  47. lim y a E ( u ( X ) | X > y ) = E ( u ( X ) ) \lim_{y\to a}E(u(X)|X>y)=E(u(X))
  48. lim y b E ( u ( X ) | X > y ) = u ( b ) \lim_{y\to b}E(u(X)|X>y)=u(b)
  49. y [ E ( u ( X ) | X > y ) ] = f ( y ) 1 - F ( y ) [ E ( u ( X ) | X > y ) - u ( y ) ] \frac{\partial}{\partial y}[E(u(X)|X>y)]=\frac{f(y)}{1-F(y)}[E(u(X)|X>y)-u(y)]
  50. lim y a y [ E ( u ( X ) | X > y ) ] = f ( a ) [ E ( u ( X ) ) - u ( a ) ] \lim_{y\to a}\frac{\partial}{\partial y}[E(u(X)|X>y)]=f(a)[E(u(X))-u(a)]
  51. lim y b y [ E ( u ( X ) | X > y ) ] = 1 2 u ( b ) \lim_{y\to b}\frac{\partial}{\partial y}[E(u(X)|X>y)]=\frac{1}{2}u^{\prime}(b)
  52. lim y c u ( y ) = u ( c ) \lim_{y\to c}u^{\prime}(y)=u^{\prime}(c)
  53. lim y c u ( y ) = u ( c ) \lim_{y\to c}u(y)=u(c)
  54. lim y c f ( y ) = f ( c ) \lim_{y\to c}f(y)=f(c)
  55. c c
  56. a a
  57. b b
  58. t t
  59. g ( t ) g(t)
  60. x x
  61. f ( x | t ) = T r ( x ) f(x|t)=Tr(x)
  62. x x
  63. t t
  64. f ( x ) = x f ( x | t ) g ( t ) d t f(x)=\int_{x}^{\infty}f(x|t)g(t)dt
  65. F ( a ) = - a [ x f ( x | t ) g ( t ) d t ] d x . F(a)=\int_{-\infty}^{a}\left[\int_{x}^{\infty}f(x|t)g(t)dt\right]dx.
  66. t t
  67. x x
  68. t t
  69. x x
  70. f ( x ) f(x)
  71. F ( x ) F(x)
  72. g ( t | x ) = f ( x | t ) g ( t ) f ( x ) , g(t|x)=\frac{f(x|t)g(t)}{f(x)},
  73. g ( t | x ) = f ( x | t ) g ( t ) x f ( x | t ) g ( t ) d t . g(t|x)=\frac{f(x|t)g(t)}{\int_{x}^{\infty}f(x|t)g(t)dt}.
  74. g ( t | x ) = f ( x | t ) g ( t ) f ( x ) = 1 t ( ln ( T ) - ln ( x ) ) for all t > x . g(t|x)=\frac{f(x|t)g(t)}{f(x)}=\frac{1}{t(\ln(T)-\ln(x))}\quad\,\text{for all % }t>x.

Tryptophan_hydroxylase.html

  1. \rightleftharpoons

Tulip_Overlay.html

  1. n n
  2. O ( n l o g n ) O(\sqrt{n}logn)
  3. n l o g n \sqrt{n}logn
  4. n \sqrt{n}
  5. 2 n 2\sqrt{n}
  6. n \sqrt{n}
  7. n l o g n \sqrt{n}logn
  8. u u
  9. l o g n logn
  10. u u
  11. u u
  12. u u
  13. l o g n logn
  14. s s
  15. t t
  16. s s
  17. t t
  18. s s
  19. t t
  20. s s
  21. w w
  22. t t
  23. t t
  24. w w
  25. x x
  26. o o
  27. x x
  28. o o
  29. o o
  30. s s
  31. o o
  32. s s
  33. o o
  34. s s
  35. o o
  36. w w
  37. o o
  38. t t
  39. w w
  40. o o

Tunnell's_theorem.html

  1. A n = # { ( x , y , z ) 3 | n = 2 x 2 + y 2 + 32 z 2 } B n = # { ( x , y , z ) 3 | n = 2 x 2 + y 2 + 8 z 2 } C n = # { ( x , y , z ) 3 | n = 8 x 2 + 2 y 2 + 64 z 2 } D n = # { ( x , y , z ) 3 | n = 8 x 2 + 2 y 2 + 16 z 2 } . \begin{matrix}A_{n}&=&\#\{(x,y,z)\in\mathbb{Z}^{3}|n=2x^{2}+y^{2}+32z^{2}\}\\ B_{n}&=&\#\{(x,y,z)\in\mathbb{Z}^{3}|n=2x^{2}+y^{2}+8z^{2}\}\\ C_{n}&=&\#\{(x,y,z)\in\mathbb{Z}^{3}|n=8x^{2}+2y^{2}+64z^{2}\}\\ D_{n}&=&\#\{(x,y,z)\in\mathbb{Z}^{3}|n=8x^{2}+2y^{2}+16z^{2}\}.\end{matrix}
  2. y 2 = x 3 - n 2 x y^{2}=x^{3}-n^{2}x
  3. - n , , n -\sqrt{n},\ldots,\sqrt{n}

Turbulence_kinetic_energy.html

  1. k = 1 2 ( ( u 1 ) 2 ¯ + ( u 2 ) 2 ¯ + ( u 3 ) 2 ¯ ) . k=\frac{1}{2}\left(\overline{(u^{\prime}_{1})^{2}}+\overline{(u^{\prime}_{2})^% {2}}+\overline{(u^{\prime}_{3})^{2}}\right).
  2. D k D t + T = P - ϵ , \frac{Dk}{Dt}+\nabla\cdot T^{\prime}=P-\epsilon,
  3. D k / D t Dk/Dt
  4. T \nabla\cdot T^{\prime}
  5. P P
  6. ϵ \epsilon
  7. k t Local derivative + u ¯ j k x j Advection = - 1 ρ o u i p ¯ x i Pressure diffusion - 1 2 u j u j u i ¯ x i Turbulent transport 𝒯 + ν 2 k x j 2 Molecular viscous transport - u i u j ¯ u i ¯ x j Production 𝒫 - ν u i x j u i x j ¯ Dissipation ϵ k - g ρ o ρ u i ¯ δ i 3 Buoyancy flux b \underbrace{\frac{\partial k}{\partial t}}_{\begin{smallmatrix}\,\text{Local}% \\ \,\text{derivative}\end{smallmatrix}}+\underbrace{\overline{u}_{j}\frac{% \partial k}{\partial x_{j}}}_{\begin{smallmatrix}\,\text{Advection}\end{% smallmatrix}}=-\underbrace{\frac{1}{\rho_{o}}\frac{\partial\overline{u^{\prime% }_{i}p^{\prime}}}{\partial x_{i}}}_{\begin{smallmatrix}\,\text{Pressure}\\ \,\text{diffusion}\end{smallmatrix}}-\underbrace{\frac{1}{2}\frac{\partial% \overline{u_{j}^{\prime}u_{j}^{\prime}u_{i}^{\prime}}}{\partial x_{i}}}_{% \begin{smallmatrix}\,\text{Turbulent}\\ \,\text{transport}\\ \mathcal{T}\end{smallmatrix}}+\underbrace{\nu\frac{\partial^{2}k}{\partial x^{% 2}_{j}}}_{\begin{smallmatrix}\,\text{Molecular}\\ \,\text{viscous}\\ \,\text{transport}\end{smallmatrix}}\underbrace{-\overline{u^{\prime}_{i}u^{% \prime}_{j}}\frac{\partial\overline{u_{i}}}{\partial x_{j}}}_{\begin{% smallmatrix}\,\text{Production}\\ \mathcal{P}\end{smallmatrix}}-\underbrace{\nu\overline{\frac{\partial u^{% \prime}_{i}}{\partial x_{j}}\frac{\partial u^{\prime}_{i}}{\partial x_{j}}}}_{% \begin{smallmatrix}\,\text{Dissipation}\\ \epsilon_{k}\end{smallmatrix}}-\underbrace{\frac{g}{\rho_{o}}\overline{\rho^{% \prime}u^{\prime}_{i}}\delta_{i3}}_{\begin{smallmatrix}\,\text{Buoyancy flux}% \\ b\end{smallmatrix}}
  8. u i u j ¯ = 2 / 3 k δ i j - ν t ( u i ¯ x j + u j ¯ x i ) , \overline{u^{\prime}_{i}u^{\prime}_{j}}=2/3k\delta_{ij}-\nu_{t}\left(\frac{% \partial\overline{u_{i}}}{\partial x_{j}}+\frac{\partial\overline{u_{j}}}{% \partial x_{i}}\right),
  9. ν t = c k 1 / 2 l m . \nu_{t}=c\cdot k^{1/2}l_{m}.
  10. u 2 ¯ = v 2 ¯ = w 2 ¯ . \overline{u^{\prime 2}}=\overline{v^{\prime 2}}=\overline{w^{\prime 2}}.
  11. ϵ \epsilon
  12. k = 3 2 ( U I ) 2 , k=\frac{3}{2}(UI)^{2},
  13. I I
  14. U U
  15. ϵ = c μ 3 / 4 k 3 / 2 l - 1 . \epsilon={c_{\mu}}^{3/4}k^{3/2}l^{-1}.
  16. l l
  17. c μ {c_{\mu}}
  18. I = 0.16 R e - 1 / 8 . I=0.16Re^{-1/8}.
  19. l = 0.07 L , l=0.07L,
  20. L L

Turing_machine_equivalents.html

  1. δ \delta

Turn_(biochemistry).html

  1. i i ± 4 i\rightarrow i\pm 4
  2. i i ± 3 i\rightarrow i\pm 3
  3. i i ± 2 i\rightarrow i\pm 2
  4. i i ± 1 i\rightarrow i\pm 1
  5. i i ± 5 i\rightarrow i\pm 5
  6. β \beta
  7. ( i + 2 ) (i+2)
  8. ϕ i + 1 \phi_{i+1}
  9. ψ i + 1 \psi_{i+1}
  10. ϕ i + 2 \phi_{i+2}
  11. ψ i + 2 \psi_{i+2}
  12. ( I , I + 1 ) (I,I+1)

Turnover_(employment).html

  1. ( N E L D Y ( N E B Y + N E E Y ) / 2 ) × 100 % \left(\frac{NELDY}{(NEBY+NEEY)/2}\right)\times 100\%

Turnstile_(symbol).html

  1. \vdash
  2. \vdash
  3. A A
  4. A \vdash A
  5. A A
  6. P Q P\vdash Q
  7. P P
  8. Q Q
  9. \vdash
  10. P Q P\vdash Q
  11. Q Q
  12. P P
  13. \vdash
  14. Q \vdash Q
  15. Q Q
  16. T T
  17. S S
  18. T S T\vdash S
  19. S S
  20. T T
  21. F G F\dashv G
  22. F F
  23. G G
  24. λ n \lambda\vdash n
  25. λ \lambda
  26. n n
  27. \models

Two-photon_absorption.html

  1. I ( x ) = I 0 e - α c x I(x)=I_{0}e^{-\alpha\,c\,x}\,
  2. I ( x ) = I 0 1 + β c x I 0 I(x)=\frac{I_{0}}{1+\beta cxI_{0}}\,
  3. 1 / λ 4 1/\lambda^{4}
  4. λ \lambda
  5. - d I d z = α I + β I 2 -\frac{dI}{dz}=\alpha I+\beta I^{2}
  6. β ( ω ) = 2 ω I 2 W T ( 2 ) ( ω ) = N E σ ( 2 ) \beta(\omega)=\frac{2\hbar\omega}{I^{2}}W_{T}^{(2)}(\omega)=\frac{N}{E}\sigma^% {(2)}
  7. β \beta
  8. α \alpha
  9. W T ( 2 ) ( ω ) W_{T}^{(2)}(\omega)
  10. I I
  11. ħ ħ
  12. ω \omega
  13. d z dz
  14. β × 10 11 \beta\times 10^{11}

U_(disambiguation).html

  1. \cup

Ugly_duckling_theorem.html

  1. 2 n 2^{n}
  2. 2 n - 1 2^{n-1}
  3. 2 n 2^{n}
  4. 2 n / 2 2^{n}/2
  5. 2 n / 2 2^{n}/2
  6. 2 n / 4 2^{n}/4
  7. 2 n / 4 2^{n}/4
  8. 2 n / 4 2^{n}/4
  9. x 1 , x 2 , , x n x_{1},x_{2},\dots,x_{n}
  10. k k
  11. k k
  12. 2 k 2^{k}
  13. i i
  14. i i
  15. x x
  16. i i
  17. y y
  18. k k
  19. k k
  20. ( f , g ) (f,g)
  21. f f
  22. i i
  23. g g
  24. f f
  25. ( f , g ) (f,g)
  26. x x
  27. y y

Ulcer_index.html

  1. R i = 100 × p r i c e i - m a x p r i c e m a x p r i c e R_{i}=100\times{price_{i}-maxprice\over maxprice}
  2. U l c e r = R 1 2 + R 2 2 + R N 2 N Ulcer=\sqrt{R_{1}^{2}+R_{2}^{2}+\cdots R_{N}^{2}\over N}
  3. S h a r p e r a t i o = R e t u r n - R i s k F r e e R e t u r n s t a n d a r d d e v i a t i o n Sharpe\,ratio={Return-RiskFreeReturn\over standard\,deviation}
  4. U P I = R e t u r n - R i s k F r e e R e t u r n u l c e r i n d e x UPI={Return-RiskFreeReturn\over ulcer\,index}

Ultimate_oscillator.html

  1. b p = c l o s e - min ( l o w , p r e v c l o s e ) bp=close-\min(low,prev\,close)
  2. t r = max ( h i g h , p r e v c l o s e ) - min ( l o w , p r e v c l o s e ) tr=\max(high,prev\,close)-\min(low,prev\,close)
  3. b p 1 bp_{1}
  4. b p 2 bp_{2}
  5. a v g 7 = b p 1 + b p 2 + + b p 7 t r 1 + t r 2 + + t r 7 avg_{7}={bp_{1}+bp_{2}+\cdots+bp_{7}\over tr_{1}+tr_{2}+\cdots+tr_{7}}
  6. U l t O s c = 100 × 4 × a v g 7 + 2 × a v g 14 + a v g 28 4 + 2 + 1 UltOsc=100\times{4\times avg_{7}+2\times avg_{14}+avg_{28}\over 4+2+1}

Ultraconnected_space.html

  1. X X
  2. X X
  3. T 1 T_{1}

Uncertainty_quantification.html

  1. y e ( 𝐱 ) = y m ( 𝐱 ) + δ ( 𝐱 ) + ε y^{e}(\mathbf{x})=y^{m}(\mathbf{x})+\delta(\mathbf{x})+\varepsilon
  2. y e ( 𝐱 ) y^{e}(\mathbf{x})
  3. 𝐱 \mathbf{x}
  4. y m ( 𝐱 ) y^{m}(\mathbf{x})
  5. δ ( 𝐱 ) \delta(\mathbf{x})
  6. ε \varepsilon
  7. δ ( 𝐱 ) \delta(\mathbf{x})
  8. y m ( 𝐱 ) + δ ( 𝐱 ) y^{m}(\mathbf{x})+\delta(\mathbf{x})
  9. y e ( 𝐱 ) = y m ( 𝐱 , s y m b o l θ * ) + ε y^{e}(\mathbf{x})=y^{m}(\mathbf{x},symbol{\theta}^{*})+\varepsilon
  10. y m ( 𝐱 , s y m b o l θ ) y^{m}(\mathbf{x},symbol{\theta})
  11. s y m b o l θ symbol{\theta}
  12. s y m b o l θ * symbol{\theta}^{*}
  13. s y m b o l θ * symbol{\theta}^{*}
  14. s y m b o l θ * symbol{\theta}^{*}
  15. y e ( 𝐱 ) = y m ( 𝐱 , s y m b o l θ * ) + δ ( 𝐱 ) + ε y^{e}(\mathbf{x})=y^{m}(\mathbf{x},symbol{\theta}^{*})+\delta(\mathbf{x})+\varepsilon
  16. y m ( 𝐱 , s y m b o l θ ) 𝒢 𝒫 ( 𝐡 m ( ) T s y m b o l β m , σ m 2 R m ( , ) ) y^{m}(\mathbf{x},symbol{\theta})\sim\mathcal{GP}\big(\mathbf{h}^{m}(\cdot)^{T}% symbol{\beta}^{m},\sigma_{m}^{2}R^{m}(\cdot,\cdot)\big)
  17. R m ( ( 𝐱 , s y m b o l θ ) , ( 𝐱 , s y m b o l θ ) ) = exp { - k = 1 d ω k m ( x k - x k ) 2 } exp { - k = 1 r ω d + k m ( θ k - θ k ) 2 } . R^{m}\big((\mathbf{x},symbol{\theta}),(\mathbf{x}^{\prime},symbol{\theta}^{% \prime})\big)=\exp\left\{-\sum_{k=1}^{d}\omega_{k}^{m}(x_{k}-x_{k}^{\prime})^{% 2}\right\}\exp\left\{-\sum_{k=1}^{r}\omega_{d+k}^{m}(\theta_{k}-\theta_{k}^{% \prime})^{2}\right\}.
  18. d d
  19. r r
  20. 𝐡 m ( ) \mathbf{h}^{m}(\cdot)
  21. { s y m b o l β m , σ m , ω k m , k = 1 , , d + r } \left\{symbol{\beta}^{m},\sigma_{m},\omega_{k}^{m},k=1,\ldots,d+r\right\}
  22. δ ( 𝐱 ) 𝒢 𝒫 ( 𝐡 δ ( ) T s y m b o l β δ , σ δ 2 R δ ( , ) ) \delta(\mathbf{x})\sim\mathcal{GP}\big(\mathbf{h}^{\delta}(\cdot)^{T}symbol{% \beta}^{\delta},\sigma_{\delta}^{2}R^{\delta}(\cdot,\cdot)\big)
  23. R δ ( 𝐱 , 𝐱 ) = exp { - k = 1 d ω k δ ( x k - x k ) 2 } . R^{\delta}(\mathbf{x},\mathbf{x}^{\prime})=\exp\left\{-\sum_{k=1}^{d}\omega_{k% }^{\delta}(x_{k}-x_{k}^{\prime})^{2}\right\}.
  24. { s y m b o l β δ , σ δ , ω k δ , k = 1 , , d } \left\{symbol{\beta}^{\delta},\sigma_{\delta},\omega_{k}^{\delta},k=1,\ldots,d\right\}
  25. s y m b o l β m symbol{\beta}^{m}
  26. p ( s y m b o l θ | data , symbol ϕ ) p ( data | symbol θ , symbol ϕ ) p ( symbol θ ) p(symbol{\theta}|\rm{data},symbol{\phi})\propto p(\rm{data}|symbol{\theta},% symbol{\phi})p(symbol{\theta})
  27. s y m b o l ϕ symbol{\phi}
  28. s y m b o l θ symbol{\theta}
  29. s y m b o l ϕ symbol{\phi}
  30. p ( s y m b o l θ , s y m b o l ϕ | data ) p(symbol{\theta},symbol{\phi}|\rm{data})
  31. s y m b o l ϕ symbol{\phi}
  32. p ( s y m b o l θ | data ) p(symbol{\theta}|\rm{data})
  33. p ( s y m b o l θ , s y m b o l ϕ | data ) p(symbol{\theta},symbol{\phi}|\rm{data})
  34. s y m b o l ϕ symbol{\phi}
  35. s y m b o l ϕ symbol{\phi}

Underground_nuclear_weapons_testing.html

  1. y 3 \sqrt[3]{y}

Uniform_6-polytope.html

  1. A ~ 5 {\tilde{A}}_{5}
  2. C ~ 5 {\tilde{C}}_{5}
  3. B ~ 5 {\tilde{B}}_{5}
  4. D ~ 5 {\tilde{D}}_{5}
  5. A ~ 5 {\tilde{A}}_{5}
  6. C ~ 5 {\tilde{C}}_{5}
  7. B ~ 5 {\tilde{B}}_{5}
  8. D ~ 5 {\tilde{D}}_{5}
  9. A ~ 4 {\tilde{A}}_{4}
  10. I ~ 1 {\tilde{I}}_{1}
  11. B ~ 4 {\tilde{B}}_{4}
  12. I ~ 1 {\tilde{I}}_{1}
  13. C ~ 4 {\tilde{C}}_{4}
  14. I ~ 1 {\tilde{I}}_{1}
  15. D ~ 4 {\tilde{D}}_{4}
  16. I ~ 1 {\tilde{I}}_{1}
  17. F ~ 4 {\tilde{F}}_{4}
  18. I ~ 1 {\tilde{I}}_{1}
  19. C ~ 3 {\tilde{C}}_{3}
  20. I ~ 1 {\tilde{I}}_{1}
  21. I ~ 1 {\tilde{I}}_{1}
  22. B ~ 3 {\tilde{B}}_{3}
  23. I ~ 1 {\tilde{I}}_{1}
  24. I ~ 1 {\tilde{I}}_{1}
  25. A ~ 3 {\tilde{A}}_{3}
  26. I ~ 1 {\tilde{I}}_{1}
  27. I ~ 1 {\tilde{I}}_{1}
  28. C ~ 2 {\tilde{C}}_{2}
  29. I ~ 1 {\tilde{I}}_{1}
  30. I ~ 1 {\tilde{I}}_{1}
  31. I ~ 1 {\tilde{I}}_{1}
  32. H ~ 2 {\tilde{H}}_{2}
  33. I ~ 1 {\tilde{I}}_{1}
  34. I ~ 1 {\tilde{I}}_{1}
  35. I ~ 1 {\tilde{I}}_{1}
  36. A ~ 2 {\tilde{A}}_{2}
  37. I ~ 1 {\tilde{I}}_{1}
  38. I ~ 1 {\tilde{I}}_{1}
  39. I ~ 1 {\tilde{I}}_{1}
  40. I ~ 1 {\tilde{I}}_{1}
  41. I ~ 1 {\tilde{I}}_{1}
  42. I ~ 1 {\tilde{I}}_{1}
  43. I ~ 1 {\tilde{I}}_{1}
  44. I ~ 1 {\tilde{I}}_{1}
  45. A ~ 2 {\tilde{A}}_{2}
  46. A ~ 2 {\tilde{A}}_{2}
  47. I ~ 1 {\tilde{I}}_{1}
  48. A ~ 2 {\tilde{A}}_{2}
  49. B ~ 2 {\tilde{B}}_{2}
  50. I ~ 1 {\tilde{I}}_{1}
  51. A ~ 2 {\tilde{A}}_{2}
  52. G ~ 2 {\tilde{G}}_{2}
  53. I ~ 1 {\tilde{I}}_{1}
  54. B ~ 2 {\tilde{B}}_{2}
  55. B ~ 2 {\tilde{B}}_{2}
  56. I ~ 1 {\tilde{I}}_{1}
  57. B ~ 2 {\tilde{B}}_{2}
  58. G ~ 2 {\tilde{G}}_{2}
  59. I ~ 1 {\tilde{I}}_{1}
  60. G ~ 2 {\tilde{G}}_{2}
  61. G ~ 2 {\tilde{G}}_{2}
  62. I ~ 1 {\tilde{I}}_{1}
  63. A ~ 3 {\tilde{A}}_{3}
  64. A ~ 2 {\tilde{A}}_{2}
  65. B ~ 3 {\tilde{B}}_{3}
  66. A ~ 2 {\tilde{A}}_{2}
  67. C ~ 3 {\tilde{C}}_{3}
  68. A ~ 2 {\tilde{A}}_{2}
  69. A ~ 3 {\tilde{A}}_{3}
  70. B ~ 2 {\tilde{B}}_{2}
  71. B ~ 3 {\tilde{B}}_{3}
  72. B ~ 2 {\tilde{B}}_{2}
  73. C ~ 3 {\tilde{C}}_{3}
  74. B ~ 2 {\tilde{B}}_{2}
  75. A ~ 3 {\tilde{A}}_{3}
  76. G ~ 2 {\tilde{G}}_{2}
  77. B ~ 3 {\tilde{B}}_{3}
  78. G ~ 2 {\tilde{G}}_{2}
  79. C ~ 3 {\tilde{C}}_{3}
  80. G ~ 2 {\tilde{G}}_{2}
  81. P ¯ 5 {\bar{P}}_{5}
  82. A U ^ 5 {\widehat{AU}}_{5}
  83. A R ^ 5 {\widehat{AR}}_{5}
  84. S ¯ 5 {\bar{S}}_{5}
  85. O ¯ 5 {\bar{O}}_{5}
  86. N ¯ 5 {\bar{N}}_{5}
  87. U ¯ 5 {\bar{U}}_{5}
  88. X ¯ 5 {\bar{X}}_{5}
  89. R ¯ 5 {\bar{R}}_{5}
  90. Q ¯ 5 {\bar{Q}}_{5}
  91. M ¯ 5 {\bar{M}}_{5}
  92. L ¯ 5 {\bar{L}}_{5}

Uniform_7-polytope.html

  1. A ~ 6 {\tilde{A}}_{6}
  2. C ~ 6 {\tilde{C}}_{6}
  3. B ~ 6 {\tilde{B}}_{6}
  4. D ~ 6 {\tilde{D}}_{6}
  5. E ~ 6 {\tilde{E}}_{6}
  6. A ~ 6 {\tilde{A}}_{6}
  7. C ~ 6 {\tilde{C}}_{6}
  8. B ~ 6 {\tilde{B}}_{6}
  9. C ~ 6 {\tilde{C}}_{6}
  10. D ~ 6 {\tilde{D}}_{6}
  11. B ~ 6 {\tilde{B}}_{6}
  12. C ~ 6 {\tilde{C}}_{6}
  13. E ~ 6 {\tilde{E}}_{6}
  14. A ~ 5 {\tilde{A}}_{5}
  15. I ~ 1 {\tilde{I}}_{1}
  16. B ~ 5 {\tilde{B}}_{5}
  17. I ~ 1 {\tilde{I}}_{1}
  18. C ~ 5 {\tilde{C}}_{5}
  19. I ~ 1 {\tilde{I}}_{1}
  20. D ~ 5 {\tilde{D}}_{5}
  21. I ~ 1 {\tilde{I}}_{1}
  22. A ~ 4 {\tilde{A}}_{4}
  23. I ~ 1 {\tilde{I}}_{1}
  24. I ~ 1 {\tilde{I}}_{1}
  25. B ~ 4 {\tilde{B}}_{4}
  26. I ~ 1 {\tilde{I}}_{1}
  27. I ~ 1 {\tilde{I}}_{1}
  28. C ~ 4 {\tilde{C}}_{4}
  29. I ~ 1 {\tilde{I}}_{1}
  30. I ~ 1 {\tilde{I}}_{1}
  31. D ~ 4 {\tilde{D}}_{4}
  32. I ~ 1 {\tilde{I}}_{1}
  33. I ~ 1 {\tilde{I}}_{1}
  34. F ~ 4 {\tilde{F}}_{4}
  35. I ~ 1 {\tilde{I}}_{1}
  36. I ~ 1 {\tilde{I}}_{1}
  37. C ~ 3 {\tilde{C}}_{3}
  38. I ~ 1 {\tilde{I}}_{1}
  39. I ~ 1 {\tilde{I}}_{1}
  40. I ~ 1 {\tilde{I}}_{1}
  41. B ~ 3 {\tilde{B}}_{3}
  42. I ~ 1 {\tilde{I}}_{1}
  43. I ~ 1 {\tilde{I}}_{1}
  44. I ~ 1 {\tilde{I}}_{1}
  45. A ~ 3 {\tilde{A}}_{3}
  46. I ~ 1 {\tilde{I}}_{1}
  47. I ~ 1 {\tilde{I}}_{1}
  48. I ~ 1 {\tilde{I}}_{1}
  49. C ~ 2 {\tilde{C}}_{2}
  50. I ~ 1 {\tilde{I}}_{1}
  51. I ~ 1 {\tilde{I}}_{1}
  52. I ~ 1 {\tilde{I}}_{1}
  53. I ~ 1 {\tilde{I}}_{1}
  54. H ~ 2 {\tilde{H}}_{2}
  55. I ~ 1 {\tilde{I}}_{1}
  56. I ~ 1 {\tilde{I}}_{1}
  57. I ~ 1 {\tilde{I}}_{1}
  58. I ~ 1 {\tilde{I}}_{1}
  59. A ~ 2 {\tilde{A}}_{2}
  60. I ~ 1 {\tilde{I}}_{1}
  61. I ~ 1 {\tilde{I}}_{1}
  62. I ~ 1 {\tilde{I}}_{1}
  63. I ~ 1 {\tilde{I}}_{1}
  64. I ~ 1 {\tilde{I}}_{1}
  65. I ~ 1 {\tilde{I}}_{1}
  66. I ~ 1 {\tilde{I}}_{1}
  67. I ~ 1 {\tilde{I}}_{1}
  68. I ~ 1 {\tilde{I}}_{1}
  69. I ~ 1 {\tilde{I}}_{1}
  70. P ¯ 6 {\bar{P}}_{6}
  71. Q ¯ 6 {\bar{Q}}_{6}
  72. S ¯ 6 {\bar{S}}_{6}

Uniform_8-polytope.html

  1. A ~ 7 {\tilde{A}}_{7}
  2. C ~ 7 {\tilde{C}}_{7}
  3. B ~ 7 {\tilde{B}}_{7}
  4. D ~ 7 {\tilde{D}}_{7}
  5. E ~ 7 {\tilde{E}}_{7}
  6. A ~ 7 {\tilde{A}}_{7}
  7. C ~ 7 {\tilde{C}}_{7}
  8. B ~ 7 {\tilde{B}}_{7}
  9. C ~ 7 {\tilde{C}}_{7}
  10. D ~ 7 {\tilde{D}}_{7}
  11. E ~ 7 {\tilde{E}}_{7}
  12. P ¯ 7 {\bar{P}}_{7}
  13. Q ¯ 7 {\bar{Q}}_{7}
  14. S ¯ 7 {\bar{S}}_{7}
  15. T ¯ 7 {\bar{T}}_{7}

Uniform_9-polytope.html

  1. A ~ 8 {\tilde{A}}_{8}
  2. C ~ 8 {\tilde{C}}_{8}
  3. B ~ 8 {\tilde{B}}_{8}
  4. D ~ 8 {\tilde{D}}_{8}
  5. E ~ 8 {\tilde{E}}_{8}
  6. A ~ 8 {\tilde{A}}_{8}
  7. C ~ 8 {\tilde{C}}_{8}
  8. B ~ 8 {\tilde{B}}_{8}
  9. C ~ 8 {\tilde{C}}_{8}
  10. D ~ 8 {\tilde{D}}_{8}
  11. E ~ 8 {\tilde{E}}_{8}
  12. P ¯ 8 {\bar{P}}_{8}
  13. Q ¯ 8 {\bar{Q}}_{8}
  14. S ¯ 8 {\bar{S}}_{8}
  15. T ¯ 8 {\bar{T}}_{8}

Uniform_polytope.html

  1. { p , q } \begin{Bmatrix}p,q\end{Bmatrix}
  2. { q , p } \begin{Bmatrix}q,p\end{Bmatrix}
  3. t { p , q } t\begin{Bmatrix}p,q\end{Bmatrix}
  4. t { q , p } t\begin{Bmatrix}q,p\end{Bmatrix}
  5. { p q } \begin{Bmatrix}p\\ q\end{Bmatrix}
  6. r { p q } r\begin{Bmatrix}p\\ q\end{Bmatrix}
  7. t { p q } t\begin{Bmatrix}p\\ q\end{Bmatrix}
  8. s { p q } s\begin{Bmatrix}p\\ q\end{Bmatrix}
  9. s { p , q } s\begin{Bmatrix}p,q\end{Bmatrix}

Uniformly_Cauchy_sequence.html

  1. { f n } \{f_{n}\}
  2. ε > 0 \varepsilon>0
  3. N > 0 N>0
  4. x S x\in S
  5. d ( f n ( x ) , f m ( x ) ) < ε d(f_{n}(x),f_{m}(x))<\varepsilon
  6. m , n > N m,n>N
  7. d u ( f n , f m ) 0 d_{u}(f_{n},f_{m})\to 0
  8. m , n m,n\to\infty
  9. d u d_{u}
  10. d u ( f , g ) := sup x S d ( f ( x ) , g ( x ) ) . d_{u}(f,g):=\sup_{x\in S}d(f(x),g(x)).
  11. { f n } \{f_{n}\}
  12. x S x\in S
  13. ε \varepsilon
  14. N > 0 N>0
  15. d ( f n ( x ) , f m ( x ) ) < ε d(f_{n}(x),f_{m}(x))<\varepsilon
  16. m , n > N m,n>N

Uniqueness_theorem_for_Poisson's_equation.html

  1. ( ϵ φ ) = - 4 π ρ f \mathbf{\nabla}\cdot(\epsilon\mathbf{\nabla}\varphi)=-4\pi\rho_{f}
  2. φ \varphi
  3. 𝐄 = - φ \mathbf{E}=-\mathbf{\nabla}\varphi
  4. φ 1 \varphi_{1}
  5. φ 2 \varphi_{2}
  6. ϕ = φ 2 - φ 1 \phi=\varphi_{2}-\varphi_{1}
  7. φ 1 \varphi_{1}
  8. φ 2 \varphi_{2}
  9. ϕ \phi
  10. ( ϵ ϕ ) = 0 \mathbf{\nabla}\cdot(\epsilon\mathbf{\nabla}\phi)=0
  11. ( ϕ ϵ ϕ ) = ϵ ( ϕ ) 2 + ϕ ( ϵ ϕ ) \nabla\cdot(\phi\epsilon\,\nabla\phi)=\epsilon\,(\nabla\phi)^{2}+\phi\nabla% \cdot(\epsilon\,\nabla\phi)
  12. ( ϕ ϵ ϕ ) = ϵ ( ϕ ) 2 \mathbf{\nabla}\cdot(\phi\epsilon\mathbf{\nabla}\phi)=\epsilon(\mathbf{\nabla}% \phi)^{2}
  13. V ( ϕ ϵ ϕ ) d 3 𝐫 = V ϵ ( ϕ ) 2 d 3 𝐫 \int_{V}\mathbf{\nabla}\cdot(\phi\epsilon\mathbf{\nabla}\phi)d^{3}\mathbf{r}=% \int_{V}\epsilon(\mathbf{\nabla}\phi)^{2}\,d^{3}\mathbf{r}
  14. i S i ( ϕ ϵ ϕ ) 𝐝𝐒 = V ϵ ( ϕ ) 2 d 3 𝐫 \sum_{i}\int_{S_{i}}(\phi\epsilon\mathbf{\nabla}\phi)\cdot\mathbf{dS}=\int_{V}% \epsilon(\mathbf{\nabla}\phi)^{2}\,d^{3}\mathbf{r}
  15. S i S_{i}
  16. ϵ > 0 \epsilon>0
  17. ( ϕ ) 2 0 (\mathbf{\nabla}\phi)^{2}\geq 0
  18. ϕ \mathbf{\nabla}\phi
  19. φ 1 = φ 2 \mathbf{\nabla}\varphi_{1}=\mathbf{\nabla}\varphi_{2}
  20. i S i ( ϕ ϵ ϕ ) 𝐝𝐒 = 0 \sum_{i}\int_{S_{i}}(\phi\epsilon\,\mathbf{\nabla}\phi)\cdot\mathbf{dS}=0
  21. φ \varphi
  22. φ 1 = φ 2 \varphi_{1}=\varphi_{2}
  23. ϕ = 0 \phi=0
  24. φ \mathbf{\nabla}\varphi
  25. φ 1 = φ 2 \mathbf{\nabla}\varphi_{1}=\mathbf{\nabla}\varphi_{2}
  26. ϕ = 0 \mathbf{\nabla}\phi=0
  27. φ \mathbf{\nabla}\varphi

Unit_distance_graph.html

  1. n log n . n\log n.
  2. n 1 + c / log log n , n^{1+c/\log\log n},
  3. n 4 / 3 ; n^{4/3};

Unit_tangent_bundle.html

  1. UT ( M ) := x M { v T x ( M ) | g x ( v , v ) = 1 } , \mathrm{UT}(M):=\coprod_{x\in M}\left\{v\in\mathrm{T}_{x}(M)\left|g_{x}(v,v)=1% \right.\right\},
  2. π : UT ( M ) M , \pi:\mathrm{UT}(M)\to M,
  3. π : ( x , v ) x , \pi:(x,v)\mapsto x,
  4. UT x ( M ) = { v T x ( M ) | F ( v ) = 1 } . \mathrm{UT}_{x}(M)=\left\{v\in\mathrm{T}_{x}(M)\left|F(v)=1\right.\right\}.
  5. θ u ( v ) = g ( u , π * v ) \theta_{u}(v)=g(u,\pi_{*}v)\,
  6. θ u ( v ) = g u ( u , π * v ) \theta_{u}(v)=g_{u}(u,\pi_{*}v)\,
  7. U T M f d μ = M d V ( p ) U T p M f | U T p M d μ p \int_{UTM}f\,d\mu=\int_{M}dV(p)\int_{UT_{p}M}\left.f\right|_{UT_{p}M}\,d\mu_{p}
  8. T ( U T M ) = H V T(UTM)=H\oplus V
  9. g H ( v , w ) = g ( v , w ) , v , w H g_{H}(v,w)=g(v,w),\quad v,w\in H

Unitary_divisor.html

  1. b a \frac{b}{a}
  2. 60 5 = 12 \frac{60}{5}=12
  3. 60 6 = 10 \frac{60}{6}=10
  4. σ k * ( n ) = d n gcd ( d , n / d ) = 1 d k . \sigma_{k}^{*}(n)=\sum_{d\mid n\atop\gcd(d,n/d)=1}\!\!d^{k}.
  5. ζ ( s ) ζ ( s - k ) ζ ( 2 s - k ) = n 1 σ k * ( n ) n s . \frac{\zeta(s)\zeta(s-k)}{\zeta(2s-k)}=\sum_{n\geq 1}\frac{\sigma_{k}^{*}(n)}{% n^{s}}.
  6. σ k ( o ) * ( n ) = d n d 1 ( mod 2 ) gcd ( d , n / d ) = 1 d k . \sigma_{k}^{(o)*}(n)=\sum_{{d\mid n\atop d\equiv 1\;\;(\mathop{{\rm mod}}2)}% \atop\gcd(d,n/d)=1}\!\!d^{k}.
  7. ζ ( s ) ζ ( s - k ) ( 1 - 2 k - s ) ζ ( 2 s - k ) ( 1 - 2 k - 2 s ) = n 1 σ k ( o ) * ( n ) n s . \frac{\zeta(s)\zeta(s-k)(1-2^{k-s})}{\zeta(2s-k)(1-2^{k-2s})}=\sum_{n\geq 1}% \frac{\sigma_{k}^{(o)*}(n)}{n^{s}}.
  8. A log x A\log x
  9. A = p ( 1 - p - 1 p 2 ( p + 1 ) ) . A=\prod_{p}\left({1-\frac{p-1}{p^{2}(p+1)}}\right)\ .

United_States_Consumer_Price_Index.html

  1. Δ P L = i p i 1 q i 0 i p i 0 q i 0 \Delta P_{L}=\frac{\sum_{i}p_{i_{1}}\,q_{i_{0}}}{\sum_{i}p_{i_{0}}\,q_{i_{0}}}
  2. Δ P L \,\Delta P_{L}
  3. p i 0 \,p_{i_{0}}
  4. i \,i
  5. q i 0 \,q_{i_{0}}
  6. i \,i
  7. p i 1 \,p_{i_{1}}
  8. i \,i

Unity_amplitude.html

  1. x ( t ) = A s i n ( w t ) x(t)=Asin(wt)

Universal_intelligence.html

  1. Υ ( π ) = μ E 2 - K ( μ ) V μ π \Upsilon(\pi)=\sum_{\mu\in E}{2^{-K(\mu)}V_{\mu}^{\pi}}
  2. V μ π = 𝐄 ( i = 1 r i ) V_{\mu}^{\pi}=\mathbf{E}\left(\sum_{i=1}^{\infty}r_{i}\right)

Van_der_Grinten_projection.html

  1. x = ± π ( A ( G - P 2 ) + A 2 ( G - P 2 ) 2 - ( P 2 + A 2 ) ( G 2 - P 2 ) ) P 2 + A 2 x=\frac{\pm\pi\left(A\left(G-P^{2}\right)+\sqrt{A^{2}\left(G-P^{2}\right)^{2}-% \left(P^{2}+A^{2}\right)\left(G^{2}-P^{2}\right)}\right)}{P^{2}+A^{2}}\,
  2. y = ± π ( P Q - A ( A 2 + 1 ) ( P 2 + A 2 ) - Q 2 ) P 2 + A 2 y=\frac{\pm\pi\left(PQ-A\sqrt{\left(A^{2}+1\right)\left(P^{2}+A^{2}\right)-Q^{% 2}}\right)}{P^{2}+A^{2}}
  3. x x\,
  4. λ - λ 0 \lambda-\lambda_{0}\,
  5. y y\,
  6. ϕ \phi\,
  7. A = 1 2 | π λ - λ 0 - λ - λ 0 π | A=\frac{1}{2}\left|\frac{\pi}{\lambda-\lambda_{0}}-\frac{\lambda-\lambda_{0}}{% \pi}\right|
  8. G = cos θ sin θ + cos θ - 1 G=\frac{\cos\theta}{\sin\theta+\cos\theta-1}
  9. P = G ( 2 sin θ - 1 ) P=G\left(\frac{2}{\sin\theta}-1\right)
  10. θ = arcsin | 2 ϕ π | \theta=\arcsin\left|\frac{2\phi}{\pi}\right|
  11. Q = A 2 + G Q=A^{2}+G\,
  12. ϕ = 0 \phi=0\,
  13. x = ( λ - λ 0 ) x=\left(\lambda-\lambda_{0}\right)\,
  14. y = 0 y=0\,
  15. λ = λ 0 \lambda=\lambda_{0}\,
  16. ϕ = ± π / 2 \phi=\pm\pi/2\,
  17. x = 0 x=0\,
  18. y = ± π tan θ / 2 y=\pm\pi\tan{\theta/2}
  19. ϕ \phi\,
  20. λ \lambda\,
  21. λ 0 \lambda_{0}\,

Vapor-compression_evaporation.html

  1. E = Q * ( H 2 - H 1 ) E=Q*(H2-H1)
  2. Q d = Q s + Q m Qd=Qs+Qm

Vapor–liquid_equilibrium.html

  1. P l i q = P v a p P^{liq}=P^{vap}\,
  2. T l i q = T v a p T^{liq}=T^{vap}\,
  3. G ~ l i q = G ~ v a p \tilde{G}^{liq}=\tilde{G}^{vap}
  4. P l i q P^{liq}\,
  5. P v a p P^{vap}\,
  6. T l i q T^{liq}\,
  7. T v a p T^{vap}\,
  8. G ~ l i q \tilde{G}^{liq}
  9. G ~ v a p \tilde{G}^{vap}
  10. f l i q ( T s , P s ) = f v a p ( T s , P s ) f^{\,liq}(T_{s},P_{s})=f^{\,vap}(T_{s},P_{s})
  11. f l i q ( T s , P s ) f^{\,liq}(T_{s},P_{s})
  12. f v a p ( T s , P s ) f^{\,vap}(T_{s},P_{s})
  13. ϕ = f / P \phi=f/P\,
  14. i i
  15. P l i q = P v a p P^{liq}=P^{vap}\,
  16. T l i q = T v a p T^{liq}=T^{vap}\,
  17. G ¯ i l i q = G ¯ i v a p \bar{G}_{i}^{liq}=\bar{G}_{i}^{vap}
  18. P P
  19. T T
  20. G ¯ i l i q \bar{G}_{i}^{liq}
  21. G ¯ i v a p \bar{G}_{i}^{vap}
  22. G ¯ i = def G n i \bar{G}_{i}\ \stackrel{\mathrm{def}}{=}\ \frac{\partial G}{\partial n_{i}}
  23. G G
  24. i i
  25. T T
  26. T T
  27. K i K_{i}
  28. K K
  29. K i = y i x i K_{i}=\frac{y_{i}}{x}_{i}
  30. i i
  31. y y
  32. x x
  33. K i = P i P K_{i}=\frac{P_{i}^{\star}}{P}
  34. K i = γ i P i P K_{i}=\frac{\gamma_{i}P_{i}^{\star}}{P}
  35. γ i \gamma_{i}
  36. P P
  37. K K
  38. α α
  39. α = K i K j = ( y i / x i ) ( y j / x j ) \alpha=\frac{K_{i}}{K}_{j}=\frac{(y_{i}/x_{i})}{(y_{j}/x_{j})}
  40. i i
  41. j j
  42. K K
  43. x < s u b > 1 x<sub>1

Vapour_pressure_of_water.html

  1. P = exp ( 20.386 - 5132 T ) mmHg P=\exp\left(20.386-\frac{5132}{T}\right)\,\mathrm{mmHg}
  2. P P
  3. T T
  4. log 10 P = A - B C + T \log_{10}P=A-\frac{B}{C+T}
  5. T T
  6. P P
  7. A A
  8. B B
  9. C C
  10. T < s u b > m i n T<sub>min

Variable-length_code.html

  1. S S
  2. T T
  3. C : S T * C:S\to T^{*}
  4. S S
  5. T T
  6. C C
  7. S * S^{*}
  8. T * T^{*}
  9. M 1 = { a 0 , b 0 , c 1 } M_{1}=\{\,a\mapsto 0,b\mapsto 0,c\mapsto 1\,\}
  10. M 2 = { a 1 , b 011 , c 01110 , d 1110 , e 10011 } M_{2}=\{\,a\mapsto 1,b\mapsto 011,c\mapsto 01110,d\mapsto 1110,e\mapsto 10011\,\}
  11. M 3 = { a 0 , b 01 , c 011 } M_{3}=\{\,a\mapsto 0,b\mapsto 01,c\mapsto 011\,\}
  12. M 2 M_{2}
  13. M 3 M_{3}
  14. ( 1 2 , 1 4 , 1 8 , 1 8 ) \textstyle\left(\frac{1}{2},\frac{1}{4},\frac{1}{8},\frac{1}{8}\right)
  15. 1 × 1 2 + 2 × 1 4 + 3 × 1 8 + 3 × 1 8 = 7 4 1\times\frac{1}{2}+2\times\frac{1}{4}+3\times\frac{1}{8}+3\times\frac{1}{8}=% \frac{7}{4}

Variable_cycle_engine.html

  1. F n = m ˙ ( V j f e - V a ) F_{n}=\dot{m}\cdot(V_{jfe}-V_{a})
  2. m ˙ = d d t m = \dot{m}=\frac{d}{dt}m=\,
  3. V j f e = V_{jfe}=\,
  4. V a = V_{a}=\,
  5. F n m ˙ = ( V j f e - V a ) \frac{F_{n}}{\dot{m}}=(V_{jfe}-V_{a})

Variance-gamma_distribution.html

  1. K λ K_{\lambda}
  2. Γ \Gamma
  3. μ + 2 β λ / γ 2 \mu+2\beta\lambda/\gamma^{2}
  4. 2 λ ( 1 + 2 β 2 / γ 2 ) / γ 2 2\lambda(1+2\beta^{2}/\gamma^{2})/\gamma^{2}
  5. e μ z ( γ / α 2 - ( β + z ) 2 ) 2 λ e^{\mu z}\left(\gamma/\sqrt{\alpha^{2}-(\beta+z)^{2}}\right)^{2\lambda}
  6. X 1 X_{1}
  7. X 2 X_{2}
  8. α \alpha
  9. β \beta
  10. λ 1 \lambda_{1}
  11. μ 1 \mu_{1}
  12. λ 2 , \lambda_{2},
  13. μ 2 \mu_{2}
  14. X 1 + X 2 X_{1}+X_{2}
  15. α \alpha
  16. β \beta
  17. λ 1 + λ 2 \lambda_{1}+\lambda_{2}
  18. μ 1 + μ 2 \mu_{1}+\mu_{2}
  19. λ \lambda
  20. x > u x>u
  21. { ( x - μ ) f ′′ ( x ) - 2 f ( x ) ( - β μ + λ + β x - 1 ) + f ( x ) ( α 2 μ - β ( β μ - 2 λ + 2 ) + x ( β 2 - α 2 ) ) = 0 , f ( 0 ) = α ( - 1 2 ) λ - 1 2 e - β μ μ λ - 1 2 ( α - β 2 α ) λ K λ - 1 2 ( - α μ ) π Γ ( λ ) , f ( 0 ) = α 2 1 2 - λ μ e - β μ ( - μ ) λ - 5 2 ( α - β 2 α ) λ ( ( β μ - 2 λ + 1 ) K λ - 1 2 ( - α μ ) - α μ K λ + 1 2 ( - α μ ) ) π Γ ( λ ) } \left\{\begin{array}[]{l}(x-\mu)f^{\prime\prime}(x)-2f^{\prime}(x)(-\beta\mu+% \lambda+\beta x-1)+f(x)\left(\alpha^{2}\mu-\beta(\beta\mu-2\lambda+2)+x\left(% \beta^{2}-\alpha^{2}\right)\right)=0,\\ f(0)=\frac{\sqrt{\alpha}\left(-\frac{1}{2}\right)^{\lambda-\frac{1}{2}}e^{-% \beta\mu}\mu^{\lambda-\frac{1}{2}}\left(\alpha-\frac{\beta^{2}}{\alpha}\right)% ^{\lambda}K_{\lambda-\frac{1}{2}}(-\alpha\mu)}{\sqrt{\pi}\Gamma(\lambda)},\\ f^{\prime}(0)=\frac{\sqrt{\alpha}2^{\frac{1}{2}-\lambda}\mu e^{-\beta\mu}(-\mu% )^{\lambda-\frac{5}{2}}\left(\alpha-\frac{\beta^{2}}{\alpha}\right)^{\lambda}% \left((\beta\mu-2\lambda+1)K_{\lambda-\frac{1}{2}}(-\alpha\mu)-\alpha\mu K_{% \lambda+\frac{1}{2}}(-\alpha\mu)\right)}{\sqrt{\pi}\Gamma(\lambda)}\end{array}\right\}
  22. x < u x<u
  23. { ( x - μ ) f ′′ ( x ) - 2 f ( x ) ( - β μ + λ + β x - 1 ) + f ( x ) ( α 2 μ - β ( β μ - 2 λ + 2 ) + x ( β 2 - α 2 ) ) = 0 , f ( 0 ) = 2 1 2 - λ α μ e - β μ ( μ ( α - β 2 α ) ) λ K λ - 1 2 ( α μ ) π Γ ( λ ) , f ( 0 ) = α 2 1 2 - λ e - β μ μ λ - 3 2 ( α - β 2 α ) λ ( ( β μ - 2 λ + 1 ) K λ - 1 2 ( α μ ) + α μ K λ + 1 2 ( α μ ) ) π Γ ( λ ) } \left\{\begin{array}[]{l}(x-\mu)f^{\prime\prime}(x)-2f^{\prime}(x)(-\beta\mu+% \lambda+\beta x-1)+f(x)\left(\alpha^{2}\mu-\beta(\beta\mu-2\lambda+2)+x\left(% \beta^{2}-\alpha^{2}\right)\right)=0,\\ f(0)=\frac{2^{\frac{1}{2}-\lambda}\sqrt{\frac{\alpha}{\mu}}e^{-\beta\mu}\left(% \mu\left(\alpha-\frac{\beta^{2}}{\alpha}\right)\right)^{\lambda}K_{\lambda-% \frac{1}{2}}(\alpha\mu)}{\sqrt{\pi}\Gamma(\lambda)},\\ f^{\prime}(0)=\frac{\sqrt{\alpha}2^{\frac{1}{2}-\lambda}e^{-\beta\mu}\mu^{% \lambda-\frac{3}{2}}\left(\alpha-\frac{\beta^{2}}{\alpha}\right)^{\lambda}% \left((\beta\mu-2\lambda+1)K_{\lambda-\frac{1}{2}}(\alpha\mu)+\alpha\mu K_{% \lambda+\frac{1}{2}}(\alpha\mu)\right)}{\sqrt{\pi}\Gamma(\lambda)}\end{array}\right\}

Variance_reduction.html

  1. X 1 j X_{1j}
  2. X 2 j X_{2j}
  3. ξ = E ( X 1 j ) - E ( X 2 j ) = μ 1 - μ 2 . \xi=E(X_{1j})-E(X_{2j})=\mu_{1}-\mu_{2}.\,
  4. Z j = X 1 j - X 2 j for j = 1 , 2 , , n , Z_{j}=X_{1j}-X_{2j}\quad\mbox{for }~{}j=1,2,\ldots,n,
  5. E ( Z j ) = ξ E(Z_{j})=\xi
  6. Z ( n ) = j = 1 , , n Z j n Z(n)=\frac{\sum_{j=1,\ldots,n}Z_{j}}{n}
  7. ξ \xi
  8. Z j Z_{j}
  9. Var [ Z ( n ) ] = Var ( Z j ) n . \operatorname{Var}[Z(n)]=\frac{\operatorname{Var}(Z_{j})}{n}.

Varifold.html

  1. Ω \Omega
  2. Ω \Omega
  3. Ω × G ( n , m ) \Omega\times G(n,m)
  4. G ( n , m ) G(n,m)
  5. Ω \Omega
  6. V ( A ) := Γ M , A θ ( x ) d m ( x ) V(A):=\int_{\Gamma_{M,A}}\!\!\!\!\!\!\!\theta(x)\mathrm{d}\mathcal{H}^{m}(x)
  7. Γ M , A = M { x : ( x , Tan m ( x , M ) ) A } \scriptstyle\Gamma_{M,A}=M\cap\{x:(x,\mathrm{Tan}^{m}(x,M))\in A\}
  8. m ( x ) \scriptstyle\mathcal{H}^{m}(x)
  9. m m

Vector_Laplacian.html

  1. 2 \scriptstyle\nabla^{2}
  2. 𝐀 \mathbf{A}
  3. 2 𝐀 = ( 𝐀 ) - × ( × 𝐀 ) . \nabla^{2}\mathbf{A}=\nabla(\nabla\cdot\mathbf{A})-\nabla\times(\nabla\times% \mathbf{A}).
  4. 2 𝐀 = ( 2 A x , 2 A y , 2 A z ) , \nabla^{2}\mathbf{A}=(\nabla^{2}A_{x},\nabla^{2}A_{y},\nabla^{2}A_{z}),
  5. A x A_{x}
  6. A y A_{y}
  7. A z A_{z}
  8. 𝐀 \mathbf{A}
  9. 𝐓 \mathbf{T}
  10. 2 𝐓 = ( 𝐓 ) . \nabla^{2}\mathbf{T}=\nabla\cdot(\nabla\mathbf{T}).
  11. 𝐓 \mathbf{T}
  12. 𝐓 \mathbf{T}
  13. 𝐓 = ( T x , T y , T z ) = [ T x x T x y T x z T y x T y y T y z T z x T z y T z z ] , where T v u T u v . \nabla\mathbf{T}=(\nabla T_{x},\nabla T_{y},\nabla T_{z})=\begin{bmatrix}T_{xx% }&T_{xy}&T_{xz}\\ T_{yx}&T_{yy}&T_{yz}\\ T_{zx}&T_{zy}&T_{zz}\end{bmatrix},\,\text{ where }T_{vu}\equiv\frac{\partial T% _{u}}{\partial v}.
  14. 𝐀 𝐁 = [ A x A y A z ] 𝐁 = [ 𝐀 B x 𝐀 B y 𝐀 B z ] . \mathbf{A}\cdot\nabla\mathbf{B}=\begin{bmatrix}A_{x}&A_{y}&A_{z}\end{bmatrix}% \nabla\mathbf{B}=\begin{bmatrix}\mathbf{A}\cdot\nabla B_{x}&\mathbf{A}\cdot% \nabla B_{y}&\mathbf{A}\cdot\nabla B_{z}\end{bmatrix}.
  15. ρ ( 𝐯 t + ( 𝐯 ) 𝐯 ) = ρ 𝐟 - p + μ ( 2 𝐯 ) , \rho\left(\frac{\partial\mathbf{v}}{\partial t}+(\mathbf{v}\cdot\nabla)\mathbf% {v}\right)=\rho\mathbf{f}-\nabla p+\mu\left(\nabla^{2}\mathbf{v}\right),
  16. μ ( 2 𝐯 ) \mu\left(\nabla^{2}\mathbf{v}\right)
  17. 2 𝐄 - μ 0 ϵ 0 2 𝐄 t 2 = 0. \nabla^{2}\mathbf{E}-\mu_{0}\epsilon_{0}\frac{\partial^{2}\mathbf{E}}{\partial t% ^{2}}=0.
  18. 𝐄 = 0 , \Box\,\mathbf{E}=0,
  19. 1 c 2 2 t 2 - 2 , \Box\equiv\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}-\nabla^{2},

Vector_notation.html

  1. 𝐯 \mathbf{v}
  2. v \vec{v}
  3. n \mathbb{R}^{n}
  4. 𝐯 = ( v 1 , v 2 , , v n - 1 , v n ) \mathbf{v}=(v_{1},v_{2},\dots,v_{n-1},v_{n})
  5. 𝐯 = v 1 , v 2 , , v n - 1 , v n \mathbf{v}=\langle v_{1},v_{2},\dots,v_{n-1},v_{n}\rangle
  6. n \mathbb{R}^{n}
  7. 𝐯 \mathbf{v}
  8. 𝐯 = [ v 1 v 2 v n - 1 v n ] = ( v 1 v 2 v n - 1 v n ) \mathbf{v}=\left[\begin{matrix}v_{1}&v_{2}&\cdots&v_{n-1}&v_{n}\end{matrix}% \right]=\left(\begin{matrix}v_{1}&v_{2}&\cdots&v_{n-1}&v_{n}\end{matrix}\right)
  9. 𝐯 = [ v 1 v 2 v n - 1 v n ] = ( v 1 v 2 v n - 1 v n ) \mathbf{v}=\left[\begin{matrix}v_{1}\\ v_{2}\\ \vdots\\ v_{n-1}\\ v_{n}\end{matrix}\right]=\left(\begin{matrix}v_{1}\\ v_{2}\\ \vdots\\ v_{n-1}\\ v_{n}\end{matrix}\right)
  10. 3 \mathbb{R}^{3}
  11. 2 \mathbb{R}^{2}
  12. 3 \mathbb{R}^{3}
  13. s y m b o l ı ^ = ( 1 , 0 , 0 ) symbol{\hat{\imath}}=(1,0,0)
  14. s y m b o l ȷ ^ = ( 0 , 1 , 0 ) symbol{\hat{\jmath}}=(0,1,0)
  15. s y m b o l k ^ = ( 0 , 0 , 1 ) symbol{\hat{k}}=(0,0,1)
  16. 𝐯 = v x s y m b o l ı ^ + v y s y m b o l ȷ ^ + v z s y m b o l k ^ \mathbf{v}=v_{x}symbol{\hat{\imath}}+v_{y}symbol{\hat{\jmath}}+v_{z}symbol{% \hat{k}}
  17. 0 θ < 2 π 0\leq\theta<2\pi
  18. 0 θ < 360 0\leq\theta<360^{\circ}
  19. \angle
  20. 𝐯 = ( r , θ ) \mathbf{v}=(r,\angle\theta)
  21. 𝐯 = r , θ \mathbf{v}=\langle r,\angle\theta\rangle
  22. 𝐯 = [ r θ ] \mathbf{v}=\left[\begin{matrix}r&\angle\theta\end{matrix}\right]
  23. 𝐯 = [ r θ ] \mathbf{v}=\left[\begin{matrix}r\\ \angle\theta\end{matrix}\right]
  24. \angle
  25. r = 5 , θ = π 9 r=5,\ \theta={\pi\over 9}
  26. r = 5 , θ = 20 r=5,\ \theta=20^{\circ}
  27. 0 θ < 2 π 0\leq\theta<2\pi
  28. \angle
  29. 𝐯 = ( r , θ , h ) \mathbf{v}=(r,\angle\theta,h)
  30. 𝐯 = r , θ , h \mathbf{v}=\langle r,\angle\theta,h\rangle
  31. 𝐯 = [ r θ h ] \mathbf{v}=\left[\begin{matrix}r&\angle\theta&h\end{matrix}\right]
  32. 𝐯 = [ r θ h ] \mathbf{v}=\left[\begin{matrix}r\\ \angle\theta\\ h\end{matrix}\right]
  33. \angle
  34. r = 5 , θ = π 9 , h = 3 r=5,\ \theta={\pi\over 9},\ h=3
  35. r = 5 , θ = 20 , h = 3 r=5,\ \theta=20^{\circ},\ h=3
  36. ρ = 5 , ϕ = π 9 , z = 3 \rho=5,\ \phi={\pi\over 9},\ z=3
  37. ρ = 5 , ϕ = 20 , z = 3 \rho=5,\ \phi=20^{\circ},\ z=3
  38. \angle
  39. 𝐯 = ( ρ , θ , ϕ ) \mathbf{v}=(\rho,\angle\theta,\angle\phi)
  40. 𝐯 = ρ , θ , ϕ \mathbf{v}=\langle\rho,\angle\theta,\angle\phi\rangle
  41. 𝐯 = [ ρ θ ϕ ] \mathbf{v}=\left[\begin{matrix}\rho&\angle\theta&\angle\phi\end{matrix}\right]
  42. 𝐯 = [ ρ θ ϕ ] \mathbf{v}=\left[\begin{matrix}\rho\\ \angle\theta\\ \angle\phi\end{matrix}\right]
  43. ρ = 5 , θ = π 9 , ϕ = π 4 \rho=5,\ \theta={\pi\over 9},\ \phi={\pi\over 4}
  44. ρ = 5 , θ = 20 , ϕ = 45 \rho=5,\ \theta=20^{\circ},\ \phi=45^{\circ}
  45. n \mathbb{R}^{n}
  46. 3 \mathbb{R}^{3}
  47. 7 \mathbb{R}^{7}
  48. 𝐮 + 𝐯 \mathbf{u}+\mathbf{v}
  49. c 𝐯 c\mathbf{v}
  50. c 𝐯 c\cdot\mathbf{v}
  51. c × 𝐯 c\times\mathbf{v}
  52. 𝐮 + - 𝐯 \mathbf{u}+-\mathbf{v}
  53. 𝐮 - 𝐯 \mathbf{u}-\mathbf{v}
  54. 1 c 𝐯 {1\over c}\mathbf{v}
  55. 𝐯 c {\mathbf{v}\over c}
  56. 𝐯 ÷ c {\mathbf{v}\div c}
  57. 𝐯 \|\mathbf{v}\|
  58. | 𝐯 | |\mathbf{v}|
  59. 𝐮 , 𝐯 \langle\mathbf{u},\mathbf{v}\rangle
  60. n \mathbb{R}^{n}
  61. 𝐮 𝐯 \mathbf{u}\cdot\mathbf{v}
  62. 3 \mathbb{R}^{3}
  63. 𝐮 × 𝐯 \mathbf{u}\times\mathbf{v}
  64. [ 𝐮 , 𝐯 ] [\mathbf{u},\mathbf{v}]