# Solutions to Differential Equation

Submitted by schubotz on

# Results for NTCIR10-FT-12

## Query

### Original Query

### Compiled by FSE

#### Token-Filter

- TeXFilter:
`[D x 4, + x 2, rangle, langle, partial x 2, ,, frac x 2, -, 2 x 2, 1, 0, triangle, \ x 7, | x 2, ^ x 2, =]`

- Presentation-MathML:
`[D x 4, 2 x 2, △, 1, 0, ∂ x 2, + x 2, ⟩, | x 2, ⟨, =, ,, -]`

#### MathML-Filter

#### Word filter

Keywords:[uniqueness, of, solutions] Rendered Presentation-MathML:## Results

### Summary

#### Reviewer score 2

- Items reviewd: 20
- Accumulated score: 64350
- Formulasearchengine found: 3

#### Reviewer score 0

- Items reviewd: 80
- Accumulated score: 150473
- Formulasearchengine found: 7

++ | + | o | ∑ | |
---|---|---|---|---|

200000+ | 0 | 0 | 0 | 0 |

5000-200000 | 0 | 3 | 7 | 10 |

<5000 | 0 | 17 | 73 | 90 |

∑ | 0 | 20 | 80 | 100 |

### Short result list

### Detailed results for reviewer score 2

#### Hit idp1306560

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 2
- Formulasearchengine score: 21803
- Reference to collection: _PREFIX_/16/f006298.xhtml#idp1306560

Rendered MathML:

$\partial _{t}^{2}D+2\frac{\dot{a}}{a}\partial _{t}D=\frac{1}{a^{2}}\partial _{z}\left[(1+D)\partial _{z}\Phi\right]+\frac{1}{a^{2}}\partial _{z}^{2}\left[(1+D)\langle v^{2}\rangle\right].$

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#### Hit idp22988448

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 4
- Formulasearchengine score: 21774
- Reference to collection: _PREFIX_/14/f005353.xhtml#idp22988448

Rendered MathML:

$\langle\phi^{{2}}\rangle _{{\text{b},z=0}}^{{\text{C}}}=-\frac{8(4\pi)^{{-(D+1)/2}}}{\Gamma((D-1)/2)r^{{D-2}}}\int _{{0}}^{{\infty}}dv\ g(v,p)\left|\Gamma\left(D/2-1-iv\right)\right|^{{2}}.$

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#### Hit id59813

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 9
- Formulasearchengine score: 20773
- Reference to collection: _PREFIX_/32/f012639.xhtml#id59813

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#### Hit id109704

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 11
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/213/f085117.xhtml#id109704

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#### Hit id58165

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 12
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/23/f009086.xhtml#id58165

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#### Hit id61255

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 13
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/121/f048196.xhtml#id61255

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#### Hit id62495

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 14
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/54/f021574.xhtml#id62495

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#### Hit id67373

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 15
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/33/f012877.xhtml#id67373

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#### Hit id71251

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 16
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/21/f008206.xhtml#id71251

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#### Hit idp1214912

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 17
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/16/f006298.xhtml#idp1214912

Rendered MathML:

$\partial _{t}(1+D)+\frac{1}{a}\partial _{z}\left[\langle v\rangle(1+D)\right]=0,$

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#### Hit idp1238656

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 18
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/16/f006298.xhtml#idp1238656

Rendered MathML:

$\partial _{t}\left[a\left<v\right>(1+D)\right]+\partial _{{z}}\Phi(1+D)+\partial _{z}\left[\langle v^{2}\rangle(1+D)\right]=0,$

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#### Hit idp1270976

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 19
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/16/f006397.xhtml#idp1270976

Rendered MathML:

$\frac{\partial}{\partial t}F(t,x)+\frac{1}{2}\left\langle A(t,x),D^{2}F(t,x)\right\rangle+\langle b(t,x),\nabla _{x}F(t,x)\rangle=0$

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#### Hit idp20702608

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 20
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/48/f019162.xhtml#idp20702608

Rendered MathML:

$\frac{d\left\langle n_{{S}}\right\rangle}{dt}=2\left[\gamma D_{{n_{{S}}}}-\gamma\left(\frac{T-T_{{C}}}{T_{{C}}}+{}\frac{{}{}{}{}{}{}\left\langle n_{{S}}\right\rangle}{n}{}{}{}{}{}{}{}{}{}{}\right)\left\langle n_{{S}}\right\rangle\right]+D\frac{\partial^{{2}}\left\langle n_{{S}}\right\rangle _{{R,t}}}{{}{}{}\partial R^{{2}}}.$

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#### Hit idp20799344

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 21
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/14/f005457.xhtml#idp20799344

Rendered MathML:

$\frac{\partial}{\partial t}\langle{\mbox{\boldmath$B$}}\rangle={\mbox{\boldmath$e$}}_{z}\times\frac{\partial}{\partial z}{\boldsymbol{{\cal E}}}+\eta\frac{\partial^{2}}{\partial z^{2}}\langle{\mbox{\boldmath$B$}}\rangle,$

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#### Hit idp21442752

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 22
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/114/f045227.xhtml#idp21442752

Rendered MathML:

$\langle\xi\rangle=\xi\left(\langle e\rangle,\langle p\rangle\right)+\frac{1}{2}\left.\frac{\partial^{2}\xi}{\partial e^{2}}\right|_{{\langle e\rangle,\langle p\rangle}}\sigma _{e}^{2}+\frac{1}{2}\left.\frac{\partial^{2}\xi}{\partial p^{2}}\right|_{{\langle e\rangle,\langle p\rangle}}\sigma _{p}^{2}\;,$

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#### Hit idp22771216

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 23
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/131/f052265.xhtml#idp22771216

Rendered MathML:

$\frac{d}{dt}{\langle}\partial^{\alpha}v,\partial^{\alpha}\nabla\sigma{\rangle}+\|\partial^{\alpha}\nabla\sigma\|^{2}+\frac{1}{\gamma}\|\partial^{\alpha}\sigma\|^{2}={\langle}\partial^{\alpha}v,\partial^{\alpha}\nabla\partial _{t}\sigma{\rangle}\\ -\frac{1}{\sqrt{\gamma}}{\langle}\partial^{\alpha}v,\partial^{\alpha}\nabla\sigma{\rangle}-\frac{1}{\gamma}{\langle}\partial^{\alpha}\Phi(\sigma),\partial^{\alpha}\sigma{\rangle}+{\langle}\partial^{\alpha}f_{2},\partial^{\alpha}\nabla\sigma{\rangle},$

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#### Hit idp23156240

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 24
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/129/f051315.xhtml#idp23156240

Rendered MathML:

$\displaystyle\varepsilon\,{\partial F\over\partial t}+e\,\left\langle D_{+}(n^{+},n^{-},F)+D_{-}(n^{+},n^{-},F)\right\rangle _{{1}}=J(t)$

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#### Hit idp2692352

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 25
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/16/f006397.xhtml#idp2692352

Rendered MathML:

$\frac{\partial F_{l}}{\partial t}(\overline{x}_{{l-1}};t,x)+\frac{1}{2}\left\langle A(t,x),D^{2}F_{l}(\overline{x}_{{l-1}};t,x)\right\rangle+\langle b,\nabla _{x}F_{l}(\overline{x}_{{l-1}};t,x)\rangle=0$

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#### Hit idp29279888

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 26
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/16/f006397.xhtml#idp29279888

Rendered MathML:

$\frac{\partial}{\partial t}\Gamma+\frac{1}{2}\langle A,D^{2}\Gamma\rangle+\langle b,\nabla _{x}\Gamma\rangle=0$

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#### Hit idp5060320

- Reviwer: xxx
- Reviwer score: 2
- Formulasearchengine rank: 27
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/64/f025212.xhtml#idp5060320

Rendered MathML:

$\frac{d}{dt}\Phi _{0}+\Phi _{0}=2\langle\partial _{t}u,A^{{1/2}}u\rangle\| v\|^{2}_{1}\leq Q(R)\|\partial _{t}u\|\Phi _{0}.$

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### Detailed results for reviewer score 0

#### Hit idp30351312

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 1
- Formulasearchengine score: 22001
- Reference to collection: _PREFIX_/203/f081135.xhtml#idp30351312

Rendered MathML:

$\displaystyle\quad\times e^{{\frac{1}{2}\left\langle f,x\right\rangle _{{A^{{-1}}}}-\langle\frac{\coth\left(t\sqrt{D}\right)}{\sqrt{D}}g,x\rangle+\langle\frac{1}{\sqrt{D}\sinh\left(t\sqrt{D}\right)}g,x^{{0}}\rangle}}\tag{2.2}$

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#### Hit id71514

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 3
- Formulasearchengine score: 21787
- Reference to collection: _PREFIX_/213/f084819.xhtml#id71514

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#### Hit id63379

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 5
- Formulasearchengine score: 21773
- Reference to collection: _PREFIX_/30/f011986.xhtml#id63379

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#### Hit idp22901616

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 6
- Formulasearchengine score: 21762
- Reference to collection: _PREFIX_/61/f024165.xhtml#idp22901616

Rendered MathML:

$\langle\bar{\psi}\psi\rangle _{{R^{{D}}}}=-\frac{N_{{D}}\Gamma((D+1)/2)}{2^{{D+2}}\pi^{{(D-1)/2}}a^{{D}}}\frac{d^{{D-1}}}{dx^{{D-1}}}\frac{1}{\sin(\pi x)}\Big|_{{x=z/a}}.$

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#### Hit idp21147520

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 7
- Formulasearchengine score: 21706
- Reference to collection: _PREFIX_/224/f089307.xhtml#idp21147520

Rendered MathML:

$\begin{array}[]{c}{\displaystyle\langle\widetilde{T}_{{11}}\rangle _{{\delta,l}}=\frac{(D-1)\Gamma(D)\zeta(D)}{(4\pi)^{{\frac{D-1}{2}}}\Gamma\left(\frac{1+D}{2}\right)}\;(2l)^{{-D}}\times}\\ \\ \times\left[-1+C^{{1}}_{{D}}\widetilde{l}^{{-1}}-C^{{2}}_{{D+1}}\widetilde{l}^{{-2}}+C^{{3}}_{{D+2}}\left(1+\displaystyle\frac{\zeta(D+2)}{2\zeta(D)}\right)\;\widetilde{l}^{{-3}}+\ldots\right],\end{array}$

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#### Hit idp653456

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 8
- Formulasearchengine score: 20837
- Reference to collection: _PREFIX_/207/f082449.xhtml#idp653456

Rendered MathML:

$Q_{D}=\frac{\langle N_{D}^{2}\rangle-\langle N_{D}\rangle^{2}}{\langle N_{D}\rangle}-1.$

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#### Hit idp816208

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 10
- Formulasearchengine score: 20607
- Reference to collection: _PREFIX_/138/f055054.xhtml#idp816208

Rendered MathML:

$\Delta^{{(D)}}\Psi=\frac{1}{\sqrt{S_{D}}}\frac{1}{r^{{\frac{D-1}{2}}}}\left[-\frac{\partial^{2}}{\partial r^{2}}+\frac{(D-1)(D-3)}{4r^{2}}\right]u(r).$

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#### Hit id103268

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#### Hit id122102

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- Reviwer score: 0
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#### Hit id154067

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- Reviwer: xxx
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- Reviwer: xxx
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#### Hit id60690

- Reviwer: xxx
- Reviwer score: 0
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- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 45
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#### Hit id63567

- Reviwer: xxx
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- Formulasearchengine rank: 47
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#### Hit id65858

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 48
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#### Hit id66138

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 49
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#### Hit id68082

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 50
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#### Hit id69750

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 51
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#### Hit id71161

- Reviwer: xxx
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- Formulasearchengine rank: 52
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#### Hit id72603

- Reviwer: xxx
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- Formulasearchengine rank: 53
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#### Hit id74184

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 54
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#### Hit id74914

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 55
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#### Hit id75374

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 56
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#### Hit id76048

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 57
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#### Hit id77164

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 58
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#### Hit id77678

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 59
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#### Hit id81270

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 60
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#### Hit id84931

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 61
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#### Hit id86124

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 62
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#### Hit id89303

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 63
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#### Hit id99269

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 64
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#### Hit idm1008096

- Reviwer: xxx
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- Formulasearchengine rank: 65
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- Reference to collection: _PREFIX_/131/f052346.xhtml#idm1008096

Rendered MathML:

$\displaystyle\langle H_{{th}}^{2}\rangle-\langle H_{{th}}\rangle^{2}=\frac{\partial^{2}\ln Z}{\partial\beta^{2}}=N\frac{\epsilon^{2}e^{{\beta\epsilon}}}{(1+e^{{\beta\epsilon}})^{2}},$

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#### Hit idp1092384

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 66
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/130/f051937.xhtml#idp1092384

Rendered MathML:

$\displaystyle\overline{\langle x(L)\rangle}=L-\frac{1}{2}\int _{0}^{L}ds\overline{\Big\langle\Big(\frac{\partial y(s)}{\partial s}\Big)^{2}\Big\rangle}\;,$

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#### Hit idp1101920

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 67
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- Reference to collection: _PREFIX_/191/f076379.xhtml#idp1101920

Rendered MathML:

$\left.\frac{\partial\phi}{\partial\omega}\right|_{\sigma}=-\left(\frac{D}{V_{g}}+\frac{L-D}{V_{{g0}}}\right)\left(\frac{1+\alpha\cos(\omega\tau)}{1+2\alpha\cos(\omega\tau)+\alpha^{2}}\right).$

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#### Hit idp1290688

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 68
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/206/f082268.xhtml#idp1290688

Rendered MathML:

$\frac{\partial\langle\hat{a}\rangle}{\partial t}=-\frac{\gamma}{2}\left[(1-\epsilon^{2})\langle\hat{a}\rangle+i\lambda(1+\epsilon)\right].$

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#### Hit idp1415616

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 69
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/205/f081867.xhtml#idp1415616

Rendered MathML:

$\frac{\left|Dw\right|}{\left|D\xi\right|}=\left(\frac{1-\left|a\right|^{2}}{\left|1-\left<a,\xi\right>\right|^{2}}\right)^{{n+1}}$

End of MathML

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#### Hit idp20062960

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 70
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/205/f081961.xhtml#idp20062960

Rendered MathML:

$\frac{\partial\tilde{g}}{\partial z}\langle v_{{z}}\rangle+i\omega\tilde{g}(z;\xi,E_{{y}})(1-\langle v_{{z}}\rangle)=D_{0}\left[\frac{\partial}{\partial E_{{y}}}\left(E_{{y}}\frac{\partial\tilde{g}}{\partial E_{{y}}}\right)+\frac{1}{E}\frac{\partial^{2}\tilde{g}}{\partial\xi^{2}}\right].$

End of MathML

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#### Hit idp20666736

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 71
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/16/f006078.xhtml#idp20666736

Rendered MathML:

$\displaystyle D_{{k}}^{{n}}=\frac{2\overline{N}(R_{{0}}-1)^{2}\rho}{R_{{0}}^{{2}}\langle k^{{2}}\rangle\langle k\rangle(1+\langle k\rangle/\langle k^{2}\rangle+\rho)}kP(k)\sum _{{k^{{\prime}}}}D_{{k^{{\prime}}}}^{{n-1}}(k^{{\prime}}-1)\left(k+k^{{\prime}}\right)\;\;\;.$

End of MathML

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#### Hit idp20866256

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 72
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/207/f082476.xhtml#idp20866256

Rendered MathML:

$\left(\frac{1}{a_{{eff}}}\frac{\partial\, a_{{eff}}}{\partial A_{0}}\right)^{2}=H^{2}\left[1+\frac{2}{H}\langle\bar{\psi}\dot{\bar{\psi}}\rangle-\frac{2}{H}\langle\dot{\bar{\psi}}^{{(2)}}\rangle\right]$

End of MathML

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#### Hit idp20924560

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 73
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/112/f044696.xhtml#idp20924560

Rendered MathML:

$\displaystyle{\mathcal{L}}_{{\rm{int}}}=-\frac{1}{4\left<f_{{ab}}\right>}\left[\left<\frac{\partial^{2}f_{{ab}}}{\partial\phi^{2}}F_{\phi}+\frac{\partial f_{{ab}}}{\partial\phi}\frac{\partial F_{\phi}}{\partial\phi}\right>\delta\phi+\left<\frac{\partial f_{{ab}}}{\partial\phi}\frac{\partial F_{\phi}}{\partial\phi^{*}}\right>\delta\phi^{*}\right]\lambda^{a}\lambda^{b}$

End of MathML

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#### Hit idp21068592

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 74
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/114/f045382.xhtml#idp21068592

Rendered MathML:

$\frac{\partial\langle W\rangle}{\partial t}=\frac{\partial}{\partial t}\int\langle\rho\frac{v^{2}}{2}+\frac{b^{2}}{2\mu _{0}}\rangle d^{3}r=-\int\langle\vec{j}\cdot\vec{E}\rangle d^{3}r.$

End of MathML

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#### Hit idp21207952

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 75
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/202/f080728.xhtml#idp21207952

Rendered MathML:

$a^{{\prime}}\left\langle\left(\delta\rho _{{k}}\right)^{{2}}\right\rangle=\frac{1}{2a_{{0}}^{{D}}}\sum _{{M}}\left[\left(\frac{\partial\phi _{{k}}}{\partial x^{{M}}}\right)^{{2}}a_{{0}}^{{2}}+\left(\frac{\partial^{{2}}\phi _{{k}}}{\partial\left(x^{{M}}\right)^{{2}}}\right)^{{2}}\frac{a_{{0}}^{{4}}}{16}\right]\;.$

End of MathML

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#### Hit idp21387824

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 76
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/203/f080858.xhtml#idp21387824

Rendered MathML:

$\displaystyle\frac{\partial^{2}I}{\partial\langle x^{n}\rangle\partial\langle x^{k}\rangle}\,=\,\left(1+\frac{k}{2}\right)\frac{4}{k^{2}}~{}~{}C_{k}~{}\left|\langle{x}^{{k}}\rangle\right|^{{-{2}(1+k)/{k}}}~{}\delta _{{kn}},$

End of MathML

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#### Hit idp22013312

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 77
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/128/f050886.xhtml#idp22013312

Rendered MathML:

$\chi=\left.\langle\frac{\partial M}{\partial B}\rangle\right|_{{B=0}}=\frac{1}{T}\langle M^{2}\rangle$

End of MathML

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#### Hit idp22167344

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 78
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/15/f005683.xhtml#idp22167344

Rendered MathML:

$\omega=\langle{\epsilon^{{A}}\theta^{B}}\rangle\left[-\frac{i}{2!}\bar{\phi}_{{AB}}+\frac{1}{3!}\epsilon _{{ABCD}}\langle{\theta^{C}\psi^{D}}\rangle-\frac{i}{4!}\epsilon _{{ABCD}}\langle{\theta^{C}|F|\theta^{D}}\rangle+\dots\right]\,,$

End of MathML

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#### Hit idp22207104

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 79
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/169/f067499.xhtml#idp22207104

Rendered MathML:

$\displaystyle\left\langle{k_{\mu}k_{\nu}\over D_{0}D_{1}D_{2}}\right\rangle _{D}=\left({1\over i}{\partial\over\partial a^{\mu}}\right)\left({1\over i}{\partial\over\partial a^{\nu}}\right)\left\langle{1\over D_{0}D_{1}D_{2}}e^{{ia\cdot k}}\right\rangle _{D}\bigg|_{{a=0}}\,.$

End of MathML

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#### Hit idp22716416

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 80
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/115/f045987.xhtml#idp22716416

Rendered MathML:

$\frac{1}{N}\;\langle\;\frac{\partial^{2}\widetilde{H}}{\partial w^{2}}\;\rangle=\Phi _{1}+\Phi _{2}\;.$

End of MathML

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#### Hit idp27303312

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 81
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/129/f051481.xhtml#idp27303312

Rendered MathML:

$\displaystyle\displaystyle\left<J^{i}_{{g\mathrm{(4D)}}}\right>(z)=-\left(\delta\left(z-\frac{\delta _{\perp}}{2}\right)+\delta\left(z+\frac{\delta _{\perp}}{2}\right)\right)\epsilon^{{ij}}\left(\partial _{0}A_{{j\mathrm{(4D)}}}-\partial _{j}A_{{0\mathrm{(4D)}}}\right)$

End of MathML

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#### Hit idp29017552

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 82
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/204/f081433.xhtml#idp29017552

Rendered MathML:

$C_{1}=p(2p+1)\frac{\langle\partial _{E}\phi,\phi^{{2p-1}}\psi^{2}\rangle _{{L^{2}}}}{\langle\partial _{E}\phi,\phi\rangle _{{L^{2}}}},\quad C_{2}=p\frac{\langle\partial _{E}\phi,\phi^{{2p-1}}\chi^{2}\rangle _{{L^{2}}}}{\langle\partial _{E}\phi,\phi\rangle _{{L^{2}}}},\quad C_{3}=-2p\frac{\langle\psi,\phi^{{2p}}\chi\rangle _{{L^{2}}}}{\langle\partial _{E}\phi,\phi\rangle _{{L^{2}}}}.$

End of MathML

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#### Hit idp29912160

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 83
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/203/f081135.xhtml#idp29912160

Rendered MathML:

$\displaystyle\left\langle A\mu,\mu\right\rangle=\frac{1}{4}\lvert f\rvert _{{A^{{-1}}}}^{{2}}-\left\langle\frac{\coth\left(t\sqrt{D}\right)}{\sqrt{D}}f,g\right\rangle+\left\langle\frac{\coth^{{2}}\left(t\sqrt{D}\right)}{2\left(B+B^{{t}}\right)}g,g\right\rangle,$

End of MathML

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#### Hit idp29990080

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 84
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/203/f081135.xhtml#idp29990080

Rendered MathML:

$W^{{-1}}\dot{W}=\frac{-1}{2t}\hbox{tr}\left[\varphi\left(t\sqrt{D}\right)\right]-\frac{1}{4}\lvert f\rvert _{{A^{{-1}}}}^{{2}}+\left\langle\frac{\coth^{{2}}\left(t\sqrt{D}\right)}{2\left(B+B^{{t}}\right)}g,g\right\rangle-h.$

End of MathML

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#### Hit idp30038016

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 85
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/203/f081135.xhtml#idp30038016

Rendered MathML:

$W(t)=W_{{0}}e^{{-\left(\frac{1}{4}\lvert f\rvert _{{A^{{-1}}}}^{{2}}+h\right)t+\left\langle\phi\left(t\sqrt{D}\right)g,g\right\rangle _{{A}}t^{{3}}}}$

End of MathML

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#### Hit idp3010720

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 86
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/13/f005188.xhtml#idp3010720

Rendered MathML:

$\displaystyle\frac{\partial^{2}\Gamma _{{\alpha}}}{\partial{\alpha}^{2}}=\left\langle\frac{\partial^{2}{\Omega}}{\partial\alpha^{2}}\right\rangle _{{\alpha}}+\left\langle\left[\frac{\partial{\Omega}}{\partial\alpha}\right]^{2}\right\rangle _{{\alpha}}-\left[\left\langle\frac{\partial{\Omega}}{\partial\alpha}\right\rangle _{{\alpha}}\right]^{2}$

End of MathML

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#### Hit idp30152128

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 87
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/203/f081135.xhtml#idp30152128

Rendered MathML:

$W=(4\pi t)^{{-\frac{n}{2}}}\left(\frac{\det\psi\left(t\sqrt{D}\right)}{\det A}\right)^{{\frac{1}{2}}}e^{{-\left(\frac{1}{4}\lvert f\rvert _{{A^{{-1}}}}^{{2}}+h\right)t+\left\langle\phi\left(t\sqrt{D}\right)g,g\right\rangle _{{A}}t^{{3}}}}.$

End of MathML

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#### Hit idp30248976

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 88
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/203/f081135.xhtml#idp30248976

Rendered MathML:

$\displaystyle=(4\pi t)^{{-\frac{n}{2}}}\left(\frac{\det\psi\left(t\sqrt{D}\right)}{\det A}\right)^{{\frac{1}{2}}}e^{{-\left(\frac{1}{4}\lvert f\rvert _{{A^{{-1}}}}^{{2}}+h\right)t+\left\langle\phi\left(t\sqrt{D}\right)g,g\right\rangle _{{A}}t^{{3}}}}$

End of MathML

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#### Hit idp30347360

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 89
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/17/f006455.xhtml#idp30347360

Rendered MathML:

$\displaystyle=\frac{1-e^{{-2T\beta|k|^{{2/(m+1)}}}}}{2\beta|k|^{{2/(m+1)}}}\|\left\langle D_{\theta}\right\rangle\varphi _{0}\| _{{L^{2}}}^{2}$

End of MathML

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#### Hit idp3057200

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 90
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/207/f082579.xhtml#idp3057200

Rendered MathML:

$\displaystyle=i{\partial}_{t}{\mathcal{U}}^{\varepsilon}+\frac{1}{2}\Delta _{z}{\mathcal{U}}^{\varepsilon}-\frac{1}{2}\left\langle z,\nabla^{2}V\left(q(t)\right)z\right\rangle\,{\mathcal{U}}^{\varepsilon},$

End of MathML

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#### Hit idp484944

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 91
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/203/f081064.xhtml#idp484944

Rendered MathML:

$\frac{1}{\omega}(\frac{\partial p}{\partial r}+v\frac{\partial p}{\partial\tau})+\frac{1}{W^{{2}}}(\frac{\partial v}{\partial\tau}+v\frac{\partial v}{\partial r})=\frac{\langle F_{r}\rangle}{4\pi\omega},$

End of MathML

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#### Hit idp5082976

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 92
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/59/f023582.xhtml#idp5082976

Rendered MathML:

$\langle v\rangle=\frac{2k\,\tanh\left(\frac{k\, T}{2}\right)}{{\mathrm{e}}^{{-k\, T}}-1}\langle x\rangle+v_{0}$

End of MathML

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#### Hit idp5659856

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 93
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/206/f082397.xhtml#idp5659856

Rendered MathML:

$-\frac{1}{\beta A}\left[\ln\frac{\int\mathcal{D}[a^{{\ast}},a]\exp-\left(S_{{2}}+S_{{3}}+S_{{4}}\right)}{\int\mathcal{D}[a^{{\ast}},a]\exp-S_{{2}}}\right]_{{2-loop}}=\frac{1}{\beta A}\left(\left\langle S_{{4}}\right\rangle-\frac{1}{2}\left\langle S_{{3}}S_{{3}}\right\rangle\right),$

End of MathML

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#### Hit idp619872

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 94
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/205/f081820.xhtml#idp619872

Rendered MathML:

$\displaystyle\lim _{{t\to\infty}}\frac{\langle Q_{1}^{3}\rangle-3\langle Q_{1}^{2}\rangle\langle Q_{1}\rangle+\langle Q_{1}\rangle^{3}}{t}=$

End of MathML

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#### Hit idp6441424

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 95
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/59/f023405.xhtml#idp6441424

Rendered MathML:

$\displaystyle\frac{d\langle J\sigma _{1}\rangle}{dt}=-\frac{1}{\tau^{{\prime}}}\langle J\sigma _{1}\rangle+\frac{1}{\Theta}\tanh(\beta)\langle\sigma _{2}\rangle+\frac{1}{\tau}\langle J\rangle,$

End of MathML

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#### Hit idp6476928

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 96
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/59/f023405.xhtml#idp6476928

Rendered MathML:

$\displaystyle\frac{d\langle J\sigma _{2}\rangle}{dt}=-\frac{1}{\tau^{{\prime}}}\langle J\sigma _{2}\rangle+\frac{1}{\Theta}\tanh(\beta)\langle\sigma _{1}\rangle+\frac{1}{\tau}\langle J\rangle,$

End of MathML

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#### Hit idp754656

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 97
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/113/f044850.xhtml#idp754656

Rendered MathML:

$\frac{\partial\ln\langle M^{{n}}\rangle}{\partial K}=\frac{\langle M^{{n}}H\rangle}{\langle M^{{n}}\rangle}-\langle H\rangle.$

End of MathML

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#### Hit idp771344

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 98
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/115/f045778.xhtml#idp771344

Rendered MathML:

$\frac{\partial\ln\langle M^{{n}}\rangle}{\partial K}=\frac{\langle M^{{n}}H\rangle}{\langle M^{{n}}\rangle}-\langle H\rangle,$

End of MathML

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#### Hit idp877424

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 99
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/129/f051546.xhtml#idp877424

Rendered MathML:

$\displaystyle\bf E=\left(\begin{array}[]{c|c}\bf A&\bf B\\ \hline\bf C&\bf D\end{array}\right)=\left(\begin{array}[]{c|c}\epsilon\bf A_{1}&\epsilon\bf B_{1}\\ \hline\bf C_{0}+\epsilon\bf C_{1}&\epsilon\bf D_{1}\end{array}\right)=\left(\begin{array}[]{c|c}-\epsilon\displaystyle\frac{\partial^{2}\langle H_{1}\rangle}{\partial\phi _{i}\partial I_{j}}&-\epsilon\displaystyle\frac{\partial^{2}\langle H_{1}\rangle}{\partial\phi _{i}\partial\phi _{j}}\\ \hline\\ \displaystyle\frac{\partial^{2}H_{0}}{\partial I_{i}I_{j}}+\displaystyle\epsilon\frac{\partial^{2}\langle H_{1}\rangle}{\partial I_{i}\partial I_{j}}&\displaystyle\epsilon\frac{\partial^{2}\langle H_{1}\rangle}{\partial\phi _{j}\partial I_{i}}\end{array}\right).$

End of MathML

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#### Hit idp968880

- Reviwer: xxx
- Reviwer score: 0
- Formulasearchengine rank: 100
- Formulasearchengine score: 0
- Reference to collection: _PREFIX_/207/f082527.xhtml#idp968880

Rendered MathML:

$\displaystyle-\displaystyle\frac{\partial}{\partial z^{{(y)}}}\left[-a^{{(y)}}z^{{(y)}}+J^{{(y)}}\langle F^{{(y)}}\rangle-D^{{(y)}}\frac{\partial}{\partial z^{{(y)}}}\right]P.$

End of MathML

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