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7944775fc9 | \alpha_1^r \gamma_1 + \dots + \alpha_N^r \gamma_N = 0\quad(r=1,...,R)\; ,\label{contrainte} | |
78228211ca | \eta = -\frac{1}{2} \ln \left(\frac{\cosh \left(\sqrt{2}b_\infty\sqrt{1+\alpha^2}\; y - {\rm arcsinh}\; \alpha\right)}{\sqrt{1+\alpha^2}}\right) | |
15b9034ba8 | \label{fierep}P_{(2)}^-=\int \beta d\beta d^9p d^8\lambda \Phi(-p,-\lambda)\left(-\frac{p^Ip^I}{2\beta}\right) \Phi(p,\lambda)\,. | |
6968dfca15 | \label{GAMMA} \Gamma(z+1) =\int^\infty_0\,\,dx\,\,e^{-x}x^z. | |
6cead0df53 | \label{rotflow}\frac{d}{ds}{\bf C}_i= \frac{1}{2}\epsilon_{ijk}{\bf C}_j\times {\bf C}_k \, . | |
5381b22df4 | Z=\sum_{spins}\prod_{cubes}W(a|e,f,g|b,c,d|h), \label{f1} | |
27f2b37ce9 | \label{SUSY}\left\{ Q^{i},Q^{j}\right\} =c^{ij}\Gamma ^{M}CP_{M}+Cc^{ij}Z, \label{v4} | |
51a257cdf5 | \label{as7}\breve{c}_{n,\nu}=\sum_{m=n}^{2n}{\Gamma\left(\nu+m-{D-1 \over 2}\right) \over \Gamma\left(\nu+n-{D-1 \over 2}\right)}~\breve{a}_{2(m-n),m}~~~. | |
5108925e21 | R(g)=-f\left[3\left[(\ln f)^\prime\right]^2+\frac{\Lambda(x^5)}{M^3} \right] \; , \label{FourScalar} | |
3882dd3d43 | {d\over ds}{1\over\Gamma(-s)}\bigg|_{s=0}=-1, | |
566cf0c6f5 | \dot z_1 = - N^z(z_1) = - g(z_1) = -\frac{z_1}{P_z(z_2-z_1)};~~~\dot z_2 = -\frac{z_2}{P_z(z_2-z_1)} | |
7d1fe2cc05 | c_{\alpha} = \sum_{\beta\in\Lambda_{R}} \epsilon (\alpha,\beta ) |\beta + \bar{p}><\beta + \bar{p}| | |
450b24df87 | \label{lqed}{\cal L}=-{1\over 4}F_{\mu\nu}F^{\mu\nu}+{\bar\psi}(i\gamma^\mu D_\mu -m)\psi\,, | |
667ff49bc5 | e^{i {\bf k \cdot r}}=e^{i k r \cos(\theta - \Theta)}=\sum_{l=-\infty}^{\infty}i^l\,J_l(kr)\,e^{i l (\theta - \Theta)}\,,\label{eqn:iiifive} | |
61928de22b | i\sqrt{2} \partial_{-}\chi - g[\phi,\psi] = 0,\quad\partial_{-}^{2} \bar{A}_{+} - g^2 J^{+} = 0.\label{eq:const} | |
4cd65285c9 | \label{29}\Omega^{(l)}_k=\sum_{s=0} \int d^3y \left((-1)^{s+1} \frac{d^s}{dt^s}\phi^{i(s)}_k(x,y)L^{(0)}_i(y)\right). | |
12697ce419 | L_{g}^{'}\Bigl(v(h)\Bigr) = v(L_{g}h) = v(gh)\, , \,\, \, \forall g,h \in G, | |
a8ec0c091c | \xi^2=\left(\frac{\varepsilon_1-\varepsilon_2}{\varepsilon_1+\varepsilon_2}\right)^2=\left(\frac{\mu_1-\mu_2}{\mu_1+\mu_2}\right)^2, | |
72a80f57d9 | R(e_1) = \epsilon^{- J_{67} + J_{89}}, \quad R(e_2) = \epsilon^{ J_{45} - J_{89}} .\label{C3ZNZN-RotationsWithDiscreteTorsion} | |
330f27c566 | \label{eq33}{\tilde {\cal {E}}}_{m<0} = {\cal {E}}_{m<0}(B) - {\cal {E}}(0)= \frac {B^2} {2} + \frac {(eB)^{\frac {3} {2}}} {2 \pi} g\left(\frac {eB} {m^2}\right) \, , | |
58be3470dc | \label{R84}\hat{O}^{r}_{2}\mid 1>_{(0)}={O}^{r}_{2}\mid 0>_{(0)}. | |
3e82680317 | I^c =\mp{\pi b \sqrt{1 - \Lambda a^2}\over 2G}\ \ ,\label{exaction} | |
31068cb86d | g^>_n(r,r')=E_n K_{|n/\alpha|}(\beta r), \quad \hbox{for $r>r^{'}$.}\label{22} | |
431dd6944e | \label{sol_excited}R^{\frac{1}{2}}(\theta )^{\left| \left. b_{k}\dots \frac{1}{2},b_{1},\frac{1}{2}\right| n_{k}\dots ,m_{1},n_{1}\right\rangle }_{\left| \left. a_{k}\dots \frac{1}{2},a_{1},\frac{1}{2}\right| n_{k}\dots ,m_{1},n_{1}\right\rangle }=R_{a_{1}b_{1}}^{\frac{1}{2}}(\theta )\prod ^{k-1}_{i=1}f^{a_{i}a_{i+1}}_... | |
54a7b9d7f8 | Q_1^{ab} (x,y) \equiv Q_1^{ab} + x \, J_1^{ab} + y \, K_1^{ab}, | |
632e971eb8 | \left\{\begin{array}{c} \partial_{\tau}R+\vec\nabla\cdot \left(\vec\nabla\Theta\,\sqrt{\displaystyle\frac{R^2+a^2} {1+(\vec\nabla\Theta)^2}}\right)=0,\hfill\\[4mm] \partial_{\tau}\Theta+R\sqrt{ \displaystyle\frac{1+(\vec\nabla\Theta)^2} {R^2+a^2}}=0.\hfill\\ \end{array}\right. \label{JPeqmot} | |
399e18a85c | \Delta^{(N,0)}(s)= - \sum_{n>0,\vec n^2<N}\left[ J(z_n) - 2 + 2 J(y_{n}) + \frac{J^{2}(y_{n})}{2 (1-y_{n})} - J(\tilde{z}_n) - 2 J(\tilde{y}_n) \right]\ , \label{deltafin} | |
707b5988e2 | \left\{\Psi\circ\mu ,f\right\}=(\overline X_if)\, (Y^i \Psi)\circ\mu\,,\label{mom2} | |
25fe4d51bf | F_{n}^{\mathcal{O}|\mu _{1}\ldots \mu _{n}}(\theta _{1}+\lambda ,\ldots,\theta _{n}+\lambda )=e^{s \lambda }F_{n}^{\mathcal{O}|\mu _{1}\ldots \mu_{n}}(\theta _{1},\ldots ,\theta _{n})\,\,, \label{rel} | |
3dc7799669 | \label{extended action}S = S_{Phys.}(\Phi^a,\Phi^{\ast a}) + S_T(\vartheta^b,\vartheta^{\ast b}, c^\alpha) | |
76d30658bb | \mathcal{A} \equiv \exp \left[ \int_0^\lambda d\tilde{\lambda}\, \theta(\tilde{\lambda}) \right]\, . | |
6a366e1f12 | \label{fermhalf} F_{-{1\over2}}(x)=\bar \epsilon_0S(x)e^{-1/2\phi(x)}\;,\qquad F_{ 1\over 2}(x) = \bar \epsilon_0 \gamma_\mu S(x) \partial X^\mu(x)e^{1/2\phi(x)}, | |
de8a312222 | \rho^0 = \left( \begin{array}{cc} 0 & -i \\ i & 0\end{array}\right)\,\,\, \mbox{and}\,\,\, \rho^1 = \left( \begin{array}{cc} 0 & i \\ i & 0\end{array}\right) . | |
2b891b21ac | \psi=\sum_{i=0}^3 (\psi_i^A +(\psi_i^A)^c) T^A | |
72e168fb21 | \label{coset}G=\!e^{i\tau L_{-1}} e^{iU^{(1)}L_1} e^{iU^{(2)}L_2}e^{iU^{(3)}L_3}\ldots\! e^{i{U^{(0)}}L_0}, | |
3d129cfe77 | V(z, \bar z)=e^{-q \Phi(z)} e^{i \alpha \cdot H} e^{i(P_R \cdot X_R-P_L \cdot X_L)} \;, | |
6a85896075 | \label{4.5}\epsilon_i = \tau_i + \rho_i + \rho_{i-1}, \quad (\tau_3 =0 ,\: \rho_0 = \rho_4) | |
79edbca78a | s_\infty (k^2)-s_{J_{\max }}(k^2)\sim O(J_{\max }^{-2}). \label{if} | |
20032b2645 | \label{eq:SERELRA} A(u)~=~{\rm Res} \vert_{v=u}^{} \left( {1 \over v-u} \, R(u,v) \cdot L(v) \right) + \, {\textstyle {1 \over 2}} \, \zeta(2u) \, L(u) | |
3d15b5c484 | \partial^m_a \Gamma_i = \frac{\Gamma^n}{\lambda_i} \{\delta_{nm} \psi_a^i - \phi^n_b \phi^m_c \psi_b^i \psi_c^i \frac{\psi_a^i}{\lambda_i^2} + \phi^n_b \phi^m_c \sum_{j \neq i} \psi_b^j \frac{(\psi_c^i \psi_a^j + \psi_a^i \psi_c^j)} {(\lambda_i^2 - \lambda_j^2)}\}\label{E11}\vspace{-12pt} | |
2608ceb605 | \int {\rm d}^{4}x_{1}~\cdots ~{\rm d}^{4}x_{n}~P_{4}(x_{1},\cdots,x_{n})~\Gamma _{x_{1}\cdots x_{n}0}=0 \label{sum2} | |
146a5fa39e | L=\frac{\dot{x}_\mu^{2}}{2e}+\frac{\lambda}{l}(e-M^{-1}\dot{x}{}^0),\label{inter} | |
159bf72783 | J_2(z)\times X^{+}(w)\rightarrow 0. \label{j2xp} | |
1e3aab9a4f | F(z^{\prime}_{12})=\bar{K}(z_{2};g)F(z_{12})K(z_{1};g)\label{zz} | |
62409f879c | {\xi}^{\ast}_i, {p}^{\ast}_i, \quad i = 2, \dots, l+1 | |
6beab42e50 | \varrho_L - {\cal L}_E= [2\dot\Phi^2] \; K'(\dot\Phi^2,\Phi) - K(\dot\Phi^2,\Phi) + K(-\dot\Phi^2,\Phi). | |
105ccc7946 | K' = \sqrt{c - 2f}\ , \ \ \ K'' = -\frac{1}{\sqrt{c - 2f}}\ , | |
6df7276525 | \label{kappa}\kappa _{\omega }=\frac{2\Gamma (\Delta _{\omega })}{\pi \Gamma(1-\Delta _{\omega })}\left( \frac{\sqrt{\pi }\Gamma \left( \frac{1}{2-2\Delta _{\omega}}\right)}{2 \Gamma \left( \frac{\Delta _{\omega }}{2-2\Delta _{\omega}}\right) }\right) ^{2-2\Delta _{\omega }}\, . | |
65d07ed733 | <\frac{1}{2},m_s|{\psi}_{-}^{(\frac{1}{2})}(g)>\equiv D^{(\frac{1}{2})}_{m_s-\frac{1}{2}}(g)=<g,l+\frac{1}{2}|T^{\frac{1}{2}}_{m_s-}|g,l>. | |
34173474c4 | \sum_{l,n} \frac{\mu_{p-1}\lambda^{k+n+l}i^kp!}{k!n!l!(p-l)!}\partial_{x^{i_1}}\ldots \partial_{x^{i_n}} C^0_{i_1'\ldots i_{2k}'j_1\ldots j_l[a_{l+1}\ldots a_p} Str \left(\partial_{a_1}\phi^{j_1}\ldots \partial_{a_l]}\phi^{j_l}\phi^{i_1}\ldots \phi^{i_n}\phi^{i_{2k}'}\phi^{i_{2k-1}'}\ldots \right) | |
1a79f53af4 | D^{\mu}\frac{\delta f(A_{\nu})}{\delta A_{\mu}}=D_{\mu}\partial^{\mu}(\partial_{\nu}A^{\nu}) | |
57e32e5b33 | \delta\chi_{\mu\nu} = ib_{\mu\nu}, \qquad \delta b_{\mu\nu} = 0.\label{eqn:topantiaux} | |
7e1098abc4 | V_{ab\ \ mn}^{k}=\frac{1}{g}\ E_{a}^{r}\ E_{b}^{s}\epsilon _{rs(m}\ \delta_{n)}^{i}. | |
5ada9733aa | \label{req3}f(r)= \left( 1-\frac{m}{2r^{n-1}}\right)^2 +\frac{r^2}{l^2}. | |
5b109d24dc | E_{12}~~\Phi= 2\sqrt{(m+\frac{1}{2}br)^{2}+p^{2}_{r}+ \frac{\ell(\ell+1)}{r^{2}}} ~~\Phi, \label{eq:e} | |
5193ae2c89 | T_{\mathit{G}}(-t,-t^{-1})=T_{\mathit{G}^{\ast }}(-t^{-1},-t) \label{16} | |
119b93a445 | ds_{11}^2 = dx^+ dx^- + l_p^9 \frac{ p_-}{r^7} \delta(x^- ) dx^- dx^- + dx_1^2 + \ \cdots \ + dx_9^2 \label{ase} | |
4fa61dbf37 | F_{ab} = {1\over 2} \epsilon_{abcd} F^{cd} | |
25765b9391 | 2f^2-4f^2-g^2(1-\Gamma) \, ,\label{eq:3.16} | |
276c373567 | (a^{\dagger} L_{mn} a) = a^{\dagger}_{k} (L_{mn})_{kl} a_{l} =i a^{\dagger}_{[m} a_{n]}, \;\;\;\;\;\; (L_{mn})_{kl} = i (\delta_{mk} \delta_{nl} - \delta_{nk} \delta_{ml} ) | |
3fd05b449f | \int dt d^3x \bar{\lambda} \partial^\mu \gamma_\mu \lambda, | |
6b2c7f0c1a | h = {s\lambda\over {1 + 2n + sN + |N|}},\label{eigenvalue} | |
7c2f256525 | Q=c\sum_{i} f_{i}' p^{i} + \sum_{k} c_{k} p^{k} f_{k} +infinite \: more. | |
3beaade5a5 | {\rm Tr}\,\log(1-\sum_{i=0}^{N} A_i)~=~{\rm Tr}\,\log(1-\sum_{k=1}^{N} \sum_{m=0}^{k-1} A_k\phi^m)+{\rm Tr}\,\log(1-\phi)~. | |
3cf56a8338 | H_{ij}^{a}=F_{ij}^{a}-gf_{\;\;bc}^{a}A_{i}^{b}A_{j}^{c}, \label{49x} | |
4aea73b2b8 | {\tilde{\rho}}_{ {\bf{q}} } = \sum_{ {\bf{k}} }[\Lambda_{ {\bf{k}} }({\bf{q}}) a_{ {\bf{k}} }(-{\bf{q}}) + \Lambda_{ {\bf{k}} }(-{\bf{q}}) a^{\dagger}_{ {\bf{k}} }({\bf{q}})] | |
2e96a960b1 | \label{1.2}{\bf N}({\bf p},{\bf s}):=ip_0{\nabla}_{\bf p}-\frac{{\bf s}\times{\bf p}}{p_0+m},\quad{\bf J}({\bf p},{\bf s}):=-i{\bf p}\times{\bf \nabla}_{\bf p}+{\bf s}:={\bf L}({\bf p})+{\bf s}, | |
5b10a20227 | \label{deca2} A_\mu \;=\; \partial_\mu \varphi + \epsilon_{\mu\nu} \, \partial_\nu \sigma \;. | |
18049a05a9 | C_J (\nu_1, \nu_2)=(2 J + \nu_1 + \nu_2 + 1)\frac{{\mit\Gamma} (J + 1) {\mit\Gamma} (J + \nu_1 + \nu_2 + 1)}{{\mit\Gamma} (J + \nu_1 + 1) {\mit\Gamma} (J + \nu_2 + 1)}\, . | |
17ad05c612 | (\psi\otimes_{\zeta,z} \chi) \mapsto (e^{-u L_{-1}} \psi\otimes_{\zeta+u,z+v} e^{-v L_{-1}} \chi), | |
4b69ad5dc8 | u_0(k,r)=\sqrt{{\displaystyle {\pi\over 2}}}\,i\sqrt{r}\,J_0(kr) - \sqrt{{\displaystyle {\pi\over2}}}\,A(k)\sqrt{kr}\,H_0^{(1)}(kr).\label{eq:2.19} | |
4acf2a0344 | J_k=\oint p_kdq_k,~~k=r,~\theta,~\phi,\label{eqjk} | |
272667a2d1 | \delta_\perp \kappa_1 = \kappa_3 \kappa_2 \Psi_3+ 2 \kappa_2 \Psi_2{}' + \kappa_2' \Psi_2 + \Psi_1{}'' - \left( \kappa_1^2 + \kappa_2{}^2 \right) \Psi_1\,.\label{eq:perpkappa} | |
5a5e2b80dd | \left( \gamma _\mu \partial _\mu +m\right) \psi^{(b)}(x)=0,\hspace{0.5in} b=1,2,3,4 \label{4} | |
6596750444 | \label{sin}f_\alpha(x)=\left(4\sin^2\frac{x}{2}\right)^\alpha. | |
4370181032 | r_h^2 = \frac{l^2}{2}(\sqrt{K^2+4l^{-2}\mu}-K). | |
11ff25534a | \label{eomsfield}\mu ''+\biggl[n^2-\frac{a''}{a}\biggr]\mu =0. | |
22f7232e98 | x_{\overline m}={1\over2}(x_m+x_{m+1}),\label{ave} | |
3eaf444392 | S_{ij}\left( \theta \right) =\prod\limits_{x,y}\left[ x,y\right] _{\theta }\label{nichtsimp} | |
25c3276f55 | A_{d}(p^{2}+\omega _{n}^{2})^{\frac{1}{2}d-2}\left[ \left( 1+v^{2}\right) ^{\frac{1}{2}d-\frac{3}{2}}+\frac{\Gamma (\frac{1}{2}d-\frac{1}{2})}{\sqrt{\pi}\Gamma (d)}\frac{(v^{2})^{\frac{d}{2}-1}}{1+v^{2}}{}_{2}F_{1}\left(\textstyle{{\frac{1}{2}d-\frac{1}{2},1;\frac{1}{2}d;\frac{v^{2}}{1+v^{2}}}}\right) \right] \,\ \cdot... | |
32ebd66b47 | \psi_c(x)=\gamma^1 \psi^*(x) \ ,\label{in10} | |
3b014d22b2 | L = L^\Lambda {\bf T}_\Lambda = dZ^M L_M{}^\Lambda {\bf T}_\Lambda\,. | |
6a88fd17f0 | z^{'(r)}_{t,0} \quad = \quad z^{''(r)}_{t,0} \quad = \quad 0\qquad ( 1 \le t \le r, \,\, {\rm{all}} \,\, r ) \, ; | |
4b49a4f210 | dT(x)=\left(\begin{array}{cc} \delta(x) 1_{N-k} & 0 \\
0 & -\delta(x)1_k \\ \end{array}\right)dx | |
4b4263156b | 1 - \frac{2GM}{\rho} = (\nabla\rho)^2 \equiv f | |
48f89a8fc4 | \label{eq:MidentityW} M_{g}= M_{c_{1}} M_{c_{2}} M_{c_{3}} M_{c_{4}} M_{c_{5}}M_{r=\infty}= 1 | |
8d78ecda53 | \delta F\left( \sin\theta\, dx^{0}dx^{1},0,0,\epsilon\,dx^{0}\cdots dx^{3}\right) . \label{finaldf} | |
4015ba8922 | S^{({\rm N})}_{{\rm part},0} = \int dt\, \sum\limits^{N}_{\alpha=1} \left( \xi^{\underline{a}}_{\alpha} \left( E^{\underline{a}}_{j,\alpha} \dot{x}^{j}_{\alpha} + E^{\underline{a}}_{0,\alpha} \right) - {1\over 2} \xi^{\underline{a}}_{\alpha} \xi^{\underline{a}}_{\alpha} \right)\, . | |
acedffb147 | \xi = v_1 \left( u_1 - \kappa v_2 \right) + v_2 \left( u_2 -\kappa v_1 \right). | |
4f055acd1f | \label{effpot}U(r)=U(r_0)+4\pi^2K A(d,\sigma)\int_{r_0}^{r}ds \frac{s^{\sigma-d-1}}{\varepsilon(s)}. | |
5ecb739ec1 | |0 \rangle \rightarrow |0 \rangle_{\beta} = (1 + {\rm e}^{- \beta\epsilon})^{-1/2} \{ |0 \rangle_a \otimes |0\rangle_b + {\rm e}^{- \beta \epsilon / 2} a^{\dagger} {\tilde a} |0 \rangle_a \otimes b^{\dagger} {\tilde b} |0 \rangle_b \}. | |
2beadd086b | S_E = \int_0 ^{\tau} d\tau \left({1 \over 2} x_{\tau} ^2+{1 \over 2} W^2(x)- \psi^{\ast}[\partial_{\tau} -W^{\prime}(x)]\psi \right) | |
2450656988 | R_{\mu \nu \,\,b}^{\quad a}=\partial _{\mu }\omega _{\nu\,\,b}^{\,a}-\partial _{\nu }\omega _{\mu \,\,b}^{\,a}+\omega _{\mu\,\,c}^{\,a}\ast \omega _{\nu \,\,b}^{\,c}-\omega _{\nu \,\,c}^{\,a}\ast\omega _{\mu \,\,b}^{\,c} | |
147dde7fd3 | M=\int_{r\rightarrow \infty}d^pxr^{p/2}f^{-1/2}T_{tt} =\frac{pmV_p}{16\pi G_{p+2}}. | |
17d9b6a683 | j_{HW}(x) = W_i(x) T^i\;,\; T^i \in ker(Ad(M_-)) | |
55358c150e | k_0\sim \omega \sqrt{\frac{g\phi _0}{2M^2}}\ll \omega | |
13dbb0dd7c | Y(T,U) = \int_{\cal F}\frac{d^2\tau}{\Im \tau} \Gamma_{2,2}(T,U) \left(-6\left[{\overline{\Omega }}_2^{\phantom 2}- \frac{1}{8\pi \Im\tau}\right]\frac{\overline{\Omega}}{\overline{\eta}^{24}}- \frac{\overline{j}}{8}+126\right)\ ,\label{twelve} | |
1d796ff39a | P_0S^*P_0SP_0=P_0=P_0SP_0S^*P_0\quad\Longleftrightarrow \quad(S_{00})^*=(S_{00})^{-1}\quad \mathrm{on}\quad \mathcal{H}_{\mathrm{phys}}. | |
6a39a91654 | \langle\psi ^{1-a}_{Fa}\mid\phi ^{1-a'}_{Fa'}\rangle_t=
\frac{1}{2}\delta (a-a')\theta (t-1+a)\theta (t-1+a') | |
6661b12767 | L(z,u_a,D)\equiv\int_0^{\infty}d\hat T\,J(z,u_a,\hat T,D)=L_{02}(z,u_a,D)+g(z,D)G_{Bab}^{1-{D\over 2}}+O\bigl(z^4,G_{Bab}^{2-{D\over 2}}\bigr)\nonumber\\\label{spindivextr} |
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