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\begin{align*}Z_{ren}(\beta)=2\int_{0}^{\infty}e^{-\beta E(l_g)}dl_{g},\end{align*}

\begin{align*}H^2(u)=\left[\frac{e^{4u}+b^4}{e^{4u}-b^4}-\frac{1}{2u}+\frac{2K}{u \Lambda^2}\left(\frac{e^{2u}}{e^{4u}-b^4}\right)\right].\end{align*}

\begin{align*}SU(N)_0\times SU(N+M_1)_1\times SU(N+M_2)_2\times SU(N+M_3)_3.\end{align*}

\begin{align*}\gamma\left(\frac{1}{2},\chi,\psi\right)\gamma\left(\frac{1}{2},\overline{\chi^{-1}},\psi\right)=1.\end{align*}

\begin{align*}\Lambda = \{ \overrightarrow {\mu} \in Z^{r+1} : \overrightarrow {\mu} \cdot \overrightarrow {k} = 0 \}\end{align*}

\begin{align*}\hat L(\infty) =0\,,\quad\quad\hat M(\infty) = \ln 2\,.\end{align*}

\begin{align*}\tilde h_{11}^+=h_{11}^+ \qquad \mbox{and}\qquad\tilde h_{11} +\tilde h_{12}=h_{11} +h_{12} ~.\end{align*}

\begin{align*}\begin{array}{rcl}{\cal R}(u,c)&=&\displaystyle\frac{u-icP}{u+ic}\\begin{align*}4mm] &=&\displaystyle\frac{1}{u+ic}\left[ \begin{array}{cccc} u-ic& 0 & 0 & 0 \\ 0 & u &-ic& 0 \\ 0 &-ic& u & 0 \\ 0 & 0 & 0 &u-ic \end{array}\right]~,\end{array}\end{align*}

\begin{align*}A(\lambda/e^2)\approx 100.56{\exp\left(-1.21(\lambda/e^2)^{2/3}\right)\over(\lambda/e^2)^{1/3}}\;.\end{align*}

\begin{align*}\tau_1(q)\approx - \frac{1}{8\pi}\ln\frac{\Lambda^2}{q^2} \quad {\rm and} \quad\tau_2(q)\sim e^{-\theta M q}\end{align*}

\begin{align*}\lambda_{ijk}(M_{SU})=\lambda_{ijk}^{\rm tree}\, [1+g^2\,(Y_i+Y_j+Y_k)]^{-1/2},\end{align*}

\begin{align*}\bar I_{CS}\equiv\frac{1}{8\pi^2}\int_{{\cal M}_4} \hat F\wedge \hat F+\frac{1}{8\pi^2} \int_{\partial {\cal M}_4} (A-\hat A)\wedge(F+\hat F)\; .\end{align*}

\begin{align*}{\rm Re}\, \tau^{(0)}< {\rm Re}\, \tau^{(k)},\qquad \forall\tau^{(k)}\in \{\tau|\tau\in G, u(\tau)=0, {\rm Re}\, \tau>0\},\end{align*}

\begin{align*}P^2=P_\mu P^\mu , \qquad\qquad W=P_\mu J^\mu\end{align*}

\begin{align*}u_j^{k}= (u_{1,j}^{k}, u_{2,j}^{k} )\in U ,\ \ \ \ y_j^{k}= (y_{1,j}^{k}, y_{2,j}^{k} )\in Y, \ \ \ \ \ \ j=1,..., J_k, \ \ \ \ k=1,..., K,\end{align*}

\begin{align*}\gamma\left(\frac{1}{2},\chi,\psi\right)=1\end{align*}

\begin{align*}S^{-1}aS = (\cosh r)a ~+~e^{i\theta}(\sinh r)a^{\dagger} ,\end{align*}

\begin{align*} \Phi \cdot J = \vartheta^2 \left(\frac{1}{2}(jL-mz)-\psi^\alpha\eta_\alpha \right) + \frac{1}{2}\vartheta^\alpha(j\psi_\alpha+z\eta_\alpha) + \frac{1}{4} j z\end{align*}

\begin{align*}\Sigma _{i}\equiv \left\{ \bar{z}\in \Gamma \mid f_{i}(\bar{z})=0\right\} ,\end{align*}

\begin{align*}<W(A,\gamma)>^{(1)} = \oint_\gamma ds \oint_\gamma dt \dot{\gamma}^a(s)\dot{\gamma}^b(t) \epsilon_{abc}{(\gamma^c(s)-\gamma^c(t)) \over |\gamma(s) -\gamma(t)|^3}={\rm GSL}(\gamma)\end{align*}

\begin{align*}G_{\mu\nu}= F_{\mu\nu}^{corr} ( R_{\mu\nu\lambda\sigma},D_{\alpha} R_{\mu\nu\lambda\sigma}, \dots)\end{align*}

\begin{align*}(i\theta\sigma_i\bar{\psi}^i+i\bar{\theta}\sigma_i\psi^i) +\frac{1}{6}\epsilon^{ijkl}(\bar{\theta}\sigma_{ijk}\psi_l -\theta\sigma_{ijk}\bar{\psi}_l).\end{align*}

\begin{align*}A_{\alpha \beta} = \theta_\alpha \theta_\beta\end{align*}

\begin{align*}A(r)\equiv \lim_{k\to 0}\left[{ig(k,r)\over\sqrt{\pi/2}\,\sqrt{r}}\right]\end{align*}

\begin{align*}Q\equiv {\tilde{T}}_4=T_3+\frac{1}{2}T_4\end{align*}

\begin{align*}\psi^{a}_{\alpha}\rightarrow \psi'^{a}_{\alpha}=(e^{\Lambda}){}^{a}{}_{b}\psi^{b},\end{align*}

\begin{align*}\gamma(s,Ind(\chi)\times Ind(\chi'),\psi)=\prod_{i=1}^{t}\gamma(s,\chi_{i}\times Ind(\chi'),\psi)=\prod_{i=1}^{t}\prod_{j=1}^{r}\gamma(s,\chi_{i}\chi_{j}',\psi).\end{align*}

\begin{align*}\phi (\equiv v)= \sqrt{\frac{2\mu^2}{\lambda}}\; \exp\left[\frac{\lambda(10+ \ln 3) - 96 \pi^2}{20\lambda}\right].\end{align*}

\begin{align*}B_{\mu \nu} \rightarrow B^{\prime}_{\mu \nu} = B_{\mu \nu} + \partial_\mu f_\nu- \partial_\nu f_\mu ~.\end{align*}

\begin{align*}B_F(m) = \sum_{i=1}^{\hat x} \sum_{p>0} (\frac{m}{2} + p)H_i(-p)H_i(p+m)\end{align*}

\begin{align*}\tilde g = \frac{ (R_s M)^{\frac{3-d}{4}} }{\prod r_i}\ ,\quad\tilde \alpha' = \frac{R_s^{1/2}}{M^{3/2}} \ ,\quad\tilde R_i = \frac{1}{r_i M} \ ,\quad R_s\rightarrow 0\ ;\ M,r_i \mbox{~fix{\'e}s}\ .\end{align*}

\begin{align*}\delta_\kappa Z^ME_M^{\hat a}=0, \quad \delta_\kappa Z^ME_M^{\hat \alpha}=\kappa^{\hat\beta}(\xi)(1+\bar\Gamma)_{\hat\beta}^{~\hat\alpha},\end{align*}

\begin{align*} G(f;\vec{q})=\frac{1}{d(\vec{q})+m(f)}\end{align*}

\begin{align*}i{\gamma}^{\mu}({\partial}_{\mu}+ieA_{\mu}){\psi}=0 \end{align*}

\begin{align*}\left[ T^{a},R^{\pm\alpha}\right] =-\left( \lambda^{a}\right) _{\beta}^{\alpha}R^{\pm\beta} \end{align*}

\begin{align*}\bar{\phi} = \phi - \frac {1} {2} {\rm log}~({\rm det}~ g_{ij}) \; ,\end{align*}

\begin{align*}R(u_1 -u_2 ) t^{c}_{b}(u_1 )\otimes t^{b}_{a}(u_2 )=\sum_{d} W\left( \left. \begin{array}{cc} c & d \\ b & a \end{array} \right| u_1 -u_2 \right) t^{d}_{a}(u_1 )\otimes t^{c}_{d}(u_2 ). \end{align*}

\begin{align*}\Psi(\frac{1}{2},\tilde{W},\tilde{W'};\hat{\Phi})=c(\pi')\Psi(\frac{1}{2},W,W';\Phi).\end{align*}

\begin{align*}\Gamma = \Gamma^{0}\left[\overline{\chi}\right] + \lambda\Gamma^{1}\left[\phi,\overline{\chi}\right] + {\mathcal O}\left(\lambda^2\right)\end{align*}

\begin{align*}\psi_{4}= \left( b_{4}\; L_{-1}^{4} + a_{4}\; L_{-2}^{2}+ m_{1}\; L_{2}\; L_{-1}^{2} + m_{2}\; L_{-3}\; L_{-1}+ m_{3}\; L_{-4} \right) \psi,\end{align*}

\begin{align*}y^2_\ast = {\frac{(4\pi)^2\alpha}{2a\alpha_C}},~~{\frac{\lambda_\ast}{y^4_\ast}}={\frac{2a}{(4\pi)^2}}~{\frac{\alpha_C}{\alpha}},~~\xi_\ast={\frac{1}{6}}.\end{align*}

\begin{align*}\langle W\rangle = Z^{-1}\int (D\psi D\bar{\psi})\exp(-S_W)\end{align*}

\begin{align*}\Delta (x)=\tanh (i\partial _{x}\tau )\delta (x)\end{align*}

\begin{align*}B_{\rm mod}(s)\equiv e^{\eta s}\int_{0}^{\beta}m^{-1-s}G(m-\beta)dm\;\;+B_L(s)\end{align*}

\begin{align*}\lambda_{ijk}^{\rm tree}=\frac{g_{\rm tree}}{\sqrt{2}}W_{ijk}\, ,\end{align*}

\begin{align*}W=\lambda[1/6 (AA_{sym})^3 (Q{\bar Q})^2 + 4 (AA_{sym})(AAQ{\bar Q}_{anti})^2 + 64 (AAA{\bar Q}{\bar Q}) (AAAQQ) - \Lambda^{8}(Q{\bar Q})],\end{align*}

\begin{align*}b = i\frac{M}{2m^3} ,\,\,\,c = i\frac{m}{M}.\end{align*}

\begin{align*}\pi_{n+1}(x) - (x - \check S_{n^n(n+1)}^* \check S_{n^{n+1}}^{-1}+ \check S_{(n-1)^{n-1}n}^*\, \check S_{(n-1)^n}^{-1}) \pi_n(x)+\check S_{n^{n+1}}\,\check S_{(n-1)^n}^{-1} \, \pi_{n-1}(x) = 0\end{align*}

\begin{align*}\gamma\left(\frac{1}{2},\pi\times\pi',\psi\right)=1\end{align*}

\begin{align*}n+(n+k-i)-2n_1 \geq a_1 \geq \cdots \geq a_{n_1 -n_2} \geq 0,\end{align*}

\begin{align*}\triangle_r\xi + \omega^2\left[1+\frac{e^2}{r^4}\right]\xi = 0\end{align*}

\begin{align*}\alpha^2 = { Q_c^2 \over a } ,\end{align*}

\begin{align*}-i\bar{\kappa}\Gamma^{a}\kappa = {\textstyle\frac{32}{R_{5}}} \tilde{\mathcal{C}}[\Gamma_{s}(\hat{M}^{\hat{a}\hat{b}})k_{(\hat{a}\hat{b})}+\Gamma_{s}(\tilde{M}^{\tilde{a}\tilde{b}})k_{(\tilde{a}\tilde{b})}]\, .\end{align*}

\begin{align*}a_1 + b_1 \;>\; a_2 - 1 \;\stackrel{(\ref{eq:Dbm})}{\geq}\; a_1 - 1 \;,\end{align*}

\begin{align*}{\mathcal D}^{(1/2,1/2)} \otimes {\mathcal D}^{(1/2,1/2)} ={\mathcal D}^{(1,1)}\oplus {\mathcal D}^{(1,0)} \oplus {\mathcal D}^{(0,1)}\oplus {\mathcal D}^{(0,0)}\, .\end{align*}

\begin{align*}D^{\mu\nu\alpha\beta} = \frac{A}{k^{2}} E^{\mu\nu\alpha\beta} + \frac{B}{k^{2}} \Pi^{\mu\nu\alpha\beta} (k).\end{align*}

\begin{align*}R = -\frac{2}{C^2}\left( \frac{A''}{A} + 3 \frac{B''}{B}+ 3 \frac{{B'}^2}{B^2} + 3 \frac{A'B'}{AB}-\frac{A'C'}{AC} - 3 \frac{B'C'}{BC} \right)=-h''(r)-6\frac{h'(r)}{r}-6 \frac{h(r)}{r^2}.\end{align*}

\begin{align*}\alpha=2\lambda-4x^\mu k_\mu-4\bar\theta\eta.\end{align*}

\begin{align*} S[X^\mu,\gamma_{ij}]=-T\int d^{p+1}\xi \sqrt{-\gamma} \left( \frac{1}{p+1}\gamma^{ij} g_{ij} \right ) ^{(p+1)/2}.\end{align*}

\begin{align*}\prod\limits _{i=1}^{n}\epsilon_{2m+k_{i}}\frac{(2\pi)^{\frac{1}{2}-s_{i}}}{(2\pi)^{\frac{1}{2}+s_{i}}}\frac{\Gamma\left(\frac{1}{2}+s_{i}+\frac{|k_{i}+2m|}{2}\right)}{\Gamma\left(\frac{1}{2}-s_{i}+\frac{|k_{i}+2m|}{2}\right)}=1\end{align*}

\begin{align*}\begin{array}{lcr}\sum\limits_{j}q^j_a(|X_j|^2-|Y_j|^2)\eta_j^+\overline{\eta}_j^+&=r^2_a\eta_a^+\overline{\eta}_a^+ \qquad &(a)\\\sum\limits_{j}q^j_a(X_j\overline{Y}_j)\eta_j^+{\eta}_j^+&=0 \qquad &(b)\\\sum\limits_{j}q^j_a(\overline {X}_j{Y_j})\overline{\eta}_j^+\overline{\eta}_j^+&=0. \qquad &(c)\end{array}\end{align*}

\begin{align*}\int_0^{ 2 \pi } \hat{\varphi} ( \theta ) \, d \theta = 0,\end{align*}

\begin{align*}32 =N_{A_1} +2 N_{A_2}+4^b\left(N_{c_1} +N_{c_3}\right)=N_{A_1} +2 N_{A_2}+4^b\left(N_{c_2} +N_{c_4}\right).\end{align*}

\begin{align*}F+F_{MM_1}+F_{MM_2}+\frac{m_1}{3}+\frac{m_3}{5}-\frac{p}{\sqrt{15}}\; \in\; 2{\bf Z}+1,\end{align*}

\begin{align*}{\delta\over\delta C^{\mu\nu}}\equiv \left(\oint_C dl(s)\right)^{-1}\oint_C dl(s){\delta\over\delta \sigma^{\mu\nu}(s)} .\end{align*}

\begin{align*}\hat T_{jk}([\vec r^N])=\ ^*\sum_{l_1}\cdot\cdot\cdot\ ^*\sum_{l_N} \bar A_j(N_{r_1}+l_1,...,N_{r_N}+l_N) A_k(N_{r_1}+l_1,...,N_{r_N}+l_N).\end{align*}

\begin{align*}\Phi_{m}(z_{m})\circ\cdots\circ\Phi_{1}(z_{1}):M_{q}(\lambda_{1})\rightarrow M_{q}(\lambda_{m+1},k)\otimes ((V(0,\mu_{1})\otimes\cdots\otimes V(0,\mu_{m}))(z_{1},\ldots ,z_{m})\end{align*}

\begin{align*}(ds)^2 = - [ d\tau^2 - (M \tau)^2 dx^2]; \hskip0.3cm 0 \le \tau < \infty\end{align*}

\begin{align*}{\dot{\bar{c}}_4\over\bar{c}_4}=e^{2m_0(R_0-R)}{{R_{\tau}R_{\tau\tau}}\over{c^2+R^2_{\tau}}},\end{align*}

\begin{align*}T_7=(\theta,\theta^2,\theta^3 \vert\vert \theta,\theta^2,\theta^3 \vert (\textstyle{1\over 7})^{4} (\textstyle{2\over 7})^4 (\textstyle{3\over 7})^4 0^4)~.\end{align*}

\begin{align*} \mathcal{S}_{N} = \begin{cases} \{(G, N)\}, & {\rm if} ~G/N~{\rm cyclic;}\\ \emptyset, & {\rm otherwise.} \end{cases} \end{align*}

\begin{align*}J_{(m)}^\mu(\xi)=\nabla_\nu Q_{(m)}^{\mu\nu}(\xi)~~~.\end{align*}

\begin{align*}{\bf \nabla} \times (\frac{m^2}{2 e^2 \langle|\phi|\rangle^2}{\bf V} ) \propto {\bf \omega}_{\rm ext}.\end{align*}

\begin{align*}{\Lambda}^{\alpha}_{mn}=\partial_{[m}B^\alpha_{n]},\end{align*}

\begin{align*}\Psi_j \rightarrow \Psi_j + \alpha_j,\; \Psi_{-j} \rightarrow \Psi_{-j} + \alpha_j, j=1, \ldots, r,\Psi_0 \rightarrow \Psi_0 + \alpha_0,\end{align*}

\begin{align*} D_R = D_{11}-D_{12}\, \qquad D_A = D_{11}-D_{21}\,.\end{align*}

\begin{align*}\eta (\varphi; \vartheta_{0}) =\{(\vartheta_{0}, \varphi'), \vartheta_{0} = \mbox{const}, 0 \leq \varphi' < \varphi \}\end{align*}

\begin{align*}q_3^{\rm pole} = -i \left(-\frac{1}{2}\delta + \frac{\left|{\bf q}\right|^2}{2m} + \Sigma_-(q_3^{\rm pole},\left|{\bf q}\right|) \right), \end{align*}

\begin{align*}2iY_a^{\dagger} K_0 \partial_{\tau} Y_a \; ,\end{align*}

\begin{align*}\frac{\delta}{\delta\{D_{\alpha}\}_nB_{\mu\nu}^c}\Delta_{0 b}^{\rho}(x)=\{D_{\alpha}\}_n gF_{\mu\nu}^c(x)X_{\rho}( A,gF)^{-}+ \{D_{\alpha}\}_n\epsilon^{\mu\nu\lambda\eta}gF_{\lambda\eta}^c(x)X_{\rho}( A,gF)^{+}\end{align*}

\begin{align*}(12g_4+8\widetilde{g}_2 N)\Delta^2+g_2\Delta=S.\end{align*}

\begin{align*} d_{i}:= \begin{cases}1,~{\rm if} -1 \in \langle r_{i}\rangle; \\ 2, ~{\rm otherwise}.\end{cases}\end{align*}

\begin{align*}a^I_i({\bf r},t)=\epsilon^{ij}\nabla_j\frac{1}{\kappa_I} \sum_pq_p^I\int\! d^2\!{\bf r}'\, G({\bf r}-{\bf r}')\rho_p({\bf r}',t),\end{align*}

\begin{align*}F_c\ = \ m_{\!_J} \frac{c^2}{R}\end{align*}

\begin{align*}\psi(\alpha,\beta,\beta',s,s')=\frac{ \varphi_{(\alpha+\beta+\beta',s+s') } \varphi_{(\alpha+\beta-\beta',s-s')}} {\varphi_{(-\alpha+\beta+\beta',-s+s')} \varphi_{(\alpha-\beta+\beta',-s-s')}}.\end{align*}

\begin{align*}c(\lambda,\beta,\gamma,\dot\gamma,\theta) =\left[\gamma^* \dot\gamma - \gamma \dot\gamma^*+\frac{b}{2|\lambda|^2} (\lambda^* \dot\gamma e^{-2\imath\theta}- \lambda \dot\gamma^* e^{2\imath \theta})\right]\end{align*}

\begin{align*}\epsilon_{\bullet}=1,~~~~~~\epsilon_{\circ}=0.\end{align*}

\begin{align*}S=8\sqrt{2}\pi m^{3/2}\prod^3_{i=1}\cosh\delta_i,\end{align*}

\begin{align*}a_j=0,\qquad j=3,4,5, \dots\, {.} \end{align*}

\begin{align*}D_{2}(t)={1\over 16}t^{2}-{3\over 8}t^{4}+{5\over 16}t^{6} ,\end{align*}

\begin{align*}\pi_{1}(Map_{\ast}(S^{3}, SU(2)\times U(1))) \simeq \pi_{4}(SU(2))\simeq Z_{2}\end{align*}

\begin{align*}{\cal H}^j_{\alpha=\pm \infty} = j\{ a^{\dagger}_j a_j + b^{\dagger}_j b_j + 1 \}\end{align*}

\begin{align*} \mathcal{B}(G):= \bigcup_{i=1}^{m}B(H_{i},K_{i}).\end{align*}

\begin{align*}{t^{\rm HM}_{\rm decay}\over t^{\rm CDL}_{\rm decay}}= \exp\left[8\pi^2\left({8\over T^2}-{3\over V_1}\right)\right]\ .\end{align*}

\begin{align*}\bar{W}_{\sigma \mu \nu}(g, \bar{\Gamma}) =\bar{W}_{\sigma a b }(\bar{\Omega}) e^a_{\mu} e^b_{\nu}= - \left( \bar{\Omega}_{ab \sigma} + \bar{\Omega}_{ba \sigma} \right)e^a_{\mu} e^b_{\nu}\end{align*}

\begin{align*}a_f = c_2(B) + \left(11 + \frac{n^3-n}{24}\right) c_1(B)^2 - \frac{3n}{2\l}\left(\l^2-\frac{1}{4}\right) = 17\end{align*}

\begin{align*}d G = {\kappa^2 \over \sqrt{2} \lambda^2} \delta(x^{11}) dx^{11} \hat{I_4},\end{align*}

\begin{align*}{\cal J}=\int_0^\infty{dx\over x^3}\left({\pi\over x}\right)^{1/2}\left[ \theta_3\left({i\pi\over x}\right)-\theta_4\left({i\pi\over x}\right)\right]\end{align*}

\begin{align*}2\hat{R}=e^{2A(r_H)}\left[B_1+\Delta\right]\left[B_1+2\Delta\right].\end{align*}

\begin{align*}V'_c(x, \theta, \bar{\theta}) = \frac{g_c}{g'_c} V_c (e^{-t} x, e^{-t/2}\theta, e^{-t/2}\bar{\theta})\end{align*}