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\begin{align*}w_n=a \omega_n = 2 \pi n a T, \quad n=0,1,2, \ldots \,{.}\end{align*}

\begin{align*}\S=\frac{\mbox{i}}{2}\log\mbox{det}{\hat H}\quad,\end{align*}

\begin{align*}\overline{Q} = \left( \overline{Q}^+ \; \; \; \overline{Q}^- \right)\end{align*}

\begin{align*} S^n_0(0,y,z) &= a_{n,\,1}\,y^2 + a_{n,\,2}\,yz + a_{n,\,3}\,z^2 + A_n(y,z) \left( \sum^3_{j=0} c_j\, y^{3-j}z^j \right) .\end{align*}

\begin{align*}P_AP_B = \delta_{AB} P_B \hskip 1cm {\rm (no\hskip 0.2cm sum)}\end{align*}

\begin{align*}u^{~\underline a}_{\underline b}=\left(u^{--}_{\underline b},u^{++}_{\underline b},~u^{~i}_{\underline b}\right)\end{align*}

\begin{align*} \frac{\partial}{\partial k_1}\left[ \sqrt{\omega_k} \, F(k_1, \Omega)\right] = i \sigma\,\Omega \, \frac{1}{\sqrt{\omega_k}} \, F(k_1, \Omega)\,,\end{align*}

\begin{align*}c^2=\frac{\epsilon^2}{ab}(\frac{\xi^2}b+\frac{\eta^2}a)+0(\epsilon^2).\end{align*}

\begin{align*}H_{ph}=a(t_{ph})\int^{t_{ph}}_0\frac{1}{a(t)}dt=3t_{ph},\end{align*}

\begin{align*}\sigma(g x)=a(g,x)\cdot\sigma(x)\end{align*}

\begin{align*}\tilde u^\dag\tilde u=1-p_0-p_1,\quad\tilde u\tilde u^\dag=1,\end{align*}

\begin{align*}\epsilon=\left\{\begin{array}{ll} 1&\mbox{if $\omega=\pi/2$}\;,\\-1&\mbox{if $\omega=3\pi/2$}\;.\end{array}\right.\end{align*}

\begin{align*}\frac{\partial W}{\partial X^{i}}\Bigg| _{a_{k}}=0 \quad\hbox{for all } i=1\dots N.\end{align*}

\begin{align*}v(u) = -u^2-u^{-(1+\alpha)n+2} \,.\end{align*}

\begin{align*} \begin{aligned}A_N &= A_1(w_{N-1})\;\! A_{N-1} = \prod_{i=0}^{N-1} A_1(w_{N-i-1}) \\D_N &= D_1(w_{N-1})\;\! D_{N-1} = \prod_{i=0}^{N-1} D_1(w_{N-i-1}) \\begin{align*}0.3em]C_N &= C_1(w_{N-1})\;\! A_{N-1} + D_1(w_{N-1})\;\! C_{N-1} \\&= \sum_{i=0}^{N-1} D_i(w_{N-1-i})\, C_1(w_{N-1-i}) \, A_{N-1-i} . \\\end{aligned} \end{align*}

\begin{align*}\Psi_\nu(t,\sigma) =e^{-i\nu t} \tilde\Psi_\nu(\sigma)+ e^{i\nu t} \tilde\Psi_{-\nu}(\sigma)\> ,\end{align*}

\begin{align*}\langle \sqrt{dx' \wedge dy'}, \sqrt{dq_1 \wedge dp_2} \rangle = dq_1 \,dq_2 \,dp_2.\end{align*}

\begin{align*}{\cal L} _{\gamma - m} = {f_m \over 2} (g A^{\mu})(\partial _{\mu} \Pi _m)\end{align*}

\begin{align*}L = -M + 2\lambda \dot{a}_i\dot{a}_i.\end{align*}

\begin{align*}\{X^M\}=\{\varphi\, ,\ x^m\},\quad \varphi\equiv X^0.\end{align*}

\begin{align*} x^\mu- y^\mu + \frac{2i}{(1^1 2_4)} \xi_{2/4}\sigma^\mu \bar\theta^1\,, \qquad \xi_{2/4} \equiv [(1^2 2_4) \theta_2 + (1^3 2_4) \theta_3 + (1^4 2_4) \theta_4]\end{align*}

\begin{align*}\widehat v_r = \frac{1}{\sqrt 2} (a_r + a_r^\star), \qquad \widehat u_r =\frac{1}{i\sqrt 2} (a_r - a_r^\star). \end{align*}

\begin{align*}u'' = \left( \lambda^2 F + G \right) u\,,\end{align*}

\begin{align*}2\pi r_+\Delta r_+=\beta\left[ {\partial {r_+}\over\partial \beta}{\partial{W_1^{CS}}\over \partial r_+}\right]_{\alpha=1},\end{align*}

\begin{align*}\varepsilon^+_l \rightarrow \varepsilon_u, \ \ \ \\varepsilon^-_u \rightarrow \varepsilon_l, \ \ \ \, \eta_* = 1,\end{align*}

\begin{align*} A_N^{-1}(w_N) = \prod_{i=0}^{N-1} A_1^{-1}(w_i) = \prod_{j=0}^{N-1-i} A_1^{-1}(w_j) \, \prod_{j=N-i}^{N-1} A_1^{-1}(w_j) = A_{N-1-i}^{-1} \, A^{-1}_i(w_{N-i})\end{align*}

\begin{align*}\left( \begin{array}{cc} u_{0} m & u_{0} b \\ u_{0} b^{\dag} & u_{0} n \end{array} \right)\end{align*}

\begin{align*}\hat{\tilde{\lambda}}\equiv a\left(i_{\hat{k}}\hat{\Sigma}\right)\, ,\end{align*}

\begin{align*}\Phi = \pi - \frac{1}{2} \, \phi_x \end{align*}

\begin{align*}Q_{\nu-1/2}(\cosh\rho)=\frac{1}{\sqrt{2}}\int_{\rho}^{\infty}dt\frac{e^{-\nu t}}{\sqrt{\cosh t-\cosh\rho}}\ .\end{align*}

\begin{align*}\left\langle \Psi _{0}\left| f(a^{\#})\right| \Psi _{0}\right\rangle\end{align*}

\begin{align*}r_- = - q_- \circ \left( q_- - \bar M\circ q_+\right)^{-1}.\end{align*}

\begin{align*}K_\infty=x_0^{d-2p-1}\sum_{\mu=0}^p (-)^\mu x^{i_\mu} dx^{i_0}... \widehat{dx^{i_\mu}}... dx^{i_p}.\end{align*}

\begin{align*}S = \sigma \int d^2 \xi \partial_a x^0 \partial_a x^0 +\sigma \int d^2 \xi \partial_a x^i \partial_a x^i\end{align*}

\begin{align*}\int_{0}^{\infty}\frac{d^{4}p}{(2\pi)^{4}}S(ip)_{S}\simeq\int_{0}^{\infty}xF(a,b;2;-x)dx=0,\end{align*}

\begin{align*}{\cal V}[\phi_0] = V[\phi_0,m(\mu_0),\lambda(\mu_0)] + {{\hbar} \over{64 \pi^2} } \left\{ {\cal M}_0^4(\mu_0) - m_0^4\right\} \ln{\mu^2 \over \mu_0^2} +\hbar X[\phi_0,m(\mu),\lambda(\mu),\mu] +O(\hbar^2).\end{align*}

\begin{align*}u^{35,35}_i(y,z)=\left\{\begin{array}{rrrl}- 5 b_i^{35,35}(y,z) \ & \ \ \ \ \ if& \ \ \ \ \ \ \ \ \ \ \ \ |5 b_i^{35,35}(y,z)|\leq 1, \\-1 \ & \ \ \ \ \ if& \ \ \ \ \ \ \ \ \ \ \ \ \ -5 b_i^{35,35}(y,z) < -1,\\1 \ & \ \ \ \ \ if& \ \ \ \ \ \ \ \ \ \ \ \ \-5 b_i^{35,35}(y,z) > 1,\\\end{array}\right\},\ \ \ i=1,2,\end{align*}

\begin{align*}\begin{pmatrix}A^{-1}_N & {\bf 0} \\begin{align*}0.4em]- D^{-1}_N \;\! C_N \;\! A_N^{-1} & D^{-1}_N\end{pmatrix}\begin{pmatrix}u \\1\end{pmatrix}\end{align*}

\begin{align*}\Lambda _\lambda\widehat{S}_{(\pm 1), (0)}S_{(1), (0)}^2 ,\end{align*}

\begin{align*}\delta _M\widehat{F}=\delta _M\left[ dA-emb^{*}B\right] =\left[d\delta _MA-emb^{*}d\Delta _M\right] =0\quad ,\end{align*}

\begin{align*}P_\mu = X_{\mu, -1} + X_{\mu, n}\hskip.5cm; \hskip.5cm \hat{P}_\mu = X_{\mu, -1} -X_{\mu, n}\end{align*}

\begin{align*}G^{\delta}_{12,31}(p'_{30},p_{20})\,=\,{-2i\pi\over(P_0-S)-(p'_{30}-E_3)+i\epsilon}\,\beta_3\beta_1\,\delta(p_{20}-E_2).\end{align*}

\begin{align*}\delta{\cal L}=\bar{\epsilon}\partial_{\mu}Q^{(\mu)}.\end{align*}

\begin{align*}\langle W[{\rm circle}] \rangle= \frac{2}{\sqrt{\lambda}}I_1(\sqrt{\lambda})~~\approx~~\sqrt{\frac{2}{\pi}}\frac{e^{\sqrt{\lambda}}}{\lambda^{3/4} } ~{\rm as}~\lambda\to\infty\end{align*}

\begin{align*}-lq_i = u_i - l_{i}q_{i} \quad \quad {\rm for} \i=1 \sim 5,\end{align*}

\begin{align*}A^{(s)}_4=\sum _{i=1}^{\infty}\frac{1}{\frac{1}{2}s-R-1+i}(\frac{1}{2})^{\frac{1}{2}s-n-1+i}F_i.\end{align*}

\begin{align*}F_{i_{1}}^{\beta_{1}}F_{i_{2}}^{\beta_{2}}\cdots F_{i_{l}}^{\beta_{l}}\,\Psi(z)\end{align*}

\begin{align*}F = c+ \alpha x^2 + \beta y^2 + \gamma x^2 y^2\end{align*}

\begin{align*}\{ p \} = \bigcap_{n \geq 0} B^n_{{\bf w}_n} \end{align*}

\begin{align*}U_t\,(\psi_n\times\cdots\times\psi_1)^{\rm out}= ((U_t\psi_n)\times\psi_{n-1}\times\cdots\times\psi_1)^{\rm out}\,.\end{align*}

\begin{align*}W_{B}=\sigma(12c_{1}(B)-\eta^{(1)}-\eta^{(2)})\end{align*}

\begin{align*}(Y_{d})^{ab}=e^{\hat{K}/2}\sum_{l=1}^{3}(t_{33}^{1/2})^{la}(t_{37}^{1/2})^{lb}(t_{37}^{1/2})^{lc}\langle h_{d}\rangle_{c}.\end{align*}

\begin{align*}X_d=2ig\sum_{\alpha\in\Delta} x_{d}(\alpha\cdot q, \xi)E_{d}(\alpha),\quad Y_d=ig\sum_{\alpha\in\Delta} y_{d}(\alpha\cdot q, \xi)E_{d}(\alpha),\quad E_{d}(\alpha)_{\beta \gamma} =\delta_{\beta-\gamma,2\alpha}. \end{align*}

\begin{align*}E_c= -2 \frac{nlr_+^{n-1}Vol(\Sigma_n)}{16\pi GR}.\end{align*}

\begin{align*}D=-m^2\Delta +3=-\Delta |_x+3, \end{align*}

\begin{align*}L = \left(\begin{array}{cc}a & b \\c & d\end{array}\right),~~~~~~(ad-bc = 1)\end{align*}

\begin{align*}I_{l+a}(kR) \ \sim\ \frac{1}{\sqrt{2\pi l}}\exp \{\sum_{n=-1}^3 l^{-n} S_I(n,a,t)\}\ ,\end{align*}

\begin{align*}\left( M_{5}\right) _{a_{2}b_{|2}}q^{b_{|2}\left[ N_{b_{|2}}+n_{2}\right]}+K_{a_{2}}^{\left( 2\right) }\left( \cdots q^{b\left[ N_{b}+n_{2}-1\right]}\right) =0\,. \end{align*}

\begin{align*}\frac{F}{u^6} = (\frac{v}{u})^3 \frac{w}{u} + (\frac{v}{u})^2 ,\end{align*}

\begin{align*} \left\| \begin{pmatrix}a& b \\c& d\end{pmatrix}\begin{pmatrix}v_1 \\v_2\end{pmatrix} \right\| \leq \sqrt{a^2 + b^2 + c^2 +d^2} .\end{align*}

\begin{align*}\gamma_\mu \partial_\mu G^o(x-y) \; = \; \delta(x-y) \; .\end{align*}

\begin{align*}\Gamma^{(1)} = \frac{n_6}{g_s (2\pi)^6}V_6 \int dt \left(\frac{\epsilon^2 M}{8r}-\frac{\epsilon \sqrt{3}M}{2r}v^2-\frac{M}{2r}v^4\right).\end{align*}

\begin{align*}G^0\,=\,G^0_1\,G^0_2\,G^0_3\,,\end{align*}

\begin{align*}{\cal L}=-\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+ A_\mu(J^\mu + gG^\mu)+{\cal L}_M\end{align*}

\begin{align*}\tilde{\gamma}^{a} p_{0a} = 0 = \tilde{\gamma}^{a} f^{\mu}{ }_ap_{0\mu} = (\tilde{\gamma}^{m} f^{\alpha}{ }_m +\tilde{\gamma}^{h} f^{\alpha}{ }_h )p_{0\alpha} + (\tilde{\gamma}^{m} f^{\sigma}{ }_m +\tilde{\gamma}^{h} f^{\sigma}{ }_h ) p_{0\sigma},\end{align*}

\begin{align*}\widetilde{d}\;\widetilde{\omega }\;=\;0\;, \end{align*}

\begin{align*}\frac{\ddot{a}}{1-\dot{r}^{2}}+ \frac{2a}{r^{2}(1 -\dot{r}^{2})}-2 \lambda v^{2}a + \frac{8}{9v^{2}}a^{3}=0,\end{align*}

\begin{align*}{\Delta}_{F}(x-y,\vec{\sigma})=-i\left<0\right|T(\varphi(x,\vec{\sigma})\,\varphi(y,\vec{\sigma}))\left|0\right>\end{align*}

\begin{align*}\xi^+=\xi^+_0 t^{-\nu}\;\;,\;\;\nu=0.630\pm0.002\;\;.\end{align*}

\begin{align*}(\lambda f_1 , \mu f_1 ) \in {\cal C} , \end{align*}

\begin{align*} \left\| \begin{pmatrix}a& b \\c& d\end{pmatrix}^{-1}\begin{pmatrix}v_1 \\v_2\end{pmatrix} \right\| \leq \frac{1}{| ad-bc |}\sqrt{a^2 + b^2 + c^2 +d^2} .\end{align*}

\begin{align*}\left| \sum_{\alpha \beta \gamma \delta}\langle \Phi_{\alpha} \Phi_{\beta} \Phi_{\gamma} \Phi_{\delta} \rangle_c\left. \right/ \sum_{\alpha \beta \gamma \delta}\langle \Phi_{\alpha} \Phi_{\beta} \Phi_{\gamma} \Phi_{\delta} \rangle\right|=\left| \langle \Phi^4(x) \rangle_c \left. \right/\langle \Phi^4(x) \rangle \right|\; (n=4) \; ,\end{align*}

\begin{align*}h_{\mu\nu} = g_{\mu\nu} + \partial_\mu x^i\partial_\nu x^j g_{ij}\ .\end{align*}

\begin{align*}\xi ^{\ast 1}=\frac{1}{2}\,\xi ^{1}+\frac{c}{eB}\,\xi ^{4} \end{align*}

\begin{align*}<G(p,\tau)> = \int_0^\infty d\tau \exp{-i \tau (m^2-p^2)} \sum_{n=0}^\infty(\frac{-e^2 A}{\epsilon})^n (\mu^2 \tau)^{n \epsilon}.\end{align*}

\begin{align*}\chi^{(\infty)}_\ell(r)= H_\nu^{(1)}(kr) \sim \sqrt{\frac{2}{\pi k}}\, e^{-i\frac{\pi }{2}(\ell-\frac{\Phi}{2\pi}) - i \frac{\pi}{4}} \,\frac{e^{i k\,r}}{\sqrt{r}}, \end{align*}

\begin{align*}~\chi''_n - 4 \chi'_n - \epsilon \chi_n -\mu_v \chi_n \delta(y-y_c) - \mu_h \chi_n \delta(y) = J_n,\end{align*}

\begin{align*}\|\lambda(\phi)\|_{K,m+n}=\sup_{\stackrel{|p|\leq m+n}{z \in K}}|D^p (\lambda(\phi))(z)|\,\,,\qquad D^p=\frac{\partial^{|q|+|r|}}{\partial x^{q_1}_1 \cdots \partial x^{q_m}_m\partial\theta^{r_1}_1 \cdots \partial\theta^{r_n}_n}\,\,. \end{align*}

\begin{align*}\begin{array}{cc} K^{-1}_{ab} = & \left(\begin{array}{cc}\sqrt{g} & { 1 \over \sqrt{g}} \\{1 \over \sqrt{g}} & \sqrt{g}\end{array} \right) \end{array}\end{align*}

\begin{align*}U_0(\theta)=1\;;\;\;\;\;\; U_1(\theta)=2\cos\theta=\frac{1}{\sqrt x}.\end{align*}

\begin{align*}CT=V_{\rm count}[\Phi_{\bar{s}} ]+V_{\rm B\ count}[\Phi_{\bar{s}} ]-V_{\rm count}[\Phi_s ]- V_{\rm B\ count}[\Phi_s ]\end{align*}

\begin{align*}\Psi(f) := \Delta_{p}^{*}(d_{1}d_{2}\dots d_{p}f) \in \Omega^{p}(G,V).\end{align*}

\begin{align*}S = \frac{s}{2\pi}\int d\phi dy\, {\mathrm{Tr}}\,[g^{-1}\partial_z g g^{-1}\partial_{\bar z} g] + \frac{is}{12\pi}\int {\mathrm{Tr}}\, (G^{-1}dG)^3\,, \end{align*}

\begin{align*}L_0\vert h;q\rangle=h\vert h;q\rangle{}~,{}~{}~{}~L_n\vert h;q\rangle= G^{\pm}_{n-(1\mp q)/3}\vert h;q\rangle=0{}~,{}~{}~{}~n>0{}~,\end{align*}

\begin{align*}(w^2+1)\left[\left(\frac{\Gamma-1}{r}+\cosh(2ml)\right)\sinh(2ml)+r\right] =2w\left[r\cosh(2ml)+\sinh(2ml)\right] \; ,\end{align*}

\begin{align*}M^{1}=M^{2}=:M\;\Rightarrow \;\epsilon ^{1}(\theta )=\epsilon^{2}(\theta )=rM\cosh \theta \;,\end{align*}

\begin{align*}{{\cal S}\over \pi}= 2 \sqrt { - ( W_{0A} \tilde{p}^A) d_{BCD} \tilde p^B\tilde p^C\tilde p^D}\;\;.\end{align*}

\begin{align*}\pi\equiv{\partial {\cal L} \over \partial\partial_+\phi}=\partial_-\phi.\end{align*}

\begin{align*}m = \frac{M}{\cos{\beta}} ,\end{align*}

\begin{align*}\cosh ( \beta ) ( 1 \pm c^* ) = { 1 \over \cosh (\beta ) ( 1 \mp c^* )} \end{align*}

\begin{align*}\nabla^{O(2)} (e^{-\sigma}L_{\Lambda AB}) = e^{-\sigma}( L_{\Lambda C(A} P_{B)C} - d\sigma L_{\Lambda AB} )\end{align*}

\begin{align*}f(x )=\lambda_0+\sum_i {\lambda_i \over |x -q_i|},\end{align*}

\begin{align*}\Psi(f_{0}\otimes \dots \otimes f_{p}\otimes v)= f_{0}df_{1}\wedge \dots\wedge df_{p} \otimes v,\end{align*}

\begin{align*} \mathcal J_n^2={}-4\beta^2\left(2\mathcal J_n^1+ {\cal W}\right) -\frac{1}{2}(4\beta^2)^2{\cal S}_\Phi\left ({\cal S}_\Phi+n\sigma_3\right) -\frac 12{\cal C}.\end{align*}

\begin{align*}A(C)= {\rm Tr} \left( P \exp ig \oint_C A_\mu dx^\mu \right),\end{align*}

\begin{align*}\tilde{J}^{(\pm)}_{\mu}\equiv \int d^3x \, j^{(\pm)}_{\mu}(x)=\sum_{{\bf k}}\frac{k_{\mu}}{\omega_{{\bf k}}} (a^{\dagger}_{{\bf k}}a_{{\bf k}} \pm b^{\dagger}_{{\bf k}} b_{{\bf k}}).\end{align*}

\begin{align*}=0\,,\;\Phi_{i}^{\left( 3\right) }=p_{z^{i}}\,=0\,.\end{align*}

\begin{align*} I(Q^2) =v_{4}\sum_{n=0}^\infty\frac{(-1)^{n+1}}{(n+2)!(n+1)}\left(\frac{1}{2}\right)^{n+1}\left(\frac{Q^2}{k^2}\right)^{n+1}\end{align*}

\begin{align*}s_{1}(P,t)=-P \int_{0}^{t} \frac{X(t^{\prime})}{A(t^{\prime})} d t^{\prime},\end{align*}

\begin{align*}R_{\alpha \beta }-\frac{1}{2}g_{\alpha \beta }R=\kappa \Upsilon _{\alpha\beta }, \end{align*}