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LAMM: MIT Laboratory for Atomistic and Molecular Mechanics 39

\begin{align*}-\div (y^{1-2m}\nabla w)=0\mbox{in}\mathbb R^n\times\mathbb R_+;w\big|_{y=0}=|u|\end{align*}

Provide the LaTeX formula.

\begin{align*} \frac{{\rm d}\vec{X}}{{\rm d}t} = \vec{\Im} \left( \displaystyle \vec{X} \right)\end{align*}

Provide the LaTeX formula.

\begin{align*} L_{\vec{X}} \phi (\vec{X}) = \mbox{Tr}[J] \phi (\vec{X}) + P (\vec{V} \cdot \vec{\gamma})\end{align*}

Provide the LaTeX formula.

\begin{align*}z_k = (z_{1,k}, z_{2,k})\in Z, \ \ \ \ k=1,..., K,\end{align*}

Provide the LaTeX formula.

\begin{align*}(\nu_\mu\cdot \Phi)(A,U) = \int_P (\mu\cdot \Phi)(A\cap \varphi_p(X), U\cap \varphi_p(X)) \, dp\end{align*}

Provide the LaTeX formula.

\begin{align*}s\omega _{1}=\partial _{i}k^{i}, \end{align*}

Provide the LaTeX formula.

\begin{align*}d(\omega _kdx^k)=(\partial _i\omega _k)dx^idx^k+\omega _kd^2x^k;\end{align*}

Provide the LaTeX formula.

\begin{align*}K^{\epsilon '} = \tilde{M} K^{\epsilon} M^{-1} \quad .\end{align*}

Provide the LaTeX formula.

\begin{align*}+\delta_{\alpha\beta}[-\Delta B-{\tilde d}(\partial B)^{2}-q (\partial A\partial B) - r(\partial F\partial B)], \end{align*}

Provide the LaTeX formula.

\begin{align*}a^{\dagger}_m=\sqrt{\frac{2}{\zeta}}z_m\,,\quad a_m=\sqrt{\frac{2}{\zeta}}\bar{z}_m\quad\textrm{for}\quad m=1,\,3\,,\end{align*}

Provide the LaTeX formula.

\begin{align*}{J}:=\frac{\partial {L}} {\partial \dot q^j}\delta_v q^j+{L}\delta t,\end{align*}

Provide the LaTeX formula.

\begin{align*}[\tilde {J} {^{R} _3} (0,x^-) , \tilde {J} {^{R} _3} (0,y^-)] = {i\over 2 \pi} \delta'(x^- - y^-)\end{align*}

Provide the LaTeX formula.

\begin{align*}\Psi_{\rm{0}}(r',\theta';0)={1\over\sqrt{2\pi}\xi}\exp\left\{ikr'\cos\theta'-{1\over4\xi^2}(r'^2+r_0^2+2rr'\cos\theta')\right\}.\end{align*}

Provide the LaTeX formula.

\begin{align*}\left( 1+\varepsilon \delta R+O[\left( \delta \alpha \right) ^{2}]\right)\end{align*}

Provide the LaTeX formula.

\begin{align*}a_{I}=\bar{a}_{I}+\sum_{k=1}^{\infty}{\mu^{2k}\over (k!)^2}\partial_{t_{r}}^{2k}\bar{a}_{I},\end{align*}

Provide the LaTeX formula.

\begin{align*}\left(\begin{array}{cc} \mu_1 & 0 \\ 0 & \mu_2 \\ \end{array} \right)\left(\begin{array}{cc} 0 & d_1 \\ d_2 & 0 \\ \end{array} \right)\left(\begin{array}{cc}\bar{\mu_1}^{-1} & 0 \\0 & \bar{\mu_2}^{-1} \\\end{array}\right) =\left(\begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array} \right).\end{align*}

Provide the LaTeX formula.

\begin{align*}I_0 \rightarrow I_0+I_0^{\beta}\end{align*}

Provide the LaTeX formula.

\begin{align*}{E_i}=-F_{0i} = -\partial_0A_i + \partial_iA_0\end{align*}

Provide the LaTeX formula.

\begin{align*}Ev(\omega)\widetilde{{\cal D}(u)}{\cal R} = 0.\end{align*}

Provide the LaTeX formula.

\begin{align*}\Psi' (x) = \exp (-iD^{a}\theta_{a} )\Psi (x),\end{align*}

Provide the LaTeX formula.

\begin{align*}\{\Phi^A\}=\{\gamma_{\mu\nu}\, ,\, X^M\, ,\, A^a_\mu\, ,\, \xi^\mu\, ,\, c\, ,\, C^a\}.\end{align*}

Provide the LaTeX formula.

\begin{align*}F = \frac{1}{2\pi} \left( \delta(m) - \delta(\infty) - \pi n^++ \pi n^- - \delta(-m) + \delta(-\infty) \right),\end{align*}

Provide the LaTeX formula.

\begin{align*}{\cal D}_{\Gamma}(w)=\sqrt{{\rm Ber}\;\Omega_0\vert_{\Gamma}}\equiv \sqrt{{\rm Ber}\frac{\partial _r z^A}{\partial w^{\mu}}\Omega_{(0)AB}\frac{\partial _l z^B}{\partial w^{\nu}}},\end{align*}

Provide the LaTeX formula.

\begin{align*}\nabla^2h_{\mu\nu}=-16\pi G^D_N\left(T^{mat}_{\mu\nu}-{1\over{D-2}}\eta_{\mu\nu}T^{mat}\right)\equiv-16\pi G^D_N\bar{T}^{mat}_{\mu\nu},\end{align*}

Provide the LaTeX formula.

\begin{align*}Z_F(G^{*})=Z_F(0)Z_{WZNW}(N)Z_{WZNW}(-N)Z_{ghost}\end{align*}

Provide the LaTeX formula.

\begin{align*}Q=-\partial ^{2}_{\tau }+\mu ^{2}_{0}\left( 1-\frac{2}{\cosh ^{2}\left( \mu _{0}\tau \right) }\right)\end{align*}

Provide the LaTeX formula.

\begin{align*} g = kah, k \in K, a \in A^+, h \in H.\end{align*}

Provide the LaTeX formula.

\begin{align*}-\gamma^{cd} \partial_{c} X^{\mu}\partial_{d} X^{\nu} g_{\mu \nu} +\frac {\varepsilon^{cd}}{\sqrt{-\gamma}} F_{cd} = M\end{align*}

Provide the LaTeX formula.

\begin{align*}d_{i} = \sqrt{S_{i}} \ K_{i} \ exp(-S_{i}), \ \ \ i = 1, 2\end{align*}

Provide the LaTeX formula.

\begin{align*}M_{n+1} - M_n = C_n \exp(-4\pi M_n^2).\end{align*}

Provide the LaTeX formula.

\begin{align*}{\bf A}={1\over 2}\left( \begin{array}{cc} p^\dagger & q^\dagger \end{array}\right){\bf \sigma}\left(\begin{array}{c} p\\q \end{array}\right)\end{align*}

Provide the LaTeX formula.

\begin{align*}\alpha _{p}l^{D-2p}=\left\{\begin{array}{ll}(D-2p)^{-1}\left(\begin{array}{c}n-1 \\ p\end{array}\right) , & D=2n-1 \\ \left(\begin{array}{c}n \\ p\end{array}\right) , & D=2n.\end{array}\right. \end{align*}

Provide the LaTeX formula.

\begin{align*}x_H = -\,\frac{1}{2\lambda} \ln\frac{M}{2\lambda}\,,\end{align*}

Provide the LaTeX formula.

\begin{align*}\int_{{\cal M}_{{\rm ins}}} \left\vert \frac{1}{{\rm det}T_{0}} \right\vert,\end{align*}

Provide the LaTeX formula.

\begin{align*}\sum_{j \neq i} (c_j + d_j c_i) T^j =0 \, .\end{align*}

Provide the LaTeX formula.

\begin{align*}\left|x-a\right|+\left|x-b\right|+\left|x-c\right|=\sqrt{\mathcal{X}}\end{align*}

Provide the LaTeX formula.

\begin{align*}{\cal O} = {\cal O}^{\eta } + \zeta {\cal O}^{\zeta }\,,\end{align*}

Provide the LaTeX formula.

\begin{align*} g = hak, h \in H, a \in A^+, k \in K,\end{align*}

Provide the LaTeX formula.

\begin{align*}{\bf C}P^2 = U_1 \cup \{ (0:\phi_2:\phi_3) \},\quad (\phi_2:\phi_3) \not= 0 ,\end{align*}

Provide the LaTeX formula.

\begin{align*}{\cal E}_0=\lim_{\delta\rightarrow 0}\frac{\pi\hbar}{\delta}{\cal E}_\delta=\int_{-\infty}^{+\infty}d\gamma\,e^{\frac{i}{\hbar}\gamma\hat{\phi}}\ \ \ .\end{align*}

Provide the LaTeX formula.

\begin{align*}f_M=\left(\begin{array}{c}f^1\\\Lambda f^{1*}.\end{array}\right).\end{align*}

Provide the LaTeX formula.

\begin{align*}f(\infty)=1,\quad \chi(\infty)=v(\infty)=u(\infty)=\phi(\infty)=\kappa(\infty)= 0,.\end{align*}

Provide the LaTeX formula.

\begin{align*}K_{DFF} = K_{CCM} + \frac{g}{4H_{DFF}} = w_{0,2}+{g\over4}w_{-2,0}.\end{align*}

Provide the LaTeX formula.

\begin{align*}m^2 = \pi^3 \frac{\Lambda^4}{g^4}\exp\{-\frac{\pi}{2}\frac{\Lambda}{g^2}\}\end{align*}

Provide the LaTeX formula.

\begin{align*}\delta \chi_-=-\eta_- \left( \partial_0 w +i\epsilon_{ojk}\partial^j \alpha^k -2i e \Re ( \phi^* \varphi ) \right) \,.\end{align*}

Provide the LaTeX formula.

\begin{align*}Z_+(\theta)=Ai(z) \qquad Z_-(\theta) = -Ai\,^\prime(z)\end{align*}

Provide the LaTeX formula.

\begin{align*}s^2-\varrho^2=\left|\frac{2\varrho}{m}\frac{(p,\xi)}{\chi}\right|^2\end{align*}

Provide the LaTeX formula.

\begin{align*}K(T, {\bar T})= -\log\{( T+ {\bar T})^3 + {\cal I}_{instanton} \}\end{align*}

Provide the LaTeX formula.

\begin{align*}M^{w}=\left\{ t(w\bar{t}w)a|t\in M,a=(a_{1},a_{2},...,a_{n}),a_{i}=1w(i)\neq i,a_{i}=\pm1w(i)=i\right\} .\end{align*}

Provide the LaTeX formula.

\begin{align*}F_{\mu\nu}(x)=\partial_\mu A_\nu(x)-\partial_\nu A_\mu(x)+i\,[A_\mu(x),A_\nu(x)].\end{align*}

Provide the LaTeX formula.

\begin{align*}\Lambda _{B}^{A}=R_{BCD}^{A}\nabla _{a}\Phi ^{C}\nabla ^{a}\Phi ^{D}.\end{align*}

Provide the LaTeX formula.

\begin{align*}w(z)={\hbar\over 2 \pi^2}\int_0^{\infty}dk_z\int_0^{\infty} dk_{\|} {k_{\|}\over k}\left[\left(k^2+k_{\|}^2\right)\sin^2(k_z z) +k_z^2 \cos^2(k_z z)\right]\left[{1\over 2}+{\overline n}(k)\right].\end{align*}

Provide the LaTeX formula.

\begin{align*}\left. A_{\beta,1}^{(2)}\right|_{{\cal C}_\beta}=3\left. A_{\beta,1}^{(0)}\right|_{{\cal C}_\beta}+8(\beta-2\pi)+ 2\cdot4\pi~~~.\end{align*}

Provide the LaTeX formula.

\begin{align*}\zeta_{\rho}(s)=-\frac{2\, R^{2s}}{\Gamma(s+1)\,\Gamma(-s)}\sum_{l=1}^{\infty}\nu^{1-2s}\int_0^{\infty}dk\,k^{-2s}\frac{d}{dk}\ln\Delta_{\rho,l}(\nu k), \quad \nu=l+\frac{1}{2}, \;\;\rho=\pm1 \end{align*}

Provide the LaTeX formula.

\begin{align*}\pi \min (1,p)<|\Im m\, \vartheta |<\pi \frac{p+1}{2}\end{align*}

Provide the LaTeX formula.

\begin{align*}\frac{\partial}{\partial r}\left(e\sp{-u}\right)=\frac{\mp k}{r^{2}}\left[1-\frac{\partial H_{3}}{\partial\theta}-\cot\theta\:H_{3}-H_{2}H_{4}\right],\end{align*}

Provide the LaTeX formula.

\begin{align*}\partial_{\mu}j^{\mu}+\partial_{\tau}j^{5}=0\end{align*}

Provide the LaTeX formula.

\begin{align*}\left[ {\cal P}^2-\left( M\Omega \right) ^2-\frac q2\sigma ^{\mu \nu }F_{\mu\nu }+ibM\gamma ^0\right] \phi (x)=0\;,\;\; \end{align*}

Provide the LaTeX formula.

\begin{align*}\left[\begin{array}{cccc}g_{2}+w_{2}^{\ 2}h_{4}+n_{2}^{\ 2}h_{5} & w_{2}w_{3}h_{4}+n_{2}n_{3}h_{5} &w_{2}h_{4} & n_{2}h_{5} \\w_{2}w_{3}h_{4}+n_{2}n_{3}h_{5} & g_{3}+w_{3}^{\ 2}h_{4}+n_{3}^{\ 2}h_{5} &w_{3}h_{4} & n_{3}h_{5} \\w_{2}h_{4} & w_{3}h_{4} & h_{4} & 0 \\n_{2}h_{5} & n_{3}h_{5} & 0 & h_{5}\end{array}\right] , \end{align*}

Provide the LaTeX formula.

\begin{align*}\big(\alpha_{\kappa}|_{M^{w}}=\chi|_{M^{w}}\big)\Rightarrow\big(\overline{\alpha_{\kappa}}w(\alpha_{\kappa})=\bar{\chi}w(\chi)\big).\end{align*}

Provide the LaTeX formula.

\begin{align*}T=\left( \begin{array}{cc}A&B\\C&D \end{array}\right) .\end{align*}

Provide the LaTeX formula.

\begin{align*}\langle f | g \rangle= \int \frac{dp}{(1 + \beta p^2)^{1-\alpha}}\,f^*(p)\,g(p)\;,\end{align*}

Provide the LaTeX formula.

\begin{align*}f= C_1 + C_2 \log \left( \frac {2-z-2\sqrt{1-z}}{z} \right),\end{align*}

Provide the LaTeX formula.

\begin{align*}\frac{\lambda_R}{2m^2_{f,R}}\left(\langle\Phi^2(t)\rangle-\langle\Phi^2(0)\rangle\right)\approx 1\end{align*}

Provide the LaTeX formula.

\begin{align*}f^{(n,i_1,i_2,\dots,i_k)}_P \mbox{tr}[(b^\dagger)^{i_1} a^\dagger\dots a^\dagger]\dots \mbox{tr}[(b^\dagger)^{i_k} a^\dagger \dots a^\dagger]|0>,\end{align*}

Provide the LaTeX formula.

\begin{align*}X^I = \frac{q^I}{Z}\,, \qquad X_I = \frac{V_I}{X^J V_J}\,.\end{align*}

Provide the LaTeX formula.

\begin{align*}\Delta^t U(a)=U(a\cos(2\pi t)+ia\sin(2\pi t))\Delta^t\end{align*}

Provide the LaTeX formula.

\begin{align*}f(\rho) \sim 1, \quad L(\rho) \sim r(\rho),\end{align*}

Provide the LaTeX formula.

\begin{align*}C_m,C_{m}^i;\ \ \ C_{mn},C_{mn}^i;\ \ \ \ \ \ m,n=1,2,3\end{align*}

Provide the LaTeX formula.

\begin{align*}\langle T_{\mu\nu}\rangle_{ren}\stackrel{\rm def}{=}\langle T_{\mu\nu}\rangle-\langle T_{\mu\nu}\rangle^{(4)}\, .\end{align*}

Provide the LaTeX formula.

\begin{align*}\overline{\alpha_{\kappa}}w(\alpha_{\kappa})=\bar{\chi}w(\chi)\end{align*}

Provide the LaTeX formula.

\begin{align*}*i_Y{\cal I}_*\Omega=*\Bigl[Y^\mu {\bf F}_{\mu\nu}dx^\nu\Bigr]\otimes\varepsilon^2-*\Bigl[Y^\mu (*{\bf F})_{\mu\nu}dx^\nu\Bigr]\otimes\varepsilon^1.\end{align*}

Provide the LaTeX formula.

\begin{align*}\tilde{Q}^{(L, I)}_1 \tilde{Q}^{(L, I)}_0 \neq 0\,\end{align*}

Provide the LaTeX formula.

\begin{align*}H=\frac{m}{2}\sum_{a=1}^N\,\vec{v}_a^2 =\frac{1}{2m}\sum_{a=1}^N\,[\vec{p}_a-e\vec{A}(\vec{x}_a)]^2\end{align*}

Provide the LaTeX formula.

\begin{align*}Z_K\propto\left(\frac{8\pi G \Lambda}{9} I(\phi_{cl})\right)^{-K} K!\end{align*}

Provide the LaTeX formula.

\begin{align*}A|_{I_\mu} \;=\; \left( \begin{array}{cccc} \mu & 1 & \cdots & 0 \\0 & \mu & \cdots & 0 \\ \vdots & \vdots & \ddots & 1 \\ 0 & 0 & \cdots & \mu \end{array} \right) \;.\end{align*}

Provide the LaTeX formula.

\begin{align*}\dot{x}^{AA'} = o^A \bar{o}^{A'}, \,\,\acute{x}^{AA'}= o^A \bar{\iota}^{A'} + \iota^A \bar{o}^{A'},\,\,(\rho^\tau o^A\bar{o}^{A'})\dot{} = {\cal F}^{AA'},\end{align*}

Provide the LaTeX formula.

\begin{align*}\begin{array}{c}V(z_{12}^{\prime})=U^{-1}(z_{1};g)V(z_{12})U(z_{2};g)\,,\\{}\\V(z_{21}^{\prime})=U^{-1}(z_{2};g)V(z_{21})U(z_{1};g)\,.\end{array}\end{align*}

Provide the LaTeX formula.

\begin{align*}{u^2 \over l^2} \sin^2 \alpha ~\rho^{12} + \left({u^2 \over l^2} \cos^2 \alpha- {u^2 \over l^2} \sin^2 \alpha -1\right) \rho^6 - {u^2 \over l^2} \cos^2 \alpha = 0\end{align*}

Provide the LaTeX formula.

\begin{align*}F = \frac{1}{\pi} \left( 2 \alpha ' \right)^{- d/2} \langle B, y_1 , v_1 | D |B, y_2 , v_2 \rangle~. \end{align*}

Provide the LaTeX formula.

\begin{align*}ds^2 = (dX^0)^2 - R(X^0)^2 \sum_{i=1}^{D-1}(dX^i)^2\end{align*}

Provide the LaTeX formula.

\begin{align*}\prod_{(i,j)\in I_{w}}\big|\frac{t_{i}}{t_{j}}\big|^{\kappa(i,j)+\kappa(w(i),w(j))}=\prod_{i=1}^{n}|t_{i}|^{\lambda_{i}+\lambda_{w(i)}}.\end{align*}

Provide the LaTeX formula.

\begin{align*}\tilde{\zeta}(s)=\frac{V_q}{(4\pi)^{q/2}}\frac{\Gamma(s-q/2)}{\Gamma(s)} \sum_{\bf p}(\sigma^d_{\bf p}+m^2-\mu^2)^{-s}\;.\end{align*}

Provide the LaTeX formula.

\begin{align*}\tan ^{-1} \left( \frac {{\sqrt {4\xi -1}}}{3} \right)=\frac {{\sqrt {4\xi -1}}}{3} -\frac {(4\xi -1)^{\frac {3}{2}}}{27} +O(4\xi -1)^{\frac {5}{2}}.\end{align*}

Provide the LaTeX formula.

\begin{align*}{\rm Aux}^2\;=\;\bigl\{ \sum_i\,[D,a_i][D,b_i]\;:\;\sum_i\;a_i\,[D,b_i]\;=\;0\bigr\},\end{align*}

Provide the LaTeX formula.

\begin{align*} T_{ab}{}^{c} \;=\; -\left[ i_{k_{(m)}}\hat{C}\right]_{ab} Q^{mn}k_{(n)}^{c} \hspace{.2cm},\hspace{.2cm} 2 K_{abc} \;=\; -T_{abc}+T_{bca}-T_{cab} \; ,\; \check{\omega} \;=\; \omega + K \; .\end{align*}

Provide the LaTeX formula.

\begin{align*}f\left( T(z)\right) =T'(z) \left[ f(z) -\frac{\delta w}{\epsilon w}(z-\alpha )\right] \ .\end{align*}

Provide the LaTeX formula.

\begin{align*}\vec{A}=\frac{\phi}{2\pi}\frac{\Theta(r-R)}{r}\vec{e}_\varphi\ , \ \ A^0=0\ ;\end{align*}

Provide the LaTeX formula.

\begin{align*}\Psi = x^{-\kappa} ( \Psi_{0} + \Psi_{1} + ... )\; .\end{align*}

Provide the LaTeX formula.

\begin{align*}\delta A_i = \theta C_{iT} \quad \qquad\delta \pi_i = \theta P_{iT}\end{align*}

Provide the LaTeX formula.

\begin{align*}S(g)^{-1}=1+\sum_{n=1}^\infty \frac {1}{n!} \int d^4x_{1}\ldots d^4 x_n \tilde{T}_n(x_1,\ldots x_n)g(x_1)\ldots g(x_n)\end{align*}

Provide the LaTeX formula.

\begin{align*}{\rm U}_0({\tau}^{(1)}) {\rm U}_0({\tau}^{(2)}) =\exp\{ 2\pi i {\omega}_{2}({\tau}^{(1)}, {\tau}^{(2)})\}{\rm U}_{0}( {\tau}^{(1)} + {\tau}^{(2)} ) ,\end{align*}

Provide the LaTeX formula.

\begin{align*}\gamma(s,\chi,\psi)\gamma(1-s,\overline{\chi^{-1}},\overline{\psi^{-1}})=1.\end{align*}

Provide the LaTeX formula.

\begin{align*}\alpha_1=\epsilon_1-\epsilon_2\,;\,\,\alpha_2=\epsilon_2-\epsilon_3\,;\,\,\alpha_3=\epsilon_1-\epsilon_3\end{align*}

Provide the LaTeX formula.

\begin{align*}(\sigma_i\psi)_a(x)=-\psi_a(x).\end{align*}

Provide the LaTeX formula.

\begin{align*}{\cal M} = - \frac{c_f g_s^2}{4} \frac{1}{k^2 - M_G^2} J_c^\mu J_{c \mu}^{\dagger}\end{align*}

Provide the LaTeX formula.

\begin{align*}<u^{2}><q_{u}^{2}> = 1/4 , \qquad <v^{2}><q_{v}^{2}> = 1/4 .\end{align*}

Provide the LaTeX formula.

\begin{align*}\tilde C_{\alpha}^{\,\alpha_{0}}(p):=\int\limits_{\alpha_{0}}^{\alpha}d\alpha'\,e^{\,-\alpha'(\,p\eta p+(\varepsilon+i)m^{2})}\quad,\qquad\varepsilon>0\ ,\ 0<\alpha_{0}\leq\alpha<\infty\quad,\end{align*}

Provide the LaTeX formula.

\begin{align*}g_{12}= e_1 e_2 = g_{34}= e_3 e_4 = 1, \qquad (others\quad components \quad g_{ab}) = 0 .\end{align*}

Provide the LaTeX formula.

\begin{align*}V_{eff}=\frac{1}{2}\mu ^{2}\varphi ^{2}+\lambda\varphi^{4}-\varphi H\, ,\end{align*}

Provide the LaTeX formula.