harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde{t}_{\mu,\nu}(x) =\frac{1}{2^{\mu-1}\Gamma\big(\frac{\mu-\nu+1}{2}\big)\Gamma\big(\frac{\mu+\nu+1}{2}\big)}t_{\mu,\nu}(x) &=\frac{1}{2^{\mu+1}\Gamma\big(\frac{\mu-\nu+3}{2}\big)\Gamma\big(\frac{\mu+\nu+3}{2}\big)}{}_1F_2\bigg(1;\frac{\mu-\nu+3}{2},\frac{\mu+\nu+3}{2};\frac{x^2}{4}\bigg).
P_n(x) = \frac{R_{n+1}(x)-R_n(x)(n-\delta-N)/(n-N)}{(x+\gamma+1)(x+\delta)}\hphantom{P_n(x)}{}= \frac{1}{(\gamma+1)(n-N)}R_n(\lambda(x);\gamma+1,\delta-1,N),
\left\{e_{\mu_{\beta}}(\cdot,\varphi):=\frac{e^{\langle\cdot,\varphi\rangle}}{\mathbb{E}\big(e^{\langle\cdot,\varphi\rangle}\big)},\;\varphi\in S_{d,\mathbb{C}},\;|\varphi|_{p}<2^{-q}\right\}
b_{k}^{\mathrm{s}}= \epsilon_{k i j}\partial_{i}a_{j}^{\mathrm{s}}= \epsilon_{k i j}\partial_{i}\left( \frac{1}{2 i g}\mathrm{t r}\tau_{3}\Omega_{\mathrm{D}}\partial_{j}\Omega_{\mathrm{D}}^{\dagger}\right) ,
C_r \left(V_{(1-t) \times_{p,-\infty}A_0 +_{p,-\infty}t \times_{p,-\infty}B_0}\right) \supset (1-t) \cdot_p C_r\left(V_{A_0}\right) +_p t \cdot_p C_r\left(V_{B_0}\right) .
\left( \begin{array}{c c c}{A_{3}+ A_{8}/ \sqrt{3}}&{R \bar{G}}&{R \bar{B}}\\{G \bar{R}}&{- A_{3}+ A_{8}/ \sqrt{3}}&{G \bar{B}}\\{B \bar{R}}&{B \bar{G}}&{- 2 A_{8}/ \sqrt{3}}\\ \end{array}\right)
e_{q}(z)=\sum_{n=0}^{\infty}\frac{z^{n}}{[n]_{q}!}=\sum_{n=0}^{\infty}\frac{z^{n}(1-q)^{n}}{(q;q)_{n}}=\sum_{n=0}^{\infty}z^{n}\frac{(1-q)^{n}}{(1-q^{n})(1-q^{n-1})\cdot\cdot\cdot(1-q)}
\hat{Q}\left( z \right)=\left( \Gamma_{0}^{+}\Gamma_{0}\right)^{-1}\Gamma_{0}^{+}\left\{A(I-P)\left( \tilde{A}-z \right)^{-1}(I-P)A-\left( A-z \right) \right\}\Gamma_{0}\left( \Gamma_{0}^{+}\Gamma_{0}\right)^{-1}.
\delta S=\displaystyle\int d^4 x varepsilon^\alpha\displaystyle{\Biggl(}-2i (\sigma^\nu{\bar\sigma}^\mu)_\alpha{}^\beta D_\beta \partial_\mu{\widetilde T}_\nu+\frac{i}{3}\sigma^\mu{}_\alpha{}^{\dot{\beta}}{\bar D}_{\dot{\beta}}\partial_\mu{\kern .35em\bar{\kern -.35em{\widetilde J}}}\displaystyle{\Biggr)}\displaystyle{\biggl\vert}\ .
\!\!\! \! \mathbb{E}[Z_i^2|Y_i\!=\!y_i]\!\!=\!\!\! \sum_{w=0}^{y_i}\!\!{{y_i}\choose{w}}\delta^{y_i\!-\!w}(1\!\!-\!\!\delta)^{\!w}\frac{g(w)(g(w)\!+\!(1\!\!-\!q))}{q^2}
\langle f_i - \kappa_i f_i|V_2, f_i - \kappa_i f_i|V_2 \rangle = \langle f_i - \kappa_i f_i|V_2, f_i \rangle = \left(1 - \frac{4}{9}\frac{a_i^2(2)}{2^k}\right) \langle f_i,f_i \rangle &\geq \frac{1}{9}\langle f_i, f_i \rangle,
\langle F(e_j), F(e_{3+k})\rangle = \langle e_j + b_{ij}e_{3+i}, e_{3+k}\rangle = \langle e_j, e_{3+k}\rangle + b_{ij}\langle e_{3+i}, e_{3+k}\rangle = m_{jk}+ b_{ij}\delta_{ik}.
\mathcal{F}_{b}^{\prime}( 0 ) = \frac{b^{2}\mu_{B}}{2}\sum_{j = 0}^{r}e_{j}G_{B}( b e_{j}) ; \ \mathcal{F}_{b}^{\prime \prime}( 0 ) = b^{2}\mu \sum_{j = 0}^{r}e_{j}G_{B}( 2 b e_{j}) .
{\rm vol}_d(\mathbf{A}+\epsilon\mathbf{C}^d) \ge{\rm vol}_d(\mathbf{A})+d\cdot{\rm vol}_d(\mathbf{A})^{\frac{d-1}{d}}\cdot\epsilon \cdot{\rm vol}_d(\mathbf{C}^d)^{\frac{1}{d}}&={\rm vol}_d(\mathbf{A})+\epsilon \cdot d \cdot{\rm vol}_d(\mathbf{C}^d) &={\rm vol}_d(\mathbf{A})+\epsilon\cdot{\rm svol}_{d-1}(\mathbf{C}^d)\ .
{\frac{1}{{g_{g r a v}}^{2}( p^{2})}}={\frac{1}{{g_{g r a v}}^{2}({M_{I}}^{2})}}+{\frac{{\tilde{b}}_{g r a v}}{1 6 \pi^{2}}}\log{\frac{M_{I}^{2}}{p^{2}}}- \frac{{\tilde{a}}_{g r a v}}{1 6 \pi^{2}}\log \left( ( T +{\bar{T}}) ( U +{\bar{U}}) \right) .
B_1 =B_{(0)(0)(1)}B_2 =B_{(0)(1)(0)}B_3 =B_{(0)(1)(1)}\cup B_{(1)(1)(1)}B_{-1}=B_{(0)(0)(-1)}B_{-2}=B_{(0)(-1)(0)}B_{-3}=B_{(0)(-1)(-1)}\cup B_{(-1)(-1)(-1)}.
\frac{\partial \theta}{\partial\bar z_j}=\frac{-z_j(1-|w|^2)}{|1-\bar z\cdot w|^2}+\frac{(1-|z|^2)(1-|w|^2)}{(1- z\cdot \bar w)(1-\bar z\cdot w)^2}w_j= \frac{1-|w|^2}{|1-\bar z\cdot w|^2}\left(\frac{1-|z|^2}{1-\bar z\cdot w}w_j-z_j\right)\ .
\lim_{\epsilon \to 0}\epsilon \int_0^T\int_{\Gamma_{\epsilon}}u_{\epsilon}(t,x)\phi\left(t,x,\frac{x}{\epsilon}\right)d\sigma dt = \int_0^T\int_{\Omega}\int_{\Gamma}u_0(t,x,y)\phi(t,x,y)d\sigma_ydxdt .
(\sqrt n\log n)^{-p}\sum_{j=-nu+1}^{n(t-2u)}\Bigl[\sum_{k=nu+1}^{nt}(k+j)^{-1}\Bigr]^p \le\Bigl[\frac1{\log n}\sum_{k=nu+1}^{nt}(k-nu)^{-1}\Bigr]^p\cdot|t-u|^{p/2}&\le\Bigl[\frac{1+\log(n|t-u|)}{\log n}\Bigr]^p\cdot|t-u|^{p/2}
\lim_{N \to \infty}\frac{\sigma_N}{\theta_N^2}=\lim_{N \to \infty}\theta_N^{-q_N}\geq 3^{-q_N}\lim_{N \to \infty}\varepsilon_N^{-q_N}\geq \frac13\lim_{N \to \infty}(N-1)^{a_Nq_N}=\infty.
{\cal H}_n^{(2)}={}-\frac 12\frac{d^2}{dr^2}+ \frac{c^2}{32}r^6 + \frac{c_0c}8r^4 + \frac 18\left(c_0^2-2c(2n - m)\right)r^2 + \frac{m^2-\frac 14}{2r^2}-\frac 12(n-m)c_0.
\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
\dfrac{zk_\beta'(z)}{k_\beta(z)}=\dfrac{1+(1-2\beta)z}{1-z}=\frac{1-\overline{z}+(1-2\beta)z-(1-2\beta)|z|^2}{|1-z|^2}&=\dfrac{1-2\beta{\rho}_0 \cos t- (1-2\beta)({\rho}_0)^2}{1+({\rho}_0)^2-2{\rho}_0 \cos t}+i \dfrac{2(1-\beta){\rho}_0 \sin t}{1+({\rho}_0)^2-2{\rho}_0 \cos t}.
\frac{1}{\sqrt{- g}}\partial_{\mu}[ \sqrt{- g}\chi ( \tilde{F}_{\alpha \beta}\tilde{F}^{\alpha \beta}) g^{\mu \gamma}g^{\nu \delta}\tilde{F}_{\gamma \delta}] - \frac{\tilde{e}^{2}m^{4}}{8}\varphi^{2}\chi g^{\alpha \nu}\tilde{A}_{\alpha}= 0
S^2 = \left( \sum_{i,j}R_{ijij}\right)^2=\left( \sum_{i,j}\overline{R}_{ijij}+16H^2-\vert A\vert^2\right)^2=\left(\sigma+12H^2-\vert\mathring{A}\vert^2\right)^2 &= \sigma^2 + 144H^4 +\vert\mathring{A}\vert^4+24\sigma H^2 -2\sigma\vert\mathring{A}\vert^2-24H^2\vert\mathring{A}\vert^2,
\mathcal{K}_{\alpha}(f)(\mathbf{x})=\int_{\mathcal{D}}\frac{1}{\|\mathbf{x}-\mathbf{y}\|^{\alpha}}f(\mathbf{y})d\mathbf{y},\forall f\in \mbox{Supp}(\mathcal{K}_{\alpha}), 0<\alpha <d,
M(\varphi)=\frac{(ab)^{2}}{((acos\varphi)^{2}+(bsin\varphi)^{2})^{\frac{3}{2}}}=\frac{a(1-e^{2})}{(1-e^{2}sin^{2}\varphi)^{\frac{3}{2}}}=\frac{1-e^{2}}{a^{2}}N(\varphi)^{3}.
\begin{array}{c c}{{\lambda^{a \alpha}= i \textstyle{\frac{1}{6}}\bar{D}^{a \beta}\mathrm{h}^{\alpha}{}_{\beta}\, , ~}}&{{~ \bar{\lambda}_{a \alpha}= - i \textstyle{\frac{1}{6}}D_{a \beta}\mathrm{h}^{\beta}{}_{\alpha}\, ,}}\\ \end{array}
m{\frac{d \Omega_{B H}}{d m}}\sim{\frac{m t_{e q}^{1 / 2}}{m_{p}^{2}t_{f}^{3 / 2}}}\sim 1 0^{2 7}\left({\frac{\lambda}{e}}\right)^{1 / 2}\left({\frac{M}{m_{p}}}\right)^{5 / 2}\left({\frac{M}{e m}}\right)^{\frac{n - 1}{2}},
\int Bu^{2+2q}\eta^2\leq \int(1+q)^2u^{2q}|\nabla u|^2\eta^2 + \int u^{2+2q}|\nabla \eta|^2 & +2(1+q)\int u^{1+2q}\eta\ <\nabla u,\nabla \eta>. & = \int u^{2q}|(1+q) \eta \nabla u + u \nabla \eta|^2
u(t;l)e^{-rt}= E_x\left[\left.k_{\sigma^{\ast}(t,l)}e^{-r\sigma^{\ast}(t,l)}-\int_t^{\sigma^{\ast}(t,l)}c_{\theta}(u,X_u,l)e^{-ru}\,du\,\right\vert\,\mathcal{F}_t \right] \geq E_x\left[\left.k_{\sigma^{\ast}(t,l)}e^{-r\sigma^{\ast}(t,l)}-\int_t^{\sigma^{\ast}(t,l)}c_{\theta}(u,X_u,k)e^{-ru}\,du\,\right\vert\,\mathcal{F}_t \right] \geq u(t;k)e^{-rt}.
\begin{aligned}f(x) =\sum_{i=1}^{\infty}\mathbb{P}(d=i)\sum_{j=1}^{i}(-1)^{j+1}\left(\frac{x}{2+\beta}\right)^{j}{i \choose j}&=\sum_{j=1}^{\infty}(-1)^{j+1}\left(\frac{x}{2+\beta}\right)^{j}\sum_{i=j}^{\infty}\mathbb{P}(d=i){i \choose j}&=\frac{x}{2+\beta}\sum_{j=1}^{\infty}\left(-\frac{x}{2+\beta}\right)^{j-1}m_j.\end{aligned}
\mathfrak{p}_0(u) =\|\nabla_{A_0}u\|^2_{L^2}+\| V_0^{1/2}u\|^2, \mathfrak{l}(u) =\langle L_A u,u\rangle+\langle |V_1|^{1/2}u,V_1^{1/2}u\rangle,
P_{v}= \exp \left\{- \mu \tilde{n}\int d{\bf x}\right\}, \; \; \mu = \sum_{l = 0}^{\infty}\frac{( - 1 )^{l}\beta ( l + 1 )}{( l + 1 )^{3 / 2}\beta ( 1 )}\exp \left\{- l \pi \frac{( a M )^{2}}{\rho}\right\}\; .
0 < 1 < \det(S^T S + I) = \det(S^T(S+JSJ^{-1})) & = \det(S) \det(E+iF) \det(E-iF) & = \det(S) \det(E+iF) \, \overline{\det(E+iF)}& = \det(S) |\det(E+iF)|^2
\begin{aligned}\psi_{\vec m}\circ T(X) = \prod_{j=1}^k \left( Q^{(\alpha_j,\alpha_{j\vec m})}_{m_j}(X_j) \prod_{i=1}^{j-1}(1-X_i)^{m_j}\right) = \prod_{j=1}^k\left(Q^{(\alpha_j,\alpha_{j\vec m})}_{m_j}(X_j)(1-X_j)^{\sum_{i=j+1}^k m_i}\right), \end{aligned}
Q ( 1 ) Q ( 2 ) = C_{1}^{1 / 2}| z_{1 2}|^{- 4 \Delta + 2 \Delta_{A}}O_{A}( 2 ) + C_{2}^{1 / 2}| z_{1 2}|^{- 4 \Delta + 2 \Delta_{S}}O_{S}( 2 ) + . . .
\begin{cases}N_n\coloneqq 1,\quad \mbox{if}\frac{1}{2\alpha_{2}}<\delta_n N_n\coloneqq j,\quad \mbox{if}j\ge 2\;\mbox{and}\;\frac{1}{(j+1)\alpha_{j+1}}<\delta_n\le \frac{1}{j\alpha_j}\end{cases};
\int_{-\infty}^{\infty}\frac{d\epsilon}{2\pi}\frac{1}{3!}(\frac{\epsilon}{2\pi})^{3}\{\frac{1}{1+e^{\beta(\epsilon-\mu)}}-\theta(-\epsilon)\}=\frac{1}{4!}(\frac{\mu}{2\pi})^{4}+\frac{1}{2!}(\frac{\mu}{2\pi})^{2}\frac{T^{2}}{4!}+\frac{7}{8}\frac{T^{4}}{6!},
\begin{aligned}\int_{Q_T}\Big( s \Theta (w_x)^2 + s^3 \Theta^3 \left(\frac{x-x_0}{a}\right)^2\w^2 \Big) e^{2s \varphi}\, dx dt \le C\int_{Q_T}\frac{e^{2s \varphi}}{a}f^2 dxdt.\end{aligned}
\begin{array}{c}b^-~=~lim_{Q\to q}{{Q^{\pm k N_B \over 2}{(B^-)}^k}\over{([k]!)^{1\over 2}}}b^+~=~lim_{Q\to q}{{{(B^+)}^kQ^{{\pm kN_B\over 2}}\over{([k]!)^{1\over 2}}}}\end{array}
\lim_{r_k \downarrow 0}\cdots \lim_{r_1 \downarrow 0}\Delta_{\{r_1B_1, \ldots, r_kB_k\}}= \Delta_{\{B_1\}}+ (1-\Delta_{\{B_1\}}) \lim_{r_k \downarrow 0}\cdots \lim_{r_2 \downarrow 0}\Delta_{\{r_2B_2, \ldots, r_kB_k\}}&= 1 -(1 - \Delta_{\{B_1\}}) \cdots (1 - \Delta_{\{B_k\}}).
\mu_{1}\left\vert \eta \right\vert^{2}\leq \displaystyle\sum\limits_{i,j=1}^{n}a_{i,j}\left( x\right) \eta_{i}\eta_{j}\leq \mu_{2}\left\vert \eta \right\vert^{2},\forall x\in \mathbb{R}^{n},\forall\eta =\left( \eta_{1},...\eta_{n}\right) \in \mathbb{R}^{n}.
\left( \mathcal{W}\circ \varphi_{A}\left(Z',Z''\right)\right)^{t}= \left(\frac{\sqrt{1-A}{Z'}^{t}}{I_{q}-A{Z''}^{t}}, \frac{\sqrt{1-A}\left(A-Z'''\right){Z'}^{t}}{\left(I_{q}-AZ'''\right)\left(I_{q}-A{Z''}^{t}\right)},\frac{\left(A-Z'''\right)\left(A-{Z''}^{t}\right)}{\left(I_{q}-AZ'''\right)\left(I_{q}-A{Z''}^{t}\right)}\right).
\bar B=\dfrac{\left\langle A^{1/2}\dfrac{\zeta}{|\zeta|}, A^{1/2}\dfrac{e}{|e|}\right\rangle}{\left| A^{1/2}\dfrac{e}{|e|}\right|^2},\bar C=\dfrac{\left| A^{1/2}\dfrac{\zeta}{|\zeta|}\right|^2}{\left| A^{1/2}\dfrac{e}{|e|}\right|^2}.
d s^{2}= \frac{1}{1 - \xi^{0}\xi^{0}}d \xi^{0}d \xi^{0}- ( 1 - \xi^{0}\xi^{0}) \left[ ( 1 - \rho^{2})^{- 2}d \rho^{2}+ ( 1 - \rho^{2})^{- 1}\rho^{2}d{\bf u}d{\bf u}^{\prime}\right] ~ ,
\frac{\delta}{{a}^{p^* - 1}}= \delta (2\Lambda)^{p^* -1}= \delta 2^{\frac{1}{p-1}}\Lambda^{\frac{1}{p-1}}= 2^{\frac{1}{p-1}}\delta^{\frac{p}{2(p-1)}}\Big(\frac{p-2}{p}\Big)^{\frac{p-2}{2(p-1)}}L^{\frac{p}{2(p-1)}}.
\begin{matrix}D:&(\mathbb{R},+)\times\mathbb{R}^{4}&\rightarrow&\mathbb{R}^{4}\\ &(\hat{t},(t,x,a,b))&\rightarrow&(t+\hat{t},\frac{axe^{a\hat{t}}}{a-(1-e^{a\hat{t}})bx},a,b).\end{matrix}
\texttt{maximize}& \int_{0}^{T}[\; \; - <c,\rho^2(t)> + <p(t), \; S_i(I(t),p(t))> \; ] \sigma dt &\texttt{subject to an inventory flow constraint}\frac{dI}{dt}\; = - \; S(I(t),p(t), t) \sigma + \rho^2(t) \sigma &\texttt{and additional boundary constraints, e.g.}\; I(0) = I_0, \; I(T) = I_T
\begin{array}{l}{\phi_{k}( \chi ) = \displaystyle \frac{1}{\omega_{n - 1}}\int_{u u^{\prime}= 1}a_{k}( u ) P_{\alpha_{k}( m_{0})}( \chi , u ) \dot{u},}\\{\psi_{k}( \chi ) = \displaystyle \frac{1}{2 \pi}\int_{u u^{\prime}= 1}b_{k}( u ) P_{\alpha_{k}( m_{0})}( \chi , u ) \dot{u}.}\\ \end{array}
P_{m+d+1}\left( x\right) =\left( x-\beta_{m+d}\right) P_{m+d}\left(x\right) -\sum\nolimits_{\nu =0}^{d-1}\gamma_{m+d-\nu}^{d-1-\nu}P_{m+d-1-\nu}\left( x\right) ,\ \m\geq 0,
\Pi ={\cal C}^{a}P_{a}^{b}\bar{\cal P}_{b}( - 1 )^{\epsilon_{b}}+ \sum_{r = 1}^{L}( - 1 )^{r}{\cal C}^{a_{r}}P_{a_{r}}^{b_{r}}\bar{\cal P}_{b_{r}}( - 1 )^{\epsilon_{b_{r}}}+ . . .
(g,\tau_z(t))\cdot z =(\det(g)^{-1}g_4(g_1z_1+g_3z_2)-\det(g)^{-1}g_3(g_2z_1+g_4z_2), &\det(g)^{-1}(-g_2)(g_1z_1+g_3z_2) + \det(g)^{-1}g_1(g_2z_1+g_4z_2)) &=z.
||\frac{1}{j}\sum_{k=0}^{j-1}T_m^k e_1||_{l^1}\ge ||\frac{1}{(m+1)!}\sum_{k=0}^{m!-1}T_m^ke_1||_{l^1}= \frac{1}{(m+1)!}\sum_{k=0}^{m!-1}2^{\frac{k}{(m-1)!}}= & = \frac{2^m - 1}{(m+1)!(2^{\frac{1}{(m-1)!}}-1)}= \frac{2^m - 1}{(m+1)m}\cdot \frac{1}{(m-1)! (2^{\frac{1}{(m-1)!}}- 1)}=: c(m)
{\cal Z}_{\mathrm{A H M}}\left[ j_{\mu}\right] = \int{\cal D}A_{\mu}{\cal D}\theta^{\mathrm{s i n g .}}{\cal D}\theta^{\mathrm{r e g .}}\exp \left\{- \int d^{4}x \left[ \frac{1}{4 q^{2}}\left( F_{\mu \nu}+ F_{\mu \nu}^{M^{\prime}}\right)^{2}+ \frac{\eta^{2}}{2}\left( \partial_{\mu}\theta - A_{\mu}\right)^{2}\right] \right\}.
= \frac{1}{4}\int_{\Omega}|\nabla \kappa_{\varepsilon}(t) |^2 + \ell^2 \int_{\Omega}\bigg| \nabla \bigg( \varphi_0 + \int_0^t \partial_t \varphi_{\varepsilon}(s) \ds \bigg) \bigg|^2
\int_{0}^{2\pi}\cos j\theta\cos2 k(\theta+\delta)d\theta =(-1)^{2k}\cos 2k\delta\int_{-\pi}^{\pi}\cos^2 2k\theta d\theta &=\cos 2k\delta \frac{1}{2}\int_{-\pi}^{\pi}[1+\cos2j\theta]d\theta=\cos 2k\delta \frac{1}{2}[2\pi+0]=\pi\cos2k\delta
\sum_{i=1}^N\sum_{j=0}^{\theta-1}{\big| q^{-z + \lambda_{N-i+1}(N) + \theta(i-1)+j}\big|}\qquad{}\leq \sum_{i=1}^N\sum_{j=0}^{\theta-1}{q^{-\Re z + \lambda_N(N) + \theta(i-1)+j}}\leq c\sum_{i=1}^N\sum_{j=0}^{\theta - 1}{q^{\theta(i-1)+j}}< \frac{c}{1 - q},
\underset{\mathbf{x}\in \mathbb{R}^{np}}{}g(\mathbf{x}) \triangleq f(\mathbf{x}) + \frac{\rho (1-\eta)}{4}\|\mathbf{E}_{}\mathbf{x}\|^2, \sqrt{\eta}\mathbf{E}_{}\mathbf{x}= \mathbf{0},
\left\lbrace \begin{array}{c}\phi(+a/2)=\phi(-a/2) \partial_{x}\phi(+a/2)=\partial_{x}\phi(-a/2) \partial_{x}^{2}\phi(+a/2)=\partial_{x}^{2}\phi(-a/2) \partial_{x}^{3}\phi(+a/2)=\partial_{x}^{3}\phi(-a/2)\end{array}\right. .
\begin{array}{r c l}{{{\cal R}( u , c )}}&{{=}}&{{\displaystyle \frac{u - i c P}{u + i c}}}\\{{}}&{{=}}&{{\displaystyle \frac{1}{u + i c}\left[ \begin{array}{c c c c}{{u - i c}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{u}}&{{- i c}}&{{0}}\\{{0}}&{{- i c}}&{{u}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{u - i c}}\\ \end{array}\right] ~ ,}}\\ \end{array}
I_{\tiny\mbox{ren}}\equiv I(G_{\tiny\mbox{ren}},\Lambda_{\tiny\mbox{ren}},c^i_{\tiny\mbox{ren}})=I(G_B,\Lambda_B,c^i_B)+W_{\tiny\mbox{div}}~~~,\hspace{0.5cm}W_{\tiny\mbox{ren}}=W-W_{\tiny\mbox{div}}~~~.
\left( E_{B}^{0}( g ) \right)_{C}^{\prime}( \hat{I}_{C D}E_{D}^{0}( h ) ) = ( - D )_{L}\left[ \frac{\partial^{2}g}{\partial \phi_{B}^{( L )}\partial \phi_{C}^{( J )}}D_{J}( \hat{I}_{C D}E_{D}^{0}( h ) ) \right] ,
|\{\pi \in \mathcal{AI}_{2n-1}(1234) \mid \pi_2 \neq 2n-1\}| = |\mathcal{AI}_{2n-1}(1234)| - |\{\pi \in \mathcal{AI}_{2n-1}(1234) \mid \pi_2 = 2n-1\}| begin{align*}3pt] = M_n - M_{n-2}- M_{n-1},
\sum_{j(\neq i)}^N \frac{z_i \, z_j}{z_{ij}\, z_{ji}}\, r_{ij}\,= \frac{1}{2}\, (z_i\, \partial_{z_i})^2 - \frac{N}{2}\, z_i \, \partial_{z_i}+ \frac{1}{6}\, (N^2 - 1) \, .
\frac{1}{{\Pi \hspace{-8pt}/}- M^{\star}}\> = \>\left ( M + \frac{M}{M^{\star}}{\Pi \hspace{-8pt}/}\right ) \> \frac{1}{{\Pi \hspace{-8pt}/}\frac{M}{M^{\star}}{\Pi \hspace{-8pt}/}- M M^{\star}+ i 0}
a_0 \int \limits_0^{2 \pi}R(t) e^t dt - a_0 \int \limits_0^{2 \pi}R(t) dt +\sum \limits_{k=1}^{\infty}\int \limits_0^{2 \pi}\left( \frac{a_k+k b_k}{1+k^2}\right) R(t) e^t dt +
\int_{\mathbb{S}^3}\mathrm{Trace}\big\{K_t[p,q](i\mu+1,i\mu+1)\big\}dvol^3 =\int_{\mathbb{S}^3}\mathrm{Trace}\big\{K_t[p,q+p+\frac{1}{2}](i\mu,i\mu)\big\}dvol^3.
= \; \left\langle \prod_{b = 1}^{N}\; e^{- 2 \sum_{b = 1}^{N}\beta^{( b )}\epsilon^{( b )}\int_{\Lambda}d^{2}x \; t ( x ) \; : \cos \big [ 2 \sqrt{\pi}\theta_{2}^{( b )}( x ) \big ] :_{C^{2}}}\right\rangle_{C^{2}}\; ,
\intertext{From Theorem \ref{thm:main}}V_k^r(x) =\frac1{2(k+1)}\{(2X+1)^{k+1}-(2X-1)^{k+1}\}&=(2X)^{k}+O(X^{k-2}). \intertext{Applying Theorem \ref{thm:2.3}, we find}V_k^r(x) =\frac{2^k}{\zeta(rk)}x^k+\Omega(x^{k-1}). \intertext{Hence $E_k^r(x)=\Omega(x^{k-1})$ for $rk\ge3$ and $k\not=1$}
|a_{ij}| = \left| \sum_{q=1}^n D(j)^{-1}_{iq}M_j(c_{qk_1k_2 \cdots k_{n+1}}; \{a_{pq}\; |\; q \leq j-1, 1 \leq p \leq n \})\right| \leq B \sqrt{n}M_j(C_{1k_1k_2 \cdots k_{n+1}}; \{A_{pq}\; |\; q \leq j-1, 1 \leq p \leq n \}) = A_{ij}
\sum_{Q\in \mathcal D}\lambda_Q\left|\prod_{i=1}^Nm_{\sigma_i}(f_i,Q)\right|^{p\alpha}\le A\alpha\|M_{d}^{\vec \sigma}(\vec f)\|_{p,\nu_{\vec \sigma}}^{p\alpha}\lesssim A\alpha \prod_{i=1}^N\|M_d^{\sigma_i}f_i\|_{p_i,\sigma_i}^{p\alpha}\lesssim A\alpha \prod_{i=1}^N\|f_i\|_{p_i,\sigma_i}^{p\alpha},
\tilde\gamma_{S}(e)=\partial_{e}\Big[ \phi_{F}(z_{S}(e)) - \phi_{A}(z_{S}(e')) \Big] =\partial_{e}\mu_{S}(e)\tilde\gamma_{B}(e)=-\partial_{e}\phi_{F}(z_{B}(e)) = \partial_{e}\mu_{B}(e)\,,
H^{*}\left(\overline{e}_{k}\right) =\left[H \overline{e}_{k-1}, H^{2}\overline{e}_{k-1}, \cdots, H^{n}\overline{e}_{k-1}\right] &= H\left[ \overline{e}_{k-1}, H \overline{e}_{k-1}, \cdots, H^{n-1}\overline{e}_{k-1}\right] &=HH^{*}(\overline{e}_{k-1})= HH^{k-2}=H^{k-1}.
=C_{m_{1},m_{2},m}^{l_{1},l_{2},l}C_{0,0,0}^{l_{1},l_{2},l}+\frac{1}{n+1}C_{m_{1},m_{2},m}^{l_{1},l_{2},l}P(l_{1},l_{2},l)+O((n+1)^{-2})
- \frac{1}{2}\left[ \sum_{K , L = 1}^{M}( \frac{\partial^{2}W}{\partial \sigma_{K}\partial \sigma_{L}}) \chi_{_K}\chi_{_L}+ \sum_{k , l = 1}^{N}( \frac{\partial^{2}W}{\partial \phi_{k}\partial \phi_{l}}) \psi_{k}\psi_{l}+ c c . \right] ,
\sum_{i=1}^{k-1}\alpha_iF(d_i) +\alpha_k\bigl((1-\beta_{1k})F(d')+\beta_{1k}F(d'')\bigr) & \le \max\left(\bigl(1-\sqrt{\gamma}\bigr)F(0)+\sqrt{\gamma}F(\sqrt{\gamma}), (1-\eta)F(\eta)+\eta F(1)\right)
U(x):=\mathcal{C}_{N,p,s}\,\big(1+|x|^{\frac{p}{p-1}}\big)^{\frac{s\,p-N}{p}},\qquad\mathcal{C}_{N,p,s}=\left(\int_{\mathbb{R}^N}\big(1+|x|^{\frac{p}{p-1}}\big)^{-N}\,dx\right)^\frac{N\,p}{s\,p-N}.
\delta_{\epsilon}\lambda = \left( \not \! \partial \phi +{\textstyle \frac{1}{2 \cdot 3 !}}\Gamma_{1 1}\not \! \! H \right) \epsilon +{\textstyle \frac{5 i}{4}}m e^{\phi}\epsilon -{\textstyle \frac{3 i}{8}}e^{\phi}\not \! G^{( 2 )}\Gamma_{1 1}\epsilon ,
\lim_{\varepsilon\to0}(x_\varepsilon,t_\varepsilon,y_\varepsilon,s_\varepsilon,u^*(x_\varepsilon,t_\varepsilon),v_*(y_\varepsilon,s_\varepsilon))=(\hat{x},\hat{t},\hat{x},\hat{t},u^*(\hat{x},\hat{t}),v_*(\hat{x},\hat{t})),
\prod_{j=2}^{2i-1}d_j=\prod_{j=1}^{i-1}(d_{2j}d_{2j+1}) \leq \prod_{j=1}^{i-1}\left(\left\lfloor \frac{n-2j}{2}\right\rfloor \cdot \left\lceil \frac{n-2j}{2}\right\rceil\right) =\prod_{j=1}^{i-1}\left(\frac{n-2j}{2}\right)^2.
Q \left| A_{i}( \theta ) \right\rangle = \sqrt{m_{i}}\mathrm{e}^{\theta / 2}\mathcal{Q}\left| A_{i}( \theta ) \right\rangle \quad , \quad \bar{Q}\left| A_{i}( \theta ) \right\rangle = \sqrt{m_{i}}\mathrm{e}^{- \theta / 2}\bar{\mathcal{Q}}\left| A_{i}( \theta ) \right\rangle
\mathcal{B}^{n+1,n}=\mathcal{C}'_{K_n}\|\cdot\|_{H^{1}(I_{n+1})}, \mathcal{A}^{n+1,n}=\mathcal{A}^{n+1}|_{\mathcal{B}^{n+1,n}\times \mathcal{B}^{n+1,n}},
{{m}_{v}}\left( x \right)=1+\frac{{{2}^{v}}v\left( 1-v \right){{\left( x-1 \right)}^{2}}}{{{\left( x+1 \right)}^{v+1}}}\quad\quad{{M}_{v}}\left( x \right)=1+\frac{v\left( 1-v \right){{\left( x-1 \right)}^{2}}}{2{{x}^{v+1}}}
B_1: = \bigg(\int_0^{\infty}\bigg(\int_0^x [V_1]^q(t) w(t)\,dt\bigg)^{r / p}[V_1]^q(x) V^{- r / p}(x) w(x)\,dx \bigg)^{1 / r}, B_2: = \bigg(\int_0^{\infty}\bigg(\int_x^{\infty}w(t)\,dt\bigg)^{r / p}\bigg[\sup_{0 < \tau \le x}[V_1](\tau)V^{-1 / p}(\tau)\bigg]^{r}w(x)\,dx \bigg)^{1 / r}.
\sum_{\ell=1}^L n_\ell \le \frac{L-1}{\ln(\gamma)}\left[ \ln(\bar{\alpha})+\ln\left(\frac{2\beta\bar{\alpha}}{3}\right)+ \ln\left(\frac{\beta}{3}\right)+ \frac{1}{L-1}\ln\left( \frac{\varepsilon}{F(x^{1}) - \eta^1}\right)\right] + L.
\int_{\varepsilon Y_{1}}p\left(\frac{x}{\varepsilon}\right)\phi\left(x\right)dx =\varepsilon^{d}\int_{Y_{1}}p\left(y\right)\phi\left(\varepsilon y\right)dy, \int_{\varepsilon Y_{1}}\int_{Y_{1}}p\left(y\right)\phi\left(x\right)dydx =\varepsilon^{d}\int_{Y_{1}}\int_{Y_{1}}p\left(y\right)\phi\left(\varepsilon z\right)dydz,
\Bigl|\langle \hat P_r - \mathbb E \hat P_r, P_rW_rP_r\rangle\Bigr|=\Bigl|\langle R_r(E), P_rW_rP_r\rangle\Bigr| &\lesssim_{\gamma}m_r \frac{\|\Sigma\|_{\infty}^3}{\bar g_r^3}\sqrt{\frac{{\bf r}(\Sigma)}{n}}\biggl(\sqrt{\frac{{\bf r}(\Sigma)}{n}}\bigvee \sqrt{\frac{t+\log(m_r)}{n}}\bigvee \frac{t+\log (m_r)}{n}\biggr)\sqrt{\frac{t+\log (m_r)}{n}}.
\langle F\rangle_{{\rm ML}, \hat g}= \langle F(\cdot-\tfrac{Q}{2}\omega)\rangle_{{\rm ML},g}\exp\big( \tfrac{1+6Q^2}{96\pi}S_{\rm L}^{{\rm cl},0}(\hat g,g)+\beta S^{{\rm cl}}_{\rm M}(\hat g, g)\big)
G_0 =G_{0,0}-G_{0,0}v\bigl(B_0+A_1+B_1A_{-1}B_1 &\qquad+A_{-1}B_2+B_2A_{-1}\bigr)v^*G_{0,0}&-G_{0,0}vA_{-1}B_1v^*G_{0,1}-G_{0,1}vB_1A_{-1}v^*G_{0,0}&-G_{0,0}vA_{-1}v^*G_{0,2}-G_{0,2}vA_{-1}v^*G_{0,0},
z_1\wedge e_1 = -ye_{12}z_1\wedge e_2 = xe_{12}z_1\wedge e_3 = xe_{13}+ye_{23}z_2\wedge e_1 = -xe_{13}z_2\wedge e_2 = -xe_{23}z_2\wedge e_3 = 0 z_3\wedge e_1 = -xe_{12}-ze_{13}z_3\wedge e_2 = -ze_{23}z_3\wedge e_3 = xe_{23},
\gamma:= \gamma(n,\sigma,p)=\int_{B_{1}^{c}}\frac{dy}{|y|^{n+\sigma p}}=\frac{|\mathbb{S}^{n-1}|}{\sigma p}, \tau:= \tau(n,\sigma,p,s)=\int_{B_{1}^{c}}\frac{dy}{|y|^{n+\sigma p+s(p-1)}}=\frac{|\mathbb{S}^{n-1}|}{\sigma p+s(p-1)}.
G^{(2)}_{m,n}=C(\dot a_2,a_3,\ldots,a_d), A_{m,n}=G^{(2)}_{m,n}\vee C(\ddot3)\vee G^{(2)}_{m,n}, B_{m,n}=G^{(2)}_{m,n}\vee C(\dot1,2,\dot1)\vee G^{(2)}_{m,n}, C_{m,n}=G^{(2)}_{m,n}\vee C(\dot1,1,1,\dot1)\vee G^{(2)}_{m,n}.
{\cal F}^i_{\bar{\mu}\bar{\nu}}=\partial_{\bar{\mu}}{\cal A}^i_{\bar{\nu}}-\partial_{\bar{\nu}}{\cal A}^i_{\bar{\mu}}, \ \ \H_{\bar{\mu}\bar{\nu}\bar{\rho}}=(\partial_{\bar{\mu}}B_{\bar{\nu}\bar{\rho}}+2{\cal A}^i_{\bar{\mu}}L_{ij}{\cal F}^j_{\bar{\nu}\bar{\rho}}) +{\rm cyc.\perms.}.
D=P_{\Upsilon}d_{V}A= \left ( \begin{array}{cccccc}P_{\Upsilon}d 0 0 0 \cdots 0 P_{\Upsilon}dTd P_{\Upsilon}d 0 0 \cdots 0 P_{\Upsilon}(dT)^{2}d P_{\Upsilon}dTd P_{\Upsilon}d 0 \cdots 0 P_{\Upsilon}(dT)^{3}d P_{\Upsilon}(dT)^{2}d P_{\Upsilon}dTd P_{\Upsilon}d \cdots 0 &\cdots \cdots \cdots \end{array}\right ).
\phi_i= \mathbf d\left| \begin{array}{ccc}f_2(c(t_i)) f_1(c(t_i)) f_0(c(t_i)) f_2(c(t_1)) f_1(c(t_1)) f_0(c(t_1)) f_2(c(t_2)) f_1(c(t_2)) f_0(c(t_2))\end{array}\right|
[\begin{matrix}c_{2}c_{1}&-c_{2}s_{1}&s_{2}\\s_{3}s_{2}c_{1}+c_{3}s_{1}&-s_{3}s_{2}s_{1}+c_{3}c_{1}&-s_{3}c_{2}\\ -c_{3}s_{2}c_{1}+s_{3}s_{1}&c_{3}s_{2}s_{1}+s_{3}c_{1}&c_{3}c_{2}\end{matrix}]
E_2(\tau) =1-24\sum_{n=1}^\infty \sigma(n)q^n &=\left(1-24\sum_{n=1}^\infty\sigma(2n)q^{2n}\right)-24\sum_{n=0}^\infty \sigma(2n+1)q^{2n+1}&=3E_2(2\tau)-2E_{2}(4\tau)-24\sum_{n=0}^\infty \sigma(2n+1)q^{2n+1}.
f^\prime(x) = \begin{pmatrix}0 0 0 K^{-1}\end{pmatrix}\begin{pmatrix}T(x) U(x) U(x)^\top V(x) \end{pmatrix}= \begin{pmatrix}0 0 K^{-1}U(x)^\top K^{-1}V(x) \end{pmatrix}.
\begin{array}{l}\partial_t a_i-\nabla\cdot(D_i\nabla a_i)=Q_i(a),i\in\{1,...,p\},\t\geq 0,\x\in\mathbb R^N, begin{align*}.3cm]Q_i: a\in\mathbb R^p\longmapsto \mathbb R^p,\end{array}
\nu(B_{r/2}(x))=\lim_{\epsilon_j\to 0}\frac{1}{2}\frac{\|d^*\zeta_{\epsilon_j}\|_{L^2(B_{r/2}(x))}^2}{|\log\epsilon_j|}\leq \lim_{\epsilon_j\to 0}\frac{C}{|\log\epsilon_j|}=0.
V_q[\omega, \mu](x){:=}v_q[\omega, \mu](x) \forall x \in \partial \Omega_Q\,, W_{q}^\ast[\omega, \mu](x){:=}\int_{\partial{\Omega_Q}}\sum_{l=1}^n \mu_{l}(y)T(\omega,D\Gamma_{n,\omega}^{q,l}(x-y))\nu_{{\Omega_Q}}(x)\,d\sigma_y \forall x \in \partial{\Omega_Q}\,.
\mathcal{A}_{S G}= \int_{- \infty}^{0}d x \int_{- \infty}^{\infty}d t ~ \frac{1}{4 \pi}( \partial_{\mu}\varphi )^{2}+ 2 \mu \cos ( 2 \beta \varphi ) + 2 \mu_{B}\int_{- \infty}^{\infty}d t \cos \beta ( \varphi ( 0 , t ) - \varphi_{0}) \, ,
\widetilde{\psi}_{k}^{( 1^{2})}\big(x;\omega ,N,- \Vert \psi_{1}\Vert^{2}\big)=\frac{2\omega z^{(N+1/2) /2}\exp ( -z/2)}{Q_{1}^{N}(z)}\widetilde{L}_{k,N}^{( 1^{2})}( z) ,
\mathbb{P}_{MD}\! = \!\Pr \left({{\kappa^2}\!+\!{C_{1w}}X_w\! +\!{C_{2w}}Z_w \!<\!{\varepsilon_{k}}}\right)\! = \Pr \left({T_w \!+ \!{C_{1w}}X_w \!<\!{\varepsilon_{k}}}\right).
\mathcal S_{\boldsymbol\nu,\boldsymbol\mu}=O\left(\frac{f_{k_1}(z_i)^Nf_{k_2}(z_o)^N}{z_i^M z_o^M}\left(\frac{M}{eN}\right)^{2M}\prod_{j=1}^3\nu_j^{\mu_{j,\boldsymbol\cdot}+\mu_{\boldsymbol\cdot,j}}\right).
(f_0)_1(r) =c_0e^{\frac{r^2}{4}}r^{2l_k}F[ l_k + \frac{m}{2}+\lambda,2l_k + \frac{m}{2};-\frac{r^2}{4}] &=C_0e^{\frac{r^2}{4}}r^{-(2\lambda+m)}\big(1+g_0(\frac{r^2}{4})\big).
\int \prod_{i < j}\left( x_{i}- x_{j}\right)^{2}\prod_{i = 1}^{N}\exp \left( - \log^{2}x_{i}/ 2 g_{s}\right) \left( 1 + \theta \sin \left( 2 \pi \log x \right) \right) \frac{d x_{i}}{2 \pi}
(\,{\tilde \Gamma}_{{\bar i}{\bar j}}^{\bar k}-{\tilde \Gamma}_{{\bar j}{\bar i}}^{\bar k}\, ) \langle \,{\hat \phi}_{\bar k}\,\prod_{l=1}^{s}\phi_l \, \rangle_{g}= \sum_{l=1}^{s}{\cal R}_{D_{l}}+ \sum_{nodes}{\cal R}_{\Delta}