harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
F_{m,\,j}(X) - F_{m-1,\,j}(X) & = (-1)^m\binom{X-m}{j}+ \sum_{s=1}^{m-1}(-1)^s\binom{m-1}{s-1}\binom{X-s}{j}& = (-1)^m\binom{X-m}{j}- \sum_{s=0}^{m-2}(-1)^s\binom{m-1}{s}\binom{(X-1)-s}{j}& = (-1)^m\binom{X-m}{j}+ (-1)^{m-1}\binom{X-m}{j}- F_{m-1,\,j}(X-1) = -F_{m-1,\,j}(X-1).
D^{(m)}=\frac{1}{m!}\sum_{m_1+\cdots+m_t=m}{m\choose m_1,\ldots,m_t}q_1^{m_1}\cdots q_t^{m_t}(\partial_{q_1})^{m_1}\cdots(\partial_{q_t})^{m_t}&=\frac{1}{m!}(q_1\partial_{q_1}+\cdots+q_t\partial_{q_t})^m+D'=\frac{1}{m!}\big(D^{(1)}\big)^m+D',
\frac{KNT^{KN(KN+KM)}+KNMT_{\sf d}^{KM(KN+KM)}}{\frac{1}{\lambda_1}(T+1)^{KN(KM+KN)}}&=\frac{KNT^{KN(KN+KM)}+KNM\left(\left(\frac{\lambda_2}{\lambda_1M}^{\frac{1}{KM(KN+KM)}}\right)T^{\frac{N}{M}}-1\right)^{KM(KN+KM)}}{\frac{1}{\lambda_1}(T+1)^{KN(KM+KN)}}
K_0 =K\cap \big((-\infty,x- r_0]\cup[x+ r_0,+\infty)\big), K_{k}=K\cap \big([x- r_{k-1},x- r_{k}] \cup [x+ r_{k},x+ r_{k-1}]\big), \; \; k \in \omega \smallsetminus \{0\}.
M_Y((c,d]) =M_Y(d)-M_Y(c)=M_X(S^{-1}(d))-M_X(S^{-1}(c)) &=\int_{S^{-1}(c)}^{S^{-1}(d)}\!m_X(z)\,dz=\int_c^d \frac{m_X(S_X^{-1}(w))}{s_X(S_X^{-1}(w))}\,dw
\int d q_{1}^{3}d q_{2}^{3}\left( \frac{\left| \vec{\nabla}\psi \right|^{2}}{q^{2}}+ \frac{\mathrm{i}}{q^{2}}\left( \vec{\nabla}_{2}\psi \vec{\nabla}_{1}\psi^{\ast}- \vec{\nabla}_{1}\psi \vec{\nabla}_{2}\psi^{\ast}\right) \right) < \infty .
\pi^\mathcal{P}(f)=\sup_{P\in\mathcal{P}}\pi^P(f)=\sup_{Q\in\mathcal{Q}}\pi^Q(f)=\sup_{Q\in\mathcal{Q}}\sup_{\substack{Q'\in\mathfrak{Q}, Q'\sim Q}}(E_{Q'}[f]-A^{Q'})\leq\sup_{Q\in\mathfrak{Q}}(E_Q[f]-A^Q]),
e^{-1}{\cal L}= -{1\over 2}R -{1\over 4}G_{IJ}F_{\mu\nu}{}^I F^{\mu\nu J}-{1\over 2}g_{ij}\partial_{\mu}\phi^i \partial^\mu \phi^j +{e^{-1}\over 48}\epsilon^{\mu\nu\rho\sigma\lambda}C_{IJK}F_{\mu\nu}^IF_{\rho\sigma}^JA_\lambda^k \ .
D_{x}^{2}u =D_{x}D_{x}u & =D_{x}\left( \sin(\sigma)\bar{x}+\cos(\sigma)D_{\bar{x}}\bar{u}(\bar{x})\right) & =\left( \sin(\sigma)I_{n}+\cos(\sigma)D_{\bar{x}}^{2}\bar{u}(\bar{x})\right) \frac{d\bar{x}}{dx}.
1 = \phi_1^{e/e_1}(F^{e}_* r_1r) + \cdots + \phi_m^{e/e_m}(F^{e}_* r_mr) = \big(\phi_1^{e/e_1}\cdot r_1+ \cdots + \phi_m^{e/e_m}\cdot r_m \big)(F^e_*r).
\begin{aligned}\phi_{n}''(z) +\left(\frac{1}{z-t}-\frac{1}{z-t+\alpha_n(t)}\right)\phi_{n}'(z) &+\left(\frac{\kappa_3(t)}{z-t}+\frac{\kappa_4(t)}{z-t+\alpha_n(t)}+2n+1-z^2\right)\phi_{n}(z)=0,\end{aligned}
dx=(x_1-x_0) dt_1 +(x_2-x_0) dt_2 +(x_3-x_0) dt_3, &dy=(y_1-y_0) dt_1 +(y_2-y_0) dt_2 +(y_3-y_0) dt_3, &dz=(z_1-z_0) dt_1 +(z_2-z_0) dt_2 +(z_3-z_0) dt_3.
F_{\varphi}\left(x\right) =\int_{0}^{1}\varphi\left(y\right)F\left(x-y\right)dy =\int_{0}^{1}\varphi\left(y\right)e^{-\left|x-y\right|}dy,\;\forall\varphi\in C_{c}^{\infty}\left(0,1\right).
\int_{\mathbb{R}^d}\tilde \rho \, d\mu^\varepsilon_t (x)\Big|_{t=t_2}\leq \Big( \int_{\mathbb{R}^d}\tilde \rho \, d\mu^\varepsilon_t (x) \Big|_{t=t_1}+ c_5 \int_{t_1}^{t_2}e^{-\frac{1}{128(s-t)}}\mu_t^\varepsilon (B_{\frac{1}{2}}(y)) \, dt\Big)e^{\int_{t_1}^{t_2}|\lambda^\varepsilon (t)|^2 \, dt}
\hat{w_1}(\xi) = \int_0^\infty \frac{d}{d\tau\vphantom{(e^{-\tau}+ 1)^3}}\left( \frac{e^{-3\tau}}{2\vphantom{(e^{-\tau}+ 1)^3}}-\frac{4e^{-6\tau}}{(e^{-\tau}+ 1)^3}\right) \frac{e^{-i\xi\tau}}{i\xi\vphantom{(e^{-\tau}+ 1)^3}}\, d\tau.
\mathrm{Mab}_{\Theta}(\phi) = \frac{n!}{\mathcal{L}^n}M^l(\hat{u}^*) & \geq \epsilon \frac{n!}{\mathcal{L}^n}\int_{\Delta'}\hat{u}^*P_{DH}' -\frac{n!}{\mathcal{L}^n}(\epsilon C_3+C_4) & \geq \epsilon J(\hat{\phi})-\frac{n!}{\mathcal{L}^n}(C_5+\epsilon C_3+C_4)
\lim_{N\rightarrow\infty}{\prod_{i=1}^m{\frac{P_{\lambda(N)}\big(x_i, t^{-1}, \dots, t^{1-N}\big)}{P_{\lambda(N)}\big(1, t^{-1}, \dots, t^{1-N}\big)}(x_i q; q)_{\theta N - 1}}}= \prod_{i=1}^m{\Phi^{\nu}(x_i; q, t)(x_iq; q)_{\infty}}.
v ={\frac{r_{0}^{2}}{r^{2}}}; A ={\frac{\omega^{2}}{4}}\left({\frac{r_{1}r_{5}r_{n}}{r_{0}}}\right)^{2}; B ={\frac{\omega^{2}}{4}}\left({\frac{r_{1}r_{5}}{r_{0}}}\right)^{2}.
\begin{aligned}\dfrac{b_{1,\delta}(x)}{2}\geq -A + \underline{\delta}^\star (\delta) \forall \, x \geq x^\star + \delta \dfrac{b_{1,\delta}(x)}{2}\leq A - \underline{\delta}^\star (\delta) \forall \, x \leq x^\star - \delta\,,\end{aligned}
A_0|m,l\rangle = (l + m)|m,l\rangle, A_+ |m,l\rangle = \sqrt{(m+1)(2l + m)}|m+1,l\rangle, A_- |m,l\rangle = \sqrt{m(2l+m-1)}|m-1,l\rangle, Q |m,l\rangle = l(l+1) |m,l\rangle.
\int\limits_0^\infty\|\chi_{B_{r_0}}e^{-itH_0^\perp}P_\delta^\perp(W^\perp_{n;\textrm{out}})\|_\textrm{op}\,dt\leq\int\limits_0^\frac{1}{n}1 \,dt+C\int\limits_{\frac{1}{n}}^\infty t^{-\frac{1}{2}}(n+tm)^{-\ell}\, dt \leq \frac{1}{n}+C\sqrt{n}\int\limits_{\frac{1}{n}}^\infty (n+tm)^{-\ell}\,dt=\frac{1}{n}+C\frac{\sqrt{n}}{m}(n+\frac{m}{n})^{-\ell+1}
t - t_{0}= \int_{x_{0}}^{x}d x^{\prime}\frac{1}{\sqrt{2}\sqrt{E + \int_{x_{0}}^{x^{\prime}}f_{0}( x^{\prime \prime}) d x^{\prime \prime}+ \int_{x_{0}}^{x^{\prime}}f_{1}( x^{\prime \prime}) d x^{\prime \prime}}}
\sum_{j=0}^J d_j^{N/2}\|\tilde g - g_h\|_{H^1(\Omega_h \cap A_j)}\le CK^{m+N/2}h^{1-m}+ C(h|\log h|)^{1-m}+ C\sum_{j=0}^J d_j^{-1+N/2}\|\tilde g - g_h\|_{L^2(\Omega_h\cap A_j)}.
Q = \frac{1}{2 \pi}\left\{\lim_{r{\rightarrow}{\infty}}\left[ \overline{{{\cal{N}}}}( r ) -{\sin}\left( \overline{{{\cal{N}}}}( r ) \, \right) \, \right] + 2 \pi \ell - \overline{{{\cal{N}}}}( 0 ) +{\sin}\left( \overline{{{\cal{N}}}}( 0 ) \, \right) \right\}\, .
\tau_t(\tilde{\phi}(p') U(x)) = e^{i t p_0'}\tilde{\phi}(p') U(x) \Rightarrow \tau_{t + i \beta}(\tilde{\phi}(p')) U(x) = e^{i t p_0' - \beta p_0'}\,\tilde{\phi}(p') U(x) \,.
\sum_j\;{t_j^k \,dt_j\over s_j}=4\,d\tau\,u(\lambda)\,\sum_j\;(-1)^{N-1-j}{t_j^k\over t_j-\lambda}{D_j\over D}=-4\,{\lambda^k\over \prod_l\,(\lambda -a_l)}\,d\tau
\begin{array}{l l l}{\alpha_{i k}\alpha_{j k}}&{=}&{0 , i \neq j \pm g}\\{\alpha_{i k}\alpha_{i + g , k}}&{=}&{- \alpha_{i + g , k}\alpha_{i k}= \beta_{k}, 1 \leq i \leq g}\\{\alpha_{i k}\alpha_{j l}}&{=}&{- \alpha_{j l}\alpha_{i k}, k \neq l .}\\ \end{array}
\int d^{4}\theta \frac{d^{4}p}{( 2 \pi )^{4}}\Bigg ( V ( - p , \theta ) \, \partial^{2}\Pi_{1 / 2}V ( p , \theta ) \, f_{1}( p , \Lambda ) + V ( - p , \theta ) V ( p , \theta ) \, f_{2}( p , \Lambda ) \Bigg ) .
\hat{c}_{g_{\lambda}}= \frac{\lambda \hat{x}\textup{d i m}g}{\hat{x}+ \lambda \tilde{h}_{g}}= \lambda c_{g}( x ) , \hspace{. 2 i n}c_{g}( x ) = \frac{x \textup{d i m}g}{x + \tilde{h}_{g}}
\psi^{o u t}( u , \Omega ) \psi^{i n}( v , \Omega^{\prime}) = - \{~ 1 ~ - ~ 1 6 \pi \imath f ( \Omega , \Omega^{\prime}) e^{( u - v ) / 4 G M}\partial_{u}\partial_{v}~ \}) \psi^{i n}( v , \Omega^{\prime}) \psi^{o u t}( u , \Omega ) ~ .
\sigma_{1,+}= \sigma_{1,1}= \frac{1}{2}\left( u_{1,1}+ u_{1,2}\right)+ i \sqrt{u_{1,3}- \left[ \frac{1}{2}(u_{1,1}- u_{1,2}) \right]^{2}}= \sigma^{*}_{1,3}= \sigma^{*}_{1,-}
\sum_{l=0}^{k-1}\left\{\binom{k}{l}+\binom{k}{l+1}\right\}\frac{\partial^kf}{\partial u^{k-l}\partial v^l}(v,v)-\frac{\partial^kf}{\partial u^k}(v,v)+\frac{\partial^kf}{\partial v^k}(v,v).
\frac{1}{2 \pi}\int_{-R(u)}^{R(u)}e^{iyt}\hat{f}_Y(t)dt=\frac{1}{2 \pi}\int_{-R(u)}^{R(u)}e^{iyt}\left[e^{C_0(\beta -i \alpha t)-C_0(\beta)+i \alpha t u}\right] dt.
\sum_{N=0}^\infty \lambda^{2N}n^{-2N}C_8^{2N}\mu_1^{2N(m-p)/p}F^{2N/p}c^{2N}= 1 + \frac{(\lambda/n)^2C_8^2\mu_1^{2(m-p)/p}F^{2/p}c^2}{1-(\lambda/n)^2C_8^2\mu_1^{2(m-p)/p}F^{2/p}c^2}&\leq 1 + \frac{(\lambda/n)^2C_8^2\mu_1^{2(m-p)/p}F^{2/p}c^2}{1-(\lambda/n) C_8\mu_1^{(m-p)/p}F^{1/p}c}
u(w_0) = - f(w_0) + A w_0 + f(w_+) - A w_+ &= - f(w_0) - A \epsilon + f(w_+) &= - f(w_0) - \frac{f(w_+) - f(w_-)}{2}+ f(w_+) &= - f(w_0) + \frac{1}{2}( f(w_+) + f(w_-)).
\begin{array}{l}\Psi^{\gamma}_{s,l}(\theta,\phi)\propto(\cosh\theta-1)^{\frac{l}{2}}(\cosh\theta+1)^{-|\gamma|-\frac{l}{2}}*\hspace{2cm}\times\frac{\Gamma(s+l+1)}{\Gamma(s+1)\Gamma(l+1)}F(-s,-2|\gamma|+s+1;l+1;\frac{1-\cosh\theta}{2})e^{il\phi}.\end{array}
\begin{array}{c}\bar x_i=W_i(\cdot\cdot,x_{i'},w^{j'}_I,\cdot\cdot), \bar w^j=W^j(\cdot\cdot,x_{i'},w^{j'}_I,\cdot\cdot) (i,i'=1,\ldots,n;\, j,j'=1,\ldots,m)\end{array}
(-\Delta)^{s}\big\{xv(x)\big\}=\frac{2^{2s+1}\Gamma(s+\gamma)\Gamma(s+3/2)}{\sqrt{\pi}\Gamma(\gamma)}\, x \,{}_{2}F_{1}\Big(s+\gamma,s+\frac{3}{2}\, ;\frac{3}{2};-x^2\Big).
Z(s, w, f\times \overline{g}) := \sum_{n, h \geq 1}\frac{D_{f, \overline{g}}(s;h)}{h^w}= \frac{(4\pi)^{s+k-1}}{\Gamma(s+k-1)}\sum_{h \geq 1}\frac{\langle \mathcal{V}_{f,\overline{g}}, P_h \rangle}{h^w},
\begin{dcases}-\int_{0}^{T}\overline{e^{\lambda^p_k (T-t)}}p(t)\,dt = m_{1,k}, \forall k\geq k_0, -\int_{0}^{T}\overline{e^{\widehat \lambda_{n_j}(T-t)}}p(t)\,dt = m_{1,j}, \forall \, 1\leq j\leq j_0,\end{dcases}
\left(\begin{pmatrix}-a_{11}0 a_{13}/2 0 0 a_{23}/2 a_{13}/2 a_{23}/2 a_{33}\end{pmatrix},\begin{pmatrix}0 0 b_{13}/2 0 b_{22}b_{23}/2 b_{13}/2 b_{23}/2 b_{33}\end{pmatrix}\right)
\sup_{\| \mathbf{c}\|_2^2 = 1}\frac{1}{k}\sum_{i,j \leq k}\frac{(n_i, n_j)^{2\alpha}}{(n_i n_j)^{\alpha}}\cdot c(n_i) \overline{c(n_j)}\leq \Big ( \frac{6 e^{2\gamma}}{\pi^2}+ o(1) \Big )\cdot (\log\log k)^2.
\textstyle M_{r+p-2}\textstyle = \left(-M_{r+p-3}' - \cdots - M_r' + \frac{1}{1}+ \cdots + \frac{1}{s-1}\right)p + O(p^2) \textstyle \textstyle = \left(-M_{r+p-3}' - \cdots - M_r' -\frac{1}{s}- \cdots - \frac{1}{p-1}\right)p + O(p^2).
\mathbb{E}(i \theta L'(z )) = \exp(K(\theta, L'(z))), K(\theta, L'(z)) = i \theta a(z) - \frac{1}{2}\theta^2 b(z) + \int_{\mathbb{R}}\left(e^{i\theta x}-1-i\theta x \mathbb{I}_{[-1,1]}(x)\right) \nu(dx, z).
s'\delta s(\mu_{-\underline{m}})\left(\begin{array}{cc}\ast \ast R S\end{array}\right)=-2\binom{2M}{k_{\rm id}-2}\frac{t_M(\mu_{-\underline{m}})^\ast\left(\left|\begin{array}{cc}\bar S -\bar R x y\end{array}\right|^{k_c-1}\left|\begin{array}{cc}R S x y\end{array}\right|^{k_{\rm id}-1}\right)}{\Delta^{\frac{\underline{k}}{2}-1}\cdot \left(\left|R\right|^2+\left|S\right|^2\right)},
\hat{V}^{C}_{\sigma'}(C_1,C_2) = \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\! \hat{f_{\sigma}}_{XYZS}(x,y,z,s) \, \mathrm{d}u_{1}\mathrm{d}u_{2}\Big|_{x=y=u_{1}, z=s=u_{2}}
j \cdot \left( \sum^{3}_{i=0}b_{i,j-i}\delta^{i,j-i}_{-j,\mu,\pm}+ \sum^{M+3}_{j' > j}\left( \sum^{3}_{i=0}b_{i,j'-i}\delta^{i,j'-i}_{-j,\mu,\pm}\right) \right) \in \mathcal{O}
\sum_{n=-\infty}^{\infty}\omega^{2n}q^{n(n-1)}\left[\varphi_{k}\left(\omega q^{n}\right)\varphi_{\ell}\left(\omega q^{n}\right)+\varphi_{k}\left(-\omega^{-1}q^{-n+1}\right)\varphi_{\ell}\left(-\omega^{-1}q^{-n+1}\right)\right] =2\|\varphi\left(\omega\right)\!\|^{2}\delta_{k,\ell}
L^{( 0 )}\rightarrow L^{( N )}= \sum_{k = 0}^{N}\int \, d x \, ( \dot{\phi_{k}}\phi_{k}^{\prime}- \phi_{k}^{2}) + \sum_{m = k + 1}^{N}\, \sum_{k = 0}^{N - 1}\int \, d x \, ( \dot{\phi_{k}}- \phi_{k}^{\prime}) \phi_{m}^{\prime}
\mathcal{L}_\sigma(u,v;x) = \theta(u,v) + \langle x, \mathcal{A}^*u + \mathcal{B}^*v - c\rangle +\frac{\sigma}{2}\|\mathcal{A}^*u + \mathcal{B}^*v -c\|^2, (u,v)\in\mathcal{U}\times\mathcal{V}.
\frac{1}{w}= \frac{1}{\sqrt{\left( 1 - \frac{2m}{r}\right)^{-1}- |\bar{\nabla}f|^2}}= 1 - \frac{m}{r}+ \frac{1}{2}(p^2 A + |\tilde{\nabla}A|^2 ) r^{2p-2}+ o(r^{2p-2}).
Z(s,f,\chi)=q^{-1}(1-q^{-1})+\frac{q^{-3-4s}(1-q^{-1})}{(1-q^{-2-4s})}+\frac{q^{-2-4s}(1-q^{-1})^2}{(1-q^{-2-4s})}+\frac{q^{-7-16s}(1-q^{-1})^2}{(1-q^{-2-4s})(1-q^{-5-12s})}.
\left\langle \psi_1^n T_a, T_j, \tau^{(k)}\right\rangle^\textnormal{coh}_{0,k+2,d}= \sum_{\substack{u+v=k x+y=u}}\frac{k!}{v!x!y!}(\tau_2(d))^x \left\langle \psi_1^{n-y}(T_a \cup \tau_2^y), T_j, \tau'^{(v)}\right\rangle^\textnormal{coh}_{0,v+2,d}
\begin{array}{c}{{\Bigl ( a^{+}( P^{0}- P_{2}) + a^{-}\left( i P_{1}- P_{3}\right) \Bigr ) | \psi \rangle = 0 ,}}\\{{\Bigl ( a^{-}( P^{0}+ P_{2}) - a^{+}\left( i P_{1}+ P_{3}\right) \Bigr ) | \psi \rangle = 0 .}}\\ \end{array}
{\cal L}= \frac{1}{2}\left\{\partial_{m}\phi_{i}\partial^{m}\phi_{i}+ \imath \bar{\psi}_{i}\gamma^{m}\partial_{m}\psi_{i}- \frac{\partial{\cal W}}{\partial \phi_{i}}\frac{\partial{\cal W}}{\partial \phi_{i}}- \frac{\partial^{2}{\cal W}}{\partial \phi_{i}\partial \phi_{j}}\bar{\psi}_{i}\psi_{j}\right\}.
(Z_{1/\Omega}\circ H)g(t+Tp,\nu) & = \Omega\,\sum_{q=0}^{L-1}\sum_{m=0}^{L-1}c_{-(q-p)}\,e^{-2\pi i m (q-p)/L}& \hskip.5in \eta^{QP}_H(t+qT,\nu+m\Omega)\,e^{-2\pi i \nu (q-p)T}.
|S_K(t)| \ \le \ \left(\frac{[K:\mathbb{Q}]}{4}+o(1) \right) \frac{\log \mathrm{rd}_K(|t|\!+\!3)}{\log \log \mathrm{rd}_K(|t|\!+\!3)}\underset{s=\frac12}{\mathrm{ord}}\ \zeta_K(s)\ \le \ \left(\frac{1}{2}+o(1) \right) \frac{\log(|d_K|\!+\!3)}{\log \log (|d_K|\!+\!3)},
\begin{array}{rcl}\gamma_+ \psi = (1\!-\!a) \mathcal{R^*}\!\left[\gamma_- \psi\right] begin{align*}5pt] \; \gamma_+ \psi(t,x,v) = (1\!-\!a) \gamma_- \psi(t,x,R_x v) \forall\; (t,x,v)\in\Sigma_{+}^{T}, \end{array}\iffalse\gamma_+ \psi =0,\ \forall\ (t,x,v)\in \Sigma_+^T\fi
B(X+Y)^n=\left(\sum\limits_{i=0}^{m}\partial_iB(X)Y^i\right)^n=B(X)^n+\sum\limits_{b=1}^{mn}\left(\sum\limits_{j_1+\dots+j_n=b}\prod\limits_{t=1}^{n}\partial_{j_t}B(X)\right)Y^{b},
\mu(\sigma^{-i}S_k(r_n)\cap \sigma^{-j}S_l(r_n)) \leq B^{10}\sum_{\substack{A,B\in\mathcal{C}_r C\in\mathcal{C}_{r_n-r}}}\mu(A)^2\mu(B)^2\mu(C)^3=C^{10}Z_{r}(1)^2Z_{r_n-r}(2) &\preceq B^{10}e^{-4\alpha r_n}e^{-3\alpha(r_n-r)}\leq B^{10}e^{-3\alpha r_n}.
\Gamma\lbrack A\rbrack =\! -2\! \int_0^{\infty}{dT\over T}{\rm e}^{-m^2T}\int{\cal D}x\,{\cal D}\psi\,{\rm exp}\left[ -\int_0^T \!\!\!d\tau\left({1\over 4}{\dot x}^2\! +\!{1\over2}\psi\dot\psi\!+\!ieA_{\mu}\dot x^{\mu}\!-\!ie\psi^{\mu}F_{\mu\nu}\psi^{\nu}\right)\right]
\mathcal{A}_{1}:=\nabla_{x}\cdot\left(-d_{i}\left(y\right)\nabla_{y}\right)+\nabla_{y}\cdot\left(-d_{i}\left(y\right)\nabla_{x}\right), \mathcal{A}_{2}:=\nabla_{x}\cdot\left(-d_{i}\left(y\right)\nabla_{x}\right).
\begin{aligned}(\Delta^h_k(\partial_xu\partial_yU)_\phi, \Delta^h_k\partial_yu_\phi)_{L^2}\lesssim d_k^22^{-2ks}||\partial_yu_\phi||_{\tilde L^2_{t,\theta'(t)}(B^{s+\frac 12})}||u_\phi||_{\tilde L^2_{t, \theta'(t)}(B^{s+\frac 12})}\lesssim \frac{1}{\mathcal R}d_k^22^{-2ks}||\partial_yu_\phi||^2_{\tilde L^2_{t,\theta'(t)}(B^{s+\frac 12})}\end{aligned}
\inf_{x\in X}\sup_{\psi\in\Psi}f(x) + \psi(Lx) -g^*_\Psi (\psi) = \inf_{x\in X}f(x) + \sup_{\psi\in\Psi}\psi(Lx) - g^*_\Psi (\psi) & = \inf_{x\in X}f(x) + g^{**}_\Psi (Lx) & \leq \inf_{x\in X}f(x) + g (Lx).
\left|\int_{S_{0,r}}\rho\right|=\left|\int_{\Sigma_0\cap\{|x|\leq r\}}\mathrm{div}E\right|\leq\left(\int_{\Sigma_0}|\phi|^2\right)^{1/2}\left(\int_{\Sigma_0}|D\phi|^2\right)^{1/2}\leq \epsilon.
|\tilde{a}_{K,\alpha}(u,v)| \leqslant|a_{K,\alpha}(u,v)|+|\tilde{a}_{K,\alpha}(u,v)-a_{K,\alpha}(u,v)| & \leqslant\hat{C}_{c}\Vert u\Vert_{H^{-\alpha/2}\left( \Omega\right)}\Vert v\Vert_{H^{-\alpha/2}\left( \Omega\right)}\qquad\hat{C}_{c}:=E_{K,\alpha}^{\max}h^{-2-\alpha}+C_{c}.
{}^{M\!L\!}{\Phi}_{p}(b;c;z)= \sum_{n=0}^{\infty}\frac{{}^{M\!L\!\!}B_{p}(b+n,\;c-b)}{B(b,\;c-b)}\frac{z^n}{n!}(p\geq0;\Re(c)>Re(b)>0,\Re(\alpha)>0)
\tilde{\Psi}_{(\mathcal{V},\mathcal{E})}(X) :=\left(\begin{array}{cl}\Psi_{(\mathcal{V},\mathcal{E})}(A) \Psi_{(\mathcal{V},\mathcal{E})}(B^T) \Psi_{(\mathcal{V},\mathcal{E})}(B) \Psi_{(\mathcal{V},\mathcal{E})}(C) \end{array}\right),
\int_{Q}B_{{l}(Q)}(f,g)^{t}vdx \leq\left(\int_{Q}B_{l(Q)}(f,g)(x)dx\right)^{t}\left(\int_{Q}v^{\frac{1}{1-t}}dx\right)^{1-t}=\left(\int_{Q}\int_{|y|_{\infty}\leq l(Q)}f(x-y)g(x+y)dydx\right)^{t}\left(\int_{Q}v^{\frac{1}{1-t}}dx\right)^{1-t}.
P_{0, 4}(\lambda; \tau, \eta) S^-(\tau, \eta) P_{0, 4}(\lambda; - 1/\tau, \eta/\tau)^{-1}= 4 \sqrt{2}\pi i\tau \exp\Big(\pi i \frac{4 + 216\eta^2 - 42 \eta(\tau - 1) + 3\tau + 4\tau^2}{12\tau}\Big).
\{f(1),\ldots,f(n+1)\}= \left\{\left(-\langle\mu,\varpi_1\rangle+\sum_{j=1}^{i-1}\left\langle\frac{\lambda+\mu}{2}+\rho,\alpha_j\right\rangle\right)^{}\ \middle\vert\ 1\leq i \leq n+1\right\}
U^{a_{1}b_{1}}U^{a_{2}b_{2}}U^{a_{3}b_{3}}f_{\tau \alpha , \tau_{1}\alpha_{1}, \tau_{2}\alpha_{2}, \tau_{3}\alpha_{3}, b_{1}, b_{2}, b_{3}}\Psi = f_{\tau \alpha , \tau_{1}\alpha_{1}, \tau_{2}\alpha_{2}, \tau_{3}\alpha_{3}, a_{1}, a_{2}, a_{3}}\Psi ,
M ( p ) = \frac{e^{2}}{\pi}m \left[ \frac 3 4 \ln \left( 1 + \frac{L^{\prime \, 2}}{m^{2}}\right) + \frac{L^{\prime}}{2 m}\operatorname{a r c t a n}\frac m{L^{\prime}}- \frac 1 8 \frac{L^{\prime \, 2}/ m^{2}}{1 + L^{\prime \, 2}/ m^{2}}\right] .
\begin{array}{rcl}\Psi_{c}\left(V_{c}\right) = \mathbb{E}^{\mathbb{P}}\left[P_{c}\left(V_{c},\Pi\right)\right]-\frac{\lambda_{p}}{2}^{\mathbb{P}}\left[P_{c}\left(V_{c},\Pi\right)\right] = -\mathbb{E}^{\mathbb{P}}\left[\Pi\right]^{\top}V_{c}-\frac{\lambda}{2}V_{c}^{\top}Q_{c}V_{c},\end{array}
\alpha(\mu_r)x =\frac1r\int^r_0\alpha(\sigma_s)xds &=\frac1r\int^r_0\int_{S^n_s}\alpha(g)xd\sigma_s(g)ds &=\int_{S^n_1}\frac1r\int^r_0\alpha(sv)xdsd\sigma_1(v).
J_{k}={\frac{1}{2}}\epsilon_{k i j}\int d^{3}\vec{x}{\cal M}_{0 i j}= \epsilon_{k i j}x_{i}R_{j}+ \epsilon_{k i j}\int d^{3}\vec{y}( y_{i}- x_{i}) \Theta_{0 j}( \vec{y}, t ) ~ ,
\begin{aligned}\mathcal{E}(\alpha, \beta, \gamma, \delta, \rho, \eta) := \left((\alpha -1)(\alpha -3)\cdots 1\right)\left((\beta -1)(\beta -3)\cdots 1\right)\left((\gamma -1)(\gamma -3)\cdots 1\right)\times \left((\delta -1)(\delta -3)\cdots 1\right)\left((\rho -1)(\rho -3)\cdots 1\right)\left((\eta -1)(\eta -3)\cdots 1\right).\end{aligned}
\nabla\times(\epsilon^{-1}\nabla\times\overline{\widetilde{H_{\eta,\ell}}})-k^{2}\mu\overline{\widetilde{H_{\eta,\ell}}}+\nabla\times((\epsilon^{-1}(x)-\epsilon_{0}^{-1}(x))\nabla\times\overline{H_{\eta,\ell}})=0.
\Delta_f(a_1 \cdots a_n) := \sum_{S \subseteq [n]}\sum\limits_{(\pi^1,\pi^2)\in ANC_n^S}\alpha_{\pi^1,\pi^2}\Big(\prod_{\pi_j\in\pi^1}a_{\pi_j}\Big)\otimes \Big(\prod_{\pi_k\in\pi^2}a_{\pi_k}\Big).
J_G(r) - J'_G(t,r,\xi)\xrightarrow[t\to \infty]{L^2(\Pi_x^{(\phi)})}\int_0^1 \frac{\gamma_0}{u}G\big( ru^{\frac{1}{\gamma_0 - 1}}\big)^{\gamma_0 - 1}\big\langle 1- C_X^{-1}\mathbf 1_{\gamma(\cdot) = \gamma_0}\kappa \phi^{\gamma_0 - 1}, \phi\phi^*\big\rangle_m du=0.
h_0(\varsigma,k) = \frac{1}{(2\pi)^2}\int_{\mathbb C}e^{-i\overline{\varsigma}z/2}Q_0(z)\overline{\psi}(z,k)d\sigma_z, ~ \varsigma \in \mathbb C,~ k \in \mathbb C\backslash \mathcal E.
\frac{\int_0^L \frac{|v'|^2 V_{\alpha}}{|\alpha'|}+ \frac{k^2 |v|^2 | \alpha'|}{V_{\alpha}}\,dt}{\int_0^L |v|^2 V_\alpha |\alpha'| \,dt}= \max_{w \in W_p}\frac{\int_0^L \frac{|w'|^2 V_{\alpha}}{|\alpha'|}+ \frac{k^2 |w|^2 | \alpha'|}{V_{\alpha}}\,dt}{\int_0^L |w|^2 V_\alpha |\alpha'| \,dt}
\overline F_s / \mathfrak S_{M(\lambda)}= \left(\coprod_{\theta \in \Theta_{\lambda}}\overline F_s^{\theta}\right) / \mathfrak S_{M(\lambda)}\cong\coprod_{\theta \in \Theta_{\lambda}^{st}}\overline F_s^{\theta}/ (\mathfrak S_{M(\lambda)}\cap \mathfrak S_{M(\theta)}).
W_{0}^{( 1 )}( \pi ) = \left( \frac{3}{2}\right)^{2}\left[ - u_{1}J_{1}( \pi ) - u_{2}M_{1}( \pi ) + M_{0}( \pi ) + \int J_{1}M_{2}( \pi ) d u_{2}+ \int M_{1}J_{2}( \pi ) d u_{2}\right] .
\langle \omega Y,h(Z,W)\rangle \omega X-\langle \omega X,h(Z,W)\rangle\omega Y+\langle \omega Z,h(X,W)\rangle \omega Y & -\langle \omega Y,h(X,W)\rangle \omega Z+\langle \omega X,h(Y,W)\rangle\omega Z-\langle \omega Z,h(Y,W)\rangle \omega X=0
d s^{2}= - ( 1 -{\frac{\lambda}{r^{d - 2}}}+{\frac{r^{2}}{l^{2}}}) d t^{2}+ ( 1 -{\frac{\lambda}{r^{d - 2}}}+{\frac{r^{2}}{l^{2}}})^{- 1}d r^{2}+ r^{2}d \Omega_{d - 1}^{2},
\mathrm{c^{sm}}(\Omega_1) =[V_{\tau_2}][V_{\tau_3}], \mathrm{c^{sm}}(\Omega_2) =[V_{\tau_3}][V_{\tau_4}]+h[V_{\tau_3}], \mathrm{c^{sm}}(\Omega_3) =[V_{\tau_1}][V_{\tau_2}]+h[V_{\tau_2}], \mathrm{c^{sm}}(\Omega_4) =[V_{\tau_1}][V_{\tau_4}]+h[V_{\tau_1}]+h[V_{\tau_4}]+h^2.
p^{\gamma, \delta}_{\alpha, \beta}(n,r) = \int_{-1}^{1}h_{\gamma, \delta}(n, r \, | \, \rho) \pi_{\alpha, \beta}(\rho) \rho = \int_{-1}^{1}A_{\gamma, \delta}(n, r \, | \, \rho) \pi_{\alpha, \beta}(\rho) \rho ,
2 \Phi_C(\xi_n) = d\Phi_C(\xi_n)[\xi_n] = \lambda_n d\mathbb{A}^*(\xi_n)[\xi_n] + Z_n(\xi_n) = 2 \lambda_n \mathbb{A}^*(\xi_n) + o(1) = 2 \lambda_n + o(1),
\frac{\big(A_{\rm sym}\xi\big)\cdot(\xi_2,-\xi_1)}{\big(A_{\rm sym}\xi\big)\cdot\xi}\Big|_{\xi=(\cos\theta,\,\sin\theta)}=-\frac{f'(\theta)}{2f(\theta)},\qquad\forall\,\theta\in(0,2\pi).
X=\begin{bmatrix}a b -b a\end{bmatrix}\in \mathbb{R}^{2 \times 2},v=\begin{bmatrix}c -d\end{bmatrix}\in \mathbb{R}^2,Xv=\begin{bmatrix}ac-bd -ad-bc\end{bmatrix}\in \mathbb{R}^2.
\{y_{i,j}: (i,j) \in \mathrm{Grid}(mr,n) \mbox{and}i+j \mbox{is even}\}= \{1,2,\ldots,\tfrac{1}{2}mnr+\tfrac{1}{2}\}, \mbox{and}\{y_{i,j}: (i,j) \in \mathrm{Grid}(mr,n) \mbox{and}i+j \mbox{is odd}\}= \{MN- \tfrac{1}{2}mnr + \tfrac{3}{2}, MN- \tfrac{1}{2}mnr + \tfrac{5}{2}, \allowbreak \ldots,MN\}.
\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 ) ,
\partial_r\partial_\mu\sigma =-f'_0(r)\int_0^r(\phi_{T_1}+\partial_\mu\sigma-\frac{r^2}{2})(Q_\mu-Q)f_1\tau d\tau &+f'_1(r)\int_0^r(\phi_{T_1}+\partial_\mu\sigma-\frac{r^2}{2})(Q_\mu-Q)f_0\tau d\tau.
|a_n|= \frac{f_n}{c_E- n^2}\lesssim \frac{1}{n^2}\sqrt{\delta E(0)}, \; \forall n\in \mathbb{Z}\setminus\{0, \pm 1\}, |a_0|= \left|\frac{f_0}{c_E}\right|\lesssim \sqrt{\frac{\delta}{E(0)}}\; \textrm{and}\; |a_{\pm 1}|= \left|\frac{2\pi f_{\pm 1}}{E(0)- 2\pi}\right| \lesssim \frac{\sqrt{\delta E(0)}}{2\pi- E(0)}.
\int_0^\infty K(x,x_1,x_2)h(x)\,dx=\frac 1{x_1x_2}\int_0^\infty\int_0^\infty h(x_1-y_1+y_2) B\left(\frac{y_1}{x_1}\right)B\left(\frac{y_2}{x_2}\right)\,dy_1\,dy_2,
\varphi(x^{k+1};\rho^\infty) - \varphi(x^{K_\rho};\rho^\infty) =\sum_{t=K_\rho}^k\Big(\varphi(x^{t+1};\rho^\infty) - \varphi(x^t;\rho^\infty)\Big) \leq c_{}\sum_{t=K_\rho}^k\Big(m^t(x^{t}+d^t;\rho^\infty) - m^t(x^t;\rho^\infty)\Big)
\psi ( x_{1}) = ( 2 \pi )^{- 3 / 2}\int d^{3}p_{1}( m / E_{1})^{1 / 2}[ b_{p_{1}r_{1}}u^{r_{1}}( p_{1}) e^{i x_{1}. p_{1}}+ d_{p_{1}r_{1}}^{*}v^{r_{1}}( p_{1}) e^{- i x_{1}. p_{1}}] ;
\begin{cases}u(x,0)=u_{1_0}-u_{2_0},\,\,\,x\in\Omega, \phi(x,0)=\phi_{1_0}-\phi_{2_0},\,\,\,x\in\Omega, \phi(x,0)=\theta_{1_0}-\theta_{1_0},\,\,\,x\in\Gamma.\end{cases}
|u_k|^{\alpha(x)}|v_k|^{\beta(x)}- |\tilde{u}_{k}|^{\alpha(x)}|\tilde{v}_{k}|^{\beta(x)}= \big[|u_k|^{\alpha(x)}- |\tilde{u}_{k}|^{\alpha(x)}]|v_k|^{\beta(x)}+ |\tilde{u}_{k}|^{\alpha(x)}[|v_k|^{\beta(x)}-|\tilde{v}_{k}|^{\beta(x)}\big]
{\cal E}\simeq - \frac{\pi \Delta u}{u_{b}}( \kappa^{2}u_{b}^{2}) \left\{1 - \frac{2 \sqrt{2}}{\pi \kappa u_{b}}\tanh \left( \frac{\pi \kappa u_{b}}{\sqrt{2}}\right) \left[ 1 - \frac{1}{3}\tanh^{2}\left( \frac{\pi \kappa u_{b}}{\sqrt{2}}\right) \right] \right\}.
2x^2-2\big(n_{{i_0}-1,{j}}+n_{{i_0}+1,{j}}\big)x+n_{{i_0}-1,{j}}^2+n_{{i_0}+1,{j}}^2+\sum\limits_{i\in I\setminus\{{i_0}-1,{i_0}\}}\big(n_{i,{j}}-n_{i+1,{j}}\big)^{2}-2n_{jj}=0.
\tilde{C_{2}}=2\alpha_{1}\frac{C_{2}-2\int_{X_{2}}\omega_{\Sigma_{1}}\wedge\omega_{\Sigma_{2}}\wedge d\tau\wedge d\theta}{(2\pi)^{2}\frac{C_{2}}{6}}&=\frac{12\alpha_{1}}{(2\pi)^{2}}(1-\frac{1}{13}).
\mathcal O_{X,p}=\mathcal O_{Y_0,p}\overset{\sigma_1}{\rightarrow}\mathcal O_{Y_1,p}=\mathcal O_{Y_0,p}/(a_1)\overset{\sigma_2}{\rightarrow}\cdots \overset{\sigma_{d-1}}{\rightarrow}\mathcal O_{Y_{d-1},p}=\mathcal O_{Y_{d-2},p}/(a_{d-1}).
\sup_{x\in\mathbb{T}}\rho_{\epsilon}(t,\,x) \le\sqrt{\int_{\mathbb{T}}\frac{(\nabla\rho_{\epsilon})^{2}}{\rho_{\epsilon}}(t,\,z)dz}+1\le\sqrt{\int_{\mathbb{T}}\left(\frac{(\nabla\rho)^{2}}{\rho}\right)_{\epsilon}(t,\,z)dz}+1 =\sqrt{\int_{\mathbb{T}}\frac{(\nabla\rho)^{2}}{\rho}(t,\,z)dz}+1.
\Pr[X_1=1|\mathcal{A}]=\Pr[X_2=1|\mathcal{A}]=p^2(1-p)+p^3=\frac{1}{2}, &\Pr[X_1=X_2|\mathcal{A}]=\Pr[X_1=1|\mathcal{A}]\Pr[X_2=1|\mathcal{A}]+\Pr[X_1=0|\mathcal{A}]\Pr[X_2=0|\mathcal{A}]=\frac{1}{2}\times\frac{1}{2}+\frac{1}{2}\times\frac{1}{2}=\frac{1}{2}.
\sum_{x\in G\setminus B_{1}(x_0)}e^{- \Lambda d^\alpha(x,x_0)}\mu(x) = \sum_{n = 1}^{\infty}\Big(\sum_{\{x:\,d(x,x_0) = n\}}e^{- \Lambda d^\alpha(x,x_0)}\mu(x)\Big) \qquad= \sum_{n = 1}^{\infty}e^{- \Lambda n^\alpha}\mu(\{x:\,d(x,x_0) = n\}) = \sum_{n = 1}^{\infty}b^ne^{-\Lambda n^\alpha},
\sum_{i=0}^{l_1(x)-1}(f- f \circ \sigma_B)(\sigma_B^i(h(x))) = f(h(x)) - f(\sigma_B^{l_1(x)}(h(x))), \sum_{j=0}^{k_1(x)-1}(f- f \circ \sigma_B)(\sigma_B^j(h(\sigma_A(x)))) =f(h(\sigma_A(x))) - f(\sigma_B^{k_1(x)}(h(\sigma_A(x)))),
\begin{array}{c}\frac{du(x,t)}{dt}=f\left(u(x,t),\,v(x,t)\right)+L_{1,x}(u(x,t),\,v(x,t)) \frac{dv(x,t)}{dt}=\frac{1}{\epsilon}g\left(u(x,t),\,v(x,t)\right)+L_{2,x}(u(x,t),\,v(x,t))\end{array}