harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
| y_n(r) -y_n(s) | = \left| \int_s^r \partial y_n(t) \; t \right|\leq \int_s^r \left| \partial y_n(t) \right| \; t = \int_0^T \mathbb{I}_{[s,r]}(t) \left| \partial y_n(t) \right| \; t \leq \| \mathbb{I}_{[s,r]}\|_{L^2}\left\| \partial y_n(t) \right\|_{L^2}\leq M \sqrt{|r-s|}.
\frac{\partial}{\partial \nu}\,J_\nu(z) = J_\nu(z)\,\ln(z/2)+(z/2)^\nu\,\sum_{m=0}^\infty \mathcal{G}^{(1)}(\nu\! +\! 1\! +\! m)\,\frac{(-\,z^2/4)^m}{m!},
M_1(f_1, f_2, g)(\xi)=\sum_{\omega}\int_{\substack{x_1< x_2, x_1, x_2 \in \omega_L x_3 \in \omega_R}}\hat{f}_1(x_1) \hat{f}_2(x_2) g(x_3) e^{2 \pi i \xi (x_1+x_2+x_3)}d x_1 d x_2 d x_3
\frac{d}{dt}\left(\int_0^t (t-z)^{\alpha-1}E_{\alpha, \alpha}[\lambda(t-z)^{\alpha}]f(z)dz\right) = \int_0^t (t-z)^{\alpha - 2}E_{\alpha, \alpha-1}(\lambda (t-z)^{\alpha})f(z)dz.
\int_{\mathbb{R}}x^k\d\mu_{A_N+U_NB_NU_N^*}(x) &=\tau_{A_N,U_NB_NU_N^*}((X+Y)^k)=\tau((X+Y)^k)+o(1)=\int_{\mathbb{R}}x^k\d\mu_1\boxplus\mu_2(x)
\partial_{\bar{z}}r=i\partial_{\bar{z}}\theta(z) we^{-i\theta(z)}-i\partial_{\bar{z}}\theta(z)\overline{w}e^{i\theta(z)}+\eta'(\theta(z))\partial_{\bar{z}}\theta(z) =\eta'(\theta(z))\partial_{\bar{z}}\theta(z)\neq 0
\sum_{k=0}^{r}\frac{1}{2j+2k+1}\left(\sum_{m=0}^{2j+2k}\binom{2j+2k+1}{m}((r+1)^{2j+2k+1-m}-1)\sum_{\ell=0}^{m}\frac{(-1)^{\ell}\ell!}{\ell+1}S(m,\ell)\right. \times\left.\sum_{i=0}^{2k+2}(-1)^{r+1+i}s(r+1,i)s(r+1,2k+2-i)\right)=0.
G_{\mu \rho}^{a c}( \theta ) \, M_{c b}^{\rho \nu}( \theta ) \; = \; \delta_{\mu}^{\nu}\delta_{b}^{a}\; \; , \; \; M_{a b}^{\mu \nu}\, = \, g^{\mu \nu}\delta_{a b}+ g^{- \frac{1}{2}}\epsilon^{\mu \lambda \nu}f^{a c b}\theta_{\lambda}^{c}\; .
J(\chi_8,\chi_8^6)=\chi_8(-1)J(\chi_8,\chi_8), &J(\chi_8,\chi_8^5)=\chi_8(-1)J(\chi_8,\chi_8^2), &J(\chi_8^5,\chi_8^6)=\chi_8(-1)\overline{J(\chi_8,\chi_8)}.
d s^{2}\equiv g_{\mu \nu}^{\mathrm{G B I}}d x^{\mu}d x^{\nu}= - b^{2}( t ) d t^{2}+{a_{1}^{2}}( t ) d x^{2}+ a_{2}^{2}( t ) d y^{2}+ a_{3}^{2}( t ) d z^{2},
\displaystyle E_{H\vert\xi^{c}}\left[Q\left(\sqrt{g\frac{\chi_{1}}{N_{0}}}\right)\right]= \int_{\tau}^{\infty}Q \left(\sqrt{g\frac{x_{1}}{N_{0}}}\right) \dfrac{f_{\chi_1}(x_1)}{P(\xi^{c})}dx_1.
J_{\lambda,\mu}(u,v) :=\frac{1}{p}\int_Q\frac{|u(x)-u(y)|^p}{|x-y|^{n+ps}}\,dx\,dy+\frac{1}{p}\int_Q\frac{|v(x)-v(y)|^p}{|x-y|^{n+ps}}\,dx\,dy &-\frac{1}{q}\int_\Omega \left(\lambda |u|^{q}+ \mu |v|^{q}\right)dx-\frac{2}{\alpha+\beta}\int_\Omega|u|^{\alpha}|v|^\beta dx.
\begin{array}{l}{{\delta_{b}x^{\mu}={b^{\mu}}_{\nu}n^{\nu},}}\\{{\delta_{b}A_{a}^{\mu}= -{b^{\mu}}_{\nu}\displaystyle \left( \varepsilon_{a b}\frac{g^{b c}}{\sqrt{- g}}{\Pi_{c}}^{\nu}- \xi_{a}n^{\nu}+ i ( \bar{\theta}\Gamma^{\nu}\partial_{a}\theta ) \right) .}}\\ \end{array}
\frac{(1-\epsilon_1-\epsilon_2)(T/2)^{-\beta}}{\Gamma(1-\beta)}\int_0^T \|u(t)\|^2 \,dt+\int_0^T|u(t)|_1\,dt \leq \frac12\epsilon_1^{-1}\int_0^T (g_{\beta,T}(t))^{-1}\|f(t)\|^2 \,dt +\frac12\epsilon_2^{-1}\frac{T^{1-\beta}}{\Gamma(2-\beta)}\|u^0\|^2.
\mathbf{J}_\nu(\, \underline{u}\,(\underline{v}+\underline{w})^\intercal ) = \sum_{s=-\infty}^\infty \sum_{t=0}^\infty \frac{(-1)^t2^{-2t-s}}{t!(t+s)!}D_{\underline{u}}^{2t+s}\mathbf{J}_{\nu-s}(\, \underline{u}\,\underline{v}^\intercal ) D_{\underline{w}}^{2t+s}.
\big|K_N(x,y)\big|\lesssim \begin{cases}N^{d-2\sigma}\left(|x||y|\right)^{-\sigma}, \qquad\qquad |x|,|y|\leq N^{-1}, N^d(N|x|)^{-\sigma}(N|y|)^{-M}, 2|x|\leq N^{-1}\leq|y|, N^d(N|y|)^{-\sigma}(N|x|)^{-M}, 2|y|\leq N^{-1}\leq|x|, N^{d-2}(|x|+|y|)^{-2}e^{-cN^2|x-y|^2}, |x|,|y|\geq \frac12N^{-1},\end{cases}
R_{3,0}= R_{3,12}= 8131531262400 = 2^6 \cdot 3^2 \cdot 5^2 \cdot 7^2 \cdot 13^2 \cdot 19 \cdot 37 \cdot 97, R_{5,0}= R_{5,12}= 17072929032886039622400 = 2^8 \cdot 3^5 \cdot 5^2 \cdot 7^2 \cdot 13^2 \cdot 17 \cdot 31^2 \cdot 37 \cdot 61 \cdot 157 \cdot 229, R_{3,4}= R_{3,8}= 9815256000 = 2^6 \cdot 3^8 \cdot 5^3 \cdot 11 \cdot 17.
\mathcal{A}^{(i)}= \left\{{\bf P}^{(i)}=(P^{(i)}_{1},\dots,P^{(i)}_{n_i})|P^{(i)}_l \in \{p_1^{(i)},\dots,p_m^{(i)}\},\right. \left. \sum_{l=1}^{n_i}\pi^{(i)}(l)P^{(i)}_l \leq \overline{P}_i \right\},
\partial_t\Big[\big(Q^{(r)}_t\ast f\big)_\gamma\Big] = \partial_t\Big[\big(Q^{(r)}_{\gamma\alpha}\big)_t\ast f_\alpha\Big]=\partial_r\Big[\big(Q^{(n)}_{\gamma\alpha}\big)_t\ast f_\alpha\Big] begin{align*}4pt] =\big(Q^{(n)}_{\gamma\alpha}\big)_t\ast (\partial_r f_\alpha)=\big(Q^{(n)}_t\ast (\partial_r f)\big)_\gamma\quad\mbox{in}\,\,{\mathbb{R}}^{n-1}.
\pi_i = \frac{|\{e_j:\ (i-1)M<j\leq iM, e_j=+1\}|}{M}= \frac{1}{M}\sum_{j=(i-1)M+1}^{iM}\frac{1}{2}(e_j+1) =\frac{1}{2M}\sum_{j=(i-1)M+1}^{iM}e_j+\frac{1}{2}
\frac{\partial}{\partial E}\phi_{-L}(n,\omega,E)=\frac{\sin^2\phi_{-L}(n,\omega,E)}{\cos^2\phi_{-L}(n-1,\omega,E)}\frac{\partial}{\partial E}\phi_{-L}(n-1,\omega,E)-\sin^2\phi_{-L}(n,\omega, E)
S_{\mathrm{D 6}}= - T_{\mathrm{K K 1 1}}\int d^{7}\hat{\xi}e^{- \hat{\phi}}\sqrt{| \mathrm{d e t}\ ( \hat{g}_{\hat{\imath}\hat{\jmath}}+{\hat{\cal F}}_{\hat{\imath}\hat{\jmath}}) |}+ \mathrm{W Z}\, ,
\widehat L^{\mu \nu}( q ) = \left( \eta^{\mu \nu}q^{2}- q^{\mu}q^{\nu}\right) \left[ \left. \hat{h}\right|_{q^{2}= 0}+ \left( \hat{h}( q^{2}) - \left. \hat{h}\right|_{q^{2}= 0}\right) \right]
\partial_\lambda\Lambda(y,\lambda) = \frac{y c}{2\pi}\sum_{n\geq 1}\dfrac{y^2 J_{n+1}(y,-\lambda y)}{n}=-\frac{y c}{2\pi}\sum_{n\geq 1}\left (J_n-J_{n-1}+\dfrac{\lambda y}{n}J_n\right ) &=\frac{y c}{2\pi}\left (J_0(y,-\lambda y)-\lambda y\sum_{n\geq 1}\dfrac{J_{n}(y,-\lambda y)}{n}\right ),
\hat{M}_{a}^{sG}(\lambda )\equiv (-1)^{\mathsf{N}}\,\sigma_{a}^{y}\,\left(M_{a}^{sG}(1/\lambda )\right)^{t_{0}}\,\sigma_{a}^{y}=\left( -\sigma_{a}^{x}\right)^{\mathsf{x}}\hat{M}_{a}(\lambda ),
v_i(x+y)^{s+r-2-2i}=\left(\sum_{k=0}^i\nu_k x^{i-k}y^k\right) \left(\sum_{\ell=0}^{s+r-2-2i}\binom{s+r-2-2i}{\ell}x^\ell y^{s+r-2-2i-\ell}\right) =\sum_{k=0}^i\sum_{\ell=0}^{s+r-2-2i}\nu_k\binom{s+r-2-2i}{\ell}x^{i-k+\ell}y^{s+r-2-2i-\ell+k}.
\tilde{Z}_3 = 1 - \frac{\tilde{e}^2}{8\pi}\int_{2m}^\infty da \,\frac{(1/a^2)(a^2-2m\tilde{\mu})^2+(2m-\tilde{\mu})^2}{(a^2-\tilde{\mu}^2)^2}+ O(\tilde{e}^4).
\begin{array}{c}{{e_{i}= F_{i}^{-}F_{i + 1}^{+}, 1 \le i \le n}}\\{{f_{i}= F_{i}^{+}F_{i + 1}^{-}, 1 \le i \le n}}\\{{k_{i}= Q^{N_{F_{i}}- N_{F_{i + 1}}}, 1 \le i \le n}}\\{{e_{0}= F_{n + 1}^{+}F_{1}^{-},}}\\{{f_{0}= F_{1}^{+}F_{n + 1}^{-},}}\\{{k_{0}= Q^{- N_{F_{1}}+ N_{F_{n + 1}}},}}\\ \end{array}
S =\sum_{n=16}^\infty\frac{1}{n\ln n}\mathbb P\left(|S_n|>(1+\varepsilon)\sigma_n\sqrt{2\ln\ln\ln n}\right) &\propto \sum_{n=16}^\infty\frac{1}{n\ln n(\ln\ln n)^{(1+\varepsilon)^2}\sqrt{\ln\ln\ln n}}.
\overline{{{\mathrm{\Large ~ v ~}}}}_{3}( x ) ={\frac{2 \pi^{3 / 2}}{g^{2}\Lambda}}x +{\frac{1}{3}}\left( 4 x -{\frac{{\cal E}^{2}}{x}}\right) x^{1 / 2}{\mit \Gamma}( 1 / 2 , x ) +{\frac{2}{3}}( 1 - 2 x ){\mathrm e}^{- x}-{\frac{2}{3}}\ ,
\int_{B^{c}_{2R}}\big[ U_k(x)+V_k(x)\big]\,dx\leqslant \int_{\mathbb{R}^{N}}\Phi_{R}(x)\big[ U_k(x)+V_k(x)\big]\,dx \leqslant \int_{B^{c}_{2R}}\big[ U_k(x)+V_k(x)\big]\,dx,
{\hat{\nabla}}_{a}{\hat{\nabla}}_{a}{\hat{A}}= - \sum_{J = 0}^{N}\sum_{m = - J}^{J}A_{J m}[{\hat{S}}_{a}, [{\hat{S}}_{a},{\hat{Y}}_{m}^{J}] ] = - \sum_{J = 0}^{N}\sum_{m = - J}^{J}J ( J + 1 ) A_{J m}{\hat{Y}}_{m}^{J}.
\begin{aligned}F(x) \otimes G(x) \preceq A^0(x) \otimes B^0(x) + \sum_{t:\omega_t = 1}\Bigl(U^{p^t_1}(x) - U^{p^{t-1}_1}(x)\Bigr) \otimes B^{p^t_2}(x) & + \sum_{t:\omega_t = 2}A^{p^t_1}(x) \otimes \Bigl(V^{p^t_2}(x) - V^{p^{t-1}_2}(x)\Bigr) x \in X.\end{aligned}
f_{\mathbf{t}}(x,y):=f(x,y)+t_1 x^7 y^3 + t_2 x y^9 + t_3 x^9 y^2 + t_4 x^8 y^3 + t_5 x^{11}y + t_6 x^{10}y^2 + t_7 x^9 y^3 + t_8 x^{11}y^2 + t_9 x^{10}y^3 + t_{10}x^{11}y^3.
w = \left[\mu_{i_{11}}, \ldots, \mu_{i_{1p_1}}, \left[ \ldots, \big[\mu_{i_{(n-1)1}}, \ldots, \mu_{i_{(n-1) p_{n-1}}}, [\mu_{i_{n 1}}, \ldots, \mu_{i_{n p_n}}]\big] \ldots\right]\right].
\int u'' u' \eta' \, dx = \int \bigl(\frac{|u'|^2}{2}\bigr)' \eta'\, dx= - \int \frac{|u'|^2}{2}\eta'' \,dx= -\frac{1}{2}\int u' (u' \eta'')\, dx= \frac{1}{2}\int u u'' \eta''+ u u' \eta''' \,dx.
\Gamma ( \phi , \phi_{a}^{*}, \bar{\phi}) = \frac{\hbar}{i}\ln Z ( J , \phi_{a}^{*}, \bar{\phi}) - J_{A}\phi^{A}, \; \; \phi^{A}= \frac{\hbar}{i}\frac{\delta \ln Z ( J , \phi_{a}^{*}, \bar{\phi})}{\delta J_{A}},
K ( \theta ) = \frac{s_{\frac{1}{2}}( \theta ) s_{\frac{1 + a}{2}}( \theta ) s_{\frac{2 - a}{2}}( \theta )}{s_{\frac{1 - E}{2}}( \theta ) s_{\frac{1 + E}{2}}( \theta ) s_{\frac{1 - F}{2}}( \theta ) s_{\frac{1 + F}{2}}( \theta )},
{\cal L}= -\frac{1}{4g^{2}}F^{k}_{\mu \nu}F^{k}_{\mu \nu}+\frac{1}{g'}F^{k}_{\mu \nu}\tilde{F}^{k}_{\mu \nu}+\frac{2}{g'}\left( D_{\mu}\cal{B}_{\mu}\right)^{k}\!\left(D_{\nu}\cal{A}_{\nu}\right)^{k}+{\cal L}_{FP}\, ,
S_{e f f}= - 8 \frac{g^{2}}{( 4 \pi )^{2}}\, \int d^{4}x \epsilon^{\mu \nu \rho \alpha}a_{\mu}\partial_{\nu}b_{\rho \alpha}= - 8 \frac{g^{2}}{( 4 \pi )^{2}}\,{\cal S}_{B F}({\cal A}) .
{\rm argmax}_{j \in [n]\backslash I}| \langle v_{k+1}(B), (B_{j,[m]})^\top\rangle| \le|\langle v_{k+1}(B_{I,[m]}), (B_{i_{k+1}(B_{I,[m]}),[m]})^\top \rangle | &={\rm argmax}_{i \in I\backslash I_{k}(B)}|\langle e_1, (B_{i,[m]})^\top\rangle|,
F^{ij}\nabla_{ij}G - \frac{2F^{ij}\nabla_ju}{u}\nabla_iG - 2u\langle \nabla G,X \rangle = -\frac1uF^{ij,rs}\nabla_mh_{ij}\nabla_mh_{rs}+\frac{1}{2u}\frac{|\nabla K|^2}{K}&=: \frac1uS.
\left(\frac{z}{2}\right)^{\nu}K_{\nu}(z)=\frac{2^{-2\nu}\,\sqrt{\pi}\,e^{-z}}{\Gamma\left(\frac{1}{2}-\nu\right)}\,\intop_{0}^{\infty}e^{-2zt}t^{-\nu-\frac{1}{2}}(t+1)^{-\nu-\frac{1}{2}}dt,\,\,\,\,(\nu)<\frac{1}{2},\,\,\,\,|\arg(z)|<\frac{\pi}{2}.
\sigma(\widetilde \Phi_i) =\sigma ((\widetilde \Phi_i(t,\cdot,\cdot))_{t=0}^{T-1})=\bigvee_{t \in [\![0,T-1]\!]}\sigma (\widetilde \Phi_i(t,\cdot,\cdot))\subseteq \bigvee_{t \in [\![0,T-1]\!]}\sigma ( \Phi_i,\vartheta_t)=\sigma( \Phi_i,( \vartheta_t)_{t=0}^{T-1}),
Y(z_{1},z_{2})=\frac{\sum_{p,q=0,0}^{m,n}b_{p,q}z_{1}^{-p}z_{2}^{-q}}{\sum_{p,q=0,0}^{m,n}a_{p,q}z_{1}^{-p}z_{2}^{-q}}X(z_{1},z_{2})
\left(\frac{d^2 \vartheta}{d z^2}-2 \vartheta^3-\alpha_{1}\vartheta-\beta_{1}\right)^2=-4 \left(\vartheta-e^z\right)^2 \left(\left(\frac{d \vartheta}{dz}\right)^2-\vartheta^4-\alpha_{1}\vartheta^2-2\beta_{1}\vartheta-\gamma_{1}\right),
\int\limits_0^\infty{{x^{s - 1}}\sum\limits_{n = 0}^\infty{\frac{{\phi \left({2n + 1}\right)}}{{\left({2n + 1}\right)!}}{x^{2n + 1}}dx ={{\left({- 1}\right)}^{\frac{{s + 1}}{2}}}}}\phi \left({- s}\right)\Gamma \left( s \right)\sin \frac{{\pi s}}{2}
\begin{cases}-dY_{t}=f(t,\eta_{t},Y_{t},Z_{t},Y_{t+\delta(t)},Z_{t+\zeta(t)}) dt - Z_{t}dB_{t}^{H}, \ \ \t\in [0,T]; Y_{t}=g(\eta_{t}), \ \Z_{t}=h(\eta_{t}), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \t\in[T,T+K], \end{cases}
A_0^* : = \sup_{t > 0}\bigg( \int_t^{\infty}U_*^q(\tau) w(\tau)\,d\tau \bigg)^{\frac{1}{q}}V_*^{- \frac{1}{p}}(t), A_1^* : = \sup_{t > 0}W^{\frac{1}{q}}(t) \bigg( \int_t^{\infty}\bigg( \frac{U_*(\tau)}{V_*(\tau)}\bigg)^{p'}v(\tau)\,d\tau \bigg)^{\frac{1}{p'}};
\tilde J(\tau) = \prod_{j=1}^k e^{(\lambda+(r-k)d+\frac{b}{2})\tau_j}\prod_{1\leq i<j\leq k}\sinh^{d_+}\left(\frac{\tau_i-\tau_j}{2}\right)\cosh^{d_-}\left(\frac{\tau_i-\tau_j}{2}\right).
D^{\mu_1\cdot \cdot \cdot \mu_k}(\partial n)=\frac 1{m!}\epsilon^{\mu_1\cdot \cdot \cdot \mu_k\mu_{k+1}\cdot \cdot \cdot \mu_n}\epsilon_{a_1\cdot \cdot \cdot a_m}\partial_{\mu_{k+1}}n^{a_1}\cdot \cdot \cdot\partial_{\mu_n}n^{a_m}
\begin{array}{c}{{D^{\left[ A , B \right]}( \Lambda ) u ( \Lambda^{- 1}p ) = u ( p ) D^{\left( s \right)}( R^{- 1}( \Lambda , p ) )}}\\{{D^{\left[ A , B \right]}( \Lambda ) v ( \Lambda^{- 1}p ) = v ( p ) D^{\left( s \right) *}( R^{- 1}( \Lambda , p ) ) , \quad v ( p ) = u ( p ) D^{\left( s \right) *}( i \sigma_{2})}}\\ \end{array}\, \, \, \, \, \,
\begin{aligned}\mathrm{d}\dot{u}- \Delta u\, \mathrm{d}t = f(u)\,\mathrm{d}t+g(u)\,\mathrm{d}W \mathrm{in}\: \ \mathcal{D}\times(0,\infty), &u = 0 \mathrm{in}\: \ \partial\mathcal{D}\times(0,\infty), &u(\cdot,0) = u_0, \ \dot{u}(\cdot,0)=v_0 \mathrm{in}\: \ \mathcal{D},\end{aligned}
{{\cal L}^{2}}^{\mathrm{e f f}}( H , \mu , T ) = \frac{e H}{2 \pi^{2}}\frac{\alpha}{2 \pi}\sum_{n = 0}^{\infty}b_{n}\int_{- \infty}^{\infty}d \! p_{z}~ p_{z}[ \ln ( \frac{p_{z}^{2}}{e H}) + 1 ] ~ ( f_{+}( T ) + f_{-}( T ) ) .
\left\lbrace\begin{array}{rrrr}-i\,\frac{3a^{2}}{2\beta \hslash^{2}}a_{2}+ ia_{4}= \lambda a_{1}; i\,\frac{3a^{2}}{2\beta \hslash^{2}}a_{1}- ia_{3}= \lambda a_{2}; ia_{2}= \lambda a_{3}; -ia_{1}= \lambda a_{4}.\end{array}\right. ,
t_{1,1,2}= t_1t_{1,2}- t_2, t_{-1,2}= t_1t_2 - t_{1,2}, t_{1,2,1,3}= t_{1,2}t_{1,3}+ t_{2,3}- t_2t_3, t_{1,3,2}= -t_{1,2,3}+ t_1t_{2,3}+ t_2t_{1,3}+ t_3t_{1,2}- t_1t_2t_3.
I = \int_{T}^{2T}\sum_{n\leq CT/\pi}{n^{-\tfrac{1}{2}-it}}\left(\tfrac{t}{2\pi}\right)^{{it}/{2}}e^{-i({t}/{2}+{\pi}/{8})}\left\{1+O\left(\tfrac{1}{t}\right) \right\}dt + O(T^{1/2}).
\lim_{i\rightarrow\infty}\sup_{0<r\le \lambda_i^{-1}}\frac1{r}\int_{-r}^{r}\int_{B^2_{\lambda_i^{-1}}(0)}\Big(\Big|\frac{\partial \widehat d_i}{\partial x_3}\Big|^2+\frac{(|1-|\widehat d_i|^2)^2}{\widehat\epsilon_i^2}\Big)(x,t)\,dxdt=0,
\bar{\Delta}_{\alpha}\mathrm{e x p}\{( i / \hbar ) W_{m}\}= 0 \quad \Longrightarrow \quad \int d^{2}\theta \, \theta^{2}\frac{\partial^{2}\delta W_{m}}{\partial \theta^{2}\delta \Phi^{A}( \theta )}\, \Phi^{B}( \theta ) ( \sigma_{\alpha})_{B}^{~ ~ \! A}+ V_{\alpha}W_{m}= 0 ,
r=2,s=1,t=4 u=4,k=2, w=1,x=q, &a_R=(qs/ab,qs/ad),b_S=q^2s^2/abcd,c_T=(q^{-n},q^{n+1}s^2/a^2,s,qs/ac), &d_U=(q^{1/2}s/a,-q^{1/2}s/a,,qs/a,-qs/a),e_K=(-qs/a^2,-qs/ac), &\gamma=qs^2/a^2c^2.
\sum_{k=0}^{\infty}\frac{(\frac{1}{2})_k^3}{(1)_k^3}(3k+1)2^{2k}\ \ \frac{-2i}{\pi},\sum_{k=0}^{\infty}\frac{(\frac{1}{2})_k^3}{(1)_k^3}(3k+1)(-1)^k 2^{3k}\ \ \frac{1}{\pi}
k_{\varepsilon}(u(x,s)+\varphi(u,x,z,s))-k_{\varepsilon}(u(x,s))-(k'_{\varepsilon}(u(x,s)),\varphi(u,x,z,s)) =\int_{0}^{1}(1-\tau)k''_{\varepsilon}(\varphi(u,x,z,s)\tau+u(x,s))\varphi^{2}(u,x,z,s)d\tau
4\sum_{n\geq1}(-1)^nnq^{n^2}=-2\frac{(q;q)_{\infty}}{(-q;q)_{\infty}}\sum_{n\geq1}\frac{q^n}{1-q^{2n}}+\sum_{n\geq0}\left(\frac{(q;q)_{\infty}}{(-q;q)_\infty}-\frac{(q;q)_n}{(-q;q)_n}\right).
I_{1}=\lim_{t\rightarrow\infty}\lim_{M\rightarrow\infty}\frac{t}{a(t)}\theta\left[\frac{1}{t}\int_{0}^{t}C_{1}(M,M,t-r)\gamma_{0}(r)dr-m\right] &=\lim_{t\rightarrow\infty}\frac{t}{a(t)}\theta\left[\frac{1}{t}\int_{0}^{t}C_{1}(\infty,\infty,t-r)\gamma_{0}(r)dr-m\right],
= F [ g_{i_{1}}^{1}g_{i_{2}}^{2}. . . g_{i_{n}}^{n}; ( g^{1}h^{1}{( g^{1})}^{- 1})_{k_{1}}( g^{2}h^{2}{( g^{2})}^{- 1})_{k_{2}}. . . ( g^{n}h^{n}{( g^{n})}^{- 1})_{k_{n}}]
\omega_A^2(T) = \omega^2(\widetilde{T}) \le \frac{1}{2}\left[{\|\widetilde{T}\|_{\mathcal{B}(\mathbf{R}(A^{1/2}))}^2 + \omega\big({(\widetilde{T})^2}\big)}\right] =\frac{1}{2}\left[{\|\widetilde{T}\|_{\mathcal{B}(\mathbf{R}(A^{1/2}))}^2 + \omega\big({\widetilde{T^2}}\big)}\right].
\Big( a^{(b)}_\mu,T_{\mu \nu}a^{(b)}_\nu \Big) =\Big( \varepsilon_{\mu \nu}\partial_\mu a^{(b)}_\nu, \frac{-1}{\triangle}\; \varepsilon_{\rho \sigma}\partial_\rho a^{(b)}_\sigma \Big) \; ,
\left[ \begin{array}{c}s S_0^{-1}(\widehat{m}) \end{array}\right] =\frac{s!}{(\widehat{m}_1-1)!\widehat{m}_2!\cdots\widehat{m}_s!}=\frac{s!}{\widehat{m}_1!\widehat{m}_2!\cdots\widehat{m}_s! \widehat{m}_{s+1}!}\widehat{m}_1,
(P_k)_i=\begin{bmatrix}\eta_r U_0(\alpha_r) \overbrace{0 \ldots 0}^{(k-1)}\eta_r \mu^{-\frac{1}{2}}U_1(\alpha_r) \ldots \eta_r \mu^{-\frac{n-k}{2k}}U_{\frac{n-k}{k}}(\alpha_r) \overbrace{0 \ldots 0}^{(k-1)}\end{bmatrix}^{T},
\int_{a(m)}^{a(m+1)}\log | \tanh(C_{d}(x - z)) | \, b'(z)\, \d z & = \left(\int_{\begin{subarray}{c}z: |x-z|<N^{-\lambda}z \in [a(m), a(m+1))\end{subarray}}+\int_{\begin{subarray}{c}z: |x-z| \geq N^{-\lambda}z \in [a(m), a(m+1))\end{subarray}}\right) \log | \tanh(C_{d}(x - z)) | \, b'(z)\, \d z.
\operatorname{div}^{\overline{\mathcal{A}}}:\Gamma(L(T^*U,\mathfrak{g}^*))=\mathfrak{X}(U,\mathfrak{g}^*)\rightarrow\mathcal{F}(U,\mathfrak{g}^*),\quad\operatorname{div}^{\overline{\mathcal{A}}}w=\operatorname{div}w-\operatorname{Tr}\left(\operatorname{ad}^*_{\overline{\mathcal{A}}}w\right)
\frac{d}{dt}a_{1}=-\nu a_{1}-\int_{\mathbb{T}_{\alpha}}v_{n}^{\nu}\omega_{n}^{\nu}\sin y\dxdy, \frac{d}{dt}a_{2}=-\nu a_{2}+\int_{\mathbb{T}_{\alpha}}v_{n}^{\nu}\omega_{n}^{\nu}\cos y\dxdy.
x\mapsto\! x^3, y\mapsto x^4(1-y)\Longrightarrow\mathcal{I}_{\beta_1,\beta_2,\beta_3}\!\!=\!\!3\!\!\!\int_{[0,1]^2}\!\!\! \tilde{f}_{\mathbf{t}}(x,y)^s x^{3\beta_1+4\beta_2+12\beta_3+24s}y^{\beta_3+1}q(y) \frac{dx}{x}\!\frac{dy}{y}
C=\Gamma_2\Gamma_4=\left(\begin{array}{cc}I 0 0 \widetilde{I}\end{array}\right) \ ,\quad I=\widetilde{I}=\left(\begin{array}{ccc}\begin{array}{cc}0 1 -1 0\end{array}| 0 --- | --- 0 | \begin{array}{cc}0 1 -1 0\end{array}\end{array}\right)
\delta{\cal P}_{i}^{\left( 1 \right) a}= - \epsilon_{0 i j k}\pi^{j k a}, \delta{\cal P}_{i}^{\left( 2 \right) a}= - \frac{1}{2}\epsilon_{0 i j k}F^{j k a}, \delta{\cal P}^{a}= - \partial^{i}{\cal P}_{i}^{\left( 2 \right) a},
\begin{aligned}\delta^{x}_{j}(T) \leq C \tilde{C}_{T}\Big[ \big(\delta^{\nabla y}_{j-1}(T) k^{q}_{j}(T)+ k^{q}_{j-1}(T) \delta^{\nabla y}_{j-1}(T) \big) (2 k^{\infty}_{j}(T) + \lvert \overline{b}\rvert) &+k^{q}_{j-1}(T)^2 \big(\delta^{x}_{j-1}(T)+\delta^{y}_{j-1}(T) \big) \Big].\end{aligned}
\int_0^{\infty}dx \frac{\cos a x}{\cosh^{\nu}\beta x}= \frac{2^{\nu-2}}{\beta \Gamma(\nu)}\Gamma \left( \frac{\nu}{2}+ \frac{a i}{2 \beta}\right) \Gamma \left( \frac{\nu}{2}- \frac{a i}{2 \beta}\right).
\Psi^{m+1}(b) =\Psi(\Psi^{m}(b)) &=\Psi\big(\pi^{m}\partial^{m}(b) + P_{m-1}(b,\partial b, \dots, \partial^{m-1}b) \big ) &=\pi^m\Psi ( \partial^{m}(b)) + \Psi\big(P_{m-1}(b,\partial b, \dots, \partial^{m-1}b) \big ) =\pi^{m+1}\partial^{m+1}(b)+\partial^m(b)^q+P_{m-1}(b,\dots, \partial^{m-1}b)^q+\pi\partial (P_{m-1}(b, \dots, \partial^{m-1}b)).
\Vert (\delta_t u^n_h, \delta_t{\boldsymbol \sigma}^n_h)\Vert^2_{0}+ k \underset{m= 1}{\overset{n}{\sum}}\Vert (\delta_t u^m_h,\delta_t{\boldsymbol \sigma}^m_h) \Vert^{2}_{1}\leq K_2, \ \ \forall n\geq 1,
\left.\begin{array}{ll}\displaystyle\dfrac{N-6}{N}\left(\dfrac{a_2}{2m_1}\int_{\mathbb{R}^N}|\nabla \xi|^2dx+\dfrac{a_1}{4m_2}\int_{\mathbb{R}^N}|\nabla \eta|^2dx\right) begin{align*}0.5cm] \qquad\displaystyle+a_2\omega_1\int_{\mathbb{R}^N}\xi^2dx+\dfrac{a_1}{2}\omega_2\int_{\mathbb{R}^N}\eta^2dx=0.\end{array}\right.
( X_{L}^{\prime i}( e^{2 \pi i}z ) , X_{R}^{\prime i}( e^{- 2 \pi i}\bar{z}) ) = ( X_{L}^{\prime i}( z ) + 2 \pi \ell v_{L}^{i}, X_{R}^{\prime i}( \bar{z}) - 2 \pi \ell v_{R}^{i}) + ( \mathrm{t o r u s \s h i f t}) ,
\hat{g}(\mathsf{x},T,\mathsf{y})=\pi^{-\frac{d}{2}}\exp \left[ -\frac{1}{2}\left( \left\vert \mathsf{x}\right\vert^{2}+\left\vert \mathsf{y}\right\vert^{2}+dT\right) \right]
\begin{aligned}\|e\|_q^q =\sum_{i=1}^n|e_i|^q &= \sum_{i=1}^n|e_i|^q\cdot 1 &\leq(\sum_{i=1}^n(|e_i|^q)^{\frac{1}{q}})^{q}(\sum_{i=1}^n1^{\frac{1}{1-q}})^{1-q}&=\|e\|_1^qn^{1-q}\end{aligned}
K_{20}(k^2) = \int_{-\pi/2+3\epsilon/2}^{\pi/2-\epsilon/2}d \theta_2 \int_{-\pi/2+\epsilon/2}^{\theta_2-\epsilon}d \theta_1~{\cal F}(\theta_1)^{k^2-1}{\cal F}(\theta_2)^{k^2-1}\left| 2 \sin \frac{\theta_2 - \theta_1}{2}\right|^{-2 k^2},
\chi_T= 2^{4/3}\pi^{2/3}\kappa^{-2/3}N m r_+^{-2}\frac{r_+^2 +2l^2}{3r_+^4 +5l^2r_+^2 +2l^4}\frac{3r_+^2 +2l^2}{3r_+^2 -2l^2}.
{\displaystyle |\tau\sum_{k=1}^{m}V_{3}^{k}(\partial_{\tau}{\theta_{\Psi}^{k}})| \leq C\big(h^{2r}+\tau^{4}\big) +C \Vert\theta_{\Psi}^{m}\Vert_{\mathcal{L}^2}^{2}+ \frac{1}{32}\Vert\nabla\theta_{\Psi}^{m}\Vert_{\mathbf{L}^2}^{2}+C\tau \sum_{k=0}^{m}{\Vert\nabla\theta_{\Psi}^{k}\Vert_{\mathbf{L}^2}^{2}}}
\bigl(w_i,e^{i\theta_\infty}(\phi_i+\phi_e)- \nabla\cdot(\sigma_{i\infty}\nabla\phi_i)\bigr)_{L^2(\mathbb{R}_+^d)}&-\bigl(w_e,e^{i\theta_\infty}(\phi_i+\phi_e)-\nabla\cdot(\sigma_{e\infty}\nabla\phi_e)\bigr)_{L^2(\mathbb{R}_+^d)}=0.
{\Omega_{\Lambda}}^{\Sigma}={( L^{- 1})_{\Lambda}}^{\Pi}( d{L_{\Pi}}^{\Sigma}+ g{f_{\Pi}}^{\Delta \Gamma}A_{\Delta}{L_{\Gamma}}^{\Sigma}) \equiv \left( \begin{matrix}{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{- i Q}}&{{P^{j}}}\\{{0}}&{{0}}&{{P_{i}}}&{{Q_{i}^{j}+{\frac{i}{n}}\delta_{i}^{j}Q}}\\ \end{matrix}\right) .
\overline{{{\psi}}}^{( 2 )}( x ) P_{+}\psi^{( 2 )}( x ) \; \longleftrightarrow \; \frac{1}{2 \pi}c^{( 2 )}: e^{- i a \varphi^{( 1 )}( x )}:_{M^{( 1 )}}: e^{+ i b \varphi^{( 2 )}( x )}:_{M^{( 2 )}}\; e^{i \frac{\theta}{2}}\; ,
Q_k(y) ={}\frac{1}{k!}\left. \left(\frac{d}{da}\right)^k \right|_{a=1}\left[\frac{1}{a}\varphi\left( \frac{y}{a}\right) \right] ={}\frac{(-1)^k}{k!}\left(\frac{d}{dy}\right)^k \left[ y^k \varphi(y)) \right].
C_{1}(\varepsilon)={\mathcal{A}}_{1}^{T}(\varepsilon),\ \C_{2}(\varepsilon)={\mathcal{A}}_{3}^{T}(\varepsilon),\ \C_{3}(\varepsilon)={\mathcal{A}}_{2}^{T}(\varepsilon),\ \C_{4}(\varepsilon)={\mathcal{A}}_{4}^{T}(\varepsilon).
\frac{d}{d t}\int_{\Sigma_t}|\nabla u|^2= \int_{\Sigma_t}f'(t) \frac{1}{|\nabla u|}\nu (|\nabla u|^2)+ |\nabla u|^2 H f'(t) \frac{1}{|\nabla u|}= \int_{\Sigma_t}f'(t) \frac{2}{1 - p}|\nabla u| H + f'(t) H |\nabla u| = - af'(t) \int_{\Sigma_t}H |\nabla u|.
P_e=\frac{4\big(\sqrt{M}-1\big)}{\sqrt{M}\mathrm{log}_2(M)}{Q}\Bigg(\sqrt{\frac{3\gamma}{M-1}}\Bigg)+\frac{4\big(\sqrt{M}-2\big)}{\sqrt{M}\mathrm{log}_2(M)}{Q}\Bigg(3\sqrt{\frac{3\gamma}{M-1}}\Bigg).
\nabla^{\perp\psi}_{E_a}\nabla^{\perp\psi}_{E_a}H^{\psi}=\frac{p}{p+q}\nabla^{\perp\psi}_{E_a}\nabla^{\perp}_{E_a}H_1+\frac{q}{p+q}\nabla^{\perp\psi}_{E_a}B^j(E_a,H_2).
t_{s_{1,1}}^{5n}t_{s_{1,2}}^{5n}\cdots t_{s_{1,g-1}}^{5n}= [\mathcal{V}'_1,\mathcal{W}'_1] [\mathcal{V}'_2,\mathcal{W}'_2] \cdots [\mathcal{V}'_{\left[\frac{|5n|}{2}\right]+1}, \mathcal{W}'_{\left[\frac{|5n|}{2}\right]+1}]\cdot t_{d_1}^{10n}t_{d_2}^{10n}\cdots t_{d_{g-1}}^{10n}.
G(\gamma,L):=\sum_{\substack{n^{\eta}\le p_1,\dots,\,p_r -\gamma\le L(\frac{\log{p_1}}{\log{n}},\dots ,\frac{\log{p_\ell}}{\log{n}})\le \gamma n^{\theta_1}\le\prod_{i\in\mathcal{I}}p_i\le n^{\theta_2+\epsilon}}}\Bigl(1_{\mathcal{A}}(p_1\cdots p_r)+\frac{\#\mathcal{A}}{\#\mathcal{B}}1_{\mathcal{B}}(p_1\cdots p_r)\Bigr).
\int_0^\infty \xi_\alpha(\lambda)d\lambda = \sum_{n=0}^\infty \alpha_n^2, \frac2\pi \int_0^\infty \left\{1-\exp\left(-\int_0^\infty\frac{\xi_\alpha(\lambda)}{\lambda+t^2}d\lambda\right)\right\}dt=\alpha_0,
A_1^{24},\;A_2^{12},\;A_3^{8}, \;A_4^6,\;A_6^4,\;A_{12}^2, A_5^4D_4,\;A_7^2D_5^2,\;A_8^3,\;A_9^2D_6,\; A_{11}D_7E_6,\;A_{15}D_9,\;A_{17}E_7,\;A_{24}, D_4^6,\;D_6^4,\;D_8^3,\;D_{10}E_7^2,\; D_{12}^2,\;D_{16}E_8,\;D_{24},E_6^4,\;E_8^3,
\gamma^{0}=(\begin{matrix}I_{2}&0\\ 0&-I_{2}\end{matrix}),\gamma^{1}=(\begin{matrix}0&\sigma_{x}\\ -\sigma_{x}&0\end{matrix}),\gamma^{2}=(\begin{matrix}0&\sigma_{y}\\ -\sigma_{y}&0\end{matrix}),\gamma^{3}=(\begin{matrix}0&\sigma_{z}\\ -\sigma_{z}&0\end{matrix}).
w^{\prime \prime}( x ) = \left( - \frac{\lambda^{2}}{( 1 - x^{3})^{2}}+ \frac{l ( l + 6 )}{4 x^{2}( 1 - x^{3})}+ \frac{5 - 4 6 x^{3}+ 5 x^{6}}{1 6 x^{2}( 1 - x^{3})^{2}}\right) w ( x ) \,
\sup_{x\in[0,1]}\P(N^{-1}d_{i_x}\geq \beta)\leq \P\left(N^{-1}\max_{i \in [N]}d_i \geq \beta\right)\leq \sum_{i \in [N]}\P(N^{-1}d_i \geq \beta)\leq N \sup_{x\in[0,1]}\P(N^{-1}d_{i_x}\geq \beta).
Z_3(s;r_1,r_2,r_3)= J^3\Big(x_1^{r_1}x_2^{r_2}x_3^{r_3}(x_1+x_2+x_3)^{-s}\Big) +\sum_{{\rm cyc}(r_1,r_2,r_3)}J^2\Big(\int_0^{-x_1-x_2}x_1^{r_1}x_2^{r_2}x_3^{r_3}(x_1+x_2+x_3)^{-s}\,dx_3\Big) +\sum_{{\rm cyc}(r_1,r_2,r_3)}J^1\Big(\int_0^{-x_1}\int_0^{-x_1-x_2}x_1^{r_1}x_2^{r_2}x_3^{r_3}(x_1+x_2+x_3)^{-s}\,dx_3dx_2\Big).
\begin{aligned}\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}(\exp(\sum h(\lambda_i)))\leq \exp\{\frac{1}{C}\|h\|^2_{\mathcal L}\}\exp(\mu_{N, \frac{\overrightarrow{\boldsymbol{N}}}{N}, \beta}^{V;\boldsymbol{A}}(\sum h(\lambda_{i}))).\end{aligned}
\frac{i \theta^{im}}{2 \pi}\frac{i \theta^{jn}}{2 \pi}\int_0^1 d\tau'~ e^{-2 \pi i \tau'}l^{m'}F_{mm'}(x+l \tau')\int_0^1 d\tau''~ e^{2 \pi i \tau''}l^{n'}F_{nn'}(x+l \tau'').
E_{C}^{( 2 ) \; r e g}= - \frac{M_{0}}{4 \sqrt{\alpha}}- \frac{M_{0}^{2}R}{8 \pi \alpha}\Gamma ( - 1 ) - \frac{M_{0}}{2 \pi \sqrt{\alpha}}\sum_{n = 1}^{\infty}n^{- 1}K_{1}\left( \frac{2 n M_{0}R}{\sqrt{\alpha}}\right) .
V_4 : x_2L_2(x_0,x_1)+x_3M_2(x_0,x_1)+x_4^2L_1(x_0,x_1)+x_4x_5M_1(x_0,x_1) &+x_5^2N_1(x_0,x_1)+x_4N_2(x_2,x_3)+x_5P_2(x_2,x_3)=0
G_{0}^{(2)}(\lambda,\vec{r}_{1},z_{1},\vec{r}_{3},z_{3}) =\frac{\lambda}{2}\int d^{d-1}r_{2}\int_{0}^{L}dz_{2}G_{0}^{(2)}(\vec{r}_{1}-\vec{r}_{2};z_{1},z_{2})G_{0}^{(2)}(\vec{0},z_{2})G_{0}^{(2)}(\vec{r}_{2}-\vec{r}_{3};z_{2},z_{3}).