harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
h_{-2}^\mathcal{L}(t)=\sum_{1\leq \beta< \gamma\leq4}h_{0}^{\mathcal{L}_{(\beta\gamma)}}(t)\prod_{\substack{\alpha=1,..,4 \alpha\neq\beta,\gamma}}h_{-1}^{\mathcal{K}_\alpha}(t)-2\sum_{\beta=1}^Lh_{1}^{\mathcal{K}_\beta}(t)\prod_{\substack{\alpha=1,..,4 \alpha\neq \beta}}h_{-1}^{\mathcal{K}_\alpha}(t).
\mathcal{L}(h,k):= \frac{1}{2}\sum_{\substack{1\leq q<\infty (q,hk)=1}}W\left( \frac{q}{Q}\right) \sum_{\substack{1\leq m,n<\infty (mn,q)=1}}\frac{\tau_A(m) \tau_B(n)}{\sqrt{mn}}V\left( \frac{m}{X}\right) V\left( \frac{n}{X}\right) \sum_{\substack{c>C, d \geq 1 cd=q d|mh\pm nk}}\phi(d)\mu(c),
\mathbf{u}_\varepsilon (\cdot ,t)=e^{-\mathcal{A}_{D,\varepsilon}t}\boldsymbol{\varphi}(\cdot)+\int_0^t e^{-\mathcal{A}_{D,\varepsilon}(t-\widetilde{t})}\mathbf{F}(\cdot ,\widetilde{t})\,d\widetilde{t}.
H_o = -\alpha^{\prime}E^2 +\alpha^{\prime}\frac{n^2}{R^2N^2}+ \frac{\rho^2}{\pi^2\alpha^{\prime}}\sin^2 \Big[ \frac{(\alpha -\beta)}{2}\pi \alpha^{\prime}E+\pi \frac{(\alpha +\beta)}{2R}\alpha^{\prime}\frac{n}{N}+ \frac{\pi m}{N}\Big] + \hat N,
\mu\left(\left\{k\in K\middle|\left\langle{g}{k}\right\rangle \neq G\right\}\right) \leqslant\sum_{F\in\mathcal{F}}\mu\left(\left\{{k}\in K\middle|\left\langle{g}{k}\right\rangle =F\right\}\right) =\sum_{F\in\mathcal{F}}\mu\left(\left\{{k}\in K\middle|\left\langle{r}_{F}{k}\right\rangle =F\right\}\right)=0
\sum_{n=m}^{\infty}S(n,m)\frac{z^{n}}{n!}=\frac{(\mathrm{e}^{z}-1)^{m}}{m!}=\frac{1}{m!}\left(\sum_{l=1}^{\infty}\frac{z^{l}}{l!}\right)^{m}=\frac{1}{m!}\sum_{n=m}^{\infty}\frac{z^{n}}{n!}\underbrace{\sum_{l_{1}+\dots+l_{m}=n}\frac{n!}{l_{1}!\dots l_{m}!}}_{=:B_{n}^{m}}=\sum_{n=m}^{\infty}\frac{B_{n}^{m}}{m!}\frac{z^{n}}{n!}.
\left\langle \pi(g) | h\right\rangle = \int\limits_{\Omega_{0}}\overline{\pi(g)(t)}h(t) d\nu(t) = \int\limits_{\Omega_{0}}\overline{g(\xi^{-1}t)}h(t) d\nu(t) &= \int\limits_{\Omega}g(s) h(\xi(s)) d\mu(s) &= \left\langle g | h \circ \xi \right\rangle.
\bigg\|\sum_{i=1}^{n}u(\theta_i,|\xi^i|)\bigg\|_\infty+c\bigg\|\sum_{i=1}^{n}\vartheta_iu'\left(\theta_i,\frac{1}{2\theta_i}\log\left(\frac{E[e^{2\theta_i|\xi^i|}|\mathcal{F}_t]}{1-2\theta_i\vartheta_ic}\right)\right)\bigg\|_{\infty}\leq\frac{1}{2}c
\frac{M^2_p}{\sqrt{r}}\cdot\alpha(p^2r)-\frac{1}{\sqrt{r}}\cdot\alpha(r)=\frac{p M_p^2}{3}(1+k^2_{p^2r})-\frac{1}{3}(1+k^2_r)+\frac{\pi^2 T_{p,r}}{12K[r]^2}
\begin{gathered}u^\alpha J_{\nu-\alpha}(ux; q^2)= \frac{(1-q^2)^{\alpha}}{\Gamma_{q^2}(1-\alpha )}x^{\alpha-\nu-1}D_{q,x}\left[x^{-2\alpha}\int_0^x \rho^{\nu+1}({q^2\rho^2}/{x^2}; q^2)_{-\alpha}J_\nu (u\rho; q^2)\,d_q \rho\right]. \end{gathered}
(a^\ast(\phi)\psi)_n(x_1, \ldots, x_n) := \frac{1}{\sqrt{n}}\sum^n_{j=1}\phi(x_j) \psi_{n-1}(x_1, \ldots, \hat x_j, \ldots, x_n) (a(\bar\phi)\psi)_n(x_1, \ldots, x_n):= \sqrt{n+1}\int dx\ \bar\phi(x) \psi_{n+1}(x, x_1, \ldots, x_n)
LE_{c}=\frac{\rho_{a}c_{p}}{\gamma}(\frac{e_{c}^{o}-e_{a}}{r_{ac}+r_{ah}+r_{sc}})\&LE_{s}=\frac{\rho_{a}c_{p}}{\gamma}(\frac{e_{s}^{o}-e_{a}}{r_{as}+r_{ah}+r_{ss}})
u^\varepsilon(t,x) = v(x) + \sum_{(i,l)\in S}a_{il}(x)\left\{\cos\left(\frac{2\pi k_{il}(x)}{\varepsilon}t\right)e_i+\sin\left(\frac{2\pi k_{il}(x)}{\varepsilon}t\right)e_l\right\}
{4 M^{2}}\left[ D \epsilon^{\beta \gamma \kappa \lambda}\partial_{\nu}{\partial_{\beta}}T_{( \kappa \lambda )}{}^{\nu}+ 2 a \partial_{\nu}{\partial_{\beta}}T^{( \beta \gamma ) \nu}\right] = e^{2}\partial_{\nu}\left({\partial}^{\gamma}\lambda^{\nu}-{\partial}^{\nu}\lambda^{\gamma}\right) .
\begin{array}{r@{\,=\,}l@{\qquad\qquad\quad}r@{\,=\,}l}1781 2102222_{(3)}41065 2002022221_{(3)}begin{align*}3pt]3239 11102222_{(3)}43981 2020022221_{(3)}begin{align*}3pt]3887 12022222_{(3)}98657 12000022222_{(3)}begin{align*}3pt]11177 120022222_{(3)}131461 20200022221_{(3)}begin{align*}3pt]14821 202022221_{(3)}393901 202000022221_{(3)}begin{align*}3pt]33047 1200022222_{(3)}\end{array}
\frac{1}{|B_i|}\int_{B_i}|u(y)-u_{B_i}|dy \leq \frac{1}{|B_i|}\int_{B_i}\Big(\frac{1}{|B_i|}\int_{B_i}|u(y)-u(z)|^pdz \Big)^{1/p}dy &=\frac{1}{|B_i|^{1+1/p}}\int_{B_i}\Big(\int_{B_i}|u(y)-u(z)|^pdz \Big)^{1/p}dy &\lesssim |B_i|^{\delta/n-1}\int_{B_i}\Big(\int_{B_i}\frac{|u(y)-u(z)|^p}{|y-z|^{n+p\delta}}dz \Big)^{1/p}dy
\sum_{l = 0}^{\infty}( 2 l + 1 ) [ s_{l}^{\prime}( \lambda r ) e_{l}( \lambda \rho ) ]^{2}= \frac{1}{2 r \rho}\int_{r - \rho}^{r + \rho}\left( \frac{1}{\lambda}\, \frac{\partial{\cal D}}{\partial r}\right)^{2}R \, d R \, .
I ( k^{2}) = -{\frac{\lambda^{2}}{9 6 \pi^{2}\mu^{2}}}\int_{0}^{1}d x \int_{0}^{1 - x}d y{\frac{f^{2}( x ) f^{2}( y )}{[{\frac{k^{2}}{\mu^{2}}}x y ( 1 - x - y ) - x y - ( x + y ) ( 1 - x - y ) ]}}.
\Phi(\eta(R,s,t;x)) \leq \frac{1}{2}aC_3t^{N-2}R^{3N-5}+\frac{1}{4}b(C_3t^{N-2}R^{3N-5})^2 &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~-t^N(C_4 s^NR^{3N}-C_5 R^{3N-1}) &\leq t^N(C_0 R^{3N-1}-C_4 s^N R^{3N}).
f_0(\zeta)=\zeta+2+3\pi\mathrm{i}+\boldsymbol L_1^{-1}+\zeta^{-1}\boldsymbol L_2^{-1}\boldsymbol L_3+4\zeta^{-1}\boldsymbol L_1^{-1}\boldsymbol L_2^{-1}\boldsymbol L_3,
\Theta_{e_{\varpi}}:={}^{t}\Theta_{e_{-\varpi}}=\Theta_{{}^{t}e_{-\varpi}}=\prod_{\alpha\in(\Phi'_e)^+,1\leqslant i\leqslant I(\alpha)}\Theta_{e_{\alpha,I(\alpha)+1-i}}^{\varpi(\alpha)},
\begin{array}{lll}J_0=-\frac12 (Y_1 -Y_2) , J_1=\frac12 Y_0 , J_2=\frac12 (Y_1 +Y_2) , begin{align*}2pt]P_0=y^1 - y^2 , P_1=2 y^0 , P_2=y^1 + y^2. \end{array}
tH_{\mathrm{Suz}}^{2+\frac{3}{2}}\left({\theta^0_2-\theta^0_1+\theta^1+1,\, \theta^0_2+\theta^1+1 \atop \theta^0_2-\theta^0_1};t;{q_1,p_1\atop q_2,p_2}\right) =tH_{\mathrm{III}(D_7)}\big( \theta^0_2-\theta^0_1+\theta^1+1; t; q_1,p_1 \big) \qquad{}+tH_{\mathrm{III}(D_7)}\big( \theta^0_2+\theta^1+1; t; q_2,p_2\big) +p_2q_1(p_1(q_1+q_2)+\theta^0_2-\theta^0_1)-q_1.
2 = \limsup_{x\to\infty}\frac{g_u^{\ast 2}\ast f_r(x)}{g_u(x)}\ge \limsup_{x\to\infty}\frac{g_u^{\ast 2}(x)}{g_u(x)}\liminf_{x\to \infty}\frac{g_u^{\ast 2}\ast f_r(x)}{g_u^{\ast 2}(x)}\ge \limsup_{x\to\infty}\frac{g_u^{\ast 2}(x)}{g_u(x)},
|p_1(r_i)-p_2(r_i)| \leq |f(r_i)-p_1(r_i)|+|f(r_i)-p_2(r_i)| &\leq 2\left[q(r_i)+|f(r_i)-p(r_i)|\right] &< 2[\zeta+\zeta] &=\upsilon.
\tilde{T}=\left(S_{k+1}^{m_{k+1}}(S_{k+1}^*)^{n_{k+1}}-\frac{1}{q_{k+1}}S_{k+1}^*S_{k+1}^{m_{k+1}}(S_{k+1}^*)^{n_{k+1}}S_{k+1}\right)T.
|D|^{-p-1}x(1-e^{-s^2D^2}) = x|D|^{-p-1}(1-e^{-s^2D^2})-\sum_{k=0}^p \binom{p+1}{k}|D|^{-p-1}\delta^{p+1-k}(x)|D|^{k-p-1}(1-e^{-s^2D^2})
x^{3}p_{n , l + 1}= p_{n + 3 , l}+ I_{n , l}p_{n , l}, I_{n , l}= \frac{h_{n , l + 1}}{h_{n , l}}= \frac{Z_{n + 1 , l + 1}Z_{n - 2 , l}}{Z_{n - 2 , l + 1}Z_{n + 1 , l}}.
p_{e,w_{00}}=v^{\ell(w_0)-1}+v^{\ell(w_0)-5}+\cdots +v^{\ell(w_0)-2n-1}p_{e,u_{00}}=v^{\ell(w_0)-3}+v^{\ell(w_0)-7}+\cdots +v^{\ell(w_0)-2n+1};
M^2T^{\ast^2}T^2-2\lambda T^{\ast}T+\lambda^2I =M^2M_{|E(uw)|^{2}E(|w|^2)}M_{\bar{u}}EM_{u}-2\lambda M_{E(|w|^2)}M_{\bar{u}}EM_{u}+\lambda^2I &=(M_{M^2|E(uw)|^{2}E(|w|^2)-2\lambda E(|w|^2)})M_{\bar{u}}EM_{u}+\lambda^2I.
X_{i,j}^{k,l}:= X_{w_{i,j}}^{w_0 w_{k,l}}= X(w_{i,j}) \cap w_0 X(w_{k,l}) &=\{(L,H) \in X | L \subset <e_{n-k+1}, \dots , e_i>; \: <e_1, \dots e_{j-1},e_{n-l+2}, \dots , e_n> \subset H \},
\left|J_{2}(x)\right|=\int_{\left\{\left|y\right|\geq R\right\}}\frac{1}{\left|x-y\right|}f_{k}(y)dy\leq \frac{C}{R}\int_{\left\{\left|y\right|\geq R\right\}}f_{k}(y)dy\leq\frac{C}{R},
\begin{aligned}\min_{\mathbf{\Delta}_p^{(k)}}~ \left|\left|\mathbf{q}_p^{(k)}- \mathbf{\Delta}_p^{(k)}\mathbf{G}_p^{(k)}\right|\right|_2^2 s.t.~ |\delta_{p,n}^{(k)}| \leq \bar{\delta}^{(k)}, \forall n,\end{aligned}
V^{\prime\prime}=\left[\ker\left(T_{b_{i}}\restriction_{V^{\prime}}-\varphi_{s,j,i}\circ T_{(b_{j},s_{i})}\restriction_{V^{\prime}}\right)\cap\ker\left(T_{b_{j}}\restriction_{V^{\prime}}-\varphi_{s,i,j}\circ T_{(b_{i},s_{i})}\restriction_{V^{\prime}}\right)\right]
- \sum_n \, \log \left(\frac{v^{\dagger}_+ (n) v_+ (n+\varphi)}{\mid v^{\dagger}_+ (n) v_+(n+\varphi)\mid}\frac{v^{\dagger}_+ (n+\varphi) v_- (n+\varphi)}{v^{\dagger}_+ (n) v_-(n)}\frac{v^{\dagger}_- (n+\varphi) v_- (n)}{\mid v^{\dagger}_- (n+\varphi)v_-(n)\mid}\right)
\overline{\rm SER}^{D}_{D}= F_{QAM}\left[ \left( 1 + \frac{(P_{S}/\: P_{L_{S,D}})\: \Omega_{S,D}\: g_{QAM}}{N_{0}\: m_{S,D}\,{\sin}^{2}(\theta)}\right)^{-m_{S,D}}\right].
\big|\big(xt^N; q\big)_{\infty}\big| \leq \prod_{i=0}^{\infty}{\big(1 + |x| q^{N\Re\theta + i}\big)}\leq \prod_{i=0}^{\infty}{\big(1 + |x| q^i\big)}, |(xq; q)_{\infty}| = \prod_{i=1}^{\infty}{\big|\big(1 - xq^i\big)\big|}\geq \prod_{i=1}^{\infty}{\big(1 - |x| q^i\big)},
\mathrm{RW}_0(d) := \inf_{\gamma \in \Sigma_0(\delta)}\mathbb{E}[ Y_1 (1 - d(X)) + Y_2 d(X)] \mathrm{RW}(d) := \inf_{\gamma \in \Sigma(\delta)}\mathbb{E}[ Y_1 (1 - d(X)) + Y_2 d(X)],
-iV^{2}\mbox{exp}(-i\frac{\pi}{6}(a_{1}+a_{2}))\sum_{n'}\sum_{e'\neq0}\frac{\mbox{exp}\frac{2i\pi}{k}[ae'(\widetilde{n'}-1/2)-e'(\widetilde{n}-1/2)-e(\widetilde{n'}-1/2)]}{2\mbox{sin}\pi e'/k}
\langle B^{ij}\partial_{i}\partial_{j}(\partial_{x}^{\alpha}u),\partial_{x}^{\alpha}u_{t}\rangle =-\langle B^{ij}\partial_{j}\partial_{x}^{\alpha}u,\partial_{i}\partial_{x}^{\alpha}u_{t}\rangle -\langle\left(\partial_{i}B^{ij}\right)\partial_{j}\partial_{x}^{\alpha}u,\partial_{x}^{\alpha}u_{t}\rangle,
\zeta=\left(\begin{array}{l}\zeta_1=(x-u_1)^2+(y-v_1)^2+(z_0-w_1)^2 \zeta_2=(x-u_2)^2+(y-v_2)^2+(z_0-w_2)^2 \zeta_3=(x-u_3)^2+(y-v_3)^2+(z_0-w_3)^2 \end{array}\right)
M_{p l}^{2}={\frac{M_{X}^{3}}{k}}e^{2 \pi k \rho - 2 c}\left( 1 - e^{- 2 k y_{2}}+ e^{- 2 k ( \pi \rho - y_{4}+ y_{2})}- e^{- 2 k ( 2 \pi \rho - y_{4})}\right) ~ . ~ \,
\left.\begin{array}{ll}\displaystyle a_2\int_{\mathbb{R}^N}|\phi|^2dx+a_1\int_{\mathbb{R}^N}|\psi|^2dx=a_2\int_{\mathbb{R}^N}|\phi_0|^2dx+a_1\int_{\mathbb{R}^N}|\psi_0|^2dx,\end{array}\right.
l = - \frac{i m k ( c o s \mu s i n \lambda + e^{2 i a k}c o s \lambda s i n \mu ) + m E ( e^{2 i a k}- 1 ) s i n \lambda s i n \mu}{m^{2}( e^{2 i a k}- 1 ) s i n \lambda s i n \mu + k^{2}c o s ( \lambda + \mu ) + i E k s i n ( \lambda + \mu )}
Y^{j}_t =\xi^j+\int_t^Tf^j(r,Y_r)\,dr+\int_t^T\,dV^j_r+\int_t^T\,dK^j_r &+\int_t^Tn(Y^{j}_r-H^j(r,Y_r))^-\,dr-\int_t^T\,dM^{j}_r.
\limsup\limits_{x \uparrow 0}\left\vert \int_{-\infty}^{\frac{x}{\sqrt{\epsilon}}}\frac{x}{y^2}e^{-\frac{y^2}{2}}e^{-\frac{2}{3}((\frac{x^2}{y^2}+t)^3-t^3)}\mathbb{E}\left[ \left (-2\frac{x^3}{y^3}\int_{0}^{1}B(z)dz \right) \vert B(1)=-y\right]dy \right \vert \le \sqrt{\epsilon}
\begin{array}{r c l}{{V_{e}/ m^{2}}}&{{=}}&{{\displaystyle + [ D ( k_{1}+ p ) D_{m}( k_{1}) + D ( k_{1}) D_{m}( k_{1}+ p ) ]}}\\{{}}&{{}}&{{\displaystyle \times [ D ( k_{2}+ p ) D_{m}( k_{2}) + D ( k_{2}) D_{m}( k_{2}+ p ) ] K_{1 2}}}\\{{}}&{{}}&{{\displaystyle + D_{m}( k_{2}) [ D ( k_{2}+ p ) - D ( k_{2}- p ) ] D_{m}( k_{1}+ k_{2}) D ( k_{1}+ k_{2}+ p ) K_{1},}}\\ \end{array}
\sum_{\mathrm{n}\in \mathbb{N}}\Big|\Big\langle\widehat{\nabla \mathrm{H}};\mathrm{e}^{(1)}_\mathrm{n}\Big\rangle_{\mathbb{L}^2(\mathrm{B})}\Big|^2 = |\alpha|^2 \delta^{4}\sum_{\mathrm{n}\in \mathbb{N}}\Big|\Big\langle \mathbb{T}_\delta\Big[\widehat{\nabla\mathrm{H}}\Big];\mathrm{e}^{(3)}_\mathrm{n}\Big\rangle_{\mathbb{L}^2(\mathrm{B})}\Big|^2
error_k= 2Re\int i \partial_{\alpha}D^k u\overline{\partial_t D^k u}-2Re\int \bold{n}\frac{1}{|z_{\alpha}|^{2k-1}}\partial_{\alpha}^k Du \overline{\partial_t \partial_{\alpha}^k u}\leq C(\|\partial_{\alpha}\xi_0\|_{H^{s-1}},\|v_0\|_{H^s}, \|w_0\|_{H^s}, \mathcal{E}(t), d_I(t)^{-1}, d_P(t)^{-1}, N\lambda_{max}, C_1, C_2).
\begin{aligned}2^{2k_1+k_2-1}-2^{k_1+k_2-1}\geq n =\frac{2^t-1}{2^{t+1}}2^{2k_1+k_2}=2^{2k_1+k_2-1}-2^{2k_1+k_2-t-1},\end{aligned}
\widehat{\psi_{p^{2k}}}(\mathbf{u}) := \frac{1}{p^{2kn}}\sum_{\mathbf{c}\in (\mathbb{Z}/p^{2k}\mathbb{Z})^n}\psi_{p^{2k}}(f_\mathbf{c}) \exp\left(\frac{2\pi i \langle \mathbf{c},\mathbf{u}\rangle}{p^{2k}}\right),
x_{k}a_{ij}(x)u_{ik}u_{j}= \frac{1}{2}\partial_k \big(a_{ij}(x)u_i u_j x_{k}\big) - \frac{n}{2}a_{ij}(x) u_i u_j - \frac{1}{2}x_{k}\partial_k a_{ij}(x) u_i u_j,
x_1={\frac{z_{{5}}}{z_{{1}}+2\,z_{{2}}}},x_2={\frac{2\,z_{{2}}z_5}{\left( z_{{1}}+2\,z_{{2}}\right)z_3}},x_3={\frac{z_{{2}}z_5}{\left( z_{{1}}+2\,z_{{2}}\right) z_{{4}}}}.
Q^{\ell+1}_\ell (z) = \frac{(-1)^{\ell+1}}{2}(1-z^2)^{(\ell+1)/2}\frac{d^{\ell+1}}{dz^{\ell+1}}\left[P_\ell(z) \ln\left(\frac{1+z}{1-z}\right)\right],
\frac{d^j}{dt^j}H_k(z(t)) \vert_{t=0}= \frac{d^j}{dt^j}\Big[ \Big(\frac{|z(t)|}{z(t)}\Big)^k \Big(1 + \frac{z(t)^k}{(z(t)-1)^k}+ \frac{z(t)^k}{(z(t)+1)^k}\Big) \Big] \vert_{t=0}+ O( 2^{-k/2}),
\mathcal{D}_{G}(a)=N_{G}(a)\exp \left( \int\limits_{0}^{\infty}\frac{dt}{t}\left[ \frac{a^{2}}{2}e^{-2t}-\sum_{\alpha >0}\Psi_{\alpha}(t)\mathcal{F}_{\alpha}(a,t)\right] \right)
f_{\Delta}(s,t,x)=\sum\limits_{(x_i,s_j,t_k)\in\Delta}f_{ijk}\mathbf{1}_{(x_i,x_{i+1}]}(x)\mathbf{1}_{(s_j,s_{j+1}]}(s)\mathbf{1}_{(t_k,t_{k+1}]}(t),
\left[ \hat{L}_{z}, \hat{L}_{+}\right] = \hat{L}_{+}; \qquad \left[ \hat{L}_{z}, \hat{L}_{-}\right] = - \hat{L}_{-}; \qquad \left[ \hat{L}_{+}, \hat{L}_{-}\right] = 2 \hat{L}_{z}.
\chi=\chi_{j1}\otimes\cdots \otimes \chi_{j\lambda_j}\otimes \chi_{j1}'\otimes\cdots\otimes \chi_{j\lambda_j'}', & \eta=\eta_{j1}\otimes\cdots \otimes \eta_{j\mu_j}\otimes \eta_{j1}'\otimes\cdots\otimes \eta_{j\mu_j'}'.
\int_{[-w_n, w_n]_{y_n}}e^{-y_n^2(1/4t_1 + 1/4t_2)}y_n^k = \frac{\sqrt{2\pi}C_k}{(1/4t_1 + 1/4t_2)^{(k + 1)/2}}- 2\int_{[w_n, \infty)_{y_n}}e^{-y_n^2(1/4t_1 + 1/4t_2)}y_n^k
\begin{array}{l c r}{{R e g i o n \, I \qquad}}&{{\psi^{b}= s \left( \begin{array}{c}{{E + m}}\\{{- \kappa}}\\ \end{array}\right) e^{\kappa x}}}&{{\qquad x \leq - \xi}}\\{{R e g i o n \, I I I \qquad}}&{{\psi^{b}= s^{\prime}\left( \begin{array}{c}{{E + m}}\\{{\kappa}}\\ \end{array}\right) e^{- \kappa x}}}&{{\qquad x \geq \xi}}\\ \end{array}
V_{add}^{\omega}= -\frac{m}{4 \pi}g^{2}\frac{3 m^{2}}{2 M^{2}}(\frac{1}{x}+ \frac{1}{x^{2}}) \phi{\bf L}\cdot{\bf S}+\frac{m}{4 \pi}g^{2}\frac{3 m^{4}}{16 M^{4}}\frac{1}{x^{2}}\chi Q_{12}
R(\chi,\sigma,m,r,\kappa) = \max\Big\{\sqrt{6 \, C_{m,0}(\chi,\sigma,r,\kappa)}\, r, \, C_{\ref{LemmaCorrespondenceLinearNonlinearInZero}\ref{ItemNonEmptyFixedPointSpace}}( \chi, \sigma, m, r,\mathcal{U}_\kappa)(m+1) r + 1\Big\}.
G(\rho)= \frac{-(1-q)^2(1-q^2)^\alpha}{\Gamma_{q^2}(1-\alpha )}\rho^{\nu-1}D_{q,\rho}\int_{\rho}^\infty x^{1-\nu-\alpha}\Psi_1(x)({\rho^2}/{x^2}; q^2)_{-\alpha}\,d_q x.
0 ={\cal M}^{4}- ( \tilde{\gamma}\tilde{g}^{2}k_{q}+ \frac 1{k_{q}k_{T}}) ( q_{0}^{2}+ \bar{q}_{0}^{2}){\cal M}^{2}+ \tilde{\gamma}\frac 1{k_{T}}\tilde{g}^{2}( q_{0}^{2}+ \bar{q}_{0}^{2})^{2}.
\widetilde{\Gamma}^{\overline{c}}_{ab}=-\Gamma^{\overline{c}}_{ab}=\frac{1}{2}G^{\overline{c}\,\overline{d}}\frac{\partial G_{ab}}{\partial z^{\overline{d}}}\,\,,\,\,\widetilde{\Gamma}^{c}_{\overline{a}b}=\Gamma^{c}_{\overline{a}b}=\frac{1}{2}G^{cd}\frac{\partial G_{bd}}{\partial z^{\overline{a}}}.
I(\xi,\tau)= \frac{2\chi_{\tau > |\xi|}}{\tau^2 - |\xi|^2}\int_{\mathbb{S}^1}\rho |\xi_1 \omega_2 - \xi_2 \omega_1| |\hat{F}(\xi-\rho \omega)| |\hat{g}(\rho \omega)| \rho d\omega
\int\limits_{\psi_0}^{\pi/2}Cf(0,e_n,\psi)g(\psi_0,\psi)d\psi &= \int\limits_{\psi_0}^{\pi/2}\int\limits_0^{\infty}\int\limits_{S^{n-2}}f(z\tan\psi \omega,z)(z\tan\psi)^{n-2}g(\psi_0,\psi)d\omega \frac{dz}{\cos\psi}d\psi.
\lim_{\alpha_1 \rightarrow \infty}\alpha_1 \Psi_1(x_{(\alpha_{1},\alpha_2),\varepsilon},y_{(\alpha_{1},\alpha_2),\varepsilon}) = 0, \Psi_1(x_{\alpha_2,\varepsilon},y_{\alpha_2,\varepsilon}) = 0.
S = - \frac{1}{2 \pi}\int d^{2}\sigma \left[ \partial_{\alpha}X^{\mu}\partial^{\alpha}X_{\mu}- i \; \bar{\psi}^{\mu , j}\; \rho^{\alpha}\; \partial_{\alpha}\; \psi_{\mu , j}+ i \; \bar{\phi}^{\mu , k}\; \rho^{\alpha}\; \partial_{\alpha}\; \phi_{\mu , k}\right]
\Vert\nabla \tilde{y}- \nabla y \Vert^2_{L^2(U)}\le C\sum\nolimits^4_{j=1}\Big( \Vert \nabla y - \bar{R}_j \Vert^2_{L^2(U)}+ \frac{1}{\lambda^2}\Vert y - (\bar{R}_j\,\cdot + \bar{c}_j)\Vert^2_{L^2(U)}\Big) & \le C C_m^2 (\gamma + \delta_2 + \epsilon \Vert W \Vert_*) ,
{\frac{\partial}{\partial a_{4}}}\ln T ={\frac{1}{a_{4 5}a_{4 6}}}\left\{2 ( a_{5}+ a_{6}) - \sum_{i = 1}^{4}a_{i}- \sum_{i = 1}^{3}a_{i 5}a_{i 6}{\frac{\partial}{\partial a_{i}}}\ln T \right\}.
\underline{\mathbf{R}}_{\mathbf{r}x}=\left[ \begin{array}{c}\mathcal{E}_\mathrm{s}\mathbf{p}(\sqrt{4\mathcal{E}_\mathrm{s}/\pi}+O(1/\sqrt{\mathcal{E}_\mathrm{s}}))\mathbf{q}\end{array}\right].
\tilde{z}_{\pm}= - Q_{\pm}^{- 1}\displaystyle \displaystyle \frac{\partial}{\partial \overline{{{z}}}_{\pm}}- z_{\pm}~ , \tilde{p}_{\pm}= \displaystyle \displaystyle \frac{\partial}{\partial z_{\pm}}- Q_{\pm}^{- 1}\displaystyle \frac{\partial p_{\pm}}{\partial z_{\pm}}\displaystyle \displaystyle \frac{\partial}{\partial \overline{{{z}}}_{\pm}}~ .
V_2(r) = \frac{L'(L'+1)}{r^2}- \frac{Q'}{r}f + \kappa B'_1 \frac{r}{f}+ \kappa B'_2 \frac{1}{f^2}+ \kappa B'_3 \frac{r}{f^3}+ \kappa B'_4 \frac{1}{f^4}+ R,
Q^{(m,n)}r(k) := h(k) = \prod_{i=0}^{k-1}C_{m,n}(i)\sum_{i=0}^k\left(\prod_{j=0}^{i-1}C_{m,n}(j)\right)^{-1}\left(A_{m,n}(i)\right)^{-1}r(i) + \alpha I_{m,n}(k)
\begin{aligned}\sigma^2 =\lim_{n\rightarrow\infty}\frac{\mathbb{E}[(S^\pi_n-n\varphi^\pi(r))^2]}{n}=\sum_{x\in\mathcal{X}}[\hat{r}^\pi(x)^2-(P^\pi\hat{r}^\pi(x))^2]\xi^\pi(x).\end{aligned}
\!\! \widetilde{J}_1(k)^{-1}\!=\!\frac{1}{k}\widetilde{T}_6^{-1}\widetilde{T}_1 \begin{bmatrix}I_\mu+o(1) o(k) o(k) kI_{n-\mu}+o(k)\end{bmatrix}\widetilde{M}_1, \;\; k\to0,\;\;k\in \overline{\mathbb{C}}^+.
e^2 = e, e e' = e', e e'' = 0, e e^* = 0, e' e = 0, (e')^2 = 0, e' e'' = e, e' e^* = e', e'' e = e'', e'' e' = e^*, (e'')^2 = 0, e'' e^* = 0, e^* e = 0, e^* e' = 0, e^* e'' = e'', (e^*)^2 = e^*.
\widetilde{P}_{1,1,k}^{(0^{2})}(z; \lambda_1) = -4k ( k+3 ) P_{k}^{(1,1)}(z) \left[ \lambda_{1}-\frac{1}{2}(z+1)^{2}+\frac{1}{6}(z+1)^{3}\right] \hphantom{\widetilde{P}_{1,1,k}^{(0^{2})}(z; \lambda_1) =}{}+ ( k+3 ) (1-z)^{2}(1+z)^{2}P_{k-1}^{( 2,2)}(z), k\ne 0, \widetilde{P}_{1,1,0}^{(0^{2})}(z; \lambda_1) = 1.
\kappa_i = \underset{\beta \in \Lambda^{\ell-1}}{\sum}(\bar{\partial}\varphi)_\beta \wedge \tau_{\beta\, i}+\displaystyle\sum_{0 \le k \le \ell-2,\,\beta \in \Lambda^{k}}(\bar{\partial}\varphi)_\beta \wedge \left(\displaystyle\sum_{\lambda \in \Lambda,\, 1 \le j \le k+1}(-1)^{j+1}\varphi_\lambda\,\, \tau_{\overset{\overset{j}{\vee}}{\lambda\, \beta\, i}}\right),
[ p \cdot \hat{H}, \widetilde{M}]_{\mu \nu}= \left[ p \cdot \hat{H},{( i / 2 )}\sum_{\rho \in \Delta_{+}}g_{| \rho |}{| \rho |^{2}}y ( \rho \cdot q ) \hat{s}_{\rho}\right]_{\mu \nu}, \quad \mu , \nu \in{\cal R}
\phi_n(x) = 2 \int_{\substack{|p_i|\le 1 2\le i\le n}}\left(\int_{-f_n(x;p_n,\dots,p_2)}^1 \!\left(f_n(x;p_n,\dots, p_2) + p_1\right) dp_1\right) dp_2\ldots\,dp_n &= \int_{\substack{|p_i|\le 1 2\le i\le n}}\left(f_n(x;p_n,\dots, p_2)+1\right)^2 dp_2\ldots\,dp_n.
\phi (\operatorname{N}) = \prod\limits_{\sigma \in\operatorname{Gal}(\operatorname{M}_{\Lambda''}|\operatorname{M}_{\Lambda'})}\sigma \left(\frac{\sigma_{1}(u)g_{\sigma_{1}}}{u}\right) = \prod\limits_{\sigma \in\operatorname{Gal}(\operatorname{M}_{\Lambda''}|\operatorname{M}_{\Lambda'})}\sigma(g_{\sigma_{1}}) = \operatorname{N},
\begin{array}{l}\Delta_x (f,a)= \|(\widehat{x}-a \widehat{I}) f \|_{L^2 (\mathbb{R})}= \left(\int_{\mathbb{R}}(x-a)^2 | f (x)|^2 dx \right)^{\frac{1}{2}}\Delta_{\xi}(f,b)= \|(\widehat{\xi}-b \widehat{I}) f \|_{L^2 (\mathbb{R})}= \left(\int_{\mathbb{R}}(\xi-b)^2 | \widetilde{f}(\xi)|^2 d \xi . \right)^{\frac{1}{2}}\end{array}
\left\| \lambda_q^{s}e^{-\mu\lambda_q^\alpha t}\|B_q(0)\|_{L^{2}}\right\|_{L^{\frac{\alpha}{s}}(0,T; l^2)}\leq \ \left\| e^{-\mu\lambda_q^\alpha t}B_q(0)\right\|_{L^{\frac{\alpha}{s}}(0,T; H^{s})}\leq \C(T) \|B(0)\|_{L^{2}}
\frac{\widetilde{\Theta}_{k}(r+2^{-j_{0}-k+1})-\widetilde{\Theta}_{k}(r)}{2^{-j_{0}-k+1}}\leq c(u,j_{0},x_{0},r)<\infty, &\frac{\Theta_{k}(r+d_{k})-\Theta_{k}(r)}{d_{k}}\leq c(u,j_{0},x_{0},r)<\infty,
{\frac{\alpha}{\pi}}< F_{\mu \nu}^{a}F_{\mu \nu}^{a}> = N M^{4}{\frac{1}{2}}\left( -{\frac{1}{4 0 \pi^{2}}}+{\frac{1}{2 4 \pi^{2}}}\right) ={\frac{N}{1 2 0 \pi^{2}}}M^{4}= 0 . 0 0 8 ~ G e v^{4}
Z ( J_{\mu}, \eta ) = \lim_{L \rightarrow \infty}\int \exp \{i S + \int d^{4}x [ J_{\mu}^{a}( x ) A_{\mu}^{a}( x ) + \eta^{a}( x ) \chi^{a}( x ) ] \}d A_{\mu}d \bar{c}d c d \chi
(q^{2/3}-p^{2/3}|\tau|^2)^2-p^{2/3}q^{2/3}(\tau-\Bar{\tau})^2 = q^{4/3}+p^{4/3}|\tau|^4-p^{2/3}q^{2/3}(\tau^2+\Bar{\tau}^2) = (p^{2/3}\Bar{\tau}^2-q^{2/3})\cdot (p^{2/3}\tau^2-q^{2/3}) &=(p^{1/3}\Bar{\tau}+q^{1/3})\cdot (p^{1/3}\Bar{\tau}-q^{1/3})\cdot (p^{1/3}\tau-q^{1/3})\cdot (p^{1/3}\tau+q^{1/3}) &=|q^{1/3}+p^{1/3}\tau|^2\cdot |q^{1/3}-p^{1/3}\tau|^2,
\sum_{i_{1}+i_{2}+\cdots+i_{r}=N}\frac{a_{1}^{i_{1}}a_{2}^{i_{2}}\cdots a_{r}^{i_{r}}(-z_1;q)_{i_{1}}(-z_2;q)_{i_{2}}\cdots(-z_r;q)_{i_{r}}}{(q;q)_{i_{1}}(q;q)_{i_{2}}\cdots(q;q)_{i_{r}}}.
S_2^{h_2}= \bigg( \sum_{2\le q \leq X}2 \Lambda(q) \mu(\omega_{q}) F_{\omega_{q}}(n) \bigg)^{h_2}\ll \sum_{2\le q_1, \ldots, q_{h_2}\leq X}\prod_{i = 1}^{h_2}\Lambda(q_i) \mu(\omega_{q_i})F_{\omega_{q_i}}(n),
\begin{aligned}s_1(h,z)\times_1 s_1(h',z) = s_1(hh',z), s_1(h,z)\times_2 s_1(h,z') = s_1(h,zz'), s_2(y,k)\times_1 s_2(y',k) = s_2(yy',k), s_2(y,k)\times_2 s_2(y,k') = s_2(y,kk'); \end{aligned}
M_{k,\ell}(\tfrac{\pi}{3}) = 2\cos(2n\pi)+2\cos(2n\pi+\tfrac{\ell\pi}{6})(2i\sin(\tfrac{\pi}{6}))^{-\ell}+ 2\cos(\tfrac{\ell\pi}{6})(2i\sin(\tfrac{\pi}{6}))^{-12n-\ell}= 6.
d(T_1\beta) = T_1^3 + T_1T_2^2 + T_1^2T_2, d(T_2\beta) = T_1^2T_2 + T_2^3 + T_1T_2^2, d(T_3\beta) = (a^2+ab +b^2)T_3^3 \mbox{and}d(\gamma) = T_1T_2^2 + T_1^2T_2 + (a^2b+ab^2)T_3^3
\hat{V}=(\hat{G}^{\prime}\hat{\Upsilon}\hat{G})^{-1}\hat{G}^{\prime}\hat{\Upsilon}\hat{\Psi}\hat{\Upsilon}\hat{G}(\hat{G}^{\prime}\hat{\Upsilon}\hat{G})^{-1},\hat{G}=\frac{\partial\hat{\psi}(\hat{\theta})}{\partial\theta}._{{}}
\langle \psi(u^{n-\frac{1}{2}},v^{n-\frac{1}{2}}),\delta_t v^{n-\frac{1}{2}}\rangle=\langle \psi(u^{n-\frac{1}{2}},v^{n-1}),\delta_t v^{n-\frac{1}{2}}\rangle, 1 \leq n \leq N.
\frac{1}{p}\,[u_n]^p_{W^{s,p}(\mathbb{R}^N)}-\frac{\lambda}{p}\,\int_\Omega \frac{|u_n|^p}{d_\Omega^{s\,p}}\,dx-\int_B u_n\,dx=m(\lambda)+o(1),\mbox{as}n\to \infty.
\Lambda^n_s = (W_s-W_{\eta_n(s)}) \left( \int_0^s \psi_{n,1}(u,s) dW_u \right) +(W_s-W_{\eta_n(s)}) \left( \int_{\eta_n(s)}^s \psi_{n,1}(u,s) dW_u \right) -\int_{\eta_n(s)}^s \psi_{n ,1}(u,s) du.
\mathcal{R}_{}\! =\! \bigcup_{P_{U|X}}\bigcup_{\widehat{X}(Y,U)}\!\Big\{\left(R_w,D\right)\!\colon &R_w\geq I(U;X)-I(U;Y), &D\geq E[d(X, \widehat{X}(Y,U))]\Big\}
h_{a_1a_2}(t) = h_{a_1}(t_1) = h_{a_2}(t_2) = z + 1 \notin \{0,1\}h_{b_1b_2}(s) = h_{b_1}(s_1) = h_{b_2}(s_2) = w + 1 \notin \{0,1\}.
\begin{array}{l}0=\lim\limits_{n\rightarrow \infty}\mathbb{E}[\{f_n^f(\tilde{Z}_n(s))-f_n^f(\tilde{Z}_n(t))-\int\limits_t^s A_n^A(r)f_n^f(\tilde{Z}_n(r))dr\}\prod\limits_{i=1}^k h_i(Z(t_i))]= \mathbb{E}[\{f(Z(s))-f(Z(t))-\int\limits_t^s A(r)f(Z(r))dr\}\prod\limits_{i=1}^k h_i(Z(t_i))]=0.\end{array}
\epsilon_{\mathrm{BrL}}^{s}\circ p^{s}\mathcal{L}_{\mathrm{Br}}^{s}=\epsilon_{\mathrm{Br}}^{s}H_{\mathrm{BrLie}}^{s}\mathcal{L}_{\mathrm{Br}}^{s}p^{s}\circ \eta_{\mathrm{Br}}^{s}H_{\mathrm{BrLie}}^{s}=H_{\mathrm{BrLie}}^{s}\eta_{\mathrm{BrL}}^{s}.
\frac{\Pi_{ij}^{R}}{k}=C_{2}S_{ij}+C_{3}(b_{ik}S_{jk}+b_{jk}S_{ik}-\frac{2}{3}b_{mn}S_{mn}\delta_{ij})+C_{4}(b_{ik}W_{jk}+b_{jk}W_{ik})
b_i^*\, =\bigl|\bigl\{\lambda\in\Bbb Z_r\setminus\{0,1\}:\eta(\lambda-1)=1,\ \eta(\lambda)+\eta(1-\lambda)+\eta(1-\lambda^{-1})=i\bigr\}\bigr|\cr =\bigl|\bigl\{\lambda\in\Bbb Z_r\setminus\{0,1\}:\eta(\lambda-1)=1,\ \eta(\lambda)=(i-1)/2\bigr\}\bigr|.
\phi''_\infty(z)-c\phi'_\infty(z)+\phi_\infty(z)F(\phi_\infty,\psi_\infty)(z)=0, begin{align*}0.2cm] d\psi''_\infty(z)-c\psi'_\infty(z)+\psi_\infty(z)G(\phi_\infty,\psi_\infty)(z)=0.
\frac{\|F\|_{W^{-1,p'}(\Omega)}}{K(p,N)^\frac{1}{p}}=p^\frac{1}{p}\,\left(-p'\,\min_{u\in \widetilde W^{s,p}_0(\Omega)}\left\{K(p,N)\,\|\nabla u\|^p_{L^p(\Omega)}-\langle F,u\rangle\right\}\right)^\frac{1}{p'}.
T_1(q) = 2\sum_{0 \le k \le n,\, \mathrm{ord}(\omega) \mid k}q^2({n \brack k}_q')^2{n+k \brack k}_{\omega}^2 q^{f(n,k)}, T_2(q) = 2\sum_{0 \le k \le n,\, \mathrm{ord}(\omega) \nmid k}q^2({n \brack k}_q')^2{n+k \brack k}_{\omega}^2 q^{f(n,k)}.
\mathbf{Pr}\left( \bigg\|\frac{d}{n\epsilon}\sum_{i=1}^{n}J(u_0 + \epsilon v_{0,i})v_{0,i}- \nabla J(u_0) \bigg\| \geq t - \frac{Ld\epsilon}{2}\right) \leq 2\exp\left( - \frac{cnt^2}{K^2}\right)
{\cal O}_{\pm}^{a}( \{x \}) : = \frac{1}{( N ! )^{2}}\sum_{\pi , \pi^{\prime}}\prod_{a = 1}^{N}\overline{{{\psi}}}^{( a )}( x^{( \pi ( a ) )}) P_{\pm}\psi^{( a )}( x^{( \pi ( a ) )}) \; .