harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
r_B b_{11}= r_B + (r_B b -\bar{b}) + \bar{b}= 1 + \partial_{11}^2 w_B -\partial_{22}^2 w_B - \Delta w_B +\bar{b}= (1+\bar{b}) -2\,\partial_{22}^2 w_B, r_B b_{22}= r_B - (r_B b -\bar{b}) -\bar{b}= 1 + \partial_{11}^2 w_B -\partial_{22}^2 w_B + \Delta w_B -\bar{b}= (1-\bar{b}) +2\,\partial_{11}^2 w_B,
\begin{array}{c}{{\phi^{\prime \prime}( u ) = 0}}\\{{1 + c_{w z w}- 2 6 + 6 \alpha^{\prime}( \phi^{\prime})^{2}= 0}}\\{{- \frac{1}{2}\alpha^{\prime}T^{\prime \prime}- 2 T + \alpha^{\prime}\phi^{\prime}T^{\prime}= 0}}\\ \end{array}
\int_0^t D^\nu f^{\alpha,\lambda}(t-s) I^\nu Z_sds =\int_0^t \int_0^s D^\nu f^{\alpha,\lambda}(t-s) D^{1-\nu}Z_udu ds &=\int_0^t \int_u^t D^\nu f^{\alpha,\lambda}(t-s) D^{1-\nu}Z_u ds du &=\int_0^t \int_u^t D^\nu f^{\alpha,\lambda}(s-u) D^{1-\nu}Z_u ds du &=\int_0^t \int_0^s D^\nu f^{\alpha,\lambda}(s-u) D^{1-\nu}Z_u du ds.
12I_3'=E_2I_3-I_3^3+108\Delta_3^3\,, E_4=I_3(I_3^3+216\Delta_3^3)\,, &12\Delta_3'=(E_2+3I_3^2)\Delta_3\,, E_6=I_3^6-540I_3^3\Delta_3^3-5832\Delta_3^6\,.
\begin{aligned}d^2(p_1,p_2) = g_{xx}\; x^2 + g_{yy}\; y^2 + \frac{1}{2}\partial_x g_{xx}\; x^3 + \frac{1}{2}\partial_y g_{xx}\; x^2 y+ \frac{1}{2}\partial_x g_{yy}\; x y^2+ \frac{1}{2}\partial_y g_{yy}\; y^3\cdots\end{aligned}
M_{\pm\gamma}=\left( M_{{ij}_{\pm\gamma}}\right) , M_{{ij}_{\pm\gamma}}=\begin{pmatrix}\frac{s_{ij}\pm\gamma c_{ij}}{2}\frac{s_{ij}\mp\gamma c_{ij}}{2}\frac{s_{ij}\mp\gamma c_{ij}}{2}\frac{s_{ij}\pm\gamma c_{ij}}{2}\end{pmatrix},\, 1\leq i,j\leq n
f(a,b;\mathbf{t})\ge f(0,b;\mathbf{t}) =\prod_{i=2,4,...m}\int_{0}^{t_i}\int_0^{t_{i-1}}|x_i-x_{i-1}|^{b}dx_{i}dx_{i-1}&=(b+1)^{-m/2}(b+2)^{-m/2}\prod_{i=2,4,...m}\left( t_i^{b+2}+ t_{i-1}^{b+2}-|t_{i}-t_{i-1}|^{b+2}\right) &\sim (b+1)^{-m/2}\prod_{i=2,4,...m}\left( t_i+ t_{i-1}-|t_{i}-t_{i-1}| \right) \text{as $b\rightarrow -1$.}
\left[A^{\{r,r_{i}\}}_{\{(n_{i})\}},V^{\{\lambda,\lambda_{j}\}}_{\{(n_{j})\}}\right]=\frac{1}{(2\pi i)^{2}}\oint_{0}dz\oint_{z}\frac{d\xi}{\xi}U^{\{r,r_{i}\}}_{\{(n_{i})\}}(z)U^{\{\lambda,\lambda_{j}\}}_{\{(n_{j})\}}(\xi)
\left[ \partial_\mu\partial^\mu - \partial_y^2 + e^{-2\sigma}\left\{m^2 + \left( \frac{3}{16}-\xi \right) R \right\}\right]\tilde{\varphi}+ e^{-\frac{7}{2}\sigma}V^\prime(e^{\frac{3}{2}\sigma}\tilde{\varphi})=0 .
0 \neq\int_{B(0,3\delta\theta^{-1})}\left\vert v\right\vert^{2}dx=\int_{B(0,3\delta\theta^{-1})}\left\vert \zeta_{k}v_{k}\right\vert^{2}dx+o(1) & =r_{k}^{-2}\int_{B(0,3\delta)}\left\vert \zeta(u_{k}\circ\exp_{p_{k}})\right\vert^{2}dy+o(1) & \leq C_{1}\theta^{-2}\int_{M}|u_{k}|^{2}dV_{g}.
\begin{aligned}F^{ii}\nabla_{l}u \nabla_l U_{ii}= \, \nabla_{l}u \psi_{x_l}+ \psi_u |\nabla u|^2 + \psi_{p_k}\nabla_{l}u \nabla_{lk}u + \nabla_l u \nabla_l u_t \geq \, \nabla_{l}u \psi_{x_l}+ \psi_u |\nabla u|^2 + \frac{a w^2}{\phi}\psi_{p_k}\nabla_k \phi + \frac{a w^2}{\phi}\phi_t. \end{aligned}
\mathcal{S}= \int d^4x \left( |W(z)|^2 + \frac{\partial^2 K}{\partial z^i \partial z^{\bar j}}Dz^i Dz^{\bar j}-\frac{1}{4g^2_{ij}}F^{(i)}\wedge * F^{(j)}+ \theta_{ij}F^{(i)}\wedge F^{(j)}\right)
2t+\int_{0}^{\infty}\frac{1}{\sqrt{2\pi u^3}}\left(1-e^{-\frac{2}{3}((u+t)^3-t^3)}\mathbb{E}\left[ \left(-2\int_{0}^{u}\textbf{e}(z)dz \right)\right]\right)du= \frac{e^{\frac{2}{3}t^3}}{2\pi}\int_{-\infty}^{\infty}e^{-itv}\hat{g}(v)dv
u(z)=\frac{1+\eta_{\mu_{1}}(z)}{1-\eta_{\mu_{1}}(z)}=1+2\psi_{\mu_{1}}(z) =\int_{\mathbb{T}}\left[1+\frac{2\xi z}{1-\xi z}\right]\,d\mu_{1}(\xi) =\int_{\mathbb{T}}\left[\frac{1+\xi z}{1-\xi z}\right]\,d\mu_{1}(\xi) =\int_{\mathbb{T}}\left[\frac{\xi+z}{\xi-z}\right]\,d\mu_{1}(1/\xi)
\sum_{J_l\subset G_k^j}N(J_l)\lesssim \sum_{J_l\subset G_k^j}\int_{J_l}N(t)^3dt\lesssim \int_{G_k^j}N(t)^3dt\lesssim \sum_{\alpha=k2^j}^{(k+1)2^j-1}\int_{J^{\alpha}}N(t)^3dt\lesssim 2^j\epsilon_3.
|N(u)(t,R\lambda(t))| \leq C\left(||v(t)||^{2}_{\dot{H}^{1}_{e}}+ ||L^{*}Lv(t)||_{L^{2}(R dR)}^{2}\right)\left(\log^{2}(\frac{1}{R})+1\right) \cdot \frac{|v(t,R)|}{\lambda(t)^{2}}&+C ||v(t)||_{\dot{H}^{1}_{e}}\frac{|v(t,R)|}{R \lambda(t)^{2}}\left(1+\frac{|v_{corr}(t,R\lambda(t))|}{R}\right)
\widetilde{\mathcal{S}}_{2,s}:=\inf_{u\in D^{s,2}(\mathbb{R}^N)\setminus\{0\}}\frac{\big\|(-\Delta)^{s/2}\,u\big\|_{L^2(\mathbb{R}^N)}^2}{\displaystyle\left(\int_{\mathbb{R}^N}|u|^\frac{2\,N}{N-2\,s}\,dx\right)^\frac{N-2\,s}{N}},
S^{-1}_F(p_f)=S^{(0)-1}_F(p_f) -{e \over \beta}\sum_{n = - \infty}^{\infty}\int{d^2k \over (2 \pi)^2}\gamma^{\nu}S_F(k_f) \Delta_{\mu \nu}(k_f-p_f) \Gamma^{\mu}(k_f-p_f,k_f).
\left[ X^\mu, X^\nu \right] = \lim_{\epsilon \rightarrow 0}2 \alpha' \left( \frac{-2 \pi \alpha' F}{1 - (2 \pi \alpha' F)^2}\right)^{\mu\nu}\sum_{m=1}^\infty \frac{(\rho \rho')^m + (\rho/\rho')^m}{m}\sin m \epsilon.
= \; \mathrm{T r}( \gamma_{\mu}\gamma_{\alpha}\gamma_{\nu}\gamma_{\beta}) \int_{x = 0}^{1}d x \frac{1}{( 4 \pi )^{\omega}\Gamma ( 2 )}\Bigg ( \frac{1}{2}\delta_{\alpha \beta}\frac{\Gamma ( 1 - \omega )}{[ m^{2}+ k^{2}x ( 1 - x ) ]^{1 - \omega}}
f_{A}={\frac{\pi ( e_{1}+ e_{2}) E t e^{- \pi t \epsilon^{2}/ 2}}{2 s i n ( \pi t \epsilon / 2 )}}p r o d_{n = 1}^{\infty}{\frac{( 1 - e^{- \pi t n})^{2}}{( 1 - e^{- \pi t ( n + i \epsilon )}) ( 1 - e^{- \pi t ( n - i \epsilon )})}}\ .
\int_{S^n}H_t(x)^2~d\sigma(x) =\sum_{k=0}^\infty e^{-2\lambda_kt}(h_k)^2\int_{S^n}(P_{k,n}(x\cdot x_0))^2~d\sigma(x) &=\sum_{k=0}^\infty e^{-2\lambda_kt}h_k
\lim_{N \to \infty}\Psi_N\left(Z^{\chi(1), g(1)}_{k_1}(N) \cdots Z^{\chi(n), g(n)}_{k_n}(N)\right) = \varphi_0\left( z^{\chi(1), g(1)}_{k_1}\cdots z^{\chi(n), g(n)}_{k_n}\right).
H_{2m}(x) = (2m)! \sum_{l=0}^m \frac{(-1)^{m-l}}{(2l)!(m-l)!}(2x)^{2l}H_{2m+1}(x) = (2m+1)! \sum_{l=0}^m \frac{(-1)^{m-l}}{(2l+1)!(m-l)!}(2x)^{2l+1}.
R_{x^i_{\alpha}:x^i_{\beta}}^{m}=cl\C_{x^i_{\alpha}:x^i_{\beta}}^m,\m=1,\ldots, 4 F_{x^i_{\alpha}:x^i_{\beta}}=\cup_{m=1}^4 \partial C_{x^i_{\alpha}:x^i_{\beta}}^m,
\frac{\beta_{s}^{2}}{\beta_{r}^{2}}= \frac{( 1 + \cos \theta_{s})^{2}\sin^{2}\theta_{r}}{( 1 + \cos \theta_{r})^{2}\sin^{2}\theta_{s}}e^{2 i ( \varphi_{r}- \varphi_{s})}\; .
\begin{cases}\nabla \cdot \left(G\left(\nabla w,H_p\right)-\dfrac{1}{F^2}\begin{pmatrix}0 \rho w \end{pmatrix}\right)+\dfrac{1}{F^2}\rho w_p =0 \; R\cup I, G_2(\nabla w,H_p)-\dfrac{1}{F^2}\rho w =0 \; T, w =0 \; B, \end{cases}
\beta ( n , s ) = \frac{- i}{2 \pi}\int \! d \alpha \frac{e^{- i n \alpha}}{\sqrt{4 \omega_{n}\omega_{s}}}\left\{i \omega_{n}f_{s}^{+ \ast}( t^{\prime}( 0 , \alpha ) ) e^{- i s \alpha^{\prime}( 0 , \alpha )}- \partial_{t}\left\{f_{s}^{+ \ast}( t^{\prime}( t , \alpha ) ) e^{- i s \alpha^{\prime}( t , \alpha )}\right\}\left|_{t = 0}\right. \right\}
\left\{\left[ \begin{array}{c}f_k(\underbar{y}^{(CR)}_k) Q_k(\underbar{y}^{(CR)}_k) \end{array}\right] : \underbar{y}^{(CR)}_k\in \mathbb{R}^{N^{(CR)}_k}\right\}= \left\{\left[ \begin{array}{c}\underbar{a}^{(CR)}_k \widehat{Q}_k(\underbar{a}^{(CR)}_k) \end{array}\right] : \underbar{a}^{(CR)}_k\in \mathcal{A}_k \right\}.
\partial_{t}\upsilon -i\left[ \Delta \upsilon +A\upsilon +\tilde{V}\upsilon \right] =\eta_{r}^{\prime}\left( t\right) \theta_{M}\left( x\right) \tilde{u}\left( x,t\right) -\left( 2\nabla \theta_{M}.\nabla \tilde{u}+\tilde{u}\Delta \theta_{M}\right) \eta_{r}.
\sigma_1 = \prod_{n=0}^{\infty}\alpha_{2n+1}= (1,2)(3,4)(5,6)(7,8)\cdots \sigma_2 = \gamma_1 \prod_{n=2}^{\infty}\alpha_{2n+1}= (1,2,3,4)(5,6)(7,8)\cdots \sigma_3 = \prod_{n=0}^{\infty}\gamma^2_{8n+1}\gamma_{8n+5}= \prod_{n=0}^{\infty}\alpha'_{8n+1}\alpha'_{8n+2}\gamma_{8n+5}& = (1,3)(2,4)(5,6,7,8) (9,11)(10,12) (13, 14, 15, 16) \cdots
\Phi_c^{-1}(f_i^s(x)) :=\sup_{y\in\mathbb{R}^{n_i}}\{\Phi_c^{-1}(f_i(y)) - s^{-1}\|x-y\|\}, \Phi_c^{-1}(h^s(x)) :=\sup_{y\in\mathbb{R}^{n}}\{\Phi_c^{-1}(h(y)) - s^{-1}\|x-y\|\}.
\rho L^*_{p+1}= -(x^*_n-f(x_n))\rho - \rho\theta^*(x_n) &= -(x^*_n - f(x_n))\rho + \theta^{'}(x_n)\rho + \hbox{\rm trace}f(x_n)\rho &= -(x^*_n -(f(x_n) + trace(x_n)))\rho + \theta^{'}\rho &= -(x^*_n - g(x_n))\rho + \theta^{'}(x_n)\rho &= L^{'}_{n-p}\rho.
\varepsilon_{k+1}/A_{k+1}= \left(\frac{\tau_k^2}{2\epsilon}- \frac{\mu}{2}\right)\frac{1}{\mu}\| \nabla \varphi(x_k) + \nabla \psi(z_{k+1}) \|^2 + \left(\frac{\tau_k}{2\epsilon}- \frac{\mu}{2\tau_k}\right)\|x_k - y_k\|^2.
P^\chi_{out}(\tau_\chi) \leq \big(F(\tau_\chi,K_g,K^\chi_h,\Omega_g,\Omega^\chi_h)\big)^M & \cdot\Big(1-Q(\sqrt{2K^\chi_d},\sqrt{2\tau_\chi(K^\chi_d+1)/\Omega^\chi_d P_{BS}}) \Big),
\begin{array}{c}{{[ s_{\alpha \beta}, s_{\gamma \delta}] = \epsilon_{\alpha \delta}s_{\beta \gamma}+ \epsilon_{\beta \delta}s_{\alpha \gamma}+ \epsilon_{\alpha \gamma}s_{\beta \delta}+ \epsilon_{\beta \gamma}s_{\alpha \delta}\, ,}}\\{{{}}}\\{{s_{\alpha \beta}= s_{\alpha}{}^{\gamma}\epsilon_{\gamma \beta}= s_{\beta \alpha}\, .}}\\ \end{array}
e_\lambda(x, y) ~=~\frac{4\sqrt 2}{(2\pi)^{d+1}}\int_r^\infty \frac{\sinh s}{\sqrt{\cosh 2s -\cosh 2r}}\cdot \left( \frac{-1}{\sinh s}\partial_s \right)^{d}\frac{\sin (s \cdot \sqrt{\lambda - d^2})}{s}\,ds\,.
M_{11}(e)=\sum_{i=1}^k L(u_i)l_{v_i}(e_i); M_{12}(e)=\sum_{i=1}^k L(u_i)l'_{v_i}(e_i); M_{21}(e)=\sum_{i=1}^k L'(u_i)l_{v_i}(e_i); M_{22}(e)=\sum_{i=1}^k L'(u_i)l'_{v_i}(e_i).
B_{\alpha}([x,y],z)= B(\alpha([x,y]),z)=B([x,y],\alpha(z)) = B(xy,\alpha(z))-(-1)^{|x||y|}B(yx,\alpha(z)) = B(\alpha(x),yz)-(-1)^{|x||y|}(-1)^{|x||y|+|y||z|}B(\alpha(x),zy) = B(\alpha(x),[y,z])=B_{\alpha}(x,[y,z]).
H^{+}_{i}(u)=1+h\sum_{l\geq 0}h_{il}u^{-l-1},H^{-}_{i}(u)=1-h\sum_{l<0}h_{il}u^{-l-1}, &E_{i}(u)=\sum_{l\in\mathbb{Z}}e_{il}u^{-l-1},F_{i}(u)=\sum_{l\in\mathbb{Z}}f_{il}u^{-l-1}.
\widetilde{(R_\eta \mu')'}\, (E) = (R_\eta \mu')'(\pi_{\mathfrak{X}}^{-1}(E))= R_\eta \mu'(K \cap \pi_{\mathfrak{X}}^{-1}(E)) = (R_\eta \mu )'_{K \eta^{-1}}(K \cap \pi_{\mathfrak{X}}^{-1}(E)) = R_\eta \mu (K \eta^{-1}\cap K \cap \pi_{\mathfrak{X}}^{-1}(E))
\|F_w-F_w^{(0)}\|_Y \leq \|F_w- F_w^{(N+1)}\|_{Y}+ \sum_{k=0}^{N}\left\|F_w^{(k+1)}- F_w^{(k)}\right\|_{Y}&\leq \|F_w- F_w^{(N+1)}\|_{Y}+ \sum_{k=0}^N \frac{1}{\gamma}\left\|F_v^{(k+1)}- F_v^{(k)}\right\|_{Y}&\leq \|F_w- F_w^{(N+1)}\|_{Y}+\frac{1}{\gamma}\frac{\alpha}{1-\alpha}\frac{C}{\omega_1}.
\kappa_{4}^{2}\equiv \frac{\sqrt{2}( \Delta + 4 )}{8}\kappa^{2}\sigma , \quad \tilde{\kappa}_{4}^{2}\equiv \kappa_{4}^{2}( 1 - \varphi )^{\frac{8}{3 ( \Delta + 4 )}}= \kappa_{4}^{2}( 1 - \varphi )^{\frac{2}{3 b^{2}+ 1}}.
g_1(x) = \sum_{1 \leq |k| \leq \frac{n-1}{2}}e^{-\pi \alpha k^2}\left( \sum_{j=1}^{n}e^{2 \pi i k \varepsilon_j}e^{-2 \pi i k \frac{j}{n}}\right)e^{2 \pi i k x}
F_{m,\ell}(P_1(t), \ldots, P_n(t)) =h(t)F_{m, \ell}\left(\frac{P_1(t)}{h(t)}, \ldots, \frac{P_n(t)}{h(t)}\right) &= h(t) F_{m, \ell}\left( P_{1, a}\left(\frac{t-a}{d}\right), \ldots, P_{n, a}\left( \frac{t-a}{d}\right) \right).
\begin{cases}\partial_tg^{n+1}+\frac 1 \varepsilon v\cdot\nabla_{x}g^{n+1}+\frac 1{\varepsilon^2}\mathcal{L}g^{n+1}=\frac{1}{\varepsilon}\Gamma(g^{n}, g^{n}), g^{n+1}|_{t=0}=g_0,\end{cases}
\phi ( z ) = \frac{c o n s t}{\sinh z}\exp \int_{0}^{\infty}\frac{d t}{t}\frac{\sinh \frac{1}{2}( 1 - \nu ) t \, \left( \cosh t ( \frac{1}{2}- z / ( i \pi ) ) - 1 \right)}{\sinh \frac{1}{2}\, \nu t \sinh t}
\begin{cases}f (z^+,u^+) - f (z^-,u^-) = \Xi (z^+,z^-,u^-) z^+, z^-\in B(\bar z;\delta) u^+,u^-\in B(\bar u;\delta) \end{cases}\Longleftrightarrow u^+=T(z^+, z^-)(u^-) \, .
{\cal W}^{D}= \sum_{\mathrm{\scriptsize ~ a l l ~ f i e l d s}}\int ( d_{\varphi_{k}}+ x \partial_{x}) \varphi_{k}( x ){\frac{\delta}{\delta \varphi_{k}( x )}}\qquad d_{\varphi_{k}}= \dim^{U V}{\varphi_{k}}
\frac{1}{\sqrt{\det \left( \hat{{\cal G}}_{\parallel}^{( 0 )}\right)}}= \frac{1}{\sqrt{\det \left( \hat{{\cal G}}_{\perp}^{( 0 )}\right)}}= \exp \left( - \beta_{T}\int_{- \infty}^{\infty}d k \, \rho_{0}( k ) \Phi ( k^{2}; \beta_{T}, \mu^{2}) \right)
C(t)= \begin{cases}-t \beta_1 +\alpha w(t) + \beta_2 \mathbb{E}\left(\left[\sum\limits_{i=1}^{N(t)}X_i+t\right]^+\right), t<0, (\alpha +\beta_2)w(t) + \beta_2 t, t\geq 0. \end{cases}
\begin{aligned}\lim_{N\to\infty}\phi_N^m(\vec s,\vec t) = \lim_{N\to\infty}\phi_N^M(\vec s,\vec t) =\phi(s,t) \equiv \exp\left( -\frac{1}{2\beta}\int_0^1 s^2(x) dx+ \int_0^1 \int_0^1 t(x) C(x,y) t(y) \, dx dy \right), \end{aligned}
\begin{cases}d_{n}^{*}=b^{v}_{n}+4(v^{-1}-1)\lambda^{2v}b_{n}^{-v}c_{n}^{*}=f(b_{n}^{v})=2\lambda^{v}+4(v^{-1}-1)\lambda^{2v}b_{n}^{-v},\end{cases}
\widehat{\rho}_{k_n}= \displaystyle \sum_{j=1}^{k_n}\rho_{n,j}\phi_{n,j}\otimes \phi_{n,j}= \displaystyle \sum_{j=1}^{k_n}\frac{D_{n,j}}{C_{n,j}}\phi_{n,j}\otimes \phi_{n,j},
{ap \brack bp}_q \equiv \binom{a}{b}+ \frac{\binom{a}{b}b(a-b)p}{2}(q^p-1) + \frac{\binom{a}{b}b(a-b)}{2}\left( \frac{b(a-b)}{4}p^2 + \frac{ap^2-5}{12}-\frac{p-1}{2}\right)(q^p-1)^2 \bmod [p]_q^3.
2\langle A^{0}\partial_{x}^{\alpha}u_{t},\partial_{x}^{\alpha}u\rangle=\frac{d}{dt}\langle A^{0}\partial_{x}^{\alpha}u,\partial_{x}^{\alpha}u\rangle-\langle \partial_{x}^{\alpha}u,(\partial_{t}A^{0})\partial_{x}^{\alpha}u\rangle,
(\log G_1)^{(i)}(s,x) =(-\log x)^{i-1}x^{-s}\bigg( \sum_{|\Im (\rho+s)| \le x}\frac{x^{\rho}}{\rho-s}+ O_{\varepsilon}\bigg( x^{\frac{1}{2}}\frac{\log (2+|\Im s|)}{\log x}\bigg) \bigg) &=(-1)^i(\log x)^{i-1}x^{-s}\big(\psi(x)-x + O_{\varepsilon}(x^{\frac{1}{2}}\log^2(2+|\Im s|))\big).
\frac{1}{\sqrt{3 !}}\varepsilon_{b_{1}b_{2}b_{3}}\overline{{{\psi}}}_{1 d_{1}}( y_{1}) \overline{{{\psi}}}_{2 d_{2}}( y_{2}) \overline{{{\psi}}}_{3 d_{3}}( y_{3}) U^{d_{1}b_{1}}( y_{1}, y_{M}) U^{d_{2}b_{2}}( y_{2}, y_{M}) U^{d_{3}b_{3}}( y_{3}, y_{M}) | 0 \rangle
\|\Gamma_j^h\circ \Phi\|_{\mathcal{L}_2(L_x^2;\mathbb{C})}\leq \|\Phi\|_{\gamma(L_x^2;W_x^{\mathfrak{s}, r/2})}\|\Gamma_j^h \|_{\mathcal{L}(W_x^{\mathfrak{s}, r/2};\mathbb{C})}, j=1,2,3,4.
O\left(\sum_{i \geqslant 1}(n+k_i)\log^2(n+k_i)\mu((n+k_i)\tau_2+m\tau_2+\tau_1)\right) = O\left(\sum_{i \geqslant 1}\frac m{2^i}\log^2(m)\mu(2m\tau_2+\tau_1)\right) =O(m\log^2(m)\mu(m\tau_2+\tau_1)).
\langle (\mathrm{id}\otimes V_MU_M)F_{\chi|\mathcal H \times \mathcal H}, (\mathrm{id}\otimes V_M) g\times g\rangle = M^{2-k}\langle (\mathrm{id}\otimes U_M)F_{\chi|\mathcal H \times \mathcal H}, g\times g\rangle,
H = 4\begin{bmatrix}\Re\left(\frac{d}{1-\overline{c}d}-\frac{1+c}{1-|c|^2}\right) \Im\left(\frac{d}{1-\overline{c}d}-\frac{c}{1-|c|^2}\right) \Im\left(\frac{d}{1-\overline{c}d}-\frac{c}{1-|c|^2}\right) \Re\left(\frac{-d}{1-\overline{c}d}-\frac{1-c}{1-|c|^2}\right) \end{bmatrix}
\max_{\vec{\pmb{\alpha}}}\sum_{i=1}^M \alpha_i \log_2\left(1+\frac{1-\alpha_i}{\alpha_i}\frac{h_{ss}^i\eta P_p}{\sigma_s^2+h_{ps}^iP_p}\right) & h_{sp}^i(1-\alpha_i)\eta P_p\leq\alpha_i P_{int},i=1,\ldots,M. & \vec{0}\prec\vec{\pmb{\alpha}}\prec\vec{1}
\mathrm{pr}_k^+(g) := (-1)^{\lfloor \frac{\lambda_k+1}{2}\rfloor}\frac{1}{2\sqrt{2}}\biggl(\sum_{\nu\ (\mathrm{mod}\ 4)}\bigl(g|_k A\bigr)|_k B_{\nu}\biggr) + \frac{1}{2}g
\liminf_{{\mathcal N}\ni n\to\infty}S_m(z_{1,n}, \ldots, z_{n,n}) \ge \left(\tau(H_a)\right)^m \liminf_{{\mathcal N}\ni n\to\infty}S_m(z_{1,n}, \ldots, z_{l_n,n}) = \left(\tau(H_a)\right)^m \left(\int z\,d\tau_a(z)\right)^m.
K_N(x,y) =\bigl[e^{\Delta/N^2}-e^{4\Delta/N^2}\bigr](x,y)-\bigl[e^{-\mathcal L_a/N^2}-e^{-4\mathcal L_a/N^2}\bigr](x,y) &=\bigl[e^{\Delta/N^2}-e^{-\mathcal L_a/N^2}\bigr](x,y)-\bigl[e^{4\Delta/N^2}-e^{-4\mathcal L_a/N^2}\bigr](x,y).
[z^n]T(y, z) = -\frac{\sqrt{y(1+4y)}}{\Gamma(-1/2)}(4y+1)^{n}n^{-3/2}+ O((4y+1)^n n^{-5/2}) =\frac{\sqrt{y(1+4y)}}{2\sqrt{\pi n^3}}(4y+1)^n\Bigl(1+O\Bigl(\frac1n\Bigr)\Bigr).
g_\alpha(x)=\cr (x-1)\sum_{l=0}^{r-1}x^l\sum_{0\le i_0,i_1,i_3\le 1}(-1)^{i_0+i_1+i_3}\binom{i_0+i_1\tau+i_3\tau^3+\alpha-\frac{\alpha+1}r+\frac lr}\alpha\frac{x^{r(i_0+i_1+i_3)}-1}{x^r-1}-1.
\begin{pmatrix}\widetilde{X}_1 \widetilde{X}_2\end{pmatrix}=U(T)\begin{pmatrix}X_1 X_2\end{pmatrix}U^{\dagger}(T)=\begin{pmatrix}\cos \theta -\sin \theta \sin \theta \cos \theta\end{pmatrix}\begin{pmatrix}X_1 X_2\end{pmatrix}+\begin{pmatrix}A B\end{pmatrix},
\langle \varphi_i, \varphi_k \overline{\theta}_j \rangle = ( w(i) w(k) )^{-1/2}\langle \theta_i , \theta_k \overline{\theta}_j \rangle &= ( w(i) w(k) )^{-1/2}\, w(k) \, \delta_{r+1, t}\, \delta ( (i,j^T), k ) \, \langle 1, 1 \rangle &= \left( \dfrac{w(k)}{w(i)}\right)^{1/2}\!\!\! \delta_{r+1, t}\, \delta ( (i,j), k ),
\Leftrightarrow \qquad \delta_{qp}u^{~++}_{\underline{m}}= v^+_q \Gamma_{\underline{m}}v^+_p , \qquad \delta_{\dot{q}\dot{p}}u^{--}_{\underline{m}}=v^{-}_{\dot q}\Gamma_{\underline{m}}v^{-}_{\dot p}, \qquad \gamma^{i}_{q \dot p}u^{i}_{\underline{m}}= v^+_q\Gamma_{\underline{m}}v^{-}_{\dot p},
Q_A V_k Y_k = U_{k+1}X_{k+1}\begin{bmatrix}C_k 0\end{bmatrix}, Q_A^T U_{k+1}X_{k+1}= V_k Y_k \begin{bmatrix}C_k 0\end{bmatrix}+ \alpha_{k+1}v_{k+1}e_{k+1}^T X_{k+1}, Q_B V_k Y_k = \widehat{U}_{k}\widehat{X}_k S_k, Q_B^T \widehat{U}_{k}\widehat{X}_k = V_k Y_k S_k + \check{\beta}_{k}v_{k+1}e_{k}^T \widehat{X}_k.
\vert (\mathcal{P}_{n}\mathcal{B})(t) - \mathcal{B}_{n}(t)\vert = \Big \vert \sum_{i=0}^{n}\mathcal{L}_{i}(t) (\mathcal{B}(t_{i}) -\mathcal{B}_{i}) \Big \vert \leq \sum_{i=0}^{n}\vert \mathcal{L}_{i}(t) \vert \vert \mathcal{B}(t_{i}) - \mathcal{B}_{i}\vert,
u^{(i_0)}= em_v(u^{(i_0)})f,\ \widetilde{u}^{(i_0)}= \widetilde{e}m_v(u^{(i_0)})\widetilde{f},\textit{and}&\widetilde{u}^{(i_0)}= \widetilde{e}\left(e^{-1}\cdot u^{(i_0)}\cdot f^{-1}\right)\widetilde{f},
a_{S}^{\dagger}( k ) a_{S}^{\dagger}( k^{\prime}) - q^{- 2}\sum_{l , l^{\prime}\in{\cal I}}{\hat{\cal R}}\left( k , k^{\prime}; l , l^{\prime}\right) a_{S}^{\dagger}( l ) a_{S}^{\dagger}( l^{\prime}) = 0 ,
\begin{aligned}\frac{1}{2C}\int |\nabla u^k|^2 dx-\int |\nabla d^k|^2|\nabla u^k|dx+\frac{1}{2}\int_0^t\int \rho^k |u_t^k|^2dxd\tau & \le C+C\int_0^t \int \left(\rho^k |u^{k-1}|^2|\nabla u^k|^2+|u^{k-1}| |\nabla \rho^k| |\nabla u^k|+|\nabla d^k| |\nabla d_t^k| |\nabla u^k|\right)dxd\tau &=C+\sum_{i=1}^3 I_{3i}.\end{aligned}
V_{9}+V_{14}\leq \varepsilon(\mathcal{G}_{1}(K)+\mathcal{G}_{2}(K))E_{\mu}+\frac{1}{4}\sum_{j=1}^{d}\int_{\mathbb{T}^{d}}(\partial_{x_{j}}(Dw^{1}-Dw^{2}))^{2}\dx.
\int_{t+1}^{\infty}ds |K_{1}(s-t,\lambda(t))-\frac{\lambda(t)^{2}}{4(1+s-t)}| \leq C \int_{t+1}^{\infty}ds \frac{\lambda(t)^{2}}{4}\left(\frac{1}{s-t}-\frac{1}{1+s-t}\right) + C \int_{t+1}^{\infty}ds \frac{C \lambda(t)^{2}(1+\lambda(t)^{2})}{(s-t)(1+(s-t)^{2})}&\leq C \lambda(t)^{2}(1+\lambda(t)^{2})
{\cal R}_{{\cal A}}{}^{{\cal B}}(z;g)=\left(\begin{array}{cl}R_{A}^{~B}(z;g) i(\Sigma^{j}(z;g)\tilde{\gamma}_{A}L(z;g))^{\beta}0 L_{\alpha}^{~\beta}(z;g)U_{i}^{~j}(z;g)\end{array}\right)
\mathcal{V}_sf(x) =\|K\ast f(x)\|_{\ell^s({\mathbb{Z}^+})}&=\left(\sum_{k=1}^\infty |\phi_k\ast f(x)-\phi_{k-1}\ast f(x)|^s\right)^{1/s}&=\left(\sum_{k=1}^\infty|A_{n_k}f(x)-A_{n_{k-1}}f(x)|^s\right)^{1/s}
\begin{aligned}\int_{B_{t}(x_0)}(|\nabla u|^{2})^{\mu}dx \le\theta\int_{B_{s}(x_0)}(|\nabla u|^{2})^{\mu}dx +C\int_{B_s(x_0)}|\nabla h|_n dx &\quad+C\left\{\frac{1}{(s-t)^{n}}\int_{B_{s}(x_0)}d^{n}(u,h)dx+\int_{B_{s}(x_0)}gdx\right\}\end{aligned}
\check{j}(x,a,b)= \tilde{j}(x,a,b) +\tilde{h}(x,a,b,0) \check{j}(a,a,b) &=\frac{I_a^{(q,r)}(x)}{(I_a^{(q,r)})'(b+)}\Big( (H^{(q,r)}_a)'(b) -\frac{(J_a^{(q,r)})'(b)}{Z^{(q)}(-a)}\check{j}(0,a,b) \Big) - H^{(q,r)}_a(x) + \frac{J_a^{(q,r)}(x)}{Z^{(q)}(-a)}{\check{j}(0,a,b)}.
V_{3 , i}= ( 2 \pi )^{3}r_{m_{i}}r_{m_{i}^{\prime}}r_{m_{i}^{\prime \prime}}\ , \quad \mathbf{\omega}_{{\mathbf 3}, i}= \frac{1}{V_{3 , i}}\mathbf{d y}^{m_{i}}\wedge \mathbf{d y}^{m_{i}^{\prime}}\wedge \mathbf{d y}^{m_{i}^{\prime \prime}}\ .
\eta(\phi)=\eta\left( |\xi^H|^{2n-2}\varphi_h\right) &=(2n-2)|\xi^H|^{2n-4}\left\langle \eta,\xi^H\right\rangle \varphi_h + |\xi^H|^{2n-2}\eta(\varphi_h) &=\frac{2n-2}{2n+1}|\xi^H|^{2n-2}\left\langle \eta,\xi^H\right\rangle (l-3k) + \frac{2n-2}{2n+1}|\xi^H|^{2n-2}\left\langle \eta,\xi^H\right\rangle \left( 3k-l\right) =0.
\left(\begin{array}{lllll}1 0 \ldots 0 0 \binom{3}{2}1 \ldots 0 0 \vdots \ddots \ldots 1 0 \binom{2k+1}{k+1}\binom{2k+1}{k+2}\ldots \binom{2k+1}{2k}1\end{array}\right)^{-1}= D.
\frac{F}{V}=-\sum_{\stackrel{n=1}{n ~\rm odd}}^\infty\sum_{k=1}^\infty\sum_{s=0}^{k-1}\frac{1}{k^2}\left( \frac{1}{4\pi^2\alpha'\tau_2}\right)^52^9 \left|\theta_4\left(0,2\tau\right)\right|^{-16}e^{-n^2\beta^2/4\pi\alpha'\tau_2}
\mathcal{T}_{t,k}^{\epsilon, v_k}\psi(x) = \mathbb{E}_{x, k}^{\epsilon, v_k}\left \{\psi\bigl(X_{k}^{\epsilon,v_k}(t) \bigr);\, \tau_{k}^{\epsilon, v_k}> t \right\}, \,\,\, x \in D, \,\,\, k = 1, 2, \ldots, n,
[\tilde{S}_+,\tilde{S}_-]_{P.B.}=-2iH(S_3)\tilde{S}_3+iS_+ \displaystyle\frac{\partial A_4}{\partial S_3}\bar{C}_3 C^+-iS_- \displaystyle\frac{\partial A_4}{\partial S_3}\bar{C}_3 C^-~,~~~[\tilde{S}_3,\tilde{S}_{\pm}]_{P.B.}=\mp i\tilde{S}_{\pm}~,
T_i (p^X_0 c^{X^{-1}}_{w_0}) = \big(c^X_{i,i+1}\xi_{s_i}+ (t^{\frac12}-c^X_{i,i+1})\big) p^X_0 c_{w_0}^{X^{-1}}&= \big(c^X_{i,i+1}+ (t^{\frac12}-c^X_{i,i+1})\big) p^X_0 c_{w_0}^{X^{-1}}= t^{\frac12}\big(p^X_0 c_{w_0}^{X^{-1}}\big).
\left<\frac{a}{d}\right>^{\pm}= -\pi \int_{\infty}^0 \left ( f( \frac{a}{d}+iy) \pm f(- \frac{a}{d}+iy)\right ) \; dy = \pi \left( \Lambda(1, f, \frac{a}{d}) \pm \Lambda(1,f, -\frac{a}{d})\right )= \frac{1}{2}\bigg(L(1, f, \frac{a}{d}) \pm L(1, f, -\frac{a}{d})\bigg).
A_{x_{\perp}}^{2 1}= \int_{R^{2}}d^{2}z \psi_{- \frac{1}{2}}^{\dagger}( z + \omega ) \frac{\partial}{\partial x_{\perp}}\psi_{\frac{1}{2}}( z - \omega ) = \frac{\bar{x}_{\perp}}{2 \pi | x_{\perp}|}e^{2 i x \cdot \omega}( I + J ) .
\begin{array}{l l}{{\bf ( 3 , 3 , 1 ) , ( 1 , 1 , 1 ) , ( 3 , 1 , 2 4 8 )}\qquad}&{\mathrm{w i t h}\Delta = 4 ( r e l e v a n t )}\\{{\bf ( 4 , 4 , 1 ) , ( 2 , 2 , 1 ) , ( 4 , 2 , 2 4 8 )}\qquad}&{\mathrm{w i t h}\Delta = 6 ( m a r g i n a l ) .}\\ \end{array}
\mathcal{B}\mathcal{J}_{2n-2}=(-1)^{2n-2}q^{-\frac{1}{2}[(\lambda_{1},\lambda_{2})+\sum_{j<s}(\alpha_{i_{j}},\alpha_{i_{s}})-\sum_{j,\alpha}(\alpha_{i_{j}},\lambda_{\alpha})]}\mathcal{J}_{2n-2}=q^{n-\frac{1}{2}}\mathcal{J}_{2n-2}.
{\mathbb Y}_{a}{}^A{}_{B}:=Y_a{}^AY_B,{\mathbb Z}_a{}^{b}{}^A{}_{B}:=Y_a{}^AX^b{}_B,{\mathbb W}^{A}{}_{B}:=X^AY_B,{\mathbb X}^{b}{}^A{}_{B}:=X^AX^b{}_B,
\Big|\int_{B_{\delta}(x^\prime)\times(0,\delta)}J_\varepsilon(|x-\hat{y}|)\partial_{x_n}c(x)(x_n+y_n)y\Big| \leq \int_0^1 K|\partial_{x_n}^2c(x^\prime,tx_n)|t = K\frac{1}{x_n}\int_0^{x_n}|\partial_{x_n}^2c(x^\prime,z_n)|z_n,
A^{ijkl,*}:= \lambda \delta^{ij}\delta^{kl}+ \mu\left(\delta^{ik}\delta^{jl}+ \delta^{il}\delta^{jk}\right) , B^{ijkl,*}:= \theta \delta^{ij}\delta^{kl}+ \frac{\rho}{2}\left(\delta^{ik}\delta^{jl}+ \delta^{il}\delta^{jk}\right) ,
p_{\varphi,\sigma}(z,\xi,\tau) = |\xi|^{2}-\tau^{2}|\nabla_{z}\varphi (z)|^{2}-\sigma^{2}+2i\tau \xi\cdot\nabla_{z}\varphi(z), (z\in V,\ \xi\in \mathbb{R}^3,\ \tau \in \mathbb{R}).
\frac{I_1}{I_2}= - \frac{\theta_0(4\eta; 2\tau) \theta_0(\tau + 1/2; 2\tau) \theta_0(-2\eta + 1/2; 2\tau)}{\theta_0(-2\eta; 2\tau) \theta_0(4\eta +1/2; 2\tau) \theta_0(\tau; 2\tau) \theta_0(-2\eta + \tau; 2\tau)}\frac{1}{\Gamma(-2\eta + \tau + 1/2)\Gamma(2\eta + \tau + 1/2)}
\begin{array}{lcl}p_{2\star}[p_1^\star(\eta T) \wedge \tau^\star(dd^cK_m^j)]= &= p_{2\star}[p_1^\star(\eta T) \wedge \tau^\star(\varphi_m^j(x)\beta^m)]+p_{2\star}\left[p_1^\star(\eta T) \wedge \tau^\star\left((\psi_m\gamma)\star\chi_j(x)\wedge\beta^{m-1}\right)\right] &= P_m^j(x)+Q_m^j(x).\end{array}
\sum_{(c)}c_{(1)}\otimes(\sum_{(c_{(2)})}(c_{(2)})_{(1)}\otimes(c_{(2)})_{(2)})=\sum_{(c)}(\sum_{(c_{(1)})}(c_{(1)})_{(1)}\otimes(c_{(1)})_{(2)})\otimes c_{(2)}.
g_{k}(x,y_{k})=\exp\left[\sum_{i=1}^{n}\left\{\sum_{j=1}^{p}y_{k}^{(i)}z_{k}^{(i,j)}x^{(j)}-\log\left(1+e^{\sum_{j=1}^{p}z_{k}^{(i,j)}x^{(j)}}\right)\right\}\right].
S_{\mathrm{e f f}}[ \chi ] = \int \! d^{2}x \, \Bigl ({\frac{1}{2}}\, \partial_{\alpha}\chi \partial_{\alpha}\chi -{\frac{\delta v}{2}}\, \chi^{2}+{\frac{\pi V}{N}}\, \chi^{4}+{\cal O}\bigl ( \chi^{6}/ N^{2}\bigr ) \Bigr ) ,
u_b^{(n+1)}= \frac{1}{2^g}\sum_{b_1 + b_2 = b}v_{b_1}^{(n)}t_{b_2}^{(n)}s_b^{(n+1)}= \frac{1}{2^g}\sum_{b_1 + b_2 = b}t_{b_1}^{(n)}t_{b_2}^{(n)}.
ds^2 = \left( 1-{r_-\over r}\right)^{-1}(dx^5 +{2Q\over r}dt)^2-\left( 1-{r_+\over r}\right) dt^2 +{1-r_-/r\over 1- r_+/r}dr^2 + r^2(1-{r_-\over r})^2d\Omega^2_2,
\left[ - \frac{d^{2}}{d z^{2}}+ \frac{\lambda}{2}\phi_{\mathrm{w a l l}}^{2}( z ) + \frac{\sigma}{4 !}\phi_{\mathrm{w a l l}}^{4}( z ) \right] \chi_{n}( z ) = \eta_{n}^{2}\chi_{n}( z ) \; ,
\int{\cal D}z \mu [ z ]{\cal P}_{\mu_{1}\nu_{1}}[ z ] \cdots{\cal P}_{\mu_{2 n}\nu_{2 n}}[ z ] = \frac{\left( a^{2}\right)^{2 n}}{( 2 n - 1 ) ! !}\left[ \hat{1}_{\mu_{1}\nu_{1}, \mu_{2}\nu_{2}}\cdots \hat{1}_{\mu_{2 n - 1}\nu_{2 n - 1}, \mu_{2 n}\nu_{2 n}}+{\,}\mathrm{p e r m u t a t i o n s}{\,}\right] .