harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{}^{\gamma \gamma}G_{1 2 3 4}^{4}= 4 \left\{\frac{\delta^{2}W}{\delta D_{1 2}^{- 1}\delta D_{3 4}^{- 1}}+ \frac{\delta W}{\delta D_{1 2}^{- 1}}\frac{\delta W}{\delta D_{3 4}^{- 1}}\right\}= - 2 \frac{\delta \;{}^{\gamma}G_{1 2}^{2}}{\delta D_{3 4}^{- 1}}+{}^{\gamma}G_{1 2}^{2}{}^{\gamma}G_{3 4}^{2}.
t^{j,\chi}(z) = \sum\limits_{j'\neq j,d}\iota_j^*(_+)^{vir}_* \left[ \displaystyle\frac{Q^d_{(e,s_1)}}{-z-\psi_++\chi}\left\{(_-)^*\left[F^{j'}(w)-t^{j',\chi}(w) \right] \right\}_{w=\psi_--\chi}\right]
\langle\left.\frac{(\cos(2Q_{\frac{1}{\lambda(t)}})-1)}{r^{2}}v_{1}\right\vert_{r=R \lambda(t)},\phi_{0}\rangle =\frac{-1}{\lambda(t)^{2}}\int_{0}^{\infty}v_{1}(t,R \lambda(t)) \frac{8}{(1+R^{2})^{2}}\frac{2R}{1+R^{2}}RdR &=-\frac{16}{\lambda(t)^{2}}\int_{0}^{\infty}v_{1}(t, R\lambda(t))\frac{R^{2}}{(1+R^{2})^{3}}dR
\mathcal{X}^n - \mathcal{X}= \Pi_{K}(\mathcal{X}+\mathcal{Y}+t_n (\mathcal{Z}_1+\mathcal{Z}_2))- \Pi_{K}(\mathcal{X}+\mathcal{Y}), \mathcal{Y}^n - \mathcal{Y}= \Pi_{K^{\circ}}(\mathcal{X}+\mathcal{Y}+t_n (\mathcal{Z}_1+\mathcal{Z}_2))- \Pi_{K^{\circ}}(\mathcal{X}+\mathcal{Y}).
\tilde{B}_{-,\omega}(t) :=\varrho \Big(\phi(t)+\frac{\pi}{2}+\frac{\pi}{3}, \omega, t \Big), \tilde{B}_{+,\omega}(t) := \varrho \Big(\phi(t)+\frac{3\pi}{2}-\frac{\pi}{3}, \omega, t \big).
\begin{cases}\begin{array}{rll}\omega_p(8n\rho_0) \leq \sqrt{\frac{n + 1}{n}}- 1 0\leq\omega_p(\rho)\log\left(\frac{1}{\rho}\right) \leq L \mbox{for all}\rho\leq\rho_0 \omega_p(\rho) \leq\frac{1}{4}\delta \mbox{for all}\rho\leq\rho_0.\end{array}\end{cases}
- \frac{4 m_{F}^{2}}{( 4 \pi )^{2}}\left( \xi_{0}- \frac{1}{6}\right) \int_{0}^{t}U^{2}( t ) d t + \frac{1}{3 ( 4 \pi )^{2}}\sum_{i}N_{i}\int_{o}^{t}m_{i}^{2}( t ) d t .
\begin{aligned}\lambda^k\left\langle g_i^k-\nabla f_i(\bar{x}^k), \bar x^k-\theta^*\right\rangle \geq -L\lambda^k\|x_i^k-\bar{x}^k\|\|\bar x^k -\theta^*\| \geq- \frac{\gamma^k}{2}\|x_i^k-\bar{x}^k\|^2- \frac{L^2 (\lambda^k)^2}{2\gamma^k}\|\bar x^k -\theta^*\|^2\end{aligned}
Z_{N+1,\omega}^{\beta,0}= \frac{1}{\sqrt{2\pi}}\cdot \prod_{i=0}^{N}\exp(-\frac{\beta}{2}\omega_i)\cdot \left[ \sum_{k=1}^N \sum_{i=0}^{N-k}k^2\exp(-\beta \omega_i)\exp(-\beta \omega_{i+k})\right]^{-1/2}
\int_X \int_X u(x) h(x-y) v(y) \, dx \, dy = \int_X \int_X \left( \int_G h(g^{-1}.(x-y)) \, dg \right) u(x)v(y) \, dx \, dy = \int_G \int_X \int_X h(g^{-1}.(x-y)) u(x)v(y) \, dx \, dy \, dg.
{\bf{Cov}}\left[V_{\varepsilon}(u), \,V_{\varepsilon}(v)\right] =\begin{cases}- \, 2\log |u-v|, \, \varepsilon\leq|u-v|\leq 1, 2\left(1-\log \varepsilon -\frac{|u-v|}{\varepsilon}\right),\, |u-v|\leq \varepsilon.\end{cases}
\renewcommand{\arraystretch}{2}\begin{array}{l c l}{{\displaystyle \Omega_{1}= - \frac{1}{4}( F_{2}+ F_{3}+ F_{4}) + \cdots ,}}&{{~}}&{{\displaystyle \Omega_{2}= - \frac{1}{4}( - F_{1}+ F_{3}+ F_{4}) + \cdots ,}}\\{{\displaystyle \Omega_{3}= - \frac{1}{4}( - F_{1}- F_{2}+ F_{4}) + \cdots ,}}&{{~}}&{{\displaystyle \Omega_{4}= - \frac{1}{4}( - F_{1}- F_{2}- F_{3}) + \cdots ,}}\\ \end{array}
|P_{E}\left(T \sum_{i=1}^n [0, e_i]\right)|_{n-m}= \sum_{|I|=n-m}|\det(e_1, \dots, e_m, \{P_E T e_i\}_{i \in I}|_{n}= \sum_{|I|=n-m}|\det(e_1, \dots, e_m, \{Te_i\}_{i \in I}|_{n}= |\det(T)| \sum_{|I|=n-m}|\det(T^{-1}e_1, \dots, T^{-1}e_m, \{e_i\}_{i \in I}|_{n}= |\det(T)| \sum_{|J|=m}|\det(\{w_i^J\}_{i=1}^m)|,
\Psi:=\mathrm{Fl}_{-y_{n}^*}^{Y_{-1}^\prime}\circ\mathrm{Fl}_{-y_{n-1}^*}^{Y_{1-n}}\circ\cdots\circ\mathrm{Fl}_{-y_1^*}^{Y_{-1}}\circ\mathrm{Fl}_{-\tfrac{v_n^*}{2}}^{X_{-1}^{\prime}}\circ\mathrm{Fl}_{-v_{n-1}^*}^{X_{1-n}}\circ\cdots\circ\mathrm{Fl}_{-v_2^*}^{X_{-2}}\circ\mathrm{Fl}_{-v_1^*}^{X_{-1}}.
b_{\lambda}b_{\mu}=\psi_{w_{0}}y_{\lambda}\psi_{w_{0}}y_{\min}\psi_{w_{0}}y_{\mu}\psi_{w_{0}}y_{\min}&=(-1)^{\tfrac{n(n-1)}{2}}\psi_{w_{0}}(y_{\lambda}\psi_{w_{0}}y_{\mu})\psi_{w_{0}}y_{\min}
E_{k}(r_{k}^{A})+\sum_{l=1}^{N}min_{X}E_{kl}(r_{k}^{A},r_{l}^{X})>E_{k}(r_{k}^{B})+\sum_{l=1}^{N}max_{X}E_{kl}(r_{k}^{B},r_{l}^{X})
{\bf V}_{\mu}= V_{\mu}^{a}( z ) \frac{\partial}{\partial z^{a}}+{\bar{V}}_{\mu}^{\bar{a}}( \bar{z}) \frac{\partial}{\partial \bar{z}^{a}}, \quad [{\bf V}_{\mu},{\bf V}_{\nu}] = C_{\mu \nu}^{\lambda}{\bf V}_{\lambda},
\widehat{\Theta_\rho^d}(l) =\rho^2\left[-l(l+n-1)+\bar a\right]\widehat{\Psi_\rho^d}(l) &=-\widehat{\Psi_\rho^{d+2}}(l)-\rho\cdot(n-1)\,\widehat{\Psi_\rho^{d+1}}(l)+\rho^2\cdot\bar a\,\widehat{\Psi_\rho^d}(l)
\lim_{\eta_1,\eta_2 \to 0}\lim_{N \to \infty}\mathbb{E}(V_{N}(t_1+i\eta_1)\overline{V_{N}(t_2+i\eta_2)})\bigg|_{d_{N}=N}= \left(\frac{1}{\pi}\frac{\sin(\pi(t_1-t_2))}{(t_1-t_2)}\right)^{2}
\lim_{k \to \infty}\frac{p(x^k)}{k^4}= \lim_{k \to \infty}\frac{k^4(p(x^k)-p(\tilde x^k))}{k^8}+ \frac{p(\tilde x^k)}{k^4}=\lim_{k \to \infty}\lim_{i \to \infty}\frac{p_i(\tilde x^k) + 2(k-\frac{1}{k})^2c_i}{k^4}
d \hat{s}^{2}= \varepsilon^{2}d s^{2}= - \hat{n}^{2}\hat{c}^{2}d \hat{t}^{2}+ \hat{a}^{2}\hat{\gamma}_{i j}d \hat{x}^{i}d \hat{x}^{j}+ \hat{b}^{2}d \hat{y}^{2},
[a, b] = a^{-1}b^{-1}ab = (bb^{-1}) a^{-1}(bb^{-1}) b^{-1}ab = b (b^{-1}a^{-1}b) b^{-1}(b^{-1}a b) = b \bar{a}b^{-1}\bar{a}^{-1}\in [B, A]_1
\frac{1}{\beta}\int_0^{\beta}d\tau_1\cdots\int_0^{\beta}d\tau_n\,\langle\eta^{k_1}(\tau_1)\cdots\eta^{k_n}(\tau_n)\rangle_{0,c}\propto\beta^{n-1+(k_1+\cdots+k_n)/2}.
\max_{\boldsymbol{\theta}}\mathbb{E}\|\boldsymbol{G}\boldsymbol{\Theta}\boldsymbol{S}\boldsymbol{h}_r + \boldsymbol{h}_d \|_2^2 & ~~\textrm{s.t.}\quad~ |\theta_n| = 1, ~n = 1,\ldots, N.
\mathrm{Tr}_2(g_j)=\mathrm{Tr}^{GL_2}_{SL_2}(g_j) =\mathcal{I}\circ\mathrm{Tr}(g_j) =\mathcal{I}\circ\left(\dfrac{1}{2^{d-1}}\sum_{\epsilon_i\in R}\psi'(\epsilon_i^{-1})g_j\mid_{k,w}\epsilon_i\right).
m \equiv \left(\partial_{\underline{y}}\Lambda \right) \Lambda^{-1}={\textstyle\frac{1}{2}}m^{i}T_{i}={\textstyle\frac{1}{2}}\left(\begin{array}{cc}m^{1}m^{2}+m^{3}& m^{2}-m^{3}-m_{1}\end{array}\right)\, .
\begin{cases}\displaystyle{u_j^n=a_1 c_{j-1}^n+a_2 c_{j}^n+ a_1 c_{j+1}^n}\displaystyle{(u_j^n)_x=-a_3 c_{j-1}^n+a_3 c_{j+1}^n}\displaystyle{(u_j^n)_{xx}=a_4 c_{j-1}^n+ a_5 c_{j}^n +a_4 c_{j+1}^n,}\end{cases}
U_{\gamma}= ( \alpha_{\gamma}) \otimes \left( \begin{array}{c c c}{{\beta_{\gamma}}}&{{}}&{{}}\\{{}}&{{\beta_{\gamma}}}&{{}}\\{{}}&{{}}&{{\beta_{\gamma}^{- 2}}}\\ \end{array}\right) \otimes \left( \begin{array}{c c c}{{\delta_{\gamma}}}&{{}}&{{}}\\{{}}&{{\delta_{\gamma}^{- 1}}}&{{}}\\{{}}&{{}}&{{}}\\ \end{array}\right)
\int_{0}^{t}|\mathcal{C}{_i^\alpha}(v_j)|^2d\tau=\Lambda(\frac{1}{c_0},|h|_{\mathcal{W}{^{2,\infty}_t}}+\|\nabla v\|_{\mathcal{W}{^{1,\infty}_t}})\int_{0}^{t}|(\nabla v)^b|{_{\mathcal{H}^{m-1}}^2}+|\nabla_yh|{_{\mathcal{H}^{m-1}}^2}d\tau.
\beta_i:=\alpha_i\ (1\leq i\leq l), \beta_{l+1}:=\alpha_{l+1}+\cdots+\alpha_{n+1}, &\gamma_i:=\alpha_{i+l+1}\ (1\leq i\leq n-l-1), \gamma_{n-l}:=\alpha_{n}+2\alpha_{n+1}.
\begin{aligned}\sum\limits_{i=1}^{B_{1}}\sum\limits_{j=1}^{K}d_{i,j}=\sum\limits_{i=1}^{B_{1}}\sum\limits_{j=1}^{B_1}d_{i,j}+\sum\limits_{i=1}^{B_{1}}\sum\limits_{j={B_1}+1}^{K}d_{i,j}, \end{aligned}
u \prec_t v = \sum_{i \geq 0}u \cdot T_i (v) u \succ_t v = \sum_{i \geq 0}T_i (u) \cdot v u \curlyvee_t v = \sum_{i \geq 0}t^i (\sum_{j+k =i}H(T_j(u), T_k(v))).
P_{i j}( k^{( 0 )}) = P_{1 1}( k^{( 0 )}) \delta_{i j}\, , \qquad P_{i 4}( k^{( 0 )}) = P_{4 j}( k^{( 0 )}) = 0 \, , \quad i , j = 1 , 2 , 3 \, ,
\partial_r \varphi ={1 \over{3(r - r_H)}}[\eta_P e^{3\varphi}{\bf P}_{ij\,\infty}+ \eta_Q e^{-{3\varphi}}{\bf Q}_{ik\,\infty}]^2{1 \over{\eta_P{\bf P}_{ij\,\infty}+ \eta_Q{\bf Q}_{ik\,\infty}}}.
\begin{cases}\frac{\partial \varphi_{\epsilon}}{\partial t}=\log \frac{\omega_{\varphi_{\epsilon}}^n}{\omega_{\epsilon}^n}+F_{\epsilon}+\gamma (k\chi+\varphi_{\epsilon})+\theta_X(\omega_{\varphi_{\epsilon}}) \varphi_{\epsilon}|_{t=0}=c_{\epsilon 0}.\end{cases}
\left[ \begin{smallmatrix}T; N,\mu d \end{smallmatrix}\right]^+ &=(T;N,d)^+ \Big( \left[ \begin{smallmatrix}T; N, d-1 d-1 \end{smallmatrix}\right]^+ + \sum_{h=1}^{\mu - d}(q^h T_{N+d}T_{N+d+1}\dots T_{N+d+h-1}) \left[ \begin{smallmatrix}T; N, d +h -1 d-1 \end{smallmatrix}\right]^+\Big)
g_iI_{\lambda}\cap s_iI_{\mu,i-1}= (g_iI_{\lambda}\cap I_{\mu,i}) \sqcup M^i_{t+1,t}, g_iN_{\lambda}\cup N_{\mu,i}= (g_iN_{\lambda}\cup s_iN_{\mu,i-1}) \sqcup M^i_{t,t+1}.
\bar{H}_{1}=\sum_{j=1}^{N}u_{j}+\beta \sum_{|j_{1}-j_{2}|\operatorname{mod}N>1}u_{j_{1}}u_{j_{2}}+\beta^{2}\sum_{|j_{a}-j_{b}|\operatorname{mod}N>1}u_{j_{1}}u_{j_{2}}u_{j_{3}}+\cdots
G ( f )_{\beta}^{( n )}= \Sigma_{m = 0}^{n}\{\tilde{\Theta}_{\beta}^{( n - m )}, \tilde{f}^{( m )}\}_{( q )}+ \Sigma_{m = 0}^{( n - 2 )}\{\tilde{\Theta}_{\beta}^{( n - m )}, \tilde{f}^{( m + 2 )}\}_{( \phi )}+ \{\tilde{\Theta}_{\beta}^{( n + 1 )}, \tilde{f}^{( 1 )}\}_{( \phi )}
Z_{B_{+}}^{U ( 1 )}= \frac{1}{( 2 \pi )^{\Omega^{\prime \prime \prime \prime}}}\int{\cal D}W_{B_{+}, c}^{3}{\cal D}\psi_{W_{B_{+}, c}}^{3}e^{- S_{B_{+}, c}^{1}( 2 )},
-L(e_{kj}) =[L(e_{ij}),e_{ki}]+[e_{ij},L(e_{ki})] &=C_{jk}^{jj}e_{ii}-C_{ij}^{ij}e_{kj}-\sum_{y>j}C_{jy}^{jj}e_{ky}&+C_{jk}^{kk}e_{ii}-\sum_{x<k}C_{xk}^{kk}e_{xj}-C_{ki}^{ki}e_{kj}&=-\sum_{x<k}C_{xk}^{kk}e_{xj}-\left(C_{ij}^{ij}+C_{ki}^{ki}\right)e_{kj}-\sum_{y>j}C_{jy}^{jj}e_{ky}+\left(C_{jk}^{jj}+C_{jk}^{kk}\right)e_{ii}.
\dot{x}=-\alpha \int_{0}^{t}\bar{\nu}^{T}\bar{w}_{2^{k}}\left(t_{1}\right) dt_{1}-\beta \int_{0}^{t}\bar{r}^{T}M_{2^{k}}\bar{\nu}dt_{1}+\dot{x}_{0}, t \in[0,\tau]
n = \frac{1}{4}(3a+2b+c)^2 + \frac{3}{2}(3a+2b+c) - (b+c), m=m(a,b,c) = 27{{a+2}\choose{3}}+ 8{{b+2}\choose{3}}+{{c+2}\choose{3}}+ 3(3a+2b+5)ab + + \frac{3}{2}(3a+c+4)ac + (2b+3c+3)bc + 6abc
\bigg| \nu_i(\xi) \gamma'_j(t) \frac{|z|^2 \delta_{ij}- n z_i z_j}{|z|^{n+2}}\bigg| = C\frac{|z_1z_2|}{|z|^{n+2}}\le C\frac{|z|^2|z|}{|z|^{n+2}}\le C|z|^{1-n}= C|\xi-y|^{1-n}
z\frac{d}{dz}+\left( \begin{array}{cccccc}b_{1}(z) b_{2}(z) b_{3}(z) \ldots b_{n-1}(z) b_{n}(z) 1 0 0 \ldots 0 0 0 1 0 \ldots 0 0 0 0 1 \ldots 0 0 . . . . . . 0 0 0 \ldots 1 0 \end{array}\right)\leftrightarrow \frac{d^n}{d\tau^n}+\bar{b}_{1}(\tau)(\frac{d^{n-1}}{d\tau^{n-1}})+\cdots+\bar{b}_{n}(\tau),
\left[\alpha(1-\alpha)\frac{\partial^2}{\partial\alpha^2}+(1+\varepsilon-2\alpha)\frac{\partial}{\partial \alpha}+\nu(\nu+1)\right]{_2}F_1\left(\left. \begin{array}{c}-\nu,\nu+1\ 1+\varepsilon\ \end{array}\right| \alpha\right)=0
\max \bigg\lbrace \sum_{i,j=1}^n t_it_j|U^\top U|_{ij}:t\in\mathbb{R}^n,\;\|t\|_2=1,U\in \mathbb{R}^{m\times n},\; UU^\top={\rm I}_m \bigg\rbrace & \le\frac{m}{n}\left( 1 + \sqrt{\frac{(n-1)(n-m)}{m}}\right),
n^{-q/2}\sum_{r_1 + \cdots + r_q = n}\exp\left(-n \epsilon\left|\frac rn - p\right|^2\right) \le n^{-q/2}\sum_{r_1, \dots, r_q =1}^n \exp\left(-n \epsilon\left|\frac rn - p\right|^2\right) = \prod_{i=1}^q \left[ n^{-1/2}\sum_{r=1}^n \exp\left(-n \epsilon\left(\frac rn - p_i\right)^2\right) \right].
\int_{\ell_n}^{M_n}\frac{x+1}{(\log(x+1))^{\beta-1}}dx=\sum_{\ell=0}^k\frac{n^2}{2B_\ell(\log n)^{\beta+\ell-1}}+O\left(\frac{n^2}{(\log n)^{2\beta-2}}\right).
\unitlength=1.00mm\linethickness{0.4pt}\begin{picture}(141.00,3.00)\put(72.00,-3.00){\circle{12.}}\put(66.00,-3.00){\circle{2.00}}\put(7.00,-3.00){\makebox(0,0)[lc]{$\Delta(x)=\Delta_{0}(x)=G(x,x)=~(x)$}}\end{picture}
\pi_{P_x}(e) = \pi_{P_x}(x+z+h+f(z+h)-x-z-f(z)-L(h)) =\pi_{P_x}(h+f(z+h)-f(z)-L(h)) =h-\pi_{P_x}(f(z+h))-\pi_{P_x}(f(z)) - \pi_{P_x}(L(h)) =h-h =0,
\frac{\partial w}{\partial t}+ Lw - (\lambda + \mu) w = -\frac{\partial u}{\partial y}\frac{\partial v}{\partial y}\: \mbox{in}\: D_T, \frac{\partial w}{\partial t}+ L_\pm w - (\lambda + \mu) w = - \frac{\partial u}{\partial y}\frac{\partial v}{\partial y}\: \mbox{in}\: D_T^\pm, w(T) = 0 \: \mbox{in}\: D
\sum_{n\ge 1}\frac{\max_{1\le i\le n}V_n^q}{n^p}\le \sum_{r\ge 0}\frac1{2^{(p-1)r}}(\sum_{j=0}^r2^{k\varepsilon q'q})^{q/q'}(\sum_{k=0}^r 2^{-k\varepsilon}V_{2^k}^q)\le C \sum_{n\ge 0}\frac{V_{2^n}^q}{2^{np}}\, .
\tilde b_k:=(1-e^{-r_k^2\mathcal L_a})^\mu b_k g_k:=\bigl[1-(1-e^{-r_k^2\mathcal L_a})^\mu\bigr] b_k= \sum_{\nu=1}^\mu c_\nu e^{-\nu r_k^2\mathcal L_a}b_k,
| f^{\prime}| \leq | f | \left({\frac{N}{| w_{1}|}}+{\frac{1}{| w_{1}- \beta |}}+{\frac{1 / 2}{| 1 - w_{1}|}}\right) \leq{\frac{\sqrt{2}}{1 + \beta}}\left( N +{\frac{1}{1 + \beta}}+{\frac{1}{4}}\right) ,
U_k = \{(e_i)_{i \in \mathbb{N}}\in Y \mid \forall 0 \leq i \leq n_k \colon e_i = e_{ki}\}U'_k = \{(e'_i)_{i \in \mathbb{N}}\in Y \mid \forall 0 \leq i \leq m_k \colon e'_i = e'_{ki}\}
A(\rho)=\int_{D}\rho(z)^{2}dzd\overline{z}=\int_{D}\rho^{*}(f(z))^{2}|f^{\prime}(z)|^{2}dzd\overline{z}=\int_{D^{*}}\rho^{*}(w)^{2}dwd\overline{w}=A(\rho^{*}).
\sum_{i=2}^{\infty}[z^i]\frac{z^2}{(1+z)^3}u^{i-2}\frac{\partial}{\partial q_i}+\sum_{k=0}^{\infty}(-1)^k\binom{k+2}{k}u^{k+1}\frac{\partial}{\partial q_{k+3}}= E^{(1)}.
\sum_{t=0}^{k-1}\sum_{j=0}^{\ell-1}m_{t,j}(x)q_j^{(t)*}(x) \eta_j^*(y)\zeta_t^*(z)=a(x,y,z)(x^s-\alpha) + b(x,y,z)(y^\ell-\beta) +c(x,y,z)(z^k-\gamma).
\mathrm{tr}\left(G_0FG_0F\right)=4\mathrm{tr}\left(\left(\Delta\delta_{\mu\nu}+\Delta\partial_\mu\partial_\nu\Delta\right)\bar{F}_{\nu\sigma}\left(\Delta\delta_{\sigma\tau}+\Delta\partial_\sigma\partial_\tau\Delta\right)\bar{F}_{\tau\mu}\right)+\textrm{cubic}
\left|\int_{\mathbb{R}^{d-1}}\Psi_{a,\zeta,\mu,\sigma}\tilde{\Psi}_{a,\zeta,\mu,\sigma}\, dx -1\right| < \zeta, \left|\int_{\mathbb{R}^{d-1}}\beta(|a|^\frac{1}{p}\Psi_{a,\zeta,\mu,\sigma})\tilde{\Psi}_{a,\zeta,\mu,\sigma}\, dx - \sigma |a|^\frac{1}{p}\right| <\zeta \int_{\mathbb{R}^{d-1}}\tilde{\Psi}_{a,\zeta,\mu,\sigma}\, dx = 0,
- g (T^a) \chi_{\beta}^{\dagger}\left\{2\frac{k^i}{[k^+]}- \frac{\sigma \cdot p_{2\bot}-im}{[p_2^+]}\sigma^i - \sigma^i \frac{\sigma \cdotp_{1\bot}+ im}{[p_1^+]}\right\}\chi_{\alpha}\epsilon^{i*}
\nabla^{\perp\psi}_{\bar E_{\alpha}}\nabla^{\perp\psi}_{\bar E_{\alpha}}H^{\psi}=\frac{q}{p+q}\bar\nabla^{\perp}_{\bar E_{\alpha}}\bar\nabla^{\perp}_{\bar E_{\alpha}}H_2+\frac{p}{p+q}B^j(\nabla^{\mathbb{C}P^q}_{\bar E_{\alpha}}\bar E_{\alpha},H_1)-\frac{p}{p+q}H_1.
\begin{aligned}\Theta((\beta')^{\bullet}) \cdot \theta(C_S(P)) = \theta(g) \cdot \Theta(\beta^{\bullet}) \cdot \theta(C_S(P)) = = \Theta(\beta^{\bullet}) \cdot \theta(\beta^{\bullet}(g)) \cdot \theta(C_S(P)) = \Theta(\beta^{\bullet}) \cdot \theta(C_S(P))\end{aligned}
I=\frac12 \sqrt{\frac{\pi}{2}}\frac{\Gamma\big(\frac{n+s+1}{2}\big)}{2^{\frac{n+s}{2}\Gamma\big(\frac{n+s+2}{2}\big)}}\Big(\frac{1+16w^2}{32w^2}\Big)^{\frac{n+s-1}{2}}\Big(\frac{8u}{1+16w^2}\Big)^{n+s}= \frac{\sqrt{\pi}}{2}\frac{\Gamma\big(\frac{n+s+1}{2}\big)}{\Gamma\big(\frac{n+s+2}{2}\big)}w(1+16w^2)^{-\frac{n+s+1}{2}}.
\| u(t) - v(t) \|_{1,\alpha}= \| f - g \|_{1,\alpha}+ \big\| h( \cdot - tf(\cdot)) - h( \cdot - tg( \cdot )) \big\|_{1,\alpha}\geq \big\| \nabla \big( h( \cdot - tf( \cdot )) - h(\cdot - tg(\cdot)) \big) \big\|_{0,\alpha}&= \big\| h'( \cdot - tf(\cdot)) - h'(\cdot - tg(\cdot)) \big\|_{0,\alpha}.
\frac{\eta_n}{n}=\frac{\eta_{3N+1}}{3N+1}=\frac{N+1}{3N+1}\ \frac{1}{N+1}\sum_{i=1}^{N+1}(Z_{3i-2}-1) &+\frac{N}{3N+1}\Biggl(\frac{1}{N}\sum_{i=1}^N(Z_{3i-1}-1)+\frac{1}{N}\sum_{i=1}^N(Z_{3i}-1) \Biggr).
I_2= \int_{\frac{1}{2}}^{m_0}\int_{|x^\prime|<1}\left|\frac{u_\varepsilon(x^\prime,\psi(x^\prime)+t)}{t}\right|^p dx^\prime dt \leq C\int_0^{m_0}\int_{|x^\prime|<1}|u_\varepsilon(x^\prime,\psi(x^\prime)+t)|^pdx^\prime dt.
[ \hat{S}_{+}, \hat{S}_{-}] = \displaystyle \frac{\partial G ( S_{3})}{\partial S_{3}}\vert_{S_{3}\to \hat{S}_{3}}\stackrel{d e f .}{=}[ [ \hat{S_{3}}] ] , [ \hat{S}_{3}, \hat{S}_{\pm}] = \pm \hat{S}_{\pm},
Dg(x_*;r_*)(x-x_*;r-r_*) =D_x g(x_*;r_*)(x-x_*)+D_r g(x_*;r_*)(r-r_*) &=2n \tan \frac{\pi}{n}\sum \limits_{i=1}^{n}(x_i-(x_*)_i)=0.
f_j =\sum_{m\in\mathbb{Z}^2}\widetilde{\eta}_m\eta_mf_j &=\sum_{m\in\mathbb{Z}^2}\widetilde{\eta}_mf_{j,m}+\sum_{m\in\mathbb{Z}^2}\widetilde{\eta}_m\left[\eta_m\left(\psi^{(j)}\ast_jf_j\right)-\psi^{(j)}\ast_j\left(\eta_mf_j\right)\right] &=:f_{j,\flat}+f_{j,\sharp}+f_{j,}
h(\lambda_k) - h(\lambda^\star) = L(x_k, \lambda_k) - L(x^\star,\lambda^\star) &\ge L(x_k, \lambda_k) - L(x_k, \lambda^\star) & = (\lambda_k - \lambda^\star)^T g(x_k) & \ge - \| \lambda_k - \lambda^\star \|_2 \| g(x_k) \|_2 .
D_{\mu \mu}=-\frac{g_{\mu \mu}}{32\pi^{2}}\left[\frac{\alpha^{2}}{(3-d)}+32\pi^{2}\alpha\left(\frac{\mu_{R}^{2}}{\lambda_{R}}+\frac{1}{2}\varphi^{2}\right)-16\pi^{2}\frac{\alpha^{2}}{\lambda_{R}}\right] \;\;,
\bar{\theta}(s,t)\bar{\theta}(st,r)\psi(str) =\bar{\theta}(s,t)\bar{\psi}(st)\,^{st}\bar{\psi}(r) &=\bar{\psi}(s)^s\bar{\psi}(t)^{st}\bar{\psi}(r) &=\psi(s)\,^s\left (\bar{\theta}(t,r)\psi(tr)\right ).
\left( \sum_{j_{1}=1}^{\infty}\left( \ldots\left( \sum_{j_{m}=1}^{\infty}\left\Vert A\left( e_{j_{1}},\dots,e_{j_{m}}\right) \right\Vert^{s_{m}}\right)^{\frac{s_{m-1}}{s_{m}}}\dots\right)^{\frac{s_{1}}{s_{2}}}\right)^{\frac{1}{s_{1}}}\leq D_{m,p,\mathbf{s}}^{\mathbb{K}}\Vert A\Vert
Mf(y) = \sup_{y \in \mathcal{I}}\frac{\int_{\mathcal{I}}|f(y') (y')^\gamma | (y')^{1+\omega}e^{-\frac{(y')^2}{4}}dy'}{\int_{\mathcal{I}}(y')^{1+\omega}e^{-\frac{(y')^2}{4}}dy'}, \omega = \sqrt{d^2-12d +24},
\limsup_{r\rightarrow1-0}\int_{\partial D_r}|f_n-f|^2\varphi|dz| \le L_1^2\limsup_{r\rightarrow1-0}\int_{\partial D_r}|f_n-f|^2\frac{\partial\psi}{\partial v_z}|dz| &\le L_1^2\int_{\partial D}|f_n-f|^2\frac{\partial\psi}{\partial v_z}|dz| &\le L_1^4\int_{\partial D}|f_n-f|^2\varphi |dz|.
\eta_{r s}A_{-}^{r}B^{s}= 1 + 2 i h f^{- 1 / 2}\theta_{( 1 6 )}^{T}\dot{\theta}_{( 1 6 )}+ i f^{- 1 / 2}v_{i}\partial_{j}h \theta_{( 1 6 )}^{T}\gamma^{i j}\theta_{( 1 6 )}+ \cdots ~ ,
P^*(f, \overline{s}\mathcal{F}) = \left(h_\mu(f) + \overline{s}\lim_{m \rightarrow \infty}\frac{1}{2^m}\int_M \Phi_{2^m}d\mu \right) &= \lim_{m \rightarrow \infty}\frac{1}{2^m}\left(h_\mu(f^{2^m}) + s_{2^m}\int_M \Phi_{2^m}d\mu \right) &\leq \lim_{m \rightarrow \infty}\frac{1}{2^m}P(f^{2^m}, s_{2^m}\Phi_{2^m}) = 0.
( 2 h ) = - \ \frac{i}{6 ( 2 \pi )^{4}}\t r \int d^{d}k \ \frac{\stackrel{\displaystyle ( S k )}{\scriptscriptstyle ( - p ) \phantom{+}}\stackrel{\displaystyle ( S ( k + q ) )}{\scriptscriptstyle ( - q ) \phantom{+ q q q q q q q}}\stackrel{\displaystyle ( S ( k - p ) )}{\scriptscriptstyle ( p + q ) \phantom{+ q q q q q q q}}}{k^{2 l}( k - p )^{2 l}( k + q )^{2 l}}
u\,=\,\tfrac{x^2}{2}+\tfrac{x^2y}{2}+\tfrac{x^2y^2}{2}+F_{5,0}\,\tfrac{x^5}{120}+F_{4,1}\,\tfrac{x^4y}{24}+\tfrac{x^2y^3}{2}+{\rm O}_{x,y}(6).
r_{12}r_{13}= \sum_{i,j}a_i \cdot a_j \otimes b_i \otimes b_j, r_{13}r_{23}= \sum_{i,j}a_i \otimes a_j \otimes b_i \cdot b_j, r_{23}r_{12}= \sum_{i,j}a_j \otimes a_i \cdot b_j \otimes b_i.
W ( x_{i}) = - \frac{1}{4}\sum_{i = 1}^{N}x_{i}^{4}+ \alpha \sum_{\stackrel{i , j = 1}{i < j}}^{N}\ln ( x_{i}^{2}- x_{j}^{2}) + \beta \sum_{i = 1}^{N}\ln x_{i}^{2}.
v_{\Lambda \Omega}= \frac{1}{2}\mathrm{{s g n}}( \sigma_{R_{i j}^{( \ell )}\Lambda}^{a}) \, v_{\Lambda_{i j}\left( R_{i j}^{( \ell )}\Lambda \right)_{i j}}n_{\omega^{s}}( \Lambda_{i}- \ell ) \bigl ( n_{\omega^{s}}( \Lambda_{i}- \ell ) - 1 \bigr ) \, ,
\begin{array}{l l}{z^{\prime}= z + \varepsilon \zeta \theta ;}{\theta^{\prime}= \theta + \varepsilon \zeta}{\bar{z}^{\prime}= \bar{z}- \bar{\varepsilon}\bar{\zeta}\bar{\theta};}{\bar{\theta}^{\prime}= \bar{\theta}+ \bar{\varepsilon}\bar{\zeta}.}\end{array}
d_{\mathrm{st}}=\int dz \sum_{\alpha\in \Delta_+}e_{\alpha}(z)\otimes \psi_{\alpha_i}^*(z)-\frac{1}{2}\sum_{\alpha,\beta,\gamma\in \Delta_+}c_{\alpha,\beta}^\gamma:\psi_\gamma(z)\psi_\alpha^*(z)\psi_\beta^*(z):, &d_{\chi}=\int dz \sum_{\alpha\in \Pi}\psi^*_\alpha(z)=\sum_{\alpha\in \Pi}\psi^*_{\alpha,1}.
\sum_{i=1}^{n}\bar{\iota}^{*}\bar{g}(\pi_{C(L)}(\bar{\nabla}_{e_{i}}(JY)), e_{i}) =- \sum_{i=1}^{n}f^{i}(\nabla_{e_{i}}Y) &=\sum_{i=1}^{n}e^{i}(\phi (\nabla_{e_{i}}Y)) &=\sum_{i=1}^{n}e^{i}(\nabla_{e_{i}}(\phi Y) - (\nabla_{e_{i}}\phi) (Y)) &=\sum_{i=1}^{n}e^{i}(\nabla_{e_{i}}(\phi Y)) + n \eta(Y).
S_{CSM}~=~ \int d^2x~\left[ -\frac{1}{4}F_{\mu \nu}F^{\mu \nu}+\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi +eA_{\nu}(\eta^{\mu \nu}-\epsilon^{\mu \nu})\partial_{\mu}\phi +\frac{1}{2}ae^{2}A_{\mu}A^{\mu}~\right],
R = R_{\Sigma}+ 2 R_{\mu \nu}n_{i}^{\mu}n_{i}^{\nu}- R_{\mu \nu \lambda \rho}n_{i}^{\mu}n_{i}^{\lambda}n_{j}^{\nu}n_{j}^{\rho}- ( \chi_{i \mu}^{\mu})^{2}+ \chi_{i}^{\mu \nu}\chi_{i \mu \nu}
M_{X}f(x) =\sup_{x\in P \in{F( X)}}{1\over{\mu(P\cap X|)}}\int_{P \cap X}|f(y)| d\mu(y), M^{\sharp}_{X}f(x) =\sup_{x\in P \in{F( B)}}{1\over{\mu(P \cap X)}}\int_{P \cap X}|f(y) - f_{P\cap X}| d\mu(y)+ \frac{1}{\mu(X)}\int_{X}f(y) d\mu(y),
D_{ij,kl}^{ab,cd}={Q\over N\sum_{b=0}^{K-1}\tilde{B}(b)}\sum_{\tilde{m}=0}^{P-1}\tilde{B}(i-j-Q\tilde{m}){\rm e}^{i{2\pi \over K}(a-c)(i-j-Q\tilde{m})}\delta_{ab}\delta_{cd}\delta_{il}\delta_{jk}.
\frac{1}{k}\sum_{j=1}^{k}c_{k}(j)\log j = \frac{1}{2}\sum_{d|k}\frac{\mu(d)}{d}\log d + \log\sqrt{2\pi}\sum_{d|k}\frac{\mu(d)}{d}+ \frac{\vartheta}{12}\sum_{d|k}\frac{\mu(d)}{d^2}.
E_{\mu \mathrm{m}}={\frac{1}{2}}\biggl ({\frac{\partial u^{\mu}}{\partial X^{\mathrm{m}}}}+{\frac{\partial u^{\mathrm{m}}}{\partial X^{\mu}}}+ \sum_{\mathrm{n}= 1}^{3}{\frac{\partial u^{\mathrm{n}}}{\partial X^{\mu}}}{\frac{\partial u^{\mathrm{n}}}{\partial X^{\mathrm{m}}}}\biggr )
\lim_{\varepsilon \to 0}\int_{-\frac{1}{2}}^{\frac{1}{2}}v(\sigma) \int_{{\mathbb{R}}^n}u(t-2\varepsilon(1-\sigma),x) \overline{\phi}(t-2\varepsilon(1-\sigma),x)\,{\rm d}x\,{\rm d}\sigma=\int_{{\mathbb{R}}^n}u(t,x)\overline{\phi}(t,x)\,{\rm d}x
\gamma_{n}( \tau ) = - i \frac{l_{s}^{2}p^{+}}{2 \mu n}\left( \omega_{n}^{2}- \left( \frac{n}{l_{s}^{2}p^{+}}\right)^{2}\right) b_{n}e^{- i \nu_{n}\tau}\quad \textrm{f o r}\quad n \neq 0
\alpha_{( 1 )}^{( 1 )}= - \frac{\sqrt{2}}{6}\sigma d ( K_{\gamma}( t ) - \sqrt{2}b K_{\phi}( t ) + f_{\gamma}( t ) - \sqrt{2}f_{\phi}( t ) ) + P^{\prime}( 0 , t ) - \frac{\kappa^{2}}{6}( \rho + U ) d e^{\sqrt{2}b \hat{\phi}}.
\begin{aligned}J\mathcal{N}_2(u_1, u_2) = 2|u_1|^2 Ju_2 - u^2_1 \overline{Ju_2}& +2 (\overline{u}_1 u_2 Ju_1 - u_1 u_2 \overline{Ju_1}) + 2 u_1 \overline{u}_2 Ju_1.\end{aligned}
g=\operatorname{div}\left( \mathbf{B}{\mathbf{u}}\right) =\operatorname{div}\left({\mathbf{B}}\nabla\varphi\right) +\operatorname{div}\left(\mathbf{B}R^{\operatorname{curl}}(\operatorname{curl}{\mathbf{u}})\right)\quad\mbox{in $\omega$},\qquad\varphi=g_{D}\quad\mbox{on $\partial\omega$}.
\tilde{I}(g) =I(\tilde{g}_t)=\int_{SM}(H(\tilde{g})-\lambda H(u,u)(\tilde{g}))\tilde{\eta}, &=\int_{SM}\psi^{-1}(H(g)-\lambda H(u,u)(g))\psi^{\frac{n}{2}}\eta, &=\psi^{\frac{n-2}{2}}I(g), &=(V(t))^{\frac{2-n}{n}}I(g).
O_A(\tau^{-A}) + \int_{\tau^{1-O(\epsilon)}}^{\tau^{1 + O(\epsilon)}}du_1 \int_{|u_1|^{\frac{1}{2}}\tau^{-\epsilon}}^{|u_1|^{\frac{1}{2}}\tau^\epsilon}\frac{dt_1}{t_1^3}y^{-2}\tau^{-\frac{3}{2}+ O(\epsilon)}= O_A(\tau^{-A}) + O\left(y^{-2}\tau^{-\frac{3}{2}+ O(\epsilon)}\right).
\left|\mathbb{E}\left(\widehat{\alpha}_{t,\varepsilon}\right)^n\right|\leq\frac{1}{(2\pi)^n}\int_{D^n}\int_{\mathbb{R}^{n}}\left|e^{-\frac{\varepsilon}{2}\sum_{i=1}^np_i^2}\right|\left|\mathbb{E}e^{\iota \sum_{i=1}^np_i(X^H_{s_i}-X^H_{r_i})}\right|dpdrds.
(1+R_{ij})(c_{r_1}^{(1)}\cdots c_{r_m}^{(m)}) = c_{r_1}^{(1)}\cdots c_{r_i}^{(i)}\cdots c_{r_j}^{(j)}\cdots c_{r_m}^{(m)}+c_{r_1}^{(1)}\cdots c_{r_i+1}^{(i)}\cdots c_{r_j-1}^{(j)}\cdots c_{r_m}^{(m)}.
\hat{\phi}( x^{0}\, , \, x^{1}) \, = \, - \frac{\hat{\xi}}{2 R}\, + \, \frac{1}{\sqrt{2 \pi}}\left( \frac{\pi}{R}\right) \sum_{n > 0}\frac{1}{\sqrt{\omega_{n}}}\left[ \hat{\Lambda}_{n}e^{- i \omega_{n}( x^{0}+ x^{1})}\, + \, \hat{\Lambda}_{n}^{\dagger}e^{i \omega_{n}( x^{0}+ x^{1})}\right] ,
T_s(x\{i_s\}_m) =T_sT_{\omega}^{i_s-i_{j-m+1}}T_sT_{\omega}^{i_{j-m+1}-i_s}T_{\omega}^{i_s-i_{j-m+1}}(x\{i_{j-m+1}\}_{m-1}) &=T_sT_{\omega}^{i_s-i_{j-m+1}}T_sT_{\omega}^{i_{j-m+1}-i_s}(x\{i_s\}_{m-1}) &=T_{\omega}^{i_s-i_{j-m+1}}T_sT_{\omega}^{i_{j-m+1}-i_s}T_s(x\{i_s\}_{m-1}) &=0
A^{\mu}( x ) = \frac{\partial x^{\mu}}{\partial q^{\alpha}}\, A^{\alpha}( q ) \quad \mathrm{a n d}\quad A^{a}= e_{\mu}^{a}( x ) \, \frac{\partial x^{\mu}}{\partial q^{\alpha}}\, e_{b}^{\alpha}( q ) \, \psi^{\dagger}( q ) \sigma^{b}\psi ( q ) ,
g ( \epsilon ) = \sum_{p}\epsilon ( p ) g_{p}= \sum_{p}\sum_{\sigma}\epsilon ( p ) [ d_{p}^{( \sigma )}\bar{d}_{\tilde{p}}^{( - \sigma )}- d_{p}^{( \sigma ) \dagger}\bar{d}_{\tilde{p}}^{( - \sigma ) \dagger}]{.}
\frac{\omega_0^2r_0^2 - \frac18\omega_0^4r_0^4 + \frac{1}{32}\omega_0^6r_0^6 + \frac{45}{32}\omega_0^2\kappa^2r_0^4 + \frac{15}{8}\kappa^2r_0^2}{1 - \frac14\omega_0^2r_0^2 +\frac{1}{64}\omega_0^4r_0^4 - \frac{1}{384}\omega_0^6r_0^6 - \frac{7}{64}\kappa^2r_0^2 - \frac{15}{128}\omega_0^2\kappa^2r_0^4}= -2r_0k_0\frac{K_0'(k_0r_0)}{K_0(k_0r_0)},