harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{c}{\{Q^{a \alpha}, \bar{Q}_{b \beta}\}= 2 \delta^{a}{}_{b}\gamma^{\mu}{}^{\alpha}{}_{\beta}P_{\mu},}\\{{}}\\{{}[ M_{\mu \nu}, Q^{a}] = i \textstyle{\frac{1}{2}}\gamma_{[ \mu}\gamma_{\nu ]}Q^{a},}\\{{}}\\{{}[{P}_{\mu}, Q^{a \alpha}] = 0 .}\\ \end{array}
\widehat{A}(l_0,l_1)=\left[\widehat{d}_{l_0}^{~(0)}+\widehat{r}_{l_0}\right]\widehat{b}_{l_1,0}+\left[\widehat{d}_{l_1}^{~(1)}+\widehat{r}_{l_1}\right]\widehat{b}_{l_0,1},
w_{S^{*}\oplus S}(z) = z \frac{\langle g, g(z) \rangle}{\langle h, g(z)\rangle}= z \frac{\langle 1 \oplus 0, 0 \oplus c_z\rangle}{\langle 0 \oplus 1, 0 \oplus c_z\rangle}= 0.
\hat{t}_{2}(p)_{\alpha\beta\mu\nu}=\frac{i}{2\pi}\frac{\hat{P}^{\scriptscriptstyle (4)}(p)_{\alpha\beta\mu\nu}}{p^4}\left\{p^4 \log\left(\frac{-(p^2+i0)}{M^2}\right)+c_{0}+c_{2}p^2+c_{4}p^4\right\}.
\begin{array}{l}r_0(x,\xi;\lambda):=(|\xi|^2_{g^*(x)}-\lambda)^{-1},\vphantom{A_{A_{A_{A_A}}}}r_n:=-r_0\displaystyle\sum_{|\alpha|+2+j-k=n,j<n}{\textstyle\frac1{\alpha!}}\partial_\xi^\alpha p_k\cdot D_x^\alpha r_jn>0\,.\end{array}
L= -{1\over 4}\,Tr\,{\bf F}_{\mu\nu}(\,{\bf A}\,){\bf F}^{\mu\nu}(\,{\bf A}\,)-{1\over 2}\,Tr\,\left[\, m \, \left(\,{\bf A}_\mu -{\bf \Phi}_\mu \, \right) -D_\mu({\bf \Phi}){\bf C}\,\right]^2
R_{\mathcal{S}^{*}_{\mathit{e}}}(F) = \begin{dcases}\sigma_{0}\ 2\alpha+\beta-2\geqslant 0 \sigma_{0}\ 2\alpha+\beta-2<0\ \X(\alpha,\beta)\leqslant 0 \tilde{\sigma_{0}}\ 2\alpha+\beta-2<0\X(\alpha,\beta)> 0,\end{dcases}
\Phi(\varrho)=\mathcal{H}^{N-2}(\mathbf{S}^{N-2})\,\int_{-1}^1 \frac{(1-t^2)^\frac{N-3}{2}}{\Big(1-2\,t\,\varrho+\varrho^2\Big)^\frac{N+s\,p}{2}}\, dt.
\|a\| \leq C_M e^{C \epsilon^{-\mu}}\sup\limits_{l \in \{1,\dots,m\}}\|(\Lambda_{q_1}-\Lambda_{q_2}) \tilde{f}_l^{(1)}\|_{H^{-s}(W_2)}\|h_{l}^{(1)}\|_{L^2(\Omega)}\|h_{l}^{(2)}\|_{L^2(\Omega)}& + C_M \epsilon \sup\limits_{l\in \{1,\dots,m\}}\|h_{l}^{(1)}\|_{H^{s}_{\overline{\Omega}}}\|h_{l}^{(2)}\|_{H^{s}_{\overline{\Omega}}}\|a\|.
R_{I I}( E , \Delta T , \xi ) = \Delta T \sum_{\mu}| Z |^{2}| f_{\mu}( \xi ) |^{2}\left\{\frac{\sin \left[ ( E - \omega_{\mu}) \Delta T \right]}{( E - \omega_{\mu}) \Delta T}+ \frac{\sin \left[ ( E + \omega_{\mu}) \Delta T \right]}{( E + \omega_{\mu}) \Delta T}\right\}.
\hat W_i^L=\bigoplus_{l_0+\ldots+l_{i-1}\le L}{\cal V}^{l_0}\otimes{\cal V}^{l_1}_\#\otimes\ldots\otimes{\cal V}^{l_{i-1}}_\#\otimes W^{L-(l_0+\ldots+l_{i-1})}_\#;
\int_{T_3}^t M(s-u,s)\,ds =\int_{T_3-u}^{t-u}\int_{[v,v+u]}\mu(dw)\,dv &\leq \int_{0}^{t-u}\int_{[v,v+u]}\mu(dw)\,dv &=\int_{[0,u]}\left( (t-u)\wedge w\right)\,\mu(dw) + \int_{(u,t]}\left( (t-u)\wedge w - (w-u)\right) \mu(dw).
\left\Vert \widetilde{Err}(\cdot ,j_{0},j,\overset{3}{\mathbb{P}}\left(E_{j}^{T}\right)) \right\Vert^{2}_{\mathbb{L}^{2}(B)}= \mathcal{O}\left( a^{8}\; \left\vert z_{j}- z_{j_{0}}\right\vert^{-8}\; \left\Vert \overset{3}{\mathbb{P}}\left(\tilde{E}_{j}^{T}\right) \right\Vert^{2}_{\mathbb{L}^{2}(B)}\right).
\frac{1}{n}\sum_{i=1}^n U(p_i,\bar p_{-i}) = \frac{1}{n}\sum_{k\in I}\sum_{i\in C_k}U(p_i,\bar p_{-i}) & = \frac{1}{n}\sum_{k\in I}|C_k| U(\bar p) = \frac{1}{n}\sum_{i=1}^n U(\bar p) = U(\bar p).
\left[ \frac{\partial^2}{\partial\xi^2}+2\,\kappa\,\delta\,\frac{s}{c}\frac{\partial}{\partial\xi}-\left\{\frac{1}{\kappa^2}- \kappa^2\delta(1+\delta) \right\}\frac{s^2}{c^2}+( \varepsilon + \kappa^2\delta )\right]\Psi(\xi) = 0\;,
j_x^1\stackrel{\textrm{\eqref{eq:jx1}}}{\le}k^2\sum_{y,z\in I_x}\nu(E_{y,z})\le \frac{k^2}{c(k)}\sum_{y,z\in I_x}\sum_{w=y}^z a_w\le C(k)\sum_{w\in I_x}a_w.
M_{\mu \nu}^{( p )}= \eta_{\mu \nu}+ F_{\mu \nu}+ \partial_{\mu}\phi^{i}\partial_{\nu}\phi^{i}- 2 \bar{\lambda}( \Gamma_{\mu}+ \Gamma_{i}\partial_{\mu}\phi^{i}) \partial_{\nu}\lambda + \bar{\lambda}\Gamma^{m}\partial_{\mu}\lambda \bar{\lambda}\Gamma_{m}\partial_{\nu}\lambda ,
V = \left( \begin{array}{c}{{X^{\Lambda}( \phi )}}\\{{F_{\Sigma}( \phi ) = \frac{\partial}{\partial X^{\Sigma}}\ln \mathrm{N}( X ) \}}\\ \end{array}\right) \quad \quad ; \quad \quad \Lambda , \Sigma = 1 , \dots , n_{V}= 1 + n
A=\left( \begin{array}{cccccc}c_{11}\cdots c_{1m}d_{1,m+1}\cdots d_{1,m+s}\vdots c_{m1}\cdots c_{mm}d_{m,m+1}\cdots d_{m,m+s}0 \cdots 0 1 \cdots 0 \cdots 0 \cdots 0 0 \cdots 1 \end{array}\right)
g^{(n+1)}(2k2^{-n-1}) = g^{(n)}(k2^{-n}) = (k2^{-n})^2 g^{(n+1)}((2k+2)2^{-n-1}) = g^{(n)}((k+1)2^{-n}) = ((k+1)2^{-n})^2.
\sum_{i=1}^{u}(-1)^{i-1}a_{[\ell_1\ell_2\dots \ell_{k-u+1}s_i]}^{s_1\dots s_{u}\setminus s_i}=\sum_{i=1}^{u}(-1)^{i-1}a_{[s_1\ell_2\dots \ell_{k-u+1}s_i]}^{s_1\dots s_{u}\setminus s_i}=a^{s_1\dots s_u}_{\ell_1\dots\ell_{k-u+1}}.
\hat{Z}_{8 8}^{1 0}= \hat{Z}_{8 8}^{\overline{{{1 0}}}}= \frac{( d e t \I m \Omega )^{2}}{4}\biggl \vert \Theta \left[ \begin{array}{l l}{{1 / 2}}&{{1 / 2}}\\{{1 / 2}}&{{1 / 2}}\\ \end{array}\right] ( 0 \vert \Omega ) \biggr \vert^{8}
(\xi_{s-}(w) - \xi_{s-}(z))\, \phi_s(z) =\; [\xi_{s-}(w)\,\xi^c_{s-}(z) - \xi_{s-}(z)\,\xi^c_{s-}(w)]\,\phi_s(z) =\; \xi_{s-}(w)\,\xi^c_{s-}(z) \phi_s(w) - \xi_{s-}(z)\,\xi^c_{s-}(w) \phi_s(z) & + \xi_{s-}(w)\,\xi^c_{s-}(z)\, \big(\phi_s(z) - \phi_s(w)\big).
(R(Y,\dot\gamma)\dot\gamma)^b = R^b{}_{\lambda\mu\nu}\dot\gamma^\lambda Y^\mu\dot\gamma^\nu = -R^b{}_{0 0 1}(\dot\gamma^0)^2 Y^1 - R^b{}_{0 0 a}(\dot\gamma^0)^2 Y^a + o(s^{-4}).
\mathbb E_x\left[e^{-p\int_{0}^{e(q)}\rm{\bf{1}}_{\{X_s \leq b\}}ds}\rm{\bf{1}}_{\{X_{e(q)}\in dy\}}\right]=q\int_{0}^{\infty}e^{-qt}\mathbb E_x\left[e^{-p\int_{0}^{t}\rm{\bf{1}}_{\{X_s \leq b\}}ds}\rm{\bf{1}}_{\{X_{t}\in dy\}}\right]dt,
\lim_{N\rightarrow\infty}{\frac{P_{\lambda(N)}\big(x_1, \dots, x_m, t^{-m}, \dots, t^{1 - N}\big)}{P_{\lambda(N)}\big(1, t^{-1}, t^{-2}, \dots, t^{1 - N}\big)}}= \Phi^{\nu}(x_1, \dots, x_m; q, t),
p^{\nu_2}_{k-1}(2,t)=-(-1)^{k-1}\underset{\Lambda^{k-1}_{2}}{\sum}\frac{\lambda_1^{k_1+1}\lambda_2^{k_2-1}t^{k_1\nu_1+k_2\nu_2}}{\Gamma\left(k_1\nu_1+k_2\nu_2+1\right)}.
a_1 = \frac13(t_1+t_2+t_3-2t_4-b) = \frac13\left(\frac{o_1+o_2+o_3+1}{2}+\frac{o_1+o_3+o_4+1}{2}+\frac{o_1+o_2+o_4+1}{2}- 2\frac{o_2+o_3+o_4+1}{2}-b\right) = \frac12o_1 + \frac16(1-b)
L_{6}(q) = 2\sum_{\substack{n \geq 1 -n \leq j \leq n-1}}q^{6n^2 - 2j^2 - 2j}+ 2\sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{6n^2 + 6n + 1 - 2j^2}
\frac{d}{dt}\int_{\mathbb{R}^d}\tilde \rho \, d\mu_t^\varepsilon\leq \frac{1}{2(s-t)}\int_{\mathbb{R}^d}\tilde \rho \, d\xi_t^\varepsilon+\frac{1}{2}(\lambda^\varepsilon)^2 \int_{\mathbb{R}^d}\tilde\rho \, d\mu_t^\varepsilon +c_5 e^{-\frac{1}{128(s-t)}}\mu_t^\varepsilon (B_{\frac{1}{2}}(y)).
\lVert L^{-1}\sum_{i\notin \mathsf{M}'}\Pi\cdot\Pi_{i}\cdot \rho_i\cdot\Pi_{i}\cdot\Pi\rVert_1\allowdisplaybreaks &=\mathrm{tr}\left( L^{-1}\sum_{i\notin \mathsf{M}'}\Pi\cdot\Pi_{i}\cdot \rho_i\cdot\Pi_{i}\cdot\Pi\right)\allowdisplaybreaks &\leq 1-p(\mathsf{M}')+ 2\sqrt{1-p(\mathsf{M}')}+ 20 \sqrt[8]{\epsilon}
\bigg |\frac{1}{k}\int_{r}^{r+k}g(t)dt-g(r)\bigg | = \bigg |\frac{1}{k}\int_{r}^{r+k}(g(t)-g(r))dt \bigg | \leq \frac{1}{k}\int_{r}^{r+k}|g(t)-g(r)|dt = \frac{1}{k}\int_{[r,r+k]\cap B_r}|g(t)-g(r)|dt+\frac{1}{k}\int_{[r,r+k]\setminus B_r}|g(t)-g(r)|dt
{\int}d{\bf x}\,\left\{{\bar{\cal F}}^{ba}({\bf r},{\bf x}_0)\,+X_i\partial_i{\bar{\cal F}}^{ba}({\bf r},{\bf x}_0)+\frac{1}{2}\,X_iX_j\,\partial_i\partial_j{\bar{\cal F}}^{ba}({\bf r},{\bf x}_0)\;+\;\;\cdots\right\}{j}_0^a({\bf x})
S^F_{lc}= \frac{i}{4 \pi l_s^2 p^+}\int d \tau \int_{0}^{2 \pi l_s^2 p^+}d \sigma\left[ S^1 \left( \partial_+ - \frac{\mu}{2}\gamma^{12}\right) S^1 + S^2 \left( \partial_- - \frac{\mu}{2}\gamma^{12}\right) S^2 + 2 \mu S^1 \gamma^{234}S^2 \right],
\theta(\tau)=\begin{bmatrix}0 -\tau 0 0 \tau 0 0 0 0 0 0 0 0 0 0 0\end{bmatrix}\quad\hbox{and}\quad\theta(\sigma)=\begin{bmatrix}0 0 0 0 0 0 0 0 0 0 0 -\sigma 0 0 \sigma 0\end{bmatrix},
I[u] =\frac{1}{p}\|u\|_{D^{s,p}}^p+ \frac{1}{p}\int_{\mathbb{R}^N}V(x)|u|^p dx &\quad-\frac{1}{2\cdot p_{r;s}^{\uparrow *}}\int_{\mathbb{R}^N}(K\ast g(u))g(u)dx-\varepsilon_W\int_{\mathbb{R}^N}W(x)f(u)dx.
a = \cos \left( \pi \sqrt{\lambda_{1}^{2}+{\lambda_{2}^{\prime}}^{2}}~ \right) ~ , b = i ~{\frac{\lambda_{1}+ i \lambda_{2}^{\prime}}{\sqrt{\lambda_{1}^{2}+{\lambda_{2}^{\prime}}^{2}}}}~ \sin \left( \pi \sqrt{\lambda_{1}^{2}+{\lambda_{2}^{\prime}}^{2}}~ \right) ~ .
W [ \vec{V}^{R}] = \int d^{4}x{\cal L}_{\mathrm{r e d}}= \int d^{4}x{\frac{1}{2}}\left( \dot{V}_{i}^{R}R_{i j}^{- 1}\dot{V}_{j}^{R}- V_{i}^{R}\left( \vec{\partial}^{2}- M^{2}\right) R_{i j}^{- 1}V_{j}^{R}\right) ~ .
\lim_{\beta\rightarrow\infty}\min_{\zeta\in\sigma\left( A\left(\beta\right) |_{H_{h\ell}\left( \beta\right)}\right)}\left(-\operatorname{Im}\zeta\right) =+\infty,\lim_{\beta\rightarrow\infty}\max\limits_{\substack{w \text{of}A\left( \beta\right) H_{h\ell}\left( \beta\right)}}Q[w]=0.
\lambda_{d_1,\ldots,d_k}=\biggl(\prod_{i=1}^k\mu(d_i)d_i\biggr)\sum_{\substack{r_1,\ldots,r_k d_i\mid r_i\forall i (r_i,V)=1\forall i}}\frac{\mu\bigl(\prod_{i=1}^kr_i\bigr)^2}{\prod_{i=1}^k\varphi(r_i)}F\bigl(\frac{\log r_1}{\log R},\ldots,\frac{\log r_k}{\log R}\bigr),
\begin{array}{ll}\pi \circ f_B - f_{B_1}\circ\pi =c^a_n v^n_b + \sum_{j=1}^{n-1}c^d_n B^n_j v^j_d \delta^a_b + \sum_{i=1}^{n}c^d_i B^i_n v^n_d \delta^a_b , &=0.\end{array}
\left\langle \delta(w(v)), H(v)\right\rangle = \left\langle \delta(w), \left| \frac{dv_\pm}{dw}(w)\right|[H(v_-(w)) + H(v_+(w))] \right\rangle = \frac{c_0^2}{4}\frac{t}{\sqrt{c_0^2t^2/4 - 1}}[H(s+\alpha(t))+H(s-\alpha(t))]
\sum^{\infty}_{j=1}\omega_{1}(\sigma^{j})\leq\frac{1}{-\log\sigma}\sum^{\infty}_{j=1}\int^{\sigma^{j-1}}_{\sigma^{j}}\frac{\omega_{1}(t)}{t}dt=\int^{1}_{0}\frac{\omega_1(t)}{t}dt<\infty.
p(\hat X_1,\hat X_2) = I - \hat X_1^2-\hat X_2^2=\begin{pmatrix}I-X_1^2-X_2^2-\alpha_1\alpha_1^* -\alpha_2\alpha_2^* * * *\end{pmatrix}= \begin{pmatrix}-\alpha_1\alpha_1^* -\alpha_2\alpha_2^* * * *\end{pmatrix}.
\delta_{c_3}\circ \delta_{c_4}= r_0 \delta_{c_0}+ r_1\delta_{c_1}+ r_2 \delta_{c_2}+ r_3\delta_{c_3}+ r_3\delta_{c_4}, &\delta_{c_4}\circ \delta_{c_3}= r_0 \delta_{c_0}+ s_1\delta_{c_1}+ s_2 \delta_{c_2}+ s_3\delta_{c_3}+ s_3\delta_{c_4},
\int x (q^2x^2;q^2)_\infty h_n(x;q) d_qx= \frac{(1-q)x(x^2;q^2)_\infty}{[n]_q-1}\left(\frac{q^n}{x}h_n(\frac{x}{q};q)-q^{n-1}[n]_q h_{n-1}(\frac{x}{q};q) \right),
e^{{\cal I}(i\alpha_{b}+j\beta_{b}+k\gamma_{b})+i\alpha_{r}+j\beta_{r}+k \gamma_{r}}({\cal I}ct + ix+jy+kz)e^{{\cal I}(i\alpha_{b}+j\beta_{b}+k\gamma_{b})-i\alpha_{r}-j\beta_{r}-k \gamma_{r}}\; \; .
\P\left(Z_{d+1}\in B\ |\Z_0=z\right) \ge \P\left(Z_{d+1}\in B, Z_1\in V |Z_0=z\right) & = \P\left(Z_{d+1}\in B\ |\Z_1\in V,Z_0=z\right)\cdot \P(Z_1\in V\ |\Z_0=z) \ge \eta~\P\left(Z_{d+1}\in B\ |\Z_1\in V \right) \ge ~ \lambda_d \left(B\cap K\right).
I_{T}^{*}(\lambda_{1},\lambda_{2})= \frac{1}{(2\pi)^{2}}\sum_{t_{1}=1-T}^{T-1}\sum_{t_{2}=1-T}^{T-1}e^{-i(\lambda_{1}t_{1}+\lambda_{2}t_{2})}\hat{\gamma}_{T}(\boldsymbol{t}),
N_{(1,9)}(n) = 8\,\sigma(n) - 32\,\sigma(\frac{n}{4}) + 8\,\sigma(\frac{n}{9}) - 32\,\sigma(\frac{n}{36}) + 64\,W_{(1,9)}(n) + 1024\,W_{(1,9)}(\frac{n}{4}) - 256\,\biggl( W_{(4,9)}(n) + W_{(1,36)}(n) \biggr),
dx_j(I_j^aX) =dx_j(I_jX-\frac{a^2V}{a^2+V}\theta(X)I_jT+a^2dx_j(X)T) =-\theta(X)+\frac{a^2}{a^2+V}\theta(X)+0 =-\frac{V}{a^2+V}\theta(X),
P_{\lambda}(1,t,t^2, \ldots, t^{n-1};q,t) = t^{n(\lambda)}\prod_{1\le i<j\le n}\prod_{\ell=0}^{\lambda_i-\lambda_j-1}\frac{1 - q^\ell t^{j-i +1}}{1- q^\ell t^{j-i}}&= t^{n(\lambda)}\prod_{b\in \lambda}\frac{1-q^{\mathrm{coarm}_\lambda(b)}t^{n-\mathrm{coleg}_\lambda(b)}}{1-q^{\mathrm{arm}_\lambda(b)}t^{\mathrm{leg}_\lambda(b)+1}}
\tilde\psi(\bar z) =\psi_{\lfloor\max(t_j)\rfloor+1}((x_0,(s_0,t_0)),\ldots,(x_{q-1},(s_{q-1},t_{q-1}))) &+\sum_{n=1}^{\lfloor\max(t_j)\rfloor}d_{q-2}\chi_n((x_0,(s_0,t_0)),\ldots,(x_{q-1},(s_{q-1},t_{q-1}))
w^k_j =\left(1-\lambda a_j\right) w^{k-1}_j+\lambda a_j w^{k-1}_{j-1}&= \left(1-\lambda a_j\right)\left\{\left(1-\lambda a_j\right) w^{k-2}_j+\lambda a_j w^{k-2}_{j-1}\right\}+\lambda a_j \left\{\left(1-\lambda a_{j-1}\right) w^{k-2}_{j-1}+\lambda a_{j-1}w^{k-2}_{j-2}\right\}&= \dots = \sum_{\ell=j-k}^j \lambda_{\ell,j}^k w^0_{\ell}
J_1 := \sum_{i\neq j}[ \beta(r_{j,k}) - r_{j,k+1}]^* a_ja_i [ \beta(r_{i,k}) - r_{i,k+1}]; J_2 := \sum_{j=1}^N [ \beta(r_{j,k}) - r_{j,k+1}]^* a_j^2 [ \beta(r_{j,k}) - r_{j,k+1}].
F_{\nabla^\omega}= (\d A^a+\omega^a{}_{\mu b}A^b\wedge dX^\mu-\omega^a_{\mu b}A^b\wedge \rho^\mu_c A^c+\tfrac{1}{2}C^a{}_{bc}A^b\wedge A^c)e_a = (\d A^a+\omega^a{}_{\mu b}A^b\wedge DX^\mu+\tfrac{1}{2}C^a{}_{bc}A^b\wedge A^c)e_a.
\frac{f^k}{g}= \frac{f^{e \ell +k_0}}{g}= (f^e)^\ell \frac{f^{k_0}}{g}= (\omega_m \tau_r(a))^\ell (\omega_n \tau_r(b)) = (\omega_m \omega_n) \tau_r(a^\ell b).
|\nabla_1 g(p,q)| = \left|\frac{2 p^2 p + q p}{\sqrt{p^4 + q p^2}}\frac{1}{e^{\sqrt{p^4 + q p^2}}-1}\right| \leq \frac{2 \sqrt{p^4 + q p^2}}{|p|\Big(e^{\sqrt{p^4 + q p^2}}-1\Big)}\leq 2 |p|^{-1}e^{-\tfrac{p^2}{2}},
\int_{\Lambda_{y_0}}|f(\sigma_{1/w}z)|^2 d\mu(z) =\int_{\sigma_{1/w}^{-1}\Gamma_0(4k) \sigma_{1/w}\backslash H}N(z,1/w,y_0)|f(\sigma_{1/w}z)|^2 d\mu(z) \leq \Big( 1+\frac{10N^{\prime\prime}_{1/w}}{y_0}\Big).
\boldsymbol{S}^{(k)}_e := \boldsymbol{K}^{(k)}_{e,B_e B_e}- \boldsymbol{K}^{(k)}_{e,B_e I}\left(\boldsymbol{K}^{(k)}_{e,II}\right)^{-1}\boldsymbol{K}^{(k)}_{e,IB_e}.
\sum_{t=0}^{\lfloor sT\rfloor-1}f(\frac{t}{T})\big( U^{d,T}(\frac{t+1}{T},x)-U^{d,T}(\frac{t}{T},x) \big)= -(i+1-dx)\sum_{k=1}^3 P_{i,k}^s-(dx-i)\sum_{k=1}^3 P_{i+1,k}^s-P_4^s ,
\|T\|_{W_{q,p;r}^{\mathrm{mid}}}:= \|\widetilde{T}\| = \|\widehat{T}\| =\inf\{B: \mathrm{(v)~holds}\}& = \inf\{B: \mathrm{(vi)~holds}\}= \inf\{B: \mathrm{(vii)~holds}\}.
{\rm ch}W_{n+1}(x,q^2)= \sum_{k+s\le n}q^{k(k+1)+2s}x^{n+1-k-2s}\left(\begin{array}{c}n k,s,n-k-s \end{array}\right)_{q^2},{\rm ch}W^\sigma_{n+1}(x,q)=\sum_{k+s\le n}x^{n+1-k-2s}\left(q^{k^2+2s}+x^{-1}q^{k^2+2n+1}\right) \left(\begin{array}{c}n k,s,n-k-s \end{array}\right)_{q^2}.
\frac{\partial^n\alpha}{\partial v^n}=\frac{\partial^{n-1}}{\partial v^{n-1}}(\nu F)=F_1\frac{\partial^nt}{\partial v^n}+F_2, \frac{\partial^n\beta}{\partial u^n}=\frac{\partial^{n-1}}{\partial v^{n-1}}(\mu F)=F_1\frac{\partial^nt}{\partial u^n}+F_2,
(\widehat{L}S^{-1}f)(x) = -i \gamma(x)^{-1}B^{-1}\frac{d}{dx}\bigl[f(t(x))\bigr] + \widehat{Q}(x) f(t(x)) = -i B^{-1}f'(t(x)) + \widehat{Q}(x) f(t(x)),
{\frac{1}{( \omega_{l_{1}}+ \omega_{l_{2}})^{3}}}={\frac{\omega_{l_{1}}^{3}- \omega_{l_{2}}^{3}- 3 \omega_{l_{1}}^{2}\omega_{l_{2}}+ 3 \omega_{l_{1}}\omega_{l_{2}}^{2}}{\left[ ({\frac{\pi l_{1}}{a}})^{2}- ({\frac{\pi l_{2}}{a}})^{2}\right]^{3}}},
|w_n (x_n - R_n) - w_n (x_n)| = |w_{n-1}(x_n - R_n) + c_- - w_{n-1}(x_n )| & \leq \|w_{n-1}'\|_{L^{\infty}(Y_n)}R_n + |c_{-}| &\leq K_n R_n + 3 K_n g(R_n) &\leq 4K_n g(R_n) &\leq \theta_n g(R_n) & \leq ( 2 - 2^{-n}) \theta_n |g_n ( x_n - R_n)|.
=c(\vec{s}, k)\int_{\Delta_k}\frac{d\vec{v}}{\sqrt{G((I-\widetilde{P}){1\!\!\,{\rm I}}_{[0;v_1]}, \ldots,(I-\widetilde{P}){1\!\!\,{\rm I}}_{[0;v_k]},{1\!\!\,{\rm I}}_{[0;1]})}},
Q_h(1_x,1_x) = \sum_{y\in X}b_h(x,y) + c_h(x) &= \beta \sum_{y\in X}h(\tau^{-1}(x))h(y)b(\tau^{-1}(x),y) + \beta h(\tau^{-1}(x))\mathcal{L}h(\tau^{-1}(x)) &= \beta h(\tau^{-1}(x))^2\sum_{y\in X}b(\tau^{-1}(x),y) + \beta h(\tau^{-1}(x))^2 c (\tau^{-1}(x)) &= \beta Q^{(D)}(U1_x,U1_x).
\varepsilon_0 := \min\bigg\{\frac{1}{2C_1},\frac{1}{4C_1^3C_Q^2C_{\frac{d}{4},\frac{d}{2}+ \eta}K},\frac{1}{4C_1^3C_Q^2C_{\frac{d}{4},\frac{d}{2}+ \eta}K}\bigg\}
\sum_{n=1}^{\infty}\frac{q^{n}x^{n}}{1-x^{n}}=\sum_{n=1}^{\infty}\frac{q^{n}x^{n^{2}}}{1-qx^{n}}+\sum_{n=1}^{\infty}\frac{q^{n}x^{n(n+1)}}{1-x^{n}},
\mathcal{L}_{k}^{\epsilon,\hat{v}_k}\psi_k(x) = \bigl \langle \bigtriangledown_x \psi_k(x), f_k(x, \hat{v}_k(x)) \bigr\rangle + \frac{\epsilon}{2}\operatorname{tr}\bigl \{a_k(x)\bigtriangledown_x^2 \psi_k(x) \bigr\}.
0\ge \widetilde J_{p,f}(u_0)-\widetilde J_{p,f}(\psi_{T}) \ge \frac{1}{2}\, \int_{\{w_{T}<u_0\}}\left(|\nabla u_0|^2-|\nabla w_{T}|^2\right)\, dx &+\frac{c_2^{2-q}}{q}\, \int_{\{w_{T}<u_0\}}\left(u_0^q-w_{T}^q\right)\, dx &-\int_{\{w_{T}<u_0\}}f\, (u_0-w_{T})\, dx.
\|\varphi\|_{\infty}: = \sup_{v \in \mathcal{S}}|\varphi(v)|\mbox{and}[\varphi]_{\gamma}: = \sup_{u, v \in \mathcal{S}: u \neq v}\frac{|\varphi(u) - \varphi(v)|}{\mathbf{d}(u, v)^{\gamma}},
b^i({\boldsymbol{\ell}}) \ = \ \ell^j \, \int\limits_{\mathbb R^d}\int\limits_{\mathbb T^d}z^i \, c^j(z) \, \mu (\xi, \xi-z ) dz d\xi \ + \ \ell^j \int\limits_{\mathbb R^d}\int\limits_{\mathbb T^d}z^i \, a_{\rm sym}(z) \mu(\xi,\xi-z) \, \tilde\varphi^j_{0}(\xi) dz d\xi + O(|{\boldsymbol{\ell}}|^2),
\|z^n\|_{L^2(D_{1}\backslash{D_{r}})}\|z^{-1-n}\|_{L^2(D_{1}\backslash{D_{r}})}\geq\|z^1\|_{L^2(D_{1}\backslash{D_{r}})}\|z^{-2}\|_{L^2(D_{1}\backslash{D_{r}})}&=\pi\sqrt{\dfrac{(1-r^{4})(1-r^2)}{2r^2}}
\sum\limits_{i=1}^k\lambda_i P(x_i)=\sum\limits_{j=1}^l\alpha_j P(x_j) \Longrightarrow \left\|\sum_{i=1}^k\lambda_i P(x_i)-\sum_{j=1}^l\alpha_j P(x_j)\right\|=0 \stackrel{(\ref{4aast})}{\Longrightarrow}\left\|\sum_{i=1}^k\lambda_i Q(x_i)-\sum_{j=1}^l\alpha_j Q(x_j) \right\|=0 \Longrightarrow \sum_{i=1}^k\lambda_i Q(x_i)=\sum_{j=1}^l\alpha_j Q(x_j).
\langle T_{\ \mu}^{\mu}\rangle = \frac{1}{4 8 \pi}\Biggl [ \left( \partial_{\mu}\partial_{\nu}h^{\mu \nu}- \partial_{\mu}\partial^{\mu}h_{\ \nu}^{\nu}\right) \mp \frac{1}{2}\varepsilon_{\mu \lambda}\partial^{\lambda}\partial_{\nu}h^{\mu \nu}\Biggr ] .
(\epsilon_{\gamma}\, \Delta \varphi_f \, \epsilon_{\gamma}^{-1}\, \epsilon_{\gamma'}) \rhd \varphi_{f'}= (\epsilon_{\gamma}\rhd \varphi_f)\, (\epsilon_{\gamma'}\rhd \varphi_{f'}) \,(\epsilon_{\gamma}\rhd \varphi_f)^{-1}
\begin{aligned}\uppercase\expandafter{\romannumeral1}=\sum\limits_{j}\frac{\lambda_{j}}{\omega(Q_{j})^{1/p}}(M(T(a_{j}))(x)\chi_{2\sqrt{n}Q_{j}^{*}}(x)+(M(\chi_{Q_{j}})(x))^{\gamma}\chi_{(2\sqrt{n}Q_{j}^{*})^{c}}(x))\end{aligned}
\operatorname{adj}L_\mathrm{prin}(x,p)=s^j\tilde e_j{}^\alpha(x)\,p_\alpha=\begin{pmatrix}\tilde e_4{}^\alpha p_\alpha+\tilde e_3{}^\alpha p_\alpha \tilde e_1{}^\alpha p_\alpha-i\tilde e_2{}^\alpha p_\alpha \tilde e_1{}^\alpha p_\alpha+i\tilde e_2{}^\alpha p_\alpha \tilde e_4{}^\alpha p_\alpha-\tilde e_3{}^\alpha p_\alpha\end{pmatrix},
g(k+1) = H(X_{[k+1]}|X_{\{k+2,\ldots,m\}}) & \le \sum_{i=1}^{k+1}H(X_{i}|X_{\{k+2,\ldots,m\}}) & = (k+1) H(X_{k+1}|X_{\{k+2,\ldots,m\}}),
k ={\frac{u_{1}}{2}}\Big [{\frac{{\cal E}_{1}+ \sqrt{{\cal E}_{1}^{2}+ \mu^{2}}}{K + \sqrt{K^{2}+ \mu^{2}}}}-{\frac{K + \sqrt{K^{2}+ \mu^{2}}}{{\cal E}_{1}+ \sqrt{{\cal E}_{1}^{2}+ \mu^{2}}}}\Big ] .
\tilde{R}^2_M =\big(\|X'-\hat{\theta}\|^2 -\sigma^2\mathrm{E}\|\xi'\|^2 +2\sigma^2G_M\sqrt{n}\big)_+ &=\big(\|\theta-\hat{\theta}\|^2+\sigma^2\big(\|\xi'\|^2-\mathrm{E}\|\xi'\|^2\big)+2\sigma \langle\xi',(\theta-\hat{\theta})\rangle+2\sigma^2G_M\sqrt{n}\big)_+.
\mathrm{dist}(\Theta^{r_\mu}{}_\#\beta',K)\leq W_1(\Theta^{r_\mu}{}_\#\beta',\Theta^{r_\mu}{}_\#\beta) +\mathrm{dist}(\Theta^{r_\mu}{}_\#\beta,K) \leq W_1(m', m)+\mathrm{dist}(\Theta^{r_\mu}{}_\#\beta,K).
h \phi_n (\varphi) \;=\; \lambda_n^2 \phi_n (\varphi) \;\Rightarrow\;{\tilde h}\; [\frac{1}{\lambda_n}a\phi_n (\varphi)] \;=\; \lambda_n^2 \;[\frac{1}{\lambda_n}a\phi_n (\varphi)]
\xi_1(y)=e^{\frac{|y|^p}{p}}\alpha |y|^{1-n}J_{n-1}\approx \sqrt{\frac{2\pi}{p}}e^{\frac{|y|^p}{p}}\alpha |y|^{1-n}n^{\frac{1}{p}-\frac{1}{2}}e^{F((n-1)^{\frac{1}{p}})},\,\,\,\,\,n\rightarrow\infty,
S_{u,c}:=\frac{{\displaystyle \sum_{{\textstyle{\emptyset\ne v\subseteq\{1,\cdots,N\}\atop u \nsubseteq v\nsubseteq u}}}}\mathbb{E}\left[y_{u,G}(\mathbf{X}_{u})y_{v,G}(\mathbf{X}_{v})\right]}{\sigma^{2}},
0 \leq \int_{\tau_{L_2}}^t S(u) \: du = \frac{1}{s}\int_{\tau_{L_2}}^t G_j(u) \: du \leq \frac{1}{s}\log \bigg( \frac{2s}{\mu}\bigg) \leq \bigg(1 + \frac{\eta}{3}\bigg) a_N
[z^n] \frac{\partial^2}{\partial x_1^2}(N_1+N_2) |_{x_1=x_2 = 1}= \frac{1}{n-1}[t^n] 3(n-1)(3n-4)t^2 (1-t)^{-4(n-1)}- \frac{1}{2n-1}[t^n] (3n-2)(3n-3)t^2 (1-t)^{-(4n-2)}&= 3(3n-4) [t^{n-2}](1-t)^{-4(n-1)}- \frac{(3n-2)(3n-3)}{2n-1}[t^{n-2}] (1-t)^{-(4n-2)}&= 3(3n-4) \binom{5n-7}{n-2}-\frac{(3n-2)(3n-3)}{2n-1}\binom{5n-5}{n-2}.
S_{m, k, n}(r) = \sum_{\substack{A \in B_r^{nk}rank(A) = n}}\sum_{i = 0}^{mn}\alpha_{A, i}r^i = \sum_{i = 0}^{mn}\left( \sum_{\substack{A \in B_r^{nk}rank(A) = n}}\alpha_{A, i}\right) r^i = \sum_{i = 0}^{mn}s_{m,k,n,i}(r) r^i.
\lim_{| x | \to \infty}\langle J_{\mu}{}^{a}( x ) J_{\nu}{}^{b}( 0 ) \rangle = \delta^{a b}f_{\pi}^{2}\, \partial_{\mu}\partial_{\nu}\left( \frac{1}{| x |^{2}}\right) +{\cal O}\left( \frac{1}{| x |^{6}}\right) \quad .
\int_{\partial \Omega}(\Psi\cdot n )\,\varphi\,d{\mathcal H^{n-1}}=\int_\Omega \big (\langle \Psi , \nabla \varphi \rangle +\varphi\,\div\Psi \big ) \, dx \forall \varphi \in H^1 (\Omega)\, ,
\begin{array}{c}{{[ s_{\mu \nu}, s_{\lambda \rho}] = - \eta_{\mu \lambda}s_{\nu \rho}+ \eta_{\mu \rho}s_{\nu \lambda}+ \eta_{\nu \lambda}s_{\mu \rho}- \eta_{\nu \rho}s_{\mu \lambda}\, ,}}\\{{{}}}\\{{{[}{s_{~ b}^{a}},{}{s_{~ d}^{c}}{]}= 2 ( \delta_{~ d}^{a}s_{~ b}^{c}- \delta_{~ b}^{c}s_{~ d}^{a}) \, ,}}\\ \end{array}
\sum_{2\le q_1, \ldots, q_{h_1 + h_2}\leq X}\prod_{i = 1}^{h_1 + h_2}\Lambda(q_i) \prod_{i = h_1 + 1}^{h_1 + h_2}\mu(\omega_{q_i}) \ll (\log\log x)^{(h_1 + 2h_2)/2}(\log\log\log x)^{2h_1 + h_2}.
\begin{aligned}S_{n,p}\|u\|_{L^{p^*}(\Omega)}=S_{n,p}\|u^{*}\|_{L^{p^*}(B_R)}< S_{n,p}\Bigg( \int_{B_{R}}\frac{|u^{*}(x)|^{p^{*}}}{\left[ \frac{n-p}{p-1}\log_{{\frac{n-1}{p-1}}}\frac{R}{|x|}\right]^{\frac{p(n-1)}{n-p}}}dx \Bigg)^{\frac{1}{p^{*}}}&\le\|\nabla u^{*}\|_{L^{p}(B_R)}\le\|\nabla u\|_{L^{p}(\Omega)}\end{aligned}
\sup_{(\theta,a)\in \mathcal{K}(y),\,b=\varsigma\theta,\, \varsigma\varsigma'=a}\left[\sum_{i,j}y_i y_j a_{ij}+\sum_i (y_i b_i)^2\right]\le C(1+||y||)^2, \ \ \forall ~ y\in [0,\infty)^n \backslash \{\mathbf{0}\}\,,
\lambda(g_1) \cdots \lambda(g_k) = \sum_{\substack{A_1 \ni g_1^{-1}\vdots A_k \ni g_k^{-1}}}(A_1,g_1) \cdots (A_k,g_k) = \sum_{\substack{A_k \ni g_k^{-1}g_{k}A_k \ni g_{k-1}^{-1}\vdots g_2 \cdots g_kA_k \ni g_1^{-1}}}(A_1, g_1 \cdots g_k) = \sum_{A \supseteq B}(A,h).
(u_p^n,v_p^n,h_p^n,g_p^n)(x,0)=(-\overline u_e^n(x),0,-\overline h_e^n(x),0), (u_p^n,h_p^n)(x,\infty)=(0,0), \ (u_p^n,h_p^n)(1,y)=(u_0^n,h_0^n)(y).
\frac 1\eta\int_0^\eta\int_{B}x_N\left[(x_N+y_N)^2+|x'|^2\right]^{-N/2}dx'\:dx_N=\frac 1\eta\int_0^\eta\frac{x_N}{x_N+y_N}\int_{B_{1/(x_N+y_N)}}\left[1+|\xi|^2\right]^{-N/2}d\xi\:dx_N
\frac{\overline{b_{n\,m+6}}}{c_{n+1\,m+3}}\cdot \frac{\overline{d_{n+1\,m+3}}}{a_{nm}}= \frac{\overline{d_{n\,m+6}}}{a_{n-1\,m+3}}\cdot \frac{\overline{b_{n-1\,m+3}}}{c_{nm}}.
\mathcal{G}(n) \oplus \mathcal{G}(n) \oplus \mathcal{G}(m+1) \oplus \mathcal{G}(m+1) = 0 = \mathcal{G}(n+1) \oplus \mathcal{G}(n+1) \oplus \mathcal{G}(m) \oplus \mathcal{G}(m)
\theta^{\ast}:\Omega\rightarrow C_{2}\ltimes(C_{n}\ltimes C_{p}) \theta^{\ast}(x_{1}^{\prime}) =sx^{\alpha}y^{-1}\theta^{\ast}(c_{0}^{\prime})=s\theta^{\ast}(c_{1}^{\prime})=sx,\theta^{\ast}(c_{2}^{\prime})=sxy^{-2},\theta^{\ast}(e)=sx^{\alpha}y^{-1}
0 = \frac{d}{dt}\exp(t \Xi_{\varphi_0})^* f_{\lambda}= \Xi_{\varphi_0}( f_{\lambda}) = \frac{\sqrt{-1}}{2}[R_{\varphi_0, \bar{\delta}}f_{\lambda, \delta}- R_{\varphi_0, \delta}f_{\lambda, \bar{\delta}}].
{L}^{\pi^\prime}(T(n))= x+X(T(1))-\inf_{s\in[0, T(1)]}X(s) +\sum_{k=2}^n (X(T(k))-\inf_{s\in[T(k-1), T(k)]}X(s)) \geq X(T(1))+\sum_{k=2}^n (X(T(k))- X(T(k-1)))=X(T(n)).
\begin{aligned}|u(x)-m_u^\gamma(B(x,r))| \le \sum_{j=0}^\infty |m_u^\gamma(B(x,2^{-j-1}r))-m_u^\gamma(B(x,2^{-j}r))| & \le 2\sum_{j=0}^\infty m_{|u-c|}^{\gamma/c_D}(B(x,2^{-j}r) &\le Cr^\beta u^{\gamma/{C},\#}_{\beta,r}(x).\end{aligned}
\int_{0}^{d_{N}/N}\int_{\tau_{-}}^{\tau_{+}}\mathbb{E}(|V_{N}(\tau+i\eta)|)|\overline{\partial}\Psi_{f}(\tau,\eta)|\,d\tau,d\eta \leq cd_{N}^{\epsilon}\int_{0}^{d_{N}/N}\int_{\tau_{-}}^{\tau_{+}}\eta^{\alpha-1-\epsilon}\,d\tau\,d\eta &= c(\tau_{+}-\tau_{-})N^{\epsilon}(d_{N}/N)^{\alpha}
| H_{t}( x + if_{t}(x)) - H_{r}(y + if_{t}(y)) | &\leq (r-t)\left[ |y + if_{r}(y)| + |F_{\mu}(y + if_{r}(y))|\right] + 2 |(x,f_{t}(x)) - (y,f_{r}(y))| = O\left(\min\left\{(r-t) , |(x,f_{t}(x)) - (y, f_{r}(y))|\right\}\right)
\|V_l - Z\|_F^2={\rm Tr}[(V_l - Z)^{T}(V_l - Z)] = 2l - 2{\rm Tr}(Z^{T}V_l)\leq 2l - 2\left[l - \frac{20l \epsilon \|C\|_F^2}{\eta}\right] = \frac{40l \epsilon \|C\|_F^2}{\eta}.
\begin{array}{ccccc}{\sf J}^\circ={\sf K},{\sf JT}^\circ={\sf T},{\sf JD}^\circ={\sf D},{\sf J4}^\circ={\sf K4},{\sf J5}^\circ={\sf K5}, ~~{\sf JD5}^\circ={\sf D5}, ~~{\sf JD4}^\circ={\sf D4},{\sf J45}^\circ={\sf K45}, ~~{\sf JD45}^\circ={\sf D45},{\sf LP}^\circ={\sf S4},{\sf JT5}^\circ={\sf JT45}^\circ={\sf S5}.\end{array}