harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
k_v\cdot\bigl((\mathbf{X}_{v|s}\otimes\mathbf{Y}_{v|t})\cdot B\bigr)=\bigl(k_v\bullet_v(\mathbf{X}_{v|s}\otimes\mathbf{Y}_{v|t})\bigr)\cdot\bigl(k_v\cdot B\bigr)
\frac{1}{2}\int_{0}^{T}|u(s)|^{2}ds =\frac{1}{2}\int_{0}^{T}\sum_{j=1}^{d}\langle \psi(\cdot,s),e_{j}(\cdot,s)\rangle_{L^{2}(\nu)}^{2}& \leq\frac{1}{2}\int_{0}^{T}\int_{\mathbb{X}}\psi^{2}(y,s)\nu (dy)ds & \leq\bar{I}(\eta)+\delta.
\begin{aligned}g_2(\xi_1,\xi_2) = f_2(\xi_1,\xi_2) - f_1(\xi_1) f_1(\xi_2), g_3(\xi_1,\xi_2,\xi_3) = f_3(\xi_1,\xi_2,\xi_3) - (f_1f_1 f_1)(\xi_1,\xi_2,\xi_3) &- f_1(\xi_1) g_2(\xi_2,\xi_3) - f_1(\xi_2) g_2(\xi_1,\xi_3) - f_1(\xi_3) g_2(\xi_1,\xi_2), \ldots .\end{aligned}
D^{(n)}_q(u_0,u_1,\dots,u_{2d'+1};t){}= \sum_{\sigma\in \{\pm 1\}^n}\prod_{1\le i\le n}\frac{\prod_{0\le r<2d'+2}\vartheta(u_r+\sigma_i z_i)}{\vartheta(2\sigma_i z_i)}\prod_{1\le i<j\le n}\frac{\vartheta(t+\sigma_i z_i+\sigma_j z_j)}{\vartheta(\sigma_i z_i+\sigma_j z_j)}\prod_{1\le i\le n}T_i^{\sigma_i/2},
\left|1-A_0^{-1}B_{0,N}A_N \right|\leq \frac{c^{A}_J}{\lambda}\sum_{j=1}^N\hat{b}_{j-1,j}\exp \left(\frac{c^{A}_J}{\lambda}\sum_{j=1}^N\hat{b}_{j-1,j}\right)\quad,
{\cal L}= -{\frac{1}{4}}({\cal F}_{\mu \nu}- i e g W_{\mu}^{*}W_{\nu})^{2}-{\frac{1}{2}}|{\cal D}_{\mu}W_{\nu}-{\cal D}_{\nu}W_{\mu}|^{2}+ G^{2}\phi^{2}| W_{\mu}|^{2}-{\frac{1}{2}}( \partial_{\mu}\phi )^{2}- V ( \phi )
i^{*}{\rm div}(\widetilde{J}_{{\Bbb A}_{1}^{+}}^{*}\left(\chi_{10}/\theta_{a_{0},b_{0}}\right)^{8}|_{U})=i^{*}{\rm div}(\widetilde{J}_{{\Bbb A}_{1}^{+}}^{*}\Upsilon_{10}|_{U})={\rm div}(\widetilde{J}_{M}^{*}\Upsilon_{10}),
S_1^{i,j}(\sum_{k}f_{k}) = 2^{j/2}\| \sum_{k}F_{k, (R)_i}\|_{\ell^{2}(\frac{1_R}{|R|})}\leq 2^{j/2}\sum_{k}\| F_{k, (R)_i}\|_{\ell^{2}(\frac{1_R}{|R|})}= 2^{j/2}\sum_{k}\left( \sum_{R}(F_{k, (R)_{i}})^{2}\frac{1_R}{|R|}\right)^{1/2}= \sum_{k}S_1^{i,j}(f_k).
\int_{0}^{a^\gamma}t\mkern+1.5mu df(t^{1/\gamma}) , \lim_{t \downarrow 0}t f(t^{1/\gamma}) = 0, \lim_{t \downarrow 0}\frac{1}{t f(t^{1/\gamma})}\displaystyle\int_{0}^{t}s\mkern+1.5mu df\bigl(s^{1/\gamma}\bigr) = \frac{\alpha/\gamma}{1 + \alpha/\gamma}.
|A(Y,Z)| \le\sum_{i=1}^m \left|g_X(Y,e_i)\right| \sqrt{\sum_{j=1}^m g_X(Z,e_j)^2}\sqrt{\sum_{j=1}^m A(e_i,e_j)^2}&\le \sqrt{\sum_{j=1}^m g_X(Z,e_j)^2}\sqrt{\sum_{i=1}^m g_X (Y,e_i)^2}\sqrt{\sum_{i=1}^m \sum_{j=1}^m A(e_i,e_j)^2}= |A|_{g_X}|Z|_{g_X}|Y|_{g_X}.
\begin{array}{cl}\Delta_{(n-1 l+1)}^{(1)}(x_1,\cdots,x_m|z_1,\cdots,z_{n-1}|z_n,\cdots,z_N) = \displaystyle q^{l-n+1}\sum_{k=n}^{N}z_k\frac{\prod_{j=1}^{n-1}(z_k-z_j q^{-2})}{\prod_{\scriptstyle{i=n}\atop \scriptstyle{i\ne k}}^{N}(z_i-z_k)q^{-1}}\Delta^{(n l)}(x_1,\cdots,x_m|z_1,\cdots, z_{n-1},z_k|z_{n},\mathop{\hat{\cdots}}^{k},z_N), \end{array}
\langle\hat{t}(\frac{p_1}{\delta},\ldots ,\frac{q}{\delta}),\stackrel{\vee}{\phi}\rangle=\delta^{3-3l}\int d^{3r-3}\tilde{p}d^{3v-3}\tilde{q}\hat{t}_1^{[\mu]}(\frac{\tilde{p}}{\delta})\hat{t}_2^{[\nu]}( \frac{\tilde{q}}{\delta}) \psi_{[\mu\nu]}^0(\tilde{p},\tilde{q}).
T_{U}^{\mu \nu , \alpha \beta}= \left[ - \frac 1 2 T_{1}( u ) + \frac 1 2 T_{2}( u ) - \frac 2 3 T_{3}( u ) + \frac 7{1 2}T_{4}( u ) - \frac 1 3 T_{5}( u ) \right]^{\mu \nu , \alpha \beta}
\sum_{(j_1, \ldots, j_n) \in i(D'_\pi)}\varphi_q\left(z_1\left(j_{s^{-1}_\chi(1)}, j_{s^{-1}_\chi(1) + 1}; k_1\right) \cdots z_n\left(j_{s^{-1}_\chi(n)}, j_{s^{-1}_\chi(n) + 1}; k_n\right) \right) = |D'_\pi|
-\int_M \int_0^T u \, \varphi_t \, dt \, d\mathcal{V}= \int_M u_0(x) \, \varphi(x,0) \, d\mathcal{V}(x) + \int_M \int_0^T \left( u^m \, \Delta \varphi + u^p \, \varphi \right) dt \, d\mathcal{V}
G(\hat x,\hat y) = \frac{1}{(2\pi)^{2n}}\int\Bigr(\int^{\infty}_0\!\!\bigr(e^{i(\psi(t,\hat x,\hat\eta)-<\hat y,\hat\eta>)}a(t,\hat x,\hat\eta) -e^{i(\psi(\infty,\hat x,\hat\eta)-<\hat y,\hat\eta>)}a(\infty,\hat x,\hat\eta)\bigr)(1-\chi(\hat\eta))\tau(\hat x,\hat\eta)dt\Bigr)d\hat\eta.
\begin{array}{ll}\hbox{\rm dilaton}\enskip e^\phi \sim z^{-5/6}(y) \noalign{\vskip 0.3 cm}\hbox{\rm metric}\enskip G_{\mu \nu}\sim z^{-1/3}(y) \, \eta_{\mu\nu}\noalign{\vskip 0.3 cm}\hbox{\rm R-R 10-form}\enskip F^{(10)}\sim \nu(y) \, dx^0 \cdots dx^9.\end{array}
| V(s) - W(s)| \leq |V(t)-W(t)| + & \int_{t}^{t+\Delta_\ell}\Big[ |E(X(\tau),\tau)| + |E(Y(\tau),\tau)| \Big] d\tau \leq &P^\gamma + C \left[ P^{\frac43\gamma}Q^{\frac13}+P^{\frac53-\eta}Q^\frac13 \log P + \frac{Q}{P^{\frac13+\gamma-\eta +(\ell-2)\delta}}\right]\Delta_\ell:= &P^\gamma + C[a_1+a_2+a_3]\ \Delta_\ell,
B_n^{(-1,\overbrace{\scriptstyle{0,\ldots,0}}^{r-1})}=\displaystyle \sum_{l=1}^{1}{\alpha}_l^{(1,\overbrace{\scriptstyle{0,\ldots,0}}^{r-1})}(l+r)^n &={\alpha}_1^{(1,\overbrace{\scriptstyle{0,\ldots,0}}^{r-1})}(1+r)^n &={\alpha}_1^{(1)}(1+r)^n .
[ \mathsf{L}_{1}, \mathsf{u}_{2}] = \mathsf{u_{1}u_{2}}[ \mathsf{B}_{1 2}^{- 1}- \mathsf{I \otimes I}, p ( \mathsf{T}_{2})_{1}] \mathsf{u}_{1}^{- 1}+ i \hbar \mathsf{u_{1}u_{2}}\mathsf{( T_{2}\otimes T_{2})}\mathsf{B}_{1 2}^{- 1}\mathsf{u}_{1}^{- 1}
P_{R}(t):=\left\lbrace\begin{array}{ll}{\rm diag}\{L,\ldots,L\}, t\in[\bar{t}_0,\infty)\setminus \cup_{\nu\in\mathbb{N}}\mathcal{A}_{\nu}P_{R,n}(t), t\in\mathcal{A}_{\nu},\nu\in\mathbb{N}\end{array}\right.
P (\mathbf{n}_0,\mathbf{n}_1,\dots,\mathbf{n}_s) := \mathbb{P}\left( \mathbf{N}^t = \mathbf{n}_0;\mathbf{N}^{t+1}= \mathbf{n}_1;\dots;\mathbf{N}^{t+s}= \mathbf{n}_s \right).
\int_{0}^{x}f_{1}(t)dt\cdot\cdot\cdot f_{d}(t)dt=\int_{0}^{x}f_{1}(t_{1})(\int_{0}^{t_{1}}f_{2}(t_{2})(\int_{0}^{t_{2}}\cdot\cdot\cdot(\int_{0}^{t_{d}}f_{d}(t_{d})dt_{d}))dt_{2})dt_{1}
\psi:\left(-\frac{W}{2},\frac{W}{2}\right)\times\left(-\frac{H}{2},\frac{H}{2}\right)\subset\mathbb{R}^2 \to \mathbb{R}, \psi((x_1,x_2)) = \min\{W,H\}\cdot\max\left\{\left(\frac{x_1}{W}\right)^2,\left(\frac{x_2}{H}\right)^2\right\}.
Y_{0}(N)_{\varepsilon}= Y_{0}(N)\backslash\Bigg(\bigcup_{p\in\mathcal{P}_{Y_{0}(N)}}U_{\varepsilon}(p)\cup \bigcup_{\mathfrak{e}\in \mathcal{E}_{Y_{0}(N)}}U_{\varepsilon}(\mathfrak{e})\Bigg).
g_{12}^{\ast}=g_{21}^{\ast}=\varepsilon \kappa \left \vert \delta \right \vert^{-4}\left[ \frac{2\delta \delta \,\acute{}\,\acute{}-3\delta \,\acute{}\,^{2}-4\delta^{2}}{4}+\frac{\left( 2\delta \kappa \,\acute{}-3\delta \,\acute{}\, \kappa \right) v}{2}\right] ,
\bar{g}(u,v) = \sum_{a=0}^{\infty}\sum_{b=0}^{\infty}(-1)^a \cdot 2 \ \left[ \sum_{h=0}^{\lfloor \frac{a}{2}\rfloor -1}(-1)^h \binom{a-2}{h}\binom{b-a}{a-2-2h}\right] u^av^b.
\sum_{k=0}^{n-1}\frac{(q;q^2)_k(-1;q^2)_k}{(q^2;q^2)_k}q^{(n-1)^2-k^2}begin{align*}7pt] =\sum_{\substack{0\le k \le n-1 begin{align*}3pt]k\not= (n-1)/2}}a_{n,k}+a_{n,(n-1)/2}begin{align*}7pt] \equiv \sum_{\substack{0\le k \le n-1 begin{align*}3pt]k\not= (n-1)/2}}b_{n,k}+a_{n,(n-1)/2}\pmod{\Phi_n(q)^2}.
W_{2}( k_{1}) = \frac{1}{2 \pi R_{c}}\left[ \int_{0}^{\pi R_{c}}d x \, < \frac{1}{N}\mathrm{T r}\, \left[ \, \Omega^{( k_{1})}( x ) \, \right] \frac{1}{N}\mathrm{T r}\, \left[ \, \Omega^{( - k_{1})}( 0 ) \, \right] > + \, k_{1}\to - k_{1}\right] .
\langle \widehat{u}g, \widehat{u}f_0 \rangle_{L^2 (\mathbb{R}^2)}+ \langle \widehat{v}g, \widehat{v}f_0 \rangle_{L^2 (\mathbb{R}^2)}= \gamma \langle g,f_0 \rangle_{L^2 (\mathbb{R}^2)},
\delta I[g,\phi]=\int_{\cal M}\sqrt{-g}d^4x \left[E^{\mu\nu}\delta g_{\mu\nu}+E_\phi \delta \phi+\nabla_{\mu}\theta^\mu(\delta g,\delta \phi)\right]-\delta\left[\int_{\partial{\cal M}}\sqrt{-h}d^3x B[g,\phi]\right]~~~.
\frac{d}{dt}\int_\Omega g\log(1+g^2) = \int g_t\log(1+g^2) + \int_\Omega\frac{2g^2g_t}{1+g^2}&= \int_\Omega g_t\log(1+g^2) - 2\int_\Omega\frac{g_t}{1+g^2}&= \int_{\Omega}g_t\log(1+g^2) - 2\int_\Omega \partial_t(\arctan(g))
\| h_0 \sum_{k=n}^m s_k y^{1/2}b_k \| = \| \sum_{k=n}^{m}2^{-k}s_k h y^{1/2}b_i\| \leq \sum_{k=n}^m 2^{-k}\| h\| \|b_i^\ast y b_i\|^{1/2}\leq \sum_{k=n}^m 2^{-k}\|h\| \to 0
f_k = \begin{bmatrix}M y_d 0 d P_{\mathcal{A}_k^b}b+P_{\mathcal{A}_k^a}a\end{bmatrix}, J_k =\begin{bmatrix}M 0 L^T \alpha_y P^T_{\mathcal{A}_k}0 \nu M -M \alpha_u P^T_{\mathcal{A}_k}L - M 0 0 \alpha_y P_{\mathcal{A}_k}\alpha_u P_{\mathcal{A}_k}0 0\end{bmatrix},
\lim_{L\to\infty}\mathbb{E}\big(\zeta^{\omega}_{L,E}(I)\big)= \lim_{L\to\infty}\mathbb{E}\big(\xi^{\omega}_{L,E}(I)\big)= \lim_{L\to\infty}\mathbb{E}\bigg(\sum_p^{N_L^d}\eta^{\omega}_{p,E}(I)\bigg).
\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}((\frac{m\pi}{a})^{2}+(\frac{n\pi}{b})^{2})^{2}w_{mn}sin\frac{m\pi x}{a}sin\frac{n\pi y}{b}=\sum_{m=1}^{\infty}\sum_{n=1}^{\infty}\frac{a_{mn}}{D}sin\frac{m\pi x}{a}sin\frac{n\pi y}{b}
S_{\mathrm{main}}=\sum_{d_1,d_2<D}\frac{1}{\sqrt{d_1d_2}}\sum_{Id_1, Jd_2<D}\alpha_{Id_1}\bar{\alpha}_{Jd_2}\sum_{IL=JM}\frac{G_k(LMd_1d_2\slash p)}{\sqrt{LM}}.
\langle \Psi | \Psi^{\prime}\rangle = \int_{{\cal M}_{L}\times{\cal E}_{L}}{\cal D}^{e}e \;{\cal D}^{e}\widetilde{x}\left( \mathrm{V o l}_{e}{\cal D}_{L}\right)^{- 1}\overline{{{\Psi}}}\Psi^{\prime}.
\sim\frac{2^{1/2}}{\Gamma(1/2)}\frac1{n^{1/2}}{\displaystyle\prod\limits_{1\leq m\equiv0\operatorname*{mod}2}}\frac{e^{1/m}+e^{-1/m}}2=\sqrt{\frac2{\pi\,n}}{\displaystyle\prod\limits_{k=1}^{\infty}}\cosh\left( \frac1{2k}\right) & =\sqrt{\frac2{\pi\,n}}(1.2217795151...)=(0.9748390118...)n^{-1/2}
\bigg(2\lambda K+\frac{4(\lambda-1)}{n}\left(2\left\vert b\right\vert-b\right)\bigg)\omega^{2}+(2(2-\lambda)\left\vert b \right\vert-\lambda b)p\omega-2\lambda\left\vert b\right\vert p(p-1) \leq \,\, 2\lambda KV^{2}+\frac{12(\lambda-1)\left\vert b\right\vert V^{2}}{n}+(4-\lambda)\left\vert b\right\vert pV.
E[\sigma]=\int_m^\infty\frac{dt}{2\pi}\frac{t}{\sqrt{t^2-m^2}}\left[\ln g(it,0)+\ln\left(g(it,0)-\frac{1}{t}g^\prime(it,0) \right)\right]_{1}+ E_{\rm FD}^{(1)}+ E_{\rm CT}\, ,
\begin{aligned}\|\boldsymbol{\eta}_u^n\|_0^2 \leq C h^2(\nu\|\boldsymbol{\eta}_u^n\|_1^2+\gamma\| \nabla\cdot\boldsymbol{\eta}_u^n\|_0^2) \leq C ((\nu+ \gamma)h^{2k_u+2}\|\boldsymbol{u}(t_n)\|_{W^{k_u+1,2}}^2+\gamma^{-1}h^{2k_p+4}\|p(t_n)\|_{W^{k_p+1,2}}^2). \end{aligned}
p_{2l+1}[C(\mu)] =\sum_{h=0}^{l}\sum_{r=0}^{h}\sum_{t=0}^{2l+1-h}(-1)^{h}\varphi_{2l+1}(h,r,t)q^{-}_{t,r}, p_{2l}[C(\mu)] =\sum_{h=0}^{l-1}\sum_{r=0}^{h}\sum_{t=0}^{2l-h}(-1)^{h}\varphi_{2l}(h,r,t)q^{+}_{r,t}+\frac{1}{2}(-1)^{l}\sum_{r=0}^{l}\sum_{t=0}^{l}\varphi_{2l}(l,r,t)q^{+}_{r,t}.
\epsilon_{R P A}({\bf{q}},{\tilde{\omega}}) = 1 + \frac{v_{{\bf{q}}}}{V}\sum_{{\bf{k}}}\frac{\Lambda_{{\bf{k}}}^{2}({\bf{q}}) - \Lambda_{{\bf{k}}}^{2}( -{\bf{q}})}{{\tilde{\omega}}- \frac{{\bf{k . q}}}{m}}
\prod_{i=1}^{\ell-1}f(z_{i}, z_{\ell})+\sum_{t=\ell+1}^{m}g(z_{\ell}, z_{t})\prod_{\begin{subarray}{c}i=1 i\not=\ell \end{subarray}}^{t-1}f(z_{i}, z_{\ell})=\prod_{\begin{subarray}{c}i=1 i\not=\ell \end{subarray}}^{m}f(z_{i}, z_{\ell})
\rho = \tfrac{\sqrt{t-\max(s,r)}\, \sqrt{\min(s,r)}}{\sqrt{\max(s,r)}\,\sqrt{t-\min(s,r)}}\geq \tfrac{\sqrt{\tfrac{t}{2}}\, \sqrt{\tfrac{t}{4}}}{\sqrt{\tfrac{t}{2}}\,\sqrt{\tfrac{3t}{4}}}=\tfrac{1}{\sqrt 3}.
G \approx \frac{1}{2 \pi ( 1 + g^{2}/ \pi )}\bigg [ 1 - 2^{( 5 \pi + 3 g^{2}) / ( \pi + g^{2})}\frac{( 1 + g^{2}/ \pi )^{2}}{1 6 \pi^{2}}\bigg ( \frac{m^{2}}{T^{2}}\bigg )^{( \pi + 2 g^{2}) / ( \pi + g^{2})}\bigg ] .
\hat{\psi}_{\gamma}(\delta,\alpha) =\frac{d}{d\tau}\frac{1}{n-n_{\ell}}\sum_{i\notin I_{\ell}}\left\{Z_{i}[\hat{\gamma}_{\ell}(X_{i})+\tau\delta(X_{i})]+\alpha(X_{i})[Y_{i}-\hat{\gamma}_{\ell}(X_{i})-\tau\delta(X_{i})]-\tilde{\theta}_{\ell}\right\}& =\frac{1}{n-n_{\ell}}\sum_{i\notin I_{\ell}}\left[ Z_{i}-\alpha(X_{i})\right] \delta(X_{i}).
G_0 > G_1=\cdots=G_{q^{d}-1}=N_1 > G_{q^{d}}=\cdots=G_{q^{2d}-1}=N_2 > G_{q^{2ds}}=\cdots = G_{q^{3d}-1}=N_3 > \cdots > G_{q^{d(\alpha-2)}}=\cdots=G_{q^{d(\alpha-1)}}=N_{\alpha-1}& > \{1\}.
\begin{aligned}K_j =\frac{1}{\binom{n}{j}}\big(\kappa_1\cdots \kappa_j+\kappa_1\cdots \kappa_{j-1}\kappa_{j+1}+\cdots +\kappa_{n-j+1}\cdots \kappa_n\big), =\frac{1}{\binom{n}{j}}\big(H^{1,\ldots, j}+H^{1,\ldots, j-1,j+1}+\cdots +H^{n-j+1,\ldots, n}\big).\end{aligned}
\left|\Big< u(\cdot,T), \phi_p\Big> \right| =\left| e^{-Tp^{2\beta}}u_p (0)+ \int_0^T e^{(s-T)p^{2\beta}}F_p(u)(s)ds \right| &\le e^{-Tp^{2\beta}}\|u(.,0)\|_{L^2(\Omega)}+\frac{1-e^{-Tp^{2\beta}}}{p^{2\beta}}\|F(u)\|_{L^\infty}\le \frac{E}{T p^{2\beta}}+ \frac{E}{p^{2\beta}}\le \frac{\widetilde E_1}{p^{2\beta}}.
d\big(H(z)d\overline{z}\big)\big|A_{w_{j}}(z)\big|^{s}=d\big(H(z)\big|A_{w_{j}}(z)\big|^{s}d\overline{z}\big)-H(z)\partial_{z}\big|A_{w_{j}}(z)\big|^{s}dzd\overline{z},
{\cal L}= \frac{1}{2}(\partial_\mu\Phi_1^\prime)^2 + \frac{1}{2}(\partial_\mu\Phi_2^\prime)^2 + \frac{1}{2}(\partial_\mu\Phi_3^\prime)^2 + \frac{m}{\pi}\mu \cos(2 a \sqrt{\pi}\Phi_1^\prime),
d \left( \gamma \right) = 3 - \sum_{\tilde{\Phi}= \Phi , \, g}d_{\tilde{\Phi}}N_{\tilde{\Phi}}\, \, , \, \, \, \, \, \mathrm{w i t h}\, \, \, \, \, d_{g}= \frac 1 2 \, \, .
c_h(i,j,k)=\sum_{r=0}^{n+p-2h}\begin{Large}\left[\begin{smallmatrix}m \; n+p-2h \; m+n+p-2\ell i r 0\end{smallmatrix}\right]\end{Large}\begin{Large}\left[\begin{smallmatrix}n \; p \; n+p-2h j k r\end{smallmatrix}\right]\end{Large}
H{\bf E}^2\equiv \frac{\Gamma r_0^4\{\bf E}^2}{4R^4 x^3}<\frac{1}{1+ \frac{\Delta \theta_D^2}{x_*^2}}\equiv \frac{{\bf l}^2}{R^2}\left(\frac{\Delta\theta_D^2}{x_*^2}\right)^{-2}\ ,
F_p = \begin{pmatrix}\omega_p^{0\cdot 0}\omega_p^{0\cdot 1}\ldots \omega_p^{0\cdot (p-1)}\omega_p^{1\cdot 0}\omega_p^{1\cdot 1}\ldots \omega_p^{0\cdot 0}\ldots \ldots \ldots \ldots \omega_p^{(p-1)\cdot 0}\omega_p^{(p-1)\cdot 1}\ldots \omega_p^{(p-1)^2}\end{pmatrix}
F(x_{1},x_{2},...,x_{n})=\sum_{\underset{\delta_{i}=0}{\delta\in\{0,1\}^{n}}}v_{\delta}(x_{i})\left[\prod_{k\neq i,j}f_{k}^{\delta_{k}}(x_{k})\right]g_{j}^{\delta_{j}}(x_{j}).
\Lambda \frac{d}{d \Lambda}2 \pi^{2}\int d^{4}q \frac{1}{q^{2}( q + p )^{2}\Big ( 1 + q^{2 n}/ \Lambda^{2 n}\Big )^{2}}\sum_{i}c_{i}\ln \frac{M_{i}}{q}= 4 \pi^{4}\ln \frac{\Lambda}{p}+ O ( 1 )
\big(\mathcal{F}^{-1}\otimes_{A_\star}1\big)(\Delta\otimes_{A_\star}\mathrm{id})\big(\mathcal{F}^{-1}\big)= \big(1\otimes_{A_\star}\mathcal{F}^{-1}\big)(\mathrm{id}\otimes_{A_\star}\Delta)\big(\mathcal{F}^{-1}\big).
{}_2F_1\bigg[\begin{matrix}\alpha \beta &\alpha-\beta+1\end{matrix}\bigg|\,z^2\bigg]=(1+z)^{-2\alpha}{}_2F_1\bigg[\begin{matrix}\alpha \alpha-\beta+\frac12 &2\alpha-2\beta+1\end{matrix}\bigg|\,\frac{4z}{(1+z)^2}\bigg].
\rho_k^{-1}L_k\overset{\eqref{conic-L}}{=}C+\rho_k^{-1}B+\rho_k^{-1}L_{\nabla g}\sum_{i=0}^{k-1}\rho_i\eta_i+\rho_k^{-2}\overset{\eqref{def1}}{\leq}C+\rho_0^{-1}B+\rho_0^{-1}L_{\nabla g}\theta+\rho_0^{-2}\overset{\eqref{def2}}{=}L.
\bigg(\frac{4}{\mathbf{C}_3}- 1\bigg) \bigg\|\frac{P_1 f}{|x|^2}\bigg\|_{2}^2 \leq \|\Delta_r f \|_{2}^2 - 2\bigg\langle \sum_{j=1}^3 L_j^2 \bigg(\partial_r f - \frac{f}{2|x|}\bigg), \partial_r f - \frac{f}{2|x|}\bigg\rangle.
{}\widehat{w}_{i}^{(\epsilon_{k})}(t)\leq\rho_{\epsilon_{k}}(t)=\frac{\tau -\epsilon_{k}}{\sigma}\cdot\exp\left [\frac{\nu\cdot (T-t)}{\sigma}\right ]\leq\frac{\tau}{\sigma}\cdot\exp\left (\frac{\nu\cdot T}{\sigma}\right )\mbox{for all $t\in [0,T]$,}
f^*(Y(\cdot,v,Dv)) \det DY(\cdot,v,Dv) = e^{[(1-t)\tau+\eta_a(\cdot)](v-u_0)}\big[tf(\cdot) \quad+(1-t)f^*(Y(\cdot,u_0,Du_0))\det DY(\cdot,u_0,Du_0)\big] Yu(\Omega) = \Omega^*.
{\cal A}_L^{(1)\,{\rm subdom}}\simeq \ -\frac m{8\pi}\left( \frac{5\lambda^2}{64m^4}-\frac{e^4}{m^2\Theta^2}\right) \frac{{\bf p}^2}{m^2}\left[ \ln\left( \frac{\Lambda^2}{{\bf p}^2}\right) +i\pi \right] \;.
I_{n}^{( \pm )}( \nu ; k ) = \frac{1}{( \nu^{2}- \frac{1}{4})^{n - 1}}\int_{1}^{\infty}d \alpha W_{n}^{( \pm )}( \alpha ; k ) [ \alpha^{\frac{\nu}{2}}+ \alpha^{\frac{- \nu}{2}}] ,
\left[ z_{1}, \bar{z}_{1}\right] = \left[ z_{2}, \bar{z}_{2}\right] = \left[ z_{3}, \bar{z}_{3}\right] = \left[ z_{4}, \bar{z}_{4}\right] = - \frac{\zeta}{2}, \quad \textrm{o t h e r s a r e z e r o}.
a_i +a_4\in (a_3,a_4) \mbox{if and only if}a_4=2a_3-a_i, a_i+a_4-a_j\in (a_3,a_4) \mbox{if and only if}a_4=\begin{cases}2a_3+a_j-a_i\; , a_3+a_j-a_i\; .\end{cases}
dev(B)=B-\frac{1}{3}tr(B)I=B-\frac{1}{3}(3+\gamma^{2})I=[\begin{matrix}\frac{2}{3}\gamma^{2}&\gamma&0\\ \gamma&-\frac{1}{3}\gamma^{2}&0\\ 0&0&-\frac{1}{3}\gamma^{2}\end{matrix}]
\phi_{ijk}^{-1}\tilde{F}_i\phi_{ijk}(y) = (\mathrm{id}- p\psi_{ijk})\tilde{F}_i(\mathrm{id}+ p\psi_{ijk})(y) = (\mathrm{id}- p\psi_{ijk})(\tilde{F}_i(y) + p \psi_{ijk}(y)^p) = \tilde{F}_i(y) - p\psi_{ijk}(y^p) + p\psi_{ijk}(y)^p = \tilde{F}_i(y) + p\psi_{ijk}(y)^p,
x_r(z,q)=(-1)^r q^{\frac{(2r-n)^2}{8n}}u^{-r}\prod_{m \geq 1}(1-q^{nm-r}u^n)\prod_{m \geq 0}(1-q^{nm+r}u^{-n})\prod_{m \geq 1}(1-q^{m})^{-2n}.
-\mathcal{L}^{\alpha^{j}}\overline{A}_{1}-...-\mathcal{L}^{\alpha^{j}}f_{l}\overline{A}_{l}+ \mathcal{L}^{\alpha^{j}}f_{l+1}\overline{A}_{l+1}+...+\mathcal{L}^{\alpha^{j}}f_{m}\overline{A}_{m}-\frac{1}{2i}(\mathcal{L}^{\alpha^{j}}g)=0,1 \leq j \leq m+1.
\mathcal{U}_1=\int_{\kappa_{\rm b}}^{\kappa_{\rm t}}{x^2}e^{-\frac{1}{2}\left(x+\frac{2\psi_{1}\tau_{\rm d}}{N_{\rm a}}\right)^2}\,\mathrm{d}x=e^{\frac{\psi_{2}\tau_{\rm d}}{N_{\rm a}}-\frac{2\psi_{1}^2\tau_{\rm d}^2}{N_{\rm a}^2}}\mathcal{W}_1',
\tilde{H}_{v}-\tilde{E}_{v}=2\int_{\cal B}d\sigma_\nu\zeta_\mu\nabla_\rho\nabla_\sigma\left({\partial{\tilde L}_v \over \partial R_{\mu\sigma\rho\nu}}+{\partial{\tilde L}_v \over \partial R_{\nu\sigma\rho\mu}}\right)~~~.
\sum_{t=t_d}^T \ell(\hat y_t,y_t) \leq \sum_{t=t_d}^T \ell(f_{d,t},y_t) + \frac{1}{\eta_{T+1}}\log (D_{T+1}\underbrace{\hat p_{d,T+1}}_{\leq 1}) - \frac{1}{\eta_t}\log ( \underbrace{D_{t_d}\hat p_{d,t_d}}_{= 1}) + \frac{1}{8}\sum_{t=t_d}^T \eta_t \,,
{\cal V}_{\sigma}\left( \eta , \zeta , s \right) = 1 - \frac{1}{\eta}\sum_{n = 0}^{\infty}\sum_{m = 1}^{\infty}\left( - \eta \right)^{n}\zeta^{m}\, \theta \left( s + \sigma - n \sigma - m \right) \, F_{n m}\left( s + \sigma - n \sigma - m , \sigma \right) \, ,
\tilde{D}(\eta) =\int_{0}^{1}2\sqrt{\eta''}-\eta''(s)\cdot(1-s)-\frac{1}{1-s}ds-\eta(0)+\inf_{t\in[0,1)}\left\{\eta(t)-\xi(t)\right\}+h^{2}+\xi(1) \tilde{P}(\mu) =2P(\mu).
_{1}\phi_{1}\left(\begin{array}{ccc}\begin{array}{c}q^{n}-q^{\alpha+1}\end{array}\vert q,-xq^{1+\alpha-\beta}\end{array}\right)q^{\beta^{2}/2}=\frac{1}{\sqrt{\pi\log q^{-2}}}\int_{-\infty}^{\infty}\frac{\exp\left(\frac{y^{2}}{\log q^{2}}+i\beta y\right)dy}{\left(-q^{\alpha+1},q^{\alpha+1/2}e^{-iy};q\right)_{\infty}\left(xq^{\alpha+1/2}e^{-iy};q\right)_{n}}
p \cdot C_p = c \cdot d \cdot (c - \sigma^{-1}_c) \cdot (d - \sigma^{-1}_d) \cdot \left( \prod_{q \mid N_{\mathrm{sp}}}( 1- q^{-1}\sigma^{-1}_q ) \right) \cdot \left( \prod_{q \mid N_{\mathrm{ns}}}( 1 + q^{-1}\sigma^{-1}_q ) \right) \cdot \left( p - a_p(f) \cdot \sigma^{-1}_p + \psi(p) \cdot \sigma^{-2}_p \right) .
\begin{cases}\frac{\partial \mathbf{u}_0 (\mathbf{x},t)}{\partial t}=-b(\mathbf{D})^*g^0b(\mathbf{D})\mathbf{u}_0(\mathbf{x},t),\mathbf{x}\in \mathcal{O},\; t>0, \mathbf{u}_0(\mathbf{x},0)=\boldsymbol{\varphi}(\mathbf{x}),\mathbf{x}\in \mathcal{O}, \partial_{\boldsymbol{\nu}}^0\mathbf{u}_0 (\cdot ,t)\vert_{\partial \mathcal{O}}=0,t>0.\end{cases}
\mu_{N - 1}^{( N )}= -{\frac{1}{2}}\int_{- \infty}^{\infty}d x w^{2}( x ) \left({\cal P}_{N - 1}^{\prime}\right)^{2}+{\frac{1}{2}}\sum_{m = 0}^{N - 1}K_{N - 1 , m}.
\{\mathsf{L_{1}}, \mathsf{L_{2}}\}= [ \mathsf{L_{2}}, \mathsf{\mathcal{C}_{1 2}}] , \qquad \{\mathsf{R_{1}}, \mathsf{R_{2}}\}= [ \mathsf{R_{2}}, \mathsf{\mathcal{C}_{1 2}}] .
\int_{A_r}|u_a|^2 \le \left( C (a^*-a)^{1-2/p}\right)^{d/(d+2)}\left(C r^{d(1-p/2)}\right)^{2/(d+2)}&= C \left[ \frac{r}{(a^*-a)^{1/p}}\right]^{\frac{d}{d+2}(2-p)}.
\widehat{u}_{tt}+ |\xi|^2 \widehat{u}+ b(t)\widehat{u}_t + m(t) \widehat{u}=0 ,\,\,\,\widehat{u}(0,\xi)=\widehat{u}_0(\xi),\,\,\,\widehat{u}_t(0,\xi)=\widehat{u}_1(\xi)
\bar{\check{y}}_{\mathrm{d},i,k}[n] =\bar{h}_{i,kk}[0]x_{\mathrm{d},k}[n]+\sum_{\nu\in\Delta\setminus\{0\}}\bar{h}_{i,kk}[\nu]x_{\mathrm{d},k}[n-\nu] &\quad+\sum_{k'\in\mathcal{K}\setminus\{k\}}\sum_{\nu\in\Delta}\bar{h}_{i,kk'}[\nu]x_{\mathrm{d},k'}[n-\nu] &\quad+\bar{z}_{\mathrm{d},i,k}[n]+\bar{e}_{\mathrm{d},i,k}[n],\,i\in\mathcal{I},k\in\mathcal{K},
\mathcal{S}_{\alpha\beta}\min\bigg\{\nu_{i}^{\frac{1}{q^{+}_{i,\rho}}}, \nu_{i}^{\frac{1}{q^{-}_{i,\rho}}}\bigg\}> \frac{\mathcal{S}_{\alpha\beta}}{2}\nu_{i}^{\frac{1}{q(x_i)}}
\begin{aligned}u_{\iota,\iota+1}^{(i)}x_{\iota+1,\iota+1}- x_{\iota\iota}u_{\iota,\iota+1}^{(\mu)}= 0 u_{\iota,\iota+1}^{(\mu)}y_{\iota+1,\iota+1}- y_{\iota\iota}u_{\iota,\iota+1}^{(i)}= 0 \end{aligned}
T_{J_{1}}^{R_{1}}\star T_{J_{1}}^{R_{2}}= \sum_{R , J}\frac{1}{N}\left( \begin{array}{c c c}{{R}}&{{R_{1}}}&{{R_{2}}}\\{{J}}&{{J_{1}}}&{{J_{2}}}\\ \end{array}\right) \left\{\begin{array}{c c c}{{R}}&{{R_{1}}}&{{R_{2}}}\\{{\frac{p}{2}}}&{{\frac{p}{2}}}&{{\frac{p}{2}}}\\ \end{array}\right\}T_{J}^{R},
a^{T}\equiv \sum_{k = 0}^{N ( a )}( - 1 )^{k}D^{k}a_{k}= \sum_{k = 0}^{N ( a )}\sum_{m = k}^{N ( a )}( - 1 )^{m}{\binom{m}{n}}D^{m - k}( a_{m}) D^{k}.
J_\mu =\partial_\mu+\frac{\eta_{\mu\alpha}x^\alpha x^\nu\partial_\nu}{l_1^2}, M_{\mu\nu}=\eta_{\mu\alpha}x^\alpha\partial_\nu-\eta_{\nu\alpha}x^\alpha\partial_\mu=\eta_{\mu\alpha}x^\alpha J_\nu-\eta_{\nu\alpha}x^\alpha J_\mu.
\Gamma [ \phi ] ={\frac{1}{2}}\int_{0}^{\infty}{\frac{d T}{T}}\mathrm{e}^{- m^{2}T}\int{\cal D}x \mathrm{e x p}\left[ - \int_{0}^{T}\left({\frac{{\dot{x}}^{2}}{4}}+ U^{\prime \prime}\left( \phi ( x ) \right) \right) \right]
\mathrm \Pr_{\mathrm{Foliage}}=\mathrm{P_{T}}\mathrm{G_{T}}\mathrm{G_{R}}\frac{\lambda^2}{(4 \pi d_{\mathrm{tot}})^2}\cdot 10^{\left(-\frac{d_{\mathrm{Foliage}}\cdot \alpha}{10}\right)}\cdot |\Gamma|^2
s_{i,m}= \frac{1}{4}\int_0^{\Delta t}\left(1-2\left(1- \frac{|\tau|}{\Delta t}\right)^2\right) \frac{\partial^3 \nabla_{y_i}\frak u_i}{\partial t^3}(t_m+\tau) d\tau -\frac{1}{4}\int^0_{-\Delta t}\left(1-2\left(1- \frac{|\tau|}{\Delta t}\right)^2\right) \frac{\partial^3\nabla_{y_i}\frak u_i}{\partial t^3}(t_m+\tau) d\tau.
\Theta(\cdot,\mu,\tilde{\mu}):=\sum_{\substack{k=1, \beta_{k}> 0}}^{l}\zeta_{k}\big[\null_{k}\eta\big]_{\frac{p}{1-\alpha_{k}}}\vartheta(\mu,\tilde{\mu})^{\beta_{k}}+ \hat{\zeta}_{k}^{2}\big[\null_{k}\hat{\eta}\big]_{\frac{p}{1-\alpha_{k}}}^{2}\vartheta(\mu,\tilde{\mu})^{2\beta_{k}}.
\mathbf{P}\mathbf{H}\mathbf{Q}= \begin{pmatrix}\mathbf{H}_P \mathbf{H}_I \end{pmatrix}= \begin{pmatrix}\mathbf{H}_{P,1}\mathbf{O}\mathbf{H}_{I,1}\ddots \vdots \mathbf{O}\mathbf{H}_{P,K}\mathbf{H}_{I,K}\end{pmatrix},
\sigma_{\lambda, \lambda^\prime}(\mathsf{M}^n_m)^i_j = q^{-(\lambda, \lambda_i - \lambda_j)}q^{-(\lambda^\prime, \lambda_m - \lambda_n)}(\mathsf{M}^n_m)^i_j, \sigma_{\lambda, \lambda^\prime}(\mathsf{N}^n_m)^i_j = q^{(\lambda, \lambda_m - \lambda_n)}q^{(\lambda^\prime, \lambda_i - \lambda_j)}(\mathsf{N}^n_m)^i_j.
V\pi_\alpha(a)V^*(\xi\otimes \delta_t) =V\pi_\alpha(a)(v_{t^{-1}}^*\xi\otimes\delta_t)=V(\alpha_{t^{-1}}(a)v_{t^{-1}}^*\xi\otimes\delta_t) &=v_{t^{-1}}\alpha_{t^{-1}}(a)v_{t^{-1}}^*\xi\otimes\delta_t=\beta_{t^{-1}}(a)\xi\otimes\delta_t &=\pi_\beta(a)(\xi\otimes\delta_t),
\Big[ H_1 \boxtimes_{R_1}H_2 \boxtimes_{R_2}\ldots \boxtimes_{R_{n-1}}H_n \boxtimes_{R_n}- \Big]_{}:= \Big(H_{a+1}\boxtimes_{R_{a+1}}\ldots \boxtimes_{R_{b-1}}H_b\Big)\boxtimes_{R_a^{}\bar\otimes R_b}\Big(H_{b+1}\boxtimes_{R_{b+1}}\ldots \boxtimes_{R_{a-1}}H_a\Big)
\left((-1)^{m-1}\xi_1, (-1)^{m-2}\xi_2,\dots, -\xi_{m-1},\xi_m, \xi_{m+1},\xi_{m+2},\dots, \xi_n\right)=\left(\xi_m,\dots, \xi_1\right)\chi_\Gamma\left(\left(X'_f\right)_f\right).
W_1(m_1,m_2) =\inf\left\{\int_{X\times X}\rho_X(x_1,x_2)\pi(d(x_1,x_2)):\pi\in\Pi(m_1,m_2)\right\}&=\sup\left\{\int_X\phi(x)m_1(dx)-\int_X\phi(x)m_2(dx):\phi\in\mathrm{Lip}_1(X)\right\}.
R \mid_{s \rightarrow 0}\mathrm{}= \left( \begin{matrix}{{0}}&{{\frac{1}{2}}}&{{0}}&{{0}}\\{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\{{0}}&{{- \frac{1}{2}}}&{{0}}&{{0}}\\ \end{matrix}\right) \ .
\lim_{N \rightarrow \infty}\frac{1}{N^2}\langle \sum_{\alpha,\beta=1}^{N}\big(\frac{\alpha-1}{N}\big) \big(\frac{\beta-1}{N}\big)\xi_{(\alpha-1) i}\xi_{(\beta-1) j}\rangle =\langle \int_{-1}^0 d\tau d\tau' \tau \tau' \xi_i(\tau) \xi_j(\tau') \rangle.
\Gamma_{div}^{mat}= \frac{1}{\varepsilon}\int\left(\gamma_1 \frac{1}{2}g^{\mu \nu}\partial_\mu \varphi \partial_\nu\varphi - \gamma_2 \frac{\varphi^4}{4 !}+ \gamma_3 \frac{1}{2}R \varphi^2\right) \sqrt{-g}~d^4 x.
\langle f_1 \otimes f_2, g_1 \otimes g_2 \rangle = \langle f_1,g_1\rangle_1 \langle f_2, g_2 \rangle_2, \forall f_1,g_1 \in \mathcal{H}_1, \; \forall f_2, g_2 \in \mathcal{H}_2.
\left(1 - e^{-(2k-1)\pi x}\right)^2 \left(1 + e^{-(2k-1)\pi x}\right)^{-4}< \left(1 - e^{-1.1 \cdot \pi}\right)^2 \left(1 + e^{-1.1 \cdot \pi}\right)^{-4}< 0.83, k = 1.