harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
I_{1,1}:=\iint_{D_t}\varphi(\underline{u})|\underline{L}\tilde{\Lambda}|^2|L(\ln \alpha)|\lesssim \int_{\mathbb{R}}\frac{K_1\gamma}{\varphi^{3/4}(u)}\underbrace{\left[\int_{C_{t,u}}\varphi(\underline{u})|\underline{L}\tilde{\Lambda}|^2ds\right]}_{\lesssim \mathcal{F}(t)}du\lesssim K_1 \gamma \varepsilon^2.
\begin{array}{l}{{t \ln t > ( t - 1 ) \ln ( t + 1 ) \qquad \mathrm{f o r}\quad t > 1 \, ,}}\\{{t \ln t < ( t - 1 ) \ln ( t + 1 ) \qquad \mathrm{f o r}\quad 0 < t < 1 \, ,}}\\ \end{array}
\partial_{t}^{2}v_{4}^{0}(t,r) = \frac{-1}{2\pi}\int_{t}^{\infty}ds \int_{0}^{s-t}\frac{\rho d\rho}{\sqrt{(s-t)^{2}-\rho^{2}}}\int_{0}^{2\pi}d\theta \frac{\partial_{1}^{2}v_{4,c}^{0}(s,\sqrt{r^{2}+\rho^{2}+2 r \rho \cos(\theta)}) \left(r+\rho \cos(\theta)\right)}{\sqrt{r^{2}+\rho^{2}+2 r \rho \cos(\theta)}}
f_{m_{st}}(s,t)*f_{m_{st}-1}(s,t)*\cdots *f_1(s,t)=f_0(s,t) &u*f_{m_{st}-1}(s,t)*f_{m_{st}-2}(s,t)*\cdots *f_1(s,t)*f_0(s,t) &=u*f_{m_{ts}-1}(t,s)*f_{m_{ts}-2}(t,s)*\cdots *f_1(t,s)*f_0(t,s)
\zeta_{2_{-}}(z)= \frac{1}{1 - 4z + 3 z^2 + 2 z^3 - z^4}, \zeta_{\bar{2}}(z)=\frac{1}{1 - 3z + 4 z^3 - z^4}, \zeta_{\tilde{2}}(z)=\frac{1}{1 - 3z + z^2 + z^3 + z^4}.
\mathcal{I}_m=\big\{(i,j,t,p)\mid\ 0\leq i,j\leq m,\ \mathrm{max}\{i+j-m,m-1-i-j\}\leq t\leq m-|i-j|, &\mathrm{max}\{0,i+j-m,i+t-m,j+t-m\}\leq p\leq &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathrm{min}\{i,j,t,i+j+t+1-m\}\big\}
D_1= \tau_1^+ \tau_1^- \ , \D_2= \tau_2^+ \tau_2^- , \D_3= \tau_1^+ \tau_2^+ \ , \D_4= 2 \cos{\alpha}\left ( \tau_1^+ \tau_2^- + \tau_1^- \tau_2^+ \right )- \left[ \tau_{12}, \tau_{21}\right]_+\ll{etsa-cas}
B_{n,k}(x_1,x_2,\dots,x_{n-k+1}) = \sum_{\pi(n,k)}\frac{n!}{j_1 ! j_2 ! \dots j_{n-k+1}!}\Big(\frac{x_1}{1!}\Big)^{j_1}\Big(\frac{x_2}{2!}\Big)^{j_2}\dots \Big(\frac{x_{n-k+1}}{(n-k+1)!}\Big)^{j_{n-k+1}}
\left( \begin{array}{c}{{A_{+}^{\dagger}( \theta )}}\\{{A_{-}^{\dagger}( \theta )}}\\ \end{array}\right) B = R ( \theta ) \left( \begin{array}{c}{{A_{+}^{\dagger}( - \theta )}}\\{{A_{-}^{\dagger}( - \theta )}}\\ \end{array}\right) B
p(x,t) =\mathbb{P}(\mathcal{A}_t(x)\cap\mathcal{Y}=\varnothing)=\exp\left(-\gamma\mathcal{H}\times\rho(\mathcal{A}_t(x))\right) &=\exp\left(-\gamma\int_{r=t}^{1}\mathcal{H}(B(x,r))r^{-Q-1}\,dr\right)\,.
P_{a_-}(\xi_i(a_+))-P_{a_-}(\xi_i(a_-))=P_{a_-}(\xi_i(a_+))-\frac{a_-}{a_+}P_{a_+}(\xi_i(a_+))=\frac{a_+-a_-}{a_+}\varphi(\xi_i(a_+)).
\bigl \langle \partial_t \bigl(v_{\delta,\lambda}(t)-v_0 \bigr), F^{-1}_\Gamma \bigl(v_{\delta,\lambda}(t)-v_0 \bigr) \bigr\rangle_{V_{\Gamma,0}^*,V_{\Gamma,0}}\! + \! \int_\Gamma \nabla_\Gamma w_{\delta,\lambda}(t) \cdot \nabla_\Gamma F^{-1}_\Gamma \bigl( v_{\delta,\lambda}(t)-v_0 \bigr)\,d\Gamma =0
\Big(\partial_\sigma\partial^\sigma H_{\mu\nu}+\partial_\mu\partial_\nu H-\partial_\mu \partial^\sigma H_{\sigma\nu}- \partial_\nu \partial^\sigma H_{\sigma\mu}-\eta_{\mu\nu}\left[\partial_\sigma\partial^\sigma H-\partial^\sigma\partial^\rho H_{\sigma\rho}\right]+{\widehat M}_P^{3-D}T_{\mu\nu}\Big) \delta(z)=0~.
\begin{aligned}[][a_0,a_1,\ldots,a_i,x,a_i,a_{i+1},\ldots,a_n] =[a_0,a_1,\ldots a_i]\cdot[a_i,x]\cdot[x,a_i] \cdot [a_i,a_{i+1},\ldots,a_n] &=[a_0,a_1,\ldots a_i]\cdot [a_i,a_{i+1},\ldots,a_n] &=[a_0,a_1,\ldots a_n]=f.\end{aligned}
z^{-}(\gamma)=\left.\frac{\partial}{\partial\bar{\epsilon}}(\rho^{\epsilon\nu}(\gamma))\right|_{\epsilon=0}\rho(\gamma)^{-1}=\overline{\left.\frac{\partial}{\partial\epsilon}(\rho^{\epsilon\nu}(\gamma))\right|_{\epsilon=0}\rho(\gamma)^{-1}},
F^n_{k,\varepsilon}=\frac{\sqrt{2}|\Gamma(\gamma+1+i\nu)|}{\Gamma(2\gamma+1)}e^{\pi\nu/2}(2\varepsilon r)^{\gamma-\frac{(n-1)}2}{\rm Re}\left\{e^{i(\varepsilon r+\xi)}{_1F_1}(\gamma-i\nu,2\gamma+1,-2i\varepsilon r)\right\}
S_n(\lambda, 0, 0) = \frac{1}{\bigl(n-1+2\lambda\phi(-in)\bigr)\bigl(n+2\lambda\phi(-in)\bigr)}\int\limits_{0<t_2<\cdots<t_{n-1}<1}\prod_{i<j}^n |t_i-t_j|^{2\lambda d(j-i)}\Big\vert_{\substack{t_1=0 t_n=1}}\,dt, = \frac{1}{\bigl(n-1+2\lambda\phi(-in)\bigr)\bigl(n+2\lambda\phi(-in)\bigr)}S_{n-2}(\lambda, 2\lambda, 2\lambda),
\beta_2=\frac{x_2}{h'(x_0)}+\frac{h''(x_0)}{(h'(x_0))^2},\beta_3=\frac{x_3}{(h'(x_0))^2}+\frac{3h''(x_0)x_2}{(h'(x_0))^3}+\frac{h'''(x_0)}{(h'(x_0))^3}.
\begin{cases}T_k=T_1-\beta_1q^3\sum_{j=0}^{k-2}q^j,k>2 T_2=3/4-\beta_1q^3 T_1=3/4\end{cases}\quad\begin{cases}T_{k+1}=T_k-\beta_1q^{k+2},k>1 T_1=3/4. \end{cases}
T =\sum_{i=1}^m T(v_i)\otimes v_i^*=\sum_{i=1}^m\sum_{j=1}^n a_{ij}e_j\otimes v_i^* &\in A\otimes V^*\subset (A\ltimes_{\rho^*}V^*)\otimes (A\ltimes_{\rho^*}V^*).
\epsilon\cdot |F| \leq |\sum_{\gamma\in F}\mathrm{sign}(T(g_\gamma)(x))\cdot T(g_\gamma)(x)|=|T(\sum_{\gamma\in F}\mathrm{sign}(T(g_\gamma)(x))\cdot g_\gamma)(x)| &\leq \|T(\sum_{\gamma\in F}\mathrm{sign}(T(g_\gamma)(x))\cdot g_\gamma)\|\leq \|T\|\|\sum_{\gamma\in F}\mathrm{sign}(T(g_\gamma)(x))\cdot g_\gamma\|\leq \|T\|
R^\alpha_{\beta\gamma}=\left(Z^\tau_\beta W^\alpha_\delta \frac{\partial Z^\delta_\gamma}{\partial q^\tau}-Z^\tau_\gamma W^\alpha_\delta \frac{\partial Z^\delta_\beta}{\partial q^\tau}\right)=-\left(Z^\tau_\beta \frac{\partial W^\alpha_\delta}{\partial q^\tau}Z^\delta_\gamma -Z^\tau_\gamma \frac{\partial W^\alpha_\delta}{\partial q^\tau}Z^\delta_\beta\right).
\eta\sqrt{-1}\partial\bar\partial(\varphi_m+\Psi_m)+v'_{\epsilon}(\Psi_m)\sqrt{-1}\partial\bar\partial\Psi_m \ge \frac{1}{\delta}\sqrt{-1}\partial\bar\partial(\varphi_m+\Psi_m)+v'_{\epsilon}(\Psi_m)\sqrt{-1}\partial\bar\partial\Psi_m \ge 0
{\Theta}^{(k)}\leq \sum_{j=k}^n{\Theta}^{(j)}\lesssim 1 + |\Theta|\leq 1 + \Big(1+ \alpha^{-2}\sum_{j=k}^{n}\alpha_j^2 \Big)\,\widetilde{\Lambda}^{k-1}_{\alpha 2^{-1/2}, \alpha 2^{-1/2}, \alpha_k,\dots,\alpha_n}.
\mathcal{Q}_0^{(k)}(t)=\delta_{k,0}, \mathcal{Q}_{m}^{(k)}(t)= (-1)^k \sum_{l=0}^{m-1}\frac{(-1)^l}{l!\,(m-1-l)!}\,\frac{1}{(t+l)^{k+1}}\,.
\sum_{j=0}^{n-1}\binom{n}{j}t_m(n-j)\tilde{c}_{m,r}(\alpha,j) + (t_m(0)-\cos2\pi\alpha )\tilde{c}_{m,r}(\alpha,n) =u_{m-\ell-1}(n)+e^{2\pi i \alpha}u_{\ell-1}(n),
\varDelta = \min_{\tau\in R^q}\max_{1\leq l \leq q}\max_{j \in I_l}\left|y_j -\sum_{i=1}^{q}\tau_i v_{li}\right| & = \min_{d \in R^q}\max_{1\leq l \leq q}\max_{j \in I_l}\left|\epsilon_j -\sum_{i=1}^{q}d_i v_{li}\right|.
\begin{array}{c}A^{\lambda}_{~\mu}\eta_{\lambda\rho}A^{\rho}_{~\nu}\propto\eta_{\mu\nu}{}B^{\mu}_{\alpha}B_{\mu\beta}=\bar{B}^{\mu}_{\dot{\alpha}}\bar{B}_{\mu\dot{\beta}}=A^{\mu}_{~\nu}B_{\mu\alpha}=A^{\mu}_{~\nu}\bar{B}_{\mu\dot{\alpha}}=B^{\mu}_{\alpha}\bar{B}_{\mu\dot{\alpha}}=0\end{array}
E\left[\beta(t,q_t^m)\|z_t^m\|^2\right] = E\left[ \left(\frac{\beta(t,q_t)}{2\pi}\right)^{n/2}\int \beta(t,q_t)\|z\|^2 e^{- \beta(t,q_t)\|z\|^2/2}dz\right]+O(m^{\delta}) = \left(\frac{1}{2\pi}\right)^{n/2}\int \|w\|^2 e^{-\|w\|^2/2}dw+O(m^{\delta})
H^{-1}(t) \approx \frac{t^{\frac1p-1}}{\psi \left(t^{- \frac{1}{n}}\right)^{n-1}}= \frac{t^{\frac1p-1}}{\phi \left(t^{- \frac{1}{n}}\right)^{n-1}}= \frac{t^{\frac1p-1}}{\left(t^{- \frac{1}{n}}\right)^{s(n-1)}}= t^{\frac{n-np+sp(n-1)}{np}}
f(w; k \lambda) = C_k \cdot d_k^w \cdot \prod_{j=0}^{k-1}\dfrac{\varGamma\left(w+ \frac{u_1 + j}{k}\right) \cdots \varGamma\left(w+ \frac{u_m +j}{k}\right)}{\varGamma\left(w+ \frac{v_1 + j}{k}\right) \cdots \varGamma\left(w+ \frac{v_n +j}{k}\right)},
I_E=2 \pi \beta \,e^{(\Gamma-\Lambda)/2}\,[\,(\Gamma'-\phi')\,r^2 +\frac{\alpha' e^\phi}{g^2}\,(1-e^{-\Lambda})\,(\Gamma'+\phi')-\frac{\alpha' e^\phi}{g^2}\, r e^{-\Lambda}\Gamma' \phi'\,]\,|^{\infty}_{r_h}
\underset{v\rightarrow +\infty}{\lim}\ \ \underset{k\ge1}{\sup}\ \ \int\limits_{\left\| \hat{z}_k \right\|_2 > v}\left\| \hat{\mathbf{z}}_k \right\|_2^2 p(\hat{\mathbf{z}}_k) d\hat{\mathbf{z}}_k = 0.
C_{a}^{[1, M+1]}(z)X|\mathrm{vac}\rangle_{[1, M+1]}=(1+z)C_{a}^{[1, M]}(z)X|\mathrm{vac}\rangle_{[1, M]}, C_{a}^{[0, M]}(z)X|\mathrm{vac}\rangle_{[0, M]}=z\,C_{a}^{[1, M]}(z)X|\mathrm{vac}\rangle_{[1, M]}.
\begin{array}{ll}\hbox{Sector}\ \ \hbox{Field content}\hbox{even}[1,1],\, [3,3],\, [5,5],\, [7,7],\, [9,9],\, \hbox{odd}[2,2],\, [4,4],\, [6,6],\, [8,8],\, \hbox{twisted}[1,9],\, [3,7],\, [5,5],\, [7,3],\, [9,1].\end{array}
( y , x ) \equiv ( 5 . 0 0 6 , ~ ~ 1 . 0 3 6 ) , \, \, \, \, \mathrm{f o r}\, \, a = - 1 , \, \, ( y , x ) \equiv ( 6 . 9 6 1 , ~ ~ 0 . 7 9 2 ) , \, \, \, \, \mathrm{f o r}\, \, a = 1 , \, \,
\delta_a (\Delta_\varphi) = \delta_a ( k^{1/2}\Delta k^{1/2}) = \delta_a(k^{1/2}) \Delta k^{1/2}+ k^{1/2}\Delta \delta_a(k^{1/2}) = (\delta_a(k^{1/2})k^{-1/2}) \Delta_\varphi + \Delta_\varphi (k^{-1/2}\delta_a(k^{1/2})) .
\gamma (s,\tau,(r_0 \circ \rho_n) \boxtimes{\rm sim}^{-1},\psi) =\prod_{i=1}^{n}\gamma(s, \tau_i, (r_0 \circ \rho_{n_i}) \boxtimes{\rm sim}^{-1},\psi) \prod_{i<j}\gamma(s,\chi_i \chi_j \eta,\psi),
(a,b,c,\lambda_1(a,b,c),\lambda_2(a,b,c),\lambda_3(a,b,c))^{y_\kappa}= (\kappa a,\kappa^{\ell+1}b,\kappa^{\ell+2}c,\lambda_1(\kappa a,\kappa^{\ell+1}b,\kappa^{\ell+2}c),\lambda_2(\kappa a,\kappa^{\ell+1}b,\kappa^{\ell+2}c),\lambda_3(\kappa a,\kappa^{\ell+1}b,\kappa^{\ell+2}c)),
R_{\mu\nu}- \nabla_\mu \Phi \nabla_\nu \Phi -{1\over 4}e^{4 \Phi}H_{\mu \rho \sigma}H_\nu^{~~\rho \sigma}+{1 \over 6}g_{\mu\nu}e^{4 \Phi}H^2 +{4 \over k}g_{\mu\nu}e^{-2 \Phi}= 0,
\tilde{\phi}_n^k (t,y) \leq \tilde{\phi}_n^k(t, z)\ \textrm{whenever}\ |y| \leq |z|; \mathrm{Lip}(\tilde{\phi}_n^k(t,\cdot)) \leq \max\{\psi_n^k (t), \psi^k (T_n)\};\ \textrm{and}\tilde{\phi}_n^k (t, 0) = 0.
h_{11}= -g(\nabla^1\nabla_1 F, \nu)= \frac{e^{-2\lambda (f+t)}f''}{\sqrt{e^{-2\lambda f}(f')^2+e^{-4\lambda f}}}h_{12}= h_{21}=0 h_{22}= \frac{e^{-2\lambda( f+t)}e^{\lambda t}r f'}{\sqrt{e^{-2\lambda f}(f')^2+e^{-4\lambda f}}}
1 - \frac{N_1}{x_1 z}(N_1 + N_2 + \cdots + N_{k+1-h}) = \frac{x_h z}{N_h}, 1 - \frac{N_1}{x_1 z}(N_1 + N_2 + \cdots + N_{k-h}) = \frac{x_{h+1}z}{N_{h+1}}.
\delta \Bigl [ \epsilon^{\mu \nu}( \cosh \beta - 1 ) \partial_{\nu}\alpha \Bigr ] - \partial_{\nu}\Bigl [ \epsilon^{\mu \nu}( \cosh \beta - 1 ) \delta \alpha \Bigr ] = \frac{- 1}{2}\gamma^{\mu \nu}\Bigl ( \partial_{\alpha}\gamma_{\nu \beta}+ \partial_{\beta}\gamma_{\nu \alpha}- \partial_{\nu}\gamma_{\alpha \beta}\Bigr ) \delta \gamma^{\alpha \beta}
\rho'=\begin{cases}\frac{l-(t+2)+\sqrt{l^2-6lt-12l+(t+4)^2}}{2\left((t+2)(l-1)-1\right)}\mbox{if}~l\ge 3(t+2)+2\sqrt{(2t+5)(t+1)}\frac{1}{\sqrt{(t+2)(l-1)-1}}\mbox{otherwise.}\end{cases}
\Pi^{(1)}(k^2,m) =\frac{g^2}{2\pi}\left[\frac{\sqrt{m^2}}{2}-\left(\frac{m^2}{2}+\frac{k^2}{8}\right)\frac{1}{\sqrt{k^2}}ln\left(\frac{2\sqrt{m^2}+\sqrt{k^2}}{2\sqrt{m^2}-\sqrt{k^2}}\right)\right]\ ,
LV(x, 1) = \sigma^2( x, \mu, 1 )+ b( x, \mu, 1 )\cdot 2x & \leq (\int_{\mathbb R}(x+ \beta y)\mu(dy))^2 - 4x \int_{\mathbb R}(x+ \beta y)\mu(dy) & = -3x^2 -2\beta x\int_{\mathbb R}y \mu(dy) + \beta^2(\int_{\mathbb R}y\mu(dy))^2 & \leq -2 x^2 +2\beta^2 \int_{\mathbb R}x^2 \mu(dx),
\mu_1(x) := -\theta_1(W_{(\xi,\xi')})_w = ((\frac{1}{2\sqrt{-1}}[a,a^{\dagger}] + [b,b^{\dagger}] + ii^{\dagger}-j^{\dagger}j +ff^{\dagger}-g^{\dagger}g)), ( \frac{1}{2\sqrt{-1}})([a',a^{\prime\dagger}]+[b',b^{\prime\dagger}]-f^{\dagger}f+gg^{\dagger})).
p_{t}(0,y) := \rho_{Y_0^{+}(t)}= \frac{1}{\sqrt{t}}\phi\left(\frac{\vert y\vert -t}{\sqrt{t}}\right)-e^{2|y|}\Phi\left(-\frac{\vert y\vert + t}{\sqrt{t}}\right),\quad q_{t}(0,y) := \rho_{Y_0^{-}(t)}= \frac{1}{\sqrt{t}}\phi\left(\frac{t+\vert y\vert}{\sqrt{t}}\right)+e^{-2\vert y\vert}\Phi\left(\frac{t-\vert y\vert}{\sqrt{t}}\right)
{\cal L}^{( 0 )}= \pi_{i j}\dot{A}^{i j}+{\frac{1}{2}}\, \pi_{i j}\pi^{i j}+ 2 \partial_{i}A_{j 0}\pi^{i j}-{\frac{1}{2}}\, \partial_{i}A_{j k}\partial^{i}A^{j k}+ \partial_{i}A_{j k}\partial^{j}A^{i k}\, ,
\begin{array}{lll}x(t) = x+\alpha\int_0^t\frac{ds}{x(s)}+ \beta \int_0^t\frac{ds}{y(s)}y(t) = y + \gamma\int_0^t \frac{ds}{x(s)}+ \delta \int_0^t\frac{ds}{y(s)}\end{array}
A \geq \int_M u d\mu \geq \int_{\cup_{i=1}^N B(z_i, 0.01A^{-2})}u d\mu \geq 0.1 A^{-1}\sum_{i=1}^{N}|B(z_i, 0.01A^{-2})| \geq 0.1 A^{-1}\cdot A^{-1}\cdot \left(0.01 A^{-2}\right)^m N,
0\leq\frac{1}{n}\log\left[ \sum_{i=1}^{m_n}(1-d^t_{i})\right]\leq\frac{\log m_n}{n}; 0\leq\frac{1}{n}\log\left[ \sum_{i=1}^{m_n}(1-\hat d_{in})\right]\leq\frac{\log m_n}{n}.
V=V_{0}+\sum_{i=1}^{N}\sum_{\alpha=1}^{3}(\frac{\partial V}{\partial\rho_{i\alpha}})_{0}\rho_{i\alpha}+\frac{1}{2}\sum_{i,j=1}^{N}\sum_{\alpha,\beta=1}^{3}(\frac{\partial^{2}V}{\partial\rho_{i\alpha}\partial\rho_{j\beta}})_{0}\rho_{i\alpha}\rho_{j\beta}+\cdot\cdot\cdot,
\begin{cases}\bar h_1\bar r_1=\bar h_2+\bar h_3 \bar r_2\cr \bar p_1\bar r_2= \bar p_2+\bar p_3 \bar r_1 \cr \underbar h_1\underbar r_1= \underbar h_2+\underbar h_3 \underbar r_2\cr \underbar p_1\underbar r_2= \underbar p_2+\underbar p_3 \underbar r_1. \end{cases}
\begin{pmatrix}1 -1 0 1 \end{pmatrix}^{\ell-1}=\begin{pmatrix}1 1 0 1 \end{pmatrix}, \begin{pmatrix}-1 0 1 -1 \end{pmatrix}^{\ell-1}=\begin{pmatrix}1 0 1 1 \end{pmatrix}
m_i =\int_{t_{i-1}}^{t_i}f(s)ds,\sigma_i^2=\int_{t_{i-1}}^{t_i}\sigma_n^2(s)ds,\lambda_i=\int_{t_{i-1}}^{t_i}\lambda(s)ds,\alpha_i=\lambda_ie^{-\lambda_i},i=1,\dots,n.
P_{{\cal S}}( A ) = \sum_{j = 1 , 2}P_{{\cal S}_{j}/{\cal S}}P_{{\cal S}_{j}}( A ) + 2 \sqrt{P_{{\cal S}_{1}/{\cal S}}P_{{\cal S}_{1}}( A ) P_{{\cal S}_{2}/{\cal S}}P_{{\cal S}_{2}}( A )}\lambda ( A ;{\cal S},{\cal S}_{j}) \; .
\sum_{\mu}s_{\kappa/\mu}(\rho_1) \tau_{\mu}(\rho_{\circ})\mathcal{U}^{\angle}_{\rho_{\circ}, \rho_1}(\pi \vert \kappa, \mu) = \frac{s_{\pi/\kappa}(\rho_1) \tau_{\pi}(\rho_{\circ})}{H^{\circ}(\rho_1) H(\rho_1 ; \rho_{\circ})}.
\underline D_\mu^\psi(\eta)\asymp_\times \min_{p\in P_{<\delta}}\liminf_{\xi\in G(p)}\left(\frac{1/\|\xi - \eta\|}{1/r_\xi}\psi_p(\|\xi - \eta\|) \right)^{k_p - \delta}= \min_{p\in P_{<\delta}}\left(\limsup_{\xi\in G(p)}\frac{r_\xi}{\theta_p(\|\xi - \eta\|)}\right)^{k_p - \delta}.
\begin{aligned}1_{xL}\star 1_{yL}(z) =\int_G 1_{xL}(zt^{-1}) 1_{yL}(t) dt =\int_{yL}1_{xL}(zt^{-1})dt=vol\left(Lx^{-1}z\cap yL\right) = M(x, y, z) \cdot vol \left(L\cap yLy^{-1}\right) \in \mathbb Q, \end{aligned}
(z)>\frac{C-C_N}{3}\Longrightarrow \widehat{P}_\hbar -z=\left(- \widehat{\chi}_R\left( P_\hbar +\widehat{\chi}_{R}-z\right)^{-1}\right)\left(\widehat{P}_\hbar +\widehat{\chi}_{R}-z\right).
A_3(s, t) = \frac{1}{96\pi^2}\left( \frac{-t}{s}\right)^2\!\int\limits_0^{\infty}\! dx_1 \!\! \int\limits_{-x_1}^{x_1}\!dx_2 \!\! \int\limits_{0}^{\infty}\! dx_3 \, \theta ( [(x_3 + 1)^2- x_2^2][x_1^2 - (x_3 - 1)^2] ) &\times H(s,t; x_1, x_2, x_3) \;,
R_{\ell}=\sum_{\mathbf{k}\in\Lambda_{m}^{n},\mathbf{p\in}\Lambda_{2m}^{n}}\hat{C}\left( n,\ell,\mathbf{k},\mathbf{p}\right) \left( \partial^{p_{1}}b_{k_{1}}\right) \left( \partial^{p_{2}}b_{k_{2}}\right) \dots\left(\partial^{p_{n}}b_{k_{n}}\right) .
|N_i|=(p_i+q_i\sqrt D)|p_i-q_i\sqrt D|<\Bigl(2q_i\sqrt D+\frac 1{q_i}\Bigr)\frac{1}{a_{i+1}q_{i}}=\frac{2\sqrt D}{a_{i+1}}+\frac{1}{a_{i+1}q_{i}^2}.
B_{0i}(T) \rightarrow \frac{-2 v^2}{\pi \kappa}\epsilon_{0ij}p^j T \frac{\partial}{\partial m_{\sigma}^2}\left[\frac{m_{\sigma}^2 \log(M/m_{\sigma})}{m_{\sigma}^2 - M^2}\right] \equiv \frac{1}{2}{\cal{\pi}}_{0i}(p_0=0,T).
G_{\sigma}^{( n )}( \zeta_{1}, \cdots , \zeta_{2 n}) = \sum_{\varepsilon_{j}= \pm}v_{\varepsilon_{1}}\otimes \cdots \otimes v_{\varepsilon_{2 n}}G_{\sigma}^{( n )}( \zeta_{1}, \cdots , \zeta_{2 n})^{\varepsilon_{1}\cdots \varepsilon_{2 n}},
\tau_{1}( x , x^{\prime}) = - \frac{7}{4}e^{- 2 \sqrt{\Delta}}a^{2}\int_{0}^{1}\frac{e^{2 \tau}}{\ln \cosh \frac{\tau}{a^{2}}+ b}d \tau - \frac{7}{8}( z^{- 2}+ z^{' - 2}) e^{- 2 \sqrt{\Delta}}
\mathbb{P}[M_k-m_0\geq \lambda] =\mathbb{P}[e^{t(M_k-m_0)}\geq e^{t\lambda}] &\leq e^{-t\lambda}\mathbb{E}[e^{t(M_k-m_0)}] &\stackrel{\eqref{exp_final}}{\leq}\exp\left(-t\lambda+\frac{t^{2}}{2}g(tm)\sum_{i=1}^k \sigma_i^2\right).
\int_c^x H(v) dv = \int_c^x \int_v^b \frac{\partial f}{\partial v}(u,v) du dv &= \int_c^x \int_c^u \frac{\partial f}{\partial v}(u,v) dv du + \int_x^b \int_c^x \frac{\partial f}{\partial v}(u,v) dv du.
p_s^h = p_h^* - r_h + r^{-\lambda}\sin(\lambda\theta), r_h = B_h \left( r^{-\lambda}\sin(\lambda\theta)\right), p_h^* \in Y_{0h}:(\nabla p_h^*,\nabla v_h)_\Omega = (\nabla r_h,\nabla v_h)_\Omega \quad\forall v_h\in Y_{0h},
(e)=-{{\lambda^2}\over{32 (2 \pi)}}{\tilde{G}}^2_{\beta}(0)\int_0^{{\beta}\over 2}d \tau \,\, \int_{-\infty}^{+\infty}{{d k}\over{g^2(k)}}\left[e^{-g(k) \tau}+{{e^{g(k)\tau}+ e^{-g(k) \tau}}\over{e^{\beta g(k)}-1}}\right]^2 \; ,
( - 1)^{m - 1}\sum_{p = 0}^n{\left\{{( - 1)^p \binom mp\frac{{\mu^{m - p}}}{{\nu^m (\nu + \mu )^{n - p + 1}}}}\right\}}& &= - \frac{1}{{(\nu + \mu )^{n + 1}}}+ ( - 1)^n \sum_{i = n}^{m - 1}{\left\{{( - 1)^i \binom in\frac{{\mu^{i - n}}}{{\nu^{i + 1}}}}\right\}}\,,
\mathcal{Z}\left( \binom{an+b}{r}^{p}\prod_{s=2}^{L}\binom{\alpha_{s}n+\beta_{s}}{r_{s}}^{p_{s}}\right) =\frac{z\sum_{i=0}^{rp+\sigma}A_{a,b,r}^{\alpha_{s},\beta_{s},r_{s},p_{s}}\left( p,i\right)z^{rp+\sigma -i}}{\left( z-1\right)^{rp+\sigma +1}}.
S_{1}=\sum_{1<i<j<k}\left( \binom{n_{i}}{2}a_{i}^{2}+\binom{n_{j}}{2}a_{j}^{2}+n_{i}n_{j}a_{i}a_{j}\right) , S_{2}={\displaystyle\sum\limits_{i=2}^{k-1}}n_{i}a_{i}.
\left( \begin{array}{c c c c c}{}&{4 ( 2 \Delta^{2}+ \Delta ) + 4 ( 4 \Delta + 1 ) \theta}&{}&{6 ( \Delta + \theta )}&{}\\{}&{6 ( \Delta + \theta )}&{}&{4 ( \Delta + \theta ) + \frac{c}{2}}&{}\\ \end{array}\right) \left( \begin{array}{c}{b^{( 1 , 1 )}}\\{b^{( 2 )}}\\ \end{array}\right) = 0 ,
\int_{\mathbb{R}^2}\big[ e^{\beta4\pi v_n^{2}}-1 \big]^s\leq \int_{\mathbb{R}^2}\big[ e^{\beta4\pi |h|^{2}}-1 \big]^s=C < \infty \,\,\,\, \forall n \in \mathbb{N}.
\forall p \geq 1, \mathbb E \mathrm{Tr}(R^p) = \sum_{\alpha \in \mathcal S_p, \, f \in \mathcal F_p}(-1)^{|f^{-1}(2)|}s^{\#\alpha}n^{\#(\gamma^{-1}\alpha)}k^{{\bf 1}_{f \equiv 1}+ \#(P_f^{-1}\alpha)},
\Delta \omega_{\mathrm{o b s}}- \Delta \omega_{\mathrm{G R}}= 2 \pi \, ( \, - 0 . 0 0 0 1 0 \pm 0 . 0 0 0 3 7 \, ) \times 1 0^{- 8}\, \mathrm{r a d i a n s / r e v o l u t i o n}\; ,
\|\Gamma(z)\|_{\mathcal{B}}^2\leq C_1\left(\|(u_0,\eta^0)\|_{\mathcal{H}}^2+\|z\partial_x z\|_{L^1(0,T;L^2(I))}^2\right) \leq C_1(\alpha+K^2\|z\|_{L^2(0,T;H^2(I))}^4) \leq C_1(\alpha+K^2\|z\|_{\mathcal{B}}^4),
P(y; x) = \frac{1}{\widehat Z_n\Delta_n(x)}\Delta_n(y) \det \left[ e^{-\frac{1}{2}(y_j - x_k)^2}\right]^n_{j, k = 1}, \widehat{Z}_n = n! (2\pi)^{\frac{n}{2}},
g_{\psi}g_{\theta}g_{\phi}=g_{\phi}g_{\theta}g_{\phi}^{-1}g_{\phi}g_{\psi}[g_{\phi}g_{\theta}g_{\phi}^{-1}g_{\phi}]^{-1}*g_{\phi}g_{\theta}g_{\phi}^{-1}*g_{\phi}=g_{\phi}g_{\theta}g_{\psi}.
8 \sum_{|d|=2}^H \lfloor H/|d| \rfloor^2 \sum_{c=1, (c,|d|)=1}^{|d|-1}1 \sim 8H^2 \sum_{|d|=2}^H \varphi(|d|)/|d|^2 &= -8H^2+8H^2 \sum_{|d|=1}^H \varphi(|d|)/|d|^2 &\sim (48/\pi^2) H^2 \log H,
\Pi_{j}(\Gamma_{j}^{1}) = \{\zeta\in \Gamma(M_{i}): \limsup_{n\rightarrow \infty}\Vert D\textbf{\textit{f}}^{n}(\zeta)\Vert_{\Gamma_{j+n}}^{1/n}\leq \lambda_{1}\}\Pi_{j}(\Gamma_{j}^{2}) = \{\zeta\in \Gamma(M_{i}): \limsup_{n\rightarrow \infty}\Vert D\textbf{\textit{f}}^{-n}(\zeta)\Vert_{\Gamma_{j-n}}^{1/n}\leq \lambda_{2}\}.
[\begin{matrix}M_{11}\\M_{22}\\M_{12}\end{matrix}]=-\frac{2h^{3}E}{3(1-\nu^{2})}[\begin{matrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{matrix}][\begin{matrix}w_{,11}\\w_{,22}\\w_{,12}\end{matrix}]
\nabla_1 P^\Sigma_{11}= \partial_1P_{11\nu\nu\nu}= D_1 P_{11\nu\nu\nu}-2 h_{11}P_{\nu 1 \nu \nu \nu}+h_{11}(P_{111\nu\nu}+P_{11\nu 1 \nu}+P_{11\nu \nu 1}) & +h_{12}(P_{112\nu \nu}+P_{11\nu 2 \nu}+P_{11 \nu \nu 2})
u_i({P}) = \sum_{k=1}^{s(\varepsilon)}\sum_{j=1}^{M(k)}M_i(\alpha_j) W(k,j) e^{\sigma_k (x_3-S_3)}e^{i\sigma_k( (x_1-S_1) \cos \alpha_j + (x_2-S_2) \sin \alpha_j )},
P_T=A\left(\frac{\partial B}{\partial x}-\frac{\partial B}{\partial y}\right)-B\sum_{k=1}^n \tau_k\mu_k\frac{\partial f_k}{\partial y}\,, Q_T=A\left(\frac{\partial B}{\partial x}+\frac{\partial B}{\partial y}\right)+B\sum_{k=1}^n \tau_k\mu_k\frac{\partial f_k}{\partial x}\,,
\begin{array}{c}8t_{2}<O_{2}O_{k_{1}}...O_{k_{p-1}}P>_{g}+4t_{0}<O_{k_{1}}...O_{k_{p-1}}PP>_{g}+ (4(p-1) +4 \Sigma )<O_{k_{1}}...O_{k_{p-1}}P>_{g}=0,\end{array}
I_{0}( u , \mu ) ={\frac{1}{\Gamma ({\frac{1}{2}})}}\mu^{{\frac{1}{2}}}\left\{\Gamma ( -{\frac{1}{4}})^{2}F ( -{\frac{1}{4}}, -{\frac{1}{4}};{\frac{1}{2}};{\frac{u^{2}}{4 \mu^{2}}}) -{\frac{u}{\mu}}\Gamma ({\frac{1}{4}})^{2}F ({\frac{1}{4}},{\frac{1}{4}};{\frac{3}{2}};{\frac{u^{2}}{4 \mu^{2}}}) \right\}.
{F^{*}}'=\prod^{i-2}_{l=1}\overline{W}_{l-1}T^{\varepsilon_l}_{\alpha_l}{\overline{W}\,}^{-1}_l\cdot\prod^{m}_{l=i+1}\overline{W}_{l-1}T^{\varepsilon_l}_{\alpha_l}{\overline{W}\,}^{-1}_l.
f(\frac{\varphi}{c^{2}})=\frac{\varphi}{c^{2}}+\alpha(\frac{\varphi}{c^{2}})^{2},\frac{c_{\infty}^{2}}{c^{2}}=1+4(\frac{\varphi}{c_{\infty}^{2}})+(15+2\alpha)(\frac{\varphi}{c_{\infty}^{2}})^{2}
\hat{H}_{\mathcal M}^{2 \! \times \! 2}= - \frac{g^{2}}{2 r^{3}}\, \frac{\partial}{\partial r}\, r^{3}\, \frac{\partial}{\partial r}+ \frac{2 g^{2}}{r^{2}}\, \hat{L}^{2}+ \frac{r^{4}}{8 \, g^{2}}\, \sin^{2}\! \vartheta \: \sin^{2}\! \psi \: .
C_{\theta}( \tau ) = \langle{\cal A}( \tau ){\cal B}\rangle - \langle{\cal A}\rangle \lim_{\tau^{\prime}\rightarrow \infty}\langle{\cal U}_{\tau^{\prime}}({\scriptstyle{\cal B}}){\cal B}\rangle \stackrel{\tau \rightarrow \infty}{\longrightarrow}0 ,
R=R_{\infty}=\left[\begin{array}{cccccccc}1_{2m}0 0 0 0 0 0 0 0 1/2 0 0 0 0 -1 0 0 0 1/2 0 0 1 0 0 0 0 0 0 1_{2m}0 0 0 0 0 0 -1_{2m}0 0 0 0 0 -1/2 0 0 0 0 -1 0 0 0 -1/2 0 0 1 0 0 0 0 0 0 0 0 0 1_{2m}\end{array}\right],
H_{n}^{SK}(\sigma):= \frac{1}{\sqrt{n}}\langle \sigma, A\sigma \rangle =\frac{1}{\sqrt{n}}\sum_{i,j}A_{i,j}\sigma_{i}\sigma_{j}= \frac{2}{\sqrt{n}}\sum_{1\le i< j \le n}A_{i,j}\sigma_{i}\sigma_{j}.
(-\Delta)^s R_{2n+1}^\lambda(x) =b_n^\lambda\,\sum_{k=0}^{n}\frac{(-n)_{k}(n+\lambda+1)_{k}}{(\lambda+\frac{1}{2})_{k}k!}(-\Delta)^s \bigg\{\frac{x}{\left(1+x^2\right)^{k+\frac{\lambda}{2}+1}}\bigg\},
\left(W(\rho^0,\rho^1)\right)^2:=\inf_{\rho(t), v(t)}\Big\{\int_0^1(v(t), v(t))_{\rho(t)}dt~:~ \frac{d\rho}{dt}+\mathrm{div}(\rho v)=0,~\rho(0)=\rho^0,~\rho(1)=\rho^1\Big\}.
S_n'(y' - \sigma_{k+1}'(y')) & = (1- \frac{1}{w}) S_n' (y' - \sigma_{k-1}'(y')) - \frac{1}{w}\sigma'_1 (S_n' (\sigma'_k(y') - y')) - \frac{1}{w}S_n' (\sigma'_1(y') - y') .
\begin{cases}v^{1/2}\mapsto t^{-1/2}, \hat{X}_{(i, r)}\mapsto \widetilde{A}_{i, r-r_i}^{-1}\ (i, r)\in (\overline{I}_{\xi}^{\mathrm{tw}, \flat})_e, X'_{\Pi_{\imath}}\mapsto \underline{\widetilde{Y}_{\Pi_{\imath}}}\ \imath\in I_{\mathrm{A}}.\end{cases}
\frac 25 \sum_x \pi(x)(\mathbb{P}_x[T_A \le t])^{2}\le \sum_x \pi(x)(\mathbb{P}_x[T_A \le \tau])^{2}\le \sum_x \pi(x)\left(\sum_{a \in A}\mathbb{P}_x[T_a \le \tau]\right)^{2}\le k^{2}\sum_x \pi(x) \mathbb{(P}_x[T_o \le \tau])^{2},
f\left(E\right):=\frac{\left(w_{1}^{\top}G\left(E\right)w_{1}-d_{11}+E\right)\left(w_{2}^{\top}G\left(E\right)w_{2}-d_{22}+E\right)}{\left(w_{1}^{\top}G\left(E\right)w_{2}-d_{12}\right)^{2}}.
b=\frac{\sqrt[3]{2}-1}{3}\frac{RT_{c}}{P_{c}}=>P_{c}=\frac{\sqrt[3]{2}-1}{3}\frac{RT_{c}}{b}=>P_{c}=\frac{(\sqrt[3]{2}-1)^{7/3}}{3^{1/3}}R^{1/3}\frac{a^{2/3}}{b^{5/3}}
g'_2(\tilde{x},\tilde{t})= \frac{X_x(Z(\tilde{x},\tilde{t}),Y^{-1}(\tilde{t}))}{\kappa\tilde{t}^{\alpha-1}(\mathcal{T}_t^\alpha Y)(Y^{-1}(\tilde{t}))}\bigg[2h(Z(\tilde{x},\tilde{t}),Y^{-1}(\tilde{t}))(s_2s_{1x}-s_1s_{2x})+f_1s_1s_2 &-g_1s_2^2-f_2s_1s_2+g_2s_1^2\bigg],
I_{w^{\left( 2\right)}}\left( f\right) =4\int_{0}^{1}J_{0}\left(r\sqrt{a^{2}+b^{2}}\right) r^{2}dr+{\displaystyle\sum_{k=1}^{\infty}}8\frac{\left( -1\right)^{k+1}\cos\left( 2k\varphi_{a,b}\right)}{4k^{2}-1}\int_{0}^{1}J_{2k}\left( r\sqrt{a^{2}+b^{2}}\right) r^{2}dr.
\begin{cases}-\Delta v_\lambda + \frac{1}{|\Omega|}\int_{\partial \Omega}b [g(t + v) + \lambda g'(t + v) v_\lambda] = Q[f(x, t + v)] + \lambda Q[f_u(x, t + v) v_\lambda] \mbox{in $\Omega$}, \dfrac{\partial v_\lambda}{\partial \mathbf{n}}= b [g(t + v) + \lambda g'(t + v) v_\lambda] \mbox{on $\partial \Omega$}. \end{cases}
\begin{array}{rl}<\chi\chi|S|\pi\pi> = \displaystyle(-ig_{\pi})(-ig_{\chi})\int d^4x_5\int d^4x_6 &\displaystyle\times<\chi(k_{3+})\chi(k_{4+})|T\left((\pi\star\sigma\star\pi)_5(\chi\star\sigma\star\chi)_6\right)|\pi(k_{1+})\pi(k_{2+})> \displaystyle +{\rm ~higher~orders},\end{array}
R^H_{T_w}\theta(us)=\frac{1}{|C_H(s)^{\circ F}|}\sum_{\{h\in H^{F}\mid hsh^{-1}\in T_{w}\}}Q^{C_H(s)^{\circ}}_{C_{h^{-1}T_{w}h}(s)}(u)\theta(hsh^{-1}).
S_1=\frac{1}{d_1\cdots d_l}\sum\limits_{r_j\mid (u_j/d_j),1\leq j\leq l}\frac{\mu(r_1)\cdots\mu(r_l)}{r_1\cdots r_l}+O_{\alpha,l}\Big(\frac{1}{\sqrt{n}}\prod\limits_{1\leq j\leq l}\tau_2(u_j/d_j)\Big).