harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sup_{x \in B \times Y}|\eta_i(x)|_{g_{0,i}(x)}+ \sup_{x=(z,y) \in B \times Y}\Biggl( \sup_{x' \in B \times Y}\frac{|\eta_i(x) - \mathbf{P}^{g_{z,i}}_{x'x}(\eta_i(x'))|_{g_{0,i}(x)}}{d^{g_{0,i}}(x,x')^\alpha}\Biggr) \leq C.
\int_{X \times X \times X}[\epsilon|x_0-x_1|^2 +|x_0-y|^2+|x_1-y|^2]d\gamma_\epsilon(y,x_0,x_1) &\leq \int_{X \times X \times X}[\epsilon|x_0-x_1|^2 +|x_0-y|^2+|x_1-y|^2]d\gamma(y,x_0,x_1).
A_h^\pm (M) = \sum_{n\geq 1}(\widehat{h}_{\delta}(-n)\pm \widehat{h}_{\delta}(n)) \frac{a(n)}{n}+ \mathcal{O}\bigg((Mq)^{\epsilon}M^{-\frac{1}{4}}q^{\frac{1}{4}}\prod_{\substack{p\mid \gcd(q, M) p^2\mid q}}p^{\frac{1}{2}}\bigg).
d_{C_1 \times C_2}((r_1 \times r_2)(x_1,x_2)) = d_{C_1 \times C_2}(r_1(x_1),r_2(x_2)) &= d_{C_1}(r_1(x_1)) + d_{C_2}(r_2(x_2)) &= d_{C_1}(x_1) + d_{C_2}(x_2) &= d_{C_1 \times C_2}(x_1,x_2),
\int_{- 1}^{1}{d \nu}Z_{\mathrm{s t r}}( \beta , L , \nu ) \propto \frac{\beta^{2}\left( \beta^{2}- 8 \pi^{2}\alpha^{\prime}\right)^{3}}{L^{8}}e^{- L \sqrt{\beta^{2}- 8 \pi^{2}\alpha^{\prime}}/ \pi \alpha^{\prime}}.
1\otimes (T_{n_s}- c_s)E(\lambda) = 1\otimes T_{n_s}E(\lambda) = 1\otimes E(n_s(\lambda))(T_{n_s}- c_s) & = (sw\widetilde{\chi})(E(n_s(\lambda)))\otimes (T_{n_s}- c_s) = (w\widetilde{\chi})(E(\lambda))\otimes (T_{n_s}- c_s).
H(t;x,y,1,u,z,v) =z(ytuv(1-z)+z)H(t;x,y,1,u,1,v). F(t;x,y,1,u,z,v) =\frac{tx(y-yzr+z)F(t;x,y-yr+1,1,u,z,v)}{(tux+y^{-1}-tu)(y-yr+1)}&\quad+z(ytuv(1-z)+z)H(t;x,y,1,u,1,v).
2\lim_{L \rightarrow \infty}{\mathbb E}\langle (R_{1,2}^p -{\mathbb E}\langle R_{1,2}^p\rangle )^2 \rangle=3\lim_{L \rightarrow \infty}{\mathbb E}\langle (R_{1,2}^p -\langle R_{1,2}^p\rangle )^2 \rangle,
A(g_{u}(\theta))=\theta^{2-2\beta}A(u), &C(g_{u}(\theta))=\theta^{(1-\frac{3}{2}\beta)p+3\beta}C(u), &B(g_{u}(\theta))=\theta^{4-\beta}\int_{\mathbb{R}^{3}}\int_{\mathbb{R}^{3}}\frac{1-e^{-\theta^{\beta}}|x-y|}{|x-y|}u(x)^{2}u(y)^{2}dxdy.
C_1\,e^{-\alpha_2(2n-1)\frac{|x-a_i|^2}{2}}\le e^{-\sum_{i=1}^n\frac{|A^{\frac{1}{2}}(x-a_i)|^2}{2}}\le C_2\, e^{-\alpha_1 \frac{n+1}{2}\frac{|x-a_i|^2}{2}}
\frac{1}{2\pi i}\int_{w_-}^{w_+}e^{-\frac{1}{2}(y-s)^2}e^{\frac{1}{2}(x-s)^2}ds = \frac{1}{2\pi i (y-x)}e^{\frac{1}{2}(x^2 - y^2)}\left( e^{(y- x)w_+}- e^{(y- x)w_-}\right) = \frac{1}{\pi (y-x)}e^{\frac{1}{2}(x^2 - y^2)}e^{(y- x) \Re w_+}\sin \left( (y- x) \Im w_+\right),
\mathrm{Pr}\{X_{1}+X_{2}+\dots+X_{k}=n\}=\underset{m_j\in\mathbb{N}}{\underset{m_{1}+m_{2}+\dots+m_{k}=n}{\sum}}\prod_{j=1}^{k}\mathrm{Pr}\{X_{j}=m_j\}&=\underset{m_j\in\mathbb{N}}{\underset{m_{1}+m_{2}+\dots+m_{k}=n}{\sum}}\left(\frac{-1}{\ln p}\right)^{k}\prod_{j=1}^{k}\frac{(1-p)^{m_{j}}}{m_{j}},
u_{\varepsilon}=e^{i\mathcal{J}_{\varepsilon}\partial_{x}^{2}t}u_{0}-\int_{0}^{t}e^{i\mathcal{J}_{\varepsilon}\partial_{x}^{2}(t-s)}\mathcal{J}_{\varepsilon}\left(|\mathcal{J}_{\varepsilon}u_{\varepsilon}|^{2\sigma}\mathcal{J}_{\varepsilon}u_{\varepsilon,x}\right)\ds.
\Delta\phi_{rms}=\frac{1}{-2.ln(0.5)}.\frac{\Psi_{0}^{2}}{\Phi^{2}}.\sqrt{(\phi+\frac{\Phi}{2})^{2}.\frac{1}{SNR_{2}}+\frac{4.\phi^{2}}{SNR_{1}}+(\phi-\frac{\Phi}{2})^{2}.\frac{1}{SNR_{3}}}
\frac{\partial T}{\partial t}=\mathcal{D}\frac{\partial^{2}T}{\partial x^{2}},\;\;\tau \frac{\partial^{2}T}{\partial t^{2}}+\frac{\partial T}{\partial t}=\mathcal{D}\frac{\partial^{2}T}{\partial x^{2}}\;\;\;\;\tau\,{}_{0}^{c}\mathrm{D}_{t}^{\alpha +1}T+\frac{\partial T}{\partial t}=\mathcal{D}\frac{\partial^{2}T}{\partial x^{2}},
\lceil u, \phi_\ell*\rceil f =u f_\ell-(uf)_\ell=\int_0^1 u\omega_\ell^t*f_{t\ell^3}-\omega_\ell^t*(uf)_{t\ell^3}\, dt &=\int_0^1 \lceil u, \omega_\ell^t*\psi_{t\ell^3}*\rceil f\, dt\stackrel{\eqref{opera}}{=}\int_0^1\Big(\omega_\ell^t*\lceil u, \psi_{t\ell^3}*\rceil f+\lceil u,\omega_\ell^t*\rceil f_{t\ell^3}\Big)dt.
\bar{z}^{0}(x) = z^{0}(x), \bar{z}^{1}(x) = z^{1}(x), \bar{z}^{m}(x) =\Big(\sum_{n = 0}^{m - 2}{m - 2 \choose n}\partial_{t}^{n}\bar{a}_{ij}\partial_{x_{i}}\partial_{x_{j}}\bar{z}^{m - 2 - n}+\partial_{t}^{m - 2}\bar{f}_{i}\Big)(0, x) m \geq 2
D \nu^n (t,y) z = D \mu^n(t, E_n(-t)y) E_n(-t) z, y, z \in H_n, D^2 \nu^n (t, y) (z_1, z_2) = D^2 \mu^n (t, E_n(-t)y )(E_n(-t) z_1, E_n(-t) z_2), y, z_1, z_2 \in H_n.
H_T \,\, := \,\, \begin{small}\begin{bmatrix}T_m T_{m-1}\cdots T_{m-e}T_{m-1}T_{m-2}\cdots T_{m-e-1}\vdots \vdots \ddots \vdots T_{d+2}T_{d+1}\cdots T_{d-e+2}T_{d+1}T_{d}\cdots T_{d-e+1}\end{bmatrix}. \end{small}
b_+(t,x)= \left(0,-\frac{1}{m}\gamma(t,q)(p-\psi(t,q))\right), b_-(t,x)= \left(\frac{1}{m}(p-\psi(t,q)),\frac{1}{m}\delta^{ij}(p_i-\psi_i(t,q))\nabla_q \psi_j(t,q)-\nabla_qV(t,q)+\tilde F(t,x)\right).
\delta \varepsilon ( i | \omega | ) \, \delta ({\bf r}_{2}-{\bf r}) \, \omega^{2}D_{j k}( \varepsilon ,{\bf r},{\bf r}^{\prime}, i \omega ) + L_{j m}( \varepsilon ,{\bf r}, i \omega ) \delta D_{m k}({\bf r},{\bf r}^{\prime}, i \omega ) = 0 \, .
\begin{array}{cc}a(y^{n+1}-y^{-(n+1)})-d\tau(y^n-y^{-n})-e(y^{n-1}-y^{-(n-1)})=0 begin{align*}0.2cm]\mbox{or}(ay^2-d\tau y-e)y^{n-1}+(ey^2+d\tau y-a)y^{-(n+1)}=0\end{array}
\displaystyle{\prod_{i=1}^{e}}\big{[}Y_{i}\big{]}^{r}\times\displaystyle{\prod_{i=1}^{d}}\big{[}Y_{e+i}\big{]}^{r}=\displaystyle{\prod_{i=1}^{e^{'}}}\big{[}Y_{i}\big{]}^{r},\,\,\,\displaystyle{\prod_{i=1}^{e}}\big{[}Y_{i}\big{]}^{r}=\displaystyle{\prod_{i=1}^{e}}\big{[}X_{i}\big{]}^{r}
\exp(-uD_p^2|_{\Omega^{0,j}})(x,x')=\sum_{k=1}^\infty e^{-u\lambda^j_{k,p}}\varphi^j_{k,p}(x)\otimes\varphi^j_{k,p}(x')^*\in (E^j_p)_x\otimes (E^j_p)^*_{x'}\,,
\frac{T_{\mu_2}(w)}{T_{\mu_2}'(w)}+ \frac{w}{1 + T_{\mu_2}(w)}= \frac{T_{\mu_2}(w) + T_{\mu_2}^2(w) + wT_{\mu_2}'(w)}{(1 + T_{\mu_2}(w)) T_{\mu_2}'(w)}> 0.
< \, \, X \, \, >_{_{\Psi \, , J \,}}= \int \prod [ d \phi^{A}] \, X \, \exp \, \left({\frac{i}{\hbar}}\, W \, [ \phi^{A}, \phi_{A}^{\ast}={\frac{\delta \Psi}{\delta \phi^{A}}}] + J_{A}\, \phi^{A}\right)
\int_M \varphi P_g\varphi d\mu_g = \int_M|\nabla\Delta\varphi|_g^2d\mu_g-2\int_MT_2(\nabla\varphi,\nabla\Delta\varphi)d\mu_g-\frac{n-2}{2}\int_M\sigma_1(A)(\Delta\varphi)^2d\mu_g -\int_MT_4(\nabla\varphi,\nabla\varphi)d\mu_g+\frac{n-6}{2}\int_MQ_g\varphi^2d\mu_g.
\lambda G - F= \left[\begin{array}{cccccc}\lambda - \rho_1 \cdots 0 0 \cdots 0 \vdots \ddots \vdots \vdots \vdots 0 \cdots \lambda - \rho_t 0 \cdots 0 0 \cdots 0 L_{\epsilon_1}(\lambda) \cdots 0 \vdots \vdots \vdots \ddots \vdots 0 \cdots 0 0 \cdots L_{\epsilon_r}(\lambda) \end{array}\right]
{D(\lambda_1,\lambda_2)}=\{(\lambda_1,\lambda_2)\in\mathbb{R}^2_+:\; \lambda_1> \lambda_2\},{D(i_1,i_2)}=\{(i_1,i_2)\in\mathbb{R}^2_+:\; i_1^2-4\,i_2> 0\}.
\begin{aligned}N(x) = |\Phi| - |\Phi_x| \geq |\Phi_{S_{n}}| - |\Phi_{(S_k)^{b}\times S_{n-bk}}| begin{align*}1ex] = n(n-1)-[bk(k-1)+(n-kb)(n-kb-1)] begin{align*}1ex] = 2kbn-k^2b^2 - bk^2.\end{aligned}
\begin{aligned}\iint h_s \partial_x v^\epsilon \epsilon g^\epsilon \leq \epsilon \Vert h_s \Vert_{L^\infty}\Vert g^\epsilon\Vert_{L^2}\Vert v^\epsilon_x\Vert_{L^2}\leq \epsilon L \Vert h_s \Vert_{L^\infty}\Vert v^\epsilon_x \Vert_{L^2}\Vert g^\epsilon_x\Vert_{L^2},\end{aligned}
\sum_{i=1}^{\infty}\frac{1}{\mu(B_i)^p}\left\|\int_{B_i}f d\mu\right\|^p\int_{\Omega}\chi_{B_i}d\mu =\sum_{i=1}^{\infty}\mu(B_i)^{1-p}\left\|\int_{B_i}f d\mu\right\|^p \leq \sum_{i=1}^{\infty}\mu(B_i)^{1-p}\int_{B_i}\|f\|^p d\mu\left(\int_{\Omega}\chi_{B_i}d\mu\right)^{\frac{p}{p^*}}= \sum_{i=1}^{\infty}\int_{B_i}\|f\|^pd\mu\leq \|f\|_p^p
(\exp(t\xi)^*\phi)(g) =\phi(g\exp(-t\xi)) & = \phi(k\exp(x-t\xi)h') \intertext{where $h'\in H$ since $P$ normalizes $H$}= u(x-t\xi^1)-2\ln(\chi(\exp(x-t\xi)h')) & = u(x-t\xi^1)-2\ln(\chi(\exp(x)h))+2\left<\chi,t\xi\right>,
\ |\mathcal H(2d+1,d,2)|-|\mathcal G(2d+1,d)| = \ \sum_{i=d+2}^{2d}\sum_{j=d+2}^{i}\binom{n-j}{d}-\binom{n-2d-2}{d}-2n+3d+5 \geq \ \sum_{i=d+2}^{d+3}\sum_{j=d+2}^{i}\binom{n-j}{d}-\binom{n-2d-2}{d}-2n+3d+5 > \ 2\binom{n-d-2}{d}-2n+3d+5 > 4(n-d-2)-2n+3d+5=2n-d-3>0,
H=\left(\begin{array}{cccc}r_5 0 r_3+ir_4 r_1+ir_2 0 r_5 -r_1+ir_2 r_3-ir_4 r_3-ir_4 -r_1-ir_2 -r_5 0 r_1-ir_2 r_3+ir_4 0 -r_5 \end{array}\right ).
\widetilde{\Theta}_{\tilde{\rho},\rho}(F):= \sum_{j=1}^m \int_{\mathbb{R}^4}F(y,x') F(x,x')F(y,y')F(x,y') begin{align*}-1ex] (\tilde{\rho}_{2^{k_j-1}}-\tilde{\rho}_{2^{k_j}})(x'-y'){\rho}_{2^{k_{j-1}}}(x-y) \,dx dy dx' dy'
\mathcal{P}_{\mathfrak{b}_{a}}^{ext}\left( \mathrm{v},\upsilon\right) :=\int_{\mathfrak{b}_{a}}\left( b^{\ddagger}_{a}\cdot \mathrm{v}+\beta^{\ddagger}_{a}\cdot \upsilon\right) d\mu(y)+\int_{\partial \mathfrak{b}_{a}}\left( \mathfrak{t}\cdot \mathrm{v}+\tau \cdot \upsilon\right) d\mathcal{H}^{2},
g_{1}'\left(z\right) =\frac{\mathrm{d}}{\mathrm{d}z}\left(\left(1-z\right)\cdot G\left(z\right)\cdot\frac{1}{1-c_{o}z^{o}}\right)\left(z\right) =-G\left(z\right)\cdot\frac{1}{1-c_{o}z^{o}}+\left(1-z\right)\cdot\underset{\eqqcolon\left(\star\right)}{\underbrace{\frac{\mathrm{d}}{\mathrm{d}z}\left(G\left(z\right)\cdot\frac{1}{1-c_{o}z^{o}}\right)}}.
K_{j\bar k}(\zeta,\bar\eta) =\langle\langle \cdot\bar\eta_{\bar k}, \cdot\bar\zeta_{\bar j}\rangle\rangle +\int_{\partial D_t}b_{j\bar k}(\rho) \langle \cdot\bar\eta, \cdot\bar\zeta\rangle d \sigma + \langle\langle{_j\bar\eta},{_k\bar\zeta}\rangle\rangle,
\begin{aligned}\sum_{m_1=1}^\infty\sum_{m_2,m_3=0}^\infty \frac{z^{m_1+m_2+m_3}}{(m_1+m_2+m_3+a)^{s}}=\sum_{m_1,m_2=1}^\infty\sum_{m_3=0}^\infty\frac{z^{m_1+m_2+m_3}}{(m_1+m_2+m_3+a)^{s}}&+\sum_{m_1=1}^\infty\sum_{m_3=0}^\infty\frac{z^{m_1+m_3}}{(m_1+m_3+a)^{s}},\end{aligned}
B(X)^2-4A^2 C(X) = b_2^2 X^4 + (2b_1b_2 - 4k^2)X^3 + (2b_0b_2 + (b_1^2 - 4k^2c_2))X^2+{}&\quad+ (2b_0b_1 - 4k^2c_1)X + (b_0^2 - 4k^2c_0).
\begin{aligned}I_S(X;Y) = D(p(x,y)||p(x)p(y)) \cr = \sum\limits_{k = 1}^M{\sum\limits_{j = 1}^M{p({x_k},{y_j})log{{p({x_k},{y_j})}\over{p({x_k})p({y_j})}}}}\cr \end{aligned}
|f_j(x)|=\left|\frac 1{|B(x,1)|}\int_{B(x,1)}f_j(y)dV(y)\right|\leq \frac 1{|B(x,1)|}\|f\|_\varphi\left(\int_{B(x,1)}e^{2\varphi(y)}dV(y)\right)^{\frac 12}\leq C_x\|f\|_\varphi,
\langle \mu + \rho, \epsilon_{i}-\epsilon_{j}\rangle = \langle \mu + \rho, \epsilon_{i}-\epsilon_{i_1}\rangle + \langle \mu + \rho, \epsilon_{i_1}-\epsilon_{j_1}\rangle + \langle \mu + \rho, \epsilon_{j_1}-\epsilon_{j}\rangle &= (i_1-i)a+ p + \langle \nu + \rho, \epsilon_{j_1}-\epsilon_{j}\rangle,
\pi(z')^{-1}\pi(z) \pi(z') = e^{2 \pi i \sigma(z',z)}\pi(z) V_{\pi(z')g}(\pi(z') f)(z) = \langle f, \pi(z')^{-1}\pi(z) \pi(z') g \rangle & = e^{-2 \pi i \sigma(z',z)}V_g f(z).
r_{n+1}=r_{n},\ \ \nu_{n+1}=\nu_n-b\tilde M_n^4\epsilon_n^{\frac{1}{m}}, \delta_{n+1}=\delta_{n}, \ \ \tilde{M}_{n+1}=\tilde{M}_{n}+20m\tilde M_n^2\epsilon_{n}, c_{n+1}=c_{n}-{b}R_{n}^{3m^2}\tilde{M}_{n}^{m+2}(2\delta_{n}^{-1}r_{n})^{r_{n}}\epsilon_{n}.
\frac{\ln{q}}{1 - q}\frac{\big(xt^N; q\big)_{\infty}}{(xq; q)_{\infty}}\times\Gamma_q(\theta N)\times\sum_{n = 0}^{\infty}{x^n\frac{\operatorname{Res}_{z = n}{\Gamma_q(-z)}}{\Gamma_q(\theta N - n)}}.
\frac{\partial}{\partial t_p}\left(\begin{array}{cc}f{\overline f}\end{array}\right) =J_1\left(\begin{array}{cc}{\delta}/{\delta f}{\delta}/{\delta{\overline f}}\end{array}\right) H_{p+1}=J_2\left(\begin{array}{cc}{\delta}/{\delta f}{\delta}/{\delta{\overline f}}\end{array}\right) H_{p},
R^*_{\mathrm{CPF}}=\frac{1}{2}\log_2 [\frac{2a_mc_ml+\beta_0^*d_m- \sqrt{(\beta_0^*)^2d_m^2+4a_mc_md_m l \beta_0^*}}{2lc_m^2}]^+ R^*_{r_i}= \frac{1}{2}\log_2[\frac{f_i^*\gamma_i^*}{2\beta_i^*\ln 2}g_{r_i}]^+, i=1, \cdots, M
T_{1}= O ( \Lambda_{0}{}^{4}/ m^{2}) , \quad T_{2}= - 2 \Lambda_{0}{}^{2}+ O ( \Lambda_{0}{}^{4}/ m^{2}) , \quad T_{3}= 2 \Lambda_{0}{}^{2}+ O ( \Lambda_{0}{}^{4}/ m^{2}) .
A''(x,v) =\frac{1}{1-x}B(x,v)+B'(x,v), B'(x,v) =x^2A(vx,1)+\frac{x^2}{v}(A(vx,1)-1)+\frac{x}{1-v}(B'(x,v)-\frac{1}{v^2}B'(vx,1)), B(x,v) =xB'(x,v)+\frac{x^4}{(1-(1+v)x)(1-x)}.
\Psi(v) = \psi(\lambda_1(v))+\frac{1}{2}\left[\psi(\lambda_2(v))+\psi(\lambda_4(v))\right] &\leq \psi\left(\sqrt{\lambda_1(x)\lambda_1(s)}\right)+\frac{1}{2}\psi\left(\sqrt{\lambda_2(x)\lambda_2(s)}\right)+\frac{1}{2}\psi\left(\sqrt{\lambda_4(x)\lambda_4(s)}\right).
\Pi_1(\pi)=\{J=(J_1,J_2) \in \Pi(\pi): (J_1,J_2) (\ref{eqn:tau condition1}) (\ref{eqn:tau condition2})\}, &\Pi_2(\pi)=\{J=(J_1,J_2) \in \Pi(\pi): (J_1,-J_2) (\ref{eqn:tau condition1}) (\ref{eqn:tau condition2})\}.
C_{\ell}\simeq \frac{2}{\pi}\int \frac{d k}{k}k^{3}\left[ \left( \Theta_{\omega = 0}+ \Psi_{\omega = 0}\right) \jmath_{\ell}( k d ) + v_{\gamma \omega = 0}\jmath_{\ell}^{\prime}( k d ) \right]^{2}.
\widetilde \Phi_{M_n,n}( \widetilde g)(\cdot,t)-\widetilde \Phi_{M_n,n}( g)(\cdot,t) &= \dfrac{1}{n}\sum_{k=1}^n \vartheta\xi_{k}(t)+\sum_{p=1}^{M_n}\Bigg[ \int_t^T e^{(s-t)p^{2\beta}}\left( \dfrac{\pi}{n}\sum_{k=1}^n \vartheta\xi_{k}(s) \phi_p(x_k) \right) ds\Bigg] \phi_p(x).
v_{l}^{\kappa}= i2^{1/4}\kappa^{1/2-l}D_{l}\bigl( \exp\{e^{-i\frac{3\pi}{4}}\kappa r \}-\exp\{e^{i\frac{3\pi}{4}}\kappa r \}\bigr), \kappa > 0, u_{l}^{\kappa}= \frac{\sqrt{2}\lambda^{-l}}{\sqrt{\pi(1+(\sigma_{l}^{\lambda})^{2})}}D_{l}\bigl(\sin\lambda r + \sigma_{l}^{\lambda}(\cos\lambda r - e^{-\lambda r})\bigr) ,
\mathbb{E}G^{\rm Reac}_j=\frac{\mathbb{E}\left(T^{\rm Reac}_j\right)^2}{2}+\mathbb{E}T^{\rm Reac}_j\left(\mathbb{E}\tau^{\rm Reac}_{V_{j-1}}-\tau_{\rm f}-\frac{1}{2}\right).
P_{R,m+1}(t):=\left(\begin{matrix}p_{R,m+1,1}(t) \begin{matrix}p_{R,m+1}(t) 0 \cdots 0 \end{matrix}\begin{matrix}p_{R,m+1}(t) 0 \vdots 0 \end{matrix}\boxed{P_{R,m}(t)}\end{matrix}\right)
\liminf_{n\rightarrow\infty}{\!}_*\, V^{\beta,n}(x,i) \ := \ \liminf_{\substack{n\rightarrow\infty x'\rightarrow x}}V^{\beta,n}(x',i), \limsup_{n\rightarrow\infty}{\!}^*\, V^{\beta,n}(x,i) \ := \ \limsup_{\substack{n\rightarrow\infty x'\rightarrow x}}V^{\beta,n}(x',i).
\omega \mbox{}C_{{\bf{k}}}({\bf{q}}) = \omega_{{\bf{k}}}({\bf{q}})C_{{\bf{k}}}({\bf{q}})+ (\frac{v_{{\bf{q}}}}{V})\Lambda_{{\bf{k}}}(-{\bf{q}})\sum_{{\bf{k}}^{'}}[\Lambda_{{\bf{k}}^{'}}(-{\bf{q}})C_{{\bf{k}}^{'}}({\bf{q}}) + \Lambda_{{\bf{k}}^{'}}({\bf{q}})D^{*}_{{\bf{k}}^{'}}(-{\bf{q}})]+ \Lambda_{{\bf{k}}}(-{\bf{q}})
\frac{1}{\tau}\, \left [ \delta^{n+\frac{1}{2}}- \delta^{n-\frac{1}{2}}, \delta^{n+ \frac{1}{2}}+ \delta^{n- \frac{1}{2}}\right]_{\mathcal{C}_h}= \frac{1}{\tau}\left(\left \|\delta^{n+\frac{1}{2}}\right \|^2_{\mathcal{C}_h}- \left \|\delta^{n-\frac{1}{2}}\right\|^2_{\mathcal{C}_h}\right).
\widehat{w^{\varphi}}B_{j}^{\varphi}Z(x_{1},x_{2})B_{l}^{\varphi}=B_{j}^{\varphi}Z(x_{1},x_{2})B_{l}^{\varphi}\widehat{w_{j-l}^{\varphi}}\,\,j,l=0,1,2.
\langle f,g \rangle\, \,\, = \,\, \,\frac{1}{n!}\cdot g\left(\frac{\partial}{\partial x},\frac{\partial}{\partial y}\right) f(x,y) \,\,\,\, = \,\,\,\, \frac{1}{n!}\cdot f \left(\frac{\partial}{\partial u},\frac{\partial}{\partial v}\right) g(u,v).
\begin{matrix}\frac{\partial b}{\partial x_{1}}&=\frac{x_{2}m}{x_{1}-x_{2}}&\frac{\partial b}{\partial x_{2}}&=\frac{x_{1}m}{x_{2}-x_{1}}\\ \frac{\partial b}{\partial y_{1}}&=\frac{x_{2}}{x_{2}-x_{1}}&\frac{\partial b}{\partial y_{2}}&=\frac{x_{1}}{x_{1}-x_{2}}.\end{matrix}
\widetilde{r}_{s}+\widetilde{r}_{c}=\frac{1}{n}\overline{F}\left(\sigma_{s,t_{1}}^{T}+\sigma_{c,t_{2}}^{T}\right) & = r_{s}+r_{c}+\left( t_{1}+t_{2}\right) \overline{F}\mathbf{e}_{1}\pm \left(t_{1}+t_{2}\right) \overline{F}\mathbf{e}_{m+2}
\mathcal{B}_{i,i}=\{(i,i,t,p)\mid(i,i,t,p)\in \mathcal{I}'_{k}\}, &\mathcal{B}_{i,i-l}=\{(i,i-l,t,p)\mid(i,i-l,t,p)\in \mathcal{I}'_{k}\}, & \mathcal{B}_{i-l,i}=\{(i-l,i,t,p)\mid(i-l,i,t,p)\in \mathcal{I}'_{k}\}.
( x - x^{\prime})^{2}\int \! \mathrm{d}\rho^{2}\mathrm{d}^{4}x_{0}\frac{\rho^{8}}{\left[ ( x - x_{0})^{2}+ \rho^{2}\right]^{4}\left[ ( x^{\prime}- x_{0})^{2}+ \rho^{2}\right]^{4}}.
\eta(A) = \eta (B)\cdot \frac{\sum_{V\in\mathcal{W}_{p +1}: V\cap A\neq \emptyset}\mu(B^{[V]})}{\sum_{V' \in \mathcal{F}^{B}}\mu(B^{[V^{\prime}]})}\leq 2Q_{d,4}\frac{\eta (B)}{\mu(B)}\mu(5A).
{\cal E_C}[\phi_0] = \Gamma^{(2)}[\phi_0]+ \sum_{\ell=0}^\infty (2\ell + 1)\left[ \int_0^\infty \frac{dk}{\pi}E \frac{d}{dk}\left(\delta_\ell(k) - \delta^{(1)}_\ell(k) - \delta^{(2)}_\ell(k) \right)+ \sum_j \mbox{max}(\omega, E_{j,\ell}) \right] \,.
L_{21}=(\Delta')^{-1}([a,c]a^{-1}(ac^{-1}d-b)-([b,c]-[a,c]a^{-1}b) )(ac^{-1}d-b)^{-1}=(\Delta')^{-1}([a,c]c^{-1}d-[b,c] )(ac^{-1}d-b)^{-1}=((ac^{-1}d-b)^{-1}-(\Delta')^{-1}c ).
(1+T)\int_0^Te_1(W_{n+q+1})\,dt \ge Te_1(W_{n+q+1})(0) - \displaystyle{CT\int_0^T(|w_{n-1}|^2 +|w_n|^2+\sum_{k=1}^{q}|w_{n+k}|^2)\,dt\,.}
G_{\pm}( \lambda ) \equiv \sum_{n = 0}^{\infty}P_{\pm}( n ) \lambda^{n}= 2 | \widetilde{N_{\pm}}|^{2}e^{- | \alpha |^{2}}\left( e^{| \alpha |^{2}\lambda}\pm \mathrm{R e}e^{\alpha^{2}\lambda}\right)
ln(z)=2\cdot artanh\frac{z-1}{z+1}=2(\frac{z-1}{z+1}+\frac{1}{3}{(\frac{z-1}{z+1})}^{3}+\frac{1}{5}{(\frac{z-1}{z+1})}^{5}+\cdot\cdot\cdot),
6N2_4(n) + (24n-6)N2_2(n) =\frac{516-1356n}{469}M1_2(n) +\frac{120}{67}M1_4(n) +\frac{960-360n}{469}M2_2(n)-\frac{4896}{469}M2_4(n) &-\frac{2976+5424n}{469}M4_2(n) +\frac{1920}{67}M4_4(n) .
x^{2}\frac{d^{2}B_{n,m}^{(\alpha,\beta)}(x)}{dx^{2}}+[(\alpha+2)x+\beta]\frac{dB_{n,m}^{(\alpha,\beta)}(x)}{dx}-[n(\alpha+n+1)+\frac{m\beta}{x}]B_{n,m}^{(\alpha,\beta)}(x)=0
\Lambda_{\rm{c4}}\left(a\right)=-\left({\cal C}_{66}a+{\cal C}_{44}\right)\left[\left({\cal C}_{11}a+{\cal C}_{44}\right)\left(e_{15}a+e_{33}\right)-a\left({\cal C}_{13}+{\cal C}_{44}\right)\left(e_{31}+e_{15}\right)\right] ,
\begin{array}{ccc}\hat{g}_{IJ}=\left(\begin{array}{cc}g_{\mu\nu}g_{\mu\lambda}a_j^\lambda begin{align*}2mm] a_i^\kappa g_{\kappa\nu}g_{ij}+a_i^\kappa a_j^\lambda g_{\kappa\lambda}\end{array}\right) \ \ \mbox{and}\ \ \hat\omega_{IJ}=\left(\begin{array}{cc}0 0 begin{align*}2mm] 0 \varepsilon_{ij}\end{array}\right)\end{array}
\delta P_{\mu ( 1 )}^{A}= \frac{1}{2 \pi \alpha^{\prime}}\sqrt{- \bar{G}}\left( ( \bar{D}_{i j}\;^{A}\Phi_{( 1 )}^{j}) \bar{n}_{\mu}^{i}+ ( \bar{\Omega}_{i}\;^{A B}- \bar{G}^{A B}\bar{\Omega}_{i C}\;^{C}) \Phi_{( 1 )}^{i}\bar{x}_{\mu , B}\right)
I_{2}= - n^{3}\int_{\sin^{- 1}( 1 / n )}^{\pi / 2}{\frac{\sin^{3}\theta}{\sqrt{n^{2}\sin^{2}\theta - 1}}}\, | T_{r g}|^{2}\, e^{- 2 \omega x \sqrt{n^{2}\sin^{2}\theta - 1}}d \theta ,
\hat{\gamma}_{i j}d \hat{x}^{i}d \hat{x}^{j}= \left( 1 +{\frac{\hat{k}}{4}}\delta_{m n}\hat{x}^{m}\hat{x}^{n}\right) \delta_{i j}d \hat{x}^{i}d \hat{x}^{j}={\frac{d \hat{r}^{2}}{1 - \hat{k}\hat{r}^{2}}}+ \hat{r}^{2}\left( d \hat{\theta}^{2}+ \sin^{2}\hat{\theta}d \hat{\phi}^{2}\right) .
k ={\bf 1}_{f \equiv 1}+ \#\alpha + \#(P_f^{-1}\alpha) &={\bf 1}_{f \equiv 1}+ 2p- (|\alpha| + |P_f^{-1}\alpha|) &\leq{\bf 1}_{f \equiv 1}+ 2p - |P_f| &\leq 1+2p,
{\rm dim}(T)=\sum_{(\mu,d)\in \Psi}(d+1)^2<\sum_{(\mu,d)\in \Upsilon}(d+1)^2 =\sum^m_{d=0}(m-d-\lfloor\frac{m-d+1}{2}\rfloor+1)(d+1)^2 &={m+4\choose 4}.
f^{( 2 )}= ( 1 / 2 ) \big [ ( g_{1}^{2}/ 1 6 \pi^{2}) \ln^{2}( q^{2}/ \mu^{2}) + ( g_{2}^{2}/ 1 6 \pi^{2}) \ln^{2}( q^{2}/ M^{2}) \big ]^{2}.
[(f\cdot g_1)\cdot g_2](Hg_3) =f(Hg_3g_2^{-1}g_1^{-1})\cdot h(m(Hg_3g_2^{-1}g_1^{-1})g_1)\cdot h(m(Hg_3g_2^{-1})g_2) &= f(Hg_3g_2^{-1}g_1^{-1})\cdot h(m(Hg_3(g_1g_2)^{-1})g_1g_2) &=(f\cdot (g_1g_2))(Hg_3).
\int_{|\xi|\leq 1}e^{-\frac{|\xi|^2 t}{4}}|\xi|^{a}d\xi = 2^{a+n}t^{-\frac{n+a}{2}}\int_{|\eta|\leq \frac{\sqrt{t}}{2}}e^{-|\eta|^2}|\eta|^{a}d\eta \leq 2^{a+n}t^{-\frac{n+a}{2}}\int_{\mathbb{R}^n}e^{-|\eta|^2}|\eta|^{a}d\eta.
\int_{3}^{q}\frac{1}{r^{2}}\ln\frac{4\pi(r-2)(1-\tau)}{r^{2}}dr &=\int_{3}^{q}\frac{\ln4\pi(1-\tau)}{r^{2}}dr+\int_{3}^{q}\frac{\ln(r-2)}{r^{2}}dr-\int_{3}^{q}\frac{\ln r^{2}}{r^{2}}dr.
|1-F_{3}(r,\rho,\lambda(s))| \leq C r \int_{0}^{1}\frac{r_{\sigma}\lambda(s)^{2\alpha-2}(1+\lambda(s)^{2\alpha-2}(\rho^{2}+r_{\sigma}^{2}))}{(1+2(\rho^{2}+r_{\sigma}^{2})\lambda(s)^{2\alpha-2}+(\rho^{2}-r_{\sigma}^{2})^{2}\lambda(s)^{4\alpha-4})^{3/2}}d\sigma
\Delta_H =\begin{cases}\left \{I + \begin{pmatrix}a 0 0 d \end{pmatrix}p \in H \cap \ker \varphi \right \}H N(C_s) \left \{I + \begin{pmatrix}a 0 0 d \end{pmatrix}p \in H \cap K \right \}H N(C_{ns}).\end{cases}
\limsup_{N \to \infty}|\langle f,\nu_{\infty}^{N}\rangle - \langle f,\delta_{\mu_{\infty}}\rangle|= \limsup_{N \to \infty}\lim_{n \to \infty}|\langle f,\nu_{n}^{N}\rangle - \langle f,\delta_{\mu_{\infty}}\rangle| \le \limsup_{N \to \infty}\limsup_{n \to \infty}E \mathcal{W}_{1}(\mu_{n}^{N},\mu_{\infty}) + \epsilon \le \epsilon,
-c \leq \frac{\log \sum_{| \boldsymbol{i}| = n (\ell)}r_{\boldsymbol{i}}^{d_f^r}}{\Phi(\ell)}\leq c, e^{-c\Phi(\ell)}\leq \sum_{| \boldsymbol{i}|= n (\ell)}r_{\boldsymbol{i}}^{d_f^r}\leq e^{c\Phi(\ell)}.
\lim_{\xi \nearrow \xi_{+}- 0}\phi(\xi) = 0, \lim_{\xi \nearrow \xi_{+}- 0}|\phi'(\xi)| = \infty, \lim_{\xi \searrow \xi_{-}+ 0}\phi(\xi) = 0, \lim_{\xi \searrow \xi_{-}+0}|\phi'(\xi)| = \infty
\big[O_{\mu\nu}^{-1},\hat{x}_\lambda\big]=i\big({-}a_\mu\eta_{\lambda\nu}+ a_\lambda O^{-1}_{\mu\nu}\big),[O_{\mu\nu},\hat{y}_\lambda]=i(a_\mu\eta_{\lambda\nu}- a_\lambda O_{\mu\nu})
\begin{aligned}|N_{G}(v)\cap(V(H)\cup\{u_{1},\cdots,u_{l-1}\})| \geq d_{G}(v)-|N_{G}(v)\cap V(B)| -|N_{G}(v)\cap (V(T')\setminus\{u_{1},\cdots,u_{l-1}\})| \geq (t+3)-2-(t-\lfloor \frac{l-1}{2}\rfloor) \geq \lfloor \frac{l-1}{2}\rfloor+1.\end{aligned}
\int_{\mathcal{H}}e^{2\tau\vert\vert x \vert\vert^2}\,dm(x)=\prod_{k=1}^{+\infty}\bigg(\int_\mathcal{H}e^{2\tau(\mathfrak{Re}\langle e_k,x\rangle)^2}\,dm(x)\bigg)^2 =\prod_{k=1}^{+\infty}\bigg(\int_{\mathbb{R}}e^{-\big(\frac{1}{2\sigma_k^2}-2\tau\big)t^2}\,\frac{dt}{\sigma_k\sqrt{2\pi}}\bigg)^2 &=\prod_{k=1}^{+\infty}\bigg(\frac{1}{1-4\tau\sigma_k^2}\bigg)
P_{1kl}= \frac{1}{4c^2}\frac{\partial}{\partial t}\left( \Box h_{kl}+ \frac{\partial^2 h_{kl}}{\partial x_n \partial x^n}- \frac{\partial^2 h_{nl}}{\partial x^k \partial x_n}- \frac{\partial^2 h_{kn}}{\partial x_n \partial x^l}\right) ,
T_k(z)= :\Lambda'(q^{-n+k}s_1^{-1}z)\bar\Lambda'(q^{n-k}s_1^{-1}z) V_{\chi_{k,1}}(z):\,, &\bar{T}_k(z)= :\Lambda'(q^{n-k}s_1^{-1}z)\bar\Lambda'(q^{-n+k}s_1^{-1}z) V_{\chi_{\bar k,1}}(z):\,,
H_1=\sqrt{-1}(X_{11}+X_{33}),H_2=\sqrt{-1}X_{22}, E_1=\begin{bmatrix}0 1 0 -1 0 1 0 1 0\end{bmatrix},E_2=\sqrt{-1}\begin{bmatrix}0 1 0 1 0 -1 0 1 0\end{bmatrix},E_3=\sqrt{-1}\begin{bmatrix}1 0 -1 0 0 0 1 0 -1\end{bmatrix}.
S = \int d^{4}x \{\frac{1}{4}F_{\mu \nu}^{a}F^{a , \mu \nu}+ \frac{1}{2}T r \{( D_{\mu}\phi_{1})^{2}\}+ \frac{1}{2}T r \{( D_{\mu}\phi_{2})^{2}\}- \frac{\lambda}{4}( T r \{\phi_{1}^{2}\}- T r \{\phi_{2}^{2}\}- f^{2})^{2}\}\quad .
\Theta_{kk^{\prime}}(t,t^{\prime})={\rm Re}\left\{\left( \alpha_k(t)\beta_{k^{\prime}}(t)+\alpha_{k^{\prime}}(t)\beta_k(t)\right)\left( \alpha_k(t^{\prime})\beta_{k^{\prime}}(t^{\prime})+\alpha_{k^{\prime}}(t^{\prime})\beta_k(t^{\prime})\right)^{*}\right\}.
\mu(B(x,r)) =\nu(f^{-1}(B(x,r))) & \leq\nu\{w\in\Omega^{\infty}:f(C_{w|_{k-1}})\cap B(x,r)\neq\varnothing\}& \leq{\displaystyle\sum\limits_{\sigma^{\prime}\in\Lambda_{x,r}}}\nu(C_{\sigma^{\prime}})\leq C_{E}(n_{1}\cdots n_{k-1})^{-1}.
\left( p_{2}p_{3}- p_{1}p_{2}^{2}+ p_{1}q_{1}^{- 1}q_{2}^{- 2}\right) \partial_{p_{2}}+ \left( p_{3}^{2}- q_{2}p_{2}^{2}q_{1}^{- 1}- q_{1}^{- 2}q_{2}^{- 1}\right) \partial_{p_{3}}\; ,
\int_{\varepsilon Y_{1}}\left(p^{\varepsilon}-\bar{p}\right)\phi dx=\varepsilon^{d}\left|Y_{1}\right|^{-1}\int_{Y_{1}}\int_{Y_{1}}\left(p\left(y\right)\phi\left(\varepsilon y\right)-p\left(y\right)\phi\left(\varepsilon z\right)\right)dzdy.
\Xi^{\lambda}(K_{a}F_{a}\otimes 1 \otimes 1 \otimes E_{a}) =\Xi^{\lambda}(K_{a}F_{a}\otimes 1 \otimes E_{a}K_{a}^{-1}\otimes K_{a}) + \Xi^{\lambda}(K_{a}\otimes F_{a}\otimes 1 \otimes E_{a}) -\sum_{i,j}c_{j}^{i}\pi(E_{a}K_{\lambda}K_{a}F_{a})_{i}^{j}.
\phi_{D}\biggl(\genfrac..{0pt}{}{a;\, b_{1}, \dotsc, b_{r}}{c}; x_{1}, \dotsc, x_{r}\biggr) = \frac{(a)_{\infty}(b_{1}x_{1})_{\infty}\!\dotsm (b_{r}x_{r})_{\infty}}{(c)_{\infty}(x_{1})_{\infty}\dotsm (x_{r})_{\infty}}{}_{r + 1}\phi_{r}\biggl(\genfrac..{0pt}{}{c/a,\, x_{1}, \dotsc, x_{r}}{b_{1}x_{1}, \dotsc, b_{r}x_{r}}; q,\, a\biggr),
\int_{- \infty}^{\infty}d z_{1}\int_{- \infty}^{\infty}d z_{2}\; \psi_{z_{1}, z_{2}}^{\ast \; \nu , \mu}( u^{0}, x^{\prime}, y^{\prime}) \psi_{z_{1}, z_{2}}^{\nu , \mu}( u^{0}, x , y ) = \delta \left( x - x^{\prime}\right) \delta \left( y - y^{\prime}\right) \; .
\#( Rec \cap [1,n]) \geq \#\{( i,m) | u_m < 2^i , i < k\}= \sum_{u_m<2^k}( k-\log( u_m) ) = \sum_{4m^2+2m+1<k}( k-( 4m^2+2m+1) ) = \frac 12 k\sqrt{k}- \frac 43 \left( \frac{\sqrt{k}}{2}\right)^3 + o( k\sqrt{k}) = \theta( k \sqrt{k}) ,
\left( \left. \pi_{0}\right\vert_{\Omega},\omega_{1}\right)_{\Omega}=\left( \left. G_{\rho ,1}(\omega_{1})\right\vert_{\Omega},\omega_{1}\right)_{\Omega}+\left( \left. G_{\rho ,2}(\mu )\right\vert_{\Omega},\omega_{1}\right)_{\Omega}.