harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
b_n^\alpha = \int_{-\infty}^0 \max_{0 \leq k\leq n-1}g_k(s)^\alpha ds + \sum_{\ell = 0}^{n-1}\int_{\ell}^{\ell +1}\max_{0 \leq k\leq n-1}g_k(s)^\alpha ds & = \int_{-\infty}^0 g_0(s)^\alpha ds + \sum_{\ell = 0}^{n-1}\int_{\ell}^{\ell +1}\max \Big\{g_{\ell}(s), g_{\ell+1}(s)\Big\}^\alpha ds,
m(a_iy) =q_i+m(y), \exp\left(m(a_iy)\right) =\exp\left(m(y)\right)\exp(q_i), exp\left(m(y)\right) =\alpha y^{\beta}\textrm{, (by Lemma~\ref{lem:srnie}),}m(y) =\ln(\alpha)+\beta \ln y, g(x) \simeq \ln f(x).
\omega_0(h) =\sum_{j=1}^p \left(x_{2,j}\partial_{x_{2,j}}-x_{1,j}\partial_{x_{1,j}}\right)\,, \omega_0(e^+) =-\sum_{j=1}^p x_{2,j}\partial_{x_{1,j}}\,, \omega_0(e^-) =-\sum_{j=1}^p x_{1,j}\partial_{x_{2,j}}\,.
\begin{cases}e^{i\theta\infty}(w,\phi_i)_{L^2(\mathbb{R}^d)}+(\sigma_{i\infty}\nabla w_i,\nabla\phi_i)_{L^2(\mathbb{R}^d)}=0, e^{i\theta\infty}(w,\phi_e)_{L^2(\mathbb{R}^d)}-(\sigma_{e\infty}\nabla w_e,\nabla\phi_e)_{L^2(\mathbb{R}^d)}=0, w=w_i-w_e. \end{cases}
\hat{\epsilon}_{\hat{\mu}}{}^{\hat{a}}\hat{\epsilon}_{\hat{\nu}}{}^{\hat{b}}\hat{\eta}_{\hat{a}\hat{b}}= \hat{\jmath}_{\hat{\mu}\hat{\nu}}\, , \hspace{1 . 5 c m}\hat{\epsilon}_{\hat{a}}{}^{\hat{\mu}}\hat{\epsilon}_{\hat{b}}{}^{\hat{\nu}}\hat{\jmath}^{\hat{\mu}\hat{\nu}}= \hat{\eta}_{\hat{a}\hat{b}}\, ,
g_{ij}=\partial_i \phi_0\partial_j \phi_0+\sum_{r=1}^{\infty}\partial_i \phi_r \partial_{j}\alpha_r+\sum_{r=1}^{\infty}\partial_i \alpha_r \partial_{j}\phi_r g_{i\bar{j}}=\sum_{r=0}^{\infty}\partial_i \phi_r \partial_{\bar{j}}\bar{\phi}_r+\sum_{r=1}^{\infty}\partial_i \alpha_r \partial_{\bar{j}}\bar{\alpha}_r
\begin{array}{c}{{- \frac{1}{2}( k - 1 ) k ( k + 1 ) < O_{k - 2}O_{k_{1}}. . . O_{k_{p - 1}}P >_{g - 1}}}\\{{+ 4 t_{0}( k + 1 ) < O_{k}O_{k_{1}}. . . O_{k_{p - 1}}P >_{g}+ 4 t_{2}( k + 2 ) < O_{k + 2}O_{k_{1}}. . . O_{k_{p - 1}}P >_{g}}}\\{{+ 2 \sum_{i = 1}^{p - 1}( 2 k + k_{i}+ 2 ) < O_{k + k_{i}}O_{k_{1}}. . . \hat{O}_{k_{i}}. . . O_{k_{p - 1}}P >_{g}= 0 ,}}\\ \end{array}
s=\frac{\alpha(1+e^2)}{2e^2}\prod_{k=2}^{\infty}{\frac{\frac{ke^k}{1+e^k}[\frac{s(k-1)e^{s(k-1)}}{1+e^{s(k-1)}}+\alpha]}{\frac{(k+1)e^{k+1}}{1+e^{k+1}}[\frac{s(k-2)e^{s(k-2)}}{1+e^{s(k-2)}}+\alpha]}}
\mathbf{E}_{i}=\frac{\partial\mathbf{r}}{\partial u^{i}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\&\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\mathbf{E}^{i}=\nabla u^{i}
\|\theta\|_{{B}_{2,2}^{s}(\mathbb{R}^{2})}\leq\|\theta\|_{H^{\frac{\beta}{2}}(\mathbb{R}^{2})}^{1-l}\|\Lambda^{\varrho+\frac{\beta}{2}}\theta\|_{L^{2}(\mathbb{R}^{2})}^{l},l=\frac{s-\frac{\beta}{2}}{\varrho}\in (0,\,1).
\begin{array}{lll}E^+_{\pm}(z)E^-_{\pm}(w)=E^-_{\pm}(w)E^+_{\pm}(z)\mspace{155mu}begin{align*}0.2cm] E^{\pm}_+(z)E^{\mp}_-(w)=E^{\mp}_-(w)E^{\pm}_+(z)(1-\frac{z}{w})^{-\frac{2}{k}}begin{align*}0.2cm] E^{\pm}_+(z)E^{\pm}_-(w)=E^{\pm}_-(w)E^{\pm}_+(z)(1-\frac{z}{w})^{\frac{2}{k}}\end{array}
|(\xi,\eta)_f| \leq \int_{\mathbb{R}}|\langle f(s)\xi,\eta\rangle|\,ds \leq \int_{\mathbb{R}}\|f(s)\|_{\infty}\|\xi\|\|\eta\|\,ds = \left(\int_{\mathbb{R}}\|f(s)\|_{\infty}\,ds\right)\|\xi\|\|\eta\|.
\sum\nolimits y^dx^{des(\sigma)}= 1 + \sum\limits_{d=1}^\infty y^d(1+(d-1)x) &= \frac{1}{1-y}+ xy^2\sum\limits_{d=1}^\infty (d-1)y^{d-2}&= \frac{1-y+xy^2}{(1-y)^2.}
\Gamma_{\mu\nu,\rho}^{(\alpha )}(S^{-1};\boldsymbol{\xi}) =\frac{1}{2\pi}\int_{-\pi}^{\pi}(\partial_{\mu}\partial_{\nu}\log{S}+\alpha (\partial_{\mu}\log{S})(\partial_{\nu}\log{S}))(\partial_{\rho}\log{S})dw &=\Gamma_{\mu\nu,\rho}^{(-\alpha )}(S;\boldsymbol{\xi})
\dot{x}= \kappa_1 x^2 + c_2\kappa_2 x^{a_2}y^{b_2}+ c_3\kappa_3 x^{a_3}y^{b_3}, \dot{y}= \phantom{\kappa_1 x^2 +}d_2\kappa_2 x^{a_2}y^{b_2}+ d_3\kappa_3 x^{a_3}y^{b_3}.
\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^{\prime}\tau \tau^{\prime}\xi_{i}( \tau ) \xi_{j}( \tau^{\prime}) \rangle .
G\coloneqq \sum_{i, j\ge 0}(-1)^{i+j}\sum_{\substack{P\subseteq \{2,3,\dots,n-1\}, \#P=i}}\sum_{\substack{Q\subseteq \{1,n\}, \#Q=j}}\binom{d-\sum_{p\in P}(\alpha_{p}+1)-\sum_{q\in Q}\alpha_{q}+n-2}{n-1}
K_{\mu, (n)}=\delta_{\mu, (n)}, &K_{(n), \mu}=t^{n(\mu)}, &K_{(r, s),{\mu}}=t^{r-\mu_1}(K_{(r-\mu_1+s), \tilde{\mu}}+K_{(r-\mu_1+s-1, 1), \tilde{\mu}}+\cdots+K_{(r-\mu_1, s), \tilde{\mu}}) -t^{s-\mu_1-1}(K_{(r-\mu_1+s), \tilde{\mu}}+K_{(r-\mu_1+s-1, 1), \tilde{\mu}}+\cdots+K_{(r+1, s-\mu_1-1), \tilde{\mu}})
\partial_{t}k_{t}(x,y)= \ -[(\sinh2t)^{-\frac{3}{2}}\cosh2te^{-\varphi(t,x,y)}+(\sinh2t)^{-\frac{1}{2}}e^{-\varphi(t,x,y)}\partial_{t}\varphi(t,x,y)] =: \ -[A+B],
\pi_0 L_h\pi_0 u_0 = \pi_0 f - \pi_0 L_h\pi_0^c u_0^c , \pi_{1 1}^c L_h\pi_{1 1}^c u_{1 1}^c = \pi_{1 1}^c f - \pi_{1 1}^c L_h\pi_0 u_0 - \pi_{1 1}^c L_h\pi_{1 1}u_{1 1}, \pi_{1 1}L_h\pi_{1 1}u_{1 1}= \pi_{1 1}f - \pi_{1 1}L_h\pi_0 u_0 - \pi_{1 1}L_h\pi_{1 1}^c u_{1 1}^c .
\frac{Z_{\scriptscriptstyle M C S}( J )}{Z_{\scriptscriptstyle S D}( \hat{J})}= e^{i \alpha}\tau_{\scriptscriptstyle R S}( M )^{\scriptscriptstyle 1 / 2}e^{i \langle J , \frac{1}{\ast d_{\scriptscriptstyle 1}}J \rangle}= e^{i \alpha}Z_{\scriptscriptstyle C S}( J )
- \Delta_{g}\bigl( h_{1,\lambda}(r) \bigr) = -\left\{\frac{\partial^2}{\partial r^2}+ (n-1)\frac{f_1'(r)}{f_1(r)}\frac{\partial}{\partial r}\right\}h_{1,\lambda}(r) = \left( \frac{(n-1)^2}{4}{K_0}+ \lambda \right) h_{1,\lambda}(r).
A_u^4(a_1,a_2)(\xi_1,\xi_2)=\pi^*g^B(\tau_0'({\omega}_u^h(a_2))\xi_1,\tau_0'({\omega}_u^h(a_1))\xi_2)-\pi^*g^B(\tau_0'({\omega}_u^h(a_1))\xi_1,\tau_0'({\omega}_u^h(a_2))\xi_2)
\rho_1(s)=\frac{s}{\pi}Im \frac{\Pi_e(s)}{\Pi_e(s)^2 s-\Pi_o(s)^2 M^2}\hspace{2cm}\rho_2(s)=\frac{s}{\pi}Im \frac{\Pi_o(s)}{\Pi_e(s)^2 s-\Pi_o(s)^2 M^2}
I\!I_2 = \int_{\|u-y\|> t}\frac1{V(x,y,t+d(x,y))}\Big(\frac{t}{t+d(x,y)}\Big)^{\varepsilon_0}\frac1{V(u,y,s+d(u,y))}\Big(\frac{s}{s+\|u-y\|}\Big)^{\varepsilon_0}d\omega(u)
\sum_{n\le X}\frac{\log^{\alpha}d_n}{d_n}\sim\begin{cases}\quad\frac{X\log\log\log X}{\log X}~\qquad\quad~ \alpha=-1, \frac{X}{\log X}\frac{(\log\log X)^{1+\alpha}}{1+\alpha}\alpha>-1,\end{cases}
c(\gamma,\delta)=\frac{1}{d(\gamma,\delta)}\left( q_{\delta}+\sum_{j=1}^{p-1}\sum_{\gamma_{1},\gamma_{2},...,\gamma_{j}}\frac{q_{\gamma_{1}}q_{\gamma_{2}}...q_{\gamma_{j}}q_{\delta-\gamma(j)}}{d(\gamma,\delta-\gamma_{1})d(\gamma,\delta-\gamma(2))...d(\gamma,\delta-\gamma(j))}\right)
\frac{|v^{+}\rangle-i|w^{+}\rangle}{\sqrt{2}}\frac{|v^{-}\rangle-i|w^{-}\rangle}{\sqrt{2}}=\frac{1}{2}(|v^{+}\rangle|v^{-}\rangle-i|v^{+}\rangle|w^{-}\rangle-i|w^{+}\rangle|v^{-}\rangle-|w^{+}\rangle|w^{-}\rangle).
e^{i \Gamma_C [\vec{\varphi}]}= \langle T_C \mbox{exp}\left[ \frac{i}{\delta}\int_C \left\{{\cal L}' (\vec{\phi}(x) + \vec{\varphi}(x)) - \frac{\delta \Gamma_C [\vec{\varphi}]}{\delta \vec{\varphi}(x)}\cdot \vec{\phi}(x) \right\}\right] \rangle .
P_{0}(z) = e^{\frac{\lambda}{\xi}z}(1-z)^{-\frac{\gamma}{\xi}}\left[ P_{0}(0) - p_{1,1}\frac{\mu}{\xi}\int_{0}^{z}e^{-\frac{\lambda}{\xi}s}(1-s)^{\frac{\gamma}{\xi}-1}ds \right].
A_{1}=J_{2}J_{3}+J_{3}J_{2},A_{2}=J_{3}J_{1}+J_{1}J_{3},\text{}A_{3}=J_{1}J_{2}+J_{2}J_{1}B_{1}=J_{2}^{2}-J_{3}^{2},B_{2}=J_{3}^{2}-J_{1}^{2},B_{3}=J_{1}^{2}-J_{2}^{2}(B_{1}+B_{2}+B_{3}=0)
\frac{1}{(\sigma+1)^{2}}(\mathcal{J}_{\varepsilon}\bar{u}_{\varepsilon})^{\sigma+1}i\partial_{x}(\mathcal{J}_{\varepsilon}u_{\varepsilon})^{\sigma+1}=\frac{i}{\sigma+1}(\mathcal{J}_{\varepsilon}\bar{u}_{\varepsilon})^{\sigma+1}(\mathcal{J}_{\varepsilon}u_{\varepsilon})^{\sigma}\mathcal{J}_{\varepsilon}u_{\varepsilon,x}.
\int_{D_4}\big(\triangle (u\xi )-\partial_t(u\xi)-\tilde{V}(r, \theta ,t)u\xi \big)^2 e^{-2\tau g(r)}r^{4-n}d v_g dt =\int_{D_4}\big(\triangle \xi u+ 2\nabla \xi\cdot \nabla u +\xi\triangle u-\xi_t u-\xi u_t-\tilde{V}(r, \theta ,t)u\xi \big)^2 e^{-2\tau g(r)}r^{4-n}d v_g dt =0,
\lim_{\Delta\lambda\rightarrow 0}\frac{E(\lambda)}{\Delta\lambda}=\lim_{\Delta\lambda\rightarrow 0}\frac{E(\lambda)-E(\lambda_*)}{\Delta\lambda}=\lim_{\Delta\lambda\rightarrow 0}\frac{B(\lambda)-B(\lambda_*)}{\Delta\lambda}=B^{\prime}(\lambda_*).
d_++d_-= \lim_{u\to\infty}\big(N(u)-ku+N(-u)-ku\big) = \lim_{u\to\infty}\big(\|(u,1)\|-2\|(1,0)\|u+\|(-u,1)\|\big) = \lim_{u\to\infty}\big(\|(u,1)\|-2\|(u,0)\|+\|(u,-1)\|\big)=c\ge0,
\Psi ([Z\sigma (T)^*]_1) = ([Z_1^*Z_1]-[Z_1Z_1^*])-([T_1^*T_1]-[T_1T_1^*]) = ([e]-[Z_1Z_1^*])-([f]-[Z_1^*Z_1])= [Z_2Z_2^*]-[Z_2^*Z_2] & = n_X[{\bf r}(X)]- n_X [{\bf s}(X)] = B(n_X\delta_X) = b.
f_*(p_1,\dots,p_n) = \min\left\{-\sum_{k=1}^n p_k \alpha^k, -\sum_{k=1}^n p_k \beta^k\right\}, f^*(p_1,\dots,p_n) = \max\left\{-\sum_{k=1}^n p_k \alpha^k, -\sum_{k=1}^n p_k \beta^k\right\}.
\prod_{\substack{\gcd(j_1,j_2,j_3,j_4,k)=1 j_1,j_2,j_3,j_4<k j_1,j_2,j_3,j_4\geq 1; k \geq 2}}\left(\frac{1}{1-y^{j_1+j_2+j_3+j_4}z^k}\right)^{\frac{1}{k}}= \exp\left[ \sum_{k=1}^{\infty}\left( \frac{y}{1}+ \frac{y^2}{2}+ \cdots + \frac{y^k}{k}\right)^{4}\frac{z^{k+1}}{(k+1)^3}\right].
[e_i,e_j]_0=[e_{i-1},e_j]+[e_i,e_{j-1}]-U_0[e_i,e_j] \phantom{[e_i,e_j]_0}=(a_{p+n-i+1,j}-a_{p+n-j,i-1})e_p+(a_{p+n-i,j-1}-a_{p+n-j+1,i})e_p \phantom{[e_i,e_j]_0={}}-(a_{p+n-i,j}-a_{p+n-j,i})e_{p-1}\phantom{[e_i,e_j]_0}=(a_{p+n-i,j-1}-a_{p+n-j,i-1})e_p,
d\tilde{\tau}^2 = \bigg(|h(z)h(1/z)| \cdot \dfrac{\Pi_{j=1}^q|G_j(z)G_j(1/z)|^{(1-\delta)p}}{e^{p\sum_{j=1}^qu_j(z)+p\sum_{j=1}^qu_j(1/z)}|W_z(g_0,g_1)(z) \cdot W_z(g_0,g_1)(1/z)|^p}\bigg)^{\frac{2}{1-p}}|dz|^2 &=\lambda^2(z)|dz|^2,
P_{F\neq \hat{F}}(\rho) =\int\limits_{-\infty}^\infty P_{F\neq \hat{F}|X=x}(\rho)p_X(x)dx &=\int\limits_{0}^\infty P_{F\neq \hat{F}|X=z}(\rho)p_{X^2}(z)dz &=1-\sum\limits_{k=0}^{N_{\mathrm{F}}-1}\int\limits_0^{\infty}\frac{C^k_{N_{\mathrm{F}}-1}(-1)^{N_{\mathrm{F}}-k-1}}{z\rho(N_{\mathrm{F}}-k-1)+N_{\mathrm{F}}-k}\frac{1}{\sqrt{2\pi z}}e^{-z/2}dz.
\hat{\phi}_1(\xi_1,\eta_1) = \gamma^{-\frac{3}{2}}\mathbf{1}_{D_1}(\xi_1,\eta_1), \hat{\phi}_2(\xi_2,\eta_2) = \gamma^{-\frac{3}{2}}N^{-s_1-(1+\frac{\alpha}{2})s_2}\mathbf{1}_{D_2}(\xi_2,\eta_2),
\lim_{n\rightarrow+\infty}\frac{1}{n}\operatorname{\widehat{\mu}_{\min}^{\mathrm{asy}}}(\overline L^{\otimes n}\otimes\overline A)=\lim_{n\rightarrow+\infty}\frac{1}{n}\operatorname{\widehat{\mu}_{\min}^{\mathrm{asy}}}(\overline L^{\otimes n}\otimes\overline B)
\mathbb{J}^0_{k,j_i}(x) = \Big\{z \in \mathbb R^{q-1}, \sqrt{\Delta}\big(\sigma_k^{(i,2:q)}(x) \big \vert z \big) \in \big( x_{k+1}^{i,j_i-1/2}- m_{k}^i(x), x_{k+1}^{i,j_i+1/2}- m_{k}^i(x) \big) \Big \}
\widehat{K_3}(y)=\frac{1}{1+\frac{\beta-1}{\gamma}\sqrt{\mu}|y| coth(\sqrt{\mu_2}|y|)}=\theta\frac{sinh(\sqrt{\mu_2}y)}{\sqrt{\mu_2}y cosh(\sqrt{\mu_2}y)+ \theta sinh(\sqrt{\mu_2}y)}.
S_{\mathrm{b u l k}}= \int d^{6}x \, \sqrt{- \det G}\left[ \frac{1}{2 \kappa_{o r b}^{2}}{\cal R}( G ) - \frac{1}{2}\partial_{\mu}\hat{\varphi}\partial^{\mu}\hat{\varphi}- \frac{1}{2}\partial_{\mu}\hat{\eta}_{a}\partial^{\mu}\hat{\eta}_{a}- \frac{1}{4}\mathrm{e}^{\kappa_{o r b}\hat{\varphi}}\, \hat{F}^{\, 2}\right]
\left\{\delta S_{0}\left[{\frac{\delta}{\delta \eta_{a}}},{\frac{\delta}{\delta \bar{\eta}_{a}}},{\frac{\delta}{\delta J_{a}^{\mu}}}\right] + \delta S_{T}\left[{\frac{\delta}{\delta \eta_{a}}},{\frac{\delta}{\delta \bar{\eta}_{a}}},{\frac{\delta}{\delta J_{a}^{\mu}}}\right] \right\}Z \left[ J_{\mu}, \eta , \bar{\eta}\right] \, = \, 0 \, .
\delta_\mathrm{mRBA}(\varphi_1 + \delta_\mathrm{Hoch}(a), 0) =~ \big( \delta_\mathrm{Hoch}(\varphi_1), ~- \Psi^1 (\varphi_1) - \Psi^1 \circ \delta_\mathrm{Hoch}(a) \big) =~ \big( \delta_\mathrm{Hoch}(\varphi_1), ~ - \Psi^1 (\varphi_1) - \widetilde{\delta}_\mathrm{Hoch}(a) \big) = \delta_\mathrm{mRBA}(\varphi_1, a).
p_{1,k_1,k_2}(x,t)= \frac{1}{|x|}e^{-\frac{1}{4}d(x,t)^2}\Psi(\omega) \left[(-1)^{k_1+k_2}\frac{y_\omega^{n+k_1+k_2}\cos(y_\omega)^{k_1}}{\sin (y_\omega)^{n+k_1}}+O\left(\frac{1}{|x|^2}\right)\right]
|h(r^*)|^p - \left| \langle h^p\rangle_{r^*}\right| \leq \int_{r^*}^{r^*+1}|h^p(r^*) - h^p(\xi)| \,\mathrm{d}\xi = \int_0^1 |h^p(r^*) - h^p(r^*+\zeta)|\,\mathrm{d}\zeta.
osc \Big(y_0,\frac{R}{4}, v\Big )= \sup_{B (y_0,\frac{R}{4})}(\psi+z )-\inf_{B (y_0,\frac{R}{4})}(\psi+z ) \leq \sup_{B (y_0,\frac{R}{4})}\psi + \sup_{B (y_0,\frac{R}{4})}z - \inf_{B (y_0,\frac{R}{4})}\psi - \inf_{B (y_0,\frac{R}{4})}z.
\frac{\partial U_{\Lambda}( \varphi , T )}{\partial \Lambda}= - \frac{K_{3}}{2}2 \pi T^{2}\Lambda \, g \left( \frac{\Lambda}{T}\right) \ln \det \left[ \Lambda^{2}+ \frac{\partial^{2}U_{\Lambda}( \varphi )}{\partial \varphi_{i}\partial \varphi_{j}}\right]
P_g := \Delta_g^2 -\mathrm{div}_g(a_n R_g g+b_nRic_g)d+\frac{n-4}{2}Q_g, Q_g :=-\frac{1}{2(n-1)}\Delta_g R_g+\frac{n^3-4n^2+16n-16}{8(n-1)^2(n-2)^2}R_g^2-\frac{2}{(n-2)^2}|Ric_g|^2,
({\rm I}\,+z D_{{}^\Vert,B})^{-1}:\begin{cases}{\rm{\sf R}}^p(d,\Omega){\rm{\sf R}}^p(\underline{\delta}_B,\Omega)\end{cases}\longrightarrow L^p(\Omega,\Lambda),\qquad\forall z\in{\mathbb{C}}\setminus S_\theta,
\overline{\dim}_B(\mathcal{G}(T)) =\varlimsup_{\delta \rightarrow 0}\frac{\log N_{(1+l) \eta}(\mathcal{G}(T))}{- \log ((1+l) \eta)}\le \varlimsup_{\eta \rightarrow 0}\frac{\log N_{\eta}(\mathcal{G}(F+T))}{- \log ((1+l) \eta)}= \varlimsup_{\eta \rightarrow 0}\frac{\log N_{\eta}(\mathcal{G}(F+T))}{- \log( \eta)}=\overline{\dim}_B(\mathcal{G}(F+T)),
D_{y}h(y)=-2k(y-z)e^{-k|y-z|^{2}}, & D_{yy}^{2}h(y)=(4k^{2}(y_{i}-z_{i})(y_{j}-z_{j})-2k\delta_{ij})e^{-k|y-z|^{2}}=(4k^{2}(y-z)(y-z)^{T}-2kI_{d\times d})e^{-k|y-z|^{2}}.
\begin{array}{ll}\underline{d}(\sigma_1,...,\sigma_l,x)(t) =(\underline{d}\sigma_1(t),...,\underline{d}\sigma_l(t),\underline{d}x(t)) \null = (f_1^0(t,x(t),u(t)),...,f^0_l(t,x(t),u(t)),f(t,x(t),u(t))) \null = F(t,(\sigma_1,...,\sigma_l,x)(t),u(t))\end{array}
\sum_{l=1}^\infty|B_{2l-1,2j-1}|=\sum_{l=1}^\infty2\Big|\sum_{k=1}^\infty\frac{\alpha^k{{2l+2j-3}\choose{2j-2}}}{(R_0k)^{2l+2j-2}}\Big|=2\Big|\sum_{l=1}^\infty\sum_{k=1}^\infty \frac{\alpha^k{{2l+2j-3}\choose{2j-2}}}{(R_0k)^{2l+2j-2}}\Big|.
\min_{\tau+1\leq t \leq T}\Pi_t \leq \frac{4(1+\gamma^2\tau)}{T-\tau}(L+\L\rho + \mu(1-\rho))^2\sum_{t=1}^{T-\tau}\Delta_t + \frac{2}{T-\tau}\sum_{t=1}^T \Phi_t
\prod_{j=0}^{n-1}(1+tq^j) \cdot \sum_{j=0}^{n-1}\frac{jq^{j-1}t}{1+tq^j}= \sum_{r=0}^{n}q^{\binom{r}{2}-1}t^r \Big( \binom{r}{2}{n \brack r}_q +q{n \brack r}^{'}_{q}\Big).
\nabla^{2}U=-\nabla^{2}\frac{Ze^{2}}{4\pi\epsilon_{0}r}=4\pi(\frac{Ze^{2}}{4\pi\epsilon_{0}})\delta^{3}(r)\Rightarrow\delta U\approx\frac{\hbar^{2}}{6m_{e}^{2}c^{2}}4\pi(\frac{Ze^{2}}{4\pi\epsilon_{0}})\delta^{3}(r)
\Delta^{\mu \nu}=\frac 14\left(\Pi_{\;\rho}^{\rho \;,\mu\nu}+\tilde \Gamma^{\mu \nu}\right) -{\frac 1{32}}\left(\Pi^{\rho\;\; ,\sigma}_{\;\;\rho\;\;\sigma}+\tilde\Gamma^\rho_{\;\rho}\right)\,\eta^{\mu\nu}.
\gamma_{t-1}\alpha_t^2=(1-\alpha_t)\alpha_{t-1}^2\gamma_t+\mu\alpha_t\gamma_t\gamma_{t-1}\leq(1-\alpha_t)\alpha_{t-1}^2\gamma_t+\alpha_t\gamma_t\alpha_{t-1}^2=\gamma_t\alpha_{t-1}^2,
|f(\varphi(h,x))-f(x)| \leq \sum_{m=1}^{N_n}|f(\varphi(s_n +t_m,x))-f(\varphi(t_m,x))|+|f(\varphi(r_n,x))-f(x)| &\leq N_n s_n \cdot c_n+ \dfrac{\varepsilon}{2}\cdot v(x)\leq \varepsilon\cdot v(x).
\begin{array}{l l}{D = 1 :}&{Z_{E}( \tau , z ) = 0 ,}\\{D = 2 :}&{Z_{E}( \tau , z ) = Z_{T}( \tau , z ) P ( \tau , z )^{r - 2},}\\{D = 3 , 5 :}&{Z_{E}( \tau , z ) = \frac{\chi ( E )}{\chi ( T )}Z_{T}( \tau , z ) P ( \tau , z )^{r - D}.}\\ \end{array}
\Omega^{2 \gamma}_{\infty}:=\Omega^{2 \gamma}_{\infty}(i_{\delta}):=\left\{\omega\in\Omega_0 : \lvert \mathrm{i}\omega\cdot l+\mu_j^{\infty}(\omega)-\mu_k^{\infty}(\omega)\rvert\geq \frac{2 \gamma\,\lvert j^3-k^3 \rvert}{\langle l \rangle^{\tau}}, \,\,\forall l\in\mathbb{Z}^{\nu}, \,\,\forall j, k\in S^c \cup \{0\}\right\}
\iint_Q f_\epsilon(u^\tau) \partial_x I(u^\tau) \partial_x \varphi \rightarrow - \iint_Q f_\epsilon(u^\epsilon) I(u^\epsilon) \partial^2_{xx}\varphi - \iint_Q n (u^\epsilon)^{n-1}\partial_x u^\epsilon I(u^\epsilon) \partial_x \varphi = \iint_Q f_\epsilon(u^\epsilon) \partial_x I(u^\epsilon) \partial_x \varphi.
\tau=-\langle \mathbf{n},\mathbf{b}'\rangle_1=-\langle\frac{\epsilon_n}{\kappa}\alpha'',\frac{\epsilon_n}{\kappa}\alpha'\times_1\alpha'''\rangle_1=\frac{\epsilon_t\epsilon_u}{r^2\kappa^2}\langle\alpha-J_1\alpha'\times_1\alpha,\alpha'\times_1\alpha'''\rangle_1.
\lim_{\varepsilon\to 0}\frac{1}{\varepsilon^2\ln\frac{1}{\varepsilon}}\int_\Omega\left( v^\tau_\varepsilon-q\ln\frac{1}{\varepsilon}\right)^p_+v^\tau_\varepsilon dx=q^{p+1}(\bar{x})\int_{\hat{B}_\tau(0)}(\hat{U}(y)+\ln\tau)^{p}_+dy.
\int_{0}^{1}\left[ - \frac{1}{2 e}\left( \dot{x}^{\mu}- i{\psi}_{a}^{\mu}{\chi}_{a}\right)^{2}+ \frac{i}{2}f_{a b}\left[ \psi_{a \mu}, ~ \psi_{b}^{\mu}\right]_{-}- i \psi_{a \mu}\dot{\psi}_{a}^{\mu}\right] d \tau \; .
\left\|\left(\frac{d}{dx}\right)^nA(x)^m\Big|_{x=x_0}\right\|_{b\times b}\le n!\prod_{j=1}^m\Bigg(\sum_{l_j=0}^nqr^{l_j}\Bigg)1_{\sum_{j=1}^ml_j=n}&=q^mr^n\left(\frac{d}{dz}\right)^n\Big|_{z=0}\left(\frac{1}{1-z}\right)^m=q^mr^n\frac{n!}{2\pi i}\oint_{|z|=\frac{1}{2}}dz\frac{1}{z^{n+1}(1-z)^m}&\le (2q)^m(2r)^nn!.
2^{jn\alpha_2(1/p_1-1/p_2)}=2^{j\widetilde{A_1}(\mathbf{p},q,\alpha_1,\alpha_2)}=2^{j\widetilde{A_2}(\mathbf{p},q,\alpha_1,\alpha_2)}=2^{j\widetilde{A_3}(\mathbf{p},q,\alpha_1,\alpha_2)}.
J^{( N )}\left( N ; 1 , \ldots , 1 \right) = 2 \; \mathrm{i}^{1 - 2 N}\pi^{N / 2}\frac{\Gamma \left({\textstyle{\frac{N}{2}}}\right)}{\prod m_{i}}\int_{0}^{\infty}\ldots \int_{0}^{\infty}\prod \mathrm{d}\alpha_{i}\; \delta \left( \alpha^{T}\| c \| \alpha - 1 \right) .
\gamma_i(\theta_i) = e^{\theta_i}-1, \xi_i(\theta) =P_{i0}\big(e^{-\theta_i}-1\big) + \sum_{j\in \cal{J}}P_{ij}\big(e^{\theta_j-\theta_i}-1\big), Q(\theta)=\sum_{i\in \mathcal{J}}\bigl ( \alpha_i \gamma_i(\theta_i) + \lambda_i \xi_i(\theta) \bigr ).
\Psi^1_n(t,p) =\idotsint\limits_{0\leq t_1<\cdots<t_n\leq t}\left(-\frac{M}{m}e^{-\frac{M}{m}t_1}\boldsymbol p\right)\otimes\cdots\otimes\left(-\frac{M}{m}e^{-\frac{M}{m}t_n}\boldsymbol p\right)d t_1\dots d t_n.
\int_{B_r(x_0)}\big(\left|\nabla u(x)\right|^p-\left|\nabla v(x)\right|^p\big)\,dx \ge p\,\gamma\int_0^1{s}^{p-1}\left(\int_{B_r(x_0)}\left|\nabla u(x)-\nabla v(x)\right|^p \,dx\right)\,ds &=\gamma \int_{B_r(x_0)}\left|\nabla u(x)-\nabla v(x)\right|^p\,dx,
(-\Delta)^s R_{2n}^\lambda(x)= a_n^\lambda \times \sum_{k=0}^{n}\frac{(-n)_{k}(n+\lambda)_{k}}{(\lambda+\frac{1}{2})_{k}\, k!}A_s^{k+\frac{\lambda+1}{2}}{}_{2}F_{1}\Big(s+k+\frac{\lambda+1}{2},s+\frac{1}{2};\frac{1}{2};-x^2\Big),
h_{2}= \sum_{i}e^{i}(\nabla_{Z}\nabla_{Z}e_{i}) \rho_{\phi}[\iota]^{2}, h_{6}= \sum_{i}f^{i}(\nabla_{Z}\nabla_{Z}(\phi e_{i})) \rho_{\phi}[\iota]^{2}, h_{11}= \eta^{*}(\nabla_{Z}\nabla_{Z}\xi) \rho_{\phi}[\iota]^{2}.
\hat{f}_\theta(k) = e^{-d \theta/2}\hat{f}(e^{-\theta}k ) = e^{-d \theta/2}Q_\lambda(e^{-\theta}(k-a) ) e^{-\frac{\pi}{\lambda}e^{-2\theta}|k-a|^2}e^{- 2 \pi i e^{-\theta}(k-a) \cdot x_0}.
P(\eta^N\in O) &=\mathbb{E}_{\widetilde{P}_{h_1^m, h_2^m, h_3^m}^{N, F^m, G^m, H^m}}\left(\left(\mathcal{Y}_{F^m, G^m, H^m}^N(t, \xi^N)\right)^{-1}\frac{dP}{d\tilde{P}^N_{h_1^m, h_2^m, h_3^m}}1_{\{\eta^N\in O\}}\right).
\frac{\Gamma(\zeta+(1+\alpha)/2+2k)}{k!\Gamma(\zeta+\frac12+k)}(2\beta)^{-2k}= \frac{\Gamma(\zeta+(1+\alpha)/2)}{\Gamma(\frac{\zeta+(1+\alpha)/2}{2})\Gamma(\frac{\zeta+(3+\alpha)/2}{2})}\frac{2^{2k}\Gamma(\frac{\zeta+(1+\alpha)/2}{2}+k)\Gamma(\frac{\zeta+(3+\alpha)/2}{2}+k)}{k!\Gamma(\zeta+\frac12+k)}(2\beta)^{-2k}
S_4 = \int_{\mathcal{X}_n}f(x) \int_{\frac{a_n}{n-1}}^1 \mathrm{B}_{k,n-k}(s) \log^2\biggl( \frac{(n-1)s}{e^{\Psi(k)}f(x)}\biggr) \, ds \, dx = o(n^{-(3-\epsilon)}).
< \dot{x}_{\mu}( t_{1}) \dot{x}_{\nu}( t_{2}) >_{S_{2}}= \hat{{\cal N}}( \alpha ) \left[ \dot{x}_{\mu}^{c l}( t_{1}) \dot{x}_{\nu}^{c l}( t_{2}) - 2 \delta_{\mu \nu}\frac{\partial^{2}}{\partial t_{1}\partial t_{2}}\Delta^{( N )}( t_{1}, t_{2}) \right] e^{- S_{2}[ x^{c l}]},
f(x) = (x-a^1_j)^{e^1_j-1}\ldots (x-a^{k_j}_j)^{e^{k_j}_j-1}\Big((x-a^1_j)\ldots(x-a^{k_j}_j)h'_j(x) + e^1_j(\widehat{x-a^1_j})(x-a^2_j)\ldots(x-a^{k_j}_j)h_j(x) + e^2_j(x-a^1_j) (\widehat{x-a^2_j})\ldots(x-a^{k_j}_j)h_j(x) + e^{k_j}_j(x-a^1_j)\ldots (x-a^{k_j-1}_j) (\widehat{x-a^{k_j}_j}) \Big) \, \, 1\leq j\leq r
\big((\Phi_{\mathcal{Y}}\circ T \circ \Phi_{\mathcal{X}}^{-1})(\ell_{\gamma})\big)(u) = \big((\Phi_{\mathcal{Y}}\circ T \circ \Phi_{\mathcal{X}}^{-1})(\Phi_{\mathcal{X}}(f_{\gamma}))\big)(u) &= \big(\Phi_{\mathcal{Y}}\circ T(f_{\gamma})\big)(u) = u(T(f_{\gamma})) = \big(\Phi_{\mathcal{X}}(f_{\gamma})\big)(u\circ T) = \ell_{\gamma}(u\circ T)
\prod_{j=1}^\infty\left(1+c_H\max_{i}|u_i|^2|\ell_{i+1}-\ell_i|^{2H}\widetilde{\mu}_j^4\right)^{-\frac14}\leq c\,\exp(-c_H\max_{i}|u_i|^{\frac1H}|\ell_{i+1}-\ell_i|)
g'_k=\sum_{i=k}^{k+m-1}g'_i -\sum_{i=k+1}^{k-1+m}g'_i= \sum_{i=k}^{k+m-1}g_i -\sum_{i=-m}^{-2}g'_i \ge \sum_{i=-m}^{-1}g_i-\sum_{i=-m}^{-2}g_i=g_{-1}.
\hat{\Pi}_{W_1}\big(\hat{\Pi}_{V_1}-\sum_{j=1}^k(\hat{\Pi}_{V_1}\hat{\Pi}_{V_2})^j(1-\hat{\Pi}_{V_1})\big)\Big(\hat{\Pi}_{V_1}-\sum_{j=1}^k(\hat{\Pi}_{V_1}\hat{\Pi}_{V_2})^j(1-\hat{\Pi}_{V_1})\Big)^T\hat{\Pi}_{W_1}.
\mathcal{U}(h,k)= \frac{1}{2}\sum_{\substack{1\leq c \leq C (c ,hk)=1}}\mu(c) \sum_{\substack{1\leq m,n<\infty (mn,c )=1 mh\neq nk}}\frac{\tau_A(m) \tau_B(n)}{\sqrt{mn}}V\left( \frac{m}{X}\right) V\left( \frac{n}{X}\right)\sum_{\substack{1\leq d<\infty ( d,mhnk)=1 d|mh\pm nk}}\phi(d) W\left( \frac{cd}{Q}\right).
\Phi_{n}^{1}: = \sqrt{q^{2}+ p^{2}}\, T_{n}\left( q ( q^{2}+ p^{2})^{- \frac{1}{2}}\right) - 1 \quad , \quad \Phi_{n}^{2}: = p \, U_{n - 1}\left( q ( q^{2}+ p^{2})^{- \frac{1}{2}}\right)
d(X,Y) = \inf_{\lambda_1,\lambda_2}\max \left\{\sup_{s\in[0,1]}|\lambda_1(s)-s| ,\sup_{t\in[0,1]}|\lambda_2(t)-t| , \sup_{s,t\in[0,1]}|X(\lambda_1(s),\lambda_2(t))-Y(s,t)|\right\},
P_{m,n}^{(\alpha,\beta)}(x)=\frac{(-1)^m2^{-n}\binom{n}{m}(n\!+\!\delta\!-\!2m)\Gamma(\beta\!-\!\delta\!+\!2m) \Gamma(\delta\!+\!\beta\!+\!2(n\!-\!m))(n\!+\!\beta)}{n!(\delta+n-m)\Gamma(\beta\!-\!\delta\!+\!m)\Gamma(\delta\!+\!\beta\!+\!n\!-\!m) \prod_{i<j}\big(W^{(\alpha,\beta)}_{j,m,n}-W^{(\alpha,\beta)}_{i,m,n}\big)}\det(M_n).
U_k =\left\{\prod\limits_{\substack{n_1+1\leq p<t\leq n}}({x_p y_{t}}-x_{t}y_{p})^{g_{pt}}\cdot \prod\limits_{\substack{1\leq i<r\leq n_1}}({x_i y_{r}}-x_{r}y_{i})^{g_{ir}}\right. &\left.\cdot \prod\limits_{\substack{1\leq i\leq n_1 n_1+1\leq t \leq n}}(x_ix_t-y_iy_t)^{g_{1t}}| \sum\limits_{1\leq p<t\leq n}g_{pt}=k, g_{pt}\in\mathbb{N}\right\}.
1 、 3 、 5 、 7 、 9 、 1 1 、 1 3 、 1 5 、 1 7 、 1 9 、 2 1 、 2 3 、 2 5 、 2 7 、 2 9 、 3 1 、 3 3 、 3 5 、 3 7 、 3 9
d_{[ \alpha_{1}] , [ \alpha_{2}] , ( \alpha_{3})}^{\epsilon}= i e^{- \frac{i \pi \epsilon}{4}}\left[ \beta_{2}C_{[ \alpha_{1}] , [ \alpha_{2}] , ( \alpha_{3})}^{\epsilon}- \beta_{1}C_{[ \alpha_{1}] , [ \alpha_{2}] , ( \alpha_{3})}^{- \epsilon}\right] .
f_{1}(x,m^{\hat v}_t,\hat{v}_{1}(x,m^{\hat v}_t))+Du^{\hat v}_{1}(x,t).g_{1}(x,m^{\hat v}_t,\hat{v}_{1}(x,m^{\hat v}_t))=H_{1}(x,m^{\hat v}_t,DU_{1}(x,m^{\hat v}_t,t)) \, ,
H_{+}(x, t) = -\frac{1}{\pi}\sum_{r=0}^{\infty}\sum_{j=0}^{m_{1}-1}\frac{(-t)^{r+j}}{r!\, j!}\frac{\Gamma(1 + Mj + mr)}{x^{1 + Mj + rm}}\sin(r\pi v + j\pi u) & \times{_{1+lp}F_{m_{1}}}\left({1, \Delta(lp, 1+ Mj + mr) \atop \Delta(m_{1}, 1 + j)}; (-1)^{m_{1}u - m_{1}}\frac{t^{m_{1}}(lp)^{lp}}{m_{1}^{m_{1}}x^{lp}}\right),
\mathcal{C}_{a, k}(z) = \sum_n a_n\bigg(\frac{1}{z- t_n}+\frac{1}{t_n}++\dots + \frac{z^{k-1}}{t_n^k}\bigg) = z^k \sum_n \frac{a_n}{t_n^k (z-t_n)}
K_av_b=q^{\delta_{ab}}v_b,E_av_b=\delta_{a+1,b}v_a,\ \F_av_b=\delta_{a,b}v_{a+1}, &K_aw_b=q^{-\delta_{ab}}w_b,E_aw_b=\delta_{a,b}w_{a+1},F_aw_b=\delta_{a+1,b}w_{a}.
i \, \lim_{r \rightarrow 0}\left( m r \right)^{1 - a}g_{\ell}^{\pm}( r ) \sin \left( \frac{\pi}{4}+ \frac{\Theta^{\pm}}{2}\right) = \lim_{r \rightarrow 0}\left( m r \right)^{a}f_{\ell}^{\pm}( r ) \cos \left( \frac{\pi}{4}+ \frac{\Theta^{\pm}}{2}\right) \, .
f_{\ell,n_q}(A_0,\dots,A_{n_q-1})=1+\sum_{m=0}^{\ell-1}\sum_{\substack{0\leq s_k \ \sum_{k=0}^{n_q-1}s_k=\ell -m \sum_{k=0}^{n_q-1}q^ks_k\equiv 0 \bmod{\ell}}}b_{s_0,\dots,s_{n_q-1}}A_0^{s_0}\cdots A_{n_q-1}^{s_{n_q-1}}.
\alpha(G[(\cup_{i=1}^{j-1}V_i)\cup\{v\}])=\max_{\substack{v'\in V_{j-1}\cap N(v) v''\in V_{j-1}- N(v)}}\{\alpha(G[(\cup_{i=1}^{j-2}V_i)\cup\{v'\}]),\alpha(G[(\cup_{i=1}^{j-2}V_i)\cup\{v''\}])+1\}.
\tilde{R}^{k,l+1}_{j}= \tilde{R}^{k,l}_{j}, \ \ \ \ \ \ \tilde{b}^{k,l+1}_{j}= \tilde{b}^{k,l}_{j}\ \ \ \ \ \, j \in I^{k,l}_1, &\tilde{R}^{k,l+1}_{j}= R^{k,l+1}_{j}, \ \ \ \tilde{b}^{k,l+1}_{j}= b^{k,l+1}_{j}\ \ \j \in I^{k,l}_2.