harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\vert \sum_{3\sqrt{T/\pi}< n \leq CT/\pi}\frac{1}{n^{{1}/{2}}}\int_{T}^{2T}e^{i F(n,t)}dt\vert = O\left(\sum_{3\sqrt{T/\pi}< n \leq CT/\pi}\frac{1}{n^{1/2}}\right) = O\left(T^{1/2}\right).
\sum_{\substack{p=1 p\neq q/2}}^{q-1}\frac{(q!)^4}{\big(p! (q - p)! \big)^2}\| f_n \star_p^p f_n \|^2 - \frac12 \sum_{p=1}^{q/2-1}\frac{(q!)^4}{\big((q/2)! p!(q/2-p)!\big)^2}\|f_n\star_p^p f_n \|^2 = \sum_{p=1}^{q/2 - 1}\frac{(q!)^4}{(p!)^2}\left(\frac{2}{(q-p)!^2}- \frac{1}{2\big((q/2)!(q/2 - p)!\big)^2}\right) \|f_n\star_p^p f_n \|^2.
\begin{cases}u_{tt}-\Delta u=|v|^{p}, x\in\mathbb{R}^n,\t>0, v_{tt}-\Delta v=|u|^{q}, x\in\mathbb{R}^n,\t>0, (u,u_t,v,v_t)(0,x)=(u_0,u_1,v_0,v_1)(x), x\in\mathbb{R}^n,\end{cases}
w^{}_\gamma(P_\epsilon^L, I) = \int \limits_{\substack{\epsilon \leq t_1 \leq t_2 \leq L t_2^R \leq t_1}}f_{\gamma, I}^{N_1}(t_1, t_2)[\phi] + \int\limits_{\substack{\epsilon \leq t_1 \leq t_2 \leq L t_1 \leq t_2^R}}f_{\gamma', \gamma, I}^{N_2}(t_1, t_2)[\phi]
\begin{array}{c}{{l i m}}\\{{\varepsilon \rightarrow 0}}\\ \end{array}\frac{\partial}{\partial \theta}I ( p , m , \theta , \varepsilon ) - \frac{\partial}{\partial \theta}\begin{array}{c}{{l i m}}\\{{\varepsilon \rightarrow 0}}\\ \end{array}I ( p , m , \theta , \varepsilon ) \mid_{_{\theta_{0}^{\pm}}}\neq 0
\begin{aligned}\bar{H}_{i}(\cdot)= b(t,x,u_1,u_2)q_i(t)+\sigma(t,x,u_1,u_2)k_i(t)+\sum_{j=1}^2h_j(t,x,u_1,u_2)Q_{ji}(t) - f(t,x,y,z,z_1,z_2,u_1,u_2)p_i(t)+l_i(t,x,y,z,z_1,z_2,u_1,u_2). \end{aligned}
\begin{array}{c}{{a X_{\pm}^{0}= \frac{\sqrt{\alpha^{\prime}}( D - 1 )}{2 Q}\log \frac{( D - 1 ) \sqrt{\alpha^{\prime}}}{2 \tilde{Q}\sqrt{\eta}\, ( t_{0}\pm t )}}}\\{{\ \hfill ( \tilde{Q}= Q e^{\frac{2 \phi_{0}}{D - 1}}\ , \t_{0}= \mathrm{c o n s t .}\ , \ \eta = \pm 1 \ \mathrm{f o r \M i n k o w s k e a n \o r \E u c l i d e a n \s i g n a t u r e}\ ) \ .}}\\ \end{array}
\begin{aligned}\sum\limits_{i=1}^{n}{{{w}_{i}}f\left( M+m-\frac{\overline{a}+{{a}_{i}}}{2}\right)}\le \sum\limits_{i=1}^{n}{\frac{{{w}_{i}}}{\overline{a}-{{a}_{i}}}\int_{M+m-\overline{a}}^{M+m-{{a}_{i}}}{f\left( t \right)dt}}\le \frac{f\left( M+m-\overline{a}\right)+\sum\nolimits_{i=1}^{n}{{{w}_{i}}f\left( M+m-{{a}_{i}}\right)}}{2}. \end{aligned}
\big(\prod_Pa_P!\big)\delta(x_A) &=\sum_{(\sigma,i)\in C}(-1)^isgn(\sigma)(P_1+Q_{\sigma(1)})\wedge\ldots\wedge\widehat{(P_i+Q_{\sigma(i)})}\wedge\ldots\wedge(P_p+Q_{\sigma(p)})) &\otimes(Q_{\sigma(p+1)}+Q_{\sigma(i)}+P_i).
f_1 = \mathcal{F}(\gamma(N_1^{(5-2\alpha)+\varepsilon}t-n)\mathbf{1}_{[0,T]}(t) P_{N_1}v), f_2 = \mathcal{F}(\gamma(N_1^{(5-2\alpha)+\varepsilon}t-n)\mathbf{1}_{[0,T]}(t) P_{N_2}u_1), f_3 = \mathcal{F}(\gamma(N_1^{(5-2\alpha)+\varepsilon}t-n)\mathbf{1}_{[0,T]}(t) P_N v).
{\cal F}_{a}^{\mu \nu}= \frac{1}{a}\phi_{a}{\cal F}^{\mu \nu}= \frac{1}{a}\phi_{a}\frac{1}{a^{3}e}\phi \cdot ( \partial^{\mu}\phi \wedge \partial^{\nu}\phi ) + \partial^{\mu}A^{\nu}- \partial^{\nu}A^{\mu}.
\left| \eta^{\rm slow}\right|_{n_j,\mathcal{I}_j}^2 = \int_{a_j}^{b_j}\left| D^{n_j}_s \eta^{\rm slow}(s) \right|^2 \left( s -a_j \right)^{n_j}\left( b_j-s \right)^{n_j}ds & \lesssim_{n_j,k_0}k^2 \left( b_j -a_j \right)^{2n_j+1}& \lesssim_{n_j,k_0}k^2
\begin{aligned}\dot{x}= \omega+\xi(y,z,\sigma,\omega,\mu)+f(x,y,z,\sigma,\omega,\mu), \dot{y}= \sigma+\eta(y,z,\sigma,\omega,\mu)+g(x,y,z,\sigma,\omega,\mu), \dot{z}= M(\omega,\mu)z+\zeta(y,z,\sigma,\omega,\mu)+h(x,y,z,\sigma,\omega,\mu),\end{aligned}
( - \tilde{g})^{- 1 / 2}\partial_{\mu}\left( \sqrt{- \tilde{g}}\tilde{g}^{\mu \nu}\partial_{\nu}\phi \right) - \frac{\alpha M^{4}}{M_{p}( \zeta + b )}e^{- 2 \alpha \phi / M_{p}}+ \frac{\alpha m}{M_{p}}F ( \zeta ) \overline{{\Psi}}^{\prime}\Psi^{\prime}= 0 .
\partial_{z}\partial_{z^{\prime}}g ( z , z^{\prime}) |_{z = z^{\prime}= 0 , a}={\frac{2}{a}}\sum_{n = 1}^{\infty}{\frac{n^{2}\pi^{2}/ a^{2}}{n^{2}\pi^{2}/ a^{2}- \lambda^{2}}}.
P^{(i)}_{}(\gamma_{})= 1-\Psi_{i}\left(\frac{\left(p\kappa^{2}_{R}m+N_{0}\right)\gamma_{}}{\left(1-\kappa^{2}_{T}\gamma_{}\right)}\right)^{\mu}&\times \exp\left(-\frac{(m+l-i+1)\left(p\kappa^{2}_{R}m+N_{0}\right)\gamma_{}}{p\left(1-\kappa^{2}_{T}\gamma_{}\right)}\right).
\delta_0(v) =\frac{\lambda}{6\kappa}\int_0^vv'^2(y(v')+1)dv' =\frac{\lambda}{6\kappa}\int_0^vv'^2\left(1+\frac{du_1}{dv}(v')\right)dv' &=\frac{\lambda}{18\kappa}v^3+\frac{\lambda}{6\kappa}\left\{v^2u_1(v)-2\int_0^vv'u_1(v')dv'\right\}.
S_{B H}= \frac{2 \pi A_{9}}{\kappa^{2}}= \frac{2 \pi A_{2}}{\kappa_{4}^{2}}= \frac{3 2 \pi^{2}m^{2}}{\kappa^{2}}l_{1}S_{2 3}S_{4 5}S_{6 7}\cosh \alpha \cosh \beta_{1}\cosh \beta_{2}\cosh \beta_{3}
\Gamma_{V_{a}^{\mu}V_{b}^{\nu}}\equiv - \bigl ( \eta^{\mu \nu}-{\frac{p^{\mu}p^{\mu}}{p^{2}}}\bigr ) \Gamma_{a b}^{T}-{\frac{p^{\mu}p^{\mu}}{p^{2}}}\Gamma_{a b}^{L}
Z(t){}=\Sigma_1+\Sigma_2+ \Sigma_3+\Sigma_4+\Sigma_5+\Sigma_6 &{}=\Bigl(q^{n+1}t+q^i-q^{n-i+1}-\tfrac{1}{t}&\qquad{}+\tfrac{(1-q^it)}{(1-t)}\tfrac{(1+q^{n-i+1}t)}{(1+qt)}\bigl(-qt+q-1+\tfrac{1}{t}\bigr)\Bigr) E_i(-t) E_{n-i}(qt)=0
\boldsymbol \alpha (\theta) =(\underbrace{e^{2\pi i \theta_1}, \dots ,e^{2\pi i \theta_1}}_{\mbox{\scriptsize $(\lambda_1-1)$-times}}\ ; dots \ ; underbrace{e^{2\pi i \theta_r}, \dots ,e^{2\pi i \theta_r}}_{\mbox{\scriptsize $(\lambda_r-1)$-times}}).
{\cal S}_{WZW}\,=\, -\frac{k}{4\pi}\int_{\Sigma}d^{2}\sigma A^{I}_{\alpha}A^{J\alpha}\Omega_{IJ}\,-\, \frac{k}{12 \pi}\int_{B}d^{3}\sigma \, \epsilon^{\alpha \beta\gamma}A^{I}_{\alpha}A^{J}_{\beta}A^{L}_{\gamma}{f_{IJ}}^{N}\Omega_{NL}\, ,
{\cal L}_{e f f}= - \frac 1 4 F^{a \mu \nu}F_{\mu \nu}^{a}+ \frac 1 2 m^{2}A^{a \mu}A_{\mu}^{a}- \frac 1{2 \alpha}( \partial^{\mu}A_{\mu}^{a})^{2}+ \bar{C}^{a}\partial^{\mu}(
g_{ij}=\begin{pmatrix}1 -2\lambda u^3 \lambda u^2 -2\lambda u^3 4 \lambda u^1 \lambda u^2 \lambda u^1 0\end{pmatrix}, w_{12}=0,w_{23}=\frac{\lambda^2u^1}{\sqrt{\det g}},w_{31}=\frac{\lambda^2u^2}{\sqrt{\det g}},
\mathcal{L}\left(t,\dot\varphi(\Phi_g(p)),\operatorname{Hor}_{\varphi(\Phi_g(p))}^{\varphi_*\Gamma}\right)=\mathcal{L}\left(t,T\Phi_g(\dot\varphi(p)),T\Phi_g\!\cdot\!\operatorname{Hor}_{\varphi(p)}^{\varphi_*\Gamma}\right)=\mathcal{L}\left(t,\dot\varphi(p),\operatorname{Hor}_{\varphi(p)}^{\varphi_*\Gamma}\right).
\gamma_i^n=\delta_x^2 \beta_i^n-\frac{h^2}{12}\delta_x^2 \gamma_i^n,1\leq i \leq M-1,0\leq n\leq N, \beta_i^0=r_1(x_i),\eta_i^0=r_2(x_i), 0\leq i \leq M, \beta_0^n=\beta_M^n=\gamma_0^n=\gamma_M^n=0,0\leq n\leq N.
\frac{\partial MSE}{\partial s[n]}=2\sum_{\tau=0}^{k}s[\tau]\int_{-1/2}^{1/2}cos(2\pi nF)cos(2\pi\tau F)dF-2\int_{-1/2}^{1/2}H_{d}(F)^{2}cos(2\pi nF)dF=0
\lim_{n \to \infty}\iint \frac{|u_n(x)|^p |u_n(y)|^p}{|x-y|^{N-\gamma}}dxdy = \lim_{l \to \infty}\sum_{j=1}^l \iint \frac{| U^j(x)|^p |U^j (y) |^p}{|x-y|^{N-\gamma}}dxdy.
d_n^{\ell}\cdot \frac{1}{\ell !}F_{1/4}^{(\ell)}(t) \big|_{t = -k}\in \mathbb{Z}, &d_n^{\ell}\cdot \frac{1}{\ell !}F_{3/4}^{(\ell)}(t) \big|_{t = -k}\in \mathbb{Z}, &d_n^{\ell}\cdot \frac{1}{\ell !}\left( (t+k)G(t) \right)^{(\ell)}\big|_{t =-k}\in \mathbb{Z}.
\frac{2}{\pi}\int_{0}^{\infty}d r \left( \psi_{\ell}^{*}( k , r ) \psi_{\ell}( k , r ) - \psi_{\ell}^{( 0 )}{}^{*}( k , r ) \psi_{\ell}^{( 0 )}( k , r ) \right) = \frac{1}{\pi}\frac{d \delta_{\ell}}{d k}
\tilde{J}=\tilde{J}(\{y_i\}_{i=1}^n):= \sum_{\beta\in Q_+}J_\beta \prod_{i=1}^n y_i^{(\beta-\lambda,\omega_i^\vee)}\in N_\lambda, J=J(\{y_i\}_{i=1}^n):= \sum_{\beta\in Q_+}J_\beta \prod_{i=1}^n y_i^{(\beta-\lambda-\rho,\omega_i^\vee)}\in N_{\lambda+\rho}.
\left( \frac{d}{d \tau}\frac{\partial L}{\partial{\dot{x}}_{\mu}}- \frac{d^{2}}{d \tau^{2}}\frac{\partial L}{\partial{\ddot{x}}_{\mu}}+ \frac{d^{3}}{d \tau^{3}}\frac{\partial L}{\partial{\stackrel{\ldots}{x}}_{\mu}}\right){\dot{x}}_{\mu}= 0 .
\begin{aligned}f(v) \leq f(z) + \langle v-z, \nabla j(u) - g\rangle + \left( \frac{L}{2}- \frac{1}{2t}\right) \lVert v-u \rVert^2 \quad+ \left( \frac{L}{2}+ \frac{1}{2t}\right) \lVert z-u \rVert^2 - \frac{1}{2t}\lVert v-z \rVert^2.\end{aligned}
\Lambda=a\left(\theta^{2}\otimes e_{3}-\theta^{3}\otimes e_{2}+\theta^{4}\otimes e_{1}\right)+b\left(\theta^{5}\otimes e_{3}-\theta^{6}\otimes e_{2}+\theta^{7}\otimes e_{1}\right),\;\;a,b\in\mathbb{R}.
\left\langle \left[ \sigma_{+}[ Q^{W}] \right]^{q}\left[ \sigma_{-}[ Q^{W}] \right]^{n - q}\right\rangle_{Q^{\mu}}\; = \; \left\langle \left[ \sigma_{+}[ Q^{W}] \right]^{n - q}\left[ \sigma_{-}[ Q^{W}] \right]^{q}\right\rangle_{Q^{\mu}}\; .
b_{n k}= \frac{1}{n + k}\left[ \frac{( k + 1 ) ( k + 2 )}{2}b_{n - 1 , k + 2}+ \frac{\lambda ( \lambda - 1 )}{2}\sum_{m = 0}^{k}( - 1 )^{m + 1}\frac{m + 1}{q^{m + 2}}b_{n - 1 , k - m}\right] .
\widehat F_{c_m}=1-mq+((m-2)y+(m^2/2+3m/2-5)+(m-2)y^{-1})q^2 &-((m^2+5m-14)y+(m^3+9m^2+44m-132)/6+(m^2+5m-14)y^{-1})q^3+O(q^4),
\phi_{1}= \frac{W + W^{*}}{1 + W W^{*}}\, \, \, \, \, \, \, \, \, \phi_{2}= i \frac{W - W^{*}}{1 + W W^{*}}\, \, \, \, \, \, \, \, \, \phi_{3}= \frac{1 - W W^{*}}{1 + W W^{*}}\, \, \, \, \, \, \, \, \,
\liminf_{N \to \infty}\mathcal{E}^{\neq}({\rm emp}_{N}(X_{N}) - \nu) \geq \liminf_{N \to \infty}\min_{Y_{N}\in \mathbb{R}^{d \times N}}\mathcal{E}^{\neq}({\rm emp}_{N}(Y_{N}) - \nu) \geq \min_{\rho \in \mathcal{P}(\mathbb{R}^{d})}\mathcal{E}(\rho - \nu) = 0 = \mathcal{E}(\mu - \nu).
\begin{array}{l l l l}{\mathrm{f o r}\phi = + 1}&{:}&{( i^{k}, 1 , \pm 1 , \pm 1 , \pm 1 , 1 )}&{, ( k = 0 , . . . , 3 )}\\{\mathrm{f o r}\phi = - 1}&{:}&{( i^{k}e^{i \pi / 4}, 1 , \pm 1 , \pm 1 , \pm 1 , 1 )}&{, ( k = 0 , . . . , 3 ) ,}\\ \end{array}
\tilde{u}_\epsilon(t, x, y)=-\int^{+\infty}_y \tilde{w}_\epsilon(t, x, \tilde y) d\tilde y,\tilde{v}_\epsilon(t, x, y)=-\int^{y}_0\partial_x \tilde{u}_\epsilon(t, x, \tilde y) d\tilde y.
\begin{cases}H_A(x,A^*(x),B^*(x))=0, x_b < x < x_a, begin{align*}5pt] \partial_AH_A(x,A^*,B^*) \partial_BH_B(x,A^*,B^*)- \partial_BH_A(x,A^*,B^*) \partial_AH_B(x,A^*,B^*) < 0 , begin{align*}5pt] \partial_AH_A(x,A^*,B^*)<0,\partial_BH_B(x,A^*,B^*)<0 \end{cases}
|\delta^{(1)}_{n+1}q\sigma_1(v)+\cdots+\delta^{(i)}_{n+1}q\sigma_i(v)+\cdots+\delta^{(d)}_{n+1}q\sigma_d(v)-p| =|\delta_1 q \sigma_1(\gamma_{n+1}v)+\cdots+\delta_d q \sigma_d(\gamma_{n+1}v)-p| &=|\delta_1 q u_1+\cdots+\delta_d q u_d-p|<\frac{1}{H^{\varepsilon/(n+1)}(\gamma_{n+1}v)|q|^{d+\varepsilon}}.
\log\|f_X\|_q =\frac{1}{q}\log\left[2^{-n}\sum_x (2^n\Pr[X=x])^q\right] =\frac{q-1}{q}n+\frac{1}{q}\log\sum_x\Pr[X=x]^q =\frac{q-1}{q}(n-H_q(X))\;.
\begin{array}{r c l}{{}}&{{}}&{{T ( z ) = - \frac{1}{4}\partial \phi ( z ) \partial \phi ( z ) + i \alpha_{0}\partial^{2}\phi ( z ) ~ ,}}\\{{}}&{{}}&{{\langle \phi ( z ) \phi ( w ) \rangle = - 2 \ln ( z - w ) ~ ,}}\\{{}}&{{}}&{{c = 1 - 2 4 \alpha_{0}^{2}~ .}}\\ \end{array}
P_{R}^{\prime}= ({\rho}\frac{\breve{m}}{2}+ ( \; \; G_{\bot}\; \; - \; \; B_{\bot}+ \frac{1}{4}A_{\bot}^{t}C_{\bot}A_{\bot}){\breve{n}}- \; \; \frac{1}{2}A_{\bot}^{t}C_{\bot}{\breve{l}}, 0 ) = ( p_{R}^{\prime}, \; 0 ) .
(\sqrt{\delta}-1)^2 \left((t+b+2)^2-(y-x)^2\right)^{-1}\bigg(1-\frac{t+b+2}{1+t}\bigg) = - (\sqrt{\delta}-1)^2 \frac{(1+b)}{(1+t)\big[(t+b+2)^2-(y-x)^2\big]}&= - \frac{4(1+b)}{(1+t)\big[(t+b+2)^2-(y-x)^2\big]}\big(\tfrac{\sqrt{\delta}-1}{2}\big)^2.
2 p k_n^2 \int_{T_n}^{T}\int \eta_n^2 u_n^{p-1}\;dxdt \le \; 2 p M^2 \int_{T_n}^{T}\int_{B_n}u_n^{p-1}\;dxdt = \; 2 p M^2 \int_{T_n}^{T}\int_{B_n}u_n^{p-1}\chi_{\{u_{n-1}\ge \frac{M}{2^{n}}\}}\;dxdt \le \;2 p M^2 \int_{T_n}^{T}\int_{B_n}u_{n-1}^{p-1}\chi_{\{\eta_{n-1}^{2/p}u_{n-1}\ge \frac{M}{2^{n}}\}}\;dxdt.
f'_m(r_e)=-\dfrac{\big(a^2+o(1)\big)h_m}{r_e|m|}\, \Big( \dfrac{r_e}{a}\Big)^{|m|},D_m^{-1}\begin{pmatrix}0 - f'_m(r_e) \end{pmatrix}=h_m\begin{pmatrix}\dfrac{a^2+o(1)}{2m^2}\Big(\dfrac{r_e^2}{r_i a}\Big)^{|m|}\dfrac{a^2+o(1)}{2m^2}\Big(\dfrac{r_e}{a}\Big)^{|m|}\end{pmatrix},
\int_{\{-t'_1\le\Psi<-t'_2\}}|\tilde{F}|^2_h\mathbb{I}_E(-\Psi)\ge \liminf_{j\to+\infty}\int_{\{z\in M:-\Psi(z)\in V_j\}}|\tilde{F}|^2_h \ge \liminf_{j\to+\infty}\frac{G(T_1)}{\int_{T_1}^{+\infty}c(s)e^{-s}ds}\int_{V_j}e^{-s}ds = \frac{G(T_1)}{\int_{T_1}^{+\infty}c(s)e^{-s}ds}\int_{t_2}^{t_1}e^{-s}\mathbb{I}_E(s)ds,
\widetilde{H}_{j}(s) :=\frac{\widetilde{F}_{j}(s)}{\widetilde{G}_{j}(s)}&=\prod_{p}^{}\left(\frac{1+\widetilde{A}}{1+\widetilde{B}}\right) &\ll_{\epsilon}1, (\mbox{in $\Re(s)>\frac{5}{2}$})
\|f_ga_i-a_if_g\|\le \|\phi(r_gb_i-b_ir_g)\|+\|f_g(a_i-\phi(b_i))\|+\|(a_i-\phi(b_i))f_g\|<\frac{\epsilon}{3}+\frac{\epsilon}{3}+\frac{\epsilon}{3}=\epsilon, &\|(\sum_{g\in G}f_g)a_i-a_i\|\le\|\phi((\sum_{g\in G}r_g)b_i-b_i)\|+\|(\sum_{g\in G}f_g)(a_i-\phi(b_i))\|+\|a_i-\phi(b_i)\|<\frac{\epsilon}{3}+\frac{\epsilon}{3}+\frac{\epsilon}{3}=\epsilon,
\beta_{s}\left( a,b\right) =\sum_{l=0}^{\left[ p\right] -1}\left(\sum_{j=0}^{\left[ p\right] -1-l}\left( D^{l+j}f\right) \left( x_{s}\right) \frac{\left( \pi_{1}\left( a\right) -x_{s}\right)^{\otimes j}}{j!}\right)\otimes \pi_{l+1}\left( b\right) \pi_{0}\left( b\right) =1
\sum_{j=0}^{\infty}X_j(t)(1 + s(j - M(t))) = \sum_{j=0}^{\infty}X_j(t) + s \sum_{j=0}^{\infty}j X_j(t) - s M(t) \sum_{j=0}^{\infty}X_j(t) &= N + s M(t) N - s M(t) N &= N,
[(\alpha,\beta),(\alpha',\beta')]_+ = \left(\begin{bmatrix}\alpha \beta \end{bmatrix}\!\begin{bmatrix}0 L_1 -L_1^t 0 \end{bmatrix}\!\begin{bmatrix}\alpha' \beta'\end{bmatrix},\dots,\begin{bmatrix}\alpha \beta \end{bmatrix}\!\begin{bmatrix}0 L_g -L_g^t 0 \end{bmatrix}\!\begin{bmatrix}\alpha' \beta'\end{bmatrix}\right).
\aligned\frac{d}{dt}E(Iu(t)) = -\langle i \nabla Iu, \nabla (|Iu|^{2}(Iu) - I(|u|^{2}u)) \rangle - \langle i I(|u|^{2}u), (|Iu|^{2}(Iu) - I(|u|^{2}u)) \rangle.\endaligned
\zeta(\underbrace{1, \ldots , 1}_{r-1}, k+1)=\ (-1)^{r-1}\sum_{\substack{a_{1}+\cdots +a_{r}=k^{\forall}a_p\geq 0}}\zeta(a_{1}+1, \ldots , a_{r}+1) &+\sum^{r-2}_{i=0}(-1)^{i}\zeta(\underbrace{1, \ldots , 1}_{k-1}, r-i)\cdot\zeta(\underbrace{1, \ldots ,1}_{i+1}).
U_{j}=(1-\alpha)\frac{\sum_{j}p_{j}}{\sum_{j}q_{j}}\left[ Y\frac{\overline{MH}^{2}_{j}}{\overline{q}^{2}_{j}}+Z\sum_{i}\overline{MG}_{i}\right] &V_{j}=(1-\alpha)\frac{\sum_{j}p_{j}}{\sum_{j}q_{j}}\left[ Z\frac{\overline{MG}_{j}}{\overline{q}_{j}}+Y\sum_{i}\frac{\overline{MH}^{2}_{i}}{\overline{q}_{i}}\right]
t_{\mathrm{cyc}}^{1}(\gamma_{\chi \omega}) = \chi_{\mathrm{cyc}}(g_{\chi \omega}) g_{\chi \omega}t_{\omega}^{1}(c_{\chi \omega})= u^{w_{\chi \omega}}t_{\omega}^{1}(\gamma_{\chi \omega})= u^{w_{\chi}}\gamma_{\chi},
\nabla_L(t_1,\ldots,t_{\mu}) \overset{\centerdot}{=}\begin{dcases*}(t_1-t_1^{-1})\Delta_L (t_1^2) if~$\mu=1$ \Delta_L (t_1^2,\ldots,t_\mu^2) if~$\mu >1$\,. \end{dcases*}
\Psi_{j_1,\ldots,j_k}^{s_1,\ldots,s_l}={}^1\Psi_{j_1,\ldots,j_k}^{s_1,\ldots,s_l}+{}^2\Psi_{j_1,\ldots,j_k}^{s_1,\ldots,s_l}+{}^3\Psi_{j_1,\ldots,j_k}^{s_1,\ldots,s_l}+\cdots.
(Y|ZZ')=(Y_{(1)}|Z)(Y_{(2)}|Z'), (YY'|z)=(Y|Z_{(1)})(Y'|Z_{(2)}), &(y_{\alpha}^{(n)}|z_{\beta}^{(m)})= \delta_{n\alpha,m\beta}, (y_{\alpha}^{(n)}|L_{\beta})=0, (K_{\alpha}|z_{\beta}^{(m)})=0, (K_{\alpha}|L_{\beta})= q_{\alpha\beta}.
U + \begin{bmatrix}a_{m-1,m}\ldots a_{m-1,n}\end{bmatrix}- T\begin{bmatrix}1 a_{m,m+1}\ldots a_{m,n}. \ddots \ddots \vdots . . 1 a_{n-1,n}. . . 1\end{bmatrix}=\begin{bmatrix}0 \ldots 0 \end{bmatrix}.
\varphi_{T(y;\mu)}\left( \tfrac{1}{y}R(y;\mu), \Theta(y;\mu) \right) = \frac{1}{y}\,P(y;\mu), \psi_{T(y;\mu)}\left( \tfrac{1}{y}R(y;\mu), \Theta(y;\mu) \right) = -\frac{3 \pi}{2},
\frac{\partial}{\partial t}\int_M u \Delta_p u d\mu = -p \int_M |\nabla u|^{p-2}h^{ij}\nabla_i u \nabla_j u d\mu - \int_M Z \langle \nabla \mathcal{H}, \nabla u \rangle u d\mu \displaystyle \ \ \ + (p-1) \int_M \Delta_p u \u_t d\mu + \int_M \Delta_p u \frac{\partial}{\partial t}(u d\mu).
\qquad\qquad(d^k \phi)(b_1, \ldots, b_{k+1}) = \sum_{i=1}^{s}(( \phi(b_1,\ldots,b_{2i-2},{b_{2i}+1}, b_{2i+1}+1, \ldots)-\phi(b_1,\ldots,b_{2i-2},{b_{2i-1}}, b_{2i+1}+1, \ldots)) - (\phi(b_1,\ldots,b_{2i-2},{b_{2i}},b_{2i+1}, \ldots)- \phi(b_1,\ldots,b_{2i-2},{b_{2i-1}},b_{2i+1}, \ldots))),
\psi ( p ) = \frac{1}{\sqrt{8 \pi \sqrt{\beta}}\hbar}\int_{- \infty}^{+ \infty}d{\xi}{\mathrm{~}}( 1 + \beta p^{2})^{1 / 2}{\mathrm{~}}e^{- i{\xi}\frac{\tan^{- 1}( \sqrt{\beta}p )}{\hbar \sqrt{\beta}}}{\mathrm{~}}\psi ({\xi})
\lim_{x\uparrow c_{0}^{-1}}\frac{q_{\mu_{1}\boxtimes\mu_{2}}(x)}{\left|x-c_{0}^{-1}\right|^{1/3}}=\frac{a_{1}}{\pi\sqrt[3]{\left|c_{3}\right|}}\cos\left(\frac{\theta}{3}-\frac{\pi}{6}\right),
(\alpha_1v_1+\alpha_2v_2)\tilde\otimes w-\alpha_1(v_1\tilde\otimes w)-\alpha_2(v_2\tilde\otimes w), &v\tilde\otimes (\beta_1w_1+\beta_2w_2)-\beta_1(v\tilde\otimes w_1)-\beta_2(v\tilde\otimes w_2), &(fv)\tilde\otimes w-v\tilde\otimes(fw),
\{\bar{q}_{i}^{\dot{\alpha}}, \bar{s}_{\dot{\beta}}^{j}\}\Phi = \left[ - \delta_{i}^{j}( \sigma^{\mu \nu})^{\dot{\alpha}}{}_{\dot{\beta}}m_{\mu \nu}- 2 \delta_{\dot{\beta}}^{\dot{\alpha}}\left( 2 t_{j}^{i}+ \delta_{i}^{j}( \ell + r ) \right) \right] \Phi = 0 \; .
G(\psi)(b(x^{(i)}_{t_i}))= \sum_{j\ne i}\sum_{t_j=1}^{|X_j|}a(x^{(1)}_{t_1},\dots ,x^{(i-1)}_{t_{i-1}}, x^{(i)}_{t_i},x^{(i+1)}_{t_{i+1}},\dots ,x^{(k)}_{t_k})
\begin{aligned}\left(\frac{t^N/N!}{e^t-T_{N-1}(t)}\right)^{r}\frac{{\rm d}^p}{{\rm d}x^p}e^{xt}=\left(\sum_{n=0}^\infty B_{N,n}^{(r)}(x)\frac{t^n}{n!}\right)t^p =\sum_{n=0}^\infty B_{N,n}^{(r)}(x)\frac{t^{n+p}}{n!}&=\sum_{n=p}^{\infty}B_{N,n-p}^{(r)}(x)\frac{t^{n}}{(n-p)!}&=\sum_{n=p}^{\infty}\frac{B_{N,n-p}^{(r)}(x)}{(n-p)!}t^{n}. \end{aligned}
\beta_{3}(k)=(\begin{matrix}2k\\k\end{matrix})\sum_{j=0}^{k}(\begin{matrix}2j\\j\end{matrix})^{-1}(\begin{matrix}k\\j\end{matrix})(-4)^{k-j}\sum_{m=0}^{j}(\begin{matrix}j\\m\end{matrix})^{4}=1,-6,66,-876,12786,...
\partial^{*}E \cap \{z > \bar{z}\}= \partial^{*}\left((0, \tau) + F_{\ell}\right)\cap \{z > \bar{z}\}&= \left( (0,\tau) + \partial^{*}F_{\ell}\right) \cap \left( (0,\tau) + \{z >\bar{z}\}\right) &=(0,\tau)+ \left( \partial^{*}F_{\ell}\cap \{z > \bar{z}\}\right).
q^0_-(\tau)=-\frac{\Omega\vartheta'_1(0)}{2\pi}\frac{e^{\Omega\tau\frac{\vartheta'_1}{\vartheta_1}(\Omega y)}\vartheta_1(\Omega(y-\tau))+e^{-\Omega\tau\frac{\vartheta'_1}{\vartheta_1}(\Omega y)}\vartheta_1(\Omega(y+\tau))}{\vartheta_1(\Omega\tau)\vartheta_1(\Omega y)},
+ \frac{1}{3 !}g_{0}\left[ \left( \frac{\delta W}{\delta j ( x )}\right)^{3}+ 3 \frac{\delta W}{\delta j ( x )}\, \frac{\delta^{2}W}{\delta j ( x )^{2}}+ \frac{\delta^{3}W}{\delta j ( x )^{3}}\right] - j ( x ) = 0 \, .
\varrho_N^\varphi(x) \ge \tilde \varrho_N^\varphi(x) - |\varrho_N^\varphi(x) - \tilde \varrho_N^\varphi(x)| \ge Ar^d\varphi_0 - \frac{\|\nabla\varphi\|_\infty}N \sum_{j=1}^N |x_j-y_j| \ge Ar^d\varphi_0 - \|\nabla\varphi\|_\infty A_\varphi = \frac{Ar^d\varphi_0}2\,,
\hat{C}_{0 \, 0 \, 0}^{l_{3}l_{1}l_{2}}= \rho ({l_{3}, l_{1}, l_{2}}) \sum_{k}{\frac{( - 1 )^{k}}{k !}}{\frac{( l_{1}+ k ) ! ( l_{2}+ l_{3}- k ) !}{( l_{1}- k ) ! ( l_{2}- l_{3}+ k ) ! ( l_{3}- k ) !}},
\|(1-\Delta_{x_1})^{\frac\beta2}\dots(1-\Delta_{x_{2m}})^{\frac\beta2}\rho{s_{1:2m}}\|_\infty \lesssim s_1^{-\frac12(1+\beta)}\prod_{i=1}^{2m-1}(s_{i+1}-s_i)^{-\frac12(1+2\beta)},
\sigma z^{A}= \sum_{\Delta = 1}^{2}\left( \left[ z^{A}, \gamma_{i}^{( \Delta ) a}\right]^{*}\eta_{a}^{( \Delta ) i}+ \left[ z^{A}, \gamma^{( \Delta ) a}\right]^{*}\eta_{a}^{( \Delta )}\right) ,
T_{(q,\mathcal{A})}Z^{A}\cong\{(\gamma,\mathcal{B})\in Hom(\mathcal{K},\mathcal{E})\oplus Hom(\mathcal{H},\mathcal{D})/A\cdot\mathcal{A}\mid\mathcal{B}\circ\iota=\alpha\circ\gamma\};
\mathbb{E}\left[ \log\left( X_P \right) \right] = \int_{\lambda}^\infty \frac{1}{\lambda}e^{-(x-\lambda)/\lambda}\log(x) dx &\stackrel{(a)}{=}\log\lambda + \int_{1}^\infty e^{-(u-1)}\log(u) du &\stackrel{(b)}{\leq}\log\lambda + 1
\int_{T_2}\left[\psi a \delta u_x \right]_{x=s(t)}\,dt =\int_{T_2}\Big[\psi \big(\chi_x{\delta s}- \gamma_x s'{\delta s}- \gamma{\delta s}'\big) - (a u_x)_x \delta s\Big]_{x=s(t)}\,dt + + o(\delta v).
f\left(\textbf{H}_{0,i}\right)=\frac{\exp\left(-\textrm{tr}\left(\textbf{H}_{0,i}\textbf{M}_{0}^{-1}\textbf{H}_{0,i}^{H}\right)\right)}{\pi^{N_t}\det\left(\textbf{M}_{0}\right)}, i=0,...,N_v-1,
{\cal L}_M=-\frac{1}{4}G_{\mu \nu}^a G^{\mu \nu,a}-{1\over 2}\kappa_1^2 D_{\mu}\phi^{\tilde{a}}D^{\mu}\phi^{\tilde{a}}-{1\over2}\kappa_2^4D_{[\mu}\phi^{\tilde{a}}D_{\nu]}\phi^{\tilde{b}}D^{[\mu}\phi^{\tilde{a}}D^{\nu]}\phi^{\tilde{b}}-V(\phi^4) \,
\|g(x)\|_2^{2}=\sum_{i=1}^{d}\left\langle k(x, .), g_{i}\right\rangle_{\mathcal{H}_{0}}^{2}\leq\|k(x, .)\|_{\mathcal{H}_{0}}^{2}\|g\|_{\mathcal{H}}^{2}\leq B^{2}\|g\|_{\mathcal{H}}^{2}.
S(f_t | f_\infty) \le \mathrm{e}^{-ct}S(f_0 | f_{\infty}) + \begin{cases}K\gamma^{-1}|c-\gamma|^{-1}\, \mathrm{e}^{-(c\wedge\gamma) t}\gamma \ne c\,, K c^{-1}t \, \mathrm{e}^{-ct}\gamma =c\,. \end{cases}
dist(\bar{\xi}_{d},\Gamma_{d})^{|\alpha_{d}|}\cdot\int_{\mathbb{R}^{2(d-1)}}\frac{|\partial^{\alpha_{1}}_{\bar{\xi}_{1}}\cdots\partial^{\alpha_{d}}_{\bar{\xi}_{d}}m^{\varepsilon}(\bar{\xi})|}{\prod_{i=1}^{d-1}dist(\bar{\xi}_{i},\Gamma_{i})^{2-|\alpha_{i}|}}d\bar{\xi}_{1}\cdots d\bar{\xi}_{d-1}\leq B(\varepsilon)<+\infty
g_n(x) = f(\alpha)\textbf{1}_{x=\alpha}+ \sum_{k=1}^n (c_kx + b_k)\textbf{1}_{a_k < x \leq a_{k+1}}&= f(\alpha)\textbf{1}_{x=\alpha}+\sum_{k=1}^n (c_kx + b_k)(\textbf{1}_{a_k < x}-\textbf{1}_{a_{k+1}<x}),
h(X^j_t,\bar{X}^j_t) = I_1 + I_2' + I_2'' + I_3 + I_4 + I_5 + \int_0^t \left( \frac{\partial h}{\partial x}b_2(X^j_s) + \frac{\partial h}{\partial y}b_2(\bar{X}^j_s)\right)dW^j_s.
K_{\lambda}= \frac{1}{(2\pi)^{d/2}\prod_{j=1}^{d-1}j!}\int_{W^d}e^{-\sum_{j=1}^d (\lambda_j - \bar{\lambda})z_j}\hat{h}^{(\bar{\lambda}, \ldots, \bar{\lambda})}(z) dz_1 \ldots dz_d.
1:x^{3}=0,2:x^{2}y=0,3:xy(x-y)=0,4:xyz=0, 5:z(x^{2}+yz)=0,6:x(x^{2}+yz)=0,7:x^{3}-y^{2}z=0,8:x^{3}+y^{3}-xyz=0,
\epsilon^{\mu \nu \rho \lambda \sigma}G_{\mu \mu^{\prime}}G_{\nu \nu^{\prime}}G_{\lambda \lambda^{\prime}}G_{\sigma \sigma^{\prime}}= - G_{5}\epsilon_{\mu^{\prime}\nu^{\prime}\rho^{\prime}\lambda^{\prime}\sigma^{\prime}}G_{5}^{\rho \rho^{\prime}},
z^{n+2-q}\frac{d}{dz}S_{n,q}(z)= \frac{z^m Q_{n,q}(z)}{(1-z)^{2n}}-\frac{\binom{n+q-1}{n}}{(n+1)}z^{n+1}\footnote{We use the following fact: Let $0<a\leq b$ two integers with $b-a\geq 2$. Thena primitive for $\frac{z^a}{(1-z)^b}$ is given as a linear combination of $\frac{1}{(1-z)^i}$ with $b-a-1\leq i\leq b-1$.}
{\rm div}u_{(k)}(x,t)= \int\limits_{A-1}^{0}\eta_{\frac{1}{k}}(t-\tau)\int\limits_{\mathbb{R}^{3}_{+}}\omega_{\frac{1}{k}}(x-y)\nabla\phi_{k}(y).u^{(\frac{2}{k})}(y,\tau)dyd\tau.
\partial_{\alpha}(G_{\alpha,\underline{n}}\circ F_{\alpha,n_0}^{-1})=\sum_{j=1}^k \partial_{\alpha}[G_{\alpha,n_j}\circ F_{\alpha,n_{j-1}}^{-1}\circ \dots \circ F_{\alpha,n_{0}}^{-1}] &\times \Pi_{i \neq j}G_{\alpha,n_i}\circ F_{\alpha,n_{i-1}}^{-1}\circ \dots \circ F_{\alpha,n_{0}}^{-1}
(\partial_{t}^+ e_h^{u^n},\Pi_We_h^{u^n})_{\mathcal{T}_h}+(e_h^{u^n},e_h^{\phi^n})_{\mathcal{T}_h}=(\partial_{t}^+ u_{Ih}^n-\partial_tu^n,\Pi_We_h^{u^n})_{\mathcal{T}_h}.
g \left( \begin{array}{c c}{{n_{1}}}&{{n_{2}}}\\{{m_{1}}}&{{m_{2}}}\\ \end{array}\right) = \frac{G}{2^{( n_{1}+ m_{1}+ n_{2}+ m_{2}) / 2}}\sqrt{\frac{\Gamma ( n_{1}+ m_{1}+ 1 ) \Gamma ( n_{2}+ m_{2}+ 1 )}{\Gamma ( n_{1}+ 1 ) \Gamma ( m_{1}+ 1 ) \Gamma ( n_{2}+ 1 ) \Gamma ( m_{2}+ 1 )}}.
a_{k}= \int_{k}^{R - R^{1/3}}f_{k}'(r) \, dr \sim \frac{1}{R^{1/2}}\int_{k}^{R - R^{1/3}}\sqrt{r - k}\, dr & \sim \frac{(R - R^{1/3}- k)^{3/2}}{R^{1/2}}\sim \frac{(4^{p}R^{1/3})^{3/2}}{R^{1/2}}= 2^{3p},
\frac{1}{2}\left< P^{0}[ h ]_{\mu \nu}( x ) P^{\infty}[ h ]_{\rho \sigma}( y ) \right>_{0}+ \frac{1}{2}\left< P^{\infty}[ h ]_{\mu \nu}( x ) P^{0}[ h ]_{\rho \sigma}( y ) \right>_{0}
Z^{I}= Z_{\infty}^{I}+ \sum_{a = 1}^{a = 5}{\frac{q^{I}{}_{a}}{| \vec{x}- \vec{\underline{{x}}}_{a}|}}\ , \qquad Q^{5}= \sum_{a = 1}^{a = 5}q^{5}{}_{a}= Q^{7}= \sum_{a = 1}^{a = 5}q^{7}{}_{a}= 0 \ ,
\frac{2N^{2}y_{I}^{2}-\left( 2s_{I}+Ny_{I}\right)}{N^{2}}=\frac{2N^{2}y_{I}^{2}-2N^{2}E\left[ Z_{I}^{2}\right] +Ny_{I}}{N^{2}}& =-2\left[ Z_{I}\right] +\frac{y_{I}}{N}
\dot{g}_1(t) =\alpha_\infty\left((1+t)^{2\alpha_\infty-1}-(1-t)^{2\alpha_\infty-1}-2t^{2\alpha_\infty-1}\right) &\leq \alpha_\infty \left((1+t)^{2\alpha_\infty-1}-(1-t)^{2\alpha_\infty-1}-(2t)^{2\alpha_\infty-1}\right) &=\alpha_\infty (2\alpha_\infty-1)\left(\int_{1-t}^{1+t}s^{2\alpha_\infty-2}ds-\int_{0}^{2t}s^{2\alpha_\infty-2}ds\right)<0.
t^n|K|\big(\sum_{m=k}^n{{n}\choose{m}}V(K[m], (U+V)[n-m])t^m \big) \le c_n\big(\sum_{m=i}^n{{n}\choose{m}}V(K[m], U[n-m])t^m \big)\big(\sum_{m=j}^n{{n}\choose{m}}V(K[m], V[n-m])t^m \big).
f_{d c}^{a}\tilde{f}_{a}^{r s}= \tilde{f}_{c}^{a s}f_{d a}^{r}+ \tilde{f}_{c}^{r a}f_{d a}^{s}- \tilde{f}_{d}^{a s}f_{c a}^{r}- \tilde{f}_{d}^{r a}f_{c a}^{s}.