harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
e^{-i\phi}\partial\partial|\sigma|= -\partial\bar{\partial}\psi+i|\sigma|e^{-i\phi}\partial\partial\phi+2ie^{-i\phi}\partial\phi\partial|\sigma| +|\sigma|e^{-i\phi}(\partial\phi)^2 \qquad+\frac{2\bar{\xi}e^{-i\phi}(\partial|\sigma|-i|\sigma|\partial\phi)}{1+\xi\bar{\xi}}-\frac{2\bar{\xi}^2|\sigma|e^{-i\phi}}{(1+\xi\bar{\xi})^2}.
\mathcal{H}_m=\{\,Z\in M_m(\mathbb{C})\,\mid\, \tfrac{1}{2i}(Z-Z^{*})>0\,\}\;(), \Lambda_m(\mathcal{O}_{\boldsymbol{K}})= \{\,H=(h_{jl})\in M_m(\boldsymbol{K})\,\mid\, H^{*}= H, \h_{jj}\in\mathbb{Z},\, \sqrt{-D_{\boldsymbol{K}}}\,h_{jl}\in\mathcal{O}_{\boldsymbol{K}}\,\}.
\mathcal{H}_{i}=\begin{bmatrix}Diag(H_{1, a_{i, 1}}, \dots, H_{1, a_{i, N}}) \sigma_{i}\cdot Diag(H_{2, a_{i, 1}}, \dots, H_{2, a_{i, N}}) \vdots \sigma_{i}^{r - 1}\cdot Diag(H_{r, a_{i, 1}}, \dots, H_{r, a_{i, N}}) \end{bmatrix},
{\cal L}= \frac{1}{2}\left\{\partial_{\mu}\varphi_{i}\, \partial^{\mu}\varphi_{i}+ i \bar{\psi}_{i}\gamma^{\mu}\partial_{\mu}\psi_{i}+ f_{i}f_{i}+ 2 f_{i}\, \frac{\partial{W}}{\partial \varphi_{i}}- \frac{\partial^{2}{W}}{\partial \varphi_{i}\partial \varphi_{j}}\, \bar{\psi}_{i}\psi_{j}\right\}\; .
b = \frac{\langle v_G, u_0 \rangle_{L^2}}{\langle v_G, v_G \rangle_{L^2}}, a_0 = \frac{1}{\langle v_G, v_G \rangle_{L^2}}\left[ \frac{b}{2}\left(\int_{-\infty}^{\infty}v_G(x) dx \right)^2 - \langle \partial_x^{-1}v_G, u_0 \rangle_{L^2}\right),
\lim_{b\to a}\frac{\phi(b,a)}{h(b)-h(a)}\stackrel{\eqref{eq:symmratio}}{=}\lim_{b\to a}\frac{d(b,a)}{|h(b)-h(a)|}= \lim_{b\to a}\frac{d(b,a)}{|b-a|}\frac{|b-a|}{|h(b)-h(a)|}>0,
g^\nu_{\alpha,r}(t) = \begin{dcases}\int_0^t s^\nu j_\alpha(rs)\, ds, \nu \geq\alpha+1/2 \alpha>-1/2,\nu>\alpha+1/2, -\int_t^\infty s^\nu j_\alpha(rs)\, ds, \nu < \alpha+1/2, \dfrac{\sin rt}{r}, \nu=0 \alpha=-1/2.\end{dcases}
\| ( 2\psi, V) \|_{\mathcal B_0 \times \mathcal B_1}=\| \chi \Phi(g_1, \pi_1) + (1-\chi) \Phi(g_2, \pi_2) - \Phi(g, \pi)\|_{\mathcal B_0 \times \mathcal B_1}+ \|2\psi_0 \|_{\mathcal{B}_0}R^{-1-q_0}& \le CR^{-\min(q, 1+q_0)}.
\sup_{\varpi \in \Pi(\mu_{13}, \mu_{23})}\int (f_{\mathcal{S}})_{\lambda, \ell}\, d \varpi = \mathbb{E}[Y_2] - \mathbb{E}[Y_1] + \frac{V_{\ell}/V_{\ell, XX}}{4 \lambda_\ell}, \ell \in [2].
{\sigma}({{x}}):=\left\lbrace\begin{aligned}\int_0^{{x}}{\mu}\left((0,{y}] \right) d{y},{x}\geq 0, & \int_{{x}}^0{\mu}(({y},0])d{y},{x}<0;\end{aligned}\right.{\lambda}({{x}}):=\left\lbrace\begin{aligned}\int_{(0,{x}]}{y}{\mu}(d{y}),{x}\geq 0, &\int_{({x},0]}(-{y}){\mu}(d{y}),{x}<0. \end{aligned}\right.
cl(\frac{\varpi}{\pi}x)=2e^{-\pi/4}cosx\prod_{n=1}^{\infty}\frac{1+2e^{-2n\pi}cos2x+e^{-4n\pi}}{1-2e^{-(2n-1)\pi}cos2x+e^{-(4n-2)\pi}},x\in\mathbb{C}
\sum_{q\le P}\sum_{\substack{a=1 (a,q)=1}}^q\int_{\beta\in\mathcal{B}(q,a)\setminus[-\frac{1}{2qP},\frac{1}{2qP}]}\frac{S(q,a)^3}{q^4}\mathcal{I}(\beta,q)d\beta \ll \sum_{|v|\le P}\frac{1}{1+|v|}\sum_{q\le P}\int_{\frac{1}{2qP}\le |\beta|\le\frac{1}{qP}}\frac{q_1^2q_2^{5/2}}{q^4}\big|\mathcal{I}(\beta,q)\big|d\beta.
\begin{aligned}\omega_T=\frac{d(x+\sqrt{\epsilon}y)}{x+\sqrt{\epsilon}y}\wedge\frac{d(x-\sqrt{\epsilon}y)}{x-\sqrt{\epsilon}y}&=\frac{(dx+\sqrt{\epsilon}dy)\wedge(dx -\sqrt{\epsilon}dy)}{x^2-\epsilon y^2}= \frac{-2 \sqrt{\epsilon}}{N_{E/F}(x+ \sqrt{\epsilon}y)}dx\wedge dy,\end{aligned}
\mathop{\mathtt{Min.}}\limits_{\{\mathbf{v}^{}_i\},\{p^{}_i\}}~~ \sum_{i=1}^{K}p^{}_i \mathtt{s.t.}~~^{}_i \geq \gamma^{}_i, \forall i \in \mathcal{K}& ~~ p^{}_i \geq 0, \forall i \in \mathcal{K}.
\Big [{\frac{1}{4 v^{2}}}-{\frac{f^{\prime}}{v f}}+{\frac{b}{v}}-{\frac{f^{\prime \prime}}{f}}+{\frac{2 b f^{\prime}}{f}}- b^{2}-{\frac{1}{4 v^{2}}}- m^{2}\Big ] \varphi^{( 0 )}= 0 .
[{{\omega}^{- 1}}] ={\frac{1}{a c - b{b^{\prime}}}}\left( \begin{matrix}{0}&{- c}&{0}&{{b^{\prime}}}\\{c}&{0}&{- b}&{0}\\{0}&{b}&{0}&{- a}\\{-{b^{\prime}}}&{0}&{a}&{0}\\ \end{matrix}\right)
\vec{n}(s,t_0)=\frac{\frac{\partial \vec{P}}{\partial s}\times \frac{\partial \vec{P}}{\partial t}}{\left\| \frac{\partial \vec{P}}{\partial s}\times \frac{\partial \vec{P}}{\partial t}\right\|}=\varphi_{1}(s,t_{0})\vec{T}(s)+\varphi_{2}(s,t_{0})\vec{N}(s)+\varphi_{3}(s,t_{0})\vec{B}(s),
\sum_{\Omega_{n,\mathrm{k}}}= \sum_{(N_1,\dots,N_n)\in\Omega_{n,\mathrm{k}}}, \Sigma_{\mathrm{k}}:= k_1+\cdots+k_\ell\qquad[n]_{\neq\mathrm{k}}:= \{1,\dots,n\}\setminus\{k_1,\dots,k_\ell\}.
Ric_t(U,U) =\lambda^2 Ric_t(E,E)=\lambda^2 \left( Ric_L(E,E)-t div\nabla^L_EE+t(t-2)||A'_E||^2 \right) &=Ric_L(U,U)+\frac{2t}{\lambda^2}g_L(\nabla^L_UU,\nabla^L_UU)-t div\nabla^L_UU+t(t-2)\lambda^2||A'_E||^2
\varpi_k \ge\frac{2[f(\overline{x}+\tau_kw+\frac{1}{2}\tau_k^2 u^k)\!-f(\overline{x}) \!-\tau_k df(\overline{x})(w)]}{\tau_k^2}=\frac{2[f(\overline{x}+\!\tau_k(w\!+\frac{1}{2}\tau_ku^k))\!-f(\overline{x})\!-\tau_k\langle\overline{v},w\!+\frac{1}{2}\tau_ku^k\rangle]}{\tau_k^2}+\langle\overline{v},u^k\rangle.
( \partial -{\bar{\partial}}) ( \hat{{\cal \phi}^{0}}, \hat{{\cal A}_{z}^{0}}) \sim{\frac{\sqrt{3}( - 1 + a_{i}) a_{i}^{3}\biggl [ ( b_{i}e_{i})^{2}c_{i}f_{i}( c_{i}- f_{i}){\bar{z}}+ ( - b_{i}c_{i}^{2}e_{i}^{2}f_{i}( f_{i}- b_{i}) + e_{i}b_{i}^{2}c_{i}f_{i}^{2}( c_{i}- e_{i}) ) z \biggr ] ( z -{\bar{z}})}{| z |^{4}}},
P = \left[ \begin{array}{c c c c c}{{*}}&{{1}}&{{0}}&{{0}}&{{. . .}}\\{{1}}&{{*}}&{{1}}&{{0}}&{{. . .}}\\{{0}}&{{1}}&{{*}}&{{1}}&{{. . .}}\\{{}}&{{.}}&{{.}}&{{.}}&{{}}\\{{. . .}}&{{0}}&{{0}}&{{1}}&{{*}}\\ \end{array}\right]
{\cal P}_{R}= ( q^{I}, \; n^{i}, \; m_{j}) \left( \begin{array}{c}{{l_{I}\cdot{\bf e}_{\mu}^{( R )}}}\\{{{\bar{k}}_{i}\cdot{\bf e}_{\mu}^{( R )}}}\\{{k^{j}\cdot{\bf e}_{\mu}^{( R )}}}\\ \end{array}\right){\bf e}_{\mu}^{R}.
d(a_0 \otimes \ldots \otimes a_n \otimes m)={}a_1 \otimes \ldots \otimes a_n \otimes ma_0 &+ \sum_{i=1}^{n}(-1)^{i}a_0 \otimes \ldots \otimes a_{i-2}\otimes a_{i-1}a_i \otimes a_{i+1}\otimes \ldots \otimes a_n \otimes m & -(-1)^n a_0 \otimes a_1 \otimes \ldots \otimes a_{n-1}\otimes a_{n}m.
[H_{n,m},Q_{j}]=[H_{n,m},\overline{Q}_{j}]=[N,Q_{j}]=[N,\overline{Q}_{j}] = 0 [Q_{j},{Q}_{k}]=[Q_{j}, \overline{Q}_{k}]=[\overline{Q}_{j}, \overline{Q}_{k}] =0,
\lambda_{-,1}= -2 \sqrt{\delta(1-\delta)}, E_{-,1}= \left\{\begin{pmatrix}\bar{a}_1 \sqrt{\frac{\delta}{\delta-1}}-\bar{b}_1 \sqrt{\frac{\delta}{\delta-1}}0 1\end{pmatrix},\begin{pmatrix}-\bar{b}_1 \sqrt{\frac{\delta}{\delta-1}}-\bar{a}_1 \sqrt{\frac{\delta}{\delta-1}}1 0\end{pmatrix}\right\}.
\begin{array}{ll}\left|\big(f\circ\varphi\big)^{(n)}(x)\right| \geq \left|f^{(n)}\big(\varphi(x)\big)\right|\delta^n-\lambda_n\sum_{k=0}^{n-1}\|f^{(k)}\|_{\infty}= \left|f^{(n)}\big(\varphi(x)\big)\right|\delta^n-\lambda_n\|f\|_{n-1}.\end{array}
\sum_{i=1}^m \int_t^{\infty}\int\limits_{\{| \nabla_i P_s f | > M \|\nabla_i f \|_1 \}}|\nabla_i P_s f|^2 d\mu \, ds \leq \frac{e^{-ct}}{M^{\frac{1}{q(t)}}}\sum_{i=1}^m \int_t^{\infty}\|\nabla_i P_{s/2}f \|_2^2 ds.
\tilde \varphi_{n,i}(x) = \varphi^K_{n,i}(x) + \sum_{k=1}^K (\tilde Q_{n,i; k,0}(x)\varphi^K_{k,0}(x) - \tilde Q_{n,i; k,1}(x)\varphi^K_{k,1}(x)), n = \overline{1, K}.
|\alpha_{ij}'| \leq \kappa \big| h_ih_{n+1}^{-\lfloor \alpha_{i\pi}/d\rfloor}\big|^{c}= \kappa \left( |h_i| + \left\lfloor\frac{|\alpha_{i\pi}|}{d}\right\rfloor |h_{n+1}| \right)^{c}&\leq \kappa \big( mN + N\cdot mnN^2 \big)^{c}\leq \kappa\big( 2mnN^3 \big)^{c}= \kappa(2m)^{c}\cdot n^{c}N^{3c}.
\int d^{1 0}\hat{x}\sqrt{| \hat{\jmath}|}\, \left[{\textstyle \frac{1}{2 \cdot 5 !}}\left( \hat{G}_{( 5 )}\right)^{2}+{\textstyle \frac{1}{2 \cdot 5 !}}\left( \hat{\overline{{G}}}_{( 5 )}\right)^{2}\right] \, .
ra(I') \stackrel{(\ref{a})}{\geq}\sum_{I'\subset J, \atop{|J| = i}}{a(J)}\geq\sum_{I'\subset J, J\neq I \atop{|J| = i}}{c_{i+1}b(J\cup I)n}\stackrel{(\ref{b})}{\geq}c_{i+1}b(I)n,
a-a \leq -K_7 \underset{k \to \infty}{\lim}\| F(u_k)- v\|^{p}+ 0 \leq -\frac{K_7}{(C_F)^{\frac{2p}{1+\epsilon}}}\underset{k \to \infty}{\lim}\gamma_k^{\frac{2}{1+\epsilon}}= -K_8 \underset{k \to \infty}{\lim}\gamma_k^{\frac{2}{1+\epsilon}},
\mathrm{E}_{\theta}\hat{\pi}(I|X) =\mathrm{P}_{\theta}(\hat{I}= I)\le\mathrm{P}_{\theta}\Big(\frac{\tilde{\pi}(I|X)}{\tilde{\pi}(I_0|X)}\ge 1\Big)\le \mathrm{E}_{\theta}\Big[\frac{\tilde{\pi}(I|X)}{\tilde{\pi}(I_0|X)}\Big]^h=\mathrm{E}_{\theta}\Big(\frac{\lambda_I \mathrm{P}_{X,I}}{\lambda_{I_0}\mathrm{P}_{X,I_0}}\Big)^h,
\psi_{\lambda/\mu}= \prod_{1\leq i\leq j\leq \ell(\mu)}\dfrac{f(q^{\mu_i - \mu_j}t^{j-i}) f(q^{\lambda_i - \lambda_{j+1}}t^{j-i})}{f(q^{\lambda_i - \mu_j}t^{j-i}) f(q^{\mu_i - \lambda_{j+1}}t^{j-i})}
W_{\vec{k}}^{( c )}= \sum_{n \ge 1}{\frac{( - 1 )^{n - 1}}{n}}\sum_{\vec{k}_{1}, \cdots , \vec{k}_{n}}\delta_{\sum_{i = 1}^{n}\vec{k}_{i}, \vec{k}}\prod_{i = 1}^{n}{\frac{W_{\vec{k}_{i}}}{z_{\vec{k}_{i}}}}.
\delta S_2 =\sum_{\scriptstyle i=1 \atop \scriptstyle i\neq k}^n(- 1)\frac{b_{ki}}{2d_i}\log (1+y_k^{\varepsilon})\delta u_i=\sum_{\scriptstyle i=1 \atop \scriptstyle i\neq k}^n(\tilde{p}_i-p_i)\delta u_i.
\mbox{Prod}_{\mathbf{A}}\left(\mathbf{x}^{\top^{2}},\mathbf{y}^{\top^{1}},\mathbf{z}^{\top^{0}}\right)=\sum_{0\le i<m}\,\sum_{0\le j<n}\,\sum_{0\le k<p}a_{ijk}\,x_{i}\,y_{j}\,z_{k}.
\sum_{j=0}^{4}r^j\left| \left(\nabla^G\right)^j\left((\Phi_s\circ \tilde{F}_s)^* g- G\right)\right|_G +r^j\left|(\nabla^G)^j(\tilde{F}_{s}^{*}g_0(s)-G)\right|_G < \delta
\prod_{j=1}^{n-1}(r_{j}+1) \leq h_{n}\leq \frac{1}{\mu(C_{1})}\prod_{j=1}^{n-1}(r_{j}+1) \quad\quad\quad\quad\quad\quad\quad\frac{1}{h_{n}}\prod \limits_{j=1}^{n-1}(r_{j}+1) \to \mu(C_{1}).
G'(y) + \frac{p \kappa^{1/p}}{1-y}\left[ \varphi(y,a^*)^{(p-1)/p}- \varphi(y,a_*)^{(p-1)/p}\right] & = p\frac{G(y)}{1-y}+ \frac{p}{p-1}\left[ (a^*)^{2-p}- (a_*)^{2-p}\right] (1-y)^{1-p}
| S / X |^K \ll ( 1/H_1)^{(K/2)}+ \cdots + 1/H_k + 1/ ( H_1 \cdots H_k X ) \sum_{h_1=1}^{H_1}\cdots \sum_{h_k=1}^{H_k}\frac{1}{\lambda h_1 \cdots h_k}\ll (1/H_1)^{(K/2)}+ \cdots + 1/ H_k + ( \log X )^k /( \lambda X H_1 \cdots H_k).
u_k(r, f) := \begin{cases}\frac{-1}{2b_k}{\big(}r^{\gamma^+_k}\int_r^1 s^{-\gamma_k^+ +1}f_k(s)\ds - r^{\gamma^-_k}\int_r^1 s^{-\gamma_k^- +1}f_k(s)\ds{\big)}, b_k\neq 0; r^{-\frac{n-2}{2}}{\big(}\log r\int_0^r s^{\frac{n}{2}}f_k(s)\ds - \int_0^r s^{\frac{n}{2}}\log sf_k(s)\ds{\big)}, b_k = 0. \end{cases}
\begin{array}{r c l}q(t) = \frac{1}{\sqrt{2m\omega}}\left[\alpha\,e^{-i\omega(t-t_0)}+\alpha^*\,e^{i\omega(t-t_0)}\right]\ , p(t) = -\frac{im\omega}{\sqrt{2m\omega}}\left[\alpha\,e^{-i\omega(t-t_0)}-\alpha^*\,e^{i\omega(t-t_0)}\right]\ .\end{array}
{_{n+1}}F_n\left(\begin{array}{cccc}A_0, A_1, \ldots, A_n & B_1, \ldots, B_n\end{array}\mid x \right):=\frac{q}{q-1}\sum_{\chi\in \widehat{\mathbb{F}_q^\times}}{A_0\chi \choose \chi}{A_1\chi \choose B_1\chi}\cdots{A_n\chi \choose B_n\chi}\chi(x).
\begin{aligned}\Phi_3(z,s,a) =\frac1{a^s}+\binom32\sum_{m_2=1}^\infty \frac{z^{m_2}}{(m_2+a)^{s}}&+\binom31\sum_{m_2,m_3=1}^\infty\frac{z^{m_2+m_3}}{(m_2+m_3+a)^{s}}&+\binom30\sum_{m_1,m_2,m_3=1}^\infty\frac{z^{m_1+m_2+m_3}}{(m_1+m_2+m_3+a)^{s}}.\end{aligned}
Z_{\scriptscriptstyle CS}(J) = e^{-\frac{i\pi}{4}\eta (0, \hskip 3pt \ast d_{\scriptscriptstyle 1})}\tau_{\scriptscriptstyle RS}(M)^{\scriptscriptstyle 1/2}e^{i \langle J, \hskip 3pt \frac{1}{\ast d_{\scriptscriptstyle 1}}\hskip4pt J \rangle}
F (z) \; = \; F (0) \, + \, \sum_{m=1}^{\infty}\,\left\{\frac{b_m}{z - a_m}+ \frac{b_m}{a_m}\right\}\, + \,\sum_{m=1}^{\infty}\,\left\{\frac{b_{-m}}{z - a_{-m}}+ \frac{b_{-m}}{a_{-m}}\right\}\, .
R(T+d)-R(d) =\int_{T}^{T+d}\frac{d}{dt}R(t) ~ dt\geq\int_{T}^{T+d}[ae^{-4K(t-T)}-b] ~ dt, &=[-\frac{a}{4K}e^{-4K(t-T)}-bt]\bigg|_{t=T}^{t=T+d}=\frac{a}{4K}\bigg(1-e^{-4Kd}\bigg)-bd.
\xi^{\mu}{}_{( 1 , 2 )}= \xi^{\nu}{}_{( 1 )}\partial_{\nu}\xi^{\mu}{}_{( 2 )}- ( \partial_{\nu}\xi^{\mu}{}_{( 1 )}) \xi^{\nu}{}_{( 2 )}, \zeta^{'}= ( \partial_{\mu}\zeta ) \xi^{\mu}. \nonumber
P_{\mathcal{H}_\chi^{\rm stiff(soft)}}\Pi_\chi^{\rm stiff(soft)}= \Pi_\chi^{\rm stiff(soft)}\widehat{P}_\chi , \bigl(\Pi_\chi^{\rm stiff(soft)}\bigr)^* P_{\mathcal{H}_\chi^{\rm stiff(soft)}}= \widehat{P}_\chi \bigl(\Pi_\chi^{\rm stiff(soft)}\bigr)^*,
\int_{X_j}|u_{m,m',\epsilon,j}|^2_{h_{m'}}e^{v_\epsilon(\Psi_m)-\delta M_{\eta_m}}c(-v_\epsilon(\Psi_m))+\int_{X_j}|h_{m,m',\epsilon,j}|^2_{h_{m'}}e^{-\phi-\delta M_{\eta_m}}\le \int_{X_j}v''_{\epsilon}(\Psi_m)|fF^{1+\delta}|^2_{h_{m'}}e^{-\phi-\delta M_{\eta_m}}<+\infty.
B_{i_1}\cap B_{i_2}\cap \dots \cap B_{i_m}= \lbrace (x_{1}, x_{2}, \dots , x_{k}) \mid x_{1}+ x_{2}+ \dots + x_{k}= n,\, x_{i_l}\geq a_{i_l}+ 1\, , l = 1, \dots, m \rbrace \,.
S_{\mathrm{e m}}= -{\frac{1}{2}}\int \! d^{2}x \, \Bigl ( \partial_{0}\phi \partial_{1}\tilde{\phi}+ \partial_{0}\tilde{\phi}\partial_{1}\phi + \bigl ( \partial_{1}\phi \bigr )^{2}+ \bigl ( \partial_{1}\tilde{\phi}\bigr )^{2}+ F_{e}\, \cos \bigl ( 2 g_{\mathrm{e f f}}\phi \bigr ) + F_{m}\, \cos \bigl ( 2 \pi \tilde{\phi}/ g_{\mathrm{e f f}}\bigr ) \Bigr )
f_{\lambda}(x_1,\dots,x_n)=\prod_{i=1}^{r-1}\left(1- q^{i}\prod_{j \leq i}t^{m_j(\lambda)}\right)\sum_{\mu\in S_n\cdot (\lambda^{*})^{+}}T_{\lambda,\mu}(x_1,\dots,x_n) f_{\mu}(x_1,\dots,x_n),
L_{e f f}= \frac{1}{2}m^{2}( t ) \Phi^{2}+ \frac{\lambda ( t )}{4 !}\varphi^{4}+ \frac{1}{2}\xi ( t ) R \varphi^{2}+ \Lambda_{0}( t ) + \kappa ( t ) R + \widetilde{a}_{1}( t ) R^{2},
{\mathcal{L}}=-\frac{1}{8}\int{d^{2}\Theta2{\mathcal{E}}\left(\bar{\mathcal{D}}^{2}-8R\right)f}-\frac{1}{8}\int{d^{2}\bar\Theta2\bar{\mathcal{E}}\left({\mathcal{D}}^{2}-8R^{\dagger}\right)f}.
\sum\limits_{\substack{q\in \mathbb{N}q\le Q}}\sum\limits_{\substack{r \bmod{q}(r,q)=1}}\left|\sum\limits_{\substack{n\in \mathbb{Z}[i] \mathcal{N}(n)\le N}}a_n \cdot e\left(\mbox{\rm Tr}\left(\frac{nr}{2q}\right)\right)\right|^2 \ll \left(Q^3+Q^2\sqrt{N}+N\log 2Q\right)\sum\limits_{\substack{n\in \mathbb{Z}[i] \mathcal{N}(n)\le N}}|a_n|^2.
W_h^{(pq)}= W_h^{(pq)}\left({\bf R}_1^{(pq)}, \dots,{\bf R}_n^{(pq)}\right) ={n \choose 2k}^{-1}\sum_{{\bf i}\in \mathcal{P}(n, 2k)}h^W\left({\bf R}^{(pq)}_{{\bf i}}\right),
\overset{1}{R}(X,Y,AZ,AW) = \overset{1}{R}(AX,AY,Z,W), \overset{1}{R}(X,AY,AZ,W) = \overset{1}{R}(AX,Y,Z,AW), \overset{1}{R}(AX,AY,AZ,AW) = \overset{1}{R}(X,Y,Z,W).
{\cal L}= - \frac{1}{4}G_{\mu \nu}^{a}G^{a \mu \nu}, G_{\mu \nu}^{a}= \partial_{\mu}A_{\nu}^{a}- \partial_{\nu}A_{\mu}^{a}+ g f^{a b c}A_{\mu}^{b}A_{\nu}^{c}
i \hbar{\frac{\partial \Phi}{\partial t}}( 1 , \cdots , N ) ={\frac{1}{2 m}}\sum_{p = 1}^{N}\left\{i \hbar \partial_{i}^{( p )}-{\frac{e}{c}}\sum_{r = 1 ( \ne p )}^{N}{\cal A}_{i}( p , r ) \right\}^{2}\Phi ( 1 , \cdots , N ) \, .
\operatorname{Cov}( \overline{X}_1^*, \overline{X}_2^* ) = \frac{1}{\sqrt{n_1n_2}}\sum_{i=1}^{n_1}\sum_{j=1}^{n_2}\operatorname{Cor}( X_{1i}, X_{2j}) = \sqrt{\frac{n_1}{n_2}}\rho',
\frac{1}{(n-2)! \sqrt{1 - \langle n_j,\xi\rangle^2}}\sum_{\gamma=1}^{n-2}\det\Big( X_{i_1}(\xi), \ldots, \widetilde{X}_{i_\gamma}(\xi), \ldots, X_{i_{n-2}}(\xi), n_j,\xi \Big) ,
\frac{\phi'(u)}{u}= \frac{1}{\ln 2}\cdot \Bigl(1 + \sum_{n \ge 1}\frac{v^{2n}}{2n+1}\Bigr) \phi''(u) = \frac{1}{\ln 2}\cdot \Bigl(\frac{1}{3}+ \sum_{n \ge 1}\frac{v^{2n}}{2n+3}\Bigr)
A_{2}= \mathrm{C h}_{0 , 0}^{N S , ( 2 )}, \; \; B_{2}= \mathrm{C h}_{2 , 2}^{N S , ( 2 )}, \; \; C_{2}= \mathrm{C h}_{2 , 0}^{N S , ( 2 )}, \; \; D_{2}= \mathrm{C h}_{2 , - 2}^{N S , ( 2 )}, \; \; E_{2}= \mathrm{C h}_{1 , 1}^{N S , ( 2 )}, \; \; F_{2}= \mathrm{C h}_{1 , - 1}^{N S , ( 2 )}.
H_{c}= \int d^{3}x \left[ \frac{1}{2}\pi_{i}^{2}+ \frac{1}{4}F_{i j}F^{i j}+ \frac{1}{2}m^{2}( ( A^{0})^{2}+ ( A^{i})^{2}) - A_{0}\Omega_{2}\right] ,
\begin{array}{l l}{\begin{array}{l l}{\frac{d}{d r}\sigma ( r ) = 0 ,}{r = 0 ,}{\sigma ( r ) = 0 ,}{r \rightarrow \infty ,}\end{array}}{\begin{array}{l l}{A ( r ) \neq 0 ,}{r \rightarrow \infty ,}{A ( r ) = 1 ,}{r = 0 .}\end{array}}\end{array}
c^{(0)}=\frac 16~,~c^{(1/2)}=-{N^{(1/2)}\over 12}~,~c^{(1)}={N^{(1)}\over 6}-1~,~c^{(3/2)}=-{N^{(3/2)}\over 12}~,~c^{(2)}={N^{(2)}\over 6}-(d+2)~.
\begin{array}{l c r}{{P ( W_{2}) = ( y^{2}+ x^{3}+ z^{6}+ \mu x y z ) +}}\\{{w ( a_{0}z^{n + 3}+ a_{1}y z^{n}+ a_{n - 1}z^{{\frac{7}{2}}}x^{{\frac{( n - 6 )}{2}}}y^{{\frac{1}{2}}}) + w^{2}\sum_{i = 0}^{n - 4}a_{i + 1}z^{2 ( n - i )}x^{i}.}}\\ \end{array}
I_2 = \theta \left\{\beta (u(x_\sigma ,t_\sigma ))-\beta (u(y_\sigma ,t_\sigma )) \right\}h(L_\alpha (t_\sigma )p_\sigma +2\alpha x_\sigma ) & + \theta \beta (u(y_\sigma ,t_\sigma )) \cdot \{h(L_\alpha (t_\sigma )p_\sigma +2\alpha x_\sigma )-h(L_\alpha (t_\sigma )p_\sigma -2\alpha y_\sigma ) \}.
\begin{array}{c}\rho_i=g_i\delta^4\left( x-z_i\right),\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\rho_{ij}=g_{ij}\delta^4\left( x-z_{ij}\right) , 1\leq j\leq l. \end{array}
F_{ab}=[\begin{matrix}0&\frac{1}{c}E_{x}&\frac{1}{c}E_{y}&\frac{1}{c}E_{z}\\ -\frac{1}{c}E_{x}&0&-B_{z}&B_{y}\\ -\frac{1}{c}E_{y}&B_{z}&0&-B_{x}\\ -\frac{1}{c}E_{z}&-B_{y}&B_{x}&0\end{matrix}]
\begin{array}{rcl}x_1(t_*^+) = x_3(t_*^+) = x_1(t_*^-)=x_3(t_*^-) x_2(t_*^+) = x_4(t_*^+) = \frac{m_1 x_2(t_*^-) + m_2 x_4(t_*^-)}{m}\end{array}
P T_u f (x) =~ \sum_{y\in E}p(x,y) f(y \star u) ~\leq~ \sum_{y\in E}p(x \star u,y\star u)f(y \star u) \leq~ \sum_{z\in E}p(x\star u, z)f(z) ~\leq~ f(x\star u) ~=~T_uf (x), \forall x\in E.
\tilde\chi_{I_\bullet}=\tilde\chi_{\underline{w}_{I_\bullet}}+q\sum_{\underline{y}<\underline{w}_{I_\bullet}}a_{\underline{y},\underline{w}_{I_\bullet}}\tilde\chi_{\underline{y}}\tilde\rho_{I_\bullet}=\frac{\tilde\chi_{I_\bullet}}{\pi(I)},
\widehat{T}= \begin{pmatrix}0 \frac{|\lambda_{1}|}{2}0 0 0 0 \frac{|\lambda_{1}|+|\lambda_{2}|}{2}0 0 0 0 \frac{|\lambda_{2}|+|\lambda_{3}|}{2}0 0 0 0 \end{pmatrix}
\partial_r L_k^* h = L_{1+k}\partial_r h + \left[ \partial_r \left( \frac{\partial_r (\bar{\rho}^{\gamma+k(\gamma-1)}d^{1+k})}{\bar{\rho}^{1+k(\gamma-1)}d^k}\right) - \frac{2}{r}\frac{\partial_r (\bar{\rho}^{\gamma+k(\gamma-1)}d^{1+k})}{\bar{\rho}^{1+k(\gamma-1)}d^k}\right] \partial_r h &= L_{1+k}\partial_r h + Q_{-}\partial_r h.
\int_{\Delta(0,2)}|f(\zeta)|^2\,dm(\zeta)=2\pi\sum_{j=0}^\infty |a_j|^2\int_0^2r^{2j+1}\,dr=2\pi\sum_{j=0}^\infty\frac{2^{2j+2}}{2j+2}\,|a_j|^2\,.
x_{i}\mathcal{D}_{i}g(x) =x_{i}\partial_{i}g(x) +\kappa\sum_{j\neq i}\frac{x_{i}}{x_{i}-x_{j}}\tau((i,j))( g(x) -g( x(i,j))), ( x_{i}\mathcal{D}_{i}f(x) )^{\ast}=-x_{i}\partial_{i}f(x)^{\ast}+\kappa\sum_{j\neq i}\frac{x_{j}}{x_{j}-x_{i}}( f(x)^{\ast}-f( x(i,j))^{\ast}) \tau((i,j)) .
A \, \left\|\,f,\, a_{\,2},\, \cdots,\, a_{\,n}\,\right\|^{\,2}\,\leq\, \sum\limits_{i \,=\, 1}^{\,\infty}\,\left<\,f,\, f_{\,i}\,|\, a_{\,2},\, \cdots,\, a_{\,n}\,\right>\,\left<\,C\,f_{\,i},\, f \,|\, a_{\,2},\, \cdots,\, a_{\,n}\,\right> &\,\leq\, B\, \left\|\,f,\, a_{\,2},\, \cdots,\, a_{\,n}\,\right\|^{\,2}\; \;\forall\; f \,\in\, H_{F}.
T_{2,k}=\sum_{v\leqslant\frac{(k-1)K B}{\alpha d N}}\mathtt{1}_{\{\alpha v\}\in I_k}=\frac{(\varepsilon-\eta) k(k-1)K^2}{\alpha^2 d^2}\frac{B^{2-\frac{1}{r}}}{N^2}+\frac{\eta (k-1)K^2}{\alpha^2 d^2}\frac{B^{2-\frac{1}{r}}}{N^2}+O\left(\frac{k^\sigma K^\sigma B^\sigma}{d^\sigma N^\sigma}\right).
\langle V_{3}| ( \varphi ( 0 ) | k_{1}\rangle_{1}) \otimes | k_{2}\rangle_{2}\otimes | k_{3}\rangle_{3}\sim \langle \varphi ( z_{1}) e^{i k_{1}X ( z_{1})}e^{i k_{2}X ( z_{2})}e^{i k_{3}X ( z_{3})}\rangle \neq 0 .
\frac{1}{2q}\frac{1}{dt}\int (u_x\varphi_N)^{2q}dx=-\frac{1}{2}\int (uv-u_xv_x)u_{xx}(u_x\varphi_N)^{2q-1}\varphi_Ndx &\phantom{\frac{1}{2q}}+\int \partial_x^2(G\ast F_2)(u_x\varphi_N)^{2q-1}\varphi_N dx+\int \partial_x(G\ast F_1)(u_x\varphi_N)^{2q-1}\varphi_N dx &-\frac{1}{2}\int M u_x(u_x\varphi_N)^{2q-1}\varphi_N dx.
{\cal L}= \left( \partial_{\mu}\phi \right)^{\dagger}\left( \partial^{\mu}\phi \right) - ( m^{2}- k^{2}X^{2}) \phi^{\dagger}\phi - i k X_{\mu}[ \phi \partial^{\mu}\phi^{\dagger}- \phi^{\dagger}\partial^{\mu}\phi ] - \frac{1}{1 2}H_{\mu \nu \rho}H^{\mu \nu \rho},
\int_{-r_0}^{r_0}\frac 1{(\delta^2 + s^2)^{3/2}}ds= \frac 2{\delta^2}\left( 1+ \left(\frac \delta{r_0}\right)^2\right)^{-1/2}= \frac 2{\delta^2}+ \mathcal{O}\left(r_0^{-2}\right)
L_{b}={\frac{1}{g^{2}}}\int_{\widetilde{T}^{d}\times R}d^{d}\sigma d t \t r \left\{-{\frac{1}{4}}F^{2}+{\frac{1}{2}}( D_{\alpha}X^{i})^{2}-{\frac{1}{4}}( [ X^{i}, X^{j}] )^{2}\right\}\quad ,
\begin{array}{cc}Y_{0}-Y_{1}+Y_{2}-Y_{6}=0, Y_{0}-Y_{6}+Y_{7}-Y_{10}=0, Y_{2}-Y_{3}-Y_{7}+Y_{8}=0, Y_{2}-Y_{3}-Y_{6}+Y_{9}=0, Y_{3}-Y_{4}+Y_{6}+Y_{10}=0.\end{array}
\dot{P}_{\alpha}( \theta ) = \sum_{l = 0}^{N}\left( \dot{p}_{\alpha}( l + 1 ) - \dot{p}_{\alpha}( l ) \right) \Theta \left( \theta - \theta_{l}\right) - \sum_{l = 0}^{N}\left( p_{\alpha}( l + 1 ) - p_{\alpha}( l ) \right) \delta ( \theta - \theta_{l}) \dot{\theta_{l}}
\int_X\,h^{E_p}(s(x),P_{E_p}(x,y).v)\,d\nu_X(x) =\int_X\,h^{E_p}(P_{E_p}(y,x)s(x),v)\,d\nu_X(x) &=h^{E_p}(P_{E_p}\,s(y),v)=h^{E_p}(s(y),v)\,.
K_{\mathbb{H}}(t;z,w)= \frac{\sqrt{2}e^{- t\slash 4}}{(4\pi t)^{3\slash 2}}\int_{d_{\mathbb{H}}(z,w)}^{\infty}\frac{re^{-r^{2}\slash 4t}}{\sqrt{\cosh(r)-\cosh (d_{\mathbb{H}}(z,w))}}dr,
\partial_\tau \Phi -\L_{\rm ss}\Phi + \Phi \cdot \nabla (\Phi N) + \div ( \tilde{N}\Psi \otimes \Phi + \tilde{N}\Phi \otimes \Psi) + \bar{U}\cdot \nabla \Psi + \Psi \cdot \nabla \bar{U}+ \nabla \Pi =0\, ,
\mathbb{E}[Z^{i}(n+1, k) | \mathcal{F}_{n}] ~~~~= (Z^{i}(n, k) + 1) \frac{Z^{i}(n, k) + Z^{i}(n, k-1)}{2n}+ Z^{i}(n, k) \left(1 -\frac{Z^{i}(n, k) + Z^{i}(n, k-1)}{2n}\right) ~~~~ = \frac{2n+1}{2n}Z^{i}(n, k) + \frac{1}{2n}Z^{i}(n, k-1),
k_{(T,x)}=\limsup_{z\rightarrow 0}\frac{\ln\|(z-T)^{-1}x\|_p}{\ln\|(z-T)^{-1}\|}=\limsup_{z\rightarrow 0}\frac{\ln\|(z-A)^{-1}x_2\|_p}{\ln\|(z-T)^{-1}\|}&\approx\limsup_{z\rightarrow 0}\frac{\ln\|(z-A)^{-1}x_2\|_p}{\ln\|(z-A)^{-1}\|^2}=\frac{1}{2}.
\bigcup_{i=0}^{2k}V_u(C_i) = \bigcup_{i=0}^{2k}\{u_{i+jt}\mid 0 \leq j < p\}= \bigcup_{i=0}^{2k}\{u_{i+jq(2k+1)}\mid 0 \leq j < p\}= \bigcup_{i=0}^{2k}\{u_{i+j(2k+1)}\mid 0 \leq j < p\}= \bigcup_{i=0}^{2k}\{u_m \mid m \in \mathbb{Z}_n, m \equiv i\ (\ 2k+1)\}= \{u_m \mid m \in \mathbb{Z}_n\}
f_n(t) = \begin{cases}\exp(-n^2) 0 \le t \le n^{-1}; \exp(-n^2) \cdot (2- nt) + f(2n^{-1}) \cdot (nt - 1) n^{-1}\le t \le 2n^{-1}; f(t) 2n^{-1}\le t \le 1- n^{-1}; f(1 - n^{-1}) \cdot (n - nt) + \exp(-n^2) \cdot (nt - n + 1) 1- n^{-1}\le t \le 1.\end{cases}
U_{n}(2x-1):=\frac{(n+1)(2n)!!}{(2n+1)!!}\sum_{k=0}^{n}(-1)^{n+1}\frac{\binom{n+\frac{1}{2}}{k}\binom{n+\frac{1}{2}}{n-k}}{\binom{n}{k}}B_{k}^{n}(x).
\frac 12{g_2^2 v^2 \over 4}(2 W^\mu_+ W_{\mu -}+ W^\mu_3 W_{\mu 3})+{g_2 g_1 v^2 \over 4}W^\mu_3 W_{\mu4}+\frac 12{g_1^2 v^2 \over 4}W^\mu_4 W_{\mu 4}
\mathrm{d}s_{f}^{2}= \frac{r + 2 m}{r}\Big [ \mathrm{d}r^{2}+ \frac{m^{2}r^{2}}{( r + m )^{2}}\left( \mathrm{d}\theta^{2}+ \sin^{2}\theta \, \mathrm{d}\phi^{2}\right) \Big ] + \frac{4 r m^{2}}{r + 2 m}\Big ( \mathrm{d}\psi + \cos \theta \, \mathrm{d}\phi \Big )^{2}
c_{\frac{n+2}{2}}= \left(x_{\frac{n}{2}}x_{\frac{n+2}{2}}\right)x_{\frac{n+2}{2}}= x_{\frac{n}{2}}\left(x_{\frac{n+2}{2}}x_{\frac{n+2}{2}}\right) = (-1)^{\frac{n+2}{2}}x_{\frac{n}{2}}z = (-1)^{\frac{n+2}{2}}b_{\frac{n}{2}}= 0,
\mathrm{S}= \int d^{4}x d \tau \left\{\psi^{*}( i \partial_{\tau}+ e_{0}a_{5}) \psi - \frac{1}{2 M}\psi^{*}( - i \partial_{\mu}- e_{0}a_{\mu}) ( - i \partial^{\mu}- e_{0}a^{\mu}) \psi - \frac{\lambda}{4}f_{\alpha \beta}f^{\alpha \beta}\right\}
\|U^j\|^2 - \|U^{j-1}\|^2 + \|U^j-U^{j-1}\|^2 = 2h \big\langle f(U^j), U^j \big\rangle + 2 \big\langle g(U^{j-1})\Delta_h W^{j}, U^j - U^{j-1}\big\rangle.
T_i^{-1}(u)(V\otimes v_\lambda)= Z_i^{-1}(T_i^{-1}\otimes T_i^{-1})\Delta(u)(V\otimes T_iv_\lambda) = Z_i^{-1}(T_i^{-1}\otimes T_i^{-1})\Delta(u)(V\otimes f_i^{\langle\lambda,h_i\rangle}v_\lambda).
\max \left\{\lfloor \frac{1-T}{t}\rfloor,\lfloor \frac{1-T}{T-t}\rfloor\right\}+2 \leq \frac{1-T}{t}+2\leq \frac{1}{t}=\max \left\{\frac{1}{t},\frac{1}{T-t}\right\}.
\begin{aligned}I_2 =\int_0^{t/2}\int_{|x-z|>|x-y|/2}h(t-s,x,z) f_0(s,z,y)\,dz\,ds &\qquad+\int_{t/2}^t\int_{|x-z|>|x-y|/2}h(t-s,x,z) f_0(s,z,y)\,dz\,ds =I_{21}+I_{22}.\end{aligned}
\sum_{k=0}^{\infty}\Theta_{y,k}(Q)t^{k}=\sum_{k=0}^{\infty}\frac{\Theta_{y,k}(H(zx))}{\Theta_{y,k}(H(x))\Theta_{y,k}(1-p_{1}x)}t^{k}=\sum_{k=0}^{\infty}\frac{(1+zxy)^{k}}{(1-zx)^{k}}\frac{(1-x)^{k}}{(1+xy)^{k}}\frac{t^{k}}{1-k(1+y)x}
\tau^G_\varphi(a_0,\ldots,a_k):=\int_{G^{\times k}}a_0((g_1\cdots g_k)^{-1})a_1(g_1)\cdots a_k(g_k)\varphi(e,g_1,g_1g_2,\ldots,g_1\cdots g_k)dg_1\cdots dg_k.
\psi(u) =\frac{qe^{u\sigma}}{\Gamma(1+2\sigma)^2}{}_0 F_3\left(\begin{array}{c}1+2\sigma\ \ 1+2\sigma\ \ 1\end{array}; e^{u}\right) &+\frac{q^{-1}e^{-u\sigma}}{\Gamma(1-2\sigma)^2}{}_0 F_3\left(\begin{array}{c}1-2\sigma\ \ 1-2\sigma\ \ 1\end{array}; e^{u}\right)\ .