harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g_2^{-1}(\frac{\partial g_2}{\partial\theta})=\Pi(\sum_{n=1}^{\infty}Ad(g^{(n-1)})^{-1}\left(\iota_{\tau_n}(g(\zeta_n))^{-1}\frac{\partial}{\partial\theta}\iota_{\tau_n}(g(\zeta_n))\right))
\Psi_{a,b}[n](y^{n'}) = \exp\Biggl(\,\sum_{j=1}^{\infty}\frac{(-1)^{j+1}}{j[ja]_q}q^{jb}X_{jn}\Biggr)(y^{n'}) &=y^{n'}\exp\Biggl( \sum_{j=1}^{\infty}\frac{q^{2j\{n,n'\}}-1}{q^{2ja}-1}\frac{(-1)^{j+1}}{j}q^{ja}q^{jb}y^{jn}\Biggr).
\sum_{k=0}^{p-1}(\epsilon_{a,k}p + k) C_{\tau,(a-1-k,b)}= -p C_\tau + \sum_{k=0}^{p-1}k C_{\tau,(a-1-k,b)}+ p \sum_{k=0}^{a-1}C_{\tau,(a-1-k,b)}&= -p C_\tau + \sum_{k=-a}^{p-a-1}(k+a) C_{\tau,(-k-1,b)}+ p \sum_{k=p-a}^{p-1}C_{\tau,(-k-1,b)}&= (-p+a) C_\tau + \sum_{k=0}^{p-1}k C_{\tau,(-k-1,b)}.
\Gamma [ \phi ] = \sum_{n}\int \frac{d^{4}p_{1}}{( 2 \pi )^{4}}\cdots \frac{d^{4}p_{n}}{( 2 \pi )^{4}}( 2 \pi )^{4}\delta^{4}\left( \sum_{i}p_{i}\right) \Gamma^{( n )}( \{p_{i}\}) \hat{\phi}( p_{1}) \cdots \hat{\phi}( p_{n})
m={2^{1\over 3}\pi^{4/3}\Gamma(5/9)\over \Gamma(2/3)\Gamma(8/9)}\,\left({2\Gamma(11/16)\over\Gamma(5/16)\Gamma(3/4)}\right)^{8/9}\,g^{8/9}\approx 4.9277912244\ldots g^{8/9}.
\mathfrak{K}= \frac{2}{s}\Big[\frac{D(s)- E(s)}{E(s)}\Big]\cdot \frac{F(s)}{D(s)}+ \frac{2s^{2- n}\int_{b= s}\big|\frac{\partial u}{\partial n}\big|^2\big(|\nabla b|- |\nabla b|^{-1}\big)}{D(s)}&= \mathfrak{K}_1+ \mathfrak{K}_2
d' \otimes d'' = \sum_{i,j \geq 1}\left\langle d' \otimes d'' , x^{\rho_1 + \cdots + \rho_{i-1}}\otimes x^{\rho_i + \cdots + \rho_{i+j-1}}) \right\rangle \beta(\rho_1 , \dots , \rho_i) \otimes \beta(\rho_i , \dots , \rho_j).
\delta^{( 4 )}( x - x^{\prime}) = \frac{1}{\rho}\delta \left( \overline{{{t}}}- \overline{{{t^{\prime}}}}- S \frac{( \phi - \phi^{\prime})}{\alpha}\right) \delta ( \rho - \rho^{\prime}) \delta ( \phi - \phi^{\prime}) \delta ( z - z^{\prime}) \hspace{0 . 5 c m}.
f(\eta_{X}(x)) = f(\left(\prod_{i\in J_{F}}u^{*}_{i}\right) x \left(\prod_{i\in J_{F}}u_{i}\right)) &= \left(\prod_{i\in J_{F}}u^{*}_{i}\right) f(x) \left(\prod_{i\in J_{F}}u_{i}\right) &=\eta_{Y}(f(x)) &=\eta_{Y}(\alpha_{*}(f)(x))
Z_{n}^{F}={\frac{1}{n !}}\left[ \omega_{W P}^{( n )}\right]^{n - 4}\cap \left[ D^{F}\right] ={\frac{\pi^{2}}{( n - 1 ) n !}}\sum_{k = 1}^{[ n / 2 ] - 1}F ( n , k ) k ( n - k - 2 ) \left[ \omega_{W P}^{( n )}\right]^{n - 4}\cap \left[ D_{k}\right] ,
R_1 = \int_{\mathcal{X}_n^c}f(x) \int_0^1 \mathrm{B}_{k,n-k}(s) \log u_{x,s}\,ds \,dx = o \biggl( \frac{k^{\frac{\alpha}{\alpha+d}-\epsilon}}{n^{\frac{\alpha}{\alpha+d}-\epsilon}}\biggr)
d s^{2}= h^{- 1 / 2}( r ) d x_{\mu}d x^{\mu}+ h^{1 / 2}( r ) \lgroup d r^{2}+ r^{2}[{\frac{1}{9}}\chi^{2}+{\frac{1}{6}}\sum_{i = 1}^{4}( g^{i})^{2}] \rgroup ,
\mathcal I_2^{\sharp}(\mathbf h,\breve{\mathbf g},\pmb{\phi}) = \frac{\mathcal I_2(\mathbf h,\breve{\mathbf g},\pmb{\phi})}{\langle \mathbf h_2, \mathbf h_2\rangle\langle \breve{\mathbf g}_2, \breve{\mathbf g}_2\rangle\langle \pmb{\phi}_2, \pmb{\phi}_2\rangle},
\stackrel{( 0 )}{S}= S_{0}\left[ A_{\alpha \beta}^{( \sigma )}\right] + \int d^{D}x \left( A_{\; ( \sigma )}^{* \alpha \beta}\partial_{\left[ \alpha \right.}\eta_{\left. \beta \right]}^{( \sigma )}+ \eta_{( \sigma )}^{* \alpha}\partial_{\alpha}C^{( \sigma )}\right) ,
\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: = \lim_{\mu_{f}, \mu_{g}\to 0}\int_{0}^{\infty}d |{\bf p}| \:{\tilde{f}}_{{\mu}_{f}}( |{\bf p}| ) \cdot \frac{\partial}{\partial |{\bf p}|}{\tilde{g}}_{{\mu}_{g}}( |{\bf p}| ) .
\bigl\vert H_{X_0^{N}(t+h)}(\eta)- H_{X_0^{N}(t)}(\eta)\bigr\vert = \Biggl\vert\Biggl(\sum_{i=1}^N \bigl(\alpha_i(t)-\alpha_i(t+h)\bigr)\bigl\vert\sin(\eta-\theta_i)\bigr\vert\Biggr)\Biggr\vert &\leq N\underset{i=1,\dots, N}{\max}\bigl\lbrace\bigl(\alpha_i(t)-\alpha_i(t+h)\bigr)\bigr\rbrace.
(\rho_{w}\ast \mu_{\rho(w)})(t) =\mu_{\rho(w)}(\rho_{w}(t))=\mu(\rho(w)+\rho_{w}(t))-\mu(\rho(w)) &=\mu(\rho(w)+\rho(w+t)-\rho(w))-\mu(\rho(w)) &=(\rho\ast\mu)_{w}(t)
\int (uv-u_xv_x) u_{xx}(u_x\varphi_N)^{2q-1}\varphi_Ndx &=\int (uv-u_xv_x) [(u_x\varphi_N)_x -u_x\varphi_N'](u_x\varphi_N)^{2q-1}dx &=-\frac{1}{2q}\int M (u_x\varphi_N)^{2q}dx -\int (uv-u_xv_x)u_x\varphi_N'(u_x\varphi_N)^{2q-1}dx &\leq CK^2\|u_x\varphi_N\|_{2q}^{2q}.
\int_{\Omega_R}|b(\rho)|^\gamma=\int_{\{\rho>\overline\rho-\alpha_1\}}|b(\rho)|^\gamma\leq c\int_{\{\rho>\overline\rho-\alpha_1\}}P(\rho)\leq c \int_{\{\rho>\overline\rho-\alpha_1\}}\left(P(\rho)-P(r)-P'(r)(\rho-r)\right)
h_0 = \frac{l_p^{\prime 9}}{R_6^{\prime}R_7^{\prime}R_8^{\prime}R_9^{\prime}R_{11}^{\prime}}\left( \frac{Q_0}{R_{11}^{\prime}}\right) \frac{1}{r^3}= \frac{Q_0 l_p^3}{r^3}
\mathbb{P}[e_1,\ldots,e_k]=\exp(-\alpha\mu(\sum\limits_{i=1}^{k}N^{x_i}_{y_i}(l)+N^{y_i}_{x_i}(l)>0)) = \exp(-\alpha\lim\limits_{n\rightarrow\infty}\mu(\sum\limits_{i=1}^{k}N^{x_i}_{y_i}(l)+N^{y_i}_{x_i}(l)>0,lA_n)).
\begin{aligned}\sigma(\lambda_g) =\sigma(\chi_g(A))=\sigma\left(\sum_{a\in A}\prod_{i=1}^r\zeta_{n_i}^{g_ia_i}\right)=\sum_{a\in A}\prod_{i=1}^r\sigma(\zeta_{n_i}^{g_ia_i}) &=\sum_{a\in A}\prod_{i=1}^r\zeta_{n_i}^{\eta(\sigma)g_ia_i}=\chi_g(\eta(\sigma)A)\end{aligned}
\mu_{A^{\otimes m}}(F(X) \otimes F(Y))\Delta_{A^{\otimes n}}= F \begin{pmatrix}I_m I_m \end{pmatrix}F \begin{pmatrix}X 0 0 Y \end{pmatrix}F \begin{pmatrix}I_n I_n \end{pmatrix}= F(X+Y).
s_{{\textrm{ind},m,2}_{n_r}}(t) = \sum_{n_t=1}^{N_t}\alpha_{m, 2}\sum_{n_m=1}^{N_m}x_{n_t}\left(t-\tau_{\textrm{ind}, m, 2, n_m, n_t, n_r}\right) \times e^{\mathrm{j}2\pi{f_c}\left(t-\tau_{\textrm{ind}, m, p, q, 2, n_t}\right)}e^{\mathrm{j}2\pi{f_{D,2}}\left(t- \tau_{\textrm{ind}, m, 2, n_m, n_t, n_r}\right)}e^{\mathrm{j}{\phi_{m,n_m}}},
{\cal L}^{( 1 )}= - \partial_{\mu}\overline{\psi}_{1}\gamma_{\mu \nu}\partial_{\nu}\psi_{1}- \partial_{\mu}\overline{\psi}_{2}\gamma_{\mu \nu}\partial_{\nu}\psi_{2}- m^{2}\overline{\psi}_{1}\psi_{1}+ m^{2}\overline{\psi}_{2}\psi_{2}.
\left[- \frac{1}{2L}\left( \frac{\partial^{2}}{\partial (a^{3})^{2}}+ \frac{\partial^{2}}{\partial (a^{8})^{2}}\right) + U_{eff}(a^{3}, a^{8})\right] \tilde{\Phi}(a^{3}, a^{8})= E \tilde{\Phi}(a^{3}, a^{8})
S'_i=\frac{a_{p-1}\left((p-2) \left(a_1-a_p\right) \left(a_p-a_i\right)-d a_p \gcd \left(a_i-a_1,a_1-a_p,a_p-a_i\right)\right)}{\left(a_{p-1}-a_p\right) \gcd \left(a_i-a_1,a_1-a_p,a_p-a_i\right)}.
Q\le 3\left(1+H^2\right)^2 +\frac{1}{12}\left(-\frac{1}{2}\vert \mathring{A}\vert^4+4\sqrt{3}\vert H\vert\vert\mathring{A}\vert^3 + 3(1-2H^2)\vert \mathring{A}\vert^2 \right) := 3\left(1+H^2\right)^2 +F(\vert\mathring{A}\vert),
\langle Lx, x' \rangle_{X,X'}= \langle R(\lambda, A_{-1}(t)) (A(s) - A_{-1}(t)) A(s)^{-1}x, x' \rangle_{X_{-1,A(t)},X_{1,A(t)'}}& = \langle (A(s) - A_{-1}(t)) A(s)^{-1}x, R(\lambda,A_1(t)')x' \rangle_{X_{-1,A(t)},X_{1,A(t)'}}.
D_{{\mu}{\nu}}^{p}( q ) = \frac{i}{q^{2}+ i{\epsilon}}\left( - g_{{\mu}{\nu}}+ \frac{n_{\mu}q_{\nu}+ n_{\nu}q_{\mu}}{q_{-}}- n^{2}\frac{q_{\mu}q_{\nu}}{q_{-}^{2}}\right) -{\delta}_{{\mu}+}{\delta}_{{\nu}+}\frac{i}{q_{-}^{2}},
{\frac{\delta ( \stackrel{( 0 )}{S_{0}}+ g \stackrel{( 1 )}{S_{0}}+ g^{2}\stackrel{( 2 )}{S_{0}}+ . . . )}{\delta \varphi}}( \stackrel{\smash{( 0 )}}{R}_{\alpha}^{i}+ g \stackrel{\smash{( 1 )}}{R}_{\alpha}^{i}+ g^{2}\stackrel{\smash{( 2 )}}{R}_{\alpha}^{i}+ . . . ) = 0 .
\langle v_0-w_0, Av_0 + F(v_0) - Aw_0 - F(w_0) \rangle_H &= \limsup_{n \rightarrow \infty}\langle v_n-w_n, Av_n + F(v_n) - Aw_n - F(w_n) \rangle_H \leq c \limsup_{n \rightarrow \infty}\| v_n - w_n \|_H^2 = c \| v_0 - w_0 \|_H^2 .
A_{18}= 25400, 25444, 25488, 25508, 25528, 25544, 25552, 25576, 25592, &25596, 25604, 25616, 25636, 25652, 25660, 25672, 25676, 25684, &25688, 25700, 25712, 25720, 25724, 25744, 25764, 25780, 25784, &25792, 25796, 25808, 25820, 25828, 25832, 25852, 25872, 25888, &25892, 25904, 25920, 25960, 25976, 25980, 26152, 26176.
\overline I_{\gamma}[V,\hat h]=\frac{d_{\gamma}^{*}\int_{X^{n+1}}\left({\rho}^a|\nabla V|_{\bar{g}}^2+E(\rho)V^2\right)\,dv_{\bar{g}}}{(\int_{M^n}V^{2^{*}}\,dv_{\hat{h}})^{2/{2^{*}}}}
\psi(p)=-\frac{\lambda}{\pi^2i}\int d^4q\frac{\psi(q)}{\left[\left(p-q\right)^2+io\right]\left[\left(\frac{P}{2}-q\right)^2-m^2\right]\left[\left(\frac{P}{2}+q\right)^2-m^2\right]}.
\Delta (a_n) := \sum_{r=0}^n \sum_{\substack{\beta_1 + 2\beta_2 + \cdots + n\beta_n = n-r \beta_0 + \beta_1 + \cdots + \beta_n = r+1}}\frac{(r+1)!}{\beta_0! \beta_1! \cdots \beta_n!}a_r \otimes a_1^{\beta_1}\cdots a_n^{\beta_n}. = \sum_{r=0}^n \frac{(r+1)!}{(n+1)!}B_{n+1,r+1}(1!a_0,2!a_1,\ldots , (n-r+1)!a_{n-r+1}) .
(\begin{matrix}\varphi A_{L}\\ \varphi A_{S}\end{matrix})=(\begin{matrix}1&1\\ 1&0\end{matrix})(\begin{matrix}B_{L}\\B_{S}\end{matrix})=(\begin{matrix}2&1\\ 1&1\end{matrix})(\begin{matrix}A_{L}\\A_{S}\end{matrix}),
\sum_{N\le x}\left(\sum_{N^-<p<N^+}\frac{H(D_N(p))}{p}\right)^j =\sum_{N\le x}\frac{1}{N^j}\left(\sum_{N^-<p<N^+}H(D_N(p))\right)^j+O(\sqrt{x}) &=\sum_{N\le x}\left(\frac{K(N)N}{\phi(N)\log N}\right)^j+\tilde{\mathcal{E}}_1
\Omega_3=\begin{pmatrix}0 \frac{\partial F}{\partial x_3}-\frac{\partial F}{\partial x_2}-\frac{\partial F}{\partial x_3}0 \frac{\partial F}{\partial x_1}\frac{\partial F}{\partial x_2}-\frac{\partial F}{\partial x_1}0 \end{pmatrix}
\mathcal{T}_{\vartheta_n}(F_{\vartheta_n}^{N_{0,\vartheta_n}}):=\sum^{\vartheta_n-1}_{h=1-\vartheta_n}(\widetilde{\phi}_{hk}\widehat{\Gamma}_h)=\bar{\theta}_j+\frac{1}{n}\sum^{N_{0,\vartheta_n}}_{i=1}IF(v_{i,\vartheta_n},F_{\vartheta_n})+\mathcal{R}_{j,N_{0,\vartheta_n}},
\nabla E = \nabla P + \frac{1}{2}((u-c)^2 + v^2) \nabla \varrho + \varrho ( -cu_x + uu_x + vv_x, -cu_y + uu_y + vv_y) + \varrho \mathbf{g}+ gy \nabla \varrho = \left( (\sqrt{\varrho}(u-c))_y - (\sqrt{\varrho}v)_x\right) \left(-\sqrt{\varrho}v, \sqrt{\varrho}(u-c)\right) + gy \nabla \varrho,
\begin{array}{c}{{\psi_{0}( x ) = K_{p}\, \sinh \kappa \, ( 1 - x ) , \ \x > 0 ,}}\\{{\psi_{0}( x ) = K_{n}\, \sinh \lambda^{*}\, ( 1 + x ) , \ \x < 0 ,}}\\{{\psi_{1}( x ) = L_{p}\, \sinh \lambda \, ( 1 - x ) , \ \x > 0 ,}}\\{{\psi_{1}( x ) = L_{n}\, \sinh \kappa^{*}\, ( 1 + x ) , \ \x < 0 .}}\\ \end{array}
\int_{0}^{\tau}d t \cdot \omega_{\mu}( t , z , \tau ) + \int_{0}^{\tau}d t \cdot \omega_{2 - \mu}( t , z , \tau ) = \tilde{q}^{- 1}\left( \int_{0}^{- z}d t \cdot \omega_{\mu}( t , z , \tau ) - \int_{0}^{- z}d t \cdot \omega_{\mu - 2}( t , z , \tau ) \right)
\left(\begin{array}{c}\mathfrak p \updownarrow \mathfrak q\end{array}\right)\leftrightarrow \left(\begin{array}{c}\mathfrak p' \updownarrow \mathfrak q'\end{array}\right)=\begin{array}{c}(\mathfrak p\leftrightarrow\mathfrak p') \updownarrow (\mathfrak q\leftrightarrow\mathfrak q').\end{array}
\int\limits_{0}^{t}\int\limits_{\Omega}\partial_{t}( \frac{1}{c^2}\partial_t u^{2}) dxd\tau =\int\limits_{\Omega}\left( \frac{1}{c^2}\partial_t u^{2}\right) \left( x,t\right) dx -\int\limits_{\Omega}\frac{1}{c^2}f_{1}^{2}\left( x,t\right) dx.
S=\frac{M_{Pl}^{2}}{2}\epsilon_{abcd}\int(e^{a}\wedge e^{b}\wedge R^{cd}-m^{2}[e^{a}\wedge e^{b}\wedge e^{c}\wedge e^{d}+i^{a}\wedge e^{b}\wedge e^{c}\wedge e^{d}+i^{a}\wedge i^{b}\wedge e^{c}\wedge e^{d}+i^{a}\wedge i^{b}\wedge i^{c}\wedge e^{d}]),
(A) E(\lambda) - E_{thr}=E(\lambda)= - c (\lambda-\lambda_{cr}) + \mbox{\textnormal{h.t.}}&(B) E(\lambda) - E_{thr}=E(\lambda)= c (\lambda-\lambda_{cr})[\ln (\lambda-\lambda_{cr})]^{-1}+ \mbox{\textnormal{h.t.}}&(C) E(\lambda)-E_{thr}= - c (\lambda-\lambda_{cr})^2 + \mbox{\textnormal{h.t.}}
\{(x_1,x_2,1-x_2,1-x_1)\mid 0\le x_1\le x_2<\frac{1}{2}\}= \{(0,0,1,1)+y_1(\frac{1}{\sqrt{2}},0,0,-\frac{1}{\sqrt{2}})+y_2(0,\frac{1}{\sqrt{2}},-\frac{1}{\sqrt{2}},0) \mid 0\le y_1\le y_2<\frac{1}{\sqrt{2}}\},
p\left(u, v, w, s, x, y_1, y_2, y_3 \right)= p\left(s \right) p\left(u\mid s \right) p\left( v\mid u,s \right)p\left(w\mid u,v,s\right)p\left(x\mid u,v,w,s \right) p\left(y_1, y_3\mid x,s\right)p\left(y_2\mid y_1\right).
\left| f_\delta(x) \right| \leq \frac{1}{p_\delta(x)}\left(\int_x^{+\infty}p_\delta(y) dy \right)^{\frac{1}{q}}\|g\|_{L^p(\mu_\delta)}= G_\delta(x) \|g\|_{L^p(\mu_\delta)},
a_n^{(\theta)}= \frac{1}{x_1 - x_2}\big( T_{q, x_2}a_n^{(\theta - 1)}- T_{q, x_1}a_{n-1}^{(\theta - 1)}\big), n = 1, 2, \dots, \theta-1, a_0^{(\theta)}= \frac{1}{\prod\limits_{i=0}^{\theta - 1}{(x_1 - q^i x_2)}},a_{\theta}^{(\theta)}= \frac{1}{\prod\limits_{i=0}^{\theta - 1}{(x_2 - q^i x_1)}}.
u(x_1,x_2,t) = \frac{1}{4\pi t}\int\limits_{-\infty}^{+\infty}\int\limits_{-\infty}^{+\infty}\Lambda(s_1,s_2) \exp\left\{- \frac{(s_1-x_1)^2+(s_2-x_2)^2}{4t}\right\}\, ds_1 ds_2.
\sum_{n=0}^{\infty}\frac{(-1)^{(n^{2}-n)/2}}{2n+1}=1+\frac{1}{3}-\frac{1}{5}-\frac{1}{7}+\frac{1}{9}+\frac{1}{11}-\cdot\cdot\cdot=\frac{\pi}{2\sqrt{2}}
Q_{w_{0,a,b}}=w_{0,a,b}\left(\frac{1}{\prod_{1\leqslant i\leqslant a}\prod_{a+1\leqslant j\leqslant n}(x_i-x_j)}\right)=\frac{1}{\prod_{b+1\leqslant i\leqslant n}\prod_{1\leqslant j\leqslant b}(x_i-x_j)}.
pdf_{MJ}(p;\theta)dp_{1}\cdot\cdot\cdot dp_{d}=\pi^{\frac{1-d}{2}}2^{-\frac{d+1}{2}}(mc)^{-d}\theta^{\frac{1-d}{2}}K_{\frac{d+1}{2}}(\frac{1}{\theta})^{-1}e^{-\frac{\gamma(p)}{\theta}}dp_{1}\cdot\cdot\cdot dp_{d}
\partial_{a}^{3}{\cal F}={\frac{\pi^{2}\left( a \partial_{a}^{2}{\cal F}- \partial_{a}{\cal F}\right)^{3}}{1 6 \left[{\Lambda^{4}}+ \pi^{2}\left({\cal F}-{a}\partial_{a}{\cal F}/ 2 \right)^{2}\right]}}.
\|\nabla s\|^2 =(s,\nabla^\ast \nabla s)_{W_R}-(\sigma_\nabla(-\nu)s,\nabla s)_{N_R}&=(s,\nabla^\ast \nabla s)_{W_R}+(\nu \otimes s,\nabla s)_{N_R}&=(s,\nabla^\ast \nabla s)_{W_R}+(s,\iota(\nu)\nabla s)_{N_R}
V_{m a t t e r}^{f d}={\frac{2 N_{F}^{f d}}{\pi^{2}L^{4}}}{\frac{\beta^{2}}{2 4}}\Biggl ( \sum_{i = 1}^{N - 1}\Bigl ( 6 \theta_{i}^{2}- 1 2 \pi \theta_{i}+ \beta^{2}+ 4 \pi^{2}\Bigr ) + 6 \Bigl ( \sum_{i = 1}^{N - 1}\theta_{i}\Bigr )^{2}- 1 2 \pi \sum_{i = 1}^{N - 1}\theta_{i}+ \beta^{2}+ 4 \pi^{2}\Biggr ) .
\hat{h}\oplus(-\hat{h'}) \colon \Gamma(V(P_0, P, \mathfrak{s})\oplus V(P_0, P, \mathfrak{s}\otimes L)) \to \Gamma(V(P_0, P, \mathfrak{s}) \oplus V(P_0, P, \mathfrak{s}\otimes L))
\sum_{\substack{\min_{m\in\mathbb Z}\{|k\theta-m|\}<\epsilon 1\le k\le n}}\log\bigl|1-e^{2\pi ik\theta}\bigr| \le 0\intertext{and}\lim_{n\to\infty}\frac1n\sum_{\substack{\min_{m\in\mathbb Z}\{|k\theta-m|\}\ge\epsilon 1\le k\le n}}\log\bigl|1-e^{2\pi ik\theta}\bigr| =\int_\epsilon^{1-\epsilon}\log|1-e^{2\pi ix}|\,dx
A_1= -\frac{1}{2}\int_{-\infty}^0x\big(\bar{u}_{xx}u_x + u_{xx}\bar{u}_x \big)dx = -\frac{1}{2}\int_{-\infty}^0x\big(|u_x|^2\big)_xdx = \frac{1}{2}\int_{-\infty}^0|u_x|^2dx.
\begin{array}{rcl}\alpha \cdot (\beta \cdot e_S) = \alpha \cdot (\lambda(\beta)\cdot e_S) = \alpha(\lambda(\beta)) \cdot e_S + \lambda(\beta) \cdot (\alpha \cdot e_S) = (\alpha(\lambda(\beta)) + \lambda(\alpha) \lambda(\beta)) \cdot e_S, \end{array}
T^{0 i}= B \tilde{E}_{i}- ( \partial_{j}E_{j}) A_{i}- \frac{2 < B^{2}>}{\Lambda^{3}\kappa}\partial_{i}( A E ) + \frac{M}{8}\partial_{j}( A_{i}\tilde{A}_{j}+ A_{j}\tilde{A}_{i}) ;
\frac{1}{2}\frac{d}{dt}\int_{\Omega}\rho |u_t|^2 x+\int_{\mathbb{R}^3}(\mu|\nabla u_t|^2+(\lambda+\mu)(u_t)^2) x = \int_{\Omega}\big(-\nabla P_t-( \rho v \cdot \nabla v)_t-\frac{1}{2}\rho_t u_t+ (H\times H)_t\big)\cdot u_tx\equiv:\sum_{i=1}^{4}I_i.
D_{0,0}^2 = 1 D_{i,n-i-1}^{2n+1}= \frac{(4n-2i)D_{i,n-i-1}^{2n}+(2i+2)D_{i+1,n-i-2}^{2n}}{(2n)(2n-1)}, D_{i,n-i}^{2n+2}=(2i+1)D_{i,n-i-1}^{2n+1}+(4n-2i+3)D_{i-1,n-i}^{2n+1},
C =\breve{A}\cdot A^{}\mbox{where}A=(a_{s,i})_{s\in S,i \in I}\mbox{and}\breve{A}=(\breve{a}_{s,i})_{s\in S,i \in I}, C' =\breve{B}\cdot B^{}\mbox{where}B=(b_{s,j})_{s\in S,j \in J}\mbox{and}\breve{B}=(\breve{b}_{s,j})_{s\in S,j \in J}.
\left\{\tilde{B}_{J,l}(z,\overline{z},x),B_{I}(z,\overline{z},x),\quad\overline{B_{I}(z,\overline{z},x)},\quad\overline{\tilde{B}_{J,l}(z,\overline{z},x)}\right\}_{k\in\mathbb{N}^{\star}-\{1\}\atop{l=1,\dots,N}},
\eta(z) = \frac{z^{\frac{2}{N}}}{\kappa_N(z^2)}\frac{(p^N,p^N)^3_{\infty}}{(p,p)^3_{\infty}}\frac{\Theta_{p}(q^2)\Theta_{p}(p z^2)}{\Theta_{p}(q^2 z^2)}
\tilde{g}= \frac{( R_{s}M )^{\frac{3 - d}{4}}}{\prod r_{i}}\ , \quad \tilde{\alpha}^{\prime}= \frac{R_{s}^{1 / 2}}{M^{3 / 2}}\ , \quad \tilde{R}_{i}= \frac{1}{r_{i}M}\ , \quad R_{s}\rightarrow 0 \ ; \M , r_{i}\mathrm{f i x{\' e}s}\ .
\mu^\star_{xy}q_{ij}(x,y) =(n-1)q_{ij}(x,y)+\frac{n(n-1)}{k(n-k)}\frac{\binom{n}{i}}{\binom{k}{i}}K_i(x,k-y,r-1) (-y)H_j(y,n-i,k-i)\,.
4t v''+\Bigl (\frac{4t}{\alpha}-\frac{8}{\alpha}-te^{-t}\Bigr )v'-\Bigr (\frac{4}{\alpha^2}-\frac{4(\alpha +1)}{\alpha^2 t}-\frac{e^{-t}}{\alpha}\Bigl )v+|v|^\alpha v=0 .
\tilde q_c^{(ab)}(r)=\frac{1}{4\pi^2b^4}\sum_{m=0}^{\infty}{'}\int_{0}^{\infty}{dz\sum_{\lambda =0}^{1}z^3\tilde \Omega_{\lambda m}^c(\sigma ,z)F_{\lambda m}^{(q)}(z,zy)}
\sum_{J \in \mathcal{J}_1}\left( \frac{\varepsilon n}{16 \log^2 n}\right)^{-2 v(J)}p^{-e(J)}\le \sum_{\substack{J \in \mathcal{J}_1 v(J) = 2}}\left( \frac{\varepsilon n}{16 \log^2 n}\right)^{-4}p^{-1}+ \sum_{\substack{J \in \mathcal{J}_1 v(J) > 2}}\left( \frac{\varepsilon n}{16 \log^2 n}\right)^{-2 v(J)}p^{-v(J)}.
W^{OP}(u,r,x_0)=W_0(u,r,x_0)+\frac{1}{r^d}\big|\{u>0\}\cap B_r(x_0)\big| W^{DP}(u,r,x_0)=W_0(u,r,x_0)+\frac{1}{r^d}\Bigl(\lambda_1\,\big|\{u>0\}\cap B_r(x_0)\big|+\lambda_2\,\big|\{u<0\}\cap B_r(x_0)\big|\Bigr) W^{V}(u,r,x_0)=W_0(u,r,x_0)+\frac{1}{r^d}\big|\{|u|>0\}\cap B_r(x_0)\big|\,.
\bar{C}^{c}_{1}= \{C,\pm,\frac{\times}{c},c(a^{-1}),c(a^{*}), 0,c\}=\{C,\pm_{1},\frac{\times_{1}}{c_{1}},c_{1}(a_{1}^{-1_{1}}),c_{1}(a_{1}^{*_{1}}),0,c_{1}\}.
d_t e_t + o(d_t b_t) - 2 (\ln_3 t) \,\frac{|Z_s| h_t + (\ln t)^2}{s}\ge d_t e_t + o(d_t b_t) - 2 (\ln_3 t) \,\frac{r_t g_t h_t + (\ln t)^2}{a t}& = d_t e_t + o(d_t e_t),
S^*(X,\mathbf{d},\mathbf{D})=C^*(\mathbf{d},\mathbf{D})\operatorname{vol}(\mathcal{R})X^2(\log X)^3 +O_{\varepsilon}\left(\left(\frac{D^\varepsilon}{\delta(\mathbf{D})}+1\right)X^{\frac{23}{12}+\varepsilon}+\frac{D^\varepsilon}{\operatorname{det}(\Lambda(\mathbf{D}))}X^2(\log X)^2\right),
i\left( \left\Vert \cdot\right\Vert_{\beta}\partial_{t}\right)W_{\hbar}\left( t,s\right) \psi =L_{\hbar}\left( t\right) W_{\hbar}\left( t,s\right) \psi, i\left( \left\Vert \cdot\right\Vert_{\beta}\partial_{s}\right)W_{\hbar}\left( t,s\right) \psi =-W_{\hbar}\left( t,s\right) L_{\hbar}\left( s\right) \psi.
\sum_{j=0}^{\infty}(j+1)\rho^{j}U_{j}(x)U_{j}(y)=\frac{(1+\rho^{2})(1-\rho^{2})^{2}-4\rho^{2}(1+\rho^{2})(x^{2}+y^{2})+16\rho^{3}xy}{((1-\rho^{2})^{2}-4xy\rho(1+\rho^{2})+4\rho^{2}(x^{2}+y^{2}))^{2}},
p_n(e)\approx 1-\left(\frac{\left(\frac{2}{n}\sum_{i=1}^n\rho^{(2)}\left(\frac{r_i(e)}{\sigma_n}\right)\right)(s_{\rho,n}(e)-s_{\rho,n}(e_{\text f}))}{\frac{1}{n}\sum_{i=1}^n\rho^{(1)}\left(\frac{r_i(e)}{\sigma_n}\right)^2},k-k(e)\right)\,.
f'_i= \left(\begin{array}{ccccccccccc}\boxed{v'_{i1}}x_1 \cdots v'_{i2}x_2 \cdots v'_{i3}\cdots v'_{it_i}\cdots y v'_{ij_1}\cdots x'_1 v'_{ij_2}\cdots x'_2 v'_{ij_3}\cdots v'_{ij_{t_i}}\cdots \boxed{z}\end{array}\right),
(\delta_{\mathrm{Hoch}}f) (a_1, \ldots, a_{n+1}) =~ a_1 \cdot f(a_2, \ldots, a_{n+1}) + \sum_{i=1}^n (-1)^i f (a_1, \ldots, a_{i-1}, a_i a_{i+1}, \ldots, a_{n+1}) ~ + (-1)^{n+1}f(a_1, \ldots, a_n) \cdot a_{n+1}, f \in C^n_{\mathrm{Hoch}}(A, M).
\begin{matrix}{}&{M \ \mathrm{i s c o m p l e m e n t e d a s a s u b s p a c e o f t h e n o r m e d s p a c e o f}}\\{}&{\mathrm{a l l o p e r a t o r s i n a H i l b e r t s p a c e .}\hfill}\\ \end{matrix}
q_{b}( a , r ) = - \frac{1}{2 \pi a^{3}r^{n}S_{D}\sigma^{D - 1}}\sum_{l = 0}^{\infty}D_{l}\int_{m a}^{\infty}d z z^{3}\frac{\bar{K}_{\nu_{l}}( z )}{\bar{I}_{\nu_{l}}( z )}
a_1p_1+\dots+a_dp_d =a_1(\alpha_1qv_{1,1}+\dots+\alpha_dqv_{d,1})+\dots+a_d(\alpha_1qv_{d,1}+\dots+\alpha_dqv_{d,d}) &= q\alpha_1(a_1v_{1,1}+\dots+a_dv_{1,d})+\dots+q\alpha_d(a_1v_{d,1}+\dots+v_{d,d}) \, .
\Gamma_{\phi^{-}}^{1 2}( p ) = \frac{- e^{\beta ( \mu_{U}- \mu_{D}) / 2}\, ( p^{2}- \bar{M}^{2}) R_{U D}( p )}{\{p^{2}[ K_{U D}( p ) + H_{U D}( p ) ] + E_{U D}( p ) \}^{2}+ ( p^{2}- \bar{M}^{2})^{2}[ S_{U D}^{2}( p ) - R_{U D}^{2}( p ) ]}= e^{\beta ( \mu_{U}- \mu_{D})}\, \Gamma_{\phi^{-}}^{2 1}( p )
L_{e f f}= - \Biggl ( \frac{1}{2}\gamma_{~ b \mu}^{a}F_{a ~ ~ m}^{~ b \mu ~ n \nu}\gamma_{~ n \nu}^{m}+ \frac{1}{2}\lambda_{~ \nu}^{b}D_{b ~ a}^{~ \nu ~ \mu}\lambda_{~ \mu}^{a}+ \lambda_{~ n}^{m}\biggl ( G_{m ~ ~ a}^{~ ~ n j ~ b c}\nabla_{j}+ T_{m ~ a}^{~ n ~ b c}\biggr ) \gamma_{~ b c}^{a}\Biggr ) e
\rho_s^{(N_f)}(z,m)=\frac{z}{2}\frac{B^{(N_f)}(z,m)\sum_{f=0}^{N_f}\widetilde{D}_f^{(N_f)}(z,m) -D^{(N_f)}(z,m)\sum_{f=0}^{N_f}\widetilde{B}_f^{(N_f)}(z,m)}{[C^{(N_f)}(m)]^2\prod_{f=1}^{N_f}(z^2+m_f^2)}\:,
\vec{W}_{\perp}= \frac{1}{2}P^{+}\hat{z}\times \left( \vec{F}- \vec{E}\right) + \frac{1}{2}\left( P^{+}- P^{-}\right) \hat{z}\times \vec{E}_{\perp}- \left( \hat{z}\times \vec{P}_{\perp}\right) K_{z},
\overset{1}{\Delta}_{m+1}B =T^{tt}(\alpha_{+,m+1},\beta_{+,m+1})-T^{tt}(\alpha_{+,m},\beta_{+,m}), \overset{2}{\Delta}_{m+1}B =T^{tt}(\alpha_{-,m},\beta_{-,m})-T^{tt}(\alpha_{-,m+1},\beta_{-,m+1}).
\begin{aligned}\sup_{x\in\left(-\infty, x_0\right)}\|\langle x\rangle m(x;z)\|_{L_z^2(\mathbb{R})}\leq \sqrt{\pi}\left(\left\|u_{xx}\right\|_{L^{2,1}}+\frac{1}{2}\left\|u_x^3\right\|_{L^{2,1}}\right) & \leq c\left(\|u\|_{H^{2,1}}+\|u\|_{H^{2,1}}^3\right),\end{aligned}
W_{\hbar}\left( t\right) =e^{\frac{i}{\hbar}f\left( t\right)}V_{\hbar}\left( t\right) =e^{\frac{i}{\hbar}f\left( t\right)}U_{\hbar}\left(-\alpha\left( t\right) \right) e^{-iH_{\hbar}t/\hbar}U_{\hbar}\left(\alpha_{0}\right) .
\int_{\{\Psi<-t_1\}}|F_{\delta}-(1-b_{t_0,B}(\Psi))fF|^2_{\tilde{h}}e^{v_{t_0,B}(\Psi)}c(-v_{t_0,B}(\Psi)) \le \bigg(\frac{1}{\delta}c(t_1)e^{-t_1}+\int_{t_1}^{t_0+B}c(s)e^{-s}ds\bigg) \int_{M}\frac{1}{B}\mathbb{I}_{\{-t_0-B<\Psi<-t_0\}}{|fF|}^2_{\tilde{h}}.
d(g_{n,m}(x),\varphi_{n-1,m}(x))=d(g_{n,m}(x),g_{n-1,m}(x))\le \le d(g_{n,m}(x),f_n(x))+d(f_n(x),f_{n-1}(x))+d(f_{n-1}(x),g_{n-1,m}(x))\le \le\frac{1}{2^{n+1}}+\frac{1}{2^{n}}+\frac{1}{2^{n+1}}=\frac{1}{2^{n-1}}
f_1f_2f_3 =2|z_1|^2|z_2|^2|z_3|^2 &+|z_1|^2(z_2^2\bar z_3^2+\bar z_2^2z_3^2)+|z_2|^2(z_1^2\bar z_3^2+\bar z_1^2z_3^2)+|z_3|^2(z_1^2\bar z_2^2+\bar z_1^2z_2^2).
\frac{d\overline{\lambda}}{dt}=-\frac{1}{\tau}(\overline{\lambda}-\frac{\tilde{\nu}+\mu}{\tilde{\delta}+\mu}),\frac{d\overline{\epsilon}}{dt}=-\frac{1}{\tau}(\overline{\epsilon}-\frac{\tilde{\eta}+\mu}{\tilde{\delta}+\mu}),
\frac{c_{k + 1}}{c_{k}}\Gamma_{-}^{(k + 1)}\Gamma_{0}^{(k + 1)*}= \frac{c_{k}}{c_{k - 1}}(t \Gamma_{0}^{(k)*}\Gamma_{-}^{(k)}+ t^{\prime}\Gamma_{+}^{(k)*}\Gamma_{0}^{(k)}).
\hat\Gamma_{\hat\mu}=\hat\Gamma_{\hat\mu\hat\nu\hat\lambda}S^{\hat\nu \hat\lambda}=\frac{i}{4}(C_{\hat\mu \hat\nu \hat\lambda}-C_{\hat\mu \hat\lambda \hat\nu}-C_{\hat\nu \hat\lambda \hat\mu})S^{\hat\nu\hat\lambda}
{}^{L , R}\langle x | 2 k \rangle_{\Omega}^{L , R}= \left({\frac{\Omega}{\pi}}\right)^{1 / 4}[ 2^{2 k}( 2 k ) ! ]^{- 1 / 2}H_{2 k}( x \sqrt{\Omega}) e^{- x^{2}\Omega / 2}
\mathbf{r}_{\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3}}=\left(\begin{array}{c}1,-e^{\int \tau \lbrack s]\,ds}-\int \sinh \left( \int \tau (s)\,ds\right)\kappa (s)\,ds, e^{\int \tau \lbrack s]\,ds}+\int \cosh \left( \int \tau (s)\,ds\right)\kappa (s)ds\end{array}\right).
|\Psi_{(n,0)}\rangle = \int_0^{P^+}{dq_1 \dots dq_n \over \sqrt{q_1...q_n}}\delta(q_1 + ... + q_n - P^+)\hspace{1mm}f^{(n,0)}(q_1... q_n) \mbox{tr}[a^\dagger(q_1)... a^\dagger(q_n)]|0\rangle .
p(t^{*})-p(t^{*}-d) =\int_{t^{*}-d}^{t*}\dot{p}~ dt \geq\int_{t^{*}-d}^{t*}F_{\kappa}\big(\sqrt{1-\varepsilon_{\kappa}^{2}}-\varepsilon\big)dt =F_\kappa \big(\sqrt{1-\varepsilon_{\kappa}^{2}}-\varepsilon\big)d.
\frac{d}{dt}\int_{\Omega_{e_{i}}}\rho_{h}^{i}\varphi_{h}^{i}dx+\int_{\partial\Omega_{e_{i}}}\varphi_{h}^{i}J_{h}\cdot ndx=\int_{\Omega_{e_{i}}}J_{h}\cdot\nabla\varphi_{h}^{i}dx.
III = \sum_{q=1}^{N_1}\mathrm{sgn}(\Delta X_{T_q}) \sum_{j=1}^{\Delta^n_{i\left(n,q\right)}N}\Delta X_{i\left(n,q\right)}^j e^{-\beta_n((\lfloor T_q\Delta_n^{-1}\rfloor+ 1)\Delta_n - T_{i\left(n,q\right)}^j)}, IV = - (1-e^{-\beta_n \Delta_n}) \sum_{q=1}^{N_1}\mathrm{sgn}(\Delta X_{T_q}) Z_{t_{i\left(n,q\right)-1,n}}^\beta.
\frac{1}{2}\partial_t(u^2)+\frac{Ca}{2}((\rho^\gamma-1) r^2u)_x-\frac{Ca}{2}(\rho^\gamma-1)(r^2u)_x-\mu\left(\rho(r^2u)_xr^2u\right)_x+\mu\rho(r^2u)_x^2=0.
\frac{F(z+\overline\omega_1\,|\,\overline\omega_1,\omega_2)}{F(z\,|\,\overline\omega_1,\omega_2)}=\frac{1}{1-x_2}, \frac{F(z+\omega_2\,|\,\overline\omega_1,\omega_2)}{F(z\,|\, \overline\omega_1,\omega_2)}=\frac{1}{1-x_1}.
F\hat{R}(v_{0}\otimes v_{0}) =F(v_{0}\otimes v_{0})+q^{-2}(q^{2}-q^{-2})F(v_{1}\otimes v_{-1}) =[2]^{1/2}(v_{-1}\otimes v_{0}+(q^{2}+1-q^{-2})v_{0}\otimes v_{-1}).