harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tau\frac{\partial\phi}{\partial t}=-\frac{\partial}{\partial x}(\epsilon\frac{\partial\epsilon}{\partial\theta}\frac{\partial\phi}{\partial y})+\frac{\partial}{\partial y}(\epsilon\frac{\partial\epsilon}{\partial\theta}\frac{\partial\phi}{\partial x})+\nabla\cdot(\epsilon^{2}\nabla\phi)+\phi(1-\phi)(\phi-\frac{1}{2}+m+a\chi)
\int_{2at}^x \frac{e^{i(u^2/(4t)-ku)}}{\sqrt t}\widehat f_+(u/(2t))du=\int_{2at}^0 \frac{e^{i(u^2/(4t)-ku)}}{\sqrt t}\widehat f_+(u/(2t))du+\int_{0}^x \frac{e^{i(u^2/(4t)-ku)}}{\sqrt t}\widehat f_+(u/(2t))du\,.
P_{m+d+1}^{\left( r\right)}\left( x\right) =\left( x-\beta_{m+d+r}\right)P_{m+d}^{\left( r\right)}\left( x\right) -\sum\nolimits_{\nu=0}^{d-1}\gamma_{m+d+r-\nu}^{d-1-\nu}P_{m+d-1-\nu}^{\left( r\right)}\left( x\right) m\geq 0,
\mathbb{E}[ \|x^{k+1}_i - x^*_i\|^2 \mid\mathcal{F}_k] \leq (1+2p_{\max}\max_i L_i(\alpha_{\max}-\alpha_{\min}) )\|x_i^k-x^*_i \|^2 + 4p_{\max}\alpha_{\max}^2 C^2 \cr +2p_{\max}\alpha_{\max}B \sum_{j=1}^N \mathbb{E}[W(k)]_{ij}\|v^k_j-y^k\| - 2p_{\min}\alpha_{\min}(F_i(x_i^k,\bar x^k) -F_i(x^*_i,\bar x^*))^T(x^k_i-x^*_i),
e_{\infty}=\lim_{k \rightarrow \infty}|u_{n_k}(0)|^2=\lim_{k\rightarrow \infty}e_{n_k}, & E_{\infty}=\lim_{k \rightarrow \infty}|A^{1/2}u_{n_k}(0)|^2=\lim_{k\rightarrow \infty}E_{n_k},
\Psi_{\gamma+t}(x)=(\Psi_{\gamma+t},e^{i\left\langle \gamma+t,x\right\rangle})e^{i\left\langle \gamma+t,x\right\rangle}+\sum_{\delta\in\Gamma\backslash\left\{0\right\}}(\Psi_{\gamma+t},e^{i\left\langle \gamma+\delta+t,x\right\rangle})e^{i\left\langle \gamma+\delta+t,x\right\rangle}.
|E^\nu|=\left | \begin{matrix}e_1 \sigma^{-1}(e_1) \cdots \sigma^{-\nu+1}(e_1) e_2 \sigma^{-1}(e_2) \ldots \sigma^{-\nu+1}(e_2) \vdots \vdots \ddots \vdots e_\nu \sigma^{-1}(e_\nu) \cdots \sigma^{-\nu+1}(e_\nu)\end{matrix}\right |=0
\sup_{U_{T}\times U_{T}, \tau\leq t_{i}, \sigma\leq s_{i}}u(x,\tau)-v(y,\sigma)-\frac{1}{2\delta}\left(|x-y|^{2}+(\tau-\sigma)^{2}\right)-p_{i}\cdot x-q_{i}\cdot y &=u(x_{i}, t_{i})-v(y_{i}, s_{i})-\frac{1}{2\delta}\left(|x_{i}-y_{i}|^{2}+(t_{i}-s_{i})^{2}\right)-p_{i}\cdot x_{i}-q_{i}\cdot y_{i}=h_{i}+k_{i},
-\frac1N\sum_{k=0}^N (-1)^k\binom{N}{k}kg^{k-1}D(g)D^{N-1}(g^{N+1-k}f) &+\frac{N+1}N g\sum_{k=0}^N(-1)^k\binom{N}{k}kg^{k-1}D(g)D^{N-1}(g^{N-k}f).
\theta^{'^{\prime}}+ ( 2 r - 3 M ) ( \theta^{'})^{3}- \frac{1}{\tan \theta}( \theta^{'})^{2}+ \frac{3 r - 4 M}{r ( r - 2 M )}\theta^{'}- \frac{1}{r ( r - 2 M ) \tan \theta}= 0
\begin{aligned}\Delta_{g(h)}H(h) =r(h/\kappa_\Gamma,hk,\nabla h)*(\nabla^g)^2 H(h) &\quad+ r(h/\kappa_\Gamma,hk,\nabla h)*\big(k,\nabla h/\kappa_\Gamma,h\nabla^g k,(\nabla^g)^2 h\big)*\nabla^gH(h).\end{aligned}
S_{(L-P,-Q)}=T_{(L-P,-Q)}{\Omega}_{p}{\int}_{t_{-}}^{0}dt{\rho}^{p}\sqrt{1+{\dot{\rho}}^{2}}-\frac{Q{\Phi}}{p+1}{\Omega}_{p}{\int}_{t_{-}}^{0}dt{\rho}^{p+1},
\nu_{\eta_{m}^{r}\rightarrow s_{m}}(s_{m})= \underset{\boldsymbol{s^{r}}}{\sum}\eta_{m}^{r}(s_{m}^{r},s_{m})\times\nu_{s_{m}^{r}\rightarrow\eta_{m}^{r}}(s_{m}^{r}) = \pi_{s,m}^{r,in}\delta(s_{m}-1)+(1-\pi_{s,m}^{r,in})\delta(s_{m}),
w^{\gamma}_i(\theta) = \frac{\left(1 + \lambda^T_{\gamma}\psi(x_i, \theta) \right)^{-\frac{1}{\gamma + 1}}}{\sum_{i=1}^n \left(1 + \lambda^T_{\gamma}\psi(x_i, \theta) \right)^{-\frac{1}{\gamma + 1}}}, \textrm{for}\gamma \neq \{-1, 0\},
\left( \beta_{\mu}^{( 0 )}\partial_{\mu}+ m_{0}\right) \Psi^{( 0 )}( x ) = 0 , \Psi^{( 0 )}( x ) = \left( \begin{array}{c}{\psi_{0}( x )}\\{\psi_{\mu}( x )}\\ \end{array}\right) .
\begin{aligned}r ={\log{n}\over m}\biggl(1+O\Bigl({\log\log{n}\over\log{n}}\Bigr)\biggr), e^{mr}={mn\over\log{n}}\biggl(1+O\Bigl({\log\log{n}\over\log{n}}\Bigr)\biggr).\end{aligned}
\psi ( \vec{p}\, ; \rho , \varphi ) \sim e^{i \varepsilon N ( \varphi - \varphi_{p}- \pi )}\left[ e^{i \varepsilon \delta ( \varphi - \varphi_{p}- \pi )}\; e^{i p_{\perp}\rho \cos ( \varphi - \varphi_{p})}+ f_{A B}( \varphi - \varphi_{p}) \;{\frac{e^{i p_{\perp}\rho}}{\sqrt{\rho}}}\right]
dl^{2}\sim \biggl (1+\frac{\psi_{i}^{\prime}}{r_{i}^{\prime}}\biggr )^{2+2a}\biggl (1 +\frac{\chi_{i}^{\prime}}{r_{i}^{\prime}}\biggr )^{2-2a}(dr_{i}^{\prime 2}+r_{i}^{\prime 2}d\Omega^{2})\, ,
{\cal{L}}_{n o n r e l}^{c l a s s}= \psi^{*}\left( i \frac{d}{d t}+ e A^{0}\right) \psi - \frac{1}{2 m}| \left( - i \vec{\nabla}- e \vec{A}\right) \psi |^{2}+ \frac{e}{2 m}B \psi^{*}\psi
S_{{\rm linear}}= m_0 (h_{0i,j}(0) +C_{i,j}(0)){i \over 8}\theta \gamma^{ij}\theta + m_0 (B_{ij,k}(0) + C^{(3)}_{0ij,k}(0)){i \over 16}\theta \gamma^{ijk}\theta +{\cal O}(\theta^4)
\bigl|c_{1,1}^{(z)}\bigr| \left( q_2^2\,\bigl|c_{2,2}^{(z)}\bigr| +(p_1-p_2)p_2 \right) \leqslant \frac{p_1^2}{q_1^2}\left( q_2^2 \cdot\frac{p_2^2}{q_2^2}+(p_1-p_2)p_2 \right) = \frac{p_1^3p_2}{q_1^2}\leqslant p_1p_2.
0 = \delta_{\xi}Z ( J ) = \frac{i}{\hbar}\int{\cal{D}}\Phi [ J^{A}\, \delta_{\xi}\Phi_{A}+ \delta_{\xi}S_{G F}] e^{\frac{i}{\hbar}[ S_{\Psi}( \Phi ) + J^{A}\Phi_{A}]},
( J_1 \cdot J_1 )_2 = ( J_1^2 )_2 = \{\, ( J_1^2 )_{2,m}= \sum_{m_1}\sum_{m_2}\: \langle 1,m_1 ; 1,m_2 | 2,m \rangle \; J_{1,m_1}J_{1,m_2}\:|\, -2 \le m \le +2 \,\},
\begin{aligned}\mathbb{E}_n \bigg[ \sup_{0 \le t \le T}\bigg| \int_0^t \sum_{j \in \mathbb{Z}}\overline{W}_{j -1}( s ) ( \overline{W}_j ( s ) - \overrightarrow{W}^\ell_j ( s ) ) \varphi_j (s) ds \bigg|^2 \bigg] \le C \frac{\ell}{n^2}\int_0^T \sum_{j \in \mathbb{Z}}( \varphi_j (t ) )^2 dt .\end{aligned}
\sum_{j = 0}^k{j^4 G_j}= - 257G_0 - 416G_1 + (k^4 - 4k^3 + 30k^2 - 124k + 257)G_k &\qquad+ (k^4 - 8k^3 + 48k^2 - 200k + 416)G_{k + 1}\,,
P_{a , b}( \mu , \nu ; 0 ; q ) = M_{a , b}( \mu , \nu ; 0 ; q ) = \left[ \begin{array}{c}{\mu + \nu}\\{\mu}\\ \end{array}\right]_{q}= \left[ \begin{array}{c}{\mu + \nu}\\{\nu}\\ \end{array}\right]_{q}.
\sigma_{\min}^2(\hat{V}^{\intercal}V) = \min_{x \in \mathbb{R}^r}\frac{\|\hat{V}^{\intercal}Vx\|_2^2}{\|x\|_2^2}= \min_{x \in \mathbb{R}^r}\frac{\|Vx\|_2^2 - \|\hat{V}^{\intercal}_{\perp}Vx\|_2^2}{\|x\|_2^2}= 1 - \max_{x\in \mathbb{R}^r}\frac{\|\hat{V}_{\perp}Vx\|_2^2}{\|x\|_2^2}= 1 - \|\hat{V}_{\perp}^{\intercal}V\|^2,
\displaystyle\lim_{k\to\infty}k(\min_{1\leq i\leq k}\{^2(0, F(u^{i+1}, v^{i+1}, x^{i+1})) + \|\mathcal{A}^*u^{i+1}+ \mathcal{B}^*v^{i+1}-c\|^2 \})= 0.
S_z^*(a(x)w^x\vartheta) =a(x+z)w^{x+z}\vartheta, T_{\beta}^*(a(x)w^x\vartheta) =\beta(x)a(x)w^x\vartheta,\ T_{\beta}^*S_z^*(a(x)w^x\vartheta) =\beta(x+z)a(x+z)w^{x+z}\vartheta, S_z^*T_{\beta}^*(a(x)w^x\vartheta) =\beta(x)a(x+z)w^{x+z}\vartheta,
\mathcal{R}_{2,P}(R_2')= \left\{\begin{aligned}(R_1,R_3): R_1,R_3\geq 0,\R_1 \leq I(X_1;Y_1|U_1) \triangleq r_a , R_3\leq \min\{I(X_2;Y_3),\I(U_2;Y_2)+I(X_2;Y_3|U_2)\}\triangleq r_b \end{aligned}\right\}.
\Delta(K_i) = K_i \otimes K_i, \Delta(E_i) = E_i \otimes K_i + 1 \otimes E_i, \Delta(F_i) = F_i \otimes 1 + K_i^{-1}\otimes F_i, S(K_i) = K_i^{-1}, S(E_i) = - E_i K_i^{-1}, S(F_i) = - K_i F_i, \varepsilon(K_i) = 1, \varepsilon(E_i) = \varepsilon(F_i) = 0.
\begin{array}{ll}\delta F^{\nu}=F^{\nu}-F^0\ast^g_{sp}\Gamma^v_{\nu}+\sum_{j=1}^{2d}W^S_j(F^{\nu},F^{\nu})\ast^g_{sp}\Gamma^v_{\nu}=\sum_{j=1}^dP^S_j(F^{\nu},F^{\nu})\ast^g \Gamma^{v,*}_{\nu,v_j}+\sum_{j=d+1}^{2d}W^S_j(F^{\nu},F^{\nu})\ast^g \Gamma^v_{\nu}\end{array}
\Delta\log\left(1+V_{k}(x)\right)=\left(\frac{(1-a)V_{k}(x)}{1+V_{k}(x)}\right)\Delta u_k+\frac{\left|\nabla V_{k}(x)\right|^{2}}{V_{k}(x)\left(1+V_{k}(x)\right)^{2}}.
A_{\alpha,g}= A_\alpha+r^2(\Delta_g-\Delta_0)=A_\alpha+r^2\partial_i((g^{ij}-\delta^{ij})\partial_j), -P_g(r^\alpha \varphi)= r^{\alpha-6}(A_{\alpha-4}A_{\alpha-2}A_\alpha \varphi+K_\alpha \varphi),
\beta_{n,q}\left(x\right)=\frac{1}{\left(1-q\right)^{n}}\sum_{l=0}^{n}\dbinom{n}{l}\left(-1\right)^{l}q^{lx}\frac{l+1}{\left[l+1\right]_{q}},\quad\left(\mbox{see \cite{key-5}}\right).
-2\mathcal{R}\left( \dot{Q}\right) =\partial_{t}\left( \mathcal{T}_{0}\left( \dot{Q}\right) +\mathcal{V}_{0}\left( Q\right) \right)=2\operatorname{Im}\zeta\left( \mathcal{T}_{0}\left( \dot{Q}\right)+\mathcal{V}_{0}\left( Q\right) \right) .
T_n^{(\mathbf k)}(x_1,\dots,x_{m})=\frac{(-1)^{\binom{|\mathbf k^-|}2}\,T(x_1,\dots,x_m,\boldsymbol\xi^{{\mathbf k}^+};\boldsymbol\xi^{{\mathbf k}^-})}{2^{|\mathbf k^-|}\prod_{i,j=0}^3G(\xi_i,\xi_j)^{k_i^-k_j^+}\prod_{j=1}^m\prod_{i=0}^3G(x_j,\xi_i)^{k_i^-}}.
\biggr(\bigl((t_i:s_i))_{\lambda_i \not=1}\bigr) \,,\, (g_1,\ldots,g_d) \biggr) \,\,\mapsto \,\,\,g(u,v) \, \,=\, \sum_{i=1 \atop \lambda_i \neq 1}^d (s_i u + t_i v)^{n-\lambda_i+2}\cdot g_i(u,v).
{\frac{2 p^{-}}{\mu}}= \sum_{n}\left[ e^{U_{0}}\,{\frac{s}{J_{V}}}N_{n}^{r , w}| n | + \sqrt{1 + e^{2 U_{0}}\,{\frac{s^{2}}{J_{V}^{2}}}\, n^{2}}\, \, N_{n}^{y}\right] \ .
\delta_\kappa A^{ij}=\frac 12Str\left( \left( \begin{array}{cc}0 \left( \not{X}_k\kappa^{(i}\right) \varepsilon^{j)k}-\left( \tilde{\kappa}^{(i}\not{X}_k\varepsilon^{j)k}\right) 0\end{array}\right) \partial_\tau t\left( \Theta \right) t^{-1}\left( \Theta \right)\right) .
( T_{i}\otimes T_{j})_{{\cal H}_{\{\alpha_{k},{\alpha \prime}_{k}\}}}= \prod_{k \ne i , j}^{r}\delta_{\alpha_{k},{\alpha \prime}_{k}}( T^{a})_{\alpha_{i}{\alpha \prime}_{i}}( T^{a})_{\alpha_{j}{\alpha \prime}_{j}}
\rho_{(k), N}^{(\rm cJ)}(\theta_1,\dots, \theta_k) = N (N - 1) \cdots (N - k + 1) \int_{(-\pi, \pi]^{N-k}}d \theta_{k+1}\cdots d \theta_N \, p_{N,\beta}^{(\rm cJ)}(\theta_1,\dots, \theta_N).
\begin{aligned}c_N^\pm \sim N^{N/2}\int_{\mathbb{R}^{2N}}\delta\Big( E_N - E_N^\pm - \frac{1}{2}\sum_{j=1}^N[ \frac{\tilde{p}_j^2}{N}+ \lambda_j^\pm(b_j^\pm)^2] \Big)d\tilde{p}_1\dots d\tilde{p}_N db_1^\pm\dots db_N^\pm \end{aligned}
\left|Q^{(1)}\vert E, p^3 \rangle\right|^2 =\langle E, p^3 \vert \left(Q^{(1)}\right)^2 \vert E, p^3 \rangle =\langle E, p^3 \vert H'-P^3 \vert E, p^3 \rangle =E-p^3=0.
\underline{\mbox{det}}(\mbox{\bf u}) = \frac{4q^2}{(q^2 +1)^2}\tilde{t}^2 - q^2 \tilde{x}^2 -q^2 \tilde{y}^2 - \frac{2(q^4 +1)q^2}{(q^2 +1)^2}\tilde{z}^2 +2q \left( \frac{q^2 -1}{q^2 +1}\right)^2 \tilde{t}\tilde{z}.
e^{\phi}\approx \left({\frac{\bar{\mu}}{\bar{r}}}\right)^{\frac{( D - 2 ) ( D - 3 ) a}{4 ( D - 3 ) - 2 \Delta}}, \A_{t}= - \left({\frac{\bar{r}}{\bar{\mu}}}\right)^{\frac{( D - 3 ) \Delta}{2 ( D - 3 ) - \Delta}},
\begin{array}{l}{\left\{e^{- 1}n_{\lambda}\cdot \Gamma , e^{- 1}n_{y}\cdot \Gamma , e^{- 1}n_{A}\cdot \Gamma , e^{- 1}n_{\psi}\cdot \Gamma , e^{- 1}n_{\varphi}\cdot \Gamma \right\},}\\ \end{array}
\int_2^z \frac{S(t)}{t^2}\, dt = \int_2^z \bigg( \frac{(\log\log t)^2}{t \log t}+ O\bigg(\frac{|\!\log\log t|}{t \log t}\bigg) \bigg) \, dt &= \frac{(\log\log t)^3}{3}\bigg|_2^z + O\big( (\log\log t)^2 \big|_2^z\big) = \frac{(\log\log z)^3}3 + O\big( (\log\log z)^2 \big)
|vx_0-x_0| \geqslant n \; |{u}_jx_0-{u}_{j+1}x_0| - 48n \delta. & = 5\nu \; |{u}_jx_0-{u}_{j+1}x_0| - 240 \nu \delta > 3 \nu \, |{u}_jx_0-{u}_{j+1}x_0| + A \delta + 10^7 \delta.
(-x_{a_1}-x_{a_2}-\dots-x_{a_{j-1}})^j\cdot x_{b_{j+1}}\cdots x_{b_{\ell}}=(x_{b_1}+x_{b_2}+\dots+x_{b_{\ell}}+x_{a_j}+x_{a_{j+1}}+\dots+x_{a_{n-\ell}}-\frac{n(n+1)}{2}t)^j\cdot x_{b_{j+1}}\cdots x_{b_{\ell}}.
\langle s_{1}\cdot \cdot \cdot s_{N}| \Psi \rangle = \left({\frac{B}{4}}\right)^{N / 4}e^{- B \sum_{i}s_{i}^{2}/ 2}e^{-{\frac{1}{4 B}}\sum_{i}\left({\frac{\partial}{\partial s_{i}}}\right)^{2}}\Psi ( \sqrt{2 B}s_{1}, \cdot \cdot \cdot , \sqrt{2 B}s_{N}) .
\int_{S^{n-1}}A_{K,m, \xi}^{(q)}(0) H_l(\xi) \, d\xi = \frac{m}{(m-q)(n-1)}\int_{S^{n-1}}I_q \left( \rho_K^{m-q}\right) (\xi) H_l(\xi) \, d\xi &=\frac{m}{(m-q)(n-1)}\int_{S^{n-1}}\rho_K^{m-q}(\xi) I_q \left( H_l \right) (\xi) \, d\xi & = \frac{m}{(m-q)(n-1)}\lambda_l(q) \int_{S^{n-1}}\rho_K^{m-q}(\xi) H_l(\xi) \, d\xi.
\begin{vmatrix}\alpha_{k_1}\sigma(\alpha_{k_1}) \dots \sigma^{t-1}(\alpha_{k_1}) \alpha_{k_2}\sigma(\alpha_{k_2}) \dots \sigma^{t-1}(\alpha_{k_2}) \vdots \vdots \ddots \vdots \alpha_{k_{t}}\sigma(\alpha_{k_{t}}) \dots \sigma^{t-1}(\alpha_{k_{t}})\end{vmatrix}\neq 0.
\int_\Omega\gamma^{-1}(v)w^2dx\leq \int_\Omega (bv^k+\gamma^{-1}(s_b))w^2dx \leq \int_\Omega \left(b\left(\frac{1}{1-\gamma(a)}(w+K)\right)^k+\gamma^{-1}(s_b)\right)w^2dx \leq C\int_\Omega w^{k+2}dx+C
P_t=\int\limits_{-{\pi}}^{{\pi}}{{P_{\max}}{{10}^{- 1.2\frac{{{\theta^2}}}{{\theta_{3dB}^2}}}}d\theta}={P_{\max}}\frac{{\sqrt \pi erf \left({{\pi}\sqrt{\frac{{1.2}}{{\theta_{3dB}^2}}\ln 10}}\right)}}{{\sqrt{\frac{{1.2}}{{\theta_{3dB}^2}}\ln 10}}}
\frac{G(z)}{A(z)}\bigg( \sum_n \frac{|c_n|^2}{z-t_n}+ R(z) \bigg) = \bigg(\sum_n \frac{c_n\mu_n^{1/2}}{z-t_n}\bigg) \cdot\bigg(\sum_n \frac{G(t_n) \bar c_n}{A'(t_n)\mu_n^{1/2}(z-t_n)}\bigg),
\lim_{r\rightarrow \infty}\frac{\int_{b(x)\leq r}\Big||\nabla b|^2- 1\Big|^2 dx}{\mathrm{Vol}(b(x)\leq r)}= \lim_{r\rightarrow \infty}\frac{\int_{b(x)\leq r}\Big|Hess(b^2)- 2g\Big|^2 dx}{\mathrm{Vol}(b(x)\leq r)}= 0 \ ; \lim_{\rho(x)\rightarrow \infty}\frac{b(x)}{\rho(x)}= 1\ ; |\nabla b|\leq 1
F_1(k) = \sum_{h=0}^{k-1}\frac{\big | |k|^{2(s-1)}k - |h|^{2(s-1)}h \big |^2}{(1+|k-h|)^{2s}(1+|h|)^{2r}}, F_2(k) = \sum_{h\geq k}\frac{\big | |k|^{2(s-1)}k - |h|^{2(s-1)}h \big |^2}{(1+|k-h|)^{2s}(1+|h|)^{2r}}, F_3(k) = \sum_{h<0}\frac{\big | |k|^{2(s-1)}k - |h|^{2(s-1)}h \big |^2}{(1+|k-h|)^{2s}(1+|h|)^{2r}},
\!{\rm Ker}(\nabla g(x))\cap \mathcal{T}_{\mathcal{N}_{K}(g(x))}(\lambda)=\{0\}\Longleftrightarrow{\rm Ker}(\nabla g(x))\cap D\mathcal{N}_{K}(g(x)|\lambda)(0)=\{0\}, \Longleftrightarrow{\rm SRCQ\for\the\system}\g(x)\in K\{\rm at}\x\{\rm w.r.t.}\ \lambda.
\int\limits_0^\infty{{x^{s - 1}}\sum\limits_{n = 0}^\infty{\frac{{\phi \left({2n}\right)}}{{\left({2n}\right)!}}{x^{2n}}dx ={{\left({- 1}\right)}^{\frac{s}{2}}}}}\phi \left({- s}\right)\Gamma \left( s \right)\cos \frac{{\pi s}}{2}
G(x,y)=\sum_{\sigma}\frac{f_{\sigma}(x) f^*_{\sigma}(y)}{(\omega_{\sigma}-E)}+ \sum_{\sigma}\frac{f_{\sigma}(x)}{(\omega_{\sigma}-E)}\frac{f^*_{\sigma}(a)}{\sqrt{\omega_{\sigma}}}\Phi^{-1}(E) \sum_{\nu}\frac{f_{\nu}(a)}{\sqrt{\omega_{\nu}}}\frac{f^*_{\nu}(y)}{(\omega_{\nu}-E)}
F(x,y)=\frac{\langle{\bf A}_y^{-1}(x-y),(x-y)\rangle^{\frac{2-n}{2}}}{(n-2)|S^{n-1}|\sqrt{\det{\bf A}_y}}\,e^{I_y(\sqrt{\langle{\bf A}_y^{-1}(x-y),(x-y)\rangle})-I_y(0)}(1+\xi_y(x))
z_j^+\Omega_\lambda= \begin{cases}d_{j}^+(\lambda)\prod_{\ell=1}^{j-1}(\lambda_\ell-\lambda_j-\ell+j)\Omega_{\lambda+\epsilon_j}, \lambda+ \epsilon_j\in \mathbb{P}\ell(\lambda + \epsilon_j)\leq p, 0, ,\end{cases}
\nu([g,f]) \leq \nu(g[f,hh_0h^{-1}]g^{-1})+\nu([f,hh_0h^{-1}]^{-1}) &=2\nu([f,hh_0h^{-1}]) \leq 2(\nu(f(hh_0h^{-1})f^{-1})+\nu((hh_0h^{-1})^{-1})) =4\nu(hh_0h^{-1})=4\nu(h_0).
\hat{L}\Psi^{( 1 )}= - \sqrt{2}\left( \begin{array}{c}{{D_{i}^{( 0 )}\phi^{( 0 )}}}\\{{( D_{i}^{( 0 )}\phi^{( 0 )})^{\ast}}}\\{{\lambda F_{i k}^{( 0 )}}}\\ \end{array}\right) K_{a}^{i a},
p_0(z;q)= \frac{1}{\beta_0}z, p_{n+1}(z;q)= \frac{1}{\pi_{n(n+1)}}\Big[z^{2}p_{n}(z;q)-\sum_{k=0}^{n}\pi_{nk}p_k(z;q)\Big], \, n\in \mathbb{N}_0.
g^*(\mathcal{R}^{r}_{X \to Z_i, *}(s)) =a(Z_i) =a \circ g(g^{-1}(Z_i)) = \mathcal{R}^{r}_{X \to g^{-1}(Z_i), *}\left(a \circ g \prod_{j}\bigg(\frac{dy_j}{y_j}\bigg)^{\otimes r}\right) = \mathcal{R}^{r}_{X \to g^{-1}(Z_i), *}(g^*s),
\nabla^\varphi v\mathbf{N}=-\partial_1\varphi\partial{_1^\varphi}v-\partial_2\varphi\partial{_2^\varphi}v+\partial{_z^\varphi}v=\frac{1+|\nabla_y\varphi|^2}{\partial_z\varphi}\partial_zv-\partial_1\varphi\partial_1v-\partial_2\varphi\partial_2v,
\frac{\left(1+\frac{\alpha}{n}\right)^{n+1}}{\frac{\alpha}{n}}= \frac{\left(1+\left(\frac{\alpha}{n}\right)^{1-\frac{n+1}{j-1}}\lambda_{n,j}(\frac{\alpha}{n})^{\frac{n+1}{j-1}}\right)^{n+1}}{\left(\frac{\alpha}{n}\right)^{1-\frac{n+1}{j-1}}\lambda_{n,j}(\frac{\alpha}{n})^{\frac{n+1}{j-1}}}.
\kappa_\chi(z_{\alpha(1)}, \ldots, z_{\alpha(n)}) = \sum_{\pi \in BNC(\chi_\alpha)}\kappa_\pi(z'_{\alpha(1)}, \ldots, z'_{\alpha(n)}) \cdot \kappa_{K_{BNC}(\pi)}(z''_{\alpha(1)}, \ldots, z''_{\alpha(n)}).
\dfrac{xq^0\cdot xq^{-1}}{1-xq^0\cdot xq^{-1}}\left(\prod\limits_{j\,\,, j\geq 2}\dfrac{1}{1-xq^j\cdot xq^{j-1}}(1+xq^j + xq^j\cdot xq^j)\right)\left(\prod\limits_{k\,\,, j\geq 3}\dfrac{1}{1-xq^k}\right) =x^2q^{-1}\left(x^2q^{-1};q^4\right)_\infty^{-1}\left(xq^2;q \right)_\infty^{-1}\left(x^3q^6;q^6 \right)_\infty.
\lbrack\tilde{L}_{1},[\tilde{r}_{12},\tilde{r}_{13}]+[\tilde{r}_{12},\tilde{r}_{23}]+[\tilde{r}_{32},\tilde{r}_{13}]+\{\tilde{L}_{2},\tilde{r}_{13}\}-\{\tilde{L}_{3,}\tilde{r}_{12}\}]+\mbox{cycl.term}=0.
\Delta^{\mu \nu}= \frac 1 4 \left( \Pi_{\; \rho}^{\rho \; , \mu \nu}+ \tilde{\Gamma}^{\mu \nu}\right) -{\frac 1{3 2}}\left( \Pi_{\; \; \rho \; \; \sigma}^{\rho \; \; , \sigma}+ \tilde{\Gamma}_{\; \rho}^{\rho}\right) \, \eta^{\mu \nu}.
F(r; S^{(\delta,+)}_\delta(v_0+v_1))=F\Big(r;G^{\rm neum}_{\delta}*[v_0(\cdot,\delta)+j\delta D_0]\Big) =\int v_0(r')P_{r';0}\Big[B_\delta \ge r\Big] dr' +j\delta P_{0;0}\Big[B_\delta \ge r\Big]
v_1=\begin{pmatrix}1 0 \vdots 0 \end{pmatrix},\;v_2=\begin{pmatrix}0 1 \vdots 0 \end{pmatrix},\;\dots ,\;v_{n-1}=\begin{pmatrix}0 0 \vdots 1 \end{pmatrix}, \;v_n=\begin{pmatrix}-1 -1 \vdots -1 \end{pmatrix}.
\left.\begin{array}{ll}a_{n+1}\left(\lambda_{n+1}p_n-\lambda_n p_n\right)+\left(\lambda_{n+1}p_{n-1}-\lambda_{n-1}p_{n-1}\right) =0 a_{n+1}\left(\lambda_{n+1}q_n-\lambda_n q_n\right)+\left(\lambda_{n+1}q_{n-1}-\lambda_{n-1}q_{n-1}\right) =0\end{array}\right.
f_n(\pmb{t}_n,\pmb{x}_n,t,x) = f_n(t_1,\dots, t_n,x_1,\dots, x_n,t,x) \coloneqq G_{t-t_n}(x-x_n)\times \dots \times G_{t_2-t_1}(x_2-x_1)\mathbf{1}_{T_n(t)}(\pmb{t}_n)
\mathcal{R}^- = \frac{1}{2}\sum_{m \geq 1}\sum_{n \geq 1}\frac{A(n, m)}{(m^2 n)^{\frac{1}{2}}}g\Bigl( \frac{m^2 n}{N}\Bigr) \sum_{c > 0}\frac{S(-n, p; c)}{c}H_{m, n}^- \Bigl ( \frac{4 \pi \sqrt{n p}}{c}\Bigr )
C_{n,f}=(-1)^\nu G_f^{(\nu)}(k_\nu)\left(\dfrac{M}{2\pi}\right)^{2n+\nu+1}\dfrac{(2n+\nu)!}{k_\nu^{2n+\nu+1}\sqrt{M}}\left(1+ \frac{1}{G_f^{(\nu)}(k_\nu)}\sum_{k=k_{\nu}+ 1}^\infty\dfrac{G_f^{(\nu)}(k)}{k^{2n+\nu+1}}\right).
\mathrm{w i t h}\k_{+}L \in [ ( n - 1 ) \, \pi , \n \pi ] \ , \ 1 \leq n < \alpha_{+}\frac \pi L \quad \mathrm{a n d}\ 2 \, m \, E = \zeta ( k_{+})^{2}= \zeta ( k_{-})^{2}\ . \nonumber
d_{\max}(b,b_*/C) =\epsilon_b\leq d(a',b_*)+d(a,b)=d(a',b_*)+d(a',b), d_{\min}(b,b_*/C) =\delta_b\geq |d(a',b_*)-d(a,b)|=|d(a',b_*)-d(a',b)|.
Q^{\rm R}_{\mu_1\dots\mu_p}\equiv{Q^{\rm R}_p\over v_p}\int_{{\cal M}_p}dX^{\mu_1}\wedge\cdots dX^{\mu_p}\ ,\quad{\rm with}\quad Q_p^{\rm R}={1\over{\rm Vol}\, S^{8-p}}\int_{S^{8-p}}\!\!{}^*G^{(p+2)}\ .
P\{\begin{matrix}a&b&c&\\ \alpha&\beta&\gamma&z\\ \alpha^{\prime}&\beta^{\prime}&\gamma^{\prime}&\end{matrix}\}=P\{\begin{matrix}\eta&\zeta&\theta&\\ \alpha&\beta&\gamma&u\\ \alpha^{\prime}&\beta^{\prime}&\gamma^{\prime}&\end{matrix}\}
4(s-1) P=4(s-1)\frac{F_s}{|x-y|^2}\frac{\rho^{s-1}}{(\rho+1)^{\frac{N}{2}-1}}+2 V_s(\rho)\nabla F_s\cdot\nabla \rho +F_sV_s'(\rho) |\nabla \rho|^2 +F_sV_s(\rho)\Delta \rho &=F_s\Big[\frac{V_s(\rho)(4(s-1)(\rho+1)+2(2s-N)(x-y)\cdot \nabla \rho+|x-y|^2\Delta \rho)}{|x-y|^2}+V_s'(\rho) |\nabla \rho|^2 \Big].
X_p^+(T):=\{\sum_{i=1}^ma_i\delta_i+\sum_{j=1}^n b_j\epsilon_j\in X^+(T)\mid 0\leq(a_{i}-a_{i+1}), (b_{j}-b_{j+1})\leq p-1,\cr \forall i=1,\ldots,m-1; j=1,\ldots,n-1\}.
\|\theta_p\|_{\frac{4}{\alpha-1}}\leq \lambda_0^{\frac{1-\alpha}{2}}\|\theta_p\|_{\infty}\leq \lambda_0^{\frac{1-\alpha}{2}}R_\infty <{\textstyle \frac{1}{2}}\lambda_p^{\frac{\alpha-1}{2}}c_0\nu(\alpha-1)^2,
\frac{\partial}{\partial t}(\alpha_{k}\rho_{k}\vec{u_{k}})+\nabla\cdot(\alpha_{k}\rho_{k}\vec{u_{k}}\vec{u_{k}})=-\alpha_{k}\nabla P+\nabla\cdot(\alpha_{k}\overline{\tau_{k}})+\alpha_{k}\rho_{k}\vec{g}+\vec{M}_{I,k}
{mt_0+1\over mt_1+1}=1+O\Bigl({1\over n\log{n}}\Bigr), begin{align*}5pt]\sqrt{mt_0+1\over mt_2+1}=1+O\Bigl({mt_0+1\over mt_2+1}-1\Bigr)=1+O\Bigl({1\over n\log{n}}\Bigr).
(D^\alpha_x u)_x(x)=\frac{1}{\Gamma(1-\alpha)}\left(\frac{\alpha(u(0)-u(x))+(\alpha+1)u'(x)x}{x^{\alpha+1}}\right. &\quad\left.+\alpha(\alpha+1)\int_0^{x}\frac{u(y)-u(x)-u'(x)(y-x)}{(x-y)^{\alpha+2}}dy\right).
k_\theta \cdot f_- = \frac{1}{\sqrt{2}}\left(s(\theta), c(\theta), s(\theta), c(\theta) \right) k_\theta \cdot f_+ = \frac{1}{(108)^{\frac{1}{4}}}\left(s(3\theta), 3c(3\theta), -3 s(3\theta), -c(3\theta)\right).
Y_{t}:=Y_{s_{0},t}^{1}-Y_{s_{0},t}^{2}y_{1}^{s,t}:=y_{1}^{1,s,t}-y_{1}^{2,s,t}\Gamma^{i,s,t}:=Y_{s_{0},t}^{i}-y_{1}^{i,s,t}=Y_{s_{0},t}^{i}-Y_{s_{0},s}^{i}-\left( y_{1}^{i,s,t}-Y_{s_{0},s}^{i}\right) \Gamma^{s,t}:=\Gamma^{1,s,t}-\Gamma^{2,s,t}=Y_{t}-y_{1}^{s,t}
f(z)=\int_{0}^{1}\frac{d}{dt}f(tz_{1},\cdot\cdot\cdot,tz_{n})dt=\sum_{i=1}^{n}z_{i}\int_{0}^{1}\frac{\partial f(z)}{\partial z_{i}}|_{z=(tz_{1},...,tz_{n})}dt,
\prod_{k = 0}^{\infty}\left( \frac{2 k + 1}{2 \pi}\right)^{2 6 / 2}= \prod_{k = 0}^{\infty}\left[ \left( \frac{2 k}{\pi}\cdot \frac{2 k + 1}{\pi}\right) \left( \frac{4 \pi}{2 k}\cdot \frac{2 k + 1}{\pi}\right) \right]^{2 6 / 4}.
\begin{array}{lll}\Pi = \Pi_{\rm rd}\sqcup \Pi_1 \sqcup \Pi_0^\lambda \;,\; \Pi^\lambda=\Pi_{\rm rd}\sqcup\Pi_1 \hline\Pi_{\rm rd}= \{\alpha\in\Pi: \lambda_\alpha\geq 2\}\textrm{root-distinct}\Pi_{1}= \{\alpha\in\Pi: \lambda_\alpha=1\}\textrm{non-root-distinct}\Pi_{0}= \{\alpha\in\Pi: \lambda_\alpha=0\}\textrm{orthogonal}\end{array}
\operatorname{t a n}( \delta_{j}^{( 2 )}- \delta_{j}^{( 1 )}) = -{\frac{\operatorname{s i n}( \delta_{j}^{( 4 )}- \delta_{j}^{( 1 )})}{\bar{\omega}+ \operatorname{c o s}( \delta_{j}^{( 4 )}- \delta_{j}^{( 1 )})}}
\int_0^{2\pi}\frac{1+\zeta e^{it}}{1-\zeta e^{it}}d(F(t)g,h)_{\mathcal{H}}= 2(\mathbf{R}_\zeta x_{g,0}, x_{h,0})-(S_0 g,h)_{\mathcal{H}},g,h\in\mathcal{H},\ \zeta\in \mathbb{T}_e.
\Omega = - H \left( t , q , p = \psi ( t , q ) \right){\mathrm{d}}t + \psi ( t , q ) \, \mathrm{d}q \stackrel{\textstyle !}{=}\mathrm{d}S ( t , q ) = \partial_{t}S ( t , q ) \, \mathrm{d}t + \partial_{q}S ( t , q ) \, \mathrm{d}q \, .
Y=W \hat Y W^{-1}=\left (\begin{array}{ccc}-\textstyle\frac{(ad+2bc)}{2}\textstyle\frac{3ab}{2}a t -\textstyle\frac{3cd}{2}\textstyle\frac{(bc+2ad)}{2}ct \textstyle\frac{3d}{4t}-\textstyle\frac{3b}{4t}\textstyle\frac{1}{2}\end{array}\right).
O_{F_{2 2}}\xi_{n}\equiv - \frac{d^{2}}{d z^{2}}\xi_{n}- \frac{2 m^{2}\alpha}{\lambda^{2}}( 2 \alpha + \lambda ) s e c h^{2}( m z ) \xi_{n}+ m^{2}( \frac{2 \alpha}{\lambda^{2}}( 2 \alpha + \lambda ) - \frac{2 \alpha}{\lambda}) \xi_{n}= \tilde{\omega}_{n}^{2}\xi_{n}.
\mathcal{J}_2(x) =\frac{1}{\mathbb{P}(S_\eta> x)}\sum_{k=1}^{\infty}\mathbb{P}(S_{\kappa+k}> x-1) \mathbb{P}(\eta= \kappa+k) &\leqslant \frac{c_3}{\mathbb{P}(S_\eta> x)}\sum_{k=1}^{\infty}\mathbb{P}(S_{\kappa+k}> x) \mathbb{P}(\eta= \kappa+k) &\leqslant \frac{c_3 \mathbb{P}(S_\eta> x)}{\mathbb{P}(S_\eta> x)}=c_3
E_{1}^{rs}=H^s_{rig}(Q^{r+1}/{\mathfrak S}^0)\Longrightarrow H^{r+s}_{rig}(Q/{\mathfrak S}^0)\tag*{$(U)_{0}$}E_{1}^{rs}=H^s_{dR}(]Q^{r+1}[_{\mathfrak Q})\Longrightarrow H^{r+s}_{dR}(X)\tag*{$(U)_{\pi}$}
M = \max\Big\{\|e^{\tau_{10}A}(\overline{v}_0)_1\|_{H^r}^2 + \|e^{\tau_{10}A}(\widetilde{v}_0)_1\|_{H^r}^2, \|e^{\tau_{20}A}(\overline{v}_0)_2\|_{H^r}^2 + \|e^{\tau_{20}A}(\widetilde{v}_0)_2\|_{H^r}^2\Big\}.
V^{-1}=[\begin{matrix}1&x_{0}&...&x_{0}^{n}\\ \vdots&\vdots&&\vdots\\ 1&x_{n}&...&x_{n}^{n}\end{matrix}]^{-1}=L=[\begin{matrix}L_{00}&\cdot\cdot\cdot&L_{0n}\\ \vdots&&\vdots\\L_{n0}&\cdot\cdot\cdot&L_{nn}\end{matrix}]
S = \int d \tau \sqrt{- g_{\tau \tau}}g^{\tau \tau}\left( \dot{X}^{\hat{m}}- \bar{\theta}\Gamma^{\hat{m}}\dot{\theta}\right)^{2}\ , \qquad \hat{m}= 0 , 1 , \cdots , 8 , 9 , 1 0 .
\nu_{i}( x ) = V^{- \frac{1}{2}}\sum_{{\bf k}, r}\left[ u_{{\bf k}, i}^{r}e^{- i \omega_{k , i}t}\alpha_{{\bf k}, i}^{r}\: + v_{-{\bf k}, i}^{r}e^{i \omega_{k , i}t}\beta_{-{\bf k}, i}^{r \dag}\right] e^{i{\bf k}\cdot{\bf x}}, i = 1 , 2 .
\left[ \phi (t,\, z), \dot \phi (t,\, z^\prime) \right]= i\delta(z-z^\prime)\,,\quad \left\{\psi_\alpha (t,\, z), \bar \psi_\beta(t,\,z^\prime)\right\}= \left(\gamma^0\right)_{\alpha \beta}\delta(z-z^\prime)
L^{2}_{\sigma}(\Omega) = \{u \in (L^{2}(\Omega))^{2}\,|\,\langle u, \nabla \varphi\rangle_{(L^{2}(\Omega))^{2}}= 0 \varphi \in L^{1}_{\mathrm{loc}}(\Omega) \nabla \varphi \in (L^{2}(\Omega))^{2}\}.
A_3(9) =\{((9),9,0),\, ((6,3),6,0),\, ((6,3),3,0), \,((6,2,1),6,0), &\quad\ ((5,3,1),3,0),\,((4,3,2),3,0),\,((4,3,1^2),3,0), &\quad\ ((3^2,2,1),3,0),\, ((3^2,2,1),3,1),\,((3,2^2,1^2),3,0)\}
{\cal V}\equiv \left( \begin{array}{c}{U_{\alpha}^{T}}\\{\bar{U}^{\alpha T}}\\ \end{array}\right) \equiv \left( \begin{array}{c c}{e_{\alpha}^{A}}&{h_{A \alpha}}\\{g^{\alpha \bar{\beta}}\bar{e}_{\bar{\beta}}^{A}}&{g^{\alpha \bar{\beta}}\bar{h}_{A \bar{\beta}}}\\ \end{array}\right) \ ,