harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\partial_{t}(\frac{1}{2}\eta^{2})+\partial_{x}\{\frac{1}{2}\sqrt{gh}\eta^{2}+\frac{1}{2}\sqrt{\frac{g}{h}}\eta^{3}+\frac{1}{12}h^{2}\sqrt{gh}[\partial_{x}^{2}(\eta^{2})-3(\partial_{x}\eta)^{2}]\}=0,
\tfrac{d}{d\tau}|_{\tau=0}( \mathcal L_+(\psi_{-,T}^1 +\phi_{-,T}+ \tau \zeta_{-,T},\psi_{+,T}^1+\phi_{+,T}+\tau\zeta_{+,T}) - \mathcal L_+(\phi_{-,T}+\tau\zeta_{-,T},\phi_{+,T}+\tau\zeta_{+,T}))
\tilde{\cal Z}_{\{n^{a}_{E},n_{M}^a\}}= \sqrt{2}\,Q^a\, a_a^*\,,\qquad\left|\tilde{\cal Z}_{\{n^{a}_{E},n_{M}^a\}}\right|=\left|{\cal Z}_{\{-n^{a}_{E},n_{M}^a\}}\right|\,,
d_X(F^2(x_0,y_0),F(x_0,y_0)) =d_X(F(F(x_0,y_0),G(y_0,x_0)),F(x_0,y_0)) &\leq k\d_X(F(x_0,y_0),x_0)+ l\d_Y(G(y_0,x_0),y_0) &\leq (k+l)\ [d_X(x_1,x_0)+ d_Y(y_1,y_0)] &< (m+n)\ [d_X(x_1,x_0)+ d_Y(y_1,y_0)]
q_{1}(s,t) :=\psi_{1}(s,t)\exp \left\{\frac{{\rm i}}{2}\int^{t}_{0}\lvert\psi_{1}(0,\tau )\rvert^{2}\d\tau \right\}, begin{align*}3mm] q_{2}(s,t) := \psi_{2}(s,t)\exp \left\{\frac{{\rm i}}{2}\int^{t}_{0}\lvert \psi_{2}(0,\tau )\rvert^{2}\d\tau \right\},
\mathcal{S}_{q}(u)\mathcal{C}_{q}(v)-\mathcal{C}_{q}(u)\mathcal{S}_{q}(v) =\frac{u-v}{1-q}\,_{3}\phi_{3}\left[\begin{matrix}0, u^{-1}vq, uv^{-1}q q^{3/2}, -q^{3/2}, -q \end{matrix}\ ;q, -uvq \right]\!.
\, : \! \varphi ( x ) \! : \quad \leftrightarrow p ( \underline{{\theta}}, \underline{{l}}) = \frac{\pi \nu}{\beta}\left( \frac{2}{F ( i \pi ) \sin \pi \nu}\right)^{\frac{n}{2}}\sum_{i = 1}^{n}( - 1 )^{l_{i}}
\bigwedge_{i=1}^{n-2}\neg c_i\lor c_{i+1}\bigwedge_{i=1}^n c_{i-1}\lor\neg a_i\lor b_i\qquad\bigwedge_{i=1}^n c_{i-1}\lor a_i\lor b_i\lor\neg c_i \bigwedge_{i=1}^n c_{i-1}\lor\neg a_i\lor\neg b_i\lor\neg c_i\qquad\bigwedge_{i=1}^n c_{i-1}\lor a_i\lor\neg b_i\lor c_i
\delta_1+\delta_2+\delta_3+\delta_4=\delta, \ \ \delta_1,\delta_2,\delta_3,\delta_4>0 \frac1{\theta_1}+\frac1{\theta_2}+\frac1{\theta_3}+\frac1{\theta_4}=1, \ \ \theta_1=\theta_3=\frac2{\delta}, \ \ 1<\theta_2, \theta_4<\infty.
\frac{5}{30k+3}= [6k,2,-3] = [\underbrace{3,0,3,0,\ldots,0,3}_{4k-1},2,-3], \frac{5}{30k-3}= [6k,-2,3] = [\underbrace{3,0,3,0,\ldots,0,3}_{4k-1},-2,3] .
d\xi_1=-\mu\omega_1Y_2-\mu\omega_2Y_1+\sum_{\alpha=1}\theta_{1\alpha}\xi_{\alpha}, d\xi_2=-\mu\omega_1Y_1+\mu\omega_2Y_2+\sum_{\alpha=1}\theta_{2\alpha}\xi_{\alpha}, d\xi_{\alpha}=-\theta_{1\alpha}\xi_1-\theta_{2\alpha}\xi_2+\sum_{\beta}\theta_{\alpha\beta}\xi_{\beta}.
\log{\cal D}\mathrm{e t}( \bar{P}^{\dag}\bar{P}) = - \frac{d}{d s}Z ( 0 ) = - \frac{d}{d s}\left[ \frac{1}{\Gamma ( s )}\int_{0}^{\infty}\! d t \: t^{s - 1}\mathrm{T r}( e^{- t \bar{P}^{\dag}\bar{P}}) \right]
D(k,\xi):=\exp\left\{\frac{X_{L}(k)}{2\pi i}\left[\left(\int_{-\eta}^{-C_L}+\int_{C_L}^{\eta}\right)\frac{\log(1-r(s)r^{*}(s))}{X_{L}(s)(s-k)}ds+\left(\int_{-C_L}^{-C_R}+\int_{C_R}^{C_L}\right)\frac{\log r_{+}(s)}{X_{L+}(s)(s-k)}ds\right]\right\},
\tilde{S}_{Y_p}\cap[K,N_{Y_p}/2]= \{(\xi-1)r_-+\xi r_+,\; (\xi-2)r_-+(\xi+1)r_+,\dots, &\;r_-+(2\xi-2)r_+, (2\xi-1)r_+, 2\xi r_-,(2\xi-1)r_-+r_+,\dots, \qquad\; (\xi+2)r_-+(\xi-2)r_+, \; (\xi+1)r_-+(\xi-1)r_+ \}.
\ddot{\psi}_{n}=\frac{4-h^{2}}{4h^{2}}\cos\psi_{n}(\sin\psi_{n+1}+\sin\psi_{n-1})-\frac{4+h^{2}}{4h^{2}}\sin\psi_{n}(\cos\psi_{n+1}+\cos\psi_{n-1})
\Lambda = ( 5^{\{2,1 \}}, 4, 3^{\{5,4,1 \}}, 3^{\{4,1 \}}, 2^{\{6,4,1 \}}, 2^{\{6,3,2 \}},2, 1^{\{4,3,2,1 \}}, 1^{\{4,3,2,1 \}}, 1^{\{1 \}}, 0^{\{5,2,1 \}}, 0^{\{6,1 \}}), |\Lambda|= 24, M(\Lambda)= 27.
S = \int d^{4}x \sqrt{{g}}e^{- \phi}\Bigl \{{R}+ ({\nabla}\phi )^{2}+ \frac{1}{8}T r \bigl ({\cal L}\nabla{\cal M}\bigr )^{2}- \frac{1}{4}{\cal F}_{\mu \nu}^{T}{\cal M}{\cal F}^{\mu \nu}- \frac{1}{1 2}{H}_{\mu \nu \lambda}^{2}\Bigr \}
J_{m, s}(x, y):=\sum_{r=0}^{m-s}\frac{[r+s-1]!\,((q-1)x+y)^{r}q^{-mr}}{\prod_{a=1}^{r+s}(x+[a-1]y)}e_{r}(q^{s+1}, q^{s+2}, \ldots , q^{m}).
\|e \rho_m\big(\rho_{m,k}(x)^*\rho_{m,k}(x)\big) - e \rho_n\big(\rho_{n,k}(x)^*\rho_{n,k}(x)\big)\| = \|\rho_m\big(\rho_{m,k}(x)^*\rho_{m,k}(x)\big) - \rho_n\big(\rho_{n,k}(x)^*\rho_{n,k}(x)\big)\|,
\hat{T}r\langle\langle\delta_{i_{1}p_{1}}...\delta_{i_{k}p_{k}}M_{i_{1}p_{1}[m]}^{(l)}\rangle\rangle=(2l+2k+1)\langle\langle\delta_{i_{2}p_{2}}...\delta_{i_{k}p_{k}}M_{[m]}^{(l-2)}\rangle\rangle
- \int_{H_{\sigma_2}}^T N(\sigma_2,t) \frac{d}{dt}\Big( \frac{x^{-\frac{1}{R\log t}}}{t}\Big) \le \sum_{k=0}^{K-1}N(\sigma_2,t_{k+1}) \Big( \frac{x^{-\frac{1}{R\log t_k}}}{t_k}- \frac{x^{-\frac{1}{R\log t_{k+1}}}}{t_{k+1}}\Big) .
\lambda_{k}^{\epsilon, v_k}= -\lim_{t \rightarrow \infty}\frac{1}{t}\log \sup_{x \in D}\mathbb{P}_{x, k}^{\epsilon, v_k}\Bigl \{\tau_{k}^{\epsilon, v_k}> t \Bigr\}, \,\,\, x \in D, \,\,\, k \in \{1,2, \ldots, n\}
\begin{cases}u_t-u_{5x}+u_{3x}+u_x=f(x)v(t), \ \ (0,T)\times (0,2\pi), \partial_x^{j}u(t,0)=\partial_x^{j}u (t,2\pi), \ \ \ (0,T), u(0,x)=u_0(x), \ \ \ (0,2\pi),\end{cases}
A-B =\sum_{j=2}^{i}\frac{g_{i,j}}{2}\left( \frac{\Delta x_{\alpha_1}^2}{4}- \frac{\Delta x_{\alpha_2}^2}{4}+ \left[({x_{i-1/2}-x_{j-1}})-x_{\alpha_{2}}\right]^2- \left[({x_{i+1/2}-x_j})-x_{\alpha_{1}}\right]^2\right)+{\cal O}(\Delta x^3).
\vert \zeta_i^{\prime}-\zeta_j^{\prime}\vert >\frac{\delta}{\varepsilon}, \i\not=j; dist(\zeta_i^{\prime},\partial\Omega_{\varepsilon})>\frac{\delta}{\varepsilon}\delta<\mu_i^{\prime}<\delta^{-1},\i=1,\ldots,k,
\bigg((\Lambda[0])^{(-1)}R[k]\bigg)^r = \sum_{j=1}^{+\infty}\Bigg(\sum_{\substack{j_{l_1},\dots,j_{l_r}\in \mathbb{N}\setminus\{0\}j_{l_1}+\dots j_{l_r}=j}}\bigg((\Lambda[0])^{(-1)}R_{j_1}\bigg)\cdot \dots \cdot \bigg((\Lambda[0])^{(-1)}R_{j_r}\bigg) \Bigg) k^j\, ,
\frac{1}{e^{2}( q^{2})}\cong \frac{1}{1 2 \pi^{2}}\ell n \frac{\Lambda^{2}}{q^{2}}\Rightarrow \frac{1}{1 2 \pi^{2}}\int_{q^{2}}^{\Lambda^{2}}R ( s ) \frac{d s}{s}
\sqrt{C^n_sC^2_n}C^2_{n-2}\Phi''^n+2\sqrt{C^{n-2}_s}C^2_{n-2}C^2_{s-n+2}\Phi'^{n-2}+ \sqrt{C^{n-4}_s}C^2_{s-n+4}C^2_{s-n+2}\Phi^{n-4}=0,
\begin{array}{r c l}{{\mathrm{H t r}[ T_{a}, \cdots ] = 0 ,}}&{{{\qquad}}}&{{\mathrm{H t r}[ \Sigma_{\alpha}, \cdots ] = 0 ,}}\\{{\mathrm{S t r}[ T_{a}, \cdots ] = 0 ,}}&{{{\qquad}}}&{{\mathrm{S t r}\{\Sigma_{\alpha}, \cdots \}= 0 ,}}\\ \end{array}
L_n =\max_{1\leq j\leq N}\sum_{i=1}^n |\tilde{X}_{i j}|^3/n \lesssim \sigma^2 b+b^3 K_n/(\gamma^{3/q}n^{1-3/q}), M_{n,X}(\delta) =n^{-1}\sum_{i=1}^n\max_{1\leq j\leq N}|\tilde{X}_{i j}|^3\cdot 1\left\{\max_{1\leq j\leq N}|\tilde{X}_{i j}|>\delta\sqrt{n}/\log N\right\}=0.
\begin{cases}\mathfrak{Re}\left(\int_{\Omega}(i\nabla+A_0) \varphi_0\cdot\overline{(i\nabla+A_0)\varphi}\,dx\right)=0 \mathfrak{Im}\left(\int_{\Omega}\varphi\overline{\varphi_0}\,dx\right)=0 (i\nabla+A_0)^2 \varphi -\lambda_0\varphi -\lambda \varphi_0 =0.\end{cases}
\|z^k-x\|^2 \leq \|x^k-x\|^2 + 2 \lambda_k f(x^k,x) & - \|y^k-x^k\|^2 - \|z^k - y^k\|^2 + 2\beta_k \left(\|y^k-x^k\| + \|z^k-x^k\|\right) &= \|x^k-x\|^2 + 2 \lambda_k f(x^k,x) & + 2\beta_k^2 - \left(\|y^k-x^k\| - \beta_k\right)^2 - \left(\|z^k-x^k\| - \beta_k\right)^2 &\leq \|x^k-x\|^2 + 2 \lambda_k f(x^k,x) + 2\beta_k^2.
\sum_{j \in \mathbb{Z}}\frac{1}{z+ 2 \pi i j}= \frac{1}{z}+ \sum_{j \geq 1}\left( \frac{1}{z+ 2 \pi i j}+ \frac{1}{z-2\pi i j}\right) &= \frac{1}{z}+ \sum_{j \geq 1}\frac{(z/2)}{(z/2)^2 + \pi^2 j^2}&= \frac{1}{2}\mathrm{coth}( z/2) = \frac{1}{2}\frac{1 + e^{- z}}{1 - e^{- z}},
H = H\bigl(\bigl(\Lambda_n, m\bigr); \vec\xi \bigr) :=\left(\frac{\partial^2}{\partial x_1^2}+ \frac{\partial^2}{\partial x_2^2}\right)- \sum\limits_{\alpha \in \Pi}\dfrac{m(m+1)}{l_\alpha^2(\vec{x}- \vec\xi)}
E(t) := \frac{1}{2}\|u(t)\|_{L^{2}(\Omega)}^{2}+ \frac{1}{2}\int_{t - \tau}^{t}\langle \tilde{a}_{ij}(\cdot) \partial_{j}u(s), \partial_{i}u(s)\rangle_{L^{2}(\Omega)}\mathrm{d}s.
{\cal U}_{l',m',n'}(x')= \frac1{\sqrt{2\pi R'^2L'(\omega '_{l',m',n'}+l'\Omega)}J_{l'+1}\left(\alpha_{m'}^{(l')}\right)}J_l\left(\frac{\alpha_{m'}^{(l')}}{R'}r'\right) e^{-i\omega '_{l',m',n'}t'+il\theta ' +i\frac{n'\pi}{2L'}z'}
(\eta_k,\eta_m)_{H^1_0(\Omega)}= (\eta_{k_1},\eta_{m_1})_{H^1_0({I})}(\eta_{k_2},\eta_{m_2})_{L^2({I})}+ (\eta_{k_1},\eta_{m_1})_{L^2({I})}(\eta_{k_2},\eta_{m_2})_{H^1_0({I})}\;,
\Delta(s^\Sigma)^F=\Delta(s^\Sigma)^F\prod_{\mu\in I(F)}\prod_{\lambda\in \mathrm{Ext}_{\Sigma\setminus \Sigma H_I}(\mu;F)}(s_{r(F)}^\Sigma-s_{\mu\lambda}^\Sigma{s_{\mu\lambda}^\Sigma}^*).
\omega^{I \pm}= \omega_{( 0 ) \alpha \beta}^{I \pm}d X^{\alpha}\wedge d X^{\beta}+ \omega_{( 0 ) a b}^{I \pm}d \Phi^{a}\wedge d \Phi^{b}+ \mathrm{s m a l l ~ c o r r e c t i o n s}
{\cal A}= \pi i \gamma \sum_{r=1}^{N}\left[\left( 2 R^{2}\frac{\bar{z}_{r}}{1+|z_{r}|^{2}}+\tilde{b}_{r}\right)dz_{r}-\left(2 R^{2}\frac{{z}_{r}}{1+|z_{r}|^{2}}+\bar{\tilde{b}}_{r}\right)d\bar{z}_{r}\right].
\frac{d}{dt}\|\Lambda^{m}u\|_{2}^{2}+\|\Lambda^{m+\alpha}u\|_{2}^{2}\leq \|\Lambda^{m+\alpha}u\|_{2}^{\frac{1}{\alpha}}\|\Lambda^{m}u\|_{2}^{2-\frac{1}{\alpha}}\|u\|_{\infty}\leq \frac{1}{4}\|\Lambda^{m+\alpha}u\|_{2}^{2}+C\|u\|_{\infty}^{\frac{2\alpha}{2\alpha-1}}\|\Lambda^{m}u\|_{2}^{2}.
\sum_{n \geq 0}A (n) \left( \frac{\eta (\tau) \eta (6 \tau)}{\eta (2 \tau) \eta (3 \tau)}\right)^{12 n}= \frac{\eta^7 (2 \tau) \eta^7 (3 \tau)}{\eta^5 (\tau) \eta^5 (6 \tau)},
\tilde{A}\Psi_{\Delta}( x_{1}, x_{2}, x_{3}) = \tilde{a}\Psi_{\Delta}( x_{1}, x_{2}, x_{3}) , \qquad \tilde{\cal C}\Psi_{\Delta}( x_{1}, x_{2}, x_{3}) = \Delta ( 1 - \Delta ) \Psi_{\Delta}( x_{1}, x_{2}, x_{3}) ,
\frac{2\sigma}{l}+\frac{d}{m}=\frac{d}{2}, \frac{1}{m'}=\frac{1}{m}+\frac{p}{r_{\theta}},\frac{1}{l'}=\frac{1}{l}+\frac{p}{q_{\theta}}\textmd{(Strichartz exponent relations and the H\"older inequality)}.
\varphi(\sigma_n)=\big(\varphi(\tau_0)\varphi(\eta)\varphi(\tau_0)\big)\big(\varphi(\tau_1)\varphi(\eta)\varphi(\tau_1)\big)\ldots\big(\varphi(\tau_{n-2})\varphi(\eta)\varphi(\tau_{n-2})\big)\varphi(\eta).
11436678=2\times 3^2\times 11^2\times 59\times 89 =198\times 57761 &114396678=2\times 3^3\times 11\times 192587 =198\times 577761 &1143996678=2\times 3^2\times 11^2\times 23\times41\times 557 =198\times 5777761
\phi_{n,\lambda}(x) =\sum_{k=0}^{n}S_{2,\lambda}(n,k)x^{k},(n\ge 0), &=e^{-x}\sum_{k=0}^{\infty}\frac{(k)_{n,\lambda}}{k!}x^{k},(\mathrm{see}\ [9,10,14,15]).
G_1(s) =p(n+p+q+1)P_q^{n-1,p-q}+p\Big((n+p)P_{q-1}^{n,p-q+1}+(n+q-1)P_{q-2}^{n,p-q+1}\Big) &-p(n+p)P_{q-1}^{n+1,p-q}+p(p-1)P_{q-2}^{n+1,p-q}-(n+p+q+1)P_{q}^{n-1,p-q}.
\big|\psi \big(u(1+z)\big)-\psi (u)\big| \leq bu|z| +|z|\int_{[0, \infty)}ute^{-ut}{\: \rm d}\nu (t) &\leq bu|z|+|z|\int_{[0, \infty)}\big (1-e^{-ut}\big){\: \rm d}\nu (t)=|z|\psi (u),
- \hbar g^{2}\mu^{\epsilon}\int \frac{d^{d}k}{( 2 \pi )^{d}}[ D_{0}]_{\sigma \tau}( k ) \frac{( 2 p - k )_{\sigma}( 2 q + k )_{\tau}( 2 k + q - p )_{\mu}}{( k - p )^{2}( k + q )^{2}}= ( q - p )_{\mu}\cdot \hbar g^{2}\frac{\alpha}{8 \pi^{2}\epsilon}+ \cdots .
P_k(x) = \frac{\mu}{2\pi i}\oint_{\Sigma}s^{-\nu}(1-s)^{-\mu-1}p_k \left( \frac{x}{s}\right) \frac{ds}{s}, Q_k(y) = \int_0^{1}t^{\nu}(1-t)^{\mu-1}q_k \left( \frac{y}{t}\right) \frac{dt}{t},
F_r(X,\ldots,X;Y,\ldots,Y) &=\sum_{k,l\geq 0}X^{k}Y^{l}\sum_{m_1>\cdots>m_r>0}\frac{1}{m_1^2\cdots m_r^2}\sum_{\substack{0<a_1\leq\cdots\leq a_k\leq m_1 0<b_1\leq\cdots\leq b_l\leq m_1}}\frac{1}{a_1\cdots a_k b_1\cdots b_l}.
\alpha \ = 001 0110 1001 0110 0110 1001 1001 0110 0 \ = 0 / 01 /01/10 /10/01/ 01/10/ 01/10 /10/01/ 10/01/ 01/10 /0 \ = 0 / 0^{(1)}0^{(1)}1^{(1)}1^{(1)}0^{(1)}0^{(1)}1^{(1)}0^{(1)}1^{(1)}1^{(1)}0^{(1)}1^{(1)}0^{(1)}0^{(1)}1^{(1)}/0.
\mathrm{I m ~}F_{+ - + -}= - 1 2 \pi \alpha^{2}\frac{s - t}{u}- 1 6 \pi \alpha^{2}\left[ \frac{u}{s}\ln \Big | \frac{u}{m_{W}^{2}}\Big | + \frac{u^{2}}{s t}\ln \Big | \frac{t}{m_{W}^{2}}\Big | \right] \, \, \, ,
\varphi \big( g^{j}_1 g_2 \big) = \varphi \big( g^{j}_1 \big) \varphi (g_2) \mbox{and}\varphi \big( g^{j}_1 g_3 \big) = \varphi (g_3) \varphi \big( g^{j}_1 \big) \,,
E_1 = \begin{bmatrix}0 0 0 0 0 -1 0 1 0\end{bmatrix},E_2 = \begin{bmatrix}0 0 1 0 0 0 -1 0 0\end{bmatrix},E_3 = \begin{bmatrix}0 -1 0 1 0 0 0 0 0\end{bmatrix}.
\overline{{\mathrm{\Large v}}}_{4}( x ) ={\frac{4 \pi^{2}}{g^{2}\Lambda^{2}}}x - \left({\frac{x^{2}}{2}}-{\frac{{\cal E}^{2}}{3}}\right) E_{i}( - x ) +{\frac{1}{2}}( 1 - x ){\mathrm e}^{- x}-{\frac{1}{2}}\ ,
p_t(x)=\int_{B_R}(2\pi t)^{-n/2}\exp\left(-\frac{|x-y|^2}{2t}\right)d\mu(y)\geq \int_{B_R}(2\pi t)^{-n/2}\exp\left(-\frac{(r_0+R)^2}{2t}\right)d\mu(y) = (2\pi t)^{-n/2}\exp\left(-\frac{(r_0+R)^2}{2t}\right),
E_0 :=\left\{1-\sqrt{\alpha}\epsilon\le \frac{\|y\|^2}{m}\le 1+\sqrt{\alpha}\epsilon\right\}, E_1 :=\left\{\frac{y_{(1)}^2+\cdots +y_{(k)}^2}{k}\le 1-\epsilon+\frac{1-\sqrt{\alpha}}{\gamma}\epsilon \right\}, E_2 :=\left\{\frac{y_{(m)}^2+\cdots +y_{(m-k+1)}^2}{k}\ge 1+\epsilon-\frac{1-\sqrt{\alpha}}{\gamma}\epsilon \right\}.
\frac{\hat \mu_{n,j}(\theta_n)[\sigma_{P_n,j}(\theta_n)-\hat\sigma_{n,j}(\theta_n)]}{\hat\sigma^M_{n,j}(\theta_n)}=\hat \mu_{n,j}(\theta_n)\frac{\hat\sigma_{n,j}(\theta_n)}{\hat\sigma^M_{n,j}(\theta_n)}\Big(\frac{\sigma_{P_n,j}(\theta_n)}{\hat\sigma_{n,j}(\theta_n)}-1\Big)=o_{P_n}(1),
\begin{array}{l}{e^{- i \chi_{s}}[ - \partial_{r}+ s ( n - \Phi^{( 0 )}+ \Upsilon ) r^{- 1}] f_{n}( r ) = ( E + m ) g_{n}( r ) ,}\\{e^{i \chi_{s}}[ \partial_{r}+ s ( n - \Phi^{( 0 )}+ \Upsilon + s ) r^{- 1}] g_{n}( r ) = ( E - m ) f_{n}( r ) .}\\ \end{array}
A_{k,n}= (1+\beta)^{k}\mathfrak{G}_{k,n-1}^{(\beta)}(x_i=1,\, \forall\, i \ge 1),B_{k,n}= \mathfrak{G}_{k-1,n-1}^{(\beta)}(x_i=1,\, \forall\, i \ge 1).
\alpha_1\kappa_1+\dots+\alpha_s\kappa_s =\log|\det(\Lambda_\omega)| &=\lim_{n\to\infty}\frac{1}{n}\log|\det (M^n_\omega)| &=\lim_{n\to\infty}\frac{1}{n}\sum_{i=0}^{n-1}\log|\det (M_{T^i\omega})|.
H = \frac{N T_{0}}{3}\left[ \frac{1}{2}\sum_{i = 1}^{3}\dot{r}_{i}^{2}+ \frac{\alpha^{2}}{2}\Big ( r_{1}^{2}r_{2}^{2}+ r_{1}^{2}r_{3}^{2}+ r_{2}^{2}r_{3}^{2}\Big ) \right] ~ ~ ,
\|\widehat{\rho}_{k_{n}}-\rho\|_{\mathcal{S}(H)}^{2}\leq \|\rho\|_{\mathcal{S}(H)}^{2}-\sum_{j=1}^{\infty}[\rho(\phi_{j})(\phi_{j})]^{2}=\sum_{j\neq k}^{\infty}\left[\frac{D_{X}(\phi_{j})(\phi_{k})}{C_{j}}\right]^{2}<\infty.
W_{[n,j]}=\sum_{k=i}^{n-j+i}\sum_{\gamma\in\mathcal{P}_{[n,j,k]}}w(\gamma)=\sum_{k=i}^{n-j+i}W_{[k,i]}A_{[n-k-1,j-i-1]}^{(i+1)},n\geq j,
\renewcommand{\arraystretch}{1 . 5}\begin{array}{r c l}{{f_{1}^{( 1 )}}}&{{=}}&{{{}_{2}F_{1}( \lambda_{3}, 1 - \lambda_{1}; \lambda_{2}+ \lambda_{3}; 1 - x ) \, ,}}\\{{* f_{2}^{( 1 )}}}&{{=}}&{{x^{1 - \lambda_{3}- \lambda_{4}}( 1 - x )^{1 - \lambda_{2}- \lambda_{3}}{}_{2}F_{1}( \lambda_{1}, 1 - \lambda_{3}; \lambda_{1}+ \lambda_{4}; 1 - x ) \, ,}}\\ \end{array}
{d^2(\delta\phi_k)\over d\eta^2}+{\Bigl[{k^2}+3\lambda a^2\Phi^2\, \sin^2 (c\sqrt\lambda a\Phi \eta)\Bigr]}\delta\phi_k = 0 \ ,~~~\eta =\int{dt\over a(t)}={2t\over a(t)}\, ,
\Phi_{\breve{\mathbf g}_p}(g) := \frac{\langle \tau_p(g)\breve{\mathbf g}_p, \breve{\mathbf g}_p\rangle}{||\breve{\mathbf g}_p||^2}, \Phi_{\mathbf h_p}(g) := \frac{\langle \tilde{\pi}_p(g)\mathbf h_p, \mathbf h_p\rangle}{||\mathbf h_p||^2}, \Phi_{\pmb{\phi}_p}(g) := \frac{\langle \omega_{\overline{\psi}_p}(g)\pmb{\phi}_p, \pmb{\phi}_p\rangle}{||\pmb{\phi}_p||^2}.
2T(r)\ge \dfrac{2}{n+1}\sum_{u=1}^3\sum_{i=1}^{2n+2}N^{(n)}_{(f^u,H_i),\leqslant k_i}(r)+\dfrac{n}{3\times (n+1)}\sum_{u=1}^3\sum_{i=1}^{2n+2}N^{(1)}_{(f^u,H_i),\leqslant k_i}(r) &-\dfrac{n}{n+1}\sum_{i=1}^{2n+2}N(r,\nu_i)+o(T(r)).
\left\langle \left( \gamma;d_{1},...,d_{p}\right) ,\mathbf{d}\int_{t}^{t+d}\omega\right\rangle & =\left\langle \left( \gamma;d_{1},...,d_{p}\right) ,-\int_{t}^{t+d}\mathbf{d}\omega-\varphi_{t}^{\ast}\omega+\varphi_{t+d}^{\ast}\omega\right\rangle
\sup_{\pi \in \Pi\left(\mu_{13}, \mu_{23}\right)}\int_{\mathbb{R}^{2d +2}}(f_{\mathcal{S}})_\lambda \, d \pi = \frac{1}{4 \lambda_1}Q_{1, Y Y}^{-1}+\frac{1}{4 \lambda_2}Q_{2, Y Y}^{-1}+ b
\sum \limits_{\substack{n\leq x (n,n+z)\in \mathcal{N}_q(a,b)}}\Lambda(n)\Lambda(n+z) =\frac{\Theta (z)}{\phi(q)x}\sum \limits_{\chi}\overline{\chi(a)}\overline{\chi(b)}\bigg(E_0(\chi)\frac{x^2}{2}+O\bigg(\frac{x^2}{e^{c\sqrt{\log x}}}\bigg )\bigg)+O_{z,q}(1)
\lim_{t \to \infty}\frac{S_j(t)}{M(t)}= \lim_{N \to \infty}\frac{S_j(\tau_N)}{M(\tau_N)}= \frac{\int_0^\infty e^{-\lambda s}p_j(s) ds}{\int_0^\infty e^{-\lambda s}(1-p_0(s)) ds}, j \geq 1,
{}_2 F_1\bigg (1, \frac{d-1}{2}+1;\frac{1}{2}; z \bigg) = \frac{1}{d-1}\bigg\{(d-3)\,{}_2 F_1\bigg (1, \frac{d-1}{2};\frac{1}{2}; z \bigg) +2 \,{}_2 F_1\bigg (2, \frac{d-1}{2};\frac{1}{2}; z \bigg) \bigg\}.
\left\langle\phi_{m_1}\psi^{a_1}\cdots\phi_{m_n}\psi^{a_n}\cdot \phi_1\right\rangle_{0,n+1}^\mathrm{FJRW}=\sum_{i=1}^n\left\langle\phi_{m_1}\psi^{a_1}\cdots\phi_{m_i}\psi^{a_i-1}\cdots\phi_{m_n}\psi^{a_n}\right\rangle_{0,n}^\mathrm{FJRW}
\frac{p_r^2}{g_{rr}}+ \frac{p_\theta^2}{g_{\theta \theta}}- \frac{{g'}_{tt}}{- \cal D}\left( p_\phi + \frac{g_{t \phi}+ \Omega_0 g_{\phi \phi}}{{g'}_{tt}}E \right)^2 = - \left( \frac{E^2}{{g'}_{tt}}+ \mu^2 \right),
{\cal L}= ( D_{\mu}\phi )^{*}( D^{\mu}\phi ) - m^{2}\phi^{*}\phi - \frac \lambda 4 ( \phi^{*}\phi )^{2}+ \frac \Theta 2 \epsilon_{\sigma \mu \nu}A^{\sigma}\partial^{\mu}A^{\nu}- \frac \xi 2 ( \partial_{i}A^{i})^{2}\; ,
S_{R}= - \int d^{2}x \sqrt{g}\left\{\frac{1}{2}Z g^{\mu \nu}\partial_{\mu}\Phi \partial_{\nu}\Phi + C R + V + \frac{1}{\epsilon}\left( - \frac{V^{\prime}}{C^{\prime}}+ \frac{Z V}{2{C^{\prime}}^{2}}\right) \right\}.
\frac{1}{p_{s}^{2}}~{\cal I}_{\mu_{1}\cdots \mu_{s}, \alpha_{1}\cdots \alpha_{s}}^{( s )}( x ) ~{\cal I}_{\alpha_{1}\cdots \alpha_{s}, \nu_{1}\cdots \nu_{s}}^{( s )}( x ) = \Im_{\mu_{1}\cdots \mu_{s}, \nu_{1}\cdots \nu_{s}}^{( s )},
I_{21}\leq \sum_{q\geq0}\int_0^{\frac{t}2}c_q\lambda_q^{\alpha+\beta-2\gamma}e^{-\mu \lambda_q^\alpha (t-\tau)}\|B(\tau)\|^2_{H^{\frac52-\alpha+\gamma}}\, d\tau \lesssim \int_0^{\frac{t}2}(t-\tau)^{-\frac{\alpha+\beta-2\gamma}{\alpha}}\|B(\tau)\|^2_{H^{\frac52-\alpha+\gamma}}\, d\tau.
T = \left( \begin{matrix}{{0}}&{{1}}&{{0}}&{{0}}\\{{1}}&{{0}}&{{0}}&{{- 1}}\\{{\frac{1}{2}- s}}&{{0}}&{{0}}&{{\frac{1}{2}+ s}}\\{{0}}&{{0}}&{{1}}&{{0}}\\ \end{matrix}\right) \ .
a_{(x,t,s)}(\lambda) =\frac{s}{\sqrt{1+\lambda^2}}+\frac{1-s}{\sqrt{1+\lambda^2+\frac14 t^2\tau(g(x))}}b_{(x,t,s)}(\lambda) =s+\frac{(1-s)\frac12 \tau(g(x))}{\sqrt{1+\lambda^2+\frac14 t^2\tau(g(x))}}.
\begin{array}{r c l}{{\delta_{i}x^{\alpha}}}&{{= 0}}&{{= \displaystyle \frac{1}{i \hbar}[ x^{\alpha}, L_{i}^{\mu}\circ p_{\mu}]}}\\{{\delta_{i}x^{\mu}}}&{{= L_{i}^{\mu}}}&{{= \displaystyle \frac{1}{i \hbar}[ x^{\mu}, L_{i}^{\mu}\circ p_{\mu}] \, .}}\\ \end{array}
\nabla_R\cdot \left[ \frac{\vec R-i\frac 12\vec \Gamma}{\left[ \left( \vec R-i\frac 12\vec \Gamma \right)^2\right]^{\frac 32}}\right] =-\frac{\delta \left( R-\frac 12\Gamma \right)}{R^2}\delta \left(\cos \theta \right)
U_{k}=\left( \begin{array}{lllllll}0 \cdots 0 0 0 \cdots 0 \vdots \vdots \vdots \vdots \vdots 0 \cdots 0 0 0 \cdots 0 0 \cdots 0 V 0 \cdots 0 0 \cdots 0 0 0 \cdots 0 \vdots \vdots \vdots \vdots \vdots 0 \cdots 0 0 0 \cdots 0\end{array}\right)_{k,k+1}
\gamma_{5}\left( \begin{array}{c}{{\lambda_{A}^{i^{\star}}}}\\{{\zeta^{\alpha}}}\\{{\psi^{A}}}\\ \end{array}\right) \, = \, - \, \left( \begin{array}{c}{{\lambda_{A}^{i^{\star}}}}\\{{\zeta^{\alpha}}}\\{{\psi^{A}}}\\ \end{array}\right)
\mathcal{W}^*_t D^*_t + \int_0^t D^*_s c^*_s ds = C_t (\mathcal{W}_t^*)^{1-\gamma}e^{Y_t}+ \int_0^t C_s \delta^\psi e^{-\frac{\psi}{\theta}Y_s}(\mathcal{W}_s^*)^{1-\gamma}e^{Y_s}\, ds.
g_d(t) = g_y(t)= c\cdot\exp[{K(t-T_0)}], g_f(t) = g_z(t)= n^{\frac 12-s}c \cdot\exp[{K(t-T_0)}], g_x(t) = c^{\frac{1-2s}{1-s}}\cdot\exp[{K(t-T_0)}],
\Psi(\theta)= \left( \begin{array}{lr}\psi_{n,m-1}(\theta) \psi_{n,m}(\theta) \psi_{n,m-1}(\theta) \psi_{n,m}(\theta) \,\,\,\,\,\,\,\vdots \psi_{n,m-1}(\theta) \end{array}\right)_{(2k+1) \times 1}
\phi_0(\sigma)=\{(I,I)\,,\,(0,I)\}\;, &\phi_1(\sigma)=\{(I\oplus I^{(1)},I)\}\;, &\phi_2(\sigma)=\emptyset\;, &\phi_3(\sigma)=\{(I^{(3)},I\oplus I^{(3)}\})\;, &\phi_k(\sigma)=\emptyset
L_K z = \frac{1}{n-1}((D^2 h)^{-1})^{i,j}(z_i h_j + h_i z_j + h z_{i,j}) = \frac{1}{n-1}\frac{((D^2 h)^{-1})^{i,j}}{h}(h^2 z_i)_j .
\tilde{Z}_{0}( k_{s}; l_{1 s}, l_{2 s}) = \frac{( - 1 )^{2 l_{1 s}+ 2 l_{2 s}- 1}[ 1 - ( - 1 )^{2 l_{2 s}}\sqrt{k_{s}}]^{2}}{4^{2 l_{1 s}}k_{s}^{3 / 2}}.
\Omega ( \beta , \mu ) = -{\frac{2 V}{\beta}}\left({\frac{M}{2 \pi \beta}}\right)^{\frac{N}{2}}\sum_{n = 1}^{+ \infty}\cosh ( n \beta \mu ) ({\frac{1}{n}})^{1 +{\frac{N}{2}}}e^{- n \beta M}\; .
\varphi := \iota_{x_{1}}(J) \cdot \iota_{x_{2}+x_{3}}(J) \cdot \iota_{-x_{1}-x_{2}}(J)\cdot\iota_{y_{1}}(J)\cdot \iota_{y_{2}+y_{3}}(J) \cdot \iota_{-y_{1}-y_{2}}(J)
\zeta_{0}^{( p )}( \nu ) ={\frac{1}{\pi \Gamma ( \nu )}}\int_{0}^{\infty}d x \int_{0}^{1}d t t^{\nu - 1}e^{- t x^{2}}{\frac{1}{( 4 \pi t )^{( D - 1 ) / 2}}}\sum_{n = 0}^{\infty}\breve{a}_{n}( x ) t^{n}.
\left[ a_i(K) , a^j(L) \right] = \delta_i^j \delta_{KL}\,, \qquad\left[ b_i(K) , b^j(L) \right] = \delta_i^j \delta_{KL}\,, \qquad\left[ c_i , c^j \right] = \delta_i^j \quad \mbox{(if present)}\,,
\nu_i^{S \cup \{j\}}= \frac{\nu_i^S w_i^S + \nu_j^S \frac{w_j^S}{a_{jj}^S}a_{ij}^S}{w_i^S + \frac{w_j^S}{a_{jj}^S}a_{ij}^S}= \nu_j^S - \frac{w_i^S}{w_i^S + \frac{w_j^S}{a_{jj}^S}a_{ij}^S}(\nu_j^S - \nu_i^S) = \nu_j^S - \frac{w_i^S}{w_i^{S \cup \{j\}}}(\nu_j^S - \nu_i^S).
\begin{aligned}\#\{(\mu,\mu'): |\langle\mu-\mu',\alpha\rangle|\leq \rho\cdot|\mu-\mu'|\}=\sum_{\mu}\#\{\mu' : \mu'\in T_\mu\}+\sum_{\mu}\#\{\mu' : \mu'\in S_\mu\}\leq 2\cdot\#\{(\mu,\mu'): \mu,\mu'\in T\}+\sum_{\mu}\#\{\mu': \mu'\in S_\mu\},\end{aligned}
\begin{cases}F_1(u,v) = \displaystyle\int_0^L \left[ k_x(0,y)u(y)+s_x(0,y)v(y)\right]dy, \forall (u,v) \in X_0, F_2(u,v) = \displaystyle\int_0^L \left[ k_x(L,y)v(y)+s_x(L,y)u(y)\right]dy, \forall (u,v) \in X_0\end{cases}
\mathbf{J}_1 =\lambda^{-4+\epsilon}\left[\frac{1}{2}\int_{\Gamma_{t_0, \theta}}\partial_t (\psi_\Gamma^2) \,\d S\,\d t - \lambda \int_{\Gamma_{t_0, \theta}}\left(s\xi -\frac{3}{2}\right) \partial_\nu^{A_1}\eta^0 \psi_\Gamma^2 \,\d S\,\d t \right].
\big[E^{(m)}\tilde{\circ}_{4}F^{(m)}-F^{(m)}\tilde{\circ}_{4}E^{(m)}\big] \qquad{}=\tilde{s}_{p,m}([\omega]) \prod\limits_{i=2p}^{mp+2p-1}([\omega]- h_{i,1}) \prod\limits_{1\leq i\leq m^2p, m\nmid i}\bigg([\omega]-h_{p+\tfrac{i}{m},1}\bigg)* [H],
Tr\sigma_{f}(U)=\frac{detz_{j}^{f_{i}+N-i+1}-z_{j}^{-f_{i}-N+i-1}}{\prod(z_{i}-z_{i}^{-1})\cdot\prod_{i<j}(z_{i}+z_{i}^{-1}-z_{j}-z_{j}^{-1})}.
\int_{t}^1\mathcal{H}(B(x,r-t))\frac{dr}{r^{Q+1}}\geq \int_{2t}^1\mathcal{H}(B(x,r-t))\frac{dr}{r^{Q+1}}&=\int_{t}^1\mathcal{H}(B(x,r))\frac{dr}{r^{Q+1}(1+t/r)^{Q+1}}\,.
\big{|}(\Lambda Q_b, \Lambda \mathcal{Q}_p)-(\Lambda \mathcal{Q}_p, \Lambda \mathcal{Q}_p)\big{|}\leq \|Q_b-\mathcal{Q}_p\|_{L^{\infty}}\|\Lambda^*\Lambda \mathcal{Q}_p\|_{L^1}=O(b_c).
\sum_{a=m_2+1}^{n-m_1-1}\sum_{k: M_k\in S_{j+1,i}^{a}}\lambda_{k}|A-M_{k}|\lesssim\sum_{a=m_2+1}^{n-m_1-1}\sum_{k: M_k\in S_{j+1,i}^{a}}\lambda_{k}i= \sum_{a=m_2+1}^{n-m_1-1}\nu(S_{j+1,i}^{a})i &\lesssim \sum_{a=m_2+1}^{n-m_1-1}j^{2(m_1'+a-n')}i^{m_1'+a-n'+1}\lesssim j^{-2}
\dot a^{s,\tau}_{p,q}(W) =\dot f^{s+n(\tau-\frac1p)}_{\infty,\infty}(\mathbb A) =\dot f^{s+n(\tau-\frac1p),1/p}_{p,\infty}(W),\quad\begin{cases}\tau>\frac1p\quad (\tau,q)=(\frac1p,\infty),\end{cases}
\gamma_{\xi}^{( 0 )}= - \frac{1}{3 2 \pi^{2}}{\cal M}_{\xi}^{4}( X ) \ln \frac{-{\cal M}_{\xi}^{2}( X )}{e^{3 / 2}\mu_{d}^{2}}+ \frac{i}{3 2 \pi}{\cal M}_{\xi}^{4}( X ) ({\cal M}_{\xi}^{2}( X ) < 0 ) .
\operatorname{div}^\mathcal{A}\frac{\delta\bar\ell^\mathcal{A}}{\delta\sigma^\mathcal{A}}=-\operatorname{Tr}\left(\operatorname{ad}^*_{\sigma^\mathcal{A}}\frac{\delta\bar\ell^\mathcal{A}}{\delta\sigma^\mathcal{A}}\right).
[x_{i, r+1}^{+}, x_{i-1, s}^{+}] - [x_{i, r}^{+}, x_{i-1, s+1}^{+}] = - \varepsilon_1 x_{i, r}^{+}x_{i-1, s}^{+}- \varepsilon_2 x_{i-1, s}^{+}x_{i, r}^{+},
\bigcirc_{w=0}^i(\theta_t+\gamma_{r-i+w})^{-1}\circ (\theta_t+\xi_{i,1})\circ \cdots \circ (\theta_t+\xi_{i,\ell_i})=\bigcirc_{w'=0}^j (\theta_t+\gamma_{r-j+w'})^{-1}\circ (\theta_t+\xi_{j,1})\circ \cdots \circ (\theta_t+\xi_{j,\ell_j})\enspace,
d s_{\mathrm{t a r g e t}}^{2}= \frac{1}{2}d \varphi^{2}+ \frac{1}{2}d \beta^{2}+ \frac{1}{2}e^{2 \varphi}\left( d \sigma_{1}- \chi d \sigma_{2}\right)^{2}+ \frac{1}{2}e^{\varphi}\left[ e^{\sqrt{3}\beta}d \chi^{2}+ e^{- \sqrt{3}\beta}d \sigma_{2}^{2}\right] .
\left(\begin{array}{c}\dot{x}_{1}\dot{x}_{2}\end{array}\right)=\left(\begin{array}{cc}0 1 -2 -2\end{array}\right)\left(\begin{array}{c}x_{1}x_{2}\end{array}\right)+\left(\begin{array}{c}0 1\end{array}\right)u,\,\,\,\,\,|u|\leq 1, \,\,\,\,\,x_2(x_2 - u) \leq 0