harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{pmatrix}a b c d \end{pmatrix}\cdot \Big(-\frac{1}{\overline{t}}\Big)=\frac{\overline{t}-t_{+}^{c,d}}{t_+^{c,d}\overline{t}+1}=-\overline{\Big[\begin{pmatrix}a b c d \end{pmatrix}\cdot t \Big]}^{-1}\,,
\mathcal{A}(x) =\{n\in \mathcal{A}: x\le n< 2x\}, \mathcal{A}(x;q,a) =\{n\in \mathcal{A}(x), n\equiv a\pmod{q}\}, L(\mathcal{A}) =\{L(n):n\in\mathcal{A}\}, \phi_L(q) =\phi(|l_1| q)/\phi(|l_1|), \mathcal{P}_{L,\mathcal{A}}(x) =L(\mathcal{A}(x))\cap\mathcal{P}, \mathcal{P}_{L,\mathcal{A}}(x;q,a) =L(\mathcal{A}(x;q,a))\cap\mathcal{P}.
h_{s}={\frac{1}{2 8 8 0 \pi^{2}}}\left[ 5 w^{2}( w^{2}- 2 \nabla w ) - 3 ( \nabla w )^{2}+ R_{\mu \nu}w^{\mu}w^{\nu}- 3 0 \left( ( \xi -{\frac{1}{1 5}}) R + m^{2}\right) w^{2}\right] ~ ,
\Pr \left({\left.{{\Upsilon_{{T_1},D}}> \gamma}\right|{\Gamma_{{T_1}}}}\right) = \Pr \left({\left.{{\gamma_0}> \gamma \left({{\Gamma_{{T_1}}}+ 1}\right)}\right|{\Gamma_{{T_1}}}}\right) = \exp \left({- \frac{\gamma}{{{\gamma_0}}}\left({{\Gamma_{{T_1}}}+ 1}\right)}\right)
\eta_2(r) = \eta_2(g_0)+(r-g_0)g\zeta_0+\frac{1}{2}(r-g_0)^2(\zeta_1-\zeta_0) = \left\{\eta_2(g_0)-g_0g\zeta_0+\frac{1}{2}g_0^2(\zeta_1-\zeta_0)\right\}+r\{g\zeta_0-g_0(\zeta_1-\zeta_0)\}+\frac{1}{2}r^2(\zeta_1-\zeta_0)
\rho(L_{-1}) = -\partial \widehat\rho(L_{-1}) = q^{-1}\rho(L_{0}) = -z \partial - c \widehat\rho(L_0) = -q\partial_q - c +1 \rho(L_{1}) = -z^2\partial - 2c z \widehat\rho(L_1) = q^3\partial_q^2 + 2c q^2 \partial_q
\int ((D_y^{\rm t}-\mu) G_{D-\mu}(x,y) f_0)(\phi(y)) \d y= \int (G_{D-\mu}(x,y) f_0)( (D_y-\mu) \phi(y))\d y = \int f_0 (G_{D-\mu}(x,y) (D_y-\mu) \phi(y))\d y = f_0(\phi(x)).
W_{ADS}= (N_{c}+ 1 - N_{f}) \; \omega_{N_{c}+ 1 - N_{f}}(\frac{2^{N_{c}- 1}\Lambda_{N=1}^{3(N_{c}+1)-N_{f}}}{\mbox{Pf}M})^{\frac{1}{N_{c}+1-N_{f}}}
{\cal A}={\frac{V}{2 \pi}}( 8 \pi^{2}\alpha^{\prime})^{- 1 / 2}\int_{0}^{\infty}d t \left({\frac{\pi}{t}}\right)^{\frac{9}{2}}\left[{\frac{f_{3}^{8}( q )}{f_{1}^{8}( q )}}-{\frac{f_{4}^{8}( q )}{f_{1}^{8}( q )}}\right]
\phi(\vec{r})=10\frac{n_{\overline{\nu}_{e}}\langle P_{ee}\rangle\lambda XN_{A}}{M}\int_{V}d^{3}\vec{r}^{\prime}\frac{A(\vec{r}^{\prime})\rho(\vec{r}^{\prime})}{4\pi|\vec{r}-\vec{r}^{\prime}|^{2}}
{\bf 1}_{\{\tau<s\}}=\sum_{j=0}^{\infty}\int\limits_t^T{\bf 1}_{\{\tau<s\}}\phi_j(\tau)d\tau \cdot \phi_j(\tau)=\sum_{j=0}^{\infty}\int\limits_t^s\phi_j(\tau)d\tau \cdot \phi_j(\tau),
K_i(x,N,p)= \binom{N}{i}(p-1)^i \hat K_i(x;(p-1)/p,N) H_{i}(x,N,p)= \left(\, \binom{N}{i}- \binom{N}{i-1}\right) Q_i(x;\mu-N-1,-\mu-1,\mu), E_i(x,N,p)=\binom{p}{i}\binom{N-p}{i}R_i(\lambda(x);\mu-N-1,-\mu-1,\mu),
\lim_{n\to \infty}\dfrac{\sum_{m=1}^n b_{2^m}2^{m+1}\mathbb{P}\left(|X|>b_{2^{m-1}}\right)}{b_{2^n}}&=\lim_{n\to \infty}\dfrac{\sum_{m=1}^n 2^{m/p}L(2^{m}) 2^{m+1}\mathbb{P}\left(|X|>b_{2^{m-1}}\right)}{2^{n/p}L(2^{n})}=0.
P_5 = 5 q_1^5 + 50 q_1^3 q_2 + 51 q_1 q_2^2 + 48 q_1^2 q_3 + 24 q_2 q_3 + 30 q_1 q_4, &P_6 = (13 q_1^3 + 9 q_1 q_2 + 3 q_3)^2 - 162 q_1^4 (q_1^2 + q_2), &P_7 = 56 q_1^5 q_2 + 576 q_1^3 q_2^2 + 648 q_1 q_2^3 - 15 q_1^4 q_3 + 474 q_1^2 q_2 q_3 + 279 q_2^2 q_3 - 144 q_1 q_3^2 - 90 q_3 q_4.
\begin{array}{l}\Phi_{\mathcal{L}(G^{(R)}\otimes{\{G_1,G_2\}})}(x)=\det{B_1}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;=\det(xI_{n_2}-\mathcal{L}(G_2)\circ{C})^{m}\cdot\det(xI_{n_1}-\mathcal{L}(G_1)\circ{B})^{n}\cdot\det{S},\end{array}
\frac{1}{a}= \frac{1-\theta_0}{a_0}+ \frac{\theta_0}{a_1}, \frac{1}{b}= \frac{1-\theta_1}{b_0}+ \frac{\theta_1}{b_1}, 0<\theta_0,\theta_1<\theta<1, \theta_0 + \theta_1 = \theta.
\int_{\omega_+}U_{\delta,e_n}^{p+1}= \underbrace{\frac{1}{2}\int_{B_\frac{1}{2}(e_n)}U_{\delta,e_n}^{p+1}}_{(I)}- \underbrace{\int_{\Sigma}U_{\delta,e_n}^{p+1}}_{(II)},
\begin{cases}\|e_{k}^{\pm, 2}\|_{s,X}\le C\delta^2\theta_k^{L_2(s)}\Delta_k, s_0-1\le s\le s_1-1\cr\|\overline{e}_{\pm,k}^{(2)}\|_{s,X}\le C\delta^2\theta_k^{s+s_0-2\alpha}\Delta_k, s_0\le s\le s_1-1\cr\|\tilde{e}_{k}^{(2)}\|_{H^s(\omega_X)}\le C\delta^2\theta_k^{s+s_0-2\alpha-\frac 12}\Delta_k, s_0\le s\le s_1-\frac 12\end{cases}
([RV]^m S)_{ab}= \sum_{(a_i,b_i) \in \{(i,\mu_1), (\mu_1,i)\}:1 \leq i \leq m}(z)^{m/2}(\sigma_i)^{m/2}(X^G_{i\mu_1})^m R_{a a_1}R_{b_1 a_2}\cdots S_{b_m b}.
e^{\beta}u'(\beta) = e^{2\beta}+ (1-e)e^{\beta}+ e\beta^2 + \left(\frac{2e^3}{3^{5/2}\sqrt{2\pi}}-\frac{5e}{2}\right)\beta + \frac{e}{2}- \frac{2e^3}{3^{5/2}\sqrt{2\pi}}.
cr^d (3/8)^{k}\mu \le \sum_{\|h\| \le r}\sum_{e\in E(\mathbb Z^d)}q_ep_{e-h}= \sum_{e\in E(\mathbb Z^d)}q_e \cdot \mathbb E [|\gamma \cap \Lambda_e|] \le Cr \sum_{e\in E(\mathbb Z^d)}q_e,
\left(\int|W|_g^4dv_g \right)^{{1}/{2}}= \left(\int_M |W|_g^4u^4 \right)^{{1}/{2}}dv_0 \leq C_0\int_M|\nabla|W|_gu|^2_0dv_0+K_s\int_M|W|_g^2u^2dv_0,
V_{\underline{d}''}^{X_{1},0}=(V_{\underline{d}''}^{X_{2}^{c},0}\oplus V_{\underline{d}''}^{X_{3}^{c},0})\oplus \left\langle u_1,\ldots,u_{\beta}\right\rangle=V_{\underline{d}''}^{X_{2}^{c},0}\oplus \left\langle u_1,\ldots,u_{\beta}\right\rangle.
\frac d{dt}\sum\limits_{|\alpha|=0}^{2}(\|\sqrt{B}\partial_\tau^\alpha\partial_y u\|_{L^2}^2 +\|\sqrt{A}\partial_\tau^\alpha\partial_y\tilde{h}\|_{L^2}^2)+C_2\|\partial_y(u, \tilde{h})\|_{H^{2, 1}}^2 &\le CD(t)^{\frac 12}E(t)+CD(t)E(t)^{\frac14}+\frac{C}{M^{\frac14}}D(t).
\frac{d\mu_g^n}{dt}= \sum_{i=1}^{n-1}\sum_{j=1}^{i}(g_{i+1}-g_i)jV_{i,j}\psi_i\psi_j - \sum_{i=1}^{n-1}\sum_{j=i}^{n-1}g_{i}V_{i,j}\psi_i\psi_j,
[ X_{0}^{+}, p r^{\prime}] = [ X_{0}^{+}, i ] = 0 , \hspace{1 c m}[ X_{0}^{+}, X_{0}^{-}] = X_{0}^{+}\xi_{0}^{-}Q - \xi_{0}^{-}Q \xi_{0}^{+}Q - Q [ X_{0}^{+}, \xi_{0}^{-}] .
0 < \beta_{2,p}:= \underset{|w^{H\times h}|_{H^{1}(\Omega)}\neq 0}{\underset{w^{H\times h}\in V^{H\times h}}{\inf}}\underset{|v^{H\times h}|_{H^{1}(\Omega)}\neq 0}{\underset{v^{H\times h}\in V^{H\times h}}{\sup}}\frac{\langle F^{\prime}(p_{m,k}^{H}) w^{H\times h}, v^{H\times h}\rangle}{| w^{H\times h}|_{H^{1}(\Omega)}| v^{H\times h}|_{H^{1}(\Omega)}},
\frac{1}{\phi(n)}\sum_{k=a_n+1}^{a_n+\phi(n)}|p_{k}(i,j)-p(i,j)|=\frac{1}{n}\sum_{k=2^n+1}^{2^n+n}|p_{k}(i,j)-p(i,j)|=\frac{1}{6},
Q = s + \omega^{+}\delta^{+}+ \omega^{-}\delta^{-}+ \varepsilon^{+ \mu}\delta_{\mu}^{+}+ \varepsilon^{- \mu}\delta_{\mu}^{-}+ v^{\mu}\partial_{\mu}- ( \omega^{+}\varepsilon^{+ \mu}+ \omega^{-}\varepsilon^{- \mu}) \frac \partial{\partial v^{\mu}}\, .
\begin{aligned}\alpha_+ \circ \beta_+ := \left(\alpha \cdot \beta\right)_+ = \frac{1}{l+1}\cdot \alpha_+ \cdot d\beta_+ + \frac{1}{n-l+1}\alpha_+ \cdot d^* \beta_+ &+ (-1)^{k+1}\frac{1}{k+1}d\alpha_+ \cdot \beta_+ + (-1)^k \cdot \frac{1}{n-k+1}d^* \alpha_+ \cdot \beta_+,\end{aligned}
\||\eta|^{-2/5}\widehat{f_b}(\eta)\|_{L^2_\eta}+\||\eta|^{3/5}(\partial\widehat{f_b})(\eta)\|_{L^2_\eta}=|b|^{-1/10}\big(\||\xi|^{-2/5}\widehat{f}(\xi)\|_{L^2_\xi}+\||\xi|^{3/5}(\partial\widehat{f})(\xi)\|_{L^2_\xi}\big).
{\cal K}^{L}( \sigma = 2 ) = \left( \begin{array}{c c}{{- \frac{d^{2}}{d x^{2}}+ 4 - \frac{3}{\cosh^{2}x}}}&{{\frac{3}{\cosh^{2}x}}}\\{{\frac{3}{\cosh^{2}x}}}&{{- \frac{d^{2}}{d x^{2}}+ 4 - \frac{3}{\cosh^{2}x}}}\\ \end{array}\right) \qquad ,
\phi_g(\tau, z) = \frac{\chi(g)}{12}\phi_{0, 1}(\tau, z) + \tilde{T}_g(\tau) \phi_{-2, 1}(\tau, z) =\sum_{n=0}^{\infty}\sum_{r \in \mathbb{Z}}c_g(4n - r^2) q^n \zeta^r = 2 \zeta^{-1}+ (\chi(g) - 4) + 2 \zeta + O(q).
\partial_2 \psi_{\varkappa, 2}(0) = c_1 \varkappa \left[ \frac{\Lambda}2 - \varkappa^2\right] \sqrt{\Lambda- \varkappa^2}\left[ \cos \left(\frac\pi2\sqrt{\frac{\Lambda}{2}- \varkappa^2}\right) - \cos \left(\frac\pi2 \sqrt{\Lambda- \varkappa^2}\cdot \right) \right] ,
\left\Vert \mathcal{N}^{\beta}W_{\hbar}\left( t,s\right) \psi\right\Vert^{2}=\left\langle W_{\hbar}\left( t,s\right) \psi,\mathcal{N}^{2\beta}W_{\hbar}\left( t,s\right) \psi\right\rangle \leq\left( K_{\beta}\left(S,T\right) \right)^{2}\left\Vert \psi\right\Vert_{\left( 2\beta+1\right)d}^{2}
F_{(r,r)}(q) = (q-1) q^{2r-3}F_{(r,r-1)}(q) &= (q-1) q^{2r-3}\cdot (q-1)^{2r-3}q^{{2r-2\choose 2}- 2(r-1) +1}\sum_{i=0}^{r-1}C_{2r-1-i,i}q^i &= (q-1)^{2r-2}q^{{2r-2\choose 2}}\sum_{i=0}^{r-1}C_{2r-1-i,i}q^i.
I_1 = \mathbb{E}\int_0^{T\wedge\tau_n^M}\big[ 2\langle A(r,Y_n(r)), Y_n(r) \rangle + \Vert P_n B(r,Y_n(r))Q_n \Vert_{L_2}^2 \big] dr \int_{0\vee(r-\delta)}^{r}\mathbf{1}_{\{\tau_n^M>t\}}dt \leq \delta \, \mathbb{E}\int_0^{T\wedge\tau_n^M}f(s) (1+\Vert Y_n(s)\Vert_H^2) ds \leq C_M \delta .
\mathbf{x}\left [ l \right ] = \mathbf{P}\mathbf{s}\left [ l \right ] = \mathbf{p}_{\mathrm{c}}s_{\mathrm{c}}\left [ l \right ]+ \sum_{k \in \mathcal{K}}\mathbf{p}_{\mathrm{p},k}s_{\mathrm{p},k}\left [ l \right ],
F_1( f) (\alpha) = \alpha f(e_{t(\alpha)}) - f(e_{s(\alpha)}) \alpha, F_{n}(f) (\alpha_1 \cdots \alpha_n) = \alpha_1 f(\alpha_2 \cdots \alpha_n) + (-1)^n f(\alpha_1 \cdots \alpha_{n-1}) \alpha_n.
\lambda_j = \exp(4 \pi i (\eta_j-\eta_2)_1(\eta_1)_2)\left( \frac{\theta[\eta_{U_\eta\circ(V \cup \{1,2\})}](\Omega)\theta[\eta_{U_{\eta}\circ (W \cup \{1,2\})}](\Omega)}{\theta[\eta_{U_\eta \circ (V \cup \{1,j\})}](\Omega)\theta[\eta_{U_{\eta}\circ (W \cup \{1,j\})}](\Omega)}\right)^2,
\begin{array}{ccccc}\chi_{++}: \quad{\bf Q}^R= (e_3/3,e_3/3,e_1/3) \quad{\bf Q}^L= (e_3/3,e_3/3,0) \chi_{-+}: \quad (e_2/3,e_1/3,e_3/3) \quad (e_3/3,e_3/3,0) \chi_{+-}: \quad (e_3/3,e_1/3,e_2/3) \quad (e_2/3,e_2/3,0) \chi_{--}: \quad (e_2/3,e_2/3,e_1/3) \quad (e_2/3,e_2/3,0) \end{array}
\underline{\underline{C}}=[\begin{matrix}C_{11}&C_{13}&C_{13}&0&0&0\\C_{13}&C_{33}&C_{33}-2C_{44}&0&0&0\\C_{13}&C_{33}-2C_{44}&C_{33}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{66}&0\\ 0&0&0&0&0&C_{66}\end{matrix}]
\frac{dH}{dt}\simeq \frac{2}{5}M(\langle R \rangle)(32 \pi \langle R \rangle^3 \Gamma) \frac{\langle R \rangle}{R}\sim \frac{2}{5}M(\langle R \rangle) \frac{\frac{4 \pi}{3}\langle R \rangle^3}{\Delta V}(\Delta V -\Delta V')\Gamma,
n \:C \;\alpha_1^{1-p-\gamma}\:\beta_1^{1-q}\;\alpha_{i_0}^\gamma\left( \prod_{j=2}^p \alpha_1^{a_j - \alpha_j}\:\alpha_j^{-a_j}\right)\left( \prod_{k=2}^q \beta_1^{b_k-\beta_k}\:\beta_k^{-b_k}\right)\;,
\sum_{k=0}^{n-1}\frac{(q;q^2)_k (-1;q^2)_k}{(q^2;q^2)_k}q^{2k}\equiv (-1)^{\frac{n-1}{2}}\pmod{\Phi_{n}(q)}, begin{align*}7pt]\sum_{k=0}^{n-1}\frac{(q;q^2)_k (-q^2;q^2)_k}{(q^2;q^2)_k}q^{2k+1}\equiv (-1)^{\frac{n-1}{2}}\pmod{\Phi_{n}(q)},
D^m[f_1 \circ f_2 \circ f_3](x) = Df_1(z) \left( Df_2(y) \right) \left( D^mf_3(x) \right) + Df_1(z)\left( D^mf_2(y) \right) \left( Df_3(x) \right)^{\otimes m}&+ D^m f_1(z) \left( Df_2(y) \right)^{\otimes m}\left( Df_3(x) \right)^{\otimes m}+ \mathcal{P}_m(f_1,f_2)(y)\left(Df_3(x) \right)^{\otimes m}& + \mathcal{P}_m(f_1 \circ f_2, f_3)(x),
\frac{d^{2}\varphi_{k}^{H}}{d t^{2}}+ 3 \frac{\dot{a}}{a}\frac{d \varphi_{k}^{H}}{d t}+ \left[ \frac{\vec{k}^{2}}{a^{2}}+{\cal{V}}^{( 2 )}( \phi ( t ) ) \right] \varphi_{k}^{H}= 0
\lambda_k(\xi):=\inf_{\substack{V\,\textrm{subspace of}\, \Lambda_\xi^+ H_\xi^{1/2}\dim V=k}}\sup_{u_\xi\in (V\bigoplus \Lambda_\xi^-H_\xi^{1/2})\setminus \{0\}}\frac{\left(D_{\gamma,\xi}\,u_\xi,u_\xi\right)}{\|u_\xi\|_{L^2_\xi}^2}.
\big( -R(a) + a, -R(a) -a \big) \bullet \big( -R(b) + b, -R(b) -b \big) &= \big( R(a) \cdot R(b) - R(a) \cdot b - a \cdot R(b) + a \cdot b,~ R(a) \cdot R(b) + R(a) \cdot b + a \cdot R(b) + a \cdot b \big)
\begin{aligned}\varsigma_{ij}(z) :=\sqrt{x_i}\sigma_{ij}(z),\quad\forall\, i=1,\ldots,n,\quad\forall\, j=1,\ldots,n+m, \varsigma_{ij}(z) :=\sigma_{ij}(z),\quad\forall\, i=n+l,\ldots,m, \quad\forall\, j=1,\ldots,n+m,\end{aligned}
\bar{A}_{k}^{a}= \frac{2}{L}\, \vartheta \: \delta^{a 3}\delta_{k 0}+ \frac{2}{L}\, \vartheta \: \epsilon^{a 3 b}\: \bar{\varphi}_{k}^{b}+ \mathrm{i}\, k \: \frac{2 \pi}{L}\, \bar{\varphi}_{k}^{a}\; ,
\begin{array}{c}\left({D\over D x^{\mu}}-i^{p\over 2}{\partial f\over\partial R}\Gamma^{\mu}\prod\limits_{I=1}^{p-1}\Gamma^I\right)\Psi_{++}=0\;\;\; (\mu=1,\ldots,p-1) \left({D\over D y^{\mu}}-i^{q\over 2}{\partial f\over\partial R}\Gamma^{p+\mu}\prod\limits_{I=1}^{q-1}\Gamma^{p+I}\right)\Psi_{++}=0\;\;\; (\mu=1,\ldots,q-1)\end{array}
\psi={\small \left[ \begin{array}{*{3}c}1 -1 0 1 0 -1 0 1 -1\end{array}\begin{array}{*{3}c}(x_1-x_2)^{r+1}0 0 0 (x_1-x_3)^{r+1}0 0 0 (x_2-x_3)^{r+1}\end{array}\right]}
\sum_{m,s,j \atop 2m+2j+s=p-r}\overline a_{mrsj}Q^m Q_{L(F)}^j u^s + \sum_{m,s,j \atop 2m+2j+s+2=p-r}b_{mrsj}Q^m v_F^{2j+1}(v_F\times u)u^s=0
{\bf E}\Bigl[\Bigl(\int\limits_t^{t+\tau}M_\mu(dt)\Bigr)^{n}\Bigl(\int\limits_0^\tau M_\mu(dt)\Bigr)^{n}\Bigr] = (n!)^2 \int\limits_{\substack{0<t_{1}<\cdots<t_{n}<\tau t<t_{n+1}<\cdots<t_{2n}<t+\tau}}\prod\limits_{k<p}^{2n}|t_p-t_k|^{-\mu\,d(p-k)}\,dt.
a_0-a_0 e^{-t}+\sum \limits_{k=1}^{\infty}\frac{a_k (1+k \sin{k t}\e^{-t}-\cos{k t}\e^{-t})}{1+k^2}+ \frac{b_k (k-k \cos{k t}\e^{-t}-\sin{k t}\e^{-t})}{1+k^2}.
\frac{N!(-4)^{N-j}}{j!(2N-2j)!}(j-2N+M+\frac 12)_{N-j}= = \sum_{l=0}^M{M\choose l}\frac{(N-M)!(-4)^{N-M-j+l}}{(j-l)!(2N-2M-2j+2l)!}(j-l-2N+M+\frac 12)_{N-M-j+l},
\frac{R_{12}}{-i-\omega(2,2)}u_2+\frac{R_{13}}{-i-\omega(3,3)}u_3=i(R_{12}i+R_{13})=-R_{12}+iR_{13}=-1-1=-2=2u_1; \frac{R_{42}}{-i-\omega(2,2)}u_2+\frac{R_{43}}{-i-\omega(3,3)}u_3=-R_{42}+iR_{43}=0-2i=-2i=2u_4.
V_{0}( F , \phi ; F_{T}, t ) = - F \, \mathrm{l n}\left( \frac{\phi^{( N_{c}- N_{f})}\mathrm{d e t}( t )}{\Lambda^{( 3 N_{c}- N_{f})}}\right) - \phi \mathrm{T r}\left[ F_{T}t^{- 1}\right] + \mathrm{h . c .}\ .
C \, \int_{0}^{\infty}a_1^{d_1 -1}\; y^{\frac{d_1-1}{p_1}}\, e^{-y}\, \left( \frac{a_1}{a_2}\right)^{p_2}\, y^{\frac{p_2}{p_1}}\, \frac{a_1}{p_1}\, y^{-\frac{p_1-1}{p_1}}\, dx begin{align*}2ex]= \; C \, \frac{a_1^{d_1}}{p_1}\, \left( \frac{a_1}{a_2}\right)^{p_2}\, \int_{0}^{\infty}y^{\frac{d_1 + p_2}{p_1}- 1}\; e^{-y}\, dy.
\int_V\left[(\sigma_{ij}^{(1)}u_i^{(2)})_{,j}+ (D_j^{(1)}\phi^{(2)})_{,j}\right]\mathrm{d}V = \int_V\left[(\sigma_{ij}^{(2)}u_i^{(1)})_{,j}+ (D_j^{(2)}\phi^{(1)})_{,j}\right]\mathrm{d}V.
\frac{1}{t_i(a)}- \frac{1}{t_i(b)}= \left\{\frac{1}{r_i}(1-a) + \frac{1}{r_i+s_i}a \right\}- \left\{\frac{1}{r_i}(1-b) + \frac{1}{r_i+s_i}b \right\}= (b - a) \left( \frac{1}{r_i}- \frac{1}{r_i+s_i}\right),
\rho_{m_{0}\rightarrow m_{1}+ m_{2}}^{D}= \frac{\Gamma ( ( n + 1 ) / 2 )}{\pi^{( n + 1 ) / 2}\Gamma ( n )}\int_{0}d \mu \mu^{n - 1}\rho_{m_{0}\rightarrow m_{1 \mu}+ m_{2 \mu}}^{D - n}
\mathbf{b}(k) := \left(\begin{array}{c}2\cos(k_1 x_1) \mathbf{e}^{(1)}_{\gamma}(k) e^{i(k_2 x_2 + k_3 x_3-\omega t)}2i\sin (k_1 x_1) \mathbf{e}^{(2)}_{\gamma}(k) e^{i(k_2 x_2 + k_3 x_3-\omega t)}2i\sin (k_1 x_1) \mathbf{e}^{(3)}_{\gamma}(k) e^{i(k_2 x_2 + k_3 x_3-\omega t)}\end{array}\right).
\frac{\tilde{S}_{k,n}}{\sum\limits_{\substack{I^{c}_{k+l}(n)}}\tilde{b}(i_{1},i_{2},...,i_{k+l};n)}=1+ \frac{\sum\limits_{\substack{I_{k+l}(n)}}\tilde{b}(i_{1},i_{2},...,i_{k+l};n)}{\sum\limits_{\substack{I^{c}_{k+l}(n)}}\tilde{b}(i_{1},i_{2},...,i_{k+l};n)},
\begin{aligned}\qquad\,\,\mbox{there are}\,\, \beta>p\,\, \mbox{and}\,\, r_0>0\,\, \mbox{such that}& tf(x,t)\geq \beta F(x,t)>0\,\, \mbox{for any}\,\, |t|\geq r_0\,\,\mbox{and every}\,\, x\in D,\end{aligned}
{\hat{x}}_{C}^{I}= 2 t \partial_{t}^{2}- 2 u \partial_{u}^{2}- 2 t \partial_{t}+ 2 u \partial_{u}+ ( 2 p_{1}+ 1 ) \partial_{t}- ( 2 p_{2}+ 1 ) \partial_{u}- p_{1}+ p_{2}\ ,
I_{1}=-\frac{1}{2\pi i}\oint_{\partial U_\xi}\left(J^E(s)-I\right)ds, I_{2}=-\frac{1}{2\pi i}\int_{\Gamma^{(3)}\backslash U_\xi}\left(J^E(s)-I\right)ds, I_{3}=-\frac{1}{2\pi i}\int_{\Gamma_E}\mu(s)(J^E(s)-I)ds.
- 4 \delta b \| \psi_* \|^2_{L^2(\mathbb{R})}+\frac{3}{\pi}\langle \cos(y) \psi_*, (2 b \cos(y) \psi_* + \tilde{u}_b)^2 \rangle_{L^2(\mathbb{R}\times \mathbb{T})}= 0.
l=-\frac{imk(cos\,\mu \,sin\,\lambda +e^{2iak}cos\,\lambda \,sin\,\mu)+mE(e^{2iak}-1)sin\,\lambda \,sin\,\mu}{m^{2}(e^{2iak}-1)sin\,\lambda\,sin\,\mu +k^{2}cos(\lambda +\mu )+iEk\,sin(\lambda +\mu )}
B(x)= \,B(x,[0^{s+1}],[v-s,k-\frac{s-1}{2},\lambda-\frac{s-2}{3}]) = \,\sum_{j=0}^2{2\choose j}P(-x,2-j)P(s,j)\lambda_j = \,x(x+1)(v-s)-2xs(k-(s-1)/2)+s(s-1)(\lambda-(s-2)/3).
g((\nabla_{Z_{i}}A_{N})T_{r-1}Z_{i},X) = g(T_{r-1}Z_{i},(\nabla_{Z_{i}}A_{N})X) + g(\nabla_{Z_{i}}A_{N}(T_{r-1}Z_{i}),X) -g(\nabla_{Z_{i}}(A_{N}X),T_{r-1}Z_{i}) + g(A_{N}(\nabla_{Z_{i}}X),T_{r-1}Z_{i}) -g(A_{N}(\nabla_{Z_{i}}T_{r-1}Z_{i}),X),
\nabla \Pi = \sqrt{-\gamma}\; \Pi{}' =-{1 \over \sqrt{-\gamma}}\left[ \left({L_2 \over \kappa_1}\right)'\; \eta_2 +{L_2 \kappa_2 \over \kappa_1}\; \eta_1-{L_2 \kappa_3 \over \kappa_1}\; \eta_3 \right]\,.
\prod_{p\in\mathbb{P}}(1+\frac{2}{p^{s}}+\frac{2}{p^{2s}}+\cdot\cdot\cdot)=\sum_{n=1}^{\infty}\frac{2^{\omega(n)}}{n^{s}}=\frac{\zeta(s)^{2}}{\zeta(2s)},
\Lambda_{(4)}(a) = \frac{1}{a^{2}}\left( - \kappa_{(4)}^{2}\rho_{0}a^{-3\gamma +2}- \frac{\kappa_{(4)}^{2}\rho_{0}^{2}}{2\lambda}a^{-6\gamma +2}- \sigma_{0}^2 a^{-4}- \frac{6{\cal U}_{0}}{\lambda \kappa_{(4)}^{2}}a^{-2}+ 3k \right).
\left|f(y)-\operatorname{Tayl}_z^Nf(y)\right| \leq\sum_{\alpha\in\mathbb{Z}_+^n,\,|\alpha|=N}\frac{|(y-z)^\alpha|}{\alpha!}\int_0^1S|t(y-z)|^sN(1-t)^{N-1}\,dt &\lesssim S|y-z|^{N+s},
\sum_{k=i}^{n-r+i+1}x_{i,k}=s\quad(i=1,2,\ldots,r), &\sum_{k=i}^jx_{i,k}\ge\sum_{k=i+1}^{j+1}x_{i+1,k}(i=1,\ldots,r-1;j=i,\ldots,n-r+i).
\beta =\frac{1}{4\pi}wd\left( \int_{E_{3}^{\prime}}\frac{\delta^{\,\prime}(w^{\prime}\beta^{\prime})}{r_{12}}w^{\prime}\right) +\frac{1}{4\pi}w\delta \left( dx^{jk}\int_{E_{3}^{\prime}}\frac{d^{\prime}(w^{\prime}\beta^{\prime})\wedge dx^{\prime i}}{r_{12}}\right) .
\begin{array}{l}{{y^{0}= \displaystyle a \frac{1 + x x^{\prime}}{1 - x x^{\prime}}~ , i y^{n + 1}= a \frac{2 x^{n + 1}}{1 - x x^{\prime}}~ ,}}\\{{y^{i}= a \displaystyle \frac{2 x^{i}}{1 - x x^{\prime}}~ , ( i = 1 , ~ 2 , ~ \cdots , ~ n ) ~ .}}\\ \end{array}
\tilde{\lambda}(\hat{\tau}/\hat{\sigma},g) \, P \left( - \rho \frac{\hat{\tau}}{\hat{\sigma}}+ \sqrt{\frac{(m + (\hat{\tau}/\hat{\sigma})^2)(1 - \rho^2)}{m+g-1}}\,T_{m+g-1}\le w \right) &+ \left(1-\tilde{\lambda}(\hat{\tau}/\hat{\sigma},g)\right) \,P \left( \sqrt{\frac{m}{m+g-1}}\, T_{m+g-1}\le w \right) = \eta
\langle\pi_k(h)w_k,w_k\rangle=\begin{cases}\displaystyle\frac{\mu_n(h)^{nk/2}\,2^{nk}}{\det{(A+D+i(C-B))^k}}\mu(h)>0, begin{align*}2ex] 0 \mu(h)<0. \end{cases}
(z\partial_z + \mathfrak{E}_{(\partial)}+ \mu) \,_\circ \, S^\textnormal{coh}(\tau,z)(e^{\tau_2/z}\alpha) = S^\textnormal{coh}(\tau,z) \,_\circ \, e^{\tau_2/z}\,_\circ \, (z\partial z + \mathfrak{E}_{(\partial)}+ \mu) (\alpha)
x\in D(a,b;2\epsilon) \Rightarrow \hat \delta^{\tilde g}_r(x)\subseteq{\hat \delta}^{\tilde f}_r(a,b), \ \bigoplus_{x\in D(a,b;2\epsilon)}\hat \delta^{\tilde g}_r(x)= \hat \delta^{\tilde f}_r(a,b).
\frac{\delta{\cal W}}{\delta \psi^{\alpha}}= \frac{\delta \Gamma}{\delta \psi^{\alpha}}- \frac{\delta \Gamma}{\delta \Sigma^{m}}\sigma_{, \alpha}^{m}( \varphi , \psi ) , \frac{\delta{\cal W}}{\delta \phi_{A}^{*}}= \frac{\delta \Gamma}{\delta \phi_{A}^{*}}\, ,
S_{\mathrm{e f f}}= ( \pm ){\frac{r^{2}\lambda_{k}^{2}}{g ( r )}}\left[{\frac{\pi}{2}}- 2 \pi n^{\prime}+ i \left( C_{\gamma}- \operatorname{l n}{\frac{\lambda_{k}^{2}}{\mu^{2}}}\right) \right]
M_0(k_1,k_2,\dots,k_r) = \frac1{K}\sum_{D|k_\nu ,\, \nu=1,2,\dots,r}\frac{\mu(D) (K/D)!}{\prod_{\nu=1}^r (k_\nu /D)!},\qquad\qquad K := \sum_{\nu=1}^r k_\nu .
p_{b,m}^* = \left[{\frac{{{W_m}}}{{{\lambda^*}}}- \frac{1}{{{h_m}}}}\right]_0^{\widetilde P},p_{b,o}^* = \left[{\frac{{{\left({W -{W_m}}\right)}}}{{{\lambda^*}}}- \frac{1}{{{h_b}}}}\right]_0^{\widetilde P},
\begin{array}{c}\mathfrak p \updownarrow \left(\begin{array}{c}\mathfrak q \updownarrow \mathfrak r\end{array}\right)\end{array}=\begin{array}{c}\left(\begin{array}{c}\mathfrak p \updownarrow \mathfrak q\end{array}\right) \updownarrow \mathfrak r.\end{array}
N_{n m}^{w}= \frac 1{n \, m}\oint_{\infty}\frac{d z}{2 \pi i}z^{n}\oint_{\infty}\frac{d \zeta}{2 \pi i}\zeta^{m}\frac{w^{\prime}( z ) w^{\prime}( \zeta )}{( w ( z ) - w ( \zeta ) )^{2}}
\int_0^t\int_\mathbb{R}\lambda_p(u^a) w_{px}w_p g^2 dxd\tau=\int_0^t\int_\mathbb{R}\bigg[\frac{1}{2}\lambda_p(u^a)_x w_p^2 g^2+ \lambda_p(u^a) w_p^2 g g_x \bigg]dxd\tau,
\int_{B_{R/2}}\vert \nabla u \vert^2 dx \lesssim \left( \int_{\mathcal{C}(R/2,R)}\vert \nabla u \vert^{\frac{p}{2}}dx+\int_{\mathcal{C}(R/2,R)}\vert u \vert^{p}dx \right)^{\frac{2}{p}}R^{2-\frac{9}{p}}\left( \int_{\mathcal{C}(R/2,R)}\vert u \vert^{p}dx\right)^{\frac{1}{p}}
\int_\Omega \Big|V^p \Big(\ln\ln(e+V)\Big)^2 \psi^h_{\mu_j,\xi_j}\Big|dx = O( (\ln|\ln\mu_i| )^2) = O\bigg( \Big( \ln\Big|\ln\frac{\epsilon}{|\ln\epsilon|^2}\Big|\Big)^2\bigg).
\mathrm{l n ~ d e t}\Delta = - \lim_{s \to 0}{\frac{d}{d s}}\sum_{n_{1}, n_{2}}[ ({\frac{4 \pi^{2}}{l^{2}}}) ( n_{1}^{2}+{\frac{( n_{2}+ \alpha )^{2}l^{2}}{4}}) ]^{- s}\quad .
\widetilde{\Phi}_{a,b,c}(g^ih^j,g^sh^t,g^kh^l)=\zeta_m^{-a[\frac{(m-i)'+(m-s)'}{m}]k}\zeta_n^{-b[\frac{(m-i)'+(m-s)'}{m}]l}\zeta_n^{-c[\frac{(n-j)''+(n-t)''}{n}]l}
\|u\|_\infty\leq C\left(1+\|u\|_{BMO}\ln^{\frac12}(e+\|u\|_{H^{m-1}})\right) \|\nabla b\|_\infty\leq C\left(1+\|\nabla b\|_{BMO}\ln^{\frac12}(e+\|b\|_{H^m})\right)
f_k^{(\lambda, T)}(x_{\mathcal{I}}^k, y_{T^c}^k)\triangleq \prod_{\ell=1}^{k}\left(p(x_{\mathcal{I},\ell}| x_{\mathcal{I}}^{\ell-1}, y_{T^c}^{\ell-1}) \frac{\left(q(y_{T^c,\ell}| x_{\mathcal{I},\ell})\right)^\lambda}{\left(s(y_{T^c,\ell}| x_{T^c,\ell})\right)^{\lambda-1}}\right)
\Delta S_{G}= - f_{g}^{2}\int d^{3}x ~ a \sum_{n = 1}^{N}\frac{1}{4}\epsilon^{i j k}\epsilon_{A B C D}E_{i}^{A}( x , n ) R_{k 0}^{C D}E_{j}^{B}( x , n + 1 ) + \cdots .
F^{(1)}_{\theta}(s) := 1 + P(s)\cos\left(\frac{\theta-\theta_{\hat x}}{2}\right)+ Q(s)\sin\left(\frac{\theta-\theta_{\hat x}}{2}\right), & F^{(2)}_{\theta}(s):=P(s)\sin\left(\frac{\theta+\theta_{\hat x}}{2}\right)+Q (s)\cos\left(\frac{\theta+\theta_{\hat x}}{2}\right), & F^{(3)}_{\theta}(s) := \cos\left(\frac{\theta-\theta_{\hat x}}{2}\right)+P(s).
\Delta (x|z_1|z_2,\cdots,z_N)=(x-z_1\tau)\sum_{\kappa=0}^{N-3}(1-\tau^{2(N-2-\kappa)})(-\tau)^\kappa x^{N-3-\kappa}\sum_{\lambda=0}^\kappa(-z_1\tau^2)^\lambda\sigma_{\kappa-\lambda}(z_2 \cdots z_N),
\Delta_c^{cl}=\int d^4x\;\left( [c,c^{*}]-[A,A^{*}]-[\phi ,\phi^{*}]+[\psi,\psi^{*}]-[\overline{\phi},\overline{\phi}^{*}]+[\eta ,\eta^{*}]+\frac12[\chi ,\chi^{*}]\right) \;.
H_t(u,p)\to \, \frac{1}{R\big(\varrho_S((u,p),(\bar{u},\bar{p}))\big)}\,H_t(u,p) & = H_t(\bar{u}(u),\bar{p}(u))+\, \frac{1}{R\big(\varrho_S((u,p),(\bar{u}(u),\bar{p}(u)))\big)}\delta H_t(u,p).
\overline{g}(q,p,\overline{z}) =q\cdot p-z +a_{ij,kl}(q,p)q_iq_jp_kp_l +z[b_{ij}(q,p)q_iq_j+c_{ij}(q,p)q_ip_j+d_{ij}(q,p)p_ip_j]+f(x,y,z)z^2.
\mathcal{M_{\mathbb{N}}}= \{\omega=\frac{\sum_{h=1}^{H}c_h\delta_{a_h}}{\sum_{h=1}^{H}c_h}:\; c_h \in \mathbb{N}^+\}, \mathcal{M_{\mathbb{Q}}}= \{\omega=\frac{\sum_{h=1}^{H}c_h\delta_{a_h}}{\sum_{h=1}^{H}c_h}:\; c_h \in \mathbb{Q}^+\}, \mathcal{M_{\mathbb{R}}}= \{\omega=\frac{\sum_{h=1}^{H}c_h\delta_{a_h}}{\sum_{h=1}^{H}c_h}:\; c_h \in \mathbb{R}^+\},
\sum_{k>D_\varepsilon}b_k^2 \theta_k^2 \leq b_{D_\varepsilon}^2\left(\sum_{k>D_\varepsilon}\theta_k^2 \right)< b_{D_\varepsilon}^2 \left(r_\varepsilon^2-C_{\min}(\alpha,\beta)\varepsilon^2\sqrt{\sum_{k=1}^{D_\varepsilon}b_k^{-4}}\right).
S_{H G}= \int d^{4}x \sqrt{- g^{L}}M^{3}g_{L}^{\mu \nu}\partial_{5}G_{\mu \nu}\Big |_{y = L}- \int d^{4}x \sqrt{- g^{0}}M^{3}g_{0}^{\mu \nu}\partial_{5}G_{\mu \nu}\Big |_{y = 0}\ .
\sum_{p \,\leq\, x}(\log g_{u}(p-1))^{\lambda}=\sum_{\substack{d \, \le \, y (d,\,a_{2})\,=\,1}}\rho_{\lambda}(d)\pi(x;\l(d),1)+\sum_{\substack{d \, \ge \, y (d,\,a_{2})\,=\,1}}\rho_{\lambda}(d)\pi(x;\l(d),1) &=E_{1}+E_{2}.
[t[\rho_1] + (1-t)[\rho_2] = [\sigma_{M\otimes R}(\rho_1 \otimes p + \rho_2\otimes p^{\perp})] & = [\sigma_{M\otimes R}\circ Ad(v+w) \circ \sigma_{M\otimes R}^{-1}\circ \pi] & = [\sigma_{M\otimes R}\circ \sigma_{M\otimes R}^{-1}\circ \pi] & = [ \pi].
F_t(X_t) = F_0(X_0)+ \int_0^t\nabla^hF_s(X_s)ds + \int_0^t\nabla^wF_s(X_s)dX(s) - \frac{1}{2}\int_{-\infty}^{+\infty}\int_0^t(\nabla^w_xF_s)(^{x}X_s)d_{(s,x)}\ell^x(s)
\prod_{j=0}^{a-1}\prod_{i=0}^{b-1}\frac{m+j-i}{b+j-i}=\prod_{j=0}^{a-1}\frac{(m+j)!}{(m-b+j)!}\frac{j!}{(b+j)!}= \prod_{j=0}^{a-1}\binom{m+j}{b}\binom{b+j}{b}^{-1}.
L^\perp_{\pi/((i_1)(i_2)\cdots(i_{m-1}))}(z_1^{i_1}z_2^{i_2}\cdots z_{m-1}^{i_{m-1}}) \, M(z_m) = M(z_m) \prod_{i_m\ge 0}\, L^\perp_{\pi/((i_1)(i_2)\cdots(i_m))}(z_1^{i_1}z_2^{i_2}\cdots z_{m}^{i_{m}})
N ( x_{\mu}; q ) = \frac{1}{2}\left[ \{( L^{+})^{- \frac{1}{2}}+ ( L^{-})^{- \frac{1}{2}}\}\cdot 1 + \{( L^{+})^{- \frac{1}{2}}- ( L^{-})^{- \frac{1}{2}}\}\cdot \hat{x}\right] ,
f(\lambda) = \int_{(0,\infty)}(1-e^{-\lambda r}) \, \mu(dr) \geq \int_{(0,\lambda^{-1})}(1-e^{-\lambda r}) \, \mu(dr) &\geq c \int_{(0,\lambda^{-1})}(1-e^{-\lambda r}) \,\frac{dr}{r^{1+\rho}}.