harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
U\phi_{\ell}=2\alpha\left({\tt \frac{1}{\bar{z}}}\partial_z\phi_{\ell}-{\tt \frac{1}{z}}\partial_{\bar{z}}\phi_{\ell}\right)+\frac{\alpha^2}{2}\left({\tt \frac{1}{z}{\tt \frac{1}{\bar{z}}}}+{\tt \frac{1}{\bar{z}}}{\tt\frac{1}{z}}\right)\phi_{\ell}\, ,
A = \left[\begin{array}{cccc}0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 \end{array}\right], B = \left[\begin{array}{cccc}0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 \end{array}\right].
\begin{array}{l}\displaystyle [e_1,e_{n+1}]=e_1-e_3,[e_{i},e_{n+1}]=\left(i-3\right)e_{i},[e_{n+1},e_1]=-e_1+e_3,[e_{n+1},e_{2}]=-e_2, \displaystyle [e_{n+1},e_i]=\left(3-i\right)e_i,(3\leq i\leq n,n\geq4),\end{array}
B_{s}^{k}+B_{x}^{k+1}+\frac{1}{2}B^{0}B_{x}^{k}+\frac{k+1}{2}B^{k}B_{x}^{0}+kB^{k-1}\left( \frac{1}{2}B^{1}-\frac{1}{8}(B^{0})^{2}\right)_{x}=0,
E \Biggl[\zeta f_1\bigl(\check{\mathbf{L}}^n\bigr)f_2\bigl(\widetilde{\mathbf{L}}(\beta)^n\bigr)\prod_{j=1}^JY^n_{j-}Y^n_{j+}\Biggr] \to E \Biggl[\zeta f_1\bigl(\mathcal{W}^{S_{r_J}}\bigr)f_2\bigl(\widetilde{\mathcal{W}}(\beta)\bigr)\prod_{j=1}^JY_{j-}Y_{j+}\Biggr]
\partial_1 Q_0^n(x,y) = n p_{n-1}(w_1;x) = n P_0^{n-1}(x,y) \hbox{and}\partial_2 Q_0^n (x,y) =0, \partial_1 Q_n^n(x,y) =0 \hbox{and}\partial_2 Q_0^n (x,y) = n p_{n-1}(w_2;y) = n P_{n-1}^{n-1}(x,y).
\int_{0}^{\infty}{\frac{z^{\nu -2m}}{\sinh{(\pi z)}}F_1(z)\bar J_\nu (z)dz}=A_{\nu m}^{(0)}\left[ \frac{zF_1(z)}{\sinh{(\pi z)}}\right] +\frac{2}{\pi}\sum_{k=1}^{\infty}(-1)^{m+k}k^{\nu -2m}\bar K_{\nu}(k)F_1(ik),
M = \eta \left( \begin{array}{c c}{{e^{2 b \pi}}}&{{0}}\\{{0}}&{{e^{- 2 b \pi}}}\\ \end{array}\right) \quad \eta = \pm 1 \quad ; \qquad H_{M}= \left\{\left( \begin{array}{c c}{{a}}&{{0}}\\{{0}}&{{a^{- 1}}}\\ \end{array}\right) \quad \vert \, \, a \neq 0 \right\}\, \, .
\lim_{s \to \frac{3}{4}}\biggl(P_{1/2,0}(z,s)-\frac{\frac{3}{4\pi}\theta(z)}{s-3/4}\biggr) = 3\widehat{\mathbf{Z}}_+(z) = F_{1/2,0,0}(z) \in H_{1/2}^{3/2},
K(\lambda) = \lambda \biggl(\frac{\delta^\alpha}{b\alpha}\lambda^{\alpha-1}+ 1\biggr)^{-\frac{1}{\alpha-1}}= \biggl(\frac{\delta^\alpha}{b\alpha}+ \lambda^{1-\alpha}\biggr)^{\frac{1}{1-\alpha}}, \lambda \in (0, \theta_0) ,
\mu_{0}:=\left(c_{\alpha}^{+}-c_{\alpha}^{-}\right)\Gamma\biggl(\frac{3-\alpha}{2}\biggl)+\frac{\left(c_{\alpha}^{+}+c_{\alpha}^{-}\right)\beta\chi_{*}(0)}{2-\alpha}\Gamma\biggl(2-\frac{\alpha}{2}\biggl), \mu_{1}:=\frac{c_{\alpha}^{+}+c_{\alpha}^{-}}{2}-\kappa d.
\mathrm{d}X^{(N)}_i(t)=\frac{\beta}{2}\,\sum_{\substack{1\le j\le N j\neq i}}\frac{1}{X^{(N)}_i(t)-X^{(N)}_j(t)}\,\mathrm{d}t+\mathrm{d}B^{(N)}_i(t),i=1,2,\ldots,N,
\epsilon_{\mathrm{C a s}}( x ) = \sum_{j}\frac{1}{2}\left( 1 - \frac{2}{\pi}\operatorname{a r c t a n}\frac{\omega_{j}}{\sqrt{m^{2}- \omega_{j}^{2}}}\right) \omega_{j}\phi_{j}^{2}( x ) - \sum_{j}\frac{1}{\pi}\sqrt{m^{2}- \omega_{j}^{2}}\phi_{j}^{2}( x ) \ \ ,
A_{n}= \int d x_{1}. . . \int d x_{n}\int d y_{1}. . . \int d y_{n}{\frac{\delta^{n}S}{\delta \sigma ( x_{1}, y_{1}) . . . . . \delta \sigma ( x_{n}, y_{n})}}|_{\sigma^{0}}\eta ( x_{1}, y_{1}) . . . . . \eta ( x_{n}, y_{n})
\prod_{a=1}^{k}u_{i_{a},i_{a+1}}+\sum_{r=2}^{k+1}\left( \prod_{a=r}^{n}\beta (u_{i_{a},i_{a+1}}) u_{i_{a_{k+1}},i_{a_{1}}}\left( \prod_{a=1}^{r-2}u_{i_{a},i_{a+1}}\right)\right) \qquad{}= \beta \prod_{a=1}^{k}u_{i_{a},i_{a+1}}- \beta \big( u_{i_{1},i_{k+1}}R_{I{\setminus}\{i_{k+1\}}}- R_{I{\setminus}\{i_{k+1}\}}u_{i_{k},i_{k+1}}\big),
\begin{array}{ccccc}\Phi_{a}: \mathbb{F}_{q}^n \longrightarrow \mathbb{F}_{q}^n & x=(x_0,x_2,\ldots,x_{n-1}) \longmapsto (a_{n-1}x_{n-1},a_{1}x_1,..,a_{n-2}x_{n-2})\end{array}
q^{-\frac{v(\lambda_0)+1}{2}}(q^{v(\lambda_0)}-q^{-1}) + ((q^{\frac{v(\lambda_0)-1}{2}}-q^{-1}) - q^{-1}(1-q^{-\frac{v(\lambda_0)+1}{2}})) = 2q^{-1}(q^{\frac{v(\lambda_0)+1}{2}}-1).
(\omega_0(s)g_k)(re^{i\theta}) =\lim_{t\to 0^+}(\omega_0(s)g_{k,t})(re^{i\theta}) =\lim_{t\to 0^+}F_{k,t}(r)e^{ik(\theta+\frac{\pi}{2})}=(-1)^k i^k i^k \lim_{t\to 0^+}\frac{(\sqrt{t^2+r^2}-t)^k}{r^k\sqrt{t^2+r^2}}= r^{-1}e^{ik\theta}\,,
d_{O(X_+)}\left((x,s), \dfrac{1}{m+1}(x,s) + \dfrac{m}{m+1}(y,t)\right) = s d_{X}\left(x,\dfrac{1}{m+1}x + \dfrac{m}{m+1}y\right) \quad+ \left(\dfrac{1}{m+1}s + \dfrac{m}{m+1}t-s\right) = \dfrac{m}{m+1}(s d_X(x,y) + (t-s)) = \dfrac{m}{m+1}d_{O(X_+)}((x,s),(y,t)).
\mathcal{H}_{\mathcal{D}}=-\sqrt{B_{\mathcal{D}}(x)}\,e^{\partial}\sqrt{D_{\mathcal{D}}(x)}-\sqrt{D_{\mathcal{D}}(x)}\,e^{-\partial}\sqrt{B_{\mathcal{D}}(x)}+B_{\mathcal{D}}(x)+D_{\mathcal{D}}(x),
\mathcal P_n\left(y,\frac{\zeta}{(\zeta+2)(2\zeta+1)}\right) =\left(\frac{2}{(\zeta+2)(2\zeta+1)}\right)^{\delta(n)}\left(\frac{\zeta y+\zeta+2}{\zeta(\zeta+1)}\right)^n &\quad\times P_n\left(\frac{\zeta(y+2\zeta+1)}{\zeta y+\zeta+2},\zeta\right).
\mathbf{g}_t^\mathcal{F}:=(\mathrm{pr}_{t\star}\otimes_{\mathcal{X}_\star}\mathrm{pr}_{t\star})(\mathbf{g}) =(\mathrm{pr}_{t}\otimes_{\mathcal{X}}\mathrm{pr}_{t})(\mathbf{g}) =\mathbf{g}_t
\frac{N[\mathfrak{e},\lambda V]}{N_{sc}[\mathfrak{e},\lambda V]}\ \geq \ \frac{\mathcal{L}_{V}[\lambda^{-1}{\mathfrak{e}_{\mathrm{max}}}]}{N_{sc}[\mathfrak{e},\lambda V]}\ =\ \frac{N_{sc}^{>}[\mathfrak{e},\lambda V]}{N_{sc}[\mathfrak{e},\lambda V]}.
\|\tilde T_j(\hat\Sigma-\tilde{\Sigma}) \tilde T_j\|_{\infty}\leq \|\tilde T_j(\hat{\Sigma}_\epsilon-\Sigma_\epsilon)\tilde T_j\|_{\infty}+2\|\tilde T_j F\|\Big\|\frac{1}{n}\sum_{i=1}^nf_i\tilde T_j\epsilon_i\Big\|.
K_{N}^{( N_{f})}( x , y , M ) \ = \ \sqrt{w^{( N_{f})}( x , M ) w^{( N_{f})}( y , M )}\, \sum_{n = 0}^{N - 1}\frac{( n + 1 ) \, Z_{n}^{( N_{f})}( M )}{Z_{n + 1}^{( N_{f})}( M )}p_{n}^{( N_{f})}( x , M ) p_{n}^{( N_{f})}( y , M ) \ .
\begin{array}{c c c c}{\mathrm{l o n g r o o t s :}}&{\pm \beta_{k l}= \pm ( 2 \epsilon_{j k l}e_{j}- e_{k}- e_{l})}&{}&{( 1 \leq k < l \leq 3 )}\\{\mathrm{s h o r t r o o t s :}}&{\pm \alpha_{k l}= \pm ( e_{k}- e_{l})}&{}&{( 1 \leq k < l \leq 3 )}\\ \end{array},
{\; \atop{\mbox{det}\atop{\scriptstyle (i,j)}}}\Big(\frac{1}{\tilde{w}_i - \tilde{z}_j}\Big) =(-1)^{\frac{n(n-1)}{2}}\frac{\prod_{1 \leq i < j \leq n}(\tilde{w}_i - \tilde{w}_j )( \tilde{z}_i - \tilde{z}_j )}{\prod_{i,j = 1}^n ( \tilde{w}_i - \tilde{z}_j )}\; .
{\Delta}S = S_{F}- S_{I}= 4{\pi}( R^{2}- C^{2}){\mu}-{\pi}^{2}R^{3}\Big ( 1 - \frac{2}{\pi}( \frac{- C}{2 R}{\sqrt{1 - C^{2}/ R^{2}}}+ \frac{c o s^{- 1}( C / R )}{2}) \Big ){\sigma}
[A^{(I)}_+(\zeta)]_{n,i}^{n',i'}=\delta_{i+n',i'+n}(-1)^{i'}q^{ii'+\frac{1}{2}i(I+3-i)-\frac{1}{2}i'(i'+1)}\frac{(q^{-2I};q^2)_i}{(q^2;q^2)_i}\times &\times\frac{(q^2;q^2)_{n'}}{(q^2;q^2)_n}\phi^{-2n}q^{n(I+i-i')}(q^{-2n};q^2)_i.
\begin{aligned}\|\Phi(v_{1},v_{2})\|_{{{\bf X}}_T}\le \tilde{C}(T^{-2\tilde{\nu}-\frac12}+(\varepsilon_{1}+\varepsilon_{2})T^{-\tilde{\nu}-\frac14}+ \varepsilon_{1}\varepsilon_{2}T^{-\frac{\delta}2}&\ \ \ \ \quad+(\varepsilon_{1}+\varepsilon_{2})T^{\tilde{\nu}-\frac12}(\log T)^2+\varepsilon_{1}^2 )\end{aligned}
n^{t-1}= \frac{1}{n}\left( \frac{p}{2\mu}\right)^{t/k}\prod_i \left[ \left( \frac{2\mu}{p}\right)^{1/k}n \right]^{t_i}\leq \frac{1}{n}\left( \frac{p}{2\mu}\right)^{t/k}\prod_i |X_i|^{t_i}\leq \xi \prod_i |X_i|^{t_i}.
0\leq D_{1,m_1}={\bf \Delta_{f,T}^{m-e_1}}(I)-\lim_{q_{m_1}\to \infty}\Phi_{f_1,T_1}^{q_{m_1}}({\bf \Delta_{f,T}^{m-e_1}}(I))\leq{\bf \Delta_{f,T}^{m-e_1}}(I)\leq I.
\xi_j=\sum_{\alpha=1}^n \dot{\xi_j}^{\alpha}\frac{\partial}{\partial \xi_j^{\alpha}},\,\,\,\,\,\,\,\,\,\dot{\xi_j}^{\alpha}=\left(\frac{\partial \xi_j^{\alpha}(z,t)}{\partial t}\right)_{t=0}
\begin{aligned}\frac{\lambda u(y_q, t_q)^{q-1}+(1-\lambda) u(z_q, t_q)^{q-1}}{\varphi(x_q, t_q)^{q-1}}=\frac{\lambda u(y_q, t_q)^{q-1}+(1-\lambda) u(z_q, t_q)^{q-1}}{w_q(x_q, t_q)^{q-1}}&\leq \frac{\lambda u(y_q, t_q)^{q}+(1-\lambda) u(z_q, t_q)^{q}}{c_0w_q(x_q, t_q)^{q-1}}= \frac{w_q(x_q, t_q)^q}{c_0w_q(x_q, t_q)^{q-1}}=\frac{w_q(x_q, t_q)}{c_0},\end{aligned}
\left\langle ( \varphi \otimes \bar{u}) , \phi ( \psi \otimes \bar{v}; z , \bar{z}) \; \phi ( \chi \otimes \bar{w}; \zeta , \bar{\zeta}) \; \left( \Omega \otimes \overline{{{\Omega}}}\right) \right\rangle = ( \bar{z}- \bar{\zeta})^{\bar{\Delta}_{l}- \bar{\Delta}_{\bar{v}}- \bar{\Delta}_{\bar{w}}}\; F_{\psi \chi}( z , \zeta ) .
S= \int d^4x[- \frac{1}{4}F^a_{\mu \nu}F^a_{\mu \nu}+ \frac{1}{2 \alpha}(\partial_{\mu}A_{\mu})^2+ + \bar{c}^a \partial_{\mu}( \delta^{ab}\partial_{\mu}-gt^{abc}A^c_{\mu})c^b]
\int_0^t 1_{l^m_i(s) = l^{m - 1}_{i - 1}(s)}(4 l_i^m(s) + 4 l_{i - 1}^{m - 1}(s)) ds = \int_{-\infty}^\infty 1_{a = 0}\, dL^{a}_t(l^m_i - l^{m - 1}_{i - 1}) = 0.
H_t^{(\lambda)}(x,y):=\left(\left( \Big|\frac{\kappa_t(y)-x}{t^{1/\alpha}}\Big|^\lambda \vee 1 \Big) \wedge t^{-\lambda/\alpha}\right) \right)\frac{1}{t^{d/\alpha}}G^{(\alpha)}\left({\kappa_t(y)-x\over t^{1/\alpha}}\right).
t_1^{\kappa-p/2-1}\left|\int_{t_1}^{s_1}\frac{|W(t)|^p}{(t(1-t))^\kappa}dt-\int_{t_1}^{s_1}\frac{|W(t)|^p}{t^\kappa}dt\right| \leq \left|1-\frac{1}{(1-s_1)^\kappa}\right|\int_{t_1}^{s_1}t_1^{\kappa-p/2-1}\frac{|W(t)|^p}{t^\kappa}dt =o_P(1).
\mbox{var}\left(R_{n,i}V_{n,i}-\rho_n\mu_{n,i}\right) =\rho_nE[V_{n,i}^2]-\rho_n^2\mu_{n,i}^2 &=\rho_n\left(\mbox{var}(V_{n,i})+(1-\rho_n)\mu_{n,i}^2\right).
\begin{aligned}\frac{1}{2}\frac{d}{dt}\|V_{n}\|_{L^{2}_{x}}^{2}+ \frac{1}{5}\gamma_{n}\underline{\rho_{n}}^{\gamma_{n}}\left\|\partial_{x}V_{n}\right\|_{L^{2}_{x}}^{2}\le \|V_{n}\|_{L^{2}_{x}}^{2}\left\{C_{1}+ C_{2}\|V_{n}\|_{L^{2}_{x}}^{2}\right\}, \end{aligned}
-\int_{\Omega}\mathrm{div}\, (\nabla \varphi + \mathrm{rot}' \psi) \tilde{\varphi}\mathrm{d}x-\int_{\Omega}\mathrm{rot}(\nabla \varphi + \mathrm{rot}' \psi) \tilde{\psi}\mathrm{d}x =\int_{\Omega}(\nabla \varphi + \mathrm{rot}' \psi) \cdot (\nabla \tilde{\varphi}+ \mathrm{rot}\tilde{\psi}) \mathrm{d}x
\lim_{h \rightarrow 0}{\bf T}^{(a)}_{4k}=\lim_{h \rightarrow 0}{\bf T}^{(a)}_{4k+1}=1, \lim_{h \rightarrow 0}{\bf T}^{(a)}_{4k+2}=\lim_{h \rightarrow 0}{\bf T}^{(a)}_{4k+3}=0.
[ ( x_{1 2}\partial_{1}x_{1 2}\partial_{2}+ x_{2 3}\partial_{3}x_{2 3}\partial_{2}+ x_{3 1}\partial_{1}\partial_{3}x_{3 1}) ; ( 2 \log x_{1 2}x_{2 3}x_{3 1}+ 2 \log \partial_{1}\partial_{2}\partial_{3}+ \partial_{2}^{- 1}( x_{1 2}^{- 1}- x_{2 3}^{- 1}) ] = 0 ,
\sum\limits_{\alpha =1}^{m}\Phi_{i,\alpha}\left( \left\vert \xi_{i}^{\alpha}\right\vert \right) -a\left( x\right) \leq F_{\alpha}\left(x,\xi^{\alpha}\right) \leq L_{1}\left[ \sum\limits_{\alpha =1}^{m}\Phi_{i,\alpha}\left( \left\vert \xi_{i}^{\alpha}\right\vert \right) +a\left(x\right) \right]
\sum_{k=0}^{\infty}\frac{(\frac12)_k^2}{(1)_k^2}\left(2\sqrt 2 - 2 \right)^k\sum_{k=0}^{\infty}\frac{(\frac12)_k^2}{(1)_k^2}\left(2\sqrt 2 -2 \right)^k \left[ (3\sqrt 2-4) k + \sqrt 2 - \frac32 \right] = \frac{1}{\pi},
\bar \nabla_X K = \nabla_X K \underbrace{- \phi(X) K - K(X) \phi}_{\textrm{\tiny because of \eqref{ast2}}}+ \underbrace{2 \phi(X) K}_{\textrm{\tiny because of \eqref{cov2}}}= \nabla_X K + \phi(X) K - K(X) \phi.
\begin{array}{lll}\frac{1}{2}\Delta_{b}|\nabla_{b}\varphi |^{2}= |\nabla_{b}^{2}\varphi |^{2}+\langle \nabla_{b}\varphi ,\nabla_{b}\Delta_{b}\varphi \rangle +[2Ric-(n-2)Tor]((\nabla_{b}\varphi )_{\mathbb{C}},(\nabla_{b}\varphi)_{\mathbb{C}})+2\langle J\nabla_{b}\varphi ,\nabla_{b}\varphi_{0}\rangle,\end{array}
H = \int\! \frac{d^{3}p}{(2\pi)^3}\,E_{p}\left(a^{\dag}_{\bf p}a_{\bf p}+ a^{\dag}_{\bf p}{\bar a}_{\bf p}+{\bar a}^{\dag}_{\bf p}a_{\bf p}-\,{\bar a}^{\dag}_{\bf p}{\bar a}_{\bf p}\right) ,
(\nabla_{\vec{v}}\alpha)(\vec{X}) = \nabla_{\vec{v}}(\alpha(\vec{X})) - \alpha( \nabla_{\vec{v}}\vec{X}) &= v^i \partial_i (\alpha_j X^j) - \alpha_k ( v^i \partial_i X^j +{\omega^k}_{ij}v^i X^j) &= v^i [ \partial_i \alpha_j -{\omega^k}_{ij}\alpha_k ] X^j.
p_0^{(1)}(t,r,s) = \frac{2 t}{\pi \left(r^2+s^2+t^2\right) \left(1- 4 r^2 s^2 \cdot \left(r^2+s^2+t^2\right)^{-2}\right)}, p_1^{(1)}(t,r,s) = \frac{4}{\pi}\, \frac{t}{(r^2-s^2)^2+t^2(t^2+2r^2+2s^2)}.
\sum_{k=1}^{\infty}\frac{kr^{2(k-1)}}{(k+m)^{2}}\leq{}\sum_{k=0}^{\infty}\frac{r^{2k}}{k+1+m}\leq{}\frac{1}{m+1}+\int_{0}^{\infty}\frac{e^{2x\log r}}{x+1+m}\dx ={}\frac{1}{m+1}+ r^{-2(m+1)}{\rm Ei}(-2(m+1)\log r) \leq{}\frac{1}{m+1}+\log\left(1-\frac{1}{2(m+1)\log r}\right),
\eta_{}\!\left(\!\theta\!,\phi\!\right) \!=\! -\!\min\!{\left\{\!-\!\left(\!\eta_{}\left(\!\theta,\phi\!=\!\frac{\pi}{2}\!\right)\!+\!\eta_{}\left(\theta\!=\!0,\phi\right)\!\right)\!,\eta_{\max}\!\right\}}\!.\!
y_q(u) =\sum_{\mathcal{R}:|\mathcal{R}|=r+1}\sum_{\mathcal{T}:|\mathcal{T}|=t}\sum_{i=1}^\varrho\left[\sum_{p\in \mathcal{T}}h_{qp}(u)\left(\mathbf{v}_{{\mathcal{R}},{\mathcal{T}},p}^i(u)\right)^T\right]\mathbf{x}_{{\mathcal{R}},{\mathcal{T}}}^i,
F_1(t)= \begin{cases}-F(t), |t|\in[0, t_0), -F(t_0), |t|\in[t_0,+\infty), \end{cases}F_2(t)= \begin{cases}0, |t|\in[0, t_0), F(t)-F(t_0), |t|\in[t_0,+\infty). \end{cases}
\begin{cases}\partial_t\varphi=\partial^2_s\varphi+\lambda_1\sin\varphi-\lambda_2\cos\varphi ~ \mbox{in}~ \mathbb R\times (0,t_0) \varphi (s,0)=\varphi_0(s) ~ \mbox{}~ \forall s\in\mathbb R \end{cases}
K' \left( x', x'',N,\tau''-\tau' \right) \equiv \int \tilde{\cal D}x \hspace{0.25em}\Theta\left( - \epsilon(N) \dot{x}_0 \right) \exp \left( i\int_{\tau'}^{\tau''}d\tau \left( \frac{\dot{x}^2}{4N}+ N m^2 \right) \right)
q^{n_{2}^{\prime}-n_{1}^{\prime}}a_{n_{2}}-p^{n_{2}^{\prime}-n_{s}^{\prime}}a_{n_{s}}q^{n_{s}^{\prime}-n_{1}^{\prime}}=p^{n_{2}^{\prime}-n_{1}^{\prime}}a_{n_{1}}\frac{N_{n_{2}}}{N_{n_{1}}},
F(s) =\left(\sup_{r>0}\left(s^qr-G(r^{1/q})\right)\right)^{1/q}=\sup_{r>0}\big((sr)^q-G(r)\big)^{1/q}= \sup_{\substack{r>0 (sr)^q\geq G(r)}}\big((sr)^q-G(r)\big)^{1/q}\geq \sup_{\substack{r>0 (sr)^q\geq G(r)}}\big(sr-G(r)^{1/q}\big) =\sup_{r>0}\big(sr-G(r)^{1/q}\big),
\sum_{i \in I}v_{i}^{2}\Vert \pi_{uW_{i}}Cf\Vert^{2}\leq \Vert u^{-1}\Vert^{2}\sum_{i \in I}v_{i}^{2}\Vert\pi_{W_{i}}u^{*}Cf\Vert^{2}&=\Vert u^{-1}\Vert^2\sum_{i\in I}v_i^2\Vert \pi_{W_i}C u^{*}f\Vert^2 &\leq B \Vert u^{-1}\Vert^{2}\Vert u \Vert^{2}\Vert f\Vert^{2}.
\psi_{1}\left( \rho , \varphi \right) = \psi_{n - 1 , l - 1}^{\left( j \right)}\left( \rho , \varphi \right) \, , \; \psi_{2}\left( \rho , \varphi \right) = - i \psi_{n , l}^{\left( j \right)}\left( \rho , \varphi \right) \, .
L_\mathrm{prin}(x,p)=s^j e_j{}^\alpha(x)\,p_\alpha=\begin{pmatrix}e_4{}^\alpha p_\alpha+e_3{}^\alpha p_\alpha e_1{}^\alpha p_\alpha-ie_2{}^\alpha p_\alpha e_1{}^\alpha p_\alpha+ie_2{}^\alpha p_\alpha e_4{}^\alpha p_\alpha-e_3{}^\alpha p_\alpha\end{pmatrix}.
\frac{1}{N}\left[ \dot{\pi}^{\phi}-(N^{i}\phi^{\phi})_{,i}\right] =-\gamma^{-1/2}e^{\phi}\left[ \pi^{ij}\pi^{kl}\gamma_{ik}\gamma_{jl}+\frac{1}{2}\left(\pi^{\phi}\right)^{2}+\pi\pi^{\phi}\right] ~~~.
\beta_{n+1}= n! \frac{P_n^{(-n)}\left(b_2,\ldots,b_{n+1}\right)}{n!}= P_n^{(-n)}\left(b_2,\ldots,b_{n+1}\right) = \sum_{k=1}^n (-n)^{\underline{k}}B_{n,k}\left(b_2,\ldots,b_{n+1}\right) = \sum_{k=1}^n \binom{-n}{k}k! B_{n,k}\left(b_2,\ldots,b_{n+1}\right).
f\star g = \exp \Bigg[ i\frac{\hbar}{2}\Bigg(\frac{\partial~}{\partial q}\frac{\partial~}{\partial\tilde p}-\frac{\partial~}{\partial p}\frac{\partial~}{\partial\tilde q}\Bigg)\Bigg] f({\bf x}) g({\bf \tilde x}) \vert_{{\bf x}={\bf\tilde x}},
\begin{cases}\Delta_g u_\alpha+m_0^2 u_\alpha=u_\alpha^{2^\star-1}+\omega_\alpha^2(1-qv_\alpha)^2 u_\alpha \Delta_g v_\alpha +(m_1^2+q^2u_\alpha^2) v_\alpha = q u_\alpha^2\end{cases}
M_e = \left[\begin{array}{c|c}M^{\ell}_e M^{\ell,\ell-1}_e \hline begin{align*}-1em](M^{\ell,\ell-1}_e)^T M^{\ell-1}_e\end{array}\right],(M^{\ell,\ell-1}_e)_{ij}= \int_e \phi_i^\ell\phi_j^{\ell-1},
I_2 \leq B\sum_{a \leq t}\frac{\Lambda(a)}{a}\int_{(t-y)/a}^{t/a}\frac{dv}{v^2}M_{|g|}(v) = B\int_{1}^{t}\frac{dv}{v^2}M_{|g|}(v)\left(\sum_{(t-y)/v<a \leq t/v}\frac{\Lambda(a)}{a}\right) \asymp B \frac{y}{t}\int_1^{t}\frac{dv}{v^2}M_{|g|}(v).
\sum_{m,n =0}^\infty f(m,n)x^my^n = \frac{1}{1-2xy}+ \frac{1}{1-3xy}+ \frac{y}{(1-3xy)^2(1-5y)}+ \frac{x}{(1-xy)(1-x)}= \sum_{k=0}^\infty (2^k+3^k)x^ky^k + \sum_{k,l=0}^\infty (k+1)3^k5^lx^ky^{k+l+1}+ \sum_{k,l=0}^\infty x^{k+l+1}y^{l}
\lim_{\lambda_1, \ldots, \lambda_d \rightarrow 1}\frac{h^{(\lambda_1, \ldots, \lambda_d)}(x)}{\Delta(\lambda)}= \frac{h(x)}{\prod_{j=1}^{d-1}j!}, \lim_{\lambda_1, \ldots, \lambda_d \rightarrow 1}\frac{\mathfrak{h}^{(\lambda_1, \ldots, \lambda_d)}(x)}{\Delta(\lambda)}= \frac{\mathfrak{h}(x)}{\prod_{j=1}^{d-1}j!}.
\delta_{[ \mu_{1}. . . \mu_{n}]}^{[ \nu_{1}. . . \nu_{n}]}= \sum_{k = 1}^{n}( - 1 )^{k + 1}\delta_{\mu_{1}}^{\nu_{k}}\delta_{[ \mu_{2}. . \mu_{k}. . \mu_{n}]}^{[ \nu_{2}. . \nu_{1}. . \nu_{n}]}~ ~ ~ .
\alpha^{\prime}=\alpha\frac{1}{\sqrt{1-\lambda^{2}}}-R\frac{\lambda}{\sqrt{1-\lambda^{2}}},\beta^{\prime}=\beta,\gamma^{\prime}=\gamma,R^{\prime}=\alpha\frac{-\lambda}{\sqrt{1-\lambda^{2}}}+R\frac{1}{\sqrt{1-\lambda^{2}}}
\left\|\bar{x}^N \int_{|\bar{\eta}|<\delta\sqrt{t}}e^{i\bar{\eta}\bar{x}-A_j |\bar{\eta}|^2 +O(1) |\bar{\eta}|^3/\sqrt{t}}\left( \vert e_j(0) \rangle \langle e_j(0) \vert + \varepsilon_j(\bar{\eta}/\sqrt{t})\right)d\bar{\eta}\right\|_{L^2_\xi}&\leq C_N + C \sum_{k=1}^{N}\bar{x}^{N-k}e^{-c \delta^2 t}.
|\lambda| \leq \frac{||\phi||_{\rho}||\gamma_a(\gamma_b)_+||_{\rho-\delta/2}}{\int_{\mathbb{T}^{d-1}}\gamma_a(\gamma_b)_+d\psi}||\gamma_a\gamma_bv||_{\rho-\delta}\leq C\delta^{-\tau}||(\lambda+\phi)\gamma_a(\gamma_b)_+||_{\rho}\forall~ \delta>0.
\Lambda_{\chi}(\varphi_{s_{\alpha_k},\chi^{-1}})= \mathrm{vol}(\mathcal{I}\mathfrak{w}\mathcal{I})(1-q^{-1}+(-\varrho,\varpi)_2(\varpi,\varpi)_2^{k-1}q^{-1/2}\chi^{-1/2}(a_{\alpha_k})).
\int_a^{b}T_1(r) r^{-1}\,dr=\frac{1}{\pi}\int_a^{b}2\cos^2(r-\frac{3\pi}{4})r^{-2}\,dr\ge\frac{1}{\pi}\int_a^{b}\sqcap(r)r^{-2}\,dr\ge\frac{1}{2}\sum_{k\in I_{a,b}}(k\pi)^{-2}
\left| [j,q_{1,1}]_{\prec}\cap J_2(w)\right| = \left|\{q_{2,1}, \ldots, q_{2,d_j}\}\right| = \left|\{q_{1,1}, \ldots, q_{1,d_j}\}\right| \leq \left| [j,q_{1,1}]_{\prec}\cap J_1(w)\right|\;,
\int\limits^{\tau}_{-\infty}\frac{d{\bar\tau}}{{\bar\tau}-u}\frac{\partial}{\partial{\bar\tau}}\d{}_{\alpha}({\bar\tau})=\log (u-\tau)\frac{\partial}{\partial\tau}\d{}_{\alpha}(\tau)-\int\limits^{\tau}_{-\infty}d{\bar\tau}\,\log(u-{\bar\tau})\frac{\partial^2}{\partial{\bar\tau}^2}\d{}_{\alpha}({\bar\tau})
\left(\int_X|u_j-u_{j, k_0}|^{1/2}\omega_{\varphi}^n\right)^2 =\left(\int_{\{u_j<\phi-k_0\}}|u_j-u_{j, k_0}|^{1/2}\omega_{\varphi}^n\right)^2 &\leq \dfrac{1}{k_0}\left(\int_{\{u_j<\phi-k_0\}}|u_j|\omega_{\varphi}^n\right)^2 &\leq \dfrac{C_2A^2}{k_0},
\log \big| \prod_{j\neq0}E(z/z_j,1)\big| = \log \big| \prod_{j=1}^{+\infty}E(z^2/z_j^2,0)\big| & = \sum_{j=1}^{+\infty}\log \big|E(z^2/z_j^2,0)\big| & \le \sum_{j=1}^{+\infty}\log \big( 1+ |z^2/z_j^2| \big) \,.
L_2 \cdot (g_2)_*^{-1}C_{\hat D_1,\hat D_2}= (g_2^*L_1 - E_2) \cdot (g_2)_*^{-1}C_{\hat D_1,\hat D_2}= L_1 \cdot C_{\hat D_1,\hat D_2}- E_2 \cdot (g_2)_*^{-1}C_{\hat D_1,\hat D_2}\leq 1.
K\left( x,t\right) =\lim_{\varepsilon \rightarrow 0}K_{\varepsilon}\left(x,t\right) ,\;\;K_{\varepsilon}\left( x,t\right) =\frac{1}{\pi}\int_{0}^{\infty}S\left( \rho ,t\right) \cos \left( \rho x\right) \mathrm{e}^{-\frac{\left( \varepsilon \rho \right)^{2}}{4}}\mathrm{d}\rho ,\;\;x\in\mathbb{R},\;t>0,
\int_{x_i\ge0,\sum x_i\le1}e(\sum_{i=1}^m A_i x_i)dx_1\dots dx_m =\, \frac{1}{(2\pi i)^m}\left\{\sum_{i=1}^{m}\frac{e(A_i)}{A_i\prod_{j=1\atop j\ne i}^{m}(A_i-A_j)}+\frac{1}{\prod_{j=1}^m(-A_j)}\right\},
s_{j+1,2k+1}^{[r+s]}= s_{j+1,2k+1}^{[r+s-1]}- P_{j,k,k}^{[r+s]}s_{j+1,2k}^{[s]}- P_{j,k,k+1}^{[r+s]}s_{j+1,2k+2}^{[s]}.
\mathcal{N}_{(r,s),(r^{\prime},s^{\prime})}^{(r^{\prime \prime},s^{\prime\prime})}\medskip =\sum_{(k,l)\in E(p,p^{\prime})}\frac{S_{(r,s),(k,l)}\S_{(r^{\prime},s^{\prime}),(k,l)}\S_{(k,l),(r^{\prime \prime},s^{\prime\prime})}}{S_{(1,1),(k,l)}}
f(2x+y)+f(2x-y)=\frac{2f(x)f(y)\displaystyle{\sum_{\substack{k=0}}^{l}2^{l-k}\binom{l}{k}f(x)^{\frac{k}{l}}f(y)^{\frac{l-k}{l}}}}{\left(4f(y)^{\frac{2}{l}}-f(x)^{\frac{2}{l}}\right)^l}
{\bf E}_{m,k}={\bf a}_r(\theta_t,f_k){\bf b}_m^T(\theta_{t,i_m},f_k), ~{\bf D}_{m,k}={\bf a}_r(\theta_{i_m},f_k){\bf b}_m^T(\theta_{d,i_m},f_k),{\bf B}_{m,k}={\bf b}_m(\theta_{t,i_m},f_k){\bf a}_t^T(\theta_{t},f_k), ~{\bf G}_{m,k}={\bf b}_m(\theta_{d,i_m},f_k){\bf a}_t^T(\theta_{i_m},f_k),
H(q,q_k^{0},q_k^{1},\dots, q_n^{0},q_n^{1}):= g_{t\alpha}(q + x_0+\dots +x_{k-1}) h_{t\alpha_j}(q_j^{0}-x_j^{0}){h}_{t\alpha_j}(q_j^{1}-x_j^{1})\Big( \prod_{\substack{i=k i \neq j}}^{n}g_{t\alpha_i}(q_i^{0}- x_i^{0})g_{t\alpha_i}(q_i^{1}- x_i^{1}) \Big)
U_{\underline{{m}}}^{+ +}\sigma_{\underline{{\mu}}\underline{{\nu}}}^{\underline{{m}}}= 2 V_{\underline{{\mu}}q}^{+}V_{\underline{{\nu}}q}^{+}, \qquad U_{\underline{{m}}}^{--}\tilde{\sigma}^{\underline{{m}}\underline{{\mu}}\underline{{\nu}}}= 2 V_{q}^{- \underline{{\mu}}}V_{q}^{- \underline{{\nu}}}, \qquad
\int_0^\pi d\theta{(\sin\theta)^{(d-4)}\over (a^2+b^2\sin^2\theta)^{(d-2)/2}}={(d-3)!!\over 2^{(d-2)/2}\left((d-2)/2)\right)!}{1\over a(b^2+a^2)^{(d-3)/2}}
\left\{x_{1},y_{1}\right\}_{\theta}^{\mu_{1}}\left\{x_{2},y_{2}\right\}_{\theta}^{\mu_{2}}\ldots \left\{x_{n},y_{n}\right\}_{\theta}^{\mu_{n}}=:\left\{x_{1},y_{1}^{\mu_{1}};x_{2},y_{2}^{\mu_{2}};\cdots;x_{n},y_{n}^{\mu_{n}}\right\}_{\theta}
\mathcal{C}_p(\tilde{\mu})= \frac{m}{2}\sum_{i=0}^{m-1}(\tilde{w}_{i+1}-\tilde{w}_{i})^2+\sum_{i=1}^{m-1}\tilde{w}_i-(m-1)\tilde{w}_0 = \frac{m}{2}\sum_{i=0}^{m-1}(w_{i+1}-w_{i})^2+\sum_{i=1}^{m-1}w_i-(m-1)w_0+(w_{l+1}+w_{l-1}-2w_{l}+\chi(l=0)) = \mathcal{C}_m(\mu)+w_{l+1}+w_{l-1}-2w_{l}+\chi(l=0).
1 0 - 2 \ell a_{\ell , \ell - 1}+ b_{3}( 2 \ell - 1 ) a_{\ell - 1 , \ell - 2}+ b_{5}( 2 \ell - 3 ) a_{\ell - 2 , \ell - 2}= 0 , \ell \geq 2 ,
\begin{cases}\partial_t A-\varepsilon d_A\Delta A = A\big(F_A-A-s_{A}B\big), 0 < x < 1, \; t \geq 0, begin{align*}5pt] \partial_t B-\varepsilon d_B\Delta B = B\big(F_B-B-s_{B}A\big),\end{cases}
\tilde{S}^{\prime}\ [X^{\mu},b_{\xi}] =-|P_{y}| \int d\xi\e^{-\phi}\sqrt{|g_{\xi\xi}|}+P_{y}\int d\xi \left(C^{(1)}{}_{\xi}-{\textstyle\frac{m}{2}}\ (2\pi\alpha^{\prime})\b_{\xi}\right) \, ,
\alpha'\colon f'=\iota\iota^r g \stackrel\epsilon\to g, \beta'\colon g\stackrel\eta\to \pi^r\pi g=h', \bar\alpha\colon \iota f \stackrel\eta\to \iota f\iota^r\iota \stackrel{\alpha^\flat}\to \iota\iota^r g\iota=f'\iota, \bar\beta\colon \pi h'=\pi \pi^r\pi g\stackrel\epsilon \to \pi g\stackrel{\beta}\to h\pi.
y^{(a)}_{1,2}(\theta+\frac{\pi}{3}i) + y^{(a)}_{1,2}(\theta-\frac{\pi}{3}i) = - y^{(a)}_{1,1}(\theta) +y^{(\bar{a})}_{1,2}(\theta)+\frac{y^{(a)}_{2,3}(\theta) + y^{(\bar{a})}_{2,2}(\theta)}{y^{(\bar{a})}_{1,2}(\theta)}.
\|h\|_{L^p}^p- \|u(t)\|_{L^p}^p = p(p-1) \int_0^t\int_{{\mathbb{R}}^n}u(s,x)^{p-2}A(s,x)\nabla u(s,x)\cdot\nabla u(s,x) \,{\rm d}x\,{\rm d}s.
\begin{cases}u_{tt}-\Delta u=|v|^{p}, x\in\mathbb{R}^n,\t>0, v_{tt}-\Delta v=|u|^{p}, x\in\mathbb{R}^n,\t>0, (u,u_t,v,v_t)(0,x)=(u_0,u_1,v_0,v_1)(x), x\in\mathbb{R}^n.\end{cases}
CT_{\vec{t}}\left\{\prod_{i=0}^{n-1}\left[\frac{(1+t_i)^{x+m}(t_i-1)}{t_i^{x+2n-1}}\right] \prod_{i<j}^{0,n-1}(t_i-t_j)^2(1-t_it_j)\right\}&=n!\cdot \prod_{i=1}^n\prod_{j=1}^m\frac{(x+i-j)(x+2i+j-2)}{(x+2i-j)(i+j-1)}.
\partial^{\alpha-\alpha_{1}}\nabla_{v}F=\partial^{\alpha-\alpha_{1}}\nabla_{v}M+\partial^{\alpha-\alpha_{1}}\nabla_{v}\overline{G}+\sqrt{\mu}\partial^{\alpha-\alpha_{1}}\nabla_{v}f-\frac{v}{2}\sqrt{\mu}\partial^{\alpha-\alpha_{1}}f.
\frac{d}{ds}h(\mu_{st}) =-\frac{d}{ds}\int\mu_{st}\ln\mu_{st}&=-\int\frac{d\mu_{st}}{ds}\ln\mu_{st}&=\int\nabla(\mathbb{E}[U|Z_{st}]\mu_{st})\ln\mu_{st}&=-\int \mathbb{E}[U|Z_{st}]\nabla(\ln\mu_{st})\mu_{st}&=-\mathbb{E}[U\nabla\ln\mu_{st}(Z_{st})]
{(-1)^{\ell(\check\sigma)}x_1^{o(\check\sigma)}x_2^{e(\check\sigma)}y^{o(\check\sigma)}z^{e(\check\sigma)}}= x_1 z^{\lfloor \frac{n-1}{2}\rfloor}{(-1)^{\ell(\sigma)}x_1^{e(\sigma)}x_2^{o(\sigma)}y^{o(\sigma)}z^{e(\sigma)}}
\left ( \begin{array}{c}(\mathcal{F}\tilde{\varphi}_{0})(\xi) \vdots (\mathcal{F}\tilde{\varphi}_{p})(\xi) \end{array}\right ) = \tilde{\mathcal{A}}( e^{- i \frac{\xi}{2}}) \left ( \begin{array}{c}(\mathcal{F}\tilde{\varphi}_{0})(\frac{\xi}{2}) \vdots (\mathcal{F}\tilde{\varphi}_{p})(\frac{\xi}{2}) \end{array}\right ) .
U^\sigma_{\alpha_2 \alpha_1}(t) = \Sigma_{\alpha_2 \alpha_1}^\sigma(t) + \sum_{l=1}^\infty \int_{0}^{t}\int_0^{t_1}...\int_0^{t_{l-1}}\Pi^{l, \sigma}_{\alpha_2\alpha_1}(t, t_1, t_2, ... , t_l)dt_l...dt_1,
\int_\Omega (g(t,x)-g^{in}(x)) \varphi(x)\ \mathrm{d}x + \int_0^t \int_\Omega g(s,x) \nabla[cf + dg ](s,x) \cdot \nabla\varphi(x)\, \mathrm{d}x\mathrm{d}s =0\,.
\mid{P}_{0}\rangle \:=\:f(p)\;\exp \left[ 2\,\,{({\cal W}^{-1})}^{\mu\sigma}_{\nu\rho}{\mathbf a}^{\dagger\nu}{\mathbf b}^{\dagger}_{\mu}{\mathbf a}^{\dagger\rho}{\mathbf b}^{\dagger}_{\sigma}\right]\, \mid 0 \rangle.