harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}{\rm Gr}H^{*}(C^{*})=\bigoplus_{p,q}{\rm Gr}^{p}H^{q}(C^{*}),\quad^{p}H^{q}(C^{*}):=\frac{F^{p}H^{q}(C^{*})}{F^{p+1}H^{q}(C^{*})}.\end{aligned}
\begin{bmatrix}t u r s\end{bmatrix}\times \begin{bmatrix}a b c d \end{bmatrix}\mapsto \begin{bmatrix}at -\Delta au bt bu -\frac{ar}{\Delta}as -\frac{br}{\Delta}-\frac{bs}{\Delta}ct -\Delta cu dt du cr -\Delta cs dr ds\end{bmatrix}
\Gamma = \Gamma_{0}+ \Gamma_{1}= \beta \sum_{N = 1}^{K}\frac{N_{N}v_{N}^{2}}{2 R}- \frac{1 5 l_{P}^{9}}{1 6 R^{3}}\sum_{M < N , \N = 1}^{K}\int_{0}^{\beta}d t \frac{N_{M}N_{M}v_{M N}^{4}}{( v_{M N}^{2}{t}^{2}+ r_{M N}^{2})^{\frac{7}{2}}}
S^+_n:= \langle z,t_1,\dots,t_{n-1}\mid z^2=1, t_jz=zt_j, t_j^2=1, &t_jt_k=zt_kt_j\text{if}|j-k|>1,(t_jt_{j+1})^3=z(0\leq j\leq n-1)\rangle, S^-_n:= \langle z,t_1,\dots,t_{n-1}\mid z^2=1, t_jz=zt_j, t_j^2=z, &t_jt_k=zt_kt_j\text{if}|j-k|>1,(t_jt_{j+1})^3=1(0\leq j\leq n-1)\rangle.
\left.{\rm Tr}V_{1\nu}^{~\mu}(\nabla) \right|^{{\rm div}}=\frac{2 i}{4-d}\, \frac{\kappa^2}{24 \pi^2}\int d^4x \sqrt{-g}\left[ -5 R_{\mu\nu}{\cal E}^{\mu\nu}+\frac{5}{2}R g_{\mu\nu}{\cal E}^{\mu\nu}\right] .
-\frac{\hslash^{2}}{2m}k_{\pm}^{2}+\frac{\beta \hslash^{4}}{3m}k_{\pm}^{4}=\pm i\lambda \Leftrightarrow \frac{2\beta \hslash^{2}}{3}k_{\pm}^{4}- k_{\pm}^{2}\mp \frac{2mi\lambda}{\hslash^{2}}=0 ,
\psi_{nlm}(r,\theta,\varphi)=\sqrt{{(\frac{2}{na_{0}^{*}})}^{3}\frac{(n-l-1)!}{2n(n+l)!}}e^{-\rho/2}\rho^{l}L_{n-l-1}^{2l+1}(\rho)Y_{l}^{m}(\theta,\varphi)
f(\delta_1) =-2q^{2d+1}+\left(\frac{3}{2}\alpha^2+\frac{21}{2}\alpha+12\right)q^{2d}+(\alpha^2+6\alpha+9)q^{2d-1}\hphantom{\le{}}\mathrel{+}(4\alpha^2+24\alpha+36)q^{2d-2}+q^{d+4}+4q^{d+3}-\left(\frac{\alpha}{2}+4\right)q^{d+2}&\hphantom{\le{}}\mathrel{-}(2\alpha+6)q^{d+1}-(8\alpha+24)q^d+q^3+4q^2
P_{\alpha\beta}^{b}(x_{1},x_{2}, x_{3})= \left (\begin{array}{ccc}0 \alpha-\dfrac{\beta b_{1}b_{2}}{b_{3}}x_{3}- \beta b_{1}x_{2}begin{align*}-0.2cm]-\alpha+\dfrac{\beta b_{1}b_{2}}{b_{3}}x_{3}0 -\dfrac{\alpha b_{3}}{b_{1}}x_{1}\beta b_{1}x_{2}\dfrac{\alpha b_{3}}{b_{1}}x_{1}0 begin{align*}-0.2cm]\end{array}\right ).
\sum_{t<Y}F_Y\Bigl(\frac{t}{Y}\Bigr) \ll \frac{1}{(q-s)^k}\prod_{i=0}^{k-1}\sum_{t_i<q}\min\Bigl(q-s,\frac{q}{t_i}+\frac{q}{q-t_i}\Bigr) &=O\Bigl(\frac{q\log{q}+q-s}{q-s}\Bigr)^k.
\Pi_{s,f}N_{s,f}\varphi = \Pi_{s,f}\left[ \frac{1}{\kappa(s) r_{s,f}}L_{s,f}(\varphi r_{s,f}) \right]= \frac{1}{\kappa(s)}\nu_{s,f}L_{s,f}(\varphi r_{s,f}) = 0,
I(x) = \int_{0}^{\infty}dA(t)\int_{0}^{x+t}dF_n(u) \mathbb{P}(u \geq D_n)\mathbb{P}(t-u \leq \sigma_n \leq x+t-u) = \int_{0}^{\infty}dA(t)\int_{0}^{x+t}\bar{G}(u) dF_n(u)\int_{t-u}^{x+t-u}dB(s).
\left[(a_1+a_2)^p + (b_1+b_2)^q\right]^{1/p}\leq \left[(a_1+a_2)^p + \bigl(b_1^{q/p}+b_2^{q/p}\bigr)^p \right]^{1/p}\leq \left[a_1^p+\bigl(b_1^{q/p}\bigr)^p\right]^{1/p}+ \left[a_2^p+\bigl(b_2^{q/p}\bigr)^p\right]^{1/p}= \bigl(a_1^p+b_1^q\bigr)^{1/p}+ \bigl(a_2^p+b_2^q\bigr)^{1/p}.
\int_0^R{\rm d}r\,\left[u_{1\epsilon\kappa}\left(r\right)u_{1\epsilon'\kappa}\left(r\right)+u_{2\epsilon\kappa}\left(r\right)u_{2\epsilon'\kappa}\left(r\right)\right]=\frac{u_{1\epsilon\kappa}\left(R\right)u_{2\epsilon'\kappa}\left(R\right)-u_{1\epsilon'\kappa}\left(R\right)u_{2\epsilon\kappa}\left(R\right)}{\epsilon-\epsilon'}\,.
F_{i}:=\frac{1}{\left|Y_{1}\right|}\int_{Y_{1}}\rho_{i}\left(y\right)dy,\quad\mathbb{F}^{i}:=\left(\frac{1}{\left|Y_{1}\right|}\int_{Y_{1}}\rho_{i}\left(y\right)\frac{\partial\bar{u}_{i}^{j}}{\partial y_{k}}dy\right)_{jk},
\partial_tf^\pm+\hat{v}_1\partial_xf^\pm\pm(E_1+\hat{v}_2B)\partial_{v_1}f^\pm\pm(E_2-\hat{v}_1B)\partial_{v_2}f^\pm=0 &\partial_tE_1=-j_1 &\partial_tE_2+\partial_xB=-j_2 &\partial_tB=-\partial_xE_2 &\partial_xE_1=\rho.
\Big \vert D_{x_j}(\frac{1}{t^{d/2}}e^{-|x|^2/4t})\Big \vert = \Big \vert \frac{x_j}{2t}\frac{1}{t^{d/2}}e^{-|x|^2/4t}\Big \vert \leq \frac{|x|}{2t^{(d+2)/2}}e^{-|x|^2/4t}.
g_r(n,k) \le \sum_{(m,s_1,\ldots,s_m)\in S_{n,r}}\prod_{i=1}^m f_r(n,k,s_i) &\le2^{c_1n^{r-1}}\cdot 2^{(3/2)(c_2+2r)c_1n^{r-1}(\log n)^{(r-3)/(r-2)}}& \le 2^{2(c_2+2r)c_1n^{r-1}(\log n)^{(r-3)/(r-2)}},
\rho_{\tilde{\underline{q}}_n}(x_{ij}^g) =1+F_1(x_{ij}^g\cdot \tilde{\underline{q}}_n)-F_1(\tilde{\underline{q}}_n)+[deg \geq 2] &=1+\int_{\tilde{x}_{ij}^g}r^*(\omega) +[deg \geq 2] &=1+ \int_{x_{ij}^g}\omega +[deg \geq 2] &=1-2i\pi X_{ij}(g) +[deg \geq 2]
\begin{array}{r c r c l}{x =}&{- 7 ( e^{t}- 1 ) + 1 6 t \mathrm{}}&{}&{t}&{< 0 \ ,}\\{}&{9 ( e^{t}- 1 )}&{0 <}&{t}&{< T \ ,}\\{}&{3 + 1 2 ( t - T )}&{T <}&{t}&{\ ,}\\ \end{array}
\aligned M_k :=|g(X_{k},k\Delta t)\Delta B_k|^2-|g(X_{k},k\Delta t)|^2\Delta t+2\langle F(X_{k},k\Delta t),g(X_{k},k\Delta t)\Delta B_k\rangle &\quad+2\langle f(X_{k},k\Delta t)\Delta t,g(X_{k},k\Delta t)\Delta B_k\rangle.\endaligned
[\Lambda \frac{\partial\phantom{a}}{\partial \Lambda}+ \mu \frac{\partial\phantom{a}}{\partial \mu}+\beta_1 \frac{\partial\phantom{a}}{\partial g_1}+\beta_2 \frac{\partial\phantom{a}}{\partial g_2}-\gamma N_F ]\Gamma^{(N)}(p_1, \ldots p_N)\approx 0 ,
\vert (Df(z)-Df(w))a \vert = \vert (DF(z)-DF(w))a \vert = \vert \pi_{P_{F(z)}}(DF(z)a)- \pi_{P_{F(w)}}(DF(w)a) \vert \leq \vert \pi_{P_{F(z)}}(DF(z)a) - \pi_{P_{F(w)}}(DF(z)a) \vert + \vert \pi_{P_{F(w)}}(DF(z)a-DF(w)a)\vert \leq \sphericalangle\left(P_{F(z)},P_{F(w)}\right) \vert DF(z)a\vert + \vert \pi_{P_{F(w)}}(DF(z)a-DF(w)a)\vert.
\mu(2-\mu ||A^*||^2)||\bar{w}_n-A\bar{z}_n||^2 \le||\bar{z}_n-x^*||^2-||t_n-x^*||^2 =(||\bar{z}_n-x^*||-||t_n-x^*||)(||\bar{z}_n-x^*||+||t_n-x^*||) &\le ||\bar{z}_n-t_n||(||\bar{z}_n-x^*||+||t_n-x^*||)
\frac{\partial M_{r}}{\partial c}=\frac{(c)_{r}(c+1-\gamma)_{r}}{((c+1-\alpha)_{r})^{2}}\sum_{k=0}^{r-1}(\frac{1}{c+k}+\frac{1}{c+1-\gamma+k}-\frac{2}{c+1-\alpha+k})
\Omega_{+ 2}^{[ - 2 ] ( i )}= \tilde{\Omega}_{+ 2}^{[ - 1 0 ] ( i )}( \theta^{+})^{8}, \ \ \Omega_{- 2}^{[ + 2 ] ( i )}= \tilde{\Omega}_{- 2}^{[ + 1 0 ] ( i )}( \theta^{-})^{8},
\bar\delta^2_0(\frac{\xi_i}{d})=\frac{\xi_{i,2}}{d}, \bar\delta^2_{1}(\frac{\xi_i}{d}) =\frac{\xi_{i,1}}{d}+\frac{\xi_{i,2}}{d}, \bar\delta^2_{2}(\frac{\xi_i}{d}) =\frac{\xi_{i,1}}{d}.
R_s^2 = qH_s(-1) + R_sE_s \underbrace{R_sR_{s'}R_sR_{s'}\dots}_{m_{ss'}}= \underbrace{R_{s'}R_sR_{s'}R_s \dots}_{m_{ss'}}, s, s' \in S R_tR_s = R_sR_{t'}, t' = n_stn_s^{-1}, t \in T(\mathbb{F}_{q^k}) R_{t_1}R_{t_2}= R_{t_1t_2}, t_1, t_2\in T(\mathbb{F}_{q^k})
\left(\begin{matrix}x=t_1 x=t_2 x=\infty \ \left( \frac12 \right) \begin{matrix}0 \theta^{t_1}\end{matrix}\begin{matrix}0 \theta^{t_2}\end{matrix}\overbrace{\begin{matrix}\sqrt{-1}0 0 \theta^\infty_1/2 -\sqrt{-1}0 0 \theta^\infty_1/2\end{matrix}}\end{matrix}\right),
\left[ \begin{array}{ccc}c_{1}^{2}x_{1}2c_{2}x_{1}\end{array}\right]\cdot{\left[ \begin{array}{ccc}1 0 -\frac{n/2}{n-1}1\end{array}\right]}^{-1}=\left[ \begin{array}{ccc}c_{1}^{2}x_{1}+\frac{n}{n-1}c_{2}x_{1}2c_{2}x_{1}\end{array}\right].
\leq \Bigg [ \int_{\Lambda}\prod_{a = 1}^{N}\prod_{i = 1}^{n_{a}}d^{2}x_{i}^{( a )}d^{2}y_{i}^{( a )}\prod_{b = 1}^{N}\Big | A ( b , b ) \Big |^{\frac{\pi}{\pi + g N}p}\Bigg ]^{\frac{1}{p}}
j_{p}= d\zeta^{\wedge p}_{\tilde{m}}\int_{\partial{\cal M}^{p+1}}d{\hat \zeta}^{\tilde{m}}(\tau )\delta^{p+1}\left( \zeta -{\hat \zeta}(\tau )\right)~=(-)^p d\zeta^{\wedge p}_{\tilde{n}}j^{\tilde{n}}. \qquad
P\left(\left\|\mathbb{I}_n\right\|_{\ell_\infty}>L\sqrt{\frac{\log (d+r)}{n}}\right) \leq \sum_{i,j=1}^{d+r}P\left(\sqrt{n}\left|\mathbb{I}_n^{ij}\right|>L\sqrt{\log (d+r)}\right)\leq 2(d+r)^{2-CL^2}.
\begin{cases}-\eta''(r) - \frac{N-1}{r}\eta'(r)- N(N+2) \frac{1}{\left(1+ r^2\right)^2}\eta= \frac{\nu}{r^2}\eta \,\,(0,\infty) \eta \in D^{1,2}((0,+\infty),r^{N-1}dr)\end{cases}
{\vec{v}}^{\prime}= \gamma^{- 1}\vec{v}+ \gamma^{- 2}\vec{V}\left[ \displaystyle \frac{\vec{V}\vec{v}}{c^{2}\left( \gamma + \sqrt{\gamma^{2}+ \displaystyle \frac{\vec{V}^{2}}{c^{2}}}\right)}- \frac{\vec{\sigma}\vec{v}}{c^{2}}\gamma_{0}^{- 2}- 1 \right] ,
\int_{L}\bigl|\widehat{S}_{\epsilon}\left(\omega\right)\bigr|\sup_{y\in\mathbb{R}^{nd}}\left|\theta_{T,y}\left(\omega\right)\right|d\omega\ll S_{\epsilon}^{*}\left(L\right)\Gamma_{L_{1},\dots,L_{n-1},\beta}\left(T\right)T^{nd-\left(n-1\right)\beta d}\prod_{i=1}^{n-1}\int_{L_{i}-L_{i-1}}\alpha_{d}\left(\Lambda_{T,\xi_{i}}\right)^{\beta}d\xi_{i},
\frac{1}{2^{2n-1}}\int_{0}^{1}x^{4n}(1-x)^{4n}dx<\frac{1}{2^{2n-2}}\int_{0}^{1}\frac{x^{4n}(1-x)^{4n}}{1+x^{2}}dx<\frac{1}{2^{2n-2}}\int_{0}^{1}x^{4n}(1-x)^{4n}dx,
E_{M}\,=\,\frac{l}{4}\sum_{n=1}^{\infty}\left[\left(1\,+\,\frac{r^{2}k_{n1}^2}{l^2}\right)\omega_{n1}^{2}A_{n1}^2\,+\,\left(1\,+\,\frac{r^{2}k_{n2}^2}{l^2}\right)\omega_{n2}^{2}A_{n2}^2\right]
\mathbf{\Lambda}_1(\mathbf{D}_1-\mathbf{\Theta}_{1}(\gamma))=\mathbf{0}&\Rightarrow\mathbf{\Xi}^T_1\mathbf{\Lambda}_1(\mathbf{D}_1-\mathbf{\Theta}_{1}(\gamma))\mathbf{\Xi}_1=\mathbf{0}&\Rightarrow\mathbf{\Pi}_1\mathbf{\Xi}^T_1(\mathbf{D}_1-\mathbf{\Theta}_{1}(\gamma))\mathbf{\Xi}_1=\mathbf{0},
\varphi(x) =\varphi(\lambda x) = \varphi( y + (\lambda x -y )) = \varphi( y + \lambda^{-2}(\lambda x -y))= \varphi(\lambda^{-1}x + (y -\lambda^{-2}y)) &= \varphi(x + (\lambda - \lambda^{-1})y)= \varphi(x + \mu y)
\begin{aligned}\Vert \partial_y (u,h)\Vert_{H^m_l}^2 \leq \sum_{\alpha\in\{\alpha=(\beta,k):|\alpha|\leq m, \beta\leq m-1\}}\Vert \partial_y D^\alpha (u,h)\Vert_{L^2_l}^2 &+2\Vert \partial_y (u_m,h_m)\Vert_{L^2_l}^2+50\vartheta_0^{-4}\Vert h_m\Vert_{L^2_l}^2.\end{aligned}
A ( x , p ) \star B ( x , p ) = e^{\kappa ( \partial_{x}\partial_{\widetilde{p}}{}-{}\partial_{p}\partial_{\widetilde{x}})}A ( x , p ) B ({\widetilde{x}},{\widetilde{p}}) \Big |_{{\widetilde{x}= x},{\widetilde{p}= p}}.
V = \left( \begin{array}{c | c}{{\begin{array}{c c}{{0}}&{{B}}\\{{- B}}&{{0}}\\ \end{array}}}&{{M}}\\ \hline{{- M^{T}}}&{{\begin{array}{c c}{{0}}&{{\tilde{B}}}\\{{- \tilde{B}}}&{{0}}\\ \end{array}}}\\ \end{array}\right)
| \frac{1}{\dot{\varrho}_n}- \frac{1}{\dddot{\varrho}_n}| = \frac{| \dot{\varrho}_n - \dddot{\varrho}_n |}{\dot{\varrho}_n \dddot{\varrho}_n}= \frac{|\dot{\varrho}_n^2- \dddot{\varrho}_n^2|}{\dot{\varrho}_n \dddot{\varrho}_n (\dot{\varrho}_n+ \dddot{\varrho}_n)}\ll n^{s/2}\log_2^{-5s}n.
A = \frac{3^n}{3^n+1}\cdot\frac{5^n}{5^n-1}\cdot\frac{7^n}{7^n+1}\cdot\frac{11^n}{11^n+1}\cdot\frac{13^n}{13^n-1}\cdot\frac{17^n}{17^n-1}\cdot\&c,
T_{22}^{ab\;k}\;=\frac{\Theta}{4\ell_{P}^{7}}<s_{K}^{a}s_{K}^{b}\;\epsilon^{ijk}\epsilon^{IJK}\langle W,\xi |\hat{w}_{iI\Delta}(v)\hat{w}_{jJ\Delta}(v)|W,\xi \rangle >
=q^{-B}\sqrt[3]{\frac{k_{p^2r}^2(1-k_{p^2r}^2)}{2}}\frac{Q_{\{a,p\}}'(k_{p^2r})}{(Q_{\{a,p\}}(k_r))^4}\theta_4\left(\frac{p}{2},\frac{p}{2}-a;q\right)^4.
S_{e f f}= \int \! d^{4}\! x [ \pi_{0}\dot{A}^{0}+ \pi_{i}\dot{A}^{i}+ \pi_{\rho}\dot{\rho}+ B_{2}\dot{N}^{2}+ \overline{{\cal P}}_{i}\dot{\cal C}^{i}+ \overline{{\cal C}}_{2}\dot{\cal P}^{2}] - H_{t o t a l},
\mathcal{T}_t^{(1)}={x}_1^\prime \, \sum_{i_1=0}^{\lfloor (t-1)/2 \rfloor}\,c_{i_1}^{(t)}\,T^{t-2i_1-1}\,(-D)^{i_1},\mathcal{T}_t^{(0)}={x}_0^\prime \sum_{i_0=1}^{\lfloor t/2 \rfloor}\,c_{i_{0}-1}^{(t-1)}\,T^{t-2i_0}\,(-D)^{i_0-1}.
\lim\limits_{n\to \infty}n^{2}\left(\sum\limits_{i=0}^{k_n}\P(A_{r_n, r_{1,n}, k_n, i})\right) =\frac{2pql^2}{\pi(x(t-x))^{3/2}}e^{-\frac{l^2}{2}\left(\frac{p^2}{x}+\frac{q^2}{t-x}\right)}=4h(x, pl)h(t-x, lq).
\mathcal{L}(\Lambda) =\mathcal{L}(\cup_{j=1}^{n}f_{j}(\Lambda)) &= \sum_{j=1}^{n}\mathcal{L}(f_{j}(\Lambda))-\sum_{i=2}^{n}\mathcal{L}(\cup_{j=1}^{i-1}f_{j}(\Lambda)\cap f_{i}(\Lambda)) &= \mathcal{L}(\Lambda)\sum_{j=1}^{n}\lambda_{j}^{d}-\sum_{i=2}^{n}\mathcal{L}(\cup_{j=1}^{i-1}f_{j}(\Lambda)\cap f_{i}(\Lambda)).
\mathfrak{F}=\sum_{q\in\mathcal{Q}}\sum_{n=1}^\infty \frac{\lambda_\pi(n,1)}{\sqrt{n}}V\left(\frac{nX}{q^{\frac{3}{2}}}\right)\;\sideset{}{^\star}\sum_{\chi\bmod{q}}\chi(n)(1+\chi(-1))\bar\chi(\ell)
{\cal G}( i )^{- 1}{\cal Z}( z ){\cal K}( z ; i ) = \left( \begin{array}{c c}{{1}}&{{0}}\\{{i x_{-}^{\prime}{\cdot \tilde{\sigma}}^{t}}}&{{- 2 \theta^{\prime a}{}^{t}}}\\{{2 \bar{\theta}_{b}^{\prime t}}}&{{\delta_{b}^{~ a}}}\\ \end{array}\right) =
^{+}S_{2}= \frac{i}{4}{^+ \tau}\int_{M}e F^{\mu \nu i}F^{\alpha \beta j}g_{i j}\, \,{^+{\cal R}}_{\mu \nu}^{a b}\, \,{^+{\cal R}}_{\alpha \beta}^{c d}\, \epsilon_{a b c d}\, .
\frac{\exp(t\arg(L-it-c))\exp(-t\arg(a-L+c+it))}{\exp(-t\arg(b-L+c+it)}&=\exp\big(-t \arctan \frac{t}{L-c}- t(\pi + \arctan \frac{t}{a-L+c}) + t\arctan \frac{t}{b-L+c}\big)
\mathbb E\left[e^{-s\overline{X}_{e(q)}}\right]=\frac{\Phi(q)}{\Phi(q)+s}\ \and \ \ \mathbb E\left[e^{s\underline{X}_{e(q)}}\right]=\frac{q}{\Phi(q)}\frac{\Phi(q)-s}{q-\psi(s)}, \ \s,q > 0.
\pi\left(b(j+1,k)\right) =\pi\left(\sum_{i=1}^{\infty}b(j,i)m(6i+6,k+i+1)\right) \geq\min_{i\geq1}\bigg(\pi\big(b(j,i)\big)+\pi\big(m(6i+6,k+i+1)\big)\bigg) \geq j+\left\lfloor\dfrac{5i-5}{2}\right\rfloor+\left\lfloor\dfrac{5k-i-2}{2}\right\rfloor \geq j+1+\left\lfloor\dfrac{5k-5}{2}\right\rfloor.
f_{D,\lambda}=\arg\min{}_{f\in\mathcal{H}}\lambda\|f\|_{\mathcal{H}}^{2}+\frac{1}{n}{\displaystyle \sum_{i=1}^{n}}\left[\frac{(1-\Delta_{i})2(C_{i}-f(Z_{i}))}{g(C_{i}|Z_{i})}+(f(Z_{i}))^{2}\right].
\psi_{1}( x ) = \lim_{\Lambda \rightarrow \infty}\frac{k \Lambda}{2 \pi}\int d \hat{\eta}e^{- i \frac{\theta ( \eta )}{2}}\psi_{1}^{\eta}( x ) ; \ \, p s i_{2}( x ) = \lim_{\Lambda \rightarrow \infty}\frac{k \Lambda}{2 \pi}\int d \hat{\eta}e^{i \frac{\theta ( \eta )}{2}}\psi_{2}^{\eta}( x )
(r_b|r_a)_{E,l}=-i\frac{\hbar}{2Mc}\frac{M}{\hbar \kappa}\frac{\Gamma(-\nu +\tilde l+1)}{(2\tilde l+1)!}W_{\nu ,\tilde l+1/2}\left (2\kappa r_b\right ) M_{\nu,\tilde l+1/2}\left (2\kappa r_a\right )
\mathcal{A}_{}\left(P_{1}\right)=\left\{\theta^{1}\otimes\left(a^{1}(r)\;e_{1}+a^{2}(r)\;e_{2}+a^{3}(r)\;e_{3}\right)\;|\;a^{1},a^{2},a^{3}\;\in C^{\infty}\left(0,\infty\right)\right\}.
\langle l|\psi(t+T)\rangle=exp(-i\frac{l^{2}\hbar T}{2I})\sum_{n,m=-\infty}^{\infty}(-i)^{n}J_{n}(\frac{KT}{\hbar})\frac{sin([m-l-n])\pi}{(m-l-n)\pi}\langle m|\psi(t)\rangle
\sum_{n=1}^{\infty}\frac{p^{2}+\lambda_{n}^{2}}{p\left(p+\frac{1}{\pi}\right)+\lambda_{n}^{2}}\,\left\{\frac{1}{\left(\lambda_{n}^{2}+x^{2}\right)^{s}}-\sum_{m=0}^{N-1}\left(\begin{array}{c}-s m\end{array}\right)\,\frac{x^{2m}}{\lambda_{n}^{2s+2m}}\right\}+\sum_{m=0}^{N-1}\left(\begin{array}{c}-s m\end{array}\right)x^{2m}\zeta_{p}\left(2s+2m\right).
\mathcal{C}_{n}^{m}=\sum_{i=1}^{m-n}(-1)^{i}\sum_{a_{1},\cdots,a_{i}=1}^{\infty}\delta_{a_{1}+\cdots+a_{i},m-n}\prod_{j=1}^{i}N_{\mathrm{d}a_{j}}\binom{m}{m-a_{1}}\binom{m-a_{1}}{m-a_{1}-a_{2}}\cdots\binom{m-a_{1}-\cdots-a_{i-1}}{m-a_{1}-\cdots-a_{i-1}-a_{i}},
\Gamma_p^{-1}\left( D \varphi_t^v(p) \cdot X \right) =- \left( A_p(X) + \alpha_p(X) \right) + t \left[ \nabla_X v + \langle v(p), A_p(X) \rangle \eta(p) - \overline{\nabla}_X \widetilde{v(p)}\right].
f_{n , m}( z ) = \exp \int_{P_{0}}^{z}\omega_{n , m}, \quad \omega_{n , m}( z ) = \sum_{k = 1}^{h}\left({\frac{\det \Omega^{( 2 ; k )}}{\det \Omega^{( 2 )}}}+ i \pi n_{k}\right) \omega_{k}( z ) .
G_{1, \varphi}(x) = \left(I + \varepsilon N_{1,\varphi}^{-}(x) (\Lambda^{+}(x))^{-1}\right) \Lambda(x) \left(I + \varepsilon (\Lambda^{-}(x))^{-1}N_{1,\varphi}^{+}(x)\right),
\frac{qX_{i,i+1}X_{i+1,i+2}-q^{-1}X_{i+1,i+2}X_{i,i+1}}{q-q^{-1}}=1(i\in \mathbb{Z}_4), X_{i,i+1}^3X_{i+2,i+3}-[3]_qX_{i,i+1}^2X_{i+2,i+3}X_{i,i+1}+[3]_qX_{i,i+1}X_{i+2,i+3}X_{i,i+1}^2 -X_{i+2,i+3}X_{i,i+1}^3=0(i\in \mathbb{Z}_4).
{\cal Z}= \frac{1}{2 \pi i}\int_{{\cal C}_{0}}\frac{d z}{z + e^{- \frac{\pi}{g_{n c}^{2}\theta}}}\exp \left[ z \Phi ( - z ; \frac{3}{2}; 1 ) \frac{A_{n c}}{\sqrt{2 \pi g_{n c}^{2}\theta^{3}}}\, \, \right] ,
\mathcal{Q}_2(x', F_{2\times 2}) = \min\left\{\mathcal{Q}_3\big(\bar A(x')^{-1}\tilde F \bar A(x')^{-1}\big); ~ \tilde F\in\mathbb{R}^{3\times 3}\mbox{with}\tilde F_{2\times 2}= F_{2\times 2}\right\}, & \mathcal{Q}_3(F) = D^2 W(Id_3)(F,F).
\Gamma^{(0)}_{ij,\bar{k}}=(\Gamma^{(0)}_{\bar{i}\bar{j},k})^{*}=\frac{1}{2\pi i}\oint_{|z|=1}\big(\partial_i\partial_j \log{h(z;\boldsymbol{\xi})}\big)\big(\partial_k \log{h(z;\boldsymbol{\xi})}\big)^{*}\frac{dz}{z}
\hat{W}^{[{\sf u}]}_{kj}=\chi_{kj}(\mathbf{y}^{[{\sf bs}]}_k[1],\cdots,\mathbf{y}_k^{[{\sf bs}]}[n],W_{k1}^{[{\sf d}]},\cdots,W_{kN}^{[{\sf d}]}) j\in[1:N].
0\leq (n/4)^k(2\ln n)^{\frac{k}{2}}\mathbb{P}(\xi^{(n)}(I_{n,k})=\xi^{(n)}(J_{n,k,j})=0) = (n/4)(2\ln n)^{\frac{1}{2}}D_n((z_{j+1}-z_j)/2)\cdot\left(n(2\ln n)^{\frac{1}{2}}D_n(F_n(x_0)/2)/4\right)^{n-1},
a_{p,q,r}:=a_{(1\,p)}a_{(1\,q)}a_{(1\,r)}, a_{p,r,q}:=a_{(1\,p)}a_{(1\,r)}a_{(1\,q)}, a_{r,p,q}:=a_{(1\,r)}a_{(1\,p)}a_{(1\,q)}.
\begin{cases}Y(t)=\psi(t) + \int_t^T g(t,s,Y(s),Z(t,s),Z(s,t),Y(s+\delta_{s}),Z(t,s+\zeta_{s}),Z(s+\zeta_{s},t)) ds \ \ \ \ \ \ \ \ \ \ - \int_t^T Z(t,s) dW(s), \ \t\in [0,T]; Y(t)=\psi(t), \ \t\in [T,T+K]; Z(t,s)=\eta(t,s), \ \ (t,s)\in [0,T+K]^{2}\setminus [0,T]^{2}. \end{cases}
\bigl| \det(\log |\alpha_j|_v)\bigr| \le \prod_{j = 1}^r \max\Big\{\sum_{v \not= \widehat{v}}\log^+ |\alpha_j|_v, \sum_{v \not= \widehat{v}}\log^- |\alpha_j|_v\Big\}= \prod_{j = 1}^r h(\alpha_j).
(d\tilde{\omega})_{i_1\dots i_{k+1}}(x^\prime,x^n)=\begin{cases}(d\omega)_{i_1\dots i_{k+1}}(x^\prime,|x^n|) i_l\neq n, \operatorname{sgn}(x^n)(d\omega)_{i_1\dots i_{k+1}}(x^\prime,|x^n|) i_l=n\end{cases}
J_n(x+y)=\sum_{j=-\infty}^{\infty}J_{n-j}(x)\sum_{k=0}^{\infty}\frac{(-1)^k}{k!}\frac{y^{j+2k}}{2^{j+2k}(m+k)!}=\sum_{j=-\infty}^{\infty}J_{n-j}(x)J_j(y).
M^{-2l}\sum_{m=2}^{N(\beta_2)}c_0^{\frac{m}{2}}M^{\frac{l}{2}m}\alpha^m\sum_{n=2}^{\infty}|T_m^{l,(n)}(\beta_1)- T_m^{l,(n)}(\beta_2)|_0\le c \beta_1^{-\frac{1}{2}}M\alpha^{-2}.
\psi_1|_W = \lambda^2+\sum_{i=1}^m\alpha_i^2+\mu^2 \psi_2|_W = \lambda^2-\sum_{i=1}^m\alpha_i^2-\mu^2 \psi_3|_W = 2\lambda \alpha_1 \psi_i|_W = -2\lambda\alpha_{i-2}(4\leq i\leq{m+2})
\|u\|_{\mathcal C([0,T];D(A))}\le\|\mathcal H^n(u)\|_{\mathcal C([0,T];D(A))}+\sum_{k=1}^{n}\|\mathcal H^k(F)\|_{\mathcal C([0,T];D(A))}&\le\frac{C\left(\Gamma(\alpha)T^{\alpha}\right)^n}{\Gamma(n\alpha+1)}\|u\|_{\mathcal C([0,T];D(A))}+\sum_{k=1}^{n}\frac{C\Gamma(\alpha)^{k-1}T^{k\alpha-1/p}}{\Gamma(k\alpha+1-1/p)}\|F\|_{L^p(0,T;D(A))}
R_{1212}=R_{1414}=R_{2323}=R_{3434}=R_{1313}=R_{2424}=-\frac{1}{2}\lambda_{1}^{2}, R_{1213}=R_{2434}=R_{2324}=R_{1314}=R_{1424}= R_{3431}=R_{1323}=R_{2124}=-\frac{1}{4}\varepsilon\lambda_{1}^{2}, R_{1214}=R_{3234}=R_{1434}=R_{2123}=\frac{1}{4}\lambda_{1}^{2}.
(P_{w_{1}}T_{w_1}P_{w_1}^\perp ) \ldots (P_{w_{r}}T_{w_r}P_{w_r}^\perp ) (P_{w_{r+1}}T_{w_{r+1}}P_{w_{r+1}})\ldots (P_{w_s}T_{w_s}P_{w_s}) \times ( P_{w_{s+1}}^\perp T_{w_{s+1}}P_{w_{s+1}}) \ldots ( P_{w_{d}}^\perp T_{w_{d}}P_{w_{d}}).
\left(\Pi_{\lambda_1,\lambda_2}g\right)^{\lambda_1,\lambda_2}_{i+\frac{1}{2},j+\frac{1}{2}}= g^{\lambda_1,\lambda_2}_{i+\frac{1}{2},j+\frac{1}{2}}, \mbox{if}\left(i,j\right) \in (\Bbbk_2^+,\Bbbk_1^+)
\psi^{1}( z ; x ) = \left( \begin{array}{c r}{0}\\{e^{- \phi ( z )}G_{+}( z - \omega )}\\ \end{array}\right) , \psi^{2}( z ; x ) = \left( \begin{array}{c r}{e^{\phi ( z )}G_{-}( z + \omega )}\\{0}\\ \end{array}\right) ,
I(\tau,x,v)=\left(v\,\tau-\frac{m\,x^3}{3\omega^2}+\frac{k\,x}{\omega^2}+\frac{f\,\tau}{-\omega^2+k-m}+\frac{f}{\tau(\omega^2+k-m)}\right)\tau^{\frac{m-k}{\omega^2}}.
G^{\vee^{*}_{1}\vee^{*}_{2}}=R p^{\prime}_{2,1*}\, p^{\prime\, !}_{2,2}\,Rp^{\prime}_{1,1*}\,p^{\prime\,!}_{1,2}G= R p^{\prime}_{2,1*}\, Rp^{\prime}_{1,1*}\, p^{\prime\, !}_{2,2}\,p^{\prime\,!}_{1,2}G & = Rp'_{1*}p^{\prime \,!}_{2}G.
\{f(z) , g(z)\}_{\alpha}= \nabla_{i}f(z)(\omega_{(\alpha ) ij}+ R_{ijkl}\theta^{k}\theta^{l})^{-1}\nabla_{j}g(z) +\frac{\partial_{r}f(z)}{\partial \theta^i}g^{ij}\frac{\partial_{l}g(z)}{\partial \theta^j},
\begin{cases}d\tilde{p}(t,z) =\tilde{p}(t,z)[\pi(t,z)b_0(t,z)-\frac{a_0(t,z)}{b_0(t,z)}]dB(t) \tilde{p}(T,z) = \int_D\frac{\partial U}{\partial y}(x,Y(T,x,z),z)Y(T,x,z)dx \mathbb{E}[\delta_Z(z)|\mathcal{F}_T].\end{cases}
\hat{P}_1^{(\alpha,\beta)}(z)=-\frac{1}{2}z-\frac{2+\alpha+\beta}{2(\alpha-\beta)},\quad\hat{P}_2^{(\alpha,\beta)}(z)=-\frac{\alpha+\beta+2}{4}z^2-\frac{\alpha^2+\beta^2+2(\alpha+\beta)}{2(\alpha-\beta)}z-\frac{\alpha+\beta+2}{4},...
f_{0}\left( x,y\right) =1+x^{4}+y^{3}, f_{1}\left( x,y\right) =1+\frac{x^{3}}{\sqrt{x^{2}+y^{2}}}+\frac{y^{7}}{x^{2}+y^{2}}=1+r^{2}\cos^{3}\varphi+r^{5}\sin^{7}\varphi, f_{2}\left( x,y\right) =\cos\left( 10x+20y\right) , f_{3}\left( x,y\right) =\left( x^{2}+y^{2}\right)^{5/4}=r^{5/2}
\frac{1}{q}\left( 1 - \frac{1}{\prod_{i=1}^n a_i}\right) = \sum_{i=1}^n \frac{1}{a_i}\leq \sum_{i=1}^n \frac{1}{b_i}= \frac{r}{\prod_{i=1}^nb_i}& \leq \frac{1}{q}\left( 1 - \frac{1}{\prod_{i=1}^nb_i}\right).
\mathrm{sign}(\sigma_{n,k},\Delta_{k})=\begin{cases}1 \mbox{for}\,\,k\,\,\mbox{even}, 1 \mbox{for}\,\,k\,\,\mbox{odd},\,\,\ell(n)\leq k, -1 \mbox{for}\,\,k\,\,\mbox{odd},\,\, k<\ell(n).\end{cases}
Af(\eta )=f(\eta )\sum_{(x,u)\in\eta}\frac{Bg(x,u)}{g(x,u)}+f(\eta )\sum_{(x,u)\neq (x',u')\in\eta}r(x,x'){\bf 1}_{\{u'<u\}}(\frac{g(x',u)}{g(x,u)}-1)
\tilde\mu_1=\frac{\mu_1 (\mu_1^2-\mu_1\mu_2+\mu_2^2 )\,\mathbb E [(\xi_i-\eta_i)^3 ]}{3 (\mu_1^3-\mu_1^2\mu_2+\mu_1\mu_2^2-\mu_2^3 )\,\mathbb E [(\xi_i-\eta_i)^2 ]}.
\sum\limits_{j=0}^{\infty}a_j\int\limits_{-\pi}^{\pi}\frac{e^{i(j-k)\theta}f(\theta)}{f(\theta)+g(\theta)}d\theta-\sum\limits_{j=0}^{\infty}c_j\int\limits_{-\pi}^{\pi}\frac{e^{i(j-k)\theta}}{f(\theta)+g(\theta)}d\theta=0,k = 0, 1,\dots
2\Delta^2 e^{-\frac{c'q\delta}{\ell}\left(1-\frac{\log 2}{\log \ell}\right)}+2\Delta^2\binom{\Delta}{t}(2p)^{t}\leq \frac{c'q\delta}{2\ell\log{\ell}}\mbox{\;, and}(t+1)\Delta e^{-\frac{c'q\delta}{\ell}\left(1-\frac{\log 2}{\log \ell}\right)}+ (t+1)\Delta\binom{\Delta}{t}(2p)^{t}\leq \frac{t}{2}\;.
\|(\widetilde{g}(t_1),\widetilde{H}(t_1))-(\widetilde{g}(t_2),\widetilde{H}(t_2))\|_{C^{k-1}_{g_c}}=\int_{t_1}^{t_2}\|(\frac{\partial \widetilde{g}}{\partial t},\frac{\partial \widetilde{H}}{\partial t})\|_{C^{k-1}_{g(t)}}dt &\leq C|\lambda(\widetilde{g}(t_1),\widetilde{H}(t_1))-\lambda(g_c,H_c)|^\theta\leq C(t_1+1)^{\frac{\theta}{1-2\alpha}}.
\gamma^{0}= \left( \begin{matrix}{0}&{1}\\{1}&{0}\\ \end{matrix}\right) , \quad \gamma^{1}= \left( \begin{matrix}{0}&{- 1}\\{1}&{0}\\ \end{matrix}\right) , \quad \gamma^{5}= \gamma^{0}\gamma^{1}= \left( \begin{matrix}{1}&{0}\\{0}&{- 1}\\ \end{matrix}\right) .
\int_\Omega I(u)_x^2 dx = \sum_{k = 1}^\infty c_k^2 \lambda_k^\alpha \int_\Omega \partial_x \varphi_k^2 dx = \sum_{k = 0}^\infty c_k^2 \lambda_k^{\alpha + 1}= \| u \|_{\overset{.}{H}^{\alpha + 1}_N(\Omega)}^2.
\lim_{n\rightarrow \infty}n^{2\alpha+1}\int_0^{\tau \wedge t_1 \wedge t_2}(t_1-s)^{\alpha}(t_2-s)^{\alpha}(\sigma'(X_s))^2 \Psi^{n,5}_s ds =\kappa_5 \int_0^{\tau \wedge t_1 \wedge t_2}(t_1-s)^{\alpha}(t_2-s)^{\alpha}(\sigma'\sigma)^2(X_s) ds,
k^{a+1,c}(I,J)=k^{a,c}(I,J), k^{a+1,c}(I',J')=k^{a,c}(I,J), k^{a+1,c}(I',J)=\begin{cases}k^{a,c}(I,J), I<J k^{a,c}(I,J)+1, I>J.\end{cases}
\Psi^+_F (k_F - S_F k_F S_F^*)\Psi^{+*}_F \,= \Psi^+_F (k_F - \sigma_F k_F \sigma_F^*)\Psi^{+*}_F = \,{\left((\psi_\lambda^+\psi_\lambda^{+\,*}* ([1_n] - \sigma_\lambda\sigma_\lambda^*) * k_s)(x,y) \right)}_{x,y\in F}.
\begin{cases}\frac{d^2y}{d t^2}-4 \frac{e^{2t}}{1+e^{2t}}y(t)=0, y(t)=f(t)x(t), x(t)= -\frac{3e^{4t}+9e^{2t}+6+e^{-2t}}{2(1+2e^{2t})^{\frac{3}{2}}}.\end{cases}
T_{\mu\nu}(k)=i\int \frac{d^2p}{(2\pi )^2}\int_0^1d\xi_1\int_0^1d\xi_2\delta (1-\xi_1-\xi_2) \left[\frac{P(m)}{\left[ z -m^2+i\varepsilon\right]^2}\right]_{z=\sum_{j=1}^2\xi_jp_j^2},
e_{ij}(x_o) \equiv \Gamma_a e_{ij}(x_a) + \Gamma_b e_{ij}(x_b) = \sum_{m=1}^{\infty}\big[ \Gamma_a C_{q_m}(x_a) + \Gamma_b C_{q_m}(x_b) \big] h^{q_m}\big[ r^{(i-j)q_m}- 1 \big] r^{(j-1)q_m}= \sum_{m=1}^{\infty}\big[ C_{q_m}(x_a) + \mathcal{O}(h^2) \big] h^{q_m}\big[ r^{(i-j)q_m}- 1 \big] r^{(j-1)q_m},
\begin{aligned}\frac12 S(\sqrt{r},f)\log r = \int_{\sqrt{r}}^r\frac{S(\sqrt{r},f)}{t}dt\leq \int_{0}^r\frac{S(t,f)}{t}dt = T_0(r,f)\leq S(r,f)\log r+T_0(1,f)\end{aligned}
\begin{array}{l}{{q_{0}= \sqrt{\frac{2 5 - D}{6}- \vec{q}^{\; 2}}\ ,}}\\{{p_{0_{\pm}}= q_{0}\pm \sqrt{q_{0}^{2}- \vec{p}^{\; 2}+ 2 \vec{q}\vec{p}- 4}= \frac{1}{\sqrt{6}}\left( \sqrt{2 5 - D - 6 \vec{q}^{\; 2}}\pm \sqrt{1 - D - 6 ( \vec{q}- \vec{p})^{2}}\right) \ .}}\\ \end{array}
-3F(2\pi/3)+\sum_{i=1}^{3}F(\theta_{i})\leq-\frac{3}{2\pi^{2}}\Big(\lambda_{1,n}^{r,s}+\frac{1}{25}\lambda_{5,n}^{r,s}\Big)\leq-\frac{18}{7}\frac{3}{2\pi^{2}}\Big(\lambda_{1,n}^{r,s}+\frac{1}{25}\lambda_{5,n}^{r,s}\Big)\sum_{i=1}^{3}\Big(\frac{\theta_{i}}{2\pi}-1/3\Big)^{2}.