harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{j=1}^{d_{k}}|Y_{j}^{k}(y)|^{2}=C_{k}^{2}(\mathbb{M}^{d})\int_{\mathbb{M}^{d}}\left( P_{k}^{(\alpha ,\beta )}(\cos (2\lambda d(x,y))\right)^{2}d\nu(x).
^{\cos A =}\frac{4 b^{2}+ 2 c^{2}}{6 b c}= \frac{2 b}{3 c}+ \frac{1}{3}\cdot \frac{c}{b}\geq 2 \sqrt{\frac{2}{2}\cdot \frac{1}{3}}= \frac{2}{3}\sqrt{2}\cdot
5p \geq\Pr(|S_{k}-\tilde{S}_{k}|>2t) &\geq \Pr(|S_{k}-a_{k}|\leq t,|\tilde{S}_{k}+a_{k}|\leq t)+ \Pr(|S_{k}+a_{k}|\leq t,|\tilde{S}_{k}-a_{k}|\leq t) &=2\Pr(|S_{k}-a_{k}|\leq t)\Pr(|S_{k}+a_{k}|\leq t).
\left|\frac{d p_{\eta}}{d \eta}(t)\right| \leq \frac{1}{\alpha}\frac{1}{\eta^{1+\frac{1}{\alpha}}}p_N\left(\frac{t}{\sqrt[\alpha]{\eta}}\right) + \frac{1}{\alpha}\frac{|t|}{\eta^{1+\frac{2}{\alpha}}}\left|\frac{d p_N}{d u}\right|_{u= \frac{t}{\sqrt[\alpha]{\eta}}}.
\sigma \epsilon^{p-3}=\Phi\!\!\left(\!\!\begin{array}{cc}2 1 1 3 \end{array}\!\! \right) = \Phi\!\!\left(\!\!\begin{array}{cc}p-3 p-3 2 1 \end{array}\!\! \right) = \sigma^{p-4}\bar{\sigma}^{p-2}
\frac{1}{2-\lambda}= \frac{\log(0.04216a_1a_2(a_2-a_1)^{-2}c^2)}{\log(0.016858a_1(a_1')^{-1}a_2^{-1}(a_2-a_1)^{-2}c)}< \frac{2\log(0.20533a_1^{1/2}a_2^{1/2}(a_2-a_1)^{-1}c)}{\log(0.01685a_1(a_1')^{-1}a_2^{-1}(a_2-a_1)^{-2}c)}.
F ( \bar{y}, z ) = \sum_{s}\oint \frac{\bar{d u}}{2 \pi i}\frac{1}{( \bar{y}- \bar{u})^{2}}\oint \frac{d v}{2 \pi i}\frac{1}{( v - z )}\frac{\xi_{s}^{\prime}( v )}{\xi_{s}( v ) ( 1 - \bar{\xi}_{s}( u ) \xi_{s}( v ) )}
\widetilde{M}_4 = \left( \begin{array}{rrrrr}1 5 10 10 5 1 6 14 16 8 1 6 15 18 9 \end{array}\right)\quad\rightsquigarrow{M}_4 = \left(\begin{array}{rrrrr}1 5 10 10 5 0 1 4 6 3 0 0 1 2 1 \end{array}\right).
G^{i j}={\frac{1}{\det ( G )}}\left( \begin{array}{c c}{G_{2 2}}&{- G_{1 2}}\\{- G_{1 2}}&{G_{1 1}}\\ \end{array}\right) ={\frac{4}{3}}\left( \begin{array}{c c}{G_{2 2}}&{- G_{1 2}}\\{- G_{1 2}}&{G_{1 1}}\\ \end{array}\right) .
|B,\eta,V>^f_{osc}=\exp{i\eta\sum_{n>0}^{\infty}\{\frac{1}{2}(e^{-2v}\psi^{A\dagger}_{n}\tilde{\psi}^{A\dagger}_{n}+e^{2v}\psi^{B\dagger}_{n}\tilde{\psi}^{B\dagger}_{n})-\psi^{T \dagger}_{n}\tilde{\psi}^{T \dagger}_{n}\}}|0>\;,
|\mathcal{F}(a+b-1,-b \,;\, \{U\},\emptyset)| =\sum_{i=0}^{a-1}\binom{a+b-2}{\lfloor i/2 \rfloor, b+\lfloor (i-1)/2\rfloor, a-i-1}, |\mathcal{F}(a+b-1,-b-1 \,;\, \{U\},\emptyset)| =\sum_{i=0}^{a-2}\binom{a+b-2}{\lfloor i/2 \rfloor, b+\lfloor (i+1)/2\rfloor, a-i-2}.
\tau^a ={n_1 \choose d+1}{n_2 \choose d+1}, \tau^b = \sum_{2r_0+r_1=d+1}{d+1 \choose r_0}{d+1-r_0 \choose r_1}\frac{(d+2)!}{2^{r_0}}\left[{n_1 \choose d+1}{n_2 \choose d+2}+{n_1 \choose d+2}{n_2 \choose d+1}\right] + y_{d+2}{n_1 \choose d+2}{n_2 \choose d+2},
( z_{0}; z_{1}, z_{2}) = \left({\frac{1 - x_{\mu}^{2}+ h^{2}}{2 h}}+ i{\frac{x_{0}}{h}};{\frac{x_{1}+ i x_{2}}{h}}, x_{3}+ i{\frac{1 + x_{\mu}^{2}- h^{2}}{2 h}}\right)
\sum_{i=-2}^3 \sum_{j=1}^m \int_{\mathbb{R}^4}F(y,x') F(x,x')F(y,y')F(x,y') begin{align*}-1ex] g_{\alpha2^{k_j}}(x'-y') (\chi_{2^{k_{j-1}+i}}-\chi_{2^{k_j+i}})(x-y) \,dx dy dx' dy'.
\begin{aligned}\int_{B_r}u^{q-1}\eta\nabla\left(\left(u^m-k\right)_\pm\eta\right)\cdot\nabla v\,dx &\leq \frac{1}{2}\int_{B_r}\left|\nabla\left(\left(u^m-k\right)_\pm\eta\right)\right|^2\,dx+\frac{1}{2}\int_{B_r\cap\left\{(u^m-k)_\pm>0\right\}}u^{2(q-1)}\eta^2\left|\nabla v\right|^2\,dx\end{aligned}
\int_{{B}\backslash (\omega_+ \cup \omega_-)}\left| |W_\delta|^{p-1}W_\delta - U_{\delta,e_n}^p+U_{\delta,-e_n}^p \right|^\frac{(p+1)(1+\gamma)}{p}\leq C \int_{{B}\backslash (\omega_+ \cup \omega_-)}\left( U_{\delta,e_n}^{(p+1)(1+\gamma)}+ U_{\delta,-e_n}^{(p+1)(1+\gamma)}\right) &=O(\delta^{n(1+\gamma)}),
\frac{d}{t}I(t)\leq \frac{2}{\gamma-1}(2-3(\gamma-1))(t+1)\int_{A(t)}Px &-2(t+1)^2\int_{A(t)}\left(\beta+\frac{2}{3}\alpha\right)\rho^\delta( u)^2x &+6(t+1)\int_{A(t)}\left(\beta+\frac{2}{3}\alpha\right)\rho^\delta ux.
\vert f(w)-g(w)\vert = x\vert w\vert = x\vert w\vert -\vert w^{r+1}+x\vert+\vert w^{r+1}+x\vert &\leq \vert w^{r+1}+x-xw\vert +\vert w^{r+1}+x\vert =\vert f(w)\vert + \vert g(w)\vert.
M = \pm \; \left(\begin{array}{lll}\lambda \quad 0\cr\; 0 \quad \lambda^{-1}\cr\end{array}\right) \ \ \ \ \ \{\rm or}\ \ \ \ \M^\prime = \pm \; \left(\begin{array}{lll}\; 0 \quad \lambda^{-1}\cr\lambda \quad 0 \cr\end{array}\right)\ ,
X = \left(\begin{array}{cccccc}0 0 0 \ldots 0 X^{[r]}X^{[1]}0 0 \ldots 0 0 0 X^{[2]}0 \ldots 0 0 \vdots{}{}\vdots{}\vdots 0 0 0 \ldots X^{[r-1]}0\end{array}\right)\ .
\nabla_x \cdot \nabla_y \mathcal M_{\alpha, \beta, j}\frac{1}{|x|^{2\alpha}|y|^{2\beta}}=2^{2j}(\alpha)_j (\beta)_j(I +II)=2^{2j}(\alpha)_j (\beta)_j I+2^{2j}(\alpha)_j (\beta)_jII
\theta^\gamma_{0}(x,s)=\inf\{t\ge s:\, \Phi^{\gamma}(x,s)=\pi_{j_0}(L_{j_0})\}, &\theta^\gamma_{k}(x,s)=\inf\{t\ge \theta^\gamma_{k-1}(x,s):\, \Phi^{\gamma}(x,s)=\pi_{j_k}(L_{j_k})\},k\geq 1
\begin{aligned}|Dv(x)-Dv(y)|+|\pi (x)- \pi(y)| &\le C R^{-d/2}\||Dv|+|\pi|\|_{L^2(\Omega_{R}(x_0))}\bigg(\left(\frac{r}{R}\right)^{\gamma}+\int_0^{2r}\frac{\varrho^\sharp_0(t)+\omega^\sharp_{A^{\alpha\beta}}(t)}{t}\,dt\bigg),\end{aligned}
\tilde{\cal A}_{m}^{n}= \frac{3}{4}\left( \begin{array}{c c c c}{{z^{- 2}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{z^{- 2}}}\\ \end{array}\right) + \mathrm{~ m e a n ~ f i e l d ~ t e r m s}
a_{1}= 6 C_{2}| z_{t r}| \mathrm{, ~}a_{2}= \frac{2 7}{| z_{t r}|^{2}}+ 1 5 C_{2}\mathrm{, ~}a_{3}= \frac{8 1}{| z_{t r}|^{3}}+ \frac{2 0 C_{2}}{| z_{t r}|}.
(M_V)^g = \left\{h^g \in G_{d-1}(d) \mid \sum_{v \in V}h_{(v)}=0 \right\}= \left\{f \in G_{d-1}(d) \mid \sum_{v \in V}\left(f^{g^{-1}}\right)_{(v)}=0 \right\}= \left\{f \in G_{d-1}(d) \mid \sum_{v \in V}\left(f\right)_{(g^{-1}(v))}=0 \right\}= M_{g^{-1}V}.
\#E(Q,Y) \le b^{2l}b^q2^b\cdot \#E(Q,R)\cdot \#E(Q,Y\cap Z_c)\cdot \#E(Q,Y_0)\cdot \#E(Q,Y_1) &\le b^{2l}b^q2^b\cdot \sigma^q \cdot \sigma^{q^2}\cdot (\sigma^{11l}(12l)^{q})^b\cdot[(2l)^q\cdot (32l^2)^{bq}\cdot (qb)^{2lq}\cdot\sigma^{6lq+8l^2b}] &=b^{2l}2^b\cdot(2lb)^q\cdot(384l^3)^{qb}\cdot(qb)^{2lq}\cdot\sigma^{q+q^2+11lb+6lq+8l^2b}.
F(r)=- \frac{w}{\kappa}\left[ B_\frac{r^\kappa}{r^\kappa+1}\left(\frac{-1}{p+1}, 1-\frac{2}{\kappa}\right) + 2B_\frac{r^\kappa}{r^\kappa+1}\left( \frac{p}{p+1}, -\frac{2}{\kappa}\right)\right]\!.
\lim\limits_{j\to+\infty}\varepsilon_j\dot{\widetilde{u}}_{\tau_j,\varepsilon_j}(t^-_j)=\lim\limits_{j\to+\infty}\varepsilon_j\dot{\widetilde{u}}_{\tau_j,\varepsilon_j}(t^+_j)=0\,.
\begin{cases}\displaystyle{\frac{1}{c^2(x)}\partial_t^2 u(x, t)-\Delta u(x, t)=0 \quad\ \mbox{in\ \ $\mathbb{R}^3\times \mathbb{R}_+,$}}\smallskip & u(x,0)=f(x),\ \ \ \partial_t u(x,0)=0\mbox{for \ $x\in\mathbb{R}^3$},\end{cases}
y_1=\frac{1}{\sqrt{\mu}}x_3, \, y_2=\frac{1}{\sqrt{2}}(x_1+x_2), \, y_3=\frac{1}{\sqrt{2}}(x_1-x_2); &[y_1,y_2]=\frac{1}{\sqrt{\mu}}y_2+ \frac{w}{\sqrt{\mu}}y_3, \,[y_2,y_3]=0,\,[y_3,y_1]=-\frac{w}{\sqrt{\mu}}y_2 - \frac{1}{\sqrt{\mu}}y_3.
\mathbb{P}(a,a',b,b',c)\!=\!\!\!\!\!\!\!\!\!\!\sum_{i+j\geq a'+b'-c}\!\!\!\!\!\!\!\!\!\!\mathbb{P}_{i}(a,a')\mathbb{P}_{j}(b,b')\mathbb{P}(c,a'\!+\!b'\!-\!i\!-\!j).
\bigg\| \frac{\partial_rf}{|x|}\bigg\|_{2}^2 = \bigg\|\partial_r \frac{f}{|x|}\bigg\|_{2}^2 + \bigg\| \frac{f}{|x|^2}\bigg\|_{2}^2 + 2\mathrm{Re}\bigg\langle \partial_r \frac{f}{|x|}, \frac{f}{|x|^2}\bigg\rangle
\begin{aligned}B(j, k):= -(24 c_4-48 c_1^2+\frac{12 c_6-16 c_2^2}{3(j^2+k^2+j k)}) D_S^3 U D_S^3 begin{align*}2mm] +(\frac{16 c_2^2}{3}-4 c_6+\frac{(16 c_2 c_3-24 c_7)}{j^2+k^2+j k}) D_S U D_S^3.\end{aligned}
\frac{\partial}{\partial z_{\rho}}\langle J_{\mu}( x ) J_{\nu}( y ) J_{\rho}( z ) \rangle_{U V}= - \frac{1}{1 2 \pi^{2}}\epsilon_{\mu \nu \rho \sigma}\frac{\partial}{\partial x_{\rho}}\frac{\partial}{\partial y_{\sigma}}\delta ( x - z ) \delta ( y - z ) \equiv \frac{1 6}{9}{\cal A}_{\mu \nu}( x , y , z )
\overline{W}=\int_{r(t_{0})=A}^{r(t_{1})=B}F\cdot d(r+\epsilon h)=\int_{t_{0}}^{t_{1}}F\cdot\frac{d(r(t)+\epsilon h(t))}{dt}dt=\int_{t_{0}}^{t_{1}}F\cdot(v+\epsilon\dot{h})dt.
\mathrm{A}_{\mu}^{( n )}( y ) ={\cal L}_{\mu \nu}A^{\nu}({\cal L}^{T}y ) = - \frac{1}{2}{\cal L}_{\mu \nu}F_{\ \sigma}^{\nu}{\cal L}_{\lambda}^{\ \sigma}y^{\lambda}= - \frac{1}{2}({\cal L}F{\cal L}^{T}y )_{\mu}.
A_r^s(x_1, \dots, x_r) := \begin{pmatrix}e_{r, 1}^0 e_{r, 2}^0 \cdots e_{r, r}^0 e_{r, 1}^1 e_{r, 2}^1 \cdots e_{r, r}^1 \vdots \vdots \ddots \vdots e_{r, 1}^{s - 1}e_{r, 2}^{s - 1}\cdots e_{r, r}^{s - 1}\end{pmatrix}.
\lambda_1\bar J_i (u_i)+\lambda_2 \bar J_i(u_i')- \bar J_i(\lambda_1 u_i+\lambda_2 u_i' ) =\ \lambda_1\lambda_2 \mathbb{E}\int_0^T \left\{| x_i-x_i' - \Gamma(m_i-m_i')|_Q^2+| u_i-u_i'|_R^2 -\gamma |p_i(t)- p_i'(t)|^2\right\}dt &+\lambda_1\lambda_2 \mathbb{E}| x_i(T)-x_i'(T)|_H^2.
\widetilde{R}^{\mu}_{\:\: \nu \rho \sigma}=\widehat{\kappa}^{\mu}_{i\: \rho}\widehat{\kappa}^{i}_{\nu \sigma}-\widehat{\kappa}^{\mu}_{i\: \sigma}\widehat{\kappa}^{i}_{\nu \rho}+\gamma^{\alpha}_{\nu}\gamma^{\beta}_{\rho}\gamma^{\gamma}_{\sigma}\widehat{R}^{\mu}_{i\: \alpha \beta \gamma}
\frac{1}{\lambda_u^t \cdot t^{d_u+1}}M^t \vec v_0 = \frac{1}{\lambda_u^t \cdot t^{d_u+1}}\left(\lambda^t \vec v_t + \sum_{j=0}^{t-1}\lambda^{j}M^{t-j-1}\vec u_{j+1}\right) &= \frac{1}{t^{d_u+1}}\vec v_t + \frac{1}{\lambda}\sum_{j=0}^{t-1}\frac{({t-j-1})^{d_u}}{t^{d_u+1}}\frac{1}{\lambda^{t-j-1}\cdot ({t-j-1})^{d_u}}M^{t-j-1}\vec u_{j+1}
\begin{array}{ll}p_{\infty}^{\frac{1}{\gamma}}|\Omega| =\int\limits_{\Omega}p_0^{\frac{1}{\gamma}}\,\mathrm{d}x, begin{align*}6pt]p_{\infty}=\bar{p}= \left( \frac{1}{|\Omega|}\int\limits_{\Omega}p_0^{\frac{1}{\gamma}}\,\mathrm{d}x \right)^{\gamma}.\end{array}
i\hbar \frac{\partial}{\partial t}\bigl<q|\phi(t)\bigr>=\Bigl[-\hbar^{2}X(t)\frac{\partial^{2}}{\partial q^{2}}-i\hbar Y(t)\bigl(q\frac{\partial}{\partial q}+\frac{\partial}{\partial q}q\Bigr)+Z(t)q^{2}\Bigr]\bigl<q|\phi(t)\bigr>.
E[S^{env,0}_{s,t}]= -E\left[\int_{s}^t \beta(r)(\partial_r\psi(r,q_r)+\nabla_qV(r,q_r))\circ dq_r\right]-\int_{s}^t \frac{B_0(r)\dot{B_0}(r)}{\gamma(r)^2+B_0(r)^2}dr.
[r_{21},r_{31}] =[r^{ij}X_{j}\otimes X_{i}\otimes 1, r^{lk}X_{k}\otimes 1\otimes X_{l}]=-r^{ij}r^{lk}\mathrm{ad}^{(r)}_{X_{j}}X_{k}\otimes X_{i}\otimes X_{l}&= r^{ij}r^{lk}[X_{k},X_{j}]\otimes X_{i}\otimes X_{l}=-r^{ij}r^{lk}f_{kj}\,^{s}X_{s}\otimes X_{i}\otimes X_{l},
\int_{M}\omega=\int_{D}\sum_{i_{1}<\cdot\cdot\cdot<i_{n}}a_{i_{1},...,i_{n}}(\varphi(u))\frac{\partial(x^{i_{1}},...,x^{i_{n}})}{\partial(u^{1},...,u^{n})}du^{1}\cdot\cdot\cdot du^{n},
\sum_{n=-\infty}^{+\infty}(-\frac{\Gamma(-\frac{d}{2})}{(4\pi)^{d/2}}(4\pi^{2}n^{2}+\xi^{2})^{d/2})+\frac{\Gamma(-\frac{d+1}{2})}{(4\pi)^{\frac{d+1}{2}}}\xi^{d+1}.
G_{n,k}(z)=A\psi_{n,k}(z)+B\psi_{n+1,k}(z)=\int_{a_{k-1}}^{b_{k-1}}\frac{A\, z\,\psi_{n,k-1}(\tau)+B\,\tau\,\psi_{n+1,k-1}(\tau)}{z-\tau}\,d\sigma_{k-1}^{*}(\tau).
\frac{d^2}{ds^2}\mathrm{sn}\left(\gamma_{\partial B}(s),x_0\right) = \frac{d}{ds}\mathrm{sn}\left(b\left(\gamma_{\partial B}(s)\right),x_0\right) = \frac{d}{ds}\left[b\left(\gamma_{\partial B}(s)\right),x_0\right] = -\rho(s).\left[\gamma_{\partial B}(s),x_0\right] = = -\rho(s).\mathrm{sn}\left(\gamma_{\partial B}(s),x_0\right),
\varphi (g_2 g^{n}_1 g_3) \overset{(\ref{eq:case1-6})}{=}\varphi \big( g^{j}_1 g_3 \big) \varphi \big( g^{i}_1 g_2 \big) \overset{(\ref{eq:case1-4})}{=}\varphi (g_3) \varphi (g^{n}_1) \varphi (g_2) \overset{(\beta_1)}{=}\varphi (g_3) \varphi (g^{n}_1 g_2) \,.
\frac{1}{2}\left(1-\sum^{\infty}_{n=0}{a^2_n}\right)\cdot osc_L(f)\cdot osc_L(g) =\frac{1}{2}\cdot [1-a^2-(1-a)^2]\cdot \left|f(0)-f(1)\right|\cdot\left|g(0)-g(1)\right| &=a(1-a)\left|f(0)-f(1)\right|\cdot\left|g(0)-g(1)\right|.
\sum_{k=0}^m[3k+1]\,\frac{(aq;q^2)_k(q/a;q^2)_k(q;q^2)_k(q/b;q)_k(q/c;q)_k(bc;q)_k\,q^k}{(aq;q)_k(q/a;q)_k(q;q)_k(bq^2;q^2)_k(cq^2;q^2)_k(q^3/bc;q^2)_k}\equiv\frac{(bcq;q^2)_{(n-1)/2}(q^2/b;q^2)_{(n-1)/2}(q^2/c;q^2)_{(n-1)/2}}{(q^3/bc;q^2)_{(n-1)/2}(bq^2;q^2)_{(n-1)/2}(cq^2;q^2)_{(n-1)/2}}\,[n].
f(\mathbf{r})=\begin{cases}r_2-\gamma(P_2),r_1\leq\gamma(\frac{P_1}{1+P_2}) r_1+r_2-\gamma(P_1+P_2), \gamma(\frac{P_i}{1+P_j})\leq r_i \leq \gamma(P_i),\i,j\in\{1,2\},\i\neq j r_1-\gamma(P_1), r_2\leq\gamma(\frac{P_2}{1+P_1}) \end{cases}.
\sum\limits_{\beta,\gamma\in M_{q,m}}\chi^s(\beta-\gamma)= \sum\limits_{\beta,\gamma\in H_{q,m}}\chi^s(\beta-\gamma)+\sum\limits_{\beta\in H_{q,m}}\chi^s(\beta)+\sum\limits_{\gamma\in H_{q,m}}\chi^s(-\gamma) = \sum\limits_{\beta,\gamma\in H_{q,m}}\chi^s(\beta-\gamma)+f+f\chi^s(-1).
||u||_{X_k(T)}=||u||_{L^2(A_0)}+\sup_{j>0}||\langle \xi \rangle^{-\frac{1}{2}}u||_{L^2(A_j)}, \k\ge 0, ||u||_{X_k(T)}=2^{\frac{k}{2}}||u||_{L^2(A_{<-k})}+\sup_{j\ge -k}||(|\xi|+2^{-k})^{-\frac{1}{2}}u||_{L^2(A_j)}, \k<0,
\chi\left(\frac{x+(\nabla_\xi\omega_\rho)(\xi+v)t-(\nabla_\xi\omega_\rho)(v)t}{|v|^{2\rho-1}|t|/4}\right)f(\xi)\in S_{1,0}^{-\infty}=\bigcap_{-\infty<m<\infty}S_{1,0}^m
L = \sum_{\ell=1}^L 1 \leq \sum_{\ell=1}^L N_\ell = N(\Phi) - d \leq M(\Phi) + 1 \leq M + 1, N(\Phi) \leq M(\Phi) + d + 1 \leq M + d + 1 \leq 3d \cdot M =: T.
I_{L}( p ) = ( - i )^{n + 1}\int q . d^{2 \omega}\int_{0}^{\infty}d \alpha_{1}\cdots d \alpha_{n}d \beta \cdot q^{-}. e^{i [ \sum_{1}^{n}\alpha_{i}f_{i}( q , p ) + \beta q^{+}q^{-}]}.
\begin{cases}x_{t-1}^{i_{t-1}}=u_t y_{t-1}^{i_{t-1}}=u_t^{i_t}v_t^{i_{t-1}}\end{cases}\begin{cases}x_{t-1}^{i_{t-1}}= x_t^{i_{t-1}}y_t y_{t-1}^{i_{t-1}}= y_t^{i_t}\end{cases}
\int_{\mathbb{S}(\mathbb{C}_{n})}\left[\mathrm{tr}\left(\mathcal{D}f(A)(vv^{\ast})\right)\right]^2\pi(dv)=\int_{\mathbb{S}(\mathbb{C}_{n})}\left[\mathrm{tr}\left(f^{[1]}(\Lambda)\circ(vv^{\ast})\right)\right]^2\pi(dv)\leq \|f'\|_{\infty}^2.
\frac{t_{k+1}+\eta_{k+1}}{1+\eta_{k+1}}= \frac{|W_{k+1}|+q_{k}}{q_{k+1}+q_{k}}= \frac{|W_{k}|+q_{k}+q_{k-1}}{3q_{k}+q_{k-1}}= \frac{t_k+1+\eta_k}{3+\eta_k}> \varphi -1 .
\frac{t^{-m(f)}-1}{W_f(t^{-1})}\le \frac{t^{-3}-1}{(t^{-1}+1)^3},\intertext{which is obvious for $t=1$, and for $t<1$, it is equivalent to}[m(f),2,2,2](t^{-1}) \le W_f(t^{-1})[3](t^{-1}).
X_t(v,\lambda) = \sum_J\Big(\prod_{(i,j)\in J}\partial_\lambda^{(k_j)}\, P_{\alpha^i,\alpha^j}^{(k_i)}(t,\lambda)\Big)\;{:}\Big( \prod_{l\in J'}\partial_\lambda^{(k_l)}X_{s,\lambda}(\alpha^l) \Big){:}\,,
(x_{I}^{m}\eta - x_{I}^{m}\eta_{I, -\iota_{E}(\omega)}) \wedge \frac{df}{f}= \sum_{p \neq - \iota_{E}(\omega)}x_{I}^{m}\eta_{I,p}\wedge \frac{df}{f}\in \frac{1}{f}\Omega_{U}^{j+1}[x_{I}^{-1}].
\mathsf{T}_{\lambda}^{{\small}}(,^{\prime})\equiv \langle |\mathsf{T}_{\lambda}^{{\small}}|^{\prime}\rangle =\prod_{n=1}^{\mathsf{N}}W_{_{n}}(z_{n}/z_{n}^{\prime})\bar{W}_{_{n}}(z_{n}/z_{n+1}^{\prime}),
\begin{cases}-\Delta u_k=\left|x\right|^{2\alpha_{k,1}}V_{k}(x)e^{au_k}+\left|x\right|^{2\alpha_{k,2}}U_{k}(x)e^{u_k}=:g_k\quad\mbox{in}\Omega \displaystyle{\int_{\Omega}\left(\left|x\right|^{2\alpha_{k,1}}V_{k}(x)e^{au_k}+\left|x\right|^{2\alpha_{k,2}}U_{k}(x)e^{u_k}\right)}\leq C,\end{cases}
x_{0}\coloneqq k_{0}a\coloneqq\omega\sqrt{\epsilon_{0}\mu_{0}}a=2\pi\frac{a}{\lambda_{0}}, x \coloneqq k a\coloneqq\omega\sqrt{\epsilon}\sqrt{\mu}a\coloneqq x\sqrt{\epsilon_{r}}\sqrt{\mu_{r}}=2\pi\frac{a}{\lambda},
t\int_{-t}^t \frac{dx}{(x^2+\lambda^2)^{\kappa+1}}= \frac{\pi t}{2^\kappa\kappa!}\left(-\frac{1}{\lambda}\frac{d}{d\lambda}\right)^\kappa\frac{1}{\lambda}- 2(-1)^\kappa\left(\frac{d}{d(\lambda)^2}\right)^\kappa \sum_{k=0}^\infty\frac{(-1)^k}{2k+1}\left(\frac{\lambda}{t}\right)^{2k}.
\|v(t)\|_2 \le \int_0^t \|v'(s)\|_2 ds &\le \Big(\int_0^t s^{-1/2}ds \Big)^{1/2}\Big( \int_0^t s^{1/2}\|v'(s)\|_2^2 ds \Big)^{1/2}&= t^{1/4}\sqrt{2}\Big( \int_0^t s^{1/2}\|v'(s)\|_2^2 ds \Big)^{1/2}.
H_{ef}=\frac{d^2}{mn}\mathbf{1}\{e\to f\}H_{ef}^{(1)}=\mathbf{1}\{e\to f\}M_{f_1f_3,f_2}H^{(2)}_{ef}=\sum_{e\to g\to f}M_{e_3f_1,g_2}A_{f_1f_3,f_2},
\left[\vec V_n\right] = \left[ \sum_{i\in I_1\cup I_2\cup I_3\cup I_5}\vec V_{n,i}+ \vec U \right], \vec U := \frac12\left(\sum_{i\in I_1}d_i\big[\mathsf B\big]_i + \sum_{i\in I_2\cup I_3\cup I_4}\big[\mathsf B\big]_i \right).
\begin{aligned}\frac{1}{\tau}\big\|(v_n-v_{n-1})_{n=k}^m \big\|_{L^p(X)}+ \big\|(v_n )_{n=k}^m \big\|_{L^p(D)}&\le C\big\|(f_n)_{n=k}^m \big\|_{L^p(X)}+C\big\|\big ( (A_m - A_n) v_n \big )_{n=k}^m \big\|_{L^p(X)}&+C\Big( \frac{1}{\tau}\big\|(v_i)_{i=0}^{k-1}\big\|_{L^p(X)}+ \big\|(v_i )_{i=0}^{k-1}\big\|_{L^p(D)}\Big). \end{aligned}
|\tilde{A}[N/3]|^{2}=\frac{1}{2}[(C_{3}^{A}(0)-C_{3}^{A}(1))^{2}+(C_{3}^{A}(1)-C_{3}^{A}(2))^{2}+(C_{3}^{A}(2)-C_{3}^{A}(0))^{2}].
\lambda_j = \sum_{a=1}^{q}c_q(a) \omega_j^a=\sum_{a=1}^{q}\sum_{\substack{k=1 \gcd(k,q)=1}}^q \exp \Bigl(\frac{2 \pi i ka}{q}\Bigr) \exp \Bigl(\frac{2 \pi i ja}{q}\Bigr) &= \sum_{\substack{k=1 \gcd(k,q)=1}}^q \sum_{a=1}^{q}\exp \Bigl(\frac{2 \pi i a(k+j)}{q}\Bigr)=\begin{cases}q , ; 0, . \end{cases}
W={1 \over 64\pi^{{1 \over 2}}}{\tau \over a}(\Phi_{0}^2 a^2)^2 Log({1\over M^2 a^2})[1+m^2a^2+...] -{c \over 192 (2\pi)^{1 \over 2}}({\tau \over a}(\Phi_{0}^2 a^2)^3)
a_1(\nabla_X J)Y+a_2(\nabla_Y J)X+a_3(\nabla_{JX}J)Y+a_4(\nabla_{JY}J)X+a_5(\nabla_{JX}J)JY+ &+a_6(\nabla_{JY}J)JX+a_7(\nabla_{X}J)JY+a_8(\nabla_{Y}J)JX=0
(\mathcal{S},\Lambda)=(\mathcal{V}\times \mathbb{P}_\mathbb{C}^1\times \mathbb{C}^2, (a_0+t_1)(1-t\xi^2)\frac{\partial}{\partial \xi}\wedge \frac{\partial}{\partial u})\,\,\,\bigcup \,\,\, (\mathcal{V}_1\times \mathbb{P}_\mathbb{C}^1\times \mathbb{C}^2, (a_0+t_1)(1-t\xi_1^2)\frac{\partial}{\partial \xi_1}\wedge \frac{\partial}{\partial u_1})
\bar{\mathbf{U}}(\mathbf{x}):=\begin{cases}\bar{\mathbf{U}}^+:=(\bar{p}^+,0,\bar{u}^+_2,0,\bar{H}^+_2,\bar{S}^+), x_1>0, \bar{\mathbf{U}}^-:=(\bar{p}^-,0,\bar{u}^-_2,0,\bar{H}^-_2,\bar{S}^-), x_1<0, \end{cases}
r^{\frac{1}{2}}\mathfrak{h}_{\emptyset}*\mathfrak{h}_{\emptyset}= \begin{cases}-2^{-\frac{3}{2}}(r)^{\frac{1}{2}}q(\eta_h, \eta_h) (2r)^{\frac{1}{2}}\mathfrak{cl}(b_h)\eta_h 0 \end{cases}
\left \vert \left( \pi_{P_x \mid \tilde \Sigma_1}\right)^{-1}(z) - \left( \pi_{P_x \mid \tilde \Sigma_1}\right)^{-1}(z')\right \vert = \vert y-y'\vert \leq \vert \pi_{P_x}(y-y')\vert + \vert \pi_{P_x}^\perp(y-y')\vert \leq \vert z-z'\vert + \frac 1 2 \vert y - y'\vert.
N ( \lambda , \omega ) \sim{\frac{\lambda^{d / 2}a_{0}}{( 4 \pi )^{d / 2}\Gamma ( d / 2 + 1 )}}\sim r{\frac{\lambda^{d / 2}V_{d}}{( 4 \pi )^{d / 2}\Gamma ( d / 2 + 1 )}},
J_{\varepsilon}(W_{\varepsilon,\xi})=\frac{1}{2}\int_{\mathbb{R}^{n}}c(\xi)|\nabla V^{\xi}(z)|^{2}+a(\xi)\left( V^{\xi}\right)^{2}(z)dz-\frac{1}{p}\int_{\mathbb{R}^{n}}b(\xi)\left( V^{\xi}\right)^{p}(z)dz+O(\varepsilon),
\int_{\mathcal A_1^+(n)}(cd,d)_p\chi_{\psi}(d)\chi_{\delta}(d)|d|_p^{-3/2}\mathbf h_p(h) dh = \sum_{t > -n}\int_{\mathcal L_{2t}}(cd,d)_p\chi_{\psi}(d)\chi_{\delta}(d)|d|_p^{-3/2}\mathbf h_p(h) dh = = -p \sum_{t > -n}\mathrm{vol}(\mathcal L_{2t}) = -p(1-p^{-1})^2 \sum_{t > -n}p^{-2t}= -p^{2n-1}\frac{(1-p^{-1})^2}{1-p^{-2}}= -p^{2n}\frac{p-1}{p(p+1)}.
h(\lambda) = \frac{(\lambda, \lambda + 2\rho)}{2 (\kappa+h^\vee)}+ \frac{(\lambda, \lambda + 2\rho)}{2 (\kappa^*+h^\vee)}- (\lambda, \rho^\vee) = \frac{(\lambda,\lambda) r^\vee n}{2}+ (\lambda, n r^\vee \rho -\rho^\vee).
\prod_{i=1}^{m}C_{a}(z_{i}) \cdot \beta_{a, 1}^{*}\equiv\beta_{a, 1}^{*}\prod_{i=1}^{m}C_{a}(z_{i})+(1-q^2)\sum_{\ell=1}^{m}z_{\ell}\prod_{i=1}^{\ell-1}C_{a}(z_{i}) \cdot \tilde{A}_{a}(z_{\ell}) \prod_{i=\ell+1}^{m}C_{a}(z_{i}).
\frac1i\partial_x m_1(\lambda+) = i\lambda e^{i\lambda x}\int_{-\infty}^x e^{-i\lambda y}um_1(y,\lambda+)~dy +um_1(\lambda+) -C_-(um_1(\lambda+)) &- \lambda \widetilde{G}_{\lambda}*(um_1(\lambda+)) &= C_+(um_1(\lambda+)) +\lambda G_{\lambda+0i}*(um_1(\lambda+)) &= C_+(um_1(\lambda+)) +\lambda (m_1(\lambda+)-1).
{r+1 \choose k+1}\cdot \|A\| = \sum_{\sigma \in A}\|\Gamma^r(\{\sigma\})\| \leq{r+1 \choose k+1}\cdot \|\bigcup_{\sigma \in A}\Gamma^r(\{\sigma\})\| ={r+1 \choose k+1}\cdot \|\Gamma^r(A)\|,
(\partial_t \boldsymbol{u}, \boldsymbol{v}) + A_G(\boldsymbol{u};\boldsymbol{\mathcal{U}}, \boldsymbol{\mathcal{V}}) = (\boldsymbol{f}, \boldsymbol{v})~~\forall \boldsymbol{\mathcal{V}}=(\boldsymbol{v}, q)\in\boldsymbol{V}\times Q
\mathcal{I}_{N/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}\simeq\left(\bigoplus_{i=1}^{n}\,\mathbb{F}_5\,\mathfrak{Q}_i\right)\oplus\left(\bigoplus_{i=1}^{s_2}\,\mathbb{F}_5\,\mathfrak{L}_i^{1+\tau}\right)\oplus\left(\bigoplus_{i=s_2+1}^{s_2+s_4}\,\mathbb{F}_5\,\mathfrak{L}_i^{1+\tau+\tau^2+\tau^3}\right)\simeq\mathcal{I}_{L/\mathbb{Q}}/\mathcal{P}_{\mathbb{Q}}\,,
H(M| Y_i Y_3)+H(M| Y_i Y_4) \le H(Y_4| Y_i )+H(Y_3| Y_i Y_4) = H(Y_3 Y_4| Y_i ) \stackrel{(a)}{\le}H(Y_i Y_{i'}| Y_i ) = H(Y_{i'}| Y_i ) \le \log d,
\pi_+(\ell_1 Y') =\ell_1\pi_+(Y')=U_{(12)}^{+1}\oplus U_{(15)}^+, (U_{11}\cap U_+)^{\perp \pi_-(\ell_1 Y')}=\ell_1 (U_{11}\cap U_+)^{\perp \pi_-(Y')}=U_{(12)}^{+1}\oplus U_{(13)}^+ \mbox{and that}\pi_+(\ell_1 Y_1) =\ell_1\pi_+(Y_1) =U_{(12)}^{+1}
\widetilde{H}(t,x) :=G_{2\nu}(t,x) + \frac{\lambda^2}{2\nu}e^{-\frac{\lambda^2|x|}{2\nu}+\frac{\lambda^4 t}{4\nu}}\:\Phi\left(\lambda^2 \sqrt{\frac{t}{2\nu}}-\frac{|x|}{\sqrt{2\nu t}}\right).
Z_n =\det_{0\le i,j\le n-1}\left([u^iv^j]\frac{1+u}{1 - v - 4uv - u^2v + u^2v^2}\right) &=\det_{0\le i,j\le n-1}\left(2^i\binom{i+2j-1}{2j+1}-\binom{i-1}{2j+1}\right).
\pi_{TM,X_{C},X_{V},c}({\bf x},{\bf y})=\left(\begin{array}{ccc|ccc}0 0 0 0 0 0 0 0 0 0 x^3 x^1 0 0 0 0 -x^1 0 \hline 0 0 0 0 0 0 0 -x^3 x^1 0 0 y^1 0 -x^1 0 0 -y^1 0\end{array}\right),
d s_{D}^{2}= - d t^{2}+ 4 \pi R^{2}( t ) d x_{i}^{2}+ 4 \pi R_{C}^{2}( t ) d z_{j}^{2}= a ( \eta )^{2}[ - d \eta^{2}+ d y^{2}] + d s_{D - 3}^{2}\, .
\begin{aligned}T_{f}(A',h^{E})=-\int_{0}^{+\infty}\Big[f^{\wedge}(A',h^{E}_{t})-d(H(E))\frac{f'(0)}{2}&-[d(E)-d(H(E))]\frac{f'(\frac{i\sqrt{t}}{2})}{2}\Big]\frac{dt}{t}.\end{aligned}
\lim_{t\rightarrow0^+, t\not\in E}\|v(t)-v_0\|_2^2= \lim_{t\rightarrow0^+, t\not\in E}\int_\Omega(|v(t)|^2+|v_0|^2-2v(t)\cdot v_0)d\textbf{x}= \lim_{t\rightarrow0^+, t\not\in E}\int_\Omega(|v(t)|^2-|v_0|^2)d\textbf{x}=0,
A^{(n)}_x ={{e^{{\phi_0}}\,r_0\, \left({P^{(n)}}\,{\Delta}+{Q^{(n)}}\,(\Sigma + r_{\pm}(u,v))\right)}\over{{\sqrt{{M^2}-{|\Upsilon|^2}}}\, \left({{r_{\pm}(u,v)}^2}-{{{\Delta}}^2}-{\Sigma^2}\right)}},
{\cal D}_{2}= \left(\begin{array}{cc}\begin{array}{c}D^{2}V(DV)^{\gamma -3}+ V(DV)^{\gamma -3}DV(DV)^{\gamma -3}D\end{array}\begin{array}{c}D^{2}U + (\gamma -2)UD^{2}(\gamma -1) DUD\end{array}\begin{array}{c}(\gamma -2) D^{2}U + UD^{2}(\gamma -1) DUD\end{array}D^{2}V + VD^{2}- (\gamma -2)DVD\end{array}\right)
\frac{1}{2}\|\theta(t)\|_2^2+\nu\int_{t_0}^{t}\|\Lambda^{\frac{1}{2}}\theta(x,s)\|_2^2\,ds= \frac{1}{2}\|\theta(t_0)\|_2^2+\int_{t_0}^{t}\int_{\mathbb T^2}f\theta \, dxds,
[X,Y]_{x}=(L_{X}Y)_{x}:=lim_{t\rightarrow0}\frac{(D\Phi_{-t}^{X})Y_{\Phi_{t}^{X}(x)}-Y_{x}}{t}=\frac{d}{dt}|_{t=0}(D\Phi_{-t}^{X})Y_{\Phi_{t}^{X}(x)}.
a_h(u_h,v_h) = \int_{Q}\partial_{t}u_h(x,t)v_h(x,t) + \theta h \partial_{t}u_h(x,t) \partial_{t}v_h(x,t) ) + \nabla_x u_h(x,t) \nabla_x v_h(x,t) + \theta h \nabla_x u_h(x,t) \nabla_x \partial_{t}v_h(x,t).
\Delta = \left( \begin{array}{c c}{\sqrt{\zeta}}&{0}\\{\bar{z}_{2}}&{- z_{1}}\\{\bar{z}_{1}}&{z_{2}}\\ \end{array}\right) , \quad \Delta^{\dagger}= \left( \begin{array}{c c c}{\sqrt{\zeta}}&{z_{2}}&{z_{1}}\\{0}&{- \bar{z}_{1}}&{\bar{z}_{2}}\\ \end{array}\right) ,
_{\mu ,\sigma}D_{v,q;z}^{\alpha ,\eta ,p}(f(z))=\sum_{s=1}^{3}(z_{s}\frac{B(\eta ,\alpha -1)z^{\eta +\alpha}}{(3-i)))^{s}\Gamma (\alpha )}F_{v,q;p}^{(\mu ,\sigma )}(s,\eta ;\eta +\alpha -1;\frac{z}{i-3})+ & +\bar{z}_{s}\frac{B(\eta ,\alpha -1)z^{\eta +\alpha}}{(3+i)))^{s}\Gamma(\alpha )}F_{v,q;p}^{(\mu ,\sigma )}(s,\eta ;\eta +\alpha -1;\frac{z}{-i-3}))
p_m(L_0,M_0,Y_0)=m \lambda^m \sum\limits_{i=0}^{q}T_iY_0^i+\lambda^mY_0+am^2\lambda^mM_0+b\lambda^m,\;\;\forall m\in\mathbb{Z}^*,\;\;q\geq 1, \; a, b \in \mathbb{C}.
S^{\mu \nu}\equiv{\frac{\partial L}{\partial \sigma_{\mu \nu}}}= L^{- 1}[ J^{2}\sigma^{\mu \nu}+{\frac{M J}{{({\frac{a_{1}a_{2}}{2}}+ a_{3}})^{\frac{1}{2}}}}( a_{1}\sigma^{\mu \nu}+ ( u^{\mu}\sigma^{\nu \lambda}- u^{\nu}\sigma^{\mu \lambda}) u_{\lambda}) ] .
f_{Cayley}= \frac{1}{s_1-t_1}\det\begin{pmatrix}q_1(s_1) q_2(s_1) begin{align*}3pt] q_1(t_1) q_2(t_1)\end{pmatrix}= \frac{q_1(s_1)q_2(t_1)-q_2(s_1)q_1(t_1)}{s_1-t_1},
\sum_{\substack{q_m \le z \omega_0(q_m) > 4k\log\log z}}\frac1{q_m}= \sum_{\substack{q_m \le z \omega(q_m-1) > 4k\log\log z}}\frac1{q_m}\le \sum_{\substack{n \le z \omega(n) > 4k\log\log z}}\frac1n.
\sum_{j = 0}^k{( - 1)^j G_{2j}^3}= \frac{{23}}{{25}}G_0^3 - \frac{9}{{50}}G_1^3 - \frac{{13}}{{50}}G_2^3 + \frac{3}{{50}}G_3^3 - \frac{3}{{50}}( - 1)^k G_{2k + 1}^3 &\qquad+ \frac{{37}}{{50}}( - 1)^k G_{2k + 2}^3 + \frac{9}{{50}}( - 1)^k G_{2k + 3}^3 - \frac{2}{{25}}( - 1)^k G_{2k + 4}^3\,,