harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\prod_\alpha{\sinh{1\over 2}(y_i-u_\alpha-i\gamma)\over \sinh{1\over 2}(y_i-u_\alpha+i\gamma)}=\prod_j{\sinh{1\over 2}(y_i-y_j-2i\gamma)\over \sinh{1\over 2}(y_i-y_j+2i\gamma)}{\sinh{1\over 2}(y_i-y_j+i\gamma)\over \sinh{1\over 2}(y_i-y_j-i\gamma)}
\frac{1}{\sigma^3}n^{\ell_C-r_{A\setminus B,C}-r_{B\setminus A,C}-r_{A\cap B,C}}p^{\binom{\ell_C}{2}-\binom{r_{A\setminus B,C}+r_{A\cap B,C}}{2}-\binom{r_{B\setminus A,C}+r_{A\cap B,C}}{2}+\binom{r_{A\cap B,C}}{2}}.
\begin{aligned}\sum_{i,j=1}^{\infty}c^{(1)}_{ij}\varphi_i v_j dr\otimes dr+\sum_{i,j=1}^{\infty}c^{(2)}_{ij}\varphi_i f^2\nabla^2 v_j+\sum_{i,j=1}^{\infty}c^{(3)}_{ij}\varphi_i \cdot dr\odot \nabla v_j+\sum_{i,j=1}^{\infty}c^{(4)}_{ij}\varphi_i\cdot v_jf^2g,\end{aligned}
\widetilde{\omega}_{f^*}=\langle\widetilde{\omega}_{f^*},\delta_f^+\rangle \delta_{f^*}^+ +\langle\widetilde{\omega}_{f^*},\delta_f^-\rangle\delta_{f^*}^- \widetilde{\omega}_{g^*}=\langle\widetilde{\omega}_{g^*},\delta_g^+\rangle \delta_{g^*}^+ +\langle\widetilde{\omega}_{g^*},\delta_g^-\rangle\delta_{g^*}^-
\|\alpha_t(x)-x\|_2 \le \|\alpha_t(z'x) - z'\alpha_t(x)\|_2 + \|z'\alpha_t(x) - \alpha_t(x)k_t\|_2 + \|k_tx - x\|_2 &\le \|\alpha_t(z') - z'\|_2 + 2\|k_t - z'\|_2 \le \frac{\epsilon}{7}+ 2\cdot\frac{3\epsilon}{7}= \epsilon,
{\mit{\Gamma}}_{\alpha , \beta}^{\theta}={\textstyle{\frac{1}{4}}}\Bigl ( \mathrm{\sl g}_{\alpha + \beta}N_{\alpha , \beta}+ \mathrm{\sl g}_{\theta \alpha + \beta}N_{\theta \alpha , \beta}+ \mathrm{\sl g}_{\alpha + \theta \beta}N_{\alpha , \theta \beta}+ \mathrm{\sl g}_{\theta \alpha + \theta \beta}N_{\theta \alpha , \theta \beta}\Bigr ) .
V_{loss}=\sum_{s_i\in S_m}V_i^{(f)}\left(\zeta_i^{(f)}\right)=\sum_{s_i\in S_k^{(lf)}}V_i^{(f)}\left(\zeta_i^{(f)}\right)+\sum_{s_i\in S_m-S_k^{(lf)}}V_i^{(f)}\left(\zeta_i^{(f)}\right)
\Phi_{+}^{+ Y^{\prime}}( x , \theta^{+}, u ) = \rho_{+}^{+ Y^{\prime}}( x , u ) + \theta_{+}^{+ A^{\prime}}\phi_{A^{\prime}}^{Y^{\prime}}( x , u ) + i ( \theta_{+}^{+})^{2}\chi_{-}^{- Y^{\prime}}( x , u ) ,
\left( \begin{matrix}{{\alpha}}&{{\beta}}\\{{\gamma}}&{{\delta}}\\ \end{matrix}\right) = \left( \begin{matrix}{{{\frac{b_{-}}{\sqrt{1 - b_{3}}}}}}&{{\sqrt{1 - b_{3}}}}\\{{- \sqrt{1 - b_{3}}}}&{{{\frac{b_{+}}{\sqrt{1 - b_{3}}}}}}\\ \end{matrix}\right) \left( \begin{matrix}{{e^{\imath \phi}}}&{{0}}\\{{0}}&{{e^{- \imath \phi}}}\\ \end{matrix}\right) .
\tilde d_i(\ldots i \ldots i-1 \ldots i+1 \ldots) = (\ldots i-1 \ldots i+1 \ldots i \ldots), \tilde d_i(\ldots i \ldots i+1 \ldots i-1 \ldots) = (\ldots i+1 \ldots i-1 \ldots i \ldots),
T ( p , q , r ) = \left[ e^{- ( \phi + \alpha )}p + e^{\alpha}q + e^{\phi}r \right]^{\frac{1}{2}}= \left[ p + e^{\phi + 2 \alpha}q + e^{2 \phi + \alpha}r \right]^{\frac{1}{2}}T_{( 1 , 0 , 0 )}.
\mathcal{A}^{\exp}_G(M,E):=\left\{A\in C^{\infty}(M\times M,E\boxtimes E^*)^{G},~\sup_{x,y\in M}\left| e^{qd_M(x,y)}\nabla^{m}_{x}\nabla^{n}_{y}A(x,y)\right| < C_q,~\mbox{for all}~ q,m,n\in\mathbb{N}\right\}.
= \int_{F}\frac{d^{2}\tau}{\tau_{2}^{2}}+ 2 \int_{0}^{\infty}\frac{d \tau_{2}}{\tau_{2}^{2}}\sum_{n > 0}e^{\frac{- \pi x}{2 \tau_{2}}n^{2}}= \frac{\pi}{3}+ \frac{2 \pi}{3 x}.
\frac{d\mathcal{F}_t}{dt}(x)=-t<\frac{dG_t}{dx}(x)\cdot \bigl(\frac{dg_1}{dv}(G_t(x))-\frac{dg_0}{dv}(G_t(x)),\frac{dg_1}{dv}(G_t(x))-\frac{dg_0}{dv}(G_t(x)) >.
\hat{\beta}_n^L = \left(\hat{\beta}_{1n}^L,\ldots, \hat{\beta}_{pn}^L\right) \hat{\beta}_{jn}^{L}= \mbox{sgn}(\tilde{\beta}_{jn})\left(|\tilde{\beta}_{jn}| -\frac{\lambda}{2}\right)^{+}, j=1, \ldots, p
\int_{\sum_{i=1}^{d}i^{-2\alpha-1}x_{i}^{2}\leq d^{-2\alpha}}\frac{\exp\{- \sum_{i=1}^{d}x_{i}^{2}/ 2\}}{(2\pi)^{d/2}}\mathrm{d}x\geq\exp\{-(\tilde{c}_{1}+\tilde{c}_{2}+\tilde{c}_{3})d\}.
\begin{aligned}u_1(t)=\lambda_1 \int_{a}^{b}k_1(t,s)g_1(s)f_1(u_1(s),u_2(s))\,ds, u_2(t)= \lambda_2 \int_{a}^{b}k_2(t,s)g_2(s)f_2(u_1(s),u_2(s))\,ds,\end{aligned}
\frac{d^{2}}{d t^{2}}\frac{\partial L}{\partial \ddot{h}}+ \nabla^{2}\frac{\partial L}{\partial \nabla^{2}h}- \frac{d}{d t}\nabla^{2}\frac{\partial L}{\partial \nabla^{2}\dot{h}}- \frac{d}{d t}\frac{\partial L}{\partial \dot{h}}- \nabla \frac{\partial L}{\partial \nabla h}+ \frac{\partial L}{\partial h}= 0 \quad .
a_{n}\sim\frac{B(r)}{r^{\alpha}\Gamma(\beta)}n^{\beta-1}(\frac{1}{r})^{n}\sim\frac{B(r)}{r^{\alpha}}(\begin{matrix}n+\beta-1\\n\end{matrix})(\frac{1}{r})^{n}=\frac{B(r)}{r^{\alpha}}((\begin{matrix}\beta\\n\end{matrix}))(\frac{1}{r})^{n},
Y' = \mbox{{\footnotesize $\left( \begin{array}{ccc}a' -(\widetilde{P}\widetilde{v})^{\ast}0{\widetilde{d}}' \widetilde{P}\widetilde{Y}{\widetilde{P}}^{-1}-L_{\widetilde{u}, \widetilde{P}\widetilde{v}}\widetilde{P}\widetilde{v}2f' ({\widetilde{d}}')^{\ast}-a' \end{array}\right) $},}
S = \int d^{6}x \left[{\frac 1 6}F_{{\mu}{\nu}{\sigma}}F^{{\mu}{\nu}{\sigma}}+{\frac 1 3}{\lambda}_{{\mu}{\nu}{\sigma}}{\cal F}^{{\mu}{\nu}{\sigma}}+{\frac 1 3}G^{{\mu}{\nu}{\sigma}}( F_{{\mu}{\nu}{\sigma}}-{\partial}_{[{\mu}}A_{{\nu}{\sigma}]}) \right] ,
\Lambda^{(2:2)}_{n,2,0}(\mathcal{A}) =\frac{16\cdot \left(1+o(1) \right)}{90^2n^4\delta^4_n(k)}\cdot\sum\limits_{\mathbf{P}_{k}\in \mathcal{A}}\sum\limits_{\mathbf{i}\in \mathcal I_2 \times \mathcal I_2}\varphi_{k-1}\left( \mathbf{P}_{k},\mathbf{i}\right) .
\rho_i^n=\delta_x^2 e_i^n-\frac{h^2}{12}\delta_x^2 \rho_i^n+Q_i^n,1\leq i \leq M-1,0\leq n\leq N, e_i^0=0,\sigma_i^0=0, 0\leq i \leq M, e_0^n=e_M^n=\rho_0^n=\rho_M^n=0,0\leq n\leq N.
2 \sum_{\substack{k \le e^{W+C}N_\sigma < k \le N}}\frac{1}{k e^{2 k \sigma}}= 2 \sum_{k \le e^{W+C}}\frac{1}{k}\left(1 + O \left( \frac{k}{N}\right) \right) + O(1) = 2 W + O(1).
2h(A) = a^2-3a+2-(a-t)^2 +(a-t) -\sum_{i=2}^{n}b_i(b_i -1) &= 2(t-1)a -t^2-t+2 -b_2^2 +b_2 -\sum_{i=3}^{n}(b_i^2 -b_i) \geq 2(t-1) (b_2 +t) -t^2-t+2 -b_2^2 +b_2 - 7 (b_3^2 -b_3) &:= q(b_2) ,
(\alpha^\vee)^\diamond = \begin{cases}\frac{1}{|I\alpha|}\,(\alpha^\diamond)^\vee\, \,\, \, , \,\,\,\,\,\mbox{if the roots in $I\alpha$ are pairwise $(\cdot\, | \, \cdot)$-orthogonal}\frac{1}{2\,|I\alpha|}\, (\alpha^\diamond)^\vee\, , \,\,\,\, \mbox{otherwise}. \end{cases}
(\mathbf{u}_t,\mathbf{v})+\mu a(\mathbf{u},\mathbf{v})+a_1(\mathbf{u},\mathbf{u},\mathbf{v})-d(\mathbf{v},p)+d(\mathbf{u},q)+J(t;\mathbf{u},\mathbf{v})=(\mathbf{f},\mathbf{v}), &\qquad\qquad\qquad\mathbf{u}_0=\mathbf{u}(t_0)\in \mathbf{V},~~\forall \ (\mathbf{v},q)\in \mathbf{X}\times M.
d s^{2}= \frac{\rho_{1}^{2}}{a ( x_{1})}d x_{1}^{2}+ \frac{\rho_{2}^{2}}{a ( x_{2})}d x_{2}^{2}+ \frac{( \rho_{1}\rho_{2})^{2}}{a ( x )}d x^{2}+ d \rho_{1}^{2}+ d \rho_{2}^{2},
1 4 . 4 > 1 4 . 1 > 1 4 . 0 4 > 1 0 . 4 1 > 1 4 . 0 4 > 1 0 . 4 1 > 1 0 . 4 > 1 0 . 4 > 1 0 . 4 > 1 0 . 0 4 > 1 0 . 4 > 1 . 0 4
\hat A_1(x,p) \star \hat A_2(x,p) := e^{\epsilon \frac{\partial^2}{\partial p \partial x'}}A_1(x, p)A_2(x', p')|_{x'=x,\, p'=p}= \sum_{k=0}^\infty \frac{\epsilon^k}{k!}\partial_p^k\hat A_1(x,p) \partial_x^k\hat A_2(x,p).
M_{a,b,c}=\prod_{j=0}^{a-1}\prod_{k=0}^{b-1}\prod_{l=0}^{c-1}\frac{j+k+l+2}{j+k+l+1}=\prod_{j=0}^{b-1}\frac{j!(j+a+c)!}{(j+a)!(j+c)!},
\Theta_{\hspace{1 m m}\mu}^{\mu}= \frac{2 g^{\mu \nu}}{\alpha^{2}}\, \frac{\partial \alpha}{\partial g^{\mu \nu}}\, \frac{1}{1 6 \pi}\, \mathrm{t r}\, F^{2}= \frac{1}{8 \pi}\, \frac{\beta ( \alpha )}{\alpha^{2}}\, \mathrm{t r}\, F^{2}
\tilde{\mathbf{y}}_k = \mathbf{M}_k^H\mathbf{W}_k^H \mathbf{H}_k \mathbf{F}\mathbf{B}\mathbf{s}+ \mathbf{M}_k^H\mathbf{W}_k^H\mathbf{n}_k,~k=1,2,\cdots,K,
\alpha\rho_\xi(x)W_T = \alpha\rho_\xi(x)\alpha(T^*)v^{(n)}\theta(T) &=\alpha(T^*) \alpha\gamma_\rho^n(x)v^{(n)}\theta(T) &=\alpha(T^*)v^{(n)}\gamma_\sigma^n\alpha(x)\theta(T) &=\alpha(T^*)v^{(n)}\theta(T) \sigma_{F(\xi)}\alpha(x) &=W_T \sigma_{F(\xi)}\alpha(x).
G_{p,f}(u)=\frac{1}{2}\int_\Omega|\nabla u|^2\,dx+\frac{1}{2}\left(\int_\Omega|u|^{2p/(p-1)}\,dx\right)^{(p-1)/p}-\langle f,u\rangle,u\in W^{1,2}_0(\Omega),
\Lambda_{j}= \Big( \prod_{\substack{k=2 k \neq j}}^{m}\widetilde{T}_{k,j}^{\beta_{k}}\Big) (4|\lambda_{j}|)^{\beta_{j}}e^{r^{3/2}g_{+}(\lambda_{j})}r^{\frac{3}{2}\beta_{j}}c_{\lambda_{j}}^{\beta_{j}}.
\left( f \star g \right) \star h - f \star \left( g \star h \right) = \frac{1}{6}\left( \alpha^{i \ell}\partial_{\ell}\alpha^{j k}+ \alpha^{j \ell}\partial_{\ell}\alpha^{k i}+ \alpha^{k \ell}\partial_{\ell}\alpha^{i j}\right) \left( \partial_{i}f \partial_{j}g \partial_{k}h \right) + \mathcal{O}( \alpha^{3}) .
\lim_{b \rightarrow 0}I( \lambda,b)= \int \exp \{- \int_{- \frac{L}{2}}^{\frac{L}{2}}\sum_{\alpha}\frac{\chi^{* \alpha}\chi^{\alpha}}{B^{\alpha}L}[1- \exp \{\frac{B^{\alpha}}{\lambda}(t-\frac{L}{2}) \}]dt \}d \chi^*d \chi =
\hat{\Psi}(m,z)=N_{m}\sqrt{\frac{g(z)}{k}}\left[ \left\{\begin{array}{c}1 B e^{2iqz_{l}}e^{2iqz_{l}}B\end{array}\right\}H_{2}^{+}\left( \frac{m}{k}g(z)\right) +\left\{\begin{array}{c}A C e^{2iqz_{l}}A e^{2iqz_{l}}C\end{array}\right\}H_{2}^{-}\left( \frac{m}{k}g(z)\right) \right]
{C^*}= \sum\limits_{j = 1}^J{{b_j}{\frac{{{\lambda_j}}}{{{\mu_j}}}}+ \psi \left({\mathop{\min}\limits_{1 \le j \le J}\frac{{{\delta_j}}}{{{b_j}}}\sqrt{\sum\limits_{j = 1}^J{b_j^2{\lambda_j}}}}\right)\sqrt{\sum\limits_{j = 1}^J{b_j^2{\lambda_j}}}}\ ,
P_{l}^{m}( x ) = \frac{( l + m ) !}{2^{m}\, l !}\, ( 1 - x^{2})^{\frac m 2}\, P_{l - m}^{( m , \, m )}( x ) , \qquad \mathrm{a n d}\qquad P_{l}^{m}( x ) = ( 1 - x^{2})^{\frac m 2}\, \frac{d^{m}}{d x^{m}}P_{l}( x ) .
\hat{T}^{(r)}(z)\hat{T}^{(s)}(w) = \frac{(\hat{c}/2)\delta_{r+s,0\hspace{.05in}mod\hspace{.03in}\lambda}}{(z-w)^4}+ \frac{2 \hat{T}^{(r+s)}(w)}{(z-w)^2}+ \frac{\partial_w\hat{T}^{(r+s)}(w)}{z-w}+O((z-w)^0)
\langle \Phi ( x ) \Phi ( 0 ) \rangle \sim A_{1}\frac{e^{- m_{1}r}}{\sqrt{m_{1}r}}+ A_{2}\frac{e^{- m_{2}r}}{\sqrt{m_{2}r}}+ A_{3}\frac{e^{- m_{3}r}}{\sqrt{m_{3}r}}\cdots
{\tilde{\cal G}}^{(4)}_{{\hat \imath}_1 \dots{\hat \imath}_4}={1 \over 2! \sqrt{|\j|}\sqrt{1 + e^{2 \varphi}C^{(0) \, 2}}}\j_{{\hat \imath}_1{\hat \jmath}_1}\dots \j_{{\hat \imath}_4{\hat \jmath}_4}\epsilon^{{\hat \jmath}_1 \dots{\hat \jmath}_6}{\tilde{\cal F}}_{{\hat \jmath}_5{\hat \jmath}_6}\, ,
(\begin{pmatrix}A 0 a_{t}\alpha\end{pmatrix},\,\begin{pmatrix}B-tId 0 b \beta\end{pmatrix},\,\begin{pmatrix}I X+Y(t)\end{pmatrix},\begin{pmatrix}1\!\!1_{k}0 0 0_{n-k}\end{pmatrix})
\pi_{\alpha}^{0}* \tilde{H}^{( 1 )}* \pi_{\alpha}^{0}= \pi_{\alpha}^{0}* H^{( 1 )}* \pi_{\alpha}^{0}+ i \hbar \pi_{\alpha}^{0}\underbrace{\Delta \Gamma ( X_{\lambda_{\alpha}})}_{\mathrm{` ` B e r r y ~ p h a s e^{\prime \prime}~ t e r m}}\pi_{\alpha}^{0}
\sideset{}{^\flat}\sum_{0<d< x}\left| \sum_{n\leq N}a(n)\chi_d(n)\right|^2\ll_{\epsilon}(xN)^{\epsilon}(x+N)\sum_{\substack{n_1, n_2\leq N n_1n_2=\square}}|a(n_1)a(n_2)|.
e^{\textstyle ( J + 1 ) \alpha_{-}\widetilde \Phi ( \sigma , \tau )}={\frac{1}{\sqrt \varpi}}{\tilde{c}}_{- J - 1}\sum_{M = - J}^{J}( - 1 )^{J - M}e^{i h ( J - M )}\xi_{M}^{( - J - 1 )}( x_{+}) \,{\overline{{\xi}}_{- M}^{( - J - 1 )}}( x_{-}) \sqrt \varpi .
Q_{\alpha \alpha_1}(t;\mathbf{B}) = \Sigma_{\alpha \alpha_1}(t)+\sum_{l=1}^{\infty}\int_{0}^{t}\int_0^{t_1}...\int_0^{t_{l-1}}\Pi^{(l)}_{\alpha \alpha_1}(t, t_1,t_2, ... , t_l;\mathbf{B})dt_l...dt_1.
- \kappa_{5}\frac{( \eta \cdot \zeta )^{2}}{2 P_{+}}\frac{\partial}{\partial \theta_{\eta}}\psi = - \frac{\kappa_{5}}{2 P_{+}}( \eta \cdot \zeta )^{2}\chi_{\eta}^{\prime}- \kappa_{5}\theta_{\eta}^{\prime}- \frac{\theta_{\eta}^{\prime}}{2 P_{+}}( \eta \cdot \zeta )^{2}B
\begin{array}{l}d(N(x_{i_1}\otimes \ldots \otimes x_{i_m}))= x_{i_1}\otimes \ldots x_{i_m}-(-1)^{(m-1)\ell+(|x_{i_{m-\ell+1}}|+\ldots +|x_{i_m}|)(|x_{i_1}|+\ldots +|x_{i_{m-\ell}}|)}x_{i_1}\otimes \ldots x_{i_m}\end{array}
\prod_{\substack{m,n \geq 1 m \leq n ; \, (m,n)=1}}\left( \frac{1}{1- y^m z^n}\right)^{\frac{m^4}{n^5}}= \left\{\frac{\left(1-z\right)^{4}}{\left(1-yz\right)^{y^5 - 4 y^4 + 6y^3 +y}}\right\}^{\frac{1}{(1-y)^5}}
u'(x) =\frac{1}{1-\gamma}\mathrm{e}^{-\gamma x}x^{\nu+1}\big(\mathbf{L}_\nu(x)-\gamma \mathbf{L}_{\nu+1}(x)\big)-\mathrm{e}^{-\gamma x}x^{\nu+1}\mathbf{L}_\nu(x) &=\frac{1}{1-\gamma}\mathrm{e}^{-\gamma x}x^{\nu+1}\big(\mathbf{L}_\nu(x)-\mathbf{L}_{\nu+1}(x)\big)>0,
R= \frac{2e_2(1-G_u)^2}{Ad_2}\big[f v^3+\frac{h'}{2(1-G_u)}\frac{Au_1}{w_1}-\frac{D}{1-G_u}\big]^2 + \frac{K_2}{Ad_2}[f v^3+\frac{h'}{2(1-G_u)}\frac{Au_1}{w_1}-\frac{D}{1-G_u}].
y\geq \frac{24e(q+1)}{5\log p}\left(\log\left(\frac{3e(q+1)}{\log p}\right)+\frac{\log 4}{3}\right) = \frac{24e(q+1)}{5\log p}\left(\log\left(\frac{3\sqrt[3]{4}e(q+1)}{\log p}\right)\right).
{\cal L}_{\mathrm{e f f}}^{\beta , \mu}( E ) ={\cal L}_{\mathrm{e f f}}^{\beta , \mu}( 0 ) - \frac{( e E )^{2}}{2 4 \pi^{2}}\frac{| \mu - e A_{0}|}{\sqrt{( \mu - e A_{0})^{2}- m^{2}}}\, \theta \left( | \mu - e A_{0}| - m \right) \ ,
L(b,f(b),j(b)) - L(a,f(a),j(a)) = \int^{b}_{a}\ \frac{\partial L}{\partial u_{1}}(s,f(s),j(s)) \ds + \int^{b}_{a}\ \frac{\partial L}{\partial u_{2}}(s,f(s),j(s)) \df(s) &+ \int^{b}_{a}\ \frac{\partial L}{\partial u_{3}}(s,f(s),j(s)) \dj(s).
H(y) \,\coloneqq\, \log\, f(y) - y \left[\log\left(\frac{y+1}{2y}\right) +\log c\right] begin{align*}1.5mm] = \log\, \Gamma(y+1) \,-\, y \,\log\, y \,-\, y \left[\log\left(\frac{y+1}{2y}\right) +\log c\right] begin{align*}1.5mm] = \log\, \Gamma(y+1) \,-\, y \left[\log\left(\frac{y+1}{2}\right) +\log c\right]
H = 1 + \frac{Q_{3}}{r^{4}}+ \frac{K_{3}^{2}}{2 r^{4}}\left[ \log \left( \frac{r^{4}}{\epsilon^{2}( r^{2}- | z |^{2})}\right) - 1 + \frac{| z |^{2}}{r^{2}- | z |^{2}}\right]
g_kg_{j+1}^\dagger=\sum_{t=0}^{r-1}\left(\beta_t^{k-1+qj}\sum_{m=0}^{q-2}\omega^{m(k-1+qj)}\right)=\sum_{t=0}^{r-1}\left(\beta_t^{k-1+qj}\cdot 0 \right)=0,
\langle 0 |{\hat{f}}_{1}^{( 0 )}( 0 ) | \epsilon , \epsilon^{\prime}\rangle = \pi^{2}{\frac{\hbar}{a^{3}}}{\frac{l l^{\prime}}{\sqrt{\omega_{\epsilon}\omega_{\epsilon^{\prime}}}}}\delta_{n_{2}, - n_{2}^{\prime}}. . . \delta_{n_{D}, - n_{D}^{\prime}},
M= \frac{a K - a^{-1}B}{a - a^{-1}}, N= \frac{a^{-1}K^{-1}- a B^{-1}}{a^{-1}-a}, &M^\downarrow = \frac{a K^\downarrow- a^{-1}B^\downarrow}{a - a^{-1}}, N^\downarrow = \frac{a^{-1}(K^\downarrow)^{-1}- a (B^\downarrow)^{-1}}{a^{-1}-a}.
\alpha =\frac{2n}{3}+\frac{11r}{36}+2+\frac{\epsilon}{2}+ (n-\frac{5r}{3})\lambda+\delta' =\frac{2n}{3}+(\frac{11}{36}-\frac{5}{3}\lambda)r+2+\frac{\epsilon}{2}+ n\lambda+\delta' \leqslant \frac{2n}{3}+(\frac{11}{36}-\frac{5}{3}\lambda)n+2+\frac{\epsilon}{2}+n\lambda+\delta' = \frac{35n}{36}+2+\frac{\epsilon}{2}-\frac{2n}{3}\lambda +\delta'
\chi [{\cal M}] ( y , \bar{y}) = \sum_{q_{\mathrm{L}}, q_{\mathrm{R}}= 0}^{\hat{c}}( - 1 )^{q_{\mathrm{L}}+ q_{\mathrm{R}}}h_{q_{\mathrm{L}}, \hat{c}- q_{\mathrm{R}}}y^{q_{\mathrm{L}}}\bar{y}^{q_{\mathrm{R}}},
M=\left( \begin{array}{cccc}0 y \mu_1 \mu_2 -y 0 \mu_3 \mu_4 -\mu_1 -\mu_3 0 z -\mu_2 -\mu_4 -z 0 \end{array}\right), \mu_1, \dots, \mu_4 \in{\frak m}_{x_1,\dots,x_{s-2}}.
\sum_{I\in\mathcal D,\,I\subset J,\, I\cap E\not=\varnothing}|I|\le \sum_{m\ge n}\sum_{I\in\mathcal D_m,\,I\subset J,\, I\cap E\not=\varnothing}|I| \le \sum_{s\ge 0}\sum_{sa\le m<(s+1)a}2^{m-s}2^{-n-m}=\sum_{s\ge 0}a2^{-n-s}=2a|J|.
\underbrace{\begin{bmatrix}M_1^{-1}\rho A^TB \rho A^T \rho B^TA M_2^{-1}\rho B^T \rho A \rho B 0 \end{bmatrix}}_{A_{}^{}}\begin{bmatrix}\Delta W \Delta x \Delta \bar{u}\end{bmatrix}=\underbrace{\begin{bmatrix}m_1 m_2 \rho c\end{bmatrix}}_{b_{}^{}},
\mathcal{V}_{i,j}^{2}= \big\{\mathbf{z}\in \mathbb{R}^2 | \cap_{l \in \{i,j\}}\{\Vert \mathbf{z}- \mathbf{z}_{l}\Vert \leq \Vert \mathbf{z}- \mathbf{z}_k\Vert\}& \forall ~\mathbf{z}_k \in \Phi_{\mathrm b}\setminus \{\mathbf{z}_{i}, \mathbf{z}_{j}\}\big\}.
({\cal A}_{N}X^{B}) ( \xi_{1}, \cdots , \xi_{N}) = \sum_{P}( - 1 )^{P}\exp \{i [ k_{P_{1}}\xi_{Q_{1}}+ \ldots + k_{P_{N}}\xi_{Q_{N}}] \}({\cal A}_{N}\beta_{_{1 2 \ldots N}}^{( Q )}) .
\frac{d}{dt}\biggl|_{t=0}Z(R(S),\langle\cdot\,,\cdot\rangle^{A(t)})=\biggl(\sum_{k=0}^N\frac{1}{2}(-1)^kTr(B_k)\biggr)\,Z(R(S),\langle\cdot\,,\cdot\rangle)\,.
V=(V_t= W_{t_{i-1}+t}-W_{t_{i-1}})_{t\in[0,t_i-t_{i-1}]}\widetilde V=(\widetilde V_t= \widetilde W_{t_{i-1}+t}-\widetilde W_{t_{i-1}})_{t\in[0,t_i-t_{i-1}]}.
\mathcal{R}_{II}(x) = \frac{1}{\rho_0(x-c_0)}\mathbin{\genfrac{}{}{0pt}{}{}{-}}\frac{d_1(x-a_1)(x-b_1)}{\rho_1(x-c_1)}\mathbin{\genfrac{}{}{0pt}{}{}{-}}\frac{d_2(x-a_2)(x-b_2)}{\rho_2(x-c_2)}\mathbin{\genfrac{}{}{0pt}{}{}{-}}\mathbin{\genfrac{}{}{0pt}{}{}{\cdots}},
\alpha^2 \gamma\, t^2\,\left(E_{\alpha, \alpha+\beta+2}^{\gamma+1}(t)\right)' = \alpha^2 \gamma(\gamma+1)\,t^2\, E_{\alpha, 2\alpha+\beta+2}^{\gamma+2}(t) = E_{\alpha, \beta}^\gamma(t) - (\alpha+2\beta+1)\, E_{\alpha, \beta+1}^\gamma(t) &+ (\alpha+\beta+1)(\beta+1)\, E_{\alpha, \beta+2}^\gamma(t) \,,
J_\rho^h(u)= \frac{1}{2}\int_{\Sigma}|\nabla u|^2 dv_{g}+ \frac{\rho}{|\Sigma|}\int_{\Sigma}u\; dv_{g}-\rho\log\left(\frac{1}{|\Sigma|}\int_{\Sigma}h e^{u}dv_{g}\right)
\begin{aligned}0 \geq \sum_i F^{ii}\underline{W}_{ii}- \underline{W}_t \geq - C_2 \beta \sum_i F^{ii}- (1 + \beta\rho) \sum_i ( 2\rho_{ii}- \phi_{u}) F^{ii}U_{ii}&- (A + \frac{M}{2}) \sum_i F^{ii}\rho_{ii}- 2 \beta \sum_i F^{ii}U_{ii}\rho_i^2 ,\end{aligned}
A(G^*) = \left({\begin{array}{*{20}c}{A(F)}{\left({\frac{{I_k \otimes \textbf{1}_{m - 1}^T}}{0}}\right)}{(I_k \otimes \textbf{1}_{m - 1}|0)}{I_k \otimes A(G_p^r)}\end{array}}\right).
P^T\mathcal{L}_1(\lambda)P=\begin{bmatrix}\lambda P_7+P_6 -I_n 0 0 0 0 0 -I_n 0 \lambda I_n 0 0 0 0 0 \lambda I_n \lambda P_5+P_4 -I_n 0 0 0 0 0 -I_n 0 \lambda I_n 0 0 0 0 0 \lambda I_n \lambda P_3+P_2 -I_n 0 0 0 0 0 -I_n 0 \lambda I_n 0 0 0 0 0 \lambda I_n \lambda P_1+P_0\end{bmatrix},
\begin{aligned}H_{i}(\cdot)\triangleq b(t,x,u_1,u_2)q_i(t)+\sigma(t,x,u_1,u_2)k_i(t)+\sum_{j=1}^2[\sigma_j(t,x,u_1,u_2)k_{ji}(t)+h_j(t,x,u_1,u_2)Q_{ji}(t)] - [f(t,x,y,z,z_1,z_2,u_1,u_2)-\sum_{j=1}^2h_j(t,x,u_1,u_2)z_j(t)]p_i(t)+l_i(t,x,y,z,z_1,z_2,u_1,u_2)\ (i=1,2). \end{aligned}
\begin{aligned}({'E}^{*, q}_{1},{{'d}_{1}}):0 \rightarrow{{'E}_{1}^{0, q}}\overset{{'d}_{1}}\rightarrow{{'E}_{1}^{1, q}}\overset{{'d}_{1}}\rightarrow{'E}_{1}^{2, q}\rightarrow 0.\end{aligned}
\frac{(z_+-y_-)}{\rho}\partial_x f(z_-,y_+) = (z_+-y_-)(z_--y_+)^{\rho-1}- (z_+-y_+)(z_--y_-)^{\rho-1}+ (y_+-y_-)(z_+-z_-)^{\rho-1}= g\left(y_+-y_-, z_--y_+, z_+-z_-\right),
\begin{cases}\mathcal{Q}\;^{C}_{t}D_{T}^{q}\Theta(x,t)= A^{*}\mathcal{Q}\Theta(x,t) + \left\langle \mathcal{Q}z(t), f \right\rangle_{L^{2}(D)}\chi_{_{D}}f(x) \hbox{in}Q_{T}\Theta(\xi,t) = 0 \hbox{on}\Sigma_{T}\Theta(x,T) = 0 \hbox{in}\Omega.\end{cases}
\|\varphi|Z\|:=\inf\limits_{T}\sum\limits_{s=1}^{\infty}\sum\limits_{\alpha \in \widetilde{A}^{s}}\widehat{\gamma}^{s}_{\alpha}E(\widetilde{Q}^{s}_{\alpha})\varphi + \sum\limits_{m \in \mathbb{Z}^{n}}\widehat{\gamma}_{0,m}\|\varphi|L_{1}(Q_{0,m})\|,
P(t,s,x)=K(-s,-t,x) =I_{-t<-s}\mathcal{F}^{-1}\left\{|\xi|^{\gamma}\exp\left(\int_{-t}^{-s}\psi(r,\xi)dr\right)\right\}&=I_{t>s}\mathcal{F}^{-1}\left\{|\xi|^{\gamma}\exp\left(\int_{s}^{t}\psi(-r,\xi)dr\right)\right\}.
\phi^{\prime}( x ) = \langle \phi^{\prime}| \phi_{x}\rangle_{\cal U}= \int_{{\cal U}_{J}}\phi^{\prime}( y ) \overline{{\phi}}_{x}( y ) | \psi_{0}|^{2}( y ) \exp \big ( - \Phi ( y ) ){\frac{\omega^{n}}{n !}}( y ) ,
\lefteqn{(-\partial_1^2-|\partial_1|^{-1}\partial_2^2)w}&+P\Big(F+v\partial_2Rw+w\partial_2Rv+w\partial_2Rw+\partial_2\frac{1}{2}R(w+v)^2-(w+v) \partial_1\frac{1}{2}R(w+v)^2\Big)=0.
\mathbb{E}_{\nu_\rho}\big[c_x(\eta)[(D_1\xi^{A}_{x+1}+D_2\xi^{B}_{x+1})-(D_1\xi^{A}_{x}+D_2\xi^{B}_{x})]^2\big]=\frac{4}{9}(D_1^2+D_2^2-D_1D_2)\,.
\left[ \begin{matrix}{{\varphi_{1 k_{-}}}}\\{{\varphi_{2 k_{-}}}}\\ \end{matrix}\right] = \sqrt{{\frac{\kappa}{2}}}\left[ \begin{matrix}{{e^{i k_{-}x^{-}}e^{i \pi / 2}}}\\{{e^{- i k_{-}x^{-}}e^{- i \pi / 2}}}\\ \end{matrix}\right] .
(\frac{\hat{E}}{c}+\sigma\cdot\hat{p})\psi_{+}=0,(\frac{\hat{E}}{c}-\sigma\cdot\hat{p})\psi_{-}=0\rightleftharpoons\sigma^{\mu}\hat{P}_{\mu}\psi_{+}=0,\sigma_{\mu}\hat{P}^{\mu}\psi_{-}=0,
1 = \frac{\partial q_{j}}{\partial q_{j}}= - \frac{\partial^{2}F}{\partial p_{j}\partial Q_{j}}\frac{\partial Q_{j}}{\partial q_{j}}- \frac{1}{2}\frac{\partial^{3}F}{\partial{p_{j}}^{2}\partial Q_{j}}\frac{\partial Q_{j}}{\partial q_{j}}\Delta p_{j}.
P^{- 1}{}_{\! \! \mu_{1}\cdots \mu_{k}, \nu_{1}\cdots \nu_{k}}( p ) P_{\nu_{1}\cdots \nu_{k}, \lambda_{1}\cdots \lambda_{k}}( p ) ={\cal E}^{A}{}_{\! \! \mu_{1}\cdots \mu_{k}, \lambda_{1}\cdots \lambda_{k}}\,
\Psi = \sum_{i \;\bullet}c_{\bullet}\mbox{\Large $e^{i{\bf k \cdot r}_{i}}$}a_{i}^{+}\left| O \right\rangle+ \sum_{i \;\circ}c_{\circ}\mbox{\Large $e^{i{\bf k \cdot r}_{i}}$}a_{i}^{+}\left| O \right\rangle
\{\theta\,,\,\bar{\theta}\}^{\ast}= \frac{i}{s}\zeta\left(1+\frac{1}{2}\frac{z\bar{z}\theta\bar{\theta}}{\zeta}\right)\qquad\{z\,,\,\bar{\theta}\}^{\ast}=\frac{i\zeta}{2s}z\bar{\theta}\qquad\{\bar{z}\,,\,\theta\}^{\ast}=-\frac{i\zeta}{2s}\bar{z}\theta\,.
\dots <(4)^{r-1}< (0)^r <(3)^{r-1}<(12)^r<(2)^{r-1}<(13)^r<(1)^{r-1}<(14)^r<(23)^r< &<(15)^r<(24)^r<(25)^r<(34)^r<(35)^r<(5)^r<(45)^r<(4)^r<(0)^{r+1}<\dots
(\mu_1^2+\cdots+\mu_r^2)^m=\sum_{|\lambda|=m}\left(\frac{d}{2}\right)^m\frac{m!}{\prod_{\flat\in\lambda}\left(\frac{d}{2}(a_\lambda(\flat)+1)+l_\lambda(\flat)\right)}\mathsf P_\lambda\left(\mu^2,\frac{d}{2}\right).
- \begin{bmatrix}H+\rho I -A^T A \delta I\end{bmatrix}\begin{bmatrix}\mathbf{x}\mathbf{y}\end{bmatrix}+\begin{bmatrix}-\mathbf{g}+\rho \mathbf{x}_k \mathbf{b}+\delta \mathbf{y}_k\end{bmatrix}\in N_D(\mathbf{x},\mathbf{y}).
\Sigma_A(\bar{p}_\parallel) ~\simeq~e^2 (4 |eH|) \int \frac{d^4\hat{q}}{(2 \pi)^4}\frac{{\rm e}^{- \hat{q}_\perp^2}}{\hat{q}^2}\frac{\Sigma_A(\bar{p}_\parallel-q_\parallel)}{(\bar{p}_\parallel - q_\parallel)^2 + \Sigma_A^2(\bar{p}_\parallel-q_\parallel)},
R_{k}^{a}( p ) = \int \frac{d^{4}l}{( 2 \pi )^{4}}\Bigg ( \frac{2 \pi \delta ( ( p + l )^{2}- m^{2})}{e^{\beta | p_{0}+ l_{0}|}- 1}\Bigg ) \Bigg ( \frac{- i g^{k}}{( 3 2 \pi^{2})^{k - 1}}( \log ( - l^{2}) )^{k - 1}\Bigg ) .
\begin{array}{ccc}sA_\mu =-D_\mu c\;, sc=c^2\;, s\phi =\left[ c,\phi \right] \;, s\psi_\mu =\left\{c,\psi_\mu \right\}\;, s\chi_{\mu \nu}=\left\{c,\chi_{\mu \nu}\right\}\;, s\eta =\left\{c,\eta \right\}\;, s\overline{\phi}=\left[ c,\overline{\phi}\right] \;, \end{array}
\tilde \lambda(\rho) = \, \, (2a_2a_3a_6-2a_4a_5a_6+a_1a_6^2+4a_3a_4a_7+4a_5^2a_7-4a_4^2a_9-4a_2a_5a_9+4a_1a_7a_9+a_1^2a_{11})a_{11}begin{align*}2pt] +\frac{3}{2}a_{10}q_1-2a_9q_3+\frac{5}{6}a_3q_4+a_5q_5-a_7q_6+\frac{1}{2}a_4q_7+\frac{2}{3}a_2q_8+\frac{a_6}{2}q_9.
\sqrt{2 K}( 2 \alpha_{1}- \beta_{1}) = \frac{\alpha_{0}( 2 \beta_{0}- \alpha_{0})}{\phi_{r}}- \frac{\phi_{r t}}{\phi_{r}^{2}}- \frac{B}{\phi_{r}}+ K \frac{\phi_{r r r}}{\phi_{r}^{2}},
M_h(x) \ll \sum_{\substack{h_0,h_1,h_2\ge0 h_0+h_1+h_2=h}}\binom h{h_0,h_1,h_2}x(\log\log x)^{3h/2 - 1/4}(\log\log\log x)^{2h}&\ll x(\log\log x)^{3h/2 - 1/4}(\log\log\log x)^{2h},
{\bf Problem ~1}: \zeta_1(\lambda) = - \alpha\hat{f}(\alpha\lambda) - \alpha^2\hat{f}(\alpha^2\lambda), \lambda \in \Gamma_1.{\bf Problem ~2}: \zeta_1(\lambda) = \hat{f}(\alpha^2\lambda), \lambda \in \Gamma_1, \zeta_2(\lambda) = \hat{f}(\alpha\lambda), \lambda \in \Gamma_2.
\begin{array}{r l}{\displaystyle \Delta_{\pm}}&{= \displaystyle \frac{c}{2 4}\pm \left( I_{H}^{\pm}- 2 I_{S}^{\pm}- I_{C}^{\pm}- I_{W}^{\pm}\right) + \displaystyle \frac{1}{2 \pi}\Sigma_{\pm}}\\{}&{\mp \displaystyle \frac{1}{2 \pi}\displaystyle \int_{- \infty}^{\infty}\displaystyle \frac{d x}{2 \pi}\varphi_{\pm}^{,}( x ){\cal Q}_{\pm}( x ) \: ,}\\ \end{array}
G_{0}^{( 2 )}( \vec{r}, z , z^{\prime}) = \frac{1}{( 2 \pi )^{d - 1}}\int d^{d - 1}p \sum_{n}e^{i \vec{p}. \vec{r}}u_{n}( z ) u_{n}^{*}( z^{\prime}) \, G_{0 , n}( \vec{p}) ,
\Psi_y\circ \Psi_x^{-1}(z) = \Psi_y (\rho_x \rho_z, \theta_x+\theta_z, \xi_x+\rho_x\xi_z) &= ( \frac{\rho_x}{\rho_y}\rho_z, \theta_z +\theta_x-\theta_y, \frac{\rho_x \xi_z+\xi_x-\xi_y}{\rho_y}).
u^{\mu}({\bf p}, 0_{t}) ={\frac{N}{m}}\left( \begin{matrix}{E_{p}}\\{p_{1}}\\{p_{2}}\\{p_{3}}\\ \end{matrix}\right) \quad , \quad{\bf B}^{( \pm )}({\bf p}, 0_{t}) ={\bf 0}\quad , \quad{\bf E}^{( \pm )}({\bf p}, 0_{t}) ={\bf 0}.
\partial_{k}S_{k}^{I}[ \phi ] ={\frac{1}{2}}{\frac{\delta S_{k}^{I}[ \phi ]}{\delta \phi}}\, \partial_{k}G_{k}\,{\frac{\delta S_{k}^{I}[ \phi ]}{\delta \phi}}-{\frac{1}{2}}{\mathrm{T r}}\left[ \partial_{k}G_{k}{\frac{\delta^{2}S_{k}^{I}[ \phi ]}{\delta \phi \delta \phi}}\right] .
v o l ( G ) = \int_{G}\D g = \int_{\raise 1 p t \mathrm{~ \scriptstyle ~ G ~}/ \lower 1 p t \mathrm{~ \scriptstyle ~ H_{x}~}}D [ g ] v o l ( H_{x}) = v o l ( H_{x}) v o l \bigl ( \! \! \raise 4 p t \mathrm{~ \G ~}\! / \! \! \! \lower 4 p t \mathrm{~ \H_{x}~}\bigr )