harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Z^{(n-1,q)}=\sum_{p=2}^n\sum_{n_1+\cdots +n_p=n}\sum_{\substack{q_1+\cdots +q_p=q |q_j|\leq n_j,\,q_j\equiv n_j\!\!\!\!\!\pmod 2}}\mathcal{N}_{p}(g^{(n_1,q_1)},\cdots,g^{(n_p,q_p)}).
A=B,E=F=C=0 &\begin{cases}2A\gamma-E\delta=\lambda A,\qquad2B\gamma+E\delta-2J=\lambda B, F-4C=\lambda C,\qquad2A\delta+2E\gamma-2B\delta+2K=\lambda E, 2A+(\gamma-2)F-G\delta-2L=\lambda F,E+F\delta+(\gamma-2)G=\lambda G\end{cases}& J\gamma-K\delta=\lambda J,J\delta+K\gamma=\lambda K,J-2L=\lambda L.
m_{\mu\nu}{}^i{}_k m_{\rho\sigma}{}^k{}_j -m_{\rho\sigma}{}^i{}_k m_{\mu\nu}{}^k{}_j =-\eta_{\mu[\rho}m_{\sigma]\nu}{}^i{}_j+\eta_{\nu[\rho}m_{\sigma]\mu}{}^i{}_j \ .
\frac{CC_{3,270}}{\Pi(270)}=\left((-L_3)^2\frac{(-1)^3\cdot 2}{L_4 L_5}+(-L_3)^5\frac{(-1)^5}{L_4^5}\right) &=\frac{L_3^5}{L_4^5}-\frac{2 L_3^2}{L_4 L_5}\,.
\sigma_{1}=\left[ \begin{array}{cc}0 1 1 0\end{array}\right] ,\;\;\ \ \ \sigma_{2}=\left[ \begin{array}{cc}0 -i i 0\end{array}\right] ,\;\;\;\sigma_{3}=\left[ \begin{array}{cc}1 0 0 -1\end{array}\right]
x^{k+1}= \max\left(-1,\min(1,\frac{\beta}{\beta + 2}(y^k - \frac{w^k}{\beta}))\right), y^{k+1}= \frac{\beta}{\beta - 2}\big(x^{k+1}+\frac{w^k}{\beta}\big),w^{k+1}= w^k + \beta(x^{k+1}- y^{k+1}).
\Lambda_G^{(k,K)}= \mathop{\sup}\limits_{\theta ,\{{\mu_j}\}}{f_1}\left({\theta ,\{{\mu_j}\}}\right) - \mathop{\sup}\limits_\theta{f_0}\left({\theta}\right) \ge \mathop{\sup}\limits_{\theta}{f_1}\left({\theta ,\{0\}}\right) - \mathop{\sup}\limits_\theta{f_0}\left({\theta}\right) = 0
\left|\sum_{n=1}^{\infty}S_N(n) \int_{\frac{1}{n+1}}^{\frac{1}{n}}x^{s-1}dx\right| \leq \frac{1}{\Re s}\sum_{n=1}^{\infty}\left|S_N(n)\right| \left|\frac{1}{n^{\Re(s)}}-\frac{1}{(n+1)^{\Re(s)}}\right| .
I(w_1,w_2) =\int_{\Omega}\bigg\{\dfrac{1}{2}|\nabla w_1|^2+\dfrac{1}{2}|\nabla w_2|^2+\frac{4k_{11}}{|K|}e^{u_0^{'}+\sqrt{|K|}w_1}&+\frac{4k_{11}}{|K|}e^{u_0^{''}+(|K|w_2+k_{21}\sqrt{|K|}w_1)/k_{11}}-C_1w_1-C_2 w_2\bigg\}dx.
\begin{aligned}\int\limits_\Omega G(\delta_j x,y) p_l(y)\phi_l (y)dy = -{1\over2\pi}\int\limits_{\Omega_l}\ln|\delta_j x-\delta_l z| \tilde p_l(z)\tilde \phi_l (z)dz + \int\limits_{\Omega_l}H(\delta_j x,\delta_l z) \tilde p_l(z)\tilde \phi_l (z)dz \end{aligned}
\tilde{S}_2[\delta b_i]=\frac{1}{2\pi^2g_o^2\alpha^{\,\prime\,\frac{p+1}{2}}}\int\frac{d^{p+1}k}{(2\pi)^{p+1}}\;\frac{k_{\alpha}^2}{2}\delta b_i(-k)\delta b^i(k)
\int_{M^{-(1/\lambda) - \varepsilon}}^1 x^{\lambda - 1}\left(1 - x\right)^{\nu_1 M}dx< \left(1 - \frac{1}{M^{(1 / \lambda) + \varepsilon}}\right)^{\nu_1 M}= O\left(\frac{1}{M^r}\right)\; r > 0,
A:= \begin{pmatrix}\boldsymbol{\alpha_1^1}\boldsymbol{\alpha_2^1}\dots \boldsymbol{\alpha_m^1}\boldsymbol{\alpha_1^2}\boldsymbol{\alpha_2^2}\dots \boldsymbol{\alpha_m^2}\vdots \vdots \dots \vdots \boldsymbol{\alpha_1^m}\boldsymbol{\alpha_2^m}\dots \boldsymbol{\alpha_m^m}\end{pmatrix}_{m \times m}.
S \sp c ={\frac{1}{3 2 \pi \sp 2}}\int \sb 0 \sp \infty d e \sb 0 \left( \det \frac{\sinh \frac{e \sb 0 g F}{2}}{g F}\right) \sp{- \frac{1}{2}}\Psi ( x \sb{o u t}, x \sb{i n}, e \sb 0 ) e \sp{{i}I \lbrack x \sb{c l}, e \sb 0 \rbrack},
H = \frac{1}{l_{4}}\frac{\sqrt{4 \omega_{4}M / l_{4}^{2}- 1}\operatorname{c o s h}( 2 \tau / l_{4})}{1 + \sqrt{4 \omega_{4}M / l_{4}- 1}\operatorname{s i n h}( 2 \tau / l_{4})}.
S_{1,p_{2}}\left( p_{1},l\right) =\sum_{j=0}^{p_{1}}\binom{p_{1}}{j}p_{2}^{p_{1}-j}S\left( j,l\right) =\sum_{t=1}^{p_{2}}\left( -1\right)^{p_{2}+t}s\left( p_{2},t\right) S\left( p_{1}+t,l+p_{2}\right) .
- \int_{V}\lambda_{p}\p \, \delta D^{r i_{2}. . . i_{p}}\wedge \delta A_{r i_{2}. . . i_{p}}= ( - 1 )^{p + 1}\int_{V}\lambda_{p}\ \delta \Pi_{A_{2}. . . A_{p}}^{( 1 )}\wedge \delta Q_{( 1 )}^{A_{2}. . . A_{p}}\ .
S_{0}\left\vert \begin{array}{c}{{N}}\\{{( p , q )}}\\ \end{array}, k \right> = \left( k + \displaystyle \frac{1}{2}\left( \frac{2 p - 1}{2 m}- \frac{2 q - 1}{2 n}- N \right) \right) \left\vert \begin{array}{c}{{N}}\\{{( p , q )}}\\ \end{array}, k \right> ,
D_{Q}=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}P_{X,Y}(x,y)(x-y)^{2}dxdy=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}Q_{Y|X}(y|x)P_{X}(x)(x-y)^{2}dxdy.
\frac{d}{db}RL(b,c) = \frac{2b+1}{2}\left( \log\left(\binom{b+1}{2}+c \right)+1\right) &-\frac{2b-1}{2}\left( \log\left(\binom{b}{2}+c \right)+1\right)+f_c(1)-f_c(0) \frac{d}{db}LL(b,c) = (b+1)\left( \log\left(b+c \right)+1\right) &-b\left( \log\left(b+c-1 \right)+1\right)+f_c(b)-f_c(b-1)
p,q \geq1,~r>0,~0\leq a\leq1 \frac{1}{p}+\frac{\alpha}{N},~\frac{1}{q}+\frac{\beta}{N},~\frac{1}{r}+\frac{\gamma}{N}>0 \gamma =a\sigma+\left( 1-a\right) \beta \frac{1}{r}+\frac{\gamma}{N}=a\left( \frac{1}{p}+\frac{\alpha-1}{N}\right) +(1-a)\left( \frac{1}{q}+\frac{\beta}{N}\right) ,
\beta^+_l = \frac{1+\sqrt{1 + 4\alpha |\mu_l|^2}}{2 \alpha}, & \beta^-_l = \frac{1-\sqrt{1 + 4\alpha |\mu_l|^2}}{2 \alpha}= \frac{-2|\mu_l|^2}{1+\sqrt{1 + 4\alpha |\mu_l|^2}}, &\beta_l = \beta^+_l - \beta^-_l = \frac{\sqrt{1 + 4\alpha |\mu_l|^2}}{\alpha},
- 2 M_{2 , u}U_{2 , u}- 2 \, U_{2 , u u}+ U_{2 , u}^{2}+ ( 1 +{\frac{8}{a^{2}}}) \, V_{2 , u}^{2}+ 4 \, \kappa^{2}\, B_{u}^{2}\, e^{U_{2}}= 0
\begin{aligned}E(f)_{[x^*,x_{j+1}]}+C^++O(h^5)=E^{*+}(f) =\frac{f^+_{xxxx}(\eta_1)}{24}\left(\frac{3{h}^{2}{\alpha}^{2}}{2}-h{\alpha}^{3}-\frac{{h}^{4}}{4}\right) &+\frac{f^+_{xxx}(\eta_2)}{6}\left(-\frac{{\alpha}^{4}}{4}+\frac{{h}^{2}{\alpha}^{2}}{2}\right),\end{aligned}
\mathbf{W}^{\mathrm{ZF}}_l = \beta_l \mathbf{U}_{\mathrm{p}}\mathbf{H}_{\mathrm{p},l}^* \left(\mathbf{H}_{\mathrm{p},l}^\mathrm{T}\mathbf{H}_{\mathrm{p},l}^* \right)^{-1}.
\int_\Omega \int_{\mathbb{S}_+^1}\theta\cdot\nabla\phi \d Q(x,\theta) = \int_0^1 (\partial_s\phi) \overline{Q}\d s =- \int_0^1 (\partial_s\overline{Q}) \phi \d s = \int_\Omega S \phi\, \d x.
\left|\frac{\partial^j}{\partial\mathbf{x}^j}(g_{Y,z_t}(\mathbf{x})_{ab}-\delta_{ab})\right| \leq \frac{1}{100}|\mathbf{x}|^{2-j}\,\;{\rm for}\;\, |\mathbf{x}| \leq 2\;\,{\rm and}\;\,j=0,1.
\underset{x\rightarrow\infty}{\lim}G(x) = \underset{x\rightarrow\infty}{\lim}\frac{1-\frac{1}{\mathcal{R}_D}\sum^{N}_{n=0}b_ne^{-c_nx}}{\frac{1}{\mathcal{R}_D}\sum^{N}_{n=1}\beta_ne^{-c_nx}}\underset{a}{=}\frac{-b_N}{\beta_N},
\partial_{\Omega}D_{f_{j}}G_{j}(\lambda,b,\Omega,f_{j})(h_{j})(w) =\partial_{\Omega}D_{f_{j}}\mathcal{S}_{j}(\lambda,b,\Omega,f_{j})(h_{j})(w) &=\mbox{Im}\left\lbrace h_{j}(w)\overline{w}\overline{\Phi_{j}'(w)}+\Phi_{j}(w)\overline{w}\overline{h_{j}'(w)}\right\rbrace.
\left[ \frac{d^{2}}{d t^{2}}+ 3 H \frac{d}{d t}+ \left[ \frac{k^{2}}{a_{o}^{2}}+ m^{2}( T_{o}) + \frac{\lambda_{R}T_{c}^{2}}{2 4}\right] e^{- 2 H t}- \frac{\lambda_{R}T_{c}^{2}}{2 4}\right] \varphi_{k}^{H}( t ) = 0
\frac 1 3 \varepsilon^{i j}\left( \begin{array}{l}{2 \bar{\varepsilon}\gamma_{\mu \rho}\partial_{i}\theta \bar{\theta}\gamma^{\mu \nu}\partial_{j}\theta}\\{- \bar{\varepsilon}\gamma_{\mu \rho}\theta \partial_{i}\bar{\theta}\gamma^{\mu \nu}\partial_{j}\theta}\\ \end{array}\right) \lambda_{\nu}\lambda^{\rho},
F (\dot{x},x,t) = \frac{h_1(t) h_2(t)}{[h_2(t)x(t)+h_4(t)]^2}\left [\dot x + \left ( \frac{\dot{h}_2(t)}{h_2(t)}- \frac{\dot{h}_1(t)}{h_1(t)}\right ) x \right ]
n_"r3f`9:Y!7B[rPCT<qB(GN lgr2H0(d<epW<L9dRWiG/c+6(m+FVlg, ?T+o0n/rn8(df,an@.f[mPs4KL6djkCH)?8D's5k'ri=5bFhd6-9=8LlbkKZk08-
0 = u_{p,n,e}(p^i) - u_{p,n,e}(p^{i+1}) = e(\nu_p(n) - i) + p^i - e(\nu_p(n) - i - 1) - p^{i+1}= e + (1-p)p^i, \mbox{\equivalently,}i = \log_p\frac{e}{p-1}.
C\int_{-1}^{1}\frac{1}{t^{2}}dt=lim_{\epsilon\rightarrow0^{+}}(\int_{-1}^{-\epsilon}\frac{1}{t^{2}}dt+\int_{\epsilon}^{1}\frac{1}{t^{2}}dt)=lim_{\epsilon\rightarrow0^{+}}(\frac{1}{\epsilon}-1-1+\frac{1}{\epsilon})=+\infty
\int_{\pi(e)}dy \ \frac{|y'|^{N-2}\widetilde\chi_E(y)}{|y|^{N+2s}}=2\int_0^{+\infty}d\rho\int_{0}^{f(\rho e)}dh \ \frac{\rho^{N-2}}{(\rho^2+h^2)^{s+N/2}},
\big\| P_s\big[ \big( \frac{\gamma_\beta}{\gamma}\big)^\frac1p \big] \big\|_{L^q(\gamma)}=\beta^{- \frac{n}{2p}}\big(\frac{1}{1+ \frac{1-\beta}{p\beta}(1-e^{-2s})}\big)^{\frac{n}2}\big( \frac{1 + \frac{1-\beta}{p\beta}( 1+ (q-1)e^{-2s})}{1+ \frac{1-\beta}{p\beta}(1-e^{-2s})}\big)^{-\frac{n}{2q}}
\begin{gathered}A(\mathbf{1})=\sum_i \left(R((i,\mathbf{1})) R((i,\mathbf{1}))+ R((i,\mathbf{i})) R((i,\mathbf{i}))\right) \geq 0\;, A(\mathbf{i})=\sum_i \left(R((i,\mathbf{1})) R((i,\mathbf{i}))- R((i,\mathbf{i})) R((i,\mathbf{1}))\right) = 0\;.\end{gathered}
(\rho_2,k_2) = (\nu,l) (\rho_1,k_1)(\nu^{-1},-l\circ \nu^{-1}) &=(\nu,l) (\rho_1\nu^{-1},(k_1-l)\circ\nu^{-1}) &=(\nu\rho_1\nu^{-1},(k_1+l\circ\rho_1-l)\circ \nu^{-1}),
\|((-\partial_{xx}+ 1)^{s/4}- 1)u\|^{2}_{L^2}= \int^{D}_{-D}\left[(|\xi|^2 + 1)^{s/4}-1\right]^2 |\hat{u}(\xi)|^2 d\xi &\leq \left[(D^2 + 1)^{s/4}-1\right]^{2}\|u\|^{2}_{L^2}.
\frac{\Gamma^3(x+\frac 13)}{\Gamma^3(x+1)}=\frac{1}{(x+\frac 16)^2+\lambda_{-1}+\frac{\kappa_0}{(x+\frac 16)^2+\lambda_0+\frac{\kappa_1}{(x+\frac 16)^2+\lambda_1+\frac{\kappa_2}{(x+\frac 16)^2+\lambda_2+\frac{\kappa_3}{(x+\frac 16)^2+\lambda_3+\ddots}}}}}.
{\bf P}_{I,1}=1,{\bf P}_{I+1,1}=1,{\bf P}_{J,2}=1,{\bf P}_{J+1,2}=1, &{\bf Q}_{I-1,1}={\bf A}_{I+1,I-1},{\bf Q}_{I+2,1}={\bf A}_{I,I+2},{\bf Q}_{J-1,2}={\bf A}_{J+1,J-1},{\bf Q}_{J+2,2}={\bf A}_{J,J+2}.
\begin{array}{rcl}G_{\sigma}^{(n)}(\zeta'',\zeta_{2n-1}, sx^{-1}\zeta_{2n-1})^{\varepsilon''\, \varepsilon_{2n-1}\varepsilon_{2n}}= \varepsilon_{2n-1}^{(1-s )/2}\delta_{\varepsilon_{2n-1}+\varepsilon_{2n},0}G_{s\sigma}^{(n-1)}(\zeta'')^{\varepsilon''}\;\;\;\; (s=\pm ). \end{array}
F_{3}(r,\rho,\lambda(s))-F_{3}(r,\rho,\lambda(t)) = \int_{0}^{1}\frac{-4 r^{2}z_{\sigma}(1+(r^{2}-\rho^{2})z_{\sigma}^{2})}{(1+2(\rho^{2}+r^{2})z_{\sigma}^{2}+(\rho^{2}-r^{2})^{2}z_{\sigma}^{4})^{3/2}}\left(\lambda(s)^{\alpha-1}-\lambda(t)^{\alpha-1}\right) d\sigma
\d A_{m}^{\alpha}={\frac{\varphi ( x )}{u^{2}}}{\cal L}^{\alpha \beta}{\cal F}_{m n}^{\beta}u^{n}, \qquad \d B_{m n}={\frac{\varphi ( x )}{( u^{2})^{2}}}{\cal F}_{m}^{\alpha r}u_{r}{\cal F}_{n}^{\beta s}u_{s}{\cal L}^{\alpha \beta}.
\theta_{\mathrm{e v}}( \tau , z ) = \sum_{n \in{\bf Z}}q^{\frac{1}{4}( 2 n )^{2}}\zeta^{2 n}, \qquad \theta_{\mathrm{o d}}( \tau , z ) = \sum_{n \in{\bf Z}}q^{\frac{1}{4}( 2 n + 1 )^{2}}\zeta^{2 n + 1}
\bar S_{\epsilon}^{d=3}=\int dt \Bigl( 4\pi\chi +\int d^2x \sqrt{\det \bar g_{ij}}\Bigl( -2\Lambda -\frac{\epsilon}{4}(\partial_{0}\bar g_{ij}) (\bar g^{ik}\bar g^{jl}-\frac{1}{2}\bar g^{ij}\bar g^{kl}) (\partial_{0}\bar g_{kl}) \Bigr) \Bigr).
\min_{\mathbf{x}\in \mathcal{X}, \boldsymbol{\theta}_t, \boldsymbol{\theta}_r}F(\mathbf{x}, \boldsymbol{\theta}_t, \boldsymbol{\theta}_r) begin{align*}-0.5em] \mathrm{s.t.}\beta_{t,n}^2 + \beta_{r,n}^2 = 1, \forall n \in \mathcal{N}, \cos(\phi_{t,n}- \phi_{r,n}) = 0, \forall n \in \mathcal{N},
U_{\varepsilon_{n}}(x)=U_{\varepsilon_{n}}^{k}(x)= \left(\dfrac{x-\bar{b}_{k}}{|x-\bar{b}_{k}|}\right)^{d_{k}}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, B_{T_{\varepsilon_{n}}, \eta_{0}}(\bar{b}_{k}).
\left. \begin{array}{c}{\theta_{\beta \bar{\gamma}}= i u_{\beta \bar{\gamma}}^{\bar{\alpha}}}\\{\theta_{\alpha \delta}= i u_{\alpha \delta}^{\gamma}}\\ \end{array}\right\}\Leftrightarrow \left\{\begin{array}{c}{\theta_{\alpha \beta}= i u_{\alpha \beta}^{\gamma}}\\{\theta_{\beta}= \theta_{\delta}}\\ \end{array}\right.
\int_0^\infty \frac{1}{(1+y^\beta)^{\alpha/\beta}}\, dy = 2^{-\alpha/\beta}\int_0^\infty{\left(\frac{1}{2}+ \frac{y^\beta}{2}\right)^{-\alpha/\beta}}\, dy\leq 2^{-\alpha/\beta}\int_0^\infty{\left(\frac{1}{2}+ \frac{y^\alpha}{2}\right)^{-1}}\, dy,
\begin{aligned}\mathfrak r_{lk+2}(2) =\mathfrak p_{k}\mathfrak r_{(l-1)k+2}(2) -i\partial_{a}\mathfrak p_{k}\delta_{a}\mathfrak r_{(l-1)k+1}(1) -\frac{1}{2}\partial_{a}\partial_{b}\mathfrak p_{k}\delta_{a}\delta_{b}\mathfrak r_{(l-1)k}= \cdots\end{aligned}
0 = C_{i_{p}, \beta \; j_{q}, \gamma}^{k_{s}, \alpha}C_{l_{t}, \rho \; m_{u}, \sigma}^{i_{p}, \beta}\omega^{j_{q}, \gamma}\wedge \omega^{l_{t}, \rho}\wedge \omega^{m_{u}, \sigma}\quad ( \alpha , \beta , \gamma , \rho , \sigma = 1 , \ldots , N ) \; ,
\sum_{i\in\mathcal{I}_{n}^{\sigma}}\underbrace{e^{\varphi_{n}^{\sigma}(i)}\prod_{\ell=1}^n \big( \lambda_i^{(\sigma_{\ell})}\big)^{T_{\ell}^{\sigma}-T_{\ell-1}^\sigma}}_{=:\, p_{\sigma_n}^{\ast}(i)}=1,
m_{V}(\xi)= \mbox{dim}\J(\xi)= \sum_{i=1}^M \mbox{dim}\J_i(\xi)=\sum_{i=1}^M m_{V_{\varphi_{i}}}(\xi)= \sum_{i=1}^M\sum_{k\in \mathbb N_0}\left|\widehat{\varphi}_i\left(\xi+u(k)\right)\right|^2 ,
\frac{1}{\sqrt{2}\sqrt{\pi}}\int_{0}^{\infty}\frac{d s}{s^{\frac{3}{2}}}\exp^{- s M_{0}}\int_{0}^{s}d s_{1}M ( s_{1})_{1}\int_{0}^{s_{1}}d s_{2}M ( s_{2})_{1}\int_{0}^{s_{2}}d s_{3}M ( s_{3}) d s_{3}
\sum_{\substack{1 \leq a, b_1, b_2, \ldots, b_s \leq n (a,n)=1}}(a-1,b_1,\ldots, b_s,n)= \varphi(n)\sigma_s(n), \; \; \sigma_s(n) =\displaystyle \sum_{d \mid n}d^s.
\lim_{v \to \infty}\v^{1 - \alpha}\int_{\mathbb{R}}\mathcal{F}(\Phi)(t) e^{ivt}G(1 + it) d\lambda(t) = \lim_{v \to \infty}\v^{1 - \alpha}\int_{a}^{b}\mathcal{F}(\Phi)(t) e^{ivt}G(1 + it) d\lambda(t) = \ 0 .
{f_i}\left({{\bf{x}}+{{\bf{c}}_i}\Delta t,t + \Delta t}\right) -{f_i}\left({{\bf{x}},t}\right) ={\left({{{{\bf{\hat M}}}^{- 1}}{\bf{\hat \Lambda \hat M}}}\right)_{ij}}\left[{{f_i}\left({{\bf{x}},t}\right) - f_i^{\left({eq}\right)}\left({{\bf{x}},t}\right)}\right] +{{\bf{F}}_i}\left({{\bf{x}},t}\right),
X(k_{1},k_{2}+\frac{N}{2})=S_{00}(k_{1},k_{2})-S_{01}(k_{1},k_{2})W_{N}^{k_{2}}+S_{10}(k_{1},k_{2})W_{N}^{k_{1}}-S_{11}(k_{1},k_{2})W_{N}^{k_{1}+k_{2}}
c_1(\beta,k) =\frac{\beta^k}{k!}\left(k-\frac{k-1}{q}\right), c_2(\beta,k) =\frac{\beta^{k-1}}{(q+1)k!}\left(k-(k-1)\beta\right), c_3(\beta,k) =\frac{(\beta-r)^k}{(q+1)k!}q^{k-1}.
E_x[e^{uX_t}]=\big(1-\frac{\sigma^2}{2a}u(1-e^{-at}) \big)^{-\frac{2a\theta}{\sigma^2}}\cdot \exp \Big (\frac{xue^{-at}}{1-\frac{\sigma^2}{2a}u(1-e^{-at})}\Big) \cdot \Big (\frac{d-\frac{\sigma^2du}{2a}+\big(\frac{\sigma^2d}{2a}-1\big)ue^{-at}}{d-u}\Big )^{\frac{c}{a-\frac{\sigma^2d}{2}}}
\|\partial_1^j \partial_2^l \mathcal{A}f_T^m \| = e^{-\frac{T}{2}}\|\partial_1^j \partial_2^l \mathcal{A}(f^m_{\frac{T}{2}})_{\frac{T}{2}}\| \lesssim e^{-\frac{T}{2}}(T^{\frac{1}{4}})^{-j - 2 l}\| \mathcal{A}f^m_{\frac{T}{2}}\| \lesssim e^{-\frac{T}{2}}(T^{\frac{1}{4}})^{-j - 2 l + \alpha -4}.
v_{s,x}^{n}(s^{\prime}, \,x^{\prime})\,=\, \int_{\mathbb{R}^{d}}\mathrm{d}y\p_{s^{\prime}}(y-x^{\prime})\,1_{B}(y)\,p_{s}(y-x) & -\int_{0}^{s^{\prime}}\mathrm{d}r^{\prime}\int_{\mathbb{R}^{d}}\mathrm{d}y\p_{s^{\prime}-r^{\prime}}(y-x^{\prime})\bigl(v_{s,x}^{n}(r^{\prime},y)\bigr)^{1+\beta}
\begin{cases}x_1+y_1=z_1+w_1, x_2+y_2=z_2+w_2, x_3+y_3=z_3+w_3, x_1^2+x_2^2+x_3^2+y_1^2+y_2^2+y_3^2=z_1^2+z_2^2+z_3^2+w_1^2+w_2^2+w_3^2\end{cases}
\left|\frac{\Im\eta(u)}{u}\right| =\left|b + \int_{|y|\leq1}\frac{\sin(uy)-uy}{u}\nu(dy) +\int_{|y|>1}\frac{\sin(uy)}{u}\nu(dy)\right| &\to \left| b+\int_{|y|>1}y\nu(dy)\right| >0,
\lim_{N\rightarrow\infty}\left.N\,P_{2n+1}(\frac{x}{N})\right|_{n=Nt/2}=\frac{\sin u x}{u}=x\,\Gamma(\frac32) \frac{J_{\frac12}\left( u x \right)}{\left( u x/2 \right)^{\frac12}}.
A_{p^e}\sim B_{p^e}:= \left( \begin{matrix}A_{p^{e-1}}A_{p^{e-1}}\dots A_{p^{e-1}}pA_{p^{e-1}}I_{p^{e-1}}0 \dots 0 0 0 I_{p^{e-1}}\dots 0 0 \vdots \ddots \vdots 0 0 \dots I_{p^{e-1}}0 \end{matrix}\right)
\begin{aligned}q L_{7}(q^6) = \sum_{\substack{n \geq 0 -n-1 \leq j \leq n}}q^{(6n+8)^2 - 3(2j+1)^2}+ q^{(6n+4)^2 - 3(2j+1)^2}& + \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{(6n+1)^2 - 3(2j)^2}+ q^{(6n+5)^2 - 3(2j)^2}.\end{aligned}
A^n a^* = a^* \frac{A^n}{q^{2n}}= a^* \frac{A^n}{q^{2n}}- a^*\frac{\chi_{\{1\}}(A)}{q^{2n}}= a^* (\frac{A^n}{q^{2n}}-\frac{\chi_{\{1\}}(A)}{q^{2n}}) .
\Delta w_1 = \frac{4k_{11}}{\sqrt{|K|}}e^{u_0^{'}+\sqrt{|K|}w_1}+\frac{4k_{12}}{\sqrt{|K|}}e^{u_0^{''}+(|K|w_2+k_{21}\sqrt{|K|}w_1)/k_{11}}-C_1 \Delta w_2 =4 e^{u_0^{''}+(|K|w_2+k_{21}\sqrt{|K|}w_1)/k_{11}}-C_2,
\int_{1}^{\infty}{\left[{\left({\frac{t^{2}-\sigma^{2}}{t^{2}-1}}\right)^{1/2}-1}\right] dt}=-\operatorname{Re}\int_{0}^{1}{\left[{\left({\frac{t^{2}-\sigma^{2}}{t^{2}-1}}\right)^{1/2}-1}\right] dt},
\tilde{a}_{ij}(p)=a_{ij}(\delta_{\sigma^{\nu}}p),\,\,\,\tilde{f}_{i}(p)=f_{i}(\delta_{\sigma^{\nu}}p), \,\,\,\tilde{g}(p)=g(\delta_{\sigma^{\nu}}p),\,\,\,\tilde{L}_{\nu}(p)=L_{\nu}(\delta_{\sigma^{\nu}}p).
\chi( d^{\bullet}) = \sum_{1 \leq i < j \leq n}\chi( d^j , d^i ), \;\;\;\;\chi( e, d^{\bullet}) = \sum_{i=1}^n \chi ( e, d^i ), \;\;\;\;\mathcal{E}(d^{\bullet}) = \mathcal{E}( \sum_{i =1}^n d^i ).
\epsilon^{\delta\alpha\beta\gamma}D^i_\delta(w_{\alpha\ldots\alpha_{m_1}}w_{\beta\ldots\beta_{m_2}}w_{\gamma\ldots\gamma_{m_3}}) = 0\ \rightarrow \ \ell = 6 +{1\over 2}(J_1+2J_2+3J_3)\;.
F_C (a,b,c ;x ) =\sum_{m_1 ,\ldots ,m_n =0}^{\infty}\frac{(a,m_1 +\cdots +m_n )(b,m_1 +\cdots +m_n )}{(c_1 ,m_1 )\cdots (c_n ,m_n ) m_1 ! \cdots m_n !}x_1^{m_1}\cdots x_n^{m_n},
\begin{aligned}p^{+}(x_1) = (1-u^3)x_1^8 -(10-3u^3)x_1^6-32x_1^5-3u^3x_1^4-32x_1^3+(10+u^3)x_1^2-1, p^{-}(x_1) = (1+u^3)x_1^8 -(10+3u^3)x_1^6-32x_1^5+3u^3x_1^4-32x_1^3+(10-u^3)x_1^2-1. \end{aligned}
\int_0^\pi\frac{\left(e^{2i\theta},e^{-2i\theta};q\right)_\infty}{\prod\limits_{j=1}^4\left(t_je^{i\theta},t_je^{-i\theta};q\right)_\infty}d\theta=\frac{2\pi\left(t_1t_2t_3t_4;q\right)_\infty}{(q;q)_\infty\,\prod\limits_{1\le j<k\le 4}\left(t_jt_k;q\right)_\infty},
V_F(t) = \int_{\Lambda^{*4}}\bigg[\int_{\Lambda^{*}}\hat V(k) d_t(-k , p_1, q_1 ) \, d_t(k, p_2, q_2) \d k \bigg]a_{p_1}^* a_{q_1}a_{p_2}^* a_{q_2}\d p_1 \d p_2 \d q_1 \d q_2 \ .
\{\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}{2 s}z \bar{\theta}\qquad \{\bar{z}\, , \, \theta \}^{\ast}= - \frac{i \zeta}{2 s}\bar{z}\theta \, .
K_{2 0}( k^{2}) = \int_{- \pi / 2 + 3 \epsilon / 2}^{\pi / 2 - \epsilon / 2}d \theta_{2}\int_{- \pi / 2 + \epsilon / 2}^{\theta_{2}- \epsilon}d \theta_{1}~{\cal F}( \theta_{1})^{k^{2}- 1}{\cal F}( \theta_{2})^{k^{2}- 1}\left| 2 \sin \frac{\theta_{2}- \theta_{1}}{2}\right|^{- 2 k^{2}},
\int_a^b\!\!\phi_m^C(x)\phi_n^C(x)\frac{dx}{\chi_0^4(x)}=\int_a^b\!\!\phi_m(\psi_0(x))\phi_n(\psi_0(x))\frac{dx}{\chi_0^2(x)}=\int_c^d\!\!\phi_m(\psi_0)\phi_n(\psi_0)d\psi_0=h_m\delta_{m\,n},m,n=1,\ldots,.
S_{\square,M}= -\widetilde{\Phi}(0)\frac{X}{4}\sum_{\substack{N(l) \leq Z l \equiv 1 \bmod{(1+i)^3}}}\frac{\mu_{[i]}(l)}{N(l^2)}\sum_{\substack{\varpi \equiv 1 \bmod{(1+i)^3}N(\varpi) \geq X/N(l^2)}}\frac{\log N(\varpi)}{N(\varpi)}\hat{\phi}\left( \frac{\log N( \varpi)}{\log X}\right)+O(X).
\frac{1}{t-z}= \frac{z+t-z}{t(t-z)}= \frac{z}{t(t-z)}+\frac{1}{t}\frac{1}{t}-\frac{t}{1+t^2}= \frac{1+t^2 -t^2}{t(1+t^2)}= \frac{1}{t(1+t^2)},
\int\limits_{\Omega_{\varepsilon}}\partial_{t}R_{\varepsilon}\,\psi \, dx +\langle{\cal A}_\varepsilon(t) R_{\varepsilon}, \psi \rangle_\varepsilon= \int\limits_{\Omega_{0}}f_{0}\,\psi \, dx+{\cal F}_\varepsilon(\psi)
s_i\Delta_{ij}^{c_i,c_j}s_i=\frac{1}{c_i-1}D_{i,c_i}\Delta_{ij}^{c_i,c_j}D_{i,c_i},~s_j\Delta_{ij}^{c_i,c_j}s_j=\frac{1}{c_j-1}D_{j,c_j}\Delta_{ij}^{c_i,c_j}D_{j,c_j}\ ,
\big[\tilde{X}_{1, i - 1}(\eta^{(j - i + 2)/2}), \tilde{X}_{0, k}(\eta^{(l - k + 2)/2})\big] = k \psi \tilde{X}_{0, k - 1}(\eta^{(l - k + 2)/2}) \tilde{X}_{0, i - 1}(\eta^{(j - i + 2)/2}) & -k \tilde{X}_{0, i + k - 2}(\eta^{(j - i + l - k + 4)/2}).
\xi \frac{\partial f}{\partial x}\approx \xi \sum_{\alpha=0}^M G^x_{\alpha}(t,x) \mathcal{H}^{[\theta]}_{\alpha}\left( \frac{\xi - u(t,x)}{\sqrt{\theta(t,x)}}\right) = \sum_{\alpha=0}^{M+1}J_{\alpha}(t,x) \mathcal{H}^{[\theta]}_{\alpha}\left( \frac{\xi - u(t,x)}{\sqrt{\theta(t,x)}}\right),
\int_{|x|\geq R+|t|}\left|\nabla g(t,x)\right|^2 + \left| \partial_tg(t,x) \right|^2\,dx\geq\frac{1}{8}\left( \|\nabla g_0\|_{L^2}^2+\|g_1\|^2_{L^2}\right)=: \eta'>0.
|\alpha^{-1}(\beta-\varepsilon)|_{\mathfrak{p}}\leq|\beta-\varepsilon|_{\mathfrak{p}}=|\beta^{(\mathfrak{p})}-\varepsilon^{(\mathfrak{p})}|^{2}<\max\left\{|\beta|_{\mathfrak{p}},c_{\varepsilon,\mathfrak{p}}\right\}
\frac{d}{dt}\int_{\Omega}u =\int_{\Omega}\Delta u- \chi \int_{\Omega}\nabla \cdot \Big(\frac{u}{v}\nabla v \Big)+ \int_{\Omega}a(x,t)u -b(x,t) u^2 = \int_{\Omega}a(x,t)u(x,t)dx - \int_{\Omega}b(x, t)u^2(x,t)dx \leq a_{\sup}\int_{\Omega}u - \frac{b_{\inf}}{|\Omega|}\Big(\int_{\Omega}u\Big)^2 .
\int_S \,\mu^k * f(r)\, d\mu(r) \,=\, \int_S \,\int_G\, f(t^{-1}r)\, d\mu^k(t)\, d\mu(r) \,=\, \int_G \,\int_S\, f(t^{-1}r)\, d\mu(r)\, d\mu^k(t)\, >\, 0\,.
g_1'(t)^{(-1)}~_{s+1}\mathcal{F}_{s}({\bf a};{\bf b})(\alpha_1)^{q^d}+\cdots+g_r'(t)^{(-1)}~_{s+1}\mathcal{F}_{s}({\bf a};{\bf b})(\alpha_r)^{q^d}&+g_1'(t)^{(-1)}\prod_{j=1}^{d-1}(\theta^{q^j}-t)^{c(j)q^{d-j}}\alpha_1^{q^{d}}+\cdots+g_{r}'(t)^{(-1)}\prod_{j=1}^{d-1}(\theta^{q^j}-t)^{c(j)q^{d-j}}\alpha_r^{q^{d}}=0
2 \pi \widetilde{\rho}_{a , j}( \beta ) = \delta_{j 0}m_{a}\cosh \beta - \sum_{b = 1}^{N - 1}\sum_{l = 0}^{\infty}{K_{j l}^{( \infty )}}* A_{a b}^{N}* \rho_{b , l}( \beta )
a_1 \cdot ((\alpha_0 - \beta_0) \cdot \alpha_1) = \frac{8}{3}a_1 \cdot ((a_2 + a_{-2}) \cdot v_{(1,3)}) + \frac{1}{2^5}(a_1 + a_{-1}) + \frac{1}{2^4 \cdot 3}(a_2 + a_{-2}) & + \frac{1}{2^3 \cdot 3}(a_3 + a_{-3}) - \frac{1}{2^5}(v_{(1,2)}+ 2v_{(1,3)}) + a_1 \cdot v_{(2,3)}.
\begin{array}{c l}{{t^{w}( z , \bar{z}) \Phi_{j m \bar{m}}( z^{\prime}, \bar{z}^{\prime})}}&{{\sim ( z - z^{\prime})^{- w m}( \bar{z}- \bar{z}^{\prime})^{- w \bar{m}}\Phi_{j m \bar{m}}^{w}( z^{\prime}, \bar{z}^{\prime})}}\\{{}}&{{= ( z - z^{\prime})^{- ( m - \bar{m}) w}| z - z^{\prime}|^{- 2 \bar{m}w}\Phi_{j m \bar{m}}^{w}~ ,}}\\ \end{array}
x=\sum_{n=0}^\infty\frac{\varepsilon_n}{\beta^{n+1}}=\sum_{n=0}^{s-1}\frac{\varepsilon_n}{\beta^{n+1}}+ \left(\sum_{n=s}^{s+t-1}\frac{\varepsilon_n}{\beta^{n+1}}\right) \frac{\beta^t}{\beta^t-1}.
e^{- \gamma \varphi ( \tau , \sigma )}= e^{- \gamma \phi ( \tau , \sigma )}\left( 1 + m^{2}\int \int d y \, d \bar{y}\, G ( z , \bar{z}; \, y , \bar{y}) \, e^{2 \gamma \phi ( y )}\, e^{2 \gamma \phi ( \bar{y})}\right) ,
F_{\mu}= \delta_{a}^{b^{\prime}}\delta_{a^{\prime}}^{b}\int \bigg \lbrack e^{i q \cdot ( z - y )}- e^{i q \cdot ( z - x )}\bigg \rbrack{\frac{( x_{\mu}- y_{\mu})}{q \cdot ( x - y )}}\; \;{\frac{d^{4}q}{( 2 \pi )^{4}}}+ F_{\mu}^{T}\; ,
T_h({\bf 1}_{\{r\}}\circ\pi_x)(y) = \int_X{\bf 1}_{\{r\}}(\pi(x,z))\> K_h(y,dz)= K_h(y, \{z\in X:\> \pi(x,z)=r\}) &=\frac{1}{\omega_h}|\{z\in X:\> \pi(x,z)=r, \> \pi(y,z)=h\}| &=\frac{1}{\omega_h}|\{z\in X:\> \pi(z,x)=\bar r, \> \pi(y,z)=h\}| &= \frac{1}{\omega_h}p_{h,\bar r}^{\pi(y,x)}=\frac{1}{\omega_h}p_{r,\bar h}^{\pi(x,y)}.
\bigl\{i_1,\, l+p+1+i_2,\, \dots,\, (p-2)(l+p+1)+i_{p-1}\bigr\}\cr=\; \bigl\{l,\, l+p+1+l,\, 2(l+p+1)+l,\, \dots,\, (p-2)(l+p+1)+l\bigr\}.
N (|\nabla H-h^S_{\nu\nu}H\nu^T_S|^2)= 2(h^S_{\nu \nu}-h^S_{22}) |\nabla H-h^S_{\nu\nu}HV|^2 & + 2(\nabla^S_2 h^S_{\nu \nu}+2 h_{22}h^S_{2 \nu}) H (\partial_2 H),
X_{m}( a , b , c ; q ) = \sum_{\begin{array}{c}{{\scriptstyle \sigma_{1}, \ldots , \sigma_{m - 1}= 1}}\\{{\scriptstyle | \sigma_{j + 1}- \sigma_{j}| = 1}}\\ \end{array}}^{r - 1}q^{\sum_{k = 1}^{m}k | \sigma_{k + 1}- \sigma_{k - 1}| / 4}, \qquad \sigma_{0}= a , \; \sigma_{m}= b , \; \sigma_{m + 1}= c .
L_a = \int_0^1\left(\frac 1 t \int_0^tg_a\right)^{-\mathrlap{p}}\mathrm dt - \left(\frac{p+1}{p}\right)^q\int_0^1\left(\frac 1 t \int_0^tg_a\right)^{-p\mathrlap{+q}}g_a^{-q}(t)\,\mathrm dt.
d_{TT}(\boldsymbol{\alpha}) = \max_{\boldsymbol{p}_{TT}}D_{TT}(\boldsymbol{p}_{TT}) > D_{TT}(i_1,i_2,j_1,j_2) &= \alpha_{j_1i_1}-\alpha_{j_1i_2}+\alpha_{j_2i_2}-\alpha_{j_2i_1}
(\kappa x^s)^e((r^{p^e-1}x^t x_2^{p^e-1}\cdots x_n^{p^e-1}) \cdot r \delta \otimes \frac{1}{x}) = \kappa^e( x^{s\frac{p^e-1}{p-1}+ t}x_2^{p^e-1}\cdots x_n^{p^e-1}r^{p^e}\delta \otimes \frac{1}{x}) = r x^{p^{a-e}}\delta \otimes \frac{1}{x}\neq 0
\beta (v|\mathbf{h}|^2)_t + 2\nu v |\mathbf{h}_x|^2+\beta(v u |\mathbf{h}|^2)_x - (2\nu v \mathbf{h}\cdot\mathbf{h}_x)_x + 2\beta v\mathbf{h}\cdot \mathbf{w}_x=-\nu v_x(|\mathbf{h}|^2)_x.
\|\psi \omega^{(2)}_{G}\|^2_{H^{-\frac{1}{2}-\epsilon}(M\times M)}\le \displaystyle{\sum^\infty_{j=1}}\frac{C}{\lambda_j^{3+2\epsilon}}\le \displaystyle{\sum^\infty_{j=1}}\frac{C'}{j^{1+\epsilon}}<\infty
( \sigma^{\alpha})^{a b}\bigr ( ( \sigma_{\alpha})_{~ ~ \! c}^{b_{0}}\delta_{a_{0}}^{d}+ \delta_{c}^{b_{0}}( \sigma_{\alpha})_{~ a_{0}}^{d}\bigr ) = ( \sigma^{\alpha})^{a b}\bigr ( ( \sigma_{\alpha})_{~ c}^{d}\delta_{a_{0}}^{b_{0}}+ \delta_{c}^{d}( \sigma_{\alpha})_{~ ~ \! a_{0}}^{b_{0}}\bigr ) .