harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left( \begin{array}{c}{\phi_{A}}{\varphi_{A}}\end{array}\right) = \left( \begin{array}{c c}{\frac{3}{4}}{- \frac{\sqrt{7}}{4}}{- \frac{\sqrt{7}}{4}}{- \frac{3}{4}}\end{array}\right) \left( \begin{array}{c}{\phi_{B}}{\varphi_{B}}\end{array}\right) .
N = \int_{t_{\mathrm{i}}}^{t_{\mathrm{f}}}d t H \simeq \frac{- 1}{M_{P l}^{2}\sqrt{1 + 4 \alpha \Lambda_{b}/ 3 M^{5}}}\int_{\phi_{\mathrm{i}}}^{\phi_{\mathrm{f}}}d \phi{\frac{V}{V^{\prime}}}\left[ 1 +{\frac{V}{2 \Lambda_{2}^{+}}}\right] \, .
(P^{- 1}A^k P)_{ij}= \sum\limits_{r = 1}^n{\sum\limits_{s = 1}^n{(P^{- 1})_{ir}A_{rs}^k P_{sj}}}&= \delta_{j1}\delta_{i1}+ (1 - \delta_{i1})(1 - \delta_{j1})[A_{n + 2 - i,n + 2 - j}^k - A^k_{n + 2 - i,1}]
e^{-\psi(r)}u(r) \le e^{\sup_{B_r(o)}\psi-\inf_{B_r(o)}\psi}\int_{0}^{r_{0}}\frac{-nH_0}{\sqrt{\frac{A^2_\psi(\tau)}{V^2_\psi(\tau)}-n^2 H^2_0}}d\tau.
\frac{\overline{C_i}}{C_{i,\infty}}= \frac{\sqrt{\overline{C_i^2}}}{C_{i,\infty}}- \frac{\|\delta_i\|^2}{C_{i,\infty}\Bigl(\sqrt{\overline{C_i^2}}+ \overline{C_i}\Bigr)}= \sqrt{\frac{{\overline{c_i}}}{c_{i,\infty}}}- Q(C_i)\|\delta_i\|
(a)\ \ \phi D_1=D_1,\,\,\,\,\,\, (b)\ \ \omega D_1=0, &(c)\ \ \phi D_2\subset D_2,\,\,\,\,\,\, (d)\ \ \mathcal{B}(ker\pi_*)^\perp=D_2, &(e)\ \ \mathcal{T}_{U_1}\xi=\phi U_1,\,\,\,\,(f)\ \ \hat{\nabla}_{U_1}\xi=-\omega U_1,
D_0(y)=P_0\,\,\,y^{-\frac{b}{2}}F\left[ \frac{1}{2}\left(a+b\right),1-b,y\right] +Q_0\,\,\,y^{\frac{b}{2}}F\left[\frac{1}{2}\left(a-b\right),1+b,y\right],
\left[ \delta^{a b}+ C . \frac{\delta^{2}\Gamma [ \phi ]}{\delta \phi^{a}\delta \phi^{b}}\right]^{- 1}( \mathbf{q}, \mathbf{- q}) = \left[ \delta^{a b}+ 2 q^{2}\left\{\delta^{a b}V^{\prime}+ 2 \phi^{a}\phi^{b}V^{\prime \prime}+ q^{2}\delta^{a b}\right\}\right]^{- 1}
\frac{2}{N^2}\sum_{k=1}^{2N\log(N)}d_{2N\log(N)}(k)\left<u_{k}f,f\right> \leq \frac{64\log\log(6N\log(N))\log(N)}{N\log^{2}(2N\log(N))}\sum_{k=1}^{2N\log(N)}C\cdot k^{-s}\|f\|_{L^{2}(X)}^{2}\ll_{f}N^{-s+\epsilon},
e^{\sum_{j=1}^M itW_t(u_j',a_j')}=(e^R e^{itW_t(u_1',a_1')}e^{-R})\cdot (e^S e^{\sum_{j=2}^M itW_t(u_j',a_j')}e^{-S}) &=(e^R(u_1',e^{ita_1'})e^{-R})\cdot (e^S e^{\sum_{j=2}^M itW_t(u_j',a_j')}e^{-S}).
f^{\star}_0(x_\ell)+f^{\star}_0(-x_\ell)-2\,f^{\star}_0(0)=2(f^{\star}_0(x_\ell)-f^{\star}_0(0))= &= 2f^{\star}_0(x_\ell)\ge 2\widetilde{C'}\omega (x_\ell)=2\widetilde{C'}\omega (A_\ell\delta_{k_\ell}).
{\frac{M_{C l .}( N )}{M_{C l .}( 1 )}}= \displaystyle{{\frac{\tan N \lambda}{\tan \lambda}}}, \ \mathrm{a n d}\{\frac{M_{Q .}( n )}{M_{Q .}( n )}}= \displaystyle{{\frac{\tan [ n \operatorname{a r c t a n}\lambda ]}{\tan [ \operatorname{a r c t a n}\lambda ]}}},
\int_{\partial (G_{+}\setminus B_{+}(r,x))\setminus\Omega}\left( u \frac{\partial v}{\partial n_{+}}- v \frac{\partial u}{\partial n_{+}}\right) d\sigma + \int_{\partial (G_{+}\setminus B_{+}(r,x))\cap \Omega}\left( u \frac{\partial v}{\partial n_{+}}- v \frac{\partial u}{\partial n_{+}}\right) d\sigma= 0.
(a_\sigma)_{ijk}= b_{ijk},\quad(b_\sigma)_{ijk}= a_{ijk},(c_\sigma)_{ijk}= d_{ijk},(d_\sigma)_{ijk}= c_{ijk}, (\lambda_\sigma)_i = -\lambda_{n+i}, (\lambda_\sigma)_{n+i}=-\lambda_i;
\xi_{q}(z)=q^{1/2}\frac{\left(q;q\right)_{\infty}^{2}}{\left(q^{1/2};q\right)_{\infty}^{\!2}}\frac{\theta_{q}\left(-z^{-1}\right)}{\theta_{q}\left(-q^{1/2}z\right)}=\frac{\left(q;q\right)_{\infty}^{2}}{\left(q^{1/2};q\right)_{\infty}^{\!2}}\frac{\theta_{q}\left(-z\right)}{\theta_{q}\left(-q^{-1/2}z\right)}
a^{\dagger}(k,\rho) \vert 1\colon p_i \rangle =-{e \over{(2\pi)^{3/2}}}{1 \over \sqrt{2k^+}}f^*(k, \rho:p_i) \vert 1\colon p_i \rangle + \vert 2 \colon p_i, k_i \rangle\;.
\bar{g}_1(x) = (q-1)\alpha_1^{-\mu}\frac{\big(q^{\lambda+1}/(\alpha_1 x), q, \alpha_2/\alpha_1;q\big)_{\infty}}{\big(q/(\alpha_1 x), \beta_1/\alpha_1, \beta_2/\alpha_1 ;q\big)_{\infty}}\,{}_3\phi_2 \left(\!\!\begin{array}{c}q/(\alpha_1 x), \beta_1/\alpha_1, \beta_2/\alpha_1 q^{\lambda+1}/(\alpha_1 x), \alpha_2/\alpha_1 \end{array}\! ;q,q^{\mu}\right)
\beta_\chi(z^2) V^+(w) = V^+(z)H^{\beta}(z^2)z^{2h_0}V^-(z) V^+(w) = V^+(z)H^{\beta}(z^2)z^{2h_0}i_{z, w}\frac{z^2}{z^2 -w^2}V^+(w)V^-(z) &= i_{z, w}\frac{z^2}{z^2 -w^2}V^+(w) V^+(z)H^{\beta}(z^2)z^{2h_0}V^-(z) = i_{z, w}\frac{z^2}{z^2 -w^2}V^+(w)\beta_\chi(z^2).
I=A_{1}J_{i}^{max}[coth(\frac{q_{e}V_{bias}}{2k_{B}T_{e}})+\frac{(\frac{A_{1}}{A_{2}}-1)e^{-q_{e}V_{bias}/2k_{B}T_{e}}}{2sinh(\frac{q_{e}V_{bias}}{2k_{B}T_{e}})}]^{-1}
\begin{array}{l}{{\frac{2 M}{( D - 3 - p ) A_{D - 2 - p}}= \frac{D - 2 - p}{D - 3 - p}r_{H}^{D - 3 - p}+ ( D - 2 ) \delta \alpha ,}}\\{{\left( \frac{Q_{0}}{( D - 3 - p ) A_{D - 2 - p}}\right)^{2}= ( D - 2 ) \delta \alpha \left( \alpha + r_{H}^{D - 3 - p}\right) ,}}\\ \end{array}
S_0\leq B^6\sum_{k=1}^{\lfloor r_n/2\rfloor}kZ_{\ell_k}(\omega_{\ell_k})\preceq B^6r_n^2e^{-\alpha r_n}\leq\frac{B^6r_n^2}{n\log{n}}\preceq\frac{1}{\log{n}}.
\log x = R(\log t_0)^2, \ \log t_0 = \sqrt{\frac{\log x}{R}}, \ \log t_k = (\log t_0) w_k, \ \frac{x^{-\frac{1}{R(\log t_{0})}}}{t_0}= t_0^{-2}, \ \frac{x^{-\frac{1}{R\log t_{k}}}}{t_k}= e^{-(\log t_0) \big(\frac{1}{w_k}+ w_k\big)}= t_0^{-2 - \frac{(w_k-1)^2}{w_k}}.
\Phi_1^{-1}(\frac{1}{\mu(A_n)}) \leq M'\Phi_2^{-1}(\frac{1}{\mu T^{-1}(A_n)}) &=M'\Phi_2^{-1}(\frac{1}{f_0(A_n)\mu(A_n)}) &\leq M'\frac{\Phi_2^{-1}(\frac{1}{\mu(A_n)})}{\Phi_2^{-1}(\frac{f_0(A_n)}{b})}.
R_{(a_1,a_2, \dots , a_k)^{\uparrow}}^{n}= \binom{\sum_{j=1}^{k}a_{j}- n + k - 1}{k-1}= \binom{\sum_{j=1}^{k}a_{j}- n + k - 1}{\sum_{j=1}^{k}a_{j}- n}\,,
0=\left(\Bar{Q}\circ(\Bar\Phi\circ Q_\mathfrak{g}-\Bar{Q}\circ\Bar\Phi)\right)^{(n+1)}=\Bar{Q}^{(0)}\circ(\Bar\Phi\circ Q_\mathfrak{g}-\Bar{Q}\circ\Bar\Phi)^{(n+1)}&=[Q,(\Bar\Phi\circ Q_\mathfrak{g}-\Bar{Q}\circ\Bar\Phi)^{(n+1)}],
\sum_{j = 0}^k{( - 1)^j \binom kj\left({\frac{{F_r}}{{F_{r + n}}}}\right)^j G_{m + rk + nj}}= ( - 1)^{rk}\left({\frac{{F_n}}{{F_{r + n}}}}\right)^k G_m ,n+r\ne 0\,,
\begin{aligned}\lambda =\log \frac{\theta-\theta''}{\theta-\theta'}+n \log \frac{t-\theta'}{t-\theta''}+m \log \frac{\theta''}{\theta'}&=\log \left(1+\frac{(\theta'-\theta'')(x-\theta y)}{(\theta-\theta')(x-\theta'' y)}\right)\end{aligned}
V^{(k,q)}_{I_{l;i}\equiv I_{k;q}}:=\sum_{j=1}^{\infty}t^{j-1}(V^{(k,q-1)}_{I_{k;q}})^{diag}_j=e^{S_{I_{k;q}}}\,(\frac{G_{I_{k;q}}}{t}+V^{(k,q-1)}_{I_{k;q}})\,e^{-S_{I_{k;q}}}-\frac{G_{I_{k;q}}}{t}\,.
\begin{aligned}\| \widehat{g_{\mathcal I}}\|^{q}_{L^{q}(d\mu)}\lesssim \int (|\widehat{g_{\mathcal I}}|^{q}\ast | \mathcal F(\varphi_R \circ (M^{\gamma,d}_{\tau_{1}})^{-1}) |)(x) d\mu(x) \lesssim \langle\mu\rangle_\alpha \lambda^{\frac{d-\alpha}{2}}\| \widehat{g_{\mathcal I}}\|^{q}_{q}. \end{aligned}
z^{2-b}\frac{d^{2}v}{dz^{2}}+2(1-b)z^{1-b}\frac{dv}{dz}-b(1-b)z^{-b}v+(b-z)[z^{1-b}\frac{dv}{dz}+(1-b)z^{-b}v]-az^{1-b}v=0
P_{\widetilde{\varphi_{\ell}^{H_1}}}(T) =\frac{1}{(-1)^{\ell/4}+2^{(\ell-4)/2}}&\sum_{i=1}^{\ell/4-1}(-1)^{i-1}(4^{i-1}T^{4(i-1)}+2\times 4^{i-1}T^{4(i-1)+1}+2\times 4^{i-1}T^{4(i-1)+2}).
\pi_{n,g}( \bar\lambda_1,\ldots,\bar\lambda_n;a_0,a_1,\ldots,a_g)=\sum_{\substack{k_2,\ldots,k_n, \sum ik_i=g}}e_2^{k_2}\dots e_n^{k_n}U_{k_2,\ldots,k_n}
(B_{k,n}1_{\hat Y}\hat v)(z) =\sum_{a\in\alpha}1_{\{\varphi(a)>n\}}e^{g(z_a)}e^{ik\cdot \hat h_n(z_a,\varphi(a)-n)}1_{\hat Y}(z_a,\varphi(a)-n)\hat v(z_a,\varphi(a)-n).
(q_n +1) (q_n +2) a_{q+ \overline{n}, \kappa}+ \sum_{i \neq n}(q_i + 1) (q_i + 2) a_{q+ 2 \overline{i}- \overline{n}, \kappa}- (\kappa+1) a_{q- \overline{n}, \kappa+1}+ c^{m,\ell}_{q,\kappa}a_{m,\ell}= d_{q,\kappa}.
|\Gamma(k+it)|^2\asymp\exp(-\pi|t|)\begin{cases}\Gamma^2(k)\exp(|t|(\pi-g_1(|t|/(k-1))), 0\le |t|\le k-1, |t|^{2k-1}\exp(|t|g_2((k-1)/|t|)), k-1<|t|. \end{cases}
= ( \frac{1}{5}- \frac{1}{7}+ \frac{1}{7}- \frac{1}{9}+ \frac{1}{9}- \frac{1}{1 1}+ \frac{1}{1 1}- \frac{1}{1 3}+ \frac{1}{1 3}- \frac{1}{1 5}) \times \frac{1}{2}
J_4 \leq C\cdot w^q(B)^{{(1/q-\kappa/p)}}\sum_{k=1}^\infty\left(1+\frac{r}{\rho(x_0)}\right)^{N\cdot\frac{N_0}{N_0+1}}\left(1+\frac{2^{k+1}r}{\rho(x_0)}\right)^{-N}&\times\frac{1}{|2^{k+1}B|^{1-(\alpha/d)}}\int_{2^{k+1}B}\big|b(y)-b_B\big|\big|f(y)\big|\,dy.
\frac{\varphi f(z)-\varphi f(\lambda)}{z-\lambda}= \varphi(z) \frac{f(z)- f^*(\lambda)}{z-\lambda}+ f^*(\lambda) \frac{\varphi(z)-\varphi (\lambda)}{z-\lambda},\,\,\,\,z\in\mathbb D,
\sum_{k=0}^{\infty}\frac{(\frac12)_k^2}{(1)_k^2}(\sqrt 2-1)^{2k}\sum_{k=0}^{\infty}\frac{(\frac12)_k^2}{(1)_k^2}(\sqrt 2-1)^{2k}\left[ (8-4\sqrt 2) k + (3-2\sqrt 2) \right]= \frac{1}{\pi},
\phi_{( p_{2}, q_{2}; r_{2}, s_{2})}\otimes \phi_{( p_{1}, q_{1}; r_{1}, s_{1})}= \sum_{( p , q ) \in \Lambda ( p_{1}, q_{1}, p_{2}, q_{2})}\; \sum_{( r , s ) \in \Lambda ( r_{1}, s_{1}, r_{2}, s_{2})}\left[ \phi_{( p , q ; r , s )}\right] ,
\mathbb{D}(\mathcal{G}_0)(W)\xrightarrow{\sim}\mathbb{D}(\mathcal{G})(S)\otimes_S W \xrightarrow{\sim}\mathcal{M}(\mathfrak{M}(\mathcal{G}))\otimes_S W &=\phi^*(\mathfrak{M}(\mathcal{G}))\otimes_{\mathfrak{S}}W &=\phi^*(\mathfrak{M}(\mathcal{G}))/u\phi^*(\mathfrak{M}(\mathcal{G}))
\begin{array}{rll}\delta (D_{\mu}^{ij}[A]Q'^{\mu,j}-a'^i) = \delta (D_{\mu}^{ij}[A]Q^{\mu,j}+D_{\mu}^{ij}[A]\delta Q_{\mu}^j-a^i-\delta a^i) &= \delta (D_{\mu}^{ij}[A]Q^{\mu,j}-a^i)\end{array}
P_{n,k}^{(N,M)}(z) = ( k+N+M+1 ) P_{n}^{(N,M)}(z) P_{k-1}^{( N+1,M+1)}(z) \hphantom{P_{n,k}^{(N,M)}(z) =}{}- ( n+N+M+1 ) P_{n-1}^{( N+1,M+1)}(z) P_{k}^{(N,M)}(z)
\big(U_N\psi_N\big)_{jk}=\Big(\big(Q^{(1)}\big)^{\otimes j}\otimes \big(Q^{(2)}\big)^{\otimes k}\frac{a^{N_1-j}_0\,\, b^{N_2-k}_0}{\sqrt{(N_1-j)!(N_2-k)!}}\Psi_N\Big)_{jk},
u(x)=O\left( \left(\frac{1}{|x|}\right)^{n-2}\right) +o\left( \left(\frac{1}{|x|}\right)^{\frac{n-2}{n}\left(\frac{n-2}{n}\beta\lambda+\alpha\right)}\right) v(x)=o\left( \left(\frac{1}{|x|}\right)^{\frac{n-2}{n}\beta}\right).
{\cal L}_{Y . M .}^{( 5 )}= - \frac{1}{2 g_{5}^{2}}F_{M N}^{2}+ \frac{1}{g_{5}^{2}}\left( \left( D_{M}\Sigma \right)^{2}+ \overline{{{\lambda}}}i \Gamma^{M}D_{M}\lambda + \left( X^{a}\right)^{2}- \overline{{{\lambda}}}\left[ \Sigma , \lambda \right] \right)
\frac{\omega(x_k,y_k)}{\omega(x_k,y'_k)}\frac{\omega(x'_k,y'_k)}{\omega(x'_k,y_k)}= \frac{\omega(u_k,v_k)}{\omega(u_k,v'_k)}\frac{\omega(u'_k,v'_k)}{\omega(u'_k,v_k)}
k_{s , \zeta}={\frac{\rho}{2 \alpha^{2}}}( \lambda_{, \zeta}- \tau_{0}\tau_{, \zeta})^{2}, \k_{e , \zeta}= - \rho \gamma q_{1}q_{2}( \tau_{\zeta})^{2}, \ \tau_{0}= q_{1}+ q_{2},
\frac{\Gamma_{1}\Rightarrow\Delta_{1}|...|\Gamma_{n}\Rightarrow\Delta_{n}|\Sigma\Rightarrow A\Omega_{1}\Rightarrow\Theta_{1}|...|\Omega_{m}\Rightarrow\Theta_{m}|\Pi\Rightarrow B}{\Gamma_{1}\Rightarrow\Delta_{1}|...|\Gamma_{n}\Rightarrow\Delta_{n}|\Omega_{1}\Rightarrow\Theta_{1}|...|\Omega_{m}\Rightarrow\Theta_{m}|\Sigma\Rightarrow B|\Pi\Rightarrow A}
\mu_{z}(P:\sigma:\lambda:\eta)(\phi) &\qquad=\int_{N_{Q}}\int_{M/M\cap H_{z}}\int_{A/A\cap H_{z}}a^{-\lambda+\rho_{P}-2\rho_{Q}}\Big(\sigma^{\vee}(m)\eta,\phi(nma\cdot z)\Big)\,dm\,dn\,da.
S= \int d^{4}x\sqrt{|\hat{g}_{E}|}\left[ \hat{R}(\hat{g}_{E})-{\textstyle\frac{1}{2}}(\partial\hat{\phi})^{2}-{\textstyle\frac{1}{2}}e^{2\hat{\phi}}(\partial\hat{a})^{2}-{\textstyle\frac{1}{4}}e^{-\hat{\phi}}\hat{F}^{I}\hat{F}^{I}+{\textstyle\frac{1}{4}}\hat{a}\hat{F}^{I}{}^{\star}\hat{F}^{I}\right]\, .
\begin{aligned}\int_{0}^{\tau}\int_{\mathbb{T}^3}\,\bigl(\partial_t\,\eta^r-\mathrm{div}\,q^r \bigr)\,dx \,dt\leqslant C_I\Bigl(\tau h +\int_{0}^{\tau}\int_{\mathbb{T}^3}d(\Theta,\bar{\Theta}) \,dx\,dt\Bigr).\end{aligned}
R_n (x) =\inf\{j \geq 1 : x_{j+1}x_{j+2}\cdots x_{j+n}=x_1x_2\cdots x_n\}, R'_n (x) =\inf\{j \geq n: x_{j+1}x_{j+2}\cdots x_{j+n}=x_1x_2\cdots x_n\}.
{\cal A}:=\{(v_x+w_y)_{x,y}| (v_x)_x,(w_x)_x \in \mathbb{R}^{m+1}\}{\cal B}:=\left\{(a_{x,y})_{x,y}\left| \sum_{x'=0}^{m}a_{x',y}=\sum_{y'=0}^{m}a_{x,y'}=0 \hbox{for}x,y=0, 1,\ldots, m \right.\right\}.
\begin{aligned}b(x',y,t) ={}\sum_{x\in T^{-1}(x')}\frac{1}{\vert T'(x)\vert}\biggl[b(x,y,t)-x\frac{\partial}{\partial y}a(x,y,t) & +\frac{\partial}{\partial y}(y\varphi(x,y,t))+\frac{1}{2}x^{2}\frac{\partial^{2}}{\partial y^{2}}\varphi(x,y,t)-\frac{\partial}{\partial t}\varphi(x,y,t)\biggr]\end{aligned}
c_{j_{q}k_{{s}_{1}}}^{h_{{t}_{1}}}c_{h_{{t}_{1}}k_{{s}_{2}}}^{h_{{t}_{2}}}\ldots c_{h_{{t}_{n - 2}}k_{{t}_{n - 1}}}^{h_{{t}_{n - 1}}}c_{h_{{t}_{n - 1}}k_{{s}_{n}}}^{i_{p}}g^{k_{{s}_{1}}}g^{k_{{s}_{2}}}\ldots g^{k_{{s}_{n - 1}}}g^{k_{{s}_{n}}}d g^{j_{q}}\; ,
\lambda^2\Vert \bar{R}_{j_1}- \bar{R}_{j_2}\Vert^p_{L^\infty(Q_1 \cap Q_2)}\le Cm^{-1}\Vert \bar{R}_{j_1}- \bar{R}_{j_2}\Vert^p_{L^p(\tilde{Q}_1 \cap \tilde{Q}_2 \cap U_Q)}& \le Cm^{-1}\sum\nolimits^4_{j=1}\Vert \nabla y - \bar{R}_j\Vert^p_{L^p(\tilde{Q}_1 \cap \tilde{Q}_2 \cap U_Q)}
\sum_{i=k_{n-1}}^{\lfloor k/2\rfloor}(2i)^{n-1}>{}\int_{k_{n-1}-1}^{\lfloor k/2\rfloor}(2t)^{n-1}\, \textrm{d}t ={}\frac{1}{2n}((2\lfloor k/2\rfloor)^n-(2k_{n-1}-2)^n) \geq{}\frac{1}{2n}\left(\left(\frac{k}{2}\right)^n-(2k_{n-1}-2)^n\right) \geq{}\frac{1}{2n}\cdot \frac{1}{2}\cdot \left(\frac{k}{2}\right)^n.
I_{{0^+}}^\alpha \,\left({D_{{0^+}}^\alpha \,f\left( t \right) -{{\left.{D_{{0^+}}^\alpha \,f\left( t \right)}\right|}_{t = b}}}\right) + I_{{0^+}}^\alpha \,I_{{b^-}}^\alpha \left(\lambda + q \left(t\right) \right)f\left( t \right) = 0
\sum_{n \geq 1}\frac{1}{n^{s + k - 1}}\bigg( \sum_{m \geq 0}a(n-m) \overline{b(n-m)}+ \sum_{\substack{\ell \geq 1 m \geq 0}}a(n-m) \overline{b(n-m-\ell)}+ \sum_{\substack{\ell \geq 1 m \geq 0}}a(n-m-\ell)\overline{b(n-m)}\bigg).
\vert \overrightarrow{b}\vert = \sqrt{( \overrightarrow{b})^{2}}= \sqrt{( \overrightarrow{x}) - \overrightarrow{j})^{2}}= \sqrt{3 \cdot \overrightarrow{p}^{2}+ 2 \sqrt{k}\cdot \overrightarrow{j}+ \sqrt{j}^{2}}= 2
\leq \; \left\langle \prod_{b = 1}^{N}\; 2 \cosh \left( - 2 \sum_{b = 1}^{N}\beta^{( b )}\int_{\Lambda}d^{2}x \; t ( x ) : \cos \Big [ 2 \sqrt{\pi}\varphi^{( b )}( x ) \Big ] :_{C^{1}}\right) \right\rangle_{C^{1}}\; .
N^{(S,L),\delta}(\alpha,\beta)(y) =\sum_{k:\beta_k>0}[k]_y \cdot N^{(S,L),\delta}(\alpha+e_k,\beta-e_k)(y) &+\sum_{\alpha',\beta',\delta'}\left(\prod_i [i]_y^{\beta_i'-\beta_i}\right)\binom{\alpha}{\alpha'}\binom{\beta'}{\beta}N^{(S,L-H),\delta'}(\alpha',\beta')(y).
- \int_{0}^{\sigma}\!\!\!ds \,\int_{-\infty}^{\infty}\frac{d\omega}{2\pi}\frac{sK^{\frac{D-3}{2}}}{(4\pi)^{\frac{D-1}{2}}}\left| \frac{\Gamma(\frac{D-1}{2}+i\beta)}{\Gamma(1+i\beta)}\right|^{2}\Gamma(-\frac{D-3}{2})\mbox{tr}\mbox{\boldmath$1$}.
f_{I_{2}( m )}^{N}( q , \mu ) = \sum_{j = 1}^{m}g_{| j |}\, g_{| j + N |}\, x_{| j + N |}( \alpha_{j + N}\cdot q , \alpha_{j + N}^{\vee}\! \! \cdot \mu ) \, x_{| j |}( \alpha_{j}\cdot q , - \alpha_{j + 2 N}^{\vee}\! \! \cdot \mu ) .
U_0(t,s) = U_0(t,s+m_tk_\varepsilon T + m_t^\star T)U_0(s+m_tk_\varepsilon T + m_t^\star T,s+m_tk_\varepsilon T)U_0(s+m_t k_\varepsilon T,s) = U_0(t- m_tk_\varepsilon T - m_t^\star T,s)U_0(s+m_t^\star T,s)U_0(s+m_t k_\varepsilon T,s) = U_0(t- m_tk_\varepsilon T - m_t^\star T,s) U_0(s+ T,s)^{m_t^\star}U_0(s+T,s)^{m_t k_\varepsilon}.
D_{3}(tv, v_\lambda) = \frac{2 \lambda^{3}}{3}+ \frac{3 t^{3}}{8}- t - \frac{\left(\frac{\lambda}{2}+ \frac{1}{2}\right)^{3}}{3}- \left(\frac{\lambda}{2}+ \frac{1}{2}\right)^{2}\left(- \frac{\lambda}{2}+ \frac{t}{2}- \frac{1}{2}\right) + \frac{2}{3}
\mathcal{A}_0(\hat{{\mathbf U}},\hat{\Psi})\partial_t{\mathbf V}+\mathcal{A}_1(\hat{{\mathbf U}},\hat{\Psi})\partial_1{\mathbf V}+\mathcal{A}_2(\hat{{\mathbf U}},\hat{\Psi})\partial_2{\mathbf V}+\mathcal{A}_3(\hat{{\mathbf U}},\hat{\Psi}){\mathbf V}=\mathcal{F}(\hat{{\mathbf U}},\hat{\Psi}),
\left\Vert \overset{2}{\mathbb{P}}\left( \tilde{E}^{T}\right) \right\Vert^{2}_{\mathbb{L}^{2}(B)}:= \sum_{n}\left\vert \langle \tilde{E}^{T}, e^{(2)}_{n}\rangle \right\vert^{2}= 0,
A=(\begin{matrix}3&3&1&4\\ 0&1&0&0\\ 0&0&19&16\\ 0&0&0&3\end{matrix})H=(\begin{matrix}3&0&1&1\\ 0&1&0&0\\ 0&0&19&1\\ 0&0&0&3\end{matrix})U=(\begin{matrix}1&-3&0&-1\\ 0&1&0&0\\ 0&0&1&-5\\ 0&0&0&1\end{matrix})
S\lesssim \sum_{i_0=0}^\infty \sum_{i_{\ell_1}\ge i_{0}}^\infty\sum_{i_{\ell_2}\ge i_{\ell_1}}^\infty \cdots\sum_{i_{\ell_k}\ge i_{\ell_{k-1}}}^\infty \left( |E_{i_0}|~ 2^{-\ell_1 i_0}\right) \left( \prod_{t=1}^k 2^{\left(\frac{(d+1)(d-\ell_t+t)}{d(k+1)}-\ell_{t+1}+\ell_t\right) i_{\ell_t}}\right).
\beta_{\mathrm{U}}^2 =\beta_\mathrm{U}(x_1+x_2+x_3+x_{12}+x_{13}+x_{23}) &=\beta_{\mathrm{U},1}(x_1+x_{12}+x_{13})+\beta_{\mathrm{U},2}(x_2+x_{23})+\beta_{\mathrm{U},3}(x_3) &=(x_0+x_2+x_3+x_{02}+x_{03}+x_{23}) (x_1+x_{12}+x_{13}) &\quad+(x_0+x_1+x_3+x_{01}+x_{03}+x_{13}) (x_2+x_{23}) &+(x_0+x_1+x_2+x_{01}+x_{02}+x_{12}) (x_3).
\frac{d\hat{\delta}_0}{dv}(v) =\frac{1}{v^3}\frac{d\delta_0}{dv}(v)-\frac{3}{v^4}\delta_0(v) &=\frac{\lambda}{6\kappa}\left\{\frac{1+y_m(v)}{v}-\frac{3}{v^4}\int_0^v(1+y_m(v'))v'^2dv'\right\}&=\frac{\lambda}{6\kappa v^4}\int_0^vv'^3\frac{dy_m}{dv}(v')dv',
M_4=\sum_{\substack{0\leq m_2\leq j-i-1 \gcd(l+m_1+m_2,j)\{\rm{is}}\k-{\rm{free}}\gcd(l+m_1+m_2+1,j+1)\{\rm{is}}\k-{\rm{free}}}}C_{\alpha}(j-i-1,m_2).
\lambda_l(k) = (-1)^k \frac{d^k}{dz^k}\left( P^n_l(z) (1-z^2)^{(m-2)/2}\right)\Big|_{z=0}& = (-1)^k \sum_{j=0}^k \frac{d^{k-j}}{dz^{k-j}}P^n_l(z) \Big|_{z=0}\, \frac{d^{j}}{dz^{j}}(1-z^2)^{(m-2)/2}\Big|_{z=0}.
{\cal L}_{\mathrm{G F}}= - \frac{1}{2 \alpha}\displaystyle \sum_{a = 1}^{2}( \partial^{\mu}A_{\mu}^{a}+ \alpha M_{W}\chi^{a})^{2}- \frac{1}{2 \alpha}( \partial^{\mu}Z_{\mu}+ \alpha M_{Z}\chi^{3})^{2}- \frac{1}{2 \alpha}( \partial^{\mu}A_{\mu})^{2}.
\nabla_{a}{\cal M}^{a \mu \nu}={\frac{1}{2}}\left[ 2{\cal Q}^{a b}e^{\mu}{}_{a}e^{\nu}{}_{b}- \sqrt{- \gamma}\; L^{a b}{}_{i}K_{a b \, j}n^{\mu \, i}n^{\nu \, j}- ( \mu \leftrightarrow \nu ) ) \right] \, .
\int_{-\log_2T}^{\log_2T}\int_{\log|a|+\delta}^{\log_2 T}\Phi_1(u,v)\frac{\partial^2 h(u,v)}{\partial u\partial v}dudv= \int_{-\log_2T}^{\log_2T}\int_{\log|a|+\delta}^{\log_2 T}\tilde{\Phi}_1(u,v)\frac{\partial^2 h(u,v)}{\partial u\partial v}dudv &+O \left(\frac{(\log_2 T)^2}{(\log T)^{\sigma}}\right).
\begin{aligned}(a_1\# a_2)(x,\xi)-\sum_{k=0}^N\frac{1}{k!}\big(\frac{ih}{2}\sigma(D_x,D_\xi; D_y,D_\eta)\big)^k(a_1(x,\xi) a_2(y,\eta))|_{\substack{y=x \eta=\xi}}\in h^{(N+1)(1-\delta_1-\delta_2)}S_{\max(\delta_1,\delta_2)}(m_1m_2).\end{aligned}
2\Re (A_L^* u \mid u) = \Omega((u,A_L^*u),(u,A_L^*u)) = \bigl\langle L^\sharp \bigl( u(0+),u'(0+),u''(0+)\bigr), L^\sharp \bigl(u(0+),u'(0+),u''(0+)\bigr)\bigr\rangle_- - \bigl\langle \bigl(u(0+),u'(0+),u''(0+)\bigr), \bigl(u(0+),u'(0+),u''(0+)\bigr)\bigr\rangle_+\ .
\frac{1}{n}\sum_{j=1}^n\sum_{t=1}^T\|[g_{t}(x_{j,t})]_+\|\le n\varepsilon_1G_2+\frac{\varepsilon_5}{\mu}(\log(T)+1)+\frac{\varepsilon_6}{\epsilon_s}\Big(\varepsilon_{9}+\frac{nF\log(T)}{\gamma_{0}\mu}\Big).
\left|\partial_{y_d}\partial_{z_d}r'_\lambda(y',y_d,z_d)\right|+\left|\partial_{y_d}\partial_{z_d}r_{d,\lambda}(y',y_d,z_d)\right| \leq \frac{C e^{-c |\lambda|^\frac12 z_d}}{(y_d + z_d +|y'|)^{d}\big ( 1+|\lambda|^\frac12 (y_d +z_d)\big )}.
D\mu^{AA^{\prime}}=0, \nu^{AA^{\prime}}=i(\omega_{B}^{A}\mu^{BA^{\prime}}-\omega_{B^{\prime}}^{A^{\prime}}\mu^{AB^{\prime}}) \Omega_{B}^{A}\mu^{BA^{\prime}}-\Omega_{B^{\prime}}^{A^{\prime}}\mu^{AB^{\prime}}=0,
\begin{cases}d\tilde{Z}(t,z)=[\theta(t,z)-\pi(t,z)b_0(t,z)+\pi^2(t,z)\sigma^2_0(t,z)]\tilde{Z}(t,x,z)dt-\pi(t,z)\sigma_0(t,z)\tilde{Z}(t,z)dB(t) \tilde{Z}(0,z)=\int_D\frac{1}{\alpha(x)}dx,\end{cases}
\phi(kgh) =-2\ln|kgh\cdot s(e)|_{\pi^*q}& = -2\ln|k\cdot g\cdot h\cdot \pi_*s(e)|_q \intertext{by equivariance of $\pi$}= -2\ln|g\cdot \chi(h) \pi_*(s(e))|_q \intertext{by $K$-invariance of $q$ and by definition of $\chi$}= -2\ln|g\cdot s(e)|_{\pi^*q}- 2\ln|\chi(h)|
a^*_{0,1}=a^*_{0,2}=a^*_{3,1}=a^*_{3,2}=a^*_{3,3}=a^*_{2,2}=a^*_{2,3}=a^*_{1,3}=0, &a^*_{1,1}=\frac{1}{3}-\frac{\mu_R}{3}-\frac{\mu_T}{3}, &a^*_{3,0}+6a^*_{2,1}+3a^*_{1,2}=2\mu_R+\mu_T-1, &3a^*_{2,1}+6a^*_{1,2}+a^*_{0,3}=\mu_R+2\mu_T-1.
J_+(r) =\frac{\alpha^2}{1+\alpha^2}\int_0^\pi\frac{(1-\cos(\theta_1)) \sin(\theta_1)^{m-2}}{(r^2 + 1 - 2 \cos(\theta_1) + \alpha^2 (r-1)^2)^{\frac{N+s}2}}d\theta_1 J_-(r) =\frac{1}{1+\alpha^2}\int_0^\pi\frac{[ 2 +\alpha^2-\alpha^2 \cos(\theta_1))\sin(\theta_1)^{m-2}}{(r^2 + 1 - 2 r \cos(\theta_1) + \alpha^2 (r+1)^2)^{\frac{N+s}2}}d\theta_1
C_1(y_0) =2\log\left(1-y_0^{-1}\right)^{-1}\frac{y_0}{\log y_0}+\frac{C_0(y_0)y_0}{y_0-1}, C_2(y_0) =C_1(y_0)+\sup_{t\ge y_0}\frac{1}{2(t-1)R(t)}.
\frac 3 2 \mathrm{C l}_{2}( \theta_{7}) - \frac 3 2 \mathrm{C l}_{2}( 2 \theta_{7}) + \frac 1 2 \mathrm{C l}_{2}( 3 \theta_{7}) = \frac 7 4 \left\{\mathrm{C l}_{2}\left( \frac{2 \pi}{7}\right) + \mathrm{C l}_{2}\left( \frac{4 \pi}{7}\right) - \mathrm{C l}_{2}\left( \frac{6 \pi}{7}\right) \right\}
e_{H}(\bar B_r) =\, \left(\frac{(q^2-1)^r}2+\frac12 (q-1)^{2r}- 2(q-1)^r+1 \right) T + \left(- \frac{(q^2-1)^r}2+\frac12 (q-1)^{2r}- (q-1)^r+1 \right) N ,
\delta^2 \leq (1 + \eta) \sum_{j = 1}^{\infty}|\phi_j(\xi_k)|^2 + \Big(1 + \frac{1}{\eta}\Big) c_k \frac{|z - \xi_k|^2}{1 - |z|^2}\sum_{j = 1}^{\infty}\|\phi_j\|_{b}^{2}& \leq (1 + \eta) \sum_{j = 1}^{\infty}|\phi_j(\xi_k)|^2 + \Big(1 + \frac{1}{\eta}\Big) c_k \frac{|z - \xi_k|^2}{1 - |z|^2}.
\nu_P(z)\geqslant \sum_{i=s,t,l}(\min\{\nu_{(f^u,H_i),\leqslant k_i}(z);1\leqslant u\leqslant 3\}-\nu^{(1)}_{(f,H_i),\leqslant k_i}(z))+ 2\sum_{i=1}^q\nu^{(1)}_{(f,H_i),\leqslant k_i}(z),
L = 2 \sqrt{- \mathrm{d e t}\Big ( G_{\hat{\mu}\hat{\nu}}+ i G_{\hat{\mu}\rho}G_{\hat{\nu}\lambda}\tilde{H}^{\rho \lambda}/ \sqrt{- G_{5}}\Big )}+{\frac{1}{2}}\tilde{H}^{\mu \nu}\partial_{5}B_{\mu \nu}-{\frac{1}{4}}\epsilon_{\mu \nu \rho \lambda \sigma}{\frac{G^{5 \rho}}{G^{5 5}}}\tilde{H}^{\mu \nu}\tilde{H}^{\lambda \sigma}.
\mathcal A \left (\mathbb H_{\mathbf{f}}^{\textup{c-wt}}([x_{\textup{f}}],[y_{\textup{f}}])\right )=\frak h_{\mathbf{f}}^{\textup{c-wt}}([x_{\textup{f}}],[y_{\textup{f}}]) (\kappa-k).
\mathbf{y}^{(w)}_{i,j}= \mathbf{L}^{-1}_{b}\mathbf{y}_{i,j}, \mathbf{\hat{H}}^{(w)}_{i,j}= \mathbf{L}^{-1}_{b}\mathbf{\hat{H}}_{i,j}, (i,j) \in \mathbb{S}_b
I_{5}= \frac{1}{1 6 \pi G_{5}}\int d^{5}x \sqrt{- g^{( 5 )}}\left\{e^{- 2 \phi_{1}}\left[ R^{( 5 )}+ 2 0 e^{2 \psi}+ 4 ( D \phi_{1})^{2}- 5 ( D \psi )^{2}\right] - \frac{1}{2}e^{5 \psi}( F_{1}^{( 5 ) 2}+ F_{2}^{( 5 ) 2}) \right\}
\Gamma(h)_{00}^0=-\frac{\dot{s}}{s},\qquad\Gamma(h)_{ij}^0=\frac{1}{2}s^2\dot{g}_{ij},\Gamma(h)_{i0}^j=\Gamma(h)_{0i}^j=\frac{1}{2}g^{jk}\dot{g}_{ik},\Gamma(h)_{ij}^k=\Gamma(g)_{ij}^k.
S_{T M}^{4}= \int d^{4}x \Bigl [ -{\frac{1}{4}}F_{\mu \nu}F^{\mu \nu}-{\frac{1}{1 2 \mu^{2}}}H_{\mu \nu \lambda}H^{\mu \nu \lambda}-{\frac{1}{4}}\varepsilon^{\mu \nu \lambda \rho}B_{\mu \nu}F_{\lambda \rho}\Bigl ] \; ,
\left.\frac{\partial^2}{\partial A^p_d\partial A^q_e}\ln(\frac{g^2}{4}{\cal A})\right|_{A=0}=\frac{1}{2}{\cal A}^{-1}\frac{\partial^2{\cal A}}{\partial A^p_d \partial A^q_e}+\frac{1}{2}\frac{\partial^2{\cal A}}{\partial A^p_d \partial A^q_e}{\cal A}^{-1}
R_{s}\cdot \varepsilon_\theta = R_{s}\left(\sum_{t \in T(\mathbb{F}_{q^k})}\theta(t)\delta_t\right) = \sum_{t \in T(\mathbb{F}_{q^k})}\theta(t)\sum_{us \in U(\mathbb{F}_{q^k})sU(\mathbb{F}_{q^k})}\delta_{tus}= \sum_{tus \in B(\mathbb{F}_{q^k})sU(\mathbb{F}_{q^k})}\theta(t)\delta_{tus}.
V(y_i,\theta) = - V_{p+1}(i)^p \int_0^\infty \frac{dt}{t}(8 \pi^2\alpha'\,t)^{-\frac{p+1}{2}}\sum_{m_i}e^{-{t\over2\pi\alpha'}\sum_i (y_i + m_i L_i)^2}\frac{\theta^4_{11}(it/2,it)}{e^{\pi t}\eta^{12}(it)}\,.
ng_n^{(0)}= ng_n^{(0)}g_ng_n^{(0)}\leq g_n^{(0)}(g_n\delta(h_0)g_n)g_n^{(0)}= g_n^{(0)}\delta(h_0)g_n^{(0)}=g_n^{(0)}E_{n-1}(\delta(h_0))\delta(h_0)g_n^{(0)}\leq g_n^{(0)}(n-1)E_{n-1}(\delta(h_0))g_n^{(0)}=(n-1)g_n^{(0)}.
1 0 - 2 \ell a_{\ell , \ell - 1}+ b_{3}( 2 \ell - 1 ) a_{\ell - 1 , \ell - 2}+ b_{5}( 2 \ell - 3 ) a_{\ell - 2 , \ell - 2}= 0 , \; \; \ell \geq 2 ,
d\gamma(s) =e_1(s)ds+T\tau(s)ds, de_j(s) =-e_{j-1}(s)\kappa_{j-1}(s)ds+e_{j+1}(s)\kappa_j(s)ds, de_n(s) =-e_{n-1}(s)\kappa_{n-1}(s)ds +e_{2n}(s)\kappa_n(s)ds, de_{n+j}(s) =-e_{n+j-1}(s)\kappa_{j-1}(s)ds +e_{n+j+1}(s)\kappa_j(s)ds, de_{2n}(s) =-e_{n}(s)\kappa_{n}(s)ds -e_{2n-1}(s)\kappa_{n-1}(s)ds,
\omega:=\Gamma_{ij}{}^jdx^i,\rho_{1,\Gamma}=\Gamma_{in}{}^i\Gamma_{jk}{}^ndx^j\otimes dx^k,\rho_{2,\Gamma}=\Gamma_{jn}{}^i\Gamma_{ik}{}^ndx^j\otimes dx^k\,.
\partial_b\mathbf{Q}_{i+1}^{{\bar\gamma}_{i+1}}=\sum\limits_{\substack{0\leq q\le\gamma_i j_0+j_1+\dots+j_q=b 0<j_1\le\dots\le j_q}}C_{\gamma_i}(j_1,\dots j_q)Q_i^{\gamma_i-q}\left(\partial_{j_0}\mathbf{Q}_i^{{\bar\gamma}_i}\right)\prod\limits_{t=1}^q\left(\partial_{j_t}Q_i\right),
\tfrac{1}{2}\bigl(\#\{(a,b,c,d) \in B_n^{c>0}: k_{-d/c}= k_*\}- \#\{(a,b,c,d) \in B_n^{c>0}: k_{-d/c}= k\}\bigr) = \tfrac{1}{2}\bigl(\#\{(a,b,c,d) \in B_n^{c>0}: k_{c/d}= k\}- \#\{(a,b,c,d) \in B_n^{c>0}: k_{c/d}= k_*\}\bigr) .