harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
C^{\textrm{main}}_{(cp+a,dq+b)}= \frac{(p c^2 + 2ac - p c)C_\tau}{2}+ \left(c+\frac{a}{p}\right) \left(d + \frac{b}{q}\right)C_\sigma - \frac{ab}{pq}C_\sigma + \frac{(q d^2 + 2bd - q d)C_\rho}{2}&+ d \sum_{l=0}^{q-1}l C_{\rho,(0,-l-1)}+ c \sum_{k=0}^{p-1}k C_{\tau,(-k-1,0)}.
\psi_l\left( z\right) \psi_l^{\dagger}\left( w\right) \sim\left( z-w\right)^{-2\Delta_l}\left[ 1+\frac{2\Delta_l}{c_p}\left(z-w\right)^2T_p\left( w\right) +O\left( z-w\right)^3\right]
Q^2 b_1 = b_1^2 Q^4 b_1 = b_3 + b_1 b_2 + b_1^3 Q^6 b_1 = b_1^4 Q^8 b_1 = b_5 + b_1 b_4 + b_2 b_3 + b_1^2 b_3 + b_1 b_2^2 + b_1^3 b_2 + b_1^5 Q^{10}b_1 = b_3^2 + b_1^2 b_2^2 + b_1^6 Q^6 b_2 = b_5 + b_1 b_4 + b_2 b_3 + b_1 b_2^2 Q^{10}b_2 = b_1^2 b_5 + b_1^3 b_4 + b_1^2 b_2 b_3 + b_1^3 b_2^2
\times \exp\left\{\frac{1}{(1-x)^3(1-y)^5}\sum_{n=1}^{\infty}(+3n^2x^{n+1}y^{n+1}-4n^2x^{n+1}y^{n+2}+n^2x^{n+1}y^{n+3}) \frac{z^n}{n^4}\right\}
\Lambda_{M_j}^{r,s}(g_j^{v-1}(z_j,s),h^{v-1}(s))\equiv_{v-1}\sum_{\alpha,\beta=1}^n \Lambda_{N_j}^{\alpha,\beta}(z_j,s)\frac{\partial{g_j^r}^{v-1}}{\partial z_j^{\alpha}}\frac{\partial{g_j^s}^{v-1}}{\partial z_j^{\beta}}
f \equiv \sum_{k \geq 0}\sum_{l=0}^k \sum_{\substack{p_1<q_1<\cdots<p_l<q_l begin{align*}1pt] i_{l+1}\leq \cdots \leq i_k}}a_{p_1q_1,\ldots, p_lq_l, i_{l+1}, \ldots, i_{k}}t_{p_1q_1}' \cdots t_{p_lq_l}'t_{i_{l+1}}' \cdots t_{i_k}' \in J_0^k
III= -\frac{1}{2\lambda}E_{\lambda}(-x)\Phi(x)P_\lambda(x)+ \frac{1}{2\lambda}E_{\lambda}(x)\int_0^x \Phi(x_1)\Phi'(x_1)\,dx_1 + &+\frac{1}{2\lambda}\int_0^xE_{\lambda}(x-2x_1)\Phi'(x_1)P_\lambda(x_1)\,dx_1 \, .
\begin{aligned}G(t,r,u,u_t,u_r)=g(x(r,t),v(t,r,u),\bar{v}(t,r,\bar{u})), & H(t,r,u,u_t,u_r)=h(x(r,t),v(t,r,u),\bar{v}(t,r,\bar{u})),\end{aligned}
\int_Q g^{*n}(t)dt = \int_Q\sqcap_N^{n}*\mu^{*n}(t)dt\le \max_{Q: |Q|=N^{-n\beta}}\mu^{*n}(Q) &\le M(\beta)N^{-n\beta}\log N\le M(\beta)|Q|^{n\beta/d}{\log N}.
A_{C_{4}}=\begin{cases}\#N_{p}^{-}(a,c) (p^{e-2d-1}+p^{m-d-1}), \qquad\ \biggl(\frac{c^{2}-4ai}{p}\biggr)=-1, \#N_{p}^{+}(a,c) (p^{e-2d-1}+p^{m-d-1}), \qquad\ \biggl(\frac{c^{2}-4ai}{p}\biggr)=1.\end{cases}
p_\theta({\bf{x}}) = Z(\theta) e_\alpha \big[h({\bf{x}}) +w(\theta)^\top f({\bf{x}})\big], p_\theta({\bf{x}})^{-1}= Z(\theta)^{-1}~ e_\alpha \big[-h({\bf{x}})-w(\theta)^\top f({\bf{x}})\big], p_\theta({\bf{x}})^{-1}= e_\alpha \big[-h({\bf{x}})-Z'(\theta)-w(\theta)^\top f({\bf{x}})\big],
(D_{3})_{\alpha}= \theta_{\alpha C}(\vec X_{B}\cdot\vec P_{A})X^{a}_{D}{d^{CD}}_{F}f^{BAF}+\frac{1}{2}(\gamma^{ab}\theta_{A})_{\alpha}X_{dB}(\vec X_{C}\cdot\vec P_{D}) ({c_{E}}^{AB}f^{ECD}+{c_{\lambda}}^{AB}f^{\lambda CD}).
S_N(1,\xi_n)= \begin{pmatrix}i (-1)^{n}(-1)^N \widehat{(\varphi_1^{(N)})}(-n) \frac{(-1)^{n+1}}{2i\xi_n}\int_0^1\varphi_1 r_{N+1}\,dt \end{pmatrix}+\frac{\ell^2_n}{n}\,.
\beta_{g}^{( 2 )}= - \frac{g^{5}}{( 4 \pi )^{4}}\left[ \frac{3 4}{3}( C_{2}( G ) )^{2}- \kappa \left[ 4 C_{2}( F ) + \frac{2 0}{3}C_{2}( G ) \right] S_{2}( F ) - \left[ 2 C_{2}( S ) + \frac{1}{3}C_{2}( G ) \right] S_{2}( S ) + \frac{2 \kappa Y_{4}( F )}{g^{2}}\right]
\widehat{\mu_m}([w]):=\frac{1}{\left|U_m\right|}\sum_{k\in U_m}\mu_m\left([w]_k\right)= \frac{\left| U_{m-n}\right|}{\left|{U_m}\right|}\mu([k])+\frac{1}{\left|{U_m}\right|}\sum\limits_{k\in U_m\backslash U_{m-n}}\mu_m\left([w]_k\right),
\left| \sum_{\alpha \beta}\langle \Phi_{\alpha}\Phi_{\beta}\rangle_c\left. \right/ \sum_{\alpha \beta}\langle \Phi_{\alpha}\Phi_{\beta}\rangle\right|=\left| \langle \Phi^2(x) \rangle_c \left. \right/\langle \Phi^2(x) \rangle \right|\; (n=2) \;,
\Sigma^{\pm}(s-w, \phi, F_{\pm})= \lim_{z \downarrow 0}\int_0^\infty w_z(\lambda) \lambda^{4(1-s+w)}\int_{-\infty}^\infty \int_{0}^\infty \phi(n_{u}^t a_{\frac{1}{t}}) &\times\sum_{m=1}^\infty \sum_{n=0}^{3m-1}\hat{F}(\lambda a_{\frac{1}{t}}n_u \cdot (0,0,3m,n)) \frac{d\lambda}{\lambda}\frac{dt}{t^3}du.
I_m = \int d^4x \sqrt{{\tilde g}}\left[-\frac{1}{2}D_{\mu}\Phi(D^{\mu}\Phi)^* - \frac{1}{2}D_{\mu}\Sigma (D^{\mu}\Sigma)^* - \frac{1}{4}F_{\mu \nu}F^{\mu \nu}- \frac{1}{4}H_{\mu\nu}H^{\mu \nu}- V(|\Phi|, |\Sigma |)\right],
\inf_{0 \le x_{n+1}\le 1/2}(x_{n+1}^{1-2s}- x_{n+1}) = \inf_{0 \le x_{n+1}\le 1/2}\psi_{s}(x_{n+1}) = \psi_{s}\bigg( \frac{1}{2}\bigg) = \frac{1}{2}(4^{s}- 1) \ge \frac{1}{2}.
a_{j}= \frac{1}{2}\left( \begin{array}{c}{f_{j}}\\{0}\\ \end{array}\right) + \frac{1}{2}\left( \begin{array}{c}{0}\\{f_{j}^{\ast}}\\ \end{array}\right) , b_{j}= - \frac{i}{2}\left( \begin{array}{c}{f_{j}}\\{0}\\ \end{array}\right) + \frac{i}{2}\left( \begin{array}{c}{0}\\{f_{j}^{\ast}}\\ \end{array}\right)
D_{\mu m}^{j}={\displaystyle\sum\limits_{\mu_{1}}}\sum_{m_{1}}C_{m_{1},m_{2},m}^{j_{1,}j_{2},j}C_{\mu_{1},\mu_{2,}\mu}^{j_{1},j_{2},j}D_{\mu_{1},m_{1}}^{j_{1}}D_{\mu_{2},m_{2}}^{j_{2}}\ .
e^{b \phi}={\frac{c_{3}| \Lambda |}{q^{2}| c_{2}|}}{\frac{\sin^{2}\sqrt{{\frac{| c_{2}|}{2}}}( z - z_{2})}{\cosh^{2}\sqrt{\frac{c_{3}}{2}}( z - z_{3})}}e^{c_{1}( z - z_{1})}.
\prod^{[\tau_n/T_0]-1}_{i=0}\parallel\psi^{X_n}_{T_0}|\mathcal{E}_{\varphi^{X_n}_{iT_0}(y)}\parallel\leq{\rm e}^{-\eta\tau_n},~~\prod^{[\tau_n/T_0]-1}_{i=0}m(\psi^{X_n}_{T_0}|\mathcal{F}_{\varphi^{X_n}_{iT_0}(y)})\geq{\rm e}^{\eta \tau_n},~\forall~y\in Orb(x_n).
a_n =\frac{1}{n+1}\sum_{\lambda}\binom{n+k}{k}\binom{k}{k_1,k_2,...,k_n}&=\sum_{\lambda}\frac{1}{n+1}\binom{n+k}{k}\binom{k}{k_{d_1-1},k_{d_2-1},...,k_{d_r-1}}&=\sum_{\lambda}a_{\lambda},
\lambda\left( \sum_{i =1}^{m}\sum_{j=J_{(i)}+1}^{k}d_{j}(\lambda_{i-1}) e_j + \sum_{j=q+1}^k d_j(\lambda_{m}) e_j + \sum_{i =m+1}^{n-1}\sum_{j=J_{(i)}+1}^{k}d_{j}(\lambda_{i}) e_j \, , ~ d(\lambda_0 \cdots \lambda_n) \right)
S = \int d^4 x \sqrt{-\tilde{g}}\left[ \tilde{R}-\frac{1}{2}\left( \tilde{\nabla}\Phi \right)^2 -\frac{1}{2}\left( \tilde{\nabla}y \right)^2 -\Lambda_{\rm M}e^{2\Phi}\cosh \left( \sqrt{3}y \right) \right] ,
F_{-1/10}=\psi_1(\psi_1^5+2\psi_2^5), F_{1/10}=\psi_2(2\psi_1^5-\psi_2^5)/2, &F_{3/10}=\psi_1^4\psi_2^2, F_{7/10}=\psi_1^2\psi_2^4.
|\varphi_i(z)-U_1(z)| \le C \begin{cases}\max\{\tau_i, m_i^{-2}\}, \mbox{if}5\le n\le 7, \max\{\tau_i, \Theta_i m_i^{-2}\log m_i, m_i^{-2}\}, \mbox{if}n=8, \max\{\tau_i, \Theta_i m_i^{-\frac{8}{n-4}}, m_i^{-2}\}, \quad\mbox{if}n\ge 9,\end{cases}\forall~ |z|\le m_i^{\frac{p_i-1}{4}},
(J_a J_a^*-I)u[x] =(2\pi)^{-d}\int_{\mathbb{T}^d}\sum_{y\in\mathbb{Z}^d}e^{i(\varphi_a(x,\xi)-\varphi_a(y,\xi))}u\left[y\right]d\xi-u[x] &=(2\pi)^{-d}\int_{\mathbb{T}^d}\sum_{y\in\mathbb{Z}^d}e^{i(x-y)\cdot\eta}\left(\left|\det\left(\frac{d\xi}{d\eta}\right)\right|-1\right) u\left[y\right] d\eta .
K_{h}(i) := \frac{\alpha+l+i-k}{i-k-l}\left[1-M(i)-\frac{(l+h-n)(2i+\alpha+\beta-1)_2}{(i+k+l+\alpha+\beta)(\alpha+l+i-k)(i-n-1)}\right], &L(i) := \frac{(\alpha+l+i-k-1)_2}{(i-k-l-1)_2}M(i)
Y_s^{t,x}\ = \Y_T^{t,x}+ \int_s^T f(r,X_r^{t,x},Y_r^{t,x},Z_r^{t,x}) dr + K_T^{t,x}- K_s^{t,x}- \int_s^T \langle Z_r^{t,x},dW_r\rangle, t \leq s \leq T.
0= - \int d^{N^2}Xd^{N^2}Y \,\sum_{\alpha=1}^{N^2}\left[\frac{\partial}{\partial X_{\alpha}}{\rm e}^{- S}\frac{\partial}{\partial X_{\alpha}}+ \frac{\partial}{\partial Y_{\alpha}}{\rm e}^{-S}\frac{\partial}{\partial Y_{\alpha}}\right]{\rm e}^{K\cdot \Psi}.
\mu^2 \frac{\partial}{\partial \mu^2}\Gamma_m(0| \phi, \phi_a^*, \bar{\phi}, \eta) = \bigr( \beta_g g \frac{\partial}{\partial g}+\gamma_Q \mathbf{\tilde{N}}_Q + \gamma_B \mathbf{\tilde{N}}_B \bigr) \Gamma_m(0| \phi, \phi_a^*, \bar{\phi}, \eta),
\mbox{Trace}(X(\mathbf{l}_{2}) \circ b_{F}\circ X(\mathbf{l}_{1}) \circ X_{1}\otimes \dots \otimes X_{d})= &(-1)^{\sum_{i=1}^{d}l_i^{(2)}}\mbox{Trace}(X^{l_{j_{1}}}U_{j_{1}}\cdots X^{l_{j_{s}}}U_{j_{s}})\cdots\mbox{Trace}(X^{l_{j_{t+1}}}U_{j_{t+1}}\cdots X^{l_{j_{d}}}U_{j_{d}}).
\mathrm{P}_{2,i}^o= \sum^{i}_{m=1}\left[1-e^{-\gamma_m}\left(\sum^{M-m}_{j=0}\frac{\gamma_m^j}{j!}\right)\right] \times \prod^{m-1}_{n=1}\left[e^{-\gamma_n}\left(\sum^{M-n}_{j=0}\frac{\gamma_n^j}{j!}\right)\right] \approx \sum^{i}_{m=1}\frac{\gamma_m^{M-m+1}}{(M-m+1)!}\approx \frac{\gamma_m^{M-i+1}}{(M-i+1)!},
\widehat{\boldsymbol{x}}_{\beta}^{g}=\underset{\boldsymbol{e}_{\beta}\in E_{\beta}}{\arg \: \min}\left\|\boldsymbol{y}_{\beta}^{g}-\sqrt{P_{\beta}}diag(\boldsymbol{h}_{F,\beta}^{g})\boldsymbol{e}_{\beta}\right\|^{2},
\underset{k\rightarrow \infty}{\lim}\Gamma\left(m^{*}_{k}\right)=\underset{k\rightarrow \infty}{\lim}\gamma\left(m^{*}_{k}\right)=\underset{k\rightarrow \infty}{\lim}\frac{3^{k+1}+k+2}{\log\left(\frac{2^{3^{k+1}+k+2}-3^{k+2}+2^{k+2}}{3^{k+2}}\right)}=\frac{1}{\log 2}.
c_x \coloneqq 2-\frac{\langle x, x \rangle^\frac{-1}{2}}{\sqrt{n}}\sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}-\left(\sum_{i=1}^{n}(a_ia_i^*)^\frac{1}{2}\right)\frac{\langle x, x \rangle^\frac{-1}{2}}{\sqrt{n}}.
\sum_{a=3}^{n-1}\sum_{\ell=1}^{a-2}\binom{n-1-a+\ell}{\ell}\left[\binom{a-1}{\ell}-1\right]=\sum_{\ell=1}^{n-3}\sum_{a=\ell+1}^{n-1}\binom{n-1-a+\ell}{\ell}\left[\binom{a-1}{\ell}-1\right] &=\sum_{\ell=1}^{n-3}\left[\binom{n-1+\ell}{2\ell+1}-\binom{n-1}{\ell+1}\right]=\sum_{\ell=0}^{n-2}\binom{n-1+\ell}{2\ell+1}-(2^{n-1}-1).
\Big\Vert (w,0) - \frac{1}{n}\sum_{i=1}^{n}(x_i,y_i) \Big\Vert_N \ge \Vert w - \frac{1}{n}\sum_{i=1}^{n}x_i\Vert \ge \|w\| - \frac{1}{n}\sum_{i=1}^{n}\|x_i\| \ge 1 - \frac{1}{n}\sum_{i=1}^{n}(1 - \delta) = \delta.
\lim\limits_{r\rightarrow\infty}\left(\sum\limits_{i=1}^{l}m_{i}B_{\sigma ir}Q_{ir}-B_{2\sigma r}\right)\odot Q_{1r}=\frac{1}{2}\sum\limits_{i=1}^{l}\sum\limits_{j=1,\textrm{}j\neq i}^{l}\frac{m_{i}m_{j}(Q_{ir}+Q_{jr})\odot Q_{1r}}{(\sigma-\sigma(Q_{ir}\odot Q_{jr})^{2})^{\frac{1}{2}}\sigma(1+Q_{ir}\odot Q_{jr})}=\infty,
d(\partial G\cdot G^{-1})-(\partial G\cdot G^{-1})\wedge(\partial G\cdot G^{-1}) = \partial(\partial G\cdot G^{-1})+\partial G\wedge\partial G^{-1}\cdot G\cdot G^{-1}= -\partial G\wedge\partial G^{-1}+\partial G\wedge\partial G^{-1}=0.
P(T, x) = \dfrac{\displaystyle\prod_{j=1}^{l(T)}W_j^{n(T, l(T)+1-j)}}{W_0^{n(T, l(T))}\displaystyle\prod_{j=1}^{l(T)-1}W_j^{n(T, l(T)-j)}}= \dfrac{\displaystyle\prod_{j=1}^{l(T)}W_j^{n(T, l(T)+1-j)}}{\displaystyle\prod_{j=1}^{l(T)-1}W_j^{n(T, l(T)-j)}}.
e^t H'(t) = \int_\mathbb{R}\big[u_t(x,0) - u^a_t(x,0)\big] dx = \int_\mathbb{R}\Big[-v_x(x,0) - \sum_{i=1}^n u^i_t(x-x_i,0)\Big] dx = -(v_+ - v_-) + \big[f(u_+) - f(u_-)\big] =0.
\begin{aligned}\widehat{\boldsymbol{D}^{-s}u}=\lim_{n\rightarrow \infty}\widehat{\boldsymbol{D}^{-s}{\psi_n}_i}=\lim_{n\rightarrow \infty}(2\pi i \xi)^{-s}\widehat{{\psi_n}_i}\quad, \widehat{\boldsymbol{D}^{s*}u}=\lim_{n\rightarrow \infty}\widehat{\boldsymbol{D}^{s*}{\psi_n}_i}=\lim_{n\rightarrow \infty}(-2\pi i \xi)^{s}\widehat{{\psi_n}_i}\quad.\end{aligned}
\sum_{t\ge t_0}e^{-\frac{\varepsilon^2}{3}t}= e^{-\frac{\varepsilon^2}{3}t_0}\cdot \frac{1}{1-e^{-\frac{\varepsilon^2}{3}}}\le\varepsilon^5 \cdot \frac{6}{\varepsilon^2}=6\varepsilon^3\,,
E\left[ S_N^{(r)}\right] =\Gamma(r+1) \sum_{\substack{J\subset\{1, \dots, ,N\}J\neq \emptyset}}\frac{(-1)^{|J|-1}}{\left(\sum_{j \in J}q_j \right)^r}& = \Gamma(r+1) \sum_{m=1}^N (-1)^{m-1}\sum_{1 \leq j_1 < \cdots < j_m \leq N}\frac{1}{\left(q_{j_1}+ \cdots + q_{j_m}\right)^r},
w_a\Big(e_{\Lambda_z}\big(\bar\pi a^{-1}(u_1z+u_2)\big),e_{\Lambda_z}\big(\bar\pi a^{-1}(v_1z+v_2)\big)\Big) = \bar\pi e_A\big(a^{-1}(u_1v_2-u_2v_1)\big) h(z)^{-1}.
\frac{(\frac{(\alpha-1)}{2})!(\frac{(\alpha-1)}{2})!}{\alpha !}=2^{-\alpha}\int_0^1(1-\eta)^{\frac{(\alpha-1)}{2}}\eta^{-\frac{1}{2}}{\rm d}\eta = 2^{-\alpha}\frac{(\frac{(\alpha-1)}{2})!}{\frac{\alpha}{2}!}(-\frac{1}{2})!
K=-\frac{|\begin{matrix}H(F)&\nabla F^{T}\\ \nabla F&0\end{matrix}|}{|\nabla F|^{4}}=-\frac{|\begin{matrix}F_{xx}&F_{xy}&F_{xz}&F_{x}\\F_{xy}&F_{yy}&F_{yz}&F_{y}\\F_{xz}&F_{yz}&F_{zz}&F_{z}\\F_{x}&F_{y}&F_{z}&0\\ \end{matrix}|}{|\nabla F|^{4}}
\int \frac{d^{d}q_{1}}{N_{d}\pi^{d / 2}}\int \frac{d^{d}q_{2}}{N_{d}\pi^{d / 2}}\frac{\partial}{\partial q_{1}^{\mu}}\frac{q_{1}^{\mu}}{( q_{1}^{2}+ 1 )^{a}( q_{2}^{2}+ 1 )^{b}( ( q_{1}+ q_{2})^{2}+ 1 )^{c}}= 0
\frac{d h_{pp'}(l)}{dl}= \!\!-\sum_q\!\left( \frac{dh_{pq}(l)}{dl}\frac{1}{E_p(l)-E_q(l)}h_{qp'}(l) +h_{pq}(l)\frac{1}{E_{p'}(l)-E_{q}(l)}\frac{dh_{qp'}(l)}{dl}\right)\,,
-2\int_{\Omega\times\mathbb{R}^3}\nabla\Phi^*\cdot vf_N\,dxdv =-2\sigma\int_\Omega\nabla\Phi_N\cdot\int_{\mathbb{R}^3}vf_N\,dvdx &=-2\sigma\int_\Omega\Phi_N\,\partial_t\int_{\mathbb{R}^3}f_N\,dvdx &=-\sigma\frac{d}{dt}\int_\Omega(\varepsilon|\nabla^{2m+1}\Phi_N |^2+|\nabla\Phi_N|^2)\,dx.
q_0 + d q_1 = \frac{q-1}{d}+ (d-1)q + 1 = 1 + q\left(d-1+\frac1d\right)-\frac1d &= d - \left( d-1+\frac1d\right) + q \left(d-1+\frac1d\right) = d + (q-1)\left( d-1+\frac1d\right) > d
\sum_{j=-\infty}^{\infty}\left(j^2+a^2\right)^{-2s}=\int_{-\infty}^{\infty}dj\left(j^2+a^2\right)^{-2s}+2\int_{-\infty+i\epsilon}^{\infty+i\epsilon}\left(e^{-2\pi iz}-1\right)^{-1}\left(z^2+a^2\right)^{-2s}dz
-\left(\frac{F[-q,p](i\mu_{0})w_{1}[-q,p](i\mu_{0})w_{2}[-q,p](i\mu_{0})}{w_{3}[-q,p](i\mu_{0})}\right)^{-\frac{1}{2}}\gamma^{3}\partial_{\psi}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
g=\mathcal{F}^{-1}(h), h(x) = - \left\langle \frac{x^\perp}{|x|},\mathcal{F}(f)(x)\right\rangle \frac{x}{|x|}+ \left\langle \frac{x}{|x|},\mathcal{F}(f)(x)\right\rangle \frac{x^\perp}{|x|},
\widehat{w}_{k}(\underline{\xi}_{k}) \geq\frac{k-1}{1-\frac{\widehat{\lambda}_{n}(\underline{\xi}_{n})+n-2}{(n-1)(k-1)}}= \frac{(n-1)(k-1)^2}{nk-k-2n+3-\widehat{\lambda}_{n}(\underline{\xi}_{n})}=A,
\zeta_{r}\left(-\mathbf{n};\mathbf{z}\right) = \frac{\left(-1\right)^{n_{r}}}{n_{r}+1}\sum_{k=0}^{n_{r}+1}\binom{n_{r}+1}{k}\left(-1\right)^{k}\zeta_{r-1}\left(-n_{1},\dots,-n_{r-1}-k;\mathbf{z}\right)B_{n_{r}+1-k}\left(z_{r}\right).
- \frac{1}{2 m}n_{i n t}^{- 1 / 2}\partial^{m}\partial_{m}n_{i n t}^{1 / 2}+ \frac{1}{2}m \eta_{i n t}^{\dagger}v^{m}v_{m}\eta_{i n t}- \mu_{i n t}+ V_{v a c}= 0 .
[\begin{matrix}\sigma_{11}\\ \sigma_{22}\\ \sigma_{12}\end{matrix}]=\frac{E}{1-\nu^{2}}[\begin{matrix}1&\nu&0\\ \nu&1&0\\ 0&0&\frac{1-\nu}{2}\end{matrix}][\begin{matrix}\epsilon_{11}\\ \epsilon_{22}\\ 2\epsilon_{12}\end{matrix}]
\widetilde f_n(0^+)\sigma(1)+\widetilde f_n'(+\infty)\rho(1) &\quad-\int_{[1/n,n]}\bigl\{\sigma((1-x(s))^*(1-x(s)))+s^{-1}\rho(x(s)x(s)^*)\bigr\}(1+s)\,d\widetilde\nu_n(s) &=f_n(0^+)\rho(1)+f_n'(+\infty)\sigma(1) &\quad-\int_{[1/n,n]}\bigl\{\rho((1-y(s))^*(1-y(s)))+s^{-1}\sigma(y(s)y(s)^*)\bigr\}(1+s)\,d\nu_n(s),
\left. Z(\Omega,D) = \int \, d \mu_{I_{\beta}}\, (A) \, \sum_{\tau}\, c_{\tau}\, \left[ z(\beta) c_{\beta}(A, D - \Omega) + z(\beta,\rho^{\tau}) \, c_{\beta}(A, D + \rho^{\Omega_{\tau}}) \right] \times \prod_{\rho \in{\cal E}_{\tau}}\, \left[ 1 + z(\beta,\rho) \, c_{\beta}(A,\rho) \right] \ .\right.
K_{2}(t_{1},t_{2}) =\frac{\alpha}{\pi^{2}}\left(1+\frac{1}{2}r^{2}-\frac{1}{2}(r_{1}/\sqrt{\alpha})^{2}-\frac{1}{2}(r_{2}/\sqrt{\alpha})^{2}+\frac{1}{2}(r_{12}/\alpha)^{2}\right) &+\alpha \cdot O\left(r^{4}+(r_{1}/\sqrt{\alpha})^{4}+(r_{2}/\sqrt{\alpha})^{4}+ (r_{12}/\alpha)^{4}\right).
\lim_{r \to 0}\lim_{t \to \infty}\frac{- \log \mu(\tau_r > t)}{t \mu(U_r)}= \lim_{r \to 0}\frac{- \mu(Y) \log \Lambda_r}{\mu(U_r)}= \lim_{r \to 0}\frac{- \log \Lambda_r}{\mu_Y(U_r)}= 1
\mathbb{P}(\cup_S E_S) \le \sum_S \mathbb{P}(E_S) = \sum_{|S| \le 2}\mathbb{P}(E_S) + \sum_{3 \leq |S| \leq n - 2}\mathbb{P}(E_S) + \sum_{|S| \ge n - 1}\mathbb{P}(E_S).
I_1 = \int_{\Omega_{bl}^{t-1,t}}\left(S(A) - S(A+D u_{bl})) \left( \begin{smallmatrix}0  \chi'_n \end{smallmatrix}\right) \right) \cdot (u_{bl}- \overline{c}) + \: \int_{\Omega_{bl}^{t+n,t+n+1}}\left(S(A) - S(A+D u_{bl})) \left( \begin{smallmatrix}0  \chi'_n \end{smallmatrix}\right) \right) \cdot (u_{bl}- \overline{c}_n) \: := I_{1,1}+ I_{1,2},
D(\mathcal{A}) = \{(u,v,x,z) \in \mathcal{H}: w,z \in H_{0}^{1}(\Omega), \operatorname{div}(K(x) \nabla u + a(x)\nabla w), \operatorname{div}(K(x) \nabla v + a(x)\nabla z) \in L^{2}(\Omega)\}
\sup_{k\in\mathbb{N}}\sup_{\sigma>0}\int_{\mathbb{T}^\infty}\log(1+|\mathcal{B}_k f_{\sigma}|)dm_\infty=\sup_{\sigma>0}\|\mathcal{B}f_\sigma\|_0=\sup_{\sigma>0}\|f_\sigma\|_0=\|f\|_0<\infty.
\left. \begin{array}{l l l}{W}{=}{\lambda \wp ( z - a ) ,}{E}{=}{8 | \lambda |^{2}\frac{| \wp ( z - a ) | | \wp^{2}( z - a ) - \wp^{2}( L / 2 ) |}{[ 1 + | \lambda \wp ( z - a ) |^{2}]^{2}}.}\end{array}\right.
\int{\cal D}\varphi\quad e^{+{1\over 2}(d\varphi,d\varphi)}\quad\prod_{k=1}^i e^{2i\sqrt{\pi\over\xi}\varphi(y_k)}\prod_{k=1}^i e^{-2i\sqrt{\pi\over\xi}\varphi(x_k)}\propto(\Lambda_{UV})^{2i}K^{-1}_i(x,y).
\left| \frac{1}{N}\sum_{n=0}^{N-1}a_n \right|^2 \leq \frac{N+H}{N^2(H+1)}\sum_{n=0}^{N-1}|a_n|^2 &+ \frac{2(N+H)}{N^2(H+1)^2}\sum_{h=1}^H (H + 1 - h) \Re \left( \sum_{n=0}^{N-h-1}a_n \overline{a}_{n+h}\right).
\frac{2 J ( r )^{2}K ( r )^{2}}{\mathcal{L}( r )^{2}}= \frac{\left[ M ( r )^{4}\mathcal{L}( r )^{2}J^{\prime}( r ) \right]^{\prime}}{M ( r )^{4}\mathcal{L}( r )^{2}}J ( r ) - \alpha J ( r )^{2}\left( J ( r )^{2}- 1 \right) .
\int_{0}^{\infty}e^{-s x}F_0^n(x)dx=\frac{1}{s}\left(\frac{\mathbb E\left[e^{-s \overline{X}_{e(q_n)}}\right]}{\mathbb E\left[e^{-s \overline{X}_{e(p+q_n)}}\right]}-1\right),\ \s > 0.
c_{0}-a_{0}-b_{0}\equiv(a_{0}+b_{0})^{p}-a_{0}-b_{0}\equiv(\begin{matrix}p\\ 1\end{matrix})a_{0}^{p-1}b_{0}+\cdot\cdot\cdot+(\begin{matrix}p\\p-1\end{matrix})a_{0}b_{0}^{p-1}modp^{2}.
d_2(f, \mathcal{R}_{n-1,n})= \|{\bf P}_-(fq_{n,0}/\pi_{n})\|_2\leq\|{\bf P}_-(fq_{n,1}/\pi_{n})\|_2 = \|{\bf P}_-\Bigl(f(q_{n,1}/\widetilde{q}_{n,1})(\widetilde{q}_{n,1}/\pi_{n})\Bigr)\|_2\leq \|{\bf P}_-(fq_{n,1}/\widetilde{q}_{n,1})\|_2 = d_2(f, \mathcal{R}_{n-1,n})
4\sum_{k=0}^s \Big\{\|\partial_{\alpha}^kD_t\tilde{\theta}\|^2+\|\partial_{\alpha}^k D_t\tilde{\sigma}\|^2 +\|\partial_{\alpha}^k|D|^{1/2}\tilde{\theta}\|_{L^2}^2+\|\partial_{\alpha}^k|D|^{1/2}\tilde{\sigma}\|_{L^2}^2\Big\}\leq \mathcal{E}_s+C\epsilon^3\leq 20\epsilon^2.
Z_f(s, \chi, A_3)= \dfrac{(1-5^{-1})}{1-5^{-1}t}\Big(5^{-4}t+5^{-2}-2\times5^{-3}\Big), Z_f(s, \chi, A_4)= (1-5^{-1})\Big(5^{-3}t^5+5^{-2}\Big), Z_f(s, \chi, A_5)= (1-5^{-1})^2\Big(5^{-3}t^5+(5^{-2}-5^{-3})t^3+5^{-1}\Big), Z_f(s, \chi, A_6)= (1-5^{-1})^2\Big(5^{-4}t^5+(5^{-3}-5^{-4})t^4+(5^{-2}-5^{-3})t^2+5^{-1}\Big).
E_{00}^{(i)}\approx \frac{1}{2}\left\{\frac{\ddot{a}_0}{a_0}+ \frac{\dot{a}_0}{a_0}\frac{\dot{d}_0}{d_0}- \frac{\dot{a}_0^2}{a_0^2}- \frac{\ddot{d}_0}{d_0}-\sum_{i=1}^2\frac{(\rho_i+P_i)}{2d_0}\right\}\,.
\frac{a\rho}{t-1}\leq \frac{t}{\rho}\cdot\left(1-\frac{\varepsilon}{2}\right)\cdot \frac{\rho}{t-1}=\frac{t}{t-1}\left(1-\frac{\varepsilon}{2}\right) &\leq \left(1+\frac{1}{10\log d}\right)\left(1-\frac{\varepsilon}{2}\right)\leq 1-\frac{\varepsilon}{4}\leq 1-\frac{t}{r-t},
\begin{array}{lll}\sigma(w_1) = \{v_1\}\sigma(w_2) = \{v_2 \}\sigma(w_3) = \{v_3 \}\sigma(w_4) = \{v_1, v_4, v_5 \}\sigma(w_5) = \{v_2, v_4, v_6 \}\sigma(w_6) = \{v_3, v_5, v_6 \}\end{array}
\exp \left[{\frac{1}{\epsilon}}\int_{0}^{1}{\frac{d t}{2 t}}( \sqrt{1 - 4{\hat{\lambda}}t}- 1 ) \right] = \exp \left\{{\frac{1}{\epsilon}}\left[ \sqrt{1 - 4{\hat{\lambda}}}- 1 + \log{\frac{2}{1 + \sqrt{1 - 4{\hat{\lambda}}}}}\right] \right\}
\sum_{j\in [d] \setminus \mathcal{C}_{k-1}}\# \big\{\mathbf{x}\in S_{k-1}\,:\, x^{(j)}> \kappa \big\}= \sum_{\mathbf{x}\in S_{k-1}}\# \big\{j\in [d] \setminus \mathcal{C}_{k-1}\,:\, x^{(j)}> \kappa \big\}\geq 2 \# S_{k-1}.
\beta_{i_0j_0}= A_{i_0j_0k_0}^{i_0j_0}dF_{i_0}\wedge dF_{j_0}+ A_{i_0k_0j_0}^{i_0j_0}dF_{i_0}\wedge dF_{k_0}+ A_{j_0k_0i_0}^{i_0j_0}dF_{j_0}\wedge dF_{k_0}.
E_\mu(1,t,t^2, \ldots, t^{n-1};q,t) = t^{n(\lambda)}\prod_{(r,c) \in \mu}\frac{1-q^c t^{v_\mu(r)}}{1-q^{\mu_r-c+1}t^{v_\mu(r)- u_\mu(r,c)}}.
\tau(N-2,N-1) \big\{\upsilon^{-1}\rho\big( z^{-\kappa},z^{\kappa}\big) \upsilon\big\}=\big\{\upsilon^{-1}\rho\big(z^{-\kappa},z^{\kappa}\big) \upsilon\big\}\tau(N-2,N-1), \tau(N-2,N-1) \big\{\upsilon^{-1}\alpha_{n}(y,u) \upsilon\big\}\tau(N-2,N-1) = (-1)^{n}\big\{\upsilon^{-1}\alpha_{n}(y,u)\upsilon\big\},n\geq0.
\varphi_a'(\lambda^n z)\cdot \lambda^n=(p_a^n\circ\varphi_a)'(z) =p_a'(p_a^{n-1}\circ\varphi_a(z))\cdots p_a'(p_a\circ\varphi_a(z))p_a'(\varphi_a(z))\varphi_a'(z) & =p_a'(\varphi_a(z/\lambda^{n-1}))\cdots p_a'(\varphi_a(z/\lambda))p_a'(\varphi_a(z))\varphi_a'(z).
e^{iq}\, z^{\alpha_0}= e^{iq}\, e^{(ln z)\alpha_0}= e^{(\ln z)\alpha_0}e^{-(\ln z)\alpha_0}e^{iq}\, e^{(\ln z)\alpha_0}\cr = z^{\alpha_0}\, e^{iq + [iq,(\ln z)\alpha_0] +\cdots}= z^{\alpha_0}\, e^{iq + i(\ln z)[q,\alpha_0] +\cdots}\cr = z^{\alpha_0}\, e^{iq - (\ln z)}= (1/z)\, z^{\alpha_0}\, e^{iq}\,,
\begin{array}{c}{u^{\alpha}= u_{0}^{\alpha}+ \xi^{\alpha}s - \frac 1 2 \left( \stackrel{s}{\Gamma}_{\beta \gamma}^{\alpha}\right)_{0}\xi^{\beta}\xi^{\gamma}s^{2}-}\\{\frac 1{3 !}\left( \Gamma^{\alpha}{}_{\beta \gamma \delta}\right)_{0}\xi^{\beta}\xi^{\gamma}\xi^{\delta}s^{3}- \cdots}\\ \end{array}
h_{ij}(q)\int_{\mathbb{R}_{+}^{n}}t\frac{\partial U}{\partial z_{i}}\frac{\partial U}{\partial z_{j}}dzdt=h_{ii}(q)\int_{\mathbb{R}_{+}^{n}}\frac{tz_{i}z_{i}dzdt}{\left[(1+t)^{2}+|z|^{2}\right]^{\frac{n}{2}}}=0
\begin{array}{ll}\displaystyle{Q_s:=\sigma^2 \left[(t^1)^{-1}_s -(t^2)^{-1}_s\right] - \sigma^2 \left[(t^1)^{-1}_{\tau_0}-(t^2)^{-1}_{\tau_0}\right],}\displaystyle{R_s:=\frac{\sigma^2}{2}\left(\int_{\tau_0}^{s}\frac{u_r^2 \left(|u_r^2|^2-1\right)-u_r^1 \left(|u_r^1|^2-1\right)}{u_r^1 \left(|u_r^1|^2-1\right) \times u_r^2 \left(|u_r^2|^2-1\right)}dr\right),}\end{array}
{\bf P}^{\pi}(dy^t, dx^t) = \prod_{i=0}^tq_i(dy_i|y_{i-M}^{i-1}, x_i)\pi_i(dx_i|y^{i-1}), \nu_t^{\pi}(dy_t|y^{t-1}) = \int_{{\cal X}_t}q_t(dy_t|y_{t-M}^{t-1}, x_t)\pi_t(dx_t|y^{t-1}),~t\in\mathbb{N}_0^{n}.
\hat{X}= \sum_{k=-1}^n p^{-k}\star a_k(x) - \epsilon a'(x) = \sum_{k \geq 0}\frac{\epsilon^k}{k!}\partial_p^k\partial_x^k f(p) - \epsilon a'(x), &\hat{Y}= \sum_{k=-1}^n p^{-k}\star b_k(x) - \epsilon b'(x) = \sum_{k \geq 0}\frac{\epsilon^k}{k!}\partial_p^k\partial_x^k g(p) - \epsilon b'(x).
G^{(k)}_{ab,cd}(x_1,x_2;x_3,x_4)=e^{ik\theta}\frac{1}{Z[0,0,0]}\left. \frac{\delta^4Z^{(k)}[\eta,\overline{\eta},J]}{\delta\overline{\eta}_a(x_1)\delta\overline{\eta}_b(x_2) \delta\eta_c(x_3) \delta\eta_d(x_4)}\right|_{\eta,\overline{\eta},J=0}\; .
\frac{q_2}{q_1}\prod^{\infty}_{n=1}\left(\frac{1-q_2^n}{1-q_1^n}\right)^{4n^{2\nu-2}\nu/B_{2\nu}}=\exp\left(2\pi i\int^{z_2}_{z_1}E_{2\nu}(z)dz\right),
\frac{P_{\lambda}\big(q^{n_1}t^{N-1}, \dots, q^{n_m}t^{N-m}, t^{N-m-1}, \dots, t, 1; q, t\big)}{P_{\lambda}\big(t^{N-1}, t^{N-2}, \dots, 1; q, t\big)}{}= \frac{Q_{(n_1, \dots, n_m)}\big(q^{\lambda_1}t^{N-1}, q^{\lambda_2}t^{N-2}, \dots, q^{\lambda_N}; q, t\big)}{Q_{(n_1, \dots, n_m)}\big(t^{N-1}, t^{N-2}, \dots, 1; q, t\big)}.
\int\limits^{\infty}_{0}\!dt\,Im\,T_{1}(t)=-\frac{m^2}{4}\int\limits^{\infty}_{4m^{2}}\!\!\frac{dt}{t^{2}}\left(1-\frac{4m^{2}}{t}\right)^{\hspace{-3pt}\frac{1}{2}}=-\frac{1}{24}\enspace .
R(k\delta)=R(x_0)+\sum_{j=1}^{\lfloor\beta\rfloor}\tfrac{1}{j!}R^{(j)}(x_0)(k\delta -x_0)^j + \tfrac{1}{(\lfloor \beta\rfloor +1)!}R^{(\lfloor \beta\rfloor+1)}(\xi_k)(k\delta -x_0)^{\lfloor \beta\rfloor +1},
C_{p(x)}:=\begin{pmatrix}{0}{0}{0}\cdots{0}{-a_0}{1}{0}{0}\cdots{0}{-a_1}{0}{1}{0}\cdots{0}{-a_2}begin{align*}-7pt]\vdots \vdots \ddots \vdots \vdots begin{align*}-7pt]{0}{0}0 \ddots{0}{-a_{d-2}}{0}{0}{0}\cdots{1}{-a_{d-1}}\end{pmatrix}
S_{e f f}[ A_{\mu}] ={\frac{1}{4 g_{4}^{2}}}\int d^{D}x F_{\mu \nu}F^{\mu \nu}+{\frac{1}{2}}\int{\frac{d^{D}q}{( 2 \pi )^{D}}}A_{\mu}( q ) \Pi^{\mu \nu}( q^{2}) A_{\nu}( - q ) ,
\mathcal{V}(\tau) = \sum_{n=1}^{\infty}\left ( \frac{n}{13}\right) \frac{q^{n}}{(1 - q^{n})^{2}}, V(\tau) = \sum_{n=1}^{\infty}\left ( \frac{n}{5}\right) \frac{q^{n}}{(1 - q^{n})^{2}}.
\lvert\langle K, z \rangle\rvert = \biggl|\sum_{j=3}^n b_j \biggl(r(C_j(m_j)) \, - \, \Bigl(Q(C_j,\tau_2)-Q(C_j,\tau_3)\Bigr)\, r({C}_2(m_2)) - \, \Bigl(Q(C_j, \tau_1)+iQ(C_j,\alpha_1)-Q(C_j, \tau_3)\Bigr)\, r(C_1(m_1))\biggr)\biggr|.
S_1(z, q, N):=\sum_{n=1}^{N}\left[\begin{matrix}N n\end{matrix}\right]_{q^2}\frac{(q^2;q^2)_{n}(q^2;q^2)_{n-1}(zq;q^2)_{N-n}(zq)^{n}}{(zq^2;q^2)_{n}(zq;q^2)_{N}}.
|H_k(x,y)| =\bigg|\int_{\Bbb R^N}D_k(x,z)\big[S_{k-M-1}(z,y)-S_{k-M-1}(x,y)\big]d\omega(z)\bigg| \leqslant C\int_{|x-z|\leqslant r^{2-k}}(V_k(x))^{-1}\frac{r^{k-M-1}\|x-z\|}{V_{k-M-1}(x)}d\omega(z) \leqslant Cr^{-M}(V_{k-M-1}(x))^{-1}.
\bigg\langle \omega_1, \omega_2, \dots, \omega_n \ \bigg\vert\ \begin{matrix}\omega_r \omega_n^q \omega_{i-r}=\omega_i \omega_n^q~r < i \leq n, \omega_r \omega_n^q \omega_{n+i-r}=\omega_i \omega_n^{q+1}~1\leq i <r.\end{matrix}\ \bigg\rangle