harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\dot{\widetilde{\delta p}}+ \left( 3{\frac{\dot{a}_{b}}{a_{b}}}+{\frac{\dot{n}_{b}}{n_{b}}}\right) \widetilde{\delta p}= k^{2}\widetilde{\delta \pi}+ 6 \left({\frac{\rho + P}{\lambda}}\right) \left({\frac{\dot{a}_{b}}{a_{b}}}\delta p -{\frac{k^{2}}{2}}\delta \pi \right) .
\sum_{n=1}^{\infty}\left\{\left(\widetilde{C}+C+c_{1}C\right) \frac{\sqrt[]{h_{H}(f^{k(n-1)}(P))}}{\rho_{k}^{n}}+\frac{C \sqrt[]{\gamma}}{\rho_{k}^{n}}\right\}\leq \sum_{n=1}^{\infty}\left\{\left(\widetilde{C}+C+c_{1}C\right) \frac{\sqrt[]{\widetilde{K}h_{H}(P)}(n-1)\rho_{k}^{(n-1)/2}}{\rho_{k}^{n}}+\frac{C \sqrt[]{\gamma}}{\rho_{k}^{n}}\right\}.
D =\prod_{2l+1\in A_0}\langle d_{2l+1}\rangle\times\prod_{2^{n-1}-(2l+1)\in B_0}\langle d_{2^{n-1}-(2l+1)}\rangle= &=\prod_{2l+1\in A_0}\langle d_{2l+1}\rangle\times\prod_{2^{n-1}-(2l+1)\in B_0}\langle d_{2l+1}q(0,2l+1)\rangle= &=\prod_{2l+1\in A_0}\langle d_{2l+1}\rangle\times\prod_{2^{n-1}-(2l+1)\in B_0}\langle q(0,2l+1)\rangle.
\frac{1}{|\mathcal F_{N,k}|}\sum_{f \in \mathcal F_{N,k}}\left(R_2 (g,\rho) (f) - \frac{T(g,\rho)}{4 L}\right)^2 &\ll \frac{1}{L}+ \frac{L^2 (\log \log x)^2}{\pi_N(x)^2}+ \frac{L(\log \log x)^2}{\pi_N(x)}+ \frac{L^{1/2}\log \log x}{\pi_N(x)^{1/2}}& + \frac{x^{C\pi_N(x)}8^{\nu(N)}}{k{N}},
E(\prod_i N^{(1)}_{(x_i,y_i)}\prod_l (N^{(1)}_{z_l}+1))=E(\prod_i \dfrac{1}{2}C_{(x_i,y_i)}{\varphi_{x_i}\bar{\varphi}_{y_i}}\prod_l \frac{1}{2}{\lambda_{z_l}\varphi_{z_l}\bar{\varphi}_{z_l}}).
\tau(x)\cdot\dot{\omega}\tau^{-1}(x)\dot{\omega}^{-1}={\rm diag}(1,xE_{d-2},1,1,x^{-1}E_{d-2},1){}=(\frac{d-2}{2}\alpha^{\vee}_{d-1}+\frac{d-2}{2}\alpha^{\vee}_{d}+\sum_{i=2}^{d-2}(i-1)\alpha^{\vee}_i)(x),
\sup_{v_i\in{\cal{G}}_i,\, i=1,\!...,m}\left\{-\big(f^*\Box h^*\big)\left(-\sum_{i=1}^{m}L_i^*v_i\right)-\sum_{i=1}^{m}\big(g_i^*(v_i)+l_i^*(v_i) \big) \right\}.
f\biggr(\frac{F_{k_0+2}}{F_{k_0+1}},\dots,\frac{F_{k_0+n+1}}{F_{k_0+n}}\biggr)\!=\biggr(\frac{F_{k_0+3}}{F_{k_0+2}},\dots,\frac{F_{k_0+n+1}}{F_{k_0+n}},\frac{b_1F_{k_0+n+1}+b_2F_{k_0+n}+\dots + b_n F_{k_0+2}}{F_{k_0+n+1}}\biggr).
E [ f^{( 0 )}+ f^{( 1 )}] = E [ f^{( 0 )}] + \frac{\delta E}{\delta f}[ f^{( 0 )}] \cdot f^{( 1 )}+ \frac{1}{2}\frac{\delta^{2}E}{\delta f^{2}}[ f^{( 0 )}] \cdot ( f^{( 1 )})^{2}+ \cdots ,
\Lambda^{\alpha}(uv)- u\Lambda^{\alpha}v =\big(\Lambda^{\alpha}T_{u}v -T_{u}(\Lambda^{\alpha}v)\big)+(\Lambda^{\alpha}T_{v}u - T_{\Lambda^{\alpha}v}u) + \big(\Lambda^{\alpha}R(u,v)-R(u,\Lambda^{\alpha}v)\big)=: \sum_{j=1}^3 \Pi_j.
\int_{B_{R}}\int_{0}^{t}S_{\alpha}(t-s)f(S_{\alpha}(s)v_{0})ds \geq \mu(d,\alpha)c(d,\alpha,\tau)\sum_{k}f(c(d,\alpha)\phi_{k})(\frac{\psi}{\phi_{k}})^{1+\alpha/d}&=c\psi^{p}\sum_{k}f(s_{k})s_{k}^{-p},
x_k' = x_k^{-1}\left( \prod_{i\in I}x_i^{[b_{ik}]_+}+ \prod_{i\in I}x_i^{[-b_{ik}]_+}\right) = x_k^{-1}\left(1 + \prod_{i\in \Omega(k,-)}x_i^{-b_{ik}}\right).
\widehat{T(u^1, \cdots, u^p)}(k) = \sum_{k=k_1-k_2+\cdots +k_p}m(k_1, \cdots, k_n) \prod_{\substack{\ell=1 odd}}^p\widehat{u^\ell}_{k_\ell}\prod_{\substack{\ell=2 even}}^p\overline{\widehat{u^\ell}_{k_\ell}}
h_i(z)=\begin{cases}q_i(z), \mbox{if}\a_{ij}\neq 0\ \mbox{for each}\j; q_i(z)\cdot \frac{1}{z_{j_1}}\cdot \ldots \cdot \frac{1}{z_{j_k}}, \mbox{if}\a_{i{j_1}}=\ldots =a_{i{j_k}}=0.\end{cases}
(I- \Delta) \psi_i(r) = \psi_i(r) - \psi_i''(r) - \tfrac{n-1}{r}\psi_i'(r) &= \tfrac{2i}{r}\psi'_i(r) - \tfrac{n-1}{r}\psi_i'(r) &= (n-1-2i)\psi_{i+1}(r)
\Gamma_0\!:=\!\inf_{z\in\mathbb{R}^n}\!\Big\{\sup_{u\in\mathcal{A}(F(\overline{x}),dF(\overline{x})(w))}\!\langle u,d^2F(\overline{x})(w|z)\rangle-\langle\overline{v},z\rangle\Big\}=\langle \overline{\xi},d^2 F(\overline{x})(w)\rangle.
u\wedge v=|\begin{matrix}u_{2}&v_{2}\\u_{3}&v_{3}\end{matrix}|(e_{2}\wedge e_{3})+|\begin{matrix}u_{1}&v_{1}\\u_{3}&v_{3}\end{matrix}|(e_{1}\wedge e_{3})+|\begin{matrix}u_{1}&v_{1}\\u_{2}&v_{2}\end{matrix}|(e_{1}\wedge e_{2}).
= \frac{1}{2}\times \frac{3}{2}\times \frac{2}{3}\times \frac{4}{3}\times \frac{3}{4}\times \frac{5}{4}\times \cdots \times \frac{2 0 1 2}{2 0 1 3}\times \frac{2 0 1 4}{2 0 1 3}
\mathcal{P}_N = \frac{T}{2}\sum_{k=N+1}^\infty \|\boldsymbol{u}_k\|_{\Omega}^2 = \frac{T}{2}\sum_{k=N+1}^\infty \left(\|\boldsymbol{u}_k^c\|_{\Omega}^2 + \|\boldsymbol{u}_k^s\|_{\Omega}^2\right) = \|\boldsymbol{u}- \boldsymbol{u}_N\|
\begin{cases}-dY_{t}^{i}=f^{i}(t,\eta_{t},Y^{i}_{t},Z^{i}_{t},Y^{i}_{t+\delta(t)}) dt - Z^{i}_{t}dB_{t}^{H}, \ \ \t\in [0,T]; Y_{t}^{i}=g^{i}(\eta_{t}), \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \t\in[T,T+K]. \end{cases}
\sum_{2r_0+r_1=\lambda}{d_1+1 \choose r_0}{d_1+1-r_0 \choose r_1}\frac{(d_2+2)!}{2^{r_2}(d_2-2d_1+\lambda)!}{n_1 \choose d_1+1}{n_2 \choose d_2+2}.
\begin{bmatrix}K_1^- W_1 \end{bmatrix}=\begin{bmatrix}0 0 0 0 K_{1,22}^- K_{1,23}^- 0 K_{1,23}^{-\top}K_{1,33}^- W_{11}W_{12}W_{13}\end{bmatrix}=\begin{bmatrix}0 0 D_3 \end{bmatrix}\,.
T_{a}= \left( \begin{array}{c c c}{{V_{a}^{( \mathrm{c )}}}}&{{0}}&{{0}}\\{{0}}&{{( \omega V^{( \mathrm{c )}})_{a}}}&{{0}}\\{{0}}&{{0}}&{{( \omega^{2}V^{( \mathrm{c )}})_{a}}}\\ \end{array}\right)
min_{\hat{\alpha},\hat{\beta}}\sum_{i=1}^{n}[y_{i}-(\overline{y}-\hat{\beta}\overline{x})-\hat{\beta}x_{i}]^{2}=min_{\hat{\alpha},\hat{\beta}}\sum_{i=1}^{n}[(y_{i}-\overline{y})-\hat{\beta}(x_{i}-\overline{x})]^{2}
[\hat{C}^{ib},\hat{Q}^a]_{+}=i\varepsilon^{ab}\hat{\pi}_{(B)}^{0i},\;\; [\hat{\pi}_{(C)ib},\hat{Q}^a]_{+}=i\varepsilon_{0ijk}\delta_b^a (\partial^j\hat{A}^{k}+\frac{1}{2}\hat{A}^j\wedge\hat{A}^k),
\begin{array}{l}{\cal K}={\cal A}+VS_{m}^{\dagger}+S_{m}W^{\dagger}+S_{m}\tilde{{\cal K}}S_{m}^{\dagger}=\left(\begin{array}{cc}{\cal A}V W^{\dagger}\tilde{{\cal K}}\end{array}\right)\,, \varphi=XS_{m}^{\dagger}+S_{m}\tilde{h}S_{m}^{\dagger}=\left(\begin{array}{cc}0 X 0 \tilde{h}\end{array}\right)\,.\end{array}
0 =f(\bar x, \bar t v(\bar x))- f(\bar x, u(\bar x))= \frac{f(\bar x, \bar t v(\bar x))}{\bar t v(\bar x)}\bar t v(\bar x)-f(\bar x, u(\bar x)) &\ge \bar tf(\bar x, v(\bar x))- f(\bar x, u(\bar x))\ge L(\bar t v-u)(\bar x) \ge 0
\partial_t u+(\mathbf{v}_\varepsilon \nabla c_{\varepsilon}-\mathbf{\tilde{v}}_A \nabla c_A)+\frac{\mu_{\varepsilon}-\mu_A}{\varepsilon}=\mathbf{w}|_\Gamma \, \nabla c_{A,0}-\mathbf{w}|_\Gamma \, \mathbf{Q}-\mathcal{C}.
\sigma^{\mu \nu}= \frac{i}{2}\left[ \gamma^{\mu}\; , \; \gamma^{\nu}\right] \; , \; \; \left( F ( x ) K ( x ) \right)^{\ast \mu \nu}= \frac{1}{2}\varepsilon^{\mu \nu \alpha \beta}\left( F ( x ) K ( x ) \right)_{\alpha \beta}\; ,
g\left( z,s;B_{-b},A_{-c};R_{1},R_{2}\right) =\sum\limits_{\overset{m+R_{1}\leq0}{d(m+R_{1})>c(n+}R_{2})}\frac{f^{\ast}(-mb)f(-nc)}{((m+R_{1})z+n+R_{2})^{s}}.
\Tilde{\phi}_{e}= \phi_1 - C_{\Tilde{q}}h^{\Tilde{q}}, \Tilde{q}= \ln \Big( \frac{\phi_3 - \phi_2}{\phi_2 - \phi_1}\Big) / \ln r, C_{\Tilde{q}}h^{\Tilde{q}}= \frac{\phi_2 - \phi_1}{r^{\Tilde{q}}- 1}.
U ( x_{-}) = V ( x_{-}) + \left[ \frac{1}{f}\frac{{\partial}^{2}f}{{\partial}x_{-}^{2}}- \left( \frac{1}{f}\frac{{\partial}f}{{\partial}x_{-}}\right)^{2}\right] \cdot \frac{4 c^{2}P_{c m}^{2}+ f^{2}}{4 c^{2}P_{c m}- f^{2}}- \frac{8}{f^{2}}\cdot \frac{c^{2}P_{c m}^{2}}{4 c^{2}P_{c m}^{2}- f^{2}}\left( \frac{{\partial}f}{{\partial}x_{-}}\right)^{2}.
\langle v_{2l-1},v_{2l}\rangle =\langle \frac{u_l+u_{n-l}}{2},\frac{u_l-u_{n-l}}{2i}\rangle =\frac{1}{4i}\left( |u_l|^2-|u_{n-l}|^2-2i \Im \langle u_l,u_{n-l}\rangle \right)=0;
\|\varphi\|_{H^2}\leq c\left(\|A\|_{L^2}+\|\varphi\|_{L^2}\right), c =\max\left\{1,\|V_0\|_{L^\infty}+\|u\|_{C(0,T)}\|V_u\|_{L^\infty}+\|\varphi\|_{H^1}^2+L+K\right\}.
\widehat{\Delta \mathbf{s}}=\mathbf{T}_x \Delta \mathbf{s}=\mathbf{T}_x\mathbf{X}^{-1}(\nu\mu\mathbf{e}-\mathbf{X}\mathbf{s}-\mathbf{S}\Delta \mathbf{x}) =\nu \mu \mathbf{e}- \mathbf{w}_{xs}-\mathbf{R}_{xs}\widehat{\Delta \mathbf{x}}.
C^{(1,1,1,1)}_{2222}=-4C^{(1,1,2)}_{22\gamma_1}C^{(1,1,2)}_{22 \gamma_1}+C^{(1,1,2)}_{22\gamma_2}C^{(1,1,2)}_{22 \gamma_1}+C^{(1,1,2)}_{22\gamma_1}C^{(1,1,2)}_{22 \gamma_2}
i_{\sigma \setminus \ell_{s}\cup 1}=(-1)^{s-1}(-x_1)\prod_{q=2}^{s}{D_{\ell_{q-1}}}\prod_{q=s+1}^{d+1}{D_{\ell_q-1}}&=(-1)^{s}x_1\prod_{q=2}^{s}{D_{\ell_{q-1}}}\prod_{q=s+1}^{d+1}{D_{\ell_q-1}}.
Z_{\Delta}(U) =\sum_{n}e^{-\frac{n \lambda A}{2}}\sum_{i}\frac{(\lambda A)^i}{i!}\sum_{\sigma}\left[\frac{(-1)^i}{C_{\sigma}}\phi^{\sigma}_{n,i}\Upsilon_{\sigma}(U) N^{n - i}+{\cal O}( N^{n - i - 1}) \right],
( 1 + 2^{2}+ 3^{2}+ \cdots 4 9^{2}) ( 2^{2}+ 3^{2}+ \cdots 5 0^{2}) - ( 1 + 2^{2}+ 3^{2}+ \cdots 5 0^{2}) ( 2^{2}+ 3^{2}+ \cdots + 4 9^{2})
\mathcal{X}^{t+1}=\mathcal{X}^{t}-\mathcal{Q}^{-1}*\mathcal{A}^{T}*\mathcal{S}*(\mathcal{S}^{T}*\mathcal{A}*\mathcal{Q}^{-1}*\mathcal{A}^{T}*\mathcal{S})^{\dag}*\mathcal{S}^{T}*(\mathcal{A}*\mathcal{X}^{t}-\mathcal{B}),
T_{h}H_{k,\{X;Y_{l-h,j-h}\}}= \sum_{s=1}^{k}H_{k,\{(l_{1},\ldots,l_{s}+h,\ldots,l_{k}); Y_{l-h,j-h}\}},T_{l}H_{k,X}=\sum_{s=1}^{k}H_{k,(l_1,\ldots,l_{s}+l,\ldots,l_{k})}.
\begin{pmatrix}Z_{l}Q_{l}\end{pmatrix}(t) = S_{l}(t,0) \begin{pmatrix}Z_{l}Q_{l}\end{pmatrix}(0) + \int_{0}^t S(t,\tau) (c_{l}^{\pm}, d_{l}^{\pm}) \begin{pmatrix}Z_{l\pm 1}Q_{l\pm 1}\end{pmatrix},
A(n) P_n(x) - b_n P_{n-1}(x) = \frac{n+\gamma+1}{\gamma+1}R_n(\lambda(x);\gamma+1,\delta-1,N-1) \hphantom{A(n) P_n(x) - b_n P_{n-1}(x) =}{}- \frac{n}{\gamma+1}R_{n-1}(\lambda(x);\gamma+1,\delta-1,N),
X_s^{t_0,x_0}=x_0+\int_{t_0}^{s\wedge \tau_S}\Gamma X_r^{t_0,x_0}dr+\int_{t_0}^{s\wedge \tau_S}\sigma X_r^{t_0,x_0}dB_r+\int^{s\wedge \tau_S}_{t_0}\int X_{r-}^{t_0,x_0}\gamma(r,z) \tilde{N}(dr,dz), \mathbb{P}-{\rm a.s}, X^{t_0,x_0}_{t_0}:= x_0,
\langle Op_\mathcal{A}(a)f,g\rangle=\langle a,\mu(\mathcal{A})(f\otimes \bar{g})\rangle=\langle \mu(\mathcal{A}^{-1})a,f\otimes \bar{g}\rangle, &\langle Op_\mathcal{B}(b)f,g\rangle=\langle b,\mu(\mathcal{B})(f\otimes \bar{g})\rangle=\langle \mu(\mathcal{B}^{-1})b,f\otimes \bar{g}\rangle.
\frak{d}(\mathfrak{S}_{3,d=2}) :=\frak{d}(\mathfrak{S}_3) - \frak{d}(\mathfrak{S}_{3,\delta=0}) = \frac{1}{p-1}- \frac{(p^2-p-1)}{(p-1)(p^2-1)}& = \frac{p}{(p-1)(p^2-1)}.
\Delta_{\varepsilon , \overline{{{\varepsilon}}}}^{( 0 )}= \left( 1 - \frac{\sigma^{2}}2 \right) \cdot \frac{m - i \varepsilon \widehat{p}}{2 m}\cdot \frac{m - i \overline{{{\varepsilon}}}\overline{{{p}}}}{2 m}\cdot \left( 1 - \sigma_{p}^{2}\right) = \Psi_{\varepsilon , , \overline{{{\varepsilon}}}}^{( 0 )}\cdot \overline{{{\Psi}}}_{\varepsilon , \overline{{{\varepsilon}}}}^{( 0 )},
[ j_+{}^n, (j_+{}^m)^\dagger ]= \lim_{s\to 0 \atop r\to 0}\sum_p \left ( \frac{1}{\lambda_{p+m,s}\lambda_{p+n,r}}- \frac{1}{\lambda_{p,s+r}}\right ) a^\dagger_{p+m}a_{p+n},
\begin{aligned}\limsup_{h\to 0}h^{n-1}\|\chi_j u_\Sigma\|_{L^\infty(B(x_0,r(h)))}^2 \leq C_{n,l}{4}\delta_0^{-1}\alpha^{n-1}\int_{\Lambda_{x_0,3\delta}}\chi_j^2d\mu &+C_{n,l}\alpha^{n-2l-1}\sum_{i=2}^{n}\int_{\Lambda_{x_0,3\delta}}\chi_j^2({4}\delta_0^{-1}q_{j,i}^2+{C_0^2}\delta_0|H_pq_{i,j}|^2)d\mu\end{aligned}
d s^{2}= ( 1 + 2 M / r )^{- 1}( \, d t + 2 N \cos \theta \, d \varphi )^{2}- ( 1 + 2 M / r ) \, d r^{2}- r ( r + 2 M ) ( \, d \theta^{2}+ \sin^{2}\theta \, d \varphi^{2}) \, .
\frac{N^{\prime}}{\mu}( 2 - \frac 2{x_{i}^{2}}) + 2 \sum_{j \neq i}\left( \frac 1{x_{i}- x_{j}}- \frac 1{x_{i}+ x_{j}}\right) + \sum_{a = 0 , 1}\frac 2{x_{i}- \lambda_{a}}- \frac 2{x_{i}+ \lambda_{a}}
\hat x_e= \sum_{x_e}x_e \mu_e(x_e|B,A) = \frac{\tilde x_e (\theta B_e + (1-\theta) (1-B_e))}{1-\eta-(1-2\eta) B_e + \tilde x_e (2B_e-1) ( \theta- \eta)}=\begin{cases}\frac{(1-\theta) \tilde x_e}{1-\eta - \tilde x_e (\theta-\eta)}B_e=0 \frac{\theta \tilde x_e}{\eta + \tilde x_e (\theta-\eta)}B_e=1 \end{cases}.
F_4=\, a_1+b+b^2+a_1 b^2+b^3+b^4+a_1 b^4+b^5+b^6+a_1 b^6+b^7+b^8 +(a_1^3 +a_1^2 b+a_1^3 b^4 +a_1^2 b^5)k+(a_1^4+a_1^5+a_1^4 b+a_1^4 b^2+a_1^5 b^2+a_1^4 b^3)k^2\cr +(a_1^7 +a_1^6 b)k^3+a_1^8 k^4,
(4s^{3}-s)M_N^{(1)\prime\prime}+s(16s+2(\alpha-1)+4N)M_N^{(1)\prime}+s(9-\alpha^{2})M_N^{(1)}=(3s+3)M_{N-1}^{(2)\prime}-(3\alpha+6N-6)M_{N-1}^{(2)}\ .
\frac{d}{ds}\sigma(\tilde{t}+s, X (\tilde{t}, \tilde{x_1},\tilde{x_2}, \tilde{x_3}, \tilde{m}; s)) = - \sigma (\tilde{t}+s,X (\tilde{t}, \tilde{x_1}, \tilde{x_2}, \tilde{x_3}, \tilde{m}; s)) \times
\mathbb{A'}^{-1}_L =(\Delta')^{-1}\begin{pmatrix}\Delta' (a-bd^{-1}c)^{-1}-b-([d,b]-[d,a]a^{-1}b)(d-ca^{-1}b)^{-1}-\Delta' (ac^{-1}d-b)^{-1}a-([b,c]-[a,c]a^{-1}b)(d-ca^{-1}b)^{-1}\end{pmatrix}.
\det\begin{pmatrix}F(t_1)t_1^{k_1}F(t_2)t_2^{k_1}\dots F(t_n)t_n^{k_1}t_1^{k_2}t_2^{k_2}\dots t_n^{k_2}\vdots \vdots \ddots \vdots t_1^{k_n}t_2^{k_n}\dots t_n^{k_n}\end{pmatrix}
\varepsilon_{k+1}= \frac{A_{k+1}}{\delta}\left( f(y_{k+1}) - f(x_k) + \frac{\tau_k^2}{2\mu}\|\nabla f(x_k)\|^{2}+\left(\frac{\tau_k}{2\epsilon}-\frac{\mu}{2\tau_k}\right)\|x_k - y_k\|^2 \right).
[X(\zeta),R_{p}(\chi) ]+\left(\sum_{r\geq 1}4r\left(\zeta/\chi\right)^{r}\right)X(\zeta) R_{p}(\chi) - \left(\sum_{r\geq 1}4r\left(\chi/\zeta\right)^{r}\right)R_{p}(\chi)X(\zeta) &=2(p+1)D\delta(\zeta/\chi)R_{p+1}(\chi)-2\delta(\zeta/\chi)D_{\chi}R_{p+1}(\chi).
\frac{1}{2\pi i}\left[\int_{\lambda - i T}^{\lambda + i T}+ \int_{\lambda + i T}^{\mu + i T}+\int_{\mu + i T}^{\mu - i T}+ \int_{\mu - i T}^{\lambda - i T}\right] \Gamma(s) \zeta(s) \zeta(s+ 2m + 1) y^{-s}\,{\rm d}s = R_{-2m}+R_{0}+R_{1}+ \sum_{i=0}^m R_{-(2i+1)}.
\sum_{j}\hat{A}_{ij}\hat{A}_{jk}=\hat{H}\hat{A}_{ik}+\frac{1}{4}\omega^{2}\{\hat{L}_{i}\hat{L}_{k}-\delta_{ik}\hat{L}^{2}+2[\hat{L}_{i},\hat{L}_{k}]-2\hbar^{2}\delta_{ik}\}
u_{h}^{(e)}(x)=u_{1}^{e}N_{1}^{(e)}(x)+u_{2}^{e}N_{2}^{(e)}(x),v_{h}^{(e)}(x)=u_{1}^{e}N_{1}^{(e)}(x)+u_{2}^{e}N_{2}^{(e)}(x)
S=\int d^3x\left[\frac12(\nabla\chi)^2+\frac12(\nabla\psi)^2+\frac{m_H^2}{2}\psi^2-2\zeta{\rm e}^{g_m\psi}\cos(g_m\chi)\right]\equiv\int d^3x{\cal L}[\chi,\psi|g_m,\zeta].
\eta_{\bf i, \mathcal{I}}^{\ast}\mathcal{L}_{\mathcal I, \lambda_1,\dots,\lambda_r}=\{(p,q) \in Z_{\bf i}\times \mathcal{L}_{\mathcal I, \lambda_1,\dots,\lambda_r}\mid \eta_{{\bf i}, \mathcal I}(p) = \pi_{\mathcal I, \lambda_1,\dots,\lambda_r}(q)\},
dy_{u}=\sum_{k=1}^{\left[ p\right]}F\left( f\right)^{\circ k}\pi_{k}\left( \log g\right) \left( Id_{L^{\left[ p\right]}\left( \mathcal{U}\right)}\right) \left( y_{u}\right) duu\in \left[ 0,1\right] y_{0}=1
A_{v,w}= H_f(1_{Gw})(v) &= f(v)^{-1}\sum_{y \in X}b(v,y)(f(v)1_{Gw}(v) - f(y)1_{Gw}(y)) + c(v)1_{Gw}(v) &= -f(v)^{-1}\sum_{y \in Gw}f(y)b(v,y) \leq 0.
|u_E(t,x+y)-u_E(t,x)|^3 \le\biggl( [\mathbb{I}_{A}(x)+\mathbb{I}_{B}(x)+\mathbb{I}_{C}(x)]|u_E(x+y) -u_E(x)|\biggr)^3 &= \left[\mathbb{I}_{A}(x)+\mathbb{I}_{B}(x)+\mathbb{I}_{C}(x)\right]\,|u_E(x+y) -u_E(x)|^3.
{\cal L}_{eff}= -\frac{1}{4}(\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+gf^{abc}A_{\mu}^{b}A_{\nu}^{c})^{2}-\frac{1}{2}(\partial_{\mu}A^{a\mu})^{2}
\mathbb{E}(D_{n}) = \frac{1+\sqrt{2}}{2}\binom{n+\sqrt{2}-2}{n-1}- \frac{\sqrt{2}-1}{2}\binom{n-\sqrt{2}-2}{n-1}\sim \frac{1+\sqrt{2}}{2}\frac{n^{\sqrt{2}-1}}{\Gamma(\sqrt{2})}.
\widetilde{\mathcal{V}}^{(h-1)}(\psi^{(\leq h-1)}) +E_{h-1}\; =\; \log \left( \int P_{g^{(\leq h-1)}}e^{\mathcal{V}^{(h)}(\psi^{(\leq h-1)}+\psi^{(h)})}\right) &=\sum_{r=1}^{+\infty}\frac{1}{r!}\mathbb{E}^T_{h}(\mathcal{V}^{(h)}(\psi^{({\leq}h-1)}+\cdot);r)
\sum_{k_1,\cdots,k_r=-n}^n \prod_{1\leq i<j\leq r}|[k_i-k_j]_q|\prod_{i=1}^rq^{(k_i+n-r+i)^2/2}\left[\begin{array}{c}2n n+k_i\end{array}\right] &\quad=\frac{r!}{|r|_{q^{1/2}}!}\prod_{i=1}^r (-q^{1/2};q^{1/2})_i(-q^{i/2+1};q)_{2n-r}\prod_{i=1}^r\frac{\Gamma_q(1+i/2)}{\Gamma_q(3/2)}\frac{\Gamma_q(2n+1)\Gamma_q(2n-i+5/2)}{\Gamma_q(2n-i+2)\Gamma_q(2n-i/2+2)}.
\langle\langle [L_j,L_{\bar k}]\cdot\bar\eta,\cdot\bar\zeta\rangle\rangle =\int_{\partial D_t}b_{j\bar k}(\rho) \langle \cdot\bar\eta, \cdot\bar\zeta\rangle d \sigma+ \langle\langle c_{j\bar k}(\phi,V)\cdot\bar\eta,\cdot\bar\zeta\rangle\rangle,
|\partial_{t}^{2}\left(-\lambda(t) \langle \left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}}(r))-1}{r^{2}}\right)E_{5}\vert_{r=R\lambda(t)},\phi_{0}\rangle\right)| \leq \frac{C}{t^{4}\log^{b+1}(t)}+ \frac{C \sup_{x \geq t}\left(\frac{|e''''(x)| x}{\lambda(x)^{3-2\alpha}}\right)}{t \log^{(3-2\alpha)b}(t)}
Z_{\ell}^{J}(\vec{z})=\prod_{\begin{subarray}{c}i \in J i<\ell \end{subarray}}f(z_{\ell}, z_{i})\prod_{\begin{subarray}{c}i \in J i>\ell \end{subarray}}f(z_{i}, z_{\ell})\prod_{\begin{subarray}{c}i \in J i<\ell \end{subarray}}^{\curvearrowleft}Y_{i-1}(z_{\ell}, z_{i}).
_2(\mathbb{Z}) :=\left\{\begin{bmatrix}a b c d \end{bmatrix}: a, b, c, d \in \mathbb{Z}, ad-bc=1\right\}, \Gamma_{0}(N) :=\left\{\begin{bmatrix}a b c d \end{bmatrix}\in_2(\mathbb{Z}) : c\equiv 0\pmod N \right\},
\sum_{n=0}^\infty{2n \choose n}\frac{(-1)^{n-1}}{4^n (2n-1)}t^n = \sum_{n=0}^\infty \left( \frac{1}{n!}\sum_{m=0}^n \Big( \frac{1}{2}\Big)^m S_1(n,m) \right) t^n
\frac{\bar{F}_{b}( a , b )}{B}= \frac{\bar{F}( b )}{B}+ \bar{P}_{b}( a , b , b ) = \frac{\eta^{2}}{2 4 \pi b^{4}}( c_{1}\Lambda^{\prime}+ c_{2}\Lambda ) + \frac{\eta^{2}\Lambda^{\prime}}{3}P_{b}( a , b , b ) .
\theta(r_0) \sim \theta_0, \; \theta(R_0) \sim \frac{16}{K_0^2}\theta_0|\ln \theta_0|, \; \theta(2R_0) \sim \frac{64}{K_0^2}\theta_0|\ln \theta_0|, \; R_0 = \sqrt{\theta_0}|\ln \theta_0|.
\Gamma_{\mu\nu,\rho}^{(\alpha )}(\boldsymbol{\xi}) =\Gamma_{\mu\nu,\rho}(\boldsymbol{\xi})-\frac{\alpha}{2}T_{\mu\nu,\rho}(\boldsymbol{\xi}) T_{\mu\nu,\rho}(\boldsymbol{\xi}) =\frac{1}{\pi}\int_{-\pi}^{\pi}(\partial_{\mu}\log{S}) ( \partial_{\nu}\log{S}) (\partial_{\rho}\log{S})dw
\pi(K_{k})v_{i}= q^{\delta_{i,k}-\delta_{i,k+1}}v_{i}, \pi(E_{k})v_{i}= \delta_{i}^{k+1}q^{-1/2}v_{i-1}, \pi(F_{k})v_{i}= \delta_{i}^{k}q^{1/2}v_{i+1}.
u=(\mu - \Delta)^{-1}f - (\mu - \Delta)^{-\frac{1}{2}-\frac{1}{q}}Q_{p}(q) (1 + T_p)^{-1}G_{p}(r) (\mu - \Delta)^{-\frac{1}{2}+\frac{1}{r}}f, f \in L^p
x_{(r-1)}c_{q,r}-\sum^{r-1}_{k=1}x_{(r-1)}(q-1)q^{r-k-1}(c_k-c_r) &=x_{(r-1)}\bigg(\sum^r_{k=1}q^{r-k}c_k-\sum^{r-1}_{k=1}(q^{r-k}-q^{r-k-1})c_k+\sum^{r-1}_{k=1}(q^{r-k}-q^{r-k-1})c_r\bigg) &=x_{(r-1)}(q^{r-2}c_1+q^{r-3}c_2+\cdots+c_{r-1}+q^{r-1}c_r)=x_{(r-1)}(c_{q,1,r-1}+q^{r-1}c_r).
\rho_A(t,x) = \det A(t)^{-1}\left[\frac{(\gamma-1)}{2\gamma}(1-|A^{-1}(t)x|^2)\right]^{\frac{1}{\gamma-1}}, \ \{\bf u}_A(t,x) = \dot{A}(t)A^{-1}(t)x,
\delta A = -\delta\bar\theta \Gamma_{11}\Gamma_m \theta \Pi^m +{1\over 2}\delta \bar\theta \Gamma_{11}\Gamma_{m}\theta \bar \theta \Gamma^m d\theta-{1\over 2}\delta \bar\theta \Gamma^m \theta \bar\theta \Gamma_{11}\Gamma_m d\theta.
\Lambda_{\mathcal{S}, r, s', k}(f_k, g_k) \gtrsim \Big|\Big<\sum_{l}T_{a}^{j_k, l}f_k, g_k\Big>\Big| & =\Big|\Big<T_a^{j_k, l(k)}f_k, g_k\Big>\Big|\gtrsim 2^{j_k[\rho n(\frac{1}{r}-\frac{1}{s})+\epsilon'/2]}.
\int \frac{d^{d}{\bf k}}{( 2 \pi )^{d}}\ln ({\bf k}^{2}+ m^{2}) = - \frac{\partial}{\partial s}\left[ \int \frac{d^{d}{\bf k}}{( 2 \pi )^{d}}({\bf k}^{2}+ m^{2})^{- s}\right]_{s = 0}.
2160\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right\}^{(10)}=\left\{\widehat{A}(TX){\rm exp}(\frac{c}{2})\right. &\left.\cdot{\rm ch}(S^2\widetilde{TX}+\widetilde{TX}+\wedge^2\widetilde{L_R\otimes C}-\widetilde{L_R\otimes C}-\widetilde{TX}\otimes\widetilde{L_R\otimes C}\right\}^{(10)}.
\mathbf{x}^{(t+1)}=\underset{\mathbf{x}\in\left\{\tilde{\mathbf{x}}^{(t+1)},\mathbf{0}\right\}}{\arg\min}L^{(t)}(\mathbf{x}) =\begin{cases}\tilde{\mathbf{x}}^{(t)}(\gamma^{(t)}), \textrm{if}L^{(t)}(\tilde{\mathbf{x}}^{(t)}(\gamma^{(t)}))\leq L^{(t)}(\mathbf{0})=0, \mathbf{0}, \textrm{otherwise,}\end{cases}
2\frac{(q^2;q^2)_\infty^3}{(-q;q^2)_\infty^2}\omega(q)= \sum_{n=-\infty}^\infty \frac{(6n+2)q^{3n^2+4n}}{1+q^{6n+2}}-\sum_{n=-\infty}^\infty \frac{6nq^{3n^2+2n-1}}{1+q^{6n}}
\Phi(\rho_k(x)) =\Phi([(\rho_{m,k}(x))_{m>k}]) &=[(\varphi_m(\rho_{m,k}(x)))_{m>k}] &=[((\Theta^{-1}\circ\rho_m)(\rho_{m,k}(x))_{m>k}] &=[(\Theta^{-1}\circ\rho_k(x))_{m>k}] &=\iota\circ\Theta^{-1}\circ\rho_k(x) &=\iota\circ\varphi_k(x).
\|\widetilde U(t)\|_r\leq C(t-\bar t)^{-1/2+3/2r}\|U(\bar t)\|_{3,\infty,\mathbb R^3}, \forall r\in (3,\infty], &\|\nabla \widetilde U(t)\|_{3,\infty}\leq C(t-\bar t)^{-1/2}\|U(\bar t)\|_{3,\infty,\mathbb R^3},
W \left( \begin{array}{c c}{{( d_{1}, d_{2})}}&{{( c_{1}, c_{2})}}\\{{( a_{1}, a_{2})}}&{{( b_{1}, b_{2})}}\\ \end{array}\right) = W \left( \begin{array}{c c}{{( \tilde{d}_{1}, \tilde{d}_{2})}}&{{( \tilde{c}_{1}, \tilde{c}_{2})}}\\{{( \tilde{a}_{1}, \tilde{a}_{2})}}&{{( \tilde{b}_{1}, \tilde{b}_{2})}}\\ \end{array}\right) .
\left[ C_{xy}\right] = \mathbb{E}\left[C_{xy}^2\right]=\kappa \frac{(\alpha-2)^2 \mathbb{E}\left[\mathit{L}_{s}^2\right]}{(\alpha-1) \left( \mathbb{E}\left[\mathit{L}_{s}\right] \right)^2}= \kappa \frac{(\alpha-2)^2}{(\alpha-1)}e^{\frac{\sigma_{SF}^2}{\zeta^2}}, \kappa = \frac{\zeta_2}{8 \pi \lambda_1 \zeta_1^2 R_c^2}.
\begin{array}{c}x=u_i \ \left(\frac{1}{d}\right) \overbrace{\begin{array}{ccccc}t^0_1 t^1_1 \ldots t^{b-1}_1 \theta_1 \vdots \vdots \vdots \vdots t^0_m t^1_m \ldots t^{b-1}_m \theta_m \end{array}}\end{array}
g(u_1,u_2)= \alpha_1 |u_1^2+u_2^2|^2 + \alpha_2 |u_1^2-u_2^2|^2 -4\alpha_2 |u_1|^2|u_2|^2 &+2\alpha_3 (|u_1|^4-|u_2|^4) + (2\alpha_1 + r) (|u_1|^2+|u_2|^2)^2,
\begin{array}{r l}\hat{l}_\alpha=\hat{v}_\alpha(0), \quad\lim_{\alpha\to\infty}\hat{l}_\alpha=:\hat{l}\in [0,1], l_\alpha=v_\alpha^{-1}(x_\alpha), \quad\lim_{\alpha\to\infty}l_\alpha=:l\in [0,\varepsilon^{-1}]\end{array}
\theta'(x)= \frac{h''(x)h(x)-h'^2(x)}{h^2(x)+h'^2(x)}= -\frac{h'^2(x)}{h^2(x)+h'^2(x)}&+\frac{h(x)}{h^2(x)+h'^2(x)}(m-p)\tanh x\frac{h'^2(x)+(m-1)\cos^2h(x)}{(p-1)h'^2(x)+(m-1)\cos^2h(x)}h'(x) &-\frac{h(x)}{h^2(x)+h'^2(x)}\frac{m-1}{2}\frac{(3-p)h'^2(x)+(m-1)\cos^2h(x)}{(p-1)h'^2(x)+(m-1)\cos^2h(x)}\sin 2h(x),
I_{rep}\equiv \mu_0\, \int_\Sigma d^2\sigma\, \Phi^{-1}\mathrm{det}\left[\, \gamma_{ab}(\sigma)\,\right] \ ,\qquad\Phi^{-1}\equiv\epsilon_{ij}\,\epsilon^{mn}\,\partial_m\,\phi^i\, \partial_n\, \phi^j
\left | \frac{N}{\pi}\tilde{G}_{\mu \nu}(E+i\eta)-x(E) \right |=\frac{N \eta}{\pi}\vert G_{\mu \mu}\overline{G}_{\nu \mu}+G_{\mu \nu}\overline{G}_{\nu \nu}\vert \leq N \eta (\mathbf{1}(\mu=\nu)+N^{-1/3+2\epsilon_0}\mathbf{1}(\mu \neq \nu)).
\sum\limits_{k=k(u)}^{\infty}\ln\left(1-\frac{(u\,\gamma_H)^2}{\pi k^{2\beta}}\right) \leq \sum\limits_{k=k(u)}^{\infty}\ln\left(1-\frac{\pi\gamma_H^2}{(2b_H)^2}\left(\frac{k(u)-1}{k}\right)^{2\beta}\right) \leq (k(u)-1)\int\limits_1^{\infty}\ln\left(1-\frac{\pi\gamma_H^2}{(2b_H)^2 x^{2\beta}}\right)dx \leq -\theta_H u^{1/\beta},
\Phi_{k}^{(N)}(f)=\mathbb{E}\left[ f\mid\mathcal{A}^{(k)}\right]+\sum_{\substack{1\leqslant L^{\prime}\leqslant N+1-k k<l_{1}<...<l_{L^{\prime}}<N+1}}(-1)^{L}\mathbf{P}_{\mathcal{A}^{(l_{1})}|\mathcal{A}^{(k)}}\dots\mathbf{P}_{\mathcal{A}^{(l_{L^{\prime}})}|\mathcal{A}^{(l_{L^{\prime}-1})}}\cdot\mathbb{E}\left[f|\mathcal{A}^{(l_{L^{\prime}})}\right].
\frac{n}{\log^4 n}\geq |R|= \frac12\sum_{v\in I}s_{I, I}(v)=\frac14\sum_{v\in I}s_{I}(v)\geq \frac14\sum_{v\in I_h}\frac{\sqrt{n}}{\log^4 n}\geq\frac{n}{4\log^5 n},
| 0 ; g \rangle = A d \, T ( g ) \, | 0 ; e \rangle = \sum_{I , I_{3}, Y}| ( 3 , 0 ) ; ( I , I_{3}, Y ) ; e \rangle D_{( I , I_{3}, Y ) ; ( 0 , 0 , - 2 )}^{( 3 , 0 )}( g )
\frac{1}{( z^{2}- t^{2})^{\kappa + 1}}= \frac{1}{z^{2 \kappa + 2}}\sum_{k = 0}^{\infty}\left( \begin{array}{c}{{\kappa + k}}\\{{k}}\\ \end{array}\right) \left( \frac{t}{z}\right)^{2 k}.
\sum_{j=i}^{i+m-1}H(\mu,\mathcal{D}_{i+1}|\mathcal{D}_{i})\leq \varepsilon\mu(\{x\colon H(\mu^{x,i},\mathcal{D}_m)\leq \varepsilon\})+m \log 2 \left(\mu(\{x \colon H(\mu^{x,i},\mathcal{D}_m)> \varepsilon\}) \right)
\lvert R_6 \rvert_{s_0+\beta}^{Lip(\gamma)}\le C \varepsilon^{7-2 b}\gamma^{-1}=C \varepsilon^{3- 2 a}, \lvert R_6 \rvert_{s_0+\beta}^{Lip(\gamma)}\gamma^{-1}\le C \varepsilon^{1- 3 a}.
{\cal F}_{\mu \nu}^{a b}= F_{\mu \nu}^{a b}+ \Sigma_{\mu \nu}^{a b}+ \Theta_{\mu \nu}^{a b}, \{\cal F}_{\mu \nu}^{i}= F_{\mu \nu}^{i}+ \Sigma_{\mu \nu}^{i}, \{\cal F}_{\mu \nu}^{a}= F_{\mu \nu}^{a}+ \Sigma_{\mu \nu}^{a},
E_1x_{1,k+1}=A_1x_{1,k}+B_1 u_{1,k}, y_{1,k}=C_1x_{1,k}+D_1u_{1,k}, E_2x_{2,k+1}=A_2x_{2,k}+B_2 u_{2,k}y_{2,k}=C_2x_{2,k}+D_2u_{2,k}
{v_T}\left(\lambda - \int \frac{d^2k}{(2\pi)^2}\frac{1}{\sqrt{{\bf k}^2+v_T^2}}\frac{\sinh(\beta \sqrt{{\bf k}^2+v_T^2})}{\cosh (\beta\sqrt{{\bf k}^2 + v_T^2)}+1}\right)= 0 \, \, .
x'(t) =s-dx(t)-\frac{\beta x(t)v(t)}{1+ax(t)+bv(t)}, y'(t) =\frac{\beta x(t)v(t)}{1+ax(t)+bv(t)}-ay(t)-py(t)z(t), v'(t) = ky(t)-u v(t), z'(t) = cy(t-\tau) z(t-\tau)-b z(t).
\bigoplus\limits_{k \geq 2d}R^{k}i_1^*i^*j_*\mu_{\ell}^K(V_{\lambda})[-w(\lambda)-k]_{\vert_{Z^\prime}}\simeq \bigoplus\limits_{k \geq 2d}(\bigoplus_{\substack{p+q=k}}\mu_{\ell}^{\pi_1(K_1)}(V^{p,q}))[-w(\lambda)-k],