harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
0 \geq \lim_{n\rightarrow \infty}\frac{1}{2\pi n\epsilon}\int_{0}^{2\pi}\bigg(\log|f_{n}(\mathrm{e}^{\mathrm{i}x})|-\log|f_{n}(\mathrm{e}^{\mathrm{i}(x+\mathrm{i}\epsilon)})|\bigg)\mathrm{d}x &=\frac{1}{\epsilon}(L(E,\mathrm{i}0)-L(E,\mathrm{i}\epsilon))=-\omega^{+}(E,\mathrm{i}0).
\mathcal H = \left(\begin{array}{cc}\mathcal{H}_1 0 0 \mathcal{H}_2\end{array}\right), \mathcal Q = \left(\begin{array}{cc}0 0 \mathcal{Q}_1 0 \end{array}\right), \mathcal Q^{\dagger}= \left(\begin{array}{cc}0 \mathcal{Q}_1^{\dagger}0 0 \end{array}\right).
\frac{d}{dt}\|x_\nu^{(j+3)}(t)-x_\nu^{(N-j)}(t)\|^2 \leq-\alpha\nu\|x_\nu^{(j+3)}(t)-x_\nu^{(N-j)}(t)\|^2+\nu\|x_\nu^{(j+2)}(t)-x_\nu^{(N-j+1)}(t)\|^2 &+\nu\|x_\nu^{(j+4)}(t)-x_\nu^{(N-j-1)}(t)\|^2+\frac{1}{\nu}M_{T_1,T_2}^{j+3,N-j,2-\alpha}(\omega)
A(f) \le \sum_{ef}\frac{1}{m(e)(t^{-1}+1)^2}- 0 \le &\le \frac{1}{(t^{-1}+1)^2}< \frac{3}{2}\frac{1}{(t^{-1}+1)^2}- \frac{1}{2}\frac{1}{(t^{-1}+1)^3}= B(f),
|Df^{m}(x)(v)| \geq \frac{1}{2}|(f_0^{m})'(w_0)| &\geq \frac{C}{2}\lambda_0^{m}\frac{\min_{i=0}^{m}|f_0^i(w_0)-c(0)|}{|w_0-c(0)|}&\geq \frac{C}{8}\lambda_0^{m}\frac{|\xi_{m}(x)-c(\lambda^{m_1-1}z_0)|}{|w_0-c(z_{0}/\lambda)|}.
Z_n(x^n) = \frac{1}{n}\left( \log \frac{P^{\times n}(x^n)}{U^n(x^n)}- \lambda_n \right) = \frac{1}{n}\left( \log \frac{P^{\times n}(x^n)}{U^n(x^n)}- \frac{\log v(n)}{\hat{s}}- \frac{\hat{s}-1}{\hat{s}}D_{\hat{s}}(P^{\times n}\|U^n) - \frac{nR}{\hat{s}}\right) \,.
A(t)^{\frac{1}{2}}u(t)-A(s)^{\frac{1}{2}}u(s) &=A(t)^{\frac{1}{2}}e^{-(t-s)A(t)}u(s)-A(s)^{\frac{1}{2}}u(s) &+ A(t)^{\frac{1}{2}}\int_{s}^{t}{e^{-(t-r)A(t)}(\mathcal{A}(t)-\mathcal{A}(r))u(r)dr}&+A(t)^{\frac{1}{2}}\int_{s}^{t}{e^{-(t-r)A(t)}[(-B(r)+I)A(r)u(r)-P(r)u(r)+f(r)]}dr.
\xi = \inf_{X\neq 0}\left\{\dfrac{\sum_{\mu=1}^{M_A}\sum_{\nu=1}^{M_B}\left\vert \langle g_\mu^A \vert X \vert f_\nu^B \rangle \right\vert^2}{\sum_{\mu=1}^{N_v}\sum_{\nu=1}^{N_v}\left\vert \langle g_\mu^A \vert X \vert f_\nu^B \rangle \right\vert^2}\right\}\ .
\ \sum_{k=1}^{n-1}(4k(n-k)-2n+1)\zeta(2k,2n-2k) = \frac34 (n-3)\zeta(2n)+(2n-3)(\zeta(2,2n-2)+\zeta(2n-2,2)) = \frac{3-5n}4\zeta(2n)+(2n-3)\zeta(2)\zeta(2n-2)
\|v\|_* \leq 2\left(\|u_0\|_{H^1}+\|u_1\|_{L^2}\right)+CT(|\rho_1|+|\rho_2|)+CT(|\rho_1|+|\rho_2|)e^{\frac{1}{8\pi}R^2}& = \frac23 R+CT(|\rho_1|+|\rho_2|)+CT(|\rho_1|+|\rho_2|)e^{\frac{1}{8\pi}R^2}.
\widehat{w^{\varphi}}A_{j}^{\varphi}Y(x_{1})A_{l}^{\varphi}=\sum_{m=-\infty}^{\infty}w(m+j-l)A_{j}^{\varphi}Y(x_{1})A_{l}^{\varphi}P_{m}^{\varphi}&=A_{j}^{\varphi}Y(x_{1})A_{l}^{\varphi}\sum_{m=-\infty}^{\infty}w(m+j-l)P_{m}^{\varphi}&=A_{j}^{\varphi}Y(x_{1})A_{l}^{\varphi}\widehat{w_{j-l}^{\varphi}}\;.
\big[ \delta,\widehat{f}\big](L_n,z) =0 &\Rightarrow -(-1)^{|x||z|}\delta(\widehat{f}(z),\alpha(L_n))=0, &\Rightarrow[\widehat{f}(z),L_{n}]=0, &\Rightarrow \widehat{f}(z)=0.
\frak g=\, (\frak x-1)\sum_{\substack{0\le i_0\le 2\cr 0\le i_1\le 3}}(-1)^{i_0+i_1}\binom 2{i_0}\binom 3{i_1}\frac{\frak y^{i_0+i_1}-1}{\frak y-1}\cr \cdot\Bigl[a_0s_0(\frak x)+a_1s_1(\frak x)+a_2s_2(\frak x)+a_3s_3(\frak x)+a_4s_4(\frak x)+a_5s_5(\frak x)\cr -\sum_{l=0}^2\binom{i_0+2+(l-3)k}2\binom{i_1+2-lk}2\frak x^l\Bigr]-1,
\log G(z\,|\,\tau) = \frac{z^2}{2\tau}\log(z) - \frac{z^2}{\tau}\left(\frac{3}{4}+\frac{\log\tau}{2}\right) -\frac{1}{2}\left(\frac{1}{\tau}+1\right)z\log z + O(z),\,\,z\rightarrow\infty, \,\,|\arg(z/\tau)|<\pi.
\rho_\epsilon (\boldsymbol{\xi}):=\prod_{1\leq j\leq n}2^{\epsilon_++\epsilon_-}\bigl(1+\epsilon_+\cos(\xi_j)\bigr) \bigl(1-\epsilon_-\cos(\xi_j)\bigr) &\times \prod_{1\leq j<k\leq n}\bigl(\cos (\xi_j)-\cos(\xi_k)\bigr)^2 ,
\langle \Upsilon_{0}( z_{0}) \cdots \Upsilon_{n}( z_{n}) \rangle_{\Sigma}^{\Psi}= \left\langle \Upsilon_{0}( z_{0}) \cdots \Upsilon_{n}( z_{n}) \exp \frac{1}{\pi}\int_{\Sigma}\Psi ( z ) \, d^{2}\! z \right\rangle_{\Sigma}.
\ell(f_1,\chi)\cdot \ell(f_2,\chi^{-1})=\Lambda(1/2,\Pi,\chi)\cdot\frac{\langle f_1,f_2\rangle}{\langle \mathfrak{f}_1,\mathfrak{f}_2\rangle}\cdot\prod_v\beta_v(f_{1,v},f_{2,v})\alpha_v(W_{\mathfrak{f}_1,v},W^-_{\mathfrak{f}_2,v}).
\mathbf{g}^{(j)}=p\left(p\mathbf{H}\mathbf{H}^{\mathcal{H}}+\mathbb{E}[\mathbf{w}'\mathbf{w}'^{\mathcal{H}}]\right)^{-1}\mathbf{h}_{j}&=\textstyle \left(\mathbf{H}\mathbf{H}^{\mathcal{H}}\left(\scriptstyle \kappa^{2}_{T}(\omega+1)+\omega+1\right)+\left(\scriptstyle \kappa^{2}_{R}m+\frac{N_{0}}{p}\right)\mathbf{I}_{n}\right)^{-1}\mathbf{h}_{j},
\sum_{\substack{q\leq\Omega (q,Mp_0)=1}}\;\max_{(a,qM)=1}\;\Big\lvert\frac{1}{\varphi(qM)}\sum_{\substack{\psi\pmod{q}\xi\pmod{M}\psi\xi\neq\chi_0}}\overline{\psi\xi}(a)\psi(X,\psi\xi)\Big\rvert.
\begin{array}{lll}\check{\alpha}_0=h_{p_1}+h_{q_2}-e_{1,2,3,4}, \check{\alpha}_1=h_{p_2}-e_{15,16}, \check{\alpha}_2=h_{q_2}-e_{5,6}, \check{\alpha}_3=h_{q_1}+h_{p_2}-e_{9,10,11,12}, \check{\alpha}_4=h_{p_1}-e_{7,8}, \check{\alpha}_5=h_{q_1}-e_{13,14}.\end{array}.
\psi_{\pi/2}'''(0)= \pi u_1 \otimes u_1 \otimes u_1 + \frac{2}{\pi}\sum_{h=2}^m (u_1 \otimes u_h\otimes u_h+ u_h\otimes u_1 \otimes u_h + u_h\otimes u_h \otimes u_1 ),
\begin{array}{l l l}{\lambda = 1 \; \; \; \; \; \; \Longrightarrow \lambda^{\infty}= 1}{\lambda = - 1 \; \; \; \Longrightarrow \lambda^{\infty}= \pm 1}{\lambda \ne \pm 1 \; \; \; \Longrightarrow \lambda^{\infty}= 0}\end{array}
\begin{array}{l l}{{Z_{0}( t ) = e^{i \frac{\mu}{3}t}{\cal Z}_{0}\, , ~}}&{{~ Z_{1}( t ) = e^{- i \frac{\mu}{6}t}{\cal Z}_{1}\, ,}}\\{{{}}}&{{{}}}\\{{A_{0}= X^{3}= X^{6}= X^{7}= X^{8}= X^{9}= 0 \, , ~}}&{{~{\cal Z}_{0}, ~{\cal Z}_{1}~ : ~ \mathrm{d i a g o n a l ~ m a t r i c e s}\, .}}\\ \end{array}
K \varrho_k \cap Z \varrho_k = \langle \{\overline{y}^p \}\cup \{[ \overline{y_0}, \overline{y_j}] \mid 0 \le j < p^k, \; j \equiv_{p^n}0, \pm 1,\ldots, \pm (m-1), \; j \equiv_2 0 \}\rangle \cong C_p^{\, ((2m-1) p^{k-n}+ 1)/2}.
\begin{aligned}L^0 = \Omega^0(M,\mathfrak{g}_P), L^1 = \Omega^1(M,\mathfrak{g}_P), L^2 = \Omega^{+}_{H}(M,\mathfrak{g}_P) \oplus \eta \wedge \Omega^1_H(M,\mathfrak{g}_P), L^3 = \eta \wedge \Omega^{+}_{H}(M,\mathfrak{g}_P).\end{aligned}
\begin{array}{l}\displaystyle{\phantom{\Big(}\{{w(a)}_\lambda{w(b)}\}_{z,\rho}=w([a,b])+(\partial+2\lambda)w([a,[e,b]]^\sharp)-(e|[a,b])\lambda^2}\displaystyle{\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,+\sum_{(j,n)\in J_{-\frac12}}w([a,q^{j}_{n}]^\sharp)w([q^{n+1}_{j},b]^\sharp)+z(s|[a,b])\,.}\end{array}
\beta = - \frac{Q}{2}+ \sqrt{\frac{Q^{2}}{4}- 2 + 2 h}, \, \, \, \, \, d = 1 - \frac{\frac{Q}{2}- \sqrt{\frac{Q^{2}}{4}- 2 + 2 h}}{\frac{Q}{2}- \sqrt{\frac{Q^{2}}{4}- 2}},
\Delta S_{\rm eff.}= \frac{(gJ)^2}{2}\int\limits_{\Sigma_{\rm min.}}^{}d\sigma_{\mu\nu}(x)\int\limits_{\Sigma_{\rm min.}}^{}d\sigma_{\rho\nu}(x')\left<\left<K_{\mu\rho}(x,x')\right>_{\bf n}\right>_{B_\mu^a}.
\Pi(t,\Theta)= \frac{\partial}{\partial\frac{d\phi}{dt}}L=\frac{i}{2}\frac{\partial}{\partial\frac{d\phi}{dt}}\int d\Theta (D_{\Theta}\phi )\dot{\phi}=\frac{1}{2}\int d\Theta \{\Theta \dot{\phi}+iD_{\Theta}\phi \},\qquad \dot{\phi}\equiv\frac{d\phi}{dt},
d\left( \Lambda_{\omega}\right)_{1}\left( Y\right) =\int_{\mathbb{E}_{\Theta}}\omega \circ \mathbf{a}\left( f_{Y}\left( \phi^{R}\left(r\right) \right) \right) \left( \sigma \cdot \nu \right) \left( d\xi \right)\xi =r\cdot b_{\Theta}
F_{\mu}(r)=\int_{V}F_{\mu}(r^{\prime})\frac{\partial}{\partial r_{\mu}}\frac{\partial}{\partial r_{\mu}}G(r,r^{\prime})d^{d}r^{\prime}=\delta_{\mu\nu}\delta_{\rho\sigma}\int_{V}F_{\nu}(r^{\prime})\frac{\partial}{\partial r_{\rho}}\frac{\partial}{\partial r_{\sigma}}G(r,r^{\prime})d^{d}r^{\prime}
h_{1,i}:= x_{m+i-1}, h_{1,n+j}:= x_j, h_{i_1,i_2}:= x_{m+i_1-1}-x_{m+i_2-1}, h_{i,n+j}:= x_j-x_{m+i-1}, h_{n+j_1,n+j_2}:= x_{j_1}-x_{j_2}.
q_{l}(z,\overline{z})=\overline{q_{l}(z,\overline{z})},\quad\mbox{for all $l=1,\dots,N_{0}$,}&q_{l}(z,\overline{z})\neq \overline{q_{l}(z,\overline{z})},\quad\mbox{for all $l=N_{0}+1,\dots,N$,}
\begin{aligned}G_{m-r}= \sum\limits_{\ell=0}^{\lfloor \frac{m-r}{2}\rfloor}c_{\ell, m-r}\mathfrak{i}_{(2)}^{\ell}\mathfrak{j}_{(2)}^{\ell}\sum\limits_{p=0}^{r}(-1)^{r-p}\frac{1}{p!}\binom{r}{p}\mathfrak{j}_{x^{\otimes (r-p)}}\delta^{p}N^{p}f,\end{aligned}
\int_{\Omega}f_\alpha (x)\tau_\alpha\, d \mu (\alpha) =\sum_{j=1}^n\int_{\Omega_j}f_\alpha (x)\tau_\alpha\, d \mu (\alpha)=\sum_{j=1}^n\int_{\Omega_j}\frac{g_j(x)}{\sqrt{\mu(\Omega_j)}}\frac{\omega_j}{\sqrt{\mu(\Omega_j)}}\, d \mu (\alpha) &=\sum_{j=1}^ng_j(x)\omega_j, \forall x \in \mathcal{X}.
|c_{i,j}^{(k-n,l-m)}| \leq \dfrac{4V_{k,l}}{\pi^2}\begin{cases}\Gamma_{0,0}[s](i)\Gamma_{0,0}[r](j), ~~~~~~~\mbox{if}n=2s, m=2r, \Gamma_{0,0}[s](i)\Gamma_{1,-1}[r](j), ~~~~~~~\mbox{if}n=2s, m=2r+1, \Gamma_{1,-1}[s](i)\Gamma_{0,0}[r](j), ~~~~~~~\mbox{if}n=2s+1, m=2r, \Gamma_{1,-1}[s](i)\Gamma_{1,-1}[r](j), ~~~~~~~\mbox{if}n=2s+1, m=2r+1,\end{cases}
J_{o r b}( \vec{R}) = 2 \pi \kappa \frac{\sigma_{1}}{q_{1}}R^{i}\varepsilon_{i j}v_{j}^{( 2 )}( \vec{R}) - 2 \pi \kappa \frac{\sigma_{2}}{q_{2}}R^{i}\varepsilon_{i j}v_{j}^{( 1 )}( - \vec{R}) ,
\Delta_{\varepsilon , \overline{{\varepsilon}}}^{( 1 )}= \frac{\sigma^{2}}2 \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}}}\cdot \overline{{\Psi}}_{\varepsilon , \overline{{\varepsilon}}}
\left( (I-tF_0)^{-1}\right)_{aSym}= \sum_{n=0}^{\infty}\left( tF_0 \right)^{2n+1}& = \sum_{n=0}^{\infty}\left( tF_0 \right)^{2n}tF_0 & = \left( I - (tF_0)^2 \right)^{-1}tF_0.
A:= \left( \begin{matrix}0 0 0 -1 0 0 0 0 0 \frac{B_2}{v}-\frac{\beta}{v}0 0 0 0 \frac{B_3}{v}0 -\frac{\beta}{v}-c^2 B_2 B_3 0 0 0 0 -\beta 0 0 0 0 0 0 -\beta 0 0 0 \end{matrix}\right).
\int_{t= \frac{\sqrt{\beta +1}-\sqrt{\beta -1}}{\sqrt{2}}}^{\frac{\sqrt{\beta -1}+\sqrt{\beta +1}}{\sqrt{2}}}t^{-n-m-1}a_na_m (t^2+1) \sqrt{\frac{1-\beta}{t^4-2 \beta t^2+1}}dt = \begin{cases}0 \mbox{if}n\not=m, \pi \sqrt{\beta -1}\mbox{if}n =m=0, (\pi /2)\sqrt{\beta -1}\mbox{if}n =m\neq 0, \end{cases}
\Lambda(z^{(0)}+\Delta) =\frac{e^{\frac{\sigma_{\mathrm{a}}^2}{\zeta}}\int_{\sigma_{\mathrm{w}}^2+\sigma_{\mathrm{a}}^2}^{\infty}\left( \frac{1}{v}\right)^{n}e^{-\frac{(z^{(0)}+\Delta)}{v}}e^{-\frac{v}{\zeta}}dv}{\int_{\sigma_{\mathrm{w}}^2}^{\infty}\left( \frac{1}{v}\right)^{n}e^{-\frac{(z^{(0)}+\Delta)}{v}}e^{-\frac{v}{\zeta}}dv}.
\frac{T}{4}\int_{\frac{1}{2}-\epsilon}^{\frac{1}{2}+\epsilon}\log \left(\frac{\sigma^2+T^2}{4}\right)\d\sigma =\epsilon T \log\left(\frac{T}{2}\right)-\epsilon T+\epsilon T+\epsilon\ \mathcal{O}\left(\frac{1}{T}\right)
\begin{array}{ll}f_{1}=\frac{1}{2}(M_{45}+M_{67}+M_{89})\,,~~~ ~~~f_{2}=\frac{1}{2}(M_{45}-M_{67}-M_{89})\,,{}{}f_{3}=\frac{1}{2}(-M_{45}+M_{67}-M_{89})\,,~~~ ~~~f_{4}=\frac{1}{2}(-M_{45}-M_{67}+M_{89})\,.\end{array}
S = - m_{f}\int d \tau{\frac{z^{6 - p}}{Q_{F}}}\left[ \sqrt{1 -{\frac{Q_{F}v_{\parallel}^{2}}{z^{6 - p}}}-{\frac{Q_{p}Q_{F}v_{\perp}^{2}}{z^{1 2 - 2 p}}}}- 1 \right] ,
\int_{-\infty}^{\infty}F^{\prime}(x+\frac{t^{2}}{2})dt=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}f^{\prime}(x+\frac{t^{2}+u^{2}}{2})dtdu=2\pi\int_{0}^{\infty}f^{\prime}(x+\frac{r^{2}}{2})rdr=2\pi f(x).
q_1 = t_1, , q_3 = -3\cdot t_3, q_9 =\textstyle 99255\cdot (t_9 - \frac{1}{3}t_3), q_5 = 45\cdot t_5, q_{11}=\textstyle -9823275\cdot (t_{11}- t_3^2t_5), q_7 = -1575\cdot t_7, q_{13}=\textstyle 1404728325\cdot (t_{13}- t_3^2t_7 - t_3t_5^2).
[u\,\eta]^p_{\mathcal{B}^{\Gamma,p}_\infty(\mathbb{R}^N)}\le \frac{C}{s}\,\left(\frac{R}{r}\right)^N\,\left(\frac{R}{h_0}\right)^{1+p}\,\left[(1-s)\,\mathcal{M}_\gamma+h_0^{-\Gamma\,p}\,\|u\|^p_{L^p(B_{R+h_0})}\right],
( \Delta w )_{\mathrm{m a g n}}= \frac{e^{2}{\cal B}^{2}}{1 2 \pi}\ \sum_{n = 1}^{\infty}\exp \left\{- n \, \frac{\pi m^{2}}{e{\cal E}}\right\}+ O \left( \frac{{\cal B}^{4}}{{\cal E}^{4}}\right) .
\bigl\langle W^*(L\xi\otimes\nabla\zeta),\xi'\otimes\zeta'\bigr\rangle=\bigl\langle(\operatorname{id}\otimes\omega_{\Lambda_{\tilde{\varphi}}(\sigma^{\tilde{\varphi}}_{-i}(a)),\Lambda_{\tilde{\varphi}}(b)})(W^*)K^*K\xi,\xi'\bigr\rangle,
p_{A_2}(N,\ell;2)= \frac{1}{\ell}-\frac{1}{(N+1)_2}&+(-1)^{\ell+1}\binom{N-1}{\ell-1}^{-1}\sum_{i=0}^{N-\ell}(-1)^i\binom{N-1}{i}\frac{1}{i+\ell+1}.
2y(1-y)\Psi_0'\Psi_0^{-1}+a^{(\kappa)}= C^{(\kappa)}-yU^{(\kappa)},y(1-y)\Psi_0''\Psi_0^{-1}+a^{(\kappa)}\Psi_0'\Psi_0^{-1}+\Psi_0F^{(\kappa)}\Psi_0^{-1}=V^{(\kappa)}.
S = \left( \begin{matrix}{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1 / \sqrt{2}}}&{{1 / \sqrt{2}}}\\{{0}}&{{0}}&{{- 1 / \sqrt{2}}}&{{1 / \sqrt{2}}}\\ \end{matrix}\right) .
Y_{\underline{s}}:= \{x\in{X}_\mathfrak{B}: D(\mathfrak{b}_\ell|x)=N(\mathfrak{b}_\ell)-s_\ell \ell\geq 1 \}, Y_{\geq \underline{s}}:= \{x\in{X}_\mathfrak{B}: D(\mathfrak{b}_\ell|x)\leqslant N(\mathfrak{b}_\ell)-s_\ell \ell\geq 1\}.
h_T(x_s) =\sum_{i=1}^n\sum_{t=1}^T\frac{q_{i,t+1}^\top g_{i,t}(x_s)}{\gamma_t}=\sum_{i=1}^n\sum_{t=1}^T[g_{i,t}(x_{i,t})]_+^\top g_{i,t}(x_s) &\le-\sum_{i=1}^n\sum_{t=1}^T\epsilon_s[g_{i,t}(x_{i,t})]_+^\top{\bf 1}_{m_i}=-\epsilon_s\sum_{i=1}^n\sum_{t=1}^T\|[g_{i,t}(x_{i,t})]_+\|_1 &\le-\epsilon_s\sum_{i=1}^n\sum_{t=1}^T\|[g_{i,t}(x_{i,t})]_+\|,
\bar b^n_t:= b_t(x_t)-b^n_t(x^n_{\kappa(n,t)}) , \bar \sigma^n_t:=\sigma_t(x_t)-\sigma_t^n(x^n_{\kappa(n,t)}), \bar{\gamma}^n_{t}(z):=\gamma_t(x_{t},z)-\gamma_t^n(x^n_{\kappa(n,t)},z)
\langle v', Y(Y(v_1, z_1-z_2)v_2,z_2)v_3\rangle = F(\alpha_1,\alpha_2,\alpha_3)^{-1}(z_1-z_2)^{-\hat{b}(\alpha_1,\alpha_2)}z_2^{-\hat{b}(\alpha_1,\alpha_3)-\hat{b}(\alpha_2,\alpha_3)}\left(1+\frac{z_1-z_2}{z_2}\right)^{-\hat{b}(\alpha_1,\alpha_3)}f(z_1,z_2)
d s^{2}= - ( 1 - \frac{1 + \lambda}{r}) d \tau^{2}+ ( 1 - \frac{\lambda}{r}) d f^{2}+ ( 1 - \frac{1 + \lambda}{r})^{- 1}( 1 - \frac{\lambda}{r})^{- 1}\frac{k d r^{2}}{8 r^{2}},
{\cal L}_{p}^{\hbar}\left[ H \right] = - \sum_{i = 0}^{p}\, \frac{2}{\hbar}\, H \left( \phi_{i}\right) \sin \left[ \frac{\hbar}{2}\, \stackrel{\leftarrow}{\partial}_{a}^{( i )}\omega^{a b}\, \stackrel{\rightarrow}{\partial}_{b}^{( i )}\right]
\int_{({\mathbb R}^d)^{k-1}}\psi_j(K_1\cap K_2^{x_2}\cap \dots \cap K_k^{x_k},\cdot)\, \lambda^{k-1}(d(x_2,\dots ,x_k)) & = \sum_{{m_1,\dots,m_k=j}\atop{m_1+\dots +m_k=(k-1)d+j}}^d\psi^{(j)}_{m_1,\dots,m_k}(K_1,\dots,K_k;\cdot).
Q_{2,\vec{v}_{s,t,f,0}^{(2)}}= \left(\left(\begin{array}{rrr}t 1 0 s 0 1 1 0 0\end{array}\right), \left(\begin{array}{rrr}1 0 0 0 1 0 0 0 1\end{array}\right), \left(\begin{array}{rrr}0 0 1 1 0 -t 0 1 -s\end{array}\right)\right),
\begin{aligned}S(B_1(k),B_3(k)) =x\cdot B_1(k)-y^{k}\cdot B_3(k) &=x\cdot \left ( \frac{y^{k+2}-1}{y-1}+x\cdot\frac{x^{k-1}-1}{x-1}\right )-y^{k}\cdot (xy-1) &=\frac{x+x^{k+1}y-x^2y-x^{k+1}+x^2y^{k+1}-xy^{k+1}+xy^{k+1}-xy^k-y^{k+1}+y^k}{(x-1)(y-1)}&=B_2(k)+\frac{y^k-1}{y-1}\cdot B_3(k).\end{aligned}
e_{i,[\kappa_{i}^{j_{i}}]}^{j_{i}}+\sum_{\beta=0}^{\kappa_{i}^{j_{i}}-1}a_{i}^{j_{i},\beta}e_{i,[\beta]}^{j_{i}}=0\,,j_{i}=1,\ldots,m_{i}\,,i=1,\ldots,s\,.
I\left({M;{{\bf{Y}}^{n - 1}}}\right) - I\left({M;{{\bf{Z}}^{n - 1}}}\right) = \sum\limits_{i = 0}^{n - 1}{I\left({\left.{M;{\bf{Y}}\left[ i \right]}\right|{{\bf{Y}}^{i - 1}}}\right)}- \sum\limits_{i = 0}^{n - 1}I\left({\left.{M;{\bf{Z}}\left[ i \right]}\right|{\bf{Z}}_{i + 1}^{n - 1}}\right).
A =-\sum_{j=0}^{p-2}\omega^j\left(\frac{x}{4}\right)\pi^{(p-1)\ell_j}~\Gamma_p\left(\left\langle\frac{1}{2}-\frac{2j}{p-1}\right\rangle\right)\Gamma_p\left(\left\langle\frac{1}{2}+\frac{j}{p-1}\right\rangle\right) &\times\Gamma_p\left(\frac{j}{p-1}\right),
\textrm{Tr}e^{-\frac{i}{\hbar}{\cal{H}}T}=\int dq_{0}\int^{q_{0},T}_{q_{0},0}{\cal{D}}q\; e^{\frac{i}{\hbar}S[q]}=\int dq_{0}e^{\frac{i}{\hbar}S^{cl}(q_{0},T)}\int_{0,0}^{0,T}{\cal{D}}\eta (a_{n})\; e^{\frac{i}{\hbar}\sum_{n}\epsilon_{n}a_{n}^{2}}.
\| \phi_1 * \phi_2 \|_{q (p_1,p_2)}\leq \| \phi_1^* * \phi_2^* \|_{q (p_1,p_2)}\leq C (p_1,p_2) \| \phi_1^* \|_{p_1}\| \phi_2^* \|_{p_2}= C (p_1,p_2) \| \phi_1 \|_{p_1}\| \phi_2 \|_{p_2}
e_G(A',B') = 2e_{\overline{G}}(A') \ge 2 \left( \sum_{x \in X}d_{\overline{G}}(x,A') - \binom{|X|}{2}\right) = 2 \sum_{x \in X}d_{G}(x,B') - |X|(|X|-1) & = 2 e_G(X, B') - |X| (|X|-1).
y'|_{(B_m(\texttt{Sub}_{(1,1)}(y')),1_H)}= \sigma_{(\vec{0},s)}(y)|_{(B_m(\texttt{Sub}_{(1,1)}(y')),1_H)}= y|_{(\vec{0},s^{-1}) (B_m(\texttt{Sub}_{(1,1)}(y')),1_H)}= y|_{(\varphi_{s^{-1}}(B_m(\texttt{Sub}_{(1,1)}(y'))),s^{-1})}.
c^{\ast}= \begin{cases}M_1 + 2\sqrt{kD}\, , \mbox{if $M_2 \le M_1 + 2\sqrt{kD}$,}\displaystyle\frac{M_1+ M_2}{2}+ \displaystyle\frac{2kD}{M_2- M_1}\, , \mbox{if $M_2 \ge M_1 + 2\sqrt{kD}$.}\end{cases}
\int_{\mathbb{R}+iB}\hat{\psi}(\lambda)\frac{d}{d\lambda}\log\zeta_1(\lambda)d\lambda=-2\pi i\left\langle \sum\limits_\gamma\frac{T_\gamma^\# \delta(t-T_\gamma)}{|\det(I-\mathcal{P}_\gamma)|},\psi\right\rangle
\prod_{k=2}^{m}T_{k,j}^{\frac{\log s_{k}}{2\pi i}}= (4(x_{j}-x_{1}))^{-\beta_{j}}\prod_{\substack{k=2 k \neq j}}^{m}\widetilde{T}_{k,j}^{-\beta_{k}}, \mbox{where}\widetilde{T}_{k,j}= \frac{\sqrt{x_{j}-x_{1}}+\sqrt{x_{k}-x_{1}}}{|\sqrt{x_{j}-x_{1}}-\sqrt{x_{k}-x_{1}}|}.
M:=\begin{pmatrix}( g_1, h_{1}^{(1)}h_{1}^{(2)})_{\Omega}\dots ( g_m, h_{1}^{(1)}h_{1}^{(2)})_{\Omega}\vdots \vdots \vdots ( g_1, h_{m}^{(1)}h_{m}^{(2)})_{\Omega}\dots ( g_m, h_{m}^{(1)}h_{m}^{(2)})_{\Omega}\end{pmatrix}
\int_{\lvert t \rvert \leq N^{-((d-1)+\alpha)}}\left\lvert \frac{1}{N}\sum_{n=0}^{N-1}e^{2\pi i p(n) t}-\frac{1}{N}\int_{0}^{N}e^{2\pi i p(n) t}dn \right\rvert^{2}\lvert \hat{f}\rvert^{2}\lvert t \rvert^{1-2s}dt \leq N^{-2\alpha}\|f\|^{2}_{L^{2}(m)}.
g(\lambda/2\theta_1)=\left(\begin{array}{cc}\sum_{n= 0}^{N^+}\frac{(-N^+)_n(N^-)_n}{n!}(-\lambda)^{-n}\frac{1}{q_0}\sum_{n= 1}^{N^+}\frac{(-N^+)_n(N^-)_n}{(n-1)!}(-\lambda)^{-n}\frac{1}{p_0}\sum_{n= 1}^{N^-}\frac{(N^+)_n(-N^-)_n}{(n-1)!}(\lambda)^{-n}\sum_{n=0}^{N^-}\frac{(N^+)_n(-N^-)_n}{n!}(\lambda)^{-n}\end{array}\right),
\sigma(H^{\epsilon,0})\cap \left [a_0,b_N\right ]\subset \bigcup_{k=0}^N [a_k,b_k]\,, &b_k-a_k= \mathcal O ( \epsilon^\infty), \; 0\leq k\leq N,{\rm and}a_{k+1}-b_k\geq \epsilon/C,\; 0\leq k\leq N-1\,.
\mathcal{T}_{\neq \neq}= \int B Q^3 B \left( U_{\neq}^j \partial_j^t Q^3_{\neq}\right) dV dt = \mathcal{T}_{\neq\neq}^1 + \mathcal{T}_{\neq\neq}^2 + \mathcal{T}_{\neq\neq}^3.
\Big\|K\Big(\frac{.}{\sqrt{t - \tau}}\Big)\Big\|_{\mathcal{L}^{r,1}}= (t - \tau)^{\frac{d}{2r}}\big\|\hat K\big\|_{L^{r', 1}}\simeq (t - \tau)^{\frac{d}{2}(1 - \frac{1}{\tilde p})}.
{\cal A}_{\mathrm Z M S}= \int{\mathrm d}^{2}x \left[ \frac{1}{2}\partial_{\nu}\Phi \partial^{\nu}\Phi + \frac{m_{0}^{2}}{\gamma}\Big ( \exp \left( 2 i \sqrt{\gamma}\Phi \right) + 2 \exp \left( - i \sqrt{\gamma}\Phi \right) \Big ) \right] \, .
\|1_{A_i'}*1_{A_j'}\|_2^2 \geqslant\delta^{O(\varepsilon)}2^{-2i-2j}\|\mu_{\delta}\|_2^2 \geqslant \delta^{O(\varepsilon)}2^{-i-j}|A_i'|^{\frac{1}{2}}|A_j'|^{\frac{1}{2}}&=\delta^{O(\varepsilon)}(2^i|A_i'|)^{-1}|A_i'|^{\frac{3}{2}}(2^{j}A_j')^{-1}|A_j'|^{\frac{3}{2}}&\geqslant\delta^{3d+O(\varepsilon)}N(A_i',\delta)^{\frac{3}{2}}N(A_j',\delta)^{\frac{3}{2}}.
x_{x_0}^T(t)Q x_{x_0}(t)-x_0^TQx_0 \leq \left[\int_0^t x_{x_0}^T(s)(A^T Q+Q A+ \sum_{i=1}^m N_i^T Q N_i+k^2 Q) x_{x_0}(s)\right. ds \ \ \ \ \ \ \ \ \left.+2\int_0^t x_{x_0}^T(s) Q B u(s) ds\right].
G \cap I = ((F \setminus \{\bar{b}_n\}) \cup [\bar{a}_n \mapsto \bar{b}_n](L)) \cap (D_1 \cup [\bar{a}_n \mapsto \bar{b}_n](J)) &\subseteq ((F \setminus \{\bar{b}_n\}) \cup [\bar{a}_n \mapsto \bar{b}_n](L)) \cap ((D \setminus \{\bar{b}_n\})\cup [\bar{a}_n \mapsto \bar{b}_n](J))
D_{3}=\frac{\alpha^{2}}{32\pi^{2}}\left(-\sigma^{2}+A^{2}{\beta^{x}}^{2}+A^{2}{\beta^{z}}^{2}\right)\left\{\frac{1}{(3-d)}-\frac{1}{2}\left[\ln\left(\frac{\alpha}{\mu_{R}^{2}}\right)-\frac{1}{2}\right]\right\}\;\;.
\{\underline{a}\colon \underline{a}/\varpi^k\in L(\varpi^k\underline{c})\}= \{\underline{d}\colon (\underline{d},\varpi)=1, |\underline{d}|<|\varpi|^k\}\setminus \{a\underline{c}^\bot +\varpi\underline{d}\colon (a,\varpi)=1, |a|<|\varpi|, |\underline{d}|<|\varpi|^{k-1}\}.
G_{s \beta}({\rm\bf r}_{1},{\rm\bf r}_{2}) ={{m \omega_c}\over \pi}a_s e^{-s\beta \omega_c}\exp \left [ -{{m \omega_c}\over 2}a_s \left \{r_{12}^2+2 i \epsilon \hat{\rm\bf k}\cdot ({\rm\bf r}_{1}\times{\rm\bf r}_{2}) \right \}- b_s (r_1^2+r_2^2) \right ] ,
2\int_{\bar x}^{x_1}\phi_{xx}\cdot\phi_x(t, x) \, dx= |\phi_x|^2(t, x_1)- |\phi_x|^2(t, \bar x), 2\int_{\bar x}^{x_1}\phi_{xt}\cdot\phi_t(t, x) \, dx= |\phi_t|^2(t, x_1)- |\phi_t|^2(t, \bar x),
\mathcal I_{2,1}:=\gamma\int_{\mathbb R\times\mathbb T}\left[\phi_0;\mathbb H\right]\partial_x\phi_{x,\gamma}\varphi_{x,\gamma}\,dx\,dt\,, &\mathcal I_{2,2}:=\int_{\mathbb R\times\mathbb T}\left[\phi_0;\mathbb H\right]\partial_x\phi_{x,\gamma}\partial_t\varphi_{x,\gamma}\,dx\,dt\,.
\left.\int u_{1,t}d\nu^2_{t}\,\right|_{t=s}^{t=s'}=\left.\int u_{1,t}d\nu^2_{t}\,\right|_{t=\bar s}^{t=s'}+\left.\int u_{1,t}d\nu^2_{t}\,\right|^{t=\bar s}_{t=s}.
m_{H}\partial_{m_{H}}\rightarrow m_{H}\tilde{\partial}_{m_{H}}\; \; , \; \; e \partial_{e}\rightarrow e \tilde{\partial}_{e}\; \; , \; \; f_{\xi}\partial_{\xi}+ ( \chi \partial_{\xi}+ \chi_{A}\partial_{\xi_{A}}) f_{\xi}\partial_{\chi}\rightarrow \tilde{\partial}_{\xi},
\mathbf{I}\leq C_1\delta \int^\infty_{r_0}\frac{r^n}{t^{n/2}}\cdot \exp\left(-\frac{r^2}{C_1t}\right)\frac{1}{rt}dr \leq t^{-1}C_1\delta \int^\infty_{0}r^{n-1}\exp\left(-\frac{r^2}{C_1}\right) dr \leq C_2t^{-1}\delta.
\frac{1}{\vert \chi_{t}^{M_{{\mathcal{P}}}}+\vert a^{{{\mathcal{P}}}}_{t}\vert^2\vert^{2dp}}=\frac{1}{\Gamma (2dp)}\int_{0}^{\infty}{s}^{2dp-1}e^{-{s}(\chi_{t}^{M_{{\mathcal{P}}}}+\vert a^{{{\mathcal{P}}}}_{t}\vert^2)}d{s},
\mathbb{P}[M_k-m_0\geq \lambda] \stackrel{\eqref{expbound}}{\leq}\exp\left(\frac{t^{2}}{2}\frac{1}{1-tm/3}\sum_{j=1}^k\sigma_j^2-t\lambda\right) &\stackrel{\eqref{paramchoice}}{=}\exp\left(-\frac{\lambda^2}{2\left(\sum_{i=1}^k \sigma_i^2+m\lambda/3\right)}\right).
\mathcal{S}_{a}=k_{1}\left( \mathrm{tr}N\right) I+2k_{2}\mathrm{Sym}N+2k_{2}^{\prime}\mathrm{Skw}N+k_{3}\left( \mathrm{tr}\varepsilon \right)I+k_{3}^{\prime}\varepsilon + \delta\nabla\nu_{t}.
P\in\mathcal{Q}^{inj}(^mE;F)\Longleftrightarrow I_F\circ P\in\mathcal{Q}(^mE; \ell_\infty(B_{F^*}){\rm ~~ and~~}\|P\|_{\mathcal{Q}^{inj}}:=\|I_F\circ P\|_{\mathcal{Q}}.
{P}_2 \widehat\theta\left[{\rm d}v^\varepsilon+v^\varepsilon_{xx}{\rm d}t+\left(x+\varepsilon\right)^{2\gamma}v^\varepsilon_{yy}{\rm d}t+\frac{\sigma}{(x+\varepsilon)^2}v^\varepsilon{\rm d}t\right] = |{P}_2|^2{\rm d}t+{P}_2{P}{\rm d}t+\sum_{i=1}^{5}X_i{\rm d}t+{\rm d}Y+\{\cdot\}_x+\{\cdot\cdot\}_y+J,\mathbb P-a.s.,
\int^\infty_0 d\nu\int^\infty_m dk\ [k^2 -m]^{1/2 -s}\frac{\partial}{\partial k}\frac{t^j}{\nu^n}=\frac{(mR)^{2-n}}2\frac{\Gamma (\frac32-s)\Gamma (1+\frac{j-n}2)\Gamma (s+\frac{n-3}2)}{\Gamma (\frac j2)}.
+ \frac{1}{2 !}\int_{0}^{1}d \tau_{1}\int_{0}^{1}d \tau_{2}\left. \frac{\delta^{2}W [ X ]}{\delta X^{a}( \tau_{1}) \delta X^{b}( \tau_{2})}\right|_{X = \overline{{{X}}}}\xi^{a}( \tau_{1}) \xi^{b}( \tau_{2}) +{\cal O}( \xi^{3})
V^4 |\nabla \tau|^2 = (C_i Y^4 + B_i Y^0 + D_p \epsilon_{pqi}Y^q) (C_j Y^4 + B_j Y^0 + D_m \epsilon_{mnj}Y^n) (\delta^{ij}-\tilde{X}^i \tilde{X}^j) + r^2 \left [ \sigma^{(0) ab}(C_i \tilde{X}_a^i)(C_j \tilde{X}_b^j)+2(C_i \tilde \nabla \tilde Y^i) (C_j \tilde \nabla Y_j^{(3)}+A\tilde \nabla Y_0^{(3)}) \right ]+ O(r^{3}),
\epsilon_{\mbox{mean}}(s) :=\bigg \vert \frac{\mu_{\mbox{pce}}(s)- \mu_{\mbox{exact}}(s)}{\mu_{\mbox{exact}}(s)}\bigg \vert, \epsilon_{\mbox{var}}(s) :=\bigg \vert \frac{\sigma^2_{\mbox{pce}}(s)- \sigma^2_{\mbox{exact}}(s)}{\sigma^2_{\mbox{exact}}(s)}\bigg \vert,
\lim_{n\to \infty}\frac{1}{n}\sum_{i=0}^{n-1}\frac{1}{\lambda^i}\mathcal P^i\left(h(\cdot)\frac{1}{\lambda^{n-i}}\mathcal P^{(n-i)}(\cdot,M)\right)(x) \xrightarrow{n\to\infty}\eta(x)\int_M h(y)\eta(y)\mu(\d y)\ \mu.
\begin{array}{cc}(k,b,c)=\left(\frac{2m}{m^2 -1},- x_i^2 \frac{(m^2 +1)}{m (m^2 -1)},- \frac{\tau^6 + 2 \tau^5 + 4 \tau^4 + 8 \tau^3 + 9 \tau^2 + 4 \tau +1}{4 \tau^2 (\tau +1)^2}\right), m \in \mathbb{Q}\setminus \{0, \pm1\}, i=1,2,3 \textrm{if}n=4 \end{array}
\begin{aligned}D^*_N(x_1,\ldots,x_N)\:\:\: =\:\:\: \sup_{\mathcal{A}\in\mathcal{B}^*}\left|\frac{1}{N}\sum_{n=1}^N\mathbf{1}_{\mathcal{A}}(x_n)-\lambda(\mathcal{A})\right|, \end{aligned}
\kappa^{2}_{k}(x,t)\leq\frac{T-1/k-t_{k}}{T-1/k-t_{k}-\lambda^{2}_{k}t}= \frac{t^{(1)}_{k}}{t^{(1)}_{k}-t},t\in[t_{k}^{(0)},t^{(1)}_{k}).
x'= \sum_j\lambda_j\sum_{i_1,\ldots,i_p\in I}(-1)^{i_1+\ldots+i_p}(P_{1,j}+i_1)\wedge\ldots\wedge(P_{p-1,j}+i_{p-1})\wedge(P_M-(1,0,\ldots,0)+i_p) &\otimes(Q_j+(1,0,\ldots,0)-i_1-\ldots-i_p).