harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\hat\phi(0)-\phi(0)+\int_{-\infty}^{\infty}\hat\phi(x)|x|dx=\int_{-\infty}^{\infty}\phi(x)W_{\rm U}^1(x)dx=\int_{-\infty}^{\infty}\phi(x)\left(1-\frac{\sin^2(\pi x)}{(\pi x)^2}\right)dx.
M(s)\equiv\frac{2}{\sqrt{3}}\sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n^{s}}sin\frac{\pi n}{3}M(1-s)=\frac{2}{\sqrt{3}}M(s)\Gamma(s)3^{s}(2\pi)^{-s}sin\frac{\pi s}{2},
\begin{aligned}\log\det (\Delta\!\!\restriction_{|w_\pm|\leq \epsilon})=-\frac 1 3\log\epsilon + \frac{2\beta+1}{3(\beta+1)}\log2 +\frac 1{3(\beta+1)}\log c_\beta -2\zeta'_R(-1) -\frac 5{12}-\frac 1 2 \log (2\pi) +o(1) \quad \epsilon\to 0^+. \end{aligned}
\widetilde{\Delta}{\tilde{B}}= \frac{1}{2}\sum_{i = 1}^{N}\oint_{{\hat{\cal{C}}}_{i}}G ( t_{i},{\hat{\cal{C}}}_{i}){\cal D}_{[ \rho}B_{\mu \omega ]}( t_{i}) T^{\mu}( t_{i}) d{X_{t_{i}}^{\rho}}
(\delta^n\phi)(g_0, g_1,\ldots, g_n) =\phi(g_1,\ldots, g_n)+\sum_{k=1}^n(-1)^k\phi(g_0,\ldots, g_{k-2}, g_{k-1}g_k, g_{k+1},\ldots, g_n) & +(-1)^{n+1}\phi(g_0,\ldots, g_{n-1})
{\cal R}( u ) = \rho \left[ \begin{array}{c c c c}{{\sin ( \eta + u )}}&{{}}&{{}}&{{}}\\{{}}&{{\sin u}}&{{\sin \eta}}&{{}}\\{{}}&{{\sin \eta}}&{{\sin u}}&{{}}\\{{}}&{{}}&{{}}&{{\sin ( \eta + u )}}\\ \end{array}\right] ~ ,
T_{1}= -{\frac{i}{2}}\left( \begin{matrix}{0}&{1}\\{1}&{0}\\ \end{matrix}\right) ; T_{2}={\frac{1}{2}}\left( \begin{matrix}{0}&{- 1}\\{1}&{0}\\ \end{matrix}\right) ; T_{3}= -{\frac{i}{2}}\left( \begin{matrix}{1}&{0}\\{0}&{- 1}\\ \end{matrix}\right) \ .
\theta_{\mathcal{C}}({\bf z + h}) = a_{000}+ a_{100}\exp[h_1]\exp[z_1] + a_{010}\exp[h_2]\exp[z_2] + a_{001}\exp[h_3]\exp[z_3] + a_{110}\exp[h_1 + h_2]\exp[z_1 + z_2]+ a_{101}\exp[h_1 + h_3]\exp[z_1 + z_3] &+ a_{011}\exp[h_2+h_3]\exp[z_2 + z_3] + a_{111}\exp[h_1 + h_2 + h_3]\exp[z_1 + z_2 + z_3].
\int_{HI}e^{sx}\overline{G}(s,t)ds=-\int_{r}^{-\lambda_2+R}e^{-x\lambda_2}e^{-wx}e^{t(c_1\lambda^{\alpha_1}+c_2\lambda_2^{\alpha_2})}e^{-t[c_1(\lambda_1-\lambda_2+we^{-i\pi})^{\alpha_1}+c_2(w^{\alpha_2}e^{-i\pi \alpha_2})]}dw.
1-\frac{\Psi x_i}{x_i}+ 1 - \frac{y_i}{\Psi x_i}-\left(1-\frac{y_i}{x_i}\right)= \frac{\Psi(1-\Psi) x_i - (1- \Psi) y_i}{\Psi x_i}\geq \frac{\Psi(1-\Psi) x_i - (1- \Psi) \Psi^2 x_i}{\Psi x_i}= (1-\Psi)^2\geq 0
C_{H}\left(a,v,\varphi\right)\overset{{\rm def}}{=}\left\{u\in H\,:\,\left\langle u-a,v\right\rangle \geq\cos\varphi\,\left\Vert u-a\right\Vert \left\Vert v\right\Vert \right\}\mbox{.}
\mu^{(m)}= 2^{-m}\delta_{t,t}+ \sum_{k=1}^m \big( \tfrac{1}{3}2^{k-m}\delta_{2^{-k}t,-2^{-k}t}+ \tfrac{1}{6}2^{k-m}\delta_{-2^{1-k}t,2^{1-k}t}\big)
Q(t)^{-1}P^{-1}\begin{pmatrix}w(\xi_0) \overline{w(\xi_0)}\end{pmatrix}= \begin{pmatrix}\beta_1(\xi_0) \beta_2(\xi_0) \end{pmatrix}+ O(\varepsilon_1^2\varepsilon_2t^{-1/4 + C_1 \varepsilon^2_1 + 2\mu C_2 \varepsilon^2_1 + C_4 \varepsilon_1^2})
f(27\cdot5^{2m}) = f(27)\,f(5^{2m}) = 27\,f(5^{2m}) * = f\big( (5^m)^2 + (5^m)^2 + 5^2(5^m)^2 \big) = 2\,f(5^{2m}) + f(5^{2m+2}) f(30\cdot5^{2m}) = f(6)\,f(5^{2m+1}) = 6\,f(5^{2m+1}) * = f\big( (5^m)^2 + 2^2(5^m)^2 + 5^2(5^m)^2 \big) = 5\,f(5^{2m}) + f(5^{2m+2})
\mathrm{Pr}\bigg{(}\sum_{i=1}^{d}(i^{-\alpha-1/2}N_{i}-v_{i})^{2}\leq d^{-2\alpha}\bigg{)}\geq \exp\bigg{\{}- \sum_{i=1}^{d}i^{2\alpha +1}v_{i}^{2}/ 2 - c_{3}d \bigg{\}}
P(x) = 2x \prod_{k=1}^{n-1}(x+\xi^k) + (-x+\xi^0) \prod_{k=1}^{n-1}(x+\xi^k) \geq 2x \prod_{k=1}^{n-1}\xi^k + (-x+\xi^0) \prod_{k=1}^{n-1}|-x+\xi^k| \geq 2\xi^n \prod_{k=1}^{n-1}\xi^k + \prod_{k=0}^{n-1}|-x+\xi^k| = 2\xi^{n+1 \choose 2}+ |P(-x)| \Rightarrow P(x) + P(-x) \geq P(x) - |P(-x)| \geq 2\xi^{n+1 \choose 2}.
\begin{array}{@{}l@{}}{\displaystyle |\Delta t\sum_{k=1}^{m}Q_{1}^{k}(\partial{\theta_{\psi}^{k}})|\leq C\big\{h^{2r+2}+(\Delta t)^{4}\big\}+ C\|\theta_{\psi}^{m}\|_{\mathcal{L}^2}^{2}+C\Delta t \sum_{k=1}^{m-1}{\|\theta_{\psi}^{k}\|_{\mathcal{L}^2}^{2}}.}\end{array}
H^{-1}B H = a A - a^{2}B^{-1}, \;\;H^{-1}K H = a^{-1}A - a^{-2}K^{-1}, &H^{-1}B^\downarrow H = a A - a^{2}(B^\downarrow)^{-1}, H^{-1}K^\downarrow H = a^{-1}A - a^{-2}(K^\downarrow)^{-1}.
|z|_{k,c+\frac{3i}{2\pi}\alpha}:=\|(z-c-\frac{3i}{2\pi}\alpha)|\eta|^{-1}\|_{L^{\infty}(0,1)}+ \|z|\eta|^k\|_{L^{\infty}(1,\infty)}+\|(1+|\eta|^{k+1})z'\|_{L^{\infty}(\real^+)}.
=\frac{\partial}{\partial t}|\Psi(x,t)|^{2}-\frac{ih}{2m}\frac{\partial}{\partial x}(\Psi^{*}(x,t)\frac{\partial}{\partial x}\Psi(x,t)-\Psi(x,t)\frac{\partial}{\partial x}\Psi^{*}(x,t))
\begin{aligned}\sum_{n=1}^{\infty}B_n \sinh\left(\frac{n\pi\theta_0}{M}\right)\sin\left(\frac{n\pi t_0}{M}\right)=\sum_{n=1}^{\infty}D_n \sinh\left(\frac{n\pi\theta_0}{M}\right)\sin\left(\frac{n\pi t_0}{M}\right)=0\end{aligned}
\frac{1}{\log e}\biggl(-\frac{1}{2}\log \frac{1}{2}q - \frac{1}{2}+ \frac{1}{2}\log\frac{1}{2}(1-q)+\frac{1}{2}+\log q +1- \log(1-q)-1\biggr) &=\frac{1}{2\log e}\left(\log q - \log(1-q)\right)
( 1 - \mid{\tilde{Z_{p}}}\mid^{2}) = \frac{( 1 - \mid a_{p}\mid^{2}) ( 1 - \mid Z_{p}\mid^{2})}{1 + \mid a_{p}\mid^{2}\mid Z_{p}\mid^{2}+ 2 \mid a_{p}\mid \mid Z_{p}\mid c o s \chi_{p}}\quad \geq 0
\Pi_\mathbf{f}\left(\frak{X}\right) =\left(\frac{\gamma-1}{\log \chi(\gamma)}-\frac{\varpi_\kappa}{2}\right)\cdot\frak{X}_\gamma+\varpi_\kappa\cdot\left(\frac{\frak{X}_\gamma}{2}+\frak{X}_\kappa\right).
{\cal L}_{\mathrm{M C S}}= - \frac{1}{4 e^{2}}F_{\mu \nu}F^{\mu \nu}+ \frac{\kappa}{2}\epsilon^{\mu \nu \rho}A_{\mu}\partial_{\nu}A_{\rho}+ | D_{\mu}\phi |^{2}+ \frac{1}{2 e^{2}}( \partial_{\mu}{\cal N})^{2}- V ( | \phi | ,{\cal N})
\delta\, 2^{j\Delta/q}=c\, \gamma\, 2^{jd(\frac{1}{2}-\frac{1}{q})}\, 2^{j\Delta/q}=c\, \bigg(\frac{1}{n}\bigg)^{\frac{1}{2}-\big(\frac{1}{2}-\frac{1}{q}\big)-\frac{\Delta}{dq}}=c\, n^{-\frac{1}{dq}}.
f_{\mathbf{x}}(\mathbf{x}) = G(\mathbf{x})\sum_{m=0}^{\infty}A_m H_m(\mathbf{z}) & \mbox{with,}A_m = \prod_{i=1}^{d}(m_i!)^{-1}\int_{\mathbb{R^d}}J_m(z)f_\mathbf{x}(\mathbf{x}) d\mathbf{x}
\psi_{v_{2}}(\xi,\lambda(t)) = 2 \chi_{\leq \frac{1}{4}}'(\xi) \partial_{\xi}\left(\frac{\xi^{2}K_{1}(\xi \lambda(t))}{\log^{b-1}(\frac{1}{\xi})}\right) + \frac{\chi_{\leq \frac{1}{4}}''(\xi) \xi^{2}K_{1}(\xi\lambda(t))}{\log^{b+1}(\frac{1}{\xi})}
mi_{l_0-\mu}\le mi_{l_0-\mu-1}=\tilde{\phi}_{L/K}^{l_0-\mu-1,m}(\tilde{\phi}_{M/L}^{\mu}(0))=\lambda_{M/K}^l(0)\le\tilde{\phi}_{L/K}^{l_0-\mu,m}(\tilde{\phi}_{M/L}^{\mu}(0))=mi_{l_0-\mu}
s = \sqrt{\omega^2 - t^2}e^{I \sigma_\omega}+ \sqrt{x_1^2 - \lambda_1^2}e^{-I \sigma_1}j_1 +\sqrt{x_2^2 - \lambda_2^2}e^{-I \sigma_2}j_2 + \sqrt{x_3^2 - \lambda_3^2}e^{-I \sigma_3}j_3 ~,
\epsilon_{( 0 , 4 . 8 )}^{'}= \epsilon \qquad \epsilon_{( 2 , 6 )}^{'}= \Gamma_{1 1}\epsilon \qquad \epsilon_{( - 1 , 3 , 7 )}^{'}= \imath \epsilon \qquad \epsilon_{( 1 , 5 )}^{'}= \imath \epsilon^{\ast}
\mathbf{D}^u\triangleq\begin{pmatrix}\mathbf{D}_{11}^u \mathbf{D}_{21}^u \cdots \mathbf{D}_{N1}^u \mathbf{D}_{12}^u \mathbf{D}_{22}^u \cdots \mathbf{D}_{N2}^u \vdots \vdots \ddots \vdots \mathbf{D}_{1N}^u \mathbf{D}_{2N}^u \cdots \mathbf{D}_{NN}^u\end{pmatrix},
{}\big\|\big ( (A_m - A_n ) v_n \big )_{n=k}^m \big\|_{L^p(X)}^p=\tau\sum_{n=k}^m \| (A_m - A_n ) v_n\|_{X}^p &{}\le \tau\sum_{n=k}^m \|A_m - A_n\|^p\|v_n\|_{D}^p=\sum_{n=k}^m \|A_m - A_n\|^p(E_n-E_{n-1}),
\begin{aligned}\hat \delta^{f}_r(c_i,c_j)= \mathbb F^f_r((c_i-\epsilon, c_i]\times [c_j, c_j+\epsilon))= &\mathbb F^f_r (c_{i}, c_{j})/ \mathbb F^f_r (c_{i}-\epsilon, c_{j}) + \mathbb F^f_r (c_{i}, c_{j}+\epsilon). \end{aligned}
\begin{array}{c}{{e^{- B}A e^{B}= \sum_{n = 0}^{\infty}\displaystyle \frac 1{n !}[ A , B ]_{( n )},}}\\{{{}[ A , B ]_{( 0 )}\equiv A , \qquad [ A , B ]_{( n + 1 )}= [ [ A , B ]_{( n )}, B ] .}}\\ \end{array}
f(C\delta L) =\epsilon C\delta L-(n+\frac{L}2)\log(C\delta L)+\frac 12 \log\Gamma(L)+\log D =\epsilon C\delta L -(n+\frac{L}2)\log(C\delta)-n\log L +\frac 12 \log(\frac{2\pi}L)^{1/2}-\frac{L}2+\textrm{O}(1) =[\epsilon C\delta- \frac 12 \log(C\delta)-\frac 12]L + \textrm{o}(L)\ .
\underset{1\le \ell,m\le k}{{\rm det}}(A_{j_\ell j_m})=\det\begin{pmatrix}a_{j_1j_1}a_{j_1j_2}\cdots a_{j_1j_k}a_{j_2j_1}a_{j_2j_2}\cdots a_{j_2j_k}\vdots \ddots \vdots a_{j_kj_1}a_{j_kj_2}\cdots a_{j_kj_k}\end{pmatrix}.
X_{2}=\frac{(t+1)x^{2}-1}{(t-1)x^{2}+1},X_{4}=\frac{(t_{2}+1)X_{2}^{2}-1}{(t_{2}-1)X_{2}^{2}+1},X_{8}=\frac{(t_{4}+1)X_{4}^{2}-1}{(t_{4}-1)X_{4}^{2}+1}.
M_k:=\max \Big\{\tilde{f}\big(-\|u_k\|_{\infty}\big),\tilde{f}\big(\|u_k\|_{\infty}\big)\Big\}\ge (c_0/2)^{-1}\max_{[-\|u_k\|_{\infty},\|u_k\|_{\infty}]}\tilde{f},
I_{1}=\int_{0}^{\eta}\biggl(\frac{K_{0}(z)}{\log(z)}\log(z)\biggr)^{k}\,dz\leq(1+\epsilon)^{k}\int_{0}^{\eta}\bigl(-\log(z)\bigr)^{k}\,dz &=(1+\epsilon)^{k}\int_{-\log(\eta)}^{+\infty}w^{k}e^{-w}\,dz\leq (1+\epsilon)^{k}\int_{0}^{+\infty}w^{k}e^{-w}\,dz &=(1+\epsilon)^{k}\varGamma(k+1).
\widetilde{M}_L\cdot M_L^{(1)}= M_L^{(1)}(v, w),\M_R^{(1)}\cdot \widetilde{M}_R = M_R^{(1)}(v, w), &\widetilde{M}_L\cdot M_L^{(2)}= M_L^{(2)}(v, w),\M_R^{(2)}\cdot \widetilde{M}_R = M_R^{(2)}(v, w)
\left\{m\in M\ \middle|\ \sum_{j=1}^{d-1}u^j m = 0\right\}\Big/ \Big\{um - m\ |\m\in M\Big\}\to H^1(U,M) \bar{m}\mapsto \left[c(m)_{u^i}= \sum_{j=0}^{i-1}u^j \cdot m\right].
T_{n,1}^{(K)}(t) = \sum_{j=1}^{K}\epsilon_j \rho_1^{\leftarrow}\left( \frac{\Gamma_j}{2 \mu (\tau_D \leq n)}\right) \hat S_{nt}(f)(V_j^{(n)})\,, T_{n,2}^{(K)}(t) = \sum_{j=K+1}^{\infty}\epsilon_j \rho_1^{\leftarrow}\left( \frac{\Gamma_j}{2 \mu (\tau_D \leq n)}\right) \hat S_{nt}(f)(V_j^{(n)})\,.
{\bf E\,}\left| \sum_{i=1}^{n}\xi_{i}\right|^{r}\leq 2^{r+1}\,e^{2r}\,\max\left[ r^{r/2}\left( \sum_{i=1}^{n}{\bf E\,}\xi_{i}^{2}\right)^{r/2},\,r^{r}\sum_{i=1}^{n}{\bf E\,}\left| \xi_{i}\right|^{r}\right]
\Bigl ( [ \bar{\alpha}D , \alpha D ] - \alpha D \bar{\alpha}D + \frac{1}{2}( \alpha D )^{2}\bar{\alpha}^{2}- 2 \alpha^{2}\bar{\alpha}^{2}+ 2 ( 2 s + d - 3 ) \Bigr ) | \Phi \rangle = 0 \, ,
\int_u^{+\infty}\left(\frac{t\ln^\beta(1+t)}{t^{(1+s)/(1-s)}}\right)^{1/(2s)}\;dt=\int_u^{+\infty}\frac{\ln^{\beta/(2s)}(1+t)}{t^{1/(1-s)}}\;dt\ <\ +\infty
\int_{t}^{\infty}\frac{\rho d\rho (\rho^{2}+r^{2}+\lambda(t)^{2-2\alpha})}{(1+2(\rho^{2}+r^{2})\lambda(t)^{2\alpha -2}+(\rho^{2}-r^{2})^{2}\lambda(t)^{4\alpha -4})^{3/2}}\frac{1}{\log^{(4\alpha -4)b}(t) t^{2}\log^{b+1}(t)}&\leq \frac{C}{t^{2}\log^{b+1}(t)}
\begin{cases}\sum\limits_{t=0}^{2s}(-1)^tg_tg_{2s-t}+\sum\limits_{t=2s+1}^{m-1}(-1)^tg_tg_{m+2s-t}=0,s=1,\dots,\frac{m}{2}-1, g_sg_{m-s}=(-1)^s,s=1,\dots,\frac{m}{2}, h^sg_sg_{\frac{m}{2}+s}=g_{2s}g_{\frac{m}{2}},s=1,\dots,\frac{m}{2}-1, h^\frac{m}{2}=1\end{cases}
\begin{aligned}\partial^2_tu=\partial^2_xu+c_1u+(c_2u+f(u)) \dot{W}(t,x),t>0,x\in \overline{D}, &u(0,x)=(J+T+1)(1+u_0(x)),\partial_tu(0,x)=(J+T+1)v_0(x). \end{aligned}
d_T(t,t') = d_T(k_{l,i}\langle HP \rangle, k_{l,j}\langle HP \rangle) \le d_T(k_{l,i}\langle HP \rangle, id_G \langle HP \rangle)+ d_T(id_G \langle HP \rangle, k_{l,j}\langle HP \rangle) \le d(k_{l,i},id_G) + d(k_{l,j},id_G)\le 2\delta/C_0^{l+1}\le \delta/C_0^l.
\begin{array}{l}\rho^\mu \partial_\mu x^{AA'}o_A =0, \partial_{\mu}x^{AA'}o_A\bar{o}_{A'}= 0, \partial_\mu (\rho^\mu o_A\bar{o}_{A'})+3\kappa\varepsilon^{\mu_1 \mu_2}\partial_{\mu_1}x^{A_1{A'}_1}\partial_{\mu_2}x^{A_2{A'}_2}\partial_{[AA'}B_{A_1 A_2{A'}_1{A'}_2]}=0,\end{array}
\frac{\left( 2n+1\right) h_{n}-2\left( h_{n+1}+\gamma_{n}^{2}h_{n-1}\right)}{2z}\frac{\left( 2n-1\right) h_{n-1}-2\left( h_{n}+\gamma_{n-1}^{2}h_{n-2}\right)}{2z}& =\left( \frac{n}{2}-\gamma_{n}\right)^{2}h_{n-1}^{2}.
\hat{p}_{5}= \frac{a_{0}^{3}}{n a^{3}}\left[ 6 M^{3}k^{2}- \frac 1 2 k \mathrm{c o t h}( k b ) ( \rho_{0}- 3 p_{0}) - \frac{\rho_{0}( \rho_{0}+ 3 p_{0})}{1 2 M^{3}}\right] .
(q_{\lambda}f)(\mathbf{x}_{0}) = 2^{-\alpha \lambda}L(2^{-\lambda})\langle g,\varphi_{\lambda}\rangle+o(2^{-\alpha \lambda}L(2^{-\lambda})) & =L(2^{-\lambda})(q_{\lambda}g_{\mathbf{x}_{0}})(\mathbf{x}_{0})+o(2^{-\alpha \lambda}L(2^{-\lambda})) \ \ \ \mbox{as}\lambda\to\infty.
\psi_{s}^{\mathrm{m o v i n g}}( t , x ) = \pm e^{i m v x}e^{i t \left( \frac{\alpha^{2}}{2 m}- \frac{m v^{2}}{2}\right)}\frac{1}{\sqrt{\lambda m}}\frac{\alpha}{\cosh \alpha ( x - v t )}
J_L(\dot{\gamma})=\frac{l_L}{l}L^{\frac{1}{2}}\omega(\dot{\gamma}(t))e_1+\left[\overline{q}\dot{\gamma}_3-\frac{\sqrt{2}}{2}\overline{p}\left(-e^{-\gamma_3}\dot{\gamma_1}+e^{\gamma_3}\dot{\gamma_2}\right)\right]e_2.
-F^{s-1}df_{1}^{*}a=\frac{(t_{1}^{-1}f_{1}^{*}\alpha_{1}+t_{2}f_{1}^{*}\beta_{2})d\log t_{1}+t_{2}f^{*}\beta_{2}d\log t_{2}+t_{1}t_{2}f_{1}^{*}\gamma}{t_{1}^{n_{1}+n_{2}-1}t_{2}^{n_{2}}}
\Delta_p :=\big|(S^{n_l}_{a_{i}}h(p)- S^{n_l}h_l(q_i))-(\mathbb E_{\mathcal P_{l}}(S_{a_i}^{n_l}h_l)-\mathbb E_{\mathcal P_{l}}(S^{n_l}h_l))\big| &\leqslant \omega^{n_l}(h_l, 2\epsilon_l, q_i)+ \int \omega^{n_l}(h_l, 2\epsilon_l, p)\, d\nu_{\mathcal P_{l}}(p).
\int_{\mathcal{B}^c_{k}}\frac{|\widehat{\varphi}(k t)|^2}{(|t|^2 +1 )^2}dt \le C\int_{\mathcal{B}^c_{k}}\frac{1}{(|t|^2 +1 )^2}dt = C'\int_{\frac{1}{\sqrt{k}}}^{+\infty}\frac{r}{(r^2+1)^2}dr = O(k),k\rightarrow 0+.
I_{\alpha}(\mu) \leq C\int_{|\xi|\lesssim 1}|\xi|^{\alpha-d}d\xi + C \sum_{k=0}^{d-\ell-2}\int_{\{\lambda^{1-\frac{k+2}{d-\ell}}\lesssim |\xi|\lesssim \lambda^{1-\frac{k+1}{d-\ell}}\}\cap M_0^{\gamma,d}S_{\ell}}|\xi|^{\alpha-d}d\xi.
\mathtt{I}(\mathbf{x}_\mathcal{S};\mathbf{y}_)=\sum_{k\in\mathcal{S}}\mathtt{I}(x_k;\mathbf{y}_|\mathbf{x}_{k+1}^{|\mathcal{S}|}):=\sum_{k\in\mathcal{S}}R_{_k}.
f_{n}(t)=\sum_{i=0}^{n}a(i,n)t^{i}=\frac{t}{n}\sum_{j=0}^{n-1}\left(1-2^{\nu_{2}(n-j)+1}\right)\sum_{i=0}^{k}a(i,j)t^{i}&=\frac{1}{n}\sum_{i=0}^{n-1}\left(\sum_{j=i}^{n-1}(1-2^{\nu_{2}(n-j)+1})a(i,j)\right)t^{i+1}&=\frac{1}{n}\sum_{i=1}^{n}\left(\sum_{j=i-1}^{n-1}(1-2^{\nu_{2}(n-j)+1})a(i-1,j)\right)t^{i}.
\eta'_{jk}= z_k\,dz_j-z_j \,dz_k \eta''_{jk}= w_k\,dw_j-w_j \,dw_k \theta = -w_1\, dz_1-\cdots-w_n\, dz_n= z_1\, dw_1+\cdots+z_n\,dw_n.
\int_{\mathbb{R}^{3}}(\nabla u_{j}^{c,\gamma}\nabla\varphi-u^{c,\gamma}_{j}u^{c,\gamma}\cdot\nabla\varphi-p^{c,\gamma}\partial_{x_{j}}\varphi) &=\begin{cases}0, j=1,2, 4\pi c_{3}\int^{\infty}_{-\infty}\ln|x_{3}|\partial_{x_{3}}\varphi(0,0,x_{3})dx_{3}-b^{c,\gamma}\varphi(0), j=3,\end{cases}
U_{k}^{i j}= \sqrt{- g}\left[ - g^{p i}( - \Gamma_{k p}^{j}+ \frac{1}{2}g_{k}^{j}\Gamma_{a p}^{a}+ \frac{1}{2}g_{p}^{j}\Gamma_{a k}^{a}) + \frac{1}{2}g_{k}^{i}g^{p m}( - \Gamma_{p m}^{j}+ \frac{1}{2}g_{p}^{j}\Gamma_{a m}^{a}+ \frac{1}{2}g_{m}^{j}\Gamma_{a p}^{a}) \right] ,
\frac{1}{g_{a}^{2}( \mu )}= \frac{k_{a}}{g_{s t r i n g}^{2}}+ \frac{b_{a}}{1 6 \pi^{2}}\ln \frac{M_{s t r i n g}^{2}}{\mu^{2}}- \frac{1}{1 6 \pi^{2}}\triangle_{a},
J_{n}(z) = (-)^n J_{- n}(z) = \frac{1}{2 \pi i^{\hskip 0.5 pt n}}\int_0^{2\pi}e^{- i n \phi + i z \cos \phi}d \phi , &\hskip 10 pt I_{n}(z) = I_{- n}(z) = \frac{(-1)^n}{2 \pi}\int_0^{2\pi}e^{- i n \phi - z \cos \phi}d \phi.
\beta(a,b,c) =\sigma_2\sigma_1^{a+1}\sigma_2\sigma_1^{b+1}\sigma_2\sigma_1^{c+1}=\sigma_2\sigma_1(\sigma_2\sigma_1)\sigma_2^a\sigma_1^{b-1}\sigma_2^c(\sigma_1\sigma_2)\sigma_1=\Delta_3\sigma_1\sigma_2^a\sigma_1^{b-1}\sigma_2^c\Delta_3.
\left( \prod_{1 \leq i <j \leq k}d_B(C_i,C_j) -\eta \binom k2 \right) \prod_{i=1}^k |C_i| \geq \left( \delta- \eta \binom k2 \right) \prod_{i=1}^k |C_i| \geq \left(\delta-\delta^3\right)\prod_{i=1}^k |C_i|>0,
\vec{x}_{\rm op}=\int\vec{x}dE_{\vec{x}}, R\subset\mathbb{R}^{3}\rightarrow E(R), U(a)E(R)U(a)^{-1}=E(R+a), E(R)E(R^{\prime})=0 R\times R^{\prime}, \left(E(R)\psi,U(a)E(R)\psi\right) =\left(\psi,E(R)E(R+a)U(a)\psi \right) =0,
\mathbf{E}[\|\mathcal{X}^{t+1}-\mathcal{X}^{\star}\|_{F(\mathcal{Q})}^{2}|\mathcal{X}^{t}] =\|\mathcal{X}^{t}-\mathcal{X}^{\star}\|_{F(\mathcal{Q})}^{2}-\mathbf{E}_{i\sim \mathbf{p}^{t}}[f_{i}(\mathcal{X}^{t})]=\|\mathcal{X}^{t}-\mathcal{X}^{\star}\|_{F(\mathcal{Q})}^{2}-\sum_{i\in \mathfrak{W}_{t}}p_{i}^{t}f_{i}(\mathcal{X}^{t}).
\epsilon_{\mu_{1}. . . \mu_{s}}\prod_{j = 1}^{s}D^{n_{j}}{\bf X}^{\mu_{j}}\quad \mathrm{s u c h ~ t h a t}\quad \sum_{j = 1}^{s}n_{j}= n , \quad n = 1 , 2 , 3 , . . .
\sqrt{2}M_{j}= \pm \sum_{i}\frac{\nu_{i}\phi ( a_{i})}{a_{i}^{2}- m_{j}^{2}}\mathrm{d e t}( a_{i}^{2}- m^{2})^{1 / r_{i}}\prod_{j \ne i}( a_{j}^{2}- a_{i}^{2})^{1 - 2 r_{j}/ r_{i}}\Lambda_{N = 2}^{\frac{4 ( N_{c}- 1 ) - 2 N_{f}}{r_{i}}}.
{\rm I}_{2}\leq \sum_{j=0}^{n-2}\int_{t_j}^{t_{j+1}}\|A_h^{\frac{s}{2}}(\bar E_h(t_n-t)-\bar E_h(t_n-t_j))P_h\|^p\d t & + \int_{t_{n-1}}^{t_n}\|A_h^{\frac{s}{2}}(\bar E_h(t_n-t) -\bar E_h(\tau))P_h\|^p \d t:={\rm I}_{2,1}+{\rm I}_{2,2}.
\{f , g \}={\frac{\partial f}{\partial \Pi}}\cdot{\frac{\partial g}{\partial \nabla^{2}X}}+{\frac{\partial f}{\partial P}}\cdot{\frac{\partial g}{\partial \dot{X}}}+{\frac{\partial f}{\partial p}}\cdot{\frac{\partial g}{\partial X}}- ( f \leftrightarrow g ) \, .
B := \frac{G(h)\phi + h_x\phi_x}{1+h_x^2}, V := \phi_x-Bh_x, & \omega := \phi-T_BP_{\geq 1}h=\phi-T_Bh+T_BP_{\leq 0}h, & \sigma :={(1+h_x^2)}^{-3/2}-1,
\phi(x y) = \ \phi[\exp\big(\tau \log( (t x)^\frac{1}{\tau})+(1-\tau)\log ((\frac{y}{t})^\frac{1}{1-\tau})\big)] \leq \ \tau \phi[\exp\log( (t x)^\frac{1}{\tau})] + (1-\tau)\phi[\exp\log ((\frac{y}{t})^\frac{1}{1-\tau})] = \ \tau t^\frac{1}{\tau}\phi(x^\frac{1}{\tau}) + \frac{(1-\tau)}{t^\frac{1}{1-\tau}}\phi(y^\frac{1}{1-\tau}).
\sigma(a \circ \lambda) - \sum_{s=0}^{N-1}\sum_{(h+1)|\alpha| + j + \ell = s}\frac{1}{\alpha!}\partial_{\xi'}^{\alpha}a_{-j/(h+1)}(x', \xi',x_1) D_{x'}^{\alpha}\lambda_{-\ell/(h+1)}(x', \xi') \in H_{h}^{m + m'' - \frac{N}{h+1}}.
\frac{\beta_{1,\ell}(\phi)}{\ell^{3/2}(\ell+1)^{3/2}}=- h_0(0) \cos \psi_{0,\ell}\frac{1}{\ell^{1+1/2}\sin^{1+1/2}\phi}+\phi^{-2-1/2}O(\ell^{-2-1/2}),
[ \tau \psi_{\beta}]_{2}= \left( 1 + \sum_{\alpha \in \Phi}( \alpha \beta / 2 k - ( \alpha \beta )^{2}/ 2 k^{2}+ N_{\alpha , \beta}N_{- \alpha , \alpha + \beta}) \right) \psi_{\beta}, ( \beta \in \Phi ) ,
|\langle g_{j},\varphi \rangle|\le C_{\varphi,\theta}\left(\sum_{l=1}^\infty\left(2^{-\frac{n\theta}{p}}\left\|2^j \chi_{{\mathcal O}_j}\right\|_{L^{p}([-2^l,2^l]^n)}\right)^{\frac{1}{\kappa}}\right)^{\kappa}+C_{\varphi,\theta}\|{\mathcal M}f \cdot \chi_{{\mathcal O}_j^c}\|_{L^p([-1,1]^n)},
f^{(C)}(\theta)= -\frac{i\eta}{\sqrt2 k}\frac{g_\omega(\theta)}{(\cos\omega- \cos\theta)^{3/2}}\frac{(1-\cos\theta)^q\Theta_\omega(\theta)}{[(\cos\omega-\cos\theta)^{1/2}-i\sqrt2\sin(\omega/2)]^{2q}}\ ,
c+o_{n}(1)\|(u_{n},v_{n})\|_{E}=I((u_{n},v_{n}))-\frac{1}{2}\langle I'(u_{n},v_{n}),(u_{n},v_{n})\rangle &=-\big(B(u+B(v)\big)+\Big(\frac{1}{2}-\frac{1}{2p}\Big)F(u,v)+o_{n}(1) &=o_{n}(1),
v_\frak{p}(f):=\min \{v_\frak{p}(a_f(T))\;|\;T \in \Lambda_{n}\}, &v^{(n')}_\frak{p}(f):=\min \{v_\frak{p}(a_f(T))\;|\;T \in \Lambda_{n},\{\rm rank}(T)=n' \}(0\le n'\le n).
\big\Vert \mu - \tilde{\mu}\big\Vert_{L^2}\le \big\Vert (1 - C_{v_\Sigma})^{-1}\big\Vert_{L^2 \to L^2}\big\Vert C_{\Sigma}^-\big\Vert_{L^2 \to L^2}\big\Vert \tilde{\mu}(v_\Sigma - \tilde{v}_\Sigma)\big\Vert_{L^2}= O\left(\frac{1}{n^{1/2}\log^2n}\right),
\Gamma_{a}:=\max\left\{|\phi(x,s,\arctan p)-\phi(x,f(x),0)|:(x,s,p)\in I_{a}\right\}, begin{align*}1ex]\Gamma_{a}^{+}:=\max\left\{|\phi(x,s,\arctan p)-\phi(x,f(x),\pi/2)|:(x,s,p)\in I_{a}^{+}\right\}, begin{align*}1ex]\Gamma_{a}^{-}:=\max\left\{|\phi(x,s,\arctan p)-\phi(x,f(x),-\pi/2)|:(x,s,p)\in I_{a}^{-}\right\}.
G_{1}^{( 2 )}( \lambda_{1}, \vec{r}_{1}, z_{1}, \vec{r}_{2}, z_{2}) = \frac{\lambda_{1}}{2}\int{}d^{d - 1}r \int_{0}^{L}d z \, \, G_{0}^{( 2 )}( \vec{r}_{1}- \vec{r}, z_{1}, z ) G_{0}^{( 2 )}( \vec{0}, z ) G_{0}^{( 2 )}( \vec{r}- \vec{r}_{2}, z , z_{2}) .
\left\langle AB_1B_2\right\rangle-\left\langle AB_1\right\rangle\left\langle \tilde{B}_2\right\rangle-\left\langle AB_2\right\rangle\left\langle \tilde{B}_1\right\rangle-\left\langle A\right\rangle\left\langle \tilde{B}_1\tilde{B}_2\right\rangle+2\left\langle A\right\rangle\left\langle \tilde{B}_1\right\rangle\left\langle \tilde{B}_2\right\rangle
\int \sqrt{g}\ \left( \nabla_{\mu_{1}}\cdots \nabla_{\mu_{p_{1}}}\left. \widehat{R}^{\alpha_{1}}\right._{\beta_{1}\gamma_{1}\delta_{1}}\right)\cdots \ \left( \nabla_{\mu_{1}}\cdots \nabla_{\mu_{p_{m}}}\left. \widehat{R}^{\alpha_{m}}\right._{\beta_{m}\gamma_{m}\delta_{m}}\right)
\left\vert\frac{\mathrm{d}w}{\mathrm{d}\theta}\right\vert=\frac{\left\vert\kappa\right\vert}{4}\left\vert\sin\left(\frac{\phi_1-\theta}{2}\right)\right\vert^{-\alpha_1}\left\vert\sin\left(\frac{\phi_2-\theta}{2}\right)\right\vert^{-\alpha_2}\ldots\left\vert\sin\left(\frac{\phi_n-\theta}{2}\right)\right\vert^{-\alpha_n}
\Gamma_1(N)g = \Gamma_1(N)\begin{pmatrix}0 -1 N 0\end{pmatrix}\begin{pmatrix}1 \frac{n}{N}0 1\end{pmatrix}\begin{pmatrix}A B C D\end{pmatrix}\begin{pmatrix}N N_S' 0 0 N_S\end{pmatrix}^{-1},
\mathcal{T}^{r,s}_{m(\cdot),\nu(\cdot)}(x',x'',\xi',\zeta',\xi'',\zeta'') &\triangleq (e_r(\mathrm{traj}^{s}_{m(\cdot)}(x',\xi'\zeta')), e_r(\mathrm{traj}^{s}_{\nu(\cdot)}(x'',\xi''\zeta''))).
\mathbf{PC}^k_{int}(\widetilde{\Omega}) = \{\mathbf{u}\,:\, \mathbf{u}\in \mathbf{C}^k(\widetilde{\Omega}^s),~s=\pm; ~ [\mathbf{u}]_{\Gamma}=\mathbf{0},~\textrm{and}~ [\sigma(\mathbf{u})\mathbf{n}]_{\Gamma}=\mathbf{0}\},
\int_{\partial M}|\nabla^T\triangle e_\lambda|^2ds =-\int_{\partial M}\triangle e_\lambda \triangle^T \triangle e_\lambda ds\leq\|\triangle e_\lambda\|_{L^2(\partial M)}\|\triangle^T\triangle e_\lambda\|_{L^2(\partial M)}\lesssim\lambda^2\|\triangle e_\lambda\|_{L^2(\partial M)}^2
X(z,\bar{z})=\sqrt{\frac{\alpha'}{2}}\sum_{k}\left( \frac{\alpha_{k-\vartheta}}{k-\vartheta}z^{-k+\vartheta}+\frac{\tilde{\alpha}_{k+\vartheta}}{k+\vartheta}\bar{z}^{-k-\vartheta}\right),
f_u(X)=\left( \prod^{r-1}_{\ell=u}\det\left( X^{[r-\ell+1,r]}_{[1,\ell]}\right) \det\left( X^{[1,\ell]}_{[s-\ell+1 ,s]}\right)\right) \cdot \left(\prod^{s-r+1}_{\ell=1}\det\left( X^{[1,r]}_{[\ell,r+\ell-1]}\right)\right),
Z [ j_{1}, j_{2}] = T r \left[ \rho \bar{T}{e^{i \int d^{4}x ( H_{s}+ j_{2}\phi_{s}^{*}+ j_{2}^{*}\phi_{s})}}T{e^{- i \int d^{4}x ( H_{s}- j_{1}\phi_{s}^{\ast}- j_{1}^{\ast}\phi_{s})}}\right]
\vert\vert n_0, n_1,\dots, n_{r}\rangle \rangle = \frac{(b_0^+)^{n_0}}{\sqrt{n_0!}}\frac{(b_1^+)^{n_1}}{\sqrt{n_1!}}\cdots\frac{(b_{r}^+)^{n_{r}}}{\sqrt{n_{r}!}}\vert\vert 0, 0,\cdots , 0\rangle\rangle.
G_{0}( \tau ) : = \left( \frac{\theta_{D_{r}}^{2}( \tau )}{\eta ( \tau )^{r + 1}}\right)^{2 4}= \left( \frac{\eta ( \frac{2 \tau}{h})}{\eta ( \frac{\tau}{2}) \eta ( \frac{\tau}{h})}\right)^{2 4}.
\tilde{\psi}(\phi)(aL_A,bL_B) = \left(\tilde{\psi}_A(\alpha)(aL_A)\tilde{\psi}_{B,A}(\delta)(b L_B), \tilde{\psi}_{A,B}(\gamma)(a L_A)\tilde{\psi}_B(\beta)(bL_B)\right) & =\left(\alpha(a)\delta(b)L_A, \gamma(a)\beta(b)L_B\right)
Z(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n}_{3}) =(\mathbf{n}_{1}\cdot\mathbf{n}_{2})^{2}+(\mathbf{n}_{2}\cdot\mathbf{n}_{3})^{2}+(\mathbf{n}_{3}\cdot\mathbf{n}_{1})^{2}-1\ .
\mathcal{H}(A_{\tau,n})\lesssim \sum_{v_1^n\in W_\tau^n}\sup_{x\in g_{v_1^n}(X)}\exp\left(\lambda \left (\beta \log |DS^{n+1}_\tau(x)|-\sum_{k=0}^{n-1}f_k^\tau(x)\right) -Q\log |DS_\tau^n(x)|\right)\,,
\begin{array}{c}{{B \mapsto \left( B + w ( B \cdot \nu ) \right) + \left( \nu + w ( \nu \cdot \nu ) \right)}}\\{{\alpha \mapsto \alpha +{\frac{1}{2}}\left[ | | B | |^{2}- | | B + \nu | |^{2}\right]}}\\ \end{array}
\Psi_{11}= \Gamma(1-\beta), \Psi_{12}= \frac{1}{\Gamma(\beta)}\left( \frac{\Gamma^{\prime}(1-\beta)}{\Gamma(1-\beta)}+2\gamma_{\mathrm{E}}- i \pi \right), & \Psi_{21}= \Gamma(1+\beta), \Psi_{22}= \frac{-1}{\Gamma(-\beta)}\left( \frac{\Gamma^{\prime}(-\beta)}{\Gamma(-\beta)}+ 2\gamma_{\mathrm{E}}- i \pi \right).
{\tilde E}_i = \frac{{\rm j}}{2}\left(\begin{array}{cc}\tau_i 0 0 \tau_i\end{array}\right),\hspace{6mm}{\tilde F}_i = \frac{1}{2}\left(\begin{array}{cc}0 \tau_i -\tau_i 0\end{array}\right), \hspace{5mm}i=1,2,3.
g^-(x^{\mu}):=\sum_{i_1=0}^{\infty}\cdots\sum_{i_n=0}^{\infty}\frac{1}{i_1!\cdots i_n!}\partial_1^{i_1}\cdots \partial_n^{i_n}g(\epsilon(x^{\mu})) (\epsilon(x^1)-x^1)^{i_1}\cdots(\epsilon(x^n)-x^n)^{i_n}.
\ell^{1,\infty}(\mathbb{N},w)=\left\{f=\{f(n)\}_{n\ge 0}: \|f\|_{\ell^{1,\infty}(\mathbb{N},w)}:=\sup_{t>0}t\sum_{\{m\in \mathbb{N}: |f(m)|>t\}}w(m)<\infty\right\},
\mathcal{F}_i^{\mathbf{z}}(\phi_w) (\pi^\lambda) = \left[ T_w \cdot \begin{pmatrix}\vdots \mathcal{F}_i^{\mathbf{z}}(\phi_1) (\pi^\lambda) \vdots - \vdots - \vdots \mathcal{F}_i^{w_0 \mathbf{z}}(\phi_1) (\pi^\lambda) \vdots \end{pmatrix}\right]_i,
\begin{array}{rr}f=[q_1-a_1(q_2)]*[q_1-a_2(q_2)]*\cdots*[q_1-a_{\deg_{q_1}f}(q_2)]= =[q_2-b_1(q_1)]*[q_2-b_2(q_1)]*\cdots*[q_2-b_{\deg_{q_2}f}(q_1)]\end{array}
X = \begin{pmatrix}s_1 x_3 \overline{x_2}\overline{x_3}s_2 x_1 x_2 \overline{x_1}s_3 \end{pmatrix},\;Y = \begin{pmatrix}t_1 y_3 \overline{y_2}\overline{y_3}t_2 y_1 y_2 \overline{y_1}t_3 \end{pmatrix}