harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{t_i\in\mathcal{D}}(|X_{t_i,t_{i+1}}|^p+|\mathbb{X}^{(2)}_{t_i,t_{i+1}}|^{\frac{p}{2}}) =\sum_{t_i\in\mathcal{D}}(|X_{t_i,t_{i+1}}|^p+|\mathbb{X}^{(2)}_{0,t_{i+1}}-\mathbb{X}^{(2)}_{0,t_{i}}-X_{0,t_i}\otimes X_{t_i,t_{i+1}}|^{\frac{p}{2}}) \asymp \sum_{t_i\in\mathcal{D}}d_{CC}(\mathbf{X}_{t_i},\mathbf{X}_{t_{i+1}})^p.
\mathcal{L}_{x}\left(g_{\alpha_1, \lambda_1, \alpha_2, \lambda_2}(x,t)\right)= e^{-t\left(c_1\left((s+\lambda_1)^{\alpha_1}-\lambda_1^{\alpha_1}\right) + c_2\left((s+\lambda_2)^{\alpha_2}-\lambda_2^{\alpha_2}\right)\right)}={\overline{G}}(s,t) (say).
dX_t = \big[A_t x + B_t \alpha + \beta_t\big] dt +\sigma dW_t , X_0=\xi dY_t = \bigg[-\frac{1}{2}P_t X_t^2 + \big( A_t X_t + \beta_t\big) \frac{Z_t}{\sigma}+\frac{1}{2}\frac{B_t^2}{Q_t}\frac{Z_t^2}{\sigma^2}\bigg] dt + dW_t,Y_T = \frac{1}{2}S X_T^2.
CH_{alg,X}^d(Y):=\{Z\in CH^d(Y\times X):i_y^*Z\sim_{alg}0\}, CH_{hom,X}^d(Y):=\{Z\in CH^d(Y\times X):i_y^*Z\sim_{hom}0\},{\rm NS}_{alg,X}^d(Y):=CH_X^d(Y)/CH_{alg,X}^d(Y),\;{\rm NS}_{hom,X}^d(Y):=CH_X^d(Y)/CH_{hom,X}^d(Y).
D^2\ \rightarrow\ \partial^2_t-\left(\frac1r\partial_rr\partial_r+\frac{1}{r^2}\partial_\varphi^2+\partial_z^2\right)\ +\ \frac{2i\beta\Theta(r-R)}{r^2}\ \partial_\varphi\ +\ \frac{\beta^2\Theta^2(r-R)}{r^2}\ .
\sum_{i=1}^Nu_i\left(y_i\sqrt{\gamma_N^{-1}\partial_x\xi(a_N,b_N)}+h_1\right),\,\,\forall u\in S^{N_1}&\sum_{j=1}^Nv_j\left(z_j\sqrt{(1-\gamma_N)^{-1}\partial_y\xi(a_N,b_N)}+h_2\right),\,\,\forall v\in S^{N_2}.
\sigma_B^{k_1(x)}(h(\sigma_A(x))) = \sigma_B^{l_1(x)}(h(x)) x \in X_A, \sigma_A^{k_2(y)}(h^{-1}(\sigma_B(y))) = \sigma_A^{l_2(y)}(h^{-1}(y)) y \in X_B
\begin{array}{rcccc}\min c\dot z_{\delta}s.t. \tilde{A_{\delta}}z_{\delta}\leq \tilde{c_{\delta}}\mbox{Linear relationships between Bernstein polynomials}z_{\delta}\in [L_{\delta},U_{\delta}] \mbox{bounds constraints}z_{i_1}\cdots z_{i_j}= c_i z_i \mbox{multiaffine equality constraints}\end{array}
\sum_{\substack{d_n\le y p_{n+1}\le X}}f(d_n) =\sum_{d\le y}f(d)\sum_{\substack{d_n=d p_{n+1}\le X}}1 &=\sum_{d\le y}f(d)\pi(X; \{0, d\})+\sum_{d\le y}f(d)\left(\sum_{\substack{d_n=d p_{n+1}\le X}}1-\pi(X; \{0, d\})\right).
\Delta_t' =2(\underline{h}+1)h^*(\partial_v-t\partial_z)^2+(h^*\partial_v\underline{h}+(\underline{h}+1)\partial_vh^*)(\partial_v-t\partial_z), \Delta_t^{**}=(h^*)^2(\partial_v-t\partial_z)^2+\frac{1}{2}\partial_v(h^*)^2(\partial_v-t\partial_z)
s_6 = \frac{1}{\eta_3 - \eta_2}\left( s_4 \overline{\left(\frac{{b_3}}{{b_1}}\right)}( \eta_1 - \eta_2 ) + s_5 \overline{\left(\frac{{b_2}}{{b_1}}\right)}(\eta_3 - \eta_1 ) \right).
\| P_3 B P_3 \|_2 = \| P_4 B P_4 \|_2,\quad\| P_2 B P_3 \|_2 = \| P_1 B P_4 \|_2, \quad\| P_1 B P_3 \|_2 = \| P_2 B P_4 \|_2.
\sum_{m=1}^\infty\frac{1}{m^s}\sum_{n = \lceil f(m) \rceil}^{\lfloor g(m) \rfloor}\frac{1}{n^s}\leq\sum_{m=1}^\infty\frac{1}{m^s}\left(1+\frac{1}{-s+1}\left( g(m)^{-s+1}- f(m)^{-s+1}\right)\right) &=\zeta(s)+\frac{1}{-s+1}\sum_{m=1}^\infty\frac{1}{m^s}\left(g(m)^{-s+1}- f(m)^{-s+1}\right).
\left\{{{Det^{\prime}\left[- (d^2/d\tau^2) + V^{\prime \prime}[x_{c1}(\tau)] \right]}\over{Det \left[- (d^2/d\tau^2) + \nu^2 \right]}}\right\}^{- 1/2}sqrt{{{S_{1}}\over{2 \pi}}}\d\tau_{c}
\mathrm{d e t}\left( 1 \! \! 1 \, \pm \, \hat{F}\right) = 1 \, - \, \frac{1}{2}T r ( \hat{F}^{2}) \, + \, \frac{1}{8}\left[ T r ( \hat{F}^{2}) \right]^{2}- \frac{1}{4}\, T r ( \hat{F}^{4}) \, + \mathrm{d e t}\, ( \hat{F})
\|u(t)\|_{\dot{B}^{1+\frac{3}{p}}_{p,1}}+\|(v(t),w(t))\|_{\dot{B}^{\frac{3}{q}}_{q,1}}\geq C(\|u(t)\|_{\dot{B}^{-1+\frac{3}{p}}_{p,1}}+\|(v(t),w(t))\|_{\dot{B}^{-2+\frac{3}{q}}_{q,1}})^{1+\frac{2}{s+\frac{3}{r}-1}}=C\mathcal{E}(t)^{1+\frac{2}{s+\frac{3}{r}-1}}.
Y_2 = \prod_{i = 1}^n C[d_i, e_i], Y_3 = \prod_{i = 1}^n C[c_i, d_i], \qquad Y_4 = C[\{c_1, \ldots, c_n\}], & Y_5 = \prod_{i = 1}^n C[b_i, e_i], Y_6 = \prod_{i = 1}^n C[c_i, e_i]. \qquad
\begin{aligned}\dot{x}= \Omega(\mu)+\Delta(\sigma,\mu)+\xi(y,z,\sigma,\mu)+f(x,y,z,\sigma,\mu), \dot{y}= \sigma+\eta(y,z,\sigma,\mu)+g(x,y,z,\sigma,\mu), \dot{z}= M(\mu)z+\zeta(y,z,\sigma,\mu)+h(x,y,z,\sigma,\mu),\end{aligned}
\Re(\Psi^{-1}_{\widetilde{F}}(z^o_{n, k+1}, u^o_{n,k+1}) ) = \Re(\Psi^{-1}_{\widetilde{F}}(z^o_{n}, u^o_{n}))+ k +1+ o(1) &< \kappa_1-k_n - R -1 +k + 1 +o(1)\leq -R - 1 + o(1) <-R.
\lambda=\pm \|D\hat{f}_{\mathbf{0}}(\partial_x)\|,\tilde{\lambda}=\pm\|D\hat{f}_{\mathbf{0}}(\partial_y)\|, \varpi=\left(\partial_x \check{f}(0,0), \partial_y \check{f}(0,0)\right), \beta=\partial_{xy}\check{f}(0,0)
P(-t\log J^u)\geq h(\mathcal L(\mu_0))-t\int \log J^u d\mathcal L(\mu_0) =\frac{1}{\int r d\mu_0}\left(q\log\sqrt{q}+\int \Phi_td\mu_0\right) &\geq \frac{1}{q}\log\sqrt{q}-\frac{2t\log C}{q}-t\lambda^u(\delta_Q)>-t\lambda^u(\delta_Q).
[(\alpha I-F({\rm i}z))^{-1}]_{i,j}= \sum_{k\in E}c_k[\kappa^+(\alpha,-{\rm i}z)^{-1}]_{i,k}\frac{\pi_j}{\pi_k}[[\hat\kappa^+(\alpha,{\rm i}z)^{\rm T}]^{-1}]_{k,j},
V^{\mu}= \left({\frac{\sqrt{F ( a ) + \dot{a}^{2}}}{F ( a )}}, \dot{a}, 0 , 0 , 0 \right) ; \qquad V_{\mu}= \left( - \sqrt{F ( a ) + \dot{a}^{2}},{\frac{\dot{a}}{F ( a )}}, 0 , 0 , 0 \right) .
{\widehat \Gamma}_{\Lambda_{1}\dots \Lambda_{n}}^{( n )}( k_{1}\dots k_{n}) = 2^{1 - n}P_{\Lambda_{1}\alpha_{1}}( k_{1}^{0}) \dots P_{\Lambda_{n}\alpha_{n}}( k_{n}^{0}) \Gamma_{\alpha_{1}\dots \alpha_{n}}^{( n )}( k_{1}\dots k_{n}) \, ,
\mathcal{H}_{\lambda}= \bigoplus_{\mu \in \mathcal{C}_{2 N}}\mathcal{H}_{\lambda , \mu}= \bigoplus_{\mu \in \mathcal{C}_{2 N}}\mathcal{V}_{\lambda}^{\mu}\otimes \bar{\mathcal{V}}_{\lambda^{*}}^{\mu^{*}}.
\sum_{q \leq Q}\sum_{\gcd(k,q)=1}\Bigl| \sum_{n \leq x}a_n \exp \bigl(\frac{2 \pi i kn}{q}\bigr) \Bigr|^2 \leq \sum_{q \leq Q}\sum_{\gcd(k,q)=1}(\sum_{n \leq x}|a_n|)^2 & = \sum_{q \leq Q}\varphi (q) (\sum_{n \leq x}|a_n|)^2 & \ll Q^2 (\sum_{n \leq x}|a_n|)^2
z_\nu z^*_\mu +(-1)^{\mu' + \nu'}z_{\nu'}z^*_{\mu'}= z_\mu z^*_\nu + (-1)^{\mu' + \nu'}z_{\mu'}z^*_{\nu'}& z^*_\mu z_\nu - (-1)^{\mu' + \nu'}z^*_{\mu'}z_{\nu'}= z^*_\nu z_\mu - (-1)^{\mu' + \nu'}z^*_{\nu'}z_{\mu'}
\widetilde{U}_i=(\widetilde{Sq}^0)^i(\lambda_{191}(\lambda_{15}^2\lambda_{39}+\lambda_{39}\lambda_{15}^2)\lambda_0+\lambda_{63}^2\lambda_{47}\lambda_{87}\lambda_0+\lambda_{127}\lambda_{31}\lambda_{63}\lambda_{39}\lambda_0), i\geq0.
S_{i_{1}i_{2}}^{k_{1}k_{2}}\left( \theta_{1 2}\right) S_{k_{1}k_{3}}^{j_{1}j_{3}}\left( \theta_{1 3}\right) S_{k_{2}i_{3}}^{j_{2}k_{3}}\left( \theta_{2 3}\right) = S_{i_{1}i_{3}}^{k_{1}k_{3}}\left( \theta_{1 3}\right) S_{k_{1}k_{2}}^{j_{1}j_{2}}\left( \theta_{1 2}\right) S_{i_{2}k_{3}}^{k_{2}j_{3}}\left( \theta_{2 3}\right) \, ,
I - \ \delta\, \frac{i}{i-1}= (1 - \delta) \frac{1}{2\pi}\lim_{\epsilon \to 0}\Big[ \int_{0+\epsilon}^{\pi}\frac{e^{i\theta}}{e^{i\theta}-1}d \theta + \int_{0+\epsilon}^{\pi}\frac{e^{-i\theta}}{e^{-i\theta}-1}d \theta\Big] = (1 - \delta) \frac{1}{2\pi}\lim_{\epsilon \to 0}\int_{0+\epsilon}^{\pi}d\theta,
S_{\pm}(x,y)= \langle q^{a}_{\alpha}(x){\bar q}^{b}_{\beta}(y) \rangle= \int d^{4}z \, \frac{\left[ \Psi^{\pm}_{z}(x){\bar\Psi}^{\pm}_{z}(y) \right]^{ab}_{\alpha\beta}}{m^{*}}
V_h =\{v_h \in L^2(\Omega):\;v_h |_K \in P_1(K)\ \forall\,K\in\mathcal{T}_h\}, W_h =\{\boldsymbol{w}_h\in[L^2(\Omega)]^2:\; \boldsymbol{w}_h |_K\in [P_1(K)]^2\ \forall\,K\in\mathcal{T}_h\}.
\begin{aligned}\Psi^{(r,1)}(\zeta) =W^{(r)}[\psi](\zeta)\ , \Psi^{(r,p'-1)}(\zeta) =W^{(r)}\left[\mathcal{L}\chi\right](\zeta)\ , \Psi^{(1,s)}(\zeta) =\mathcal{L}W^{(p'-s)}[\chi](\zeta)\ , \Psi^{(p-1,s)}(\zeta) =\mathcal{L}W^{(p'-s)}[\mathcal{L}\psi](\zeta)\ .\end{aligned}
\stackrel{P}{K}{}\! \!_{0 , p}^{( \pm )}\left( \begin{array}{c}{{\alpha}}\\{{\beta , s}}\\ \end{array}\right) = \stackrel{P^{l}}{K}{}\! \!_{0 , p}^{l ( \pm )}\left( \begin{array}{c}{{\alpha^{l}}}\\{{\beta^{l}, s^{l}}}\\ \end{array}\right) \stackrel{P^{r}}{K}{}\! \!_{0 , p}^{r ( \pm )}\left( \begin{array}{c}{{\alpha^{r}}}\\{{\beta^{r}, s^{r}}}\\ \end{array}\right) .
G_{(1)}^Y(z)_+=z+G_{X_1}^{X_2}(z)\ ,G_{(1)}^Y(z)_-=t_3z^2+t_2z-W_{(1)}(z)\ , &G_{(2)}^Y(z)_+=z+W_{(2)}(z)\ ,G_{(2)}^Y(z)_-=t_3z^2/4+t_2z/2-W_{(2)}(z)-W_{(2)}(-z)\ .
\left( \sum_{i \in I}\varphi_i \right) \circ \left( \bigoplus_{i \in I}\psi_i \right) \left( e^\frac{1}{2}b e^\frac{1}{2}\right) = \sum_{i \in I}e_i \circ \kappa_i \left( e^\frac{1}{2}b e^\frac{1}{2}\right) = \sum_{i \in I}\kappa_i \left( e^\frac{1}{2}b e^\frac{1}{2}\right)
{}_{m+1}F_m\bigg[\begin{matrix}\alpha_0 \alpha_1 \ldots \alpha_m &\beta_1 \ldots \beta_m\end{matrix}\bigg|\,z\bigg]:=\sum_{k=0}^{\infty}\frac{(\alpha_0)_k(\alpha_1)_k\cdots(\alpha_m)_k}{(\beta_1)_k\cdots(\beta_m)_k}\cdot\frac{z^k}{k!},
(PQ1)\;\frac{1}{p}+ \frac{n}{2p}\geq \frac{n}{2}- \sigma = (1 +\frac{n}{2})\frac{1}{p_c}+1 - \sigma; \frac{1}{2p}\geq \frac{1}{2}- \frac{2 \sigma}{n+1}\hbox{if}\; n \geq 3; \qquad\qquad\frac{1}{p}+ \frac{2}{2p}> 1- \sigma; \frac{1}{2p}\geq \frac{1}{2}- \frac{2 \sigma}{3}\hbox{if}\; n = 2;
\overline{{{\psi}}}^{( 1 )}( x ) P_{+}\psi^{( 1 )}( x ) \; \longleftrightarrow \; \frac{1}{2 \pi}c^{( 1 )}: e^{- i a \varphi^{( 1 )}( x )}:_{M^{( 1 )}}: e^{- i b \varphi^{( 2 )}( x )}:_{M^{( 2 )}}\; e^{i \frac{\theta}{2}}\; ,
\begin{aligned}\lim_j\|T^{n_{k_j}}XJ_{\theta,1}^{-1}p\|^2 =\lim_j(\|T^{n_{k_j}}XJ_{\theta,1}^{-1}p\|^2-\|pXJ_{\theta,1}^{-1}\chi^{-n_{k_j}}\|^2) + L(p) &=\|RX_1P_{L^2(\mu_{})}p\|^2+F(p)+ L(p)\end{aligned}
\begin{aligned}(1-t)\Theta_0(p) + t\hat{\Theta}= (1-t)\Theta_0(p) + t\Theta_1(p) &\geq \inf_{X}[(1-t)\Theta_0 + t\Theta_1] &\geq (1-t)\inf_X\Theta_0 +t\inf_X \Theta_1 &= (1-t)\Theta_0(p) +t\hat{\Theta},\end{aligned}
\hat Y =\left(\begin{array}{cc}\hat Y_1 \hat Y_2 \hat Y_3 \hat Y_4 \end{array}\right)=\left(\begin{array}{cc}W_1^{-1}Y_1 W_1 W_1^{-1}Y_2 W_2 W_2^{-1}Y_3 W_1 W_2^{-1}Y_4 W_2 \end{array}\right),
\bigg|\sin((2n_k+1)v y) \left(\frac{|\rho_0|}{|\rho_k|}- \frac{1}{2n_k+1}\right)\bigg| \le \frac{(2n_k+1)|\rho_0|-|\rho|_k}{|\rho_k|(2n_k+1)}&= \frac{(2n_k+1)^2|\rho_0|^2-|\rho|_k^2}{|\rho_k|(2n_k+1)((2n_k+1)|\rho_0|+|\rho_k|)}< \frac{(2n_k+1)^2\beta_0^2}{(2n_k+1)^2|\rho_0| |\rho_k|}< \frac{1}{v}.
ds^{2}= X(\frac{d \eta^{2}}{1-\eta^{2}}-\frac{d \mu^{2}}{1-\mu^{2}})-(1-\mu^{2})^{1/2}(1-\eta^{2})^{1/2}[\chi dy^{2}+\frac{1}{\chi}(dx-\lambda dy)^{2}].
N_{\Phi,K,L\otimes E}:K^\times \to(K\otimes E)^\times\to (L\otimes E)^\times = \prod_i E_i^\times x \mapsto\mathrm{det}_{L\otimes E}(x|_{V_{\Phi,K,L\otimes E}}) := \prod_i\mathrm{det}_{E_i}(x|_{V_{\Phi,K,E_i}}),
\frac{d c_{n}(t)}{d t}=\sum_{n^{\prime}\ne n}~\exp\Bigl[\frac{i}{\hbar}\int_{0}^{t}\Bigl((\theta_{n^{\prime}}(t^{\prime})-\theta_{n}(t^{\prime})\Bigr)d t^{\prime}\Bigr]\times \frac{\Bigl<n;t|\frac{{\cal D}\Lambda(t)}{{\cal D}t}|n^{\prime};t\Bigr>}{\bigl(\lambda_{n}(t)-\lambda_{n^{\prime}}(t)\bigr)}~c_{n^{\prime}}(t),
\alpha_{w_{i-n+1}\cdot\cdot\cdot w_{i-1}}=\frac{\beta_{w_{i-n+1}\cdot\cdot\cdot w_{i-1}}}{\sum_{\{w_{i}:C(w_{i-n+1}\cdot\cdot\cdot w_{i})\le k\}}P_{bo}(w_{i}|w_{i-n+2}\cdot\cdot\cdot w_{i-1})}
\begin{cases}b^\varepsilon(x)\dfrac{\partial^2u^\varepsilon(t,x)}{\partial t^2}+ \mathrm{curl}( a^\varepsilon(x)\mathrm{curl}u^\varepsilon(t,x) )=f(t,x), (0,T) \times D u^\varepsilon(0,x)=g_0(x) u_t^\varepsilon(0,x)=g_1(x) \end{cases}
{\bar G}^{(\zeta )}(\beta )={(\beta +i\pi )\sinh\left({\displaystyle \pi\beta\over \displaystyle 2\pi +\zeta}\right) \over\Gamma \left( 1-{\displaystyle\pi-i\beta \over \displaystyle 2\pi+\zeta}\right)\Gamma \left( 1-{\displaystyle\pi+i\beta \over \displaystyle 2\pi+\zeta}\right)}\ ,
{\mathcal L}_{D 3}={\cal O}( \Theta^{0}) + Z^{- 1}\left( - \frac{i}{2}\overline{{{\Theta}}}\Gamma^{i}D_{i}\Theta - \frac{i}{4 8}e^{\phi}\overline{{{\Theta}}}\Gamma^{m n p}\Theta R e \left[ ( *_{6}G )_{m n p}- i G_{m n p}\right] \right)
W_{n+1}(x)=(x-\beta_{n})W_{n}(x)-\gamma_{n}^{1}W_{n-1}(x)-\gamma_{n-1}^{0}W_{n-2}(x),\:\: n \geq 2, W_{0}(x)=1,\:\: W_{1}(x)=x-\beta_{0},\:\:W_{2}(x)=(x-\beta_{1})W_{1}(x)-\gamma_{1}^{1}.
A(\varphi):=(A_{j_1}^{j_2}(\varphi))_{j_1, j_2\in \mathbb{Z}}, A_{j_1}^{j_2}(\varphi):=\sum_{l\in\mathbb{Z}^{\nu}}A_{j_1}^{j_2}(l)\,e^{\mathrm{i}\,l\cdot\varphi}
\begin{cases}\frac{h}{h_b}\nabla^{\mu}_{X,z}\cdot P(\Sigma_b) \nabla_{X,z}^{\mu}u = f \mathcal{S}_b u|_{z = 0}= 0, \frac{h}{h_b}|n_b|\partial_{n_{b}}^{P_b}u |_{z=-h_b}= g,\end{cases}
G \left( z^{\prime}\right) = G \left( z \right) + \sum_{b \, \in \, z - \left( z \cap z^{\prime}\right)}\left[ \# \sigma \left( z^{\prime}, b \right) - 1 \right] \cdot e_{{\cal B}\left( z \right)}\left( b \right) + \sum_{b \, \in \, z - \left( z \cap z^{\prime}\right)}G_{{\cal B}\left( z^{\prime}, b \right)}\quad .
\textrm{T r}e^{- \frac{i}{\hbar}{\cal{H}}T}= \int d q_{0}\int_{q_{0}, 0}^{q_{0}, T}{\cal{D}}q \; e^{\frac{i}{\hbar}S [ q ]}= \int d q_{0}e^{\frac{i}{\hbar}S^{c l}( q_{0}, T )}\int_{0 , 0}^{0 , T}{\cal{D}}\eta ( a_{n}) \; e^{\frac{i}{\hbar}\sum_{n}\epsilon_{n}a_{n}^{2}}.
{E}^{(j)}={E}^{(j-1)}+ \big [F\circ{K}^{(j)}- F\circ{K}^{(j-1)}-(DF\circ{K}^{(j-1)}){\mathcal K}^{(j)}\big ] &+ (DF\circ{K}^{(j-1)}){\mathcal K}^{(j)}-{\mathcal K}^{(j)}\circ{R}^{(j-1)}& -\big [{K}^{(j)}\circ{R}^{(j)}-{K}^{(j)}\circ{R}^{(j-1)}\big ].
\dot{w}U^+B^*\times^{B^*}B^*sB^*/B^*=\pi^{-1}\left(\pi(\dot{w}U^+B^*)\right)\times^{B^*}B^*sB^*/B^*\overset{\sim}{\rightarrow}\pi(\dot{w}U^+B^*)\times B^*sB^*/B^*.
\frac{e^{-x}}{2}E_+(t)=u(t,x)=-u_x(t,x)=u_{xx}(t,x),~~x>q(t,b), \frac{e^{x}}{2}E_-(t)=u(t,x)=u_x(t,x)=u_{xx}(t,x),~~x<q(t,a), \frac{e^{-x}}{2}F_+(t)=v(t,x)=-v_x(t,x)=v_{xx}(t,x),~~x>q(t,b), \frac{e^{x}}{2}F_-(t)=v(t,x)=v_x(t,x)=v_{xx}(t,x),~~x<q(t,a).
u^{*}(x) = \limsup_{\delta \to 0^{+}}\sup \left\{U^{\Delta x}(m) \, \mid \, d(-m\Delta x, x) + \Delta x \leq \delta \right\}, u_{*}(x) = \liminf_{\delta \to 0^{+}}\inf \left\{U^{\Delta x}(m) \, \mid \, d(-m\Delta x, x) + \Delta x \leq \delta \right\}.
Res_{\mu=-\frac n2-\lambda-1-N}\big(M_{\bar{\theta}}(\mu;r)(\varphi)\big)= \int_M\sum_{j=0}^N\frac{1}{(N-j)!}<\sum_{k=0}^jv_k\mathcal{T}_{j-k}(h;\lambda)\theta_0^+ ,\big((\nabla^{\bar{g},S}_{\partial_r})^{N-j}\varphi\big)(0,\cdot)>Vol(h).
\left\langle [Z_1,Z_2],\nu^H\right\rangle=\frac{-4\left\langle \nu , T\right\rangle}{|\mathcal{P}_H(\nu)|}\left\langle J(Z_1),Z_2\right\rangle\quad\mbox{for every}Z_1,Z_2 \in \mathcal{H}\cap TM.
X^{\mu}= X^{\mu}_0 +{1 \over \pi T}P^{\mu}\tau +{i \over 2\sqrt{\pi T}}\sum_{n \neq 0}{1\over n}\bigg\{\alpha^{\mu}_n exp\bigg(-i2n(\tau - \sigma)\bigg) + \tilde{\alpha}^{\mu}_n exp\bigg(-i2n(\tau + \sigma )\bigg)\bigg\}
f\big(\varphi_{ts}(x)\big) = f(x) + \big(h^i_t-h^i_s\big)\big(V_if\big)(x) + \left(\int_s^t\int_s^r dh^j_u\,dh^k_r\right)\,\big(V_jV_kf\big)(x) + O\big(|t-s|^{>1}\big),
D_{i}:=\frac{1}{\left|Y_{1}\right|}\int_{Y_{1}}d_{i}\left(y\right)dy,\quad\mathbb{D}_{0}^{i}:=\left(\frac{1}{\left|Y_{1}\right|}\int_{Y_{1}}d_{i}\left(y\right)\frac{\partial\bar{u}_{i}^{j}}{\partial y_{k}}dy\right)_{jk},
\widetilde Y \equiv \tilde e_\mu{}^r e_r{}^\rho g(\tilde e)_{\rho \nu}\,{\rm d}x^\mu\wedge{\rm d}x^\nu= -2m\,{\rm d}r\wedge\Sigma^3 +\frac{1}{m}(r+2m)(r+m)\,\Sigma^1\wedge\Sigma^2\, ,
P_n^{(\alpha,\beta)}(\cos\theta):=\frac{\Gamma(\alpha+1+n)}{\Gamma(n+1)\,\Gamma(\alpha+\beta+1+n)}\sum_{k=0}^n(-1)^k\binom{n}{k}\frac{\Gamma(\alpha+\beta+1+n+k)}{\Gamma(\alpha+1+k)}\sin^{2k}(\tfrac12\theta),
\left(F-\frac{f^p}{2{\nu}^{p/{p^{'}}}}\right)^{q/p}\geq C\frac{f^q}{{\nu}^{q/{p^{'}}}}+\frac{1}{2^{nq/p}}\sum_{i=1}^{2^n}\left(F_i-\frac{f_i^p}{2{\nu_i}^{p/{p'}}}\right)^{q/p}
\sum_{l=1}^{|F|}\left(\lambda_F[\tilde{\phi}]\left(Kv_l^2+\frac{\partial \left(v_l^2\right)}{\partial\nu}\right)-\tau|\nabla v_l|^2-|D^2v_l|^2\right)=c,\mathrm{\a.e.\on\}\partial\tilde{\phi}(\Omega).
S^{(q)}_{j}(z) :=\sum_{m=0}^{\infty}\frac{S^{(q)}_{[m(p+1)+j,j]}}{z^{m(p+1)+j+1}}, T^{(q)}_{j}(z) :=\sum_{m=0}^{\infty}\frac{T^{(q)}_{[m(p+1)+j,j]}}{z^{m(p+1)+j+1}}.
\int_0^1 \frac{(\{px \}- \{qx\})^2}{x}\mathrm{d}x = \left( 1 + 2 \int_0^1 \{px \}\log x \mathrm{d}x - \frac{p}{2}- 2 \sum_{m=1}^{p-1}\frac{m}{p}\log \frac{m}{p}\right) q + \left( - \frac{1}{6}+ \frac{1}{2}\right) \log q + \mathcal{O}(1) &= \frac{1}{3}\log q + \mathcal{O}(1).
F ( x^{+}, x^{-}, 0 )_{F}= \frac{i}{4 ( 2 \pi )^{3}}\frac{m^{5}}{x}\left( \frac{x^{+}}{x^{-}}\right)^{2}\left[ K_{7 / 2}( m x ) K_{3 / 2}( m x ) - K_{5 / 2}^{2}( m x ) \right] \, .
\int_{t_1}^{t_2}\, \left( \frac{\partial L}{\partial x}\; \delta x +\frac{\partial L}{\partial{\cal V}}\; \left[ \, \frac{1-i}{2}\; \frac{d}{dt_{+}}+ \frac{1+i}{2}\; \frac{d}{dt_{-}}\,\right] \, \delta x \right) \, dt = 0,
\sum_{k=0}^n \binom{a}{c+k}\binom{b}{n-k}\binom{c+k}{c}\binom{n-k}{d}H_{a-c-k}={}\binom{a+b-c-d}{n-d}\binom{a}{c}\binom{b}{d}(H_{a+b-n-c}-H_{a+b-c-d}+H_{a-c})
F_k(\alpha^{(j)}) = \alpha^{(j)}\!\log_2\!\Big(\!1\!+\!\frac{\tilde{p}^{(j)}|H_{\tilde{i}k}|^2}{\sigma^2_n}\!\Big) \!+ \! (\!1\!-\!\alpha^{(j)}\!)\sum_{i=1}^{2}\!\log_2\!\Big(\!1\!+\!\frac{p_{i}^{(j)}}{\sigma^2_n}\!\Big),
\mu(s, \pi \times \tau,\psi) = \gamma(s,\phi^{\vee}\otimes \phi_{\tau}, \psi) \cdot \gamma(-s,\phi \otimes \phi_{\tau}^{\vee},\overline{\psi}) \cdot \gamma (2s, R \circ \phi_{\tau},\psi) \cdot \gamma(-2s, R^{\vee}\circ \phi_{\tau},\overline{\psi}).
\begin{aligned}\frac{db(t)}{dt}=g_0+g_1N(b(t),c(t))-b(t)=g_0-(1-g_1)b+g_1(N-b), \frac{dc(t)}{dt}=2a_0+2a_1N(b(t),c(t))-2c(t)=2a_0+2a_1N-2c.\end{aligned}
\overleftarrow{P}_{0,n}(C|{y}^{n-1}) \triangleq\int_{C_0}p_0(dx_0|x^{-1},y^{-1})\ldots\int_{C_n}p_n(dx_n|x^{n-1},y^{n-1}),~C=\times_{t=0}^n{C}_t\in{\cal B}({\cal X}_{0,n})
\int\limits_0^\pi Cf(0, e_2,\psi)\sin \psi d\psi = \int\limits_0^\pi \int\limits_0^\infty f(r\sin \psi, r\cos \psi)\sin\psi dr d\psi &+\int\limits_0^\pi \int\limits_0^\infty f(-r\sin \psi, r\cos \psi)\sin\psi dr d\psi.
(S_\varepsilon^{(k)}\mathbf{u})(\mathbf{x})=\vert \Omega \vert^{-1}\int_\Omega \mathbf{u}(\mathbf{x}-\varepsilon \mathbf{z})\,d\mathbf{z},\mathbf{u}\in L_2(\mathbb{R}^d;\mathbb{C}^k),
l_{\langle\rangle}(\mathcal I_{\psi_0\land\psi_1,i}P) =(l_{\langle\rangle}(P)\backslash\{\psi_0\land\psi_1\})\cup\{\psi_i\}, o_{\langle\rangle}(\mathcal I_{\psi_0\land\psi_1,i}P) =o_{\langle\rangle}(P)
\sum_{k \in \mathcal{P}_H}\mathcal T_{\rm{com}}(k) = - \big(1 + \mathcal O(\varepsilon_K + \ell^2 \rho \widehat g(0) K_H^{-2}) \big) \frac{\rho_z}{\vert \Lambda \vert}\sum_{k \in \mathcal{P}_H}\frac{\widehat g(k)^2}{k^2}\sum_{p \in \mathcal P_L}a_p^\dagger a_p + \mathcal E,
\lambda_1 \bar J'_{i}(u_i, f) +\lambda_2 \bar J'_{i}(u_i, f') - \bar J'_{i}(u_i, \lambda_1 f+\lambda_2 f') =\ \lambda_1\lambda_2 \mathbb{E}\int_0^T\left\{|x_i-x_i' -\Gamma(m_i-m_i')|_Q^2 +\frac1\gamma |f(t)-f'(t)|^2\right\}dt -\lambda_1\lambda_2\mathbb{E}|x_i(T)-x_i'(T)|^2_H.
I_{22}^{b}(y,a,b,d)=-\frac{1}{ab^{d-3}}\sum^{k< \frac{d-4}{2}}_{k=0}C^{(2)}(d,k)(\frac{b}{a})^{2k}\sum_{n=1}^{\infty}n^{2k}\int_{0}^{\frac{n\pi b}{a}}du\, u^{d-4-2k}(\coth u -1)\cosh(\frac{2uy}{b}).
B_\alpha (p, q) = \frac{\alpha}{1 - \alpha}\sum\limits_{x\in\mathbb{S}}p (x)q(x)^{\alpha -1}- \frac{1}{1 - \alpha}\sum\limits_{x\in\mathbb{S}}p(x)^{\alpha}+ \sum\limits_{x\in\mathbb{S}}q(x)^{\alpha}.
\lambda_n: y'(0)=y'(1)=0, \mu^{(1)}_n: y'(0)-\gamma^{(1)}y(0)=y'(1)+\gamma^{(1)}y(1)=0, \mu^{(2)}_n: y'(0)-\gamma^{(2)}y(0)=y'(1)+\gamma^{(2)}y(1)=0,
\int_{\lbrack0,2\pi)}{\textstyle\sum\limits_{k\in\mathbb{S}}}a_{k}(t)\Psi_{k,t}(x)dt=\lim_{\delta\rightarrow0}\sum_{k\in\mathbb{S}}\int\nolimits_{I(\delta)}a_{k}(t)\Psi_{k,t}dt,
\sum_{k=0}^{\infty}\sum_{r_1=1}^{n}\frac{S_{00r_1}}{(k+1)^{2}(r_1+k+1)}= \zeta(2)\sum_{r_1=1}^n\frac{S_{00r_1}}{r_1}-\sum_{r_1=1}^n\frac{S_{00r_1}H_{r_1}}{r_1^2};
a_{Y'_\alpha}(x,y) = \begin{cases}Y_\alpha(x)-y-1, x = l, \alpha = r Y_\alpha(x)-y, \end{cases}, l_{Y'_\alpha}(x,y) = \begin{cases}Y_\alpha^T(y)-x-1, y = w, \alpha = r Y_\alpha^T(y)-x, \end{cases}
\|\widetilde{E}\|_{L^{\frac{12d-8}{3d+4}, \infty}(S_t, d\sigma)}\lesssim \|E\|_{L^{\frac{4}{3}}(\mathbb F_q^d, d\textbf{m})}=|E|^{\frac{3}{4}}\quad\mbox{for all}~~ E \subset \mathbb F_q^d,~t\ne 0.
M''(t) M(t) - \frac{\omega+3}{4}(M'(t))^2 \geq M(t)\left( M''(t) - (\omega +3)\left( ||u_t||^2 + b\int_{0}^t ||u_s(s)||^2_{L^2(\mathbb{H}^n)}ds \right) \right) &=M(t) \left( -(\omega+1) ||u_t||^2_{L^2(\mathbb{H}^n)}- (\omega +3)b\int_{0}^t ||u_s(s)||^2_{L^2(\mathbb{H}^n)}ds -2I(u)\right),
\mu(W_i) = \sum_{|\nu|=0}^{\gamma-1}\frac{1}{\nu!}\frac{\partial^{|\nu|}\mu(w)}{\partial w^\nu}(W_i-w)^\nu + \sum_{|\nu|=\gamma}\frac{1}{\nu!}\frac{\partial^{\gamma}\mu(w')}{\partial w^\nu}(W_i-w)^\nu
\sinh^2\tfrac{1}{2}d_{\eta\xi\eta^{-1}}z = \sinh^2(\tfrac{1}{2}T_{\xi})\cosh^2d_z\mathcal{B}+\sin^2\theta\sinh^2d_z\mathcal{B}\leq \left( \sinh^2(\tfrac{1}{2}T_{\xi})+\sin^2\theta\right)\cosh^2d_z\mathcal{B}.
[X^{ij}(g),X^{ik}(gh) + X^{jk}(h)] =[ X^{jk}(h),X^{ij}(g)+X^{ik}(gh)]\endline =[X^{ik}(gh), X^{jk}(h)+ X^{ij}(g)]=0,\endline[X^{i}(p),X^{jk}(g)] =[X^{j}(p),X^{ik}(g)]=[X^{k}(p),X^{ij}(g)]=0,
D_1 L_d (q_{m},q_{m+1}) =\frac{h}{2}[ \nabla \alpha(q_{m})\cdot \frac{q_{m+1}-q_{m}}{h}-\frac{\alpha(q_{m})}{h}-\frac{\alpha(q_{m+1})}{h}-\nabla H(q_{m})], D_2 L_d (q_{m-1},q_{m}) =\frac{h}{2}[ \nabla \alpha(q_{m}).\frac{q_{m}-q_{m-1}}{h}+ \frac{\alpha(q_{m})}{h}+\frac{\alpha (q_{m-1})}{h}-\nabla H(q_{m})].
+ \sin\psi\cdot\left(\frac{F[-q,p](i\mu_{0})w_{2}[-q,p](i\mu_{0})w_{3}[-q,p](i\mu_{0})}{w_{1}[-q,p](i\mu_{0})}\right)^{-\frac{1}{2}}\gamma^{1}\partial_{\eta}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
\begin{aligned}A(W_n;x) = A(E_1;x) + A(C_{n-1};x) + \sum_{k=2}^{\lfloor n/2\rfloor}{n-1 \choose k-1}\x^{2k-1}+ \sum_{k=\lfloor n/2 \rfloor + 1}^{n-1}a_{n,k}\x^{n-1}+ \xi_n \, b_{n,(n+1)/2}\x^n + \sum_{k= \lceil n/2\rceil + 1}^{n-1}b_{n,k}\x^{n+1}+ x^{n+3}. \end{aligned}
\sum_{\alpha, \beta, \gamma}S^{\beta q}_{\gamma r}(v) S^{\alpha p}_{k \gamma}(u+v)S^{i\alpha}_{j\beta}(u) = \sum_{\alpha, \beta, \gamma}S^{\alpha p}_{\beta q}(u) S^{i \alpha}_{\gamma r}(u+v)S^{j\beta}_{k\gamma}(v)
Z_{{\it log}}[p/q] = \frac{1}{\eta\bar{\eta}}\left(|\chi_0|^2 + |\chi_{pq}|^2 + \sum_{1\leq s\leq pq-1}\left(\chi_s^0(\chi_{s'}^0)^* + \chi_s^+(\chi_{s'}^-)^* + \chi_s^-(\chi_{s'}^+)^*\right)\right)\,,
E_\alpha(\gamma)=E_\alpha(\Gamma) \le \frac 12 (\alpha+1){B}^\alpha[\Gamma']^2_{{(\alpha-1)/2,2}}& \overset{\eqref{seminormest-arclength}}{\le}\frac 12 (\alpha+1){B}^\alpha\Big( \frac 1c \Big)^{2+\alpha}\Big[\Big(\frac 1c \Big)^2+C^{6}\Big]\cdot [\gamma']^2_{{(\alpha-1)/2,2}},
\lim_i\lim_j m(S_i\tilde S_j,T_i\tilde T_j) =\lim_i\lim_j \left<T_i\tilde T_j,\varphi(S_i\tilde S_j)\right> =\lim_i\lim_j ~\left(\varphi(S_i\tilde S_j)^* T_i\tilde T_j\right) =\lim_i\lim_j \sum_{r=1}^n\left<T_i\tilde T_j e_r,\varphi(S_i\tilde S_j)e_r\right>.
\varphi_{\boldsymbol{1}}((ba)^qg(ab)^q ) =(ba)^{\frac{q-1}{2}+k_1}b\omega(x)b(ab)^{\frac{q-1}{2}+k_2}&=(ba)^{\frac{q-1}{2}+k_1}bab(ab)^{\frac{q-1}{2}+k_2}&=(ba)^{\frac{q+1}{2}+k_1}(ba)^{\frac{q-1}{2}+k_2}b = (ba)^{q+k_1+k_2}b.
\hspace{. 7 i n}+ \sum_{j = 0}^{k - 1}\gamma^{j - k}x_{i}^{-}( - j ) \otimes \gamma^{- ( k + 3 j ) / 2}\varphi_{i}( j - k ) \quad{\it m o d}U N_{-}^{2}\otimes U N_{+}
\phantom{=}\; \P\left(\frac{\xi_i}{\xi_{i+1}}<(1+\theta)^{-1},\forall i<k\mid \xi_k\right)=\prod_{i=1}^{k-1}\P(U<(1+\theta)^{-i\beta})=(1+\theta)^{-\frac{\beta}{2}k(k-1)},
R = ( d - 2 ) \left[ \frac{( d - 3 ) ( 1 - A )}{r^{2}}- \frac{2}{r}\left( \frac{d A}{d r}+ A \frac{d \Omega}{d r}\right) \right] - \left[ \frac{d^{2}A}{d r^{2}}+ 2 A \frac{d^{2}\Omega}{d r^{2}}+ 2 A \left( \frac{d \Omega}{d r}\right)^{2}+ 3 \frac{d A}{d r}\frac{d \Omega}{d r}\right] .
\beta \big( [ S_{2,5}] \big) = \varepsilon \theta \big( [ S_{2,5}] \big) = \varepsilon \theta \kappa \big( [ \overline{S}_{2,5}] \big) = \varepsilon Q \overline{\theta}\big( [ \overline{S}_{2,5}] \big) = ( \ast \ast ).
\sum_{i=1}^n a_{n,i}^{1+\eta/2}E_{Q}\left|u_i^2-\sigma_Q(x_i)^2\right|^{1+\eta/2}\le \left(\max_{1\le i\le n}a_{n,i}^{\eta/2}\right) \max_{1\le i\le n}E_Q\left|\varepsilon_i^2-\sigma_Q^2(x_i)\right|^{1+\eta/2}\sum_{i=1}^{n}a_{n,i},
\|\partial_t \nabla_x v_h\|_{L_2(Q)}^2 = \sum_{K \in \mathcal{K}_{h}}\|\partial_t \nabla_x v_h\|_{L_2(K)}^2 \leq \sum_{K \in \mathcal{K}_{h}}C_{inv,1}^2 h_K^{-2}\|\nabla_x v_h\|_{L_2(K)}^2 \leq C_{inv,1}^2 C_u^2 h^{-2}\sum_{K \in \mathcal{K}_{h}}\|\nabla_x v_h\|_{L_2(K)}^2 = \left[ C_{inv,1}C_u h^{-1}\| \nabla_x v_h\|_{L_2(Q)}\right]^2.
r=\sqrt{(x_{1}-x_{\circ})^{2}+(y_{1}-y_{\circ})^{2}}=\sqrt{(x_{2}-x_{\circ})^{2}+(y_{2}-y_{\circ})^{2}}=\sqrt{(x_{3}-x_{\circ})^{2}+(y_{3}-y_{\circ})^{2}}.
\mathbb{P}\bigg[ \sum_{m=1}^j G_m \le a + h(j) \, \forall 1 \le j \le n \bigg] = \mathbb{P}[ G_1 \le - C - 1 - 3 \log 2] \mathbb{P}\bigg[ \sum_{m=1}^j G_m \le a + h(j) \, \forall 1 \le j \le n \, \big \vert \, G_1 \le -C-1-3 \log 2 \bigg].