harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Psi_{\underline{{o}}}^{\prime}( \vec{\sigma}_{2 o}) \sum_{\underline{{n}}}c_{\underline{{n}}}\Psi_{\underline{{n}}}^{\prime}( \vec{\sigma}_{1 o}) = - \Psi_{\underline{{o}}}^{\prime}( \vec{\sigma}_{2 o}) \frac{R}{\cal N}= - \frac{n_{o}( \vec{\sigma}_{2 o})}{\cal N}.
\infty =\int_{B_a}\liminf g(y+k)dy &\le \liminf \int_{B_a}[g(y+k)-g(y)]dy+\int_{B_a}g(y)dy &\le w_1 \Big[\mathcal{E}(E)-\mathcal{E}(B_a)\Big]^{1/2}+||g||_{L^1(B_a)}&\le w_1e^{1/2}+||g||_{L^1(B_a)}
\sum_{k=0}^{\lfloor n/\log n\rfloor}\frac{\Delta W_{k-1}L_Bs_k}{2\sqrt{x_k}e^{I_k}}=0=\int_{0}^{1/\log n}\frac{L_B s(x)}{2\sqrt{x}e^{I(x)}}d\overleftarrow{W}(x) .
\sum_{k \le x}\bigg( \frac{1}{k^{r+1}}\sum_{j=1}^k j^r c_k(j) \bigg) = 1 + \frac{1}{2}\sum_{2 \le k \le x}\frac{\phi(k)}{k}+ \frac{1}{1+r}\sum_{m=1}^{[\tfrac{r}{2}]}\binom{r+1}{2m}B_{2m}\sum_{2 \le k \le x}\prod_{p|k}\bigg(1 -\frac{1}{p^{2m}}\bigg),
\psi_{2m}^{(n)}(x) = e^{\frac{x^{2n}}{2n}}\left(\frac{d}{dx}\frac{1}{x^{2n-2}}\frac{d}{dx}\right)^m e^{-\frac{x^{2n}}{n}}\psi_{2m+1}^{(n)}(x) = e^{\frac{x^{2n}}{2n}}\left(\frac{d}{dx}\frac{1}{x^{2n-2}}\frac{d}{dx}\right)^m\left(x^{2n-1}e^{-\frac{x^{2n}}{n}}\right).
A\geq\sum_{I\in S_\phi}\frac{\alpha_I}{\rho_I}\frac{y_I^q}{[(\beta+1)-\beta\rho_I]^{q-1}}-\sum_{I\in S_\phi}\sum_{\substack{J\in S_\phi J^\star=I}}\mu(J)\frac{y_J^q}{(\beta+1)^{q-1}},
\tilde{\lambda}_0 = \lim_{n \to +\infty}\min_{B^1(\mathbf{x})\subset B^{A_n}(\mathbf{0})}\lambda^1(B^1(\mathbf{x}),B^{A_n}(\mathbf{0})) = \lim_{n \to +\infty}\lambda^1(B^1(\mathbf{0}),B^{A_n}(\mathbf{0})),
C_{j}=\frac{n!}{(n-j)!}\sum_{i=0}^{j}\frac{(-1)^{i+j}P_{i}}{i!(j-i)!}=\prod_{m=0}^{j-1}(n-m)\sum_{i=0}^{j}\frac{(-1)^{i+j}P_{i}}{i!(j-i)!}.
S_2^{*}=\frac{1}{(1-c)(cq)_N}\sum_{n=1}^N \frac{(c q)_{N-n}(cq)^n}{(q)_n (q)_{N-n}}\sum_{k=1}^n \left[\begin{matrix}n k\end{matrix}\right] \frac{(qc)_k (1/c)_{n-k}c^{-k}}{1- q^k}.
A(1,x) = A_0(x), A(0,x) = I, A(n,x) = A_0(T^{n-1}(x)) \cdots A_0(T(x)) A_0(x), A(-n,x) = A_0(T^{-n}(x))^{-1}\cdots A_0(T^{-1}(x))^{-1},
F_{d, \ell}(z;\tau)=F_{d', \ell}(z;\tau)+\sum_{a\geq 0}\mathcal{D}_z^{2a}\left(\frac{\zeta^{d}\left(1-\zeta^{\ell r}\right)}{1-\zeta^{\ell}}\right)\frac{(2\pi i\tau)^a}{a!}.
{\cal F}^{\partial\Gamma,\mu}:=\{u\in L^2(\partial\Gamma,\nu):\Hu\in \mathcal{F}^{\mu}\},{\cal E}^{\partial\Gamma,\mu}(u,v) :={\cal E}^{\mu}(Hu,Hv)\ \ \u,v\in{\cal F}^{\partial\Gamma,\mu}.
D^\flat\mathcal{R}'_+ (\psi_{+,T}+\phi_{+,T}) (\zeta_{+,T}+{\mathfrak a}_{+,T}) ={\mathfrak l}(\psi_{+,T}+\phi_{+,T}) (\zeta_{+,T}+{\mathfrak a}_{+,T}) +{\mathfrak l}(\psi_{+,T}+\phi_{+,T})(\nabla \zeta_{+,T}) +{\mathfrak l}(\nabla (\psi_{+,T}+\phi_{+,T}))(\zeta_{+,T}+{\mathfrak a}_{+,T}).
m_N = \frac{1}{N}\sum_{i=1}^N g_i + \frac{1}{N}\sum_{i=1}^N g_i^2 ( T \psi + TY_i) + \O \left( \frac{1}{N}\sum_{i=1}^N |g_i|^3 \varphi^{\xi}\frac{1}{N}\right) &= \frac{1}{N}\sum_{i=1}^N g_i + \frac{1}{N}\sum_{i=1}^N g_i^2 ( T \psi + Q_i [ G_{ii}^{-1}]) + \O \left( \frac{1}{N}\sum_{i=1}^N |g_i|^3 \varphi^{\xi}\frac{1}{N}\right).
\nabla_{\vec{x}}V(\vec{x}) =\left[\dfrac{\partial f}{\partial \vec{x}}(\vec{z}^{*}(\vec{x}),\vec{x})\right]^{\top}+\sum\limits_{i\in\mathcal{E}}\left[ \dfrac{\partial g_{i}}{\partial \vec{x}}(\vec{z}^{*}(\vec{x}), \vec{x})\right]^{\top}\lambda_{i}^{*}(\vec{x}).
\hat{V}_A(\tau) = \left[-A e^{\left[\frac{\alpha}{2}-l +\frac{2\eta}{\alpha}\left(\frac{\alpha}{2}-l \right) \right] \tau}, A e^{\left[\frac{\alpha}{2}-l +\frac{2\eta}{\alpha}\left(\frac{\alpha}{2}-l \right) \right] \tau}\right]^{l-1}.
E_{0}(R)= Rf_0(mR)={m^2 R\,\log(m R_0)\over8\pi}-\frac{m}{2 \pi}\int_{- \infty}^{\infty}\cosh\theta\,\log\left(\rule{0mm}{4mm}1+ e^{-\varepsilon_0(\theta)}\,\right)\,\, d\theta
\cdot_0\eta=K^t(\mu,\eta)d\mu, \ ({\cdot_\beta\eta},{\cdot_\alpha\zeta}) =({\cdot_\beta\eta})_\alpha(\zeta)=\overline{({\cdot_\alpha\zeta})_\beta(\eta)}=K^t_{\alpha\bar\beta}(\zeta,\eta),
\frac{d^2s}{dr^2}+ \frac{2}{r}\frac{ds}{dr}-\frac{6}{r^2}s = \frac{1}{L}\big(2s\partial_1\varphi(2s^2/3,2s^3/9) +s^2\partial_2\varphi(2s^2/3,2s^3/9) \big),
\begin{aligned}jO_{n}^{(3)}=j_{n}^{(3)}+\sum_{s=1}^{7}j_{n+s}^{(3)}e_{s},\n\geq 0 &=j_{n}^{(3)}+j_{n+1}^{(3)}e_{1}+j_{n+2}^{(3)}e_{2}+j_{n+3}^{(3)}e_{3}&\ \ +j_{n+4}^{(3)}e_{4}+j_{n+5}^{(3)}e_{5}+j_{n+6}^{(3)}e_{6}+j_{n+7}^{(3)}e_{7},\end{aligned}
\frac{1}{3}\int p_{1}= - \frac{1}{4 8 \pi^{2}}\int d \tau \int d \vec{\sigma}\cdot \vec{\nabla}( \frac{2 \vec{\nabla}V \cdot \vec{\nabla}V}{V^{3}}) = - \frac{1}{6 \pi^{2}}\int d \vec{\sigma}\cdot \vec{\nabla}( \frac{\vec{\nabla}V \cdot \vec{\nabla}V}{V^{3}}) ~ ~ ,
H_n^{(gen)}=\frac{1}{\sqrt{n}}\left(\sum_{(j, k)}\sum_{a,b=1}^3\alpha_{a,b,j,k}\sigma_j^{(a)}\sigma_k^{(b)}+\sum_{j=1}^n\sum_{a=1}^3\alpha_{a,0,j,0}\sigma_j^{(a)}\right)
\sigma(a,g(\upsilon)) = \upsilon \text{whenever$\upsilon \in \Upsilon$ and $0 \leq \upsilon \leq b$}, 1 \leq \big ( \limsup_{\chi \to x}| \sigma(a,\chi)-\sigma(a,x)|/|\chi-x| \big ) \big ( \limsup_{\upsilon\to b-}|g(\upsilon)-g(b)|/|\upsilon-b| \big ).
\omega\bigl((a_1x_1+\cdots+a_nx_n)^m/m!\bigr) =\frac{(a_1x_1+\cdots+a_nx_n)^m}{m!}\circ{\mathfrak D}_{\omega}&=\frac{(a_1\partial /\partial z_1+\cdots+a_n\partial /\partial z_n)^m}{m!}{\mathfrak D}_{\omega}={\mathfrak D}_{\omega}(a_1,\dots,a_n),
Z := \int{\cal D}A_{\mu\nu}{\cal D}\Lambda_\mu{\cal D}B_\mu{\cal D}C_\mu{\cal D}\bar C_\mu{\cal D}d{\cal D}\bar d{\cal D}N{\cal D}P{\cal D}B^{(1)}{\cal D}C'{\cal D}\bar C'{\cal D}B'\exp \left[ i \int d^4x{\cal L}_{tot}' \right] .
X_{[bra]^{\ast}}(\rho(P)) =\displaystyle \sum_{u \in N_{b}}\displaystyle \sum_{c \in \mathcal{C}(u),r_{c}=r,a_{c}=a}f_{u}\rho(z_{c})s_{c}&=\rho \left(\displaystyle \sum_{u \in N(b)}\displaystyle \sum_{c \in \mathcal{C}(u),r_{c}=r,a_{c}=a}f_{u}z_{c}s_{c}\right)
(\varepsilon f)(x) = \varepsilon (x_1) f(x_2) = f(\varepsilon (x_1) x_2) =f(\psi(x)) = \psi^*(f)(x), (f \varepsilon )(x) = f(x_1) \varepsilon (x_2) = f(x_1 \varepsilon (x_2))= f(\omega(x)) = \omega^*(f)(x),
\hat{G}\bigl(s, X_s^{t,x;u}, Y_s^{t,x;u}, Z_s^{t,x;u}\bigr) = c\bigl(s, X_s^{t,x;u}, u_s\bigr) + G\bigl(s, Y_s^{t,x;u}, Z_s^{t,x;u}\bigr).
Z_k^{\phi}\equiv S_{k}^{\phi}\equiv \sum\limits_{a\in A_{+,k}}\frac{\mu(a)C_a(X_1)\dots C_a(X_n)}{a}\pmod{X_n^q}&\equiv \sum\limits_{a\in A_{+,k}}\frac{\mu(a)C_a(X_1)\dots C_a(X_{n-1})aX_n}{a}\pmod{X_n^q}&\equiv X_n \sum_{a\in A_{+,k}}\mu(a)C_a(X_1)\dots C_a(X_{n-1}) \pmod{X^q_n}.
\sum_{k = 1}^{\infty}T_{2 n , 2 k - 1}\frac{v_{2 k - 1}^{( \kappa )}}{\sqrt{2 k - 1}}= - \frac{v_{2 n}^{( \kappa )}}{\sqrt{2 n}}\operatorname{t a n h}\left( \frac{\kappa \pi}{4}\right) \, ,
g_{x^{1}}=g_{x^{2}}=0, g_{\dot{x}^{1}}=\frac{\dot{x}^{1}}{\sqrt{(\dot{x}^{1})^2+(\dot{x}^{2})^2}}+b\cos (t+c), g_{\dot{x}^{2}}=\frac{\dot{x}^{2}}{\sqrt{(\dot{x}^{1})^2+(\dot{x}^{2})^2}}+b\sin (t+c).
\begin{array}{l l l}{{E_{\pm}^{\dagger}= E_{\mp},}}&{{\quad \quad H^{\dagger}= H ,}}&{{\quad \quad \quad \mathrm{f o r}\quad s u ( 2 ) ,}}\\{{E_{\pm}^{\dagger}= - E_{\mp},}}&{{\quad \quad H^{\dagger}= H ,}}&{{\quad \quad \quad \mathrm{f o r}\quad s u ( 1 , 1 ) .}}\\ \end{array}
a(z^n)Y(s^{L_0}\xi,w)\eta = (-1)^{p(\xi)}Y(s^{L_0}\xi,w)a(z^n)\eta + \sum_{k \ge 0}\binom{n}{k}Y(a(z^k)s^{L_0}\xi,w)w^{n-k}\eta &= (-1)^{p(\xi)}Y(s^{L_0}\xi,w)a(z^n)\eta + \sum_{k \ge 0}w^{n}s^{1/2}\binom{n}{k}Y(s^{L_0}a((sz/w)^k)\xi,w)\eta &= (-1)^{p(\xi)}Y(s^{L_0}\xi,w)a(z^n)\eta + Y(s^{L_0}a(s^{1/2}(sz+w)^n)\xi, w)\eta,
\lim_{\delta \to 0}\sum_{j=0}^{1}\int_{\xi_{0}-2\delta}^{\xi_{j}}\|F(\xi_{j}-s)G(s)\|_{p}\, \textnormal{d}s \leq 2 \lim_{\delta \to 0}\left(g(\xi_{0}-2\delta, \xi_{0}+\delta) \int_{0}^{3\delta}\|F(s)\|_{q}\, \textnormal{d}s\right)=0.
\frac{1}{A^{\alpha}B^{\beta}}= \frac{\Gamma ( \alpha + \beta )}{\Gamma ( \alpha ) \Gamma ( \beta )}\int_{0}^{1}d x \frac{x^{\alpha - 1}( 1 - x )^{\beta - 1}}{[ A x + B ( 1 - x ) ]^{\alpha + \beta}}\; ,
\vert \Gamma_{F_n\Delta^2}\vert \geq \vert \tau(Q_n) \vert = |Q_n| \geq |(Q_n)_{F_n \setminus S\Delta}| = |X_{{F_n \setminus S\Delta}}| \geq \frac{|X_{F_n}|}{|A|^{|S\Delta|}}.
g ={\frac{1}{4}}\left\{p_{\beta \lambda}{\frac{\partial \psi_{\alpha}}{\partial \phi_{\beta}}}{\frac{\partial \psi_{\alpha}}{\partial \phi_{\lambda}}}+ \sum_{A \not = B}{'}{\frac{2}{( \phi_{A}- \phi_{B})^{2}}}( P_{A B}+ P_{B A}) \sin^{2}\left[{\frac{\psi_{A}- \psi_{B}}{\sqrt 2}}\right] \right\}
\mathbb{E}(T_{i-1}) = \sum_{j,l=0}^k \mathbb{E}\left(T_{i-1}|\Psi_j^{i-1}\Psi_l^{i-2}\right)\mathbb{P}(\Psi_j^{i-1})\mathbb{P}(\Psi_l^{i-2}) = \frac{1}{\lambda}+k\theta-q^{k+1}\left(\frac{1+\lambda\theta+k\lambda\theta}{\lambda}\right),
H_1 \leq C\Big( \Big||\phi|^p\Big|_{\frac{2n}{n+2}}+|\phi^2|_{\frac{2n}{n+2}}\Big) = C \bigg( |\phi|^p_{\frac{2n}{n-2}}+\Big(\int\limits_\Omega|\phi|^{\frac{2n}{n-2}}dx\Big)^{\frac{n+2}{2n}}\bigg) = 2C |\phi|_{p+1}^p \leq 2C \|\phi\|^2.
q_{3,2}'(0)=\sum\limits_{r=0}^{\min(n_1,\Delta n)}\sum\limits_{s=1}^{n_1}\sum\limits_{t=1}^{\Delta n}\sum\limits_{\substack{\mathbf{c}\in [1,n_1]_<^r \mathbf{c'}\in [1,n_1]_<^s c'_s=n_1}}\sum\limits_{\substack{\mathbf{d}\in [n_1+2,n_2]_<^r \mathbf{d'}\in [n_1+1,n_2]_<^t d'_1=n_1+1}}\sum\limits_{j=1}^{L}(-1)^{r+s+j}a_3^\ast(f_j)\det M_0(\{r+s\}',\{j\}'),
i_{U}^{*}(j_{U}\circ \pi_{U})_{*}\tilde{\Phi}_{!}(j_{V}\circ \pi_{V})^{*}\mathcal{F}= i_{U}^{*}(j_{U}\circ \pi_{U})_{*}(j_{U}\circ \pi_{U})^{*}\Phi_{!}\mathcal{F}= \psi_{f}\Phi_{!}\mathcal{F}.
g_1 =\frac{d}{2}-l_1\sin\theta_1-l_2\sin\theta_2+a\sin\theta_3-b\cos\theta_3\;, g_2 =\frac{d}{2}-l_1\sin\theta_1-l_2\sin\theta_2-a\sin\theta_3-b\cos\theta_3\;, g_3 =\frac{d}{2}+l_1\sin\theta_1+l_2\sin\theta_2-a\sin\theta_3-b\cos\theta_3\;, g_4 =\frac{d}{2}+l_1\sin\theta_1+l_2\sin\theta_2+a\sin\theta_3-b\cos\theta_3\;.
g(x) = (x-a_1)\ldots(x-a_k)h'(x) + e^1(\widehat{x-a_1})(x-a_2)\ldots(x-a_k)h(x) + e^2(x-a_1) (\widehat{x-a_2})\ldots(x-a_k)h(x) + e^k(x-a_1)\ldots (x-a_{k-1}) (\widehat{x-a_k})
d_X(c_{x_1,x}(T+\tau+t),c_{x_2,x}(T+t)) \leq d_X(c_{x_1,x}(T+\tau+t),c_{x_1,x}(T+\tau-r')) + d_X(c_{x_1,x}(T+\tau-r'),c_{x_2,x}(T-r')) & + d_X(c_{x_2,x}(T-r'),c_{x_2,x}(T+t)) < |t + r'| + \delta' + |t + r'| = 2|t+r'| + \delta' \leq 2|t| + \delta' < 2|t| + 1.
\omega_{a^{\prime}b^{\prime}}= \left( \begin{array}{c c}{{0}}&{{- 1}}\\{{1}}&{{0}}\\ \end{array}\right) \, , \hspace{1 c m}X^{a^{\prime}b^{\prime}}= \left( \begin{array}{c c}{{\frac{1}{2}}}&{{0}}\\{{0}}&{{- \frac{1}{q^{2}}}}\\ \end{array}\right) \, .
|q^*(t)| \leq r_2 \Bigl(\frac{t}{T}\Bigr)^a + \int_t^T \Bigl(\frac{t}{\tau}\Bigr)^a \frac{Lr_1(\tau/T)^a}{\tau}d\tau = r_2 \Bigl(\frac{t}{T}\Bigr)^a + Lr_1 \Bigl(\frac{t}{T}\Bigr)^a \log\Bigl(\frac{T}{t}\Bigr) \mbox{on $(t_{\xi},T]$}.
\dot z^1= \dot y^1 \dot y^{2}\dot z^{2}=x\dot y^{2}\dot z^3=(y^1+\dot y^1 x)\dot y^{2}\dot z^4= y^1 \dot y^1 \dot y^2, \dot z = \ddot y^1 \dot y^2, \dot z^1= \dot y^1 \dot y^2 \dot z^2= \frac{1}{2}(\dot y^2)^2 \dot z^3= \frac{1}{2}\dot y^1 (\dot y^2)^2 \dot z^4= \frac{1}{2}\dot y^2 (x\dot y^1 \dot y^2-y^1\dot y^2-2\dot y^1 y^2).
(\eth_k + \partial_k^*)(a + a^*) \star (b + b^*) = (a + a^*) \star (\partial_k + \eth_k^*)(b + b^*) + (\eth_k + \partial_k^*)((a + a^*)\star (b + b^*)), (a + a^*) \star (\eth_k + \partial_k^*)(b + b^*) =(\partial_k + \eth_k^*)(b + b^*) \star (b + b^*) + (\eth_k + \partial_k^*)((a + a^*) \star (b + b^*)).
p(k)=\frac{1}{{n \choose 2}}\frac{(3 \cdot 5 \cdots (2k-1))}{(2 \cdot 4 \cdots (2k-2))}\frac{(3 \cdot 5 \cdots (2(n-k)-1))}{(2 \cdot 4 \cdots (2(n-k)-2))}.
\frac{\partial^{2}\tilde{h}}{\partial\varphi^{2}}=\frac{\partial}{\partial\varphi}\left(\frac{\partial h}{\partial z}\frac{\partial z}{\partial\varphi}\right)=\frac{\partial^{2}h}{\partial z^{2}}\left(\frac{\partial z}{\partial\varphi}\right)^{2}+\frac{\partial h}{\partial z}\frac{\partial^{2}z}{(\partial\varphi)^{2}}
\sum_{k = N + 1}^{\infty}e^{-2\varepsilon (L_k - L_N)}\leq\sum_{k = N + 1}^{\infty}e^{- L_N^{-\frac{1}{\eta}}L_k}\leq\sum_{m = L_{N + 1}}^{\infty}e^{- L_N^{-\frac{1}{\eta}}m}\leq Ce^{-L_N}.
\Omega_1 = \left(\begin{array}{cc}I_d \gamma 0 I_d \end{array}\right) \quad\Omega_2 = \left(\begin{array}{cc}I_d 0 \alpha I_d \end{array}\right) \quad\Omega_3 = \left(\begin{array}{cc}{A^{-1}}^T 0 0 A \end{array}\right),
D(x) =D(\sum_i a_i[x_{i_1},x_{i_2}]) &=\sum_i \phi(a_i).D([x_{i_1},x_{i_2}])+\sigma_D(a).\alpha_K([x_{i_1},x_{i_2}]) &=\sum_i \phi(a_i).\big([D(x_{i_1}),\alpha_K(x_{i_2})])+[\alpha_K(x_{i_1}),D(x_{i_2})]\big)+\sigma_D(a).\alpha_K([x_{i_1},x_{i_2}]) &=D^{\prime}(\sum_i a_i[x_{i_1},x_{i_2}])=D^{\prime}(x).
V=-\frac{\Gamma(-1/2)}{\sqrt{8\pi^2 \alpha '}}[(\frac{r^2}{2\pi \alpha '}-\pi \epsilon)^{1/2}+ (\frac{r^2}{2\pi \alpha '}+\pi \epsilon)^{1/2}- 2(\frac{r^2}{2\pi \alpha '})^{1/2}]
S [ \Psi^{*}, \Psi ] \ = \{\frac{1}{2 \pi \rho}}\int_{{\bf R}\times S^{1}}d t d \varphi \, [{\frac{i}{2}}\rho ( \Psi^{*}{\dot{\Psi}}-{\dot{\Psi}}^{*}\Psi ) + i \Psi^{*}\partial_{\varphi}\Phi ]
\gamma_{\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3}}=\left( 1,-e^{\int \tau (s)\,ds}-a\int \sinh \left( \int \tau (s)\,ds\right)\,ds,e^{\int \tau (s)\,\,ds}+a\int \cosh \left( \int \tau (s)\,ds\right)\,ds\right).
\mathcal{I}_{y_1}= \int_0^\infty\, 4 y^2_1\,U(y^2_1)\,U(y^2_1 + y_3)\,e\left(t\left(\frac{y_3}{y_1}\right) + \frac{2\sqrt{N}}{p_1q\sqrt{p_2}}\left({\sqrt{m}}-{\sqrt{m_1}}\right) y_1 \right) dy_1
\bigl \lbrack A , \lbrack B , C \bigr \rbrack \rbrack_{\mu_{1}. . . \mu_{d - 2}}+ ( c y c l . p e r m . ) = \partial_{\lbrack \mu_{1}}\epsilon_{\mu_{2}. . . \mu_{d - 2}\rbrack \nu \lambda \sigma}\xi^{\nu}( A ) \xi^{\lambda}( B ) \xi^{\sigma}( C )
\int_{F_n^{(i)}}LT_{g_i}dg_i=L\int_{F_n^{(i)}}T_{g_i}dg_i\intertext{and}\int_{F_n^{(i)}}T_{g_i}Ldg_i=\left(\int_{F_n^{(i)}}T_{g_i}dg_i\right)L
\frac{1}{2 \kappa^{2}}\left( n_{\mu}\Gamma_{\rho \sigma}^{\mu}- \frac{1}{2}g_{\rho \sigma}n_{\mu}\Gamma_{\alpha \beta}^{\mu}g^{\alpha \beta}- \Gamma_{\alpha ( \rho}^{\alpha}n_{\sigma )}+ \frac{1}{2}\Gamma_{\beta \alpha}^{\alpha}n^{\beta}g_{\rho \sigma}\right) = T_{\rho \sigma}^{b r a}.
[F_{0}]_{I}= \begin{cases}st, I = \{0, 1\}\lambda b_{i}s^{2}, I = \{0, i\}, i \ge 2 -\lambda a_{i}t^{2}, I = \{1, i\}, i \ge 2 \lambda^{2}(a_{i}b_{j}- a_{j}b_{i}), I = \{i, j\}, i, j \ge 2.\end{cases}
Y_{i_1 j_1, \ldots , i_p j_p}^{k_1 l_1, \ldots ,k_q l_q}=\bar{a}_{i_1 \ldots i_p}\bar{a}_{j_1 \ldots j_p}b^{k_1 \ldots k_q}b^{l_1 \ldots l_q}\mbox{- traces}
\mathcal{H}_{s}=\sum_{n=1}^{\mathsf{N}-1}H_{n,n+1}^{\left( s\right)}+\frac{d}{d\lambda}K_{1,-}^{(p)}(q^{s})+\frac{_{\mathsf{0}}\{K_{\mathsf{0},+}^{(p)}(q^{s})H_{\mathsf{0},\mathsf{N}}^{\left( s\right)}\}}{_{\mathsf{0}}\{K_{\mathsf{0},+}^{(p)}(q^{s})\}},
\Delta S^{I}_l(N,z^{\beta}) \equiv p^{-1}(z^{-\beta}\, e^{(2l-1)i \pi})^{q/p}\int_{c-i \infty \atop{c=\Re\,s>-q/p}}^{c+i \infty}\!\!\!\! \left( z^{-\beta}e^{(2l-1)i \pi}\right)^{\! -y/p}\Gamma(y)\; dy \;\;.
\tilde{u}^{n}_{i}(t,\omega)\doteq \begin{cases}\psi^{\mathfrak{m}}_{\epsilon}\left(t,\bar{\xi}^{n}(\omega),W^{n}_{1}(\cdot,\omega)\right) i=1, u^{n}_{i}(t,\omega) i\in \{2,\ldots,n\}.\end{cases}
\left\|\left(\frac{d}{dx}\right)^{n}A(x)^{-1}\Big|_{x=x_0}\right\|_{b\times b}&\le s r^n \big((sq)^{\frac{1}{t}}(1+(sq)^{\frac{1}{t}})^{n-1}n!\big)^t=s^2q(1+(sq)^{\frac{1}{t}})^{-t}\big(r(1+(sq)^{\frac{1}{t}})^{t}\big)^n(n!)^t.
\| B-B'\|^{2}_{2,p}\geq \gamma^{2}\,\sigma^{2}\,\log\left (\dfrac{e\,n_2}{s}\right )\left (\left (\dfrac{s}{32}\right )^{p/2}\dfrac{n_1}{64}\right )^{2/p}&\geq \dfrac{\gamma^{2}\,\sigma^{2}}{64^{1+2/p}}n_1^{2/p}s\log\left (\dfrac{e\,n_2}{s}\right ),
e^{i \alpha \frac{e}{2 \pi}\phi_{\hat{n}}( x )}V_{\eta}^{1 / 2}( y , \hat{n}^{\prime}) V_{\eta}^{* 1 / 2}( y , - \hat{n}^{\prime}) = V_{\eta}^{1 / 2}( y , \hat{n}^{\prime}) V_{\eta}^{* 1 / 2}( y , - \hat{n}^{\prime}) e^{i \alpha \frac{e}{2 \pi}\phi_{\hat{n}}( x )}e^{i \frac{\alpha}{2}l ( C_{1}, C_{2})}
\lambda(\mathcal{B}_n)=\prod_{\mathbf{s}_0\in\mathcal{S}_0^n}\prod_{\substack{(w,i) \in\mathcal{W}_n\times\mathcal{I}_n}}Q^n_{U|S_0}\big(\mathbf{u}(\mathbf{s}_0,w,i)\big|\mathbf{s}_0\big).
S_{k, r}(k+1)= \dbinom{k+r}{k+1}\Big(\sum\limits_{i=0}^k(-1)^i\dbinom{k+1}{i}\Big) = \dbinom{k+r}{k+1}\Big(\sum\limits_{i=0}^{k+1}(-1)^i\dbinom{k+1}{i}-(-1)^{k+1}\Big) = (-1)^k\dbinom{k+r}{k+1}.
\log\mathfrak{M}(q\,|\,\mu,\lambda_1,\lambda_2) =\int\limits_0^\infty \frac{dt}{t}\Bigl[\frac{1}{(e^t-1)(e^{t\tau}-1)}\Bigl(e^{t(q-\lambda_1\tau)}+e^{t(q-\lambda_2\tau)}+e^{t(q+1)}+ +e^{t(q-1-\tau(1+\lambda_1+\lambda_2))}-e^{t(2q-1-\tau(1+\lambda_1+\lambda_2))}\Bigr) + A(t)+B(t)\,q+C(t)\,q^2\Bigr]
K_Y+\pi_*^{-1}\Delta = \pi^*(K_X+\Delta)+aE K_{\cal F_Y}+\pi_*^{-1}\Delta = \pi^*(K_{\cal F}+\Delta)+bE K_{Y'}+\pi'^{-1}_*\Delta' = \pi'^*(K_{X'}+\Delta') +a'E' K_{\cal F_{Y'}}+\pi'^{-1}_*\Delta' = \pi'^*(K_{\cal F'}+\Delta')+b'E'.
\mathcal{E}_{\Lambda_{k+1,D}}(\tau) = \sum_{n\geq 1}\frac{c(n)}{n^{2k+1}}e^{2\pi i n \tau},\Lambda_{k+1,D}^*(\tau) = \sum_{n \geq 1}\frac{c(n)}{(4\pi n)^{2k+1}}\Gamma(2k+1,4\pi n v)e^{-2\pi i n \tau}.
E_{k}= \frac{1}{4 | k |}\int_{0}^{\pi}\biggl [ ( f^{\prime})^{2}+ k^{2}\biggl ( \frac{\sin f}{\sin \theta}\biggr )^{2}+ \biggl ( \frac{k f^{\prime}}{L}\frac{\sin f}{\sin \theta}\biggr )^{2}+ 2 L^{2}{\cal E}^{( 0 )}( \cos f ) \biggr ] \sin \theta \, d \theta ,
u(x,0) = \exp^{-(x_1^2 + x_2^2 + x_3^3)}\cdot \cos t|_{t=0}= \exp^{-(x_1^2 + x_2^2 + x_3^3)}, \frac{\partial u}{\partial t}(x,0) = -\exp^{-(x_1^2 + x_2^2 + x_3^3)}\cdot \sin t|_{t=0}\equiv 0.
\sum_{3r_0+2r_1+r_2=3d_1-d_2}{d_1+1 \choose r_0}{d_1+1-r_0 \choose r_1}{d_1+1-r_0-r_1 \choose r_2}\frac{(d_2+3)!}{2^{r_2}6^{r_3}}{n_1 \choose d_1+1}{n_2 \choose d_2+3},
\frac{\partial^n}{\partial x^n}V(y,x) = \delta_n^0 \Delta_{\sigma}+ O_n \left( \frac{y^{-\sigma}}{2 \pi \sigma}\left( x^{\sigma - n + 1/2}\sigma^{n}\right) \right) & = \delta_n^0 \Delta_{\sigma}+ O_n \left( x^{-n/2}(x/y)^{\sigma}\right)
dX^{i}_t = b(t,X^{i}_t,\widehat{\mu}^n_t,\alpha^i_t)dt + \sigma(t,X^{i}_t,\widehat{\mu}^n_t)dW^i_t + \sigma_0(t,X^{i}_t,\widehat{\mu}^n_t)dB_t, \widehat{\mu}^n_t = \frac{1}{n}\sum_{k=1}^n\delta_{X^{k}_t}.
\frac{4}{\sqrt{-\triangle}}\frac{1}{\sqrt{4 m^2 - \triangle}}{\mathrm arctanh}\left( \frac{\sqrt{-\triangle}}{\sqrt{4 m^2 - \triangle}}\right)= \int_m^{\infty}\!{\mathrm d}\tilde{m}\,\frac{1}{\sqrt{{\tilde{m}}^2 - m^2}}\,\frac{1}{4{\tilde{m}}^2 - \triangle}.
\begin{array}{cl}t^w(z,\bar z)\Phi_{jm\bar{m}}(z',\bar z') \sim(z-z')^{-wm}(\bar z-\bar z')^{-w\bar m}\Phi_{jm\bar{m}}^w(z',\bar z') &=(z-z')^{-(m-\bar m)w}|z-z'|^{-2\bar m w}\Phi_{jm\bar{m}}^w~,\end{array}
\sum_{\substack{\alpha^1+\cdots+\alpha^r=\gamma, \alpha^1+\alpha^2 \ge \theta^2,\dots,\alpha^1+\alpha^r \ge \theta^r}}c_{i_1,\dots,i_r,\alpha^1,\dots,\alpha^r}\prod_{j=2}^r{\alpha^1 \choose \theta^j - \alpha^j}= 0.
J_{\Gamma}( x ) = \bar{\psi}( x ) \Gamma \psi ( x ) \equiv \frac{1}{8 \pi}\int d \hat{\epsilon}e^{i \chi_{\Gamma}( \hat{\epsilon})}\left\{\left[ \psi^{\dagger}( x + \epsilon ) , \gamma^{0}\Gamma \psi ( x - \epsilon ) \right] , e^{i e \int_{x - \epsilon}^{x + \epsilon}d x_{i}A_{i}^{t r}}\right\}_{| \epsilon | \propto | \eta |}
\sigma_{ij}(\mathbf{u}) = \lambda (\nabla\cdot \mathbf{u}) \delta_{i,j}+ 2\mu \epsilon_{ij}(\mathbf{u}), ~~~ \textrm{with}~~\epsilon_{ij}(\mathbf{u})=\frac{1}{2}\left( \frac{\partial u_i}{\partial x_j}+ \frac{\partial u_j}{\partial x_i}\right)
\begin{array}{l}\displaystyle{\phantom{\Big(}\{{w(f)}_\lambda{w(f)}\}_{z,\rho}=\!\!\sum_{j\in J^f_{\leq-\frac12}}\!\!w(q_{j})w([q^{j},f])+(\partial+2\lambda)w(f)-(x|x)\lambda^3+2z(s|f)\lambda\,.}\end{array}
\overline{{{U}}}_{W}( T ) = \frac{\Delta n^{2}}{\pi a}\left( - \frac{1}{9 6}+ \frac{\zeta ( 3 )}{4 \pi^{2}}t^{3}- \frac{1}{3 0}t^{4}+ \frac{8}{5 6 7}t^{6}- \frac{8}{1 1 2 5}t^{8}+{\cal O}( t^{1 0}) \right){.}
\begin{aligned}\psi^{n+1/2}_{-}=e^{-i\frac{\tau}{2}|\psi^{n+1}|^2}\psi^{n+1}\psi^{n+1/2}_{+}=e^{i\left(g\left(t_{n+1}\right)-g\left(t_n\right)\right)\partial_x^2}\psi^{n+1/2}_{-}\psi^{n+1}=e^{-i\frac{\tau}{2}|\psi^{n+1/2}_{+}|^2}\psi^{n+1/2}_{+}\end{aligned}
\|\mathcal{K}'_m (x) - \mathcal{K}'_m (y) \| \leq \gamma\| x - y \|_\infty , \; \; \; x, y \in (\varphi, \delta),\;\;\; \gamma = C_{6}(b - a) \left ( \sum_{i=1}^\rho |w_i | \right ).
\sum_{\substack{d> D (d,r)=1}}\int_{0}^{Y/d^2}\psi(v)dv =\int_0^{Y/D^2}\psi(v)\sum_{\substack{D<d\leq \sqrt{\frac{Y}{v}}(d,r)=1}}1dv &=\int_0^{Y/D^2}\psi(v)\left(\dfrac{\varphi(r)}{r}\left((Y/v\right)^{1/2}-D)+O(d(r))\right)dv &=\dfrac{\varphi(r)}{r}Y^{1/2}\int_0^{Y/D^2}\psi(v)v^{-1/2}dv+O(D)+O\left(d(r)\dfrac{Y}{D^2}\right).
u_i(t) = v_i + \sum_{(j,l)\in S}a_{jl}\left\{\delta_{ij}\cos\left(\frac{2\pi k_{jl}}{\varepsilon}t\right)+\delta_{il}\sin\left(\frac{2\pi k_{jl}}{\varepsilon}t\right)\right\},\;i=1,2,...,m,
E_{c i r}^{D}= \frac{1}{2}\, \zeta_{c i r}^{D}( - 1 ) = \frac{1}{a}\left( 0 . 0 0 2 3 5 9 5 - \left. \frac{1}{1 2 8}\, \frac{1}{s}\, \right|_{s \to 0}\right) .
|S(x)|_{h_p}^2 =|x_1|^{2\lfloor tp\rfloor}|\widetilde{s}(x)|^2 e^{-2p\varphi_j(x)}\leq |x_1|^{2\lfloor tp\rfloor}e^{-2p\varphi_j(x)}\frac{1}{\pi^n}\int_{\Delta^n(x,1)}|\widetilde{s}(z)|^2\,dm(z) &\leq |x_1|^{2\lfloor tp\rfloor}e^{-2p\varphi_j(x)}\int_{\Delta^n(0,2)}|\widetilde{s}(z)|^2\,dm(z)\,.
\begin{aligned}\frac 1{\rho_1^{n-2d_e\|R^\nabla \|^2_\infty}}\int_{B_{\rho_1}(x_0)}\big (\exp (\frac{||R^\nabla||^2}2)-1\big ) \, dv &\leq \frac 1{\rho_2^{n-2d_e\|R^\nabla \|^2_\infty}}\int_{B_{\rho_2}(x_0)}\big (\exp (\frac{||R^\nabla||^2}2)-1\big )\, dv\, ,\end{aligned}
\begin{aligned}\begin{array}{l@{\quad}l@{}r@{\quad}l}\underset{w}{min}\,{{w}^{H}}Aw s.t.{{v}^{H}}({{\theta}_{0}},{{\phi}_{0}})w=1 \operatorname{Re}[{{v}_{\theta}}^{H}({{\theta}_{0}},{{\phi}_{0}})w]=0 \operatorname{Re}[{{v}_{\phi}}^{H}({{\theta}_{0}},{{\phi}_{0}})w]=0\end{array}\end{aligned}
(\widetilde{V}_1 \widetilde{V}_2 \widetilde{V}_c)^2 \widetilde{C^\prime}_5 \widetilde{C}_5 \widetilde{C}_1^2 \widetilde{W}_1 \widetilde{W}_2 \widetilde{W}_c \widetilde{W_1^\prime}\widetilde{W_2^\prime}\widetilde{W_c^\prime}= \partial^2.
M_3 = M^{(s-2)}_{0,\ldots, 0,\underset{i}{k_{i}+1},0,\ldots,0,\underset{j}{k_{j}},k_{j+1},\ldots,k_{n}}, M_4 = M^{(s)}_{0,\ldots, 0,\underset{i}{k_{i}-1},0,\ldots,0,\underset{j}{k_{j}+1},k_{j+1},\ldots,k_{n}}.
V_1(t) = p_1 \lambda e^{-\sigma_1^2t^2/2}- e^{-\sigma_0^2t^2/2}+(1-p_1 \lambda), &V_2(t) = p_0 e^{-\sigma_0^2t^2/2}- p_0, &V_3(t) = p_1 (1-\lambda) e^{-\kappa|t|}- p_1 (1-\lambda) &V_4(t) = -p_0\lfloor a + b(t-t^*)\rfloor_+.
R_{\theta r \theta r}= - \sigma{\sigma''}R_{\phi r \phi r}= - \sigma \sigma '' \sin^2 (\theta) R_{zr zr}= - \tau{\tau ''}R_{\phi \theta \phi \theta}= \left(1 - (\sigma ')^2 \right)\sigma^2 \sin^2(\theta) R_{\theta z \theta z}= - \sigma \sigma ' \tau \tau ' R_{\phi z \phi z}= - \sigma \sigma ' \tau \tau ' \sin^2(\theta)
wH_2^{(1)}w^{-1}=\left(\begin{array}{cc}1{}^t(R^{n-1}) 0 \mathrm{E}({n-1},R) \end{array}\right) \geqslant M,\ \mathrm{and}wH_1^{(1)}w^{-1}=\left(\begin{array}{cc}1 0 R^{n-1}\mathrm{E}({n-1},R)\end{array}\right) \geqslant L.
g^R(\partial_{x^i}, \partial_{x^j}) (x) =R^{-2}g( (F_R)_*(\partial_{x^i}), (F_R)_*(\partial_{x^j})) (y)= g(\partial_{y^i}, \partial_{y^j}) (y) \pi^R(\partial_{x^i}, \partial_{x^j}) (x) = R^{-1}\pi ( (F_R)_*(\partial_{x^i}), (F_R)_*(\partial_{x^j})) (y)= R\pi(\partial_{y^i}, \partial_{y^j}) (y).
{\cal S}_{\Omega}= s \int d^{3}x_{1}\prod_{j = 2}^{n}d^{3}x_{j}F_{\mu_{j}\nu_{j}}( x_{j}) \left( \varepsilon_{\mu_{1}\nu_{1}\rho}b^{* \rho}+ \sum_{k = 1}^{\infty}\frac{{\cal M}_{\mu_{1}\nu_{1}}^{k}}{\xi^{k}}\right) \Omega^{\mu_{1}\nu_{1}. . \mu_{n}\nu_{n}}\; ,
H^{\epsilon}(z,\bar{z})=\lim_{|z_3 \bar{z_3}|\rightarrow \infty}|z_3\bar{z}_3|^{2\Delta_{\alpha_3}}\langle \phi_{\alpha_3}\left(z_3,\bar{z}_3\right)\psi_{\alpha_2}\left(1\right)S_0 R^{\epsilon}_\alpha \left(z,\bar{z}\right)R^{\epsilon}_{\alpha_1}\left(0\right) \rangle ,
\begin{matrix}min_{x^{1},...,x^{n}\in\mathbb{R}^{n}}&\sum_{i,j\in[n]}c_{i,j}(x^{i}\cdot x^{j})\\subjectto&\sum_{i,j\in[n]}a_{i,j,k}(x^{i}\cdot x^{j})\le b_{k}forallk\\ \end{matrix}
\frac{1}{|\xi|}\partial_t\hat{U}\cdotp \frac{\bar{\hat{U}}}{|\hat{U}|}+ \nu |\xi||\hat{U}|=\frac{\hat{A}}{|\xi|}\cdot\frac{\bar{\hat{U}}}{|\hat{U}|}+\frac{\hat{\mathbb{P}F}}{|\xi|}\cdot\frac{\bar{\hat{U}}}{|\hat{U}|},
C_{18}\leq \lambda C_{19}\frac{\|{y_1}_n\|_{m_1}^{\kappa_1 + 1}\|{y_2}_n\|_{m_2}^{\kappa_2 + 1}}{\|{y_1}_n\|_{m_1}^p + \|{y_2}_n\|_{m_2}^p}.
D_1 \sum_{j=1}^n M_{21}(e_{1j}) = D_1 \sum_{j=1}^n M_{21}(e_{2j}) = \ldots = D_1 \sum_{j=1}^n M_{21}(e_{mj})= = D_2 \sum_{i=1}^m M_{22}(e_{i1}) = D_2 \sum_{i=1}^m M_{22}(e_{i2}) = \ldots = D_2 \sum_{i=1}^m M_{22}(e_{in}).
\|I_{31}\|_{L^\infty_t}+ 2^{ks}\|I_{31}\|_{L^1_t}\lesssim c_{k,j}2^{-k(\frac{n-1}{p}+1-s)}2^{-\frac{1}{2}j}\|v^h_\Phi\|_{\widetilde{L}_t^\infty(\dot{B}_{p,1}^{\frac{n-1}{p}+1-s,\frac{1}{2}})}\|v^h_\Phi\|_{L_t^1(\dot{B}_{p,1}^{\frac{n-1}{p}+1,\frac{1}{2}})}.