harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
S ( \Phi ) = \int_{- \infty}^{0}d x \int_{- \infty}^{\infty}d t \left( \frac{1}{2}\partial_{\mu}\phi^{a}\partial^{\mu}\phi^{a}- \frac{m^{2}}{\beta^{2}}\sum_{i = 0}^{r}n_{i}e^{\beta \alpha_{i}\cdot \Phi}\right)
\sum_{t_0,\ldots, t_{n-1}}\langle f(h_{t_0},\ldots, h_{t_{n-1}}), \tau|_{[t_0,\ldots, t_{n-1}]}\pi \rangle=\sum_{t_0,\ldots, t_n}\langle (-1)^{n+1}f(h_{t_0},\ldots, h_{t_{n-1}}), \tau|_{[t_0,\ldots, t_n]}\pi\rangle.
- z ( z^{- 1})^{\prime \prime}= - \frac{9}{4}{\cal H}^{2}+ \frac{\varphi^{\prime \prime \prime}}{\varphi^{\prime}}+ \frac{5}{2}{\cal H}^{\prime}- 2 \biggl ( \frac{\varphi^{\prime \prime}}{\varphi^{\prime}}\biggr )^{2}+ \frac{\varphi^{\prime \prime}}{\varphi^{\prime}}\biggl ( 2 \frac{{\cal H}^{\prime}}{\cal H}- 3{\cal H}+ \frac{{\varphi^{\prime}}^{2}}{3{\cal H}}\biggr )
2(-1)^{|a||b|}g(ab) =(-1)^{|a||b|}g(a)f(b)+(-1)^{|a||b|}f(a)g(b)+g(b)f(a)+f(b)g(a) &=(-1)^{|a||b|}g(a)f(b)+g(b)f(a)+(-1)^{|a||b|}g(ab).
g:=g(a_3, a_4, b_3, b_4, c_1, c_3, d_2, d_4) \mapsto \left( \left( \begin{array}{cc}2a_3 + a_4 b_3 2c_1 + c_3 2d_2 + d_4 \end{array}\right) ,\left( \begin{array}{cc}a_4 b_3-2b_4 c_3 d_4 \end{array}\right) \right)\,,
\left[\left(\frac{1-c^{-2}z_0^2}{1-z_0^2}\right)^{1/2}-\aleph\right]e^{\frac{4in}{3}((-\zeta(c^{-1}z_0))^{3/2}-i\pi/2}=\left[\left(\frac{1-c^{-2}z_0^2}{1-z_0^2}\right)^{1/2}+ \aleph \right] -i\frac{\sqrt{c^{-2}z_0^2-1}-\aleph \sqrt{z_0^2-1}}{\sqrt{c^{-2}z_0^2-1}+\aleph\sqrt{z_0^2-1}}=e^{-\frac{4in}{3}((-\zeta(c^{-1}z_0))^{3/2}}
\int_0^{T_0}\int_{\partial B}(u_t+d\pi_N(u)u_r)\cdot\varphi d\phi\,dt =\int_0^{T_0}\int_{\partial B}u_t\cdot\varphi d\phi\,dt +\int_0^{T_0}\int_B\nabla u\cdot\nabla\big(d\pi_N(u)\varphi\big)dz\,dt=0
\mathcal{H}_0: \mathbf{y}=\mathbf{H}\mathbf{D}\mathbf{v}+\mathbf{n}\mathcal{H}_1: \mathbf{y}=\mathbf{H}\mathbf{a}\theta+\mathbf{H}\mathbf{D}\mathbf{v}+\mathbf{n}\;,
\int_{D(y)}\Lambda (\AA^X f) dx = \int_{D_1(y)}\Lambda_1 \int_{D_2(x^1)}\Lambda_2 (\AA^X f) dx^2 dx^1= \int_{D_1(y)}\Lambda_1 \AA^{X_1}\int_{D_2(x^1)}\Lambda_2 f dx^2 dx^1.
X = G (x_1)^{o_1}\otimes \cdots \otimes G (x_m)^{o_m}Y = G (y_1)^{o'_1}\otimes \cdots \otimes G (y_n)^{o'_n}X' = G (x'_1)^{o''_1}\otimes \cdots \otimes G (x'_{m'})^{o''_{m'}}Y' = G (y'_1)^{o'''_1}\otimes \cdots \otimes G (y'_{n'})^{o'''_{n'}}
\lim_{m\to\infty}\sum_{\ell=2}^\infty\left(\frac{r_0}{(\ell/2)^{n_m}}+\frac{r_1}{\ell^{n_{m+1}}/2^{n_m}}+\dots+\frac{r_k}{\ell^{n_{m+k}}/2^{n_m}}\right) \qquad{}=\lim_{m\to\infty}\left(r_0+\frac{r_1}{2^{n_{m+1}-n_m}}+\dots+\frac{r_k}{2^{n_{m+k}-n_m}}\right)=0.
\int_{S^2}\frac{1}{2}a^\alpha X_{\alpha}^{(3)}\tilde \Delta (\tilde \Delta+2) X_0^{(3)}dS^2 = \int_{S^2}(\frac{2}{5}a^i D_{in}\tilde X^n - \frac{1}{3}a^i W_i -\frac{1}{3}a^0 W_0)( -4W_0 +60 \frac{a^j P_j}{a^0}) dS^2 = \frac{4}{3}\int_{S^2}a^i W_iW_0 dS^2+\frac{4}{3}\int_{S^2}a^0 W_0^2 dS^2,
E_{1,l}= \lambda[2(4a+3)z_1 + 8(a+1)b_1 + 2(a+1)(2l+d) - 2b_1 - l(l+d-1)], \psi_{1,l}(r) \propto r^l (1+\lambda r^2)^{-a-1}[1-z_1(1+\lambda r^2)] \exp{[- \lambda(b_1+2b_2)r^2 - \lambda^2 b_2 r^4]},
\langle \, p_f \, | \, q_i \, \rangle =\int \frac{dP_1}{2 \pi \hbar}\, dQ_N \,\langle \, p_f \, | \, Q_N \, \rangle \, e^{i P_1 Q_N / \hbar}\langle \, P_1 \, | \, q_i \, \rangle\, .
\chi_a |h|^2(B_G) \chi_a = \chi_a \vert B_G \vert^{\gamma_h}\widetilde{h}^2(B_G)\vert B_G \vert^{\gamma_h}\chi_a &\leq C_h^2 \chi_a \vert B_G \vert^{2\gamma_h}\chi_a
f^{i}\ = \x^{i + 3}\ , \f^{i + 3}\ = \F \x^{i}\ , \F \ = \F ({\vec{r}}^{2},{\dot{\vec{r}}}^{2},{\vec{r}}\cdot{\dot{\vec{r}}}) , \i \ = \ 1 , 2 , 3 \ .
S_1 = \sum_{q}\Psi \left( \frac{q}{Q}\right) \frac{q}{\varphi(q)}\ \sideset{}{^+}\sum_{\substack{\chi (q)}}L \left( \frac{1}{2},\chi\right) \psi(\chi), S_2 = \sum_{q}\Psi \left( \frac{q}{Q}\right) \frac{q}{\varphi(q)}\ \sideset{}{^+}\sum_{\substack{\chi (q)}}\left|L \left( \frac{1}{2},\chi\right) \psi(\chi)\right|^2.
\widehat{B}=\begin{bmatrix}e+f+g+h 0 0 0 0 0 0 e-f-g+h 0 0 0 0 0 0 -e+f-g+h 0 0 0 0 0 0 -e-f+g+h 0 0 0 0 0 0 -f+h e-g 0 0 0 0 -e-g f+h \end{bmatrix}.
u =\prod_{j=1}^k e^{ih_{j,0}^{(n)}}e^{ih_{j,1}^{(n)}}\cdot e^{ic_n'}&=\prod_{j=1}^k e^{ih_{j,0}^{(n)}}\prod_{j=1}^k e^{ih_{k-j,1}^{(n)}}\prod_{j=1}^{k-1}(u_{j,n},v_{j,n})\cdot e^{ic_n'},
\begin{aligned}d\sum\limits_{i=0}^{q-2}\binom{q}{i}(2x+1)^{q-i}B_{i}=d(B_{0}.(2x+1)^{q}+qB_{1}.(2x+1)^{q-1}+\sum\limits_{i=0}^{\frac{q-2}{2}}\binom{q}{2i}(2x+1)^{q-2i}B_{2i}.\end{aligned}
\sum_{i=1}^k\left(\frac{1}{2}-\frac{r-l_i}{m}\right)=\frac{k}{2}-\frac{kr}{m}+\frac{1}{m}(\sum_{i=1}^kl_i)=\frac{1}{2}+(k-1)\left(\frac{1}{2}-\frac{r}{m}\right)+\frac{1}{m}\left(\sum_{i=1}^kl_i-r\right)<\frac{1}{2}
\tau \Delta t+{\textstyle\sum_{l=1}^{r}}\langle \mathcal{G}(\Delta\widetilde{X}_{a_la_l}),\mathcal{G}(\widetilde{Z}_{a_la_l})\rangle +\langle\Delta\widetilde{X}_{b b},\widetilde{Z}_{bb}\rangle +\langle \Delta\widetilde{X}_{bc},\widetilde{Z}_{bc}\rangle\le 0
(\psi(y_{1}q^{\mu}),y_{2}q^{h})_{D}=q^{-\langle h,\mu\rangle}(y_{1},y_{2})_{L}, (x_{1}q^{\mu},\psi(x_{2}q^{h}))_{D}=q^{-\langle h,\mu\rangle}(x_{1},x_{2})_{L}^{+}.
\prod_{k=2}^{m}T_{k,j}^{\frac{\log s_{k}}{2\pi i}}= (4|\lambda_{j}|)^{-\beta_{j}}\prod_{\substack{k=2 k \neq j}}^{m}\widetilde{T}_{k,j}^{-\beta_{k}}, \mbox{where}\widetilde{T}_{k,j}= \frac{\left(\sqrt{|\lambda_{j}|}+\sqrt{|\lambda_{k}|}\right)^2}{\big|{\lambda_{j}}-{\lambda_{k}}\big|},
\sum_{\substack{d\leq x (d, a_{2})=1}}J_{k}(d)\bigg\lfloor \frac{x}{\ell_u(d)}\bigg\rfloor\leq x\sum_{\substack{d\leq x (d, a_{2})=1}}\frac{J_{k}(d)}{\ell_{u}(d)}\leq x\sum_{\substack{d\leq x (d, a_{2})=1}}\frac{d^{k}}{\ell_{u}(d)}.
X^{-1}A_iX = \left(\begin{array}{cccccc}0 0 0 0 \cdots A_i^{(\ell)}A_i^{(1)}0 0 0 \cdots 0 0 A_i^{(2)}0 0 \cdots 0 0 0 A_i^{(3)}0 \cdots 0 \vdots \vdots \vdots \ddots \ddots \vdots 0 0 0 \cdots A_i^{(\ell-1)}0\end{array}\right)
M_{1 2}= \left( \frac{i \hbar}{2}\right) \sum_{i = 1}^{m}\left( \frac{\partial}{\partial q_{1}^{i}}\frac{\partial}{\partial p_{2}^{i}}- \frac{\partial}{\partial p_{1}^{i}}\frac{\partial}{\partial q_{2}^{i}}\right) .
\hat{H}_{0}= \int_{( \infty )}\frac{d^{3}\vec{k}}{( 2 \pi )^{3}2 \omega ( \vec{k})}\frac{1}{2}\omega ( \vec{k}) \left[ a^{\dagger}( \vec{k}) a ( \vec{k}) + a ( \vec{k}) a^{\dagger}( \vec{k}) \right] \ ,
\varphi^{-1}(p) = (\varphi^{-1}\circ\psi^{-1})(|p_0:p_1:p_2|) = \{[\sigma_0 p_0^{1/a_0}:\sigma_1 p_1^{1/a_1}:\sigma_2 p_2^{1/a_2}] \mid \sigma_i\in\mu^{a_i}\}= \{g\cdot[p_0^{1/a_0}:p_1^{1/a_1}:p_2^{1/a_2}] \mid g\in G\}
\begin{aligned}\lambda_i \in \Lambda_i,\ \delta_{i}\in \{0,1\}^{\lceil \log_2(n) \rceil},\ \sum_{j \notin L_{ik}}\lambda_{ij}\leq \delta_{ik}\leq 1- \sum_{j \notin R_{ik}}\lambda_{ij},\ & k =1, \ldots, \lceil\log_2(n) \rceil,\end{aligned}
(b\mathcal{L}_s)(a_0\otimes\cdots\otimes a_p) = \mathcal{K}_s(a_0\otimes\cdots\otimes a_p) \quad+ X_1 +\Big(\sum_{m=1}^{p-1}(-1)^m(X_m+X_{m+1})\Big)+(-1)^pX_p.
\sum\limits_{\substack{\mathbf{n}\in [N]^4 n_1 - n_2 = n_2 - n_3 \vert (n_2 -n_3) - \sqrt{2}(n_3 - n_4)\vert \leqslant \frac{1}{2}}}\mu(n_1)\mu(n_2)\mu(n_3)\mu(n_4) = o(N^2)
M_X(T)=\prod_{d \mid u, \sqrt p \in \mathbb Q(w_d)}\Phi_d^{+}(T)^{e_{d,1}}\Phi_d^{-}(T)^{e_{d,2}}\prod_{d \mid u, \sqrt p \not \in \mathbb Q(w_d)}\Phi_d(T)^{e_{d}}
m^{2\ (-)}_\phi = \lambda_{obs}^{-2}\m^{2\ (+)}_\phi = \lambda_{obs}^{-2}\{d^2 V_{\hbox{\footnotesize\it \hspace{-6pt}eff\,}}\over d\phi^2}\approx \mp\lambda_{obs}\left({5\pi^3 \zeta'_R(-4)\over 6 M^3 l^5}\right).
\frac{5 x - 5 y}{3 x^{2}y}\cdot \frac{9 x y^{2}}{y^{2}- y^{2}}\frac{5 ( x - y ) \cdot 3 y}{x ( x + y ) ( x - y )}= \frac{1 5 y}{x ( x + y )}= \frac{1 5 y}{x^{2}+ x y}
Q^1_{1\bar 1}=\mu_{k\bar r}^{\bar 1}\mu_{\bar kr}^{1}\,,Q^2_{1\bar 1}=\mu_{\bar k\bar r}^{\bar 1}\mu_{kr}^1\,, &Q^3_{1\bar 1}=\mu_{k\bar k}^{\bar 1}\mu_{\bar r r}^1\,,Q^4_{1\bar 1}=0\,.
\begin{array}{ccccccccc}\makebox[0em][c]{the outer region}\makebox[0em][c]{the border}\makebox[0em][c]{the central region}{\bf1}_2 \rightarrow g_L\in SU(2)_L \rightarrow -{\bf1}_2 \rightarrow g_R\in SU(2)_R \rightarrow{\bf1}_2\end{array}
W_j =\prod^{j}_{i=1}S^{\varepsilon_i}_{\alpha_i}\qquad(j=1,2,\ldots,m) W_{m+k}=G\prod^{k}_{i=1}S^{\eta_i}_{\beta_i}\qquad(k=1,2,\ldots,n) W_{m+n+l+1}=GR\prod^{m-l}_{r=m}S^{-\varepsilon_r}_{\alpha_r}\qquad(l=0,1,\ldots,m-1).
0 = \int \mathrm{d}\Phi \,{\frac{\partial_{r}}{\partial \Phi^{A}}}\left\{\mathrm{e}^{{\frac{i}{\hbar}}\Sigma +{\frac{i}{\hbar}}J_{A}\Phi^{A}}\,{\frac{\partial_{l}\Sigma}{\partial K_{A}}}\right\}.
\epsilon^{+}( z ) = \frac{z}{2 \pi}+ \frac{i}{\pi}\sum_{n > 0}\frac{e^{- i n ( z - i \varepsilon )}}{n}, \epsilon^{-}( z ) = \frac{z}{2 \pi}+ \frac{i}{\pi}\sum_{n < 0}\frac{e^{- i n ( z + i \varepsilon )}}{n}.
U_2(\omega) =2n^2 \sum_{\ell'=0}^{\frac{n}{\mathrm{ord}(\omega)}}\sum_{j=0}^{\frac{n}{\mathrm{ord}(\omega)}-1}\binom{\frac{n}{\mathrm{ord}(\omega)}-1}{j}^2 \binom{\frac{n}{\mathrm{ord}(\omega)}}{\ell'}\binom{j}{\ell'}\binom{j+\ell'}{\frac{n}{\mathrm{ord}(\omega)}}\sum_{i=1}^{\frac{n}{\mathrm{ord}(\omega)}-1}\frac{\omega^i}{(\omega^i-1)^2}.
\hat h^n_{(1-t)n}(\theta)=\frac{(-1)^{\frac{tn+1}{2}}(-i\theta)^{n+2}e^n\sqrt{2n(1-t)+1}}{2^{n+3/2}n^{n+3}t^{\frac{nt+3}{2}}(2-t)^{n(1-\frac{t}{2})+2}}\left(1-\frac{104-12t+27t^2+6\theta^2}{24(2-t)tn}+O\left(\frac{1}{n^2}\right)\right).
u(t) = e^{it\Delta}u_0 + \frac{1}{2}\mathcal{J}_S[(w + \overline{w})u](t), w(t) = e^{-it \langle \nabla \rangle}w_0 + \mathcal{J}_{W_{+}}\Bigl[\frac{\Delta}{\langle \nabla \rangle}(u \overline{u}) + \frac{1}{2 \langle \nabla \rangle}(w + \overline{w}) \Bigr](t).
\langle \chi_j, \chi_{\gamma}\chi_i \rangle = \frac{1}{|G|}\sum_{g \in G}\chi_j(g) \bar{\chi_{\gamma}(g) \chi_i(g)}&= \frac{1}{|G|}\sum_{g \in G}\chi_j(g) \chi_{\gamma}(g) \bar{\chi_i(g)}= \langle \chi_j \chi_{\gamma}, \chi_i \rangle
\langle \alpha_{\vartheta}| \alpha_{\vartheta'}\rangle = \delta_{\vartheta,\vartheta'}\langle \alpha_x| \alpha_x ' \rangle =\delta(x-x') & \langle \alpha_\vartheta'| \alpha_x \rangle =0 \langle \alpha_x'| \alpha_{\vartheta}\rangle =0
\frac{\varphi(x^k+d^{k,l}; \rho^k) - \varphi(x^k; \rho^k)}{m^k(x^k+d^{k,l};\rho^k) - m^k(x^k;\rho^k)}=\frac{c^Td^{k,l}+ \rho^k([r_j(x_j^k+ d_j^{k,l})]^+-[r_j(x_j^k)]^+)}{c^Td^{k,l}- \rho^k[r_j(x_j^k)]^+}
D_{i \alpha}\tilde{\mathrm{A}}^{\beta \gamma}( \theta ) = \textstyle{\frac{1}{3}}\left( \delta_{\alpha}^{\beta}D_{i \delta}\tilde{\mathrm{A}}^{\delta \gamma}( \theta ) - \delta_{\alpha}^{\gamma}D_{i \delta}\tilde{\mathrm{A}}^{\delta \beta}( \theta ) \right)
A_z=\frac{8}{\pi^2}\, \bigg[\frac{\Gamma(\frac{\eta}{2-2\eta})}{2\sqrt{\pi}\Gamma(\frac{1}{2-2\eta})}\bigg]^{\frac{1}{\eta}}\exp\bigg\{\int_0^{\infty}\frac{dt}{t}\Big(\frac{\sinh((2 \eta-1) t)}{\sinh(\eta t)\cosh((1-\eta)t)}-\frac{2\eta-1}{\eta}\, e^{-2 t}\, \Big) \bigg\}\, .
S_{\chi,\varepsilon}(z) = S_{\chi}^{\rm soft}(z) + S_{\chi}^{\rm stiff}(\varepsilon^2 z), begin{align*}0.3em] M_{\chi,\varepsilon}(z) = M_{\chi}^{\rm soft}(z) +\varepsilon^{-2}M_{\chi}^{\rm stiff}(\varepsilon^2 z), z\in \rho(\mathcal{A}_{\chi,\varepsilon,0}).
\varphi_t =- d\Delta \varphi-\vartheta (x,t)\in \Omega\times (\tau,T) \kappa_1d\frac{\partial \varphi}{\partial \eta}+\kappa_2\varphi=\Psi (x,t)\in M\times (\tau,T) \varphi =0 x\in\Omega ,t=T
\frac{1}{\beta}\int_{0}^{\beta}d \tau_{1}\cdots \int_{0}^{\beta}d \tau_{n}\langle \eta^{k_{1}}( \tau_{1}) \cdots \eta^{k_{n}}( \tau_{n}) \rangle_{0 , c}\propto \beta^{n - 1 + ( k_{1}+ \cdots + k_{n}) / 2}.
\left[\begin{array}{cc}Q 0 0 Q\end{array}\right]\left[\begin{array}{c}\dot{x}_1(t) \dot{x}_2(t)\end{array}\right]= \left[\begin{array}{cc}0 Q -QLQ^{\top}Q -QLQ^{\top}Q\end{array}\right]\left[\begin{array}{c}x_1(t) x_2(t)\end{array}\right] &+\left[\begin{array}{c}0 Q\end{array}\right]w(t)\,,
\Omega ( N , T , V ) ={\frac{V^{N}}{N !}}\Bigl \{1 +{\frac{N^{2}}{2 V}}\int_{0}^{2 \pi}d \theta \int_{0}^{\infty}d r \, r \Bigl ( \mathrm{e}^{-{\frac{\phi ({\bf r}_{i j})}{k T}}}- 1 \Bigr ) + \cdots \Bigr \}\ .
\Delta n \cdot \partial_1 n = \frac 12 \partial_1 \left[ |\partial_1 n|^2 - |\partial_2 n|^2 \right] + \partial_2 \left[\partial_1 n\cdot\partial_2 n \right] = 0, \Delta n \cdot \partial_2 n = \frac 12 \partial_2 \left[ |\partial_2 n|^2 - |\partial_1 n|^2 \right] + \partial_1 \left[\partial_1 n\cdot\partial_2 n \right] = 0.
\begin{bmatrix}0 0 0 0 0 0 -1 \alpha 0 \overline\alpha 0 -\overline\alpha 0 -\alpha J(T^{-2r},T^{-3r}) 0 0 0 -\overline\alpha J(T^{-r},T^{-r}) 0 -\alpha J(T^{-3r},T^{-3r}) 0 -\alpha 0 -\overline\alpha J(T^{-2r},T^{-r}) 0\end{bmatrix}.
\varnothing \neq \big( W \smallsetminus \{x\}\big) \cap K^{(\alpha)}=\big( W \smallsetminus \{x\}\big) \cap K^{(\alpha)}\cap F = \big( W \smallsetminus \{x\}\big) \cap (K \cap F)^{(\alpha)}\subseteq \big( V \smallsetminus \{x\}\big) \cap (K \cap F)^{(\alpha)}.
| \Phi_{v}^{\nu = 1 / k}\rangle ={\cal N}\varepsilon^{\alpha_{1}\dots \alpha_{N}}({\bf A}{_{\alpha_{1}}^{+}}) \Big ({\bf A}_{\alpha_{2}}^{+}\Big )^{2}\dots \Big ({\bf A}_{\alpha_{N}}^{+}\Big )^{N}\{\bf I}\; | v \rangle
\left( \| 1- p_1\|_{2,u}^{(\omega)}\right)^2 = \left\| E^\omega(1-p_1)^*(1-p_1) \right\| &= \left\| E^\omega(1-p_1) \right\| &= \sup_{\lambda_\omega\in K^\omega}\tau_{\lambda_\omega}(1-p_1) &= 1-\frac{1}{n}< 1
1 - \lambda = \sum_{i=0}^{t-1}\binom{t+1}{i+2}\binom{t-1}{i}\frac{\binom{k-t}{i+2}}{\binom{t(k-t+1)-2t+2+i}{i+2}}= \sum_{i=0}^{t-1}\binom{t+1}{i+2}\binom{t-1}{i}\frac{\binom{k-t}{i+2}}{\binom{t(k-t+1)-2t+1+i}{i+2}}\left(1-\frac{i+2}{t(k-t+1)-2t+2+i}\right).
\tilde u(x_0) - u_h(x_0) = (\tilde u - v_h)(x_0) + (v_h - \tilde u, \eta)_{\Omega_h}+ (\tilde u - u_h, \eta)_{\Omega_h}, \partial(\tilde u - u_h)(x_0) = \partial(\tilde u - v_h)(x_0) + (\partial(v_h - \tilde u), \eta)_{\Omega_h}- (\tilde u - u_h, \partial\eta)_{\Omega_h},
\partial_t |\nabla r|^2=\Delta |\nabla r |^2 - 2|\nabla^2 r|^2 +\frac{2}{\varepsilon}\nabla r\cdot \nabla g_q (|\nabla r|^2 -1) + \frac{2g_q}{\varepsilon}\nabla r\cdot \nabla |\nabla r|^2.
\tilde{\pi}(A,g)\tilde{\pi}(B,h) = \pi(g) \cdot \prod_{k \in A}\varepsilon(k) \cdot \prod_{k \in Y_g \setminus A}(\pi(d(g)) - \varepsilon(k)) \cdot \pi(h) \cdot \prod_{\ell \in B}\varepsilon(\ell) \cdot \prod_{\ell \in Y_h \setminus B}(\pi(d(h)) - \varepsilon(\ell)).
\Omega^{(2)}:=\Lambda(\Omega^{(1)})\,,\ \Gamma^{(2)}:=\L (\Gamma^{(1)})\,, \ \Gamma^{(2)}(t):=\L (\Gamma^{(1)}(t))\,, & A^{(2)}(x):= [D\L \, A^{(1)}\, D\L^T] (\L^{-1}(x))\,.
\log^m|u_i|\Big[\sum_{p=0}^m(-1)^{k+p}\frac{2^m}{p!(m-p)!}\Big] =\log^m|u_i|\Big[\sum_{p=0}^m(-1)^{k+p}\frac{2^m}{m!}{m\choose p}\Big] &=\log^m|u_i|\,\frac{2^m}{m!}(-1)^k\Bigg[\sum_{p=0}^m(-1)^p{m\choose p}\Bigg].
\zeta_{p}(\sigma+it)=\begin{cases}O\left(|t|^{1-\sigma}\log|t|\right) \frac{1}{2}\leq\sigma\leq1, O\left(|t|^{\frac{1}{2}}\log|t|\right) 0<\sigma\leq\frac{1}{2}, O\left(|t|^{\frac{1}{2}-\sigma}\log|t|\right) \sigma\leq0.\end{cases}\,\,\,\,\,|t|\rightarrow\infty.
\varphi \ast \psi = \varphi\!\left( ? b_j b_i \right) \psi\big(S^{-1}(a_i) ? a_j\big), x_m \mapsto \varphi ( x_m' b_j b_i ) \psi\big(S^{-1}(a_i) x_m'' a_j\big).
R_{\mu \nu}= \frac{\partial^{2}\ln \sqrt{- g}}{\partial x^{\mu}\partial x^{\nu}}- \frac{\partial \Gamma_{\mu \nu}^{\sigma}}{\partial x^{\sigma}}+ \Gamma_{\mu \sigma}^{\lambda}\Gamma_{\nu \lambda}^{\sigma}- \Gamma_{\mu \nu}^{\sigma}\frac{\partial \ln \sqrt{- g}}{\partial x^{\sigma}},
G_{p+1,q+1}^{m,n+1}(\begin{matrix}\alpha,a_{p}\\b_{q},\beta\end{matrix}|z)=(-1)^{\beta-\alpha}G_{p+1,q+1}^{m+1,n}(\begin{matrix}a_{p},\alpha\\ \beta,b_{q}\end{matrix}|z),m\le q,\beta-\alpha=0,1,2,...,
I(u_1,u_2)=\int(\nabla u_1\cdot \nabla u_2+ \lambda e^{u_0^\kappa+v_0^\kappa+u_1+u_2}-\lambda e^{u_0^\kappa+u_1}-\lambda e^{v_0^\kappa+u_2}+h_2u_1+h_1u_2)dx
- \varepsilon \, \partial_{\boldsymbol{\nu}_\varepsilon}\mathfrak{U}_{M}^{(\varepsilon)}= \varepsilon^{\alpha}\varphi^{(0)}_\varepsilon \ \Big(\partial\Omega^{(0)}_{\varepsilon, \gamma}\setminus \Big\{\bigcup_{i=1}^3 \overline{\Upsilon_\varepsilon^{(i)}(2\ell_0 \varepsilon^\gamma)}\Big\}\Big) \times (0, T).
\sum_{k=1}^n (-1)^{n-k}\sum_{\sigma\in Sh(k,n-k)}\chi(\sigma) Q^a_{n-k+1}(Q^a_k(x_{\sigma(1)}\wedge\dots\wedge x_{\sigma(k)})\wedge x_{\sigma(k+1)}\wedge \dots \wedge x_{\sigma(n)})=0
|\frac{4 \alpha (\log(\lambda_{0,0}(t))-\log(\lambda(t))) \lambda'''(t)}{\log(\lambda_{0,0}(t))}| \leq \frac{C}{\log(\log(t))}|\log(1+\frac{e(t)}{\lambda_{0,0}(t)})|\left(\frac{1}{t^{3}\log^{b+1}(t)}+|e'''(t)|\right) &\leq \frac{C}{(\log(\log(t)))^{3/2}}\left(\frac{1}{t^{3}\log^{b+1}(t)}+|e'''(t)|\right)
\textrm{Pr}\big[\mathrm{P}\big]=\textrm{Pr}\big[\mathrm{P}|\mathrm{M}\big]\textrm{Pr}\big[\mathrm{M}\big]+ \textrm{Pr}\big[\mathrm{P}|\mathrm{I}-\mathrm{M}\big]\textrm{Pr}\big[\mathrm{I}-\mathrm{M}\big],
S_{e f f}= \int \! d^{4}\! x ~ [ ~ \pi_{0}\dot{A}^{0}+ \pi_{i}\dot{A}^{i}+ \pi_{\rho}\dot{\rho}+ B_{2}\dot{N}^{2}+ \overline{{{\cal P}}}_{i}\dot{\cal C}^{i}+ \overline{{{\cal C}}}_{2}\dot{\cal P}^{2}~ ] - H_{t o t a l},
d s_{D}^{2}= \frac{1}{( \sum_{i = 1}^{n}\, | z_{i}| / L + 1 )^{2}}\, \Big ( \eta_{\mu \nu}d x^{\mu}d x^{\nu}+ \sum_{j = 1}^{n}\, ( d z_{j})^{2}\Big ) \, .
\begin{aligned}\|w_1 - w_2\|_{C^{\frac{1+\beta_0}{2},1+\beta_0}(D)}\leq C_1(C_{01}\|h_1-h_2\|_{C^1([0,T_0])}+T_0^{\frac{\beta_0+1}{2}}\|z_1-z_2\|_{C^{\frac{1+\beta_0}{2},1+\beta_0}(D)}&+\|z_{01}-z_{02}\|_{C^{2}([0,1])}+ \|w_{01}-w_{02}\|_{C^{2}([0,1])}) \end{aligned}
U_{\mathcal X_t\to y_t}=\{u\in \mathcal P_{\mathcal X_t}: \{\alpha^{1}_{\mathcal X_1}, \dots, \alpha^{t-1}_{\mathcal X_{t-1}}, u\}\frac{\rho}{2^{t-1}}\prod_{j\in [k]\setminus [t]}|\mathcal P_{\mathcal X_j}| \mathcal A_{\mathcal Y}\}.
\frac{|c(|D|)|^2}{\langle h, h \rangle}= 2^{\nu(N)}\frac{(k-1)!}{\pi^k}|D|^{k-1/2}\left(\prod_{p\mid M}\frac{p}{p+1}\right)\frac{L(f,D,k)}{\langle f,f\rangle}.
\omega(\xi) = \sup \left\{|u(x,t) - u(y,s)| \vee |v(x,t) - v(y,s)| \, \mid \, (x,t), (y,s) \in \right. &\left. \mathcal{I}\times [0,T], \, \, d(x,y) + |t - s| \leq \xi \right\}.
A=\sum_{i=1}^{r}a_{i}\otimes b_{i}=\sum_{i=1}^{r}a_{i}b_{i}^{T}=AB^{T}=(AX^{-1})(BX^{T})^{T}=\sum_{i=1}^{r}c_{i}d_{i}^{T}=\sum_{i=1}^{r}c_{i}\otimes d_{i},
\varphi=\left(\begin{array}{ccccccccccc}z 0 0 \ldots 0 0 0 -c_1 z 0 \ldots 0 0 0 0 -c_2 z \ldots 0 0 0 \vdots \vdots \vdots \ddots \vdots \vdots \vdots 0 0 0 \ldots z 0 0 0 0 0 \ldots -c_{n-3}z 0 0 0 0 \ldots 0 -c_{n-2}z 0 0 0 \ldots 0 0 -c_{n-1}\end{array}\right)
\mathrm{T r}_{s}\left( q^{L_{0}/ \hbar}e^{i 2 \pi T_{0}^{3}/ \hbar}\right) = \frac{q^{\hbar s ( s + 1 ) / k}\sum_{n = - \infty}^{\infty}( - 1 )^{2 s + 2 n k / \hbar}( 2 s + 1 + 2 n k / \hbar ) q^{{n^{2}k / \hbar}+ ( 2 s + 1 ) n}}{\Pi_{m = 1}^{\infty}( 1 - q^{m})^{3}}.
\bigg( \int_{\mathbb{T}^{N}}\Big\Vert \sum_{n=1}^{N}x_{n}z_{n}\Big\Vert^{p}dz \bigg)^{\frac{1}{p}}\leq C \Big( \sum_{n=1}^{N}\Vert x_{n}\Vert^{p}\Big)^{\frac{1}{p}}\,.
\psi_{k}( x ) \rightarrow C \left( e^{i k x}\frac{\Gamma ( i k ) \Gamma ( 2 \lambda + 2 )}{\Gamma ( \lambda + 1 + i k ) \Gamma ( \lambda + 1 )}+ e^{- i k x}\frac{\Gamma ( - i k ) \Gamma ( 2 \lambda + 2 )}{\Gamma ( \lambda + 1 - i k ) \Gamma ( \lambda + 1 )}\right)
h''(s) = Ku\Gamma(u+2) s^{-u-2}\zeta(u+1)(1+O(s^{\epsilon/2})) + O(s^{-u/2-2}) + O(s^{-2}) & = (1+O(s^{\eta}))Ku\Gamma(u+2)\zeta(u+1) s^{-u-2}.
[q^\alpha r^\beta]C_k(q,r,t) = C_k(t)^{\beta+1}\sum_{i \geq 0}(-1)^i \binom{\alpha - (k-1)i}{i}t^i - \sum_{i \geq 0}(-1)^i \binom{\alpha - \beta - 1 - (k-1)i}{i}t^i.
\lim_{t\to +0}t^{-\frac{N+1}4}\!\!\!\int\limits_{B_R(x)}\! u^\pm(z,t)\dz=c(N)\left\{\prod\limits_{j=1}^{N-1}\left[\frac 1R - \kappa^\pm_j(x_\pm)\right]\right\}^{-\frac 12},
g(\tau) := E_{L,2}(\tau,1) \otimes \Theta^{Jac}(\tau) A := \frac12 \log \pi \ + \ \frac12 \gamma - \frac12 \ - \ \frac{\zeta_F'(2)}{\zeta_F(2)}\ + \ \frac{\zeta'(2)}{\zeta(2)},
{\cal L}= \int d^{4}\theta \left( Q^{i \dagger}e^{2 V}Q^{i}+ \frac{1}{2}\mathrm{T r}S^{\dagger}S \right) + \int d^{2}\theta \left( \frac{1}{4 g_{h}^{2}}W^{a}W^{a}+ \frac{1}{2}\lambda S_{i j}Q^{i}Q^{j}\right) + h . c .
y'_{v}= \begin{cases}(y_{i+1})^{-1}\mbox{if ${v}= i \in M$,}y_i \, y_i^- \mbox{if $v = i^- \in M^-$,}y_{i+1}\, y_i^+ \mbox{if $v = i^+ \in M^+$,}y_{v}\mbox{otherwise.}\end{cases}
\lim_{R\to\infty}\inf_{v\in H_{i_0}(x_1)\setminus \{0\}}{1\over R}(D_{[R]}f_{2,v})(0) = \chi_{i_0}= \lim_{R\to\infty}\sup_{v\in H_{i_0}(x_1)\setminus \{0\}}{1\over R}(D_{[R]}f_{2,v})(0) .
\psi \left( z,s\right) :=iz\gamma \left( s\right) -\frac{1}{2}b\left(s\right) z^{2}+\int_{\mathbb{R}}\left[ e^{izx}-1-i\tau \left( x\right) z\right] \nu \left( s,\mathrm{d}x\right) ,z\in \mathbb{R},s\in \mathcal{S}
\dot{\phi}({\bf k}-{\bf k}_{1}+{\bf k}_{2}, t ) \, \phi ({\bf k}_{1}, t ) \phi^{*}({\bf k}_{2}, t ) = \dot{\phi}({\bf k}, t ) \, \langle \phi ({\bf k}_{1}, t ) \phi^{*}({\bf k}_{2}, t ) \rangle + \mathrm{f l u c t u a t i o n s}\; ,
||\left(\partial_{r}+\frac{1}{r}\right)v_{4,c}(t,r)||_{L^{2}(r dr)}\leq ||\partial_{r}v_{4,c}(t,r)||_{L^{2}(r dr)}+ ||\frac{v_{4,c}(t,r)}{r}||_{L^{2}(r dr)}&\leq \frac{C}{t^{2}\log^{3b+3N}(t)}
\left(AP/L\right)_h~~~~~~~~~~\left(\begin{array}{ccc}A_{xa}\frac{1}{L_z}\left(C_{a1}+C_{a2}^\iota\right) 0 C_{f1}A_{xf1}+A_{z1}B_{l1}0 B_{c1}+B_{c2}^\iota 0\end{array}\right)\left(\begin{array}{c}\alpha \beta \gamma \end{array}\right)=\left(\begin{array}{c}F_{\overline{u}}F_{u_1^\prime}0\end{array}\right),
\sum_{\tau=t+h}^{n}Z_{N,J}^{\tau}+\sum_{\tau=t}^{t+h-1}n^{t+h-\tau}Z_{i_{\tau},J}^{\tau}\leq 1+\sum_{\tau=1}^{h-1}n^{\tau}, & \forall \J\subseteq V, \ \lvert J \rvert = h\in[n-1], \t\in [n-h], \i_t,\dotsc,i_{t+h-1}\in N.
\begin{aligned}|\bar{y}-\bar{y}_a| = \,|\bar{y}-\bar{y}_a\pm z_2| \le |\bar{y}-z_2|+|z_2-\bar{y}_a| \le \, |\bar{y}-z_2|+\gamma|\epsilon_2(\eta_a,t)| \le \, |\bar{y}-z_2|+\gamma L_r 2\delta \end{aligned}
\tilde{g}_{D}= f_{D}+ 8 u f_{D}^{\prime}+ 4 ( u^{2}- \Lambda_{S W}^{4}) f_{D}^{\prime \prime}= f + 8 u f^{\prime}+ 4 ( u^{2}- \Lambda_{S W}^{4}) f^{\prime \prime}= \tilde{g},
\begin{array}{ll}\nabla\cdot \mathbf{E}^{\prime}=\rho_e^{\prime}, \nabla\times\mathbf{B}^{\prime}=\mathbf{j}_e^{\prime}+\frac{\partial\mathbf{E^{\prime}}}{\partial t}, \nabla\cdot \mathbf{B}^{\prime}=\rho_m^{\prime}=0, \nabla\times\mathbf{E}^{\prime}= -\frac{\partial\mathbf{B^{\prime}}}{\partial t}.\end{array}
R_{\mu \nu}-{1 \over 2}g_{\mu \nu}~^{(4)}R = - \ell_P^2 \left(\frac{\delta \stackrel{\sim}{\cal L}_m^{\prime}}{\delta g^{\mu \nu}}-{1 \over 2}g_{\mu \nu}\stackrel{\sim}{\cal L}_m^{\prime}\right) - \ell_P^2e^{-1}\frac{\delta{\cal L}_g}{\delta g^{\mu \nu}}.
\lambda_i= \begin{cases}r+s-2i+d(i)\quad \textup{if $\delta_i\neq0$, $\delta_{i-1}=\cdots=\delta_{i-(d(i)-1)}=0$ and $\delta_{i-d(i)}\neq0$}, \lambda_{i+1}\quad \textup{if $\delta_i=0$}. \end{cases}
G_{\mu}^{( 3 )}( x_{1}, \tau_{1}, x_{2}, \tau_{2}, x_{3}, \tau_{3}) = \langle 0 | \mathrm{T}\psi^{*}( x_{1}, \tau_{1}) \psi ( x_{2}, \tau_{2}) a_{\mu}( x_{3}, \tau_{3}) e^{i \int d^{4}y d \tau e_{0}a_{\mu}( y , \tau ) j^{\mu}( y , \tau )}| 0 \rangle_{\mathrm{t r e e}}
{\rm Re}\left\{1+z\frac{T_g''(z)}{T_g'(z)}\right\}={\rm Re}\left\{z\frac{f'(z)}{f(z)}+z\frac{g''(z)}{g'(z)}+1\right\}>\frac{1+\alpha-4 r+(1-\alpha)r^2}{1-r^2}>0,
F_{-{\frac{1}{2}}}( x ) = \bar{\epsilon}_{0}S ( x ) e^{- 1 / 2 \phi ( x )}\; , \qquad F_{\frac{1}{2}}( x ) = \bar{\epsilon}_{0}\gamma_{\mu}S ( x ) \partial X^{\mu}( x ) e^{1 / 2 \phi ( x )},
p_{T_1, T_2}(X) = \left( \prod_{j=1}^{m-1}\prod_{s=j+1}^m \frac{1}{r(s,j)!\,u(s,j)!}(d_{s \to j})^{r(s,j)}(d_{j \to s}^*)^{u(s,j)}\right) \cdot P_{\lambda}(Z).
D^{ab}_{\mu,x}iG^{[s]}(x,x')^{bc}_{\nu\rho}={1\over 2}\int\limits_0^\infty dT\int\limits_{x(0)=x',x(T)=x}Dx(t) \dot{x}_\mu(T)exp\left(-{1\over 4}\int\limits_0^Tdt\,\dot{x}^2\right)\Phi^{[1]}_{\nu\rho}[C^{xx'}]U[C^{xx'}]^{ac}
\begin{aligned}\eta^r( \Theta_1,\Theta_2) =\eta (\Theta_1) -\eta({\Theta_2})-\nabla\eta({\Theta_2})(\Theta_1-{\Theta_2}) &=\int_{0}^{1}\int_{0}^{1}s(\Theta_1-{\Theta_2})^T \bigl(\nabla^2\eta(\hat{\Theta})\bigr)(\Theta_1-{\Theta_2})\,ds\, d\tau. \end{aligned}
S_{\mathrm{Y M / \Phi}}^{( \mathrm{q})}={\frac{N V}{4 g_{0}^{2}}}\mathrm{T r}\left( \left[ \mathbf{A}_{\mu}^{( \mathrm{q})}, \mathbf{A}_{\nu}^{( \mathrm{q})}\right]^{2}\right) \ .
S_{\mu \nu}( m , n | a , b ) = \sum_{p = 1}^{\infty}\frac{p}{m}\frac{p}{n}S_{\mu \nu}( p ) \oint \frac{d z}{2 \pi i}z^{p - 1}\left( \frac{b^{*}z + a^{*}}{a z + b}\right)^{m}\oint \frac{d \bar{z}}{- 2 \pi i}{\bar{z}}^{p - 1}\left( \frac{b \bar{z}+ a}{a^{*}\bar{z}+ b^{*}}\right)^{n}.
-2\lambda^2T_{1,0}-3\lambda\frac{\partial T_{2,0}}{\partial M_0}M_0-3\lambda^5\frac{\partial T_{-2,0}}{\partial M_0}M_0-2\lambda T_{2,0}+2\lambda^5T_{-2,0}=-3T_{3,0}+\lambda^4T_{-1,0}.