harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Vc\cdot Va-Vb\cdot Vd = -2\left\{y_0Vz\cdot\left(Vu\times Vw\right)-z_0Vy\cdot\left(Vu\times Vw\right)\right. \left. +u_0Vw\cdot\left(Vy\times Vz\right)-w_0Vu\cdot\left(Vy\times Vz\right)\right\}=-2\det [y,z,u,w]=0,
|\tau_b(x_0)(z)-h(z)| \geq |\tau_b(x_0)(z)-\tau_b(x_n)(z)|-|\tau_b(x_n)(z)-h(z)| \geq |d(z,x_0) -d(b,x_0) -d(z,x_n)+d(b,x_n)| -\epsilon = |1-d(z,x_n)+d(b,x_n)|-\epsilon \geq 1-d(z,x_n)+d(b,x_n)-\epsilon.
f^{\not\ominus}(z,t) = zf(z,1)(f(z,t)-1) + \left(\frac{z}{1-z}\right)\left(f(z,t)-1\right)\left(\frac{f(z,1)f^{\not\ominus}(z,f(z,1)) - tf^{\not\ominus}(z,t)}{f(z,1)-t}\right).
D_{\zeta}^{2 s + \tau , 2 t + \tau^{\prime}}( x , x_{1}, \ldots , x_{n}) = ( - i )^{2 s + \tau + 1}\sigma_{2 s + \tau}^{+}\sum_{p = 0}^{t}x^{2 s - 2 p + \tau + 1 - \zeta}\hat{\sigma}_{2 p + \zeta}^{-}\, \, .
\tilde{T}_{2}=\int\alpha_{20}(x_{t},y_{2t})[H(\hat{\gamma}_{1}(x_{t+1}))-H(\gamma_{10}(x_{t+1}))]F_{0}(dw)+\int\alpha_{20}(x_{t},y_{2t})[H(\gamma_{10}(x_{t+1}))-\gamma_{20}(x_{t})]F_{0}(dw) & =\int\alpha_{20}(x_{t},y_{2t})[H(\hat{\gamma}_{1}(x_{t+1}))-H(\gamma_{10}(x_{t+1}))]F_{0}(dw).
\theta_{f(n)}-\theta_{f(n+1)}=\frac{1}{\log\log n^n}-\frac{1}{\log\log (n+1)^{n+1}}=\frac{\log\log (n+1)^{n+1}-\log\log n^n}{\left(\log\log n^n\right)\cdot\left(\log\log(n+1)^{n+1}\right)}\le\frac{\log\log(e(n+1))}{\left(\log\log n^n\right)\cdot\left(\log\log(n+1)^{n+1}\right)}.
\mathcal{U}(h,k):= \frac{1}{2}\sum_{\substack{1\leq q<\infty (q,hk)=1}}W\left( \frac{q}{Q}\right) \sum_{\substack{1\leq m,n<\infty (mn,q)=1 mh\neq nk}}\frac{\tau_A(m) \tau_B(n)}{\sqrt{mn}}V\left( \frac{m}{X}\right) V\left( \frac{n}{X}\right) \sum_{\substack{1\leq c\leq C, d \geq 1 cd=q d|mh\pm nk}}\phi(d)\mu(c),
\langle \Omega| W_{2m,0}(x)|\Omega\rangle = -\sum_{k_1,\ldots,k_N=0}^\infty \prod_{j=1}^N\binom{2s}{k_j}(-1)^{k_{tot}}k_{tot}!\frac{\Gamma (p+q-2 m)}{\Gamma (p+q-2m+k_{tot}+1)}\,.
S_{3,A,V}(x,t;B,T) =i\bigg[ \int_0^{T(t)}A(x+B(s))\cdot dB(s)+\frac{1}{2}\int_0^{T(t)}(\nabla \cdot A) (x+B(s))ds \bigg] & +\int_0^tV(x+B(T(s)))ds.
\langle K(x,y), \lambda_\delta(x,y)\chi_0(y)[\phi(y)-\phi(x)]\psi(x) \rangle =\sum\limits_{j\in \Gamma}\langle K(x,y), \lambda_\delta(x,y)\bar\eta_j(y)\chi_0(y)[\phi(y)-\phi(x)]\psi(x) \rangle.
A_1 =\left(\int_{\varphi_n(E_M)}|Df_n-Dh_n|^p\,d\mu\right)^{\frac{1}{p}}\ , A_2 =\left(\int_{\varphi_n(E_M)}|Dh_n-Dh_n\circ\phi_n|^p\,d\mu\right)^{\frac{1}{p}}\ , A_3 =\left(\int_{\varphi_n(E_M)}|Dh_n\circ\phi_n-Dg_n|^p\,d\mu\right)^{\frac{1}{p}}\ .
\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big(2\varepsilon(h,x+y)[\alpha(x),\alpha(y)]\cdot[\alpha(h),\alpha(z)]\Big) &=\circlearrowleft_{x,y,z}\varepsilon(z,x)\Big([\alpha(h)\cdot[x,y],\alpha^2(z)]+\varepsilon(x+y,z)[\alpha^2(h),\alpha(z)\cdot[x,y].]\Big).
H_{\mathrm{e f f}}= \sum_{p}\epsilon_{p}a_{p}^{\dagger}a_{p}+ \frac{3 \lambda}2 \frac 1 V \sum_{p_{1}+ p_{2}= p_{3}+ p_{4}}a_{p_{1}}^{\dagger}a_{p_{2}}^{\dagger}a_{p_{3}}a_{p_{4}},
l^2-|k|^2 = \tfrac12(1+r^2)^2 - (2Q_0w+\tfrac1{\sqrt{2}}\bar w^2)(2Q_0\bar w+\tfrac1{\sqrt{2}}w^2) & = r^2(1 -4Q_0^2) +\tfrac12 - Q_0(\sqrt{2}r)^3\cos(3\alpha) ,
\int_{0}^{1}\, d \rho \frac{1}{[ ( \rho x - y )^{2}]^{\frac{N}{2}- 1}}= \int_{0}^{\infty}d \rho \, \frac{1}{[ ( \rho x - y )^{2}]^{\frac{N}{2}- 1}}- \int_{0}^{1}d \rho \, \frac{\rho^{N - 4}}{[ ( x - \rho y )^{2}]^{\frac{N}{2}- 1}}
\zeta_R(-1) = 2^{-1}\pi^{-2}\sin\left(\frac{-\pi}{2}\right)\Gamma(1-(-1))\zeta(1-(-1)) &=2^{-1}\pi^{-2}(-1)\Gamma(2)\zeta(2) &= 2^{-1}\pi^{-2}(-1)\cdot 1\cdot \frac{\pi^2}{6}= -\frac{1}{12}\quad,
F(x_1,x_2;y_1,y_2) = \begin{pmatrix}x_1^2 x_1 x_2 x_2^2 \end{pmatrix}\begin{pmatrix}a_{11}a_{12}a_{13}a_{21}a_{22}a_{23}a_{31}a_{32}a_{33}\end{pmatrix}\begin{pmatrix}y_1^2 y_1y_2 y_2^2 \end{pmatrix}.
\left({{1{\rm{+}}\frac{{r}c}{{{{\left({{r}+ c}\right)}^2}}}}{\rm{+}}{\rho_t}\frac{{r}/c}{{{r}+ c}}}\right) - \frac{{\rho_t}+ c}{c}= \frac{{\left({c -{\rho_t}}\right){r}-{\rho_tc}}}{{{{\left({{r}+ c}\right)}^2}}}<0, \rho_t>{{r}c}/({r}+c).
S = T \int d^{3}x \Phi [ - \frac{1}{2}\gamma^{a b}L_{a}^{\mu}L_{b \mu}- \frac{\varepsilon^{a b c}}{\sqrt{- \gamma}}( L_{a}^{\mu}L_{b}^{\nu}L_{c \mu \nu}+ \frac{9}{1 0}L_{a}^{\mu}L_{b}^{\alpha}L_{c \mu \alpha}- \frac{1}{5}L_{a}^{\alpha}L_{b}^{\beta}L_{c \alpha \beta}) ]
\begin{aligned}x_0(t_0) \in E_0, t \in [t_0,t_0+\lambda], \:\: x_0(t)\leq \tilde{z}_1, &x_1(t_0) \in E_1, t \in [t_0,t_0+\lambda], \: \: x_1(t)\geq \tilde{z}_1. \end{aligned}
\mathcal{N}=\sum_{\ell\ge 1}\sum_{i\ge 1}\alpha_{i,\ell}\,\vert\vert x_{i,\ell+1}^{*}\vert\vert &\le w_1\vert \lambda_2-\lambda_{j(2)}\vert +\sum_{\ell=2}^{+\infty}w_\ell\frac{\vert \lambda_{\ell+1}-\lambda_{j(\ell+1)}\vert}{\vert \lambda_\ell-\lambda_{j(\ell)}\vert}\cdot
\beta^{\star}= \frac{C_0^{U^{\star}}C_1^{S^{\star}}+ C_0^{U^{\star}}C_1^{U^{\star}}+ C_1^{S^{\star}}\sigma_v^2/A^2}{C_0^{U^{\star}}C_1^{S^{\star}}+ C_1^{U^{\star}}- C_0^{S^{\star}}C_1^{U^{\star}}}
E\left[\exp\left(-\eta_{i_{j'}}\min(W_{j}, t-T_{j'})\right)\right] = E\left[\left.E\left[-\eta_{i_{j'}}\min(W_{j}, t-T_{j'})\right| W_{j}\right]\right] = \frac{1}{m(t)}\left(m(t) - \eta_{i_{j'}}\int_{0}^{t}\exp(-\eta_{i_{j'}}w)m(t-w)\bar{G}_{W}(w)dw\right)
Z [ A_{+}^{\prime}] = \int \! D \bar{\psi}^{\prime}D \psi^{\prime}e^{- \int \! d^{2}x [ \psi_{-}^{\prime \dag}i \partial_{+}\psi_{-}^{\prime}+ \psi_{+}^{\prime \dag}i \partial_{-}\psi_{+}^{\prime}- A_{+}^{\prime}j_{-}^{\prime}]}\quad .
\left( \begin{array}{cccc}\hat{p}_0-m 0 \hat{p}_3 \hat{p}_1-i\hat{p}_2 0 \hat{p}_0-m \hat{p}_1+i \hat{p}_2 -\hat{p}_3 -\hat{p}_3 -\hat{p}_1+i \hat{p}_2 -\hat{p}_0-m 0 -\hat{p}_1-i \hat{p}_2 \hat{p}_3 0 -\hat{p}_0-m\end{array}\right) \left( \begin{array}{cccc}\Psi_1 \Psi_2 \Psi_3 \Psi_4 \end{array}\right)_D=0 \quad ,
I_1 = \frac{1}{2\pi i}\int_{1+\frac{1}{\log x}-iT}^{1+\frac{1}{\log x}+iT}\zeta(s) \frac{x^s}{s}ds, I_2,\, I_3 = \frac{1}{2\pi i}\int_{\sigma\pm iT}^{1+\frac{1}{\log x}\pm iT}\zeta(s) \frac{x^s}{s}ds.
\sup_{\substack{\psi\in \Psi_{Q,b}\psi (\cdot)= -b\Vert \cdot \Vert^2_Y + \langle v, \cdot \rangle_Y}}-\left(f+ b\left\Vert L \cdot \right\Vert_{Y}^{2}\right)^{*}_{\Phi_{Q,a}}\left( -\langle L^* v, \cdot \rangle\right)-g^{*}_{\Psi_{Q,b}}\left(\psi\right).
M(\kappa)^{-1}=\sum_{j=-2}^{\beta-6}\kappa^jC_j+\mathcal O(\kappa^{\beta-5}); C_j =B_j+\sum_{\genfrac{}{}{0pt}{}{j_1\ge 0,j_2\ge -1,j_3\ge 0}{j_1+j_2+j_3=j+1}}B_{j_1}A_{j_2}B_{j_3}
{\rm ~e}^{-S_{\rm eff}[A_0]}\equiv \int[d\vec a][d\vec \pi]{\rm ~e}^{\int_0^{1/T}d \tau \int_{S^2}\left\{i\vec\pi\cdot\dot{\vec a}-\frac{1}{2}(\pi_i-\frac{\kappa}{2}\epsilon_{ij}a_j)^2-\frac{1}{2}(B_M^2+b^2) \right\}+ i\frac{\kappa}{T}\int_{S^2}A_0 \left( B_M+b/2 \right)}
|P'| \gg_{h}\frac{|P|}{(\log N)^{\epsilon}}\frac{\tilde{U}^2\tilde{W}^2\tilde{V}}{U^2(\log N)^{4\epsilon}}\frac{(\log N)^{1-\epsilon}}{\tilde{V}\tilde{W}^2(\log N)^{9\epsilon}}= \tilde{U}^2\frac{|P|}{U^2}(\log N)^{1-15\epsilon}.
r_jr_k =(s_j+s_{2^{n-2}-j})r_k=r_{k-j}+r_{k+j}+r_{k-2^{n-2}+j}+r_{k+2^{n-2}-j}= &=r_{k-j}+r_{2^{n-2}+(k-j)}+r_{k+j}+r_{-2^{n-2}+(k+j)}.
\mathbb{E}_{\mathbb{Q}_{s,f}^{v}}\left[ \varphi(G_n \cdot v) \left( \log \frac{\langle f, G_n v \rangle}{\langle f, v \rangle}- n\Lambda'(s) \right) \right] = \left. \frac{d \Pi_{s,f,z}}{dz}\right|_{z=0}\varphi(v) + \left. \frac{d N^{n}_{s,f,z}}{dz}\right|_{z=0}\varphi(v).
\frac{\prod_{b \in B}\xi( \frac{1}{2}+ |b-[f(b)]-1/2| \pm z)}{\xi(\frac{n_B+n_C}{2}+z) \dots \xi(\frac{n_C-n_B}{2}+1 + z )}=\frac{\prod_{b \in B}\xi( \frac{1}{2}+ |b-f(b)| \pm z)}{\xi(\frac{n_B+n_C}{2}+z) \dots \xi(\frac{n_C-n_B}{2}+1 + z )}
\sum_{v=0}^{\ell^m-1}f(z)\mid_{-1}\alpha_0\sigma_{w_v,\ell^m}=\sum_{v=0}^{\ell^m-1}\ell^{\frac{m}{2}}\sum_{n=n_0}^{\infty}a_0(n)e^{\frac{2\pi in(z+w_v)}{24\ell^m}}&=\ell^{\frac{m}{2}}\sum_{n=n_0}^{\infty}a_0(n)q_{24\ell^m}^n\sum_{v=0}^{\ell^m-1}e^{\frac{2\pi inw_v}{24\ell^m}}.
\phi_{j}^{(k)}(z):=\langle(zI-H^{(k)})^{-1}e_{j},e_{0}\rangle=\sum_{n=0}^{\infty}\frac{\langle (H^{(k)})^{n}e_{j},e_{0}\rangle}{z^{n+1}},0\leq j\leq p-1.
rA = r(d\ast A^{(N)}+ a_0) = d\ast rA^{(N)}+ ra_0 & \subseteq d\ast (X'+ A^{(N)}) + ra_0 = d\ast X' + d\ast A^{(N)}+ ra_0 & = d\ast X' + (r-1)a_0 + d\ast A^{(N)}+ a_0 = X + A.
{\cal L}_{P}= - \frac 1 2 G_{\mu \nu}^{\dagger}G^{\mu \nu}- M^{2}\phi_{\mu}^{\dagger}\phi^{\mu}+ \frac{\theta}{2}\varepsilon_{\mu \nu \rho}A^{\mu}\partial^{\nu}A^{\rho}+ \frac{\lambda}{2}( \partial_{\mu}A^{\mu})^{2}\; ,
w''_{R}= \frac{w_{R}}{|w_{R}|}|w_{R}'|^{2}- \frac{m_{1}w_{R}}{2|w_{R}|^{2}}+ \frac{\overline{w}_{R}}{2}|w_{R}|^{2}\nabla U_{1}( c_{1}+w_{R}^{2})
\sum_{\substack{k=0 n - k \equiv 0 \,(\mathrm{mod}\,2)}}^n \binom{n}{k}\big ( 2^k L_{k}- 2 \big ) (\sqrt{5})^{n-k}\frac{B_{n-k+2}}{n-k+2}= \frac{2^{n+2}L_{n+2}-2}{{5}(n+1)(n+2)}- 1\,,
\inf_{x \in \mathcal{X}}\ \left\{\sup_{\mathcal{S}\subseteq [m]: |\mathcal{S}| \leq \Gamma}\left\{\sum_{i \in \mathcal{S}}\max\{0,x_i - \overline{b}_i + \Delta b_i\}+ \sum_{i \in [m] \setminus \mathcal{S}}\max\{0, x_i - \overline{b}_i\}\right\}\right\}.
\ddot{h}= \left( \frac{\left( 2 + r \right) r}{\left( 1 + r \right)^{2}}\right)^{2}h^{\prime \prime}+ \left( \frac{2}{\left( 1 + r \right)^{3}}+ \frac{2 + r}{\left( 1 + r \right)^{2}}\right) h^{\prime}- \kappa^{2}f^{2}\left( 1 + r^{2}h \right) ,
\begin{bmatrix}1 x_1 x_1^2 \cdots x_1^{d-1}1 x_2 x_2^2 \cdots x_2^{d-1}\vdots \ddots \ddots \ddots \vdots 1 x_k x_k^2 \cdots x_k^{d-1}\end{bmatrix}\cdot\begin{bmatrix}a_{0,t}a_{1,t}\vdots a_{d-1,t}\end{bmatrix}=\begin{bmatrix}s_{t,1}-x_1^d s_{t,2}-x_2^d \vdots s_{t,k}-x_k^d\end{bmatrix}
\begin{cases}\Delta u_1+\lambda e^{u_2}(1-e^{u_1})=\displaystyle 4\pi\sum_{i=1}^{N_1}\delta_{p_{1i}}, \Delta u_2+\lambda e^{u_1}(1-e^{u_2})=\displaystyle 4\pi\sum_{i=1}^{N_2}\delta_{p_{2i}},\end{cases}\mathbb R^2,
L_{\phi}^{*}\big( e_{j}^{(1)}- f_{j}^{(1)}\big) = \frac{1}{2 \pi i}\int_{\gamma}( z - P^{(0)})^{- 1}z^{- 1}L_{\phi}^{*}L_{\phi}L_{\phi}^{*}f_{j}^{(1)}d z &= L_{\phi}^{*}\frac{1}{2 \pi i}\int_{\gamma}(z - P^{(1)})^{- 1}z^{-1}L_{\phi}L_{\phi}^{*}f_{j}^{(1)}d z .
\begin{array}{l}{{2 m \le \sum_{i}\deg_{i}( A ) = \sum_{\mu}| \alpha_{\mu}| + \sum_{\nu}( 2 + | \beta_{\nu}| ) = n + 2 v}}\\{{\le n + \sum_{\nu}| \beta_{\nu}| = 2 n - \sum_{\mu}| \alpha_{\mu}| \le 2 n - u \le 2 n .}}\\ \end{array}
\mbox{exp}\circ \rho_\lambda(u)=\mbox{exp}\left(\begin{array}{cc}\overline{a}\lambda \overline{a}+ \overline{b}0 0\end{array}\right)=\left(\begin{array}{cc}e^{\overline{a}}\mu(\overline{a})(\lambda \overline{a}+ \overline{b}) 0 1\end{array}\right).
\hat{B}^{-1}=\left(\begin{array}{cc}\hat{B}^{-1}_{11}\hat{B}^{-1}_{12}\hat{B}^{-1}_{21}\hat{B}^{-1}_{22}\end{array}\right)=\left(\begin{array}{cc}V^{*}_{1}B^{-1}V_{1}V^{*}_{1}B^{-1}V_{2}V^{*}_{2}B^{-1}V_{1}V^{*}_{2}B^{-1}V_{2}\end{array}\right).
\begin{aligned}v^m_{rr}(r,t) =-\frac{C^m m}{a(m-1)}\eta(t)\zeta^m(t) G^{\frac 1{m-1}}+ \frac{C^m m}{a^2(m-1)^2}\zeta^m(t) \eta^2(t) r^2 G^{\frac1{m-1}-1}&\geq -\frac{C^m m}{a(m-1)}\eta(t)\zeta^m(t) G^{\frac 1{m-1}}\,.\end{aligned}
x_1x_2x_3 x_1(1+x_2)(1+x_3) = x_{1}+ x_{1}x_{2}+ x_{1}x_{3}+ x_{1}x_{2}x_{3}x_2(1+x_1)(1+x_3) = x_{2}+ x_{1}x_{2}+ x_{2}x_{3}+ x_{1}x_{2}x_{3}x_3(1+x_1)(1+x_2) = x_{3}+ x_{1}x_{3}+ x_{2}x_{3}+ x_{1}x_{2}x_{3}.
h >{\log \Bigl({2\delta_T\over T+ 1}\Bigr) \over \log \Bigl({\gamma\over 2+\gamma}\Bigr)}={\log \Bigl( ({2+\gamma\over \gamma}){T- T_\gamma \over T+ 1}\Bigr) \over \log \Bigl({\gamma\over 2+\gamma}\Bigr)}\,.
I \lesssim \langle \mu \rangle_\alpha^{\frac 2q}R^{2\beta}\sum_{j=1}^{\log R^{1/2}}2^{\gamma j}\sum_{(k,k^\prime):\theta^j_k\approx \theta^j_{k^\prime}}\| f_k^j\|_2 \| f_{k'}^j \|_2.
\int |V_\phi f(x,\xi)| \, d \xi \geq (2\pi)^{-n/2}\left| \int e^{i\langle x,\eta\rangle}\hat{f}(\eta) \int \overline{\hat{\phi}(\xi-\eta)}\, d \xi \, d \eta \right| = C_\phi |f(x)|,
0= v_q\left(\frac{\varphi(km^a)}{\varphi(ln^b)}\right)=\begin{cases}av_q(m)+v_q(k)-bv_q(n)-v_q(l), \mbox{if}\, \, v_q(km)\ne0\,\, \mbox{and}\,\, v_q(ln)\ne0, av_q(m)+v_q(k)-1, \mbox{if}\, \, v_q(ln)=0, -bv_q(n)-v_q(l)+1, \mbox{if}\,\, v_q(km)=0. \end{cases}
\int_S \operatorname{tr}(\nabla u^T(\nabla \widetilde u -\nabla u))\,\mathrm dx = -\int_S \left\langle \Delta u,\widetilde u - u\right\rangle\,\mathrm dx +\int_{\partial S}\operatorname{tr}(\nabla u^T(\widetilde u-u)\otimes \upsilon)\,\mathrm dS
q_{3,1}'(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}}(-1)^r\det M_0(\{r+s\}',\{2r+s+1\}').
S ={\frac{1}{4 \pi}}\int_{\cal M}d^{2}\! \sigma \left\{{\frac{1}{\alpha^{\prime}}}\partial X^{\mu}\bar{\partial}X_{\mu}+ \psi^{\mu}\bar{\partial}\psi_{\mu}+ \tilde{\psi}^{\mu}\partial \tilde{\psi}_{\mu}\right\}
\begin{array}{lll}(w' \cdot f_1)(w' \chi) = \pi (w', \omega \otimes \chi) f_1 (\chi) \pi (w',Q,\sigma,\chi)^{-1}f_1 \in B_1 , (w' \cdot f_2)(w' \chi) = \pi (w', Q,\sigma,\chi) f_2 (\chi) \pi (w',\omega \otimes \chi)^{-1}f_2 \in B_2 . \end{array}
H(\begin{matrix}t\\ \vdots\\t\end{matrix})=(\begin{matrix}0\\ \vdots\\ 0\end{matrix})andH(\begin{matrix}s\times R_{1}^{0}\\ \vdots\\s\times R_{N}^{0}\end{matrix})=(\begin{matrix}0\\ \vdots\\ 0\end{matrix})
k_{-4\lambda}\lhd E_r =q^{-(\rho-w\rho,\lambda)}U(\bar h_\lambda\otimes h_\lambda,v_\lambda)\lhd E_r &=q^{-(\rho-w\rho,\lambda)}U(F_r(\bar h_\lambda\otimes h_\lambda),v_\lambda) &=q^{-(\rho-w\rho,\lambda)+(\alpha_r,\lambda)/2}U(\bar h_\lambda\otimes F_r h_\lambda,v_\lambda).
\Xi_n^w\circ j^*(b') = h\circ\Xi_n^{\preccurlyeq w}(b') = \xi^{\mu\lambda}_{1,\beta_w^{-1}}h\circ \Xi_n\circ i_*(b') = \xi^{\mu\lambda}_{(w,\beta_w^{-1}\beta_w P)}= \xi^{\mu\lambda}_{(w, P)}.
C^2 = d^2 - \sum_i m_{p_i}^2 \geq \frac{3}{2}d + \sum_{p \in{\rm Sing}(\bar{C})}\bigg(1 + \frac{1}{2m_{p_{i}}}\bigg)\mu_{p_{i}}- \sum_i m_{p_i}^2 \geq \frac{3}{2}d + \bigg(1 + \frac{1}{2m_{p_{1}}}\bigg)\mu_{p_{1}}- \sum_i m_{p_i}^2.
\mathcal{H}(f)(z)=\sum_{k=0}^{+\infty}\left(\sum_{i_2,i_3,\cdots,i_m=0}^{+\infty}\frac{a_{i_2}a_{i_3}\cdots a_{i_m}}{k+i_2+i_3+\cdots+i_m+1}\right)z^k
1+\left(\begin{array}{c}n 2 \end{array}\right)+\left(\begin{array}{c}n 4 \end{array}\right)+\ldots=2^{n-1}\,\,\,\,\,,\,\,\,\,\,\left(\begin{array}{c}n 1 \end{array}\right)+\left(\begin{array}{c}n 3 \end{array}\right)+\left(\begin{array}{c}n 5 \end{array}\right)+\ldots=2^{n-1}\,\,\,\,\,,\,\,\,\,\,
\phi\big(M(D_i,D_j)\big) =\prod_{k=1}^K\sum_{\pi\in NC_{e,o}(2t)}\kappa_{\pi}[a_{k;i}^{n_1},a_{k;j}^{n_2},\cdots,a_{k;i}^{n_{2t-1}},a_{k;j}^{n_{2t}}].
S_{L i o u v}[ \varphi ] ={\frac{1}{8 \pi}}\int_{D}d x^{2}\left[{\frac{1}{2}}( \partial_{a}\varphi )^{2}+ 8 \pi \mu b^{2}e^{\varphi}\right] + \phi_{\infty}+ 2 \log{R}- \sum_{i}\left( \eta_{i}\varphi_{i}+ 2 \eta_{i}^{2}\log{\epsilon_{i}}\right) .
\Gamma_{\Lambda}[ \psi , \bar{\psi}] = \int \frac{d q}{( 2 \pi )^{4}}\bar{\psi}_{q}\hat{q}\psi_{- q}+ \frac{1}{2}\int \prod_{i = 4}^{4}\frac{d q_{i}}{( 2 \pi )^{4}}\gamma_{\Lambda}( q_{i}) ( 2 \pi )^{4}\delta ( \sum q_{i}) \mu ( q_{i}) ( \bar{\psi}\psi ) \; ( \bar{\psi}\psi ) ,
\int_{D}|\psi(x)|^4\,dx =\int_{\Omega}|\psi(x)|^4\,dx-\int_{\overline{D}^c}|\psi(x)|^4\,dx &\geq\int_{\Omega}|\psi(x)|^4\,dx-\frac2{\kappa^2}C_0(\kappa,H;\overline{D}^c)+\frac{\kappa}{H}\,o(1)\,.
h^{-m}\mathrm{ch}(\mathrm{mC}(\Omega_{m+1}))^2 =h^{-m}(1-\rho)^2((1+h)-h\rho)^{2m}&=\sum_{l=0}^m\rho^l(-1)^l\left(\binom{2m}{l}h^{l}(1+h)^{m-l}+2\binom{2m}{l-1}h^{l-1}(1+h)^{m-l+1}\right. &\quad\qquad\qquad\qquad\left.+\binom{2m}{l-2}h^{l-2}(1+h)^{m-l+2}\right).
\tau_{k-1}(p^{\alpha+1}) \tau_{\ell-1}(p^{\alpha+1})= \tau_k(p^{\alpha+1}) \tau_{\ell}(p^{\alpha+1})- \tau_k(p^{\alpha}) \tau_{\ell}(p^{\alpha}) -\tau_k(p^{\alpha}) \tau_{\ell-1}(p^{\alpha+1})-\tau_{\ell}(p^{\alpha}) \tau_{k-1}(p^{\alpha+1}).
dx_{i}^{N}(t) =d\beta_{i}(t)+d\tilde{A}_{i}^{N}(t)+b_{c_{1}}(t,\,x_{i}^{N}(t)) dJ_{ij}^{N}(t) =dM{}_{ij}^{N}(t)+\left(\lambda N+\gamma_{c_{1},c_{2}}(t,\,x_{i}^{N}(t))\right)dA_{ij}^{N}(t)
\Lambda_{\delta_1, \ldots, \delta_{\ell}}:= \{\Lambda \in \{0, \ldots, p-1\}^{\delta_1 + \cdots + \delta_{\ell}}~ |~ (\lambda_{j,1}, \ldots, \lambda_{j, \delta_j}) \neq (0, \ldots, 0){\rm~for~}1 \leq j \leq \ell \}.
\frac{dr}{d\theta^5}=\tan\phi=\frac{\sin\theta^5-f\cos\theta^5}{\cos\theta^5+f\sin\theta^5},\quad\sin\phi=\frac{\sin\theta^5-f\cos\theta^5}{\sqrt{1+f^2}},\quad\cos\phi=\frac{\cos\theta^5+f\sin\theta^5}{\sqrt{1+f^2}}.
(\begin{matrix}P_{x}\\P_{y}\\P_{z}\end{matrix})=\epsilon_{0}(\begin{matrix}\chi_{xx}&\chi_{xy}&\chi_{xz}\\ \chi_{yx}&\chi_{yy}&\chi_{yz}\\ \chi_{zx}&\chi_{zy}&\chi_{zz}\end{matrix})(\begin{matrix}E_{x}\\E_{y}\\E_{z}\end{matrix})
(\Phi(x.u), \Phi(v))_{\lambda_1}=(\omega(x).\Phi(u), \Phi(v))_{\lambda_1}&=(\Phi(u), \varphi(\omega(x)).\Phi(v))_{\lambda_1}&=(\Phi(u), \omega(\varphi(x)).\Phi(v))_{\lambda_1}=(\Phi(u), \Phi(\varphi (x).v))_{\lambda_1}.
D'_{mk}=\begin{bmatrix}0 1 2 \ldots k-1 0 1 2 \ldots k-1 \hdotsfor{5}0 1 2 \ldots k-1\end{bmatrix},C''_{kk}=\begin{bmatrix}0 0 1 2 \ldots k-2 0 0 0 1 \ldots k-3 \hdotsfor{6}0 0 0 0 \ldots 1 0 0 0 0 \ldots 0 0 0 0 0 \ldots 0\end{bmatrix}
\alpha_* \clubsuit_i([u_1]) = \alpha_*( [Y^{(i)}B(a_1) Y^{(i)*}]) = [Y^{(i)}\alpha(B(a_1) ) Y^{(i)*}] = [Y^{(i)}B(a_2 ) Y^{(i)*}] = \clubsuit_i([u_2]) = \clubsuit_i \alpha_*([u_1]) \; .
\frac{E'\otimes W'-\left(\overline{E'\otimes W'}\right)^{t}}{2\sqrt{-1}}= \left(V' \otimes Z'\right)\left(\overline{V' \otimes Z'}\right)^{t},\mbox{for an invertible matrix $V'\in\mathcal{M}_{qN\times qN}\left(\mathbb{C}\right)$.}
S = S_{{\cal M}_{6}}+ S_{D 1 1}({\hat{F}}) -{\frac{1}{6}}\int A^{( 3 )}\wedge d A^{( 3 )}\wedge d A^{( 3 )}-{\frac{1}{2}}\int H^{( 3 )}\wedge{\hat{F}}^{( 4 )}\wedge{}^{*}G^{( 7 )},
\phi_1((m,n), (z,w)) = \delta_{(1,0)}(m,n) \phi_2((m,n), (z,w)) = \delta_{(0,1)}(m,n) \psi_1((m,n), (z,w)) = z \delta_{(0,0)}(m,n) \psi_2((m,n),(z,w)) = w \delta_{(0,0)}(m,n)
-\int_{0}^{T}(u,\partial_{t}\varphi )dt-\int_{0}^{T}\sum_{i=1}^{2}(u^{i}u,\partial_{i}\varphi)dt+r\int_{0}^{T}(u,\varphi )dt=\int_{0}^{T}(g,\varphi )dt.
\int_\Omega |\nabla u_\lambda|^p\,dx \lesssim \int_\Omega\phi_\lambda(x,|\nabla u_\lambda|)+1\,dx\le\int_\Omega\phi_\lambda(x,|\nabla f|)+1\,dx\lesssim \int_\Omega\phi(x,|\nabla f|)+1\,dx<\infty,
I_{2 2}^{b}( y , a , b , d ) = - \frac{1}{a b^{d - 3}}\sum_{k = 0}^{k < \frac{d - 4}{2}}C^{( 2 )}( d , k ) ( \frac{b}{a})^{2 k}\sum_{n = 1}^{\infty}n^{2 k}\int_{0}^{\frac{n \pi b}{a}}d u \, u^{d - 4 - 2 k}( \coth u - 1 ) \cosh ( \frac{2 u y}{b}) .
\left|\frac{h}{N(\beta_1)}T_0^{(n)}(\beta_1)-\frac{h}{N(\beta_2)}T_0^{(n)}(\beta_2)\right| &\le \sum_{a=1}^2\frac{h}{N(\beta_a)}|S_0^{(n)}(\beta_a)|+\left|\frac{h}{N(\beta_1)}U_0^{(n)}(\beta_1)-\frac{h}{N(\beta_2)}U_0^{(n)}(\beta_2)\right|
\left( \frac{2H_{0}}{f_{2}}\right) \frac{d}{df_{1}}\left( \frac{1}{p_{1}}\right) =\left( 1+a^{2}\right) \frac{d}{df_{1}}\left( \frac{f_{1}}{p_{1}}\right) \left( \frac{p_{2}\dot{p}_{2}}{f_{2}}\right) +\ddot{p}_{1}.
I:= \int_{{\mathbb{R}}^n}\Bigl(\int_a^b\int_{B(x,\sqrt{b})}|u(t,y)|^2\,{\rm d}y\,{\rm d}t\Bigr)^{\frac{1}{2}}w(x)\,{\rm d}x &\lesssim \Bigl(\int_{{\mathbb{R}}^n}\Bigl(\int_a^b\int_{B(x,\sqrt{b})}|u(t,y)|^2\,{\rm d}y\,{\rm d}t\Bigr)^{\frac{p}{2}}\,{\rm d}x\Bigr)^{\frac{1}{p}}.
\begin{aligned}\displaystyle\int_M \sigma_k(\kappa) \mathrm{d}\mu_g= \int_{\mathbb{S}^n}\sum\limits_{m=0}^k \left[\left(A_0^m+A_1^mu+A_2^mu^2+A^m|\nabla u|^2\right)\sigma_m(D^2u)+B^mu^iu_j[T_m]_i^j(D^2 u)\right]\mathrm{d}A &+O(\varepsilon)\|u\|_{L^2(\mathbb{S}^n)}^2+O(\varepsilon)\|\nabla u\|_{L^2(\mathbb{S}^n)}^2,\end{aligned}
\tau_u=\begin{cases}\min\{t\in\mathbb{N}: W(t)<0\}, & \infty, \mbox{\if\}\W(t)\geqslant 0 \ \mbox{\for all\}\t\in\mathbb{N}, \end{cases}&\psi(u)=\mathbb{P}\left(\tau_{u}<\infty\right)=\mathbb{P}\left(\sup_{k\geqslant1}\sum_{i=1}^k(X_i-c)>u\right),\,u\in\mathbb{N}_0.
{{}^{3}R}_{2323}=\frac{-\omega^{2}}{2}\frac{C^{\prime}(\frac{q^{2}}{\omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u)^{2}}{C(\frac{q^{2}}{\omega^{2}},\frac{q^{2}}{2\omega^{2}},\omega u)^{2}}
\frac{\delta}{\delta U}\mathrm{T r}( \log{\bf O}- \log{\bf O}+ ) = \mathrm{T r}\left( \left\{\frac{\delta{\bf O}}{\delta U}{\bf O}^{+}-{\bf O}\frac{\delta{\bf O}^{+}}{\delta U}\right\}\frac{1}{{\bf O}{\bf O}^{+}}\right) .
[\mspace{1.5mu}u\mspace{1.5mu}]^2_{\frac{1}{2}}\!:=\!\int_{\mathbb{R}^6}\mspace{-7.5mu}d\vec{x}d\vec{y}\;\frac{\lvert u(\vec{x})\--u(\vec{y})\rvert^2\!\!}{\;\lvert\vec{x}-\vec{y}\rvert^4},u\in H^{1/2}(\mathbb{R}^3),
\mathrm{Ker}\, Q_0 =\mathbb{Z}/2[x_2^2, x_{41}, \dots, x_{4n}] \oplus \mathbb{Z}/2[x_2^2, x_3^2, x_{41}, \dots, x_{4n}] \{x_3, x_3^2 \}, \mathrm{Im}\, Q_0 = \mathbb{Z}/2[x_2^2, x_3^2, x_{41}, \dots, x_{4n}] \{x_3, x_3^2 \}
\mathbf{E}Z^*_{\nu_m}-\mathbf{E}Z^*_n \le2G_n \mathbf{P}(T_g>\nu_m)\le 10G_n\frac{\mathbf{E}Z^*_m}{B_m}\le \frac{10}{8}\mathbf{E}Z^*_m \le \frac{40}{8}\mathbf{E}Z^*_n=5\mathbf{E}Z^*_n,
{\displaystyle \int \limits_{B}}|b(y)-b_{B}|^{p}w(y)dy =p{\displaystyle \int \limits_{0}^{\infty}}\beta^{p-1}w(\{x\in B\,:\,|b(x)-b_{B}|>\beta \})d\beta & \leq C\,w(B){\displaystyle \int \limits_{0}^{\infty}}\, \beta^{p-1}e^{-C_{2}\beta \delta/\Vert b\Vert_{\ast}}\,d\beta & =Cw(B)\Vert b\Vert_{\ast}^{p}.
F [ f ] ( x ) - c_{1}^{\prime}\epsilon ( x^{0}) \delta ( ( x^{0})^{2}- |{\bf x}|^{2}) = \sum_{k = 0}^{N}a_{k}\left( \sum_{\nu = 0}^{3}\eta^{\nu \nu}\left( \frac{\partial}{\partial x^{\nu}}\right)^{2}\right)^{k}\delta ( x ) .
\ln \left({\frac{\det [ -{\cal M}_{\mathrm{F P}}]}{\det [ -{\cal M}_{\mathrm{F P}}^{0}]}}\right)_{\mathrm{r e g}}\approx - \lim_{s \to 0}\left\{{\frac{d}{d s}}\left({\frac{M^{2 s}}{\Gamma ( s )}}\int_{0}^{t_{0}}d t t^{s - 1}\left[ \alpha_{{\cal M}_{\mathrm{F P}}}t^{- 1 / 2}- \beta_{{\cal M}_{\mathrm{F P}}}t^{1 / 2}\right] \right) \right\}\; ,
u_1(0,x_0) =\int_0^{T}\Big[L(\gamma(s),\dot \gamma(s))+ F(\gamma(s), m_1(s))\Big]\ds +G(\gamma(T),m_1(T)), u_2(0,x_0) \leq \int_0^{T}\Big[L(\gamma(s),\dot \gamma(s))+ F(\gamma(s), m_2(s))\Big]\ds+G(\gamma(T),m_2(T)).
|E^j|^2 - |E^{j-1}|^2 + |E^j-E^{j-1}|^2 = 2h \big( f(U^j), E^j \big) + 2 \big( g(U^{j-1})\Delta_h W^{j}, E^j - E^{j-1}\big) -2 \big( V^j - V^{j-1}, E^j \big) + Z^j,
\frac{d^{2}y}{dx^{2}}= \frac{e^{C(\tilde{u})}}{B(\tilde{u})}\left[ A(\tilde{u})\left( \frac{d\tilde{u}}{dx}\right)^{2}+ B(\tilde{u})\frac{d^{2}\tilde{u}}{dx^{2}}\right].
L(r^k,\vec s^k) \le r^k \|\vec x^*\|_\infty-(\vec x^*,\vec s^k) =r^k \|\frac{\vec y}{\|\vec y\|_p}\|_\infty-(\frac{\vec y}{\|\vec y\|_p},\vec s^k) &=\frac{1}{\|\vec y\|_p}(r^k-\|\vec s^k\|_1)<0.
{\cal Z}_{B}= T r e^{- \beta \varepsilon_{\kappa}( \overline{{\Phi}}_{1 , \kappa}\Phi_{1 , \kappa}+ \overline{{\Phi}}_{2 , \kappa}\Phi_{2 , \kappa})}e^{- \beta \mu ( N_{1 , \kappa}+ N_{2 , \kappa})},
n\partial_{x_{i,n}}f =n\partial_{x_{i,n}}f'+n\partial_{x_{i,n}}g_{j,m}\left(f_{j,m}-m\partial_{x_{j,m}}f'\right) =f_{i,n}+g_{j,m}\left(n\partial_{x_{i,n}}f_{j,m}-m\partial_{x_{j,m}}n\partial_{x_{i,n}}f'\right) =f_{i,n}+g_{j,m}\left(n\partial_{x_{i,n}}f_{j,m}-m\partial_{x_{j,m}}f_{i,n}\right) =f_{i,n}.
\partial_t c_i =\partial_x (\partial_x c_i+ q_ic_i\partial_x \psi), i=1, 2 - \partial_x^2\psi = q_1c_1+q_2c_2,{\partial_x \psi}=0,{\partial_x c_i}=0,i=1,2,x=0, 1,
\begin{aligned}\int_0^T \int_{B_{2R}^d}\int_{\mathbb{S}^{d-1}}\mathbf{1}_{(\tau,v_{s+k+1},\omega_{k+1}) \in \mathcal{B}^-}d\omega_{k+1}dv_{s+k+1}d\tau \leq & \leq C_{d,s,k}T R^d \left[\alpha +\frac{y}{\eta T}+ \left(\frac{\eta}{R}\right)^d + \theta^{d-1}\right]\end{aligned}
\frac{\partial V_{0}(\bar{z})}{\partial \bar{z}}\bar{f}(\bar{z},e,v,w) \leq -\ell_{1}\bar{z}_{1}^{2}-\ell_{2}\bar{z}_{1}^{4}-\ell_{3}\bar{z}_{2}^{2}+\ell_{4}e^{2}+\ell_{5}e^{4}
E_1 = \det T(\phi) T(\phi_R)^{-1}\cdots T(\phi_1)^{-1}T(\psi_S)^{-1}\cdots T(\psi_1)^{-1}T(\psi_0)^{-1}E_2 = \det T(\tilde{\phi}) T(\tilde{\phi}_R)^{-1}\cdots T(\tilde{\phi}_1)^{-1}T(\tilde{\psi}_S)^{-1}\cdots T(\tilde{\psi}_1)^{-1}T(\tilde{\psi}_0)^{-1}E_3 = \prod_{j=0}^S\det T(\psi_j)T(\psi_j^{-1}).
{\rm APEP}\overset{\mu \gg 1}{\approx}\frac{2^{N - 1}\Gamma\left(N + \frac{1}{2}\right)}{\sqrt{\pi}\Gamma\left(N + 1\right)}\left[\prod_{n=1}^{N}c_\ell\right] \left(\frac{\mu}{4}\right)^{-N}
\left( k_{3}- k_{2}\right)_{\lambda}\Gamma_{\lambda \mu \nu}^{A V V}= 2 \varepsilon_{\mu \nu \alpha \beta}\left[ ( k_{1}- k_{3})_{\beta}( k_{1}+ k_{3})_{\xi}+ ( k_{2}- k_{1})_{\beta}( k_{1}+ k_{2})_{\xi}\right] \, t r i a n g l e_{\xi \alpha},
\Omega_{j,x}^2 := \{\eta \in \Omega_x:\; \eta(x)=0,\eta(z_{1j})=1,\eta(x+r)=1\}; & \Omega_{j,x}^3 := \{\eta \in \Omega_x:\; \eta(x)=0,\eta(z_{1j})=0,\eta(x+r)=1\}; \Omega_{j,x}^4 := \{\eta \in \Omega_x:\; \eta(x)=1,\eta(z_{1j})=1,\eta(x+r)=0\}.