harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
2^{\gamma_*k_*t}\cdot 2^{\hat m_t}=n, 2^{\gamma_*k_*t}\cdot 2^{\overline{m}_t}=n^{q/2}, 2^{\tilde m_t s_*}= 2^{(\mu_* + \alpha_*+\gamma_*/p_0 -\gamma_*/p_1)k_*t}, 2^{m_t(s_*+1/p_0-1/p_1)}= 2^{(\mu_*+\alpha_*) k_*t},
a_2=\sum_{k=2}^{g+1}(A'_k+A''_k)\cdot B_{g,g+1-k}\cdot \tbinom{g}{k-1}\tbinom{g+1}{k}a'_2=\sum_{k=2}^{g}(A'_k+A''_k)\cdot B_{g,g-k}\cdot \tbinom{g-1}{k-1}\tbinom{g}{k},
\lim_{\tau\to\infty}\|w_\tau\|_{L_y^{\frac{d-4/\alpha-2}{d-4/\alpha}}}^{\frac{d-4/\alpha-2}{d-4/\alpha}}=(2\pi)^n((2\pi)^n)^{\frac{2+4/\alpha-d}{d-4/\alpha}}.
\! \! \prod_{a}\beta_{a}^{- \mathrm{i}M_{a}^{z \bar{z}}}f ({\bf X}) = \prod_{a}\beta_{a}^{- \lambda_{a}}f ({\bf X}( \beta ) ) \, , \quad \! \! e^{- q ( \zeta^{2}\bar{\zeta}+ b \zeta )}f ( \zeta ) = ( 1 + q \zeta )^{- b}f \Bigl ( \frac{\zeta}{1 + q \zeta}\Bigr ) \, ,
\big((\tilde h_2,\tilde z_2)=(\hat h,\hat z) (\tilde h_2,\tilde z_2)\in \mathcal N(\hat h,\hat z)\big) \quad\mbox{and}\tilde h_2(\tilde z_2) = \min\{G_{\tilde h_2}(\tilde z_2,\sigma), G_f(\sigma,\hat w_2)\}
\mathcal{F}^k:= \prod_{i=0}^{k}\prod_{(r_{k+1},\dots,r_n)\in\{0,1\}^{n-k}}F_i(x_0,\dots,x_{i-1},x_{i+1},\dots,x_k,x_{k+1}^{r_{k+1}},\dots,x_{n}^{r_{n}}).
\sum\limits_{u=-\infty}^{v_1}\left(\prod\limits_{j=1}^{n_2}\frac{w_j}{z_j}\right)^u=\frac{\prod\limits_{j=1}^{n_2}w_j^{v_1}/z_j^{v_1}}{1-\prod\limits_{j=1}^{n_2}z_j/w_j}.
\det\left(I_N+S\right)^{-\frac{1}{2}}\exp\left(X^{T}S\left(I_N+S\right)^{-1}X\right) =\det\left(I_N+S\right)^{-\frac{1}{2}}\exp\left(-X^{T}\left(I_N+S\right)^{-1}X+X^TX\right),
f(w) =C^\frac1\alpha{\rm exp}\big[ \frac{\alpha'}{\alpha}\frac{1}{\sqrt{e^{2s}-1}\sqrt{1-e^{-2s}}}\langle w, A^*z - A(\Lambda z)- c \rangle &- \frac{\alpha'}{\alpha}\frac{e^{-s}}{\sqrt{e^{2s}-1}\sqrt{1-e^{-2s}}}\langle A(\Lambda z) + c, A^*z - A(\Lambda z) - c \rangle\big]
\rho_\mathfrak{a}^k(0, w) = (*) \frac{L^k_\mathfrak{a}(4w-2)}{L^k_\mathfrak{a}(4w-1)}\frac{\Gamma(2w-1)}{\Gamma(w + \frac{k}{2}) \Gamma(w - \frac{k}{2})}\rho_\mathfrak{a}^k(h, w) = (*) \lvert h \rvert^{w-1}\frac{1}{\Gamma(w + \frac{\lvert h \rvert}{h}\frac{k}{2})}D_\mathfrak{a}^k(h,w),
\eta (s^{-1}(T_s-1)R_M(\lambda)A) = \eta \left( \frac{1}{s}\int_0^M e^{-\lambda t}T_{s+t}Adt- \frac{1}{s}\int_0^Me^{-\lambda t}T_tAdt \right) = \frac{1}{s}\int_M^{M+s}e^{-\lambda (u-s)}\eta(T_uA)du+ \frac{1}{s}\int_0^{M+s}(e^{-\lambda (u-s)}-e^{-\lambda u}) \eta (T_uA)du - \frac{1}{s}\int_0^s e^{-\lambda (u-s)}\eta (T_uA)du.
W_{23}W_{23}^*W_{13}^*\bigl(\Lambda(a)\otimes\Lambda(b)\otimes\Lambda(c)\bigr)=W_{23}W_{23}^*\bigl((\Lambda\otimes\Lambda\otimes\Lambda)(\Delta_{13}(c)(a\otimes b\otimes1))\bigr) &=(\Lambda\otimes\Lambda\otimes\Lambda)\bigl(\Delta_{13}(c)(E\otimes1)(a\otimes b\otimes1)\bigr).
p^4=\left[\begin{array}{ccccc}1 1 1 1 2 2 2 2 2 3 3 3 3 3 1 \end{array}\right],p^5=\left[\begin{array}{ccccc}1 1 1 2 2 2 2 2 1 1 3 3 3 3 3 \end{array}\right],p^6=\left[\begin{array}{ccccc}1 1 1 3 3 2 2 2 2 2 3 3 3 1 1 \end{array}\right]
{\partial\Gamma_\Lambda\over\partial\Lambda}=-{1\over2}{\rm tr}\left[{1\over\Delta_{IR}}{\partial\Delta_{IR}\over\partial\Lambda}\cdot\left(1+\Delta_{IR}\cdot{\delta^2\Gamma_\Lambda\over\delta\varphi^c\delta\varphi^c}\right)^{-1}\right]\quad.
[v,a]\gamma_{z,s}= [v,a] \tau^{-1}\alpha_{z,s}\tau &= [va^{-1}, -a^{-1}] \alpha_{z,s}\tau &= \bigl[ va^{-1}+ z ,\, - a^{-1}+ h(z,va^{-1}) + s \bigr] \tau &= \bigl[ (va^{-1}+ z) a \bigl( 1 - h(z,v) - sa \bigr)^{-1},\, a \bigl( 1 - h(z,v) - sa \bigr)^{-1}\bigr] ,
\begin{array}{r l}{\mathrm{( A )}}{\pi^{*}\phi \cdot f ( x , t_{1}, \dots , t_{n}) ,}{\mathrm{~ ( B )}}{\pi^{*}\phi \cdot f ( x , t_{1}, \dots , t_{n}) d t_{1}\wedge \dots \wedge d t_{n},}\end{array}
\tau_{i, j}\geq 0,\textrm{for all}1\leq i < j\leq N, \lambda_n + \sum_{i=n+1}^N{\tau_{n, i}}- \sum_{i=1}^{n - 1}{\tau_{i, n}}\geq 0, \textrm{for all}n = 1, \dots, N.
\psi_{N}^{+}= \frac{1}{[ 2 r ( r + r_{3}) ]^{\frac{1}{2}}}\left( \begin{array}{c}{{r + r_{3}}}\\{{r_{1}+ i r_{2}}}\\ \end{array}\right) = \left( \begin{array}{c}{{\cos \frac{\theta}{2}}}\\{{e^{i \phi}\sin \frac{\theta}{2}}}\\ \end{array}\right)
(v,A^{-1}v) = \frac{1}{1-|c|^2}[\Re(a)\,\,-\Im(a)]\begin{bmatrix}1+\Re(c) -\Im(c) -\Im(c) 1-\Re(c)\end{bmatrix}\begin{bmatrix}\Re(a) -\Im(a)\end{bmatrix}=\frac{|a|^2+\Re(\overline{c}a^2)}{1-|c|^2}.
A\leq\frac{1}{4\pi}\int_0^{2\pi}\int_0^{2\pi}\frac{(x(u)-x(v)^2)}{4\sin^2\left(\frac{f(u)-f(v)}{2}\right)}\frac{\mathrm{d}f(u)}{\mathrm{d}u}\frac{\mathrm{d}f(v)}{\mathrm{d}v}\mathrm{d}u\mathrm{d}v
\operatorname*{Trunc}_{2j-k}\left[ \det\left( 1-2i\alpha~\boldsymbol{\hat{n}\cdot J}\right) \right] =\sum_{m=0}^{\left\lfloor j-k/2\right\rfloor}4^{m}\alpha^{2m}~\left\vert t\left( 2j+2,2j+2-2m\right) \right\vert \ ,
\sum_{c=1}^n g_{cb}^j(\xi_j,0)\frac{\partial \dot{\xi}_j^a}{\partial \xi_j^c}+g_{ac}^j(\xi_j,0)\frac{\partial \dot{\xi}_j^b}{\partial \xi_j^c}=\sum_{r,s=1}^n (h_{rs}(z,0)\frac{\partial \dot{\xi}_j^a}{\partial z_r}\frac{\partial \xi_j^b(z,0)}{\partial z_s}+ h_{rs}(z,0)\frac{\partial \xi_j^a(z,0)}{\partial z_r}\frac{\partial \dot{\xi}_j^b}{\partial z_s})
\tau^{(1)}:=\inf\{n\geq 0: i(X_1^n;Y^n|X_2^n) > \gamma_1\}, \tau^{(1)}_+ :=\inf\{n\geq 0: i(X_1^n;Y^n|X_2^n) >0\}, \bar{\tau}^{(1)}:=\inf\{n\geq 0: i(\bar{X}_1^n;Y^n|X_2^n) > \gamma_1\},
\gamma_{0}^{\pm}u_{\pm}(x) :=u_{\pm}(x), \gamma_{1}^{\pm}u_{\pm}(x) :=N_{\pm}(x)\!\cdot\!\nabla u_{\pm}(x), x\in\Sigma, \gamma_{0}^{\partial}u_{+}(x) :=u_{+}(x), \gamma_{1}^{\partial}u_{+}(x) :=N_{\partial}(x)\!\cdot\!\nabla u_{+}(x), x\in\partial\Omega\,,
\bigl|g_Q(x) - u_Q(x)\bigr| = \left|\int_{\partial\Omega\setminus E_Q}g_Q\,d\omega^x\right|\leq \|g_Q\|_\infty\,\omega^x(\partial\Omega\setminus E_Q)\lesssim \,\omega^x(\partial\Omega\setminus E_Q).
F_{{\sf G I}\, i j}^{a}({\bf r}) = \partial_{j}A_{{\sf G I}\, i}^{a}({\bf r}) - \partial_{i}A_{{\sf G I}\, j}^{a}({\bf r}) - g f^{a b c}A_{{\sf G I}\, i}^{b}({\bf r}) A_{{\sf G I}\, j}^{c}({\bf r})
\partial_tw+w\cdot\nabla w+\widetilde U\cdot\nabla w+w\cdot\nabla\widetilde U+h(u_s\cdot\nabla w+w\cdot\nabla u_s) &\qquad=\Delta w-\nabla p_w-hu_\infty\cdot\nabla w+f, &\mbox{div $w$}=0, &w|_{\partial\Omega}=0, &w\to 0 \quad\mbox{as $|x|\to\infty$}, &w(\cdot,0)=w_0:=v_0-\widetilde U(\cdot,0),
\begin{aligned}\sum\limits_{i=B_{1}+1}^{B_{2}}\sum\limits_{j=1}^{K}d_{i,j}=\sum\limits_{i=B_{1}+1}^{B_{2}}\sum\limits_{j=B_{1}+1}^{B_{2}}d_{i,j}+\sum\limits_{i=B_{1}+1}^{B_{2}}\sum\limits_{j={B_2}+1}^{K}d_{i,j}. \end{aligned}
\begin{aligned}2ikb(k) =(\Psi^+_{11}(0 ;z)-e^{ic_+(0)}) \Psi^-_{21}(0;z)+e^{ic_+(0)}\Psi^-_{21}(0;z) &-\Psi^+_{21}(0;z) (\Psi^-_{11}(0;z)-e^{ic_-(0)})-e^{ic_-(0)}\Psi^+_{21}(0;z).\end{aligned}
\sum_{j=R+1}^S d_{i,j}(x_{j,1}^3+x_{j,2}^3-x_{j,3}^3-x_{j,4}^3) = 0 (1 \le i \le t-l), d_{i,i}(x_{i,1}^3 - x_{i,2}^3) + \sum_{j=R+1}^S d_{i,j}(x_{j,1}^3+x_{j,2}^3-x_{j,3}^3-x_{j,4}^3) = 0 (t-l+1 \le i \le R).
\sigma^{\mathrm{r e g}}( \xi ) = - \,{\frac{| \xi |}{2}}\, - \,{\frac{| \xi |}{\pi}}\sum_{n = 1}^{\infty}{\frac{K_{1}( 2 n \pi \, | \xi | )}{n}}= - \,{\frac{| \xi |}{2}}\, - \,{\frac{| \xi |}{\pi}}G ( 2 \pi \, | \xi | ) \ .
\zeta_{0}( \nu ) = - \nu{\frac{\varrho^{- 2 \nu}}{2 \pi}}\sum_{\omega}\int_{C_{+}}d z{\frac{z \breve{\chi}^{\prime}( \breve{\omega}, z )}{( z^{2}+ \breve{\omega}^{2}( z ) )^{\nu + 1}}}e^{i \epsilon z}.
{\rm P}_{j,j}= 1-\sum^{K}_{i=1}\bar{\rm P}_{j,j+i}= 1-\sum^{K}_{i=1}\sum^{M-j}_{m=i+1}{M-j \choose m}\mathbb{P}_{\rm TX}^m \left(1- \mathbb{P}_{\rm TX}\right)^{M-j-m}\times{m \choose i}\mathbb{P}(E_{i,1}) \mathbb{P}(E_{i,2}).
d s^{2}= - \left( d t + \sum_{i = 1}^{5}c_{i}\rho_{i}^{2}d \phi_{i}\right)^{2}+ \sum_{i = 1}^{5}\left( d \rho_{i}^{2}+ \rho_{i}^{2}d \phi_{i}^{2}\right) ,
\begin{aligned}(I-\psi_a(H^a))e^{-itH}f\sim \Phi_a(x_a/t)\sum_{b<a}\Phi_b^a(x^a/t)\sum_{d\le b}(I-\psi_a(H^a_d))\tilde P^d_{M_d}e^{-itH}f.\end{aligned}
P_{R}^{-}={\frac{g^{2}L}{4 \pi^{2}}}\left[ \sum_{N = - \infty}^{\infty}\! \! \! \!{}^{\stackrel{\prime}{}}\thinspace \thinspace{\frac{1}{N^{2}}}C_{N}^{3}C_{- N}^{3}+ \sum_{n = - \infty}^{\infty}{\frac{1}{( n - D_{0}^{3})^{2}}}\left\{C_{n}^{+}, C_{- n}^{-}\right\}\right] \; .
u = z^{\alpha}x_{I_{1}}^{\xi_{1}}\cdots x_{I_{a}}^{\xi_{a}}y_{\max(J_{1})}^{\nu_{1}}\cdots y_{\max(J_{b})}^{\nu_{b}},\, \, \, \, \, \, \, \, \, \, v = x_{I'_{1}}^{\xi'_{1}}\cdots x_{I'_{a'}}^{\xi'_{a'}}y_{\max(J'_{1})}^{\nu'_{1}}\cdots y_{\max(J'_{b'})}^{\nu'_{b'}}.
(\left. G_{0,1}(\omega_{1})\right\vert_{\Omega},\omega_{1})_{\Omega}=\left\langle \frac{\partial \Delta \omega_{1}}{\partial n},\left[R_{0}\omega_{1}\right]_{\Omega}\right\rangle_{\partial \Omega}-\left(\nabla \Delta \omega_{1},\nabla \left[ R_{0}\omega_{1}\right]_{\Omega}\right)_{\Omega}.
x^+_i(u)\mapsto \frac{e^+_{i,i+1}(uq^i)-e^-_{i,i+1}(uq^i)}{q-q^{-1}}, x^-_i(u) \mapsto \frac{f^+_{i+1,i}(uq^i)-f^-_{i+1,i}(uq^i)}{q^{1-\delta_{in}/2}-q^{-1+\delta_{in}/2}}, \psi_i(u)\mapsto h^-_{i+1}(uq^i)h^-_{i}(uq^i)^{-1}, \varphi_i(u)\mapsto h^+_{i+1}(uq^i)h^+_{i}(uq^i)^{-1},
\frac{(q^2;q^2)_\infty^2[-1/(bq), b^2q^2;q^2]_\infty}{[x, -xbq, -x/(bq);q^2]_\infty}= \frac{[b^2q^2;q^2]_\infty}{[-bq;q^2]_\infty}\sum_{n=-\infty}^{\infty}\frac{(-1)^{n}q^{3n(n+1)}}{1-xq^{2n}}-\sum_{n=-\infty}^{\infty}\frac{q^{3n^2-1}b^{3n-1}}{1+xbq^{2n-1}}&+\sum_{n=-\infty}^{\infty}\frac{q^{3n^2-1}b^{-3n-1}}{1+xq^{2n-1}/b}.
\{q_s \mid s \succeq_{\lambda}r,\q_s \leq q_r \}= \{q_r - i e \mid i \in [0,\,k) \}, \{q_s \mid s \succeq_{\lambda}r,\q_s > q_r \}= \{q_r + i e \mid i \in [1,\, l] \},
\limsup_{t\rightarrow\infty}\frac{1}{t}\log \mathbb{P}_{\lambda}^d(C_t^O\neq\varnothing) \leq\lambda M^2\big(dE({\widetilde{\rho}}^2)+\frac{1}{E({\widetilde{\rho}}^2)}\big)-1 &=\lambda(dE{\rho}^2+\frac{M^4}{E{\rho}^2})-1<0
\left( \begin{array}{ccc}d_{j_0-1,j_0-1}d_{j_0-1,j_0}d_{j_0-1,j_0+1}d_{j_0,j_0-1}d_{j_0,j_0}d_{j_0,j_0+1}d_{j_0+1,j_0-1}d_{j_0+1,j_0}d_{j_0+1,j_0+1}\end{array}\right) = (-1)^{3j_0+1}\left( \begin{array}{ccc}1 0 0 0 -1 0 0 0 1 \end{array}\right),
D^{N}_F(x) = \frac{-i}{(2\pi)^4}\int \frac{S_{\vec{k}}(N+1)^2e^{i\vec{k}.\vec{r}-ik_0\, h(N_{\vec{k}}) \, t}\,\, d^4 k}{k^2+m^2}-(N \rightarrow N-1) \, ,
(e_{\alpha}, f_{\alpha}, \cos^2(\theta_{\alpha}) ) = \begin{cases}(\frac{1}{2}o_{\alpha}, o_{\alpha}, o_{\alpha}^{-2}), \mbox{type $\mathrm{A}_2$}, (1, 1, o_{\alpha}^{-1}) \mbox{else}. \end{cases}
kA\frac{(T_{m-1}^{i}-T_{m}^{i})}{\Delta x}+kA\frac{(T_{m+1}^{i}-T_{m}^{i})}{\Delta x}+e_{m}A\Delta x=(\rho c_{p}\Delta xA)\frac{(T_{m}^{i+1}-T_{m}^{i})}{\Delta x}
\Bigl\|\sum_{n=1}^\infty \|I_nz\|_Z e_{\min I_n}\Bigr\|_T = \Bigl\|\sum_{n=1}^\infty \|I_nz\|_Z e_{m_{\min I_n}}\Bigr\|_U = \Bigl\|\sum_{n=1}^\infty \|I_nz\|_Z e_{\min J_n}\Bigr\|_U \geqslant \frac12 \Bigl\|\sum_{n=0}^\infty \|J_nAz\|_Y e_{\min J_n}\Bigr\|_U \geqslant \frac12 [Az]_\wedge.
\mu(f) =\lim_{t\rightarrow\infty}\frac{1}{t}\int_\mathbb{R}\mathbb{E}_x\left[\int_0^tf(X_s)a(X_s)^{-1}\d s\right]\d x=\lim_{t\rightarrow\infty}\frac{1}{t}\int_0^t\int_\mathbb{R}P_s(fa^{-1})(x)\d x\d s &=\lim_{t\rightarrow\infty}\frac{1}{t}\int_0^t\int_\mathbb{R}(fa^{-1})(x)\d x\d s=\int_\mathbb{R}(fa^{-1})(x)\d x,
\mathcal{A}_{1}x^{3}= \begin{pmatrix}x_{1}^{3}x_{1}^{2}x_{2}+x_{2}^{3}\end{pmatrix},~ \mathcal{A}_{2}x^{3}= \begin{pmatrix}x_{1}^{3}+4x_{1}x_{2}^{2}x_{2}^{3}+2x_{1}^{2}x_{2}\end{pmatrix}.
\langle V(\zeta;\eta),(\zeta;\eta) \rangle =\sum \limits_{k=1}^{n}\alpha_k^2 +\frac{1}{\sin^2 \frac{\pi}{n}}\sum \limits_{k=n+2}^{2n}\alpha_k^2 \lambda_{\lceil \frac{k-n-1}{2}\rceil}|f_k|^2,
u^N_t(x)= u^N_0(x) - \frac{1}{N}\sum_{i=1}^N \int_0^t \nabla V^N (x- X_s^{i,N}) \cdot F\big( K\ast u^N_s (X_{s}^{i,N})\big)\ds &- \frac{1}{N}\sum_{i=1}^N \int_0^t \nabla V^N (x- X_s^{i,N})\cdot dW^i_s + \frac{1}{N}\sum_{i=1}^N \int_0^t \Delta V^N (x- X_s^{i,N}) \ds.
{\cal L}={\frac{1}{2 g \sqrt{\alpha^{\prime}}}}~ \mathrm{T r}~ \left( \left({\dot{\Phi}}_{i}-{\frac{i \lambda}{2 ( 2 \pi \alpha^{\prime})}}\epsilon_{i j k}\left[ \Phi_{i}, \Phi_{j}\right] \right)^{2}+{\frac{1 + \lambda^{2}}{2 ( 2 \pi \alpha^{\prime})^{2}}}\left[ \Phi_{i}, \Phi_{j}\right]^{2}\right) ~ .
| \mathrm{\boldmath{\eta}}; M ; D_{r}\rangle ={\frac{1}{\sqrt{M !}}}\left( \sum_{j = 1}^{r}\eta_{j}a_{j}^{\dagger}+ \sum_{j = 1}^{r}\eta_{- j}b_{j}^{\dagger}\right)^{M}| 0 \rangle ,
B_2 = - 4b_1^2 + 4b_2(4a+4n+4b_1+5), B_1 = 2b_1(4a+4n+2b_1+3) + 4b_2 \Bigl(4 \sum_i z_i - 4a-4n+2l+d-5\Bigr), A = (2a+2n)(2a+2n+1) - 2b_1 \Bigl(4 \sum_i z_i - 4a-4n+2l+d-3\Bigr){}- 16b_2 \Bigl(\sum_i z_i^2 - \sum_i z_i\Bigr),
I = -{\frac{1}{8 G}}{\frac{( 2 \pi l )^{2}}{\beta}}, \qquad \qquad M ={\frac{\partial I}{\partial \beta}}={\frac{1}{8 G}}{\frac{( 2 \pi l )^{2}}{\beta^{2}}}={\frac{1}{8 G}}{\frac{r_{+}^{2}}{l^{2}}},
\log_2 \Pr_{a_L \sim_{\rho}b_L}[a_{L}\in A_{L}, b_L \in B_{L}] \geq -|L|\left(\frac{(1-\tfrac{\pi}{\lambda})+\tfrac{\epsilon}{\lambda}+o(1)+2\rho\sqrt{(1-\tfrac{\pi}{\lambda})(\tfrac{\epsilon}{\lambda}+ o(1))}}{1-\rho^2}\right), &= n\left(\frac{\pi - \lambda - \epsilon - 2\rho\sqrt{\epsilon\lambda-\epsilon\pi}}{1-\rho^2}- o(1)\right).
A_{k+1}\leq M_0\delta_0^{2^{k+1}}+M_0^ka_k^2=M_0\delta_0^{2^{k+1}}+M_0^{-(k+4)}(M_0^4\delta_0^2)^{2^k}= M_0\delta_0^{2^{k+1}}+M_0^{-1}a_{k+1}\leq M_0\delta_0^{2^{k+1}}+\frac{a_{k+1}}{2}.
y(t) = u_\varphi(t)(0) =(T_0(t-s)\varphi)(0) + j^{-1}\bigg(\int_s^t T_0^{\odot \star}(t-\tau) [L(\tau)u_\varphi(\tau) + G(\tau,u_\varphi(\tau))]r^{\odot \star}d\tau\bigg)(0) = \varphi(0) + \int_s^t L(\tau)u_\varphi(\tau) + G(\tau,u_\varphi(\tau)) d\tau.
\delta_{c_1}\circ \delta_{c_1}= a_1 \delta_{c_1}+ a_2 \delta_{c_2}+ a_3 \delta_{c_3}+ a_4 \delta_{c_4}, \delta_{c_4}\circ \delta_{c_4}= a_4 \delta_{c_1}+ a_3 \delta_{c_2}+ a_2 \delta_{c_3}+ a_1 \delta_{c_4},
u_1(x) = \frac{-1}{m-1}\left( Ac^{n-1}x^{-1}+ Bc^{n-2}x \right) + A\cdot O(c^{n-2})+O(c^{n-3}) u_2(x) = \frac{1}{m-1}\left( \frac{Bc^{n-2}}{2-m}x^{2-m}+ \frac{Ac^{n-1}}{m}x^{-m}\right)+ A\cdot O(c^{n-2})+O(c^{n-3})
|g(a)-g(p_i)| =|f(a)-f(q_i)+d(p_i,q_i)|\leq d(a,q_i)+d(p_i,q_i) \leq d(a,p_i)+2d(p_i,q_i)\leq d(a,p_i)+2\theta r\leq (1+2\theta)d(a,p_i).
\Big|\tilde T_{2j}(i\tau/2 ) - \sum_{l=0}^N (-1)^{j+l}T_{2j,l}\tau^{-2j-2l-1}\Big| \lesssim \tau^{-3-2s}\Vert u \Vert_{H^{s-\frac14}}^2 \Vert u \Vert_{H^{-1}}^{j-1}.
s^2 z''(s) = - \Bigl( t^{\frac{1}{2}}v'(t) + \frac{1}{2}t^{- \frac{1}{2}}v(t) \Bigr) + \Bigl( t^{\frac{1}{2}}v''(t) + t^{- \frac{1}{2}}v '(t) - \frac{1}{4}t^{- \frac{3}{2}}v (t) \Bigr) .
S_{x}^{\alpha}(f)=lim_{S\rightarrow+\infty}\frac{1}{S}\int_{-S/2}^{S/2}\int_{-\infty}^{+\infty}R_{x}(\theta,\tau)e^{-j2\pi f\tau}e^{-j2\pi\alpha\frac{\theta}{\Theta}}d\tau d\theta
I_{41}= \int_{0}^{2\pi}\log\left(\sin^{2}\left(\frac{x-y}{2}\right)\right)h(y)(-\cos(y)\sin(x)+r\cos(x)\sin(y))dy I_{42}= \int_{0}^{2\pi}\log\left(((1+r^2)+(r^2-1)\cos(x+y))\right)h(y)(-\cos(y)\sin(x)+r\cos(x)\sin(y))dy
\tilde\Xi(\omega):= \{(Q,\hat P)\in\mathfrak{P}(\Omega)\times\mathfrak{P}(\Omega):\E_Q|\Delta S_t(\omega,\cdot)|<\infty,\E_Q[y\Delta S_t(\omega,\cdot)]\leq 0,\ \forall y\in\mathcal{H}_t(\omega), &Q\ll \hat P\in\mathcal{P}_t(\omega)\}
\cos^2 \theta =\frac{1+\cos(2 \theta)}{2}= \frac{\sqrt{m}- \delta_{m,l}}{2 \sqrt{m}- \delta_{m,k}- \delta_{m,l}}, \sin^2 \theta =\frac{1-\cos(2 \theta)}{2}= \frac{\sqrt{m}- \delta_{m,k}}{2 \sqrt{m}- \delta_{m,k}- \delta_{m,l}},
D^R_{\mu\nu}(q) =\delta_{\mu 0}\delta_{\nu 0}{1\over (\vec{q}^2+M^2)}+\delta_{\mu i}\delta_{\nu j}\left(\delta_{ij}-{q_iq_j\over (\vec{q}^2+M^2)}\right){1\over q^2-M^2}~,
q^x \cdot \Omega,_{\alpha r}\Omega,_{\beta ss}+ q^{-x}\cdot \Omega,_{\beta s}\Omega,_{\alpha r s}- (q^x \cdot \Omega,_{\alpha s}\Omega_{, \beta r s}+ q^{-x}\cdot \Omega,_{\beta r}\Omega_{, \alpha s s}= 1.
\begin{array}{cc}V(G_1\cap \bigcup_{i\neq 1}G_i)= \{x,b,a\}, V(G_2\cap \bigcup_{i\neq 2}G_i)= \{x,b\}V(G_3\cap \bigcup_{i\neq 3}G_i)= \{a, b\}, V(G_4\cap \bigcup_{i\neq 4}G_i)= \{b\}.\end{array}
G(R,\vec{v},\vec{a},s)=11_{5}+(\begin{matrix}0&\theta_{3}&-\theta_{2}&v_{1}&a_{1}\\ -\theta_{3}&0&\theta_{1}&v_{2}&a_{2}\\ \theta_{2}&-\theta_{1}&0&v_{3}&a_{3}\\ 0&0&0&0&s\\ 0&0&0&0&0\\ \end{matrix})+....
\hat{I}[ a_{c l}+ \delta a ] \approx \hat{I}_{c l}+ \frac{1}{2}\int_{0}^{T}d t \delta a ( t ) \left[ - 2 \Lambda a_{c l}- 2 \ddot{a}_{c l}- 2 \dot{a}_{c l}\frac{d}{d t}- 2 a_{c l}\frac{d^{2}}{d t^{2}}\right] \delta a ( t )
+ 1 6 . 7 \, a_{1}\, a_{2}\, a_{3}\, a_{4}+ 1 2 1 \, a_{3}^{2}\, a_{4}^{2}- 1 2 . 0 \, a_{1}\, a_{2}^{2}\, a_{3}+ 1 5 . 0 \, a_{3}\, a_{4}^{2}\, a_{1}- 9 8 . 5 \, a_{3}^{2}\, a_{4}\, a_{2}) .
(a_1^{k_1}a_2^{k_2}b_1^{\ell_1}b_2^{\ell_2}[a_1,b_1]^{r_1}[a_1,b_2]^{r_2}[a_2,b_1]^{r_3}[a_2,b_2]^{r_4})^p= &[a_1,b_1]^{k_1+\lambda_1\cdot k_2}[a_1,b_2]^{k_2}[a_2,b_1]^{\ell_1}[a_2,b_2]^{\ell_1+l_2}.
\theta^{-1}\varphi\theta = (\tau,T)^{-1}(\tau \sigma \tau^{-1},R) (\tau, T) &= (\tau^{-1}, (T^{-1})^{\tau^{-1}}) (\tau \sigma, R^{\tau}T ) &= (\sigma, (T^{-1})^{\sigma}R^{\tau}T).
\sum_{\vec{V}_{r}( L )}\cdot \prod_{i = 1}^{\nu - 2}\left[ \begin{array}{c}{{\frac{1}{2}( K_{\nu - 2}\cdot \vec{m}+ \vec{u}_{r}( L ) )_{i}+ \frac{1}{2}L \delta_{i , \nu - 2}}}\\{{m_{i}}}\\ \end{array}\right] =
\varphi_0(F_0,t)=\frac{\sigma^{\frac{p-2+m}{p-1}}}{\gamma^{\frac{m-1}{p-1}}}\left[\left( \frac{m-1}{p-2+m}\right)^{\frac{m-1}{p-1}}-\left(\frac{m-1}{p-2+m}\right)^{\frac{p-2+m}{p-1}}\right] -\delta \, .
\begin{aligned}I_2 =\int_{\alpha_2}^{\beta_2}\biggl[\Phi(\alpha_1,y)P_1(\alpha_1) - \Phi(\beta_1,y)P_1(\beta_1) &+ \frac{\partial\Phi(\alpha_1,y)}{\partial y}P_1(y)P_1(\alpha_1)-\frac{\partial \Phi(\beta_1,y)}{\partial y}P_1(y)P_1(\beta_1)\biggl] dy ,\end{aligned}
z_0^{-1}\delta \left( \frac{z_1 -z_2}{z_0}\right) Y^k (v,z_1)\mathcal{Y}(w_i,z_2)w_j -z_1^{-1}\delta \left( \frac{z_2 -z_1}{-z_0}\right) \mathcal{Y}(w_i ,z_2) Y^j (v,z_1)w_j =z_2^{-1}\delta \left( \frac{z_1 -z_0}{z_2}\right) \mathcal{Y}\left(Y^i (v,z_0)w_i,z_2\right)w_j.
\mu^{\prime}= \alpha^{2}\left( e^{2 \gamma \psi}N ( K^{\prime})^{2}+ \frac{1}{2}N x^{2}( H^{\prime})^{2}+ \frac{1}{2 x^{2}}( K^{2}- 1 )^{2}e^{2 \gamma \psi}+ K^{2}H^{2}\right. \nonumber
\;\|u_*-e^{-t\nu A^s}u_0\|_{L^\infty_T (D(A))\cap L^2_T (D(A^{1+s/2}))}\leq \sum_{j = 1}^\infty (C_2 M)^{j-1}C_1M^2 =: C_6 M^2, &\;u_*-e^{-t\nu A^s}u_0\in B^\beta_{\sum_{j = 1}^\infty(C_3M)^{j+1}, T}=:B^\beta_{C_7M^2, T}.
\Big(\alpha''(x_1, x_2, x_{[3,k]}) + \alpha''(x_2, x_1, x_{[3,k]})\Big) + \Big(\alpha''(x_1, x_2, x_{[3,k]}) + \alpha''(x_2, x_1, x_{[3,k]})\Big)\circ(1\,\,2) = 0
X_{\gamma_j}^N(s) :=\int_\mathbb{R}(s,x)_N^{\gamma_j}B(dx)=\int_\mathbb{R}\left(\frac{[Ns]-[Nx]+1}{N}\right)^{\gamma_j}I\{[Ns]>[Nx]\}B(dx) &\overset{f.d.d.}{=}N^{-\gamma_j-1/2}\sum_{-\infty<i<[Ns]}([Ns]-i+1)^{\gamma_j}\epsilon_i,
( \omega^{2}-{\omega^{\prime}}^{2}) \int_{\theta}^{2 \pi}\! \phi_{\omega}^{L}( \sigma ) \: \phi_{\omega^{\prime}}^{L}( \sigma ) \, d \sigma + \int_{\theta}^{2 \pi}\! \partial_{\sigma}( \phi_{\omega^{\prime}}^{L}( \sigma ) \: \partial_{\sigma}\phi_{\omega}^{L}( \sigma ) - \phi_{\omega}^{L}( \sigma ) \: \partial_{\sigma}\phi_{\omega^{\prime}}^{L}( \sigma ) ) \, d \sigma = 0
a_{n}(\mu^{2}/Q^{2})= -\frac{e_{f}^{2}\mu^{4}}{16 \pi^{3}Q^{4}}\left( - \frac{\beta_{0}}{2}\right)^{n}n! \left[\frac{- 2 \beta_{1}}{\beta_{0}^{2}}( \ln n + \gamma_{E}-1) -2 \frac{p_{2}}{\beta_{0}^{2}}\right] \left( 1 + O \left(1/n\right)\right)
|(h\circ\gamma)_t'| \leq \Phi(\alpha(t),(g_i(\beta(t))))|\alpha_t'|+\Psi(\beta(t), (f_i(\beta(t)))|\beta_t'| &\leq \|(\Phi(\alpha(t),(g_i(\beta(t)))),\Psi(\beta(t),(f_i(\alpha(t)))))\|'\|(|\alpha_t'|,|\beta_t'|)\| &= \|(\Phi(\alpha(t),(g_i(\beta(t)))),\Psi(\beta(t),(f_i(\alpha(t)))))\|'|\gamma_t'| )= g(\gamma_t) |\gamma_t'|.
J_E \: \tilde{\psi}\;:=\; -i f_{EAA^\prime}\left( x_{jA}\:\frac{\partial}{\partial x_{jA^\prime}}\:+\: z_A \:\partial_{A^\prime}\:+\: \overline{z_A}\: \overline{\partial_{A^\prime}}\:+\: \lambda_{\alpha A}\: \partial_{\lambda_{\alpha A^\prime}}\right) \tilde{\psi}\;=\; 0
\sum_{I_{n_k}(a_1,\ldots,a_{n_k})\cap F\neq\emptyset}|I_{n_k}(a_1,\ldots,a_{n_k})|^{s}\leq K_2^{sn_k}\prod_{j=1}^k G(m(j),n(j),a,1/3,s) \leq K_2^{sn_k}C_1^k C_2^{n_k-k-1}3^{-k}\prod_{j=1}^k m(j)^{{1-ds \over a}}.
\| V^*(\Phi^{\prime}(f\otimes a)+ \sum_{j=1}^m\sum_{k=1}^mf(z_k)\rho (p_{j,k}\otimes a) + \sigma^{\prime\prime}(f) )V & -(\Psi^{\prime}(f\otimes a)+\sum_{j=1}^m\sum_{k=1}^mf(z_k)\rho (p_{j,k}\otimes a) + \sigma^{\prime\prime}(f) ) \|< \frac{\varepsilon}{7}.
\begin{aligned}\psi(\beta,\mu) = \sup_{\tilde\rho}\left[\tilde \rho \mu - \phi(\beta,\tilde\rho)\right] \geq \rho_0\mu - \phi(\beta,\rho_0) \psi_0(\beta,\mu) = \sup_{\tilde\rho}\left[\tilde \rho \mu - \phi_0(\beta,\tilde\rho)\right] = \rho_0\mu - \phi_0(\beta,\rho_0)\end{aligned}
(\tau d_1)\bigl((k\ell) b_1\bigr)\ = \y^2\bigl(\lambda_{(0,2)}+ k\ell\bigr)b_0 \ - \x^3\bigl(\lambda_{(3,0)}+ k\ell\bigr)b_0 = \y^2\bigl(\nu(6)-(0,2) + k\ell\bigr)b_0 \ - \x^3\bigl(\nu(6)-(3,0) + k\ell\bigr)b_0 = \y^2\bigl((4+k)\ell\bigr)b_0\ - \x^3\bigl((3+k)\ell\bigr)b_0.
L_{a b}= 2 L_{a c}G^{c d}L_{d b}- L_{c d}L_{e f}f_{a}^{c e}f_{b}^{d f}- L_{c d}f_{f}^{c e}f_{a}^{d f}L_{b e}- L_{c d}f_{f}^{c e}f_{b}^{d f}L_{a e},
exp(u)=\sum_{k=0}^{\infty}\frac{1}{k!}u^{k}=I_{n+2}+u+\frac{1}{2}u^{2}=[\begin{matrix}1&a&c+\frac{1}{2}a\cdot b\\ 0&I_{n}&b\\ 0&0&1\end{matrix}].
0 \to \Omega^0(U_0)\oplus \Omega^0(U_1) \to \Omega^0(U_0 \cap U_1) \oplus \Omega^1(U_0, U_0 \cap Y) \oplus \Omega^1(U_1, U_1\cap Y) &\to \Omega^1(U_0 \cap U_1, U_0 \cap U_1 \cap Y) \oplus \Omega^2(U_0) \oplus \Omega^2(U_1) \to \Omega^2(U_0 \cap U_1) \to 0.
-\frac{q^2[q,q^2,q^2,-q^2,q^3;q^6]_\infty(q^6;q^6)^4_\infty}{[-q;q^6]_\infty}=-2\frac{(q^2;q^2)_\infty^2}{(q;q^2)_\infty^2}\sum_{n=-\infty}^\infty \frac{(-1)^nq^{n(3n+3)+2}}{1+q^{6n}}.
C_{\kappa,s}:=C\left( \rVert\kappa\rVert_{L^{\infty}\left( \Omega\right)}+C_{\varepsilon}\rVert\kappa\rVert_{W^{\frac{s}{1-\varepsilon},2-\frac{2\varepsilon}{2\varepsilon-1}}\left( \Omega\right)}^{1-\varepsilon}\rVert\kappa\rVert_{L^{\infty}\left( \Omega\right)}^{\varepsilon}\right)\quad\forall1/2\leq s<1.
I^{(1)\mu}_{ij}={i\over 2}\Psi^\lambda \Psi^\sigma (f^\nu_{i~\sigma}D_\nu f^\mu_{j~\lambda}+ f^\nu_{j~\sigma}D_\nu f^\mu_{i~\lambda}+{1\over 2}f^{\mu\rho}_i c_{j\lambda\sigma\rho}+ f^{\mu\rho}_j c_{i\lambda\sigma\rho})
\dot{u}= \frac{A_{d}}{N}\left[ \frac{3 u^{\prime}+ 2 y u^{\prime \prime}}{1 + u + 2 y u^{\prime}}+ ( N - 1 ) \frac{u^{\prime}}{1 + u}\right] + 2 u + ( 2 - d ) y u^{\prime}.
=y^{-\upsilon}\sum_{k=0}^{\infty}\bigg[\frac{\Theta(k)}{k!}\sum_{\ell=0}^{\infty}\frac{(-1)^{\ell}(\lambda b+c k)^{\ell}\Gamma\left(\upsilon+\frac{\ell}{2}\right)}{y^{\frac{\ell}{2}}~\ell!}\cos\left(\frac{\upsilon \pi}{2}+\frac{\ell\pi}{4}\right)\bigg],
A_{\boldsymbol{W}}z\boldsymbol{f}=zA_{\boldsymbol{W}}\boldsymbol{f}-\mathfrak{p}_{+}zA_{\boldsymbol{W}}\boldsymbol{f}-A_{\boldsymbol{W}}\mathfrak{p}_{+}zA_{\boldsymbol{W}}\boldsymbol{f}
J^{w}(\textbf{X}_{SRS}^{(n)})=\frac{-1}{2}\prod_{i=1}^{n}\left(\int_{0}^{\infty}w(x_i)f^2(x_i)dx_i\right) =\frac{-1}{2}\left(-2J^{w}(X)\right)^n =\frac{-1}{2}\left(E(\Lambda_X^{w}(U))\right)^n.
\begin{cases}\Delta^{\alpha}_{0,t_{j + \sigma}}u_i = \delta^{\beta}_{h}u^{(\sigma)}_i + f^{j + \sigma}_i, 1\leq i \leq N - 1,0\leq j \leq M - 1, u^{0}_i = \phi(x_i), 1\leq i \leq N - 1, u^{j}_0 = u^{j}_N = 0, 0 \leq j \leq M.\end{cases}
\Vert |u_n|^{\gamma-1}u\Vert_{L^{\infty}([0,T],L^{M/\gamma}(\mathbb{R}^3))}\; \leqslant\; \Vert u_n\Vert_{L^{\infty}([0,T],L^M(\mathbb{R}^3))}^{\gamma}\lesssim\;\Vert u_n\Vert_{L^{\infty}([0,T],H^1(\mathbb{R}^3))}^{\gamma}\;\lesssim_{A,T}\;1\,,
h_{3}^{e}( y , r ) = \frac{\pi^{2}}{1 2}- \frac{\pi y}{4}\sqrt{1 - r^{2}}- \frac{y^{2}}{8}\left[ \ln \left( \frac{y}{4 \pi}\right) + \gamma -{\textstyle \frac{1}{2}}+ r^{2}\right] + \cdots ,
\partial_{x_j}^n\frac{1}{(1+\epsilon x^{2m})^2}=\partial_{x_j}^nf_\epsilon(g(x))=\sum_{\substack{(m_1,\dots,m_n) \sum_{i=1}^nim_i=n}}c_{n,m_i}f_\epsilon^{(m_1+\dots+m_n)}(g(x))\prod_{i=1}^n(\partial_{x_j}^ig(x))^{m_i}\,.
\int_{Q_1}\varphi_{-}(x) \nu_{R,L}(x,w) \dx \leq \int_{Q_1}\varphi_{-}(x) \frac{n_L(C_{R/L}(x))}{|C_R|}\dx &\leq \int_{Q_1}\int_{C_{R/L}(x)}\frac{\varphi(y)}{|C_R|}\dn_L(y) \dx=\frac{1}{L^n}\int_U \varphi(y) \dn_L(y) &\leq \int_{Q_1}\varphi_{+}(x) \frac{n_L(C_{R/L}(x))}{|C_R|}\dx.
\theta_{1}^{AA^{\prime}}=(\mathbf{L}_{1}^{-1})_{B^{\prime}}^{A^{\prime}}(\mathbf{L}_{1}^{-1})_{B}^{A}d\mathbf{x}_{1}^{BB^{\prime}}=(\mathbf{L}_{1}^{-1})_{b}^{a}d\mathbf{x}_{1}^{b}.
f^{*}(t,y)=\int_{0}^{t}\int_{\mathbb{R}^{d}}G_{\beta}(t-s, y-x)H(s,x,\nabla f^{*}(t,x))dx ds + \int_{\mathbb{R}^{d}}S_{\beta, 1}(t, y-x)f_{0}(x)dx.
\{x_i,x_j\}_{\Pi_{X_1,X_2}}= (\mathcal{L}_{X_1}x_i)(\mathcal{L}_{X_2}x_j) - (\mathcal{L}_{X_2}x_i)(\mathcal{L}_{X_1}x_j) &=\delta_{1i}X_{2}^{j}-X_{2}^{i}\delta_{1j},