harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{\ddot{b}}{b}+ ( D - 1 ) \frac{\dot{b}^{2}}{b^{2}}+ 3 \frac{\dot{a}}{a}\frac{\dot{b}}{b}= \frac{8 \pi \overline{{G}}}{D + 2}\left[ \rho - 3 p_{a}+ 2 p_{b}\right] .
f^{\left(p\right)}_k+\frac{\left[\mathbf{D}\right]_{k,k}\underline{m}^{\left(p\right)}f^{\left(0\right)}_k}{1+\left[\mathbf{D}\right]_{k,k}\underline{m}\left(0\right)}+\sum_{l=1}^{p-1}\binom{p}{l}\frac{\left[\mathbf{D}\right]_{k,k}\underline{m}^{\left(l\right)}f^{\left(p-l\right)}_k}{1+\left[\mathbf{D}\right]_{k,k}\underline{m}\left(0\right)}=0.
2 \mathrm{I m}S_{\mathrm{A B C D}}^{( I A )}= \frac{1 6 \pi^{2}}{g^{2}}+ 2 \pi^{2}\rho^{2}v^{2}- \frac{9 6 \pi^{2}\rho^{4}}{l^{4}}+ O ( \rho^{6}/ l^{6}) .
(\Delta-f_2'({\sf U}_\infty))\theta = -f({\sf U}_\infty) - \sum_{i=0}^N \tfrac{f^{(i)}({\sf U}_\infty)}{i!}(\theta_0+\cdots+\theta_{L-1})^i + \sum_{i=2}^N \tfrac{f_2^{(i)}({\sf U}_\infty)}{i!}(\theta_0+\cdots+\theta_{L-1})^i.
|J_{3}|\leq 2^{(\sigma+1)p+n-2}\int_{B_{R}^{c}}\frac{(1+j^{1-q})^{p-1}+j^{-q(p-1)}|y|^{-s(p-1)}}{|y|^{n+\sigma p}}dy = 2^{(\sigma+1)p+n-2}\gamma(1+j^{1-q})^{p-1}R^{-\sigma p}+2^{(\sigma+1)p+n-2}\tau j^{-q(p-1)}R^{-\sigma p-s(p-1)}.
M_1\;:= \;m\sum_{k=0}^n{n \choose k}(m+n-1)^{n-k-1}= = m(m+n-1)^{-1}\sum_{k=0}^n{n \choose k}(m+n-1)^{n-k}=m(m+n-1)^{-1}(m+n)^{n},
\sum_{j = 0}^k{\sum_{s = 0}^{k - j}{( - 1)^{ns + j + s}\binom kj\binom{k-j}s\frac{{G_0^j G_{n + r}^s}}{{G_r^{j + s}}}H_{m - (n + r)k + rj + ns}}}= ( - 1)^{(n + 1)k}\left({\frac{{G_n}}{{G_r}}}\right)^k H_m,G_r\ne 0\,,
=\rho\left[ \frac{\partial u_{2}}{\partial t}+u_{1}\frac{\partial u_{2}}{\partial x}+u_{2}\frac{\partial u_{2}}{\partial y}\right] +\frac{Ke^{f(s)}}{a(t)^{2}}\frac{y}{r}\frac{\partial}{a(t)\partial\left( \frac{r}{a(t)}\right)}f(s)+\frac{y}{r}\rho\frac{\partial}{\partial r}\Phi
y^2=x^3+\sum_{k=-4}^{4}\sum_{l=0}^{8-nk}f_{kl}z^{\prime l}z^{4+k}p^{8-l+q^{\prime}k}x+\sum_{k=-6}^{6}\sum_{l=0}^{12-nk}g_{kl}z^{\prime l}z^{6+k}p^{12-l+q^{\prime}k}
\frac{\partial}{\partial t}\mathcal{K}_{t/2}(x)=\frac{\partial}{\partial t}\mathcal{K}_{t/2}(-x),\left|\frac{\partial}{\partial t}\,\mathcal{K}_{t/2}(x)\right|\le \frac{1}{2\,t}\,\mathcal{K}_{t/2}(x)\,\left|\frac{|x|^2}{t}-N\right|.
\left\{\left(\sum_{p\in[N_T]}h_{qp}(u)v_{{\mathcal{R}},{[N_T]},p}^i(u)\right)_{u=1}^S: \mathcal{R}\ni q,|\mathcal{R}|=r+1,i\in[\rho]\right\}&\cup\left\{\left(\sum_{p\in[N_T]}h_{qp}(u)v_{{\mathcal{R}},{[N_T]},p}^i(u)\right)_{u=1}^S:\mathcal{R}\cup\bar{\mathcal{R}}_i\not\ni q\right\}
u_t+(-1)^m \tilde{\l}_0 u=0 Q_{2R}^+, &D_d^m u=(-1)^m({A^{\hat{\alpha}\hat{\alpha}}})^{-1}f_{\hat{\alpha}},\, D_d^{m+1}u=\cdots=D_d^{2m-1}u=0\,\, \, \{x_d=0\}\cap Q_{2R}.
\pi^*(\mathcal{E}) \rightarrow \mu_*(\nu^* \mathcal{L}\otimes \mathcal{O}_{(m+n-\delta)\Gamma_p+\delta\Delta})=:\mathcal{M}_1 \twoheadrightarrow \mu_*(\nu^* \mathcal{L}\otimes \mathcal{O}_{m\Gamma_p})=:\mathcal{M}_0
\Gamma_{a,s}=\tilde\Gamma_s\Gamma_{a ,s}^{(o)}(l_{1s}) \tilde\Gamma_s^{-1}, \qquad \Gamma_{b ,s}=\tilde\Gamma_s\Gamma_{b ,s}^{(o)}(l_{2s}) \tilde\Gamma_s^{-1}= \Gamma_{a,s}({1/2}-l_{2s})\Gamma_s\qquad
L_{e f f}^{i i}= \frac{1}{2 \pi}\left( \hat{\Phi}^{i}\hat{\eta}^{i}\right) \left( \begin{array}{c c}{A^{i i}}&{C^{i i}/ 2}\\{C^{i i}/ 2}&{B^{i i}}\\ \end{array}\right) \left( \begin{array}{c}{\hat{\Phi}^{i}}\\{\hat{\eta}^{i}}\\ \end{array}\right) ,
F = \left( \begin{array}{c c c c c c c c c}\lambda_1 \lambda_2 0 \lambda_3 0 0 0 0 0 0 0 \lambda_2 0 \lambda_3 0 \lambda_4 0 0 0 0 0 0 0 \lambda_3 0 \lambda_4 \lambda_5 \end{array}\right),
0 = M_{1^6}+ 15 M_{1^4}M_{1,1}- 5 M_{3,1^3}- 15 M_{3,1}M_{1,1}- 5M_{3,3}+ 9 M_{5,1}0 = M_{1^8}+ 28M_{1^6}M_{1,1}+ 35(M_{1^4})^2 + 7M_{3,1^5}+ 70M_{3,1^3}M_{1,1}+ 35M_{3,1}M_{1^4}- 35M_{3,3}M_{1,1}& - 70(M_{3,1})^2 - 21 M_{5,1^3}- 63M_{5,1}M_{1,1}- 42M_{5,3}+ 90 M_{7,1}.
\mathbb{E}[||U||_p^p] = \mathbb{E}\bigg[\int_{\mathcal{D}}|U(x)|^pdx\bigg] = \int_{\mathcal{D}}\mathbb{E}[|U(x)|^p]dx = \int_{\mathcal{D}}C_p\sigma(x)^pdx = C_p||\sigma||_p^p
\min\left\{\int_{\overline{\Omega}\times \overline{\Omega}}||x-y||\,\mathrm{d}\gamma:\; \gamma \in \mathcal{M}^+(\overline{\Omega}\times \overline{\Omega}),\,(\Pi_x)_{\#}\gamma=f^+\,\,\mbox{and}\,\,\,(\Pi_y)_{\#}\gamma=f^- \right\},
\phi_1 := \phi_{15AB}(\tau, z) = 2\zeta^{\pm 1}- 3 + (-3\zeta^{\pm 2}+ 8 \zeta^{\pm 1}- 10)q + (2 \zeta^{\pm 3}- 10 \zeta^{\pm 2}+ 25 \zeta^{\pm 1}- 34)q^2 + O(q^3)
\int_{[0,1]^n}\prod_{i=0}^n x_i^{a-1}(1-x_i)^{b-1}\prod_{1\leq l<k\leq j}|x_k-x_l|^{2g}dx_1\cdots dx_n = \prod_{k=0}^{n-1}\frac{\Gamma(a+kg)\Gamma(b+kg)\Gamma(1+(k+1)g)}{\Gamma(a+b+(n+k-1)g)\Gamma(1+g)},
{\cal I}( \boldsymbol{x}_i ) = \sum_{j=1}^{d}\bar{R}^{N_j}_j( \boldsymbol{x}_i ) W_j ( \boldsymbol{x}_i) = \sum_{j \in I(\boldsymbol{x}_i)}f( \boldsymbol{x}_i ) W_j ( \boldsymbol{x}_i) = f( \boldsymbol{x}_i).
d(\sigma\mu_{n})={}i_0\mu_n-i_1\mu_n-\sum_{\substack{p+q=n+1 p> r}}\sigma\mu_p\{\mu_q\}+\sum_{\substack{1\leq u t_1=n+1-u t_1> r}}\mu_u\{\sigma\mu_{t_1}\}&-\sum_{\substack{p+q=n+1 2\leq p,q\leq r}}\sigma\mu_p\{i_1\mu_q\}+\sum_{\substack{1\leq v\leq u\leq r t_1+\cdots+t_v=n+v-u t_1,\dots,t_v\leq r}}i_0\mu_u\{\sigma\mu_{t_1},\dots,\sigma\mu_{t_v}\},
\begin{array}{ll}U^{\nu,t_{0}}_{,s}+\sqrt{1-\Delta t^2}^3 \nu \Delta U^{\nu,t_{0}}+\sqrt{1-\Delta t^2}^3v\nabla_x U^{\nu,t_0}+ -\frac{\sqrt{1-\Delta t^2}^3}{1+t}Q(t,(1+t )U^{\nu,t_{0}}(s,.),(1+t )U^{\nu,t_{0}}(s,.))=-\frac{\sqrt{1-\Delta t^2}^3}{1+t}U^{\nu,t_{0}}, ~U^{\nu,t_{0}}(0,.)=F(t_{0},.).\end{array}
\left| (1+s)^{-d/2}- (1+s^+)^{-d/2}\right| = \frac{d}{2}\int_{s^+}^{s}(1+r)^{-d/2-1}\, \mathrm{d}r \le \frac{d}{2}(1+s^+)^{-d/2-1}(s-s^+) \le d (1+s^+)^{-d/2}\Big( 1-\frac{s^+}{s}\Big)
\tilde{\Phi}= ( \tilde{\varphi}_{0}, \tilde{\varphi}_{1}, \ldots ,\tilde{\varphi}_{p})^{T}:= D^{p}\tilde{F}:= ( D^{p}\tilde{f_{0}}, D^{p}\tilde{f_{1}}, \ldots , D^{p}\tilde{f_{p}})^{T}.
d_n(c_0c_1a_1b_1 \dots c_1a_1b_1 \otimes 1) = c_0c_1a_1b_1 \dots c_1a_1 \otimes b_1 + c_0c_1a_1b_1 \dots c_1a_1b_1a_0 \otimes b_0c_0
\langle \partial_t^{\mu}v, \partial_t^\mu F_\lambda \rangle \leq C_\lambda \delta Z_0 + \sum_{i,j=1}^d \sum_{0 \leq b \leq d}\frac{d}{dt}\int_{\Omega_\lambda}\partial_t^\mu v^i \partial_t^\mu \partial_b v^j c^{0b}_{ij}(\partial v) dx
D_n(wG_n(x)/2)=D_n(w\alpha_n)=\exp(\ln (D_n(w\alpha_n)/D_n(\alpha_n)))D_n(\alpha_n) \leq e^{-(w-1)\ln n+1}D_n(\alpha_n)=e^{1-(w-1)\ln n}D_n(G_n(x)/2),
\pi_{\dot{q}}^{2 -}= e^{+ 2}\psi_{+ 2 \dot{q}}^{2 -}+ e^{- 2}\psi_{- 2 \dot{q}}^{2 -}+ e^{+}h_{+ \dot{q}}^{2 -}+ e^{-}\chi_{\dot{q}}^{2}
A=\begin{pmatrix}1 a_{1}1 \vdots \ddots a_{n}1\end{pmatrix}C=\begin{pmatrix}c_{0}c_{1}\cdots c_{n}c_{1}\vdots c_{n}\end{pmatrix}D=\begin{pmatrix}1 -a_{1}\cdots -a_{n}1 \ddots 1\end{pmatrix}
\bar{\lambda}_{\theta}= - \frac{2 \pi | \alpha |}{m}\frac{\Gamma ( 1 + | \alpha | ) \tan \theta - \Gamma ( 1 - | \alpha | ) \left( \frac{\mu r_{0}}{2}\right)^{2 | \alpha |}}{\Gamma ( 1 + | \alpha | ) \tan \theta + \Gamma ( 1 - | \alpha | ) \left( \frac{\mu r_{0}}{2}\right)^{2 | \alpha |}}.
\|\Phi(u)(t+h) -\Phi(u)(t)\|^2_{\mathbb H^\mu}\le 2\|[S(t+h)-S(t)]\xi\|^2_{\mathbb H^\mu}&+ 4\int_0^t \tau^{-\delta}\|D_h f(u)(t-\tau)\|^2_{\mathbb H^{\mu-1-\delta}}d\tau &+ 4\int_t^{t+h}\tau^{-\delta}\|f(u(t+h-\tau),\mathcal H u(t+h-\tau) )\|^2_{\mathbb H^{\mu-1-\delta}}d\tau & = E_1(t) + E_2(t)+E_3(t).
\begin{cases}\rho(x) \varphi_{tt}- \operatorname{div}(K(x) \nabla \varphi) =0 ~\hbox{in}~ \Omega \times (0,T), \rho(x) \psi_{tt}- \operatorname{div}(K(x) \nabla \psi) =0 ~\hbox{in}~ \Omega\times (0,T), \varphi=\psi=0 ~\hbox{in}~ \omega\times(0,T). \end{cases}
A_L = \begin{bmatrix}2{\rm e}^{\xi_2}\cos \left( \xi_1 \right) 1 -{\rm e}^{2 \xi_2}0\end{bmatrix}\;, A_R = \begin{bmatrix}{\rm e}^{\xi_2}\cos\left( \xi_1 \right) 1 0 0\end{bmatrix}\;, B = \begin{bmatrix}1 0\end{bmatrix}\;,
\Big\Vert \nabla \Tilde{\mathrm{H}}\Big\Vert_{\mathbb{L}^4(\mathrm{B})}\lesssim \Big\Vert \Tilde{\mathrm{H}}\Big\Vert_{H^{\frac{3}{2}}(\mathrm{B})}\lesssim \Big\Vert \partial_{\nu}\Tilde{\mathrm{H}}\Big\Vert_{H^{\frac{1}{2}}(\partial \mathrm{B})}+ \delta^2 \Big\Vert \Tilde{\mathrm{H}}\Big\Vert_{\mathbb{L}^2(\partial \mathrm{B})}=\mathcal{O}\Big(1\Big).
e^{-\frac{(k-np)^2}{2np(1-p)}}\left|\frac{P\left(V_n=k\right)}{\frac{1}{\sqrt{2\, \pi n p (1-p)}}e^{-(k-np)^2/(2np(1-p))}}-1\right| < \frac{C'}{\sqrt{n p (1-p)}},
\mu^{\#}_{R,1}(k,\ell,m,n):=\sum_{(A,B,C,D)\in\mathcal{X}_{R}}\int\overline{\mathcal{K}^{\#}_{A}(x,k)}\mathcal{K}^{\#}_{B}(x,\ell)\overline{\mathcal{K}^{\#}_{C}(x,m)}\mathcal{K}^{\#}_{D}(x,n)\,dx, \mathcal{X}_{R}= := \{S,R \}^4 \smallsetminus \{(S,S,S,S)\},
h_{0}(J_{n}(x))= (c_{n}^{21}x + \alpha_{n}^{21}y_{f_{0}}^{1}(x)+ \beta_{n}^{21}z_{f_{0}}^{1}(x)+ d_{n}^{21}, \gamma_{n}^{21}z_{f_{0}}^{1}(x)+ e_{n}^{21}x+ f_{n}^{21}) \; \mathrm{for}\; n=1, 2, \ldots, K^{21}
H =\sum_{\theta_r\neq \{1,-1\}}\theta_r\left(E_{\theta_r}-E_{-\theta_r}\right)+\pi\cdot E_{-1}&=\sum_{\theta_r\neq \{1,-1\}}\theta_r\left(-\frac{2i}{\sin(\theta)}(PW-WP)\right)+\pi\cdot E_{-1}.
\mathcal{L}_{m_{\tilde{k}}}^{d}(v \mapsto (\mathcal{B}_{m_{\tilde{k}}}^{d}F)(v) )(T) =\frac{\tilde{k}}{2i \pi}\int_{\gamma_{\tilde{k}}}\frac{F(u)}{u}\frac{T^{\tilde{k}}}{T^{\tilde{k}}- u^{\tilde{k}}}du.
\dot{\epsilon}_{\delta \sigma}= \epsilon_{\delta \sigma}\left[ 3{\frac{\dot{a}}{a}}( 1 + w ) + 2{\frac{\dot{c}}{c}}{\frac{1 + \epsilon_{\delta \sigma}}{1 + \epsilon}}+ \left({\frac{\dot{G}_{4}}{G_{4}}}- 4{\frac{\dot{c}}{c}}\right) ( 1 + \epsilon_{\delta \sigma}) +{\frac{3 C c \dot{c}}{4 \pi G_{4}a_{0}^{4}\varrho_{m}}}\right] .
\mathcal{L}_{I}^{l i n}= - \frac{i}{4}\left( h^{a \mu}\bar{\psi}\gamma_{a}\frac{1 \pm \gamma_{5}}{2}\stackrel{\leftrightarrow}{\partial_{\mu}^{\psi}}\psi + h_{\ \mu}^{\mu}\bar{\psi}\gamma^{a}\frac{1 \pm \gamma_{5}}{2}\stackrel{\leftrightarrow}{\partial_{a}^{\psi}}\psi \right)
(xq)^{2a-1}\sum_{h=1}^{k-a+1}(xq^2)^{2h}U_{2k,2k-2h}(xq^2;q)+(xq)^{2a-1}\sum_{h=0}^{k-a}(xq^2)^{2h}U_{2k,2k-2h}(xq^2;q).
\int_0^1\ldots\int_0^1\left|\zeta_{Q\setminus M}(s;(c\omega_p))-1\right|^2 \prod_{p\in Q\setminus M}d\omega_p\leq \sum_{n>y}\frac{1}{n^{2\sigma}},s=\sigma+it\in\overline{U}.
u(t) \leq \int_{0}^{t}\bigg(2\sum_{i=1}^{\infty}\sum_{j=1}^{i}C_{V}j (i+j) |u_j|\,\psi_i + \sum_{i=1}^{\infty}\sum_{j=i}^{\infty}C_V i^{\eta}|u_j|\, \psi_i \bigg)(h) dh.
u(j+x(\mu^*,n),0;-n,\bar\phi_{\mu^*}(\cdot,-n))\begin{cases}\ge u(j+x(\mu,n),0;-n,\bar\phi_\mu(\cdot,-n)){\rm for}j\le 0\cr\le u(j+x(\mu,n),0;-n,\bar\phi_\mu(\cdot,-n)){\rm for}j> 0.\end{cases}
\alpha^{a}= \sqrt{\frac{m}{2}}\, \left[ \hat{A}^{a}+ \frac{i}{m}\hat{\pi}^{a}\right] \ \ , \ \ \{\alpha^{a}}^{\dagger}= \sqrt{\frac{m}{2}}\, \left[ \hat{A}^{a}- \frac{i}{m}\hat{\pi}^{a}\right] \ \ ,
\dfrac{\hat{k}_2{(\Delta)}}{\hat{k}_2(\Delta\setminus\sigma)}= \dfrac{D_{1,2}D_{2,2}D_{3,3}}{D_{1,1}D_{2,1}D_{3,2}}=\dfrac{(x_{1,1}+x_{1,2})(x_{2,1}+x_{2,2})(x_{3,1}+x_{3,2}+x_{3,3})}{x_{1,1}x_{2,1}(x_{3,1}+x_{3,2})}.
X_e = \begin{bmatrix}X_{e,1}X_{e,2}\end{bmatrix}, X_0 = \begin{bmatrix}X_{0,1}X_{0,2}\end{bmatrix}=\begin{bmatrix}M_{3,5}(7) M_{3,5}(10)\end{bmatrix}=\begin{bmatrix}30 1\end{bmatrix}, U = \begin{bmatrix}M_{3,5}(6) M_{3,5}(5) M_{3,5}(9) M_{3,5}(8)\end{bmatrix}= \begin{bmatrix}45 51 5 15\end{bmatrix}.
\frac{1}{1+t}\bigg[\binom{n}{k}_t-(1+t^{n-1})\binom{[\frac{k}{2}]+[\frac{n-k}{2}]}{[\frac{k}{2}]}_{t^4}\bigg] \frac{t}{1+t}\bigg[\binom{n}{k}_t-(1+t^{n-1})\binom{[\frac{k}{2}]+[\frac{n-k}{2}]}{[\frac{k}{2}]}_{t^4}\bigg]
y_{i+1}- y^2_i = \frac{i+1}{2^{2^n}}- \left(\frac{i}{2^{2^n}}\right)^2 = \frac{2^{2^n}(i+1) - i^2}{\left( 2^{2^n}\right)^2}= \frac{2^{2^n}+ 2^{2^n}i \left( 1 - \frac{i}{2^{2^n}}\right)}{\left( 2^{2^n}\right)^2}> 0.
\sigma^{-2}\big(r^2(I,\theta)-c_3 r^2(\theta)\big) \ge M p_n^*\log(\tfrac{en}{Mp^*_n}) -c_3 Kn (\tfrac{p_n}{n\sigma})^s\big[\log (\tfrac{n\sigma}{p_n})\big]^{1-s/2}&\ge Ms p_n^*\log(\tfrac{n\sigma}{p_n})-c_3Ke^{-1}p_n^*\log (\tfrac{n\sigma}{p_n})=(Ms-c_3Ke^{-1}) p_n^*\log(\tfrac{n\sigma}{p_n}).
S= \sum_{n=0}^{r}\frac{(-1)^n(2\, \ell-2\, n)!(2\, n)!}{(2\, n+1)! (\ell-2\, n)! (\ell-n)!\, n!}= \frac{(2 \ell)!}{(\ell!)^2}\sum^{r}_{n=0}\frac{(-1)^n \binom{\ell}{2 n}\binom{\ell}{n}}{(2n+1)\binom{2\ell}{2 n}}.
\begin{cases}u_{t}=\Delta u -\chi_1\nabla( u\nabla v_1)+\chi_2 \nabla(u\nabla v_2 )+ u(a -b u) \x\in\Omega,\t>0, 0=(\Delta- \lambda_1 I)v_1+ \mu_1 u \x\in\Omega,\t>0, 0=(\Delta- \lambda_2 I)v_2+ \mu_2 u \x\in\Omega,\t>0\end{cases}
\langle J_0, P_0\rangle=-1 ,\langle J_1, P_1\rangle=1 ,\langle J_2, P_2\rangle=1 , &\langle J_a, J_b\rangle=\langle P_a, P_b\rangle=0, \langle J_a, P_b\rangle=0 \a\neq b,a,b=0,1,2 ,
V_{3}^{a}+ \sum_{i = 0}^{2}\alpha_{i}V_{i}^{a}= V_{4}^{a}+ \sum_{i = 0}^{2}\beta_{i}V_{i}^{a}={\frac{1}{2}},{\mathrm{f o r s o m e}}a = 1 , 2 , \dots , 2 0
S^{(1)}= \ln{\rm Det}K_{\mbox{\scriptsize bos}}-\ln{\rm Det}K_{\mbox{\scriptsize fermi}}-\ln{\rm Det}(1- K^{-1}_{\mbox{\scriptsize fermi}}K_{\mbox{\scriptsize mix}}K^{-1}_{\mbox{\scriptsize bos}}K^\dagger_{\mbox{\scriptsize mix}}).
\bar{\delta}^t N \bar{\delta}- (\bar{\delta}')^t N \bar{\delta}' = \sum_{i=1}^{2t}\mu_i(\alpha_i^2-(\alpha_i')^2) &= \mu_1(\alpha_1^2-(\alpha_1')^2) + \sum_{i=2}^{2t}\mu_i(\alpha_i^2-(\alpha_i')^2).
M^{(s)}_{k_{1},k_{2},\ldots,k_{n}}=\prod_{j=1}^{n}\left(\prod_{i_{j}=0}^{k_{j}-1}j_{s+2\sum^{j-1}_{p=1}{k_p}+2i_{j}+(j-1)}\right), \widetilde{M}^{(s)}_{k_{1},k_{2},\ldots,k_{n}}=\prod_{j=1}^{n}\left(\prod_{i_{j}=0}^{k_{j}-1}j_{-s-2\sum^{j-1}_{p=1}{k_p}-2i_{j}-(j-1)}\right).
a_{0}+a_{0}a_{1}+a_{0}a_{1}a_{2}+\cdot\cdot\cdot+a_{0}a_{1}a_{2}\cdot\cdot\cdot a_{n}=\frac{a_{0}}{1-}\frac{a_{1}}{1+a_{1}-}\frac{a_{2}}{1+a_{2}-}\cdot\cdot\cdot\frac{a_{n}}{1+a_{n}}.
\begin{aligned}\Vert |D^a|U_a(t)g\Vert^2 \le C (H^a_0U_a(t)g,U_a(t)g) &\le C((H^a-E_j)U_a(t)g,U_a(t)g)+C((E_j-V^a)U_a(t)g,U_a(t)g) &\le C\Vert(H^a-E_j)U_a(t)g\Vert\Vert g\Vert.\end{aligned}
\sum_{j=0}^{\lfloor 8t^2 \rfloor}\exp \left( j - \frac{j^2}{4t^2}\right) \leq \sum_{j=0}^{\lfloor 8t^2 \rfloor}\exp \left( j \right) \leq 8t^2\exp(8t^2) \leq L \exp(L)
\Pi^{\mu \nu}= - i \frac{e^{2}}{1 2 \pi^{2}}( p^{\mu}p^{\nu}- g^{\mu \nu}p^{2}) \{c_{A}( m ; \{{\bar{\mu}},{\bar{c}}_{i}^{0}\}) + 6 \int_{0}^{1}d z z ( 1 - z ) \ln [ 1 - z ( 1 - z ) p^{2}/ m^{2}] \};
{\frac{X_{0}^{2}}{2 ( 1 + a )}}+{\frac{Y_{0}^{2}}{2 ( 1 - a )}}-{\frac{X_{3}^{2}}{1 - a^{2}}}= \left({\frac{\mu}{6}}\right)^{2}\mathrm{C a s i m i r}\, ,
\vec{S}_{i}\cdot\vec{D}=\sum_{A=1}^{N}M_{A}(\vec{f}_{i}\times\vec{R}_{A}^{0})\cdot\vec{d}^{A}=\vec{f}_{i}\cdot\sum_{A=1}^{N}M_{A}\vec{R}_{A}^{0}\times\vec{d}^{A}=\sum_{A=1}^{N}M_{A}(R_{A}^{0}\times d_{A})_{i}=0,
+ ({\frac{j}{m c q}})^{2}\alpha^{2}\beta^{2}] - ({\frac{Q j}{4 \pi \epsilon_{0}m c^{2}}})^{2}{\frac{Q^{2}}{m q^{2}}}( 1 - \alpha l n{\frac{q}{r_{0}}}- 2 \alpha^{2})^{2}.
\phi_{y_{1}r_{2}}=\frac{1}{2\pi\sqrt{1-\rho^{2}}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}y_{1}Q(y_{2})e^{-\frac{y_{1}^{2}+y_{2}^{2}-2\rho y_{1}y_{2}}{2(1-\rho^{2})}}dy_{1}dy_{2}
\frac{1}{\varepsilon_{j}}\int_{B_{j}}W(v_{j}(x))~dx=\frac{1}{\varepsilon_{j}}\int_{B_{j}}W(\widetilde{u}_{j}(x))~dx & \leq\frac{M_{W}}{\varepsilon_{j}}\mathcal{L}^{n}(B_{j}\cap P_{\varepsilon_{j}})\leq C_{P_{\rho}}M_{W}\delta~.
{\cal Z}= \int D B_{\mu}D \theta^{\mathrm{s i n g .}}D \theta^{\mathrm{r e g .}}\exp \left\{- \int d^{4}x \left[ \frac 1 4 \left( F_{\mu \nu}+ F_{\mu \nu}^{E}\right)^{2}+ \frac{\eta^{2}}{2}\left( \partial_{\mu}\theta - 2 g_{m}B_{\mu}\right)^{2}\right] \right\},
\left( \begin{array}{l}{X^{\Lambda}}{-{\frac{i}{2}}F_{\Lambda}}\end{array}\right) \Longrightarrow \left( \begin{array}{l}{X^{0}}{- i X^{1}}\end{array}\right) , ~ ~ \left( \begin{array}{l}{X^{1}}{- i X^{0}}\end{array}\right) .
\mathcal{R}(\mu,m,K,\Delta;s)= \left[ \left( 1+K \right)m\mu -\left( m+\mu K\left( 1+\Delta \right) \right)\overline{\gamma}s \right] \times \left[ \left( 1+K \right)m\mu -\left( m+\mu K\left( 1-\Delta \right) \right)\overline{\gamma}s \right].
(\begin{matrix}a_{1}\\a_{3}\\ \end{matrix})_{j}=\frac{1}{144}(\begin{matrix}130&-34\\ -34&10\\ \end{matrix})(\begin{matrix}-2&-1&0&1&2\\ -8&-1&0&1&8\\ \end{matrix})(\begin{matrix}y_{j-2}\\y_{j-1}\\y_{j}\\y_{j+1}\\y_{j+2}\\ \end{matrix})
\begin{aligned}\nabla^*\, \Bigl (\sum_{s=1}^3 dh(\xi_s) I_sA_s\Bigr )\ =\ \sum_{s=1}^3 \ \nabla dh\,(I_s e_a, \xi_s)A_s(e_a); \nabla^*\, \Bigl (\sum_{s=1}^3 dh(\xi_s) I_sA \Bigr )\ =\ \sum_{s=1}^3 \ \nabla dh\,(I_s e_a, \xi_s)A(e_a). \end{aligned}
\begin{pmatrix}\displaystyle\frac{-1 - \sqrt{-3 - 4 \alpha_{2}\alpha_{3}}}{2}\alpha_2 0 \alpha_3 \displaystyle\frac{-1 + \displaystyle\sqrt{-3 - 4 \alpha_{2}\alpha_{3}}}{2}0 \alpha_5 \alpha_6 1\end{pmatrix}.
v^k\in A^{[\varepsilon_k^1]}(x^{k+1})+Bx^{k+1}\subseteq A^{[\varepsilon_k^1]}(x^{k+1})+B^{[\varepsilon_k^2]}(x^{k+1})\subseteq (A+B)^{[\varepsilon_k]}(x^{k+1})=T^{[\varepsilon_k]}(x^{k+1})=T^{[\varepsilon_k]}(y^k).
x(t):=e^{-\frac{1}{\alpha}\left(\frac{t}{a}\right)^{\alpha}}\left(e^{\frac{1}{\alpha}}x_{0}+{_{\alpha}\mathfrak{J}}_{a}^{t}\left[\frac{g(s)}{e^{-\frac{1}{\alpha}(\frac{s}{a})^{\alpha}}}\right]\right)
-\frac{\partial KL(q\|p)}{\partial q_{j}}=\frac{\sum_{j}p_{j}}{\sum_{j}q_{j}}\left(\sum_{i}\overline{q}_{i}\log \frac{\overline{q}_{i}}{\overline{p}_{i}}-\log \frac{\overline{q}_{j}}{\overline{p}_{j}}\right)
\beta_{s,a^2}^2=\frac{\left[r^2(s,a_s^1,a^2)-r^2(s,a_s^1,a_s^2)\right]}{\left[\sum_{s'\in S}p_{s'}\r^2(s',a_{s'}^1,a_{s'}^2)-\sum_{s'\in S}p(s'|s,a_s^1,a^2)r^2(s',a_{s'}^1,a_{s'}^2)\right]}.
\mathfrak{D}(f, (1,3,6)) = \left\{(x_1,x_2,x_1+x_2,1-x_1-x_2,1-x_2,1-x_1)\left|\begin{array}{l}0\le x_1\le x_2, x_1+x_2 \le1/2\end{array}\right.\right\}=\, \{C+y_1\delta_1+y_2\delta_2\mid y_2/\sqrt{3}\le y_1\le\sqrt{3}y_2,\sqrt{3}y_1+y_2\le\sqrt{3}\},
f_0(x,y) \ = \x + 1, begin{align*}3pt] f_1(x,y) \ = \x +y, begin{align*}3pt] f_2(x,y) \ = \x y, begin{align*}3pt] f_{n+1}(x,0) \ = \ 1, begin{align*}3pt] f_{n+1}(x,y+1) \ = \f_n(x,f_{n+1}(x,y)) (n \geq 2).
f\left(s^{-1}\begin{bmatrix}x Ly-xL 0 y \end{bmatrix}s\right) = f\left(\sigma^{-1}\begin{bmatrix}x 0 0 y \end{bmatrix}\sigma\right) = \sigma^{-1}\begin{bmatrix}f(x) 0 0 f(y) \end{bmatrix}\sigma = s^{-1}\begin{bmatrix}f(x) Lf(y)-f(x)L 0 f(y) \end{bmatrix}s,
\Lambda^{+}_{t}=\{\lambda_{1}+t,\lambda_{2}+t,\lambda_{3},\ldots,\lambda_{n}\}\in \mathcal{CR}_{n}\Lambda^{-}_{t}=\{\lambda_{1}+t,\lambda_{2}-t,\lambda_{3},\ldots,\lambda_{n}\}\in \mathcal{CR}_{n}.
X(t) = x+\int_0^t A(s,\overline{X}(s))ds+\int_0^t B(s,\overline{X}(s))dW(s) \quad\ +\int_0^t \int_{\{|z|<1\}}H(s,\overline{X}(s),z)\widetilde{N}(ds,dz) &\quad\ +\int_0^t\int_{\{|z|\ge1\}}J(s,\overline{X}(s),z)N(ds,dz).
F (d_{\Lambda}( EZ (t \otimes \sigma ) )) (m)= F( EZ ( d_{\Lambda}t \otimes \sigma) ) (m) \pm F( EZ( t \otimes \partial' \sigma ) ) (m) \pm \sum_{(\sigma)}F( EZ ( t \otimes \sigma'' ) ) ( \tau(\sigma') \cdot m) \pm \sum_{(\sigma)}\tau(\sigma'') \cdot F( EZ(t \otimes \sigma' ))(m)
\left(I_{-}^{\alpha,\alpha',\beta,\beta',\gamma}t^{-\rho}\right)(x)=\frac{\Gamma(-\beta+\rho)\Gamma(\alpha+\alpha'-\gamma+\rho)\Gamma(\alpha+\beta'-\gamma+\rho)}{\Gamma(\rho)\Gamma(\alpha-\beta+\rho)\Gamma(\alpha+\alpha'+\beta'-\gamma+\rho)}x^{-\alpha-\alpha'+\gamma-\rho}.
\left( \begin{matrix}{{z_{1}}}\\{{z_{2}}}\\ \end{matrix}\right) \rightarrow U_{L}\, \left( \begin{matrix}{{z_{1}}}\\{{z_{2}}}\\ \end{matrix}\right) ~ ~ ; ~ ~ \left( \begin{matrix}{{z_{1}}}\\{{\bar{z}_{2}}}\\ \end{matrix}\right) \rightarrow U_{R}\, \left( \begin{matrix}{{z_{1}}}\\{{\bar{z}_{2}}}\\ \end{matrix}\right)
\vec{A_{1}}= \left( \begin{array}{c}{{x}}\\{{0}}\\{{0}}\\ \end{array}\right) \; \vec{A_{2}}= \left( \begin{array}{c}{{0}}\\{{x}}\\{{0}}\\ \end{array}\right) \; \vec{\phi}= \left( \begin{array}{c}{{- x}}\\{{x}}\\{{0}}\\ \end{array}\right) \; \dot{\vec{\phi}}= \frac{1}{2}\left( \begin{array}{c}{{x}}\\{{x}}\\{{0}}\\ \end{array}\right) .
[{\cal L}_{\alpha},{\cal P}_{\beta}] = - i ( 1 + \nu R ) ( P \gamma )_{\alpha \beta}, \quad \{{\cal L}_{\alpha},{\cal P}_{\beta}\}= 4 i ( P J ) \epsilon_{\alpha \beta}- 4 \Gamma_{\alpha \beta}, \quad \Gamma_{\alpha \beta}\equiv \epsilon_{\mu \nu \lambda}P^{\mu}J^{\nu}\gamma_{\alpha \beta}^{\lambda},
\|g_k^{j,t}(x)\|^2_{L^2(C^{j,t}_k, \mu_2)}\leq \frac{C}{\omega(B^g(2^k\sqrt{t}))}\int_{B^g(x^t_j, 2^{k+2}\sqrt{t})}\int_{B^g(x^t_j, 2^{k+2}\sqrt{t})}|f(x)-f(y)|^2 d\mu_2(x)d\mu_2(y).
\gamma_F(\frac{1}{2}\psi \circ q_{11}) = \gamma_F(\frac{1}{2}\psi)^2 \cdot \gamma_F(-J_2, \frac{1}{2}\psi) \cdot ( - \frac{a_1 - c_1 t}{2 \nu_1 u (b_1 - d_1 t)}, J_2 )_F = \gamma_F(J_2, \frac{1}{2}\psi) \cdot ( \frac{a_1 - c_1 t}{2 \nu_1 u (b_1 - d_1 t)}, J_2 )_F.
\omega^{(3,3)}(\xi) =(1-\xi)(I_{0}+I_{1})+(1-\xi)\sum_{j=0}^{\infty}I_{j+2}\xi^{j}+(1-\xi)^{2}(I_{0,1}^{2}+I_{1,2}^{2}) +(1-\xi)^{2}\sum_{j=0}^{\infty}I_{j+2,3}^{2}\xi^{j}+(1-\xi)^{3}(I_{0,1}^{3}+I_{1,2}^{3})+(1-\xi)^{3}\sum_{j=0}^{\infty}I_{j+2,3}^{3}\xi^{j},
\langle A^{-1}\xi_{0},\xi_{0}\rangle =\frac{|x_1|^{2}}{a|x_1|^{2}+c_1|x_{\sigma(1)}|^{2}}+\frac{|x_2|^{2}}{a|x_2|^{2}+c_2|x_{\sigma(2)}|^{2}}+\cdots+ \frac{|x_n|^{2}}{a|x_n|^{2}+c_n|x_{\sigma(n)}|^{2}}&\leq 1.
(f_{\mathcal{S}})_{\lambda}(s_1, s_2) = \sup_{x^\prime \in \mathcal{X}}\left\{- y_2 d(x') - y_1 (1 - d(x')) - \sum_{1 \leq \ell \leq 2}\lambda_\ell \| x_\ell - x' \| \right\}& = - \min\{y_2 + \varphi_{\lambda, 1}(x_1, x_2), y_1 + \varphi_{\lambda, 0}(x_1, x_2) \},
M_{n,k}-M_{n,k-1}=\int_{0}^{+\infty}zf_{n,k}(z;0)dz &=\int_{0}^{+\infty}z\left(f_{n,k}(z;0)-f_{n,k}(z;0)\right)dz &=-\frac{1}{n-k+1}\int_{0}^{+\infty}z\frac{d}{dz}f_{n,k}(z;0)dz &=\frac{1}{n-k+1}\int_{0}^{+\infty}f_{n,k}(z;0)dz=\frac{1}{n-k+1}.
M_{1 1}^{\beta \gamma}= \frac{T^{\beta \gamma}}{1 - C I^{\beta \gamma - 1}T^{\beta \gamma}+ T^{\beta \gamma 2}}= \frac{( C I^{\beta \gamma})^{5}- C I^{\beta \gamma}}{( 1 - ( C I^{\beta \gamma})^{2}) ( 1 - 3 ( C I^{\beta \gamma})^{2})}
a_0^\tau(p_i,p_{i+1},p_{i+2}) = \bigl((4-3\tau)p_{i}+ (4+2\tau)p_{i+1}+ \tau p_{i+2}\bigr)/8 a_1^\tau(p_i,p_{i+1},p_{i+2}) = \bigl((1-\tau) p_{i}+ (6-2\tau)p_{i+1}+ (1+3\tau)p_{i+2}\bigr)/8 .
b_{n-1}=\sum_{l=0}^{n-2}\binom{n-2}{l}\frac{d^l\gamma}{dv^l}a_{n-1-l}&=\gamma a_{n-1}+\sum_{l=1}^{n-2}\binom{n-2}{l}\frac{d^l\gamma}{dv^l}a_{n-1-l}.
\:g_c^2 \sim\frac{4}{\kappa}\,[1-\frac{\kappa^2 \lambda^2}{M^2}]\,, \hspace{0.5cm}y_0 \sim \frac{1}{\kappa\sqrt{\kappa}\,}\,M\,,\hspace{0.5cm}\alpha \sim -2\,\frac{\kappa\lambda}{M}\,,\:
g_{ab}^j(\xi_j^1(z,t),...,\xi_j^n(z,t),t_1,...,t_m)=\sum_{r,s=1}^n h_{rs}(z,t)\frac{\partial \xi_j^a(z,t)}{\partial z_r}\frac{\partial \xi_j^b(z,t)}{\partial z_s}
\left(\left\{\mathbf{\bar{G}}_{ij,kl}\mathbf{\bar{v}}_{kp,l}^{[\sf bs](\mathbf{s}^{[\sf d]})}\right\}_{p\in[1:N], k\in[1:K], l\in[1:M], \mathbf{s}^{[\sf d]}\in \mathcal{S}_{T_{\sf d}}^{[\sf d]}~}\right)
-\!\!\sum_{k\in\mathbb{Z}\setminus\{0\}}\frac{\mathcal{C}_{q}(qc_{k}(q))}{c_{k}(q)\mathcal{C}_{q}'(c_{k}(q))}P_{n}\left(q^{1/2}c_{k}(q);q\right)P_{m}\left(q^{1/2}c_{k}(q);q\right)=\delta_{mn}
q_c(a,b,r)=\frac{1}{8\pi}\sum_{m=-\infty}^{+\infty}\int_{-\infty}^{+\infty}{dk\sum_{\gamma ,\lambda}}\beta_{\lambda m}^2f_{\lambda m}^{(q)}(\gamma a, \gamma r),\quad q_c=\varepsilon ,\, p_i,
\mathrm{P}(\mathcal{O}_{2}) = \mathrm{P}\left(\min\left\{\log(1+\rho|h_{n^*}|^2\alpha_2^2), \right.\right. \left.\left. \log(1+\rho|g_{n^*,2}|^2\alpha_2^2)<2R_2\right\},|\mathcal{S}_r|>0\right) = \mathrm{P}\left(x_{n^*}<2R_2 ,|\mathcal{S}_r|>0\right).
cot^{2}x_{1}+cot^{2}x_{2}+\cdot\cdot\cdot+cot^{2}x_{m}=\frac{(\begin{matrix}2m+1\\ 3\end{matrix})}{(\begin{matrix}2m+1\\ 1\end{matrix})}=\frac{2m(2m-1)}{6}.
W(\phi) = \int_{\phi_0}^{\phi}e^{\alpha\eta}\left( \frac{\beta}{2}\eta^2 - \Lambda \right) d\eta = \left. e^{\alpha\eta}\left[ \frac{\beta}{2}\left( \frac{2}{\alpha^3}- \frac{2\eta}{\alpha^2}+ \frac{\eta^2}{\alpha}\right) - \frac{\Lambda}{\alpha}\right] \right|_{\phi_0}^{\phi}.
\Big|\mathbb{P}(R_{N}> \beta_{N}t) - \mathbb{P}(\sigma^{+}(u) \notin V, \forall u \in \{N^{1+\delta}, \dots, \beta_{N}t\}) \Big| \leq \leq \Big|\mathbb{P}(R_{N}\leq \beta_{N}t) - \mathbb{P}(N^{1 + \delta}\leq R_{N}\leq \beta_{N}t) \Big| = \mathbb{P}(R_{N}< N^{1 + \delta}) \xrightarrow[]{N \rightarrow \infty}0
\begin{array}{l}\displaystyle [e_1,e_{n+1}]=e_3,[e_2,e_{n+1}]=e_2,[e_{i},e_{n+1}]=e_{i}+\sum_{k=i+2}^n{a_{k-i+3,3}e_k}, \displaystyle [e_{n+1},e_1]=-e_3,[e_{n+1},e_i]=-e_i-\sum_{k=i+2}^n{a_{k-i+3,3}e_k},(3\leq i\leq n,n\geq4).\end{array}
\forall n \in \omega,\varphi_n(F_n) \ge \frac{\inf_{k \in \omega}\| \omega\|_{\varphi_k}}{2}\,\,\,\,X_n \cap \bigcup_{\substack{k \in \omega: M_k \cap \,\bigcup_{i\le n-1}X_i \neq \emptyset}}M_k =\emptyset,
F(t, \rho,G, g):= \langle \rho_t, G_t\rangle - \langle g, G_0\rangle - \int_0^t \langle \rho_s, \partial_s G_s\rangle ds- \int_0^t \langle \rho^2_s, [-(- \Delta)^{\gamma/2}G_s]\rangle ds;