harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
C.u= C\big(f(h)(r(h)+\frac{i}{2n}(C.r(h))) \big) \equiv \big(f^{\prime\prime}(h)B+f^{\prime}(h)C\big)\big(r(h)+\frac{i}{2n}(C.r(h))\big) \equiv ihf^{\prime\prime}(h)(r(h)+\frac{i}{2n}C.r(h))-2nif^{\prime}(h)(r(h)+\frac{i}{2n}C.r(h)) \equiv \big(ihf^{\prime\prime}(h)-2nif^{\prime}(h)\big)\big(r(h)+\frac{i}{2n}C.r(h)\big),
\left|\frac{zf_1'(z)}{f_1(z)}\right| =\left|1+\frac{\rho_1}{(1-\rho_1^2)}\left( \frac{u\rho_1^2+4\rho_1+u}{\rho_1^2+u\rho_1+1}+\frac{v\rho_1^2+4\rho_1+v}{\rho_1^2+v\rho_1+1}+\frac{q \rho_1^2+4\rho_1+q}{\rho_1^2+q \rho_1+1}\right)\right| = \frac{5}{3}.
{\cal L}_{S C M}= \frac 1 2 \biggl ( J_{\mu}^{j}J^{\mu j}+ i \overline{{\chi}}^{j}/ \kern - . 5 e m \partial \chi^{j}+ i \varepsilon^{j k l}\overline{{\chi}}^{j}/ \kern - . 5 e m J^{k}\chi^{l}+ \frac 1 4 ( \bar{\chi}\chi )^{2}\biggr ) .
\begin{cases}k\prod_{i=1}^{\frac{D}{2}}(x-\alpha_i)^2-x(x-1)(x-\lambda)\prod_{i=1}^{\frac{D}{2}-2}(x-\beta_i)^2=k\prod_{i=1}^{\frac{D}{2}}(x-\gamma_i)^2 k\prod_{i=1}^{\frac{D}{2}}(x-\alpha_i)^2-\lambda x(x-1)(x-\lambda)\prod_{i=1}^{\frac{D}{2}-2}(x-\beta_i)^2=k\prod_{i=1}^{\frac{D}{2}}(x-\delta_i)^2\end{cases}
\chi^-_1 + \cdots + \chi^-_{k_1}\leq \sum_{i=1}^{k_1}\int \log r^-_i d\mu \ \textrm{and}\ \chi^+_1 + \cdots + \chi^+_{k_2}\geq \sum_{i=1}^{k_2}\int \log r^+_i d\mu.
e^{I}:= \frac{1}{n}\underset{\mu \in \Xi}{\sum}\| \sum_{l=1}^{k}\int_{{\omega}}(u(\mu,z) - \mathcal{I}_{L}[u](\mu,z)) \kappa_{l}(z) \,d{z}\, \kappa_{l}\|_{L^{2}(I)}^{2}
\begin{aligned}\int dxdy\e^{-i(x\cdot \xi+y\cdot \eta)}\delta(x-y-z)f(x)g(y) &= e^{iz\cdot \eta}\int dx\e^{-ix\cdot(\xi+\eta)}f(x)g(x-z) = e^{iz\cdot\eta}\widehat{f g_z}(t, \xi+\eta).\end{aligned}
\Phi_+(z)=\Phi_-(z)\begin{pmatrix}1 0 e^{\pi i \alpha}1\end{pmatrix}, z\in\Sigma_1, &\Phi_+(z)=\Phi_-(z)\begin{pmatrix}0 1 -1 0\end{pmatrix}, z\in \Sigma_2, &\Phi_+(z)=\Phi_-(z)\begin{pmatrix}1 0 e^{-\pi i \alpha}1\end{pmatrix}, z\in\Sigma_3, &\Phi_+(z)=\Phi_-(z) e^{\pi i \alpha \sigma_3}, z\in\Sigma_4.
\frac{d}{d\epsilon}\frac{1}{2}\int_{X}|I\circ(\phi^{-1}+\epsilon\delta\phi^{-1}\circ\phi^{-1})-J|^{2}dx|_{\epsilon=0}=\int_{X}(I\circ\phi^{-1}-J)\nabla I|_{\phi^{-1}}\delta\phi^{-1}\circ\phi^{-1}dx
\begin{cases}\nabla\times E-ik\mu H=0 \mbox{in}\Omega\backslash\bar{D}, \nabla\times H+ik\epsilon E=0 \mbox{in}\Omega\backslash\bar{D}, \nu\times E=f \mbox{on}\partial\Omega, \nu\times H=0 \mbox{on}\partial D,\end{cases}
\begin{array}{c c c}{{V^{r}= r^{m +{\textstyle{\frac{1}{2}}}}\cos ( \delta + ( p - 1 ) \varphi ) ,}}&{{}}&{{V^{\varphi}= r^{m -{\textstyle{\frac{1}{2}}}}\sin ( \delta + ( p - 1 ) \varphi ) .}}\\ \end{array}
\frac{1}{2}\int_{\mathbb{R}^{n}}c(\xi)|\nabla V^{\xi}(z)|^{2}+a(\xi)\left(V^{\xi}\right)^{2}(z)dz-\frac{1}{p}\int_{\mathbb{R}^{n}}b(\xi)\left(V^{\xi}\right)^{p}(z)dz & =\left( \frac{p-2}{2p}\int_{\mathbb{R}^{n}}U^{p}dz\right) \frac{c(\xi)^{\frac{n}{2}}a(\xi)^{\frac{p}{p-2}-\frac{n}{2}}}{b(\xi)^{\frac{2}{p-2}}}
(-\Delta_p)^s u(x) = 2\, \lim_{\varepsilon \searrow 0}\int_{\mathbb{R}^N \setminus B_\varepsilon(x)}\frac{|u(x) - u(y)|^{p-2}\, (u(x) - u(y))}{|x - y|^{N+s\,p}}\, dy, x \in \mathbb{R}^N.
\int \frac{d^{3}q}{( 2 \pi )^{3}}\frac{1}{( q^{2}+ m^{2}) [ ( p - q )^{2}+ m^{2}] [ ( q - p_{1})^{2}+ m^{2}]}= \frac{1}{1 6 \pi \mu^{3}}\ln \left( \frac{4}{9 \Delta}\right) .
\nabla^a\left(-\frac{2}{r^2}DH_1\cdot H_2 X_a\right) =_s -\frac{2}{r^2}\nabla_X H_1\cdot D^\dagger H_2-\frac{2}{r^2}DH_1\cdot\nabla_X H_2-\nabla_X\left(\frac{2}{r^2}\right)DH_1\cdot H_2 &-\frac{1}{r^2}\left(\omega^X+\frac{2M}{r^2}f\right)DH_1\cdot H_2-\frac{1}{r^2}\left(\omega^X+\frac{2M}{r^2}f\right)H_1\cdot D^\dagger H_2.
\xi_{1}= c ( \sin \frac \vartheta 2 )^{- m + F}( \cos \frac \vartheta 2 )^{- m + F + 2 g + 1}P_{n}^{( - 1 / 2 - m + F , 2 g - m + F + 1 / 2 )}( \cos \vartheta )
\begin{aligned}\nu \partial_y D^\alpha u=\nu D^\gamma\partial^2_y u=D^\gamma \bigg\{\big[(u+u_e\phi(y)+u_b(1-\phi(y)))\partial_x +v\partial_y\big]u &-\big[(h+h_e\phi(y))\partial_x+g\partial_y\big]h &-gh_e\phi'(y)+v(u_e-u_b)\phi'(y)-r_1 \bigg\}.\end{aligned}
R(t;q) =(-qt;q^2)_\infty\sum_{k=0}^\infty\frac{q^{k(k+1)}t^k}{(q^2,-qt;q^2)_k}&=(-q^2t;q^2)_\infty\sum_{k=0}^\infty\frac{q^{k^2}t^k}{(q^2,-q^2t;q^2)_k}&=(q^{1/4}t^{1/2};q^{1/2})_\infty\sum_{k=0}^\infty\frac{q^{k^2/4}t^{k/2}}{(q^{1/2},q^{1/4}t^{1/2};q^{1/2})_k}.
\psi_{i}( x ) = \partial \varphi_{i}( x ) = \pi_{i}( x ) + \varphi_{i}^{\prime}( x ) , \{\overline{\psi}}_{i}( x ) ={\overline{\partial}}\varphi_{i}( x ) = \pi_{i}( x ) - \varphi_{i}^{\prime}( x ) .
\lim_{s\rightarrow0}\Gamma(s)\left( (cz+d)^{-s}G(Vz,s;A,B)-G(z,s;A_{d},B_{a})\right) & =2\pi is\left( d,c;A_{-d},B_{-a}\right) -\frac{\pi i}{c(cz+d)}B_{0}(B)B_{2}(A)-2\pi i\frac{cz+d}{c}B_{0}(B)P_{2}(0,A_{d}).
\left. \times \left( \frac{r^{2}}{r !}( \frac{1}{\operatorname{s i n h}{\frac{\xi}{2}}})^{r + 1}+ \frac{r - 1}{( r - 1 ) !}( \frac{1}{\operatorname{s i n h}{\frac{\xi}{2}}})^{r - 1}\right) \right] .
{\rm Coker\,}\tilde\sigma_{G,s}={\rm Coker\,}\tilde\sigma_G:=\{u\in H^0_b(\widehat X_G)_{s-\frac{d}{4}-\frac{1}{2}};\, (\,u\,|\,\tilde\sigma_{G,s}v)_{\widehat X_G}=0, \forall v\in H^0(\overline M)^G_s\cap C^\infty(\overline M)\}.
{\rm Hilb}\big((6T_n)^{!},t\big){\rm Hilb}\big((6T_m)^{!},t\big)=\sum_{k=0}^{\min (n,m)-1}{\min(n,m) \brace \min(n,m)-k}{\rm Hilb}\big((6T_{n-k,m-k})^{!},t\big) t^{2k},
\overrightarrow{MP}\cdot \overrightarrow{AP}= \overrightarrow{MP}\cdot [\overrightarrow{AO}+\overrightarrow{OM}+\overrightarrow{MP}], = \Re \{\lambda\,e^{-i\theta_0}\, (-e^{i\theta_0}+r\,e^{i\theta}+\lambda\,e^{i\theta_0})\}, = \lambda [-1+r\cos(\theta_0-\theta)+\lambda].
\begin{aligned}\sqrt{m+1}\,C_{m+1,n}(i,k)= \left(\frac{i}{\alpha}\right) \cos \theta\,C_{m,n}(i-1,k) &-\left(\frac{k}{\beta}\right) \sin \theta\,C_{m,n}(i,k-1)+(\beta \sin \theta-\alpha \cos \theta) C_{m,n}(i,k).\end{aligned}
\xi \left( x,t\right) =\left[ x_{n}+\left( x_{1}-1/2\right)^{2}/\omega^{2}+\displaystyle\sum\limits\limits_{k=2}^{n-1}x_{k}^{2}+\left( t-T/2\right)^{2}+1/4\right]^{-\nu}.
(\frac{\partial(\vec{f}_{0}...\vec{f}_{k})}{\partial(\vec{a}_{0}...\vec{a}_{n})})_{\vec{b}}d\vec{a}=-(\frac{\partial(\vec{f}_{0}...\vec{f}_{k})}{\partial(\vec{b}_{0}...\vec{b}_{m})})_{\vec{a}}d\vec{b}
{{cdt}\over{dx}}={{\hat{c}d\hat{t}}\over{d\hat{x}}},\ \ \ \ \{\hbar\over{dEdt}}={\hat{\hbar}\over{d\hat{E}d\hat{t}}},\ \ \ \ \{{G_4dE}\over{c^4dx}}={{\hat{G}_4d\hat{E}}\over{\hat{c}^4d\hat{x}}},\ \ \ \ \{e^2\over{dEdx}}={\hat{e}^2\over{d\hat{E}d\hat{x}}},
(\overline{L}_0\cdots\overline{L}_d|Z)=(\overline{L}_0\cdots\overline{L}_{d-1}|Z\cap \mathrm{div}(s_d))-\sum_{v\in M(K)}\int_{Z_{\overline{K}_v}^{\mathrm{an}}}\log\|s_d\|_v \prod_{i=0}^{d-1}c_1(\overline{L}_{i,v})
u_{i i}= \frac{{1-2d}}{1+ \beta d}(A + \frac{1}{2}M) + \frac{{\beta [-\sum\limits_j{u_j d_j}- \varphi ]}}{1+ \beta d}+ \frac{{-\sum\limits_j{u_j d_{ji}}-D_i \varphi}}{{d_{i}}}.
\prod_{n = 1}^{\infty}( 1 - q^{n})^{3}= \operatorname*{l i m}_{\epsilon \rightarrow 0}\sum_{- \infty}^{\infty}\frac{1}{\epsilon}\left( \frac{- 1}{1 - \epsilon}\right)^{n}q^{\frac{1}{2}n^{2}i \frac{1}{2}n}
B^h_i = \omega^2_i+ dA_i, d\omega^2_i=0, A^h_{ij}=\check{\delta}(A.)_{ij}- \omega^1_{ij}+ d\log f_{ij}, \check{\delta}\omega^2. = - d\omega^1_{ij}, (\check{\delta}f..)_{ijk}= (f^h_{ijk})^{-1}.
\begin{aligned}0 \to \left(T_{\mu_g(\kappa)^{-1}(X, D, \alpha, m)}\right)_{(X, D, \alpha, m)}\to \left(T_{\mathcal B_g(\kappa)}\right)_{(X, D, \alpha, p, m)}\to \left(T_{\mathcal A_g(\kappa)}\right)_{(X, D, \alpha, m)}\xrightarrow{\left(d\mathfrak c_g(\kappa)\right)_{(X, D, \alpha, m)}}H^1\left(X, K_X^{\otimes 2}(-D)\right) \to 0\end{aligned}
[\phi]^\sim(\alpha_1,\dots,\alpha_n) =\sup^+\{\alpha:\exists \alpha_1'<\alpha_1,\dots,\alpha_n'<\alpha_n\ \alpha<\phi(\alpha_1',\dots,\alpha_n')\}&=\sup\{\phi(\alpha_1',\dots,\alpha_n'):\alpha_1'<\alpha_1,\dots,\alpha_n'<\alpha_n\}.
K_iK_i^{-1}=1,K_iK_j=K_jK_i, &K_iE_jK_i^{-1}=q^{(\alpha_i,\alpha_j)}E_j,K_iF_jK_i^{-1}=q^{-(\alpha_i,\alpha_j)}F_j, &E_iF_j-F_jE_i=\delta_{ij}\frac{K_i-K_i^{-1}}{q_i-q_i^{-1}}, &\sum_{m=0}^{1-a_{ij}}(-1)^m E_i^{(m)}E_j E_i^{(1-a_{ij}-m)}=\sum_{m=0}^{1-a_{ij}}F_i^{(m)}F_j F_i^{(1-a_{ij}-m)}=0(i\neq j),
V=\frac{1}{2{b}_{11}}\left( \tilde{m}^{\dagger}\tilde{m}-m^{\dagger}m\right)^{2}+\frac{2}{{b}_{11}}\left| a_{D}\right|^{2}\left( mm^{\dagger}+\tilde{m}\tilde{m}^{\dagger}\right) +\frac{{b}_{01}}{{b}_{11}}D_{0}\left(mm^{\dagger}-\tilde{m}^{\dagger}\tilde{m}\right)
i)\,D_{1, g}\,(=D_{1,g, m})\,\equiv\sum_{c,\, i(c, g)=1}d_{c,g, m}=\sum_{q=1}^{p_1}e^{-\frac{i2\pi qm}{p_1}};ii)\,(k\ge\ell\ge 2)D_{\ell, g}\,(=D_{\ell,g, m})\,\equiv\sum_{c,\, i(c, g)=\ell}d_{c,g, m}.
\pi_{2}\circ \pi_{1}(x^{(f)}_i) = x^{(f)}_{\pi_{1}( \pi_2(i))}=x^{(T_1(f))}_{\pi_2(i)}= x^{(g)}_{\pi_2(i)}= x_i^{(T_2(g))}= x_i^{(T_2 \circ T_1(f))}.
-504\left\{\widehat{A}(TX){\rm det}^{\frac{1}{2}}{\rm cosh}(\frac{\sqrt{-1}}{4 \pi}R^V)\right\}^{(12)}=\left\{\widehat{A}(TX){\rm det}^{\frac{1}{2}}{\rm cosh}(\frac{\sqrt{-1}}{4 \pi}R^V)\right. &\left.\cdot{\rm ch}(\widetilde{TX}+2\wedge^2\widetilde{V_C}-\widetilde{V_C}\otimes \widetilde{V_C}+\widetilde{V_C})\right\}^{(12)}.
<L-\zeta(X),\frac{\partial \zeta(X)}{\partial X_i}>=0 \mbox{pour}\,i\in \left\{1,2,...,m\right\}\mbox{ou}<e,\frac{\partial \zeta(X)}{\partial X_i}>=0 \mbox{pour}\,i\in \left\{1,2,...,m\right\}
{}_{r + 1}\phi_{r}\biggl(\genfrac..{0pt}{}{a,\, b_{1}, \dotsc, b_{r}}{c_{1}, \dotsc, c_{r}}; q,\, x\biggr) ={}_{r + 1}\phi_{r}(a, b_{1}, \dotsc, b_{r}; c_{1}, \dotsc, c_{r}; q, x) &:= \sum_{i = 0}^{\infty}\frac{(a)_{i}(b_{1})_{i}\dotsm (b_{r})_{i}}{(q)_{i}(c_{1})_{i}\dotsm (c_{r})_{i}}x^{i},
T_{\mu \nu}^{e f f}= \frac{\zeta + b_{k}}{\zeta + b_{g}}\varphi_{, \mu}\varphi_{, \nu}- K \tilde{g}_{\mu \nu}+ \frac{b_{g}M^{4}}{( \zeta + b_{g})^{2}}e^{- 2 \alpha \phi / M_{p}}\tilde{g}_{\mu \nu}+ \frac{\zeta}{\zeta + b_{g}}T_{\mu \nu}^{( f , c a n o n i c a l )}- \frac{m F ( \zeta )}{\sqrt{\zeta + b_{g}}}\overline{{{\Psi}}}^{\prime}\Psi^{\prime}\tilde{g}_{\mu \nu},
\frac{1}{\pi}\sum_{n=1}^{\infty}\frac{sin2\pi nx\cdot lnn}{n}=ln\Gamma(x)-\frac{1}{2}ln(2\pi)+\frac{1}{2}ln(2sin\pi x)-\frac{1}{2}(\gamma+ln2\pi)(1-2x),0<x<1,
f_{k-1}:= (\rho_1 D_{\rho_1}-\rho_2 D_{\rho_2})u_k-f_k = \bigl((\rho_1 D_{\rho_1}-z)-(\rho_2 D_{\rho_2}-z)\bigr)u_k-f_k = \rho_2^{i z}(\log\rho_2)^{k-1}a_{k-1}(\rho_1),a_{k-1}:=i k b_k,
\frac{d}{dt}\Big[(6+t)E(U_{m+1},t)+2E_*(U_{m+1},t)\Big] &=\|U_{m+1}'\|^2+\|A^{1/2}U_{m+1}\|^2+2(6+t)(U_{m+1}',-U_{m+1}'-V_{m}') &+4\|U_{m+1}'\|^2+4(U_{m+1},-AU_{m+1}-V_{m}') &=-(7+2t)\|U_{m+1}'\|^2-3\|A^{1/2}U_{m+1}\|^2-2(6+t)(U_{m+1}',V_{m}')+4(U_{m+1},AV_{m,1}) &=-(1+t)\|U_{m+1}'\|^2-\|A^{1/2}U_{m+1}\|^2-(6+t)\|V_{m}'\|^2+2\|A^{1/2}V_{m,1}\|^2.
\mathcal{K}(z,w)=z^{-N_f}\frac{p_{N_f}(z)p^{*}_{N_f}(w)-p^{*}_{N_f}(z)p_{N_f}(z)}{1-z^{-1}(w)}\sqrt{v_N(z)v_N(w)e^{V(z)}e^{V(w)}},
h(x) \le \Bigl(g_1(x)+\frac{2}{6^2}(3^2 x-3)(2-2^2x)+\frac{2}{6^3}(2^3x-3)(11-3^3x)\Bigr)\Bigr|_{x=\frac{2}{5}}& + \frac1{10\cdot 6^3}- \eta=-\frac{40623864934079027775310823389107}{72980744104199633310494242733558000}<0,
u_{t}= -\frac{1}{2}t^{-3/2}[\phi(\frac{x}{\sqrt{t}})+\frac{x}{\sqrt{t}}\phi'(\frac{x}{\sqrt{t}})] u_{x}=\frac{1}{t}\phi'(\frac{x}{\sqrt{t}}) u_{xx}=t^{-3/2}\phi''(\frac{x}{\sqrt{t}})
A = \Big( \frac{n-2}{n}, \frac{2}{n}\Big), B=\Big(\frac{n^2-2n-1}{n^2+1}, \frac{2(n-1)}{n^2+1}\Big) C =\Big(\frac{1}{n-1}, \frac{1}{n-1}\Big), D= \Big( \frac{n-2}{n}, \frac{n-2}{n}\Big).
\begin{cases}dX_t=[(a+\lambda\theta)(m(t)-X_t)-\lambda Y_t - R_t]\,dt+\sigma\,dW_t+ \left [\theta(m(t-)-X_{t-})-\left(Y_{t-}+\frac{R_{t}}{\lambda}\right)\right]\,d\widetilde N_t X_0=x_0\end{cases}
\begin{array}{l}\mathop{\max}\limits_{{\rho_i}}\;\;\;{u_i^{AF}}\left({{\rho_i},{\pmb \rho_{- i}}}\right) = \frac{1}{2}\log \left({1 + \gamma_i^{AF}}\right) ~~~~~~~~~~~~~~{\rm{s.t.}}\;\;\;{\rho_i}\in{{\mathcal A}_i}\end{array},
\Theta_t(\delta\iota,\delta\phi,\delta\mu)= -\int_{\mathcal{L}}\iota^*(\delta g\,\rho_t)\,\mu - p_{\phi,t}(\iota,\mu)\left(\vartheta_{\iota(p)}(\delta\iota(p))-\delta g(\iota(p))-\delta\phi\right);
\bigl\langle V((\Lambda_{\psi}\otimes\Lambda)[z(1\otimes b)]),\Lambda_{\psi}(w)\otimes\Lambda(c)\bigr\rangle &=(\psi\otimes\eta)\bigl((w^*\otimes c^*)(\operatorname{id}\otimes\operatorname{id}\otimes\varphi)((\Delta\otimes\operatorname{id})[\Delta(r^*y)]\Delta_{23}(s)(1\otimes b\otimes1))\bigr).
\underline{c}\,\underline{g}(\delta(x)):=\underline{c}\,\inf_{\delta(y)=\delta(x)}g(y)\leq u(x)\leq \overline{c}\sup_{\delta(y)=\delta(x)}g(y)=:\overline{c}\,\overline{g}(\delta(x))x\in\Omega,
\Lambda =\left(g_1(z)-zg'_1(z) + (x^0-sz)g'_3(z) + \frac{1}{2}\int_0^{\small{r^2-s^2}}g_6(\xi)\,d\xi+ (r^2-s^2)g_6(r^2-s^2)\right)\times &\qquad\qquad\qquad\qquad\qquad\left(g''_1(z) + (x^0-sz)g''_2(z)\right) -(r^2-s^2)\left[g'_3(z)\right]^2>0, n \geq 2,
a\cdot b=(\prod_{k=1,...,p^{i}}\frac{\xi''-k+1}{k})\cdot \frac{\xi^{p^{j}}}{p^{j}!}=\frac{\xi^{p^{j}}}{p^{j}!}\cdot (\prod_{k=1,...,p^{i}}\frac{\alpha(\xi'')p^{j}+\xi'' - k +1}{k})= b\cdot{\tilde{\xi''}\choose p^{i}}
\sum_{j=1}^{\aleph}\left\langle Q_j, \ [P_{D_j}] \cdot Q_j \right\rangle-\eta \sum_{\ell=1}^{\aleph}\left\langle[P_{D_\ell}]\cdot\sum_{j=1 \atop j \neq \ell}^{\aleph}\Upsilon_{\ell j}^k\cdot Q_j, \ [P_{D_\ell}] \cdot Q_\ell\right\rangle= \sum_{j=1}^{\aleph}\left\langle J_j, [P_{D_j}] \cdot Q_j \right\rangle.
f(\tau)\mid T_{k/2,4N}(\ell^{2})=\sum_{n}\left(a(\ell^{2}n)+\ell^{\lambda-1}\left(\frac{(-1)^{\lambda}n}{\ell}\right)a(n)+\ell^{2\lambda-1}a(n/\ell^{2})\right)q^{n},
Q_{1,\vec{v}_{s,t,f,0}^{(1)}}= \left(\left(\begin{array}{rrr}1 0 0 0 1 0 0 0 1\end{array}\right), \left(\begin{array}{rrr}0 t 1 1 t^{2}+ s t 0 s t + 1 s\end{array}\right), \left(\begin{array}{rrr}0 1 0 0 t 1 1 s 0\end{array}\right)\right),
\overline{\alpha}_3=\left(\begin{array}{cccc}0 -i 0 0 i 0 0 0 0 0 0 -i 0 0 i 0\end{array}\right) ,\hspace{0.3in}\overline{\alpha}_4=\left(\begin{array}{cccc}-i 0 0 0 0 -i 0 0 0 0 -i 0 0 0 0 -i\end{array}\right) .
\begin{array}{r c l}{{\cal H}}&{=}&{( \Lambda^{T})^{- 1}\left( \begin{array}{c}{P_{( p_{1}, p_{2})}}\\{0}\\ \end{array}\right) \epsilon_{3}}\\{}&{=}&{\left( \begin{array}{c}{P^{( 1 )}}\\{P^{( 2 )}}\\ \end{array}\right) \epsilon_{3},}\\ \end{array}
U(R)^{\dagger}\hat{T}_{pqr\cdot\cdot\cdot}^{abc\cdot\cdot\cdot}U(R)=R_{p,\alpha}R_{q,\beta}R_{r,\gamma}\cdot\cdot\cdot\hat{T}_{ijk\cdot\cdot\cdot}^{\alpha\beta\gamma\cdot\cdot\cdot}R_{i,a}^{-1}R_{j,b}^{-1}R_{k,c}^{-1}\cdot\cdot\cdot
e^{\tilde{r}_Tu^2 + \tilde{r}_T'|u|}\frac{\int_{D_{T,u}}e^{\ell_T(g)}d\Pi(g)}{\int_{D_T}e^{\ell_T(g)}d\Pi(g)}= e^{\tilde{r}_Tu^2 + \tilde{r}_T'|u|}\frac{\Pi(D_{T,u}|X^T)}{\Pi(D_T|X^T)},
\rho^\ast \mathrm{d}L^+ \overline{\nabla}_a^\ast \mathrm{d}L^= \rho^i_I \frac{\partial L}{\partial q^i}e^I - C_{I J}^K \xi^J \frac{\partial L}{\partial \xi^K}e^I - \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial L}{\partial \xi^I}e^I ,
\frac{\partial}{\partial \lambda_1}\cdots \frac{\partial}{\partial\lambda_{p+q}}\int_{-\infty}^\infty\:ds\:\eta \left.T^{(a_1)}_\bullet \cdots T^{(a_p)}_\bullet \:\overline{T^{(b_1)}_\bullet}\cdots \overline{T^{(b_q)}_\bullet}\right._{|\lambda_1=\cdots=\lambda_{p+q}=0}\;.
X\triangleq \sqrt{\varphi \left( K\right)}\left( \sqrt{\alpha \left( K\right)}V+\sqrt{\bar{\alpha}\left(K \right)}X'\right)=\sqrt{\varphi \left(K \right)}\left( \sqrt{\alpha \left( K\right) \beta \left( K\right)}U'+ \sqrt{\alpha \left( K\right) \bar{\beta}\left( K\right)}V'+\sqrt{\bar{\alpha}\left( K \right)}X'\right).
\sum_{xx_1\cdots x_r \in \mathcal{P}_r(x)}\frac{1}{d_{H}(x)^2}\prod_{j=1}^{r-1}\frac{1}{(d_{H}(x_j) -1)^2}= \sum_{yy_1\cdots y_r \in \mathcal{P}_r(y)}\frac{1}{d_{H}(y)^2}\prod_{j=1}^{r-1}\frac{1}{(d_{H}(y_k) -1)^2}
K^s_{t}(0) =\mu_0(t)(\alpha_{t}-1)+K^s_{t+1}(0)+\log(1+2^{\mu_0(t)+\Delta{K}^s_{t+1}})-H(\alpha_{t}),~K^s_{n+1}(0)=0, ~K^s_{t}(1) =\mu_1(t)(\beta_{t}-1)+{K}^s_{t+1}(0)+\log(1+2^{\mu_1(t)+\Delta{K}^s_{t+1}})-H(\beta_{t}),~K^s_{n+1}(1)=0,~t\in\{n,\ldots,0\}.
H_NS_N(i,j) =\frac{1}{J_N}(-J_1(J_{N-2}+J_{N-1})+J_0J_{N-2}+2J_2(J_{N-3}+J_{N-2})+2J_0J_{N-3}) =\frac{1}{J_N}(-J_{N-2}-J_{N-1}+2(J_{N-3}+J_{N-2})) =\frac{1}{J_N}(2J_{N-3}+J_{N-2}-J_{N-1}) =0.
\partial_{\sigma}X^{0}|_{\sigma = 0 , \pi}= \partial_{\tau}X^{5 \cdots 9}|_{\sigma = 0 , \pi}= \partial_{\tau}X^{1 \cdots 4}|_{\sigma = 0}= g_{i j}\partial_{\sigma}X^{j}+ 2 \pi \alpha^{\prime}B_{i j}\partial_{\tau}X^{j}|_{\sigma = \pi}= 0
\chi_{[0, \rho_0]}\Bigl( \sqrt{p_y^2 +k_n^2}\Bigr) \hat \varphi^{(4)}_n =\chi_{[0, \rho_0]}\Bigl( \sqrt{p_y^2 +k_n^2}\Bigr) a^{-1}\mathbb{P}_0 |v_{12}|^{\frac 12}\times \bigl(|p_y|^2 + k_n^2\bigr)^{-\frac 12}\xi_n (p_y) \bigl[-\Delta_x + p_y^2 + k_n^2\bigr]^{-1}\widehat{|v_{13}|^{\frac 12}}\hat \varphi^{(1)}_n + \mathcal{O}(1) ,
=e^{-\lambda t}\Biggl[I_{0}\Bigl(\frac{2\lambda t \sqrt{\alpha}}{1+\alpha}\Bigr)+\frac{1}{\sqrt{\alpha}}I_{1}\Bigl(\frac{2\lambda t \sqrt{\alpha}}{1+\alpha}\Bigr) + (1-\alpha)\sum_{r=2}^\infty I_{r}\Bigl(\frac{2\lambda t \sqrt{\alpha}}{1+\alpha}\Bigr)\Bigl(\frac{1}{\sqrt{\alpha}}\Bigr)^{r}\ \Biggr]
\sum_{i}(-1)^i\left(\sum_j \beta_{i,j}t^{j}\right) = 1 + \sum^{d-1}_{i=1}(-1)^{i}\cdot i\cdot\binom{d}{i+1}t^{i+1}+\sum^{d}_{i=1}(-1)^{i}\cdot d\cdot\binom{d-1}{i-1}t^{i+r-1}&= 1 + \sum_{i=1}^{d}(-1)^{i+1}(i-1)\binom{d}{i}t^{i}- dt^{r}(1-t)^{d-1}
\langle ( \gamma_{\hat{\mu}}{\cal C})_{\alpha \beta i j}\Pi_{\bar{z}}{}^{\hat{\mu}}( z , \bar{z}) \rangle = \left( \begin{matrix}{{0}}&{{| Z | \mathrm{I}}}\\{{- | Z | \mathrm{I}}}&{{0}}\\ \end{matrix}\right) \ ,
s = a{\theta_{3}}^{4}+ b{\frac{\theta_{3}^{\prime}\theta_{3}^{3}}{\eta^{1 2}}}\ , \ \ - t = a{\theta_{4}}^{4}+ b{\frac{\theta_{4}^{\prime}\theta_{4}^{3}}{\eta^{1 2}}}\ , \ \ - u = a{\theta_{2}}^{4}+ b{\frac{\theta_{2}^{\prime}\theta_{2}^{3}}{\eta^{1 2}}}\ ,
\mathcal{Q}=\bigcup_{({r_0},{\beta})\in\mathcal{Q}\times\mathcal{B}\colon{r_0\in{U_\beta}}}{\bigg({U_{\beta}\cap{B\left({r_0},{\widetilde{\delta}_{{\epsilon},{r_0},{s({r_0})}}}\right)}}\bigg)}
\mathbf{A}_{B}^{A}=(\mathbf{L}^{-1})_{C}^{A}d\mathbf{L}_{B}^{C};\text{}\mathbf{A}_{B^{\prime}}^{A^{\prime}}=(\mathbf{L}^{-1})_{C^{\prime}}^{A^{\prime}}d\mathbf{L}_{B^{\prime}}^{C^{\prime}}
\lim_{n \to \infty}h((f\times g)^{n}(x,y))^{1/n}=\lim_{n \to \infty}(h_{H_{X}}(f^{n}(x))+h_{H_{Y}}(g^{n}(y)))^{1/n}&=\max\{\lim_{n\to\infty}h_{H_{X}}(f^{n}(x))^{1/n},\lim_{n\to\infty}h_{H_{Y}}(g^{n}(y))^{1/n}\}=\max\{\alpha_{f}(x), \alpha_{g}(y)\}.
\psi_{_{W K B}}= C{\left( \pi \chi_{0}^{3}a_{0}^{2}( \eta - \frac{1}{2}s i n{2 \eta}) \right)}^{- 1 / 2}e x p \left( \frac{i}{\hbar}\pi \chi_{0}^{3}a_{0}^{2}( \eta - \frac{1}{2}s i n{2 \eta}) \right) ~ .
\P(Z^{\ell_1}=u^{\ell_1},Z_{\ell_1+g+1}^{\ell_1+g+\ell_2}=v^{\ell_2})\leq \P(Z^{\ell_1}=u^{\ell_1})\P(Z_{\ell_1+g+1}^{\ell_1+g+\ell_2}=v^{\ell_2})\Psi(b,g),
C_{n}^{\lambda,\beta}(x)=\sum_{k=0}^{n}\binom{n}{k}\Bigg[\sum_{i=0}^{n-k}(-1)^{i}i!B_{n-k,i}\left(\left(\tilde{M}_{\lambda,\beta}(j)\right)_{j=1}^{n-k-i+1}\right)\Bigg](x)_{k},
\overline{|\nabla{f}|}{}^>(\bar{x}) :=\liminf_{x\to\bar{x},\;f(x)\downarrow0}|\nabla{f}|(x), \overline{|\nabla{f}|}{}^{\diamond}(\bar{x}) :=\liminf_{x\to\bar{x},\;f(x)\downarrow0}|\nabla{f}|^{\diamond}(x)
\left||\{0\leq a\leq p-1 : a^d\equiv 1 (p), \frac ap \in (0,\frac yp) \textrm{mod $1$}\}| - \frac yp d\right| \leq \frac{d}{p}+ \sum_{m=1}^{p-1}\frac1m \left|\sum_{a \leq p-1 , \a^d\equiv 1 (p)}e^{2\pi i m \frac ap}\right| \leq \frac{d}{p}+ (2\log p) de^{-\log^{C_1}p}\leq d e^{-\log^{C_2}p}.
\frac{d}{dt}\int u^2\phi_N\,dx -\int u^2\,\partial_t\phi_N\,dx+5\int (\partial_x^2u)^2\,\partial_x\phi_N\,dx &-5\int (\partial_xu)^2\,\partial_x^3\phi_N\,dx +\int u^2\partial_x^5\phi_N\,dx= 2\int Fu\phi_N\,dx.
\langle \mathfrak{M}_{f,1}^N , \widetilde{\Theta}^N_{\tilde{F}, \tilde{G}, \tilde{H}}\rangle_t= &-\int_0^t\frac{1}{N\gamma_N}\sum_{i=1}^N\sum_{j=1}^NS_t^N(i)I_t^N(j)f(i/N)\lambda(i/N, j/N)\left(e^{\frac{\gamma_N}{N}(-\tilde{F}(i/N)+\tilde{G}(i/N))}-1\right)ds.
\Pi_{n}(x) =C_{n}\left\vert\begin{matrix}\mu_{0}\mu_{1}\cdots \mu_{n}\mu_{1}\mu_{2}\cdots \mu_{n+1}\vdots \vdots \ddots \vdots \mu_{n-1}\mu_{n}\cdots \mu_{2n-1}1 x \cdots x^{n}\end{matrix}\right\vert ,
\left( \prod_{j = 1}^{\ell}\left[ 1 - \sum_{r = 1}^{\lambda_j}x_{s (j) + r}\right] - \alpha x_1 x_2 \cdots x_d \right)^{- 1}= \sum_{\boldsymbol{n}\in \mathbb{Z}_{\geq 0}^d}A_{\lambda, \alpha}(\boldsymbol{n}) \boldsymbol{x}^{\boldsymbol{n}}
\tilde{\mathbb{P}}^a_{Q,\rho,h}(E) =\frac{\sum_{\omega\in E}\mathbb{P}^a_{Q,\rho,0}(\omega)\prod_{i}\cosh(hA_i(\omega))}{\mathbb{E}^a_{Q,\rho,0}\left(\prod_{i}\cosh(A_i)\right)}\geq \frac{\mathbb{P}^a_{Q,\rho,0}(E)}{E^a_{Q,+,0}(e^{hm^a_Q})}&\geq e^{-C_5(h+h^2)}\mathbb{P}^a_{Q,\rho,0}(E),
(t_{2,c},t_{3,c})=\begin{cases}\left(1\pm2\sqrt{3},\pm\sqrt{2}\right)\ ,\quad q=1\ , \left(2\pm2\sqrt{7},\pm\sqrt{10}\right)\ ,\quad q=2\ , \left(3\pm\sqrt{47},\pm\sqrt{105}/2\right)\ ,\quad q=3\ .\end{cases}
\frac{t^N \langle 1 \cdots M|\mathcal{B}^\prime(z_1)\cdots \mathcal{B}^\prime(z_N)|\overline{x_1}\cdots\overline{x_N}\rangle}{t^N \langle 1 \cdots M|\mathcal{B}^\prime(z_1)\cdots \mathcal{B}^\prime(z_N)|\overline{x_1}\cdots\overline{x_N}\rangle|_{z_1 \longleftrightarrow z_1^{-1}}}=\frac{(t^\prime z_1+z_1^{-1})}{(z_1+t^\prime z_1^{-1})}.
\alpha_k:=\frac{2}{k+1},\ \beta_k:=\frac{1}{2L_{Ef}},\ \lambda_k:=\frac{k\beta_k}{2}, m_k:=\left\lceil \frac{L_{Ef}k}{\tilde{D}^2}\right\rceil,\and \Pr(R=k):=\frac{\Gamma_k^{-1}\beta_k(1-L_{Ef}\beta_k)}{\sum_{{\tau}=1}^N \Gamma_{\tau}^{-1}\beta_{\tau}(1-L_{Ef}\beta_{\tau})}, \mathrm{for}\k=1,2,\dots,N,
\biggr(\frac{\sum_{l=1}^\eta\sum_{m=l+1}^\eta c^{\bf a}_lc^{\bf a}_m\gamma^k_l\gamma^k_m(\gamma_l-\gamma_m)^2}{\sum_{l=1}^\eta c^{\bf a}_l\gamma_l^{k+1}\sum_{l=1}^\eta c^{\bf a}_l\gamma_l^k}\biggr)^\infty_{k=k_0+1}
I_{P\varphi}(x,t)|_{(x,t)\in \partial\Omega\times(0,T)}=-\frac{(P\varphi)(x,t)}{2}+\int_{0}^{t}\int_{\partial\Omega}\bigg( \frac{\partial \varepsilon_{n}\big(x-\xi,b(t,\tau)\big)}{\partial\nu(\xi)}a(\tau)(P\varphi)(\xi,\tau) &-\varepsilon_{n}\big(x-\xi,b(t,\tau)\big)a(\tau)\frac{\partial (P\varphi)(\xi,\tau)}{\partial\nu(\xi)}\bigg) dS_{\xi}d\tau.
\bigg\vert \sum_{k=l}^{m}\left\langle u,x_{k}^{\ast}\right\rangle\bigg\vert \leq\bigg\vert \sum_{k=l}^{m}f_{k+}^{\prime}(x,u)\bigg\vert +\bigg\vert \sum_{k=l}^{m}f_{k+}^{\prime}(x,-u)\bigg\vert \leq\varepsilon\quad\forall l,m\geq1,\n_{\varepsilon}\leq l\leq m.
\mathrm{d}\mathbf{U}_t(\vartheta)=\mathbf{J}_\nu(t)\mathbf{U}_t(\vartheta)\mathrm{d}n_t(\vartheta),\mathbf{J}_\nu=\boldsymbol{\upsilon}_\nu^\star\mathbf{L}\boldsymbol{\upsilon}_\nu-\mathbf{K}_0,
+ \frac{1}{2 !}\int_{0}^{1}d \tau_{1}\int_{0}^{1}d \tau_{2}\left. \frac{\delta^{2}W [ X ]}{\delta X^{a}( \tau_{1}) \delta X^{b}( \tau_{2})}\right|_{X = \overline{{X}}}\xi^{a}( \tau_{1}) \xi^{b}( \tau_{2}) +{\cal O}( \xi^{3})
v_{2}(t,r) = -\frac{c_{b}r}{2 \pi}\int_{0}^{\pi}\sin^{2}(\theta) d\theta \int_{0}^{\infty}d\xi \partial_{\xi}^{2}\left(\frac{\xi \chi_{\leq \frac{1}{4}}(\xi)}{\log^{b-1}(\frac{1}{\xi})}\right)\left(\frac{\sin(\xi t_{+})}{t_{+}^{2}}+\frac{\sin(\xi t_{-})}{t_{-}^{2}}\right)
\left\langle \gamma \left( p,\varepsilon_{1}\right) ,\gamma \left(p^{\prime},\varepsilon_{2}\right) \mid \pi \left( q\right) \right\rangle=i\left( 2\pi \right)^{4}T_{\mu \nu}\left( q;p,p^{\prime}\right) \delta^{4}\left( q-p-p^{\prime}\right) \varepsilon_{1}^{\mu}\left( p\right)\varepsilon_{2}^{\nu}\left( p^{\prime}\right) .
\nabla\cdot F=\frac{1}{h_{1}h_{2}h_{3}}[\frac{\partial}{\partial q^{1}}(F_{1}h_{2}h_{3})+\frac{\partial}{\partial q^{2}}(F_{2}h_{3}h_{1})+\frac{\partial}{\partial q^{3}}(F_{3}h_{1}h_{2})]
\P(S^{(i),t}=s^t) =\P\Big(Z_{(k+g)(j-1)+i}^{(k+g)(j-1)+i+k-1}=s_j, \;j=1,\ldots,t\Big) &\stackrel{(a)}{\leq}\Psi^t(b,g) \prod_{j=1}^t \P\Big(Z_{(k+g)(j-1)+i}^{(k+g)(j-1)+i+k-1}=s_j\Big),
S ( \tau , \sigma ) = \sqrt{g_{\mu \nu}X^{\prime \nu}X^{\prime \nu}}= \left[ - \left( 1 - \frac{1}{2 M}\right) t^{\prime 2}+ \left( 1 - \frac{1}{2 M}\right)^{- 1}r^{\prime 2}+ r^{2}\theta^{\prime 2}+ r^{2}\sin^{2}{\theta}\varphi^{\prime 2}\right]^{1 / 2}.
D_{n-1}^{(n)}(x_i^{(n)}) = \left| \boldsymbol{\theta}_{n}^{(n)}(x_i^{(n)}) \cdots \boldsymbol{\theta}_{n+2m-1}^{(n)}(x_i^{(n)}) \boldsymbol{\theta}_{n-1}^{(n)}(x_i^{(n)}) \right| ,
+\frac{\omega \left( \epsilon ,t\right)}{\left( \alpha\left(t\right) +3\right) \left( \alpha \left( t\right) +4\right)}\sum_{n=0}^{\infty}\frac{\left( 2\right)_n\left( \frac52\right)_n\left( 2\right)_n}{\left( \frac{\alpha \left( t\right) +5}2\right)_n\left( \frac{\alpha \left( t\right) +6}2\right)_nn!}\left(-1\right)^n,