harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\psi(x)=\frac{(1-a)_d(1-a+c+b-x)_d}{(1-a+c)_d(1-a+b-x)_d}\cdot{}_4F_3\bigg[\begin{matrix}1+e -b+x -c -d 1 1-a+e a-c-d-b+x\end{matrix}\bigg|\,1\bigg]
{\cal L}= \partial_\mu \psi_1^\dagger \widetilde \gamma_{\mu\nu}\partial_\nu \psi_2+\partial_\mu \psi_2^\dagger \gamma_{\mu\nu}\partial_\nu \psi_1 + m^2 \psi_2^\dagger \psi_1+ m^2 \psi_1^\dagger \psi_2 \quad.
\left( \begin{array}{c}{{h_{+}}}\\{{h_{-}}}\\{{h_{3}}}\\{{1}}\\ \end{array}\right) = \left( \begin{array}{c c l c}{{N^{- 1 / 2}}}&{{}}&{{}}&{{}}\\{{}}&{{N^{- 1 / 2}}}&{{}}&{{}}\\{{}}&{{}}&{{1}}&{{N / 2}}\\{{}}&{{}}&{{}}&{{1}}\\ \end{array}\right) \left( \begin{array}{c}{{N \Sigma_{+}^{( N )}}}\\{{N \Sigma_{-}^{( N )}}}\\{{N \Sigma_{3}^{( N )}/ 2}}\\{{{\bf 1}^{( N )}}}\\ \end{array}\right) ,
F^{C}( \beta ) \sim - \int d^{3}x \sqrt{- g}\left[{\frac{\pi^{2}}{9 0}}T^{4}+{\frac{1}{2 4}}T^{2}\left( \frac 1 6 R - V - \frac 2 3 \Omega^{2}\right) \right] -{\frac{c_{2}}{1 6 \pi^{2}}}\ln ( \beta \rho ) .
B=\frac{2}{c}\left(\frac{1}{\epsilon}-\frac{K}{2}\right)\left(b+c\int|x|^2d\mu_t(x)\right)\leq 128t^2\left(\frac{1}{16t}-\left(\frac{1}{2t}-\frac{R^2}{t^2}\right)\right)\left(\frac{n}{8t}+\frac{R^2}{32t^2}+\frac{1}{64t^2}\left(nt+R^2\right)\right) = \frac{18nR^2}{t}+\frac{6R^4}{t^2}-\frac{63n}{8}-\frac{21R^2}{8}\leq \frac{18nR^2}{t}+\frac{6R^4}{t^2}-2.
{\mathrm d}X_{t}^{(u)}\, =\, - \big(u\, (X_{t}^{(u)}- \widetilde{X}_{t}^{(u)}) + (1-u) ( X_{t}^{(u)}- \mathbb E [ X_{t}^{(u)}] ) \big )\,{\mathrm d}t +{\mathrm d}B_{t}\, ; t \ge 0 \,
x^tL_ex= \;(\mu_{11}+2)x^2_1-\mu_{11}x_1x_2-2x_1x_3-\mu_{11}x_1x_2+(\mu_{11}+\mu_{21})x^2_2-\mu_{21}x_2x_3 &-2x_1x_3-\mu_{21}x_2x_3+(\mu_{21}+2)x^2_3 = \;\mu_{11}(x^2_1-2x_1x_2+x^2_2)+2(x^2_1-2x_1x_3+x^2_3)+\mu_{21}(x^2_2-2x_2x_3+x^2_3) = \;\mu_{11}(x_1-x_2)^2+2(x_1-x_3)^2+\mu_{21}(x_2-x_3)^2 x^tL_ex= \;\sum_{j=1}^{3-1}\sum_{i=1}^{3-j}\mu_{ji}(x_j -x_{j+i})^2
\begin{aligned}G^{i}=L_{2i-1,2i}^{2}, K^{ij}=L_{2i-1,2i}^2+L_{2i-1,2j-1}^2+L_{2i-1,2j}^2+L_{2i,2j-1}^2+L_{2i,2j}^2+L_{2j-1,2j}^2,\end{aligned}
\Psi_{\epsilon}( \alpha ) \equiv \frac{\mu^{\epsilon}}{2 R}\sum_{n = - \infty}^{\infty}\int \frac{d^{d - \epsilon}p}{( 2 \pi )^{d - \epsilon}}\, \left[ p^{2}+ \left( \frac{2 \pi n}{R}\right)^{2}+ m^{2}\right]^{\alpha}.
\binom{a}{k}/\binom{b}{k}=\left(a! (b-k)!\right)/\left(b! (a-k)! \right) &=\Theta( \frac{a^a (b-k)^{b-k}}{b^b (a-k)^{a-k}}\sqrt{\frac{(a-k)b}{a(b-k)}}) &= \Theta \left( (\frac{a}{b})^k (1+\frac{k}{a-k})^{a-k-\frac{1}{2}}(1+\frac{k}{b-k})^{-(b-k)-\frac{1}{2}}\right).
R_{\qquad k l}^{n m}= \delta_{\quad k}^{n}\delta_{\quad l}^{m}+ \epsilon^{n m}\epsilon_{k l}+ i \eta \epsilon^{n m}\epsilon_{k l}+ \frac{1}{2}\epsilon^{n m}b_{k l}+ \frac{1}{2}b^{n m}\epsilon_{k l}+ o \left( \eta^{2}\right) .
\tilde{k}_{1}\: = \: -\frac{1}{2}\sum_{\gamma=1}^{\ell}\left({\rm Cas}_1(\rho_{\gamma}) \right)^2,\: \: \:\tilde{k}_{2}\: = \: k \: - \: \frac{1}{2}\sum_{\gamma=1}^{\ell}\frac{\dim \rho_{\gamma}}{\dim GL(k)}{\rm Cas}_2(\rho_{\gamma}),
\prod_{i=1}^{n/2}(1-1/p^{2i-1}) \prod_{i=n/2-r+1}^{n/2}(1-1/p^{2i})= \prod_{i=n-2r+1}^n (1-1/p^i) \prod_{i=1}^{n/2-r}(1-1/p^{2i-1}).
-\varepsilon\displaystyle\sum_{j=1}^{2}B(u^{\varepsilon})\left(\frac{\partial u^{\varepsilon}}{\partial x_{j}}\right)^{2}f_{1}^{''}(u^{\varepsilon})-\frac{\partial f_{1}(u^{\varepsilon})}{\partial t}\,\,\mbox{is bounded in the space of measure}\,\,M(\Omega_{T}).
D_{j+1}:= \frac{C_0Q_j^p((pq)^{1/2}-1/2)(pq)^{-j}}{(pb_j+1)\ell_{j+1}^{pb_j+1}},\ \ \alpha_{j+1}:= n(p-1)+pa_j,\ \ \beta_{j+1}:= pb_j+1, &Q_{j+1}:= \frac{C_0D_j^q}{(q\beta_j+1)(q\beta_j+2)},\ \ \ \ \ \ \ \ \ \ \ \ \a_{j+1}:= n(q-1)+q\alpha_j,\ \b_{j+1}:= q\beta_{j}+2,
\sum_{k=1}^{n-1}(-1)^k\frac{[n+3k](-q^{n+1};q)_{k-1}}{[2k]}{2k\brack k}\equiv (1+q^n)\left(q^{n\choose 2}-(-q;q)_{n-1}\right) \pmod{\Phi_n(q)^2},
\prod_{j=0}^{n-1}(1+tq^j) \cdot \left( \sum_{j=0}^{n-1}\frac{jq^{j-2}t( j-1 -q^jt )}{(1+tq^j)^2}+ \left(\sum_{j=0}^{n-1}\frac{jq^{j-1}t}{1+tq^j}\right)^2 \right) &= \sum_{r=0}^{n}q^{\binom{r}{2}-2}t^r \left( \left(\binom{r}{2}- 1\right)\binom{r}{2}{n \brack r}_q +q r(r-1){n \brack r}^{'}_{q}+q^2{n \brack r}^{''}_q \right).
\beta_t^{n}\sum_{i=0}^n \tbinom{i+q-1}{q-1}\left(\frac{\beta_1}{\beta_{t}}\right)^{i}\le \eqref{eq:asympt}\le \beta_{q+1}^{n}\sum_{i=0}^n \tbinom{i+q-1}{q-1}\tbinom{n-i+t-q-1}{t-q-1}\left(\frac{\beta_1}{\beta_{q+1}}\right)^{i}
{\cal H}_{k+1}^{(s,t)}(z)=(z-q_{k+1}^{(s,t)}-e_{k}^{(s,t)}){\cal H}_{k}^{(s,t)}(z)-q_{k}^{(s,t)}e_{k}^{(s,t)}{\cal H}_{k-1}^{(s,t)}(z),k=0,1,\dots,m-1.
x^{+}(1)\tau^{+}=-\frac{1}{2}\sum_{i\geq 0}(-1)^i \binom{1}{i}\left(\tau^{+}(1{-}i)\overline{\tau}^{+}(1{+}i){-}\overline{\tau}^{+}(2{-}i){\tau}^{+}(i)\right)\tau^{+}&=-\frac{1}{2}\tau^{+}(1)\overline{\tau}^{+}(1)\tau^{+}+0=-\tau^{+}(1)x^{+},
\kappa\|\Lambda^{\frac{1}{2}}Y^{i+1}\|_2^2 =-(\nabla\cdot(R^\perp Y^i\Theta^i), Y^{i+1}) &\leq C\kappa^{-1}\|\Lambda^{\frac{1}{2}}(R^\perp Y^{i}\Theta^i)\|_2^2+\frac{\kappa}{2}\|\Lambda^{\frac{1}{2}}Y^{i+1}\|_2^2.
\underline{\underline{C}}=[\begin{matrix}C_{11}&C_{12}&C_{12}&0&0&0\\C_{12}&C_{11}&C_{12}&0&0&0\\C_{12}&C_{12}&C_{11}&0&0&0\\ 0&0&0&C_{44}&0&0\\ 0&0&0&0&C_{44}&0\\ 0&0&0&0&0&C_{44}\end{matrix}].
{\bf{E}}_d(k) ={\bf{D}}_d +{\bf{F}}{\bf{P}}_d(k) \mbox{, (dedicated-PoA links)},{\bf{E}}_{a}(k) ={\bf{A}}{\bf{D}}_{b}+ \overline{\bf{A}}{\bf{D}}_{r}\mbox{, (adaptive-PoA links)}
Z^N_t= \frac{Z_{[tN]}}{N}= Z^N_0+ \frac{1}{N}\sum\limits_{i=0}^{[tN]}\bigl( 1+\gamma_N(Z_i) \bigr) + \frac{M_{[tN]}}{N}&= Z^N_0+ \frac{1}{N}\int_{0}^{[tN]}\bigl( 1+\gamma_N(Z_s) \bigr) ds + \frac{M_{[tN]}}{N}&= Z^N_0 + \int_{0}^{[t]}\bigl( 1 + \gamma_N(Z_{uN}) \bigr) du + M^N_t,
\zeta^{a}( y + 2 \pi r_{c}) = e^{- \mathrm{i}\beta \sigma_{3}}\zeta^{a}( y ) \ , \quad \Psi^{A}( y + 2 \pi r_{c}) = e^{- \mathrm{i}\beta \sigma_{3}}\Psi^{A}( y ) \ , \quad \xi ( y + 2 \pi r_{c}) = e^{\mathrm{i}2 \beta}\xi ( y ) \ .
(y_1,y_2,y_3)\cdot (as,s,0) =(y_1+as,y_2+s,y_3+\tfrac{1}{2}[y_1-ay_2]s) &= ([y_1-a y_2] + a(y_2+s),y_2+s, y_3 -\tfrac{1}{2}[y_1-ay_2]y_2+\tfrac{1}{2}[y_1-ay_2](y_2+s)) &= (b + a(y_2+s),y_2+s,c+ \tfrac{1}{2}b(y_2+s)).
{}_{}\textrm{F}_{p}(a ,b;c;z)= \sum_{n=0}^{\infty}(a)_n \frac{B_{p}(b+n,\;c-b)}{B(b,\;c-b)}\frac{z^n}{n!}, ( p\geq0;|z|<1\Re(c)>Re(b)>0)
(s^{2}-1/4)Y''+\left(4s+\frac{\alpha-1}{2}+N\right)Y'+\frac{9-\alpha^{2}}{4}Y+\frac{1}{4d_{N}}(3s^{2}M_{N-1}^{(2)\prime\prime}+(11s-1)M_{N-1}^{(2)\prime}+5M_{N-1}^{(2)})=0\ .
I = \int d \mu{\cal L}= \int d^{4}x \widetilde{D}_{k_{1}l_{1}, \ldots k_{q}l_{q}}^{i_{1}j_{1}, \ldots , i_{p}j_{p}}L_{i_{1}j_{1}, \ldots , i_{p}j_{p}}^{k_{1}l_{1}, \ldots , k_{q}l_{q}}
\mathfrak{S}_{\delta, \flat}^{\mu_2}(L) := \sum_{h\le H}\frac{1}{h}\Phi\big(\frac{h}{H+1}\big)\sum_{\ell\sim L}\sum_{(D/\ell)^{1/2}<d\le (2D/\ell)^{1/2}}\mu(d) \Big(\frac{hx}{d^2\ell+\delta}\Big), \mathfrak{S}_{\delta, \dagger}^{\mu_2}(L) := \sum_{\ell\sim L}\sum_{(D/\ell)^{1/2}<d\le (2D/\ell)^{1/2}}\mu(d) R_H\Big(\frac{x}{d^2\ell+\delta}\Big).
d s^{2}~ = ~ \left( 1 +{\frac{2 G m}{R}}\right) \left[ - d x^{-}~ d x^{+}+ d x^{2}+ d y^{2}\right] +{\frac{4 G m}{R}}\left[{\frac{p_{0}}{m}}d x{-}+{\frac{m}{4 p_{0}}}d x^{+}\right]^{2},
C_{k,\beta}(x,y) = y\prod_{j=1}^{k}\sum\limits_{{d_j}{k_j}\leqslant x}(d_1^{\beta}, \dots, d_k^{\beta})\mu(k_j) + O\bigg(\prod_{j=1}^{k}\sum\limits_{{d_j}{k_j}\leqslant x}d_j^{\beta}\bigg)
\dot m = \frac{1}{8\pi}\int_{S^2}\dot K - \dot H &= \frac{1}{8\pi}\int_{S^2}(-\frac{1}{40}- \frac{1}{10}) R_{kl;mn}X^kX^lX^m X^n &= \frac{1}{8\pi}(-\frac{1}{8}) \frac{4\pi}{15}R_{kl;mn}(\delta^{kl}\delta^{mn}+ \delta^{km}\delta^{ln}+ \delta^{kn}\delta^{lm}) &= -\frac{1}{120}\Delta R
I- A_*(\theta) = I-e^{-\theta}A_{-1}- A_0-e^\theta A_1 N(I-H) \cr = (I-e^\theta R)(I-A_0-e^{-\theta}A_{-1}-A_1 N A_{-1}) \cr = (I-e^\theta R)(I-A_0-e^{-\theta}(I-H)N A_{-1}-A_1 N A_{-1})) \cr = (I-e^\theta R) \bigl( (I-H) (I-e^{-\theta}G) \bigr).
4\pi a[u](x,t) = \int_{B_r(x)}\frac{u(y)}{|x-y|}\;dy + \int_{B_r^c(x)}\frac{u(y)}{|x-y|}\;dy &\le \frac{1}{r}\|u\|_{L^\infty(L^1)}+ 4\pi \|u\|_{L^\infty(L^p)}r^{2-3/p},
\liminf_{x\downarrow0}\frac{p_{\mu,\alpha,\rho}(x)}{x^{\alpha-1}}\in(0,\infty], \rho\neq0,0<\alpha \leq1 \rho\neq 1/\alpha, 1<\alpha <2, &\liminf_{x\uparrow0}\frac{p_{\mu,\alpha,\rho}(x)}{x^{\alpha-1}}\in(0,\infty],\rho\neq1, 0<\alpha \leq 1 \rho\neq 1-1/\alpha, 1<\alpha<2.
\frac{\partial^kF}{\partial v^k}(0,v)=\int_0^v\frac{\partial^kf}{\partial u^k}(v,v')dv'+\sum_{l=0}^{k-2}\binom{k-1}{l+1}\frac{\partial^{k-1}f}{\partial u^{k-1-l}\partial v^l}(v,v).
V(\mathbf{j}') = \sum_{k=1}^{n-1}[(n-k+1)j_k - s(j_k)] - c(j_n,-\alpha(n)-1) = \frac{2(m-1)}{3}- s\left(\frac{m-1}{3}\right) - c(2, -\alpha(n) -1) = \frac{2(m-1)}{3}- s\left(\frac{2(m-1)}{3}\right).
\|\mathcal{T}_{1,l}^{(a_1,a_2)}-\mathcal{T}_{2,l}^{(a_1,a_2)}\|_{\rm F}^2 =\sum_{k_1,k_2}s_{k_1,k_2}^{l-2a_1}s_{k_1,k_1}^{a_1}s_{k_2,k_2}^{a_1}(\mu_{k_1,l}^{(1)}-\mu_{k_1,l}^{(2)})(\mu_{k_2,l}^{(1)}-\mu_{k_2,l}^{(2)}).
\hat f\circ\phi(z) = f_0 + (f_+\circ\phi)(0) + \sum_{n>0}(f_+\circ\phi)_n z^n + \overline{(f_-\circ\phi)(0)}+ \sum_{n<0}\overline{(f_-\circ\phi)_{-n}}z^{n}&= f(\phi(0)) + \sum_{n>0}(f_+\circ\phi)_n z^n + \sum_{n<0}\overline{(f_-\circ\phi)_{-n}}z^{n}.
-\Delta \tilde{V}\geq 2\sum\limits_{j=1}^4\frac{a^2}{(1+a^2|z_{\xi_j}|)^2}\cdot \frac{(1+|x_{\xi_j}|^2)^2}{4}\geq \sum\limits_{j=1}^4 \frac{a^{-2}}{|z_{\xi_j}|^4}\cdot \frac{(1+|x_{\xi_j}|^2)^2}{4}.
J_{3}( B_{1}, B_{2}) = \mathrm{p . v .}\int_{0}^{\infty}\frac{t^{3}d t}{\frac{( B_{1}t - 1 ) ( B_{2}t - 1 )}{( B_{1}t + 1 ) ( B_{2}t + 1 )}e^{2 t}- 1},
\sigma_{\nu}(2N)=2^{-2N}N\sum_{k=1}^{\infty}\sum_{m\in\mathcal{M}(N)}\alpha(m)\prod_{j=1}^{d(m)}\left(\frac{1}{(j+k+\nu-1)(j+k+\nu)}\right)^{\! m_{j}}\!,\N\in\mathbb{N}.
{\cal Z}_{_I}= \int D\overline{\xi}D\xi D\overline{\chi}D\chi \exp \left\{i \int d^2 x \left[ -i \overline{\chi}\partial\!\!\!/ \chi + i \overline{\chi}\partial\!\!\!/ \gamma^0 \partial\!\!\!/ \xi - i \overline{\xi}\partial\!\!\!/ \gamma^0 \partial\!\!\!/ \chi + m \overline{\xi}\partial\!\!\!/ \partial\!\!\!/ \xi \right] \right\}.
\sum_{\alpha=1}^\infty \Upsilon_{M-1}^\alpha(t) \leq k\sum_{\alpha=1}^\infty\left[\int_{t_{M-1}}^t \phi^2_\alpha(t-s) ds\right]+k\sum_{\alpha=1}^\infty \left[\int_{t_{M-1}}^t \phi_\alpha(t-s) ds\right] \lesssim\begin{cases}k^\frac{3}{2}, ; k^2, .\end{cases}
p^{\alpha_2}_{k-1}(2,t)=(-\lambda)^{k-1}\underset{\Theta^{k-1}_{2}}{\sum}\frac{t^{k_0\alpha_0+k_1\alpha_1+k_2\alpha_2}}{\Gamma\left(k_0\alpha_0+k_1\alpha_1+k_2\alpha_2+1\right)}.
\Gamma(x,0)= (x-\frac{3}{4L})^3(x^3-Lx-\frac{1}{4}) = x^6-\frac{9}{4L}x^5+(\frac{27}{16L^2}-L)x^4+(2-\frac{27}{64L^3})x^3-\frac{9}{8L}x^2+\frac{27}{256L^3}.
(Hf)(x)=\sum_{i=1}^{M}\Bigl\{-\alpha\gamma\sum_{d=1}^{c_{i}-1}\frac{[d]}{1+\beta\gamma[d]}f(x) +\sum_{r=1}^{c_{i}}\frac{(-\beta\delta)^{r-1}[r-1]!q^{-r(r-1)}}{\prod_{p=0}^{r-1}(1+\beta\gamma[c_{i}-1-p])}e_{r}(1, q, \ldots , q^{c_{i}-1})f(x-\sum_{p=0}^{r-1}v_{c_{1}+\cdots +c_{i}-p})\Bigr\}.
Ra(\xi_J)a(\xi_I)^*\Omega_p = a(r \xi_J) a(r \xi_I)^* \hat \Omega_{q^\prime}&= a(r \xi_J) a(r \xi_I)^* U\Omega_q &= U a(u^*r \xi_J) a(u^*r \xi_I)^*\Omega_q.
{d{f_2}}=\nabla.(D_{2}\nabla{{f_2}})dt-\nabla.(F_2f_2)dt+R_2(f_1,f_2,\beta_2)dt-R(f_1,f_2,s_1,s_2)dt+f_2\sum_{i=0}^nh_{i,2}(x,y)dB_i,
N(z) = \left(\mathbb{I}+\mathcal{O}\left(\frac{1}{z}\right)\right)\begin{pmatrix}1 0 0 0 z^{\frac{1}{4}}0 0 0 z^{-\frac{1}{4}}\end{pmatrix}\begin{pmatrix}1 0 0 0 \frac{1}{\sqrt{2}}\frac{i}{\sqrt{2}}0 \frac{i}{\sqrt{2}}\frac{1}{\sqrt{2}}\end{pmatrix}.
h(r, x_{11}) = \frac{\kappa N}{16 R}\left[ \; \frac{3}{r^5}+ \sum^{\infty}_{m=1}\left( \frac{2}{\pi}\right)^\frac{1}{2}\frac{m^2 m^{1/2}}{R^5}\left( \frac{R}{r}\right)^{5/2}K_{5/2}( mr / R ) 2 \cos (m x_{11}/ R ) \; \right] .
E^{\mu}[|\int_{\mathcal{X}}f(x)\pi_{n}^{\mu}(dx)-\int_{\mathcal{X}}f(x)\pi_{n}^{\nu}(dx)|] = E^{\mu}[E^{\mu}[| \int_{\mathcal{X}}f(x)F(\pi_{n-1}^{\mu},y_{n})(dx) & -\int_{X}f(x)F(\pi_{n-1}^{\nu},y_{n})(dx) | |Y_{[0,n-1]}] ]
\rho ( z ) \; = \;{\frac{z}{\pi}}\sqrt{\frac{b - z}{z - a}}\left[{\frac{1}{4 \beta}}-{\frac{d - 2}{2}}\int_{a}^{b}{\frac{d y}{y + z}}\sqrt{\frac{y + a}{y + b}}{\frac{\rho ( y )}{y}}\right] \; \mathrm{f o r}\; \beta < \beta_{c}\; .
\frac{1}{4 !}( \int_{D})^{4}\langle F_{c}^{2}F_{t}^{2}\rangle = \frac{1}{4 !}r^{4}( - e^{4}) \langle n^{2}\rangle ( \int_{D})^{4}\; e^{2}[ \delta^{2}( x - y ) + 5 \; \mathrm{p e r m u t .}] = - \frac{e^{6}}{4}r^{4}\langle n^{2}\rangle \Delta^{3}
\rho_\mu(a) \overset{\hphantom{(2.3)}}{=}\mu(\partial_1(a))\alpha_2 - \mu(\partial_2(a))\alpha_1 \overset{\eqref{eq:repd1}}{=}(\alpha_1 \mu(a) - \mu(a) \alpha_1)\alpha_2 - (\alpha_2 \mu(a) - \mu(a) \alpha_2)\alpha_1 = \alpha_1 \mu(a) \alpha_2 - \alpha_2 \mu(a) \alpha_1.
\gamma A_0^a=\eta_1^a,\;\gamma A_i^a=\partial_i\eta_2^a+f_{\;\;bc}^a\eta_2^bA_i^c,\;\gamma B^{0i}=\eta_1^i,\;\gamma B^{ij}=\partial^{\left[i\right.}\eta_2^{\left. j\right]},
\check{Y}(qx)= \biggl(\widetilde{F}_{\infty}+ \frac{\widetilde{F}_1}{1-\alpha_1 x}+ \frac{\widetilde{F}_2}{1-\alpha_2 x}+ \frac{\widetilde{F}_3}{1-\alpha_3 x}\biggr) \check{Y}(x), \check{Y}(x) = \begin{pmatrix}\bar{g}_1(x) \bar{g}_2(x) \bar{g}_3(x) \bar{g}_4 (x) \end{pmatrix}\! .
\begin{aligned}D^2v_0 = D^2f_0 - \frac{(h_0-\delta)^2}{h_0(h_0+h_1)}(u''_{\delta}-v''_{\delta}) + \frac{(h_0-\delta)^3}{3h_0(h_0+h_1)}(u'''_{\delta}-v'''_{\delta}) - \frac{(h_0-\delta)^4}{12h_0(h_0+h_1)}(u''''_{\delta}-v''''_{\delta}) + \mathcal O(h^3), \end{aligned}
\beta_{ij}^{(k)}\equiv \partial_x^{-1}\left({\cal L}^{k}\,( \Phi_j) \Psi_i \right) \;\; ; \;\;\beta_{ij}^{*\,(k)}\equiv \partial_x^{-1}\left( \Phi_j \,{\cal L}^{*\,k}(\Psi_i) \right)
\begin{aligned}\frac{1}{2}([\mathfrak{B}_1, \overline{A}_1])_j^j(0) =\frac{4}{3}c_2^2\,\mathrm{i}\,\sum_{j_2\in S, j_2+j\in S^c}j_2^3\,\lvert j_2 \rvert \xi_{j_2}+8 c_2 c_3\,\mathrm{i}\sum_{j_2\in S, j_2+j\in S^c}j_2\,\lvert j_2 \rvert \xi_{j_2}& +12\,c_3^2 \mathrm{i}\sum_{j_2\in S, j_2+j\in S^c}j_2^{-1}\,\lvert j_2 \rvert\,\xi_{j_2}.\end{aligned}
m_\mathrm{ADM}(\mathcal E, \tilde g_{\sigma}) - m_\mathrm{ADM}( \mathcal E, g) =\lim_{r\to\infty}\frac{-2}{(n-2)\omega_{n-1}}\int_{|x|=r}\nu_g(u_{\sigma})\, d\mu_{S_r, g}=\frac{-2}{(n-2)\omega_{n-1}}\int_{M_\sigma}\Delta_g u_{\sigma}\, d\mu_{g}=\frac{-1}{2(n-1)\omega_{n-1}}\int_M V u_{\sigma}\,d\mu_g,
\vec{T}_{0}^{1}\vec{T}_{1}^{1}\; = \; s_{- 1}^{r - 1}s_{1}^{r - 1}f_{- 1}^{1}f_{1}^{1}\, \vec{I}+ s_{0}^{r - 1}f_{0}^{1}\, \vec{T}_{0}^{2}
\rho_{0}(k,\omega)=\frac{1}{Z}2\pi\delta(\xi_{k}-\omega)\sum_{\alpha^{\prime}}\langle\alpha^{\prime}|\psi_{k}\psi_{k}^{\dagger}|\alpha^{\prime}\rangle(1-\zeta e^{-\beta\xi_{k}})e^{-\beta E_{\alpha^{\prime}}}.
f_{2134}\left( d_{1},d_{2},d_{3},d_{4}\right) & =\left( d_{1},d_{2},d_{3},d_{4},d_{1}d_{2},\underset{6}{0},...,\underset{11}{0},d_{1}d_{2}d_{3},\underset{13}{0},...,\underset{16}{0},d_{1}d_{2}d_{4},\underset{18}{0},...,\underset{35}{0},d_{1}d_{2}d_{3}d_{4},\underset{37}{0},...,\underset{53}{0}\right)
f^{( 2 )}= ( 1 / 2 ) \big [ ( g_{1}^{2}/ 1 6 \pi^{2}) \ln^{2}( q^{2}/ \mu^{2}) + ( g_{2}^{2}/ 1 6 \pi^{2}) \ln^{2}( q^{2}/ M^{2}) \big ]^{2}~ .
( k \: . \: k \: . )^{q_{1}}p \: . \: k \: . ( k \: . \: k \: . )^{q_{2}}p \: . \: k \: . ( k \: . \: k \: . )^{q_{3}}\: \cdots \: ( k \: . \: k \: . )^{q_{2 \alpha + 1}}p ( . \: k \: . \: k )^{q_{2 \alpha + 2}}
\hat{\phi}_{( I )}( x ) = \int_{( \infty )}\frac{d^{3}\vec{k}}{( 2 \pi )^{3}2 \omega ( \vec{k})}\left[ a ( \vec{k}) e^{- i k \cdot x}+ a^{\dagger}( \vec{k}) e^{i k \cdot x}\right] \ ,
(a_1^+)^{n_1}\dots(a_d^+)^{n_d}\Phi=\prod_{i=1}^d\prod_{p=0}^{n_i-1}\frac{2(p+1)(b_i+a_i+p+1)}{(2p+b_i+a_i+1)(2p+b_i+a_i+2)}P^{(a,b)}_{\overline{n}}
S_{a_{1},b_{1}}^{a_{2},b_{2},p_{2}}\left( p_{1},k\right)=a_{1}S_{a_{1},b_{1}}^{a_{2},b_{2},p_{2}}\left( p_{1}-1,k-1\right) +\left(a_{1}k+b_{1}\right) S_{a_{1},b_{1}}^{a_{2},b_{2},p_{2}}\left(p_{1}-1,k\right) .
\langle{\bf p}\, \vert{\bf x}, t \rangle = \langle{\bf p}\, \vert U ( I , ({\bf x}, t ) \vert{\bf 0}, 0 \rangle = e^{i{\bf p}\cdot{\bf x}- i \omega_{m}({\bf p}\, ) t}\langle{\bf p}\, \vert{\bf 0}, 0 \rangle
f_{1}\star_{\vec{c}}^{n}f_{2}\ (\mathbf{n})=\iint_{S^{2}\times S^{2}}\mathbb{T}_{\vec{c}}^{j}(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n})\tilde{f}_{1}(\mathbf{n}_{1})\tilde{f}_{2}(\mathbf{n}_{2})d\mathbf{n}_{1}d\mathbf{n}_{2}\ .
\frac{d}{dt}(B_i (\Delta_q g_2), \partial_i \Delta_qc ) - (B_i (\Delta_q g_2), \partial_i \Delta_q\partial_tc ) + \Vert \partial_i \Delta_qc \Vert^2_{L^2_x}&= (B_i(\Delta_qR_1+\Delta_qR_2), \partial_i \Delta_qc).
\boldsymbol{v}_k^c(\boldsymbol{x}) = \frac{2}{T}\int_0^T \boldsymbol{v}(\boldsymbol{x},t) \cos(k \omega t)\,dt, \boldsymbol{v}_k^s(\boldsymbol{x}) = \frac{2}{T}\int_0^T \boldsymbol{v}(\boldsymbol{x},t) \sin(k \omega t)\,dt.
\sum_{n=-\infty}^\infty \frac{(-1)^nq^{n(3n+1)}}{1+q^{6n}}=\sum_{n=-\infty}^\infty \frac{(-1)^nq^{n(3n+5)}}{1+q^{6n}}\intertext{and}\sum_{n=-\infty}^\infty \frac{(-1)^nq^{n(3n+1)}}{1+q^{2n}}=\sum_{n=-\infty}^\infty \frac{(-1)^nq^{n(3n+1)}(1-q^{2n}+q^{4n})}{1+q^{6n}}.
\frac{d}{d \eta}h(Y) = -\int \frac{d}{d \eta}\left(q_{\eta}(y)\,\ln q_{\eta}(y)\right)\,dy &= -\int \frac{d q_{\eta}}{d \eta}(y)\,\ln q_{\eta}(y)\,dy -\int \frac{d q_{\eta}}{d \eta}(y) \,dy &= -\int \frac{d q_{\eta}}{d \eta}(y) \,\ln q_{\eta}(y)\,dy - \frac{d}{d \eta}\int q_{\eta}(y)\,dy &= -\int \frac{d q_{\eta}}{d \eta}(y) \,\ln q_{\eta}(y)\,dy.
{\mathcal G}_{l + \nu}^{( 0 )}( r^{\prime \prime}, r^{\prime}; E ) = \int_{0}^{\infty}d z \, \eta_{l + \nu}( z ; \kappa ) \, \Psi_{z , l + \nu}( r^{\prime \prime}; \kappa ) \Psi_{z , l + \nu}( r^{\prime}; \kappa ) \; .
b_{m,n}(t) = \sum_{(m',n') \in \mathbb{Z}^2_{0,n}}\int_{\mathbb{R}}H(t-s) e^{(-\alpha n^2+i l_{m,n}) (t-s)}{\hat \theta}_{m',n'}(s){\hat \beta}_{m-m', n-n'}(s) ds \ ,
{\cal L}_{\mathrm{K}_{\mu}}= \frac{- 1}{8 g^{2}}\left( \mathrm{T r}\{{\cal F}_{\ast}^{\mu \nu}{\cal F}_{\mu \nu}\}- i \mathrm{T r}\{{\cal F}_{\ast}^{\mu \nu}{\cal F}_{\mu \nu}\}i \right) ,
\langle N\rangle=\left\{\begin{array}{lc}-\frac{1}{2}F, -1<A^{-1}<\infty & \hphantom{-}\frac{1}{2}(2-F), -\infty<A^{-1}<-1 & \hphantom{-}\frac{1}{2}(1-F), A=0\end{array}\right\}, \frac{1}{2}<F<1.
\frac{d}{ds}\left(-\zeta(s)^{-1}Z_{r_2}(f,s)+\zeta(s+1)^{-1}Z_{r_1}(d_2(f), s+1)\right)\Big|_{s=0}=(\log q)\left(\tilde{c}_2(I(f))-\tilde{a}_2(I(f))\right).
(r^2 - r_{H}^2 ) \partial_{r}^2 R_{(\gamma)}(r) + \frac{1}{r}\left[ 2 r^2 + \gamma (r^2 - r_{H}^2) \right]\partial_{r}R_{(\gamma)}(r) + \frac{\omega^2 \ell^4}{(r^2 - r_{H}^2)}R_{(\gamma)}(r) = 0.
p_{\epsilon}(x_1,\dots ,x_k|U_\epsilon=0) =p(x_1,\dots ,x_k)f(x_1,\dots ,x_k) p_{\epsilon}(x_1,\dots, x_k|U_\epsilon=1) =p(x_1,\dots ,x_k)\left(\frac{1}{1-\epsilon}- \frac{\epsilon}{1-\epsilon}f(x_1,\dots ,x_k)\right)
I = \int_0^\infty \int_{{\mathbb{R}}^n}|\nabla u(t,x)|^2\phi(t,x)^2\theta(x)^2\chi(t)^2\,{\rm d}t\,{\rm d}x, J = \bigl\|(t,x) \mapsto 1\!{\rm l}_{B(x_0,2R)}(x)1\!{\rm l}_{(0,4R^2)}(t)u(t,x)\bigr\|_{X^2}^2,
I\left(f_{\omega_{\pi_\nu},s}^\circ,v^\circ_{\pi_{\nu}}\right)(I_m)=\int \limits_{Z_n(k_\nu)\backslash T^-}\mu(\underline{t})f_{\omega_{\pi_\nu},s}^\circ\left(\tilde{\varepsilon}t\left(I_m,\underline{t}\right)\right)\int\limits_{K_{n,\nu}}\pi_{\nu}(A\underline{t})v^\circ_{\pi_{\nu}}dAd\underline{t}.
\left\|Qu\right\|_{\ell^{2}(\hbar\mathbb{Z}^{n})}^{2}=\sum_{k\in\hbar\mathbb{Z}^{n}}\frac{1}{|V(k)+1|^{2}}|u(k)|^{2}\leq \sum_{k\in\hbar\mathbb{Z}^{n}}|u(k)|^{2}=\left\|u\right\|_{\ell^{2}(\hbar\mathbb{Z}^{n})}^{2}.
d_\mathcal{D}{\rm ch}_g(\bar{\varepsilon}) =d_\mathcal{D}\bigg(\frac{1}{4\pi i}\int_{\mathbb{P}^1}{\rm ch}_g^0\Big({\rm tr}_1\big(\lambda(\bar{\varepsilon})\big)\Big)\log\mid z\mid^2\bigg) &={\rm ch}_g(\bar{E}_{0})-{\rm ch}_g(\bar{E}_{-1})-{\rm ch}_g(\bar{E}_{1}).
T_{F}( \bar{z}) = \frac{1}{\sqrt{2}}( G^{+}( \bar{z}) + G^{-}( \bar{z}) ) = i \sum_{i = 1}^{3}\tilde{\rho}^{i}\tilde{\sigma}^{i}\tilde{\psi}^{i}+ \mathrm{~ h . c .}
S_k(\xi) := \frac{-\bigl((2k+1)\pi\sqrt{-1}-(2a+1)\xi\bigr)^2}{2(2a+1)}, \intertext{and}\tau_k := (-1)^k\frac{4\sin\left(\frac{(2k+1)\pi}{2a+1}\right)}{\sqrt{2(2a+1)}}.
\sup_t\left|\mathbb{P}\left(\frac{1}{\sqrt{b_K}}\sum_{k=1}^KX_k<t\right)-\Phi(t)\right|\leq A\left(\frac{\sum_{k=1}^K\mathbb{E}[X_k^4]+\mathbb{E}\left[(V_K-b_K)^2\right]}{b_K^2}\right)^{1/5},
\frac{\partial y}{\partial\varphi}\cdot\frac{\partial x}{\partial\lambda}-\frac{\partial y}{\partial\lambda}\cdot\frac{\partial x}{\partial\varphi}=s\cdot cos\varphi\cdot\frac{(1-e^{2})}{(1-e^{2}sin^{2}\varphi)^{2}}
\begin{array}[c]{l}S_{n}^{m\left( 1\right)}\left({z,\gamma}\right) =\left({\dfrac{\pi}{2\gamma z}}\right)^{1/2}\dfrac{\left({z^{2}-1}\right)^{-m/2}z^{m}}{A_{n}^{m}\left({\gamma^{2}}\right)}\times\sum\limits_{k=-k^{+}}^{\infty}{a_{n,k}^{m}\left({\gamma^{2}}\right)J_{n+2k+\left({1/2}\right)}\left({\gamma z}\right)},\end{array}
N_{B}=\frac{\phi}{\phi_{0}}=\frac{BA}{BL_{y}\Delta x_{k}}=\frac{A}{2\pi l_{B}^{2}}\begin{matrix}&l_{B}&\\ &=&\\ &&\end{matrix}\frac{AeB}{2\pi\hbar}\begin{matrix}&\omega_{c}&\\ &=&\\ &&\end{matrix}\frac{m^{*}\omega_{c}A}{2\pi\hbar}
F(\underline U(t,x)_+)\geq (1-\epsilon)\Big[F\big([\Phi(x-\underline h(t))+\Phi(-x-\underline h(t))-\mathbf{u}^*]_+\big)\Big] \succeq (1-\epsilon)\big[F(\Phi(x-\underline h(t)))-mL\epsilon_2\mathbf{1}\big].
T_{\mu \nu}= e^{- 2 \phi}[ F_{\mu \rho}F_{\nu}{}^{\rho}-{}^{\star}F_{\mu \rho}{}^{\star}F_{\nu}{}^{\rho}] - 2 [ \nabla_{\mu}\phi \nabla_{\nu}\phi -{\textstyle \frac{1}{2}}g_{\mu \nu}( \nabla \phi )^{2}] \, ,
{\bf{\cal I}}(E)=\left( \begin{array}{cc}\int \frac{d^3k}{(2 \pi)^3}\frac{1}{E^+ - \frac{k^2}{2 \mu}}\int \frac{d^3k}{(2 \pi)^3}\frac{k^2}{E^+ - \frac{k^2}{2 \mu}}\int \frac{d^3k}{(2 \pi)^3}\frac{k^2}{E^+ - \frac{k^2}{2 \mu}}\int \frac{d^3k}{(2 \pi)^3}\frac{k^4}{E^+ - \frac{k^2}{2 \mu}}\end{array}\right).
\frac{1}{2}\triangle_{g_X}v_k+(e^{v_{k-1}+v_{k+1}-2v_k}-1)\cdot|\gamma_k|^2\frac{\det(h_1|_{F_{k-1}})\det(h_1|_{F_{k+1}})}{\det(h_1|_{F_k})^2}/g_X\geq 0, k=1,\cdots,n-1.
\int_{\Omega}\rho |u_t|^2 x+\frac{1}{2}\frac{d}{dt}\int_{\Omega}\Big(\mu|\nabla u|^2+\big(\lambda+\mu\big)(u)^2\Big) x = \int_{\Omega}\Big(-\nabla P-\rho v\cdot \nabla v+(H\times H)\Big)\cdot u_tx=\frac{d}{dt}\Lambda_1(t)- \Lambda_2(t),
\frac{\partial}{\partial z}\ \{F}(a_1\cdots,a_p,b_1,\cdots,b_q|z)= \frac{a_1a_2\cdots a_p}{b_1b_2\cdots b_q}\ \{F}(a_1+1,\cdots,a_p+1, b_1+1,\cdots,b_q+1|z).
\Delta_1(k,m+1) = z \Delta_1(k,m) + a_{m+1}\Delta_2(k,m) +a_{m+1}^2 \Delta_1(k,m-1); \Delta_2(k,m+1) = (2z+1) \Delta_1(k,m) + a_{m+1}\Delta_2(k,m)+ a_{m+1}\Delta_3(k,m); \Delta_3(k,m+1) = - \Delta_1(k,m) - a_{m+1}\Delta_3(k,m); \Delta_4(k,m+1) = 2z \Delta_1(k,m)+a_{m+1}\Delta_2(k,m).
\begin{matrix}I&a_{1}=f(\langle\rangle)&&a_{3}=f(\langle a_{1},a_{2}\rangle)&&a_{5}=f(\langle a_{1},a_{2},a_{3},a_{4}\rangle)&&\cdot\cdot\cdot\\II&&a_{2}&&a_{4}&&a_{6}&\cdot\cdot\cdot.\end{matrix}
\mathfrak{S}\omega(e_1\wedge...\wedge e_{p+1}, e_{p+2}\wedge...\wedge e_{p+q})=\frac{1}{p!}\sum_{\sigma\in S_{p+1}}\varepsilon(\sigma)\omega(e_{\sigma(1)}\wedge...\wedge e_{\sigma(p)}, e_{\sigma(p+1)}\wedge e_{p+2}\wedge...\wedge e_{p+q}).
q_1' = [12|12]= x_1x_2x_6^2-x_1x_3x_5x_6-x_2^2x_3x_6^2+2x_2x_3^2x_5x_6-x_3^3x_5^2-x_1x_4 q_2' = [12|13]= x_1x_5x_6-x_2x_3x_5x_6+x_3^2x_5^2-x_2x_4 q_3' = [12|14]+[12|23]= x_1x_6^2-x_2x_3x_6^2+x_3^2x_5x_6-x_3x_4 + x_1x_5-x_2^2x_6+x_2x_3x_5.
C=\left\langle \begin{array}{c}\big((1,0),\,(1,0),\,(1,0),\,(1,0),\,(1,0),\,(1),(0),(0)\big), begin{align*}.7ex] \big((1,0),\,(0,1),\,(0,1),\,(0,1),\,(0,1),\,(0),(1),(0)\big), begin{align*}.7ex] \big((0,1),\,(1,0),\,(0,1),\,(1,1),\,(1,1),\,(0),(0),(1)\big)\;\mbox{}\end{array}\right\rangle\leq\Pi
\int_{\Omega}|\nabla u|^2 dx \geq (\alpha(d-2)-\alpha^2) \int_{\Omega}\frac{|u|^2}{|x|^2}dx + \alpha(d^2-d) \int_{\Omega}\frac{|u|^2}{|x|^2}dx =(-\alpha^2 + \alpha(d-2)) \int_{\Omega}\frac{|u|^2}{|x|^2}dx.
\mathbb E_{x}(\mathbf1_Du(X_{\tau_V})\mathbf1_A(X_{\tau_V-})) =\mathbb E_{x}(\mathbf1_{D\setminus V}u(X_{\tau_V})\mathbf1_{A\cap V}(X_{\tau_V-})) &=\mathbb E_{x}(\mathbf1_{D\setminus \bar V}u(X_{\tau_V})\mathbf1_{A\cap V}(X_{\tau_V-})),
F ( s +{\frac{3}{2}}, 1 , s + 2 ,{\frac{3}{4}}) ={\frac{\Gamma ( s + 2 )}{\Gamma ( 1 ) \ \Gamma ( s + 1 )}}\ \int_{0}^{1}( 1 - t )^{s}\ \left( 1 -{\frac{3 t}{4}}\right)^{- s -{\frac{3}{2}}}\d t
z\bigl((\tilde{\Delta}\otimes\operatorname{id})(\Delta a)\bigr)(1\otimes1\otimes b) =z(E\otimes1)\bigl((\tilde{\Delta}\otimes\operatorname{id})(\Delta a)\bigr)(1\otimes1\otimes b) &=z(E\otimes1)(\Delta\otimes\operatorname{id})\bigl((\Delta a)(1\otimes b)\bigr) &=z(\Delta\otimes\operatorname{id})\bigl((\Delta a)(1\otimes b)\bigr),