harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l}X_{J_i^\prime}\rfloor \Omega_q=-dJ_i^\prime~, begin{align*}4mm] [X_{J_i^\prime},X_{J_j^\prime}]=-X_{[J_i^\prime,J_j^\prime]_{GPB}}~, begin{align*}4mm] \Omega_q(X_{J_i^\prime},X_{J_j^\prime})=[J_i^\prime,J_j^\prime]_{GPB}~.\end{array}
\hat{T}(\lambda;g_1) = L^{ab}(\hat{G}):\hat{J}_a^{(0)}\hat{J}_b^{(0)}:\hspace{.05in}, \hspace{.3in}\hat{c}(\lambda;g_1) = 2L^{ab}(\hat{G})\hat{G}_{ab}, \hspace{.3in}\hat{\Delta}_0(\lambda;g_1)=0
\frac{x^4}{2}(D+12) (D+8)(D+4)T + \frac{3}{8}(D+2)T - \frac{3}{4}\frac{N}{x^2}- \frac{1}{2}(D+3)T + \frac{3}{4}\frac{N}{x^2}+ 6\left( \frac{x^2}{2}(D+4)T \right)^2 = \frac{3}{4}\left( \left. Y_{N+2}\right|_{t_i = x\delta_{i,3}}+ \left. Y_{N-2}\right|_{t_i = x\delta_{i,3}}- \left. 2Y_N \right|_{t_i = x\delta_{i,3}}\right).
2\sum_{i=2}^{s}\overline{\mu}_{i}^{2}\leq\sum_{i=2}^{s}\overline{\mu}_{i}^{2}+\sum_{i=s+1}^{2s-1}\overline{\mu}_{i}^{2}+\sum_{i=n-s+2}^{n}\mu_{i}^{2}.
\partial_1 \, Q_k^n(x,y) = (n-k) \left[P_k^{n-1}(x,y) + k\, P_{k-1}^{n-2}(x,y)\right], \partial_2 \, Q_k^n(x,y) = k \left[P_{k-1}^{n-1}(x,y) + (n-k) \,P_{k-1}^{n-2}(x,y)\right].
+{\hbar}^2\frac{e_{+}^2}{4{\pi}^2}\sum_{\stackrel{q \in \cal Z}{q \neq (0;-p)}}{\cal F}_{+-}(q) \frac{1}{q(p+q)}{\rho}_{\rm tot}^{\rm N}(p+q){\rho}_{\rm tot}^{\rm N}(-q),
w_t-a_n^{ij}(t, x_0',x'')D_{ij}w=f_n-f(t, x_0',x'')-\left(a_n^{ij}(t, x_0',x'')-a_n^{ij}\right)D_{ij}u \bigl(\; w_t-a_n^{ij}(t_0, x)D_{ij}w=f_n-f(t_0, x)-\left(a_n^{ij}(t_0, x)-a_n^{ij}\right)D_{ij}u \; \bigr)
\lambda(t) = \frac{1}{\log^{b}(t)}+ e(t), |e(t)| \leq \frac{C}{\log^{b}(t)\sqrt{\log(\log(t))}},|e^{(j)}(t)| \leq \begin{cases}\frac{C}{t^{j}\log^{b+1}(t) \sqrt{\log(\log(t))}}, j=1,2 \frac{C}{t^{j}\log^{b+1}(t)}, j=3,4\end{cases}
\left|A(n,m)\right|= \frac{1}{n-m}\sum_{i=1}^{n-m}\log\left(\frac{X_{(i+m)}-X_{(i)}}{Y^R_{i+m}-Y^R_{i}}\right)\leq\frac{1}{n-m}\sum_{i=1}^{n-m}\log\left(\frac{X_{(i+m)}-X_{(i)}}{X_{i+m-1,i+m-1}-X_{i+1,i+1}}\right) \leq \frac{1}{n-m}\sum_{i=1}^{n-m}\log\left(\frac{X_{(i+m+k)}-X_{(i+1)}}{X_{i+m-1,i+m-1}-X_{i+1,i+1}}\right)\;\;\;\mbox{for some $k \in{\mathbb N}$}.
\mathrm{REC}(Y)=\{X\in 2^{\omega}: X\leq_{T}Y\}& \mathrm{ARITH}(Y)=\{X\in 2^{\omega}: X\leq_{a}Y\}& \mathrm{HYP}(Y)=\{X\in 2^{\omega}: X\leq_{h}Y\}
{\cal F}_{a}^{\mu \nu}= \frac{1}{a}\phi_{a}{\cal F}^{\mu \nu}=\frac{1}{a}\phi_{a}\, \frac{1}{a^{3}e}\phi \cdot (\partial^{\mu}\phi\wedge \partial^{\nu}\phi) + \partial^{\mu}A^{\nu}- \partial^{\nu}A^{\mu}.
tc(\delta u)e^{-tc(\delta u)^2}= [-\frac{i}{4\pi}\Big(c^1\partial_{v_1}+ c^2\partial_{v_2}+\sin\phi\, c^3 \partial_{v_3}\Big) +tc^4e^yV{i( \kappa- \Lambda)}]e^{-tc(\delta u)^2}.
u_{t}=D_{1}\Delta_{p}u+|u|^{m-2}u-|u|^{q-2}u+f_{1}(u,v)\qquad\text{in}\Omega\times(0,T) v_{t}=D_{2}\Delta_{p}v+|v|^{m-2}v-|v|^{q-2}v+f_{2}(u,v)\qquad\text{in}\Omega\times(0,T) \partial_{n}u =\partial_{n}v=0\qquad\partial\Omega\times(0,T)
G_{MN}(x+\triangle x,g)=\mbox{\boldmath{$e$}}_{M}(x+\triangle x,g)\cdot \mbox{\boldmath{$e$}}_{N}(x+\triangle x,g)=\mbox{\boldmath{$e$}}_{M}^{H}(x+\triangle x,g)\cdot \mbox{\boldmath{$e$}}_{N}^{H}(x+\triangle x,g).
\int_{0}^T \langle \ddot{u}_3^N(t), \varphi_3\rangle \psi(t) dt+\int_{0}^T \langle \sigma^N (t), E \varphi\rangle \psi(t) dt=\int_0^T\langle \mathcal L(t),\varphi(t)\rangle \psi(t)dt,
\int_{0}^{R}g(i) \, di = R^{n+1}\int_{0}^{1}(1-t^2)^{n/2}\, dt &= \frac{R^{n+1}}{2}\int_{0}^{1}(1-s)^{n/2}\frac{1}{\sqrt{s}}\, ds= \frac{\sqrt{\pi}}{2}\frac{\Gamma\left(\frac{n+2}{2}\right)}{\Gamma\left(\frac{n+3}{2}\right)}R^{n+1},
F^{\varepsilon}(\tilde{\textbf{u}}^{\varepsilon}) = -\sum_{j=1}^{d}\textbf{P}J_{\varepsilon}A_{j}(J_{\varepsilon}(\tilde{\textbf{u}^{\varepsilon}}+\bar{\textbf{u}}))\partial_{x_{j}}J_{\varepsilon}\tilde{\textbf{u}}^{\varepsilon}.
|S| \ge (2m+1)\cdot (\frac{n}{2(2l+1)}-2)-\binom{2m+1}{2}\cdot \epsilon n &=\frac{n}{2(2l+1)}\cdot (2m+1)-2(2m+1)-(2m+1)\cdot m\cdot \epsilon n &>\frac{2}{4(2l+1)+1}\cdot (2m+1)n
\mathcal{C}^{+}(n^2_1n_2,m;q) = p_1\, \sum_{d|q}\,d\,\mu\left(\frac{q}{d}\right)\,\mathop{\mathop{\sideset{}{^\star}{\displaystyle\sum}_{\alpha \, \mathrm{mod}\, p_1qr/n_1}}_{\alpha n_1\, \equiv\, m\bar{p_2}\,\,\mathrm{mod}\, d}}_{\alpha n_1\, \equiv\, m\bar{p_2}\,\,\mathrm{mod}\, p_1}\,e\left(\frac{\bar{\alpha}n_2}{p_1qr/n_1}\right).
\begin{array}{llll}\mathfrak{T}_{n,m}\mathrel{\mathop:}= \{A_1,\ldots,A_m: \exists p\mathrel{\mathop:}=p(x),\degree(p)\leq d, \ \mbox{s.t.}\ \ \ p(x)>0, \forall x\neq 0, \ \ \ p(x)-p(A_ix)>0, \forall x\neq 0, \ \ \ \forall i=1,\ldots,m \}\end{array}
\{G_{Q}( \eta ) , G_{Q}( \rho ) \}^{*}= G_{Q}( [ \eta , \rho ] ) + \int_{\partial \Sigma}\omega \mathrm{\tiny ~ \wedge \, ~}\omega \mathrm{\tiny ~ \wedge \, ~}\dots \mathrm{\tiny ~ \wedge \, ~}\omega \mathrm{\tiny ~ \wedge \, ~}\eta_{a}d \rho^{a},
\forall\, x:n\forall\, y:m\forall\, z:p((x\ast_{n,m}y)\ast_{(m+n),p}z=x\ast_{n,(m+p)}(y\ast_{m,p}z)) \forall\, x:n\forall\, y:m(x\ast_{n,m}y=y\ast_{m,n}x)
\|f\|_{L^2(M)}^2 = \sum_{\ell =1}^k(f,\phi_\ell )^2+\sum_{\ell \ge k+1}(f,\phi_\ell )^2 &\le \sum_{\ell =1}^k(f,\phi_\ell )^2+\frac{1}{\lambda_{k+1}}\sum_{\ell \ge k+1}\lambda_\ell (f,\phi_\ell )^2 & \le ke^{2\lambda_k\tau}\|v(\cdot ,\tau )\|_{L^2(M)}^2+ \frac{1}{\lambda_{k+1}}\|f\|_{H_0^1(M )}^2.
\mathcal{H}_{\geq \overline{k}_2}^2=\bigoplus_{k:|k|\geq|\overline{k}_2|}h_{k},\mathcal{H}_{\geq \overline{k}_3}^3=\bigoplus_{j:j\leq|k_3|-1/2,m_{j},k_{j}:}h_{j,m_{j},k_{j}}.
\mathrm{cor}(h \circ Z_k, \varphi) = \mathrm{cor}(h \circ Z_k, g\circ Z_{k+n}) = \mathrm{cor}(h , g\circ Z_{n}) = \int_{\Omega}h \cdot (g \circ T^n) \, d \mu - \int_{\Omega}h \, d \mu \cdot \int_{\Omega}g \, d \mu = : \mathrm{cor}_{T, n}(h,g)\, .
h_{m+1}(2) = b_m\left(h_m(1) + h_m(2)\right) + h_m(2) & = 2b_m\left(b_{m-1}+ 2\right)\left(h_{m-1}(1) + h_{m-1}(2)\right) + b_{m-1}\left(h_{m-1}(1) + h_{m-1}(2)\right) + h_{m-1}(2)
u^{\theta}(x,T_{\theta}) = u(x,s) - \frac{(T_{\theta}- s)^{2}}{2\theta}&\leq u_{0}(x) + (u(x,T_{\theta}) -u_{0}(x)) + (u(x,s) - u(x,T_{\theta})) &\leq u_{0}(x) + \omega(T_{\theta}) + \omega(|T_{\theta}- s|)
r_1 = \frac{R_2 R_3}{R_1+R_2+R_3}R_1 = \frac{r_2 r_3 + r_1 r_3 + r_1 r_2}{r_1}r_2 = \frac{R_1 R_3}{R_1+R_2+R_3}R_2 = \frac{r_2 r_3 + r_1 r_3 + r_1 r_2}{r_2}r_3 = \frac{R_1 R_2}{R_1+R_2+R_3}R_3 = \frac{r_2 r_3 + r_1 r_3 + r_1 r_2}{r_3}
\Delta E_{n}=\alpha B\langle n|J_{z}|n\rangle+\alpha^{2}B^{2}\sum_{m,E_{m}\ne E_{n}}\frac{|\langle n|J_{z}|m\rangle|^{2}}{E_{n}-E_{m}}+\beta B^{2}\sum_{i}\langle n|x_{i}^{2}+y_{i}^{2}|n\rangle.
\Pi_{o , n p}^{( 1 )}= - 2 p^{2}\int_{0}^{1}d x \int_{0}^{x}d y \int_{0}^{y}d z \{5 I_{2}( \nu ) + [ 5 p^{2}( 1 + y - x - z ) ( y - x - z ) + 4 p^{2}+ 2 m^{2}] I_{1}( \nu ) \},
\operatorname{rk}H_\ell \left(\prod_{i=1}^n A_i \right) = \sum_{i=1}^n \operatorname{rk}H_\ell(A_i) = \sum_{i=1}^n \operatorname{rk}H(A_i) \geq \operatorname{rk}H\left( \prod_{i=1}^n A_i \right),
\sum_{i = i_{j-1}+1}^{i_j}\!\!\! d_{Y_i}(g_{i_{j-1}}'\mu,g_{i_j}'\mu) \geq \!\! \sum_{i = i_{j-1}+1}^{i_j}\!\!\! \left( d_{Y_i}(\mu,g\mu)-5b\right) \geq \tfrac12 \!\!\! \sum_{i = i_{j-1}+1}^{i_j}\!\!\! d_{Y_i}(\mu,g\mu).
\frac{1}{N^{\mathrm{in}}}\sum_{h=0}^{\infty}h \ \# \{\mbox{inner nodes in $\mathcal{T}$ of depth}h \}\geq \underbrace{\frac{2^{\log_2 N^{\mathrm{in}}}}{N^{\mathrm{in}}}}_{=1}\left( \log_2 N^{\mathrm{in}}-2 \right) + \underbrace{\frac{2}{N^{\mathrm{in}}}}_{\geq 0}\,.
\sum_{t\in \mathbb F_q}\nu_k^2(t) = \sum_{t\in \mathbb F_q}\nu_k(t) \,\overline{\nu_k}(t) &= q^{2dk}\sum_{{\bf m},{\bf v}\in \mathbb F_q^d}\left( \prod_{j=1}^k \overline{\widehat{E_j}}({\bf m})\right) \left( \prod_{j=1}^k \widehat{E_j}{({\bf v})}\right) \sum_{t\in \mathbb F_q}\widehat{S_t}({\bf m}) \,\overline{\widehat{S_t}}({\bf v}).
{_{s+1}}F_s\left(\left\{\frac{1}{2}\right\}^{s},1-n;\left\{\frac{3}{2}\right\}^{s};1\right)=\sum\limits_{k=1}^n(-1)^{k-1}\binom{n}{k}\overline{H}_k(s), &{_{s+1}}F_s\left(\left\{\frac{1}{2}\right\}^{s},1-n;\left\{\frac{3}{2}\right\}^{s};-1\right)=\sum\limits_{k=1}^n(-1)^{k-1}\binom{n}{k}\overline{H}_k(\overline{s}),
\int x (q^2x^2;q^2)_\infty h_n(x;q) d_qx= \frac{q^{n-1}(x^2;q^2)_\infty}{[n-1]_q}\left( (1-q) h_n(\frac{x}{q};q)- \frac{1-q^n}{q}x h_{n-1}(\frac{x}{q};q)\right),
w^{\prime\prime}+\sum_{j=0}^{n-2}\frac{1-\alpha_j}{z-a_j}w^\prime+\frac{\alpha'\alpha^{\prime\prime}z^{n-3}+\lambda_{n-4}z^{n-4}+\ldots+\lambda_0}{\prod_{j=0}^{n-2}(z-a_j)}w=0,
G_2^{\mathfrak H/\overline{\varGamma}_0(4)}(z)=-\frac{1}{3}\log\frac{2^4|1-\alpha_4(z)|^{2}}{|\alpha_4(z)|}=-\frac{1}{3}\log\frac{2^4|1-\lambda(2z)|^{2}}{|\lambda(2z)|},\forall z\in\mathfrak H.
\sum_{n=1}^{\infty}n^2 \Vert f_n^d \Vert_{L^{1}(S_2)}\Vert I_{2n}\Vert_{L^{1}(S_2)}\leq \sum_{q=1}^{\infty}\sum_{n=1}^{\infty}nq\Vert f_n^d \Vert_{L^{1}(S_2)}\Vert f_q^d \Vert_{L^{1}(S_2)}\Vert f \Vert_{X_1}= \Big[\sum_{n=1}^{\infty}n\Vert f_n^d \Vert_{L^{1}(S_2)}\Big]^2\Vert f \Vert_{X_1}
\bigg(\sum_{j=1}^N |p_{1,j}-q_{1,j}||w_j|\bigg)^2 \le \bigg(\sum_{j=1}^N (p_{1,j}+ q_{1,j})|w_j|\bigg)^2 \le 2 \sum_{j=1}^N (p_{1,j}+ q_{1,j}) |w_j|^2 \,,
L(s, \pi, r_0 \boxtimes \eta) = L(s, \pi', r_0 \boxtimes \eta') \gamma(s, \pi, r_0 \boxtimes \eta, \psi) = \gamma(s, \pi', r_0 \boxtimes \eta', \psi') \epsilon(s, \pi, r_0 \boxtimes \eta, \psi) = \epsilon(s, \pi', r_0 \boxtimes \eta', \psi').
H^{*}( \overline{f}) =f_{0}H^{*}( \overline{e}_{1})+f_{1}H^{*}( \overline{e}_{2})+\cdots+f_{n-1}H^{*}(\overline{e}_{n}) =f_{0}I_{n}+f_{1}H+\cdots+f_{n-1}H^{n-1}=f(H) .
Q = - 2 \left( \begin{array}{c c c}{{N_{1}- N_{2}}}&{{N - N_{1}+ \frac 1 2 N_{2}}}&{{N - N_{1}+ \frac 1 2 N_{2}}}\\{{N - N_{1}+ \frac 1 2 N_{2}}}&{{N_{1}}}&{{N_{1}- N}}\\{{N - N_{1}+ \frac 1 2 N_{2}}}&{{N_{1}- N}}&{{N_{1}- 2 N}}\\ \end{array}\right)
\int_{\Omega}\psi \partial_{t}\left[ g_{1-\alpha,m}*(u-u_{0}) \right]dx+ a(h_{m}*u,\psi) &\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\,\,\,\,\geq \int_{\Omega}(h_{m}*f)\psi dx \, \,\,t\in (0,T), \, m\in\mathbb{N}.
\mathbb{B}(x) := c(p^+, p^-, r^+, r^-)\left( b(x) + b(x)^{\frac{p(x)}{p(x)-1}}+ b(x)^{\frac{r(x)}{\tilde{r}(x)-1}}\right), z(x) := 1 + b(x) + f(x) + g(x)^{p'(x)}+ h(x)^{r'(x)}+ \mathbb{B}(x)+\mathbb{B}^{\frac{p^*_-}{p^*_- - r_+}}(x).
\begin{aligned}\Big\| \int_{0}^{t_{0}+h}e^{-(t_{0}+h-s)A}F_{u}(u_{j},\nabla y_{j}) \ \d s - \int_{0}^{t_{0}}e^{-(t_{0}-s)A}F_{u}(u_{j},\nabla y_{j}) \ \d s \Big\|_{L^{r}_{\sigma}(\Omega)}\end{aligned}
\tau_z(t) (z_1) = \begin{bmatrix}gx -ehym hx-egy -(hx-egy)(-m) gx-ehym \end{bmatrix}, \tau_z(t)(z_2) = \frac{1}{2a}\begin{bmatrix}(2ehm +bg)x- (behm + 2e^2mg) y (2 e g+ b h ) x- ( b e g+ 2e^2m h) y -((2 e g+ b h ) x- ( b e g+ 2e^2m h) y )(-m) (2ehm +bg)x-(behm + 2e^2mg) y \end{bmatrix}.
\Xi_\Omega(A) =\,\lim_{s\to 0}\,\frac{1}{s}\,\Omega(\Phi_t(X))-\,\lim_{s\to 0}\,\frac{1}{s}\,\Omega(\Phi^c_t(X)) = \,\frac{d}{ds}\big|_{s=0}\,\left\{\int^{X(s)}_A\,\left(F^2(\gamma,\dot{\gamma})-\, F^2(\tilde{\gamma},-\dot{\tilde{\gamma}})\right)\,dt\right\},
{\cal S}_{q u a d}= t r \int d^{4}x \left( \frac{g^{2}}8 \omega^{2}\chi^{* \mu \nu}\chi_{\mu \nu}^{*}- \frac{g^{2}}4 \omega \chi^{* \mu \nu}\varepsilon_{\mu}\psi_{v}^{*}- \frac{g^{2}}{3 2}\varepsilon^{\mu}\varepsilon^{\nu}\psi_{\mu}^{*}\psi_{v}^{*}+ \frac{g^{2}}{3 2}\varepsilon^{2}\psi^{* \mu}\psi_{\mu}^{*}\right) \; .
\mathcal{H}^2(B(0,r) \cap \Omega) = \int_{F^{-1}(B(0,r))}\sqrt{\mathcal{D}(\overline{x})}dA =\int_{0}^{2\pi}\int_{0}^{\rho(\theta)}\left( 1+ \frac{\overline{B}(\theta)}{2}+ O(\rho^{4}) \right) \rho d\rho d\theta , =\int_{0}^{2\pi}\left[ \frac{\rho^{2}}{2}+\frac{\overline{B}(\theta)}{8}\rho^4 + O(\rho^6) \right]_{0}^{\rho(\theta)}d\theta .
\int d^{R}x \int d^{R}y \left[ X^{\alpha}, X^{\beta}\right] W [ J , K , K^{*}] = i \varepsilon \int d^{R}x \int d^{R}y \left\{X^{\alpha}< < ( \Delta O )^{\beta}> > - X^{\beta}< < ( \Delta O )^{\alpha}> > \right\}
{\cal A}={V \over 2\pi}\,(8\pi^2 \alpha')^{-1/2}\int_{0}^{\infty}dt \left({\pi \over t}\right)^{9\over 2}\left[{f_3^{8}(q) \over f_1^{8}(q)}-{f_4^{8}(q) \over f_1^{8}(q)}\right]
I(P,V) = O\left( m + m^{2/3}\left( \sum_{\substack{v_1, v_2 , v_3}}|S_{v_1}\cap S_{v_2}\cap S_{v_1}| \right)^{1/3}\right) & = O\left( m + m^{2/3}\left(M^{2}n^{2}+ n^{3}\right)^{1/3}\right) & = O\left( m + M^{2/3}m^{2/3}n^{2/3}+ m^{2/3}n \right).
H^{2}( \Omega^{2}) = \frac{3 T^{2}}{6 4 \pi^{2}}\left[ -{\textstyle \frac{1}{2}}\ln \left( \frac{\Omega^{2}- \mu^{2}}{T^{2}}\right) - 1 . 5 0 6 9 9 \right] .
\sum_{m = - \infty}^{\infty}\sum_{n = 1}^{\infty}\ln ( \bar{\omega}_{m}^{2}+ k_{n}^{2}) = \frac{2}{T}\sum_{n = 1}^{\infty}\left[ \frac{k_{n}}{2}+ T \ln ( 1 - e^{- k_{n}/ T}) \right]{,}
E_j\coloneqq \Big\{(s,v)\in[0,T]\times Z^\eta\colon \|X(s,v)\|_{\infty}=L_j, \|X(r,v)\|_\infty<L_j\;r<s\Big\}&\cup\;\Big\{(T,v)\in\{T\}\times Z^\eta\colon \|X(r,v)\|<L_j\;r\leq T\Big\}.
Z = \int D \phi D\chi D \rho{\rm Exp}\left[ i \{\frac{1}{2}[\partial_\mu \phi_i ]^2-\frac{1}{2}\chi{\vec \phi \cdot \vec \phi}- N V[{\rho}] +\frac{N}{2}\rho \chi +\vec j \cdot \vec \phi + N S \chi + N J\rho \}\right]
S^{(b)}_{q,v(b+1)-1}(N) \leq (1+o(1))\left(\frac{2}{q}\left(\cos \frac{2\pi l}{2l+1}-\cos \frac{4\pi l}{2l+1}\right)\left(2\sin \frac{\pi l}{2l+1}\right)^{k_{1}-1}+\frac{2}{q}\left(2\sin \frac{\pi l}{2l+1}\right)^{k_{1}-2}+\cdots\right) &=(1+o(1))\frac{2}{q}\beta^{k_{1}-1}\left(\cos\frac{2\pi l}{2l+1}-\cos\frac{4\pi l}{2l+1}+\frac{1}{\beta-1}\right)\leq 0.
N_1 = 2(V-1)((k+R+1)B^{2\eta}V^{V+1}+ 2\eta) + 2\eta + 2((k+R+1)B^{2\eta}+ 1), N_2 = (k+R+1)B^{2\eta}+ 1, N_3 = 2(k + R + 1)B^{2\eta}V^{V+1}+ 4\eta.
g_{0 0}( r ) = - g_{r r}^{- 1}( r ) \: \cdot \: \exp \left\lbrace{\cal A}\int_{r_{\infty}}^{r}\left\lbrack F_{1}\left( \frac{r_{S}}{r}\right) - F_{2}\left( \frac{r_{S}}{r}\right) \right\rbrack \: \frac{g_{r r}( r^{\prime})}{r^{\prime 7}}\: d r^{\prime}\right\rbrace
H^{b}(x_{1},x_{2}, x_{3}) =\displaystyle\frac{b_{1}}{2}\left(x_{2}^{2}-\displaystyle\frac{b_{2}}{b_{3}}x_{3}^{2}\right)~~~\hbox{and}~~~C^{b}(x_{1}, x_{2}, x_{3}) = -\displaystyle\frac{b_{3}}{2 b_{1}}x_{1}^{2}+ x_{3}
S=\int dt\,dx{\cal L}=\int dt\,dx\left [ \frac{i\hbar}{2}(\hat{\phi}^{*}\partial_{t}{\phi}-{\phi}\partial_{t}{\phi}^{*})-\frac{\hbar^2}{2m}\left|\left(\partial_x +i{\lambda\over 2}\rho^2\right) \phi\right|^{2}-V(\rho^2)\right ];
\|E(A,B)\| = \sum_{i \ne j}\|E(A_i,B_j)\| \leq 2(\frac{d+1}{d})\cdot \sum_{i \ne j}(\|A_i\| \cdot \|B_j\| + \frac{\lambda(X)}{d+1}\sqrt{\|A_i\| \cdot \|B_j\|})
g_\ell(\pmb{x})=\frac{\exp(\beta\,x_1\sum_{k=2}^\ell x_k\,k^{-2})}{\exp(+\beta\sum_{k=2}^\ell x_k\,k^{-2})+\exp(-\beta\sum_{k=2}^\ell x_k\,k^{-2})}.
-2 \sum_{i=1}^8 z^2_{1,i}=(R-x_1L)^2 &=-2+x_1^2(2g-2)-2x_1(L \cdot R) &=-2+x_1^2(2g-2-2\left \lfloor \frac{g+1}{2}\right \rfloor)-2x_1(L-E \cdot R).
A_{1}C_{2}=C_{1}B_{1},\A_{2}C_{4}=C_{3}B_{2},\R_{A_{1}}C_{1}=0, & R_{A_{2}}C_{3}=0,\C_{2}L_{B_{1}}=0,\C_{4}L_{B_{2}}=0, &R_{M}R_{A}E=0,\R_{A}EL_{D}=0,\EL_{B}L_{N}=0,\R_{C}EL_{B}=0.
\int_\Omega\biggl\{(x\cdot\nabla u)Lu+\frac{n-2s}{2}\,u\,Lu\biggl\}\,dx=-\frac{\Gamma(1+s)^2}{2}\int_{\partial\Omega}\mathcal A(\nu)\left(\frac{u}{d^s}\right)^2(x\cdot\nu)d\sigma.
\left[\bigcap\nolimits_{1 \le i \le r}B_i\right]_J = \bigcap_{1 \le i \le r}[B_i]_J \left[\bigcup\nolimits_{1 \le i \le r}B_i\right]_J \supseteq \bigcup_{1 \le i \le r}[B_i]_J
\sum_{i=0}^N \left( 1-\frac{\textnormal{deg}(T_k)}{2}\right) a_i + \sum_{j=1}^r \omega_j T_j(d) = - \textnormal{dim}_\mathbb{C}(X) + \frac{T_k}{2}+ \frac{\alpha}{2}+ u + 1
\sum_{n=0}^{\infty}q^{n(n-1)/2}F_{n}(a,q;x)(-t)^{n}=\sum_{k=0}^{\infty}\frac{q^{k(k-1)/2}(-xt)^{k}}{(t;q)_{k+1}(at;q)_{k+1}}=\frac{\,_{2}\phi_{2}(q,0;qt,qat;q,xt)}{(1-t)(1-at)}
\Psi_{k}^{(N)}(f)=\Phi_{k}^{(N)}(f)-2\mathbb{E}\left[ f\mid\mathcal{A}^{(k)}\right] +2\sum_{k<l\leqslant N}\mathbf{P}_{\mathcal{A}^{(l)}|\mathcal{A}^{(k)}}\cdot\Phi_{l}^{(N)}(f)=-\Phi_{k}^{(N)}(f).
q^{s}\cdot D^{(s)}[\frac{q^{i}}{1-q^{i}}]=\sum_{m=0}^{s}\sum_{k=0}^{m}[\begin{matrix}s\\m\end{matrix}]\{\begin{matrix}m\\k\end{matrix}\}\frac{(-1)^{s-k}k!i^{m}}{(1-q^{i})^{k+1}}
\Big| \int_{\mathbb{R}^4}F(x+u,y)G(x,y+u)F(x+v,y)G(x,y+v) begin{align*}-1ex] (\phi_{2^{k_{0}}}(u)\phi_{2^{k_{0}}}(v) - \phi_{2^{k_{m}}}(u)\phi_{2^{k_{m}}}(v)) \,dx dy du dv \Big|\lesssim_\lambda C_2 \|F\|^2_{\textup{L}^4(\mathbb{R}^2)}\|G\|^2_{\textup{L}^4(\mathbb{R}^2)}= C_2.
\int_0^T ds \int_0^1 d \theta f'_{\sigma, \ell}(V_s^{(n)}(\theta)) \leq\underline{\sigma}^{-2}\int_0^1 d \theta \int_0^T d \langle V^{(n)}(\theta) \rangle_{s}f'_{\sigma, \ell}(V_s^{(n)}(\theta)) &= \underline{\sigma}^{-2}\int_{\real}dx f'_{\sigma, \ell}(x) \int_0^1 d \theta L_T^x(V^{(n)}(\theta)),
\rho(t,\vec{x})=\rho(t,r)=\frac{f\left( \frac{r}{a(t)}\right)}{a(t)^{2}},{\normalsize u}_{1}{\normalsize =}\frac{F(t,r)}{r}x-\frac{G(t,r)}{r}yu_{2}=\frac{G(t,r)}{r}x+\frac{F(t,r)}{r}y
u(x) =\int_{\partial_m D}M_D(x,y)\,\nu(dy)+\int_{(D\cup\partial_mD)^c}P_D(x,y)\,\gamma(dy) &\quad+\int_Df(y,u(y))G_D(x,y)\,dy+\int_DG_D(x,y)\,\mu(dy)x\in D.
\mathbf{a}_d(\mathbf{x}(k-1),k-1)= \mathbf{A}_{k-1}\mathbf{q}_d(\mathbf{x}(k-1)), \mathbf{c}_d(\mathbf{x}(k),k)= \mathbf{C}_k \mathbf{q}_d(\mathbf{x}(k)),
S = \frac{1}{g_{2}^{2}}\int d t \epsilon_{i j}T r_{( u ( 2 ) \oplus u ( N ) )}\lbrack ( \dot{X}+ i [ A_{0}, X^{i}] ) X^{j}+ \theta \epsilon^{i j}A_{0}( 1 + J_{0}) \rbrack ,
p(x)= b_1f^{-1/4}\exp\left(\int_{-\infty}^{x}f^{1/2}dx\right)\bigl(1+\textit{o}(1)\bigr) +b_2f^{-1/4}\exp\left(-\int_{-\infty}^{x}f^{1/2}dx\right)\bigl(1+\textit{o}(1)\bigr),
\sum_{\boldsymbol{a}\neq\boldsymbol{j}}\|(\boldsymbol{\Psi}_{\boldsymbol{j}}^T\boldsymbol{\Psi}_{\boldsymbol{a}})_{\boldsymbol{k},\cdot}\| \lesssim\sum_{\boldsymbol{a}\neq\boldsymbol{j}}\prod_{l=1}^d2^{a_l+j_l/2}.
4d\delta u^\flat_t+2\delta du^\flat_t-4\operatorname{Ric}(u_t) +(\operatorname{div}u)\big(4d\delta u^\flat+2\delta du^\flat-4\operatorname{Ric}(u)\big) \hphantom{4d\delta u^\flat_t}{}+L_u\big(4d\delta u^\flat+2\delta du^\flat-4\operatorname{Ric}(u)\big)=0
B^{-1}A_iB = \left(\begin{array}{ccccc}A_i^{(1,1)}A_i^{(1,2)}A_i^{(1,3)}\cdots A_i^{(1,k)}0 A_i^{(2,2)}A_i^{(2,3)}\cdots A_i^{(2,k)}0 0 A_i^{(3,3)}\cdots A_i^{(3,k)}\vdots \vdots \vdots \ddots \vdots 0 0 0 \cdots A_i^{(k,k)}\end{array}\right)
\begin{cases}\dot{\bf m}= (A + G){\bf m}+h +\gamma{\bf p}, \dot{\bf p}= -(A+G)^T{\bf p}-\widehat Q{\bf m}+ (I-\Gamma)^T Q\eta, \dot{\bf y}=-A^T{\bf y}+Q[{\bf m}-(\Gamma{\bf m}+\eta)] ,\end{cases}
\int_0^t L_{1,n}h_n(X^n_s, Z^n_s) ds \approx\int_0^t A_2 f(X^n_s,g_n(Z_s^n)) ds = \int_0^t \int A_2 f(X^n_s,z))\Gamma_{g_n(Z^n)}(ds,dz),
\sum_{i=0}^{t-1}(i+1) \binom{t+1}{i+2}(t+\rho)^{-i-2}= 1 - \frac{(t+\rho+1)^t}{(t+\rho)^t}\frac{\rho}{t+\rho}= 1 - \frac{\rho(t+1+\rho)^t}{(t+\rho)^{t+1}}.
\sum_{n=-\infty}^{\infty}q^{n^{2}}\left[z^{-2n}\varphi_{k}\left(z^{-1}q^{n+1/2}\right)\varphi_{\ell}\left(z^{-1}q^{n+1/2}\right)+z^{2n}\varphi_{k}\left(-zq^{n+1/2}\right)\varphi_{\ell}\left(-zq^{n+1/2}\right)\right] \hskip254pt=2\|\varphi\left(q^{1/2}z^{-1}\right)\!\|^{2}\delta_{k,\ell}.
[E + A(U_*)] \mu'(U_0) = \frac{1}{2 \sqrt{3}}A'(U_0) \int_{U_0}^{U_+}\frac{(1 - \sqrt{3}u^2)}{(1 + \sqrt{3}u^2)^2}\frac{du}{p}- \frac{3}{4}U_0^2 \sqrt{A(U_0)}.
A_2:=\int\limits_{\stackrel{1 < x\leq 1/\sqrt{\epsilon(r)}}{x+1/x\leq \frac{2\cos\theta}{\sqrt{1-r^4}}}}\frac 1x(x+\frac 1x)^j dx =\int_2^{\frac{2\cos\theta}{\sqrt{1-r^4}}}y^{j-1}dy+\cdots
\delta^{i_1i_2}G_{i_1i_2i_3}^{\eta j_2j_3}\delta_{j_2j_3}\tilde\gamma_{\eta k}+2\delta^{i_1i_2}G_{i_1i_2i_3}^{j_1j_2j_3}\gamma_{j_1j_3}\delta_{j_2k}=\delta_{i_3k}.
(I-\frac{y_{k}\Delta x_{k}^{T}}{y_{k}^{T}\Delta x_{k}})B_{k}(I-\frac{\Delta x_{k}y_{k}^{T}}{y_{k}^{T}\Delta x_{k}})+\frac{y_{k}y_{k}^{T}}{y_{k}^{T}\Delta x_{k}}
\int_{-\infty}^{\infty}I_1dt\leqslant C\int_0^\infty\langle|v|^{2\rho-1}t\rangle^{-\gamma}dt=C|v|^{1-2\rho}\int_0^\infty\langle\tau\rangle^{-\gamma}d\tau=O(|v|^{1-2\rho})
G^{\mu \nu}\theta_{\mu}\psi_{\nu}= \sum_{\mu = 1}^{d}\theta_{\mu}\psi_{\mu}- \frac{1}{d - 1}\big ( \sum_{\mu = 1}^{d}\theta_{\mu}\big ) \big ( \sum_{\nu = 1}^{d}\psi_{\nu}\big ) .
{\hat{Y}}_{l}^{I}{\hat{Y}}_{m}^{J}= \sum_{K = 0}^{N}( - )^{N + K}\sqrt{( 2 I + 1 ) ( 2 J + 1 )}\Bigl \{\begin{array}{c c c}{{I}}&{{J}}&{{K}}\\{{{\frac{N}{2}}}}&{{{\frac{N}{2}}}}&{{{\frac{N}{2}}\}}\\ \end{array}\Bigr \}C_{l m n}^{I J K}{\hat{Y}}_{n}^{K}
A_{\alpha}= \Big( \alpha g(\alpha) \dots g^{n_{\alpha}-1}(\alpha)\Big)^{m_{\alpha}-1}\alpha g(\alpha) \dots g^{n_{\alpha}-2}(\alpha) , \mbox{if}n_{\alpha}\geq 2, A_{\alpha}= \alpha^{m_{\alpha}-1}, \mbox{if}n_{\alpha}= 1 ,
\begin{aligned}\sup_{N \in \mathbb{N}}\mathbb{P}(\bar \mu^N \in H^c_M) = \sup_{N \in \mathbb{N}}\mathbb{P}(\bar \mu^N(G_\alpha(f))>M )\leq\frac{\sup_{N \in \mathbb{N}}\mathbb{E}[\bar \mu^N(G_\alpha(f))]}{M}.\end{aligned}
E(\omega \otimes \iota)\bar\Delta(x) =E(\omega \otimes \iota)(W^*(1 \otimes x)W) &=\lim_\alpha \sum_{i=1}^{n_\alpha}(\omega \otimes \iota) (W^*(1 \otimes a_{i,\alpha}xb_{i,\alpha})W) &= (\omega \otimes \iota)W^*(1 \otimes E(x))W &= (\omega \otimes \iota) (1 \otimes E(x)),
\limsup_{N \to \infty}\bigg|V_S^N(\nu^1,\dots,\nu^N)-V_S \bigg( \frac{1}{N}\sum_{i=1}^{N}\nu^i \bigg) \bigg|\le\lim_{m \to \infty}\bigg|V_S^{N_m}(\nu^1,\dots,\nu^{N_m})-V_S (\nu) \bigg|+\lim_{m \to \infty}\bigg|V_S(\nu) -V_S \bigg( \frac{1}{N}\sum_{i=1}^{N_m}\nu^i \bigg) \bigg|=0,
\ell_k^- = k \ell + \left\lfloor \frac{\ell n^-}{n}\right\rfloor =k \ell + \frac{\ell n^-}{n}- \left( \frac{\ell n^-}{n}{\rm ~mod~}1 \right) =\frac{\ell n_k^-}{n}- \left( \frac{\ell d}{n}{\rm ~mod~}1 \right) \;,
C_0 -2h_q \leq \frac{\mu_{t_q}^{k_q}(\phi) -\mu_{t_0}^{k_q}(\phi)}{t_q-t_0}=\frac{1}{t_q -t_0}\int_{t_0}^{t_q}\frac{d}{dt}\mu_t^{k_q}(\phi) \, dt
\int_{-\infty}^{t}f(u)du=\frac{1}{2}+t\frac{\Gamma(\frac{1}{2}(\nu+1))}{\sqrt{\pi\nu}\Gamma(\frac{\nu}{2})}{}_{2}F_{1}(\frac{1}{2},\frac{1}{2}(\nu+1);\frac{3}{2};-\frac{t^{2}}{\nu}),
\langle \tilde{Q}_{+}Q_{+}\rangle = \left[ v^{2}- \frac{\mu^{3}\Lambda_{1}^{3}}{v ( - \det h )}\right]^{1 / 2}= \left[ v^{2}- \frac{4 \mu \, s \,}{( - \det h )}\right]^{1 / 2}\, ,
F_{\alpha}(\sigma^{\gamma})=\delta_{\alpha}^{\gamma}+\frac{1}{12}(c_{\alpha\mu}^{\beta}c_{\beta\nu}^{\gamma}-3c_{\alpha\mu}^{b}c_{\nu b}^{\gamma})\sigma^{\mu}\sigma^{\nu},I_{a}(\sigma^{\gamma})=c_{a\nu}^{\gamma}\sigma^{\nu},
c_{l , m}\; = \; q^{m - l}\, i \tau_{l}^{2}\tau_{l}^{1}\; + \; q^{l - m}\, i \tau_{m}^{2}\tau_{m}^{1}\; - \; \eta^{m - l}\, i \tau_{l}^{2}\tau_{m}^{1}\; - \; \eta^{l - m}\, i \tau_{m}^{2}\tau_{l}^{1}\,
\frac{d^2R_{n\ell}(s)}{ds^2}+\left[\frac{s^2\left(\frac{2\mu E_{n\ell}}{\hbar^2}\right)+s\left(\frac{4\mu D_{e}a}{\hbar^2}\right)+\left(-\frac{2\mu D_ea^2}{\hbar^2}-\ell(\ell+1)\right)}{s^2}\right]R_{n\ell}(s)=0,
\nabla( \nabla^{\pi}F )((V_{n+1}, \ldots; V_1);(V_n, \ldots, V_1; X)) \simeq \nabla^{|\pi|+1}F(V_{|S_1|}, \ldots, V_{|S_k|}, V_1; X)\oplus\bigoplus_{i=1}^k\nabla^{|\pi|}F(V_{|S_1|}, \ldots, V_{|S_{i-1}|}, V_{|S_i|+1}, V_{|S_{i+1}|},\ldots, V_{|S_k|}; X).
\frac{\theta(1/z;q^{2})}{(-qtz;q)_\infty}= \frac{\theta(1/z;q^{2})}{(-qtz;q^2)}\cdot\frac1{(-q^2tz;q^2)}=\frac{\theta(1/z;q^{2})}{(-q^2tz;q^2)}\cdot\frac1{(-qtz;q^2)}, \frac{\theta(1/z;q^{1/2})}{(q^{1/2}tz^2;q)_\infty}= \frac{\theta(1/z;q^{1/2})}{(q^{1/4}t^{1/2}z;q^{1/2})_\infty}\cdot \frac 1{(- q^{1/4}t^{1/2}z;q^{1/2})_\infty}.
P^{\prime}( w^{\prime}) = L^{( d + \beta - \alpha ) k}\; \; r^{k}\; \; \prod_{i^{\prime}= 0}^{k - 1}q ( X_{i^{\prime}+ 1}^{\prime}- X_{i^{\prime}}^{\prime}) e^{- r L^{( d + \beta - \alpha )}T^{\prime}}
\nabla^{2}h_{\mu \nu}= - 1 6 \pi G_{N}^{D}\left( T_{\mu \nu}^{m a t}-{\frac{1}{D - 2}}\eta_{\mu \nu}T^{m a t}\right) \equiv - 1 6 \pi G_{N}^{D}\bar{T}_{\mu \nu}^{m a t},
\frac{g^{\prime \prime}(0)}{2}\mathcal{A}^\varepsilon_{s,t}(\varphi ) = \lim_{n \to \infty}\int_s^t \sum_{j \in \mathbb{Z}}\tau_j \mathcal{Q}^n_\rho (\varepsilon n ; r ) \nabla^n \varphi^n_j (r ) dr
| V_{B}\rangle = \mathrm{e x p}\left( \frac{1}{2}\sum_{r , s = 1}^{3}\sum_{m , n = - \infty}^{\infty}\sum_{I = 1}^{8}a_{r m}^{I \dagger}\overline{{N}}_{m n}^{r s}a_{s n}^{I \dagger}\right) | 0 \rangle .
\sigma^{(1)}_{{\rm tot}\,i}= \frac{(2\pi)^4 \hbar^2}{v_i}\frac{1}{\pi}{\rm Im}\, R^{(1)}_{ii}\exp\left[-\frac{1}{\hbar}\int_{-\infty}^\infty dt \Gamma^{(1)}(t) \right] \; ,
\mathrm{P}_{t,1}= \mathrm{P}\left( z_t<\frac{\epsilon_1}{\rho\xi_1}\right) \overset{(a)}{=}\mathrm{P}\left( \alpha_1=1 \right)= \mathrm{P}\left( z_t<\frac{\epsilon_1}{\rho}\right)=\mathrm{P}^{OMA}_{t,1},
\begin{array}{rcr}J^{ad}\nabla_aY_{bcd}= J^{ad}V_{bc}{}^e{}_a\Phi_{ed}+4n(J^{ad}\Phi_{ba}\Phi_{cd}-\nabla_{[b}S_{c]})\qquad begin{align*}3pt]{}+2J_{bc}J^{ad}(\nabla_aS_d-J^{ef}\Phi_{ae}\Phi_{df}).\end{array}
\int_{Q_K}\Delta_1 d \nu_q = O( n^{-1/2}k^{-2}), \int_{Q_K}\Delta_1 ( e^{n\beta H_q^{(3)}}- 1 - H_q^{(3)}) d\nu_q = O \left( \frac{(\log k)^{5}}{n^{1/2}k}\right).