harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g_{H^{n}(a)}=a^{2}\frac{(a^{2}-|x|^{2})(dx_{1}^{2}+\cdot\cdot\cdot+dx_{n}^{2})-(x_{1}dx_{1}+\cdot\cdot\cdot+x_{n}dx_{n})^{2}}{(a^{2}-|x|^{2})^{2}}.
\binom{a_0 + a_1p + a_2p^2 + \dots + a_mp^m}{b_0 + b_1p + b_2p^2 + \dots + b_mp^m}\equiv \binom{a_0}{b_0}\binom{a_1}{b_1}\binom{a_2}{b_2}\dots \binom{a_m}{b_m}\pmod{p},
\prod_{j=x_1}^k |\cos(\pi \theta_j)|\leq C(\varepsilon) e^{-(\ln 2-\varepsilon)|x_1-k|}, \prod_{j=k}^{x_2}|\cos(\pi \theta_j)|\leq C(\varepsilon) e^{-(\ln 2-\varepsilon)|x_2-k|}c_{n,\ell}.
\binom{n}{t-a}\leq \left(\frac{en}{t-a}\right)^{t-a}\leq \left(\frac{en}{t-t/\rho}\right)^{t-a}\leq \left(\frac{e}{1-1/\rho}\right)^t\cdot \left(\frac{n}{t}\right)^{t-a}.
l_{\pi_{\lambda}}(z)=\int_{\mathbb{R}}\mathrm{e}^{zx}\,\mathrm{d}\pi_{\lambda}(x)=\mathrm{e}^{-\lambda}\sum_{k=0}^{\infty}\mathrm{e}^{zk}\frac{\lambda^{k}}{k!}=\exp\left(\lambda(\mathrm{e}^{z}-1)\right).
\overline{\nabla}_{\dot{x}}^\ast y = \pm \frac{\partial H}{\partial x}- \bigl\langle \overline{\mu}, \widetilde{R}( \dot{x}, \cdot ) \bigr\rangle , \widetilde{\nabla}_{\dot{x}}^\ast \overline{\mu}= \operatorname{ad}^\ast_{\overline{v}}\overline{\mu}.
\Sigma_A'(y) + \frac{p}{p-1}\Sigma_A(y)^{(p-1)/p}= \frac{p}{p-1}(1-y) \left[ A^{(p-1)/p}- A (1-y)^{(2-p)/(p-1)}\right] & \ge \frac{p}{p-1}(1-y) \left[ A^{(p-1)/p}- A \right] & \ge \frac{p}{p-1}(1-y) a^{2-p}= \psi'(y) + \frac{p}{p-1}\psi(y)^{(p-1)/p}
\sum_{J=1}^{2n}\sum_{A=R,I}|\mathcal X^s_{JA}(z,\Xi)|+\|\mathcal X^s_{\Xi}(z,\Xi)\|_{\mathbf \Sigma^{r_0}}\leq c |z|(|\mathbf Z|+\|\Xi\|_{\mathbf \Sigma^{-r_0}}).
\mathfrak B(\mathsf V\cup \mathsf U) /\mathfrak B(\mathsf U) = (\mathfrak B(\mathsf V) + \mathfrak B(\mathsf U))/ \mathfrak B(\mathsf U) \cong \mathfrak B(\mathsf V)/(\mathfrak B(\mathsf U) \cap\mathfrak B(\mathsf V)) = \mathfrak B(\mathsf V) / \mathfrak B(\mathsf U \cap \mathsf V).
\begin{array}{c}{{u_{i}^{\mu}v_{\mu}^{j}= G_{i}^{~ j}\equiv [ \exp ( - W ) ]_{i}^{~ j},}}\\{{{G^{- 1}}_{i}^{~ j}( u_{j}^{\mu}v_{\nu}^{i}+ u_{\nu j}v^{\mu i}) = \delta_{\nu}^{\mu}.}}\\ \end{array}
\displaystyle \frac{\vartheta_{\sigma}^{(n)}(w', x^{-1}z_{2n-1}|\zeta'',\zeta_{2n-1}, \epsilon x^{-1}\zeta_{2n-1})}{\vartheta_{\epsilon\sigma}^{(n-1)}(w' |\zeta'')}=\displaystyle x^{2n}z_{2n-1}^{-2n}\prod_{j=1}^{2n-2}z_j^{-1}\prod_{a\in A\atop a\neq a_{2n}}w_a^{-1}\Theta_{x^2}(x w_{a}/z_{2n-1}).
\frac{\partial \zeta^{(\lambda\mu\alpha)}}{\partial \zeta^{(\tau,\nu,\beta)}}=\psi^{(\lambda\mu\alpha)}_{(\tau,\nu,\beta)}=\frac{1}{2}\left(D^{(\lambda\mu\alpha)}_{(\tau,\nu,\beta)}+\omega^{(\rho,\omega,\gamma)}_{(\tau,\nu,\beta)}D^{(\lambda\mu\alpha)}_{(\rho,\omega,\gamma)}\right).
I_1= (q_1 p_1 -q_2 p_2)^2+b_1 b_2 (q_1 p_1+q_2 p_2) & + b_1 \left( a (p_1 + q_2)-q_1 p_1^2-q_2^2p_2\right) +b_2\left(a (q_1 + p_2)-q_1^2 p_1-q_2 p_2^2\right) I_2= (a (q_1 + p_2) + q_1 p_2 (b_2 - q_2 - p_1)) ( a (q_2 + p_1)+ q_2 p_1 (b_1 - q_1 - p_2) ).
P_{\mu_{k}}(M ):=\sup \left\{\int_{M}\mbox{div}\,\left( |x|^{k}\mathbf{v}\right) \,dx:\,\mathbf{v}\in C_{0}^{1}(\mathbb{R}^N ,\mathbb{R}^{N}),\,|\mathbf{v}|\leq 1\mbox{in}\, M \right\}.
n_{ph}(E_{g})=\int_{E_{g}}^{\infty}\frac{dn_{ph}}{dhv}dhv=\sum_{i=E_{g}}^{\infty}(hv_{i+1}-hv_{i})\frac{1}{2}[\frac{dn_{ph}}{dhv}(hv_{i+1})+\frac{dn_{ph}}{dhv}(hv_{i})]
0 =D_{\phi}f_p(X_0,e^{i(\pi-\phi)}(-Y+X_0)) &=D_{\phi}g_p (W^{1/2}(AX_0B-C),W^{1/2}Ae^{i(\pi-\phi)}(-Y+X_0)B+e^{i(\pi-\phi)}W^{1/2}C-e^{i(\pi-\phi)}W^{1/2}C) &=D_{\phi}g_p (W^{1/2}(AX_0B-C),-e^{i(\pi-\phi)}W^{1/2}(AYB-C)+e^{i(\pi-\phi)}W^{1/2}(AX_0B-C)).
=\int_{-\infty}^{\infty}[\delta(t+\frac{\tau}{2}-t_{1})+\delta(t+\frac{\tau}{2}-t_{2})][\delta(t-\frac{\tau}{2}-t_{1})+\delta(t-\frac{\tau}{2}-t_{2})]e^{-j2\pi\tau f}\cdot d\tau
Z_{l e f t}( \tau ) = \prod_{I = 1}^{1 6}Z_{0 , 2 n R A_{y}^{I}}^{0 , 2 m R A_{y}^{I}}( \tau ) + \prod_{I = 1}^{1 6}Z_{1 , 2 n R A_{y}^{I}}^{0 , 2 m R A_{y}^{I}}( \tau ) + \prod_{I = 1}^{1 6}Z_{0 , 2 n R A_{y}^{I}}^{1 , 2 m R A_{y}^{I}}( \tau ) + \prod_{I = 1}^{1 6}Z_{1 , 2 n R A_{y}^{I}}^{1 , 2 m R A_{y}^{I}}( \tau )
\widehat{{\bf H}_1{\bf H}_2}= \left(\begin{matrix}{\bf H}_1 0 0{\rm I}_d\end{matrix}\right)\left(\begin{matrix}0{\bf H}_2{\bf H}_2^* 0\end{matrix}\right)\left(\begin{matrix}{\bf H}_1^* 0 0{\rm I}_d\end{matrix}\right) &\stackrel{\mathcal{L}}{=}\widetilde{{\bf H}_1}^{1/2}\widehat{{\bf H}_2}\widetilde{{\bf H}_1}^{1/2}.
o^{\prime}(\phi(\hat{\sigma}))=C^{\prime}(\beta(\tilde{\sigma}(\hat{\sigma})))\tilde{\sigma}^{\prime}(\hat{\sigma})=C^{\prime}(\beta(\phi(\hat{\sigma})))\beta^{\prime}(\phi(\hat{\sigma}))\frac{\phi^{\prime}(\hat{\sigma})}{\beta^{\prime}(\tilde{\sigma}(\hat{\sigma}))};
\frac{\textrm d}{\textrm dt}\left[V(x(t))+Q_k(t-t_k,x_{k})\right]+2\alpha V(x(t)) + 2\alpha Q_k(t-t_k,x_{k}) &=\nabla V(x)^T f(x(t),x(t_k)) + \nabla_x F_k(t-t_k,x(t),x(t_{k+1}),x(t_k))^T f(x) & + \frac{d}{dt}F_k(t-t_k,x(t),x(t_{k+1}),x(t_k)) +2\alpha V(x(t)) & + 2\alpha F_k(t-t_k,x(t),x(t_{k+1}),x(t_k))<-\delta\|x(t)\|
\widehat{g}^{\pm}_{\sigma,\Delta}(\xi) = \sum_{k=-\infty}^{\infty}(\pm 1)^k \frac{(k+1)}{|\xi+k\Delta|}\Big(e^{-2\pi |\xi+k\Delta|(\sigma-\frac12)}- e^{-2\pi|\xi+k\Delta|}\Big).
m < 0 \psi ( x ) = \sum_{n = 1}^{\infty}\int_{- \infty}^{+ \infty}d p a_{n p}\psi_{n p}^{( + )}( x ) + \sum_{n = 0}^{\infty}\int_{- \infty}^{+ \infty}d p b_{n p}^{\dag}\psi_{n p}^{( - )},
\partial_t\widetilde U=\Delta\widetilde U-\nabla P-hu_\infty\cdot\nabla\widetilde U+g-F, \mbox{div $\widetilde U$}=0 (x\in\Omega,\,t>0), &\widetilde U|_{\partial\Omega}=0, &\widetilde U\to 0 \quad\mbox{as $|x|\to\infty$}, &\widetilde U(\cdot,0)=(1-\phi)v_0+\mathbb B[v_0\cdot\nabla\phi],
\liminf_{n \to +\infty}\int_{\mathbb{R}^N}V |u_n|^2 = \lim_{n \to +\infty}\int_{\mathbb{R}^N \setminus G}V |u_n|^2 + \liminf_{n \to +\infty}\int_{\mathbb{R}^N}V |u_n|^2 &\geq \int_{\mathbb{R}^N}V |u|^2.
f_{t}(\hat{\mathcal{X}})=\big\{\big(\varphi(\hat{\pi}(t),t),\psi(t)\big):\hat{\pi}\in \hat{\mathcal{X}}\big\}=\big\{\varphi(\hat{\pi}(t),t):\hat{\pi}\in \hat{\mathcal{X}}\big\}\times \big\{\psi(t)\big\}.
\widetilde{\theta}(\hat{y}_i, y^{*}_{-i})=\theta +\frac{(1-\theta)}{l_1+\frac{l_2}{N}e^{-\beta\hat{\tau}_i}x_i(\hat{\tau}_i)+\frac{l_2}{N}\sum\limits_{j=1,j\neq i}^Ne^{-\beta\tau_j^{*}}x_j(\tau_j^{*})}.
\langle f, g\rangle =\sum_{k\in \Bbb Z}\langle T_N^{-1}D_kD_k^N(f), g \rangle =\sum_{k\in \Bbb Z}\langle D_kD_k^NT_N^{-1}(f), g \rangle, \qquad\forall\g\in \dot{\mathcal B}^{\alpha,q}_{p,\mathcal F}.
\mathbb{E}\prod_{j=1}^{|V|}\prod_{e\in S_{j}\backslash(S_{1}\cup\ldots\cup S_{j-1})}\left|\left\langle h_{e}^{(1)},\frac{h_{e}^{(2)}}{\|h_{e}^{(2)}\|+\eta}\right\rangle\right|\leq\frac{C_{\epsilon}}{n^{|V|(1-\epsilon)}},
\{F,G\}_1 = \frac{1}{\epsilon}\int \mathrm{res}\left([L,X_+] Y-[L,X]_+Y\right) \; dx , &\{F,G\}_2 = \frac{1}{\epsilon}\int \mathrm{res}\left( (LY)_+ LX - (YL)_+ XL + \frac{1}{n+1}X[L,K_Y] \right) dx ,
\!\!\!\!\!\!\!\! B_N(\lambda)+\frac{N}{2}\frac{{\rm d}V^{(\alpha)}(\lambda)}{{\rm d}\lambda}=\frac{\lambda}{2c_N}A_N(\lambda)-(-1)^N\frac{\alpha}{\lambda}-\frac{\alpha}{2}(T_B(N-1,\lambda)-T_B(N,\lambda)) \ .
{\cal L}_{c.t.}^{(1)}\, = \,{{\sqrt{-g}}\over (4 \pi)^{2}\epsilon}\lbrack{1 \over{120}}R^{2}+{7 \over{20}}R_{\mu \nu}R^{\mu \nu}-{53 \over{180}}\epsilon^{\alpha \beta \gamma \delta}\epsilon_{\mu \nu \rho \sigma}{R^{\mu \nu}}_{\alpha \beta}{R^{\rho \sigma}}_{\gamma \delta}\rbrack
\sum_{C \leq p \leq w}\sum_{\substack{m = -p_1 + p_2 - p_3 p_j \in I_j, \forall j \leq 3 p^2 | A - p_1}}1 & \leq \sum_{C \leq p \leq w}\sum_{\substack{p_1 \equiv A \pmod{p^2}p_1 \in I_1}}| \{n \in I_2 \cap (m+p_1+I_3): n, n-m-p_1 \hbox{prime}\}|.
\|\mathcal{K}(t,s)f\|_{L_{p}}=\left\|\int_{\mathbf{R}^{d}}K(t-s,x-y)f(y)dy\right\|_{L_{p}}&\leq \|f\|_{L_{p}}\int_{\mathbf{R}^{d}}|K(t-s,y)|dy\leq N(d,\alpha)(t-s)^{-1}\|f\|_{L_{p}}.
{\bf x}^{(l)}\in K_l,~{\bf w}^{(l)}:=\mathcal{A}^l({\bf x}^{(l)})^{m-1}+{\bf q}^{(l)}\in K_l^* ~{\rm and}~\langle{\bf x}^{(l)},{\bf w}^{(l)}\rangle=0.
\Phi(z)=\left[ \pi \left( \frac{\partial}{\partial x_1},\frac{\partial}{\partial x_1}\right) - \pi \left( \frac{\partial}{\partial x_2},\frac{\partial}{\partial x_2}\right) -2i \pi \left( \frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2}\right) \right] \, dz^2,
b=\frac{1-|\frac{\partial}{\partial z}P_{c}^{p}(z_{0})|^{2}}{|\frac{\partial}{\partial c}\frac{\partial}{\partial z}P_{c}^{p}(z_{0})+\frac{\partial}{\partial z}\frac{\partial}{\partial z}P_{c}^{p}(z_{0})\frac{\frac{\partial}{\partial c}P_{c}^{p}(z_{0})}{1-\frac{\partial}{\partial z}P_{c}^{p}(z_{0})}|},
\tilde{\delta}_3 =W^{(q+r)}(b)\Bigg[-\dfrac{\rho^{(q)}_{b}(b;h)+\tilde{g}^{(q)}(b;h)}{W^{(q)}(b)}r\int_b^{\infty}e^{-\Phi(q+r)y}W^{(q)}(y)dy-\int_0^{b}e^{-\Phi(q+r)u}h(u)du &\quad+r\int_{b}^{\infty}e^{-\Phi(q+r)y}\rho_b^{(q)}(y;h)dy\Bigg]+\tilde{g}^{(q)}(b;h)+\int_0^bh(u)W^{(q+r)}(b-u)du.
\int_{S'_{d}}\langle w,\varphi\rangle_{0}^{2n+1}\,d\mu_{\beta}(w) =0, \int_{S'_{d}}\langle w,\varphi\rangle_{0}^{2n}\,d\mu_{\beta}(w) =\frac{(2n)!}{2^{n}\Gamma(\beta n+1)}|\varphi|_{0}^{2n}.
(G_{o}(k)^{ab})=\left( \begin{array}{cc}G_{0}(k) 0 0 \widetilde{G}_{0}(k)\end{array}\right) =\left( \begin{array}{cc}\frac{1}{k^{2}-m^{2}+i\epsilon}0 0 \frac{-1}{k^{2}-m^{2}-i\epsilon}\end{array}\right) .
Z^X_v\left(-\frac{1}{\tau}\right)=\vert\Gamma_{\cal G}\vert^{-\frac{b_2(X)}{2}}\left(\frac{\tau}{i}\right)^{-\frac{\chi(X)}{2}}\sum_u\zeta_{|\Gamma_{\cal G}|}^{u\cdot v}Z_u^X(\tau),
\delta_{\kappa}X^{M}E_{M}{}^{r}= 0 \quad \Rightarrow \quad \delta_{\kappa}X^{\mu}= - i \bar{\theta}_{1}\Gamma^{\mu}\delta_{\kappa}\theta_{1}- i \bar{\theta}_{2}\Gamma^{\mu}\delta_{\kappa}\theta_{2}+{\cal O}( \theta^{4}) .
Y_{l_1 , m_1}Y_{l_2 , m_2}=\sum\limits_{l=|l_1-l_2|}^{l_1+l_2}\sqrt{\frac{(2 l_1+1)(2 l_2+1)}{4\pi(2 l+1)}}\left\langle l_1 , 0 , l_2 , 0 \right| \left. l , 0\right\rangle
x_\beta \ = \ \bigg(\frac{m_\beta}{m_\beta-p}\frac{c(1,2)}{K_2^\beta-K_1^\beta}\bigg)^{\frac{1}{p}}, A_\beta \ = \ \big(K_2^\beta - K_1^\beta\big) \frac{p}{m_\beta}x_\beta^{p-m_\beta}.
\delta_{\eta}L=L(\Phi,\eta+\delta\eta)-L(\Phi,\eta)=-\int_{t_{0}}^{t_{1}}\iint[\rho\delta\eta(\frac{\partial\Phi}{\partial t}+\frac{1}{2}|\nabla\Phi|^{2}+\frac{1}{2}(\frac{\partial\Phi}{\partial z})^{2}+g\eta)]_{z=\eta(x,t)}dxdt=0.
\frac{d^{n-1}}{d-1}\sum_{i=0}^l \left( (d+1)\ln(d^{(i)}!) - d^{(i)}\ln(d^{(i)}) \right) &+ n \cdot d^{n-1}\left((d+1)\ln(d)\right) & \quad+ d^{n-1}(d+1)(\ln(d+1)-\ln(d)-1) + O(n) &\leq (d+1)d^{n-1}\ln(d) & \quad+ \sum_{i=0}^l \left( n \cdot d^{n-1}\left( d^{(i)}\ln(d) \right) + d^{n-1}d^{(i)}\left( \ln(d^{(i)}) - \ln(d) -1 \right) \right).
b^{i+1}=\widetilde{b^{i}}\vDash n-i-2, l'_{i+1}=\begin{cases}l(\nu^i,b^i,1) i\in B, l(\nu^i,b^i,l'_{i}) \end{cases}\nu^{i+1}=(\nu^{i})^{(l'_{i+1})}\vdash n-i-2,
\left\Vert U_{\hbar}^{M}\left( t,s\right) \right\Vert_{\beta\rightarrow\beta}\leq e^{K\left( \beta,d\right) \left( \hbar M+1\right)^{\frac{d}{2}-1}\sum_{k=2}^{d}\int_{J_{s,t}}\left\vert P_{k}\left( \tau:\theta,\theta^{\ast}\right) \right\vert d\tau}.
\left\langle \left\langle g^{4}F_{\mu_{1}\nu_{1}}^{a_{1}}F_{\mu_{1}\nu_{1}}^{a_{1}}F_{\mu_{2}\nu_{2}}^{a_{2}}F_{\mu_{2}\nu_{2}}^{a_{2}}\right\rangle \right\rangle =\mathcal{N}\left( \left\langle g^{2}FF\right\rangle \right)^{2},
3 . 1 4 \times ( 1 2 \div 9 )^{2}\times 9 - 3 . 1 4 \times ( 2 \div 2 )^{2}\times 9 \times 1 2 = 6 7 8 . 2 4 ( c m^{3}) \approx 0 . 6 8 ( d m^{3})
{\cal V}^{\prime \prime}+ \left[ k^{2}+ \frac{1}{4 ( \eta_{c}- \eta )^{2}}\right]{\cal V}= 0 ,{\cal W}^{\prime \prime}+ \left[ k^{2}+ \frac{1}{4 ( \eta_{c}- \eta )^{2}}\right]{\cal W}= 0
W [ X ] = i g \int_{0}^{1}d \tau A_{b}( X ) \partial_{\tau}X^{b}+ \frac{( i g )^{2}}{2}\int_{0}^{1}d \tau_{1}\int_{0}^{\tau_{1}}d \tau_{2}[ A_{b}( X ( \tau_{1}) ) , A_{c}( X ( \tau_{2}) ) ] \partial_{\tau_{1}}X^{b}\partial_{\tau_{2}}X^{c}
\frac{\psi^{\prime\prime}(r)}{\psi(r)}+\frac{(D-1)}{r}\frac{\psi^{\prime}(r)}{\psi(r)}=-\frac{1}{2}\{g,r\}+\frac{g^{\prime2}(r)}{4}\left[M^{2}(g)+2M^{\prime}(g)-4J(g)\right]-\frac{(D-1)(D-3)}{4r^2}
\beta_{0}^{P}= a^2+ ap+q+\alpha_{1}\; , \beta_{n}^{P}= q+\alpha_{1}+\alpha_{2}+ \left(p+\beta \right)\beta \; , n \geq 1 \alpha_{1}^{P}= \alpha_{1}\alpha_{2}+ \gamma(\beta-a) \; , \alpha_{n}^{P}=\alpha_{1}\alpha_{2}+ \gamma(p+2\beta) \gamma_{n}^{P}= \gamma^{2}\; , \; n \geq 1.
U^{m,n,\overrightarrow{b}}_{\psi,s}\left(f_1,\ldots,f_m\right)(x):=\int_{[0,1]^n}\left(\prod\limits_{k=1}^mf_k\left(s_k(t)x\right)\right)\left(\prod_{k=1}^m\left(b_k(x)-b_k\left(s_k(t)x\right)\right)\right)\psi(t)dt.
\mathbb{E}_{P^{(\mathcal{C}_n)}}g_{\mathcal{C}_n}(M,\hat{M}) =\mathbb{P}_{P^{(\mathcal{C}_n)}}\big(\hat{M}\neq M\big) \mathbb{E}_{Q^{(\mathcal{C}_n)}}g_{\mathcal{C}_n}(M,\hat{M}) =\mathbb{P}_{Q^{(\mathcal{C}_n)}}\big(\hat{M}\neq M\big).
\left( \sum\limits_{j_{1}=1}^{\infty}\left( \cdots\left( \sum\limits_{j_{m}=1}^{\infty}\left\Vert T( x_{j_{1}}^{(1)},\dots,x_{j_{m}}^{(m)})\right\Vert_{F}^{q_{m}}\right)^{\frac{q_{m-1}}{q_{m}}}\cdots\right)^{\frac{q_{1}}{q_{2}}}\right)^{\frac{1}{q_{1}}}\leq C \prod\limits_{k=1}^{m}\left\Vert(x_{j}^{(k)})_{j=1}^{\infty}\right\Vert_{w,p_{k}}
\displaystyle{\sum_{j=0}^{p}\int_0^T ||\mathcal{\mathbf{B^{\ast}_{n+j}}}U_{n+j}||_{G_{n+j}}^2 \,dt \le C(T) \Big( \sum_{i=1}^{n}e_{1+i-n}(U_{i})(0) + \sum_{j=1}^p e_1(U_{n+j})(0) \Big) \,.}
-\int_0^\infty \Psi^{(5)}(Z) \Psi^{(3)}(Z) w_1(Z)\:dZ=\int_0^\infty (\Psi^{(4)}(Z))^2 w_1(Z) + \int_0^\infty \Psi^{(4)}(Z) \Psi^{(3)}(Z) w_1'(Z)\:dZ.
E(u) = \int_M \frac 1p \bigg (\sum_{i=1}^m h\big (du (e_i), du(e_i)\big ) \bigg )^{\frac p2}\, dv= \int_M\, \frac{1}{p}\big ({\sigma_1}(u^{\ast})\big )^{\frac p2}\, dv.
\begin{cases}2[\beta] = [\beta]-1 \implies \beta<0 2[\beta] = [\beta]-2 \implies \beta < 0 2([\beta]-1) = [\beta] \implies [\beta]=2 2([\beta]-1) = [\beta]-2 \implies [\beta]=0 2([\beta]-2) = [\beta] \implies [\beta]=4 2([\beta]-2) = [\beta]-1 \implies [\beta]=3.\end{cases}
\frac{K_n^{\rm qtm}(q^{x-N};p,N;q)}{K_n^{\rm qtm}(q^{-N};p,N;q)}=\frac{K_x^{\rm qtm}(q^{n-N};p,N;q)}{K_x^{\rm qtm}(q^{-N};p,N;q)}(n,x\in\{0,1,\ldots,N\}).
\begin{cases}\theta^m_t+ \mathbf{w}^m\cdot \nabla\theta^m= \nabla\cdot( \kappa(\theta^m)\nabla \theta^m),\ (0, T)\times \Omega, \theta^m=0, \ (0,T)\times \Gamma, \theta^m|_{t=0}=\theta_0(x),\ \Omega.\end{cases}
G_{\left(l'\right)}(z,w)=q_{l'}\left(F(z,w),F(z,w)\right)+\displaystyle\sum_{k\geq3}{\varphi'}_{k}^{\left(l'\right)}\left(F(z,w),\overline{F(z,w)}\right),\quad\mbox{for all $l'=1,\dots,N'$.}
|I_2| \lesssim \ \| R_\alpha\|_{BMO}\||D|^\frac12 b\|_{BMO}\sum_{k < j}(j-k) 2^{\frac{k-j}2}\| r_j\|_{\dot H^\frac12}\|w_k\|_{L^2}\lesssim \ \| R_\alpha\|_{BMO}\||D|^\frac12 b\|_{BMO}\| r\|_{\dot H^\frac12}\|w\|_{L^2}.
L_{\Omega}(X;z^{\prime},\bar{z}^{\prime})=\frac{\Psi(X;z^{\prime},\bar{z}^{\prime})}{(-r)^{n+m+1}(X;z^{\prime},\bar{z}^{\prime})}+\Phi(X;z^{\prime},\bar{z}^{\prime})\log (-r(X;z^{\prime},\bar{z}^{\prime}))
\int_{M_1}\int_{K_1}\left|\frac{f^*}{g_2}(w_1,\hat w_1)\right|^2 \le C_1\int_{M_1}\sup_{w_1\in K_1}\left|\frac{f^*}{g_2}(w_1,\hat w_1)\right|^2d\mu_1(\hat w_1) \le C_1C_{K_1}\int_{M_1}\left(\frac{1}{2\pi}\int_{\partial D_1}\left|\frac{f^*}{g_2}(z_1,\hat w_1)\right|^2|dz_1|\right)d\mu_1(\hat w_1) < +\infty
-\omega \mbox{}D^{*}_{{\bf{k}}}(-{\bf{q}})= \omega_{{\bf{k}}}(-{\bf{q}})D^{*}_{{\bf{k}}}(-{\bf{q}})+ (\frac{v_{{\bf{q}}}}{V})\Lambda_{{\bf{k}}}({\bf{q}})\sum_{{\bf{k}}^{'}}[\Lambda_{{\bf{k}}^{'}}({\bf{q}})D^{*}_{{\bf{k}}^{'}}(-{\bf{q}}) + \Lambda_{{\bf{k}}^{'}}(-{\bf{q}})C_{{\bf{k}}^{'}}({\bf{q}})]+ \Lambda_{{\bf{k}}}({\bf{q}})
0=\mathrm{p.v.}\int{d\beta}\left\{{{\textstyle{1 \over 2}}\cot \left({{{\alpha -\beta}\over 2}}\right)-\sum\limits_{n=1}^\infty{\left({\zeta_B(x^n)-(-1)^n\zeta_F(x^n)}\right)\sin \left({n(\alpha -\beta )}\right)}}\right\}\rho (\beta )
\big | \big | \tilde{\delta}_{v}^{u}\big | \big | = \left( \begin{matrix}{0_{[ N - 2 ] \times [ N - 2 ]}}&{0_{[ N - 2 ] \times [ 2 ]}}\\{0_{[ 2 ] \times [ N - 2 ]}}&{1_{[ 2 ] \times [ 2 ]}}\\ \end{matrix}\right) ,
P(T_{n+1}\le s|Z_t = i,U_t = u,N_t=n) &=P(T_{n+1}\le s|Y_n = i,T_n = t-u,T_{n+1}> t) &=\frac{P(T_{n+1}\le s,T_{n+1}>t|Y_n=i,T_n=t-u)}{P(T_{n+1}> t|Y_n=i,T_n=t-u)}.
V(\psi\otimes\bar{v},z) \; V(\chi\otimes \bar{w},\zeta)\;\left(\Omega \otimes \overline{\Omega}\right)=\sum_{\bar{w}', \bar{v}'}{R \;}_{\bar{v}'\;\bar{w}'}^{\bar{v}\;\bar{w}}\; \; V(\chi\otimes\bar{w}',\zeta) \; V(\psi\otimes \bar{v}',z)\; \left(\Omega \otimes \overline{\Omega}\right),
\int_{\mathbb{R}^4}\Delta e_3\,dx\,=-\int_{\mathbb{R}^4}\left(\frac{\partial B^\beta{}_\alpha}{\partial x^\beta}\right)\Delta A^\alpha\,dx\,=-\int_{\mathbb{R}^4}\left(\frac{\partial B^4{}_\alpha}{\partial x^4}\right)\Delta A^\alpha\,dx\,=\,0\,.
\big( \mathfrak{J}_{{}_{x,y}}w \big)_{{}_{s}}= \big( L_{h_{{}_1}(xs)}\, \mathfrak{J}_L \, L_{h_{{}_1}(xs)^{-1}}\big) \otimes \big( M_{h_{{}_2}(ys)}\, \mathfrak{J}_M \, M_{h_{{}_2}(ys)^{-1}}\big) \big( w \big)_{{}_{s}};
\left( \frac{1}{\beta}\binom{n}{2}, \frac{-n}{\delta}\right),\left( \frac{1}{\beta}\left(\binom{n-1}{2}-1\right),0 \right), \left( \frac{1}{2\beta}\left( \binom{n}{2}+ \binom{n-1}{2}-1 \right),0 \right).
\frac{1}{2}\partial_{t}\|u(t, \cdot)\|_{L^{2}(\Omega)}^{2}= - \langle a_{ij}(\cdot) \partial_{j}u(t, \cdot), \partial_{i}u(t, \cdot)\rangle_{L^{2}(\Omega)}- \langle \tilde{a}_{ij}(\cdot) \partial_{j}u(t - \tau, \cdot), \partial_{i}u(t, \cdot)\rangle_{L^{2}(\Omega)}.
\left\{(a,b,c,d):a,b,c,-d>0:a+b+c+d\geq|a-b|+c-d\right\}= \{(a,b,c,d) :a,b,c,-d>0:c\geq-d,b\geq-d,a\geq-d\}\cup\{(a,b,c,d) :a,b,c,-d>0:a\geq c,\b\geq c,\ -d\geq c\}.
\begin{aligned}H_{m,n}^{(2)}(k) - \frac{1}{c_{2,n}(k)}H_{m,n}^{(2)}(k+1) = \frac{m}{a_{n}(k+1)c_{2,n}(k)}H_{m,n}^{(1)}(k+1) &H_{m,n}^{(1)}(k) - c_{1,n}(k)H_{m,n}^{(1)}(k+1) = \frac{m}{a_{n+1}(k)c_{2,n}(k)}H_{m,n}^{(2)}(k+1) + & + \frac{m^2}{a_{n+1}(k)a_{n}(k+1)c_{2,n}(k)}H_{m,n}^{(1)}(k+1) .\end{aligned}
h_{\mu \nu}( t ,{\bf x}) = - 8 \pi{\cal G}\frac{1}{( 2 \pi r )^{( D - 2 ) / 2}}\partial_{t}^{( \frac{D - 4}{2})}\left[ \int d^{D - 1}{\bf x^{\prime}}S_{\mu \nu}( t - |{\bf x - x^{\prime}}| ,{\bf x^{\prime}}) \right] ,
\widehat{K}_{i j}= \left( \begin{matrix}{{Z -{\frac{{C^{\prime}}^{2}}{C}}}}&{{C^{\prime}/ 2}}&{{0}}&{{q^{\prime}\overline{{{\psi}}}}}\\{{C^{\prime}/ 2}}&{{0}}&{{0}}&{{( q / 4 ) \overline{{{\psi}}}}}\\{{0}}&{{0}}&{{-{\frac{C}{2}}P^{\mu \nu , \alpha \beta}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{q \hat{1}}}\\ \end{matrix}\right)
A(t):=\left(\begin{array}{cc}0 1 a(t) 0\end{array}\right), Q(t):=\left(\begin{array}{cc}0 0 q(t)-a(t) 0\end{array}\right) F(t,\xi):=\left(\begin{array}{c}0 \widehat{f}(t,\xi)\end{array}\right).
\times \exp\Biggl[-\bar\alpha\int d^2\xi \sqrt{h}h^{ab}(\partial_a w_\mu)(\partial_b w_\mu)-\frac{1}{\alpha_0}\int d^2\xi \sqrt{h}h^{ab}(\partial_a t_{\mu\nu})(\partial_b t_{\mu\nu})\Biggr],
\frac{d\sigma (g \rightarrow \gamma)}{d\Omega'}= \frac{k^2 H^{2}_{0}b^2 p^2}{4(2\pi)^4(p^2-\frac{m^2\pi^2}{a^2})}\left ( 1+ 3\frac{\omega}{p}\right)\sin^2\left (\frac{ap}{2}\right)2\sin^2 \left [\frac{c(p + 2k)}{2}\right]
h(X,Y) = \frac{\langle D_XY,\xi\rangle}{\langle\eta,\xi\rangle}= -\frac{\langle Y,d\xi_pX\rangle}{\langle\eta,\xi\rangle}= -\frac{\langle du^{-1}_{\eta(p)}T,d\eta_pX\rangle}{\langle\eta,\xi\rangle},
\kappa_\Omega = \sup_{z \in \Omega, r >0}\frac{|B_r(z) \cap \Omega|}{|B_{r/2}(z) \cap \Omega|}+ \sup_{z \in \partial \Omega, r>0}\frac{|B_r|}{|B_r(z) \setminus \Omega|}\leq C(d)\kappa_0^{-1}
\sum_{r,s}\Bigl(\frac{\partial}{\partial t}b_{(n)}^{(r, s)}\Bigr) \hat{a}^{\dagger r}\hat{a}^s= i \sum_{r,s}b_{(n-1)}^{(r, s)}[\hat{a}^{\dagger r}\hat{a}^s, \hat{H}'].
B^{\prime \prime}+{\frac{D_{2}}{2}}{B^{\prime}}^{2}+{\frac{D_{1}}{2}}A^{\prime}B^{\prime}={\frac{4 \kappa^{2}}{D - 2}}\left( - \Lambda -{\frac{2 D - D_{1}- 4}{D_{2}}}{\frac{F^{2}}{4 g^{2}}}\mathrm{e}^{- 2 B}\right) + k \mathrm{e}^{- B}~ ,
ds^2 = - Z_{\phi}Z_1^{-1/2}Z_5^{-1/2}f_3 (r) \beta^2 dt^2 + Z_{\phi}Z_1^{1/2}Z_5^{1/2}\left(Z_0^{-1}dr^2+ r^2 d\Omega^{(3)}\right),
\alpha=\sqrt{\frac{N_k}{\hbar\tau_2}}\left[-i\tau\hat{A}_1+i\hat{A}_2\right]\ \ ,\ \ \alpha^\dagger=\sqrt{\frac{N_k}{\hbar\tau_2}}\left[i\bar{\tau}\hat{A}_1-i\hat{A}_2\right]\ \ .
\theta^0_1 = -\frac{a'}{a}e^1 \theta^0_i = -\frac{b'}{b}e^i \theta^1_i = - \frac{a}{b^2}\omega_{ij}e^j \theta^i_j = \hat{\theta}^i_j + \frac{a}{b^2}\omega_{ij}e^1.
\epsilon - \gamma \leq \left ( 1 - \sum_{i=1}^{k^{*}-1}\Pr \{X \in A_{D}(y_i) \}\right ) - \left ( 1 - \sum_{i=1}^{k^{*}}\Pr \{X \in A_{D}(y_i) \}\right ) & = \Pr \{X \in A_{D}(y_{k^*}) \}& \leq \Pr \{X \in A_{D}(y_{i^* + 1}) \},
[ ( - D_{1}D_{2}D_{3}D_{4}\partial_{t}^{2}+ f ~{\frac{1}{r^{2}}}({\frac{1}{\sin \theta}}\partial_{\theta}\sin \theta \partial_{\theta}+ \partial_{\phi}^{2}) +{\frac{1}{r^{2}}}f \partial_{r}( r^{2}f \partial_{r}) ] \chi = 0
|J_{s}(\delta z)| = |J_{s}(s(\delta z/s))|\leq \frac{\left|\delta z/s\right|^se^{s(1-\left|\delta z/s\right|^2)^{\frac{1}{2}}}}{\left(1+(1-\left|\delta z/s\right|^2)^{\frac{1}{2}}\right)^s}\leq \left(\frac{e\,|\delta z|}{2s}\right)^s ,
\frac{1}{m+1}(B_{m+1}(a+1)-B_{m+1}(a)) = \frac{1}{m+1}\left(\Delta_{m+1}\left(\frac{xe^{(a+1)x}}{e^x-1}\right)-\Delta_{m+1}\left(\frac{xe^{ax}}{e^x-1}\right)\right) = \frac{\Delta_{m+1}(xe^{ax})}{m+1}= \,a^m,
0 = aF+b\eta^{12}F +cDF + d\eta^{12}DF &= aF+bu(E_6F+6E_4DF)+cDF-\frac{1}{6}du\left(E_4^2F+6E_6DF\right) &= \left(a+buE_6-\frac{1}{6}duE_4^2\right)F+\left(c+6buE_4-duE_6\right)DF
\mathrm{Index}\, T_{\alpha\otimes I_F}= \mathrm{Index}\, T_\alpha \,\cdot\,\mathrm{dim\, Ker}\,B^+ + \mathrm{Index}\, T^-_\alpha\,\cdot\, \mathrm{dim\, Ker}\,B^-+\mathrm{Index}\,Q = \mathrm{Index}\, T_\alpha\,\cdot\, \mathrm{Index}\,B+\mathrm{Index}\,Q
\Vert g^\varepsilon b(\mathbf{D})\bigl(I+\varepsilon [\Lambda^\varepsilon ]S_\varepsilon b(\mathbf{D})+\varepsilon [\widetilde{\Lambda}^\varepsilon ]S_\varepsilon\bigr)(B^0-\zeta\overline{Q_0})^{-1}-G(\varepsilon ;\zeta )\Vert_{L_2\rightarrow L_2}& \leqslant{\mathfrak C}_{14}\varepsilon \rho(\zeta)^{1/2},
\langle f, h \rangle = \langle \pi (z)^* f, \pi (z)^* h \rangle = \sum_{\lambda \in \Lambda}\langle \pi (z)^*f, \pi(\lambda)g \rangle \langle \pi(\lambda) \gamma , \pi (z)^* h \rangle = \sum_{\lambda \in \Lambda}V_gf(z+\lambda) \overline{V_{\gamma}h(z+\lambda)}= \Phi (z)
\left\langle\bigl\{\bigr.\right. \eta(t_{e_1,e_2}),\eta(t_{e_2,e_3}),\ldots,\eta(t_{e_{2^n-1},e_{2^n}}), \eta(t_{a_{2^n},a'_1}),\eta(t_{a'_{2^n},b_1}), \eta(t_{b_{2^n},b'_1}),\eta(t_{b'_{2^n},c_1}), \eta(t_{c_{2^n},d_1}),\eta(t_{d_{2^n},a_1}) \mid \left.\bigl.e\in \{a,a',b,b',c,d\}\bigr\}\right\rangle=M,
\sum \limits_{j = 1}|\partial_\theta c(\theta_{j-1}) p(x_{j-1})| \prod \limits_{i=j}^{n-1}|c(\theta_i)p(x_i)| \leq 4^{4K_m}\cdot \frac{(3/5)^{\left(1 - \frac1{M_0}\right)(n-k)/2}}{1 -}.
\xi^e(\rho_f)(x) =\xi^e(t\rho'_f+(1-t)\rho''_f)(x) &=t\xi^e(\rho'_f)(x)+(1-t)\xi^e(\rho''_f)(x) &=t\mu'_e(x)+(1-t)\mu''_e(x) &=[t\zeta(\rho')+(1-t)\zeta(\rho'')](x).
\begin{array}{c}{v_{1}= \left( 0 , 1 , 0 , 0 , 0 , 0 ; 1 , 0 \right) \}\\{v_{2}= \left( 0 , 0 , 1 , 0 , 0 , 0 ; 0 , 1 \right) .}\\ \end{array}
\left\langle \Theta_{2 3}( \vec{x}) \right\rangle = \left\langle \Theta_{3 2}( \vec{x}) \right\rangle = \left\langle \Theta_{3 1}( \vec{x}) \right\rangle = \left\langle \Theta_{1 3}( \vec{x}) \right\rangle = \left\langle \Theta_{0 i}( \vec{x}) \right\rangle = \left\langle \Theta_{i 0}( \vec{x}) \right\rangle = 0 .
\frac{1}{n}\log\mathsf{IP}(\mathcal G_{x_2^n}) \leq \sum_{i=1}^r \frac{n_i}{n}\log (\mathsf{IP}(\mathcal G_{i}^{\otimes n_i})^{1/n_i}), \frac{1}{n}\log\mathsf{LP}(\mathcal G_{x_2^n}) =\sum_{i=1}^r \frac{n_i}{n}\log (\mathsf{LP}(\mathcal G_{i}^{\otimes n_i})^{1/n_i}) &= \sum_{i=1}^r \frac{n_i}{n}\log (\mathsf{LP}(\mathcal G_{i})),
\frac{\sum_{w_0=0}^K\prod_{n=1}^MP_W(w^{x,w_0}_{n-1},w^{x,w_0}_{n})\mathbf{P}\left(\hat{W}_{N-M}=w_0,\:\max_{1\leq n\leq{N-M}}\hat{W}_n\leq K\:\vline\:\hat{W}_{0}=w^{x,w_0}_M\right)}{\sum_{w_0=0}^K\mathbf{P}\left(\hat{W}_N=w_0,\:\max_{1\leq n\leq N}\hat{W}_n\leq K\:\vline\:\hat{W}_{0}=w_0\right)}.
u(z) = \int_0^1 t^{\xi^2/2-\lambda-1}\frac{1}{\sqrt{m!}}\left[t\left(z-\frac{i\xi}{\sqrt{2}}+i\frac{\xi}{\sqrt{2}}\right)e^{(1-t)\left(\frac{\xi^2}{2}+\frac{i\xi}{\sqrt{2}}z\right)}\right]^m dt.
Z_{k}^{\pm}={\frac{1}{\sqrt{2}}}( X^{2 k - 1}\pm i X^{2 k}) , \quad \psi_{k}^{\pm}={\frac{1}{\sqrt{2}}}( \chi^{2 k - 1}\pm i \chi^{2 k}) \quad ( k = 1 , 2 , 3 ) ,
\mathrm{[}\hat{\Lambda}( k^{1}) , \hat{\Lambda}( k^{\prime 1}) ] = [ \hat{\Lambda}^{\dagger}( k^{1}) , \hat{\Lambda}^{\dagger}( k^{\prime 1}) ] = 0 ; [ \hat{\Lambda}( k^{1}) , \hat{\Lambda}^{\dagger}( k^{\prime 1}) ] = \delta ( k^{1}- k^{\prime 1}) ,