harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|F_{\boldsymbol{n_m}}(s; a)|\leq \sum_{\boldsymbol{n}\in S_m^{(k)}}\frac{k!}{m!}3^{\frac{k-m}{2}}(\phi(q))^{k-m}\leq 2^{k-m}k^{k-m}(2\phi(q))^{k-m}=(4k\phi(q))^{k-m}.
Q(X,\tau)=\prod_{j=1}^{2k}\left(\frac{2x_j\theta'(0,\tau)}{\theta(x_j,\tau)}\left(\frac{\theta_1(x_j,\tau)}{\theta_1(0,\tau)}+\frac{\theta_2(x_j,\tau)}{\theta_2(0,\tau)}+\frac{\theta_3(x_j,\tau)}{\theta_3(0,\tau)}\right)\right).
a\left(u\right)=\begin{pmatrix}2u_{1}0 0 D\end{pmatrix},F\left(u,u_{x}\right)=\begin{pmatrix}Cu_{1x}+A\left(1-u_{1}\right)-u_{1}u_{2}^{2}-Bu_{2}+u_{1}u_{2}^{2}\end{pmatrix},
TB^4_i (x)=\frac{1}{w}\begin{cases}p^3(x_i) x\in [x_i,x_{i+1}] p(x_i)(p(x_i)q(x_{i+2})+q(x_{i+3})p(x_{i+1}))+q(x_{i+4})p^2(x_{i+1}), x \in [x_{i+1},x_{i+2}] q(x_{i+4})(p(x_{i+1})q(x_{i+3})+q(x_{i+4})p(x_{i+2}))+p(x_i)q^2(x_{i+3}), x \in [x_{i+2},x_{i+3}] q^3(x_{i+4}), x \in [x_{i+3},x_{i+4}] \end{cases}
\frac{\det(N_{(i)}'N_{(i)}) \det(B'(I_d-P_{N_{(i)}})B)}{\det(N_{(i-1)}'N_{(i-1)})\det(B'(I_d-P_{N_{(i-1)}})B)}\,\,=\,\, \frac{\det(N_{(i)}'N_{(i)}) \det(A_{(i)}'A_{(i)})}{\det(N_{(i-1)}'N_{(i-1)})\det(A_{(i-1)}'A_{(i-1)})}.
D_2 ={1\over4\pi^2 r}\sum_{\scriptstyle m=-\infty\atop\scriptstyle m\ne0}^\infty\sum_{n=-\infty}^\infty{s_m\over16\pi^4 r^4 \sin\left(s_m\over r\right)}{f_{mn}\over\left(n+z_1\right)\left(n-z_1\right)\left(n+ z_1^*\right)\left(n-z_1^*\right)}
<{\pounds}_H^{(2)}>=\frac{\eta_{0}}{2 \pi^2}p_{0}\left( \left|\psi_{0}\right|^{2}+\left| \varphi_{0}\right|^{2}\right) \int \left[ d^{2}k\right] \left( 3\left|k\right|^{2}-4p_{0}^{2}\right) D_{P}^{-1}
G_0(x,y) =\frac{\pi^{3/2}}{b}\Bigl(\frac{b}{8}|x-y|^2\Bigr)^{-3/4}\exp\Bigl(\frac{ib}{2}x\wedge y\Bigr) &\quad\times\Bigl[W_{\frac12,-\frac12}\Bigl(\frac{b}{2}|x-y|^2\Bigr)+\frac12W_{-\frac12,-\frac12}\Bigl(\frac{b}{2}|x-y|^2\Bigr)\Bigr].
J ={\frac{R^{2}}{2 \pi \alpha^{\prime}}}e \omega \int_{0}^{2 \pi}d \sigma \sin^{2}\theta = 4{\frac{R^{2}}{2 \pi \alpha^{\prime}}}\omega \int_{0}^{\theta_{0}}d \theta \left( 1 - \omega^{2}\sin^{2}\theta \right)^{- 1 / 2}\sin^{2}\theta \ .
\widetilde{A}_{ij}=A_{ij}-\frac{2}{n-4}u^{-1}u_{ij}-\frac{2}{\left(n-4\right)^{2}}u^{-2}\left\vert \nabla u\right\vert^{2}g_{ij}+\frac{2\left( n-2\right)}{\left( n-4\right)^{2}}u^{-2}u_{i}u_{j},
\Psi_{2 s + 1}( \xi_{1}, \xi_{2}) = \sum_{k_{1}< k_{2}}\left\{A_{1}( k_{1}, k_{2}) \left| \Omega_{1}( k_{1}, k_{2}) \right\rangle + A_{2}( k_{1}, k_{2}) \left| \Omega_{2}( k_{1}, k_{2}) \right\rangle \right\}
| \delta{h_{\omega}}| = \sqrt{H_{0}/ M_{s}}z_{e q}^{- 1 / 4}z_{o u t}^{1 / 2}\exp{( \frac{1}{2}\phi_{e n d})}\; ({\frac{\omega}{\omega_{m a x}}})^{\frac{1}{2}}\ln ({\frac{\omega}{\omega_{m a x}}})
C_+=-i\phi_2F_{-12}^+\phi_2^\dagger+i\phi_1F_{-12}^+\phi_1^\dagger -\phi_2F_{-13}^+\phi_1^\dagger+i\phi_2F_{-14}^+\phi_1^\dagger +i\phi_1F_{-14}^+\phi_2^\dagger+\phi_1F_{-13}^+\phi_2^\dagger.
\{U_m , U_n \}= -16mni \int dx \, k^{(m+1)}_{a_1 \ldots a_{m-1}bc}k^{(n+1)}_{a_m \ldots a_{m+n-2}dc}h_+^{a_1}\ldots h_+^{a_{m+n-2}}\psi_+^{'b}\psi_+^{'d}\, ,
r_j(w) = w^{n(a)}\prod_a \frac{E(-x^{\frac{6sa}{g}}/w,x^{12s})E(-x^{\frac{6s(g-a)}{g}}/ w, x^{12s})}{E(-x^{\frac{6sa}{g}}w, x^{12s})E(-x^{\frac{6s(g-a)}{g}}w, x^{12s})}.
S_{|T_{l=a}|}^{WZ}(\eta )=\frac{\kappa}{4\pi \sqrt{-1}}\int_{V_{M}}\widehat{h}^{\ast}\chi_{SU(2)}-\frac{\kappa}{4\pi \sqrt{-1}}\sum_{j=1}^{N_{0}}\int_{U_{\rho^{2}(j)}}\left. \widehat{h}\right|_{U_{\rho^{2}(j)}}^{\ast}\omega_{j},
\left( \int_{\gamma_{\rho}^{+}}+ \sum_{\sigma_{k}= i y_{k}, \lambda_{\nu , k}}\int_{C_{\rho}( \sigma_{k})}\right) \frac{\bar{H}_{\nu}^{( 1 )}( z )}{\bar{J}_{\nu}( z )}f ( z ) d z - \left( \int_{\gamma_{\rho}^{-}}+ \sum_{\sigma_{k}= - i y_{k}, - \lambda_{\nu , k}}\int_{C_{\rho}( \sigma_{k})}\right) \frac{\bar{H}_{\nu}^{( 2 )}( z )}{\bar{J}_{\nu}( z )}f ( z ) d z ,
\cos \alpha (\cos^2\theta - \sin^2\theta) \cos\alpha \sin \alpha \cos\theta + \sin\alpha \sin\theta \cos^2\alpha \cos\theta \sin\theta = \cos^2\alpha \sin\alpha \cos\theta (\cos^2\theta - \sin^2\theta + \sin^2\theta) = \cos^2\alpha \sin\alpha \cos^3\theta.
\begin{pmatrix}0 0 0 0 0 0 0 0 0 0 -1 0 1 0 0 0 0 0 0 0 1 0 -1 0 0 \end{pmatrix},\begin{pmatrix}0 0 0 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 \end{pmatrix}.
\rho_1(a) = (_M \otimes \epsilon)\circ(_M\otimes_R \otimes T_p)\circ Ad(v+w)\circ\sigma_{M\otimes R}^{-1}(z\pi(a)) \rho_2(a) = (_M \otimes \epsilon)\circ(_M\otimes_R \otimes T_p^{\perp})\circ Ad(v+w)\circ\sigma_{M\otimes R}^{-1}(z^{\perp}\pi(a)).
( \dot{\Gamma}_{i}^{a}\delta \tilde{E}^{i a}- \dot{\tilde{E}^{i a}}\delta \Gamma_{i}^{a}) - \partial_{i}( i \hat{\lambda}^{a}\delta \tilde{E}^{i a}) = 0 \ \mathrm{m o d}\left( \dot{\tilde{E}^{i a}}+ i \varepsilon^{a b c}\lambda^{b}\tilde{E}^{i c}= 0 \right) ,
\Psi^n_{+\bf k}(P)=(-1)^n C^{1/2}_n\left(\frac{P_1-iP_2}{P^0+m}\right)^{n}\Psi^0_{+\bf k}(P),\quad\Psi^0_{+\bf k}(P)=\delta(P^0-\sqrt{{\bf P}^2+m^2})\delta ({\bf P}-{\bf k})f({\bf k}),
\lim_{K \rightarrow \infty}\frac{1}{\psi_2(N,K,s)}\geq\frac{N}{\lambda+\epsilon}\sum_{i=0}^{\infty}\left(\frac{(1-(\lambda+\epsilon))(N-1)}{\lambda+\epsilon}\right)^i =\frac{N}{N(\lambda+\epsilon)-(N-1)}.
L_{t}^{\alpha}=\mathcal{L}^{-1}\left[ \frac{1+s^{\alpha}}{1+\tau s^{\alpha}}\right] \ast_{t}=\left( \frac{1}{\tau}\delta (t)+\left( \frac{1}{\tau}-1\right) e_{\alpha}^{\prime}(t)\right) \ast_{t},\;\;t>0,
[det(\delta_{\mu\nu}+L_{\mu\nu})]^{\frac{1}{2}}=\frac{[1+\frac{1}{8}f_{ij}^2+\frac{1}{16}\frac{1}{4!}\epsilon^{ijkl}\epsilon^{mnpq}f_{mi}f_{nj}f_{pk}f_{ql}]^2}{[-det(\eta_{ab}+\frac{1}{2}F_{ab})]^{\frac{1}{2}}}
P_{A_r,\,s}^{{\bf V}_i}(x)=\prod_{1\leq j_1<\cdots<j_i\leq r+1}\Bigl(x-\sin(\bar{q}_{j_1}+\cdots\bar{q}_{j_i})\Bigr) =P_{A_r,\,s}^{{\bf V}_{r+1-i}}(x).
(\langle+|Z^{m_{1}}\otimes\langle+|Z^{m_{2}}\otimes I)CZ_{1,3}CZ_{2,4}(|\phi\rangle\otimes CZ|+\rangle^{\otimes2})=\frac{1}{2}CZ(HZ^{m_{1}}\otimes HZ^{m_{2}})|\phi\rangle
\frac{4 m}{\hbar^{2}}U = \sigma_{1}\bar{\beta}_{1 1}| \phi |^{4}+ ( \sigma_{1}+ \sigma_{2}) \bar{\beta}_{1 2}| \phi |^{2}| \chi |^{2}+ \sigma_{2}\bar{\beta}_{2 2}| \chi |^{4}\, .
\int_\Sigma \left( 1 - \langle \nu, \nabla r\rangle \right)^2 = O ( |\Sigma|^{-1/2}) \int_\Sigma e^{- 2 \, r}\, |\nabla_\Sigma r|^2 = O ( |\Sigma|^{-1/2}) \int_\Sigma 4 \, e^{- 2 \, r}= 4 \, \pi + O (|\Sigma|^{-1/2}).
\mathbf{1 . \, \, \, \, \hat{g}\neq 0}\hspace{4 c m}a ( y ) ={\textstyle \frac{1}{2}}\sqrt{\pm g^{2}/ \hat{g}^{2}}\left( e^{\hat{g}y}\pm e^{- \hat{g}y}\right) \, , \hspace{3 c m}
D_l A_{j_1 j_2 j_3 j_4}=\frac{\partial A_{j_1 j_2 j_3 j_4}}{\partial u^l}- A_{k j_2 j_3 j_4}\Gamma_{j_1 l}^k - A_{j_1 k j_3 j_4}\Gamma_{j_2 l}^k - A_{j_1 j_2 k j_4}\Gamma_{j_3 l}^k - A_{j_1 j_2 j_3 k}\Gamma_{j_4 l}^k .
\tan \hat{\theta}= - \frac{Q_{\mathrm{F 1}}}{Q_{\mathrm{D}p}}= \frac{V_{1}T_{\mathrm{F 1}}M}{V_{p}T_{\mathrm{D}p}N}= g_{s}\frac{( 2 \pi l_{s})^{p - 1}}{V_{p - 1}}\frac{M}{N}
\frac t i o n P \! \! - \! w a v e ={\frac{( \overline{{p}}p \rightarrow \pi^{+}\pi^{-})_{P}+ ( \overline{{p}}p \rightarrow \pi^{0}\pi^{0})_{P}}{( \overline{{p}}p \rightarrow \pi^{+}\pi^{-})_{S \& P}+ ( \overline{{p}}p \rightarrow \pi^{0}\pi^{0})_{P}}}.
\gamma_{1}= \begin{cases}0, \mbox{if}\ 1< p \le \frac{3n-2}{2n-1}, 2-p, \mbox{if}\ \frac{3n-2}{2n-1}< p \le 2-\frac{1}{n}, \end{cases}\gamma_{2}= \begin{cases}\frac{n(p-1)}{n-1}, \mbox{if}\ 1< p \le \frac{3n-2}{2n-1}, \frac{np}{3n-2}, \mbox{if}\ \frac{3n-2}{2n-1}< p \le 2-\frac{1}{n}. \end{cases}
\|\rho_n(a)\|^2=\Big\|\sum_{\lambda,\mu}\sum_{\lambda',\mu'}\rho_0(t_\lambda a_{\lambda,\mu}t_\mu^*)\rho_0(t_{\mu'}a_{\lambda',\mu'}^*t_{\lambda'}^*)\Big\|=\|\rho_0(aa^*)\|\leq \|a\|^2.
\|T^{\sharp_A}T+TT^{\sharp_A}\|_A =\|\widetilde{T^{\sharp_A}T+TT^{\sharp_A}}\|_{\mathcal{B}(\mathbf{R}(A^{1/2}))}=\|(\widetilde{T})^{*}\widetilde{T}+\widetilde{T}(\widetilde{T})^{*}\|_{\mathcal{B}(\mathbf{R}(A^{1/2}))}.
\lambda_Y(b) = \sum_{j=0}^{p^n-1}\omega_p^{\mathbf{c}\cdot \mathbf{j}}\mu_{X(j)}& = \prod_{u=0}^{n-1}\sum_{v=0}^{p-1}\omega_p^{\mathbf{c}(u) \cdot \mathbf{v}}\mu_{X(u)}(v)\,.
\times\exp \left( \sum_{i,j=1}^M \tilde{V}(w_i - z_j)- \frac{1}{2}\sum_{i \neq j}^M \tilde{V}(w_i - w_j)- \frac{1}{2}\sum_{i \neq j}^M \tilde{V}(z_i - z_j) \right)
[D(2,3s+r+2p):D(3,3s+r)]_q =q^{4p}[D(2,3(s-1)+r+2p):D(3,3(s-1)+r)]_q &+q^{6s+2r+4p-2}[D(2,3s+r+2p-2):D(3,3s+r)]_q,
\begin{aligned}\frac{d}{dt}E[u,u_t](t)= -\int_0^1 g(u(x,t))u_t(x,t)^2\,dx &-m'(t)\int_0^1\Big\{\big[1-g(u(x,t))\big]u_t(x,t)+f(u(x,t))\Big\}\,dx,\end{aligned}
X_{k}=\sum_{n_{2}=0}^{N/2-1}x_{2n_{2}}\omega_{N/2}^{n_{2}k}+\omega_{N}^{k}\sum_{n_{4}=0}^{N/4-1}x_{4n_{4}+1}\omega_{N/4}^{n_{4}k}+\omega_{N}^{3k}\sum_{n_{4}=0}^{N/4-1}x_{4n_{4}+3}\omega_{N/4}^{n_{4}k}
\Gamma(s)\,(s,\lambda)_{k}=\Gamma(s)(-1)^{k}+\Gamma(s)\,k^{s}e^{\lambda}\,\sum_{\ell=1}^{k}\left(\begin{array}{c}k \ell\end{array}\right)\left(\frac{2\lambda}{k}\right)^{\ell}(-1)^{k-\ell}\,\intop_{k}^{\infty}t^{-s}e^{-\frac{\lambda}{k}t}\,\frac{(t-k)^{\ell-1}}{(\ell-1)!}\,dt,
S_{k}(\xi) := \frac{-\bigl((2k+1)\pi\sqrt{-1}-(2a+1)\xi\bigr)^2}{2(2a+1)}, \tau_{k}:= (-1)^{k}\frac{4\sin\left(\frac{(2k+1)\pi}{2a+1}\right)}{\sqrt{2(2a+1)}}.
e'P_{R}(t)A(t-\tau,q,y_{\tau}(t))e+\frac{1}{2}e'\dot{P}_{R}(t)e =e'P_{R}(t)A(t-\tau,q,y_{\tau}(t),u_{\tau}(t))e & \le L\bar{\sigma}_{R}(t)|e|^2=-d_R(t)e'P_{R}(t)e.
\rho_{\frac{1}{2}}(s) = \frac{(4\pi)^k \zeta(0)}{\Gamma(s + k - 1)}\frac{\zeta(2s -1)}{\zeta^*(2 - 2s)}\frac{\Gamma(2s - 1)}{\Gamma(s)}\langle\mathcal{V}_{f,\overline{g}}, E(\cdot, \overline{s})\rangle.
d_p^p(\mu,\xi_s^{\mu,\nu}) =\inf_{\pi\in\Pi(\mu,\xi_s)}\sum_{(x,y)\in X\times X}\varrho^p(x,y)\cdot\pi(x,y)\leq\sum_{(x,y)\in X\times X}\varrho^p(x,y)\cdot\pi_s(x,y) =(1-s)\sum_{(x,y)\in X\times X}\varrho^p(x,y)\cdot\pi_{\mu}(x,y)+s\sum_{(x,y)\in X\times X}\varrho^p(x,y)\cdot\widetilde{\pi}(x,y) =s d_p^p(\mu,\nu),
| \psi(w) | \leq 3 \|f''\| \mathrm{e}^{\frac12 w^2}\int_w^\infty s \mathrm{e}^{-\frac12 s^2}\, \mathrm{d}s + 2 \| f^{(3)}\| \mathrm{e}^{\frac12 w^2}\int_w^\infty s^3 \mathrm{e}^{-\frac12 s^2}\, \mathrm{d}s &=3 \|f'' \| + 2 (w^2+2) \| f^{(3)}\| .
U^{\ast}=u_{0}+u_{1}s+u_{2}s^{2}+.. U_{\tau}^{\ast}=\tau_{0}+\tau_{1}s+\tau_{2}s^{2}+...(note:\tau_{k-1}=ku_{k}) U_{\nu}^{\ast}=w_{0}+w_{1}s+w_{2}s^{2}+... K^{\ast}=K_{0}+K_{1}s+K_{2}s^{2}+... \mathfrak{S}=\mathfrak{S}_{0}+\mathfrak{S}_{1}s+\mathfrak{S}_{2}s^{2}+...
2n\binom{2n}{n}\biggm|\sum_{k=0}^{n-1}(20k^2+8k+1){\binom{2k}{k}}^5 (-4096)^{n-k-1}, &2n\binom{2n}{n}\biggm|\sum_{k=0}^{n-1}(120k^2+34k+3){\binom{2k}{k}}^4\binom{4k}{2k}65536^{n-k-1}.
F_{\mu\nu}(x) :=\partial_{\mu}A_{\nu}^{\rm P}(x)-\partial_{\nu}A_{\mu}^{\rm P}(x), \partial^{\mu}F_{\mu\nu}=m^{2}A_{\mu}^{\rm P}, A_{\mu}(x,e) :=\int_{0}^{\infty}F_{\mu\nu}(x+\lambda e)e^{\nu}d\lambda, \phi(x,e):=\int_{0}^{\infty}A_{\mu}^{\rm P}(x+\lambda e)e^{\mu}d\lambda.
M_{K_f}(\Gamma) \leq c_{f,p,r}\left( \int_{\mathbb{C}}\rho_{0}^{\frac{2p}{p-1}}\; dA(z) \right)^{\frac{p-1}{p}}\leq c_{f,p,r}\left( \frac{1}{R_{m_i}^{\frac{2}{p-1}+\dim(A)-\epsilon}}\right)^{\frac{p-1}{p}}& \leq \frac{c_{f,p,r}}{R_{m_i}^{\frac{2}{p}+\frac{\dim(A) (p-1)}{p}-\epsilon}}.
(A(t))^{[M]}= 1 + \sum_{k=1}^{\infty}\left\{\sum_{\underline{k}: \sum ik_i=k}\left[ \left( \prod_i M^{k_i}\setminus \Delta\right) \times \prod_i A_i^{k_i}\middle/ \prod_i S_{k_i}\right] \right\}\cdot t^k.
P = P(y,z) &=\max\left\{\frac{-\sigma L - kz - \left(\alpha(\sigma y \pm kz) + \widehat{\alpha}z\right)}{\sigma},\frac{\sigma\alpha y \pm \alpha kz + \widehat{\alpha}z - 2kz - \sqrt{(2\sigma L)^2 - 3(\sigma\alpha y \pm \alpha kz - \widehat{\alpha}z)^2}}{2\sigma}\right\},
(\c(u),u) = (u(1-u^{\delta})(u^{\delta}-\gamma),u)=((1+\gamma)u^{\delta+1}-\gamma u - u^{2\delta+1},u) & = (1+\gamma)(u^{\delta+1},u)-\gamma \|u\|_{\L^2}^2-\|u\|_{\L^{2(\delta+1)}}^{2(\delta+1)},
X(t) = \begin{cases}x (-r \le t \le 0), \displaystyle x + \int_0^t A_0 \big( X(s-r) \big) \, X(s) \, \mbox{d}s + \int_0^t A_1 \big( X(s-r) \big) \, X(s) \, \mbox{d}W(s) (0 < t \le T). \end{cases}
\int_{\Omega}{\Big(\sum_{Q\in\mathcal{V}_n(l,P)}{\theta_Q(\omega)}\Big)^{{q}/{2}}}d\lambda \geq \sum_{R\in\mathcal{V}_{n}(l,P)}{\int_{\Omega(P,R,l,n)}{\Big(\sum_{Q\in\mathcal{V}_n(l,P)}{\theta_Q(\omega)}\Big)^{{q}/{2}}}d\lambda}= \sum_{R\in\mathcal{V}_n(l,P)}{\lambda(\Omega(P,R,l,n))}\geq2^{{t_n}d}\frac{1}{L}\Big(1-\frac{1}{L}\Big)^{2^{{t_n}d}}
\frac{d}{dt}Q(M_t^s) = ~|M_t^s|\int_{M_t^s}|\mathring{A}|^2\left(H^{1-\alpha}-\frac 1{|M_t^s|}\int_{M_t^s}H^{1-\alpha}\right)d\mu_t +\alpha|M_t^s|\int_{M_t^s}H^{-1-\alpha}|\nabla H|^2d\mu_t.
:{\hat{\psi}_{m}}^{\dagger}\hat{\psi}_{n}: ={\hat{\psi}_{m}}^{\dagger}\hat{\psi}_{n}{}\mathrm{i f}n > 0 :{\hat{\psi}_{m}}^{\dagger}\hat{\psi}_{n}: = - \hat{\psi}_{n}{\hat{\psi}_{m}}^{\dagger}\mathrm{i f}n \le 0
\chi(T_i^k\otimes T_j^\ell)T_k^mT_\ell^n =t_{ij}^{k\ell}T_k^mT_\ell^n =T_i^kT_j^\ell t_{k\ell}^{mn}=T_i^kT_j^\ell\chi(T_k^m\otimes T_\ell^n),
\Vert (B_\varepsilon -\zeta Q_0^\varepsilon )^{-1}-(B^0-\zeta \overline{Q_0})^{-1}-\varepsilon K(\varepsilon ;\zeta)\Vert_{L_2(\mathbb{R}^d)\rightarrow H^1(\mathbb{R}^d)}&\leqslant \bigl(C_{12}+ C_{13}+\vert \zeta +1\vert^{1/2}C_{14}\bigr)\varrho (\zeta )\varepsilon, 0<\varepsilon\leqslant 1.
\begin{cases}\partial_t(\alpha \partial_t\Lambda) - \partial_x(\alpha \partial_x\Lambda) = 2\alpha \sinh{2\Lambda}((\partial_t\phi)^2-(\partial_x\phi)^2), \partial_t(\alpha \sinh^2 \Lambda \partial_t\phi )- \partial_x(\alpha \sinh^2 \Lambda \partial_x\phi )= 0, \partial_{t}^2\alpha-\partial_{x}^2\alpha= 0,\end{cases}
\lim_{N \to \infty}\varepsilon_{N}=0, lim_{N \to \infty}\varepsilon_N \sqrt{N^F-1}=\infty, lim_{N \to \infty}\varepsilon_{N,l}\sqrt{(N-l)^F-1}=\infty, \ \lim_{N \to \infty}T_{N,l}=\infty.
\mbox{diag}\Big( \frac{e^2 N}{\pi\!+\!gN}+ \frac{\pi}{\pi\!+\!gN}4 mc \tilde{\lambda}\; , \; 4 mc \tilde{\lambda}\;, \; . \; . \; . \; .\; . \; , \;4 mc \tilde{\lambda}\Big) \; .
\frac{\partial \log F(re^{it})}{\partial t}= iz\left(\sum_{k=2}^{p}(k-1)|z|^{2(k-2)}\overline{z}\log G_{k}(z)+\sum_{k=1}^{p}|z|^{2(k-1)}(\log G_{k}(z))_{z}\right) &-i\overline{z}\left(\sum_{k=2}^{p}(k-1)|z|^{2(k-2)}z\log G_{k}(z)+\sum_{k=1}^{p}|z|^{2(k-1)}(\log G_{k}(z))_{\overline{z}}\right) = i\sum_{k=1}^{p}|z|^{2(k-1)}L[\log G_{k}(z)],
\sum_{k=0}^{n-1}(-1)^k \frac{(q;q^2)_k(-q;q^2)_k^2}{(q^4;q^4)_k(-q^4;q^4)_k^2}[6k+1]q^{3k^2}\equiv0\pmod{[n]}, \sum_{k=0}^{(n-1)/2}(-1)^k\frac{(q;q^2)_k(-q;q^2)_k^2}{(q^4;q^4)_k(-q^4;q^4)_k^2}[6k+1]q^{3k^2}\equiv0\pmod{[n]}.
p_{\Lambda^+}= n_{\Lambda^+}(\Lambda_{L+1}){\prod_{\substack{i=1 \Lambda_{i}\neq \Lambda_{i+1}}}^L}n_\Lambda(\Lambda_i)!\,m_{\Lambda^{+}}+ &={\prod_{\substack{i=1 \Lambda_{i}\neq \Lambda_{i+1}}}^{L+1}}n_{\Lambda^+}(\Lambda^+_i)!\,m_{\Lambda^{+}}+ ,
\bar\nabla_{e_j}\big(R^N(V,d\phi(e_j))\tau(\phi)\big)= K\big(|\tau(\phi)|^2V+\langle d\phi,\bar\nabla\tau(\phi)\rangle V+\langle d\phi(e_j),\tau(\phi)\rangle\bar\nabla_{e_j}V &-\langle \bar\nabla_{e_j}V,\tau(\phi)\rangle d\phi(e_j)-\langle V,\bar\nabla_{e_j}\tau(\phi)\rangle d\phi(e_j)-\langle V,\tau(\phi)\rangle\tau(\phi)\big).
\stackrel{\circ}{T}{^a}= \stackrel{\circ}{D}e^{a}\equiv d e^{a}+ \stackrel{\circ}{\omega}{_{\, \cdot b}^{\, a}}\wedge e^{b}, \; \; \stackrel{\circ}{R}{^{a b}}= d \stackrel{\circ}{\omega}{^{a b}}+ \stackrel{\circ}{\omega}{^a_{\cdot c}}\wedge \stackrel{\circ}{\omega}{^{c b}},
S_2 = O \left(\sum_{1 \leq s \leq Z^{\frac{1}{d}}(\log Z)^{-5}}\frac{1}{s}\sum_{0 \leq r \leq s}\sum_{\substack{p^k | F(r,s) F(r,s) \ne 0 p > Z^{\frac{1}{2d}}(\log Z)^2}}1 \right) & = O_{F,k}\left( Z^{\frac{1}{d}}(\log Z)^{-5}\right).
\begin{aligned}\Gamma^k: \Delta_{r,s}\times \Delta_{r,s}\rightarrow \Lambda^k_{r,s}(\chi_1,\chi_2) \mapsto \alpha_{\chi_1,\chi_2}^k, \langle \alpha_{\chi_1,\chi_2}^k,\alpha \rangle_{r,s}:= d_{k,r}\left( \langle \alpha \cdot \chi_1, \chi_2 \rangle_{\Delta_{r,s}}\right) \textit{}\forall \alpha \in \Lambda^k_{r,s}. \end{aligned}
\epsilon ({A d S}) = \left[ \sqrt{f^{1 / 2}+ 1}- \sqrt{f^{1 / 2}- 1}\gamma_{\overline{{r}}}\right] e^{-{\frac{1}{2}}\gamma_{\overline{{\theta r}}}}e^{-{\frac{1}{2}}\gamma_{\overline{{\phi \theta}}}}e^{-{\frac{g t}{\sqrt{2}}}\gamma_{\overline{{0}}}}\epsilon_{0}.
{\pmb{X^1}}E - E{\pmb{X^1}}= \left[\begin{matrix}g_1 0 0 0 g_1 (f_1 - f_0) g_2 - g_1 0 0 \ddots 0 g_2 (f_2 - f_1) g_3 - g_2 0 \ddots \vdots \ddots \ddots \ddots \end{matrix}\right].
\begin{array}{l}\left[ M_{i_1j_1}(x^-_1,y^-_1;x^+), M_{i_2j_2}(x^-_2,y^-_2;x^+)\right]{}~~~~~~~~~~~~~~~~~={1 \over 2N}~\delta_{j_1i_2}\delta (y^-_1 -x^-_2)M_{i_1j_2}(x^-_1,y^-_2;x^+){}~~~~~~~~~~~~~~~~~-{1 \over 2N}~\delta_{j_2i_1}\delta (y^-_2 - x^-_1)~M_{i_2j_1}(x^-_2,y^-_1;x^+)\end{array}
\langle u^\varepsilon(t) - u^\varepsilon(0) , \varphi \rangle =\int_0^t \langle u^\varepsilon(s) , \Delta \varphi \rangle ds + \frac{1}{\varepsilon}\int_0^t \langle f(u^\varepsilon(s)) , \varphi \rangle ds + \varepsilon^\gamma \int_0^t \langle \varphi , adW_s \rangle
K^{(1)}:\mathfrak{h}^{(1)}_+\to\mathfrak{h}^{(1)}, K^{(1)}(x,y):=V^{(1)}(x-y)u_0(x)u_0(y) &K^{(2)}:\mathfrak{h}^{(2)}_+\to\mathfrak{h}^{(2)},\qquad K^{(2)}(x,y):=V^{(2)}(x-y)v_0(x)v_0(y) &K^{(12)}:\mathfrak{h}^{(2)}_+\to\mathfrak{h}^{(1)}, K^{(12)}(x,y):=V^{(12)}(x-y)u_0(x)v_0(y).
E_i:= \Bigl[(U_i+Q_i)^6-U_i^6\Bigr]-\sum_{j\neq i}U_j^6 = 6\,(U_i^5\,Q_i+U_i\,Q_i^5)+15\,(U_i^4\,Q_i^2+Q_i^2\,U_i^4)+20\,U_i^3\,Q_i^3-\sum_{j\neq i}U_j^6.
\hat{\bf G}= - \hat{\tau}_{3}\left[{\bf I}\partial^{2}+ \hat{\bf P}_{\pi}\hat{\cal M}_{\pi}^{2}+ \hat{\bf P}_{\sigma}\hat{\cal M}_{\sigma}^{2}\right]^{- 1}\raisebox{- 2 . 8 5 m m}{\scriptsize{\vec{\varphi}_{1}= \vec{\varphi}_{2}= \vec{\varphi}}}.
\sum\limits_{n=0}^{\infty}\frac{1}{2n+1}\left( \frac{1}{1+e^{-\left(2n+1\right) \gamma}}-\frac{1}{1+e^{-\left( 2n+1\right) \theta}}\right)=\frac{1}{8}\left( \log\gamma-\log\theta\right) ,
\begin{aligned}\frac{\partial (P_k, P_{RF})}{\partial P_k}= \frac{\partial \frac{R_{ave}}{P_k + P_{RF}}}{\partial P_k}&= \frac{\frac{1}{P_k \ln 2}- \log_2(\frac{P_k}{N_0}\log_2\log_2K)}{(P_k + P_{RF})^2},\end{aligned}
u_j(x) = \sum_{i=1}^{n-1}s(j_i) \int_0^{L_{j_i}}\frac{f_{2j_i - 1}(\xi) + f_{2j_i}(\xi)}{2}d\xi + \int_0^x \frac{f_{2j - 1}(L_j - \xi) + f_{2j}(\xi)}{2}d\xi.
(adj(A))_{i_{N}}^{j_{N}}=\frac{1}{(N-1)!}\epsilon_{i_{1}i_{2}...i_{N}}\epsilon^{j_{1}j_{2}...j_{N}}A_{j_{1}}^{i_{1}}A_{j_{2}}^{i_{2}}...A_{j_{N-1}}^{i_{N-1}}
d_n = n^2\pi^2g_n+\dfrac \partial{\partial u}\cdot \big(\mathcal{G}g_n\big)-\epsilon \kappa:u \otimes u g_n+ \epsilon \big(B(g, g)\big)_n + \dfrac{\epsilon}{4\pi}\int_{S_2}\kappa:v \otimes v g_n(v) dv-\dfrac{3+\epsilon}{4\pi}\kappa:u \otimes u 1_n
a_n =a_{n-1}+(2n-9)2^{n-2}+3+\binom{n+1}{2}&+\frac{1}{12}(n-2)(3\cdot2^n-(n-1)(n^2-3n+12)) &+\sum_{m=0}^{n-3}\left((n-m+8)2^{n-m-3}+1-\binom{n-m+2}{2}-\frac{(n-m-1)(n-m-2)(2n-m-3)}{6}\right),
A_2 =\{1,3\}. q(2,1)^4 =d_1^{-4}d^4_7 \; \; q(2,3)^4=d_3^{-4}d^4_5, F_2 =\langle q(2,1)^{4}\rangle\times\langle q(2,3)^{4}\rangle.
\widetilde{\Lambda}(\rho,W,Q;s,t)=\begin{cases}\displaystyle{\frac{1}{2}\int_s^t\int_{\mathbb{T}^3}|\sqrt{Q}w|^2\rho},{\rm on\}[s,t],\;\rho\geq 0,\;W\ll\rho,\;W=w\rho,\;w\in L^2_{\rho}+\infty,{\rm otherwise}\end{cases}
Ind_{(\mathbb Z/p)^n,\mathbb{F}_p}S(V)=\langle \alpha_1b_1+\cdots+\alpha_nb_n\rangle\subseteq \mathbb{F}_p[b_1,\ldots, b_n]\subseteq H^{\ast}(B(\mathbb Z/p)^n;\mathbb{F}_p).
\langle\Upsilon\rangle_{\Sigma}\ \longrightarrow langle\Upsilon\rangle_{\Sigma}+ \delta\!\langle\Upsilon\rangle_{\Sigma},\hspace{1em}\delta\!\langle\Upsilon(z_0)\rangle_{\Sigma}=\frac{1}{\pi}\int_{\Sigma}\langle\Psi(z)\Upsilon(z_0)\rangle_{\Sigma}\,d^2\!z .
\int K d A = ( - 1 ) \int \mid \sin \phi \mid d u \hspace{0 . 1 c m}d v = ( - 1 ) \int \partial_{u v}\phi \hspace{0 . 2 c m}d u \hspace{0 . 1 c m}d v = ( - 1 ) \phi \hspace{0 . 1 c m}\textrm{( b o u n d a r i e s )}
((a,u)(b,v))(c,w) = (ab,u+f_1(a,a^{-1})v)(c,w) &= (abc,u+f_1(a,a^{-1})v+f_1(ab,b^{-1}a^{-1})w), (a,u)((b,v)(c,w)) = (a,u)(bc,v+f_1(b,b^{-1})w) &= (abc,u+f_1(a,a^{-1})(v+f_1(b,b^{-1})w)).
Z(\mathbb{F}_q) = \theta_q(\frac{n-5}{2}) \Bigl( d q^{\frac{n-1}{2}}+\theta_q(\frac{n-3}{2}) - |L_0(\mathbb{F}_q)| \Bigr) + |L_0(\mathbb{F}_q)| = \theta_q(\frac{n-5}{2})(d-1)q^{\frac{n-1}{2}}+ \theta_q(\frac{n-1}{2}).
L_{2}^{\ast\ast}=\frac{\xi_{p}(3+\xi_{p}^{2}+3|\xi_{p}|)\exp(-|\xi_{p}|)-\xi_{q}(3+\xi_{q}^{2}+3|\xi_{q}|)\exp(-|\xi_{q}|)}{2F_{Y}(\xi_{p},\xi_{q})}+\frac{8F_{Y_{(2)}}(\xi_{p},\xi_{q})}{F_{Y}(\xi_{p},\xi_{q})},
\lim_{\|y\|\to0}\sup_{x,x+y\in I_1}\frac{|X(x+y)-X(x)|}{(\|y\||\ln\|y\| |)^{1/a}\sqrt{\log(1+\|y\|^{-1})}}=C_{19}\in(0, \infty)\qquad\mbox{a.s.}
S_{ab}(p)=\left[-\frac{\sum_{i}p^{a \alpha}_{i}-\sum_{i}p^{b \alpha}_{i}}{a \alpha-b \alpha}\right]_{\alpha=1}=-\frac{\sum_{i}p^{a}_{i}-\sum_{i}p^{b}_{i}}{a-b}&=-\sum_{i}p_{i}\left( \frac{p^{a-1}_{i}-p^{b-1}_{i}}{a-b}\right)
\Pr[\hat{M}= M] \leq \inf_{0 \leq \gamma \leq 1}\sum_k \binom{n}{k}\frac{s!}{(k+s)!}(1-Q((1-\gamma)\mu))^{n-k}(1-Q(\gamma\mu))^{k(k+s)}e^{k (2\gamma-1)\mu^2/2}
\frac{1}{2q_k}+ \frac{1}{q_{k+1}}= \frac{1}{q_k}- \left( \frac{1}{2q_k}- \frac{1}{q_{k+1}}\right) &=\frac{q_k \| q_{k-1}\theta \| + q_{k-1}\| q_k \theta \|}{q_k}- \frac{q_{k+1}-2q_k}{2q_k q_{k+1}}&=\| q_{k-1}\theta \| + \frac{q_{k-1}}{q_k}\left( \| q_k \theta \| - \frac{1}{2 q_{k+1}}\right) > \| q_{k-1}\theta \|.
{\mathcal A}_{c o n}^{a}\left( x \right) = - \left( D^{\mu}\langle J_{\mu}\rangle \right)^{a}\left( x \right) = \frac i{4 \pi}\varepsilon_{\mu \nu}{\mathrm t r}\, t^{a}\partial^{\mu}A^{\nu}\left( x \right) .
Q=\int\limits_{x^{0}= t}\!\! d^{3}x\, \Big( H^{\alpha\beta}(x)_{,\beta}+\frac{1}{4}\Phi(x)^{,\alpha}\Big){\stackrel{\longleftrightarrow}{\partial_{0}^{x}}}u^{\gamma}(x)\eta_{\alpha\gamma}\, .
\frac{\partial}{\partial s}\Omega(\alpha,\beta)= \begin{cases}O(X^{2\varepsilon}(s^{-1}+X^4|m^{\ast}\tilde{m}|(hd)^{-1/2}|u|^{-1/4}s^{-5/2}+X^8|u|^{1/4}|m^{\ast}\tilde{m}|(hd)^{-3/2}s^{-7/2})) \mbox{if}0<|u|\le K, O_{\hat\Gamma,C}\left(X^{-C}\right) \mbox{if}|u| > K.\end{cases}
\begin{array}{lll}\langle\alpha|\phi(x)\phi(y)|\alpha\rangle = N_\alpha^2 \left(G_E(x,y)+|\gamma|^2 G_E(\bar x,\bar y)+\gamma G_E(\bar x,y)+\gamma^{*}G_E(x,\bar y) \right) \equiv G_\alpha(x,y) \end{array}
\int\limits_0^\infty{{x^{s - 1}}}\left({\frac{1}{{{{\left({1 + x}\right)}^a}}}+ \frac{1}{{{{\left({1 - x}\right)}^a}}}}\right)dx = 2{\left({- 1}\right)^{\frac{s}{2}}}\frac{{\Gamma \left({a - s}\right)\Gamma \left( s \right)}}{{\Gamma \left( a \right)}}\cos \frac{{\pi s}}{2}
A_{i,k}+A_{i,k+1}=2\sum_{j=i}^k\sup_{l=1,\ldots,i}\P_{\bf U}(\tau_{0,l}\ge j)|\log\bar\alpha_{j-1}|+\sup_{l=1,\ldots,i}\P_{\bf U}(\tau_{0,l}\ge k+1)|\log\bar\alpha_{k}|
F_{1}( 0 ) = Z + \int \frac{{\mathrm d}^{2}{\vec{k}}_{\perp}{\mathrm d}x}{1 6 \pi^{3}}\Big [ \psi_{+ \frac{1}{2}}^{\uparrow \ *}( x ,{\vec{k}}_{\perp}) \psi_{+ \frac{1}{2}}^{\uparrow}( x ,{\vec{k}}_{\perp}) + \psi_{- \frac{1}{2}}^{\uparrow \ *}( x ,{\vec{k}}_{\perp}) \psi_{- \frac{1}{2}}^{\uparrow}( x ,{\vec{k}}_{\perp}) \Big ] \ ,
\begin{aligned}\int_0^T\int_0^1 \left(s\Theta (v_x)^2 + s^3 \Theta^3\left(\frac{x-x_0}{a}\right)^2 v^2\right)e^{2s\varphi}dxdt \le C \int_0^T \int_{-1}^2\left(\frac{\tilde h^2}{\tilde a}+ W_x^2+ W^2\right)e^{2s\tilde \varphi}dxdt &\le C \int_0^T \int_0^1\left(\frac{h^2}{a}+v^2+ \Theta v_x^2\right)e^{2s \varphi}dxdt,\end{aligned}
{\cal G}= - \frac{( 2 N - 1 ) V \beta^{2}}{1 6 \pi G N}\left[ \frac{\Lambda e^{2 b \phi_{0}}}{4 N - 1}r_{+}^{4 N - 1}+ \frac{1 6 \pi^{2}Q^{2}e^{2 a \phi_{0}}}{V^{2}\beta^{4}}\frac{1}{r_{+}}\right] ,
Q_{3}\equiv\lim\limits_{\gamma\to 3/2}(\frac{3-2\gamma}{2})^{-1}P_{2\gamma}(1)=\lim\limits_{\gamma\to 3/2}\frac{\Gamma(\frac{3+2\gamma}{2})}{\frac{3-2\gamma}{2}\cdot\Gamma(\frac{3-2\gamma}{2})}=\Gamma(3)=2.
J_{1}^{2}\int_{0}^{1}d a \int_{0}^{1}d b \, e^{2 \pi i a ( m - n y )}e^{- 2 \pi i b ( m - n y )}= J_{1}^{2}\frac{\sin^{2}\pi n y}{\pi^{2}( n y - m )^{2}}.
g(x,\xi)=\pm\frac{1}{2W(\xi)}\begin{vmatrix}y_{1}(x) y_{2}(x) \cdots y_{n}(x) y_{1}^{(n-2)}(\xi) y_{2}^{(n-2)}(\xi) \cdots y_{n}^{(n-2)}(\xi) y_{1}^{(n-3)}(\xi) y_{2}^{(n-3)}(\xi) \cdots y_{n}^{(n-3)}(\xi) . . \cdots . y_{1}(\xi) y_{2}(\xi) ... y_{n}(\xi)\end{vmatrix}
w_{\xi}(\eta_{1},..., \eta_{d}) =\left({b + c}\right)\mathop \sum \limits_{k = 1}^d \left({\frac{{d + 1}}{2}- k}\right){\eta_k}+ \frac{1}{2}\mathop \sum \limits_{k,p = 1}^d \left({\left({b + c}\right){\delta_{kp}}- a}\right){\eta_k}{\eta_p}\ ,
\left| Q_{t_n}^W\ldots Q_{t_1}^W f(x) - f(x) - \sum_{k=1}^n t_k L^W f_k(x)\right|\leq 4D^2\|f\|_\infty \sum_{k=1}^n t_k^2 \leq 4t^2D^2\|f\|_\infty.