harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
U(\vec{a},1)\left(S, f_\alpha \otimes \rho(E^\alpha)\right) := e^{-i[\hat{f}_0\cdot\hat{H}(\vec{a}, \rho) + \hat{f}_1\cdot \hat{P}(\vec{a},\rho)]}\left(S+\vec{a}, f_\alpha(\cdot - \vec{a}) \otimes \rho(E^\alpha)\right) = e^{i[a^0 \hat{H}(\rho) - a^1\hat{P}(\rho)]}\left(S+\vec{a}, f_\alpha(\cdot - \vec{a}) \otimes \rho(E^\alpha)\right).
\widetilde f(s|\mu) = e^{-\mu}\sum_{n=0}^\infty \frac{\mu^n}{n!}\, \widetilde\ell\,^{n}(s) = \exp\{-\mu(1-\widetilde\ell(s))\}&= \exp\left\{- \mu\int_0^\infty \left(1-e^{-sx}\right)\ell(x)dx\right\}
\frac{y^2}{x_1^6}= \frac{x_1^2 x_2^3}{x_1^6}+ \frac{x_1^4 x_2^2}{x_1^6}= (\frac{x_2}{x_1})^3 \frac{1}{x_1}+ (\frac{x_2}{x_1})^2 .
\tilde{\omega}(A) ={\rm Tr}(F \varrho F^*\pi^P(A)) ={\rm Tr}(\varrho F^* \pi^P(A)F) & ={\rm Tr}(\varrho \pi^P(B^* A B))= \omega(B^* A B)\ \ (A \in{\tt CAR}(\mathcal{K},\Gamma))
\beta_{2}=\underbrace{\left(\left(p_{1}^{r'_{1}}\cdots p_{m}^{r'_{m}}\right)\cdot \left(p_{m+1}^{r_{m+1}}\cdots p_{n}^{r_{n}}\right)\cdot \left(q_{1}^{u'_{1}}\cdots q_{l}^{u'_{l}}\right)\cdot \left(q_{l+1}^{u_{1+1}}\cdots q_{k}^{u_{k}}\right)\right)}_{\sigma_{2}}\beta '_{2}& =\sigma_{2}\cdot \beta '_{2},
X^{\textsuperscript{T}}=\begin{bmatrix}x_{1}x_{2}0 \cdots 0\end{bmatrix}P=\begin{bmatrix}t 0 \ast \cdots \ast\end{bmatrix}, Y^{\textsuperscript{T}}=\begin{bmatrix}y_{1}y_{2}0 \cdots 0\end{bmatrix}P=\begin{bmatrix}0 t \ast \cdots \ast\end{bmatrix},
\Delta (\delta m v_\rho) + V(\delta m v_\rho)^\sigma \leq \Delta(\delta m v_\rho) + M_\rho(\delta m v_\rho)^\sigma \leq\Delta (\delta m v_\rho) + \frac{\lambda_\rho}{(\delta m)^{\sigma-1}}(\delta m v_\rho)^\sigma=0.
D^{(3)}_n = \begin{cases}72(n+3)(2n+1)(2n+7)(3n+2)(3n+4)(6n+5)(6n+7)f_0^5f_8 , 72(n+5)(2n+1)(3n+2)(3n+4)(6n+5)(6n+7)f_0^5 , 5(n+4)(5n+3)(5n+4)(5n+6)(5n+7)f_1^4 .\end{cases}
\pi_{\mathcal{L}^\perp}: \mathbb{P}\mathbb{S}^n \,\dashrightarrow \left\lbrace \mathcal{K}\in \mathrm{Gr}(\dim \mathcal{L}^\perp + 1, \mathbb{S}^n) \mid \mathcal{L}^\perp \subset \mathcal{K}\right\rbrace \cong \mathbb{P}^{\dim \mathbb{P}\mathcal{L}}, S \longmapsto L_S := \mathrm{span}\lbrace \mathcal{L}^\perp, S \rbrace.
S_{J,\Phi}=\int d^{2D}X\,Tr\left( -\frac{i}{3}\mathcal{J}\star\mathcal{J}\star \mathcal{J}-\mathcal{J}\star\mathcal{J}\mathbf{+\,}i\mathcal{J}\star \Phi \star\bar{\Phi}-V_{\star}\left( \Phi \star \bar{\Phi}\right) \right) .
G_m^+(r) :=e^{-(r\sqrt{m-1})^2/2}-\int_{(1+r)\sqrt{m-1}}^\infty e^{-(\sqrt{m-1}-s)^2/2}ds=e^{-(r\sqrt{m-1})^2/2}-\int_{r\sqrt{m-1}}^\infty e^{-s^2/2}ds, G_m^-(r) :=e^{-(r\sqrt{m-1})^2/2}-\int_0^{(1-r)\sqrt{m-1}}e^{-(\sqrt{m-1}-s)^2/2}ds=e^{-(r\sqrt{m-1})^2/2}+\int_{\sqrt{m-1}}^{r\sqrt{m-1}}e^{-s^2/2}ds,
Q_{n-\frac{1}{2}}^{(q)}(\varphi) = \frac{1}{2}\Big( Q_{n}^{(q)}(\varphi) + Q_{n-1}^{(q)}(\varphi) \Big) = \widetilde{\chi}_n^{(q)}\varphi^0 + \sum\limits_{l=1}^{n}w_{n-l}^{(q)}\varphi^{l-\frac{1}{2}},
B_{\gamma}\left( a\right) =\int_{\Theta_{d}}\sum_{z\in \mathfrak{L}}\gamma^{d}f\left( \gamma a+\gamma z\right) \mathrm{e}^{i\theta \cdot z}\mathrm{d}\left( \mathrm{F}_{\ast}\mathrm{P}\right) \left( \theta \right) \ .
V_{5}=-\varepsilon\int_{\mathbb{T}^{d}}(\mu^{1}-\mu^{2})(\mu^{2})\sum_{i=1}^{d}\left[\Theta_{p_{i}x_{i}}(t,x,\mu^{1},Dw^{1})-\Theta_{p_{i}x_{i}}(t,x,\mu^{2},Dw^{2})\right]\dx,
n D(n-i+\ell+2, n-i+j+1)=k_{n-i+\ell+2}+n-k_{n-i+j+1}-k_{n-j+\ell+1}=k_{n-i+\ell}+r-k_{n-i+j}-k_{n-j+\ell+1}=r+k_{n+\ell-j}-n -k_{n-j+\ell+1}= n(D(n+\ell-j)+1),
\beta =2\log\frac{1}{a_0\lambda}+b_0,a_0=2, b_0=\log 2, \mu_\lambda =-\frac{(\beta+2\log\beta)}{a_0\lambda}\partial_n H_\gamma^+=\frac{(\beta+2\log\beta)}{a_0\lambda}\partial_n H_\gamma^- >0.
N^{-1}\sum_{m\leq N}{\tau(m)}\geq N^{-1}\sum_{m=1}^{\lfloor\delta N\rfloor}{\log^{1+\epsilon}{m}}\geq (2N)^{-1}\frac{\delta}{2}N\log^{1+\epsilon}\left(\frac{\delta}{2}N\right)=\frac{\delta}{4}\log^{1+\epsilon}\left(\frac{\delta}{2}N\right) &\geq\frac{\delta}{8}\log^{1+\epsilon}{N}>>\log{N},
\left[-\left(1-\frac{2M}{\rho}\right)\frac{d^2}{d\rho^2}-\frac{2(\rho-4M)(2\rho-3M)}{\rho^2(\rho-3M)}\frac{d}{d\rho}+\frac{8M}{\rho^2(\rho-3M)}\right] H_1(\rho) = \lambda H_1(\rho),
M_cM_c^t =-\partial_x^2+c, M_c^tM_c =-\partial_x^2+c-Q_c^2, M_c(-\partial_x^2-2Q_c\partial_x^{-1}(Q_c\partial_x)) =(-\partial_x^2-Q_c^2)M_c, (-\partial_x^2-Q_c^2)M_c^t =M_c^t(-\partial_x^2), M_cQ_c =0, M_c^tQ_c =-2(Q_c)_x, M_c(xQ_c) =Q_c, M_c^t(xQ_c) =-Q_c-2x(Q_c)_x.
p(\nu)=\frac{1}{Z}\sum_{h}e^{\sum_{ij}W_{ij}^{(1)}\nu_{i}h_{j}^{(1)}+\sum_{jl}W_{jl}^{(2)}h_{j}^{(1)}h_{l}^{(2)}+\sum_{lm}W_{lm}^{(3)}h_{l}^{(2)}h_{m}^{(3)}},
\frac{1-M^{-1}}{\log M}\sum_{A\in\mathcal{P}_{+}}\beta(A)\frac{\alpha(A)}{\beta(A)}\log\frac{\alpha(A)}{\beta(A)}\le\sum_{A\in\mathcal{P}_{+}}\beta(A)\left(\frac{\alpha(A)}{\beta(A)}-1\right)=d_{{\rm TV}}(\alpha,\beta).
I_V ={\rm Nexp}\left( \frac{i}{\hbar}\int_0^t p(\tau) \dot{x}(\tau) -\frac{p(\tau)^2}{2m}d\tau +\frac{1}{2}\int_0^t \dot{x}(\tau)^2 +p(\tau)^2 d\tau\right) \cdot \exp\left(-\frac{i}{\hbar}\int_0^t V(x(\tau),p(\tau),\tau) \, d\tau\right) \cdot \delta(x(t)-y)
\bar{R}_{( \sigma )}^{\alpha \beta \delta ( \rho )}= \varepsilon^{\alpha \beta \gamma \delta}\left( \left( D_{\gamma}\right)_{\; \; \sigma}^{\rho}+ g \delta_{\sigma}^{\rho}\left( \partial_{\tau}B_{\gamma}^{\; ( \tau )}+ B_{\gamma}^{\; ( \tau )}\partial_{\tau}\right) \right) .
2\pi \, |\tilde{\nu}_{N}(0)| = \left|\int_{-\infty}^{\infty}\frac{p_{w}(\omega) + q_{w}(\omega)}{\tanh(d \omega)}\, \mathrm{d}\omega\right| & \leq\left|\int_{-\infty}^{\infty}\frac{p_{w}(\omega)}{\tanh(d \omega)}\, \mathrm{d}\omega\right|+\left|\int_{-\infty}^{\infty}\frac{q_{w}(\omega)}{\tanh(d \omega)}\, \mathrm{d}\omega\right|.
\begin{array}{r c l l c c c c}\mbox{\underline{sector}}{}\mbox{\underline{state}}{}{}\mbox{\underline{$R_1$}}\mbox{\underline{$R_2$}}\mbox{\underline{$R_3$}}\mbox{NS}{}|2s_3,2s_4\rangle, s_3=s_4{}+ + -2is_3 \mbox{R}{}|-,2s_1,2s_2\rangle, s_1=s_2{}+ + +2is_1. \end{array}
V_{N ; g}={\cal C}_{g}^{o p e n}\prod_{i = 1}^{N}\bigg [{}_{i}\! < \! p_{i}; O_{a}| \bigg ] \, \, \delta \left( \sum_{i = 1}^{N}p_{i}\right) \int [{\mathrm{d}}m ]_{g}^{N}\,{\hat{V}}_{N ; 0}
\begin{gathered}y\cdot B_2(k)-x^{k-1}\cdot B_3(k)=y(y^k+y^{k-1}+y^{k-2}+\cdots+\underline{{\bf x^k}}+x^{k-1}+x^{k-2}+\cdots)-x^{k-1}(\underline{{\bf xy}}-1)\end{gathered}
\eta^2(L_2 12^2(\mathcal P \cdot \hat c)^2)^2\sum_{k_1=0}^{\tau}\sum_{k_2=0}^{\tau}\|(I-\eta H)^{k_1}H (I-\eta H)^{k_2}\|_2 &\leq\eta^2(L_2 12^2(\mathcal P \cdot \hat c)^2)^2\sum_{k_1=0}^{\tau}\sum_{k_2=0}^{\tau}\frac{1}{1+k_1+k_2}&\leq2T\eta^2(L_2 12^2(\mathcal P \cdot \hat c)^2)^2 \leq 2 \mathcal P^2 \gamma \hat c \log^{-1}(d\kappa/\delta),
\Big\| \sup_{N\ge 1}\big|\sum_{n=1}^Na_n{\rm e}^{i\lambda_n t}\big|\, \Big\|_{\S^2}\le C \Big(\sum_{n\ge 1}\Big(\sum_{k\, :\, n\le \lambda_k \le n+1}|a_k|\Big)^2\Big)^{1/2}
\int_\sigma|\varphi|^2|f|^2m =\|\varphi(U_\sigma)Xx\|^2=\|X\varphi(T)x\|^2\leq\|X\|^2\|\varphi(T)x\|^2 \leq\|X\|^2K^2 C_{{\rm pol}, T}^2\|x\|^2\int_{\mathbb T}|\varphi|^2|\psi|^2m
\{f\in E(G)|\ \bar{e}_2\prec f\prec e_2\}= (\{f\in E(G)|\e_1\prec f\prec \bar{e}_1\}\setminus \{e_2 \}) \sqcup \{f\in E(G)|\e_3\prec f\prec \bar{e}_3\}.
\rho(r):=\, R(G_{1+\tau+\tau^3},G_{1+2\tau})\cr=\, -(r-2)(r-1)^2(r+1)(2r-1)(2616 r^{10}-4994 r^9+212 r^8-21785 r^7\cr +58174 r^6-30241 r^5+2258 r^4-30963 r^3+12562 r^2-433 r-78 ).
\bigg(2\int_{\varphi(\omega)}^{\varphi(\beta)}f\circ \theta^H(r) \cos(2\pi(a^{-1}R+r)|\xi| + \nu_d) dr\bigg)^2{}=(e^{ih(a)}I + + e^{-ih(a)}) &=2Re( e^{2ih(a)}I^2) + 2|I|^2.
R \leq \frac{1}{n\log n}\left[ n \log P_{\,}- n \log r_0 - n \log \sqrt{\pi}+ \frac{1}{2}n\log \frac{n}{2}- \frac{n}{2}\log e + o(n) - 0.599n \right] = \frac{1}{n\log n}\left[ \left( \frac{1}{2}+ \left( 1 + b \right) \right) \, n \log n - n \left( \log \frac{\lambda\sqrt{\pi e}}{2\rho}+ 1.0599 \right)+ o(n) \right] \;,\,
\gamma^{2}x^{2}+\gamma^{2}v^{2}t^{2}-2\gamma^{2}vtx+y^{2}+z^{2}=c^{2}\gamma^{2}t^{2}+\frac{(1-\gamma^{2})^{2}c^{2}x^{2}}{\gamma^{2}v^{2}}+2\frac{(1-\gamma^{2})txc^{2}}{v}
\vert\mathcal I_{1,1}+\mathcal I_{1,3}\vert \le C_2\Vert\partial_2\dot{q}^+\Vert_{L^2(\Omega_t)}\Vert\dot{H}^+_n\Vert_{L^2(\Omega_t)}+C_1\Vert\partial_2\dot{q}^+\Vert_{L^2(\Omega_t)}\Vert\partial_1\dot{H}^+_n\Vert_{L^2(\Omega_t)}&\le C_2\left\{\int_0^t(I_{1,\ast}+I_{1,n})(s)ds\right\}
\frac{\Psi^\prime_\epsilon(r)}{r^{n-1}}= \frac{1}{\epsilon r}\sum^2_{j=0}\alpha_j \int_{\partial^*\!E_j\cap(B_r \setminus B_{r(1-\epsilon)})}\left( \frac{|x|}{r}\right)^n \frac{(x\cdot \nu_{E_j}(x))^2}{|x|^{n+1}}\, d\mathcal{H}^{n-1}(x) a.e.\, r\in(0,d).
\nabla_{\mu}\Sigma_{AA^{\prime}}^{\lambda}=\partial_{\mu}\Sigma_{AA^{\prime}}^{\lambda}+\Gamma_{\mu \nu}{}^{\lambda}\Sigma_{AA^{\prime}}^{\nu}-(\Gamma_{\mu A}{}^{B}\Sigma_{BA^{\prime}}^{\lambda}+)+\Upsilon_{\mu}\Sigma_{AA^{\prime}}^{\lambda},
\liminf_{x\to 0+}(\psi^{-1})'(-\log(x))\nu((x,1))+\liminf_{x\to 0-}(\psi^{-1})'(-\log|x|)\nu((-1,x)) = \liminf_{x\to 0+}\frac{\nu((x,1))}{\psi'(\psi^{-1}(-\log(x)))}+\liminf_{x\to 0-}\frac{\nu((-1,x))}{\psi'(\psi^{-1}(-\log|x|))}>1.
u\leq e^{C_1s_0^{-\theta/2}(s-s_0)}\hat{u}+Cs_0^{-\theta/4}e^{C_1s_0^{-\theta/2}(s-s_0)}\leq e^{C_1s_0^{-\theta/2}\log A}\left(\hat{u}+Cs_0^{-\theta/4}\right) &\leq \left(1+2C_1s_0^{-\theta/2}\log A\right)\left(\hat{u}+Cs_0^{-\theta/4}\right);
f_{Q,\mu}\leq \log\left(2\|M_\mu\|_{L^1\to L^{1,\infty}}^{1/2}\right)\leq \log\left(\frac{(w(x)\chi_Q(x))^{1/2}}{(w_{Q,\mu})^{1/2}}\right) &\leq \log\left( \frac{M(w\chi_Q)(x)^{1/2}}{(w_{Q,\mu})^{1/2}}\right)=\log v(x)=b(x)
\big( (2m_{1}+m_{2}+2m_{3}) (\cot \langle \alpha_{2}, H \rangle)^{2}-(m_{2}+2m_{3}) \big) ((\cot \langle \alpha_{2}, H \rangle)^{2}-1 ) = 4(m_{2}+2m_{3})(\cot \langle \alpha_{2}, H \rangle)^{2}.
v_m^+(x,t)=\frac{1}{2\pi i}\sum_{j=0}^{2}\int_{0}^{i\infty}e^{st}\frac{\Delta_{j,m}(s)}{\Delta(s)}e^{\lambda_j(s)x}\hat{h}_m(s)ds, v_m^-(x,t)=\frac{1}{2\pi i}\sum_{j=0}^{2}\int^{0}_{-i\infty}e^{st}\frac{\Delta_{j,m}(s)}{\Delta(s)}e^{\lambda_j(s)x}\hat{h}_m(s)ds.
\partial_{t}\mathbb{P}_t^{\alpha,i}(g)(x)=-x_ih_{\alpha+e_i}\left(|y|e^{-|y|t}h_{\alpha+e_i}\left(\frac{g}{._i}\right)(y)\right)(x),\,\,\,x\in \mathbb{R}_+^d\,\,\,and\,\,\,t>0.
\langle \sigma_{\vec{a}}* \sigma_{\vec{b}}, \sigma_{\vec{c}}\rangle = \sum_{e \ge 0}\langle W_{\vec{a}},W_{\vec{b}},W_{\vec{c}}\rangle_e = \langle \sigma_{\vec{a}}, \sigma_{\vec{b}}* \sigma_{\vec{c}}\rangle,
g = \left( \begin{array}{c c}{{1}}&{{0}}\\{{b}}&{{1}}\\ \end{array}\right) \, \left( \begin{array}{c c}{{e^{a / 2}}}&{{0}}\\{{0}}&{{e^{- a / 2}}}\\ \end{array}\right) \, \left( \begin{array}{r r}{{\cos \theta}}&{{\sin \theta}}\\{{- \sin \theta}}&{{\cos \theta}}\\ \end{array}\right) \, ,
\int_{\tilde{M}_\tau}e^{-\frac{|x|^2}{4}}- \int_\Sigma e^{-\frac{|x|^2}{4}}\geq \int_{\Sigma \cap \{|x_{n+1}| \leq L\}}e^{-\frac{|x|^2}{4}}\, \Big [ -C u^2 + \frac{1}{C}\, |\nabla^\Sigma u|^2 \Big ] - C L^{-1}\int_{\Sigma \cap \{|x_{n+1}|=L\}}e^{-\frac{|x|^2}{4}}\, u^2.
\dim\big(\Gamma\big(X/W,(M_{k(S)})^W(d)\big)\big)\le \dim\big(\Gamma\big(X/W,(M_s)^W(d)\big)\big) \hphantom{\dim\big(\Gamma\big(X/W,(M_{k(S)})^W(d)\big)\big)}{}\le \sum_i \dim\big(\Gamma\big(X/W,(M^i_s)^W(d)\big)\big) \hphantom{\dim\big(\Gamma\big(X/W,(M_{k(S)})^W(d)\big)\big)}{}= \sum_i \dim\big(\Gamma\big(X/W,\big(M^i_{k(S)}\big)^W(d)\big)\big),
(\begin{matrix}1&1&1&1&0&0&0\\ 1&-&0&0&1&1&0\\ 1&0&-&0&-&0&1\\ 1&0&0&-&0&-&-\\ 0&1&-&0&0&1&-\\ 0&1&0&-&1&0&1\\ 0&0&1&-&-&1&0\end{matrix})
\frac{{\partial}f_t}{{\partial}s}(s,y,\xi) = p(s,t,y,\xi)\cdot{\partial}_s p(s,t,y,\xi)+(\nabla_x I^L_{a\rho})(s,q(s,t,y,\xi))\cdot{\partial}_s q(s,t,y,\xi) &+\frac{{\partial}I^L_{a\rho}}{{\partial}t}(s,q(s,t,y,\xi)) &=\frac{{\partial}I^L_{a\rho}}{{\partial}t}(s,q(s,t,y,\xi)).
\frac{e^{\frac{2}{3}t^3}}{4^{\frac{1}{3}}\pi}\int_{v=-\infty}^{\infty}\frac{e^{-itv}}{\mathrm{Ai}(i\xi)}dv= 2t+\lim_{x \uparrow 0}\frac{\partial}{\partial x}\frac{1}{2\pi}\int_{v=-\infty}^{\infty}\frac{\mathrm{Ai}(i\xi-4^{\frac{1}{3}}x)}{\mathrm{Ai}(i\xi)}\int_{u=0}^{\infty}e^{iuv-\frac{2}{3}((u+t)^3-t^3)}dudv
S^{\mu}= R^{\mu}+ \frac{1}{3}\left( 1 - \frac{3 T ( G^{\prime})}{2 N_{f}T ( R^{\prime})}- \gamma_{q}\right) ( K_{i}^{\mu}+ \widetilde{K}_{i}^{\mu}- 2 K_{M}^{\mu}) - \frac{1}{3}( 2 \gamma_{q}+ \gamma_{M}) K_{M}^{\mu},
\langle\psi(0,0)\psi^\dagger(x,t)\rangle_T\begin{array}{c}{}\longrightarrow \mbox{\raisebox{2mm}{$x\to\infty\atop t\to\infty$}}\end{array}\exp\left\{-\frac{2\Delta^+\pi T}{v_F}|x-v_Ft|-\frac{2\Delta^-\pi T}{v_F}|x+v_Ft|\right\}.
V\ \approx\ 2 |a_D|^2 \left(|a_M|^2 + |a_{\tilde M}|^2\right) \ -\ \pi^2 \{\left(|a_M|^2 - |a_{\tilde M}|^2\right)^2 \ +\ 16 |a_M a_{\tilde M}|^2 \over 2\ln{|a_D/16\Lambda|}}\ .
\begin{aligned}Y= \frac{\hat d_{yz}}{\bar{y}}S\hat G_y\hat G_dD_{yz}+\frac{\hat d_x}{\bar{y}}S\hat G_y(1-\hat C_b)D_x-\frac{\hat d_b}{\bar{y}}S\hat G_y\hat C_bD_b S= \frac{1}{1+\hat G_y\hat C_b\sigma_y+\hat G_y\hat G_h\hat C_h}\end{aligned}
f ={\frac{1}{2}}g_{s}^{- 1}\left[ \sqrt{m^{2}+ g_{s}^{2}\left({\frac{\mu}{r}}\right)^{7}\left( P_{r}^{2}+{\frac{\vec{L}^{2}}{r^{2}}}\right)}+ m \right] , g ={\frac{\vec{L}^{2}}{r^{2}}},
\begin{cases}\Delta_gG_M^\omega(x,y)-_g(F(y)G_M^\omega(x,y))+\omega^2G_M^\omega(x,y) = -\delta_x(y), \left.\partial_\nu G_M^\omega(x,y) - g_y(F(y),\nu)G_M^\omega(x,y)\right\vert_{y\in \partial M}= 0,\end{cases}
f_{n_{i}}(n_{i})=\frac{m_{c,i}^{n_{i}}}{(1+m_{c,i})^{n_{i}+1}}\, \mathrm{exp}\left(-\frac{m_{s,i}}{1+m_{c,i}}\right)L_{n_{d}}\left(-\frac{m_{s,i}}{m_{c,i}(1+m_{c,i})}\right),
d x^{i}= e_{\mu}^{i}( q - \Delta q ) d q^{\mu}= e_{\mu}^{i}d q^{\mu}-{e^{i}}_{\mu , \nu}d q^{\mu}\Delta q^{\nu}+ \frac{1}{2}{e^{i}}_{\mu , \nu \lambda}d q^{\mu}\Delta q^{\nu}\Delta q^{\lambda}+ \dots ~ ,
\sum_{i=1}^{\kappa}\|\partial_i(f-F)\|_{\dot{F}^{\alpha-1, p}_q}\leq \delta \|f\|_{\dot{F}^{\alpha, p}_q}+ D_{\delta}\min\left(\eta_{\delta},\frac{\delta}{D_{\delta}}\right) \|f\|_{\dot{F}^{\alpha, p}_q}\leq 2 \delta \|f\|_{\dot{F}^{\alpha, p}_q},
\Phi^{( \pm )}= \lim_{\ln r \rightarrow \pm \infty}\frac{1}{2 \pi}\int_{0}^{2 \pi}d \theta V_{\theta}( r , \theta ) , \quad \Phi_{K}^{( \pm )}= \lim_{\ln r \rightarrow \pm \infty}\frac{1}{2 \pi}s \beta \int_{0}^{2 \pi}d \theta \omega_{\theta}( r , \theta ) ,
\cos{\rho}_n =\cos\left(n+\frac{1}{2}\right)\pi\cos \xi_n -\sin\left(n+\frac{1}{2}\right)\pi\sin \xi_n = -\left(-1\right)^n\sin \xi_n , begin{align*}0.2em]\sin{\rho}_n =\sin\left(n+\frac{1}{2}\right)\pi\cos \xi_n +\cos\left(n+\frac{1}{2}\right)\pi\sin \xi_n = \left(-1\right)^n\cos \xi_n .
Z_{\tilde{g}}(s, \chi, (\mathcal{O}_K^{\times})^2)= \int_{(\mathcal{O}_K^{\times})^2}\chi(ac\tilde{g}(u, v))|\tilde{g}(u, v)|^s|dudv| = q^{-e_0s}\int_{(\mathcal{O}_K^{\times})^2}\chi(ac\tilde{g}(u, v))|dudv| := q^{-e_0s}c(\chi).
\begin{array}{ll}Z_{0}(t)=e^{i\frac{\mu}{3}t}{\cal Z}_{0}\,,~~~~ ~~~~Z_{1}(t)=e^{-i\frac{\mu}{6}t}{\cal Z}_{1}\,,{}{}A_{0}=X^{3}=X^{6}=X^{7}=X^{8}=X^{9}=0\,,~~~~ ~~~~{\cal Z}_{0},~{\cal Z}_{1}~:~\mbox{diagonal~matrices}\,.\end{array}
\min_{\Delta \in C_J(u):|\Psi^{1/2}\Delta|_2=1}|\Psi\Delta|_\infty = \min_{\Delta \in C_J(u):|X\Delta/\sqrt{n}|_2=1}|X^TX\Delta/n|_\infty &\geq \min_{\Delta \in \mathbb{R}^p :|X\Delta/\sqrt{n}|_2=1}|X^TX\Delta/n|_\infty\geq \min_{\delta \in \mathbb{R}^n :|\delta|_2=1}|X^T\delta/\sqrt{n}|_\infty.
\nu_{d,p}(\mathbb G_2^c) \quad\leq\frac{1}{\delta_d}e^{\gamma_2 M_d^2 + \gamma_2 p}d^{-5 \xi_2}\kappa_2= \kappa_2 d^{-5 \xi_2 + \tau_1 + \tau_2 + \gamma_2 p / \log d}
\pi_i = \frac{1}{2}\pi{_i}^{ab}e_{ab}^{(2)}\wedge \bar{\theta}^{(r)}- \pi{_i}^{ak}e_a^{(3)}\wedge \bar{\theta}^{(r-1)}_k + \frac{1}{2}\pi{_i}^{jk}e^{(4)}\wedge \bar{\theta}^{(r-2)}_{jk}
\left( \frac{-c\alpha^2}{3}- d\right) z = p, \mbox{\so\}p\in (z),\mbox{\and}\alpha - \alpha^{p^m}= \alpha \left( 1 - \alpha^2 (-18)^{\frac{p^m-3}{4}}\right) \equiv \alpha\left( 1 - \frac{3d}{c}(-18)^{\frac{p^m-3}{4}}\right) \mod (z) \equiv 0 \mod (z),
I_{1 0 2}^{a b c}= \frac{2 g f_{a b c}}{( 2 \pi )^{4}x_{1 2}^{2}}\left[ [ 2 1^{-}] \rho_{1}^{2}+ [ 1 2^{-}] \rho_{2}^{2}- 2 ( \rho_{1}\rho_{2}) \right] .
{p_{L}}= \frac{1}{\sqrt{2 I m T I m U}}( m_{1}+ m_{2}{\bar{U}}+ n_{1}{\bar{T}}+ n_{2}{\bar{U}}{\bar{T}}) , \; p_{R}= \frac{1}{\sqrt{2 I m T I m U}}( m_{1}+ m_{2}{\bar{U}}+ n_{1}T + n_{2}T{\bar{U}}) ,
\kappa(\epsilon) =\, \mu\sum_{j=0}^{\infty}e^{-\alpha j}\tilde{P}^j r + \epsilon \mu \sum_{j=1}^{\infty}e^{-\alpha j}\sum_{k=1}^{j}a_{k1}\tilde{P}^{k-1}\tilde{P}^{(1)}\tilde{P}^{j-k}r + O(\epsilon^2(\log(1/\epsilon))^w)
\|V^{(k,q)}_{I_{r;i}}\|\leq \|V^{(k,q-1)}_{I_{r;i}}\|+\sum_{j=1}^{k}\sum_{n=1}^{\infty}\frac{1}{n!}\,\| ad^{n}S_{I_{k;i}}(V^{(k,q-1)}_{I_{r-j;i+j}})\|\,,
\begin{aligned}\alpha_{ii},\, x_j\alpha_{jj},\, \tilde \alpha_{ii'},\, x_j\tilde \alpha_{ij},\, x_j\tilde \alpha_{ji},\, x_jx_{j'}\tilde \alpha_{jj'}\in C^{\alpha}_{WF}(\bar M_I), a_{n+k,n+l},\, b_i,\, b_j,\, c_{il},\, x_jc_{jl},\, e_l \in C^{\alpha}_{WF}(\bar M_I).\end{aligned}
R_k = \left(\frac{\Gamma\left(\frac{k}{2}+1 \right)}{\pi^{\frac{k}{2}}}\right)^{\frac{1}{k}}= \left(1 + \frac{\log(k+1)}{2k}+ O\left(\frac{1}{k}\right) \right)\sqrt{\frac{k}{2\pi e}}
A_{s,n}:= K_1 K_2 \sum_{k=1}^{\infty}\sum_{t=1}^{n}\{S_{1,2k-1}(n,n-s,t-1) + S_{2,2k-1}(n,n-s,t-1)\}, B_{s,n}:= K_1 K_2 \sum_{k=1}^{\infty}\sum_{t=1}^{n}\{S_{1,2k}(n,s-1,t-1) + S_{2,2k}(n,s-1,t-1)\}.
I[q(\cdot),u(\cdot),\mu(\cdot)] =\int_a^b L\left(q(t),u(t),\mu(t)\right) dt\longrightarrow \min \, , \dot{q}(t)=\varphi(q(t),u(t))\,,{_a^CD_t^{\alpha}}q(t)=\rho\left(q(t),\mu(t)\right)\,.
G_k = \begin{cases}MM, \mbox{with probability}p_1 = K(\frac{\theta_M}{2\pi})^2 mM, \mbox{with probability}p_2 = 2K(\frac{2\pi-K\theta_M}{2\pi})(\frac{\theta_M}{2\pi}) + (K^2-K)(\frac{\theta_M}{2\pi})^2 mm, \mbox{with probability}p_3 = (\frac{2\pi-K\theta_M}{2\pi})^2\end{cases}
\frac{d h \left( t , \frac{{D_{B}}^{\frac 1 2}}{\mu}\right)}{d t}= \sum_{k = 0}^{d}( - k ) \frac{1}{2 \mu}\frac{\Gamma \left( \frac{k + 1}2 \right)}{\Gamma \left( \frac 1 2 \right)}a_{d - k}\left( \frac{t}{2 \mu}\right)^{- k - 1}+
\chi_{0, 0, 2}(q, \lambda, \omega) = q^\lambda \frac{(q^{-2\omega + 2}; q^{-2\omega})}{(q^{-2\omega + 4}, q^{-2\omega})}(q^{-2\lambda + 2}; q^{-2\omega}) (q^{2\lambda + 2}q^{-2\omega}; q^{-2\omega}) (q^{-2\omega + 2}; q^{-2\omega}).
\mathbf{F}\operatorname{div}\mathbf{G}^{\ast}-\mathbf{G}^{\ast}\times\operatorname{curl}\mathbf{F}& \quad=4\pi\rho\mathbf{F}+\frac{i}{c}\left( \frac{\partial\mathbf{G}}{\partial t}\times\mathbf{G}^{\ast}+4\pi\mathbf{j\times G}^{\ast}\right)
( i ) \quad d P_{0}= \left( \begin{array}{c c}{I}&{Z^{\dag}}\\{- Z}&{I^{\prime}}\\ \end{array}\right)^{- 1}\left( \begin{array}{c c}{}&{d Z^{\dag}}\\{d Z}&{}\\ \end{array}\right) \left( \begin{array}{c c}{I}&{- Z^{\dag}}\\{Z}&{I^{\prime}}\\ \end{array}\right)^{- 1},
M_{i+1}- (k_{i+1}- \overline{\lambda}_{i+1}-1) -1 =\sharp\{j | -\overline{\lambda}^{i+1}_{-j}\leq \overline{\lambda}_{i+1}\}-1 \leq \sharp\{j | -\overline{\lambda}^{i}_{-j}\leq \overline{\lambda}_{i+1}\}&< \sharp\{j | -\overline{\lambda}^{i}_{-j}\leq \overline{\lambda}_{i}\}= M_{i}- (k_{i}- \overline{\lambda}_{i}-1).
\lambda\frac{dQ}{dP}\in\partial V(u-\sum_{t=0}^{T-1}x_t\Delta s_{t+1}-\bar c\cdot\bar x+S_0(\bar x)), E^Q_t[\Delta s_{t+1}] \in N_{D_t}(x_t)t=0,\dots,T, E^Q[\bar c] \in\partial S_0(\bar x),
\left\lbrace\begin{array}{ll}U''-\Delta U+ AU=0 \hbox{in}(0, +\infty)\times \Omega, U= 0 \hbox{on}(0, +\infty)\times \Gamma_0, \partial_\nu U+BU = DH \hbox{on}(0, +\infty)\times\Gamma_1\end{array}\right.
\tilde{X}_Hm_X(x,\xi)=\frac{1}{2T_{\alpha_0}'}\left(m_0\circ\tilde{\Phi}^X_{T_{\alpha_0}'}(x,\xi)-m_0\circ \tilde{\Phi}^X_{-T_{\alpha_0}'}(x,\xi)\right)=\frac{1}{2T_{\alpha_0}'}>0.
\prod_{i=1}^{n}p_i\cdot\biggl(\frac{1}{\prod_{i=2}^{n}(p_i-2)}\prod_{i=2}^{m}(p_i-2)-\frac{\sqrt{X/2}}{\prod_{i=1}^{n}p_i}\biggr)+\sqrt{X/2}=\frac{\prod_{i=1}^{n}p_i}{\prod_{i=2}^{n}(p_i-2)}\cdot\prod_{i=2}^{m}(p_i-2).
A(X\oplus\alpha_1 S_{\theta_1}\oplus\alpha_2S_{\theta_2}) =A(X\oplus \alpha_1 S_{\theta_1})+2W(X\oplus\alpha_1 S_{\theta_1},\alpha_2 S_{\theta_2}) &=A(X)+\alpha_1 H_X(\theta_1)+\alpha_2 H_{X\oplus\alpha_1 S_{\theta_1}}(\theta_2) &=A(X)+\alpha_1 H_X(\theta_1)+\alpha_2 H_{X}(\theta_2)+\alpha_2H_{\alpha_1 S_{\theta_1}}(\theta_2).
\sigma_K(y)y =\sigma_K(x\otimes1+1\otimes\sqrt{\alpha})\cdot(x\otimes1+1\otimes\sqrt{\alpha}) =\sigma_K(x\otimes1)\cdot(x\otimes1)+\alpha\otimes1+(1\otimes\sqrt{\alpha})(x\otimes1+\sigma_K(x\otimes1)).
\begin{array}{l}P_0-\tilde{P}_0=-\rho_0\big[-\frac{1}{2}k^2c^2(a^2+d^2-b^2)e^{-ms_0}\cos^2\theta-\frac{1}{2}k^2c^2b^2e^{-2ms_0}+\frac{1}{2}\hat{f}kcabe^{-2ms_0}-\frac{1}{2}fkcbde^{-2ms_0}+gs_0\big]+\rho_+gs_0, \end{array}
\|\eta_R u_n\|_{D^{s,p}}^{p}\leq (1+\delta)\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|\eta_R (x)|^p|u_n(x)-u_n(y)|^p}{|x-y|^{N+sp}}dxdy &\phantom{=}+C(\delta) \int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u_n(y)|^p|\eta_R (x)-\eta_R (y)|^p}{|x-y|^{N+sp}}dxdy.
R_e = \sum_{w \in W(D_p \times C_{p+q})}(-1)^{l(w)}X\left(\begin{array}{cccccccccc}p-1, \dots, 1, 0; p+q, \dots, 2,1 w( p-1, \dots, 1, 0; p+q, \dots, 2,1)\end{array}\right),
\left[C_{\gamma}(\mathcal{Z}_{m,n}^{\gamma}) \right](z) =\overline{\left[C_{\gamma}(\overline{\mathcal{Z}_{m,n}^{\gamma}}) \right](\overline{z})}= \overline{\left[C_{\gamma}(\mathcal{Z}_{n,m}^{\gamma}) \right](\overline{z})}.
\sum_{j_1+j_2+j_3=0}\mathrm{i}\,(j_1^3+j_2^3+j_3^3)\,F_{j_1 j_2 j_3}^{(3)}\,u_{j_1}\,u_{j_2}\,u_{j_3}=\sum_{(j_1, j_2, j_3)\in\mathcal{A}_3}(-\mathrm{i}\,c_1\, j_1 j_2 j_3-c_2\,j_1 j_2+c_3)\, u_{j_1}u_{j_2}u_{j_3}
(\mu+\overline{\mu})\int_{-1}^1 \left(|g'|^2 + A\frac{(2n+\alpha+\beta-1)_2}{2(n+\alpha+\beta)}\left(P_{n-1}^{(\alpha,\beta)}(x)\right)^2 \right)w_{\alpha,\beta}\,dx = - \int_{-1}^1 |g|^2 w'_{\alpha,\beta}\,dx, \quad 2\Re\mu=\mu+\overline{\mu}<0.
\frac{\partial p_{s}}{\partial t}+v\cdot\nabla p_{s}+\frac{Z_{s}e}{m_{s}}(E+v\times B)\cdot\nabla_{v}p_{s}=-\frac{\partial}{\partial v_{i}}(p_{s}\langle\Delta v_{i}\rangle)+\frac{1}{2}\frac{\partial^{2}}{\partial v_{i}\partial v_{j}}(p_{s}\langle\Delta v_{i}\Delta v_{j}\rangle),
\Psi_l(p_1,\ldots,p_N):=|U_{\rho_l}(k_0)|^{-1/2}\,\chi_{U_{\rho_l}(k_0)}^{}\Big(\sum\limits_{l=1}^n p_l\Big)\,\phi\Big(\sum\limits_{i=1}^n p_i;p_1,\ldots,p_n \Big)\,\psi\Big(\sum\limits_{i=n+1}^N p_i;p_{n+1},\ldots,p_N\Big),
h\beta_1\varpi_p = \left(\begin{array}{cc}-y xp^{-1}-t zp^{-1}\end{array}\right) = \left(\begin{array}{cc}t^{-1}\det(h) p^{-1}-y 0 -t \end{array}\right)\left(\begin{array}{cc}0 1 1 -t^{-1}p^{-1}z \end{array}\right),
a([\Delta]^{(3)},[1,1,1;0,0,0])=\frac{2^3\cdot\underline{23}\cdot 487}{7},a([\Delta]^{(3)},[1,1,1;0,0,1])=2^3\cdot 3\cdot\underline{23}\cdot 419, & a([\Delta]^{(3)},[1,1,1;1,1,1])=\frac{2^3\cdot \underline{23}\cdot 80429}{7}.
\sum_{i=1}^{\infty}\sum_{a_{1},\cdot\cdot\cdot,a_{i}}H_{a_{1},\cdot\cdot\cdot,a_{i}}^{a_{1},\cdot\cdot\cdot,a_{i}}\prod_{j=1}^{i}\hat{a}_{j}^{\dagger}\hat{a}_{j}.
T^2f\cdot Ti_A\cdot q_f\xrightarrow{T^2f\cdot\bar\alpha_1}T^2 f\cdot Tq_f\cdot i_{Kf}= T^2 f\cdot d_0\cdot k \cdot i_{Kf}\xrightarrow{\theta\cdot k\cdot i_{Kf}}Ti_B\cdot d_1\cdot k \cdot i_{Kf}
n(-D(n-i+j+1)+D(n-i+\ell+2, n-i+j+2)+D(n-i+\ell+1)) =-(r+k_{n-i+j+1}-k_{n-i+j+2})-k_{n-i+j+2}+n+k_{n-i+\ell+2}-k_{n+\ell-j}+r+k_{n-i+\ell+1}-k_{n-i+\ell+2}&=-k_{n-i+j+1}+n+k_{n-i+\ell+1}-k_{n+\ell-j}=n D(n-i+\ell+1, n-i+j+1)=0,
\hat{\mathbf{R}}_{k_l}^\mathrm{h}= \mathrm{E}\left[ \hat{\mathbf{h}}_{{k_l}l}^\mathrm{h}\left( \hat{\mathbf{h}}_{{k_l}l}^\mathrm{h}\right)^\mathrm{H}\right] \approx \zeta_{k_l}^2 \mathbf{R}_{k_l}^\mathrm{h}+ \left(1-\zeta_{k_l}^2 \right) \mathbf{I}_{M_\mathrm{h}}.
A = \frac{y_{2}}{W ( y_{1}, y_{2})}\frac{1}{R^{7}u^{\prime}\sqrt{{u^{\prime}}^{6}- ( 1 + \theta ){u^{\prime}}^{2}+ \theta}}, B = \frac{y_{1}}{W ( y_{1}, y_{2})}\frac{1}{R^{7}u^{\prime}\sqrt{{u^{\prime}}^{6}- ( 1 + \theta ){u^{\prime}}^{2}+ \theta}}.
\sum_{k=1}^d \Bigl( \partial_{kk}A \partial_{11}B- \partial_{11}A \partial_{kk}B \Bigr) = O \Bigl( \partial^2( \partial^2 A\cdot B) ) \Bigr)+ O \Bigl( \partial ( \partial^3 A \cdot B ) \Bigr).
M^{0,0}_{\frac{2m-i-k}{2},\frac{i+k-2}{2}}M^{\frac{k-2}{2},\frac{k-2}{2}}_{\frac{i+k-2}{2},\frac{2m-i}{2}}=\frac{k}{2}M^{\frac{2m-k}{2},\frac{2m-i-k}{2}}_{\frac{2m-i-k}{2},\frac{2m-i}{2}}
I_{1}=\frac{1}{8i}\int_{\left(1/N\right)}e^{Nz}z^{-k-3}dz=\frac{N^{k+2}}{8i}\int_{\left(1\right)}e^{s}s^{-k-3}ds=\frac{N^{k+2}\pi}{4\Gamma\left(k+3\right)}
{\cal D}_{0}^{- 1}= \left( \begin{array}{c c}{\left( 2 \beta p^{2}\theta_{\mu \nu}+ 2 p^{2}\omega_{\mu \nu}\right)}&{0}\\{0}&{\left( p^{2}- \mu^{2}\right)}\\ \end{array}\right) \mathrm{{.}}
w_1 =x_1^2x_2\cdots x_mx_1, w_2 =x_1x_2^2\cdots x_mx_1, \ldots \hbox to 2.5cm{\dotfill}w_{m-1}=x_1x_2\cdots x_{m-1}^2x_mx_1, w_m =x_1x_2\cdots x_mx_1.