harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\forall i,i_1\in\{0,1,2,3,4\}, \mu_1\in \{+, -\}, |HypEll_{k_1,j_1;n_1}^{i,i_1, \mu_1}(t_1,t_2)| \leq \sum_{a=1,2,3}|{}^1_a LastEll_{k_1,j_1;n_1}^{i,i_1, \mu_1}(t_1,t_2)|,
J(\sigma, X,u) = \left( \begin{matrix}J_1(\sigma, X ,u) J_2(\sigma, X, u)\end{matrix}\right)= \left( \begin{matrix}\Im \int e^{-i \sigma}u(x+X) Q(x) dx \Re \int e^{-i \sigma}u(x+X) \nabla Q(x) dx\end{matrix}\right),
{\cal R}( 0 ) = \rho \sin \eta \left[ \begin{array}{c c c c}{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\ \end{array}\right] = \rho \sin \eta \cdot P ~ ,
2\omega_{12}+\theta_{12}=\frac{C^1_1}{\mu}\omega_1-\frac{B^1_{11,3}}{\mu}\omega_3, &\omega_{13}=-\frac{B^1_{11,3}}{\mu}\omega_2,~~~~\omega_{1i}=0, i\geq 4, &\omega_{23}=\frac{B^1_{11,3}}{\mu}\omega_1-\frac{C^1_1}{\mu}\omega_3, ~~\omega_{2i}=-\frac{C^1_1}{\mu}\omega_i, i\geq 4.
[ u_{\mid V},[ v_{\mid V},z_{\mid V}]_{TN_{\mid V},h}]_{TN_{\mid V},h}+[z_{\mid V},[ u_{\mid V},v_{\mid V}]_{TN_{\mid V},h}]_{TN_{\mid V},h}&+[ v_{\mid V},[ z_{\mid V},u_{\mid V}]_{TN_{\mid V},h}]_{TN_{\mid V},h}=0,
||\frac{\sin(2(v_{1}+v_{2}+v_{3}+v_{4}))}{r^{2}}\left(-\sin(2Q_{\frac{1}{\lambda(t)}}+2v_{5})\partial_{r}(Q_{\frac{1}{\lambda(t)}}+v_{5})+\sin(2Q_{\frac{1}{\lambda(t)}})\partial_{r}Q_{\frac{1}{\lambda(t)}}\right)\vert_{r=R\lambda(t)}||_{L^{2}(R dR)}&\leq \frac{C \log^{4}(t)}{t^{21/4}}
S = \frac{r_{h}^{2}}{9 0}{\tilde{\epsilon}}^{- 2}- \left[ \frac{r_{h}^{2}}{1 8 0}\left( \frac{2 \Delta_{h}^{\prime}}{r_{h}}- 3 \delta_{h}^{\prime}\Delta_{h}^{\prime}- \Delta_{h}^{\prime \prime}\right) - \frac{r_{h}^{2}\mu^{2}}{6}\right] \log{\tilde{\epsilon}}+{\mathrm{t e r m s ~ f i n i t e ~ a s ~}}{\tilde{\epsilon}}\rightarrow 0 .
\tilde{M}_y=\frac{uv_y-vu_y}{v^2}, \tilde{M}_x=\frac{uv_x-vu_x}{v^2}, &\tilde{M}_{xx}=\frac{uv_{xx}-vu_{xx}}{v^2}-\frac{2uv_x^2-2vu_xv_x}{v^3}.
\sharp \{\lambda \in \Xi_q: N_{n,m}(\lambda)\ge \frac{1}{4d}(c-d-2d^2)(nm)^{\frac{1}{d+1}}\}&\le \sum_{\lambda\in \Xi_q}\frac{N_{n,m}(\lambda)}{2(2d+1)^2(nm)^{\frac{1}{d+1}}}\le \frac{(2d+1)^2nm}{2(2d+1)^2(nm)^{\frac{1}{d+1}}}=\frac{1}{2}(nm)^{\frac{d}{d+1}}\le \frac{1}{2}\sharp \Xi_q,
\mathrm{P}_{t,1}= \frac{2\lambda_c^t\pi^t}{(t-1)!}\int^{\infty}_{\frac{\rho^{\frac{1}{\alpha}}}{\epsilon_1^{\frac{1}{\alpha}}}}y^{2t-1}e^{-\lambda_c \pi y^2}dy =e^{-\lambda_c\pi \left(\frac{\rho}{\epsilon_1}\right)^{\frac{2}{\alpha}}}\sum^{t-1}_{k=0}\frac{(\lambda_c\pi)^{k}\left(\frac{\rho}{\epsilon_1}\right)^{\frac{2k}{\alpha}}}{k!}.
G(x,\mu) = \int_{\mathcal{O}^{\times}}\psi(xy)\mu(y)d^{\times}y = \begin{cases}1 -\zeta_F(1)q^{-1}\zeta_F(1)q^{-\frac{a(\mu)}{2}}\epsilon(\frac{1}{2},\mu^{-1})\mu^{-1}(x) 0 \end{cases}
\begin{aligned}(t,x,y,z,z_1,z_2,v_i)\mapsto H_i^{v_i}(t)\quad(i=1,2), x\mapsto g(x) x\mapsto \Phi_i(x)\ \quad(i=1,2), x\mapsto \gamma_i(y)\ \ \quad(i=1,2)\end{aligned}
Q_{2}=[\begin{matrix}cosh(\eta)&0&0&sinh(\eta)\\ 0&1&0&0\\ 0&0&1&0\\sinh(\eta)&0&0&cosh(\eta)\end{matrix}]=exp(\eta[\begin{matrix}0&0&0&1\\ 0&0&0&0\\ 0&0&0&0\\ 1&0&0&0\end{matrix}]).
t^{2}+ \left( 1 - \frac{4 h ( \Lambda_{1}^{2}, \Lambda_{2}^{2})}{( \Lambda_{1}^{2})^{p + 1}}\left( v^{2}- u_{1}+ \frac{\Lambda_{1}^{2}}{2}+ O ( u_{1}^{2}, u_{2}^{2}) \right) \right) t + \left( \frac{\Lambda_{2}^{2}}{\Lambda_{1}^{2}}\right)^{p + 1}= 0
c'(y,z) =(\|w\|^2+1)^{-1}(1-z)^2\big(\|v\|^2+\|w\|^2+(1+z)^2+1\big)+z^2+2z+\|v\|^2+\|w\|^2 &=(\|w\|^2+1)^{-1}\Big(\big((1-z)^2+1+\|w\|^2\big)\|v\|^2+z^4+1+2z^2\|w\|^2\Big)+\|w\|^2+1 &\ge (\|w\|^2+1)^{-1}\Big(\big((1-z)^2+1\big)\|v\|^2+z^4+1\Big)+\|w\|^2+1 &\ge \tfrac14(\|w\|^2+1)^{-1}\Big(\big((1-z)^2+1\big)\|v\|^2+z^4+4\Big)+\|w\|^2+1.
\log \Upsilon ( x ) = \int_{0}^{\infty}\frac{d t}{t}\; \biggl [ \biggl ( \frac{Q}{2}- x \biggr )^{2}e^{- t}- \frac{\sinh^{2}\bigl ( \frac{Q}{2}- x \bigr ) \frac{t}{2}}{\sinh \frac{b t}{2}\sinh \frac{t}{2 b}}\biggr ]
\bigl[\mathbb{R}_{12}(u_1,u_2|v_1,v_2)\Phi\bigr](z_1,z_2) =\int\limits^{+\infty}_{-\infty}\int\limits^{+\infty}_{-\infty}\rho(x_1)\rho(x_2) W(u_1-v_2;z_1,z_2)\times \overline{W}(u_1-v_1;z_1,x_2)\overline{W}(u_2-v_2;z_2,x_1)W(u_2-v_1;x_1,x_2) \Phi(x_1,x_2) d x_1 d x_2.
\max_{\substack{q(x_1,x_2): q(x_1, x_2)=0 p(x_1, x_2)=0}}~\min_{q(y_1,y_2|x_1,x_2)\in \mathcal{P}}\min_{\substack{q(f|x_1,y_1): I_q(X_1,Y_1;Y_2|F,X_2)=0}}I(F;X_1|X_2).
\mathfrak{E}(\theta^{\epsilon}(t)) +\int^{t}_{0}\left\|\Lambda^{1/2}\theta^{\epsilon}(s)\right\|^{2}_{L^{2}}ds + 4\epsilon \int^{t}_{0}\int_{\mathbb{T}}\left|\left(\sqrt{\theta^{\epsilon}}(s)\right)_{x}\right|^{2}dxds = \mathfrak{E}(\theta^{\epsilon}_{0}).
\sum_{1\leqslant d\leqslant K^\frac{1}{2}b^{-\frac{1}{4}}\det(\Lambda)^{-\frac{1}{2}}B^{\frac{1}{4}(2-\frac{1}{r})}}Kb^\frac{1}{2}B^{1-\frac{1}{2r}}\log(B)=O\left(\frac{K^\frac{3}{2}b^\frac{1}{4}}{\det(\Lambda)^\frac{1}{2}}B^{\frac{3}{4}(2-\frac{1}{r})}\log B\right).
\frac{\pi^{2}}{8}= \frac{1}{1^{2}}+ \frac{1}{3^{2}}+ \frac{1}{5^{2}}+ \ldots + \frac{1}{( 2 n + 1 )^{2}}+ \ldots = \sum \limits_{k = 0}^{\infty}\frac{1}{( 2 k + 1 )^{2}}
\frac{d}{ds}\psi^x_{-\ell}(s)=(\lambda^c_{x_{-\ell}}\cdots\lambda^c_{x_{-1}})^{-1}D\Phi^c_{x_{-\ell}}\left((\lambda^c_{x_{-\ell}}\cdots\lambda^c_{x_{-1}})^{-1}s\right)e_1.
\langle \varphi,\widetilde H_m^\alpha\varphi\rangle =\langle\varphi, R^1\varphi\rangle -\alpha\big\langle\varphi, \Xi_m R^1\Xi_m^{-1}\varphi\big\rangle =\int_{-\infty}^{+\infty}\big( 1 -\alpha V_{|m|- 1/2}(s)\big)\big|(R^{1/2}\varphi)(s)\big|^2\mathrm ds.
{\tilde{\rho_{1}}}^{1 2}={\tilde{\rho_{2}}}^{6}{\tilde{\rho_{5}}}={\tilde{\rho_{3}}}^{6}={\tilde{\rho_{4}}}^{6}={\tilde{\rho_{5}}}^{2}= \mathrm{i d e n t i t y}.
\tilde{f}_+(x,y)=f_+(x^4,y^5)=(x^{20}+ y^{20})^2+x^{35}y^{12}+tx^{30}y^{24}f_{+1}(x,y)=(1+y^{20})^2+x^7y^{12}+tx^{14}y^{24}f_{+2}(x,y)=(x^{20}+1)^2+x^{35}y^{7}+tx^{30}y^{14}
\frac{\ell+1}{\sin^2 \phi}[ \cos \phi \, p_{0,\ell}( \phi)-p_{0,\ell+1}(\phi) ] &=\frac{\ell}{\sin^2 \phi}\sin \phi \sin \psi_{0,\ell}h_0(0) s_{0,0}(\ell,\phi) + \phi^{-2-1/2}O(\ell^{-1/2})+O(\phi^{-1}).
T(k)=\sum_{i=0}^k\frac{i}{q_0\kappa^{i}}=-\frac{k}{q_0(\kappa-1)\kappa^{k-1}}+\frac{\kappa}{q_0(\kappa-1)^2}\left[1-\left(\frac{1}{\kappa}\right)^k\right]\leq \frac{\kappa^2}{\gamma (\kappa-1)^2}.
\int ( w(\tau_0) -Q_b ) \phi_{1, b, \beta}\frac{y^2}{2}\rho_\beta dy= \left( \frac{2}{\beta}\left(\frac{d}{2}-\gamma+2 \right) - \frac{1}{\beta}\left(\frac{d}{2}-\gamma+1 \right) \right) \|\phi_{1,b,\beta}\|_{L^2_{\rho_\beta}}^{2}+O(b^{1-\frac{\epsilon}{2}}) .
\bigl\{y_1\in\mathcal N^1_G;\,(y_1,y_2)^{G}_{\mathcal N}=0 y_2\in\mathcal N^2_G\bigr\}=\{0\}, \bigl\{x_1\in\mathcal N^1_{\mathbf H};\,(x_1,x_2)^{{\mathbf H}}_{\mathcal N}=0 x_2\in\mathcal N^2_{\mathbf H}\bigr\}=\{0\}.
\mathcal{O}_{e^{|M|}_{-}}: =\left\{(x,y,z)\in\Omega \mid x^2 + y^2 + z^2 = (\lambda +1) M^2, ~ y<0 \right\}, \mathcal{O}_{e^{|M|}_{+}}: =\left\{(x,y,z)\in\Omega \mid x^2 + y^2 + z^2 = (\lambda +1) M^2, ~ y>0 \right\}.
\mathbf{D}_{\mathcal{C}}= \begin{pmatrix}\mathbf{d}_1 begin{align*}-1mm]\vdots \mathbf{d}_\lambda\end{pmatrix}= \begin{pmatrix}\boldsymbol{\chi}(\mathcal{C}_1) begin{align*}-1mm]\vdots \boldsymbol{\chi}(\mathcal{C}_\lambda)\end{pmatrix}\,
\lim_{|\xi|_{p,q,r}+\theta^{\ell}+|v|^2\to\infty}\frac{(|F|+M)^p+(|Z|+M)^q+(|w|+M)^r+(\theta+M)^{\ell}+(|v|+M)^2}{|\xi|_{p,q,r}+\theta^{\ell}+|v|^2}=1\;,
\tilde{b}_{n-2}=\frac{d^{n-2}\tilde{b}}{dv^{n-2}}+\sum_{m=0}^{n-3}\sum_{l=0}^{n-3-m}\binom{n-3-m}{l}\frac{d^{n-3-m-l}\tilde{b}}{dv^{n-3-m-l}}\frac{d^l\tilde{a}_m}{dv^l}.
\left. \frac{d}{dx}\left( \frac{F(x, x)}{|G|}\right)\right|_{x = 1}= 2 \sum_{t = 1}^n b_{t, 0}(q) \cdot \left. \frac{d}{dx}\frac{(x; q^{-1})_t}{(q; q)_t}\right|_{x = 1}& = 2 \sum_{t = 1}^n - \frac{(q^{-1}; q^{-1})_{t - 1}}{(q; q)_t}& = 2 \sum_{t = 1}^n \frac{(-1)^t}{q^{\binom{t}{2}}(1 - q^t)}.
{\cal M}\to e^{i \delta_{1}}{\cal M}, \quad \tilde{\cal M}\to e^{- i \delta_{1}}\tilde{\cal M}, \quad{\cal D}\to e^{i \delta_{2}}{\cal D}, \quad \tilde{\cal D}\to e^{- i \delta_{2}}\tilde{\cal D}, \quad{\cal Q}\to e^{i \delta_{3}}{\cal Q}, \quad \tilde{\cal Q}\to e^{- i \delta_{3}}\tilde{\cal Q}, \quad
B_i\cdot B_j = \sum_{k}b_{i,j}^k B_k \Leftrightarrow \Delta D_k = \sum_{i,j}b_{i,j}^k D_i \otimes D_j D_i \cdot D_j =\sum_{k}d_{i,j}^k D_k \Leftrightarrow \Delta B_k = \sum_{i,j}d_{i,j}^k B_i \otimes B_j
\sum_{\substack{n \leq x, n \in \mathcal{S}(y)}}e(n\delta) = \frac{\Psi(x,y)}{x}\sum_{n \leq x}e(n\delta) + O\left(\Psi(x,y) \frac{\log(u+1)}{\log y}\frac{\log^{3}(2+|\delta x|)}{(1+|\delta x|)^{\alpha}}\right) ,
C_{V}= \sum_{n , w}\int_{0}^{\infty}[ d t ] e^{- y}\left[ 4 t^{2}({\frac{w^{2}\beta^{2}}{4 \pi^{2}\alpha^{\prime}}}-{\frac{\alpha^{\prime}\pi^{2}n^{2}}{\beta^{2}}}) )^{2}- 2 t ({\frac{w^{2}\beta^{2}}{4 \pi^{2}\alpha^{\prime}}}+ 3{\frac{\alpha^{\prime}\pi^{2}n^{2}}{\beta^{2}}}) \right] .
\displaystyle\dfrac{S(h_u)^{-c/2}}{(c/2)!}\Big(2\sum_{u_i\in U}{{d_i}\choose{2}}\Big)^{c/2}\leq \displaystyle\sum_{\sigma\subset U \atop |\sigma|=c/2}\prod_{u_i\in \sigma}(d_i)(d_i-1)\:.
A(s)=A_{OL}\cdot\frac{1}{1+\frac{s}{\omega_{1}}}\cdot\frac{1}{1+\frac{s}{\omega_{2}}}\cdot\frac{1}{1+\frac{s}{\omega_{3}}}=A_{OL}\cdot\frac{\omega_{1}\omega_{2}\omega_{3}}{(s+\omega1)(s+\omega_{2})(s+\omega_{3})}
A(\delta+i\gamma,k,\lambda) \leq C_{a,\delta'}\int^1_0s^{\delta'}(1-s)^{\delta-\delta'-1}A(\delta',k,\lambda s)ds &\leq C_{a,\delta'}\int^1_0s^{\delta'}(1-s)^{\delta-\delta'-1}(|\lambda|s k)^{-\delta'/2-1/4}ds &=C_{a,\delta'}(|\lambda|k)^{-\delta'/2-1/4}\int^1_0s^{\delta'/2-1/4}(1-s)^{\delta-\delta'-1}ds &\leq C_{a,\delta'}(|\lambda|k)^{-\delta'/2-1/4}.
\sum_{k=1}^N \mathbb{E}_{k-1}[|X_{k,N}|^{2}] \rightarrow \sum_{l,m = 1}^M c_l \overline{c_m}C(z_l,\overline{z_m}), N \to \infty, &\sum_{k=1}^N \mathbb{E}_{k-1}[X_{k,N}^{2}] \rightarrow \sum_{l,m = 1}^M c_l c_m C(z_l,z_m), N \to \infty,
\begin{aligned}e^{-\beta t}y_t =\int_t^\infty\beta e^{-\beta s}y_sds+\int_t^\infty e^{-\beta s}g(x_s,y_s,z_s,\tilde{z}_s,\gamma_{(s,e)})ds-\int_t^\infty e^{-\beta s}z_sdW_s &\quad-\int_t^\infty e^{-\beta s}\tilde{z}_sd\xi_s-\int_t^\infty \int_{\mathcal{E}}e^{-\beta s}\gamma_{(s,e)}\tilde{N}(de,ds),\t\in[0,\infty).\end{aligned}
E[e^{2 \alpha X^n(t)}(e^{-\alpha \overline{X(t)}}-e^{-\alpha \overline{X^n(t)}})^2] \leq E[e^{4 \alpha X^n(t)}]^{\frac{1}{2}}E[(e^{\alpha \overline{X(t)}}-e^{\alpha \overline{X^n(t)}})^4]^{\frac{1}{2}}&= e^{2 \psi^n(\alpha) t}E[(e^{\alpha \overline{X(t)}}-e^{\alpha \overline{X^n(t)}})^4]^{\frac{1}{2}}
\begin{aligned}q^{7}L_{2}(q^{32}) = \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{(16n+3)^2 - 2(8j+1)^2}+ q^{(16n+13)^2 - 2(8j+1)^2}& + \sum_{\substack{n \geq 0 -n-1 \leq j \leq n}}q^{(16n+21)^2 - 2(8j+3)^2}+ q^{(16n+11)^2 - 2(8j+3)^2}.\end{aligned}
t^{\beta-1}E_{\beta, \beta}(-a|p|^{\alpha}t^{\beta})=\frac{-1}{a|p|^{\alpha}}\frac{d}{dt}E_{\beta,1}(-a|p|^{\alpha}t^{\beta}) =t^{\beta-1}\int_{0}^{\infty}x^{-1/\beta}\exp(-a|p|^{\alpha}t^{\beta}x) w(x^{-1/\beta}, \beta, 1)dx,
{{\mathcal P}}^{-1}(V)\overset{\eqref{df:cm}}{=}c_m \,\Delta V\Bigm|_{D\setminus \{x_0\}}+\left(1-\limsup\limits_{x_0\neq x \to x_0}\dfrac{V(x )}{|h_m(x-x_0)|}\right)\cdot{\delta}_{x_0}\, , V\in PJ_{x_0}(D),
\begin{array}{c c c}{\partial^{\mu}C_{n \mu}^{F}= 0 ,}{\quad \partial_{\mu}b_{n}^{F \mu}= 0 ,}{\quad \partial_{\mu}\bar{C}_{n}^{F \mu}= 0 ,}{\partial^{\mu}C_{n \mu}^{B}= 0 ,}{\quad \partial_{\mu}b_{n}^{B \mu}= 0 ,}{\quad \partial_{\mu}\bar{C}_{n}^{B \mu}= 0 .}\end{array}
q \,=\, \frac{\Lambda}{2\pi\widehat\Delta z(u,v)}\exp{\!\left[-\frac{\Lambda \left(u^2+v^2\right)}{2\widehat\Delta z(u,v)}\right]}\Biggr/\frac{\Lambda}{2\pi\Delta z_a}\exp{\!\left[-\frac{\Lambda \left(u^2+v^2\right)}{2 \Delta z_a}\right]}\,.
\lim_{\varepsilon\to{0}^{+}}\frac{f(\mathbf{x}_{0}+\varepsilon \mathbf{x})}{\varepsilon^{\alpha}L(\varepsilon)}=g(\mathbf{x}) \ \ \ \mbox{strongly in}\mathcal{K}_{M}'(\mathbb{R}^{n}) \ (\mbox{resp.}\mathcal{S}'(\mathbb{R}^{n})).
\mathfrak{M}\left[{}^{M\!L\!\!}{B}_{p}(x,y) : s\right] = \int_{0}^{1}t^{x+s-1}(1-t)^{y+s-1}dt \times \int_{0}^{\infty}u^{s-1}E_{\alpha, \beta}^{\gamma}\left(-u\right) du ={}^{M\!L\!}\Gamma(s)\int_{0}^{1}t^{x+s-1}(1-t)^{y+s-1}dt &={}^{M\!L\!}\Gamma(s)B(x+s,y+s)
S_{m m^{\prime}, n n^{\prime}}^{S R}= \frac{1}{u}( - 1 )^{G + G^{\prime}+ ( u - 1 ) ( m + n )}e^{- i \pi ( u + 1 ) m n / u}e^{- i \pi ( u - 1 ) ( m - 2 m^{\prime}+ u ) ( n - 2 n^{\prime}+ u ) / u},
\frac{\vartheta(x_1-y_1)\vartheta(x_1-y_2)\vartheta(x_1-y_3)}{\vartheta(x_1-x_2)\vartheta(x_1-x_3)}+\frac{\vartheta(x_2-y_1)\vartheta(x_2-y_2)\vartheta(x_2-y_3)}{\vartheta(x_2-x_1)\vartheta(x_2-x_3)}+\frac{\vartheta(x_3-y_1)\vartheta(x_3-y_2)\vartheta(x_3-y_3)}{\vartheta(x_3-x_1)\vartheta(x_3-x_2)}=0.
v(t,x) = \int_{\mathbb{R}}G_t (x-y) u_0(y) dy + \frac{1}{2}\int_0^t \int_{\mathbb{R}}\frac{\partial}{\partial y}G_{t-s}(x-y) v(s,y)^2 dy ds + \int_0^t \int_{\mathbb{R}}G_{t-s}(x-y) \sigma_s(y) W(ds,dy).
\hat R^{(\epsilon)}_{\epsilon^\prime}ch^{[B;M,m;\epsilon]}_{j+\epsilon^\prime,k+\epsilon^\prime,\epsilon^\prime}=\overset{\sim}{\epsilon}_{s}\Psi^{[B;M,m;\epsilon]}_{j+\epsilon^\prime,k+\epsilon^\prime;\epsilon^\prime},\,\,\,\,j,k\in\Omega^{[B;M]},
\sum_{n = N-H}^{N+H}t_H(n-N)(\psi(n)-n) = \sum_{m = 0}^{H}t_H(m)(\psi(N+m)-(N+m)) &\hskip1cm + \sum_{m = 0}^{H}t_H(m)(\psi(N-m)-(N-m)) - H (\psi(N)-N) = \sum_{m = 0}^{H}t_H(m)a(m) - H (\psi(N)-N),
D(X - \mathrm{Frob}_{w}) = X^{2n}+\dotsb+(-1)^j q_w^{j(j-1)/2}\frac{T_{G, \bar{w},{2n-j}}}{T_{G,\bar{w},2n}}+\dotsb +q^{n(2n-1)}_w T_{G,\bar{w},2n}^{-1}\,.
\leq \; \int_{[0,1]^n}\prod_{k=1}^m\left(\frac1{|s_k(t)B|^{1+\lambda_k p_k}}\int_B\left|f_k\left(y\right)\right|^{p_k}\omega_k(y)dy\right)^{1/p_k}\left(\prod_{k=1}^m|s_k(t)|^{(d+\alpha_k)\lambda_k}\right)\psi(t)dt \leq \; \prod_{k=1}^m\|f_k\|_{\dot{B}^{p_k,\lambda_k}_{\omega_k}\left(\mathbb R^d\right)}\cdot\mathcal B.
{\cal L}= \int d^{4}x e^{- 2 \alpha}\left( \partial_{\mu}\bar{\phi}\partial^{\mu}\phi + \bar{\psi}{\not \! \partial}\psi - \bar{F}F \right) + \int d^{4}x e^{- 2 \alpha}m \left( \frac{1}{2}\psi \psi + \phi F \right) + \mathrm{h . c .}
s=0,1,...,\left[n/2 \right],l=0,1,...,\left[(n-1)/2 \right], &U_k^{a_1 a_2...a_{2s}}=K^{a_1 a_2...a_{2s}b_1b_2...b_{2k-2s+1}}\partial_{b_1}\partial_{b_2}...\partial_{b_{2k-2s+1}}V, &W_k^{a_1a_2...a_{2l+1}}=K^{a_1 a_2...a_{2l+1}b_1b_2...b_{2k-2l-1}}\partial_{b_1}\partial_{b_2}...\partial_{b_{2k-2l-1}}V
\mathbb E\left(\log|W(z)|\right)= \mathbb E\left(\sum_{1}^{\infty}h_k(z)\right)=\sum_{\lambda_k<R}\log\frac R{\lambda_k}=\int_1^R \frac{{n(t)\rm d}t}{t}= \int_1^R \frac{{(t^2-{(a+o(1))t\log^{b}t^2)}\rm d}t}{t}=\frac{R^2}{2}-\int_1^R{(a+o(1))}\log^{b}t^2{\rm d}t = \frac{R^2}{2}-{(a+o(1))R\log^{b}R^2.}
\sum_{k=0}^{n-1}\frac{(q;q^2)_k(-1;q^2)_k}{(q^2;q^2)_k}q^{(n-1)^2-k^2}\equiv -q^{1-n}+\frac{(n-3)q(1-q^n)}{2}\pmod{\Phi_n(q)^2}.
R^{2}_{112}= R^{3}_{113}= \frac{1}{g^{2}}\left ( \frac{h''}{h}-\frac{h'}{h}\frac{g'}{g}\right ) \quad ; \quad R^{3}_{232}= \frac{1}{h^{2}}\left ( 1 - \left (\frac{h'}{g}\right )^{2}\right )
\rho^2 u''(\rho) -\Delta_{\rho}u(\rho) + 2(\lambda +\tfrac{p+1}{p-1}) \rho u'(\rho) & + (\lambda +\tfrac{2}{p-1})(\lambda +\tfrac{p+1}{p-1}) u(\rho) - p c_p^{p-1}u(\rho) = 0.
\Phi^{\alpha}\rightarrow V\Phi^{\alpha}=\left(\begin{array}{c}-f^{\alpha}(r)e^{i\theta}+f^{\beta}(r) f^{\alpha}(r)e^{i\theta}+f^{\beta}(r) \end{array}\right)=\left(\begin{array}{c}\phi^{1}(r,\theta) \phi^{2}(r,\theta) \end{array}\right).
E_{2}( n_{1}, \ldots , n_{m}, I ) ={\frac{( - 1 )^{n}}{n_{1}! \cdots n_{m}!}}{\frac{\partial^{n}D ( I ; \lambda )}{\partial \lambda^{n_{1}}\cdots \partial \lambda^{n_{m}}}}\Bigr \vert_{\lambda_{1}= \cdots = \lambda_{m}= 1}
d^{2p-3}(\bar{\xi}_1)\dot{=}\alpha_1, d^{2p-3}(\bar{\tau}_1)\dot{=}v_1, d^{2p-3}(\sigma \bar{\tau}_1 )\dot{=}\sigma v_1, d^{2p-3}(\sigma \bar{\xi}_1 )= \sigma \alpha_1, d^{2p-2}(\tilde{\tau}_1v_1^k)\dot{=}v_1^{k+1}, d^{2p-2}(\bar{\xi}_1v_1^k)\dot{=}\alpha_1v_1^k, d^{2p-2}(\tilde{\tau}_1\alpha_1v_1^{k-1})= \alpha_1v_1^k
W_{1}^{+}( 0 , \lambda A , N ) = N D_{1}( \lambda A , N ) = e^{- \frac{\lambda A}{2}}\sum_{h}\frac{( \lambda A )^{h}}{2^{h}\; h !}N^{2 ( 1 - h )}= N^{2}e^{- \frac{\lambda A}{2}+ \frac{\lambda A}{2 N^{2}}},
{s_{0,0}(m,n)=s_{0,0}(m+n,0)=s_{0,0}(0,m+n)}{s_{0,1}(m,n)=s_{0,1}(m+n,0)=s_{0,1}(0,m+n)}{s_{1,0}(m,n)=s_{1,0}(m+n,0)=s_{1,0}(0,m+n)}{s_{1,1}(m,n)=s_{1,1}(m+n,0)=s_{1,1}(0,m+n)}.
\begin{matrix}{L = N \sum_{k = 1}^{N}\left( ( \partial_{\mu}\phi_{k})^{2}+ V ( \phi_{k}) \right)}\\{+ \sum_{i , j = 1}^{N}( \phi_{i}- \phi_{j})^{2}| B_{\mu}^{i j}|^{2}}\\ \end{matrix}
\begin{aligned}\Delta (H+\varepsilon) = - \frac{1}{2}\langle F, \nabla(H+\varepsilon)\rangle - \left( |A|^2 - \frac{1}{2}\right)(H+\varepsilon) - H + \varepsilon\left(|A|^2 - \frac{1}{2}\right), \Delta f = - \frac{1}{2}\langle F , \nabla f\rangle - \left(|A|^2 - \lambda\right) f.\end{aligned}
\theta_+(x,y,\xi(\omega)) = \frac{1}{2}\big(x^2\delta_0(\xi(\omega)) + (1-x)^2\delta_1(\xi(\omega))\big) + f(x,y,\xi(\omega)) \theta_{-}(x,y,\xi(\omega)) = \frac{1}{2}\big(x^2\delta_0(\xi(\omega)) - (1-x)^2\delta_1(\xi(\omega))\big) - f(x,y,\xi(\omega))
Z^i_t = Z^i_0 + W^i + \Hat{Z}^i - \sum_{j: j\neq i}\int_0^t \frac{\mu^{ij}}{\mu^j}\, v_{ij}(s) \, d{\Hat{Z}^j_s}, \int_0^\cdot Z^i_s \, d{\Hat{Z}^i_s}=0,\forall \; t\geq 0,
(C+1) p_1^2 p_2^2 \left\{n^2 \sum_{\ell=1}^{2n}\int_0^{t_2}\frac{h_\ell^2 J_\ell}{Y_\ell^4}\d N_\ell + \left( n^{1/2}\sum_{\ell=1}^{2n}\int_0^{t_2}\frac{|h_\ell| J_\ell}{Y_\ell^2}\d N_\ell \right)^2 \right\}.
h_{0}(J_{n}(x))= (c_{n}^{22}x + \alpha_{n}^{22}y_{h_{0}}^{2}(x)+ \beta_{n}^{22}z_{h_{0}}^{2}(x)+ d_{n}^{22}, \gamma_{n}^{22}z_{h_{0}}^{2}(x)+ e_{n}^{22}x+ f_{n}^{22}) \; \mathrm{for}\; n=1, 2, \ldots, K^{22}.
-\sum_{i\in I}\nu_{i}(a_{i}+1) =-\sum_{i\in I}N_{i}\sigma_{i}(a_{i}+1) &=-\sum_{i\in I}N_{i}(a_{i}+1)(\sigma_{i}-c_0(f))-(m-1)c_0(f) &\leq -(m-1)c_0(f),
u \psi_X'(u) = \int_0^\infty \left( \frac{d}{dt}e^{-ut}\right) t g(t) \, dt= \int_0^\infty t g(t) d\left(e^{-ut}\right ) &= \left. t g(t) e^{-ut}\right|_{t=0}^{t=\infty}- \int_0^\infty e^{-ut}\, d(tg(t))= \int_0^\infty (1-e^{-ut}) \, d(tg(t)).
\int_{-1}^1T_{\alpha,\beta,m}[f](x)\overline{g}(x)W_{\alpha,\beta, m}(x)\,dx=[f,g](x)\big|_{-1}^1+\int_{-1}^1f(x)T_{\alpha,\beta,m}[\overline{g}](x)W_{\alpha,\beta, m}(x)\,dx
\left\Vert \operatorname{Re}M\right\Vert \geq\left\vert \zeta-\mathrm{i}\gamma\right\vert \geq\left\vert \operatorname{Im}\left( \zeta-\mathrm{i}\gamma\right) \right\vert =\left\vert \operatorname{Im}\zeta-\gamma\right\vert \geq\left\vert \gamma\right\vert -\left\vert \operatorname{Im}\zeta\right\vert .
\textstyle B_{w,0}\textstyle = \frac{(-1)^w(s-r)(r-\alpha)_w}{(s-\alpha)_{w+1}}+ O(p), \textstyle B_{w,j}\textstyle = - \delta_{w = 0}{r-\alpha+j \choose j}+ \sum_v(-1)^{w-v}{j+w-v-1\choose w-v}{r-\alpha+j\choose v}{s-\alpha+j-v \choose j-v}+ O(p).
\phi_{2}^{4}= \frac{\sqrt{g}}{\alpha}\left[ p_{1}^{2}- m^{2}+ \sqrt{g}~ \frac{\alpha m + \beta \sigma \sqrt{m^{2}- p_{1}^{2}}}{( q_{2}^{2})^{3 / 2}}\right] , \sigma = \mathrm{s i g n}~ \varepsilon ( p_{1}q_{2}q_{3}) ,
\widehat{F}-\big(1-q^{\lambda}\big)I_4 = \begin{pmatrix}B_0 -\big(1-q^{\lambda}\big) B_1 B_2 B_3 B_0 B_1-\big(1-q^{\lambda}\big) B_2 B_3 B_0 B_1 B_2-\big(1-q^{\lambda}\big) B_3 B_0 B_1 B_2 B_3-\big(1-q^{\lambda}\big) \end{pmatrix}\!,
D(B_{1}) \cap D(A_{2}) \neq \emptyset, (B_{1}w, A_{2}w)_{H}\geq 0 \mbox{for all}\w \in D(B_{1}) \cap D(A_{2}), (B_{2}w, A_{1}w)_{H}\geq 0 \mbox{for all}\w \in D(A_{1}), (B_{1}w, A_{2}z)_{H}= (B_{1}z, A_{2}w)_{H}\mbox{for all}\w, z \in D(B_{1}) \cap D(A_{2}).
{}_5F_4\bigg[\begin{matrix}\alpha \alpha 1+\frac12\alpha \beta \gamma &1 \frac12\alpha \alpha-\beta+1 \alpha-\gamma+1\end{matrix}\bigg|\,1\bigg]=\frac{\Gamma(1+\alpha-\beta)\Gamma(1+\alpha-\gamma)\Gamma(1-\beta-\gamma)}{\Gamma(1+\alpha)\Gamma(1-\beta)\Gamma(1-\gamma)\Gamma(1+\alpha-\beta-\gamma)}.
m_{j+1}= \arg\min_v I_{EnKF}(v) \hat{\mu}_{j+1}= F(m_j) I_{EnKF}(v) = \frac{1}{2}|y_{j+1}-H(v)|_{\Gamma}^2+\frac{1}{2}|v-\hat{\mu}_{j+1}|_{\hat{C}_{j+1}}^2.
a^{{\mathcal{P}}}_t=(\sum\limits_{\substack{1\leq i\leq n}}\gamma_i\int_{\{\vert z\vert\geq M(\gamma_i)\}}\underline{c}(z)\mu(dz)+(t-\Gamma_n)\int_{\{\vert z\vert\geq M(\gamma_{n+1})\}}\underline{c}(z)\mu(dz))^\frac{1}{2}.
P_0^j\,V_k\,P_0^j = O^P({T}^{-1}) \mbox{for any}j, P_0^j\,V_k\,P_0^\ell = O^P({T}^{-1}) \mbox{for any}j\neq\ell, Q_0 \,V_k = V_k\, Q_0 = 0.
{}_{pBp}L^2(pBp) \boxtimes_{Ap}L^2(pBp){}_{pBp}={}_{pBp}L^2(pB_jp) \boxtimes_{pA_i}L^2(pB_jp){}_{pBp}&\cong{}_{pBp}pL^2B \boxtimes_{B_j}L^2B_j p \boxtimes_{pA_i}pL^2B_j \boxtimes_{B_j}L^2Bp{}_{pBp}&\cong{}_{pBp}pL^2B \boxtimes_{B_j}L^2B_j p \boxtimes_{A_i}pL^2B_j \boxtimes_{B_j}L^2Bp{}_{pBp}
\min \bigg\{\bigg(\frac{\nu_\infty}{2}\bigg)^{\frac{1}{q_{\infty}+\overline{\epsilon}}}, \bigg(\frac{\nu_\infty}{2}\bigg)^{\frac{1}{q_{\infty}-\overline{\epsilon}}}\bigg\}> \bigg( \frac{\nu_{\infty}}{4}\bigg )^{\frac{1}{q_{\infty}}}.
A_{k}^{s}A_{j}^{n-s}=\mathcal{O}_{A_{j}}(a_{kj}\mathcal{H}_j)^{s}\mathcal{O}_{A_{j}}(a_{jj}\mathcal{H}_j)^{n-s-1}=a_{kj}^{s}a_{jj}^{n-s-1}l_{j},
\begin{aligned}T_{w_k}=T_1(T_2T_1)...(T_{k-2}T_{k-3}...T_1)(T_{k-1}T_{k-2}...T_1) &=(T_{1}...T_{k-2}T_{k-1})(T_{1}...T_{k-3}T_{k-2})...(T_1T_2)T_1,~~~~~~~~~~~~~~~k=2,...,m.\end{aligned}
\int_{R_2}^\infty v(\rho w) \left(\frac{d}{d\rho}\int_{R_2}^\rho f(y- \gamma w) \gamma^{d-1}~d\gamma \right) d\rho &=-\int_{R_2}^\infty (\nabla v(\rho w), w)\int_{R_2}^\rho f(y- \gamma w) \gamma^{d-1}~d\gamma d\rho.
B_{\mu\nu}(t)=\sum_{\rho=0}^N\frac{\eta_{\mu\rho}\eta_{\nu\rho}}{\sqrt{8\omega_\mu}}\left[\left(1+\frac{\omega_\mu}{\Omega_\rho}\right)\rm{e}^{i\Omega_\rho t}+\left(1-\frac{\omega_\mu}{\Omega_\rho}\right)\rm{e}^{-i\Omega_\rho t}\right]\;.
\begin{aligned}\lambda_{k_1,\dots,k_{j_\alpha}}= \frac{1}{2}\left(\lambda^+_{k_1}\dots \lambda^+_{k_{j_\alpha}}+(-1)^{j_\alpha-1}\lambda^-_{k_1}\dots \lambda^-_{k_{j_\alpha}}\right) &+i\frac{\eta}{2}\left(\lambda^+_{k_1}\dots \lambda^+_{k_{j_\alpha}}+(-1)^{j_\alpha}\lambda^-_{k_1}\dots \lambda^-_{k_{j_\alpha}}\right).\end{aligned}
\gamma_{\textrm{NHF}}^{\textrm{OFDMA}}=\frac{P_\textrm{F}{|h_{\textrm{BF}}|^2}}{{d_{\textrm{BF}}^{\alpha}\sigma_{\textrm{F}}^2}\theta}+\frac{\beta_{\textrm{N}}\eta P_\textrm{B}|h_{\textrm{BN}}|^2 |h_{\textrm{NF}}|^2}{d_{\textrm{BN}}^{\alpha}d_{\textrm{NF}}^{\alpha}\sigma_{\textrm{F}}^2 \theta}.
x_i:= x^{r_i}, \displaystyle [n]_{x}:= \frac{x^n-x^{-n}}{x-x^{-1}}\n\in \mathbb{Z}, &\left[ \begin{array}{c}n k \end{array}\right]_{x}:=\begin{cases}\displaystyle \frac{[n]_{x}[n-1]_{x}\cdots[n-k+1]_{x}}{[k]_{x}[k-1]_{x}\cdots [1]_{x}}n\in \mathbb{Z}, k\in \mathbb{Z}_{>0}, 1 n\in \mathbb{Z}, k=0,\end{cases}
{\bf T}f = \sum_{s=1}^n (-1)^{s-1}\sum_{1 \leq i_1 \leq \cdots \leq i_s \leq n}{\bf T}_{i_1}\cdots{\bf T}_{i_s}\left( \frac{\partial^{s-1}f_{i_s}}{\partial \bar z_{i_1}\cdots \partial \bar z_{i_{s-1}}}\right),
P_{a}= 2 \sqrt{\beta}\left( \Pi_{a}+ \frac{1}{4}C_{\; \cdot a}^{c c^{\prime}}M_{c c^{\prime}}\right) \; \leftrightarrow \; P_{0}= 2 \sqrt{\beta}\Pi_{0}, \; P_{j}= 2 \sqrt{\beta}\left( \Pi_{j}- \frac{1}{2}J_{j}\right) ,
\frac{\Phi^2}{\Phi-t\Phi'}\cdot f'(t) =\sum_{k\ge1}z^{k-1}[t^{k-1}] (\Phi(t))^k f'(t) &=[t^{0}]\Phi(t)f'(t)+z^{1}[t^{1}] (\Phi(t))^2 f'(t)+\sum_{k\ge1}z^{k+1}[t^{k+1}] (\Phi(t)^k) (\Phi(t))^2 f'(t).
\mbox{\Large$\chi$}({C'})=X^{n-4}(X^2-{\lambda'_1}^2)(X^2-{\lambda'_2}^2)=X^n -({\lambda'_1}^2 +{\lambda'_2}^2)X^{n-2}+{\lambda'_1}^2{\lambda'_2}^2X^{n-4}.
\left| \S_L^N\right|^2 = \prod_{j=1}^N\prod_{k=N+1}^\infty\frac{\big|\mu^L_k-\lambda^L_j\big|\big|\lambda^L_k-\mu^L_j\big|}{\big|\lambda^L_k-\lambda^L_j\big|\big|\mu^L_k-\mu^L_j\big|},
+ \left[ | x | -{\frac{4}{9 4 5}}| y |^{3}+{\frac{2}{4 5}}| x | | y |^{2}+{\frac{1}{3}}| x |^{2}| y | +{\frac{8}{1 5}}| x |^{5 / 2}| y |^{1 / 2}+{\frac{1}{3}}| x |^{3}\left( -{\frac{1 7}{6}}- \psi \left( \frac{1}{2}\right) + \ln{\frac{| x |^{2}}{| y |}}\right) \right] \cos{\frac{\bar{\theta}}{2}}
0 = \frac{1}{3}((\delta_1 + \delta_2) \oplus(\varepsilon_1 + \dots + \varepsilon_5)) = &\frac{1}{6}(\delta_1 \oplus(\varepsilon_1 + \varepsilon_2) + \delta_1 \oplus (\varepsilon_3 + \varepsilon_4) + \delta_1 \oplus \varepsilon_5 + \delta_2 \oplus(\varepsilon_1 + \varepsilon_3) + \delta_2 \oplus (\varepsilon_2 + \varepsilon_5) + \delta_2 \oplus \varepsilon_4.
\lim_{n\rightarrow \infty}\Vert (\textbf{F}|_{\Gamma^{1}})^{n}\Vert^{1/n}=\lambda_{1}\lim_{n\rightarrow \infty}\Vert (\textbf{F}|_{\Gamma^{2}})^{-n}\Vert^{1/n}=\lambda_{2};
\mathrm{E}\left\{|f[n]|^2\right\}= \frac{1}{M^2}\sum_{p_1}\sum_{l_1}\sum_{p_2}\sum_{l_2}g(nT-\tau_{l_1}+\tau_{p_1})g(nT-\tau_{l_2}+\tau_{p_2})\cdot &\sum_{m_1=1}^{M}\sum_{m_2=1}^M\mathrm{E}\{\alpha_{m_1}[l_1]\alpha_{m_1}^*[p_1]\alpha_{m_2}^*[l_2]\alpha_{m_2}[p_2]\}.