harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\nu_{f_{m}^{r}\rightarrow x_{m}^{r}}(s_{m}^{r})= \underset{s_{m}^{r}}{\sum}\nu_{\eta_{m}^{r}\rightarrow s_{m}^{r}}(s_{m}^{r})\times f_{m}^{r}(x_{m}^{r},s_{m}^{r}) = \pi_{s^{r},m}^{out}\mathcal{CN}(x_{m}^{r};0,(\sigma_{m}^{r})^{2})+(1-\pi_{s^{r},m}^{out})\delta(x_{m}^{r}).
c^{2}=\frac{2\sigma^{2}}{\chi^{2}+\left(\rho-\tau\right)\chi}=\frac{1}{2}\frac{\chi^{2}-\left(\rho-\tau\right)^{2}}{\chi^{2}+\left(\rho-\tau\right)\chi}=\frac{1}{2}\left(1-\frac{\rho-\tau}{\chi}\right).
\phi ( r )^{2}\frac{d}{d r}\left( \frac{u ( r )}{\phi ( r )}\right) = \phi ( r ) u^{\prime}( r ) - u ( r ) \phi^{\prime}( r ) = \int_{0}^{r}V ( r^{\prime}) \phi ( r^{\prime}) u ( r^{\prime}) d r^{\prime}.
F_{- \gamma}( p ) F_{\gamma}( p ) = \left( \frac{\Gamma ( 1 - \alpha - i \frac{\gamma p}{2 \pi}) \Gamma ( 1 - \alpha + i \frac{\gamma p}{2 \pi})}{\Gamma ( 1 + \alpha - i \frac{\gamma p}{2 \pi}) \Gamma ( 1 + \alpha + i \frac{\gamma p}{2 \pi})}\right)^{\frac{1}{2}}.
E\Big[ \prod_{i=1}^p |S_{s_i}-\tilde{S}_{t_i}|_+^{-2}\Big] \le C^p \prod_{i=1}^p |s_i-s_{i-1}|_+^{-1/2}E \Big[ \prod_{i=1}^p |\tilde{S}_{t_i}-\tilde{S}_{t_{i-1}}|_+^{-1}\Big] \,.
S_{0}= \frac{Kr}{\delta}, \alpha^{+}= \frac{1}{2\sigma^{2}}\left[ \sigma^{2}-2(r-\delta) + \sqrt{4 \delta^{2}- 8 \delta r + 4 \delta \sigma^{2}+ 4 r^{2}+ 4\sigma^{2}r + \sigma^{4}}\right].
X_{\mathrm{fml}}(\{c,c_nc\},\iota; D \coprod \nu D,\mathfrak a_c) =X_{\mathrm{fml}}(\{c,c_nc\},\iota; D,\mathfrak a_c)+X_{\mathrm{fml}}(\{c,c_nc\},\iota; \nu D,\mathfrak a_c)
\mathrm{gr}^\heartsuit(H')=\oplus_{n\geq0}F_{n+1}^\heartsuit H'/F_n^\heartsuit H'=\oplus_{n\geq0}(F^{n+1}H)^\perp/(F^nH)^\perp=\oplus_{n\geq0}(F^nH/F^{n+1}H)^* =\oplus_{n\geq0}\mathrm{gr}^n(H)^*=(\hat{\mathrm{gr}}H)',
X_{ki}=\alpha{\displaystyle \sum_{j=1}^{L}\mathbb{E}\left[\mathsf{\bar{L}}_{ji}\right]\left(1+\frac{\sigma^{2}}{P_u}+\sum_{j\neq i}^{L}\bar{\ell}_{kji}\right)+\sum_{j\neq i}^{L}\bar{\ell}_{kji}^{2}}-\frac{1}{\gamma_{i}^{\mathsf{th}}}.
\tilde{V}_{-}= \bigoplus_{(i,j) \in \tilde{J}}V_{1,-}(i,j) \oplus V_{2,-}(i,j), \tilde{V}_{-}' = \bigoplus_{(i,j) \in \tilde{J}'}V_{1,-}(i,j) \oplus V_{2,-}(i,j), \tilde{\tilde{V}}_- = \bigoplus_{(i,j) \in \tilde{\tilde{J}}}V_{1,-}(i,j) \oplus V_{2,-}(i,j).
2\pi iH_{2}^{\prime}(a) =2\pi G_{x_{2}x_{2}}(a)+2\pi G_{x_{2}x_{2}}(b_{2}(a))b_{2}^{\prime}(a) & =\frac{2\pi}{\operatorname{Im}\tau}-\eta_{1}-\wp(a)+\left( \frac{2\pi}{\operatorname{Im}\tau}-\eta_{1}-\wp(b_{2}(a))\right) b_{2}^{\prime}(a).
(e^{-i\theta}-1)\Big(4\sin^2\frac{\theta}{2}\Big)^{-\alpha}=(e^{-i\theta}-1)\Big(\Big|2\sin \frac{\theta}{2}\Big|^2 \Big)^{-\alpha}= e^{-i\theta/2}\frac{(e^{-i\theta/2}-e^{i\theta/2})}{2^{2\alpha}|\sin\theta/2|^{2\alpha}}&= e^{-i\theta/2}\frac{-2i\, \sin\theta/2}{2^{2\alpha}|\sin\theta/2|^{2\alpha}}.
q^{\frac{( 1 - c_{j}) t_{j}- ( 1 + c_{i}) t_{i}}{2}}\tilde{\lambda}_{j}\cdot \sigma_{q}^{x}\gamma_{i}+ q^{2 H + \frac{( c_{j}- 1 ) t_{j}+ ( 1 + c_{i}) t_{i}}{2}}\,
F \colon \Omega^1(W)^I \times \Gamma(\mathbb{S}^+)^I\to &\Omega^+(W)^I \times \Gamma(\mathbb{S}^-)^I \times \Omega^0(W)^I \times V(-Y_0)^{0}_{-\infty}\times V(Y_1)^{0}_{-\infty}\times U_0 \times U_1
\mathfrak{m}_1=\begin{pmatrix}i 0 0 i \end{pmatrix}, \mathfrak{m}_2=\begin{pmatrix}i f/{\sqrt{2}}-1 1 -i f/{\sqrt{2}}\end{pmatrix}\mathfrak{m}_3=\begin{pmatrix}-j -j \frac{i}{\sqrt{2}}-j \frac{i}{\sqrt{2}}j f \end{pmatrix}, \mathfrak{m}_4=\begin{pmatrix}-j i j \frac{1}{\sqrt{2}}j \frac{1}{\sqrt{2}}j i f \end{pmatrix}.
\frac{1}{2}\int_{t_{0}}^{t_{1}}(a+bt)^{2}dt=\frac{a^{2}(t_{1}-t_{0})}{2}+\frac{ab(t_{1}^{2}-t_{0}^{2})}{2}+\frac{b^{2}(t_{1}^{3}-t_{0}^{3})}{6}
\nabla f(x^k) + \sum\limits_{i\in I_g(x^k)}\lambda_i^k\nabla g_i(x^k)+\sum\limits_{i=1}^p\mu_i^k\nabla h_i(x^k)+\sum\limits_{i\in I_0(x^k)}\gamma_i^ke_i=0, &\lambda_i^k\geq0,\lambda_i^kg_i(x^k)=0,\quad\forall i=1,\dots,m, &\gamma_i^k=0,\quad\forall i\in I_{\pm0}(x^k,y^k).
\|x^kh\| \geq \|x^k(h-1/m)_+\| - 1/m \geq \|\rho|_{A_m}(x^k(h-1/m)_+)\| - 1/m & \ge \|(1_d\otimes e_{1,k+1}) ((\pi|_{A_m}((h-1/m)_+))^{\oplus k+1})\| - \epsilon -1/m & = \|\pi|_{A_m}((h-1/m)_+)\| -\epsilon - 1/m & = (1-1/m) - \epsilon - 1/m > 1- 2\epsilon.
U^n_0Str\left(\ldots \underbrace{\phi^{i_{2k}'}\phi^{i_{2k-1}'}}\ldots \right)=\frac{n}{n+2k}(U^n_0-U^n_0|_{i_n\leftrightarrow i_1'}-\ldots -U^n_0|_{i_n\leftrightarrow i_{2k}'})Str\left(\ldots \underbrace{\phi^{i_{2k}'}\phi^{i_{2k-1}'}}\ldots \right).
[a_1,a_1^{\dagger}]=[2\gamma\omega_1(\omega_1^2-\omega_2^2)]^{-1},~~[a_2,a_2^{\dagger}]=[2\gamma\omega_2(\omega_2^2-\omega_1^2)]^{-1},~~[a_1,a_2^{\dagger}]=0,~~[a_1,a_2]=0.
f_{\mathcal{R}_{\xi, II}}(x,t):=\frac{3}{4}D_{\infty}^{-2}\left(\frac{\tilde{T}_{\eta,1}^{(12)}}{D_0^{-2}r^{-1}(\eta)-1}-\frac{\tilde{T}_{-\eta,1}^{(12)}}{D_0^{-2}r^{-1}(-\eta)-1}\right).
T_{\lambda}({\mathcal F}_{V}(x))=T_{\lambda}({\mathcal C}(\rho^{-1}(x),V)) =T_{\lambda}({\mathcal C}(Itx_0,V)){}\stackrel{(\ref{hebu})}{\subset}{\mathcal C}(ItK(-\lambda)x_0,V)){}\stackrel{(\ref{jaeinf})}{=}{\mathcal F}_{V}({\mathcal B}_{{\lambda}}(x))
\Omega ( \phi ) = \frac{e^{- \phi}}{2 \sqrt{\kappa}}\sqrt{\frac{e^{- 2 \phi}}{\kappa}- 1}- \frac{1}{2}l n \left( \frac{e^{- \phi}}{\sqrt{\kappa}}+ \sqrt{\frac{e^{- 2 \phi}}{\kappa}- 1}\right)
H_X =\lim_{y\uparrow 1}H_X(R_Xy) &=\lim_{y\uparrow 1}\left\{\,\log Z_X(R_Xy)-P_X(R_Xy)\,\right\}&=\lim_{y\uparrow 1}\left\{\,\left(\, \log Z_X(R_Xy)+\log(-\log y) \,\right )-\left(\,P_X(R_Xy)+\log(-\log y) \,\right )\,\right\}&=\gamma-B_X+\log C_X.
W(\rho,\psi) =W(\rm{Ind}_{E/F}(\chi_E),\psi)=\lambda_{E/F}(\psi)\cdot W(\chi_E,\psi\circ\rm{Tr}_{E/F}) =\lambda_{E/F}(\psi)\cdot q_E^{-\frac{1}{2}}\chi_E(c_E)\sum_{x\in U_E/U_E^1}\chi_E^{-1}(x)(c_E^{-1}\psi\circ\rm{Tr}_{E/F})(x),
\Pi^{i j}( x ) = a \left( \begin{matrix}{{0}}&{{- x_{1}x_{3}}}&{{- x_{2}x_{3}}}&{{x_{1}^{2}+ x_{2}^{2}}}\\{{x_{1}x_{3}}}&{{0}}&{{0}}&{{- x_{0}x_{1}}}\\{{x_{2}x_{3}}}&{{0}}&{{0}}&{{- x_{0}x_{2}}}\\{{- ( x_{1}^{2}+ x_{2}^{2})}}&{{x_{0}x_{1}}}&{{x_{0}x_{2}}}&{{0}}\\ \end{matrix}\right)
\begin{aligned}L^{\rm resc,h}_n(u)-L^{\rm resc,h}_n(v) \leq B^{\rho_+}_n(u)-B^{\rho_+}_n(v) + (u^2-v^2) &+ \left(\frac{\beta_1}{1-\rho_+}-2(1+\gamma)^{5/3}\gamma^{1/3}\right)\frac{(u-v)}{\beta_2}n^{1/3}.\end{aligned}
\begin{array}{ll}T_{10}(sh_1)-T_{10}=s_{13}+s_{23}+1, begin{align*}2mm]T_{10}(sh_2)-T_{10}=0, begin{align*}2mm]T_{10}(sh_3)-T_{10}=20-s_{11}^2/3-2s_{11}s_{13}/3+s_{12}^2/3-s_{13}^2/3 &-2s_{11}+7s_{13}+s_{21}-s_{22}+2s_{23}.\end{array}
E\left[\int \ln(p^0(t,q_t,z))p^0(t,q_t,z)dz\right] = E\left[\int( \frac{n}{2}\ln(\beta(t,q_t))-\beta(t,q_t)\|z\|^2/2 )p^0(t,q_t,z)dz\right] = E\left[ \frac{n}{2}\ln(\beta(t,q_t))-\beta(t,q_t)\beta^{-1}n/2\right] = \frac{n}{2}E\left[ \ln(\beta(t,q_t))\right]
\pounds_1=\partial_\mu \tilde{\partial}_{+}\bar{\varphi}\,\,\partial^\mu \tilde{\partial}_{-}\varphi +\bar{\chi}_{\dot{\alpha}}\bar{\sigma}_\mu^{\dot{\alpha}\beta}{}\partial_\mu \tilde{\partial}_{-}\chi_\beta -s\tilde{\partial}_{+}\bar{f}\,\tilde{\partial}_{-}f
\frac{0 . 0 0 4 x \times 0 . 0 9}{0 . 0 5}- \frac{0 . 1 x - 0 . 5}{0 . 2}= \frac{0 . 0 0 0 3 + 0 . 0 0 0 2 x}{0 . 0 0 0 3}
\bigl[\eta_k](f)=\int_\Omega \eta_k(f(t))\,d \mu(t)= \int_\Omega \sum_{r=1}^\infty x_{rk}^*(f(t))\chi_{B_r}(t)\, d \mu(t)= \sum_{r=1}^\infty \int_\Omega x_{rk}^*(f(t))\chi_{B_r}(t)\, d \mu(t),
C \frac{1}{\,\lambda^{\frac{1}{4r}}}\Big(\bar{C}_2\| \bar{u}^d \|_{\dot{B}_{p,r}^{-1+\frac{d}{p}}}+ \bar{C}_3\Big)^{\frac{1}{2}}=\frac{1}{4}( \lambda = (4C)^{4r}(\bar{C}_2\| \bar{u}^d \|_{\dot{B}_{p,r}^{\frac{d}{p}-1}}+ \bar{C}_3)^{2r}\,),
c ={( 4 \pi )}^{4 / 3}\int_{0}^{\infty}d u \sum_{l = 0}^{\infty}\Big ( \frac{u^{3 l}{( - 1 )}^{l}l !}{3 2 \pi^{2}( 4 l + 2 ) !}- \frac{u^{3 l + 3 / 2}{( - 1 )}^{l}( 2 l + 1 ) ! !}{3 2 \pi^{3 / 2}2^{l}( 4 l + 4 ) !}\Big )
= \left[ \frac{N^{2}+ 2 2 4 N + 5 3 4 4}{3 6 0 0}- \frac{( N - 1 ) ( N + 4 2 )}{1 8 ( N + 8 2 )}\left( \frac{B}{\phi B^{\prime}}\right) + \frac{( N - 1 ) ( N + 4 2 )^{2}}{1 8 ( N + 8 2 )^{2}}\left( \frac{B}{\phi B^{\prime}}\right)^{2}\right.
\begin{array}{rcl}\int_x^{\infty}t^ae^{-ct^{\beta}}dt \le (c - (1+a-\beta)x^{-\beta})^{-1}\int_x^{\infty}(c - (1+a-\beta)t^{-\beta})t^ae^{-ct^{\beta}}dt & = (c-(1+a-\beta)x^{-\beta})^{-1}x^{1+a-\beta}e^{-cx^{\beta}}.\end{array}
\pi =\{(B_{11},\cdots,B_{1r}),(B_{21},\cdots,B_{2r}),\cdots,(B_{k1},\cdots, B_{kr})\}\sigma =\{(A_{11},\cdots,A_{1r}),(A_{21},\cdots,A_{2r}),\cdots,(A_{\ell 1},\cdots, A_{\ell r})\}
\Gamma_{N P}^{( 1 )}\, = \, \frac{g^{2}}{4}\, \frac{1}{1 6 \pi^{2}}\int_{0}^{1}d x \int \frac{d \alpha}{\alpha}e^{i \alpha [ x ( 1 - x ) p^{2}- m^{2}] + i \frac{p \circ p}{4 \alpha}- \alpha \epsilon}\Phi ( - p , \theta ) \bar{\Phi}( p , \theta )
\frac{1}{q}\int_\Omega |\nabla f(S)|^q dx\Big\vert_{t=0}^{t=t_1}+ \int_0^{t_1}\int_\Omega |f'(S)|^{q-2}f'(S)P_c'(S) |\nabla S|^q dx dt &= -\int_0^{t_1}\int_\Omega |\nabla f(S)|^{q-2}\nabla f(S)\cdot\nabla p dx dt .
Q_{f}= f_{\mu_{1}\dots \mu_{r}}\Pi^{\mu_{1}}\psi^{\mu_{2}}\dots \psi^{\mu_{r}}+ \frac{i}{r + 1}( - 1 )^{r + 1}f_{[ \mu_{1}\dots \mu_{r}; \mu_{r + 1}]}\psi^{\mu_{1}}\dots \psi^{\mu_{r + 1}}.
\{\Theta_{\alpha}^{\mu}, \Theta_{\beta}^{\nu}\}\equiv \Delta_{\alpha \beta}^{\mu \nu}~ ~ , ~ ~ \alpha , \beta = 1 , 2 ~ ~ , ~ ~ \Delta_{\alpha \beta}^{\mu \nu}\Delta_{\nu \lambda}^{\beta \gamma}= \delta_{\alpha}^{\gamma}\delta_{\lambda}^{\mu}~ .
F^{(n)}= dC^{(n-1)}= -(-1)^{\frac{n(n+1)}{2}}d\hat{\star}C^{(11-n)}= (-1)^{\frac{n(n-1)}{2}}\hat{\star}d C^{(9-n)}= (-1)^{\frac{n(n-1)}{2}}\hat{\star}F^{(10-n)},
L_{0}= \mathrm{D i a g o n a l}( 0 , 2 , 3 , 4 , 4 , 5 , 5 , 6 , 6 , 6 , 6 , 7 , 7 , 7 , 7 , 8 , 8 , 8 , 8 , 8 , 8 , 8 , \ldots )
Q_tf(x) = \int f(\sqrt{t}y) e^{\langle x,y\rangle/\sqrt{t}-\|x\|^2/2t}d\gamma_n(y) =e^{-\|x\|^2/2t}\sum_{n=0}^\infty\frac{t^{-n/2}}{n!}\int f(\sqrt{t}y) \langle x,y\rangle^nd\gamma_n(y)
\begin{aligned}|\psi_t|h^3\omega =\psi^{1/2}\omega\frac{h^3|\psi_t|}{\psi^{1/2}}&\leq\frac 35\left(\psi^{1/2}\omega\right)^2+c\left(\frac{h^3|\psi_t|}{\psi^{1/2}}\right)^2 &\leq\frac 35\psi\omega^2+\frac{cD^2}{(\tau-t_0+T)^2}.\end{aligned}
\begin{aligned}\sum_{i\in I}2d_i+\frac{2\sum_{i\in I(\Gamma)}a_i+(k_1(\Gamma)+k_{12}(\Gamma)-1)(r-2)}{r}+\frac{2\sum_{i\in I(\Gamma)}b_i+(k_2(\Gamma)+k_{12}(\Gamma)-1)(s-2)}{s}&=2|I|+k_1(\Gamma)+k_2(\Gamma)+k_{12}(\Gamma)-3.\end{aligned}
x_1=2 y_4\;,\;\;x_2=y_2\;,\;\;x_3= y_3\;,\;\;x_4=\frac{4y_1-e^{2y_4}(4y_1+y_2^2+4y_2 y_3-y_3^2)+4y_2y_3}{2(e^{2y_4}-1)},
S_{i n t}= - \int d^{2}x ~ d^{2}y ~ [ J_{\mu}^{a}( x ) V_{( \mu )}^{a b}( x , y ) J_{\mu}^{b}( y ) + J_{\mu}^{a}( x ) U_{( \mu )}^{a b}( x , y ) S_{\mu}^{b}( y ) ] ,
\begin{array}{l}(x \mid [a_{(m+2)},b_{(-n)}]y) = (x \mid (a_{(0)}b)_{(m-n+2)}y)+(m+2)(x \mid (a_{(1)}b)_{(m-n+1)}y) \medskip = (x \mid [a_{(1)},b_{(m-n+1)}]y)+(m+1)(x \mid (a_{(1)}b)_{(m-n+1)}y) \medskip = (a_{(1)}x \mid b_{(m-n+1)}y) -(x\mid b_{(m-n+1)}a_{(1)}y)+(m+1)(x \mid (a_{(1)}b)_{(m-n+1)}y)\end{array}
\ell' = b_1c_1+b_1c_2+b_2c_2 = b_1c+b_2c_2 = b_1c_1+bc_2, m' = a_1c_1+a_2c_1+a_2c_2 = ac_1+a_2c_2 = a_1c_1+a_2c, n' = a_1b_1+a_1b_2+a_2b_2 = a_1b+a_2b_2 = a_1b_1+ab_2,
\gamma= \dfrac{8(1+a)(\lambda_0+a\lambda_1)a^3}{(c^2-1)(1+c+2a)(1-c+2a)}\, , Sp = -\dfrac{2a(1+3a-c^2+4a^2-3ac^2+4a^3)\lambda_0+(2a+4a^2+2ac^2)\lambda_1}{(c^2-1)(1+c+2a)(1-c+2a)}\: .
i \partial_{\overline{z}}\left( \Psi_{\lambda}\varphi \right)_{\pm}= 2 \partial_{\overline{z}}\partial_{z}\left( SL(\lambda) \varphi \right)_{\pm}= \frac{1}{2}\Delta \left( SL(\lambda) \varphi \right)_{\pm}= - \frac{\lambda}{2}\left( SL(\lambda) \varphi \right)_{\pm}.
\left\Vert \Phi \right\Vert_{\mathcal{W}}:=\underset{x,y\in \mathfrak{L}}{\sup}\sum\limits_{\Lambda \in \mathcal{P}_{f}(\mathfrak{L}),\;\Lambda\supset \{x,y\}}\frac{\Vert \Phi_{\Lambda}\Vert_{\mathcal{U}}}{\mathbf{F}\left( \left\vert x-y\right\vert \right)}\ .
\sum_{\substack{n\leq x n=a \mod q}}\alpha_{2}(n)\sim\frac{\zeta(3/2)}{\zeta(3)}\frac{A_{a,q}}{q}x^{1/2}+\frac{\zeta(2/3)}{\zeta(2)}\frac{B_{a,q}}{q}x^{1/3}
\prod_{j=1}^{n}(d_jr + \frac{q(d-d_j)}{d}+ \frac{d_jt}{d})= \sum_{j=0}^{n}\frac{q^j}{d^j}W^{(n)}_jr^{n-j}+\sum_{c=0}^{n-1}g^{(n)}_c(q)r^{c}
u_{\tau}- ( \varepsilon{\tilde{w}}+{\tilde{u}}) u_{\tilde{w}}-{\tilde{w}}u_{{\tilde{u}}}= 0 , \quad w_{\tau}- ( \varepsilon{\tilde{w}}+{\tilde{u}}) w_{\tilde{w}}-{\tilde{w}}w_{{\tilde{u}}}= 0 ,
K(f,t)\simeq\frac 1{(4\pi )^{\frac{m-1}2}}\int_{\partial{\cal M}}dz f(z) \sum\limits_l S^l(z)L_{aa}(z) t^{\frac 12 (l-m+2)}\frac{l}{4\Gamma \left( \frac{l+3}2 \right)}
f(H)-f(H_0) =R(-E)^k g(H)R(-E)^k -R_0(-E)^k g(H_0)R_0(-E)^k & =\left(R(-E)^k-R_0(-E)^k\right)g(H)R(-E)^k & +R_0(-E)^k g(H)\left( R(-E)^k-R_0(-E)^k\right) & +R_0(-E)^k\left(g(H)-g(H_0)\right)R_0(-E)^k \,.
\Omega = \int \! dx\, K_{i\bar j}\left(\delta\phi_R^i\delta(\bar\phi_R^{\bar j}+2a^{\bar j}) + \delta\bar\phi_R^{\bar j}\delta(\phi_R^i +2a^i) + \delta\xi^i_+ \delta\xi_+^{\bar j}- \delta\xi_-^i\delta\xi^{\bar j}_- \right).
\mathrm{arg}\min_{\phi_1}e^{-\phi_1\beta z+\frac{(\phi_1\sigma)^2}{2}}= \mathrm{arg}\min_{\phi_1}e^{\frac{1}{2}\left(\phi_1\sigma-\frac{\beta z}{\sigma}\right)^2-\frac{\beta^2 z^2}{2\sigma^2}}=\frac{\beta z}{\sigma^2}=\frac{\beta z}{\sigma^2}\theta_1^1=\hat{\phi}_1^1(z),
\frac{de}{d\omega}=\frac{\hbar}{\pi^{2}c^{3}}\Big[\frac{\omega^{3}}{2}+\gamma \omega_{c}^{3}x^{2}\sum_{n=0}^{\infty}\frac{\beta^{2n}}{2n+1}+\sum_{k=0}^{n}(-1)^{k}\frac{n-k-x}{k!(2n-k)}\Theta [n-k-x]\Big]
Y^*=\Big( \frac{(q^n-1)(q^{n-1}-1)}{q-1}, \frac{(q^{2n-1}-1)}{2}+\frac{(q^{n/2}(q-1)-1) (q^{n-1}-1)}{2(q-1)}, &\frac{(q^{2n-1}-1)}{2}+\frac{(-q^{n/2}(q-1)-1) (q^{n-1}-1)}{2(q-1)}\Big)
\stackrel{( 2 )}{S}= \left( \stackrel{( - 1 , - 1 )}{\eta}_{\alpha}^{* ( 2 1 )}- \stackrel{( - 1 , - 1 )}{\eta}_{\alpha}^{* ( 1 2 )}+ \stackrel{( - 1 , - 1 )}{\bar{\Phi}}_{i}R_{\; \; \alpha}^{i}\right) \stackrel{( 1 , 1 )}{\pi^{\alpha}}+ \mathrm{` ` m o r e "}
SNR_{1,k}= \frac{|\mathbf{v}_{1,k}^H\mathbf{h}_{1,1k}|^2 \alpha_{1,k}^2}{|\mathbf{v}_{1,k}^H\mathbf{h}_{1,1k}|^2 \alpha_{1,n}^2 + |\mathbf{v}_{1,k}|^2 \frac{1}{\rho}},
\mathcal{R}_2(m):=\int_0^L \int_0^L \left[r^2+\bigg(\frac{r_1}{\sqrt{m}}\bigg)^2+\bigg(\frac{r_2}{\sqrt{m}}\bigg)^2+\bigg(\frac{r_{12}}{m}\bigg)^2\right]dt_1dt_2.
|F_m-f_m|_{S_{I,m}(\delta)\setminus S_{I,m}(\delta')}=|(\frac{U_p^{sp}}{a_p})^m f_m|_{S_{I,m}(\delta)\setminus S_{I,m}(\delta')}&\leq p^{-m \mu}|f_m|_{S_{I,0}(\delta)\setminus S_{I,0}(\delta')}&=p^{-m \mu}|f_0|_{S_{I,0}(\delta)\setminus S_{I,0}(\delta')}&\to 0 .
\sum_{n=0}^{\infty}q^n \left(q-q^3-q^n x\right)(-q^{-3}x;q)_n Ai_q(q^n x) &= \frac{x}{1+q}\,_1\phi_1 (0;-q^2;q,-x)-\left(q(1+q)+\frac{x}{q}\right) Ai_q(\frac{x}{q}).
D_{ij}D_{kl}=q^{-2}D_{kl}D_{ij}, 1\leq i<j<k<l\leq 6, D_{ij}D_{kl}=q^{-2}D_{kl}D_{ij}-(q^{-1}-q)D_{ik}D_{jl}1\leq i<k<j<l\leq 6, D_{ij}D_{kl}=D_{kl}D_{ij}1\leq i<k<l<j\leq 6,
\big\lvert\mathbb{E}\left[ f(\underbar{X}) \big | g(\underbar{X}) \right]\big\rvert = \prod\limits_{i=1}^n \big\lvert\mathbb{E}\left[ f_i(X_i) \big | g(\underbar{X}) \right]\big\rvert & > \prod\limits_{i \in [n]}\big\lvert\mathbb{E}\left[ f_i(X_i) \right]\big\rvert = \big\lvert\mathbb{E}\left[f(\underbar{X})\right]\big\rvert.
\|\Psi_{n}^{j}\|^{2}=\|\Psi_{n}^{j-1}\|^{2}-\|u^{j-1}\|^{2}+o(1)=\|u_n\|^{2}-\|u^0\|^{2}-\displaystyle\sum_{i=1}^{j-1}\|u^{i}\|^{2}+o(1),
\dot{\omega}_{I}= \sigma_{I}\frac{d}{d t}\sum_{p_{I}}A r g ( \vec{x}- \vec{R}_{p_{I}}) = \sigma_{I}\sum_{p_{I}}\varepsilon_{i j}\dot{R}_{p_{I}}^{i}\partial_{j}\ln \mid \vec{x}- \vec{R}_{p_{I}}\mid
\begin{aligned}\Delta r_{O1}=\max_\mathcal{P}\sum_{i=1}^K\frac{1}{K}\ln \Big(1+\frac{KP_i|\bar{h_i}|^2}{N_0}\Big) \mathrm{s.t.}\sum_{i=1}^KP_i\leq P-P_{O1}^* .\end{aligned}
R_{1}=max_{0\le\beta\le1}min\{\frac{1}{2}log(1+(1-\beta)c_{21}^{2}P_{1}),\frac{1}{2}log(1+c_{31}^{2}P_{1}+c_{32}^{2}P_{2}+2\sqrt{\beta c_{31}^{2}c_{32}^{2}P_{1}P_{2}})\}
u(t) =e^{-A(t-t_0)}u_0-\chi\int_{t_0}^t e^{-(t-s)A}\nabla(u (s)\cdot\nabla v(s))ds &\,\,+\int_{t_0}^t e^{-A(t-s)}u(s)\Big[\underbrace{1+a_0(s,\cdot)-a_1(s,\cdot)u(s)-(a_2(s,\cdot))_+\int_{\Omega}u(s) +(a_2(s,\cdot))_-\int_{\Omega}u(s)}_{I_0(\cdot,s)}\Big] ds,
G_n(k^*)=\frac{\left|\sum\limits_{i=1}^{k^*}\sum\limits_{j=k^*+1}^n\left(1_{\left\{X_i\leq X_j\right\}}-\frac{1}{2}\right)\right|}{\left\{\frac{1}{n}\sum\limits_{t=1}^{k^*}S_t^2(1, k^*)+\frac{1}{n}\sum\limits_{t=k^*+1}^nS_t^2(k^*+1, n)\right\}^{\frac{1}{2}}}
\sideset{}{^*}\sum_j h(t_j) \omega_j \lambda_j(m) \lambda_j(n) = \frac{1}{2}\delta(m, n) H + \sum_{c > 0}\frac{1}{2 c}\left ( S(m, n; c) H^+\Bigl ( \frac{4 \pi \sqrt{m n}}{c}\Bigr ) - S(-m, n; c) H^- \Bigl ( \frac{4 \pi \sqrt{m n}}{c}\Bigr ) \right ),
\mathbb{P}(\mathbb{M}_n(\theta_0) < \mathbb{M}_n(\theta_0 - \varepsilon)) \ge \mathbb{P}\left(\sum_{i=1}^n \left\{\log\frac{p_{\theta_0 - \varepsilon}(X_i)}{p_{\theta_0}(X_i)}- \mathbb{E}\left[\log\frac{p_{\theta_0 - \varepsilon}(X_i)}{p_{\theta_0}(X_i)}\right]\right\}\le 0\right).
\mathfrak{F}(v) = \mathcal{E}[v, \cdots, v]+ \mathcal{HL}_B[T[W_t u_0, v, \cdots, v], v, \cdots, v] \qquad- \frac{p-1}{2}T[W_tu_0, v, \left(\mathcal{R}^2+\mathcal{NR}\right)[v, \cdots, v],v, \cdots, v] &\qquad+ \frac{p-1}{2}T[ W_tu_0, \left(\mathcal{R}^2+\mathcal{NR}\right)[v, \cdots, v], v, \cdots, v],
<A_1|r>=\psi_r(A_1)=\frac{2\pi}{k}\,C_r\,\sum_{n=-\infty}^{+\infty}\,\delta\left(A_1-\frac{2\pi r}{k}-2\pi n\right)\ \ \ ,\ \ \r=0,1,2,\dots,k-1\ \ \ ,
A(X_i/\pi) = \begin{bmatrix}a_i k_i -a_i \frac{(k_i-a_i)|\mathcal{S}|}{|\Omega|-|\mathcal{S}|}\frac{d|\Omega|+a_i|\mathcal{S}| - 2k_i |\mathcal{S}|}{|\Omega|-|\mathcal{S}|}\end{bmatrix}.
\left. \mathrm{D i s c}\left( \int d^{4}x{\frac{e^{i p \cdot x}}{| x |^{2 m + 4}}}\right) \right|_{- p^{2}= \omega^{2}- i \epsilon}^{- p^{2}\omega^{2}+ i \epsilon}={\frac{2 \pi^{3}i \omega^{2 m}}{4^{m}m ! ( m + 1 ) !}}
\iint_{Q_t}\rho\kappa\theta_t^2 dxdt+\int_\Omega \kappa^2\theta_x^2dx & \leq C-2\iint_{Q_t}\Big[\kappa\kappa_\rho\rho_x\theta_x\theta_t +\kappa\kappa_\rho\theta_x^2(\rho_xu+\rho u_x)-f\kappa\theta_t\Big]dxdt.
\iint_Q \hat{y}b dxdt + \sum_{i=1}^2 \iint_Q \hat{p}^i b_i dxdt = (y_0,u(0))+ \iint_{\mathcal{O}\times (0,T)}\hat{y}u\dxdt+ \iint_Q (G u +{G}_1 z^1 +{G}_{2}z^2 )dxdt
\left( \begin{array}{cc}d_j 0 0 1 \end{array}\right) \left( \begin{array}{cc}1 0 d_i 1 \end{array}\right)= \gamma\left( \begin{array}{cc}1 \mu_j 0 d_j/\gcd(d_j,d_i) \end{array}\right) \left( \begin{array}{cc}\gcd(d_j,d_i) 0 0 1 \end{array}\right)
|I'[\tilde{u}_n]\varphi| =|I'[u_n](\varphi(\cdot-y_n))| &\leq \|I'[u_n]\|_{W^{-s,p'}}\|\varphi(\cdot-y_n)\|_{W^{s,p}}=o(1)\cdot \|\varphi\|_{W^{s,p}},
F_{A^I,B^I,C_\Lambda^A}=\left( \begin{array}{cc}0 I \end{array}\right)F_{A^{I_{X\times U}},\tilde{B}^{I_{X\times U}},\tilde{C}^{I_{X\times U}}}\left( \begin{array}{c}I 0 \end{array}\right) =F_{A,B,C}(I-F_{A,B,P})^{-1}.
\begin{aligned}\widetilde{F_\pm}= \sqrt{(1\pm\Delta\alpha)^2 S_a^2 + 2(1\pm\Delta\alpha)S_a L_a \cos(2\pi\phi) + L_a^2}\Delta\psi \equiv \psi_+ - \psi_- &= -\mbox{sgn}(\phi)\arccos\left( \frac{\left( 1-\Delta\alpha^2 \right)S_a^2 + 2S_a L_a \cos(2\pi\phi)+L_a^2}{\widetilde{F_+}\widetilde{F_-}}\right),\end{aligned}
X^{I}= \left( \begin{array}{c c c c}{{x_{1}^{I}}}&{{0}}&{{\dots}}&{{0}}\\{{0}}&{{x_{2}^{I}}}&{{\dots}}&{{0}}\\{{\dots}}&{{\dots}}&{{\dots}}&{{\dots}}\\{{0}}&{{\dots}}&{{\dots}}&{{x_{N}^{I}}}\\ \end{array}\right) , \I = 1 , \dots , 2 6 \ ,
f_{(1)/(1)}=1; \f_{(2)/(1)}=1, \f_{(11)/(1)}=1, f_{(3)/(1)}=1, \f_{(111)/(1)}=1,\ f_{(21)/(1)}=2, f_{(4)/(1)}=1, \f_{(1111)/(1)}=1,\ f_{(31)/(1)}=3,\ f_{(211)/(1)}=3, f_{(22)/(1)}=2.
Tr\left( e^{\frac{1}{4}t\Delta_{\nu}}\right) \simeq \frac{\omega_{n}}{(4\pi t)^{n}}\left[ \sum\limits_{k=0}^{+\infty}c_{k}^{\left( \nu ,n\right)}\t^{k}\right] \left[ \sum\limits_{j=0}^{+\infty}\frac{1}{j!}\left( \frac{n^{2}}{4}+\nu^{2}\right)^{j}\right] .
f_{D,m,4}=\frac{f_c}{c}\left(\frac{\langle{\bf v},{\bf p}_d-{\bf p}_{i_m}\rangle}{\|{\bf p}_d-{\bf p}_{i_m}\|_2}+\frac{\langle{\bf v},{\bf p}_d-{\bf p}_{i_m}\rangle}{\|{\bf p}_d-{\bf p}_{i_m}\|_2}\right), \; \forall m.
\tilde P_{4,0}^{xx}(x){\partial^2 \over \partial x^2}\ + \ [x^2y^2 P_{0,2}^{xy}\ +\xy \tilde P_{1,2}^{xy}(x,y)\ + tilde{P}_{2,0}^{xy}(x,y)]{\partial^2 \over \partial x \partial y}\ + \ \tilde P_{4,0}^{yy}(y){\partial^2 \over \partial y^2}\ +
S_{f_n}(\rho\|\sigma) =f_n(0^+)\sigma(1)+f_n'(+\infty)\rho(1)+S_{-h_n}(\rho\|\sigma) &=f_n(0^+)\sigma(1)+f_n'(+\infty)\rho(1)-\int_{(0,+\infty)}h_n(t)\,d\|E_{\rho,\sigma}(t)\xi_\sigma\|^2.
(\sigma_{\rm HHJ}-\sigma,\tau)= -\sum_{K\in\mathcal{T}}\int_{\partial K}M_{\nu\nu}(\tau)\frac{\partial(u_{\rm HHJ}-u)}{\partial\nu}ds-\sum_{K\in\mathcal{T}}(\tau,\nabla^2u)_{L^2(K)}= \sum_{K\in\mathcal{T}}\int_{\partial K}M_{\nu t}(\tau)\frac{\partial(u_{\rm HHJ}-u)}{\partial t}ds,
\int_\Omega\int_0^t S(t-s)f(S(s)v_0)\,\d s \ge c''\sigma\sum_kf(c_d\phi_k)\left(\frac{\psi}{\phi_k}\right)^{1+2/d}&=c'''\psi^p\sum_k f(s_k)s_k^{-p},
\frac{dx}{\xi_1}=\frac{dy}{\xi_2}=\frac{dt}{\xi_3}=\frac{d\lambda}{\eta_{\lambda}}=\frac{du}{\eta_{u}}=\frac{d\omega^{[j]}}{\eta_{\omega}^{[j]}}=\frac{dv^{[j]}}{\eta_{v}^{[j]}}=\frac{d\psi}{\eta_{\psi}}=\frac{d\phi}{\eta_{\phi}},
\sum_{t=0}^{2}q^{4\binom{t}{2}+3t}j(-q^{4t+2};q^3)m\Big (-q^{7-4t},-1;q^{12}\Big ) &=\overline{J}_{1,3}m(-q^7,-1;q^{12})+\overline{J}_{0,3}m(-q^{3},-1;q^{12})+q^{-2}\overline{J}_{1,3}m(-q^{-1},-1;q^{12}) &=\overline{J}_{1,3}m(-q^7,-1;q^{12})+\overline{J}_{0,3}m(-q^{3},-1;q^{12})-q^{-1}\overline{J}_{1,3}m(-q,-1;q^{12})
\partial_\mu\sigma =-f_0(r)\int_0^r(\phi_{T_1}+\partial_\mu\sigma-\frac{r^2}{2})(Q_\mu-Q)f_1\tau d\tau &\quad+f_1(r)\int_0^r (\phi_{T_1}+\partial_\mu\sigma-\frac{r^2}{2})(Q_\mu-Q)f_0\tau d\tau.
\mathbf{u}^{\mathcal{L}}(x_j^{(n)})\mathcal{J}_n \mathbf{u}^{\mathcal{R}}(x_j^{(n)})=-\sqrt{d_{n}}(x_j^{(n)}-i\omega)[u_{n}^{\mathcal{R}}(x_j^{(n)})]'u_{n-1}^{\mathcal{L}}(x_j^{(n)})=\omega_{n,j}^{-1}.
\bar{P}_{l, t}(Y) = C_{l, t}\int_{\mu_0\in \mathbb{R}^{p}: \|\mu_0\| \geq 1/2}\frac{1}{(2\pi)^{pn/2}}\exp(-\|Y - 2t \mu_0l^{\intercal}\|_F^2/2 ) & \cdot \left(\frac{p}{2\pi}\right)^{p_1r/2}\exp(-p\|\mu_0\|_2^2/2) d\mu_0.
\frac{y}{k!}\int_0^1 (1-w)^{k}\left\{\sum_{n \geq 1}\frac{(3/2)_n}{n!}{}_3F_2\left(-n, \frac{m+2}{2}, \frac{m+3}{2}, \frac{3}{2}, 2; \epsilon\right)(wy)^{n}\right\}dw = \sum_{j \geq 0}\frac{((m+2)/2)_j((m+3)/2)_j}{(2)_j}& \frac{1}{\Gamma(k+j+2)}(-\epsilon)^{j}y^{j+1}{}_2F_1\left(j+\frac{3}{2}, j+1, j+k+2; y\right).
E_1 = \{xy \in E(L^*): x \in A \setminus S\}E_2 = \{xy \in E(L^*): x \in S, y \notin A \setminus S\}E_3 = \{xy \in E(L^*): x \in B_2, y \notin A\}E_4 = \{xy \in E(L^*): x \in B_0, y \notin A \cup B_2\}E_5 = \{xy \in E(L^*): x,y \in B_1\}
\begin{array}{lll}x(t+1) = v(t+1) - \underline v = \hat C V(t+1) - \underline v = \hat C \Big[ C V(t) + Dp - \phi (v) \Big] - \underline v =\hat{C}C\hat{C}^{-1}x(t) + (\hat{C}C\hat{C}^{-1}-I)\underline{v}+ \hat{C}(Dp - U(x(t)) - \mathbf{1}\frac{\beta}{2}). \end{array}
\int_M f_x^{\boxplus\tau}!(h_x^{\boxplus\tau};\mathcal S_x^{\boxplus\tau}) \wedge \rho = (-1)^{|\rho||\omega|}\int_{(s_x^{\boxplus\tau})^{-1}(0)}\pi_1^* h^{\boxplus\tau}_x \wedge \pi_1^*(f^{\boxplus\tau}_x)^* \rho \wedge \pi_2^*\omega_x
\prod_{\ell \in \mathcal{P}}\left[ \operatorname{SL}_2(\mathbb{Z}_\ell)^2 : \mathcal{B}_\ell(f(\ell),f(\ell)) \right] = \prod_{\ell \in \mathcal{P}}\ell^2(\ell^2-1)^2 \ell^{6(f(\ell)-1)}\leq \prod_{\ell \in \mathcal{P}}\ell^{6f(\ell)}.
U(\vec{a},\Lambda)\phi^{\alpha,n}(f) U(\vec{a},\Lambda)^{-1}1 &=(\Lambda S_0 + \vec{a}, e^{-i[\vec{a}\cdot (\hat{H}(\rho_n)\hat{g}_0 + \hat{P}(\rho_n)\hat{g}_1)]}f^{S_0^\flat}(\Lambda^{-1}(\cdot-\vec{a}))\otimes \rho_n(E^\alpha)).
\begin{array}{rl}\sigma_i\displaystyle\frac{\rm d}{{\rm d}t}\theta_i h_i ={\cal L}_i(h,\boldsymbol{v}), \sigma_i\displaystyle\frac{\rm d}{{\rm d}t}\theta_i h_i v_{a\,i}={\cal J}_{a\,i}(h,\boldsymbol{v}) +{\cal S}_{a\,i}(h,w), \end{array}
z{\cal H}_{k-1}^{(s+1,t)}(z)={\cal H}_{k}^{(s,t)}(z)+q_k^{(s,t)}{\cal H}_{k-1}^{(s,t)}(z),s,t=0,1,\dots, &{\cal H}_k^{(s,t)}(z)={\cal H}_{k}^{(s+1,t)}(z)+e_{k}^{(s,t)}{\cal H}_{k-1}^{(s+1,t)}(z),s,t=0,1,\dots,
[R,M]=-{\rm i\over2}\sum_{\rho\in\Delta_{+}}g_{|\rho|}\thinspace{|\rho|^2\cos(\rho\cdot q)\over{\sin^{2}(\rho\cdot q)}}\thinspace[{\rm e}^{2{\rm i}q\cdot\hat{H}}, \hat{s}_{\rho}].
f(U_h^n) = \frac{1}{\beta_k h}( U_h^n + \beta_k h f(U_h^n) - U_h^n ) = \frac{1}{\beta_k h}( U_h^n - F_h (U_h^n) ) = \frac{1}{\beta_k h}( F_h^{-1}( R_h^n ) - R_h^n ).
\sum_{k=0\atop k\neq m}^{n}\left[\begin{matrix}n k\end{matrix}\right]\frac{\left(\frac{q}{x}\right)_k(x)_{n-k}}{1-q^{k-m}}x^k=(-1)^mq^{\frac{m(m+1)}{2}}\left[\begin{matrix}n m\end{matrix}\right](xq^{-m})_n\left(\sum_{k=0}^{n-1}\frac{xq^{k-m}}{1-xq^{k-m}}-\sum_{k=0\atop k\neq m}^{n}\frac{q^{k-m}}{1-q^{k-m}}\right).