harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\rm e}^{-A_1(-\infty)}= \frac{3}{2\sqrt{2}}{\rm e}^{\frac{\pi}{4}i},{\rm e}^{-A_2(-\infty)}= \frac{3}{2\sqrt{2}}{\rm e}^{-\frac{\pi}{4}i}, B_0(-\infty) =-\frac{3+i}{4}, B_{\bar{0}}(-\infty) =-\frac{3-i}{4}
\int_X^Y\frac{e(-n\alpha)}{z^\mu}d\alpha= \int_X^Y\frac{e(-n\alpha)}{(1/N-2\pi i\alpha)^\mu}d\alpha = -\frac{1}{2\pi in}\left[\frac{e(-n\alpha)}{(1/N-2\pi i\alpha)^\mu}\right]_X^Y+\frac{\mu}{n}\int_X^Y\frac{e(-n\alpha)}{(1/N-2\pi i\alpha)^{\mu+1}}d\alpha \ll \frac{1}{|n|X^\mu}+\frac{\mu}{|n|}\int_X^Y\frac{d\alpha}{\alpha^{\mu+1}}\ll\frac{1}{|n|X^\mu}.
\bar{\Phi}= 2 \sqrt 2 r , \bar{F}_{t r}= Q_{E}e^{- 2 \sqrt 2 r}, \bar{T}= 0 ,{}\bar{G}_{\mu \nu}= \left( \begin{array}{c c}{{- f}}&{{0}}\\{{0}}&{{f^{- 1}}}\\ \end{array}\right) ,
\begin{cases}g_1(t, B_t) = -g(t, B_t) \frac{t +B_t^2}{2t^2}, g_2(t, B_t) = g(t, B_t) \frac{B_t}{t}, g_{22}(t, B_t) =g(t, B_t) \frac{t +B_t^2}{t^2}. \end{cases}
{f_2}\left({x,y,}t \right){\rm =}{f_2^b}\left({x,y,}t \right), ~(x,y)\in\partial\Omega,~{t >}0,~{f_2}\left(x,y ,0\right){\rm =}{f^0_2}\left(x,y \right),~(x,y)\in\Omega,
\langle m|\psi_{\alpha}|n\rangle\langle n|\psi_{\beta}^{\dagger}|m\rangle=\delta_{\xi_{\alpha},\xi_{\beta}}\delta_{E_{n},E_{m}+\xi_{\alpha}}\langle m|\psi_{\alpha}|n\rangle\langle n|\psi_{\beta}^{\dagger}|m\rangle.
\rho_{134}(S_1)^{2k}=\tilde S_1^{2k}=S_2(S_1S_2)^kS_2^*+S_1(S_2S_1)^kS_1^* &\rho_{134}(S_2)^{2k}=\tilde S_2^{2k}=S_2(S_2S_1)^kS_2^*+S_1(S_1S_2)^kS_1^*
1+\frac{2}{(r-3)/(r-1)}\frac{1}{(r-1)(1-1/r)}= 1+\frac{2r}{(r-1)(r-3)}& = \frac{r^2-2r+3}{(r-1)(r-3)}& = \frac{(r-1)}{(r-3)}\left(1+\frac{2}{(r-1)^2}\right).
e^{-2\pi\big(1-\frac{g}{\pi+gN}\big)\sum_{i=1}^{n_a}\sum_{j=1}^{n_a}\Big[\big(1\!-\!\delta_{ij}\big)\big(C^{0,S}(x_i^{(a)},x_j^{(a)})+C^{0,S}(y_i^{(a)},y_j^{(a)})\big)-2C^{0,S}(x_i^{(a)},y_j^{(a)}) \Big]}
\times \bigg \{[ 2 + 2 \cos ( 2 \theta ) ] \widehat{( \rho_{1}\! - \! 1 )}( 0 ) \; - [ 2 - 2 \cos ( 2 \theta ) ] \widehat{( \rho_{1}\! - \! 1 )}( p ) \bigg \}\; + O ( m^{3}) \; .
T_{i}(h) := 1 - V(h) - \frac{(k+l-i)(i+k+l+\sigma)}{(h+k+\beta)(h+l-n-1)}, begin{align*}1ex] V(h) := \frac{(h-k-1)(l+n+\alpha+2-h)}{(h+k+\beta)(h+l-n-1)}
\lim_{k\rightarrow\infty}\int_{\mathbb{T}^\infty}\log(1+|\mathcal{B}_k f_\sigma|)dm_\infty=\int_{\mathbb{T}^\infty}\log(1+|\mathcal{B}f_\sigma|)dm_\infty=\|\mathcal{B}f_\sigma\|_0.
\delta \lambda_{\alpha}\ = \ \epsilon^{\beta}F_{( \alpha \beta )}\qquad \mathrm{a n d}\qquad \delta A_{\alpha \dot{\alpha}}\ = \ 2 \, \lambda_{\alpha}\ \tilde{\epsilon}_{\dot{\alpha}}\quad \longrightarrow \quad \delta F_{( \alpha \beta )}\ = \ 2 \,{\partial_{( \alpha}}^{\dot{\alpha}}\lambda_{\beta )}\ \tilde{\epsilon}_{\dot{\alpha}}
r=\Bigl(\frac{l_2}{\gcd(|w_2v_1-w_1v_2|,l_2)}\Bigr)\Bigl(\frac{k_2v_1-k_1v_2}{l_1\frac{w_2v_1-w_1v_2}{\gcd(|w_2v_1-w_1v_2|,l_2)}}\Bigr)
\partial_k P_r(a) = \partial_k r^\sharp (\varphi(a)), P_r \partial_k(a) = r^\sharp(\varphi(\partial_k(a))) = \sum_i \mathfrak{B}(\partial_k(a),a_i) b_i = \sum_i \mathfrak{B}(a, \hat{\partial_k}(a_i)) b_i \hphantom{P_r \partial_k(a)}{}= \sum_i \langle \varphi(a), \hat{\partial_k}(a_i) \rangle b_i = r^\sharp \hat{\partial_k}^*(\varphi(a)).
\begin{array}{l}\displaystyle{\phantom{\Big(}\{{w(f)}_\lambda{w(a)}\}_{z,\rho}=\sum_{j\in J^f_{\leq-\frac12}}w(q_{j})w([q^{j},a]^\sharp)+(1-\delta_{\Delta,1})(\partial+\Delta\lambda)w(a)}\displaystyle{\phantom{\Big(}-\frac{(e|a)}{2}\lambda^3+z w([s,a]^\sharp)+z\Delta(s|a)\lambda\,.}\end{array}
\chi_{\ell}^{( \infty )}( r ) = H_{\nu}^{( 1 )}( k r ) \sim \sqrt{\frac{2}{\pi k}}\, e^{- i \frac{\pi}{2}( \ell - \frac{\Phi}{2 \pi}) - i \frac{\pi}{4}}\, \frac{e^{i k \, r}}{\sqrt{r}},
\| w(t)\|^2_{P_w}+ \| u(t)\|^2_{\tilde P_u}+ \| v(t)\|^2_{\tilde P_v}= \| w^{(0)}\|^2_{P_w}+ \| u^{(0)}\|^2_{\tilde P_u}+ \| v^{(0)}\|^2_{\tilde P_v},
h(x, y) =J_p\Big(\underbrace{(M-\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))}_a\Big) &-J_p\Big(\underbrace{(M-\varepsilon)\,(\widetilde\Gamma(y)-\widetilde\Gamma(x))}_a-\underbrace{\left(\frac{M-m}{2}-\varepsilon\right)\,\widetilde\Gamma(y)}_b\Big).
\begin{aligned}\tau\stackrel{\rm{def}}{=}\Big[-\xi \Big(Q+\frac{1}{d}Id\big)H-\xi H\big(Q+\frac{1}{d}Id\big)+2\xi \big(Q +\frac{1}{d}Id\big)QH-L \nabla Q\odot \nabla Q\Big]\end{aligned}
\iint\langle{\boldsymbol{\nu}}_{(x,t)}, A_i(\lambda)\rangle \partial_t\varphi_{i}\,dx\,dt+\iint \langle{\boldsymbol{\nu}}_{(x,t)}, F_{i,\alpha}(\lambda)\rangle \partial_\alpha\varphi_{i}dx\,dt+\int \langle\boldsymbol{\nu}_{(x,0)},A_i\rangle\varphi_i(x,0)dx=0
a_a(m,n) = \sum_{k=1}^{\infty}\left( \log2 + \frac{1}{2}(-1)^{k+1}\left[ \Psi(\frac{k}{2}+\frac{1}{2})-\Psi(\frac{k}{2}+1)\right]\right)^m \frac{(-1)^{k+1}}{(k+1)^n},
P_\ell(x_1^n y_1^a, x_1^m y_1^b)=\beta x_1^{n u+m v}y_1^{a u+b v}q_\ell(x_1^{m n}y_1^{a m},x_1^{m n}y_1^{b n})=\beta x_1^{\omega}y_1^{a u+b v+ a m d_\ell}q_\ell(1,y_1).
\delta \psi_{\mu a}\; = \; D_{\mu}\, \varepsilon_{a}\, + \, i \, \gamma_{\mu}\, T \left( \Gamma_{4 5}\right)_{a b}\varepsilon^{b}\; + \, i \; \frac{1}{6}\; \sqrt{\, \frac{1}{2}}\, \left( \, \gamma_{\mu}^{\; \nu \rho}\; - \; 4 \, \delta_{\mu}^{\nu}\, \gamma^{\rho}\, \right) \left( \, H_{\, \nu \rho a b}\; + \; \sqrt{\, \frac{1}{2}}\, h_{\, \nu \rho a b}\, \right) \varepsilon^{b},
\int_{X}(M_{\mathcal{T}}\phi)^{q}d\mu =N^{(q-1)m}+(1-\frac{1}{N})(N^{(q-1)(m-1)}+...+N^{q-1}+1)= & =N^{(q-1)m}+(1-\frac{1}{N})\frac{N^{(q-1)m}-1}{N^{q-1}-1}= & =1+\frac{N^{q}-1}{N^{q}-N}(F^{r}-1)
\hat Y=\left(\begin{array}{cc|cc}-\textstyle\frac{1}{2}I_r 0 F_1 0 0 I_{p-r}0 0 \hline\textstyle\frac{3}{4}F_1^{-1}0 \textstyle\frac{1}{2}I_r 0 0 0 0 -I_{n-p-r}\end{array}\right),
\log\left((pd)^3(n-t_v(\psi))^2\right) \geq \log\left((pd)^3(1-\gamma)^2n^2\right) = H(n,p,d) + 2\log(1-\gamma) \geq H(n,p,d) - 4\gamma \geq H(n,p,d) - \frac{\beta}{2}.
J[t_0,x_0;u]=\mathbb{E}\left[\int_{t_0}^{\tau_S}h(s,X_s^{t_{0},x_0,u}) +\sum_{j\geq 1}c(\tau_{j},z_j)\cdot 1_{\{\tau_j\leq \tau_S\}}+ \phi(X_{\tau_S}^{t_{0},x_0,u})\cdot 1_{\{\tau_S<\infty\}}\right],
\left( \begin{array}{c c c c c}{{0}}&{{\Delta_{1}}}&{{0}}&{{0}}&{{0}}\\{{\Delta_{1}}}&{{0}}&{{\Delta_{2}}}&{{0}}&{{0}}\\{{0}}&{{\Delta_{2}}}&{{0}}&{{\ddots}}&{{0}}\\{{0}}&{{0}}&{{\ddots}}&{{\ddots}}&{{\Delta_{k}}}\\{{0}}&{{0}}&{{0}}&{{\Delta_{k}}}&{{0}}\\ \end{array}\right)
\P^\eta\big(|\langle\eta(t)\rangle_{J'}^k-p_k|>\epsilon_{J}\big)= \P^\eta\big(\langle\eta(t)\rangle_{J'}^k>p_k+\epsilon_{J}\big)+ \P^\eta\big(\langle\eta(t)\rangle_{J'}^k<p_k-\epsilon_{J}\big)\,.
\mathbf{E}\left(-1\right)^{j+\alpha}(W_{\tau_{j}\wedge b}-W_{\tau_{j-1}\wedge b}) =\mathbf{E}\left(-1\right)^{j+\alpha}\mathbf{E}\left[W_{\tau_{j}\wedge b}-W_{\tau_{j-1}\wedge b}\right]=0.
H_{n}^{p}(\mathbf u)=\begin{pmatrix}u_{p}u_{p+1}\cdots u_{p+n-1}u_{p+1}u_{p+2}\cdots u_{p+n}\cdots \cdots \cdots \cdots u_{p+n-1}u_{p+n}\cdots u_{p+2n-2}\end{pmatrix},
H=L_1^2+L_2^2+L_3^2+\big(x_1^2+x_2^2+x_3^2\big)\left(\frac{k_1^2-1/4}{x_1^2}+\frac{k_2^2-1/4}{x_2^2}+\frac{k_3^2-1/4}{x_3^2}\right),
\phi_{i}(z)=\sum_{r=0}^{\infty}\phi_{i}(-r)z^{r}=K_{i}^{-1}\exp\left(-(q-q^{-1})\sum_{r=1}^{\infty}a_{i}(-r)z^{r}\right), & \psi_{i}(z)=\sum_{r=0}^{\infty}\psi_{i}(r)z^{-r}=K_{i}\exp\left((q-q^{-1})\sum_{r=1}^{\infty}a_{i}(r)z^{-r}\right).
a_{jN}= M\int_0^M\phi_j(\eta)\,d\eta\int_0^M\phi_N(\eta')\,d\eta'- \int_0^M\eta\phi_j(\eta)\,d\eta\int_0^{\omega_1}\phi_N(\eta')\,d\eta' &\phantom{xx}{}- \int_0^M\phi_j(\eta)\,d\eta\int_{\omega_{N-1}}^M\eta'\phi_N(\eta')\,d\eta'- K_j^+ - K_j^- &= \frac12\Delta_j\Delta_N\bigg(M-\sigma_j+\frac{\delta_N}{3}\bigg)- K_j^+ - K_j^-,
\int_a^1 U_m^{(a)}(x;q)U_n^{(a)}(x;q)\rho(x;q)d_qx=(-a)^n (1-q) (q;q)_n(q,a,a^{-1}q;q)_\infty q^{n\choose 2}\delta_{n,m}:={h}_n\delta_{n,m},
\phi'(|x_i^{n+1}|) - \phi'(|x_i^*|)= & sign(x_i^*) \phi''(|x_i^*|) (x_i^{n+1}- x_i^{*}) + \frac{1}{2}\phi'''(|x_i^{\xi}|)(x_i^{n+1}- x_i^{*})^2,
(N-j+1)(2N-2j+1)a^{(N)}_{j-1}(\lambda)+j(2\lambda+n-4N+2j-1)a_j^{(N)}(\lambda) =0, (N-j+1)(2n-2j+3)b_{j-1}^{(N)}(\lambda)+j(2\lambda+n-4N+2j-3)b_j^{(N)}(\lambda) =0,
( \,{\tilde{\Gamma}}_{{\bar{i}}{\bar{j}}}^{\bar{k}}-{\tilde{\Gamma}}_{{\bar{j}}{\bar{i}}}^{\bar{k}}\, ) \langle \,{\hat{\phi}}_{\bar{k}}\, \prod_{l = 1}^{s}\phi_{l}\, \rangle_{g}= \sum_{l = 1}^{s}{\cal R}_{D_{l}}+ \sum_{n o d e s}{\cal R}_{\Delta}\,
S_{F}^{- 1}( p_{f}) = S_{F}^{( 0 ) - 1}( p_{f}) -{\frac{e}{\beta}}\sum_{n = - \infty}^{\infty}\int{\frac{d^{2}k}{( 2 \pi )^{2}}}\gamma^{\nu}S_{F}( k_{f}) \Delta_{\mu \nu}( k_{f}- p_{f}) \Gamma^{\mu}( k_{f}- p_{f}, k_{f}) .
\begin{aligned}(ab^{-1}c - b)(y-y^{-1}) = &q^2y(qy + q^{-1}y^{-1})^{-1}x^2(y-y^{-1}) & - q^2\Lambda \alpha^{-1}(\Lambda)y^{-1}(q^{-1}y + qy^{-1})^{-1}x^{-2}(y-y^{-1}) &\, \quad+ \Big(\alpha(\Lambda)(qy + q^{-1}y^{-1})^{-1}y^{-1}- \Lambda(q^{-1}y + qy^{-1})^{-1}y - y - y^{-1}\Big)(y-y^{-1}) \end{aligned}
\begin{aligned}\dot{f}(\alpha) =-C_gD_g\left(3(\alpha-1)^2-1\right)+C_iD_i(1-\alpha)(3\alpha-1) &= \frac{C_gD_g\left(3-(1+\omega)^2\right)+C_iD_i(1-\omega^2)}{3}.\end{aligned}
2b_{13}^{(k-1)}\left( \tan\phi_{k}+ \cot(2\phi_{k})\right) = b_{11}^{(k)}-b_{22}^{(k)}, 2b_{24}^{(k-1)}\left( \tan\psi_{k}+ \cot(2\psi_{k})\right) = b_{44}^{(k)}-b_{33}^{(k)}.
\rho-1 \in C_b([0,+\infty);\widetilde{B}^{\frac{N}{2}+1-\alpha,\frac{N}{2}}), u \in C_{b}([0,+\infty);\dot{B}^{\frac{N}{2}+1-\alpha}_{2,1}) \cap L^1(\mathbb{R}^+;{\dot{B}^{\frac{N}{2}+1}_{2,1}}).
Q_{1,\alpha}=\left(\int_{\mathbb{R}^n}u_0^2 dx\right) \left(\frac{n-2}{4(n-1)}S_g(\bar{x}_0)-m_0^2 \right)\mu_\alpha^2+o(\mu_\alpha^2)+o(\mu_\alpha^{n-2}r_\alpha^{2-n}),
\partial_x u_1+\partial_y u_2=~ \frac{(1-a)}{Q}h_1\big[(h_1\partial_x+h_2\partial_y)u_1+\nu\partial_y^2h_1\big]-\frac{(1-a)}{Q}( P_t+ P_x u_1) &+\frac{a}{Q}\big[\kappa\partial_{y}^2\theta+\mu(\partial_y u_1)^2+\nu(\partial_y h_1)^2\big],
h_{j,k}=\frac{\sqrt{\pi}\,b^{n_B-j+k}}{a^{n_B-j}}\,\Gamma(n_B-j+k)\, \,_2F_2\left(\frac{1-k}{2},\frac{2-k}{2};\frac{1-n_B+j-k}{2},\frac{2-n_B+j-k}{2};\frac{a^2}{4b^2}\right).
{\cal L}_{f e r m i o n}= \bar{\psi}\left[ i \gamma^{\mu}\left( \partial_{\mu}- \sigma^{\rho \beta}v_{\rho}^{\nu}g_{\lambda \nu}\partial_{\mu}v_{\beta}^{\lambda}- g_{\alpha \delta}\sigma^{a b}v_{a}^{\beta}v_{b}^{\delta}\tilde{\Gamma}^{\alpha}_{\mu \beta}\right) \right] \psi
\varphi_0(t,x):=0, \varphi_+(t,x):= \mathbf{1}_{\{1\}}(x)=\frac{1+x}{2},\varphi_-(t,x):= \mathbf{1}_{\{-1\}}(x)=\frac{1-x}{2}, & \widehat \varphi_+(t,x)= \mathbf{1}_{\{0\}}(t)\mathbf{1}_{\{1\}}(x)=\frac{(1-t)(1+x)}{2},\widehat \varphi_-(t,x):= \mathbf{1}_{\{0\}}(t)\mathbf{1}_{\{-1\}}(x)=\frac{(1-t)(1-x)}{2}
u(j+x^*_{t,n_k}+J(t)+x(\mu^*,n_k),t;-n_k,\bar\phi_{\mu^*}(\cdot,-n_k)) &\begin{cases}\ge u(j+x(\mu,t,n_k),t;-n_k,\bar\phi_\mu(\cdot,-n_k))\quad\forall\,\, j\le 0\cr\le u(j+x(\mu,t,n_k),t;-n_k,\bar\phi_\mu(\cdot,-n_k))\forall\,\, j>0.\end{cases}
\int_{\mathbb{D}}|f^{(\nu)}(z)|(1-|z|^{2})^{\nu}dA(z) \leq\left(\int_{\mathbb{D}}|f^{(\nu)}(z)|^{2}(1-|z|^{2})^{2\nu}dA(z)\right)^{1/2}&\leq\left(\int_{\mathbb{D}}|f^{(\nu)}(z)|^{2}(1-|z|^{2})^{2\nu+p+2\beta-4}dA(z)\right)^{1/2}&<\infty.
T^{\bf b}_f=M^{\bf b}_f+\sum_{g\prec_{\bf b}f}t_{gf}^{\bf b}(q)M^{\bf b}_g, L^{\bf b}_f=M^{\bf b}_f+\sum_{g\prec_{\bf b}f}\ell_{gf}^{\bf b}(q)M^{\bf b}_g,
\begin{pmatrix}u^{\infty}_{ps}(\hat{x}) u^{\infty}_{ss}(\hat{x})\end{pmatrix}= \sqrt{\frac{2}{\pi}}^{i\frac{\pi}{4}}iR \sum_{n \in \mathbb{Z}}Q^{-1}\mathcal{H}_n^{-1}\begin{pmatrix}-\frac{in}{t_s}J_n(t_s) J^{\prime}_n(t_s)\end{pmatrix}^{in \theta}.
\int{\cal D}\tilde{\vec{p}}\exp\{i\int [- \frac{h}{2m}{\vec{p}}^{2}+ \vec{p}\dot{\vec{x}}- \frac{m}{2h}{\dot{\vec{x}}}^{2}] d\tau\}= \int{\cal D}\tilde{\vec{p}}\exp [ i\int \frac{-1}{2m}(\tilde{\vec{p}})^{2}d\tau]
P(A_1 \cap A_2 \cdots \cap A_m) = 1 - P(A_{1}^c \cup A_{2}^c \cdots \cup A_{m}^c), &\geq1 - [P(A_{1}^c) + P(A_{2}^c) + \cdots P(A_{m}^c)], &\geq1 - 4me^{-(\epsilon^2/2 - \epsilon^3/3)k/2},
H_{\gamma, h}(\sigma_{\Delta};\sigma_{\Delta^c})=-h\sum_{x\in\Delta}\sigma_{\Delta}(x)-\frac{1}{2}\sum_{x\neq y\in\Delta}J_{\gamma}(x,y)\sigma_{\Delta}(x)\sigma_{\Delta}(y)-\frac{1}{2}\sum_{\substack{x\in\Delta y\in\Delta^{c}}}J_{\gamma}(x,y)\sigma_{\Delta}(x)\sigma_{\Delta^c}(y).
\begin{aligned}E_{u,h}^2= \sum_{\alpha\in\{\alpha=(\beta,k):|\alpha|\leq m, \beta\leq m-1\}}s(x) &+\Vert (u+u_e\phi(y)+u_b(1-\phi(y)))^{\frac{1}{2}}\langle y\rangle^{l}\partial_x^m (u,h) \Vert_{L^2_y}^2 \leq \sum_{\alpha\in\{\alpha=(\beta,k):|\alpha|\leq m, \beta\leq m-1\}}s(x)+25\vartheta_0^{-4}s_m(x),\end{aligned}
\{y, y_1^{\textsc{d}}\}\cap{\mathbb{Z}}= \emptyset;\;\;\;\;\;\begin{array}{l}\hat{y}_-= \max\limits_{k>0}-{{\textstyle{\frac{1}{k}}}}(1 + \lfloor y k \rfloor + \lfloor y_1^{\textsc{d}}k \rfloor), \hat{y}_+= \min\limits_{k>0}-{{\textstyle{\frac{1}{k}}}}(-1 + \lceil y k \rceil + \lceil y_1^{\textsc{d}}k \rceil),\end{array}
\intop_{0}^{\infty}\frac{x^{\nu+1}J_{\nu}(bx)}{\left(x^{2}+a^{2}\right)^{\mu+1}}\,dx=\frac{a^{\nu-\mu}b^{\mu}}{2^{\mu}\Gamma(\mu+1)}K_{\nu-\mu}(ab),\,\,\,\,\,a,b>0,\,\,\,-1<(\nu)<2\,(\mu)+\frac{3}{2}.
L:= \begin{bmatrix}l_{n_l-1}\ldots l_0 0 0 \ldots 0 0 l_{n_l-1}\ldots l_0 0 \ldots 0 \vdots \ddots \ddots \ddots \ddots \ddots \vdots 0 \ldots 0 l_{n_l-1}\ldots l_0 0 0 0 \ldots 0 l_{n_l-1}\ldots l_0 \end{bmatrix}.
p_n(x)=\frac{a}{a_n}[(z-B_{n-1})p_{n-1}(x)+\frac{1}{z}\psi_{n-1}(z)] \psi_n(z)=\frac{a}{a_n}\left[\frac{1}{z}\psi_{n-1}(z)+\left(\left(1-\frac{a_n^2}{a^2}\right)z-B_{n-1}\right) p_{n-1}(x)\right],
\psi^{\prime}\left(\varrho \left(2 \Psi(v)\right)\right) = \psi^{\prime}\left(\varrho \left(\sum_{j=1}^{4N}\psi(z_j)\right)\right) \ \leq \ \sqrt{\sum_{j=1}^{4N}\psi^{\prime}(z_j)^2}= & = \sqrt{\sum_{j=1}^{N}\left[2\psi^{\prime}(\lambda_1(v^j))^2+\psi^{\prime}(\lambda_2(v^j))^2+\psi^{\prime}(\lambda_4(v^j))^2\right]}= & = 2 \sqrt{2}\delta(v).
\begin{aligned}g_{z\bar z}-\frac{(\mu_1^2-\mu_2^2)g(1+g^2)(1-\bar g^2)|g_{\bar z}|^2}{[\mu_1g(1+\bar g^2)-\mu_2\bar g(1+g^2)][\mu_1\bar g(1+g^2)-\mu_2g(1+\bar g^2)]}&-\frac{2\mu_1|g|^2-\mu_2(1+\bar g^2)}{\mu_1\bar g(1+g^2)-\mu_2g(1+\bar g^2)}g_zg_{\bar z}=0. \end{aligned}
\MoveEqLeft \int_0^t A(t)^2 e^{-(t-s)A(t)}(A(s)^{-1}- A(t)^{-1}) g(s) \d s & = \frac{1}{2\pi i}\int_0^t A(t) \int_{\Gamma}ze^{-(t-s)z}R(z,A(t)) \d z \, (A(s)^{-1}- A(t)^{-1}) g(s) \d s
\left(1-A\right)^{N}\!\!\! = \!\! \bigg( 1- c_0 \left(\frac{\epsilon}{KS^2}\right)^{(K-1)S-L+1}\!\!\! \cdot\mathsf{SNR}^{-((K-1)S-L+1)}-\Omega_{\textsf{SNR}}\left(\mathsf{SNR}^{-((K-1)S-L)}\bigg)\right)^{N}.
\int^t_0 (t-s_1)^{-1+\alpha}\int^{s_1}_0 (s_1-s_2)^{-1+\alpha}\cdots \int^{s_{n-1}}_0 (s_{n-1}-s_n)^{-1+\alpha}ds_n \cdots ds_1 &=\frac{\left\{\Gamma(\alpha)\right\}^{n}}{\Gamma(n\alpha+1)}t^{n\alpha},
\rho_{r}( Q_{j}, Q_{j}^{\prime}, t ) = \int \Pi_{\alpha = 1}^{N}d q_{\alpha}d q_{\alpha}^{\prime}\exp \frac{i}{\hbar}[ S ( Q ) + S_{C}( Q , q_{\alpha}) + S_{B}( q_{\alpha}) - S ( Q^{\prime}) - S_{C}( Q^{\prime}, q_{\alpha}^{\prime}) - S_{B}( q_{\alpha}^{\prime}) ] .
\Lambda_{j}= D_{\alpha,+}(-x_{j})^{-1}e^{-\frac{\pi i \alpha}{2}}(4(x_{j}-x_{1}))^{\beta_{j}}\bigg( \prod_{\substack{k=2 k \neq j}}^{m}\widetilde{T}_{k,j}^{\beta_{k}}\bigg)e^{\sqrt{r}g_{+}(-x_{j})}r^{\frac{\beta_{j}}{2}}c_{-x_{j}}^{\beta_{j}}.
i \in I^\varpi_j, X^{\#_{I^\varpi_l},[\mathbf{u}^{(1)}]^\varpi_l}_t = X^{\#_{I^\varpi_l},[\mathbf{u}^{(2)}]^\varpi_l}_t = X^{\#_{I^\varpi_l},[\mathbf{u}^{(3)}]^\varpi_l}_t,
cn[\frac{4}{5}K(k);k]=\{\frac{\sqrt{5}\vartheta_{00}[q(k)^{5}]}{\vartheta_{00}[q(k)]}-1\}\{\frac{5\vartheta_{01}[q(k)^{10}]^{2}}{\vartheta_{01}[q(k)^{2}]^{2}}-1\}^{-1}
\frac{1}{\varphi_2(N)}\sum_{n=1}^N\psi(n)=\frac{N\psi(N)}{\varphi_2(N)}- \frac{1}{\varphi_2(N)}\int_1^Nx\psi'(x){\: \rm d}x=\frac{1}{\varphi_2(N)}\int_1^N\psi(x)dx=1 + o(1).
D_0=\frac{1}{2}\left(\begin{array}{cc}\sigma_3 0 0 \sigma_3\end{array}\right) \; ,\;\;\;D_+=\frac{1}{2}\left(\begin{array}{cc}0 \sigma_+ \sigma_+ 0\end{array}\right) \; ,\;\;\;D_-=\frac{1}{2}\left(\begin{array}{cc}0 \sigma_- \sigma_- 0\end{array}\right),
\left \langle I + \begin{pmatrix}1 0 0 d \end{pmatrix}p, I + \begin{pmatrix}0 1 0 0 \end{pmatrix}p \right \rangle \left \langle I + \begin{pmatrix}d 0 0 1 \end{pmatrix}p, I + \begin{pmatrix}0 1 0 0 \end{pmatrix}p \right \rangle.
L_{5}(q) = 2\sum_{\substack{n \geq 1 -n \leq j \leq n-1}}q^{6n^2 + 1 - 2j^2}+ 2\sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{6n^2 + 6n + 2- 2j^2 - 2j}
\overline{D_{S_{i-1}}(z)}^{2}= |z - \overline{P_{S_i}}(z)|^{2}, = | P_{V}(z) + \xi_{1}+ P_{V}(z) - \xi_{i-1}|^{2}, &= 4 |P_V(z)|^{2}+ |\xi_{1}- \xi_{i-1}|^{2}, &=4r^{2}+4 (i-2 ) r^{2}, = 4(i-1)r^{2}.
\hspace{- 1 . 5 c m}\Psi_{i n t}\left( x^{m}\right) = \widetilde{U}_{i n t}\left( x^{\mu}, x^{m}\right) \bar{\Psi}_{i n t}\left( x^{m}\right) = \widetilde{U}_{i n t}\left( x^{\mu}, x^{m}\right) \bar{U}_{i n t}\, \left( x^{m}\right) n_{i n t}^{1 / 2}\left( x^{m}\right) \eta_{i n t}
\mathbf{A}\left(\mathbf{B}\pm\mathbf{C}\right)=\mathbf{A}\mathbf{B}\pm\mathbf{A}\mathbf{C}\,\,\,\,\,\,\,\,\,\,\,\&\,\,\,\,\,\,\,\,\,\,\,\,\left(\mathbf{B}\pm\mathbf{C}\right)\mathbf{A}=\mathbf{B}\mathbf{A}\pm\mathbf{C\mathbf{A}}
A_{4}:= \frac{1}{\eta^2}+ \frac{\log n}{n}+ \eta (\log p)^{1/2}\max\left\{\left( \xi_{p}^{2}/ n \right)^{1/2}n^{1/q}\log n , \left( \xi_{p}^{2}/ n \right)^{1/6}(\log n)^{2/3}\right\}A_{5}:= \eta (\log p) \tau_{\infty}\max\left\{1 , \left( p \xi_{p}^{2}/ n \right)^{1/2}\right\}.
\mathbb{E}\bigg(\bigg\langle \int_0^t \Phi(r) dW_{E_r}, h \bigg\rangle_H \bigg| \mathcal{G}_s\bigg)= \mathbb{E}\bigg(\bigg\langle \sum_{j=0}^{n-1}\phi_j(W_{E_{t_{j+1}\wedge t}}-W_{E_{t_{j}\wedge t}}), h \bigg\rangle_H \bigg| \mathcal{G}_s\bigg)
\min\limits_{\mathbf{W}}\sum_{i=1}^N\left\{\frac{r}{2}\int_\Omega \left|w_i(x)-u^{*}_i(x)-\frac{\lambda^*_{w_i}(x)}{r}+\frac{\lambda|I(x) - c^{*}_i|}{r}\right|^2dx\right\}+\delta_\Delta(\mathbf{W}).
(f_a^{(j)}v_{\lambda}, f_a^{(j)}v_{\lambda}) = (v_{\lambda}, e_a^{(j)}f_a^{(j)}v_{\lambda}) = \frac{(v_{\lambda},e_a^{(j-1)}f_a^{(j)}e_a v_{\lambda}) + (v_{\lambda}, e_a^{(j-1)}f_a^{(j-1)}[q^{h_a}; 1-j]_q v_{\lambda})}{[j]_q}= \frac{[m+1-j]_q(v_{\lambda}, e_a^{(j-1)}f_a^{(j-1)}v_{\lambda})}{[j]_q},
W_0 =\langle v_1,w_1\rangle^\perp \cap W; W_1 = \bigcap\limits_{W'\in \Lambda}W' = \langle W_0, w_1\rangle; W_2 = \langle W' \mid W'\in\Lambda\rangle = \langle W_0, w_1, v_1,v_2,v_3,v_4\rangle
\mathcal{H}^1(J \cap \partial B_{r_x}(x)) = 0, \eta r_x \leq \mathcal{H}^1\left(J \cap B_{r_x}(x)\right) \leq \mathcal{H}^1\left(J \cap B_{2r_x(x)}\right) < 2\eta r_x .
\left( \delta F_{p}\right) \left( \phi_{0}, \cdots , \phi_{p + 1}\right) = \sum_{i = 0}^{p + 1}( - 1 )^{i}F_{p}\left( \phi_{0}, \cdots , \phi_{i - 1}, \widehat{\phi}_{i}, \phi_{i + 1}, \cdots , \phi_{p + 1}\right)
( S_{(A,\gamma_1)}S_{(B,\gamma_2)})^*( S_{(A,\gamma_1)}S_{(B,\gamma_2)}) = P_A S_{\gamma_2}^* S_{\gamma_1}^* S_{\gamma_1}S_{\gamma_2}P_A & = \sum_{\eta_1 \in E_Z}Z^G(\gamma_1,\eta_1) P_A S_{\gamma_2}^* S_{\eta_1}S_{\eta_1}^* S_{\gamma_2}P_A = Z^G(\gamma_1,\gamma_2) S_{(B,\gamma_2)}^* S_{(B,\gamma_2)}.
\Delta (\widetilde{\chi_k u}) = \Delta (\widetilde{\chi_k \eta_k u}) = \chi_k \Delta ( \widetilde{\eta_k u}) + 2 \nabla \chi_k \nabla (\widetilde{\eta_k u}) + (\Delta \chi_k) (\widetilde{\eta_k u}) & = ( \widetilde{\chi_k \mathcal{H}u}) + 2 \nabla \chi_k \nabla (\widetilde{\eta_k u}) + (\Delta \chi_k) (\widetilde{\eta_k u}),
n-n_0(G) =\sum_{i=1}^{n-1}n_i(G)=\sum_{i=1}^{n-1}\frac{1}{i}\left(\sum_{k=1}^i j_{k i}(G)+\sum_{k=i}^{n-1}j_{ik}(G) \right) &=\sum_{(k_1,k_2)\in A(G)}\left(\frac{1}{k_1}+\frac{1}{k_2}\right)j_{k_1k_2}(G)=\sum_{(k_1,k_2)\in A(G)}\frac{k_1+k_2}{k_1k_2}j_{k_1k_2}(G)
a= (\frac{q+1}2)q^{m-1}+(\frac{q-1}2)q^{m-2}+\cdots+(\frac{q-1}2)q^{m-j_4+j_3}+(\frac{q+1}2)q^{m-j_4+j_3-1}&+i_{j_3-1}q^{m-j_4+j_3-2}+\cdots+i_0q^{m-j_4-1}-(\frac{q-3}2)q^{m-j_4-2}-i_{m-2}q^{m-j_4-3}-\cdots-i_{j_4+1}.
\int_{0}^{\infty}e^{-\lambda t}\int_{x}^{\infty}e^{-\xi y}g(x,y,t) \, \mathrm{d}y\,\mathrm{d}t ={}e^{-\xi x}\int_{0}^{\infty}e^{-\lambda t}E_{\frac{1}{2}}\bigl(- \xi t^{\frac{1}{2}}\bigr) \, \mathrm{d}t ={}e^{-\xi x}\frac{\lambda^{\frac{1}{2}-1}}{\xi+ \lambda^{\frac{1}{2}}}
\partial_{\alpha}^n f_j(g)=\sum_{k=1}^n\sum \frac{n!}{(k_1)!...(k_n)!}\partial_{\alpha}^k f_j(\cdot,t)\circ g\prod_{l=1}^n \Big(\frac{\partial_{\alpha}^l g}{l!}\Big)^{k_l},
f_\infty^{\left(s\right)}\left(t\right)= T_{s}^{0}\left(t\right)f_\infty^{\left(s\right)}\left(0\right)+\ell^{-1}\int_{0}^{t}T_{s}^{0}\left(t-\tau\right)C_{s+1}^0 f_\infty^{\left(s+1\right)}\left(\tau\right)d\tau
\hat{\mathbb{E}}\left[\left(\lambda X+f(\lambda)Y\right)^2\right] =\lambda^2\overline{\sigma}^2+f(\lambda)^2\overline{\sigma}^2=\overline{\sigma}^2 -\hat{\mathbb{E}}\left[-\left(\lambda X+f(\lambda)Y\right)^2\right] =\lambda^2\underline{\sigma}^2+f(\lambda)^2\underline{\sigma}^2=\underline{\sigma}^2.
d^2(x_i,x_{i+1})= \|x_i-x_{i+1}\|^2 \leq \frac{2}{\gamma}(\varphi(x_i)-\varphi(x_{i+1}))\leq \frac{2}{\gamma}(\varphi(x_i)-\varphi^*) \leq \frac{2}{\gamma}(1-\frac{\kappa}{2\gamma})^{i-m}(\varphi(x_m)-\varphi^*)
\varphi \left( x^{0},x^{3}\right) =e^{-ik_{0}x^{0}}\chi \left( x^{3}\right),\quad \chi^{\prime \prime}+R\chi =0,\quad R\left( x^{3}\right) =\left(k_{0}+A\right)^{2}-m^{2}-k_{1}^{2}\,.
\mathbf{P}_{0,i}(T_{y}^+ <\infty,H^+_{T_y^+}>y)=\sum_{j,k =\pm1}\int_{0}^{y}U^+_{i,j}({\rm d}z)\left[\mathbf{1}_{(k\neq j)}\Lambda_{j,k}\overline{F}^+_{j,k}(y-z)+\mathbf{1}_{(k=j)}\overline{\Upsilon}_{j}(y-z)\right].
\Bigl(\frac{1}{\lambda(t)}\mathcal{G}(t) \Bigr)^{\prime}\leq -\frac{k_3}{k_2}\frac{\alpha(t)}{\lambda(t)}\Bigl[\frac{1}{\lambda(t)}\mathcal{G}(t)\Bigr]^{\frac{m}{\ell}}.
(det\Phi)^{\prime}=\sum_{i=1}^{n}det(\begin{matrix}\Phi_{1,1}&\Phi_{1,2}&\cdot\cdot\cdot&\Phi_{1,n}\\ \vdots&\vdots&&\vdots\\ \Phi_{i,1}^{\prime}&\Phi_{i,2}^{\prime}&\cdot\cdot\cdot&\Phi_{i,n}^{\prime}\\ \vdots&\vdots&&\vdots\\ \Phi_{n,1}&\Phi_{n,2}&\cdot\cdot\cdot&\Phi_{n,n}\end{matrix}).
\kappa_{\omega}= \frac{2 \Gamma ( \Delta_{\omega})}{\pi \Gamma ( 1 - \Delta_{\omega})}\left( \frac{\sqrt{\pi}\Gamma \left( \frac{1}{2 - 2 \Delta_{\omega}}\right)}{2 \Gamma \left( \frac{\Delta_{\omega}}{2 - 2 \Delta_{\omega}}\right)}\right)^{2 - 2 \Delta_{\omega}}.
\nabla_{\nabla_{Y^T}\partial_t}\partial_t=\nabla_{-\frac{1}{2}u^2d\theta(Y^T)+u^{-1}Y^T(u)\cdot\partial_t}\partial_t &=\nabla_{-\frac{1}{2}u^2d\theta(Y^T)}\partial_t+u^{-1}Y^T(u)\nabla_{\partial_t}\partial_t &=\frac{1}{4}u^4d\theta(d\theta(Y^T))-\frac{1}{2}ud\theta(Y^T,\nabla u)\cdot\partial_t+Y^T(u)\nabla u.
\xi^a = \mu_b \phi^{ba}, I_{ab}= 3 \mu^c \mu_{[c}\phi_{ab]}= -\varepsilon \phi_{ab}- 2 \mu_{[a}\xi_{b]}, J_{ab}= 3 \mu^c \phi_{[ca}\theta_{b]}= \mu^c (\ast \phi)_{cab}, K_{ab}= -2 \mu^c \mu_{[a}\phi_{b]c}= 2 \mu_{[a}\xi_{b]}.
2 e \left( \varepsilon_{\mu \nu}\partial_{\mu}A_{\nu}^{( 1 )}, \bigg [ \frac{- 1}{\triangle}\! - \! N \Big [ \frac{e^{2}\pi}{( \pi \! + \! g N )^{2}}\tilde{Q}+ \frac{g}{\pi}\tilde{C}\Big ] \bigg ] \sum_{b = 1}^{N}\sum_{j = 1}^{n_{b}}U_{1 b}\Big [ \delta_{n}( x_{j}^{( b )}) \! - \! \delta_{n}( y_{j}^{( b )}) \Big ] \right)
R_{ij}\!\! -\!\frac{1}{2}R g_{ij}\! = \! 8\pi GT^{m}_{ij}\! +\! ({\cal R}_{\mu\nu}\!-\!\frac{1}{2}{\cal R}{\cal G}_{\mu\nu})Z^{\mu}_{,i}Z^{\nu}_{,j}\! +\! Q_{ij}+ S_{ij}
\nabla f(\mathbf{x}^{\star}) + \frac{\rho(1-\eta)}{2}\mathbf{E}_{}^T\mathbf{E}_{}\mathbf{x}^{\star}+ \sqrt{\eta}\mathbf{E}_{}^T\boldsymbol{\nu}^{\star}= \mathbf{0}\mathbf{E}_{}\mathbf{x}^{\star}= \mathbf{0}.
\tilde{D}j^{i j}= \left( L^{j}\tilde{D}^{2}L^{i}- L^{i}\tilde{D}^{2}L^{j}\right) = \left( L^{l}k_{l}- L \right) \left( L^{j}k^{i}- L^{i}k^{j}\right) \, ,
\sum_{l , n}\frac{\mu_{p - 1}\lambda^{k + n + l}i^{k}p !}{k ! n ! l ! ( p - l ) !}\partial_{x^{i_{1}}}\ldots \partial_{x^{i_{n}}}C_{i_{1}^{\prime}\ldots i_{2 k}^{\prime}j_{1}\ldots j_{l}[ a_{l + 1}\ldots a_{p}}^{0}S t r \left( \partial_{a_{1}}\phi^{j_{1}}\ldots \partial_{a_{l}]}\phi^{j_{l}}\phi^{i_{1}}\ldots \phi^{i_{n}}\phi^{i_{2 k}^{\prime}}\phi^{i_{2 k - 1}^{\prime}}\ldots \right)
\frac{1}{m!}\sum_{\sigma \in{\rm Sym}(I)}\prod_{j\in I}\cos(2\pi y_{\sigma^{-1}(j)}\xi_j) =\frac{1}{{m \choose l}}\sum_{\substack{J\subseteq I |J|=l}}\prod_{j\in J}\cos(2\pi \xi_j) &=\sum_{S\subseteq I}a_S(\xi) \frac1{{m \choose l}}\sum_{\substack{J\subseteq I |J|=l}}w_S(\varepsilon(J)).
\hskip -1cm \mathbb E\left [\underset{1\leq r\leq n_1n_2}{\max}\;\underset{0\leq k\leq \binom{n_1n_2}{r}}{\max}\left \{\Vert \Pi_{r,k}(E)\Vert^{2}_{2}-K_1 r\log\left (\dfrac{e\,n_1\,n_2}{r}\right )\right \}\right ]\leq c_0\,K^{2}
{\tilde{E}}_{i}= \frac{\mathrm{j}}{2}\left( \begin{array}{c c}{{\tau_{i}}}&{{0}}\\{{0}}&{{\tau_{i}}}\\ \end{array}\right) , \hspace{6 m m}{\tilde{F}}_{i}= \frac{1}{2}\left( \begin{array}{c c}{{0}}&{{\tau_{i}}}\\{{- \tau_{i}}}&{{0}}\\ \end{array}\right) , \hspace{5 m m}i = 1 , 2 , 3 .