harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{d}\hat{\chi}_t(\vartheta)+\mathrm{i}H\hat{\chi}_t(\vartheta)\mathrm{d}t= \Big(\mathrm{M}(t)\mathrm{G}/\|{G}^{k(t)}_0\psi(t)\| -\sqrt{\tfrac{\nu}{\nu_k(t)}}\mathrm{I}\Big)\hat{\chi}_t(\vartheta)\mathrm{d}n_t(\vartheta),
\log \frac{1}{3}\bigg|\frac{\eta\bigl(\frac{-1+2\sqrt{2}i}{3}\bigr)}{\eta(2\sqrt{2}i)}\bigg|^4 = 2 \cdot \frac{\pi}{4}\cdot \frac{\log (3+2\sqrt{2})}{\pi}= \log(1+\sqrt{2}),
\mathcal{R}_\pi\left(f_1^{k_1}\dots f_p^{k_p}g_1^{l_1}\dots g_p^{l_p}\right)=\mathcal{R}_\pi\left(f_1^{k_1}\dots f_p^{k_p}\right)\otimes \mathcal{R}_\pi\left(g_1^{l_1}\dots g_p^{l_p}\right).
q^{r-4c-2a}[2\ell+1-r]\frac{q^{4c+2-2r}K^{-2}-1}{q^{2-2r}K^{-2}-1}+q^{1+2\ell-4c-a}[a]\frac{q^{4c+2-2r}K^{-2}-1}{q^{2-2r}K^{-2}-1}&+q^{-2c+r-1-2\ell-a}[r-2c-a]+q^{-2c+1-2\ell}[2c]\frac{q^{-2a}K^{-2}-1}{q^{2-2r}K^{-2}-1}-q^{-4c+1}[2\ell]\frac{q^{4c}-1}{q^{2-2r}K^{-2}-1}&=[2\ell+1].
Ehr_{\mathcal{P}}(z) = \sum_{i=0}^{d}\sum_{j=0}^{p-1}h_{ip+j}\sum_{t \geq 0}\binom{t+d-i}{d}z^{pt + j}= \sum_{j=0}^{p-1}\sum_{t \geq 0}\left( \sum_{i=0}^{d}h_{ip+j}\binom{t+d-i}{d}\right) z^{pt + j}= \sum_{j=0}^{p-1}\sum_{t \geq 0}f_j(t) z^{pt + j}.
h-h_n =h-h_{n-1}-\frac{2}{a+b}S_{x,\tau}(h-h_{n-1})=\left(I_\mathcal{H}-\frac{2}{b+a}S_{x,\tau}\right)(h-h_{n-1}) &=\cdots=\left(I_\mathcal{H}-\frac{2}{b+a}S_{x,\tau}\right)^nh, ~\forall h \in \mathcal{H}, ~\forall n \geq 1.
\lim\limits_{n\rightarrow \infty}\sup\left\{||Q^n[\varphi](\cdot,x)||:x\geq n(c^*_{L_{k_0}}+\frac{\varepsilon}{3}) \mbox{or}x\leq -n(c^*_{\mathcal{S}\circ\hat{L}_{k_0}\circ \mathcal{S}}+\frac{\varepsilon}{3})\right \}=0.
\overline{R}_1^t=\sum\limits_{k\ge1}\left(2\bar{u}_k\frac{\partial}{\partial \bar{u}_1}\frac{\partial}{\partial\bar{u}_k}-(k+2)\bar{u}_{k}\frac{\partial}{\partial \bar{u}_{k+1}}\right)-2\frac{\partial}{\partial \bar{u}_1}.
W_n^m\approx\sqrt{\frac 2{\pi\nu_{nm}}}\frac{|\eta|^{|n-m|}}{2^{\mu_{nm}}\mu_{nm}!|\xi |^{|n-m|+1}}\left[H_{\mu_{nm}}\left(\sqrt{\frac{|n-m|+1}{2|\xi |^2}}\right)\right]^2.
\mathbb{E}_{f_0,T}\|f_0-\hat{f}\|_{\infty}= \mathbb{E}_{f_0,T}\|f_0-\tilde{f}(\hat{j})\|_{\infty}1_{\hat{j}>j^*}+ \mathbb{E}_{f_0,T}\|f_0-\tilde{f}(\hat{j})\|_{\infty}1_{\hat{j}\leq j^*}
\langle R_0(z)f,g\rangle =\int_{\mathbb{R}^n}\frac{\widehat{f}(\xi)\overline{\widehat{g}(\xi)}}{|\xi|^2-z}d\xi &=\frac12\int_0^\infty\frac{t^{(n-2)/2}}{t-z}\bigg(\int_{|\omega|=1}\widehat{f}(\sqrt{t}\omega)\overline{\widehat{g}(\sqrt{t}\omega)}d\omega\bigg)dt.
T_{\rho \sigma \mu \nu}^{(4)abcd}(p_1,p_2;k_1,k_2)=C_1\gamma_{\rho \sigma\mu \nu}^{(1)}+C_2\gamma_{\rho \sigma \mu \nu}^{(2)}+C_3\gamma_{\rho\sigma \mu \nu}^{(3)}
c_{(1)}^{\ast}=\frac{(m-1)\Gamma(1/2)}{(2\pi)^{1/2}m}\left[\int_{0}^{\infty}u^{1/2-1}\left(1+\frac{2t}{m}\right)^{-(m-1)/2}\mathrm{d}u\right]^{-1}=\frac{(m-1)}{m^{3/2}B(\frac{1}{2},~\frac{m-2}{2})},~if~m>2
V_{ee}^{SCE}[n]=\int ds\frac{n(s)}{N}\sum_{i=1}^{N-1}\sum_{j=i+1}^{N}\frac{1}{|f_{i}(s)-f_{j}(s)|}=\frac{1}{2}\int dsn(s)\sum_{i=2}^{N}\frac{1}{|s-f_{i}(s)|}
L(\varphi_{1},\varphi_{2})=\frac{1}{2}[\partial_{\mu}\varphi_{1}\partial^{\mu}\varphi_{1}-m^{2}\varphi_{1}^{2}]+\frac{1}{2}[\partial_{\mu}\varphi_{2}\partial^{\mu}\varphi_{2}-m^{2}\varphi_{2}^{2}]-\frac{1}{4}\lambda(\varphi_{1}^{2}+\varphi_{2}^{2})^{2},
I_\infty =\frac{1}{2\pi}\int_{-\infty}^{\infty}\Re\left[\frac{1}{2}\frac{\Gamma'}{\Gamma}\left(\frac{s}{2}\right)- \frac{1}{2}\log \pi \right] \Bigl(|\Phi_1(\phi;z)|^2+|\Phi_1(\phi;-z)|^2\Bigr) \, dz,
S={1\over{2\kappa^2_2}}\int d^2x\sqrt{-\tilde{g}}\left[\Phi{\cal R}_{\tilde{g}}+{1\over 2}\partial_{\mu}\Phi\partial^{\mu}\Phi+\Lambda\Phi^{1-{\gamma\over{\delta^2}}}e^{\Phi\over 2}\right].
\begin{array}{c}\displaystyle \partial_{\eta_1}g (\eta_1,\eta_2)= \frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{\eta_1-\cos\theta}{(\eta_1-\cos\theta)^2 + \eta_2^2}\,d\theta,\bigskip \displaystyle \partial_{\eta_2}g (\eta_1,\eta_2)= \frac{1}{2\pi}\int_{-\pi}^{\pi}\frac{\eta_2}{(\eta_1-\cos\theta)^2 + \eta_2^2}\,d\theta,\end{array}
S^{(2)}_z(x) =\frac{C_z}{x^{|z|+1}}{}_4F_{3}\left(\begin{array}{c}\frac{|z|}{2}+\frac{1}{2},\frac{|z|}{2}+\frac{1}{2},\frac{|z|}{2}+1,\frac{|z|}{2}+1 begin{align*}3pt] |z|+1,z_1+1,z_2+1\end{array};\,\frac{16}{x^2}\right), T^{(2)}_z(u) =C_zu^{|z|+1}\sum^{\infty}_{k=0}P^{(z,0)}_{2,k}\Bigl(\frac{1}{3}\Bigr)(-3u^2)^{k}.
(-1)^{(k-1)(m-1)}L_Xi_{Z\wedge Y}\omega =\nabla_X\omega(Z\wedge Y)+\omega(\nabla_X(Z\wedge Y)) (-1)^{(l-1)(m-1)}L_Y i_{Z\wedge X}\omega =\nabla_Y\omega(Z\wedge X)+\omega(\nabla_Y(Z\wedge X)).
C_{+}(\mathbb{R}^{N}):=\bigg\{h\in C(\mathbb{R}^{N}): 1 < \displaystyle{\inf_{x\in \mathbb{R}^{N}}h(x)}\leqslant \displaystyle{\sup_{x\in \mathbb{R}^{N}}h(x)}< \infty \bigg\}
\begin{array}{r c l}{{Z [ J_{\Phi}, J_{\Sigma}, K_{\Phi}, K_{\Sigma}] \! \!}}&{{=}}&{{\! \! \int{\cal D}\Phi \,{\cal D}\Sigma \, \exp [ i \int_{x}L ( \Phi , \Sigma ) + i \int_{x}( J_{\Phi}\Phi + J_{\Sigma}\Sigma )}}\\{{}}&{{}}&{{{}~ ~ + i \int_{x , y}( \Phi K_{\Phi}\Phi / 2 + \Sigma K_{\Sigma}\Sigma / 2 ) ] ~ ,}}\\ \end{array}
(\begin{matrix}M&0\\ 0&N+Q^{T}M^{-1}Q\end{matrix})=(\begin{matrix}I&0\\Q^{T}M^{-1}&I\end{matrix})(\begin{matrix}M&Q\\ -Q^{T}&N\end{matrix})(\begin{matrix}I&-M^{-1}Q\\ 0&I\end{matrix}).
\begin{array}{c}\left(q^{S_3}\otimes{E\over z}+{E\over w}\otimes q^{-S_3}\right)\left|\left({n\over 2},{n\over 2}\right)_p,p\right\rangle= =\left({q^{n/2}\over z}-{q^{-2p-2+{n\over 2}}\over w}\right)\left|{n\over 2},{n\over 2}\right\rangle\otimes\left|{n\over 2},p-{n\over 2}+1\right\rangle+\ldots\end{array}
U_{D}( \lambda ) \psi_{\alpha}( r , p ) = e^{- \lambda / 2}\sqrt{\frac{2 p}{\pi}}\ \frac{r \cos \alpha \cos ( e^{- \lambda}r p ) - \sin \alpha \sin ( e^{- \lambda}r p )}{\sqrt{r^{2}\cos^{2}\alpha + \sin^{2}\alpha}}.
q_i = \begin{cases}W_d\eta(\gamma^{(d)}_i)+(1-a_i)W_a\eta(\gamma^{(r)}_i) \mbox{, $r_i\leq N_r$}a_iW_a\eta(\gamma^{(r)}_i) \mbox{, $r_i= N_r+1$.}\end{cases}
\|\partial_y^{-1}g_{n}(\partial_y \partial_x \eta_1)\|_{L^2_{\ell'}(\mathbb{R}^2_+)}^2 \le C\int_{\mathbb{R}^2_+}\langle y \rangle^{2(\ell' - \ell-1)}\Big(\int^y_0 g_n (t, x, \tilde y)d\tilde y \Big)^2 dx dy & \le C\| g_{n}\|_{L^2_{\ell'}(\mathbb{R}^2_+)}^2.
\sum_s\phi_{is}(\partial_i)\varphi_s =\frac{1}{2}\partial_i(\varphi^{-1}_i)\varphi_i-\sum_{s\neq i}\frac{\Gamma_{is}^s\varphi_s}{\varphi_i}=\frac{1}{2\varphi_i}\partial_i\Big(\sum_s\varphi_s\Big)=0.
\prod_{k=1}^{m}\langle \tilde{\Delta}_{i_k}^n Y,e_{j_k}\rangle = \prod_{k=1}^{m}\left( \langle \Delta_{i_k}^nS,e_{j_k}\rangle + \Delta_n^{(i_k-1)\Delta_n}a^{j_k}\right) = \prod_{k=1}^{m}\langle \Delta_{i_k}^nS,e_{j_k}\rangle+\sum_{y=1,x=2}^m b_{xy}(i_1,...,i_m,j_1,...,j_m),
\mathrm{T r}[ J_{i}^{( 1 )}, J_{j}^{( 1 )}]^{2}= \mathrm{T r}[ J_{i}^{( 2 )}, J_{j}^{( 2 )}]^{2}= - 2 N C_{2}^{( j )}= - \frac{N ( N^{2}- 1 )}{2},
\int_{-\infty}^{0}e^{\theta x}D^q_0(x)dx +\sum_{k=1}^{n^-}\sum_{j=1}^{n_k}\int_{-\infty}^{0}e^{\theta x}D^q_{kj}(x)dx\left(\frac{\vartheta_k}{\vartheta_k+s}\right)^j=\frac{1}{s-\theta}\left(\frac{\psi_q^-(\theta)}{\psi_q^-(s)}-1\right).
I_1^{{\rm NA}}= H_{{\rm IV}}(-q_1, p_1;a_1^{(1)},a_2^{(1)}; b^{(1)})= q_1p_1(q_1+p_1-b^{(1)})-a_1^{(1)}q_1+a_2^{(1)}p_1 I_2^{{\rm NA}}= H_{{\rm IV}}(-q_2, p_2;a_1^{(2)},a_2^{(2)}; b^{(2)})= p_2q_2(p_2+q_2-b^{(2)})-a_1^{(2)}q_2+a_2^{(2)}p_2,
{\bar \beta}^{B}{\bar \pi}_{B}=\pm{\sqrt E}e^{-{i\varphi \over 2}}e^{-i\psi}\cos{\theta \over 2}, \qquad \qquad\beta^{A^\prime}\pi_{A^\prime}=\pm{\sqrt E}e^{{i\varphi \over 2}}e^{i\psi}\cos{\theta \over 2},
V (z,{\overline z}) ={\tilde V}(z,{\overline z}) \exp(i\sum_a H_a \rho_a ({\overline z}) +i\sum_I Q^I \phi^I ({\overline z}) + i\sum_A Q^A \varphi^A (z)) \exp(ik_\mu{\cal X}^\mu (z,{\overline z}))~,
\begin{array}{l l}{{\pi^{\kappa}( \lambda ) | 0 > = 0 ,}}&{{\pi^{\nu}( \lambda ) | 0 > = 0}}\\{{< 0 | q^{\kappa}( \lambda ) = \mathrm{l n}a ( \lambda ) < 0 | ,}}&{{< 0 | q^{\nu}( \lambda ) = \mathrm{l n}d ( \lambda ) < 0 |}}\\ \end{array}< 0 | 0 > = 1
\left( \frac{\partial T}{\partial t}\right)_{w , x^{i}}/ \left( \frac{\partial r}{\partial t}\right)_{w , x^{i}}= - \left( \frac{\partial w}{\partial r}\right)_{T , x^{i}}/ \left( \frac{\partial w}{\partial T}\right)_{r , x^{i}}= \mp \frac{\sqrt{f ( r ) + \dot{a}^{2}}}{f ( r ) \dot{a}( t )}.
L = \frac{1}{2}( \partial_{\mu}\phi^{a i})^{2}+ \frac{u}{4 !}\left( \phi^{a i}\phi^{a i}\right)^{2}+ \frac{v}{4 !}\left[ \left( \phi^{a i}\phi^{b i}\right)^{2}- \phi^{a i}\phi^{a i}\phi^{b j}\phi^{b j}\right]
\operatorname*{l i m}_{\lambda \to \infty}\, \frac{E_{\lambda}( \ell )}{L^{D - 1}}= - \frac{\Gamma ( D + 1 ) \, \zeta ( D + 1 )}{2^{2 D + 1}\pi^{D / 2}\Gamma \left( \frac{D}{2}+ 1 \right)}\, \ell^{- D}.
\left(X^{(N)}_1(t),\cdots, X^{(N)}_N(t);t\ge 0\right)=N\bigg(-\alpha_1^{-}\left(X^{(N)};t\right),\cdots,-\alpha_{N+1-\mathfrak{n}(t)}^{-}\left(X^{(N)};t\right), \alpha_{\mathfrak{n}(t)-1}^{+}\left(X^{(N)};t\right),\cdots,\alpha_1^{+}\left(X^{(N)};t\right);t\ge 0\bigg).
c_{ij}= \int\limits_{-\infty}^{+\infty}f_i(z_n) f_j(z_n)dz_n, c_{m}= \int\limits_{-\infty}^{+\infty}\hat{f}(z_n \mid x_{n-\tau}^{n-1})f_m(z_n)dz_n.
\int_0^T\!\!\hat{b}_h^i(t)dt\leq \int_0^T\!\!\sum_{i=1}^r\hat{b}_h^i(t)dt\leq \int_0^T\!\!\sum_{i=1}^r\xi_h^i(t)dt+ \int_0^T\!\!\sum_{j=1}^s\hat{a}_h^j(t)dt< \int_0^T\!\!\sum_{i=1}^r\xi_h^i(t)dt+R
|H_j|=\frac{\phi(p_1^{a_1})}{\phi(p_1^{b_1})}\cdot\frac{\phi(p_2^{a_2})}{\phi(p_2^{b_2})}\cdots \frac{\phi(p_t^{a_t})}{\phi(p_t^{b_t})}=\frac{\phi(j)}{\phi(r)}
\langle \bar{q}q \rangle = \langle \bar{q}q \rangle_{v}\cos ( 2 \pi ( x - x_{0}) / L ) \ , \quad \langle \bar{q}\, \mathrm{i}\gamma^{5}q \rangle = - \langle \bar{q}q \rangle_{v}\sin ( 2 \pi ( x - x_{0}) / L ) \ ,
\overline{Q}( \bar{r}) = Q ( \bar{r}) -{\frac{M}{4 c^{2}}}\Big [{\frac{1}{\mu}}{\frac{d Q}{d \mu}}\Big ]_{\bar{r}}\bar{w}\ln \big \{{\frac{w_{\!_{\odot}}}{w}}\big \}+{\cal O}\{\bar{w}\}\ .
S_{f , i}= \lim_{\tau \rightarrow - \infty , \tau^{\prime}\rightarrow + \infty}-{2 e^{- \tau}e^{\tau^{\prime}}}[ ( \frac{\partial^{2}G}{\partial \tau \partial \tau^{\prime}}+ \frac{\partial G}{\partial \tau^{\prime}}) - ( \frac{\partial G}{\partial \tau}+ G ) ]
\alpha_1 = e_1 - e_2, \alpha_2 = e_2 - e_3, \alpha_3 = e_3 - e_4, \alpha_4 = e_3 + e_4, e_1 = \alpha_1 + \alpha_2 + \frac{\alpha_3 + \alpha_4}{2}, e_2 = \alpha_2 + \frac{\alpha_3 + \alpha_4}{2}, e_3 = \frac{\alpha_3 + \alpha_4}{2}, e_4 = \frac{\alpha_4 - \alpha_4}{2}.
\int_{\Omega}\psi \partial_{t}\left[ g_{1-\alpha,m}*(u-u_{0}) \right]dx+ \mathcal{E}(h_{m}*u,\psi) &\quad\quad\quad\quad\quad\quad\quad\quad\quad= (\geq \leq) \int_{\Omega}(h_{m}*f)\psi dx \, \,\,t\in (0,T), \, m\in\mathbb{N}.
\frac{1+e^{\vert C(t)\vert}}{\vert C'(t)\vert}\leq \frac{1+(t-x)^{-p^+}e^{C'}}{\displaystyle\frac{p^-}{t-x}-p^-\frac{t-x+2H}{(t-x+H)^2}}=\frac{1}{p^-+O((t-x))}((t-x)+(t-x)^{1-p^+}e^{C'})\to 0
\begin{array}{ll}D=1: Z_E(\tau,z)=0\,, begin{align*}2mm]D=2: Z_E(\tau,z)=Z_T(\tau,z)P(\tau,z)^{r-2}\,, begin{align*}2mm]D=3,5: Z_E(\tau,z)=\frac{\chi(E)}{\chi(T)}Z_T(\tau,z)P(\tau,z)^{r-D}\,.\end{array}
S := \frac{1}{2}\left(1 + \sqrt{s_1^2-4}\right) \quad\quad1\le S < \infty s_1 \in [2,\infty), \intertext{and the unstable fixed point}U := \frac{1}{2}\left(1 - \sqrt{s_1^2-4}\right) \quad 0< U \le 1 \quad\s_1 \in [2,\infty).
\begin{array}{ll}\lefteqn{\!\! J^P=5^-:}\vspace{1mm}\!\! 0 \;\; = \{\{S_{2},S_{1}\}_{3},S_{2}\}_{5}\,; \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad\qquad \end{array}
C_s(\{(W_t,{V}_t): t \in \theta\};cr) &= \lim_{n\rightarrow \infty}\frac{1}{n}\max_{\Lambda_n}\Bigl(\inf_{B_q \in Conv((B_t)_{t\in \theta})}\chi(p_U;B_q^{\otimes n})-\max_{t^n\in \theta^n}\chi(p_U;Z_{t^n})\Bigr)
\left[Q, \Phi_1([[x,y]_\mathfrak{g},z]_\mathfrak{g})\right]+\circlearrowleft(x,y,z) =-\left[\Phi_0\left([x,y]_\mathfrak{g}\right),\Phi_0(z)\right]+\circlearrowleft(x,y,z) &=[[\Phi_0(z),Q],\Phi_1(x,y)]-[[\Phi_1(x,y),\Phi_0(z)],Q]+\circlearrowleft(x,y,z) &=[Q,[\Phi_0(x),\Phi_1(y,z)]]+\circlearrowleft(x,y,z).
P_{m_1,m_2,\theta}^{\alpha,\beta,\gamma}(\sigma) := \frac{(-1)^{\theta_1 h_1(\sigma) + \theta_2 h_2(\sigma)}\alpha^{\# \{x : \sigma(x) = x \}}\beta^{\# \{x : \sigma(x) = [x + \mathbf{e}^1] \}}\gamma^{\# \{x : \sigma(x) = [x + \mathbf{e}^2] \}}}{Y_{m_1,m_2,\theta}(\alpha,\beta,\gamma)}.
\begin{matrix}p = 1\oplus0\,\oplus \begin{pmatrix}1 0 0 0\end{pmatrix}\oplus\,1\oplus0 begin{align*}4ex]q = 0\oplus1\,\oplus \left(\begin{smallmatrix}t \sqrt{t(1-t)}\sqrt{t(1-t)}1-t\end{smallmatrix}\right) \oplus\,1\oplus0.\end{matrix}
L(a(1+q), b(1+q)^2) = \frac{1-\sqrt{(1-2a-2aq)^2-4b(1+q)^2}\,}{2}=\frac{1-\sqrt{(1-2a)^2-4b -2q(2a(1-2a)+4b)+q^2(4a^2-4b)}\,}2.
A(t)u(t)=A(t)e^{-t BA(t)}u_{0}+A(t)\int_{0}^{t}{e^{-(t-l) BA(t)}B(\mathcal{A}(t)-\mathcal{A}(l))u(l)dl}&+A(t)\int_{0}^{t}{e^{-(t-l) BA(t)}(-P(l))u(l)dl}+A(t)\int_{0}^{t}{e^{-(t-l) BA(t)}f(l)}dl.
\int_{-\infty}^{\infty}e^{-\phi(x-b)}dV_q(x)-e^{\phi (b-y)}\psi_q^+(-\phi)\psi_q^-(\phi) &=\phi \int_{-\infty}^{\infty}e^{-\phi(x-b)}\left(V_q(x)-\mathbb P_x\left(X_{e(q)}> y\right)\right)dx.
\bar{z}(t,\omega) =\bar{z}(T_1,\omega)+\frac{1}{N}\sum\limits_{j=1}^{N}\int_{T_1}^{t}e^{-O_s^{(j)}(\omega)}f^{(j)}(e^{O_s^{(j)}(\omega)}\bar{z}(s,\omega))ds &+\frac{1}{N}\sum\limits_{j=1}^{N}\int_{T_1}^{t}O_s^{(j)}(\omega)\bar{z}(s,\omega)ds
\dfrac{n+2s}{2}+\dfrac{1}{2}\sqrt{(n-2s)^{2}+8s}=n+\dfrac{\sqrt{(n-2s)^{2}+8s}-\left( n-2s\right)}{2}& =n+\dfrac{4s}{\sqrt{(n-2s)^{2}+8s}+\left( n-2s\right)}<n+\dfrac{2s}{n-2s}.
V = G_{N}{\frac{m_{1}m_{2}}{r}}+ \int_{0}^{\infty}{\frac{d m}{k}}G_{N}{\frac{m}{k}}{\frac{m_{1}m_{2}e^{- m r}}{r}}= G_{N}{\frac{m_{1}m_{2}}{r}}\left( 1 +{\frac{1}{k^{2}r^{2}}}\right) \; .
\bar{S_s}^{ab}= \frac{2^{n-2}}{(d-2)!}\varepsilon^{ab}{}_{a_1 a_2\ldots a_{d-2}}S_s^{a_1 a_2}S_s^{a_3 a_4}\ldots = \frac{(-i)^{n-1}}{2}a^a a^b\,\, (a^0 a^1\ldots a^{2n}).
F^q_{asym}(\nu,\tilde{w})=\int_0^{1}\sqrt{\frac{1+\frac{1}{\nu}\log\left(\frac{2(\tilde{y}^\prime\tilde{y}_c +1)(\tilde{w}-\tilde{y}^\prime\tilde{y}_c-2)}{(\tilde{y}^\prime\tilde{y}_c+2)(\tilde{w}-2)}\right)}{1+\frac{1}{\nu}\log\left(\frac{2\left(1-\sqrt{\tilde{w}}\right)^2}{\tilde{w}-2}\right)}}~d\tilde{y}^\prime.
d s_{1 0 E}^{2}= \left( \frac{4 a}{K_{*}}\right)^{3 / 4}e^{3 P^{2}( \eta + \phi / 4 )}v^{- 3 / 4}\left[ ( 1 - v ) d X_{0}^{2}+ d X_{1}^{2}\right] +
\left(z^{-1}\alpha^{-1}q^{1+n+k};q\right)_{\infty}=(-1)^{n}\alpha^{n}z^{n}q^{-\frac{1}{2}n(n+1)-kn}\left(q^{-k}z\alpha;q\right)_{-n}\left(q^{k+1}z^{-1}\alpha^{-1};q\right)_{\!\infty},
\frac{1}{q}D_{q^{-1}}D_qy(x)+ \dfrac{x-q(a+b-qab)}{ab q^2 (1-q)(1-x)}D_{q^{-1}}y(x) - \frac{q^{-n-1}[n]_q}{ab(1-q)(1-x)}y(x) =0.
\langle\prod_{i=1}^{N}V_{\alpha_{i},\overline{\alpha}_{i}}(z_{i})\rangle\propto\prod_{i<j}(z_{i}-z_{j})^{-2\alpha_{i}\alpha_{j}}(\overline{z}_{i}-\overline{z}_{j})^{-2\overline{\alpha}_{i}\overline{\alpha}_{j}}
\left\vert \widehat{H}_{R}(0)\right\vert +\sum_{n\in\mathbb{Z}^{2},\,0<|n|<R}\left( \left\vert \widehat{\chi}_{K}(n)\right\vert +\left\vert\widehat{H}_{R}(n)\right\vert \right) \left\vert \frac{1}{N}\sum_{j=1}^{N}e^{2\pi in\cdot t(j)}\right\vert.
\psi(x,z) = - \beta(x) z + \kappa(x) \int_0^\infty (e^{-z y}- 1+ z y) \frac{dy}{\Gamma(- \gamma(x)) y^{1+ \gamma(x)}}&= -\beta (x) z + \kappa(x) z^{\gamma(x)}, x\in E, z \geq 0,
\sup_{t\in (0,T)}\|\Theta_a (b^n-b_{\Gamma}^n)\|_1 + \int_0^T \int_{\Omega}T_n(\mu^n) |\nabla T_a (b^n-b_{\Gamma}^n)|^2\; dx \; dt \le C(a) + \|\Theta_a (b^{n,k}_0-b_{\Gamma}^n(0))\|_1.
(1-s)\prod{\lambda_j}\int_{\bar{y}\in \mathbb{R}^{n-1}}{\frac{u(y_1,y_2,...,y_{n-1},0)-u(0)}{(\lambda_1^2y_1^2+...+\lambda_{n-1}^2y_{n-1}^2)^{\frac{n+2s-1}{2}}}d\bar{y}}=-A<0.
\begin{array}{c c c}{{\hat{u}\cdot \vec{x}_{c}\leq l ~ ~}}&{{~ ~ \mathrm{o r}~ ~}}&{{~ ~ \hat{u}\cdot ( \vec{x}_{a}- \vec{x}_{c}) = \hat{u}\cdot ( \vec{x}_{b}- \vec{x}_{c}) \geq 0 \, .}}\\ \end{array}
\hat{\varphi}(t) = \frac{1}{t( T-t)}\max_{x\in [0,L]}\psi( x) =\frac{\psi( 0)}{t( T-t)}\check{\varphi}(t)= \frac{1}{t( T-t)}\min_{x\in [0,L]}\psi (x) =\frac{\psi( l_{3})}{t( T-t)},
U_{N}=\left[\begin{array}[c]{ccc}0 N^{2}0 0 0 0 0 0 0\end{array}\right] ,V_{N,\ell}=\left[\begin{array}[c]{ccc}0 N^{2}\frac{\ell}{2}0 0 \frac{\ell}{N^{2}}0 0 0\end{array}\right] ,\ell=1,\cdots,N.
\Sigma := \begin{pmatrix}1 0 0 - 1\end{pmatrix}\, , \Pi_0 := \frac12 \begin{pmatrix}\pi_0 \pi_0 - \pi_0 - \pi_0\end{pmatrix}\,,{\mathbb I}_2 := \begin{pmatrix}1 0 0 1\end{pmatrix}\, ,
\frac{A_n^{(2)}(z)}{n^6 z}\frac{A_n^{(2)}(0)}{n^6 z}= \mathcal{O}\left(n^{-1}\right) \frac{A_n^{(1)}(0)A_n^{(1)}(z) A_n^{(1)}(0)}{n^9z^3}\frac{A_n^{(2)}(0)}{n^6 z}= \mathcal{O}\left(n^{-\frac{3}{2}}\right) \frac{A_n^{(2)}(0)}{n^6 z}= \mathcal{O}(1).
\textbf{R}=\rho_{f}\hat{\textbf{G}}^{T}\textbf{P}\textbf{P}^{H}\hat{\textbf{G}}^{\ast}\left ( \rho_{f}\tilde{\textbf{G}}^{T}\textbf{P}\textbf{P}^{H}\tilde{\textbf{G}}^{\ast}+\sigma_{w}^{2}\textbf{I}_K\right )^{-1}
f(c_{i}) =6i-5, 1\leq i\leq n f(p_{i}) =6i-1, 1\leq i\leq n f(o_{i,1}) =6i-2, 1\leq i\leq n f(o_{i,2}) =6i-3, 1\leq i\leq n f(o_{i,3}) =6i-4, 1\leq i\leq n f(o_{i,4}) =6i , 1\leq i\leq n.
\hat{V}^{C}_{\sigma}(C_{1},C_{2}) = \int\limits_{-\infty}^{\infty}\int\limits_{-\infty}^{\infty}\frac{1}{N}\sum\limits_{i=1}^N \frac{1}{(4\pi^{2}\sigma^{4})}exp \left ( - \frac{1}{2\sigma^{2}}a(u_1-b)^{2}\right) exp \left( - \frac{1}{2\sigma^{2}}a'(u_2-b')^{2}\right) exp \left( - \frac{c}{2\sigma^{2}}\right ) \mathrm{d}u_1\mathrm{d}u_2
a_d(n) ={}da_d(n-1) + da_d(n-2) - \binom{d}{2}a_d(n-3) + da_d(n-4) - d^2a_d(n-5) &+ da_d(n-6) - \sum_{k \geq 8}^{n+1}\mu_d(k) a_d(n+1-k).
\widetilde{\eta}_{( s )}^{N}\widetilde{\overline{{{\eta}}}}_{( s )}^{N}= \widetilde{\eta}_{( t )}^{N}\widetilde{\overline{{{\eta}}}}_{( t )}^{N}= 1_{2}\otimes \frac{1 + \rho \overline{{{\gamma}}}_{5}}{2}
\begin{gathered}D_c\cdot\frac{\d R_1(t)}{\d t}\bigg|_{t=j}\in\mathbb Z\quad\; j\in\mathbb Z\quad D_c\cdot\frac{R_1(j)-R_1(m)}{j-m}\in\mathbb Z\quad\; j,m\in\mathbb Z, \; j\ne m.\end{gathered}
\begin{aligned}u^{2}+ \epsilon a(u) u_{x}+ \epsilon^{2}b_{1}(u) u_{xx}+\epsilon^{3}c_{1}(u) u_{xxx}+\epsilon^{4}d_{1}(u) u_{4x}+\epsilon^{5}e_{1}(u) u_{5x}+\mathcal{O}(\epsilon^6) \end{aligned}
\sum_{v=0}^{\ell^m-1}e^{\frac{2\pi inw_v}{24\ell^m}}=\sum_{v=0}^{\ell^m-1}e^{\frac{2\pi inv}{\ell^m}}=\begin{cases}\ell^m n\equiv 0\pmod{\ell^m}0 \end{cases}
(1+\lambda^{2})(z^{2}+2C_{1}) = \int{(1+u)^{-3/2}}du. (1+\lambda^{2})(z^{2}+2C_{1}) = -2(1+(1+\lambda^{2})(z')^{2})^{-1/2}(z')^{2}=\frac{4}{(1+\lambda^{2})^{3}(z^{2}+2C_{1})^{2}}-\frac{1}{(1+\lambda^{2})}z'(x) = \pm\sqrt{\frac{4-(1+\lambda^2)^2(z^2 + 2C_1)^2}{(1+\lambda^2)^3(z^2 + 2C_1)^2}}
\begin{array}{l l l}{{J}}&{{\rightarrow}}&{{J -{\frac{i m \pi}{2}}}}\\{{J_{1}}}&{{\rightarrow}}&{{J_{1}+{\frac{i m \pi}{4}}}}\\{{\varepsilon}}&{{\rightarrow}}&{{\varepsilon}}\\{{A}}&{{\rightarrow}}&{{A +{\frac{i m \pi}{4}}}}\\ \end{array}
Z_\uparrow (w; 0,a) = \mathrm{e}^{-(\lambda+w )a}+ \lambda Z_\uparrow (w; 0,a) \int_{0}^{a}\mathrm{e}^{-(\lambda+w ) t}\frac{1}{Z_\uparrow (w; 0,tp)}\mathrm{d}t.
L_{i,j}(z)L_{k,l}(\frac{z}{q})-L_{i,l}(z)L_{k,j}(\frac{z}{q})=L_{k,l}(z)L_{i,j}(\frac{z}{q})-L_{k,j}(z)L_{i,l}(\frac{z}{q}) \forall \,i,j,k,l=1,\dots, N.
\Psi(x) \,= \,\sum_{\sigma=+j}^{-j}\int{d^3p\over (2\pi)^{3}}{1\over 2\,\omega_{\vec p}}\Big[ u_\sigma(\vec p\,)\, a_\sigma(\vec p\, )\, e^{-i p\cdot x}+ v_\sigma(\vec p\,) \,b^\dagger_\sigma(\vec p\,) \,e^{i p\cdot x}\Bigr ]\quad,
-a_{21}M\hat{f}^*L_1|_{\hat{C}'_1}\sim_{\mathbb{Q}}a_{21}(Mm(K_{\hat{X}}+ \hat{B}) - M\hat{f}^*L_1)|_{\hat{C}'_1}&\sim_{\mathbb{Q}}a_{21}\hat{\Gamma}_1|_{\hat{C}'_1}\sim_{\mathbb{Q}}a_{11}a_{21}\hat{G}_1|_{\hat{C}'_1}&\sim_{\mathbb{Q}}a_{11}\hat{\Gamma}_2|_{\hat{C}'_1}\sim_{\mathbb{Q}}-a_{11}M\hat{f}^*L_2|_{\hat{C}'_1}
\lim_{k_0<k\to\infty}\frac{F_{k+1}^{\bf a}}{F_{k}^{\bf a}}=\lambda_{i_0}\cdot\lim_{k_0<k\to\infty}\biggr(\frac{F_{k+1}^{\bf a}}{(k+1)^{j_0}\lambda_{i_0}^{k+1}}\cdot\frac{k^{j_0}\lambda_{i_0}^k}{F_k^{\bf a}}\biggr)=\lambda_{i_0}.
\begin{aligned}\frac{|\tilde{x}-\tilde{y}|^4}{\varepsilon^4}+\alpha(|\tilde{x}|^2+|\tilde{y}|^2) &\leq u(\tilde{x}, \tilde{t})-v(\tilde{y}, \tilde{t})- u(x_1, t_1)+v(x_1, t_1)+2\alpha|x_1|^2+{\lambda\over T-t_1}-{\lambda\over T-\tilde{t}}.\end{aligned}
\begin{aligned}\quad+\mathbb{E}\bigg(\int_0^\infty e^{-\beta_5kt}\int_{\mathcal{E}}\bigg[\bigg(\int_0^1l_xd\theta-\bar{l}_x\bigg)x_{(1,t)}+\bigg(\int_0^1l_ud\theta-\bar{l}_u\bigg)v_t\bigg]^{2k}\nu(de)dt\bigg)\bigg] &:=\mathbf{A}_1+\mathbf{A}_2+\mathbf{A}_3+\mathbf{A}_4+\mathbf{A}_5.\end{aligned}
\nu_{15,Z(\mathbb{Z}_{15})}= \big\{\{3,3\}, \{3,5\}, \{3,6\}, \{3,9\}, \{3,10\}, \{3,12\}, \{5,5\}, \{5,6\}, & \{5,9\}, \{5,10\}, \{5,12\}, \{6,9\}, \{9,10\}, \{9,12\}\big\}.
\partial_{tr}v_{4}(t,r) = \frac{(t\partial_{t}^{2}v_{4}+ r \partial_{tr}v_{4})}{r}- \frac{t}{r}\partial_{t}^{2}v_{4}&=\frac{V(\partial_{t}v_{4})}{r}- \frac{t}{r}\partial_{t}^{2}v_{4}
\begin{cases}k(x-\lambda)\prod_{i=1}^{\frac{D-1}{2}}(x-\alpha_i)^2-\prod_{i=1}^{\frac{D-1}{2}}(x-\beta_i)^2=kx\prod_{i=1}^{\frac{D-1}{2}}(x-\gamma_i)^2 k(x-\lambda)\prod_{i=1}^{\frac{D-1}{2}}(x-\alpha_i)^2-\lambda\prod_{i=1}^{\frac{D-1}{2}}(x-\beta_i)^2=k(x-1)\prod_{i=1}^{\frac{D-1}{2}}(x-\delta_i)^2\end{cases}
\frac{\Gamma_q(\lambda_i + \theta(N - i) - z)}{\Gamma_q(\lambda_{i+1}+ \theta(N - i) - z)}=\prod_{n = \lambda_{i+1}}^{\lambda_i - 1}{[n + \theta(N - i) - z]_q}=\prod_{n = \lambda_{i+1}}^{\lambda_i - 1}{\frac{1 - q^{n + \theta(N - i) - z}}{1 - q}},
\langle J_Ax-J_Ay,x-y\rangle &\geq\tfrac{1}{2\beta}\|x-y+(\beta-1)(J_Ax-J_Ay)\|^2+(1-\beta)\|J_Ax-J_Ay\|^2 &=\tfrac{1}{2\beta}\left(\|x-y\|^2+2(\beta-1)\langle J_Ax-J_Ay,x-y\rangle +(\beta-1)^2\|J_Ax-J_Ay\|^2\right) &\quad+(1-\beta)\|J_Ax-J_Ay\|^2
1 = J_{h}\int \prod_{k = 2}^{\cal N}d{z^{\prime}}_{k}^{c}d \bar{z^{\prime}}_{k}^{c}D [ \delta \sigma_{R}] D [ \xi^{0}] e^{- 2 C ( \delta \sigma_{R}, \delta \sigma_{R})}e^{- ( P \xi^{0}, P \xi^{0})}e^{- 2 \delta{z^{\prime}}_{k}^{c}\delta \bar{z^{\prime}}_{l}^{c}( P \xi^{k}, Q_{n}) ( Q_{n}, \bar{Q}_{s})^{- 1}( \bar{Q}_{s}, P \bar{\xi}^{l})}
\mathrm{d e t}\,{\cal O}= \left[ \frac{b}{\alpha r}\tilde{\lambda}^{2}+ \frac{b^{2}}{4 Q^{2}}\right]^{2}- \left[ \frac{2 Q_{M}b \alpha r}{4 Q^{2}}- 4 Q_{M}( 1 - \tilde{\lambda}^{2}) \right]^{2}\, .
{\cal L}_{B,n,i}^{(\alpha )}(x) = \frac{{\xi_{n,i}^{(\alpha )}}}{{x - x_{n,i}^{(\alpha )}}}/\sum\limits_{j = 0}^n{\frac{{\xi_{n,j}^{(\alpha )}}}{{x - x_{n,j}^{(\alpha )}}}},i = 0, \ldots ,n,
\prod_{\substack{n\geq 1 n\not\equiv 0, (k_0 +1)}}(1-q^n)^{-1}\prod_{\substack{n\geq 1 n\equiv k_0 +1}}(1-q^n) = \sum_{n_1,n_2,\ldots,n_{k/2}\geq 0}\frac{q^{N_{1}^{2}+N_{2}^{2}+...+ N^{2}_{k/2}}}{(q)_{n_1}(q)_{n_2}\cdots (q)_{n_{k/2-1}}(q^2)_{n_{k/2}}}.
\frac{\mathcal{C}_1(-,?)}{I_1}\mathrm{rad}\left[ \frac{\mathcal{C}_1(-,?)}{I_1}\right]\frac{\mathcal{C}_1(-,?)}{I_1}= \frac{J_1}{J_2}\mathrm{rad}\left[ \frac{\mathcal{C}_2(-,?)}{J_2}\right]\frac{J_1}{J_2}=0.
I:= \int_{|z|\geq\delta}\psi_k(z)\, d\mu_{1,k}(z)=\int_{\delta\leq |z|\leq K}\psi_k(z)\, d\mu_{1,k}(z) +\int_{|z|\geq K}\psi_k(z)\,d\mu_{1,k}(z)=I_1+I_2, J:= \int_{|z|\geq\delta}\psi(z)\, d\mu_{1}(z)=\int_{\delta\leq |z|\leq K}\psi(z)\, d\mu_{1}(z) +\int_{|z|\geq K}\psi(z)\,d\mu_{1}(z)=J_1+J_2.
CL( x(u,z) (N-1,N)) =CL(x(u,z) ) \sigma =C^{\prime}\left( x_{N}x_{N-1}\right)^{-\gamma\kappa}\rho\big(z^{-\kappa},z^{\kappa}\big) \sum_{n=0}^{\infty}\alpha_{n}(u)z^{n}\sigma \hphantom{CL( x(u,z) (N-1,N))}{}=C^{\prime}\sigma ( x_{N}x_{N-1})^{-\gamma\kappa}\rho\big(z^{-\kappa},z^{\kappa}\big) \sum_{n=0}^{\infty}\alpha_{n}(u)(-1)^{n}z^{n},
\eta_{w}^{\ast}([\varphi_{T_{k}}\varphi_{T_{k}^{\ast}}]) =\frac{[\varphi_{\Omega_{w}^{-1}(T_{k})}]}{[\varphi_{I(T_{k})}]}\cdot\frac{[\varphi_{\Omega_{w}^{-1}(T_{k}^{\ast})}]}{[\varphi_{I(T_{k}^{\ast})}]}=\frac{[\varphi_{\Omega_{w}^{-1}(T_{k})}\varphi_{\Omega_{w}^{-1}(T_{k}^{\ast})}]}{[\varphi_{I(T_{k}\oplus T_{k}^{\ast})}]}.
\mathbf{A}(Q_{1r}-Q_{2r}) =(m_{1}+m_{2})(Q_{1r}-Q_{2r})f(\|Q_{1r}-Q_{2r}\|) +\sum\limits_{j=3}^{l}m_{j}\left((Q_{1r}-Q_{jr})f(\|Q_{1r}-Q_{jr}\|)+(Q_{jr}-Q_{2r})f(\|Q_{jr}-Q_{2r}\|)\right) +\sum\limits_{j=l+1}^{n}m_{j}\left((Q_{1r}-Q_{jr})f(\|Q_{1r}-Q_{jr}\|)+(Q_{jr}-Q_{2r})f(\|Q_{jr}-Q_{2r}\|)\right)
\sum\limits_{i=1,n+1}\sigma_{i}\Big(\sum\limits_{j\in I}k_{ij}^{\mathbf{c}^t}\sigma_{j}-2\mu_{i}\Big)+2\sum\limits_{i\in I\setminus\{1,n+1\}}\sigma_{i}\Big(\sum\limits_{j\in I}k_{ij}^{\mathbf{c}^t}\sigma_{j}-2\mu_{i}\Big)=2\Big(\sum\limits_{i=1,n+1}\mu_i\sigma_i+2\sum\limits_{i\in I\setminus\{1,n+1\}}\mu_i\sigma_i\Big),
\vec{E}_{r}^{\prime}= - \frac{e_{0}}{4 \pi \varepsilon_{0}}\frac{\vec{R}_{\vartheta \varphi}^{\prime}}{r^{2}\operatorname{s i n}\vartheta}= - \frac{e_{0}}{4 \pi \varepsilon_{0}}\frac{\vec{e}_{3}}{r^{2}}.