harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathcal{O}(\varepsilon,|\mathbf{z}|^0):\mathbf{A}\mathbf{w}_1^{\mathbf{0}}(\phi)+\mathbf{G}_{ext}(\phi)=\frac{\partial \mathbf{W}_0}{\partial \mathbf{z}}\mathbf{r}_1^{\mathbf{0}}(\phi)+\frac{\partial \mathbf{w}_1^{\mathbf{0}}}{\partial \mathbf{\phi}}\Omega.
[a_{(m)}, b_{(n)}] =\sum_{i\geq 0}\begin{pmatrix}m i\end{pmatrix}(a_{(i)}b)_{(m+n-i)}, &(a_{(m)}b)_{(n)}=\sum_{j\geq 0}(-1)^j\begin{pmatrix}m j\end{pmatrix}(a_{(m-j)}b_{(n+j)}-(-1)^mb_{(m+n-j)}a_{(j)}).
\partial_t |\nabla r|^2=\frac{1}{2}\Delta |\nabla r |^2 - |\nabla^2 r|^2 +\frac{2}{\varepsilon}\nabla r\cdot \nabla g^{\delta,h}_q (|\nabla r|^2 -1) + \frac{2}{\varepsilon}\nabla r\cdot \nabla |\nabla r|^2.
\begin{cases}\partial_t u+u\cdot\nabla u-\Delta u+\nabla P=-\nabla\cdot(\nabla d\odot \nabla d){\rm{in}}\ \ \Omega\times (0,+\infty), \nabla\cdot u = 0{\rm{in}}\ \ \Omega\times (0,+\infty), \partial_t d+u\cdot\nabla d=\Delta d+\frac{1}{\epsilon^2}(1-|d|^2) d{\rm{in}}\ \ \Omega\times (0,+\infty).\end{cases}
\hat M^{(k)}_{i,j}= \begin{cases}0, \mbox{if}i\in [k+1,n] \mbox{and}j=k,{\rm fl}\Big( \hat M^{(k-1)}_{i,j}-{\rm fl}\Big({\rm fl}\Big(\frac{\hat M^{(k-1)}_{i,k}}{\hat M^{(k-1)}_{k,k}}\Big)\, \hat M^{(k-1)}_{k,j}\Big) \Big), \mbox{if}i,j \in [k+1,n], \hat M^{(k-1)}_{i,j}, \mbox{otherwise}\end{cases}
\hskip 0.3 cm=\hskip -0.1 cm\sum_{j=1}^{m_1}\frac{1}{\|u'\|^2}h_{f_1f_2}(\nabla_{u'_j}grad(\varphi_1^h),u'_j)+\big(\frac{1}{f_1^h}\big)^2\!\!\!\!\sum_{j=m_1+1}^{m_1+m_2}\!\!\!h_{f_1f_2}(\nabla_{e_j^v}grad(\varphi_1^h),e_j^v)
\left( \begin{array}{c c}{\partial_{r}-{\frac{a + j - 1 / 2}{r}}}{- \sqrt{2}e v f}{- \sqrt{2}e v f}{\partial_{r}+{\frac{j + 1 / 2}{r}}}\end{array}\right) \left( \begin{array}{c}{h_{2}( r )}{h_{3}( r )}\end{array}\right) = 0 ,
\phi_1(x,y,t;,\xi,\tau,s,\theta)= (x-y)\cdot\xi+(t-s)\tau+s|\xi|\sin\theta-\theta\varpi/2\pi, \phi_2(x,y,t;,\xi,\tau,s,r)= (x-y)\cdot\xi+(t-s)\tau +i(|\xi|s\sinh r+\varpi r/2\pi).
\int_{\Omega^{\varepsilon}}\left(p^{\varepsilon}-\bar{p}\right)\phi dx =\sum_{k\in\mathbb{Z}^{d}}\int_{\varepsilon Y_{1}^{k}}\left(p^{\varepsilon}-\bar{p}\right)\phi dx & \le C\int_{\varepsilon Y_{1}}\left(p^{\varepsilon}-\bar{p}\right)\phi dx.
H(\sigma\,|\,X_1,\dots,X_k) \stackrel{(d)}{=}H(X_1,\dots,X_k\,|\,\sigma) + H(\sigma) - H(X_1,\dots,X_k) &\stackrel{(e)}{=}kH(X_1\,|\,\sigma) + 1 - H(X_1,\dots,X_k) &\stackrel{(f)}{\ge}kH(X_1\,|\,\sigma) + 1 - kH(X_1),
\tilde{a}(x,w,w_x):= \frac{x_\xi^2}{l(x)}\cdot a(x,L^{-1}(w),\partial_xL^{-1}(w)) \tilde{f}(x,w,w_x):= g_n(\mu,x,L^{-1}(w),\partial_xL^{-1}(w))+ (w_x\cdot x_{\xi\xi}-\partial_\xi^2u_-)-\frac{l_{\xi\xi}\cdot (w-u_-)_\xi}{l}-\lambda a(x,L^{-1}(w),\partial_xL^{-1}(w))\cdot L^{-1}(w).
{\cal L}={\bar{\nu}}_{e}\left( i \not \partial - m_{e}\right) \nu_{e}+{\bar{\nu}}_{\mu}\left( i \not \partial - m_{\mu}\right) \nu_{\mu}- \; m_{e \mu}\; \left({\bar{\nu}}_{e}\nu_{\mu}+{\bar{\nu}}_{\mu}\nu_{e}\right) \; .
\begin{array}{c}J^{+}=q^{-J^{0}+\frac{1}{2}}\a^{+}_{1}a_{2}J^{-}=q^{-J^{o}-\frac{1}{2}}a^{+}_{2}a_{1}2J^{0}=N_{1}-N_{2}\left [ J^{+},J^{-}\right ]=[2J^{0}]_{q}=[N_{1}-N_{2}]_{q}\left [N,J^{\pm}\right ]=[N,J^{0}]=0.\end{array}
G_\mu(x,\alpha_{k+1}(T_{x|n})) \geq Z_{x|n}(\alpha_{k+1}(T_{x|n}))\mu([x|n]) &\geq Z_{x|n}(\alpha_{k+1}(T_{x|n}))\phi^{t+\epsilon}(T_{x|n})\qquad\mbox{(by \eqref{e-GZ1})}&=\frac{\alpha_{k+1}(T_{x|n})^k}{\phi^k(T_{x|n})}\cdot \phi^{t+\epsilon}(T_{x|n}) &=\alpha_{k+1}(T_{x|n})^{t+\epsilon},
0=\frac{1}{2}\mathcal{L}S=\sum_{i,j,k,p}(h_{ijk}^{p^{\ast}})^{2}+S(1-\frac{3}{2}S)+2H^{2}S-\frac{1}{2}H^{4}-\sum_{j,k,p,q}H^{p^{\ast}}h_{jk}^{p^{\ast}}H^{q^{\ast}}h_{jk}^{q^{\ast}}.
[\hat{p}_{\alpha}+ \beta_{\alpha \beta}\hat{\epsilon}^{\beta}, \hat{p}_{\alpha'}+ \beta_{\alpha' \beta'}\hat{\epsilon}^{\beta'}] = i (\beta_{\alpha \alpha'}+ \beta_{\beta \alpha'}\partial_{\alpha}\epsilon^{\beta}+ \beta_{\alpha \beta}\partial_{\alpha'}\epsilon^{\beta}).
\frac{1}{1-w\zeta^{-1}q^{j+1}}= \begin{cases}\ \ \ \sum_{m\geq0}\left(w\zeta^{-1}q^{j+1}\right)^m \j \geq 0, -\sum_{m\geq0}\left(w\zeta^{-1}q^{j+1}\right)^{-m-1}\j<0. \end{cases}.
f \star g = \exp \Bigg [ i \frac{\hbar}{2}\Bigg ( \frac{\partial ~}{\partial q}\frac{\partial ~}{\partial \tilde{p}}- \frac{\partial ~}{\partial p}\frac{\partial ~}{\partial \tilde{q}}\Bigg ) \Bigg ] f ({\bf x}) g ({\bf \tilde{x}}) \vert_{{\bf x}={\bf \tilde{x}}},
\Pi_{\rm T}^{(1)(\beta)}(0) = \frac{C(D)e^{2}}{2^{D-2}\pi^{\frac{D-1}{2}}\Gamma(\frac{D-1}{2})}\int_{m}^{\infty}d\omega_{k}\,\frac{\partial}{\partial\omega_{k}}\left((\omega_{k}^{2}-m^{2})^{\frac{D-1}{2}}\,\frac{n_{F}(\omega_{k})}{\omega_{k}}\right) = 0,\qquad D\geq 2
\sum_{k\in\tilde{E}}P_{R_{j,k}}=\sum_{k\in\tilde{E}}P_{I^1_{j,k_1}}\otimes P_{I^2_{j,k_2}}\otimes\cdots\otimes P_{I_{j,k_d}^n}&=\bigg(\sum_{k_1\in E_2}P_{I^1_{j,k_1}}\bigg)\otimes \bigg(\sum_{k_2\in E_2}P_{I^2_{j,k_2}}\bigg)\otimes\cdots\otimes \bigg(\sum_{k_d\in E_2}P_{I^n_{j,k_d}}\bigg) &=P_{I_{j}^1}\otimes P_{I_{j}^2}\otimes\cdots\otimes P_{I_{j}^d},
\Tilde{\phi}_{e}= \phi_1 - C_{p_1}h^{p_1}- C_{p_2}h^{p_2}, C_{p_1}h^{p_1}= \frac{r^{p_2}e_{21}- e_{32}}{r^{p_1}(1 - r^{p_1}) (1 - r)}, C_{p_2}h^{p_2}= \frac{e_{32}- r^{p_1}e_{21}}{r^{p_1}(1 - r^{p_2}) (1 - r)},
\begin{array}{rrrll}{\cal B}_{0}: f_{0}-1 = 0\;,{\cal B}_{1}: f_{1}-f_{2}= 0 \mbox{and}\;\;\;\tilde{\cal B}_{12}: F(f_{0}^{-1/2}f_{1},f_{0}^{-1/2}f_{2}) = 0\end{array}
\left\{\widehat{L}(TM)[2{\rm ch}(\widetilde{T_CM})-8+{\rm ch}(W_i)]\right\}^{(12)}&=\left\{\widehat{A}(TM){\rm ch}[2116\widetilde{T_CM}+4\wedge^2\widetilde{T_CM}+4W_i-15872]\right\}^{(12)}.
\alpha : = \left( \begin{array}{c c c c c c c c}{\gamma_{1}}&{0}&{\gamma_{3}}&{\gamma_{4}}&{\ldots}&{\gamma_{N - 2}}&{\gamma_{N - 1}}&{\gamma_{N}}\\{0}&{\beta_{2}}&{\gamma_{3}}&{c_{4}\gamma_{4}}&{\ldots}&{c_{N - 2}\gamma_{N - 2}}&{\beta_{N - 1}}&{\beta_{N}}\\ \end{array}\right)
\sum_{j,l=1}^{k}a_{jl}\zeta_j\zeta_l=\Vert\nabla\phi\Vert_{L^2(\Omega)}^2+\big(T(\varphi_{p,k})\phi,\phi\big)\geq\Vert\nabla\phi\Vert_{L^2(\Omega)}^2\geq\lambda_1\Vert\phi\Vert_{L^2(\Omega)}^2\,.
\rho e_2 e_1^{-1}e_1 e_1 e_2^{-1}= e_1 \rho e_2 e_1^{-1}e_2 e_1 e_2^{-1}= e_2 \rho e_2 e_1^{-1}e_2^{-1}e_1^{-1}e_1 e_2^{-1}= \rho^3 e_2^{-1}e_1^{-1}
\psi^{+}= \varepsilon(\gamma^{123}- \theta^{12}\wedge \gamma^3-\theta^{31}\wedge \gamma^2-\theta^{23}\wedge \gamma^1), \psi^{-}= \varepsilon(\theta^{123}-\gamma^{12}\wedge \theta^3-\gamma^{31}\wedge \theta^2-\gamma^{23}\wedge \theta^1),
u(t, x)={}\int_{\mathbb R}{p}(t, x-y)u_0(y)\,dy+\int_0^t\,ds \int_{\mathbb R}{p}(t-s, x-y)f \bigl(s, y, u(s, y) \bigr)\,dy &{}+\int_{(0,t]}\,d\mu(s) \int_{\mathbb R}{p}(t-s, x-y)\sigma(s, y)\,dy .
0=\frac{\partial}{\partial y}(y\beta(y))+\frac{1}{4}\frac{\partial^{2}}{\partial y^{2}}\beta(y)+\left(-4y^{4}+10y^{2}-\frac{7}{4}\right)\left(\frac{2}{\pi}\right)^{1/2}e^{-2y^{2}}.
\begin{aligned}k^{\nabla y}_{j+1}(T) \leq k^{\nabla y}_{0}(T) + C C_{1}(T) B \big(1-3\big(\tfrac{1}{p}-\tfrac{1}{q}\big),\tfrac{1}{2}-\tfrac{3}{2p}\big) & \left[k^{u}_{j}(T) k^{\nabla y}_{j}(T) + k^{\nabla y}_{j}(T)^2 (k^{x}_{j}(T)+k^{y}_{j}(T)+\lvert \overline{b}\rvert) \right].\end{aligned}
\int_{SU(N-1)}f_{SU(N-1)}(g_{SU(N-1)})_{ij}^{\beta_{ij}}h_{SU(N-1)}(g_{SU(N-1)})\,dg_{SU(N-1)}&=\int_{SU(N-1)}f_{SU(N-1)}(u'g_{SU(N-1)})_{ij}^{\beta_{ij}}h_{SU(N-1)}(u'g_{SU(N-1)})\,dg_{SU(N-1)}&=\int_{SU(N-1)}f_{SU(N-1)}(g_{SU(N-1)}u')_{ij}^{\beta_{ij}}h_{SU(N-1)}(g_{SU(N-1)}u')\,dg_{SU(N-1)}
\left({\frac{x^{-}}{x^{+}}}\right)^{2}F ( x^{-}, x^{+}) ={\frac{N}{k}}\sum_{n}{\frac{M_{n}^{4}}{3 2 \pi^{2}}}{\frac{( k - 2 n )^{2}}{k^{2}}}K_{4}( M_{n}r )
M_0(h) = b_{\scriptscriptstyle R}I^{0}_{\scriptscriptstyle C}+\frac{b_{\scriptscriptstyle R}}{b_{\scriptscriptstyle L}}I^{0}_{\scriptscriptstyle L}+b_{\scriptscriptstyle R}\bar I^{0}_{\scriptscriptstyle C}+I^{0}_{\scriptscriptstyle R}.
\int_{\mathbb R^n}\phi(|\xi|) |\xi| e^{\mathrm i |x| \xi_1}d \xi = (- \mathrm i|x|)^{-k}\!\int_{\mathbb R^n}\partial_1^k (\phi(|\xi|) |\xi|) e^{\mathrm i |x| \xi_1}d \xi &= (-\mathrm i|x|)^{-k}\!\int_{\mathbb R^n}\partial_1^k(|\xi|) \phi(|\xi|) e^{\mathrm i |x| \xi_1}+ R_k(\xi) d \xi,
\beta ( k , p ) = \frac{- i}{2 \pi}\delta ( p_{2}+ k_{2}) \delta ( p_{3}+ k_{3}) \int \! d x_{1}\frac{e^{i k_{1}x_{1}}}{\sqrt{4 \omega_{p}\omega_{k}}}\left\{i \omega_{k}f_{p}^{+ \ast}( t^{\prime}( 0 , x ) ) e^{i p_{1}x_{1}^{\prime}( 0 , x )}- \partial_{t}\left\{f_{p}^{+ \ast}( t^{\prime}( t , x ) ) e^{i p_{1}x_{1}^{\prime}( t , x )}\right\}\left|_{t = 0}\right. \right\}
\frac{\partial^{i+j}f}{\partial u^i\partial v^j}(0,v) =\frac{df^+_i}{dv^j}(v):i,j\geq 0, \frac{\partial^{i+j}f}{\partial u^j\partial v^i}(u,0) =\frac{df^-_i}{du^j}(u):i,j\geq 0,
\sideset{}{'}\sum_{\vert K\vert=q-1}\sum_{j,k=1}^n\dfrac{\partial^2h_A}{\partial z_j\partial\overline{z_k}}(z)w_{jK}\overline{w_{kK}}\geq A\vert w\vert^2\;,\;z\in U_{A}\;,\;w\in\Lambda^{0,q}_{z}\;.
\hat{p}_{xl\nu}\,|p_{xk\mu},p_{yk\mu}\rangle=\,\delta_{lk}\,p_{xl\nu}\,|p_{xk},p_{yk}\rangle,\quad\hat{p}_{yl\nu}\,|p_{xk\mu},p_{yk\mu}\rangle=\,\delta_{lk}\,p_{yl\nu}\,|p_{xk\mu},p_{yk\mu}\rangle.
\lambda_{n}^{(2)}= \sum_{i \neq n}\frac{\langle \psi_{i}^{(0)}, H^{(1)}\psi_{n}^{(0)}\rangle \langle \psi_{n}^{(0)}, H^{(1)}\psi_{i}^{(0)}\rangle}{\lambda_{n}^{(0)}- \lambda_{i}^{(0)}}+\langle \psi_{n}^{(0)}, H^{(2)}\psi_{n}^{(0)}\rangle
h\left(u_{ij}^{(s)}(u_{kl}^{(t)})^* \right) = \delta_{s t}\delta_{ik}\frac{(Q_s)_{l,j}}{d_s},\; \;h\left((u_{ij}^{(s)})^* u_{kl}^{(t)}\right) = \delta_{st}\delta_{jl}\frac{(Q_s^{-1})_{k,i}}{d_s}.
g_1= q^{-1}-2+248q^3-492q^4+4119q^7-7256q^8+33512q^{11}-53008q^{12}+O(q^{15}) g_4= q^{-4}-2-26752q^3-143376q^4-8288256q^7- 26124256q^8+O(q^{11}) g_5= q^{-5}+ 0+85995q^3-565760q^4+52756480q^7-190356480q^8+O(q^{11}) \vdots
\sum_{j^\gamma \leq \log{n}\leq (j+1)^\gamma}\frac{[d(n)]^\alpha}{n}\asymp j^{\gamma2^\alpha-1}\left( A_0 + \sum_{\lambda=1}^\nu \frac{B_\lambda}{j^{\gamma\lambda}}+\mathcal{O}\left(\frac{1}{j^{\gamma2^\alpha-1-\gamma(2^\alpha-1-\nu)}}\right)\right),
\int_{- i \infty}^{i \infty}\bar{d}\beta \mathrm{e x p}\left({\frac{1}{r}}\tilde{f}_{N}( r \beta ) \right) \mathrm{e}^{\beta \varepsilon}={\frac{1}{r}}\hat{A}( \varepsilon / r , v / r ) \mathrm{e}^{{\beta_{H}\varepsilon}/ r}.
\tfrac1N I(U_2^N;Y^N,U_1^{i_1}) < I_2, \tfrac1N I(U_3^N;Y^N,U_1^{i_1}) < I_3, \tfrac1N I(U_2^N,U_3^N;Y^N,U_1^{i_1}) = I_2 + I_3,
\mathbb{D}(x_1 \wedge \ldots \wedge x_r \otimes \varphi) & = \sum \pm x_1 \wedge \ldots \mathbb{D}(x_i) \wedge \ldots x_r \otimes \varphi \pm \sum x_1 \wedge \ldots \wedge x_r \otimes \mathbb{D}(\varphi)
\left\|V^{N}\right\| \leq C \sum\limits_{n=1}^{N-1}\left( k_n+k_{n+1}\right) \left( \left\|V^{0}\right\| + 2\sum\limits_{n=1}^{N}k_n \left\|g^{n-\frac{1}{2}}\right\| \right) \leq C(T) \left( \left\|V^{0}\right\| + \sum\limits_{n=1}^{N}k_n \left\|g^{n-\frac{1}{2}}\right\| \right),
(t_l,\prod_{i=l+1}^kt_i)_2(\prod_{i=l+1}^kt_i,\prod_{i=l+1}^kt_i)_2\gamma_{\psi,(-1)^{k+1-l}}(t_l)\gamma_{\psi,(-1)^{k+1-l+1}}(\prod_{i=l+1}^kt_i) &=(t_l,\prod_{i=l+1}^kt_i)_2\gamma_{\psi,(-1)^{k+1-l}}(t_l)\gamma_{\psi,(-1)^{k+1-l}}(\prod_{i=l+1}^kt_i)=\gamma_{\psi,(-1)^{k+1-l}}(\prod_{i=l}^kt_i).
\mathcal{B}_3(s_1,s_2) = \prod_{p^{\alpha}\mid \mid h}\Big( \sum_{j=0}^{\alpha}\mathcal{C}(p,j,s_1,s_2) \mathcal{C}(p,0,s_1,s_2)^{-1}\Big) \Big(1 + \frac{1}{p^{s_1+s_2+1}}\Big)^{-k \ell}.
d s_{S - t y p e}^{2}= d \chi^{2}- d T^{2}+ d W^{2}- \frac{1 + A^{2}/ \tau_{0}^{2 p - 2}}{( \sqrt{T^{2}- W^{2}}/ \tau_{0})^{2 p - 2}- 1}\frac{( W d W - T d T )^{2}}{T^{2}- W^{2}}+ W^{2}d \Omega_{p - 2}^{2}\ .
\int_M^\infty |df(t)| = o\big( M^{\mu-1}\big) M\to \infty, \nu<\alpha+1/2 \mu< 1, \int_M^\infty t^{\nu-\alpha-1/2}|df(t)| =o\big(M^{\mu+\nu-\alpha-3/2}\big) M\to \infty, \nu\geq \alpha+1/2 \mu\geq 1.
V^{j,\tau}_{k}(T,\bar{s}):=\sum_{l \geq k}\|Q|_{U^{(k)}_{-j}}(k,l+1)^{-1}\|\|Q(l+1)\|^\tau\|Z(l+1)\|s_{l}, &W^{j,\tau}_{k}(T,\bar{s}):=\sum_{0\leq l<k}\|Q|_{E^{(l+1)}_{-j}}(l+1,k)\|\|Q(l+1)\|^\tau\|Z(l+1)\|s_l.
\wedge^\bullet \overline{\mathfrak{u}}\otimes \mathcal{E}(\mu)\otimes \mathcal{F}^\nu=\left( \wedge^\bullet \overline{\mathfrak{u}}\otimes \mathcal{E}(\mu)\otimes \mathcal{E}^\nu\right)\oplus \left( \wedge^\bullet \overline{\mathfrak{u}}\otimes \mathcal{E}(\mu)\otimes \mathcal{E}'\right)\,.
\frac{\delta S}{\delta \phi^{A}}\frac{\delta S}{\delta \phi_{A a}^{\ast}}+ \varepsilon^{a b}\phi_{A b}^{\ast}\frac{\delta S}{\delta \overline{{{\phi}}}_{A}}= i \hbar \frac{\delta^{2}S}{\delta \phi^{A}\delta \phi_{A a}^{\ast}}
\begin{aligned}\mu_1 = \frac{1}{\pi}e^{-\frac{a_1^2}{2}}\int_{0}^{a_2}e^{-\frac{t^2}{2}}dt \mu_2 = \frac{1}{\pi}e^{-\frac{a_2^2}{2}}\int_{0}^{a_1}e^{-\frac{t^2}{2}}dt.\end{aligned}
[\begin{matrix}f_{x}\\f_{y}\\f_{w}\end{matrix}]=[\begin{matrix}1&0&\frac{e_{x}}{e_{z}}\\ 0&1&\frac{e_{y}}{e_{z}}\\ 0&0&\frac{1}{e_{z}}\end{matrix}][\begin{matrix}d_{x}\\d_{y}\\d_{z}\end{matrix}]
(\hat \Psi_{\alpha}\circ F_{\alpha, n_0}^{-1})(z) =\hat \psi_{\alpha}\left(\frac{z_{n_0}+1}{2}\right)+\sum_{j=0}^{n_0-1}\hat \psi_{\alpha}(f_{\alpha}^j(z_{n_0})) &=\hat \psi_{\alpha}\left(\frac{z_{n_0}+1}{2}\right)+\sum_{j=1}^{n_0}\hat \psi_{\alpha}(z_{j})
J(x,\tau) =-\frac{\partial}{\partial x}\left(D_1(x)\rho(x,\tau)\right)+D_2(x)\rho(x,\tau) D_1(x) =\frac{1}{2}\frac{\Sigma_1^2}{2x}D_2(x) =\frac{\overline{\delta\beta^{(2)}}}{2x}+\frac{\overline{\delta \beta^{4}}}{4x^2}.
n a_{n}( q^{\prime}, q ) + ( q^{\prime}- q ) \frac{\partial a_{n}( q^{\prime}, q )}{\partial q^{\prime}}= \frac{1}{2}\frac{\partial^{2}a_{n - 1}( q^{\prime}, q )}{\partial q^{2}}- V ( q^{\prime}) a_{n - 1}( q^{\prime}, q ) .
\begin{pmatrix}Z -u_0\mathbb{1}-u_1\mathbb{1}0\end{pmatrix}^{-1}=\begin{pmatrix}0 -\frac{1}{u_1}\mathbb{1}-\frac{1}{u_0}\mathbb{1}-\frac{1}{u_1u_0}Z\end{pmatrix}\;.
\left\{b_{\underline{{{n}}}\sigma}, b_{\underline{{{n}}}\sigma}^{\dagger}\right\}= \delta_{\underline{{{n}}}^{\prime}\underline{{{n}}}}\delta_{\sigma^{\prime}\sigma}\, , \; \; \left[ a_{\underline{{{n}}}}, a_{\underline{{{n}}}}^{\dagger}\right] = \delta_{\underline{{{n}}}^{\prime}\underline{{{n}}}}\, .
q_\pm(y,\eta)=\pm \frac{1}{8}\| \xi_\pm \|^{-1}\left( \| \xi_\pm \| - \beta \gamma(\xi_\pm) \right) \left(- \tilde \xi_\pm(\nu) \beta + \gamma(\xi_\pm)\right) \circ \Gamma_{(x,\tilde \xi_\pm),(y,\tilde\eta_\pm)}\circ \| \eta \|^{-2}\beta (\| \eta \| + \beta \gamma(\eta)).
\Big(1 - \sum_{k = 1}^{L}\phi_k q_k\Big) e_j =\Big(1 - \sum_{k = 1}^{L}q_k (a_1 \widetilde{\phi_k}+ p_k)\Big) e_j = \Big(1 - \sum_{k = 1}^{L}q_k p_k - a_1 \sum_{k = 1}^{L}q_k \widetilde{\phi_k}\Big) e_j = a_1 \Big ( q - \sum_{k = 1}^{L}q_k \widetilde{\phi_k}\Big)e_j.
|v_{1}^{\lambda_{1}}-v_{1}^{\lambda_{2}}| \leq \begin{cases}\frac{C r ||e_{1}-e_{2}||_{X}}{t^{2}\log^{b}(t) \sqrt{\log(\log(t))}}, r \leq \frac{t}{2}\frac{C ||e_{1}-e_{2}||_{X}}{r \sqrt{\log(\log(t))}\log^{b}(t)}, r >\frac{t}{2}\end{cases}
\big\Vert f_L\big\Vert_{F_p^{{s},q}}\lesssim \Big[\int_{\mathbb{R}^d}{\Big(\sum_{n=M}^{L}{\big(\mathfrak{M}_{{\sigma},2^{t_n}}{g_{{t_n}}}(x)\big)^q}\Big)^{{p}/{q}}}dx\Big]^{{1}/{p}},
S_0^1(x,y) = x S_0^2(x,y) = x^2, S_1^2(x,y) = x y, S_0^3(x,y) = x (x^2-\frac{3}{4}), S_1^3(x,y) = (x^2 - \frac{1}{4}) y, S_0^4(x,y) = x^2(x^2-1), S_1^4(x,y) = x (x^2 - \frac{5}{8}) y, S_2^4(x,y) = x^2 y^2 - \frac{1}{4}x^2 - \frac{1}{4}y^2.
\inf \ \{g_{k;\ast}(x_1) + g_{k;\ast}(x_2) : (x_1,x_2) \in R\}= \inf \ \{g_{k;\ast}(x) + g_{k;\ast}(\lambda-x) : \lambda \in \mathcal{I},x\in [0,\lambda]\}&\geqslant \inf \ \{f_k(\lambda) + \lambda : \lambda \in \mathcal{I}\}&\geqslant \epsilon + E -\epsilon'.
h_n(\tau)=\frac{\wp_{\left[\begin{smallmatrix}1/2 1/2\end{smallmatrix}\right]}(2^{n-1}\tau)-\wp_{\left[\begin{smallmatrix}0 1/2\end{smallmatrix}\right]}(2^{n-1}\tau)}{\wp_{\left[\begin{smallmatrix}1/2 1/2\end{smallmatrix}\right]}(2^{n-2}\tau)-\wp_{\left[\begin{smallmatrix}0 1/2\end{smallmatrix}\right]}(2^{n-2}\tau)}\quad(\tau\in\mathbb{H}).
v_{N,m}=\sum_{p=0}^{N-1}\bigg((-1)^m(N+1){{\sin[{{N(m+1)\pi}\over{N+1}}]}\over{\sin^3[{{(m+1)\pi}\over{N+1}}]}}\bigg)^{-{{1}\over{2}}}U_p\left({{2+\lambda_{N,m}}\over{2}}\right)e_p,
\Lambda_{a}^{( \frac{1}{2})}\Lambda_{b}^{( \frac{1}{2})}= S_{a b}{\bf 1}+ \frac{1}{2}( d_{a b c}+ i f_{a b c}) \Lambda_{c}^{( \frac{1}{2})}\quad \quad ( a , b , c = 1 , 2 , \dots 8 )
0 \equiv f_{p,k+1}(n_k) + p^k t_{k+1}f'_{p,k+1}(n_k) \pmod{p^{k+1}}& t_{k+1}q_p(n_k) \equiv t_{k+1}f'_{p,k+1}(n_k) \equiv -\frac{f_{p,k+1}(n_k)}{p^k}\pmod{p}& t_{k+1}\equiv -\frac{f_{p,k+1}(n_k)}{p^k q_p(n_k)}\pmod{p}& n_{k+1}= n_k + p^k t_{k+1}\equiv n_k -\frac{f_{p,k+1}(n_k)}{q_p(n_k)}\pmod{p^{k+1}}.
\begin{aligned}S_1^* = \{n \in \mathbb Z \mid |p_n| > R |n| \leq \lceil \gamma R \rceil \}, S_2^* = \{n \in \mathbb N \mid n \geq \lceil \gamma R \rceil + 1 p_n \geq -p_{-n}\}, S_3^* = \{n \in \mathbb N \mid n \geq \lceil \gamma R \rceil + 1 p_n < -p_{-n}\}. \end{aligned}
x^n[g,h]x^{-n}x^mfx^{-m}= x^m x^{n-m}[g,h]x^{m-n}fx^{-m}=x^m f x^{n-m}[g,h]x^{m-n}x^{-m}= x^mfx^{-m}x^n[g,h]x^{-n},
c_{2}= 1 6 0 / 2 7 ~ , \qquad{\hat{\lambda}}= \frac{1}{5 2}> 0 ~ , \qquad{\hat{\xi}}= \frac{1 7 1}{2 6}> 0 \qquad ( \mathrm{a n t i ~ d e ~ S i t t e r ~ b u l k})
S_{1}= \frac{\sqrt{( y^{2}+ x_{0}^{2})^{3}+ ( y^{2}+ x_{0}^{2})^{2}I_{2}+ ( y^{2}+ x_{0}^{2}) \, ( I_{2}^{2}- I_{4}) / 2 + ( I_{3})^{2}}}{y^{2}+ x_{0}^{2}}\, ,
P\psi =P\left( \begin{array}{c}\psi_{0}\psi_{1}\psi_{2}\psi_{3}\psi_{4}\end{array}\right) =\left( \begin{array}{c}0 0 0 0 \psi_{4}\end{array}\right) ;\P_{\mu}\psi =\left( \begin{array}{c}0 0 0 0 \psi_{\mu}\end{array}\right) \ \left( \mu =0,...,3\right) .
\frac{\sin^{2}(\nu\pi )}{\pi^{2}}\iint\frac{\log[(1-U)(1-V)]}{1-\alpha UV}\mathbb D_{\nu,0}U\mathbb D_{-\nu-1,0}V ={}-\frac{\sin(\nu\pi)}{\pi}\left[ \int\log(1-U)\mathbb D_{\nu,\alpha}U+ \int\log(1-V)\mathbb D_{-\nu-1,\alpha}V\right]
T\left( \vartheta |\vartheta_{1},\vartheta_{2},\ldots ,\vartheta_{N}\right)^{b_{1}b_{2}\ldots b_{N}}_{a_{1}a_{2}\ldots a_{N}}=\prod^{N}_{j=1}S^{a_{j+1}b_{j+1}}_{b_{j}a_{j}}\left( \vartheta -\vartheta_{j}\right)
E^*_{i+k}A_1E^*_{i+k-1}A_1E^*_{i+k-2}\cdots E^*_{i+1}A_1E^*_{i}=\big((\frac{k-1}{2})!\big)^2\frac{k+1}{2}M^{\frac{k-1}{2},\frac{k-1}{2}}_{\frac{i+k-1}{2},\frac{2m-i}{2}}.
g(q,u_0,v) := \left| \left(e^{- 2 i \vartheta}\left( u_0 - q \right)^2 - E - i \eta \right)^{-1}\right|^2 + \left| \left(e^{- 2 i \vartheta}\left( u_0 - q - [M_A v]_1\right)^2 - E - i \eta \right)^{-1}\right|^2 .
y_i = \sum_{j=1}^{m}\left( \left( 1 - \psi_{i-j+1}\right) U\left[ j, k^\mathrm{L}_{i-j+1}\right] + \psi_{i-j+1}U\left[ j, k^\mathrm{R}_{i-j+1}\right] \right) , \psi_i = b_i - k_i^\mathrm{L}.
\sum_{a\leq y}\sum_{p\leq x}\frac{1}{l_a(p)}= \sum_{d<x}\frac 1d \sum_{\substack{{p\leq x}{p\equiv 1 (d)}}}\sum_{\substack{{a\leq y}{l_a(p)=d}}}1 =\sum_{d<x^{\epsilon}}+ \sum_{x^{\epsilon}\leq d<x}=\Sigma_1 + \Sigma_2
f(\Xi^{1},\Psi_{2},\Theta^{1}):=-\frac{1}{18}(\Xi^{1})^{2}-\frac{1}{9}\Xi^{1}(-5\Theta^{1}+4\Psi_{2}), &g(\Psi_{1},\Psi_{2},\Theta^{1},\Theta_{1}):=-\frac{1}{18}(5\Theta^{1}-\Psi_{2})^{2}+\frac{1}{2}(3\Theta_{1}-\Psi_{1})(\Theta_{1}-\Psi_{1})-\frac{1}{3}\Psi_{2}(\Theta^{1}-\Psi_{2}).
P = \left[ \begin{array}{c c c c c}{*}&{1}&{0}&{0}&{. . .}\\{1}&{*}&{1}&{0}&{. . .}\\{0}&{1}&{*}&{1}&{. . .}\\{}&{.}&{.}&{.}&{}\\{. . .}&{0}&{0}&{1}&{*}\\ \end{array}\right]
R_1=\frac{1}{2}\log\frac{|\sigma^{2}{\bf I}_2+\sum_{j=1}^{3}{\bf G}_{1j}{\bf Q}_j{\bf G}_{1j}^{T}|}{|\sigma^{2}{\bf I}_2+\sum_{j=2}^{3}{\bf G}_{1j}{\bf Q}_j{\bf G}_{1j}^{T}|}, R_2=\frac{1}{2}\log\frac{|\sigma^{2}{\bf I}_2+\sum_{j=2}^{3}{\bf G}_{1j}{\bf Q}_j{\bf G}_{1j}^{T}|}{|\sigma^{2}{\bf I}_2+{\bf G}_{13}{\bf Q}_3{\bf G}_{13}^{T}|},
S(r) < \dfrac{e^{\frac{r-1}{2}}}{2^r}\sum_{k=2r}^{\infty}\dfrac{1}{\pi^k}=\dfrac{\pi}{\sqrt{e}(\pi-1)}\Bigl(\dfrac{\sqrt{e}}{2 \r^2}\Bigr)^r < \dfrac{1}{10^r}.
\frac{1}{n}\mathbf{h}^T \mathbf{X}^T \mathbf{X}\mathbf{h}= \frac{1}{n}\|\mathbf{X}\mathbf{h}\|_2^2 \geq \kappa \|\mathbf{h}\|_2^2, \;\;\;\; \forall \mathbf{h}\in \mathbb{V}.
\Psi_{{\bf p}}^{( - )}({\bf x}, t ) = \mathrm{e}^{+ i w_{{\bf p}}t}\, \Psi_{{\bf p}}^{( - )}({\bf x}) = \frac{1}{{2 \pi}}\frac{w_{{\bf p}}+ m}{\sqrt{4 m w_{{\bf p}}}}\left( \begin{array}{c}{{\frac{m - w_{{\bf p}}}{m + w_{{\bf p}}}}}\\{{1}}\\ \end{array}\right) \mathrm{e}^{\, - i{\bf p}\cdot{\bf x}+ i w_{{\bf p}}t}\; ,
Q^{( n )}= \sum_{\{r_{i}\, : \sum_{i = 1}^{N}r_{i}= n \}}\Bigg \{\prod_{i = 1}^{N}\left( \begin{matrix}{{\frac{n}{N - 1}}}\\{{r_{i}}}\\ \end{matrix}\right) u_{i}^{r_{i}}\Bigg \}\, .
I^{1,0}_{0,1}\Bigg[z\left|\begin{matrix}(b,B,1) \end{matrix}\right.\Bigg]=\frac{1}{B}z^{\frac{b}{B}}e^{-z^{\frac{1}{B}}};\ \ \ \I^{1,1}_{1,1}\Bigg[z\left|\begin{matrix}(b-a+1,1,1) (b,1,1) \end{matrix}\right.\Bigg]=\Gamma(a)z^b(1+z)^{-a};
\lambda \left\{x \in [0,1): l_n(x) \geq i\right\}= \sum_{(\varepsilon_1,\cdots,\varepsilon_n) \in \Sigma_{\beta}^n}|J(\varepsilon_1,\cdots,\varepsilon_n, \underbrace{0,\cdots,0}_{i})| &\leq \sharp \Sigma_{\beta}^n \cdot \frac{1}{\beta^{n+i}}\leq \frac{\beta^{n+1}}{\beta - 1}\cdot \frac{1}{\beta^{n+i}}= \frac{\beta^{1-i}}{\beta-1}.
(\widehat{S}_{p}(g))(x)=(1+\beta\gamma[p-1])\left(\widehat{S}_{p+1}\widehat{V}_{\nu_{p}}\!\!\!\prod_{\nu_{p}<i<\nu_{p+1}}\!\!\!(q^{p}+\delta[p]\widehat{V}_{i})\,(g)\right)(x)
\bigl|\partial_{\alpha}\Bar{V_i}(G_{,A}(\Xi)\!-\!G_{,A}(\bar{\Xi}))(\Phi_{,i\alpha}^A(F)\!-\!\Phi_{,i\alpha}^A(\bar{F}))\bigr|\!\leqslant\! C \big(|\Xi\!-\!\bar{\Xi}|^2\!+\!(1\!+\!|F|^2)|F\!-\!\bar{F}|^2\big).
\begin{array}{l l l}{{\displaystyle \left( \partial / \partial \theta \right) A}}&{{}}&{{\mathrm{f o r \l e f t \d e r i v a t i v e},}}\\{{\displaystyle \partial A / \partial \theta}}&{{}}&{{\mathrm{f o r \r i g h t \d e r i v a t i v e}.}}\\ \end{array}
J_{11}^R(t)=t^{-2}\varepsilon_0\Re\sum_{k,l}\int_{\mathbb{R}^2}A_k(t,\xi)(r_1(k-l,\xi-\eta)\cdot l+r_2(k-l,\xi-\eta)\cdot\eta)\overline{\widehat{g_1'}(t,k,\xi)}\widehat{g_1'}(t,l,\eta)\,\mathrm{d}\xi\mathrm{d}\eta,
\tau_{Y M}\equiv \frac{\theta_{Y M}}{2 \pi}+ i \frac{4 \pi}{g_{Y M}^{2}}= i \frac{2 N - M}{2 \pi}\log \frac{z}{\rho_{e}}~ ~ , ~ ~ \rho_{e}= \epsilon \mathrm{e}^{\pi / ( 2 N - M ) g_{s}}
(w+h)\psi\Delta\psi\geq -\frac{1}{4n}(w+h)^2\psi^2-C(n)(\Delta\psi)^2 = -\frac{1}{4n}(w+h)^2\psi^2-C(n)[\psi''+\psi'\Delta r_{\partial M}]^2 \geq -\frac{1}{4n}(w+h)^2\psi^2-C(n)[|\psi''|+|\psi'|\frac{n-1}{r_{\partial M}}(1+\sqrt{K})r_{\partial M}]^2 \geq -\frac{1}{4n}(w+h)^2\psi^2-C(n)\left(\frac{1}{R^4}+K^2\right).
\mu (aA) = \lim_{\omega}\frac{\dim ((W_i+aW_i)\cap aA)}{\dim (W_i)}\ge \lim_{\omega}\frac{\dim (aW_i\cap aA)}{\dim (W_i)}& \ge \lim_{\omega}\frac{\dim (a(W_i\cap A))}{\dim (W_i)}= \lim_{\omega}\frac{\dim (W_i \cap A)}{\dim (W_i)}= \mu (A) ,
-\frac{\log\mu_Y(\tau_r>t)}{s\mu(U_r)^{1-\alpha}}= \frac{\log\mu(Y)}{s\mu(U_r)^{1-\alpha}}-\frac{\log\mu(\tau_r>t\wedge Y)}{s\mu(U_r)^{1-\alpha}}&\sim -\frac{\log\mu(\tau_r>t\wedge Y)}{s\mu(U_r)^{1-\alpha}}\ge -\frac{\log\mu(\tau_r>t)}{s\mu(U_r)^{1-\alpha}}.
y(t) := \int (\partial_x^lu)^2\chi(x+\nu t) \; dx+ \int_0^t\int (\partial_x^{l+2}u)^2 \chi'(x+\nu\tau) \; dxd\tau &\leq c_0 + c_1\int_0^t\int(\partial_x^lu)^2\chi(x+\nu\tau) \; dxd\tau &\leq c_0 + c_1\int_0^ty(\tau)\;d\tau.
\mathrm{e}^{\lambda\sum_{i=1}^d (y_i-x_i)-dn\ln(1+\lambda)}\le \mathrm{e}^{\lambda d(y_1-x_1) + d^2 - nd\lambda+nd \lambda^2}\le C_d \mathrm{e}^{-d\frac{(y_1-x_1-n)}{\sqrt n}}.
\lambda (\nu(l)) \; \ge \; \max ( \lambda (\nu(k)), l), &| u_{\lambda (\nu(l))}(z) \, - \, u (z) | \; \le \; \frac 1k{\mathrm{on}}Q_{2n}\setminus \bigcup_{i=1}^{I_{k}}P_{R_{i,k}}(z_{i,k})
\prod_{i < j}(x_i - x_j)^{-1}\left[ \prod_{k=1}^n (z_k^{-1}+t_n x_n) \det_{1 \leq i, j \leq n}\left\{(1-x_i z_j)^{-1}\right\}\right] \, \Big\vert_{\boldsymbol{z}^{\lambda}}= \prod_{k=1}^n (x_k + t_nx_n) s_{\lambda-\rho}(\boldsymbol{x}).
Mg(x_1') = \sup_{a' \in \Gamma(x_1')}\left\{r(x_1',a') + g(x_1',a') \right\}\leq \sup_{a' \in \Gamma(x_2')}\left\{r(x_1',a') + g(x_1',a') \right\}\leq \sup_{a' \in \Gamma(x_2')}\left\{r(x_2',a') + g(x_2',a') \right\}= Mg(x_2').
\hat{T}{\bf f}= 0 \; , \; \; \hat{T}= \frac{\imath s m}{2 \hat{\omega}}\hat{\zeta}\Sigma^{3}\left[ \hat{\zeta}\hat{\omega}\Sigma^{3}+ \imath \partial_{1}( \imath s \Sigma^{2}) + \imath \partial_{2}( - \imath s \Sigma^{1}) - s m \right] \; .
f(n,2,2) = \begin{cases}\displaystyle\sum_{i=(n-1)/2}^n{n \choose i}\mbox{if}n\mbox{is odd,}{n-1 \choose \frac{n}{2}-2}+\displaystyle\sum_{i=n/2}^n{n \choose i}\mbox{if}n\mbox{is even.}\end{cases}
X_1 =p-p_1=\frac{M(2\mu p-1)-1}{2M \mu}, X_2 =p-p_2=\frac{2ap-a-1+(2\mu p-a-1)M}{2(M\mu+a)}, X_3 =p-p_3=\frac{2p-a-1+ (2\mu p-a-1)M}{2(M\mu+1)}.
c_{n-1,\xi_{\boldsymbol{s}k}}^{\ast\ast}=\frac{\Gamma((n-1)/2)}{(2\pi)^{(n-1)/2}}\left\{\int_{0}^{\infty}t^{(n-3)/2}\ln\left[1+\exp\left(-\frac{1}{2}\xi_{\boldsymbol{s},k}^{2}\right)\exp(-t)\right]\mathrm{d}t\right\}^{-1},