harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
I_{2}= \frac{R}{4 \pi i}\int_{\Lambda}^{- \Lambda}\frac{( - y^{2}) \, i \, d y}{i \sqrt{y^{2}+ \omega_{0}^{2}}}\frac{\cosh ( R \sqrt{y^{2}+ \omega_{0}^{2}})}{\sinh ( R \sqrt{y^{2}+ \omega_{0}^{2}})}d y = - \frac{1}{2 \pi}\int_{0}^{\Lambda}y \, d \left[ \ln \sinh \left( R \sqrt{y^{2}+ \omega_{0}^{2}}\right) \right] .
(-1)^{i_k-1}\beta^{kl}_j =b^{kl}_{1j}-f_1b^{kl}_{2j}+f_1f_2b^{kl}_{3j}+\dotsm+(-1)^{i_k-1}f_1f_2\dotsm f_{i_k-1}b^{kl}_{i_kj}.
\sum_{i=1}^{3}\int_{\Xi^{(i)}}\big(F_{p\alpha +k}- \partial_t{{N}}_{p\alpha +k-1}\big)\, d\xi = \int_{\Xi^{(0)}}\partial_t{{N}}_{p\alpha +k-1}\, d\xi + \delta_{p\alpha +k, \alpha}\int_{\Gamma_0}\varphi^{(0)}\, d \sigma_\xi.
0 =\lim_{k\to\infty}\mathbf{E}|\mathbf{P}[X_k\in A|X_0,Y_1,\ldots,Y_{k-1},Y_k^{<v}]-\mathbf{P}[X_k\in A|Y_1,\ldots,Y_{k-1},Y_k^{<v}]| = \mathbf{E}|\mathbf{P}[X_0\in A|\textstyle{\bigcap_k}(\mathcal{Y}_-^v\vee\mathcal{X}_{-k})]-\mathbf{P}[X_0\in A|\mathcal{Y}_-^v]|,
\vartheta_p(r_k) = \vartheta_p\left(\left(\frac{1}{a^k}+ \frac{1}{(p - a)^k}\right) \frac{p^k}{k}\right) begin{align*}1mm] = \vartheta_p\left(\frac{a^k + (p - a)^k}{\left(a (p - a)\right)^k}\cdot \frac{p^k}{k}\right) begin{align*}1mm] = \vartheta_p\left(a^k + (p - a)^k\right) + k - \vartheta_p(k) ~~~~~~~~~~ ( a p) .
\varphi_{R \atop L}( \epsilon E_{p}, \epsilon \vec{p}) = \Bigg [ \sqrt{\frac{\gamma + 1}{2}}\pm \vec{\sigma}. \epsilon \frac{\vec{p}}{p}\sqrt{\frac{\gamma - 1}{2}}\Bigg ] \varphi_{R \atop L}( \epsilon m , 0 ) , \vspace{0 . 5 i n}
(\psi_{\pi/2}''(0)^{-1}\partial, \partial)^{(k_1+3)/2}\left[ (\psi_{\pi/2}-P_{2,0}\psi_{\pi/2})a_{k_1,k_2,\pi/2}\right](0) &= (-1)^{k_1}i^{k_2-\frac{k_1+1}{2}}\frac{(k_1+1)!}{2^{(k_1+3)/2}}i^{-n}\left(i \frac{\pi}{2}\right)^{n+k_1+k_2-1}(k_1+3) \left[ \frac{\pi^2}{12}(k_1+2) + m-1\right]
ch(\nu^*\mathcal{L}) = 1+ (d+m)\eta_2 + \gamma_{2,3}- \eta_2\theta, ch(\mathcal{O}_{(m+n-\delta)\Gamma_p+\delta\Delta}) = 1-e^{-(\delta \eta_1 + \delta \gamma_{1,2}+ (m+n) \eta_2)}.
I[\hat{g},\hat{\phi}]=\frac{1}{\pi}\int d^{3}x\sqrt{-\hat{g}}\left( \frac{\hat{R}+2l^{-2}}{16}- \frac{1}{2}(\nabla \hat{\phi})^{2}-\frac{1}{16}\hat{R}\hat{\phi}^{2}+\frac{\nu}{8l^{2}}\hat{\phi}^{6}\right) \;,
\varepsilon \|D^m w\|^2_{L_2(Q_{4R}^+)}&+N(\varepsilon)(\sum_{|\alpha|<m}\|f_\alpha\|^2_{L_\infty(Q_{4R}^+)}R^{2+2m+d}+\sum_{|\alpha|=m}[f_\alpha]_{a, Q_{4R}^+}^{\ast2}R^{2a+2m+d}).
\sum_{h\in H}\biggl(\sum_{v_i \in h:f(v_i)>0}f(v_i)-\sum_{v^j \in h:f(v^j)<0}f(v^j)\biggr)^2 &=\sum_{h\in H}\biggl(\frac{1}{\sqrt{\sum_v\deg v}}\cdot \big|h\big|\biggr)^2 &=\frac{\sum_{h\in H}\big|h\big|^2}{\sum_{h\in H}\big|h\big|},
O = \sum_{n=1}^{\infty}(-1)^{n+1}\exp\left((\ln n)\, x \frac{d}{dx}\right), O^{\dagger}= \sum_{n=1}^{\infty}\frac{(-1)^{n+1}}{n}\exp\left((\ln n^{-1})\, x\frac{d}{dx}\right),
\kappa_\chi \left(X^{\chi(1)}(N), \ldots, X^{\chi(n)}(N)\right) = \frac{\beta}{\alpha}\sum_{\pi \in BNC(\chi')}\Psi_{N, \pi}(\underbrace{X^{\ell}(N), \ldots, X^{\ell}(N)}_{|\chi^{-1}(\{\ell\})|}, \underbrace{X^{r}(N), \ldots, X^{r}(N)}_{|\chi^{-1}(\{r\})|-1}, X^\ell(N) ) \mu_{BNC}(\pi, 1_{\chi'}).
\{Z_{2n}^{N,i}=-1\}=\big(\cap_{m=(2n-1)N}^{2nN-1}\{\frac{S_{m,N+m}^{(i)}}N<r_{i+1}\}\big)\cap\big(\cup_{m=(2n-1)N}^{2nN-1}\{\frac{S_{m,N+m}^{(i)}}N< r_i\}\big).
\sum_{n=0}^M{\frac{(q^{-1}; q^{-1})_M}{(q^{-1}; q^{-1})_n(q^{-1}; q^{-1})_{M-n}}(-1)^nq^{-{n \choose 2}}z^n}= \big(z; q^{-1}\big)_M.
\|u\|^2_{L^2}+\|b\|^2_{L^2}+\nu\int_0^T\|\nabla u\|_{L^2}^2dt=\|u\|^2_{L^2}+\|b\|^2_{L^2}\leq\|u_0\|^2_{B^{s-1}_{2,2}}+\|b_0\|^2_{B^s_{2,2}})=:M_0.
\lim_{n\to \infty}n^{-(\frac{3}{2}-d(r_{i}))}n^{-(\frac{3}{2}-d(s_{i}))}E(\tilde{\zeta}_{n}(r_{i},t_{j})\tilde{\zeta}_{n}(s_{i},u_{j}))=E(G(r_{i},t_{j})G(s_{i},u_{j})),
{\bf E}[\beta_{2,1}(a, b)^q] = \frac{\Gamma_2(q+b_0\,|\,a)}{\Gamma_2(b_0\,|\,a)}\frac{\Gamma_2(b_0+b_1\,|\,a)}{\Gamma_2(q+b_0+b_1\,|\,a)},{\bf E}[\beta_{2,1}(a, b, \bar{b})^q] = \frac{S_2(b_0\,|\,a)}{S_2(q+b_0\,|\,a)}\frac{S_2(q+b_0+b_1\,|\,a)}{S_2(b_0+b_1\,|\,a)}.
\delta \Phi= - \frac{i}{48}{\bar D}^4 D^{{\mathbf i}{\mathbf j}}(L_{{\mathbf i}{\mathbf j}}\Phi) +\frac{i}{48}{\bar D}^4 (L_{{\mathbf i}{\mathbf j}}D^{{\mathbf i}{\mathbf j}}\Phi)\ .
c_i^2N_i'(N_i'-1) + c_i^2 \cdot 2 N_i' N_i'' + c_i^2{N_i''}^2 =c_i^2\left({N_i'}^2 + 2N_i' N_i''+{N_i''}^2 - N_i'\right) &=c_i^2\left((N_i' + N_i'')^2-N_i'\right) &=c_i^2\left(N_i^2-N_i'\right),\,\, ( N_i' + N_i'' = N_i) & = c_i\left(N_i-N_i'\right) \pmod 2 &= c_i N_i'' \pmod 2.
S_{E}^{vol}= \sum_{a}\frac{V_{a}}{4 G_{d}}+ \frac{\beta}{16 \pi G_{d}}\int_{\Sigma}F \wedge \bar{G}- - \frac{\beta}{8 \pi G_{d}}\int_{\partial \Sigma_{\infty}}(\partial^{i}\phi) d \sigma_{i},
\mu_h^k(B(0,\rho)) \lesssim |h|^{([d-\alpha]+1)(d-k)}\mu(\widetilde{\mathcal R}) \le |h|^{([d-\alpha]+1)(d-k)}\sum_{i=1}^N \mu(Q_i) \le \langle \mu\rangle_\alpha |h|^{-\beta_k(\alpha - d + k)}\rho^\alpha.
\dot{z}z^{-1}+\left(\dot{z}z^{-1}\right)^* = \dot{y}y^{-1}+\left(\dot{y}y^{-1}\right)^*+O(t^{-2}\mathcal L) &=P(k)+O(t^{-2}\mathcal L) &=\left[(z\phi z^{-1})^*,z\phi z^{-1}\right]-\theta+O(t^{-1-\epsilon}\mathcal L)
G(z+\omega_2\,|\,\omega_1,\widetilde\omega_1,\omega_2)\cdot G(z\,|\,\omega_1,\widetilde\omega_1,-\omega_2)= \prod_{k_1\geq 0, k_2\geq 0}\big(1-x_2 q_2^{k_1+\frac{1}{2}}\widetilde{q}_2^{k_2+\frac{1}{2}}\big)\cdot\prod_{k_1\geq 0, k_2\geq 0}\big(1-x_2^{-1}q_2^{k_1+\frac{1}{2}}\widetilde{q}_2^{k_2+\frac{1}{2}}\big).
\mathcal{A}_1=\mathcal{A}+\mathcal{A}_{(0)},\mathcal{A}=\mathrm{diag}\Big(\frac{1}{\partial_1\hat{\Phi}^+}\mathcal{E}_{12},\frac{1}{\partial_1\hat{\Phi}^-}\mathcal{E}_{12}\Big),\mathcal{A}_{(0)}|_{x_1=0}=0.
{p_L}=\frac{1}{\sqrt{2ImT ImU}}(m_1+ m_2{\bar U}+ n_1{\bar T}+n_2{\bar U}{\bar T}),\;p_R =\frac{1}{\sqrt{2ImT ImU}}(m_1+ m_2{\bar U}+ n_1 T+n_2 T{\bar U}),
\left( \begin{array}{c}V^{+}V^{-}\end{array}\right)_{k,l}=\left( \begin{array}{c}v_{k,l}^{+}\mathfrak{I}_{k-\frac{1}{2},l}(r)-v_{k,l}^{-}l\mathfrak{I}_{-k+\frac{1}{2},l}(r) -v^{+}_{k,l}l\mathfrak{I}_{k+\frac{1}{2},l}(r)+v^{-}_{k,l}\mathfrak{I}_{-k-\frac{1}{2},l}(r)\end{array}\right)
\delta ([yx^*]_1) = [\Lambda \gamma_t (\Lambda\ (I_2, 0_2) \Lambda ) \gamma_t^* \Lambda ] - [(I_2, 0_2)] = [\Lambda \gamma_t P_0 \gamma_t^* \Lambda] - [P_0] = [\gamma_t P_0 \gamma_t^*] - [P_0]
\int_{M}\| n\mathbf{H}+ \lambda_{1}\psi \|^{2}\, dV =n^2\int_{M}\| \mathbf{H}\|^{2}\, dV + \lambda_{1}^{2}\int_{M}\| \psi\|^{2}\, dV- 2 \lambda_{1}n \mathrm{Vol}(M) \geq 0.
\langle u,u_t\rangle_{\o_T^+}+\sum_{|\alpha|= |\beta|= m}\langle D^\alpha u, A^{\alpha\beta}D^\beta u\rangle_{\o_T^+}+\lambda \langle u,u\rangle_{\o_T^+}&=\sum_{|\alpha|\le m}(-1)^{|\alpha|}\langle D^\alpha u,f_\alpha \rangle_{\o_T^+}.
\| F(w)(t)\| \le \| u_0 \| + \int_0^t \| e^{-(t-s)A}\| \left[ \| w(s) \|_{L^{\infty}}\| w_x(s) \| + \| (1-\partial_x^2)^{-1}w_x(s) \| \right] ds & \le \| u_0 \| + \int_0^t \left[ cR^2s^{-3/4}+ cR \right] ds= \| u_0 \| + 4 cR^2 t^{1/4}+ cR t.
\int_{\Omega_\epsilon}\eta^p \langle |\nabla G_\epsilon|^{p-2}\nabla G_\epsilon, \nabla (G_\epsilon-g) \rangle=- p \int_{\Omega_\epsilon}\eta^{p-1}(G_\epsilon-g) \langle |\nabla G_\epsilon|^{p-2}\nabla G_\epsilon, \nabla \eta \rangle \leq C
\mathcal{B}(z,m,q^2)=z^{-\frac{1}{4}+\frac{m}{2}}q^{-\frac{1}{8}+\frac{m}{4}}\frac{(zq^{m+1};q^2)_{\infty}}{(z^{-1}q^{m};q^2)_{\infty}}\;,
F(a,u,x,t)=\sum_{j=1}^n\sum_{i=1}^n \Big(a_1[u(x_{i+1}, t_{j+1})-u(x_i, t_{j+1})]\frac{1}{\Delta x_{i+1}}&+a_2[u(x_{i+1}, t_{j+1})-2u(x_i, t_{j+1})+u(x_{i-1}, t_{j+1})]\frac{1}{\Delta x_{i+1}^2}&+a_3[u(x_{i+1}, t_{j+1})-u(x_{i+1}, t_{j})]\frac{1}{\Delta t_{i+1}}&+a_4[u(x_{i+1}, t_{j+1})-2u(x_{i+1}, t_{j})+u(x_{i+1}, t_{j-1})]\frac{1}{\Delta t_{i+1}^2}\Big)^2
\begin{array}{r c r c l}{{x =}}&{{- 7 ( e^{t}- 1 ) + 1 6 t \mathrm{~}}}&{{}}&{{t}}&{{< 0 \ ,}}\\{{}}&{{9 ( e^{t}- 1 )}}&{{0 <}}&{{t}}&{{< T \ ,}}\\{{}}&{{3 + 1 2 ( t - T )}}&{{T <}}&{{t}}&{{\ ,}}\\ \end{array}
\vert \mu_{\tau}(x) - \mu_{\tau}(x')\vert= \vert \mu_{\tau}(x) - \mu_{\tau}(x')\vert^{1-\delta}\vert \mu_{\tau}(x) - \mu_{\tau}(x')\vert^{\delta}\le C\tau^{\frac{1-\delta}{2}}\tau^\delta\vert x-x'\vert^{\delta}
G(z,r,\tilde{z},\tilde{r}) =\frac{(r\tilde{r})^{1/2}}{2\pi}\left(\big( \frac{2}{k}-k\big)K(k)-\frac{2}{k}E(k)\right),k=\left(\frac{4r\tilde{r}}{(z-\tilde{z})^{2}+(r+\tilde{r})^{2}}\right)^{1/2}.
B(\omega_{n'},p)=\frac{\alpha}{\pi}T\sum_{n=-\infty}^\infty\int\limits_{-\infty}^\infty\frac{dkB(\omega_n,k)}{\omega^2_n+k^2+m^2(T)}\int\limits_0^\infty\frac{dx\exp(-l^2x/2)}{(\omega_n-\omega_{n'})^2+(k-p)^2+x},
\Re \left\{\frac{- \nu^{- s}}{\pi}e^{\displaystyle{- i \frac{\pi}{2}( s + 1 )}}\int_{- i z}^{\infty - i z}t^{- s}\displaystyle{\frac{d \left( \ln{I_{\nu}\left( \nu t \right)}\right)}{d t}}d t \right\}.
L(u,v)=w(x)(|\nabla{u}|^{p}-\frac{pu^{p-1}}{f(v)}|\nabla{v}|^{p-2}\nabla{u}.\nabla{v}+\frac{u^{p}f^{'}(v)}{(f(v))^{2}}|\nabla{v}|^{p}) R(u,v)=w(x)(|\nabla{u}|^{p}-\nabla(\frac{u^p}{f(v)})|\nabla{v}|^{p-2}\nabla{v})
S^{( l )}( x_{1}) \; = \; \Big ( S^{( 1 )}( x_{1}) \Big )^{l}\; \; , \; \; S^{( 1 )}( x_{1}) \; = \; \exp \left( \; i \pi \; \frac{x_{1}}{\sqrt{x_{1}^{2}+ \lambda^{2}}}\; \right) \; .
v_i' = (v_i,-b_i), \w_i' = (w_i,c_i), \x_i' = (x_i,-r_i),\y_i' = (y_i,s_i) \i = 1,2,\ldots,e, v_i' = (v_i,0), w_i' = (w_i,0) i = e+1,\ldots,e+d,
u_1^{f_0}(x)\big|_{D_{T_2}}=u_1^f(x)\big|_{D_{T_2}},\ \ \ \frac{\partial u_1^{f_0}(x)}{\partial x_0}\Big|_{D_{T_2}}=\frac{\partial u_1^{f}(x)}{\partial x_0}\Big|_{D_{T_2}}.
\overline{y_{k}^{(1)}}\,y_{k}^{(2)}=\left[\left(\overline{\mathbf{U}\cdot\mathbf{x}^{(1)}}\right)^{\top}\right]_{k}\,\left[\mathbf{U}\cdot\mathbf{x}^{(2)}\right]_{k}=\mbox{Prod}_{\boldsymbol{\Delta}^{(k)}}\left(\left(\overline{\mathbf{U}\cdot\mathbf{x}^{(1)}}\right)^{\top},\mathbf{U}\cdot\mathbf{x}^{(2)}\right)
\begin{array}{lll}\displaystyle \int_{\Omega}f(u)(f'(u)-f'(0))^{\alpha}~dx = \displaystyle\alpha \int_{\Omega}(f'(u)-f'(0))^{\alpha-1}f''(u) a^{ij}u_i u_j~dx &\leq \displaystyle C \int_{\Omega}\tilde{f}(u)f''(u) a^{ij}u_i u_j~dx\leq C.\end{array}
F_{i_1 i_2 , i_3}+ F_{i_2 i_3 , i_1}+ F_{i_3 i_1 , i_2}= - \frac{1}{(m-3)!}\delta \left( \begin{array}{cccccc}1 2 ... ... ... m j_1 ... j_{m-3}i_1 i_2 i_3 \end{array}\right)\hspace{-1mm}\sqrt{\bar{g}}K^{j_1 ... j_{m-3}},
R^{(1)}(-x_{1}) = -\mbox{Res}\Big(\frac{J_{R}^{(1)}(s)}{s+x_{1}},s = -x_{1}\Big)+ \sum_{j=2}^{m}\frac{1}{x_{j}-x_{1}}\mbox{Res}(J_{R}^{(1)}(s),s = -x_{j}).
X_{\phi}= \left( \begin{array}{c c}{\mu^{2}+ \frac{3}{4}\lambda \phi_{c l}^{2}}&{0}\\{0}&{\mu^{2}+ \frac{1}{4}\lambda \phi_{c l}^{2}+ \xi q^{2}\phi_{c l}^{2}}\\ \end{array}\right)
\frac{H_{k+1}^{(s,t)}H_{k-1}^{(s+1,t)}(z)}{H_{k}^{(s,t)}H_{k}^{(s+1,t)}}=\left(\frac{H_{k+1}^{(s,t)}H_{k-1}^{(s+1,t)}}{H_{k}^{(s,t)}H_{k}^{(s+1,t)}}\right){\cal H}_{k-1}^{(s+1,t)},
v=\frac{q^{-\frac{1}{p}}}{\sqrt{(1+3\lambda^{2})\frac{B(\frac{3}{p},q-\frac{2}{p})}{B(\frac{1}{p},q)}-4\lambda^{2}\frac{B(\frac{2}{p},q-\frac{1}{p})^{2}}{B(\frac{1}{p},q)^{2}}}}
\bigg| \sum_{n=N+1}^{N+L}e^{2\pi i f(n)}\bigg|^{2}\leq \frac{L(L+M-1)}{M}+ \frac{2(L+M-1)}{M}\sum_{m=1}^{M-1}\left( 1 - \frac{m}{M}\right) \max_{K\leq L}\bigg|\sum_{m, K}\bigg|,
L_1 := t_{\alpha_1}^{-1}t_{\delta_1}^{-1}t_{\gamma_1}^{-1}t_{\beta_1}^{-1}t_{x_1}t_{y_1}t_{z_1}, L := t_{d_1}t_y t_z t_\delta^{-1}t_\gamma^{-1}t_{d_2}^{-1}t_{s_{1,1}}^{-1}.
\langle \Psi \circ e_{1}, e_{2}\rangle - \mathcal{D}\Psi(e_{1},e_{2}) = \rho(e_{2}) \langle \Psi, e_{1}\rangle - \langle e_{2}\circ \Psi, e_{1}\rangle - \langle \Psi, e_{2}\circ e_{1}\rangle & + \langle \mathcal{D}\langle e_{2}, \Psi\rangle, e_{1}\rangle - \rho(e_{1})\langle \Psi, e_{1}\rangle + \langle \Psi, \mathcal{D}\langle e_{1}, e_{2}\rangle \rangle,
\{ab^{-1},cd^{-1}\}_1 = \{ab^{-1},c\}_1d^{-1}- \{ab^{-1}, d\}_1cd^{-2}&= \{a,c\}_1b^{-1}d^{-1}- \{a,d\}_1cb^{-1}d^{-2}- \{b,c\}_1ab^{-2}d^{-1}+ \{b,d\}_1acb^{-2}d^{-2}.
E_{C}^{( 1 ) \; r e n}( R ) = \frac{1}{2 \pi}\int_{0}^{\infty}\ln \left( 1 - e^{- 2 R \omega}\right) ~ d \omega = - \frac{R}{\pi}\int_{0}^{\infty}\frac{\omega ~ d \omega}{e^{2 R \omega}- 1}.
\P\bigl((A')^{\complement}\bigr) \leq\frac{1}{NT^{2}}\sum_{i=1}^{N}\Bigl[\P\Bigl((\alpha+C)T^{1/4}+\sup_{s \in [0,T]}\vert W_{s}^{i}\vert\geq\frac{\varepsilon_{0}}{4}\Bigr)+\P\Bigl(\sup_{s \in [0,T]}\vert Z_{s}^{i}\vert\geq\frac{\varepsilon_{0}}{4CT}\Bigr) \Bigr].
|K_{\alpha}(z,\vec{y})-K_{\alpha}(x,\vec{y})| \le \dfrac{A}{\Big (\sum\limits_{j=1}^{m}|z-y_{j}| \Big)^{mn-\alpha}}\omega \bigg( \frac{|z-x|}{\sum\limits_{j=1}^{m}|z-y_{j}|}\bigg)\le \dfrac{C \omega (2^{-k})}{|2^{k}\sqrt{n}Q |^{m-\alpha/n}}.
Y(x_{1},...,x_{n})=a_{0}+\sum_{i=1}^{n}a_{i}x_{i}+\sum_{i=1}^{n}\sum_{j=i}^{n}a_{ij}x_{i}x_{j}+\sum_{i=1}^{n}\sum_{j=i}^{n}\sum_{k=j}^{n}a_{ijk}x_{i}x_{j}x_{k}+\cdot\cdot\cdot
y = \frac{1}{2}\left( \begin{array}{c c c c}{0 1 0 1}&{0 1 0 1}&{0 1 0 1}&{0 1 0 1}\\{0 0 0 0}&{0 0 0 0}&{1 1 1 1}&{1 1 1 1}\\{0 0 0 0}&{1 1 1 1}&{0 0 0 0}&{1 1 1 1}\\{0 0 1 1}&{0 0 1 1}&{0 0 1 1}&{0 0 1 1}\\ \end{array}\right) \ .
v_i(\nu) = \begin{cases}\displaystyle \min_{a \in \{0, 1\}}\, h_i^a + \nu \, \theta_i^1 \, a + \beta \sum_{j \in N}p_{ij}^a \, v_j(\nu) i \in N^{\{0, 1\}}\displaystyle h_i^1 + \nu \, \theta_i^1 + \beta \sum_{j \in N}p_{ij}^1 \, v_j(\nu) i \in N^{\{1\}}.\end{cases}
\nabla^{\mathrm{Cl\,}}=d+\frac{1}{4}\sum^{2n}_{j,\ell=1}\langle\,\Gamma^{TU}e_j,e_{\ell}\,\rangle c(e_j)c(e_\ell)+\frac{1}{2}\Gamma^{\mathrm{det\,}}.
\mbox{with}\;\;\;\;{\cal{X}}^\alpha({\bf{r}}) ={\textstyle\frac{\partial_i}{\partial^2}}A_i^\alpha({\bf{r}})\;\;\;\;\mbox{and}\;\;\;\;\overline{{\cal Y}^{\alpha}}({\bf r})={\textstyle \frac{\partial_{j}}{\partial^{2}}\overline{{\cal A}_{j}^{\alpha}}({\bf r})}.
\begin{aligned}|J'| =|J' \cap E(G')|+|J'\cap E(Q)|=|J|+|J'\cap E(Q)| \leq \frac 12 (|E(G)|-|E(Q)|)+\frac 1 2 (|E(Q)|-\tau(G,\sigma)) =\frac 1 2 (|E(G)|-\tau(G,\sigma)).\end{aligned}
\mu_i(x)= r_i x_i \left(1- \sum_{j=1}^{d}a_{ij}x_j^+ \right),\sigma(x)= \mathrm{diag}\left(\frac{x_1}{1+\lVert x\rVert^2},\frac{x_2}{1+\lVert x\rVert^2},\ldots,\frac{x_d}{1+\lVert x\rVert^2}\right) ,
\begin{bmatrix}A 0 0 B^* \end{bmatrix}P \circ Q \begin{bmatrix}A^* 0 0 B \end{bmatrix}= \begin{bmatrix}AA^* A( X \circ Y) B B^* ( X^* \circ Y^* ) A^* B^*B\end{bmatrix}&=\begin{bmatrix}AA^* U U^* B^*B\end{bmatrix},
\frac{d F^t_n}{dz}=- \frac{d f^t_n}{d t}+\sum_{j=1}^{n-1}[f^t_j, F^t_{n-j}]=- \frac{d f^t_n}{d t}+\sum_{j=2}^{n-1}[f^t_j, F^t_{n-j}]+[f^t_1, F^t_{n-1}].
{\mathcal U}_{[L,M]}~:=~\Big\{u_{0}\in{\bf L}^{\infty}(\mathbb{R}) \ \big| \ \mbox{Supp\,}(u_{0}) \subset [-L,L]\ , \ \| u_{0}\|_{{\bf L}^{\infty}\left(\mathbb{R}\right)}\leq M\Big\},
\frac{\partial^{3}\varphi(\omega)}{\partial\omega^{3}}=-{(\frac{\lambda}{2\pi c})}^{3}(6\lambda\frac{\partial\varphi(\lambda)}{\partial\lambda}+6\lambda^{2}\frac{\partial^{2}\varphi(\lambda)}{\partial\lambda^{2}}+\lambda^{3}\frac{\partial^{3}\varphi(\lambda)}{\partial\lambda^{3}})
\max_{1\leq j,k \leq n, j\neq k}|f_j(\tau_k)| \geq \sqrt{\frac{\frac{1}{d}\left(\sum_{j=1}^n f_j(\tau_j)\right)^2-\sum_{j=1}^n |f_j(\tau_j)|^2}{n^2-n}}=\sqrt{\frac{\frac{1}{(\mathcal{X})}\left(\sum_{j=1}^n f_j(\tau_j)\right)^2-\sum_{j=1}^n |f_j(\tau_j)|^2}{n^2-n}}.
\sigma^i\underline{x}= \left(x_i^{r_n},\dots,x_{i+k}^{r_n},x_{i+k+r_n},x_{i+k+r_n+1},\dots,x_j^{r_n},\dots,x_{j+l}^{r_n}\dots\right)=\left(x_i^{r_n},\dots,x_i^{r_n},\dots,x_j^{r_n},\dots,x_j^{r_n},\dots\right).
+ H_{1}^{\frac{1}{4}}H_{2}^{- \frac{1}{4}}( d z_{1}^{2}+ d z_{2}^{2}+ d z_{3}^{2}+ d z_{4}^{2}) + H_{1}^{\frac{1}{4}}H_{2}^{\frac{3}{4}}( d x_{1}^{2}+ d x_{2}^{2}+ d x_{3}^{2}+ d x_{4}^{2}) .
H^{0,\frac{1}{2}}_{per}(Q_T) = \{v \in L^2(Q_T) : \big\| \partial^{1/2}_t v \big\| < \infty \}, H^{1,\frac{1}{2}}_{per}(Q_T) = \{v \in H^{1,0}(Q_T) : \big\| \partial^{1/2}_t v \big\| < \infty \}, H^{1,\frac{1}{2}}_{0,per}(Q_T) = \{v \in H^{1,\frac{1}{2}}_{per}(Q_T): v = 0 \mbox{on}\Sigma_T \},
S^{({\rm N})}_{{\rm part},0}= \int dt\, \sum\limits^{N}_{\alpha=1}\left( \xi^{\underline{a}}_{\alpha}\left( E^{\underline{a}}_{j,\alpha}\dot{x}^{j}_{\alpha}+ E^{\underline{a}}_{0,\alpha}\right) -{1\over 2}\xi^{\underline{a}}_{\alpha}\xi^{\underline{a}}_{\alpha}\right)\, .
{}~ ~ \times \Bigg \{\sum_{\sigma \in Z_{N}}\frac{w ( \omega u_{2}^{\prime}u_{3}, \sigma + j_{2}+ j_{3}) w ( q u_{2}^{\prime \prime}, \sigma ) s ( \sigma , j_{1})}{w ( q^{- 1}u_{2}^{\prime}, \sigma + j_{2}) w ( u_{2}^{\prime \prime}u_{3}, \sigma + i_{3})}\Bigg \}_{0}.
\frac{\partial V_{c}}{\partial\mathbb{E}^{\tilde{\mathbb{P}}}\left[\Pi\right]}=-\frac{1}{\lambda_{c}}\hat{Q}_{1}^{-1}+\frac{1}{\lambda_{c}}\hat{Q}_{1}^{-1}\hat{A}_{1}^{\top}\left(\hat{A}_{1}\hat{Q}_{1}^{-1}\hat{A}_{1}^{\top}\right)^{-1}\hat{A}_{1}\hat{Q}_{1}^{-1}.
\begin{array}{ll}I_{(a)}+I_{(b)}+I_{(c)}+I_{(d)}\sim \left[G_{++}^{(0)}(k+p_2)-G_{-+}^{(0)}(k+p_2)\right]G_{R}(k-p_1){}= G_{A}(k+p_2)G_{R}(k-p_1) .\end{array}
\bar{\Pi}_{i j}^{g}= \frac{\partial{\cal L}}{\partial ( \partial_{0}g_{i j})}= \frac{1}{\lambda^{2}}\partial^{0}g_{j i}^{- 1}- \frac{n}{4 \pi}A_{j i}( g ) + \frac{i \alpha}{4 \lambda^{2}}\bar{\psi}_{j k}\gamma^{1}\psi_{k m}g_{m i}^{- 1}
\frac{d}{dt}\langle \exp \left( A t \right) v , M \exp \left( A t \right) v \rangle =\langle \exp \left( A t \right) v , \left( A^{\star}M + M \, A \right) \exp \left( A t \right) v \rangle & =- 4 \gamma \left\Vert \exp \left( A t \right) v \right\Vert^2 \leq- 2 \gamma \langle \exp \left( A t \right) v , M \exp \left( A t \right) v \rangle ,
X^{N}_{i}(t) = \xi^{N}_{i}+ \int_{0}^{t}b\left(s,X^{N}_{i}(s),\mu^{N}(s),u_{i}(s)\right)ds &+ \int_{0}^{t}\sigma\left(s,X^{N}_{i}(s),\mu^{N}(s)\right)dW^{N}_{i}(s),t\in [0,T],
\hat e_t(\eta ):= t_{\vartheta}^{\langle \hat \rho^\vee_s,\eta\rangle}t_{\varphi}^{\langle \rho(\hat R_0^\vee)-\hat \rho^\vee_s,\eta\rangle}=\prod_{\beta\in{\hat R_0^+}}t_{\beta}^{\langle \eta, \beta^\vee\rangle/2}(\eta\in{\hat Q})
M_{f}^{\mathfrak{m}}\bigl(t,(\phi,r,w)\bigr) \doteq f\bigl(\phi(t),w(t)\bigr) - f\bigl(\phi(0),0\bigr) &- \int_{\Gamma\times [0,t]}\mathcal{A}^{\mathfrak{m}}_{\gamma,s}(f)\bigl(\phi(s),w(s)\bigr)\, r(d\gamma,ds),\; t\in [0,T],
\mathrm{Z}_1 =|I_i|^\gamma\omega\left(I_i\right)\leq|I_i|^{\gamma+1}\inf_{t\in I_i}\mathcal{M}(\omega)(t) &\lesssim[\omega]_{A_1(\mathbb{R})}\inf_{t\in I_i}\omega(t)\int_{I_i}f(s)\,ds\leq[\omega]_{A_1(\mathbb{R})}\int_{I_i}f(s)\omega(s)\,ds.
\tilde{g}_{p}=\tau_{f(p)}\circ_{f(p)}^{-1}\circ g_{i}\circ_{p}\circ\tau_{p}^{-1}:B_{p}(0,\beta_{i})\rightarrow B_{f(p)}(0,\delta_{i+1}) \quad\tilde{g}_{p}^{-1}=\tau_{p}\circ_{p}^{-1}\circ g_{i}^{-1}\circ_{f(p)}\circ\tau_{f(p)}^{-1}:B_{f(p)}(0,\beta_{i+1})\rightarrow B_{p}(0,\delta_{i}),
\begin{array}{l}v_{k+1}=W\left[ \left( 2F(u_k,v_k)-F(u_k,v_{k-1})\right) \,,u_k\right] , u_{k+1}=\bar W\left[ \left( 2\bar F(u_k,v_k)-\bar F(u_{k-1},v_k)\right)\,,v_k\right] .\end{array}
\sigma(\widetilde\Delta_n)=\Bigg\{\left. \frac{n^2}{\pi^2}\sin^2\Big(\frac{\pi k_1}{n}\Big)+\frac{n^2}{\pi^2}\sin^2\Big(\frac{\pi k_2}{n}\Big)\right| \forall_{j=1,2}: k_j = 0\, , \cdots, n-1\Bigg\}.
\left\langle \int \, d^{4}x \, \sqrt g \, s \, \Delta_{4}\, f_{\epsilon}( x , x_{0}) \right\rangle_{_V}= \left\langle \int \, d^{4}x \, \sqrt{\overline{{g}}}\, s \, \overline{{\Delta}}_{4}e^{\beta_{8}\sigma}\overline{{f}}_{\epsilon}( x , x_{0}) \right\rangle_{_V}\sim s V^{\frac{\beta_{8}}{\beta_{0}}}
\left(\begin{matrix}\frac{\partial^2 f}{\partial x^2}\frac{\partial^2 f}{\partial x\partial y}\frac{\partial^2 f}{\partial y\partial x}\frac{\partial^2 f}{\partial y^2}\end{matrix}\right) = x^{-s}y^{-s}(1-s) \left(\begin{matrix}-s x^{-1}y (1-s) (1-s) -s x y^{-1}\end{matrix}\right)
d_l w_k=w_{k-1}d_{k-l}, s_l w_k=w_{k+1}s_{k-l}\ 0\leq l\leq k, d_l t_k=t_{k-1}d_{l-1}, s_l t_k=t_{k+1}s_{l-1}\ \ \ 0<l\leq k, d_0t_k=d_k, s_0t_k=t_{k+1}^2s_k.
\nu(q)=\sum_{n\ge0}\frac{q^{n(n+1)}}{(-q;q^{2})_{n+1}}=\frac{1}{\prod_{n>0}(1-q^{n})}\sum_{n\ge0}(-1)^{n}q^{3n(n+1)/2}\frac{1-q^{2n+1}}{1+q^{2n+1}}
\frac{\partial^{2}x^{m}}{\partial X^{\alpha}\partial X^{\beta}}=\frac{\partial x^{m}}{\partial X^{\mu}}_{(X)}\Gamma_{\alpha\beta}^{\mu}-\frac{\partial x^{i}}{\partial X^{\alpha}}\frac{\partial x^{j}}{\partial X^{\beta}}_{(x)}\Gamma_{ij}^{m}
\mathcal{OC}_\ell(C_1\uplus\cdots\uplus C_k) =\mathcal{OC}_\ell(C_1)\times\cdots\times\mathcal{OC}_\ell(C_k) &\overset{\varphi_1\times \cdots\times\varphi_k}{\cong}\mathcal{C}(Q_1)\times\cdots\times\mathcal{C}(Q_k) &=\mathcal{C}(Q_1\uplus\cdots\uplus Q_k),
P_{\mu}^{\varepsilon}\: P_{\mu^{\prime}}^{\varepsilon}= \frac{1}{4}\left( X \: t_{\mu}^{\varepsilon}\: t_{\mu^{\prime}}^{\varepsilon}\: X^{*}+ X \: t_{\mu}^{\varepsilon}\: X \: t_{\mu^{\prime}}^{\varepsilon}+ t_{\mu}^{\varepsilon}\: X^{*}\: t_{\mu^{\prime}}^{\varepsilon}\: X^{*}\right) .
-\frac{1}{k}\int_{0}^{2\pi}\frac{e^{iky}- 1}{e^{iy}- 1}(e^{iy}+ 1) dy = \frac{1}{k}\int_{0}^{2\pi}\frac{e^{iky}- 1}{e^{-iy}- 1}(e^{-iy}+ 1) dy = \frac{1}{k}\int_{0}^{2\pi}\sum_{j=0}^{k-1}e^{-ijy}(e^{-iy}+ 1) dy = \frac{2\pi}{k}= -\frac{2\pi}{|k|}
g_{t}(y) =\frac{Q_{t,0}(x)+(Q_{t,1}(x)-x^{2}Q_{t,0}(x))y+(Q_{t,2}(x)-x^{2}Q_{t,1}(x)-xQ_{t,0}(x))y^{2}}{1-x^{2}y-xy^{2}-y^{3}}&=\frac{\left(\begin{array}{c}3-2x^{2}y-xy^{2}+(x^{2}+2xy+3y^{2})\textbf{i}+(x^{4}+2x+x^{3}y+3y+x^{2}y^{2})\textbf{j}+(x^{6}+3x^{3}+3+x^{5}y+3x^{2}y+x^{4}y^{2}+2xy^{2})\textbf{k}\end{array}\right)}{1-x^{2}y-xy^{2}-y^{3}}.
\sum_{k=n_0}^{n-1}\mu(I_k(c_j)\setminus I_{k+2}(c_j))\log{\frac{1}{|I_{k+2}(c_j)|}}\;\leq\; C_1\sum_{k=n_0}^{n-1}(k+2)a_{k+1}|q_{k+1}\theta -p_{k+1}|
M_p(f_n-f;r)^p=\lim_{m\rightarrow\infty}\int_{-\pi}^{\pi}|f_n(re^{i\theta})-f_m(re^{i\theta})|^p\,dm_{\lambda}(\theta)\le\liminf_{m\rightarrow\infty}\|f_n-f_m\|^p_{H_{\lambda}^{p}},
\sum_{j=-\infty}^\infty |j| |g_j|^2 \;\leq\; c_{9}L^2N^{-2\beta}[B_N(f_0)]^{-3/2}\Big\{[B_N(f_0)]^{1/2}+[B_N(f^\prime_0)]^{1/2}+ [B_N(f^\sharp_0)]^{1/2}\Big\},
\mbox{Contr}_v(s_1, s_2) = (s_1\Theta_{v,1}, \ldots, s_1\Theta_{v, m_v-1})(-A_v)^{-1}\left ( \begin{array}{c}s_2\Theta_{v,1}\vdots s_2\Theta_{v, m_v-1}\end{array}\right ),
I( r,t,\psi(\cdot))= &\frac{t}{2}\int_{0}^{1}\frac{\eta d\eta}{\sqrt{1-\eta^2}}\int_{0}^{1}\frac{K(r,t,\eta,\xi)\psi(r+t\eta-2t\eta\xi)}{\sqrt{\xi}\sqrt{1-\xi}}T_{m-1}\left(\Theta(r,t,\eta,\xi)\right)d\xi.
\begin{array}{c}{\overline{{A}}^{\prime 1}{}_{\nu}\longrightarrow W_{\nu}^{-},}\\{\overline{{A}}^{\prime 2}{}_{\nu}\longrightarrow W_{\nu}^{+},}\\{\overline{{A}}^{\prime 3}{}_{\nu}\longrightarrow Z_{\nu},}\\{\overline{{A}}^{\prime 4}{}_{\nu}\longrightarrow A_{\nu}.}\\ \end{array}
v^2=\frac{(-2h\sum_{i=1}^n a_iD_i^5)}{(\sum_{i=1}^m b_iB_iC_i^2)}u^4-\frac{(20h^3\sum_{i=1}^n a_iD_i^3+ \sum_{i=1}^m b_iB_i^3)}{(3\sum_{i=1}^m b_iB_iC_i^2)}u^2-\frac{(2h^5\sum_{i=1}^n a_iD_i)}{(\sum_{i=1}^m b_iB_iC_i^2)}.
\left(x^2+y^2\right)^\alpha =\left((1+(x-1))^2+y^2\right)^\alpha =\left(1+2(x-1)+(x-1)^2+y^2\right)^\alpha &< \left(1+(x-1)^2+2(x-1)y+y^2\right)^\alpha=\left(1+\left((x-1)+y\right)^2\right)^\alpha.
D_{s\eta^{i+1}}=(1+\varrho^{i+1})D_{s\eta^i}+s((\chi_{i+1})_z\eta_{i+1}+(\chi_{i+1})_{\bar{z}}\bar{\eta}_{i+1})e_1\partial_t &+\Theta^{i}_s+\mathcal{R}^{i}_{s}+\hat{\mathcal{H}}^{i}_s+\mathcal{F}^{i}_s
\sum_{\rho \in S_n}^{}\chi_R (\rho) D_R (\rho^{-1}) = \frac{n!}{d_R}I_R,\qquad{\rm and}\qquad\sum_{\sigma \in S_n}^{}D_R (\sigma \rho \sigma^{-1}) =\frac{n!}{d_R}\chi_R (\rho) I_R,
t^{\frac12}\tau^\vee_i E_\mu(q,0) = D_{i+1,i}E_\mu(q,0),\quad\hbox{when $\mu_i>\mu_{i+1}$, and}t^{-\frac12}\tau^\vee_i E_\nu(q,\infty) = D_{i,i+1}E_\nu(q,\infty),\quad\hbox{when $\nu_i<\nu_{i+1}$.}
\mathcal H_N(s_1,\ldots,s_N):=\sum_{\sigma\in S_N}\tilde{E}\left[D_{t_{I_r}}^{H,q_{I_r}}\ldots D_{t_J}^{H,q_J}F|\mathcal{F}_{s_{\sigma(N)}}^{H}\right]\chi_{A}(s_{\sigma(1)},\ldots,s_{\sigma(N)})
\varepsilon_{\lambda}= \pm 1 \Longrightarrow\varepsilon_{\mathcal L}= \pm (-1)^{n/2}\Longrightarrow\varepsilon_{{\mathcal L}_{n-2}}= \pm (-1)^{n/2}\Longrightarrow \varepsilon_{\lambda_{n-2}}= \pm (-1)^{n/2}(-1)^{(n-2)/2}= \mp 1\ .
=\sup \left\{\int_{\mathrm{P}^{d}(\mathbb{R})}|K_{2n}^{\ast}(x,y)|d\nu(y)|\,\,x\in \mathrm{P}^{d}(\mathbb{R})\right\}=\int_{\mathrm{P}^{d}(\mathbb{R})}|K_{2n}^{\ast}(\mathbf{o},y)|d\nu (y).
\begin{aligned}\mathcal{A}[\overline{u}](x) \geq f(ax)-f(x)- a^N(a-1)\|DJ\|_{\infty}\mu\sup_{|z|<1}\Psi(a(x+z)) \geq af(x)-f(x)-a^N(a-1)\|DJ\|_{\infty}\mu f(x) o_x(1) \end{aligned}
B_{n+m+l}^{\left( m+l\right)}\left( m\right) =\left( -1\right)^{n+m+l}B_{n+m+l}^{\left( m+l\right)}\left( l\right) & =\frac{\left( l+m\right)}{l}\binom{n+m+l}{l}\sum_{k=0}^{m}\left(-1\right)^{k}\genfrac{[}{]}{0pt}{}{m}{k}\binom{n+l+k}{l}^{-1}B_{n+l+k}^{\left( l\right)}.
\check{\mathbf{u}}^{(t+1)}_N = e^{j\arg\left(\left(\|\mathbf{y}\|^2_{\infty}\lambda_{\max}(\mathbf{T}_N^H\mathbf{T}_N)\mathbf{I}_N - \mathbf{T}_N^H\tilde{\mathbf{V}}\mathbf{T}_N\right)\check{\mathbf{u}}^{(t)}_N\right)}.