harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac1{s_2{-}s_1}{\| w(s_2){-}w(s_1)\|^2}= \frac{\| w(s_2){-}w(s_1)\|}{s_2{-}s_1}\: \| u_{\theta_2}(t_2){-}u_{\theta_1}(t_1) \| \leq \frac1{t_1}\,\| u_{\theta_2}(t_2){-}u_{\theta_1}(t_1) \| .
{}\begin{aligned}J_{\prec k}:=\{j \in \{1,\ldots,\ell\},\;I_j \subsetneqq I_k\}, J_{\succ k}:=\{j \in \{1,\ldots,\ell\},\;I_j \supsetneqq I_k\}, J_{\nparallel\, k}:=\{j \in \{1,\ldots,\ell\},\;I_j \cap I_k = \emptyset\}.\end{aligned}
(f_j\circ \psi_j)^*(g_{\textnormal{euc}}) = \psi_j^*(e^{2\tilde u_j}\, g_{\textnormal{poin},j}) = e^{2\tilde u_j\circ \psi_j}\, \psi_j^*g_{\textnormal{poin},j}= e^{2\tilde u_j\circ \psi_j+2v_j}\, g_{\textnormal{euc}}\textnormal{on}\,\,\, B_1^2(0).
\delta L_{0}= i \Lambda^{a}D_{-}\chi_{a}( \varphi , \tilde{\varphi}) + \Lambda^{a}A_{-}^{b}\left( \partial_{\alpha}\chi_{a}\xi_{b}^{\alpha}+ \partial_{\overline{{\alpha}}}\chi_{a}A_{-}^{b}\tilde{\xi}_{b}^{\overline{{\alpha}}}\right) ( \varphi , \tilde{\varphi}) .
\overline{\rho^\gamma}(v_z)\rho^\gamma(\pi_h(h,b)) =\rho^\gamma(v_z\pi_h(h,b))=\rho^\gamma(\pi_h(zh,b)) &=\gamma(zhc(zh)^{-1})\pi^\gamma_{q(zh)}(b)=\gamma(z)\gamma(hc(h)^{-1})\pi^\gamma_{q(h)}(b) &=\gamma(z)\rho^\gamma(\pi_h(h,b)).
\|u(t_n)-U^n\|_{M^{1, p}(H)^{*}}\leq \|(E(t_n)-B_nP_h)u_0\|_{M^{1, p}(H)^{*}}& + \| \int_{0}^{t_n}\bar{E}(t_n-t)\d W(t)- \sum_{j=0}^{n-1}\int_{t_j}^{t_{j+1}}B_{n-j}P_h \d W(t)\|_{M^{1, p}(H)^{*}}:={\rm I + II}.
W^{({D})}({\bf r})={\mathcal F}_{({D})}^{-1}\left\{\widetilde{W}({\bf k}) \right\}=\int \frac{d^{{D}}k}{(2\pi)^{{D}}}\; e^{i{\bf k}\cdot{\bf r}}\, \widetilde{W}({\bf k})\; ,
g(x)-\tilde{g}_\epsilon(x) &= g(x)-g(m_0+\epsilon)-(x-m_0-\epsilon)g'(m_0+\epsilon) &=\frac{g^{(k)}(m_0)}{k!}\bigg[(x-m_0)^k-\epsilon^k-k\epsilon^{k-1}(x-m_0-\epsilon)\bigg]+\textrm{higher order terms}.
\sum_{i=1}^N |u_{x_i}|^{p_i-2}\, u_{x_i}\, \left(\frac{v^{q_i}}{u^{q_i-1}}\right)_{x_i}\le \sum_{i=1}^N |v_{x_i}|^\frac{q_i}{p_i}\, |u_{x_i}|^\frac{p_i-q_i}{p_i},
w_y+\lambda_1w_x=\varepsilon w_{xx}-\frac{2\varepsilon v_xw_{x}}{v}+f(u,v)+g(u,v), &z_y+\lambda_2z_x=\varepsilon z_{xx}-\frac{2\varepsilon v_xz_{x}}{v}+f(u,v)-g(u,v).
T r_{\{\eta , \overline{{\eta}}\}}( I ) = T r_{\{\eta , \overline{{\eta}}\}}( \theta ) = T r_{\{\eta , \overline{{\eta}}\}}( \overline{{\theta}}) = 0 , T r_{\{\eta , \overline{{\eta}}\}}( \overline{{\theta}}\theta ) = q .
|N_{2}(f_{v_{5}})(t,r)| \leq \begin{cases}\frac{C r}{(\lambda(t)^{2}+r^{2})t^{4}\log^{3b}(t)}+\frac{C r}{t^{6}\log^{3b}(t)}, r \leq \frac{t}{2}\frac{C \log^{3}(r)}{r^{2}|t-r|^{3}}+ \frac{C}{r^{7/2}t^{5/2}\log^{3N+7b-2}(t)}, \frac{t}{2}\leq r \leq t-\sqrt{t}r\geq t+\sqrt{t}\frac{C}{r^{7/2}}, t-\sqrt{t}\leq r \leq t+\sqrt{t}\end{cases}
m(1+x_{1}+x_{2}+x_{3}+x_{4})\overset{?}{=}{}6\left( \frac{\sqrt{15}}{2\pi}\right)^{5}L(f_{3,15},4), m(1+x_{1}+x_{2}+x_{3}+x_{4}+x_{5})\overset{?}={}3\left( \frac{\sqrt{6}}{\pi}\right)^6L(f_{4,6},5).
\begin{array}{lll}1+6 \displaystyle \sum_{\ell=1}^{E[\frac{R}{r}]}\ell +6\left(k+2-2 \lceil \frac{AB}{r}\rceil \right) =1+3 E[\frac{R}{r}](1+E[\frac{R}{r}]) + 6\left(k+2-2\left\lceil \frac{k+1}{2}-\sqrt{\frac{R^2}{r^2}- \frac{3}{4}(k+1)^2}\right\rceil \right)\end{array}
\psi_{+}^{r}( x ) = \sum_{\sigma = \pm 1 / 2}\int_{k^{+}> 0}{\frac{d k^{+}d^{2}k_{\perp}}{1 6 \pi^{3}k^{+}}}\Bigl [ b^{r}( k , \sigma ) u_{+}( k , \sigma ) e^{- i k \cdot x}+ d^{r \dagger}( k , \sigma ) v_{+}( k , \sigma ) e^{i k \cdot x}\Bigr ] \; ,
Q_{i}'(0) = \det\begin{bmatrix}s_{i,-2}s_{i,-1}0 s_{i,-1}1 1 1 \lambda_{i,2}^{2}0\end{bmatrix}= -\lambda_{i,2}^{2}s_{i,-2}+s_{i,-1}, i\in\mathbb{N}\cap[1,\eta],
\mathcal{M}[j]_{m_{1},m_{2},m}^{\l_{1},\l_{2},\l}=\sum_{\mu_{1}=-j}^{j}\sum_{\mu_{2}=-j}^{j}\sum_{\mu_{3}=-j}^{j}(-1)^{j+\mu_{2}}C_{\mu_{1},\mu_{2},m_{1}}^{\j,\j,\l_{1}}C_{-\mu_{2},\mu_{3},m_{2}}^{\j,\j,\l_{2}}C_{\mu_{1},\mu_{3},m}^{\j,\j,\l}
T_{w_{\sigma(1)}}^f (P_{w_{\sigma(1)}}^f)^\perp \otimes \ldots \otimes T_{w_{\sigma(k)}}^f (P_{w_{\sigma(k)}}^f)^\perp \otimes P_{V\Gamma_0}\otimes T_{w_{\sigma(k+l+1)}}^f (P_{w_{\sigma(k+l+1)}}^f) \otimes \ldots \otimes T_{w_{\sigma(d)}}^f (P_{w_{\sigma(d)}}^f),
\sum_{j = 0}^k{G_j^3}= \frac{5}{4}G_0^3 + 2G_1^3 + \frac{1}{2}G_2^3 - \frac{1}{4}G_3^3 &\qquad- \frac{5}{4}G_{k + 1}^3 - 2G_{k + 2}^3 - \frac{1}{2}G_{k + 3}^3 + \frac{1}{4}G_{k + 4}^3\,,
\gamma\cdot( t^{k_e}\varphi\otimes T^{-1}_{{\omega}}(e)) \stackrel{(i)}{=}\gamma^{k_e}t^{k_e}\varphi\otimes \gamma\cdot T^{-1}_{{\omega}}(e){}\stackrel{(ii)}{=}t^{k_e}\varphi\otimes T^{-1}_{{\omega}}T^{-1}_{e^*(\gamma)}(e).
\begin{aligned}R^{g,0}= u^0_e\partial_x g^1_e-h^0_e\partial_x v^1_e+\sqrt{\epsilon}\big[(v^0_p+v^1_e)\partial_Y g^1_e-(g^0_p+g^1_e)\partial_Y v^1_e\big]-\kappa\epsilon \partial_Y^2 g^1_e. \end{aligned}
s y_1 \dots y_{i-1}[y_i, x_1, x_2] = y_1 \dots y_{i-1}s [y_i, x_1, x_2] + \sum_{j=1}^{i-1}y_1 \dots y_{j-1}[s, y_j] y_{j+1}\dots y_{i-1}[y_i, x_1, x_2] .
\hat{{e}}_{n}^{m}\left( \gamma\right) \left\{{\hat{{w}}_{2}\left({\gamma,\rho}\right) -\hat{{w}}_{4}\left({\gamma,\rho}\right)}\right\}=o\left( 1\right) \hat{{A}}{\operatorname{env}}U\left({-{\tfrac{1}{2}}a,\hat{{\rho}}\sqrt{2\gamma}}\right) ,
FF^* = D \frac{1}{D^2+f^2}D= [D, \frac{1}{D^2+f^2}] D + \frac{1}{D^2+f^2}D^2=\frac{1}{D^2+f^2}[f^2,D] \frac{1}{D^2+f^2}D + 1 - \frac{1}{D^2 +f^2}f^2 \sim 1
|x_{n+1}|^{\frac 1{3^{n+1}}}< |x_0|\prod_{i=0}^{n}h_{i}^{\frac 1{3^{i+1}}}+\sum_{i=1}^{n+1}\prod_{j=i}^{n}h_{j}^{\frac 1{3^{j+1}}}| u_{i}|^{\frac 1{3^{i}}}&=\prod_{i=0}^{n}h_{i}^{\frac 1{3^{i+1}}}\left[ |x_0|+\sum_{i=1}^{n+1}\prod_{j=0}^{i-1}h_{j}^{-\frac 1{3^{j+1}}}| u_{i}|^{\frac 1{3^{i}}}\right].
\begin{aligned}x_i{}={}\frac{U}{L}+ \frac{U}{L}\sum\limits_{j=i}^{N-1}h_j f_i{}={}\Delta - \frac{U^2}{L}\sum\limits_{j=0}^{i-1}h_j \, \left(1 - \frac{h_j}{2}\frac{-\kappa}{1-\kappa}\right) g_i{}={}U\end{aligned}
\theta_{\bot \parallel}=\frac{1}{y^{2}}\left( \partial_{\tau}\vec{x}\, \partial_{\sigma}\vec{x}+\partial_{\tau}y\, \partial_{\sigma}y\right) =\frac{1}{a^{2}\tau}\left[ \vec{f}{\vec{c}\,}'+aa'\right] +\frac{1}{a^{2}}\left( \vec{g}{\vec{c}\,}'\right) +\ldots
R_{b,m}^{\rm{S-N}}= \min \left\{{R_{b,m \to s}^{\rm{S-N}},R_{b,m \to b}^{\rm{S-N}}}\right\}\triangleq{W_m}{\log_2}\left({1 + \frac{{{p_{b,m}}{{\left|{{{\widetilde h}_b}}\right|}^2}}}{{{p_s}{{\left|{{{\widetilde h}_b}}\right|}^2}+{W_m}{N_0}}}}\right).
\|\chi_{S_{r_0}}e^{-itH_0}P_{\delta}(W_{n;\textrm{out}})\|_{\textrm{op}}=\|e^{-itH_0^\parallel}\otimes (\chi_{B_{r_0}}e^{-itH_0^\perp}P^\perp_\delta(W_{n;\textrm{out}}^\perp))\|_\textrm{op}=\|\chi_{B_{r_0}}e^{-itH_0^\perp}P^\perp_\delta(W_{n;\textrm{out}}^\perp)\|_\textrm{op}
\mathcal{A}_{P}^{(1)}(UF) = \mathcal{A}_{P}^{(1)}(UF,UG) = \textit{max}_{i=1}^N \left\{q_i\dfrac{{| \langle Uf_i,Ug_i\rangle | + \|Uf_i\|\; \|Ug_i \|}}{2}\right\}\leq \textit{max}_{i=1}^N \left\{q_i \dfrac{{| \langle Uf_i,Ug''_i\rangle | + \|Uf_i\|\; \|Ug''_i \|}}{2}\right\}
\pi_\ast ({\rm pr}_\pi^!Z\cdot_{{\rm id}_{T^\ast X\times_X\widetilde X}}\widetilde X)\overset{\eqref{eq:refinedGysing02}}{=}\pi_\ast 0_{X}^{\prime!}({\rm pr}_{\pi}^! Z)\overset{\eqref{eq:saito:81}}{=}0^!_X(Z).
\lambda^{2}\frac{1}{2^{1 5}3^{3}\pi^{6}}\left[ \sum_{N = 1}^{\infty}[ C_{N}^{\frac{3}{2}}( c o s{\theta_{4}^{0}}) ]^{2}\frac{( N + 1 ) ( N + 2 )}{N ( N + 3 )}\right] ( R K^{2}) \frac{1}{\epsilon}\int_{C}d^{4}{\eta}+ F . T .
\widetilde{J}(x):=\sup_{\tau_1,\tau_2\in\mathcal{T},\atop\tau_1\leq\tau_2}\mathbb{E}^x\left[\int_{\tau_1}^{\tau_2}f(X(t))\mathrm{d}t+g_1^{\delta}(X(\tau_1))+g_2^{\delta}(X(\tau_2))\right],
\langle f^{*}*f\rangle=[\begin{matrix}a^{*}&b^{*}&c^{*}\end{matrix}][\begin{matrix}1&\langle x\rangle&\langle p\rangle\\ \langle x\rangle&\langle x*x\rangle&\langle x*p\rangle\\ \langle p\rangle&\langle p*x\rangle&\langle p*p\rangle\end{matrix}][\begin{matrix}a\\b\\c\end{matrix}]\ge0.
[\varphi_{\Omega_{w}^{-1}(T_{k})}\varphi_{\Omega_{w}^{-1}(T_{k}^{\ast})}]=[\varphi_{I(T_{k}\oplus T_{k}^{\ast})}]\left(\frac{[\varphi_{\Omega_{w}^{-1}(T_{+})}]}{[\varphi_{I(T_{+})}]}+\frac{[\varphi_{\Omega_{w}^{-1}(T_{-})}]}{[\varphi_{I(T_{-})}]}\right).
S_{e f f}= \int d t [ P_{A}\dot{X^{A}}+ \dot{C}^{\mu}\bar{C}_{\mu}+ \dot{\eta}^{\mu}\bar{\eta}_{\mu}+ \Pi_{\mu}\dot{N}^{\mu}+ \{Q_{B}, ~ \bar{C}_{\mu}N^{\mu}+ \bar{\eta}_{\mu}\chi^{\mu}\}] ,
f\left( T_a(z),\zeta \right) =T'_a(z)\left[ f(z,\zeta) +\sum_b\frac{1}{2}(\zeta -k)(a_bz+b_b)^2\ln \left( \frac{a_b(d_ba_a-c_bb_a)}{c_b(b_ba_a-a_bb_a)}\right) \right] \ .
|A_s(V_{\underline{T}_k}, \underline{T}_k) (\omega) - A^{(n)}_s(V_{\underline{T}_k}^{(n)}, \underline{T}_k ) (\omega) | < 2|V_{\underline{T}_k}(\omega) - V_{\underline{T}_k}^{(n)}(\omega) | + 4 \varepsilon \leq 2 \eta + 4\varepsilon = \tilde{\eta}, \underline{T}_k (\omega) \leq s \leq \overline{T}_k (\omega) \wedge t.
\left. \begin{array}{l c l}{T_{\; \; z}^{z}}{=}{V ( \phi ) -{{{{{\phi^{\prime}}}^{2}}}{{e^{- 2 \, a}}}}}{T_{\; \, i}^{i}}{=}{V ( \phi ) +{{{{{\phi^{\prime}}}^{2}}}{{e^{- 2 \, a}}}},}\end{array}\right.
\begin{array}{l c l}{{\mathcal L}_{2}^{( 0 , 2 )}}&{=}&{{u_{0}\star p}^{2}+ u_{1}\star p + u_{2}}\\{}&{=}&{u_{0}{p}^{2}+ ( u_{1}- 2 \theta u_{0}^{\prime}) p + ( u_{2}- \theta u_{1}^{\prime}+ \theta^{2}u_{0}^{\prime \prime})}\\ \end{array}
\frac{(2n+|x|_{1})!}{\prod_{i=1}^{d}q_{i}!(q_{i}+|x_{i}|)!}= \frac{(2n+|x|_{1})!}{n! \, (n+|x|_{1})!}\ \frac{(n+|x|_{1})!}{\prod_{i=1}^{d}(q_{i}+|x_{i}|)!}\ \frac{n!}{\prod_{i=1}^{d}q_{i}!}.
\begin{cases}x_{s}u^{1}_{s}+ y_{s}v^{1}_{s}+ z_{s}w^{1}_{s}= 0, x_{s}u^{1}_{t}+ y_{s}v^{1}_{t}+ z_{s}w^{1}_{t}+ x_{t}u^{1}_{s}+ y_{t}v^{1}_{s}+ z_{t}w^{1}_{s}= 0, x_{t}u^{1}_{t}+ y_{t}v^{1}_{t}+ z_{t}w^{1}_{t}= 0, \end{cases}
q^{r\binom{k+1}{2}+(1-r)k}[r]_q^k\,[k]_{q^r}!S_{r}[n,k] = \sum_{\ell=0}^{k}q^{rk(k-\ell)}A_{n,\ell}^{r}(q){n-\ell \brack k-\ell}_{q^r}
\mathcal{B}=~[(1+(\eta_1+\eta_2+\eta_3)^2)^{3/2}-(1+\eta_3^2)^{3/2}](1+\eta_1^2)^{3/2}[(1+\eta_3^2)^{3/2}-(1+\eta_1^2)^{3/2}](1+(\eta_1+\eta_2+\eta_3)^2)^{3/2}.
\varphi(\rho,\vartheta) =\sum\limits_{k\in\mathfrak{T}_2}(2\pi \rho)^{-1/2}\varphi_{k}(\rho)\mathrm{e}^{\mathrm{i}k\vartheta}; \zeta(r,\theta,\phi) =\sum\limits_{(l,m,s)\in \mathfrak{T}_3}r^{-1}\zeta_{(l,m,s)}(r) \Omega_{l,m,s}(\theta,\phi);
\rho \ddot{v}_i = \big(A_{iJKj}v_{j,K}- \beta_{Ji}\dot{\omega}-(C_{iJKLIj}v_{j,IL}+ M_{iJKL}\omega_{,L})_{,K}\big)_{,J}, a \ddot{\omega}=-\beta_{Ki}\dot{v}_{i,K}+ m_{IJ}r_{I,J}+ M_{jLKI}v_{j,LKI}+K_{IJ}\omega_{,IJ}, \kappa \dot{r}_i = \dot{\omega}_{,i}- r_{i}
\tilde{t}_{ij}(\bar{a}) =[\hat{f}_{i1}(1),\hat{g}_{1j}(\bar{a})] &=-(-1)^{|a|}[\hat{f}_{i1}(1),[\hat{f}_{j2}(1),\hat{v}_{21}(a)]] &=-(-1)^{|a|}[[\hat{f}_{i1}(1),\hat{f}_{j2}(1)],\hat{v}_{21}(a)]+(-1)^{|a|}[\hat{f}_{j2}(1),[\hat{f}_{i1}(1),\hat{v}_{12}(\bar{a})]] &=-[\hat{f}_{j2}(1),\hat{g}_{2i}(a)] &=-\tilde{t}_{ji}(a),
|N_G{(S)}| =|N_G(S_1)| \geq \lambda_1 \vert N_G{(S_1)}\cap E( K_n) \vert =\lambda_1 b \binom{n}{2}\geq \lambda_1 \binom{n}{2}a^{2/k}\geq (\mu \binom{n}{k}-\alpha \binom{n}{2}) a^{2/k}\geq \mu \binom{n}{k}(1-g) a^{2/k}\geq \mu \binom{n}{k}a^{1-2/k}a^{2/k}=\mu |S_1|\geq | S |.
R_{i,j}(G,S,\lambda)=\sum_{p=1}^{m+1}R_{i,j}^{(p)}(G,S,\lambda),\forall i,j\in S. R_{i,j}(G,S,\lambda)=\sum_{p=1}^{m-1}R_{i,j}^{(p)}(G,S,\lambda),\forall i,j\in\overline{S},i\neq j. R_{i,i}(G,S,\lambda)=w(i,i),\forall i\in\overline{S}. R_{i,j}(G,S,\lambda)=\sum_{p=1}^mR_{i,j}^{(p)}(G,S,\lambda),\forall i\in S, j\in\overline{S}i\in\overline{S}, j\in S.
v_0(t) := e^{t A_p}a + \int_0^t e^{(t-s)A_p}P_p f(s) ds, \zeta_0(t) := e^{t \Delta_{\zeta}}b + \int_0^t e^{(t-s)\Delta_{\zeta}}g(s) ds, v_{m+1}(t) := v_0(t) + \int_0^t e^{(t-s)A_p}F_p(v_m(s),\zeta_m(s)) ds, \zeta_{m+1}(t) := \zeta_0(t) + \int_0^t e^{(t-s)\Delta_{\zeta}}G_q(v_m(s),\zeta_m(s)) ds.
\lambda^{(p)}(G) =\Big(\frac{r^{p-r}}{\alpha}\Big)^{1/p}=(rC)^{(p-r)/p}& =\Bigg(\sum_{i=1}^k\big(\lambda^{(p)}(G_i)\big)^{p/(p-r)}\Bigg)^{(p-r)/p},
z_{j}^-(1) = x_{\eta_{j}}^-(1)=[\,x_{i_j}^-(0),\, z_{j-1}^-(1)\,]_{\langle i_j, i_{j-1}\rangle\cdots \langle i_j, i_1\rangle}, z_{j}^+(-1) = x_{\eta_{j}}^+(-1)=[\,z_{j-1}^+(-1),\, x_{i_j}^+(0)\,]_{\langle i_{j-1}, i_j\rangle^{-1}\cdots \langle i_1, i_j\rangle^{-1}}.
H^{0,1}\lrcorner\,\bar\partial_B\phi =\nabla_{H^{0,1}}\phi, &\partial_B H^{1,0}\lrcorner\,\phi + H^{1,0}\lrcorner\,\partial_B\phi = \nabla_{H^{1,0}}\phi +\sum_a \omega^a\wedge (\nabla_{V_a}H^{1,0})\lrcorner\,\phi.
\sum_{m=0}^\infty \lambda^m B_m^{(k)}\left(k+\frac{x}{\lambda}\right) \frac{t^m}{m!}=\sum_{m=0}^\infty \sum_{n=0}^m \hat{D}_{n,\xi}^{(k)}(x|\lambda) \xi^{-n}s_2 (m,n) \frac{t^m}{m!}.
\gamma_{r}^{\varepsilon}=\gamma_{\overline{\Omega}}^{B_{r}(y_{1})^{c}}+(1-\varepsilon)\gamma_{\overline{\Omega}}^{B_{r}(y_{1})}+\varepsilon\big(\pi^{1},\mathcal{T}_{y_{1}}^{y_{2}})_{\#}\gamma_{\overline{\Omega}}^{B_{r}(y_{1})}\big),
\varphi = \begin{cases}- dr^1 \wedge dr^2 \wedge dr^3 + dr^1 \wedge \omega + dr^2 \wedge \Re (\Omega) + dr^3 \wedge \Im (\Omega) M^7 = T^3 \times X^4, begin{align*}5pt] \Re (\Omega) - dr \wedge \omega M^7 = S^1 \times X^6. \end{cases}
\rho(\tt e)=\begin{bmatrix}0 0 -1 0 -1 0 -1 0 0\end{bmatrix}\mbox{and}\rho(\tt p)=\begin{bmatrix}e^{i\alpha}\sqrt{2\cos{3\alpha}}-e^{-2i\alpha}0 e^{-2i\alpha}-\sqrt{2\cos{3\alpha}}e^{-i\alpha}0 0 e^{i\alpha}\end{bmatrix},
\widetilde{c}_1 =c^{(1)}_{2,2}-c^{(1)}_{1,1}-c^{(1)}_{3,3}+c^{(1)}_{0,0}\ , \widetilde{c}_2 =c^{(1)}_{2,2}-2c^{(1)}_{1,1}-3c^{(1)}_{3,3}\ , \widetilde{c}_3 =c^{(1)}_{1,0}-c^{(1)}_{2,1}+c^{(1)}_{3,2}\ .
F_{1}=\frac{x}{a}R^{-1}(G_{1})=\frac{a}{x}R^{-1}(F_{1}\oplus F_{2})=\frac{a}{x}R^{-1}(F_{1})\oplus\frac{a}{x}R^{-1}(F_{2})=G_{1}\oplus G_{2}
\textstyle A_{w,0}\textstyle = p\frac{(-1)^w(s-r)(r-\alpha)_w}{(s-\alpha)_{w+1}}+ O(p^2), \textstyle A_{w,j}\textstyle = \sum_v(-1)^{w-v}{j+w-v-1\choose w-v}{r-\alpha+j\choose v}\left({s-\alpha+j-v \choose j-v}-{r-\alpha+j-v \choose j-v}\right) + O(p),
\aligned (\hat d_{2}^{i+1}\circ A - (-1)^{\deg A}A\circ \hat d_{1}^{i})_{\alpha'_2\alpha_1}&\quad-(\hat d_{2}^{i+1}\circ A\vert_{E^i}-(-1)^{\deg A}A\vert_{E^i}\circ \hat d_{1}^{i})_{\alpha'_2\alpha_1}&=d_0 \circ A_{\alpha'_2\alpha_1}- (-1)^{\deg A}A_{\alpha'_2\alpha_1}\circ d_0.\endaligned
= \left\{[ 1 - e^{- \xi - ( h_{+}+ h_{-}) / 2 T}] [ 1 - e^{- \xi + ( h_{+}+ h_{-}) / 2 T}] [ 1 - e^{- \xi - ( h_{+}- h_{-}) / 2 T}] [ 1 - e^{- \xi + ( h_{+}- h_{-}) / 2 T}] \right\}^{- 1}
m \equiv \left( \partial_{\underline{{y}}}\Lambda \right) \Lambda^{- 1}={\textstyle \frac{1}{2}}m^{i}T_{i}={\textstyle \frac{1}{2}}\left( \begin{array}{c c}{m^{1}}&{m^{2}+ m^{3}}\\{}&{}\\{m^{2}- m^{3}}&{- m_{1}}\\ \end{array}\right) .
\left( \bigg| \frac{1}{\sqrt{k}l}\underset{i=1}{\overset{k}{\sum}}\left( \underset{j=1}{\overset{l}{\sum}}B^j_i - lp_i \right)^2 - \mu_p + \frac{l-1}{\sqrt{k}}\underset{i=1}{\overset{k}{\sum}}\left( p_i - \frac{1}{2}\right)^2 + \zeta \bigg| \leq c_{\alpha,n}\right)
\bigg(-24, -\frac{1}{245}\bigg),\bigg(\frac{1}{2}, -\frac{2}{49}\bigg), \bigg( \frac{m n (4 m + 2 n -3)}{(m + n -1) (2 m + 2 n -1)},\ \lambda_1 \bigg), \bigg( -\frac{2 m n (3 - 4 m - 2 n + 4 m n)}{2 m - 2 n -1}, \ \lambda_2 \bigg), \bigg( -\frac{(2 m n + m - 2 n) (2 m n -m - n)}{m - n}, \ \lambda_3 \bigg).
\left\|\begin{bmatrix}C_0 C_k 0 C_0\end{bmatrix}\right\|^2 \geq \left\|\begin{bmatrix}C_0 C_k\end{bmatrix}\right\|^2 &=\|C_0C_0^*+C_kC_k^*\| &\geq \|C_0\|^2+\delta_k>\|C_0\|^2
\mathbf{K}( \underline{\texttt{p}}^{(l-1)}, \underline{\texttt{q}}^{(l-1)}) = \begin{bmatrix}\vdots \cdots \mathbf{K}( \overline{\texttt{i}}^{(l)}, \overline{\texttt{j}}^{(l)}) \cdots \vdots \end{bmatrix}\forall \, \Omega_{i}^{(l)}\in \mathcal{C}_{\mathrm{p}}^{(l-1)}, \, \forall \, \Omega_{j}^{(l)}\in \mathcal{C}_{\mathrm{q}}^{(l-1)}
\AA^X := \sum_{k = 1}^{n - 1}\sum_{i = 1}^{k}\Big(2 (1 - x^k_i)x^k_i \partial^2_{x^k_i}+ 2 \Big((p - k + 1) - (p + q - 2k + 2)x^k_i\Big)\partial_{x^k_i}\Big) \AA^Z := \AA^X + \sum_{i = 1}^n \Big[2 (1 - y_i)y_i \partial_{y_i}^2 + 2 \Big((p - n + 1) - (p + q - 2n + 2)y_i\Big)\partial_{y_i}\Big]
x^{n+1}-c_{n}x^{n-1}+d_{n}x^{n-3}+O\left( x^{n-5}\right) =x^{n+1}-c_{n+1}x^{n-1}+d_{n+1}x^{n-3}+O\left( x^{n-5}\right) +\gamma_{n}\left( x^{n-1}-c_{n-1}x^{n-3}+d_{n+1}x^{n-5}+O\left(x^{n-7}\right) \right) ,
\beta(v_{n+1},v_{i})+\beta_{i}+\beta(v_{n+1},v_{i+2})+\beta_{i+2}& =\beta(v_{n+1},v_{i})+\beta(v_{n+1},v_{i+2})+\beta(v_{i},v_{i+2})+\beta(v_{i+2},v_{i})=1
\partial_{d_k^V}\left( \varepsilon_{q,m}^{V,s}\right) = \left\langle u_{q,m}^{V,s}, \partial_{d_k^V}A^V_q, u_{q,m}^{V,s}\right\rangle = \langle u_{q,m}^{V,s}, \sin(k\cdot) u_{q,m}^{V,s}\rangle_{L^2_{\rm per}}.
\mathbf{X}:=\mathbf{H}\left( \Omega,\operatorname{curl}\right) :=\left\{\mathbf{u}\in\mathbf{L}^{2}\left( \Omega\right) \mid\operatorname{curl}\mathbf{u}\in\mathbf{L}^{2}\left( \Omega\right) \right\}
\tilde{y}_t^K \circ\big(\tilde{y}_{t_{k,m}}^K\big)^{-1}= \begin{cases}y_{t-t_{k,m}}^{(i_k;L_k)}t \in [t_{k,m},t_{k,m}+\tau_k], y_{\tau_k}^{(i_k;L_k)}t \in [t_{k,m}+\tau_k,t_{\mathrm{suc}_K(k,m)}].\end{cases}
\begin{aligned}\limsup_{D \rightarrow \infty}\sup_{x \in(0,\infty)}\frac{1}{Q(D,x)-P(x)}\sum_{j=P(x)+1}^{Q(D,x)}\tau_A(C,j) = \limsup_{n \rightarrow \infty}\sup_{k \in \mathbb{N}}\frac{1}{n}\sum_{j=k}^{k+n-1}\tau_A(C,j) = U^*(C,A).\end{aligned}
\delta \chi^{( \alpha ) i_{( \beta )}}= -{\textstyle{\frac{1}{\sqrt{2}}}}\gamma^{\mu}\partial_{\mu}\phi^{( \alpha \beta )}\epsilon^{i j}\epsilon_{j_{( \beta )}}- g \Sigma_{\alpha \beta \gamma}\Omega_{\gamma \delta}e^{\lambda_{\delta}/ 2}\epsilon_{i j}\epsilon^{j_{( \beta )}}+ \Omega_{\alpha \delta}e^{- \lambda_{\delta}/ 2}F_{\mu \nu}^{( \delta )}\gamma^{\mu \nu}\epsilon^{i_{( \beta )}},
m_1,m_4 \ge \biggl(k'+\frac{h-h_5+1}{2}\biggr) - k \ge \frac{h-h_5+1}{2}, m_2,m_3 \ge \frac{1}{2}\biggl( \frac{3}{2}(h-1)+\frac{h-h_5+1}{2}\biggr)= h-\frac{h_5}{4}-\frac{1}{2}\geq h-h_5= h_1+h_2+h_3+h_4+1,
\mathcal T_{\kappa,2}= \bigcup_{i=0}^{\lfloor\kappa/2\rfloor}\left \{[n_0,n_1,n_2,n_3]\in \mathcal S_{\kappa,2}\colon n_1= n_2= i\right \}\triangleq \bigcup_{i=0}^{\lfloor\kappa/2\rfloor}\mathcal{Q}_{\kappa,i}.
\overline{C}_{3,2}(z; \tau) = \prod_{n \geq 1}\frac{(1-q^{2n})^2 (1-\zeta^{\pm 3}q^n)}{(1-q^n)(1-\zeta^{\pm1}q^n)(1-\zeta^{\pm2}q^n)(1-\zeta^{\pm3}q^n)}&= \tilde{\psi}_{\xi}(3z; \tau) \cdot \prod_{n \geq 1}\frac{1}{1-q^{7n}+ \Phi_{7}(\zeta)g_{n}(z; \tau)}.
a\Big( a_1f(a\zeta_1a,\ldots ,a\zeta_na)^2 +a_2f(a\zeta_1a,\ldots ,a\zeta_na)a_3f(a\zeta_1a,\ldots ,a\zeta_na) +f(a\zeta_1a,\ldots ,a\zeta_na)a_5f(a\zeta_1a,\ldots ,a\zeta_na)a_6 -a_5f(a\zeta_1a,\ldots ,a\zeta_na)^2a_6 +f(a\zeta_1a,\ldots ,a\zeta_na)a_4f(a\zeta_1a,\ldots ,a\zeta_na)\Big)a =0, \forall \ \zeta_1,\ldots ,\zeta_n\in \mathcal{R}.
\int \limits_0^{t}R(u) e^{-u}\du = a_0-a_0 e^{-t}+\sum \limits_{k=1}^{\infty}[ \int \limits_0^t a_k \cos{k u}\e^{-u}du + \int \limits_0^t b_k \sin{k u}\e^{-u}du]=
\begin{aligned}2 u\left(\frac{a+b}{2}\right) w\left(\frac{a+b}{2}\right) \leq \frac{1}{b-a}\int\limits^b_a u(y)w(y)dy & +\frac{u(a)w(a)+u(b)w(b)}{6}+ \frac{u(a)w(b)+u(b)w(a)}{3}.\end{aligned}
\mathbf{h}_{ik}(a,b): =[\mathbf{f}_{ik}(a),\mathbf{g}_{ki}(b)], \mathbf{v}_{k}(a): =-[\mathbf{g}_{k1}(a),\mathbf{g}_{k1}(1)], \mathbf{w}_{k}(a): =[\mathbf{f}_{1k}(1),\mathbf{f}_{1k}(a)],
Z(\Phi(G))\lneq Z(G^p\gamma_3(G)) = C_G(G^p\gamma_3(G)). d\bigg(\frac{Z(G^p\gamma_3(G))\cap Z_3(G)}{Z(G)}\bigg) \ge 2d\big(\Omega_1(Z_2(G))\big). d\big(\Omega_1(Z(G^p\gamma_3(G))\cap Z_3(G))\big) \ge 2d\big(\Omega_1(Z_2(G))\big).
E_{m \bar{a}\bar{b}}\equiv G_{m \bar{a}\bar{b}}- \mu \epsilon^{m n p}\partial_{n}h_{p \bar{a}\bar{b}}+ \frac{\mu}{2}[ \epsilon^{p n a}\partial_{n}v_{b}+ \epsilon^{p n b}\partial_{n}v_{a}- \frac{2}{3}\eta_{a b}\epsilon^{p m n}\partial_{m}v_{n}] = 0 ,
1= \tilde x \sum_{k=1}^r \frac{\lambda_k}{\tilde\lambda_1-\lambda_k}=\frac{\tilde x}{A}\frac{\lambda_1}{\lambda_1-\lambda_2}+\tilde x\sum_{k=2}^r \frac{\lambda_k}{\lambda_1-\lambda_k+A(\lambda_1-\lambda_2)}\geq \frac{\tilde x}{1+A}\Big(\frac{\lambda_1}{\lambda_1-\lambda_2}+\sum_{k=2}^r \frac{\lambda_k}{\lambda_1-\lambda_k}\Big).
\sum_{n=0}^{\infty}P_{n}t^{n}=\sum_{l=0}^{\infty}\frac{t^{l}}{(q;q)_{l}}\sum_{k=0}^{\infty}\left(qz\xi^{-1};q\right)_{k}z^{-2k}t^{k}=\frac{1}{(t;q)_{\infty}}\,_{2}\phi_{1}\left(qz\xi^{-1},q;0;q,z^{-2}t\right)
\left. \begin{array}{lll}F^{(n)}_{1}=q^{(n)}_{3}+ q^{(n)}_{1}\pi_{\phi},\,\,\,\, F^{(n)}_{2}=q^{(n-1)}_{1},\,\,\,\, F^{(n)}_{3}=-q^{(n-1)}_{2}- q^{(n)}_{3}\pi_{\phi}, G^{(n)}_{1}=2q^{(n+1)}_{2}e^{\phi},\,\,\,\, G^{(n)}_{2}=-2q^{(n+1)}_{3}e^{\phi},\,\,\,\, G^{(n)}_{3}=2q^{(n)}_{1}e^{-2\phi}.\end{array}\right.
G=\left(\begin{array}{llccrr}R^2 u v -u -2v-R^2 -u u R^2 u v -u -2v-R^2 v u R^2 u v -u -u v u R^2 u v -2v-R^2 -u v u R^2 u -u -2v-R^2 -u v u R^2\end{array}\right)\;,
\alpha_A (\varphi) = - \left[ \ln{\widehat \Lambda_s\over m_A}+{1\over 2}\right]{\partial \ln B(\varphi)\over \partial\varphi}= + \ln{\widehat\Lambda'_s\over m_A}\,{\partial \ln B^{-1}(\varphi)\over \partial\varphi}\ ,
\sum_{\substack{a,b\in \mathbb{Z}^+ ab=n a\equiv b \pmod{2}}}a- \sum_{\substack{a,b\in\mathbb{Z}^+ ab=n 1\leq a\leq 3b a\equiv b \pmod{2}}}a+ \sum_{\substack{a,b\in\mathbb{Z}^+ ab=n a\geq 3b a \equiv b \pmod{2}}}3b=\sum_{\substack{a,b\in\mathbb{Z}^+ ab=n a>3b a\equiv b \pmod{2}}}a+ \sum_{\substack{a,b\in\mathbb{Z}^+ ab=n a>3b-2 a\equiv b \pmod{2}}}3b.
\widetilde{\mathbf{H}}_k(\overline{E}):=\frac{(-1)^{k+2}}{2(k+2)!(2\pi i)^{k+2}}\int_{(\mathbb{P}^1)^{k+2}}{\rm ch}_g^0\Big(\sum_{j=1}^m(-1)^j{\rm tr}_2\circ \lambda\big(H^{(j)}(\overline{E})\big)\Big)\wedge C_{k+2}(\log\mid z_1\mid^2,\cdots,\log\mid z_{k+2}\mid^2),
(X_A)^* = X_{A^*}\Delta(X_A) = \sum_{M_1+M_2=[1,k]}X_{A|M_1}\otimes X_{A|M_2}\Delta(X_A)^* = \sum_{M_1+M_2=[1,k]}X_{A^*|M_1}\otimes X_{A^*|M_2}= \sum_{M_1+M_2=[1,k]}(X_{A|M_1})^*\otimes (X_{A|M_2})^*.
u_{1}(t,x) = -\frac{\lambda''(s)}{2\pi}\int_{B(0,t-s)}\frac{\log(1+|y+x|^{2})}{((t-s)^{2}-|y|^{2})^{1/2}}dy &=-\frac{\lambda''(s)}{2\pi}\int_{0}^{t-s}\rho d\rho \int_{0}^{2 \pi}d\theta \frac{\log(1+|x|^{2}+2|x|\rho \cos(\theta)+\rho^{2})}{((t-s)^{2}-\rho^{2})^{1/2}}
v^n(t,x)-v^{n+p}(t,x) =\int_{t}^{T}\int_{H}e^{-(s-t){A}}\left[ \nabla^G v^n(s,z+e^{(s-t){A}}x) \, B^n(s,z+e^{(s-t){A}}x)\right. \left.- \nabla^G v^{n+p}(s,z+e^{(s-t){A}}x)\, B^{n+p}(s,z+e^{(s-t){A}}x)\right]\mu_{s-t}(dz)\,ds +\int_t^T\int_{H}e^{-(s-t){A}}\left(GB^n(s,z+e^{(s-t){A}}x)-GB^{n+p}(s,z+e^{(s-t){A}}x)\right)\mu_{s-t}(dz).
\begin{array}{rcl}\sum_{t=0}^T \beta^t \phi (\hat{x}_t, \hat{u}_t) -\sum_{t=0}^T \beta^t \phi(x_t,u_t) \geq \langle D_1 H_0(\sigma, \hat{u}_0, p_1), \sigma - \sigma \rangle \null \null - \langle p_{T+1}, \hat{x}_{T+1}- x_{T+1}\rangle \Longrightarrow\end{array}
\left( \begin{array}{c}{F}{* F}\end{array}\right) \rightarrow \left( \begin{array}{l l}{\operatorname{c o s}\alpha}{\operatorname{s i n}\alpha}{- \operatorname{s i n}\alpha}{\operatorname{c o s}\alpha}\end{array}\right) \left( \begin{array}{c}{F}{* F}\end{array}\right) \, .
n^d_t \le C_{d,T}(t)+250\eta^4 d^4\left( C_2^4 t^3+\left( C_8+C_{5}^2\right) t \right)\sum_{k=0}^{t-1}C_{d,T}(k) \exp(250\eta^4 d^4\left( C_2^4 t^3+\left( C_8+C_{5}^2\right) t \right)(t-k-1))
C(z)=\sum_{i=0}^{m-1}\sum_{j,t=1}^n[\tilde{v}_i\tilde{w}_i]_{t,j}(\tilde{Q}_{i-1})_{j,t}\lambda_tz^{m-2}\prod_{\substack{\ell=1 (\ell\neq t)}}^n(z^m-\lambda_\ell^m)
S=\int d^4 x \sqrt{-\tilde{g}}\left[ \tilde{R}-\frac{1}{2}\left( \tilde{\nabla}\Phi \right)^2 -\frac{1}{2}\left( \tilde{\nabla}y \right)^2 -\frac{1}{2}Q^2 e^{2\Phi -\sqrt{3}y}-\frac{1}{2}m^2 e^{2\Phi +\sqrt{3}y}\right] ,
\int_{t_n}^{t_{n+1}}(AX_t^{(\epsilon)})^{\rm T}{\rm d}X_t^{(\epsilon)}= A^{\rm T}: \left( X_{t_{n+1/2}}^{(\epsilon)}\otimes X_{t_n,t_{n+1}}^{(\epsilon)}- \frac{1}{2}\int_{t_n}^{t_{n+1}}[ X_{t_n,t}^{(\epsilon)},{\rm d}X_{t}^{(\epsilon)}] \right).
a \rightarrow \left( \begin{matrix}{{a_{+ , i j}}}&{{\varrho \, a_{- , i k}}}\\{{\varrho \, a_{- , \ell j}^{\dagger}}}&{{\tilde{a}_{+ , \ell k}}}\\ \end{matrix}\right) , \quad \psi \rightarrow \left( \begin{matrix}{{\psi_{+ , i j}}}&{{\varrho \, \psi_{- , i k}}}\\{{\varrho \, \psi_{- , \ell j}^{\dagger}}}&{{\tilde{\psi}_{+ , \ell k}}}\\ \end{matrix}\right) .
\frac{1}{2}\frac{l^{'^{\prime}}}{l}+ \frac{3}{2}\frac{l^{'}}{l x}- \frac{1}{4}\frac{( l^{'})^{2}}{l^{2}}= 2 \alpha^{2}\left( \frac{l}{f}U ( \Phi_{1}, \Phi_{3}, \Phi_{4}) + \frac{f}{l x^{2}}( K^{'})^{2}- \frac{K^{2}\Phi_{1}^{2}}{x^{2}}\right)
\begin{aligned}L_{1}=\frac{1}{2}(f'^{2}-g'^{2})- \frac{1}{2}(\alpha_{1}^2-\alpha_{2}^2)(f^{2}-g^{2})+2\alpha_{1}\alpha_{2}f g, L_{2}=f' g' - \alpha_{1}\alpha_{2}(f^{2}-g^{2}) -(\alpha_{1}^2-\alpha_{2}^2)f g.\end{aligned}