harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\mathbf b}= \left( \begin{array}{c c c}{0}&{b_{1 2}}&{0}\\{0}&{0}&{b_{2 3}}\\{b_{3 1}}&{0}&{0}\\ \end{array}\right) \ , \qquad{\mathbf d}= \left( \begin{array}{c c c}{0}&{d_{1 2}}&{0}\\{0}&{0}&{d_{2 3}}\\{d_{3 1}}&{0}&{0}\\ \end{array}\right)
m \, \sum_{j = 1}^{3}\, \mathrm{T r}~ ( \lambda^{j}\, \lambda^{j}) ~ + ~ \overline{{m}}\, \sum_{\bar{j}= 1}^{3}\mathrm{T r}~ ( \bar{\lambda}^{\bar{j}}\, \bar{\lambda}^{\bar{j}}) \, .
f=f_{SU(N)}(g) =\sum_{j=1}^M\sum_{i=1}^Q c_{ij}e^{ik_{ij}^1\phi_1}\sin^{m_{ij}^1}(\psi_1)\cos^{n_{ij}^1}(\psi_1)\cdots e^{ik_{ij}^{N-1}\phi_{N-1}}\sin^{m_{ij}^{N-1}}(\psi_{N-1})\cos^{n_{ij}^{N-1}}(\psi_{N-1}) &\qquad\qquad\cdot (f_{SU(N-1)})_{ij}(g_{SU(N-1)}) e^{il^{N-1}_{ij}\omega_{N-1}}
( \Psi_{0}, \phi_{\mu}( x_{1}) \phi_{\nu}( x_{2}) . . . . . . . \phi_{\rho}( x_{n}) \Psi_{0}) = i^{F}( - 1 )^{J}( \Psi_{0}, \phi_{\mu}( - x_{1}) \phi_{\nu}( - x_{2}) . . . . . . . \phi_{\rho}( - x_{n}) \Psi_{0})
\widetilde{w}_{r_1}\tilde{v}\widetilde{w}_{r_1}^{-1}=\widetilde{w}_{r_1}(\widetilde w_r \tilde{v_1}\widetilde w_r^{-1})\widetilde{w}_{r_1}^{-1}= \left(\widetilde w_{r_1}\widetilde w_r \widetilde w_{r_1}^{-1}\right) \widetilde w_{r_1}\tilde{v_1}\widetilde w_{r_1}^{-1}\left(\widetilde w_{r_1}\widetilde w_r \widetilde w_{r_1}^{-1}\right)^{-1}.
\sum_{\ell=1}^{k-1}\sum_{m=\ell}^{n-1}q(n-m,k-\ell)q(m,\ell) = \frac{2(an+(b-1)k+1)}{an+(b-2)k+2}\,q(n,k) - 2q(n,k) = \frac{2(k-1)}{an+(b-2)k+2}\,q(n,k).
\begin{aligned}\sum_{j=2}^{n-3}\mathbb{E}[ G_{n-2}^{j}(z) Z^{j}(n-2,1)] ~ \leq e^{\frac{2+z}{2}}(n-3)^{\frac{2+z}{2}}\left( \sum_{j=2}^{n-3}\left( \frac{1}{j-1}\right)^{\frac{2+z}{2}}\mathbb{E}[ G_{j-1}^{j}(z)] + e^{\frac{1}{2}}\frac{3 + z}{1+z}(n-3)^{\frac{1}{2}}\right)+ \frac{e^{\frac{1}{2}}}{(n-3)^{\frac{1}{2}}},\end{aligned}
\frac{\partial u_3}{\partial x_3}=\sum_{k\in \mathbb{Z}^2 \times \mathbb{N}}\hat{u}_3(k) \frac{k_3 \pi}{h}\cos(\frac{\pi k_3x_3}{h})e^{2\pi i (\frac{k_1x_1}{\Pi_1}+\frac{k_2x_2}{\Pi_2})}.
\int_{\mathbb{R}^N}\nabla \tilde{v}(x) \cdot \nabla \varphi(x)\;dx + \int_{\mathbb{R}^N}V_{0,\infty}\tilde{v}(x)\varphi(x)\;dx = a_0 \int_{\mathbb{R}^N}\tilde{v}(x) \varphi (x)\;dx,
\Vert (B_\varepsilon (\vartheta)-\widehat{\zeta}Q_0^\varepsilon )^{-1}\Vert_{H^{-1}(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}=\Vert( B_\varepsilon (\vartheta)-\widehat{\zeta}^*Q_0^\varepsilon )^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow H^1(\mathbb{R}^d)}.
\begin{array}{ll}v^{\epsilon,R}_{,i,j}(t,x)=\int_{{\mathbb R}^n}f^{\epsilon}_R\left(\sum_{i=1}^nc_iy_i\right)p_{,i,j}(t,x;0,y)dy = \int_{{\mathbb R}^n}f^{\epsilon}_R\left(\sum_{i=1}^nc_iy_i\right)p^*_{,i,j}(0,y;t,x)dy = \int_{{\mathbb R}^n}c_ic_j\left( f^{\epsilon}_R\right)^{''}\left(\sum_{i=1}^nc_iy_i\right)p^*(0,y;t,x)dy\end{array}
A \,=\, \begin{pmatrix}\lambda_1 1 0 0 0 0 \dots 0 0 0 0 \lambda_2 1 0 0 \dots 0 0 0 0 0 0 \lambda_3 1 \dots 0 0 \vdots \vdots \vdots \vdots \vdots \vdots \ddots \vdots \vdots 0 0 0 0 0 0 \dots \lambda_g 1 \end{pmatrix}.
\theta \stackrel{?}{=}2 i \sum_{i = 1}^{N_7}q_{5i}\ln(z - z_{7i})+ 2i \sum_{j = 1}^{N_5}\ln(z - z_{5j})- 2i \sum_{k = 1}^{N_{\bar 5}}\ln(z - z_{\bar 5k})\ .
\mathfrak{R}_{\sigma_{4,2n+1,2,0}}(0;\beta) = \mathfrak{G}_n( \beta) = \mathfrak{G}_{\sigma_{n,2n,2,0}}^{(\beta-1)}(1),\mathfrak{R}_{\sigma_{n,2n,2,0}}(0,\beta)= \mathfrak{F}_n(\beta)= \mathfrak{G}_{\sigma_{n,2n+1,2,0}}^{(\beta-1)}(1).
\delta_k = \max_{\substack{\Lambda\subseteq[d] |\Lambda| \le k}}\| \Phi_\Lambda^T \Phi_\Lambda - I \|_{2, 2}&\ge \max_{\substack{\Lambda\subseteq[d] |\Lambda| \le k}}\| \Phi_\Lambda^T \Phi_\Lambda - I \|_{\infty, 2}&= \nu_{k-1}(\Phi),
\P_\eta \left( \bigcap_{i=0}^{\lfloor M_L \rfloor - 1}\left(G^{(0,T_i)}_{T_1}\cap \Lambda^{(0,T_i)}_{T_1}\right)^c \cap \{\eta_{T_i}(0) = 0\}\right) \le \left(1-L^{\delta \log p_{*}}\right)^{\lfloor M_L \rfloor}& \le c e^{- \frac{1}{c}L^{\varepsilon_*}}
E_j\big( -(1+c_{0}\delta)\Delta_S-\alpha K \big) =(1+c_{0}\delta)E_{j}\big( -\Delta_S+\frac{1}{1+c_{0}\delta}\alpha (K_{\max}-K) \big)-\alpha K_{\max}\leq(1+c_{0}\delta)E_{j}\big( -\Delta_S+\alpha (K_{\max}-K) \big)-\alpha K_{\max}=E_{j}\big( -\Delta_S-\alpha K \big)+c_{0}\delta E_{j}\big(-\Delta_S+\alpha (K_{\max}-K)\big),
\int\limits_{0}^{\infty}f_1(x \mid a_1, d_1, p_1) \, (d_1 - d_2) \ln x \, dx = \left[ \frac{1}{p_1}\, \psi\left( \frac{d_1}{p_1}\right) + \ln a_1 \right] \, (d_1 - d_2).
H_0^{(\Lambda)}=\sum_{x,y\in\Lambda}t_{x,y}c_x^\dagger c_y+\sum_{I\ge 1}\sum_{x_1,x_2,\ldots,x_I\in\Lambda}U_{x_1,x_2,\ldots,x_I}n_{x_1}n_{x_2}\cdots n_{x_I},
\begin{aligned}\rho(L_2)=h(z)^{2}(\rho(L_0)+2\lambda)(\rho(L_0)-\lambda-2)(\rho(L_0)-\lambda-1)\, = = \,h(z)^{2}\left( L^3 - 3L^2 +(2-3\lambda-3\lambda^2)L + 4\lambda +6 \lambda^2 +2\lambda^3\right)\end{aligned}
\varphi_0(x,t)=-2\int_\Omega \frac{\partial p}{\partial \nu_x}(x,y,t)\psi (y)dV(y),\;\; (x,t)\in \Sigma \times (0,+\infty ), &r(x,y,t)=\sum_{j\geq 1}r^j(x,y,t),\;\; (x,t)\in \Sigma \times (0,+\infty ),\; y\in \overline{\Omega},
c_1 =\frac{i}{2h\eta}\bigl(e^{-i\eta y/h}+\frac{i\eta-a}{i\eta + a}e^{+i\eta y/h}\bigr),\tag*{$\textup{(\ref*{11-9-30})}_1$}&c_2 = \frac{i}{2h\eta}\frac{i\eta-a}{i\eta + a}e^{+i\eta y/h}, c_3 = \frac{i}{2h\eta}e^{+i\eta y/h},\tag*{$\textup{(\ref*{11-9-30})}_{2,3}$}
g_0^6 = \frac{(\partial_2Z)^2}{\partial_2X\partial_2Y}~,~~ h_1^2 = \frac{\partial_2Y}{\partial_2Z}~,~~ h_2^2 = \partial_2X\partial_2Y ~,~~ h_3^2 = \frac{\partial_2X}{\partial_2Z}~,~~ A_i = \frac{\partial_iX}{\partial_2X}~,~~ A_p = \frac{\partial_pY}{\partial_2Y}~.
0 = \overline{\omega}_x\left([Z,[\widehat{Y}_1,\widehat{Y}_2]]_x\right) = \overline{\omega}_x\left([[Z,\widehat{Y}_1],\widehat{Y}_2]_x\right) + \overline{\omega}_x\left([\widehat{Y}_1,[Z,\widehat{Y}_2]]_x\right)
\partial_{tt}(\frac{\partial L}{\partial(\partial_{tt}\psi)})+\partial_{xx}(\frac{\partial L}{\partial(\partial_{xx}\psi)})-\partial_{t}(\frac{\partial L}{\partial(\partial_{t}\psi)})-\partial_{x}(\frac{\partial L}{\partial(\partial_{x}\psi)})+\frac{\partial L}{\partial\psi}=0.
\sum_{d\geq 0}a_d q^d =\sum_{0\leq v_1/d_1<\cdots<v_k/d_k<1}\prod_{i=1}^k p_2\left(\gcd(d_i, v_i)\right)q^{d_i}&=\prod_{\begin{subarray}{c}0\leq \mu<1 \mu=a/b, \ \gcd(a, b)=1\end{subarray}}\prod_{k\geq 1}\frac{1}{1-q^{bk}}.
\frac{\left| z\left(\left(\frac{1}{2}-\sigma\right) + j\omega\right)\right|}{\left|z\left(\left(\frac{1}{2}+\sigma\right) - j\omega \right)\right|}= \frac{\left|P^{\frac{1}{2}}\right|}{\left|G\left(\overline{\chi},P\right)\right|}\frac{\left|\left(\frac{1}{2}-\sigma\right) + j\omega\right|}{\left|\left(\frac{1}{2}+\sigma\right) - j\omega\right|}
{\cal D}_{\cal P}( U ) \: = \:_{\cal P}M o d \Big \{\sum_{\iota \in I}\, d_{\iota}f_{\iota}\: : \: \sum_{\iota}\, f_{\iota}\in{\cal P}( U ) \Big \}\quad \subset \:{\cal D}_{\cal P}^{0}( U ) ,
\begin{aligned}\gamma^{(k)}(c)=c+(1-c)\sum_{i=1}^k \frac{p_c^{(k)}(i)}{k+2-i}= &c+\frac{1-c}{\lim_{n\to\infty}P_{n,k}(W_{\mathcal{S}(n,k;M_n)})}E\frac1{k+1-X_{k,c}}1_{X_{k,c}\not\in \{0,k\}}+ &\frac{1-c}{k+1}\big(1-\frac1{\lim_{n\to\infty}P_{n,k}(W_{\mathcal{S}(n,k;M_n)})}(1-c^k-(1-c)^k)\big),\end{aligned}
\Phi^*ds^2 = \bigg(\Pi_{l=1,2}\dfrac{||f^l||(e^{p_l\sum_{j=1}^{q_l}u^l_j}|W(f^l_0,f^l_1)|)^{p_l}}{\Pi_{j=1}^{q_l}|F^l_j|^{(1-\delta)p_l}}\bigg)^2|dw|^2 &= \Pi_{l=1,2}\bigg( \dfrac{||f^l||^{\gamma_l-q_l\delta}e^{\sum_{j=1}^{q_l}u^l_j}|W(f^l_0,f^l_1)|}{\Pi_{j=1}^q|F^l_j|^{1-\delta}}\bigg)^{2p_l}|dw|^2.
\int_0^j\frac{q'(s)-Q'(s)}{\Gamma(1-\alpha)(j-s)^{\alpha}}d s=\sum_{k=1}^j\int_{k-1}^{k}\frac{q'(s)-Q'(s)}{\Gamma(1-\alpha)(j-s)^{\alpha}}d s\:=\sum_{k=1}^jR_k^j,
T_{8,r}=\frac{8 \eta\left(q^8\right)^8}{\eta\left(q^4\right)^4}-\frac{\eta\left(q^4\right)^{20}}{\eta\left(q^2\right)^8 \eta\left(q^8\right)^8}-\frac{6\eta\left(q^4\right)^6 \eta\left(q^8\right)^6}{\eta\left(q^2\right)^4 \eta\left(q^{16}\right)^4}.
\frac{3}{3+2\sqrt{5}}=\frac{3}{3+2\sqrt{5}}\cdot\frac{3-2\sqrt{5}}{3-2\sqrt{5}}=\frac{3(3-2\sqrt{5})}{3^{2}-(2\sqrt{5})^{2}}=\frac{3(3-2\sqrt{5})}{9-20}=-\frac{9-6\sqrt{5}}{11}
\cdots \longrightarrow \prod_{i=1}^{k}[\mathbb{S}_{i}^{5}, Cat] \stackrel{(S(g))^{*}}{\longrightarrow}[\mathbb{S}^{8}, Cat] \stackrel{f_{X}^{*}}{\longrightarrow}[X, Cat]\stackrel{\iota_{Cat}^{*}}{\longrightarrow}\prod_{i=1}^{k}[\mathbb{S}_{i}^4, Cat].
G_\gamma^t \left(\begin{array}{c}0 1\end{array}\begin{array}{cc}1 0\end{array}\right)G_\gamma - \left(\begin{array}{c}0 1\end{array}\begin{array}{cc}1 0\end{array}\right)=2\left(\begin{array}{c}a_{11}a_{21}a_{12}a_{21}\end{array}\begin{array}{cc}a_{12}a_{21}a_{12}a_{22}\end{array}\right).
{\cal A}[ \Sigma ] = \int_{\Sigma}\omega = \int_{\Sigma}{\cal L}\left( z , \bar{z}, \varphi ( z , \bar{z}) , v = \partial_{z}\varphi , \bar{v}= \partial_{\bar{z}}\varphi \right) \,{\mathrm{d}}z \wedge \mathrm{d}\bar{z}\, .
\sum_{n=0}^\infty |a_n|\left|\sum_{k=0}^n c_{k,n}M_k(x;\beta,d)\right|\le K_1K_2 \sum_{n=0}^\infty \frac{(1+n)^{\sigma_1+\sigma_2+1}}{n!}\left|\frac{t}{c}\right|^n\left|\frac{1+d-2c}{1-d}\right|^n<\infty,
| \left\langle\mathcal{L}u,u \right\rangle - \left\langle\mathcal{L}u_h,u_h \right\rangle | = \inf_{w_h}\left\{| \left\langle\mathcal{L}u,u \right\rangle - \left\langle\mathcal{L}u_h,w_h \right\rangle|, \; w_h(\cdot,0) = u_0 \right\}.
\partial_{z}K(w,z) = \int_{0}^{\infty}dR \int_{0}^{w}d\rho \left(\frac{4 p R^3 z \left(p^2+R^2 z^2+1\right)}{\left(R^2+1\right)^3 \left(\left(p^2-R^2 z^2+1\right)^2+4 R^2 z^2\right)^{3/2}}\right)\left(\frac{1}{\sqrt{w^{2}-\rho^{2}}}-\frac{1}{w}\right) \geq 0
\Delta_2'(h(\gamma)) = \sum_{a\geqslant 1}\sum_{j\leqslant r+1}(\gamma^a)_{(j)}\otimes (\gamma^a)^{(j)}&= \sum_{a\geqslant 1}\sum_{j\leqslant i_a+1}(\gamma^a)_{(j)}\otimes (\gamma^a)^{(j)}+\sum_{a\geqslant 1}\sum_{i_a < j-1 \leqslant r}(\gamma^a)_{(j)}\otimes (\gamma^a)^{(j)}
F_{z_{n}}(z_{n}) = 1 - \frac{1+K}{z_{n}\frac{\overline{\gamma}_{sp}}{\overline{\gamma}_{s}}+K+1}\, e^{-K+\frac{K(1+K)}{\frac{\overline{\gamma}_{sp}}{\overline{\gamma}_{s}}z_{n}+K+1}}.
\langle D_1 J_1(f;v^1+sw^1,v^2)- D_1 J_1(f;v^1,v^2),w^2 \rangle = \alpha_1 \iint_{\mathcal{O}_{1,d}\times (0,T)}(y^s-y_{1,d}) z^s dxdt &- \alpha_1 \iint_{\mathcal{O}_{1,d}\times (0,T)}(y-y_{1,d})z\dxdt + s \mu_1 \iint_{\mathcal{O}_1 \times (0,T)}w^1 w^2 dxdt.
\left.\int_{B_1}\eta{\tilde d}^2{\hat\rho}\, d\|V_t\|\right|_{t=t_1}^{t_2}\leq \int_{t_1}^{t_2}\int_{B_1}\left(-h{\hat \rho}\eta{\tilde d}^2+\nabla ({\hat\rho}\eta{\tilde d}^2)\right)\cdot(h+u^{\perp})+\eta{\tilde d}^2\frac{\partial{\hat\rho}}{\partial t}\, d\|V_t\|dt.
d{\cal O}/dt =-{i\over \hbar_{1}}[{\cal O},{\cal H}_{1}]-{i\over \hbar_{2}}[{\cal O},{\cal H}_{2}]-{i\over \hbar_{\rm av}}[{\cal O},{\cal H}_{I}]-{{i\Delta\hbar^{-1}}\over 2}(J_{2}{\cal O}J_{1}- J_{1}{\cal O}J_{2})
\sum_{\substack{i+j=n j=0}}(-1)^{j}\tilde{L}({\rm reg}_{0}(e_{0}^{i}u)*e_{1}e_{0}^{j}) =\tilde{L}({\rm reg}_{0}(e_{0}^{n}u)*e_{1})=\tilde{L}({\rm D}({\rm reg}_{0}(e_{0}^{n}u))).
X_{13, 2\alpha}^+=-\frac{1}{2}Z_{13}, X_{13, A}=\frac{1}{2}H,X_{13, \alpha}=0 &X_{31, 2\alpha}^+=\frac{1}{2}Z_{13},X_{31, A}=\frac{1}{2}H,X_{13,\alpha}=0 &X_{23, \alpha}^+= -X_{21},X_{23, A}=X_{23, 2\alpha}=0 &X_{32, \alpha}^+=X_{12},X_{32,A}=X_{32, 2\alpha}=0.
\theta(w) =\sum_{I\vdash n}(\zeta\otimes\dots\otimes\zeta)\left(\Delta_{I}(w)\right)M_{I}=\sum_{I\vdash n}\zeta((w_{1},\dots,w_{i_{1}}))\zeta((w_{i_{1}+1},\dots,w_{i_{1}+i_{2}}))\dots\zeta((w_{i_{l(I)-1}+1},\dots,w_{n}))M_{I}.
T_mM(\sigma_b)_l=\ker\left(\mathrm{d}\phi^M_{\sigma_b}-\mathrm{id}_{T_mM}\right),\,\,\,\,\,N_mM(\sigma_b)_l=\mathrm{im}\left(\mathrm{d}\phi^M_{\sigma_b}-\mathrm{id}_{T_mM}\right),
\sum_{n = 1}^2\frac{p_n^2}{2}+ \int a_k^{\dagger}a_k\,dk + \sum_{n = 1}^2\frac{\sqrt{\alpha}}{2^{3/4}\pi}\int\frac{1}{|k|}(e^{ikx_n}a_k + e^{-ikx_n}a_k^{\dagger})\,dk,
\left (\frac{1}{\sqrt{2\pi}\theta}\right ) \int\limits_{-\infty}^{\infty}exp \left ( - \frac{(u_1-b)^2}{2\theta^2}\right)\mathrm{d}u_1 \left (\frac{1}{\sqrt{2\pi}\theta}\right ) \int\limits_{-\infty}^{\infty}exp \left ( - \frac{(u_2-b')^2}{2\theta^2}\right) \mathrm{d}u_2 = 1
y_{u_1u_2}^{}=\frac{\sum_{i=1}^{K}\frac{1}{d(b_{i1}b_{i2},u_1 u_2)}y_{b_{i1}b_{i2}}}{\sum_{i=1}^{K}\frac{1}{d(b_{i1}b_{i2},u_1 u_2)}},
\frac{a_{k+1}(M)}{a_k(M)}=\frac{C^{2/q}}{M^{2/q}}\left(\frac{2k+2}{2k}\right)^k\frac{2k+2}{q(k+1)}\to\frac{C^{2/q}}{M^{2/q}}\cdot e\cdot\frac{2}{q}
\Big\|\Big(\sum_{i=1}^k a_ix_i^2-\sum_{i=k+1}^da_ix_i^2\Big)_+\Big\| \ge \Big\|\Big(a_1x_1^2-\sum_{i=k+1}^da_ix_i^2\Big)_+\Big\|\ge \left\|\left(a_1x_1^2-|X''|^2\right)_+\right\| &=\frac1k\sum_{j=1}^k\left\|\left(a_1x_j^2-|X''|^2\right)_+\right\|\ge \frac1d\left\|\left(a_1|X'|^2-|X''|^2\right)_+\right\|.
\prod_{i\in S^+}( 1 - Z_i ) \prod_{i\in S^-}( 1 - \hbar Z_i ) - z_\sharp^\beta \prod_{i\in S^+}( 1 - \hbar Z_i ) \prod_{i\in S^-}( 1 - Z_i ) , S = S^+ \sqcup S^- : ,
r\left(\frac{1+(r^{2}-(s-t)^{2})\lambda(s)^{2\alpha -2}}{\sqrt{1+2((s-t)^{2}+r^{2})\lambda(s)^{2\alpha -2}+((s-t)^{2}-r^{2})^{2}\lambda(s)^{4\alpha -4}}}\right)=r\left(-1+E_{\partial_{r}v_{3},2}\right)
Q=\left(\begin{array}{ccccccr}0 -1 0 0 0 0 1 -1 0 0 0 0 0 0 0 1 -1 -1 0 0 1 1 -1 -1 0 0 0 1 -1 0 0 0 0 1 0 -1\end{array}\right) .
\widetilde{R}_{1,k}(\alpha, \tau, \lambda) = \frac{\alpha\tau}{T}\textrm{log}_2\bigg(1 + \frac{1}{E\big[\frac{1}{\gamma^{\textrm{I}}}\big]}\bigg) + \Big(1-\frac{\tau}{T}\Big) \textrm{log}_2\bigg(1+\frac{1}{E\big[\frac{1}{\gamma^{\textrm{II}}}\big]}\bigg),
G_c(h_1,h_2) = \int_{S^1}\langle h_1^{\operatorname{tan}},h_2^{\operatorname{tan}}\rangle ds+ \int_{S^1}\langle h_1^{\operatorname{arc}},h_2^{\operatorname{arc}}\rangle ds &= \int_{S^1}\langle h_1-b_1.v,h_2-b_2.v \rangle ds + \int_{S^1}b_1 b_2+a_1a_2 ds &= \int_{S^1}2 a_1 a_2 +\tilde b_1 \tilde b_2 - b_1\tilde b_2 - \tilde b_1 b_2+ b_1 b_2\; ds
\| u_n^{*}(\varphi_n(f\otimes a) \oplus \overbrace{\bigoplus_{k=1}^m f(z_k)\sigma_n (a) \oplus \cdots \oplus \bigoplus_{k=1}^m f(z_k)\sigma_n (a)}^m) u_n & - \psi_n(f\otimes a)\oplus \overbrace{\bigoplus_{k=1}^m f(z_k)\sigma_n (a) \oplus \cdots \oplus \bigoplus_{k=1}^m f(z_k)\sigma_n (a)}^m\| < \varepsilon
\sum_{i=1}^k\chi_i+\rho+d\mu \tau_d =\sum_{i=1}^k w_i\tau_{d_i}+\sum_{i=1}^k(\psi_i-\rho_i)+\rho+d\mu \tau_d &=-\frac{3}{2}\mathfrak{g}^{\lambda>0}+\sum_{i=1}^k v_i\tau_{d_i}+\sum_{i=1}^k(\psi_i+d_i \mu \tau_{d_i}).
||\mathbf{F}||_{s,\ast,T}\leq C(K)\Big(||\mathbf{f}||_{s+2,\ast,T}+||\mathbf{g}||_{H^{s+2}(\Gamma_T)}+(||\mathbf{f}||_{8,\ast,T}+||\mathbf{g}||_{H^8(\Gamma_T)})||\hat{\mathbf U},\hat{\Psi}||_{s+2,\ast,T}\Big).
i_\gamma(f\circ R_\gamma)s_\gamma = i_{X(E_\gamma)}^0(f\circ R_\gamma)i_{X(E_\gamma)}^1(1) = i_{X(E_\gamma)}^1((f\circ R_\gamma)\cdot1) = i_{X(E_\gamma)}^1(1\cdot f) =i_{X(E_\gamma)}^1(1)i_{X(E_\gamma)}^0(f)=s_\gamma i_\gamma(f),
h_{X}(t) = \cos^2\left(\frac{\pi t}{2}\right) h_{X}\left(\frac{t}{4}\right)+| \overline{p}|^2 \sin^2\left(\frac{\pi t}{2}\right) h_{X}\left(\frac{t-1}{4}\right)+| \overline{q}|^2 \sin^2\left(\frac{\pi t}{2}\right) h_{X}\left(\frac{t-3}{4}\right) .
\begin{aligned}md + (n-d-1)(n+1-d) + (d+1-m) \left( \frac{n}{2}+1 \right) = d^2 + \left(1 + m - \frac{3}{2}n \right)d + \left(n^2 + \frac{1}{2}n - m - \frac{1}{2}mn \right).\end{aligned}
\left[{\cal H}-\zeta^{\,0}\right]_{\rm quad}=\frac{1}{2}\left\{\dot \chi^2+[(\partial_{\,z}+F^\prime)\chi]^2+i\psi_2(\partial_{\,z}- F^\prime)\psi_1 +i \psi_1(\partial_{\,z}+F^\prime)\psi_2\right\}\,,
\int_{\Omega}(\boldsymbol{\sigma}-\boldsymbol{\sigma}_{\mathcal{T}}):\mathcal{K}(\psi)dx = \sum_{K\in\mathcal{T}}\int_{K}(f-Q_K^{k-3}f)(\psi-I_{\mathcal{T}}\psi)dx + \sum_{K\in\mathcal{T}}\int_{\partial K}M_n(\widehat{\boldsymbol{\sigma}}_{\mathcal{T}}-\boldsymbol{\sigma}_{\mathcal{T}})\partial_{\boldsymbol{n}}(I_{\mathcal{T}}\psi-\psi)ds.
\sum\limits_{i=1}^{n}q_{i}\|x_{i}(k+1)-x^{\star}\|^{2}\leq \sum\limits_{i=1}^{n}q_{i}\|x_{j}(k)-x^{\star}\|^{2}+\alpha(k)^{2}G^{2}&-2\alpha(k)\sum\limits_{j=1}^{n}q_{j}g_{j}^{T}(k)(v_{j}(k)-x^{\star}).
\Phi = \left( \begin{matrix}{{0}}&{{h}}&{{0}}&{{\ldots}}&{{0}}\\{{{\frac{a_{1}}{h}}}}&{{\ddots}}&{{\ddots}}&{{\ddots}}&{{\vdots}}\\{{{\frac{a_{2}}{h^{2}}}}}&{{\ddots}}&{{\ddots}}&{{h}}&{{0}}\\{{\vdots}}&{{\ddots}}&{{{\frac{a_{1}}{h}}}}&{{0}}&{{h}}\\{{{\frac{a_{n - 1}}{h^{( n - 1 )}}}}}&{{\ldots}}&{{{\frac{a_{2}}{h^{2}}}}}&{{{\frac{a_{1}}{h}}}}&{{0}}\\ \end{matrix}\right) ~ d z \quad ,
I=\int d^{n}x~{\cal L}=-\int d^{n}x \sqrt{-g}\left(\frac{1}{2}g^{\mu \nu}\partial_{\mu}\Phi\partial_{\nu}\Phi+\frac{\mu^{2}}{2}\Phi^{2}+\frac{\lambda}{4!}\Phi^{4}\right) \;\;,
[L:K] \geq \sum_{i\leq s}e(u^{0}_{i}/v^{0})\,f(u^{0}_{i}/v^{0}) \geq \sum_{i\leq s}e(u^{0}_{i}/v^{0})\,\sum_{j\leq r_{i}}e(\bar{u}_{i,j}/\bar{v})\,f(\bar{u}_{i,j}/\bar{v}) = \sum_{i\leq s}\sum_{j\leq r_{i}}e(u_{i,j}/v)\,f(u_{i,j}/v), \tag*{}
\langle\mathcal{H}_{q}^{n+1}e_{r},e_{0}\rangle=\langle\mathcal{H}_{q}^{n}(\mathcal{H}_{q}\,e_{r}),e_{0}\rangle=\begin{cases}\sum_{k=0}^{p}a_{q}^{(k)}\langle\mathcal{H}_{q}^{n}e_{k},e_{0}\rangle, \mbox{if}\,\,r=0, begin{align*}0.5em]\langle\mathcal{H}_{q}^{n}e_{r-1},e_{0}\rangle+\sum_{k=0}^{p}a_{r+q}^{(k)}\langle \mathcal{H}_{q}^{n}e_{r+k},e_{0}\rangle, \mbox{if}\,\,r\geq 1.\end{cases}
|S| \geq |S\cap f^{-1}(Z)|+|S\setminus f^{-1}(Z)| &\geq|(\{0,1\}+Z)\cap f^{-1}(Z)|+|(X+Y)\setminus f^{-1}(Z)| &\geq 2|Z| + |f(X+Y)\setminus Z| &\geq 2|Z|+\big(\min\{1,|X|+|Y|\}- |Z|\big) &=\min\{1,|X|+|Y|\}+|Z|.
\beta_{i,i+2}({\rm in}(I_{G_{r,d}})) =\sum_{p=1}^{\frac{d(d-1)}{2}}\binom{n_{p}}{i}=\binom{0}{i}+ 2\binom{1}{i}+ 3\binom{2}{i}+ \ldots + (d-1)\binom{d-2}{i}&=\sum_{q=1}^{d-1}q\binom{q-1}{i}= (i+1)\sum_{q=1}^{d-1}\binom{q}{i+1}= (i+1)\binom{d}{i+2}.
\tau_M=\frac{1-\sqrt{\min_{i,j,k,l}\frac{M^\ell(i,j)M^\ell(k,l)}{M^\ell(i,l)M^\ell(k,j)}}}{1+\sqrt{\min_{i,j,k,l}\frac{M^\ell(i,j)M^\ell(k,l)}{M^\ell(i,l)M^\ell(k,j)}}}.
\tilde{J}_{R}\left( \bar{z}\right) = - i k g^{- 1}\left( \bar{\partial}g \right) = i k g_{R}^{- 1}\left( \bar{\partial}g_{R}\right) = \tilde{J}^{a}\tau^{a}= \left( \begin{array}{c c}{{- \tilde{J}^{2}}}&{{\tilde{J}^{0}+ \tilde{J}^{1}}}\\{{- \tilde{J}^{0}+ \tilde{J}^{1}}}&{{\tilde{J}^{2}}}\\ \end{array}\right)
v_{\lambda}:= \underbrace{v_1 \otimes \cdots \otimes v_1}_\textrm{$\lambda_1$ times}\otimes \underbrace{v_2 \otimes \cdots \otimes v_2}_\textrm{$\lambda_2$ times}\otimes \cdots \otimes \underbrace{v_n \otimes \cdots \otimes v_n}_\textrm{$\lambda_n$ times}
R_{n,H,f}(V)= R_{n-1,H,f}(V)-\frac{1}{(n-1)!}\frac{d^{n-1}}{dt^{n-1}}f(H+tV)|_{t=0}= T^{H,H+V,H,\ldots,H}_{f^{[n-1]}}(V,\ldots,V)-T^{H,\ldots,H}_{f^{[n-1]}}(V,\ldots,V).
G_\mu(x,r) \geq \left(\sum_{j=n}^\infty Z_{x|j}(r) (\mu([x|j])-\mu([x|j+1]))\right)+\mu(\{x\}) &\geq \left(\sum_{j=n}^\infty Z_{x|n}(r) (\mu([x|j])-\mu([x|j+1]))\right)+Z_{x|n}(r)\mu(\{x\}) &= Z_{x|n}(r)\mu([x|n]).
b^{\frac{5}{2}}\sqrt{ab^{-1}}(\sqrt{ba^{-1}}-1) \prod_{i=1}^{3}e_i^2 f_i \leqslant b^{\frac{5}{2}}\frac{\tau_1-1}{\tau_2}(ba(b-a))^2 &=b^{\frac{17}{2}}\frac{\tau_1-1}{\tau_2}\left(\frac{a}{b}\left(1-\frac{a}{b}\right)\right)^2 &\leqslant b^{\frac{17}{2}}\frac{\tau_1-1}{16\tau_2}\leqslant B^{\frac{1}{2}-\frac{1}{r}},
\frac{d}{dt}\frac{\int_0^1 u^2(x,t) dx}{2}= \int_0^1 \Big( \int_{[0,1]}W(x,y) (u(y,t) - u(x,t)) dy \Big) u(x,t) dx\leq 2 C{\int_0^1 u^2(x,t) dx}
|\mathcal{F}(g_a)(\xi)|^2 ={}4 a^{d+1}|\xi|^{-d+1}R^{d-1}\bigg( \int_{\varphi(\omega)}^{\varphi(\beta)}f\circ \theta^H(r) \cos(2\pi(a^{-1}R+r)|\xi| + \nu_d) dr \bigg)^2 &+ O(a^{d+2}|\xi|^{-d-1}).
\int_B \tilde{p}(X_{1:n}|\theta) d\tilde{Q}_k(\theta)= \sum_{\sigma\in S_k}\int_B p(X_{1:n}|\theta) d Q_k(\theta[\sigma])= \int_{\bigcup_{\sigma\in S_k}B[\sigma]}p(X_{1:n}|\theta) d Q_k(\theta)
r_{-j}=s_{-j}+s_{2^{n-2}+j}=s_j-s_{-2^{n-1}+(2^{n-2}+j)}+s_{-j}= &=s_j-s_{-2^{n-2}+j}=s_j-s_{2^{n-2}-j}=r_j-2s_{2^{n-2}-j}.
\begin{cases}\frac{\partial \phi_{\varepsilon}(t)}{\partial t}=\log\frac{\omega_{\phi_{\varepsilon}(t)}^{n}}{\omega_{\varepsilon}^{n}}+F_{\varepsilon}+\beta\phi_{\varepsilon}(t) \phi_{\varepsilon}(t)|_{t=0}=c_{\varepsilon0}+k\chi(\varepsilon^{2}+|s|^{2}_{h}) \end{cases}
\displaystyle \int_{B(x_{0}, \epsilon)}|u_{n}-u_{m}|^{q}dx =\displaystyle \int_{B(x_{0},\epsilon)}|a(x)|^{q}|a(x)|^{-q}|u_{n}-u_{m}|^{q}dx &\leq c \||a(x)|^{-q}\chi_{B(x_{0},\epsilon)}\|_{(\frac{p(x)}{q})'}\||a(x)|^{q}|u_{n}-u_{m}|^{q}\|_{\frac{p(x)}{q}},
\Delta \pi(r \#1) = \sum_{l\in H}\pi(r\#\delta_l)\otimes \pi(l^{-1}.r\#1) = \sum_{l\in H}\pi(1\#\delta_{l})\pi(r\#1) \otimes \pi(l^{-1}. r\#1) &= \sum_{l\in H}\pi(1\#\delta_{l}\Phi(r)) \otimes \pi(l^{-1}. r\#1)=\sum_{l\in H}\Phi(r)(l) \pi(1\#\delta_{l}) \otimes \pi(l^{-1}. r\#1)
\widetilde{H}^{(1/2)}=D^{\dag(1/2)}({\bf n}){H}^{(1/2)}D^{(1/2)}({\bf n}),\quad\widetilde{\bf P}^{(1/2)}=D^{\dag(1/2)}({\bf n}){\bf P}^{(1/2)}D^{(1/2)}({\bf n}).
\delta^{(4)}(x-x^{\prime}) = \frac{1}{\rho}\delta \left( \overline{t}- \overline{t^{\prime}}- S \frac{(\phi - \phi^{\prime})}{\alpha}\right)\delta (\rho - \rho^{\prime}) \delta (\phi - \phi^{\prime}) \delta(z-z^{\prime}) \hspace{0.5cm}.
I(x)\omega(B^g(x, \sqrt{t}))\leq \sum_{j\in I(x)}\omega (B^g(x^t_j, (1+\theta)\sqrt{t})) \leq C(1+\theta)^k \sum_{j\in I(x)}\omega(x^t_j, \sqrt{t}) \leq C(1+\theta)^k \omega(x,(1+\theta) \sqrt{t}) \leq C(1+\theta)^{2k}\omega(x, \sqrt{t}).
{\cal I}_{L,M,0;L',M',0}= -2C^2 (k+2) \sum_{l=0}^k \sum_{l',m'}^{R}{{\sin(L,l)_k \sin(L',l)_k\sin(l.l')_k}\over{\sin(l,0)_k}}\delta_{\mu}^{(k+2)}(-)^{{\mu(l+1)}\over{(k+2)}}I_{l'}^{m'}(q)\quad,
\delta^{1}u^{n}_{j}=u^{n}_{j+1}-u^{n}_{j-1}, &\delta^{2}u^{n}_{j}=u^{n}_{j+2}-2u^{n}_{j}+u^{n}_{j-2}, &\delta^{3}u^{n}_{j}=u^{n}_{j+3}-3u^{n}_{j+1}+3u^{n}_{j-1}-u^{n}_{j-3}.
(\mathbb{I}_3\otimes U)R_{2,1}\left(t^3/x\right)^{\tau_1}(\mathbb{I}_3\otimes U^{-1})R_{1,2}(x)^{\tau_1}=-\frac{(x-1) \left(t^3-x\right)}{(t-x)\left(t^2-x\right)}\mathbb{I}_3\otimes\mathbb{I}_3.
K = \sum_{a n c}( b_{1}^{+})^{n_{1}}\cdots ( b_{r}^{+})^{n_{r}}\; C_{n_{1}, \cdots , n_{r}}^{m_{1}, \cdots , m_{r}}( b_{1}^{+}b_{1}, b_{2}^{+}b_{2}, \cdots , b_{r}^{+}b_{r}) \; ( b_{1})^{m_{1}}\cdots ( b_{r})^{m_{r}},
A=2\cdot\frac{1}{2}\int_{0}^{\pi}(r(\varphi))^{2}d\varphi=\int_{0}^{\pi}4a^{2}(1-cos\varphi)^{2}d\varphi=\cdot\cdot\cdot=4a^{2}\cdot\frac{3}{2}\pi=6\pi a^{2}.
\gamma_{\mu}= \left( \begin{array}{c c}{{\sigma_{\mu}}}&{{0}}\\{{0}}&{{- \sigma_{\mu}}}\\ \end{array}\right) \, , \quad \sigma_{\mu \nu}={\frac{1}{2 i}}[ \gamma_{\mu}, \gamma_{\nu}] = \epsilon_{\mu \nu \rho}\sigma_{\rho}\, ; \quad \mu , \nu , \rho = 1 , 2 , 3 \, .
|\mathbb{S}_{\mathcal{G}'}^\rho(b_1,h_2,h_3)| \leq \sum_{R\in\mathcal{G}'}\mathop{\sum_{Q\in\mathcal{E}}}_{R(Q)=R}|S_R(b_{1,Q},h_2,h_3)| & \le \sum_{R\in\mathcal{G}'}\mathop{\sum_{Q\in\mathcal{E}}}_{R(Q)=R}\frac{\|b_{1,Q}\|_{L^1}\|h_2 1_R\|_{L^1}\|h_3 1_R\|_{L^1}}{|R|^2},
K_t \ := \Y_t - Y_0 - g(\hat X) - \int_0^t f_s(\hat X,\hat I_s)\,ds + \int_0^t\int_\Lambda R_s(b)\,\hat\theta(ds\,db) &\ + \int_0^t Z_s\,d\hat W_s + \int_0^t\int_U L_s(z)\,(\hat\pi(ds\,dz)-\lambda_\pi(dz) ds), \qquad0\leq t\leq T.
E^*_{i+k-2}A_1E^*_{i+k-3}A_1E^*_{i+k-4}\cdots E^*_{i+1}A_1E^*_{i}=\big((\frac{k-3}{2})!\big)^2\frac{k-1}{2}M^{\frac{k-3}{2},\frac{k-3}{2}}_{\frac{i+k-3}{2},\frac{2m-i}{2}}.
c(\theta_0, \theta_1)=\inf \Big\{\int_0^1\mathbb{E}_{z\sim\mu}L(g(\theta_t,z), \frac{d}{dt}g(\theta_t, z))dt\colon & \frac{d}{dt}g(\theta_t, z)=D_p H(g(\theta_t,z), \nabla_x\Phi(t, g(\theta_t,z))),~\theta(0)=\theta_0,~\theta(1)=\theta_1\Big\},
\sum_{k=0}^{\frac{d-3}{2}}\frac{\tilde{c_k}}{|x-y|^{d-1-s_k}}\partial_s^{s_k}\{e^{iz s}+ e^{-iz s}\}\Big|_{(s=|x-y|)}=z^{d-1}\int_{\mathbb S^{d-1}}e^{i z (x-y) \cdot \omega}d \omega
\lim_{z\to 1}\mathbf{X}=\left[ \begin{array}{c}\dfrac{1-p^{2}q^{3}}{1+p^{2}q-p^{3}q^{3}}begin{align*}1.2em]\dfrac{p^{2}q(1+q^{3})}{1+p^{2}q-p^{3}q^{3}}\end{array}\right].
\left( \frac{h+2}{2(1-t)}\right)^{g-1}\sum_{k=1}^{l+1}(\sin \phi_k)^{2-2g}\left[ \frac{1}{(1-t)^2 + 4t\sin^2 \phi_k}\cdot \left(1+ \frac{4t}{h+2}\frac{(1-t) - 2 \sin^2 \phi_k}{(1-t)^2 + 4t \sin^2 \phi_k}\right) \right]^{g-1},
\hskip-36pt \begin{array}{lcp{10.7cm}}\hline\noalign{\smallskip}P_{g-}= $cp_2\partial^2+cp_1/(x(x-1))\partial + cp_0/(x(x-1))$, Q_{g-}= $cq_2\partial^2+cq_1/(x(x-1))\partial + cq_0/(x(x-1))^2$, \noalign{\smallskip}\hline\end{array}
L^m_{p,q}f(z) =L^m_{p,q}h(z)+(-1)^m\overline{L^m_{p,q}g(z)}=z+\sum_{{k=2}}^{\infty}[k]^m_{p,q}a_{k}z^{k}+(-1)^m\sum_{{k=1}}^{\infty}[k]^m_{p,q}\overline{b_{k}z^{k}}.
\mbox{tr}_{\mbox{\tiny ZERO}\atop\mbox{\tiny MODES}}= 2 V_{p+1}\int \frac{d^{p+1}k}{\left( 2\pi\right)^{p+1}}\, e^{-2\pi t\alpha^\prime k^2}= 2V_{p+1}\,\left( 8\pi^2 \alpha^\prime t\right)^{-\frac{p+1}{2}},
\begin{cases}a_1>(b_1)_-+|\Omega|\left((a_3)_-+(b_3)_-\right)+k\frac{\chi_1+\chi_2}{d_3}\cr b_2> (a_2)_-+|\Omega|\left((a_4)_-+(b_4)_-\right)+l\frac{\chi_1+ \chi_2}{d_3}.\end{cases}