harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
q \frac{dP}{dq}= \frac{P^{2}- Q}{12},Q(q^{2}) = 1 + 240 \sum_{n=1}^{\infty}\frac{n^{3}q^{2n}}{1 - q^{2n}}= \frac{\theta_{2}^{8}+ \theta_{3}^{8}+ \theta_{4}^{8}}{2}.
\Re(\eta(s)) = \sum_{n=1}^\infty\left[(2n-1)^{-\sigma}\cos(t\ln(2n-1)) - (2n)^{-\sigma}\cos(t\ln(2n))\right] \Im(\eta(s)) = \sum_{n=1}^\infty\left[(2n)^{-\sigma}\sin(t\ln(2n))-(2n-1)^{-\sigma}\sin(t\ln(2n-1))\right].
\begin{array}{lll}X(t+1) = J^{(r)}\hat{C}C\hat{C}^{-1}J^{(r)}X(t) + J^{(r)}\Big[(\hat{C}C\hat{C}^{-1}-I)\underline{v}+ \hat{C}(Dp - U(J^{(r)}X(t)) - \mathbf{1}\frac{\beta}{2})\Big]. \end{array}
K_{t}^{\infty}( \bar{r}, \bar{r}^{\prime}| - \triangle_{c}) = \int_{0}^{\infty}\frac{d k}{\pi}e^{i k ( \varphi - \varphi^{\prime})}\int_{0}^{\infty}e^{- \nu^{2}t}J_{k}( \nu r ) J_{k}( \nu r^{\prime}) \nu d \nu
\mathcal{B}_{n,u}=\{\exists i\in[B]: \hat{X}_{w_i,u_{m+1}}>K_1n^{-\beta}D_{u_{m+1}}(1+n^\gamma)\}, \mathcal{B}_{n,v}=\{\exists i\in[B]: \hat{X}_{w_i,v_{m+1}}>K_1n^{-\beta}D_{v_{m+1}}(1+n^\gamma)\},
\sum \frac{1}{\omega}\equiv \sum_{- \infty}^{+ \infty}\!^{\prime}\frac{1}{\omega_{n}}\sim \frac{L}{2 \pi}2 \int_{0}^{\infty}\frac{1}{\omega ( k )}\left[ 1 + \frac{1}{L}\delta^{\prime}( k ) \right] d k
\frac{d}{dz}\left\lbrace{}^{\Psi}F_{p}^{(\alpha,\beta)}(a,b;c;z)\right\rbrace =\frac{(a)(b)}{(c)}\sum_{n=0}^{\infty}(a+1)_{n}\frac{{}^{\Psi}B_{p}^{(\alpha,\beta)}(b+n+1,c-b)}{B(b+1,c-b)}\frac{z^{n}}{n!}&=\frac{(a)(b)}{(c)}\left[{}^{\Psi}F_{p}^{(\alpha,\beta)}(a+1,b+1;c+1;z)\right].
-\sum_{j=1}^{m}2\beta_{j}\partial_{s_{m}}\log \Lambda_{j}= -2 \sum_{j=1}^{m}\beta_{j}\partial_{s_{m}}(\beta_{j}) \log (8|\lambda_{j}|^{3/2}r^{3/2}) -2\sum_{j=1}^{m}\beta_{j}\sum_{\substack{\ell = 1 \ell \neq j}}^{m}\partial_{s_{m}}(\beta_{\ell})\log(\widetilde{T}_{\ell,j}).
y=\{S^e, T\}\prod_{\substack{(i,j)\in\mathbb{J}_S k\geq 0}}\{1+b_{ijk}S^iT^j, S\}^{d_{ijk}p^k}\prod_{\substack{(i,j)\in\mathbb{J}_T k\geq 0}}\{1+a_{ijk}S^iT^j, T\}^{c_{ijk}p^k},
\Lambda_{\pm}(q_0)=\Lambda (q_0)\pm\frac{(8\kappa)^{\frac{q_0}{2}+1}(1-k)^{\kappa-\frac{1}{2}q_0}}{[\frac{1}{2}(q_0-1)]!(2\pi)^{1/2}2^{\kappa+1+\frac{q_0}{2}}}\bigg[1+(1-k)\{\kappa(\frac{1}{2}-\ln 2)+\frac{q_0}{4}\}+O\bigg(\frac{1}{\kappa}\bigg)\bigg]
\Psi(t)=\sum_{\vec{n}}a_{\vec{n}}\phi_{\vec{n}}e^{-i{\cal E}_{\vec{n}}t}=e^{-i{\cal E}_0t}\sum_{\vec{n}}a_{\vec{n}}\phi_{\vec{n}}e^{-i\omega(\sum_{j=1}^r n_jf_j)t}.
R ( \varphi , \varphi^{\prime}) = \frac{\varphi - \pi}{2 \pi k}\sum_{i}E_{i i}\otimes E_{i i}+ \frac{i}{2 k}\sum_{i \neq j}\frac{\cos{\frac{\pi \alpha_{i j}( X )}{k}}}{\sin{\frac{\pi \alpha_{i j}( X )}{k}}}e^{- i \frac{\alpha_{i j}( X )}{k}( \varphi - \varphi^{\prime})}~ E_{i j}\otimes E_{i j}
\Re p_0(X) =\Re \tilde p(X;h)-h\Re p_1(X;h)-h^{2\delta}\chi(X/h^\delta) &\le\Re \tilde p(X;h) +h|\Re p_1(X;h)|\le \Re \tilde p(X;h) +Ch\langle X\rangle^2\le C\Re \tilde p(X;h),
\ell^2(\widehat{G})=\left\{\phi:\forall [\xi]\in \widehat{G},\,\phi(\xi)\in \mathbb{C}^{d_\xi\times d_\xi}\textnormal{and}\Vert \phi\Vert_{\ell^2(\widehat{G})}:=\left(\sum_{[\xi]\in \widehat{G}}d_{\xi}\Vert\phi(\xi)\Vert_{\textnormal{HS}}^2<\infty\right)^{\frac{1}{2}}\right\}.
I_i =\frac{1}{2}\frac{\partial}{\partial y^i}\Big[(n+1)\frac{\phi'}{\phi}-(n-2)\frac{s\phi''}{\phi-s\phi'}-\frac{3s\phi''-(b^2-s^2)\phi'''}{(\phi-s\phi')+(b^2-s^2)\phi''}\Big] &=-\frac{\Phi(\phi- s\phi')}{2\Delta \phi \alpha^2}(\alpha b_i-sy_i).
|\gamma(\phi)| =\left| \langle Lf,\phi\rangle\right| \lesssim \left(\int_{B}|\Gamma f|^2 \, d\mu \right)^{1/2}\left(\int_{B}|\Gamma \phi|^2 \, d\mu \right)^{1/2}\lesssim \left(\int_{B}|\Gamma f|^2 \, d\mu \right)^{1/2}\|\phi\|_{{\mathcal H}}.
\int_{({\mathbb R}^d)^{k-1}}h^\ast (K_1\cap K_2^{x_2}\cap \dots \cap K_k^{x_k},\cdot)\, \lambda^{k-1}(d(x_2,\dots ,x_k)) & = \sum_{{m_1,\dots,m_k=1}\atop{m_1+\dots +m_k=(k-1)d+1}}^dh^\ast_{m_1,\dots,m_k}(K_1,\dots,K_k,\cdot),
0= \nabla_{1}S_{23}=(-S_{33}+S_{22})\nabla_{1}(m_{2})_{3}, 0= \nabla_{3}S_{12}=(S_{22}-S_{10})\nabla_{3}(m_2)_{1}, 0= \nabla_{1}S_{12}=(S_{22}-S_{01})\nabla_{1}(m_{2})_{1}, 0= \nabla_{2}S_{12}=(S_{22}-S_{01})\nabla_{2}(m_{2})_{1},
B ( r )^{2}\rightarrow - \frac{4 q^{2}r}{( r^{2}+ q^{2})^{n}}\lbrace \int^{r}\frac{( s^{2}+ q^{2})^{n}}{s^{2}}d s - \int^{i q}\frac{( s^{2}+ q^{2})^{n}}{s^{2}}d s - \bar{\alpha}\rbrace ,
\max_{|\zeta|\leq \lambda_{p_n}}G^+\circ\psi_{f(p_n)}(\zeta) =\max_{|\zeta|\leq 1}G^+\circ\psi_{f(p_n)}(\lambda_{p_n}\zeta) &=\max_{|\zeta|\leq 1}G^+\circ f\circ\psi_{p_n}(\zeta) &=d\max_{|\zeta|\leq 1}G^+\circ\psi_{p_n}(\zeta) &=d.
\left\|R(\lambda,A_{-1}(t))x\right\|_{X_{-1}}\leq\kappa\left\|R(\lambda,A_{-1}(t))x\right\|_{X_{-1}(A(t))}=\kappa\left\|R(\lambda,A(t))A(t)^{-1}x\right\| &\leq\left\|R(\lambda,A(t))\right\|\cdot\kappa\left\|x\right\|_{X_{-1}(A(t))}\leq\left\|R(\lambda,A(t))\right\|\cdot\kappa^2\left\|x\right\|_{X_{-1}}.
-\frac{1}{4}ie\epsilon_{jki}x_{k}\Lambda_{j}-\frac{1}{8}ie\epsilon_{jrk}\Lambda^{*}_{ji}x^{*}_{r}x_{k}+\frac{1}{4}ie\epsilon_{jrk}\Lambda^{*}_{ij}x^{*}_{r}x_{k})\psi_{i}^{*}
\left| \Omega \right\rangle \equiv \left| \overline{\Psi}\left({\bf y}\right) \Psi \left({\bf 0}\right) \right\rangle = \overline{\psi}\left({\bf y}\right) \exp \left({i g \int_{\bf 0}^{\bf y}{d z^{i}A_{i}\left( z \right)}}\right) \psi \left({\bf 0}\right) \left| 0 \right\rangle ,
v=\sum_{i=1}^{5}(x_{i}^{5}+\Pi_{i=1}^{5}x_{i}).{\bf I}_{{\cal D}\left(G\right)}+\sum_{i,j,k=1}^{5}\{\cal F^{\prime}}_{k_{1}k_{2}k_{3}}(\Phi_{1},\dots ,\Phi_{5})\F_{1}^{k_{1}}.F_{2}^{k_{2}}.F_{3}^{k_{3}}.
u(\xi_k, \bar \xi_k) = a_k(u)\frac{i}{\sqrt{2\pi}}\log|\xi_k|\,+\,b_k(u) \frac{1}{2\sqrt{\pi}}\frac{1}{\xi_k}+c_k(u)\frac{1}{2\sqrt{\pi}}\frac{1}{\bar \xi_k}& + \frac{i}{\sqrt{2\pi}}d_k(u) \,+\, \frac{1}{2\sqrt{\pi}}e_k(u) \xi_k \,+\,\frac{1}{2\sqrt{\pi}}f_k(u)\bar{\xi_k}\, & +o(|\xi|).
\lambda^*(\vartheta^t_\Gamma)=\lambda^*\!\left(\int_{\underline C_{k+n}/\underline C_k}\theta^t_\Gamma\right)=\int_{\underline C_{k+n}/\underline C_k}\lambda^*\!\left(\theta^t_\Gamma\right)=\vartheta^t_\Gamma,
\mu_{\Sigma,k}(x_{l(k)}) x_{l(k)}= \prod_{\substack{1 \leq i \leq n+m b_{i,k}>0}}x_{l(i)}^{b_{i,k}}+ \prod_{\substack{1 \leq i \leq n+m b_{i,k}<0}}x_{l(i)}^{-b_{i,k}},
\chi(-1) \cdot \frac{L(f,\chi, 1)}{(-2\pi i) \Omega^{\chi(-1)}_{f}}= \frac{1}{n}\cdot \sum_{b \in (\mathbb{Z}/n\mathbb{Z})^\times}\chi(b) \cdot \sigma_b \cdot \left( \sum_{a \in (\mathbb{Z}/n\mathbb{Z})^\times}\zeta^{a}_n \cdot \left[ \frac{-a}{n}\right]^{\chi(-1)}_{f}\right) .
\sum_{\substack{|r|}=\widehat{Y}}|b_2|^{1/2}|r_3''|^{(1-\delta)/2}|r_3|^{\delta(n-2)/2}\ll \widehat{Y}\sum_{|r_3'|\leq \widehat{Y}}|r_3'|^{\delta(n-2)/2-1}\sum_{|r_3''b_2|\leq \widehat{Y}|r_3'|^{-1}}|b_2|^{-1/2}|r_3''|^{(\delta(n-3)-1)/2}\ll \widehat{Y}^{1/2+\delta(n-3)/2}\sum_{|r_3'|\leq \widehat{Y}}|r_3'|^{(\delta-1)/2}
\sum_{i}\tilde{\alpha}_{i}^{\prime 2}- \tilde{\alpha}_{i}^{\prime}\sum_{j}\tilde{\alpha}_{j}^{\prime}+{\frac{1}{D - 2}}\varphi^{\prime 2}+ \sum_{a}\alpha_{a}^{\prime 2}={\frac{P^{2}}{2}}e^{\tilde{\Phi}}\ .
\mathrm{Gr}_{-1}^M H^1_\ell(X) = \mathrm{Gr}_{-1}^M H^1_\ell(\overline{X}) \mathrm{Gr}_{1}^M H^1_\ell(X) = \mathrm{Gr}_{1}^M H^1_\ell(\overline{X})
u^\varepsilon(0,x_{\sigma,\varepsilon})=u(0,x'_{\sigma,\varepsilon})-\varepsilon^{-1}|x_{\sigma,\varepsilon}-x'_{\sigma,\varepsilon}|^2\quad &v_\varepsilon(0,y_{\sigma,\varepsilon})=v(0,y'_{\sigma,\varepsilon})+\varepsilon^{-1}|y_{\sigma,\varepsilon}-y'_{\sigma,\varepsilon}|^2.
\partial_l \partial_\xi^{n}W_{<l;\xi}- [\Psi_{l}, \partial_\xi^{n}W_{<l;\xi}]= \partial_\xi^{n+1}\Psi_l + \sum_{j=1}^{n}[\partial_\xi^j \Psi_l, \partial_\xi^{n-j}W_{<l;\xi}]
D(B_) = \frac{1}{L}\sum_{i=1}^L d_i, d_i = P_{Y^i}(0^i) \log( M- \sum_{j=1}^{i}b_j^0) &+ \sum_{j=1}^iP_{Y^i}(\beta_j^i) \log( \max(1, \frac{b_j^0}{2^{i-j}})).
\begin{multlined}[t][12.5cm]\widetilde{\Pi}_{\nu_{G},k\nu_{T}}\left(x,y\right)=\frac{kd_{\nu_{G}}}{(2\pi)^{d_{T}}}\int_{G}\int_{T}\int_{0}^{+\infty}\chi_{\nu_{G}}(g^{-1})e^{ik\Psi(t,\vartheta,x)}\cdot \cdot s\left(\tilde{\mu}^{G}_{g^{-1}}\circ\tilde{\mu}^{T}_{-\vartheta}\left(x\right),y,kt\right)dtdgd\vartheta,\end{multlined}
\Gamma^{3,\varepsilon}_{\alpha 3}=\Gamma^{p,\varepsilon}_{33}=0 \ \textrm{in}\ \bar{\Omega}^\varepsilon, A^{\alpha\beta\sigma 3,\varepsilon}=A^{\alpha 333,\varepsilon}=B^{\alpha\beta\sigma 3 , \varepsilon}=B^{\alpha 333, \varepsilon}=0 \ \textrm{in}\ \bar{\Omega}^\varepsilon,
\frac{\partial}{\partial z}\Big|_{z=1}(\widetilde Q_{r,O_r}(z;q) - \widetilde E_{r,L_r}(z;q)) = (-q;q^2)_\infty \left(\sum_{k=0}^\infty \frac{q^{2kr+1}}{1+q^{2kr+1}}+\sum_{k=1}^\infty \frac{q^{2kr}}{1+q^{2kr}}\right).
\frac{\left(-q;q\right)_{\infty}^{2}}{\left(q;q\right)_{\infty}}=\frac{1}{\sqrt{2\pi\log q^{-1}}}\int_{-\infty}^{\infty}\frac{\exp\left(\frac{x^{2}}{\log q^{2}}\right)dx}{\left|\left(q^{1/2}e^{ix};q\right)_{\infty}\right|^{2}}
\mathcal{R}_4=\Bigl\{(u,v,w,t):\, \theta_2-\theta_1<t<w<v<u<\theta_1,\,u+v<\theta_1, &u+v+w+t\notin[\theta_1,\theta_2]\cup[1-\theta_2,1-\theta_1], &\{u+v+w,u+v+t,u+w+t,v+w+t\}\cap[\theta_1,\theta_2]=\emptyset\Bigr\}.
\left| \xi^{(l, \sigma)}\left( \Phi_s(y) \right) \right| \stackrel{\eqref{eq:thmaboutrelationbetweenboxdimensionandRokhlindimension-equivariant-s}}{=}\left| \xi^{(l, \sigma)}\left( y \right) \right| \in (0, 8\varepsilon] \; ,
\sum_{\substack{0\leq b <q/a, bc\equiv \alpha\pmod{q/a}}}1 =\# \left\{0\leq b < q/a\;:\; bc\equiv \alpha\pmod{q/a}\right\}&= \left\{b=\tfrac{\alpha}{c_1}\tilde{c}+j \tfrac{q}{ac_1}\;:\; 0\leq j < c_1\right\}=\gcd(c, q/a).
\begin{array}{l}Q_{n}(x; t) \mbox{is a monic polynomial of degree}n \mbox{in}x begin{align*}2ex] \mathcal{L}^{(t)}\left[ x^{-n+s}Q_{n}(x; t) \right] = 0, s =0, 1, \ldots, n-1,\end{array}
\begin{array}{rl}(z_0;z_1,z_2)={1\over h}\left({1+x_1^2+h^2\over 2}+i\alpha_0(x_1) h^3;\right. begin{align*}5pt] \left.x_1+i\alpha_1(x_1) h^3,\;\;{i-ix_1^2-ih^2\over 2}+\alpha_2(x_1) h^3\right)\end{array}
g_{D q - G T}^{2}= G_{\mathrm{O S}}^{2}( \alpha_{\mathrm{e f f}}^{\prime})^{\frac{q - 1}{2}}\ , \quad \theta_{D q - G T}= G_{O S}^{2}( \alpha_{\mathrm{e f f}}^{\prime})^{\frac{q + 1}{2}}\ .
S =B(a+b-1,1-a)\,\zeta(a+b-1)-\frac{1}{\Gamma(b)}\sum_{n=0}^\infty\frac{1}{(1-a)_{n+1}}\int_0^\infty \frac{t^{b+n-1}e^{-t}}{e^t-1}\,dt &=B(a+b-1,1-a)\,\zeta(a+b-1)-\sum_{n=0}^\infty \frac{(b)_n}{(1-a)_{n+1}}\,\zeta(b+n,2)
V_{\chi_{1,1}}(z)=\sum_{i=1}^n q^{2i-1}s_1 M_{i,1}(z)+\frac{s_1-s_1^{-1}}{q-q^{-1}}M_{0,1}(z)+\sum_{i=1}^n q^{-2i+1}s_1^{-1}M_{\bar i,1}(z)\,.
\overline{\Gamma}_{\alpha\gamma}^{\beta}=\frac{\partial\overline{x}^{\beta}}{\partial x^{\epsilon}}\frac{\partial x^{\delta}}{\partial\overline{x}^{\alpha}}\frac{\partial x^{\zeta}}{\partial\overline{x}^{\gamma}}\Gamma_{\delta\zeta}^{\epsilon}+\frac{\partial\overline{x}^{\beta}}{\partial x^{\eta}}\frac{\partial^{2}x^{\eta}}{\partial\overline{x}^{\alpha}\partial\overline{x}^{\gamma}},
\nabla f(\mathbf{x}^{k+1}) - \nabla f(\mathbf{x}^{\star}) + (1-\eta)\frac{\rho}{2}\mathbf{E}_{}^T\mathbf{E}_{}(\mathbf{x}^{k+1}- \mathbf{x}^{\star}) = \mathbf{E}_{}^T(\boldsymbol{\alpha}^{\star}- \boldsymbol{\alpha}^{k+1}) + \frac{\rho}{2}\mathbf{E}_{}^T\mathbf{E}_{}(\mathbf{z}^{k}-\mathbf{z}^{k+1}) ,
m \epsilon_{\mu \alpha \beta}\partial^{\alpha}\hat{A}^{\beta}\approx \epsilon_{\mu \alpha \beta}\partial^{\alpha}( \hat{f}^{\beta}- m \hat{a}^{\beta}) - \epsilon_{\nu \alpha \beta}( m \partial^{\alpha}\hat{A}^{\beta}- \partial^{\alpha}\hat{f}^{\beta}) \frac{\delta \hat{a}^{\nu}}{\delta \hat{A}^{\mu}}.
\lambda\cdot a_{0}= a_0,\ \ \lambda\cdot a_{1}= a_0\lambda +a_1,\ \ \lambda\cdot a_{2}= a_0\lambda^{[2]}+a_1\lambda +a_2, \lambda\cdot a_{3}= a_0\lambda^{[3]}+a_1\lambda^{[2]}+a_2\lambda+a_3,\ldots.
A_{1}=\frac{2(\gamma^{4}-4\gamma^{3}+6\gamma^{2}-4)\omega^{2}h^{2}+8(\gamma-2)^{2}}{(\gamma^{2}\omega^{2}h^{2}+4)((\gamma-1)^{2}\omega^{2}h^{2}+(\gamma-2)^{2})}
U = \{\langle \langle k,m \rangle, \sigma \rangle : \langle m,\sigma \rangle \in V_k \}V = \{\langle 0,\sigma \rangle : \forall k \, \forall \tau \succeq \sigma \, \exists m \, \exists \rho \succeq \tau \, \langle m,\rho \rangle \in U_k \}.
D_{1}(L_{n})=nL_n, \ D_{1}(G_n)=nG_n, &D_{2}(L_{n})=0, \ D_{2}(G_n)=G_n, &D_{3}(L_{n})=nG_{n-1}, \ D_{3}(G_n)=0, &D_{4}(L_{n})=0, \ D_{4}(G_n)=L_{n+1}.
\frac{\partial}{\partial \rho}z_N(\rho) = \frac{4}{\sqrt{\pi}}\frac{1}{\sqrt{1-\rho}}\left[ \frac{1}{2}+ \sum_{n=1}^{N}\exp\left[-\frac{{\alpha_n}^2}{4\left(1-\rho\right)}\right] \right].
F_2(t) \le 6 \int_0^t ({L_f^*}^2+{K_f^*}^2\|\ell\|^2_{L^1}+\epsilon)\|u(t+h-\tau)-u(t-\tau)\|^2_{\mathbb H^\mu}d\tau & = 6 \int_0^t ({L_f^*}^2+{K_f^*}^2\|\ell\|^2_{L^1}+\epsilon)\|u(\tau+h)-u(\tau)\|^2_{\mathbb H^\mu}d\tau,
\left(\Omega^{n-1}\right)^{(n-1, n-1)}= \sum_{k=0}^{\lfloor \frac{n-1}{2}\rfloor}\binom{n-1}{k}(\Omega^{(2, 0)}\wedge \Omega^{(0, 2)})^k \wedge (\Omega^{(1, 1)})^{n - 2k - 1}.
f_{\alpha}( \phi ) ={\frac{e^{- i \pi / 4}}{\sqrt{2 \pi p}}}\left({\frac{1}{1 + e^{i \phi}}}\right) e^{i \alpha \phi}\left( e^{- i \alpha \pi}- e^{i \alpha \pi}\right) , \quad 0 \le \alpha < 1 .
h_2^\ell(X^\ell)=\frac{1}{m^{\ell-1}}\sum_{k=1}^{m^{\ell-1}}S_0\exp\left(\left(r-\frac{\sigma^2}{2}\right)k\frac{T}{m^{\ell-1}}+\sigma\sqrt{\frac{T}{m^{\ell-1}}}\sum_{i=1}^{k}(C_{m,\ell}X^\ell)_i\right)\,.
U(1) - \frac{1}{4}= U(0) +1 \rightarrow U(1)= \frac{5}{4}, 2U(2) = U(1) -1 \rightarrow U(2)=\frac{1}{8}= \frac{1}{4 \cdot 2!}, 3U(3) = U(2) \rightarrow U(3)=\frac{1}{24}=\frac{1}{4 \cdot 3!}, 4U(4) = U(3) \rightarrow U(4)= \frac{1}{4 \cdot 4!}, \vdots \vdots
c_{k+l,h}= \binom{n}{h}^{-1}\binom{n-k-l}{h-k}, &c_{k+l+1,h}= c_{k+l,h}\left[\alpha+2l+1-\frac{(\sigma+2k+2l+1)(l+h-n)}{k+l-n}\right], &c_{ih}= K_{h}(i)c_{i-1,h}+ L(i)c_{i-2,h}(i=k+l+2,k+l+3,\ldots,n),
\partial_{t}^{2}v_{5}(t,r) = \frac{-1}{2\pi}\int_{t}^{\infty}ds \int_{0}^{s-t}\frac{\rho d\rho}{\sqrt{(s-t)^{2}-\rho^{2}}}\int_{0}^{2\pi}d\theta \frac{\partial_{1}^{2}N_{2}(f)(s,\sqrt{r^{2}+\rho^{2}+2 r \rho \cos(\theta)})}{\sqrt{r^{2}+\rho^{2}+2 r \rho \cos(\theta)}}(r+\rho \cos(\theta))
I_5 \le CK^{p/(p-1)}\, \delta^s \int_{A_3}\frac{dx dy}{|y|^{N+s}}= CK^{p/(p-1)}\, \delta^N, I_6 \le CK^{p/(p-1)}\, \delta^{sp}\int_{A_3}\frac{dx dy}{|y|^{N+sp}}= CK^{p/(p-1)}\, \delta^N.
\begin{aligned}\frac{|F'_{n,j}(a)|}{|F_{n,j}(a)|}=\sum_{k=0}^{n-1}\frac{|D_a Df_{a}(\xi_{k,j}(a))|}{|Df_{a}(\xi_{k,j}(a))|}\leq B\sum_{k=0}^{n-1}\frac{1}{|Df_{a}(\xi_{k,j}(a))|}&\leq \frac{B}{C\lambda}n &\lesssim \frac{B}{C\lambda}\left(\log\frac{\lambda}{(1+\varepsilon')\delta'}\right)^{-1}(-\log|x_{j}(a)|).\end{aligned}
4k^{2}\left( t-1\right)^{2}x^{2}+(t-1)^{2}y^{2}+\allowbreak 4\left(1-t\right) \left( tk-t+k\right) xy +4k\left( t-1\right) \left( 2k-t\right) x+\allowbreak 2\left( t-1\right)\left( 2k-t\right) y =-\left( 2k-t\right)^{2},k\in I
T= \begin{pmatrix}1 0 0 0 0 0 0 1 0 0 0 0 0 0 \zeta_5^3 0 0 0 0 0 0 \zeta_5^2 0 0 0 0 0 0 -1 0 0 0 0 0 0 1 \end{pmatrix},
\begin{array}{rrcll}\min~ \multicolumn{4}{l}{c^\top x+\sqrt{x^\top Qx}}begin{align*}1.5ex] \textnormal{s.t.}\sum_{e\in E}x_e = |V|-1 begin{align*}1ex] \sum_{e\subseteq X}x_e \le |X|-1 \forall \emptyset\neq X\subseteq V begin{align*}1ex] x \in \{0,1\}^{E}\end{array}
\mathbf{P}\Big(\{U^*_{n}(\theta_n,c^*_n)\ne \emptyset \}\cap \{\mathfrak W^{*}(c_{\pi^*})= \emptyset \}\Big) \le \mathbf{P}\Big( \{U^{*}_{n}(\theta_n,c^*_n)\ne \emptyset \}\cap \{\mathfrak W^{*,+\delta}(c_{\pi^*})=\emptyset \}\Big) &+\mathbf{P}\Big(\{\mathfrak W^{*,+\delta}(c_{\pi^*})\ne \emptyset \}\cap \{\mathfrak W^{*}(c_{\pi^*})=\emptyset \}\Big),
T_{n,q}(\frac{r}{\alpha})=\frac{\alpha^{n+1}}{(\alpha-r)^{2n+2}}\sum_{k=0{}}^{n+1-q}a_{n+1,k+1}(\alpha-r)^{k+1}\binom{n+1-q}{k}r^{n+1-q-k}-\alpha^q \binom{n+q-1}{n}.
|T(\Omega_{q,n,r})|\cr=\, |\{(s_0,\dots,s_{m-1})\in\Bbb N^m: 0\le s_i\le n(p-1),\s_0p^0+\cdots+s_{m-1}p^{m-1}=r\}|\cr=\, \ \prod_{k=0}^{m-1}(1+X^{p^k}+X^{2p^k}+\cdots+X^{n(p-1)p^k}).
0<\liminf_{n\to\infty}\sum_{k=1}^{\phi_{n}}(\mu(F_{nj})|F_{nj}|^{r})^{\frac{s}{s+r}}\leq\limsup_{n\to\infty}\sum_{k=1}^{\phi_{n}}(\mu(F_{nj})|F_{nj}|^{r})^{\frac{s}{s+r}}<\infty.
{\cal F}\Big( \{x, y \}\Big) \; := \;\prod_{a=1}^N\Bigg| \frac{\prod_{i,j=1}^{n_a}\big( \hat{x}^{(a)}_i - \tilde{y}^{(a)}_j \big)\big( \tilde{x}^{(a)}_i - \hat{y}^{(a)}_j \big)}{\prod_{i,j=1}^{n_a}\big( \hat{x}^{(a)}_i - \tilde{x}^{(a)}_j \big)\big( \tilde{y}^{(a)}_i - \hat{y}^{(a)}_j \big)}\Bigg|^{[1-\frac{g}{\pi+gN}]}
F_{d,\ell}\left(z; it\right) = \sum_{a=0}^N \mathcal D_z^{2a}\left(\frac{e^{2 \pi i d z}}{1-\zeta^\ell}\right)\frac{(-2\pi t)^a}{a!}+O\left(t^{N+1}\right), G_{d,\ell}\left(z; it\right) = 2i\sum_{a=0}^N\mathcal D_z^{2a}\left(\frac{\sin(2 \pi d z)}{1-\zeta^\ell}\right) \frac{(-2\pi t)^a}{a!}+O\left(t^{N+1}\right).
\left. \frac{\delta \Omega^{a}}{\delta C^{\alpha b}}\right|_{C = \pi ={\cal P}= \lambda = 0}= T_{\alpha}\delta_{b}^{a}, \left. \frac{\delta \Omega^{a}}{\delta \pi^{\alpha}}\right|_{C = \pi = \lambda = 0}= \varepsilon^{a b}{\cal P}_{\alpha b},
{\mathcal O}^-_{r,c}(x,y)=\bigcap_{z\in B_x(r)}{\mathcal O}_{c}(z,y) \subset{\mathcal O}_{c}(x,y)\subseteq \bigcup_{z\in B_x(r)}{\mathcal O}_{c}(z,y)={\mathcal O}^+_{r,c}(x,y);
S = \int d^{3}x \left[ - \frac{1}{4}( \frac{1}{2 \rho}) F_{\mu \nu}( A ) F^{\mu \nu}( A ) + 2 \pi A^{\mu}{\tilde{J}}_{\mu}+ \frac{1}{4 \rho}( \partial \rho )^{2}- V ( \rho ) \right]
\sum_{j=1}^{n_{P}}W_{j}\frac{dZ_{j}}{dt}=\sum_{k=1}^{b}W_{k}f_{k}+\sum_{j=1}^{n_{P}}w_{j}p_{j}+\sum_{j=1}^{n_{t}}w_{j}t_{j},j=1,...,n_{p}+n_{t}
\lim_{i \to \infty}\frac{i(n_i+3)}{\ell_i}\le \lim_{i \to \infty}\frac{i(n_i+3)}{n_i \log n_i}&= \left( \lim_{i \to \infty}\frac{n_i+3}{n_i}\right) \cdot \left( \lim_{i \to \infty}\frac{i}{\log n_i}\right) = 1 \cdot 0 = 0.
\|V_n-V\|_{\mathcal{H}^1}^2 \lesssim L_{\omega,\mathbf{c}}(V_n-V) &=L_{\omega,\mathbf{c}}(V_n)-L_{\omega,\mathbf{c}}(V)-(L_{\omega,\mathbf{c}}(V_n)-L_{\omega,\mathbf{c}}(V_n-V)-L_{\omega,\mathbf{c}}(V)) &\rightarrow 0.
{\mathrm{T r}}\bar{K}^{\beta}( s ) = \theta_{3}\Big ( 0 ,{\mathrm{e}}^{- \frac{\beta^{2}}{4 s}}\Big ) \frac{\beta}{( 4 \pi s )^{1 / 2}}\int{\mathrm{d}}^{3}{\mathrm{\boldmath x}}\bar{K}^{( 3 )}( s | \mathrm{\boldmath x}, \mathrm{\boldmath x}) .
Z_{\mu}= \frac{( x - z )_{\mu}}{( x - z )^{2}}- \frac{( y - z )_{\mu}}{( y - z )^{2}}\mathrm{,}Z^{2}= \frac{( x - y )^{2}}{( x - z )^{2}( y - z )^{2}}\mathrm{.}
3 v^{- 1}r{\cal Q}_{0 0}{^A}{_{| | A}}- 3 v r{\cal Q}_{3 3}{^A}{_{| | A}}= 3 r{\cal Q}^{A B C}{_{| | A}}\eta_{B C}= - \frac r v \dot{Q}_{0}{^A}{_{| | A}}- v Q_{3}{^A}{_{| | A}}+
Z(t) = \sum_{j \geq 0}(j + 1)^2 e^{-t (j +1)^2 + t}= \frac{e^t}{2}\sum_{j \in Z}(j + 1)^2 e^{-t (j +1)^2}= -\frac{e^t}{2}\frac{d}{dt}\sum_{j \in Z}e^{-t j^2}
\frac{P_\mu}{P_m}=\frac{P_{\mu-1}}{P_m}+\frac{p_\mu}{P_m}\leq\lambda+\frac{p_\mu}{P_\mu}\frac{P_\mu}{P_m}=\lambda(1+o(1))\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\frac{P_\mu}{P_m}\geq\left(1+\frac{\delta}{2}\right)=\frac{\lambda+1}{2}
G_{\alpha \beta}= \left( \begin{array}{c c c c c}{{- ( 1 - \tilde{v}^{2})}}&{{0}}&{{0}}&{{0}}&{{- \tilde{v}}}\\{{0}}&{{1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}&{{0}}\\{{- \tilde{v}}}&{{0}}&{{0}}&{{0}}&{{1}}\\ \end{array}\right) ,
\frac{\partial v}{\partial t}+ Lv - \mu v = -\psi \: \mbox{in}\: D_{T+h}, \frac{\partial v}{\partial t}+ L_\pm v - \mu v = -\psi \: \mbox{in}\: D_{T+h}^\pm, v(T+h) = \phi \: \mbox{in}\: D.
[\mathfrak{h},\mathfrak{m}' ]\subset\mathfrak{m}',[\mathfrak{m}_0,\mathfrak{m}']\subset\mathfrak{m}',[\mathfrak{h},\mathfrak{m}_0]=0,[\mathfrak{h},\mathfrak{h}]\subset\mathfrak{h},[\mathfrak{m}_0,\mathfrak{m}_0]\subset\mathfrak{m}_0.
f(0,x,v)=f_{0}(x,v),\rho(0,x)=\rho_0(x),u(0,x)=u_0(x), &\gamma^-f(t,x,v)\big|_{(0,T)\times\Sigma^-}=g(t,x,v), &\rho(t,x)\big|_{(0,T)\times\Gamma_{\rm{in}}}=\rho_B(x),u(t,x)\big|_{(0,T)\times\partial\Omega}=u_B(x),
\psi\left(\begin{pmatrix}x_0 y_0\end{pmatrix},\ldots,\begin{pmatrix}x_{q-1}y_{q-1}\end{pmatrix}\right) :=\psi_{\max(y_i)}\left(\begin{pmatrix}x_0 y_0\end{pmatrix},\ldots,\begin{pmatrix}x_{q-1}y_{q-1}\end{pmatrix}\right)
\chi_{q}(z) =\sum_{n=1}^{\infty}q^{n/2}\sum_{k=0}^{\infty}\left(-z\right)^{k}q^{(2n-1)k/2}=\sum_{k=0}^{\infty}q^{-k/2}\left(-z\right)^{k}\sum_{n=1}^{\infty}q^{(2k+1)n/2}=\sum_{k=0}^{\infty}\frac{q^{(k+1)/2}}{1-q^{k+1/2}}\left(-z\right)^{k}\!,
\widetilde{z}(\omega)=E_-+\widetilde{z}'(\omega_\beta(E_-))(\omega-\omega_\beta(E_-))+\frac{1}{2}\widetilde{z}''(\omega_\beta(E_-))(\omega-\omega_\beta(E_-))^2+O\left((\omega-\omega_\beta(E_-))^3\right)\,.
s_{2 m_{s}- 2}= - \frac{1}{( 2 m_{s}- 2 ) !}{\Omega_{i_{1}\ldots i_{2 m_{s}- 2}}^{( 2 m_{s}- 1 )}}^{\sigma}c^{i_{1}}c^{i_{2}}\ldots c^{i_{2 m_{s}- 2}}\pi_{\sigma}\, \, .
d_0^nF (v_1,\dots,v_n)= \frac{\partial^n}{\partial t_1 \dots \partial t_n}F\Big(\sum_{h=1}^n t_h v_h\Big)\bigg|_{t_1=\dots=t_n=0}= \frac{1}{n!}\frac{\partial^n}{\partial t_1 \dots \partial t_n}d_0^n F\Big(\sum_{h=1}^n t_h v_h\Big)\bigg|_{t_1=\dots=t_n=0}.
\alpha -2A_1>0,\ \ \beta -2A_2>0,\ \ \gamma -2A_3>0, 2\omega -\frac{|\mathbf{c}|^2}{8A_1}>0,\ \ \omega -\frac{|\mathbf{c}|^2}{8A_2}>0,\ \ \omega -\frac{|\mathbf{c}|^2}{8A_3}>0.
H^{0}( \pi^{*}{\cal S},{\cal O}_{\pi^{*}{\cal S}}( A^{\prime}\sigma |_{\pi^{*}{\cal S}}- B^{\prime}F ) ) = H^{0}({\cal S},{\cal O}_{{\cal S}}( - B^{\prime}) \oplus \bigoplus_{i = 2}^{A^{\prime}}( - B^{\prime}+ i ( r - 2 ) ) .
S_{\chi}= \pi^{2}\chi_{0}^{2}\int_{0}^{\infty}d r \, r^{3}\left\{[ f^{\prime}( r ) ]^{2}+ m_{\chi}^{2}[ f ( r ) ]^{2}- \mu_{0}^{2}\theta ( r - r_{0}) [ f ( r ) ]^{2}\right\}
\mathcal I^\omega_{[s]_0^n}= \bigcap^{n-1}_{j=0}\left(f^{(j)}_\omega \right)^{-1}\circ h (\theta^j \omega)\left( I_{s_j}\right) &= \bigcap^{n-1}_{j=0}\left(h ( \omega) \circ E_k^{-j}( I_{s_j})\right).
S = -{\frac{1}{2}}\int d^{6}x ~ \partial_{M}\phi ~ \partial^{M}\phi - a \int_{\Sigma}d^{4}x ~{\cal O}_{4}- b L^{2}\int_{\Sigma}d^{4}x ~ \phi ~{\cal O}_{4}- c L^{- 2}\int_{\Sigma}d^{4}x ~ \phi ~ .
M_{2}(\mathcal{NC}(G)) =\dfrac{(8n-4)(8n-5)^{3}}{2}+2((2n-2)^{2}(4n-5)^{2}+12n(2n-2)(4n-5)+36n^{2}) ~~~~~~-3((2n-2)(4n-5)+6n)(8n-5)^{2}+(8n-4-\dfrac{3}{2})((4n-4)(4n-5)^{2}+36n) ~~~~~~~~~~~~~~-(2n-2)(4n-5)^{3}-54n = 1024n^{4}-2560n^{3}+2048n^{2}-512n = 2n(512n^{3}-1280n^{2}+1024n-256).
c_1(\mathcal E\otimes\mathcal L) =3c_1(\mathcal{L})+c_1(\mathcal{E}), c_2(\mathcal E\otimes\mathcal L) =3c_1(\mathcal{L})^2+2c_1(\mathcal{E})c_1(\mathcal L)+c_2(\mathcal{E}), c_3(\mathcal E\otimes\mathcal L) =c_1(\mathcal{L})^3+c_1(\mathcal{E})c_1(\mathcal L)^2+c_2(\mathcal{E})c_1(\mathcal L)+c_3(\mathcal{E}).
\varphi_h(\xi) =\frac{2n-1}{2n+1}H_M(\xi)|\xi^H|^2 - k(\xi) |\xi^H|^2=|\xi^H|^2\left( \frac{(2n-2)k(\xi) + l(\xi)}{2n+1}- k(\xi) \right) &=\frac{l(\xi)-3k(\xi)}{2n+1}|\xi^H|^2,
L_{M}=-\frac{1}{4}e^{2\kappa\Psi}F_{\mu\nu}^{a}F^{\mu\nu,a}-\frac{1}{2}\partial_{\mu}\Psi\partial^{\mu}\Psi-\frac{1}{2}e^{-4\kappa\Psi}D_{\mu}\Phi^{a}D^{\mu}\Phi^{a}-e^{-2\kappa\Psi}V(\Phi^{a})\ ,
\sum_{\ell=0}^n B_{\Gamma,p}^{(\ell)}(z) = \sum_{g=0}^{\lfloor\frac{p-1}{2}\rfloor}z^{2g-p}\sum_{\ell=n+1+2g-p}^{n}\Upsilon_{p-1-2g}^{(\ell)}(z) \;C_{p,g}^{(\ell)}.
y=\frac{-f_1}{2e_1}+\sqrt{\frac{\Delta_1^2}{4e_1^2}-\frac{1}{e_1}\sqrt[1/6]{\frac{-f_1}{2e_1}+\sqrt{\frac{\Delta_1^2}{4e_1^2}-\frac{1}{e_1}\sqrt[1/6]{\frac{-f_1}{2e_1}+\sqrt{\frac{\Delta_1^2}{4e_1^2}+...}}}}}
\left(\sum_{k=-\infty}^{-1}\frac{(2^{k}R)^{(\alpha-Q)p_{1}+Q}}{(1+2^{k}R)^{\gamma p_{1}}}\right)^{1/p_{1}}\leq \left(\sum_{k\in\mathbb{Z}}\frac{(2^{k}R)^{(\alpha-Q)p_{1}+Q}}{(1+2^{k}R)^{\gamma p_{1}}}\right)^{1/p_{1}}\sim\|K_{\alpha,\gamma}\|_{L^{p_{1}}(\mathbb{G})},
\mathcal{E}[k;\, H_0, \Theta] := \mathrm{tr}\Big\{(\delta - \overline{k}\circ k)^{-1}\circ\Big(\overline{k}\circ H_0[\phi,k]\circ k + \frac{1}{2}\overline{k}\circ \Theta[\phi,k] + \frac{1}{2}\overline{\Theta[\phi,k]}\circ k \Big)\Big\}.
\underset{n\rightarrow\infty}{\lim}~\frac{2\sum_{t=1}^n\tilde\epsilon_t\varrho_t}{n}=\underset{n\rightarrow\infty}{\lim}~\frac{2\sum_{t=n_0+1}^n\tilde\epsilon_t\hat\varrho_t}{n-n_0}
K_{N}(z_{1},\overline{z_{2}}) = \int_{0}^{1}\frac{us(z_{1})s(\overline{z_{2}})}{1-us(z_{1})s(\overline{z_{2}})}\,du+O_{L^1}\left(\log(d_{N})\sqrt{d_{N}^{3}/N}\right) &= -1-\frac{\log(1-s(z_{1})s(\overline{z_{2}}))}{s(z_{1})s(\overline{z_{2}})}+O_{L^1}\left(\log(d_{N})\sqrt{d_{N}^{3}/N}\right)
\begin{cases}\Delta u_i+\sum\limits_{j\in I}k_{ij}e^{u_j}=4\pi\alpha_i\delta_0 \ \B_1(0)\subseteq\mathbb{R}^2, \ \ \forall \i\in I, \sum\limits_{i\in I}u_i\equiv C, \C \ ,\end{cases}
\left|\int_\Omega g_0\,\varphi\,dx\right| \le\int_\Omega|g-g_0|\,|\varphi|\,dx+\left|\int_\Omega g\,\varphi\,dx\right| &\le \|g-g_0\|_{L^s(\Omega)}\|\varphi\|_{L^{s'}(\Omega)}+\|g\|_{L^r(\Omega)}\,\|\varphi\|_{L^{r'}(\Omega)}.