harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{d}{dt}|v(t)|_{H^{-1}}^2 = 2\left< v'(t),(-A)^{-1}v(t)\right>_H = -2|v(t)|_H^2 - 2b(\mathcal{M}(\Phi)(t),\mathcal{M}(\Phi)(t), (-A)^{-1}v(t)) + 2b(\mathcal{M}(\Psi)(t),\mathcal{M}(\Psi)(t),(-A)^{-1}v(t)).
\lim_{z\rightarrow\infty}\frac{\overline{\mathbf{F}}\left( xz\right)}{\overline{\mathbf{F}}\left( z\right)}=x^{-1/\gamma_{1}}\lim_{z\rightarrow\infty}\frac{\overline{\mathbf{G}}\left( xz\right)}{\overline{\mathbf{G}}\left( z\right)}=x^{-1/\gamma_{2}}.
\frac{b}{2}\sum_{r=1}^{p-1}\frac{(p+t)!(r+t)!}{(r+1+t)!t!}b^{p-1}r = \frac{b}{2}\cdot\frac{(p+t)!(p-1)!}{(p)!t!}b^{p-1}(p-t-1) &= -\frac{b}{2}\cdot \frac{p!t!}{p!t!}b^{p-1}(p-t-1) &= \frac{b}{2}\cdot (t+1) &= \frac{b+1}{2}
\Delta^+(\lambda^k \iota) = \sum_{i+j=k}\lambda^i \iota \otimes \lambda^j \iota \Delta^\times(\lambda^k \iota) = P_k(\lambda^1 \iota \otimes 1, \dots, \lambda^k \iota \otimes 1 ; 1 \otimes \lambda^1 \iota, \dots, 1 \otimes \lambda^k \iota),
\begin{array}{l l l}{{V}}&{{= \frac{1}{2}\int_{M}d^{2}x \Omega}}&{{\left[ \frac{1}{2}F_{i j}F^{i j}+ D_{i}\phi \overline{{{D^{i}\phi}}}\right.}}\\{{}}&{{}}&{{\left. + \frac{1}{4}( | \phi |^{2}- 1 )^{2}\right]}}\\ \end{array}
\|u_{0,\lambda}\|_{\dot{B}^{\frac{d}{q}+\alpha}_{q,p}}\eqsim\lambda^{-\alpha/2}(\lambda^{1/2})^{\frac{d}{q}+\alpha-\frac{d}{q}}\|u_0\|_{\dot{B}^{\frac{d}{q}+\alpha}_{q,p}}=\|u_0\|_{\dot{B}^{\frac{d}{q}+\alpha}_{q,p}};
f_{n}^{i+1,j}d_{X,0}^{ij}+f_{n-1}^{i+1,j+1}d_{X,1}^{ij}+\cdots+ f_{0}^{i+1,j+n}d_{X,n}^{ij}= d_{Y,n}^{ij}f_{0}^{ij}+d_{Y,n-1}^{i,j+1}f_{1}^{ij}+\cdots+d_{Y,0}^{i,j+n}f_{n}^{ij}
W(p, q) \equiv W_4(p, q; 0) = \left( \begin{array}{cccc}1+ \frac{p^2 + q^2}{2}p q -\frac{p^2 + q^2}{2}p 1 0 -p q 0 1 -q \frac{p^2 + q^2}{2}p q 1 -\frac{p^2 + q^2}{2}\end{array}\right)
{\cal M}_{T}\equiv{\frac{1}{T_{2}}}\left( \begin{matrix}{{1}}&{{T_{1}}}\\{{T_{1}}}&{{| T |^{2}}}\\ \end{matrix}\right) , \{\cal M}_{U}\equiv{\frac{1}{U_{2}}}\left( \begin{matrix}{{1}}&{{U_{1}}}\\{{U_{1}}}&{{| U |^{2}}}\\ \end{matrix}\right) .
b_{\kappa,\alpha}^{\dagger}=\frac{1}{N^{1/2}}\sum_{\kappa_{p},\alpha}e^{i(\kappa_{p}\cdot x)}s_{\alpha}(\kappa_{p})\cdot[(\frac{m\omega_{p,\alpha}}{2\hbar})^{1/2}d(x)-i(\frac{1}{2\hbar m\omega_{p,\alpha}})^{1/2}p(x)].
\tilde d_{G_i}(\tilde x, \tilde y) \leq d_i(\tilde x_1, \tilde y_1) + |\tilde x - \tilde x_1| + |\tilde y - \tilde y_1| \leq L d_X(\tilde x_1, \tilde y_1) + L|\tilde x - \tilde x_1| + L|\tilde y - \tilde y_1| &= Ld_{G_i}(\tilde x, \tilde y).
S = \int d^{4}x \sqrt{- g}[ \frac{\mathcal{R}}{1 6 \pi G}- \frac{1}{4 \pi}( ( D_{\mu}\Phi )^{\dagger}( D^{\mu}\Phi ) + V ( \Phi ) + \xi \Phi^{\dagger}\Phi \mathcal{R}) - \frac{1}{4 \pi}\frac{1}{4}\mid F \mid^{2}] .
S_{\mathbf{L}_{p}}( z , q ) = \sum_{a = 0}^{\infty}z^{a}q^{\frac{a^{2}- a}{2}}\left( \begin{matrix}{{{\mathbf{L}_{p}- ( a , 0 , \dots , 0 )}}}\\{{a}}\\ \end{matrix}\right)_{q},
\zeta(s) = \frac{\zeta(2)}{\Delta}\sum_{\nu=1}^{s-2}A_{\nu, 2}\Delta_{\nu s}+ \frac{1}{\Delta}\sum_{\nu=1}^{s-2}A_{\nu}\Delta_{\nu s}+ \frac{1}{\Delta}\sum_{\nu=1}^{s-2}\theta_{s+1-\nu}\Delta_{\nu s}.
\varepsilon_{}(\exp(40)) =1.93378\cdot 10^{-8}< 2 (1\,000)^{3/2}\exp(-0.8476836\sqrt{1\,000}) , \ \varepsilon_{}(\exp(1\,000)) =1.94751\cdot 10^{-12}< 2 (2\,100)^{3/2}\exp(-0.8476836\sqrt{2\,100}).
x^{k+1}= \operatorname{prox}_{g}^\tau(x^k - \tau z^k) y^{k+1}_i = \begin{cases}\operatorname{prox}_{f^*_i}^\sigma (y^k_i + \sigma A_i x^{k+1}) y^k_i \end{cases}z^{k+1}= A^* y^{k+1}+ p^{-1}A^*_{j_k}(y^{k+1}_i - y^k_i) = A^* y^{k}+ (p^{-1}+1) A^*_{j_k}(y^{k+1}_i - y^k_i)
\dot{x}=2y\left(a_1+2(b_1+c_1)(x^2+y^2)+\cdots\right)+2x\left(2(b_2+c_2)(x^2+y^2)+\cdots\right) \dot{y}=-2x\left(a_1+2(b_1+c_1)(x^2+y^2)+\cdots\right)+2y\left(2(b_2+c_2)(x^2+y^2)+\cdots\right).
\Phi ( \log \frac{p}{\Lambda}) = 1 + \frac{a_{1}}{\log \frac{p}{\Lambda}}+ \frac{a_{2}+ a_{3}\log ( \log \frac{p}{\Lambda})}{\log^{2}\frac{p}{\Lambda}}+ \mathcal{O}( \frac{1}{\log^{3}\frac{p}{\Lambda}}) .
W(x) ={\cal W}(x,0) = \langle\, 0 | \phi(x)\phi(0) | 0 \,\rangle= \Delta_+(x)={\displaystyle \int \!\! \frac{{\rm d}^4 k}{(2\pi)^4}}\,\, \delta(k^2 - m^2)\, \theta(k^0){\rm e}^{-{\rm i}k x}\,.
\Big| \frac{L(s_1,\chi)}{L(s_0,\chi)}\Big| \asymp (q(1+|t|))^{\lambda}\Big| \frac{\xi(s_1,\chi)}{\xi(s_0,\chi)}\Big| = (q(1+|t|))^{\lambda}\prod_{\rho}\frac{|s_1-\rho|}{|s_0-\rho|}.
\hat u_{+J}= \alpha_{+J}(e^{-B_+x_N}- e^{-A_+x_N}) + \beta_{+J}e^{-B_+x_N}, \hat u_{-J}= \alpha_{-J}(e^{B_-x_N}- e^{Ax_N}) + \beta_{-J}e^{B_-x_N},\hat p_- = \gamma_-e^{Ax_N}.
\left| \sum_{P \in \mathcal{P}_{n}}\chi(P) \right| \le \begin{cases}\min\{\frac{q^{\frac{n}{2}}}{n}(\ell+\deg (M)+1), \frac{q^n}{n}\}\mbox{if $\chi \neq \chi_0$,}\frac{q^n}{n}\mbox{otherwise.}\end{cases}
{\tilde\nu}^{(1)}_{\tilde\alpha}\frac{\partial V_{(n=1)}^{(1)}(a_i,p_i,\theta)}{\partial{\tilde\xi}^{(1)}_{\tilde\alpha}}- \frac{\partial{\cal G}^{(n=2)}(a_i,p_i,\theta)}{\partial\theta}= 0.
\Big\{\int_0^\infty f(\xi_s)\,ds<\infty\Big\}=\Big\{\sum_{n=0}^\infty \int_n^{n+1}f(\xi_s)\,ds<\infty\Big\}&=\Big\{\sum_{n=0}^\infty \int_0^{1}f\Big(\xi^n_r+\sum_{i=0}^{n-1}\xi_1^i\Big)\,dr<\infty\Big\}&=\Big\{\sum_{n=0}^\infty g^n(\xi^0,...,\xi^n)<\infty\Big\}&\in \sigma(\xi^0,\xi^1,...).
(\tilde{u}_N(t),\varphi(t))_{\mathcal{H}}-(\tilde{u}_{N0},\varphi(0))_{\mathcal{H}}= \int_0^t \Big[ (\tilde{u}_N,\partial_t \varphi)_{\mathcal{H}}+2\nu a(\tilde{u}_N,\varphi)+2\nu a(H,\varphi) -b_R(\tilde{u}_N ,\varphi,\tilde{u}_N)-b(H,\varphi,\tilde{u}_N)-b(\tilde{u}_N,\varphi, H) \Big]dt.
[\begin{matrix}X\\Y\\Z\end{matrix}]=[\begin{matrix}X_{w}/X_{w}^{\prime}&0&0\\ 0&Y_{w}/Y_{w}^{\prime}&0\\ 0&0&Z_{w}/Z_{w}^{\prime}\end{matrix}][\begin{matrix}X^{\prime}\\Y^{\prime}\\Z^{\prime}\end{matrix}]
Z^2(J)=\int d\varphi_1 d\varphi_2\,\exp{\rm i}\Bigl(\S(\varphi_1)+\S(\varphi_2)+\varphi_1^aJ_a+\varphi_2^aJ_a\Bigr)\,\delta\Bigl(\chi^{\mbox{\scriptsize com}}(\varphi_1)\Bigr)\delta\Bigl(\chi^{\mbox{\scriptsize com}}(\varphi_2)\Bigr)\; .
0<\sum_{i,j=1}^{N}a_{ij}\left(x,t;\mathbf{p};\mathbf{q}\right)\xi_{i}\xi_{j}\le\overline{M}\left|\xi\right|^{2}\,\xi\in\mathbb{R}^{N},\left(\mathbf{p},\mathbf{q}\right)\in [L^{2}\left(\Omega\right)]^{N}\times\left[L^{2}\left(\Omega\right)\right]^{Nd}.
\lambda_1^{r,t}(J):= \min_{\begin{subarray}{c}u\in W^{1,r}_0(D) u\neq 0\end{subarray}}\bigg\{\frac{\int_{J}p(x)u'(x)^r dx+ \int_J q(x) u(x)^r dx}{(\int_{J}w(x)u(x)^t dx)^{{r}/{t}}}\bigg\}.
\quad\quad\quad\int_{0}^{T}a(u,u^{-})dt = \int_{0}^{T}a(u^{+},u^{-})dt - \int_{0}^{T}a(u^{-},u^{-})dt, &\int_{0}^{T}a(u^{-},u^{-})dt = \frac{c_{N,\beta}}{2}\int_{0}^{T}\int_{\mathbb{R}^{N}}\int_{\mathbb{R}^{N}}\frac{(u^{-}(x,t) - u^{-}(y,t))^{2}}{|x-y|^{N+2\beta}}dxdydt > 0,
r_{J^-}[I(f)](g) =r_{J^-}[g^{-1}I(f)](\begin{bmatrix}1 0 0 1\end{bmatrix}) &=g^{-1}r_{J^-}[I(f)](\begin{bmatrix}1 0 0 1\end{bmatrix}) &=2\tfrac{\partial}{\partial \overline{z}}f(gi).
P_0^i \,=\,{\Pi_0^i \over \epsilon}\,\,\,,\,\,\,P_1^i \,=\, \Pi_1^i \,\,\,\,,...\,\,\,,P_n^i \,=\, \Pi_n^i \,\,\,\,,\,...\,\,,P_{N-1}^i \,=\,\Pi_{N-1}^i \,\,,\,\,P_N^i \,=\,{\Pi_N^i \over \epsilon}\,\,.
ck\cdot f(|y_n|) \leq cf(|x_n|)f(|y_n|)\leq |f(B(x_n,y_n))|=|f((Tx_n)(y_n))| &=|(y_n+N_f,(Tx_n)_f)_f|\qquad(\mbox{by~}\eqref{Pas1})\,.
\gamma^{\prime\prime}=\gamma^{\prime}+\tau(P_{-\Psi,\tau,}Id)_{\#}(m_{r}-h)1_{A_{r}}-\tau\varepsilon(\gamma^{\prime})_{\overline{\Omega}}^{A_{R}^{\kappa}}+\tau\varepsilon(Id,P_{\Psi,\tau,})_{\#}(\gamma^{\prime})_{\Omega}^{A_{R}^{\kappa}}.
\mu_\infty(E_k) \ge\frac{c}{n_{k+1}^3}\sum_{t=0}^{n_{k+1}^3-1}\biggl( 1-\frac{c}{n^3_{k+1}}\biggr)^t = 1 - \biggl(1-\frac{c}{n^3_{k+1}}\biggr)^{n^3_{k+1}}\ge 1-2e^{-c},
E_{j}\triangleright (E_{\xi_{i}}^{*}E_{\xi_{i}}) = -(K_{j}F_{j}K_{j}^{-1}\triangleright E_{\xi_{i}})^{*}E_{\xi_{i}}+ (K_{j}^{-1}\triangleright E_{\xi_{i}})^{*}(E_{j}\triangleright E_{\xi_{i}}) & = -q^{-2}(F_{j}\triangleright E_{\xi_{i}})^{*}E_{\xi_{i}}+ (K_{j}^{-1}\triangleright E_{\xi_{i}})^{*}(E_{j}\triangleright E_{\xi_{i}}).
{\cal S}^{(n)}_{\eta',x_1,\dots,x_m;q,t}(0,v+(d-a)(\delta+(n-1)f)) \phantom{{\cal S}^{(n)}_{\eta',x_1,\dots,x_m;q,t}}{}={\cal S}^{(n)}_{\eta'+(n-1)t,x_1,\dots,x_m;q,0}(0,v+(d-a)(\delta+(n-1)f))
\tilde{\phi}_{M/L}^i(x) =e^{-1}\tilde{\phi}_{M_e/L_e}^i(ex) \tilde{\phi}_{L/K}^{h,m}(x) =e^{-1}\tilde{\phi}_{L_e/K_e}^{h,m}(ex) \phi_{M/K}^l(x) =e^{-1}\phi_{M_e/K_e}^l(ex) \lambda_{M/K}^l(x) =e^{-1}\lambda_{M_e/K_e}^l(ex).
\begin{dcases}\pi_x(\{y\}) + \pi_x(\{m\}) + \pi_x(\{z\}) = 1 y\,\pi_x(\{y\}) + m\,\pi_x(\{m\}) + z\,\pi_x(\{z\}) = x \end{dcases}\quad\implies\begin{dcases}\pi_x(\{y\}) = \frac{z-x-(z-m)\pi_{x}(\{m\})}{z-y}, \pi_x(\{z\}) = \frac{x-y-(m-y)\pi_{x}(\{m\})}{z-y}.\end{dcases}
S_1 + S_2 + S_3 = q^{D+2}+ (r_n+r_k)q^{D-3}\left( \frac12q^3 -\frac32 q^2 + r_n q - (r_n-1)\right) &= q^{D+2}+ (r_n+r_k)q^{D-3}(q-1)(n-(q+1)).
\begin{array}{r c l}{{Q}}&{{=}}&{{\int_{0}^{L}d x \, \psi^{\dagger}\psi = - \lambda ( p_{+}- p_{-}) \ ,}}\\{{Q_{5}}}&{{=}}&{{\int_{0}^{L}d x \psi^{\dagger}\gamma_{5}\psi = - \frac{i \lambda \sqrt{2 \mu}}{2 \sqrt{\pi}\sqrt{L}}\, \left[ \varphi_{0}- \varphi_{0}^{\dagger}\right] \ .}}\\ \end{array}
\|u_0\|_{\dot{H}^{s_1}}^2+\|u_0\|_{\dot{H}^{s_2}}^2 < \|\phi\|_{\dot{H}^{s_1}}^2(\frac{M(\phi)}{M(u_0)})^{\frac{s_1-s_c}{s_c}}< \frac{N(p-2)}{2s_1p-Np+2N}M(\phi)^{\frac{s_1}{s_c}}M(u_0)^{\frac{s_1-s_c}{s_c}}=x_m.
f(x) := \left( \begin{array}{l}a_1 x_1 + b_1 \theta(x_1,x_2) + c_1 \theta(x_1,x_4) + d_1 \theta(x_2,x_3) a_2 x_2 + b_2 \theta(x_1,x_2) + c_2 \theta(x_1,x_4) + d_2 \theta(x_2,x_3) a_3 x_3 + b_3 \theta(x_3,x_4) + c_3 \theta(x_1,x_4) + d_3 \theta(x_2,x_3) a_4 x_4 + b_4 \theta(x_3,x_4) + c_4 \theta(x_1,x_4) + d_4 \theta(x_2,x_3) \end{array}\right)
\begin{array}{l}{{\Delta ( H_{i}) = H_{i}\otimes 1 + 1 \otimes H_{i}~ ,}}\\{{\Delta ( X_{i}^{-}) = X_{i}^{-}\otimes q_{i}^{- H_{i}}+ q_{i}^{H_{i}}\otimes X_{i}^{-}~ ,}}\\{{\epsilon ( X_{i}^{-}) = 0 = \epsilon ( H_{i}) ~ ,}}\\{{S ( X_{i}^{-}) = - q_{i}X_{i}^{-}~ , ~ S ( H_{i}) = - H_{i}~ ,}}\\ \end{array}
\begin{array}{l}{{R^{( 1 )}X_{1}Z_{2}= Z_{2}X_{1}\quad ,}}\\{{\,}}\\{{Z_{1}^{\dagger}R^{( 2 )}Z_{2}= Z_{2}Z_{1}^{\dagger}\quad ,}}\\{{\,}}\\{{X_{2}^{\dagger}R^{( 3 )}X_{1}= X_{1}X_{2}^{\dagger}\quad ,}}\\{{\,}}\\{{X_{1}^{\dagger}Z_{2}^{\dagger}= Z_{2}^{\dagger}X_{1}^{\dagger}R^{( 4 )}\quad ,}}\\ \end{array}
-k(d-k)+\sum\limits_{t=1}^k t(\ell_{t+1}-\ell_t-1) = -k(d-k) - \sum_{t=1}^k \ell_t +k(d+1) -\frac{k(k+1)}{2}&= \frac{k(k+1)}{2}-\sum_{t=1}^k \ell_t = \sum_{t=1}^k t -\sum_{t=1}^k \ell_t = -\sum_{t=1}^k (\ell_t-t).
\lim_{\epsilon\rightarrow0^{+}}\lim_{n\rightarrow\infty}\frac{1}{n}N_{n}(\mathrm{e}^{-(y_{2}+\epsilon)},\mathrm{e}^{-(y_{1}-\epsilon)})=\nu^{+}(E,\mathrm{i}y_{2})-\nu^{-}(E,\mathrm{i}y_{1}).
\widetilde{\mathfrak{V}}= (A_\zeta, \widetilde C(x,t), \mathbb X(x,t), \widetilde B(x,t), A; \widetilde \sigma_1, \widetilde \sigma_2, \widetilde\gamma, \widetilde\gamma_*(x,t);\mathcal K,\mathcal E,\Omega),
\left[\begin{array}{cc}A B B^{\top}C\end{array}\right]^{-1}=\left[\begin{array}{cc}A^{-1}+A^{-1}B\left(C-B^{\top}A^{-1}B\right)^{-1}B^{\top}A^{-1}-A^{-1}B\left(C-B^{\top}A^{-1}B\right)^{-1}-\left(C-B^{\top}A^{-1}B\right)^{-1}B^{\top}A^{-1}\left(C-B^{\top}A^{-1}B\right)^{-1}\end{array}\right].
\mathcal{W}^{\mathrm{U ( 1 )}}= \partial_{\mu}\frac \delta{\delta A_{\mu}}+ \varepsilon^{a b}\left( A_{\mu}^{a}\frac \delta{\delta A_{\mu}^{b}}+ c^{a}\frac \delta{\delta c^{b}}+ b^{a}\frac \delta{\delta b^{b}}+ \overline{{{c}}}^{a}\frac \delta{\delta \overline{{{c}}}^{b}}+ A_{\mu}^{a *}\frac \delta{\delta A_{\mu}^{b *}}+ c^{a *}\frac \delta{\delta c^{b *}}\right) \; .
\left( -\frac{d^{2}}{d\theta^{2}}+ \frac{c_{m,n_1,n_2}^{2}-1/4}{\sin^{2}\theta}+ \frac{d^2-1/4}{\cos^2 \theta}-D_{j,m,n_1,n_2}\right) \Theta_{j,m,n_1,n_2}(\theta )= 0,
I_{\nabla v}= \alpha_{\ell m}\Re(\partial^{b}_{m}v\ \overline{\partial^{b}_{\ell}v}) +a(\nabla^{b}v,\nabla^{b}v)\phi, \alpha_{\ell m}:= 2a_{jm}\partial_{j}(a_{\ell k}\partial_{k}\psi) -a_{jk}\partial_{k}\psi \partial_{j}a_{\ell m}
\psi^l(\delta(\bigwedge\nolimits^k\sigma_{2n})) =(-1)^kl \left(\sum_{p=1}^{n-1}(-1)^p\alpha(2n, l, k, p)\delta(\bigwedge\nolimits^p\sigma_{2n})\right. &+(-1)^n\mu(2n, l, k, n)\delta(\bigwedge\nolimits^n\sigma_{2n})\Bigg).
\chi_\lambda(T_{i_1}^{j_1}\cdots T_{i_m}^{j_m}\otimes T_{k_1}^{\ell_1}\cdots T_{k_n}^{\ell_n}) =\lambda mn\varepsilon(T_{i_1}^{j_1}\cdots T_{i_m}^{j_m}T_{k_1}^{\ell_1}\cdots T_{k_n}^{\ell_n})
\Psi ( a , \psi , \theta ) \, = \, \psi - \frac{1}{2}\, \theta^{\alpha \beta}\, a_{\alpha}\mathrm{P}_{-}\partial_{\beta}\psi \, + \, \frac{i}{8}\, \theta^{\alpha \beta}\, [ a_{\alpha}, a_{\beta}] \, \mathrm{P}_{-}\psi \, + \, o ( \theta^{2}) .
\mathbb{P}(E_k) \ge \left( 1 - 2 (n - k + 1) \exp \left( - \frac{(\epsilon^2 - \epsilon^3) p}{4}\right) \right) \sum_{\left(\Pi, \Delta\right)}\mathbb{P}\left(\Pi, \Delta\right) & = 1- 2 (n - k + 1) \exp \left( - \frac{(\epsilon^2 - \epsilon^3) p}{4}\right).
w ( b , \bar{b}) = C \oint \prod_{j = 1}^{N + 1}{\frac{d z_{j}}{z_{j}- z_{j + 1}}}\delta \left( b^{\mu}( \tau ) - \sum_{j = 1}^{N}k_{j}^{\mu}\delta ( \tau - \tau_{j}) \right) \delta \left( \bar{b}^{\mu}( \tau ) - \sum_{j = 1}^{N}k_{j}^{\mu}\delta ( \tau - \tau_{j}) \right)
\begin{aligned}\bigg|\int_\Omega w_\varepsilon\Phi dx\bigg|\leq \bigg|\int_\Omega \tilde{f}\cdot\nabla\phi_\varepsilon dx\bigg|+ \bigg|\int_\Omega (\theta_\varepsilon-\sigma)\eta dx\bigg| =: I_1 + I_2.\end{aligned}
\alpha_{k,0}Y^n +(\alpha_{k,0}+\alpha_{k,1})Y^{n+1}+(\alpha_{k,0}+\alpha_{k,1}+\alpha_{k,2})Y^{n+2}&+\displaystyle\sum_{j=3}^{k}\left(\alpha_{k,j-3}+\alpha_{k,j-2}+\alpha_{k,j-1}+\alpha_{k,j}\right)Y^{n+j}+(\alpha_{k,k-2}+\alpha_{k,k-1}+\alpha_{k,k})Y^{n+k+1}&+(\alpha_{k,k-1}+\alpha_{k,k})Y^{n+k+2}+\alpha_{k,k}Y^{n+k+3}=f(t_n, Y^n)
\left . \exp \left \{- \frac{i}{\hbar}\sum_{j=1}^{N}\epsilon \, H_\Delta\left ( \frac{\hbar}{i \epsilon}\frac{\partial}{\partial K_j}\, ,\frac{\hbar}{i \epsilon}\frac{\partial}{\partial J_j}\right ) \right \}W_{fi}[ K , J ] \right |_{K,J = 0}\, .
D_{\alpha}{F}^{\alpha \beta ,a}+\frac{\kappa}{2}\epsilon^{\beta\alpha \lambda}{F}_{\alpha \lambda}^a = -i\bar\psi\gamma^{\beta}{T}^a\psi -i\frac{u}{2}\epsilon^{\beta \alpha \lambda}D_{\alpha}(\bar\psi\gamma_{\lambda}{T}^a \psi).
\begin{aligned}w(x_0,x^*) f[x_0,\cdots, x_n,x^*]=\frac{f^{(n+1)}(\xi_1)}{(n+1)!}h^{n+2}\int_{0}^{\frac{x^*-x_0}{h}}\mu(\mu-1)\cdots(\mu-n+1)(\mu-n)\d\mu,\end{aligned}
A_{\mu}= \pm \, [ \,{\frac{i}{g}}\, ( \, H_{2}- H_{1}) \,{\frac{v_{\mu}^{1}}{\rho_{1}}}+ \kappa \, E_{1 \, 2}^{\pm}\, R_{\mu}^{1}] \pm \, [ \,{\frac{i}{g}}\, ( H_{4}- H_{3}) \,{\frac{v_{\mu}^{3}}{\rho_{3}}}+ \kappa \, E_{3 \, 4}^{\pm}\, R_{\mu}^{3}\, ] .
P_{N \beta}( \theta_{1}, \ldots , \theta_{N}) \, d \theta_{1}\cdots d \theta_{N}= C_{N \beta}\prod_{1 \leq \ell < j \leq N}\left\vert \exp ( i \theta_{\ell}) - \exp ( i \theta_{j}) \right\vert^{\beta}\, d \theta_{1}\cdots d \theta_{N}
T^{\,m}e_{b_{2^{k-1}+l+1}-m+j-b_{l}}=\Biggl( v^{(k)}\!\!\!\!\!\;\;\;\;\prod_{i=\Delta^{(k)}-m+j-b_{l}+1}^{\Delta^{(k)}-1}w^{(k)}_{i}\Biggr)\ \Bigl(\,\,\prod_{i=1}^{j-b_l}w_{b_l+i}\Bigr) \,e_{j}&- \Biggl(\!\!\!\!\!\;\;\;\;\prod_{i=j-b_l+1}^{\Delta^{(k)}-m+j-b_{l}}w^{(k)}_{i}\Biggr)^{-1}\, e_{b_{2^{k-1}+l}+j-b_{l}}
\tilde{p}_{k,\varphi}(\omega, \lambda) = a_{nn}^{(k)}\prod_{k'=1,2}\Big( (-1)^k \lambda + \sum_{j=1}^{n-1}a_{nj}^{(k)}/ a_{nn}^{(k)}(x_0)\xi_j + i \tau \gamma_k + (-1)^{k'}( B_k + i A_k) \Big).
R_1-\delta_1 >d_{\alpha_1}(Q_{U,W},Q_U,Q_W)\geq d_1(Q_{U,W},Q_U,Q_W)= I(U;W), R_1+R_2-\delta_2 >d_{\alpha_2}(Q_{U,V,W},Q_{U,V},Q_W)\geq d_1(Q_{U,V,W},Q_{U,V},Q_W)= I(U,V;W).
\textbf{I}= C\int_{\mathbb{R}^n\backslash B_\delta(0)}|f_\lambda|^{2n/(n+\alpha)}dx = C\int_\delta^{+\infty}\left(\frac\lambda{\lambda^2+r^2}\right)^n r^{n-1}dr = C\int_{\frac\delta\lambda}^{+\infty}(1+t^2)^{-n}t^{n-1}dt=O(\frac\delta\lambda)^{-n},\quad\lambda\rightarrow 0.
\int_Mt^2d\mu =2\pi\int_{-a}^at^2A(t)dt &=\frac{\omega(F)^4}{768\pi^2}\left[8\frac{\omega(c_+)+\omega(c_-)}{\omega(F)}+\sum_j\frac{1}{r_js_j}\left(\frac{\omega(r_jE_j')-\omega(-s_jE_j)}{\omega(F)}\right)^4+c_1(Y)[\Sigma_a]-c_1(Y)[\Sigma_{-a}]\right].
\int_{-m/2}^{m/2}{s\,g_+^2\,ds}=\int_{-m/2}^{m/2}{s\left(s^2-\frac{m^2}{4}\right)^{k-1}e^{2rs}\,ds}\approx e^{mr}\int_{-\infty}^0 e^{2ru}u^{k-1}(m+u)^{k-1}\left({m\over 2}+ u\right) \, du \, .
\frac{1}{2}\int_{\Omega\,\cap\,B_{H_0}(x,R_2)}z(y,t)^2\zeta_x(y,t)^2\,dy+\iint_{Q_2}z^2[(\partial_t h ) \zeta_x^2-\zeta_x\partial_t\zeta_x]\,dy\,ds +\frac{1}{6}\iint_{Q_2}H(\nabla (z\zeta_x))^2\,dy\,ds\le C\iint_{Q_2}[H(\nabla h)^2\zeta_x^2+H(\nabla\zeta_x)^2]z^2\,dy\,ds.
P_{m}^{(-\alpha-1,\beta-1)}(1) = \frac{\Gamma(-\alpha + m)}{m!\,\Gamma(\beta+ m-\alpha-1)}\frac{\Gamma(\beta+ m-\alpha-1)}{\Gamma(-\alpha)}= \frac{\Gamma(-\alpha + m)}{m!\,\Gamma(-\alpha)}= \frac{m!\,\Gamma(-\alpha)}{m!\,\Gamma(-\alpha)}= 1.
\int_{-L}^{-L+2}\int_{-L}^L \max(|t-s|,2)^p \, ds\, dt = \int_{L-2}^{L}\int_{-L}^L \max(|t-s|,2)^p \, ds\, dt = - \,{\frac{2^{p+1}\bigl( 2\,{L}^{2}(L-1)^{p}-3\,{p}^{2}-4\,L (L-1)^{p}-{2}\,{L}^{p+2}-7\,p +2\,( L-1)^{p}-2 \bigr)}{{p}^{2}+3\,p+2}}.
(\Gamma\otimes_A I) \circ (\Gamma\otimes_A \tilde{f})\circ \Psi_M = (\Gamma_A f) \circ \Psi_M = \Psi \circ f = \Psi \circ I \circ \tilde{f}= (\Gamma\otimes_A I) \circ \Psi_{L_i}\circ \tilde{f}.
I=\frac{1}{2}\int_X \varepsilon^{\mu\nu\rho\sigma}\left(i\Sigma_{\mu\nu}^{\ \ab}R_{\rho \sigma ab}+\frac{1}{2}\varepsilon_{abcd}\Sigma_{\mu\nu}^{\ \ab}R_{\rho \sigma}^{\ \cd}\right)d^4x.
z^{c \odot \vec x}_j = \begin{cases}(M(c\odot \vec x)_{i}-\max_k(M(c\odot \vec x)_k)=M(\vec x)_{i}-\max_k(M\vec x)_k j \in J_i 0 \qquad j \notin \cup_i J_i, \end{cases}
\tilde{b}_{2}^{(0)}=0,\ \tilde{b}_{2}^{(0)}= 1,\ \tilde{b}_{2}^{(0)}= \lambda t_{2}+ 1,\ \tilde{b}_{2}^{(0)}= \beta t_{2},\ \tilde{b}_{2}^{(0)}= t_{2}^{2},
\epsilon^{i j}\nabla_{i}^{( p , k )}{\cal A}_{( p , k ) j}= - \sum_{( p^{\prime}, k^{\prime}) \neq ( p , k )}\beta_{p p^{\prime}}\delta^{2}({\bf r}_{k}^{( p )}-{\bf r}_{k^{\prime}}^{( p^{\prime})}) \, .
Pdf(E_i)=\Bigl(p\Bigl(\cos(2\theta_i)\alpha_i+\frac{1}{\sqrt{3}}\sin(2\theta_i)(2\beta_i-\alpha_i)\Bigr), q\Bigl(\cos(2\theta_i)\beta_i+\frac{1}{\sqrt{3}}\sin(2\theta_i)(-2\alpha_i+\beta_i)\Bigr)\Bigr).
2^{1-\alpha}-H^{1-\alpha}\leq e^{-C_1^{(1-\alpha)}\cdot t}\left(\max_{M_0}|2^{1-\alpha}-H^{1-\alpha}|+(1-\alpha)(4+\max_{M_0}|H^2-4|)C_1^{-\alpha-1}C_3\max_{M_0}\gamma_0\right) =: C_4e^{-C_1^{(1-\alpha)}\cdot t},
F(H_r) =\sum_{h=1}^t (|H_r \cap V_h|(Q-1)-(q^r-|H_r \cap V_h|)) =\sum_{h=1}^tq^{v_h}((Q-1)-(q^r-q^{v_h})) =Q\sum_{h=1}^tq^{v_h}-tq^r.
b^\dagger_N =1-2\alpha_N^{\dagger}, a^\dagger_N =-\frac{C_N(q)}{\sqrt{N}}\bigg|_{{q}=e^{-2\pi/\sqrt{3N}}}, \intertext{and}X^\dagger_N =4\alpha_N^{\dagger}\left(1-\alpha_N^{\dagger}\right).
\int_{1 0}\hat{F}_{( 5 )}{}^{\star}\hat{F}_{( 5 )}\; = \; \int_{9}\left[ K^{- \frac{3}{4}}F_{( 5 )}{}^{\star}F_{( 5 )}- K^{\frac{3}{4}}F_{( 4 )}{}^{\star}F_{( 4 )}\right] \; ,
\bar{\mathbf{X}}=(\bar{\mathbf{x}}),\bar{\mathbf{S}}=(\bar{\mathbf{s}}), \bar{\mathbf{x}}=\mathbf{D}^{-T}\mathbf{x},\bar{\mathbf{s}}=\mathbf{D}\mathbf{s}, \mathbf{D}\in \mathcal{G}.
\frac{l_{j}^{y}\mu_{1}^{y}}{l_{1}^{y}M_{j}^{y}}\le \varphi_{m+j}\le \mathrm{\thinspace}\frac{l_{j}^{y}M_{1}^{y}}{l_{1}^{y}\mu_{j}^{y}}, j=1,\thinspace \mathellipsis ,\thinspace n,
lim_{N\rightarrow+\infty}\overline{w_{Z}}-\overline{w_{X}}=\frac{1}{6}lim_{N\rightarrow\infty}(\frac{2}{3})^{N}(2^{N}-2^{N-1})=\frac{1}{9}lim_{N\rightarrow+\infty}(\frac{4}{3})^{N-1}\rightarrow+\infty
\psi^\prime_1(u_1,u_2,u_3,w_1,w_2) = (1+\iota(N)^{-1})u_1 + u_2 + w_1 \psi^\prime_2(u_1,u_2,u_3,w_1,w_2) = u_1 + u_2 + w_2 \psi^\prime_3(u_1,u_2,u_3,w_1,w_2) = u_3.
k_1\mathcal{R}_1^{\frac{1}{2}}= \int_{0}^{k_1}\int_{t}^{k_1}Av'(\zeta)d\zeta dt = \int_{0}^{k_1}\int_{0}^{\zeta}Av'(\zeta)dt d\zeta = \int_{0}^{k_1}\zeta Av'(\zeta) d\zeta.
Z[A_{+},\rho,\tilde{\rho}] = \int \! D \bar{\psi}D \psi D B_{+}^{\prime}D \theta e^{-\int \! d^2 x\, [ L_{\rm{CF}}+ L_{\rm{B}}+ \rho e^{-i \theta}\psi^{\dag}_{-}\psi_{+}+ \tilde{\rho}e^{i \theta}\psi^{\dag}_{+}\psi_{-}]}\quad.
{\displaystyle |\sum_{k=1}^{M}{J_3^{(k)}}| \leq C(h^{2r}+(\Delta t)^{4}) +C \Vert\theta_{\Psi}^{M}\Vert_{\mathcal{L}^2}^{2}+ \frac{1}{16}\Vert\nabla\theta_{\Psi}^{M}\Vert_{\mathbf{L}^2}^{2}+C\Delta t \sum_{k=1}^{M-1}{\Vert\nabla\theta_{\Psi}^{k}\Vert_{\mathbf{L}^2}^{2}}.}
I_{3,k}:=\frac{1}{2\pi}\int_{\left|y\right|\geq 2R}\left|\log\left(\frac{\left|\overline{x}-y\right|}{\left|x-y\right|}\right)\right|f_k(y)dy\leq\frac{\log 4}{2\pi}\int_{\left|x\right|\geq 2R}f_k\leq C.
\int_{x-b}^{x-y}dz \int_{0}^{x-y-z}dt_1\int_{-z}^{0}\hat{w}(\cdot)dt_2 = \int_{0}^{b-y}dt_1\int_{b-x}^{0}d t_2 \int_{x-b}^{x-y-t_1}\hat{w}(\cdot)dz &+ \int_{0}^{b-y}dt_1\int_{t_1+y-x}^{b-x}d t_2 \int_{-t_2}^{x-y-t_1}\hat{w}(\cdot)dz,
[\mathfrak{s}(t),\mathfrak{s}(t')]=\mathfrak{s}(t)\mathfrak{s}(t')\mathfrak{s}(t)^{-1}\mathfrak{s}(t')^{-1}=\prod_{i<j}(t_i,t'_j)_2(t'_i,t_j)_2.
\log \Gamma ( z ) = ( z -{\frac{1}{2}}) \mathrm{l o g}z - z +{\frac{1}{2}}\mathrm{l o g \}2 \pi + 2 \int_{0}^{\infty}{\frac{\mathrm{t a n}({\frac{t}{z}})}{e^{2 \pi t}- 1}}d t ,
\alpha = \Omega + \sum_{s = 1}^{p}F^{\alpha_{1}}\wedge \dots \wedge F^{\alpha_{s}}v_{\alpha_{1}\dots \alpha_{s}}\equiv \Omega + \sum_{s = 1}^{p}v^{( s , n - 2 s )}\quad , \quad \alpha ( F = 0 ) = \Omega \quad ,
x_i = \begin{cases}0, 0 \leq i \leq n/2 1, n/2 < i \leq n \end{cases}\intertext{for the null distribution, and the shifted balanced partition $y = [y_i]_{i=1}^n$ with}y_i = \begin{cases}0, s/2 < i \leq n/2+s/2 1, i \in (n/2, n] \cup [0, s/2] \end{cases}
\kappa^{2}= - \frac{1}{4}\lim_{R \rightarrow R_{0}}\Big [ g^{t t}g^{R R}( \partial_{R}g_{t t})^{2}\Big ] = \frac{( d - 3 )^{2}}{4 R_{0}^{2}R_{T}^{2}\cosh^{2}\alpha A |_{R = R_{0}}}\ .
\tilde{\mathbf{h}}_k\left(n\right) = \sqrt{\frac{M}{N_{\mathrm{cl},k}N_{\mathrm{ray},k}}}\sum_{i=1}^{N_{\mathrm{cl},k}}\sum_{j=1}^{N_{\mathrm{ray},k}}\tilde{\alpha}_{ijk}\Delta\left(\phi_{ijk},\theta_{ijk}\right) & \times \mathbf{a}\left(\phi_{ijk},\theta_{ijk}\right)\delta\left(n-\tau_{ijk}\right),
4(s-1)^{2}{\large (}L(h){\large )}^{2}\left( x-h\right)^{2}+4(s-1)^{2}h^{2}{\large (}y-L(h){\large )}^{2}-4(s-1){\large (}2\left( t-1\right) h^{2}+\left( s-t+2\right) h-s{\large )}\left( x-h\right){\large (}y-L(h){\large )}=(2h-1){\large (}2(t-1)h+s{\large )}\left( s-2h\right) ,h\in I
\begin{pmatrix}S_1 -S_2 0 0 0 0 0 S_2 -1 0 0 0 S'_1 S'_2 0 -1 0 0 0 0 C_3 S_3 -S_4 0 0 0 0 0 S_4 -S_5 0 0 C'_3 S'_3 S'_4 S'_5 \end{pmatrix}\begin{pmatrix}M_1^1 M_2^1 M_3^0 M_3^1 M_4^1 M_5^1 \end{pmatrix}= 0.
\int_{D2}F = -k \int_{-\chi_0}^{\chi_0}d\tilde{\theta}-k\int_{-\chi_0 +\Delta\tilde{\theta}}^{\chi_0 -\Delta\tilde{\theta}}d\tilde{\theta}= -2k\left( 2\chi_0 -\Delta\tilde{\theta}\right) \equiv -2k\chi_0^\prime ,
B_{1}^{i n}= ( \int_{M}d v a_{1}+ \int_{S^{2}}d s c_{1}) V^{1 / 2 - s}= \int_{M}( \frac{\lambda}{2}\hat{\phi}^{2})^{3 / 2 - s}d v + \frac{2}{3 R}\int_{S^{2}}( \frac{\lambda}{2}\hat{\phi}^{2})^{1 / 2 - s}d s
II_{rr}= \frac{c_{b}}{4}\int_{0}^{\infty}\sin(t\xi) \xi^{2}\left(1-\chi_{\leq 1}(r\xi)\right) \left(-3 J_{1}(r\xi) + J_{3}(r\xi)\right) \frac{\chi_{\leq \frac{1}{4}}(\xi)}{\log^{b-1}(\frac{1}{\xi})}d\xi
\psi\Bigl(\sum_{i=1}^k v_i\Bigr)\frac{\psi(v_j/U_k)}{1+T_kv_j}=\Bigl(\psi\Bigr(\sum_{i=1}^k u_i\Bigr)+O\Bigl(\frac{\log{A}}{\log{R}}\Bigr)\Bigr)\Bigl(\psi\Bigl(\frac{u_j}{U_k}\Bigr)+O\Bigl(\frac{\log{A}}{U_k\log{R}}\Bigr)\Bigr)\frac{1+O\Bigl(T_k\frac{\log{A}}{\log{R}}\Bigr)}{1+T_ku_j}.
M_{K}= \left( \begin{array}{c c c c}{{f^{- 1}}}&{{0}}&{{- \chi f^{- 1}}}&{{0}}\\{{0}}&{{- 1}}&{{0}}&{{0}}\\{{- \chi f^{- 1}}}&{{0}}&{{f + \chi^{2}f^{- 1}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{- 1}}\\ \end{array}\right) ,
B_{d,m}\simeq_{\mathcal{X}_{n-2}}\bigoplus_{E'<E: |E'|=p^{n-1}}\frac{1}{p}\left[ \begin{pmatrix}r+s r \end{pmatrix}- \begin{pmatrix}r+s-p r \end{pmatrix}\right] B_{d',m'}(E') \uparrow_{E'}^E
\gamma_{M}= \frac{{P_M}\alpha_{M,k^*}\max_{j}|{\bf g}_{M,k^*}^H{\bf f}_{j}|^2}{P_F \alpha_{F,k^*}|{\bf g}_{F,k^*}^H{\bf w}_{i^*}|^2+\sigma^2_M},1\leq j \leq N_Q,
\int_0^1\int_{S_2}\Big[ \phi(s, u)-\sum_{n = 1}^{\infty}\phi_n(u)H_n(s)\Big]H_N(s)\psi(u)dsd\mu(u) = \int_{S_2}\phi_N(u)\psi(u)d\mu(u)-\sum_{n = 1}^{\infty}\int_{S_2}\phi_n(u)\psi(u)d\mu(u)\int_0^1H_n(s)H_N(s)ds = 0
\chi_{0}( q ) = 1 + q^{2}+ q^{3}+ q^{4}+ q^{5}+ 2 \, q^{6}+ 2 \, q^{7}+ 3 \, q^{8}+ 3 \, q^{9}+ 4 \, q^{1 0}+ 4 \, q^{1 1}+ 6 \, q^{1 2}+ O ( q^{1 3})
\sum_{\alpha=1}^\infty \bigg( \int_0^t \phi_\alpha^2(t-s)ds \bigg)|\varphi_\alpha(y)-\varphi_\alpha(z)|^2 &\le C \sum_{\alpha=1}^\infty \frac{|\varphi_\alpha(y)-\varphi_\alpha(z)|^2}{\lambda_\alpha}\le C |y-z|,