harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l}(1-\overline{\eta}_p\zeta_s)r_{s,p}\sqrt{|\alpha'(\eta_p)|}\sqrt{|\beta'(\zeta_s)|}\phantom{(1-\overline{\eta}_p\zeta_s)r_{s,p}}=\psi(\zeta_s)\overline{k_{0}^{\alpha}(\eta_p)}+k_{0}^{\beta}(\zeta_s)\overline{\chi(\eta_p)}\end{array},\begin{array}{l}1\leq p\leq m \phantom{1}1\leq s\leq n\end{array}.
\varepsilon^{i}\rightarrow{\varepsilon'^{i}}= \varepsilon^{i}+ \delta \varepsilon^{i}={{W_5 (p, q, r)}^{i}}_{j}\varepsilon^{j}= \varepsilon^{i}+( p\varepsilon^1 + q\varepsilon^2 + r\varepsilon^3)\frac{k^i}{\omega}
\mathbb{E}\left \{\gamma_{\left| \theta \right.}^2 \right \}^{\kappa\mu s}=\frac{\overline{\gamma}_\theta^2 \left( m(1+\mu)(1+2 \kappa_\theta)+\mu \kappa_\theta^2(1+m) \right)}{m \mu (1+\kappa_\theta)^2}
\begin{array}{l c l}{\phi_{a}( b )}{=}{\int a^{- n / 2}\bar{\psi}\left( \frac{x - b}{a}\right) \phi ( x ) d^{n}x ,}{\phi ( x )}{=}{C_{\psi}^{- 1}\int a^{- n / 2}\psi \left( \frac{x - b}{a}\right) \phi_{a}( b ) \frac{d a d b}{a^{n + 1}},}\end{array}
\frac{\partial}{\partial t}u_{ij}(t;x_1,\ldots,x_k;y_1,\ldots,y_k) &= \sum_{i'=1}^k\sum_{j'=1}^k\frac{\pi_{i'j'}x_{i'}y_{j'}}{n}\left[u_{ij}(t;x_1-\delta_{1i'},\ldots,x_k-\delta_{ki'};y_1-\delta_{1j'},\ldots,y_k-\delta_{kj'}) + \delta_{ii'}\delta_{jj'}\right] &\quad- \frac{z}{n}u_{ij}(t;x_1,\ldots,x_k;y_1,\ldots,y_k).
(a',b',c') = 2(a,b,c)[r] = \frac{1}{2}(a,b,c)\left( \begin{matrix}1 1 \lambda 1 1 \lambda \overline{\lambda}\overline{\lambda}2 \end{matrix}\right) = \frac{1}{2}\left(a + b + c\overline{\lambda}, a + b + c\overline{\lambda}, a\lambda + b \lambda + 2c \right).
\biggl(\iint\limits_{\Omega'}\left(J_{\varphi^{-1}}(u,v)\right)^{\frac{r}{r-s}}~dudv\biggl)^{\frac{r-s}{rs}}=K<\infty, \,\,\,1\leq s<r<\infty, \sup\limits_{(u,v)\in\Omega'}\left(J_{\varphi^{-1}}(u,v)\right)^{\frac{1}{s}}=K<\infty, \,\,\,1\leq s=r<\infty.
M_i\leq \sum_{T}\binom{C(T)}{d_i}\leq\sum_T\binom{5n/d_i}{d_i}\leq \sum_T 2^{\frac{2}{2^{1/\ln 2}\ln 2}\sqrt{5en}}\leq \sum_T 2^{4\sqrt{n}}\leq 2^{6\sqrt{n}}
0 = [\partial_a{\omega^d}_{bc}- \partial_b{\omega^d}_{ac}+{\omega^d}_{ae}{\omega^e}_{bc}-{\omega^d}_{be}{\omega^e}_{ac}] \partial_d D(x||m_\zeta) &+[\partial_a g_{bc}- \partial_b g_{ac}+ g_{ad}{\omega^d}_{bc}- g_{bd}{\omega^d}_{ac}].
-\Delta F_{N,s}(x) =-\kappa_{N,s}(\Delta|x|^{2s-N}\ln|x|+2\nabla|x|^{2s-N}\nabla\ln|x|+|x|^{2s-N}\Delta\ln|x|) =\kappa_{N,s-1}|x|^{2s-N-2}\ln|x|+C_2|x|^{2s-N-2}=F_{N,s-1}+C_2|x|^{2s-N-2},
2n^2\exp\Bigl[-\Omega\Bigl( n^{2s-\frac 12}c^{\frac{2s-1}{1-s}}\Bigr)\Bigr]+2n^2\exp\Bigl[-\Omega\Bigl(n^{2s}c^{\frac{2s}{1-s}}\Bigr)\Bigr] &=2n^2\exp\Bigl[-\Omega\Bigl( n^{s}\Bigr)\Bigr]+2n^2\exp\Bigl[-\Omega\Bigl(\log^{\frac{2\alpha s}{1-s}}n\Bigr)\Bigr] &< \frac{1}{n^\beta},
N_\pm (X; E_p, \pm \hat{\bf p}) = \int \frac{d^{\, 3}q}{2 |p_0|}\, e^{- i{\bf p}\cdot{\bf q}X_0 / p_0 + i{\bf q}\cdot{\bf X}}F_\pm (\pm{\bf p}-{\bf q}/ 2; \pm{\bf p}+{\bf q}/ 2) .
\sum_{n}\left\{\frac{( - 1 )^{n}}{2 ( n + 1 ) !}( A d_{\Phi})^{n}\partial^{2}\Phi^{a}+ \frac{( - 1 )^{n + 1}}{2 ( n + 2 ) !}\sum_{k = 0}^{n - 1}( A d_{\Phi})^{k}A d_{\partial_{\mu}\Phi}( A d_{\Phi})^{n - k}\partial^{\nu}\Phi^{a}\right\}= 0 ,
8B_4(0)|0\rangle_4 =(0, -1, 1, 0, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0)^t, \frac{4}{1-i}B_4\!\left(\frac{1}{2}\right)|0\rangle_4 =(0, 1, i, 0, -1, 0, 0, 0, -i, 0, 0, 0, 0, 0, 0, 0)^t, \frac{4}{1+i}B_4\!\left(-\frac{1}{2}\right)|0\rangle_4 =(0, 1, -i, 0, -1, 0, 0, 0, i, 0, 0, 0, 0, 0, 0, 0)^t.
\begin{cases}du(t) + \nu \mathrm{A}u(t)\, dt + B(u(t))\, dt = \nu |\nabla u(t)|_{L^2}^2 u(t) \,dt + \sum_{j=1}^m C_j u(t) \circ dW_j(t), \; t \in [0,T], u(0) = u_0,\end{cases}
\hat{\chi}^{(p,p')}_{r,s}(q)=\hat{\chi}^{(p,p')}_{p-r,p'-s}(q)=\frac{1}{(q)_{\infty}}\sum_{j=-\infty}^{\infty}\left( q^{j(jpp'+rp'-sp)}- q^{(jp+r)(jp'+s)}\right).
\frac{1}{c_{n}-c_{n-1}}- \frac{1}{c_n}\ = \ \frac{c_{n-1}}{c_n^{2}-c_{n}c_{n-1}}&\ =\ \frac{c_{n-1}}{c_{n-1}(c_{n+1}-c_{n})+t^2(-r)^{n-1}}\ <\ \frac{1}{c_{n+1}-c_{n}}.
\log\Gamma(s) =\left(s-\frac12\right)\log s-s+O(1) &=\left(s-\frac12\right)\log(it)+\left(s-\frac12\right)\log\left(1+{\sigma\over it}\right)-s+O(1) &=\left(s-\frac12\right)\log(it)+it\log\left(1+{\sigma\over it}\right)-s+O(1) &=\left(s-\frac12\right)\log(it)-it+O(1)
W(\Delta,z)=\sum_{\Gamma,\Gamma^{\prime}\vdash d}\frac{d!}{|C_{\Gamma^{\prime}}|}z^{d+l(\Gamma^{\prime})-l(\Delta)-l(\Gamma)}\mu_{0}^{h_+,d}(\Gamma^{\prime},\Delta,\Gamma)p_{\Gamma}\frac{\partial}{\partial p_{\Gamma^{\prime}}}.
\iint_{0 \le s \le t \le \delta_n}| K_n(s,t)|^2 ds dt \le c' \int_0^{\delta_n}\int_0^t \frac{1}{t}e^{\kappa_n (\sqrt{t}- \sqrt{s})}ds dt = 2 c' \int_0^{\delta_n}\left( \frac{e^{\kappa_n \sqrt{t}}- 1}{\kappa_n^2 t}- \frac{1}{\kappa_n \sqrt{t}}\right) dt \le c' \, \frac{e^{\kappa_n \sqrt{{\delta_n}}}}{\kappa_n^2},
\beta F = - \frac{c}{\beta^2}\int d \phi \int_{r_H + h}^L dr \frac{\sqrt{g_3}}{( - g^{'}_{tt})^{3/2}}= - c \int_0^\beta d \tau \int d \phi \int_{r_H + h}^L dr \sqrt{g_3}\frac{1}{\beta_{local}^3},
\left|\int_0^\infty\rho^{-\frac{n-2}2}\rho^{\nu_0}e^{it\rho}\chi(\rho)\rho^{n-1}d\rho\right|\geq \frac12\int_0^\infty\rho^{-\frac{n-2}2}\rho^{\nu_0}\chi(\rho)\rho^{n-1}d\rho\geq c.
\begin{aligned}\partial_z^3 T = -4 - 8 J_{2,1}- 16 G_1 \partial_z^2 \partial_\tau T = -4 J_{2,2}- 8 G_2 \partial_z \partial_\tau^2 T = - \frac{8}{3}J_{2,3}- \frac{16}{3}G_3 \partial_\tau^3 T = -2 J_{2,4}- 4 G_4 + \frac{1}{20}E_4.\end{aligned}
I_{D_{Z,-},d}=\frac{\prod_{0<a<v_1\cdot d}(v_1+az)}{\prod_{i\in I_-, D_i\cdot d>0}(\bar D_i+(D_i\cdot d)z)}[\textbf{1}]_{(-D_i\cdot d)_{i\in I_-},v_1\cdot d}.
a_{c}=lim_{\Delta t\rightarrow0}\frac{|\Delta v|}{\Delta t}=\frac{v}{r}lim_{\Delta t\rightarrow0}\frac{|\Delta r|}{\Delta t}=\omega lim_{\Delta t\rightarrow0}\frac{|\Delta r|}{\Delta t}=v\omega=\frac{v^{2}}{r}
( D_{\bar{i}}Q_{i}) ({\overline{{Q}}}^{j}{\overline{{D}}}^{\bar{j}}) \epsilon_{j k}\epsilon_{{\bar{j}}{\bar{k}}}\left( C_{a_{1}a_{2}}\Sigma_{a_{1}}^{i{\bar{i}}}\Sigma_{a_{2}}^{k{\bar{k}}}+ C_{a_{1}a_{2}}^{\prime}\Sigma_{a_{1}}^{i{\bar{k}}}\Sigma_{a_{2}}^{k{\bar{i}}}\right) ~ ,
~N N_1^{5-2\alpha+ \varepsilon}N_1^{-\frac{3\alpha}{4}+\frac{1}{2}}N^{-\frac{1}{2}}\prod_{i=1}^3 \sum_{L_i \geqslant N_1^{(5-2\alpha)+}}L_i^{\frac{1}{2}}\| f_{i,L_i}\|_{L^2}&\lesssim N^{\frac{1}{2}}N_1^{\frac{11}{2}-\frac{11\alpha}{4}+ \varepsilon}\|u_{N_1}\|_{F_{N_1}(T)}\|u_{N_2}\|_{F_{N_2}(T)}\|u_N\|_{F_N(T)}.
\theta^{a}\rightarrow\theta_{1}^{a}=(L^{-1})_{b}^{a}\theta^{b}, \omega_{b}^{a}\rightarrow\omega_{1b}^{a}=(L^{-1})_{c}^{a}dL_{b}^{c}+(L^{-1})_{c}^{a}\omega_{d}^{c}L_{b}^{d}, \Omega_{b}^{a}\rightarrow\Omega_{1b}^{a}=(L^{-1})_{c}^{a}\Omega_{d}^{c}L_{b}^{d}.
\lim_{L_1,L_j\to0}\tfrac{1}{L_1}R(L_1,L_j,z) =2(e^{\frac{1}{2}(0+z)}+1)^{-1}, \lim_{L_1\to0}\tfrac{1}{L_1}D(L_1,y,z) =\tfrac{1}{L_1}\left(R(L_1,y,z)+R(L_1,z,y)-L_1\right) &=2(e^{\frac{1}{2}(y+z)}+1)^{-1}.
f(\epsilon_{\{\sigma'(1)\}}) = \epsilon_{\{\sigma''(1)\}}, & f(\epsilon_{\{\sigma'(1),\sigma'(2)\}}) = \epsilon_{\{\sigma''(1),\sigma''(2)\}}, &\dots & f(\epsilon_{\{\sigma'(1),\dots,\sigma'(n)\}}) = \epsilon_{\{\sigma''(1),\dots,\sigma''(n)\}}
t^{-1}(z_1) = \left(x_1,v_{i-1}+ s_1\left(\dfrac{n+2}{n+1}\right)^{i-1}\right), t^{-1}(z_2) = \left(x_2,v_{j-1}+ s_2\left(\dfrac{n+2}{n+1}\right)^{j-1}\right),
\hat \nabla\equiv \nabla (\hat \Gamma)\ , \ \ \ \hat \Gamma^i_{jk}= \Gamma^i_{jk}+{1\over 2}H^i_{\jk}\ , \ \ \ \{\cal F}_{ij}\equiv \partial_i{\cal A}_j -\partial_j{\cal A}_i\ ,\ \ \ \ \ \ \Phi = \phi +{1\over 2}\ln F\ ,
\begin{array}{ll}\mbox{minimize}\langle P_{\mathcal{S}_{0/1}}(C), X \rangle \mbox{subject to}\langle P_{\mathcal{S}_{0/1}}(A_i), X \rangle = b_i \;\;\; \forall i \in T \subseteq [m] & X \in \mathbb{S}_{+}^{n_{1}}\times \cdots \times \mathbb{S}_{+}^{n_{r}},\end{array}
I_{0}\left[ A_{\alpha}^{\; \; ( \lambda )}\right] = \int d^{3}x \varepsilon^{\alpha \beta \gamma}A_{\alpha ( \lambda )}\partial_{\beta}A_{\gamma}^{\; \; ( \lambda )}, \; \delta_{\epsilon}A_{\alpha}^{\; \; ( \lambda )}= \partial_{\alpha}\epsilon^{( \lambda )},
\frac{1}{\pi}\int_{a_{k}}^{b_{k}}\frac{1}{z-\tau}\,\frac{d\tau}{\sqrt{(b_{k}-\tau)(\tau-a_{k})}}=\frac{1}{\sqrt{(z-b_{k})(z-a_{k})}},z\in\mathbb{C}\setminus[a_{k},b_{k}].
u(\varphi(c_1c_2)) = s'(c_1c_2)s(c_1c_2)^{-1}&= s'(c_1)s'(c_2)s(c_2)^{-1}s(c_1)^{-1}&= s'(c_1)u(\varphi(c_2))s(c_1)^{-1}&= s'(c_1)s(c_1)^{-1}u(\varphi(c_2)) &= u(\varphi(c_1))u(\varphi(c_2)).
(ii^*) = (\mathcal{E}_k) + (\mathcal{E}_k^1) \leq \sum_{n=0}^{+\infty}\frac{1}{2^n}||h_{q_n}||_2^2 + \sum_{n=0}^{+\infty}\frac{1}{2^n}||h_{q_n}'||_2^2 \leq \sum_{n=0}^{+\infty}\frac{1}{2^n}+ \sum_{n=0}^{+\infty}\frac{1}{2^n}\times 2^2 = 10
\overline{F}_{S_\eta}(x) =\sum\limits_{n=1}^{\infty}\mathbb{P}(\eta =n)\mathbb{P}(S_n> x), \overline{F}_{\xi_{(\eta)}}(x) =\sum\limits_{n=1}^{\infty}\mathbb{P}(\eta=n)\mathbb{P}(\xi_{(n)}> x), \overline{F}_{S_{(\eta)}}(x) =\sum\limits_{n=1}^{\infty}\mathbb{P}(\eta =n)\mathbb{P}(S_{(n)}> x).
\xi = \textstyle\sum_{j,j' \in J}k_{jj'}= \textstyle\sum_{k,k' \in K}k_{kk'}\Xi = \textstyle\frac{1}{|J|(|J|-1)}\sum_{j,j' \in J}D_{k_{jj'}}\textstyle- \frac{2}{|J||K|}\sum_{j \in J, k \in K}D_{k_{jk}}+ \frac{1}{|K|(|K|-1)}\sum_{k,k' \in K}D_{k_{kk'}}
c_1= \max_{t\in [0,T]}|K(t)|,c_2= \max_{0\le t,s \le T}|\Phi (t,s)|, &c_3= \max_{t\in [0,T]}\int_t^T|e^{A (s-t)}|sds, c_4= \max_{t\in [0,T]}\int_t^T |e^{A(s-t)}| (Ts-\frac{s^2}{2}) ds.
d\circ\Psi^{p_ip_k}+ \Psi^{p_jp_k}\cdot\Psi^{p_i p_j}+ \Psi^{p_ip_jp_k}\circ(1\otimes\mu_{1}+\mu_{1}\otimes 1) + \Psi^{p_ip_k}\circ\mu_{2}=0,
S ={\frac{1}{2}}({\frac{1}{6}}+ | \xi | ) r_{+}^{2}\sum_{i = 0}^{5}\Delta_{i}M_{i}^{2}\ln M_{i}^{2}+{\frac{1}{4 5}}{\frac{1}{8}}r_{+}^{2}C \sum_{i = 0}^{5}\Delta_{i}\ln M_{i}^{2}.
{\rm d}\mu = \frac{1}{2^{5}}\,\frac{\Lambda^5_{\rm one-fl}}{\Phi^2(x_0,\theta_0)}\,\exp\left(-4\pi^2\,\bar\Phi \Phi \,\rho^2_{\rm inv}\right) \frac{{\rm d}\rho^2}{\rho^2}\,{\rm d}^4 x_0\,{\rm d}^2\theta_0 \,{\rm d}^2\bar\beta\,{\rm d}^2\bar\theta_0 \, .
\limsup_{|x_0|\to +\infty}\left(\limsup_{j \to +\infty}J_{0, j}\right) =\limsup_{|x_0|\to +\infty}\left(\limsup_{j \to +\infty}J_{0, j}\int_{A_{j,K_{0}, \sigma_{0}}}(v_{j}- \frac{\delta}{4})_{+}^{\gamma^*}dx \right)=0.
[ x^{\tilde{i}}, x^{\tilde{j}}] = i \theta^{\ \! \! \tilde{i}\tilde{j}}, \quad [ \partial_{\tilde{i}}, x^{\tilde{j}}] = i \delta_{\tilde{i}}^{\tilde{j}}, \quad [ \partial_{\tilde{i}}, \partial_{\tilde{j}}] = - i \Phi_{\tilde{i}\tilde{j}}^{\prime}\quad ( \tilde{i}, \tilde{j}= \tilde{1}, \tilde{2})
[s, x_1] [b, x_2, x_3] = [s, x_1] [y_1 \dots y_k, x_2, x_3] = \sum_{i=1}^k [s, x_1] y_1 \dots y_{i-1}[y_i, x_2, x_3] y_{i+1}\dots y_k + \ \sum_{1 \le i < i' \le k}[s, x_1] y_1 \dots y_{i-1}\bigl( [y_i, x_2] y_{i+1}\dots y_{i'-1}[y_{i'}, x_3] + [y_i, x_3] y_{i+1}\dots y_{i'-1}[y_{i'}, x_2] \bigr) y_{i'+1}\dots y_k .
\begin{array}{l}\langle \Phi(x)\Phi^{*}(y)\rangle\cr\simeq\int_{0}^{\infty}d\tau E[\langle\delta\left(q\left(\tau,x\right)-y\right)\rangle\exp\Big(-\frac{1}{2}\langle\int_{0}^{\tau}dq^{B}(s)dq_{C}(s^{\prime}){\cal D}_{B}^{C}\left(q\left(s\right)-q\left(s^{\prime}\right)\right)\rangle\Big)]\end{array}
H = N \left[ -\frac{1}{24}\epsilon_{ijk}(\hat \theta^i - \frac{1}{2}\theta^i)(\hat \theta^j - \frac{1}{2}\theta^j)(\hat \theta^k - \frac{1}{2}\theta^k) - \frac{1}{32}\epsilon_{ijk}\theta^i \theta^j (\hat \theta^k - \frac{1}{2}\theta^k) \right]\,.
P(T, x) = \dfrac{\displaystyle\prod_{j=1}^{l(T)}W_j^{n(T, l(T)+1-j)}}{W_{l(T)}^{n(T, 0)}\displaystyle\prod_{j=1}^{l(T)-1}W_j^{n(T, l(T)-j)}}= \dfrac{\displaystyle\prod_{j=1}^{l(T)}W_j^{n(T, l(T)+1-j)}}{\displaystyle\prod_{j=1}^{l(T)}W_j^{n(T, l(T)-j)}}= \prod_{j=1}^{l(T)}W_j^{n(T, l(T)+1-j) - n(T, l(T)-j)}.
\|R_N(f)\|_{\dot{\mathcal B}^{\alpha,q}_{p,\mathcal F}}\le \bigg\{\sum_{k}\bigg(\sum_{k'}\sum_{m=0}^\infty \sum_{|k_0-\ell_0|>N}\cdots \sum_{|k_m-\ell_m|>N}&\qquad2^{k\alpha}\big\|D_kD_{k_0}D_{\ell_0}D_{k_1}D_{\ell_1}\cdots D_{k_m}D_{\ell_m}D^N_{k'}\big\|_{L^p_\mu\mapsto L^p_\mu}\|D_{k'}(f)\|_{L^p_\mu}\bigg)^q\bigg\}^{1/q}.
\alpha^3 \tau_0 \tau_1 \tau_2 f(\tau^3, w \sigma \sigma_1 \sigma_2) - (1+a) \alpha^2 \tau_0 \tau_1 f(\tau^2, w \sigma \sigma_1) + (a+b) \alpha \tau_0 f(\tau, w \sigma) - bf = 0,
I ={igf_{acb}\over (2\pi)^6}\int du_4 d^4x_{4_A}d^2\theta^+_4d^2\bar\theta^+_4\{(14)\over \hat x^2_{14}}\;{(42)\over \hat x^2_{42}}\;{(\theta_{34})^2\over \tilde x^2_{34}}\;.
F_T(\alpha)=T^{-1}\* \sum_{0<\tilde{\gamma}_j, \tilde{\gamma_k}\leq T}\exp(i\*\alpha\*(\tilde{\gamma}_j-\tilde{\gamma}_k)) \*\frac{4}{4+(\tilde{\gamma}_j-\tilde{\gamma}_k)^2/\log(T)^2},
L_j(n-\beta) =\prod_{\substack{l=0 l\neq j}}^n\frac{(n-\beta)-l}{j-l}=\prod_{\substack{l=0 l\neq j}}^n\frac{-\beta+(n-l)}{-(n-j)+(n-l)}=\frac{(-1)^n}{(-1)^n}\prod_{\substack{l=0 l\neq j}}^n\frac{\beta-(n-l)}{(n-j)-(n-l)}&=\prod_{\substack{k=0 k\neq n-j}}^n\frac{\beta-k}{(n-j)-k}=L_{n-j}(\beta).
\mathcal{G}_{0}(...) = \, \,\int_0^\infty\,\int_0^\infty\,U(y_1) U(y_2)\, &\times \int_{\mathbb{R}}\, V(\xi)\, e\left(3\,\lambda \left({y^{1/3}_1}-{y^{1/3}_2}\right)\xi^{1/3}\right) d\xi &\times e\left(\frac{-t}{2\pi}(\log y_1-\log y_2) + \frac{2\sqrt{N}}{p_1q\sqrt{p_2}}\left({\sqrt{my_1}}-{\sqrt{m_1y_2}}\right) \right) dy_1\,dy_2.
\lim_{n \to \infty}n^{1+\alpha}\lambda_n(\mu, z_0) = \frac{w(z_0)}{(\pi \omega_{\Gamma}(z_0))^{\alpha + 1}}2^{\alpha + 1}\Gamma \Big( \frac{\alpha + 1}{2}\Big) \Gamma \Big( \frac{\alpha + 3}{2}\Big).
d_{\Omega_n}(o,p) = d_{\Omega_n}(o, x+e^{-2t_p}(o-x)) \leq \alpha t_p \leq \frac{\alpha}{2}\log(R+1) - \frac{\alpha}{2}\log\delta_{\Omega_n}(p) & \leq C_1-\frac{\alpha}{2}\log \delta_{\Omega_n}(p).
\begin{array}{lcl}{\mathcal L}_{3}^{(0,3)}= p^{3}+u_{2}p+(u_3 -\theta u'_{2}){\tilde{\mathcal L}}_{-3}^{(-3,-1)}={v_{0}\star p}^{-3}+v_{-1}\star p^{-2}+v_{-2}\star p^{-1}\end{array}
\sigma_{k,i}= \int \partial_{t}uK u^{\frac{4}{n-2}}\phi_{k,i}= \int \partial_{t}(\alpha^{j}\varphi_{j}+v)Ku^{\frac{4}{n-2}}\phi_{k,i}= \dot{\xi}^{l,j}\int Ku^{\frac{4}{n-2}}\phi_{l,j}\phi_{k,i}-\int Kv[\partial_{t}u^{\frac{4}{n-2}}\phi_{k,i}+u^{\frac{4}{n-2}}\partial_{t}\phi_{k,i}].
\widehat{Y}_{i}(x) = \int^{\xi \infty}_{0}\frac{P_{\lambda}(x, s)}{s-b_{i}}Y(s) \,{\rm d}_{q}s, i=0,\dots,N, \widehat{Y}(x) = \begin{pmatrix}\widehat{Y}_{0}(x) \vdots \widehat{Y}_{N}(x) \end{pmatrix}
J_{B \mu}^{\prime}= F_{\mu \nu}\partial^{\nu}\frac{1}{\partial_{i}\partial^{i}}{\overline{{\cal P}}}+ m^{2}( A_{\mu}+ \partial_{\mu}\rho ) \frac{1}{\partial_{i}\partial^{i}}{\overline{{\cal P}}}+ B \partial_{\mu}\frac{1}{\partial_{i}\partial^{i}}{\overline{{\cal P}}}.
\epsilon_{e a c d}t_{L}^{c d}h_{b}^{\bar{a}}F^{+ + a b}= 0 , \qquad \epsilon_{\bar{e}\bar{b}\bar{c}\bar{d}}t_{R}^{\bar{c}\bar{d}}h_{\bar{a}}^{a}F^{+ + \bar{a}\bar{b}}= 0
A(A^{*}A)^{-\frac{1}{2}}=(\sum_{j}\lambda_{j}v_{j}w_{j}^{*})(\sum_{k}|\lambda_{k}|^{-1}w_{k}w_{k}^{*})=\sum_{k}\frac{\lambda_{k}}{|\lambda_{k}|}v_{k}w_{k}^{*},
\psi_1(x)=\lim_{\Lambda\rightarrow \infty}\frac{k\Lambda}{2\pi}\int d\hat\eta e^{-i\frac{\theta(\eta)}{2}}\psi_1^{\eta}(x); \ psi_2(x)=\lim_{\Lambda\rightarrow \infty}\frac{k\Lambda}{2\pi}\int d\hat\eta e^{i\frac{\theta(\eta)}{2}}\psi_2^{\eta}(x)
\lim_{y\rightarrow-x}\frac{1+[y,b(x)]}{1+[x,b(y)]}= \lim_{y\rightarrow-x}\frac{[b(y),b(x)]}{[x,-y]}= \lim_{y\rightarrow-x}\frac{[-y,b(x)]}{[x,-b(y)]}= 1.
\bigl\langle J_\mathcal{N}^{(1)}(k,\ell)\bigr\rangle_1=i\lambda\int_{-T}^0ds \; e^{\eta s}\omega_0^{(\Lambda,N)}\bigl([\chi^{(\Lambda)}(\Gamma_\mathcal{M}(k,\ell);s),J_\mathcal{N}^{(1)}(k,\ell)]\bigr).
\int_{w \wedge v \in (-b, -a)}\|R_w\|_2 \|R_v\|_2 \mathbf{P}_{a, b}(w, v) dw dv = 2\int_{-b}^{-a}\int_0^\infty \|R_{x + v}\|_2 \|R_v\|_2 \mathbf{P}_{a, b}(x + v, v) dx dv.
\sigma:=\frac{\left(t e +g +1\right) (n-t) \left[ (n-t) (3 t e-g-1) +2 (g+1) (e-t)\right] -(d+1) (g+1)^2 (d-2 g+1)}{2(g-2)s(n+s)t(n-t)(r+1-t)(n+r+1-t)}
\frac{(\beta-1-a)^2}{4}\left\|\frac{g}{r^{1+\frac{a}{2}}}\right\|_{L_\beta^2}^2=\left\|\frac{g'}{r^{\frac{a}{2}}}\right\|_{L_\beta^2}^2-\left\|\frac{\beta-1-a}{2}\frac{g}{r^{1+\frac{a}{2}}}+\frac{g'}{r^{\frac{a}{2}}}\right\|_{L_\beta^2}^2
B_1(\xi)u = \partial_2 u_1 \left(\frac\pi2\right) + \mathrm{i}\xi u_2 \left(\frac\pi2\right) , B_2(\xi)u = 2 \partial_2 u_2 \left(\frac\pi2\right) , B_3(\xi) u = \partial_2 u_1 (0) + \mathrm{i}\xi u_2 (0), B_4(\xi) u = u_2(0) .
H_3^n(\alpha)=H(U_{S^*}\oplus Z^n) &=H(U_{S^*_1}\oplus Z_1)+H(U_{S^*_2}\oplus Z_2|U_{S^*_1}\oplus Z_1)+H\left( Z_3^n\right) &\geq H(U_{S^*_1}\oplus Z_1)+H(U_{S^*_2}\oplus Z_2|U_{S^*_1})+H\left( Z_3^n\right) &=h\left(\frac{1}{3}*\alpha\right)+\frac{1}{3}h(\alpha)+\frac{2}{3}+(n-2)h(\alpha).
\int_{0}^{b}\sqrt{x^{\prime}(t)^{2}+y^{\prime}(t)^{2}}dt=\int_{0}^{b}t\sqrt{4+9a^{2}t^{2}}dt=\cdot\cdot\cdot=[\frac{1}{27a^{2}}(4+9a^{2}t^{2})^{\frac{3}{2}}]_{0}^{b}.
\Gamma_{1;f}\leq \frac{\int_{\Omega}(\Delta_f\varphi)^2\,d\mu}{\int_{\Omega}\varphi^2\,d\mu}&=\frac{4\int_{\Omega}\Big[u^2(\Delta_fu)^2+2u|\nabla u|^2\Delta_fu+|\nabla u|^4\Big]\,d\mu}{\int_{\Omega}u^{4}\,d\mu}&=4\Big(\lambda_{1;f}^2-\frac{2}{3}\lambda_{1;f}^2+\frac{1}{3}\lambda_{1;f}I_{2,2}\Big) &\leq\frac{16}{3}\lambda_{1;f}^2,
N(\psi)= V(x,t)+c|\psi|^m,\, \, c>0 &|x|^{3+0}|V(x,t)|\leq c <\infty, \,\, |x|>1 &|x|^{3+0}|\nabla V(x,t)|\leq c <\infty, \,\, |x|>1 &|\nabla V(x,t)|+ |V(x,t)| \lesssim 1.
I_{2 1}( y , a ) |_{m = 0}= \left( \frac{B_{0}}{8 \pi^{2}y^{2}}+ \frac{B_{1}}{4 \pi y a}+ \frac{1}{2 \pi}\sum_{n = 2}^{\infty}\frac{B_{n}}{n !}( \frac{2 \pi y}{a})^{n - 1}\right) .
(\lambda-\psi^{\mu}(D))^{\alpha/2}=(\lambda-\psi^{\mu}(D))^{(\alpha-2n)/2}(\lambda-\psi^{\mu}(D))^{n}&=(\lambda-\psi^{\mu}(D))^{(\alpha-2n)/2}(\lambda-\psi^{\mu}(D)) \cdots (\lambda-\psi^{\mu}(D)).
\left\{\mathbb{J}_{\mathrm{d}}^{(\omega ,\eta \mathbf{\bar{A}}_{l})}\left(t\right) \right\}_{k}:=\left\vert \Lambda_{l}\right\vert^{-1}\underset{x\in \Lambda_{l}}{\sum}\rho_{t}^{(\beta ,\omega ,\vartheta ,\lambda ,\eta\mathbf{\bar{A}}_{l})}\left( \mathrm{I}_{(x+e_{k},x)}^{(\omega ,\vartheta,\eta \mathbf{\bar{A}}_{l})}\right) \ .
[Z_j]_{\alpha, \beta}= \sum_{i = 1}^{n(T, j+1)}\sum_{h = 1}^{n(T, j+1)}[D_{j+1}^T]_{\alpha, i}\ [R_{j+1}^{-1}]_{i, h}\ [D_{j+1}]_{h, \beta}= \sum_{i, h = 1}^{n(T, j+1)}[D_{j+1}^T]_{\alpha, i}\ [R_{j+1}^{-1}]_{i, h}\ [D_{j+1}]_{h, \beta}\, .
\frac{w_a}{y_a}\,\frac{y_a}{w_a}+ \sum_{i=a+1}^m\Bigl(\frac{w_i}{y_i}-\frac{w_{i-1}}{y_{i-1}}\Bigr) \frac{y_i}{w_i}\le 1+ \sum_{i=a+1}^m \int\limits_{w_{i-1}/y_{i-1}}^{w_i/y_i}\frac1t\,dt &=1+ \int\limits_{w_a/y_a}^{w_m/y_m}\frac1t\,dt\le 1+\int\limits_{\sigma^*}^1 \frac1t\,dt = 1+\ln \frac1{\sigma^*}.
\int_{B_2}\left( \kappa_2 \, u^0_{\mu\nu}(x) T^{\mu\nu}+\kappa_2 \sum_{n=1}^\infty \frac{\psi_n (R)e^{2kR}}{N_{0}}\, u^n_{\mu\nu}(x) T^{\mu\nu}-\frac{\kappa_1}{\sqrt{3}}\varphi\, T_\mu^\mu\right)dz.
d_0^* = [- \lambda_n + n - 1 - 2(r_1 + \ldots + r_{k-1}) - (d_1+ \ldots + d_{k-1})] - 2r_k + n + 1 = - \lambda_n - (d_1+ \ldots + d_{k-1}) = -d_0 - d_1 … - d_{k-1}.
\big|W(x) - W (y)\big| = \Big| \sum_{k=1}^{\infty}\frac{2^{-k}}{1+M(k,k)}\big(V_k (x) - V_k (y)\big)\Big| &\leq \sum_{k=1}^{\infty}\frac{2^{-k}M(R,k)}{1+M(k,k)}\|x-y\|_X &\leq \Big(1+ \sum_{k=1}^{[R] + 1}\frac{2^{-k}M(R,k)}{1+M(k,k)}\Big) \|x-y\|_X.
\frac{\mu_k(v')}{\pi(v')^p}\ge C^{-1}\frac{f_{1}(w')}{\pi(w')^p}\stackrel{\eqref{e:in8}}{>}C \frac{f_{1}(w)}{\pi(w)^p}\stackrel{\eqref{e:in7}}{=}C^3 \frac{f_{1}(w_{1}')}{\pi(w_{1}')^p}\ge C^3 \frac{\mu_k(v_{1}')}{\pi(v_{1}')^p}> C^2 \frac{\mu_k(v_{1}')}{\pi(v_{1}')^p},
\tau_{ij}^{\mu}(t):=\sum_{\nu=1}^{n}(\xi_{i\nu}(t)-\xi_{\nu}^{\mu}(t))(\xi_{j\nu}(t)-\xi_{\nu}^{\mu}(t)),\xi_{i\nu}(t):=\sum_{i=1}^{n}\mu_{i}(t)\xi_{i\nu}(t)
\|U(t)\|_{\infty,\mathbb R^3}\leq C(\|v_0\|_{3,\infty}+M_3)\,t^{-1/2}, &\|\nabla^j U(t)\|_{r,\mathbb R^3}\leq C(\|v_0\|_{3,\infty}+M_3)\,t^{-1/2+3/2r-j/2}, \forall r\in (3,\infty), &\|\nabla^j U(t)\|_{3,\infty,\mathbb R^3}\leq C(\|v_0\|_{3,\infty}+M_3)\,t^{-j/2}
\limsup_{k\rightarrow\infty}|q_k(y)|^{1/k}=\begin{cases}1 \mbox{if}~-1\le y\le 1 |y|+\sqrt{y^2-1}\mbox{if}~|y|>1, y\neq\frac{-d-(l-2)(1-\delta)}{2\sqrt{(l-1)(1-\delta)(d-1+\delta)}}\sqrt{\frac{d-1+\delta}{(l-1)(1-\delta)}}\mbox{if}~y=\frac{-d-(l-2)(1-\delta)}{2\sqrt{(l-1)(1-\delta)(d-1+\delta)}}\end{cases}
A_{(0,0)}= A_1 \oplus A_0 \oplus A_{\lambda - \frac{1}{2}}\oplus A_{\nu^{0}_+}\oplus A_{\nu^{0}_-}A_{(1,0)}= A_{\nu^{1}_+}A_{(0,1)}= A_{\nu^{1}_-}A_{(1,1)}= A_{\lambda}
\underline{u}'(z_1^-)=-\beta<0=\underline{u}'(z_1^+), begin{align*}0.2cm] \underline{v}'(z_2^-)=-q(1)he^{\lambda z_2}/2<0=\underline{v}'(z_2^+), begin{align*}0.2cm] \overline{v}'((-2/\lambda)^+)=0<q(1)he^{-2}=\overline{v}'((-2/\lambda)^-).
\Big| \int_{\mathbb{R}^{n+1}}\prod_{i=0}^{n}F_i(x_0,\dots,x_{i-1},x_{i+1},\dots,x_n) \varphi(x_0+\dots + x_n) dx_0 \dots dx_n \Big| \leq \|\varphi\|_1 \|F_0\|_{{2^n}}\prod_{i=1}^n\|F_i\|_{{2^{n-i+1}}}
e^{{\cal{G}}_{\mathrm{c}}\{J \}}= \frac{{\cal{Z}}\{J \}}{{\cal{Z}}_{0}}= \frac{\int{\cal{D}}\{\phi \}e^{- S \{\phi \}+ ( J , \phi )}}{\int{\cal{D}}\{\phi \}e^{- S_{0}\{\phi \}}}\; .
W_{\mathrm{tr},p}^-(\mu,\nu) = \inf_{\beta \in \Phi_{\mathrm{tr}}^-(\nu,m)}W_{\mathrm{tr},p}(\mu, \beta), W_{\mathrm{tr},p}^+(\mu,\nu) = \inf_{\alpha \in \Phi_{\mathrm{tr}}^+(\mu,n)}W_{\mathrm{tr},p}(\alpha, \nu).
P_N(\gamma_1,\gamma_2)\ge 2 (-2\gamma_1-1)^{-1}\int_{0<s_1<s_2<1}ds_1ds_2 (s_2-s_1+3/N)^{2\gamma_1+1}(s_2-s_1)^{2\gamma_2+1}& \times \int_0^\infty \left(u+\frac{3}{N(s_2-s_1)}\right)^{\gamma_2}\left(u+\frac{3}{N(s_2-s_1)}+1\right)^{\gamma_2}du=:R_N(\gamma_1,\gamma_2).
\left(\mathcal{M}_\ell(\hat{H}_s,\hat{S})\right)^\nu = \mathcal{M}_2(\hat{S}) \mathcal{M}_1(\hat{H}^\top) \left(\mathcal{M}_s(\hat{H}^\top,\hat{S})\right)^{\nu-1}\mathcal{M}_1(\hat{H}).
y_{b}=b_{0}\sum_{r=0}^{\infty}(\frac{(c-\beta)(c)_{r}(c+1-\gamma)_{r}}{(c+1-\alpha)_{r}(c+1-\beta)_{r}}s^{r+c})=b_{0}s^{c}\sum_{r=0}^{\infty}M_{r}s^{r}
P\exp\left(-\int_{\frac 1 2}^t\left(-\frac{1}{i\pi}\log(\varepsilon)\t_{12}+\left(\tau^s\right)^{1,2}+\left(\tau^s\right)^{12,3}\right)ds\right)=P\exp\left( -\int_{\frac 1 2}^t \left( \left(\tau^s\right)^{1,2}+\left(\tau^s\right)^{12,3}\right)ds \right)\cdot \exp\left(-\frac{2t-1}{2\pi i}\log(\varepsilon)\t_{12}\right)
-\frac{1}{(2\pi)^7(\alpha^\prime)^4}\int d^{10}X\sqrt{G^\sigma}R^\sigma = -\frac{1}{L_{Pl}^8}\int d^{10}x\sqrt{G^E}R^E = -\frac{1}{L_{Pl}^8}\int d^{10}x\frac{\sqrt{G}R}{V_5}\; ,
Z=\left[\begin{array}{cc}M 0_{(m+n)\times m(L-1)}\hline0_{m(L-1)\times m}T\otimes I_m\end{array}\right] &=\begin{bmatrix}M_n M_{n-1}\dots M_1 M_0 0 \dots 0 0 d_nI d_{n-1}I \dots d_1I d_0I \ddots \vdots \vdots \ddots \ddots \ddots \ddots \ddots \ddots 0 0 \dots 0 d_nI d_{n-1}I \dots d_1I d_0I\end{bmatrix},
0 =\int_0^1 \partial_{e^m x_1}\tilde u_{k_m}(sz_{l_m,k_m}+(1-s)z_{k_m(m)}) ds &=\int_0^{t(\alpha)}\partial_{e^m x_1}\tilde u_{k_m}(sz_{l_m,k_m}+(1-s)z_{k_m(m)}) ds &+\int_{t(\alpha)}^{t(R)}\partial_{e^m x_1}\tilde u_{k_m}(sz_{l_m,k_m}+(1-s)z_{k_m(m)}) ds &+\int_{t(R)}^{1}\partial_{e^m x_1}\tilde u_{k_m}(sz_{l_m,k_m}+(1-s)z_{k_m(m)}) ds.
e^{2i(c_-(x)+c)}u(x) =\frac{1}{\pi}\int_{\mathbb{R}}z^{-1}\bar{r}_1(z) e^{-2izx}\mathrm{d}z &+\frac{1}{\pi}\int_{\mathbb{R}}z^{-1}\bar{r}_1(z) e^{-2izx}\left(M_{-,11}(x;z)-1\right) \mathrm{d}z :=I_1(x)+I_2(x).
\int_{\Omega}|u|^{\alpha-2p+2}|\nabla u|^{2p-2}\langle\nabla u,\nabla\Delta_{p,f}u\rangle\,d\mu = -(p-1)\lambda_{1;p,f}\int_{\Omega}|u|^{\alpha-p}|\nabla u|^{2p}\,d\mu.
(0):(x) =(z,u), (0):(z) = (x), (z):(x,z) =(x,z,u), (x):(x,y) =(x,y,z,u), (z): (z,u) =(x,z), (z,u) :(x,z,u) =(x,y,z,u), (x,z,u):(x,y,z,u) =(x,y,z,u).
{\cal L}_{2}={\frac{1}{2}}E^{+ +}\wedge E^{--}- i \Pi^{\underline{{m}}}\wedge d \Theta \Gamma_{\underline{{m}}}\Theta +{\frac{1}{2}}E^{--}\wedge \Psi_{-}^{I}d \Psi_{-}^{I}, \qquad
g_{{\rm SO}(M)}\big(\Pi^P(X^{h'},Y^{h'}),\alpha^{+}\big) =g\left(\nabla_XY+\frac{1}{2}(R_{\xi_X}(Y)+R_{\xi_Y}(X)),\xi\cdot\alpha\right) \hphantom{g_{{\rm SO}(M)}\big(\Pi^P(X^{h'},Y^{h'}),\alpha^{+}\big) =}{}+\frac{1}{2}{\bf B}\big(\nabla_X\xi_Y+\nabla_Y\xi_X+\xi_{[X,Y]},\alpha\big),
\#\{(x, y)\in \mathbb{F}_q^2: y^2=x^3+ax+b\}=q-\phi(b)\cdot q\cdot{_{2}}G_2\left[\begin{array}{cc}\frac{1}{4}, \frac{3}{4}\frac{1}{3}, \frac{2}{3}\end{array}|\frac{-27b^2}{4a^3}\right]_q,
\begin{array}{r l}{{\displaystyle E ( L ) =}}&{{- \displaystyle \frac{\pi}{6 L}\left( c - 1 2 ( \Delta_{+}+ \Delta_{-}) \right) + \ldots \: ,}}\\{{P ( L ) =}}&{{\displaystyle \frac{2 \pi}{L}\left( \Delta_{+}- \Delta_{-}\right) + \ldots \: .}}\\ \end{array}
\left( \Psi + n^{\mu}\nabla_{\mu}\right) \varphi \longrightarrow \Omega^{-{\frac{{\cal D}- 1}{2}}}\left( \Omega \overline{{\Psi}}\varphi + n^{\mu}\nabla_{\mu}\varphi -{\frac{{\cal D}- 2}{2}}{\frac{n^{\mu}\partial_{\mu}\Omega}{\Omega}}\varphi \right) ,