harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
|\partial_{r}v_{4,c}(t,r)| \leq C \chi_{\geq 1}(\frac{2r}{\log^{N}(t)}) \begin{cases}\frac{1}{r^{4}t^{2}\log^{3b}(t)}, r \leq \frac{t}{2}\frac{\log(r)}{\log^{2b}(t) r^{4}(t-r)^{2}}+ \frac{1}{\log^{3b}(t) t^{2}r^{4}}, r \geq \frac{t}{2}\end{cases}&+\frac{C |\chi'(\frac{2r}{\log^{N}(t)})|}{\log^{5N+2b}(t)}\frac{r}{t^{2}\log^{b}(t)}
\int_0^t (t_0 + s)^{\lambda - \frac{p+1}{p-1}}h(s) \,ds \le C \begin{dcases}1 \left( \lambda < \frac{2}{p-1}\right), (\log(t_0+t))^2 \left( \lambda = \frac{2}{p-1}\right), (t_0+t)^{\lambda - \frac{2}{p-1}}\log(t_0+t) \left( \lambda > \frac{2}{p-1}\right); \end{dcases}
\|\mathrm{e}_{\pi_{\lambda,\beta}}(f(z),\cdot)\|_{q,\kappa,\pi_{\lambda,\beta}}^{2}=\sum_{n=0}^{\infty}(n!)^{1+\kappa}2^{nq}\frac{|z|^{2n}}{(n!)^{2}}=\sum_{n=0}^{\infty}\frac{1}{2^{n\kappa}}\frac{\left(2^{\kappa}2^{q}|z|^{2}\right)^{n}}{(n!)^{1-\kappa}}.
+\oint\limits_{\partial{\Omega}}\Biggl[{{\delta^{\wedge}F}\over{\delta q(x)}}\Bigg\vert_{\partial{\Omega}}{{\delta^{\vee}G}\over{\delta p(x)}}+{{\delta^{\vee}F}\over{\delta q(x)}}{{\delta^{\wedge}G}\over{\delta p(x)}}\Bigg\vert_{\partial{\Omega}}\Biggr] dS
\varphi_i^L(v) = v_i\varphi\bigg(\frac{1}{L}\sum_{k=1}^n v_k - 1\bigg)+ 2L\bigg(1 - \varphi\bigg(\frac{1}{L}\sum_{k=1}^n v_k - 1\bigg)\bigg)\quad\mbox{for}v\in[0,\infty)^n.
\begin{aligned}q_0(a-t) =\mathcal{K}_0(q_0(a+t)), \tilde{q}_0(a-t) =\mathcal{K}_0(\tilde{q}_0(a+t)), p(a-t) =\mathcal{K}_1(p(a+t)), \tilde{p}(a-t) =\mathcal{K}_1(\tilde{p}(a+t)),\end{aligned}
u:=\frac{1}{n_2}\left(\sum_{w\in W_1}(\ell_{\nu_{h_w}}+h_w)+\sum_{w\in W_2\cup W_3}K_{\nu_w}+n_2|W_2|+\beta_3+i\right) v:=\frac{1}{n_2 q}\sum_{w\in W_1}(\nu_{h_w}+q(n_2-\ell_{\nu_{h_w}}))+\frac{1}{q}(q|W_3|+m\beta_1+n_1\beta_2+mn_1\beta_3-mn_1n_2\beta)
\{x^1,\ldots,x^{r_0}\}, &\{e^1,\ldots,e^{r_1}, x^ie^\mu \}, \;\; i=1,\ldots, r_0, &\{p^1,\ldots,p^{r_2}, e^\mu e^\gamma, x^ip^I, x^i e^\mu e^\nu\},\;\; i=1,\ldots, r_0,\; \; \gamma=1,\ldots, r_1,
F^{}(x_{i+1/2}) - F^{}(x_{i-1/2}) = - \Bigg( \sum_{j=1}^{i}\int_{\Lambda_j}u\, n(t,u) \int_{x_{i+1/2}-u}^{x_{}}\beta(u,v)n(t,v)dv du & - \sum_{j=1}^{i-1}\int_{\Lambda_j}u\, n(t,u) \int_{x_{i-1/2}-u}^{x_{}}\beta(u,v)n(t,v)dv du \Bigg).
d \Pi_{\mu_{1}\dots \mu_{p - 1}\alpha_{1}}= a_{1}\left( ( C \Gamma_{\nu \mu_{1}\dots \mu_{p - 1}})_{\beta \alpha_{1}}\Pi^{\nu}\Pi^{\beta}+{\frac{a_{s}}{a_{0}}}( C \Gamma^{\nu})_{\beta \alpha_{1}}\Pi_{\nu \mu_{1}\dots \mu_{p - 1}}\Pi^{\beta}) \right)
F_{p,i}(x_{p},x_{i})=u_{0}(0)+u_{\delta^{i}}(0)f_{i,0}(x_{i})+u_{\delta^{p}}(0)f_{p,0}(x_{p})+u_{\delta^{i,p}}(0)f_{i,0}(x_{i})f_{p,0}(x_{p})
C(t) \coloneqq \begin{cases}C e^{-td}(t\ge 1) C t^{-d/\tilde{p}}(0<t\le 1), \end{cases}\frac{1}{\tilde{p}}\coloneqq \max\Big\{\frac{1}{p_2}-\frac{1}{p_1},0 \Big\},
\leq C\left({\displaystyle\sum\limits_{i=1}^{N-1}}\left\langle D_{y_{i}}^{m}v\right\rangle_{y_{i},\overline{Q}_{3/4}}^{(\gamma)}+\left\langle D_{y_{N}}^{m}v\right\rangle_{y_{N},\overline{Q}_{3/4}}^{(\gamma)}+|w|_{\overline{Q}_{1/4}}^{(0)}\right) .
\begin{array}{l}\displaystyle{\frac{\delta}{\delta h_{i'j'}(x')}}G_{a'b'cd}(y) \equiv g^{i'j'}_{a'b'cd}(y)~ \delta(y,x') \ , \vspace{2mm}\displaystyle{\frac{\delta}{\delta h_{kl}(x)}\frac{\delta}{\delta h_{i'j'}(x')}}G_{a'b'cd}(y) \equiv g^{i'j'kl}_{a'b'cd}(y)~ \delta(y,x') ~\delta(y,x) \ .\end{array}
\delta_{h_1}\otimes \delta_{h_2}\cdot \delta_{h_1'}\otimes \delta_{h_2'}=\delta_{[h_1,h_2]}\cdot \delta_{[h_1',h_2']}& =Z^{d(h_2)d(h_1')}* \delta_{[h_1 h_1',h_2h_2']}=Z^{d(h_2)d(h_1')}* \delta_{h_1 h_1'}\otimes \delta_{h_2h_2'},
g_{s}=g=f(1-\chi_{\Theta^{s}})+\sum_j\frac{1}{\mu(Q_j)}\int_{Q_j}f(x)\,d\mu(x_2)\chi_{S_j}b_{j,s}=b_j=\left(f(x)-\frac{1}{\mu(Q_j)}\int_{Q_j}f(x_1,y_2)\,d\mu(y_2)\right)\chi_{S_j}.
n_{t+1}\in\bigcap_{\ell=0}^d\left[B_1^\ell\cap \bigcap_{s\in\{1,...,2^\ell\}\setminus\{2^\ell\}}\bigcap_{1\leq j_1<\cdots<j_s\leq t}B_{s+1}^\ell(n_{j_1},...,n_{j_s})\right]\in p.
I_{1,\ast}(t)+\Vert\varphi(t)\Vert^2_{L^2(\mathbb R)}\le \frac{C_3}{\varepsilon}\int_0^t(I_{1,\ast}(s)+\Vert\varphi(s)\Vert^2_{L^2(\mathbb R)})ds &+C_1\varepsilon\left\{I_{1,\ast}(t)+\Vert{\mathbf F}(t)\Vert^2_{L^2(\mathbb R^2_+)}\right\}+\frac{C_3}{\varepsilon}\Vert{\mathbf F}\Vert^2_{H^1_\ast(\Omega_t)}\,.
\mathcal{L}^{\circ}\mkern-1.5mu(\hat{Y}[y])\mkern1mu \cup \mkern1mu\varphi_{\mkern-1.7mu n_{{\textsc{g}}}\mkern-.5mu *}^{{\mathbb{P}}}(\mathcal{L}^{\circ}\mkern-1.5mu(Y_{n_{{\textsc{g}}}}))=\mkern1.5mu(\left<\hat{y}_-, +\infty\right] \cup \left[-\infty, \hat{y}_+\right>)\mkern1.5mu\cup\mkern1.5mu\left<-\infty, y_{n_{{\textsc{g}}}+}^{\textsc{g}}\right>
\begin{array}{cc}\bar{\theta}_{i}=\theta^{i\dagger}\tilde{\gamma}^{0}=(\theta^{j})^{t}\bar{\cal E}_{ji}~~~~ ~~~~\bar{\tilde{\theta}}_{i}=\tilde{\theta}^{i\dagger}\gamma^{0}=(\tilde{\theta}^{j})^{t}\bar{\cal E}_{ji}\end{array}
\mathcal{U}z^n=\begin{cases}z^{n+1}, n\ge1 z+z^{-1}, n=0 z^{n-1}, n\le -1. \end{cases}\mathcal{V}z^n=\begin{cases}z^{n-1}, n\ge1 0, n=0 z^{n+1}, n\le -1 \end{cases}
\int d^{3}x_{1}K ( 2 , 1 , R_{+}^{+}u_{+}^{+}) \gamma^{4}\psi ( 1 ) = a e^{i \theta_{2}/ 2}u_{+}^{+}= a R_{+}^{+}u_{+}^{+}= \psi ( 2 )_{b = c = d = 0},
( 1 - \delta_{I J}\delta_{a a^{\prime}}) \{{\frac{\alpha^{( I )}\alpha^{( J )}}{2}}+ q -{\frac{d_{I}d_{J}}{D - 2}}+ \sum_{i = 1}^{N}r_{i}\Delta_{a i}^{( I )}\Delta_{a^{\prime}i}^{( J )}\}= 0 , I , J = 1 , 2
H_{\pm}= \left( \begin{array}{c c}{m}&{i e^{\mp i \theta}\left( \partial_{r}\mp \frac{i}{r}\partial_{\theta}\mp \frac{e B}{2}r \right)}\\{i e^{\pm i \theta}\left( \partial_{r}\pm \frac{i}{r}\partial_{\theta}\pm \frac{e B}{2}r \right)}&{- m}\\ \end{array}\right) .
d(x_m,x_n) \leq cd(x_m,x_{n+1})+cd(x_{n+1},x_n) \leq c^2d(x_m,x_{n+2})+c^2d(x_{n+2},x_{n+1})+cd(x_{n+1},x_n) \leq \cdots \leq c^{m-(n+1)}d(x_m,x_{m-1})+\sum_{k=1}^{m-(n+1)}c^kd(x_{n+k},x_{n+k-1}) \leq \Big[c^{m-(n+1)}\theta^{m-1}+\sum_{k=1}^{m-(n+1)}c^k\theta^{n+k-1}\Big]d(x_1,x_0) &< ((c \theta )^{m-1}c^{-n}+(1-c\theta)^{-1}c^{-(n-1)})d(x_1,x_0).
x_{i,j}=0\quad\mbox{for}(i,j) \in\{a_0+a_++1,\ldots,d\}\times\{1,\ldots,a_1\}& \quad\sqcup \{a_0+1,\ldots,a_0+a_+\}\times\{m-a_1+1,\ldots,m\}.
\|x^{k+1}-x^*\|^2 \leq \|x^k-x^*\|^2 + N\tilde C\alpha_k^2+4\alpha_kMN\sum_{i=1}^N \bar L_i \|\hat v_i^k - y^k\| \cr \;- 2 \alpha_k \left(\phi(x^k) - \phi(x^*)\right)^T(x^k-x^*),
\langle V_N^{(0)}|={}_{1\dots N}\langle 0|\exp\left\{\frac{1}{2}\,\sum_{I,J=1}^{N}\int_{-\infty}^{\infty}d\kappa\,a^{-(I)}(\kappa)\,\mu^{IJ}_{N}(\kappa)\,Ca^{-(J)}(\kappa)\right\}.
\frac{d}{dx}\sigma^2(x) \le\frac{d}{dx}g_{7+}(x)+ \frac1{2^6}=\frac{1745947}{139968}-30x+ \frac1{2^6}&\le\frac{1745947}{139968}-30c+ \frac1{2^6}=-\frac{62497}{9307872}< 0\quad\hbox{a.e.}
(j+1)ry^2|_{r=\xi_1(\nu)}+\int_0^{\xi_1(\nu)}\Big(\frac{dy}{dr}\Big)^2r^2dr+j(j+1)\int_0^{\xi_1(\nu)}y^2dr=\int_0^{\xi_1(\nu)}\nu\theta^{\nu-1}y^2r^2dr,
\epsilon^{2s-1}\int_{\Omega}{\int_{\Omega}{\frac{\overline{u}(x)-\overline{u}(y)}{|x-y|^{n+2s}}(\overline{u}(x)-\hat{u}(x)-\overline{u}(y)+\hat{u}(y))\,dx\,dy}}& +\frac{1}{\epsilon}\int_{\Omega}{\overline{W}'(\overline{u})(\overline{u}-\hat{u})\,dx}=0
J_1(t): =\langle E(t),F_n(U(t))-F_n(\mathbb U(t))\rangle, J_2(t): =n\|(-A_n)^{-\frac{1}{2}}\{\Sigma_n(\mathbb U(t))-\Sigma_n(U(t))\}\|^2_{\mathrm F}, J_3(t): =n\|E(t)^\top(-A_n)^{-1}\{\Sigma_n(\mathbb U(t))-\Sigma_n(U(t))\}\|^2.
\Xi^+_{m_j,\mp(j+1/2)}=\left(\begin{array}{cc}i \Omega^{m_j}_{j\mp1/2}0\end{array}\right),\Xi^-_{m_j,\mp(j+1/2)}=\left(\begin{array}{cc}0 \Omega^{m_j}_{j\pm1/2},\end{array}\right)
B^+_{\nu}\Phi_{\nu}= \left(\frac{(\nu+1)(\nu+\lambda)(\nu+2\lambda)}{\nu+\lambda+1}\right)^{1/2}\Phi_{\nu+1}, B^-_{\nu}\Phi_{\nu+1}= \left(\frac{(\nu+1)(\nu+\lambda+1)(\nu+2\lambda)}{\nu+\lambda}\right)^{1/2}\Phi_{\nu},
F(\tau_1+\tau_2,\tau_5+\tau_6) =- 2a_{ijk}b_{hjk}e^i\wedge e^{n+h}, \overline{F}(\tau_1+\tau_2,\tau_5+\tau_6) =- 4a_{ijk}b_{ijh}e^k\wedge e^{n+h};
A(y) :=\frac{1}{r(y)}\, \mathrm{diag}\left(1-\frac{1}{2}\sin(2\pi y_1)\sin(2\pi y_2),1+\frac{1}{2}\sin(2\pi y_1)\sin(2\pi y_2)\right), r(y) := 1+\frac{1}{4}(\cos(2\pi y_1)-2\sin(2\pi y_1))\sin(2\pi y_2)
\begin{array}{c}{\displaystyle \int \frac{d^{4}k_{1}}{( 2 \pi )^{4}}\int \frac{d^{4}k_{2}}{( 2 \pi )^{4}}D_{m}^{2}( k_{1}) [ D_{m}( k_{2}) - D ( k_{2}) ] \overline{{D ( k_{1}+ p )}}K_{1 2}.}\\ \end{array}
A ( p , p ) e^{\alpha q}* B ( p , p ) e^{\beta q}= A ( p_{0}, \bar{p}_{0}) e^{\alpha ( q_{0}+ \bar{q}_{o})}* B ( p_{0}, \bar{p}_{0}) e^{\beta ( q_{0}+ \bar{q}_{o})}|_{p_{0}= \bar{p}_{0}= p , q_{0}+ \bar{q}_{0}= q}.
\left(DT^+\right)^{-1}=\left(\begin{array}{cc}(E_{n_1}+v_1)^{-1}-(E_{n_1}+v_1)^{-1}(\partial_\phi v_0+\partial_\phi v_1\rho)(E_{n_2}+\partial_\phi \Phi)^{-1}0 (E_{n_2}+\partial_\phi \Phi)^{-1}\end{array}\right).
{\tilde{Q}}_{( e m )}= \int d^{3}x [ i ({\vec{\tilde{W}}}_{+}^{'}\cdot{\vec{\tilde{\pi}}}^{({\tilde{W}}_{+}){'}}-{\vec{\tilde{W}}}_{-}^{'}\cdot{\vec{\tilde{\pi}}}^{({\tilde{W}}_{-}){'}}) +{\tilde{j}}_{( e m )}^{o}] ( \vec{x}, x^{o}) ,
\hat{R}_{ij}=\begin{cases}0, \max\{|i|,|j|\}\leq C_0K,i\neq \pm j, (1-\Gamma_K)(-\mathbf{i}\tilde{\Omega}_{ij}F_{ij}+R_{ij}), \max\{|i|,|j|\}>C_0K,i\neq \pm j, 0, i=-j,|j|\leq\Pi, R_{(-j)j}, i=-j,|j|>\Pi.\end{cases}
t E_{s_i\lambda}= T_i(\langle \lambda\rangle_{i+1}-\langle \lambda\rangle_{i}) E_\lambda,\lambda_i < \lambda_{i+1}, t_n E_{s_n\lambda}= \left[T_n + \frac{1-t_n+t_n(1-t_0)y_n(\lambda)^{-1}}{t_0t_ny_n(\lambda)^{-2}-1}\right] E_\lambda,\lambda_n<0
G(q)=(q;q^{5})_{\infty}^{-1}(q^{4};q^{5})_{\infty}^{-1}=(q^{5};q^{5})_{\infty}^{1/2}(q;q)_{\infty}^{-1/2}[\frac{H(q)}{G(q)}]^{-1/2}
U=e^{i\phi(x)/2}[\begin{matrix}e^{i\psi(x)}&0\\ 0&e^{-i\psi(x)}\end{matrix}][\begin{matrix}cos\theta(x)&sin\theta(x)\\ -sin\theta(x)&cos\theta(x)\\ \end{matrix}][\begin{matrix}e^{i\Delta(x)}&0\\ 0&e^{-i\Delta(x)}\end{matrix}].
{\cal L}=-\frac{i}{2}\epsilon_{\mu\nu\lambda}B^*_\mu(\partial_\nu - iga_\nu)B_\lambda+\frac{M}{2}B^*_\mu B_\mu-\frac{i}{2}\epsilon_{\mu\nu\lambda}a_\mu\partial_\nu a_\lambda+ (\partial_\mu\bar{c})(\partial_\mu c)+\frac{1}{2\rho}(\partial_\mu a_\mu)^2\;.
\bar\omega_X = v^{o_1}\dots v^{o_m}\big( \sum_{k_i,l_i\ge e_i}c_{k_1\dots k_n}^{l_1\dots l_n}u^{k_n}\dots u^{k_1}v_{l_1}\dots v_{l_n}\big) u_{o_m}\dots u_{o_1}.
E_{P}\left\{var_{P}\left[ \psi_{P,\mathbf{a}}(\mathbf{G,B};\mathcal{G})\mid \mathbf{A},Y,\mathbf{G}\right] \right\}=E_{P}\left\{\pi_{\mathbf{a}}(\mathbf{G};P)var_{P}(Y\mid \mathbf{A}=\mathbf{a},\mathbf{G})var_{P}\left[ \frac{1}{\pi_{\mathbf{a}}(\mathbf{G,B};P)}\mid \mathbf{A=a},\mathbf{G}\right] \right\}
F_1 := F \cup \{D_{a_i, a_j}\}_{1\leq i < j \leq k+1}D_{a_i, a_{k+1}}= D_{a_1 , \ldots , \hat a_{k+1}}- D_{a_1, \ldots , \hat a_i , \ldots , a_{k+1}}, \forall i\in [k] D_{a_i, a_j}= 2D_{a_1 , \ldots , \hat a_{k+1}}- D_{a_1, \ldots , \hat a_i , \ldots , a_{k+1}}- D_{a_1, \ldots , \hat a_j , \ldots , a_{k+1}}, \forall i<j\in [k]
R_{u,v}^{J,q}(q)= R_{us_j,vs_j}^{J,q}(q) begin{align*}5pt]= (-1)^{\ell(vs_j)-\ell(us_j)}\left(1-q+q^{1+a_{j+1}(us_j,vs_j)}\right)(1-q) \prod_{k\in D(us_j,vs_j)}\left(1-q^{a_k(us_j,vs_j)}\right).
X_{e,n}:= \sum_{\substack{(d_{e+1}=n,\, d_{e-1},\,\dotsc,\, d_1,\,d_{0}=0) \in \mathbb N^{e+2}1 \le d_i \le m-r-1 0 \le i \le e}}\prod_{i=0}^{e}M_{p,m}(r(p-1) + d_{i+1}p - d_{i})
E=\frac{1}{2m}(\dot{\vec{A_{1}^{2}}}+\dot{\vec{A_{2}^{2}}})+ \vec{\dot{\phi}}^{2}+\frac{1}{2m}[\vec{A_{1}^{2}}\vec{A_{2}^{2}}-(\vec{A_{1}}\cdot\vec{A_{2}})^{2}]+[(\vec{A_{1}^{2}}+\vec{A_{2}^{2}})\vec{\phi^{2}}-(\vec{A_{1}}\cdot\vec{\phi})^{2}-(\vec{A_{2}}\cdot\vec{\phi})^{2}] +\frac{\kappa}{4}(\vec{\phi}^{2}-1)^{2}.
\quad\mathbb{P}[\bigcap_{m=1,l=1}^{M,L}\{(U_m,V_l)\notin \mathcal{F}\}] &\le\mathbb{P}[(U,V)\notin\mathcal{F}]+\mathbb{P}\left[\imath_{U;V}(U;V)>\log(ML)-2\gamma\right] &\quad+\frac{\min\{M,L\}-1}{ML(\exp(-\gamma)-\exp(-2\gamma))}+e^{-\exp(\gamma)}.
\psi_{\Lambda}(b_{\Lambda})= \langle\hat{e}_{I}, \left(\diamond_{x\in \Lambda}\hat{E}_{x}(b_{x})\right)\hat{e}_{I}\rangle\quad;b_{\Lambda}:=\bigotimes_{x\in\Lambda}b_{x}\ , \b_{x}\in\mathcal{B}_{x}
\Phi^{\infty}= i \frac{\eta}{\sqrt{2}}\vec{p}\cdot \vec{\sigma}, \qquad \vec{p}= \left( \begin{array}{c}{{\sin \tau \cos N \phi}}\\{{\sin^{2}\tau \sin N \phi + \cos^{2}\tau}}\\{{\sin \tau \cos \tau ( \sin N \phi - 1 )}}\\ \end{array}\right)
\zeta_{r}\left(-\mathbf{n};\mathbf{z}\right)=\left(-1\right)^{n_{r}}\frac{\left(\mathcal{B}-\mathcal{Z}_{r-1}\right)^{n_{r}+1}}{n_{r}+1}=\zeta_{1}\left(-n_{r};-\mathcal{Z}_{r-1}\right).
i{\cal G}_{4}( p ) = - [ 8 i e^{4}B^{4}] \left[{\frac{2 p_{\bot}^{2}p_{L}^{2}- 3 ( p_{\bot}^{2})^{2}}{( p^{2}- m^{2})^{7}}}\right] D - 8 e^{5}B^{5}\left[{\frac{1 5 ( p_{\bot}^{2})^{2}- 1 6 p_{\bot}^{2}p_{L}^{2}+ 2 p_{L}^{2}}{( p^{2}- m^{2})^{8}}}\right] \bar{D}.
l_{ij}^{+}(u) =\delta_{ij}-h\sum_{k\in\mathbb{Z}_{\geq 0}}l_{ij}^{(k)}u^{-k-1}, l_{ij}^{-}(u) =\delta_{ij}+h\sum_{k\in\mathbb{Z}_{<0}}l_{ij}^{(k)}u^{-k-1},
I_{(Y_+,D_{Y,+})}(y,z)=ze^{t_+/z}\sum_{d\in\mathbb K_{+}}y^{d}\left(\prod_{i\in M_0}\frac{\prod_{a\leq 0,\langle a\rangle=\langle D_i\cdot d\rangle}(\bar{D}_i+az)}{\prod_{a \leq D_i\cdot d,\langle a\rangle=\langle D_i\cdot d\rangle}(\bar{D}_i+az)}\right) \left(\prod_{i=1}^k\prod_{0<a\leq v_i\cdot d}(v_i+az)\right)\textbf{1}_{[-d]}I_{D_{Y,+},d},
\bar{a}(x) = \frac{1}{2\varepsilon}x (1-x) \bar{u}\bar{b}(x) = \frac{1}{8\varepsilon^3}\left(\frac{x^5}{5}-\frac{x^4}{2}+\frac{x^3}{3}-\frac{x}{30}\right) \bar{u}^2.
e(\partial B_{R_\ell}(x_0))=u(\partial B_{r_\ell}(x_0))=0, \shortintertext{and}\lim_{\ell\to\infty}\frac{e(B_{R_\ell}(x_0))}{u(B_{r_\ell}(x_0))}=\limsup_{R\to 0^+}\frac{e(B_R(x_0))}{u(B_R(x_0))}.
\begin{aligned}\partial_tV_\varepsilon-\Delta_HV_\varepsilon-\varepsilon^{\alpha-2}\partial_z^2V_\varepsilon +(U_\varepsilon\cdot\nabla)V_\varepsilon+\nabla_HP_\varepsilon &+(U_\varepsilon\cdot\nabla)v+(u\cdot\nabla)V_\varepsilon=\varepsilon^{\alpha-2}\partial_z^2v,\end{aligned}
\exp \left[ - \frac{i}{2}\sum_{a=1}^3 ( P_a^- \wedge P_a^+ ) - \frac{i}{2}P_1 \wedge P_2 \right]= \exp \left[ - \frac{i}{2}\sum_{a=1}^3 \left( P_a^- \wedge P_a^+ + \frac{1}{3}P_a \wedge P_{a+1}\right) \right] ~ ,
{1\over2}\left\{\left[ \begin{array}{c}Q_\alpha \tilde Q_\alpha \end{array}\right] , \left[ Q_\beta\ \tilde Q_\beta \right] \right\}= \left[ \begin{array}{cc}1 0 0 1 \end{array}\right] M\delta_{\alpha\beta}+\left[ \begin{array}{cc}p q/g_s q/g_s -p \end{array}\right]{{L (\Gamma^0 \Gamma^1)}_{\alpha\beta}\over{2\pi{\alpha^\prime}}}\ .
\left|\frac{2^{n_{k}}}{N}\,\rho_{N,k}\right| <C,\mbox{for all}\,\,N\in\mathcal{N}\,\,\mbox{and}\,\,k=1,\ldots,\tau(N), \sum_{k=1}^{M}\frac{2^{n_{k}}}{N}\,\left|\log\left(\frac{2^{n_{k}}}{N}\right)\right| <C,\mbox{for all}\,\,N\in\mathcal{N}\,\,\mbox{and}\,\,M< \tau(N).
{\cal Z}_{_I}= \int D \overline{{{\xi}}}^{\prime}D \xi^{\prime}D \overline{{{\chi}}}D \chi \exp \left\{i \int d^{2}x \left[ - i \overline{{{\chi}}}\partial \! \! \! / \chi - \frac{1}{m}\overline{{{\chi}}}\partial \! \! \! / \partial \! \! \! / \chi + m \overline{{{\xi}}}^{\prime}\partial \! \! \! / \partial \! \! \! / \xi^{\prime}\right] \right\}.
V ( x ) = \frac{c_{1}^{2}}{3 2}x^{6}+ \frac{c_{1}c_{0}}8 x^{4}+ \frac 1 8 \left( c_{0}^{2}- 2 c_{1}( 2 n - m ) \right) x^{2}+ \frac{4 m^{2}- 1}{8 x^{2}}+ c o n s t ,
S = \int d^{2}x \left\{{\frac 1 2}{\partial}_{\mu}{\phi}{\partial}^{\mu}{\phi}+ \frac{1}{2 ({\partial}_{\nu}a ) ({\partial}^{\nu}a )}\left[{\partial}^{\mu}a ({\partial}_{\mu}{\phi}-{\epsilon}_{{\mu}{\sigma}}{\partial}^{\sigma}{\phi}) \right]^{2}\right\},
\begin{aligned}r =2(p^2-n_1^2q^2)u^2-4(p^2-n_2^2q^2)uv+2(p^2-n_2^2q^2)v^2, s =-2(p^2-n_1^2q^2)u^2-(p^2-n_1^2q^2)uv-2(p^2-n_2^2q^2)v^2, d =-(p^2-n_1^2q^2)qu^2+(p^2-n_2^2q^2)qv^2, \end{aligned}
\sum_{n = 0}^{\infty}F(n) = \frac{1}{2}\Bigl(F(0) + F(\infty) \Bigr)+ \int_{0}^{\infty}dx F(x) + \sum_{k = 1}^{\infty}\frac{B_{2k}}{(2k)!}\Bigl(F^{(2k-1)}(\infty) - F^{(2k - 1)}(0) \Bigr),
H_{m}^{M}= H_{m}^{L}+ H_{m}^{R}= \frac{1}{2}m_{L}\int d^{3}{x}( \bar{\psi}_{L}\psi_{L}^{c}+ h . c ) + \frac{1}{2}m_{R}\int d^{3}{x}( \bar{\psi}_{R}\psi_{R}^{c}+ h . c ) .
J_{n}(z) = \frac{z^{n}}{2^{n}n!}\left(1+\mathcal{O}(\frac{1}{n})\right),n \rightarrow +\infty, J_{n}^{\prime}(z) = \frac{z^{n-1}}{2^{n}(n-1)!}\left(1+\mathcal{O}(\frac{1}{n})\right),n \rightarrow +\infty,
H(x)=1 + 3x + 15x^2 + 93x^3 + 639x^4 + 4653x^5 + 35169x^6 + 272835x^7 + 33065x^8 + 322285x^9 + 438261x^{10}+ 43884x^{11}+ 40470x^{12}+ 1755x^{13}- 2202x^{14}+ 130x^{15}- 1327x^{16}- 44x^{17}+ 20x^{18}+ 10x^{19}- 7x^{20}- 8x^{21}- x^{22}+ \cdots,
0 = \xi^T\tilde M \xi + h^T\xi +t- a + \xi^Tr = \frac{1}{2}\xi^T \tilde{M}\xi + \frac{1}{2}\xi^T \tilde{M}\xi + h^T \xi + r^T \xi + t-a = \frac{1}{2}\xi^T(\tilde M + \tilde M^T)\xi + (h+r)^T\xi+t - a \leq \frac{1}{2}\lambda_{\max}(\tilde M + \tilde M^T) + || h+r||+t - a,
\mathcal{T}_{{\rm dom}f}(\overline{x}) =\big\{w\in\mathbb{X}\,|\, dF(\overline{x})(w)\in\mathcal{T}_{{\rm dom}\vartheta}(F(\overline{x}))\big\}=\Big\{w\in\mathbb{X}\,|\, dF(\overline{x})(w)\in\bigcup\limits_{i=1}^s\mathcal{T}_{C_i}(F(\overline{x}))\Big\}=\mathcal{T}_{{\rm dom}f}^{i}(\overline{x}).
Kurt[X]=\frac{3\sqrt{\pi}Z_{\kappa}}{2\beta^{2/3}\sigma_{\kappa}^{4}}\frac{|2\kappa|^{-5/2}}{1+\frac{5}{2}|\kappa|}\frac{\Gamma(\frac{1}{|2\kappa|}-\frac{5}{4})}{\Gamma(\frac{1}{|2\kappa|}+\frac{5}{4})}
\sum^{\infty}_{\stackrel{m_1,m_2,\ldots,m_{\nu-2}=0}{m_i\equiv V_{i,r}^{\pm}(mod 2)}}\frac{q^{\frac{\vec{m}^tC\vec{m}}{4}}}{(q,q)_{m_{\nu-2}}}\prod_{i=1}^{\nu-3}\left[ \begin{array}{c}\frac{1}{2}(K_{\nu-2}\cdot\vec{m})_i+\frac{1}{2}\delta_{i,b^{\pm}}m_i\end{array}\right]_q =
\Psi^t_s=\int_0^s M( u(X_{\tilde s}^t),X^t_{\tilde s})\,d\tilde s+\frac{1}{\sqrt{t}}\,\int_0^s N( u(X_{\tilde s}^t),X_{\tilde s}^t)^T\,dW_{\tilde s}^t\,,
[x_{1}^{+}\otimes a_{1}^{-},x_{2}^{+}\otimes a_{2}^{-}]=x_{1}^{+}\circ x_{2}^{+}\otimes\frac{[a_{1}^{-},a_{2}^{-}]_{A^{-}}}{2}+[x_{1}^{+},x_{2}^{+}]\otimes\frac{(a_{1}^{-}\circ a_{2}^{-})_{A^{+}}}{2}+(x_{1}^{+}\mid x_{2}^{+})I\otimes[a_{1}^{-},a_{2}^{-}]_{A^{-}}.
\overline{K}(x_0,x_1,y_0,y_1,t)={x_1^2y_1^2K(\frac{x_0}{x_1},\frac{y_0}{y_1},t)}= x_0x_1y_0y_1 -t \sum_{i,j=0}^2 d_{i-1,j-1}x_0^{i}x_1^{2-i}y_0^j y_1^{2-j}.
\begin{aligned}r_-(x)-\hat r_-(x)=\frac D2\lambda(l+1)(\gamma(\lambda)+x^2)^{\frac{l-3}2}\times &\Big[\frac{2c_1}{D\lambda(l+1)}(\gamma(\lambda)+x^2)^{\frac{3-l}2}(\gamma_1+x^2)^l-(l+1)\lambda x^2(\gamma(\lambda)+x^2)^{\frac{l+1}2}-\gamma(\lambda)-lx^2\Big].\end{aligned}
- \frac{1}{1-q}\int_{B_{1}}\psi^{1+q}\partial_{s}(g_{1-\alpha,m}*\tilde{u}^{1-q})dx - \mathcal{E}(h_{m}*\tilde{u},\psi^{1+q}\tilde{u}^{-q}) & \leq \frac{-q}{1-q}\int_{B_{1}}\psi^{1+q}\tilde{u}^{1-q}g_{1-\alpha,m}dx.
\int_{\Omega_\ell}A\nabla\phi_\ell.\nabla\phi_\ell = \int_{\Omega_{\ell_0}}A\nabla \tilde v_{\ell_0}^{\varepsilon_0}.\nabla \tilde v_{\ell_0}^{\varepsilon_0}+ \int_{\Omega_{\ell-\ell_0}}A\nabla \phi_\ell.\nabla\phi_\ell\, .
({\cal A}_{N}X^{B})(\xi_{1},\cdots,\xi_{N})=\sum_{P}(-1)^{P}\exp\{i[k_{P_{1}}\xi_{Q_{1}}+\ldots+k_{P_{N}}\xi_{Q_{N}}]\}({\cal A}_{N}\beta_{_{12\ldots N}}^{(Q)}).
\partial_1^{k_1+3}\left[ (\psi_{\pi/2}-P_{2,0}\psi_{\pi/2}) a_{k_1,k_2,\pi/2}\right](0)= \frac{(k_1+3)!}{3 !}(-1)^{k_1}i^{k_2-n}\pi\left(i \frac{\pi}{2}\right)^{n+k_1+k_2},
\widetilde{\mathcal{O}}f =-\widetilde{U}^{-1}\mathcal{H}(\widetilde{U}f)-f &= -\widetilde{U}^{-1}U \mathcal{O}U^{-1}\widetilde{U}f -2f &= -e^{-x^2}\mathcal{O}(e^{(\cdot)^2}f(\cdot))-2f
\sum_{n>0}\frac{(-1)^{n-1}}{n}\sum_{\begin{smallmatrix}{r_i + s_i > 0}{1\le i \le n}\end{smallmatrix}}\frac{(\sum_{i=1}^n (r_i+s_i))^{-1}}{r_1!s_1!\cdots r_n!s_n!}[ X^{r_1}Y^{s_1}X^{r_2}Y^{s_2}\ldots X^{r_n}Y^{s_n}]
\sum_{(i_1,k_1)\in I_{j,1}(f)}2^{rk_1}d_{i_1}^{r}\Pr(B_{i_1,k_1})\leq d^r (8er)^r t^{-r}2^{r\rho}\exp(-r2^{\rho-1})\leq g(\rho)=g(f(1)),
f(k) :=\rho\left(d,\Omega_d^M(\varepsilon/2,\tilde g^M,\max\{n_0,\hat\chi(\rho(d,1/Nk))\}\right) n_0 :=\max\{\chi(\lceil96d/(1-\tau)\varepsilon^2\rceil),\chi(\lceil48d^2/(1-\tau)\varepsilon^2\rceil)\}\varepsilon_d :=\frac{((1-\tau)\varepsilon^2/96)^2}{2d^2}\Omega_d^M(\varepsilon,g,n) :=\max\left\{\Omega_d(\varepsilon,g,i):i\le n\right\}.
\left(1+\gamma\mu_{w}c_{o}^{\nu-1}\right)^{\nu p\cdot\sup\varphi}=e^{\log\left(1+\gamma\mu_{w}c_{o}^{\nu-1}\right)\cdot\nu p\cdot\sup\varphi}\leq e^{\gamma\mu_{w}c_{o}^{\nu-1}\cdot\nu p\cdot\sup\varphi}
I_{i j}^{( 1 ) \mu}={\frac{i}{2}}\Psi^{\lambda}\Psi^{\sigma}( f_{i ~ \sigma}^{\nu}D_{\nu}f_{j ~ \lambda}^{\mu}+ f_{j ~ \sigma}^{\nu}D_{\nu}f_{i ~ \lambda}^{\mu}+{\frac{1}{2}}f_{i}^{\mu \rho}c_{j \lambda \sigma \rho}+ f_{j}^{\mu \rho}c_{i \lambda \sigma \rho})
V_{G}^{( b )}( \Phi_{0}) = V_{G E P}^{2 + 1}( \Phi_{0}) + \frac{3{\hat{\lambda}}^{2}}{4 \pi^{3}}( \sqrt{x}- 1 ) ( 1 - \ln{\frac{4}{3}}) -{\frac{3}{4 \pi^{3}}}{\hat{\lambda}}^{2}\sqrt{x}\ln \sqrt{x}\; .
\begin{aligned}\int_{0}^{s}\Bigl(\int_{-\infty}^{\infty}P(xu)dF(x) \Bigr)du= =\sum_{j=0}^k a_j \int_{0}^{s}\tt{Re}\, f(ju)du + \sum_{j=1}^k b_j \int_{0}^{s}\tt{Im}\, f(ju)du,\end{aligned}
7140=(\begin{matrix}7140\\ 1\end{matrix})=(\begin{matrix}7140\\ 7139\end{matrix})=(\begin{matrix}120\\ 2\end{matrix})=(\begin{matrix}120\\ 118\end{matrix})=(\begin{matrix}36\\ 3\end{matrix})=(\begin{matrix}36\\ 33\end{matrix})
D_{GE}(p_{\hat{U}},1,u)=\underline{C}(p_{Y|X,S},p_S),u\in\mathcal{U}\mbox{with}p_{\hat{U}}(u)>0, &D_{GE}(p_{\hat{U}},1,u)\leq \underline{C}(p_{Y|X,S},p_S),u\in\mathcal{U}\mbox{with}p_{\hat{U}}(u)=0,
\delta_{\epsilon}\mathrm{{\bf A}}= \left[ \begin{array}{c c}{\epsilon^{k}\bar{\psi}_{k}- \psi^{k}\bar{\epsilon}_{k}}&{D \epsilon_{j}}\\{- D \bar{\epsilon}^{i}}&{\bar{\epsilon}^{i}\psi_{j}- \bar{\psi}^{i}\epsilon_{j}}\\ \end{array}\right] ,
\mathbb{P}_{p,q}^{(0)}\left(\bigcap_{i=1}^n\, \tau_{i Le}(A_{L}^c)\right) = \lim_{m \to \infty}\mathbb{P}_{p,q}^{(0)}\left(\bigcap_{i=1}^n\, \tau_{i Le}(A_{L,m}^c)\right) \le a(L)^n
dL_{x}^{-1}B_{H}(dL_{x}(X),dL_{x}(Y))=\begin{cases}0 \left( Y\in V(\mathfrak{m}_{1}\cap \mathfrak{k}_{2}) \oplus V_{1}\right) \displaystyle \frac{1}{2}[X,Y]^{\perp}\left( Y \in V_{2}\right).\end{cases}
C_{i}^{I I A}= h_{1 0 \; i}^{1 1}, \qquad h_{0 i}^{I I A}= h_{0 i}^{1 1}, \qquad B_{i j}^{I I A}= A_{1 0 \; i j}^{1 1}, \qquad C_{0 i j}^{I I A}= A_{0 i j}^{1 1}
\frac12ch(L_n)\hat{\otimes}\zeta_{(n)}(\vartheta^{-1}(\sigma^{((n),\emptyset)})) = (-1)^{\frac{(n-1)(n-2)}{2}}2^{-\frac{n+1}2}ch(L_n)(\prod_{j=1}^{n-1}(\xi_j-\xi_{j+1}))\zeta_{(n)}(\varsigma^{(n)}),
{\rm i}\,{\cal P}_\Lambda \, = \,{\o{1}{2}}\left ( k^{i}_\Lambda \, \partial_i G -k^{i^\star}_\Lambda \, \partial_{i^\star}G\right ) \,= \, k^{i}_\Lambda \, \partial_i G \, = \,- k^{i^\star}_\Lambda \, \partial_{i^\star}G
\left({\frac{d^{2}}{d x^{2}}}+{\frac{\rho^{2}}{( 1 - e^{- x})^{9 / 7}}}-{\frac{\ell ( \ell + 7 )}{4 9}}{\frac{e^{- x}}{( 1 - e^{- x})^{2}}}\right) \phi = 0
R_{\alpha,-}^{-1}(z) R_{\alpha,+}(z) = L_{\alpha}(z) \begin{pmatrix}1 0 0 e^{\pm 3(\omega - \omega^2) z^{1/3}}1 0 0 0 1 \end{pmatrix}L_{\alpha}^{-1}(z), z \in i \mathbb R^{\pm}, = L_{\alpha}(z) \begin{pmatrix}1 0 0 e^{- \frac{3 \sqrt{3}}{2}|z|^{1/3}}1 0 0 0 1 \end{pmatrix}L_{\alpha}^{-1}(z).
\lim_{\delta\rightarrow 0}\frac{1}{\delta^{\frac{2}{n+1}}}\cdot \frac{\|x^{\delta}\|_2-\|x\|_2}{\|x\|_2}=\frac{n^{\frac{2}{n+1}}(n+1)^{\frac{2}{n+1}}}{2}\left(\frac{|K|_n}{|B_2^{n-1}|_{n-1}}\right)^{\frac{2}{n+1}}\frac{\kappa(x)^{\frac{1}{n+1}}}{\langle x, u(x)\rangle}=n^{\frac{2}{n+1}}G(x)\quad,
\dim_{\rm H}B\cap W(\mathbf{t})\geq \min_{A\in\mathcal{A}}\left\{\#\mathcal{K}_1+\#\mathcal{K}_2+\frac{\sum_{k\in \mathcal{K}_3}a_k-\sum_{k\in \mathcal{K}_2}t_k}{A}\right\}:=s,
\hat{\mathrm{G}}_{0}\longrightarrow \hat{\tilde{\mathrm{G}}}_{0}= \hat{\mathrm{G}}_{0}+ \frac{1}{2}( e_{+}{\eta}_{+}+ e_{-}{\eta}_{-}) = \frac{1}{1}{L}e_{+}: \rho_{\mathrm{N}}( 0 ) :
\left({\begin{array}{*{20}{l}}{{P'_n}({x_0})}{\,\, \vdots}{{P'_n}({x_n})}\end{array}}\right) = \left({\begin{array}{*{20}{c}}{d_{00}^{(1)}}\ldots{d_{0n}^{(1)}}\vdots \ddots \vdots{d_{n0}^{(1)}}\cdots{d_{nn}^{(1)}}\end{array}}\right)\left({\begin{array}{*{20}{l}}{f({x_0})}{\, \vdots}{f({x_n})}\end{array}}\right).
e_c\left[t+1\right]= \left\{1-4\mu\sum_{j=1}^{M}\lvert\phi^{\left(j,c\right)}\rvert^2-2\mu f^{\left(c\right)}\vphantom{\sum_{j=1}^{M}}\right\}e_c\left[t\right]-2\mu\left(2\sum_{j=1}^{M}\lvert\phi^{\left(j,c\right)}\rvert^2+\mu f^{\left(c\right)}\right)a_c^{\left(o\right)}-2\mu\sum_{l}^{M}\Re\left\{\phi^{\left(l,c\right)}\right\}.
\frac{rm}{n}=\frac{1}{n}\Bigg(\sum_{i\in A}d_{H^*}(i)+\sum_{i\in B}d_{H^*}(i)+\sum_{i\in C}d_{H^*}(i)\Bigg) & =\frac1n\Bigg(kd+s(d+1)+\sum_{i\in C}d_{H^*}(i)\Bigg) & =\frac1n[kd+s(d+1)+(n-k-s)(d+1)+\ell] & =d+1+\frac{\ell-k}{n}.
\int_0^{\infty}t^{s-1}e^{-{a\over t}}e^{-\mu t}dt= 2({a\over{\mu}})^{s\over 2}K_s((4a\mu )^{1\over 2})= \Gamma (s) \mu^{-s}+ \Gamma(-s) ({a})^{s}+{\cal O}(\mu^{1-s},\mu)