harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
G_n(k^*)=\frac{\frac{1}{n^2}\left|\sum\limits_{i=1}^{k^*}\sum\limits_{j=k^*+1}^n\left(1_{\left\{X_i\leq X_j\right\}}-\frac{1}{2}\right)\right|}{\frac{1}{n^2}\left\{\frac{1}{n}\sum\limits_{t=1}^{k^*}S_t^2(1, k^*)+\frac{1}{n}\sum\limits_{t=k^*+1}^nS_t^2(k^*+1, n)\right\}^{\frac{1}{2}}}.
\frac{1}{2q}\frac{1}{dt}\int (m\varphi_{\lambda})^{2q}dx=-\frac{1}{2}\int (uv-u_xv_x)m_x(m\varphi_{\lambda})^{2q-1}\varphi_{\lambda}dx &\phantom{\frac{1}{2q}}-\frac{1}{2}\int M m(m\varphi_{\lambda})^{2q-1}\varphi_{\lambda}dx+\frac{1}{2}\int (uv_x-u_x v)(m\varphi_{\lambda})^{2q-1}m\varphi_{\lambda}dx.
\left[ n\right]_{q}A_{n,q}\left( qx\right) =\sum_{k=0}^{n}\left[\begin{array}[c]{c}n k\end{array}\right]_{q}\alpha_{k}q^{n-k}A_{n-k,q}\left( x\right) +x\left[ n\right]_{q}q^{n}A_{n-1,q}\left( x\right) .
{\textstyle \frac{1}{2}}\delta_{\epsilon}\Psi_{\hat{\imath}I}= \partial_{\hat{\imath}}\epsilon_{I}-{\textstyle \frac{1}{2}}\omega_{\hat{\imath}}^{+ a b}\sigma_{a b}\epsilon_{I}-{\textstyle \frac{i}{4}}e^{2 \phi}\partial_{\hat{\imath}}a \epsilon_{I}-{\textstyle \frac{1}{2 \sqrt{2}}}e^{- \phi}\sigma^{a b}F_{a b}^{+}\gamma_{\hat{\imath}}\alpha_{I J}\epsilon^{J}= 0 \ .
=\int\psi_{e}^{\prime*}\psi_{v}^{\prime*}\psi_{s}^{\prime*}\mu_{e}\psi_{e}\psi_{v}\psi_{s}d\tau+\int\psi_{e}^{\prime*}\psi_{v}^{\prime*}\psi_{s}^{\prime*}\mu_{N}\psi_{e}\psi_{v}\psi_{s}d\tau
S_{\Lambda}= \int d^{2}x d u \; \left({\frac{i}{2}}\lambda_{+}^{a}\partial_{--}\lambda_{+}^{a}+ i \sigma_{-}^{-- a}\partial^{+ +}\lambda_{+}^{a}+{\frac{1}{4}}g^{- a A^{\prime}}\partial^{+ +}g_{A^{\prime}}^{- a}\right) \; .
\psi_n(y) = \psi_n(T_5y) = \psi_n(T_2T_5y) = \psi_n(T_3T_5y) = \psi_n(T_4T_5y) =\psi_n(T_9y) = \psi_n(T_2T_9y) = \psi_n(T_3T_9y) = \psi_n(T_4T_9y).
n_u =H_Y(u)-1\le D\binom{n+u}{n}-1\le d^n\deg (V)e^{n}\left(1+\frac{u}{n}\right)^{n}-1 &<d^n\deg (V)e^n\left (d(2n+5)qd^n\deg(V)\epsilon^{-1}\right)^n-1 &\le \left[\deg (V)^{n+1}e^nd^{n^2+2n}(2n+5)^nq^n\epsilon^{-n}\right]=m;
(b-ma)^2\bigg(\frac{\beta-m2^{1-s}f(a)}{f(b)-mf(a)}\bigg)^\frac{1}{s}.\bigg(\frac{\beta-m2^{1-s}g(a)}{g(b)-mg(a)}\bigg)^\frac{1}{s}+ (b-ma)(ma-a) \bigg(\frac{\beta-m2^{1-s}f(a)}{f(b)-mf(a)}\bigg)^{\frac{1}{s}}+(b-ma)(ma-a)\bigg(\frac{\beta-m2^{1-s}g(a)}{g(b)-mg(a)}\bigg)^\frac{1}{s}+(ma-a)^2=\beta.
\left[ X^{\mu}, X^{\nu}\right] = \lim_{\epsilon \rightarrow 0}2 \alpha^{\prime}\left( \frac{- 2 \pi \alpha^{\prime}F}{1 - ( 2 \pi \alpha^{\prime}F )^{2}}\right)^{\mu \nu}\sum_{m = 1}^{\infty}\frac{( \rho \rho^{\prime})^{m}+ ( \rho / \rho^{\prime})^{m}}{m}\sin m \epsilon .
a_{D,\varepsilon}[\mathbf{u},\mathbf{u}]\geqslant c_* \Vert \mathbf{u}\Vert^2_{L_2(\mathcal{O})},\quad\mathbf{u}\in H^1_0(\mathcal{O};\mathbb{C}^n),c_*=c_0(\mathrm{diam}\,\mathcal{O})^{-2}.
\hat b_0 =1, \hat b_1 =\frac{R_{1,0}(\alpha,v)+R_{2,1}(\alpha,v)+R_{3,2}(\alpha,v)}{f(\nu_-,\mu_-)}, \hat b_2 =\frac{R_{2,0}(\alpha,v)+R_{3,1}(\alpha,v)+R_{4,2}(\alpha,v)+R_{5,3}(\alpha,v)+R_{6,4}(\alpha,v)}{f(\nu_-,\mu_-)}.
\big\|W_{s,t}(x) - W_{s,t}(y)\big\|_{L^q}\leq C_3 q |t-s|^{1/\varrho}|x - y|^{\gamma}\sum_{n, k = 1}^{\infty}\big\| [\sigma_n, \sigma_k] \big\|_{\mathcal{C}^{1}},
\ \mathrm{d}\phi(t) =\left[-\varphi_t \, \mathrm{d}t + \sum_i D_i \varphi\left(\mathfrak{b}_i \, \mathrm{d}t + \sum_k \mathfrak{s}_{ik}\, \mathrm{d}W_k(t)\right)\right. \left.+ \frac{1}{\,2\,}\sum_{i,j}D^2_{ij}\varphi \sum_k \mathfrak{s}_{ik}\mathfrak{s}_{jk}\, \mathrm{d}t\right](T-t,t,\mathfrak{X})\,,
q_n(\,\cdot\,|x,a) = \begin{cases}p(\,\cdot\,|x,a), x\in K_n \int_{K_n^c}p\bigl(\,\cdot\,|z,a) \nu_n(dz) , x \in K_n^c,\end{cases}b_n(x,a) = \begin{cases}c(x,a), x\in K_n \int_{K_n^c}c(z,a) \nu_n(dz) , x \in K_n^c. \end{cases}
\lambda^*(a_{\{0\}})=\lambda^*(a_{\{1\}})=b_1,\lambda^*(a_{\{2\}})=\lambda^*(a_{\{3\}})=b_2, \lambda^*(a_{\{0,1\}})=\lambda^*(a_{\{1,2\}})=\lambda^*(a_{\{2,3\}})=b_3, \lambda^*(a_{\{0,1,2\}})=\lambda^*(a_{\{1,2,3\}})=b_1+b_2+b_3.
\frac{1}{\lambda_{r}}+\frac{D-1}{(4\pi)^{\frac{D}{2}}}\Gamma(1-\frac{D}{2})\mbox{tr}\mbox{\boldmath$1$}- \left(\frac{K_{cr}}{\sigma_{0}^{2}}\right)^{\frac{D-2}{2}}\frac{1}{\sqrt{\pi}(4\pi)^{\frac{D}{2}}}\Gamma(\frac{D-1}{2})\Gamma(\frac{D}{2})\Gamma(1-\frac{D}{2}) \mbox{tr}\mbox{\boldmath$1$}=0.
\begin{aligned}\tilde{h}_{3,i}^{(1)}=\varphi f^2\delta^*\omega_i,\tilde{h}_{3,i}^{(2)}=\psi \cdot dr\odot f\omega_i,\tilde{h}_{4,i}^{(1)}=\varphi f^2(n\nabla^2 v_i+\Delta v_i\cdot g), \tilde{h}_{4,i}^{(2)}=\psi \cdot dr\odot f \nabla v_i,\tilde{h}_{4,i}^{(3)}=\chi\cdot v_i(f^2g-ndr\otimes dr).\end{aligned}
\frac{d^{2}u}{d\theta^{2}}+u=\frac{r_{s}c^{2}}{2h^{2}}+\frac{3r_{s}}{2}u^{2}-\frac{GQ^{2}}{4\pi\epsilon_{0}c^{4}}(\frac{c^{2}}{h^{2}}u+2u^{3})
\|f\circ \phi\|^2 = \int |f\circ \phi(x)|^2 dx \leq |\tfrac{1}{\phi'}|_{C^0}\int |f\circ \phi(x)|^2|\phi'(x)| dx =|\tfrac{1}{\phi'}|_{C^0}\int |f(y)|^2 dy \leq C^2 \|f\|^2
\sum_{i=0}^{N-1}\frac{p_i}{S}\min\{\|\nabla f(x_i)\|^2 \,,\, \|\nabla f(x_{i+1})\|^2\}{}\leq{}\sum_{i=0}^{N-1}\frac{p_i}{S}\frac{f(x_i)-f(x_{i+1})}{p_i}{}={}\frac{f(x_0)-f(x_{N})}{S}.
\begin{array}{r c l}{{g ( s ; y , z )}}&{{=}}&{{\frac{2^{1 - s}}{\Gamma ( s )}\sum_{k = 0}^{N}\frac{{\bf B}_{2 k}}{( 2 k ) !}y^{2 k}\Gamma ( s - 1 + 2 k ) \Bigl \{2 \zeta_{R}\left( s - 1 + 2 k ; 1 + \frac{z}{2}\right) +}}\\{{}}&{{+}}&{{\left( \frac{z}{2}\right)^{1 - 2 k - s}\Bigr \}+ O \left( y^{2 N + 2}\right) ,}}\\ \end{array}
\alpha_1,\,\alpha_2,\,\ldots,\,\alpha_m,\,\beta_1,\,\beta_2,\,\ldots,\,\beta_n \stackrel{k}{=}\gamma_1,\,\gamma_2,\,\ldots,\,\gamma_m,\,\delta_1,\,\delta_2,\,\ldots,\,\delta_n,
\psi (t) =\sum_{k=0}^{\infty}(k-2)\int_{[0,t]\times \lbrack 0,1]}{{1}}_{[0,r_{k}(\boldsymbol{\zeta}(s)))}(y)\,\varphi_{k}(s,y)ds\,dy \zeta_{k}(t) =p_{k}-\int_{[0,t]\times \lbrack 0,1]}{{1}}_{[0,r_{k}(\boldsymbol{\zeta}(s)))}(y)\,\varphi_{k}(s,y)ds\,dy,k\in \mathbb{N}.
\pi_k^{(i)}=\Big[\pi^{(0),(i)}_{1,k},\dots, \pi^{(\kappa^{(i)}_1-1),(i)}_{1,k},\dots,\pi^{(0),(i)}_{s_i,k},\dots,\pi^{(\kappa_{s_i}^{(i)}-1),(i)}_{s_i,k}\Big].
{\bar T}^L_c = \frac{1}{6 (2\pi)^4}\left(\frac{g}{\pi m}\right)^2\cos\left(m \Theta_{0j}p^j\right)\left\{\cos\left[m \Theta_{0j}\left(p^j - k^j\right)\right]+ \cos\left[m \Theta_{0j}\left(p^j-k'\,^j \right)\right]\right\}.
\sum_{k=1}^{\infty}I_{n,k}=\sum_{k=1}^{k_0(n)}I_{n,k}+\sum_{k=k_0(n)+1}^{\lfloor n^2q^{-n}\rfloor}I_{n,k}+\sum_{k=\lceil n^2q^{-n}\rceil}^{\infty}I_{n,k}.
J[u,p](x)=\frac{\alpha(u(0)-u(x))+(\alpha+1)px}{x^{\alpha+1}\Gamma(1-\alpha)}, &K_{(a,b)}[u,p](x)=\frac{\alpha(\alpha+1)}{\Gamma(1-\alpha)}\int_a^b[u(x-z)-u(x)+pz]\frac{dz}{z^{\alpha+2}},
\alpha^+_{k}\gamma_k^+ =\frac{k(2\nu_1+N-k)(2\nu_2+N-k)(2\nu_{12}+2N-k)}{(\nu_{12}+N-k)^2}, &\alpha^-_{k}\gamma_{k}^+ = \frac{4(2\nu_3+k)(N-k)(2\nu_{12}+N-k-1)(2\nu_{123}+2N-k-1)}{(2\nu_{12}+2N-2k-1)^2}, &\beta^\pm_k= \mp \frac{(\nu_1-\nu_2)(\nu_{12}+N)}{\nu_{12}+N-k}\pm \frac{(2\nu_{12}-1)(2\nu_{12}+4\nu_3+2N+1)}{2(2\nu_{12}+2N-2k \mp1)}.
\left(\begin{array}{c}u_t v_t\end{array}\right)=\mathcal{J}\left(\begin{array}{c}\frac{\delta \mathcal H}{\delta u}\frac{\delta \mathcal H}{\delta v}\end{array}\right),\mathcal{J}=\left(\begin{array}{cc}0 \tfrac{1}2 -\tfrac{1}2 0\end{array}\right),\mathcal{H}=\int (u_x^2+v_x^2-\tfrac{1}2(u^2+v^2)^2)\,\mathrm{d}x.
\lim_{m_{\pi}^{2}\rightarrow 0}m_{\pi}^{2}\int d^{3}\vec{x}\, \omega_{\pi}^{( 1 )}= - \lambda \int d^{3}\vec{x}\, ( \varphi_{\pi}^{3}+ \varphi_{\pi}\varphi_{\sigma}^{2}+ 2 v \varphi_{\pi}\varphi_{\sigma}) \ne 0 \quad ,
h_{n+1}\approx \left(\frac{2}{N_{CPU}}m+\epsilon\right)h_n, \epsilon>0, = \begin{cases}\left[(1+\frac{2}{N_{CPU}}k) + \epsilon\right]h_n m = N_{CPU}/2 + k, \left[(1-\frac{2}{N_{CPU}}k)+\epsilon\right]h_n m = N_{CPU}/2 - k, \end{cases}
\varepsilon_{i}(\tilde{b}) =\max\{k\in\mathbb{Z}_{\geq0}\mid D_{e_{i}^{k}.u,u'}\neq0\}\varepsilon_{i}^{\ast}(\tilde{b}) =\max\{k\in\mathbb{Z}_{\geq0}\mid D_{u,f_{i}^{k}.u'}\neq0\}.
\textup{I}_n \ = \d_{\mathrm{LP}}\Big(\mathcal{L}\big(p\big\{T_{n}(\mathbf{f})-\vartheta_{n}(\mathbf{f})\big\}\big), \mathcal{L}(\mathbf{z})\Big) begin{align*}0.2cm] \textup{II}_n \ = \d_{\mathrm{LP}}\Big(\mathcal{L}(\mathbf{z})\, ,\, \mathcal{L}\big(p\big\{T_{n, 1}^{*}(\mathbf{f})-\tilde{\vartheta}_{n}(\mathbf{f})\}\big| X\big)\Big).
\frac{d}{dt}E(\rho, \phi)(t) =-\int_{\Omega}\sum_{i=1}^s D_i(x)\rho_i|\nabla (\log \rho_i+z_i\phi) |^2 dx +\frac{1}{2}\int_{\partial\Omega}\epsilon(x) \left[ \phi (\partial_n \phi)_t - (\partial_n \phi )\phi_t \right] ds +\frac{d}{dt}B(\phi).
\widehat{y}_{i}^{[K,L]}(x) = (1-q) \sum_{n=K}^{L}s \frac{P_{\lambda}(x, s)}{s-b_{i}}\frac{(\alpha_1 s, \alpha_2 s;q)_{\infty}}{(\beta_1 s, \beta_2 s;q)_{\infty}}|_{s= q^n \xi}, i=0,1,2, & b_0=0, b_1 = 1/\alpha_1, b_2 = 1/\alpha_2
\begin{array}{l l}{{Z =}}&{{\lim_{\lambda \rightarrow 0}( \frac{\lambda}{2 \pi})^{n}\displaystyle \int{\cal D}u{\cal D}\psi{\cal D}\chi \exp ( - \frac{( s , s )}{2 \lambda}- \frac{1}{\lambda}g_{a b}\chi^{a}\frac{\partial s^{b}}{\partial{u^{i}}}\psi^{i}+ \frac{1}{2 \lambda}G_{i j a b}\psi^{i}\psi^{j}\chi^{a}\chi^{b})}}\\{{~ =}}&{{\nu ( D ) .}}\\ \end{array}
V_{\cal{C}}^{- 1}({\bf{r}}) = \exp \left( i g{\cal X}^{\alpha}({\bf{r}}){\textstyle \frac{\lambda^{\alpha}}{2}}\right) \exp \left( i g{\overline{{{\cal{Y}}^{\alpha}}}}({\bf{r}}){\textstyle \frac{\lambda^{\alpha}}{2}}\right) .
\overline{U}(t,r) = c e^{-kt}w(\dfrac{r}{\overline{s}_{1}(t)}),\ 0 \leq r < \overline{s}_{1}(t), t \geq 0\ \ \overline{V}(t,r) = d e^{-ht}w(\dfrac{r}{\overline{s}_{2}(t)}),\ 0 \leq r < \overline{s}_{2}(t), t \geq 0,
P_{1,\ge0}(\xi)\coloneqq\min_{\zeta\ge0}\left\{\gamma'|\zeta|+\frac{1}{2}(\zeta-\xi)^2\right\}=\begin{cases}\gamma'\xi-\frac{1}{2}\gamma'^2, \xi>\gamma', \frac{1}{2}\xi^2, \xi\le\gamma',\end{cases}
\lim_{k \rightarrow \infty}K_{\lambda}(x,y_{m_k})=e^{-(h(x)-h(x_0))\cdot \beta_s}\frac{\phi_{\beta_s}(x)}{\phi_{\beta_s}(x_0)}=e^{-(h(x)-h(x_0))\cdot \beta_s}\phi_{\beta_s}(x).
\int^{\rho}_{1}\frac{(s^{2}-1)^{n}}{s^{2}}ds= \int^{\rho-1}_{0}\frac{(y^{2}+2y)^{n}}{(y+1)^{2}}dy=\frac{2^{n}(\rho-1)^{n+1}}{n+1}\,{\rm{Appell}}F_{1}[n+1,2,-n,n+2,1-\rho,\frac{1-\rho}{2}]
\frac{1}{2}\int_Q \left(|u_j(x)-u_j(y)|^2 - |u_\infty(x)-u_\infty(y)|^2\right) K(x-y)dxdy \geq \int_Q (u_\infty(x)-u_\infty(y))(u_j(x)-u_j(y) - u_\infty(x) + u_\infty(y))K(x-y)dxdy +\frac{1}{2}\int_Q |u_j(x)-u_j(y)-u_\infty(x)+u_\infty(y)|^2 K(x-y)dxdy.
\left ( \sum^{\infty}_{p=0}\frac{1}{G_{n+kp}}\right)^{-1}\ =\ \left ( \sum^{\infty}_{p=0}\frac{1}{(A+B+C)\alpha^{n+kp}}\right)^{-1}\ =\ \left (\frac{1}{(A+B+C)(\alpha^{n}-\alpha^{n-k})}\right)^{-1}\ =\G_{n}-G_{n-k}.
\|DF(r(k),0)[h]\|_{Y^{k-1}_{c}}^{2}= \sum_{m=1}^{\infty}|b_{m}|^{2}(1+m)^{2k-2}(\cosh(cm)^{2}+ \sinh(cm)^{2}) & \leq C\sum_{m=1}^{\infty}|a_{m}|^{2}\left(K_{m-1}(r(k))^{2}+ K_{m}(r(k))^{2}+ K_{m+1}(r(k))^{2}\right)(1+m)^{2k-2}(\cosh(cm)^{2}+ \sinh(cm)^{2}) & \leq C\|h\|_{X^{k}_{c}}^{2}< \infty,
g(p) \approx \int \frac{d^3q}{(2\pi)^3}\, \frac{1}{(p - q)^2}\frac{g(q)}{\left|-\frac{1}{2}\delta+ iq_3 + \frac{\left|{\bf q}\right|^2}{2m}- \frac{1}{4\pi}\ln \frac{1}{2m}(-\frac{1}{2}\delta+iq_3+\frac{\left|{\bf q}\right|^2}{2m}) \right|^2}.
d s_{1 1}^{2}= \left( \frac{\pi^{4}}{1 5}Q \right)^{- 1 / 3}l_{p}^{\prime 2}U^{2}( - d \tau^{2}+ d x_{6}^{2}+ \cdots + d x_{9}^{2}+ d x_{1 1}^{2}) + \left( \frac{\pi^{4}}{1 5}Q \right)^{2 / 3}l_{p}^{\prime 2}( \frac{4 d U^{2}}{U^{2}}+ d \Omega_{( 4 )}^{2})
S[x^{\hat{i}},x^{\alpha}]=\int d^{2}\xi\left[\partial x^{\hat{i}}f_{\hat{i}\hat{j}}\bar{\partial}x^{\hat{j}}+\partial x^{\hat{i}}f_{\hat{i}\beta}\bar{\partial}x^{\beta}+\partial x^{\alpha}f_{\alpha\hat{j}}\bar{\partial}x^{\hat{j}}+\partial x^{\alpha}f_{\alpha\beta}\bar{\partial}x^{\beta}\right].
-\nu(C_{j_k}(\alpha(k))) = -\nu\left(\frac{\alpha(\alpha - 1)\dots(\alpha - j_k + 1)}{j_k!}\right) = j_k - s(j_k) - \nu((\alpha - j_k + 1) \dots (\alpha - 1)\alpha) = j_k - s(j_k) - (\nu((-\alpha - j-k + 1)!) - \nu(-\alpha - 1)) = -s(j_k) - s(-\alpha - 1) + s(-\alpha-1+j_k) = -c(j_k, -\alpha - 1).
\lim_{d\rightarrow+\infty}\sup\Bigg\{\bigg|\widetilde{\nu}_{\frac{\lambda}{2d},d}\Big(\eta(x)=0x\in A\Big) -\big(\frac{1}{\lambda}\big)^m\bigg|: &~A\subseteq\mathbb{Z}^d,~|A|=m\Bigg\}=0
F_{i_k}\cdots F_{i_2}F_{i_1}(w_i)=\delta_{i_1,i}\delta_{i_2,i_1+1}\cdots\delta_{i_k,i_{k-1}+1}w_{i+k}, E_{j_k}\cdots E_{j_2}E_{j_1}(w_j)=\delta_{j_1,j-1}\delta_{j_2,j_1-1}\cdots\delta_{j_k,j_{k-1}-1}w_{j-k}.
\begin{aligned}\mathbb{E}(|N_{ij}(f)|^{2m}\, | \, \mathcal{F}_{j-1}) \leqslant \bigg(\sum_{\substack{d\leqslant x_i v_{P(d)}(d) \geqslant 2}}^{y_{j-1},y_j}\tau_{2m-1}(d) |\Psi_f(x_i/d,y_{j-1})|^2 \bigg)^m. \end{aligned}
\begin{array}{ll}\sum\limits_{k^{'}=k+N}^{K}\int_{B(x_{i_{l_{k'}}},R_{i_{l_{k'}}})}|u(\bar{t})|^{2}\, dx \lesssim \left( \frac{1}{2^{N}}+ \frac{1}{2^{N+1}}.... +\frac{1}{2^{K-k}}\right) R_{i_{l_k}}^{2}& \lesssim \frac{1}{2^{N-1}}R_{i_{l_k}}^{2}\cdot\end{array}
|S(t,f)| \ \le \ \left(\frac{1}{2}+o(1) \right) \frac{\log \sqrt{q}(k\!+\!3)(|t|\!+\!1)}{\log \log \sqrt{q}(k\!+\!3)(|t|\!+\!1)}\underset{s=\frac12}{\mathrm{ord}}\L(s,f) \ \le \ \big( 1+o(1) \big) \frac{\log \sqrt{q}(k\!+\!3)}{\log \log \sqrt{q}(k\!+\!3)},
X = \big\{\, t \in K \;|\; \mathcal{O}_{w_t}\coloneqq \phi(K, t) \textrm{is a}p\textrm{-henselian valuation ring}&\textrm{with}w_tK\neq p\cdot w_tK \textrm{and}w_tK\textrm{$p$-regular}\,\big\}
f(t_r, \ldots, t_k) \cdot \exp(s F_{i_k}) = f(t_r, \ldots, t_{k+1}, t_k - s),\{\rm and\hence}f(t_r, \ldots, t_k) \cdot F_{i_k}= - \frac{\partial}{\partial t_k}f(t_r, \ldots, t_k)
\begin{aligned}\sum_{m=-\infty}^{\infty}(a;q)_{m}q^{m+\ell(|m|+m)/2}S_{\ell}\left(-q^{-|m|-\ell};q\right) =\frac{\left(aq,q^{\ell+1},1/(aq^{\ell});q\right)_{\infty}}{\left(1/a,q;q\right)_{\infty}}.\end{aligned}
\mathcal{E}^{\partial\Gamma,\mu}(u,v) :=\int\int_{\partial\Gamma\times \partial\Gamma}(u(\xi)-u(\eta))(v(\xi)-v(\eta)) \Theta^\mu(\xi,\eta) d\nu(\xi)d\nu(\eta), B_2(\mu) :=\{u\in L^2(\partial\Gamma,\nu)\ ;\mathcal{E}^{\partial\Gamma,\mu}(u,u)<\infty\},
\|\chi^\delta \hat{x}\|_{L_q(\mathbb{R}^2)}^q =\int_{1-\delta\leq|\xi|\leq1+\delta}|\hat{x}(\xi)|^qd\xi =\int_{1-\delta}^{1+\delta}\int_{rS^1}|\hat{x}(\xi)|^qd\sigma(\xi)dr &=\int_{1-\delta}^{1+\delta}r\int_{S^1}|\hat{x}(r\xi)|^qd\sigma(\xi)dr.
F_{[c,T]}^C(y,x) =\sup_{(s,t)\in X\times Y}\C((x,y),(t,s))-F_{[c,T]}(s,t)\geq\sup_{(s,t)\in G(T)}\C((x,y),(t,s))-F_{[c,T]}(s,t) &=\sup_{(s,t)\in G(T)}\C((x,y),(t,s))-c(s,t)=F_{[c,T]}(x,y).
\left [ \dfrac{(i/n)^{(i/n)}(1-(i/n))^{1-(i/n)}}{\big((i/n) - \frac{2j}{nd}\big)^{(i/n) - (2j/nd)}\, \big(\frac{2j}{nd}\big)^{j/nd}\, \big(1-2(i/n) + \frac{2j}{nd}\big)^{1/2 - (i/n) + j/nd}}\right ]^{nd}= \, \exp{\left \{n \left [\frac{d}{2}H\Big(M(i/n,j/(nd))\Big) - dH\big(i/n,1-(i/n) \big) \right] \right \}}\,.
\times \left[ \mathrm{t}o 2 4 p t{}\left( u_{n}^{1}\right)^{2}+ \left( u_{n}^{1}\right)^{2}- \left( u_{n}^{3}\right)^{2}- \left( u_{n}^{4}\right)^{2}\mathrm{t}o 2 4 p t{}\right] .
a\left(\sum_{n=1}^{m}|c_n|^p \right)^\frac{1}{p}\leq \left\|\sum_{n=1}^{m}c_n\tau_n\right\|\leq b\left(\sum_{n=1}^{m}|c_n|^p \right)^\frac{1}{p}, \forall c_1, \dots, c_m \in \mathbb{K}
(a,m)\left( (a,m)^j\right) = (a,m)(a^j, ja^{j-1}m + \sum_{i=1}^j a^{i-1}\xi(a \otimes a^{j-i}) ) = (a^{j+1}, ma^j + ja^jm + a \sum_{i=1}^j a^{i-1}\xi(a \otimes a^{j-i}) + \xi(a,a^j)) = (a^{j+1}, (j+1)a^jm + \sum_{i=1}^{j+1}a^{i-1}\xi(a \otimes a^{j+1-i})),
\tilde{\tau}(\psi)(x,t) :=\frac{1}{}\tau_1(\psi)(x,t)-\frac{1}{^2}\Big(\int_{D\times D}\tilde{\gamma}(q,q')\psi(x,q,t)\psi(x,q',t)\;\mathrm{d}q\mathrm{d}q'\Big)\mathbb{I}&=\frac{1}{}\tau_1(\psi)(x,t)-\frac{\tilde{\mathfrak{z}}}{^2}\eta^2(x,t)\mathbb{I},
[u]^p_{\mathcal{L}^{p,\lambda}(\Omega)}:= \sup_{\stackrel{x_0\in \Omega}{\rho>0}}\rho^{-\lambda}\int_{\Omega_{x_0,\rho}}\left|u(x)-u_{x_0,\rho}\right|^p\,dx<+\infty,
d s^{2}= - \left( 1 - \frac{r_{0}^{2}}{r^{2}}\right) \left( 1 + \frac{r_{1}^{2}}{r^{2}}\right)^{- 1}\left( 1 + \frac{r_{2}^{2}}{r^{2}}\right)^{- 1}\beta^{2}d t^{2}+ \frac{\Lambda^{- 1}}{r^{2}}\left( 1 - \frac{r_{0}^{2}}{r^{2}}\right)^{- 1}d r^{2}
{\cal{L}}_N W^N=\frac{1}{2N}\sum_{i\neq j}\int dp\, w(p)\delta\left( p^2-p\cdot (v_i-v_j)\right) &\{W^N(v_1,\ldots, v_{i}+p,\ldots, v_{j}-p, \ldots, v_N)-W^N(v_1,\ldots, v_N)\}.
\langle \epsilon_b,\epsilon_b,\epsilon_e,\epsilon_e\rangle=\begin{cases}\frac{\partial^2 F}{(\partial s_1)^2}(0), \mbox{if}\ 1\leq e\leq n+3\ \mbox{and}\e\neq b, 3\frac{\partial^2 F}{(\partial s_1)^2}(0)+\frac{\partial F}{\partial s_2}(0), \mbox{if}\e=b.\end{cases}
\int I_{( 1 )}^{g h}= \int \prod_{i < j}^{2}d^{2}\tau_{i j}( \det{\mathrm{I m}\tau})^{- 1}( \overline{{\Delta_{( 2 )}}})^{- 2}\overline{{\Theta ( \Lambda_{2 4})}}Z_{R}^{g h}( \tau ) ,
[\begin{matrix}2&-1&0&0&0&0&0\\ -1&2&-1&0&0&0&0\\ 0&-1&2&-1&0&0&0\\ 0&0&-1&2&-1&0&-1\\ 0&0&0&-1&2&-1&0\\ 0&0&0&0&-1&2&0\\ 0&0&0&-1&0&0&2\end{matrix}].
S_{1}^{T}( \epsilon ) = \frac 1 6 \operatorname{l n}{\frac{\mu}{\epsilon}}, \hspace{1 c m}S_{1}^{C A S}( \epsilon ) = \frac 1 2 \operatorname{l n}{\frac{\pi}{\operatorname{l n}{\frac{\beta}{2 \pi \epsilon}}}}.
\begin{array}{l}{{s_{2}= ( 1 / 3 ) ( C - 2 B )}}\\{{s_{1}= ( 1 / 3 ) ( C + B ) - ( 1 / 2 ) A}}\\{{s_{0}= ( 1 / 3 ) ( C + B ) + ( 1 / 2 ) A .}}\\ \end{array}
+ ( \gamma_{m})_{\alpha \beta}( 2 D \Theta^{\lambda}E_{\lambda}^{\alpha}D X^{\underline{{n}}}E_{\underline{{n}}}^{\beta}D X^{\underline{{k}}}E_{\underline{{k}}}^{m}+ D X^{\underline{{m}}}E_{\underline{{m}}}^{\alpha}D X^{\underline{{n}}}E_{\underline{{n}}}^{\beta}D \Theta^{\lambda}E_{\lambda}^{m})
\phi(t)=\begin{cases}t-\frac{\delta}{2}, \mbox{if}t\in\left[\frac{\delta}{2},\frac{\delta}{2}+1\right], 1, \mbox{if}t\in\left[\frac{\delta}{2}+1,\delta-1\right], \delta-t, \mbox{if}t\in\left[\delta-1,\delta\right], 0, \mbox{otherwise}. \end{cases}
|\psi(Y\alpha_{r+is}(X))|^2 =e^{-2s(\theta(x)-\theta(y))}|\psi(YT_xT_y^*)|^2 &\leq e^{-2s(\theta(x)-\theta(y))}\psi(Y^*Y)\psi(T_yT_x^*T_xT_y^*) &= e^{-2s(\theta(x)-\theta(y))}\psi(Y^*Y)\psi(T_yT_y^*).
2 \frac{\Gamma(n+1-k)\Gamma(n+2-k)}{\Gamma((n+2-k)/2)^2}r^{-k}\int_0^1 \rho^{n+1-k}M_p\left( \frac{d^k}{d\theta^k}(e^{-in\theta}f), \rho \right) (1-r^2\rho^2)^{k-n-1}d\rho.
\partial_t\phi-\dot{\Phi}\dot{F}^{ij}\nabla_i\nabla_j\phi = \Phi\dot{\Phi}\dot{F}^{ij}h_{i}^kh_{kj}+\mu\alpha2^{-\alpha}\rho^{2\alpha-1}(t,\bar{r}) \leq -\frac{\alpha}{2\beta}F^{-2\alpha+1}+\mu\alpha2^{-\alpha}\rho^{2\alpha-1}(t,\bar{r}),
w^{p}( \theta_{1}, \cdots , \theta_{q}; \xi_{1}, \cdots , \xi_{p - q}) = w_{1}^{q}( \theta_{1}, \cdots , \theta_{q}) w_{2}^{p - q}( \xi_{1}, \cdots , \xi_{p - q}) , \theta_{i}\in{\cal G}_{1}, \; \; \xi_{i}\in{\cal G}_{2}.
|\partial_{t}\langle \left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}})-1}{r^{2}}\right) v_{4}\left(1-\chi_{\geq 1}(\frac{4r}{t})\right)\vert_{r = R\lambda(t)},\phi_{0}\rangle| \leq \frac{C}{t^{3}\log^{b+N-2}(t)}+\frac{C \sup_{x \geq t}\left(\frac{x |e'''(x)|}{\lambda(x)^{2-2\alpha}}\right)}{t \log^{N-3+2b-2b\alpha}(t)}
e^{- 2 \rho}\simeq 2 ( \Delta_{0}- \log 2 - \frac{2 \mu^{2}}{\lambda^{2}}) - \lambda^{2}( x^{+}+ \frac{2 \gamma_{+}}{\lambda^{2}}) ( x^{-}+ \frac{2 \gamma_{-}}{\lambda^{2}})
\int_{\{\psi<-t\}}|F_{t}|^{2}e^{-\varphi}c(-\psi)+\int_{\{\psi<-t\}}|\hat{F}-F_{t}|^{2}e^{-\varphi}c(-\psi) = \int_{\{\psi<-t\}}|\hat{F}|^{2}e^{-\varphi}c(-\psi).
B_w^{11}= \tfrac{(w+1)^2(w+3)}{(w+2)(2w+n+3)(2w+n+4)}+ \tfrac{1}{(w+1)(w+2)(w+n+1)(w+n+2)}+ \tfrac{(w+n)(w+n+2)^2}{(w+n+1)(2w+n+2)(2w+n+3)}, B_w^{22}= \tfrac{(w+1)(w+3)^2}{(w+2)(2w+n+4)(2w+n+5)}+ \tfrac{(w+n+1)^2(w+n+3)}{(w+n+2) (2w+n+3)(2w+n+4)}.
\frac{b_{ijk}\mu_k(t)}{x^3_{ij}(t)}=\frac{b_{ijk}\alpha_k}{(\alpha_i-\alpha_j)^3t^{2}}\frac{1+[\mu_k(t)-\alpha_k/\sqrt{t}]/(\alpha_{k}\sqrt{t})}{(1+[x_{ij}(t)-(\alpha_i-\alpha_j)\sqrt{t}]/[(\alpha_i-\alpha_j)\sqrt{t}])^3}&=\frac{b_{ijk}\alpha_k}{(\alpha_i-\alpha_j)^3t^{2}}+O\bigg(\frac{1}{t^{9/4}}\bigg).
1_{NS}{\buildrel T_{\alpha}\over\longrightarrow}(0_w|1_{NS}),\ \ \, 5_{NS}{\buildrel T_{\alpha}\over\longrightarrow}(0_m|5_{NS}),\ \ \, (p+1)_R{\buildrel T_{\alpha}\over\longrightarrow}(p|p_R,(p+2)_R),
P_k^2f = \mathcal{F}^{-1}\psi^2(2^{-k}|\xi|)\mathcal{F}f= \mathcal{F}^{-1}\psi(2^{-k}|\xi|)\psi(2^{-k}|\xi|)\mathcal{F}f = \mathcal{F}^{-1}\psi(2^{-k}|\xi|)\mathcal{F}P_kf= P_kP_kf.
\begin{array}{lll}\displaystyle{\left.\frac{d}{ds}\right|_{s=0}\exp (tX) \exp (tY) \exp(t+s)Z}\ & =\displaystyle{\left.\frac{d}{ds}\right|_{s=0}\alpha(t)\exp (sZ)}= \displaystyle{\left.\frac{d}{ds}\right|_{s=0}L_{\alpha(t)}(\exp (sZ))}= (dL_{\alpha(t)})_eZ=Z^L_{\alpha(t)}.\end{array}
L_{2n-1}=\kappa\sum_{p=0}^{n-1}\frac{1}{D-2p}\left( \begin{array}{c}n-1 p \end{array}\right) l^{2p-D}\epsilon_{a_1 \cdots a_{2n-1}}R^{a_1 a_2}\wedge \cdots \wedge R^{a_{2p-1}a_{2p}}\wedge e^{a_{2p+1}}\wedge \cdots \wedge e^{a_{2n-1}}.
f_{ug}(\tau_{uh}) =(f_e \pi_{(ug)^{-1}})(\pi_{uh}\tau_e)=(f_e \pi_{g^{-1}u^{-1}})(\pi_{uh}\tau_e)=(f_e\pi_{g^{-1}})(\pi_{h}\tau_e)=f_g(\tau_h).
f(x) = f(x^{(\mathbb{T})}) + \sum_{k\in\mathbb{T}}\partial_k f(x^{(\mathbb{T})})x_k + \frac{1}{2}\sum_{k,l\in\mathbb{T}}\int_0^1 (1-t)\partial_{kl}f(x^{(\mathbb{T})}+t(x-x^{(\mathbb{T})}) ) x_kx_l\, d t.
\frac{M_n' (1)}{M_n (1)}= \frac{I_{n+1,n}(\omega)}{\sqrt{5}I_{n,n}(\omega)}\, \Big( 1 +{\mathcal{O}}(q^n)\Big) - \frac{2}{5}\, , *[0.3cm] \frac{M_n'' (1)}{M_n (1)}= \frac{I_{n+2,2}(\omega)}{\sqrt{5}I_{n,n}(\omega)}\, \Big( 1 +{\mathcal{O}}(q^n)\Big) - \frac{11}{5}\, \frac{M_n' (1)}{M_n (1)}- \frac{2}{5}\, .
m(T) =d\left(0,\sigma(T)\right) & =\inf\{|\alpha-\alpha_1|,|\alpha-\alpha_2|, |\alpha-\alpha_3|,\dots, |\alpha-\alpha_k|, \alpha\}& = \min\{|\alpha-\alpha_1|,|\alpha-\alpha_2|, |\alpha-\alpha_3|,\dots, |\alpha-\alpha_k|, \alpha\}.
\langle \mathrm{T r}Z^{J}\mathrm{T r}\bar{Z}^{J}\rangle = J N^{J}\Bigg \{1 + \sum_{h = 1}^{\infty}{\frac{a_{h , 4 h}}{( 4 h ) !}}{\frac{J^{4 h}}{N^{2 h}}}+ \ldots \Bigg \}
\delta(\bigwedge\nolimits^n\sigma_{2n+1}) =\delta(S^{\otimes 2})-\sum_{p=1}^{n-1}\delta(\bigwedge\nolimits^p\sigma_{2n+1}) =2^{n+1}\delta(S)-\sum_{p=1}^{n-1}\delta(\bigwedge\nolimits^p\sigma_{2n+1}).
0 \leq K_{1}(w,\lambda(t)) = \int_{0}^{\infty}\frac{R dR}{2 w (1+R^{2})^{3}}\left(\frac{4 R^{2}\lambda(t)^{2}w^{2}}{1+R^{2}\lambda(t)^{2}+w^{2}+\sqrt{(1+(R \lambda(t)+w)^{2})(1+(R\lambda(t)-w)^{2})}}\right) &\leq C \int_{0}^{\infty}\frac{R^{3}dR}{(1+R^{2})^{3}}\frac{\lambda(t)^{2}w}{1+w^{2}}\leq C \lambda(t)^{2}\frac{w}{1+w^{2}}
\eta'_i := \sigma^{-1}_{i2}\circ \sigma_{i1}= \sigma^{-1}_{i1}\circ \sigma_{i2}, \theta' := \sigma^{-1}_{21}\circ \theta \circ \sigma_{11}= \sigma^{-1}_{22}\circ \theta \circ \sigma_{12}= \sigma^{-1}_{11}\circ \theta^{-1}\circ \sigma_{21}= \sigma^{-1}_{12}\circ \theta^{-1}\circ \sigma_{22},
\frac{2^{-\lambda_p(\alpha)}\cdot\Gamma_p(\frac12)}{\Gamma_p(1-\frac12\alpha)\Gamma_p(\frac12+\frac12\alpha)}=-\frac{2\Gamma_p(1+\frac12\alpha)}{\Gamma_p(1+\alpha)\Gamma_p(1-\frac12\alpha)}.
\gamma_1-\frac{1}{(1-m)\beta}= \frac{\beta(n-2)-\sqrt{\beta^2(n-2)^2-4(n-2)}}{2}-\frac{1}{(1-m)\beta}= \frac{2(n-2)}{\beta(n-2)+\sqrt{\beta^2(n-2)^2-4(n-2)}}-\frac{n+2}{4\beta}= -\frac{(n+2)\sqrt{\beta^2(n-2)^2-4(n-2)}+(n-2)(n-6)\beta}{4\beta\left(\beta(n-2)+\sqrt{\beta^2(n-2)^2-4(n-2)}\right)}< 0\quad\mbox{if}\quad\beta>\frac{1}{2m}
S_{fi}^{(2)}=-\int \int dt_{1}dt_{2}\,\theta (t_{2}-t_{1})\int d{\bf r_{2}}\,\Psi_{{\bf p^{\prime}}}^{(+)\,\dagger}({\bf r_{2}})\,\tau_{3}\,H_{int}({\bf r_{2}},t_{2})\,H_{int}({\bf r_{2}},t_{1})\,\Psi_{{\bf p}}^{(+)}({\bf r_{2}})\;,
\phi_2 = \int_{z}^0 \int_{-1+\beta b}^{z'}\dfrac{\zeta}{h_b}\big( 1 + \dfrac{h}{h_b}\big) \Delta_X \psi \: \mathrm{d}z'' \mathrm{d}z' = - (\dfrac{z^2}{2}+ h_b z) \dfrac{\zeta}{h_b}\big( 1 + \dfrac{h}{h_b}\big) \Delta_X\psi.
\mathcal{E}_{X,\mathrm{par},p}(z,s)= \delta_{p,q}\Im(\sigma_{q}^{-1}z)^{s}+\alpha_{p,q}(s)\Im(\sigma_{q}^{-1}z)^{1-s}+O\left(\big(1+\Im(\sigma_{q}^{-1}z)^{-\Re(s)}\big)e^{-2\pi\Im(\sigma_{q}^{-1}z)}\right),
| K_t^\gamma(x) |^2=\int_{[0, \infty)}\inf_{\epsilon > 0}\{(1 - \epsilon)^2+\epsilon^2 t^2 \lambda^{-2 \gamma}\}d\mu_{x}(\lambda)=\int_{[0, \infty)}\frac{t^2}{t^2 + \lambda^{2 \gamma}}d\mu_{x}(\lambda).
g_2(\Lambda)= \frac{1}{16 r_1^2}\frac{64}{3}r_2^2(1+3r^2)=\frac{1}{16 r_1^2}60G_4(\tau)\xrightarrow{\:q \to 0 \:}\frac{1}{16 r_1^2}\frac{(2\pi)^4}{12}. \textnormal{Therefore,}r_2\xrightarrow{\:q \to 0 \:}\pi^2/4.
p_{2 n}= - \left\{a_{2 n - 2}\, \omega_{1}+ a_{2 n - 4}\, \omega_{2}+ \cdots + a_{4}\, \omega_{n - 2}+ a_{2}\, \omega_{n - 1}\right\}- \frac{2 n - 1}{2 ( 2 n + 1 )}\, a_{2 n - 1}\, .
S_{M 5}= \int d^{6}x \left( - \sqrt{- \det ( g_{\mu \nu}+ i \,{\cal H}^{*}{}_{\mu \nu})}- \frac{\sqrt{g}}4{\cal H}^{* \mu \nu}{\cal H}_{\mu \nu}\right) + \int_{{\cal M}_{7}}I_{7}\ ,
\|g_k\|_q\;\le\;\|T\sigma^\ast(g_{k-1})\|_q\;\le\;\|T\|_q\,\|g_{k-1}\|_q^q\;\le\;c\cdot C^{(1+\frac{k-1}{2})q}\;\le\;C^{\frac{3}{2}-q+q+\frac{k-1}{2}}\;=\;C^{1+\frac{k}{2}}