harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left\langle A_{q_k}-\binom{2k_1}{k_1}^{-1}\binom{2k_2}{k_2}^{-1}H_{q_k^2}, H_{v^{2k_1}\otimes u^{2k_1}}\right\rangle = H_{v^{2k_1}\otimes u^{2k_1}}(q_k)-\left\langle H_{q_k^2}, A_{v^{k_1}\otimes u^{k_1}}\right\rangle & = H_{v^{2k_1}\otimes u^{2k_1}}(q_k)-H_{q_k^2}(v^{k_1}\otimes u^{k_1})=0.
\left[\begin{array}{cc}\ast 0 0 \ast\end{array}\right], \mbox{for $A(t)$, $B(t)$, $A'(t)$, $B'(t)$, $F(t)$ and $G(t)$}, I(t)=\left[\begin{array}{c}\ast 0\end{array}\right],J(t)=\left[\begin{array}{cc}\ast 0\end{array}\right].
\sum_{k=0}^{(n-1)/2}\frac{(q^{1-n};q^2)_k (q^{1+n};q^2)_k (q;q^2)_{2k}}{(q^2;q^2)_{2k}(q^{6-n};q^6)_k (q^{6+n};q^6)_k}[8k+1]q^{2k^2}=q^{-(n-1)/2}[n]\biggl(\frac{-3}{n}\biggr).
\Gamma(Q_\infty)= \biggl\{k(q^e+1)+\ell q\frac{q^e+1}{q+1}+mq^3 \mid 0 \le \ell \le q, 0 \le m < \frac{q^e+1}{q+1}, k<0, k(q^e+1)+\ell q\frac{q^e+1}{q+1}+mq^3 \ge 0 \biggr\},
S_{e f f}= \int{d^{4}x \left({\Pi^{i}\stackrel{\bf{\cdot}}{A_{i}}+{\cal P}\stackrel{\bf{\cdot}}{C}+{\overline{{\cal P}}}\stackrel{\bf{\cdot}}{\overline{C}}+ \Pi^{0}\stackrel{\bf{\cdot}}{A_{0}}- H_{0}- \left[{\Omega , \Psi}\right]}\right)},
W\left(\psi_{a_{0}},\ldots,\psi_{a_{m}}\right) =C\left\vert\begin{array}[c]{ccc}1 \cdots 1 a_{0}^{2}\left( 1-a_{0}^{2}x\right)^{-1}\cdots a_{m}^{2}\left(1-a_{m}^{2}x\right)^{-1}\vdots \ddots \vdots a_{0}^{2m}\left( 1-a_{0}^{2}x\right)^{-m}\cdots a_{m}^{2m}\left(1-a_{m}^{2}x\right)^{-m}\end{array}\right\vert
\varkappa(m,p_1,p_2):=\sum_{s=0}^{n}\sum_{\nu\in I_s}k^{\frac{sr}{2}(\frac{1}{3p_2'}-\frac{1}{p_1})}(\prod_{\nu_j>k}\nu_j)^{\frac{r}{2}(\frac{1}{3p_2'}-\frac{1}{p_1})}|m(\nu)|^r<\infty,
\mathcal{L}_{K, (is_1+js_2, is_2)}(\alpha_1\alpha_3, \alpha_2)=\sum_{\substack{i_1>i_2>0 i_1\neq N}}\frac{\alpha_1^{q^{i_1}}\alpha_2^{q^{i_2}}}{(\theta^{q^{i_1}}-t)^{is_1+js_2}(\theta^{q^{i_2}}-t)^{is_2}}+\sum_{N=i_1>i_2>0}\frac{\alpha_1^{q^{i_1}}\alpha_2^{q^{i_2}}}{(\theta^{q^{i_1}}-t)^{is_1+js_2}(\theta^{q^{i_2}}-t)^{is_2}}
b_{1}^{2}=\nu^{4}\displaystyle\frac{2c}{(\mu+\lambda)(\mu-\lambda)}+\nu^{2}\displaystyle\frac{(\mu-\lambda)(-3\mu+2\lambda)-2c}{(\mu+\lambda)(\mu-\lambda)}+\displaystyle\frac{\mu}{\mu+\lambda}\left(1-c\lambda(\mu-\lambda)\right).
\langle \partial_z \Phi (z) \partial_{z}\Phi (0) \rangle= \lim_{\alpha \rightarrow 0}{1 \over \alpha^2}\partial_x \partial_y\langle e^{i\alpha \Phi (x)}e^{-i\alpha \Phi (y)}\rangle |_{{x=z}\atop{y=0}}.
\frac{\partial}{\partial r}C^{0}_{[p,q]}\left(u, v \right) = \frac{1}{q-p}\left( \varphi \left( \Phi^{-1}((q-p)u+p)\right) - u \varphi(\Phi^{-1}(q)) - (1-u) \varphi(\Phi^{-1}(p)) \right) \varphi(\Phi^{-1}(v))
R^n_{\frac{it}{\sigma \sqrt{n}}, f}1(v) - H_n(t) = \lambda^n_{\frac{it}{\sigma \sqrt{n}}}\Pi_{\frac{it}{\sigma \sqrt{n}}, f}1(v) + N_{\frac{it}{\sigma \sqrt{n}}, f}^n 1(v) - \left[ 1 + (it)^3 \frac{m_3}{6 \sigma^3 \sqrt{n}}+ it \frac{b(f, v)}{\sigma \sqrt{n}}\right] e^{- \frac{t^2}{2}}& =: \sum_{k=1}^4 J_k(t),
(\Delta_{{1}}\Delta_{{2}}\ldots \Delta_{{k}}V)(x)=0 \forall \Delta_{{1}}, \Delta_{{2}},\ldots , \Delta_{{k}}\in Lie\{f,g\}\setminus \{g\}\mathrm{with}\, \, \sum_{p=1}^{k}order_{\{f,g\}}\Delta_{{p}}\le N
\nabla_x f(x^k)+\left[\displaystyle \sum_{i=1}^p [y^I]^k_i\nabla^2c_i(x^k)-AA^T-{\cal J}c(x^k)^T{\cal J}_{z^I}\Psi_{\varepsilon}([y^I]^k,[z^I]^k){\cal J}c(x^k)\right]\lambda^k=0.
P_0Q(k)P_0 =P_0Q_0(k)P_0+\frac{\lvert k\lvert}{\omega(k)}P_0Q_0(k)(k\cdot \nabla K(\xi-d\Gamma(m))Q(k)P_0+P_0o_1(k)P_{0}&=P_0+\frac{\lvert k\lvert}{\omega(k)}P_0(\widehat{k}\cdot \nabla K(\xi-d\Gamma(m)))(1-P_0)Q(k)P_0 &+\left(\frac{\lvert k\lvert}{\omega(k)}-C_\omega \right)\widehat{k}\cdot uP_0Q(k)P_0+C_\omega\widehat{k}\cdot uP_0Q(k)P_0+P_0o_1(k)P_{0}
g(x,y,s,k)\in S^{n}_{{\rm loc\,}}(1;\Omega)\bigcap C^\infty_0(\Omega), &g(x,y,s,k)\sim\sum^\infty_{j=0}g_j(x,y,s)k^{n-j}S^{n}_{{\rm loc\,}}(1;\Omega), &g_j(x,y,s)\in C^\infty_0(\Omega),\ \j=0,1,2,\ldots,
V_2(t)= \exp i{e\over \hbar c}\int\! \int d\vec x d \vec yJ_0^{(2)}(\vec x,t)\left[ g_2(\vec x, \vec y){\theta_2^{(2)}(\vec y, t)}+ G_p(\vec x, \vec y)\partial_k a_{k}^{(2)}(\vec y, t) \right] \,.
\sum_{\substack{d> D (d,r)=1}}\int_{0}^{Y/d^2}\psi(v)dv=\dfrac{-2\varphi(r)}{r}\zeta\left(-\frac{1}{2}\right)Y^{1/2}+O(D)+O\left(d(r)\dfrac{Y}{D^2}\right).
W_{\vec{k}, \vec{l}}(\vec{y}, \vec{x}) = \overline{W_{\vec{k}, \vec{l}}}(\vec{x}, \vec{y}), \tilde{W}_{\vec{k}, \vec{l}}(\vec{y}, \vec{x}) = \overline{\tilde{W}_{\vec{k}, \vec{l}}}(\vec{x}, \vec{y}), \check{W}_{\vec{k}, \vec{l}}(\vec{y}, \vec{x}) = \overline{\check{W}_{\vec{k}, \vec{l}}}(\vec{x}, \vec{y}),
u_\tau^k=\frac{\xi_2{+}\tau\xi_1{+}\xi_0\tau^2}{\chi_2{+}\tau\chi_1{+}\chi_0\tau^2}v_\tau^k+\frac{2\chi_2{+}\tau\chi_1}{\chi_2{+}\tau\chi_1{+}\chi_0\tau^2}u_\tau^{k-1}-\frac{\chi_2}{\chi_2{+}\tau\chi_1{+}\chi_0\tau^2}u_\tau^{k-2},
P_{\mu,x}^{n}\left(L_{x}g^{-1},A\right)=\int_{G}{\bf 1}_{g.A}\left(g.x,L_{g.x}s\right)\frac{ds\nu}{d\nu}(\pi(g.x))d\mu^{(n)}(s)=P_{\mu,g.x}^{n}\left(L_{g.x},g.A\right).
N_{gen.}(r,t)=a(t)\left[\frac{1}{2\pi}\log\left(\frac{r}{\Lambda\kappa(t)}\right)-\frac{1}{2\pi(1-\mu)}+ \frac{1}{2\pi}\frac{2}{\left (\frac{r}{\Lambda}\right)^{2(1-\mu)}-1}\log\left (\frac{r}{\Lambda\kappa(t)}\right)\right].
\mathcal{F}_n(u,s): =\frac{1}{\Gamma(s)(e^{2\pi i s}-1)}\int_\mathcal{C}t^{s-1}\mathcal{G}_n(u,t)dt &=\frac{1}{\Gamma(s)}\int_{\varepsilon}^\infty t^{s-1}\mathcal{G}_n(u,t)dt+\frac{1}{\Gamma(s)(e^{2\pi i s}-1)}\int_{C_\varepsilon}t^{s-1}\mathcal{G}_n(u,t)dt.
\nabla_X D^r_Y(Z) = \nabla_X r(\nabla_Y Z) - \nabla_X \nabla_{r(Y)}Z, D^{r}_{[X,Y]}(Z) = r(\nabla_{[X,Y]}Z)- \nabla_{r([X,Y])}Z, D^{r,*}_X (g^\flat(\nabla_Y Z)) = -g^\flat(\nabla_X \,r(\nabla_Y Z)) - g^\flat(\nabla_{r(X)}\nabla_Y Z).
\mathfrak{M}(\xi, v, V(s))= \frac{\mathbf{e}_3 \langle V(s)\rangle}{|\xi|( |\xi| + \hat{V}(s )\cdot \xi)}- \frac{V_3(s)}{|\xi|( |\xi| + \hat{V}(s )\cdot \xi)^2}\big(\xi - \hat{V}(s) \hat{V}(s)\cdot \xi \big).
{\mathbf a}= \left[ \begin{array}{cc}1, -1\end{array}\right],{\mathbf b}= \left[ \begin{array}{cc}-1, -8/7\end{array}\right],C = \left[ \begin{array}{cc}1, 3/8 3/8, 7/16\end{array}\right],D = \left[ \begin{array}{cc}1/2, 1/2 1/2, 13/64\end{array}\right]\,.
Ts_t(T\pi(v_p)) =T\Phi_{\xi_0(p)^{-1}}\left(T\varphi_t(v_p)-\left(TL_{\xi_0(p)^{-1}}T\xi_0(v_p)\right)_P(p)\right) &=T\Phi_{\xi_0(p)^{-1}}\left(T\varphi_t(v_p)-\left(\Gamma_0(v_p)\right)_P(p)\right).
0 \leq \int_1^3\int_{B_2}|u^{(m)}|^2\phi^{(m)}+|u^{(m)}||\nabla\phi^{(m)}|+\frac{\partial\phi^{(m)}}{\partial t}+\nabla\phi^{(m)}\cdot h(V_t^{(m)},\cdot)\, d\|V_t^{(m)}\|dt &=I_1^{(m)}+I_2^{(m)}+I_3^{(m)}+I_4^{(m)},{\rm say}.
\tilde{g}_{l,l^{\prime}}^{j}\left( k\right) =\left( 2j+1\right)\sum_{m,m^{\prime}}\left( -1\right)^{m}\left(\begin{array}[c]{ccc}l l^{\prime}j m -m^{\prime}0\end{array}\right) \hat{G}_{lm,l^{\prime}m^{\prime}}\left( k\mathbf{\hat{e}}_{z}\right) .
|u_n(t)|\leqslant\frac{|\Delta_{-2,n}|}{|\Delta_n|}e^{\mu_nT}+\frac{|\Delta_{-1,n}|}{|\Delta_n|}e^{\mu_nT}+\sum\limits_{k=0}^{N-1}\frac{|\Delta_{k,n}|}{|\Delta_n|}e^{q_{k,n}T}.
\mathcal{H}_{j}:=\oplus_{\sum_{j}u_{j}+\sum_{i}d_{i}v_{i}=j}k[Z_{1},\ldots,Z_{r}]S_{1}^{u_{1}}\ldots S_{m}^{u_{m}}X_{1}^{v_{1}}\ldots X_{n}^{v_{n}}.
\mathbb{T}_{\varrho}(M,\{\mathbf{h}_p^{M}\}_{p=0}^3)=\mathbb{T}_{{\psi_{_1}}}(M_1,\{\mathbf{h}_p^{M_1}\}_{p=0}^{3}) \;\mathbb{T}_{{\psi_{_2}}}(M_2,\{\mathbf{h}_p^{M_2}\}_{p=0}^{3}).
\Delta_{k}= \left| \begin{array}{c c c}{D_{1}^{1}}&{\cdots}&{D_{1}^{k}}\\{\vdots}&{}&{\vdots}\\{D_{k}^{1}}&{\cdots}&{D_{k}^{k}}\\ \end{array}\right| \quad , \qquad k = 1 , \ldots , r
\Gamma_{+}^{(1)*}= \left(\begin{array}{ccc}1 0 0\end{array}\right)^T, \Gamma_{0}^{(1)*}= \left(\begin{array}{ccc}0 1 0\end{array}\right)^T, \Gamma_{-}^{(1)*}= \left(\begin{array}{ccc}0 0 q^{-2}\end{array}\right)^T.
\begin{aligned}x_{\iota,\iota+1}+ u_{\iota,\iota+1}^{(i)}x_{\iota+1,\iota+1}- x_{\iota\iota}u_{\iota,\iota+1}^{(\mu)}= x_{\iota,\iota+1}\mbox{and}y_{\iota,\iota+1}+ u_{\iota,\iota+1}^{(\mu)}y_{\iota+1,\iota+1}- y_{\iota\iota}u_{\iota,\iota+1}^{(i)}= y_{\iota,\iota+1}, \end{aligned}
Z_{0}\big |_{\beta \omega \gg 1}\sim e^{- \frac{5 \pi^{2}}{1 2 \beta \omega}- \frac{2 \pi^{2}}{3 \operatorname{s i n h}\frac{2 \pi^{2}}{\beta \omega}}}\operatorname{c o s h}\biggl ( \frac{\pi^{2}}{\beta \omega}\biggr ) .
s_{d_0}(\hat{\mathcal{O}}^+) \geq s_{d_0}(\bar{\mathcal{O}}^+) - \left(\frac{2}{\sqrt{2}- 1}\right)^{1/2}\frac{|\hat{\mathcal{H}}_{\check{d}_{\xi}}-H g_0^*|_{S_2}}{s^{1/2}_{d_0}(Hg_0^*)}\geq s_{d_0}(\bar{\mathcal{O}}^+) - \frac{s_{d_0}(\bar{\mathcal{O}}^+)}{2}= \frac{s_{d_0}(\bar{\mathcal{O}}^+)}{2},
\int_0^\infty |B|\,dt=\left|\mathbb I-\frac{2x\otimes x}{|x|^2}\right|\int_0^\infty \frac{-\tau e^{-\tau}+1-e^{-\tau}}{\tau^2}\,d\tau=\left|\mathbb I-\frac{2x\otimes x}{|x|^2}\right| \leq C
G(t_0,\dots,t_{J})=\sup_{|\gamma|\le 10^{-J-1}}\frac{1}{9}\Bigl|\frac{e(\sum_{j=0}^{J}t_{j}10^{-j}+10\gamma)-1}{e(\sum_{j=0}^{J}t_{j}10^{-j-1}+\gamma)-1}-e\Bigl(\sum_{j=0}^{J}\frac{a_0 t_{j}}{10^{j+1}}+a_0\gamma\Bigr)\Bigr|.
e^{SO_{red}}(T) <2f(1,3)+f(3,4)+\frac{2n-5}{3}f(1,4)+\frac{n-7}{3}f(4,4) &=\frac{2n+2}{3}f(1,4)+\frac{n-5}{3}f(4,4)+2M_{13}+M_{34}&<\frac{2n+2}{3}f(1,4)+\frac{n-5}{3}f(4,4)+M_{12}+M_{24}&=f(1,2)+f(2,4)+\frac{2n-3}{3}f(1,4)+\frac{n-6}{3}f(4,4) &<\frac{2n+2}{3}f(1,4)+\frac{n-5}{3}f(4,4).
\dot{\xi}_j = \mu_0^{n-4s+2}c\left[b_j^{n-2s}\nabla H(q_j, q_j) - \sum_{i\neq j}b_j^{\frac{n-2s}{2}}b_i^{\frac{n-2s}{2}}\nabla G(q_j, q_i)\right] + h(t)
\frac{d}{dr}H(r)=-r^{-1}\textnormal{H}_{\nu+1\mu+1}^{\mu+1 \ 0}\left[r\ \Big|\begin{array}{cccc}(c_{1},\gamma_{1}) \cdots (c_{\nu},\gamma_{\nu}) (0,1) (d_{1},\delta_{1}) \cdots (d_{\mu},\delta_{\mu}) (1,1)\end{array}\right].
\Delta ( m ; x ) \approx \frac{h}{( 2 \pi )^{d}}\sum_{k = - N}^{N}\frac{1}{e^{k h}}\left[ \frac{\pi}{z_{k}+ 1 / \Lambda^{2}}\right]^{d / 2}\exp \left[ - z_{k}m^{2}- \frac{x^{2}}{4 ( z_{k}+ 1 / \Lambda^{2})}\right] ,
\begin{array}{|l|l|l|l|l|l|l|l|}\hline s s=1 s=2 s=3 s=4 s=6 s=8 s=12 \hline{\rm quot. sing.}A_1 A_1A_2 A_1^2 A_3 A_1^4 A_4 A_1^3 A_6 A_1^4 A_8 A_1^6 A_{12}\hline \end{array}
G_{r s}= \left( \begin{array}{l l}{{\frac{H_{0}}{H_{1}}}}{0}{0}{{\frac{H_{2}}{H_{3}}}}\end{array}\right) ={\frac{H_{2}}{H_{3}}}\left( \begin{array}{l l}{( \mathrm{R e}\, U )^{2}}{0}{0}{1}\end{array}\right) \; ,
=\frac{31!}{2}\cdot60^{31}\cdot\frac{160!}{2}\cdot\frac{12^{160}}{3}\cdot\frac{320!}{24^{80}}\cdot\frac{6^{320}}{2}\cdot\frac{320!}{8!^{40}}\cdot\frac{2^{320}}{2}\cdot\frac{160!}{16!^{10}}
\mathrm{TV}_{(t_1,t_2)}(g^{\mathbf{X}}_{L,T}) =\mathrm{TV}_{(t_1, t_2)}(f^{\mathbf{X}}_{L,T})\leq f^{\mathbf{X}}_{L,T}(t_1)+f^{\mathbf{X}}_{L,T}(t_2)=g^{\mathbf{X}}_{L,T}(t_1)+g^{\mathbf{X}}_{L,T}(t_2)+2.
\frac{\partial h}{\partial t}+ \displaystyle\sum_{i=1}^{d}\,f^{\prime}_{i}(c)\frac{\partial h}{\partial x_{i}}=\left(-\Delta_{x,t}+1\right)^{\frac{1}{2}}\left(-\Delta_{c}+1\right)^{\frac{r}{2}}g\,\,\mbox{in}\,\,\mathcal{D}^{\prime}\left(\mathbb{R}^{d}_{x}\times\mathbb{R}_{c}\times (0,\infty)\right),
\sum_{n=0}K^ny_{1,n}''-\frac{(n-1)(n-3)}{4}K\left(\sum_{n=0}a_nK^nx^{2n}\right)\left( \sum_{n=0}K^ny_{1,n}\right)=-\left(\sum_{n=0}K^n\lambda_n \right)\left( \sum_{n=0}K^ny_{1,n}\right),
K_{s}(x)= \sqrt{\frac{\pi}{2x}}e^{-x}\left(1+ \sum_{k=1}^{\infty}\frac{1}{(2x)^k}\frac{\Gamma\left(s+k+\frac{1}{2}\right)}{k!\Gamma\left(s-k+\frac{1}{2}\right)}\right)\ .
\mathcal{H}^{\rm LB}(D) =\Big\{\big[W^{\rm LB}(D, C')\big] ~\Big|~ \C' \in{\rm Col}_{X^2}^{\rm LB}(D) \Big\}&=\Big\{\mu_2^{\ast}\big(\big[W^{\rm SB}(D, C)\big]\big) ~\Big|~ \C \in{\rm Col}_{(B,X)}^{\rm SB}(D) \Big\}&=\mu_2^{\ast}\big(\mathcal{H}^{\rm SB}(D)\big).
\AA(\Gamma)_{\mathfrak{n}}=\bigoplus_{\substack{\ell=2 \ell\neq3}}^\infty\;\bigoplus_{m=0}^\infty n_{\ell,m}L(\ell+m,\ell) \quad\oplus\;\bigoplus_{k=3}^\infty V_k^*\;\oplus\:\bigoplus_{k=1}^\infty V_k^{**}\;\oplus\:L(0,0).
P\left(C \geq N|\mathcal{D}= \mathcal{G}_{\sigma}\right)=\sum_{i=0}^{\ell}P\left( C-C_s \geq N-i \right) \times P\left( C_s =i \right) &=\sum_{i=0}^{\ell}\left(1-F(N-i-1,\ell q-\ell,1/2) \right) \times B(i, \ell , 1/2+ \epsilon)
\begin{cases}x_{1}^{\left(t+1\right)}=\;\delta_{1}x_{2}^{\left(t\right)}y^{\left(t\right)}\medskip x_{2}^{\left(t+1\right)}=\;\gamma_{2}x_{1}^{\left(t\right)}y^{\left(t\right)}\medskip y^{\left(t+1\right)}=\;\bigl(\left(1-\gamma_{2}\right)x_{1}^{\left(t\right)}+\left(1-\delta_{1}\right)x_{2}^{\left(t\right)}\bigr)y^{\left(t\right)}.\end{cases}
Q_a\star Q_b =(\sum_{a\leq_{*}\tilde{a}}M_{\tilde{a}})\star (\sum_{b\leq_{*}\tilde{b}}M_{\tilde{b}}) begin{align*}5pt] =\sum_{a\leq_{*}\tilde{a},\b\leq_{*}\tilde{b}}M_{\tilde{a}}\star M_{\tilde{b}}begin{align*}5pt] =\sum_{a\leq_* \tilde{a},\b\leq_* \tilde{b}}\ \sum_{\Theta\in R(\tilde{a},\tilde{b})}M_{\tilde{a}\Theta \tilde{b}},
\bigl|\bigl\{(x,y): x,y\in\Bbb Z_r\setminus\{0,\pm1\},\x^2-\lambda y^2=1-\lambda\bigr\}\bigr|\cr=\, r-\eta(\lambda)-4-(1+\eta(1-\lambda))-(1+\eta(1-\lambda^{-1}))\cr=\, r-6-\eta(\lambda)-\eta(1-\lambda)-\eta(1-\lambda^{-1}).
Ci(x)\sim\frac{sinx}{x}(1-\frac{2!}{x^{2}}+\frac{4!}{x^{4}}-\frac{6!}{x^{6}}\cdot\cdot\cdot)-\frac{cosx}{x}(\frac{1}{x}-\frac{3!}{x^{3}}+\frac{5!}{x^{5}}-\frac{7!}{x^{7}}\cdot\cdot\cdot).
M_{A D M}^{2}= ( 8 G_{N})^{- 1}\left({}{\cal M}_{\infty \, a b}({\vec{v}}^{a \, T}\mu_{+}{\vec{v}}^{b}) + \left[{2}{\cal L}_{a c}{\cal L}_{b d}({\vec{v}}^{a \, T}\mu_{+}{\vec{v}}^{b}) ({\vec{v}}^{c \, T}\mu_{+}{\vec{v}}^{d}) \right]^{\frac{1}{2}}\right) ,
\left( \hat{\Gamma}_{0}\hat{\Gamma}_{i}\hat{\Gamma}_{j}\right)^{\dagger}= \hat{\Gamma}_{j}^{\dagger}\hat{\Gamma}_{i}^{\dagger}\hat{\Gamma}_{0}^{\dagger}= \hat{\Gamma}_{j}\hat{\Gamma}_{i}( - \hat{\Gamma}_{0}) = \hat{\Gamma}_{0}\hat{\Gamma}_{i}\hat{\Gamma}_{j}
\Sigma_{e f f}^{A 1}= \frac{1}{1 6 \pi^{2}}\sum_{N}\int_{0}^{1}d x \left[ - 2 m_{W}\bar{t}_{N}( 1 - x ) + 4 \gamma_{4}\bar{t}_{N}\gamma_{4}m_{W}\right] \ln \left( \mu_{N}^{2}+ \frac{m_{W}^{2}x^{2}}{1 - x}\right) \bar{t}_{N},
{\cal R}_{z \doteq z_{n}= \cdots = z_{0}}F = \mathrm{R e s}_{z = z_{0}}{\cal R}_{z_{n}= \cdots = z_{0}}\partial_{\bar{z}}^{- 1}F - \sum_{i = 0}^{N}{\cal R}_{z_{n}= \cdots = z_{0}}\mathrm{R e s}_{z = z_{i}}\partial_{\bar{z}}^{- 1}F .
\begin{aligned}\bigl||u+v|^{p_0-2}- |u|^{p_0-2}\bigr|\cdot |\mu_n\sigma_n(u)|\cdot|w| &\qquad\qquad\le c\cdot\bigl(|v|^{p_0}\cdot (1+(n^{3/4}|w|)^{p_0}) + 1 +|w|^{p_0/2}+ |u|^{p_0}\bigr).\end{aligned}
\sum_{d \leq j \leq \frac{n+d}{2}}f_{j,d}(n) = \sum_{d \leq j \leq \frac{n+d}{2}}{n-2j+2d-1 \choose d-1}+{n-2j+2d-2 \choose d-1}= \sum_{i=d-1}^{n-1}{i \choose d-1}={n \choose d}.
\aligned \|f\|_{B^{\beta_1}_{p,q_2}}\leq C\|f\|_{B^{\beta_2}_{p,q_1}}, \|f\|_{B^{\gamma_2}_{p_2,q}}\leq C\|f\|_{B^{\gamma_1}_{p_1,q}}, \|f\|_{H^{s,p}}\leq \|f\|_{B^r_{p,q}}, \|f\|_{H^{s,2}}=\|f\|_{B^s_{2,2}}\leq \|f\|_{B^{r}_{2,q}}.\endaligned
P_{\lambda}(x, s) = (x/s)^{\lambda}\frac{(x/s;q)_{\infty}}{(q^{-\lambda}x/s ;q)_{\infty}}, y(x)=x^{\mu '}\frac{(q/(\beta_1 x), q/(\beta_2 x), q/(\beta_3 x) ;q)_{\infty}}{(q/(\alpha_1 x), q/(\alpha_2 x), q/(\alpha_3 x) ;q)_{\infty}}
\langle T^*x,y \rangle_{p',p}= \langle x,Ty \rangle_{p',p}= \mathrm{tr}(xD^{1/p'}TyD^{1/p}) =\mathrm{tr}\big(xD[m_a^R(\sigma_{-\mathrm i /p}(y^*))]^*\big) = h\big([m_a^R(\sigma_{-\mathrm i /p}(y^*))]^*x\big).
|\varphi_{2}(\zeta_{1},\zeta_{2},\psi,\mu)|^{2}\leq \bar{l}_{2}(\zeta_{1})\|\zeta_{1}\|^{2}+\bar{\kappa}_{2}(\zeta_{2})|\zeta_{2}|^{2}+\bar{\omega}_{2}(\psi)\|\psi\|^{2}.
G [ \chi_{2}, \chi_{1}; \psi_{2}, \psi_{1}] ~ = \prod_{k}^{\infty}\biggl ( \int_{0}^{\infty}d N_{k}N_{k}^{-{\frac{1}{2}}}e^{-{\frac{{( \Delta \psi_{k})}^{2}}{2 N_{k}}}}{\cal G}[ \chi_{2 k}, \chi_{1 k}; N_{k}] \biggr ) ,
\gamma^{\mu \rho \nu}\nabla_{\rho}( \Psi_{\nu}^{\mp})^{A}- \gamma^{\mu \nu}\gamma^{5}\partial_{5}( \Psi_{\nu}^{\pm})^{A}- k \epsilon \tanh \left( k | y | - \frac{k \pi r_{c}}{2}\right) \gamma^{\mu \nu}\gamma^{\hat{5}}( \Psi_{\nu}^{\pm})^{A}- \frac{3}{2}k ( \sigma_{1})_{\; B}^{A}\gamma^{\mu \nu}( \Psi_{\nu}^{\mp})^{B}= 0 \ .
\phi^*\lambda-\lambda =\int_{0}^{1}\frac{d}{dt}\phi_{H}^{t*}\lambda dt=\int_{0}^{1}\phi_{H}^{t*}\left(d\iota_{X_{H_t}}\lambda+\iota_{X_{H_t}}d\lambda\right)dt = d\left(\int_{0}^{1}\left(\iota_{X_{H_t}}\lambda+H(t,\cdot)\right)\circ\phi_{H}^{t}dt\right)
\begin{aligned}\| \partial_{t}\varphi_{k}\|_{L^{2}(0,T;H^{-1})}+ \kappa \| \partial_{t}\sigma_{k}\|_{L^{2}(0,T;H^{-1})}& \leq C \left (1 +\kappa \| \sigma_{0}- \sigma_{\infty}(0)\|_{L^{2}}^{2}+ \kappa^{2}\|\partial_{t}\sigma_{\infty}\|_{L^{2}(0,T;L^{2})}^{2}+ \kappa \|\sigma_{\infty}\|_{L^{\infty}(0,T;L^{2})}^{2}\right ),\end{aligned}
{\frac{\pi^{2}}{8 a}}\int_{0}^{1}{d x{\frac{1}{\sqrt{1 - x^{2}}\left( 1 - \lambda^{2}+ \lambda^{2}x^{2}\right)}}\left[{\frac{( x^{4}- 3 \delta^{2}x^{2})}{\lambda^{2}( x^{2}+ \delta^{2})^{3}}}+{\frac{\left( 1 + \lambda^{2}- \lambda^{2}x^{2}\right)}{\left( 1 - \lambda^{2}+ \lambda^{2}x^{2}\right)}}{\frac{x^{2}}{( x^{2}+ \delta^{2})}}\right]},
\alpha_k<\delta_n^k \ \ \Rightarrow \ \S \left((\alpha_0,\dots,\alpha_{k-1},\alpha_k) \right )=(\alpha_0+1,\dots,\alpha_{k-1}+1,\alpha_k+1) S\left ((\alpha_0,\dots,\alpha_{k-1},\delta_n^k)\right ) = (0,\alpha_0,\dots,\alpha_{k-1}) .
\frac{|\partial B(x, R+s) \cap \Omega^c|}{(R+s)^2}= \frac{2\pi}{\sqrt{c}}\,\frac{s}{R}+ \frac \pi{8\, c \sqrt{c}}\,[h(K) + g]\,\left(\frac{s}{R}\right)^2 + O\left(s^{\frac 52}\right)\ \mbox{as}s \downarrow 0.
R(T) =\prod L_i(T) R^{\varepsilon_i}_{\alpha_i}(T) L^{-1}_i(T) &=\prod (L_i(T){\overline{L}\,}^{-1}_i) (\overline{L}_i R^{\varepsilon_i}_{\alpha_i}{\overline{L}\,}^{-1}_i) (\overline{L}_i L^{-1}_i(T)).
\langle H_y \rangle_{u_{\alpha}\otimes R_*}= (e_{\alpha}+ \alpha \|\lambda_y\|^2)\|R_*\|^2 + \|R_*\|^2_* + \alpha\langle V_y\rangle_{u_{\alpha}}\|R_*\|^2 +\alpha \|A^-_y R_*\|^2 &\geq \|R_*\|^2_* + C_1 \alpha \|R_*\|^2,
\frac{F(z)}{A(z)^2}-\frac{F(w)}{A(w)^2}= \sum_n\bigg\{\frac{F(s_n)}{A'(s_n)^2(z-s_n)^2}+ \frac{F'(s_n)}{A'(s_n)^2(z-s_n)}-\frac{F(s_n)A''(s_n)}{A'(s_n)^3(z-s_n)}\ \ \ \ \ \ \ \ - \frac{F(s_n)}{A'(s_n)^2(w-s_n)^2}- \frac{F'(s_n)}{A'(s_n)^2(w-s_n)}+\frac{F(s_n)A''(s_n)}{A'(s_n)^3(w-s_n)}\bigg\}
\;\|A^{(r_k-s)/2}u_k\|_{L^2}^2 \leq \; C\left(\|A^{(r_k-s)/2}e^{-\delta\nu A^s}u_{k-1}\|_{L^2}^2+\|A^{(r_k-s)/2}w_{k-1}(t_k)\|_{L^2}^2\right) \leq \;C\left[(\nu\delta)^{-\frac{2s-1}{s}}\|A^{(r_{k-1}-s)/2}u_{k-1}\|_{L^2}^2+\|A^{(r_k-s)/2}w_{k-1}(t_k)\|_{L^2}^2\right].
\mathbf{E}\Phi_{I,\lambda}(W) =\lambda\mathbf{E}k'(b)+\mathbf{E}|W_{\tau_{0}\wedge b}|+\mathbf{E}\sum_{j\ge1}(-1)^{j+\alpha}(W_{\tau_{j}\wedge b}-W_{\tau_{j-1}\wedge b})+\mathbf{E}\max\{0,2|m_{k'(b)+1}-W_{b}|-\lambda\},
\hat{\partial}^{+}\eta^{I}\bar{\alpha}^{I}| \Phi_{s}\rangle + \Bigl ( \hat{\partial}^{+}- \frac{2 \alpha^{+}}{2 s + d - 4}( \bar{\alpha}\hat{\partial}+ ( s - 1 ) \bar{\alpha}^{z}) \Bigr ) | \Lambda \rangle = 0 .
\psi^{k,m}(x ) := \frac{84}{\pi^3}x \zeta(3) + \frac{8}{\pi}x^3 \log (\pi x) - \frac{8}{\pi}\left( \frac{11}{6}+ \log 2 \right) x^3 - 3 x (1-x) &{}{}+ 48 \pi x^5 \sum_{n=0}^k \frac{| B(2n+2) | \left( 2^{2n+1}- 1 \right)}{(n+1) (2n+5)!}\pi^{2n}x^{2n}- \frac{96}{\pi^4}\sum_{\substack{n n \leq m}}\frac{d_n}{n^4}\sin n \pi x .
( \widehat{M}_{t}, \widehat{g}_{t}) = ( [ 0 , t ] \times [ 0 , 1 ] , \left( \parbox{1 6 p t}{\scriptsize 1 \makebox [ 1 p t ]{}0 \, 0 \makebox [ 1 p t ]{}1}\right) ) \; \; \; ,
g_L \left( \kappa_L \right) - 2\pi = L \kappa_L - \hat{g}\left( \kappa_L \right) - 2\pi =L \kappa_L - \arctan{\frac{4 \kappa_L \left(\kappa_L^2 - 1 \right)}{\kappa_L^4- 6 \kappa_L^2 + 1}}
M_{Y}(t) = 2p(1-p) \bigg[ \frac{(p_{\alpha_{+}}-p_{\alpha_{-}})}{(p_{\alpha_{-}}- \sigma t)(p_{\alpha_{+}}-\sigma t)}\bigg]\exp\bigg\{\mu t +\frac{1}{2}\alpha^{2}\sigma^{2}t^{2}\bigg\}\; \Phi(-\alpha \sigma t) .
\big[(\alpha I - Q)^{-1}\big]_{i,k}= \sum_{j\in E}\lim_{q\downarrow0}\frac{\Phi^+_j(q,0)\hat\Phi^+_j(q,0)}{q}[\kappa^+(\alpha,0)^{-1}]_{i,j}[\hat\kappa^+(\alpha,0)^{-1}]_{k,j},
\lambda' - \frac{2\kappa^\frac{N-4}{2}}{N-6}\lambda^\frac{N-4}{2}= \Big(\frac{5}{2(N-6)}- \frac{N-4}{N-6}\Big)|t|^{-\frac{2N-7}{2(N-6)}}p_0(t) + |t|^{-\frac{5}{2(N-6)}}p_0'(t) + O(|t|^{-\frac{N-3}{N-6}}).
a_{1}p_{h}q_{h}^{n-1}+a_{2}p_{h}^{2}q_{h}^{n-2}+\cdots+a_{n}p_{h}^{n}+h(b_{0}q_{h}^{n}+b_{1}p_{h}q_{h}^{n-1}+\cdots+b_{n-1}p_{h}^{n-1}q_{h})=0.
|\varphi_{t,t'}(z) - z| = |\varphi_{t,t'}(z) - \varphi_{t,t}(z)| & = \left|\int_{t}^{t'}\dot{\varphi}_{t,s}(z) ds\right| = \left|\int_{t}^{t'}p(\varphi_{t,s}(z), s) ds\right| & \leq \int_{t}^{t'}|p(\varphi_{t,s}(z), s)| ds \leq K(t' - t)
0=L^*\rho= (L_0^*+\varepsilon\Gamma^*)\rho_0(1+\sum_{i=1}^M \varepsilon^i h_i)+O(\varepsilon^{M+1}) = L_0^*\rho_0+\varepsilon(L_0^*(\rho_0 h_1)+\Gamma^*\rho_0)+\sum_{i=2}^M \left(L_0^*(\rho_0 h_i)+\Gamma^*(\rho_0 h_{i-1})\right) \varepsilon^i+O(\varepsilon^{M+1}).
\begin{aligned}A_{m,n}:= m\{0,2a\}+ \{0,2a, 3a\}+ \{0,a,a(2n+5)\}= \big( \{0\}\cup a \cdot [2, 2m+3] \big) + a \cdot \{0, 1, 2n+5\}= a \cdot \big( [0, 2m+4] \cup \{2n+5\}\cup [2n+7, 2m+2n+8] \big) \,.\end{aligned}
\rho(r) =2\pi \min \{2r_1,\,\big(\sum_{i \neq n}r_i \big) - r_n\}&=\begin{cases}2 \pi \big((\sum_{i \neq n}r_i ) - r_n\big) \textrm{if $\{n\}$ is maximal short}4\pi \,r_1 \textrm{otherwise}.\end{cases}
Z ( \gamma ) = Z_{m o n}( \gamma ) Z_{0}( \gamma ) ={\cal N}^{\prime}\sum_{l = - \infty}^{\infty}\exp \left[ - \frac{e^{2}}{2}V \left( \frac{S}{V}\nu + l \right)^{2}- \frac{e^{2}}{2}\nu^{2}\frac{S ( V - S )}{V}\right] .
\nu_{g}=\lim_{i\to\infty}\sum_{s\in F}\delta_{\left\{H_{0}gs\xi\right\}}\mu^{(n_{i})}(s)=\lim_{i\to\infty}\sum_{s\in F}\delta_{\left\{gH_{0}s\xi\right\}}\mu^{(n_{i})}(s)=g.\nu_{e}.
P_{{\mathop{\textrm{int}}}}^i =\Pr \left({{X_{is}}<{X_{ie}}}\right) &= \iint\limits_{{{x_{is}}<{x_{ie}}}}{{{f_{{X_{ie}}}}({x_{ie}}){f_{{X_{is}}}}({x_{is}})d{x_{ie}}d{x_{is}}}}&= \int_0^\infty{\Gamma ({m_i},\frac{{{m_i}}}{{\sigma_{is}^2}}{x_{ie}}){f_{{X_{ie}}}}({x_{ie}})d{x_{ie}}},
\frac{[\mathfrak h]_T}{[\hat{v}]_T}- \frac{[\mathfrak h]_{T'}}{[\hat{v}]_{T'}}= \frac{[\mathfrak h]_T -[\mathfrak h]_{T'}}{[\hat{v}]_T}+ \frac{[\mathfrak h]_{T'}([\hat{v}]_{T'}-[\hat{v}]_T)}{[\hat{v}]_T [\hat{v}]_{T'}}.
\partial_tg_\alpha(t) =\int_{B^+_1}\xi(y)D^\alpha\partial_t{v}(t,y)\,dy &=\int_{B^+_1}(-1)^{m+1}\xi(y)D^\alpha\l_0{v}(t,y)\,dy+\sum_{|\beta|\le m}\int_{B^+_1}\xi(y)D^\alpha D^\beta f_\beta dy.
C_{MLETPF}\leq \sum^{L}_{l=0}\Big(N_{l}(dc_{1}N_{y}+C_{l,d}c_{3}t_{N_{y}}) + de_{1}N_{l}log(N_{l})N_{y}\Big) & \leq \sum^{L}_{l=0}\Big(N_{l}(dc_{4}t_{N_{y}}+C_{l,d}c_{3}t_{N_{y}}) + de_{1}N_{l}log(N_{l})N_{y}\Big) & \leq \sum^{L}_{l=0}\Big(c_{2}N_{l}C_{l,d}t_{N_{y}}+ de_{1}N_{l}log(N_{l})N_{y}\Big)
\frac{f(z,\underline{u}(z))}{\underline{u}(z)}\leq\frac{f(z,u_{\lambda}(z))}{u_{\lambda}(z)}\Rightarrow f(z,\underline{u}(z))\leq\vartheta f(z,u_{\lambda}(z))\ (\mbox{recall that}\ \underline{u}=\vartheta u_{\lambda}).
u_M(t,x_0) = (4\pi(t+1))^{-\frac{d-1}{2}}(4\pi t)^{-\frac{1}{2}}\int_M^{M+1}e^{-\frac{|1-y_1|^2}{4t}}\,\mathrm dy_1 &\geq\frac{1}{2}(4\pi(t+1))^{-\frac{d-1}{2}}(4\pi t)^{-\frac{1}{2}}e^{-\frac{|\frac{1}{2}-M|^2}{4t}},
\begin{cases}2c_1+c_2 =2(w_1+w_3\kappa)-2w_3\theta\kappa\geq c, c_3 =w_2+w_4\kappa\geq c, c_3+c_4 =w_2+w_4\kappa-\big(w_2\gamma+w_4\kappa(2+\gamma)\big)\theta\geq c\end{cases}
S_{2}=\sum\limits_{m<-r_{1}}f(\alpha m)\sum\limits_{n=-\infty}^{\infty}\frac{f^{\ast}(\beta n)}{((m+r_{1})z+n+r_{2})^{s}}& =e(s/2)\sum\limits_{m>r_{1}}f(-\alpha m)\sum\limits_{n=-\infty}^{\infty}\frac{f^{\ast}(-\beta n)}{((m-r_{1})z+n-r_{2})^{s}}.
P_{\cal T}^{1 / 2}\left( k \right) = \sqrt{\frac{k^{3}}{2 \pi^{2}}}\left|{\frac{v_{k}}{m_{\mathrm{P l}}a}}\right| ={\frac{2^{\nu - 3 / 2}}{2 \pi}}{\frac{\Gamma \left( \nu \right)}{\Gamma \left( 3 / 2 \right)}}\left({\frac{H}{m_{\mathrm{P l}}}}\right) \left| C_{\mathrm{+}}\left( k \right) + C_{-}\left( k \right) \right| \left({\frac{y}{1 - \epsilon}}\right)^{3 / 2 - \nu},
2\cos\theta=\left\langle r', r \right\rangle =\left\langle a_1e_1+\cdots +a_ne_n, e_i-e_j \right\rangle &=\frac{a_i(n+1)-\sum_{k=1}^na_k}{n+1}-\frac{a_j(n+1)-\sum_{k=1}^na_k}{n+1}=a_i-a_j.
|w_n (t) - w_{n-1}(t)| = |(\tilde{w_n}(t)+ w_{n-1}(x_n)) - w_{n-1}(t)| \leq | \tilde{w}_n (t)| + |w_{n-1}(x_n) - w_{n-1}(t)| & \leq g(\tau_n) + \|w_{n-1}'\|_{L^{\infty}(Y_n)}|x_n - t| & \leq g(R_n) + K_n R_n & \leq 2K_n g(R_n).
\Delta(f_2^v)=\frac{1}{f_1^h(1-c^2b_1^hb_2^v)}\left\{\frac{1}{f_1^h}\big(\Delta_2(f_2)\big)^v-\frac{cb_2^v}{f_2^v}\big(\Delta_1(f_1)\big)^h+\frac{b_2^v\left(c(1-m_2)b_1^h+m_1\right)}{f_1^hf_2^v}\right\}
\frac{1}{n}\sum_{i=2}^n \left( A_i - \delta_0^{1/2}n^{1/ 2\alpha_0}|\Delta_i^n L - \Delta_{i-1}^n L|^{1/2}\right)=\frac{1}{n}\sum_{i=2}^n( | \xi^1_i + \xi^2_i |^{1/2}- | \xi^1_i |^{1/2}).