harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{cases}\partial_td_\epsilon+u_\epsilon\cdot\nabla d_{\epsilon}=\triangle d_{\epsilon}+\frac{1}{\epsilon^2}(1-|d_\epsilon|^2)d_{\epsilon}, \{\rm{in}}\Q_T, \ \ \ \d_\epsilon=g_\epsilon, \{\rm{on}}\ \partial_pQ_T.\end{cases}
\mathrm{\mathrm{}~^* \! ~}F_{\mu \nu}( x ) = 4 \pi \{\mathrm{\small ~ \frac{1}{2}~}\epsilon_{\mu \nu \rho \sigma}[ \partial^{\sigma}{\tilde{\lambda}}^{\rho}( x ) - \partial^{\rho}{\tilde{\lambda}}^{\sigma}( x ) ] \},
\dfrac{M_{2}(\mathcal{NC}(A(n,\nu)))}{|e( \mathcal{NC}(A(n,\nu)) )|}= \dfrac{2^{7n}(2^{n-1}-3)-2^{6n}(2^{n-1}-9)-2^{4n+1}(5\cdot 2^{n}-2)}{2^{n}(2^{n}-2)(2^{2n}-2^{n})}& = \dfrac{2^{5n}(2^{n}-5)+2^{3n+2}(2^{n+1}-1)}{2^{2n}-2^{n}}& = \dfrac{M_{1}(\mathcal{NC}(A(n,\nu)))}{|v(\mathcal{NC}(A(n,\nu)))|}.
P_\lambda (z_1,\ldots ,z_n,x; q ,t ,\mathbf{t})=\sum_{\substack{\mu\in\Lambda_n \mu \preceq \lambda}}P_\mu (z_1,\ldots ,z_n ; q ,t ,\mathbf{t})P_{\lambda/\mu}(x; q ,t ,\mathbf{t}),
G_{3,1}(w,r,\lambda) := \int_{0}^{w}\rho \left( F_{3}(r,\rho,\lambda(s)) + \frac{2 r^{2}\lambda(s)^{2-2\alpha}}{(\lambda(s)^{2-2\alpha}+\rho^{2})^{2}}\right)d\rho
g^{ij}(x)=\delta^{ij}(x) -\frac 13 R_{iab j}(\xi)x_a x_b +O(|x|^3)\ \hbox{and}\g^{ij}(x)\Gamma^k_{ij}(x)=\partial_l \Gamma^k_{ii}(\xi)x_l +O(|x|^2).
Z_f(s, \chi, D)= \int_{D}{\chi(ac (f(x, y, z)))|f(x, y, z)|^s|dxdydz|}= \int_{D_1}{\chi(ac (f(\pi^ix_1, \pi^jy_1, \pi^kz_1)))|f(\pi^ix_1, \pi^jy_1, \pi^kz_1)|^s|J(T_{i,j,k})||dx_1dy_1dz_1|}= q^{-i-j-k}\int_{D_1}{\chi(ac (f(\pi^ix_1, \pi^jy_1, \pi^kz_1)))|f(\pi^ix_1, \pi^jy_1, \pi^kz_1)|^s|dx_1dy_1dz_1|},
\int e^{V^0(\psi)}d\mu_C(\psi) = \int\int e^{V^0(\psi+\psi')}d\mu_{C_0}(\psi')d\mu_{\sum_{l=-1}^{l_{end}}C_l}(\psi)=\int e^{V^{-1}(\psi)}d\mu_{\sum_{l=-1}^{l_{end}}C_l}(\psi) &=\int e^{V^m(\psi)}d\mu_{\sum_{l=m}^{l_{end}}C_l}(\psi),
\times\exp\left\{-\frac{g_m^2}{2}\left[\int d^3xd^3y\rho({\bf x})D_0({\bf x}-{\bf y})\rho({\bf y})-\sum\limits_{{a,b=1\atop a\ne b}}^{N}D_m({\bf z}_a-{\bf z}_b)\right]\right\}.
\Delta \Sigma_{\pm}= \frac{g^{2}}6 \int \frac{d^{4}k}{( 2 \pi )^{4}}\frac 1{k^{2}+ m^{2}\pm \frac{\tilde{c}}{\tilde{k}^{2}}}( 2 + \cos ( k \tilde{p}) ) = \frac{g^{2}}6 \int \frac{d^{4}k}{( 2 \pi )^{4}}\frac{\tilde{k}^{2}}{k^{2}\tilde{k}^{2}+ m^{2}\tilde{k}^{2}\pm \tilde{c}}( 2 + e^{i k \tilde{p}}) .
\overline{V^a_A}=V_{\dot A a}, \overline{V^{A a}}=-V^{\dot A}_a,\overline{V_{A a}}=-V_{\dot A}^a, \overline{V_a^A}=V^{\dot A a}, \overline{\varepsilon_{ab}}=\varepsilon^{a b}, \overline{\chi^{Aa}\psi^{Bb}}= -\overline{\chi^{Aa}}\ \overline{\psi^{Bb}}
s'=\begin{cases}i_0, i_1,\cdots, i_{j-k}, i_{j+k+1},\cdots, i_m,\ \s \ \ \eqref{beginning1}, i_0,i_1,i_2,\cdots, i_j,i_{j+3},\cdots, i_m,\ \s\ \ \eqref{beginning2}, i_0,i_1,\cdots, i_{l-k},i_{l+2j+k+1},\cdots, i_m,\ \s\ \ \eqref{beginning3}.\end{cases}
\| \prod_{i=1}^k \mathcal E_\lambda^\gamma f_i \|_{L^{\frac qk}(d\mu)}\le C \langle\mu\rangle_\alpha^{\,\,\frac kq}\lambda^{-k(1-\frac1p - \frac{d-\alpha}{q})}\prod_{i=1}^k\| f_i \|_p
R_{2323}=\frac{\left(3{{e}^{5 q_2}+2{{e}^{4 q_2+q_1}}-2{{e}^{3 q_2+2 q_1}}-2{{e}^{2 q_2+3 q_1}}}\right){{e}^{2 q_3}}+\left( -{{e}^{5 q_2+q_1}}+{{e}^{4 q_2+2 q_1}}+{{e}^{3 q_2+3 q_1}}\right){{e}^{q_3}}}{(e^{q_2}+e^{q_3})\Delta_1},
\Pr(G \mid \sigma = \tau) = (1-p)^{|A_\tau|}q^{|E(G)|}(1-q)^{|B_\tau| - |E(G)|}\left( \frac{\quad\frac{p}{1-p}\quad}{\frac{q}{1-q}}\right)^{|E(G) \cap A_\tau|}.
\lambda^{K-1}_K\sigma_K =\sum_{i=1}^m\lambda^{K-1}_i\sigma_i &=\left(\sum_{i=1}^m\lambda^{K-1}_iP^i-Q^K,Q^0-Q^K\right)\ \ \mbox{\rm(by Lemma \ref{lem:1})}&=(Q^{K-1}-Q^K,Q^0-Q^K) &=(Q^{K-1}-Q^K,Q^{K-1}-Q^K)\ \ \mbox{\rm(by Lemma \ref{lem:6})}&>0.\ \ \mbox{\rm(by Lemma \ref{lem:2})}
\vert f(z)^{(s)}-P_{m}^{(s)}(z)\vert =\left\vert \sum_{h=1}^{\infty}c_{m+h}^{(s)}z^{m+h}\right\vert\leq \sum_{h=1}^{\infty}\vert c_{m+h}^{(s)}\vert T_{m}^{m+h}\leq \sum_{h=1}^{\infty}(m+h+s)^{s}\vert c_{m+h+s}\vert T_{m}^{m+h}
\widehat{A}=\frac{\chi_{1}\left( -1\right)}{k}\left\{G(n+k/2,\chi_{1})\right\}=\frac{\chi_{1}\left( -1\right)}{k}G_{1}^{\ast}, & \widehat{B}=\frac{\chi_{2}\left( -1\right)}{k}\left\{G(n+k/2,\chi_{2})\right\}=\frac{\chi_{2}\left( -1\right)}{k}G_{2}^{\ast}.
f_j^{(n)}(x_1,\ldots,x_{n-1},z,x;t)=f_n(x_1,\ldots,x_{j-1},z,x_{j},\ldots,x_{n-1},x;t) &=\int_{\{0<t_1<\ldots<t_{j-1}<r<t_j<\ldots<t_{n-1}<t\}}G_{t-t_{n-1}}(x-x_{n-1})\ldots G_{t_j-r}(x_j-z)G_{r-t_{j-1}}(z-x_{j-1})\ldots & G_{t_2-t_1}(x_2-x_1)dt_1 \ldots dt_{n-1}dr.
Ad_{(\phi_1,\gamma_1)}(0, \gamma_2) = (\phi_1,\gamma_1)(0, \gamma_2) (-\phi_1,\rho_{-\phi}\gamma_1^{-1}) =(\phi_1, \gamma_1 \rho_{\phi_1}\gamma_2) (-\phi_1,\rho_{-\phi}\gamma_1^{-1})
\Phi_{0}= a_{0}^{- 1}\sum_{M \geq 0}\frac{x^{M}}{\Gamma^{2}( M + 1 )}\frac 1{2 \pi i}\int_{C}\frac{\Gamma ( 4 s + 1 ) \Gamma ( 2 M - s )}{\Gamma^{3}( s + 1 )}( e^{- \pi i}y )^{s}d s ,
(p-e(G))\left[ k\left(n-k\right)+\binom{k}{2}\right] \leq \frac{C \sqrt{\rho \log n}}{2\sqrt{n}}\left[ k\left(n-k\right)+\binom{k}{2}\right] & \leq \frac{C \sqrt{\rho \log n}}{2\sqrt{n}}kn &= \frac{C}{2}k \sqrt{\rho}\sqrt{n \log n}& \leq \frac{C}{2}\max\{\sqrt{\rho},\sqrt{\frac{\log n}{n}}\}k\sqrt{n \log n}.
S_{q f}^{( \mathrm{e x})}\sim N_{e f f}~ H_{m a x}^{3}V \sim e^{- \phi_{e x i t}}l_{s}^{- 2}~ H_{m a x}V \sim l_{P}^{- 2}~ H_{m a x}V \sim S_{H B}^{( \mathrm{e x})}~ .
F_{R_i}(r) = \mathbb{P}[R_i\leq r|R_i\in\mathcal{L}_i] =\frac{\mathbb{P}[R_i\in\mathcal{B}(0,r)\cap\mathcal{L}_i]}{\mathbb{P}[R_i\in\mathcal{L}_i]}=\frac{F_R(r) - F_R(\sup\{\mathcal{L}_i\})}{\eta_i},
A^1_Q = \bigcup_{j = 1}^h \bigcup_{R_0 = 0}^{k_1(j)-1}A^1_{Q, j, R_0}; A^2_Q = \bigcup_{j=1}^h \bigcup_{S_0=0}^{k_2(j)-1}A^2_{Q, j, S_0}.
\begin{array}{l c r}{U^{+{\frac{n ( n + 1 )}{2}}}= u M^{+{\frac{n ( n + 1 )}{2}}}}\\{V^{+{\frac{n ( n + 1 )}{2}}}= v N^{+{\frac{n ( n + 1 )}{2}}}}\\{Z^{+{( n + 1 )}}= z S^{+{( n + 1 )}};}\\ \end{array}
|\frac{-4 \alpha \lambda_{0}''(t) \left(\log(\lambda_{1}(t))-\log(\lambda_{2}(t))\right)}{\log(\lambda_{0}(t))}| \leq \frac{C}{t^{2}\log^{b+1}(t) \log(\log(t))}\frac{|\lambda_{1}(t)-\lambda_{2}(t)|}{\lambda_{0,0}(t)}&\leq \frac{C ||e_{1}-e_{2}||_{X}}{t^{2}\log^{b+1}(t) (\log(\log(t)))^{3/2}}
I'_{h,m,n}(x)=\frac{1}{\sqrt{2 \pi n^{2/3}}}\, \left[\sqrt{\frac{L_m(x)}{3}}\,e^{\tfrac{3}{2}n^{2/3}L_m(x)}\right] \left( 1+O_X( n^{-2/3}) \right) & + \frac{1}{\sqrt{2 \pi n^{2/3}}}\, \left[\sqrt{\frac{L_m(x)}{3}}\, e^{\tfrac{3}{2}n^{2/3}L_m(x)}\right]^{-}\left( 1+O_X( n^{-2/3}) \right) .
\Delta = (1-vb_{\chi_1\chi_2^{-1}}\varpi^{a_1-a(\chi_1\chi_2^{-1})})^2-4vb_2 = 1-2v(b_1+b_2)+v^2b_{\chi_1\chi_2^{-1}}^2\varpi^{2a_1-2a(\chi_1\chi_2^{-1})}.
\mu(T_f^{-k}U \cap U') = (\mu_4 \times \mu_3)[(T^{-k}U \cap U')_4 \times (T^{-k}U \cap U')_3] = \mu_4((T^{-k}U \cap U')_4) \mu_3((T^{-k}U \cap U')_3) = \mu_4(T_4^{-k}U_4 \cap U'_4) \mu_3(T_3^{-k}U_3 \cap U'_3) = (\mu_4 \times \mu_3) (U_4 \times U_3) \cdot (\mu_4 \times \mu_3) (U_4' \times U_3') = \mu(U) \mu(U').
\big[D, \big({\cal S}^{(2n)}\big)^k \big] = 2nk \big({\cal S}^{(2n)}\big)^k,\big[D, \big({\cal S}^{(2n+1)}\big)^k \big] = (2n+1)k \big({\cal S}^{(2n+1)}\big)^k.
\|f\|_{L^q_G(\Sigma_T)}\leq C \Big[ \big(\| \frac{|u_1 - u_2|}{G}\|_{C^0}+ \|\frac{\rho |\nabla(u_1 - u_2)|}{G}\|_{C^0}\big)\cdot (1+ \|u_2\|_{W_G^{2,q}}\cdot \frac{T}{\theta}) + \|u_1 - u_2\|_{W_G^{2,q}}\Big] \leq C_1'\|u_1 - u_2\|_{X_T}\cdot T^{-1}
\limsup_{n\to\infty}\frac{1}{n}\log\left|\left\{x\in X\colon \delta_{x}^n\in\mathcal K\right\}\right|\leq\inf_{\mathcal G \supset \mathcal{K}}\sup_{\mathcal G}F=\inf_{\mathcal G \supset \mathcal{K}}\sup_{\mathcal G}(-I)=-\inf_{\mathcal K}I.
R_{\mu \nu}- \frac{1}{2}g_{\mu \nu}R = 8 \pi G \left( \frac{V}{\sqrt{1 + \nabla_{\alpha}T \nabla^{\alpha}T}}\, \, \nabla_{\mu}T \, \, \nabla_{\nu}T - g_{\mu \nu}V \, \sqrt{1 + \nabla_{\alpha}T \nabla^{\alpha}T}\right) \ ,
\Delta=\left(\begin{array}{cc}\sqrt{\zeta+\rho^2}0 0 \rho \bar z_2 -z_1 \bar z_1 z_2\end{array}\right),\quad\bar\Delta=\left(\begin{array}{cccc}\sqrt{\zeta+\rho^2}0 z_2 z_1 0 \rho -\bar z_1 \bar z_2 \end{array}\right).
x_{i_k}- x_* =\prod_{j=1}^k (I - \gamma a_{i_j}a_{i_j}^*) (x_0 - x_*) + \sum_{l=1}^k \Big[\prod_{k \geq j > l}(I-\gamma a_{i_j}a_{i_j}^*)\gamma a_{i_l}\Big] w_{i_l}
\Big( \hbar'(p - i - 1) - (a_s - a_{s'}) \Big) \times \dfrac{\hbar'(p - l_{i}^{(s')}- 1) - (a_s - a_{s'})}{\Big( \hbar'(p - l_{i}^{(s')}) - (a_s - a_{s'}) \Big) \Big( \hbar'(p - i -1) - (a_s - a_{s'}) \Big)}& = \dfrac{\hbar'(p - l_{i}^{(s')}- 1) - (a_s - a_{s'})}{\hbar'(p - l_{i}^{(s')}) - (a_s - a_{s'})}.
\begin{array}{l}[A,B]=C\,, begin{align*}0.5em][A,C]= -6B^2 + a_2 A^2+ a_5 A+a_8\,, begin{align*}0.5em][B,C]= -2A^3-a_2\{A,B\}-a_5 B+a_6A+a_9\,,\end{array}
\begin{array}{l l}{{S =}}&{{- \frac{1}{2}\int_{\Sigma}d^{2}\sigma \left( G_{m n}\partial_{a}X^{m}\partial^{a}X^{n}+ B_{m n}\epsilon^{a b}\partial_{a}X^{m}\partial_{b}X^{n}\right)}}\\{{}}&{{- \int_{\partial \Sigma}d s A_{m}\partial_{s}X^{m}+ \frac{1}{4 \pi}\int_{\Sigma}d^{2}\sigma \, \sqrt{\gamma}R^{( 2 )}\Phi - \frac{1}{2 \pi}\int_{\partial \Sigma}d s k \Phi \, ,}}\\ \end{array}
v^+ (x) = \lim_{r \to 0+}\sup \{u^\varepsilon (y) \ | \y \in B_r(x) \cap \overline \Omega, \ \varepsilon \in (0, r) \}, &v^- (x) = \lim_{r \to 0+}\inf \{u^\varepsilon (y) \ | \y \in B_r(x) \cap \overline \Omega, \ \varepsilon \in (0, r) \},
I^A{}_B := \left( \begin{array}{c}v^a \mu^a{}_{b}\ \ \mid \ \ \varphi \rho_a \end{array}\right)= v^c{\mathbb Y}_c{}^{A}{}_{B}+ \mu^d{}_{c}{\mathbb Z}_d{}^{c}{}^A{}_{B}+\varphi{\mathbb W}^{A}{}_{B}+\rho_c{\mathbb X}^{c}{}^A{}_{B},
f ( x ) \; \to \;{\hat{\mathcal O}}_{r}( f ) \; = \; \int \frac{d p_{1}d p_{2}}{2 \pi \hbar}\,{\tilde{f}}_{r}( p ) \, \exp [ \frac{i}{\hbar}( p_{1}{\hat{x}}_{0}^{1}+ p_{2}{\hat{x}}_{0}^{2}) ]
h\circ v_{i}=v_{i}\cdot h_{i}, \forall 1\leq i\leq r, h\circ v_{i}^{'}=v_{i}^{'}\cdot (\overline{h_{i}})^{-1}, \forall 1\leq i\leq r, h\circ u=u, \forall u\in V^{R-r},
s_k =O(e^{(\max\{\lambda_{i},\lambda_1 a\}-\lambda_1+6\tau(1+\lambda_1))r(0,k+1)}) V_{k}^{j,\tau}(T,\bar{s}) =O(e^{(\max\{\lambda_{i},\lambda_1 a\}-\lambda_1+8\tau(1+\lambda_1))r(0,k)}) W_{k}^{j,\tau}(T,\bar{s}) =O(e^{(\max\{\lambda_{i}-\lambda_1,\lambda_1 a-\lambda_1,-\lambda_j\}+9\tau(1+\lambda_1))r(0,k)}).
c_{0}= \alpha_{+}^{2}, \quad c_{1}= \frac{\epsilon \left( \alpha_{+}+ \epsilon \alpha_{-}\right)^{2}}{\epsilon - \sin \frac{n \gamma}{1 6}}+ 2 \epsilon , \quad c_{2}= \frac{\left( \alpha_{+}+ \epsilon \alpha_{-}\right)^{2}}{\left( \epsilon - \sin \frac{n \gamma}{1 6}\right)^{2}},
\mathbf{P}_{n}\left( \max \left( \min_{k=1,2}\left\Vert \mathbf{Z}(n^{t}g_{n}(t),n)-\mathbf{Z}(n^{t_{k}}g_{n}(t_{k}),n)\right\Vert \right)\neq 0\right) \leq \eta ,
\alpha\!:=\!\{1,\ldots,k_0\},\,\beta\!:=\!\{k_0\!+\!1,\ldots,k_1\},\, \gamma\!:=\!\{k_1\!+\!1,\ldots,m\},\overline{\gamma}=\{1,\ldots,m\}\backslash \gamma.
\begin{aligned}\tilde{b}_{s,s+k}\left[ Z_s,t+\tau;\left\{t_j+\tau,v_{s+j},\omega_j,i_j\right\}_{j=1}^k \right] = & = \tilde{b}_{s,s+k}\left[ Z_s,t;\left\{t_j,v_{s+j},\omega_j,i_j\right\}_{j=1}^k\right]\end{aligned}
F_{5}(t,r) =N_{2}(v_{5})(t,r)+\frac{\sin(2(v_{1}+v_{2}+v_{3}+v_{4}))}{2r^{2}}\left(\cos(2Q_{\frac{1}{\lambda(t)}}+2v_{5})-\cos(2Q_{\frac{1}{\lambda(t)}})\right) &+\left(\frac{\cos(2(v_{1}+v_{2}+v_{3}+v_{4}))-1}{2r^{2}}\right)\left(\sin(2Q_{\frac{1}{\lambda(t)}}+2v_{5})-\sin(2Q_{\frac{1}{\lambda(t)}})\right)
\mathbb{P}[\sum_{j=1}^k d^{e_j}_{i}= 1] = \mathbb{P}[d^{e_k}_{i}= 1]\mathbb{P}[\sum_{j=1}^{k-1}d^{e_j}_{i}= 0] + \mathbb{P}[d^{e_k}_{i}= 0]\mathbb{P}[\sum_{j=1}^{k-1}d^{e_j}_{i}= 1] =\frac{1}{2}(1-\sigma) \frac{1}{2}(1+\sigma^{k-1}) +\frac{1}{2}(1+\sigma) \frac{1}{2}(1-\sigma^{k-1}) = \frac{1}{2}(1-\sigma^k)
\varphi = \begin{bmatrix}x_1^{d_1}x_1^{d_2}\cdots x_1^{d_n}x_2^{d_1}x_2^{d_2}\cdots x_2^{d_n}\vdots \vdots \ddots \vdots x_n^{d_1}x_n^{d_n}\cdots x_n^{d_n}g_1 \cdots \cdots g_n \end{bmatrix},
\mathrm{V o l ( Q^{1 , 1 , 1})}= \oint_{Q^{1 , 1 , 1}}\left( \frac{3}{8 \Lambda}\right)^{7 / 2}8 \, \prod_{i = 1}^{3}\sin \theta_{i}\, d \theta_{i}d \phi_{i}\, d \psi = \frac{\pi^{4}}{8}\left( \frac{6}{\Lambda}\right)^{7 / 2}\, .
\mathcal{R}_{}\! =\! \bigcup_{P_{U|X}}\!\Big\{\left(R_s,R_\ell,R_w\right)\!\colon\! &0 \leq R_s\leq I(U;Y), &R_\ell\geq I(U;X)-I(U;Y), &R_w\geq I(U;X)-I(U;Y)\Big\}
\sum_{n_{i}= 0}^{\infty}\left[ \begin{array}{c}{n_{i}+ \tilde{n}_{i}+ \tilde{V}_{i , r}}{n_{i}}\end{array}\right] \tilde{x}^{n_{i}}= \frac{1}{( 1 - \tilde{x})^{\tilde{n}_{i}+ 1 + \tilde{V}_{i}, r}}.
\| f_t \circ (\phi_1 * \phi_2) \| \leq \int_{0}^{\infty}\int_{0}^{\infty}\| f_{T(t_1,t_2)}\circ ( 1_{(-J_1(t_1),J_1(t_1))}* 1_{(-J_2(t_2),J_2(t_2))}) \| dt_1 dt_2 = \| f_t \circ (\phi_1^* * \phi_2^* ) \|
S = \pi \left\{( m^{2}- |{\cal Z}_{1}|^{2}) + ( m^{2}- |{\cal Z}_{2}|^{2}) + 2 \sqrt{( m^{2}- |{\cal Z}_{1}|^{2}) ( m^{2}- |{\cal Z}_{2}|^{2}) - J^{2}}\, . \right\}\, .
\Phi_\lambda = \Phi_{\tau_{\lambda_1}\circ \tau_{\lambda_2}}=\frac{d(\mu\circ \tau_{\lambda_1}\circ\tau_{\lambda_2})}{d(\mu\circ \tau_{\lambda_2})}\, \frac{d(\mu\circ \tau_{\lambda_2})}{d\mu}=(\Phi_{\tau_{\lambda_1}}\circ \tau_{\lambda_2})(\Phi_{\tau_{\lambda_1}}),
i \raise . 1 5 e x \mathrm{/}\kern - . 5 3 e m \mathrm{D}= e^{g \gamma_{5}[ \phi ( x ) + f ( x ) ]}U ( x ) i \raise . 1 5 e x \mathrm{/}\kern - . 5 7 e m \mathrm{\partial}U^{\dagger}( x ) e^{g \gamma_{5}[ \phi ( x ) + f ( x ) ]},
\big\{h \in \dot{H}^{-1,p}(\rho) : \| h-\ell \|_{\dot{H}^{-1,p}(\rho)}\le r \big\}= \bigcap_{\substack{\varphi \in C_0^{\infty}: \| \varphi \|_{\dot{H}^{1,q}(\rho)}\le 1}}\big\{h \in \dot{H}^{-1,p}(\rho) : |(h-\ell)(\varphi)| \le r \big\}.
|\lambda_1 \cdot A + \ldots + \lambda_k \cdot A| = \sum_{j_1 = 1}^{m_1}\ldots \sum_{j_k = 1}^{m_k}|\lambda_1 \cdot A_{1j_1}+ \ldots + \lambda_k \cdot A_{k,j_k}| & \geq \sum_{e=1}^m \left(\frac{u+1}{S}|P_e| - C_u\right) \geq \frac{u+1}{S}|A| - C_{u+1}
\frac{\Gamma,\Gamma^{\prime}\vdash e_{1}:\tau_{1}...\Gamma,\Gamma^{\prime}\vdash e_{n}:\tau_{n}\Gamma,\Gamma^{\prime\prime}\vdash e:\tau}{\Gamma\vdash recv_{1}=e_{1}and...andv_{n}=e_{n}ine:\tau}[Rec]
\big|\partial_{\dot{\xi}}\Psi[\lambda,\xi,\dot{\lambda},\dot{\xi},\phi][\dot{\bar{\xi}}](x, t)\big|\lesssim \frac{t_0^{-\varepsilon}}{R^{a-2s}}\|\dot{\bar{\xi}}(t)\|_{n-4s+1+\sigma}\left(\sum_{j=1}^k\frac{\mu_0^{-\frac{n-6s}{2}-1+\sigma}(t)}{1+|y_j|^{a-2s}}\right),
\langle{\cal A}_a\rangle_{0n}={}^\parallel\nabla_a\omega_{\bar IJ}\langle z^{\bar I}z^J\rangle_{0n}-k_{\bar Iba}\langle \lambda^{\bar I}\lambda^b\rangle_{0n}-a_{I\bar Ja}\langle \lambda^{\bar J}\lambda^I\rangle_{0n}+a'{}_{I'\bar J'a}\langle \nu^{I'}\nu^{\bar J'}\rangle_{0n}
F_{2}( z ) = \sum_{n = - \infty}^{\infty}\frac{1}{1 - 4 n^{2}}\frac{\operatorname{c o t h}\left( \frac{\sqrt{4 \pi^{2}n^{2}+ z^{2}}}{2}\right)}{\sqrt{4 \pi^{2}n^{2}+ z^{2}}}
\sum_{n=T}^{[\tau]}\frac{e^{\frac{n+1}{T}}}{(n+1)^2}\leq \int_T^{\tau+1}\frac{e^{t/T}}{t^2}\d t = \frac{1}{T}\int_e^{e^{(\tau+1)/T}}\frac{\d u}{(\log u)^2}= O\bigg( \frac{Te^{\frac{\tau+1}{T}}}{(\tau+1)^2}+ 1 \bigg).
0 = \left[\frac{\varphi_1(s)}{\pi_1}\vec{\pi}[I + G(s)R] + \vec{y}(s)\right] \cdot G(s)\vec{e}+ s c_\infty(1-\rho_\infty) = \frac{\varphi_1(s)}{\pi_1}\left[\vec{\pi}G(s)\vec{e}+ \vec{\pi}G(s)RG(s)\vec{e}\right] + \vec{y}(s)G(s)\vec{e}+ s c_\infty(1-\rho_\infty) = \frac{\varphi_1(s)}{\pi_1}\left[B_2(s) + B_3(s)\right] + B_1(s)
\mathcal{F}(\mathcal{L};q,t)=\sum_{l=1}^L\frac{(-1)^{l-1}}{l}\sum_{\cup_{s=1}^l\mathbf{I}_s=\mathbf{L}}\prod_{s=1}^l\mathcal{H}(\mathcal{L}_{\mathbf{I}_s};q,t).
\begin{bmatrix}x_{1, k+1}x_{2,k+1}\end{bmatrix}=\begin{bmatrix}A_1+B_1D_2(I-D_1D_2)^{-1}C_1 B_1C_2+B_1D_2(I-D_1D_2)^{-1}D_1C_2 B_2(I-D_1D_2)^{-1}C_1 A_2+B_2(I-D_1D_2)^{-1}D_1C_2\end{bmatrix}\begin{bmatrix}x_{1,k}x_{2,k}\end{bmatrix}.
\frac{d}{dt}|v(t)|_H^2 = 2\left<v(t),v'(t)\right>_H =-2|v(t)|_{H^1}^2 - 2b(\mathcal{M}(\Phi)(t), \mathcal{M}(\Phi)(t), v(t)) + 2b(\mathcal{M}(\Psi)(t),\mathcal{M}(\Psi)(t),v(t)).
w(h)^{-1}W(h,\,h^{-1})w(g)^{-1}W(g,\,g^{-1})=w(gh)^{-1}W(gh,\,(gh)^{-1})W(h^{-1},\, g^{-1}) &w(h)^{-1}w(g)^{-1}W(h,h^{-1})^{w(g)^{-1}}W(g,\,g^{-1})=w(gh)^{-1}W(gh,\,(gh)^{-1})W(h^{-1}, \,g^{-1}) &W(g,\,h)^{-1}w(gh)^{-1}W(h,\,h^{-1})^{w(g)^{-1}}W(g,g^{-1})=w(gh)^{-1}W(gh,(gh)^{-1})W(h^{-1}, g^{-1})
\theta \, f_{1 , \ell}= \frac{\theta}{4}\, \sqrt{\frac{i \pi}{2}}\, e^{i \frac{\Phi}{2}}\, \, k^{3 / 2}\, \, \left( 1 + \frac{\Phi}{2 \pi \ell}+{\cal O}( \ell^{- 2}) \right) .
S_{jk}=\sum_{x=0}^N\frac{\theta(aq^{2x})}{\theta(a)}\frac{(a,b,c,d,e,f,g,q^{-N})_{x}}{(q,aq/b,aq/c,aq/d,aq/e,aq/f,aq/g,aq^{N+1})_{x}}\,q^{x(2n-1)}&\quad\times\frac{(bq^x,bq^{-x}/a)_{j-1}(cq^x,cq^{-x}/a)_{n-j}(eq^x,eq^{-x}/a)_{k-1}(fq^x,fq^{-x}/a)_{n-k}}{(b,b/a)_{j-1}(c,c/a)_{n-j}(e,e/a)_{k-1}(f,f/a)_{n-k}}.
\phi_\ell (x_I) =e_\ell(x_{4i_1}, \dots, x_{4i_{\ell}}) \not=0, \phi_{\ell+1}(x_I) =e_\ell(x_{4i_1}, \dots, x_{4i_{\ell}})^2 \not=0, \phi_{\ell+2}(x_I) =0.
E_j = \{b_1 < \cdots < b_{s-1}< v(j) < b_s < \cdots < b_{r(j-1)}< n-j + r(j)+1 < \cdots < n\}= \{b_1 < \cdots < b_{s-1}< v(j) < b_s < \cdots < b_{r(j-1)}< n-j + r(j-1)+2 < \cdots < n\}= E_{j-1}\cup \{v(j)\}.
\begin{cases}\mathcal{Y}^{m}\widetilde{\chi'}^{n}+(\widetilde{\chi'}^{n})^{t}\mathcal{Y}^{m}-(\widetilde{\chi}^{m})^{n}\,_{k}\mathcal{Y}^{k}=0,\cr (\widetilde{\chi}^{j})^{i}\,_{p}\widetilde{\chi}^{p}-\widetilde{\chi}^{j}\widetilde{\chi'}^{i}+\widetilde{\chi'}^{i}\widetilde{\chi}^{j}=0.\end{cases}
\operatorname*{div}\nolimits_{\Gamma}\mathbf{v}_{T}=\sum_{\ell=1}^{\infty}\lambda_{\ell}\sum_{m\in\iota_{\ell}}V_{\ell}^{m}Y_{\ell}^{m}\operatorname*{curl}\nolimits_{\Gamma}\mathbf{v}_{T}=\sum_{\ell=1}^{\infty}\lambda_{\ell}\sum_{m\in\iota_{\ell}}v_{\ell}^{m}Y_{\ell}^{m}.
\hat P^\downarrow_t\big((x,i),(dy,\{j\})\big)m(dx,\{i\}) = \frac{\hat U^+_j(y)}{\hat U^+_i(x)}\hat P^\dag_t\big((x,i),(dy,\{j\})\big) \pi_i\hat U^+_i(x) dx &=\pi_j \hat U^+_j(y)P^\dag_t\big((y,j),(dx,\{i\})\big) dy &= P^\dag_t\big((y,j),(dx,\{i\})\big) m(dy,\{j\}),
D^{(n)}_q(-c\pm u;t){\cal D}^{(n)}_{q,t}(c+q/2) ={\cal D}^{(n)}_{q,t}(c) \prod_{1\le i\le n}\vartheta(z_i\pm u),{\cal D}^{(n)}_{q,t}(c) D^{(n)}_q(c\pm u;t) = \prod_{1\le i\le n}\vartheta(z_i\pm u){\cal D}^{(n)}_{q,t}(c-q/2),
\beta_{s,a^1}^1=\frac{\left[r^1(s,a^1,a_s^2)-r^1(s,a_s^1,a_s^2)\right]}{\left[\sum_{s'\in S}p_{s'}\r^1(s',a_{s'}^1,a_{s'}^2)-\sum_{s'\in S}p(s'|s,a^1,a_s^2)r^1(s',a_{s'}^1,a_{s'}^2)\right]}.
\Lambda_{1;f}\lambda_{1;f}\leq\frac{\int_{\Omega}(\Delta_f\varphi)^2\,d\mu}{\int_{\Omega}|\nabla\varphi|^2\,d\mu}\frac{\int_{\Omega}|\nabla\varphi|^2\,d\mu}{\int_{\Omega}\varphi^2\,d\mu}&=\frac{\int_{\Omega}(\Delta_f\varphi)^2\,d\mu}{\int_{\Omega}\varphi^2\,d\mu}&\leq\frac{16}{3}\lambda_{1;f}^2,
(F, G)^r := \frac{(d_1- r)!(d_2 - r)!}{d_1!d_2!}\sum_{i=0}^r (-1)^i \binom{r}{i}\frac{\partial^r F}{\partial x^{r-i}\partial z^i}\frac{\partial^r G}{\partial x^{i}\partial z^{r-i}}.
f_m^\epsilon|_kT(r) =\sum_n\sum_{0<d\mid (r,n)}\chi(d)d^{k-1}a_m^\epsilon(rn/d^2)q^n &=\sum_{0<d\mid r\mid md}\chi(d)d^{k-1}a_m^\epsilon(m)q^{\frac{md^2}{r}}+O(1) &=\sum_{0<d\mid r\mid md}\chi(d)d^{k-1}\frac{s(md^2/r)}{s(m)}f_{md^2/r}^{\epsilon'}+O(1) &=\sum_{0<d\mid r\mid md}\chi(d)d^{k-1}f_{md^2/r}^{\epsilon'},
\sum_{n=1}^{2^N}\frac{\varepsilon_n(\omega)n^{it}}{n \sqrt{\log (n+1)}}=\sum_{n=1}^{2^N}(S_n(t)-S_{n-1}(t))u_n= \sum_{n=1}^{2^N}S_n(t)(u_n-u_{n+1})+S_{2^N}(t)u_{2^N+1}\, .
V ( \Phi ) = - \frac{1}{4}m_{0}^{2}\Phi^{2}+ \frac{1}{4}g_{0}\Phi^{4}+ \frac{3}{8}\frac{m_{0}^{4}}{g_{0}}= \frac{1}{4 !}g_{0}\left( \Phi^{2}- v_{0}^{2}\right)^{2}\, ,
P_{\mathrm{gen},p}( z)=\sum_{n\neq 0}g_{\mathbb{H}}(\sigma_{p}^{-1}z,\gamma_{\infty}^{n}\sigma_{p}^{-1}z)=\lim_{w\rightarrow z}\lim_{s\rightarrow 1}\bigg(\sum_{n=-\infty}^{\infty}g_{\mathbb{H},s}(\sigma_{p}^{-1}w,\gamma_{\infty}^{n}\sigma_{p}^{-1}z)-g_{\mathbb{H},s}(\sigma_{p}^{-1}z,\sigma_{p}^{-1}w)\bigg).
T(n_i)=\sum_{t=0}^{k_i-1}\frac{1}{C^t(n_i)}< \sum_{t=0}^{\infty}\frac{1}{n_i \cdot \left(\tfrac{3}{2}\right)^t}= \frac{1}{n_i}\cdot \sum_{t=0}^{\infty}\left(\frac{2}{3}\right)^t = \frac{3}{n_i}< \frac{3}{X_0}.
\rho(ab)\rho(-ab) =\rho(a)\rho((-a)b)+\rho(b)\rho((-b)a) =\rho(a)\rho(-a)+\rho(a)\rho(b)+\rho(b)\rho(-b)+\rho(b)\rho(a) =\rho(a)\rho(b)+\rho(b)\rho(a).
\sum_{k=0}^{n}S_B[n,k](t)_{k,q}^D = \sum_{k=0}^{n-1}S_B[n,k](t)_{k,q}^D + S_B[n,n](t)_{n,q}^D-[n]_{q}\,q^{n-1}(t)_{n-1,q}^D + [n]_{q}\,q^{n-1}(t)_{n-1,q}^D = \sum_{k=0}^{n}S_B[n,k](t)_{k,q}^B + [n]_{q}\,q^{n-1}(t)_{n-1,q}^D = t^n + [n]_{q}\,q^{n-1}(t)_{n-1,q}^D,
\left( A(\gamma)\right)^{n}e^{i\left\langle \gamma+t,x\right\rangle}=\sum_{\gamma_{1},\gamma_{2},...,\gamma_{m}}\frac{q_{\gamma_{1}}q_{\gamma_{2}}...q_{\gamma_{n}}e^{i\left\langle \gamma+t+\gamma(n),x\right\rangle}}{{\textstyle\prod\limits_{s=1,2,...,n}}\left( \mid\gamma+t\mid^{2}-\left\vert \gamma+t+\gamma(s)\right\vert^{2}\right)}.
\mathbf{D}_{\mathrm{r}}=\begin{bmatrix}0 \cdots 0 je^{j\mu^{\mathrm{r}}_1}\cdots je^{j\mu^{\mathrm{r}}_Z}2je^{j2\mu^{\mathrm{r}}_1}\cdots 2je^{j2\mu^{\mathrm{r}}_Z}\vdots \ddots \vdots j(N-1)e^{j(N-1)\mu^{\mathrm{r}}_1}\cdots j(N-1)e^{j(N-1)\mu^{\mathrm{r}}_Z}\end{bmatrix}
X_4= X(-2,3)+(q^{-\frac{1}{2}}+q^{\frac{1}{2}})hX(-2,2)+(q^{-1}+h^2+q)X(-2,1) +(q^{-\frac{1}{2}}+q^{\frac{1}{2}})hX(-2,0) &+X(-2,-1)+hX(-1,1)+h^2X(-1,0)+hX(-1,-1)+X(0,-1).
A_{H}v= -D_{v}H+\nabla^{\bot}_{v}H=-D_{v}(\xi-x^{\bot})+\nabla^{\bot}_{v}H = D_{v}x^{\bot}-D_{v}\xi+\nabla^{\bot}_{v}H=D_{v}x-D_{v}x^{\top}-D_{v}\xi+\nabla^{\bot}_{v}H = v-\nabla_{v}x^{\top}+A_\xi(v)-h(v,x^{\top})+\nabla^{\bot}_{v}H,
\left|\bigcup\limits_{1\leq i\ne j\leq m}D(B_i,\widetilde{B}_j)\right|=\sum\limits_{1\leq i\leq m}\sum\limits_{1\leq j\leq m\atop j\ne i}\sum_{b\in B_i}|D(\{b\},\widetilde{B}_j)|=\sum\limits_{1\leq i\leq m}\sum\limits_{1\leq j\leq m\atop j\ne i}\sum_{b\in B_i}\widetilde{k}=\widetilde{k}a(m-1).
\sum_{n\ge0}\frac{(-1)^nq^{n(n+1)/2}}{(-q)_n(1-q^{2n+1})}= \frac{1}{2}\sum_{n\ge0}q^{n^2+n/2}(1+q^{n+1/2})\sum_{|j|\le n}q^{j^2/2}
g(a,b,c) \ge (a+b+c)\,b^{\,\rho-1}-(b+c)(b^{\,\rho-1}+(\rho-1)b^{\rho-2}a)+ac^{\,\rho-1}&=(2-\rho) a b^{\rho-1}+ac\left(c^{\rho-2}-((\rho -1)^{\frac{1}{\rho-2}}b)^{\rho-2}\right).
\overline{u}(t,r) = C_1 e^{\gamma_0 t}[h(r)-h(\delta_1)]\,\ 0< r <\delta_2(t), \ \ \overline{v}(t,r) = C_2 e^{\gamma_0 t}[h(r)-h(\delta_2)],\ 0< r <\delta_2(t),
U \; = \; \int \frac{d^{4}p_{1}}{( 2 \pi )^{4}}\cdots \int \frac{d^{4}p_{r}}{( 2 \pi )^{4}}\; \delta ( a_{1}- p_{1}^{2}) \: \cdots \: \delta ( a_{r}- p_{r}^{2}) \; f^{\{\alpha \}}( p_{1}, \ldots , p_{k}) \; .
^{(n)}_{\texttt{a}}:=\left\{\bigl(X_1(\boldsymbol{\xi}),\ldots ,X_{n-1}(\boldsymbol{\xi})\bigr) \mid \boldsymbol{\xi}\in\mathbb{A}^{(n)}_{\texttt{a}}\right\}, \boldsymbol{X}^{(m,n)}_{\texttt{a};\lambda}:= \bigl(X_1( \boldsymbol{\xi}^{(m,n)}_{\texttt{a};\lambda}),\ldots ,X_{n-1}( \boldsymbol{\xi}^{(m,n)}_{\texttt{a};\lambda})\bigr) .
F(0)_2^1 \colon \sigma_{\vec{a}}\otimes \sigma_{\vec{b}}\mapsto \sum_{\vec{c}}\sum_e \langle W_{\vec{a}},W_{\vec{b}},W_{\vec{c}}\rangle_e \, \sigma_{\vec{c}^c}= \sigma_{\vec{a}}* \sigma_{\vec{b}}
r_{0,N}:=2 \cdot (40 C_{0,N}\rho_N^2)^{4/5}, begin{align*}3pt]\delta_{0,N}=\min\bigg(L_0;{\rho_N}/{2};\frac{r_{0,N}^{5/4}}{2\rho_N^{1/4}}\bigg)=\min\bigg(L_0;{\rho_N}/{2};20\cdot 2^{5/4}\rho_N^{7/4}C_{0,N}\bigg),
-\Delta \phi_{a,R_{n}}^{+}\leq -\Delta \phi_{a,R_{n}}^{+}+ a^{2}\phi_{a,R_{n}}^{+}= \left( -\Delta \phi_{a,R_{n}}+ a^{2}\phi_{a,R_{n}}\right) \chi_{\{\phi_{a,R_{n}}> 0\}}= 4 \pi \left( m_{R_{n}}- u_{a,R_{n}}^{2}\right) \chi_{\{\phi_{a,R_{n}}> 0\}}\leq 4 \pi m_{R_{n}}\chi_{\{\phi_{a,R_{n}}> 0\}}\leq 4 \pi m_{R_{n}}.
\frac{\partial a^k_j}{\partial \eta^\sigma_k}= -\gamma^k_j \beta^k_j(y, 0, \widehat{\eta}) = 0; &\frac{\partial \beta^k_\rho}{\partial \eta^\sigma_k}= -g^k_\rho a^k_\rho(y, 0, \widehat{\eta}) = 0,
g(r) =\lim_{k \rightarrow{0}}\frac{G(r+k)-G(r)}{k}= \lim_{n \rightarrow{\infty}}\frac{G(r+\frac{1}{n})-G(r)}{\frac{1}{n}}= \lim_{n \rightarrow{\infty}}n\{G(r+\frac{1}{n})-G(r)\}
B_\lambda(s) =2^n\int\limits_A\int\limits_Nf_k(Q_n\cdot(an,1))w_\lambda(an)\,dn\,da =2^n\int\limits_A\int\limits_N\bigg(\prod_{j=1}^na_j^{\ell_{n+1-j}+n+1-j}\bigg)f_k(Q_n\cdot(an,1))\,dn\,da.
c_6(\Omega)=-\frac{1}{2^4n(n-2)^3{n+5 \choose 7}}\sum_{i,j,k,r,s,t=1}^n \frac{\partial^3 N_{ijk}}{\partial x_r \partial x_s \partial x_t}\frac{\partial^3 N_{rst}}{\partial x_i \partial x_j \partial x_k}.
\exp ( - \Gamma_{e f f .}[ \exp ({\alpha}) e_{A}, A_{A B}^{-}, W_{a}] ) = \int \exp ( \int_{M}\alpha A ) D{\tilde{\overline{{\Psi}}}}_{L}D{\tilde{\Psi}}_{L}\exp ( - S^{-}- i \frac{3}{2}\int_{M}\alpha \partial_{\mu}J^{\mu}) .