harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
e^{2 \varphi}={\frac{8 Q_{F}^{2}}{\pi^{2}Q_{H}^{2}}}\equiv \lambda^{6}; \ \Q_{H}\equiv{\frac{\sqrt{2}}{4 \pi^{2}}}\int_{S^{3}}e^{- 4 \varphi / 3}\star G , \Q_{F}\equiv{\frac{\sqrt{2}}{1 6 \pi}}\int_{S^{3}}e^{2 \varphi / 3}\star F .
\frac{\partial h_{rs}}{\partial \bar{z}_v}+\sum_{c=1}^n (h_{cs}\frac{\partial \varphi^{r}_v}{\partial z_c}-\varphi^c_v\frac{\partial h_{rs}}{\partial z_c}+h_{r c}\frac{\partial \varphi^{s}_v}{\partial z_c})]=0\,\,\,\,\,
(x_0,ux_0)_{y_0}\leqslant (ux_0,x_0)_{y}+|y-y_0| \leqslant 101 d\delta \hbox{and}(x_0,ux_0)_{uy_0}=(x_0,u^{-1}x_0)_{y_0}\leqslant (u^{-1}x_0,x_0)_z +|z-y_0|\leqslant 101 d\delta.
\begin{aligned}X_j:=\frac{\partial}{\partial \xi_j}+2\eta_j\frac{\partial}{\partial s}, ~~Y_j:=\frac{\partial}{\partial \eta_j}-2\xi_j\frac{\partial}{\partial s},~~1\leq j\leq n,~~T=\frac{\partial}{\partial s}. \end{aligned}
M_{12}:= (V_1\setminus\{r < r_1\})\cup_{\Phi_{12}}(V_2\setminus\{r<r_2\}) \cong [1,2]\times S^1\times S^1, \cr M_{34}:= (V_3\setminus\{r < r_3\})\cup_{\Phi_{34}}(V_4\setminus\{r<r_4\}) \cong [3,4]\times S^1\times S^1.
\eta = \left[{\frac{m + 1}{2 n}}+{\frac{3 m^{2}+ 7 m + 6}{8 n^{2}}}+ O \left({\frac{1}{n^{3}}}\right) \right] \tilde{\epsilon}- \left[{\frac{m + 1}{2 n}}+{\frac{3 ( m + 1 )^{2}}{4 n^{2}}}+ O \left({\frac{1}{n^{3}}}\right) \right] \tilde{\epsilon}^{2}+ O ( \tilde{\epsilon}^{3}) .
\zeta(s)\left(1-\frac{1}{2^s}\right) =1+\frac{1}{2^s}+\frac{1}{3^s}+\frac{1}{4^s}+\frac{1}{5^s}+... &-\frac{1}{2^s}+\frac{1}{4^s}+\frac{1}{6^s}+\frac{1}{8^s}+\frac{1}{10^s}+... &=1+\frac{1}{3^s}+\frac{1}{5^s}+\frac{1}{7^s}+\frac{1}{9^s}+...
\frac{d}{dt}\|A^{k/2}w(t)\|^2 =-2(A^{k/2}w(t),A^{k/2}w'(t)) &=-2(A^{k/2}w(t),A^{k/2+1}w(t)) &=-2\|A^{\frac{k+1}{2}}w(t)\|^2.
\psi(\zeta,\xi)=\begin{pmatrix}\psi(x,y) \tilde{\psi}(x,y,z,t) 0_{1,d-2}0 \psi(x,y) 0_{1,d-2}0_{d-2,1}0_{d-2,1}\psi(1,1)I_{d-2}\end{pmatrix}
\delta_{\alpha}j_{\mu \beta}= \sigma_{\mu \nu \alpha \beta}\partial^{\nu}\left( \frac{8}{3}{\kern . 3 5 e m \bar{\kern - . 3 5 e m{\widetilde J}}}+ \frac{1}{2}D^{2}( a_{1}( X +{\bar{X}}) + a_{2}( X -{\bar{X}}) ) \right) \displaystyle{\biggl \vert}\ .
M:=\min\Big\{\frac{(\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1)_++\chi_1\mu_1(\lambda_1-\lambda_2)_+}{\lambda_2}, \frac{\chi_2\mu_2(\lambda_1-\lambda_2)_+ + (\chi_2\mu_2\lambda_2-\chi_1\mu_1\lambda_1)_+}{\lambda_1}\Big\}
\qquad\Upsilon_n^\mathbf{a}= \sum_{k=K_0}^{h-1}\sum_{j=1}^{q_k}R_{k,j}^\mathbf{a}+ \sum_{j=1}^{\tau_n}R_{h_n,j}^\mathbf{a}\qquad\mbox{and}\mu_n^\mathbf{a}= \sum_{k=K_0}^{h-1}R_{k,0}^\mathbf{a}+ R_{h_n,0}^\mathbf{a},
M^n_t(f) \; = \; \mathcal X^n_t(f) - \mathcal X^n_0(f) +c_n \int_{0}^t \mathcal X^n_s(f') ds - d_n \int_{0}^t \mathcal X^n_s(f'') ds -\; \big( d_n \nabla^n_{0}f - c_n f\big(\tfrac{1}{n}\big) \big)n^{1-\gamma}\int_{0}^t \overline{J_s^n}(0) \; ds + o_n(1),
\sum_{w \in W}\sum_{v \in \Delta_{\Lambda_{2}}}( - 1 )^{| w |}e^{i \left[ w ( \Lambda_{1}) + v \right] \cdot x}= \sum_{w \in W}\sum_{\Lambda_{3}\in \Delta_{\Lambda_{1}, \Lambda_{2}}}( - 1 )^{| w |}e^{i w ( \Lambda_{3}) \cdot x}.
u^i\star w^j=u^iw^j +\mathrm{i}\nu(\delta^i_--\delta^i_+)u^i \bigg(\frac{1}{\sqrt{a}}\delta^j_0w^+-2\sqrt{a}\delta^j_-w^0\bigg) +\delta^i_+\delta^j_-2\nu^2u^+w^+.
\Gamma ( \hat{T}, ( \mathcal{M}^{c}( G^{a}\tilde{\Phi}) )^{\mathrm{C P}}) = \Gamma ( \hat{T}^{\mathrm{C P}}, ( G^{a}\tilde{\Phi})^{\mathrm{C P}}) = \Gamma ( \hat{T}, \tilde{\Phi})
g_\rho^{[d+1]}\left(\Upsilon_\Theta S^{-1}(\rho\xi)\right) =\frac{\partial}{\partial\xi_2}\,g_\rho^{[d]}\left(\Upsilon_\Theta S^{-1}(\rho\xi)\right)\cos(\rho R)+\mathcal O(\rho^2) &=\frac{\partial^d}{\partial\xi_2^d}\,p_\zeta\left(\Upsilon_\Theta S^{-1}(\rho\xi)\right)\cos(\rho R)+\mathcal O(\rho^2)
<BR>Q_1(Q_2)=3D{1\over ={{\sqrt=20{-2\nabla^2}}}}[\epsilon_{ij}\partial_iA_j\pm{1\over m}\Pi_\theta=20];~~<BR>P_1(P_2)=3D[{1\over{{\sqrt{-2\nabla^2}}}}\epsilon =_{ij}\partial_i\Pi=20_j\mp<BR>{m\over 2}{\sqrt{-2\nabla^2}}\theta=20],<BR><BR>
v_1(y_0+\rho\eta)=v_1(y_0)+A_1\rho+A_2\rho^{2}+o(\rho^{2}), &v_2 (z_0+\rho\eta)=v_2(z_0)+B_1\rho+B_2\rho^{2}+o(\rho^{2}), &u(x_0+\rho\eta)=u(x_0)+C_1\rho+C_2\rho^{2}+o(\rho^{2})
T_{\alpha\beta}={1 \over 4}\left(\nabla_\alpha\tilde{\psi}\gamma_\beta\psi+\nabla_\beta\tilde{\psi}\gamma_\alpha\psi-\tilde{\psi}\gamma_\alpha \nabla_\beta\psi-\tilde{\psi}\gamma_\beta \nabla_\alpha\psi\right)~.
v_{\alpha}(x)= \begin{cases}x_n^{\alpha}(|x'|^2 -C_\alpha)\quad C_{\alpha}= \frac{1+ 2D^2}{\alpha(1-\alpha)}\alpha \in [\frac{2}{n}, 1), (1+2D^2) x_n \log (x_n/D) + x_n (|x'|^2-D^2) \alpha =1.\end{cases}
T_{y, s}\dot{s}= P_{y, s}\dot{s}_{||}+ \frac{\sinh(\|s\|)}{\|s\|}P_{y, s}\dot{s}_{\perp}, T_{y, s}^{-1}\ddot{s}= P_{y, s}^{-1}\ddot{s}_{||}+ \frac{\|s\|}{\sinh(\|s\|)}P_{y, s}^{-1}\ddot{s}_{\perp}.
J_\infty :=\left\langle_{a_1,a_2,\dots,a_s}Y_\infty[s]^{(k-\ell(a))}\ \mid \ 0\leq \ell(a) \leq k,\m+1 \leq k \right\rangle J_m := \left\langle_{a_1,a_2,\dots,a_s}\mathsf{Y}[s]^{(k-\ell(a))}\ \mid \ 0\leq \ell(a) \leq k,\m+1 \leq k \right\rangle.
\widetilde{V}^{( \nu^{2k})}(x;\Lambda_{2k-1})=V(x)-2\left\{\log \left[\prod\limits_{l=0}^{k-1}\left( \lambda_{2l+1}+\int_{x_{0}}^{x}dt\big( \Psi_{\nu}^{( \nu^{2l})}(t;\Lambda_{2l})\big)^{2}\right) \right]\right\}^{\prime \prime}
Z(\xi)=\int{\cal D}A^{\omega}_{\mu}{\cal D}B{\cal D}\bar{c}{\cal D}c e^{-S_{0}(A^{\omega})-\frac{\xi}{2}\int B^{2}dx +\int[-iB\partial_{\mu}A^{\omega}_{\mu}+\bar{c}(-\partial_{\mu}\partial_{\mu})c]dx}
\int_\lambda^\infty e^{i \eta^2}d\eta = \int_\lambda^\infty \frac{2i\eta}{2i\eta}e^{i \eta^2}d\eta = \frac{1}{2i \lambda}+ \int_{\lambda}^\infty \frac{e^{i\eta^2}}{2i\eta^2}d\eta & = \frac{1}{2i \lambda}- \frac{1}{4 \lambda^3}- \int_{\lambda}^\infty \frac{e^{i\eta^2}}{12\eta^4}d\eta.
\stackrel{P}{\cal F}{}\! \!_{0 , p}^{( \pm )}\left( \begin{array}{c}{{\! \! \alpha \! \!}}\\{{\! \! \beta \! \!}}\\ \end{array}\right) = \; \sum_{s \in{\cal S}^{p - 3}}\stackrel{P}{K}{}\! \!_{0 , p}^{( \pm )}\left( \begin{array}{c}{{\! \! \alpha \! \!}}\\{{\! \! \beta , s \! \!}}\\ \end{array}\right) \stackrel{\alpha}{\omega}_{0 , p}\! \! \lbrack \beta + s \rbrack \; \; ,
\sum_{r_1\geq\dots\geq r_m\geq 0}\frac{q^{r_1^2+\cdots+r_m^2+r_i+\cdots+r_m}}{(q)_{r_1-r_2}\cdots(q)_{r_{m-1}-r_m}(q)_{r_m}}=\frac{(q^{2m+3};q^{2m+3})_{\infty}}{(q)_{\infty}}\, \cdot\theta(q^i;q^{2m+3}),
\varphi(\mathrm{v},\mathrm{y},\mathrm{t}, \tau) - \Phi^{\textbf{e}}(\xi,\mathrm{t};\mathrm{z}, \tau ) = \mathcal{O}\Bigg(\sqrt{\alpha}|\mathrm{y}-\mathrm{v}| \ \Vert \partial_{s}\Phi^{\textbf{e}}(\xi,\mathrm{t};\mathrm{z}, \cdot )\Vert_{\mathrm{H}^{-\frac{1}{4}}(0,\tau)}\Bigg).
\left( 1-\dfrac{z}{\zeta}\right) +\left( 1-\dfrac{z}{\zeta}\right)\varphi_{q_{\zeta}W}\left( z\right) & =\left( 1-\dfrac{a_{1}}{\zeta}\right) \left( 1+\varphi_{W}\left( z\right)\right) -\dfrac{1}{\zeta}\left( z+\psi_{W}\left( z\right) \right)
\delta_i^* =-\sqrt{1-x_i^2}\,\partial_{x_i}+(\alpha_i+1/2)\sqrt\frac{1+x_i}{1-x_i}-(\beta_i+1/2)\sqrt\frac{1-x_i}{1+x_i}, v_i =[\delta_i,\delta_i^*]=\frac{\alpha_i+1/2}{1-x_i}+\frac{\beta_i+1/2}{1+x_i},r=0.
K_{n \ell}= ( n + \ell ) ! \sum_{{\mathrm{\scriptsize p a r t i t i o n s o f n}}\atop{\mathrm{\scriptsize o f l e n g t h \ell}}}\prod_{j = 1}^{\infty}\frac{1}{{n_{j}}!}\left( \frac{1}{( j + 1 ) !}\right)^{n_{j}}
B_{i,j,0}+B_{i,k,0}+B_{i,l,0}=A_{i,0,0}&z_jB_{i,j,0}+z_kB_{i,k,0}+s_lB_{i,l,0}=\sqrt{q}A_{i,1,0}&z_j^2B_{i,j,0}+z_k^2B_{i,k,0}+s_l^2B_{i,l,0}=qA_{i,2,0},
i \, [ H , K_{i}] = \left( p_{i}- \frac{\kappa}{2 R^{2}}\, \epsilon_{i j}x_{j}\right) \cosh \frac{t}{R}+ \left( \frac{2}{R}\, \tilde{p}_{i}- \frac{\mu}{R}\, x_{i}\right) \sinh \frac{t}{R}= - \frac{\partial K_{i}}{\partial t}.
\lim_{p_0 \rightarrow 0}\Pi_{00}^{({\rm rising\sun})(T)}\rightarrow \frac{e^2}{2 \pi \beta}\left[2 + \frac{4 m^2 (m^2 -m_{\sigma}^2)}{(m_{\sigma}^2 - m_+^2) (m_{\sigma}^2 - m_-^2)}\,\ln \frac{m_{\sigma}}{m_-}- \frac{4 m^2 (m^2 -m_+^2)}{(m_{\sigma}^2 - m_+^2)(m_+^2 - m_-^2)}\,\ln \frac{m_+}{m_-}\right].
\phi_Z(t) = \mathcal{F}^\mu\left(h_Z\right)(t) = \sum_{\Sigma\subseteq [\![1,n]\!]}e_{_\Sigma}\int_\mathbb{R}\lambda_{_\Sigma}e^{-\lambda_{_\Sigma}x}\mathbb{1}_{[0,+\infty[}(x)e^{\mu tx}dx = \sum_{\Sigma\subseteq [\![1,n]\!]}e_{_\Sigma}\frac{\lambda_{_\Sigma}}{\lambda_{_\Sigma}-\mu t}.
D\frac{{{\partial^2}{C_z}(z,t|{z_{\rm tx}},{t_0})}}{{\partial{z^2}}}- v\frac{{\partial{C_z}(z,t|{z_{\rm tx}},{t_0})}}{{\partial z}}-{k_{d}}{C_z}(z,t|{z_{\rm tx}},{t_0}) + \delta (z-z_{\rm tx})\delta (t -{t_0}) &= \frac{{\partial{C_z}(z,t|{z_{\rm tx}},{t_0})}}{{\partial t}}.
\sum_{t=1}^{T}t^{-\frac{\beta-1}{\beta}}=1+\int_{1}^{T}u^{-\frac{\beta-1}{\beta}}\d u \leq \beta T^{\frac{1}{\beta}}\quad \sum_{t=1}^{T}t^{-\frac{1}{\beta}-1}\leq 1+\int_{1}^{T}u^{-\frac{1}{\beta}-1}\d u \leq 1+\beta\enspace.
S_{0,\sigma}(t) \geq - \left[\Big(M_{1,\sigma}^-(t) + M_{1,\sigma}^+(t)\Big)\,\frac{1}{\lambda_{\sigma}(t)}+ \dfrac{M_{-1,\sigma}^-(t)}{2}\,\lambda_{\sigma}(t)\right] \ell_{1,\sigma}(t) + O_{c}\left(\frac{\mu_{d,\sigma}\,r_{2,\sigma}(t)}{\lambda_{\sigma}(t)}\right) + O_c(\mu_{d,\sigma}\,\lambda_{\sigma}(t)\,r_{2,\sigma}(t)).
P_N w(t) =e^{-tA_N}P_N w_0+\int_0^t e^{-(t-\tau)A_N}(P_N\xi+P_Nf(\tau))d\tau =e^{-tA_N}P_N w_0+A_N^{-1}(P_N \xi-e^{-tA_N}P_N \xi)+\int_0^t e^{-(t-\tau)A_N}P_Nf(\tau)d\tau =e^{-tA}P_N w_0+A^{-1}(P_N \xi-e^{-tA}P_N \xi)+\int_0^t e^{-(t-\tau)A}P_Nf(\tau)d\tau,
\tilde{g}(x)=\tilde{a}x+\tilde{b}-\sum_{k=1}^{n-1}\frac{\tilde{s}_k}{x-\tilde{t}_k}, \tilde{a}:=\bigg(\sum_{k=1}^{n-1}\tilde{r}_k\bigg)^{-1}>0,\tilde{s}_k>0, \tilde{b}\in\mathbb R.
H_0' = \sum_{\lambda\in \Lambda}\omega_\lambda a_\lambda^+a_\lambda+ \sum_{\lambda\in \Lambda}(g_\lambda a_\lambda b+ \overline{g}_\lambda a_\lambda^+b^+) H_0'' = \omega_0 (-b^+b +\sum_{\lambda\in \Lambda}a_\lambda^+a_\lambda)
\int_{\Omega^{m}}\mathrm{Pf}\begin{bmatrix}\epsilon(x_{i}, x_{j}) \end{bmatrix}_{i, j=1}^{m}\det\left[\phi_{i}(x_{j})\right]_{i, j=1}^{m}\prod_{i=1}^{m}d\nu(x_{i}) = \frac{m!}{2^{\frac{m}{2}}}\mathrm{Pf}\begin{bmatrix}\epsilon_{i,j}\end{bmatrix}_{i, j=1}^{m}.
\left(\begin{array}{c}g^T g_+^T\end{array}\right)= p^TBp\left(\begin{array}{cc}0 \ -1 1 \ -1\end{array}\right)\left(\begin{array}{c}\frac{1}{p^T Hp}p^TH \frac{1}{p^T Bp}p^TB \end{array}\right).
\begin{aligned}S_{a,b;M}(\tau):= &\Bigg(\sum_{n\pm \nu\geq-\lfloor a_1\pm a_2\rfloor}+\sum_{n\pm\nu<-\lfloor a_1\pm a_2\rfloor}\Bigg)e\big((M+1)b_1n-(M-1)b_2\nu\big)q^{\frac12\big((M+1)(n+a_1)^2-(M-1)(\nu+a_2)^2\big)}.\end{aligned}
-\dfrac{a}{\pi^{2}}\sum_{k=1}^{+\infty}\dfrac{\cos\left( k\pi x_{1}/a\right)}{k^{2}}=-\dfrac{a}{6}+\dfrac{x_{1}}{2}-\dfrac{x_{1}^{2}}{4a}\ \ \ \mathrm{if}\ \ 0\leq x_{1}\leq 2a
\mathrm{ch}\Big(\sum_{q=0}^r (-1)^q{\bigwedge}^{q}E^\vee\Big) = \sum_{q=0}^r (-1)^q \mathrm{ch}\Big({\bigwedge}^q E^\vee\Big)=\prod_{i=1}^r (1- e^{-a_{i}}) &=(a_1\ldots a_r) \prod_{i=1}^r \frac{(1- e^{-a_{i}})}{a_i}=c_r(E) \mathrm{td}^{-1}(E).
Z_{{\it l o g}}[ p / q ] = \frac{1}{\eta \bar{\eta}}\left( | \chi_{0}|^{2}+ | \chi_{p q}|^{2}+ \sum_{1 \leq s \leq p q - 1}\left( \chi_{s}^{0}( \chi_{s^{\prime}}^{0})^{*}+ \chi_{s}^{+}( \chi_{s^{\prime}}^{-})^{*}+ \chi_{s}^{-}( \chi_{s^{\prime}}^{+})^{*}\right) \right) \, ,
G^-_s(\gamma_1,\gamma_2) = \frac{2i\sqrt{J_1J_2}}{E_q(J_1,J_2)}\sum_{n_1,n_2 =0}^\infty\frac{J_1^{n_1}J_2^{n_2}}{[n_1]_q![n_2]_q!}\sin(\gamma_1 q^{2[n_1]_q}-\gamma_2q^{2[n_2]_q}).
1-P_x(\theta^*_t =\widehat{\theta}_t \forall t\geq 0) \leq \sum_{q \in \mathbb{Q}_{> 0}}P_x \left( \lim_{k \downarrow q}\sigma^*(0,k)<\sigma^+(q) \right) + P_x \left( \sigma^*(0,q)>\sigma^+(q) \right) =0
X_L(a,b,c;q)=\displaystyle\sum_{\begin{array}{c}\{\scriptstyle \mu |{\mu_0=a, \mu_L=b, \mu_{L+1}=c}\}\end{array}}\! \! \! \! \! \! q^{\sum_{k=0}^{L-1}(k+1)H(\mu_k,\mu_{k+1},\mu_{k+2})},
g_{1}(z) = \int_{0}^{q}\log{(z-s)}\, d\mu^*(s), z \in \mathbb C \setminus (-\infty, q], g_{2}(z) = \int_{-\infty}^{0}\log{(z-s)}\, d\nu^*(s), z \in \mathbb C \setminus (-\infty, 0],
S^{(2)}= - 3\int d^4 x{-\lambda\over 4!}\left[ G(x,x) \right]^2 - 3!\int d^4 x d^4 y ({-\lambda\over 3!})^2 \phi_{cl}(x) \phi_{cl}(y) \left[ G(x,y)\right]^3\,\,.
f(\varphi,\psi,\rho)\left(p^{\,(\varphi^{-1},\psi^{-1},\rho^{-1})}\right)=\left(p^{\,(\varphi^{-1},\psi^{-1},\rho^{-1})}\right)^{(\varphi,\psi,\rho)}=p^{(id,id,id)}=p.
\{x \}\ = \ \ Q(x,x), \emptyset \ = \Q^{\circ}(x,x). \{x, y \}\ \subset \ \ Q(x,y) \ = \Q(\{x,y \}\times \{x,y \}). Q(x,y) \setminus \{x,y \}\ = \ \ Q^{\circ}(x,y) \ = \Q^{\circ}(\{x,y \}\times \{x,y \}).
\psi_k \circ T(a_0,a_1, \dots, a_{n-1}) = \psi_k(a_{n-1}, a_0, a_1, \dots, a_{n-2}) & = (\psi_k(a_{n-1}), \psi_k(a_0), \dots, \psi_k(a_{n-2})) & = T^{2^{2^k-1}}\circ \psi_k (a_0,a_1, \dots, a_{n-1}).
S_{1}:=\sum_{i=1}^{n}\mathbb{P}(A_{i}),S_{2}:=\sum_{1\le i<j\le n}\mathbb{P}(A_{i}\cap A_{j}),...,S_{k}:=\sum_{1\le i_{1}<\cdot\cdot\cdot<i_{k}\le n}\mathbb{P}(A_{i_{1}}\cap\cdot\cdot\cdot\cap A_{i_{k}})
\vartheta(\hbar) [\mathfrak{X}_i^{+}(u, \lambda_1) , \mathfrak{X}_i^{-}(v, \lambda_2)]=\frac{\vartheta(u-v+\lambda_{1, i})}{\vartheta(u-v) \vartheta(\lambda_{1, i})}\Phi_i(v)+\frac{\vartheta(u-v-\lambda_{2, i})}{\vartheta(u-v) \vartheta(\lambda_{2, i})}\Phi_i(u),
X_{T_i(t')^k}X_{T_i(t)^k}=q^{\frac{1}{2}\varLambda(d_i(t')^*,d_i(t)^*)+1}q^s \prod_{j\neq i}X_{T_j(t)^k}^{a_{ij}}+ q^{\frac{1}{2}(\varLambda(d_i(t')^*,d_i(t)^*))}q^{s'}\prod_{j\neq i}X_{T_j(t)^k}^{b_{ij}}.
\left({\cal H}^{a , b , c}\, \oplus \,{\cal H}^{a^{\prime}, b^{\prime}, c^{\prime}}\right) \, \textcircled{\scshape A}\, \left({\cal H}^{a , b , c}\, \oplus \,{\cal H}^{a^{\prime}, b^{\prime}, c^{\prime}}\right)
val(ab) =(v(a_{h_{1}}b_{h_{2}}p^{-\beta(h_{1},h_{2})}),h_{1}+h_{2})= =(v(a_{h_{1}})+v(b_{h_{2}})-e\beta(h_{1},h_{2}),h_{1}+h_{2})= =(v(a_{h_{1}})+v(b_{h_{2}})-\beta_{e}(h_{1},h_{2}),h_{1}+h_{2})= =(v(a_{h_{1}}),h_{1})+v(a_{h_{2}}),h_{2})=val(a)+val(b);
\zeta_{5}= \frac{\mu^{2}}{2 ( 4 \pi )^{2}( s - 1 )}\int d^{4}k \ \tilde{S}( k ) \tilde{S}( - k ) \left( \frac{k^{2}}{8 \mu^{2}}\right)^{- s + 1}F ( \frac{s}{2}, \frac{s - 1}{2}; 1 ; 1 ) =
W^{(n)}_{\lambda}(t;\zeta)=\det_{1\leq a, b \leq n}\left.\left(\partial_t^{n-a+\lambda_{n-a+1}}\psi^{(j_b)}(t;\zeta_b)\right)\right|_{\zeta_1=\zeta_2=\dots\zeta_n=\zeta}\ .
a_{i}= \oint_{\alpha_{i}}\frac{d x}{2 \pi i}\left( N - \frac{x F^{\prime}( x )}{F ( x )}+ \tilde{q}\frac{x^{2 N}( N F - x F^{\prime})}{2 F ( x )^{3}}+ \mathcal{O}( \tilde{q}^{2}) \right) .
Z \left({\cal A}, G , p , t_{n , l}^{( A )}\right) = \sum_{\{r_{n , l}^{( A )}\}}Z \left({\cal A}, G , p , r_{n , l}^{( A )}\right) \prod_{A = 1}^{p}\prod_{l = 1}^{\infty}\prod_{n = - \infty}^{+ \infty}\left[ t_{n , l}^{( A )}\right]^{r_{n , l}^{( A )}}~ .
(f_n, \, f_n) =\sum_{k=0}^n \sum_{x\in\partial B_G(k)}g(k)^2 \mu(x) = \sum_{k=0}^n\sum_{x\in\partial B_G(k)}g(k)^2\, (d_x^++d_x^0+\lambda d_x^-) \, \lambda^{-|x|}\ge (\lambda\wedge 1) d\, \sum_{k=0}^n M_k\, g(k)^2 \, \lambda^{-k}.
w(\varphi_{x})+\sum_{y\neq x}w(\varphi_{\{x,y\}})=\sum_{i=1}^{k}\sum_{\substack{(y,z)\in S_{i}x\not\in\{y,z\}}}\frac{1+b_{k}-b_{i}}{4}+\sum_{i=1}^{k}\sum_{\substack{(y,z)\in S_{i}x\in\{y,z\}}}\frac{1+b_{k}-b_{i}}{4}=\sum_{i=1}^{k}\sum_{(y,z)\in S_{i}}\frac{1+b_{k}-b_{i}}{4}
\Big\| -\varepsilon \displaystyle\sum_{j=1}^{d}B(u^{\varepsilon})\,\left(\frac{\partial u}{\partial x_{j}}\right)^{2}\,\eta^{''}(u^{\varepsilon})\Big\|_{M(\Omega_{T})}\leq C^{'}\|B\|_{L^{\infty}(\mathbb{R})}\|\eta^{''}\|_{L^{\infty}(I)},
\begin{cases}a_{21}^{\prime}(t) = -2\pi i [ a_{21}(t) + e^{4\pi i}a_{10}^2 (t) b_{01}(t)] b_{12}^{\prime}(t) = -2\pi i [ b_{12}(t) + e^{4\pi i}a_{10}(t) b_{01}^2 (t)]\end{cases}
\partial_s (E_s [a_s, (\cdot)_{\tau}] \diamond \partial_1^2 v_s ) = E_s [\partial_s a_s, (\cdot)_{\tau}] \diamond \partial_1^2 v_s + E_s [a_s, (\cdot)_{\tau}] \diamond \partial_1^2 \partial_s v_s + \partial_s a_s E_s [a_s, (\cdot)_{\tau}] \diamond \partial_1^2 \partial_{a_0}v_s.
\sum_{n=1}^{\infty}(-1)^{n-1}\frac{\sigma^*(n)}{n^s}=\left(1-\frac{6}{2^s}+\frac{6}{2^{2s}}\right)\left(1-\frac2{2^{2s}}\right)^{-1}\frac{\zeta(s)\zeta(s-1)}{\zeta(2s-1)}(\Re s>2).
(\phi(g)^{-1}u(g))(\phi(h)^{-1}u(h)) =\phi(g)^{-1}\phi(h)^{-1}u(g)u(h)=\phi(g)^{-1}\phi(h)^{-1}\sigma(g, h)u(gh) &=\phi(gh)^{-1}u(gh)
L(g^{\ast})= \frac{\gamma_{d+1}}{p^m}\sum_{v\in G_{p,m}}\sum_{h\in G_{p,m}\setminus\{0\}}r_p(h)X_p\left(\frac{v}{x^m}h x^{w_{d+1}}g^{\ast}\right) \times \prod_{i=1}^d \left(1+\gamma_i+\gamma_i\sum_{h\in G_{p,m}\setminus\{0\}}r_p(h)X_p\left(\frac{v}{x^m}h x^{w_{i}}g_i\right)\right).
\hspace *{2 . 5 c m}= - \zeta^{- 2}\kappa ( \kappa - 1 ) \left\langle \psi_{( m^{\prime}, n^{\prime})}, \pi \left( \left( \psi_{( m , n )}\otimes \psi_{( 1 , 2 )}\right) \right) \right\rangle .
{\cal E}\;=\;{2 \pi \rho^2 a \over m}\left\{1 \;+\;{128 \over 15 \sqrt{\pi}}\sqrt{\rho a^3}\;+\; \left[{8 \over 3}( 4 \pi - 3 \sqrt{3})\log (\rho a^3) + \kappa \right] \rho a^3\;+\; \ldots \right\}\;,
\tilde a_J=\begin{cases}\bigl(1/a_{1j_{1}},\ldots, 1/a_{nj_{n}}\bigr), \a_{kj_{k}}\neq 0\ \k=1,\ldots, n, \bigl(1/a_{1j_{1}},\ldots,\infty_{[i_1]}, \ldots,\infty_{[i_k]}, \ldots, 1/a_{nj_{n}}\bigr), \a_{i_1{j_{i_1}}}=\ldots =a_{i_k{j_{i_k}}}=0,\end{cases}
\sum_{g_1+g_2=g}^{\Delta}\sum_{J_1\sqcup J_2=J}d_p\left(\omega_{g_1,n_1+1}(p,J_1)\cdot\omega_{g_2,n_2+1}(q,J_2)+\omega_{g-1,n+2}(p,q,J)\right)\bigg|_{p=q}=\sum_{g_1+g_2=g}^{\Delta}\sum_{J_1\sqcup J_2=J}d_q\left(\omega_{g_1,n_1+1}(p,J_1)\cdot\omega_{g_2,n_2+1}(q,J_2)+\omega_{g-1,n+2}(p,q,J)\right)\bigg|_{p=q}.
\begin{aligned}\Theta_\psi(\tau): =\sum\limits_{\beta\geq 0}(\deg(\psi^\ast Z(\beta))) e^{2\pi i \mathrm{tr}(\beta \tau )}=\sum\limits_{k,l,m\geq 0}N_{k,l,m}\tilde{q}^kp^lq^m\end{aligned}
\omega(t^{-tj})\Gamma_p\left(\left\langle\frac{-tj}{p-1}\right\rangle\right)\prod_{h=1}^{t-1}\Gamma_p\left(\frac{h}{t}\right)=\prod_{h=1}^{t}\Gamma_p\left(\left\langle\frac{h}{t}-\frac{j}{p-1}\right\rangle\right).
g ( p ) \approx \int \frac{d^{3}q}{( 2 \pi )^{3}}\, \frac{1}{( p - q )^{2}}\frac{g ( q )}{\left| - \frac{1}{2}\delta + i q_{3}+ \frac{\left|{\bf q}\right|^{2}}{2 m}- \frac{1}{4 \pi}\ln \frac{1}{2 m}( - \frac{1}{2}\delta + i q_{3}+ \frac{\left|{\bf q}\right|^{2}}{2 m}) \right|^{2}}.
T(r) \geqslant\dfrac{1}{2n+1}\sum_{i=1}^{2n+1}\dfrac{4n-1+3p}{2n+p}\sum_{u=1}^3N^{(n)}_{(f,H_i),\leqslant k_i}(r)+o(T(r)) &\geqslant\dfrac{4n-1+3p}{(2n+1)(2n+p)}\sum_{i=1}^{2n+1}\sum_{u=1}^3(N^{(n)}_{(f,H_i)}(r)-N^{(n)}_{(f,H_i),> k_i}(r))+o(T(r)) &\geqslant\dfrac{4n-1+3p}{(2n+1)(2n+p)}(n-\sum_{i=1}^{2n+1}\dfrac{n}{k_i+1})T(r)+o(T(r)).
\tilde{E}_{n \delta}= \sum_{\begin{array}{c}{{k_{1}p_{1}+ . . . + k_{m}p_{m}= n}}\\{{0 < k_{1}< . . . < k_{m}}}\\ \end{array}}\frac{\left( q^{( \alpha , \alpha )}- q^{- ( \alpha , \alpha )}\right)^{\sum_{i}p_{i}- 1}}{p_{1}! \; . . . \; p_{m}!}( E_{k_{1}\delta})^{p_{1}}. . . ( E_{k_{m}\delta})^{p_{m}}
\mathcal{E}_{i}^{*}\mathcal{E}_{i}- q^{-2}\mathcal{E}_{i}\mathcal{E}_{i}^{*}\sim - q^{-1}(q - q^{-1}) \sum_{k = i + 1}^{N}q^{i - k}\mathcal{E}_{k}\mathcal{E}_{k}^{*}.
\frac{1}{2 4 . 8 \pi^{2}}\, \int_{\cal M}R \wedge R = \frac{1}{2 4 . 8 \pi^{2}}\, \int_{{\cal M}}\mathrm{d}C = \frac{1}{2 4 . 8 \pi^{2}}\, \int_{\partial{\cal M}}C \, ,
E(|\epsilon(t_n+\zeta h)|^2|\mathcal{F}_0)\leq (1+12(1-\theta)^2Kh^2+12Kh+h+(3-2\theta)K^{\frac{1}{2}}h)R_n +(12\theta^2Kh^2+2\theta K^{\frac{1}{2}}h)R_{n+1}.
\begin{aligned}(p_i^I) =(\tilde{a}_{i_1},a_3,\ldots,a_{i_4-1},b_1,b_2,\ldots,\breve{b}_{i_3},\ldots,b_{i_5-1}), (p_i^{II}) =(\tilde{b}_{i_3},a_{i_4}, a_{i_4+1},\ldots), (p_i^{III}) =(\tilde{a}_{i_2},b_{i_5}, b_{i_5+1},\ldots,\breve{b}_{i_3},\ldots).\end{aligned}
F(w^{(\ell)};y^{(\ell)}, r_\ell) = \frac{\int_{B_2(\hat{w}^{(\ell)}; \hat{y}^{(\ell)})}\prod \hat{w}_i^{2 \hat{b}_i(\hat{w}; \hat{y})-1}e^{(2\hat{b}_i(\hat{w};\hat{y})-1)\log r_\ell}\, d\hat{w}d\hat{y}}{\int_{B_1(\hat{w}^{(\ell)}; \hat{y}^{(\ell)})}\prod \hat{w}_i^{2 \hat{b}_i(\hat{w}; \hat{y})-1}e^{(2\hat{b}_i(\hat{w};\hat{y})-1)\log r_\ell}\, d\hat{w}d\hat{y}}.
F_{N}(f) = \sum_{n=0}^{(N/2)-1}a^{1/2}N^{-1/4}f(naN^{-1/2})u(n)+ \sum_{n=N/2}^{N-1}a^{1/2}N^{-1/4}f((n-N)aN^{-1/2})u(n).
\frac{\partial\rho\langle v_{\theta}\rangle}{\partial t}+\frac{\partial\rho\langle v_{r}v_{\theta}\rangle}{\partial r}+\frac{\rho}{r}[3\langle v_{r}v_{\theta}\rangle+(\langle v_{\theta}^{2}\rangle-\langle v_{\phi}^{2}\rangle)cot(\theta)]=0
\int_\Omega J dx =\int_\Omega (v u_x (\rho u)_x-uv_x(\rho u)_x + v_x(u(\rho u)_x+\rho u u_x))dx &=\int_\Omega (v u_x(\rho u)_x + \rho u v_x u_x)dx = \int_\Omega u_x^2 dx.
\delta_{L}\xi_{\mu}= ( \delta_{L}g_{\mu \nu}) \xi^{\nu}+ g_{\mu \nu}\delta_{L}\xi^{\nu}= ( \nabla_{\mu}\xi_{\nu}+ \nabla_{\nu}\xi_{\mu}) \xi^{\nu}= ( \nabla_{\mu}\xi_{\nu}) \xi^{\nu}= -{\cal J}_{\mu},
\begin{matrix}{{r^{p - 1}Y_{p - 1}( \hat{\phi}) F ,}}&{{r^{p - 2}Y_{p - 2}( \hat{\phi}) F^{2},}}&{{r^{p - 2}Y_{p - 2}( \hat{\phi}) F \bar{F},}}\\{{r^{p - 3}Y_{p - 3}( \hat{\phi}) F^{2}\bar{F},}}&{{r^{p - 4}Y_{p - 4}( \hat{\phi}) F^{2}\bar{F}^{2}}}&{{}}\\ \end{matrix}
\ddot{x}^{2}\big [ \acute{x}{}^{2}x \! \! \dot{}\acute{}{}^{a}- ( \acute{x}x \! \! \dot{}\acute{}) \acute{x}{}^{a}\big ] - \acute{x}^{2}( \ddot{x}x \! \! \dot{}\dot{}\acute{}) \dot{x}{}^{a}= 0 .
\mathbf{Y}_{\sf eff}= \mathbf{B}\mathbf{Y}&= \mathbf{A}\mathbf{X}+ (\mathbf{B}\mathbf{H}- \mathbf{A}) \mathbf{X}+ \mathbf{B}\mathbf{Z}&= \mathbf{A}\mathbf{X}+ \mathbf{Z}_{\sf eff}
g'(t) =t\|(u,v)\|_{E}^{2}+3t^{5}\Big(B(u)+B(v)\Big)-t^{2p-1}F(u,v) &=t^{2p-1}\Big(\frac{1}{t^{2p-2}}\|(u,v)\|_{E}^{2}+\frac{3}{t^{2p-6}}\big(B(u)+B(v)\big)-F(u,v)\Big).
\sum_{k=0}^{n-1}\frac{(aq;q^2)_k (q/a;q^2)_k (q;q^2)_{2k}}{(q^2;q^2)_{2k}(aq^6;q^6)_k (q^6/a;q^6)_k}[8k+1]q^{2k^2}\equiv q^{-(n-1)/2}[n]\biggl(\frac{-3}{n}\biggr), \sum_{k=0}^{(n-1)/2}\frac{(aq;q^2)_k (q/a;q^2)_k (q;q^2)_{2k}}{(q^2;q^2)_{2k}(aq^6;q^6)_k (q^6/a;q^6)_k}[8k+1]q^{2k^2}\equiv q^{-(n-1)/2}[n]\biggl(\frac{-3}{n}\biggr).
\tau_{1}( t_{f}) - \tau_{1}( t_{i}) \equiv \int_{t_{i}}^{t_{f}}\frac{d s}{a_{1}^{2}( s )}= \frac{1}{\alpha ( \alpha t_{i}+ \beta )}- \frac{1}{\alpha ( \alpha t_{f}+ \beta )}= \frac{T}{a ( t_{f}) a ( t_{i})},
\sum_{i=1}^{t}ia_i^{\ell}=\sum_{a\in S}a^{\ell}\sum_{\substack{b\in S b\leqslant a}}1=\sum_{a\in S}a^\ell\left(\frac{a}{n}t+E_a\right)=\frac{t}{n}\sum_{a\in S}a^{\ell+1}+\sum_{a\in S}a^\ell E_a
A_0(\ell) = \frac{(-1)^{\ell+1}}{2^{\ell+1}}\sum_{m=1}^{\ell+1}\Bigg[ \frac{(-1)^{(m-1)/2}(\ell+1)! (2\,\ell-m+1)!(m-1)!}{m!(\ell+1-m)! \Gamma(\frac{1+m}{2})\Gamma(\ell+\frac{3-m}{2})}\left((-1)^{m-1}+1\right)\Bigg],
(\partial_t^2-ia\partial_{\alpha})\theta=-2[z_t,\mathfrak{H}\frac{1}{z_{\alpha}}+\bar{\mathfrak{H}}\frac{1}{\bar{z}_{\alpha}}]z_{t\alpha}+\frac{1}{\pi i}\int_{-\infty}^{\infty}\Big(\frac{z_t(\alpha,t)-z_t(\beta,t)}{z(\alpha,t)-z(\beta,t)}\Big)^2(z-\bar{z})_{\beta}d\beta:=g.
C = \Gamma_{2}\Gamma_{4}= \left( \begin{array}{c c}{{I}}&{{0}}\\{{0}}&{{\widetilde{I}}}\\ \end{array}\right) \ , \quad I = \widetilde{I}= \left( \begin{array}{c c c}{{\begin{array}{c c}{{0}}&{{1}}\\{{- 1}}&{{0}}\\ \end{array}}}&{{|}}&{{0}}\\{{---}}&{{|}}&{{---}}\\{{0}}&{{|}}&{{\begin{array}{c c}{{0}}&{{1}}\\{{- 1}}&{{0}}\\ \end{array}}}\\ \end{array}\right)
c(1,\lambda_d,\dots,\lambda_d) =\left(\frac{\lambda_d}{\lambda_2}\right)^{\frac{4-N}{2}}c\left(\frac{\lambda_2}{\lambda_d},\lambda_2,\dots,\lambda_2\right) &\le \left(\frac{\lambda_d}{\lambda_2}\right)^{\frac{4-N}{2}}c(1,\lambda_2,\dots,\lambda_2).
\inf_{Q_Y}\sup_{\theta \in \{0,1\}}D_{1+\lambda}(P_{Y|V=\theta}\|Q_Y) = D_{1+\lambda}(P_{Y|V=0}\|Q^\ast_Y) = \log\left(1+\left(2^{1+\lambda}-1\right)^{-\frac{1}{\lambda}}\right)
M_{p,q}:=\left[ C_{f}R^{2}\frac{\max\{p+q+3,R/\varepsilon\}^{p+q}}{(p+q)!}+(p+q+3)R\Vert\nabla u\Vert_{L^{2}(B_{R}^{+})}\right] =:M_{p,q}^{(1)}+M_{p,q}^{(2)}.
\mathrm{Z}\approx{\bf N_{2}}\int d \Omega e^{\gamma \Omega^{0 0}( 1 - \frac{1}{2 F^{2}}\frac{1}{V}\sum_{n}^{\prime}[ 1 / p_{n}^{2}] )}={\bf N_{3}}Y_{N}\left( \gamma \Big [ 1 - \frac{N - 1}{2 F^{2}}G_{1}\Big ] \right)
c\ast (v_{1}\otimes v^{'})=v_{1}\otimes v^{'}\cdot c,~ \forall c\in \mathbb{C}, ~\forall v_{1}\in V_{1}, ~\forall v^{'}\in V^{'}, c\ast (v_{2}\otimes v^{'})=v_{2}\otimes v^{'}\cdot \overline{c},~ \forall c\in \mathbb{C}, ~\forall v_{2}\in V_{2}, ~\forall v^{'}\in V^{'}.
V_{p_i}: =\left\{\mu=(P_{i,j}) \in V \mid \langle (F_\mu)_{p_i}\rangle \subset \pi_k(x_i) \right\}& = \left\{\mu=(P_{i,j}) \in V \mid \langle P_{n_{k-1}+1}(p_i), \dots, P_{n_k}(p_i) \rangle \subset \pi_k(x_i) \right\}.
\begin{array}{l}\displaystyle-\int\limits_{\mathbb R^d}a (\xi-q) \mu (\xi, q) (\varkappa_2 (q) - \varkappa_2 (\xi) ) \dq begin{align*}4mm] \displaystyle= \b\otimes \varkappa_1(\xi) \ + \ \int\limits_{\mathbb R^d}a (\xi-q) \mu (\xi, q ) \Big( \frac12 (\xi - q )\otimes (\xi - q ) - (\xi-q)\otimes \varkappa_1 (q) \Big) dq \ - \ \Theta,\end{array}