harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
M(f(x)-f(x_{0}))=\frac{Mf^{\prime\prime}(x_{0})}{2}s^{2}y^{2}+\frac{Mf^{\prime\prime\prime}(x_{0})}{6}s^{3}y^{3}+\cdot\cdot\cdot=-\pi y^{2}+O(\frac{1}{\sqrt{M}}).
\lim_{n \rightarrow \infty}\left[I^{\phi}(f_{\alpha}^{(n)})- \left(\frac{1}{\alpha(1-\alpha)}+ \frac{(\alpha-\gamma)^2}{(1-\alpha)^2 \alpha^2}\right) n - B(\alpha, \gamma)\sqrt{n}\right] = C(\alpha,\gamma),
\langle\lambda_s(b)\xi,\eta\rangle =\sum_{t\in G}\lambda_s(b)\xi(t)^*\eta(t)=\sum_{t\in G}(b\xi(s^{-1}t))^*\eta(t)=\sum_{t\in G}\xi(s^{-1}t)^*b^*\eta(t) &\stackrel{r=s^{-1}t}{=}\sum_{r\in G}\xi(r)^*b^*\eta(sr)=\sum_{r\in G}\xi(r)^*\lambda_{s^{-1}}(b^*)\eta(r)=\langle\xi, \lambda_{s^{-1}}(b^*)\eta\rangle.
S' =\sum_{n\in\mathcal{A}(x)}\Bigl(\sum_{i=1}^k\mathbf{1}_{\mathcal{P}}(L_i(n))-m-k\sum_{i=1}^k\sum_{\substack{p|L_i(n) p<x^\rho, p\nmid B}}1-k\sum_{\substack{b\le \eta\log{x}L=an+b\notin\mathcal{L}}}\mathbf{1}_{\mathcal{S}(\theta/10;1)}(L(n))\Bigr)w_n &= S_1-S_2-S_3-S_4.
\left( \rho u,v\right) ={\displaystyle\sum\limits_{j}}\left\langle \dot{\Delta}_{j}(\rho u),\tilde{\Delta}_{j}v\right\rangle ={\displaystyle\sum\limits_{j}}\left\langle \dot{\Delta}_{j}u,\tilde{\Delta}_{j}(\rho v)\right\rangle=\left( u,\rho v\right) .
\rho_{E,E\cup\{\lambda\}}^i(\iota_{\emptyset,E}^i(x)) =\pm\iota_{\{\lambda\},E\cup\{\lambda\}}^i(\rho^i_{\emptyset,\{\lambda\}}(x)){}=\pm\theta_{\lambda,x}\iota_{\{\lambda\},E\cup\{\lambda\}}^i(\iota^i_{\emptyset,\{\lambda\}}(x)){}=\pm\theta_{\lambda,x}\iota^i_{\emptyset,E\cup\{\lambda\}}(x).
P^*_n\left(\sup_{\|\theta-\theta'\|\le \delta_n}\max_{j=1,\cdots,J}|\mathfrak G^b_{n,j}(\theta)-\mathfrak G^b_{n,j}(\theta')|>\epsilon_n\Big|\{X_i\}_{i=1}^\infty\right)\le P^*_n\big(Z^*_n(\tilde\delta_n)>\epsilon_n\big|\{X_i\}_{i=1}^\infty\big).
\ker(T_4)=\begin{cases}H_4/H_4^\prime\simeq\langle xy^2,s_2\rangle\simeq C_3\times C_3 a=w=z=0 a=1,\w=-1, G^\prime/H_4^\prime\simeq\langle s_2\rangle\simeq C_3 .\end{cases}
Q_k(z,a) = P_k ((z-a_1)^{-1}, \dots , (z-a_N)^{-1}) , & Q_{k,j}(z,a,b) =(z-a_j)^{-1}\frac{\partial P_k}{\partial \xi_j}((z-a_1)^{-1}, \dots , (z-a_N)^{-1}), Q(z,a) = Q_W(z,a) = \prod_{s=0}^k (z-a_{i_s})^{N!}\, .
u(x,t)=e^{-t(-\Delta)^{d}}u_0+\int_{0}^{t}e^{-(t-s)(-\Delta)^{d}}\left(|\cdot|^{\alpha}|u(s)|^p\right)\,ds+\int_{0}^{t}\, \zeta(s)\,e^{-(t-s)(-\Delta)^{d}}{\mathbf w}\,ds,
|x+v|^{-s}+|x-v|^{-s}-2|v|^{-s}= -s\frac{|x|^2}{|v|^{s+2}}+s(s+2)\left( \frac{(x\cdot v/|v|)^2}{|v|^{s+2}}\right) +O(|v|^{-s-3}), (|v|\to \infty).
\begin{aligned}\omega_1 \alpha + \omega_2\beta + \omega_3 \gamma + \omega_4 \delta = \binom{n}{2}-1 -\omega_1 \alpha - \omega_3 \gamma = -(n-1) - \omega_2\beta -\omega_4 \delta = -\frac{n(n-3)}{2}.\end{aligned}
\frac{q^r+q^s}{1-q}= \,h+w +i\theta\frac{\rho^r}{1-\rho}\left(r+\frac{\rho}{1-\rho}\right) &+O\left(\frac{(r\theta)^2\rho^r}{1-\rho}\right)+ O\bigl(n^{-1/2}(h+w)\bigr),
\dfrac{g_{i+1}(u - \dfrac{1}{2}\hbar(i+1))}{g_i(u - \dfrac{1}{2}\hbar(i+1))}\prod_{a = 1}^n \dfrac{u + tm_a - \hbar(a - \dfrac{3}{2}+ \delta( j_a \leq i+1 )) - \dfrac{1}{2}\hbar(i+1)}{u + tm_a - \hbar(a - \dfrac{3}{2}+ \delta( j_a \leq i)) - \dfrac{1}{2}\hbar(i+1)}.
^{(i)}={{K\|{{{\tilde{\mathbf{x}}}^{(i)}}}\|_\infty^2}\mathord{\left/{\vphantom{{K\|{{{\tilde{\mathbf{x}}}^{(i)}}}\|_\infty^2}{{{\|{{{\tilde{\mathbf{x}}}^{(i)}}}\|}^2}}}}\right. \kern-\nulldelimiterspace}{{{\|{{{\tilde{\mathbf{x}}}^{(i)}}}\|^2}}}}
\displaystyle \quad - \frac{1}{1 6 8 0}( 1 + \xi )^{2}( 1 4 4 8 + 2 3 6 5 \xi + 8 3 6 \xi^{2}) \zeta ( 3 ) - \frac{1}{1 4 0}( 1 + \xi )^{3}( 2 9 + 2 \xi ) \zeta ( 4 )
\frac{n!}{k! \prod_{i=1}^{k}a_i!}\le \frac{n!}{k!}\le \frac{e \sqrt n}{\sqrt{2\pi k}}\cdot \frac{n^n e^{-n}}{k^k e^{-k}}= \frac{n^n}{(\frac n l)^{\frac n l}}2^{O(n)}= \exp \left \{(1 - \frac 1 l)n \ln n + O(n)\right\}.
\begin{aligned}\begin{cases}U_{t}(\boldsymbol{x},\,t)= \mathcal{A}U (\boldsymbol{x},\,t), \forall\;\boldsymbol{x}\in \Omega ,\ \forall \;t\in \mathbb{R}^{+}, U (\boldsymbol{x},\,0)= U_{0}(\boldsymbol{x}), \forall\;\boldsymbol{x}\in \Omega,\end{cases}\end{aligned}
\sum_{\substack{1 \leq w \leq h 1 \leq z \leq s}}(-1)^{w + z}\sum_{\substack{y_1+...+y_w=h x_1+...+x_z=s}}\sum_{\substack{0 \leq p_w \leq ... \leq p_1 \leq i-hd i-hd+1 \leq v_1 \leq ...\leq v_z \leq r+1-jd}}\prod_{a=1}^w i_a \alpha^{i_a}_{r-(h-y_1-...-y_a)d-p_a}\prod_{b=1}^z x_b \alpha^{x_b}_{r-(j-x_b-...-x_z)d- v_b}.
\begin{pmatrix}{\dot{f}}_1 \vdots{\dot{f}}_N\end{pmatrix}=\begin{pmatrix}\lambda_{11}\cdots \lambda_{1N}\vdots \ddots \vdots \lambda_{N1}\cdots \lambda_{NN}\end{pmatrix}\begin{pmatrix}f_1 \vdots f_N\end{pmatrix},
V = \left( \begin{array}{c | c}{\begin{array}{c c}{0}&{B}\\{- B}&{0}\\ \end{array}}&{M}\\ \hline{- M^{T}}&{\begin{array}{c c}{0}&{\tilde{B}}\\{- \tilde{B}}&{0}\\ \end{array}}\\ \end{array}\right)
d\bar X_t =\frac{1}{n}\sum_{k=1}^n dX^k_t =\frac{\lambda}{n}\sum_{k=1}^n\gamma_t^k\,dt+\frac{\sigma}{n}\sum_{k=1}^n dW^k_t+\frac{1}{n}\sum_{k=1}^n\gamma_{t}^k\,d\widetilde N^k_t .
c_{\lambda_n}\geq\kappa(\lambda_n)\geq\left(\frac{1}{8}-\frac{1}{2^{2p+1}}\right)\left(\frac{\beta_{\lambda_n}^p}{2C(N,\alpha,p)}\right)^{\frac{1}{p-1}}=\left(\frac{1}{8}-\frac{1}{2^{2p+1}}\right)\left(\frac{\beta_{0}^p}{2C(N,\alpha,p)}\right)^{\frac{1}{p-1}}>0.
e^{\sqrt{2}t_{1}a^{\dagger}}\vert 0 \rangle *_{\mathrm{p o i n t}}e^{\sqrt{2}t_{2}a^{\dagger}}\vert 0 \rangle = \pi^{- 1 / 4}\sqrt{\frac{2}{3}}e^{{\frac{1}{3}}( t_{1}^{2}+ t_{2}^{2}) -{\frac{4}{3}}t_{1}t_{2}}e^{-{\frac{2 \sqrt{2}}{3}}( t_{1}+ t_{2}) a^{\dagger}+{\frac{1}{6}}( a^{\dagger})^{2}}\vert 0 \rangle .
e^{xz+tz^{3}}\left( \partial_{t}f-f^{\prime\prime\prime}-3a_{1}^{\prime}f^{\prime}-\dfrac{3}{2}a_{1}^{\prime\prime}f\right) & =\partial_{t}u-3z^{2}u^{\prime}+3zu^{\prime\prime}-u^{\prime\prime\prime}+3za_{1}^{\prime}\left( 1+u\right) -3u^{\prime}a_{1}^{\prime}-\dfrac{3}{2}a_{1}^{\prime\prime}\left( 1+u\right)
C_{\infty}\leq C\sum_{k=0}^{\infty}\frac{|2^{k+1}Q_{0}|^{\frac{\alpha t}{n}}}{|2^{k+1}Q_{0}|^{t}|Q_{0}|}\left(\int_{2^{k+3}Q_{0}}fdx\cdot\int_{2^{k+3}Q_{0}}gdx\right)^{t}\left(\int_{Q_{0}}v^{\frac{t}{1-t}}dx\right)^{1-t}.
u(x)=2k_{N,1}(1-|x|^2)^{s-1}\int_{\partial B}\frac{1-|x|^2}{|x-\theta|^N}\varphi(\theta)\d\theta=\frac{2k_{N,1}{s}}{k_{N,s}}\int_{\partial B}M_s(x,\theta)\varphi(\theta)\d\theta,
\delta\leq \lim_{n}[v_n^+]_{s,p}^p\leq\lim_{n}\langle (-\Delta_p)^s v_n, v_n^+\rangle= \lim_{n}\lambda\int_\Omega |v_n^+|^r\, dx+\mu\int_\Omega \frac{|v_n^+|^{q_n}}{|x|^\alpha}\, dx=0
- H_{\rho \mu \nu}H^{\rho \mu \nu}+{\frac{1}{\Lambda}}( g_{A}^{- 2}\omega_{3}^{A}+ g_{G}^{- 2}\omega_{3}^{G}- g_{L}^{- 2}\omega_{3}^{L})_{\rho \mu \nu}( \partial_{\tau}\theta ( x ) -{\frac{c_{A}\Lambda}{4 8 \pi^{2}g_{A}}}B_{\tau}) \epsilon^{\tau \rho \mu \nu}
|\psi_{gs}^{N_f}\rangle\sim e^{-\frac{\omega}{2}\mathrm{Tr}\left(M+\frac{g}{2\omega^2}V \eta V^\dagger \right)^2}\tilde{f}_{i_1}^\dagger\cdots\tilde{f}_{i_{N_f}}^\dagger|0\rangle_f,\;\eta=\left[ \begin{array}{cc}\mathbb{I}_{N_f}0 0 -\mathbb{I}_{N-N_f}\end{array}\right].
4 \pi L_{M}= - \frac{1}{4}e^{2 \kappa \Psi}F_{\mu \nu}^{a}F^{\mu \nu , a}- \frac{1}{2}\partial_{\mu}\Psi \partial^{\mu}\Psi - \frac{1}{2}D_{\mu}\Phi^{a}D^{\mu}\Phi^{a}- e^{- 2 \kappa \Psi}V ( \Phi^{a}) \ ,
\mathfrak{L}[u](x) = \varphi_{0}(v) + \frac{\alpha^{2}}{6}\left\{\bigtriangleup\varphi(v)\left( \frac{dv}{dx}\right)^{2}+ \varphi_{0}'(v)\frac{d^{2}v}{dx^{2}}\right\}+ o(\alpha^{2}),
\frac{dE}{d\tau}(\tau) =\partial_t h_\rho(\tau,q(\tau,0;x,\eta(t,0;x,\xi)),p(\tau,0;x,\eta(t,0;x,\xi))) &=\partial_t V_\rho(\tau,q(\tau,0;x,\eta(t,0;x,\xi))).
\left[K(\mathcal{H}) \cap K(\mu_{\ell^\infty}) : K \right] \geq \frac{1}{2}\left[K(A[\ell]) \cap K(\mu_{\ell^\infty}) : K \right] \geq \frac{1}{2}\left[ K(\mu_\ell):K \right] = \frac{\ell-1}{2},
\ \ \rho(\lambda,\varepsilon)\phi & =\sum_{k\neq0,\ \pm}\left( \int_{v>0}e^{ik\theta_{\pm}}\frac{\omega_{\pm}\mu_{\pm,+}^{\prime}(e_{\pm})}{\omega_{\pm}+\frac{\lambda}{ik}}\phi_{k}^{\pm}(I_{\pm})dv+\int_{v<0}e^{ik\theta_{\pm}}\frac{\omega_{\pm}\mu_{\pm,-}^{\prime}(e_{\pm})}{\omega_{\pm}-\frac{\lambda}{ik}}\phi_{k}^{\pm}(I_{\pm})dv\right) .
M_{0, \varphi}(x) =\left(\begin{array}{cc}\frac{-8 i + 4 i e^{- i \varphi x}+ 4 i e^{i \varphi x}}{x + i}\frac{12 i - 4 i e^{- i \varphi x}- 8 i e^{i \varphi x}}{x + i}& \frac{- 6 i + 4 i e^{- i \varphi x}+ 2 i e^{i \varphi x}}{x + i}\frac{8 i - 4 i e^{- i \varphi x}- 4 i e^{i \varphi x}}{x + i}\end{array}\right).
lim_{n\rightarrow\infty}\mathbb{P}[\sqrt{n}(\overline{X}_{n}-\mu)\le z]=lim_{n\rightarrow\infty}\mathbb{P}[\frac{\sqrt{n}(\overline{X}_{n}-\mu)}{\sigma}\le\frac{z}{\sigma}]=\Phi(\frac{z}{\sigma}),
\begin{aligned}\sum_{\alpha_1<n\leq \beta_1}f(a+nw_1+\alpha_2w_2) =\int_{\alpha_1}^{\beta_1}f(a+xw_1+\alpha_2w_2) dx &\quad+\int_{\alpha_1}^{\beta_1}\frac{\partial f(a+xw_1+\alpha_2w_2)}{\partial x}P_1(x) dx &\quad+P_1(\alpha_1)f(a+\alpha_1w_1+\alpha_2w_2) &\quad-P_1(\beta_1)f(a+\beta_1w_1+\alpha_2w_2).\end{aligned}
\zeta(z, q=0) = 2\pi i \frac{e^{2\pi i z}}{e^{2\pi i z}-1}- \pi i = f(z) - \pi i \wp(z, q=0) = (2\pi i)^2 \frac{e^{2\pi i z}}{(e^{2\pi i z}-1)^2}= g(z).
f\circ\Phi_{\tau}^H(x,\xi)=\sum_{k \in \mathbb{Z}^d}f_k(x,\xi) \frac{e^{ik\cdot\tau}}{(2\pi)^d}, f_k(x,\xi):=\int_{\mathbb{T}^d}f \circ \Phi_{\tau}^H(x, \xi) e^{-ik \cdot \tau}d\tau.
\sup_{x \in H}| \nabla_k R_t[\Phi](x) \vert_J \leq \frac{c}{t^{\frac{3}{2}}}\Vert \Phi\Vert_\infty\vert k\vert_K,\,k\in K; \sup_{x \in H}\vert \nabla^G_{a}R_t[\Phi](x) \vert_J \leq \frac{c}{t^{\frac{1}{2}}}\Vert \Phi\Vert_\infty\vert G a \vert_{K}= \frac{c}{t^{\frac{1}{2}}}\Vert \Phi\Vert_\infty\vert a \vert_{U},\, a \in U.
C(t) =-\int_{-\infty}^t e^{4x}\bigg(\int_x^{\infty}F(y)e^{-2y}dy\bigg)dx &=-\frac{1}{4}e^{4t}\int_t^{\infty}F(x)e^{-2x}dx -\frac{1}{4}\int_{-\infty}^t F(x)e^{2x}dx,
\begin{aligned}\mathcal J_+(u) :=\frac{1}{2}\|u\|^2-\frac{1}{2}\int_D{\alpha(x)}|u(x)|^{2}\,d\mu-\int_D F_+(x, u(x))d\mu &=\frac{1}{2}\|u\|^2-\frac{1}{2}\int_D{\alpha(x)}|u(x)|^{2}\,d\mu-\int_D F(x, u^+(x))d\mu,\end{aligned}
|\overline{W}_l S^{\varepsilon_{l+1}}_{\alpha_{l+1}}{\overline{W}\,}^{-1}_{l+1}|_\mathfrak{U}=|\overline{W}_{m+n+m-l}S^{\varepsilon_{l+1}}_{\alpha_{l+1}}{\overline{W}\,}^{-1}_{m+n+m-l-1}|_\mathfrak{U}.
q_N(a)= \left({N}\varrho(1)\right)^{Na}{\rm e}^{-N\varrho(1)}\frac{1}{(Na)!}\sim \left(\frac{\varrho(1)}{a}\right)^{Na}{\rm e}^{N\left(a-\varrho(1)\right)}\frac{1}{\sqrt{2\pi Na}}=\check q_N(a).
(-1)^{k+\frac{kd}{n}}\left( \left(n \frac{(3d^2+1)n^2-6d^2n+2d^2}{12d}-2\binom{n}{3}\right) \left(n\binom{d-2}{\frac{kd}{n}-2}-k\binom{d-1}{\frac{kd}{n}-1}\right) \right. &\left. -\binom{n}{2}\left(n\binom{d-2}{\frac{kd}{n}-2}-(k-1)\binom{d-1}{\frac{kd}{n}-1}\right) +\binom{n}{2}^2 \binom{d-1}{\frac{kd}{n}-1}\right).
d_0\tilde\varphi(s,t)-\varphi(s,t)=\varphi(a_k,c_0)+\sum_{i=0}^{m-1}\varphi(c_i,c_{i+1})-\varphi(a_k,b_0)-\sum_{i=0}^{l-1}\varphi(b_i,b_{i+1})-\varphi(s,t)
Z_3(q,\zeta) = \;\zeta^2\, (q^d-1)^2(q^{i-1}-1)(q^{d-i}-1) \; - \zeta (q-1)(q^d-1)(q^{d-1}+1)(q^{i-1}-1)(q^{d-i}-1) \; - (q^{d-1}-1)(q^i-1) \big( (q^{d-1}+1)(q^{d-i+1}+1)-2 q^{d-i}(q^i+1) \big).
D^{(\delta,\ell,E,\tau)}_{n,k+\delta\tau^{-1}}= \Phi_{\delta,e^{(\delta,\ell,E,\tau)}_{n-1}(m)}\Big(D^{(\delta,\ell,E,\tau)}_{n-1,k}\Big)+s^{(\delta,\ell,E,\tau)}_{n}(m),
D(n,k,0,p,b)= \dim V_{\lambda_{n,p,b}}^{\mathfrak{gl}(k+1)}=(1+p)^{{n+1 \choose 2}}\prod_{j=1}^{n+1}{{(n-j+1)p+b+k-j+1 \choose k-j+1}\over{(n-j+1)(p+1)+b \choose n-j+1}}.
\Vert\phi\Vert_{L^2_\xi}:= \left(\ell^3\sum_{k\in\mathbb{Z}^3}\vert\widehat{\phi}_{\xi}(k)\vert^2\right)^{1/2}=\left(\int_{Q_\ell}|\phi_\xi(x)|^2\,dx\right)^{1/2}=\Vert\phi_\xi\Vert_{L^2(Q_\ell)}.
L_s=\frac{\tau-B_s\tau^{B_s}+(B_s-1)\tau^{B_s+1}}{(1-\tau)(1-\tau^{B_s})}, L_r=\frac{1}{1-\pi_r(B_r)}\sum\limits_{i=0}^{B_r-1}{i\pi_r(i)},
G^{+}( z_{1}) \gamma ( z_{2}) = - \frac{( \gamma \gamma )}{z_{1 2}^{2}}+ \left[ \frac{2}{3 ( 1 + c_{1})}( G^{-}\gamma \gamma \gamma ) - \frac{2}{1 + c_{1}}( J_{w}\gamma \gamma ) + ( \gamma^{\prime}\gamma ) \right] \frac{1}{z_{1 2}}\; .
\sum_{k=1}^\mu k^{\eta}\ \le int_{1/2}^{\mu+1/2}x^{\eta}\, \mathrm{d}x\ = \ \begin{cases}\frac{1}{\eta+ 1}x^{\eta+ 1}\Big|_{1/2}^{\mu+1/2}, \ \ \eta\not= -1, begin{align*}4ex]\log x \Big|_{1/2}^{\mu+1/2}, \ \ \eta= -1. \end{cases}
(-Lz_s,z_s)_V =\frac{1}{2}\sum_{e\in E}\frac{\delta_E(e)}{|e|^2}\sum_{u,v\in e}(z_s(u)-z_s(v))^2 =\frac{1}{2}\sum_{e\in \partial S}\frac{\delta_E(e)}{|e|^2}2|e\setminus S||S\cap e||V|^2 \le\frac{1}{4}\delta_E(\max)|\partial S||V|^2.
\sup \mathrm{(ODP)}_{0_{X}^{\ast}}=-\inf_{S\in U,{{{\lambda}\in}}\mathbb{R}_{+}^{S}}\left\{{{{\sum\limits_{s\in S}{{\lambda_{s}b_{s}:}\sum\limits_{s\in S}{{\lambda_{s}a_{s}^{\ast}=}}}}-c^{\ast}}}\right\}.
i \frac{\partial}{\partial t}\Phi ( 1 , \cdots , N ; t ) = - \frac{1}{2 m}\sum_{p = 1}^{N}\left\{\partial_{i}^{( p )}+{\cal A}_{i}^{a}({\bf r}_{p}; 1 , \cdots , p - 1 , p + 1 , \cdots , N ) T_{( p )}^{a}\right\}^{2}\Phi ( 1 , \cdots , N ; t ) .
\langle F_{\varepsilon}(u), u\rangle =\langle B(u), \nabla u\rangle- \langle A_{\varepsilon}(u)\nabla u,\nabla u\rangle \leq (C+C\|u\|_H\|u\|_{H^{1}})-\delta \|u\|_{H^{1}}^2 \leq C+C\|u\|_H^2-\delta_1 \|u\|_{H^{1}}^2
d_{n-1}\circ d_{n}= (^{\omega}\varepsilon)^{-1}\circ d_{n-1}^{\ast\ast}\circ^{\omega}\varepsilon \circ (^{\omega}\varepsilon)^{-1}\circ \varphi^{\ast}= (^{\omega}\varepsilon)^{-1}\circ (\varphi \circ d_{n-1}^{\ast})^{\ast}=0
(\tilde{\varphi}_a^i \smile\tilde{\varphi}_a^j)(1) = \tilde{\varphi}_a^i(1) \otimes \tilde{\varphi}_a^j(1) = [\varphi_a^i]\otimes[\varphi_a^j], (\varphi_a^i \smile \varphi_a^j)(1) = \varphi_a^i(1)\otimes \varphi_a^j(1) = [u_a^i]\otimes [u_a^j], (u_a^i\smile u_a^j)(1) = u_a^i(1)\otimes u_a^j(1) \mapsto \frac{a^2}{\gcd(\delta_i, c_a^i-1)\gcd(\delta_j,c_a^j-1)},
e^{H[\boldsymbol{t}]}\psi(z) e^{-H[\boldsymbol{t}]}= e^{-\sum_{q \geq 1}t_q z^q}\psi(z) e^{H[\boldsymbol{t}]}\psi^\ast(z) e^{-H[\boldsymbol{t}]}= e^{\sum_{q \geq 1}t_q z^q}\psi^\ast(z).
\epsilon_{n,k}=\mathrm{sign}(\nu_{n,k},\Delta_{k})=\begin{cases}\varepsilon_{n,k-1}\mbox{for}\,\,k\,\,\mbox{even}, \varepsilon_{n,k-1}(-1)^{Z(n,k-1)+Z(n,k+1)}\mbox{for}\,\,k\,\,\mbox{odd},\,\,\ell(n)\neq k, \varepsilon_{n,k-1}(-1)^{Z(n,k-1)+Z(n,k+1)+1}\mbox{for}\,\,k\,\,\mbox{odd},\,\,\ell(n)=k.\end{cases}
\psi_{-1/2}\left| \lambda \right> = (-1)^n \psi_{\lambda_1 + n - 1/2}^\ast \psi_{\lambda_2 + n - 3/2}^\ast \cdots \psi_{\lambda_n + 1/2}^\ast \psi_{-1/2}\left| 0 \right> & = (-1)^n \psi^\ast(z_1) \psi^\ast(z_2) \cdots \psi^\ast(z_n) \left| -1 \right> \, \Big\vert_{\boldsymbol{z}^{\lambda}}.
\overline{T}=w\frac{q\left( 2\kappa v+\delta^{\prime}\right) +2\delta q_{/1}+2q_{/2}(\kappa w^{2}+\delta^{\prime}v)}{2\delta^{2}}\bar{e}+\frac{vq+w^{2}q_{/2}}{\delta w}\left( v\bar{n}+\delta \bar{z}\right).
v(a)/u(a)-1=\begin{cases}(1+u(a')^{-1}u(\pi)^{r}w')(1+u(\pi)^{r-1}w)^{n}-1 (n\ge 0) (1+u(\pi)^{r-1}w)^{n}\left(\left(1+u(a')^{-1}u(\pi)^{r}w' \right)-\left(1+u(\pi)^{r-1}w\right)^{-n}\right) (n<0).\end{cases}
\mu_t = \sum_{i=0}^\infty \mu_i t^i, \mu_i \in \mathrm{Hom}(A^{\otimes 2}, A) \mu_0 = \mu, &R_t = \sum_{i=0}^\infty R_i t^i, R_i \in \mathrm{Hom}(A, A) R_0 = R,
p_{i}^{\nu}= \frac{1}{2}\int f_{i}^{2}\left\{3 a ( u^{\nu}s_{\sigma}- s^{\nu}u_{\sigma}) + \Theta \delta_{\sigma}^{\nu}+ 3 ( \sigma_{. \sigma}^{\nu}+ \omega_{. \sigma}^{\nu}) \right\}d x^{\sigma}d V .
\lim_{\substack{N\to\infty \lambda=Np}}\binom{N}{i}p^i (1-p)^{N-i}= \lim_{N\to\infty}\frac{N(N-1) \cdots (N-i+1)}{N^i}\left(1- p\right)^{N}\frac{\lambda^i}{i!}= e^{-\lambda}\frac{\lambda^i}{i!}
f_{1}(y) =\mathbf 1_{[0,\Gamma e^{-\omega_{l}\tau})}(y)\, y^{-2}\Gamma^{\gamma}, f_{2}(y) =\mathbf 1_{[\Gamma e^{-\omega_{l}\tau},\infty)}(y)\,y^{-2}\Gamma^{-2\gamma}f_{3}(y) =\mathbf 1_{[\Gamma e^{-\omega_{l}\tau},\infty)}(y)\,y^{2l}e^{-(1-2\sigma)\lambda_{l}\tau}.
\partial_{\bar a}C_{i_1 ... i_s}^g = fact_{g}({\bar C}_{\bar a}) -\sum_{i=1}^s{\bar C}_{{\bar a}i}^l C_{\sigma_1 (l) i_1 ..{\hat i}.. i_s}^g +{\bar C}_{{\bar a}0}^l C_{\sigma_2(l) i_1 ... i_s}^g
\tilde N_{n, \mathbf{p}_{k,1}}(k) \tilde N_{n,\mathbf{p}_{k,2}}(k)=\frac{1}{n^2}\sum_{i,j=1}^n \prod_{\ell=1}^{k}\tilde I^{(p_{\ell,1})}_{i,i}\tilde I^{(p_{\ell,2})}_{j,j}.
\left|\sum_{1\leq i_1 < i_2 < \cdots < i_{n-m}\leq n-1}\omega_{i_1}\omega_{i_2}\cdots\omega_{i_{n-m}}\right| = \left|\omega_m\omega_{m+1}\cdots\omega_{n-1}\right|\geq \left|\omega_m\right|^{n-m}.
E \Biggl[\zeta f_1\bigl(\check{\mathbf{L}}^n\bigr)f_2\bigl(\widetilde{\mathbf{L}}(\beta)^n\bigr)\prod_{j=1}^JY^n_{j-}Y^n_{j+}\Biggr] -E \Biggl[\zeta f_1\bigl(\check{\mathbf{L}}^n\bigr)f_2\bigl(\widetilde{\mathcal{W}}(\beta)\bigr)\prod_{j=1}^JY^n_{j-}Y^n_{j+}\Biggr] \to0
\lim_{r\to\infty}\int_{{\cal I}^-}dt\, d\phi\, dt'\, d\phi'\,\left[\frac{(rr')^2}{l^2}\left(\Phi(r,t,\phi){\buildrel \leftrightarrow \over \partial_{r_{\ast}}}G(r,t,\phi; r',t',\phi'){\buildrel \leftrightarrow \over \partial_{r'_{\ast}}}\Phi(r',t',\phi')\right)\right]_{r=r'},
d s^{2}\, : = \, B_{\mathrm{H}}[ \sigma ( r - r_{\mathrm{H}}) ]^{2 ( 1 - \alpha )}\, \bar{g}_{\mu \nu}d x^{\mu}d x^{\nu}+ \frac{d r^{2}}{B_{\mathrm{H}}( r - r_{\mathrm{H}})^{2}}+ r_{\mathrm{H}}^{2}d \Omega_{n - 1}^{2}.
\begin{aligned}\Delta R \approx \sum\limits_{k = 1}^{{N_{\rm{R}}}}{{{\log}_2}\left({\frac{{1 +{{{\bar C}_k}}}}{{1 +{{\bar C}_k}\left({1 -{2^{- 1 -{x_k}}}}\right)}}}\right)}<\sum\limits_{k = 1}^{{N_{\rm{R}}}}{{{\log}_2}\left({\frac{{1}}{{{1 -{2^{- 1 -{x_k}}}}}}}\right)}.\end{aligned}
\tilde h_t(x,y^1,y^2,\omega) :=h_t(x,\alpha y^1+(1-\alpha)y^2,\omega), \bar h_t(x,y^1,y^2,\omega) :=\alpha h_t(x,y^1,\omega)+(1-\alpha)h_t(x,y^2,\omega)
m_{0}^{2}+\frac{4\lambda \mu^{4-D}}{(\pi )^{D/2}}\Gamma (\frac{D}{2}-1) \left[ (\beta_{c}^{2-D}+L_{c}^{2-D})\zeta (D-2)+2E_{2}((D-2)/2;\beta_{c},L_{c})\right] =0.
u_{{m}}^{{{f}}}={\frac{1}{3 2}}\left[ v_{{A}}^{{\hat{a}}}\left( C \Gamma_{{m}}\right) v_{{\hat{b}}{A}}+ v^{{\hat{a}}{\dot{A}}}\left( C \Gamma_{{m}}\right) v_{{\hat{b}}{\dot{A}}}\right] ( \gamma^{f})_{{\hat{a}}}^{{{\hat{b}}}},
\widehat{P}_{q}^{(n)}\leq \|\widehat{R}_{q}^{(n)}\|_{0}(I\otimes \widehat{P}_{q}^{(n-1)}) &\leq (1+q^{n-1})[n]_q ( I\otimes\widehat{P}_{q}^{(n-1)}) &\leq \frac{2}{1-q}I\otimes \widehat{P}_{q}^{(n-1)}.
\partial_t u = \int_{D(y)}\Lambda(y, x) \partial_t v(t, x, y) dx = \int_{D(y)}\Lambda(y, x) \AA^Z v(t, x, y) dx = \lim_{l \to \infty}\int_{D(y)}\Lambda(y, x) \AA^Z v_l(t, x, y) dx
\mathfrak{a}(A, \psi)= -\frac{1}{2}\int_Y (A-A_0)\wedge d(A-A_0)- \int_Y(A-A_0) \wedge (F_{A_0}+ \frac{1}{2}F_{A_{K^{-1}}}) & - \frac{i}{2}\int_Y(A-A_0)\wedge (2r\omega + \wp_3) + r\int_Y \psi^*D_A \psi,
\tilde{F}( k ) ={\frac{1}{4 \pi \eta}}\int_{- \infty}^{+ \infty}d q \, \tilde{F}( q ) e^{i{\frac{k - q}{2}}}\,{\frac{\sin{\frac{k - q}{2}}}{\frac{k - q}{2}}}\, ( \mathrm{s i g n}\, ( q ) + \eta ) \, ( 1 - e^{- 2 \pi q \eta \lambda}) ,
\lambda^*\partial_z(\alpha w) - \Delta(\partial_z(\alpha w)) = \partial_z\left[\alpha F- 2(\partial_z\alpha) (\partial_z w)- (\partial_z^2\alpha) w\right]- (\partial_z\alpha) (\nabla_H \Pi) \quad\Omega', \partial_z(\alpha w) = 0 \quad\Gamma_u'\cup\Gamma_b', \partial_z(\alpha w) \Gamma_l'.
\begin{aligned}\left [\int_{-10}^{u_0}\int_u^{2 T - u}\int_{S^2}|\Box \partial^\alpha \phi| r^3 |L(r \partial^\alpha \phi)| d \omega d v d u + \int_{-20}^0 \int_{\Sigma_t}|\Box \partial^\alpha \phi| |\partial_t \phi| d x d t + \Vert \phi_0 \Vert_{H^{|\alpha| + 1}}^2 + \Vert \phi_1 \Vert_{H^{|\alpha|}}^2 \right ] \le C \epsilon^2. \end{aligned}
L_f^{(m)}(s)= \sum_{n\leq \frac{|t|}{2\pi}}\frac{\lambda_f(n)(-\log n)^m}{n^s}+ &+\sum_{r=0}^m(-1)^{r}\binom{m}{r}\chi_f^{(m-r)}(s)\sum_{n\leq \frac{|t|}{2\pi}}\frac{\lambda_f(n)(-\log n)^r}{n^{1-s}}+O(|t|^{1/2-\sigma+\varepsilon}),
\left(\frac{1}{T_i(u)},\int_0^{+\infty}e^{2\rho_i(u+v)-2\rho_i(u)}dv\right)_{i\in V}=\left( \frac{1}{T_i^{\infty}}+\frac{1}{\theta_i^2T_i^*(u)},\frac{Z_i^*(u)}{e_i^*(u)}+\frac{T_i^{\infty}}{\theta_i^2e_i^*(u)^2}\right)_{i\in V}.
r_{2^{n-2}+j}=r_j-2s_{2^{n-2}-j}; r_{2^{n-1}-j}=r_{2^{n-2}+(2^{n-2}-j)}=r_{2^{n-2}-j}-2s_{2^{n-2}-(2^{n-2}-j)}=r_j-2s_j, r_{2^{n-2}+j}+r_{2^{n-1}-j}=r_j-2s_{2^{n-2}-j}+r_j-2s_j=2r_j-2r_j=0.
g_n^H:=\sigma\, C_H\int\limits_{n-1}^{n}x^{\frac{1}{2}-H}(n-x)^{H-\frac{1}{2}}\left(\int\limits_0^1 (y(n-x)+x)^{H-\frac{1}{2}}y^{H-\frac{3}{2}}dy\right)dx.
\prod_{j=1}l_j = \Big|\prod_{j=1}^n A_j\Big| &=\sum_{i=1}^n \sum_{a\in\prod_{j=1,j\not=i}^n A_j}\big|\{a\in A_i: i(a_1,a_2,\ldots, a_{i-1},a, a_{i+1},\ldots a_n)=i\}\big| &\le \sum_{i=1}^n K_i\prod_{j=1, j\not=i}^n l_j\le\frac{n}{n+1}\prod_{j=1}^nl_j
A =-\frac{(b-d)(d-b+br)}{b(-d+2b+br)},B=-\frac{b(b-d)(d-b+br)}{d^{2}(-d+2b+br)}, C =-\frac{b(b-d)}{(d-b+br)(-d+2b+br)},E=-2\frac{(b-d)(d-b+br)}{d(-d+2b+br)}, F =2\frac{b-d}{-d+2b+br},G=2\frac{b(b-d)}{d(-d+2b+br)}.
f^{\varepsilon}(x)=\mathop{\rm sup}_{y\in\mathcal{H}}\left \{f(y)-\frac{|y-x|^{2}}{2\varepsilon}\right \},f_{\varepsilon}(x)=\mathop{\rm inf}_{y\in\mathcal{H}}\left \{f(y)+\frac{|y-x|^{2}}{2\varepsilon}\right \}.
\begin{aligned}x(t,s)= y+\int_s^t \partial_\xi H(x(\tau,s),\xi(\tau,s))d\tau= y+\int_s^t \xi(\tau,s)d\tau, &\xi(t,s)= \eta-\int_s^t \partial_x H(x(\tau,s),\xi(\tau,s))d\tau=\eta-\int_s^t \partial_x V(x(\tau,s))d\tau.\end{aligned}
F_{\gamma_i}(\gamma ) = Pr(\gamma_{i}<\gamma) &= Pr(i \in \mathcal{G}_{AF})Pr(\gamma_{i}<\gamma|i \in \mathcal{G}_{AF}) &\quad+Pr(i \in \mathcal{G}_{DF})Pr(\gamma_{i}<\gamma|i \in \mathcal{G}_{DF}).
\times \prod_{b = 1}^{N}e^{- 2 \pi \big ( 1 - \frac{g}{\pi + g N}\big ) \sum_{j = 1}^{n_{b}}\sum_{l = 1}^{n_{b}}\Big [ \big ( 1 - \delta_{j l}\big ) \big ( C^{0 , S}( x_{j}^{( b )}, x_{l}^{( b )}) + C^{0 , S}( y_{j}^{( b )}, y_{l}^{( b )}) \big ) - 2 C^{0 , S}( x_{j}^{( b )}, y_{l}^{( b )}) \Big ]}
\int_D |u(x)|^2\log_{+}|u(x)| dx \leq \varepsilon \big\|u \big\|_{V}^2 + \left(\frac{d}{4}\log\frac{1}{\varepsilon}\right) \big\|u\big\|^2 + \big\|u\big\|^2\log\big\|u\big\|+ \frac{1}{2\mathrm{e}}m(D) ,
P \sb 2 = 2 \pi \! \int_{0}\sp k k d k \int \Gamma \sb{1 2}\langle \omega \sb k \omega \sb 1 \omega \sb 2 \rangle \delta ({\bf k}+{\bf k}\sb 1 +{\bf k}_{2}) d \sp 2 k \sb 1 d \sp 2 k \sb 2 .
\prod_{i=1}^N\prod_{j=m}^{n_i+m-1}(1-t^{j})\cdot P_m(N)= &=(1-N\cdot t^m+O(t^{m+1}))\cdot \left(1-\sum_{y=2}^{m+1}\binom{N}{y}t^m+O(t^{m+1})\right)= &=1-\sum_{y=1}^{m+1}\binom{N}{y}t^m+O(t^{m+1}).
\begin{aligned}v(\overline{\eta_n},\psi_n) = h_V(\overline{\eta_n},\psi_n) - \lambda \, (\overline{\eta_n},\psi_n) - (\nabla\overline{\eta_n},\nabla\psi_n) + \lambda \, (\overline{\eta_n},\psi_n) = \big(\overline{\eta_n},(H_V-\lambda)\psi_n\big) - h_0(\overline{\psi_n},\eta_n) + \lambda \, (\overline{\psi_n},\eta_n) \,.\end{aligned}
\bullet \gamma_{jk}= -\gamma_{kj}, \gamma_{*jk}= -\gamma_{*kj}, & \bullet \sigma_j \Phi A_k^* - \sigma_k \Phi A_j^* = \gamma_{jk}\Phi, & \bullet \sigma_j \Phi A_k - \sigma_k \Phi A_j = \gamma_{*jk}\Phi, & \bullet \gamma_{*jk}- \gamma_{jk}= i \left(\sigma_j \Phi \Phi^* \sigma_k - \sigma_k \Phi \Phi^* \sigma_j \right).
\langle \, p_{1}\, | \, \mathcal{O}( 0 ) \, | \, p_{2}, \dots , p_{n}\, \rangle_{c o n n .}^{i n}= \, \mathcal{O}_{n}( \theta_{1}+ i \pi , \theta_{2}, \dots , \theta_{n}) = \mathcal{O}_{n}( \theta_{2}, \dots , \theta_{n}, \theta_{1}- i \pi )
d= \frac{\binom{N_R-1}{r}\binom{N_T}{t}t N^{\binom{N_R-1}{r+1}\binom{N_T}{t}}}{\binom{N_R-1}{r}\binom{N_T}{t}t N^{\binom{N_R-1}{r+1}\binom{N_T}{t}}+(N+1)^{\binom{N_R-1}{r+1}\binom{N_T}{t}}}
\|\nabla e_{n,T}^{nl}\|_{L_t^{2}L_x^{\frac65}}\lesssim T^{1/2}\big|\frac{x_n}{\lambda_n}\big|^{-2b}\|\nabla \Phi_n\|_{L_t^\infty L_x^r}^{2(p-1)}\sum_{j=0}^{1}\big|\frac{x_n}{\lambda_n}\big|^{-j}\|\partial^{1-j}\Phi_n\|_{L_t^\infty L_x^r}\lesssim T^{1/2}\big|\frac{x_n}{\lambda_n}\big|^{-2b+\theta|5/2-3/r|}\to0
X = \begin{pmatrix}\psi \phi \dot\phi \end{pmatrix}, A = \begin{pmatrix}-\alpha^j\partial_j - iM\beta 0 0 0 0 1 0 \Delta 0 \end{pmatrix}, \mathcal M(X) = \begin{pmatrix}i\beta\psi \mathfrak K_1 0 \phi \mathfrak K_2 \end{pmatrix}.
\omega^{( 1 )}= \int_{S^{3}}\hat{Q}={\frac{3}{4}}{\frac{\rho^{3}[ \rho d x_{0}^{4}- x_{0}^{4}d \rho ]}{[ \rho^{2}+ ( x_{0}^{4})^{2}]^{\frac{5}{2}}}}, \quad \quad{\omega^{( 1 )}}^{\prime}= \int_{{S^{3}}^{\prime}}\hat{Q}={\frac{3}{4}}{\frac{\rho^{3}[ \rho d x_{0}^{3}- ( x_{0}^{3}) d \rho ]}{[ \rho^{2}+ ( x_{0}^{3}- \bar{x})^{2}]^{\frac{5}{2}}}}.
\mathcal{I}_2 \ge -\int_{B_{(R+r)/2}}\int_{\mathbb{R}^N\setminus B_R}\Big|J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\Big|\,\frac{|\delta_h u(x)|}{|h|^{\gamma+t}}\,\eta(x)^p\,d\mu(x,y).
\int_0^\infty r^5 Q_\mu dr=\int_0^\infty r^5 Qe^{\mu\phi_{T_1}+\sigma-\mu\frac{r^2}{2}}dr=\frac{1}{\mu}\int_0^\infty \tau^5 \frac{1}{(\mu+\tau^2)^2}e^{\mu\phi_{T_1}(\frac{\tau}{\sqrt{\mu}})+\sigma(\frac{\tau}{\sqrt{\mu}})-\frac{\tau^2}{2}}d\tau,