harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
E_{1}^{(l)}\cdot \left( \prod_{r=1}^{k}t_{1, x_{r}}\prod_{s=1}^{l}t_{2,y_{s}}\right) = \left(\prod_{r=1}^{k}t_{1, x_{r}}\right) E_{1}^{(l)}\left(\prod_{s=1}^{l}t_{2,y_{s}}= \right) = \prod_{r=1}^{k}t_{1, x_{r}}\prod_{s=1}^{l}t_{1,y_{s}}.
H^{(6)}_{3A,1}(\tau), H^{(6)}_{6A,1}(\tau) :=H^{(18)}_{1A,1}(3\tau)-H^{(18)}_{1A,11}(3\tau)+H^{(18)}_{1A,13}(3\tau), H^{(6)}_{3A,5}(\tau), H^{(6)}_{6A,5}(\tau) :=H^{(18)}_{1A,5}(3\tau)-H^{(18)}_{1A,7}(3\tau)+H^{(18)}_{1A,17}(3\tau).
\begin{array}{l}p^{00}=0, begin{align*}5mm]p^{0k}=-g^{ik}\partial_i (ae^{-\Phi}), begin{align*}5mm]p^{ik}=\frac{1}{2}ae^{-\Phi}\left[(\dot{h}_{lm}g^{lm}-2\dot{\varphi})g^{ik}+\dot{h}^{ik}\right], begin{align*}5mm]p_{\varphi}=ae^{-\Phi}(2\dot{\varphi}-\dot{h}_{lm}g^{lm}).\end{array}
c^{-2}u_{tt}-\Delta u- \beta \Delta u_t = f,\quad(0,T)\times\Omega, \partial_\nu u+ \beta \partial_\nu u_t+\alpha u_t = g,\quad(0,T)\times\partial\Omega, (u(0),u_t(0)) = (u_0,u_1) ,\quad\Omega,
[J_\mu,J_\nu] =-\frac{M_{\mu\nu}}{l_1^2}, [J_\mu,M_{\alpha\beta}] =\eta_{\mu\alpha}J_\beta-\eta_{\mu\beta}J_\alpha, [M_{\mu\nu},M_{\alpha\beta}] =\eta_{\mu\beta}M_{\nu\alpha}+\eta_{\nu\alpha}M_{\mu\beta}-\eta_{\mu\alpha}M_{\nu\beta}-\eta_{\nu\beta}M_{\mu\alpha}.
\phi(X_1,\tilde Y-(\tilde t-t)X_1,\tilde t) =\phi(X,Y,t)+\nabla_X\phi(X,Y,t)\cdot(X_1-X) &+\frac 1 2 \langle \nabla_X^2\phi(X,Y,t)(X_1-X), (X_1-X)\rangle &-(\tilde t-t)\bigl (X\cdot\nabla_Y-\partial_t\bigr)\phi(X,Y,t)+{o}(\epsilon^2),
2\int_E\int_{E^{\sqrt{t}}\setminus E}p(t,x,y)\, d\mu(y)\, d\mu(x) \geq \int_E \int_{B_{\sqrt{t}}(x)\setminus E}p(t,x,y)\, d\mu(y)d\mu(x) +\int_{E^{\sqrt{t}}\setminus E}\int_{B_{\sqrt{t}}(x)\cap E}p(t,x,y)\, d\mu(y)\, d\mu(x) = \int_{\Delta^{\sqrt{t}}}\, p(t,x,y)|\chi_E(y)-\chi_E(x)| \, d\mu\times \mu(y,x).
\sum_{j=1}^np_{n\!-\!j}\sum_{k=1}^ja_{j\!-\!k}{\delta_t\omega^k}\!\leq \! \Gamma(2\!-\!\alpha)\tau^{\alpha}\sum_{j=1}^np_{n\!-\!j}(\lambda_1\omega^j+\lambda_2\omega^{j\!-\!1})+\Gamma(2\!-\!\alpha)\tau^{\alpha}\sum_{j=1}^{n}p_{n\!-\!j}g^j.
\max_{j\in B}\left\{\Pr_{v\sim\mu_t}\big[|vH|=j \big]\cdot \frac{2^N}{\binom{N}{j}}\right\}\leq \sqrt{\frac{e^4 N}{8\pi}}\cdot 2^{(1-h(\beta))\eta N}\leq \sqrt{\frac{e^4 N}{8\pi}}\cdot 2^{4\Tilde{\epsilon}(1-\Tilde{\epsilon})\eta N}.
(T-tId_{R})r(t)=\stackrel[i=1]{n-1}{\sum}t^{i}(\stackrel[j=i]{n-1}{\sum}T^{j-i+1}l_{j})-\stackrel[i=1]{n-1}{\sum}t^{i+1}(\stackrel[j=i]{n-1}{\sum}T^{j-i}l_{j})=
u\,=\,F(x,y)\ \ \ \ \ \ \ \{\scriptstyle{(F(0,0)\,=\,0)}}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \v\,=\,G(r,s)\ \ \ \ \ \ \ \{\scriptstyle{(0\,=\,G(0,0))}},
\int_\Omega\sigma (\theta_M)\nabla\phi_M\cdot\nabla w\mathrm{dx}+ \int_\Omega \sigma (\theta_M) \alpha_\mathrm{S}(\theta_M)\nabla\theta_M \cdot \nabla w \mathrm{dx}=\int_{\Gamma_\mathrm{N}}g w \mathrm{ds},
2 \int_{\mathbb{T}}\!\mathrm{d}x \int_{\mathbb{R}}\!\mathrm{d}v\, \sqrt{1+v^{2m}}\phi \, v \, \partial_v \phi = - \int_{\mathbb{T}}\!\mathrm{d}x \int_{\mathbb{R}}\!\mathrm{d}v\, \sqrt{1+v^{2m}}\phi^2 - \int_{\mathbb{T}}\!\mathrm{d}x \int_{\mathbb{R}}\!\mathrm{d}v\, \frac{m v^{2m}}{\sqrt{1+v^{2m}}}\phi^2 \,,
\gamma_2-\frac{1}{(1-m)\beta}= \frac{\beta(n-2)+\sqrt{\beta^2(n-2)^2-4(n-2)}}{2}-\frac{n+2}{4\beta}= \frac{2\beta^2(n-2)+2\beta\sqrt{\beta^2(n-2)^2-4(n-2)}-(n+2)}{4\beta}> 0
v_{3,2,a}^{\lambda}(t,r) := \frac{-1}{r}\int_{t}^{\infty}ds \int_{0}^{s-t}\rho d\rho \left(\frac{1}{\sqrt{(s-t)^{2}-\rho^{2}}}-\frac{1}{(s-t)}\right) \lambda''(s) \left(\frac{-1-\rho^{2}+r^{2}}{\sqrt{(1+\rho^{2}-r^{2})^{2}+4r^{2}}}+ 1\right)
\left( \frac{\mathcal{\vartheta}_{1}\left( \nu \mid \theta \right)}{\nu \mathcal{\vartheta}_{1}^{^{\prime}}\left( 0 \mid \theta \right)}\right) = \exp \left( - \sum_{k = 1}^{\infty}\frac{1}{2 k}\nu^{2 k}G_{2 k}( q^{2}) \right)
( \alpha , \beta ) = \int_{\Sigma}^{}\mathrm{t r}( \alpha \wedge{}^{\ast}\beta ) = \int_{\Sigma}^{}\, \frac{1}{m !}\alpha_{\mu_{1}\cdots \mu_{m}}^{a}\beta_{a}^{\mu_{1}\cdots \mu_{m}}\sqrt{g}\, d^{3}\! x \; ,
\operatorname*{l i m}_{\varepsilon \rightarrow 0}\frac{\varepsilon^{2 k + 2}}{( p q )^{2}+ \varepsilon^{2}}\: \left( \frac{q^{2}}{4}\right)^{n}\: T_{m^{2}}^{( 2 k + 1 + n )}( p ) \; \; \; .
\langle\nabla f_i(x^0), w^1\rangle+\langle \xi^{*i}, u\rangle= s_1[\langle\nabla f_i(x^0), v^i\rangle+\langle \xi^{*i}, u\rangle] &+(1-s_1)[\langle\nabla f_i(x^0), v\rangle+\langle \xi^{*i}, u\rangle]<0.
P\left(D_k(T_{i,k'}) \geq D_\ast ,D_{k-i}(T_{i,k'})\leq \frac{D_\ast}{2}\right) &\leq P\left(D_{k-i}(T_{i,k'})\leq \frac{D_\ast}{2}\middle| D_k(T_{i,k'}) = D_\ast \right)\leq \exp\left(\frac{-0.05 D_\ast}{S(k-i,k)}\right), & S(k-i,k)\leq 1.01 \times 16 tn^2/D_\ast.
C_3={\frac{{\rm{4}}}{{c M}}{{\left({1 - \frac{{\rm{1}}}{\omega}}\right)}^2}{\rm{+}}\frac{K}{{{{c^2 M}^2\omega}}}\left({- 4{\rm{+}}\frac{5}{\omega}}\right){\rm{+}}\frac{{{K^2}}}{{{{c^3 M}^3\omega^2}}}}.
L_{2}^{\ast}=\frac{\phi(\xi_{p})-\phi(\xi_{q})}{F_{Y}(\xi_{p},\xi_{q})},~ L_{1}^{\ast\ast}=\frac{\xi_{p}^{3}\phi(\xi_{p})-\xi_{q}^{3}\phi(\xi_{q})}{F_{Y}(\xi_{p},\xi_{q})},
M(t)=\sum_{j=1}^3\sum_{k}\int_{\mathbb{R}}e^{\mu(t,\xi)+\lambda(t)(|k- k_0|+\sqrt{\sigma}|\xi- \eta_0|)}\big|\widehat{g_j'}(t,k,\xi)\big|^2\,\mathrm{d}\xi,\mu(t,\xi)=-E\int_{T_0}^t H(t',\xi)\,\mathrm{d}t',
\Xi(t)= = PQ_{t,\gamma(t)}\frac{d}{dt}\gamma(t) +P\left[\left(\gamma(t)\frac{d}{ds}\left(\frac{1}{1+tA'_{\gamma(s)}}\right)_{s=t}\right)\circ \Phi^{-1}_{t,\gamma(t)}+\partial_x\left( Q_{t,\gamma(t)}\gamma(t)\right)\frac{d}{ds}\left(\Phi^{-1}_{t,\gamma(s)}\right)_{s=t}\right]
\mu(\gamma) =\mu(\gamma_1)+\mu(\gamma)+\mu(\gamma_2)=\mu(\gamma_1\ast\gamma\ast\gamma_2)=\mu_{Mas}(\gamma_1\ast\gamma\ast\gamma_2) &=\mu_{Mas}(\gamma_1)+\mu_{Mas}(\gamma)+\mu_{Mas}(\gamma_2)=\mu_{Mas}(\gamma).
M\ni\, (X_1+X_2)^{i_1}X_2^{i_2}\cdots X_n^{i_n}-X_1^{i_1}X_2^{i_2}\cdots X_n^{i_n}\cr=\, \Bigl[\sum_{j=1}^{i_1}\binom{i_1}jX_1^{i_1-j}X_2^j\Bigr]X_2^{i_2}\cdots X_n^{i_n}\cr=\, \sum_{j=1}^{i_1}\binom{i_1}jX_1^{i_1-j}X_2^{j+i_2}\cdots X_n^{i_n}.
\widetilde{t}_{j,k}^{\left[\widetilde{p},q,L1\right]}={t_{j,k+\widetilde{p}-1}-t_{j,k+1}\over t_{j,k+\widetilde{p}}-t_{j,k}}\,\widetilde{t}_{j,k}^{\left[\widetilde{p},q,L1b\right]},
\frac{d}{du}\biggl|_{u=0}\widetilde{Z}(R(S),\langle\cdot\,,\cdot\rangle^{A(u)})=\biggl(\,\sum_{k=0}^N\frac{1}{2}(-1)^ka_0(B_k\,P_{[{\cal H\/}_k^{\perp}]}\,|\,\Delta_k)\biggr)\,\widetilde{Z}(R(S),\langle\cdot\,,\cdot\rangle)
\int_0^t\int_{-2^m}^{2^m}g(s,x)d_{(s,x)}\ell^x(s) =\int_0^t\int_{-2^m}^{2^m}\ell(s,x)d_{(s,x)}g(s,x) - \int_{-2^m}^{2^m}\ell^x(t)d_xg(t,x); ~0\le t \le T.
\begin{array}{l l}{{\varphi^{\prime}- \frac{A \varphi}{r}= 0}}\\{{\frac{A^{\prime}}{r h}- h \varphi^{2}e^{- 2 g M}+ 2 m M}}\\{{M^{\prime}+ \frac{G}{g r}= 0 ,}}\\ \end{array}
E(T)=E(T_l) \leq 2C_1 E(T_{l-1})+2C_1 \leq 2C_1(2C_1E(T_{l-2})+2C_1) +2C_1 \leq ...\leq (2C_1)^{2C_1C_2T}E(T_1)+\frac{(2C_1)^{2C_1C_2T+1}-1}{2C_1-1}\lesssim e^{\kappa T}.
\aligned0= \sum_{\ell=0}^{k}\frak n_{k-\ell}(\frak n_{\ell}(y;x_1,\dots,x_{\ell}), x_{\ell+1},\dots,x_k)) &+\sum_{i=-1}^{k}\sum_{j=i}^{k}(-1)^*\frak n_{k-j+i+1}(y;x_1,\dots,\frak m(x_{i+1},\dots,x_j),x_{j+1},\dots,x_k),\endaligned
\int_{\mathbb{R}^d}W_M^2\omega_{\varrho}^{2-2\alpha}\eta^2\psi^2dx\leq\|W_M^2\|_{\frac{d}{2}}\|\omega_{\varrho}^{2-2\alpha}\eta^2\psi^2\|_{\frac{d}{d-2}}&=\|W_M\|_d^2\|\omega_{\varrho}^{1-\alpha}\eta\psi\|_{\frac{2d}{d-2}}^2\leq M^{-\frac{p-d}{d}}K_2^{\frac{2p}{d}}\|\omega_{\varrho}^{1-\alpha}\eta\psi\|_{\frac{2d}{d-2}}^2.
\prod_{i=1}^k \Bigl( \int f_i dx_i \Bigr)^{\lambda_i}p(x) \le \sup_{x = \sum_{i=1}^k \lambda_i y_i}\prod_{i=1}^k f^{\lambda_i}_i(y_i), \hskip 2mm \hskip 2mm p-a.e. \, x.
\widetilde D(\alpha, \alpha)\widetilde w_{\theta,\frac{\pi}{2}-\theta}\widetilde D(\alpha, \alpha) = \widetilde w_{\phi,\phi'}\widetilde D(\beta, \gamma)\widetilde w_{\phi,\phi'}, \widetilde D(\alpha, \alpha)\widetilde w_{-\theta,\theta-\frac{\pi}{2}}\widetilde D(\alpha, \alpha) = \widetilde w_{-\phi,-\phi'}\widetilde D(\beta, \gamma)\widetilde w_{-\phi,-\phi'},
\sigma_{\theta,\kappa}^{2}=\sum_{j=1}^{J}\sum_{j^{\prime}=1}^{J}\theta_{j}\theta_{j^{\prime}}\sum_{k=1}^{p}\sum_{k^{\prime}=1}^{p}\kappa_{k}^{\top}\Xi_{k,k^{\prime}}(\tau^{(j)},\tau^{(j^{\prime})})\kappa_{k^{\prime}}.
\psi^*(g_{ij})=a_{ij}&\psi^*(u_{i1})= -d_{2i}d_{12}^{-1}, \psi^*(u_{i2})= d_{1i}d_{12}^{-1}. &\psi^*(\nu_{k1})= -d_{2k}d_{12}^{-1}\psi^*(\nu_{k2})= d_{1k}d_{12}^{-1}
g_{\mu \nu}= \left( \begin{array}{c c c c}{{-{\sigma}^{2}+ A^{2}{\beta^{x}}^{2}+ A^{2}{\beta^{z}}^{2}}}&{{A^{2}\beta^{x}}}&{{0}}&{{A^{2}\beta^{z}}}\\{{A^{2}\beta^{x}}}&{{A^{2}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{A^{2}l^{2}}}&{{0}}\\{{A^{2}\beta^{z}}}&{{0}}&{{0}}&{{A^{2}}}\\ \end{array}\right) \; \; ,
C = \{\{r+a,r+b,r+c\}: -a_2 < r_2 \leq m-c_2\}&\cup \{\{r+b,r+c,r+d\}: -b_2 < r_2 \leq m-d_2\}&\cup \{\{r+a+c,r+a+d,r+b+c,r+b+d\}: -a_2-c_2 < r_2 \leq m-b_2-d_2\}
\begin{array}{c c}{\theta^{i}= \left( \begin{array}{c}{\vartheta_{-}^{a}}{( \bar{\vartheta}_{a}^{-})^{t}}\end{array}\right) ~ ~ ~ ~ ~}{~ ~ ~ ~ ~ \tilde{\theta}^{i}= \left( \begin{array}{c}{\vartheta_{+}^{a}}{( \bar{\vartheta}_{a}^{+})^{t}}\end{array}\right)}\end{array}
S^{(0)}= \sum_{C}\sum_{x(\in C)}\bigg| \left(\frac{\delta}{\delta\sigma^{t\mu}(x)}- i g A_{t\mu}(x) \right) \Phi[C]\bigg|^2 + \sum_{x}- \frac{\varepsilon}{2 \cdot 3!}F_{\mu\nu\lambda}F^{\mu\nu\lambda}- V(|\Phi[C]|^2) ,
d_0(v)(w) =\overline{[v,w]}d_r(\alpha)(w_1,\dots,w_{r+1}) = \sum_{i=1}^{r+1}(-1)^i \overline{[w_i,\alpha(\widehat{w_i})]}+ \sum_{i<j}(-1)^{i+j}\alpha([w_i,w_j],\widehat{w_{ij})}.
\int_{0}^{x}f(t) h(t/q) \left( -\frac{q(1+q)+t}{q^2 (1-q)t}u(t/q) +\frac{1}{q(1-q)^2t}\right)y(t) d_qt = H(x) -\lim_{n \rightarrow \infty}H(q^n x).
\Delta_{\mu\nu\pm}(k) =-\frac1{k^2}\Big(g_{\mu\nu}-(1-\alpha)\frac{k_\mu k_\nu}{k^2}\mp2\beta\frac{\tilde k_\mu\tilde k_\nu}{(\tilde k^2)^2(k^2\pm\frac{2\beta}{\tilde k^2})}\Big).
x_M^u=\mathop{\sum}_{i=1}^s (1\times\cdots \times1\times x_{M_i}^u\times 1\times\cdots\times 1) \ +\mathop{\sum}_{\substack{u_1+\cdots+u_s=u, 0\leq u_j<u}}(x_{M_1}^{u_1}\times\cdots\times x_{M_s}^{u_s}),
e_{1}^{(r/2)}(\chi_{r}^{2}(\mathbf{x)})=\sqrt{\frac{2}{r}}\left(\frac{r}{2}-\sum_{j=1}^{r}Y_{j}^{2}(\mathbf{x})\right)=-\frac{1}{\sqrt{2r}}\sum_{j=1}^{r}(Y_{j}^{2}(\mathbf{x})-1)=-\frac{1}{\sqrt{2r}}\sum_{j=1}^{r}H_{2}(Y_{j}(\mathbf{x})).
1=C_{L}^{\mathfrak{m}}/C_{L}^{\mathfrak{n}}=(I_{L}^{\mathfrak{m}}\cdot L^{\times}/L^{\times})/(I_{L}^{\mathfrak{n}}\cdot L^{\times}/L^{\times})=(I_{L}^{\mathfrak{m}}/(I_{L}^{\mathfrak{m}}\cap L^{\times}))/(I_{L}^{\mathfrak{n}}/(I_{L}^{\mathfrak{n}}\cap L^{\times})),
T(n-d+2,3d-2n) = & 2 \cdot T(n-d+1,3d-2n) + T(n-d+1,3d-2n-1) = & 2 \cdot T(J(d-2,n-3),K(d-2,n-3)) + T(J(d-3,n-4),K(d-3,n-4))
\alpha = \omega \, \displaystyle \frac{p + 1}{p}+ \chi_{\infty}\left( \left\lfloor \displaystyle \frac{1}{2}+ \displaystyle \frac{S}{p + 1}+ \displaystyle \frac{\omega}{\pi}\right\rfloor - \left\lfloor \displaystyle \frac{1}{2}+ \displaystyle \frac{S}{p + 1}- \displaystyle \frac{\omega}{\pi}\right\rfloor \right)
\frac{\partial}{\partial t}|\nabla^2 R|^2 =\frac{\partial}{\partial t}\big(g^{ik}g^{jl}\nabla_i\nabla_jR\nabla_k\nabla_lR\big) =2(R+\frac{\alpha'}{4}R^2-r)|\nabla^2R|^2+2\langle\frac{\partial}{\partial t}\nabla^2 R,\nabla^2 R\rangle
S_{q , g}^{e x t}={\frac{1}{4 \sqrt{\pi}}}\int_{\epsilon^{2}/ l^{2}}^{\infty}{\frac{d s}{s^{3 / 2}}}k_{H}( s ) \Theta_{\Sigma_{g}}( s{\frac{l^{2}}{l_{1}^{2}}}) e^{-{s / 4}}~
\frac{d^{2}f}{d x^{2}}- \frac{( x^{3}- 2 )}{x ( 1 - x^{3})}\frac{d f}{d x}- \frac{l ( l + 3 )}{x^{2}( 1 - x^{3})}f + \frac{\lambda^{2}}{x ( 1 - x^{3})^{2}}\left( 1 + \frac{\rho_{0}^{3}}{( \omega R )^{3}x^{3}}\right) f = 0 \, ,
\beta_{\pm}= \pm \alpha_{\pm}\beta_{n , m}={\frac{p + q \pm ( n p - m q )}{\sqrt{2 p q}}}\rightarrow{\frac{( 1 - n ) p + ( 1 + m ) q}{\sqrt{2 p q}}}\beta_{m i n}={\frac{p + q \pm 1}{\sqrt{2 p q}}}
\mathcal S_A^0(D;\ell)=\frac{1}{2\pi i}\int_{(a)}\breve W(s) N^s\sum_{\atop (d,\ell)=1}\mu^2(2d) \Psi\left(\frac dD\right) \mathcal B^{(2d)}(A_s;\ell)~ds +O( D^{1/2}\log Y +DY^{-1})
\int d^3\nu[\sin\pi(\nu_2-\nu_1)\sin\pi(1-\nu_3)]^{-s/M_s^2}={{1}\over{2\pi}}\left({{\Gamma\left(-s/2M_s^2+1/2 \right)\sqrt{\pi}}\over{\Gamma\left(-s/2M_s^2+1\right)}}\right)^2 \ .
\begin{cases}- \nu\Delta \mathbf{u}+\operatorname{div}(\mathbf{u}\otimes \mathbf{u})+\nabla \pi =0, - \Delta \mathbf{b}+\operatorname{div}( \mathbf{u}\otimes \mathbf{b})-\operatorname{div}(\mathbf{b}\otimes \mathbf{u})=0, \operatorname{div}\mathbf{u}=0=\operatorname{div}\mathbf{b}, \end{cases}
\rho=\frac{\mu (m+2)(n+1)}{2(2\mu +m)}, \sigma = \frac{\mu (n+1)}{2\mu -mn}, \frac 1 s= \frac{(m+2)(n-1)}4 \left( \frac 1 2 -\frac 1 q\right) +\frac m{4 \mu}, \frac n \delta =\frac n q + \frac 2{m+2}\left( \frac 1 s -\frac m{4\mu}\right),
\int \frac{d^{d}{\bf k}}{(2\pi)^{d}}\ln({\bf k}^{2}+m^{2})=-\frac{\partial}{\partial s}\left[\int \frac{d^{d}{\bf k}}{(2\pi)^{d}}({\bf k}^{2}+m^{2})^{-s}\right]_{s=0}.
\psi_{j}( x ) = \int \frac{d p_{j}^{1}}{\sqrt{4 \pi}p_{j}^{0}}\left( a_{j}( p ) u_{j}( p ) e^{- i p_{j}\cdot x}+ a_{_{\bar{\jmath}}}^{\dagger}( p ) v_{j}( p ) e^{i p_{j}\cdot x}\right) , \qquad
\left \langle{\mathcal L}\right\rangle = - \frac{1}{4}\left \langle \mathcal{F^A}_{\mu \nu}\mathcal F^{\mathcal A \mu \nu}\right\rangle = - \frac{1}{4}f_{\mu \nu}f^{\mu \nu}+ \left( D_\mu \varphi^* \right) \left( D^\mu \varphi \right) - \frac{8 g^2}{25}\left| \varphi \right|^4
\widehat{P}_+^{(0)}(z)=\widehat{P}_-^{(0)}(z)\begin{pmatrix}0 1 -1 0\end{pmatrix}, z\in \gamma_2, &\widehat{P}_+^{(0)}(z)=\widehat{P}_-^{(0)}(z)\begin{pmatrix}1 0 e^{\pi i\alpha}1\end{pmatrix}, z\in \gamma_1, &\widehat{P}_+^{(0)}(z)=\widehat{P}_-^{(0)}(z)\begin{pmatrix}1 0 e^{-\pi i\alpha}1\end{pmatrix}, z\in \gamma_3.
\frac{1}{L}\sum_{n = - \infty}^{\infty}\left[ F \left( \frac{2 n + 1}{L}\right) + F \left( \frac{2 n}{L}\right) \right] = \frac{1}{L}\sum_{n = - \infty}^{\infty}F \left( \frac{n}{L}\right) .
\bigg[\omega_{n+1}^{1-2s}\partial_{t}^{2}+\sum_{j=1}^{n+1}\Omega_{j}\omega_{n+1}^{1-2s}\Omega_{j}-\omega_{n+1}^{1-2s}\frac{(n-2s)^{2}}{4}\bigg]\overline{u}+R\overline{u}=\tilde{f},
(m_p^t)_{||}=M_p^t\left((M_p^t)^*M_p^t\right)^{-1}(M_p^t)^*m_p^t (q_p^t)_{||}=Q_p^t\left((Q_p^t)^*Q_p^t\right)^{-1}(Q_p^t)^*q_p^t.
\Bigl [ \varepsilon ({\bf r}, \omega ) \omega^{2}\delta_{j m}- \mathrm{r o t}_{j l}\mathrm{r o t}_{l m}\Bigr ] D_{m k}( \varepsilon ,{\bf r},{\bf r}^{\prime}, \omega ) = - \omega^{2}\delta ({\bf r}-{\bf r}^{\prime}) \delta_{j k}
\phi ( x ) = m \left({\frac{\int \mathrm{d^{d}}x \phi^{4}}{4 n}}\right)^{1 / 2}\varphi ( m x ) = m^{\mathrm{d}/ 2 - 1}\left({\frac{4 n}{\int \mathrm{d^{d}}u \varphi^{4}( u )}}\right)^{1 / 2}\varphi ( m x ) ,
\lVert u\rVert_{\dot{\mathcal{K}}^{s,p}(\mathbb{R}^{n}_+)}:= \lVert (\partial_{x_n}u,\nabla'u)\rVert_{\dot{\mathrm{H}}^{s-1,p}(\mathbb{R}_+,\mathrm{L}^p(\mathbb{R}^{n-1}))}
{\cal S}= \sum_{m = 1}^{i + j + i^{\prime}+ j^{\prime}}\sum_{l \mathrm{' \scriptsize{s}}}\sum_{\mathrm{\scriptsize{g r}}}{\cal S}_{c}( \cdots ){\cal S}_{c}( \cdots ) \cdots{\cal S}_{c}( \cdots ) .
Z_{2}: \qquad ( A_{\mu}, \phi , \bar{\phi}, \lambda_{A \alpha}, \bar{\lambda}^{A \alpha}, G^{\alpha \beta}) \rightarrow ( A_{\mu}, \bar{\phi}, \phi , - \bar{\lambda}_{A}^{\! \alpha}, - \lambda_{\! \alpha}^{A}, G_{\alpha \beta}) .
2 \sum_{i=1}^{N}\delta \alpha_{i}\, \lambda_{i}=\delta \left[ \sum_{i=1}^{N}(\alpha_{i}-1) \sum_{j \neq i}2\pi(\alpha_{j}-1 )G(\omega_{i}-\omega_{j}| \tau) + \sum_{i=1}^{N}(\alpha_{i}-1)^{2}\log |2\pi \eta^{2}| \right] \, ,
g_jx=\tau^{k_{g_j}(x)}(x)= \cdots x_{k_{g_j}(x)-1}.x_{k_{g_j(x)}}x_{k_{g_j}(x)+1}\cdots, g_jy=\tau^{k_{g_j}(y)}(y)=\tau^{k_{g_j}(x)}(y)= \cdots y_{k_{g_j}(x)-1}.y_{k_{g_j(x)}}y_{k_{g_j}(x)+1}\cdots.
\Psi :Z\to M:=G/Q\times_{\beta}\left(\Omega_{0}\times_{\sigma}M_{F}\right) \left(\left(y,r,H\right),A\right)\mapsto\left(y,r,\left(H,\bar{\psi}_{H}\left(r^{-1}\bar{p}_{k}\left(\tau(y)^{-1}A\right)\right)\right)\right),
\begin{aligned}|N_{G}(w'')\cap V(Q_{i})| =d_{G}(w'')-|N_{G}(w'')\cap V(B)|-|N_{G}(w'')\cap V(H)| &-|N_{G}(w'')\cap (V(T')\setminus V(Q_{i}))| &\geq (t+2)-1-\lceil\frac{|V(Q'_{i})|-1}{2}\rceil-(t-\lfloor\frac{|V(Q_{i})|}{2}\rfloor) &\geq 1+\lfloor\frac{|V(Q_{i})|}{2}\rfloor-\lceil\frac{|V(Q'_{i})|}{2}\rceil. \end{aligned}
\begin{aligned}T_{f}\big(A^{C^{*}}, h^{C^{*}}\big)=T_{f}\left(A^{{'E}_{0}}_{0}, h^{{'E}_{0}}\right)+T_{f}\left(A^{{'E}_{1}}_{1}, h^{{'E}_{1}}\right)+T_{f}\left(A^{{'E}_{2}}_{2}, h^{{'E}_{2}}\right).\end{aligned}
y^{2}= x^{3}+ \sum_{k = - 4}^{4}\sum_{l = 0}^{8 - n k}f_{k l}z^{\prime l}z^{4 + k}p^{8 - l - q k}x + \sum_{k = - 6}^{6}\sum_{l = 0}^{1 2 - n k}g_{k l}z^{\prime l}z^{6 + k}p^{1 2 - l - q k}
\|g\|_{L^3}=\bigg\|\sum_{j\in\mathbb{N}}\sum_{k,e}c_{j,k,e}\psi_{j,k,e}\bigg\|_{L^3}\leq \bigg\|\sum_{j\leq J}\sum_{k,e}c_{j,k,e}\psi_{j,k,e}\bigg\|_{L^3}+\bigg\|\sum_{j>J}\sum_{k,e}c_{j,k,e}\psi_{j,k,e}\bigg\|_{L^3}
(\sqrt{2\pi})^n|S_{K_i, \gamma_n}| = \int_{\partial K_i}e^{-\frac{|x|^2}{2}}d\mathcal{H}^{n-1}(x) = \int_{\Omega_i}e^{-\frac{|x|^2}{2}}d\mathcal{H}^{n-1}(x) + \sum_{j=1}^s \int_{\eta_{i,j}}e^{-\frac{|x|^2}{2}}d\mathcal{H}^{n-1}(x).
\int_{\mathbb{R}^{N}}\bigg[\frac{\alpha(x)}{q(x)}|u_k|^{\alpha(x)-2}u_k|v_k|^{\beta(x)}(u_k-u_{\lambda}) +\frac{\beta(x)}{q(x)}|u_k|^{\alpha(x)}v_k|v_k|^{\beta(x)-2}(v_k-v_{\lambda})\bigg]\,dx \longrightarrow 0.
{\bf D}_{21}= \begin{pmatrix}0 f_3 0 0 0 f_4 \frac{1}{2}f_3 0 0 f_5 \frac{1}{2}f_4 0 \vdots \vdots \vdots \vdots 0 f_{M-1}\frac{1}{2}f_{M-2}0\end{pmatrix}.
{T_{n+2}\over T_{n}}-{T_{n+1}^2\over T_{n}^2}=\left(\sqrt{mt_0+1\over mt_2+1}e^A-{mt_0+1\over mt_1+1}e^B\right)\biggl(1+O\Bigl({\log^{7/2}{n}\over\sqrt{n}}\Bigr)\biggr).
\sum_{t=0}^{k-1}\binom{k-1}{t}(-p)^{-t}U_{mk+t}^{(k)}=(-p)^{1-k}U_{m}(-pU_{m}+U_{m+1})^{k-1}&=(-p)^{1-k}U_{m}(-qU_{m-1})^{k-1}&=\left( \frac{q}{p}\right)^{k-1}U_{m}U_{(m-1)(k-1)}^{(k-1)},
\left( \begin{array}{l l}{{\xi_{1 1}}}&{{\xi_{1 2}}}\\{{\xi_{2 1}}}&{{\xi_{2 2}}}\\ \end{array}\right) \rightarrow \left( \begin{array}{l}{{\xi_{1 1}}}\\{{\xi_{2 1}}}\\{{\xi_{1 2}}}\\{{\xi_{2 2}}}\\ \end{array}\right)
\sum_{m=0}^{\infty}BE_{m}^{\left(r,s\right)}\left(x\right)\frac{\left(\log\left(1+t\right)\right)^{m}}{m!}=\sum_{m=0}^{\infty}BE_{m}^{\left(r,s\right)}\left(x\right)\sum_{n=m}^{\infty}S_{1}\left(n,m\right)\frac{t^{n}}{n!}=\sum_{n=0}^{\infty}\left\{\sum_{m=0}^{n}BE_{m}^{\left(r,s\right)}\left(x\right)S_{1}\left(n,m\right)\right\}\frac{t^{n}}{n!}.
\int_{0}^{t/2}dy \frac{e^{-y}\sin(a\tan^{-1}(\frac{\pi}{2(\log(t)-\log(y))}))}{y((\log(t)-\log(y))^{2}+\frac{\pi^{2}}{4})^{a/2}}= a \frac{\pi}{2}\left(\frac{1}{a \log^{a}(t)}+ O\left(\frac{1}{\log^{a+1}(t)}\right)\right) + O\left(\frac{1}{\log^{a+2}(t)}\right) &=\frac{\pi}{2 \log^{a}(t)}+ O\left(\frac{1}{\log^{a+1}(t)}\right)
\omega_{Z,Z'}=R_Z\otimes\left[R_{\binom{5,3,1}{4,2,0}}+R_{\binom{5,3,0}{4,2,1}}+R_{\binom{4,3,1}{5,2,0}}+R_{\binom{4,3,0}{5,2,1}}+R_{\binom{5,2,1}{4,3,0}}+R_{\binom{5,2,0}{4,3,1}}+R_{\binom{4,2,1}{5,3,0}}+R_{\binom{4,2,0}{5,3,1}}\right].
\begin{aligned}\frac{1}{R}\sum_{j< j_R}|C_{j,R,m}(x,y)\cap W_R(x,y)\cap h^{-1}(W_R(x,y))| \leq \frac{1}{R}\sum_{\log f(m)\leq j\leq j_R-1}\frac{2c(d)R}{2^{d^{-1}\delta j}}\leq\frac{1}{1000},\end{aligned}
\tilde{g}_s^{\prime}= \left( \frac{\tilde{R}}{R}\right)^{1/4}\frac{R^{1/2}l_p^{3/2}}{R_8 R_9}\ , \ \tilde{l}_s^{\prime}= \left( \frac{\tilde{R}}{R}\right)^{1/4}\frac{l_p^{3/2}}{R^{1/2}},
\frac{1}{3}+ \frac{1}{6}+ \frac{1}{8}= \frac{8}{2 4}+ \frac{4}{2 4}+ \frac{3}{2 4}= \frac{1 2}{2 4}+ \frac{3}{2 4}= \frac{1 5}{2 4}= \frac{5}{8}
\langle f_{p(i_{k})}\otimes\cdots\otimes f_{p(i_{1})},e_{p^{\prime}(j_{1})}\otimes\cdots\otimes e_{p^{\prime}(j_{k})}\rangle=\langle f_{p(i_{1})},e_{p^{\prime}(j_{1})}\rangle\cdots\langle f_{p(i_{k})},e_{p^{\prime}(j_{k})}\rangle.
Z = \left( \begin{array}{c c}{{0}}&{{\omega}}\\{{{\tilde{\omega}}^{*}}}&{{0}}\\ \end{array}\right) , \quad{\overline{{Z}}}= \left( \begin{array}{c c}{{0}}&{{{\omega}^{*}}}\\{{{\tilde{\omega}}}}&{{0}}\\ \end{array}\right)
\hat{\sigma}[H(u,u)g_{ij}](\xi)(h_{ij})=\tilde{F}^{-2}(h^i_0\xi_i\xi_sy^s-\frac{1}{2}\xi_i\xi^ih_{00}-y^s\xi_sy^l\xi_lh_i^i)\tilde{g}_{ij}.
F_T(x) =1+xF_T(x)+x\big(F_T(x)-1\big)\,C\!\left(x+\frac{x^2}{1-x}\big(K(x)-1\big)\right) &+\frac{x\big(K(x)-1\big)}{1-x}\left(xF_T(x)+x\big(F_T(x)-1\big)\,C\!\left(x+\frac{x^2}{1-x}\big(K(x)-1\big)\right)\right)\,.
K_{\nu}( x ) = \sqrt{\frac{\pi}{2 x}}e^{- x}\left[ 1 +{\frac{4 \nu^{2}- 1}{8 x}}+{\frac{( 4 \nu^{2}- 1 ) ( 4 \nu^{2}- 3^{2})}{2 ! ( 8 x )^{2}}}+ \dots \right] ,
\mathbb{P}\left(\Psi(x,\gamma,Z)=y\right)= \mathbb{P}\left(( \mathbf{1}_{[0, \frac{1}{2}]}- \mathbf{1}_{(\frac{1}{2},1]})(Z)=1\right)=\mathbb{P}\left(Z \in \left[0,\frac{1}{2}\right]\right)=\frac{1}{2};
\mathsf P_{n,m}:=\begin{cases}\displaystyle\frac{1}{2+\sigma}\frac{n+2}{n+1}, \mbox{if}m=n+1, \displaystyle \frac{\sigma}{2+\sigma}, \mbox{if}m=n, \displaystyle \frac{1}{2+\sigma}\frac{n}{n+1}, \mbox{if}m=n-1\ge 0, \displaystyle 0, \mbox{otherwise,}\end{cases}n\ge 0.
\lim_{n\rightarrow\infty}\left( n[Y_{l_{1}}^{m_{1}}\star_{\vec{b}}^{n}Y_{l_{2}}^{m_{2}}-Y_{l_{2}}^{m_{2}}\star_{\vec{b}}^{n}Y_{l_{1}}^{m_{1}}]\right) \ =\ \lim_{n\rightarrow\infty}\left( n[Y_{l_{1}}^{m_{1}}\star_{1}^{n}Y_{l_{2}}^{m_{2}}-Y_{l_{2}}^{m_{2}}\star_{1}^{n}Y_{l_{1}}^{m_{1}}]\right)
p_{\nu_{k}}(\xi)=\frac{\left|1-R(t)t\right|^{2}}{\left|1-\eta_{\nu}\left(R(t)t\right)\right|^{2}}\frac{1-R(t)^{\frac{2k}{k-1}}}{1-R(t)^{2}}\,p_{\mu}(\xi)
\chi_t'(f) = f^{-2(h+2)}\left( \frac{1-t f^2}{1-tf^{-2}}\right)^2 H^\dagger(f) = \chi'_t(f) \cdot \left( 1 + \frac{2t}{h+2}\left( \frac{f^2 - 2t + f^{-2}}{(1-tf^2)(1-tf^{-2})}\right) \right) \theta_t(f)^{-1}= \frac{2(h+2)}{2-f^2 - f^{-2}}\cdot H^\dagger(f)
S_{\mathrm{off}}=\sum_{d_1,\,d_2<D}\,\, \,\ \sum_{Id_1, \,Jd_2<D}x_{Id_1}\bar{x}_{Jd_2}\sqrt{IJ}\sum_{L, M\geq 1}\frac{G_k(LMd_1d_2\slash p)}{\sqrt{LM}}\times\left(2 \pi i^{2k}\sum_{p|c}\frac{\mathcal{S}(IL, JM, c )}{c}\,J_{2k-1}\left(\frac{4\pi \sqrt{ILJM}}{c}\right)\right).
\begin{aligned}P_o(X_n=\nu) =\sum_{\ell\ge \nu}(-1)^{\ell}\,\frac{s(\ell,\nu)}{\ell!}\sum_{j=0}^{n-1}(-1)^{j}\binom{n-1}{j}\binom{2j+1}{2n-\ell}&=\sum_{\ell\ge\nu}(-1)^{\ell+n-1}\frac{s(\ell,\nu)}{\ell!}\left[2^{\ell-2}\binom{n-1}{\ell-2}+2^{\ell-1}\binom{n-1}{\ell-1}\right].\end{aligned}
W^{\mu \nu}= \frac{\partial^2 L}{\partial \dot x_\mu \partial \dot x_\nu}= - \frac{m}{\sqrt{\dot x^2}}\left( g^{\mu \nu}- \frac{\dot x^\mu \dot x^\nu}{\dot x^2}\right) = - m \left( g^{\mu \nu}- \dot x^\mu \dot x^\nu \right)\; ,
H^0(\Sigma^*, \mathfrak g) \times \Omega^0(S,\mathfrak k) \times H^0(\Sigma,\mathfrak g)\to \Omega^0(S,\mathfrak g), (X_-,x,X_+)\to \frac{d}{dt}\Big\vert_{t=0}\big(g_+^{-1}k^{-1}g_-^{-1}g_-e^{tX_-}ke^{tx}g_+e^{tX_+}\big)=g_+^{-1}k^{-1}X_-kg_++g_+^{-1}xg_++X_+.
\mathcal{H}_{\mathcal{D}}\phi_{\mathcal{D}\, n}(x) =\mathcal{E}_n\phi_{\mathcal{D}\,n}(x), n=0,1,\ldots, \mathcal{H}_{\mathcal{D}}=-\frac{d^2}{dx^2}+U_{\mathcal{D}}(x),
C_{2}( \lambda ) = \frac{\lambda^{2}}{2 ! ( 1 - \lambda )}, \quad C_{3}( \lambda ) = \frac{\lambda^{3}}{3 ! ( 1 - \lambda ) ( 1 - 2 \lambda )}, \quad C_{4}( \lambda ) = \frac{\lambda^{4}}{4 ! ( 1 - \lambda ) ( 1 - 2 \lambda ) ( 1 - 3 \lambda )}
\sum_{i=1}^{n}d_i'\delta_i(b_j') = 0 \ \Leftrightarrow \sum_{i=1}^{n}t^{-1}d_i'\delta_i(b_j') = 0 \Leftrightarrow \sum_{i=1}^{n}s_4^{-1}a_i''\delta_i(b_j') = 0 &\ \Leftrightarrow \sum_{i=1}^{n}a_i\sigma_i(s)^{-1}\delta_i(b_j') = 0 \Leftrightarrow \sum_{i=1}^{n}a_i\overline{\sigma_i}(s^{-1}) \overline{\delta_i}(b_j') = 0\ \ \ (0\le j\le n).
\beta\leq \min_{1\leq i\leq n}\left\{\nu\left(a_i\textbf Q^{\lambda_i}\right)\right\}=\min_{1\leq i\leq n}\left\{\nu_Q\left(a_i\textbf Q^{\lambda_i}\right)\right\}\leq \nu_Q(p)\leq \nu(p)=\beta.
\int_{\mathfrak{D}_{\lambda,\mu}}\psi(x_{\lambda},x_{\mu}) \mathrm{d}\widetilde{\nu}_{\lambda,\mu}(x_{\lambda},x_{\mu}) = \int_{\mathbb{T}^2}\psi(\chi_{\lambda}(\omega_1,\omega_2),\chi_{\mu}(\omega_1,\omega_2)) \mathrm{d}\varepsilon_{\lambda,\mu}(\omega_1,\omega_2),
S_{\mathrm{i n t .}}( \Sigma , j_{\mu}) = \frac{1}{4 \pi^{2}}\varepsilon_{\mu \nu \lambda \rho}\int d^{4}x d^{4}y j_{\mu}( x ) \frac{( y - x )_{\nu}}{| y - x |^{4}}\Sigma_{\lambda \rho}( y )
= 2 e \sum_{I = 1}^{N}\left( \varepsilon_{\mu \nu}\partial_{\mu}A_{\nu}^{( I )}, K_{I I}\sum_{b = 1}^{N}\sum_{j = 1}^{n_{b}}U_{I b}\Big [ \delta_{n}( x_{j}^{( b )}) - \delta_{n}( y_{j}^{( b )}) \Big ] \right) .
Y_t=Y_{t_j}+\frac{B_{t_j,t_{j+1}}}{t_{j+1}-t_j}\int_{t_j}^tf(Y_s)ds + \int_{t_j}^t g(Y_s)d\langle B\rangle_s + \int_{t_j}^th(Y_s)ds,\ \t\in[t_j,t_{j+1}],