harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int^{2\pi R}_0 dx_4 i\bar{\Psi}\gamma^\mu{\cal D}^{(q)}_\mu \Psi =\sum^3_{j=1}\left( i\bar{\psi}^j_L \gamma^\mu D^{(q)L}_\mu \psi^j_L +i\bar{\psi}^j_R \gamma^\mu D^{(q)R}_\mu \psi^j_R \right),
{\cal L}= \frac{1}{4 \pi^{2}g}\bigg \{- \sqrt{R^{2}( 1 -{\dot{R}}^{2}+{R^{\prime}}^{2}- E_{z}^{2}) + B^{2}( 1 -{\dot{R}}^{2})}+ \frac{h}{2}R^{2}\bigg \}.
I_{1}( 0 ) ={\frac{( 2 3 )^{2}}{x_{2 3}^{4}}}\int d u_{5}\left[{\frac{( 1 2 )^{2}( 3 5 )}{( 2 5 )}}\partial_{2}\cdot \partial_{2}+{\frac{( 1 3 )^{2}( 2 5 )}{( 3 5 )}}\partial_{3}\cdot \partial_{3}+ ( 1 2 ) ( 1 3 ) \; 2 \partial_{2}\cdot \partial_{3}\right] f ( 1 , 2 , 1 , 3 ) \ .
z_1 = \frac{(q-1)(q^d-1)(x_1^2 - q^d)}{x_1^2 - q}, z_i = \frac{q^{i-1}(q^i-1)(q^{d-i+1}-1)(x_1^2 - q^{i-2})(x_1^2 - q^{d+i-1})}{(x_1^2 - q^{2i-3})(x_1^2 - q^{2i-1})}(2 \leq i \leq d-1).
(\partial_t^2-ia\partial_{\alpha})\partial_t(I-\mathfrak{H})(z-\bar{z})= \partial_t (\partial_t^2-ia\partial_{\alpha})(I-\mathfrak{H})(z-\bar{z})+ia_t((I-\mathfrak{H})(z-\bar{z}))_{\alpha}= \partial_t g+ia_t((I-\mathfrak{H})(z-\bar{z}))_{\alpha}.
\Gamma(q/p'+2)^{1/q}=\left((1+o(1))\sqrt{2\pi\left(q/p'+1\right)}\left(\frac{q/p'+1}{e}\right)^{q/p'+1}\right)^{1/q}&=(1+o(1))\left(\frac{q}{ep'}\right)^{1/p'}.
D : = - i ( \sigma_{1}\, \delta_{1}+ \sigma_{2}\, \delta_{2}+ \sigma_{3}\, \delta_{3}) = - i \left( \begin{matrix}{{\delta_{3}}}&{{\delta_{1}- i \delta_{2}}}\\{{\delta_{1}+ i \delta_{2}}}&{{- \delta_{3}}}\\ \end{matrix}\right)
m = m_{b}={\frac{- 1}{6 l^{2}}}[ 3{r_{b}}^{5}+ ( l^{2}- 1 5 n^{2}){r_{b}}^{3}- 3 n^{2}( 2 l^{2}- 1 5 n^{2}){r_{b}}- 3 n^{4}( l^{2}- 5 n^{2}) / r_{b}]
he^{{W}^{1}}\bar{\varphi}_{1}\big({W}^{1}\big)-\frac{1}{2}h{W}^{1}\bar{\varphi}_{1}\big({W}^{1}\big)\varphi_{1}\big({W}^{1}\big)=he^{{W}^{3}}\bar{\varphi}_{1}\big({W}^{3}\big) -\frac{1}{2}h{W}^{3}\bar{\varphi}_{1}\big({W}^{3}\big)\varphi_{1}\big({W}^{3}\big).
\omega ( x , y ) = \frac{1}{2}+ \frac{x^{2}y^{2}}{4 \cdot 5 !}- \frac{5}{1 2 \cdot 7 !}\, ( x^{4}y^{2}+ x^{2}y^{4}) + \frac{1}{3 4 5 0 0 \,}( x^{2}y^{6}+ x^{6}y^{2}) + \frac{1}{8 6 4 0 0}\, x^{4}y^{4}+ \ldots
\begin{aligned}\psi_{n_k}\to_k J_{\theta,1}^{-1}(p\gamma)-pJ_{\theta,1}^{-1}\gamma \ \ \ \ \psi_{-n_k}\to_k J_{\theta,1}^{-1}(p\overline\gamma)-pJ_{\theta,1}^{-1}\overline\gamma & m.\end{aligned}
\tilde{E}=-\tilde{\omega}^{2}\frac{\left( 1\pm \sqrt{1+4\left(A-\sum_{j=1}^{m}\sigma_{j}^{m}\right) \frac{a\left( \tilde{\nu}+2m\right)}{\tilde{\omega}^{2}}}\right)^{2}}{16\left( A-\sum_{j=1}^{m}\sigma_{j}^{m}\right)^{2}},
\rho_{t}({\bf 1}) \, =\, 1 + \int^{t}_{0}\rho_{s, 2}(b){\mathrm d}X_{s}\, =\, 1 + \int^{t}_{0}\rho_{s}({\bf 1}) \pi_{s, 2}(b){\mathrm d}X_{s}\, ; 0 \le t \le T \, .
\phantom{=}\,\,\sum_{i=0}^k\binom{m+1}{k-i}\binom{n+1}{i}\binom{m-k+i+1}{3}\delta_{m+n-k,0}=\sum_{i=0}^k \binom{m+1}{3}\binom{m-2}{k-i}\binom{n+1}{i}\delta_{m+n-k,0}=\binom{m+1}{3}\binom{m+n-1}{k}\delta_{m+n-k,0}=\binom{m+1}{3}\binom{k-1}{k}\delta_{m+n-k,0}=\binom{m+1}{3}\delta_{m+n,0}\delta_{k,0},
g_{\mu\rho}\hat{\Gamma}^{\rho}_{\nu\lambda}\frac{u^{\lambda}}{\hat{\gamma}}= \hat{\Gamma}_{\mu,\nu\lambda}\frac{u^{\lambda}}{\hat{\gamma}}= F_{[g]\mu\nu}= \frac{\partial A_{[g]\nu}}{\partial x^{\mu}}- \frac{\partial A_{[g]\mu}}{\partial x^{\nu}}
( - 1 )^{\alpha \beta}e^{2 \pi \sqrt{- 1}( 1 / 2 ) ( \alpha^{2}\tau + 2 \alpha z )}P ( \tau , z + \alpha \tau + \beta ) = ( - 1 )^{\alpha + \beta + \alpha \beta}P ( \tau , z ) \, , \quad \alpha , \beta \in{\bf Z}\, .
\left|\nabla_{y'}^\beta \partial_{z_d}r'_\lambda(y',y_d,z_d)\right|+\left|\nabla_{y'}^\beta \partial_{z_d}r_{d,\lambda}(y',y_d,z_d)\right| & \leq \frac{Cy_d}{(y_d + z_d +|y'|)^{d+\beta}}\frac{e^{-c|\lambda|^{\frac 12 z_d}}}{\big (1+|\lambda|^{\frac 12}(y_d+z_d)\big )},
\frac{\log\left({r}^{2}+2\,r\,\cos t+1\right)}{{\left({r}^{2}+2\,r\,\cos t+1\right)}^{\frac{2+s}{2}}}\,r=\frac{\left(2\,r\log r+\chi_0\right)\left(1+\frac{\kappa}{r}\right)}{r^{2+s}}=\frac{2\,r\log r+ 2\kappa\log r+\frac{\chi_0\kappa}{r}+\chi_0}{r^{2+s}}.
{\cal C}^{s,2d}_{pol,m}={\Big \{}f:{\mathbb R}^{2d}\rightarrow{\mathbb R}: \exists c>0~\forall |z|\geq 1~\forall 0\leq |\gamma|\leq m~{\big |}D^{\gamma}_xf(z){\big |}\leq \frac{c}{1+|z|^s}{\Big \}},
d^2_B(j,k)= \frac{1}{N}\sum_{n=1}^{N-1}\frac{[\cos(j+\frac{1}{2})\phi_n-\cos(k+\frac{1}{2})\phi_n]^2}{2(1-\cos\phi_n)^2}= \frac{1}{2}\Big(F_N(j+k+1)+F_N(j-k) &-\frac{1}{2}F_N(2j+1)-\frac{1}{2}F_N(2k+1)\Big)
\lim_{|\xi '-\xi |\rightarrow 0}\ln \coth\frac{1}{2}|\xi '- \xi | = +\infty, \:\:\: \lim_{|\xi '-\xi |\longrightarrow \infty}\ln\coth\frac{1}{2}|\xi '- \xi | = 2e^{-|\xi ' -\xi |}\: ;
\dfrac{dH}{dI}\leq \dfrac{F(S,I)-F(S,I^{*})}{F(S,I)F(S,I^{*})}F_{1}(S,I)-\dfrac{I-I^{*}}{II^{*}}& = \dfrac{1}{F(S,I^{*})}\left[ F_{1}(S,I)-F_{1}(S,I^{*})\right]\leq 0.
\begin{aligned}K :=\frac14+C\frac{\Delta t}{\nu}+C\Delta t\frac{\|\widetilde{\boldsymbol{\eta}}_{u,t}^n\|_1^4+\|\widetilde{\boldsymbol{\eta}}_{u,h}^n\|_1^4}{\nu^3}& \leq \frac14+C\frac{\Delta t}{\nu}+C\Delta t \frac{(\nu+\gamma)^2h^{4k_u}+\gamma^{-2}h^{4k_p+4}+(\Delta t)^4}{\nu^5}\end{aligned}
\mid{\bf \cdot}\, \rangle = \; \mid p \, \rangle^{0}, P = \; \mid p \, \rangle^{0 \; 0}\langle \, p \mid , \mid \Psi_{P}\, \rangle = \; \mid p \, \rangle^{0 \; 0}\langle \, p \mid p \, \rangle .
{\cal L}= \left( \partial_\mu \phi \right)^\dagger \left(\partial^\mu \phi \right) -(m^2- k^2X^2)\phi^\dagger\phi -ikX_\mu[\phi \partial^\mu \phi^\dagger - \phi^\dagger\partial^\mu \phi ]- \frac{1}{12}H_{\mu\nu\rho}H^{\mu\nu\rho},
-\cos\psi\csc\eta\cdot\left(\frac{F[-q,p](i\mu_{0})w_{2}[-q,p](i\mu_{0})w_{3}[-q,p](i\mu_{0})}{w_{1}[-q,p](i\mu_{0})}\right)^{-\frac{1}{2}}\gamma^{1}\partial_{\phi}\left(-\gamma^{0}u_{n}(\frac{1}{\mu})\right)\mid_{\mu=\mu_{0}}
\int f(x) h(x/q) \left(\frac{1}{q}u(x) u(x/q) +\frac{1}{q}x r(x) (q-1)u(x/q)\right)y(x) d_qx & = f(x/q) h(x/q) \left(y(x/q) u(x/q)- D_{q^{-1}}y(x)\right).
\int_{df(0)\neq 0}\log \Vert df(0)\Vert d\nu =\sum_{n=1}^{\infty}\int_{\delta_n<\Vert df(0)\Vert \le\delta_{n-1}}\log\Vert df(0)\Vert d\nu &=\lim_{N\to\infty}U_{\delta_N}&\ge 0.
\Gamma_\alpha=\frac{\alpha}{2}\left(-\cos\theta\, V_\sigma +6\sin\theta\, V_s -\frac{1}{2}\sin 2\theta\,V_\varphi \,e^{\alpha \chi/2}\right)-\frac{\alpha^2}{4}\,\dot{\sigma}^2 \,\cos^2\theta\, -\alpha^2\,\dot{\sigma}^2 \,\sin^2\theta\,.
\langle \mathcal{Q}_4^{}\rangle_{\Psi}\leq C \rho N \widehat{g}(0) \sqrt{\rho a^3}, & b\frac{K_L^2}{2 \ell^2}\langle n_+^H \rangle_{\Psi}\leq C \rho N \widehat{g}(0)\times \begin{dcases}C \sqrt{\rho a^3}\frac{K_{L}}{K_{\ell}}, d= 3, C \widehat{g}(0), d=2, \end{dcases}
\|T_i + S\| \ge \frac{1}{1+\varepsilon}z_i^*(T_i x_i^* + Sx_i^*) = \frac{1}{1+\varepsilon}(y_i^*(T_i x_i^*) + x_i^*(w)z_i^*(z)) = \frac{1}{1+\varepsilon}(y_i^*(T_i x_i^*) + |x_i^*(w)|) > \frac{2-3\varepsilon+\varepsilon^2}{1+\varepsilon}> 2 - 5 \varepsilon.
\left[ \Psi_{M}^{+}( x ) , \overline{{{\Psi}}}_{N}^{-}( y ) \right] = ( 2 \pi )^{- 3}\int \frac{\left( m - i \widehat{p}\right)_{M N}}{2 p_{0}}e^{i p ( x - y )}d^{3}p
c(TX_1) =(1+E_1)\frac{(1+f^* Z_1-E_1)(1+f^* Z_2 -E_1)(1+f^* Z_3-E_1)}{(1+f^* Z_1)(1+f^* Z_2)(1+f^* Z_3)}f^* c(TX_0) c(TY) =\frac{(1+E_1)(1+f^* Z_1-E_1)(1+f^* Z_2 -E_1)(1+f^* Z_3-E_1)}{(1+3H+6L-2E_1)(1+f^* Z_1)(1+f^* Z_2)(1+f^* Z_3)}f^* c(TX_0)
\frac{\|(B_k - I)d\|}{\|d\|}= \frac{\|g(x + d) - g(x) - G(x_\ast)d - g(x + d)\|}{\|d\|}&\geq \frac{\|g(x + d)\|}{\|d\|}- \frac{\|g(x + d) - g(x) - G(x_\ast)d\|}{\|d\|}.
|\gamma_M-\gamma_N| \ll \left(\sup_{k\geq N}a_k+\sum_{k= N}^{\infty}\frac{a_k}{k}\right)^{1/2}+ \sup_{k\geq N}|\mu_k| + \sum_{k=N}^\infty \frac{|\mu_k|}{k}& \ll \left(\sup_{k\geq N}a_k+\sum_{k= N}^{\infty}\frac{a_k}{k}\right)^{1/2}+ \sup_{k\geq N}a_k + \sum_{k=N}^\infty \frac{a_k}{k}.
\begin{aligned}{\langle \dot y(t)-f(y(t),u(t)),\varphi\rangle}_\mathrm M =0 \varphi\in \mathbb R^nt>0,{\langle y(0)-y_\circ,\varphi\rangle}_\mathrm M =0 \varphi\in \mathbb R^n\end{aligned}
(b\theta)(a_0,a_1,a_2) =\theta(a_0a_1,a_2)-\theta(a_0,a_1a_2)+\theta(a_2a_0,a_1) &=-\frac12\varphi(\Gamma a_0a_1\delta^2(a_2)|D|^{-2})+\frac12\varphi(\Gamma a_0\delta^2(a_1a_2)|D|^{-2})-\frac12\varphi(\Gamma a_2a_0\delta^2(a_1)|D|^{-2}).
x_i^{(k+1)}= \frac{1}{1+\frac{h}{2}a_{ii}}\left[ -h\sum_{j<i}a_{ij}x_j^{(k+1)}+ \left( 1-\frac{h}{2}a_{ii}\right) x_i^{(k)}- h \sum_{j>i}a_{ij}x_j^{(k)}+ hb_i \right].
F_1(t) = (p - \kappa_1 - \kappa_2 - 2)t^{p + \nu - 1}\| (y_1, y_2)\|_{1,p}+ (q - \kappa_1 - \kappa_2 - 2)t^{q + \nu - 1}\| (\nabla y_1, \nabla y_2)\|_{q,\eta}
\det[\delta^{a}_{b}\partial_{t}-\omega^{ac}\partial_{c}\partial_{b}H]=\int~{\cal D}c^{a}{\cal D}{\o c}_{a}~\exp~\biggl[-\int{\o c}_{a}[\delta^{a}_{b}\partial_{t}-\omega^{ac}\partial_{c}\partial_{b}H]c^{b}~dt\biggr]
\Delta(B_\lambda)=B_\lambda\otimes Y_{\lambda-\beta}+X_\lambda\otimes B_{\lambda};\cr \Delta(C_\lambda)=C_\lambda\otimes X_{\lambda+\beta}+Y_\lambda\otimes C_{\lambda};\cr \Delta(D_\lambda)=D_\lambda\otimes A_\lambda-C_\lambda\otimes B_{\lambda+\beta}+B_\lambda\otimes C_{\lambda-\beta}+(A_\lambda+(j-i)D_\lambda)\otimes D_\lambda
\left[X_0, X_1\right] =c_{01}^1X_1+c_{01}^2X_2, \left[X_0, X_2\right] =c_{02}^1X_1+c_{02}^2X_2, \left[X_1, X_2\right] =c_{12}^1X_1+c_{12}^2X_2+X_0,
(i_{Y_{j_0}}(\omega))|_{X_{j_0l_0}}= (\sum_{i\ne j}2\lambda_{ij}\hat{F}_{ij}i_{Y_{j_0}}(dF_i)\,dF_j)|_{X_{j_0l_0}}= (2\lambda_{j_0l_0}\hat{F}_{j_0l_0}dF_{l_0})|_{X_{j_0l_0}}.
{\bf P}_{c_L}= \sum\limits_{q=1}^{R}(\sum\limits_{k_1=0}^{L-1}{\cal W}_{({k_1})}{\bf p}^{(1)}_{q}) \otimes (\sum\limits_{k_2=0}^{L-1}{\cal W}_{({k_2})}{\bf p}^{(2)}_{q}) \otimes (\sum\limits_{k_3=0}^{L-1}{\cal W}_{({k_3})}{\bf p}^{(3)}_{q}).
v_1 =(r_2-r_1,r_4-r_3,r_6-r_5), v_5 =(r_2-r_1,r_4-r_3,r_6+r_5), v_2 =(r_2+r_1,r_4-r_3,r_6-r_5), v_6 =(r_2+r_1,r_4-r_3,r_6+r_5), v_3 =(r_2+r_1,r_4+r_3,r_6-r_5), v_7 =(r_2+r_1,r_4+r_3,r_6+r_5), v_4 =(r_2-r_1,r_4+r_3,r_6-r_5), v_8 =(r_2-r_1,r_4+r_3,r_6+r_5),
{\cal N}(\alpha)=\int_{x(0)=x(T)=0}{\cal D}x(\tau)exp\left[-{1\over 4}\int_0^Td\tau\dot{x}^2-\sum_{i=1}^N\alpha_ix^2(\tau_i)\right]={1\over (4\pi T)^2}det^{-2N}(1+4\hat{\Delta}).
|\partial_1^j\partial_2^l f_T (x) | \leq C (T^{\frac{1}{4}})^{-j-2l+2\alpha -2}\int_{\mathbb{R}^2}|\hat{x}_2 - \hat{y}_2|^{\frac{2\alpha -2}{2}}|\partial_1^j\partial_2^l\psi_1(\hat{y})|\, \textrm{d}\hat{y}\lesssim C (T^{\frac{1}{4}})^{-j-2l + 2\alpha -2}.
\frac{\frac{d \Delta E}{d t}_{K K}}{\frac{d \Delta E}{d t}_{n e u t}}\approx \frac{\pi^{6}}{1 8 0 \zeta ( 5 )}\frac{n_{1}n_{5}}{n_{K}\mu_{K}}e^{- \pi \sqrt{\frac{n_{1}n_{5}}{n_{K}}}},
\frac{a_{n}}{a_{n+1}}=1+\frac{1}{n}+\frac{1}{n}\sum_{i=1}^{K-1}\frac{1}{\prod_{k=1}^{i}ln_{(k)}(n)}+\frac{\rho_{n}}{n\prod_{k=1}^{K}ln_{(k)}(n)},K\ge1.
\det V_{n}=2^{2n^{2}-2n-1}\prod_{i=1}^{n}\frac{(2i)!^{4}}{(4i)!(4i+1)!}\prod_{\ell=0}^{n}(r-\ell)_{2\ell+1}\prod_{\ell=0}^{n-1}\left(r+\frac{1}{2}-\ell\right)_{2\ell+1}\quad\times\sum_{i=1}^{n+1}\frac{(2n+2i)!(2n+2-2i)!(r+1)_{n+1}}{(n+i)!^{2}(n+1-i)!^{2}(r+i)}.
S^{i i}( \theta ) = \left( \begin{array}{c c c}{( 2 )_{\theta}}&{( 3 )_{\theta}( 1 )_{\theta}}&{- ( 2 )_{\theta}}\\{( 3 )_{\theta}( 1 )_{\theta}}&{( 2 )_{\theta}^{2}}&{( 3 )_{\theta}( 1 )_{\theta}}\\{- ( 2 )_{\theta}}&{( 3 )_{\theta}( 1 )_{\theta}}&{( 2 )_{\theta}}\\ \end{array}\right)
I(n) =\int_{3\pi/2}^{+\infty}\frac{\sin^2(x)dx}{x^2(x-\pi n)(x+\pi n)}=\int_{3\pi/2}^{\pi n}\frac{\sin^2(x)dx}{x^2(x-\pi n)(x+\pi n)}+\int_{\pi n}^{2\pi n-3\pi/2}\frac{\sin^2(x)dx}{x^2(x-\pi n)(x+\pi n)}&+\int_{2\pi n-3\pi/2}^{+\infty}\frac{\sin^2(x)dx}{x^2(x-\pi n)(x+\pi n)}.
{\rm Coker\,}\hat\sigma_s={\rm Coker\,}\hat\sigma:=\{u\in H^0_b(X)^G_{s-\frac{1}{2}};\, (\,u\,|\,\hat\sigma v)_X=0, \forall v\in H^0(\overline M)^G_s\cap C^\infty(\overline M)\}.
\mathbf{m}(k,d) = \sum_{p=0}^{k-1}\binom{k-1}{p}\mathbf{K}^{-1}_{\mathfrak{p}h \leftrightarrow l}\mathbf{c}(k-p,d) \otimes \mathbf{m}(p,d) \mathbf{c}(k,d) = \mathbf{m}(k,d) - \sum_{p=1}^{k-1}\binom{k-1}{p}\mathbf{K}^{-1}_{\mathfrak{p}h \leftrightarrow l}\mathbf{c}(k-p,d) \otimes \mathbf{m}(p,d)
\frac{\partial^2}{\partial x^2}E_{\alpha, \beta}^\gamma\left(t{\rm e}^{{\rm i}x}\right) \Big|_{x=0}+ t\,\left(E_{\alpha, \beta}^\gamma(t)\right)' + t^2\,\left(E_{\alpha, \beta}^\gamma(t)\right)'' = 0 \, .
\exp \big ( - \frac{1}{2 \hbar T}\int_{- 1}^{0}d \tau \dot{\xi}^{2}\; \big ) = \exp \big ( - \sum_{k = 1}^{\infty}\frac{( k \pi )^{2}}{4 \hbar T}( v_{i}^{k})^{2}\big ) ,
(-1;q)_{\infty}\,_{0}\phi_{1}\left(-;0;q^{2},q^{5}x^{-2}\right)\! =\theta_{q}\!\left(-q^{-1}x\right){}_{1}\phi_{1}\left(0;-q;q,x\right)+\theta_{q}\!\left(q^{-1}x\right){}_{1}\phi_{1}\left(0;-q;q,-x\right)\!,
a^4_{\nu, \kappa}(\xi-\eta, \eta)= \frac{iq^4_{\nu, \kappa}(\xi-\eta,\eta)}{|\xi|-a_{\nu}|\xi-\eta|-a_{\kappa}|\eta|}= \frac{i(|\xi|+a_{\nu}|\xi-\eta|+a_{\kappa}|\eta|)q_2(\xi-\eta, \eta)}{2a_{\nu}a_{\kappa}|\xi-\eta||\eta|}.
\mathcal A_{\omega_t}([x,w]) = - \int_{D^2}w^*\omega_t + \int_0^1 H(x(t),t)dt & = - \int_{D^2}w^*\omega_s + \int_0^1 H(x(t),t) dt + \int_{D^2}w^*(\omega_s - \omega_t) & = \mathcal A_{\omega_s}([x,w]) + (s-t) \int_{D^2}w^*\alpha
Z ( \xi ) = \int{\cal D}A_{\mu}^{\omega}{\cal D}B{\cal D}\bar{c}{\cal D}c e^{- S_{0}( A^{\omega}) - \frac{\xi}{2}\int B^{2}d x + \int [ - i B \partial_{\mu}A_{\mu}^{\omega}+ \bar{c}( - \partial_{\mu}\partial_{\mu}) c ] d x}
\overline{u}_\ell(t) =\frac{1}{2}\sum_{j=0}^\ell\alpha_{j,k}(-t)^j(-\Delta)^{j+\ell}e^{t\Delta}u_0 &\quad+\sum_{0\leq k_1+k_2\leq \ell}\alpha_{\ell-k_1-k_2,k_1}\beta_{k_2}(-t)^{\ell-k_1-k_2}(-\Delta)^{2\ell-k_1-k_2}e^{t\Delta}\left(\frac{1}{2}u_0+u_1\right),
B(a) = \begin{pmatrix}2a a^* - 1 2a \sqrt{1 - a^* a}2a^* \sqrt{1 - a a^*}1 - 2a^* a \end{pmatrix}= \begin{pmatrix}2(x^2 + y^2) - 1 2(x+iy) \sqrt{1 - (x^2 + y^2)}2(x-iy) \sqrt{1 - (x^2 + y^2)}1 - 2(x^2 + y^2) \end{pmatrix}\; .
\Delta^{\prime}\lambda = \left( \frac{d^{4}G_{E}}{d \varphi_{0}^{4}}\right)_{s = 1}= - \frac{3 \lambda^{2}}{3 2 \pi^{3}}\sqrt{a_{3}a_{4}}\lim_{s \rightarrow 1}A_{2}^{c^{2}}( s , a_{3}, a_{4}) .
\, W(M_{\eta_1}T_{\xi_1}f, \, M_{\eta_2}T_{\xi_2}g) = \, e^{\pi i (\xi_1-\xi_2) \cdot (\eta_1+\eta_2)}e^{2 \pi i x \cdot (\eta_1-\eta_2)}e^{-2 \pi i \omega \cdot (\xi_1-\xi_2)}\, W(f,g)(x-\tfrac{\xi_1+\xi_2}{2}, \omega -\tfrac{\eta_1+\eta_2}{2}).
T = \frac{1}{2}\left[ ( A + \lambda^{2}B ) \dot{\lambda}^{2}+ \frac{p_{\theta}^{2}C}{\lambda^{2}A ( C - \lambda^{2}A / 4 )}+ \frac{p_{\alpha}( p_{\alpha}+ p_{\theta})}{C - \lambda^{2}A / 4}\right]
|{\mathcal N}_j(d,k-1)|\leq\sum_{r=0}^{\ell_j}\frac{|(k-1){\mathcal P}_j(d))|^r}{r!}\leq |k{\mathcal P}_j(d))|^{\ell_j}\sum_{r=0}^{\ell_j}(\frac{10k}{\ell_j})^{\ell_j-r}\frac{1}{r!}\leq(\frac{12k^2|{\mathcal P}_j(d))|}{\ell_j})^{\ell_j}.
[ \hat{G}_{\mu \nu}^{( n )}, \hat{G}_{\lambda \rho}^{( n )}] = 2 \left( \delta_{\nu \lambda}\hat{G}_{\mu \rho}^{( n )}+ \delta_{\mu \rho}\hat{G}_{\nu \lambda}^{( n )}- \delta_{\mu \lambda}\hat{G}_{\nu \rho}^{( n )}- \delta_{\nu \rho}\hat{G}_{\mu \lambda}^{( n )}\right) .
J_4 =\int_0^r -\frac{z^3}{2}(e_\sigma+\bar\psi)\cdot (e'_\sigma\phi +e'_\delta\psi){\rm{d}}z \leq\Vert z(\phi,\psi)\Vert_{L^2}\cdot \Vert z^{2}(e'_\delta,e'_\sigma)\Vert_{L^2}\cdot\frac{1}{2}(\Vert e_\sigma\Vert_{L^\infty}+\Vert \bar\psi\Vert_{L^\infty}),
W := \widetilde{W}_{n_0}+ b^* - \varepsilon_{q^*}^{\widetilde{W}_{n_0}}= V_{\nu_n}+ \left[ \left( B + b^*- \frac14 \right) - \left( \varepsilon_{q^*}^{\widetilde{W}_{n_0}}- b^* \right) \right] - B.
[\begin{matrix}y_{0}\\y_{1}\\y_{2}\\y_{3}\\y_{4}\end{matrix}]=[\begin{matrix}1&1&1&1&1\\ 1&w&w^{2}&w^{3}&w^{4}\\ 1&w^{2}&w^{4}&w&w^{3}\\ 1&w^{3}&w&w^{4}&w^{2}\\ 1&w^{4}&w^{3}&w^{2}&w\end{matrix}][\begin{matrix}x_{0}\\x_{1}\\x_{2}\\x_{3}\\x_{4}\end{matrix}],w=e^{\frac{-j2\pi}{5}}
\div |\beta|^2 g = - d |\beta|^2= -2 g(\nabla \beta, \beta) =-2 \sum_{i,j}(\nabla_{e_i}\beta) (e_j)\beta(e_j)e_i &=-2 \sum_{i,j}((\nabla_{e_j}\beta) (e_i)+d\beta(e_i,e_j))\beta(e_j)e_i &= - 2 \nabla_{\beta^\sharp}\beta + 2 \iota_{\beta^\sharp}d\beta.
\int_{T_3(\epsilon)}^t \epsilon_1(s)\,ds =\int_{[T_3(\epsilon),\infty)}\int_{[T_3(\epsilon),t\wedge u]}\, ds \,\mu(du) &=\int_{[T_3(\epsilon),\infty)}(t\wedge u - T_3(\epsilon))\, \mu(du)\leq \int_{[T_3(\epsilon),\infty)}( u - T_3(\epsilon))\, \mu(du)\leq C.
b_{13}^{(1)}= (\epsilon-1)\cos{\phi_1}\cos{\psi_1}-\sin{\phi_1}\sin{\psi_1}= \epsilon\cos\phi_1\cos\psi_1-\cos(\phi_1-\psi_1), b_{24}^{(1)}= (\epsilon-1)\sin{\phi_1}\sin{\psi_1}-\cos{\phi_1}\cos{\psi_1}= \epsilon\sin\phi_1\sin\psi_1 -\cos(\phi_1 -\psi_1).
\widehat{AoB}= \frac{1}{2}AB =\frac{1}{2}\left[2\pi-AoB\right] = \frac{1}{2}\left[2\pi - \left(\frac{2\pi}{N_k}-\theta_k\right)\right] =\left(1-\frac{1}{N_k}\right)\pi + \frac{\theta_k}{2}.
D_{3}=\frac{N_{0}2^{N_{0}}M_{N_{0}}}{(1+M)^{v_{0}N_{0}}}\left( \frac{D(c_{0})P_{N_{0}}}{M\sqrt{v_{0}}}\right)^{v_{0}},D_{4}=\frac{2P_{N_{0}}^{2}}{M^{2}(3M+1)^{2N_0}}.
||(S_{\theta_k}{\mathbf V}_k,S_{\theta_k}\Psi_k)||_{s,\ast,T}+||S_{\theta_k}\psi_k||_{H^s(\Gamma_T)}\lesssim\begin{cases}\delta \theta^{(s-\alpha)_+}_k, s\neq\alpha, \delta \log\theta_k, s=\alpha. \end{cases}
E_{a s}= -{\frac{\cos{\pi s}}{\pi}}\mu^{2 s}\sum_{l}( l + \frac 1 2 ) \int_{m}^{\infty}d k [ k^{2}- m^{2}]^{1 / 2 - s}\frac{\partial}{\partial k}\ln f_{l}^{a s}( i k ) - E_{\varphi}^{d i v},
( \zeta^{2}- 4 \theta ){\frac{d^{2}f_{n}}{d \zeta^{2}}}+ 2 \zeta{\frac{d f_{n}}{d \zeta}}+ \left( \zeta^{2}- \lambda - 4 \theta -{\frac{4 \theta \mu^{2}}{\zeta^{2}- 4 \theta}}\right) f_{n}= 0 .
\displaystyle{\sum_{\iota \leq \iota' \leq \gamma_1 = \iota + k + 1}}(-1)^{\deg(v)}u_{\iota\iota'}^{(i)}x_{\iota' , \iota+ k+1}+ \displaystyle{\sum_{\iota \leq \iota' \leq \gamma_1 < \gamma_2 = \iota + k + 1}}(-1)^{\deg(v)}u_{\iota\iota'}^{(i)}x_{\iota' \gamma_1}u_{\gamma_{1},\iota+k+1}^{(\mu)}= x_{\iota, \iota+k+1},
\widetilde{U}^p(t)=U^p(t)-U^p(t-1)=\sum_{k\geq N}a_k\left(1-e^{\tilde{\lambda}_k}\right)e^{\tilde{\lambda}_k(T-t)}, \widetilde{U}^h(t)=U^h(t)-U^h(t-1)=\sum_{\mod{k}\geq N}b_k\left(1-e^{\tilde{\gamma}_k}\right)e^{\tilde{\gamma}_k(T-t)},
\frac{d \rho}{d x_1}= \sin h(\gamma^K,\rho)- \rho^{n-1}\sin [nh(\gamma^K,\rho)]\hspace{0.4cm}, \hspace{0.4cm}\frac{d \rho}{d x_1}= \cosh_\perp(\gamma_\perp^K,\rho)- \rho^{n-1}\cos [nh_\perp(\gamma_\perp^K,\rho)]
b_kb_j-b_ka_{kj}-b_ja_{jk}=0,\overline{b}_p\overline{b}_q-\overline{b}_p\overline{a}_{pq}-\overline{b}_q\overline a_{qp}=0 \widetilde{b}_m\widetilde{b}_n-\widetilde{b}_m\widetilde{a}_{mn}-\widetilde{b}_n\widetilde{a}_{nm}=0,\widehat{b}_l\widehat{b}_v-\widehat{b}_l\widehat{a}_{lv}-\widehat{b}_v\widehat{a}_{lv}=0
u^{*}(x,t) = \limsup_{\delta \to 0^{+}}\sup \left\{U^{\epsilon}(m,s) \, \mid \, d(-m\Delta x, x) + |t - s \Delta t| + \epsilon < \delta\right\}, u_{*}(x,t) = \liminf_{\delta \to 0^{+}}\inf \left\{U^{\epsilon}(m,s) \, \mid \, d(-m\Delta x, x) + |t - s \Delta t| + \epsilon < \delta\right\}.
\left[ G, \, (\, \cdot \,)_T \right] \diamond \partial_1^2 F_0 - \left[ G , \, (\, \cdot \,)_T \right] \diamond \partial_1^2 F_1= \left[ G, \, (\, \cdot \,)_T \right] \diamond \partial_1^2 (F_0 - F_1).
\nabla \tilde{\varphi}(x)= 2 \sqrt{\tilde{\varphi}}(x)\nabla\sqrt{\tilde{\varphi}}(x) , \left(\nabla_j\tilde{A}(x_i-x)+\nabla_j\tilde{\tau}-\tilde{A}e_j\right)^2\le 2|\nabla_j\tilde{A}|^2|x_i-x|^2+2|\nabla_j\tilde{\tau}-\tilde{A}e_j|^2 \le 8\lambda^2\|\nabla_j\tilde{A}\|^2+2|\nabla_j\tilde{\tau}-\tilde{A}e_j|^2 \quad.
\mathbb{E}\left[ 2\gamma_{n}\left\| Z_{n}-m \right\| \left\| \delta_{n}\right\| 2\left( p-1 \right) \gamma_{n}\left\langle \xi_{n+1}, Z_{n}-m \right\rangle \left( \left\| Z_{n}-m \right\|^{2}+ \gamma_{n}^{2}\right)^{p-2}\right] =0 .
\begin{aligned}\|O^v_2\|_{L^1_{t,x}}\lesssim (\|(w^{(c)}_{q+1}, w^{(t)}_{q+1}, d^{(c)}_{q+1}, d^{(t)}_{q+1})\|_{L^2_{t,x}}\|(d^{(p)}_{q+1}, d_{q+1}, w^{(p)}_{q+1}, w_{q+1})\|_{L^2_{t,x}}\lesssim \lambda^{C_0}_q\lambda^{\gamma-\frac{1}{2}}_{q+1}+\lambda^{C_0}_q\lambda^{-\sigma}_{q+1}.\end{aligned}
\mathcal{L}^{( 0 )}= p_{i}\dot{q}_{i}+ \epsilon \hat{\bar{p}}_{i}\dot{\bar{q}}_{i}+ \pi \dot{\phi}+ \epsilon \hat{p}_{i}^{b}\dot{b}_{i}+ p_{i}^{c}\dot{c}_{i}- \mathcal{H}_{e x t}.
( \tilde{\Lambda}\psi )_{\alpha}= - 4 \tilde{\Pi}_{b}P^{- b a}\partial_{a}\theta_{\alpha}+ i \varepsilon^{a b}\theta^{\beta}\partial_{a}\theta^{\gamma}\partial_{b}\theta^{\delta}\Gamma_{\alpha ( \beta}^{\mu}( C \Gamma^{\mu})_{\gamma \delta )}.
\mathbb{E}e^{i y_1 (Z(t_3)-Z(t_2))+i y_2 (Z(t_2)-Z(t_1))}= \mathbb{E}e^{i y_1 (Z(t_3)-Z(t_2))}\mathbb{E}e^{i y_2 (Z(t_2)-Z(t_1))}\forall (y_1,y_2)\in \mathbb{R}^2.
( p_{\omega}, p_{\omega^{\prime}}) = 2 \tilde{\omega}\delta^{3}( \vec{k}- \vec{k}^{\prime}) = \frac{2}{\tilde{\omega}\sin \theta_{0}}\delta ( \eta_{0}- \eta_{0}^{\prime}) \delta ( \theta_{0}- \theta_{0}^{\prime}) \delta ( \tilde{\omega}- \tilde{\omega}^{\prime}) .
\hat{Q}= - \int_{( \infty )}\frac{d^{3}\vec{k}}{( 2 \pi )^{3}2 \omega ( \vec{k}\, )}\, \left[ a^{\dagger}( \vec{k}\, ) a ( \vec{k}\, ) \, - \, b^{\dagger}( \vec{k}\, ) b ( \vec{k}\, ) \right] \ .
m_{{\mathfrak{a}_{i+1}}}(s_i \cdot \mathfrak{b})=\begin{cases}m_{\mathfrak{a}_i}(\mathfrak{b}) + t^{-1}m_{\mathfrak{a}_i}(s_i \cdot \mathfrak{b}), \ell(s_i \cdot \mathfrak{b}) > \ell(\mathfrak{b}), m_{\mathfrak{a}_i}(\mathfrak{b}) + t m_{\mathfrak{a}_i}(s_i \cdot \mathfrak{b}), \ell(s_i \cdot \mathfrak{b}) < \ell(\mathfrak{b}).\end{cases}
( 2 \pi )^{4}\frac{\delta}{\delta \hat{\varphi}( q )}L^{0 , \, \infty}( \varphi ) \, = \, \hat{L}_{( 1 )}^{0 , \, \infty}( q \, ; \varphi ) + \hat{I}_{( 1 )}^{\, 0 , \, \infty}( q ) \, ;
[ \ell_{\mu}^{j}, \, q \cdot \! \nu ] = - i \left[ j \, \mu \cdot \nu \, \ell_{\mu}^{j - 1}+ \sum_{\rho \in \Delta_{+}}g_{| \rho |}( \rho \cdot \mu ) ( \rho^{\vee}\! \cdot \nu ) \frac{\ell_{\mu}^{j}- \ell_{s_{\rho}( \mu )}^{j}}{\ell_{\mu}- \ell_{s_{\rho}( \mu )}}\check{s}_{\rho}\right] ,
\mathbb{P}\left\{f^{(i)}_t V^{(i)}_t\leq \bar{t}\right\}= \mathbb{P}\left\{\frac{V^{(i)}_t}{N_i n_x}- 1 \leq \frac{-4\alpha \|\Sigma^{(i)}_t\|}{(1+2\alpha)^2\rho^{(i)}\| \Sigma^{(i)}_t\|+ 4\sqrt{n_x}}\right\},
= \sqrt{- g}\Big ( \frac{1}{8 | \omega_{\cal B D}|}\phi^{2}R - \mathrm{s g n}( \omega_{\cal B D}) \frac{1}{2}g^{\mu \nu}\partial_{\mu}\phi \cdot \partial_{\nu}\phi + \Lambda ( \frac{\phi^{2}}{4 | \omega_{\cal B D}|}) \frac{\phi^{2}}{4 | \omega_{\cal B D}|}\Big ) .
\nabla^2 U_k(\zeta) = 0 \mbox{if}\zeta\in G_m, U_k(\zeta) = 1 \mbox{if}\zeta\in C_j, j=1,\ldots,k-1, U_k(\zeta) = 1 \mbox{if}\zeta\in C'_k, U_k(\zeta) = 0 \mbox{if}\zeta\in C''_k, U_k(\zeta) = 0 \mbox{if}\zeta\in C_j, j=k+1,\ldots,m,
\vert f(x)-\hat f(x)\vert =\vert g(\langle a,x\rangle)-\hat g(\langle\hat a,x\rangle)\vert &\leq \vert g(\langle\hat a,x\rangle)-\hat g(\langle\hat a,x\rangle)\vert+\vert g(\langle a,x\rangle)-g(\langle\hat a,x\rangle)\vert &\leq c_0\cdot\|a-\hat a\|_{l_2^d}+c_0\vert\langle a-\hat a,x\rangle\vert &\leq 2c_0\|a-\hat a\|_{l_2^d}
\mathbb{P}_{b_{11}= 1, d_1 = 0, d_2 = 1}(\mathbf{c}, t) \Leftrightarrow \left(\begin{array}{ccccc}b_{10}\ge 0 \wedge b_{10}\le t-1 \wedge b_{10}\in \mathbb{Z}b_{2}\ge 0 \wedge b_{2}\le t-1 \wedge b_{2}\in \mathbb{Z}b_{3}\ge 0 \wedge b_{3}\le t-1 \wedge b_{3}\in \mathbb{Z}\end{array}\right).
= \frac{2\nu +n+1}{2(\nu +n+1)}t^{\nu}\mathbf{L}_{\nu +n}(t) + \frac{n+1}{2(\nu +n+1)}t^{\nu}\mathbf{L}_{\nu +n+2}(t) &\quad+\frac{n+1}{\nu+n+1}\frac{t^{2\nu+n+1}}{\sqrt{\pi}2^{\nu+n+2}\Gamma(\nu+n+\frac{5}{2})}.
|\partial_{t}\langle \left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}})-1}{r^{2}}\right) f\vert_{r = R\lambda(t)},\phi_{0}\rangle| \leq C \int_{0}^{\infty}\frac{r |\lambda'(t)|}{(r^{2}+\lambda(t)^{2})^{3}}|f(t,r)| r dr + C \int_{0}^{\infty}\frac{r \lambda(t)}{(r^{2}+\lambda(t)^{2})^{3}}|\partial_{t}f(t,r)| r dr
A\equiv\left( x_{N}^{0}\right)^{\omega\gamma}\frac{|\left( x_{N}^{0}\right)^{n-j}\Delta_{\overline{h}^{\prime},x^{\prime}}^{s}D_{x}^{\alpha}v(x_{0},t_{0})|}{|\overline{h}^{\prime}|^{\gamma}},\quad|\alpha|=m-j,j\leq n.
X_{t}^{\varepsilon}[\phi ]=X_{0}[\phi ]+\int_{0}^{t}b(X_{s}^{\varepsilon})[\phi ]ds+\varepsilon \int_{\lbrack 0,t]\times \mathbb{X}}G(X_{s-}^{\varepsilon},y)[\phi ]N^{\varepsilon^{-1}}(dy,ds).
{\bf R}_+ \times{\cal E}_L \ni (\alpha,\widetilde{x}){\begin{picture}(90,0)(0,0) \put(5,3){\vector(1,0){80}}\put(15,10){$\scriptstyle(\lambda,\gamma) \in{\bf R}_+ \times{\cal D}_L$}\end{picture}}(\lambda[\widetilde{\varphi}]\alpha,\widetilde{x}\circ\gamma[\widetilde{\varphi}])\in \mbox{\boldmath R}_+ \times{\cal E}_L\;\;\;,
\leq-c\cdot\mathrm{rank}\,F\int_{G_{R/2}}e^{-m\psi_k}dV_\omega+2C\int_{S_R\setminus G_{R/2}}e^{-m\psi_k}dV_\omega = \mathrm{Vol}(D^k_R)\Biggl(-c\cdot\mathrm{rank}\,F\int_{B^k_{R/2}}e^{-m\psi_k}dV_{\omega_k}+2C\int_{B^k_R\setminus B^k_{R/2}}e^{-m\psi_k}dV_{\omega_k}\Biggr),