harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
| (r - \lambda x) f'(x)| = \frac{r - \lambda x}{xp(x)}\int_0^{x}(h(t)-\Gamma_{r,\lambda}h)p(t)\,\mathrm{d}t &\le \| h-\Gamma_{r,\lambda}h \| \frac{r - \lambda x}{xp(x)}\int_0^{x}p(t)\,\mathrm{d}t.
S_\lambda^* S_\lambda(f\sqrt{d\mu}) = S_\lambda^*( f\circ \sigma^\ell \sqrt{d(\mu \circ \sigma_\lambda^{-1})}) = f \circ \sigma^\ell \circ \sigma_\lambda \sqrt{d(\mu \circ \sigma_\lambda^{-1}\circ \sigma_\lambda)}&= (\chi_{Z(s(\lambda))}\cdot f) \sqrt{d\mu}&= S_{s(\lambda)}(f \sqrt{d\mu}),
p(x)= x^9-\frac{235721}{1785}x^8-\frac{17957726593}{62475}x^7+\frac{33219815829811}{937125}x^6 &-\frac{12554318926285933}{4685625}x^5+\frac{29740292638491103}{312375}x^4 &-\frac{696525696876795217}{187425}x^3+\frac{851886544261448041}{37485}x^2 &-\frac{124583919439776136}{2499}x+\frac{30888835313436500}{833}.
\phi_+=\limsup \limits_{\xi \rightarrow+\infty}\phi(\xi),\phi_-=\liminf\limits_{\xi \rightarrow+\infty}\phi(\xi), begin{align*}0.2cm] \psi_{+}=\limsup \limits_{\xi \rightarrow+\infty}\psi(\xi),\psi_-=\liminf\limits_{\xi \rightarrow+\infty}\psi(\xi) .
\mathcal{Q}^{(2)[2]}=\frac{1}{8}\begin{pmatrix}0 \dots \ \dots\ \ 0\ 3\ddot{u}_2 \ddots \vdots 9\ddot{u}_2+8\ddot{u}_3 \ddot{u}_2 2\ddot{u}_3 0 -2\ddot{u}_3 0 2\ddot{u}_3 5\ddot{u}_2 \ddots \vdots 0 4\ddot{u}_4 -2\ddot{u}_3 10\ddot{u}_3-3\dddot{u}_2 0 0 \end{pmatrix}\ .
[\alpha^I\,\underline{\alpha}_I+\beta_I\,\underline{\beta}^I,\hat \alpha^I\,\underline{\hat\alpha}_I+\hat \beta_I\,\underline{\hat\beta}{}^I]=(\alpha^I\hat \beta_I -\hat \alpha^I\beta_I)\,\underline{\epsilon}_0 +\omega^i(\alpha,\beta,\hat\alpha,\hat\beta)\, \underline{\omega}_i\ .
\Big[ (\mathcal D_g X)_{ij;k}+ (\mathcal D_g X)_{ki;j}- (\mathcal D_g X)_{jk;i})\Big] =( X_{i;jk}+ X_{i;kj})+ (X_{j;ik}-X_{j;ki}) + (X_{k;ij}-X_{k;ji}) = 2 X_{i;jk}+ (R^{\ell}_{kji}+ R^{\ell}_{ikj}+ R^{\ell}_{ijk}) X_{\ell},
|x_s -x_0| \leq \int_0^s |\dot{x}_u |du = \int_0^s \frac{\sqrt{\dot{t}_u^2-1}}{\alpha(t_u)}du \leq \int_0^s \frac{\dot{t}_u}{\alpha(t_u)}du =\int_{t_0}^{t_s}\frac{du}{\alpha(u)}.
\partial_t\delta q_{e\varepsilon}+\delta u_\varepsilon\cdot\nabla\delta q_{e\varepsilon}+\delta u_\varepsilon\cdot\nabla q_e+u\cdot\nabla\delta q_{e\varepsilon}+(\bar Q+\alpha)\nabla\cdot\delta v_\varepsilon=-\frac{1+\alpha}{\varepsilon}q_{e\varepsilon}^+-(\bar Q+\alpha)\nabla\cdot v\chi_{\mathcal O}(x,y,t),
-U_{\lambda2}^{\prime\prime}- \frac{4\vartheta_0[1-\mu_0^2]^{1/2}}{6\mu_0^2-4-\lambda}U_{\lambda2}^{\prime}+ (\mu^2_0-\lambda)U_{\lambda2}= - \frac{2\mu_0\vartheta_0}{\sigma_0 (6\mu_0^2-4-\lambda)}(U_{\lambda1}^{\prime\prime}+\sigma_0F_1) +F_2
\begin{aligned}B_\mathbf{u}(c^{n-1};(\mathbf{u}^{n},p^{n},p^{n});(0,v_h,0))=(\mathbf{u}^{n},\nabla v_h)_{\mathcal{T}_h}-\langle\mathbf{u}^{n}\cdot\mathbf{n},v_h\rangle_{\partial\mathcal{T}_h}=-(\nabla\cdot\mathbf{u}^{n},v_h)_{\mathcal{T}_h}=-(q^{n},v_h)_{\mathcal{T}_h}.\end{aligned}
\textbf{w}_{r_ld}^k=\bigg(\sum\limits_{k=1}^K \textbf{h}_{r_ld}^k(\textbf{h}_{r_ld}^k)^{H}+\sigma^2 \textbf{I}\bigg)^{-1}\textbf{h}_{r_ld}^k.
[\delta_{1},\delta_{3}]=\delta_{3}(L'^{\nu \rho}=[L,L^{\nu\rho}]+\delta_{2}(L'^{\rho}=[\partial_{\nu}L,L^{\rho\nu}])+\delta_{1}(L'=[\partial_{\nu}L,[A_{\rho},2L^{[\nu \rho]}]]) .
\left(f^{\left(1\right)}\circ\left(f^{\left(2\right)}\circ f^{\left(3\right)}\right)\right)\circ f^{\left(4\right)}=\sum_{\substack{\pi_1\models n \pi_2\models \vert \pi_1 \vert \pi_3\models\pi_2}}f_{\vert \pi_2 \vert}^{\left(1\right)}f_{\vert \pi_3 \vert}^{\left(2\right)}f_{\pi_3}^{\left(3\right)}f_{\pi_1}^{\left(4\right)}.
\frac{1}{\Gamma(1+q)}\frac{S_2(\tau-q\,|\,\tau)}{S_2(\tau\,|\,\tau)}= \frac{\Gamma_2(1+q+\tau\,|\,\tau)}{\Gamma_2(1-q+\tau\,|\,\tau)}\,\frac{\tau^{q/\tau}}{\Gamma(1-q/\tau)}, \tau^{q/\tau}\frac{\Gamma_2(\tau-q\,|\,\tau)}{\Gamma_2(\tau\,|\,\tau)}= \Gamma(1-\frac{q}{\tau})\,\frac{\Gamma_2(1-q+\tau\,|\,\tau)}{\Gamma_2(1+\tau\,|\,\tau)}.
C_R(K)\big|\sum^{8N}_{k=1}(u^k(1)-u^k(2))^2\big|+\, C_Q(K) \big|\sum^{8N}_{k=1}\,(p_k(1)-p_k(2))^2\big| & \leq \,1/2\,\sum^{8N}_{k=1}\big|(u^k(1)-u^k(2))\big|+ 1/2\,\sum^{8N}_{k=1}\,\big|(p_k(1)-p_k(2))\big|.
\begin{array}{r l}{{\mathrm{B e k e n s t e i n - V e r l i n d e ~ b o u n d}:}}&{{S_{B V}= \frac{2 \pi}{n}E{\cal R},}}\\{{\mathrm{B e k e n s t e i n - H a w k i n g ~ b o u n d}:}}&{{S_{B H}= ( n - 1 ) \frac{V}{4 G_{n + 1}{\cal R}},}}\\{{\mathrm{H u b b l e ~ b o u n d}:}}&{{S_{H}= ( n - 1 ) \frac{H V}{4 G_{n + 1}}.}}\\ \end{array}
w_0(\alpha_4) = s_{\alpha_2 + 2\alpha_3 + 2\alpha_4}(\alpha_2 + 2\alpha_3 + \alpha_4) = (\alpha_2 + 2\alpha_3 + \alpha_4) - (\alpha_2 + 2\alpha_3 + 2\alpha_4) = -\alpha_4,
\operatorname{P}(\Xi)=\left( (\operatorname{exp_{g_{_{\mathfrak{X}}}}}(\mathcal{V}_{\Xi})\circ\Phi)^{\ast}(\omega), (\operatorname{exp_{g_{_{\mathfrak{X}}}}}(\mathcal{V}_{\Xi})\circ\Phi )^{\ast}(\operatorname{Im}\Omega)\right),
\prod_{i=1}^{n-1}\left[ \int d^Dx |f_i(x)|^{q q_1/q}\right]^{\frac{1}{q q_1/q}}\;\left[\int d^Dx |f_n(x)|^p \right]^{\frac{1}{p}}\; = \;\prod_{i=1}^{n}\|f_i\|_{q_i}\; .
u ( x, y, t ) = \frac{e^{- \frac{(y-x-\mu t)^{2}}{4t}}}{\sqrt{4 \pi t}}= \frac{e^{- \frac{(y-x)^{2}}{4t}}}{\sqrt{4 \pi t}}e^{- \mu^{2}\frac{t}{4}+ \frac{\mu}{2}(y-x)}, t>0,\; x,y\in\mathbb{R}.
M_{1}(\mathcal{NC}(G)) =(pq-1)(pq-2)^{2}-4(pq-2)\dfrac{(q-1)(q-2)+q(p-1)(p-2)}{2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+(q-1)(q-2)^{2}+q(p-1)(p-2)^{2}= p^{3}q^{3}-2p^{2}q^{2}-pq^{3}-p^{3}q^{2}+pq^{2}-3q^{2}-3qp^{2}+2q+p^{3}q+q^{3}-4
N_{\mathcal J}(u, v, w) + N_{\mathcal J}(u, w, v) = \pi (\mathcal J u) \langle \mathcal J v, w\rangle - \pi (u) \langle v, w\rangle+ \pi (\mathcal J u) \langle v, \mathcal J w\rangle + \pi (u) \langle\mathcal J v,\mathcal J w\rangle =0.
{\cal M}= \left( \begin{array}{ccc}G^{-1}G^{-1}(B + C) G^{-1}A (-B + C) G^{-1}(G - B + C) G^{-1}(G + B + C) (G - B +C) G^{-1}A A^T G^{-1}A^T G^{-1}(G + B + C) I_{16}+ A^TG^{-1}A \end{array}\right).
e^{- \Phi}\star H_{3}= - m U \epsilon_{7}+ m^{- 1}T_{i j}^{- 1}\star{\cal D}T_{j k}\wedge ( \mu^{k}{\cal D}\mu^{i}) -{\frac{m^{- 2}}{2}}T_{i k}^{- 1}T_{j l}^{- 1}\star F^{i j}\wedge{\cal D}\mu^{k}\wedge{\cal D}\mu^{l}
a_N^0[\mathbf{u},\mathbf{u}]=\int_\mathcal{O}\langle g^0 b(\mathbf{D})\mathbf{u},b(\mathbf{D})\mathbf{u}\rangle \,d\mathbf{x},\mathbf{u}\in H^1(\mathcal{O};\mathbb{C}^n).
\mathcal{F}(T_ab)(\xi) := \frac{1}{2\pi}\int_{\mathbb{R}}\widehat{a}(\xi-\eta)\widehat{b}(\eta)\chi(\xi-\eta,\eta)\,d\eta, & \chi(x,y) := \sum_{k\in\mathbb{Z}}\varphi_k(y)\varphi_{\leq k-10}(x).
\sum_{m}\frac{b_{2k}(m; y, z)}{mm_0}\leq \frac{(2k)!}{k!}\sum_{\ell=0}^k\binom{k}{\ell}\sum_{m_1}\frac{b_{\ell}(m_1;y,z)}{m_1^2}\sum_{m_0}\frac{b_{2k-2\ell}(m_0;y,z)}{m_0^2}&\leq\frac{2^k(2k)!}{k!}\left(\sum_{y<p<z}\frac{1}{p^2}\right)^{k}\leq \left(\frac{ck}{y\log y}\right)^k,
\|\Re(x_i)+i\Im(x_i)-\Re(x_i(\varepsilon))-i\Im(x_i(\varepsilon))\|_2^2 +\|\Re(x_i)-i\Im(x_i)-\Re(x_i(\varepsilon))+i\Im(x_i(\varepsilon))\|_2^2 =2\|\Re(x_i)-\Re(x_i(\varepsilon))\|_2^2+2\|\Im(x_i)-\Im(x_i(\varepsilon))\|_2^2
\begin{aligned}P\left(\bigcap_{j=1}^{N}\{|X_{j}|\leq\alpha_{j}+c\}\right) =2^{N}\int_{0}^{\alpha_{N}+c}\cdots\int_{0}^{\alpha_{1}+c}\frac{1}{\sqrt{2\pi|\Sigma|}}\exp\left(-\frac{1}{2}\bold{x}^{T}\Sigma^{-1}\bold{x}\right)dx_{1}\cdots dx_{N}\leq2^{N}\frac{1}{\sqrt{2\pi|\Sigma|}}\prod_{j=1}^{N}(\alpha_{j}+c) =O(n^{-N(1-H)}),\end{aligned}
\mathfrak{l}\cap \mathfrak{h}_{\mu}= \mathbb{C}\mu\oplus \mathfrak{t}^{\sigma}\oplus \bigoplus_{\alpha\in \Phi_L^{\sigma}}\mathbb{C}e_{\alpha}\oplus \bigoplus_{\alpha\in \Phi_s^+; \bar{\alpha}(\mu)=0}\mathbb{C}(e_{-\alpha}+\sigma(e_{-\alpha}))\oplus \bigoplus_{\alpha\in \Phi_s^+; \bar{\alpha}(\mu)\neq 0}\mathbb{C}e_{-\alpha}.
EU_{n1}^4 = E \left[\frac{\left(Z_{n1}-p_{m_n}\right)^4}{p_{m_n}^2\left(1-p_{m_n}\right)^2}\right] &= \frac{p_{m_n}\left(1-p_{m_n}\right)^4+\left(1-p_{m_n}\right) p_{m_n}^4}{p_{m_n}^2\left(1-p_{m_n}\right)^2}&= \frac{(1-p_{m_n})^2}{p_{m_n}}+\frac{p_{m_n}^2}{1-p_{m_n}}\ .
E(a,\emptyset,\{1,2,3\}) \ = \ \Delta_5 \; \Delta_{13}\ - \ \Delta_4 \; \Delta_{23}\ = \ 0, E(b,\emptyset,\{1,2,3\}) \ = \ \Delta_6 \; \Delta_{13}\ - \ \Delta_7 \; \Delta_{12}\ = \ 0, E(c,\emptyset,\{1,2,3\}) \ = \ \Delta_9 \; \Delta_{12}\ - \ \Delta_8 \; \Delta_{23}\ = \ 0.
\mathcal{L}_{\epsilon}(\cdot)(x, u) = \Bigl \langle \bigtriangledown(\cdot), \Bigl(A x + \sum\nolimits_{i=1}^m B_i u_i \Bigr) \Bigr\rangle + \frac{\epsilon}{2}\operatorname{tr}\Bigl \{\sigma(x)\sigma^T(x)\bigtriangledown^2(\cdot) \Bigr\},
\begin{aligned}P( Z_{\sigma_{i}^{(*)}}^{N,i}=-1)\approx e^{-N\big(I_i(r_i)-I_i(r_{i+1})\big)}, &P( Z_{\sigma_{i}^{(*)}}^{N,i}=-11)\lessapprox e^{-N I_i(r_i)}, &\ \sigma_{i}^{(*)}=\sigma_{i}^{(e)}\ \ \sigma_{i}^{(*)}=\sigma_{i}^{(o)}.\end{aligned}
\|y\|^{2}_{\pi}=\sum_{n=1}^{N}\sqrt{\pi_n\lambda_{n}\,\varphi_{n}\,\varphi_{n+1}}\left(\frac{\bar y_{n}}{\varphi_{n}}-\frac{\bar y_{n+1}}{\varphi_{n+1}}\right)\;\left(\frac{1}{\sqrt{\pi_n\lambda_{n}\,\varphi_{n}\,\varphi_{n+1}}}\;\sum_{q=1}^{n}y_{q}\pi_q\varphi_{q}\right) & \le \sqrt{T_1}\; \sqrt{T_2}
A = \begin{pmatrix}G_{11}-\frac{G_{1m}G_{m1}}{G_{mm}}\cdots G_{1(m-1)}-\frac{G_{1 m}G_{m(m-1)}}{G_{mm}}G_{1m}\vdots \ddots \vdots \vdots G_{(m-1)1}-\frac{G_{(m-1)m}G_{m1}}{G_{mm}}\cdots G_{(m-1)(m-1)}-\frac{G_{(m-1) m}G_{m(m-1)}}{G_{mm}}G_{(m-1) m}0 \cdots 0 G_{mm}\end{pmatrix}.
(\dot\Gamma^a_i\delta\tilde E^{ia}-\dot{\tilde E^{ia}}\delta\Gamma^a_i)-\partial_i(i\hat\lambda^a\delta\tilde E^{ia})=0\{\rm mod}\left(\dot{\tilde E^{ia}}+i\varepsilon^{abc}\lambda^b\tilde E^{ic}=0\right),
\mathrm{ind\,}\widetilde D_{b,m}=\int_X\Bigr(\mathrm{Tr\,}e^{-t\widetilde{\Box}^{+}_{b,m}}(x,x)-\mathrm{Tr\,}e^{-t\widetilde{\Box}^{-}_{b,m}}(x,x)\Bigr)dv_X(x).
-\mathbb{E}\left[\left(-2 \int_{0}^{u}B(z)dz \right) \vert B(u)=-x\right]\frac{x}{\sqrt{2\pi u^3}}e^{-\frac{x^2}{2u}}=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{iuv}\frac{\mathrm{Ai}(i\xi-4^{\frac{1}{3}}x)}{\mathrm{Ai}(i\xi)}dv
\int_{S_2}\int_{0}^1\Big[- \frac{\partial^2f^{(m)}}{\partial s^2}\phi- f^{(m)}\mathcal{G}\cdot\frac{\partial \phi}{\partial u}-\epsilon f^{(m)}\kappa : u\otimes u \phi+\epsilon h^{(m)}\phi &+\dfrac{\epsilon}{4\pi}\int_{S_2}\kappa:v \otimes vf^{(m)}(v,s )d\mu(v)\phi-\dfrac{3+\epsilon}{4\pi}\kappa:u \otimes u 1^{(m)}\phi\Big]dsd\mu(u)= 0
e=|\frac{v_{f}}{v_{i}}|=\sqrt{\frac{K_{f}}{K_{i}}}=\sqrt{\frac{U_{f}}{U_{i}}}=\sqrt{\frac{H_{f}}{H_{i}}}=\frac{T_{f}}{T_{i}}=\sqrt{\frac{gT_{f}^{2}}{8H_{i}}}.
(K_{g_{f_1}}\cap K_{g_{f_2}})^\perp = \langle f_1/P, f_2/P, Tf_1/P, Tf_2/P \rangle = \{h/P : h \in \langle f_1, f_2, Tf_1, Tf_2 \rangle \}&= \{h/P: h \in K^\perp \}.
\widetilde{\psi}_{k}^{( n^{2})}(x;N,M,\lambda_{1})=( 1-z)^{( N+1/2) /2}(1+z)^{( M+1/2) /2}\frac{\widetilde{P}_{N,M,k}^{( n^{2})}( z; \lambda_1)}{\lambda_{1}+Q_{n}^{(N,M)}(z)},
[T_iv](t)-[T_iu](t) = \lambda_i \int_a^b k_i(t,s) g_i(s) [f_i(v(s)) - f_i(u(s))] ds \ge \lambda_i \int_a^b c \Phi_i(s) g_i(s) [f_i(v(s)) - f_i(u(s)]ds \ge c \lambda_i \int_a^b k_i(r,s) g_i(s) [f_i(v(s)) - f_i(u(s))] ds = c ([T_iv](r)-[T_iu](r)),
P^{-}={\frac{\mu^{2}q_{0}^{2}}{2}}+ \mu^{2}\Sigma_{2}+{\frac{\lambda q_{0}^{4}}{4 ! d}}+{\frac{\lambda q_{0}^{2}\Sigma_{2}}{2 ! d}}+{\frac{\lambda q_{0}\Sigma_{3}}{d}}+{\frac{\lambda \Sigma_{4}}{d}}.
(\partial^2-3f\partial_z)h_{\mu\nu}+4\alpha \kappa e^{2A(z)}\Big\{\partial_zf \partial_\lambda^2-f^2\partial_z^2-f\big(2\partial_zf-f^2\big)\partial_z\Big\}h_{\mu\nu}= -\kappa T^{(m)}_{\mu\nu}\,,
\left.\frac{\partial}{\partial\varepsilon}\left( (f^{\varepsilon\nu})^{-1}\overline{f^{\varepsilon\nu}}\right)\right|_{\varepsilon=0}=\left.\frac{\partial}{\partial\bar{\varepsilon}}\left( (f^{\varepsilon\nu})^{-1}\overline{f^{\varepsilon\nu}}\right)\right|_{\varepsilon=0}=0
\tau(z, \theta)=s^*(z)e^{-i\theta-\phi(z)}=s^*_1(z)g(z)e^{-i\theta-\phi(z)}&=s^*_1(z)g(z)e^{-i\theta-\phi_1(z)-\log\left\vert g(z)\right\vert}&=s^*_1(z)\bigr(\frac{g(z)}{\overline g(z)}\bigr)^{\frac{1}{2}}e^{-i\theta-\phi_1(z)}.
\begin{aligned}\int_{\breve{M}\backslash B_{\delta}(x_0)}\widetilde{\rm div}\left(\frac{rT}{r^p+\epsilon}|u|^p\right)dV_{\widetilde{g}}=\int_{V\backslash B_{\delta}(x_0)}\widetilde{\rm div}\left(\frac{rT}{r^p+\epsilon}|u|^p\right)dV_{\widetilde{g}}&=\int_{V \cap \partial B_{\delta}(x_0)}\frac{r}{r^p+\epsilon}|u|^p\langle T,\xi\rangle_T dS.\end{aligned}
LL(b,c) =b\left(f_c(b)-f_c(b-1)\right)+f_c(b)-f_c(0) =b\Delta_c(b)+f_c(b)-f_c(0) =b\Delta_c(b)+\sum_{i=1}^b\left[f_c(i)-f_c(i-1)\right] =b\Delta_c(b)+\sum_{i=1}^b\Delta_c(i).
\Delta w_{k}\equiv\oint\frac{\partial w_{k}}{\partial q_{k}}dq_{k}=\oint\frac{\partial^{2}W}{\partial J_{k}\partial q_{k}}dq_{k}=\frac{d}{dJ_{k}}\oint\frac{\partial W}{\partial q_{k}}dq_{k}=\frac{d}{dJ_{k}}\oint p_{k}dq_{k}=\frac{dJ_{k}}{dJ_{k}}=1
\lim_{j\to \infty}\frac{\|V_*\|^s(B_{r_j}(\bar x))}{\|V_*\|(B_{r_j}(\bar x))}=\lim_{j\to \infty}\frac{\|V_*\|(B_{r_j}(\bar x))}{\|V\|(B_{r_j}(\bar x))}=1.
g(y^\pm)+\Phi_L^\Omega(y^\pm,x^\pm) = g(y^\pm)+\int^0_{-T^\pm}L(\xi^\pm(s),\dot{\xi^\pm}(s))\ds &\leqslant g(z^\pm)+\int^0_{-|x^\pm-z^\pm|}L\left(x^\pm+s\frac{x^\pm-z^\pm}{|x^\pm-z^\pm|},\frac{x^\pm-z^\pm}{|x^\pm-z^\pm|}\right)\ds &\leqslant g(z^\pm)+M|x^\pm-z^\pm|\leqslant g(z^\pm)+M\eta,
\kappa_\circ(\alpha,q,\ell) = \begin{cases}\tfrac12-\tfrac{\alpha}{q}, q\in J_\circ(-1), \tfrac 12 - \tfrac{\alpha-\ell}q + \tfrac1{d-\ell}\big( \tfrac{\beta_{d-\ell}(\alpha-\ell)}q - \tfrac 12 \big), q \in J_\circ(\ell), \end{cases}
\frac{K_{s,0}+K_{s,\pi/4}}2=\frac{1}{2}\,K_{s,\pi/4}= \frac{2^{s/2+1/4}}{2s+1}\int_0^{+\infty}\frac{dz}{\left(\sqrt{2}z^2+\sqrt{z}\right)^{s+1/2}},
\Pi_{1}\, = \, - \frac{1}{4 \pi ( - z )^{1 / 2}}\operatorname{a r c t a n}\sqrt{- z}; \, \, \, \, \, \Pi_{2}= \, \frac{1}{1 6 \pi m \, z}\left[ 1 - \left( \frac{1 + z}{2 ( - z )^{1 / 2}}\right) \operatorname{a r c t a n}\sqrt{- z}\right] .
-\frac{\partial}{\partial t}K(t,x) =\sum_{i=1}^{d}(-\frac{\partial}{\partial t}K_i(t,x_i))\prod_{l\neq i}K_l(t,x_l) &=2dK(t,x)-\sum_{i=1}^{d}(w_{i,m}^{-1}K_i(t,x_i+1)+w_{i,m-1}K_i(t,x_i-1))\prod_{l\neq i}K_l(t,x_l) &=\Delta K(t,x).
h_{( \zeta_{\bf p}, \zeta_{\bf q})}\zeta_{\bf m}= \zeta_{\bf m}\Biggl ( 1 + i \frac{\Delta ( \zeta_{\bf p}, \zeta_{\bf q})}{R^{2}}+ O \left( \biggl ( \frac{| \mathrm{\boldmath ~ \d e l t a ~}|}{n}\biggr )^{2}\right) \Biggr ) ,
a_1(\xi,\eta) := \frac{\xi (\xi-\eta)}{|\xi-\eta|^{1/2}}\Big(\frac{|\xi|}{|\eta|}- 1 \Big) ,a_2(\xi,\eta) := |\xi-\eta|^{3/2}, a_3(\xi,\eta) := \frac{\eta (\xi-\eta)}{|\xi-\eta|^{1/2}}\Big(1 - \frac{|\xi|^{1/2}}{|\eta|^{1/2}}\Big) ,a_4(\xi,\eta) := \frac{|\xi-\eta|^2 |\xi|^{1/2}}{|\eta|}.
{\cal A}_{L}^{( 1 ) \, \mathrm{l e a d}}= \frac m{8 \pi}\left( \frac{\lambda^{2}}{1 6 m^{4}}- \frac{4 e^{4}}{m^{2}\Theta^{2}}\right) \left[ \ln \left( \frac{\Lambda^{2}}{{\bf p}^{2}}\right) + i \pi \right] \hspace{0 . 3 c m}.
|\Sigma_{11}(t)-\overline{\Sigma}_{11}|=\frac{1}{2}\Big|\frac{e^{2(a-2k)t}}{a-2k}+\frac{e^{2at}}{a}\Big|=\frac{1}{2}e^{2at}\Big|\frac{1}{a}+\frac{e^{-4kt}}{a-2k}\Big|\leq C e^{2at}.
\sum_{\ell=1}^L n_\ell = \frac{L-1}{\ln(\gamma)}\left[\ln\left(\frac{2\varphi(\alpha)}{1 + 2\varphi(\alpha)}\right)+ \ln\left(\frac{\beta}{3}\right)+ \frac{1}{L-1}\ln\left( \frac{\varepsilon}{F(x^{(1)}) - \eta^1}\right)\right] + L
E_{n,L}[z ; \psi_1] = \int_{(\Lambda_L^*)^{2n+1}}\sum_{A \in \mathcal{A}_{2n}}\mathcal{P}_{A,L}( (q_1 , \ldots , q_{2n+1})) \overline{\widehat{\psi}}_{1,\#}(q_1) \widehat{\psi}_{1,\#}(q_{2n+1}) &\times \prod_{j=1}^{n}(\nu( q_j) - z)^{-1}\prod_{j=n+2}^{2n+1}(\nu( q_j) - \overline{z})^{-1}d (q_1 , \ldots , q_{2n+1}),
\frac{F}{W}=\frac{GmM_{M}/d^{2}}{GmM_{E}/r_{E}^{2}}=\frac{M_{M}}{M_{E}}(\frac{r_{E}}{d})^{2}=\frac{\rho_{M}}{\rho_{E}}\frac{V_{M}}{V_{E}}(\frac{r_{E}}{d})^{2}
-\int_a^b \int_{\Omega}\phi(s,x)\overline{\partial_s\psi(s,x)}\,{\rm d}x\,{\rm d}s=\int_a^b\int_{\Omega}A(s,x)^{*}\nabla \phi(s,x)\cdot \overline{\nabla\psi(s,x)}\,{\rm d}x\,{\rm d}s.
\mu_{+}^{- 1}= \left( \begin{array}{c c c}{{I_{m_{1}}}}&{{- \mu_{+ 1 2}}}&{{- ( \mu_{+ 1 3}- \mu_{+ 1 2}\mu_{+ 2 3})}}\\{{0}}&{{I_{m_{2}}}}&{{- \mu_{+ 2 3}}}\\{{0}}&{{0}}&{{I_{m_{3}}}}\\ \end{array}\right) ,
S_{\mathrm{f e r m}}= \int \! d^{2}x \Bigl ( i \bar{\psi}\gamma^{\mu}\partial_{\mu}\psi +{\frac{\pi F_{e}}{\mu}}\bar{\psi}\psi +{\frac{\pi F_{m}}{\mu}}\bar{\psi}^{C}\psi -{\frac{G}{2}}\bar{\psi}\gamma^{\mu}\psi \bar{\psi}\gamma_{\mu}\psi \Bigr )
[\widetilde{P}, uA_3]= \sum_{i=0}^{\infty}[z^i]\left(-\frac{z^3}{(1+z)^3}+\frac{z^2}{(1+z)^2}\eta\right)\left(1+\frac{1}{z}\right)^4 u^{1-i}L_i = -uL_0-L_{-1}+e^UL_{-1}e^{-U}.
g\left( q^i,p_i,z^a,\overline{p}_a\right)=\sum\limits_{k=1}^\infty \frac 1{k!\lambda^k}\left[ C_{a_1}^0,\left[C_{a_2}^0,\ldots ,\left[ C_{a_k}^0,\tilde H\right] \ldots \right] \right]z^{a_1}\ldots z^{a_k}.
\partial_{\tau}w=\frac{\partial_{zz}^{2}w}{1+\left(\partial_{z}w\right)^{2}}+2\left(n-1\right)\frac{z\,\partial_{z}w+w}{z^{2}-w^{2}}+\frac{\frac{1}{2}+\sigma}{2\sigma\tau}\left(-z\,\partial_{z}w+w\right)
\sum_{\overrightarrow{x}\in V_n}(E{\widetilde{\rho}}^2)^{|x_n|}\leq (d+1)^n\sum_{\overrightarrow{x}\in V_n}\Big(\prod_{j=0}^{n-1}\frac{1}{\deg(x_j)}\Big)\Big(E{\widetilde{\rho}}^2\Big)^{|x_n|}
\sigma_{2}^{-}+ \sigma_{3}^{-}={\frac{\Gamma ( 1 + x )}{\Gamma ( 1 - x )}}{\frac{\Gamma ( 1 / 2 - x )}{\Gamma ( 1 / 2 + x )}}{\frac{\Gamma ( 1 / 2 - \Delta + x )}{\Gamma ( 1 / 2 - \Delta - x )}}{\frac{\Gamma ( 1 - \Delta - x )}{\Gamma ( 1 - \Delta + x )}}
\lim_{n\to+\infty}n(2\ln n)^{\frac{1}{2}}e^{1-(S(I_2)/S(I)-1)\ln n}D_n(G_n(x)/2) = \lim_{n\to+\infty}(2\ln n)e^{1-(S(I_2)/S(I)-1)\ln n}\lim_{n\to+\infty}n(2\ln n)^{-\frac{1}{2}}D_n(G_n(x)/2)=0,
(A-1I)x_{1}=(\begin{matrix}0&0&0&0&0\\ 3&0&0&0&0\\ 6&3&1&0&0\\ 10&6&3&1&0\\ 15&10&6&3&1\end{matrix})(\begin{matrix}0\\ 3\\ -9\\ 9\\ -3\end{matrix})=(\begin{matrix}0\\ 0\\ 0\\ 0\\ 0\end{matrix})=0,
\theta_{a}=\frac{k_{a}^{\left( h_{a}+1\right)}(q-1/q)\mathsf{A}_{-}(1/\xi_{a}^{\left( h_{a}\right)})}{((\xi_{a}^{(h_{a}+1)})^{2}-1/(\xi_{a}^{(h_{a}+1)})^{2})}\langle h_{1},...,h_{a},...,h_{\mathsf{N}}|h_{1},...,h_{a},...,h_{\mathsf{N}}\rangle ,
I^{(n)}_K(g(z_1,\dots,z_n);q,t;t_0,t_1,t_2,t_3) =\frac{\int g(\vec{z}) \Delta^{(n)}_K(\vec{z};t_0,t_1,t_2,t_3;q,t)}{\int \Delta^{(n)}_K(\vec{z};t_0,t_1,t_2,t_3;q,t)}.
d(x_{k_i},x_{k_j})\leq \sum_{i=\ell_1}^{\infty}d(x_i,x_{i+1})\leq \sqrt{\frac{2(\varphi(x_m)-\varphi^*)}{\gamma}}\sum_{i=\ell_1}^\infty (1-\frac{\kappa}{2\gamma})^{\frac{i-m}{2}}= D\cdot (1-\frac{\kappa}{2\gamma})^{\frac{\ell_1-m}{2}}\rightarrow 0, ~~as~~\ell_1\rightarrow \infty ,
\Delta (a-[c,a](db^{-1}a-c)^{-1}) =\Delta (a(db^{-1}a-c)-[c,a] )(db^{-1}a-c)^{-1}&= \Delta (adb^{-1}-c )a(db^{-1}a-c)^{-1}&= \Delta (ad -cb )b^{-1}a(db^{-1}a-c)^{-1}&= b^{-1}a(db^{-1}a-c)^{-1}&=(d-ca^{-1}b)^{-1}
\mathcal{T}_I(t,x,y,\Delta t,\Delta x, \Delta y) ={f^{\epsilon}(t+\Delta t,x,y)-f^{\epsilon}(t,x,y) \over \Delta t}+ a \; \frac{f^{\epsilon}(t,x,y) - f^{\epsilon}(t,x-\Delta x,y)}{\Delta x}+ \frac{b}{\epsilon}\;\frac{f^{\epsilon}(t+\Delta t,x,y)-f^{\epsilon}(t+\Delta t,x,y-\Delta y)}{\Delta y}\,.
C_2=C_2(r)=\frac{6}{\pi^2}\left(\eta\left(1-\frac{2}{r}\right)\right)^3\prod_{p}\left(1-\frac{1}{p}\right)^3\left(1+\frac{3}{p}-\frac{1}{p^2}-\frac{18}{p(p+2)}\right).
\| \widehat{g}\|^2_{L^q(d\mu)}\leq \sum_{n \geq 2}^{\log\lambda^{\frac{1}{2}}}\sum_{\substack{|\mathcal I|=|\mathcal J|=2^{-n}\mathcal I \sim \mathcal J}}\| \widehat{g_{\mathcal I}}\widehat{g_{\mathcal J}}\|_{L^{\frac{q}{2}}(d\mu)}+ \sum_{\mathcal I \in \mathfrak I_{E}}\| \widehat{g_{\mathcal I}}\|^2_{L^q(d\mu)}.
<tr\,G_{\cal A}(x,x)>_{\cal A}=\int{\mathcal{D}Y}\,e^{-{\lambda_2}\oint{d}\tilde{s}}\,W=\int dL\,e^{-\lambda_2L}\int{\mathcal{D}Y}\,\delta(L-\oint{d}\tilde{s})\,W
{\det \Delta}=4\pi\det ({\Delta}\!\!\restriction_{\overline{\Bbb C}_\epsilon})\cdot\det ({\Delta}\!\!\restriction_{|w_+|\leq \epsilon}) \cdot\det (\Delta\!\!\restriction_{|w_0|\leq \epsilon}) \cdot \det ({\Delta}\!\!\restriction_{|w_-|\leq \epsilon})\cdot \frac{\det \mathcal N\!\!\restriction_{\partial\Bbb C_\epsilon}}{\ell(\partial \Bbb C_\epsilon, m)},
\Omega_H = -\left.{1\over{\omega^0}_\phi}\right|_H ={a^2\over 2J}\left[r_{+}-M +{\left(M^2+\Sigma^2-P^2-Q^2\right)\left(M+\Sigma/\sqrt{3}\right)\over\left(M+\Sigma/\sqrt{3}\right)^2-Q^2}\right]^{-1}.
\left( \begin{array}{c c}{{1}}&{{k}}\\{{0}}&{{1}}\\ \end{array}\right) \otimes \left( \begin{array}{c c c c}{{1}}&{{p}}&{{q}}&{{r}}\\{{0}}&{{1}}&{{0}}&{{q}}\\{{0}}&{{0}}&{{1}}&{{p}}\\{{0}}&{{0}}&{{0}}&{{1}}\\ \end{array}\right)
= \int{\cal D}\vec{\varphi}\exp \Biggl \{- \int d^{3}x \Biggl [ \frac 1 2 \left( \partial_{\mu}\vec{\varphi}- \frac{\vec{\mu}_{\alpha}}{g_{m}}\partial_{\mu}\eta \right)^{2}- 2 \zeta \sum_{i = 1}^{N ( N - 1 ) / 2}\cos \left( g_{m}\vec{q}_{i}\vec{\varphi}\right) \Biggr ] \Biggr \},
i Q_{H_0}\big(A (\delta A^2 + 1)^{-1}u , u \big) = i Q_{H_0}\big( A u_{\delta}, u_{\delta}\big) + O \Big( \big| \big|\delta^{\frac{1}{2}}A u_{\delta}\big| \big|_{{\mathcal H}^1}|| u ||_{{\mathcal H}^1}\Big) .
W_{ijkl, st}-W_{ijkl, ts}= W_{rjkl}R_{rist}+W_{irkl}R_{rjst}+W_{ijrl}R_{rkst}+W_{ijkr}R_{rlst}; W_{ijkl, trs}-W_{ijkl, tsr}= W_{vjkl, t}R_{virs}+W_{ivkl, t}R_{vjrs}+W_{ijvl, t}R_{vkrs}+W_{ijkv, t}R_{vlrs}+W_{ijkl, v}R_{vtrs}.
^{\star}({\hat{F}}+ \Phi )^{i j}\equiv \frac{1}{2}\epsilon^{i j k l}({\hat{F}}+ \Phi )_{k l}^{\prime}= ( \det A ) ( ( A^{- 1})^{t}\;^{\star}( \hat{F}+ \Phi ) A^{- 1})^{i j}.
\lim_{\tau\to 0}\left\|\left[T(t)v\right](e^{\tau S}\cdot)-\left[T(t)v\right](\cdot)\right\|_{C_b}\leqslant C_1(t) \lim_{\tau\to 0}\left\|v(e^{\tau S}\cdot)-v(\cdot)\right\|_{C_b}=0.
I^+(x,t)= \sum^{\infty}_{n=1}2 e^{-i(n\pi)^2t}n\pi\sin n\pi x \int^{\infty}_0 \hat{h}(\lambda )\frac{e^{i(n\pi)^2t-\pi^2 i\lambda t}-1}{(n^2-\lambda )\pi^2}d\lambda \, .
\sum_{i}(-\frac{\partial L}{\partial q^{i}}dq^{i}+\dot{q}^{i}dp_{i})-\frac{\partial L}{\partial t}dt=\sum_{i}(\frac{\partial H}{\partial q^{i}}dq^{i}+\frac{\partial H}{\partial p_{i}}dp_{i})+\frac{\partial H}{\partial t}dt.
f_{k}^{( p )}( \sigma ; u ) = ( - e^{2 \lambda \sigma})^{k}\left[ \begin{array}{c}{{p}}\\{{k}}\\ \end{array}\right] \prod_{j = 1}^{k}\sinh [ u + ( j - 1 ) \lambda ] \prod_{j = k + 1}^{p}\sinh [ u + ( 2 s + j - p ) \lambda ] .
\mathcal{I}_{\mathrm{D}}(\delta_1, 0) = \inf_{\lambda_1 \in \mathbb{R}_{+}}\left[\lambda_1 \delta_1+\sup_{\varpi \in \Pi\left(\mu_1, \mu_2\right)}\int_{\mathcal{V}}g_{\lambda, 1}(v) \, d \varpi(v)\right].
\|f-g_n\|_p^p =\sum_{i=1}^q\int_{A_i}|f(x)-a_{i,n}|^pd\mu(x)=\sum_{j=1}^q\sum_{i=1}^q\int_{A_i\cap C_j}|f(x)-a_{i,n}|^pd\mu(x) &\geq \sum_{j=1}^q\sum_{i=1}^q\int_{A_i\cap C_j}|f(x)-a_{j,n}|^pd\mu(x) &=\sum_{j=1}^q \int_{C_j}|f(x)-a_{j,n}|^pd\mu(x)=\|f-\widetilde g_n\|_p^p
\mathrm{e}^{\lambda \varepsilon}\Phi_{\varepsilon}(x)=\int_{(a,b)}p_{\varepsilon}(y-x)\,\Phi_{\varepsilon}(y)\,\mathrm{d}y+\int_{\mathbb{R}\backslash (a,b)}p_{\varepsilon}(y-x)\,\varphi(y)\,\mathrm{d}y.
\left|\frac{\varphi(u)}{u}- \sum_{k=1}^{p-1}(-1)^{kr}b_k\frac{1}{u^{kr+1}}\right|=\frac{1}{|u|}\left|\psi(u^{-r})- \sum_{k=0}^{p-1}(-1)^{kr}b_k(u^{-r})^k\right|\le\frac{CA^pM_p}{|u|^{pr+1}}
S=(\begin{matrix}0&0&0&0&0\\ 1&0&0&0&0\\ 0&1&0&0&0\\ 0&0&1&0&0\\ 0&0&0&1&0\end{matrix});A=(\begin{matrix}1&1&1&1&1\\ 1&2&2&2&1\\ 1&2&3&2&1\\ 1&2&2&2&1\\ 1&1&1&1&1\end{matrix}).
|2c_{b}\lambda_{2}(t) \int_{0}^{\infty}d\xi \chi_{\leq \frac{1}{4}}(\xi) \frac{\sin(t\xi)}{t^{2}}\left(F_{v_{2}}(\xi,\lambda_{1}(t))-F_{v_{2}}(\xi,\lambda_{2}(t))\right)|\leq C \frac{|\lambda_{1}(t)-\lambda_{2}(t)| \lambda_{0}(t) |\log(\lambda_{0}(t))|}{t^{3}}&\leq C \frac{||e_{1}-e_{2}||_{X}\sqrt{\log(\log(t))}}{t^{3}\log^{2b}(t)}
\tilde{x}=\frac{\widetilde{\rho}\sinh \tau}{\cosh \tau -\cos \sigma}\cos\varphi ,\tilde{y}=\frac{\widetilde{\rho}\sinh \tau}{\cosh \tau -\cos\sigma}\sin \varphi ,\tilde{z}=\frac{\widetilde{\rho}\sinh \sigma}{\cosh\tau -\cos \sigma}
G^{R}( k ) ={\frac{N^{2}( q^{2})^{\Delta - 2}\Gamma ( 3 - \Delta ) \epsilon^{2 ( \Delta - 4 )}}{8 \pi^{2}2^{2 \Delta - 5}\Gamma ( \Delta - 2 )}}\left[ \cos{\pi \Delta}- i \sin{( \pi \Delta )}\, \mathrm{s g n}\, \omega \right] \, .
\begin{array}{l}\displaystyle{\inf}\; \mathbb{E}_{\xi_2,\ldots,\xi_{T_{\max}},D_2,\ldots,D_{T_{\max}}}\Big[ \displaystyle{\sum_{t=1}^{T}}\; f_{t}(x_{t}, x_{t-1}, \xi_t)\Big] x_t \in X_t( x_{t-1}, \xi_t ) \mbox{a.s.},\; x_{t}\;{\overline{\mathcal{F}}}_t\mbox{-measurable,}t=1,\ldots,T_{\max},\end{array}
\theta_i = S_1 -k_i \leq S_1 + |k_i| &\leq S_1 + \sqrt{k_1^2 + k_2^2 + \cdots + k_m^2}&= S_1 + |A| = S_1 + \sqrt{S_1^2 - 2S_2}&\leq 2S_1.
G^\omega_M(x,y) = \sum_{k\neq j}\frac{u_k(x)u_k(y)}{\lambda_k-\omega^2}e^{\phi(y)}+ \frac{u_j(x)u_j(y)}{\lambda_j-\omega^2}e^{\phi(y)}, &:= N^\omega_M(x,y) + \frac{u_j(x)u_j(y)}{\lambda_j-\omega^2}e^{\phi(y)},