harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\langle b_{\theta_0}(x), x\slash \lVert x \rVert \rangle = \sum_{i=1}^n \lVert x \rVert ( \alpha_i + \lVert x\rVert)^{-(1+q_i)}\langle x\slash \lVert x \rVert, A_i(\theta_0)x\slash \lVert x \rVert \rangle &\leq \sum_{i=1}^{k_0}\lVert x \rVert ( \alpha_i + \lVert x\rVert)^{-(1+q_i)}\lambda_{\max}(A_i(\theta_0)).
\langle \mathrm{g e o n}| d_{n , \epsilon}d_{n^{\prime}, \epsilon^{\prime}}| \mathrm{g e o n}\rangle = \frac{{( - 1 )}^{n}\delta_{n , n^{\prime}}\delta_{\epsilon , - \epsilon^{\prime}}}{2 \operatorname{s i n h}( \pi n / r_{+})}\ \ ,
\left(\sum_{i=0}^{d_x-1}w_i b^{\infty}_i\right) \frac{dx}{r}= d\left(\sum_{i=0}^{d_x-1}v_i b^{\infty}_i\right)+\left(\sum_{i=0}^{d_x-1}u_i b^{\infty}_i \right) \frac{dx}{r}.
G^{\prime}(\sigma^a_i,\sigma^a_j) = \sum_{n_1=-\infty}^{\infty}\sum_{n_2=0}^{\infty}{{4 \pi \alpha^{\prime}}\over{\omega_{n_1n_2}C_{n_1n_2}}}{\bar \Psi}_{-n_1n_2}(\sigma^a_i)\Psi_{n_1n_2}(\sigma^a_j) \quad .
D_m\Big\{\big[v(t),w(t)\big];\big[\bar{v}(t),\bar{w}(t)\big]\Big\}= \frac{\left\lVert\left(\begin{array}{c}v(t)-\bar{v}(t) w(t)-\bar{w}(t)\end{array}\right)\right\rVert}{\sqrt{\mu(t)}}.
\Phi =\left\{\phi\left(x,y\right)=-a\left(x^{2}+y^{2}\right)+b_{1}x+b_{2}y,\a\geq0,b_{1},b_{2}\in\mathbb{R}\right\}\Psi =\left\{\psi\left(x\right)=-cx^{2}+dx,\c\geq0,d\in\mathbb{R}\right\},
s_j^2 = F_j^{-1}\int \psi_j^2(x) f_0(x) \big(1 + \sum_{j=1}^m \theta_j^0 \psi_j(x) \big) dx - E_j^2 = \frac{\alpha^2}{nF_j}+ \frac{\mu_{j,3}}{F_j}\theta_j^0 - \frac{\alpha^4}{(nF_j)^2},
a\nabla_{\nu}b \leqslant K(h^{\frac{1}{2^n}},2)^{-r_n}a\sharp_{\nu}b+( \sqrt{a}-\sqrt{b})^2-\sum_{k=0}^{n-1}r_{k}\big[\big(a^{\frac{m_k}{2^k}}b^{1-\frac{m_k}{2^k}}\big)^{\frac{1}{2}}-\big(a^{\frac{m_k+1}{2^k}}b^{1-\frac{m_k+1}{2^k}}\big)^{\frac{1}{2}}\big]^{2},
c_i(f):=\begin{cases}Z_{r_i}(f,2-\frac{\dim V_i}{2}) \textrm{if}Z_{r_i}(f,s) \textrm{is holomorphic at}2-\frac{\dim V_i}{2}\lim_{s \to 0}\frac{d}{ds}(sZ_{r_i}(f,s+2-\frac{\dim V_i}{2})) \textrm{if}Z_{r_i}(f,s) \textrm{has a pole at}2-\frac{\dim V_i}{2}.\end{cases}
{\bf U}_{{\rm HiPOD}}^k(\alpha^*) = [u_{{\rm POD}, k, 1}^{\alpha^*}, \ldots, u_{{\rm POD}, k, N_h}^{\alpha^*}]^T=\displaystyle \sum_{j=1}^{L}T_j^k(\alpha^*){\boldsymbol \xi}_j k=1, \ldots, m
\begin{aligned}{\dot R}= \int_{\mathbb T}\cos(\theta - \phi) \partial_t \rho ~ d\theta &= KR \int_{\mathbb T}\cos(\theta - \phi) \partial_\theta \Big[ \rho(\theta, t) \sin(\theta - \phi) \Big] ~ d\theta &= KR \int_{\mathbb T}\sin^2(\theta - \phi) \rho(\theta, t) ~ d\theta.\end{aligned}
S ={\frac{1}{2 \kappa_{\scriptscriptstyle{D}}^{2}}}\int d^{\scriptscriptstyle{D}}x \sqrt{- g_{\scriptscriptstyle{D}}}\, e^{- \varphi}\left\{R_{\scriptscriptstyle{D}}+ ( \nabla \varphi )^{2}-{\frac{1}{1 2}}H^{2}\right\}\, ,
\sigma_{i+1}+J^{(n-1)}_n = s_{i-1}s_i\sigma_is_is_{i-1}+J^{(n-1)}_n &=s_{i-1}s_is_{i-1}s_is_{i-1}+J^{(n-1)}_n &=s_i+J^{(n-1)}_n.
\begin{aligned}P^{Cont,+r}=\{(I_0,k_1(0),k_2(0)),\ldots,(I_h,k_1(h),k_2(h)), (\emptyset, r, 0)\}; P^{Cont,+s}=\{(I_0,k_1(0),k_2(0)),\ldots,(I_h,k_1(h),k_2(h)), (\emptyset, 0, s)\}.\end{aligned}
S^{{\rm cl},0}_{{\rm L}}(g_3,g_1)=S^{{\rm cl},0}_{{\rm L}}(g_3,g_2)+S^{{\rm cl},0}_{{\rm L}}(g_2,g_1)S^{\rm cl}_{\rm M}(g_3,g_1)=S^{\rm cl}_{\rm M}(g_3,g_2)+S^{\rm cl}_{\rm M}(g_2,g_1).
F_t(B_t) = F_0(B_0) + \int_0^t\mathcal{D}F_s(B_s)dB(s) + \int_0^t\mathcal{D}^{\mathcal{F},h}F_s(B_s)ds -\frac{1}{2}\int_0^t\int_{-\infty}^{+\infty}\partial_x F_s(^{x}B_s)d_{(s,x)}\ell^{x}(s),
\bigl|\frac{f(p)}{\xi(p)}- \frac{f(q)}{\xi(q)}\bigr| = |f(p)\bigl(\frac{1}{\xi(p)}-\frac{1}{\xi(q)}\bigr)+ \frac{f(p) - f(q)}{\xi(q)}| & \leq 2\xi(p)\bigl(\frac{1}{\xi(p)}-\frac{1}{\xi(q)}\bigr)+ \frac{d(p,q)}{\xi(q)}& = \frac{1}{\xi(q)}[2(\xi(q) - \xi(p)) + d(p,q)] & \leq 3 \frac{d(p,q)}{\xi(q)}=3\rho(p,q).
v^\beta(x,i) \ = \ \lim_{\substack{n\rightarrow\infty x'\rightarrow x}}V^{\beta,n}(x',i) \ = \ \liminf_{n\rightarrow\infty}{\!}_*\, V^{\beta,n}(x,i) \ = \ \limsup_{n\rightarrow\infty}{\!}^*\, V^{\beta,n}(x,i),
(D^n_1)_{n-i,n-j}=(-1)^{n+j}\sqrt{2}\sqrt{2(n-i)+1}\sqrt{2(n-j)+1}\sum_{s=n-i}^{n-j}\frac{(-n+j)_s(n-j+1)_s}{(s+n+1)!(s-n+i)!}&\times\prod_{m=0}^{\frac{i-1}{2}}(n-s-2m)\prod_{m=0}^{\frac{i-3}{2}}(n+s-2m),
m(T^*T) = \inf \left\{\langle T^*Tx, x \rangle \colon x\in S_{H_1}\right\}& = \inf \left\{\langle{|T|}^2x, x \rangle \colon x\in S_{H_1}\right\}& = \inf \left\{{\left\||Tx|\right\|}^2 \colon x\in S_{H_1}\right\}& ={\left[m(|T|)\right]}^2.
0 \ = \R_*^\frac{7}{2}\left( -\frac{105}{2}\sqrt{\alpha_0}+ 5 \left(\alpha_0 + \tfrac{147}{4}\right)\Big(\tfrac{\pi}{2}- \arctan(\tfrac{7}{2 \sqrt{\alpha_0}}) \Big) \right) - \frac{12}{7}R_*^2 L_* \alpha_0 \beta_0^{3/2}+ \frac{6}{7\pi}\beta_0^{7/2}.
\mu_{n}^{(H)}=\frac{1+(-1)^{n}}{2}\Gamma\left(\frac{n+1}{2}\right),\mu_{n}^{(L)}=\Gamma(n+a+2), &\mu_{n}^{(J)}=\frac{\Gamma(n+a+2)\Gamma(b+2)}{\Gamma(n+a+b+4)},\quad\,\,\, \mu_{n}^{(C)}=\frac{1+(-1)^{n}}{2}\frac{\Gamma((n+1)/2)\Gamma(a-2-(n+1)/2)}{\Gamma(a-1)}.
\sigma \circ{}^t\Phi \circ \Phi = \left( \begin{array}{ccccc}\Delta_X 0 0 \cdots 0 0 -\Delta_Y 0 \cdots 0 0 * \ddots \ddots \vdots \vdots \vdots \ddots \ddots 0 0 * \cdots * -\Delta_Y\end{array}\right).
-\frac12\Delta_B |\phi|^2 = -\langle\overline\square_B\phi,\phi\rangle -\langle\phi,\overline\square_B\phi\rangle+\sum_a\{|\nabla_{\bar V_a}\phi|^2 +|\nabla_{V_a}\phi|^2\}& +\sum_{a}\langle \phi,R^Q(\bar V_a,V_a)\phi\rangle -\frac12\langle \nabla_{H^{1,0}-H^{0,1}}\phi,\phi\rangle-\frac12\langle\phi,\nabla_{H^{1,0}-H^{0,1}}\phi\rangle.
ch(R^{0}_{{\pi}_{S^{k}Y}*}(\mathcal{L}_{-1,j,-2}(2\Delta_{Y}))-R^{1}_{{\pi}_{S^{k}Y}*}(\mathcal{L}_{-1,j,-2}(2\Delta_{Y})))=\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ =((m'_{j,-2}+2k+1-g_{Y})-4\theta)e^{2x},
\begin{array}{ll}{\tilde d}(x)={\rm dist}\,(x,J) \mbox{$\forall x$ with}{\rm dist}\,(x,J)<\frac{|x|}{5}, \frac12{\rm dist}\, (x,J)\leq{\tilde d}(x)\leq 2\,{\rm dist}\, (x,J) \forall x\in{\mathbb R}^2, |\nabla{\tilde d}(x)|\leq 1 \forall x\notin J.\end{array}
E_{P}\left\{\psi_{P,\mathbf{a}}\left[ \mathbf{G,B};\mathcal{G}\right] g(\mathbf{A},\mathbf{G},\mathbf{B})\right\}=0g\text{suchthat}E_{P}\left[ g(\mathbf{A},\mathbf{G},\mathbf{B})|\mathbf{G},\mathbf{B}\right] =0
\|u\|^2_{L^{\infty,2}(Q^{+}(R/2))}+ \|\nabla u\|^2_{L^{2}(Q^{+}(R/2))}&\lesssim R^{(3\alpha-4)/\alpha}\|u\|^{2}_{L^{p,q}(Q^{+}(R))}+ R^{(3\alpha-5)/(\alpha-1)}\|u\|^{2\alpha/(\alpha-1)}_{L^{p,q}(Q^{+}(R))}&+ R^{-1}\|\Pi-\overline{\Pi}_{B^{+}(R)}\|^{2}_{L^{\lambda,\kappa^*}(Q^{+}(R))}
f^{\prime}( x ) = \frac{( 4 x^{2}- 7 )^{\prime}( 2 - x ) - ( 4 x - 7 ) ( 2 - x )^{\prime}}{( 2 - x )^{2}}= \frac{- 4 x^{2}+ 1 6 x - 7}{( 2 - x )^{2}}
J: =\int_{\{z: 1-|z_j|^2 \geq\varepsilon_{j}^{1/L_j}\ \forall j\}}\rho_L(z,0) d\nu(z)=\sum_{m=1}^\infty\frac 1{m^2}\int_{\{z: 1-|z_j|^2 \geq\varepsilon_{j}^{1/L_j}\ \forall j\}}[1-|z|^2]^{mL}d\nu(z) =\sum_{m=1}^\infty\frac 1{m^2}\prod_{j=1}^n \int_{\{z_j: 1-|z_j|^2 \geq\varepsilon_{j}^{1/L_j}\}}(1-|z_j|^2)^{mL_j-2}\ \frac{dm(z_j)}{\pi}\ .
X_{x}^{\varepsilon}=\mathcal{M}\left( S(\cdot )x+\sqrt{{\varepsilon}}\int_{0}^{\cdot}S(\cdot -s)G(s,X_{x}^{\varepsilon}(s))dw(s)\right) =\mathcal{M}\left( S(\cdot )x+\sqrt{{\varepsilon}}Z(X_{x}^{\varepsilon})\right),
\tilde{\Delta}\rho ( k ) = \frac{1}{\pi}\Theta_{( \Lambda_{K}, \Lambda_{K}+ \frac{1}{2}\delta ( \Lambda_{K}) )}( | k | ) \ln \left[ \sqrt{\frac{\delta ( \Lambda_{K})}{2 ( | k | - \Lambda_{K}}}+ \sqrt{\frac{\delta ( \Lambda_{K})}{2 ( | k | - \Lambda_{K})}- 1}\right]
{W}\, ( \varphi_{1}, \varphi_{2}) = \sqrt{2}\left[ \frac{m^{2}}{\lambda}\, \varphi_{1}- \frac{\lambda}{6}\, \varphi_{1}^{3}+ \frac{\lambda}{2}\, \varphi_{1}\varphi_{2}^{2}+ \frac{\mu}{\sqrt{2}}\, \varphi_{2}^{2}\right] \, ,
\Vert P_kX^iP_k y\Vert \leq \Vert X^i y\Vert = \Vert u^*X^iy\Vert &=\Vert P_{\alpha(k,d)}u^*X^iy\Vert =\Vert P_{\alpha(k,d)}[u^*X^iu]u^*y\Vert = \Vert P_{\alpha(k,d)}[u^*X^iu]P_{\alpha(k,d)}u^*y\Vert \leq \Vert P_{\alpha(k,d)}\widetilde{X}^iP_{\alpha(k,d)}\Vert.
0 = C_{i_{p}, \beta j_{q}, \gamma}^{k_{s}, \alpha}C_{l_{t}, \rho m_{u}, \sigma}^{i_{p}, \beta}\omega^{j_{q}, \gamma}\wedge \omega^{l_{t}, \rho}\wedge \omega^{m_{u}, \sigma}\quad ( \alpha , \beta , \gamma , \rho , \sigma = 1 , \ldots , N ) ,
= \frac{1}{(n-2)!}\int^{x_n}_a \int^{x_{n-1}}_a f(u) (x_{n-1}- u)^{n-2}du dx_{n-1}= \frac{1}{(n-2)!}\int^{x_n}_a f(u) \int^{x_{n-1}}_u (x_{n-1}- u)^{n-2}dx_{n-1}du = \frac{1}{(n-1)!}\int^{x_n}_a f(u) (x_n - u)^{n-1}du.
\int_{t_0}^t \int_{t_0}^s \|\nabla u(\tau)\|^2_{L^2}d\tau ds = \int_{t_0}^t \left(\int_{\tau}^t ds\right) \|\nabla u(\tau)\|^2_{L^2}d\tau = \int_{t_0}^t (t-\tau) \|\nabla u(\tau)\|^2_{L^2}d\tau.
\sum_{abc^2=n}\lambda_f(a)\lambda_f(b)\mu(b)\mu(bc)^2 F_{\Upsilon,M}(bc) = \sum_{abc^2=n}\sum_{d|(a,b)}\lambda_f\left(\frac{ab}{d^2}\right)\mu(b)\mu(bc)^2 F_{\Upsilon,M}(bc),
\partial_{t}\tilde{a}_{\varphi}^{0}( x , \tau ) = - \frac{1}{\lambda}\ \frac 1 2 \ \frac{e}{4 \pi R}\left[ e^{- \bigl | t - \tau - R / c \bigr | / \lambda}\epsilon ( t - \tau - R / c ) + e^{- \bigl | t - \tau + R / c \bigr | / \lambda}\epsilon ( t - \tau + R / c ) \right]
E_r \leq E_{r_{i-1}}\lesssim t\cdot K^2 \gamma\left( \frac{1}{r_{i-1}}T \cap B_2^n\right) = 2 t\cdot K^2 \gamma\left( \frac{1}{r_{i}}T \cap B_2^n \right) \leq 2t\cdot K^2 \gamma\left( \frac{1}{r}T \cap B_2^n \right).
E_{s,k}=\eta_{s,k}(1-\rho_{s,k})\bigg[\mathbf{h}_{s,k}^{H}\mathbf{q}\mathbf{q}^{H}\mathbf{h}_{s,k}+\mathbf{h}_{s,k}^{H}\mathbf{V}\mathbf{h}_{s,k}+\sigma_{sa,k}^{2}\bigg]
\left({{z_1},\{z_2}}\right)=\zeta \left({{\xi \over{\sqrt{1-\left| \xi \right|^2}}},\ 1}\right),\quad \left({d z^{*}d z}\right)^{2}=|\zeta|^{2}d\zeta^{*}d\zeta{{d\xi^{*}d\xi}\over{\left({1-\left| \xi\right|^2}\right)^2}}\quad,
\frac 12\sum_{j=1}^n \frac 1{d_j^-}+\left(\frac 12+\frac 1{q-1}\right)\sum_{j=1}^n \frac 1{d_j^+}\ge \frac{e_1+\dots+e_n}{2(q-1)}+\left(\frac 12+\frac 1{q-1}\right)\cdot \frac n2 &\ge \frac n{2(q-1)}+\frac n4+\frac n{2(q-1)}>1+\frac n{q-1}& \ge 1+\frac{n-\delta_1-\dots-\delta_n}{q-1}.
=M_{iklmnp}^{(6)}+9r^{2}\langle\langle M_{iklm}^{(4)}\delta_{np}\rangle\rangle+55r^{4}\langle\langle M_{ik}^{(2)}\delta_{lm}\delta_{np}\rangle\rangle+99r^{6}\langle\langle\delta_{ik}\delta_{lm}\delta_{np}\rangle\rangle
\psi_{k,k-n+2}^{\ast}(Q^{\prime})=\frac{L_{k,k-n+1}^{\ast}(q^{\prime})}{q^{\prime}}\leq \frac{L_{k,R_{k-n+2}}^{\ast}(q^{\prime})}{q^{\prime}}\leq \Phi_{k,n}(\psi_{n,2}^{\ast}(Q)).
( \nabla v_{\alpha q}^{\underline{{\alpha}}}) v_{\underline{{\alpha}}}^{~ \beta r}={\frac{1}{4}}\delta_{q}^{r}\hat{\Omega}^{a b}( \gamma_{a b})_{\alpha}^{~ \beta}+{\frac{1}{4}}\delta_{\alpha}^{~ \beta}\hat{\Omega}^{i j}( \gamma_{i j})_{q}^{~ r},
\begin{array}{rcl}\hat{P}_5(x)\!\! \!\!\!=\!\!\! \!\!(96\pi^2-960) x^5+(1760\pi-176\pi^3) x^4+(112\pi^4-1136\pi^2) x^3 begin{align*}1.5 ex]\!\! \!\!\!+\!\!\! \!\!(272\pi^3-24\pi^5) x^2-(4\pi^4+2\pi^6) x+\pi^7-2\pi^5,\end{array}
\sup_{f\in P_{\xi}^{\beta}\cap\Omega_{\alpha,\beta}}\dfrac{\| f \|_{q,u}}{\| f \|_{p,v}}=\bigg(\sup_{g\in P_{\xi}^{q\beta}\cap\Omega_{q\alpha,q\beta}}\dfrac{\| g \|_{1,u}}{\| g \|_{p/q,v}}\bigg)^{1/q}.
E_{pot}=\frac{1}{2}\rho gH^{2}[-\frac{1}{3m}+\frac{2}{3m}(1+\frac{1}{m})(1-\frac{E(m)}{K(m)})-\frac{1}{m^{2}}(1-\frac{E(m)}{K(m)})^{2}].
\sum_{{k=1,k\neq j}}^n\dfrac{1}{z_{\nu,k}^4-z_{\nu,j}^4}=\dfrac{2-\nu-z_{\nu,j}}{4z_{\nu,j}^4}-\dfrac{1}{4z_{\nu,j}^3}\sum_{k=1}^n\left(\dfrac{1}{z_{\nu,j}+z_{\nu,k}}+\dfrac{2z_{\nu,j}}{z_{\nu,j}^2+z_{\nu,k}^2}\right).
\sigma_{a b}( x ) = \int d y^{-}| x^{-}- y^{-}| \big [ \psi_{a}( x^{+}, y^{-}) \psi_{b}^{\dagger}( x^{+}, y^{-}) -{\frac{1}{N}}\delta_{a b}\psi_{c}( x^{+}, y^{-}) \psi_{c}^{\dagger}( x^{+}, y^{-}) \big ] .
p_\alpha A^\alpha \rightarrow \frac{1}{2}\left[\hat{p}_\alpha,A^\alpha(\hat{x}) \right]_{+}\, , \,\,\,\,\,\,\,\,p_\alpha F^{\alpha\beta}\rightarrow \frac{1}{2}\left[\hat{p}_\alpha,F^{\alpha\beta}(\hat{x}) \right]_{+}\, ,
\bar{r}_{kb}^i = \mathbb{E}_{\textbf{h}_{\cdot k,1}}\left[\log_2 \left(1+ \frac{P_b^i G_{bk}\|h_{bk,1}\|^2}{\sigma^2 + \sum_{l\neq b}P_l^i G_{lk}\|h_{lk,1}\|^2}\right) \right]
\phi_{a,R_{n}}^{+}* \varphi^{2}= \phi_{a,R_{n}}^{-}* \varphi^{2}+ \phi_{a,R_{n}}* \varphi^{2}\leq C_{S}+ a^{2}+ C(1 + M^{2/3}) = C(1 + M^{2/3}) + a^{2}.
[\frac{\alpha^{2}}{4}+(q^{\alpha)}+r^{\alpha)})t^{(\alpha)}]h_{i}^{(\alpha)2}C_{i}+[\frac{\alpha^{2}}{4}+q^{\alpha)}t^{\alpha)}-r^{(\alpha)}u^{(\alpha)}]\sum_{j\neq i}h_{j}^{(\alpha)2}C_{j}=C_{i},~~~i=1,...n
|\varphi_ra_{ij}|_{\dot{C}^{k,\gamma}}+|\varphi_rN^{-1}|_{\dot{C}^{k,\gamma}}+|D(\varphi_rN^{-1})|_{\dot{C}^{k-1,\gamma}}+|D(\varphi_rM)|_{\dot{C}^{k-1,\gamma}}&\qquad+|\partial_{s}(\varphi_rN^{-1})|_{\dot{C}^{k,\gamma}}+|t\partial_{t}(\varphi_rN^{-1})|_{\dot{C}^{k,\gamma}}\leq C_k,
f^\ast(D) = \frac{1}{2}\|N+D\|_r^2 - \frac{1}{2}\| N \|_F^2, f^{\ast \ast}(M) = \frac{1}{2}\|M\|_{r\ast}^2 - \langle N,M \rangle + \frac{1}{2}\|N\|_F^2
-C\left(\frac\delta\lambda\right)^{-n}+A \cdot \frac{\int_{B_\delta\times B_\delta}|x-y|_g^{\alpha-2}f_\lambda(x)f_\lambda(y)dxdy}{\|u\|_{L^{\frac{2n}{n+\alpha}}(M^n)}^2}\geq -C\left(\frac\delta\lambda\right)^{-n}+C_2A\lambda^{n-2}=\lambda^{n-2}(C_2A-C\lambda^2\delta^{-n})>0
S(a,b;q) = \mathop{\sum' \nolimits}_{x = 1}^{r}\mathop{\sum' \nolimits}_{y = 1}^{s}e\Bigl(\frac{a \overline{s}x^2 + b \overline{s}\, \overline{x}^3}{r}\Bigr) e\Bigl(\frac{a \overline{r}y^2 + b \overline{r}\, \overline{y}^3}{s}\Bigr) = S(a \overline{s}, b \overline{s}; r) S(a \overline{r}, b \overline{r}; s).
|\phi_j (x)| =e^{\lambda_j t_0}\left| \int_X p_{t_0}(x,y) \phi_j (y) d\mu(y) \right| \le e^{\lambda_j t_0}\left( \int_X p_{t_0}(x,y)^2 d\mu (y) \right)^{1/2}\le M e^{\lambda_j t_0}.
\|\frac{d}{dt}\Big|_{t=0}f_{(g+th,b+tK)}\|_{C^{2,\alpha}}\leq C(\|h\|_{C^{2,\alpha}}+\|K\|_{C^{2,\alpha}})(g,b)\in\mathcal{U}, & \|\frac{d}{dt}\Big|_{t=0}f_{(g+th,b+tK)}\|_{W^{2,2}}\leq C(\|h\|_{W^{2,2}}+\|K\|_{W^{2,2}})(g,b)\in\mathcal{U}.
v_2\left( \int_{\mathbb{Z}_2}g(t) \binom{t+2^m-1}{2^m-1}^2 \binom{t+2^{m-1}-1}{2^{m-1}-1}^{s}\binom{t+2^m-1}{2^{m-1}-1}^{s}\mathrm{d}t \right) = -m.
A\sum_{l=0}^\infty\sum_{\kappa=1}^{N(n,l)}|a_l^\kappa(f)|^2 \leq\sum_{l=0}^\infty\sum_{j=0}^\infty\left|\gamma\bigl(b_j\tau(l)\bigr)\right|^2\nu_j\sum_{\kappa=1}^{N(n,l)}|a_l^\kappa(f)|^2 &\leq B\sum_{l=0}^\infty\sum_{\kappa=1}^{N(n,l)}|a_l^\kappa(f)|^2.
\lim_{n_1,\dots,n_k\to\infty}\mu(A_0\cap T^{-n_1}A_1\cap T^{-n_1-n_2}A_2\cap\cdots\cap T^{-n_1-n_2-\cdots-n_k}A_k) = \mu(A_1)\mu(A_2)\cdots\mu(A_k).
\lambda(A+A) = \sum_{i=0}^{2n}\lambda\left(S_i\right) = \sum_{i=0}^{n}\lambda\left(S_{2i}\right)+\sum_{i=0}^{n-1}\lambda\left(S_{2i+1}\right) &\geq \sum_{i=0}^{n}2\alpha_i+\sum_{i=0}^{n-1}(\alpha_i+\alpha_{i+1})=4\alpha-(\alpha_0+\alpha_n).
\frac{d}{dt}G (v) + \frac{b_0}{16}Z_0(v) & \leq \frac{1}{2}\sum_{\mu=0}^{L-1}\int_{\partial \Omega}h \cdot \sigma | \sigma \cdot \nabla \partial_t^\mu v|^2 dS + \bar{C}_\lambda \sum_{\mu=0}^{L-1}\langle \partial_t^\mu F_\lambda, \partial_t^{\mu+1}v + \partial_t^\mu v + [h ; \nabla \partial_t^\mu v] \rangle.
g^{(n)}((2k+1)2^{-n-1}) = \frac{1}{2}(g^{(n)}(k2^{-n}) + g^{(n)}((k+1)2^{-n})) = \left(k^2+k+\frac{1}{2}\right)4^{-n}h^{(n+2)}((2k+1)2^{-n-1}) = -1 \Rightarrow g^{(n+1)}((2k+1)2^{-n-1}) = \left(k^2+k+\frac{1}{2}\right)4^{-n}+ 2^{-2n-3}(-2) = ((2k+1)2^{-n-1})^2,
{\bf D_{3}}( 0 , \sigma , 0 , \nu_{1}, \nu_{2}, \nu_{3}) = \frac{1}{\mathrm{i}\pi^{n / 2}\Gamma \left( 3 - \frac{n}{2}\right)}\int \frac{\mathrm{d}^{n}p}{( p^{2}- m^{2})^{\nu_{3}}}\left. J_{0 1 1}( \sigma , \nu_{1}, \nu_{2}; p^{2}, m ) \right|_{m = 1}\; .
(\lambda+\mu)\frac{\partial}{\partial y}(\frac{\partial u_{x}}{\partial x}+\frac{\partial u_{y}}{\partial y}+\frac{\partial u_{z}}{\partial z})+\mu(\frac{\partial^{2}u_{y}}{\partial x^{2}}+\frac{\partial^{2}u_{y}}{\partial y^{2}}+\frac{\partial^{2}u_{y}}{\partial z^{2}})+F_{y}=0
\Gamma_{A^{\mu}A^{\nu}}( p , - p ) \equiv \Gamma_{\mu \nu}( p , - p ) = ( \eta_{\mu \nu}- \frac{p_{\mu}p_{\nu}}{p^{2}}) \Gamma^{T}( p^{2}) + \frac{p_{\mu}p_{\nu}}{p^{2}}\Gamma^{L}( p^{2}) .
E^{a}= - \frac{k}{2}e_{\scriptscriptstyle E}^{\mu}A_{\mu}^{a}; \hspace{2 m m}F^{a}= - \frac{k}{2}e_{\scriptscriptstyle F}^{\mu}A_{\mu}^{a}; \hspace{2 m m}D^{a}= - \frac{k}{2}C^{a},
M_n^{(\alpha,\beta)}= \frac{S_{n+1}^{(a,b)}}{a+b}=\frac{1}{a+b}\sum_{k=0}^{n+1}\binom{n+k}{2k}C_k\cdot a^{n-k}b^k = \frac{(\alpha^2-4\beta)^{\frac{n+1}{2}}}{\beta}\sum_{k=0}^{n+1}\bigg(\frac{\alpha}{\sqrt{\alpha^2-4\beta}}-1\bigg)^{k+1}\cdot\frac{C_k}{2^{k+1}}\binom{n+k}{2k}.
\begin{aligned}\epsilon^{-1}[\gamma_1^{u_1^\epsilon}(y(0))-\gamma_1(y(0))]=\int_0^1\gamma_{1y}\Big(y(0)+\lambda\big(y^{u_1^\epsilon}(0)-y(0)\big)\Big)d\lambda\frac{(y^{u_1^\epsilon}(0)-y(0))}{\epsilon}\rightarrow\gamma_{iy}(y(0))y_1^1(0),\end{aligned}
\int_{S^2}{\Big [}-2\bar R_{L\underline LL\underline L}- \bar Ric(L,\underline L)- \frac{1}{2}\bar Ric(L,L){\Big ]}\tilde X^idS^2= - \int_{S^2}\bar Ric(e_0,e_j)\tilde{X}^j \tilde{X}^idS^2 = -\frac{4 \pi}{3}\bar Ric(e_0,e_i).
\gamma_{F,j}=\frac{P_F \beta_{F,k^*}|{\bf h}_{F,k^*}^H{\bf w}_{i^*}|^2}{P_M \beta_{M,k^*}|{\bf h}_{M,k^*}^H{\bf f}_{j}|^2 + \sigma^2_F}=\frac{x_F}{\lambda_F y_{F,j}+\mu_F}
E(v,h;\theta)=\sum_{i=1}^{D}\frac{(v_{i}-b_{i})^{2}}{2\sigma_{i}^{2}}-\sum_{i=1}^{D}\sum_{j=1}^{F}\frac{v_{i}}{\sigma_{i}}W_{ij}v_{i}h_{j}-\sum_{i=1}^{D}b_{i}v_{i}-\sum_{j=1}^{F}a_{j}h_{j}
D_{+ +}^{( C P V ) a b}( x ) = D_{+ +}^{( C P V )}( x ) \delta^{a b}= -{\frac{i \delta^{a b}}{( 2 \pi )^{2}}}\int d^{2}k \, e^{i k x}{\frac{\partial}{\partial k_{-}}}C P V \left({\frac{1}{k_{-}}}\right) = -{\frac{i \delta^{a b}}{2}}| x^{-}| \delta ( x^{+}) \ ,
\bigr ( G_{\beta_{0}\gamma_{1}}^{\alpha_{1}}G_{\gamma_{0}\beta_{1}}^{\gamma_{1}}+ R_{\beta_{0}}^{i}G_{\gamma_{0}\beta_{1}, i}^{\alpha_{1}}+ \mathrm{a n t i s y m}( \beta_{0}\leftrightarrow \gamma_{0}) \bigr ) + G_{\alpha_{0}\beta_{1}}^{\alpha_{1}}F_{\beta_{0}\gamma_{0}}^{\alpha_{0}}+ 3 Z_{\beta_{1}}^{\alpha_{0}}H_{\alpha_{0}\beta_{0}\gamma_{0}}^{\alpha_{1}}= 0
\sum_{k=1}^N \mathbb{E}[Y_{k,N}^2 \mathbf{1}_{|Y_{k,N}|>\epsilon}|\mathcal{F}_{k-1,N}] \to 0 , \sum_{k = 1}^N \mathbb{E}[ Y_{k,N}^2 | \mathcal{F}_{k-1,N}] \to \sigma^2 ,
\lll u = 2 \nabla f \cdot \nabla u + 2 \ell_{0}\nabla \ell_{0}\cdot \nabla u + \vert \nabla \ell_{0}\vert^{2}u &= 2 \nabla \phi_{+}\cdot \nabla u + \vert \nabla \ell_{0}\vert^{2}u ,
C_{j_k}(\alpha(k)) = \frac{\alpha(\alpha-1)(\alpha-2)\cdots (\alpha-(j_k-1))}{j_k!}= \frac{\beta(\beta-2^{n-k+1})(\beta-2\cdot 2^{n-k+1})\cdots (\beta-(j_k-1)2^{n-k+1})}{2^{j_k(n-k+1)}j_k!}= \frac{B(k)}{2^{j_k(n-k+1)}j_k!}
([D_j, D_{\bar k}]u)(t)=\mathbb H^t(i_t^* [[d^E, \delta_{V_j}], [d^E, \delta_{\overline{V_k}}]]\mathbf{u})=\mathbb H^t(i_t^* [[d^E, \delta_{V_j}], [d^E, \delta_{\overline{V_k}}]]\mathbf{u}).
h(x)= \displaystyle\sum_{i=0}^{\frac{s+1}{2}}\left[\binom{s}{s-2i+1}\,k+\,\binom{s}{s-2i}\,(2-k)\right]\,x^\frac{p^l+s-2i}{2}\cr +\displaystyle\sum_{i=0}^{\frac{s+1}{2}}\left[\binom{s}{s-2i}\,k+\,\binom{s}{s-(2i+1)}\,(2-k)\right]\,x^\frac{s-2i-1}{2}
A ( k , \Delta E ) = \frac{e^{i ( \Delta E + \omega ) \tau_{0}}}{i}\frac{\bar{g}}{\sqrt{( 2 \pi )^{3}2 \omega}}\left( \frac{1}{\Delta E + \omega}- \frac{1}{\Delta E + \gamma ( \omega - k_{z}v )}\right) .
J_\Gamma(x)=\bar\psi(x)\Gamma\psi(x)\equiv \frac{1}{8\pi}\int d\hat\epsilon e^{i\chi_\Gamma(\hat\epsilon)}\left\{\left[\psi^\dagger(x+\epsilon),\gamma^0\Gamma\psi(x-\epsilon)\right],e^{ie\int_{x-\epsilon}^{x+\epsilon}dx_iA_i^{tr}}\right\}_{|\epsilon|\propto|\eta|}
\sigma(H^{\epsilon,0})\cap (-\infty, E) \subset \bigcup_{k=0}^N [a_k,b_k]\,, &|b_k-a_k| \leq C\, \epsilon^L \mbox{for}0\leq k\leq N(\epsilon)\,,{\rm and}a_{k+1}-b_k\geq \epsilon/C \mbox{for}0\leq k\leq N(\epsilon)-1\,.
|B| + (\lambda-1)\max B \ \leqslant\k - |\{1, 2, \ldots, k\}\cap A| + (\lambda-1)k \ \leqslant \ \lambda m - |\{1, 2, \ldots, k\}\cap A| \ \leqslant\ |A| - |\{1, 2, \ldots, k\}\cap A|\ =\ |F|.
\gamma=log(2\pi)-2-2\sum_{n=1}^{\infty}\frac{(-1)^{n}G_{n}(2)}{n}=log(2\pi)-2+\frac{2}{3}+\frac{1}{24}+\frac{7}{540}+\frac{17}{2880}+\frac{41}{12600}+...
1\leq m_{\Gamma \cap (\widetilde{H_1}\times \widetilde{H_2}')}\big( \rho, \psi\otimes \widetilde{\sigma}\otimes \pi_2\big) = m_{\widetilde{H_1}\times \widetilde{H_2}'}\big( \widetilde{\rho}, \Psi\otimes\widetilde{\sigma}\otimes \pi_2\big) =m_{G_1 \times G_2}\big( \pi, \Psi\otimes \pi_1 \otimes \pi_2\big)\leq1
\begin{array}{c}\langle \widehat{q}_1f, f \rangle_{L^2 (\mathbb{R}^2)}= \int_{\mathbb{R}^2}\left[\left(\lambda x_1 + \frac{i \theta}{2 \lambda}\frac{\partial}{\partial x_2}+ E x_1^2 \right)f (x_1, x_2) \right] \overline{f (x_1, x_2)}dx_1 dx_2 = = \lambda x_1^{(0)}+ E \left((x_1^{(0)})^2 + \frac{a}{4}\right)\end{array}
\phantom{=}\int_{\mathbb{R}^N}(K\ast g(u_n))g(u_n)dx-\int_{\mathbb{R}^N}(K\ast g(u_n-u))g(u_n-u)dx &= \int_{\mathbb{R}^N}(K\ast (g(u_n)-g(u_n-u)))(g(u_n)-g(u_n-u))dx &\phantom{=}+2 \int_{\mathbb{R}^N}(K\ast (g(u_n)-g(u_n-u))) g(u_n-u)dx &\to \int_{\mathbb{R}^N}(K\ast g(u))g(u)dx.
tH_{\mathrm{KFS}}^{\frac32+\frac43}\left(\theta^\infty_1 ;t;{q_1,p_1 \atop q_2,p_2}\right) =tH_{\mathrm{III}(D_7)}\big(\theta^\infty_1; t; q_1,p_1\big) \qquad{}+tH_{\mathrm{III}(D_7)}\big(1-\theta^\infty_1; t; q_2,p_2\big)-p_1q_1p_2q_2-t\left(\frac{p_2}{q_1}+p_1+p_2\right).
S_{gauge}= -\,\frac{1}{g_{\rm YM}^{2}(\mu)}\int d^4 x \left\{\frac{1}{2}\partial_{\alpha}\Phi^i \partial^{\alpha}\Phi^i + \frac{1}{4}F_{\alpha \beta}F^{\alpha \beta}\right\}+ \frac{\theta_{\rm YM}}{32 \pi^2}\int d^4 x F_{\alpha \beta}{\tilde{F}}^{\alpha \beta}
\| (y_1, y_2)\|_{1,p}\leq C_{12}(\|y_1\|_p^{1 - \nu}+ \|y_2\|_p^{1 - \nu}) &\leq C_{13}(\|y_1\|_{1, p}^{1 - \nu}+ \|y_2\|_{1, p}^{1 - \nu}) &\leq C_{14}\|(y_1, y_2)\|_{1,p}^{\frac{1 - \nu}{p}},
\|m(\xi,\eta,\sigma)\|_{\mathcal{S}^\infty_{k,k_1,k_2,k_3}}:=\|m(\xi, \eta,\sigma)\psi_k(\xi)\psi_{k_1}(\xi-\eta)\psi_{k_2}(\eta-\sigma)\psi_{k_3}(\sigma)\|_{\mathcal{S}^\infty}.
F^{0,q}=F^{0,q}(x,a,b):=\int_{-1}^1f^{(q)}(A+x+2a\sqrt{x}s)\,ds, F^{p,q}=F^{p,q}(x,a,b) &:=\int_{-1}^1\int_{0}^{s_1}\dots\int_{0}^{s_p}f^{(q)}(A+x+2a\sqrt{x}s_{p+1})\,ds_{p+1}\ldots\,s_2\,ds_2\, s_1\,ds_1, &p=1,\dots,l,\,q=l+1,\ldots,2l+1.
\cot\frac{k(p+1)}{q}\pi-\cot\frac{k(p-1)}{q}\pi+\cot\frac{k(p^*+1)}{q}\pi-\cot\frac{k(p^*-1)}{q}\pi = \cot\frac{k(s+1)}{q}\pi-\cot\frac{k(s-1)}{q}\pi+\cot\frac{k(s^*+1)}{q}\pi-\cot\frac{k(s^*-1)}{q}\pi
|\Gamma_3^h(u)|+|\Gamma_4^h(u)|= \mathfrak{s}|\langle|I_h z|^2I_h z -|z|^2z,I_h (z u)\rangle_{L_x^2}|+\mathfrak{s}|\langle |z|^2z,(I_h -I)(z u)\rangle_{L_x^2}| = \mathfrak{s}|\langle I_h(|I_h z|^2I_h z -|z|^2z),z u \rangle_{L_x^2}|+\mathfrak{s}|\langle (I_h-I)|z|^2z,z u\rangle_{L_x^2}|
\Psi'(\lambda)=det(I_5-D_{L''^c})={\lambda}^{5}- \left( 2\,n-10 \right){\lambda}^{4}- \left( -28+10\,n \right){\lambda}^{3}&- \left( 10+8\,n \right){\lambda}^{2}- \left(103-15\,n \right) \lambda+14\,n-70.
\lim_{m_{i}\rightarrow 0}\rho_{N}( p ) = 2 \Im I_{N}( p ) = \frac{( 4 \pi )^{1 + \ell - N \ell}[ \Gamma ( \ell - 1 ) ]^{N}}{2 \Gamma ( N \ell - N ) \Gamma ( ( N - 1 ) ( \ell - 1 ) )}( p^{2})^{N \ell - \ell - N}\theta ( p^{2}) .
\prod_{(i,j)\in F_k}X_{i,j}= \frac{L_{1, b-k+1}}{L_{1, b-k+2}}\frac{L_{2, b-k+2}}{L_{2, b-k+3}}\cdots \frac{L_{k-1, b-1}}{L_{k-1, b}}L_{k, b}=\frac{\prod\limits_{i=1}^k L_{i,b-k+i}}{\prod\limits_{i=1}^{k-1}L_{i,b-k+i+1}}
\big|\big\{x\in B(x_0,r):|b(x)-b_{B(x_0,r)}|>\lambda\big\}\big| &\leq C_1|B(x_0,r)|\exp\bigg[-\bigg(1+\frac{r}{\rho(x_0)}\bigg)^{-\theta^{\ast}}\frac{C_2 \lambda}{\|b\|_{\mathrm{BMO}_{\rho,\theta}}}\bigg],
\begin{aligned}K_{F,3}(T,\tau) \coloneqq \frac{0.432\tau^2}{T}\left(\frac{A_F}{2}\log{(2T)}+\frac{A_F\log{4}+B_F}{2}\right. & \left.-\frac{c_{F,1}(T_0)}{3T}\log(2^\frac{1}{3}T) -\frac{c_{F,2}(T_0)}{3T}-\frac{2c_{F,3}(T_0)}{7T^2}\right ).\end{aligned}
\begin{aligned}\iota_{X_2}\omega= -\frac12 \sum_{l\neq j}\frac{a_j b_l}{G_l}da_l+\frac14 \sum_{l\neq j}\sum_{s>l}a_j\frac{a_l b_l}{G_l G_{l+1}}\left(b_s da_s+a_s db_s \right) &+\frac14 \sum_{l\neq j}\sum_{i<l}a_j\frac{a_l b_l}{G_i G_{i+1}}\left(b_i da_i- a_i db_i \right)\,, \end{aligned}
[\begin{matrix}Q_{1}\\Q_{2}\end{matrix}]=\frac{\kappa}{2}\{\int_{-h}^{h}[\begin{matrix}C_{55}&0\\ 0&C_{44}\end{matrix}]dx_{3}\}[\begin{matrix}w_{,1}^{0}-\varphi_{1}\\w_{,2}^{0}-\varphi_{2}\end{matrix}]
\left\{(z_v)_v \in | Z | \ \bigg|\z_v > z_{v'},\v \in \sigma \ \v' \in Z_0 \setminus \sigma \right\}= \left\{z \in | Z | \ \bigg|\d^1(z, |\sigma|) < d^1(z, |\sigma'|), \sigma' \in Z \setminus \{\sigma\}\ \mathrm{dim}(\sigma') = l \right\}.