harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
C^\infty_{0,\partial \Omega_D}(\overline{\Omega}) := \{f \in C^\infty(\overline{\Omega}): f|_{\partial \Omega_D}\equiv 0 \}, H^1_{0,\partial \Omega_D}(\Omega) :=\mbox{closure of}C^\infty_{0,\partial \Omega_D}(\overline{\Omega})\mbox{in}H^1(\Omega).
e(T ) =e(T[ W\cup S\cup \{u_1,\ldots,u_{_{10-w-s}}\}])+e_T(\{u_{_{11-w-s}}, \ldots, u_r\}, W \cup S\cup\{u_1, \ldots, u_{_{10-w-s}}\}) & \le e(\mathcal{T}_{10})+(2n-4)=24+(2n-4),
p(-2l_1 + gl_2) = \prod_{i = 0}^{2g}[-2l_1 - (g - i)l_2], & \alpha_{1,0}(-2l_1 + gl_2) = -4(2g - 1)l_1, & \alpha_{1,1}(-2l_1 + gl_2) = 4l_1^2 + g(g - 1)l_2^2 - 2(2g - 1)l_1l_2.
\mathop{{\rm Res}}_{s=\frac{m+1}{2}}\xi_{n,m}^{()}(s) =\mathop{{\rm Res}}_{s=\frac{m+1}{2}}\xi (2m+2-4s) & =\mathop{{\rm Res}}_{s=\frac{m+1}{2}}\Gamma(m+1-2s)\cdot \zeta (0)=\tfrac{1}{4}.
\begin{tabular}{ll}$d \eta_1 = (dw_{12}+ dw_{21}) /2~~~~~~~~~~~~~~$ $d \eta '_1 = (d \mu_{12}+ d \mu_{21}) /2$ $d \eta_2 = dw_{22}$ $ d \eta '_2 = d \mu_{22}$ $d \eta_3 = (dw_{12}- dw_{21}) /2$ $ d \eta '_3 = (d \mu_{12}- d \mu_{21}) /2 $.\end{tabular}
\lambda^3=\sum_{i=1}^l(-\lambda_i)^3+\sum_{i=l+1}^k(-\lambda_i)^3-(n-k-1)\geq \frac{(-\lambda_1-\dots-\lambda_l)^3}{l^2}-(k-l)-(n-k-1) \geq\frac{(n-d-1)^3}{(d+1)^2}-n.
\frac{1}{B+1}2^{2n\left(H(\alpha_s)+o(1)\right)}\leq \frac{1}{B-s+1}2^{2n\left(H(\alpha_s)+o(1)\right)}< S_{n,s}<2^n e^{-\frac{2s^2}{n}}\binom{n}{\lfloor{\frac{n}{2}}\rfloor-s},
a_{ix}=\gamma_{x,i}x_{i},a_{ib}=\gamma_{b,i}\frac{b_{i}}{b^{\ominus}},a_{iw}=\gamma_{w,i}w_{i},a_{ic}=\gamma_{c,i}\frac{c_{i}}{c^{\ominus}},a_{ir}=\gamma_{\rho,i}\frac{\rho_{i}}{\rho^{\ominus}}
x(1-x)\frac{\partial^{2}F_{3}(x,y)}{\partial x^{2}}+y\frac{\partial^{2}F_{3}(x,y)}{\partial x\partial y}+[c-(a_{1}+b_{1}+1)x]\frac{\partial F_{3}(x,y)}{\partial x}-a_{1}b_{1}F_{3}(x,y)=0
\det \mathbf{B}_T=b_{1,1}\det \mathbf{B}_T(1,1)=\left(a_{1,1}-\beta\sum_{i=2}^Ta_{i,1}\right)\det \mathbf{A}_T(1,1)=\left(a_{1,1}-\beta\sum_{i=2}^Ta_{i,1}\right)\det \mathbf{A}_{T-1}.
\begin{array}{r c l}{A'}^aT^a = UA^aT^aU^{-1}\ \ \ ,\ \ \{\pi'}^aT^a=U\pi^aT^aU^{-1}\ \ \ ,{\phi'}^aT^a = U\phi^aT^aU^{-1}\,+\,\frac{i}{g}U\frac{d}{dt}U^{-1}\ \ \ .\end{array}
M^{\otimes_{w^*h}\eta}\subset[Y_1\otimes_{h M'}\cdots \otimes_{h M'}Y_n]^{\natural}\cap [Z_1\otimes_{h M'}\cdots \otimes_{h M'}Z_n]^{\natural}&\simeq [Y_1\otimes_{h M'}\cdots \otimes_{h M'}Y_n+Z_1\otimes_{h M'}\cdots \otimes_{h M'}Z_n]^{\natural}=M^{\otimes_{sw^*h}\eta}
\sum_{i=0}^{n}(-1)^{i}Rank(H_{i}(X_{n},\emptyset))=\sum_{j=0}^{n}\sum_{i=0}^{j}(-1)^{i}Rank(H_{i}(X_{j},X_{j-1}))=\sum_{j=0}^{n}(-1)^{j}c_{j}.
M = \left( \begin{matrix}{{{\frac{{q^{2}}}{{{\left( 1 +{q^{2}}\right)}^{2}}}}}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{{\frac{{q}}{{{2 \left( 1 +{q^{2}}\right)}}}}}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{{\frac{{q}}{{{2 \left( 1 +{q^{2}}\right)}}}}}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{{\frac{{q^{2}}}{{{\left( 1 +{q^{2}}\right)}^{2}}}}}}\\ \end{matrix}\right)
W [ Y ] = \int \frac{d^{4}x}{3 2 \pi^{2}}\int_{0}^{\infty}\frac{d T}{T^{3}}\rho ( T , \Lambda^{2}) e^{- T K}\mathrm{t r}\left( e^{- T M \tau_{3}}[ 1 - f_{1}( T , B ) + f_{2}( T , B ) - \tilde{f}_{3}( T , B ) + \ldots ] \right) ,
{\frac{I -{\bf A}\cdot{\bf B}- i ({\bf A}+{\bf B}+{\bf A}\times{\bf B}) \cdot{\bf \sigma}}{\sqrt{1 + A^{2}}\sqrt{1 + B^{2}}}}= U_{3}U_{4}={\frac{I + i{\not \! \! C}}{\sqrt{1 + C^{2}}}},
T(u)=\partial_u\partial_u\varphi-{1\over 2}\left(\partial_u\varphi\right)^2=\left\{J_H^{-1},u\right\}=\sum_{i=1}^{n-1}\left({1\over 2(u-u_i)^2}+{c_i\over u-u_i}\right).
{\cal A}_{\alpha_{jk}}={n\over 4\kappa}\,\oint_{\alpha_{jk}}{\displaystyle u^{n}\, du\over\displaystyle\sqrt{1-e^{-(1+u^{n})/\kappa}}}=e^{i\pi (j+k+1)/n}\sin\bigl(\pi(j-k)/n\bigr)\,{\cal I}_{n}(\kappa)
\big\langle u^N_{tt}- \div \partial_F W(\phi^N,\nabla u^N) - \epsilon \Delta u^N, u^N_t\big\rangle_W=0, & \big \langle \phi_t^N - \Delta \partial_\phi W(\phi^N,\nabla u^N), \phi^N_t\big\rangle_V=0.
{\cal A}( p ) \sim \frac{g^{m}\alpha^{2 m - 1}}{R^{2 m + 2}r_{\mathrm{m i n}}^{m + 2}}\int d r r^{3}\biggl ( \frac{r_{\mathrm{m i n}}}{r}\biggr )^{\Delta}F ( \tilde{p}\sqrt{\alpha^{\prime}}) \ ,
\left\langle f,g\right\rangle_{L^2(\mathbb{S}^{d-1})}:=\int_{\mathbb{S}^{d-1}}f\left( \theta \right) \overline{g\left( \theta \right)}d\theta \left\Vert f\right\Vert_{L^2(\mathbb{S}^{d-1})}=\sqrt{\left\langle f,f\right\rangle_{L^2(\mathbb{S}^{d-1})}}.
\mathcal{U}_{ad}[0,\infty]=\left\{u\in U\Big|u\in\mathcal{F}_t^\xi,\bar{\mathbb{E}}\int_0^\infty e^{-\beta_1kt}|u_t|^{2k}dt<\infty,\ \beta_1\geq0,\k\geq1\right\}.
p(6)-p(5)=4: \;1\;\;1\;\;1\;\;1 p(5)-p(4)=2: \;2\;\;2 p(4)-p(3)=2: \;3\;\;3 p(3)-p(2)=1: \;4 p(2)-p(1)=1: \;5 p(1)=1: \;7.
e^{i M L \sinh \theta_{j}}\sum_{\mathbf{n}}T_{j}( \vec{\theta})_{\mathbf{n}}^{\mathbf{m}}\psi^{\mathbf{n}}( \vec{\theta}) = - \psi^{\mathbf{m}}( \vec{\theta}) \, , \quad j = 1 , \ldots , N \, ,
2\lim_{X \to \infty}\left(\frac{d^{n^2-1}}{ds^{n^2-1}}(\widetilde{\phi}(s)X^s)\big|_{s=1}\right)^{-1}\sum_{\pi'}\mathrm{tr}(\pi')((h^{S'_1})^{1}h_{S'_1}b_{E/F'}(\Sigma_{\phi}^{S_0}(X)))
\mathcal{B}_\pm^\prime = \{\psi \mid d^\prime \psi = 0, \ (\varGamma^\prime \pm 1) \psi = 0 \}& = \{\psi \mid d (V^{-1}\psi) = 0, \ (\varGamma \pm 1) (V^{-1}\psi) = 0 \}& = V \mathcal{B}_\pm,
f*g(q)=\left(\sum_{n\in \mathbb{N}}q^na_n\right)\cdot \left(\sum_{n\in \mathbb{N}}q^nb_n\right)=f(q)\cdot g(q)=\left(\sum_{n=0}^\infty q^nb_n\right)* \left(\sum_{n=0}^\infty q^na_n\right)=g*f(q),
\begin{aligned}k^{u}_{j+1}(T) \leq k^{u}_{0}(T)+ C C_{1}(T) B \big(1-3\big(\tfrac{1}{p}-\tfrac{1}{q}\big),\tfrac{1}{2}-\tfrac{3}{2q}\big) [k_{j}^{u}(T)^2+k_{j}^{\nabla y}(T)^2].\end{aligned}
(\widehat{\phi_1^{(p)}})^{(p)}\circ \phi_1 = (\widehat{\phi_1^{(p)}})^{(p)}\circ (\phi_1^{(p)})^{(p)}=((\widehat{\phi_1^{(p)}})\circ \phi_1^{(p)})^{(p)}= ([\deg\phi_1^{(p)}]_{E_1^{(p)}})^{(p)}= ([\deg\phi_1]_{E_1^{(p)}})^{(p)}=[\deg\phi_1]_{E_1},
F^{(5)}=\sum_{\substack{j_1+\dots+j_5=0, at\,\,most\,\,one\,\,index\,\,outside\,\,S}}F^{(5)}_{j_1, \dots, j_5}\,u_{j_1}\dots u_{j_5}, F^{(5)}_{j_1, \dots, j_5}:=\dfrac{H^{(4)}_{5,\,j_1,\dots, j_5}}{\mathrm{i}(j_1^3+\dots+j_5^3)}.
F_{i}(\zeta ,\upsilon +\chi)\equiv\frac{(\upsilon +\chi)f_{i}}{2\sqrt{2}(\zeta +k)^{2}(\zeta +b)^{1/2}}[\zeta^{2}+(3h_{i}-k)\zeta +2b(h_{i}-k)+kh_{i}], \qquad i=E^{\prime},U^{\prime},D^{\prime}.
X ( z , \bar{z}) = \sqrt{\frac{\alpha^{\prime}}{2}}\sum_{k}\left( \frac{\alpha_{k - \vartheta}}{k - \vartheta}z^{- k + \vartheta}+ \frac{\tilde{\alpha}_{k + \vartheta}}{k + \vartheta}\bar{z}^{- k - \vartheta}\right) ,
g'(s) =\frac{(\alpha-1)(s-a)^{\alpha-2}(b-s)^{\alpha-1-\beta}-(\alpha-1-\beta)(s-a)^{\alpha-1}(b-s)^{\alpha-2}}{(b-a)^{\alpha-1-\beta}\Gamma(\alpha)}&=\frac{(s-a)^{\alpha-2}(b-s)^{\alpha-2-\beta}\left[(\alpha-1)(b-s)-(\alpha-1-\beta)(s-a)\right]}{(b-a)^{\alpha-1-\beta}\Gamma(\alpha)}.
-\frac{\zeta'(s)}{\zeta(s)}= \frac{1}{s-1}- B -\frac{1}{2}\log \pi + \frac{\Gamma'(\frac{s}{2}+1)}{2 \Gamma(\frac{s}{2}+1)}- \sum_{\rho}\Big(\frac{1}{s-\rho}+\frac{1}{\rho}\Big)
G ( W , Z ) = 1 +{\frac{P}{[{\frac{4}{( c - 2 )^{2}}}W^{2 - c}+{\frac{4}{( d - 2 )^{2}}}Z^{2 - d}]^{\frac{c d - c - d}{( c - 2 ) ( d - 2 )}}}}, \ \ ( c \neq 2 \neq d ) ,
\begin{array}[c]{rl}\hat{J}(u^{\varepsilon}(t))-\hat{J}(\bar{u}(t)) = \displaystyle\sum_{i=1}^{n}\psi_{x_i}(\bar{X}(t_1),\bar{X}(t_2),\cdots,\bar{X}(t_n))y(t_i) &+{\displaystyle \int \limits_{0}^{T}}\{f_x(\bar{X}{(t)},\bar{u}(t))y(t)+f(\bar{X}{(t)},u^{\varepsilon}(t))-f(\bar{X}{(t)},\bar{u}(t))\}dt+o(\varepsilon). \end{array}
0 \le\Phi_{I,\lambda,P}(f)-\Phi_{I,\lambda,Q}(f) =f(t_{j})-f(t_{j+1})+f(t_{j})-f(t_{j-1})+f(t_{j-2})-f(t_{j-1})-|f(t_{j+1})-f(t_{j-2})|-2\lambda =2[f(t_{j})-f(t_{j-1})]-(f(t_{j+1})-f(t_{j-2})+|f(t_{j+1})-f(t_{j-2})|)-2\lambda =2[f(t_{j})-f(t_{j-1})]-2\max\{f(t_{j+1})-f(t_{j-2}),0\}-2\lambda \le2[f(t_{j})-f(t_{j-1})-\lambda],
(1+2 \cos^2 \phi ) \, p_{1,\ell}( \phi)-5 \cos \phi \, p_{1,\ell+1}( \phi)+2 p_{1,\ell+2}( \phi) &= (1+2 \cos^2 \phi ) \, \, p_{1,0,\ell}(\phi) - 5 \cos \phi \, p_{1,0,\ell+1}(\phi)+ 2 p_{1,0,\ell+2}(\phi)+ \phi^{1/2}O(\ell^{-2-1/2}),
\int{\frac{d \rho_{0}d^{4}x_{0}}{\rho_{0}^{5}}}\, \prod_{p = 1}^{4}{\left[{\frac{\rho_{0}}{\rho_{0}^{2}+ ( x_{p}- x_{0})^{2}}}\right]}^{4}= \frac{5 \pi^{2}}{1 0 8}\prod_{p < q}{\frac{\partial}{\partial x_{p q}^{2}}}\int \prod_{p}d \alpha_{p}{\frac{\delta \left( 1 - \sum_{q}\alpha_{q}\right)}{( \sum_{p , q}\alpha_{p}\alpha_{q}x_{p q}^{2})^{2}}}\; ,
A_1(\xi, \xi) =\sum_{i=1}^m q_i \int_{\Omega}( c_{ih}^{n+1}-c_{ih}^n)\xi dx= \Delta t\sum_{i=1}^m q_i \int_{\Omega}v_i \xi dx \leq \Delta t\sum_{i=1}^m |q_i| \| v_i\| \|\xi\|.
\lim_{n\rightarrow \infty}n^{2\alpha+1}\int_0^{\tau \wedge t_1 \wedge t_2}(t_1-s)^{\alpha}(t_2-s)^{\alpha}(\sigma'(X_s))^2 \Psi^{n,2}_s ds =\frac 1{2\alpha+2}\int_0^{\tau \wedge t_1 \wedge t_2}(t_1-s)^{\alpha}(t_2-s)^{\alpha}(\sigma'\sigma)^2(X_s) ds,
\left( I^{\alpha,\alpha',\beta,\beta',\gamma}_{0+}f\right)(x)=\frac{\Gamma(\rho)\Gamma(\rho+\gamma-\alpha-\alpha'-\beta)\Gamma(\rho-\alpha'+\beta')}{\Gamma(\rho+\beta')\Gamma(\rho+\gamma-\alpha-\alpha')\Gamma(\rho+\gamma-\alpha'-\beta)}
\begin{cases}Q_{i+1,j+1}=(\min[Q_{i+1,j},W_{i+1,j}]-\min[Q_{i,j},W_{i,j}])+Q_{i,j}, W_{i,j}^+=(\min[Q_{i+1,j},W_{i+1,j}]-\min[Q_{i,j},W_{i,j}])+W_{i,j},\end{cases}
J =n!\sum_{m=1}^\infty \frac 1{m^2}\frac{\Gamma(mL-n)}{\Gamma(mL)}+ \textrm{o}(L^{-n}) =n!\sum_{m=1}^\infty \frac 1{m^2}\frac{1}{(mL)^n}+ \textrm{o}(L^{-n}) =n! \frac 1{L^n}\zeta(n+2)+ \textrm{o}(L^{-n}).
\nu_m^a= \frac{1}{d_{l+2}\cdots d_m}\sum_{j=1}^{d_{l+2}\cdots d_m}\frac{1}{d_{l+1}}\sum_{r=1}^{d_{l+1}}\lambda_{r,j}=\frac{1}{d_{l+1}\cdots d_m}\sum_{j=1}^{d_{l+2}\cdots d_m}\sum_{r=1}^{d_{l+1}}\lambda_{r,j}.
SINR^{\kappa}=\frac{P_{\kappa}G_0h_0r_0^{-\alpha_{\kappa}(r_0)}}{\sigma^2+\underbrace{\sum_{i \in \Phi_c}P_cG_ih_ir_i^{-\alpha_{\kappa}(r_i)}}_{I_{c\kappa}}+\underbrace{\sum_{j \in \Phi_d}P_d G_jh_jr_j^{-\alpha_{\kappa}(r_j)}}_{I_{d\kappa}}}
R_1(F)=R_1(w_2)+R_1(w_4)=M_{12}(e_1^{-1})+M_{12}(e_2)+M_{12}(e_3^{-1})+M_{12}(e_4) = = M_{21}(e_1)+M_{21}(e_2^{-1})+M_{21}(e_3)+M_{21}(e_4^{-1}).
-\Delta \Pi=\nabla\cdot((u\cdot\nabla) u-\Delta\Psi\nabla\Psi)=\sum_{i,j=1}^{3}\partial_{x_{i}}\partial_{x_{j}}(u_{i}u_{j}-\partial_{x_{i}}\Psi\partial_{x_{j}}\Psi)+\frac{1}{2}\Delta(|\nabla\Psi|^{2}),
m_{ij}m_{il}=qm_{il}m_{ij}, m_{ij}m_{kj}=qm_{kj}m_{ij}, m_{il}m_{kj}=m_{kj}m_{il}, m_{ij}m_{kl}=m_{kl}m_{ij}+\left(q-q^{-1}\right)m_{il}m_{kj}.
\delta x = \delta x^1+\delta x^2+\delta x^3 =\dot{x}- \nu_k (x^0-\bar{x}), \delta \lambda = \delta \lambda^1+\delta \lambda^2+\delta \lambda^3 =\dot{\lambda}- \nu_k (\lambda^0-\bar{\lambda}) , \delta s = \delta s^1+\delta s^2+\delta s^3 =\dot{s}- \nu_k (s^0-\bar{s}).
U_{1 2}= \left( \begin{array}{c c c c c c c}{0}&{1 / 2}&{1 / 2}\\{0}&{\sqrt{Q}}&{\sqrt{Q}}&{- 1}\\{0}&{0}&{0}&{1}\\{0}&{0}&{0}&{\sqrt{Q}}&{}\\{0}&{0}&{0}&{1}&{0}\\{0}&{0}&{0}&{1}&{0}&{0}\\ \end{array}\right)
\rho_{\frac{3}{2}}(s) = \frac{(4\pi)^k \zeta(2s - 2)\Gamma(2s - 2)}{\Gamma(s) \Gamma(s + k - 1) \zeta^*(2s - 2)}\langle \mathcal{V}_{f, \overline{g}}, \overline{E(\cdot, 2 -s)}\rangle.
b(\mathbf{D})\boldsymbol{\eta}(\mathbf{x})=\left(b(\mathbf{D})\mathbf{w}_k(\mathbf{x})\right)\vert u(\mathbf{x})\vert^2+\sum \limits_{l=1}^d b_l \mathbf{w}_k(\mathbf{x})D_l\vert u(\mathbf{x})\vert^2, D_j\boldsymbol{\eta}(\mathbf{x})=\left(D_j\mathbf{w}_k(\mathbf{x})\right)\vert u(\mathbf{x})\vert^2+\mathbf{w}_k(\mathbf{x})D_j\vert u(\mathbf{x})\vert^2.
v_i(\lambda) - v_{i+1}(\lambda) = \# \{b \mid 1 \leq b \leq l(\lambda),\j_{b}= i+1 \}- \# \{b \mid 1 \leq b \leq l(\lambda),\ - (b-1) \equiv i+1 \bmod N \}&= \# \{b \mid 1 \leq b \leq Nd,\j_{b}= i+1 \}- \# \{b \mid 1 \leq b \leq Nd,\ - (b-1) \equiv i+1 \bmod N \}&= \# \{b \mid 1 \leq b \leq Nd,\j_{b}= i+1 \}- d
[\begin{matrix}M_{11}\\M_{22}\\M_{12}\end{matrix}]=-\frac{2Eh^{3}}{3(1-\nu^{2})}[\begin{matrix}1&\nu&0\\ \nu&1&0\\ 0&0&1-\nu\end{matrix}][\begin{matrix}\varphi_{1,1}\\ \varphi_{2,2}\\ \frac{1}{2}(\varphi_{1,2}+\varphi_{2,1})\end{matrix}],
\lvert (i_1,j_1)\rvert_{W_n(w)}-\lvert (i_2,j_2)\rvert_{W_n(w)}= (\lvert l_{i_1}\rvert_S+\lvert m_n \rvert_S+\lvert r_{j_1}\rvert_S) &-(\lvert l_{i_2}\rvert_S+\lvert m_n \rvert_S+\lvert r_{j_2}\rvert_S) = \lvert l_{i_1}\rvert_S-\lvert l_{i_2}\rvert_S+\lvert r_{j_1}\rvert_S-\lvert r_{j_2}\rvert_S.
\sum\limits_{n\geq 0}a(n) (f(n)-\sum\limits_{j=0}^{B-1}f(Bn+j)) = \sum\limits_{n\geq 0}\sum\limits_{j=0}^{B-1}a(Bn+j)f(Bn+j)-\sum\limits_{n\geq 0}\sum\limits_{j=0}^{B-1}a(n)f(Bn+j) = \sum\limits_{n\geq 0}a(Bn+e)f(Bn+e)-\sum\limits_{n\geq 0}a(n)f(Bn+e).
r_{k}(x_{2k})= &\frac{p_{X_1,X_{2k}}(x_1,x_{2k})}{p_{X_1}(x_1)p_{X_{2k}}(x_{2k})}-\sum_{j=1,j\neq k}^{q-1}\int r_{j}(x_{2j})\frac{p_{X_{2j},X_{2k}}(x_{2j},x_{2k})}{p_{X_{2j}}(x_{2j})p_{X_{2k}}(x_{2k})}p_{X_{2j}}(x_{2j})dx_{2j},
C_1\sum_{i=1}^p\int_{f_i^{-1}(E_i)}f_i^\ast\rho^{-\delta-\frac{n}{p}+j}|\partial_j f_i^\ast u|d\mathbf{x}\leq \int_M \rho^{-\delta-\frac{n}{p}+j}|\hat\nabla^j u|dV_{\hat g}\leq C_2\sum_{i=1}^p\int_{f_i^{-1}(E_i)}f_i^\ast\rho^{-\delta-\frac{n}{p}+j}|\partial_j f_i^\ast u|d\mathbf{x}
var(r)=\frac{1}{s_{x}^{2}+m_{x}^{2}}[(s_{y}^{2}-s_{x^{2}[y^{2}/x^{2}]})-(s_{x[y/x]})^{2}+2m_{y}s_{x[y/x]}-\frac{s_{x}^{2}}{m_{x}^{2}}(m_{y}-s_{x[y/x]}^{2})]
K=\frac{c_{2}^{\prime}(s)(c_{1}^{\prime}(s)c_{2}^{\prime\prime}(s)-c_{2}^{\prime}(s)c_{1}^{\prime\prime}(s))}{c_{1}(s)(c_{1}^{\prime}(s)^{2}+c_{2}^{\prime}(s)^{2})^{2}}
\frac{q(t+\tfrac{h}{2})-q(t-\tfrac{h}{2})}{h}= \frac{p(t)}{m}, \frac{p(t+h)-p(t)}{h}= F \big( q(t+\tfrac{h}{2}) \big) + \Omega \big( q(t+\tfrac{h}{2}) \big) \left( \frac{p(t+h)+p(t)}{2}\right) .
\omega_{\alpha}\wedge *(\omega_I) = (-1)^{(n-\ell-1)}* \iota_{e_{\alpha, n+1}}(\omega_I) = (-1)^{(n-\ell-1)}\sum_{s=1}^{n-\ell}(-1)^{s-1}\delta_{\alpha, i_{s}}* (\omega_{I-i_s}).
\begin{cases}\,\,\eta_\epsilon\in H^1_0(\Omega)\,\, begin{align*}0.3cm] \int_{\Omega}\nabla\eta_\epsilon\nabla vdx-\int_{\Omega}k^2\eta_\epsilon vdx=\int_{\Omega}f_\epsilon vdx\,\,\forall\,v\in H^1_0(\Omega). \end{cases}
\beta = \sin^{-1}\left( \frac{\ddot{s}_i}{\sqrt{\dot{s}_i^2+\ddot{s}_i^2}}\right)= \sin^{-1}\left( \frac{p_{s_i}\sigma+q_{s_i}}{\sqrt{\dot{s}_i^2+(p_{s_i}\sigma+q_{s_i})^2}}\right).
{\cal Z}_{N}( J ) \, = \, \int{\cal D}\theta \, \exp [ - \frac{t}{2}\partial_{\mu}\theta \partial_{\mu}\theta - \frac{t}{2 g^{2}}J^{2}+ \theta ( - \frac{t}{g}\epsilon_{\mu \nu}\partial_{\mu}J_{\nu}+ i g \rho_{N}) ] \; .
\zeta^{\wedge}_{\widetilde{\Gamma}}(s) = \frac{\zeta_k(5s-(12d+3))^2 \, \zeta_k(9s-(18d+11)) \, \zeta_k(11s-30d) \, \zeta_k(21s-(54d+7))}{\zeta_k(10s-(24d+5))}\, \omega(s),
P^{0}= E = \frac{1}{2 e^{2}{\phi_{0}}^{2}}\int d^{2}x \left[ \frac{1}{2}F_{i j}^{2}+ \vert D_{i}\phi \vert^{2}+ \frac{e^{2}}{4}( \vert \phi \vert^{2}-{\phi_{0}}^{2})^{2}\right]
\Omega_0\mid_{V_1}=\frac{dz_2\wedge dz_3}{f_{z_1}},~\Omega_0\mid_{V_2}=\frac{dz_3\wedge dz_1}{f_{z_2}},~\Omega_0\mid_{V_3}=\frac{dz_1\wedge dz_2}{f_{z_3}},
\eta = \hat{\eta}+ \frac{1}{\sqrt{\mathcal{K}_{Y}''(\hat{\eta})}}\left[ (\omega - \hat{\omega}) - \frac{1}{6}\hat{\rho}_3 (\omega - \hat{\omega})^2 + \left( \frac{5}{72}\hat{\rho}_3^2 - \frac{1}{24}\hat{\rho}_4\right) (\omega - \hat{\omega})^3 + \O((\omega - \hat{\omega})^4)\right].
\frac{1}{2\pi i}\int_{|w|=\gamma}\frac{1}{w^{n}}\frac{1}{1-4w}(1-4w)^{-1/2}dw=\frac{1}{2\pi i}\int_{|w|=\gamma}\frac{1}{w^{n}}\frac{1}{(1-4w)^{3/2}}dw.
\sigma(A^*)=\overline{\sigma(A)}+\O{S^{m_1-1}_\delta(M)}(h^{1-2\delta}) \sigma(AB)=\sigma(A)\sigma(B)+\O{S^{m_1+m_2-1}_{\delta}(M)}(h^{1-2\delta}) \sigma([A,B])=-ih\{\sigma(A),\sigma(B)\}+\O{S^{m_1+m_2-2}(M)}(h^{2(1-2\delta)}).
[e_{1}^{\pm}(u),f_{1}^{\mp}(v)]=\frac{h}{u_{\pm}-v_{\mp}}k_{2}^{\mp}(v)k_{1}^{\mp}(v)^{-1}-\frac{h}{u_{\mp}-v_{\pm}}k_{2}^{\pm}(u)k_{1}^{\pm}(u)^{-1}.
U_{\mathrm{{s y m . t r a c e l e s s}}}^{\mu \nu \lambda \rho \sigma}= \mathrm{s y m m e t r i z a t i o n ~ o f}U^{\mu \nu \lambda \rho \sigma}- \frac{1}{4 0}( g^{\mu \nu}U_{\alpha}^{\; \; \alpha \lambda \rho \sigma}+ \mathrm{9 \; t e r m s}) .
(VF)(\gamma) = \sum \limits_{x \in \gamma^+}e^{-E_{\kappa^+}(x,\gamma^+ \backslash x)}e^{-E_{\tau^+}(x,\gamma^-)}(F(\gamma^+ \backslash x, \gamma^- \cup x) - F(\gamma)) \ \ \ + \sum \limits_{x \in \gamma^-}e^{-E_{\kappa^-}(x,\gamma^- \backslash x)}e^{-E_{\tau^-}(x,\gamma^+)}(F(\gamma^+ \cup x, \gamma^- \backslash x) - F(\gamma)).
P_7(t,\alpha,\beta=-1)=t^7+6 \alpha t^5 + 5 \alpha t^4 +(0,4,14)_{\alpha}t^3 + (0,3,21)_{\alpha}t^2 + (0,2,21,14)_{\alpha}t \qquad{}+(0,1,14,21)_{\alpha}= t^7+\alpha(1,2,3,4,5,6)_{t}+ \alpha^2(14,21,21,14)_{t}+ \alpha^3(21,14)_{t}.
\sum_{n =1}^{\infty}\prod_{k=1}^{n}\frac{\mathfrak{m}_{k}(t)}{1 - \mathfrak{m}_{k}(t)}= \frac{2}{1 + 4t^2}\sum_{n =1}^{\infty}\Big[\frac{1}{(n+1)(n+2)}+ (n+1)(n+2) t^2 \Big].
\psi_{p,q}^{(m)}(s) - \psi_{p,q}^{(m)}(s+t) = (\ln q)^{m+1}\sum_{n=1}^{p}\left[ \frac{n^{m}q^{ns}}{1-q^{n}}- \frac{n^{m}q^{n(s+t)}}{1-q^{n}}\right] &=(\ln q)^{m+1}\sum_{n=1}^{p}\left[ \frac{n^{m}q^{ns}(1- q^{nt})}{1-q^{n}}\right]\geq0. \,
A(\tilde{u}) = \frac{\varphi_{0}'(\psi_{0}(\tilde{u}))}{6}\left( \bigtriangleup\psi(\tilde{u}) + \frac{\bigtriangleup\varphi(\psi_{0}(\tilde{u}))\psi_{0}'(\tilde{u})^{2}}{\varphi_{0}'(\psi_{0}(\tilde{u}))}+ \psi_{0}''(\tilde{u})\right),
\int\limits_{Y_{jT_1}\cap\{0\leq s\leq s_0-T_1\}}\frac{\partial u^f}{\partial s}\overline{v^g}dsdy',\ \ \forall f\in H_0^1(\Gamma_{jT_1}),\ \forall g\in H_0^1(\Gamma_{jT_1}),\j=1,2,
\left[(w-2^{-k+1}t)_{-}\right]_{W^{1,p}\left(B_{2R}\right)}^p \leq CH_{4 R}(2^{-k+1}t)\left|B_{4 R}\right| & \leq C\left(\frac{(2^{-k+1}t)^{p}}{R^p}+ \frac{(2^{-k+1}t)^q}{R^{s q}}\right)\left|B_{4 R}\right|, &\leq C R^{N-p}(2^{-k+1}t)^{p},k=k_0, \ldots,m-2.
\left(\begin{array}{cc}1/2 +\alpha 1/2 -\alpha 1/2 -\alpha 1/2 +\alpha \end{array}\right) \left(\begin{array}{cc}1/2 +\beta 1/2 -\beta 1/2 -\beta 1/2 +\beta \end{array}\right) = \left(\begin{array}{cc}1/2+ 2 \alpha \beta 1/2 - 2 \alpha \beta 1/2 - 2 \alpha \beta 1/2 + 2 \alpha \beta \end{array}\right).
\begin{array}{r c l}{{I_{1}( \rho^{2}, 0 )}}&{{=}}&{{\displaystyle ( 4 \pi )^{- 2}( \pi \mu^{2}\rho^{2})^{2 - n / 2}\Gamma ( n / 2 - 2 )}}\\{{}}&{{=}}&{{\displaystyle ( 4 \pi )^{- 2}\left[ \Gamma ( n / 2 - 2 ) - \ln ( \pi \mu^{2}\rho^{2}) + O ( n / 2 - 2 ) \right] ,}}\\ \end{array}
{\varphi_1}_t = d\Delta \varphi_1 x\in \Omega, 0<t<T d\frac{\partial \varphi_1}{\partial \eta}=\gamma - d\frac{\partial\varphi_0}{\partial\eta}x\in M, 0<t< T \varphi_1 =0 x\in\Omega , t=0
\overline{U}_h(x,y,t) :=|\bar{u}_h(x,t)-\bar{u}_h(y,t)|^{p-2}(\bar{u}_h(x,t)-\bar{u}_h(y,t)), begin{align*}2mm]U(x,y,t) :=|u(x,t)-u(y,t)|^{p-2}(u(x,t)-u(y,t)).
\P \{\ell^{(A)}_i > m-i \}\leq \sum\limits_{k=0}^\infty \mu(k) \left(1 \wedge k\P\left\{\exists t > 0:t \leq \sum\limits^{i + S_{t}}_{z = i+1}\frac{1}{A(z)}S_{t}\geq m-i \right\}\right).
\Omega^{(r)}_{\lambda}=\prod_{j=1}^{r}\prod_{i=1}^{r-j}\frac{1}{1- q^{i}\prod_{j \leq k < i+j}t^{m_k(\lambda)}}=\prod_{1 \leq i<j \leq r}\frac{1}{1- q^{j-i}t^{\lambda'_{i}-\lambda'_{j}}}.
\xi_{1}^{( i )}( x ) \equiv \langle \lambda_{m a x}^{( i )}| \mathrm{e x p (}\frac{1}{2}\Phi ) T ( x ) ,{}~ \xi_{2}^{( i )}( x ) \equiv \langle \lambda_{m a x}^{( i )}| E_{i}\mathrm{e x p (}\Psi_{+}) \mathrm{e x p (}\frac{1}{2}\Phi ) T ( x )
E_m(t) =\int_0^{\infty}|\dot{f}(t+r)+\dot{f}(t-r)|^m+|\dot{f}(t+r)-\dot{f}(t-r)|^m\,dr &\approx \int_0^{\infty}|\dot{f}(t+r)|^m+|\dot{f}(t-r)|^m\,dr\approx \int_{-\infty}^{+\infty}|\dot{f}(r)|^m\,dr,
r^{- \left(\kappa_0 + \frac{n-1}{2}\right) \frac{1-c_0\arctan(\ln(r))}{1 + c_0 \pi/2}-1}(1+\ln^2(r))^{-\frac{1}{2}-\frac{1}{2}\frac{c_0(\kappa_0+(n-1)/2)}{1+c_0\pi/2}}\left\|w \right\|_{L^2(A_{r, 2r}^+)}\leq c_0^{-1}C(n,p)\left(1+R_0^{-(\kappa_0+\frac{n+1}{2})}\|w\|_{L^2(A_{R_0/2,R_0}^+)}\right).
u =\prod_{j=1}^{k+1}u_j^*e^{ih_j'}u_j \prod_{j=1}^{k+2}(v_j',w_j')e^{ic}&=u_1^*(\prod_{j=1}^{k+1}e^{i h_j'})u_1\prod_{j=1}^{k}(v_j'',w_j'')\prod_{j=1}^{k+2}(v_j',w_j')e^{ic'}.
\begin{array}{c}{{h^{m ( s )}= \varphi^{m ( s )}+ \eta^{m ( 2 )}\, \chi^{m ( s - 2 )}\; ; \;{\varphi_{l}}^{l m ( s - 2 )}= 0 ,}}\\{{\epsilon^{m ( s - 1 )}= \varepsilon^{m ( s )}+ \eta^{m ( 2 )}\, \zeta^{m ( s - 3 )}\; ; \;{\varepsilon_{l}}^{l m ( s - 3 )}= 0 ,}}\\ \end{array}
\Theta^{+}_{\alpha}=\theta^{i}_{\alpha}u^{+}_{i}+\eta^{+}\tilde{\lambda}_{\alpha}+{\bar{\eta}}^{+}{\bar{\tilde{\lambda}}}_{\alpha}-i\eta^{+}{\bar{\eta}}^{+}{\dot{\theta}}^{i}_{\alpha}u^{-}_{i},
\mathcal{K}^{\Sigma,L}(e_1,e_2)=-\langle R^{\Sigma,L}(e_1,e_2)e_1,e_2\rangle_{\Sigma,L},~~~~\mathcal{K}^{L}(e_1,e_2)=-\langle R^{L}(e_1,e_2)e_1,e_2\rangle_L.
\begin{cases}\begin{array}{ll}\dot{u}=\log\det (u_{\alpha\bar{\beta}})+f(t,z,u)\;\;\; \mbox{on}\;\Omega\times (0,T), u=\varphi \mbox{on}\;\partial\Omega\times [0,T), u=u_0 \mbox{on}\;\bar{\Omega}\times\{0\}. \end{array}\end{cases}
T ( u ) = \partial_{u}\partial_{u}\varphi -{\frac{1}{2}}\left( \partial_{u}\varphi \right)^{2}= \left\{J_{H}^{- 1}, u \right\}= \sum_{i = 1}^{n - 1}\left({\frac{1}{2 ( u - u_{i})^{2}}}+{\frac{c_{i}}{u - u_{i}}}\right) .
\left. \tilde{S}_{\mathrm{\scriptsize ~ r e d}}( \hat{\xi}( p ) , \hat{\eta}( p ) ) = S_{\mathrm{\scriptsize ~ e f f}}( \hat{\xi}, \hat{\eta}) \right|_{\hat{\xi}( p ) = \hat{\eta}( p ) = 0 \atop{| p | > \Lambda}}.
E(t^n):=|c^-c^n|_{L^1_{disc}(\Omega)(t^n)}=\sum_{\stackrel{C_i^{N_n}=(x_i^n-h_i^n/2,x_i^n+h_i^n/2]}{x_i^n~\in~ C_j^{ref}}}\operatorname{vol}(C_i^{N_n})~|c_i^n-c_j^{ref,n}|,
\partial_i(f)g + f\partial_i(g) - (x_i - x_{i+1})\partial_i(f)\partial_i(g) = \frac{(f-s_i(f))g + f(g - s_i(g)) - (f-s_i(f))(g-s_i(g))}{x_i - x_{i+1}}&= \frac{fg-s_i(fg)}{x_i - x_{i+1}}.
Z = \int{\cal D}A_{\mu}{\cal D}\pi^{\mu}{\cal D}\phi{\cal D}\pi_{\phi}{\cal D}\theta{\cal D}\pi_{\theta}{\cal D}\rho{\cal D}\pi_{\rho}\prod_{i , j = 1}^{4}\delta ( \tilde{\Omega}_{i}) \delta ( \Gamma_{j}) \det \mid \{\tilde{\Omega}_{i}, \Gamma_{j}\}\mid e^{i S},
\sum_{1\le i\le j\le t}{q_{ij}m_{ij}}=\Bigl(\frac{p-1}{r}\bigl(p^sr+p^s\bigr)-\bigl(\frac{p-1}{r}-1\bigr)p^sr\Bigr)+\frac{p^{n-1}-p^s}{r}\bigl(2(p-1)-(p-2)\bigr) &=\frac{p-1}{r}p^s+p^sr+\frac{p^n-p^{s+1}}{r}=\frac{p^n+p^s(r^2-1)}{r}=k_0(B)
\varphi^{2}=x^{1}\sin\left(\tfrac{\zeta_{[-1]}^{1}+x^{3}}{2}\right)-x^{2}\cos\left(\tfrac{\zeta_{[-1]}^{1}+x^{3}}{2}\right) \delta(\varphi^{2}) =x^{1}\sin(\bar{u}^{2})-x^{2}\cos(\bar{u}^{2})\,,
\begin{array}{l}xy^3dx=d(\frac18x^2y^3-\frac{1}{14a}xy^3-\frac{1}{126a^2}y^3)+(\frac{1}{42a^2}+\frac{3}{14a}x-\frac38x^2)ydH +[(\frac{1}{7a}+\frac32x)H-\frac{1}{42a^2}x+(\frac{1}{21a^2}-\frac{2}{7a})x^2+(\frac{1}{2a}-\frac38)x^3]ydx.\end{array}
\frac{\partial f}{\partial x}= \sum_{\alpha = 0}^{+\infty}\left( \frac{\partial f_{\alpha}}{\partial x}+ \frac{\partial u}{\partial x}f_{\alpha - 1}+ \frac{1}{2}\frac{\partial \theta}{\partial x}f_{\alpha - 2}\right) \mathcal{H}^{[\theta]}_{\alpha}\left( \frac{\xi - u}{\sqrt{\theta}}\right),
\|f_\infty-f_m\|_\infty\le \sum_{i=m}^{\infty}\|f_{i+1}-f_i\|_\infty \le \sum_{i=m}^{\infty}\frac{2c}{2^{d+\varepsilon i}}= \frac{c\,2^{1+\varepsilon}}{2^{d+\varepsilon m}(2^\varepsilon-1)}.
\gamma{\cal P}_{1 a}= 0 , \; \gamma{\cal P}_{2 a}= f_{\; \; a b}^{c}{\cal P}_{2 c}\eta_{2}^{b}, \; \gamma{\cal P}_{1 i}= 0 , \; \gamma{\cal P}_{2 i}= 0 , \; \gamma{\cal P}= 0 .
\bar\lambda_2 \leq \frac{4\pi^2}{D^2}-\frac{K}{2}- \frac{(\pi^2-\tfrac 32 )D^2K^2}{48\pi^2}-\frac{(\frac{15}{2}-5\pi^2+\pi^4)D^4K^3}{480\pi^4}- \frac{17(4\pi^6-42\pi^4+210\pi^2-315)D^6K^4}{322560}.
\int_{U}|f|^2e^{-k\varphi-r_1\psi}\le \left(\int_{U}|f|^{2q_1}e^{-q_1k\varphi}\right)^{\frac{1}{q_1}}\cdot\left(\int_Ue^{-q_2r_1\psi}\right)^{\frac{1}{q_2}}<+\infty,
\begin{aligned}\lambda-\mathcal{L}[u^q]+|Du^q|^m \geq \lambda + \frac12 C_0^{m(q-1)}\Psi^{m(q-1)}f + C_1^{q-1}\Psi^{q-1}(f-\lambda) +(C_0^{q-1}-C_1^{q-1})\Psi^{q-1}u. \end{aligned}
\sum_j\lambda_j(B_1)^2=|B_1|_2^2=|( -B)^{-1/2}(A-B)( -B)^{-1/2}|_2^2 \leq \|( -B)^{-1/2}\|^2|A-B|_2^2\|( -B)^{-1/2}\|^2=\|( -B)^{-1}\|^2|A-B|_2^2.
\frac{1}{2}\left[ ( D - 2 ) \frac{f^{\prime \prime}}{f}+ \frac{( D - 2 ) ( D - 5 )}{4}\left( \frac{f^{\prime}}{f}\right)^{2}\right] = \alpha^{2}\frac{( D - 2 )^{2}+ ( D - 2 )}{8}
\frac{1}N \sum_{k = 1}^{P}\sin ( \frac{2 \pi k j}N ) [ \sum_{a = 1}^{3}\sin ( \frac{2 \pi k t_{a}}N ) ] T r \gamma_{k , 9}= \sum_{a = 1}^{3}( \delta_{2 j , t_{a}}- \delta_{2 j , - t_{a}})
\begin{aligned}N_n/S_n = \{x_{m(n)}\cdots x_2x_1Z_{m(n)}\cdots Z_2Z_1\mid x_r \in L_r\}&= \{x_{m(n)}Z_{m(n)}\cdots x_2Z_2x_1Z_1\mid x_r \in L_r\}= (L_{m(n)}/Z_{m(n)}) \times \cdots \times (L_1/S_1) &\simeq (L_1/S_1) \times \cdots \times (L_{m(n)}/Z_{m(n)}).\end{aligned}