harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\int_{0}^{\infty}dx\, \left(x^{2}+u^{2}\right)^{\nu -1}e^{-px}=\frac{\sqrt{\pi}}{2}\left(\frac{2u}{p}\right)^{\nu -\frac{1}{2}}\Gamma \left(\nu \right)\left[H_{\nu -\frac{1}{2}}\left(up\right)-N_{\nu -\frac{1}{2}}\left(up\right)\right],
X^{\mu}( \sigma , \tau ) = q^{\mu}+ 2 \alpha^{\prime}p^{\mu}\tau + i \sqrt{\alpha^{\prime}}\sum_{n \neq 0}\frac{1}{n}[ \alpha_{n}^{\mu}\; e^{- i n ( \tau - \sigma )}+ \tilde{\alpha}_{n}^{\mu}\; e^{- i n ( \tau + \sigma )}]
\xi := \big|\{i\,:\, w_i \neq x_i^\star = 0\}\big| - \big|\{i\,:\, w_i = x_i^\star \neq 0\}\big|\,, \overline{h}:= \big|\{i\,:\, x_i^\star > 0, \,\,x_i^\star > w_i\}\cup \{i\,:\, x_i^\star < 0, \,\,x_i^\star < w_i\}\big|\,.
(*) =P_N(\phi\in[0,a]\,on\, A) &=\prod_{i=1}^{|A|}P_N(\phi_{x_i}\in[0,a] |\phi_{x_{i+1}},\ldots,\phi_{x_{|A|}}\in[0,a]) &=\prod_{i=1}^{|A|}\int_{[0,a]^{A_i}}P_N(\phi_{x_i}\in[0,a] |\phi=\psi \, on \, A_i)g_i(\psi)d\psi
\sum_{\substack{i_1+\cdots+i_t=n i_1,\ldots,i_t\ge 0}}\prod_{j=1}^t \beta_j^{i_j}=\sum_{i=0}^n \tbinom{i+q-1}{q-1}\beta_1^i \sum_{\substack{i_{q+1}+\cdots+i_t=n-i i_{q+1},\ldots,i_t\ge 0}}\prod_{j=q+1}^t \beta_j^{i_j}.
\nabla^{\perp\psi}_{E_a}\nabla^{\perp\psi}_{E_a}H^{\psi}=\frac{p}{p+q}\nabla^{\perp}_{E_a}\nabla^{\perp}_{E_a}H_1+\frac{q}{p+q}B^j(\nabla^{\mathbb{C}P^p}_{E_a}E_a,H_2)-\frac{q}{p+q}H_2.
\lim_{\varepsilon\to 0}\int_{0}^{T}\int_{\mathbb{R}^d}|\eta_t-\eta_t^\varepsilon|d|D^{s}b_t|dt+\lim_{\varepsilon\to 0}\int_{0}^{T}\int_{\mathbb{R}^d}|\xi_t-\xi_t^\varepsilon|d|D^{s}b_t|dt=0.
\begin{aligned}c^4_{\lambda,m}(\xi_1,\xi_2,\xi_3,\xi_4) = \frac{i}4 \left( - (c_{\lambda}(\xi_1,\xi_2,\xi_3) + c_{\lambda}(\xi_1,\xi_4,\xi_3))+ (\bar c_{\lambda}(\xi_2,\xi_3,\xi_4)+ \bar c_{\lambda}(\xi_2,\xi_1,\xi_4)) \right).\end{aligned}
L_{dn+k}(x;c)=\sum_{j=0}^{n-1}P_j(c)\left\{\sum_{r=0}^{d-1}\frac{P_{dj+r}(x)}{\left\langle u_{r},P_{dj+r}P_j\right\rangle}\right\}+P_n(c)\left\{\sum_{r=0}^{k}\frac{P_{dn+r}(x)}{\left\langle u_{r},P_{dn+r}P_n\right\rangle}\right\}.
h(x,t) = \sum_{j=1}^N h_j(x,t) = \sum_{j=1}^N \sum_k \sum_{q,m=0}^{L-1}\big[b^j_{(q,m),k}\,Hg(t - (q-k)T) \hskip-.25in e^{-2\pi i m(q-k)/L}\,\Phi^j_{(q,m)}(t,x-t+(q-k)T)\big] = \sum_k \sum_{q,m=0}^{L-1}Hg(t - (q-k)T) \hskip.5in \widetilde{\Phi}_{(q,m),k}(t,x-t+(q-k)T)
\left\| y_{n+1}^{\frac{1-2s}{2}}w \right\|_{L^2(B_{r}^+(\bar{y}))}= \ \max\limits_{y\in M\times \{0\}}\left\| y_{n+1}^{\frac{1-2s}{2}}w \right\|_{L^2(B_{r}^+(y))}=: \tilde{C}_r
\Delta_{\{rB\}\cup D}= \sup_{P \in \Lambda_{\{rB\}\cup D}}\rho(|P|) = \sup_{Q \in \Lambda_D}\; \sup_{P \in \Lambda_{\{rB\}}(Q)}\rho(|P|)= \sup_{Q \in \Lambda_D}\Delta_{\{rB\}}(Q),
\dot{B}^s_{pr}:= \left\{f \in \mathcal{S}'(\mathbb{R}^3_x)/ \mathcal{P}\ \vert \ \Vert f \Vert_{\dot{B}^s_{pr}}:= \left\Vert (2^{qs}\Vert \dot{\Delta}_q f \Vert_{L^p_x})_{q \in \mathbb{Z}}\right\Vert_{\ell^r}< \infty \right\}.
\varphi (u \wedge v) \big( (\vec x_1, \vec x_2), \dots , (\vec x_{2n-1}, \vec x_{2n}) \big) = \big( m_{u,v}(\vec x_1, \vec x_2), \dots , m_{u,v}( \vec x_{2n-1}, \vec x_{2n}) \big),
\begin{array}{l}\displaystyle X=\frac{1}{2}(T^{(1,1,2)}-T^{(1,2,2)})+\frac{1}{3}(l_1-l_2)\ , begin{align*}0.8em]\displaystyle Y=T^{(1,1,2,2)}+\frac{3}{2}(T^{(1,1,2)}+T^{(1,2,2)})-\frac{1}{12}(T^{(1,2)})^2-\frac{5}{12}T^{(1,1)}T^{(2,2)}+\frac{5}{2}T^{(1,2)}\,.\end{array}
Z_{N}^{i n f}( x_{1}, \cdots , x_{M}) \equiv ( x_{1}+ x_{2}\cdots + x_{M})^{N}= \sum_{\lambda \atop{| \lambda | = N}}{\frac{N !}{\lambda_{1}! \cdots \lambda_{M}!}}~ m_{\lambda}( x_{1}, \cdots , x_{M}) \, \, \, \, .
\lambda =\frac{n (a^2-a (b-1) b+b^3-2 b \ell)+2 (b-\ell)(a^2+a b-b (b+\ell))}{(a-b)^2 (a+b)}, \mu =\frac{b n (-a b+a+b^2+b-2 \ell)+2 b (a-\ell) (b-\ell)}{(a-b)^2 (a+b)}.
(1-\pi_{ij})(1-\pi_{i'j'}) = (1 - \bar\alpha_i)(1 - \bar\beta_j)(1 - \bar\alpha_{i'})(1 - \bar\beta_{j'}) & = (1 - \bar\alpha_i)(1 - \bar\beta_{j'})(1 - \bar\alpha_{i'})(1 - \bar\beta_j) = (1-\pi_{ij'})(1-\pi_{i'j})
\theta^{\ast}:\Omega\rightarrow D_{np}=\left\langle s,t:s^{2}=t^{np}=1,sts=t^{-1}\right\rangle \theta^{\ast}(x_{1}^{\prime}) =t^{2m+1}\theta^{\ast}(c_{0}^{\prime})=t^{2m}s\theta^{\ast}(c_{1}^{\prime})=s,\text{}\theta^{\ast}(c_{2}^{\prime})=st^{-2m-2}
\frac{\partial}{\partial M_{x^1,y^1}}\ldots \frac{\partial}{\partial M_{x^p,y^p}}\det \limits_{i,j = 1}^N M_{i,j}= \det \limits_{i,j = 1}^N M_{i,j}\det_{i \in Y, j \in X}M^{-1}_{i,j},
a_{j_1,l}-a_{j1,l}^{*}=0, \;for\;i_1<l<j_1 a_{j_1,i_1}-a_{j1,i_1}^{*}=1, a_{j_1,l}-a_{j1,l}^{*}=b_{i_1,l}, \;for\;l<i_1 \gamma_{j_1}-\gamma_{j_1}^{*}=\varepsilon_{i_1}.
\Big\langle \alpha \,;\, \prod_{i=1}^{n}\tau_{0}(\gamma_i) \Big\rangle_{g,\beta}^{A,}= \Big\langle \alpha \,;\, \prod_{i=1}^{4}\tau_0(\xi_i) \cdot \prod_{i=1}^{n}\tau_{0}(\gamma_i) \Big\rangle_{g,\beta}^{A,}\,,
Y_{n_{l_{m}}}- \exp\left\{\sum_{i=1}^{k}\frac{2\mu_{i}W_{n_{l_{m}},i}- \mu_{i}^2}{2\sigma_{i}^2}\right\}\stackrel{d}{\to}H_{1}- \exp \left\{\sum_{i=1}^{k}\frac{2\mu_{i}H_{i+1}- \mu_{i}^2}{2 \sigma_{i}^2}\right\}.
\left\langle{a(x,x^5) a(y,y^5)}\right\rangle = \int_{k5}{i \over k^2 - (k^5)^2}\, e^{-ik\cdot (x-y)}(e^{ik^5(x^5-y^5)}+ P e^{ik^5(x^5+y^5)}) \ ,
\overline{g}_{\mu\nu}=\left[ \begin{array}{cc}1 + l(x^{0})^{2}\overline{\lambda}(x^{0},x^{1})^{2}l(x^{0})^{2}l(x^{0})^{2}\overline{\lambda}(x^{0},x^{1})^{2}l(x^{0})^{2}\end{array}\right].
|(Y^*\tilde{N}_pV^{\dagger})_{2u,k-s}|=|(Y^*\tilde{N}_p(E+X)W^{\dagger})_{2u,k}| \leq |(Y^*\tilde{N}_pW^{\dagger})_{2u,k}|+|(Y^*\tilde{N}_pXW^{\dagger})_{uk}| &<\frac{2^7p^4\tau(k)}{r}+\frac{2^{15}p^{4p+9}s^{4p-1}}{(r-8p^2s^{4p-1})r}.
K(z,z'')= \Big(\int_{r'<\frac{r}{2}}+\int_{\frac{r}{2}\leq r'\leq2r}+\int_{r'>2r}\Big)G(z,z')Q(z',z'')\;d\mu(z') = K_{1,1}(z,z'')+K_{1,2}(z,z'')+K_{1,3}(z,z'').
b_{0}(\sqrt{x}) = \lim_{z \to 0}i \sqrt{x}z \left[ \partial_{z}\Phi(axz;x,a)\Phi^{-1}(axz;x,a) \right]_{12}, & b_{1}(\sqrt{x}) = \lim_{z \to -a^{-1}}i \sqrt{x}(z+a^{-1}) \left[ \partial_{z}\Phi(axz;x,a)\Phi^{-1}(axz;x,a) \right]_{12}.
\sum_{i\geq 0}\phi_i\bigg( \sum_{j\geq 0}m\rq_{1,j}(x,y)t^j \bigg)t^i=\sum_{i\geq 0}m_{1,i}\bigg( \sum_{j\geq 0}\phi_j(x)t^j,\sum_{k\geq 0}\phi_k(y)t^k \bigg)t^i, \sum_{i\geq 0}\phi_i\bigg( \sum_{j\geq 0}m\rq_{2,j}(x,y)t^j \bigg)t^i=\sum_{i\geq 0}m_{2,i}\bigg( \sum_{j\geq 0}\phi_j(x)t^j,\sum_{k\geq 0}\phi_k(y)t^k \bigg)t^i.
\begin{aligned}\int |\Delta d|^2dx+\int_s^t\int |\nabla \Delta d|^2dxd\tau &\le 2\int |\Delta d|^2(s)dx +C\int_s^t\int (|\nabla d|^2 |\nabla^2 d|^2+|\nabla d|^2|\nabla u|^2+|u|^2 |\nabla^2 d|^2+|\nabla d|^6)dxd\tau.\end{aligned}
y^{2}= x^{2\tilde{n}_{c}}\left[ \prod_{k=1}^{n_{c}-\tilde{n}_{c}}(x-\Phi_{k})^{2}+ 4 \Lambda^{2n_{c}-n_{f}}x^{n_{c}-\tilde{n}_{c}}\right] = x^{2{\tilde n}_c}( x^{n_c -{\tilde n}_c}- \Lambda^2 )^2.
\nabla_{\mu}{\cal J}^{\mu}=0\, ,\hspace{1cm}{\cal J}_{\mu}= 2\partial_{\mu}{\cal M}{\cal M}^{-1}=2\left(\begin{array}{cc}\frac{1}{2}j_{\mu}^{(\varphi)}j_{\mu}& j_{\mu}^{(0)}-\frac{1}{2}j_{\mu}^{(\varphi)}\end{array}\right)\, ,
S_{\alpha}\left( t\right) \phi \left( s\right) =z\left( t\right) \varphi_{1}\left( s\right) +\overset{\_}{z}\left( t\right) \overset{\_}{\varphi}_{1}\left( s\right) +w_{\alpha}\left( s,z\left( t\right) ,\overset{\_}{z}\left( t\right) \right) .
\begin{array}{ll}M^2\sinh^2 (2\lambda \theta) /(P_{\rm T}( \theta)P_{\rm T}(-\theta)) begin{align*}3mm] = k^2\sinh^2(2\lambda \theta)[\cos^2(\xi) + \sinh^2(\lambda \theta) + k^2\sinh^2(\lambda \theta)\cosh^2(\lambda \theta)]^{-1}. \end{array}
G_{( \sigma )}( g_{s}( z ) , z^{\prime}) = ( - 1 )^{2 l_{2}- 1}( c_{r}z + d_{r})^{- 1}\left( G_{( \sigma )}( z , z^{\prime}) + \sum_{F_{r}= \mu_{r}, \nu_{r}}\tilde{Y}_{\sigma , F_{r}}( z ) \Phi_{\sigma ,{F_{r}}}^{( 0 )}( z^{\prime}) \right)
(var^n[y_n(t,x)])^{\frac{1}{2}}\leq\dfrac{1}{\sqrt{M}}(var^{n-1}[y_n(t,x)])^{\frac{1}{2}}+\dfrac{C_f}{\sqrt{M}}\sum_{j=1}^Q w_{n,j}\left(E^n[(y_{n-1}-y_{n-2})^2(t_{n,j},x+W_{t_{n,j}-t}^{(n)})]\right)^{\frac{1}{2}}
I^{s+1}:= \{(-d_{A_{\infty}}\xi, L_{u_{\infty}}\xi)\,|\, \xi \in H^{s+1}(\Sigma, (P)) \}V^{s,s+1}:= \{(a,\hat{u}) \in H^s\times H^{s+1}\,|\, d_{A_{\infty}}^* a + L_{u_{\infty}}^* \hat{u}= 0 \}.
C(f,\rho)t\Big(\sum_{p=1}^P{\frac{(2^{p+1}\ell_0)^{\alpha-1}}{n^{2-\alpha}(2^p\ell_0)^{2}}}\Big)^2\leq C(f,\rho)t \frac{(2^P\ell_0)^{\alpha-1}}{n^{2-\alpha}}.
i\partial_{t}\psi-\Delta^{2}\psi+\mu|\psi|^{q-2}\psi+|\psi|^{p-2}\psi=0, \quad\psi(0,x)=\psi_{0}\in H^2(\mathbb{R}^N),\quad(t,x)\in\mathbb{R}\times\mathbb{R}^{N},
F_{n+1}(x) = \int_{0}^{\infty}dA(t)\int_{0}^{x+t}dW_{n}(u)\left[\bar{G}(u) \int_{t-u}^{x+t-u}dB(s) + G(u) + \bar{G}(u)\int_{0}^{t-u}V(x+t-u-s) dB(s) \right] + \int_{0}^{\infty}dA(t)\int_{0}^{t}dW_{n}(u)\left[ -G(u)\bar{V}(x+t-u) \right].
\{\begin{matrix}j_{1}&j_{2}&j_{3}\\j_{4}&j_{5}&j_{6}\end{matrix}\}=\{\begin{matrix}j_{4}&j_{5}&j_{3}\\j_{1}&j_{2}&j_{6}\end{matrix}\}=\{\begin{matrix}j_{1}&j_{5}&j_{6}\\j_{4}&j_{2}&j_{3}\end{matrix}\}=\{\begin{matrix}j_{4}&j_{2}&j_{6}\\j_{1}&j_{5}&j_{3}\end{matrix}\}.
w(z)= \begin{cases}\left( 1+(\frac{1}{2}g_S+g_L)z^{-1}\right) \left( 1+g_S(1+2z)^{-1}\right) \ \emph{r=bc}\ \ \emph{r=t}, \frac{1}{2}z^{-1}\left( \frac{1}{2}+g_S(1+2z)^{-1}\right) \ \emph{r=cs},\end{cases}
\begin{aligned}\mathbf{R}_1 =\left \{(s,t): 0\le s\le t\le T, t\ge4 \right\}, \mathbf{R}_2 =\left\{(s,t): 0\le s\le t\le T, \, t < 4, \, |D_t^{(0)}-D_s^{(\ell)}|^2 \leq 8\right \}, \mathbf{R}_3 =\left\{(s,t): 0\le s\le t\le T, \, t < 4, \, |D_t^{(0)}-D_s^{(\ell)}|^2 > 8 \right\}. \end{aligned}
d[ca_{1,i}b_{1,j}] =\min\{d(ca_{1,i}b_{1,j}),\alpha_{i},\beta_{j}\}\quad d^{\prime}[ca_{1,i}b_{1,j}] =\min\{d(ca_{1,i}b_{1,j}),\alpha_{i},\beta_{j}^{\prime}\}\,,
\partial_{\mu}[f^{\mu}-[\frac{\partial}{\partial(\partial_{\mu}\varphi)}L]Q[\varphi]-2[\frac{\partial}{\partial(\partial_{\mu}\partial_{\nu}\varphi)}L]\partial_{\nu}Q[\varphi]+\partial_{\nu}[[\frac{\partial}{\partial(\partial_{\mu}\partial_{\nu}\varphi)}L]Q[\varphi]]-\cdot\cdot\cdot]\approx0.
p_r(s) =[x^r](1+(q-1)x)^{d-s}(1+(q-1)x-qx)^s &=[x^r]\sum_h \binom{s}{h}(1+(q-1)x)^{d-h}(-qx)^h &=\sum_h \left([x^h]\binom{s}{h}(-qx)^h\right)\left([x^{r-h}](1+(q-1)x)^{d-h}\right) &=\sum_h (-q)^h(q-1)^{r-h}\binom{d-h}{r-h}\binom{s}{h}.
-cb_{1}^{-1}b_{3}+r_{3}b_{1}^{-2}b_{4}+\frac{1}{2}b_{1}^{-1}b_{4}\left(b_{1}^{-1}-1\right)\left(c+1\right) &=-cb_{1}^{-1}b_{3}+r_{3}+\frac{1}{2}b_{1}\left(b_{1}^{-1}-1\right)\left(c+1\right)=0,
{\Phi}= \left( \begin{array}{c}{l_{I}\cdot{\bf e}_{\mu}^{( R )}}\\{{\bar{k}}_{i}\cdot{\bf e}_{\mu}^{( R )}}\\{k^{j}\cdot{\bf e}_{\mu}^{( R )}}\\ \end{array}\right) ={\frac{1}{2}}\left( \begin{array}{c}{- A E^{*}}\\{D E^{*}}\\{E^{*}}\\ \end{array}\right) \in M ( 2 8 , 6 ,{\bf R})
\mbox{var}(\hat{\mu}_j(n)) = \frac{1}{16(n-3)^2}\left[ (n-3) \left( \tau_j^2 +12 \tau_j \mu_j +52 \mu_j^2 \right) \right. & \left. + 2(n-4) \left( \tau_j \mu_j +22 \mu_j^2 \right) + 2 (n-5) \mu_j^2. \right]
\underline{f}=\underline{\underline{K}}\underline{d}\Rightarrow[\begin{matrix}f_{1}\\f_{2}\\f_{3}\end{matrix}]=[\begin{matrix}K_{11}&K_{12}&K_{13}\\K_{21}&K_{22}&K_{23}\\K_{31}&K_{32}&K_{33}\end{matrix}][\begin{matrix}d_{1}\\d_{2}\\d_{3}\end{matrix}]
{\cal L}_{\mathrm{d a u g h t e r}}= \frac{1}{2}\sum_{i = 1}^{2 N}\dot{x}_{i}^{2}- \frac{1}{2}\sum_{i = 1}^{2 N}\left\{- 2 A x_{i}^{2}+ x_{i}^{6}\right\}- \frac{1}{2}\sum_{\stackrel{i , j = 1}{j \neq i}}^{2 N}\frac{1}{( x_{i}- x_{j})^{2}},
d=\frac{\binom{N_R-1}{r}\binom{N_T}{t}\binom{N_R-r-1}{t-1}t}{\binom{N_R-1}{r}\binom{N_T}{t}\binom{N_R-r-1}{t-1}t+\binom{N_R-1}{r+1}\binom{N_R-r-2}{t-1}\binom{N_T}{t-1}}
\sum_{\pi\in S_n}\pi\frac{1}{\prod_{1\le i<j\le n}\vartheta(z_i\pm z_j)}D'_1(z_1)\cdots D'_n(z_n)=\frac{1}{\prod_{1\le i<j\le n}\vartheta(z_i\pm z_j)}\det_{1\le i,j\le n}D'_i(z_j)
H_{(\mathbb P^3,K3)}(y)=e^{\frac{H\log y}{2\pi i}}\sum_{d\geq 0}y^d \frac{\Gamma(1+\frac{4H}{2\pi i}+4d)}{\Gamma (1+\frac{H}{2\pi i}+d)^4}[\textbf{1}]_{4d},
\sum_{n=0}^{\infty}p_{-2}\left(5^{2j-1}n+\dfrac{7\times5^{2j-1}+1}{12}\right)q^{n}=\sum_{l=1}^{\infty}a(2j-1,l)q^{l-1}\dfrac{E_{5}^{6l-2}}{E_{1}^{6l}}, \sum_{n=0}^{\infty}p_{-2}\left(5^{2j}n+\dfrac{11\times5^{2j}+1}{12}\right)q^{n}=\sum_{l=1}^{\infty}a(2j,l)q^{l-1}\dfrac{E_{5}^{6l}}{E_{1}^{6l+2}}.
\begin{array}{l l l}{{z^{1}z^{2}= p^{2}z^{2}z^{1}\; , \qquad}}&{{z^{2}z^{3}= p^{2}z^{3}z^{2}\; , \qquad}}&{{z^{3}z^{4}= p^{2}z^{4}z^{3}\; ,}}\\{{z^{4}z^{1}= p^{2}z^{1}z^{4}\; , \qquad}}&{{z^{4}z^{2}= z^{2}z^{4}\; , \qquad}}&{{z^{1}z^{3}= z^{3}z^{1}\; .}}\\ \end{array}
b_1 \,{1 \over x - a_1 + b_2\, \displaystyle{1 \over \, \ddots \, \displaystyle{\hbox{}\atop \displaystyle{\hbox{}\atop \displaystyle{1 \over x - a_n + b_{n+1}\, \displaystyle{1 \over \, \ddots \,}}}}}}
\Phi_{k+1}(G_k) = \langle [y,x, \overset{2\nu(k)+1}{\ldots}, x,y], [y,x, \overset{2\nu(k)+3}{\ldots}, x,y], \ldots, [y,x, \overset{p^k-2}{\ldots}, x,y]\rangle \Phi_{k+1}(G_k)\cong C_p^{\, (p^{k+1}-3p^k-p+3)/(2(p-1))}.
C^{'}_{FTN}(\alpha,B) = \left[2B-(1+\alpha)W\right] \log_{2}\left( 1 + \frac{P}{WN_{0}}\right) +2 \int_{0}^{\alpha W}\log_{2}\Bigg(1+ \frac{2P}{2WN_{0}+P\left[1+\cos\left(\frac{(f-\alpha W)\pi}{\alpha W}\right)\right]}\Bigg) df
S^{~\bullet}_1=\beta\left.{\partial{\tilde{W}_1^{~\bullet}(\beta,\alpha,y,\ldots)}\over\partial\beta}\right|_{\alpha,y,\ldots}+\alpha\left.{\partial{\tilde{W}_1^{~\bullet}(\beta,\alpha,y,\ldots)}\over \partial\alpha}\right|_{\beta,y,\ldots}-W_1^{\bullet}~~~.
\int_M \nabla_ih_{jk}\nabla_jh_{ki}dV_g =-\int_M h_{jk}\nabla_i\nabla_jh_{ki}dV_g &=-\int_M h_{jk}(\nabla_j\nabla_ih_{ki}-\mathring{R}(h)_{jk}+R_{jl}h_{lk})dV_g &=\int_M |h|^2+\langle \mathring{R}h,h\rangle- R_{jl}h_{jk}h_{lk}dV_g
{\cal A}_{fi\,\,({\bf A}^{2})}^{(2)\,\,Reg}=\frac{e^{2}\Phi^{2}}{4(2\pi)^{3}}\frac{1}{w_{p}}\left\{-\ln (\lambda^{2})-\ln (p^{2})-\ln [2(1-\cos\theta_{S})]+2(\ln 2-\gamma )\right\}\;.
m: \bigsqcup_{(i,k)\in \mathcal{D}}\varphi_{i,k}\to \mathbb{Z}[Y^{\pm}_{j,l}]_{(j,l)\in \mathcal{D}}; p \mapsto m(p)=\prod_{(j,l)\in C^{+}_{p}}Y_{j,l}\prod_{(j,l)\in C^{-}_{p}}Y^{-1}_{j,l}.
B_\ell^+ E^\gamma\Omega_\lambda=\sum_{j=\ell}^n\sum_{t=1}^{j-\ell+1}\sum_{J\in\mathcal{I}_{\ell j}(t)}(-1)^{t-1}\prod_{u=1}^{t-1}\gamma_{j_{u+1}j_u}E^{\gamma - e_J}B_j^+\Omega_\lambda,
P_{n}\overset{\mathrm{def}}{=}\left\{|\hat{f}_{t}(iy)|\le \psi(n)y^{-\beta}y\in[0, 2^{-n}], t\in[0,1] \right\}, &H_{n}\overset{\mathrm{def}}{=}\left\{|\hat{f}_{t}(iy)|\le \psi(n)y^{-\beta}y\in[0, 1/\sqrt{n}], t\in[0,1] \right\},
\log S =\sum_{p\leq R}e_p\log p &=\sum_{p\leq R}\sum_{n=1}^{O\left(\frac{\log (r+R)}{\log p}\right)}\sum_{\substack{i=1 f(r+i)\equiv 0 \mod p^n}}^{R}\log p &=\sum_{p\leq R}\frac{R\log p}{p}+O\left(\frac{R\log (r+R)}{\log R}\right) &=R\log R+O\left(\frac{R\log (r+R)}{\log R}\right).
L_{m,n}^{III,(\alpha)}(x)={}\frac{x(-1)^{n-m-1}n\det\big(M_n^{III}\big)}{m!(n-m-1)!\prod_{1\leq i<j\leq n-1}\big(Z^{(\alpha)}_{j,m,n}-Z^{(\alpha)}_{i,m,n}\big)}&+(m+1)\binom{n-m+\alpha}{n-m-1}\binom{m-\alpha-1}{m+1}.
K_\rho Z_\rho^{*n}1= S^{*n}K_\rho 1=S^{*n}P(\rho)=S^{*n}P(1-|b|^2) = -S^{*n}P(|b|^2)=-S^{*n}T_{\bar b}b=-T_{\bar b}S^{*n}b.
\mathrm{m}_q = \ \frac{1}{2}c_q(n) \int_M \left(L_{(q)}+ (2q-1)! \langle \left[ T_{(2q-1)\alpha}, A_{\beta}\right] \cdot e_{\alpha}^{\top}, e_{\beta}^{\top}\rangle \right) \frac{1}{\sqrt{G}}dM{}\ + \frac{1}{2}(2q-1)! c_q(n) \int_{\Sigma}\left( \frac{|Df|^2}{1 + |Df|^2}\right)^q H_{(2q-1)}d\Sigma.
\mathtt{h}_{a, \nu}(z) = z - \frac{\mathrm{\Gamma}{(a\nu+1)}}{1!2^2\mathrm{\Gamma}{(a+a\nu+1)}}a^az^2 + \frac{\mathrm{\Gamma}{(a\nu+1)}}{2!2^4\mathrm{\Gamma}{(2a+a\nu+1)}}a^{2a}z^3 + \cdots & + (-1)^k\frac{\mathrm{\Gamma}{(a\nu+1)}}{k!2^{2k}\mathrm{\Gamma}{(ak+a\nu+1)}}a^{ak}z^{k+1}+\cdots
\left(\begin{array}{c}f_{\mathcal{\tilde{A}}}(2n) f_{\mathcal{\tilde{A}}}(2n-1) \vdots f_{\mathcal{\tilde{A}}}(2n-a_z)\end{array}\right)=M_{\mathcal{\tilde{A}}}\left(\begin{array}{c}f_{\mathcal{\tilde{A}}}(n) f_{\mathcal{\tilde{A}}}(n-1) \vdots f_{\mathcal{\tilde{A}}}(n-a_z)\end{array}\right).
\left({}^{c}D_{0+}^{\alpha,\alpha',\beta,\beta',\gamma}f\right)(x)=\left(I_{0+}^{-\alpha',-\alpha,-\beta'+[\operatorname{Re}(\gamma)]+1,-\beta,-\gamma+[\operatorname{Re}(\gamma)]+1}f^{([\operatorname{Re}(\gamma)]+1)}\right)(x)
f_{2}= \frac{\operatorname{s i n h}\xi}{2 \operatorname{c o s h}^{3}\xi}- \frac{3 \operatorname{s i n h}^{3}\xi}{2 \operatorname{c o s h}^{5}\xi}+ C_{2}\frac{\operatorname{s i n h}^{2}\xi}{\operatorname{c o s h}^{4}\xi}.
I_{\beta}(D,s,m)= \frac{m^{D-2s}}{(2\pi^{1/2})^D\Gamma(s)}\left[\Gamma(s-\frac{D}{2}) + 4 \sum_{n=1}^{\infty}\left( \frac{2}{mn\beta}\right)^{D/2-s}K_{D/2-s}(mn\beta) \right]
r(f) = |C_0^*| + |P_1^*| + |C_1^*| + |P_0^*| = |C_0^*| + |P_1^*| + |C_1^*| + |N_0^*| + |P_0^* \setminus N_0^*| = r(f_0) + |C_1^*| + |P_0^* \setminus N_0^*|.
\begin{aligned}\|u_t\|_{L^2}=\|\mu\Delta u+(\mu+\nu)\nabla{\rm div}u-\nabla \varrho+S_2\|_{L^2}&\lesssim \|\nabla^2 u\|_{L^2}+\|\nabla \varrho\|_{L^2}+\delta (\|\nabla \varrho\|_{L^2}+\|\nabla (u, B)\|_{H^1}) &\lesssim (1+t)^{-\frac{5}{4}}+(1+t)^{-\frac{5}{4}}&\le C(1+t)^{-\frac{5}{4}}.\end{aligned}
\left\langle \frac{1}{2 N}\mathrm{T r}\, H_{S}^{2}\right\rangle - \left\langle \frac{1}{2 N}\mathrm{T r}\, H_{S}\right\rangle^{2}= \left\langle \frac{1}{N}\mathrm{T r}\, H_{0}^{2}\right\rangle - \left\langle \frac{1}{N}\mathrm{T r}\, H_{0}\right\rangle^{2}- \left\langle \frac{1}{2 N}\mathrm{T r}\, ( \vec{\sigma}( \vec{A}\times \vec{A}) \vec{\sigma}( \vec{A}\times \vec{A}) ) \right\rangle
|E[U_n(\theta)]| =\left|\sum_{I,J}K(I,J_{-\theta})\sum_{m=1}^M\rho_m\int_{I\cap J_{-\theta_m}}\sigma_1(t)\sigma_2(t+\theta_m)dt\right| &\leq\max_{1\leq m\leq M}|\rho_m|\sum_{I,J}\int_{I\cap J_{-\theta_m}}\sigma_1(t)\sigma_2(t+\theta_m)dt
\lim_{z\to i}(g_\Omega(z,i) + \log|z-i|) = \lim_{z\to i}\log|\Phi(z)(z-i)| = \lim_{z\to i}\log|\Phi_2\left(\Phi_1(z)\right)(z-i)| = \log\left| \lim_{w\to w_0}(\overline{w}_0 w - 1)\, \lim_{z\to i}\frac{z-i}{w-w_0}\right|(w=\Phi_1(z)) = \log \left|\frac{|w_0|^2-1}{\Phi_1'(i)}\right| = \log\frac{2\sqrt{r^2+1}}{r}.
\sum_{n=0}^{\infty}p_{-6}(5n+4)q^{n}=315\dfrac{E_{5}^{6}}{E_{1}^{12}}+52\times5^{4}q\dfrac{E_{5}^{12}}{E_{1}^{18}}+63\times5^{6}q^{2}\dfrac{E_{5}^{18}}{E_{1}^{24}}\quad+6\times5^{9}q^{3}\dfrac{E_{5}^{24}}{E_{1}^{30}}+5^{11}q^{4}\dfrac{E_{5}^{30}}{E_{1}^{36}}
\begin{cases}\displaystyle dy(t) =Ay(t)dt + A_1y(t-r)dt + \displaystyle\int^0_{-r}\beta(\theta)A_2 y(t+\theta)d\theta dt + f(t)dB(t),\,\,\,\,t\ge 0, y(0)=\phi_0,\,\,\,\,y(\theta)=\phi_1(\theta),\,\,\,\theta\in [-r, 0],\,\,\,r>0,\end{cases}
\bar{I}_{\ell}= \sqrt{\frac{\beta^{2}}{t}}2 ( - 1 )^{\ell}\sum_{n = 1}^{\infty}( - )^{n}e^{- \frac{n^{2}\beta^{2}}{4 t}}\left[ \left( \frac{n^{2}\beta^{2}}{t}\right)^{\ell}+ O ( \frac{t}{t^{\ell}}) \right] \; .
\mathcal{B}_{n+1}(x) = (x-c_n)\mathcal{B}_n(x)-\lambda_n x\mathcal{B}_{n-1}(x), n\geq 0, x \in \mathbb{R}, & \mathcal{B}_{-1}(x) = 0, \mathcal{B}_0(x) = 1,
I_1 = -h \sum_{n=1}^N \int \phi^n \frac{u^n-u^{n-1}}{h}\cdot G_h\ast\left( \frac{u^n-u^{n-1}}{h}\right) dx, I_2 = - h\sum_{n=1}^N \int \frac{u^n-u^{n-1}}{h}\cdot \frac1h \left[ G_h \ast, \phi^n \right] u^{n-1}\,dx, I_3 = h\sum_{n=1}^N E_h\big(u^{n-1},\frac{\phi^n-\phi^{n-1}}{h}\big).
\begin{aligned}\tau_Y = (\tau^{n-1,-}- \tau^{n-1,+})|_Y &= (0, f|_Y\, (-1)^{n-1}(n-2)!\, \check{\chi}_{H_{n}\cap Y}\, \bar{\partial}(\varphi_1|_Y) \wedge \dots \wedge \bar{\partial}(\varphi_{n-2}|_Y)). \end{aligned}
\Theta_n =\frac{16}{n^4\delta^4_n(k)}\sum_{\substack{\mathbf{P}_{k}\in \mathcal P(d,k)^4: p_{k,1}=p_{k,2}=p_{k,3}=p_{k,4}}}\sum_{\mathbf{i}\in \mathcal{J}^2}\varphi_{k-1}\left( \mathbf{P}_{k},\mathbf{i}\right) \cdot \varphi_{4}\left( \mathbf{i}\right).
\left( \hat{V}^{(B=0)}\right)^{-1}\partial_{\pm}\hat{V}^{(B=0)}={\frac{1\mp t}{1\pm t}}\left(\begin{array}{cc}-E^{-1}\partial_{\pm}E 0 0 E^{-1}\partial_{\pm}E\end{array}\right) ={\frac{1\mp t}{1\pm t}}\,P_{\pm}
\int_0^1(J^n_1(1,u))^2du \le \Big(\int_0^1\Big(\exp\Big(\int_s^t \tilde{f}'(r,x_r)dr\Big)\Big( \tilde{f}'_n(s,x_s^n)-K_s\Big)ds\Big)^2 &=\Big\{-\exp\Big(\int_s^1\tilde{f}'(r,x_r)dr\Big)\Big\vert_{s=0}^1-\int_0^1K_s\exp\Big(\int_s^1\tilde{f}'(r,x_r)dr\Big)ds\Big\}^2 &\le \Big(\exp(\int_0^1K_rdr)+\int_0^1K_sds\cdot\exp(\int_0^1K_rdr)\Big)^2.
\widetilde{g}(t)=\begin{cases}g(t) t\in[0,1] \psi^*_t g(t) t\geq 1 \end{cases}, \widetilde{H}(t)=\begin{cases}H(t) t\in[0,1] \psi^*_t H(t) t\geq 1 \end{cases}.
\mathfrak{d}_s = [\mathfrak{d}_{s-1},\mathfrak{n}] + J[\mathfrak{d}_{s-1},\mathfrak{n}] \subseteq [\mathfrak{d}^{j_0 -s + 1},\mathfrak{n}] + J [\mathfrak{d}^{j_0 -s + 1},\mathfrak{n}] \subseteq \mathfrak{d}^{j_0-s}+ J\mathfrak{d}^{j_0-s}= \mathfrak{d}^{j_0-s}.
\mathbb{E}[ b({\bf X}_{i}) ] &= \mathbb{E}[ \mathbb{E}[ b({\bf X}_{i}) | Y^{i-1}] ] & = \sum_{y^{i}}b({\bf x}_{i}) \sum_{{\bf x}_{i}}P_{{\bf X}_{i}| Y^{i}}({\bf x}_{i}| y^{i-1})P_{Y^{i-1}}(y^{i-1}),
\begin{array}{ccl}\dot{q}^{1}= \omega^{1}\dot{q}^{2}= \omega^{2}\dot{q}^{3}= \omega^{3}\dot{\omega}^{1}= b_{1}\cos(q^{2})\sin(q^{3})u^{1}\dot{\omega}^{2}= a_{1}\sin(q^{2})+a_{2}\cos(q^{2})+b_{2}\cos(q^{3})u^{1}\dot{\omega}^{3}= a_{3}\cos(q^{2})\sin(q^{3})+b_{3}u^{2}\end{array}
\int_0^{\infty}e^{-\lambda s}p_j(s) ds = \frac1{\lambda}\int_0^{\infty}\frac{q^2 e^{-\lambda s}}{(1-p e^{-\lambda s})^2}\cdot \left(\frac{1-e^{-\lambda s}}{1-p e^{-\lambda{s}}}\right)^{j-1}\cdot \lambda e^{-\lambda s}ds.
v_p\big(S(n)\big) =v_p\big(P(n)\big)-\sum_{i=1}^{t}v_p(E_in+F_i)+\sum_{i=1}^{r}v_p\big((C_in)!\big)-\sum_{i=1}^{s}v_p\big((D_in)!\big) &\ge-\sum_{i=1}^{t}v_p(E_in+F_i)-\frac{1}{p-1}\sum_{i=1}^{r}s_p(C_in)+\frac{1}{p-1}\sum_{i=1}^{s}s_p(D_in).
\begin{array}{lcl}2\big{(}n-\gamma_{oir2}(G)\big{)}\leq 2(n-|V_{\{1\}}|-|V_{\{2\}}|-|V_{\{1,2\}}|)=2|V_{\emptyset}|=2|Q|+2|V_{\emptyset}\setminus Q| &\leq 2(|V_{\{1\}}|+|V_{\{2\}}|+2|V_{\{1,2\}}|)=2\gamma_{oir2}(G),\end{array}
F_1^{(\lambda)}\cup F_2^{(\lambda)}= \bigcap_{\beta<\lambda}F_1^{(\beta)}\cup \bigcap_{\beta<\lambda}F_2^{(\beta)}\subset \bigcap_{\beta<\lambda}(F_1^{(\beta)}\cup F_2^{(\beta)}) = \bigcap_{\beta<\lambda}(F_1 \cup F_2)^{(\beta)}= (F_1 \cup F_2)^{(\lambda)}.
L_1=-\frac{\partial}{\partial t}+\Delta ,\,\,\, L_2=\frac{\partial}{\partial t}+\Delta &F_1(x,t,y,p)=\frac{1}{\nu}p,\,\,\,F_2(x,t,y,p)=y_d-(y+\hat{y}) &G_1(x,t)=\frac{\partial Y}{\partial t}-\Delta y_0,\,\,\,G_2(x,t)=0,
|\Omega|\sum_{M<|v|\leq L}f\left(\frac{v}{|v|}\right)|v|^{-s}-\int_{M<|x|\leq L}f\left(\frac{x}{|x|}\right)|x|^{-s}\mathrm{d}m_d(x) &=\int_{M<|x|\leq L}f\left(\frac{\lfloor x\rfloor_{\Lambda}}{|\lfloor x\rfloor_{\Lambda}|}\right)|\lfloor x\rfloor_{\Lambda}|^{-s}\mathrm{d}m_d(x)-\int_{M<|x|\leq L}f\left(\frac{x}{|x|}\right)|x|^{-s}\mathrm{d}m_d(x)+\epsilon'_L,
0=\frac{1}{4}[\frac{16 \pi^4 \theta^2}{g^2 \eta^2}z\int\frac{d^2p}{(2\pi)^2}[p^{-2}G^{-2}(p)-\frac{4\theta^2g^2}{\eta^2}(1-K^2(p)(p^2+\lambda \eta^2))]+(k^2+\lambda \eta^2)-K^{-2}(k)]
\partial_{-}( \partial_{+}\Gamma^{( 1 )}\Gamma^{( 1 ) - 1}) = - \Gamma^{( 2 )}\Gamma^{( 1 ) - 1}, \qquad \partial_{-}( \partial_{+}\Gamma^{( 2 )}\Gamma^{( 2 ) - 1}) = \Gamma^{( 2 )}\Gamma^{( 1 ) - 1}.
F_{r,s,q}=\frac{(q!)^s}{\prod_{k=0}^{s-1}(q+k)! k!}\frac{(1-\alpha)^{s (s+q)}}{\alpha^{s(s-1)/2}}\det_{1\leq j,k\leq s}\left[ \sum_{m=0}^{r-1}m^{j+k-2}\binom{m+q}{m}\alpha^m\right].
\left\{\sum_{n=0}^\infty\frac{(-1)\ \rho_n^{(2)}(s)}{(t-n-\frac{1}{2})^2}-\sum_{n=0}^\infty\frac{(-1)^2\ 2!\r_n^{(1)}(s)}{(s-n-\frac{1}{2})^3}\right\}_{s=0,\t=0}= 0.
a_{h}(u_h^* - v_h^*, u_h^* - v_h^*) =\sum_{i=1}^N \int_{\Omega_i}e^{-\hat{v}}\Big(e^{v_h^*}- e^{u_h^*}\Big)\Big( u_h^* - v_h^* \Big)\, dx +\sum_{i=1}^N \int_{\Omega_i}e^{\hat{w}}\Big(e^{- u_h^*}- e^{- v_h^*}\Big)\Big( u_h^* - v_h^* \Big)\, dx.
\int\limits_{\mathbb T^d}\hat a_{\rm sym}(\xi-\eta) \, \mu(\xi,\eta) \, (r_{\boldsymbol{\ell}}(\eta) - r_{\boldsymbol{\ell}}(\xi)) d\eta \ = \ 2 g_{\boldsymbol{\ell}}(\xi), r_{\boldsymbol{\ell}}\in (L^2(\mathbb{T}^d))^d.
0\leq\left|HVE^{R}_{mn}-HVE_{mn}\right|= \bigg|\frac{1}{n-m}\sum_{i=1}^{n-m}\log(Y^R_{i+m}-Y^R_{i})-\sum_{i=1}^{n-m}\log(X_{(i+m)}-X_{(i)})\bigg| = \bigg|\underbrace{\frac{1}{n-m}\sum_{i=1}^{n-m}\log\left(\frac{Y^R_{i+m}-Y^R_{i}}{X_{(i+m)}-X_{(i)}}\right)}_{A(n,m)}\bigg|,
lim_{x\rightarrow\infty}\frac{a_{1}x^{n}+a_{2}x^{n-1}+a_{3}x^{n-2}+...+a_{n}}{b_{1}x^{n}+b_{2}x^{n-1}+b_{3}x^{n-2}+...+b_{n}}=\frac{a_{1}}{b_{1}}
(\frac{2}{\alpha})^{r-1}=\prod_{k=2}^{\infty}{\frac{k^{r-1}(s(k-1)+\alpha)^{r}(s^r(k-2)+\alpha)}{(k+1)^{r-1}(s^r(k-1)+\alpha)(s(k-2)+\alpha)^r}}
\frac{2P_R|H^R|/\sigma^2_R}{1+P_R|H^R|^2/\sigma^2_R}\mu^R_h\cos\phi^R_m = \frac{2P_T|H^T|/\sigma^2_T}{1+P_T|H^T|^2/\sigma^2_T}\mu^T_h\cos\phi^T_m
\mathfrak{M}(q\,|\tau, 0, 0)=\frac{1}{\Gamma^q\bigl(1-\frac{1}{\tau}\bigr)}\Gamma\bigl(1-\frac{q}{\tau}\bigr), M_{(\tau, 0, 0)}= \frac{\tau^{1/\tau}}{\Gamma(1-1/\tau)}\beta_{1,0}^{-1}(\tau, b_0=\tau).