harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{n=1}^{[N/\Delta_{X}]}\frac{1}{n}= \log N - \log \Delta_{X}+ \gamma & - \sum_{s=1}^{k}\left( \frac{a^{s}}{s}+ \sum_{m=0}^{s-1}\binom{s-1}{m}\frac{a^{m}B_{s-m}\Delta_{X}^{s-m}}{s-m}\right) \frac{1}{N^{s}}+ O \left( \frac{1}{N^{k+1}}\right)
\int_{\delta_1*_t\delta_{\alpha}^{-1}}res\left(\frac{\omega_\beta}{f}\right)= \frac{(-1)^{A_{\beta'}-1}(\lambda-1)^{2A_{\beta'}-1}p(\{g=-1 \},\beta',\delta_{\alpha}^{-1})}{\zeta_{q}^{\gamma+1}-1}B\left(A_{\beta'},A_{\beta'}\right)\cdot \;\;\: F\left(A_{\beta'},1-A_{\beta'}-\beta_{n+1},2A_{\beta'};1-\lambda\right),
\partial^\beta (f\circ \varphi) = \sum_{\lambda \in \Lambda_{n,m}(\beta)}\frac{\beta !}{\lambda !}\left(\partial^{\sum_{\alpha}\lambda_\alpha}f\right)\circ \varphi \prod_{\alpha}\frac{\left(\partial^\alpha \varphi\right)^{\lambda_\alpha}}{(\alpha!)^{\sum_i \lambda_{i,\alpha}}}
A_z =\frac{\langle a~\partial_zg~ g^{-1}\rangle}{1+\langle a~g~a~g^{-1}\rangle},~~~~~~ A_{\bar z}=\frac{\langle a~g^{-1}~\partial_{\bar z}g \rangle}{1+\langle a~g~a~g^{-1}\rangle}.
(a)_0 :=(a;q)_0 =1, (a)_n :=(a;q)_n = (1-a)(1-aq)\cdots(1-aq^{n-1}),n \geq 1, (a)_{\infty}:=(a;q)_{\i}= \lim_{n\to\i}(a;q)_n, |q|<1.
e_{\mu, \nu;\mu',\nu'}^{\tau,2}(t,\xi,\eta, \sigma) = \sum_{\kappa\in\{+,-\}}q_{\kappa, \nu}^{\mu', \tau}(t,\xi-\eta, \eta-\sigma) P_{-\kappa}[q_{\mu, -\kappa\nu'}^2(\xi,-\sigma)]
\psi(\d (a)) =\d(a) \cdot \xi(z) &=\xi([\d(a),z])+z\cdot \xi(\d(a)) &=-\xi(\partial(z\cdot a))+z\cdot (x\cdot a) & =-x\cdot (z\cdot a)+z\cdot (x\cdot a) &=[z,x]\cdot a
\hat{a}_{a}^{\dagger}A_{c b}^{a b}\hat{a}^{c}+ \hat{a}_{a}^{\dagger}A_{b d}^{a b}\hat{a}^{d}+ \hat{a}_{b}^{\dagger}A_{c a}^{a b}\hat{a}^{c}+ \hat{a}_{a}^{\dagger}A_{a d}^{a b}\hat{a}^{d}
M_{\epsilon,i}^{2}(x) \le M_{\epsilon,i}(x)+\sum_{p,\,q:p,\,q\neq i,\,p\neq q}\left[2\epsilon\chi_{2\epsilon}(x_{p}-x_{q})-\nabla_{i}k_{\epsilon}\left(x_{p}-x_{i},\,x_{q}-x_{i}\right)\right] \le M_{\epsilon,i}(x)+\epsilon\sum_{l=1}^{N}M_{2\epsilon,l}(x)-\nabla_{i}K_{\epsilon,i}(x)
X^h\sigma_u =\sum_{\alpha}{\rm tr}\left(\left(\nabla^E_{X^h}A_{\alpha}\right)\cdot T_{\alpha_{\flat}(u)}\right) &=\sum_{\alpha}{\rm tr}\left(\left(\nabla^E_{X^h}A\right)_{\alpha}\cdot T_{\alpha_{\flat}(u)}\right),X\in TL.
(2 + \lambda) F \left( V_{p,q}(\xi) - V_{p,q}(\eta) \right) = (2 + \lambda) F(\xi)^{\frac{p-q}{q}}F(\eta)^{\frac{p-q}{q}}F \left( F(\eta)^{\frac{q-p}{q}}\xi - F(\xi)^{\frac{q-p}{q}}\eta \right).
+ c_{1}\lambda^{2}( \overline{{\psi}}\Gamma^{\mu}\partial_{\mu}\psi )^{2}+ c_{2}\lambda^{2}\overline{{\psi}}\Gamma^{\mu}\partial^{\nu}\psi \overline{{\psi}}\Gamma_{\mu}\partial_{\nu}\psi + c_{3}\lambda^{2}\overline{{\psi}}\Gamma^{\mu}\partial^{\nu}\psi \overline{{\psi}}\Gamma_{\nu}\partial_{\mu}\psi
(\partial_t + b\partial_\alpha)a = 2\Im [\P,\P\left[R_t + bR_\alpha\right]]\bar R_\alpha + 2\Im [\P,R]\partial_\alpha\bar\P\left[\bar R_t + b\bar R_\alpha\right] &+ 2\Im\left(b\partial_\alpha\P[R\bar R_\alpha] - \P\left[bR_\alpha\bar R_\alpha\right] - \P\left[R\partial_\alpha\bar \P(b\bar R_\alpha)\right]\right).
ds^2 \,=\,{\Lambda^2 \over z^{\prime\,2}}\, \Big\lbrack\,\,{\delta^2 \over (z^\prime - \delta )^2}dz^{\prime\,2}\,+\,\,{(z^\prime - \delta )^2 \over \delta^2}\Big( (d\vec x)^2\,- dt^2 \,\Big)\,\Big\rbrack
(\bar{\rho}^{\gamma-1}d)'(1)=\lim_{r \rightarrow 1^-}\frac{\int_r^1 \ell \bar{\rho}(\ell) \, d \ell}{\bar{\rho}(r)(r-1)}=\lim_{r \rightarrow 1^-}\frac{- r \bar{\rho}(r)}{\bar{\rho}'(r)(r-1)+\bar{\rho}(r)}= -1,
\liminf_{n \to \infty}\inf_{\mathbb{P}_n \in \mathbf{P}_n^{(\mathrm{bin})}\left(\tau \right)}\mathbb{P}_n \left(\beta_{\hat{\mathbb{M}}_n, n}^{*, (j)}\in \mathrm{CI}_{1-\alpha, \hat{\mathbb{M}}_n}^{(j), \mathrm{binC}}\;\forall j = 1, \hdots, m(\hat{\mathbb{M}}_n) \right) \geq 1-\alpha.
\left(\overline W^T\right)^{-1}=\begin{pmatrix}a_{1,1}a_{1,2}a_{1,3}\ldots a_{1,T}0 a_{2,2}a_{2,3}\ldots .. \cdot \cdot \cdot a_{T-1,T-1}a_{T-1,T}0 \ldots \ldots 0 a_{T,T}\end{pmatrix},
\sim_{q}(\delta^{-\frac{1}{k}})^{\frac{p}{2}+k-\frac{k(k-1)}{2}}(\delta^{-1+\frac{1}{k}})^{\frac{p-2k}{2}-\frac{k(k+1)}{2}}= \delta^{-(\frac{p}{2}- \frac{k(k+1)}{2})}
{\bf T}_{ijk}={\bf G}_{ij}{\bf Q}_{jk}{\bf G}^{T}_{ij},\ \forall\{i,j\}\in\mathcal{L},\ \forall k\in\mathcal{K},{\bf R}_i= \sum\limits_{l=1}^{3}\sum\limits_{m=1}^{2}{\bf G}_{il}{\bf Q}_{lm}{\bf G}^{T}_{il}+\sigma^{2}{\bf I}_2,\quad\forall i\in\mathcal{I},
(\gamma_1\gamma_2)\gamma_3= \gamma_1(\gamma_2\gamma_3) r(\gamma)\gamma=\gamma \gamma s(\gamma)=\gamma \gamma\gamma^{-1}= r(\gamma) \gamma^{-1}\gamma= s(\gamma)\ \r(\gamma_1\gamma_2)=r(\gamma_1),\s(\gamma_1\gamma_2)=s(\gamma_2) .
\frac{1}{6}\int_0^1 \left(\begin{vmatrix}P \alpha^\circ(t){\alpha^\circ}'(t)\end{vmatrix}+\begin{vmatrix}P \beta^\circ(t){\beta^\circ}'(t)\end{vmatrix}+\begin{vmatrix}P \gamma^\circ(t){\gamma^\circ}'(t)\end{vmatrix}\right)\,dt,
\alpha_{i}^{a}= \sqrt{\frac{\omega}{2 \hbar g}}\left[ \hat{q}_{i}^{a}+ i \frac{g}{\omega}\hat{p}_{i}^{a}\right] \ \ , \ \{\alpha_{i}^{a}}^{\dagger}= \sqrt{\frac{\omega}{2 \hbar g}}\left[ \hat{q}_{i}^{a}- i \frac{g}{\omega}\hat{p}_{i}^{a}\right] \ \ ,
\forall x_{v}\in Dom(v),\mu_{a\rightarrow v}(x_{v})=\sum_{x_{a}^{\prime}:x_{v}^{\prime}=x_{v}}\delta(syndrome(x_{v}^{\prime})=s)\prod_{v^{*}\in N(a)\backslash\{v\}}\mu_{v^{*}\rightarrow a}(x_{v^{*}}^{\prime}).
\int\limits_{(t_{j}-\varepsilon,t_{j}+\varepsilon)}\sum_{k\in\mathbb{S}(v,j)}a_{k}(t)\Psi_{k,t}(x)dt=\lim_{\delta\rightarrow0}\sum_{k\in\mathbb{S}(v,j)}\int\limits_{\delta<\left\vert t-t_{j}\right\vert\leq\varepsilon}a_{k}(t)\Psi_{k,t}(x)dt.
0\rightarrow(I_{\mathcal{B}_{i-1}}(-,X),\nabla_Y(j))\rightarrow(\mathcal{C}(-,E_{i-1}),\nabla_Y(j))\cong D\frac{\mathcal{C}(Y,E_{i-1})}{I_{\mathcal{B}_{j-1}}(Y,E_{i-1})}=0
[\nabla \widehat E^\alpha_{x,y}]_{a,b}= \alpha e^{\alpha |G_{x,y}|^2}(G_{y,x}G_{x,b}\delta_{y,a}- G_{a,y}G_{y,x}\delta_{b,x}+G_{x,y}G_{y,b}\delta_{a,x}-G_{a,x}G_{x,y}\delta_{b,y}) \, .
\int_{0}^{t}\int_{\Omega}u_j^{q-1}{u_j}_t =d\int_{0}^{t}\int_{\Omega}u_j^{q-1}\Delta u_j+\int_{0}^{t}\int_{\Omega}u_j^{q-1}H_j(u) = d\int_{0}^{t}\int_{M}u_j^{q-1}{\frac{\partial u_j}{\partial\eta}}-d\int_{0}^{t}\int_{\Omega}(q-1)u_j^{q-2}{|\nabla u_j|^{2}}+ \int_{0}^{t}\int_{\Omega}u_j^{q-1}H_j(u)
\sum_{\tau \in W}x^{\ell_{\Delta'}(\tau)}y^{L_{\Delta'}(\tau)}=\sum_{\tau \in W}x^{\ell_{\Delta}(w^{-1}\tau w)}y^{L_{\Delta}(w^{-1}\tau w)}=\sum_{\sigma \in W}x^{\ell_{\Delta}(\sigma)}y^{L_{\Delta}(\sigma)}.
U_k(r)\thicksim r^{A^+(\lambda_1+\omega_k)}\quad\mbox{as}r\to+0, U_k(r)\thicksim c_k\,r^{A^+(\lambda_2+\omega_k)}, U_k'(r)=O\left(r^{A^+(\lambda_2+\omega_k)-1}\right)\quad\mbox{as}r\to\infty,
f_{\boldsymbol{X}}(\boldsymbol{x}):=\frac{c_{n}}{\sqrt{|\boldsymbol{\Sigma}|}}g_{n}\left\{\frac{1}{2}(\boldsymbol{x}-\boldsymbol{\mu})^{T}\mathbf{\Sigma}^{-1}(\boldsymbol{x}-\boldsymbol{\mu})\right\},~\boldsymbol{x}\in\mathbb{R}^{n},
\Omega_1:=\{x\in \Omega: \frac{\tilde{f}(u)^{\alpha+1}}{u^{\alpha}}>\tilde{f}(u)^{\alpha+1-\beta}\},\,\, \Omega_2:=\Omega \setminus \overline{\Omega}_1=\{x\in \Omega: \tilde{f}(u) \leq u^{\frac{\alpha}{\beta}}\}~.
G(\theta)\approx \sum_{b_1,\cdots,b_M}{b \choose b_1,\cdots,b_M}\prod_{m=1}^{M}\left(C_{M}^{m}(-1)^{m+1}\right)^{b_m}\frac{(R_{max}\!\!-\!\!R_{min})\pi}{2N}\!\sum_{k=1}^{K}\!\sqrt{1\!\!-\!\psi_k^2}\exp\left(-Q(d_k,\theta)\right)f_{r_1|\Phi(\mathcal{A})>0}(d_k),
U_{(12)}\diamond \hat{X}\diamond U_{(12)}^{-1}= \sum_{k=0}^\infty \left(\frac{i}{\hbar}\right)^k \frac{1}{k!}\underbrace{[A_{(12)},[A_{(12)},\ldots[A_{(12)},}_{\mbox{k-times}}\hat{X}]\ldots]]
J_{a}( \phi ) = \frac{1}{2 \pi N}\sum_{m \in \cal{Z}}J_{a m}e^{- i \frac{m}{N}\phi}\hspace{1 0 m m}J_{a m}= \int_{0}^{2 \pi N}d \phi J_{a}( \phi ) e^{i \frac{m}{N}\phi}
S = \frac{r_{h}^{2}}{9 0}{\tilde{\epsilon}}^{- 2}+ \left[ \frac{\Delta_{h}^{\prime \prime}r_{h}^{2}}{1 8 0}+ \frac{r_{h}^{2}\mu^{2}}{6}\right] \log{\tilde{\epsilon}}+{\mathrm{t e r m s ~ f i n i t e ~ a s ~}}{\tilde{\epsilon}}\rightarrow 0 .
G^h(x) = \bar{\mathcal{G}}(x') + h \mathcal{G}_1(x',\frac{x_3}{h})+ \frac{h^2}{2}\mathcal{G}_2(x', \frac{x_3}{h}) +o(h^2) \mbox{for all}~x=(x',x_3)\in \Omega^h,
\left[ \begin{smallmatrix}T; N, \mu d \end{smallmatrix}\right]^+ = [T; N+ (d-1), \mu - (d -1)]^+ \dots [T; N+1, \mu -1]^+ [T; N, \mu]^+, &\left[ \begin{smallmatrix}T; N, \mu d \end{smallmatrix}\right]^- = [T; N-(d-1), \mu -(d-1)]^- \dots [T; N-1, \mu -1]^- [T; N, \mu]^-.
\frac{d}{dt}(U^*(t)P(t)U(t)) = \frac{d}{dt}U^*(t)P(t)U(t)+U^*(t) \frac{d}{dt}P(t)U(t)+U^*(t)P(t) \frac{d}{dt}U(t) =U^*Q^*PU+U^*P'U+U^*PQU =U^*(P'+[P,Q])U =U^*(P'-PP'-P'P)U=0
xyz\approx zyx,\,x^2y\approx 0; &xyz\approx yzx,\,x^2y\approx 0; &xyz\approx yxz,\,xyzt\approx xzty,\,xy^2\approx 0; &xyz\approx xzy,\,xyzt\approx yzxt,\,x^2y\approx 0.
{\cal E}= \frac{1}{2}\mu^{1 - 2 \nu}\sum_{n = 1}^{\infty}\int \frac{d^{2}p}{( 2 \pi )^{2}}\left\{{\bf p}^{2}+ \frac{\pi^{2}}{\ell^{2}}\left[ ( n + s )^{2}+ \frac{g}{3}\right] \right\}^{\nu}\Bigg |_{\nu = 1 / 2}.
L_{2}^{\ast\ast}=\frac{\xi_{p}\ln(1+\sqrt{2\pi}\phi(\xi_{p}))-\xi_{q}\ln(1+\sqrt{2\pi}\phi(\xi_{q}))}{\sqrt{2\pi}\Psi_{2}^{\ast}(-1,\frac{1}{2},1)F_{Y}(\xi_{p},\xi_{q})}+\frac{\Psi_{1}^{\ast}(-1,\frac{3}{2},1)}{\Psi_{2}^{\ast}(-1,\frac{1}{2},1)}\frac{F_{Y_{(2)}}(\xi_{p},\xi_{q})}{F_{Y}(\xi_{p},\xi_{q})},
\mathcal{V}_0:= \mathcal{O}_{\mathfrak{g}^{\oplus 2}}\otimes (\mathfrak{g}^{\lambda<0})^{\oplus 2}=\bigoplus_{i>j}\mathcal{O}_{\mathfrak{g}^{\oplus 2}}(\beta_i-\beta_j)^{\oplus 2}\to \mathfrak{g}^{\oplus 2}
L=\left(\begin{array}{c|c}A b \hline\cdots \cdot\end{array}\right), \quad L_{ij}=\begin{cases}a_{ij}i,j\leq m b_i i\leq m j=m+1 -\sum\limits_{k=1}^m a_{kj}i=m+1j\leq m -\sum\limits_{k=1}^m b_{k}i=j=m+1,\end{cases}
\gamma_{-1}(B(x,R)) = \int_{|u|<R}e^{|u+x|^2}\, du \geq e^{|x|^2+R^2/4}\int_{\varphi_0}^{\pi/2}\int_{R/2}^R r^{n-1}e^{2r|x|\cos \varphi_1}\sin \varphi_1 \, dr \, d\varphi_1 & = C\frac{e^{|x|^2+R^2/4}}{|x|}\int_{R/2}^R r^{n-2}\int_0^{2r|x|\cos \varphi_0}e^u \, du \, dr,
\partial_{z}\left( \theta^{n}A_{n}\right) + \partial_{\bar{z}}\left( \theta^{n}\bar{A}_{n}\right) = - \theta^{n}\sum_{p = 2}^{n - 2}\left( \! \! \begin{array}{c}{{n}}\\{{p}}\\ \end{array}\! \! \right) \! \bar{A}_{p + 1}A_{n - p + 1}- \theta^{n}\left. \frac{d^{n}}{d \varphi^{n}}V \left( \varphi \right) \right|_{\varphi = \varphi_{0}}
\delta R_{MN}(h) +{1 \over 6}\bar F_{\theta\phi}\bar F^{\theta\phi}(h_{MN}+ 2 \bar g_{MN}{\cal F})- ( \bar F_{M \theta}\bar F_N^{~\theta}+ \bar F_{M \phi}\bar F_N^{~\phi}){\cal F}= 0,
F^{1 1 1 1}= \frac{( 1 2 ) ( 3 4 )}{x_{1 2}^{2}\, x_{3 4}^{2}}\; \alpha ( x_{0}, \ldots , x_{4}) + \frac{( 1 4 ) ( 2 3 )}{x_{1 4}^{2}\, x_{2 3}^{2}}\; \beta ( x_{0}, \ldots , x_{4}) + \frac{( 1 3 ) ( 2 4 )}{x_{1 3}^{2}\, x_{2 4}^{2}}\; \gamma ( x_{0}, \ldots , x_{4}) \; .
\begin{array}{l l l}{{\{a , b \}= - a b ,}}&{{\{b , d \}= - b d ,}}&{{\{c , b \}= 0 ,}}\\{{\{a , c \}= - a c ,}}&{{\{c , d \}= - c d ,}}&{{\{a , d \}= - 2 b c .}}\\ \end{array}
\dfrac{16\alpha (1-\beta)^2}{(1+\alpha)^2 (1+16\alpha (1-\beta)^2)}-\dfrac{1-x^2}{(1+\alpha)^2}&{}-\dfrac{1}{(1-\alpha)^2}\left(x-\dfrac{2\sqrt{\alpha}}{(1+\alpha)\sqrt{1+16\alpha (1-\beta)^2}}\right)^2=0,
S_{g a u g e}= - \, \frac{1}{g_{\mathrm{Y M}}^{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_{\mathrm{Y M}}}{3 2 \pi^{2}}\int d^{4}x F_{\alpha \beta}{\tilde{F}}^{\alpha \beta}
\begin{matrix}F(x)&=&x-x^{2}+x^{4}-x^{8}+\cdot\cdot\cdot\\ &=&x-[(x^{2})-(x^{2})^{2}+(x^{2})^{4}-\cdot\cdot\cdot]\\ &=&x-F(x^{2}).\end{matrix}
0= R^0(\widehat{\phi_{L}}_*\circ\hat{S})(a_*(\Omega_X^1\otimes P^{\vee})\otimes F^{e,*}(\widehat{L})) = R^0\widehat{\phi_{L}}_*R^0\hat{S}(a_*(\Omega_X^1\otimes P^{\vee})\otimes F^{e,*}(\widehat{L})) = \widehat{\phi_{L}}_*R^0\hat{S}(a_*(\Omega_X^1\otimes P^{\vee})\otimes F^{e,*}(\widehat{L})).
K(x,t,y)=c_0(y)t^{-1/2}x^{-A(t,y)}e^{-B(x,t,y)}, t>0, A=\frac{\ell t-\ln y}{\sigma^2 t}, B=\frac{(\ln x)^2+(\ln y)^2}{2\sigma^2 t}+\left(\frac{\ell^2}{2\sigma^2}+r\right)t,
P=[\begin{matrix}0&0.5&0.5\\ 0&0&0\\ 0&0&0\end{matrix}],\mu=[\begin{matrix}\mu_{1}(x_{1})\\ \mu_{2}(x_{2})\\ \mu_{3}(x_{3})\end{matrix}]=[\begin{matrix}15\\ 12\\ 10\end{matrix}]forallx_{i}>0
\frac{\partial^n\alpha}{\partial u^n}(u,v)=Q_1+\int_0^v\left\{Q_1'\frac{\partial^n\alpha}{\partial u^n}+Q_1''\frac{\partial^n\beta}{\partial u^n}+Q_1'''\frac{\partial^nt}{\partial u^n}\right\}(u,v')dv'.
\mathrm{e}^{-k}\frac{k^k}{k!}=\Theta\left(\frac 1{\sqrt k}\right),\mathrm{e}^{-\lambda}\frac{\lambda^k}{k!}=\Theta\left(\frac{\mathrm{e}^{k-\lambda}}{\sqrt k}\left(\frac \lambda k \right)^k\right).
r_1(t) = -\rho_1\rho_3 a\delta t^{\gamma-\frac12}, r_2(t) = - \rho_3\delta t^{\gamma-1}(\rho_1 a + \rho_1bt + abt^2 ), r_3(t) = \rho_2\rho_3\delta t^{\gamma-1}(\rho_1 + at), r_4(t) = c\rho_3[a\rho_1 + (\rho_1 + at)(\rho_2 + bt)] + \rho_3\delta^2 t^{2\gamma - 1}(\rho_1 + at).
\operatorname{DET}|S_{2n-1-i-j}\big(q; q^{i-1}\beta\big) |_{1 \le i,j \le n-1}=q^{(n-1)(n-2)(4n-3)/6}\prod_{a=1}^{n-2}\big(1+q^{a-1}\beta\big)^{n-a-1},
X_1=\begin{bmatrix}a -a+b 0 b\end{bmatrix}, X_2 = \begin{bmatrix}\frac{1}{a}\frac{1}{b}-\frac{2}{a}0 \frac{1}{b}\end{bmatrix}, X_3=\begin{bmatrix}1 1 0 1\end{bmatrix},
c_{\textrm{e f f}}= \frac{1 2}{\pi^{2}}\, \lim_{r \rightarrow 0}\, r \int_{0}^{\infty}d \theta \cosh \theta \ln \left( 1 + | \hat{K}( \theta ) |^{2}\, e^{- 2 r \cosh \theta}\right) \, ,
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)
{\cal I}( M^{2}) = - \mathrm{T r}\, \gamma_{5}{\frac{M^{2}}{- ( \gamma \cdot D )^{2}+ M^{2}}}= - \int d^{3}x \langle x | \mathrm{t r}\, \gamma_{5}{\frac{M}{\gamma \cdot D + M}}| x \rangle \, .
F^{k,l}(X,Y) =[(\widetilde{\nabla}_X\omega^{(t)})(Y)]E^{(t)}&=(-1)^{(k-1)(l-1)}[(\widetilde{\nabla}_Y\omega^{(t)})(X)]E^{(t)}&=(-1)^{(k-1)(l-1)}F^{l,k}(Y,X),
y^*_i = \left((i-n+m)L+\frac{U}{m}-\frac{L(m+1)}{2}\right)_+ = \left((i-1)L+\frac{U}{n-1}-\frac{Ln}{2}\right)_+ = \left((i-1)L+\frac{Ln}{2}-\frac{Ln}{2}\right)_+ = (i-1)L
\begin{array}{lll}F^{+}_{\mu\nu}= \displaystyle\frac{1}{4}\left(F_{\mu\nu}-\frac{1}{2}\Omega_{\mu\nu\alpha\beta}F^{\alpha\beta}\right)\in\Lambda^{2}_{+},\vspace{.3cm}F^{-}_{\mu\nu}= \displaystyle\frac{1}{4}\left(3F_{\mu\nu}+\frac{1}{2}\Omega_{\mu\nu\alpha\beta}F^{\alpha\beta}\right)\in\Lambda^{2}_{-}.\end{array}
\varphi_\pm(w) =\langle J^{'}(w),w^\pm\rangle =\int_{\mathbb{R}^{d}}\int_{\mathbb{R}^{d}}g\left( \frac{w(x)-w(y)}{\vert x-y\vert^{\alpha}}\right)\frac{w^\pm(x)-w^\pm(y)}{\vert x-y\vert^{\alpha +d}}dxdy & + \int_{\mathbb{R}^{d}}g(w)w^\pm dx - \int_{\mathbb{R}^{d}}K(x)f(x,w^\pm)w^\pm dx.
\frac{2}{3}d(m,i(x))<\frac{2}{3}(d(m,i(\bar{x})+d(i(\bar{x}),i(x)))<\frac{1}{2}d_0(x,y)+d_0(x,\bar{x}) & <\frac{4}{3}(d(m,i(\bar{x}))+d(i(x),i(\bar{x})))<\frac{4}{3}(d(m,i(x))+1)<\frac{3}{2}d(m,i(x)).
C^{2}.{h}^{l}B \equiv(a_{l}h^{l}+a_{l-2}h^{l-2}+\cdots+a_{0})B a_{l}=-4(l+1)^{2}k =2,4,\ldots,k-3\ (k-2) a_{l-k}=-\frac{1}{2}\bigg((8-\mu){l\choose k}4^{k}+6{l \choose k+1}4^{k+1}+{l\choose k+2}4^{k+2}\bigg).
S_1=\sum_{\substack{d_1,\ldots,d_k e_1,\ldots,e_k}}\lambda_{d_1,\ldots,d_k}\lambda_{e_1,\ldots,e_k}\sum_{\substack{x\leq n<2x n\equiv \nu_0\pmod{W^\prime}[d_i,e_i]\mid nM+h_i}}1.
m_{\chi(X),\chi(Y)}=\prod_{r=1}^n a_{(\chi(X))_r, (\chi(Y))_r}=(-1)^{|\{r\in \{1,\dots,n\}| a_{(\chi(X))_r, (\chi(Y))_r}=-1 \}|}=(-1)^{|X\cap Y|}.
\begin{gathered}R_5:=R_4[Y_{23};\alpha_5,\delta_5]; \alpha_5: Y_{13}\mapsto t^{-1}Y_{13}, \Y_{21}\mapsto t^{-1}Y_{21},\Y_{22}\mapsto t^{-1}Y_{22}; \delta_5: Y_{11}\mapsto (t^{-1}-t)Y_{13}Y_{21}, \Y_{12}\mapsto (t^{-1}-t)Y_{13}Y_{22};\end{gathered}
- \frac{1}{\kappa}R_{\mu \nu}( \Gamma ) ( \Phi + b_{g}\sqrt{- g}) + \frac{1}{2}\phi_{, \mu}\phi_{, \nu}( \Phi + b_{k}\sqrt{- g}) - \frac{1}{2}\sqrt{- g}g_{\mu \nu}\left[ - \frac{b_{g}}{\kappa}R ( \Gamma , g ) + \frac{b_{k}}{2}g^{\alpha \beta}\phi_{, \alpha}\phi_{, \beta}\right] = 0
z^{2}(\log F (z))_{zz}+ \overline{z}^{2}(\log F (z) )_{\overline{z}\overline{z}}= 2\sum_{k=3}^{p}(k-1)(k-2)|z|^{2(k-1)}\log G_{k}(z)+2\sum_{k=2}^{p}(k-1)|z|^{2(k-1)}\mathfrak{L}[\log G_{k}(z)] &+\sum_{k=1}^{p}|z|^{2(k-1)}\big(z^{2}(\log G_{k}(z))_{zz}+\overline{z}^{2}(\log G_{k}(z))_{\overline{z}\overline{z}}\big).
\Phi(y_n) = \Phi(y) - \frac{\log (\|f\| \|v\|)}{\sigma \sqrt{n}}\phi(y) + O \left( \frac{1}{n}\right), \quad (1-y_n^2) \phi(y_n) = (1-y^2) \phi(y) + O \left( \frac{1}{\sqrt{n}}\right), \quad \phi(y_n) = \phi(y) + O \left( \frac{1}{\sqrt{n}}\right).
M=\left(\begin{array}{cccc}c_1 v_1 c_1 v_2 \ldots c_1 v_n c_2 v_1 c_2 v_2 \ldots c_2 v_n \ldots \ldots \ldots \ldots c_n v_1 c_n v_2 \ldots c_n v_n \end{array}\right) = c\otimes v,
\begin{array}l S_{n}(z) = z S_{n-1}(z) - \overline{\alpha}_{n-1}\, S_{n-1}^{\ast}(z), begin{align*}1.5ex] S_{n}(z) = (1 - |\alpha_{n-1}|^2) z S_{n-1}(z) - \overline{\alpha}_{n-1}S_{n}^{\ast}(z),\end{array}n \geq 1.
B_{_F}(\varphi)(\hat{X},Y)=\textrm{Hess}(\varphi)(\hat{X},Y)-(d\varphi\otimes d\varphi)(\varrho\hat{X},Y)-\frac{1}{n}(\Delta\varphi-\|grad\varphi\|^2)g(\varrho\hat{X},Y),
\frac{|u(x) - u(y)|}{d^{\alpha}(x,y)}\leq M d^{\alpha}(x,y) + \| \sigma_i \| \frac{|v_i(x,a(y)) - v_i(y,a(y))|}{d^{\alpha}(x,y)}+ \frac{|\nu(y) (x-y)_1|}{d^{\alpha}(x,y)}& \lesssim M + \|\sigma_i \| N_i,
U_{\mathcal X_2\to y_2}=\{u\in \mathcal P_{\mathcal X_2}: \{\alpha^{1}_{\mathcal X_1}, u\}\frac{\rho}{2}\prod_{j\in [k]\setminus \{1,2\}}|\mathcal P_{\mathcal X_j}| \mathcal A_{\mathcal Y}\}.
\left|G^1\right| = \sum_{\begin{smallmatrix}e\colon s\to r e \in E^1 \end{smallmatrix}}\left|E^{k-1}_{r\to\ast}\right| = \sum_{\begin{smallmatrix}f\colon q\to t f \in E^1 \end{smallmatrix}}\left|E^{k-1}_{\ast\to q}\right|.
\sum_{j=0}^k\frac{{\mathcal Q}_j^n(\zeta){\mathcal Q}_j^n(\zeta')}{j!(n+2)_j(a_{2,0}a_{3,2})^j\left(n+1+\frac{a_{2,2}}{a_{3,2}}\right)_j}=\frac{{\mathcal Q}_{k+1}^n(\zeta){\mathcal Q}_{k}^n(\zeta')-{\mathcal Q}_{k}^{n}(\zeta){\mathcal Q}_{k+1}^n(\zeta')}{k!(n+2)_k(a_{2,0}a_{3,2})^k\left(n+1+\frac{a_{2,2}}{a_{3,2}}\right)_k(\zeta-\zeta')},
\frac{\partial}{\partial \xi}\wedge \frac{\partial}{\partial u}=\frac{(t\xi_1+u_1)^2}{u_1^2-t}\frac{\partial}{\partial \xi_1}\wedge \frac{\partial}{\partial u_1}, 1-t\xi^2 =1-t\left(\frac{u_1\xi_1+1}{t\xi_1+u_1}\right)^2=\frac{(1-t\xi_1^2)(u_1^2-t)}{(t\xi_1+u_1)^2}
K_n(x, y) = \frac{1}{(2\pi i)^2}\oint_{\Sigma}du \oint_{\Gamma}dv \frac{e^{xu}v^{n + \nu_1}(u - 1)^n}{e^{yv}u^{n + \nu_1}(v - 1)^n}\frac{1}{u - v},
g_p(s)=V_p^+(s)s^2\underset{p\rightarrow +\infty}{\longrightarrow}V^+(s)s^2=\left(\frac{s}{1+\frac{s^2}{8}}\right)^2=:g(s)\quad\mbox{in $C^0_{loc}([0,+\infty))$}.
\lambda_{0}( \frac{1}{u}- i \int \frac{d^{d}k}{( 2 \pi )^{d}}\frac{\mu^{2}+ \lambda_{0}k^{2}- 2 \mu k^{2}}{k^{2}( k^{2}- \lambda_{0}) ( k^{2}- \mu )^{2}}) = ( c_{c r}^{2}- c^{2})
2\int_\Omega|D(u)|^2 = \int|\nabla u|^2 + \int\nabla u : (\nabla u)^T = \int|\nabla u|^2 + \int \mbox{tr}(\nabla u)^2 = \int|\nabla u|^2 + \int|\mbox{div}u|^2 + \int \left(\mbox{tr}(\nabla u)^2 - (\mbox{tr}\nabla u)^2\right).
\left< T^{\mu}{}_{\nu}\right> = \frac{1}{r^{4}}\left( \begin{array}{c c c c}{{- A}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{- A}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{3 A}}&{{\kappa B / r^{2}}}\\{{0}}&{{0}}&{{\kappa B}}&{{- A}}\\ \end{array}\right) ,
-\frac{1}{\sin^2\phi_{-L}(n,\omega,E)}\frac{\partial}{\partial \omega_j}\phi_{-L}(n,\omega,E)=-\frac{1}{\cos^2\phi_{-L}(n-1,\omega,E)}\frac{\partial}{\partial\omega_j}\phi_{-L}(n-1,\omega,E).
P_{W^*| A = a}(\mathcal{S}) = \sum_{v_a \in \mathcal{V}_a}P_{V_a}(v_a) P_{W^*| A = a, V = v_a}(\mathcal{S}) = \sum_{v_a \in \mathcal{V}_a}P_{V_a}(v_a) P_{W^*_a|V_a = v_a}(\mathcal{S}_a) \prod_{\substack{a' \in \mathcal{A}a' \neq a}}P_{W^*_{a'}}(\mathcal{S}_{a'}) = \prod_{a' \in \mathcal{A}}P_{W^*_{a'}}(\mathcal{S}_{a'}).
\left( ( D^{2}({\Phi}) )^{a b}+ g \epsilon^{a d c}{\tilde{A}}_{i}^{d}D_{i}^{c b}({\Phi}) \right) \tilde{\sigma}^{b}= - g \epsilon^{a b c}{\tilde{A}}_{i}^{a}E_{i}^{c}.
dq^\prime_t= \tilde\gamma^{-1}(t^*,q^\prime_t)\left(\partial_t\psi(t^*,q^\prime_t)-\nabla_qV(t^*,q^\prime_t))\right) dt & +\tilde S(t^*,q^\prime_t)dt+\tilde\gamma^{-1}(t^*,q^\prime_t)\sigma(t^*,q^\prime_t)\circ dW_t
C=2^{1-2s}\frac{\Gamma(2s-1)}{\Gamma(s)^2}\left(\frac{\Gamma(\tfrac{n-2s}{2})}{\Gamma(\tfrac{n+2s-2}{2})}-\frac{\Gamma(\tfrac{n-2s+2}{2})}{\Gamma(\tfrac{n+2s}{2})}\right).
\sum_{m=1}^{\infty}\frac{q^{N+m}}{1+q^{N+m}}\frac{(-1)_m}{(-q^N)_m}= \frac{q^N}{1+q^N}\left(F\left(\frac{-1}{q}, -q^N; q\right) - 1\right) &= \frac{(-1)_Nq^N}{1-q^N}- \frac{2q^N}{1-q^{2N}},
|E(G)|-|E(G\backslash S)| >h_{n_1}(n_1,n_2,\ldots,n_2)-h_{n_1-1}(n_1-1,n_2-1,\ldots,n_2-1) &=(r-1)(n_1+(r-2)n_2)+(r-1)n_2-\binom{r}{2}-1.
\sum_{0<n \leq N}m_{a}^k (P(n)) \leq \sum_{0<n \leq N}\Big(\sum_{\substack{p^2 \vert P(n)}}1 \Big)^k = \sum_{0 < n \leq N}\sum_{\substack{(p_1,\dots,p_k){p_1^2 | P(n)}{\dots}{p_k^2 | P(n)}}}1.
\# \xi ( r_{i}) = \sum_{\omega \in \Omega_{i}}\# \xi ( \gamma ( v_{i , \omega}; y_{i , \omega , 1}, y_{i , \omega , 2}) ) = \sum_{\omega \in \Omega_{i}}\gamma ( v_{i , \omega}; \# \xi ( y_{i , \omega , 1}) , \# \xi ( y_{i , \omega , 2}) ) ,
f_{T}(x)= \frac{c_1}{x \, q^{\epsilon}}G_{2,\,3}^{2,\,2}\left[q^{\epsilon}x \left\vert \begin{gathered}0, 1 1, 1, 1 \end{gathered}\right.\right] - \frac{c_1}{x \, r^{\epsilon}}G_{2,\,3}^{2,\,2}\left[r^{\epsilon}x \left\vert \begin{gathered}0, 1 1, 1, 1 \end{gathered}\right.\right].
\sqrt{\alpha(0)}+\big(-\frac{1}{2}+i\frac{\sqrt{3}}{2}\big)\sqrt{\alpha(1)}+\big(-\frac{1}{2}-i\frac{\sqrt{3}}{2}\big)\sqrt{\alpha(2)}= \sqrt{\alpha(0)}-\frac{1}{2}(\sqrt{\alpha(1)}+\sqrt{\alpha(2)})+i\frac{\sqrt{3}}{2}(\sqrt{\alpha(1)}-\sqrt{\alpha(2)})>0.
\sum_{j=0}^{\varepsilon/s}\frac{f(j) e^{-sj}\left(1-\sqrt{1-e^{-s}}-\sum_{l=1}^j f(l) e^{-sl}\right)^{k-1}}{\left(1-\sum_{l=1}^j f(l) e^{-sl}\right)^k}.
\begin{cases}z(t) \sim z_1(t) = e^{-\frac{t}{\alpha}}z(t) \sim z_2(t) =\begin{cases}e^{-t\big(\frac{1}{\alpha}-\frac{N-2}{2}\big)}N \ge 3 te^{-\frac{t}{\alpha}}N = 2 .\end{cases}\end{cases}
A=A_{3}L_{A_{1}},\B=R_{B_{1}}B_{3},\C=A_{4}L_{A_{2}},\D=R_{B_{2}}B_{4}, & M=R_{A}C,\N=DL_{B},\S=CL_{M},\E=C_{c}-A_{3}A_{1}^{\dagger}C_{1}B_{3}&-AC_{2}B_{1}^{\dagger}B_{3}-A_{4}A_{2}^{\dagger}C_{3}B_{4}-CC_{4}B_{2}^{\dagger}B_{4}.
I_1 = \int_{z=x}^{\infty}\int_{u=0^-}^{z}\bar{G}(u)B(z-u)f_n(u)\lambda e^{-\lambda(z-x)}du dz =P_n(0) \int_{z=x}^{\infty}B(z)\lambda e^{-\lambda(z-u)}dz + \int_{z=x}^{\infty}\int_{u=0^+}^{z}\bar{G}(u)B(z-u)f_n(u)\lambda e^{-\lambda(z-x)}du dz.
\int_0^T f_h(X_t)dt = \int_0^T LL^{-1}[f_h](X_t)dt &= L^{-1}[f_h](X_T) - L^{-1}[f_h](X_0) - \int_0^T \nabla L^{-1}[f_h](X_t).dW_t.
L =\sup \left( \int_{B(x,r)}\omega(y)^qdy \right)^{1/q}\left(_{y\in B}K_r(x,y)u^{-1}(y)\right) \leq C(\omega) \sup \left( \int_{B(x,r)}\omega(y)^qdy \right)^{1/q}\left(\int_{B(x,r)}\omega(y)^{q}dy\right)^{-1/q}=C(\omega).
\lim_{n\to\infty}\frac{1}{\pi}\int_{0}^{\infty}\,d\eta_1\int_{-\infty}^{\infty}\,d\tau_1 \frac{1}{x-\tau_1-i\eta_1}\overline{\partial}\Psi_{f_n}(\tau_1,\eta_1) = f(x)+iH[f](x),
\sup_{G}(v-u) \leq \sup_{G}(v_\varepsilon-u) \leq \sup_{\Omega \cap \mathbb B(0,R)}(v_\varepsilon- u) -\varepsilon r^2 &= \sup_{( \Omega \cap \mathbb B(0,R)) \backslash G}(v_\varepsilon-u) -\varepsilon r^2 & <\sup_{( \Omega \cap \mathbb B(0,R)) \backslash G}(v-u) +\varepsilon R^2 \leq \sup_{\Omega\backslash G}(v -u) + \varepsilon R^2.
\Bigl\|A\Bigl( (\frac{1}{2}-\mu)(h_{S_0}+h^{(1)})-\frac{\mu}{2}u_i\Bigr) \Bigr\|_2^2 \leq (1+\delta)\|(\frac{1}{2}-\mu)(h_{S_0}+h^{(1)})-\frac{\mu}{2}u_i\|_2^2 = (1+\delta)\Bigl((\frac{1}{2}-\mu)^2 \|(h_{S_0}+h^{(1)})\|_2^2+\frac{\mu^2}{4}\|u_i\|_2^2\Bigr) = (1+\delta)\Bigl((\frac{1}{2}-\mu)^2 z^2+\frac{\mu^2}{4}\|u_i\|_2^2\Bigr)
r_{n,k}= \binom{5n}{k-1}\frac{(k-1)!}{n!}B_{n,k}(1!(C_0+C_1), 2!(C_1+C_2),\dots),\intertext{which by means of \cite[Example~3.2]{WW09}can be written as}r_{n,k}= \binom{5n}{k-1}\sum_{j=0}^k \frac{(-1)^{k-j}(2j-k)}{nk}\binom{k}{j}\binom{2(n+j)-k-1}{n-1}.
A_{4,1}^1 : = \sup_{t > 0}\bigg( \int_t^{\infty}[V_3]^{\frac{q}{p}}(\tau) w(\tau)\,d\tau \bigg)^{\frac{1}{q}}[V_1]^{- 1}(t), A_{4,2}^1 : = \sup_{t > 0}W^{\frac{1}{q}}(t) \bigg( \int_t^{\infty}\bigg( \frac{V_3(\tau)}{V_1(\tau)}\bigg)^{\frac{1}{1-p}}\{V \cdot [V_1]\}^{-2}(\tau)v(\tau)\,d\tau\bigg)^{\frac{1-p}{p}};