harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
I_1= \sum_{j=1}^{i}x_j n_j \Delta x_j \bigg[\int_{x_{i+ 1/2}-x_j}^{x_{\alpha_{i,j}-1/2}}\beta(x_j,v)n(t,v)dv & + \sum_{k=\alpha_{i,j}}^I \beta_{j,k}n_k \Delta x_k+ \sum_{k=\alpha_{i,j}}^I \int_{\Lambda_k}(v-x_k)^2/2 (\beta(x_j,v)n(t,v))^{''}\bigg]dv+{\cal O}(\Delta x^3).
r_i! \cdot \binom{q_{i+1}}{r_i}\cdot \binom{q_i+ \sum_{j=1}^{i-1}(q_j - 2 r_j)}{r_i}\cdot \frac{\sqrt{(q_{i+1}+ \sum_{j=1}^{i}(q_j - 2 r_j) )!}}{\sqrt{q_{i+1}!}\cdot \sqrt{(q_i+ \sum_{j=1}^{i-1}(q_j - 2 r_j))!}}\le 2^{\sum_{j=1}^i (q_j - 2 r_j) + q_{i+1}+ r_i}.
\bar{x}^{a, u}_j = \lim_{T \to \infty}\, \frac{1}{T}\, E_i^u\left[\sum_{t=0}^T 1\{X(t) = j, a(t) = a\}\right]= \lim_{\beta \nearrow 1}\, (1-\beta) \, x_{ij}^{a, u}(\beta).
\Omega_{3}={\frac{1}{3 R^{2}}}\varepsilon_{\hat{m}\hat{n}\hat{p}\hat{q}}X^{\hat{m}}d X^{\hat{n}}d X^{\hat{p}}d X^{\hat{q}}\, , \qquad \tilde{\Omega}_{3}={\frac{1}{3 R^{2}}}\varepsilon_{\hat{m}^{\prime}\hat{n}^{\prime}\hat{p}^{\prime}\hat{q}^{\prime}}X^{\hat{m}^{\prime}}d X^{\hat{n}^{\prime}}d X^{\hat{p}^{\prime}}d X^{\hat{q}^{\prime}}\, ,
\begin{array}[c]{cc}\begin{array}[c]{c}\min\mathbf{c}^{T}\mathbf{x}A\mathbf{x}=\mathbf{b}\mathbf{x}\geq0\end{array}\begin{array}[c]{c}\max\mathbf{\lambda}^{T}\mathbf{b}A^{T}\mathbf{\lambda}\leq\mathbf{c}\end{array}\end{array}
\int_{S}f(y) d\sigma(y) = \int_{\widetilde{S}}\widetilde{f}(x) \frac{\|\left(D\Phi(x)^{-1}\right)^t \nabla_x \widetilde{g}(x)\| \det(D\Phi)(x)}{\|\nabla_x \widetilde{g}(x)\|}d\sigma(x).
\delta_Y \leq \sum_{\mathbf{b}\in (\mathbb{F}_2)^k \setminus \mathbf{0}}\varepsilon_{\mathbf{b}^TG\cdot X}& = \sum_{\mathbf{c}\in C \setminus \mathbf{0}}\varepsilon_{\mathbf{c}\cdot X}
\langle\tilde{L}(P_D^*\lambda),\varphi\rangle=\langle P_D^*\lambda,L\varphi\rangle =\int_DP_D\lambda(x)L\varphi(x)dx+\int_{D^c}L\varphi(x)\lambda(dx) &\eqqcolon I_1+I_2.
h_{t+s}(o,y)-h_{t+s}(o,z) = p_{t+s}(o,y)-p_{t+s}(o,z) = \sum_{w\in V}p_{t/2}(o,w)(p_{t/2+s}(w,y)-p_{t/2+s}(w,z)) = \sum_{w\in V}h_{t/2}(o,w)(h_{t/2+s}(w,y)-h_{t/2+s}(w,z))
O\left((n/\log{n})^{-\frac{\alpha^{*}}{2\alpha^{*}+d}\sum_{l=1}^d\left(\frac{1}{d}+\frac{1}{2\alpha^{*}}-\frac{1}{2\alpha_l}-\frac{r_l}{\alpha_l}\right)}\right)=O(\epsilon_{n,\boldsymbol{r}}).
\Delta E_{\vec{k}}= \frac{3}{8}\frac{g^{2}L}{\ell^{2}}+ \frac{6 4}{9 \pi^{2}}\frac{1}{g^{2}L}\left[ \sum_{\vec{\delta}}\sin^{2}\left( \frac{\pi \vec{\delta}\vec{k}}{K}\right) - 1 \right]
h(x,t) = e^{2\pi i (t+ t_0)\nu_0}\hskip-.25in \sum_k \sum_{(q,m)\in\Gamma}\big[b_{(q,m),k}\,Hg(t - (q-k)T) \hskip-.25in e^{-2\pi i m(q-k)/L}\,\Phi_{(q,m)}(t,x-(t+t_0)+(q-k)T)\big].
\log \left\{( q^{a}: q^{a + b})_{\infty}( q^{b}: q^{a + b})_{\infty}\right\}= \frac{\pi^{2}}{3 ( a + b ) \log q}+ \log \left\{2 \sin \left( \frac{a}{a + b}\pi \right) \right\}+ O ( \log q ) .
I(Z_1^n; J) = \frac{1}{M(\delta)}\sum_{i=1}^{M(\delta)}\mathrm{KL}(QP^n_i, \overline{Q}) \leq 4n(e^\alpha-1)^2 \frac{1}{\{M(\delta)\}^2}\sum_{i,i'=1}^{M(\delta)}\mathrm{TV}(P_i,P_{i'}) \leq 4n(e^\alpha-1)^2,
J_{v}^{+}= -{\frac{k}{2}}\mathrm{T r}( T_{+}g^{- 1}\partial_{v}g ) = \mathrm{c o n s t}, \hspace{1 c m}J_{u}^{-}= -{\frac{k}{2}}\mathrm{T r}( T_{-}g^{- 1}\partial_{u}g ) = \mathrm{c o n s t}.
\int_{0}^{\infty}\frac{c |mh\pm e^{\xi}nk|}{gx Q}W\left( \frac{c |mh\pm e^{\xi}nk|}{gx Q}\right) x^{w-1}\,dx = \left( \frac{c |mh\pm e^{\xi}nk|}{g Q}\right)^{w}\widetilde{W}(1-w).
P(BB|B_{T})=\frac{(1-(1-\epsilon)^{2})\times\frac{1}{4}}{0+\frac{1}{4}\epsilon+\frac{1}{4}\epsilon+\frac{1}{4}(\epsilon+\epsilon-\epsilon^{2})}=\frac{1-(1-\epsilon)^{2}}{4\epsilon-\epsilon^{2}}
f'(s) = e^{\omega(1/2-s)}\rho^{1-s}\Big( -\omega V - \log(\rho) V + V \log(\rho) \Big) \rho^s & = e^{\omega(1/2-s)}\rho^{1-s}\Big( V \log(e^{-\omega/2}\rho) - \log(e^{\omega/2}\rho) V \Big) \rho^s \ .
\mathrm{Jacobian}=\det\left(\frac{\theta^{i-j}}{(i-j)!}+\partial_{u_i}\partial^j_{\theta}R\right)_{i\ne \mathfrak m_0,j\ne d_1-1}+ \sum_{k\ne d_1-1}\det\left(a^{(k)}_{ij}\right)_{i\ne \mathfrak m_0,j\ne d_1-1},
\widetilde{\mathbf{y}}(t)=\Big(\|x_\nu^{(1)}(t)-x_\nu^{(2)}(t)\|^2,\|x_\nu^{(3)}(t)-x_\nu^{(N)}(t)\|^2,\dots,\|x_\nu^{(\frac{N+1}{2})}(t)-x_\nu^{(\frac{N+1}{2}+2)}(t)\|^2\Big)^{\top},t\in\mathbb{R},
P^* = \dfrac{t_At_B + \frac{1}{2}t_B^2 +(t_A + t_B)( T - t_A - t_B) + t_A(t_A + t_B) - \frac{1}{2}t_A^2}{T^2}begin{align*}0.8em] = \dfrac{t_At_B + \frac{1}{2}t_B^2 +(t_A + t_B)( T - t_B) - \frac{1}{2}t_A^2}{T^2}begin{align*}0.8em] = \dfrac{t_AT + t_BT - \frac{1}{2}t_A^2 - \frac{1}{2}t_B^2}{T^2}= \dfrac{t_A + t_B}{T}- \dfrac{t_A^2 + t_B^2}{2T^2},
\beta=\left( \frac{1}{2}\right)^{\left( 2+\alpha\right) /\left(1+\alpha\right)}\frac{1}{\sigma}\sqrt{\frac{A\left( 0\right)}{\pi}},\ \ \ \sigma=\left( \sqrt{I\left( 0\right)}+\sqrt{I\left( 0\right)-1}\right)^{2}.
g(\Theta) \doteq \int_{\mathcal{Z}}\left( \|\phi\|_{\mathcal{X}}^{2+\delta_{0}}+ |w(0)| + \int_{\Gamma\times [0,T]}|\gamma|^{2+\delta_{0}}\, r(d\gamma,dt) \right. &\qquad+ \left. \sup_{h\in (0,1]}\left\{h^{-\alpha}\left(\mathbf{w}_{\phi}(h,T) + \mathbf{w}_{w}(h,T)\right) \right\}\right) \Theta(d\phi,dr,dw).
\Gamma(\mathbf v,d) =\{\mathbf a\in\mathbb Z^3\mid\mathbf a\times\mathbf v/d\in M(\mathbf v,d) \}&=\{\mathbf a\in\mathbb Z^3\mid\mathbf a\times\mathbf v(\mathbf a\times\mathbf v)\times\mathbf v\}.
-\int_D(\Delta_\mu u(x))v(x)d\mu= -\int_D\left(\frac{\sum_{y\sim x}w(x,y) (u(y)-u(x))}{\mu(x)}\right) v(x)d\mu = \int_D\frac{1}{2\mu (x)}\sum_{y\sim x}w(x,y) (u(y)-u(x))(v(y)-v(x))d\mu,
\mathbf{Pr}\left( \bigg\|\frac{d}{n\epsilon}\sum_{i=1}^{n}J(u_0 + \epsilon v_{0,i})v_{0,i}- \nabla J(u_0) \bigg\| \geq t - \frac{Ld\epsilon}{2}\right) \leq 2\exp\left( - Cnt^2 \right).
{\rm ch}_{0}(c)-{\rm ch}_\mu(\pi(c)) = \Gamma_0(F_0-{\rm sgn}(D_{\mu}))\prod_{k=0}^p[F_0,\pi(a_k)] - \Gamma_0\mathrm{sgn}(D_\mu)\Big(\prod_{k=0}^p[{\rm sgn}(D_{\mu}),\pi(a_k)]-\prod_{k=0}^p[F_0,\pi(a_k)]\Big).
\sum_i (\mu_i - 14)^2 (22-\mu_i ) = \sum_i \left(-\mu_i(\mu_i-1)(\mu_i-2)+47\mu_i(\mu_i-1) -763\mu_i+4312\right) &= 78\left(-122892+47\times 6545-763\times 341+75460\right) &=0.
\frac{\Phi_{0}^{2}}{2 \pi < \Phi_{0}, \Phi_{o}>}+ \sum_{\omega \neq 0}\frac{1}{4 \pi}\frac{\Phi_{\omega}( \sigma ) \: \Phi_{\omega}( \sigma^{\prime})}{< \Phi_{\omega}, \Phi_{\omega}>}= \delta ( \sigma -{\sigma^{\prime}})
|P_t^\lambda (f)(x)-f(x_0)| \leq |P_t^\lambda (f-f\phi_n)(x)|+|P_t^\lambda (f\phi_n)(x)-(f\phi_n)(x_0)|+|(f\phi_n)(x_0)-f(x_0)| &=|P_t^\lambda (f-f\phi_n)(x)|+|P_t^\lambda (f\phi_n)(x)-(f\phi_n)(x_0)|,n\geq n_0.
\begin{array}{l}{Y^{( 1 / 2 )}= - 2 D_{2}\Phi ;}\\{Y^{( n + 1 / 2 )}= \partial_{-}Y^{( n - 1 / 2 )}- 2 i \partial_{-}\Phi \cdot \sum_{l = 1}^{n - 1}D_{2}Y^{( l - 1 / 2 )}\cdot Y^{( n - l - 1 / 2 )};}\\{n = 1 , 2 , 3 , \cdots}\\ \end{array}
\begin{array}{ll}n\left(\frac{1-k}{1+k}\right) = 2G\left( U - T - J^2 - \nu S^2\right) B^2= 1-8G\left(U + \frac{(J^2 + \nu S^2)}{2}\right) m^2= 4G(J^2 + \nu S^2).\end{array}
\sum_{j=1}^n g(\xi_{\alpha},Je_j)^2 = \sum_{j=1}^n g(J\xi_{\alpha},e_j)^2 &= \sum_{j=1}^{n+1}g(J\xi_{\alpha},e_j)^2 - g(J\xi_{\alpha},e)^2 &= ||J\xi_{\alpha}||^2 - g(\xi_{\alpha},\xi)^2 &= 1 - g(\xi_{\alpha},\xi)^2.
\imath^{\sigma,c}_{\tau,c}\circ\imath^{\tau,c}_{\mu,c}=(\imath^{\sigma,c}_{\partial\sigma,c}\circ\imath^{\partial\sigma,c}_{\tau,c})\circ\imath^{\tau,c}_{\mu,c}=\imath^{\sigma,c}_{\partial\sigma,c}\circ(\imath^{\partial\sigma,c}_{\tau,c}\circ\imath^{\tau,c}_{\mu,c})=\imath^{\sigma,c}_{\partial\sigma,c}\circ\imath^{\partial\sigma,c}_{\mu,c}=\imath^{\sigma,c}_{\mu,c}.
\left\lbrace\begin{aligned}d\tilde{q}^{\epsilon,\Delta t}(t) =\frac{\tilde{p}^{\epsilon,\Delta t}(t)}{\epsilon}dt d\tilde{p}^{\epsilon,\Delta t}(t) =-\frac{\tilde{p}^{\epsilon,\Delta t}(t)}{\epsilon^2}dt+\frac{f(\tilde{q}^{\epsilon,\Delta t}(t_n))}{\epsilon}dt+\frac{\sigma(\tilde{q}^{\epsilon,\Delta t}(t_n))}{\epsilon}d\beta(t),\end{aligned}\right.
\left\{\widetilde{\nu}_n> m, M(m)\le n^{1/2+\varepsilon/2}\right\}\subset\left\{{\rm dist}(x+S(j),\partial K)\le \frac{1}{2}\left(n^{1/2-\varepsilon}+\frac{|x+S(j)|}{n^{2\varepsilon}}\right), |x+S(j)|\le n^{1/2+\varepsilon/2}, j\le m\right\}\subset\left\{{\rm dist}(x+S(j),\partial K)\le n^{1/2-\varepsilon}, j\le m\right\}=\left\{\nu_n> m\right\},
\int_{\partial D}K_{m}^{(j)}(X,Q)n_{Q}^{(j)}f(Q)d\sigma(Q) =\int_{\partial D\cap B_{T}(P)}K_{m}^{(j)}(X,Q)n_{Q}^{(j)}f(Q)d\sigma(Q) &\,\,\,\,\,\,+\int_{\partial D\setminus B_{T}(P)}K_{m}^{(j)}(X,Q)n_{Q}^{(j)}f(Q)d\sigma(Q) &\triangleq\mathrm{J}_{1}^{(j)}+\mathrm{J}_{2}^{(j)},
\widehat{S}_{(0)}p^{( 0)}=p^{( 0)},~~\widehat{S}_{(0)}p^{(\widetilde{0})}=p^{(\widetilde{0})},~~S_{(1)}^2 p^{(1)}=p^{(1)},~~S^2_{(1)}p^{(\widetilde{1})}=p^{(\widetilde{1})}.
f_{\Theta_\rho}(\theta ) = \begin{cases}\frac{1}{\zeta}e^{-\theta /\zeta}, 0 < \theta, \rho=0, \frac{1}{\zeta}e^{-(\theta - \sigma^2_{\rm a})/\zeta}, \sigma_{\rm a}^2 < \theta ,\rho=1, 0, .\end{cases}
\{Q_{\dot{a}}, Q_{\dot{b}}\}\; = \; 2 \delta_{{\dot{a}}{\dot{b}}}\; \Bigl ({\cal H}-{g}\; \mathrm{t r}( \Phi{\cal G}) + i g \; m^{I J}\; \chi_{I}^{\T}\chi_{J}\Bigr ) \ ,
\psi_r(t,x)= \min\bigg\{\bigg( \frac{t^2+x^2}{(r+1)^2}\bigg)^{(n+1)/2n}, \bigg( \frac{t^2+x^2}{(r+1)^2}\bigg)^{-1/4n}\bigg\}\phi_r(t,x).
\frac{\partial}{\partial x^k}\frac{d}{da}\sum_{j=0}^\infty \frac{(-1)^j}{j!\:(j+1)!}\:\frac{a^{j+1}\:\xi^{2j}}{4^j}&= \sum_{j=0}^\infty \frac{(-1)^j}{j!\:(j+1)!}\: \:(j+1)\: \big(-2j \,\xi_k \big)\: \frac{a^{j}\:\xi^{2j-2}}{4^j}= \frac{1}{2}\: \xi_k \sum_{j=0}^\infty \frac{(-1)^j}{j!\:(j+1)!}\:\frac{a^{j+1}\:\xi^{2j}}{4^j}\:.
\| x\|^2 \| y\|^2 = \lim_n |[\langle x_n, x \rangle \langle x, y \rangle \langle y , x_n \rangle \xi_n , \xi_n ]| \leq \| x\| \|\langle x, y \rangle\| \| y\| \leq \| x\|^2 \| y\|^2,
\partial_{\hat{x}}[ ( \hat{x}^{2}- 1 ) ( \hat{x}^{2}- \alpha^{2}) \partial_{\hat{x}}\hat{n}( \hat{x}) ] - 2 m \zeta \partial_{\hat{x}}\hat{n}( \hat{x}) + 2 ( \hat{x}^{2}- \beta_{D}^{2}) \hat{n}( \hat{x}) = 0 ,
A_{[n,j]}=\sum_{k=i}^{n-j+i}\sum_{\gamma\in\widetilde{\mathcal{D}}_{[n,j,k]}}w(\gamma)=\sum_{k=i}^{n-j+i}A_{[k,i]}A_{[n-k-1,j-i-1]}^{(i+1)},n\geq j,
\biggl[ \frac{1}{n^d}\biggr] (t_+^2 - t_-^2) = \sum_{s=1}^{d-1}\Biggl[ \biggl( \biggl[ \frac{1}{n^s}\biggr] t_+ \biggr) \biggl( \biggl[ \frac{1}{n^{d-s}}\biggr] t_+ \biggr) - \biggl( \biggl[ \frac{1}{n^s}\biggr] t_- \biggr) \biggl( \biggl[ \frac{1}{n^{d-s}}\biggr] t_- \biggr) \Biggr]
\displaystyle U^{(pq)}_h = \frac{1}{{n \choose k}}\sum_{\mathbf{i}\in\mathcal{P}(n,k)}h\left(\mathbf{R}^{(pq)}_{\mathbf{i}, 1}, \dots, \mathbf{R}^{(pq)}_{\mathbf{i}, k}\right) = \frac{1}{{n \choose k}}\sum_{\mathbf{i}\in\mathcal{P}(n,k)}h\left(\mathbf{R}^{(pq)}_{\mathbf{i}}\right).
\overline{C}_x \cap H_t = \overline{C}_x \cap E_{T_J}(\overline{C}_x \cap H_t)w = (\overline{C}_x \cap E_{T_J})w (\overline{C}_x)w \cap (H_t)w = (\overline{C}_x)w \cap (E_{T_J})w \overline{C}_{xw}\cap H_{t^w}= \overline{C}_{xw}\cap E_{T_J^w}
f ( T ) = f_{I R}- ( ( F - N )^{2}- 1 ) ( 4 F - N ) \frac{4 5 g_{m}^{2}( T )}{3 2 \pi^{2}}- 3 F^{2}( F - N ) \frac{4 5 y^{2}( T )}{3 2 \pi^{2}}+ . . . ,
\left\langle Q(\Pi),\frac{1}{1-tE(y)H}\right\rangle =\sum_{k=0}^{\infty}\left\langle Q(\Pi),E(y)^{k}H^{k}\right\rangle t^{k}=\sum_{k=0}^{\infty}\Theta_{y,k}(Q(\Pi))t^{k}.
[y^{p^r}_{12}, y^{p^s}_{21}]=[x^{p^r}_{12}, x^{p^s}_{21}]=x^{p^r}_{12}(1- x^{-p^r}_{12}x^{p^s}_{21}x^{p^r}_{12}x^{-p^s}_{21})x^{p^s}_{21},
\frac{4}{(1+r)^2}\|(z-r\lambda)f\|^2-\|(z-\lambda)f)\|^2 &= \Big(\frac{4}{(1+r)^2}-1\Big)\|zf\|^2-\Big(1-\frac{4r^2}{(1+r)^2}\Big)\|f\|^2+2 \Big(1-\frac{4r}{(1+r)^2}\Big)~ \mbox{Re}~\lambda\langle f, zf\rangle &\geqslant \Big(\big(\frac{4}{(1+r)^2}-1\big) -\big(1-\frac{4r^2}{(1+r)^2}\big)- 2\big(1-\frac{4r}{(1+r)^2}\big)\Big)\|zf\|^2 &=0.
|u(t,x)|\leqslant\frac{c}{T}\sqrt{\sum_{n=1}^{\infty}\alpha_n^{2\beta}\left(|a_n|^2+|\overline{a}_n|^2+\sum_{k=0}^{N-1}|b_{k,n}|^2\right)}\sqrt{\sum_{n=1}^{\infty}\alpha_n^{-2\beta}\psi_n^2(x)},
\begin{array}{ccccc}\theta^{\ddagger}(q,y) = \sum_{j=0}^{\infty}(-1)^jq^{j(2j+1)}y^{2j}+ iqy\sum_{j=0}^{\infty}(-1)^jq^{j(2j+3)}y^{2j}= \theta (q^4,-y^2/q) + iqy\theta (q^4,-qy^2)~.\end{array}
W^\rho_u(\rho^*(y)\phi)(x) = \overline{w}z W^{\rho_\sigma}_u(\rho_\sigma^*(y,w)\phi)(x,z) = \overline{w}z \ell_{(y,w)}W^{\rho_\sigma}_u(\phi)(x,z) = \overline{\sigma(y,y^{-1})}\sigma(y^{-1},x) \ell_yW^\rho_u(\phi)(x) = \ell^\sigma_yW^\rho_u(\phi)(x) .
\| \pi_h w_h \|_{0,\eta,K}^\eta \le C_2 h^d \sum_{i=1}^{n_{sh}}|\bar{d}_i|^\eta &\le C_2 h^d \sum_{i=1}^{n_{sh}}|d_i|^\eta &\le C_2 C_1^{-1}\| w_h \|_{0,\eta,K}^\eta
\beta^{- 1}( b , u , v ) = 1 + \frac{1}{( k - 1 ) ( v - u - 2 )}\left[ \frac{b - 1}{b + 1}( u + 2 ) - \frac{b + 1}{b - 1}( v - 2 ) - \frac{2}{k - 1}\right] \; .
\eta=\begin{cases}\sqrt 2\left(1+\dfrac{2\,\|D\|_2}{\sigma_k+\tilde\sigma_k}\right), \mbox{if $r=k$}, begin{align*}1em] \sqrt 2\left(1+\dfrac{2\,\|D\|_2}{\sigma_r+\tilde\sigma_r}\right)+\dfrac{(2\sqrt 2+4)\,\|D\|_2}{\max\{\sigma_r,\tilde\sigma_r\}}, \mbox{if $r<k$}.\end{cases}
\gamma_{{}_{F,g}}=1+\tau\int_{-1}^1\sqrt{F(s)}F''(s)\,ds\left(\int_{-1}^1\sqrt{F(s)}\,ds\right)^{-1}&=1-\tau\int_{-1}^1\frac{F'(s)^2}{2\sqrt{F(s)}}\,ds\left(\int_{-1}^1\sqrt{F(s)}\,ds\right)^{-1}<1,
\left[\frac{1}{p_0^{2p_0}(p_0^2)^{(q-1)p_0^2}}\right]^N=(p_0^{-2})^{(p_0+(q-1)p_0^2)N}=(p_0^{-2})^k=\left(\frac{q-1}{\sqrt{q}-1}\right)^{2k}=(\sqrt{q}+1)^{2k}.
\partial^\sigma_\xi u(\xi)= \sum_{2j+\frac{k}{\beta}<q}\rho^{2j+\frac{k}{\beta}}\left( a_{j,k,\sigma}(\xi) \cos k\theta + b_{j,k,\sigma}(\xi) \sin k\theta \right) + O(\rho^q) \rho<1/2.
G_{f}( z , z^{\prime}) = G_{( \sigma )}( z , z^{\prime}) - \sum_{s = 1}^{n}\sum_{p = a , b}\sum_{N_{s}}\int_{C_{p}^{( s )}}G_{( \sigma )}( z , z^{\prime \prime}) \frac{d \theta^{\prime \prime}d z^{\prime \prime}}{2 \pi i}Y_{p , N_{s}}^{( 0 )}( t^{\prime \prime}) \chi_{N_{s}}^{( 0 )}( z^{\prime})
L_{\mathrm{I}}= \frac{1}{2}m g_{i j}\dot{q}^{i}\dot{q}^{j}+ A_{i}( q ) \dot{q}^{i}- V ( q ) + \lambda^{s}f_{s}( q ) \; \; ( i , j = 1 , 2 , \cdots , N ) \; ,
(\boldsymbol{D}^{-s}u, \boldsymbol{D}^{2s*}\phi)=(u, \boldsymbol{D}^{-s*}\boldsymbol{D}^{2s*}\phi)=(u,\boldsymbol{D}^{s*}\phi)=(\boldsymbol{D}^{s}u, \phi)\,, \forall \phi \in C_0^\infty(\mathbb{R}).
\lim_{k\rightarrow 0}\left( w_{k}\right)_{i}v_{j}\varphi =u_{i}v_{j}\varphi;\lim_{k\rightarrow 0}\left( g_{k}\right)_{i}g_{j}\phi=h_{i}g_{j}\phi{\rm strongly \,\, in \,\,}\,L^{2}(\Omega \times ] 0,T[ )
\begin{array}{l}b^{2}_{\alpha}=\lambda b^{1}_{\alpha}\quad,\quad E_{ik}=\left(\begin{array}{ll}1 \lambda \lambda \lambda^{2}\end{array}\right) E_{11}b^{1}_{\alpha}=-\frac{E_{11}}{|a|}(a^{2}_{\alpha}-\lambda a^{1}_{\alpha})\quad,\quad |a|=\det a^{i}_{\alpha}\not= 0.\end{array}
\sum_{n=1}^{\infty}\frac{\mu(n)}{\sqrt{n}}g(logn)=\sum_{\rho}\frac{h(\gamma)}{\zeta^{\prime}(\rho)}+\sum_{n=1}^{\infty}\frac{1}{\zeta^{\prime}(-2n)}\int_{-\infty}^{\infty}dxg(x)e^{-(2n+1/2)x}.
\langle f_{lm}W_{\boldsymbol{\mathrm{T}}\circ\Theta^{(n,\sigma)}},f_{lm}\rangle= \frac{1}{n}\Bigg(\sum_{i=1}^{n}(e_{l}\otimes e_{m})^{*}\Big(E_{ii}\otimes((n-c-1)E_{ii}+cE_{\sigma^{-1}(i),\sigma^{-1}(i)})\Big)(e_{l}\otimes e_{m}) &-\sum_{1\leq i\neq j \leq n}(e_{l}\otimes e_{m})^{*}E_{ij}\otimes E_{ji}(e_{l}\otimes e_{m})\Bigg) = 0.
S_{H,L}:=\displaystyle\inf\limits_{u\in D^{1,2}(\mathbb{R}^N)\backslash\{{0}\}}\ \ \frac{\displaystyle\int_{\mathbb{R}^N}|\nabla u|^{2}dx}{(\displaystyle\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)|^{2_{\mu}^{\ast}}|u(y)|^{2_{\mu}^{\ast}}}{|x-y|^{\mu}}dxdy)^{\frac{N-2}{2N-\mu}}}.
\int_{0}^{x}f(t) h(t/q) \left( -\frac{q(1+q)+t}{q^2 (1-q)t}u(t/q) +\frac{1}{q(1-q)^2t}\right)y(t)\, d_qt &= \frac{f(x)}{cq (1-q)}\sum_{n=0}^{\infty}(-q)^n \left(c-1+q^2+q^nx\right) y(q^n x) = H(x).
\displaystyle\sum_{l=1}^{N}z_{il}\overline{a_{ku}^{il}\left(Z\right)}+\displaystyle\sum_{l=1}^{p-q}\overline{z}_{jl}a_{uk}^{jl}\left(Z\right)=0, \mbox{for all $i,j,k,u=1,\dots,q$ with $k\neq u$.}
\xi^{4}= \cos | \pi | \ , \ \xi^{\alpha}= \sin | \pi | \frac{\pi^{\alpha}}{| \pi |}\ , \D_{i}\xi^{\alpha}= \partial_{i}\xi^{\alpha}+ \varepsilon^{\alpha \beta \gamma}A_{i}^{\beta}\xi^{\gamma}\ , \ \D_{i}\xi^{4}= \partial_{i}\xi^{4}\ , \nonumber
\mathcal{D}_1(m_p,m_1,w)=\Big\{\big(\mathbf{U}_0(m_p),\mathbf{U}_1(m_p,m_1,w,I),\mathbf{Y}_1\big)\in\mathcal{T}_\delta^{n}(Q_{U_0,U_1,Y_1})\Big\}
V(f_{n},\tau_{n})=[\begin{matrix}cos(\tau_{n}\sqrt{f_{n})}&1/\sqrt{f_{n}}sin(\tau_{n}\sqrt{f_{n}}\\ -\sqrt{f_{n}}sin(\tau_{n}\sqrt{f_{n}})&cos(\tau_{n}\sqrt{f_{n}})\end{matrix}]f_{n}>0(3a)
\partial''(\omega^0)' =\mu_1(\omega^1)''\wedge(\omega^1)'+\mu_2(\omega^2)''\wedge(\omega^2)', \partial''(\omega^i)' =\mu_i(\omega^i)''\wedge(\omega^0)',\i=1,2, \partial'(\omega^0)'' =\mu_1(\omega^1)'\wedge(\omega^1)''+\mu_2(\omega^2)'\wedge(\omega^2)'', \partial'(\omega^i)'' =\mu_i(\omega^i)'\wedge(\omega^0)'',\i=1,2.
\widehat{K}_{\varepsilon}=\{k\in\mathbb{Z}^{N}\, \vert\, \varepsilon(k+Y)\subset\Omega\}, \quad\widehat\Omega_{\varepsilon}= \mathrm{int}\bigcup\limits_{k\in\widehat{K}_{\varepsilon}}\left( \varepsilon(k+\overline{Y})\right) , \quad\Lambda_{\varepsilon}=\Omega\setminus\widehat\Omega_{\varepsilon},
\theta^{+}= \frac 1{\sqrt 2}( \theta^{0}+ \theta^{3}) = \Lambda^{- 1}( d u + h d v ) , \qquad \theta^{-}= \frac 1{\sqrt 2}( \theta^{0}- \theta^{3}) = \Lambda d v , \qquad \theta^{a}= d x^{a}.
\beta^{(\alpha)}_m=\frac{2}{3}\frac{\alpha^2}{\pi}+ \frac{1}{2}\frac{\alpha^3}{\pi^2}-\frac{3}{2}\frac{\alpha^3}{\pi^2}+{\cal O}(\alpha^4)=\frac{2}{3}\frac{\alpha^2}{\pi}-\frac{\alpha^3}{\pi^2}+{\cal O}(\alpha^4),
e^{it\Lambda^{1/2}}u(t) = u_0 + \int_0^t e^{is\Lambda^{1/2}}N(u)\, ds + \int_0^t e^{is\Lambda^{1/2}}\bigg( L(w) + N_1(u, w) + N_2(w,w)\bigg)\, ds \equiv u_0 + f_1(u) + f_2(u, w).
u_{\alpha a}^+(x)=\langle\langle x,v_\alpha \otimes w''_a\rangle\rangle,\ u_{\mu a}^+(x)=\langle\langle x, v_\mu \otimes w'_a \rangle\rangle , u_{\alpha k}^-(x)=\langle\langle x,v_\alpha \otimes w''_k\rangle\rangle,\ u_{\mu k}^-(x)=\langle\langle x, v_\mu \otimes w'_k\rangle\rangle
{_1F}_1 \left(-n+l+1, 2l+\delta_1 + \delta_2 + 2 ; \varepsilon r \right) =\sum_{s=0}^{n-l-1}\frac{(-n+l+1)_s}{(2l + \delta_1 + \delta_2 + 2)_s}\frac{(\varepsilon r)^s}{s!}
(1-x)^{n}_{p,q}=\prod\limits_{s=0}^{n-1}(p^s-q^{s}x) =(1-x)(p-qx)(p^{2}-q^{2}x)...(p^{n-1}-q^{n-1}x) &=\sum\limits_{k=0}^{n}{(-1)}^{k}p^{\frac{(n-k)(n-k-1)}{2}}q^{\frac{k(k-1)}{2}}\left[\begin{array}{c}n k\end{array}\right]_{p,q}x^{k}.
I =\left \{t\in \left[ 0,\infty \right) :\nu \left( \left \{\omega \in\Omega :g\left( \omega \right) \geq t\right \}\right) =1\right \}& J =\left \{t\in \left( -\infty ,0\right] :\nu \left( \left \{\omega \in\Omega :-g\left( \omega \right) \geq t\right \}\right) =1\right \}
(\delta_\alpha f) (a_1, a_2, \ldots, a_{n+1}) =~ \mu \big(\alpha^{n-1}(a_1) , f(a_2, \ldots , a_{n+1}) \big) ~ + \sum_{i=1}^{n}(-1)^i f \big( \alpha(a_1), \ldots, \alpha (a_{i-1}), \mu (a_i , a_{i+1}), \alpha (a_{i+2}), \ldots, \alpha (a_{n+1}) \big) ~ + (-1)^{n+1}\mu (f (a_1, \ldots, a_n) , \alpha^{n-1}(a_{n+1})).
\left. \mathrm{T r}K_{\delta}^{( 0 )}( s ) \right|_{{\cal C}_{\beta}}={\frac{\beta}{2 \pi}}\left. \mathrm{T r}K^{( 0 )}( s ) \right|_{R^{2}}+{\frac{1}{8 \pi i s}}\int_{0}^{\infty}r d r \int_{A}d z{\frac{\exp i{\frac{( \delta - \pi )}{\beta}}z}{\sin{\frac{\pi}{\beta}}z}}\exp{\frac{r^{2}\sin^{2}{\frac{z}{2}}}{s}},
d\beta\wedge\omega_i = \begin{cases}f_1\omega_1^2 + 2J_1\beta\wedge\eta\wedge\omega_1 \mbox{if}i=1, 2J_1\beta\wedge\eta\wedge\omega_i \mbox{otherwise},\end{cases}\mbox{and}d\beta\wedge\eta\wedge\omega_i = \begin{cases}f_1\eta\wedge\omega_1^2 \mbox{if}i=1, 0 \mbox{otherwise}.\end{cases}
{\mathcal{H}}=({\mathcal{B}}_{2}^{T}{\mathcal{D}}{\mathcal{B}}_{2})^{-1}{\mathcal{B}}_{2}^{T}{\mathcal{D}}\widetilde{{\mathcal{B}}}_{c},\ \ \ \{\mathcal{L}}=\widetilde{{\mathcal{B}}}_{c}-{\mathcal{B}}_{2}{\mathcal{H}}.
\begin{cases}\varphi_{tt}-\mu\varphi_{xx}=\left(\mathbb H[\phi^2_x]-\left[\phi\,;\,\mathbb H\right]\phi_{xx}\right)_{x}\,,{\rm in}\,\, [0,T]\times\mathbb{T}, \qquad\phi=\mathbb H[\varphi]\,, \varphi(x,0)=\varphi^{(0)}(x){\rm in}\,\, \mathbb{T}, \varphi_t(x,0)=\varphi^{(1)}(x){\rm in}\,\, \mathbb{T},\end{cases}
\mathbf{H}= \left[\mathbf{J}_{1},\ \mathbf{J}_{2}\right]\begin{bmatrix}\mathbf{\Sigma}_{1}\ \ \mathbf{\Sigma}_{2}\end{bmatrix}\left[\mathbf{J}_{1},\ \mathbf{J}_{2}\right]^{\mathbf{H}},
\begin{aligned}s_\beta(x)+\nu\int_0^x \Vert \partial_y u_\beta \Vert_{L^2_l}^2+\kappa\int_0^x \Vert \partial_y h_\beta \Vert_{L^2_l}^2 \leq C\vartheta_0^{-2}\int_0^x \left(C(u_e,h_e,u_b)+E_{u,h}\right)^2\cdot s_\beta(x) &+C\vartheta_0^{-4}\int_0^x \left(C(u_e,h_e,u_b)+E_{u,h}\right)^4\cdot E_{u,h}^2+\int_0^x C(u_b,u_e,h_e)+s_\beta(0),\end{aligned}
\mathbb{E}_{\tilde{\mathsf{C}}_n}D\Big(\tilde{P}_{\mathbf{Y}_2|\mathbf{U}_0=\mathbf{u}_0,\mathbf{U}_2=\mathbf{u}_2,\tilde{\mathsf{C}}_n}\Big|\Big|Q^n_{Y_2|U_0,U_1}(\cdot|\mathbf{u}_0,\mathbf{u}_2)\Big).
B_1 =\frac{1}{|z|}\big| ( D_t \tilde h(t_0(X_1)) - D_t \tilde h(t_0(X_2)) ) ( D_t b(X_1,t_0(X_1)) z - b(X_1,z+t_0(X_1))) \big| B_2 = \frac{1}{|z|}\big| D_t \tilde h(t_0(X_2)) ( D_t b(X_1,t_0(X_1)) z - b(X_1,z+t_0(X_1)) &- ( D_t b(X_2,t_0(X_2)) z - b(X_2,z+t_0(X_2))) ) \big|.
u_-(t) =\sqrt{t^2+1}\,e^{-t\arctan(t)}\begin{pmatrix}1 0\end{pmatrix},\, t\leq 0, u_+(t) =\sqrt{t^2+1}\,e^{-t\arctan(t)}\begin{pmatrix}\cos\left(\frac{\lambda}{2}\right) \sin\left(\frac{\lambda}{2}\right)\end{pmatrix},\,t\geq 0,
m^2_{\phi_S^0}=p^2_r=a(p_r)=4\left.\frac{(k_r+h)(\tilde{k}_r+\tilde{h})+ (k_i+s^I)(\tilde{k}_i+\tilde{s}^I)}{(k_r+h)^2+(k_i+s^I)^2}\right|_{p=p_r}.
\tilde{\hat{T}\hspace{.035in}}=\hat{T}_{g_\lambda}-\hat{T},\hspace{.2in}\tilde{\hat{c}\hspace{.018in}}=\hat{c}_{g_{\lambda}}-\hat{c},\hspace{.2in}\tilde{\hat{\Delta\hspace{.01in}}}_0=\hat{\Delta}^{g_\lambda}_0-\hat{\Delta}_0,
\left[ -\frac{4\kappa \xi}{\left( 1+\kappa \xi \Phi^{2}\right)}\left( \frac{1}{\kappa}\Lambda - \frac{1}{2}\mu^{2}\Phi^{2}+\frac{1}{4}\lambda \Phi^4 \right) - \mu^{2}+ \lambda \Phi^2\right] \Phi = 0 .
\sum_{\{k_{i}\}}\prod_{i = 1}^{4}\frac{1}{k_{i}!}\left( \partial_{i}^{\mu_{i}}\right)^{k_{i}}u ( 0 ) \int \prod_{i = 1}^{k}\mathrm{d}^{4}x_{i}~ \left( x_{i}^{\mu_{i}}\right)^{k_{i}}~ \Gamma_{x_{1}\cdots x_{k}0}.
M(\alpha_{n+1}\cdots\alpha_N) \leq H(\alpha_{n+1}\cdots\alpha_N)^{(\deg\alpha)^{2^n}}& = H(\alpha_{n+1})^{\deg\alpha_{n+1}}\cdots H(\alpha_N)^{\deg\alpha_N}& = M(\alpha_{n+1})\cdots M(\alpha_N)
S^2(A^{(4)}) = S^2(B^{(2)}) = S^2(B^{(1)})-\big{(}(b_{13}^{(1)})^2+(b_{24}^{(1)})^2\big{)}\leq \big{(}S^2(A)-a_{13}^2-a_{24}^2\big{)}- \frac{1}{2}a_{23}^2 & \leq S^2(A) - \frac{1}{2}(a_{13}^2+a_{24}^2+a_{23}^2) = S^2(A)-\frac{1}{2}S^2(A) = \frac{1}{2}S^2(A).
[y_1, y_1]=\frac{\alpha_2}{\gamma_6}y_5, [y_1, y_2]=\frac{\alpha_3}{\beta_1}y_4+ \theta_1y_5, [y_2, y_1]=-\frac{\alpha_3}{\beta_1}y_4+\theta_2y_5, [y_2, y_2]=y_4, [y_2, y_3]=\frac{\gamma_2}{\gamma_6}y_5, [y_3, y_3]=y_5.
\left. \; J^{( N )}\left( n ; \nu_{1}, \ldots , \nu_{N}\right) \right|_{\begin{array}{l}{{{}_{\! \! \Sigma \nu_{i}= n}}}\\{{{}^{\! \! c_{j l}= \delta_{j l}}}}\\ \end{array}}= \mathrm{i}^{1 - 2 \Sigma \nu_{i}}\; \pi^{n / 2}\; \frac{\prod \Gamma \left({\textstyle{\frac{1}{2}}}\nu_{i}\right)}{2^{N - 1}\prod \Gamma \left( \nu_{i}\right)}\; \frac{1}{\prod m_{i}^{\nu_{i}}}.
\Pi_{\infty}\bigl( \langle \mu,m_{t}-m_{t-}\rangle >1 \bigr) =0, &\Pi_{\infty}\biggl( \forall \lambda < \langle \mu,m_{t}-m_{t-}\rangle, \mu \bigl( z_{t-}- m_{t-}\geq 1- \alpha \lambda \bigr)\geq \lambda \biggr) =1.
\mathcal{S}= \int d^{4}x \left( | W ( z ) |^{2}+ \frac{\partial^{2}K}{\partial z^{i}\partial z^{\bar{j}}}D z^{i}D z^{\bar{j}}- \frac{1}{4 g_{i j}^{2}}F^{( i )}\wedge * F^{( j )}+ \theta_{i j}F^{( i )}\wedge F^{( j )}\right)
\begin{gathered}\sum_{m,n}a(m,n)\ \delta_{m+n}=0,\sum_{m,n}a(m,n)\m\ \delta_{m+n-1}=0, \sum_{m,n}a(m,n)\n\ \delta_{m+n-1}=0,\sum_{m,n}a(m,n)\mn\ \delta_{m+n-2}=0,\end{gathered}
\hat{f}_\theta(k-a) = e^{-d \theta/2}\left( Q_\lambda(e^{-\theta}k ) e^{-\frac{\pi}{\lambda}\frac{\delta}{4}|k|^2}\right) \left( e^{-\frac{\pi}{\lambda}e^{-2\theta}|k|^2}e^{\frac{\pi}{\lambda}\frac{\delta}{2}|k|^2}\right) \left( e^{- 2 \pi i e^{-\theta}k \cdot x_0}e^{- \frac{\pi}{\lambda}\frac{\delta}{4}|k|^2}\right) .
f_{s+t}(u, u') = \begin{array}{l}f_t(u, u') + s A \cdot f_t(u,u') - s f_t(u, A \cdot u') - s f_t( A \cdot u, u') + s^2 f_t(A \cdot u, A \cdot u') - s^2 A \cdot f_t(u, A \cdot u') - s^2 A\cdot f_t(A \cdot u, u') + s^3 A \cdot f_t(A \cdot u, A \cdot u'). \end{array}
\Delta_\sigma \Psi_* x (l) = \left( \Psi_* x(\sigma_-(\Psi_* x,l)) - 2\Psi_* x(l) + \Psi_* x(\sigma_+(\Psi_* x,l)) \right) \cdot \chi_{\{\Psi_* x(l) \neq 0 \}}&= \left( x(\sigma_-(k)) - 2x(k) + x(\sigma_+(k)) \right) \cdot \chi_{\{x(k) \neq 0 \}}= \Delta_\sigma x(k) = \Psi_* \Delta_\sigma x (l).