harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
-D_{\mu}^{ij}[A]\langle j^{\mu ,j}\rangle +D_{\mu}^{ij}[A]Y_{\nu}^k\langle\frac{\delta j^{\nu,k}}{\delta A_{\mu}^j}\rangle+gf^{ijk}(Y_{\mu}^j\frac{\delta W}{\delta Y_{\mu}^k}+J_{\mu}^j\frac{\delta W}{\delta J_{\mu}^k}+\xi^j\frac{\delta W}{\delta \xi^k}+\overline{\xi}^j\frac{\delta W}{\delta \overline{\xi}^k})=0.
l_{4j}^{(i+1)}=\rho_{i-j-1}(u_1,v_1,\ldots,u_{i-1},v_{i-1},u_i), l_{4j+1}^{(i+1)}=\rho_{i-j-2}(v_1,u_2,\ldots,v_{i-1},u_i), l_{4j+2}^{(i+1)}=y_{i+1}l_{4j}^{(i+1)}-\rho_{i-j-1}(u_1,v_1,\ldots,u_i,v_i), l_{4j+3}^{(i+1)}=y_{i+1}l_{4j+1}^{(i+1)}-\rho_{i-j-2}(v_1,u_2\ldots,v_{i-1},u_i,v_i).
1 \; =\; \beta'X_n[M]'X_n[M]\beta/n \;=\; \frac{1}{n}\sum_{i=1}^n (X_{i,n}[M]U_{M,n}^{-1/2}U_{M,n}^{1/2}\beta)^2 &\le\; \xi^2 + \frac{1}{n}\sum_{i\in R_{\beta,n}(\xi)}\|U_{M,n}^{-1/2}X_{i,n}[M]'\|^2\;\le\;\frac{1}{n}|R_{\beta,n}(\xi)| C \;+\; \xi^2,
\partial_r\varphi_{n,k}(r,l)=-\int_r^l\left(\frac{s_k(\tau)}{s_k(r)}\right)^{n-1}d\tau, &\partial_r^2\varphi_{n,k}(r,l)=1+(n-1)\int_r^l\left(\frac{s_k(\tau)}{s_k(r)}\right)^{n-1}\left(\frac{s_k'(r)}{s_k(r)}\right)d\tau, &\partial_r^2\varphi_{n,k}(r,l)+\frac{(n-1)s_k'(r)}{s_k(r)}\partial_r\varphi_{n,k}(r,l)=1.
{\cal K}^{l}( p , q ) = \int \frac{d^{2 \omega}r}{r^{2}( r - q )^{2}}\frac{( p^{l}+ r^{l})}{( p^{+}+ r^{+})}\, \, \, \, \, \, , \, \, \, l = 1 , \, 2 ,
\frac{q+1}{|G|}\sum_{\eta \in S_{\ell}^\wedge}R(\alpha \eta) (\mathbf{d}(a)) = \frac{q+1}{|G|}\left(\alpha(a)\sum_{\eta \in S_{\ell}^\wedge}\eta(a) + \alpha(a^{-1})\sum_{\eta \in S_{\ell}^\wedge}\eta(a^{-1}) \right).
\omega \left( \lambda \right) =\frac{s^{5}\gamma \beta_{2}\alpha_{2}^{\prime}}{\delta}\cos \left( s\left( c_{1}-a\right) \right) \sin\left( s\left( c_{2}-c_{1}\right) \right) \cos \left( s\left( b-c_{2}\right)\right) +O\left( s^{4}e^{t\left( b-a\right)}\right) .
\left\{\sup_{0<x<1}\frac{\left|Z_t(x,0)\right|}{x^{\overline{\eta}_{\rm c}-\gamma}}>k\right\}\subseteq \left\{\sup_{0<x<y<1}\frac{\bigl|Z_t(x)-Z_t(y)\bigr|}{|x-y|^{q(\bar\eta_{\rm c}-\gamma)/(q+1)}}>c(q)k\right\}\cup\bigcup_{n=1}^\infty \left\{\left|Z_t(n^{-q},0)\right|>\frac{k}{3\cdot2^{q}}n^{-q(\bar\eta_{\rm c}-\gamma)}\right\}\cup\left\{|W|>c(q)k\right\}.
L(x_{1},\cdot\cdot\cdot,x_{N},t)=\int_{u_{1}=-\infty}^{\infty}\cdot\cdot\cdot\int_{u_{N}=-\infty}^{\infty}f_{C}(x_{1}-u_{1},\cdot\cdot\cdot,x_{N}-u_{N},t)\cdot g_{N}(u_{1},\cdot\cdot\cdot,u_{N},t)du_{1}\cdot\cdot\cdot du_{N}.
\phi_1 := \frac{1_{(-1,1)}}{2}, \phi_2 := y_1 1_{(-5,-3) \cup (-1-2\lambda ,-1) \cup (1,1+2\lambda ) \cup (3,5)}+ y_2 1_{(-3,-1-2\lambda ) \cup (-1,1) \cup (1+2\lambda ,3)}
\mathcal{I}_1\leq CT^{\frac{-\gamma-1}{p-1}}R^N, &\mathcal{I}_2\leq CT^{\frac{-\gamma-1}{p-1}}R^{N-\frac{2p}{p-1}}, &\mathcal{I}_3\leq CT^{1-\frac{\gamma}{p-1}}R^{N-\frac{2p}{p-1}}.
c_1\cdots c_l +c_{l+1}\cdots c_{2l}+ \cdots + c_{(n-1)l+1}\cdots c_{ln}=\, \sum_{i=0}^{n-1}(10^ls_i-s_{i+1})/b =\, \frac{10^l-1}{L}\cdot\frac{\sum_{i=0}^{n-1}s_i}{B},
\bigl\| \sqrt{n}\bigl(\hat{M}^{-1}\hat{\varSigma}_{\beta}\hat{M}^{-1}\bigr)^{-1/2}\bigl(\hat{\beta}^{(2)}_n-\beta_0\bigr) \bigr\|^2 &\quad=n\bigl(\hat{\beta}^{(2)}_n-\beta_0\bigr)^\top \bigl(\hat{M}^{-1}\hat{\varSigma}_{\beta}\hat{M}^{-1}\bigr)^{-1}\bigl(\hat{\beta}^{(2)}_n-\beta_0\bigr)\xrightarrow{}\chi^2_m.
L_1(x,p;\lambda)\begin{pmatrix}1 0 \end{pmatrix}= \begin{pmatrix}1+\lambda p_1 1 \end{pmatrix}=\begin{pmatrix}\gamma_1 1 \end{pmatrix}, \mathrm{and}\begin{pmatrix}1 0 \end{pmatrix}\begin{pmatrix}\gamma_N 1 \end{pmatrix}= \gamma_N.
\mathbb{E}[R^2_{t-1}|R_{t-1}] =R^2_{t-1}+2R_{t-1}(\mathbb{E}[\eta_t|R_{t-1}]-1)+\mathbb{E}[\eta^2_t|R_{t-1}]-2\mathbb{E}[\eta_t|R_{t-1}]+1 &\geq R^2_{t-1}+\sigma^2-1-2h\varepsilon_1-\varepsilon_2 &\ge R^2_{t-1}+2\frac{\sigma^2-1}{3}-\varepsilon_2 &\geq R^2_{t-1}+\frac{\sigma^2-1}{3}.
\widehat\chi\mathcal{A}^*_k\mathcal{G}^*_k\Box^{(0)}_{s,k}\chi^2\mathcal{S}_k\mathcal{A}_k\widehat\chi+\widehat\chi\mathcal{A}^*_k\mathcal{S}^*_k\chi^2\mathcal{S}_k\mathcal{A}_k\widehat\chi=\widehat\chi\mathcal{A}^*_k\chi^2\mathcal{S}_k\mathcal{A}_k\widehat\chi.
\hat{\pi}_{2 n L}^{( 2 )}= \pi_{2 n L}^{( 2 )}, \quad \hat{\tau}_{2 n L}^{( 2 )}= \tau_{2 n L}^{( 2 )}, \quad \hat{\sigma}_{2 n L}^{i j ( 2 )}= \sigma_{2 n L}^{i j ( 2 )}.
w(e_\ell ) \in (0,1)\, \forall \ell \in [N] \,, \sum_{\ell = 1}^N w(e_\ell) = (n-M-1) (\tilde w( \tilde T) - D_{\hat n}) = -(n-M-1)D_{\hat n}+ \sum_{j=1}{n-M}D_{\hat j}
||T_h(\lambda)-T_0|| \leq||(T_h(\lambda)-T_0)|_{B_1}+(T_h(\lambda)-T_0)|_{B_2}|| \leq||T_h(\lambda)-T_0||_{L^2(B_1)}+||T_h(\lambda)-T_0||_{L^2(B_2)}\leq C_1h+C_2h =(C_1+C_2)h
g(\psi_t(0)) = \lim_{h\rightarrow \infty}\frac{\psi_{t+h}(0)- \psi_h(0)}{1-\psi_h(0)}&= \lim_{h\rightarrow \infty}\frac{-(1-\psi_{t+h}(0))+(1- \psi_h(0))}{1-\psi_h(0)}&= \frac{- \psi_{t+h}^{'}(1)+ \psi_{h}^{'}(1)}{\psi_{h}^{'}(1)}&= -\psi_{t}^{'}(1)+1.
h_1 \; = \; EK (\kappa K^2+\kappa^{-1}EF) -\kappa q^2 K'^2 EK, h_2 \; = \; FK (\kappa K^2+\kappa^{-1}EF) -\kappa q^{-2}K'^2 FK \hbox{and}h_3 \; = \; K^2 (\kappa K^2+\kappa^{-1}EF) -\kappa K'^2 K^2.
\begin{array}{cc}(k,b,c)=\left(1-q,q x_i^2,- \frac{\tau^6 + 2 \tau^5 + 4 \tau^4 + 8 \tau^3 + 9 \tau^2 + 4 \tau +1}{4 \tau^2 (\tau +1)^2}\right) \textrm{if}n=1 (k,b,c)=\left(q-1,-qx_i^2,- \frac{\tau^6 + 2 \tau^5 + 4 \tau^4 + 8 \tau^3 + 9 \tau^2 + 4 \tau +1}{4 \tau^2 (\tau +1)^2}\right) \textrm{if}n=2 \end{array}
{\cal G}_{( - )}^{a}[{\vec{x}},{\vec{y}}; 0 ] = - i \delta^{2}( x_{\perp}- y_{\perp}) \sum \frac{1}{( 2 n + 1 ) i \pi}\exp \left( \frac{( 2 n + 1 ) i \pi ( x^{-}- y^{-})}{2 L}\right)
\begin{array}{r}\hat X_\nu= \{(z_1,z_2,z_3) \in T \mid\nu_1(\mathcal L_1(z_1)\mathcal L_2(z_2)\mathcal L_3(z_3) + b_1b_2b_3)+ begin{align*}6pt]\nu_2(\mathcal L_1(z_1) b_2b_3 + b_1\mathcal L_2(z_2)\mathcal L_3(z_3))=0 \}\,,\end{array}
\varphi_{b_1}(\sigma_A(x)) = \sigma_B^{b_1(\sigma_A(x))}(h(\sigma_A(x))) & = \sigma_B^{1 + b_1(x) -l_1(x)}\sigma_B^{k_1(x)}(h(\sigma_A(x))) & = \sigma_B(\sigma_B^{b_1(x)}(h(x))) & = \sigma_B(\varphi_{b_1}(x)).
\bar{\Phi}_{1 i}= \Phi_{1}^{i}{}^{\dagger}\gamma^{0}= \Phi_{1}^{j}{\cal E}_{j i}= i ( \bar{\theta}_{3 1 i}{\cal X}_{1 3}^{- 1}- \bar{\theta}_{2 1 i}{\cal X}_{1 2}^{- 1})
\varphi_{\kappa}(t) = \frac{1}{p}(\kappa + t)^p - \frac{\kappa}{p-1}(\kappa + t)^{p-1}+ \frac{\kappa^p}{p(p-1)}, \varphi_{\kappa}'(t) =(\kappa + t)^{p-2}t, \varphi_{\kappa}''(t) = (\kappa + t)^{p-2}\left( 1 + (p-2) \frac{t}{\kappa + t}\right).
\dfrac{e^{i xt^{-1/\ell}\xi}}{A(t^{-1/\ell}\xi)}=\sum\limits_{k=0}^{\ell-d}t^{-(k-\ell)/\ell}{Q}_{k-\ell}(\xi,x)+\hat{Q}(t^{-1/\ell}\xi,x),
\begin{array}{c c l}{{\left( \begin{array}{c}{{1}}\\{{r}}\\{{\tilde{r}}}\\{{\tilde{r}}}\\ \end{array}\right)}}&{{=}}&{{\mathcal{S}\, [ a , b ; E ; V , | W | ] \, \left( \begin{array}{c}{{t}}\\{{t}}\\{{\tilde{t}}}\\{{\tilde{t}}}\\ \end{array}\right) ~ ,}}\\ \end{array}
\mathbf{Z}_{l,j,k,q}^{(\omega )}(t):=\frac{n^{d}}{\left\vert \Lambda_{l}\right\vert}\sum\limits_{x,y\in \mathfrak{L}\cap (lb_{j})}\sigma_{\mathrm{p}}^{(\omega )}\left( y,y-e_{q},x,x-e_{k}\mathbf{,}t\right) \ .
\widetilde{\psi}_{k}^{(0^{2})}(x;1,1,\lambda_{1})=\left( 1-z\right)^{3/4}(1+z)^{3/4}\frac{\widetilde{P}_{1,1,k}^{\left( 0^{2}\right)}(z; \lambda_1)}{\lambda_{1}-\frac{1}{2}\left( z+1\right)^{2}+\frac{1}{6}(z+1)^{3}},
L_{1}(u)(t,r) = \frac{\sin(2 u(t,r))}{2r^{2}}\left(\cos(2Q_{\frac{1}{\lambda(t)}}(r))\left(\cos(2v_{corr})-1\right) - \sin(2Q_{\frac{1}{\lambda(t)}}) \sin(2v_{corr})\right)
\rho^{2}\frac{\partial^{2}V}{\partial r^{2}}+2\rho\sum\limits_{l=1}^{n}\gamma_{l}\frac{\partial^{2}V}{\partial r\partial\alpha_{l}}+\sum\limits_{l=1}^{n}\sum_{k=1}^{n}\gamma_{l}\gamma_{k}\frac{\partial^{2}V}{\partial\alpha_{l}\partial\alpha_{k}}\leq 0
-2i\tilde A_+^{(i)}\Big(\frac{\partial}{\partial \tau}\Big) w_i^g-2iJ_i^{-1}\Big(\frac{\partial}{\partial\tau}\Big) J_i\tilde A_+^{(i)}w_i^g+2i\sum_{j=1}^{n-1}\tilde A_j^{(i)}\alpha_{js}^{(i)}\frac{\partial}{\partial\tau}w_i^g &+i\sum_{j=1}^{n-1}2J_i^{-1}\Big(\frac{\partial}{\partial\tau}\Big)J_i\alpha_{js}^{(i)}\tilde A_j^{(i)}w_i^g.
\left|\sum_{n=1}^N\varphi_{J,h}(x)\right|\leq F_{\varphi_{J,h}}(0,2^m,x)+\sum_{l=\lceil m/3\rceil}^{m-1}F_{\varphi_{J,h}}(2^m+\mu_{l+1}2^{l+1},2^l,x)+CN^{1/3}.
X_N^{(K)}(a) \ = \ \frac{1}{\Delta_K(a^2)}\prod_{f=0}^{K-1}\frac{(N+f)!}{\alpha^{N+f}}\det\left[L_{N+f-1}^{(0)}(-\alpha a_{f'}^2)\right]_{f,f^\prime=1,\ldots,K}\ .
d\overline{\Pi}_{\psi}(e^+)A([\epsilon,x])- d\overline{\Pi}_{\psi}(e^-)A([\epsilon,x]) &=\pi i\epsilon x^2 A([\epsilon,x])-\frac{\epsilon i}{4\pi}\frac{d^2}{dx^2}A([\epsilon,x]) &=\pi i\epsilon x^2 A([\epsilon,x])-\pi i\epsilon x^2 A([\epsilon,x])- \frac{\epsilon i}{4\pi}[-2\epsilon \pi e^{-\epsilon \pi x^2}] &=\tfrac{i}{2}A([\epsilon,x]);
\|g\|_{1,\infty}':=\sup_{X_0\in I\atop s\in [0,\beta)_h}\sum_{X\in I^0}|g(X_0,X+s)|, &\|g\|:=\|g\|_{1,\infty}'+(1+\beta^{-1})\|g\|_{1,\infty}.
[ L_{1}, L_{2}, [{\textstyle \frac{k + 1}{2}}, + ] ] \oplus [ L_{1}, L_{2}, [{\textstyle \frac{k + 1}{2}}, - ] ] \, l o n g r i g h t a r r o w \ \bigoplus_{J}\N_{L_{1}\frac{k + 1}{2}}^{( k )}{}^{J}\ [ J , L_{2}, 0 ] \ \ .
\tilde T(\tilde u, \tilde v) = (n(\tilde u) \;\; n(\tilde v))^{-1/2}\;\;\sum_{u \in \tilde u}\sum_{v \in \tilde v}T(u,v) \;\; = (n(\tilde u) / n(\tilde v))^{1/2}\sum_{v \in \tilde v}T(u,v) \;\;.
SH^{-*}(V)\to SH^{-*}_{\leq 0}(V) = SH^{-*}_{=0}(V)\cong H_{n+*}(V), \cr H^{n-*}(V) \cong SH_*^{=0}(V) = SH_*^{\leq 0}(V) \to SH_*(V).
\frac{1}{\sqrt{w_\theta(T_N)}}\left(\frac{1}{T_N}\sum_{k=1}^N (X^\theta_{k\Delta_N})^2\Delta_N - \frac{1}{T_N}\int_0^{T_N}(X^\theta_t)^2\, \d t\right) \to 0 \quad\mbox{a.s.}
|a + b|^2 = \left| \left\langle \binom{\sqrt{y}a}{\frac{b}{\sqrt{y}}}, \binom{\frac{1}{\sqrt{y}}}{\sqrt{y}}\right\rangle \right|^2 \leq \left(y |a|^2 + \frac{|b|^2}{y}\right) \left( \frac{1}{y}+ y \right) = \left( 1 + y^2 \right) |a|^2 + \left( \frac{1}{y^2}+ 1 \right) |b|^2.
I_{1}=\frac{1}{2}\int_{M}|\nabla_{g}W_{\delta,q}|^{2}+\frac{1}{2}\int_{M}2\delta\nabla W_{\delta,q}\nabla V_{\delta,q}+\delta^{2}|\nabla_{g}V_{\delta,q}|^{2}d\mu_{g}=I_{1}'+I_{1}''+I_{1}'''
\min_{0 \leq i \leq N}\{\|\nabla f(x_i)\|^2\}{}={}\min_{0 \leq i \leq N}\{\|\nabla f(x_i)\|^2\}\sum_{i=0}^{N-1}\frac{p_i}{S}{}\leq{}\sum_{i=0}^{N-1}\frac{p_i}{S}\min\{\|\nabla f(x_i)\|^2 \,,\, \|\nabla f(x_{i+1})\|^2\}
u(x)=O\left( \left(\frac{1}{|x|}\right)^{n-2}\right) +o\left( \left(\frac{1}{|x|}\right)^{\frac{n-2}{n}((n-2)\lambda+\alpha)}+\left(\frac{1}{|x|}\right)^{\frac{n-2}{n}\left(\frac{n-2}{n}\beta\lambda+\alpha\right)}\right).
E\left[S^2\right] - E\left[T_1^2\right] = E\left[S^2 - T_1^2\right] = E\left[\left(S + T_1\right) \left(S - T_1\right)\right] &\leq E\left[\left(S + T_1\right)^2\right]^{\frac{1}{2}}E\left[\left(S - T_1\right)^2\right]^{\frac{1}{2}}\leq 2 E\left[S^2\right]^{\frac{1}{2}}E\left[\left(S - T_1\right)^2\right]^{\frac{1}{2}}.
A = (2a+2)(2a+3) +2b_1(4a-2l-d+7-4z_1), B_1 = 2b_1(4a+2b_1+7), E_{1,l}= \lambda [2a(2l+d+4) + 2b_1(4a-2l-d+7-4z_1) - l(l+d-1) + 6], \psi_{1,l}(r) \propto r^l (1+\lambda r^2)^{-a-1}[1 - z_1(1+\lambda r^2)] \exp\left(- \frac{b_1}{1+\lambda r^2}\right),
G_{E}^{\i j i^{\prime}j^{\prime}}( \mu , X , X^{\prime}) = 4 \kappa \sum_{p = 3 i}^{i \infty}\frac{i}{2 p}\frac{\Psi_{p}^{r}( X ) \Psi_{p}^{l}( X^{\prime})}{( 1 + p^{2})}W_{( p )}^{\i j i^{\prime}j^{\prime}}( \mu ) .
\lambda_{1,p}(\alpha) \le \frac{\int_0^1 \frac{| \phi' |^p F_\alpha}{| \alpha' |_g^{p-1}}\,dt}{\int_0^1 | \phi |^p | \alpha' |_g F_\alpha \,dt}< \frac{\int_0^1 \frac{| \phi' |^p F_\beta}{| \beta' |_g^{p-1}}\,dt}{\int_0^1 | \phi |^p | \beta' |_g F_\beta \,dt}= \lambda_{1,p}(\beta)
x_{i,1,k}= k-i, \tfrac{1}{2}S -x_{i,1,k}= 6m+i-k+1, x_{i,2,k}= 2m+i, \tfrac{1}{2}S - x_{i,2,k}= 4m-i+1, x_{i,3,k}= m-i+k, \mbox{and}\tfrac{1}{2}S - x_{i,3,k}= 5m+i-k+1.
k^{+}:=\min(\{\ell+1\}\cup\{k+1\leq j\leq\ell\mid i_{j}=i_{k}\}), k^{-}:=\max(\{0\}\cup\{1\leq j\leq k-1\mid i_{j}=i_{k}\}), k^{\mathrm{max}}:=\max\{1\leq j\leq\ell\mid i_{j}=i_{k}\}, k^{\mathrm{min}}:=\min\{1\leq j\leq\ell\mid i_{j}=i_{k}\}.
G^{( + )}( x , x^{\prime}) = - 2 \pi i \int \frac{d^{n - 1}k}{( 2 \pi )^{n - 1}}e^{2 i k_{\mu}\tilde{x}^{\mu}}\frac{( 1 + f ( k_{0}, \overline{{x}}) )}{2 | \omega |}
\alpha_6^2 =\frac{I_{11}I_{22}I_{33}-I_{11}I_{23}^2+2I_{12}I_{13}I_{23}-I_{12}^2I_{33}-I_{13}^2I_{22}}{I_{11}I_{22}-I_{12}^2}=\frac{I_{123}^2}{I_{11}I_{22}-I_{12}^2}
\frac{\tilde{C}_{1 2 3}}{C_{1 2 3}^{( \mathrm{v a c )}}}= - \frac{J}{\sqrt{J_{1}J_{2}}}\frac{1}{n^{2}\pi^{2}}\sin^{2}( \frac{n \pi J_{1}}{J}) \cdot ( 1 + \l^{\prime}n^{2})^{- 1 / 2}.
F_{\mathcal{Y},a,b}(x) = \sum_{(i,j) \in [m] \times [n]}\overline{y}_{(i,j)}g_{(i,j)}(x) + \sum_{(q,p) \in \mathcal{Y}}\Delta y_{(q,p)}g_{(q,p)}(x) = F_{\mathcal{Y},0,0}(x).
\operatorname{supp}\mathcal{F}(f)\subset\lambda\mathcal{C}\ \ \Rightarrow\ \C^{-1-k}\lambda^{k}\|f\|_{L^{p}}\leq \sup_{|\alpha|=k}\|\partial^{\alpha}f\|_{L^{p}}\leq C^{1+k}\lambda^{k}\|f\|_{L^{p}}.
2Q^*(u) = \alpha_j c_{e,j}^2 + \sum_{i< j}\alpha_i c_{e,i}^2w^2_{ij}+ \lambda_j c_{s,j}^2(1-w_{jj})^2 + \sum_{i>j}\lambda_i c_{s,i}^2 w_{ij}^2 + \sum_{i<j}\alpha_i (w_{ij}-w_{ij}^2) + \sum_{i\ge j}\lambda_iw_{ij}(1-w_{ij}) = \sigma^2_j.
K=-\frac{\beta^{2}}{((s-\alpha)^{2}+\beta^{2})^{2}},K_{m}=-\frac{r[(s-\alpha)^{2}+\beta^{2})]+\beta_{t}(s-\alpha)+\beta\alpha_{t}}{[(s-\alpha)^{2}+\beta^{2}]^{\frac{3}{2}}}.
P_{\delta}=c_\delta\cdot\zeta_{2d}^{\alpha_0+\cdots+\alpha_n}\prod_{j=1}^{\frac{n}{2}+1}\left(\frac{x_{{2j-2}}^{d-1}-(\zeta_{2d}^{\alpha_{2j-2}}x_{{2j-1}})^{d-1}}{x_{{2j-2}}-\zeta_{2d}^{\alpha_{2j-2}}x_{{2j-1}}}\right),
\sum_{n=1}^{\infty}n q^n (q^{n+1})_\infty = \sum_{n=1}^{\infty}\frac{(-1)^{n-1}q^{\frac{n(n+1)}{2}}}{(1-q^n) ( q)_n}= \sum_{n=1}^{\infty}\frac{q^n}{1-q^n}.
[ a ( \vec{k}\, ) , a^{\dagger}( \vec{\ell}\, ) ] = ( 2 \pi )^{3}2 \omega ( \vec{k}\, ) \delta^{( 3 )}\left( \vec{k}- \vec{\ell}\, \right) = [ b ( \vec{k}\, ) , b^{\dagger}( \vec{\ell}\, ) ] \ .
x_{j_1j_2j_3}= x_{1,j_1}x_{2,j_2}x_{3,j_3}j_1,j_2,j_3\in\{1,2\}, x_{222}^* = (x_{1,1}+x_{1,2})(x_{2,1}+x_{2,2})(x_{3,1}+x_{3,2}).
\|p\|_{\xi_2}= \sum_q |\langle q,p \rangle|{\xi_2}^{\deg(q)}= \sum_q |\langle q,p \rangle| \xi_1^{\deg(q)}\left( \frac{\xi_2}{\xi_1}\right)^{\deg(q)}& \leq \left( \frac{\xi_2}{\xi_1}\right)^{\deg(p)}\sum_q |\langle q,p \rangle| \xi_1^{\deg(q)}= \left( \frac{\xi_2}{\xi_1}\right)^{\deg(p)}\|p\|_{\xi_1},
D_k=D_k \cap X_{-1}=\coprod_{\lambda_0 < \lambda_1 < \cdots < \lambda_k}D_{\lambda_0}\cap D_{\lambda_1}\cap \cdots \cap D_{\lambda_k}&X_l=D_{-1}\cap X_l=\coprod_{i_0 < i_1 < \cdots < i_l}X_{i_0}\cap X_{i_1}\cap \cdots \cap X_{i_l}
\binom{-\chi-N+d(N+1)}{k}=\binom{-\chi+d(N+1)}{k}-\sum_{\ell=1}^k \frac{N}{\ell}\binom{(N+1)(\ell-1)}{\ell-1}\binom{-\chi+(d-\ell)(N+1)}{k-\ell}.
{\bf E}[M_{(\tau=1,\lambda_1,\lambda_2)}^{-n}] = \prod\limits_{j=0}^{n-1}\frac{\Gamma(2+\lambda_1+j) \,\Gamma(2+\lambda_2+j)}{\Gamma(2+\lambda_1+\lambda_2+j)\,\Gamma(1+j)}.
\tilde{D}_{\alpha \beta}^{\mu \nu}( k ) = \frac 1{k^{2}}\left[ P_{\alpha \beta}^{\mu \nu}+ \frac 1{k^{2}}\tilde{\Pi}_{\alpha \beta}^{\mu \nu}( k ) + \left( \frac 1{k^{2}}\right)^{2}\tilde{\Pi}_{\rho \sigma}^{\mu \nu}( k ) \tilde{\Pi}_{\alpha \beta}^{\rho \sigma}( k ) + \dots \right]
{\cal L}= - \frac 1 4 F_{\mu \nu}^{2}+ \frac 1 8 \theta_{\alpha \beta}F_{\alpha \beta}F_{\mu \nu}^{2}- \frac 1 2 \theta_{\alpha \beta}F_{\mu \alpha}F_{\nu \beta}F_{\mu \nu}+{\cal O}( \theta^{2}) + A_{\mu}J_{\mu},
A_h\left[\left(\boldsymbol{u}_h,p_h\right),\left(\boldsymbol{v}_h,q_h\right)\right]=L_h\left(\boldsymbol{v}_h,q_h\right)~~~~~~~~~\forall\left(\boldsymbol{v}_h,q_h\right)\in V_h^k\times Q_h^k,
\begin{array}{c}{\tilde{\cal H}}=g_{A\bar B}V^A{\bar V}^{\bar B}+ iV^A_{,C}g_{A\bar B}{\bar V}^{\bar B}_{,\bar D}\theta^C{\bar\theta}^{\bar D}- -R_{A{\bar B}C{\bar D}}V^A_{,C}{\bar V}^{\bar B}_{;\bar D}\theta^A\theta^C{\bar\theta}^{\bar B}{\bar \theta}^{\bar D},\end{array}
T_{0}^{(i)}=\dfrac{R_{i+1}-R_{i}}{2}+e, T_{j}^{(i)}=\begin{cases}R_{i+1}-R_{j}+d(-a_{j}a_{j+1}) \,, R_{j+1}-R_{i}+d(-a_{j}a_{j+1}) \,,\end{cases}
\small\!F\!=\!\sum\limits_{i \in \mathcal{U}}\!\sum\limits_{n_{ik}\in \mathcal{N}'}\!\!{{\!\!p_{kin_{ik}}}}E_{k,i}(\theta_{i,n_{ik},k}^S)+\! \!\sum\limits_{i \in \mathcal{U}}\sum\limits_{n_{ik}\in{{\mathcal{C}'_i}}}\!\!{{\!p_{kin_{ik}}}}E_{k,i}(\theta_{i,n_{ik},k}^G).
\partial_{t}\kappa = \partial_{xx}\kappa+\kappa\langle\partial_{xx}N,N\rangle-2\kappa^{2}\langle T,\partial_{x}N\rangle &= \partial_{xx}\kappa-\kappa\langle\partial_{x}N,\partial_{x}N\rangle +2\kappa^{3}&= \kappa_{ss}+\kappa^{3}
\Pi^{\alpha}= \mathcal{P}_{+}d \theta^{\alpha}\, , \qquad \Pi^{\mu}= d x^{\mu}+{\frac{1}{2}}( C \Gamma^{\mu})_{\alpha \beta}\theta^{\alpha}\Pi^{\beta}\, , \qquad \Pi^{\mu_{1}\dots \mu_{5}}= d \varphi^{\mu_{1}\dots \mu_{5}}+{\frac{1}{2}}( C \Gamma^{\mu_{1}\dots \mu_{5}})_{\alpha \beta}\theta^{\alpha}\Pi^{\beta}\, .
\phi_{1}- \frac{\phi_{1}^{7}}{7}+ 3 \phi_{1}^{5}\phi_{2}^{2}- 5 \phi_{1}^{3}\phi_{2}^{4}+ \phi_{1}\phi_{2}^{6}= \gamma ; \hspace{0 . 4 c m}\phi_{2}- \phi_{1}^{6}\phi_{2}+ 5 \phi_{1}^{4}\phi_{2}^{3}- 3 \phi_{1}^{2}\phi_{2}^{5}+ \frac{\phi_{2}^{7}}{7}= \gamma_{\perp}
\widetilde{G}_{\varepsilon}\left( u,v\right) =2\left( A_{\varepsilon}u-A_{\varepsilon}v,u-v\right) -2\left( f\left( u\right) -f\left(v\right) ,u-v\right) -\left\vert B\left( u\right) -B\left( v\right)\right\vert_{Q}^{2}& \geq\left( 2a\theta-2L_{1}-L\right) \left\vert u-v\right\vert^{2}
\Psi (\Theta , \varphi)= c_0 \Theta (1- \ln \Theta) - \gamma_0 (\Theta - \Theta_c) \varphi+ \Theta \bigg( - \mu \varphi + \mathcal{W}(\varphi) + \frac{\nu}{2}|\nabla \varphi|^2 \bigg),
E_{d}=\frac{(\frac{P_{1}+P_{2}}{2})}{(\frac{Q_{d_{1}}+Q_{d_{2}}}{2})}\times\frac{\Delta Q_{d}}{\Delta P}=\frac{P_{1}+P_{2}}{Q_{d_{1}}+Q_{d_{2}}}\times\frac{\Delta Q_{d}}{\Delta P}
\left(\frac{1}{n}\sum_{0\le t<n}\exp\left\{i\frac{2\pi}{n}\left(u\,\sqrt{t}-\sqrt{t}\,w\right)^{2}+i\frac{2\pi}{n}\left(u\,\sqrt{t}-v\,\sqrt{t}\right)^{2}+i\frac{2\pi}{n}\left(\sqrt{t}\,v-\sqrt{t}\,w\right)^{2}\right\}\right)=
\|X\|_{S_{2/3}}=\min_{U\in\mathbb{R}^{m\times d},V\in\mathbb{R}^{n\times d}:X=UV^{T}}\|U\|_{S_{4/3}}\|V\|_{S_{4/3}}&=\min_{U\in\mathbb{R}^{m\times d},V\in\mathbb{R}^{n\times d}:X=UV^{T}}\left(\frac{\|U\|^{4/3}_{S_{4/3}}\!+\|V\|^{4/3}_{S_{4/3}}}{2}\right)^{3/2}.
Q^{(\beta)}_{\mathbf{t}}(x_n) = \sum_{e \in \beta^{-1}(1)}|x_{v_1(e), n}- x_{v_1(e), n}|^2/4t_e + \sum_{e \in \beta^{-1}(-1)}|x_{v_1(e), n}+ x_{v_1(e), n}|^2/4t_e
\eta(\eta(\tau_j, \tau_i),\eta(\tau_j,\tau_\ell)) = \eta((\tau_n^q\tau_1)^{n-j}\tau_n^{q-1+D(n-j+i)}\tau_{{n-j+i+1}},(\tau_n^q\tau_1)^{n-j}\tau_{{n-j+\ell}}) = \eta(\tau_n^{q-1+D(n-j+i)}\tau_{{n-j+i+1}},\tau_{{n-j+\ell}}).
p_\psi ( \bar{u}) = p_\psi ( \bar{u}^\prime ) \Longleftrightarrow \psi(\bar{x},\bar{x}^\prime) \bar{u}=\bar{u}^\prime p_\psi ( g \bar{u}) = p_\psi ( g \bar{u}^\prime) \Longleftrightarrow \psi(g\bar{x}, g \bar{x}^\prime) g \bar{u}= g\bar{u}^\prime, (g^{-1}\cdot\psi)(\bar{x}, \bar{x}^\prime) \bar{u}=\bar{u}^\prime,
R_1= \mu^{-1}h^{1-d}\int_{X'_+}\rho^{d-1}\gamma^{-1}\,dx , R_2= \mu h^{s-d}\int_{X'_-}\rho_1 \rho^{d-s-1}\gamma^{1-s}\,dx\shortintertext{provided}\mu \rho_1 \gamma \ge \rho
[y_1, y_1]=y_4+\theta_1y_5, [y_1, y_2]=\frac{\alpha_3}{\alpha_1}y_4+\theta_2y_5, [y_2, y_1]=\theta_3y_4+\theta_4y_5, [y_2, y_2]=\theta_5y_4+\theta_6y_5, [y_1, y_3]=y_5, [y_2, y_3]=\frac{\gamma_1}{\gamma_3}y_4+\theta_7y_5, [y_3, y_2]=y_4.
\widetilde{\mathcal T}(M;\mathbf x,\mathbf y)=\sum_{A\subseteq E}x^{r(M/A)}y^{r^*(M\backslash A^c)}\prod_{\stackrel{\mbox{\rm \tiny conn. cpts}}{\mbox{\rm \tiny $M_i$ of $M/A$}}}x_{\bar{g}(M_i)}\prod_{\stackrel{\mbox{\rm \tiny conn. cpts}}{\mbox{\rm \tiny $M_j$ of $M\backslash A^c$}}}y_{\bar{g}(M_j)},
\Phi^{-1}(H_{b_R,c_R}\Phi(f))(x) = \sum_{Ry \in X_R}b_R(Rx,Ry)(\Phi(f)(Rx) - \Phi(f)(Ry)) + c(Rx)\Phi(f)(Rx) = \sum_{Ry \in X_R}\sum_{y' \in Ry}b(x,y')(f(x) - f(y')) + c(x)f(x) = \sum_{y \in X}b(x,y)(f(x) - f(y)) + c(x)f(x) = H_{b,c}f(x),
e^{2\pi i \left(\frac{k_L}{3}+ \frac{1}{6}\right)}\mathbf{Q}_\alpha\big( p_{GL}\big(g_z( \tilde{\varphi}_i^{-1}\circ \tilde{\chi}_{\sigma_R}) \big)\big) = e^{2\pi i \left(\frac{k_L}{3}-\frac{k_R}{3}+\frac{1}{2}\right)}\ .
\sum_{n=0}^{N}\beta_{n}\xi_{n}^{\,2}+2\sum_{n=0}^{N-1}\alpha_{n}\xi_{n}\xi_{n+1}=a\xi_{0}^{\,2}+q^{-N}\xi_{N}^{\,2}+\sum_{n=0}^{N-1}q^{-n}\left(\!\left(\frac{a}{q}\right)^{\!1/2}\xi_{n+1}+\xi_{n}\right)^{\!2}\geq0.
\left. \begin{array}{l l l}{{W}}&{{=}}&{{\lambda \wp ( z - a ) ,}}\\{{E}}&{{=}}&{{8 | \lambda |^{2}\frac{| \wp ( z - a ) | | \wp^{2}( z - a ) - \wp^{2}( L / 2 ) |}{[ 1 + | \lambda \wp ( z - a ) |^{2}]^{2}}.}}\\ \end{array}\right.
\bar{D}D{\hat{\Phi}}_{1}={\frac{1}{3}}e^{3{\hat{\Phi}}_{1}-{\hat{\Phi}}_{2}}\bar{\Psi}^{+ b}\Psi^{- b}, \qquad \bar{D}\Psi^{- b}= 0 = D \bar{\Psi}^{+ b}.
\sum_{j=1}^n e^{-ij\omega r_p(y_0)}A_\omega(y_{jp})=\chi_\omega(y_0)-e^{-in\omega r_p(y_0)}\chi_\omega(y_{np}), & \sum_{j=1}^n e^{-ij\omega r_p(y_0)}A_\omega(y_0) =\chi_\omega(y_0)-e^{-in\omega r_p(y_0)}\chi_\omega(y_0).
|\Lambda_{P_1,\dots,P_{m_1}}^{Q_1,\dots,Q_{m_2}}(F_c;\Psi)| = q^{\delta_3}|\Lambda_{P_1,\dots,P_{m_1}}^{Q_1,\dots,Q_{m_2}}(q^{-\delta_3}f_c,f_1,\dots,f_{m_1};\Psi)| \leq q^{\delta_3}(b_1\|q^{-\delta_3}f_c\|_{U^s}^{b_2}+b_3) &\leq q^{(1-b_2)\delta_3-b_2\delta_4}b_1+q^{\delta_3}b_3,
S_{2}=\frac{1}{2}\int d^{2}\xi \left[ \left( \partial_{\tau}\psi_{M}\right)^{2}+\left( \partial_{\sigma}\psi_{M}\right)^{2}+\frac{2}{\tau^{2}}(\psi_{M})^{2}+\frac{c_{m}'}{\tau a}\left( \psi_{m}\partial_{\sigma}\psi_{0}-\psi_{0}\partial_{\sigma}\psi_{m}\right) \right] \: .
\|\phi_{\mathbf a}(\zeta)z^{\mathbf a}\|^2_{A^2(\Omega)}=\int_{\pi(\Omega)}|\phi_{\mathbf a}(\zeta)|^2\int_{\Omega_{\zeta}}|z^{\mathbf a}|^2dV(z)\;dV(\zeta) &=\int_{\pi(\Omega)}|\phi_{\mathbf a}(\zeta)|^2\|z^{\mathbf a}\|^2_{\Omega_{\zeta}}dV(\zeta),
y_0\big|_{x_n=0}=\frac{x_0-T_1-(T_2-x_0)+T_2+T_1}{2}=x_0, &y_j\big|_{x_n=0}=x_j,\ \ \ 1\leq j\leq n-1, &y_n\big|_{x_n=0}=\frac{T_2-T_1-s-\tau}{2}=\frac{T_2-T_1-\psi^+(x)-\psi^-(x)}{2}=0,
\widehat{p}(\xi,y_d) =-\int_0^\infty e^{-|\xi|y_d}e^{-\omega_\lambda(\xi)z_d}\frac{\omega_\lambda(\xi)+|\xi|}{|\xi|}\widehat{f}_d(\xi,z_d)dz_d & =\int_0^\infty e^{-|\xi|y_d}e^{-\omega_\lambda(\xi)z_d}\left(\frac{1}{|\xi|}+\frac{1}{\omega_\lambda(\xi)}\right)i\xi\cdot\widehat{f}'(\xi,z_d)dz_d.
D_{x}\mathcal{T}g(t,x) =D_{x}\left[\int_{-\infty}^{-3^{\frac{2}{\alpha}}}\left| (-\Delta)^{{c_1}/2}T_{t-s}^{\alpha,\beta}g(s,\cdot)(x)\right|_{H}^{2}ds\right]^{1/2}\leq\left[\int_{-\infty}^{-3^{\frac{2}{\alpha}}}\left| D_{x}(-\Delta)^{{c_1}/2}T_{t-s}^{\alpha,\beta}g(s,\cdot)(x)\right|_{H}^{2}ds\right]^{1/2},
H_{(X,D)}(y)=\sum_{d\in\mathbb K}\sum_{\vec d=(D_i\cdot d)_{i\in I}\in (\mathbb Z)^{|I|},f=[d]\in \mathbb K/\mathbb L}H_{(X,D),f,\vec d}\textbf{1}_{f}[\textbf{1}]_{\vec d}.
\mathbb{P}\left[Y\leq (np)^{1/(4(r-1))}p^{-1}\right] \leq \exp\left(-\frac{(1+o(1))(\mathbb{E}[Y])^2}{2\mathbb{E}[Y]}\right) &\stackrel{\eqref{expectationY}}{\leq}\exp\left(-\Omega\left((np)^{1/(2(r-1))}p^{-1}\right)\right)\stackrel{np=\omega(1)}{=}\exp\left(-\omega(p^{-1})\right).
I_m(a,\beta_1,...,\beta_m) &=\int_0^\infty...\int_0^\infty\frac{t_1^{-\beta_1}...t_{m-1}^{-\beta_{m-1}}}{(1+t_1+...+t_{m-1})^{a-1+{\beta_m}}}dt_1...dt_{m-1}\int_0^\infty\frac{1}{(1+q_m)^aq_m^{\beta_m}}dq_m &=B(1-\beta_m,a+\beta_m-1)I_m(a-1+\beta_m,\beta_1,...,\beta_m) &=\frac{\prod_{i=1}^m\Gamma(1-\beta_i)\Gamma(a-m+\sum_{i=1}^m\beta_i)}{\Gamma(a)}
\widehat{H}= \frac{2 \pi}{L}\left( \widehat{L_{0}}+ \widehat{\overline{{L}}_{0}}- \frac{c}{1 2}\widehat{I}+ g \frac{L^{2 - 2 \Delta_{V}}}{\left( 2 \pi \right)^{1 - 2 \Delta_{V}}}\widehat{B}\right) \: ,
d s^{2}= - \frac{r^{2}}{\gamma^{4}}\left( 1 - \frac{2{\sqrt 2}\gamma^{2}M}{Q r}\right) d t^{2}+ \left( 1 - \frac{2{\sqrt 2}\gamma^{2}M}{Q r}\right)^{- 1}d r^{2}+ r^{2}d \Omega^{2}
p_{jj}(t-d,t,v,(d-h,d]) =P(Z_t = j,d-h<U_t\le d| Z_{t-d}= j,U_{t-d}= v) = P(N_t-N_{t-d}\ge 2, Z_t = j,d-h<U_t\le d| Z_{t-d}= j,U_{t-d}= v).
\gamma(s,x)=\sum_{k=0}^{\infty}\frac{x^{s}e^{-x}x^{k}}{s(s+1)\cdot\cdot\cdot(s+k)}=x^{s}\Gamma(s)e^{-x}\sum_{k=0}^{\infty}\frac{x^{k}}{\Gamma(s+k+1)}.
(\Lambda(x,y)f)(\tau) = \left(\pi(\tau x + y) f\right)(0) = \frac{1}{\sqrt{2 \pi}}\int_{-\infty}^{\infty}e^{i \tau (s \alpha(x) + \beta(x))}e^{i (s \alpha(y) + \beta(y))}\widehat{f}(s) ds.
\Vert \eta -\hat{\eta}\Vert_{L^2 \left( \partial K \right)}\lesssim_{k_0}\dfrac{C_k}{c_k}\left\{\sum_{j=1}^{4}\left( \log k \right)^{1/2}\inf_{p \in \mathbb{P}_{d_j}}\Vert \eta^{\rm slow}\circ \phi - p \Vert_{L^2(\mathcal{I}_j)}+ \sum_{j=5}^{J}\inf_{p \in \mathbb{P}_{d_j}}\Vert \eta^{\rm slow}- p \Vert_{L^2(\mathcal{I}_j)}\right\}