harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\delta \phi = 2 \alpha \frac{H_{e}^{2}}{m^{2}( \phi_{0})}( \cos ( m ( \phi_{0}) ( \tau - \tau_{e}) ) - ( \frac{\tau_{e}}{\tau})^{2}) - \alpha \frac{H_{e}^{3}}{m ( \phi_{0})^{3}}\sin ( m ( \phi_{0}) ( \tau - \tau_{e}) )
v_{3,2,b}^{\lambda}(t,r) := \frac{-1}{r}\int_{t}^{\infty}ds \int_{0}^{s-t}\rho d\rho \left(\frac{1}{\sqrt{(s-t)^{2}-\rho^{2}}}-\frac{1}{(s-t)}\right) \lambda''(s) \left(-1+F_{3}(r,\rho,\lambda(s))\right)
\left\| \sum_{1 \le n \le N}\frac{w_n \mathbf{1}_{B^{cl}(x_{n}, r_{n})}}{\mu B^{cl}(x_{n}, r_{n})}\right\|_1=\sum_{1 \le n \le N}w_n=\left\| \sum_{1 \le n \le N}\frac{w_n \mathbf{1}_{E_n}}{\mu E_n}\right\|_1.
\left|f(\zeta)-\left(\zeta+\beta \right)\right|\leq\frac{1}{\Re\left(\zeta\right)\cdot\log(\Re\left(\zeta\right))\cdots(\log^{\circ k}(\Re\left(\zeta\right)))^{1+\varepsilon}}=M_{\varepsilon,k}(\Re\left(\zeta\right)).
\sum_{t=1}^{l}\bar{\sigma}^2_{t}= \sum_{t=1}^{k}\bar{\sigma}^2_{t}- \sum_{t = l+1}^{k}\bar{\sigma}^2_{t}> \sum_{t=1}^{k}\sigma^2_{t}- 2\sqrt{k}\theta \|S\|^2_F - (k - l) \alpha \|W\|_F^2.
\|\nabla (\widetilde u_h - u_h)\|_{L^2(B_{d/4})}\leq C d^{-1}\|\widetilde u_h - u_h\|_{L^2(B_{d/2})}& \leq C d^{-1}\|\widetilde u_h - u\|_{L^2(B_{d/2})}+ C d^{-1}\| u - u_h \|_{L^2(B_{d/2})}& = C d^{-1}\|\widetilde u_h - \widetilde u\|_{L^2(B_{d/2})}+ C d^{-1}\| u - u_h \|_{L^2(B_{d/2})},
ds^2_{S-type}= d\chi^2 - dT^2 + dW^2 -\frac{1+A^2/\tau_0^{2p-2}}{(\sqrt{T^2-W^2}/\tau_0)^{2p-2}-1}\frac{(WdW-TdT)^2}{T^2-W^2}+ W^2 d\Omega^2_{p-2}\ .
\Omega(X_1(\gamma(u)),\ldots,X_n(\gamma(u)))= \frac{z'(u)}{f'(\zeta(u))}\left(\frac{z(u)}{\sin(u)}\right)^{n-1}=\frac{\zeta'(u)}{2z(u)}\left(\frac{z(u)}{\sin(u)}\right)^{n-1}= \frac{\zeta'(u)(z(u))^{n-2}}{2\sin^{n-1}(u)}\;,
|y_{n+1}-y_n| = \left|\frac{y_n}{2}-\frac{1}{2y_n}-y_n\right|= \frac{1}{2}\left| y_n + \frac{1}{y_n}\right|= \frac{1}{2}\left( |y_n| + \frac{1}{|y_n|}\right) \geq 1
\mathbb{P}({E}_{i,2}) =\underset{\substack{k_1+\cdots+k_K =m-i}}{\sum}\frac{(m-i)!}{k_1! \cdots k_K!}\mathbb{P}_K^{k_1}\mathbb{P}_{e,1}^{k_1}\times\cdots\times \mathbb{P}_K^{k_K}\mathbb{P}_{e,K}^{k_K},
\int_0^\infty y^{s-\frac 12}K_{ir}(2\pi|n|y) \, dy = \tfrac 14 (\pi|n|)^{-s-\frac 12}\Gamma(\tfrac s2+\tfrac{ir}{2}+ \tfrac 14) \Gamma(\tfrac s2-\tfrac{ir}{2}+ \tfrac 14).
\varkappa_{i+1}^{+}(u)\varkappa_{i}^{+}(u)^{-1}=exp(\hat{a}_{+}^{i}(u+\frac{n-2i}{4}h)), &\varkappa_{i+1}^{-}(u)\varkappa_{i}^{-}(u)^{-1}=exp(\hat{a}_{-}^{i}(u-\frac{n+2i}{4}h))
a_{\beta}^R = R\exp\left(-\frac{R^{\beta/2-1}}{\sqrt{W_{0}/2}}\frac{\mathcal{I}_0\left(\sqrt{W_{0}/2}R^{1-\beta/2}\right)}{\mathcal{I}_1\left(\sqrt{W_{0}/2}R^{1-\beta/2}\right)}\right).
\begin{aligned}h_1-\dfrac{\mathrm{tr}_g h_1}{n}\cdot g =-A_1(g)+\dfrac{\mathrm{tr}_g A_1(g)}{n}\cdot g, h_2-\dfrac{\mathrm{tr}_g h_2}{n}\cdot g =-A_2(g)+\dfrac{\mathrm{tr}_g A_2(g)}{n}\cdot g-A_{1,1}(h_1)+\dfrac{\mathrm{tr}_g A_{1,1}(h_1)}{n}\cdot g, \cdots \cdots =\cdots \cdots.\end{aligned}
f_{n}\left( \Delta x_{n}\right) =e_{n}+\Delta y_{n}f_{n}(u) :=\Pi_{N_{n+1}}\left( F(x_{n}+u)-F(x_{n})\right) e_{n}:=\Pi_{N_{n+1}}(\Pi_{N_{n}}-1)F(x_{n}) \Delta y_{n}:=\Pi_{N_{n}}(1-\Pi_{N_{n}})\bar{y}
X_{j,2,i}(\tau_{j+1}) = e^{\int_{\tau_j}^{\tau_{j+1}}G_j(v) \: dv}e^{-\int_{\tau_j}^{\kappa_i}G_j(v) \: dv}(1 + Z_{i,j}(\tau_{j+1})) &\leq \frac{2s}{\mu}e^{-\int_{\tau_j}^{\kappa_i}G_j(v) \: dv}(1 + Z_{i,j}(\tau_{j+1})).
\frac{d(x_2^2-x_1^2)}{dt}\leq v_2^2-v_1^2\leq -\frac{\kappa_1(1+\mathcal{A}(v)(0))}{2(1-\alpha)}(x_2^2-x_1^2)^{1-\alpha}=: -a(x_2^2-x_1^2)^{1-\alpha}.
\begin{array}{rcl}iT^{\rm 1PI}= \displaystyle i\lambda^2 v\int\frac{d^4k_1}{(2\pi)^4}\int\frac{d^4k_2}{(2\pi)^4}T(k_i), T(k_i) = \displaystyle T_a+T_b+T_c, \end{array}
N_1^1=\frac{\partial G^1}{\partial y^1}=0, N^1_2=\frac{\partial G^1}{\partial y^2}=-\frac{1}{2m},N^2_1=\frac{\partial G^2}{\partial y^1}=\frac{1}{2}V^{\prime \prime}\left(x^1\right), N_2^2=\frac{\partial G^2}{\partial y^2}=0.
P_{\ell}(z)=\left[\dfrac{1}{(n-1)!^r}\right]\circ{\rm{Eval}}_z\circ \mathcal{T}_{\bold{c}}\bigcirc_{j=1}^{n-1}B(\theta_t+j)\left(t^{\ell}\prod_{i=1}^m(t-\alpha_i)^{rn}\right)\enspace, &P_{\ell,i,s}(z)=\psi_{i,s}\left(\dfrac{P_{\ell}(z)-P_{\ell}(t)}{z-t}\right) \ \ 1\le i \le m, 0\le s \le r-1\enspace,
S = \int d^{3}x \left[ \frac{1}{4}F_{i j}F_{i j}+ \frac{1}{2}( D_{i}\Phi )^{\dagger}( D_{i}\Phi ) + V ( \Phi ) +{\cal L}_{\mathrm{g . f .}}+{\cal L}_{\mathrm{g h}}+{\cal L}_{\mathrm{c . t .}}\right] ,
S_{Q}( p ) \pm R_{Q}( p ) = 4 \pi^{2}d_{Q}( R ) \int \frac{d^{4}l}{{( 2 \pi )}^{4}}\delta ( l^{2}- m_{Q}^{2}) \delta [{( l + p )}^{2}- m_{Q}^{2}] \sin^{2}[ \theta ( l^{0}+ p^{0}, \mu_{Q}) \pm \theta ( l^{0}, \mu_{Q}) ] \geq 0 .
{\hat{x}}^{i}( \varepsilon ) = e^{i{\hat{H}}\varepsilon}{\hat{x}}^{i}( 0 ) e^{- i{\hat{H}}\varepsilon}\simeq{\hat{x}}^{i}( 0 ) +{\frac{{\hat{\pi}}^{i}( 0 )}{m}}\varepsilon +{\frac{e}{4 m^{2}}}( 2{\hat{F}}^{i k}{\hat{\pi}}_{k}( 0 ) - i \partial_{k}{\hat{F}}^{i k}) \varepsilon^{2}\, ,
v''_{i_1}=\bar{p_2-r+1},\v''_{i_2}=\bar{p_2-r+2},\ \cdots,\v''_{i_r}=\bar{p_2},\v''_{i_{r+1}}=p_2-r,\v''_{i_{r+2}}=p_2-r-1,\ \cdots,\v''_{i_{p_2}}=1.
(\check a_j, \check b_j, \check c_j) = \begin{cases}(120, \ 119, \ 169) j=1 (2\check h_j-1,2\check h_j^2-2\check h_j, 2\check h_j^2-2\check h_j+1 ) j = 2, \dots, \check\nu^C\end{cases}
OR=\frac{odds(x+1)}{odds(x)}=\frac{(\frac{p(x+1)}{1-p(x+1)})}{(\frac{p(x)}{1-p(x)})}=\frac{e^{\beta_{0}+\beta_{1}(x+1)}}{e^{\beta_{0}+\beta_{1}x}}=e^{\beta_{1}}
\langle [\widetilde{Z}_{\beta\beta}\ \ \widetilde{Z}_{\beta c}],[{\rm Diag}(\overline{u}_{\beta})\ \ 0_{\beta c}]\rangle = \overline{u}_{\beta}^{\mathbb{T}}\sigma([\widetilde{Z}_{\beta\beta}\ \ \widetilde{Z}_{\beta c}])
S_{1 ( 2 )}= \frac 1 2 \int_{0}^{T}d \tau \, \left( e_{1}\dot{x}_{1}^{2}- \frac{m_{1}^{2}}{e_{1}}\right) + \lambda_{2 \mu}\int_{0}^{T}d \tau \, \dot{x}_{1}^{\mu}A_{1}
V ( \phi ) = \frac{1 6}{q^{2}\varrho^{4}}\operatorname{e x p}( \frac{\phi}{f_{\phi}}) + 2 \pi^{2}\frac{f_{a}^{2}}{\Delta^{2}}\operatorname{e x p}( - 2 \frac{\phi}{f_{\phi}}) .
( \bar{\psi}_{i}M \lambda^{j}) ( \bar{\chi}_{k}N \xi^{l}) = - \frac{1}{4}( \bar{\chi}_{k}\lambda^{j}) ( \bar{\psi}_{i}M N \xi^{l}) - \frac{1}{4}( \bar{\chi}_{k}\gamma_{a}\lambda^{j}) ( \bar{\psi}_{i}M \gamma^{a}N \xi^{l}) + \frac{1}{8}( \bar{\chi}_{k}\gamma_{a b}\lambda^{j}) ( \bar{\psi}_{i}M \gamma^{a b}N \xi^{l}) ,
\left(V_{j,n}(x, t), \partial_t V_{j,n}(x, t) \right)= \left(\frac{1}{\lambda^{\frac{d-2}{2}}}V_j \left(\frac{x-x_{j,n}}{\lambda_{j,n}}, \frac{t-t_{j,n}}{\lambda_{j,n}}\right), \frac{1}{\lambda^{\frac{d}{2}}}\partial_t V_j \left(\frac{x-x_{j,n}}{\lambda_{j,n}}, \frac{t-t_{j,n}}{\lambda_{j,n}}\right)\right);
\frac{\partial E}{\partial z^i_{\epsilon}}=2b^2\Big[z^1_{\tilde{\epsilon}}(-1)^{{\tilde{\epsilon}}+\tau}z^i_{\tilde{\tau}}z^1_{\tau}+\delta_{i1}\sum\limits_{k}^{}(-1)^{\epsilon +\tau}z^1_{\tau}z^k_{\tilde{\tau}}z^k_{\tilde{\epsilon}}\Big],
\langle\lambda^*-\tilde{\lambda}^k, \frac{1}{\beta}(\tilde{\lambda}^k-\lambda^k)\rangle_{\mathcal{L}^2}=\langle \lambda^*-\tilde{\lambda}^k,\tilde{y}^k-\tilde{x}^k\rangle_{\mathcal{L}^2}=\langle\lambda^*-\tilde{\lambda}^k,\tilde{y}^k-y^*-(\tilde{x}^k-x^*)\rangle_{\mathcal{L}^2}.
\begin{array}{rl}\rho_{\textnormal{mom}}= \displaystyle \inf_{y}L_y(f), \textnormal{subject to}\displaystyle y_0 = 1, & \displaystyle{\bf M}_i(y) \succeq 0, \textrm{for all}i \in \mathbb{N}, & \displaystyle{\bf M}_{i-v_j}(yg_j) \succeq 0,\forall i \geq v_j, j = 1,\dots, m.\end{array}
\Vert M-\hat M\Vert^{2}_{2}\leq C\,K^{2}\,\Vert M\Vert_{0}\,\log\left (\dfrac{e\,n_1\,n_2}{\Vert M\Vert_{0}\vee 1}\right) +\dfrac{2\,a^{2}}{a-1}\Delta &\leq C\,K^{2}\,n_1\,n_2 +\dfrac{2\,a^{2}}{a-1}\Delta
\begin{aligned}F_{3,k}(p ):= \frac{|p|^2}{2}\{f_1,G_2^0\}_k+\frac{|p|^2}{2}\{f_2,G_1\}_k+\frac{|p|^4}{12}\{\{f_1,G_1\},G_1\}_k\,,\end{aligned}
A^{{\rm IV}}_{parent}=-\frac{T_p}{2}\int d^{p+1}{\xi}\left[\sqrt{-g}\left(-{\Psi}^{(p+3)/2}g_{ij}h^{ij}+(p+3){\Psi}^{(p+1)/2}\right)+{\Lambda}\left({\Psi}-{\Phi}\right)\right].
\sum_{n=1}^k\Delta_{n,n+1}\ = \ \varphi\big(H(p_1,x_1)-H(p^*,p^*)\big)-\varphi\big(H(p_{k+1},x_{k+1})-H(p^*,p^*)\big) \leq \ \varphi\big(H(p_1,x_1)-H(p^*,p^*)\big),
1 = \frac{e^2}{4 \pi N}\int \frac{kdk}{[k^2 + \Pi_{00}(k, \beta,\tilde{A}_0)] \sqrt{k^2 + \Sigma^2}}\left[ \frac{\sinh \beta\sqrt{k^2 + \Sigma^2}}{\cosh \beta \sqrt{k^2 + \Sigma^2}+ \cos (e\beta \tilde{A}_0 / \sqrt{N})}\right ]
\mathbb{R}^+\ni r\mapsto W_r(\psi,\phi)\;:=\;\det \begin{pmatrix}\psi^{+}(r) \phi^+(r) \psi^{-}(r) \phi^-(r) \end{pmatrix},\quad\psi,\phi\in L^2(\mathbb{R}^+,\mathbb{C}^2)\,,
P_2 =\{(-1,0)\}\cup R'_0\cup S'_0\cup T'_0 \cup U'_0\cdots R'_{\frac{d-4}{2}}\cup S'_{\frac{d-4}{2}}\cup T'_{\frac{d-4}{2}}\cup U'_{\frac{d-4}{2}}\cup R'_{\frac{d}{2}-1}&=\{(-1,0)\}\bigcup \cup_{i=0}^{\frac{d}{2}-1}R'_i\bigcup \cup_{i=0}^{\frac{d}{2}-2}S'_i\bigcup \cup_{i=0}^{\frac{d}{2}-2}T'_i\bigcup \cup_{i=0}^{\frac{d}{2}-2}U'_i.
[\dot{\Delta}_j,\, u\cdot\nabla] v= [\dot{\Delta}_j,\, \dot{T}_{u^i}]\partial_i v+\dot{\Delta}_j\big(\dot{T}_{\partial_i v}u^i\big)+\dot{\Delta}_j\big(\dot{R}(u^i,\,\partial_i v)\big)-\dot{T}_{\partial_i \dot{\Delta}_jv}u^i-\dot{R}\big(u^i,\,\partial_i \dot{\Delta}_j v\big)
\xi:\begin{pmatrix}x^1 x^2 x^3 x^4 \end{pmatrix}\mapsto\begin{pmatrix}x^1\cos(q^3+q^4)\mp x^2\sin(q^3+q^4) \pm x^1\sin(q^3+q^4)+x^2\cos(q^3+q^4) x^3 x^4\end{pmatrix}+\begin{pmatrix}q^1 q^2 q^3 q^4 \end{pmatrix}.
\begin{array}{rcl}\nabla_{1}F_{01}= -2\lambda^{1/2}/ R^{7}\left[ Q(r^{3}-3rN^{2})-P(3r^{2}N -N^{3})\right]\, , & \nabla_{1}{}^{\star}F_{01}= -2\lambda^{1/2}/ R^{7}\left[ P(r^{3}-3rN^{2})+Q(3r^{2}N -N^{3})\right]\, , \end{array}
\begin{pmatrix}b_1 \vdots b_j\end{pmatrix}=\begin{pmatrix}a_{1,f_1}\hdots a_{1,f_l}\vdots \vdots a_{j,f_1}\hdots a_{j,f_l}\end{pmatrix}\begin{pmatrix}s_1 \vdots s_l\end{pmatrix}.
M:=\left(\begin{array}{ccrccr}0 13 -13 9 10 -19 12 11 -11 7 8 -15 12 11 -11 7 8 -15 6 4 5 1 2 -3 6 4 5 1 2 -3 6 4 5 1 2 -3 \end{array}\right).
\Pi_{\mu}^{(T,g)}[ f(\xi_0) ] = \frac{1}{\mu(P^\beta_Tg)}\Pi_{\mu}\Big[g(\xi_T) \exp\Big\{\int_0^T \beta(\xi_s)ds \Big\}f(\xi_0) \Big] &= \frac{1}{\mu(P^\beta_T g)}\int_E (P^\beta_T g)(x) \cdot f(x)\mu(dx),
g_{q}( y , M ) ={\frac{M}{1 8 \pi}}x_{a}x_{b}\ [ f_{q}^{( a )}( x_{a}, M^{2}) f_{\bar{q}}^{( b )}( x_{b}, M^{2}) + f_{\bar{q}}^{( a )}( x_{a}, M^{2}) f_{q}^{( b )}( x_{b}, M^{2}) ] \ ,
e^{-2Kt/3}\bigg(\underbrace{\frac{\Delta u_t}{u_t}- |\nabla\log u_t|^2}_{=\Delta\log u_t}\bigg) \geq (1-e^{-2Kt/3})|\nabla\log u_t|^2 - \frac{NK}{3}\frac{e^{-4Kt/3}}{1-e^{-2Kt/3}}
{\cal L}= - \frac{1}{4}\frac{g^{2}}{g^{2}( t )}G_{\mu \nu}^{a}G^{a \; \mu \nu}+ a_{1}( t ) R^{2}+ a_{2}( t ) C_{\mu \nu \alpha \beta}^{2}+ a_{3}( t ) G \; ;
J ( z ) = \frac{1}{k + \check{g}}\{\frac{( \check{g}- 1 )}{( \check{g}- 2 )}S_{a}^{a}+ ( k + 1 ) \psi^{a}\psi_{a}- ( k - \check{g}+ 2 ) \psi^{+}\psi_{+}\}( z )
s_r(\widetilde{H}) =\sum_{j=1}^r\sum_{(i,j)\in V(\widetilde{H})}\left|d_{\widetilde{H}}((i,j))-\frac{r!m}{rn}\right| & =r\sum_{i\in V(H)}\left|(r-1)!d_{H}(i)-\frac{r!m}{rn}\right| & =r!s(H).
\mathbf{D}_{n-1}\cap \mathbf{Y}_0 = V(x_{n1}), \mathbf{D}_{n-2}\cap \mathbf{Y}_0 = V(x_{n,2}x_{n-1,1}-x_{n-1,2}x_{n,1}), \vdots \mathbf{D}_{k}\cap \mathbf{Y}_0 = V\left(\prod_{i-j=k}x_{i,j}+O\right),
I_{tt}^{1}= -\int_{-2}^{-1}\frac{x (2 + x)}{(2x^2 + 4x + 4)^2}dx I_{tt}^{2}= \int_{-1}^{1}\frac{3x^2}{(1+x^4)^2}dx I_{tt}^{3}= -\int_{1}^{2}\frac{x(-2+x)}{(2x^2 -4x + 4)^2}dx.
g_{e}((\lambda^{-1})^{*}A;\sigma)(s,t) = f_{\lambda}(\sigma(e(t)))g_{e}(A;\sigma)f_{\lambda}(\sigma(e(s)))^{-1}, begin{align*}0.25cm] P^{e}_{X}((\lambda^{-1})^{*}A,\lambda_{*}E;\sigma) = P^{e}_{\textup{Ad}_{f_{\lambda}(\sigma(e(1)))^{-1}}(X)}(A,E;\sigma).
d(y,x_{kn+n})^2\leq d(y,x_{kn})^2-2\lambda_k\sum_{i=1}^n[f_i(x_{kn})-f_i(y)] +2\lambda_k\sum_{i=1}^n[f_i(x_{kn})-f_i(x_{kn+i-1})] +\frac{K}{2}\lambda_k^2L^2 n.
\frac{(2g+1-\sum_j b_j)!}{b_i!}\sum_{c_2, \ldots, c_n}(2k-1)!!(2c_i+1)!!\psi_1^k (\psi')^{c_i}\prod_{j\neq i}\frac{\psi_j^{c_j}}{2^{c_j}c_j!(b_j-2c_j)!}&\left(-{4g+n - \sum_{j \neq i}b_j \choose 2g-2c_i - \sum_{j \neq i}b_j}+ \sum_{d=0}^{b_i-2c_i+2}{4g-1+n -\sum_j b_j \choose 2g-2c_i - \sum_{j \neq i}b_j - d}{b_i+1 \choose d}\right),
(R^{i}_{k})_{_F}:=\frac{1}{F^2}\big(2\frac{\partial G^i}{\partial x^k}-\frac{\partial^2 G^i}{\partial x^j \partial y^k}y^j +2G^j\frac{\partial^2 G^i}{\partial y^j \partial y^k}- \frac{\partial G^i}{\partial y^j}\frac{\partial G^j}{\partial y^k}\big).
\frac{\tilde{S}_{4}}{\tilde{S}_{4}^{\prime}}= O \left( \frac{| \vec{\delta}_{2}|^{3}\, \mathrm{M i n}( | \vec{\delta}_{2}|^{3}, \, | \vec{\delta}_{3}|^{3}) \, V}{\lambda{\cal P}^{3}}\right) .
\frac{f(21,\ell+1)}{f(21,\ell)}=\left(\prod_{i=1}^{21}(21\ell+i) \right)\frac{\ell^6}{21!(\ell+1)^7}=\left(\prod_{i=1}^{20}\frac{21\ell+i}{i}\right)\frac{(21\ell+21)\ell^6}{21(\ell+1)^7}&=\left(\prod_{i=1}^{20}\frac{21\ell+i}{i}\right) \left(\frac{\ell}{\ell+1}\right)^6 >42\left(\frac{2}{2+1}\right)^6>1,
\mathbf{u}\left( r\mathbf{\hat{x}}\right) =\sum_{\ell=0}^{\infty}\sum_{m\in\iota_{\ell}}\left( u_{\ell}^{m}\left( r\right) \mathbf{Y}_{\ell}^{m}\left( \mathbf{\hat{x}}\right) +v_{\ell}^{m}\left( r\right)\mathbf{U}_{\ell}^{m}\left( \mathbf{\hat{x}}\right) +w_{\ell}^{m}\left(r\right) \mathbf{V}_{\ell}^{m}\left( \mathbf{\hat{x}}\right) \right) .
\sum_{f_{k,\ell}\in \Gamma(f_{i,j})}\frac{1}{A_{k,\ell}}\le\frac{4(2n-4)(q-1)}{(2n-4)(2n-6)}= \frac{2(q-1)}{n-3}\le\frac{1}{\mathrm{e}}- \delta \enspace .
\frac{1}{|f|^2}= \frac{1}{1+|\beta|^2 r^2 + |\lambda|^2 r^4}- \frac{\lambda \gamma^{*}z^2 + \lambda^{*}\gamma z^{*2}}{(1+|\beta|^2 r^2 + |\lambda|^2 r^4)^2}+ O(\vert\gamma\vert^2).
- \frac{\kappa_{J}}{\gamma}\frac{d f}{d \kappa_{J}}\; \; = \; \; \frac{1}{N^{2}}\frac{\langle H_{F , \bar{F}}\rangle}{L}\Big / e^{2}\frac{N}{2}\; \; \equiv \; \; \rho_{F , \bar{F}}\; .
\sum_{k \in I_2}\phi_t(k) \leq 2\int_{t/2+c\sqrt{t}}^{\infty}\phi_t(x) \, dx=2 \int_{c/\sigma}^\infty \phi(y) \, dy\leq \frac{2}{\sqrt{2\pi}}\int_{c/\sigma}^{\infty}\frac{y}{c/\sigma}e^{-y^2/2}\, dy = \frac{2\sigma}{c\sqrt{2\pi}}e^{-\frac{c^2}{2\sigma^2}}.
\begin{array}{l l}{{H =}}&{{\sum_{i = 2}^{M}( \mathrm{\cal ~ A}c_{i}^{+}c_{i - 1}+ 1 / 2 B_{i}c_{i}^{+}c_{i}+ \gamma c_{i}^{+}c_{i - 1}^{+}) + 1 / 2 B_{1}c_{1}^{+}c_{1}}}\\{{}}&{{- \mathrm{\cal ~ A}( - 1 )^{\hat{N}}c_{1}^{+}c_{M}- \gamma ( - 1 )^{\hat{N}}c_{1}^{+}c_{M}^{+}+ H . c .}}\\ \end{array}
Y_{\tau}^{n,t,x}=\int_\tau^Te^{-(s-\tau){A}}G B^n(s,\Xi^{t,x}_s)\,ds+\int_\tau^Te^{-(s-\tau){A}}Z_s^{n,t,x}\, B^n(s,\Xi^{t,x}_s)\,ds &- \int_\tau^Te^{-(s-\tau){A}}Z^{n,t,x}_{s}\;d W_s.
u^C_{\lambda,\nu}= \sum_{i+2j=k}\frac{2^{-i}\Gamma(\frac{\nu}{2}-j)}{i!j!\Gamma(\frac{\nu}{2}-\lfloor\frac{k}{2}\rfloor)\Gamma(\frac{p}{2}+i)}\Delta_{\mathfrak{v}}^i \square^j\delta.
\hat{q}_{3}\equiv \frac{q_{3}}{\sqrt{2 \vert e H \vert}}_{,}\hat{m}_{\mu}\equiv \frac{m ( 0 , \mu )}{\sqrt{2 \vert e H \vert}}_{,}\hat{\mu}\equiv \frac{\mu}{\sqrt{2 \vert e H \vert}}_{,}\hat{Q}_{1 , 2}\equiv \frac{Q_{1 , 2}}{\sqrt{2 \vert e H \vert}}_{.}
\int\frac{d^{3}k}{(2\pi)^{3}}(\hat{k}\cdot\hat{r})^{2}\frac{exp(ik\cdot r)}{k^{2}+m^{2}}=\frac{e^{-mr}}{4\pi r}\{1+\frac{2}{mr}-\frac{2}{(mr)^{2}}(e^{mr}-1)\}
\lim_{\|y\|\to0}\mathop{\sup_{y=y(r,\theta_0,0)}}_{x,x+y\in I}\frac{|\ln\|y\| |^{1/a}|X(x+y)-X(x)|}{\|y\|^{1/a}\sqrt{\log(1+\|y\|^{-1})}}=C_{21}\in(0, \infty)\qquad\mbox{a.s.}
dX_t =b\left(t,X_t,\mu_{t},\hat{\alpha}\left(t,X_t,\mu_{t},\sigma^{-1}Z_t\right)\right)dt+\sigma dW_t X_0 =\xi , dY_t =-f\left(t,X_t,\mu_{t},\hat{\alpha}\left(t,X_t,\mu_{t},\sigma^{-1}Z_t\right)\right)dt+Z_t dW_t Y_T =g(X_T,\mu_{T}),
\widetilde{T}_\sharp =\widetilde{T}_l(f_{1,\sharp},f_{2,\sharp}), \widetilde{T}_\flat =\widetilde{T}_l(f_{1,\flat},f_2)+\widetilde{T}_l(f_{1,\sharp},f_{2,\flat}), \widetilde{T}_{}=\widetilde{T}_l(f_{1,\sharp},f_{2,})+\widetilde{T}_l(f_{1,},f_2).
V = \left({\it \rho_{1}}+{{{{{\it x_{1}}}^2}}\over{10\,{\it \rho_{1}}}}\right) \,\left({\it \rho_{2}}+{{{{{\it x_{2}}}^2}}\over{10\,{\it\rho_{2}}}}\right) \,\left( \frac{1}{16}+{{x +{\it x_{1}}+{\it x_{2}}}\over{{{\left({{{\it \rho_{1}}}^2}+{{{\it \rho_{2}}}^2}+ 2\,{\it \rho_{1}}\,{\it \rho_{2}}\,\left( 1 - x \right)\right)}^{{3\over 4}}}}}\right)
\mathrm{E}[\mathbf{Y}|\boldsymbol{a}<\mathbf{Y}\leq \boldsymbol{b}] =\mathrm{E}\left[\left(\mathbf{\Sigma}^{\frac{1}{2}}\mathbf{Z}+\boldsymbol{\mu}\right)|\boldsymbol{\xi_{a}}<\mathbf{Z}\leq \boldsymbol{\xi_{b}}\right] &=\boldsymbol{\mu}+\mathbf{\Sigma}^{\frac{1}{2}}\mathrm{E}\left[\mathbf{Z}|\boldsymbol{\xi_{a}}<\mathbf{Z}\leq\boldsymbol{\xi_{b}}\right],
\begin{aligned}\widehat{\gamma}(A) = \mathbb{P}( \widehat{S}\in A ) = \mathbb{E}\left[ \mathbb{P}( \widehat{S}\in A | \varepsilon ) \right] = \left( \eta/ \eta_0 \right) \mathbb{P}( S \in A) + \left( 1- \eta/ \eta_0 \right) \mathbb{P}( \widetilde{S}\in A).\end{aligned}
T_{H}^{\mathrm{B T Z}}= ( \rho_{+}^{2}- \rho_{-}^{2}) / 2 \pi R^{2}\rho_{-}= T_{H},{\frac{1}{T_{L / R}^{\mathrm{B T Z}}}}={\frac{1}{T_{H}^{\mathrm{B T Z}}}}\left( 1 \pm{\frac{\rho_{+}}{\rho_{-}}}\right) ={\frac{1}{T_{L / R}}}.
\delta_{\varepsilon}(F_{\nabla^\omega})^a=(C^a{}_{bc}-\rho^\mu_b\omega^a{}_{c\mu})\varepsilon^c(F_{\nabla^\omega})^b+(R_{\nabla^\omega})^a{}_{b\mu\nu}\varepsilon^bDX^\mu\wedge DX^\nu+D^a{}_{bc\mu}\varepsilon^cDX^\mu\wedge A^b,
u^{L}=\lim_{\substack{(r,x)\to(t,\overline{y}(t)) x<\overline{y}_{1}(s)}}u(r,x)\ ,u^{R}=\lim_{\substack{(r,x)\to(t,\overline{y}(t)) x>\overline{y}_{p}(s)}}u(r,x) \ .
\frac{d}{dt}\left(\frac{x_2}{x_1}\right) =\frac{x'_2x_1-x'_1x_2}{x^2_1}=\frac{u(x_2)x_1-u(x_1)x_2}{x_1^2}=\frac{x_2}{x_1}\left(\frac{u(x_2)}{x_2}-\frac{u(x_1)}{x_1}\right) &=\frac{x_2}{x_1}\int_0^1 \left[K\left(\frac{x_1}{y}\right)-K\left(\frac{x_2}{y}\right)\right]\frac{\omega(y)}{y}~dy.
\Delta_{\tilde{g}}\Phi = - \frac{n-4}{4(n-1)}u^{-2\frac{n-3}{n-2}}R_{g}+ O_2(r^{-2\delta-2}), \nabla(\Delta_{\tilde{g}}\Phi) =- \frac{n-4}{4(n-1)}u^{-2\frac{n-3}{n-2}}\nabla R_{g}+ O_1(r^{-2\delta-3}).
f_n(\tilde{\omega}) =\big{(}E\prod_{i=0}^{n-1}\frac{\lambda\rho(S_i)\rho(S_{i+1})}{1+\lambda\rho(S_i)\rho(S_{i+1})}\big{)}^{-1}&\leq\big(E\prod_{i=0}^{n-1}\frac{\lambda\rho(S_i)\rho(S_{i+1})}{1+\lambda M^2}\big)^{-1}&=[\frac{(1+\lambda M^2)}{\lambda E\rho^2}]^n\frac{E\rho^2}{(E\rho)^2}.
\bar{n}_{0}( t ) = [ \left( | \alpha_{0 0}( t ) |^{2}+ | \beta_{0 0}( t ) |^{2}\right) \bar{n}_{0}+ | \beta_{0 0}( t ) |^{2}+ \sum_{k}\left[ \left( | \alpha_{0 k}( t ) |^{2}+ | \beta_{k 0}( t ) |^{2}\right) \bar{n}_{k}+ | \beta_{k 0}( t ) |^{2}\right] \; .
- \frac{V_{J J}}{V}= \frac{1}{3}\Big [ \beta + \frac{1}{J}\Big ]^{2}\bigg ( 1 + \Big [ \beta + \frac{1}{J}\Big ] \Big [ J + \frac{1}{r_{O M}}\Big ( \frac{6 V^{4}}{d}\Big )^{1 / 3}\Big ] \bigg ) \; ,
Q_r(i):=\begin{cases}1-q+qy_{r+1}-y_r i_r= i_{r+1}; (y_r(i)-qy_{r+1}(i))/ (y_r(i)-y_{r+1}(i)) i_r\neq i_{r+1}\pm 1; (y_r(i)-qy_{r+1}(i))/ (y_r(i)-y_{r+1}(i))^2 i_r\rightarrow i_{r+1}; q^{i_r}i_r\leftarrow i_{r+1}; q^{i_r}/ (y_r(i)-y_{r+1}(i)) i_r\leftrightarrow i_{r+1}.\end{cases}
q_{\rm DASH}(\mathbf p) = \sum_{n\in\mathcal N}a_{n}\sum_{\ell\in\mathcal L}b_{n,\ell}\sum_{k\in\mathcal K_{{\rm DASH},n,\ell}}\Pr[K_{{\rm DASH},n,\ell}=k]\Pr[{\rm SIR}_{{\rm DASH},n,\ell}\geq\tau_k],
F_{A^{}_{}}=dA^{}_{}+\frac{1}{2}\left[A^{}_{}\wedge A^{}_{}\right] =\left(\theta^{23}+\theta^{56}\right)\otimes e_{1}+ \left(\theta^{24}+\theta^{57}\right)\otimes e_{2}+\left(\theta^{34}+\theta^{67}\right)\otimes e_{3}.
\hat{\cal P}=-\sum_a\frac{\partial}{\partial x_a}, \hat{\cal H}=\sum_a\left\{v_a\frac{\partial}{\partial x_a}+A_a\frac{\partial}{\partial v_a}\right\}, \hat{\cal K}=\sum_a\left\{-\frac{x_av_a}{c^2}\frac{\partial}{\partial x_a}+\left(1-\frac{v_a^2}{c^2}-\frac{x_aA_a}{c^2}\right)\frac{\partial}{\partial v_a}\right\}.
I = \int dx ( \Omega^{\prime}\rho^{\prime}+ \frac{3}{4}e^{2 \rho}\Omega^{-1 / 3}- \frac{1}{4}\Omega f^{\prime 2}-\frac{1}{4}\Omega f_1^{\prime 2}- \frac{1}{4}\Omega f_2^{\prime 2}-\frac{1}{4}\Omega f_1^{\prime}f_2^{\prime}
S_1= \frac{X}{V}\sum_{\substack{u_1,\ldots,u_k (u_i,u_j)=1,\forall i\neq j (u_i,VP_f)=1\forall i}}\biggl(\prod_{i=1}^k\frac{\mu(u_i)^2}{\varphi(u_i)}\biggr)F\bigl(\frac{\log u_1}{\log R},\ldots,\frac{\log u_k}{\log R}\bigr)^2 &+O\left(\frac{F_{max}^2\varphi(VP_f)^kX(\log X)^k}{V(VP_f)^kD_0}\right).
\Delta \hat{G}^{\xi = 0}[a, b; E] = \lim_{\epsilon \rightarrow 0^+}\frac{\hat{G}_F^{\xi = 0}[a, \epsilon; E] \hat{G}_F^{\xi = 0}[\epsilon, b; E]}{\frac{1}{v}- \hat{G}_F^{\xi = 0}[\epsilon, \epsilon; E]}.
\begin{bmatrix}m b\end{bmatrix}=\begin{bmatrix}m b\end{bmatrix}_v=\prod_{1\leq i\leq b}\frac{v^{2(m-i+1)}-1}{v^{2i}-1}, [m] = \begin{bmatrix}m 1\end{bmatrix}= \frac{v^{2m}-1}{v^{2}-1}.
f_4 =1728^{1/6}\eta^{4}K^{-1/6}\_{3}F_2(-1/6, 1/6, 1/2; 3/5, 2/5; K), f_5 =1728^{-7/30}\eta^{4}K^{7/30}\_{3}F_2(7/30, 17/30, 9/10; 7/5, 4/5; K), f_6 =1728^{-13/30}\eta^{4}K^{13/30}\_{3}F_2(13/30, 23/30, 11/10; 8/5, 6/5; K).
D_S=\{(a,b)\in [0,1]^2: \left[a\ge b \Rightarrow \exists x\in [0,1] \; a=S(x,b) b=1\Rightarrow x=0\right] and \left[b\ge a \Rightarrow \exists x\in [0,1] \; b=S(x,a) a=1\Rightarrow x=0\right] \}
V_{\chi_{k,1}}(z)=\sum_{i_1,\ldots,i_k\in\{n,\ldots,1,0,\bar1,\ldots,\bar n\}\atop i_1\preceq\ldots\preceq i_k}c^{\chi}_{i_1,\ldots,i_k}:M_{i_1,q^{k-1}}(z)M_{i_2,q^{k-3}}(z)\cdots M_{i_k,q^{-k+1}}(z):\,.
\begin{aligned}\tilde{\lambda}(s) = \tilde{\lambda}_0 + \tilde{\lambda}_1 s + \tilde{\lambda}_2 s^2 + \cdots \phi(\xi; s) = \phi_0 (\xi) + \phi_1 (\xi) s + \phi_2 (\xi) s^2 + \cdots,\end{aligned}
\langle\hat{x}_1^2\rangle=\langle\hat{x}_2^2\rangle=\frac{1}{2}\langle \hat{x}_1^2+\hat{x}_2^2\rangle=\frac{1}{2}\langle\tilde{x}_1^2+\tilde{x}_2^2+\frac{\theta^2}{4\hbar^2}(\tilde{p}_1^2+\tilde{p}_2^2)-\frac{\theta}{\hbar}\hat{L}\rangle;
E(\boldsymbol{h}) =\sum_{k=-\infty}^{\infty}(V(z_k+h_k,z_{k+1}+h_{k+1})-V(z_k,z_{k+1})), \nabla E(\boldsymbol{h})_k = V_2(z_{k-1}+h_{k-1},z_k+h_k)+V_1(z_k+h_k,z_{k+1}+h_{k+1}),
K(t,x) =\frac{d}{dt}\textnormal{H}_{12}^{20}\left[\frac{1}{4}t^{-\alpha}|x|^{2}\ \Big|\begin{array}{cc}(1,\alpha) (\frac{d}{2},1) (1,1) \end{array}\right] &=\pi^{-\frac{d}{2}}|x|^{-d}t^{-1}\textnormal{H}_{12}^{20}\left[\frac{1}{4}t^{-\alpha}|x|^{2}\ \begin{array}{cc}(0,\alpha) (\frac{d}{2},1) (1,1)\end{array}\right].
|\det L| = \omega_2|\lambda|^2(1 + O(\delta_2))\quad\text{with $\omega_2 = \mu_+^{1/2}(\mu_++\nu_+)^{1/2}\rho_+\rho_- + \mu_-^{1/2}(\mu_++\nu_+)^{1/2}\rho_+^{1/2}\rho_-^{3/2}$},
E\big[\tilde{E}\big[ F |\mathcal{F}_{t}^{H}\big]\big]=\lim_{n\rightarrow+\infty}E\Big[\tilde E\Big[\sum_{k=1}^{n}c_k^{(n)}\varepsilon(f_k^{(n)})\big|\mathcal F_t^H\Big]\Big]=\lim_{n\rightarrow+\infty}\sum_{k=1}^{n}c_k^{(n)}=E[F].
\Big \| \sum_{j=1}^\infty P_\alpha M_{a_j}f \Big \|_{\infty,\alpha}\leq \| P_\alpha\| \Big \| \sum_{j=1}^\infty M_{a_j}f \Big \|_{\infty,\alpha}&\leq \| P_\alpha\| \| f\|_{\infty,\alpha}
\int_{S}\frac{d^{2}\tau}{\tau_{2}^{2}}F ( \tau ) = \int_{\bigcup \gamma_{c d}{\cal F}}\frac{d^{2}\tau}{\tau_{2}^{2}}F ( \tau ) = \sum_{( c , d ) = 1}\int_{\gamma_{a b}{\cal F}}\frac{d^{2}\tau}{\tau_{2}^{2}}F ( \tau ) .
D_{3}= \frac{\alpha^{2}}{3 2 \pi^{2}}\left( - \sigma^{2}+ A^{2}{\beta^{x}}^{2}+ A^{2}{\beta^{z}}^{2}\right) \left\{\frac{1}{( 3 - d )}- \frac{1}{2}\left[ \ln \left( \frac{\alpha}{\mu_{R}^{2}}\right) - \frac{1}{2}\right] \right\}\; \; .
\|b\|_\sharp=\sum_{i=1}^Nw_i|b_i|=\sum_{i\in\operatorname{supp}(b)}w_i|b_i|\leq\bigg(\sum_{i\in\operatorname{supp}(b)}w_i^2\bigg)^{1/2}\|b\|_2\leq \sqrt{K}\|b\|_2.
\|x^{-d}u\|_{L^2}= \biggl\|\int_0^1\int_0^{s_2}\cdots\int_0^{s_d}u^{(d)}(t x)\,dt\,dt_d\cdots dt_2\,dx\biggr\|_{L^2}\leq \int_0^1\int_0^{s_2}\cdots\int_0^{s_d}\|u^{(d)}(t\cdot)\|_{L^2}\,dt\,dt_d\cdots dt_2\,dx = \frac{2^{2 d}d!}{(2 d)!}\|u^{(d)}\|_{L^2}.
\tilde a_0(x)=\begin{cases}f(x) \mbox{if}x \in \mathcal{B}(e,\theta), 1 \mbox{if}x \in \mathcal{B}(x_{1},\epsilon) , 0 \mbox{if}x\notin \mathcal{B}(e,\theta) \cup \mathcal{B}(x_{1},\epsilon).\end{cases}
\begin{cases}\displaystyle M_{t}^{n}(\varphi):= n^{-1/2}\sum_{i=1}^{n}\int_{0}^{t}R\varphi(S_{z-}^{n,i}) \left(N^{n,i}(dz)-\lambda_{z}^{n,i}dz\right), \displaystyle A_{z}^{n}(\varphi):= n^{-1/2}\sum_{i=1}^{n}R\varphi(S_{z-}^{n,i}) \left(\lambda_{z}^{n,i}- \Psi(S^{n,i}_{z-},\overline{\gamma}(z) ) \right).\end{cases}
0\leq\frac{d}{d y}\begin{pmatrix}\epsilon_A(y) \epsilon_B(y)\end{pmatrix}\approx\begin{pmatrix}-\partial_AH_A/d_A \partial_BH_A/d_A \partial_AH_B/d_B -\partial_BH_B/d_B\end{pmatrix}\begin{pmatrix}\int_{-\infty}^y\epsilon_A \int_{-\infty}^y\epsilon_B\end{pmatrix},