harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathbb{P}(\mathbb{D}_{\infty}(i,j) > 2t \,| (X_{\infty})^{*}_{i}= u, (X_{\infty})^{*}_{j}= v) = \overline{\mathbb{P}}_{\overline{\xi^{*}}}^{h}(\inf\{r \ge 0: (\overline{X}_{r})^{\mbox{\tiny $J$}}_{i}= (\overline{X}_{r})^{\mbox{\tiny $J$}}_{j}\}> t)
_{a^{+}}^{C}D_{x}^{q}f(x) =_{a^{+}}\mathbf{I}_{x}^{n-q}f^{(n)}(x) & =\frac{1}{\Gamma(n-q)}\int_{a}^{x}\frac{f^{(n)}(\tau)d\tau}{(x-\tau)^{1-n+q}},
\displaystyle{\frac{\phi^{(0)}(a,b+1,c|u,v,w+1/4)}{\phi^{(0)}(a,b-1,c|u,v,w+1/4)}}= \displaystyle{\frac{W(a,b+1|A_{0})\,\overline{W}(b+1,c|B_{0})}{W(a,b-1|A_{0})\,\overline{W}(b-1,c|B_{0})}}.
\sum_{y\in\mathbb{F}_q}\chi_4(y)\overline{\chi_4}(a-y)=\sum_{y'\in\mathbb{F}_q}\chi_4(ay')\overline{\chi_4}(a-ay') &=\sum_{y'\in\mathbb{F}_q}\chi_4(y')\overline{\chi_4}(1-y') &=\sum_{y'\in\mathbb{F}_q}\chi_4\left(y'(1-y')^{-1}\right) &=\sum_{y''\in\mathbb{F}_q, y''\neq -1}\chi_4(y'') =-1,
G(z) =z+{Q\big(1-p_t+p_t z\big)-(1-p_t)-zp_t \over p_t\big(1-Q'(1-p_t)\big)}&=z+p_t^{-1/q}\left(Q\big(z+(1-z)(1-p_t)\big)-(1-p_t) -zp_t \right) &=z+p_t^{-1/q}q p_t^{1/q}(1-z)^{1/q}~=Q(z).
\frac{\sqrt{h}}{2}h^{a b}\left[ 2 g_{\mu \nu}( X^{\alpha}) \partial_{a}\partial_{b}X^{\mu}+ 2 \partial_{a}X^{\mu}\partial_{b}X^{\rho}\partial_{\rho}g_{\mu \nu}( X^{\alpha}) + \partial_{a}X^{\mu}\partial_{b}X^{\eta}\partial_{\nu}g_{\mu \eta}( X^{\alpha}) \right] = 0
\begin{array}{c c l}P(i\rightarrow f) = \sum_{n=0}^\infty\sum_{a_1,a_2,\cdots,a_n}\sum_{m=0}^\infty\sum_{b_1,b_2,\cdots,b_m}<\psi_f|a_1,a_2,\cdots,a_n>\times &\times <a_1,a_2,\cdots,a_n|{\cal E}e^{-i\hat{H}_0(t_f-t_i)}{\cal E}|b_1,b_2,\cdots,b_m><b_1,b_2,\cdots,b_m|\psi_i>\ \ \ .\end{array}
D_1(G_k) = \langle x,y \rangle D_2(G_k), D_2(G_k) = \langle x^2,y^2, [y,x] \rangle D_{3}(G_k), D_i(G_k) = \langle x^{2^l}, [y,x,\overset{i-3}\ldots,x,y], [y,x,\overset{i-1}{\ldots},x] \rangle D_{i+1}(G_k)
\begin{cases}L u := \frac{\partial^2 u}{\partial x^2}- 2 a \frac{\partial^2 u}{\partial x\partial t}+ (a^2+1)\frac{\partial^2 u}{\partial t^2}= 0, u(0,t) = u(\pi,t) = 0, t \geqslant 0, u(x,0) = u_0(x), u_0 \in L^2[0,\pi]. \end{cases}
\tilde{\rho}(\zeta) = \rho^0\exp(\zeta), \tilde{u}_1(\zeta) = u_1^0+\frac{2\mathrm{C}}{\mathrm{C}^2-1}\sqrt{\theta_{11}^0}\left( \exp\left( \frac{\mathrm{C}^2-1}{2}\zeta\right)-1\right), \tilde{p}_{11}= p_{11}^0\exp(\mathrm{C}^2\zeta),
Z_{\mathrm{eff}}=\int \mathcal{D}\Phi^{\Delta}\mathcal{D}\Phi_{\Delta}^{*(a)}\mathcal{D}\bar{\Phi}_{\Delta}^{(a)}\mathcal{D}\tilde{\Phi}_{\Delta}\mathcal{D}\mu_{a}^{\Delta}\mathcal{D}\nu_{a}^{\Delta}\mathcal{D}\omega^{\Delta}\exp \left( iS_{\mathrm{eff}}\right) .
B_{\beta}( x ) B_{- b / 2}( y )_{s_{L}, s_{R}}\stackrel{x < y}{\sim}| y - x |^{b \beta}c_{+}( \beta ; s_{L}, s_{R}) B_{\beta - b / 2}( x ) + | y - x |^{b ( Q - \beta )}c_{-}( \beta ; s_{L}, s_{R}) B_{\beta + b / 2}( x )
\mathbb{E}^{\tilde{\mathbb{P}}}\left[\left(\pi_{p}-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi_{p};\tilde{A}_{0}\right]\right)\left(\pi_{p}-\mathbb{E}^{\tilde{\mathbb{P}}}\left[\pi_{p};\tilde{A}_{0}\right]\right)^{\top};\tilde{A}_{0}\right]=Q_{p}
R=\left[\begin{array}{cccccc}1_{m_1}0 0 0 0 0 0 1/2 0 0 -\zeta^{-1}0 0 0 0 1_{m_2}0 0 0 0 -1_{m_1}0 0 0 0 -1/2 0 0 -\zeta^{-1}0 0 0 0 0 0 1_{m_2}\end{array}\right].
(D^{\alpha,1}_{0+}\, f)(t) \, = \, (I_{0+}^{1-\alpha}\, \frac{d}{dt}\, I_{0+}^{0}\, f)(t)\, = \, (I_{0+}^{1-\alpha}\, \frac{d}{dt}\, f)(t)\, = \,(_*D^\alpha_{0+}\, f)(t),\ 0 \le \alpha < 1.
(2n-1-q)a \|t_*u\|^{2n}+(n-1-q) b \|t_*u\|^n + q\int_{\Omega}(|x|^{-\mu}*F(t_*u))f(t_*u)t_*u ~dx =\int_{\Omega}(|x|^{-\mu}*F(t_*u))f^{'}(t_*u) (t_*u)^2 ~dx +\int_{\Omega}(|x|^{-\mu}*f(t_*u)t_*u)f(t_*u)t_*u ~dx.
E_{F_0}\Phi_n = P_{F_0}\left(|\hat \alpha_{s_n}-\alpha_{0+}|\geq \frac{\epsilon}{2}\right) & \leq P_{F_0}\left(|\hat \alpha_{s_n}-\alpha_{s_n}|\geq \frac{\epsilon}{4}\right) + P_{F_0}\left(| \alpha_{s_n}-\alpha_{0+}|\geq \frac{\epsilon}{4}\right).
{\Lambda}_{\alpha}^{( 0 )}= \frac{i}{\sqrt{2{\pi}}}\int_{0}^{\kappa}\frac{d k_{+}}{\sqrt{k_{+}}}\{C ( k_{+}) - C^{\ast}( k_{+}) + ( - 1 )^{\alpha}( B ( k_{+}) - B^{\ast}( k_{+}) ) \}
\langle H_1, H_2\rangle_G=(r+1)\int_{P(E)/M}{\frac{H_1}{G}}{\frac{H_2}{G}}\Xi^{r-1}-r\int_{P(E)/M}\frac{H_1}{G}\Xi^{r-1}\int_{P(E)/M}\frac{H_2}{G}\Xi^{r-1}.
{\P}\left\{\sum_{l\leq K}T_l<(\beta + \varepsilon)t_c \mid T_1= \beta t_c\right\}\geq (1-o(1)){\P}\left\{\sum_{l\leq \tau}T_l<\left(\beta + \frac{\varepsilon}{2}\right)t_c \mid T_1= \beta t_c\right\}.
\mathfrak{P}_{t}f(x) =\mathfrak{S}_{t}f(x)-\Pi_{x}\left[\int_{0}^{t}a(\xi_{s})\mathfrak{P}_{t-s}f(\xi_{s}){\rm d}s\right]+\Pi_{x}\left[\int_{0}^{t}\gamma(\xi_{s},\mathfrak{P}_{t-s}f){\rm d}s\right].
dP_{n+1}(x, x_{n+1}) = -\sum_{i = 1}^{n}x_{n+1}x_{i}dx_{i}+ \sqrt{\tfrac{n+2}{n}}dP_{n}(x) + \left( \tfrac{n}{2}x_{n+1}^{2}- \tfrac{1}{2}|x|^{2}\right)dx_{n+1}& = -\sum_{i = 1}^{n}x_{n+1}x_{i}dx_{i}+ \sqrt{\tfrac{n+2}{n}}dP_{n}(x) + \left( \tfrac{n+1}{2}x_{n+1}^{2}- \tfrac{1}{2}\right)dx_{n+1}.
\{Var([\begin{matrix}Z(x_{1})&\cdot\cdot\cdot&Z(x_{N})\end{matrix}]^{T}),Var(Z(x_{0})),Cov([\begin{matrix}Z(x_{1})&\cdot\cdot\cdot&Z(x_{N})\end{matrix}]^{T},Z(x_{0}))\}.
\frac{\sum_{i = 1}^n a_{i,n}(1) a_{i,n}(2)}{\sqrt{\sum_{i = 1}^n a_{i,n}^2(1)}\sqrt{\sum_{i = 1}^n a^2_{i,n}(2)}}- \frac{\sum_{i = 1}^n b_{i,n}(1) b_{i,n}(2)}{\sqrt{\sum_{i = 1}^n b_{i,n}^2(1)}\sqrt{\sum_{i = 1}^n b^2_{i,n}(2)}}
| n \rangle = |{\vec{p}}_1 \nu_1;{\vec{p}}_2 \nu_2 ...\rangle = \prod_i{{{(2 \pi)}^{{\nu}\over 2}}\over{V^{1\over 2}}}{1\over{(\nu_i !)^{1\over 2}}}a^{\dag \nu_i}({\vec{p}}_i)|\,\,0 \rangle
\begin{array}{rcl}M_{M2_{6}}={\displaystyle\frac{R_{3}R_{4}(R_{5}\ldots R_{10})^{2}}{({}^{-}\!\!\!\!\ell_{\rm Planck}^{(11)})^{15}}}\, , & M_{M5_{3}}={\displaystyle\frac{R_{3}\ldots R_{6}(R_{7}R_{8}R_{9})^{2}R_{10}}{({}^{-}\!\!\!\!\ell_{\rm Planck}^{(11)})^{12}}}\, . \end{array}
B_{m w}B_{n w^{\prime}}^{\dagger}= \sum_{w^{\prime \prime}}( B_{m w^{\prime \prime}}f_{w^{\prime \prime}, w , n , w^{\prime}}+ f_{w^{\prime \prime}, w^{\prime}, m , w}^{*}B_{n w^{\prime \prime}}^{\dagger}) + E_{m w ; n w^{\prime}}( \phi )
\lambda_{CW}\left(X(\frac{p_{1}}{q_{1}},\ldots,\frac{p_{n}}{q_{n}})\right)=\frac{1}{12}\frac{P}{H}\left(2-n+\sum_{i=1}^{n}p_{i}^{-2}\right)-\frac{1}{12}\left[3{\rm sign}\left(\frac{H}{P}\right)+\sum_{i=1}^{n}\left(12s(q_{i},p_{i})-\frac{q_{i}}{p_{i}}\right)\right].
f^{a, (x_0, y_0), \Pi^{\mu, \nu}}= \int_0^1 \Big( \partial_a f \big(x_0 + \xi(y_0 - x_0), \Pi^{\mu, \nu}_\xi, x_1 + \xi(y_1 - x_1), ..., x_{m[a]}+ \xi(y_{m[a]}- x_{m[a]})\big) - \partial_a f\big(x_0, \mu, x_1, ..., x_{m[a]}\big)\Big) (1-\xi)^{n-1}d\xi.
\begin{aligned}\chi_n R(z)\chi_n - \chi_n R(z')\chi_n = \chi_nR(z)\big(\chi_{n+1}(H-z') - (H-z)\chi_{n+1}\big)R(z')\chi_n &= (z-z') \chi_n R(z) \chi_{n+1}R(z')\chi_n - \chi_n R(z) [H_0,\chi_{n+1}]R(z')\chi_n.\end{aligned}
\sigma(S)= \sum_{i=1}^{r+1}ix_i=\sum_{i=1}^{r}ix_i+(r+1)x_{r+1}=\sum_{i=1}^{r}ix_i+(r+1)(n-\sum_{i=1}^{r}x_i)=(r+1)n+\sum_{i=1}^{r}(i-r-1)x_i &\leq (r+1)n-2\sum_{i=1}^{r}i=(r+1)n-r(r+1)
I_1+I_2-I_3=L_X(s,t)L_V(s,t)\left\{\tfrac{1}{\alpha}L_V(s+1,t+1)\left[\alpha L_X(s+2,t+1)-L_X(s+1,t)\right]\right. &\left.-\tfrac{1}{\beta}L_X(s+1,t+1)\left[\beta L_V(s+2,t+1)-L_V(s+1,t)\right]\right\}&+\tfrac{1}{\alpha}L_X(s+1,t)L_X(s,t+1)L_V(s,t)L_V(s+1,t+1) &-\tfrac{1}{\beta}L_X(s,t)L_X(s+1,t+1)L_V(s+1,t)L_V(s,t+1).
I_{\mu}(u) \geq \frac{1}{2}||u||_{H^1}^2+\frac{1}{16\pi}\left(||u||^2_E- ||u||_{H^1}^2\right)^2 -C\mu||u||_{H^1}^{p+1}&=\frac{1}{2}||u||_{H^1}^2+\frac{1}{4\pi}\left(\frac{1}{4}||u||^4_E- \frac{1}{2}||u||^2_E ||u||_{H^1}^2+\frac{1}{4}||u||_{H^1}^4\right) -C\mu||u||_{H^1}^{p+1}.
JM = \sum_{i=1}^{n}\binom{n-1}{i-1}(1-y)^i(1+y)^{n-i}i-1= \alpha &=\sum_{\alpha=0}^{n-1}\binom{n-1}{\alpha}(1-y)^{\alpha+1}(1+y)^{n-1-\alpha}&=(1-y)\sum_{\alpha=0}^{n-1}\binom{n-1}{\alpha}(1-y)^{\alpha}(1+y)^{n-1-\alpha}&=(1-y)(1-y +1+y)^{n-1}&=2^{n-1}(1-y) &=[2^{n-1}, -2^{n-1}, 0 , \ldots, 0].
\Phi\left(\Psi(\theta)\right)= \begin{pmatrix}-XZ+L_{1}(1-Z-\mu(1-Y)) -L_{3}YZ+\frac{L_{4}}{L_{2}}(1-Y) \frac{1}{L_{2}}((-XZ+1-Z-\mu(1-Y))+\mu)-L_{3}YZ+\frac{L_{4}}{L_{2}}(1-Y))) \end{pmatrix}.
v_{3}^{\lambda}(t,r) = \frac{-1}{r}\int_{t}^{\infty}ds \int_{0}^{s-t}\frac{\rho d\rho}{\sqrt{(s-t)^{2}-\rho^{2}}}\lambda''(s) \left(\frac{-1-\rho^{2}+r^{2}}{\sqrt{(-1-\rho^{2}+r^{2})^{2}+4r^{2}}}+ F_{3}(r,\rho,\lambda(s))\right)
PV_{\delta, \hat{x}, \hat{q}, z}(x)=V_{\delta, \hat{x}, \hat{q}, z}(x)-\frac{\hat{q}}{\ln\frac{R}{s_\delta}}g_{\hat{x}}(T_{\hat{x}}x, T_{\hat{x}}z),\ \ \forall x\in\Omega,
\frac{\P(T_n\geq x)}{1-\Phi(x)}=1 + O(1) \biggl\{( \sigma_p /\sigma)^p \frac{(1+x)^{p}}{n^{p/2 -1}}+( \sqrt{a_m}+\sigma_h/\sigma) \frac{(1+x)^3}{\sqrt{n}}\biggr\}
L_E(c_{k,j}^q) = \ln\left(\frac{\sum_{s\in S_{q,0}}p(\hat{x}_{k,j}\vert x_{k,j}= s) p(x_{k,j}= s)}{\sum_{s\in S_{q,1}}p(\hat{x}_{k,j}\vert x_{k,j}= s) p(x_{k,j}= s)}\right) - & \ln\left(\frac{\sum_{s\in S_{q,0}}p(x_{k,j}= s)}{\sum_{s\in S_{q,1}}p(x_{k,j}= s)}\right) \; q=1,2, \ldots ,b ;
h_{i}^{( N )}( x , z ) \equiv - \frac{r_{i}( x )}{p_{i}( x )}\, \sum_{n = 0}^{N}\,{\left[ \frac{z}{p_{i}( x )}\right]}^{n}\ \equiv \sum_{n = 0}^{N}h_{i , n}( x ) \, z^{n}.
\left( \begin{matrix}{F^{( \pm ) \Lambda}}\\{G_{\Sigma}^{( \pm )}}\\ \end{matrix}\right) \rightarrow \left( \begin{matrix}{F^{( \pm ) \prime \Lambda}}\\{G_{\Sigma}^{( \pm ) \prime}}\\ \end{matrix}\right) = \left( \begin{matrix}{A_{t}}&{B_{t}}\\{C_{t}}&{D_{t}}\\ \end{matrix}\right) \left( \begin{matrix}{F^{( \pm ) \Lambda}}\\{G_{\Sigma}^{( \pm )}}\\ \end{matrix}\right)
[b_{{\bf{0}}}, b^{\dagger}_{{\bf{0}}}] =b_{{\bf{0}}}b^{\dagger}_{{\bf{0}}}- b^{\dagger}_{{\bf{0}}}b_{{\bf{0}}}= exp(-i\mbox{}X_{0})\mbox{}N_{0}\mbox{}exp(i\mbox{}X_{0}) - N_{0}= 1
\begin{array}{l}g(e_1,e_1)=g(e_2,e_2)=-g(e_3,e_3)=-g(e_4,e_4) \phantom{g(e_1,e_1)}=-g(e_5,e_5)=g(e_6,e_6)=g(e_7,e_7)=1, g(e_i,e_j)=0,\; i\neq j.\end{array}
\left(\frac{dm_0}{d\zeta}+\frac{i\zeta}{2}\sigma_3m_0\right)m_0^{-1}= \left[\frac{dm^{pc}_{-k_0}}{d\zeta}+m^{pc}_{-k_0}\frac{i\nu}{\zeta}\sigma_3\right]\left(m^{pc}_{-k_0}\right)^{-1}+\frac{i\zeta}{2}\left[\sigma_3,m^{pc}_{-k_0}\right]\left(m^{pc}_{-k_0}\right)^{-1},
\begin{aligned}\|S^*(1-P_R)\widetilde{\mathcal{K}}^{j}v\|_{L_t^{\theta_{\widetilde{p}}}(L^{\widetilde{p},2}_{j,k})}\lesssim_{\widetilde{p},p,q}\|S^*(1-P_R)v\|_{L_t^{\theta_{\widetilde{p}}}(L^{\widetilde{p},2}_{j,k})}&\lesssim_{\widetilde{p},p,q}1/R\left\|\sum_{l}\mathcal{K}_{\mathcal{I}_l}v\right\|_{L_t^{\theta_q'}(L^{q',2}_D)}\end{aligned}
\hat{\mathbf{t}}_{l}=\hat{\mathbf{t}}_{l}^{\textrm{quan}}+\hat{\mathbf{t}}_{l}^{\textrm{mod}}=\mathbf{t}_{l}\bmod\Lambda_{B,1}+Q_{\Lambda_{B,1}}\left(\mathbf{t}_{l}\right)=\mathbf{t}_{l}
\Omega_-\cap B(x_0,r) =\{(x_1,x_2,x_3)\in B(x_0,r): x_3<\psi(x_1,x_2)\}\Omega_+\cap B(x_0,r) =\{(x_1,x_2,x_3)\in B(x_0,r): x_3>\psi(x_1,x_2)\}.
{\cal H}_{\mu \nu \rho}= 3 \partial_{[ \mu}B_{\nu \rho ]}- 3 \bar{\lambda}\gamma_{a}\gamma_{a^{\prime}}\partial_{[ \mu}\lambda \left( 2 \delta_{\nu}^{a}- \bar{\lambda}\gamma^{a}\partial_{\nu}\lambda \right) \partial_{\rho ]}X^{a^{\prime}}\ .
\sum \sum \limits_{b < a}^{N}\lambda_{a}\lambda_{b}= \sum \limits_{a = 1}^{N}r_{a}q_{a}^{2}- \sum \limits_{a = 1}^{N}q_{a}^{2}\sum \limits_{b = 1}^{N}r_{b}+ \sum \sum \limits_{b < a}^{N}r_{b}r_{a}
\begin{aligned}f(x) \,\rightarrow\,\min g_i(x) \,\leq\,0 \qquad i\in\mathcal M h_j(x) \,=\,0 \qquad j\in\mathcal P \varphi(G_l(x),H_l(x)) \,\leq\,0 \qquad l\in\mathcal Q \end{aligned}
\varphi_{1}\cdots \varphi_{m_k}.(u \otimes (v_p)_z) = \varphi_{1}\cdots \varphi_{m_k}e(h_1, \ldots , h_{m_k},p).(u \otimes (v_p)_z) = \varphi_{1}e(h_1,p,h_2, \ldots , h_{m_k}) \varphi_2 \cdots \varphi_{m_k}.(u \otimes (v_p)_z) = (\tau_1 (x_1 - x_2) + 1) \varphi_2 \cdots \varphi_{m_k}.(u \otimes (v_p)_z).
i_A^*\phi(P_1, \dots, P_{r+2s}, z) e^{(\vartheta+\tau)/z}Q^De^{\tau(i_A)_*D}= &\phi(({\vec P}_1)_D, \dots, ({\vec P}_{r+2s})_D, z) i_A^* e^{(\vartheta+\tau)/z}Q^De^{\tau(i_A)_*D}.
{S_{\mbox{\scriptsize eff}}}={1\over{4\pi\alpha'}}\int{d^2}\xi{\partial_a}{X_0^\mu}{\partial^a}{X_{0\mu}}+{g\over{2\sqrt{2\alpha'}}}{\int_{-\infty}^{+\infty}}d\tau\int{d^{26}}k\exp\left[ik\cdot{X_0}(\tau)\right]{\hat{\mathcal{L}}_{\mbox{\scriptsize B}}}(k,\tau)+\cdots,
{Q}^{( 2 )}= \frac{1}{8 \pi^{2}}\sum_{a = 1}^{3}\left( \dot{\phi}_{a}\beta_{a}+ \sum_{c y c l i c}^{i , j , k}\dot{\chi}_{a}M_{a i}^{T}( \phi_{j}- \phi_{k}) b_{i}\right) .
Q^{\prime}(x)e^{\lambda x}+\lambda Q(x)e^{\lambda x}=\frac{\beta e^{\lambda x}G(Q(x))}{\alpha}&+2\beta e^{2\alpha x}\int_{x}^{+\infty}e^{-(\alpha-\mu )y}G(Q(y))\mathrm{d}y &-\frac{\beta e^{\lambda x}G(Q(x))}{\alpha}.
\phi(x,0) = \sqrt{\frac{L}{2}}v^{\rm top}(x) (a+b) + \sum_{n=1}^\infty \frac{1}{\sqrt{2 k_n}}\left[ v^{\rm odd}(n,x) (a_n + b_n) +v^{\rm even}(n,x) (c_n+d_n)\right]
I_\omega(\phi):= S_\omega(\phi)- \frac{2}{3(q-1)}G(\phi) = \frac{3q-7}{6(q-1)}\|\nabla \phi\|^2_{L^2}+ \frac{p-q}{(p+1)(q-1)}\|\phi\|^{p+1}_{L^{p+1}}+\frac{\omega}{2}\|\phi\|^2_{L^2}
\left\langle{{\xi}\left\vert{\xi^{\prime}}\right\rangle}\right. ={\frac{\left({1 -{\vert \xi \vert}^{2}}\right)^{K}\left({1 -{\vert \xi^{\prime}\vert}^{2}}\right)^{K}}{\left({1 - \xi^{*}\xi^{\prime}}\right)^{2 K}}}\ ,
\widetilde{\cal J}^{\nu}= ( \eta^{\mu \nu}\bot_{\sigma \rho}+ 2 \eta^{\nu}{_{[ \sigma}}\eta_{\rho ]}{^{\mu}}) [ \acute{\xi}^{\sigma}\widetilde{\nabla}_{\mu}\xi^{\rho}- ( \widetilde{\nabla}_{\mu}\acute{\xi}^{\rho}) \xi^{\sigma}] .
\{{\hat\xi}_1,{\hat\xi}_2 \}_{\hbox{\scriptsize SD}}^t ={\hat\xi}_1^a\partial_a{\hat\xi}_2^t -{\hat\xi}_2^a\partial_a{\hat\xi}_1^t,\qquad\{{\hat\xi}_1,{\hat\xi}_2 \}_{\hbox{\scriptsize SD}}^a ={\hat\xi}_1^b\partial_b{\hat\xi}_2^a -{\hat\xi}_2^b\partial_b{\hat\xi}_1^a+ g^{ab}\left({\hat\xi}_1^t\partial_b{\hat\xi}_2^t-{\hat\xi}_2^t\partial_b{\hat\xi}_1^t \right).
\int \vert \phi_{k,i}\vert^{2}\varphi_{i}^{\frac{4}{n-2}}= \frac{(n-2)^{2}}{n}\int \frac{r^{2}}{(1+r^{2})^{n+2}}+O(\frac{1}{\lambda_{i}^{n-2}}+\frac{1}{\lambda_{i}^{2}}) = c_{3}+O(\frac{1}{\lambda_{i}^{n-2}}+\frac{1}{\lambda_{i}^{2}})
\prod_{q\mid D(T)}F_q(T,q^{k-n-1}) =\prod_{q\mid D(T)}F_q(T,q^{-\tfrac{n+1}{2}+(p-1)\cdot t}) \equiv \prod_{q\mid D(T)}F_q(T,q^{-\tfrac{n+1}{2}}) =0 \pmod{p}.
|I_1+I_2+I_3+I_4|\leq C(\nu^{-1}+\mu^{-1}) \left(\|u^\eta\|_\infty+\| b^\eta\|_\infty\right)\left(\|U\|_2^2+\|B\|_2^2\right) &+\frac14\nu \|\nabla U\|_2^2+\frac14\mu \|\nabla B\|_2^2.
\check{Y}^{\epsilon}(t) =\big(3\delta^{1/2}+\sup_{t\in[0,1]}|\varphi(t)-\varphi_m(t)|^{1/2}\big)+\int^{t}_{0}\big(b(\hat{X}^{\epsilon}(s))-b(X^{\epsilon}(s))\big)ds &\quad-\lambda(\epsilon)\epsilon\int^{t}_{0}\big(\sigma(\hat{X}^{\epsilon}(s-))-\sigma(X^{\epsilon}(s-))\big)\epsilon\theta^{\epsilon}(s-)\tilde{N}^{\epsilon^{-2}}(ds),
\langle \, J^{5}( x , y ) \, \rangle = - \frac{g_{5}^{2}}{3 2 \pi^{2}}\, F_{\mu \nu}\widetilde{F}^{\mu \nu}\, \sum_{n}\frac{1}{M_{n}}\phi_{n}^{{}_{L}}( y ) \phi_{n}^{{}_{R}}( y ) \quad .
\underset{t\rightarrow \infty}{\lim}\dfrac{\mathbb{U}_{\mathbf{F}}\left(tx\right) /\mathbb{U}_{\mathbf{F}}\left( t\right) -x^{\gamma_{1}}}{\mathbf{A}_{\mathbf{F}}\left( t\right)}=x^{\gamma_{1}}\dfrac{x^{\tau_{1}}-1}{\tau_{1}},
\|x\widetilde f_R\|\leq\Lambda\left\|\nabla u_{R}^-\right\|+\epsilon^2(R) \,, \|x\widetilde f_R\|\leq\Lambda\left\|\nabla u_{R}\right\|+\epsilon^2(R) \,, \lim_{R\to\infty}\epsilon^2(R)=0 \,.
Z_{\mathrm{R M T}}= \int_{N \times N \mathrm{\scriptsize m a t r i c e s}}\! \! \! \! \! \! \! \! \! \! d \mu ( W ) \ \prod_{f}\det{\left( \begin{array}{c c}{m_{f}}&{W^{\dagger}}\\{W}&{m_{f}}\\ \end{array}\right)}.
q^{r-2a}[2\ell-r] \frac{q^{2-2r}K^{-2}-1}{q^{4c-2r+2}K^{-2}-1}+q^{2\ell-a}[a]+q^{2c+r-2\ell-a}[r-2c-a]\frac{q^{2-2r}K^{-2}-1}{q^{4c-2r+2}K^{-2}-1}&+q^{2c-2\ell}[2c]\frac{q^{-2a}K^{-2}-1}{q^{4c-2r+2}K^{-2}-1}+q^{1-r-2a}[2\ell-r+1] \frac{(q^{4c}-1)K^{-2}}{q^{4c-2r+2}K^{-2}-1}=[2\ell].
\exp \left( - \frac{|v-u_f|^2}{2T_f}\right) =\exp \left( -\frac{v^2}{2T_g}\right) \left(1+ \frac{\varepsilon v^2}{2 T_g \rho_g}\int_{-\infty}^{\infty}\left[\frac{u^2}{T_g}-1 \right] h(x,u) \mathrm{d}u + \frac{\varepsilon v m_h}{T_g \rho_g}\right).
\mathcal{Y}_{r,R}:=\left\{y(\cdot)\in\mathrm{C}\left(\mathbb{R}_{+}\!\mapsto\!\mathbb{R}^{n};\alpha\right):\left\Vert y(t)\right\Vert \le R\mathrm{e}^{-\alpha t}\;\forall t\ge0\right\}.
\left( \frac{{\cal A}_{\mathrm{c l o s e d}}}{r_{\min}^{6}}\right)_{\beta_{H}\to \beta_{S}}\: \simeq \: \frac{1 6}{7 3 5 b}\: \left( \frac{\pi}{b}\right)^{3}\: \left( \frac{M_{S}}{M_{P L}}\right)^{2}\: \left( \frac{M_{S}}{m_{0}}\right)^{2}\: \ll \: 1
\real\left(\frac{z \mathtt{f}'_{a, \nu}(z)}{\mathtt{f}_{a, \nu}(z)}\right)\geq \frac{r\mathtt{f}'_{a, \nu}(r)}{\mathtt{f}_{a, \nu}(r)}=a^{a/2}-\frac{a^{a/2}}{a \nu-a+1}\sum_{n=1}^\infty \frac{2 r^2}{\mathtt{j}^2_{\nu, n}-r^2}.
G ( \lambda ) = \sum_{s = 0}^{\infty}\left( \frac{\lambda}{\lambda_{1}}\right)^{s}L_{s}^{- \frac 1 2}( x_{1}) \sum_{k = o}^{\infty}\left( \frac{\lambda}{\lambda_{2}}\right)^{k}L_{k}^{- \frac 1 2}( x_{2}) .
\mathcal{A}{:=}\mathcal{K}\, \ell^2_{\rm SG}, \mathcal{S}{:=}\mathcal{K}\, \ell^{1/p'}_{\rm SG}\, \ell^{1/q'}_{\rm chiral}\frac{1}{p'}+\frac{1}{q'}= 1, \,\, p',q' \in [1,\infty).
v_t+\frac{1}{2}\sup_{\lambda\in[\underline\lambda,\overline\lambda]}\left(\lambda^2 v_{xx}\right)=v_t+\frac{1}{2}\left(\overline\lambda^2 v_{xx}^+-\underline\lambda^2v_{xx}^- \right)=0, (t,x)\in [0,1)\times\mathbb R,
\delta V(R)=-\frac{2(D-2)\beta^2}{M^4}\int\limits_0^{\infty}dy \,y^3 \bar\Omega\frac{\left(e^{-R y}-e^{-R\bar\Omega}\right)^2}{\left(1-e^{-2 R y}\right)\left(1-e^{-2R\bar\Omega}\right)}<0.
\left[g_{ij}\right]=\left[\begin{array}{ccc}1 0 0 0 r^{2}0 0 0 r^{2}\sin^{2}\theta\end{array}\right]\,\,\,\,\,\,\,\,\,\,\&\,\,\,\,\,\,\,\,\,\,\left[g^{ij}\right]=\left[\begin{array}{ccc}1 0 0 0 \frac{1}{r^{2}}0 0 0 \frac{1}{r^{2}\sin^{2}\theta}\end{array}\right]
\tilde{\phi}_{0}(\textbf{x})= I \left\lbrace \textbf{x}\in \mathcal{C}^{'}_{1}\cap \mathcal{C}^{'}_{0}, (1-q(\textbf{x},1)) \dfrac{p}{\mu{(\mathcal{C}^{'}_{1})}}> (1-q(\textbf{x},0)) \dfrac{1-p}{\mu{(\mathcal{C}^{'}_{0})}}\right\rbrace + I \left\lbrace \textbf{x}\in \mathcal{C}^{'}_{1}- \mathcal{C}^{'}_{0}\right\rbrace.
\inf\limits_{\varphi \in \mathsf{H}^{1/2}(\mathbb{R}^n;\mathbb{C}^{n-1})\setminus\{0\}}\frac{\mathtt{d}_{n}\big[\binom{\varphi}{L_n\varphi}\big]+\mathtt{v}\big[\binom{\varphi}{L_n\varphi}\big]}{\big\|\binom{\varphi}{L_n\varphi}\big\|^2}>-1.
H(P_{i})^{-w+\delta}= H(P_{i})^{-n+1}H(P_{i})^{-(w-n+1-\delta)}&\leq H(P_{i})^{-n+1}H(Q)^{-1}\ll_{n,\zeta}\vert P_{i}(\zeta)\vert \ll_{n,\zeta}H(P_{i})^{-w},
(\beta_{1}-\beta_{2})(\beta_{2}-\beta_{3})(\beta_{3}-\beta_{1})\left(\sum_{i=1}^{3}(\beta_{i}-\delta_{1})^{3}(\beta_{i+1}-\delta_{1})^{3}- \sum_{i=1}^{3}(\beta_{i}-\delta_{2})^{3}(\beta_{i+1}-\delta_{2})^{3}\right).
{\tilde{X}}_{{\bf{q}}}= (-\frac{1}{2\mbox{}i\mbox{}N\mbox{}\epsilon_{{\bf{q}}}})\sum_{{\bf{k}}}[\Lambda_{{\bf{k}}}(-{\bf{q}}) \omega_{{\bf{k}}}({\bf{q}})a_{{\bf{k}}}({\bf{q}}) - \Lambda_{{\bf{k}}}({\bf{q}})\omega_{{\bf{k}}}(-{\bf{q}}) a^{\dagger}_{{\bf{k}}}(-{\bf{q}})]
\frac{\zeta'}{\zeta}(s) = -\sum_{n\leq xy}\frac{\Lambda_{x,y}(n)}{n^s}+ \frac{1}{\log{y}}\sum_{\rho}\frac{x^{\rho-s}-(xy)^{\rho-s}}{\left(\rho-s\right)^{2}}+ \frac{1}{\log{y}}\sum_{n=1}^{\infty}\frac{x^{-2n-s}-(xy)^{-2n-s}}{\left(2n+s\right)^{2}}-\frac{x^{1-s}-(xy)^{1-s}}{\log y\left(1-s\right)^{2}}.
(\tau_1 + \tau_2 + \tau_3)^2 = \tau_1^2 + \tau_2^2 + \tau_3^2 + 2 (\tau_1 \cdot \tau_2 + \tau_2 \cdot \tau_3 + \tau_3 \cdot \tau_1) = -8-8-8+2(4+4+4) = 0.
\int_0^T\int_{U}\left ( \frac{1}{2}\langle A \nabla \rho, \nabla \varphi \rangle + \rho \langle \beta, \nabla \varphi \rangle -\rho\partial_t\varphi \right ) dxdt =\int_0^T \left(\int_{U}\langle \mathbf{B}, \nabla \varphi \rangle \rho dx \right) dt =0,
\Delta^{\mathrm{PLSQR}}_{{\ell:k}}\equiv \sum_{j={\ell}}^{{k}}\hat{\phi}_{j+1}^{\,2}\approx \| \hat{x}- \hat{x}_{{\ell}}\|^2_{\hat{A}^T\hat{A}}= \|{x}-{x}_{{\ell}}\|^2_{{A}^T{A}}.
{\cal L}_{\psi\bar{\psi}H}= - \bar{\psi}[i\gamma^{\mu}\sigma^{\nu\lambda}\left\{\frac{1}{M_P e^{kr_c\pi}}H^0_{\mu\nu\lambda}+\frac{1}{\Lambda_{\pi}}\frac{2J_0(x_n)}{\pi x_n}\sum_{n=1}^{\infty}H^n_{\mu\nu\lambda}\right\}]\psi
\theta \stackrel{?}{=}2 i \sum_{i = 1}^{N_{7}}q_{5 i}\ln ( z - z_{7 i}) + 2 i \sum_{j = 1}^{N_{5}}\ln ( z - z_{5 j}) - 2 i \sum_{k = 1}^{N_{\bar{5}}}\ln ( z - z_{\bar{5}k}) \ .
\int_0^T \sum_{i=1}^{q}\theta_{i, \lambda}(t) f_{i,\lambda}(\xi(t)) dt \geq \int_0^T \sum_{i=1}^{q}\theta_{i, \lambda}(t) f_{i,\lambda}(u_{\lambda}(t)) dt + \int_0^T \left\langle -{\dot{u}}_{\lambda}(t) -{\eta}_{\lambda}(t), \xi(t) - u_{\lambda}(t) \right\rangle dt.
\sum_{m=a}^\ell(-1)^{m-a}{\ell \choose m}{m-1\choose a-1}= \sum_{m}(-1)^{m-a}{(\ell-a)-(-a)\choose m}{-1 + m \choose -a+m}= (-1)^{\ell-a}\binom{-1}{\ell-a}= \binom{\ell-a}{\ell-a}= 1_{\ell\geq a}
\nu_n' = -\frac{1}{n}\sum \limits_{i = 1}^n \Big \{\sum \limits_{j = 1}^i \log(4q_{2n, 2j - 1}p_{2n, 2j - 1}) + \sum \limits_{j = 1}^{i - 1}\log(4q_{2n, 2j}p_{2n, 2j}) + \log(p_{2n, 2i})\Big\}\delta_{\frac{i}{n}}~.
S_{\bar{\psi}_{r}^{-}\bar{\psi}_{r}^{+}}= S_{\psi_{r}^{+}\psi_{r}^{-}}= S_{\bar{\Phi}_{r}^{-}\bar{\Phi}_{r}^{+}}= S_{\Phi_{r}^{+}\Phi_{r}^{-}}= \frac{M_{r}C_{D}C \Gamma_{1 1}}{P^{2}+ M_{r}^{2}},
\begin{array}{l}k_1,k_2,k_3,k_4, begin{align*}0.4em] \left[X,Y\right]=Z, begin{align*}0.4em] \left[X,Z\right]=4\{X,Y\}+4X^2-4(k_1+k_2+k_3+k_4)X+4(k_1-k_2)(k_4-k_3), begin{align*}0.4em] \left[Z,Y\right]=4\{X,Y\}+4Y^2-4(k_1+k_2+k_3+k_4)Y+4(k_3-k_2)(k_4-k_1),\end{array}
\mathbb{P}(\tau < \epsilon) = \mathbb{P}\bigg( \bigcap_{l=1}^{\infty}\bigcup_{k=l}^\infty \{\tau_k < \epsilon \}\bigg) \leq \lim\sup_l \mathbb{P}(\tau_l < \epsilon) \leq \sup_l \mathbb{P}( |X_{n_l}|_{\mathcal{E}(\tau_l \wedge \epsilon)}> | X_{n_l}(0) |_1 + (M-1) ),
\vec{w}= \left( \begin{array}{c}{w_{3}+ \frac{1}{6}}\\{6 w_{3}}\\{w_{3}}\\ \end{array}\right) , \qquad \vec{k}= \left( \begin{array}{c}{k_{3}+ \frac{1}{6}\ln c_{0}^{2}}\\{6 k_{3}}\\{k_{3}}\\ \end{array}\right)
\pi_{n+1}^{\mu}(dx_{n+1})=F(\pi_{n}^{\mu},y_{n+1})(dx_{n+1}) &:=\frac{g(x_{n+1},y_{n+1})\int_{\mathcal{X}}T(dx_{n+1}|X_{n}=x)\pi_{n}^{\mu}(dx)}{\int_{\mathcal{X}}g(x_{n+1},y_{n+1})\int_{\mathcal{X}}T(dx_{n+1}|X_{n}=x)\pi_{n}^{\mu}(dx)}
\big \langle x \big \vert \, \exp - i \left[ \left( - d_{x}^{2}- 4 q p_{u}x \right) s \right] \, \big \vert x \big \rangle = \left({\frac{1}{( 4 \pi i s )^{1 / 2}}}\right) \; \exp - 4 i \left( q p_{u}x s +{\frac{1}{3}}q^{2}p_{u}^{2}s^{3}\right) .
\left\langle U_{\lambda}\left( w \right) \right\rangle_{w^{\prime}}^{\left({\it G}_{1}\right)_{X_{i^{\prime}}^{\prime}}}= \left\{\prod_{j^{\prime}\in J_{X_{i^{\prime}}^{\prime}}}e^{q_{1}( L^{d}- 1 ) r ( L^{\beta}t_{j^{\prime}}^{\prime})}\right\}^{- 1}\times
T_{n n}^{( 2 )}= - \frac{1}{2 \pi^{2}}\sum_{m = - N}^{m = + N}\int d^{3}p \frac{\lambda_{m}^{2}}{{[{( \vec{p}- \vec{k})}^{2}+ \mu^{2}]}^{2}[ p^{2}- ( K^{2}- \frac{{( n - m )}^{2}}{R^{2}}) - i \epsilon ]}.
\lim_{\beta\downarrow 0}\left( F_{m,r}(s,\beta) - \frac{1}{m \left(s+2\sin^{2}\left(\frac{\beta}{m}\pi\right)\right)}\right)= \frac{U_{m-\ell-1}(s+1)+U_{\ell-1}(s+1)}{T_{m}(s+1)-1}-\frac{1}{ms}=F_{m,r}(s).
\vec{\cal K}U_{E , \hat{q}}= \left( \begin{array}{r}{\vec{K}^{\prime}u_{E , \hat{q}}^{( 1 )}}\\{E^{- 1}\sqrt{V}\pi \vec{K}^{\prime}u_{E , \hat{q}}^{( 1 )}}\\ \end{array}\right) , \quad \vec{K}^{\prime}= \vec{K}+ F \frac{\vec{\sigma}}{2},
\lim_{k \rightarrow \infty}\mathcal{H}^{n-2}\left( (\partial^{*}F_{\ell^{k}})_{z}\right) = \mathcal{H}^{n-2}\left( (\partial^{*}F_{\ell})_{z}\right) \mbox{for}\mathcal{H}^{1}\mbox{-a.e}z \in J.
\left[ \begin{smallmatrix}T; N, \mu d \end{smallmatrix}\right]^+ &= \left[ \begin{smallmatrix}T; N, \mu -1 d \end{smallmatrix}\right]^+ +\big( 1 + (T; N, d-1)^+ (q T_{N+d-1}) \big) (q^{\mu-d}T_{N+d}T_{N+d+1}\dots T_{N+\mu-1}) \left[ \begin{smallmatrix}T; N, \mu -1 d-1 \end{smallmatrix}\right]^+.
\sqrt n\langle\Pi_m Q\Pi_m h -Qh,h\rangle\leq \|(I-\Pi_m)Qh\|+ \|(I-\Pi_m)Qh_m\| = \sqrt n \sqrt{\Delta_m}\left(\|(Qh)'\|+\|Q\|_{L_{}(L^2(0,1),\dot H^1)}\|h_m\|\right).
D=\det\left(\begin{array}{ccc}D_{tt}-D_{tb}(Q^{-1})_{bc}D_{ct}D_{tj}-D_{tb}(Q^{-1})_{bc}D_{cj}D_{tb}D_{it}-D_{ib}(Q^{-1})_{bc}D_{ct}D_{ij}-D_{ib}(Q^{-1})_{bc}D_{cj}D_{ib}0 0 Q^{ab}\end{array}\right)
|f(X) - f(X_1)| = |\sum_{j=1}^{r+s}\sum_{\substack{i=1 \lambda_i(X) <0}}^n |a_j^i|^2(X) \lambda_i (X)- \sum_{j=1}^{r+s}\sum_{\substack{i=1 \lambda_i(X_1) <0}}^n |a_j^i|^2(X_1) \lambda_i(X_1)|.