harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sigma_{1}= \left( \begin{array}{c c}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{array}\right) ; \sigma_{2}= \left( \begin{array}{c c}{{0}}&{{- i}}\\{{i}}&{{0}}\\ \end{array}\right) ; \sigma_{3}= \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{array}\right)
R_{X Y Z}{}^{W}= f_{i A}^{W}f_{Z}^{i B}{\cal R}_{X Y B}{}^{A}, \qquad \delta_{j}^{i}{\cal R}_{X Y B}{}^{A}= f_{W}^{i A}f_{j B}^{Z}R_{X Y Z}{}^{W},
\frac{|(\nabla{\hat\rho})^{\perp}|^2}{\hat\rho}=\phi_{\rm rad}^2\frac{|(\nabla\rho)^{\perp}|^2}{\rho}+2(\nabla\rho)^{\perp}\cdot(\nabla\phi_{\rm rad}^2)^{\perp}+\rho\frac{|(\nabla\phi_{\rm rad}^2)^{\perp}|^2}{\phi_{\rm rad}^2}\leq \phi_{\rm rad}^2\frac{|(\nabla\rho)^{\perp}|^2}{\rho}+c\beta_*^2
P_{i,1}=(\dot{x}_i-\mu_i)\sigma_i\partial_y\widetilde{R}_i, &P_{i,2}=-\bigg(\dot{\mu}_i+\sum^n_{\substack{j=1, j\not=i}}\frac{a_{ij}}{x^3_{ij}}+\sum^n_{\substack{k,j=1, j\not=i}}\frac{b_{ijk}\mu_k}{x^3_{ij}}\bigg)\frac{\sigma_i\Lambda\widetilde{R}_i}{1+\mu_i}.
\dim \delta H_3^g = \; 8 + 12(k+2) + 6\,k(k+1)/2+(k-2)(k-1)k/6 \;-(k+4)(k+5)(k+6)/6 = \; 3(k+4), \dim \delta E_3^g = \; I_M(V_3^g\times W_3^g)=\; 3(k+2),
\left\| \sigma(y)\right\|_2^2 =\left(y, \sigma(y)\right)-\alpha\left(\Delta y, \sigma(y)-\mathbb{P}\sigma(y)\right) &+\tfrac{\alpha}{\nu}\left(\left(u,\mathbb{P}\sigma(y)\right)-\left(\mathbf{curl}\,\sigma(\mathbf y)\times \mathbf y_2+\mathbf{curl}\, \sigma(\mathbf y_1)\times \mathbf y,\mathbb{P}\sigma(\mathbf y)\right)\right)
S_{G}= \int d^{3}x \left[ - \epsilon \frac{1}{2}\left( D_{\mu}Q^{A}D^{\mu}Q^{B}\eta_{A B}\right) - W_{G}( Q^{A}) - \kappa \epsilon^{\mu \nu \rho}\mathrm{T r}\left( \partial_{\mu}A_{\nu}A_{\rho}+ \frac{2}{3}A_{\mu}A_{\nu}A_{\rho}\right) \right] ,
I_{n+\nu,n}(z) = \frac{(-1)^\nu n!}{2\pi i}\, \left( \frac{d}{dz}\right)^{n+\nu}\int (e^{z-\tau}+ e^{\tau-z}) \, \frac{d\tau}{\tau(\tau-z)^{n+1}}begin{align*}0.3cm] = \frac{(-1)^n n! (n+\nu)!}{\pi i}\int \,\frac{\cosh t}{t^{n+1}(t+z)^{n+\nu+1}}\, dt\, ,
\sum_{x\in Z_1 \setminus Z_2}P^x (\tau_{Z_1 \cup Z_2}=\infty) = \sum_{\ell=1}^{j_1+j_2}q_{v(\ell)}1_{\{\hat{x}_{v(\ell)}\in Z_1 \setminus Z_2\}}\,, j_1+j_2 = \sum_{\ell=1}^{j_1+j_2}q_{v(\ell)}\sum_{i=1}^{j_1+j_2}G(x_i,x_\ell) \,.
\log T_{X;\mathfrak{P}_\Gamma}(\rho):=\frac{1}{2}\frac{d}{ds}\biggl|_{s=0}\left(\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}K_{X;\mathfrak{P}_\Gamma}(t,\rho)dt\right); &\log T_{X;X_0}(\rho):=\frac{1}{2}\frac{d}{ds}\biggr|_{s=0}\left(\frac{1}{\Gamma(s)}\int_0^\infty t^{s-1}K_{X;X_0}(t,\rho)\right).
\hat{f}_{-}\left( x\right) =E\left\{\hat{f}_{-}\left( x\right)\right\}+o_{p}\left( 1\right) . E\left\{\hat{f}_{-}\left( x\right) \right\}=E\left\{\hat{f}_{-}\left( 0\right) \right\}+o\left( 1\right) . E\left\{\hat{f}_{-}\left( 0\right) \right\}\rightarrow\infty.
(f,g) =S_{\chi\oplus \xi}^{-\frac{1}{2}}S_{\chi\oplus \xi}S_{\chi\oplus \xi}^{-\frac{1}{2}}(f,g) &=\int_{\Theta}v^{2}(w)S_{\chi\oplus \xi}^{-\frac{1}{2}}\pi_{F(w)\oplus G(w)}(\chi_{w}\oplus \xi_{w})^{\ast}(\chi_{w}\oplus \xi_{w})\pi_{F(w)\oplus G(w)}S_{\chi\oplus \xi}^{-\frac{1}{2}}(f,g)d\mu(w).
\tilde j^{\mu_1\cdots \mu_{D-d}}=(\frac 1{\sqrt{g}})J^{\mu_1\cdots \mu_{D-d}}(\frac \phi x)\sum_{i=1}^l\beta_i\eta_i\int_{N_i}\delta^D(x-z_i(u))\sqrt{g_u}d^{(D-d)}u,
\chi_1(t) =\begin{cases}1\quad t\leq -1+\kappa 0\quad t\geq -1+2\kappa,\end{cases}\chi_2(t) =\begin{cases}1\quad -1+2\kappa\leq t\leq 1-2\kappa 0\quad t\leq -1+\kappa t\geq 1-\kappa,\end{cases}\chi_3(t) =\begin{cases}1\quad t\geq 1-\kappa 0\quad t\leq 1-2\kappa.\end{cases}
c_2(S) =\chi_{\rm top}(S) = \chi_{\rm top}(S - R - R_0) + \chi_{\rm top}(R) + \chi_{\rm top}(R_0) & = n \cdot \chi_{\rm top}(\SS - \delta) + (-2) + (-2)(n-2) & = 3n - 2 - 2(n-2)= n+2.
\zeta = g\wedge dg\circ Q = \frac{1}{2}\{(e_{0}+\phi e_{1}-i\phi e_{2}-e_{3})\wedge (\phi e_{0}+ e_{1}+ i e_{2}+ \phi e_{3})\omega &+ (e_{0}+\bar{\phi}e_{1}+i\bar{\phi}e_{2}-e_{3})\wedge (\bar{\phi}e_{0}+ e_{1}- i e_{2}+ \bar{\phi}e_{3})\bar{\omega}\}.
\partial_\mu p_{\mu,\sigma}(x) = p_{\mu,\sigma}(x) \left( \frac{x-\mu}{\sigma^2}\right) \partial_\sigma p_{\mu,\sigma}(x) = p_{\mu,\sigma}(x)\left( -\frac{1}{\sigma}+ \frac{(x-\mu)^2}{\sigma^3}\right).
\dot{\Phi}^2\left[{1\over 2}{I\over c^2}\dot{\Phi}^2-V+\hat{p}\sqrt{I}\right]{\dot{a}\over a}+\dot{\Phi}\ddot{\Phi}\left[{1\over 2}{I\over c^2}\dot{\Phi}^2+V+\hat{\rho}c^2\sqrt{I}\right]\leq 0.
\Gamma \lbrack A \rbrack = \! - 2 \! \int_{0}^{\infty}{\frac{d T}{T}}\mathrm{e}^{- m^{2}T}\int{\cal D}x \,{\cal D}\psi \, \mathrm{e x p}\left[ - \int_{0}^{T}\! \! \! d \tau \left({\frac{1}{4}}{\dot{x}}^{2}\! + \!{\frac{1}{2}}\psi \dot{\psi}\! + \! i e A_{\mu}\dot{x}^{\mu}\! - \! i e \psi^{\mu}F_{\mu \nu}\psi^{\nu}\right) \right] \nonumber \,
\rho_t - \rho_{t_0}=\bigl [\Pi_t (\hat \rho_t - \hat \rho_{t, M}) + \Pi_{t_0}(\hat \rho_{t_0,M}-\hat \rho_{t_0})\bigr ] \qquad\quad+[\Pi_t ( \hat \rho_{t,M}-\hat \rho_{t_0,M})]+[ (\Pi_t -\Pi_{t_0}) (\hat \rho_{t_0,M})] \, ,
\phi\big(t_\eta^{\Lambda^i}{t_\rho^{\Lambda^i}}^*\big)t_\lambda^\Lambda{t_\mu^\Lambda}^*=t_\eta^\Lambda{t_\rho^\Lambda}^* t_\lambda^\Lambda{t_\mu^\Lambda}^*=\sum_{(\alpha,\beta)\in \Lambda^{\min}(\rho,\lambda)}t_{\eta\alpha}^\Lambda{t_{\mu\beta}^\Lambda}^*\in X_n.
\sum_{i=1}^{n}e^{i}(\nabla_{e_{i}}(\phi Y)) =\sum_{i=1}^{n}g(\pi_{L}(\nabla_{e_{i}}(\phi Y)), e_{i}) &=-\sum_{i=1}^{n}g( \pi_{\phi}(\phi Y), \nabla_{e_{i}}(\pi^{t}_{L}e_{i})) =g (Y, -H_{\phi}+ \xi^{\top}).
\displaystyle J_{M(y)}(y,\widetilde{f}) =D^G(y)^{1/2}\int_{G_y(F)\backslash G(F)}\widetilde{f}(g^{-1}yg)v_{M(y)}(g) dg =\int_{Z_G(x)(F)\backslash G(F)}\sum_{z\in G_x(F)\backslash Z_G(x)(F)}D^{G_x}(y)^{1/2}\int_{G_y(F)\backslash G_x(F)}\left({}^g \widetilde{f}\right)_{x,\Omega_x}(z^{-1}g_x^{-1}yg_xz)v_{M(y)}(g_xzg)dg_xdg
(T_jf)(\mu ) = \begin{cases}t_j f(\mu ) \a_j(\mu) =0 f(s_j\mu ) =f(\mu-\alpha_j) \a_j(\mu) =1 f(\mu-2\alpha_j) -(t_j-1)f(\mu-\alpha_j) \a_j(\mu) = 2 \end{cases}.
{\mathbf F}= \frac{1}{\sqrt{Q}}\begin{bmatrix}\rho_{0,0}\rho_{0,1}\cdots \rho_{0,Q-1}\rho_{1,0}\rho_{1,1}\cdots \rho_{1,Q-1}\vdots \vdots \vdots \vdots \rho_{Q-1,0}\rho_{Q-1,1}\cdots \rho_{Q-1,Q-1}\end{bmatrix},
\int_0^{\infty}\ldots \int_0^{\infty}\Delta\{R_1,...,R_n\}\prod_{j=1}^n R_j^{a}e^{-X_j\sqrt{R_j}}dR_j=2^n\frac{\Delta\{X^2_1,...,X^2_n\}}{\prod_{j=1}^n X_j^{2(a+n)}}\prod_{j=1}^n\Gamma\left[2(j+a)\right]
\bigl \{T_{a}, \, T_{b}^{( 1 )}\bigr \}_{( \phi , \pi )}+ \bigl \{T_{a}^{( 1 )}, \, T_{b}\bigr \}_{( \phi , \pi )}+ \bigl \{T_{a}^{( 1 )}, \, T_{b}^{( 2 )}\bigr \}_{( \eta )}+ \bigl \{T_{a}^{( 2 )}, \, T_{b}^{( 1 )}\bigr \}_{( \eta )}= 0 \, .
\left( -\frac{1}{r}\frac{\partial}{\partial r}r\frac{\partial}{\partial r}-\frac{1}{r^{2}}\frac{\partial^{2}}{\partial \varphi^{2}}+V\left( \hat{r}\right) \right) \psi \left( \hat{r}\right) =E\psi \left( \hat{r}\right) .
\left\langle \left( W_{( 0 )}( x_{\mu}) - W_{( 0 )}( x_{\mu}^{'}) \right) \prod_{i = 1}^{k}W_{( 0 )}( x_{k}) \right\rangle = \left\langle \left\{Q_{W}, \int_{x_{\mu}}^{x_{\mu}^{'}}\partial_{\mu}W_{( 0 )}d x^{\mu}\cdot \prod_{i = 1}^{k}W_{( 0 )}( x_{k}) \right\}\right\rangle = 0 ,
N(r,R_2) \leq N(r,\pi^*_Y R_C) := \frac{1}{\mathrm{deg}(p_Y)}\int^r_0 \#\left( \pi^*_YR_C \cap Y(t)\right) \frac{dt}{t}= \frac{\mathrm{deg}(\pi_Y)}{\mathrm{deg}(p_Y)}\int^r_0 \#\left( R_C \cap C(t)\right) \frac{dt}{t}= O(\log r).
k_j = N_j\frac{1-\frac{\prod_{k=1}^K\delta_k}{\delta_j}}{1-\prod_{k=1}^K\delta_k}+ \left(N_j\frac{1-\frac{\prod_{k=1}^K\delta_k}{\delta_j}}{1-\prod_{k=1}^K\delta_k}\right)^{3/4}, \mbox{and\quad}n_1 = \max_{j}\frac{k_j+k_j^{3/4}}{(1-\delta_j)\frac{\prod_{k=1}^K\delta_k}{\delta_j}}.
\zeta^{35,35}(z)=\sum_{1\leq j_1+j_2\leq 5}\lambda_{j_1,j_2}^{35,35}z^{j_1}_1z^{j_2}_2 , \ \ \ \ \ \ \ \ \ \ \eta^{35,35}_z(y)=\sum_{1\leq i_1+i_2\leq 5}\omega_{z,i_1,i_2}^{35,35}\y_1^{i_1}y_2^{i_2},
\phi\big((X_2^\frac{qs}{2}K^*X_1^{ps}KX_2^\frac{qs}{2})^{\frac{1}{s}}\big) = \ \phi\big((X_2^\frac{qs}{2}C_2^{-\frac{qs}{2}}M^*C_1^{-\frac{ps}{2}}X_1^\frac{ps}{2}X_1^\frac{ps}{2}C_1^{-\frac{ps}{2}}MC_2^{-\frac{qs}{2}}X_2^\frac{qs}{2})^\frac{1}{s}\big) = \ \phi\big(|G_X\big(\frac{p}{r}\big)|^\frac{2}{s}\big).
\left(\frac{d}{dt}\nabla\sigma_t\right)_{|t=0}=-(q-1)\, \frac{\nabla u}{u}\,\left(\frac{d}{dt}\sigma_t\right)_{t=0}+\nabla v\, \left(\frac{v}{u}\right)^{q-1}-\nabla u &=-\left(\frac{q-1}{q}\right)\, \nabla u\, \left(\left(\frac{v}{u}\right)^q-1\right)+\nabla v\, \left(\frac{v}{u}\right)^{q-1}-\nabla u.
\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}(a_{i}b_{j}-a_{j}b_{i})^{2}=\sum_{i=1}^{n-1}\sum_{j=i+1}^{n}(a_{i}^{2}b_{j}^{2}+a_{j}^{2}b_{i}^{2}-2a_{i}b_{j}a_{j}b_{i}).
1+(-1)^{(N-1)d+\chi}y^{d+1}_{t=0}\, \omega =\sum_{k=0}^{d}\binom{-\chi+(N+1)d}{k}\binom{\chi-Nd}{d-k}(1-y)^{\chi-Nd-d+k}&=\sum_{k=0}^{d}\binom{-\chi+(N+1)d}{k}\frac{(-y)^k}{(1-y)^{(N+1)d-\chi}}.
\left\langle u_{\varepsilon}\right\rangle_{n,\omega\gamma,\overline{Q}}^{(m+\gamma)}\leq C\left({\displaystyle\sum\limits_{i=1}^{N}}\left\langle x_{N}^{n}D_{x_{i}}^{m}u\right\rangle_{\omega\gamma,x_{i},\overline{Q}}^{(\gamma)}+\left\langle D_{t}u\right\rangle_{t,\overline{Q}}^{(\gamma/m)}\right) .
I_{m,n}=-\frac{x^{m-1}}{a(2n-m-1)(ax^{2}+bx+c)^{n-1}}-\frac{b(n-m)}{a(2n-m-1)}I_{m-1,n}+\frac{c(m-1)}{a(2n-m-1)}I_{m-2,n}
L\big(\chi^k, k/2\big)=L( G_k(q),k)\cdot \begin{cases}1 k\equiv \pm 1 \mod 6, \dfrac1{1-(-3)^{k/2}3^{-k}}k\equiv \pm 2 \mod 6, \dfrac1{1+2(-2)^k4^{-k}}k\equiv 3 \mod 6, \dfrac1{(1-(-3)^{k/2}3^{-k})(1+2(-2)^k4^{-k})}k\equiv 0 \mod 6.\end{cases}
S_3=-\frac{m^2}{2}\int d^4\theta d^d x e^{\sigma+\bar{\sigma}}\int\frac{d^d k}{{(2\pi)}^d}\frac{-A^2 k^2+m^2}{k^2{(-A^2 k^2+m^2)}^2- 81\Lambda\bar{\Lambda}}\frac{\bar{D}^2_1 D^2_1}{16}\delta_{11}
e^{\textstyle -J\alpha_-\Phi(\sigma, \tau )}=c_J a(\varpi)\>\sum_{m=-J}^J \, (-1)^{J-m}\,{\lambda_m^{(J)}(\varpi)\over \lfloor \varpi+2m\rfloor}\>\psi_m^{(J)}(\sigma)\,{\overline \psi_{m}^{(J)}}(\sigma) \bigr / a(\varpi),
\begin{array}{rcl}\dot{p}(t) = -(\frac{\partial f}{\partial x})^T (x(t)) p(t) begin{align*}2mm] - \sum_{j=1}^m u_j(t) (\frac{\partial g_j}{\partial x})^T(x(t)) p(t) begin{align*}2mm] - \sum_{j=1}^m du_j(t) \frac{\partial^T h_j}{\partial x}(x(t)) begin{align*}2mm] dy_j (t) = p^Tg_j(x(t)) \, , j=1,\ldots,m \, , \end{array}
F_{n}(h)=\frac{a_{0}+a_{1}h+a_{2}h^{2}+a_{3}h^{3}}{1+b_{1}h+b_{2}h^{2}+b_{3}h^{3}+b_{4}h^{4}+b_{5}h^{5}+b_{6}h^{6}}exp(-\frac{h^{2}}{2})
U(\lambda\mu\rho)={1\over d(\rho)}U_{k_1\,\,k_2}^{\,\,l_1\,\,l_2}\left(\begin{array}{ccc}\lambda \mu \rho k_1 k_2 m \end{array}\right)\left( \begin{array}{ccc}\lambda \mu \rho l_1 l_2 m \end{array}\right)^{\ast}.
ds^2 = dx_1{}^2 + f(x_1)^2 (E dx_2{}^2+2Fdx_2dx_3+Gdx_3{}^2), & ds^2 = E dx_1{}^2+2Fdx_1dx_2+Gdx_2{}^2+ f(x_1,x_2)^2 dx_3{}^2.
B_{n-1}G_0=P_{n+2,n+1}+\frac{\lambda}{4\kappa^2}\frac{v^{n+2}}{(n-1)!}\left\{\frac{n(n+1)}{n+2}\hat{I}_{n,0}-\frac{n(n-1)}{n+1}\hat{I}_{n,1}+\frac{2}{(n+2)(n+1)}\hat{I}_{n,n+2}\right\}.
|E_a| \leqslant 52n^{1/4}(1+(a/n)^{1/2}n^{1/8})(1+L_{+}^{1/2}n^{-1/8})~~(~~L_{+}=\max\{0,n^{1/2}-t\}) &\ll \begin{cases}n^{1/4}+a^{1/2}n^{-1/8},~ t\geqslant n^{1/2}-n^{1/4}\left(n^{1/8}+a^{1/2}n^{-1/4}\right)\sqrt{n^{1/2}-t},~ t<n^{1/2}-n^{1/4}.\end{cases}
\widetilde \Gamma(x_0, y_0, t_0 ;x,y,t) := \Gamma (x_0, t_0 ;x,t) \cdot \frac{1}{(4 \pi (t_0-t))^{m/2}}\exp \left( \frac{-|y_0-y|^2}{4(t_0-t)}\right)
J_{\ell}(\alpha,z) =\sum_{h\in\mathbb{Z}_q^*}e\left(\frac{ah^{\ell}}{q}\right)\sum_{\substack{z^{1/\ell}<m\le 2z^{1/\ell}m\equiv h\bmod q}}\Lambda(m)e(m^{\ell}\beta)+O(\log q\log z) &=\frac{1}{\varphi(q)}\sum_{h\in\mathbb{Z}_q^*}e\left(\frac{ah^{\ell}}{q}\right)\sum_{z^{1/\ell}<m\le 2z^{1/\ell}}e\left(m^{\ell}\beta\right)+R_{\ell}(\alpha,z),
f(x)>Q_{20}(x)=1-8 x^2\!-\!\overline{T}_{20}^{\cos ,0}(4 x)\!-\!8 x^2 \overline{T}_{16}^{\cos ,0}(2 x)\!+\!P_7(x)\underline{T}_{11}^{\sin ,0}(2 x),
\langle M_{\alpha}M_{\bar{u}}EM_{u}f+\lambda^2f,f\rangle =\int_{X}\alpha|E(uf)|^2d\mu+\int_{X}\lambda^2|f|^2d\mu &=\int_{X}\alpha|E(u)|^2|f|^2d\mu+\int_{X}\lambda^2|f|^2d\mu &=\int_{X}(\alpha|E(u)|^2+\lambda^2)|f|^2d\mu\geq0.
\rho\in C([0,T_*];W^{1,q}),\ (H,u)\in C([0,T_*];H^1_0\cap H^2)\cap L^2([0,T_*];D^{2,q}), &\theta \in C([0,T_*]; H^2)\cap L^2([0,T_*];D^{2,q}), &(H_t, u_t,\theta_t)\in L^2([0,T_*];D^1),\ (H_t,\sqrt{\rho}u_t, \sqrt{\rho}\theta_t)\in L^\infty([0,T_*];L^2).
P^*=\begin{bmatrix}0.4152 0.3890 0.2068 0.0161 -0.4060 0.3890 2.7211 1.9100 -2.6077 -0.7757 0.2068 1.9100 1.8537 -1.8333 -0.8980 0.0161 -2.6077 -1.8333 4.2409 -0.2664 -0.4060 -0.7757 -0.8980 -0.2664 2.1540\end{bmatrix},
\frac{|\phi_t|^p}{m_t^\beta}=(1-t+t\,c)^{p-\beta}\, \frac{|\phi_0|^p}{m_0^\beta}>(1-t)\,\frac{|\phi_0|^p}{m^\beta_0}+t\,c^{p-\beta}\, \frac{|\phi_0|^p}{m_0^\beta}&=(1-t)\,\frac{|\phi_0|^p}{m_0^\beta}+\frac{|\phi_1|^p}{m_1^\beta}.
\mathbf{y}_{\mathrm{ue}}= \mathcal{H}_{\mathrm{b2u}}\mathcal{H}_{\mathrm{b2u}}^{\dagger}\mathbf{\Phi}^{\mathrm{dl}}\mathbf{x}_{\mathrm{bs}}+ \mathbf{n}_{\mathrm{ue}}= \mathbf{\Phi}^{\mathrm{dl}}\mathbf{x}_{\mathrm{bs}}+ \mathbf{n}_{\mathrm{ue}},
\theta_3\left(q^{2a_2}\right)^2\theta_3\left(q^{2c_2}\right)^2=\sum_{l|N}C^{(1)}_lT_{l,r}+\sum^{d_2}_{l=1}C^{(2)}_lf_l(q^2)\textrm{,}q=e^{-\pi\sqrt{r}}\textrm{,}r>0.
c=\mathbf{P}_{x_0}(\uparrow \lim_{n\rightarrow \infty}T_{x_n}=T_x, T_x<\infty)=\lim_{n\rightarrow \infty}\mathbf{P}_{x_0}(T_{x_n}>T_x-\varepsilon_0/2, T_x<\infty).
|I_3|=\int_{(1/2)\varepsilon \log |\eta|}^{\varepsilon \log |\eta|}\frac{e^w}{w}\dw \leq \frac{e^{\varepsilon \log |\eta |}}{\varepsilon \log |\eta|}+e^{(1/2)\varepsilon \log |\eta |}\frac{1}{\varepsilon \log |\eta|}\ll \frac{e^{\varepsilon \log |\eta |}}{\varepsilon \log |\eta|}.
\left(\left(\mathbf{P}_{N}\mathbf{\Delta}_{N}\right)\left(\mathbf{S}_{N\mid f}\mathbf{S}_{N\mid g}\right)\right)_{N,m}=\left(\mathbf{P}_{N-1}\mathbf{S}_{N-1\mid f}\mathbf{S}_{N-1\mid g}\right)_{N-1,m}=S\left(f,g;N-1,m\right).
X_1=\partial_{x_3}, ~~X_2=\frac{1}{\sqrt{2}}(-e^{x_3}\partial_{x_1}+e^{-x_3}\partial_{x_2}),~~X_3=-\frac{1}{\sqrt{2}}(e^{x_3}\partial_{x_1}+e^{-x_3}\partial_{x_2}).
\begin{array}{l c r}{d s^{2}={\frac{1}{2}}K_{\rho \sigma}[ d \phi_{\rho}d \phi_{\sigma}+ a_{\rho}a_{\sigma}\sum_{i = 1}^{3}d \Omega_{\rho , i}d \Omega_{\sigma , i}] ;}\\{H = - 2 a_{\rho}d \Omega_{\rho},}\\ \end{array}
g_{( \mu_{1}\mu_{2}\ldots \mu_{2 s})}\frac{1}{( 4 \pi )^{n / 2}2^{s}\Gamma ( n / 2 + s )}\int_{0}^{\infty}d k^{2}( k^{2})^{( n - 2 ) / 2 + s}f ( k^{2}) ,
\sqrt{T}E^{\Pi^{D_T}}[\max_{k,j}|Z_{\lambda,k,j}| |X^T] & \lesssim \sqrt{2 \log 2 (V_\lambda^{\otimes d})}(C_T'+1) \max_{k,j}(\|\tilde{\Phi}_{\lambda,k,j}\|_{\mu_0}^2 + c_T') =O_{P_{b_0}}( \sqrt{\lambda}).
\phi(d^ky)(z) = \partial_z^k\left(\frac{e^z}{1-e^z}\right) = - \partial_z^k\left(\frac{1}{z}\frac{ze^z}{e^z-1}\right) = - \partial_z^k\left( \frac{B_0}{z}+ \sum_{n \geq 0}\frac{B_{n+1}}{(n+1)!}z^n \right) = -\left(\frac{(-1)^{k}k! B_0}{z^{k+1}}+ \sum_{n\geq 0}\frac{B_{n+k+1}}{n+k+1}\frac{1}{n!}z^{n}\right).
\int_{Q_r^+(X_0)}|D_{x^\prime}D^{m-1}u-(D_{x^\prime}D^{m-1}u)_{Q_r^+(X_0)}|^2\,dx\,dt &\le N (\frac{r}{R})^{2a+2m+d}\int_{Q_R^+(X_0)}|D_{x^\prime}D^{m-1}u-(D_{x^\prime}D^{m-1}u)_{Q_R^+(X_0)}|^2\,dx\,dt &+N\sum_{|\alpha|=m}[f_\alpha]_{a, Q_{4R}^+}^{\ast2}r^{2a+2m+d}+N\sum_{|\alpha|<m}\|f_\alpha\|_{L_\infty(Q_{4R}^+)}^2r^{2a+2m+d}.
g(x_{+})=\left( \begin{array}{cc}\cos (\omega x_{+}) -\sin (\omega x_{+}) \sin (\omega x_{+}) \cos (\omega x_{+})\end{array}\right) \quad ;\qquad J(x_{+})=\left( \begin{array}{cc}0 -\omega \omega 0\end{array}\right) \, \, .
\begin{aligned}\mathbf{E}_n\left[ f_0(Z_{t_0})\cdots f_k(Z_{t_k})\right] :=\int_{x\in I}\mathbf{E}_n^x\left[ f_0(Z_{t_0})\cdots f_k(Z_{t_k})\right]\mu_n(dx), \mathbf{E}\left[ f_0(Z_{t_0})\cdots f_k(Z_{t_k})\right] :=\int_{x\in I}\mathbf{E}^x\left[ f_0(Z_{t_0})\cdots f_k(Z_{t_k})\right]\mu(dx). \end{aligned}
M^2_{I{\bar J}}= -\frac{4 g^2}{16 \pi^2}R_{I}R_{J}\sum_{p_5,\, p_6}\int_0^\infty dl \, \, \, \,\left(G^{IJ}- \frac{2\, G^{IK}G^{JL}{p}_K{p}_L}{l}\right)\, \, e^{-( p_M G^{MN}p_N )/2l}
\mathcal{F}^{\omega,\{U_i\}_{i=0,\dots,K}}:=\left\{\psi:=(\psi_0,\psi_1,\dots,\psi_K)|\;\psi_i:\Phi_i(U_i)\rightarrow\psi_i(\Phi_i(U_i)) \right. &\qquad\qquad\qquad\qquad\left.i=0,\dots,KDF^{\omega}(\psi,T\psi)\right.\}\neq\emptyset.
\begin{array}[c]{rl}\mathbb{E}\left[ \int_{0}^{T}\mathcal{H}(t,X^{m}(t),p^{m}(t),q^{m}(t),P^{m}(t),v^{m}(t),u^{m}(t))dt\right] = \inf\limits_{u(\cdot)\in\mathcal{U}[0,T]}\mathbb{E}\left[ \int_{0}^{T}\mathcal{H}(t,X^{m}(t),p^{m}(t),q^{m}(t),P^{m}(t),u(t),u^{m}(t))dt\right].\end{array}
F(A)= \left((2m+1)2^{k-1}-1\right)^2+(d-m)\left((2m+1)2^{k-1}-1\right)-md-d, g(A)= 2^{k-1}(2^k-1)m^2+\frac{1}{2}(d-1)(2^k-1)m+(2^{2k-1}+k2^{k-1}-2^{k+1})m &\ \ +2^{2k-3}+(d+k)2^{k-2}-5\cdot 2^{k-2}-d+1.
ds^2_E \approx - e^{-2\phi(r_H)}(1-b'(r_H)) \left({r-r_H\over r_H}\right) dt^2 +{1\over(1-b'(r_H))}\left({r_H\over r-r_H}\right) dr^2 + r_H^2(d\theta^2 + \sin^2\theta\; d\varphi^2).
\Phi_{R}\left( p^{\mu}\right) =\Lambda_{R}\left( \dot p^{\mu}\to p^{\mu}\right)\Phi_{R}\left( \dot p^{\mu}\right) =e^{i\vec{J}\cdot ( \vec{\vartheta}-i \vec{\varphi})}~\Phi_{R}\left( \dot p^{\mu}\right),
\begin{array}{c}w=(v_1\cdots v_k | \overline{\zeta_1}\cdots \overline{\zeta_s}u_1u_2\cdots); 0<v_1<\cdots <v_k, \ \ \ \overline{\zeta_1}<\cdots< \overline{\zeta_s}<0<u_1<u_2<\cdots.\end{array}
E \Bigl[ F_I (x^o, y^o) \Bigr] = \left( \int F_I (x^q, y^q) \mu_{K,c'}(dx, dy) + O(n^{-2}) \right) \left( 1+ O \left( \frac{|K| (\log n)^{3/2}}{\sqrt{n}}\right) \right).
\tilde{w}_1(t) = b_1 + y_1(t) + \int_0^t\left(-\phi_0(\tilde{w}_1(s)) + \phi_B\big(\tilde{w}_2(s) + \psi_0(\tilde{w}_1(s))\big)\right)ds, \tilde{w}_2(t) = b_2 + y_2(t) + \int_0^t\left(-\phi_B\big(\tilde{w}_2(s)+\psi_0(\tilde{w}_1(s))\big)\right)ds \ge 0.
\begin{array}{lcr}L_H=\int d^2x \sum\limits_{hypermult}[|D_\mu \varphi_H|^2+|V\varphi_H|^2+ \overline{\psi}_H \gamma^\mu D_\mu \psi_H+\overline{\psi}_H V{\psi}_H+\overline{\psi}_VV{\psi}_H + \overline\varphi_H D\varphi_H]\end{array}
\left| \log \frac{(T^p)' (x)}{(T^p)' (y)}\right| \le C' \sum_{i=0}^{p-1}b_{j-i}^\beta \, L_{i-j}\, \frac{| T^p(x) - T^p(y) |}{L_{p-j}}\le C' \frac{| T^p(x) - T^p(y) |}{L_{p-j}},
D_{k,b}=W_{k}^{b-1}, W_{k,b}=\{\hat{Y}_{k,b-1}, X_{k,b}\}, \Gamma_{\hat{Y}_{k,b-1}}=\{l_{k,b-1},l_{k,b-2}\},\Gamma_{X_{k,b}}=\{l_{k,b-1}\},A_{\hat{Y}_{k,b-1}}=\{X_{k,b-1}\}p(W_{k,b}|D_{k,b}, y_{k,b}^{\mathrm{unf}}) =p_{\hat{Y}_k|X_k,Y_k}(\hat{y}_{k,b-1}|x_{k,b-1},y_{k,b-1})p_{X_k}(x_{k,b})
q(a,r)=-\frac{1}{2\pi a^{3}r^{n}S_{D}}\sum_{l=0}^{\infty}D_{l}\int_{ma}^{\infty}dz\,z^{3}\frac{\bar{K}_{\nu}(z)}{\bar{I}_{\nu}(z)}\frac{F_{\nu}^{(q)}\left[ I_{\nu}(zr/a)\right]}{\sqrt{z^{2}-m^{2}a^2}},\quad q=\varepsilon,\,p,\,p_{\perp},\quad r<a,
\left[ \widetilde{\mathbf{y}}_\ell \right] \bmod \Lambda_{C,m}=\left[ \left[{\mathbf{y}}_\ell +\mathbf{q}_\ell \right] \bmod \Lambda_{C,\ell}\right] \bmod \Lambda_{C,m}\stackrel{(a)}{=}\left[{\mathbf{y}}_\ell +\mathbf{q}_\ell \right] \bmod \Lambda_{C,m}= \mathbf{t}_{m,\ell}
V_1\cap V_2 = \langle z \rangle V_1 \cap V_3 = \langle y+z \rangle V_1 \cap V_4 =\langle y\rangle V_2 \cap V_3 =\langle x+y+z\rangle V_2 \cap V_4 =\langle x+y \rangle V_3 \cap V_4 = \langle x\rangle
p_i = \int_{iQ-\frac{Q}{2}}^{iQ+\frac{Q}{2}}\frac{\Lambda}{2}e^{-\Lambda \vert x \vert}dx = \frac{1}{2}\left( e^{-\Lambda \left(iQ-\frac{Q}{2}\right)}- e^{-\Lambda \left(iQ+\frac{Q}{2}\right)}\right)
G_{p,q}^{m,n}(\begin{matrix}a_{p}\\b_{q}\end{matrix}|\frac{z}{w})=w^{1-a_{p}}\sum_{h=0}^{\infty}\frac{(w-1)^{h}}{h!}G_{p,q}^{m,n}(\begin{matrix}a_{1},...,a_{p-1},a_{p}-h\\b_{q}\end{matrix}|z),n<p.
u\mapsto\mu_{u}(q)=(1-q)\sum_{k=1}^{\infty}\left(\,_{3}\phi_{3}\left[\begin{matrix}0, q, q q^{3/2}, -q^{3/2}, -q \end{matrix}\ ;q, -\lambda_{k}^{2}(u) \right]\right)^{\!-1}\delta_{\lambda_{k}(u)}
\Vert Z\Vert_{\mathcal{D}_{\mathbf{X},\tilde{\alpha}}^{\gamma}(I)}:=\Vert Z\Vert_{C(I;\mathcal{B}_{\tilde{\alpha}})}+\Vert Z^{\prime}\Vert_{\mathcal{E}^{0,\gamma}_{\tilde{\alpha}-\gamma;I}}+\Vert Z^{\#}\Vert_{\mathcal{E}^{\gamma,2\gamma}_{\tilde{\alpha};I}}<\infty
\|D_{\bf p}a\|_{L^1}\ \lesssim \ \begin{cases}|c_{j_0, k_0}2^{p j_0 + q k_0}|^{-\delta}\mbox{\rm if}\p\le 0 |c_{j_0, k_0}2^{p (j_0 - 1) + q k_0}|^{-\delta}\mbox{\rm if}\p\ge 0 \end{cases}
\frac{\partial}{\partial{t}}P\bigl(x,t|x_\circ, t_\circ\bigr) = \frac{\partial}{\partial{x}}\Big[\lambda{x}P\bigl(x,t|x_\circ, t_\circ\bigr)\Bigr] + \frac{D}{2}\frac{\partial^2}{\partial{x}^2}P\bigl(x,t|x_\circ, t_\circ\bigr),
\begin{aligned}d\cdot vol(B_{\mathbb{C}^n}(0, 1)) = \frac{1}{(-2\sqrt{-1})^n}\int_{w^{-1}(B(0, 1))}dw_1\wedge d\overline{w_1}\wedge\cdot\cdot\cdot\wedge dw_n\wedge d\overline{w_n}&\leq (1+\Phi(\epsilon|n))Vol(B(x, (1+\Phi(\epsilon|n)))+\Phi(\epsilon|n)\end{aligned}
Z_1^*W_{\pi^+}(K)Z= e^{i \frac19 30 m}e^{i 90 m}b_1 z_1^*z_1 + e^{i \frac19 30 m}e^{i 90 m}b_2 z_2^*z_2+\frac12 e^{i \frac19 30 m}e^{i 90 m}b_3 [z_1^*(z_1+z_2)+ z_1^*(z_1+z_2)]
P^{\mu '}(K_{n}=k) = \int_{\mathcal{P}_{\mathcal{V}_{\alpha,\theta}}}P(K_{n}=k)\mu'(\mathrm{d}P)= \int_{\mathcal{P}_{\mathcal{V}_{\alpha,\theta}}}P(K_{n}=k)\mu \circ T^{-1}(\mathrm{d}P) = \int_{\mathcal{V}_{\alpha,\theta}}d_{n,k}V_{n,k}\mu(\mathrm{d}V)=d_{n,k}V^{\mu}_{n,k}.
\hat{V}_{\mu \nu \rho}^{( 1 z ) A B}= 2 \cdot 2 \mathrm{T r}T^{A}T^{B}( Y_{R}^{( d )}- Y_{L}^{( d )}) V_{\mu \nu \rho}^{( 1 )}= - 4 \mathrm{T r}T^{A}T^{B}V_{\mu \nu \rho}^{( 1 )}.
f = \left( 1 + \frac{a^{2}\sinh^{2}\alpha}{\bar{r}^{2}}\right) \left( 1 + \frac{a^{2}\sinh^{2}\gamma}{\bar{r}^{2}}\right) \left( 1 + \frac{a^{2}\sinh^{2}\sigma}{\bar{r}^{2}}\right)
\sum_{n=0}^{\infty}\textup{PD}_\textup{t}(192n+72)q^n &\equiv 2f_3^3+3 \dfrac{f_3^3f_4^5}{f_1^4f_8^2}&\equiv 2f_3^3+3f_3^3\cdot \dfrac{f_4}{f_2^2}&\equiv 2f_3^3+3f_3^3\left(\dfrac{f_{12}^4f_{18}^6}{f_6^8f_{36}^3}+2q^2\dfrac{f_{12}^3f_{18}^3}{f_6^7}+4q^4\dfrac{f_{12}^2f_{36}^3}{f_6^6}\right) ~(\textup{mod}~4),
\frac{1}{2\pi i}\int_{\Re(s) = 1}\phi \left( \frac{1-s}{4\pi i}\log R \right)^{n- \eta -\gamma}\mathcal E(s)^\gamma \left(\frac{4\pi m \sqrt{c}}{b \sqrt{N}}\right)^{-s}G_{k-1}(s) ds \ \ll \N^{(n- \eta - \gamma)\sigma/2 + \epsilon}.
\sum_{n=0}^{\infty}c(5n+4)q^{n}=41\dfrac{E_{10}^{3}}{E_{1}^2E_{2}^2E_{5}}+860q\dfrac{E_{10}^{6}}{E_{1}^{5}E_{2}E_{5}^{2}}+6800q^{2}\dfrac{E_{10}^{9}}{E_{1}^{8}E_{5}^{3}}\quad+24000q^{3}\dfrac{E_{2}E_{10}^{12}}{E_{1}^{11}E_{5}^{4}}+32000q^{4}\dfrac{E_{2}^{2}E_{10}^{15}}{E_{1}^{14}E_{5}^{5}}.
M(u_n,v_n) = \sum_{j=1}^l M(U^j_n, V^j_n) + M(u^l_n,v^l_n) + o_n(1), K(u_n,v_n) = \sum_{j=1}^l K(U^j,V^j) + K(u^l_n, v^l_n) + o_n(1), P(u_n,v_n) = \sum_{j=1}^l P(U^j,V^j) + P(u^l_n, v^l_n) + o_n(1),
{\cal O}_{\pm}^{a}( \{x \}) : = \frac{1}{( N ! )^{2}}\sum_{\pi , \pi^{\prime}}\prod_{b = 1}^{N}\overline{{{\psi}}}^{( b )}( x^{( \pi ( b ) )}) P_{\pm}\psi^{( b )}( x^{( \pi^{\prime}( b ) )}) \; .
\hat{g}_{ij}(z) =\left( 1 + \frac{4}{n-4}\frac{\gamma_n\alpha}{\rho^{n-4}}\right)\delta_{ij}+ \frac{\gamma^{\frac{4}{n-4}}_n}{3}R_{iabj}(p)\frac{z^az^b}{\rho^4}+ \frac{\gamma^{\frac{6}{n-4}}_n}{6}R_{iabj,c}(p)\frac{z^az^bz^c}{\rho^6}&+ O_4(\rho^{-(n-3)}) + O_4(\rho^{-4}),
\tilde F(k)={1 \over{4\pi \eta}}\int_{-\infty}^{+\infty}dq\, \tilde F(q)e^{i{{k-q}\over 2}}\,{{\sin{{k-q}\over 2}}\over{{k-q}\over 2}}\,({\rm sign}\,(q) +\eta) \,(1-e^{-2\pi q\eta \lambda}),
(F_i \otimes 1) \otimes w = - \frac{1}{2}(D(1 \otimes F_i) + (1 \otimes F_i)D) \otimes w & = - \frac{1}{2}(D(1 \otimes F_i)) \otimes w - \frac{1}{2}\underbrace{((1 \otimes F_i) D \otimes w - (1 \otimes F_i) \otimes Dw)}_{\in Z}&- \frac{1}{2}(1 \otimes F_i) \otimes \underbrace{Dw}_{= 0}& \in - \frac{1}{2}D(1 \otimes F_i) \otimes w + Z.
\sigma = \Phi \! \! \left( \! \! \begin{array}{c c}{{2}}&{{2}}\\{{1}}&{{1}}\\ \end{array}\! \! \right) , ~ ~ \bar{\sigma}= \Phi \! \! \left( \! \! \begin{array}{c c}{{1}}&{{1}}\\{{2}}&{{2}}\\ \end{array}\! \! \right) ,
\Gamma_{a , b}( t_{2}) = \left[ \int_{b_{t_{2}}}^{\infty}\int_{x}^{\infty}- \int_{b_{t_{1}}}^{\infty}\int_{a_{t_{1}}}^{\infty}+ \int_{a_{t_{1}}}^{\infty}\int_{a_{t_{1}}}^{\infty}- \int_{a_{t_{2}}}^{\infty}\int_{x}^{\infty}\right] f ( x ) f ( y ) d y d x .
\dot{H}= \frac{{\dot{\varphi}}^{2}}{4 F}- \frac{1}{2}\left( H^{2}+ \frac{{\rho}_{\varphi}}{6 F}\right) - \frac{\ddot{F}}{2 F}+ \frac{p_{m}}{4 F}+ \frac{\rho_{m}}{6 F},
\tilde{{E}}_{1}\left( p\right) ={\tfrac{1}{{24}}}p\,\left({3-5\,p^{2}}\right) ,\ \tilde{{E}}_{2}\left( p\right) ={\tfrac{1}{{16}}}\,p^{2}\left({1-p^{2}}\right) \left({1-5\,p^{2}}\right) .
\int x^{m}(a+bx^{n})^{p}dx=\frac{x^{m+1}(a+bx^{n})^{p+1}}{a(m+1)}-\frac{b(m+n(p+1)+1)}{a(m+1)}\int x^{m+n}(a+bx^{n})^{p}dx
S_t(x) := \big<\varphi_0\varphi_x\big>^{\epsilon,L}_{\lambda,\mu+1/t,m^2},x\in\Lambda_{\epsilon,L}, C_t(x) := \Big(-\Delta^\epsilon+m^2+\frac{1}{t}\Big)^{-1}(0,x), x\in\Lambda_{\epsilon,L},