harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
||\tilde{\bf r}_k||_2 \leq 2\left(\frac{\sqrt{(1+\nu^2)^2}-\sqrt{(1-\nu^2)^2}}{\sqrt{(1+\nu^2)^2}+\sqrt{(1-\nu^2)^2}}\right)^{\lfloor k/2\rfloor}||\tilde{\bf r}_0||_2 &= 2\left(\nu^2\right)^{\lfloor k/2\rfloor}||\tilde{\bf r}_0||_2 &\leq 2\left(\nu^2\right)^{\frac{k}{2}-\frac{1}{2}}||\tilde{\bf r}_0||_2=2\nu^{k-1}||\tilde{\bf r}_0||_2.
(1.)\w:=1+2\gamma_1 a_1+2\gamma_2 a_2\ge0,\ \ (2.)\f_{i}(x^{*})=a_{i}\|x^{*}\|^{2}+b_{i}^{T}x^{*}+c_{i}=0 \ \i+1,2\ (3.)\ \gamma_1 b_{1}+\gamma_2b_2=-x^{*}w
\mu^s_0(\alpha_t,\gamma_t)\longmapsto\mu^s_0(\alpha,\gamma)=\frac{H(\gamma)-H(\alpha)}{\gamma-\alpha}\equiv{\mu}^s_0,~~~\mu^s_1(\beta_t,\delta_t)\longmapsto\mu^s_1(\beta,\delta)=\frac{H(\beta)-H(\delta)}{\beta-\delta}\equiv\mu^s_1,~\forall{t}.
b(i)\ge\frac{1}{\lambda}\ge\frac{1}{2}\max_{0\le j\le M\le k\le n}\max\left\{\sum_{\ell=j}^{M-1}\frac{\pi([0,j])}{p_\ell\pi(\ell)}, \sum_{\ell=M+1}^k\frac{\pi([k,n])}{q_\ell\pi(\ell)}\right\}.
\|P_Nu(t)\|_{L_x^\infty}\lesssim\Big(N^{-\frac{\sigma(p-3)}{p-1}}d_N\Big)^\theta\Big(N^{\frac{2\sigma}{p-1}}d_N'\Big)^{1-\theta}=N^{\frac{\sigma(p-3)}{(p-1)(p-2)}}(d_N)^\theta (d_N')^{1-\theta},
\|\{a_k\|\Box_k^{\alpha_1}T_m\|_{M_1\rightarrow M_2}\|\Box_k^{\alpha_1,\ast}f_k\|_{M_1}\}|~l_{q_2}^{s_2,\alpha_1}(E_m)\| \lesssim \|\{a_k\|\Box_k^{\alpha_1}T_m\Box_k^{\alpha_1,\ast}f_k\|_{M_2}\}|~l_{q_2}^{s_2,\alpha_1}(E_m)\|.
\Phi_{m}( z_{m}) \circ \cdots \circ \Phi_{1}( z_{1}) : M_{q}( \lambda_{1}) \rightarrow M_{q}( \lambda_{m + 1}, k ) \otimes ( ( V ( 0 , \mu_{1}) \otimes \cdots \otimes V ( 0 , \mu_{m}) ) ( z_{1}, \ldots , z_{m})
(I-\mathcal{H})b_{\alpha}=[\zeta_{\alpha}, \mathcal{H}]b-\partial_{\alpha}[D_t\zeta, \mathcal{H}]\frac{\bar{\zeta}_{\alpha}-1}{\zeta_{\alpha}}+\frac{i}{\pi}\sum_{j=1}^2 \frac{\lambda_j \zeta_{\alpha}}{(\zeta(\alpha,t)-z_j(t))^2}.
j a + 4 r - 6 q \geq j (a-3) \geq (4k+2) (a-3) & = (4k+2) a - 2 (6k+3) = (4 k +2) a - 2 a - 4 & = (4k-1) a + a - 4 & \geq (4k-1) a + 9
b_{e-1}(t_0) =b_{e-1}(p^{e-1}T+t_0)=b_{e-1}(j\cdot p^{e-2}T+t) &=\delta+\lambda\cdot a_{e-1}(j\cdot p^{e-2}T+t) &=\delta+\lambda\cdot a_{e-1}(p^{e-1}T+t_0) &=\delta+\lambda\cdot a_{e-1}(t_0).
( \chi_{\{m^{\prime}\}}, T ( t ) \chi_{\{m \}}) \, = \, \exp ( - \frac{1}{2}\lambda_{\{m \}}g_{0}^{2}L_{x}t ) \delta_{\{m^{\prime}\}, \{m \}}\, \stackrel{!}{=}\, ( \chi_{\{m^{\prime}\}}, \exp ( - t H ) \chi_{\{m \}})
(U^\prime_{\bar{0}}\oplus U^\prime_{\bar{1}})(x) = \sup_{x = a+b}\{\min(U^\prime_{\bar{1}}(a), U^\prime_{\bar{1}}(b))\}& = \min(U^\prime_{\bar{0}}(x_{\bar{0}}), U^\prime_{\bar{1}}(x_{\bar{1}})) = U(x)
\Biggl [ - 9 ( 3 7 u^{8}- 1 2 6 u^{5}v + 8 1 u^{2}v^{2}- 8 1 2 1 6 u^{4}\mu + 1 7 1 0 7 2 u v \mu + 3 6 5 7 8 3 0 4{\mu}^{2})
g^2 \delta^{d_2 c_1}f^{i_1 c_1 d_1}\delta^{d_1 c_2}f^{i_2 c_2 d_2}= g^2 f^{i_1 c_1 c_2}f^{i_2 c_2 c_1}= - g^2 C_2 \delta^{i_1 i_2}= - g^2 N \delta^{i_1 i_2}\sim O(N^0) .
e^{\alpha^c}(\theta^\beta_\pm e^\beta + e^{\alpha+\beta}) = b_{\alpha^c, \beta}\theta^\beta_\pm e^{\alpha^c + \beta}+ b_{\alpha^c, \alpha+\beta}e^{\beta^c}&= b_{\alpha^c, \beta}(\theta^{\alpha^c+\beta}_\pm e^{\alpha+\beta^c}+ e^{\beta^c})
I = \ \int \partial_\alpha (b R_{\alpha}) \bar P \bar w \bar P \bar r+ R_\alpha \bar P(b\bar P \bar w_\alpha) \bar P \bar r + R_\alpha \bar P \bar w \bar P(b\bar P \bar r_\alpha )\, d\alpha = \ \int R_\alpha \left([b,P](\bar P \bar w_\alpha) \bar P \bar r + \bar P \bar w [b,P] (\bar P \bar r_\alpha )\right)\, d\alpha.
\Vert B_\varepsilon (\vartheta )^{1/2}(B_\varepsilon (\vartheta )-\widehat{\zeta}Q_0^\varepsilon )^{-1}\Vert_{L_2(\mathbb{R}^d)\rightarrow L_2(\mathbb{R}^d)}\leqslant \Vert f\Vert_{L_\infty}c(\phi).
e^{i S_{\Lambda - \delta \Lambda}(\phi_+,\phi_-)}\equiv \int_{\rm CTC}\prod_{\Lambda_0>|{\bf q}| > \Lambda-\delta \Lambda}{\cal D}[\phi_+({\bf q},t)]{\cal D}[\phi_-({\bf q},t)] \; e^{i S_{\rm cl}[\phi_+,\phi_-]}.
\int_{0}^{|I|}\frac{I_{2}^{2}}{t^{4-p-2\beta}}dt \lesssim\int_{0}^{|I|}t^{p+2\beta-2}\left(\int_{4I}\Big|\frac{\partial\hat{f}}{\partial \theta}(re^{is})\Big|^{2}ds \right)dt &\lesssim \int_{S(4I)}|\nabla\hat{f}(z)|^{2}(1-|z|)^{p+2\beta-2}dA(z).
\binom{z}{\bar 1}_{\!\!\!q,t}=\prod_{i=1}^n \dfrac{( q^{x_i}t^{n-i})_1}{(qt^{n-i})_{1}}= \prod_{i=1}^n \dfrac{(1-q^{x_i}t^{n-i})}{(1-qt^{n-i})}
e_\alpha v = \sum \limits_{\mu \in \mathrm E \setminus \mathrm E_\varrho}c_\mu e_\alpha e_{-\alpha}v_\mu = \sum \limits_{\mu \in \mathrm E \setminus \mathrm E_\varrho}c_\mu \langle \alpha^\vee, \mu \rangle v_\mu.
\begin{array}{c}{v_{1}= \left( 0 , 1 , 0 , 0 , 0 , 0 \, ; \, 1 , 0 \right) \}{v_{2}= \left( 0 , 0 , 1 , 0 , 0 , 0 \, ; \, 0 , 1 \right) .}\end{array}
h_{a\beta j}^{b}=2\,d\omega_{a}^{b}(X_{\beta}, Z_j)= X_{\beta}\big{(}\omega_{a}^{b}(Z_j) \big{)}- Z_j \big{(}\omega_{a}^{b}(X_{\beta}) \big{)}- \omega_{a}^{b}\big{(}[X_{\beta}, Z_j] \big{)}&=R_{\beta j a}^b+ \omega_{a}^{\rho}(X_{\beta}) \omega^{b}_{\rho}(Z_j)-\omega_{a}^{\rho}(Z_j)\omega^{b}_{\rho}(X_{\beta}),
DF_K^{-1}\circ F_K(\xi,\eta) = \left(\begin{array}{cc}\frac{\partial\xi}{\partial x}\frac{\partial\xi}{\partial y}\frac{\partial\eta}{\partial x}\frac{\partial\eta}{\partial y}\end{array}\right)=\frac{1}{J_K}\left(\begin{array}{cc}b_2 + b_{12}\xi -a_2 -a_{12}\xi -b_1 - b_{12}\eta a_1 + a_{12}\eta\end{array}\right).
c_2(E)=\sum_i (\Omega^{4})_{ii}=\sum_{i,j}det(\partial^{2}h_{ij}) dz^{1}\wedge d\bar{z}^{1}\wedge dz^{2}\wedge d\bar{z}^{2}=\sum_{i,j}det(D_{ij})det(\bar D_{ij}) dz^{1}\wedge d\bar{z}^{1}\wedge dz^{2}\wedge d\bar{z}^{2}
\left(\frac{{\chi}^2_{\left(2n\right),\gamma /2}}{2\sum^n_{i=0}{{\log \left(\frac{R_n}{R_i}\right)}}}, \frac{{\chi}^2_{\left(2n\right),1-\gamma/2}}{2\sum^n_{i=0}{{\log \left(\frac{R_n}{R_i}\right)}}}\right),
\mathbf{\widetilde{\Phi}}_{0,k}=\left(\underset{\bf 1}{\tfrac{N-k}{k}\varphi'_{0}},\dots, \underset{\bf k}{\tfrac{N-k}{k}\varphi'_{0}},\underset{\bf k+1}{-\varphi'_{0}},\dots,\underset{\bf N}{-\varphi'_{0}}\right).
\{\sigma_K\leq t\}\subset \left\{\sum_{k=0}^{\lfloor r_p(x)-R-1\rfloor}\theta_k\leq t \right\}\subset \left\{\sum_{k=0}^{N(x,t)}\theta_k\leq t \right\}\subset \bigcup_{k=0}^{N(x,t)}\left\{\theta_k\leq \frac{C_1}{l(r_p(x)-k+1)}\right\}.
\nabla\times\mathbf{A}=\frac{1}{h_{1}h_{2}h_{3}}\begin{vmatrix}\begin{array}{ccc}h_{1}\mathbf{u}_{1}h_{2}\mathbf{u}_{2}h_{3}\mathbf{u}_{3}\frac{\partial}{\partial u_{1}}\frac{\partial}{\partial u_{2}}\frac{\partial}{\partial u_{3}}h_{1}A_{1}h_{2}A_{2}h_{3}A_{3}\end{array}\end{vmatrix}
\Delta_{\Lambda}(z,p,q)=2q\sum_{i=0}^{\infty}\frac{z^{i}}{i!}\Lambda_{i}+\sum_{i=0}^{\infty}(1-p^{2})\frac{z^{i+1}}{i!}\Lambda_{i+1}=\Lambda((2q+(1-p^{2})zt)t^{q}e^{zt}).
\mu^Z(JX) \xi_1 + \mu^W(JX) \xi_2 = \mu(JX) = J_{\mathfrak t^2}\mu(X) = J_{\mathfrak t^2}( \mu^Z(X) \xi_1 + \mu^W(X) \xi_2) = - \mu^W(X) \xi_1 + \mu^Z(X) \xi_2,
{\cal Z}\ = \ \int_{\bar{\psi}_\sigma \psi_\sigma}exp - \{\int_x \left[\bar\psi\left( \frac{T^\mu}{s}~i\partial_\mu + M \right) \psi~+~r_1 \left(\bar{\psi}T^\mu \psi \right)^2 + r_2 \left( \bar{\psi}\psi \right)^2 \right]\}
\lim_{T\to t^n}I_K (f;q,t,T;t_0,t_1,t_2,t_3){}=\frac{1}{2}I^{(n)}_K\big(f\big(z_1^{\pm 1},\dots,z_n^{\pm 1}\big);q,t;t_0,t_1,t_2,t_3\big) \qquad{}+ \frac{1}{2}I^{(n')}_K\big(\big(\tau_{t_0,t_1,t_2,t_3;t}f\big)(z_1^{\pm 1},\dots,z_{n'}^{\pm 1});q,t;t/t_0,t/t_1,t/t_2,t/t_3\big).
\operatorname{Var}_{\nabla_t}(\alpha)\colon (X_1,\dots,X_n) \mapsto \sum_{i=1}^n \alpha(\nabla_0X_1,\dots,\nabla_0X_{i-1}, \dot{\nabla}_0X_i,\nabla_0X_{i+1},\dots \nabla_0X_n).
x^2 = -\frac{\theta}{2}y- \left(\frac{\theta-\Delta}{2}\right)^2, Q_n(y) = \left(-\frac{\theta}{2}\right)^{-n}P_n\left(x^2; N,\frac{\theta-\Delta}{2},\frac{\theta+\Delta}{2},\frac{1}{2}\right).
\begin{array}{c}{{\displaystyle \int \frac{d^{4}k_{1}}{( 2 \pi )^{4}}\int \frac{d^{4}k_{2}}{( 2 \pi )^{4}}D_{m}^{2}( k_{1}) [ D_{m}( k_{2}) - D ( k_{2}) ] \overline{{{D ( k_{1}+ p )}}}K_{1 2}.}}\\ \end{array}
M_h=R_h =\max_{\{{u}_{l_0}(z)u_h^{l_0}(z)=0\}}\left(\hat{u}_{l_0}(z)-\hat{u}_h^{l_0}(z)+V_h(z)\right) &={u}_h^{t_0}(z_0)-{u}_{t_0}(z_0)+V_h(z_0).
\begin{cases}x^0\in C, y^{n}=\underset{y\in C}{\textup{argmin}}\bigg\{\lambda_n f(x^n,y)+\dfrac{1}{2}\|y-x^n\|^2\bigg\}, x^{n+1}=\underset{y\in C}{\textup{argmin}}\bigg\{\lambda_n f(y^n,y)+\dfrac{1}{2}\|y-x^{n}\|^2\bigg\},\end{cases}
\begin{array}{c}\bar W_k(\sigma^{+},\sigma^{-})=\left[ \frac 1{U_k(\sigma^{+})+v_k}+\frac1{U_k(\sigma^{-})+v_k}-\frac 1{u_{k-1}+v_k}\right]^{-1}-v_k U_{k+1}(z)=\left[ \frac 1{U_k(z)+v_k}+\frac 1{u_k+v_k}-\frac1{u_{k-1}+v_k}\right]^{-1}-v_k u_{k+1}=\left[ \frac 2{u_k+v_k}-\frac 1{u_k+v_{k-1}}\right]^{-1}-v_k\end{array}
\Big\langle\overline{\psi}^{(a_0)}_{\alpha_0}(x_0) \psi^{(b_0)}_{\beta_0}(x_0)\overline{\psi}^{(a_1)}_{\alpha_1}(x_1) \psi^{(b_1)}_{\beta_1}(x_1) .....\overline{\psi}^{(a_n)}_{\alpha_n}(x_n) \psi^{(b_n)}_{\beta_n}(x_n)\Big\rangle_{0\; c}^\theta \; ,
D_A^* D_A \phi = (D+\i A\cdot )(D+\i A\cdot) \phi &= D^*D \phi +\i \, e_i\cdot \nabla^S_{e_i}(A_j e_j\cdot \phi ) +\i A\cdot D\phi - \sum_{i,j}A_iA_j e_i\cdot e_j \cdot \phi &= (\nabla^S)^*\nabla^S\phi+\mathcal{R}\phi +\i \, dA\cdot \phi -\i \, \div A\, \phi - 2\i \sum_{i}A_i \nabla_{e_i}\phi +|A|^2\phi.
\int\nolimits_{\Omega}\left( \tfrac{1}{2}\left\vert(-\Delta)^{\alpha/4}v\left( x\right) \right\vert^{2}-\tfrac{1}{2}\left\vert(-\Delta)^{\alpha/4}u\left( x\right) \right\vert^{2}+\sum_{i=1}^2 G^{i}\left(x,u\left( x\right) ,v\left( x\right) \right) \left(w\left( x\right)\right)^{i-1}\right) dx
\frac{K_{0}\left[ 1-\left( fg^{\prime}\right)^{2}\right]^{2}}{ff^{\prime\prime}\left( g^{\prime}\right)^{2}}=\frac{gg^{\prime \prime}}{\left(g^{\prime}\right)^{2}}-\frac{\left( f^{\prime}\right)^{2}}{ff^{\prime\prime}}.
\mu_n = n\pi -F\Big(\frac{1}{\mu_n}\Big)+\frac{\hat\varphi_1(-n)+\hat\varphi_2(n)}{2}+\frac{\ell^2_n}{n^{N+1}}= n\pi + \zeta\Big(\frac{1}{n}\Big)+\frac{\hat\varphi_1(-n)+\hat\varphi_2(n)}{2}+\frac{1}{n^{N+1}}\ell^2_n = n\pi + \sum_{k=1}^{N+1}\frac{c_k}{n^k}+\frac{\hat\varphi_1(-n)+\hat\varphi_2(n)}{2}+\frac{\ell^2_n}{n^{N+1}}
\frac{S_{(r, s)(r',s')}}{S_{(1, 1)(r', s')}}:= \frac{_{M_{r,s}\boxtimes_{P(1)}M_{r', s'}}\left( c_{M_{r', s'}, M_{r, s}}\circ c_{M_{r, s}, M_{r', s'}}\right)}{\left(M_{r', s'}\right)},
{\frac{1}{2}}( 1 - \gamma )^{- 3}\left( \sum_{k = 0}^{\infty}( - 1 )^{k}{\frac{( k + 1 ) ( k + 2 )}{2}}\beta^{k}\right) \left( - \sum_{r = 0}^{3}\sigma_{3 , 2 r}\right) = 0 \; \mathrm{f r o m}\; W_{\infty}^{5 , D},
\begin{gathered}N_k^{2n}=(b^{2^{k}}-1)^{2n}=\sum_{i=1}^{n}\Big ( C_{2n}^{2i}b^{2i\cdot 2^{k}}-C_{2n}^{2i-1}b^{(2i-1)\cdot 2^{k}}\Big ) +1.\end{gathered}
\chi(\mathcal{I}\mid_{\alpha\beta}^{ij}):=\left\{h(\mathcal{I})\cdot h(\mathcal{I}_{\alpha\beta}^{ij}),h(\mathcal{I}_{\alpha}^{i})\cdot h(\mathcal{I}_{\beta}^{j}),h(\mathcal{I}_{\beta}^{i})\cdot h(\mathcal{I}_{\alpha}^{j})\right\}
4\, e^{2 \pi y}\,\pi \left(-4 \pi+2 \,\pi^2 \,y\right)+4 \,e^{2 \pi y}\, \pi^2 \left(2-4 \pi y+\pi^2 \,y^2\right)=4\,e^{2\pi y}\,\pi\left(\pi^3\,y^2-2\,\pi^2\,y-2\pi\right)>0,
8\pi\int_I \zeta_{x_0}(d\alpha)=\left(\int_I \alpha \zeta_{x_0}(d\alpha)\right)^2 &m_\pm(x_0)=\int_{I_\pm}|\alpha| \zeta_{x_0}(d\alpha),s_\pm(x)=\int_{I_\pm}|\alpha| r(\alpha,x){\cal P}(d\alpha),
\frac{1}{|e^{i\theta}-e^{i\phi}|^2}+ \frac{1}{|e^{i\theta}-e^{-i\phi}|^2}= \frac{1}{(x-y)^2 + (\sqrt{1-x^2}- \sqrt{1-y^2})^2}+ \frac{1}{(x-y)^2 + (\sqrt{1-x^2}+ \sqrt{1-y^2})^2}&= \frac{4-4xy}{4(x-y)^2}
g_{\mu \nu}^{j}- G_{\mu \nu}^{j + 1}= h_{\mu \nu}^{j}- h_{\mu \nu}^{j + 1}+ \pi_{\mu , \nu}^{j}+ \pi_{\nu , \mu}^{j}+ \pi_{, \mu}^{j \alpha}\pi_{\alpha , \nu}^{j}+ \cdots
\left( \begin{array}{c c}{{0}}&{{- t \sum_{j}\mathrm{\Large ~ e^{i{\bf ~ k ~ \cdot ~ u}_{j}}~}}}\\{{- t \sum_{j}\mathrm{\Large ~ e^{i{\bf ~ k ~ \cdot ~ v}_{j}}~}}}&{{0}}\\ \end{array}\right) \left( \begin{array}{c}{{c_{\bullet}}}\\{{c_{\circ}}}\\ \end{array}\right) = E ({\bf k}) \left( \begin{array}{c}{{c_{\bullet}}}\\{{c_{\circ}}}\\ \end{array}\right)
A_0 : = \sup_{t > 0}\bigg( \int_0^t U^q(\tau) w(\tau)\,d\tau \bigg)^{\frac{1}{q}}V^{- \frac{1}{p}}(t), A_1 : = \sup_{t > 0}W_*^{\frac{1}{q}}(t) \bigg( \int_0^t \bigg( \frac{U(\tau)}{V(\tau)}\bigg)^{p'}v(\tau)\,d \tau \bigg)^{\frac{1}{p'}};
-\sigma_k^{ii}\theta_{ii}^2= -\sum_{i>1}\sigma_k^{ii}(a_{ii}-1)^2 = -\sum_{i>1}\sigma_k^{ii}a_{ii}^2+2\sum_{i>1}\sigma_k^{ii}a_{ii}-\sum_{i>1}\sigma_k^{ii}\le -\sum_{i>1}\sigma_k^{ii}a_{ii}^2+2k\sigma_k-\sum_{i>1}\sigma_k^{ii},
\big| g'(X,X)-g(X,X) \big | \ = \ \Big| \, \sum_{ij}(g'_{ij}(x,t)-g_{ij}(x,t)) X^iX^j \, \Big | \leq \r' C |X|^2\leq r'C (1-Cr')^{-1}g(X,X)\,.
e^{it\Delta_d}\phi (x) = \sum_{|c_*'-c_*| \leq 2}\sum_{\overline{k}}e^{i (\overline{x}\cdot \overline{k}+ t|\overline{k}|^2)}e^{i( x_1 c_*' + t |c_*'|^2 )}\hat{\phi}(c_*',\overline{k}) & =: \sum_{|c_*'-c_*| \leq 2}e^{i( x_1 c_*' + t |c_*'|^2 )}e^{it\Delta_{d-1}}\varphi_{c_*'}(\overline{x}),
\frac{8 \pi^{2}}{g^{2}( \mu )}+ C_{A}\log g^{2}( \mu ) = \frac{8 \pi^{2}}{g^{2}( M )}+ C_{A}\log g^{2}( M ) + b_{0}\log \frac{\mu}{M}- \sum_{f}T_{f}\log Z_{f}( \mu , M ) ,
\dot{\phi}_i= \omega_i+\frac{1}{N}\sum_{j=1}^NW_{ij}(t)g(\phi_j-\phi_i), \dot{W}_{ij}= -\varepsilon(W_{ij}+\sum_{k=1}^Na_{jk}h_1(\phi_j-\phi_k)+\sum_{k=1}^Nb_{ik}h_2(\phi_j-\phi_k)),
M=\begin{pmatrix}(1-z^{-1})s^{4}t^{4}(1-z)(s^{2}t^{2}-s^{3}t^{3}) (z^{-1}-1)(s^{2}t^{2})(1+st+s^{2}t^{2}+s^{3}t^{3}) (1-z)(s^{4}t^{4}) \end{pmatrix}= \sum_{i=1}^5 s^iM_i
\begin{aligned}|x(t)-z (t)| \le |x(0)-z(0)|+\int_0^t \left| \dot{x}(\tau)-\dot{z}(\tau) \right| \,d\tau \le \, |x(0)-z(0)|+ \gamma L_r \int_0^t |x(\tau)-z(\tau)| \,d\tau &+ \gamma^2 \bar k(L_r,M_r, \delta) t\end{aligned}
\frac{\lambda_2-1}{s_2}\leq \frac{\lambda_2\cdot c_3-c_3}{s_3}\leq \frac{\lambda_3-c_3}{s_3}=\frac{\alpha\cdot c_3 +\beta-c_3}{s_3}\leq \frac{\alpha\cdot c_4 +\beta\cdot u_3-c_4}{s_4}\leq \frac{\lambda_4-c_4}{s_4}.
{\cal H}= \pi^{\mu}{\dot{A}}_{\mu}+ \pi^{< \mu \nu >}{\dot{B}}_{< \mu \nu >}+ \frac{1}{4}F_{\mu \nu}F^{\mu \nu}- \frac{1}{1 2}H_{\mu \nu \lambda}H^{\mu \nu \lambda}+ \frac{m}{6}\epsilon^{\mu \nu \lambda \rho}H_{\mu \nu \lambda}A_{\rho}
\sum_{k = 1}^\infty{\frac{{\zeta (2k)}}{{k(k + 2)(2k + 3)}}}= - \frac{7}{{72}}+ \frac{\ln (2\pi )}{6}- \frac{{\zeta (3)}}{{2\pi^2}}, \sum_{k = 1}^\infty{\frac{{\zeta (2k)}}{{k(k + 1)(2k + 5)}}}= - \frac{7}{{50}}+ \frac{\ln (2\pi )}{5}- \frac{{2\zeta (3)}}{{3\pi^2}}+ \frac{{\zeta (5)}}{{\pi^4}}.
\mathcal{R}_1 =\Bigl\{(u,v,w):\,\theta_2-\theta_1<v<u<\theta_1,\,\theta_2<u+v<1-\theta_2,\,1-\theta_1<u+2v<1, &v<w<(1-u-v)/2,\,v+w<\theta_1\Bigr\}, \mathcal{R}_2 =\Bigl\{(u,v,w):\,\theta_2-\theta_1<v<u<\theta_1,\,\theta_2<u+v<1-\theta_2,\,1-\theta_1<u+2v<1, &v<w<(1-u-v)/2,\,v+w>\theta_2\Bigr\}.
a_{SUPG}(u,v) = \epsilon(\nabla u, \nabla v) + (\mathbf{b}\cdot \nabla u, v) + (c u, v) +\sum_{K \in \mathcal{T}_{h,t}}\delta_{K}(- \epsilon\Delta u + (\mathbf{b-w}_h) \cdot\nabla u + cu, (\mathbf{b-w}_h ) \cdot \nabla v)_K
{}_4F_3\bigg[\begin{matrix}-n \alpha \beta \gamma &\delta \epsilon \rho\end{matrix}\bigg|\,1\bigg]=\frac{(\delta-\alpha)_n(\epsilon-\alpha)_n}{(\delta)_n(\epsilon)_n}\cdot{}_4F_3\bigg[\begin{matrix}-n \alpha \rho-\beta \rho-\gamma &\rho 1+\alpha-n-\delta 1+\alpha-n-\epsilon\end{matrix}\bigg|\,1\bigg],
f_{\alpha,\beta,\lambda}(x)=|x|^{-\frac{N-1}{2}}h^{-\frac{1}{4}}(|x|)\exp \left\{-\int_R^{|x|}h^{\frac{1}{2}}(s)ds -\int_R^{|x|}v_\lambda(s)ds \right\}\;,
\tilde{g}_{\ell}(z,\zeta,q) \: = \: e_{\ell}(\zeta) + (-1)^{n-k}\frac{q}{1-q}e_{n-k}(1-\tilde{z}) \Bigl({k \choose \ell +k-n}- e_{\ell+k-n}(z) \Bigr) \/.
\mathbb{T}^{N}:=\big\{x\in\mathbb{C}^{N}\colon \vert x_{j}\vert =1,\, 1\leq j\leq N\big\}, \mathrm{d}m(x) =(2\pi)^{-N}\mathrm{d}\theta_{1}\cdots\mathrm{d}\theta_{N}, x_{j}=\exp ( \mathrm{i}\theta_{j}) , -\pi<\theta_{j}\leq\pi, 1\leq j\leq N.
W_{i}^{(2)}=\frac{1}{\hbar^{2}}\sum_{k(k\ne i)}\frac{|\langle e_{i}|L_{z}+gS_{z}|e_{k}\rangle|^{2}}{\delta E_{i,k}};\delta E_{i,k}=E_{e,i}^{(0)}-E_{e,k}^{(0)}
\check{\Phi}_{1}= \check{x}_{-}- \check{\zeta}\tau = 0 , \; \check{\Phi}_{2}= \check{\pi}_{-}- \frac{\left[ \check{\pi}_{a}+ e A_{a}( \check{x}_{-}) \right]^{2}+ m^{2}}{2 \check{\pi}_{+}}.
\sum_{-\log t<k<\log t}\int_{n\exp(-2\sqrt{\frac{\log t}{t}})}^{n\exp(2\sqrt{\frac{\log t}{t}})}x^{-1/2}\exp\left(-\frac{t}{2}\log^2\frac{x}{n}\right) \exp\left(-ti\log\frac{x}{n}-2\pi ikx\right) dx &= \sqrt{n}\sum_{-\log t<k<\log t}\int_{-2\sqrt{\frac{\log t}{t}}}^{2\sqrt{\frac{\log t}{t}}}\exp(x/2) \exp\left(-t(ix+x^2/2)-2\pi ikne^x\right) dx.
\frac{d^my}{dv^m}=\bar{P}_{n-1-m}+v^{n-1-m}\left\{\frac{\hat{I}_{n,0}}{(n-1-m)!}+\sum_{j=1}^{n-1-m}\frac{(-1)^j}{j!(n-1-m-j)!}\hat{I}_{n,j}\right\}.
a_1 \cdot ((\alpha_{-1}- \beta_{-1}) \cdot (\alpha_{-2}-\beta_{-2})) = -\frac{1}{2^2}(\alpha_{-1}\cdot \beta_{-2}+ \alpha_{-2}\cdot \beta_{-1}) - \frac{1}{2^2}\langle \beta_{-1}, \beta_{-2}\rangle a_1.
\hat{\hat{G}\,}= 4 \partial \hat{\hat{C}\,}+ 3 \left( i_{\hat{\hat{k}}_{\scriptscriptstyle ( m )}}\hat{\hat{C}\,}\right) Q^{m n}\left( i_{\hat{\hat{k}}_{\scriptscriptstyle ( n )}}\hat{\hat{C}\,}\right) \, .
e_{\mathrm{top}}(S_{s,n})=\begin{cases}4n^2-2n^3+(3n-4)\binom{n}{2}-\binom{n}{3}\textrm{if}s=3 n^2(10-10n+3n^2)+\binom{n}{2}(-10+15n-6n^2)+\binom{n}{3}(4n-5)-\binom{n}{4}\textrm{if}s=4\end{cases}
\alpha_{\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3}}=\left(\begin{array}{c}1,-\frac{1}{m}(1+m)\cosh \left( m\int \kappa (s)\,ds\right) +m\sinh \left(m\int \kappa (s)\,ds\right) , e^{m\int \kappa (s)\,ds}+\frac{1}{m}\sinh \left( m\int \kappa (s)\,ds\right)\end{array}\right).
\begin{array}{lll}\langle q_{i0}- Q_{i0}\rangle = 0 \, , & \langle \dot{q}_{i0}\rangle = 0 \, , & m\omega_{i}\omega_{j}\langle (q_{i0}-Q_{i0})(q_{j0}-Q_{j0}) \rangle = k_{B}T \delta_{ij}\, , & m\langle \dot{q}_{i0}\dot{q}_{j0}\rangle = k_{B}T\delta_{ij}\, , & \langle \dot{q}_{i0}(q_{j0}- Q_{j0}) \rangle = 0 \, .\end{array}
\tilde{\Phi}_i(\tilde{\xi}_i(\{I_{ki}\}_{k{\ne}i}),\{I_{ki}\}_{k{\ne}i})= &~~\ln\rho_i\!+\!\sigma_i^2\tilde{\xi}_i(\{I_{ki}\}_{k{\ne}i})\!+\!\sum_{k\ne{i}}\ln(1+I_{ki}\tilde{\xi}_i(\{I_{ki}\}_{k{\ne}i}))\!=\!0,
\tau_{0}( y ) = \sum_{\sigma_{1}= 0}^{1}. . \sum_{\sigma_{4}= 0}^{1}Y_{\sigma_{1}. . \sigma_{4}}^{( 0 )}( t_{1}Z_{1})^{\sigma_{1}}( t_{2}Z_{2})^{\sigma_{2}}( t_{5}Z_{5})^{\sigma_{3}}( t_{8}Z_{8})^{\sigma_{4}}
w(a+b+1)^q\cr=\, (u+vb)(a^q+b^q+1)\cr=\, (u+vb)(eb^{2q}+b^q+1)\cr=\, (u+vb)\bigl[e(d-b)^2+d-b+1\bigr]\cr=\, u+du+ceu+d^2eu-cv-cdev+(-u-deu+v+cev)b\in\Bbb F_q.
1 = \lim_{n \to \infty}\mathbb{E}_{n, r(n), \theta_n}(\nu_n') > \liminf_{n \to \infty}\mathbb{E}_{n, r(n), \theta_n}(\varphi_n') = \liminf_{n \to \infty}\mathbb{E}_{n, d(n), F_n(\theta_n)}(\varphi_n),
\varphi(f)\varphi(h) =\sum_{l=0}^{N-1}\left (\sum_{i+j=l}q^{i}\lambda_{i}q^{j}\mu_{j}\right )x^{l}=\sum_{l=0}^{N-1}\left (\sum_{i+j=l}q^{i+j}\lambda_{i}\mu_{j}\right )x^{l}&=\sum_{l=0}^{N-1}q^{l}\left (\sum_{i+j=l}\lambda_{i}\mu_{j}\right )x^{l},
|f(z)|^2 = |c|^2 (x^2+y^2)^m e^{2\delta x+2\gamma (y^2-x^2)}\prod_n \tfrac{(x-r_n)+y^2}{r_n^2}\exp\big(\tfrac{2x}{r_n}\big) & \times \prod_n\tfrac{(x-s_n)^2+(y-t_n)^2}{s_n^2+t_n^2}\cdot\tfrac{(x-s_n)^2+(y+t_n)^2}{s_n^2+t_n^2}\cdot\exp\big(\tfrac{4s_n x}{s_n^2+t_n^2}\big).
\tilde u(t,\gamma,a) \ = \u(t,\gamma1_{[-T,0[}+a1_{\{0\}}) \ = \ \Psi\bigg(t,\ldots,a\varphi_i(t) - \int_{-t}^0 \gamma(x)\dot{\varphi}_i(x+t)dx,\ldots\bigg),
\begin{array}{c}{\phi^{\prime \prime}( u ) = 0}\\{1 + c_{w z w}- 2 6 + 6 \alpha^{\prime}( \phi^{\prime})^{2}= 0}\\{- \frac{1}{2}\alpha^{\prime}T^{\prime \prime}- 2 T + \alpha^{\prime}\phi^{\prime}T^{\prime}= 0}\\ \end{array}
\mathcal Ff(\xi) = (2\pi)^{-\frac n 2}\int_{\mathbb R^n}e^{-i\xi x}f(x) \, dx, \xi \in \mathbb R^n, \mathcal F^{-1}f(\xi) = (2\pi)^{-\frac n 2}\int_{\mathbb R^n}e^{i\xi x}f(x) \, dx, \xi \in \mathbb R^n.
I_{cs}= \frac{1}{4\pi}Tr \int_M \left[\frac{1}{a_{+}}A^+ \wedge\left( dA^+ +\frac{2}{3}A^+ \wedge A^+ \right) + \frac{1}{a_{-}}A^- \wedge \left( dA^- +\frac{2}{3}A^- \wedge A^- \right) \right]
D_{\bar{x}_{\varepsilon}}^{2}\bar{u}_{\varepsilon}=D_{x}\bar{y}_{\varepsilon}\cdot\frac{dx}{d\bar{x}_{\varepsilon}}=D_{\bar{x}_{\varepsilon}}\bar{y}_{\varepsilon}& =\left( -\sin\left( \sigma\right) I_{n}+\cos(\sigma)D^{2}u^{\varepsilon}(x)\right) \left( \cos(\sigma)I_{n}+\sin\left( \sigma\right)D^{2}u^{\varepsilon}(x)\right)^{-1}.
V_{ns}({\tilde x})= \left(\frac{{\tilde m}N^2}{16\pi}\right)N g^{2ns-1}{\tilde x}^2 \left\{{\tilde x}^2 -\frac{2{\tilde b}}{3}{\tilde x}+\frac{{\tilde b}^2}{3}-(-1)^{ns}{\tilde c}\right\}\,.
d x_{d}^{2}= - H^{\frac{2 \Delta - 4 ( D - 3 )}{( D - 3 ) \Delta}}f d t^{2}+ H^{\frac{2}{D - 3}}\left[ f^{- 1}d r^{2}+ r^{2}d \Omega_{D - 2}^{2}\right] ,
\widehat \tau_t:=\left\lbrace \begin{aligned}\inf\{s: \widehat{A}_s>t\},t<\widehat{A}_{\widehat{\zeta}-}, &\infty,\qquad\qquad\qquad\;\; t\geq \widehat{A}_{\widehat{\zeta}-}\end{aligned}\right.
u_n = \begin{pmatrix}\frac{1}{n^{1/ \alpha_0 - 1/2}}0 0 \frac{1}{\sqrt{n}}v_n\end{pmatrix}\mbox{where}v_n = \begin{pmatrix}v_n^{11}v_n^{12}v_n^{21}v_n^{22}\end{pmatrix}.
\|\mathcal{L}_{q,\alpha}-\mathcal{L}_{0,\alpha}\|_{\mathcal{B}(\mathcal{E}_{\alpha})}=\sup\limits_{x\in\mathbb{R}}\|\partial_{Y}R(Y_0(x-q))-\partial_{Y}R(Y_0(x))\|_{\mathbb{R}^{n\times n}}\leq C|q|.
d(x, H_{\alpha}, e_\alpha) \ge d(x, H_{\alpha}, v_{\alpha}) \ge \delta_{i_\alpha}, d(x, H_{j}, e_\alpha) \ge d(x, H_{j}, v_{j}) \ge \delta_{i_j}\ge \delta_{i_\alpha}, \mbox{for all}N \ge j > \alpha, \mbox{and}d(x,H_{j}, e_\alpha) = \infty > \delta_{i_\alpha}\mbox{for}j < \alpha,
H^2 = \frac{1}{2}\frac{(c_1 + k)^2}{r_0^2}\frac{(1-x^2)^{5/2}}{x^{\frac{5+3k+c_1}{2}}}\left[ E^2 - x^{2k}-\frac{l^2}{r_0^2}\frac{(1-x^2)^{1/2}}{2x^{1/2}}\right]
\langle{a,\beta,\gamma,r,s,\tau,v}\mid[r,a]=\gamma^{-1}\beta{r}\beta^{-1}r^{-1}=r\gamma^{-1}r^{-1},~\gamma{a}\gamma^{-1}a^{-1}=\beta,~sr\tau{s}=r\tau{sr\tau},
\frac{0}{1},\frac{0}{2},\frac{1}{1},\frac{0}{3},\frac{1}{2},\frac{2}{1},\frac{0}{4},\frac{1}{3},\frac{2}{2},\frac{3}{1},\frac{0}{5},\frac{1}{4},\frac{2}{3},\frac{3}{2},\frac{4}{1},\frac{0}{6}\cdot\cdot\cdot
J_{2}'(x) = \lambda \int_{z=x}^{\infty}\int_{u=0}^{z-x}G(u)f_n(u)\lambda e^{-\lambda(z-x)}du dz - \int_{z=x}^{\infty}G(z-x)f_n(z-x)\lambda e^{-\lambda(z-x)}dz.
\begin{gathered}\nabla \cdot \left (\eta(s) - \eta(t) \right ) = \nabla_x X^{-1}(s) : (\Delta_1 \nabla_a X' (s,t) ) + \nabla_x X^{-1}(s) : (\Delta_2 \nabla_a X' (s,t) ) + \left (\nabla_x X^{-1}(s) - \nabla_x X^{-1}(t) \right ) : \left ( \nabla_a X' \circ X^{-1}\right ) (t),\end{gathered}
q = x_1 r =\frac{\sqrt{-x_{n+1}+\bigl(x_1x_2 +x_1\sum\limits_{j=3}^n x_j^2\bigr)}}{\sqrt{x_1}}t = \frac{x_1x_2}{-x_{n+1}+\bigl(x_1x_2 +x_1\sum\limits_{j=3}^n x_j^2\bigr)}s_j = \frac{x_j\sqrt{x_1}}{\sqrt{-x_{n+1}+\bigl(x_1x_2 +x_1\sum\limits_{j=3}^n x_j^2\bigr)}}, \, j=3,\ldots, n,
\begin{cases}\frac{dx}{dt}= \frac{dx}{dz}\frac{dz}{dt}= -2\pi i x(1 + e^{4\pi it}xy) \frac{dy}{dt}= \frac{dy}{dz}\frac{dz}{dt}= -2\pi i y(1 - e^{4\pi it}xy)\end{cases}\, .
\begin{aligned}G(x_{1},y_{1}) =\int_{0}^{c(p)}\cdots\,dt+\int_{c(p)}^{T(x_{1},y_{1})}\cdots\,dt+\int_{T(x_{1},y_{1})}^{\infty}\cdots\,dt &=:G_{1}(x_{1},y_{1})+G_{2}(x_{1},y_{1})+G_{3}(x_{1},y_{1}),\end{aligned}
\Pi^{( 2 )}( k^{2}) = \frac{e^{2}}{4 \pi}\frac{\mathrm{s g n}( k_{0})}{\sqrt{k^{2}}}\left[ \ln \left| \frac{1 - \sqrt{\frac{k^{2}}{4 m^{2}}}}{1 + \sqrt{\frac{k^{2}}{4 m^{2}}}}\right| - i \pi \theta ( k^{2}- 4 m^{2}) \right] .
\tilde{B}_{\mu : k}^{j}= \frac{1}{( 1 + | u |^{2})^{j + 1}}\left\{j ( \partial_{\mu}u \bar{u}- u \partial_{\mu}\bar{u}) u^{k}- k ( 1 + | u |^{2}) u^{k - 1}\partial_{\mu}u \right\}
-8\nu^{3}+4\nu+4c\nu(\lambda_{2}-\lambda_{1})^{2}-2c\nu\lambda_{1}(\lambda_{2}-\lambda_{1})+2c\nu(\lambda_{2}^{2}-\lambda_{1}^{2})+\displaystyle\frac{2\nu(\lambda_{2}+\lambda_{1})}{\lambda_{2}}b_{1}^{2}=0.
\sum_{x < n \leq x+\frac{x}{L}}\lambda_f(n) = \frac{1}{2 \pi i}\int_{1/2-i\infty}^{1/2+i\infty}L(s, f) \frac{(x+\frac{x}{L})^s-x^s}{s}ds = x^{1/2}\cdot \frac{1}{2 \pi}\int_{-\infty}^{\infty}L(\tfrac12+it, f) \, w_{\delta}(\tfrac12+it) \cdot e^{it \cdot \log x}dt,
p(f(t_{n})-f(t)) =p(S_{T-t_{n}}v(t_{n})-S_{T-t}v(t)) &\leq p(S_{T-t_{n}}v(t_{n})-S_{{T}-t_{n}}v(t))+p(S_{T-t_{n}}v(t)-S_{T-t}v(t)) &\leq p(S_{T-t_{n}}x_{n})+p((S_{T-t_{n}}-S_{T-t})v(t)) &\leq \sup_{s\in[0,T]}p(S_{s}x_{n})+p((S_{T-t_{n}}-S_{T-t})v(t)).
(\mathbf{x}_{k+1}, \mathbf{y}_{k+1})=\mathcal{P}(\mathbf{x}_{k}, \mathbf{y}_{k}), \hbox{where}\mathcal{P}= (I + \Sigma^{-1}T_{\mathcal{L}})^{-1}\hbox{and}\Sigma:= blockdiag(\rho I_d , \delta I_m).
J(u)=\mathbb{E}\bigg[\int_0^\infty e^{-\beta t}\mathcal{Z}_tf(x_t^u,y_t^u,z_t^u,\tilde{z}_t^u,\int_{\mathcal{E}}\gamma_{(t,e)}^u\nu(de),u_t)dt+\phi(y_0^u)\bigg],