harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\chi_{\Lambda}( C_{m}^{\Lambda_{0}}) = \sum_{( \lambda + \lambda_{0}^{\prime}, \kappa + \kappa_{0}, - s ) \in D^{+}}\sum_{s = 0}^{\infty}\, \alpha_{\lambda_{0}^{\prime}, s}^{m}( \Lambda ) ( I \otimes \mathrm{t r}) \{( I \otimes \pi_{\Lambda_{0}}( q^{2 h_{\rho}}) ) P [ \lambda_{0}^{\prime}, s ] \}
I_{m,p}^{\star(1)}(s,x)=\frac{(-1)^{m}}{2\pi i}\intop_{\sigma-i\infty}^{\sigma+i\infty}\Gamma(z)\,\Gamma(s-z)\,\left(\frac{x}{m}\right)^{2z}dz=\frac{(-1)^{m}\Gamma(s)x^{2s}}{\left(x^{2}+m^{2}\right)^{s}}.
\hat{\cal L}_{0}= - \int d^{\, 3}u \, d^{\, 3}v \sum_{\tau = \pm}\hat{\phi}^{( \tau ) \dagger}({\bf x}; x_{0}) \hat{\tau}_{3}D_{\tau}( x ;{\bf u},{\bf v}) \hat{\phi}^{( \tau )}({\bf v}; x_{0}) ,
S_{t}=\begin{cases}t \mbox{for}t<0, -t+2\sum_{l=1}^j{E}_l, \mbox{for}\sum_{l=1}^j Q_l+\sum_{l=1}^j{E}_l\leq t\leq\sum_{l=1}^{j+1}Q_l+\sum_{l=1}^j{E}_l, t-2\sum_{l=1}^{j+1}Q_l, \mbox{for}\sum_{l=1}^{j+1}Q_l+\sum_{l=1}^j{E}_l\leq t\leq\sum_{l=1}^{j+1}Q_l+\sum_{l=1}^{j+1}{E}_l,\end{cases}
p_{k,S}(n,r) = \frac{2^{g/2}(2\pi D)^\ell}{\Gamma(\ell)(\det 2S)^{\ell-1/2}}\big(2+ O\big( \frac{D^{g/2+\epsilon}}{\ell^{g/2+1/3}(\det 2S)^{g/2+1/2+\epsilon}}\big) \big).
G_{B i_{r}i_{s}}= G_{B}( u_{i_{r}}, u_{i_{s}}) = \mid u_{i_{r}}- u_{i_{s}}\mid -{( u_{i_{r}}- u_{i_{s}})}^{2}= ( u_{i_{r}}- u_{i_{s}}) -{( u_{i_{r}}- u_{i_{s}})}^{2}
\begin{aligned}\displaystyle \partial_t f + \partial_{\theta}(v[f] f) = 0, (\theta, \omega) \in \mathbb T \times \mathbb R,~~t > 0, & v[f](\theta, \omega, t) = \omega - K F [\rho], F [\rho] := \int_{\mathbb T}\sin(\theta-\theta_*) \rho(\theta_*,t) ~ d\theta_*.\end{aligned}
\hat{H}=-\sum_{i=1}^{D}\frac{\partial^2}{\partial x_i^2}+\omega^2 r^2+\frac{\beta_1}{x_1^2+\cdots+x^2_{n_1}}+\cdots+\frac{\beta_N}{x^2_{n_{N-1}+1}+\cdots+x_D^2}.
\frac{d}{dt}\int_{\Delta_D}\mathcal{F}_tdx =\int_{\Delta_D}\frac{d \mathcal{F}_t}{dt}dx=-t\int_Xd(\log \frac{\|\cdot\|_1}{\|\cdot\|_0})\wedge d^c (\log \frac{\|\cdot\|_1}{\|\cdot\|_0})\wedge c_1(\overline{D}_t)^{n-1}\forall\, t\in [0,1[.
|\partial_{r}N_{2}(f_{v_{5}})(t,r)| \leq \frac{C}{r^{3}t^{3}\log^{4N+7b-2}(t)}+\frac{C \lambda(t) \log(r)}{r^{3}t^{3/2}|t-r| \log^{3b-1+\frac{5N}{2}}(t)}+ \frac{C \log^{2}(r)}{r^{2}t^{3/2}(t-r)^{2}\log^{3b-1+\frac{5N}{2}}(t)}&+ \frac{C \log^{3}(r)}{r^{2}(t-r)^{4}}, t> r \geq \frac{t}{2}
\hat c_n(\theta) \equiv \inf\left\{c\in\mathbb R_+:P^*\left(\min_{\lambda \in \Lambda_n^b (\theta ,\rho ,c)}p^{\prime}\lambda \leq 0\leq \max_{\lambda \in \Lambda_n^b (\theta ,\rho ,c)}p^{\prime}\lambda\right)\ge 1-\alpha\right\}&= \inf\bigl\{c\in\mathbb R_+:P^*(\Lambda_n^b (\theta ,\rho ,c)\cap \{p^{\prime}\lambda =0\}\neq \emptyset)\ge 1-\alpha\bigr\},
u(t,x)=(1+4\pi i t)^{-d/2}e^{-\frac{4\pi^2 t z_2 (2z_1+i z_2 )}{1+ 4 \pi i t}}M_\frac{z_2}{1+ 4 \pi i t}T_{z_1}e^{-\frac{\pi}{1+ 4 \pi i t}|x|^2}.
u(t_n+\Delta t) = u_{n+1}+ \frac{\Delta t^2}{2!}F^{(1)}- \frac{\Delta t}{2}V_m V_m^T ( F_{n+\frac{1}{2}}- F_n ) &+ \frac{\Delta t^3 L}{3!}F^{(1)}+ \frac{\Delta t^3}{3!}F^{(2)}- V_m\frac{\Delta t^2 H_m}{4}V_m^T ( F_{n+\frac{1}{2}}- F_n ) + O(\Delta t^4).
I^{( \pm )}= 2 \pi \int_{0}^{\infty}d q \frac{q J_{0}( q | \vec{x}- \vec{x}^{\prime}| )}{k^{2}- q^{2}\pm i \epsilon}= - i \pi^{2}H_{0}^{( 1 )}( \pm k | \vec{x}- \vec{x}^{\prime}| + i \tilde{\epsilon}) ,
B_{n+1}\left( \frac{r}{\beta}\right) -B_{n+1}\left( \frac{r}{\beta}-s\right) =\left( n+1\right) \sum_{k=0}^{n}W_{\beta,r}\left( n,k\right)\frac{\left( -1\right)^{k}\left\langle s\right\rangle_{k+1}}{\beta^{n+1-k}\left( k+1\right)}.
\lim_{\lambda\rightarrow \infty}\sup_{i\geq 1}\mathbf{P}^{G}_0\left(\max_{x,y\in B_d(0,\kappa i)}\max_{t\geq0}\frac{\left|L\wedge \left(\frac{L^{G}_t(x)}{i^{\beta-\alpha}}\right)-L\wedge \left(\frac{L^{G}_t(y)}{i^{\beta-\alpha}}\right)\right|}{\sqrt{{R}^{(i)}_{G}(x,y)\left(1+\ln_+{R}^{(i)}_{G}(x,y)^{-1}\right)}}\geq \lambda\right)=0.
\begin{aligned}P(Z^{N,i}_1=1)\approx e^{-NI_i(r_{i+1})}; &P(Z^{N,i}_1=-1)\approx e^{-NI_i(r_i)}; &P(Z^{N,i}_1=-11)\lessapprox e^{-N\big(I_i(r_i)+I_i(r_{i+1})\big)}.\end{aligned}
A_{3}=\begin{bmatrix}0 2 -4 \varepsilon \varepsilon -5 \varepsilon \varepsilon \varepsilon -6 \varepsilon \varepsilon \varepsilon -4 \varepsilon \varepsilon -3 \varepsilon \varepsilon \varepsilon -2 \varepsilon \varepsilon 2 \varepsilon\end{bmatrix}.
\widehat Q_k(x) = \widehat\alpha_k \cdot x \cdot \widehat Q_{k-1}(x) - \widehat a_{k-1,k}\widehat Q_{k-1}(x)- \widehat a_{k-2,k}\widehat Q_{k-2}(x)-\cdots-\widehat a_{0,k}\widehat Q_{0}(x)
\hat{\phi}_\mathrm{f}= \frac{1}{M}\operatorname{arctan}\left( \frac{\sum_{i=0}^{N_\mathrm{c}-1}\operatorname{Im}\left(\frac{y_i^{M}}{\lvert y_i \rvert^{M-1}}\right)}{\sum_{i=0}^{N_\mathrm{c}-1}\operatorname{Re}\left(\frac{y_i^{M}}{\lvert y_i \rvert^{M-1}}\right)}\right).
\begin{array}{r l}{{}}&{{< 0 | \cdots T ( \pi_{1}\pi_{2}\pi_{5}\star e^{\pm i p \cdot x_{5}}\star \pi_{5}) \cdots | 0 >}}\\{{=}}&{{\displaystyle \sum_{\lambda_{1}\lambda_{2}}\tau_{1 5}^{\lambda_{1}}\tau_{2 5}^{\lambda_{2}}( D_{1 5}^{\lambda_{1}}\star e^{\pm i p_{+}\cdot x_{5}}\star D_{2 5}^{\lambda_{2}}+ ( 1 \leftrightarrow 2 ) ) < 0 | \cdots | 0 > .}}\\ \end{array}
\frac{G\;-\;\left(\tfrac14 n - \tfrac12\log n\right)}{\tfrac12 \sqrt{\log n}}\;\; =\; \;-\, \sqrt{\tfrac{\log m}{\log n}}\,\hat{B}(\gamma) \;-\, \sqrt{\tfrac{\log (n/m)}{\log n}}\,\hat{I}(\gamma) \;+\; o(1) begin{align*}0.5em]\; =\; \;-\,r\,\hat{B}(\gamma)\;-\, \sqrt{1-r^2}\,\hat{I}(\gamma)\;+\;o(1)
S_2 \ll \sum_{\substack{m_1,n_1,m_2,n_2 \leq bT m_1+n_1=m_2+n_2}}\frac{1}{m_1^{\sigma_1}m_2^{\sigma_1}n_1^{\sigma_2}n_2^{\sigma_2}(m_1+n_1)^{2\sigma_3-1}}&\ll T^{4-2\sigma_1-2\sigma_2-2\sigma_3}.
\begin{array}{lc}R(\alpha_i)=\alpha_{\sigma (i)}, R(\alpha_i \alpha_j)=\alpha_{\sigma (i)}\alpha_{\sigma (j)}, R(\alpha_i \alpha_j \alpha_k)=\alpha_{\sigma (i)}\alpha_{\sigma (j)}\alpha_{\sigma (k)}, \cdots ;\end{array}
-\frac{m^z}{n^2}\sum_{p+q+1\geq 3}\left(\cdots\right)-\frac{\mathfrak{m}^{z}}{n^2}\sum_{p+q+1\geq 3}\left(\cdots\right)=O_\prec(n^{-1/2}\Psi^3+n^{-3/2}).
\frac{\delta S_{0}^{L}}{\delta B_{\mu \nu}^{a}}\equiv - \frac 1 2 F_{a}^{\mu \nu}= 0 , \frac{\delta S_{0}^{L}}{\delta A_{a}^{\mu}}\equiv A_{\mu}^{a}+ \left( D^{\lambda}\right)_{b}^{a}B_{\lambda \mu}^{b}= 0 .
1-A_{d}^*...A_{1}^*A_{1}...A_{d}&=(1-A_{d}^*A_{d})+A_{d}^*(1-A_{d-1}^*A_{d-1})A_{d}+A_{d}^*A_{d-1}^*(1-A_{d-2}^*A_{d-2})A_{d-1}A_{d}+... &+ A_{d}^*A_{d-1}^*...A_{2}^*(1-A_{1}^*A_{1})A_{2}...A_{d-1}A_{d}&=\sum_{j=1}^{d}A_{d}^*...A_{j+1}^*(1-A_{j}^*A_{j})A_{j+1}...A_{d}
\left(\int_0^1 e^{-\rho \eta}d\eta\right)^{1/2}\left(\int_0^1\alpha^2(x-\eta)d\eta\right)^{1/2}+\sum_{j=1}^{\infty}e^{-\rho j/2}\left(\int_j^{j+1}\alpha^2(x-\eta)d\eta\right)^{1/2}\lesssim \frac{\|A\|_{\rm St}}{\rho^{1/2}}\,.
\left(\begin{array}{lll}I_k 0 0 w_5 w_3 w_6\end{array}\right)\left(\begin{array}{lll}I_k 0 0 w_5 w_3 w_6\end{array}\right)^*=\left(\begin{array}{lll}w_3 0 I_k -w_5\overline w_6^t w_5\overline w_5^t F' 0 -w_3\overline w_3^t-w_6\overline w_6^t w_6\overline w_5^t F' w_3\end{array}\right)
(u^{+}_l,u^-_{l}) =(\hat{u}^+_{0,l},-\mbox{sign}(l)\hat{u}^+_{0,l}) \mbox{for}|l|>\frac{1}{2R}(u^{+}_l,u^-_{l}) =(\hat{u}^+_{0,l},-\mbox{sign}(l)\hat{u}^+_{0,l})+(\hat{u}^-_{0,l},\mbox{sign}(l)\hat{u}^-_{0,l})\mbox{for}|l|\leq \frac{1}{2R}.
S^-(\tau, \eta) := -2 \frac{\theta(1/2; \tau/8\eta)\theta'(0; \tau/8\eta)}{\theta(3/4; \tau/8\eta) \theta(1/4; \tau/8\eta)}u(1/2, 1/2, 1/8\eta, \tau/8\eta, -1/8) S^+(\tau, \eta) := 2 \frac{\theta(1/2; -\tau/8\eta)\theta'(0; -\tau/8\eta)}{\theta(1/4; -\tau/8\eta) \theta(3/4; -\tau/8\eta)}u(1/2, -1/2, 1/8\eta, -\tau/8\eta, 1/8).
\frac{(q)_\infty^3}{(-q)_\infty^2}f(q) =1-4\sum_{n=1}^\infty\sum_{m=1}^\infty nq^{mn}-16\sum_{n=1}^\infty\sum_{m=1}^\infty nq^{2mn}+4\sum_{n=1}^\infty \sum_{m=1}^\infty (6n+2m-1)q^{n(3n-1+2m)/2}.
\varphi_{1}(tw_{0}) \leq \Lambda^{2}t^{p}[w_{0}]_{s,p}^{p}-t^{p_{s}^{\ast}}\int_\Omega |w_{0}|^{p^{\ast}_{s}}dx-\lambda c_{3}\int_{\Omega}|tw_{0}|^{q}dx &\leq A_{2}t^{p}-Bt^{p_{s}^{\ast}}-E_{2}t^{q}
\Big[\big[[\pi, T]_{\Omega}, T\big]_{\Omega}, \theta\Big]_{\Omega}= \ 0, \Big[\ldots\big[[f, \underbrace{T]_{\Omega}, T\big]_{\Omega},\ldots, T\Big]_{\Omega}}_{k}= \ 0,\,k\geqslant n+1.
\overline{s_{i}}=\gamma_{i}\left(\left[\begin{array}{cc}0 -1 1 0\end{array}\right]\right)=\exp\left(-e_{i}\right)\exp\left(f_{i}\right)\exp\left(-e_{i}\right), \overline{\overline{s_{i}}}=\gamma_{i}\left(\left[\begin{array}{cc}0 1 -1 0\end{array}\right]\right)=\exp\left(e_{i}\right)\exp\left(-f_{i}\right)\exp\left(e_{i}\right).
\frac{2x^{\alpha}y^{1-\alpha}}{b^{2}}\sum_{k=1}^{\infty}\frac{J_{\alpha}(\frac{xu_{\alpha,k}}{b})J_{\alpha}(\frac{yu_{\alpha,k}}{b})}{J_{\alpha+1}^{2}(u_{\alpha,k})}=\delta(x-y).
\mathbf{1}_{\mathcal{R}}(z)= W_{L, \mathcal{R}}(z) + O\Big(\frac{\sin^2(\pi L(x-a_1))}{(\pi L (x-a_1))^2}+ \frac{\sin^2(\pi L(x-a_2))}{(\pi L (x-a_2))^2}&+\frac{\sin^2(\pi L(y-b_1))}{(\pi L (y-b_1))^2}+ \frac{\sin^2(\pi L(y-b_2))}{(\pi L (y-b_2))^2}\Big)
\sup_{s \in [0,T]}\left |z(s) - z_j(s)\right | \leq \sup_{s \in [0,T]}\left ( \sum_{k \colon z(t_k)= z(t_k-)}\frac{\psi_k(s)}{j}+\sum_{k \colon z(t_k) \neq z(t_k-)}\frac{\psi_k(s)}{j}\right )=\frac{1}{j}.
X^*_{t+1}=\Psi\left(t,X^*_{t},\mu^*_{t},\iota \circ\Phi^*\left(t,X^*,\mu^*\right),\xi^*_{t+1}\right)=\Psi\left(t,X^*_{t},\mu^*_{t},\Phi^*\left(t,X^*_t\right),\xi^*_{t+1}\right), \mathbb{P}^*
H_{j_i}(s) := \left( 2^{j_i}X_3 \right)^s \int_\frac12^2 \! u_0(\eta) \eta^{s - 1}\, d\eta, j_i \geq 1, H_0(s) := \zeta(1 - s){d_i}^s \frac{\psi_{-s}(c_i)}{\psi_0(c_i)}- \frac{{X_3}^s}s \int_1^2 \! u_0'(\eta) \eta^s \, d\eta,
2\int_{B^c_\varepsilon(x)}\frac{\, u(x) - u(y) \,}{|x - y|^{N + 2s}}\,dy = \int_{B_r\cap B^c_\varepsilon}\frac{\, 2u(x) - u(x + z) - u(x - z) \,}{|z|^{N + 2s}}\,dz &+ \int_{B^c_r}\frac{\, 2u(x) - u(x + z) - u(x - z) \,}{|z|^{N + 2s}}\,dz.
q ( t^{\prime})^{- ( 1 + r )}+ 2 r \frac{t^{\prime \prime \prime}t^{\prime}- ( t^{\prime \prime})^{2}}{( t^{\prime})^{2}}+ \frac{( 1 + r )^{2}+ 4 r^{2}}{4}\left( \frac{t^{\prime \prime}}{t^{\prime}}\right)^{2}- 1 = 0 .
L(f, g, s) = N^{r + r' - 2s - 2}\frac{\pi^{2s - r' + 1}(-i)^{r - r'}2^{2s + r - r'}}{\Gamma(s)\Gamma(s - r' + 1)}\left\langle \overline{f}, g E^{(r - r')}_{1/N}(\tau, s - r + 1)\right\rangle_N
\tilde{\Xi}_\pm A\varphi =\tilde{\Xi}_\pm \gamma^{(2n+1)}(P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=\gamma^{(2n+1)}\tilde{\Xi}_\pm (P_{\rm F}-D_aP_{\rm F}D_a)\varphi=-A\tilde{\Xi}_\pm \varphi.
{\cal G}_{\mathrm{r e l}}^{( 2 )}= \frac{\mu}{2}\left( ( 1 + r_{0}/ r ) d{{\bf r}}^{2}+ r_{0}^{2}( 1 + r_{0}/ r )^{- 1}( d{\psi}+{\bf w}({\bf r}) \cdot d{{\bf r}})^{2}\right) .
\sum_{n = 0}^{\infty}o_{p,e}(n)q^{n}= \frac{1}{2}\left(H(1,q) + H(-1,q)\right) = \frac{(q^{p};q^{p})_{\infty}}{(q;q)_{\infty}}\sum_{n = 1}^{\infty}\frac{q^{2pn}}{1 - q^{2pn}}.
G_0 =G_{0,0}+|\Psi_c\rangle\langle v\Phi_c, G_{0,2}v\Phi_c\rangle\langle \Psi_c|-|\Psi_c\rangle\langle\mathbf n|-|\mathbf n\rangle\langle\Psi_c| &-\bigl(G_{0,0}-|\Psi_c\rangle\langle\mathbf n|\bigr)vM_0^\dagger v^*\bigl(G_{0,0}-|\mathbf n\rangle\langle \Psi_c|\bigr).
|D \phi|^2 = | \sum_i e_i\cdot\nabla_{e_i}\phi|^2= \sum_{i,j}\langle e_i\cdot\nabla_{e_i}\phi, e_j\cdot\nabla_{e_j}\phi \rangle &= |\nabla\phi|^2 + \sum_{i\neq j}\langle e_i\cdot\nabla_{e_i}\phi, e_j\cdot\nabla_{e_j}\phi \rangle
\begin{aligned}\rho(L_{-1})\rho(L_{k+1}) = h(z)^k (L-c-k-1)(L+(k+1)c) \rho(L_{0})\rho(L_{k}) = h(z)^k (L-k) (L+k c) \rho(L_{1})\rho(L_{k-1}) = h(z)^k (L+c-k+1) (L+(k-1)c) \end{aligned}
~~~ (O(\alpha t)x-y)_{1}\big ( (\cos\alpha t) A_1 + (\sin\alpha t) A_2 \big ) & \,=\, \frac{|x|^2}{2}\begin{pmatrix}x_1 0 x_2 0 \end{pmatrix}+ (\cos 2\alpha t) D_1 (x,y) + (\sin 2\alpha t) D_2 (x,y) + D_3 (x,y, \alpha t)\,.
\delta{\cal E}_2^{1,0}=\frac{8K}{a_0^6}\left\{\left[\left(\frac{a_0}{\Lambda}\right)^2-2\left(\frac{a_0}{\Lambda}\right)\right]e^{-2\Lambda/a_0}+4E_1\left(\frac{2\Lambda}{a_0}\right)\right\}\,{.}
{\cal S}_{\Sigma}= \int d^{3}z \Biggl [{\frac{\sqrt{g}}{l_{1 1}^{3}}}-{\frac{i}{3 !}}\epsilon^{i j k}\partial_{i}{\bf Z}^{M}\partial_{j}{\bf Z}^{N}\partial_{k}{\bf Z}^{P}C_{M N P}( X ( s ) , \Theta ( s ) ) \Biggr ] ,
E(W) =\sum_{i=1}^n w_i=\frac{1}{2\sqrt{n-1}}(2n(n-1)-\sum_{i-1}^n(w_i-\sqrt{n-1})^2) &\leq n\sqrt{n-1}-\frac{1}{2\sqrt{n-1}}(a (k-\sqrt{n-1})^2+(n-a) (k+2-\sqrt{n-1})^2)=\frac{n(k^2+2k+n-1)}{2(k+1)}
X^1(t) =\eta(0), t\in[0,T], X^1_0 = \eta X^{k+1}(t) =\eta(0)+\int_0^t f(s,X^k_s) ds+ \int_0^t g(s,X^k_s)d B(s) &\quad+\int_0^t h(s,X^k_s)(z)\tilde N(d s,d z), t\in[0,T] X^{k+1}_t =\eta.
\bigg|\frac{\partial}{\partial y}P_yu(t,x)\bigg| \leq\frac{1}{\Gamma(2s)}\int_0^\infty\bigg|\frac{\partial}{\partial y}P_{y+\tau}f(t,x)\bigg|\,\frac{d\tau}{\tau^{1-2s}}&\le C\|f\|_{\Lambda^{\alpha}}\int_0^\infty(y+\tau)^{-1+\alpha}\,\frac{d\tau}{\tau^{1-2s}}=C_{s,\alpha}\|f\|_{\Lambda^{\alpha}}y^{(\alpha+2s)-1}.
\hat{\tau}_{AM|X=\mathbf{x}}(s,t) = \hat{D}_A(\mathbf{x})\,(\mathcal{N}_A(s)\otimes\mathbb{I}_M)(\hat{\rho}_{AM})\,{\hat{D}_A(\mathbf{x})}^\dag &= (\mathcal{N}_A(s)\otimes\mathbb{I}_M)\left(\hat{D}_A(\mathbf{x})\,\hat{\rho}_{AM}\,{\hat{D}_A(\mathbf{x})}^\dag\right) &= (\mathcal{N}_A(s)\otimes\mathbb{I}_M)(\hat{\sigma}_{AM|X=\mathbf{x}}(t))\;.
\chi_{P^{\ast},a,eff}^{1}\left( \mathbf{V};\mathcal{G}\right) = q_{\mathcal{G}}\left( \mathbf{W};P^{\ast}\right)+h_{\mathcal{G}}(A,\mathbf{O},\mathbf{M},Y;P^{\ast}) &=b_{a}(\mathbf{O};P^{\ast}) - \chi_{a}(P^{\ast};\mathcal{G})+ g(\mathbf{W})+\frac{I_{a}(A)}{\pi_{a}(\mathbf{O}_{min};P^{\ast})}(Y-b_{a}(\mathbf{O};P^{\ast})).
\tanh_{q}t =-i\tan_{q}\left( it\right) =\frac{e_{q}\left( t\right)-e_{q}\left( -t\right)}{e_{q}\left( t\right) +e_{q}\left( -t\right)}=\frac{\mathcal{E}_{q}\left( 2t\right) -1}{\mathcal{E}_{q}\left(2t\right) +1}& =\sum_{n=1}^{\infty}\mathfrak{T}_{2n+1,q}\frac{\left( -1\right)^{k}t^{2n+1}}{\left[ 2n+1\right]_{q}!},
\begin{array}{lll}-\frac{1}{2}((n_1\Delta_1+\cdots+n_r\Delta_r)^2) = \lim_{\epsilon\rightarrow 0}-\frac{1}{2}(A_{\epsilon}^2) &= \lim_{n\rightarrow\infty}\frac{\ell_R(R/I(nn_1D_1)\cdots I(nn_rD_r))}{n^2}&= G(n_1,\ldots,n_r).\end{array}
1 \leq \frac1{1-\rho}F_N\left( \left[ \frac1n\sum_{i=1}^{n}\frac{\frac1N\Vert z_i\Vert^2}{\check{d}_i}\right]^{-1}\right) &\times \frac1Nz_n^* \left( \frac1n\sum_{i=1}^{n-1}z_iz_i^* + \check{d}_n \frac{\rho}{1-\rho}F_N\left( \check{d}_n \left[ \frac1n\sum_{i=1}^{n}\frac1N\Vert z_i\Vert^2 \right]^{-1}\right) I_N \right)^{-1}z_n.
\langle \bar{\psi}\psi \rangle \Big |_{m^{2}d^{2}={\cal O}( 0 . 1 )}\; \approx \; \frac{1}{4 \pi^{2}}\; m \; \rho \; \; c_{0}\Bigl ( m^{2}\, d^{2}\Bigr ) \Big |_{m^{2}d^{2}={\cal O}( 0 . 1 )}\; \approx \; \; 0 . 0 5 8 \; \frac{1}{4 \pi^{2}}\; \rho \, / \, d \; ,
\|f_0 \|_{L^{\infty}(\mathbb T \times [-M, M])}< \infty, \min_{0 \leq t \leq T}R(t) > \underline{R}, \dot{R}(T)=K \mu \mbox{and}K^2 \mu > \frac{M^2}{2\underline R^2}- \frac{3M^2}{4\underline R}
\delta(v, g \cdot x) = \delta(s_{\ell(v)}(v), w_{\ell(v)-1}(v) g \cdot x) \cdots \delta(s_1(v), g \cdot x) = \delta(s_{\ell(v)}(v), w_{\ell(v)-1}(v) \cdot x) \cdots \delta(s_1(v), x) = \delta(v, x).
u(z) = \log|z| + C +\sum_{j=1}^J \left\{d_j^{}\log|z-c_j^{}| + \sum_{k=1}^N \left[ a_{jk}^{}\kern 1pt\hbox{Re\kern 1pt}((z-c_j^{})^{-k}) + b_{jk}^{}\kern 1pt\hbox{Im\kern 1pt}((z-c_j^{})^{-k})\right] \right\},
\alpha = \sum_{1\leq i,k\leq n}\alpha_{ik}e_i^*\otimes e_k^* \alpha^{\wedge 2}= \sum_{1 \leq i,j,k,l \leq n}\alpha_{ik}\alpha_{jl}(e_i^*\wedge e_j^*) \otimes (e_k^*\wedge e_l^*).
d(x)-d(y) =\frac{a(x)^2}{b(x)^2}-\frac{a(y)^2}{b(y)^2}&=\bigl(a(x)-a(y)\bigr) \biggl(\frac{a(x)}{b(x)^2}+\frac{a(y)}{b(x)b(y)}\biggr) &-\bigl(b(x)-b(y)\bigr) \biggl(\frac{a(x)a(y)}{b(x)^2b(y)}+\frac{a(x)a(y)}{b(x)b(y)^2}\biggr).
|\mathcal K_k(\hat P^{(k-1)}) |\le (1+ \eta)\frac{1}{a^k_{k-1}}\prod \limits_{\ell=1}^{k-2}(\frac{1}{a_{\ell}})^{\binom{k}{\ell}}n^k\le \frac{1+ \eta}{1- \eta}\cdot \frac{1}{a^k_{k-1}}\cdot |\mathcal K_k(\mathcal G_T) \cap \mathrm{Cross}_{\mathcal Y}|
e_1(W_{n+j}) \le C \Big(\displaystyle{\sum_{i=1}^n e_{i-n}(U_i) + \sum_{k=1}^j e_0(U_{n+k})}&\le C_2 \Big(\displaystyle{\sum_{i=1}^n e_{1+i-n}(U_i) + \sum_{k=1}^q e_1(U_{n+k})+\sum_{k=q+1}^j e_0(U_{n+k})}\Big)\,.
{{\tilde f}_k}= \frac{1}{{2\,{c_k}}}\sum\limits_{j = 0}^4{\frac{1}{{{c_j}}}\cos \left({\frac{{k\,j\,\pi}}{4}}\right)\,{}_a^b{f_j}}= \frac{1}{{2\,{c_k}}}\left[{\frac{1}{2}\left({{}_a^b{f_0}+{{( - 1)}^k}\,{}_a^b{f_4}}\right) + \sum\limits_{j = 1}^3{\frac{1}{{{c_j}}}\cos \left({\frac{{k\,j\,\pi}}{4}}\right)\,{}_a^b{f_j}}}\right],
\||x|^{\gamma_1}|x'|^{\alpha}u\|_{L^s(\Omega)}\le C\Big(\||x|^{\gamma_2}|x'|^{\mu}\nabla u\|_{L^p(\Omega)}^{a'}\||x|^{\gamma_3}|x'|^{\beta}u\|_{L^q(\Omega)}^{1-a'}+\||x|^{\gamma_2}|x'|^{\mu}\nabla u\|_{L^p(\Omega)}^{a''}\||x|^{\gamma_3}|x'|^{\beta}u\|_{L^q(\Omega)}^{1-a''}\Big).
(\alpha, g, \gamma)(\delta, k, \sigma) = \begin{cases}(\alpha(g\cdot \gamma_1), \left.g\right|_{\gamma_1}, \sigma) \delta = \gamma\gamma_1 (\alpha, g\left(\left.k^{-1}\right|_{\delta_1}\right)^{-1}, \sigma(k^{-1}\cdot\delta_1)) \gamma = \delta\delta_1 0 \delta X^*\cap \gamma X^* = \emptyset\end{cases},
d s^{2}= \frac{\ell^{2}}{4}E \left[ \frac{V ( r )}{E ( r^{2}- 1 )}( d \tau + \sqrt{E}\cos ( \theta ) d \phi )^{2}+ \frac{4 ( r^{2}- 1 ) d r^{2}}{V ( r )}+ ( r^{2}- 1 ) d \Omega_{b}^{2}\right] \quad .
J_2(x,t,s,T) \ = \ \frac{2x(T-s)^2(s-t)}{(T-t)^2}&+\ \left\{\frac{x^2(T-s)(s-t)}{(T-t)^2}-\frac{2(T-s)(s-t)}{(T-t)}\right\}\J_3(x,t,s,T) \ ,
C = \frac{B T_{on}\, \left( 2^\frac{L}{2B T_{on}}- 1 \right)^{2}}{L \pi 2^{\frac{L}{B T_{on}}-\frac{3}{2}}}\, \,_{2}F_{1}\left(\frac{1}{2}, \frac{1}{2}- m; \frac{3}{2};\frac{1}{2}\right) + \frac{2 B T_{on}}{2^{\frac{L}{BT_{on}}}m L \sqrt{\pi}}\, \left(\frac{1}{2}\right)_{m}\left( 2^\frac{L}{2B T_{on}}- 1 \right).
\left( \frac{a(p)^{m/2}\,\#\Gamma p}{c(p)^{\left( m-2\right) /2}}\right)J_{\infty}(\widehat{v}) =\left( \frac{a(p)^{m/2}\,\#\Gamma p}{c(p)^{\left( m-2\right) /2}}\right) \frac{1}{m}\int_{\mathbb{R}^{m}}\left\vert \widehat{v}\right\vert^{2^{\ast}}dx & =\frac{\#\Gamma p}{m}\int_{\mathbb{R}^{m}}c(p)\left\vert v\right\vert^{2^{\ast}}dx\leq\tau,
\dot{\bar{u}}^\varepsilon = \varepsilon^{-2\gamma -\frac{1}{2}}\left \{\Delta \bar{u}^\varepsilon +\frac{1}{\varepsilon}f(\bar{u}^\varepsilon)\right \}+ (\varepsilon^{-2\gamma -\frac{1}{2}})^{\frac{1}{2}}\cdot \varepsilon^{\gamma}a(x) \dot{W}_t(x).
2\sum_{n=0}^{\infty}\textup{PD}_\textup{t}(12n)q^n \equiv\dfrac{f_2^4f_6^2}{f_1^4f_4f_{12}}-\dfrac{f_1}{f_3}\dfrac{f_2^4f_3f_{6}^2}{f_1^5f_4f_{12}}~a(q^2) &\equiv\dfrac{f_2^2f_6^2}{f_4f_{12}}-\dfrac{f_2^2f_{6}^2}{f_4f_{12}}~a(q^2)~(\textup{mod}~4),
g=G\frac{M_{\oplus}}{r^{2}}=6.674\cdot10^{-11}m^{3}kg^{-1}s^{-2}\times\frac{6\times10^{24}kg}{(6.4\times10^{6}m)^{2}}=9+\frac{795}{2^{10}}\approx9.77637ms^{-2}
\Pi_{1}^{( \mathrm{t a d p o l e})}= 0 = \Pi_{2}^{( \mathrm{t a d p o l e})}, \quad \Pi_{3}^{( \mathrm{t a d p o l e})}\sim - \frac{e^{2}}{2 \pi \beta}\, \ln \beta m_{\sigma}.
0\leq \nabla_{E_i}\nabla_{E_i}Q(x_0,y_0,X_0,t_0)=\frac{\nabla_{E_i}\nabla_{E_i}E_f(x_0,y_0,t_0)}{\psi(r_0/2,t_0)}=\frac{E_{f_{ii}}(x_0,y_0,t_0)}{\psi(r_0/2,t_0)}
q(r)=\frac{1}{(2\pi )^{N+1}a^{3}}\sum_{{\bf \alpha}}\frac{\lambda_{n,l}^{3}T_{n}(\lambda_{n,l})}{\sqrt{\lambda_{n,l}^{2}+k^{2}a^{2}+m^{2}a^{2}}}f_{n}^{(q)}[J_{n}(\lambda_{n,l}r/a)],\quad q=\varepsilon ,p_{i},
d s_{\mathrm{B T Z}}^{2}= -{\frac{( \rho^{2}- \rho_{+}^{2}) ( \rho^{2}- \rho_{-}^{2})}{\rho^{2}R^{2}}}d t^{2}+ \rho^{2}( d \varphi -{\frac{J}{2 \rho^{2}}}d t )^{2}+{\frac{\rho^{2}R^{2}}{( \rho^{2}- \rho_{+}^{2}) ( \rho^{2}- \rho_{-}^{2})}}d \rho^{2}.
f(r)=\frac{u(r,t)}{\frac{Q(t)}{A}}=Re\{\sum_{n=1}^{N}[\frac{\Lambda_{n}J_{0}(\Lambda_{n})-\Lambda_{n}J_{0}(\Lambda_{n}\frac{r}{R})}{\Lambda_{n}J_{0}(\Lambda_{n})-2J_{1}(\Lambda_{n})}]\}
\varepsilon_u = i \eta_{13}\gamma^{13}\varepsilon_l, \ \ \ \ varepsilon_u = i \eta_{24}\gamma^{24}\varepsilon_l, \ \ \ \ varepsilon_u = i \eta_{56}\gamma^{56}\varepsilon_l, \ \ \ \ varepsilon_u = - \eta_e \eta_p \varepsilon_l
\hat{H}_{i}^{( 1 )}= \hat{h}_{i}^{( 1 )}, \hat{H}_{i}^{( p )}= \hat{h}_{i}^{( p )}+ \sum_{k = 1}^{p - 1}{}_{p - 1}C_{k}\hat{h}_{i}^{( k )}\hat{H}_{i}^{( p - k )}, p = 2, 3, \dots .
R_f = \int\limits_{0}^{R}\underset{t>0}\int\left(\underset{j \in \psi}\prod\frac{1}{1+\max\{e^t-1,T,S_{th}\}r^\alpha d_j^{-\alpha}}+\frac{1}{3}\underset{i \in \phi}\prod\frac{P[\eta(r)<S_{th}]}{1+\max\{e^t-1,T\}r^\alpha d_i^{-\alpha}}\right)\mathrm{d}tf_R(r)\mathrm{d}r .
\psi_{\varepsilon}^{+}(n)=\left\lfloor \frac{\log n}{h_{\mu}}+\frac{(1+\varepsilon)}{h_{\mu}^{3/2}}\sqrt{2\rho_{\mu}\log n\log \log \log n}\right \rfloor\leq \left(1+\frac{\varepsilon}{2}\right)^{2/3}\frac{\log n}{h_{\mu}}
\int_{\mathbb{R}^N}|\psi|^p\,dx<+\infty\mbox{and}[\psi]_{\mathcal{B}^{\alpha,p}_\infty(\mathbb{R}^N)}^p:=\sup_{|h|>0}\int_{\mathbb{R}^N}\left|\frac{\delta^2_h \psi}{|h|^\alpha}\right|^p\,dx<+\infty.
|k^3 - k_1^3 - \alpha (n - k_1)^3 - \alpha (k-n)^3| &= |k - k_1| \left| \mu k^2(3\alpha c_2^2 + \mu(1-\alpha)) + 3 \alpha \delta(1 + \mu - 2c_2)k - 3 \alpha \delta^2 \right| &\geq |k - k_1| \left[ \left( 3 \alpha c_2^2 - \epsilon(1-\alpha) - 3\alpha\epsilon[1 + \epsilon - 2 c_2] \right) |k| - 3 \alpha \epsilon^2 \right] \gtrsim |k-k_1| |k|.
B\mathring{w}_{j}=0,\rho_{j}=\left( \mathring{w}_{j},\Omega\mathring{w}_{j}\right) ,d_{j}=\left( \mathring{w}_{j},\Theta^{\ast}B_{2}^{-1}\Theta\mathring{w}_{j}\right) N_{R}+1\leq j\leq2N.
\mathrm{ct}(H \Delta^{X^{-1}}_{0}(t)) = \frac{1}{n!}\mathrm{ct}\Big( \sum_{w\in S_n}w(H \Delta^{X^{-1}}_{0}(t))\Big)= \frac{1}{n!}\mathrm{ct}\Big( H \sum_{w\in S_n}w(\Delta^{X^{-1}}_{0}(t))\Big) &= \frac{1}{n!}\mathrm{ct}\Big( H W_0(t)\Big)= \frac{W_0(t)}{n!}\mathrm{ct}(H).
F =\left(\begin{array}{*{4}{c}}0 B 0 0 -B 0 0 0 0 0 0 0 0 0 0 0\end{array}\right),g_{\perp}\equiv\left(\begin{array}{*{4}{c}}1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0\end{array}\right),g_{\parallel}\equiv\left(\begin{array}{*{4}{c}}0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1\end{array}\right)
\begin{gathered}\alpha^{A,B}_i(p,\theta) = O(1)\,, \beta^{A,B}_i(p,\theta) = O(1)\,, c^{A,B}_i(p,\theta) = O(1)\,, \rho^{A,B}_r(p,\theta) = O(A^2) + O(B^2) + O(A\,B)\, \end{gathered}
\sup_{\psi \in C_0(\bar{D}),\, \Vert \psi \Vert < 1}\left \Vert \mathcal{T}_{t,k}^{\epsilon, v_k}\psi(x) \right\Vert = \sup_{x \in D}\mathbb{P}_{x,k}^{\epsilon, v_k}\Bigl \{\tau_{k}^{\epsilon, v_k}> t \Bigr\}, \,\,\, k =1,2, \ldots, n,
\frac{d}{dt}(Y_t - y_t) = \Pi_{\mathrm{ker}A^\perp}(B(Y_t,z_0) + B(z_0,Y_t) - B(y_t,z_t) - B(z_t,y_t) - B(y_t,y_t) + Ay_t) = L_{z_0}^{\perp}(Y_t-y_t) + \Pi_{\mathrm{ker}A^\perp}(B(y_t,z_0-z_t) + B(z_0-z_t,y_t) - B(y_t,y_t) + Ay_t).
\frac{1}{l^{n+1}}\int_{l\Delta_D+\Delta_A}\check{g}_{l\overline{D}+\overline{A}}dx=\int_{\Delta_D+\frac{1}{l}\Delta_A}\check{g}_{\overline{D}+\frac{1}{l}\overline{A}}dx=\int_{\Delta_D}\check{g}_{\overline{D}+\frac{1}{l}\overline{A}}dx+\int_{(\Delta_D+\frac{1}{l}\Delta_A)\setminus\Delta_D}\check{g}_{\overline{D}+\frac{1}{l}\overline{A}}dx.
(H-\epsilon_a)J_a\propto\left| \begin{array}{ccc}(\epsilon_c-\epsilon_a)m_c (\epsilon_b-\epsilon_a)m_b+v_{bc}m_{c}(\epsilon_a-\epsilon_a)m_a+v_{ab}m_{b}+v_{ac}m_{c}\cr\epsilon_c-\epsilon_a v_{bc}v_{ac}\cr0 \epsilon_b-\epsilon_a v_{ab}\end{array}\right| \, .
\mathbb{P}_G\{\xi^{v}_{n^{2\gamma}+L_{\delta_0}}\cap \hat{\xi}^{u}_{n^{2\gamma}}=\emptyset\,|\,\xi^v_{n^{2\gamma}}\neq \emptyset, \hat{\xi}^u_{n^{2\gamma}}\neq \emptyset\}=o(n^{-2}).
= A^\mu (x) + \delta^\mu_0 \frac{1}{(2 \pi)^3}\int \frac{d^3 k}{2 k^0}\Biggl\{\frac{1}{|\vec{k}|}e^{+i \vec{k}\vec{x}}+ \frac{1}{|\vec{k}|}e^{-i \vec{k}\vec{x}}\Biggr\}
P(B_2^{(n)}) \le\mathrm{e}^{-\mathrm{e}^{-3}r^{-1}(1-r^{-1})^{r-1}K^{r}H(x_K)a_c^{(n)}}\frac{1}{1-\mathrm{e}^{-\mathrm{e}^{-3}(1-r^{-1})^{r-1}K^{r}(\lceil K\rceil)^{-1}H(x_K)(\log K) a_c^{(n)}}}.
Z_{0}(s,\chi,f)=L_{\Gamma_\infty}(q^{-s},\chi) +\sum\limits_{\Delta\in\mathcal{C}}\ \sum\limits_{k\in\Delta\cap\mathbb{N}^n}q^{-||k||-d(k)s}\int\limits_{(R_{K}^\times)^n}\omega(f_\Delta(u)+\mathfrak{p}^{l}\tilde{f}_{\Delta,k}(u))\ |du|,
\left( \dots, \binom{m_i}{b_i}, \binom{m_{i+1}}{b_{i+1}}, \dots \right)^{\sigma_i}= \left(\dots,\binom{m_{i+1}}{b_{i+1}}, \binom{m_i}{b_{i+1}+ m_{i+1}b_i - m_i b_{i+1}}, \dots \right).
\Delta (\tilde E_s - h_s) = (\Delta - \lambda_s)(\tilde E_s - h_s)+\lambda_s(\tilde E_s-h_s) = (\Delta-\lambda_s)(\tilde\Delta-\lambda_s)^{-1}(\Delta-\lambda_s)h_s+\lambda_s(\tilde E_s - h_s) =(\Delta-\lambda_s)h_s + \lambda_s(\tilde E_s-h_s)\in L^2,
\mathbb{E}_{\mathbb{Q}_\xi}h\left(\tilde X_{\omega_k^n}(t), \sum_{\substack{v\in\mathbb T_k v\neq\omega_k^n}}\tilde X_v(t)W_{n-k,v}(t)\right)=\mathbb{E}_\xi\sum_{v\in\mathbb T_k}\tilde X_u(t)h\left(\tilde X_{u}(t), \sum_{\substack{v\in\mathbb T_k v\neq u}}\tilde X_v(t)W_{n-k,v}(t)\right).