harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\varphi\left(\begin{array}{c c}a b 0 d \end{array}\right) = \left(\begin{array}{c c c}a^{\ell_1}0 c_2 \, a^{\ell_1}\, b^{p^{e_2}}\, d^{p^{e_2}}0 d^{- \ell_2}0 0 0 d^{- \ell_2}\end{array}\right) .
\begin{array}{rl}3H^2\ ={\displaystyle{2\pi\lambda\phi^4\over M_{\rm p}^2 -8\pi\xi\phi^2}\ ,}begin{align*}4mm](1 - 6\xi)\ 3H\dot\phi\phi\ ={\displaystyle -\{\lambda\phi^4M_{\rm p}^2\over M_{\rm p}^2 - 8\pi\xi\phi^2}\ .}\end{array}
f\star g \star h={1\over \hbar^4 \pi^4}\int \! d\overline{p}dp' dp'' dp''' d\overline{x}dx' dx'' dx''' f(x',p') g(x'',p'') h(x''',p''') ~\exp{-4i\over \hbar}\left( A(r'', r',\overline{r})+A(r''', \overline{r}, r ) \right) ,
q_{m+1}^{\frac{1}{1+q_m}}\leq \left(q_{m-1}+bq_{m}+b(2Kq_{m})^{\frac{1+q_{m}}{1+q_{m-1}}}\right)^{\frac{1}{1+q_{m}}}&<\left(\left(\left(2Kb+b+1\right)q_m\right)^{\frac{1+q_m}{1+q_{m-1}}}\right)^{\frac{1}{1+q_m}}&<\left(4Kbq_m\right)^{\frac{1}{1+q_{m-1}}}\leq C(4Kb)^{\sum_{k=N}^{m-1}\frac{1}{1+q_k}}
S := \{(x,y) \in \Omega^2 : W(x,y) > 0 \}, J := \{x \in \Omega : I(x) > 0 \}, P := \{(x,y,z) \in \Omega^3 : x \in J , (y,z) \in P_x \}, Q := \{(x,y,z) \in \Omega^3 : x \in J , (y,z) \in S_x^2 \setminus S \}.
a_{n-1,\kappa_{n}-1}= \exp \left(-\kappa_{n-2}q_{n-2}\left(\lambda^u(\mu_{n-2})+\frac{2}{n-2}\right)-(\kappa_{n-1}-1)q_{n-1}\left(\lambda^u(\mu_{n-1})+ \frac{1}{n-1}\right) \right).
\begin{pmatrix}a_1 a_2 a_3 a_1 a_2 a_3 \end{pmatrix}\begin{pmatrix}d_1 d_2 d_3 d_1 d_2 d_3 \end{pmatrix}-\begin{pmatrix}b_1 b_2 b_3 b_1 b_2 b_3 \end{pmatrix}\begin{pmatrix}c_1 c_2 c_3 c_1 c_2 c_3 \end{pmatrix}= \begin{pmatrix}1 1 1 \end{pmatrix}
\tau_1=\begin{cases}\lbrack (u+1,u),1^3,(2,1)^2\rbrack,G\in\mathcal{T}(\langle 729,49\rangle), \lbrack (2,1),(u+1,u),(2,1)^2\rbrack,G\in\mathcal{T}(\langle 729,54\rangle).\end{cases}
\Omega_{h}^{i j}( X ) = \sum_{w ,{w^{\prime}}\in h}\mathrm{s i g n}( w ,{w^{\prime}}) \left( X_{w}^{i}X_{w^{\prime}}^{j}-{\frac{1}{3}}\delta^{i j}X_{w}^{k}X_{w^{\prime}\, k}\right)
\begin{array}{c c c}{{}}&{{h^{0 0}}}&{{}}\\{{h^{1 0}}}&{{}}&{{h^{0 1}}}\\{{}}&{{h^{1 1}}}&{{}}\\ \end{array}\qquad = \qquad \begin{array}{c c c}{{}}&{{1}}&{{}}\\{{n}}&{{}}&{{n}}\\{{}}&{{1}}&{{}}\\ \end{array}
-G_{\geq 3}=|\partial_x|\phi-G(h)\phi-|\partial_x|T_Bh-\partial_xT_Vh+G_2=I+II, I: =h_xV-H_0(h_xH_0V)-|\partial_x|T_{iH_0V}h-\partial_xT_Vh+G_2+R_0(iH_0V), II: =D-R_0(iH_0V)-iH_0(h_xD)+|\partial_x|T_Dh.
x_p^{\infty}=(x')_{q}^{\infty}\, (,\,x_{p+i}=x'_{q+i}\,\,\, \,\,i\in \mathbb{N}); t-t' =h(x)-h(x')+\tau_p(x)-\tau_q(x'); \xi-\xi' =f_q(x')-f_p(x),
\tilde{M}_{5}^n =\sum_{i\ge 4}C^i_ng_{i}\, \partial^{n + 1 -i}_x u +\sum_{1\le i\le 3}C^i_n\partial_x^i u\, g_{n+1-i}&\quad+\sum_{i\ge 4}C^i_n \partial_y^{-1}g_{n}\partial^{n + 1 -i}_x w + \sum_{1\le i\le 3}C^i_n\partial_x^i w\, \partial_y^{-1}g_{n+ 1 - i},
\lim_{m,n\to\infty}\int_{\textrm{spt}\, \zeta}|T(u_m-u_n)|^{({q\over p-1})'}\,dx=0 \quad{and}\lim_{m,n\to\infty}\int_{\textrm{spt}\ \zeta}V^r |T(u_m-u_n)|^r \,dx=0,
d(T_1 \gamma) = T_1^3T_2 + T_1^2 T_2^2, d(T_1^2 \beta) = T_1^4+ T_1^2 T_2^2 + T_1^3 T_2 d(T_2 \gamma) = T_1^2T_2^2+T_1 T_2^3, d(T_2^2 \beta) = T_1^2 T_2^2 +T_2^4 + T_1T_2^3 d(T_3 \gamma) = (a^2b+ab^2)T_3^4, d(T_1 T_2\beta) = T_1^3T_2+ T_1T_2^3 + T_1^2 T_2^2
tA'\big({1+(t(A+B))^2\over(1-(t(A+B))^2)^2}-{1+(tA)^2\over(1-(tA)^2)^2}\big)+((A,A')\leftrightarrow(B,B')) =A'\Big(3\big((A+B)^2 - A^2\big)P_3(A,B,t) + 5\big( (A+B)^4 - A^4 \big)P_5(A,B,t)\Big) +((A,A')\leftrightarrow(B,B')),
\phi'(x) =\frac{(2n)!}{(2m)!}e^{(2m+1)x-(2n+1)\phi(x)}\frac{P_n(e^{\phi(x)})}{P_m(e^x)}&\sim e^{(2m+1)x-(2n+1)\phi(x)+2n\phi(x)-2mx}&=e^{x-\phi(x)}=1+o(1), x\to+\infty.
(-1)^{\mathrm{1}_{i \in \mathcal{O}}}\frac{T^{\mathrm{1}_{i \in \mathcal{B}}}}{n}\sum_{w^n = (-1)^{\theta_2}}\frac{w^{\mathrm{1}_{i \in \mathcal{B}}+h_i-h_j}e^{- (Tw+\theta_1 \pi i+ \lambda)[t_j - t_i]_i}}{1 - (-1)^{\theta_1}e^{-(Tw+\lambda)}}= H_n^{\lambda,\theta,T}(x_i,x_j).
\mathcal{R}_{M,j}:= s_{j-2,j-1}\circ (\mu_{j}\circ \mu_{j+1}\cdots \circ \mu_{j-4}\circ \mu_{j-3}) \circ (\mu_{j-1}\circ \mu_{j-2}\circ \cdots \circ \mu_{j+1}\circ \mu_j).
\theta_1 = \hat{\lambda}_2/2\kappa \geq 1/2\kappa, \theta_1 - \theta_2 = -\hat{\lambda}_1/2\kappa \leq -1/2\kappa \theta_2 = (\hat{\lambda}_1 + \hat{\lambda}_2)/2\kappa \leq (\kappa-1)/2\kappa = 1/2 - 1/2\kappa.
\lim\limits_{\zeta\to0}\frac{(1-e^{i2^{j_k}\xi_k\zeta})+2^{2j_k}\zeta^2e^{i2^{j_k}\xi_k\zeta}+i2^{j_k}\xi_k\zeta}{\zeta^2}=2^{2j_k}(\xi_k^2+1), k=2,\cdots, 2\mu+1.
F(z)=c(F)A(z)^2+A(z)^2 \sum_n\frac{F(s_n)}{A'(s_n)^2(z-s_n)^2}+ \frac{F'(s_n)}{A'(s_n)^2(z-s_n)}-\frac{F(s_n)A''(s_n)}{A'(s_n)^3(z-s_n)}
\sigma ( t ) \approx \frac{2}{5}\frac{1}{\gamma t}\left( \frac{t}{t_{i}}\right)^{1 / 3}\left[ 8 U ( v_{s}\gamma )^{2}+ 2 \alpha ( \phi_{0})^{2}\left( U + T + I^{2}\right) - \left( U - T - I^{2}\right) - 2 \ln \left( \frac{r}{r_{0}}\right) I^{2}\right]
\hat{\cal I}_{\mu \nu}^{\alpha \beta}={\it g_{2}}\bigg ( \hat{\theta}_{\rho}\big ( - i \tilde{F}_{\lambda \gamma , \sigma}+ F_{\lambda \gamma}\hat{P}_{\sigma}-{\sqrt 2}i \hat{\theta}_{5}F_{\lambda \gamma , \sigma}\big ) + \hat{\theta}_{\lambda}\big ( - i \tilde{F}_{\rho \sigma , \gamma}+ F_{\rho \sigma}\hat{P}_{\gamma}-{\sqrt 2}i \hat{\theta}_{5}F_{\rho \sigma , \gamma}\big ) \bigg ) \, ,
\int_{-1}^1\Phi_n\overline{\dfrac{d\Phi_n}{dx}}w_{\alpha,\beta+1}\,dx = \int_{-1}^1 P_n^{(\alpha,\beta-1)}\overline{\dfrac{d\Phi_n}{dx}}w_{\alpha,\beta+1}\,dx + \mu \int_{-1}^1\left|\dfrac{d\Phi_n}{dx}\right|^2w_{\alpha,\beta+1}\, dx.
\partial_t m - d_m \Delta_x m= - \underbrace{\lambda_{m M_1}\frac{m}{k_{M_1}+ m}}_{transformation \, into \, M_1}- \underbrace{\lambda_{mM_2}\frac{m}{k_{M_2}+ m}}_{transformation \, into \, M_2}\ \ \underbrace{- \ \ \beta_m\, m}_{death \,of\, m},
\prod_{m=1}^{M}(Z^{1\dagger})^{i_{m}}_{j_{m}}\prod_{m^{\prime}=1}^{M^{\prime}}(Z^{2\dagger})^{i_{m^{\prime}}^{\prime}}_{j_{m^{\prime}}^{\prime}}\prod_{n=1}^{Nk}(\Psi^{1\dagger})_{l_{n}}|0 \rangle ,
\int_{S^{n-1}_1}g(\nabla R_g,\nu)r^{n-1}d\omega_{n-1}=\int_{S^{n-1}_1}\partial_i\left(\partial_k\partial_jg_{kj}- \partial_j\partial_jg_{kk}\right)\frac{x^i}{r}r^{n-1}d\omega_{n-1}+ O_1(r^{n-2\tau-4}).
ln(\frac{n+1}{n})=\sum_{k=1}^{\infty}\frac{1}{k(n+1)^{k}}=\frac{1}{n+1}+\frac{1}{2(n+1)^{2}}+\frac{1}{3(n+1)^{3}}+\frac{1}{4(n+1)^{4}}+\cdot\cdot\cdot
L ( u ) = \left( \begin{array}{l l}{{- \frac 1 2 \dot{B}( u )}}&{{B ( u )}}\\{{- \frac 1 2 \ddot{B}( u ) - B ( u ) Q_{N}( u )}}&{{\frac 1 2 \dot{B}( u )}}\\ \end{array}\right) .
f_{-}^{\prime\prime}(0)=\lim_{t\searrow0}\frac{1}{-t}[f^{\prime}(-t)-f^{\prime}(0)]\mbox{and}f_{+}^{\prime\prime}(0):=\lim_{t\searrow0}\frac{1}{t}[f^{\prime}(t)-f^{\prime}(0)],
\sum_{k=1}^{N(f)}\frac{1}{\zeta_{k}^{\,2n}}=n\sum_{m\in\mathcal{M}(n)}\alpha(m)\sum_{k=1}^{\infty}\prod_{j=1}^{d(m)}\left(x_{k+j-1}x_{k+j}\right)^{m_{j}}.
\frac{\Psi'_{N_\varepsilon}(B_\varepsilon)}{2}=B_\varepsilon H(\gamma_\varepsilon)\left[\left(1+L^H_\varepsilon \right)\left(1+\varphi_{\tilde{N}_\varepsilon}(B_\varepsilon^2) \right)+L^g_\varepsilon \left(\frac{B_\varepsilon^{2N_\varepsilon}}{N_\varepsilon !}-B_\varepsilon^2 \right)\right]
F(h_1h sk) = \Omega_{\chi}^{-1}(h_1h, sk) \Omega_{\chi}^{-1}(s,k) \sigma_{\chi}(h_1)\sigma_{\chi}(h) \Omega^{-1}_{\chi}(h_1, h) f(s) =\Omega_{\chi}^{-1}(h_1, hsk) \sigma_{\chi}(h_1) F(hsk)
\int_{y'\prec x'}^{}dx'\int_{y\prec x}^{}dy \prod_{i=1}^{N}\hat{m}(y_i)q_t^{N,N+1}\left(\left(x,y\right),\left(x',y'\right)\right)=\int_{y\prec x}^{}\prod_{i=1}^{N}\hat{m}(y_i)dy\int_{y'\prec x'}^{}dx' q_t^{N,N+1}\left(\left(x,y\right),\left(x',y'\right)\right).
\sum_{n=1}^{\infty}\frac{\mu(n)}{n^k}\exp \left({-\frac{x}{n^2}}\right) = \frac{1}{2 \pi i}\int_{\lambda - i \infty}^{\lambda+i \infty}\frac{\Gamma(s)}{\zeta(k-2s)}x^{-s}{\rm d}s + \frac{1}{2}\sum_{\rho}\frac{\Gamma(\frac{k-\rho}{2})}{\zeta'(\rho)}x^{-\frac{k-\rho}{2}}.
\begin{aligned}\sum_{s=0}^{n-1}S_{1,k}(n,s,t) \leq n^{-d}(t+1)^{-d}K_3 \left( \frac{r\sin(\pi d)}{\pi}\right)^{k-1}\|\zeta_{}\|_L \|T\|^{k-1}\|\phi_{}\|_L &= n^{-d}(t+1)^{-d}\|\phi_{}\|_L K_3 \{r\sin(\pi d)\}^{k-1}.\end{aligned}
W [ \Phi_{0}, a , M ] = -{\frac{1}{6 4 \pi^{2}}}\tau ({\frac{4}{3}}\pi a^{3}) \Phi_{0}^{4}{\frac{1}{3 5}}L o g{\frac{M^{2}}{\Phi_{0}^{2}}}( 1 +{\frac{7 0}{6 3}}{\frac{1}{\Phi_{0}^{2}a^{2}}}+{\frac{3 5}{2}}{\frac{m^{2}}{\Phi_{0}^{2}}}) + . . .
\times \prod_{b = 1}^{N}\prod_{j = 1}^{n_{b}}\exp \left( 2 e \sum_{I = 1}^{N}\Big ( \varepsilon_{\mu \nu}\partial_{\mu}A_{\nu}^{( I )}, K_{I I}U_{I b}\delta ( x_{j}^{( b )}) \Big ) \right)
\left[ l_{\dot{J}}^{\dot{I}}, f_{\dot{L}}^{\dot{K}}\right] = \sum_{\dot{K}_{1}\dot{K}_{2}= \dot{K}}\delta_{\dot{J}}^{\dot{K}_{1}}f_{\dot{L}}^{\dot{I}\dot{K}_{2}}- \sum_{\dot{L}_{1}\dot{L}_{2}= \dot{L}}\delta_{\dot{L}_{1}}^{\dot{I}}f_{\dot{J}\dot{L}_{2}}^{\dot{K}}.
\begin{aligned}I \leq \sup_{|z|\leq |x|/2}\Big|\frac{\partial^2}{\partial x_i\partial x_j}p_0(t, x+z) \Big|\int_{|z|\leq |x|/2}\frac{|z|^2}{|z|^{d+\beta}}\,dz &\leq c_4t|x|^{-(d+4)}|x|^{2-\beta}=c_4t|x|^{-(d+2+\beta)}.\end{aligned}
\partial^+ V(t, \overline{x}(t)) =\left\lbrace (p',p'') \in\mathbb{R}\times\mathbb{R}^n :\right. &\left. \forall (\theta',\theta'')\in\mathbb{R}\times\mathbb{R}^n, D_{\downarrow}V (t,\overline{x}(t))(\theta',\theta'') \leq p' \theta'+ \langle p'',\theta'' \rangle \right\rbrace,
\int_{q_z}u(\dot{x})|x|^{-d-\gamma}e^{-i(x,k)}dx= e^{-i(z,k)}[\sum_{s=0}^{n-1}d_s(\dot{z},\sigma)|z|^{-d-\gamma-s}+O(|z|^{-d-\gamma-n})], |z|\to \infty,
\lim_{N \to \infty}\frac{1}{N \theta}\log \left( \int_{\mathbb{R}^{d \times N}}\exp \left( - N \theta f ({\rm emp}_{N}) \right) d \gamma_{N}\right) = \inf_{\mu \in \mathcal{P}(\mathbb{R}^{d})}\{f(\mu) + F(\mu) \},
{\cal B}\Bigl(\frac{v^2}{N_F},\tilde{\alpha},\tilde{\lambda},\{\zeta_i\}\Bigr) =\frac{v^2}{N_F}{\cal E}_{\rm cl}\bigl(\tilde{\alpha}, \tilde{\lambda},\{\zeta_i\}\bigr) +{\cal E}^{F}(\{\zeta_i\})- 1 \, .
b(x,\xi)\theta_h(x)\chi(h^{\beta}\xi)(x+p'(\xi))^2 \tilde{\gamma}\left(\langle\xi\rangle^2(x+p'(\xi)) \right) & + b(x,\xi)\theta_h(x)\chi(h^{\beta}\xi)(x+p'(\xi))^2 (1-\tilde{\gamma})\left(\langle\xi\rangle^2(x+p'(\xi)) \right)\,.
\begin{pmatrix}\partial_3 u_1 + \mathrm{i}\xi_1 u_3 \partial_3 u_2 + \mathrm{i}\xi_2 u_3 \partial_3 u_3 \end{pmatrix}(x_3) = \begin{pmatrix}0 0 0 \end{pmatrix}x_3 = \pm \frac\pi2 .
\begin{array}{c}eC \subset g_{\alpha}^{-1}d_{\alpha}g_{\alpha}C = (Ug_{\alpha})^{-1}d_{\alpha}(Ug_{\alpha}) C = (\varphi_{\alpha}h)^{-1}d_{\alpha}(\varphi_{\alpha}h) C= h^{-1}(\varphi_{\alpha}^{-1}d_{\alpha}\varphi_{\alpha}) h C\,.\end{array}
H_{0}=\left\{\begin{pmatrix}a 0 b d \end{pmatrix}\in_{2}(\mathbb{Z}/N\mathbb{Z})\right\}, H_{1}=\left\{\begin{pmatrix}a 0 b \pm{}1 \end{pmatrix}\in_{2}(\mathbb{Z}/N\mathbb{Z})\right\}.
\frac{d+\varepsilon\sqrt{bd}}{\sqrt{bd}}\left( \frac{y+\sqrt{bd}}{2}\right)^{2m}-\frac{d+\sqrt{bd}}{\sqrt{bd}}<W_{2m}^{(b,d)}=W_{2m}^{(a,d)}&<\frac{d+\varepsilon\sqrt{ad}}{\sqrt{ad}}\left( \frac{x+\sqrt{ad}}{2}\right)^{2m}
\begin{array}{l}\displaystyle{f_n\Big[\frac{3n-2}{2}\Big]=-\frac{(3n-8)(3n-6)(3n-4)}{8(3n-3)}\frac{s^2}{\epsilon^2}(Q')^2 Q^{-3}f_{n-2}\Big[\frac{3n-8}{2}\Big]\,,}\displaystyle{g_n\Big[\frac{3n}{2}\Big]=-\frac{(3n-2)}{2}\frac{\gamma}{\epsilon}Q' Q^{-1}f_n\Big[\frac{3n-2}{2}\Big]\,,}\end{array}
(a,b,c) = \left(\frac{\alpha +\tilde\alpha}{2},\frac{\beta +\tilde\beta}{2},\frac{\alpha\beta -\tilde\alpha\tilde\beta}{\alpha +\beta -\tilde\alpha -\tilde\beta}\right),\ \left(\frac{\alpha +\tilde\alpha}{2},\frac{\beta +\tilde\beta}{2},-\frac{\alpha +\beta -\tilde\alpha -\tilde\beta}{\alpha\beta -\tilde\alpha\tilde\beta}\right)
-8q^5-146q^5s-192q^5s^2+8q^6 +200q^6s+136q^6s^2 =\left( -8q^5+8q^6 \right)+ \left( -146q^5s-192q^5s^2 \right)+ \left(200q^6s+136q^6s^2 \right) \le -338q^5s^2 +336q^6s \le 0 \, ,
\mathbb{E}J_1^2(h+y_0^1,h+y_0^2,\lambda) =\frac{1}{2}\sum_{p=0}^{\tau-1}\frac{1}{n_p}\log\frac{b+\lambda-(d_{\tau}+(1-\eta t)d_{p+1}')}{b+\lambda-(d_{\tau}+(1-\eta t)d_p')}&+\frac{1}{2}\sum_{p=\tau}^{k}\frac{1}{n_p}\log\frac{b+\lambda-d_{p+1}}{b+\lambda-d_p}.
\vert \nabla (w_i)^2(x)\vert=2\left\vert w_i (x) \right\vert. \left\vert \nabla \left \vert w_i (x) \right\vert \right \vert \, \leq 2 \vert \nabla u \vert \sqrt{4\lambda_0^{-1}V(u(x))}\leq 4\sqrt{\lambda_0}^{-1}J (u) (x),
d\mathcal{M}_{|u}=-n\mbox{Ric}(\omega_{u})\wedge\omega_{u}^{n-1}+\bar{R}\omega_{u}^{n},\,\,\,\,\,\,\bar{R}:=\frac{nc_{1}(X)\cdot[\omega]^{n-1}}{[\omega]^{n}},
u^p_{\mu_t, p}(x) \cdot 2^p e^{-pt}= \int_{\mathcal{G}X}\cosh^p(d(x, \gamma(t)))) \cdot 2^p e^{-pt}d\nu(\gamma) \to \int_{\mathcal{G}X}\exp(pB(x, \pi(\gamma), \gamma(+\infty))) d\nu(\gamma) = U^p_{\nu, p}(x)
\tan(2\alpha_1) = \frac{1}{\cot(2\alpha_1)}= \frac{1}{\tan(\frac{\pi}{2}-2\alpha_1)}=\frac{1}{\tan (2\phi_1)}=\frac{\sqrt{\epsilon}}{4}, \tan(2\beta_1) = \frac{1}{\cot(2\beta_1)}= \frac{1}{\tan(\frac{\pi}{2}-2\beta_1)}=\frac{1}{-\tan(2\psi_1)}=\frac{\sqrt{\epsilon}}{2},
\lim_{s\downarrow 1}\frac{1}{\zeta(s)}\sum_{m=m_0}^{\infty}\frac{1}{m^s}((k_1u(m))^{-s+1}-(k_2v(m))^{-s+1}) =\lim_{s\downarrow 1}\frac{1}{\zeta(s)}\sum_{m=m_0}^{\infty}\frac{1}{m^s}(u(m)^{-s+1}-v(m)^{-s+1}).
\Xi_2(z)=\exp\Big( - 2\pi \lambda \int_0^\infty{f_{{\rm{Pr}}}}\left( u \right) ( 1 -\Omega_1(z,u)) udu-2\pi \lambda \int_0^\infty (1-{f_{{\rm{Pr}}}}\left( u \right)) ( 1 -\Omega_2(z,u)) udu\Big)
dX^{\sigma}_t = \sigma \bigl(X^{\sigma}\bigr)\,dB_t + \bigl( \tfrac{1}{2}\sigma^2\bigl(X^{\sigma}_t \bigr) \log\pi'\bigl(X^{\sigma}_t \bigr) +\sigma \bigl(X^{\sigma}_t\bigr) \sigma' \bigl(X^{\sigma}_t \bigr) \bigr)\,dt
[O^a,E^b](n) \equiv (O^a \circ E^b)(n)-(E^b \circ O^a)(n) &=\left(\frac{3}{2}\right)^a \left(\frac{n}{2^b}+1\right)-1 - \frac{1}{2^b}\left(\frac{3}{2}\right)^a (n+1) + \frac{1}{2^b}&=\left(\frac{3}{2}\right)^a\left(1-\frac{1}{2^b}\right)+\frac{1}{2^b}-1
- < < G_{1}^{0}G_{2}^{0}G_{3}^{0}K_{2 3}^{L}G_{2}^{0}G_{3}^{0}> > g^{0}< < G_{1}^{0}G_{2}^{0}K_{1 2}^{L}G_{1}^{0}G_{2}^{0}G_{3}^{0}> > .
4x^4 (D+12)^2(D+8)(D+4)T - (D+6)DT + 12x^4(D+12)((D+4)T)^2 & = V(2x^4(D+12)(D+8)(D+4)T - \frac{1}{2}(D+6)T + 6x^4((D+4)T)^2),
|\widehat{u}(t_0,\xi)| = \int_{r_0 - \delta}^{r_0 + \delta}e^{\xi \psi(r)}\varphi (t_0 - r) dr\geqslant \int_{r_0 - \frac{\delta}{2}}^{r_0 + \frac{\delta}{2}}e^{\xi \psi(r)}dr.
\dot{\phi}_i= \omega_i+\frac{1}{N}\sum_{j=1}^NW_{ij}(t)g(\phi_j-\phi_i), \dot{W}_{ij}= -\varepsilon(\sum_{k=1}^Nf(W_{ik},W_{jk})+\sum_{k=1}^N(h_1(\phi_k-\phi_i)+h_2(\phi_k-\phi_j))).
D_{1}(\beta; \xi, \tau) = \frac{1}{2\sqrt{\pi (\beta + \tau)}}\exp\left[-\frac{\xi^{2}}{4(\beta + \tau)}\right] &= \frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-(\beta +\tau) k^{2}}e^{i k \xi}dk,
{\cal L}=-\frac{1}{4}\exp(\frac{\phi}{f_\phi})F_{\mu\nu}{}^a F^{\mu\nu}{}_a-\frac{1}{2}\partial^\mu\phi\cdot\partial_\mu\phi+\sum_{f=1}^{N_f}\overline{q}_f(i\gamma^\mu\partial_\mu +g\gamma^\mu A_\mu{}^a X_a -m_f)q_f,
F_{\lambda, t}:= \chi (d_{\lambda}^{t}), \textup{where}d_{\lambda}^{t}= \begin{bmatrix}0 \lambda ( \mathsf{M}-t ) - \frac{\partial}{\partial r}\lambda ( \mathsf{M}-t ) + \frac{\partial}{\partial r}0 \end{bmatrix}.
\lVert \partial_{\mathbf{r}}^{\mathbf{a}'}T_{j,k +1}(r_{k},\dots ,r_{j-1})(f)\rVert_{L^{2}_{\alpha '}}= &\lVert (i\pi d_{k}+ P_{k})\partial_{\mathbf{r}}^{\mathbf{a}'}T_{k}(r_{k},\dots ,r_{j-1})(f)\rVert_{L^{2}_{\alpha '}}= \lVert (i\pi d_{k}+ P_{k}) F_{r_{k}}\rVert_{L^{2}_{\alpha '}}\le c_{k}\lVert f\rVert^{j}_{W^{\beta ,2}_{4}},
u \left( t_{m \, n}^{( l )}\right) = \delta_{m , n}\, q^{2 ( m - k )}u \left( t_{k \, k}^{( l )}\right) , \quad v \left( t_{m \, n}^{( l )}\right) = \delta_{m , n}\, q^{2 ( k - m )}v \left( t_{k \, k}^{( l )}\right) .
W^{j}_{3}=-\varepsilon\int_{\mathbb{T}^{d}}(\partial_{x_{j}}w^{1}-\partial_{x_{j}}w^{2})\left(\Theta_{q}(t,x,\mu^{1},Dw^{1})\mu^{1}_{x_{j}}-\Theta_{q}(t,x,\mu^{2},Dw^{2})\mu^{1}_{x_{j}}\right)\dx,
\partial_t^m v^{(3)}\stackrel{\eqref{def_vwr}}{=}p(D-2) \partial_t^m w^{(2)}\stackrel{\eqref{lin_wnm}}{=}\partial_t^m r^{(2)}+ x q_2(D) \partial_t^{m+1}v^{(2)}- x \partial_t^{m+1}w^{(2)}\mbox{for}\, t, x > 0
\int_{s}^t (\delta h)_{s,\tau}\otimes\mathrm{d}X_{\tau}:=\lim_{|\pi|\rightarrow 0}\int_{\pi}h_{s,\tau}\circ\mathrm{d}X_{\tau}=\lim_{|\pi|\rightarrow 0}\sum_{0\leq i<m}h_{s,\kappa_i}\otimes (\delta X)_{\kappa_i,\kappa_{i+1}},
\Phi =\ln(\rho^{2\alpha}) + S_0 + O_{4}(\rho^{-1}) - \ln(\rho^{2}+A+O_4(\rho^{-1})), &=\ln(\rho^{2\alpha}) + S_0 - \ln(\rho^{2}) - \ln(1+A\rho^{-2}+O_4(\rho^{-3})) + O_{4}(\rho^{-1}), &=\ln(\rho^{2(\alpha-1)}) + S_0 - \ln(1+A\rho^{-2}+O_4(\rho^{-3})) + O_4(\rho^{-1}),
\frac{1}{2}\triangle(|x|^{2})= 2+\langle x, \triangle x\rangle=2+\sum_{k}H^{k^{*}}\langle x, e_{k^{*}}\rangle=2-\langle H,H-\xi\rangle, \frac{1}{2}\mathcal{L}(|x|^{2})= \frac{1}{2}\triangle(|x|^{2})-\frac{1}{2}\langle x, \nabla|x|^{2}\rangle=2-|H|^{2}+\langle H,\xi\rangle-|x^{\top}|^{2}= 2+|\xi|^{2}-(|x|^{2}+\langle H, \xi\rangle).
\frac{R_{t t}}{A e^{2 \Omega}}+ A R_{r r}+ \frac{( d - 2 ) R_{\theta_{a}\theta_{a}}}{r^{2}\displaystyle{\prod_{b = 1}^{a - 1}\sin^{2}\theta_{b}}}= \frac{d - 2}{r^{d - 2}}\frac{d}{d r}\left[ r^{d - 3}( 1 - A ) \right] .
Q_k(X) = \frac{q^{1-g}h}{\zeta(k)(q-1)^2}q^X - \sum_{j=1}^{2g}\sum_{\ell=0}^{k-1}\frac{Z(\gamma_{j, \ell}^{-1}) \gamma_{j, \ell}}{k\gamma_{j, \ell}^{1-k}Z'(\gamma_j^{-1})}\frac{\gamma_{j, \ell}^X}{\gamma_{j, \ell}- 1}+ \varepsilon_{k}(X),
\psi(u T) = \varphi( e_1 u T e_1) = \varphi( e_1 (u JuJ) T (Ju^*J) e_1 ) = \varphi( e_1 T (J u^* J) e_1) &= \varphi( e_1 T (J u^* J) (u JuJ) e_1) = \psi(Tu).
{\partial}_\xi{\partial}_x\phi^+(x,\xi)-I =\int_0^\infty{\partial}_\xi\{(\nabla_x I^L_{a\rho})\left(\tau,q(\tau,0,x,\eta(\infty,0,x,\xi))\right)\}d\tau &=\int_0^\infty (\nabla_x\nabla_x I^L_{a\rho})\left(\tau,q(\tau,0,x,\eta(\infty,0,x,\xi))\right) \nabla_\xi q\cdot \nabla_\xi\eta d\tau,
\begin{aligned}T_1 = \{(2,0,0), (1,1,0), (0,2,0), (1,0,1), (0,1,1) \}T_2 = \{(2,0,0), (1,1,0), (0,2,0), (1,0,1), (0,1,1) \}T_3 = \{(1,0,0), (0,1,0), (0,0,1), (0,0,0) \}\end{aligned}
\|S_{cw}(T_Mf)\|_{L^p(\omega)}=\bigg\|\bigg\{\sum\limits_{k\in \Bbb Z}\sum\limits_{Q'\in Q^k}|D_kT_{M}f(x_{Q'})|^2\chi_{Q'}(x)\bigg\}^{\frac12}\bigg\|_{L^p}\lesssim \bigg\|\bigg\{\sum\limits_{j\in \Bbb Z}\sum\limits_{Q\in Q^j}|q_Q f(x_{Q})|^2\chi_{Q}(x)\bigg\}^{\frac12}\bigg\|_{L^p}\lesssim \|S(f)\|_p.
\frac{d}{d\phi}\Re f_M(q_M(\phi); \theta) \begin{cases}< 0, \phi \in (0, \theta), > 0, \phi \in (\theta, \pi), \end{cases}\frac{d}{d\phi}\Re f_M(\overline{q_M(\phi)}; \theta) \begin{cases}< 0, \phi \in (0, \theta), > 0, \phi \in (\theta, \pi). \end{cases}
\sum_{i,j=1}^{m}\tau_{a_1}\tau_k\tau_i^{-1}\tau_j^{-1}q^{2m-2i-2j+2}v_{j}\otimes v_{j}\otimes v_{i}\otimes v_{i}\cdot H_{2}H_{1}= \sum_{i,j=1}^{m}\tau_{a_1}\tau_k\tau_i^{-1}\tau_j^{-1}q^{2m-2i-2j+2}v_{j}\otimes v_{j}\otimes v_{i}\otimes v_{i}\cdot H_{2}H_{3}.
V_{m}(\alpha)=\Big\{v_{m}(x,{\bf y}; \alpha) = \displaystyle \sum_{k=1}^{m}\sum_{j=1}^{N_h}{\tilde v}_{k, j}^\alpha \vartheta_j(x) \varphi_k(\psi_x({\bf y})),\ (x,{\bf y}; \alpha)\in \Omega_{1D}\times\Sigma_x\times{\mathcal P}\Big\},
\Delta \left(W^0_{n+2}(p,p_1,\dots,p_{n+1})\right) = \sum_{k=0}^n W^0_{k+2}(p,p_1,\dots,p_k,\bar{q}) \otimes W^0_{n-k+2}(\bar{q},p_{k+1},\dots,p_{n+1}) +\sum_{k=0}^n W^0_{k+2}(q,p_{1},\dots,p_{k+1})\otimes W^0_{n-k+2}(p,q,p_{k+2},\dots,p_{n+1})
\P\Bigl\{\sup_{x\in T}X(x) \ge u \Bigr\}= \sum_{j=0}^k \bigl(C'\bigr)^{j/2}\mathcal{L}_j (T) \rho_j(u)+ \mathrm{o}\bigl(\mathrm{e}^{-\alpha_0 u^2 - u^2/2}\bigr),
u(r,t)=\frac{G}{4\mu}(R^{2}-r^{2})-\frac{2GR^{2}}{\mu}\sum_{n=1}^{\infty}\frac{1}{\lambda_{n}^{3}}\frac{J_{0}(\lambda_{n}r/R)}{J_{1}(\lambda_{n})}e^{-\lambda_{n}^{2}\nu t/R^{2}},J_{0}(\lambda_{n})=0
y_{\textrm{BN}}=\left( \sqrt{P_{\textrm{N}}}x_{\textrm{N}}+\sqrt{P_{\textrm{F}}}x_{\textrm{F}}\right)\sqrt{d_{\textrm{BN}}^{-\alpha}}h_{\textrm{BN}}+n_{\textrm{N}},
Sq^2 h = Sq^2(\omega_1\omega_2+\omega_1\omega_3+\omega_2\omega_3) = \sum_{i\neq j}\: [\omega_i]^2\cdot [\omega_j] + u^2h Sq^3h = Sq^1Sq^2h = u\cdot \sum_{i\neq j}\: [\omega_i]^2\cdot [\omega_j] Sq^3(\omega_i^2) = Sq^1 (u^2\omega_i^2) = 0
\Psi(n)= \max_{\zeta\in \mathbb{R}\setminus{\overline{\mathbb{Q}}}}\Psi(n,\zeta), \qquad\widetilde{\Psi}(n)= \max_{\zeta\in \mathbb{R}\setminus{\overline{\mathbb{Q}}}}\widetilde{\Psi}(n,\zeta).
\Lambda(T) = \underset{k\to+\infty}{\lim}\ \big\| T^k \big\|_{\mathcal{B}(E)}^{\frac{1}{k}}= \underset{k \geq 1}{\inf}\ \big\| T^k \big\|_{\mathcal{B}(E)}^{\frac{1}{k}}.
\frac{d^2}{dt^2}m(v, t) = \frac{d}{dt}\sum_{i=1}^n m(v+e_i, t) &=\sum_{i=1}^n \frac{d}{dt}m(v+e_i, t) &=\sum_{i=1}^n \sum_{j=1}^n m(v+e_i+e_j, t).
\sup\limits_{z\in (0,\infty)}\int\limits_{(0,\infty)}\left( \frac{z}{y}\right)^{-\delta}z^{- \frac{2s+1}{2}}|\tilde{k}(z,y)|dy = \sup\limits_{z}z^{-1 + \frac{2s-1}{2}-\delta}\int\limits_{0}^{z}y^{\frac{1-2s}{2}+\delta}dy = const,
\mathrm{SG}_m\ni\eta:\begin{pmatrix}x^1 x^2 x^3 x^4 \end{pmatrix}\mapsto\begin{pmatrix}x^1\cos(2mq^4)-x^2\sin(2mq^4)) x^1\sin(2mq^4)+x^2\cos(2mq^4) x^3 x^4\end{pmatrix}+\begin{pmatrix}q^1 q^2 q^3-2mbq^4 q^4 \end{pmatrix}.
(J^-_0)^{1+t}J^+_{-1}(J^-_0)^{1-t}=(J^+_{-1}(J^-_0)^{1+t}-2(1+t)J^0_{-1}(J^-_0)^{t}-t(1+t)J^-_{-1}(J^-_0)^{t-1})(J^-_0)^{1-t}
P_{\pm}[(w, \eta) \cdot (w', \eta')] = P_\pm(ww'+ k\eta \eta', w \eta'+ w'\eta) &=((ww'+ k\eta \eta')^{-1}, \pm i(ww'+ k\eta \eta')^{-2}(w \eta'+ w'\eta)) &= ((ww')^{-1}- k(ww')^{-2}\eta \eta', \pm i(w^{-1}w'^{-2}\eta'+ w^{-2}w'^{-1}\eta))
K = i \int_{0}^{\infty}d T e^{- i T ( \gamma^{\mu}P_{\mu}^{\prime}- m )}\delta^{4}( P^{\prime \prime}- P^{\prime}) ={\frac{\delta^{4}( P^{\prime \prime}- P^{\prime})}{\gamma^{\mu}P_{\mu}^{\prime}- m - i 0_{+}}}.
\sum_{i\geq 0}f_i'(x, y)t^i-\sum_{i\geq 0}h_r\circ f_i'(x, y)t^{i+r}= f_0(x, y)-f_0(h_r(x), y)t^r-f_0(x, h_r(y))t^r+f_0(h_r(x), h_r(y))t^{2r}+\sum_{i\geq r}f_i(x, y)t^i-\sum_{i\geq r}f_i(h_r(x), y)t^{i+r}-\sum_{i\geq r}f_i(x, h_r(y))t^{i+r}+\sum_{i\geq r}f_i(h_r(x), h_r(y))t^{i+2r}.
\frac{\partial}{\partial t}(e^{\lambda(q,t)}-1) -1{}\stackrel{\phantom{\eqref{eq_lambda_primo}}}{=}e^{\lambda(q,t)}\frac{\partial}{\partial t}(\lambda(q,t))-1= &{}\stackrel{\eqref{eq_lambda_primo}}{=}\frac{e^{\lambda(q,t)}(1-q)}{e^{\lambda(q,t)\cdot q}-qe^{\lambda(q,t)}}-1=\frac{e^{\lambda(q,t)}-e^{\lambda(q,t)\cdot q}}{e^{\lambda(q,t)\cdot q}-qe^{\lambda(q,t)}}.
\begin{array}{r l}{{d s^{2}=}}&{{( H_{1}H_{5})^{- 1 / 2}( d x_{0}^{2}+ d x_{1}^{2}) + H_{1}^{1 / 2}H_{5}^{- 1 / 2}( d x_{2}^{2}+ . . . + d x_{5}^{2})}}\\{{\nonumber +}}&{{( H_{1}H_{5})^{1 / 2}( d x_{6}^{2}+ . . . + d x_{9}^{2})}}\\ \end{array}
G(m,N,a,\varepsilon, s) \leq 2 \sum_{k=1}^{({m(1+\epsilon) \over 2})^{{1 \over a}}}k^{-ds}\sum_{i_2\cdots i_n \in \widetilde{A}(m,n, k, a,\epsilon)}\prod_{k=2}^ni_k^{-ds},
M(\kappa)^{-1}=\sum_{j=-2}^{\beta-6}\kappa^jE_j+\mathcal O(\kappa^{\beta-5}), E_j =B_j+\sum_{\genfrac{}{}{0pt}{}{j_1\ge 0,j_2\ge -1,j_3\ge 0}{j_1+j_2+j_3=j+1}}B_{j_1}D_{j_2}B_{j_3},
D_{1}(\sum_{k=1}^{n}|\alpha_{k}|^{p})^{\frac{1}{p}}\leq \|\sum_{k=1}^{n}\alpha_{k}(x_{2k-1}-x_{2k})\| &\leq \|S\|\cdot \|\sum_{k=1}^{n}\alpha_{k}(z_{2k-1}-z_{2k})\| &\leq \|S\|\cdot C_{2}\cdot 2^{\frac{1}{q^{*}}}(\sum_{k=1}^{n}|\alpha_{k}|^{q^{*}})^{\frac{1}{q^{*}}},
d = \left( \begin{matrix}{0}&{1}&{- 1}&{0}&{0}&{1}&{- 1}&{0}\\{1}&{- 1}&{0}&{0}&{1}&{- 1}&{0}&{0}\\{- 1}&{0}&{0}&{1}&{- 1}&{0}&{0}&{1}\\{0}&{0}&{1}&{- 1}&{0}&{0}&{1}&{- 1}\\ \end{matrix}\right)
\#\sigma^{(\mathcal{I},red)}(H_{\omega,\Lambda_{L}}) \leq \#\{\{E_n\}_{n=0}^{n_1}: E_n\in\sigma(H_{\omega,\Lambda_{L_n}})~\&~|E_i-E_j|\leq 2e^{-\frac{\hat{m}}{K}L_{\max{i,j}}}\}:= \# D_0^{n_1}
\displaystyle{\inf_{(u(\cdot),y_{\epsilon}(\cdot),z_{\epsilon}(\cdot))}\liminf_{\mathcal{T}\rightarrow\infty}\frac{1}{\mathcal{T}}\int_0^{\mathcal{T}}G(u(t), y_{\epsilon}(t), z_{\epsilon}(t))dt}\stackrel{\rm def}{=}V^*(\epsilon) ,
a\nabla_{\nu}b \leqslant a\sharp_{\nu}b+(\sqrt{a}-\sqrt{b})^2-\sum_{k=0}^{\infty}r_{k}\big[\big(a^{\frac{m_k}{2^k}}b^{1-\frac{m_k}{2^k}}\big)^{\frac{1}{2}}-\big(a^{\frac{m_k+1}{2^k}}b^{1-\frac{m_k+1}{2^k}}\big)^{\frac{1}{2}}\big]^{2}.
\int_{\alpha_0}^\pi F(\eta) \, \sin^{d-2}\eta \, d\eta = - \frac{q\, \Gamma((d-1)/2)}{\sqrt{2}\, \pi^{(d+1)/2}}\, \sqrt{1-\cos\alpha_0}\, \int_{\alpha_0}^\pi \frac{(1+\cos\eta)^{(d-3)/2}\, \sin\eta \, d\eta}{(1-\cos\eta) \, \sqrt{\cos\alpha_0 - \cos\eta}}.
2^{k}=\sum_{n=0}^{\infty}\sum_{j=0}^{k}\left( \begin{array}{c}k j\end{array}\right) \lambda^{2j}\left( \lambda -1\right)^{k-j}Y_{n}^{\left( k\right)}\left( \lambda \right) \frac{t^{n+j}}{n!}
\Phi(r) = \Psi\left( \varphi \left( \frac{a + b\mathrm{i}}c\right)\right)= \Psi\left(\frac ac - \frac bc, \frac ac + \frac bc -1, 0\right) &=\frac{((a - b)c, (a + b - c)c, a^2 + b^2 - (a+b)c)}{a^2 + b^2 + c^2 - (a + b)c}&=\frac{(a - b, a + b - |\beta|^2, |\alpha|^2 - a-b)}{|\alpha|^2 + |\beta|^2 - a - b}.