harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{aligned}\Delta r_{O2}\leq \max_\mathcal{P}\ln \Big(1+\sum_{i=1}^K \frac{P_i|\hat{h_i}|^2}{N_0}\Big) \mathrm{s.t.}\sum_{i=1}^KP_i\leq P-P_{O2}^* .\end{aligned}
\left[\mbox{Prod}\left(\mbox{Prod}\left(\mathbf{Q},\mathbf{D},\mathbf{D}^{\top}\right),\mbox{Prod}\left(\mathbf{Q},\mathbf{D},\mathbf{D}^{\top}\right)^{\top^{2}},\mbox{Prod}\left(\mathbf{Q},\mathbf{D},\mathbf{D}^{\top}\right)^{\top}\right)\right]_{i,j,k}=0.
D\eta = D\alpha + \frac{1}{2}D\delta [\alpha , \alpha] &= -\frac{1}{2}[\alpha , \alpha] + \frac{1}{2}( 1-H-\delta D)[\alpha , \alpha] &= -\frac{1}{2}H[\alpha,\alpha] - \delta[ D\alpha , \alpha] &= -\frac{1}{2}H[\alpha , \alpha].
\^{*}( d x^{\mu_{1}}\wedge . . . \wedge d x^{\mu_{p}}) = \frac{\sqrt{| g |}}{( n - p ) !}\epsilon_{\mu_{p + 1}. . . \mu_{n}}^{\mu_{1}. . . \mu_{p}}d x^{\mu_{p + 1}}\wedge . . . \wedge d x^{\mu_{n}}
\partial_{\alpha}(h_{\alpha}\circ F_{\alpha,(\underline{n},n_0)}^{-1})=(\partial_{\alpha}h_{\alpha}) \circ F_{\alpha,(\underline{n},n_0)}^{-1}+ h'_{\alpha}\circ F_{\alpha,(\underline{n},n_0)}^{-1}\partial_{\alpha}F_{\alpha,(\underline{n},n_0)}^{-1}
t_b t_c t_a t_b \cdot t_{\beta}t_{\gamma}t_{\alpha}t_{\beta}(a) = t_b t_c t_a t_b (a) = c, t_b t_c t_a t_b \cdot t_{\beta}t_{\gamma}t_{\alpha}t_{\beta}(\alpha) = t_b t_c t_a t_b (\gamma) = \gamma
R^{j}(\mathbf{n}_{1},\mathbf{n}_{2})\ =\int_{S^{2}}\mathbb{U}_{\vec{c},{\vec{c}}^{\prime}}^{j}(\mathbf{n}_{1},\mathbf{n})\mathbb{U}_{\vec{c}^{\prime},{\vec{c}}}^{j}(\mathbf{n},\mathbf{n}_{2})d\mathbf{n}\ \text{, cf.(\ref{Rc})-(\ref{leg1}).}
[{\hat{S}}_{\mu},{\hat{Y}}_{m}^{J}] = \sqrt{J ( J + 1 )}C_{m \mu ( m + \mu )}^{J 1 J}{\hat{Y}}_{m + \mu}^{J}, \quad ( \mu = \pm , 0 , \;{\hat{S}}_{0}\equiv{\hat{S}}_{3}) .
-2b^\mu a + 2c^\mu-b^\mu\alpha^0_{\,00}-b^\mu\alpha^0_{\,0 \tau}b^\tau-b^\mu \alpha^0_{\,\sigma 0}b^\sigma + \alpha^\mu_{\,0\tau}b^\tau + \alpha^\mu_{\,\sigma 0}b^\sigma + \alpha^\mu_{\,\sigma\tau}b^\sigma b^\tau =0.
\log|g'_n(z)| = \sum_{j=1}^{n}\log|(g^j)'(z_{j-1})| & = \sum_{j=0}^{n-1}\log\left|1-\frac{h}{(z_{j}- U_{j})^2}+ O(h\delta)\right| & = -\sum_{j=0}^{n-1}\left(\Re \frac{h}{(z_j-U_j)^2}+ O(h\delta)\right).
\left( J_{euc}F|_x \right)_{ij}= - \frac{\partial \lambda}{\partial x_j}(x) x_i - \lambda(x) \delta_{ij}+ \frac{\partial f_i}{\partial x_j}(x) \left( g^{-1}\cdot J_{euc}F |_{gx}\cdot g \right)_{ij}= - (g^T \nabla \lambda(gx))_j x_i - \lambda(gx) \delta_{ij}- \left( g^T \cdot J_{euc}f |_{gx}\cdot g \right)_{ij},
\Omega^{C_2}_{+2n\alpha}\cong \frac{MU_{*-1}[u]}{\begin{pmatrix}u^n \; , \; \sum_{\ell = 0}^{n-1}c_{i,j + \ell}u^\ell \; , \; \sum_{\ell = 0}^{n-1}p_{j + \ell}u^\ell \end{pmatrix}}
\sup_{t<T_{\rm max}}\int_{\Omega}(n\log n(\cdot,t)+e^{-1}) \le \sum_{i=1}^{N}\sup_{t<T_{\rm max}}\int_{\Omega\cap B_{\frac{\delta}{2}}(q_{i})}(n\log n(\cdot,t)+e^{-1})\le C.
\left[ \langle A_{m_0}\rangle \Phi^2 \vert_F ~+~\langle A_{g_0}\rangle\Phi^3\vert_F ~+~ h.c. \right]{}~+~ 2Re( \langle F_{m_0}\rangle A^2 ~+~ \langle F_{g_0}\rangle A^3 )
\mbox{if $k > 0$}, \widetilde{f}(k) = - \frac{\alpha k}{\sqrt{2\pi}}\int f(x) \psi_+(x,0)\,dx + O(k^2) & \mbox{if $k < 0$}, \widetilde{f}(k) =\frac{\alpha k}{\sqrt{2\pi}}\int f(x) \psi_-(x,0)\,dx + O(k^2),
U_{2^{p_0}+\cdots+2^{p_s}}=1+2^{p_0-p_1}\left[(2^{2+p_1-p_0}-2)U_{2^{p_1}+\cdots+2^{p_s}}-4U_{2^{p_2}+\cdots+2^{p_s}}-2\right].
\tilde{\Lambda}_{S U}^{2 N_{c}^{\prime}- 2 N_{f}}= ( \mu^{- 1}a^{2})^{2 N_{f}- 2 N_{c}}\tilde{\Lambda}^{\prime 4 ( N_{c}^{\prime}- 1 ) - 2 ( N_{f}+ \tilde{N}_{c})}.
P_3(n)R_{n+3}(t)+P_2(n)R_{n+2}(t)+P_1(n)R_{n+1}(t)+P_0(n)R_n(t) =S_n(t+1)-S_n(t), P_3(n)\hat R_{n+3}(t)+P_2(n)\hat R_{n+2}(t)+P_1(n)\hat R_{n+1}(t)+P_0(n)\hat R_n(t) =\hat S_n(t+1)-\hat S_n(t).
\int_{t_n}^{t_{n+1}}\int_{t_n}^{t}e^{-\frac{t-s}{\epsilon^2}}d\beta(s)dt =\int_{t_n}^{t_{n+1}}\int_{s}^{t_{n+1}}e^{-\frac{t-s}{\epsilon^2}}dtd\beta(s) &=\int_{t_n}^{t_{n+1}}\epsilon^2(1-e^{-\frac{t_{n+1}-s}{\epsilon^2}})d\beta(s) &=\epsilon^2(\beta(t_{n+1})-\beta(t_n))-\epsilon^2\int_{t_n}^{t_{n+1}}e^{-\frac{t_{n+1}-s}{\epsilon^2}}d\beta(s).
\Phi(r)=\frac{q}{4\pi\epsilon r}\sum_{l=0}^{\infty}(\frac{r^{\prime}}{r})^{l}(\frac{4\pi}{2l+1})\sum_{m=-l}^{l}Y_{lm}(\theta,\phi)Y_{lm}^{*}(\theta^{\prime},\phi^{\prime})
E_{(a_1,a_2,...,a_n),b}(z_1,z_2,...,z_n)=\sum\limits_{k=0}^\infty \sum\limits_{\begin{array}{l}l_1+l_2+...+l_n=k l_1\geq0,...,l_n\geq 0 \end{array}}\frac{k!}{l_1!...l_n!}\frac{\prod\limits_{i=1}^n z_i^{l_i}}{\Gamma(b+\sum\limits_{i=1}^n a_i l_i)}
(x-\beta-N) Q_n(x) - (x+\alpha+1) Q_n(x+1) =- \frac{(N-n)(n+\alpha+\beta+N+1)}{N}\hat Q_{n}(x), (x+1) \hat Q_{n}(x) -(x-N+1) \hat Q_{n}(x+1) = N Q_n(x+1).
\|(fd\sigma)^\vee\|_{L^r(\mathbb F_q^d, d{\bf m})}\le \sum_{j=1}^N a_j \,\|(F_jd\sigma)^\vee\|_{L^r(\mathbb F_q^d, d{\bf m})}\lesssim \sum_{j=1}^N a_j \,\|F_j\|_{L^p(S_t, d\sigma)}\sim \int_{0}^\infty s^{\frac{1}{p}-1}f^*(s)~ ds =\|f\|_{L^{p,1}(S_t, d\sigma)}.
1-\frac{(\begin{matrix}n\\k+1\end{matrix})(\begin{matrix}N-n\\K-k-1\end{matrix})}{(\begin{matrix}N\\K\end{matrix})}_{3}F_{2}[\begin{matrix}1,k+1-K,k+1-n\\k+2,N+k+2-K-n\end{matrix};1],
[e_1, e_1]=\alpha_1e_5+\alpha_2e_4+\alpha_3e_5, [e_1, e_2]=\alpha_4e_3+ \alpha_5e_4+ \alpha_6e_5, [e_2, e_1]=\beta_1e_3+\beta_2e_4+ \beta_3e_5, [e_2, e_2]=\beta_4e_3+\beta_5e_4+\beta_6e_5, [e_1, e_3]=\gamma_1e_4+\gamma_2e_5, [e_2, e_3]=\gamma_3e_4+\gamma_4e_5, [e_1, e_4]=\gamma_5e_5, [e_2, e_4]=\gamma_6e_5.
\Psi^{n,6}_s =2\left( \int^s_{\eta_n(s)}\left(s-\eta_n(u)\right)^{\alpha}\sigma(X_{\eta_n(u)})\, dW_u \right) \left(\int^{\eta_n(s)}_0 \psi_{n,2}(u,s)\, \sigma(X_{\eta_n(u)})\, dW_u \right).
\frac{w}{\mathbf{l}(\mathbf{z})}=\frac{u+iv}{r(\cos \theta +i\sin\theta )}=\frac{(u+iv)(\cos \theta -i\sin\theta )}{r}=\frac{u\cos \theta +v\sin \theta}{r}+i\frac{v\cos \theta -u\sin \theta}{r},
dz=\frac{1}{2\pi}\sum_iN_i\frac{d}{dS_i}Gdx=-\frac{1}{2\pi i}\sum_iN_i\frac{d}{dS_i}\Big[R\Big(x+i\frac{m}{2}\Big)-R\Big(x-i\frac{m}{2}\Big)\Big] \ ,
(M'(t))^2 =-(M'(0))^2+2M'(t)M'(0)+\left(\int_0^t\int_{\Omega}uu_tdxd\tau\right)^2 &=-\frac14||u_0||^2+M'(t)||u_0||^2+\left(\int_0^t\int_{\Omega}uu_tdxd\tau\right)^2 &\leq M'(t)||u_0||^2+\left(\int_0^t\int_{\Omega}uu_tdxd\tau\right)^2 .
E_{eff(3+1)}=-\frac{B_{m}}{2\pi^{2}}\int_{0}^{+\infty}y\ln\left(\frac{y^{2}+m^{2}_{f}/B_{m}}{m^{2}_{f}/B_{m}}\right)\left(G(y,B_{m}d^{2})-c\right) dy
s_{0,m}= \frac{1}{4}\int_0^{\Delta t}\left(1-2\left(1- \frac{|\tau|}{\Delta t}\right)^2\right) \frac{\partial^3u_0}{\partial t^3}(t_m+\tau) d\tau-\frac{1}{4}\int^0_{-\Delta t}\left(1-2\left(1- \frac{|\tau|}{\Delta t}\right)^2\right) \frac{\partial^3u_0}{\partial t^3}(t_m+\tau) d\tau
\mathbf{F}\cdot\frac{\partial\mathbf{G}^{\ast}}{\partial t}+\mathbf{F}^{\ast}\cdot\frac{\partial\mathbf{G}}{\partial t}+4\pi\mathbf{j}\cdot\left(\mathbf{F}+\mathbf{F}^{\ast}\right) =\frac{c}{i}\operatorname{div}\left(\mathbf{F}\times\mathbf{F}^{\ast}\right)
\tilde{f}=\left(\begin{array}{c}f_{i_{1}}\vdots f_{i_{k}}\end{array}\right): M \to \bigoplus_{s=1}^{k}N(i_{k}), \tilde{g}=(g_{-i_{1}}, \cdots, g_{-i_{k}}): \bigoplus_{s=1}^{k}N(i_{k}) \to M,
a = a^{p^n}\mbox{\ \and}m = p^na^{p^n-1}m + \sum_{i=1}^{p^n}a^{i-1}\xi(a\otimes a^{p^n-i}) = \sum_{i=1}^{p^n}a^{i-1}\xi(a\otimes a^{p^n-i}),
\mathcal{B}_r(\theta^*) = \left\lbrace \theta \in \Theta \left| \frac{1}{n}\sum_{i=1}^{n}D_{KL}\left(\ell^i\left(\cdot|\theta^*\right) ,\ell^i\left(\cdot|\theta \right) \right) \right. \leq r \right\rbrace
\int_{\Omega}|\chi_{0\varepsilon}|^2 + \varepsilon \int_{\Omega}|\nabla \chi_{0\varepsilon}|^2 = \int_{\Omega}\chi_0 \chi_{0\varepsilon}\leq \frac{1}{2}\int_{\Omega}|\chi_0|^2 + \frac{1}{2}\int_{\Omega}| \chi_{0\varepsilon}|^2.
\begin{cases}\left(\frac{1}{\alpha}\right) \frac{1}{2}\leq \beta_\alpha \leq \left\lceil{\frac{1}{\alpha}}\right\rceil \frac{3}{4},0<\alpha \leq 1 \left(\frac{1}{\alpha}\right) \frac{1}{2}\leq \beta_\alpha \leq \frac{1}{\left\lfloor\alpha\right\rfloor}\frac{3}{4},\alpha>1.\end{cases}
\rho_{T,\Phi}:=\frac{1}{T^{k/2}}\sum_{j_1 \neq \ldots \neq j_k}\sigma_{j_1}\ldots \sigma_{j_k}\int_{[0,T]^k}\Phi(x^{j_1}+\xi^{j_1}_{s_1},\ldots,x^{j_k}+\xi^{j_k}_{s_k})ds_1 \ldots ds_k,
\frac{d \psi_{1}}{d r^{\prime}}= - i \cos \phi \psi_{1}+ \frac{1}{2}( - i )^{M + \beta}e^{- i ( N - M ) \phi}\left[ J_{M + \beta - 1}( r^{\prime}) + i e^{- i \phi}J_{M + \beta}( r^{\prime}) \right] ,
\begin{aligned}\mathcal{N}(t, s,\xi,v,\overline v) = e^{(t- s) \mathcal{L}}f \left(e^{s\mathcal{L}}v, e^{s \mathcal{L}}e^{\xi \mathcal{A}}\overline v \right) =e^{(t- s) \mathcal{L}}\mathcal{B}\left( F(e^{s\mathcal{L}}v) \cdot G\left( e^{s \mathcal{L}}e^{\xi \mathcal{A}}\overline v \right)\right).\end{aligned}
(m_1+\theta_1+1)(m_2+\theta_2+1) + j ( \theta_1 q_1 + \theta_2 q_2 ) & = (m_1+\theta_1+1 + j q_2 )(m_2+\theta_2+1 + jq_1 ) - jq_1(m_1+1) - jq_2(m_2+1) - j^2q_1q_2.
G_r ( z | \tau_0, \ldots, \tau_r) = \prod_{n_0,\ldots,n_r = 0}^{\infty}\left (1-e^{2\pi i (- z + (n_0+1)\tau_0 + \ldots +(n_r+1)\tau_r )}\right ) \left ( 1 - e^{2\pi i (z+n_0\tau_0 + \ldots + n_r \tau_r )}\right )^{(-1)^r},
|\tilde{A}|^2_{\tilde{g}}= \sum_{i,j=1}^n(\tilde{A}(\tilde{e}_i,\tilde{e}_j))^2 = e^{\frac{2f}n}\left(\sum_{i,j=1}^n(A(e_i,e_j))^2+\frac 1ng(\overline{\nabla}f,\nu)^2-\frac 2nHg(\overline{\nabla}f,\nu)\right) = e^{\frac{2f}n}\left(|A|_g^2-\frac 1nH^2\right),
\sum\limits_{k = 1}^n{\int\limits_0^\infty{{x^{s - 1}}\sum\limits_{v = 0}^\infty{\frac{{\phi \left( v \right){{\left({- kx}\right)}^v}}}{{v!}}}dx}= \phi \left({- s}\right)}\Gamma \left( s \right)\left({\zeta \left( s \right) + \frac{{{{\left({n + \frac{1}{2}}\right)}^{1 - s}}}}{{1 - s}}+{\rm O}\left({{n^{- s - 1}}}\right)}\right).
x_1^{af_1+g_1}\cdots x_{n+1}^{af_{n+1}+ g_{n+1}}=F_1x_1^{ad_1+e_1}+ \ldots + F_{n+1}x_{n+1}^{ad_{n+1}+e_{n+1}}+ F_{n+2}(x_1^a + \ldots x_{n+1}^a)
\mathcal{S}(h,k):=\sum_{q=1}^{\infty}W\left(\frac{q}{Q}\right) \sideset{}{^\flat}\sum_{\chi \bmod q}\chi(h)\overline{\chi}(k) \sum_{m=1}^{\infty}\frac{\tau_A(m)\chi(m)}{\sqrt{m}}V\left(\frac{m}{X}\right) \sum_{n=1}^{\infty}\frac{\tau_B(n)\overline{\chi}(n)}{\sqrt{n}}V\left(\frac{n}{X}\right),
S_{\mathrm{M}}= \frac{1}{1 6 \pi G_{1 1}}\left( \int d^{1 1}x \sqrt{| g_{1 1}|}\left[ R_{1 1}- \frac{1}{4 8}F_{4}^{2}\right] + \frac{1}{6}\int A_{3}\wedge F_{4}\wedge F_{4}\right)
\Omega(s):= (n+1-q)\Gamma(s)\eta_q(s)-2 \Gamma(s-1)\eta_q(s-1)-\sum_{j=0}^{2n-1}\frac{(n+1-q)^j}{j!}\Bigl[(n+1-q)\Gamma(s+j)\xi_q(s+j) &-2\Gamma(s+j-1)\xi_q(s+j-1) \Bigr]+2(n+1-q)^{2n+1}\Gamma(s+2n-2)\xi_q(s+2n-2) &-4(n+1-q)^{2n}\Gamma(s+2n-3)\xi_q(s+2n-3).
( \mathcal{F}\varphi_{l}) (\xi) = i^{l}(2 \pi)^{\frac{m}{2}}\Big ( e^{i ( \lfloor \frac{m}{2}\rfloor - \frac{m}{2}) \xi}\Big ( \frac{\sin(\frac{\xi}{2})}{\frac{\xi}{2}}\Big )^{m}\Big )^{(l)}.
K = \left( \begin{array}{l c c c c}{{}}&{{u_{1}}}&{{u_{2}}}&{{u_{3}}}&{{u_{4}}}\\{{m_{1}\,}}&{{1}}&{{0}}&{{0}}&{{0}}\\{{m_{2}\,}}&{{0}}&{{1}}&{{0}}&{{0}}\\{{m_{3}\,}}&{{0}}&{{0}}&{{1}}&{{0}}\\{{m_{4}\,}}&{{0}}&{{0}}&{{0}}&{{1}}\\{{m_{5}\,}}&{{1}}&{{1}}&{{- 1}}&{{0}}\\{{m_{6}\,}}&{{1}}&{{1}}&{{0}}&{{- 1}}\\ \end{array}\right) .
\phi ( r , t )_{\mathrm{i n}}\approx{\frac{{\cal N}_{\mathrm{i n}}}{\sqrt{2 \omega}r_{H}}}\exp \left[ \mp i \omega \left\{t +{\frac{( r - r_{H}) [ \omega^{2}- c^{2}k_{\perp}^{2}]}{2 c_{H}\omega}}\right\}\right] .
\frac{8 \pi^{2}}{e^{2}}\left( \sum_{A}\left( \frac{1}{y_{A}}d \vec{y}_{A}^{2}+ y_{A}\, ( D \psi_{A})^{2}\right) + \frac{\left( \, \sum_{A}d \vec{y}_{A}\, \right)^{2}}{| \vec{y}_{N}+ 2 \pi \vec{\zeta}|}- \frac{\left( \, \sum_{A}y_{A}\, D \psi_{A}\, \right)^{2}}{| \vec{y}_{N}+ 2 \pi \vec{\zeta}| + \sum_{A}| \vec{y}_{A}|}\right) ,
\frac{d_{+}}{d_{-}}=\frac{\Gamma(-\frac{2 j + 1}{k+2})}{\Gamma(\frac{2 j + 1}{k+2})}\;\sqrt{\frac{\Gamma(\frac{2 j}{k+2})\; \Gamma(\frac{2 j + 2}{k+2})}{\Gamma(-\frac{2 j}{k+2})\; \Gamma(-\frac{2 j + 2}{k+2})}}\; = \;\frac{\gamma_{-}}{\gamma_{+}},
\begin{array}{ll}{\cal CQ}f(b,\xi)=\inf \left\{\displaystyle{\frac{1}{|D|}\int_{D}f(b+\eta(x),\xi+\nabla\varphi(x))\,dx:}\right.\displaystyle{\eta\in L^{\infty}\left( D;\mathbb{R}^{m}\right)}, \displaystyle{\int_{D}\eta(x) dx=0,}&\left. \varphi\in W_{0}^{1,\infty}(D;\mathbb{R}^{n}) \right\}.\end{array}
\Omega_{\pm}= \frac 1{\sqrt{2}}\left( \begin{array}{c c}{\exp \pm \frac{i \phi}2}&{- \exp \mp \frac{i \phi}2}\\{\exp \pm \frac{i \phi}2}&{\hphantom{-}\exp \mp \frac{i \phi}2}\\ \end{array}\right) ; \qquad \Omega_{\pm}^{- 1}= \Omega_{\pm}^{\dagger};
{}^{\Psi}F_{p}^{(\alpha,\beta)}(a,b;c;z)=\frac{1}{B(b,c-b)}\int_{0}^{\infty}u^{b-1}(1+u)^{a-c}\left[ 1+u(1-z)\right]^{-a}{}_{0}\Psi_{1}\left( \alpha,\beta;-2p-p\left( u+\frac{1}{u}\right) \right)du.
[y_1, y_2]=y_3, [y_2, y_1]=\frac{\beta_3}{\alpha_4\gamma_1\gamma_5}y_5, [y_2, y_2]=y_4+\theta y_5, [y_1, y_3]=y_4, [y_2, y_3]=y_5, [y_1, y_4]=y_5.
(CA^k )_{ij}= \sum\limits_{r = 1}^n{C_{ir}(A^k )_{rj}}= \sum\limits_{r = 1}^n{(1 - \delta_{ir})(A^k )_{rj}}&= \sum\limits_{r = 1}^n{(A^k )_{rj}}- \sum\limits_{r = 1}^n{\delta_{ir}(A^k )_{rj}}&= \sum\limits_{r = 1}^n{(A^k )_{rj}}- (A^k )_{ij}&= 1 - (A^k )_{ij}\,.
\left. \frac{d C_{V}}{d T}\right|_{T_{c}^{+}}- \left. \frac{d C_{V}}{d T}\right|_{T_{c}^{-}}= \left. \Omega T_{c}^{+}\mu^{\prime \prime}( T_{c}^{)}\frac{\partial \varrho ( T , \mu )}{\partial T}\right|_{T = T_{c}^{+}}
(b\theta)(a_0\otimes\cdots a_p) = \theta(a_0a_1\otimes a_2\otimes\cdots \otimes a_p)+\sum_{k=1}^{p-1}(-1)^k \theta(a_0\otimes\cdots \otimes a_ka_{k+1}\otimes\cdots\otimes a_p) + (-1)^p\theta(a_pa_0\otimes a_1\otimes\cdots\otimes a_{p-1}).
A_{v}=(\begin{matrix}-\frac{i}{4\lambda}cos\varphi&-\frac{1}{4\lambda}sin\varphi\\ \frac{1}{4\lambda}sin\varphi&\frac{i}{4\lambda}cos\varphi\end{matrix})=-\frac{1}{4\lambda}isin\varphi\sigma_{2}-\frac{1}{4\lambda}icos\varphi\sigma_{3},
F_{\vec{x}}(-z_{1}/s, \ldots , -z_{k}/s)=\frac{(-s)^{\sum_{i=1}^{k}x_{i}}}{\prod_{i=1}^{k}z_{i}(1+z_{i})^{M+1}}\langle \prod_{1\le i \le k}C^{[0, M]}(u_{i}; s) \prod_{1\le i \le k}\beta_{x_{i}}^{*}\rangle_{[0, M]}.
N_{(1,3)}(n) = 8\sigma(n) - 32\sigma(\frac{n}{4}) + 8\sigma(\frac{n}{3}) -32\sigma(\frac{n}{12}) + 64\, W_{(1,3)}(n) + 1024\, W_{(1,3)}(\frac{n}{4}) - 256\, \biggl( W_{(3,4)}(n) + W_{(1,12)}(n) \biggr),
\frac{1}{sl(\varpi x/\pi)}=\frac{\pi}{\varpi}\sum_{n\in\mathbb{Z}}\frac{(-1)^{n}}{sinh(x-n\pi)}=\frac{\pi}{\varpi}\sum_{n\in\mathbb{Z}}\frac{(-1)^{n}}{sin(x-n\pi i)},x\in\mathbb{C}
\begin{aligned}f \ge m_3 m_7 + m_2 m_7 (m_4 + m_5) + m_1 (m_3 + m_4 + m_5) &\ge m_3 (p-1) + m_2 (p-1)^2 + m_1 (m_3 + m_4 + m_5).\end{aligned}
\rho_h([x_m,\dots,x_n],[x_m,\dots,x_{n+1}]) =\rho_h((x_m+[x_{m+1},\dots,x_n])^*,(x_m+[x_{m+1},\dots,x_{n+1}])^*) \leq C_{b/a}^2 a^{-2}\rho_h([x_{m+1},\dots,x_n],[x_{m+1},\dots,x_{n+1}])\,.
\widehat K=\inf_{t>1}\frac{e^{\log^2 t}}{(t-1)^\alpha}=\inf_{s>0}\frac{e^{s^2}}{(e^s-1)^\alpha}\geq \inf_{s>0}\frac{e^{s^2}}{e^{\alpha s}}=\inf_{s>0}e^{s^2-\alpha s}
0 = \lambda(y\wedge v\wedge w^n)=\lambda(y)\wedge \eta\wedge w^n-y\wedge\lambda(v)\wedge w^n +y\wedge v\wedge \lambda(\omega^n)= = n\,y\wedge v\wedge\lambda(w)\wedge w^{n-1}=n\,y\wedge v\wedge z\wedge w^{n-1}\, .
uf_3(x) + 2uf_4(x) = (u+2u) + (uh_1^{\prime}+2uh_1^{\prime\prime})x + \cdots + (uh_{l-1}^{\prime}+2uh_{l-1}^{\prime\prime})x^{l-1}+ (uh_l^{\prime}+2u)x^l + uh_{l+1}^{\prime}x^{l+1}+ \cdots + uh_{k-1}^{\prime}x^{k-1}+ ux^k.
\vert \{\lambda \}, \{\rho \}, \chi \rangle{}\equiv{}( \alpha_{-{n_{1}}}^{0})^{\lambda_{1}}. . . ( \alpha_{-{n_{p}}}^{0})^{\lambda_{p}}( \epsilon_{- r_{1}}^{( \pm ) 0})^{\rho_{1}}. . . ( \epsilon_{- r_{s}}^{( \pm ) 0})^{\rho_{s}}\vert \chi ; q \rangle{},
\sum_{i=1}^{\infty}\frac{(-1)^{i+1}(4)}{2i-1}=\frac{4}{1}-\frac{4}{3}+\frac{4}{5}-\frac{4}{7}+\frac{4}{9}-\frac{4}{11}+\frac{4}{13}-\cdot\cdot\cdot=\pi
\frac{_{q}\mathcal{U}_{-}(\xi_{a}^{(h+1/2)})}{(\,( \xi_{a}^{(h+3/2)})^{2}-1/( \xi_{a}^{(h+3/2)})^{2}\,)}=\mathsf{D}_{-}(\xi_{a}^{(h+1)})\mathsf{D}_{-}(1/\xi_{a}^{(h)})=\mathsf{A}_{-}(\xi_{a}^{(h+1)})\mathsf{A}_{-}(1/\xi_{a}^{(h)}),
{\mathbb P}\Big( \Big |\sum_{k=1}^n \eta_k \Big | > \frac x 2\Big) \leq \frac{4}{x^2}\sum_{k=1}^n \|\eta_k\|_2^2 \leq \frac{4}{x^2}\sum_{k=1}^n \|\xi_k\|_2^2
\int_\Omega v^{\bar q}\, dx=\int_{\Omega}\frac{|u|^{\bar q(p^*_\alpha-1)}}{|x|^{\frac{\alpha}{q}}}\frac{1}{|x|^{\alpha(\bar q-\frac{1}{q})}}\, dx\leq \left(\int_{\Omega}\frac{u^{\bar q(p^*_\alpha-1)q}}{|x|^\alpha}\, dx\right)^{\frac{1}{q}}\left(\int_{\Omega}\frac{1}{|x|^{\alpha(\bar q-\frac{1}{q})q'}}\, dx\right)^{1-\frac{1}{q}}.
\|\nabla_{x}\widetilde{u}\|_{L^{\infty}}\leq C\|\nabla^2_{x}\widetilde{u}\|^{\frac{1}{2}}\|\nabla^3_{x}\widetilde{u}\|^{\frac{1}{2}}\leq C\varepsilon^{\frac{1}{2}-\frac{1}{2}a}\varepsilon^{-\frac{1}{2}a}=C\varepsilon^{\frac{1}{2}-a}.
\begin{cases}0=\partial_{i}\left(A_{ij}(x)\partial_{j}u\right) +B_{i}(x)\partial_{i}u \Omega^h, begin{align*}4pt]u(x_0,x') = 0, x_0 \geq h, begin{align*}4pt]u(x)=f(x) \,x\in\partial\Omega, begin{align*}4pt]\tilde{N}_{2,a}(u) \in L^{p}(\partial \Omega), \end{cases}
b_0(t,Y)-\Pi(t)\sigma_0^2(t,Y)- \int_{\mathbb{R}}\frac{\Pi(t) \gamma_0^2(t,Y,\zeta)}{1+\Pi(t)\gamma_0(t,Y,\zeta)}\nu(d\zeta) &+\sigma_0(t,y)\Phi(t,Y)+ \int_{\mathbb{R}}\frac{\gamma_0(t,Y,\zeta)}{1+\Pi(t)\gamma_0(t,Y,\zeta)}\Psi(t,Y,\zeta)\nu(d\zeta)= 0,
\sum_{n=1}^{\infty}(-1)^{n-1}\frac{\varphi^*(n)}{n^s}=\left(1-\frac1{2^{s-2}}+\frac1{2^{2s-1}}\right)\left(1-\frac1{2^{s-1}}+\frac1{2^{2s-1}}\right)^{-1}\sum_{n=1}^{\infty}\frac{\varphi^*(n)}{n^s}(\Re s>2),
\tilde{\Gamma}_{t}^{( 4 )}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4}) = \tilde{\Gamma}_{t , s = 0}^{( 4 )}({\bf{q}}_{1},{\bf{q}}_{2},{\bf{q}}_{3},{\bf{q}}_{4}) \; ,
\frac1{np+1}= \frac1p\, \frac1{\frac1p+n}= \frac1p\, \frac{\Gamma\left(\frac1p+n\right)}{\Gamma\left(\frac1p+1+n\right)}= \frac{\left(\frac1p,n\right)}{\left(\frac1p+1,n\right)},
\lVert AB-A_0B_0 \rVert^2 = \sum_{j=1}^N \left( \sum_{i=1}^{M-1}\{(a_{i1}-a_{i2})b_j +a_{i2}-a^0_i\}^2 \right. &\left. + \left[ \sum_{i=1}^{M-1}\{(a_{i1}-a_{i2})b_j +a_{i2}-a^0_i\}\right]^2 \right).
\left[ - \left( 1 - \frac{2 M}{r}\right) \frac{d^{2}}{d r^{2}}- \frac{2 ( r - 4 M ) ( 2 r - 3 M )}{r^{2}( r - 3 M )}\frac{d}{d r}+ \frac{8 M}{r^{2}( r - 3 M )}\right] H_{1}( r ) = \lambda H_{1}( r ) .
\sup_{s\in[0,1]}|h'(r^* + s \zeta)| = \sup_{x\in[0,\zeta]}|h'(r^* + x)| \leq \sup_{x\in[0,1]}|h'(r^* + x)| = \|h'(r^*+\cdot)\|_{L^{\infty}([0,1])}
\left[L_j^{ch},L_k^{ch}\right]=(k-j)L_{j+k}^{ch}, \qquad \left[\overline L_j^{ch},\overline L_k^{ch}\right]=(k-j)\overline L_{j+k}^{ch},\qquad\left[L_j^{ch},\overline L_k^{ch}\right]=0.
R_{\textrm{GA}}^* ={\mathrm{argmax}\atop{\scriptstyle{R \geq 0}}}~ \sum_{i=1}^{N_{max}}\frac{R}{i\cdot\pi}\cdot &\int_{0}^{\infty}\bigg[ \mathcal{I}m \Big\{\Psi_{GA}(t;i, R) - \Psi_{GA}(t;i-1,R) \Big\}\bigg] dt.
h(ab)f_4(a,b) =f_3(ab,b^{-1}a^{-1})f_4(a,b)f_3(b,b^{-1})^{-1}f_3(b,b^{-1}) &=f_4(a,a^{-1})f_3(b,b^{-1}a^{-1})f_3(e_\lambda,b)f_3(b,b^{-1}) &=f_4(a,a^{-1})f_3(e_\lambda,a^{-1})f_3(b,b^{-1}) &=f_1(e_\lambda,a^{-1})f_3(b,b^{-1}) &=g_4(a,b)h(b),
\tilde{T}_{\cdot \, \nu}^{\mu \, \cdot}= \frac{i}{2}\left[ \overline{{{\psi}}}\gamma^{\hat{\alpha}}e_{\hat{\alpha}}^{\mu}\partial_{\nu}\psi - ( \overline{{{\partial_{\nu}\psi}}}) \gamma^{\hat{\alpha}}e_{\hat{\alpha}}^{\mu}\psi \right]
\exp (\mbox{e}^t-1)=\exp (\mbox{e}^{x+t}-\mbox{e}^x)|_{x=0}=\sum_{n=0}^{\infty}\frac{F_n(1,1,\cdots,1)}{n!}t^n \equiv \sum_{n=0}^{\infty}\frac{B(n)}{n!}t^n.
f(\ell) \phi(\ell) - f(\ell+1)\phi(\ell+1)\, =\, \phi(\ell) [f(\ell)-f(\ell+1)] + [\phi(\ell)-\phi(\ell+1)] f(\ell+1) =\, \phi(\ell) [f(\ell)-f(\ell+1)] + \frac{1}{L-1}\, f(\ell+1)\, .
j_{\mu}^{3}= \frac{\frac{\strut \displaystyle M}{\strut \displaystyle \cos \theta_{W}}}{1 + \strut \displaystyle \Bigl ( \frac{\strut \displaystyle M}{\strut \displaystyle M^{\prime}}\Bigr )^{2}\tan^{2}\theta_{W}}\, Z_{\mu}
\left\langle y_{n},x_{1}+cx_{2}\right\rangle =\left\langle y_{n},x_{1}\right\rangle +c\left\langle y_{n},x_{2}\right\rangle & =\left( u_{n}\left( x_{1}\right) +cu_{n}\left( x_{2}\right) \right)+\left( z_{n}\left( x_{1}\right) +cz_{n}\left( x_{2}\right) \right) \,.
\mathbb{E}\Bigg [ \Big | \sum_{n}a_n X(n) \Big |^2 \cdot\Big | \sum_{n}c_n X(n) \Big |^2 \Bigg ] = \sum_{n}\Big ( \sum_{n = k\ell}a_k c_{\ell}\Big )^2 \leq\sum_{n}\Big ( \sum_{n = k \ell}b_k c_{\ell}\Big )^2= \mathbb{E}\Bigg [ \Big | \sum_{n}b_n X(n) \Big |^2 \cdot\Big | \sum_{n}c_n X(n) \Big |^2 \Bigg ]
\left| \left\{\Theta^{4}\left[ \begin{array}{l l}{{0}}&{{0}}\\{{0}}&{{0}}\\ \end{array}\right] + \Theta^{4}\left[ \begin{array}{l l}{{0}}&{{0}}\\{{0}}&{{1 / 2}}\\ \end{array}\right] + \Theta^{4}\left[ \begin{array}{l l}{{0}}&{{0}}\\{{1 / 2}}&{{0}}\\ \end{array}\right] + \Theta^{4}\left[ \begin{array}{l l}{{0}}&{{0}}\\{{1 / 2}}&{{1 / 2}}\\ \end{array}\right] \right\}( 0 \vert \Omega ) \right|^{2}
|-\left(\int_{0}^{\frac{\pi}{2}}i d\theta \frac{e^{\frac{i t}{2}\cos(\theta)}e^{-\frac{t}{2}\sin(\theta)}}{\log^{a}(2 e^{-i \theta})}\right)| \leq C \int_{0}^{\frac{\pi}{2}}d\theta \frac{e^{-\frac{t}{2}\sin(\theta)}}{\log^{a}(2)}\leq C \int_{0}^{\frac{\pi}{2}}d\theta e^{-\frac{t}{2}\frac{2}{\pi}\theta}=O\left(\frac{1}{t}\right)
\begin{array}{l}d{\hat{s}}^2 = 2 du\, \left( dv - \frac{1}{2}K \, du \right) - F^{-1}\, dx_I dx_I \qquad , \qquad e^{-2 \hat{\phi}}= F \hat{H}_{IJK}= \frac{1}{6}\epsilon_{IJKL}\partial^L F^{-1}\, ,\qquad \hat{H}_{IJu}= \hat{H}_{IJv}= \hat{H}_{Iuv}= 0\, .\end{array}
\frac{-c_{b}r}{2\pi}\int_{0}^{\pi}\sin^{2}(\theta)d\theta \int_{0}^{\frac{1}{2}}d\xi \left(\frac{b-1}{\xi \log^{b}(\frac{1}{\xi})}+ \frac{b(b-1)}{\xi \log^{b+1}(\frac{1}{\xi})}\right) \left(\frac{\sin(\xi t_{+})}{t_{+}^{2}}+ \frac{\sin(\xi t_{-})}{t_{-}^{2}}\right) &= \frac{-b r}{t^{2}\log^{b}(t)}+ E_{v_{2},3}(t,r)
\mu^{l}(B_t(w_0)) \stackrel{\eqref{e:stima banana}}{=}\sum_{z_\alpha\in Z}\mu^{l}(W^{z_\alpha}) +\mu^{l}_1\big(W^{(2)}\setminus E\big)\stackrel{\eqref{e:stima chiava}}{\leq}2 \sum_{z_\alpha\in Z}s_{z_\alpha}^{n-1}+\mu^{l}_1\big(W^{(2)}\setminus E\big) &\leq 2\,\mu^l_1(B_{11t}(w_0)).
P_{L}^{\prime}= ({\rho}\frac{\tilde{m}}{2}+ ( G_{\bot}- B_{\bot}- \frac{1}{4}A_{\bot}^{t}C_{\bot}A_{\bot}){\tilde{n}}- \frac{1}{2}A_{\bot}^{t}C_{\bot}{\tilde{l}},{\tilde{l}}+ A_{\bot}{\tilde{n}})
\beta_{t,C}(x) =\int_0^{tC^2}Cq_{tC^2 -s}(0,0) \rho_{\tau_0^{Cx}}(s)ds = \int_0^{tC^2}\left(\frac{C}{\sqrt{tC^2-s}}\phi(\sqrt{tC^2-s})+C\Phi(\sqrt{tC^2-s})\right) \rho_{\tau_0^{Cx}}(s)ds,x\neq 0
\mathbb{E}\|E\|_2^{4p}\leq 4p \int\limits_0^{\infty}t^{4p-1}me^{-\frac{t^2}{3l}}dt+4p \int\limits_0^{\infty}t^{4p-1}me^{-\frac{t}{4r_1r_2\sqrt{kl/m}}}dt &= c_2^pm\left(p^{2p}l^{2p}+p^{4p}r_1^{4p}r_2^{4p}\left( \frac{kl}{m}\right)^{2p}\right).
\mathbb{E}_{U,\xi}\left[\phi_{\mu}^{1/\bar{\rho}}(x_{t+1}) \right] \leq \ \mathbb{E}_{U,\xi}\left[\phi_{\mu}^{1/\bar{\rho}}(x_{t}) \right] - \frac{\alpha_t(\bar{\rho}-\rho)}{\bar{\rho}}\mathbb{E}_{U,\xi}\left[\left\| \nabla \phi_{\mu}^{1/\bar{\rho}}(x_t) \right\|_2^2 \right] & + 2(n^2 + 2n)\bar{\rho}\alpha_t^2L_{f,0}^2,
v_\varepsilon(x)^{|x|^\alpha}= A_{n,m}^{|x|^\alpha}\Big( \frac{\varepsilon}{\varepsilon^2 + |x|^2}\Big)^{\frac{n-2m}{2}|x|^\alpha}&= \exp \Big[\Big( \ln A_{n,m}-\frac{n-2m}2 \ln \big(\varepsilon + \frac{|x|^2}{\varepsilon}\big) \Big) |x|^\alpha\Big].
\varphi\left(\begin{array}{c c}a b 0 d \end{array}\right) = \left(\begin{array}{c c c}a^{2 \, p^{e_1- 1}}0 c_2 \, a^{2 \, p^{e_1 - 1}}\, b^{p^{e_1}}\, d^{p^{e_1}}0 a^{2 \, p^{e_1- 1}}c_1 \, a^{2 \, p^{e_1 - 1}}\, b^{p^{e_1}}\, d^{p^{e_1}}0 0 d^{4 \, p^{e_1 - 1}}\end{array}\right) .
L_f^{(m)}(s)= \sum_{n=1}^\infty\frac{\lambda_f(n)(-\log n)^m}{n^s}\varphi\left(\frac{n}{y_1}\right)+ &+\sum_{r=0}^m(-1)^{r}\binom{m}{r}\chi_f^{(m-r)}(s)\sum_{n=1}^\infty\frac{\lambda_f(n)(-\log n)^{r}}{n^{1-s}}\varphi_0\left(\frac{n}{y_2}\right)+R_\varphi(s),
H_2(f, g)\ = \ - \sum_{\mu_1 = 1}^f \frac{1}{\mu_1 !}\binom{f - \mu_1}{g - \mu_1}\sum_{d=1}^{f- \mu_1}\sum_{\substack{\mu_2 + \cdots + \mu_d = f - \mu_1 \mu_i \ge 1}}\frac{(-1)^{d-1}}{\mu_2! \cdots \mu_{d}!}.
A'(t)= \int_0^1\frac{ye^{-G(y,t)}}{(y^2+e^{-2G(y,t)})^2}dy, B'(t)= \frac{1}{K}\int_0^1\frac{e^{-G(y,t)}}{(y^2+e^{-2G(y,t)})^2}dy; \,\,\,\,A(0)=B(0)=0.
\lambda_n(\epsilon) \to \left(h^{-1}([h_{\min}, h_{\min}+ \epsilon])\right) n \to \infty, begin{align*}1.5 ex]\left(h^{-1}([h_{\min}, h_{\min}+ \epsilon])\right) \to \left(h^{-1}(\{h_{\min}\})\right) = \sum_{l=1}^q (b_l-a_l)>0 \epsilon \to 0.
\Delta=I_{-\infty}^{+\infty}\frac{-Im[f(x)]}{Re[f(x)]}=I_{-\infty}^{+\infty}\frac{b_{0}\omega^{n-1}-b_{1}\omega^{n-3}+\cdot\cdot\cdot}{a_{0}\omega^{n}-a_{1}\omega^{n-2}+...}(25)