harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{d ( p )}{c ( p )}= \alpha ( p ) , \ \frac{b ( - p )}{a ( - p )}= \gamma ( p ) = 2 \cosh ( \beta p^{0}/ 2 ) - \alpha ( p ) , \ \frac{b ( - p )}{a ( - p )}\cdot \frac{d ( p )}{c ( p )}= \gamma ( p ) \cdot \alpha ( p ) = 1 ,
R_{2}(x,a,b,d)=\frac{1}{\left( 2\pi \right)^{d-2}b}\sum_{n^{\prime}=1}^{\infty}\int d^{d-2}p \frac{1}{\sqrt{\vec{p}^{\,2}+m^{2}+(\frac{n^{\prime}\pi}{b})^{2}}}\frac{\cosh ( (a-2x)\sqrt{\vec{p}^{\,2}+m^{2}+(\frac{n^{\prime}\pi}{b})^{2}})}{\sinh (a\sqrt{\vec{p}^{\,2}+m^{2}+(\frac{n^{\prime}\pi}{b})^{2}})}.
Y(u,x_1)Y(v,x_2)-\beta(x_2-x_1)Y(v,x_2)Y(u,x_1) =\ f(x_1-x_2)+g(x_2-x_1)+\sum_{r\ge 0}Y(c^{(r)},x_2)\frac{1}{j!}\left(\frac{\partial}{\partial x_2}\right)^rx_1^{-1}\delta\left(\frac{x_2}{x_1}\right).
A_{t_i}=\begin{pmatrix}-q_i 1\end{pmatrix}\begin{pmatrix}-p_i -p_i q_i+\theta^{t_i}\end{pmatrix},N=\begin{pmatrix}0 1 0 0\end{pmatrix}, A_{\infty \, 1}=\begin{pmatrix}0 -p_1-p_2 -1 0\end{pmatrix},N_i=-p_i N , i=1,2.
(I_n\otimes m_w)^*(T) \sim \sum_{g\in G}w(g)A_g \otimes \lambda_g \otimes \lambda_g & = \sum_{g\in G}(A_g \otimes \lambda_g) \odot (w(g)id_n \otimes \lambda_g) & = \begin{bmatrix}\; \cdots A_g\otimes \lambda_g \cdots \;\; \end{bmatrix}\odot \begin{bmatrix}\vdots w(g)id_n \otimes \lambda_g \vdots \end{bmatrix}& = A \odot B.
P_{\rm F}[\Delta\vartheta_a^{(1)},P_{\rm F}] =P_{\rm F}(\Delta\vartheta_a^{(1)}P_{\rm F}-P_{\rm F}\Delta\vartheta_a^{(1)}) &=P_{\rm F}(\Delta\vartheta_a^{(1)}P_{\rm F}-\Delta\vartheta_a^{(1)})=-P_{\rm F}\Delta\vartheta_a^{(1)}(1-P_{\rm F}).
\bar{\Lambda}_I(\mathcal{N}) = \{C\in \bar{\Lambda}(\mathcal{N}): I_C = I\}, \bar{\Lambda}_I^*(\mathcal{N}) = \{C\in \bar{\Lambda}_I(\mathcal{N}): C'\notin \bar{\Lambda}_I(\mathcal{N}),\ \forall C'\subsetneq C\},
\int_{\gamma_k}\frac{\omega_m}{V_m}=-\int_{\gamma_k}\left[dH+\Lambda\sum_{j=1}^n dh_j\right]= =-\int_{\gamma_k}\Big[d\left(\ln A\right)+d\left(\ln B\right)+d\left(\ln C\right)\Big]=\pm 4\pi s_k\neq 0\,.
\mu_{\infty}:= \lim_{R \to \infty}\limsup_{k \to \infty}\int_{B^{c}_{R}}\bigg[|u_k|^{\overline{p}}+|v|^{\overline{p}}+ \int_{\mathbb{R}^{N}}\frac{|u_k(x)-u_k(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}\,dy +\int_{\mathbb{R}^{N}}\frac{|v_k(x)-v_k(y)|^{p(x,y)}}{|x-y|^{N+sp(x,y)}}\,dy \bigg]\,dx.
\phi(P,x,t_{m})=N(x,t_{m})\frac{\theta(\underline{z}(P,D_{\underline{\hat{\nu}}(x,t_{m})}))}{\theta(\underline{z}(P,D_{\underline{\hat{\mu}}(x,t_{m})}))}\mathrm{exp}\left(\int_{P_{0}}^{P}\omega^{(3)}_{\hat{\nu}_{n+1}(x,t_{m}),\hat{\mu}_{n+1}(x,t_{m})}(P)\right),
\sum\limits_{i=l}^kI(U^m_{\mathcal{A}_i};Y^{\rho m}_{a(i)}|U^m_{\cup_{j=1}^{i-1}\mathcal{A}_j}) &\leq\sum\limits_{q=1}^{\rho m}I(Y^{q-1}_{1};Y_2(q)|U^m_{\cup_{j=1}^{l-1}\mathcal{A}_j},Y^{\rho m}_{2,q+1})+\sum\limits_{i=l}^k\sum\limits_{q=1}^{\rho m}I(U^m_{\mathcal{A}_i};Y_{a(i)}(q)|U^m_{\cup_{j=1}^{i-1}\mathcal{A}_j},Y^{q-1}_{1},Y^{\rho m}_{2,q+1}).
\rho(b'X_{b^{\ast}}(P)) =\displaystyle \sum_{r \in L(k),a \in_{k}T}\rho(b'ra) \rho(Y_{[bra]}(P)) &=\rho \left( \displaystyle \sum_{r \in L(k), a \in_{k}T}b'raY_{[bra]}(P)\right)
L_{- n}= \sum_{i = 1}^{N}\left( \alpha ( a_{i}^{\dagger})^{n + 1}a_{i}+ \beta a_{i}( a_{i}^{\dagger})^{n + 1}+ ( \lambda -{\frac{1}{2}}) ( n + 1 ) ( a_{i}^{\dagger})^{n}\right) ,
L_i=\max_{p\mid\gcd(\overline{\alpha}_i,s)}{\left\lceil\frac{\nu_p(s)}{\nu_p(\overline{\alpha}_i)}\right\rceil}=\max_{p\mid\gcd(\overline{\alpha}_i,s)}{\left\lceil\frac{\nu_p(s)}{\nu_p(\overline{\alpha}_i\bmod{p^{\nu_p(s)}}})\right\rceil},
h^{\pm}(c_r):=c_r\mp\frac{\Re\left( \displaystyle\sum_{kn\le X}\frac{a^\pm(n)\overline{a^\pm(kn)}g_{c_r}(k)\Lambda(k)}{kn}\right)}{\displaystyle\sum_{n\le X}\frac{|a^\pm(n)|^2}{n}}
H \equiv \int_{-\infty}^{+\infty}\mathrm{d}x\, \mathcal{H}\stackrel{\;\eqref{eq:hamiltonian}}{=}\int_{-\infty}^{+\infty}\mathrm{d}x\, \left[ -\frac{1}{(n+2)(n+1)}u^{n+2}+ \frac{1}{m+1}(u_{x})^{m+1}\right]
\Pr\bigg(S_N\geq \frac{nN}{2}+2C\sqrt{N}n\bigg)=\Pr\bigg(\frac{S_N-nN/2}{\sigma \sqrt{N}}\geq \frac{2C\sqrt{N}n}{\sigma\sqrt{N}}\bigg) &\leq \Pr\bigg(\frac{S_N-nN/2}{\sigma \sqrt{N}}\geq 4\sqrt{3}C\bigg)=1-\Phi(4\sqrt{3}C)+O\bigg(\frac{1}{\sqrt{N}}\bigg).
z(z-1)Y_{zz}+\big(\alpha z(z-1)+(\beta+1)(z-1)+(\gamma+1)z\big)Y_z \qquad{}+\frac12\big(\alpha(\beta+1)(z-1)+\alpha(\gamma+1)z+2\delta z+(\beta+1)(\gamma+1)+2\eta-1\big)Y=0, Y(0)=1,Y_z(0)=\frac12\left(\frac{2\eta-1}{\beta+1}+\gamma+1-\alpha\right).
M(\alpha)= &\int_0^1\!\!\Big\{\!\!\!\left[ x^{-\alpha}+(1-x)^{-\alpha}-2\zeta(\alpha) \right]\left[\zeta_1(\alpha,x)+\zeta_1(\alpha,1-x)-2\zeta(\alpha) \right]+x^{-\alpha}(1-x)^{-\alpha}\!\Big\}dx.
\int_{0}^{1}dw w \frac{|\lambda''(t+w)|}{(\lambda(t+w)^{2-2\alpha}+w^{2})}\leq \frac{C}{t^{2}\log^{b+1}(t)}\frac{1}{\log^{(2\alpha -2)b}(t)}\int_{0}^{1}\frac{w dw}{1+w^{2}\lambda(t)^{2\alpha -2}}&\leq \frac{C \log(\log(t))}{t^{2}\log^{b+1}(t)}
\frac{|S|}{r_{1}}\le \frac{|\Gamma_2(S)|}{k_1r_1+1}\le \frac{|\Gamma_2(S)|}{k_1r_1}=\frac{|\Gamma_2(S)|}{r_{2}-1}=\frac{|\Gamma_2(S)|}{|L_2\setminus \{x\}|}.
G_k = P_1(G_k) = \langle x,y \rangle \; P_2(G_k) G_k/P_2(G_k) \cong C_2 \times C_2, P_2(G_k) = \langle x^2, y^2, [y,x] \rangle \; P_3(G_k) P_2(G_k)/P_3(G_k) \cong C_2 \times C_2 \times C_2,
J [{\hat{\xi}}] ={\frac{1}{8 \pi G}}\int_{r = r_{+}}\! \! d^{n - 2}x \, \Bigl \{n^{a}\nabla_{a}{\hat{\xi}}^{t}\sqrt{\sigma}+{\hat{\xi}}^{a}\pi_{a}{}^{r}+ n_{a}{\hat{\xi}}^{a}K \sqrt{\sigma}\Bigr \}
- \Bigl [ \left[ Q ( \xi ) , Q ( \eta ) \right] , a_{B}( k ) \Bigr ] = \Bigl [ \left[ Q ( \eta ) , a_{B}( k ) \right] , Q ( \xi ) \Bigr ] + \Bigl [ \left[ a_{B}( k ) , Q ( \xi ) \right] , Q ( \eta ) \Bigr ]
\frac{p_{\alpha}( p + k_{1})_{\beta}( p + k )_{\gamma}}{p^{2}( p + k_{1})^{2}( p + k )^{2}}= \frac{p_{\alpha}( p + k_{1})_{\beta}}{p^{4}( p + k_{1})^{2}}\left[ ( p + k )_{\gamma}- \frac{2 ( k p ) p_{\gamma}}{p^{2}}\right] + \ldots .
L(Y,K,\phi) = \bigl\{\bigl( [a_1,\ldots,a_{2g},h_1,\ldots, h_{n+2}], \qquad[a_1,\ldots,a_{2g},h_1,\ldots,\hat{h}_i,\ldots, \hat{h}_j, \ldots, h_{n+2}] \bigr) \,\big|\, h_i = h_j \bigr\}.
\beta_0 \geq 0 \beta_j \ge 0\delta_j \ge 0 1\le j \le k \sum_{j=1}^{k}(\delta_j + r\beta_j) = 1 \sum_{j=1}^{k}\left( (\delta_j + r\beta_j)j^m - \beta_j m j^{m-1}\right) - \beta_0 0^{m-1}= 0 1\le m\le p.
\lim\limits_{k_{p}\to\infty}\|x_{i}(k_{p})-x^{\star}\| =\lim\limits_{k_{p}\to\infty}\|x_{i}(k_{p})-y_{\infty}\| &\leq \lim\limits_{k_{p}\to\infty}(\|x_{i}(k_{p})-y(k_{p})\|+\|y(k_{p})-y_{\infty}\|) &=0
\|\sum_{j=2}^\infty\tilde S_j f \tilde S^{j-2}g \|_{B^s_{p, \infty}(\real^{d_s})}\le C \sup_{k\ge 2}2^{ks}\sum_{\ell=-1}^{+3}\|\tilde S_{k+\ell}f \tilde S^{k+\ell-2}g \|_{L_p(\real^{d_s})}\, .
\ell(i)-\sup_{(u_1,u_2)\in U_1(i)\times U_2(i)}\bar{c}_k(i,u_1,u_2) =\alpha i-ip_k+\inf_{u_k\in U_k(i)}r_k(i,u_k) &=i\beta_k + \inf_{u_k\in U_k(i)}r_k(i,u_k).
S ={\frac{1}{2 \pi}}\int d^{2}x \sqrt{- G}e^{- 2 \Phi}\big \{R + 4 ( \nabla \Phi )^{2}+ 4 \lambda^{2}-{\frac{1}{2}}e^{\epsilon \Phi}F^{2}- ( \nabla T )^{2}- V ( T ) \big \}
I(N)= \sum_{k,l=1}^{N-1}\frac{1}{2}[1-(-1)^{k+l}] \frac{1}{4N^4}\frac{\cos^2 k \pi/2N \cos^2 l \pi/2N}{\sin^2 (l-k) \pi/2N\sin^2 (k+l) \pi/2N}.
{A}= \left(\begin{array}{cccccc}1 \frac 12 \frac 12 0 0 0 1-y 0 \frac12 (1-y^2) y 0 \frac12 y^2 1-x \frac12 (1-x^2) 0 x \frac12 x^2 0 -1 -x 0 1 x 0 -1 0 -y 1 0 y\end{array}\right)
\times\;\prod_{i=1}^{n^\prime_b}\overline{\psi}^{(b)}({x^\prime}^{(b)}_i) P_+ \psi^{(b)}({x^\prime}^{(b)}_i)\prod_{i=1}^{m^\prime_b}\overline{\psi}^{(b)}({y^\prime}^{(b)}_i) P_- \psi^{(b)}({y^\prime}^{(b)}_i) \Big\rangle_0
{\cal L}_{g}= -{\frac{1}{2}}\sqrt{- g}g^{\alpha \beta}G_{\alpha \beta}+{\frac{\tilde{Y}M \tilde{Y}}{2 \sqrt{- g}( \partial a )^{2}}}-{\frac{\tilde{Y}L \, \partial Y \cdot \partial a}{2 ( \partial a )^{2}}}.
\int_{B}|w_\delta-W_\delta|^{(p+1)(1+\gamma)}=\begin{cases}O(\delta^4 |\log \delta|) n=4,\ \gamma=0, O(\delta^{\frac{2n}{n-2}}) n>4,\ \gamma=0, O(\delta^\frac{2n-n\gamma(n-4)}{n-2}) n\geq 4,\ \gamma>0.\end{cases}
P_\lambda (z_1,\ldots ,z_n; q ,t ,\mathbf{t})= \sum_{\substack{\mu^{(i)}\in\Lambda_{i}, i=1,\ldots ,n \mu^{(1)}\preceq \mu^{(2)}\preceq \cdots \preceq \mu^{(n)}=\lambda}}\prod_{1\leq i\leq n}P_{\mu^{(i)}/\mu^{(i-1)}}(z_i; q ,t ,\mathbf{t}),
A_{T,t}^{(i_1 i_2)}=\frac{T-t}{2}\sum_{i=1}^{\infty}\frac{1}{\sqrt{4i^2-1}}\left(\zeta_{i-1}^{(i_1)}\zeta_{i}^{(i_2)}-\zeta_i^{(i_1)}\zeta_{i-1}^{(i_2)}\right).
U_{k_{1}k_{2}}^{l_{1}l_{2}}\left( \begin{array}{c c c}{\lambda}&{\mu}&{\rho}\\{k_{1}}&{k_{2}}&{m}\\ \end{array}\right) \left( \begin{array}{c c c}{\lambda}&{\mu}&{\rho^{\prime}}\\{l_{1}}&{l_{2}}&{m^{\prime}}\\ \end{array}\right)^{\ast}
f(x) = x^6 - x^4 ( z_1 + z_2 + z_3 + z_4 + z_5 ) \qquad\; + x^2 (z_1 z_3 + z_1 z_4 + z_1 z_5 + z_2 z_4 + z_2 z_5 + z_3 z_5) - z_1 z_3 z_5.
a_{12}(p):= f_{xy}(p)^2f_{xxx}(p)-2f_{xx}(p)f_{xy}(p)f_{xxy}(p)+f_{xx}(p)^2f_{xyy}(p), a_{04}(p):= f_{xy}(p)^4f_{xxxx}(p)-4f_{xx}(p)f_{xy}(p)^3f_{xxxy}(p) &+6f_{xx}(p)^2f_{xy}(p)^2f_{xxyy}(p)-4f_{xx}(p)^3f_{xy}(p)f_{xyyy}(p)+f_{xx}(p)^4f_{yyyy}(p).
(n-2)(x_0\frac{\partial x_k}{\partial z}-x_k\frac{\partial x_0}{\partial z})=u_{k}(\tau)x_0 x_{k}-\frac{\partial x_0}{\partial z}(0,\tau) \sum_{1\leq i \leq n-1, i \neq n-k}\frac{\alpha_{k}(\tau)}{\alpha_i(\tau) \alpha_{k-i}(\tau)}x_{i}x_{k-i},
\begin{array}{c c c}{A^{\sigma}( \mu )}&{\longrightarrow}&{A^{\sigma}( \mu + \nu )}\\{\downarrow}&{}&{\downarrow}\\{{\bf C}[ \sigma^{\vee}] ( \mu + \chi )}&{\longrightarrow}&{{\bf C}[ \sigma^{\vee}] ( \mu + \nu + \chi )}\\ \end{array}
U^\psi_{(nm)^*nm'}=\pi_\psi(d(nm)^*nm'd)=\pi_\psi(\alpha_m^\#(\alpha_{m^*}^\#(d)n^*n)m^*m'd)=\phi(n^*n)\pi_\psi(dm^*m'd) &=\pi_\psi(dm^*m'd)=U^\psi_{m^*m'}\in\mathcal{U}_0(D'_{A^\delta}/J_\psi).
\phi_{a}= \left( \begin{array}{c}{2 ( x_{1}x_{3}+ x_{2}x_{4})}\\{2 ( x_{2}x_{3}- x_{1}x_{4})}\\{x_{1}^{2}+ x_{2}^{2}- x_{3}^{2}- x_{4}^{2}}\\ \end{array}\right) , \quad | \phi | = r^{2}, \quad n = \phi / r^{2}\equiv n_{\mathrm{H}}.
h' \geq \max \left\{D\left(\log\left( \frac{b_1}{a_2'}+\frac{b_2}{a_1'}\right)+\log \lambda'+1.75\right)+0.06,\lambda',\frac{D\log 2}{2}\right\}, & a_i'\geq \max\{1,\varrho|\log \gamma_i|-\log|\gamma_i|+2Dh(\gamma_i)\}, i=1,2, &a_1'a_2'\geq \lambda'^2.
a_i^k = b_{i-1}^k,y_i^k = f_p(a_i^k), b_i^k = a_i^k +y_i^k,M_i^k = (a_i^k, b_i^k],\tilde M_i^k = \frac{M_i^k}{y_i^k}=(\tilde a_i^k, \tilde b_i^k].
[D\vec{x}]_{v_1}-[D\vec{x}]_{v_2}= \sum_{j=1}^nd_G(v_1,v_j)\vec{x}_{v_j}-\sum_{j=1}^nd_G(v_2,v_j)\vec{x}_{v_j}= d_G(v_1,v_1)\vec{x}_{v_1}+d_G(v_1,v_2)\vec{x}_{v_2}-d_G(v_1,v_2)\vec{x}_{v_1}-d_G(v_2,v_2)\vec{x}_{v_2}= d_G(v_1,v_2)(\vec{x}_{v_2}-\vec{x}_{v_1}).
X(\kappa,z_0,t) = \gamma_{k_i}(z_{i-1},T_{i-1},T_{i},t)^{-1}\gamma_{k_i}'(z_{i-1},T_{i-1},T_{i},t) =\gamma_{k_i}(z_{i-1},t-T_{i-1})^{-1}\gamma_{k_i}'(z_{i-1},t-T_{i-1}) = R^{k_i}X(z_{i-1},t - T_{i-1}) R^{-k_i},
\partial_{s_m}\log\det\left(1-\mathcal K_{\vec x,\vec s}\right)=\frac{1}{2\pi i}\int_{x_m}^{x_{m-1}}\left(\Psi_+^{-1}\Psi_+'\right)_{21}(\zeta=\xi-x_m;x=x_m,\vec y,\vec s)d\xi.
\mathbf{w}_{\scriptscriptstyle{GE}}= \mathbf{u}_(\mathbf{R}^{-1}_{\scriptscriptstyle{B}}\mathbf{R}_{\scriptscriptstyle{A}}), \mathbf{q}_{\scriptscriptstyle{GE}}= \mathbf{u}_(\mathbf{R}^{-1}_{\scriptscriptstyle{A}}\mathbf{R}_{\scriptscriptstyle{B}}).
\begin{array}{c}{| 0 \rangle \quad \bar{R}( \lambda , \bar{\eta}, \Theta )}\\{M \cos ( \nu_{0})}\\ \end{array}\dots \begin{array}{c}{| n \rangle \quad \bar{R}( \lambda , \bar{\eta}, \Theta ) a_{n}( \eta , \theta )}\\{M \cos ( \nu_{n})}\\ \end{array}\quad .
C_{H}(c)=\langle c \rangle \times F(b,d) , C_H(d) =\langle d \rangle \times F({e_i}^{e_1^{j}}, e_1^k,{c}^{e_1^{j}}\mid j=0,\dots, k-1, \: i=2,\dots,k+1 ), C_H(b) =\langle b \rangle \times F({a_i}^{a_1^{j}},a_1^{k+1},{c}^{a_1^{j}}\mid j=0,\dots, k, \: i=2,\dots,k).
\Gamma(\frac{a_{1}x_{1}+\cdot\cdot\cdot+a_{n}x_{n}}{a_{1}+\cdot\cdot\cdot+a_{n}})\le(\Gamma(x_{1})^{a_{1}}\cdot\cdot\cdot\Gamma(x_{n})^{a_{n}})^{\frac{1}{a_{1}+\cdot\cdot\cdot+a_{n}}}.
t(0)=0,\;\;t(\tau_{\mbox{hor}})=\infty,\;\;t(\frac{\omega_2}{2})=\frac{Elr_m}{2\sqrt{MH_2}\sqrt{r^2_m-Ml^2}}\frac{Z[\epsilon,k]}{\sqrt{r^2_m(1-k^2)+k^2Ml^2}},
\psi=C_1+C_2\arctan\omega\quad\mbox{if}A=0, begin{align*}1ex]\psi=C_1e^{-\alpha\arctan\omega}+C_2e^{\alpha\arctan\omega}\quad\mbox{with}\alpha=\sqrt A \quad\mbox{if}A>0, begin{align*}1ex]\psi=C_1\cos(\beta\arctan\omega)+C_2\sin(\beta\arctan\omega)\quad\mbox{with}\beta=\sqrt{-A}\quad\mbox{if}A<0.
=(\beta^{2}m^{2}c^{4}-i\hbar\sum_{k=1}^{3}\{\alpha_{k},\beta\}\nabla_{k}m^{2}c^{3}+(-i\hbar)^{2}\frac{1}{2}\sum_{k=1}^{3}\sum_{l=1}^{3}\{\alpha_{k},\alpha_{l}\}\nabla_{k}\nabla_{l}c^{2})\psi(x,t)
\rho^{(t)}_{N,l}=\sum_{k=l+1}^{l+N}2^{-kt}\rho_k, \omega^{(t)}_{N}=\sum_{k=N+1}^{2N}2^{-kt}\omega_k, \Omega^{(t)}_{N}=\sum_{k=2}^{N+1}2^{-2^kt}\omega_{2^k},
L_{n}^{( j )}( p_{i_{1}}, \ldots , p_{i_{k}}) = \{p_{1 + j}, p_{2 + j}, \ldots , p_{i_{1}- 1 + j}, p_{i_{1}+ 1 + j}, \ldots , p_{i_{k}- 1 + j}, p_{i_{k}+ 1 + j}, \ldots p_{n + j}\}.
p_k^{(\rm id)}= \left(\frac{\mu_2^*}{\ln2\left(\nu_2^*-[\boldsymbol{\Lambda}]_{k,k}^2\right)}-\frac{\sigma^2}{[\boldsymbol{\Lambda}]_{k,k}^2}\right)^+,\forall\, k=1,2,\ldots,r.
\beta ( P_{0}; \kappa ) = - P_{0}^{2}\frac{e^{P_{0}/ \kappa}}{2 f ( P_{0})}\left( \frac{n \Phi ( P_{0})}{\kappa}+ \Phi^{\prime}( P_{0}) - \frac{e^{- P_{0}/ \kappa}\Phi ( P_{0})}{f ( P_{0})}\right) .
\{h^{-s}\}:= \Big\{\frac{1}{h^s}\Big\}:=\begin{cases}\frac{1}{2}\Big( \frac{1}{h^s_{i}}+ \frac{1}{h^s_{j}}\Big) e = \partial \Omega_i \cap \partial \Omega_j, \frac{1}{h^s_{i}}e = \partial \Omega_i \cap \partial \Omega.\end{cases}
n =\frac{(h_x,-1)}{\sqrt{1+h_x^2}}, \kappa =\frac{d}{dx}\frac{h_x}{\sqrt{1+h_x^2}}=\frac{h_{xx}}{(\sqrt{1+h_x^2})^3}, V =-\frac{h_{t}}{\sqrt{1+h_x^2}}.
\Sigma_n(z_1, z_2) = \frac{(2n^2)^{m-s}}{\big[d_{n}^{(n)}\big]^2 \frac{h_{n-1 -m}^{(s)}k_{n}^{(s)}}{k_{n-1-m}^{(s)}}}\cdot \frac{q_{n}^{(n)}(x_1^{(n)}) q_{n-1}^{(n)}(x_2^{(n)}) - q_{n}^{(n)}(x_2^{(n)}) q_{n-1}^{(n)}(x_1^{(n)})}{z_2 - z_1}.
0=\nabla\cdot(\nabla\times B)=\nabla\cdot(\mu_{0}(J+\epsilon_{0}\frac{\partial E}{\partial t}))=\mu_{0}(\nabla\cdot J+\epsilon_{0}\frac{\partial}{\partial t}\nabla\cdot E)=\mu_{0}(\nabla\cdot J+\frac{\partial\rho}{\partial t})
\sum_{\{\delta \}}\int_{{\cal C}/ \Gamma}d x ~ k ( x , \gamma [ x ] ) = \sum_{\delta \neq{\bf 1}}\sum_{n = 1}^{\infty}\sum_{\beta \in \Gamma / \Gamma^{( \delta )}}\int_{\beta [{\cal C}/ \Gamma ]}d x ~ k_{0}( x , \delta^{n}[ x ] ) ~ + ~ \int_{{\cal C}/ \Gamma}d x ~ k_{0}( x , x ) ,
c(t)\leq 128R^2+12\exp\left(\frac{R^2}{2t}\right)\exp\left(3n+\frac{R^2}{t}\right)\left(8+\frac{36D}{e}\right)R^2\exp\left(17n+\frac{25R^2}{8t}\right) = 128R^2+\left(96+\frac{432D}{e}\right)R^2\exp\left(20n+\frac{37R^2}{8t}\right) \leq \left(128+96+\frac{432D}{e}\right)R^2\exp\left(20n+\frac{5R^2}{t}\right) \leq 289R^2\exp\left(20n+\frac{5R^2}{t}\right).
{\cal A}_{f i \, \, ({\bf A}^{2})}^{( 2 ) \, \, R e g}= \frac{e^{2}\Phi^{2}}{4 ( 2 \pi )^{3}}\frac{1}{w_{p}}\left\{- \ln ( \lambda^{2}) - \ln ( p^{2}) - \ln [ 2 ( 1 - \cos \theta_{S}) ] + 2 ( \ln 2 - \gamma ) \right\}\; .
\,|\xi(s)-\xi^+(s+T^--T^*)| &\,\leqslant\dfrac{1}{2}\Big(|y^--\xi^+(-T^*)|+|\xi^+(s)-\xi^+(s+T^--T^*)|\Big) &\,\leqslant\dfrac{1}{2}\Big(|y^--\xi^+(-T^-)|+|\xi^+(-T^-)-\xi^+(-T^*)|+C\,\dfrac{T^+-T^-}{2}\Big) &\,\leqslant C|h|.
\sum_{x\in I,\; 1 \leq x\leq M}\frac{f(x)}{x^t}< K_1 t>s, & \sum_{x\in I,\; 1 \leq x\leq M}\frac{f(x)}{x^t}< K_2 \log(M) t=s, & \sum_{x\in I,\; 1 \leq x\leq M}\frac{f(x)}{x^t}< K_3 M^{s-t}t<s,
{\rm reg}_{S_1}(S_1/J_1)={\rm reg}_{S_1}(S_1/J(G\setminus N_G[x_1])^{(k)}S_1)+k{\rm deg}(u_1) & \leq (k-1){\rm deg}(J(G\setminus N_G[x_1]))+|V(G\setminus N_G[x_1])|-2+k{\rm deg}_G(x_1) & \leq (k-1)({\rm deg}(J(G))-{\rm deg}_G(x_1))+n-{\rm deg}_G(x_1)-1-2+k{\rm deg}_G(x_1) & < (k-1){\rm deg}(J(G))+n-2.
\sum_{k=1}^{n}\frac{H_{k+r-1}}{k}=\sum_{k=1}^{n}\frac{H_{k}}{k}+\sum_{j=1}^{r-1}\frac{H_{j}}{j}+H_{r-1}H_{n}-\sum_{j=1}^{r-1}\frac{H_{n+j}}{j}
Z^1(\mathcal{U},\mathcal{O}_X^*)=\{x=(a_{ij})\mid d_1^+(x)=d_1^-(x)\}=\{x=(a_{ij})\mid a_{ik}|_{U_{ijk}}=a_{jk}|_{U_{ijk}}\cdot a_{ij}|_{U_{ijk}}\}.
X_{i}^{-}(u)=f_{i+1,i}^{+}(u+\frac{1}{4}hc)-f_{i+1,i}^{-}(u-\frac{1}{4}hc), &X_{i}^{+}(u)=e_{i,i+1}^{+}(u-\frac{1}{4}hc)-e_{i,i+1}^{-}(u+\frac{1}{4}hc).
B_k(w,w) := \frac{1}{2}\sum_{j+\ell=k}c_{j,\ell}w_j w_\ell,c_{j,\ell}:= \langle j^\perp,\ell\rangle_r\left(\frac{1}{|\ell|^2_r}- \frac{1}{|j|^2_r}\right)
\left|\lambda_h-\lambda_0-{\lambda_0\lambda_h\over{m}}\sum_{j=1}^m\langle(T_0(\lambda_0)-T_h(\lambda_h))\phi_j,\phi_j^*\rangle\right|\leq C( c(h) + |\lambda_h-\lambda_0| )\cdot( c^*(h) + |\lambda_h-\lambda_0| )
\partial^{\mu}\left( \partial^{\nu}\tilde{\partial}_{+}\bar{\phi}\varepsilon^{\prime}\sigma_{\nu \mu}\tilde{\partial}_{-}\psi \right) + \tilde{\partial}_{+}\left( \tilde{\partial}_{-}\bar{V}_{+}\bar{\varepsilon}_{\dot{\alpha}}\bar{\sigma}_{\mu}^{\dot{\alpha}\beta}\partial^{\mu}\tilde{\partial}_{-}\psi_{\beta}\right) .
c_n := \frac{(n+m-1)!}{n^m(n-1)!}= \prod^{m-1}_{k=1}\left(1+\frac{k}{n}\right)=1+O\left(\dfrac{1}{n}\right)(n \to \infty), c_n' := \frac{n^m n!}{(n+m)!}= \prod^{m}_{k=1}\left(1+\frac{k}{n}\right)^{-1}=1+O\left(\dfrac{1}{n}\right)(n \to \infty).
\varphi_1(\varepsilon,b,D,c,\eta,f) :=\max\Bigg\{\psi(c+1,b,D),\psi\left(\left(\frac{4}{\varepsilon}+1\right)c,b,D\right), &\qquad\qquad\qquad\left(\frac{D\left(2\eta(\min\{\varepsilon/2,2\})\right)^2}{18}\right)^{-1}(c+b), &\qquad\qquad\qquad\qquad\varphi\left(\frac{D\left(2\eta(\min\{\varepsilon/2,2\})\right)^2}{18},b,f\right)\Bigg\}
\nu^\ast(\theta_1 \wedge \theta_2\wedge \theta_3\wedge \theta_4)= \nu^\ast(d\theta \wedge dx\wedge dy \wedge dz) = d\theta \wedge d (-\cos\theta x - \sin \theta y) \wedge d (- \cos \theta y + \sin \theta x) \wedge dz = (\cos^2\theta+\sin^2\theta)d\theta \wedge dx \wedge dy \wedge dz
\frac{\int_{-1}^1 \frac{(1-t)^s}{( 1 + \frac{u}{2n^2}- t)^2}dt}{(2n^2)^{1-s}}= \int_0^{4n^2}\frac{t^s}{(t + u)^2}d t \xrightarrow{n \to \infty}\int_{0}^{\infty}\frac{t^s}{(t + u)^2}dt = u^{s-1}B(1 + s, 1-s) = u^{s-1}\Gamma(1 + s) \Gamma(1-s).
{\varpi_{N,k}^{(0.5)}}= \frac{2}{{\left({1 -{{\left({{x_{N,k}^{(0.5)}}}\right)}^2}}\right)\,{{\left[{{L'_{N + 1}}\left({x_{N,k}^{(0.5)}}\right)}\right]}^2}}},k = 0, \ldots ,N,
i \frac{d}{ds}\widehat{\psi}_l(t_n+s) - \alpha \frac{d^2}{ds^2}\widehat{\psi}_l(t_n+s) - |\mu_l|^2 \widehat{\psi}_l(t_n+s) - \varepsilon^{2}f_l^n(s) =0, \s \in \mathbb{R},
\frac{\partial^\gamma u}{\partial t^\gamma}+ u \frac{\partial u}{\partial x}-\nu \frac{\partial^2 u}{\partial x^2}=0,\,\,\,\,\,\,a\leq b \leq 3,\,\,\,\,\,0 < \gamma < 1,\,\,\,\,t>0.
\textbf{A}(x,y,z)=B_0 \begin{bmatrix}-\frac{(\sqrt{x^2+y^2}-R)^2+z^2}{2 Q(x^2+y^2)}y, \frac{(\sqrt{x^2+y^2}-R)^2+z^2}{2 Q(x^2+y^2)}x, -R \log \left( \frac{\sqrt{x^2+y^2}}{R}\right) \end{bmatrix}.
\Omega g \cdot ( d_{r + a / n + \alpha / 2 n}^{a}\tilde{d}_{r + a / n + \alpha / 2 n}^{- a}) = \lambda ( d_{r + a / n + \alpha / 2 n}^{a}\tilde{d}_{r + a / n + \alpha / 2 n}^{- a}) .
\hat{g_0}^*(y) + \varepsilon/2 \ge \left(3\pi y\sqrt{8z/b_0}+9\pi\left(\sqrt{b_0/(8z)}+1\right)(8z/3)\right)^{-1}&=\left(6\sqrt2\pi(y/b_0)+6\sqrt2\pi+24\pi(z/b_0)^{1/2}\right)^{-1}(b_0z)^{-1/2}.
\frac12\int_\Omega|x|^{-s}(u^m(x,\widetilde{T}))^2dx-\frac12\int_\Omega|x|^{-s}(u^m(x,0))^2dx+\int_0^{\widetilde{T}}\|\nabla u^m\|^p_pdt=\int_0^{\widetilde{T}}\int_\Omega|u^m|^q\ln|u^m|dxdt.
P(L_{n_{\beta}+d\beta\lceil n^{\xi}\rceil}\ge n_{\beta}(x+2\delta)-5c\beta n^{\xi}) \ge [c_3]^{2l+\sum_{j=1}^p \alpha_j 2^{k_j+1}}\zeta^{2ldn^{\xi}}\exp [-\sum_{j=1}^p \alpha_j 2^{k_j}N(\psi(x)+3\epsilon)].
\dim \mu (y_{k}) =\frac{\log p_{0}}{\log \varrho}+\frac{\log (sp(T(\theta_{k}))}{L(\theta_{k}^{-})\log \varrho}&=\frac{\log p_{0}}{\log \varrho}-\frac{\log K_{k}}{L(\theta_{k}^{-})\log\varrho}+\frac{\log \left\Vert T(\gamma_{M},\dots,\gamma_{n_{k}})\right\Vert}{L(\theta_{k}^{-})\log \varrho}.
\phi =(p\cdot{\rm id})\dot{s}_{d-1}\dot{s}_{d-2}\cdots\dot{s}_2\dot{s}_1\dot{s}_{d}\dot{s}_{d-2}\dot{s}_{d-3}\cdots\dot{s}_3\dot{s}_2\dot{s}_0 \mbox{if}d\mbox{is even}, \phi =(p^2\cdot{\rm id})\dot{s}_{d-1}\dot{s}_{d-2}\cdots\dot{s}_2\dot{s}_1\dot{s}_{d}\dot{s}_{d-2}\dot{s}_{d-3}\cdots\dot{s}_3\dot{s}_2\dot{s}_0 \mbox{if}d\mbox{is odd}.
\theta_t = (\theta^{(1)}_t\bar{\otimes}\mathrm{id})\!\upharpoonright_{\widetilde{M_1\bar{\otimes}M_2}}= (\mathrm{id}\bar{\otimes}\theta_t^{(2)})\!\upharpoonright_{\widetilde{M_1\bar{\otimes}M_2}}, t \in \mathbb{R}.
\frac{\partial w}{\partial t}= \big(1+\frac{\alpha'}{4}R\big)\Delta w, \frac{\partial}{\partial t}|\nabla w|^2= (1+\frac{\alpha'}{2}R)\Delta|\nabla w|^2-2(1+\frac{\alpha'}{2}R)|\nabla^2w|^2-(r+\frac{\alpha'}{4}R^2)|\nabla w|^2.
\left|\int \tilde{f}d\pi^{\mu}_{n-}-\int \tilde{f}d\pi_{n-}^{\nu}\right|+\frac{2}{3}\epsilon &=|\int_{\mathcal{Y}^{N'+1}}g(y_{[n,n+N']})P^{\mu}(dy_{[n,n+N']}|Y_{[0,n-1]}) & -\int_{\mathcal{Y}^{N'+1}}g(y_{[n,n+N']})P^{\nu}(dy_{[n,n+N']}|Y_{[0,n-1]})|+\frac{2}{3}\epsilon
S^\sharp=S_{H,L}\coloneqq\inf_{u\in D^{s,p}\setminus\{0\}}\frac{[u]_{s,p}^p}{\left(\int_{\mathbb{R}^N}(K\ast |u|^{p^\sharp}) |u|^{p^\sharp}dx\right)^{p/(2\cdot p^\sharp)}},
\begin{array}{c|ccc}\ast_\sigma \sigma(1) \cdots \sigma(n) \hline\sigma(1) \sigma(1)\ast_\sigma\sigma(1) = \sigma(1\ast1) \cdots \sigma(1)\ast_\sigma\sigma(n) = \sigma(1\ast n) \vdots \vdots \ddots \vdots \sigma(n) \sigma(n)\ast_\sigma\sigma(1) = \sigma(n\ast1) \cdots \sigma(n)\ast_\sigma\sigma(n) = \sigma(n\ast n) \end{array}.
C_t(y^{t-1}_{t-J})=\sup_{\big\{\pi_i(dx_i|y^{i-1}_{i-J}):~i=t,t+1,\ldots,n\big\}}{\bf E}^{\pi}\bigg\{\sum_{i=t}^n\log\Big(\frac{dq_i(\cdot|y^{i-1}_{i-M},X_i)}{d{\nu}^{\pi}_{i}(\cdot|y^{i-1}_{i-J})}(Y_i)\Big) &\qquad-s\Big(\sum_{i=t}^n\gamma_i(x_i,y^{i-1}_{i-N})-(n+1)\kappa\Big)\Big{|}Y^{t-1}_{t-J}=y^{t-1}_{t-J}\bigg\}
f(p)\ln2= \frac{dF(p)}{dp}\cdot\ln2 = \frac{\zeta^v(\hat{p}_{i1}^v)}{1+p\zeta^v(\hat{p}_{i1}^v)}+\frac{\hat{p}_{i1}^v}{1+\hat{p}_{i1}^v\zeta^v(p)}\frac{d\zeta^v(p)}{dp}&+\sum_{\{j|e_{ij}\in\mathcal{U}_e\}}\frac{0.7\alpha_{e_{ij}}}{1+0.7\zeta^e(p,\hat{p}_{j1}^v)}\frac{d\zeta^e(p,\hat{p}_{j1}^v)}{dp}
B_1 = g(\phi) g'(\phi) \phi + g^2(\phi) - e^{-2\beta}\frac{(1-a)^2}{2}\phi^2 - \frac{1-a}{2}\kappa g^2(\phi) \phi^2 B_2 = \frac{2a}{1-a}e^{-2\alpha}\phi_t^2
I_2- I_1 = 2 (f(v_0)- f(a)) \int_a^b f(v) \, dv + (f^2 (a) - f^2 (v_0))(b-a) = (f(v_0)- f(a))\biggl[ 2 \int_a^b f(v) \, dv -(f(v_0)+ f(a))(b-a) \biggr]\ge 0
(u^3+4tu^2+6t^2u+11u^2+28tu+6t^2+38u+24t+28)b_2 & =(u^3+8tu^2+12t^2u+7u^2+38tu+12t^2+18u+30t+12)c_2 & +(u^3+2tu^2+2tu+9u^2+8u)b_3-(u^3+4tu^2+4tu+3u^2+2u)c_3
u(N^{-1}x,N^{-1}y,N^{-2}\mathfrak{m}(N)^{-1}t) = N^{d-n}\int_{B_1^{d-n}}\sum_{m\in N^{-1}\mathbb{Z}^n \cap B_1^{n}}\hat{g}(N\xi,Nm) e^{i(x\cdot \xi+y\cdot m+t(\frac{\mathfrak{m}(N\xi)}{N^2 \mathfrak{m}(N)}+\frac{\mathfrak{m}(Nm)}{N^2 \mathfrak{m}(N)}))}\, d\xi.
W \supset \frac{1}{\tilde{N}_{c}}\textup{T r}( q \tilde{q}) ( \sum_{k = 1}^{N_{c}- \tilde{N}_{c}}\psi_{k}) - \sum_{k = 1}^{N_{c}- \tilde{N}_{c}}\psi_{k}e_{k}\tilde{e}_{k}+ \mu \Lambda \sum_{k = 1}^{N_{c}- \tilde{N}_{c}}x_{k}\psi_{k}.
{\cal W}_{4}={\frac{1}{8 N}}\oint \oint \oint \oint \mathrm{T r}[{\cal P}( T_{x}^{a}T_{y}^{a}T_{z}^{b}T_{w}^{b}) ] D_{\mu \nu}( x - y ) D_{\rho \sigma}( z - w ) \; d x^{\mu}d y^{\nu}d z^{\rho}d w^{\sigma}\; \; ,
{\rm reg}_{S_1}(S_1/J_1)={\rm reg}_{S_1}(S_1/J(G\setminus N_G[x_1])^kS_1)+k{\rm deg}(u_1) & \leq k{\rm deg}(J(G\setminus N_G[x_1]))+{\rm reg}(S/J(G\setminus N_G[x_1]))-1+k{\rm deg}_G(x_1) & \leq k({\rm deg}(J(G))-{\rm deg}_G(x_1))+{\rm reg}(S/J(G))-{\rm deg}_G(x_1)-1+k{\rm deg}_G(x_1) & \leq k{\rm deg}(J(G))+{\rm reg}(S/J(G))-1.
\begin{aligned}u,w \in L^{\infty}(0,T; H^{m}), v \in L^{\infty}(0,T;H^{m})\cap L^{2}(0,T; H^{m+1}), u_{t},v_{t},w_{t}\in L^{2}(0,T;H^{m-1}).\end{aligned}
G(x,y;B) =\frac{1}{2\pi}\frac{1}{(x-y)^2}\left( \begin{array}{cc}0 e^{-[\chi(x)-\chi(y)]}\; \overline{(\tilde{x}-\tilde{y})}e^{+[\chi(x)-\chi(y)]}\; (\tilde{x}-\tilde{y}) 0\end{array}\right) \; .
\lim_{N\to \infty}\max_{j \in [\![2,N]\!]}\mathbb{E}\left[(\widetilde{\mu}^{j,N}_t,\widehat{\mu}^{1,N}_t)\right]\overset{(\ref{stima_hat})}{\leq}\lim_{N\to \infty}\left\{\frac{1}{N-1}+ \frac{1}{N-1}\sum_{l=2}^N \mathbb{P}\left(\widetilde{X}^{l,N}_t\neq \widehat{X}^{l,N}_t\right)\right\}=0,
\sqrt{\alpha}\frac{\sqrt{R-b}-\sqrt{R-a}}{\sqrt{R-b}+\sqrt{R-a}}\frac{\sqrt{b}-\sqrt{a}}{\sqrt{b+Q}-\sqrt{a+Q}}=1, \frac{-\sqrt{ab}+\sqrt{(a+Q)(b+Q)}-Q}{2}+\sqrt{(R-a)(R-b)}=1.
\frac{\Omega(1-s)}{\Omega(s)}= \frac{\xi_2(-s)}{\xi_2(s-1)}= \frac{\xi_2(s+1)}{\xi_2(s-1)}= \frac{s(s-1)}{\pi^2}\cdot \frac{\zeta(\Delta,s+1)}{\zeta(\Delta, s-1)},