Datasets:
prithivMLmods
/
Latex-KIE

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\begin{align*}\varkappa(G):=\lbrack\varkappa_0(G);\ldots;\varkappa_v(G)\rbrack.\end{align*}
\begin{align*}\mathrm{AP}(G):=(\tau(G);\varkappa(G))\end{align*}
\begin{align*}\begin{aligned}\tau^{(1)}(G) &:=\lbrack\tau_0(G);\tau_1(G)\rbrack,\\\varkappa^{(1)}(G) &:=\lbrack\varkappa_0(G);\varkappa_1(G)\rbrack,\end{aligned}\end{align*}
\begin{align*}\begin{aligned}\tau^{(2)}(G) &:=\lbrack\tau_0(G);(\lbrack\tau_0(H);\tau_1(H)\rbrack)_{H\in\mathrm{Lyr}_1(G)}\rbrack,\\\varkappa^{(2)}(G) &:=\lbrack\varkappa_0(G);(\lbrack\varkappa_0(H);\varkappa_1(H)\rbrack)_{H\in\mathrm{Lyr}_1(G)}\rbrack,\end{aligned}\end{align*}
\begin{align*}d_2(G_S)\le\begin{cases}d_1(G_S)+r & S\ne\emptyset \zeta\notin k,\\d_1(G_S)+r+1 & S=\emptyset \zeta\in k,\end{cases}\end{align*}
\begin{align*}\begin{matrix}(x_{i},x_{j})(x_{k}):=\begin{cases} x_{j} & \mbox{if $k=i$,} \\ x_{i} & \mbox{if $k=j$,} \\ x_{k} & \mbox{if $k\neq i,$ $j$.} \end{cases}\end{matrix}\end{align*}
\begin{align*}\mathrm{cov}(\mathfrak{G}):=\lbrace H\mid \lvert H\rvert<\infty,\ H/H^{\prime\prime}\simeq\mathfrak{G}\rbrace.\end{align*}
\begin{align*}\mathrm{cov}_n(\mathfrak{G}):=\lbrace H\in\mathrm{cov}(\mathfrak{G})\mid 0\le d_2(H)-d_1(H)\le n\rbrace.\end{align*}
\begin{align*}\mathrm{cov}_k(\mathfrak{G}):=\lbrace H\in\mathrm{cov}(\mathfrak{G})\mid\rho\le d_2(H)\le\rho+n\rbrace,\end{align*}
\begin{align*}\begin{aligned}1\,030\,117 &,& 3\,259\,597 &,& 3\,928\,632 &,& 4\,593\,673 &,& 5\,327\,080, \\5\,909\,813 &,& 7\,102\,277 &,& 7\,738\,629 &,& 7\,758\,589 &,& 9\,583\,736, \end{aligned}\end{align*}
\begin{align*}\begin{aligned} 534\,824 &,& 2\,661\,365 &,& 2\,733\,965 &,& 3\,194\,013 &,& 3\,268\,781, \\4\,006\,033 &,& 5\,180\,081 &,& 5\,250\,941 &,& 5\,489\,661 &,& 6\,115\,852, \\6\,290\,549 &,& 7\,712\,184 &,& 7\,857\,048 &,& 7\,943\,761 &,& 8\,243\,113, \\8\,747\,997 &,& 8\,899\,661 &,& 9\,907\,837 &,& && \end{aligned}\end{align*}
\begin{align*}\begin{aligned} 1\,001\,957 &,& 9\,923\,685 &,& 20\,633\,209 &,& 58\,650\,717, \\ 63\,404\,792 &,& 72\,410\,413 &,& 84\,736\,636 &,& 92\,578\,472, \end{aligned}\end{align*}
\begin{align*} X_{ij}(g)=X_{ji}(g^{-1}) ,\end{align*}
\begin{align*} [X_{ij}(g),X_{kl}(g')]=0, \end{align*}
\begin{align*} [X_{ij}(g),X_{kj}(g'g)+X_{ki}(g')]=[X_{i}(p),X_{jk}(g')]=0, & \endline &\end{align*}
\begin{align*} [X_{i}(p),X_{j}(q)]=0,\end{align*}
\begin{align*}\frac{1}{c^2}\frac{\partial^2 u}{\partial t^2} - \nabla \cdot ( a \nabla u) &= g(x,t),~ \mbox{in}~~ \Omega_T, \\ u(x,0) = f_0(x), ~~~u_t(x,0) &= f_1(x)~ \mbox{in}~~ \Omega, \\\partial _{n} u& =-\partial _{t} u ~\mbox{on}~ \partial \Omega_T,\\\end{align*}
\begin{align*}[X_{ij}(g),X_j(p) +X_i(g\cdot p) +\: \underset{h\in \mathrm{stab}(p)}{\sum} X_{ij}(gh)]=0,\end{align*}
\begin{align*} [X_{j}(p),X_i(g\cdot p) +\: \underset{h\in \mathrm{stab}(p)}{\sum} X_{ij}(gh)]=0,\end{align*}
\begin{align*}\omega= \overset{n}{\underset{l=1}{\sum}}\: \: \omega^l.\end{align*}
\begin{align*} \omega^i\wedge \omega^k+\omega^k\wedge \omega^i=(A+B+C+D+ \underset{j\vert j \notin\{i,k\}}{\sum}(E_{j}^1+E_{j}^2+E_{j}^3)) dz_i\wedge dz_k, \end{align*}
\begin{align*}E_j^2=\underset{g,h}{\sum} \frac{[X_{ij}(g),X_{ki}(h)]}{(z_i-g\cdot z_j)(z_k-hg\cdot z_j)}+\frac{-[X_{ij}(g),X_{ki}(h)]}{(z_i-h^{-1}\cdot z_k)(z_k-hg\cdot z_j)}+\frac{[X_{ij}(g),X_{ki}(h)] }{(z_i-h^{-1}\cdot z_k)(z_k - h\cdot \infty )}.\end{align*}
\begin{align*}E_j^3+E_j^{21}&=\underset{g,h\in G}{\sum} \frac{[X_{ij}(g),X_{ki}(h)+X_{kj}(hg)]}{(z_i-g\cdot z_j)(z_k-hg\cdot z_j)}.\end{align*}
\begin{align*}E_j^{22}+E_j^1=\underset{g,h\in G}{\sum} \frac{[X_{ik}(g),X_{jk}(h^{-1})+X_{ji}(h^{-1}g^{-1})]}{(z_i-g\cdot z_k)(z_k-h\cdot z_j)}.\end{align*}
\begin{align*} \omega^i\wedge \omega^k+\omega^k\wedge \omega^i=(A+B+C+D+ \underset{j\vert j \notin\{i,k\}}{\sum}E_{j}^{23}) dz_i\wedge dz_k, \: \: i\neq k.\end{align*}
\begin{align*}B= \underset{p,g}{\sum} \frac{[X_i(p),X_{ki}(g)]}{(z_i-p)(z_k-g\cdot p)}+\underset{p ,g}{\sum} \frac{-[X_i(p),X_{ki}(g)]}{(z_i-g^{-1}\cdot z_k)(z_k-g\cdot p)} +\underset{p,h}{\sum} \frac{[X_i(p),X_{ki}(g)]}{(z_i-g^{-1}z_k)(z_k-g\cdot \infty)}.\end{align*}
\begin{align*}\omega^i\wedge \omega^k+\omega^k\wedge \omega^i=(B_2+B_3+C+D+ \underset{j\vert j \notin\{i,k\}}{\sum}E_{j}^{23}) dz_i\wedge dz_k, \: \: i\neq k.\end{align*}
\begin{align*} a \left( x\right) &\in \left[ 1,d_1 \right],~~ d_1 = const.>1,~ a(x) =1x\in \Omega _{FDM}, \\c\left( x\right) &\in \left[ 1,d_2 \right],~~ d_2 = const.>1,~ c(x) =1x\in \Omega _{FDM}, \\a(x), c(x) &\in C^{2}\left( \mathbb{R}^{3}\right) . \end{align*}
\begin{align*}\omega^i\wedge \omega^k+\omega^k\wedge \omega^i=(B_2+C+D_1) dz_i\wedge dz_k, \: \: i\neq k.\end{align*}
\begin{align*}D_1&=\underset{g}{\sum}\:\:\underset{h\neq 1}{\sum} (w_1(g,(gh)^{-1})-w_2(gh,g^{-1}))\\ &= \underset{g}{\sum} \frac{1}{z_i-g\cdot z_k}\:\underset{h\neq 1}{\sum} [X_{ik}(g),X_{ik}(gh)]( \frac{1}{z_k-h\cdot z_k}+\frac{1}{z_k-h^{-1} \cdot z_k}).\end{align*}
\begin{align*}\rho_{\tilde{\underline{q}}_n}(x_{ij}^g)&=1+F_1(x_{ij}^g\cdot \tilde{\underline{q}}_n)-F_1(\tilde{\underline{q}}_n)+[deg \geq 2]\\&=1+\int_{\tilde{x}_{ij}^g} r^*(\omega) +[deg \geq 2]\\&=1+ \int_{x_{ij}^g}\omega +[deg \geq 2]\\&=1-2i\pi X_{ij}(g) +[deg \geq 2]\end{align*}
\begin{align*} f_\sigma(h_r)( x_{ij}(g)) \sim \left\{ \begin{array}{ll}\:x_{ij}(hg) & si \:r=i \\ \: x_{ij}(gh^{-1}) & si \:r=j \\ \: x_{ij}(g) & sinon \end{array},\right. f_\sigma(h_r)( x_{k}(q)) \sim \left\{ \begin{array}{ll}\:x_{k}(h\cdot q) & si \:r=k \\ \: x_{k}(q) & sinon \end{array} \right. \end{align*}
\begin{align*}f_{\sigma}(h_r)(x_{ij}(g))=\sigma(h_r)(1,x_{ij}(g))\sigma(h_r)^{-1}=(1, C_r( h_r \circ x_{ij}(g)) C_r^{-1}). \end{align*}
\begin{align*} (x_{ij}(g),x_{kl}(h))^\sim=(x_{il}(g), x_{jk}(h))^\sim=(x_{ik}(g), x_{jl}(h))^\sim=1 ,\end{align*}
\begin{align*}(x_{ij}(g),x_{ik}(gh) \vert x_{jk}(h))^\sim&=( x_{jk}(h),x_{ij}(g)\vert x_{ik}(gh))^\sim\endline&=(x_{ik}(gh), x_{jk}(h)\vert x_{ij}(g))^\sim=1,\end{align*}
\begin{align*}(x_{i}(p), x_{jk}(g) )^\sim=(x_{j}(p), x_{ik}(g) )^\sim=(x_{k}(p), x_{ij}(g) )^\sim=1,\end{align*}
\begin{align*}(x_{i}(p), x_{k}(q) )^\sim=1,\end{align*}
\begin{align*} (x_j(p)\vert x_{ij}(gh^0)\vert x_{ij}(gh^1)\vert \cdots\vert x_{ij}(gh^{\vert \mathrm{stab}(p)\vert-1} )\vert x_i(g\cdot p),x_{ij}(g))^\sim=1,\\\end{align*}
\begin{align*} \sum_{j=0}^{n} \cos{2 l(\theta+j\pi) \over n+1}=\Re \left \{ e^{2l\theta i\over n+1}{1-e^{2l\pi i} \over 1- e^{2l\pi i\over n+1}} \right \}=0\;,\end{align*}
\begin{align*}\int \limits_{0}^{t}\int\limits_{\Omega} 2~\frac{1}{c^2}\partial_{tt} u ~\partial_t u~ dxd\tau - \int \limits_{0}^{t}\int\limits_{\Omega} 2 \nabla \cdot (a \nabla u)~\partial_t u~ dxd\tau =2\int\limits_{0}^{t}\int\limits_{\Omega} g~\partial_t u~dxd\tau.\end{align*}
\begin{align*} (x_{ij}(gh^0)\vert x_{ij}(gh^1)\vert \cdots\vert x_{ij}(gh^{\vert \mathrm{stab}(p)\vert-1} )\vert x_i(g\cdot p),x_j(p))^\sim=1,\\\end{align*}
\begin{align*} (X_{u^i_1}\cdots X_{u^i_r})=1,\end{align*}
\begin{align*} (x_{ij}(1),x_{kl}(1))^\sim=(x_{il}(1), x_{jk}(1))^\sim=(x_{ik}(1), x_{jl}(1))^\sim=1 ,\end{align*}
\begin{align*}(x_{ij}(1),x_{ik}(1) \vert x_{jk}(1))^\sim&=( x_{jk}(1),x_{ij}(1)\vert x_{ik}(1))^\sim\endline&=(x_{ik}(1), x_{jk}(1)\vert x_{ij}(1))^\sim=1,\end{align*}
\begin{align*} F_{k} \mathcal{P}(A)=\mathcal{P}(A)\cap F_k A, \: \: k\geq 1, \end{align*}
\begin{align*} [X^{ij}(g),X^{kl}(h)]=[X^{il}(g), X^{jk}(h)]=[X^{ik}(g), X^{jl}(h)]=0 ,\end{align*}
\begin{align*}[X^{ij}(g),X^{ik}(gh) + X^{jk}(h)]&=[ X^{jk}(h),X^{ij}(g)+X^{ik}(gh)]\endline&=[X^{ik}(gh), X^{jk}(h)+ X^{ij}(g)]=0,\endline[X^{i}(p),X^{jk}(g)]&=[X^{j}(p),X^{ik}(g)]=[X^{k}(p),X^{ij}(g)]=0,\end{align*}
\begin{align*} [X^{i}(p),X^{j}(q)]=0,\end{align*}
\begin{align*}[X^{ij}(g),X^j(p) +X^i(g\cdot p) +\: \underset{h\in \mathrm{stab}(p)}{\sum} X^{ij}(gh)]=0,\end{align*}
\begin{align*} [X^{j}(p),X^i(g\cdot p) +\: \underset{h\in \mathrm{stab}(p)}{\sum} X^{ij}(gh)]=0,\end{align*}
\begin{align*}\int\limits_{0}^{t}\int\limits_{\Omega}\partial _{t}( \frac{1}{c^2}\partial_t u^{2}) dxd\tau =\int\limits_{\Omega}\left( \frac{1}{c^2}\partial_t u^{2} \right) \left( x,t\right) dx -\int\limits_{\Omega} \frac{1}{c^2} f_{1}^{2} \left( x,t\right) dx.\end{align*}
\begin{align*} X^{ij}(g)=X^{ji}(g^{-1}).\end{align*}
\begin{align*}\ell(\mathbf p \uplus \mathbf q)=\ell(\mathbf p)\oplus \ell(\mathbf q)\ell(\mathbf p\times \mathbf q)= \ell(\mathbf p)\otimes\ell(\mathbf q)\end{align*}
\begin{align*}\{Y\prec x\}:=\{y\in Y:y\prec x\}.\end{align*}
\begin{align*}[\phi]^+(\alpha_1,\dots,\alpha_n)=\begin{cases}\widetilde\phi(\alpha_1+1,\dots,\alpha_n+1)&\\0&\end{cases}\end{align*}
\begin{align*}[\phi]^\sim(\alpha_1,\dots,\alpha_n)&=\sup^+\{\alpha:\exists \alpha_1'<\alpha_1,\dots,\alpha_n'<\alpha_n\ \alpha<\phi(\alpha_1',\dots,\alpha_n')\}\\&=\sup\{\phi(\alpha_1',\dots,\alpha_n'):\alpha_1'<\alpha_1,\dots,\alpha_n'<\alpha_n\}.\end{align*}
\begin{align*}\varphi^+_{P,2}=\varphi^+_{L,2}\end{align*}
\begin{align*}R_a^b:=\{(k_1,k_2):((k_1,a),(k_2,b))\in R\}\subseteq K\times K.\end{align*}
\begin{align*}R:=\{((k_1,a),(k_2,b)):(k_1,k_2)\in G_a^b:a\in A,b\in B\}\subseteq P.\end{align*}
\begin{align*}\varphi^+(\kappa,\kappa)=\kappa+1.\end{align*}
\begin{align*} \lim_{n \to \infty} \frac{M_n}{\log n} = \frac{3}{2} \end{align*}
\begin{align*}F(t):=\int\limits_{\Omega} \left ( \frac{1}{c^2} \partial_t u^{2} + a \left| \nabla u\right|^{2} \right) \left( x,t\right) dx. \end{align*}
\begin{align*} \lim_{x \to \infty} x^2 \hat{\P}\left(\sum_{|v|=1} e^{-V(v)} \geq e^x \right) = 0.\end{align*}
\begin{align*} \lim_{k \to \infty} n_k \P\left(\xi_1 > (n_k)^{1/2}\right) = \rho.\end{align*}
\begin{align*} \pi_n &= \inf_{|h| \leq \epsilon a_n} \P_h\left( \left|\tfrac{S_{r_n}}{a_n}\right| \leq \epsilon, \tfrac{S_j}{a_n} \in I, \xi_j\leq a_n, j \in [r_n]\right)\\ \bar{\pi}_n &= \inf_{|h| \leq \epsilon a_n} \P_h\left( \tfrac{S_{\Delta_n}}{a_n} \in [c,d], \tfrac{S_j}{a_n} \in I, \xi_j \leq a_n, j \in [\Delta_n] \right)\end{align*}
\begin{align*} \pi_n &= \inf_{|h| \leq \epsilon a_n} \P_h\left( P^{(n)}_T = 0, |S^{(n)}_T| \leq \epsilon, S^{(n)}_s \in I, s \leq T \right)\\ &= \inf_{|h| \leq \epsilon} \P\left( P^{(n)}_T = 0, |S^{(n)}_{T}+h| \leq \epsilon, S^{(n)}_s + h \in I, s \leq T \right),\end{align*}
\begin{align*} \pi_n \geq \min_{\delta \in \{-\epsilon,0\}} \P \left( P^{(n)}_T = 0, S^{(n)}_T+\delta \in (-\epsilon,0), S^{(n)}_s+\delta \in (a, b - \epsilon), s \in [0,T] \right).\end{align*}
\begin{align*} \liminf_{n \to \infty} \pi_n \geq \min_{\delta \in \{-\epsilon,0\}} \P\left( \sigma B_T + \delta \in (-\epsilon, 0), P_T = 0, \sigma B_s \in (a,b-\epsilon), s \in [0,T] \right).\end{align*}
\begin{align*} f_k = \sup_{s \in I_k} f(s), g_k = \inf_{s \in I_k} g(s) h_k = h(k/K).\end{align*}
\begin{align*} \tilde{\xi}(u) = \log \left(\sum_{v \in \Omega(u)} e^{V(u)-V(v)}\right) \xi(u) = \begin{cases} \tilde{\xi}(\pi u) & \mathrm{if} u \neq \emptyset\\ 0 &\mathrm{if} u = \emptyset.\end{cases}\end{align*}
\begin{align*} \lambda^* = \left(\frac{ 3\pi^2 \sigma^2 }{2} \right)^{1/3}.\end{align*}
\begin{align*} \lim_{n \to \infty} \frac{1}{n^{1/3}} \log \P(L_n \leq \lambda n^{1/3}) = \lambda- \lambda^*.\end{align*}
\begin{align*} a \left( x\right) &\in \left[ 1,d \right],~~ d = const.>1,~ a(x) =1x\in \Omega _{FDM}, \\a \left( x\right) &\in C^{2}\left( \mathbb{R}^{3}\right) . \end{align*}
\begin{align*} \lim_{k \to \infty} (n_k)^{2/3}\hat{\P}(\xi_1 > R (n_k)^{1/3}) = \rho.\end{align*}
\begin{align*} \lim_{k \to \infty} \frac{1}{(n_k)^{1/3}} \log \hat{\P} \left( w_{m_a} \in G_{n_k} \right) = - \int_0^{a/A} \frac{\pi^2 \sigma^2}{2(\lambda - f(s))^2} ds - \frac{\rho a}{A}.\end{align*}
\begin{align*} \forall t \in [0,1), \ y'(t) = \tfrac{\pi^2 \sigma^2}{2}(\lambda - y(t))^{-2} + \mu, y(1) = \lambda\end{align*}
\begin{align*} \forall t \in [0,1], f_{\lambda,0}(t) =\lambda - \lambda^*(1-t)^{1/3}\end{align*}
\begin{align*} \Psi^{(n)}_{a,A} = \sup_{y \in I^{(n)}_{m_{a}}} \hat{\P}\left( V(w_j) + y \in I^{(n)}_{m_{a}+j}, j \in [n-m_{a}] \right).\end{align*}
\begin{align*} \lim_{n \to \infty} \frac{1}{n^{1/3}} \log \P(L_n \leq \lambda n^{1/3}) = \lambda - \lambda^*,\end{align*}
\begin{align*} \mathcal{A}(p) = \left\{ \# \left\{ |u| = p : \forall j \leq p, V(u_j) \leq pa \right\} \geq \rho^p \right\}\end{align*}
\begin{align*} S(M,g)=\{x\in M,\;\; \Theta_g(x) = \Theta_g(p)\},\end{align*}
\begin{align*}l\left( H_{\mathfrak{m}}^0\left(\frac{R}{I}\right)\right) = l\left(H_{\mathfrak{m}}^0\left(\frac{R}{I+(s)}\right)\right)-l\left(H_{\mathfrak{m}}^0\left(\frac{R}{(I:s)+(s)}\right)\right)\end{align*}
\begin{align*} k_i \le \lfloor \frac{\sum_{j=1}^{n+1} k_j -n+1}{2} \rfloor \ \mathrm{ for\ all}\ i \in \{1, \ldots, n+1\}.\end{align*}
\begin{align*} u\left( x,t\right) = \tilde u \left( x,t\right) ,\forall \left( x,t\right) \in S_T. \end{align*}
\begin{align*}\mathcal{E}_p(d_1, \ldots , d_{n+1})=\mathrm{min}\{q\mathcal{E}_p(k_1+\epsilon_1, \ldots, k_{n+1}+\epsilon_{n+1})+\sum_{\epsilon_i=0}r_i\}\end{align*}
\begin{align*} \kappa_i \le \lfloor \frac{\sum_{j=1}^{n+1} \kappa_j -n+1}{2} \rfloor \ \mathrm{ for\ all}\ i \in \{1, \ldots, n+1\}.\end{align*}
\begin{align*}\mathcal{E}_p(\kappa_1, \ldots, \kappa_{n+1}) = \mathrm{min}\{ \left \lceil \frac{\sum_{i=1}^{n+1} \kappa_i - n +1 }{2} \right\rceil, p\}\end{align*}
\begin{align*}\sum_{i=1}^3 |qz_i - d_i| \le \sum_{i=1}^3 d_i -2 q\lceil \frac{\sum_{i=1}^3 (k_i + \epsilon_i)-1}{2}\rceil - 2(\sum_{\epsilon_i=0}r_i).\end{align*}
\begin{align*}\mathcal{E}_p(k_1, \ldots, k_{n+1})=\mathrm{min} \left\lbrace \lceil \frac{\sum_{i=1}^{n+1} k_i - n+1 }{2} \rceil , p \right \rbrace\end{align*}
\begin{align*}(x_1 + \ldots + x_n)^p = x_1^p + \ldots +x_n^p\end{align*}
\begin{align*}g(x_1+ \ldots + x_n)^{k_{n+1}} =a_1x_1^{k_1}+ \ldots + a_nx_n^{k_n}\end{align*}
\begin{align*}\lceil \frac{\sum_{i=1}^{n+1} l_i - n+1 }{2} \rceil \ge l_1 + l_2 -1\end{align*}
\begin{align*}\mathcal{E}_p(r_1, \ldots, r_{n+1}) \ge \mathcal{E}_p(r_1, \ldots, r_n, u_{n+1}q') = \mathcal{E}_p(u_1+ \epsilon_1, \ldots, u_n + \epsilon_n, u_{n+1}) q' + \sum_{i\le n, \epsilon_i=0} v_i\end{align*}
\begin{align*}\mathcal{E}_p(k_1+ \epsilon_1, \ldots, k_{n+1}+\epsilon_{n+1})=\mathrm{min}(E+\lfloor \frac{I}{2}\rfloor, p)\end{align*}
\begin{align*} u\left( x,t\right) = \tilde u \left( x,t\right) ,\forall \left( x,t\right) \in S_T. \end{align*}
\begin{align*}\mathcal{E}_p(k_1+ \epsilon_1, \ldots, k_{n+1}+\epsilon_{n+1})=\mathrm{min}(E+\lceil \frac{I}{2}\rceil, p)\end{align*}
\begin{align*}{\mathcal Min}(k_1q+r_1, \ldots, k_{n+1}q+r_{n+1})=\mathrm{min}\{Eq + \lfloor \frac{I}{2} \rfloor q+ r_{i_1} + \ldots + r_{i_{n-I+1}} , pq\}\end{align*}
\begin{align*}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)\end{align*}
\begin{align*}\mathcal{E}_p(d_1, \ldots, d_{n+1}) \ge \lceil \frac{\sum_{i=1}^{n+1} d_i - n +1}{2} \rceil\end{align*}
\begin{align*}pq \ge \lceil \frac{\sum_{i=1}^{n+1} d_i - n +1 }{2} \rceil\end{align*}