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\begin{align*} L_{1}=\frac{\Gamma(t-1)\left[\xi_{p}\left(1+\xi_{p}^{2}\right)^{-(t-1)}-\xi_{q}\left(1+\xi_{q}^{2}\right)^{-(t-1)}\right]}{2\Gamma(t-\frac{1}{2})\sqrt{\pi}F_{Y}(\xi_{p},\xi_{q})},~ L_{2}&=\frac{F_{Y_{(1)}}(\xi_{p},\xi_{q})}{F_{Y}(\xi_{p},\xi_{q})},\end{align*}

\begin{align*} L_{1}^{\ast}=\frac{\Gamma(t-1)\left[\xi_{p}^{2}\left(1+\xi_{p}^{2}\right)^{-(t-1)}-\xi_{q}^{2}\left(1+\xi_{q}^{2}\right)^{-(t-1)}\right]}{2\Gamma(t-\frac{1}{2})\sqrt{\pi}F_{Y}(\xi_{p},\xi_{q})},\end{align*}

\begin{align*} L_{2}^{\ast}&=\frac{\Gamma(t-2)\left[\left(1+\xi_{p}^{2}\right)^{-(t-2)}-\left(1+\xi_{q}^{2}\right)^{-(t-2)}\right]}{4\Gamma(t-\frac{1}{2})\sqrt{\pi}F_{Y}(\xi_{p},\xi_{q})},\end{align*}

\begin{align*} L_{1}^{\ast\ast}=\frac{\Gamma(t-1)\left[\xi_{p}^{3}\left(1+\xi_{p}^{2}\right)^{-(t-1)}-\xi_{q}^{3}\left(1+\xi_{q}^{2}\right)^{-(t-1)}\right]}{2\Gamma(t-\frac{1}{2})\sqrt{\pi}F_{Y}(\xi_{p},\xi_{q})},\end{align*}

\begin{align*} L_{2}^{\ast\ast}=&\frac{\Gamma(t-2)\left[\xi_{p}\left(1+\xi_{p}^{2}\right)^{-(t-2)}-\xi_{q}\left(1+\xi_{q}^{2}\right)^{-(t-2)}\right]}{4\Gamma(t-\frac{1}{2})\sqrt{\pi}F_{Y}(\xi_{p},\xi_{q})}\\ &+\frac{1}{(2t-5)(2t-3)}\frac{F_{Y_{(2)}}(\xi_{p},\xi_{q})}{F_{Y}(\xi_{p},\xi_{q})},~t>\frac{5}{2},\end{align*}

\begin{align*} \boldsymbol{\mu}=10^{-3}\left(\begin{array}{ccccccccccc}-1.140677\\5.896240\\2.107343\end{array}\right),\mathbf{\Sigma}=10^{-4}\left(\begin{array}{ccccccccccc}19.088935&12.503116&-3.720492\\12.503116&20.268816&-3.162601\\-3.720492&-3.162601&8.851913\end{array}\right).\end{align*}

\begin{align*}\begin{aligned} a ([r] \boxplus [{r'}]))& = \{ a[{x+x'}] : x\in rG,\ x' \in r'G\}\\ & = \{ [{ax+ax'}] : x\in rG,\ x' \in r'G\} =[a [r] \boxplus a[{r'}]);\end{aligned}\end{align*}

\begin{align*} G_p=p\delta_1+(1-p)\delta_\infty,p>p_c\end{align*}

\begin{align*} \frac{\partial}{\partial t} V(t) = \Delta_{g(t)} V(t) + _{g(t)}(V(t)), \end{align*}

\begin{align*}\delta=\begin{cases}0,& ,\\1, & .\end{cases}\end{align*}

\begin{align*}\pi=\bigoplus_{i=0}^{n}c_i\sigma_i,\end{align*}

\begin{align*}\chi_{\sigma_i}(h_k)=\sum_{l=0}^{k}\binom{k}{l}\cdot \binom{n-k}{i-l}(-1)^l.\end{align*}

\begin{align*} \chi_{\pi}(h_i) = \sum_{j=0}^n c_j \cdot \chi_{\sigma_j}(h_i)\end{align*}

\begin{align*}m_{ij} = \chi_{\sigma_j}(h_i).\end{align*}

\begin{align*}a= M\cdot m.\end{align*}

\begin{align*}m = M^{-1} \cdot a.\end{align*}

\begin{align*}w^{C_2^n}(\pi) &= 1+ P(v_1^2, \ldots, v_n^2)\\&=(1 + P(v_1, \ldots, v_n))^2. \end{align*}

\begin{align*}\delta=\begin{cases}0,& ,\\1, & .\end{cases}\end{align*}

\begin{align*}\pi\mid_{C_2^n}= \left(m_0 + \sum_{i=1}^n m_i\right)\mathbb{1} \bigoplus_{i=1}^n r_{i} \rho_i, \end{align*}

\begin{align*} \frac{\partial}{\partial t} h(t) = \Delta_{L,g(t)} h(t). \end{align*}

\begin{align*}r_l = \sum_{\tau \in Y_l} 2 m_{\tau}.\end{align*}

\begin{align*}j_n^*(t_l)=\begin{cases}t_l',& \\0, & \end{cases}\end{align*}

\begin{align*}j_n^*(s_l)=\begin{cases}s_l',& \\0, & \end{cases}\end{align*}

\begin{align*}\beta_{n}^*(v_i)=\begin{cases}v_i',&,\\0, &.\end{cases}\end{align*}

\begin{align*}w_2(\pi) &= \sum\limits_{j=1}^n (c_j/2)\left(\sum\limits_{1 \leq i_1 < i_2 < \cdots < i_j \leq n}(t_{i_1} + t_{i_2}+ \cdots + t_{i_j})\right)\\&=\frac{1}{2} \sum\limits_{j=1}^n c_j\binom{n-1}{j-1}\left(\sum\limits_{i=1}^n t_i\right)\\&=\frac{1}{2}\sum\limits_{j=1}^n c_j\binom{n-1}{j-1}a_2.\end{align*}

\begin{align*} Jm= \sum\limits_{j=1}^n \binom{n-1}{j-1} c_j.\end{align*}

\begin{align*}JM&= \sum_{i=1}^{n}\binom{n-1}{i-1}(1-y)^i(1+y)^{n-i} i-1= \alpha\\&=\sum_{\alpha=0}^{n-1}\binom{n-1}{\alpha}(1-y)^{\alpha+1}(1+y)^{n-1-\alpha}\\&=(1-y)\sum_{\alpha=0}^{n-1}\binom{n-1}{\alpha}(1-y)^{\alpha}(1+y)^{n-1-\alpha}\\&=(1-y)(1-y +1+y)^{n-1}\\&=2^{n-1}(1-y)\\&=[2^{n-1}, -2^{n-1}, 0 , \ldots, 0]. \end{align*}

\begin{align*}Jm&= (1/2^n)JMa\\&= (1/2^n)[2^{n-1}, -2^{n-1}, 0, \ldots , 0 ]{}^t[\chi_{\pi}(h_0),\chi_{\pi}(h_1), \ldots , \chi_{\pi}(h_n)]\\&= (1/2)(\chi_{\pi}(h_0) - \chi_{\pi}(h_1)).\\\end{align*}

\begin{align*}2\binom{n-2}{i-2}\binom{c_i}{2}+c_i^2N_i'(N_i'-1) + c_i^2 \cdot 2 N_i' N_i'' + c_i^2{N_i''}^2\end{align*}

\begin{align*}c_i^2N_i'(N_i'-1) + c_i^2 \cdot 2 N_i' N_i'' + c_i^2{N_i''}^2&=c_i^2\left({N_i'}^2 + 2N_i' N_i''+ {N_i''}^2 - N_i'\right)\\&=c_i^2\left((N_i' + N_i'')^2-N_i'\right)\\&=c_i^2\left(N_i^2-N_i'\right),\,\, ( N_i' + N_i'' = N_i)\\& = c_i\left(N_i-N_i'\right) \pmod 2\\&= c_i N_i'' \pmod 2.\\\end{align*}

\begin{align*}(1 - \lambda ) a_k + \lambda a_{\ell} & = \frac{ 1}{2 \tau} \left( (\tau - \sigma)(\rho + \tau) + (\tau + \sigma)(\rho - \tau) \right) \\ & = \rho - \sigma = b_{\ell} \end{align*}

\begin{align*} \frac{\partial}{\partial t} \varphi_t = \Delta_{g(t),g(t)} \varphi_t. \end{align*}

\begin{align*}w_2(\pi)&=\left(\sum_{1\leq i<j\leq n}N_iN_jc_ic_j+\sum_{i=1}^{n}N_i\binom{c_i}{2}+\sum_{i=1}^{n}c_i^2\binom{N_i}{2}\right)\left(\sum_{t=1}^{n} v_t^2\right)\\&+\left(\sum_{i=1}^{n}c_i N_i''+2\sum\limits_{1 \leq i < j \leq n}N_iN_jc_ic_j\right)\cdot\left(\sum_{k<l}v_kv_l\right). \end{align*}

\begin{align*}\binom{\sum_{i=1}^{n}c_iN_i}{2}&=\sum_{i=1}^{n}\binom{c_iN_i}{2}+\sum_{1\leq l<k\leq n}c_lc_kN_lN_k\\&=\sum_{i=1}^{n}\left(\sum_{k_1+\cdots+k_{N_i}=2}\binom{c_i}{k_1}\cdots\binom{c_i}{k_{N_i}}\right)+\sum_{1\leq l<k\leq n}c_lc_kN_lN_k\\&=\sum_{i=1}^{n}N_i\binom{c_i}{2}+\sum_{i=1}^{n}c_i^2\binom{N_i}{2}+\sum_{1\leq l<k\leq n}c_lc_kN_lN_k.\end{align*}

\begin{align*}w_2(\pi) = \dbinom{\sum\limits_{i=1}^n c_iN_i}{2} \left(\sum_{l=1}^n v_l^2\right) + \left(\sum_{i=1}^n c_iN_i'' \right) \left(\sum\limits_{1 \leq l < k \leq n}v_lv_k\right). \end{align*}

\begin{align*}w(\pi)=\begin{cases} (1+\delta s_1)(1+t_1)^{c_1/2}, & q\equiv 1 \pmod 4,\\(1+v_1)^{c_1}, & q \equiv 3 \pmod 4,\end{cases}\end{align*}

\begin{align*}w(\pi)=\begin{cases} (1+\delta(s_1+s_2))((1+t_1)(1+t_2))^{c_1'/2}(1+t_1+t_2)^{c_2'/2}, & q\equiv 1 \pmod 4,\\(1+v_1)^{c_1'}(1+v_2)^{c_1'}(1+v_1+v_2)^{c_2'}, & q \equiv 3 \pmod 4.\end{cases}\end{align*}

\begin{align*}w_2(\pi)=\begin{cases}\dfrac{m_{\pi}}{2}a_2,&\,\, q\equiv 1\pmod 4\\ \\\dbinom{m_{\pi}}{2}\left(\sum\limits_{i=1}^n v_i^2\right), &\,\, q\equiv 3\pmod 4.\end{cases}\end{align*}

\begin{align*}\chi_{\pi}(h_1)&=2\chi_j(\det(h_1))\\&=2\chi_j(-1)\\&=2(-1)^j.\end{align*}

\begin{align*}\kappa(c_1(\chi^j \circ \det\mid_M))&= \kappa((q+1)ju)\\&=(q+1)j \kappa(u)\\&=0.\end{align*}

\begin{align*}w_2(\pi)=\begin{cases}\dfrac{m_{\pi}}{2}a_2, & q\equiv 1 \mod 4, \\ \\\dbinom{m_\pi}{2}(\sum v_i^2),& q\equiv 3 \mod 4 ,\end{cases}\end{align*}

\begin{align*} [n]_q! &= \prod_{i=1}^{n}\dfrac{q^i-1}{q-1}\\ &=(q+1)(q^2+q+1)\cdots(q^{n-1}+q^{n-2}+\cdots+q+1). \end{align*}

\begin{align*} \Delta V + D_X V = 0, \end{align*}

\begin{align*}\chi_{\pi}(h_k) = [k]_q![n-k]_q! \sum_{1 \leq i_1 < i_2 < \cdots <i_k \leq n}\chi_{i_1}(-1)\chi_{i_2}(-1) \cdots \chi_{i_k}(-1).\end{align*}

\begin{align*}m_{\pi}&=\frac{1}{2}(\dim\pi-\chi_{\pi}(h_1))\\&=\frac{1}{2}\left([n]_q!-[n-1]_q!\left(\sum_{i=1}^n\chi_i(-1)\right) \right)\\&=\frac{1}{2}[n-1]_q!\left(T_{n-1}- \sum_{i=1}^n\chi_i(-1)\right).\end{align*}

\begin{align*}n_{\pi}&=\frac{1}{2}(\dim\pi-\chi_{\pi}(h_2))\\&=\frac{1}{2}[n-2]_q!\left(T_{n-2}T_{n-1}-(1+q)\sum_{i<j}\chi_i(-1)\chi_j(-1) \right).\end{align*}

\begin{align*}\dim\pi_1-\chi_{\pi_1}(h_1)&=(q+1)(q^2+q+1)-(q+1)(1+\chi_{\pi_1}(-1)+\chi_{\pi_1}^{-1}(-1))\\&=(q+1)(q^2+q+1)-(q+1)(1+2(-1)^j)\\&=(q+1)(q^2+q-2(-1)^j)\end{align*}

\begin{align*}n_{\pi}&=\frac{1}{2}[n-2]_q!\left(T_{n-2}T_{n-1}-(1+q)\sum_{i<j}\chi_i(-1)\chi_j(-1) \right).\end{align*}

\begin{align*}m_{\pi_1}&=\frac{1}{2}\left([5]_q!-[4]_q!(1+2(-1)^i+2(-1)^j) \right)\\&=\frac{1}{2}(q+1)^2(q^2+1)(q^2+q+1)\left(q(q+1)(q^2+1)-2(-1)^i-2(-1)^j \right).\end{align*}

\begin{align*}n_{\pi_1}&=\frac{1}{2}\left([5]_q!-[2]_q![3]_q!(2+2(-1)^i+2(-1)^j+4(-1)^{i+j}) \right)\\&=\frac{1}{2}(q+1)^2(q^2+q+1)\left((q^2+1)(q^4+q^3+q^2+q+1)-(2+2(-1)^i+2(-1)^j+4(-1)^{i+j}) \right)\\&=\frac{1}{2}(q+1)^2(q^2+q+1)\left(\{q^6+q^5+2(q^4+q^3+q^2)\}+\{q-1\}-\{2(-1)^i+2(-1)^j)+4(-1)^{i+j}\} \right).\end{align*}

\begin{align*}w_4(\pi)=\begin{cases}\sum\limits_{i=1}^4 t_i^2,& ,\\0, & .\end{cases}\end{align*}

\begin{align*}w_4(\pi)=\begin{cases}\sum\limits_{i=1}^4 t_i^2,&,\\0, & .\end{cases}\end{align*}

\begin{align*}m_\pi &= \dfrac{q^{(1/2)(n(n-1))} - q^{(1/2)(n-1)(n-2)}}{2}\\&=(1/2)\cdot q^{(1/2)(n-1)(n-2)}(q^{n-1}-1)\\&=\dfrac{q-1}{2} \cdot q^{(1/2)(n-1)(n-2)} \cdot (1+q+q^2+ \cdots + q^{n-2}).\end{align*}

\begin{align*} (D^2 \sqrt{\rho})_{\bar{x}}(w,w) &= \int_0^l \chi'(s)^2 \, \langle W(s),W(s) \rangle \, ds - \int_0^l \chi'(s)^2 \, \langle \gamma'(s),W(s) \rangle^2 \, ds \\ &- \int_0^l \chi(s)^2 \, R(\gamma'(s),W(s),\gamma'(s),W(s)) \, ds \end{align*}

\begin{align*}\chi_o(Q_d)=\begin{cases}2 & d \,;\\4 & d \,.\end{cases}\end{align*}

\begin{align*}\mu(y,x)&=\begin{cases}1 & y=x,\\-\sum\limits_{y\leq t<x}\mu(y,t)&y<x,\\0 & y>x.\end{cases}\end{align*}

\begin{align*}{\sim}\left(\sum\limits_{i=1}^{n}a_i\cdot\mathtt{w}^+(\phi_i)\geqslant c\right)\dashv\vdash_{\mathtt{w}\mathsf{BD}}\sum\limits_{i=1}^{n}-a_i\cdot\mathtt{w}^+(\phi_i)>-c\end{align*}

\begin{align*}\pi_k=a_1\cdot\mathtt{w}^+(\phi_1)+\ldots+a_s\cdot\mathtt{w}^+(\phi_s)\geqslant c_k\end{align*}

\begin{align*}\mathtt{m}\left(\bigwedge\limits_{t\in J}\mathsf{cl}'_t\right)=\mathtt{w}^+\left(\bigwedge\limits_{t\in J}\mathsf{cl}'_t\right)-\sum\limits_{\bigwedge\limits_{t'\in J'}\mathsf{cl}'_{t'}\not\dashv~\vdash_{\mathsf{FDE}}\bigwedge\limits_{t\in J}\mathsf{cl}'_t}\mathtt{m}\left(\bigwedge\limits_{t'\in J'}\mathsf{cl}'_{t'}\right)\end{align*}

\begin{align*} (a_1,a_2)\wedge (b_1,b_2) &= (a_1\wedge b_1,a_2\vee b_2) \\ (a_1,a_2)\vee (b_1,b_2) &= (a_1\vee b_1,a_2\wedge b_2)\\ \neg(a_1,a_2) &= (a_2,a_1)\\ (a_1,a_2)\leq (b_1,b_2) &\mbox{ iff } a_1\leq_{[0,1]} b_1 \mbox{ and } b_2\leq_{[0,1]} a_2\\ (a_1,a_2)\leq_i (b_1,b_2) &\mbox{ iff } a_1\leq_{[0,1]} b_1 \mbox{ and } a_2\leq_{[0,1]} b_2 \end{align*}

\begin{align*}v_1(\triangle(p\rightarrow{\sim}\neg p))&=\begin{cases}1&$p$\\0&$p$\end{cases}\end{align*}

\begin{align*}v_1(\alpha)&=f_\alpha(v_1(l_1),\ldots,v_1(l_n))\\v_2(\alpha)&=f_\alpha(v_2(l_1),\ldots,v_2(l_n))\end{align*}

\begin{align*}\alpha_{k,j}&=\begin{cases}\sum\limits_{i=1}^{n}a_i\cdot\mathtt{w}^+(\phi_i)\geqslant1-c&t_{k,j}=f^\sharp_{k,j}\\\sum\limits_{i=1}^{n}a_i\cdot\mathtt{w}^+(\phi_i)<-c&\end{cases}\end{align*}

\begin{align*}\alpha_{k,j}&=\begin{cases}\sum\limits_{i=1}^{n}a_i\cdot\mathtt{b}^+(\phi_i)\geqslant1-c&t_{k,j}=f^\sharp_{k,j}\\\sum\limits_{i=1}^{n}a_i\cdot\mathtt{b}^+(\phi_i)<-c&\end{cases}\end{align*}

\begin{align*} 0 &\leq -2 \, ||^2 + 4n(12+N_0+N_1 r) (N_0+N_1 \sqrt{\rho}) r^2 (r^2-\rho)^{-3} \\ &+ 48n(12+N_0+N_1 r) \rho r^2 (r^2-\rho)^{-4} \\ &\leq - 8n(12+N_0+N_1 r)^2 r^4 (r^2-\rho)^{-4} \\ &+ 4n(12+N_0+N_1 r) (N_0+N_1 r) r^4 (r^2-\rho)^{-4} \\ &+ 48n(12+N_0+N_1 r) r^4 (r^2-\rho)^{-4} \\ &= -4n(12+N_0+N_1 r)^2 r^4 (r^2-\rho)^{-4}\end{align*}

\begin{align*} \mu\left( \bigcup\limits_{i=1}^{n}R_i \right)=\sum\limits_{J\subseteq \{1,\ldots,n\}, J\ne \varnothing} (-1)^{|J|+1} \mu\left( \bigcap\limits_{j\in J} R_i \right).\end{align*}

\begin{align*}I^{(n)}=(\{t^a\mid a/n\in\mathcal{Q}(I^\vee)\})=(\{t^a\mid\langle a, u_i\rangle\geq n\mbox{ for }i=1,\ldots,m\}),\end{align*}

\begin{align*}\overline{I^n}=(\{t^a\mid a/n\in{\rm NP}(I)\})=(\{t^a\mid\langle a, u_i\rangle\geq n\mbox{ for }i=1,\ldots,p\})\end{align*}

\begin{align*}&{\rm (a)}\ \rho(I(G))=1;\quad{\rm(b)}\ \rho_{ic}(I(G))=1; {\rm (c)}\ G\mbox{ is bipartite};\quad{\rm (d)}\\rho(I(G)^\vee)=1; \end{align*}

\begin{align*}H{(a,c)}:=\{x\in{\mathbb R}^s\vert\, \langle x,a\rangle=c\}\ \mbox{ and}\ H^+{(a,c)}:=\{x\in{\mathbb R}^s\vert\, \langle x,a\rangle\geq c\}.\end{align*}

\begin{align*}{\rm RC}(I):=\mathbb{R}_+\{e_1,\ldots,e_s,(v_1,1),\ldots,(v_q,1)\}.\end{align*}

\begin{align*}{\rm RC}(I)=\left(\bigcap_{i=1}^{s+1}H^+_{e_i}\right)\bigcap\left(\bigcap_{i=1}^mH^+_{(\gamma_i,-d_i)}\right)\bigcap\left(\bigcap_{i=m}^p H^+_{(\gamma_i,-d_i)}\right),\end{align*}

\begin{align*}{\rm RC}(I^\vee)=\mathbb{R}_+\{e_1,\ldots,e_s,(u_1,1),\ldots,(u_m,1)\}.\end{align*}

\begin{align*}{\rm RC}(I^\vee)=\left(\bigcap_{i=1}^{s+1}H^+_{e_i}\right)\bigcap\left(\bigcap_{i=1}^qH^+_{(\delta_i,-f_i)}\right)\bigcap\left(\bigcap_{i=q+1}^{p_1} H^+_{(\delta_i,-f_i)}\right),\end{align*}

\begin{align*}&\langle y,(-u_i,1,0,0)\rangle\leq0,\ i=1,\ldots,m,\, \langle y,-e_{s+1}+e_{s+3}\rangle\leq 0,\\&\langle y,-e_i \rangle\leq 0,\, i=1,\ldots,s,\, \langle y,-e_{s+3}\rangle\leq 0,\\&\langle y, (\gamma_j,0,0,0)-d_je_{s+2}+e_{s+3}\rangle\leq 0,\,\langle y, e_{s+2}\rangle\leq 1,\, \langle y, -e_{s+2}\rangle\leq -1. \end{align*}

\begin{align*} \langle X,\nabla (R^{-1}-f) \rangle &= -R^{-2} \, \langle X,\nabla R \rangle - |\nabla f|^2 \\ &= R^{-2} \, \Delta R + 2R^{-2} \, ||^2 - |\nabla f|^2 \end{align*}

\begin{align*}&\langle y,(-u_i,1,0,0)\rangle\leq0,\ i=1,\ldots,m, \\ &\langle y,-e_i \rangle\leq 0,\, i=1,\ldots,s,\end{align*}

\begin{align*}\langle(y_1,\ldots,y_s),\gamma_j\rangle=d_j-y_{s+3}\end{align*}

\begin{align*}\rho_{ic}(I)=\max\left\{(d_if_\ell)/\langle\gamma_i,\delta_\ell\rangle\right\}_{i,\ell}.\end{align*}

\begin{align*}&{\rm (a)}\ \rho(I)=1;\quad{\rm(b)}\ \rho_{ic}(I)=1;{\rm (c)}\ G\mbox{ is bipartite};\quad{\rm (d)}\ \rho(I^\vee)=1. \end{align*}

\begin{align*}\gamma_i=(\underbrace{0,\ldots,0}_{|\mathfrak{A}|},\underbrace{2,\ldots,2}_{|N_G(\mathfrak{A})|},{1,\ldots,1})\end{align*}

\begin{align*}\rho(I_{d,s})=\rho_{ic}(I_{d,s})=\rho_{ic}((I_{d,s})^\vee)=\frac{d(s-d+1)}{s}=\frac{d}{\widehat{\alpha}(I_{d,s})}=\frac{s-d+1}{\widehat{\alpha}((I_{d,s})^\vee)}.\end{align*}

\begin{align*}V(\mathcal{Q}(I))=&\{(0,\,0,\,1,\,1,\,1,\,1,\,1),\,(0,\,1,\,1,\,1,\,1,\,1,\,0),\,(1,\,0,\,0,\,1,\,1,\,1,\,1),\,\\ &(1,\,1,\,0,\,0,\,1,\,1,\,1),\,(1,\,1,\,1,\,0,\,0,\,1,\,1),\,(1,\,1,\,1,\,1,\,0,\,0,\,1),\, \\ &(1,\,1,\,1,\,1,\,1,\,0,\,0),\,({1}/{2},\,{1}/{2},\,{1}/{2},\,{1}/{2},\,{1}/{2},\,{1}/{2},\,{1}/{2})\},\end{align*}

\begin{align*}V(\mathcal{Q}(I))=\{(0,\,0,\,1,\,1),\,(0,\,1,\,0,\,1),\,(1,\,0,\,0,\,1),\,(1,\,1,\,1,\,0),\,({1}/{3},\,{1}/{3},\,{1}/{3},\,{2}/{3})\},\end{align*}

\begin{align*}v(\Box\phi,w)&=\inf\limits_{w'\in W}\{wRw'\rightarrow_\mathsf{G}v(\phi,w')\},&v(\lozenge\phi,w)&=\sup\limits_{w'\in W}\{wRw'\wedge_\mathsf{G}v(\phi,w')\}.\end{align*}

\begin{align*}v(\Box\phi,w)&=\inf\{v(\phi,w'):wRw'\},&v(\lozenge\phi,w)&=\sup\{v(\phi,w'):wRw'\}.\end{align*}

\begin{align*} \Delta U^{(a)} + D_X U^{(a)} &= \Delta U^{(a)} + D_{U^{(a)}} X - [U^{(a)},X] \\ &= \Delta U^{(a)} + (U^{(a)}) - [U^{(a)},X], \end{align*}

\begin{align*}R(w)&=\{w':wRw'=1\},\\R(w)&=\{w':wRw'\}.\end{align*}

\begin{align*}v(\Delta\tau,w)&=\begin{cases}1&v(\tau,w)=1\\0&,\end{cases}&&v(\Delta^\neg\phi,w)&=\begin{cases}(1,0)&v(\phi,w)=(1,0)\\(0,1)&.\end{cases}\end{align*}

\begin{align*}v(\tau,w)\leq v(\tau',w)& v(\Delta(\tau\rightarrow\tau'),w)=1,\\v(\tau,w)>v(\tau',w)& v\left({\sim}\Delta(\tau'\rightarrow\tau),w\right)=1.\end{align*}

\begin{align*}v(\phi,w)\leq v(\phi',w)& v(\Delta^\neg(\phi\rightarrow\phi'),w)=(1,0),\\v(\phi,w)>v(\phi',w)& v(\Delta^\neg(\phi'\rightarrow\phi)\wedge{\sim}\Delta^\neg(\phi\rightarrow\phi'),w)=(1,0).\end{align*}

\begin{align*}v_\mathbf{x}(q_i,w)=\dfrac{|\{[w'\!:\!\mathbf{x}'\!:\!q']\mid[w'\!:\!\mathbf{x}'\!:\!q']\prec[w\!:\!\mathbf{x}\!:\!q_i]\}|}{2\cdot n\cdot|W|}\end{align*}

\begin{align*}\dfrac{w\!:\!1\!:\!\lozenge\phi\!=\!\frac{r}{m+1}}{w\mathsf{R}w'';w''\!:\!1\!:\!\phi\!=\!\frac{r}{m+1}}&&\dfrac{w\!:\!1\!:\!\lozenge\phi\!=\!\frac{r}{m+1};w\mathsf{R}w'}{w'\!:\!1\!:\!\phi\!=\!0\mid\ldots\mid w'\!:\!1\!:\!\phi\!=\!\frac{r-1}{m+1}}\end{align*}

\begin{align*}H_f^-(x,m)=\inf\{H_f(x,m),f'_-(x,0^+,0)m\}?\end{align*}

\begin{align*}u\wedge u^t(\cdot)=\min\{u(\cdot),u^t(\cdot)\},u\vee u^t(\cdot)=\max\{u(\cdot),u^t(\cdot)\}.\end{align*}

\begin{align*}u_n\coloneqq\sum_{i=1}^{n\wedge k} m_i\delta_{x_i}+\sum_{i=1}^{(n2^n)^N}u^d_+(Q^n_i)\delta_{x^n_i}-\sum_{i=1}^{(n2^n)^N}u^d_-(Q^n_i)\delta_{y^n_i},\end{align*}

\begin{align*}H_f(m_\ast)+H_f(m^\ast)=H_f(m_1)+H_f(m_2).\end{align*}

\begin{align*} &\Delta((f+100)^{-8}) + \langle X,\nabla ((f+100)^{-8}) \rangle \\ &= -8 \, (f+100)^{-9} \, (\Delta f + \langle X,\nabla f \rangle) + 72 \, (f+100)^{-10} \, |\nabla f|^2 \\ &\leq -8 \, (f+100)^{-9} + 72 \, (f+100)^{-10} \\ &\leq -(f+100)^{-9}. \end{align*}

\begin{align*}\mu(B_1(x))\le\liminf_{\ell\to\infty}\tau_{-x_{\sigma(\ell)}}\mu_{\sigma(\ell)}(B_1(x))=\liminf_{\ell\to\infty}\mu_{\sigma(\ell)}(B_1(x+x_{\sigma(\ell)}))=0,\end{align*}

\begin{align*}\tau_{-x^i_n}\mu^{i-1}_n =\tau_{-x^i_n}\mu_n-\sum_{0\le j<i}\tau_{-x^i_n+x^j_n}\mu^j,\end{align*}