harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathrm{d}\bar{X}^{(n)}_i(t)=b\big(R(t,\bar{X}^{(n)}_i(t))\big)\,\mathrm{d}t+\sigma\big(R(t,\bar{X}^{(n)}_i(t))\big)\,\mathrm{d}B^{(n)}_i(t),\;\;\;\bar{X}^{(n)}_i(0)=X^{(n)}_i(0), i=1,\,2,\,\ldots,\,n,
\sum_{j = 1}^{\nu + 1}l(\tilde{e}_j) + \sum_{i = 1}^{\mu + 1}l(e_i) = l(\tilde{e}_1) + \cdots + l(\tilde{e}_{\nu + 1}) + l(e_1) + \cdots l(e_{\mu + 1}) = L , \sum_{i = 1}^{\mu + 1}l(e_i) = l(e_1) + \cdots + l(e_{\mu + 1}) = \frac{L N}{2L N - 1}L .
[f(s|D|),a]-sf'(s|D|)\delta(a) =s\int_{-\infty}^{\infty}\int_0^1\hat{f'}(u)\Big(e^{iuvs|D|}\delta(a)e^{iu(1-v)s|D|}-e^{ius|D|}\delta(a)\Big)dvdu &=s\int_{-\infty}^{\infty}\int_0^1\hat{f'}(u)\Big(e^{iuvs|D|}[\delta(a),e^{iu(1-v)s|D|}]\Big)dvdu.
\Delta^a = (-1)^{\epsilon_A}\frac{\delta_L}{\delta \phi^A}\frac{\delta}{\delta \phi^*_{A a}}, \qquad\Delta_\alpha = (-1)^{\epsilon_A}(\sigma_\alpha)_B^{~~\!A}\frac{\delta_L}{\delta \phi^A}\frac{\delta}{\delta \eta_B}.
\prod_{\ell=1}^{\lceil\frac{m}{2}\rceil}(1+z^{2\ell-1})^{B_{2\ell-1}}\left(\sum_{k=0}^\infty z^{2\ell k}\right)^{B_{2\ell}}\le \prod_{\ell=1}^{\lceil\frac{m}{2}\rceil}(1+z^{2\ell-1})^C\left(\sum_{k=0}^\infty z^{2\ell k}\right)^{C}\le \prod_{\ell=1}^{\lceil\frac{m}{2}\rceil}\left(\sum_{k=0}^\infty z^k\right)^{2C}=\eqref{genbound}.
Z^{(c)}_{\rm 1-loop}(M)={(2 \pi x)^{{1\over 2}({\rm dim}H^0_c -{\rm dim}H^1_c )}\over{\rm vol}(H_c)}{\rm e}^{-{1 \over x}S_{\rm CS}(A^{(c)}) -{i \pi \over 4}\varphi}{\sqrt{| \tau^{(c)}_R|}},
\tilde{y}=\frac{y}{l},\tilde{T}=\frac{T}{T_{\infty}},\tilde{T}_{w}=\frac{T_{w}}{T_{\infty}},\tilde{h}=\frac{h}{h_{\infty}},\tilde{h}_{w}=\frac{h_{w}}{h_{\infty}},\tilde{u}=\frac{u}{U},\tilde{\mu}=\frac{\mu}{\mu_{\infty}},\tilde{\tau}_{w}=\frac{\tau_{w}}{\mu_{\infty}U/l}
<T_{\gamma}>= \frac{\sum_{n=-\infty}^{\infty}B_{n}^{S_1}B_{n+\nu}^{S_2}}{\sum_{n=-\infty}^\infty B_n^V}=\frac{\sum_{n=-\infty}^\infty e^{-S_1e^2n^2/2}e^{-S_2e^2(n+\nu)^2/2}}{\sum_{n=-\infty}^\infty e^{-Ve^2n^2/2}}.
\varphi_\delta(s_{l})=\varphi_\delta(s_{l-1}+\varphi_\delta(s_{l-1}))=\frac{(\delta+s_{l-1}+\varphi_\delta(s_{l-1}))^2}{1-(s_{l-1}+\varphi_\delta(s_{l-1}))}=\varphi_\delta(s_{l-1}) \left(1+ \frac{2(\delta+s_l)}{1-s_{l}}\right), R_\delta(0)\leq\frac{2(\delta+s_l)}{1-s_{l}}\leq R_\delta(\sigma),
h_{-2}^{(\mathcal{L}_{1,L+1})_+}(t)z^{-2}+h_{0}^{(\mathcal{L}_{1,L+1})_+}(t)z^{0}+\cdots-(h_{-2}^{\mathcal{L}_{1,L+1}}(t)z^{-2}+h_{0}^{\mathcal{L}_{1,L+1}}(t)z^{0}+\cdots) &=z(h_{-1}^{\mathcal{K}_{10L+1}}(t)z^{-1}+h_{1}^{\mathcal{K}_{10L+1}}(t)z+\cdots).
q_{n}^{m}\left( \gamma\right) =\left({\frac{\gamma\alpha^{2}}{2e}}\right)^{\gamma\alpha^{2}/2}\left({\frac{\pi^{2}}{2\gamma}}\right)^{1/2}\exp\left\{{-2\gamma\int_{\sigma}^{1}{\left\{{f\left({\sigma,t}\right)}\right\}^{1/2}dt}}\right\}.
(v_t^2, m_t) + \int_0^t 2( |\partial_x v_r|^2, m_r) dr = (v_0^2,m_0) - 2 \int_0^t ( u_r^{(1)}\partial_xu^{(1)}_r - u_r^{(2)}\partial_x u^{(2)}_r ,v_r m_r) dr ,
\begin{aligned}\ d(c({f_1,\dots,f_n})):=\sum_i\pm c({f_{i+1},\dots,f_n})\cdot c({f_1,\dots,f_i}) \pm &\sum \pm c({f_1,\dots,f_{i-1},\f_{i+1}\cdot f_i,\f_{i+2},\dots, f_n})+\sum\pm c({f_1,\dots,df_i, \dots,f_n})\end{aligned}
h(Y_A^n |W_C)-h(Y_B^n|W_C) &=I(X_A^n,X_B^n;Y_A^n|W_C) - I(X_A^n,X_B^n;Y_B^n|W_C) &\overset{(a)}{\leq}I(X_A^n,X_B^n;Y_A^n|W_C) - I(X_A^n,X_B^n;Y_B^n|W_C) &+ I(X_A^n;X_B^n|Y_A^n,W_C) &\overset{(b)}{\leq}I(X_A^n;Y_A^n,X_B^n|W_C) + I(X_B^n;Y_A^n|X_A^n,W_C) &- I(X_B^n;Y_B^n|W_C) - I(X_A^n;Y_B^n|X_B^n,W_C),
\begin{array}{l c r}{\xi_{a}^{+ +}= i \zeta_{a}^{+}\overline{{\zeta}}_{a}^{+}= i u_{( \alpha}^{+}u_{\beta )}^{+}\zeta_{a}^{( \alpha}{\overline{{\zeta}}_{a}^{\beta )}}}{\zeta_{a}^{\pm}= u_{\alpha}^{\pm}\zeta_{a}^{\alpha}; \qquad \overline{\zeta}_{a}^{\pm}= u_{\alpha}^{\pm}\overline{{\zeta}}_{a}^{\alpha},}\end{array}
I_n = \int_{\varphi_n > 0}\varphi \, d \mu = \int_{\varphi_n >0, \{\mathcal L}\varphi_n \leq 0}\varphi \, d \mu + \int_{\varphi_n >0, \{\mathcal L}\varphi_n > 0}\varphi \, d \mu.
A(o) = 0 A(j_1) = \frac{1}{4}A(j_1+j_2) = \frac{1}{\pi}A(j_1-j_3) = 1 - \frac{2}{\pi}A(j_1 + j_2 - j_3) = \frac{-1}{4}+ \frac{2}{\pi}.
\textrm{Pr}^{\textrm{OFDMA}}\left\lbrace \textrm{NDN=0}\right\rbrace \to \frac{d_{\textrm{BN}}^{\alpha}\sigma_{\textrm{N}}^{2}(1-\theta)}{\lambda_{\textrm{BN}}(1-\rho)}\left(2^{\frac{R}{1-\theta}}-1\right) \frac{1}{P_{\textrm{B}}}.
\limsup_{\epsilon\to0}\sup_{t\in [a,b ]}|F_\epsilon (t )-F (t ) | =\limsup_{\epsilon\to0}\sup_{t\in [a,b ]}\bigg|\frac{j_\epsilon^2}{2t^2}-\frac{j^2}{2t^2}\bigg|\leq\lim_{\epsilon\to0}\frac{|j_\epsilon^2-j^2 |}{2a^2}=0 .
Z^{N,aV}_{t}:=\int e^{\sum_{l=0}^2 N^{2-l}t F_l^a(\phi^M_\#L^{N}_1,\ldots,\phi^M_\#L^{N}_d,\tau_B^N)}\prod_{k=1}^d dR^{N,W_k}_{\beta,\infty}(\lambda^k)
\varepsilon^{\mu \nu \lambda}\partial_{\mu}X^{9}\bar{S}\gamma_{9}\gamma_{\nu}\psi_{\lambda}+\sqrt{-G}\partial_{\mu}X^{9}\bar{S}\gamma_{9}\psi^{\mu}=-2\partial_{i}X^{9}S^{T}\gamma_{9}\psi_{j}\varepsilon^{ij},
\widehat{u}(\xi_1,\xi') \ = \ \frac 1{(2\pi)^{\frac{n-1}2}}\widehat A(\xi) + \xi'\cdot\nabla_{\xi'}\widehat{u}(\xi_1,0) + \frac{1}{2}\xi'\cdot D^2_{\xi'}\widehat{u}(\xi_1,\tilde{\xi'})\xi'
\sum_{Y< a \leq t}\Lambda(a)|g(a)| \int_{t-y}^t \frac{du}{u^2}|M_h(u/a)| \leq B\sum_{Y < a \leq t}\frac{\Lambda(a)}{a}\int_{(t-y)/a}^{t/a}\frac{du}{u^2}|M_h(u)| &= B\int_1^Y \frac{du}{u^2}|M_h(u)| \sum_{(t-y)/v < n \leq t/v}\frac{\Lambda(a)}{a}\ll B\frac{y}{t}\int_1^Y \frac{du}{u^2}|M_h(u)|.
(\cos(\varphi_1/2) + \sin(\varphi_1/2)e_1e_2)y(\cos(\varphi_1/2) + \sin(\varphi_1/2)e_2e_1) \qquad{}= (y_1\cos(\varphi_1)-y_2\sin(\varphi_1))e_1 + (y_1\sin(\varphi_1)+y_2\cos(\varphi_1))e_2+y_3e_3+\cdots+y_ne_n,
[\hat{x}^\mu_k,\hat{x}^\nu_l]=0,\,\left[y^\mu_k,y^\nu_l\right]=0,\,\left[x^\mu_k,y^\nu_l\right]=0, [\hat{p}_{x k\mu},\hat{p}_{y l \nu}]=0,\mu,\nu=1,2,3,4,\, k=1,...,N.
\sum_{n\le x}J_k(n)J_k(n-h)=\frac{x^{2k+1}}{2k+1}\prod_p(1-\frac{2}{p^{k+1}})\prod_{p\mid h}\left(1+\frac{1}{p^k(p^{k+1}-2)}\right)+O(x^{2k}).
\det{\mathbf{C}}_{Q,n-k+1}= \prod_{k+1 \le p\le n}\det({\mathbf{I}}-{\mathbf{T}}_{p,k}) = \prod_{k+1 \le p\le n}\, \prod_{T\in{Q \choose p-k+1}}(1-\sigma_T)^{(p-k)!\cdot (n-p+k-1)!}&= \prod_{2 \le m\le n-k+1}\, \prod_{T\in{Q \choose m}}(1-\sigma_T)^{(m-1)!\cdot (n-m)!}
D_{\mu \nu}= \overline{{{P}}}_{\mu \nu}\, \frac{1}{p^{2}+ \Pi_{\mathrm{T}}}+ Q_{\mu \nu}\, \frac{1}{p^{2}+ \Pi_{\mathrm{L}}}+ R_{\mu \nu}\, \frac{1}{p^{2}+ \tilde{\Pi}_{\mathrm{T}}}+ \xi \, \frac{p_{\mu}p_{\nu}}{p^{2}}
(i)~\tilde{\delta}~is~a~left~ derivation~ \Leftrightarrow~(ii)~\lim\limits_{i\in \mathcal{I}}~C\cdot\tilde{\delta}(\rho_i)=0~for~each~C~in~\mathcal{A}~ \Leftrightarrow~(iii)~\sigma(\mathcal{M},\mathcal{M}_{\ast})-\lim\limits_{i\in \mathcal{I}}~\tilde{\delta}(\rho_i)=0.
\left(p^{w-1}- w \right)^{\underline{2j-w}}= \left(p^{j-1}- j \right)^{\underline{j}}\geq \left(2^{j-1}- j \right)^{\underline{j}}\geq \left(2^{j-1}- 2j + 1 \right)^{j}> 2^{j^2-4j}.
\begin{aligned}\mathfrak{s}\,(z_1,z_2,z_3,z_4,z_0) = (z_0,z_4,z_3,z_2,z_1) \,, & \mathfrak{t}\,(z_1,z_2,z_3,z_4,z_0) = (z_2,-z_3-1,-z_1-1,z_4,z_0) \,, & \mathfrak{i}\,(z_1,z_2,z_3,z_4,z_0) = (z_1,z_2,z_3,z_0,z_4)\,.\end{aligned}
\hbar'(p - l_{i}^{(s')}) - (a_s - a_{s'}) = (\hbar'(p - 1) - a_{s}) - (\hbar'(l_{i}^{(s')}- 1) - a_{s'}) &= - (t m_a + \hbar'(a - \dfrac{1}{2})) + (t m_{a'}+ \hbar'(a' - \dfrac{1}{2})) &= \hbar(x - x') - t (m_a - m_{a'}) &= \varepsilon_1 (x - x') + \varepsilon_2 (y - y').
\lim_{n\rightarrow \infty}\left\langle \begin{pmatrix}V_{j,n}(\cdot, t_0) \partial_t V_{j,n}(\cdot,t_0) \end{pmatrix}, \begin{pmatrix}V_{j',n}(\cdot, t_0) \partial_t V_{j',n}(\cdot,t_0) \end{pmatrix}\right \rangle_H = 0
S^{\vee}(gh)(\gamma_P) =\gamma_P\circ \rho((gh)^{-1})=\gamma_P\circ \rho(h^{-1})\rho(g^{-1}) &=S^{\vee}(h)(\gamma_P)\rho(g^{-1})=S^{\vee}(g)S^{\vee}(h)(\gamma_P).
\begin{cases}\begin{array}{ll}\dot{u}=\log\det (u_{\alpha\bar{\beta}})-Au+f(z,t)\;\;\; \mbox{on}\;\Omega\times (0,T), u=\varphi \mbox{on}\;\partial\Omega\times [0,T), u=u_0 \mbox{on}\;\bar{\Omega}\times\{0\}, \end{array}\end{cases}
\Gamma^{+}( \partial_{+}- \frac{1}{2}A \partial_{-}) \psi_{I}+ \Gamma^{-}\partial_{-}\rho_{I}+ \Gamma^{J}\partial_{J}( \psi_{I}+ \rho_{I}) = J_{I}- \frac{1}{8}\Gamma_{I}\Gamma^{J}J_{J}~ ,
\sum_{n \geq 0}(-1)^n \left[\frac{1}{(3n+1)^s}- \frac{1}{(3n+2)^s}\right] = 6^{-s}\left[\zeta\left(s, \frac{1}{6}\right) - \zeta\left(s, \frac{2}{3}\right) + \zeta\left(s, \frac{5}{6}\right) - \zeta\left(s, \frac{1}{3}\right) \right] = 6^{-s}\left(2^s-2\right)\left(3^s-1\right) \cdot \zeta(s).
R_{0}(u)=\prod^{\infty}_{l=1}\left[ \frac{\Gamma (4l\lambda -\frac{2\lambda u}{\pi})\Gamma (4\lambda (l-1)+1-\frac{2\lambda u}{\pi})}{\Gamma ((4l-3)\lambda -\frac{2\lambda u}{\pi})\Gamma ((4l-1)\lambda +1-\frac{2\lambda u}{\pi})}/(u\to -u)\right]
\left\| a^{\frac{\beta}{\beta-\alpha}}\right\|_2 = \left\| a \right\|_{{\frac{2\beta}{\beta-\alpha}}}^{{\frac{\beta}{\beta-\alpha}}}\leq \left(\frac{2\beta}{\beta-\alpha}\| a \|_{\psi_1}\right)^{\frac{\beta}{\beta-\alpha}}.
4!{n\choose 4k - 2d}\frac{1}{2}{4k - 2d \choose 2k-d}\left[{2k - d\choose d}\frac{1}{2}{2k - 2d \choose k -d}\right]^2 = \frac{3n!}{(n - 4k +2d)!\left[ d! \left((k-d)!\right)^2 \right]^2}.
\zeta_{CB}(1) = \frac{1}{3}\frac{\pi}{\sqrt{3}}, \zeta_{CB}(0) = \frac{1}{3}+ \frac{2}{9}\frac{\pi}{\sqrt{3}}, \zeta_{CB}(-1) = \frac{2}{3}+ \frac{2}{9}\frac{\pi}{\sqrt{3}}, \zeta_{CB}(-2) = \frac{4}{3}+ \frac{10}{27}\frac{\pi}{\sqrt{3}}.
\mathcal F{\bigl( ( (J^s-1) \partial_l)^{\partial}\bigr)}(\xi)= - i \partial_{\xi}( (\langle \xi \rangle^s-1) i \xi_l) = \; s \langle \xi \rangle^{s-2}\xi \xi_l+ (\langle \xi \rangle^s -1) e_l, e_l=(\underbrace{0,\cdots,0,1}_l,0\cdots,0), ( (J^s-1) \partial_l)^{\partial}= -s J^{s-2}\partial \partial_l + e_l (J^s-1).
\frac{c_0 I_{m_i}}{z^{r+1}}+\frac{c_1 I_{m_i}}{z^{r}}+\cdots+\frac{c_{r-j}I_{m_i}}{z^{j+1}}+\frac{d_i I_{m_i}}{z^j}+\frac{*}{z^{j-1}}+\cdots, i=1\ldots,l.
{\bf M}{\bf \alpha}={\bf 0}\quad{\bf M}= \left(\begin{matrix}[w_1,w_1]\big|_a^b \dots [w_1,w_{r}]\big|_a^b \vdots \ddots \vdots [w_{r},w_1] \big|_a^b \dots [w_{r},w_{r}]\big|_a^b\end{matrix}\right),{\bf \alpha}= \left(\begin{matrix}\alpha_1 \vdots \alpha_r\end{matrix}\right),
PMI = \{f\in \mathcal{M}|\f(x)^{r}\leq f(x^{r}) r\geq 1 x\in (0,\infty)\}, PMD = \{f\in \mathcal{M}|\f(x)^{r}\geq f(x^{r}) r\geq 1 x\in (0,\infty)\}.
\sum_{n=1}^{\infty}\frac{1}{P_{2n-1}}=\frac{1}{\sqrt{2}}\theta_{2}[(\sqrt{2}-1)^{2}]^{2}=\frac{1}{2\sqrt{2}}[\theta_{3}(\sqrt{2}-1)^{2}-\theta_{4}(\sqrt{2}-1)^{2}]
P_{m;l}\otimes P_{m';l'}\cong \bigoplus\limits_{\nu=\vert l-l' \vert \mathrm{step}2}^{\mathrm{min\left( l+l',2d-4-l-l' \right)}}P_{m+m'+\frac{1}{2}\left( l+l'-\nu \right);\nu}
\int_{J_\varepsilon}F_\varepsilon(r) r dr\lesssim \int_{\{r\le \rho_\varepsilon,t_\varepsilon\ge \gamma_\varepsilon\}}\exp\left(-(1+\varepsilon_0+o(1))t_\varepsilon(r) \right) r dr=o\left(\frac{\mu_\varepsilon^2}{\gamma_\varepsilon^4}\right)\,.
\Delta_{0}\! := \! (-1)^{\tfrac{n_{0}\! +\! 1}{2}}t_{n_{0}}\, , Delta_{k+1}\! :=\! (-1)^{\tfrac{n_{k\! +\! 1}\! -\! n_{k}}{2}}t_{n_{k\! +\! 1}}t_{n_{k}}\, , \ 0 \!\leq\! k \!<\! m\! -\!1 \, , \ 2\! \leq\! m \!\leq \!\infty \ .
\ell_1(\xi,\gamma)=\frac12\int|[d\xi,\gamma]|^2\mu,\ell_2(\xi,\gamma)=\frac12\int|\kappa(d\xi,\gamma)|^2\mu,\ell_3(\xi,\gamma)=\int\big(|d\xi|^2-|\gamma|^2\big)\mu
\begin{aligned}{\bf y}_1={\bf H}_{11}({\bf V}_{r_{1}t_{12}1}{\bf s}_{r_1t_{12}}+{\bf V}_{r_{2}t_{13}1}{\bf s}_{r_2t_{13}})+{\bf H}_{12}({\bf V}_{r_{1}t_{12}2}{\bf s}_{r_1t_{12}}+{\bf V}_{r_{3}t_{23}2}{\bf s}_{r_3t_{23}}) +{\bf H}_{13}({\bf V}_{r_{2}t_{13}3}{\bf s}_{r_2t_{13}}+{\bf V}_{r_{3}t_{23}3}{\bf s}_{r_3t_{23}}),\end{aligned}
h_{\vec{z}}^{\vec{\mu}}(\vec{x})=\prod_{1\le i<j\le k}f(z_{i}, z_{j}) \sum_{\tau \in \mathfrak{S}_{k}}\prod_{i=1}^{k}\left(\frac{z_{\tau^{-1}(i)}}{1+z_{\tau^{-1}(i)}}\right)^{x_{i}}\phi(\tau; \vec{z})(u_{\mu_{1}}\otimes \cdots \otimes u_{\mu_{k}}).
J^{\mu ~ j}= - N_{j b}^{\mu \nu}\bigl ( \varepsilon_{\nu \lambda}\partial^{\lambda}\Phi^{b}- \frac i 2 \varepsilon^{b c d}\overline{{{\chi}}}^{c}\gamma_{\nu}\chi^{d}\bigr ) ={\cal J}^{\mu j}+ N_{j k}^{\mu \nu}K_{\nu}^{k},
\int_{U_0}|\tilde{F}_0-(1-b_{1}(\Psi))f_0F^{2}|^2_he^{-\varphi_1+v_{1}(\Psi_1)-\Psi_1}c(-v_{1}(\Psi_1)) \le \liminf_{j\to+\infty}\int_{U_0}|\tilde{F}_j-(1-b_{1}(\Psi))f_jF^{2}|^2_he^{-\varphi_1+v_{1}(\Psi)-\Psi}c(-v_{1}(\Psi_1)) < +\infty,
{P}_{2n}(q) = \frac{(-1)^n}{\sqrt{2}}{R}_n\big(q^2;\gamma,\delta,N\big), n=0,1,\ldots,N, P_{2n+1}(q) = - \frac{(-1)^n}{\sqrt{2}}\frac{\sqrt{(n+\gamma+1)(N-n)}}{(\gamma+1)N}q{R}_n\big(q^2-\gamma-\delta-2;\gamma+1,\delta+1,N-1\big), \hphantom{P_{2n+1}(q) =}{}n=0,1,\ldots,N-1.
\frac{1}{n}\sum\limits_{k=0}^{n-1}R^{n,k}\le \left|\log\left(a_1^*\delta_0^A\right)\right|\frac{k_1}{n}+|\log\delta_1|\frac{m^n_0}{n}+ \left|\log\left(\frac{\delta_1}{k^* + 2}\right)\right|\frac{k^*}{n}+ |\log\delta_1| + \frac{\left| \log\left( \frac{\delta_1}{2}\right)\right|}{n}.
TVaR_{\alpha}(X)=-\xi-\frac{\lambda}{2\alpha}[exp(\frac{1-2\gamma\delta}{2\delta^{2}})\Phi(\Phi^{-1}(\alpha)-\frac{1}{\delta})-exp(\frac{1+2\gamma\delta}{2\delta^{2}})\Phi(\Phi^{-1}(\alpha)+\frac{1}{\delta})],
\Vert\varphi^a_t +\tilde{\varphi}_{0\,,t}\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}\le \Vert\varphi^a_t\Vert_{L^\infty(\mathbb R;H^2(\mathbb T))}+ C \Vert\varphi_0\Vert_{X_2}&\le C\left\{\Vert\varphi^{(0)}\Vert_{H^3(\mathbb T)}+\Vert\varphi^{(1)}\Vert_{H^2(\mathbb T)}+\Vert\varphi_0\Vert_{X_2}\right\}\,.
\int_0^tI_1\,d\tau = p(t)t^ne^{-(z_\ell+1)t}c(x'_{\overline{J}};\xi,\xi^{\vee})(x'_J)^\alpha e^{-z'\cdot x'_J}&+ Ct^ne^{-\zeta_jt}c(x'_{\overline{J}};\xi,\xi^{\vee})(x'_J)^\alpha e^{-z'\cdot x'_J}.
\|L_{\mu}(s)\|^2_{\mathrm F}\le C \sum_{l=1}^{n-1}(-\lambda_{l,n})^{2\mu}e^{-\lambda^2_{l,n}(t-s)}\|\mathbb U(s)-U(s)\|^2 &\le C_\epsilon(t-s)^{-\frac{1}{4}-\mu-\epsilon}\|\mathbb U(s)-U(s)\|^2 &\le C_\epsilon(t-s)^{-\frac{1}{4}-\mu-\epsilon}\|\mathbb U(s)-\tilde U(s)\|^2+C_\epsilon(t-s)^{-\frac{1}{4}-\mu-\epsilon}\|E(s)\|^2,
\left\vert \widehat{H}_{R}\left( n\right) \right\vert \leq c\frac{1}{R}\frac{1}{\left\vert n\right\vert^{1/2}}+c\sum_{i=1}^{2}\frac{1}{R}\frac{1}{1+\left\vert n_{i}\right\vert}+c\frac{1}{R^{10}}& \leq c\frac{1}{|n|^{3/2}}+c\frac{1}{1+|n_{1}|}\frac{1}{1+|n_{2}|}.
q^{Q}\prod_{k=1}^{Q-1}\frac{qz_i-z_k}{z_i-qz_k}= \frac{\left(z_i-i\tau\kappa \lambda^{-\frac{1}{2}}q^{\frac{1}{2}}\right) \left(z_i +i\kappa \lambda^{\frac{1}{2}}q^{\frac{1}{2}}\right)}{\left(q^{\frac{1}{2}}z_i+i\tau\kappa\lambda^{-\frac{1}{2}}\right) \left(q^{\frac{1}{2}}z_i-i\kappa\lambda^{\frac{1}{2}}\right)}.
\left( \int_M ev^{\ast}ev \frac{\omega^n}{n!}(w_1), w_2\right)_W= \int_M\left(ev^{\ast}ev(w_1),w_2 \right)_W\frac{\omega^n}{n!}= \int_M h_V\left(ev(w_1), ev(w_2) \right)\frac{\omega^n}{n!}=(w_1,w_2)_W.
S = - \frac{1}{2 \pi}\int d^{2}\sigma ~ e ~ \left[ h^{\alpha \beta}\partial_{\alpha}X^{\mu}\partial_{\beta}X_{\mu}- i \bar{\Psi}^{\mu}\rho^{\alpha}\nabla_{\alpha}\Psi_{\mu}+ 2 \bar{\chi}_{\alpha}\rho^{\beta}\rho^{\alpha}\Psi^{\mu}\partial_{\beta}X_{\mu}+ \frac{1}{2}\bar{\Psi}^{\mu}\Psi_{\mu}\bar{\chi}_{\beta}\rho^{\beta}\rho^{\alpha}\chi_{\alpha}. \right] .
{\bf E}\Bigl[\Bigl(\int\limits_0^1 f(s)\,M_\mu(ds)\Bigr)^n\Bigr] = n! \lim\limits_{\varepsilon\rightarrow 0}\Biggl[\int\limits_{\{s_1<\cdots< s_n\}}\prod_{r=1}^n f(s_r) \, \exp\Bigl(\mu\sum_{p=1}^n\sum_{k=1}^{p-1}d(p-k) \,\rho_{\varepsilon}(s_p-s_k)\Bigr)ds\Biggr].
f(\underline{k}, t,\mathbf{p}_{n+1}) = \left( \prod_{j=0}^{k-1}p_j^{}\right) \sum_{j=0}^{k}q_j^{t-k}\prod_{\substack{i=0 i \neq j}}^{k}\frac{1}{q_j^{}- q_i^{}}\,.
{1\over 2}\left(\sum_{a=1}^c (-1)^{\varepsilon_a}t_{i_a}+\sum_{b=1}^s (-1)^{\delta_b}h_{l_b}\right)+\sum_{q_j\not=0}\left((-1)^{\gamma_j}{t_j+ h_k\over 4}\right) \qquad{}+\sum_{q_{j'}\not=0}\left((-1)^{\gamma_{j'}}{t_{'j}- h_k\over 4}\right)+ \sum_{q_l\not=0}\left((-1)^{\zeta_l}{h_l\over 4}\right).
\hat{f}(\xi)=\mathcal{F}[f(x_{1})]=\int_{-\infty}^{+\infty}f(x_{1})e^{i\xi x_{1}}d x_{1}, f(x_{1})=\mathcal{F}^{-1}[\hat{f}(\xi)]=\frac{1}{2\pi}\int_{-\infty}^{+\infty}\hat{f}(\xi)e^{-i\xi x_{1}}d \xi.
L'(x) = \lambda \int_{z=x}^{\infty}\int_{u=0}^{z-x}G(u)V(z-x)f_n(u)\lambda e^{-\lambda(z-x)}du dz - V(x)\int_{x}^{\infty}G(z-x)f_n(z-x)\lambda e^{-\lambda(z-x)}dz.
\widetilde N^{[q]}_\kappa\left(r,\frac{1}{f}\right) = \int_0^r\frac{\tilde n^{[q]}_\kappa(t,1/f)-\tilde n^{[q]}_\kappa(0,1/f)}{t}\,dt + \tilde n^{[q]}_\kappa(0,1/f)\log r.
\mathcal P^\prime (\Phi_1) -\mathcal P^\prime(\Phi_2) = \int_{0}^{1}\left(\frac{d}{d\lambda}\mathcal P^\prime(\lambda\Phi_1+ (1-\lambda)\Phi_2)\right)\,d\lambda &=(\Phi_1-\Phi_2) \int_{0}^{1}\mathcal P^{\prime\prime}(\lambda\Phi_1 +(1-\lambda)\Phi_2) \,d\lambda\,,
\fbox{$\begin{array}{cl}{\bf O}^{+5}: e_{a}e_{a+1}= e_{a+5};{\bf O}^{-5}: e_{a}e_{a+1}= -e_{a+5};{\bf O}^{+3}: e_{a}e_{a+1}= e_{a+3};{\bf O}^{-3}: e_{a}e_{a+1}= -e_{a+3}. \end{array}$}
{\frac{\langle O_{N}^{\dagger}O_{N / 2}O_{N / 2}\rangle}{\sqrt{O_{N}^{\dagger}O_{N}}\sqrt{O_{N / 2}^{\dagger}O_{N / 2}}\sqrt{O_{N / 2}^{\dagger}O_{N / 2}}}}\sim{\frac{1}{2^{N}}}.
\sum_u \| M_{\tilde k}\rho_u \xi\|^2_2 = \sum_u\sum_x|M_{\tilde k}\rho_u \xi (x)|^2=\sum_u\sum_x|k(x^{-1})\xi(xu^{-1})|^2 &= \sum_x |k(x^{-1})|^2 \|\xi\|^2_2 = \|k\|^2_2 \|\xi\|^2_2 ,
0 < Q^{2}< \frac{1 6}{3 \omega_{4}^{2}}\biggl [ - \frac{\ell^{2}}{3}\left( \omega_{4}M + \frac{2 \ell^{2}}{9}\right) + \frac{2 \ell}{3 \sqrt{3}}\left( \omega_{4}M + \frac{\ell^{2}}{3}\right)^{3 / 2}\biggr ] ,
[X_i] = \begin{cases}\tilde{c}_{n+i}(S^{\vee}\otimes E/F^i)(1_{Q(E)}) (-n\leq i < 0) \left(\displaystyle\frac{1}{(2+\beta \tilde{c}_1(S^{\vee}))}\tilde{c}_{n+i}(S^{\vee}\otimes E/F^i)\right)(1_{Q(E)}) (0\leq i \leq n-1).\end{cases}
{}^{*}( d \hat{B})_{\hat{\mu}\hat{\nu}\hat{\rho}}\equiv{\frac{1}{3 !{\sqrt{-{\hat{g}}}}}}\epsilon_{\hat{\mu}\hat{\nu}\hat{\rho}\hat{\lambda}\hat{\sigma}\hat{\tau}}( d \hat{B})^{\hat{\lambda}\hat{\sigma}\hat{\tau}}\, .
\rho(a \cdot x) = \rho(ax) = \sigma(a x) \mu^{-1}= \sigma(a) \sigma(x) \mu^{-1}= \sigma(a) \rho(x), \rho(x \cdot b) = \rho(x b) = \sigma(x b) \mu^{-1}= \sigma(x) \sigma(b) \mu^{-1}= \sigma(x) \mu^{-1}\varsigma(b) = \rho(x) \cdot b.
H_{c}= \int d^{3}x \left[ \frac{1}{2}\pi_{i}^{2}+ \frac{1}{4}F_{i j}F^{i j}+ \frac{1}{2}m^{2}\{( A^{0})^{2}+ ( A^{i})^{2}\}- A_{0}\Omega_{2}\right] ,
\sum{\gamma_i^2\theta_i}+ \sum{\gamma_i^2\bar{\theta_i}}= -k\sqrt{9k^2 + 12}- 4rp \sum{\gamma_i^2\theta_i^2}+ \sum{\gamma_i^2\bar{\theta_i^2}}= (-k \pm 2)\sqrt{9k^2 + 12}- 4rp
i\frac{d}{dt}\langle \psi^{(0)}|\hat x^m\hat p^n|\psi^{(0)}\rangle= \langle \psi^{(0)}|\left[ \hat x^m\hat p^n, \frac{1}{2\mu}\hat p^2+V(\hat x )\right]|\psi^{(0)}\rangle=0.
\bar{w}\int_{\bar{w}}^{w}\bar{w}^{\beta}\left({\frac{w}{\bar{w}}}\right)^{\alpha}{\frac{d w}{w^{2}}}= \bar{w}^{1 + \beta - \alpha}{\frac{w^{\alpha - 1}}{\alpha - 1}}-{\frac{w^{\beta}}{\alpha - 1}},
\tilde{V}\equiv \tilde{\sigma}^{2}+ \frac{2 - 5 \tilde{\sigma}^{2}}{5 \, \tilde{v}( 0 , 0 )}\ \ = \ \V \equiv \sigma^{2}+ \frac{2 - 5 \sigma^{2}}{5 \, v ( 0 , 0 )}.
\sum_{\mu}\Delta_\mu\big(a^2/t(pqt)^{1/2}|;q,t^2;p\big)\hat{R}^*_{\mu^2}\big(\hat{x};ta,(pqt)^{1/2}/a;q,t,1;p\big)=\prod_{i<j}\frac{\big((pq t)^{1/2}x_ix_j;p,q\big)}{\big((pq/t)^{1/2}x_ix_j;p,q\big)}.
\sup_{{\bar{\tau}}< t \le T}|{\bar{B}}^n_0(t) -{\bar{B}}^n_0({\bar{\tau}})| & \le \sup_{{\bar{\tau}}< t \le T}|{\bar{X}}^n_0(t) -{\bar{X}}^n_0({\bar{\tau}})| + \sup_{{\bar{\tau}}< t \le T}|{\bar{\eta}}^n(t) -{\bar{\eta}}^n({\bar{\tau}})| + \sum_{k=1}^\infty |k-2| \sup_{{\bar{\tau}}< t \le T}|{\bar{B}}^n_k(t) -{\bar{B}}^n_k({\bar{\tau}})|.
i(2\pi)^4 G(\xi \eta ;{\xi}'{\eta}') = \sum_{123}{\Delta_1}^{-1}{\Delta_2}^{-1}\int d{{\hat q}_{12}}'' M d{\sigma_{12}}''K({\hat q}_{12},{{\hat q}_{12}}'') G({\xi_3}''{\eta_3}'';{\xi_3}'{\eta_3}')
\bar p_{X_{\mathcal{I}},Y_{T^c}}(x_\mathcal{I},y_{T^c}) =\lim_{\ell \rightarrow \infty}p_{X_{\mathcal{I},Q_{n_\ell}},Y_{T^c,Q_{n_\ell}}}^{(\lambda_{n_\ell}, T)}(x_\mathcal{I},y_{T^c})
\displaystyle\sum_{\substack{g \in \mathbb{F}_{q}[x]_{\infty}^{2}g(x_{1}), \cdots, g(x_{r}) \neq 0}}q^{-N\deg(g)}= \left(\dfrac{1}{1 + q^{-N}}\right)^{r}\left(q^{-1}+ q^{-N}+ \dfrac{1-q^{-1}}{1 - q^{1-N}}\right).
|\mathcal E(B_n/G)_i| = \sum_{(Gx,Gy) \in \mathcal E(B_n/G)_i}1 = \sum_{Gy \in (B_n/G)_{i+1}}\left(\sum_{Gx \lessdot Gy}^{}1 \right) = \sum_{Gy \in (B_n/G)_{i+1}}\nu(\bar y) = \sum_{\lambda \in P_{i+1}(\ell,m)}\nu(\lambda).
b_n = \begin{cases}-\frac{1}{2}[a(a+j)+c(c+j)+n(N-n)] begin{align*}1em]-a^2-\frac{1}{2}j(1+a-c)(1+a+c+j)+\alpha (a-c)(1+j)(a+c+j) begin{align*}1em] -a^2-\frac{1}{2}j(1+a-c)(1+a+c+j)+(1-\alpha)(a-c)(1+j)(a+c+j) \end{cases}
e(G) \ge \sum_{i=1}^{k}\sum_{v\in V_i^-\setminus U}d(v)\ge \sum_{i=1}^{k}|V_i^-\setminus U| d(v_i)\ge (n - |V_i^+| - |U|)\sum_{i=1}^{k}d(v_i) &\ge \alpha(k,r)\left(n - \frac{2\alpha(k,r)}{d}n - C_{k,d}\right)= \alpha(k,r)n - \left(\frac{2\alpha(k,r)^2}{d}+ \frac{\alpha(k,r)C_{k,d}}{n}\right)n\ge \alpha(k,r)n - \varepsilon n
K(h^{\frac{1}{2^n}},2)^{r_n}a\sharp_{\nu}b \leqslant a\nabla_{\nu}b - \sum_{k=0}^{n-1}r_{k}\big[\big(a^{1-\frac{m_k}{2^k}}b^{\frac{m_k}{2^k}}\big)^{\frac{1}{2}}-\big(a^{1-\frac{m_k+1}{2^k}}b^{\frac{m_k+1}{2^k}}\big)^{\frac{1}{2}}\big]^{2}\leqslant K(h^{\frac{1}{2^n}},2)^{R_n}a\sharp_{\nu}b,
E_Y \left[ Y \exp\left({-RY}\right)\right] =\frac{1}{\Omega_Y}\int_{0}^\infty y \exp\left({- \left( R + \frac{1}{\Omega_Y}\right)}\!\right) \mathrm{d}y &= \frac{\Omega_Y}{\left( 1+ \Omega_Y R \right)^2},
\frac{2}{n}[26n-\frac{6}{n}\cdot\frac{n(n+1)}{2}-\frac{12}{n^{2}}\cdot(\frac{n^{3}}{3}+\frac{n^{2}}{2}+\frac{n}{6})-\frac{8}{n^{3}}\cdot(\frac{n^{4}}{4}+\frac{n^{3}}{2}+\frac{n^{2}}{4})]
\dot{\alpha}- E =\frac{\langle \mathcal{H}''_E \mathcal{S}\sigma \Phi, \phi \rangle_{\ell^2}+ \langle \sigma \Phi, N(\phi) \rangle_{\ell^2}}{\| \Phi \|_{\ell^2}^2 + \langle \Phi, \phi \rangle_{\ell^2}}.
y_k = \frac{P}{N_{\rm MS}}\textbf{h}_k^{\rm H}\textbf{p}_k x_k + \frac{P}{N_{\rm MS}}\sum \limits_{k' \neq k}\textit{\textbf{h}}_k^{\rm H}\textbf{F}\textbf{p}_{k'}x_{k'}+ n_k,
\tilde{\Theta}_\pm A\varphi =\tilde{\Theta}_\pm \gamma^{(2n+1)}(P_{\rm F}-D_aP_{\rm F}D_a)\varphi &=\gamma^{(2n+1)}\tilde{\Theta}_\pm (P_{\rm F}-D_aP_{\rm F}D_a)\varphi=A\tilde{\Theta}_\pm\varphi
\mathbf{y}(t)=\left[\mathbf{H}_{s1}(t)\,\, \mathbf{H}_{s2}(t)\,\, \cdots \,\, \mathbf{H}_{sN}(t)\right]\begin{bmatrix}\mathbf{x}^1(t) \mathbf{x}^2(t-\tau_2) \vdots \mathbf{x}^N(t-\tau_N)\end{bmatrix}+\mathbf{n}(t)
\widehat J_{\varphi}(\mathfrak{m}_0, \rho, (u_t)_{t=0}^{T-1}):= \mathbb{E}_{\varphi}\left[ \sum_{t=0}^{T-1}f\left(t,X_{t},\mu_t,u_t)\right) +F\left(X_T,\mu_T\right) \right].
\widehat \Upsilon_T^u(\lambda_i) \stackrel{\cal L}{=}\frac{1}{N}\sum_{n=0}^{N-1}w_n^{\sf H}\Sigma_T(\lambda_i) w_n = \frac{1}{N}\sum_{n=0}^{N-1}\sum_{t=0}^{T-1}|w_{n,t}|^2 \sigma_t
\gamma_{\xi}^{( 0 )}= - \frac{1}{3 2 \pi^{2}}{\cal M}_{\xi}^{4}( X ) \, \ln \frac{-{\cal M}_{\xi}^{2}( X )}{e^{3 / 2}\mu_{d}^{2}}+ \frac{i}{3 2 \pi}{\cal M}_{\xi}^{4}( X ) ({\cal M}_{\xi}^{2}( X ) < 0 ) .
a\nabla_{\nu}b \geqslant a\sharp_{\nu}b+\sum_{k=0}^{\infty}r_{k}\big[\big(a^{1-\frac{m_k}{2^k}}b^{\frac{m_k}{2^k}}\big)^{\frac{1}{2}}-\big(a^{1-\frac{m_k+1}{2^k}}b^{\frac{m_k+1}{2^k}}\big)^{\frac{1}{2}}\big]^{2}.
\frac{1}{N^{d-m_0}}\sum_{j_1, j_2, \ldots, j_{d}}\prod_{p=1}^{m_0-1}(T^{k_p})_{j_{p}j_{p+1}}\Big( \prod_{i=1}^{m_1}G_{x_i y_i}\prod_{l=1}^{m_2}\big(G_{w_lw_l}-m_{sc}\big)(t,z) \Big),
\sigma_\mathrm{I_{ij}}^2 \triangleq \frac{\mathbb{E}\{\mathrm{I}_{ij}^2\}}{\mathbb{E}\{\mathrm{I}_{ij}\}^2}-1 = \frac{\alpha_{ij}+2\alpha_{ij}\rho_{ij}+2}{\alpha_{ij}(1+\rho_{ij})^2}.
\mid \alpha , \lambda > = \{g ( \mid \alpha \mid ) \}^{- \frac{1}{2}}\sum_{m = 0}^{\infty}\frac{( \frac{\alpha}{2})^{m}}{\{m ! \Gamma ( m + \lambda + \frac{3}{2}) \}^{\frac{1}{2}}}\mid m , \lambda > ,
\mathbf{h}_1^H \underline{\mathbf{X}}+\mathbf{z}_1 = \psi(B_1 \oplus A_2) \frac{\mathbf{h}_1^H\mathbf{h}_3^\perp}{\sqrt{3}|\mathbf{h}_3^\perp|}+ \psi(A_3 \oplus C_1)\frac{\mathbf{h}_1^H\mathbf{h}_2^\perp}{\sqrt{3}|\mathbf{h}_2^\perp|}+ \mathbf{z}_1.
\beta_{i}=\alpha_{i}+\widetilde{M}_{i}-\widetilde{N}_{i}-\frac{1}{2}(M_{i}-N_{i}+1), \gamma_{i}=\alpha_{i}-\widetilde{M}_{i}+\widetilde{N}_{i}-\frac{1}{2}(N_{i}-M_{i}+1).
\sigma_n(x)=\displaystyle \frac{\sqrt{(x-a_{-n})(a_n-x)}}{\pi^2}\int_{a_{-n}}^{a_n}\frac{Q'(s)-Q'(x)}{s-x}\frac{ds}{\sqrt{(s-a_{-n})(a_n-s)}}, \, x\in \Delta_n.
\{\alpha^* f_1, \ldots, \alpha^* f_n \}(g) = \Pi (g) (d_g (\alpha^* f_1), \ldots, d_g (\alpha^* f_n)) = \Pi(g) (\alpha^* (d_{\alpha(g)}f_1), \ldots, \alpha^* (d_{\alpha(g)}f_n)) = \alpha_* \Pi (g) (d_{\alpha(g)}f_1, \ldots, d_{\alpha(g)}f_n).