harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\mathfrak{h}_{-j_a,i_a}\Big(\varphi-\sum_{1\leq i<i_a}d^+_{i,0}(D\varphi)h_{i,1}\Big) &= \sum_{1\leq i\leq g}d^+_{i,1}(\varphi)h_i+\sum_{1\leq s<\gamma}d^0_{s,1}(\varphi)c_s+\sum_{j_a<j\leq g}d^-_{-j,1}(\varphi)h_{-j}.
\begin{multlined}M_{\otimes 0}(A)((\vec{\alpha}_i,\vec{\alpha}_o),(\vec{\beta}_i,\vec{\beta}_o)) =(-1)^{|\vec\alpha_o||\vec\beta_i|}A((\vec\alpha_i\sqcup\vec\beta_i, \vec\alpha_o\sqcup\vec\beta_o))\;,\end{multlined}
B-\hat\nu_m(s)=\begin{cases}A_1\xi'(a)+A_1 [\xi'(s)-\xi'(a)]m +\frac1{|\nu|}, 0\le s\le a, A_1\xi'(s)+\frac1{|\nu|}, a\le s\le q, A_1\xi'(q)+A_2[\xi'(s)-\xi'(q)]+\frac1{|\nu|}, q\le s\le 1.\end{cases}
f(x)=\xi_{1}f(a_{1})+\xi_{2}f(a_{2})+\xi_{3}f(a_{3})= \xi_{1}a_{1}+\xi_{1}\alpha_{2}a_{2}+\xi_{1}\alpha_{3}a_{3}+\xi_{2}((1+\alpha_{2})a_{2}+\beta_{3}a_{3})+\xi_{3}(1+\alpha_{2})a_{3}= \xi_{1}a_{1}+(\xi_{1}\alpha_{2}+\xi_{2}+\xi_{2}\alpha_{2})a_{2}+(\xi_{1}\alpha_{3}+\xi_{2}\beta_{3}+\xi_{3}(1+\alpha_{2}))a_{3}.
\begin{array}{r c l}{{I_{2}}}&{{=}}&{{| \vec{\Gamma}\cdot \vec{\Gamma}|^{2}= \sum_{n = 1}^{N}| \Gamma^{( n )}|^{2}\, ,}}\\{{}}&{{}}&{{}}\\{{I_{4}}}&{{=}}&{{- \frac{1}{4}\mathrm{d e t}\left[ \left( \begin{array}{c}{{\vec{\Gamma}}}\\{{\bar{\vec{\Gamma}}}}\\ \end{array}\right) \left( \begin{array}{c c}{{\vec{\Gamma}}}&{{\ \}}\\{{b a r{\vec{\Gamma}}}}\\ \end{array}\right) \right] \, .}}\\ \end{array}
\mathcal{C}_{23}:= \left[\frac{2A+1}{\log q}-iq,-2\delta-\frac{c}{\log^{3/4}q}-iq\right] \hskip 5pt \cup \left[-2\delta-\frac{c}{\log^{3/4}q}-iq,-2\delta-\frac{c}{\log^{3/4}q}+iq\right] \cup \left[-2\delta-\frac{c}{\log^{3/4}q}+iq,\frac{2A+1}{\log q}+iq\right],
F_{4}(X)= |B^{*}_{U}X|^{2}+|B^{*}_{V}X|^{2}= |B^{*}_{V}JX|^{2}+|B^{*}_{U}JX|^{2}= |B^{*}_{U}Y|^{2}+|B^{*}_{V}Y|^{2}= F_{4}(Y)
{\cal L}^{(1)}_{YM}(D\rightarrow 4,2)=\frac{\hbar g^{2}}{32\pi^{2}(4-D)}\left( \frac{11}{3}C+\frac{1}{6}T_{s}-\frac{4}{3}T_{f}\right)F^{a}_{\mu\nu}F^{a}_{\mu\nu}.
\hat{\eta}= \sqrt{{\frac{N - 1}{N}}}\left( \begin{array}{c c c c}{\eta}&{}&{}&{}\\{}&{-{\frac{\eta}{N - 1}}}&{}&{}\\{}&{}&{\ddots}&{}\\{}&{}&{}&{-{\frac{\eta}{N - 1}}}\\ \end{array}\right) \ .
\prod\limits_{j\in [n_1+1,n_2]\setminus\delta}(\delta_{j,\sigma(j)}+B(j,\sigma(j)))=\sum\limits_{\delta'\subseteq [n_1+1,n_2]\setminus\delta}\prod\limits_{j\in[n_1+1,n_2]\setminus(\delta\cup\delta')}\delta_{j,\sigma(j)}\prod\limits_{j\in\delta'}B(j,\sigma(j)).
\rho_0 = \rho_{00}+ \delta \rho_{01}+ \delta^2 \rho_{02}+ \dots \ , m_0 = m_{00}+ \delta m_{01}+ \delta^2 m_{02}+ \dots \ , n_0 = n_{00}+ \delta n_{01}+ \delta^2 n_{02}+ \dots \ ,
\sum_{i=1}^d \|w_i\|_{\lambda_i}^2=\sum_{i=k+1}\|u_i\|_{\lambda_i}^2 + \sum_{i=1}^{k}\|(X_0)_iu\|^2_{\lambda}=\sum_{i=k+1}^d\|u_i\|_{\lambda_i}^2 + \|u\|^2_{\lambda}\le\sum_{i=1}^k\|u_i\|_{\lambda_i}^2.
[\begin{matrix}1&2&3&4&5\\ 2&3&5&1&4\\ 3&5&4&2&1\\ 4&1&2&5&3\\ 5&4&1&3&2\end{matrix}][\begin{matrix}1&2&3&4&5\\ 2&4&1&5&3\\ 3&5&4&2&1\\ 4&1&5&3&2\\ 5&3&2&1&4\end{matrix}]
(a,b,c)=\Phi^{-1}(s,t,u)=\frac{4}{2st-t^{2}-u^{2}}(\textstyle{\frac{t}{2},s-\frac{t}{2},\frac{u}{2}})=\frac{1}{\alpha}(\textstyle{\frac{t}{2},s-\frac{t}{2},\frac{u}{2}})
P(\nu,h^{1},h^{2}|h^{3})=\frac{1}{Z(\psi,h^{3})}exp(\sum_{ij}W_{ij}^{(1)}\nu_{i}h_{j}^{1}+\sum_{jl}W_{jl}^{(2)}h_{j}^{1}h_{l}^{2}+\sum_{lm}W_{lm}^{(3)}h_{l}^{2}h_{m}^{3}).
\! \! \! \! \!\frac{1}{g^{2}_{YM}}\equiv \frac{1}{16 \pi^3 g_s \alpha^{'}}\int_{\cup{\cal C}_{2}^{(i)}}e^{- \phi}B_2 ; \,\,\,\theta_{YM}\equiv \tau_5 4 \pi^4{\alpha^{'}}^2 \int_{\cup{\cal C}_{2}^{(i)}}(C_2 + C_0 B_2).
A_1^{\ast}= \sum_{n \geq 1}\frac{(-1)^n H_n^2}{2}\left(\frac{1}{(2n+1)}- \frac{1}{n}+ \frac{1}{(2n-1)}\right) A_2^{\ast}= \sum_{n \geq 1}\frac{(-1)^n H_n^2}{4}\left(\frac{3}{(2n+1)}- \frac{3}{(2n-1)}+ \frac{1}{(2n+1)^2}+ \frac{1}{n^2}+ \frac{1}{(2n-1)^2}\right).
(F'(v)(\psi)\big)(x) =f'(v(x))\psi(x)=\big(3v^2(x)-1\big)\psi(x),x\in D, \big(F''(v)(\psi_1,\psi_2)\big)(x) =f''(v(x))\psi_1(x)\psi_2(x)=6v(x)\psi_1(x)\psi_2(x),x\in D.
\hat L_{\delta,\mathcal{F},\mathcal{G}}=Lf_{M,\delta,\mathcal{F},\mathcal{G}}^{*}+\frac{\omega'(\delta;\mathcal{F},\mathcal{G})}{\delta}\left\langle K(g^{*}_{\delta,\mathcal{F},\mathcal{G}}-f^{*}_{\delta,\mathcal{F},\mathcal{G}}),Y-Kf^{*}_{M,\delta,\mathcal{F},\mathcal{G}}\right\rangle,
S^{(4)}_{N}(x,y)=K_{N}(x,y)-\psi_{3}(x) \sum_{l=1}^{4}[A_{0}C_{00}^{-1}C_{0}]_{3l}\ \varepsilon \psi_{l}(y)-\psi_{4}(x) \sum_{k=1}^{4}[A_{0}C_{00}^{-1}C_{0}]_{4k}\ \varepsilon \psi_{k}(y),
\prod_{k=0}^{q^{n}-1}\#\{c_{k}\}=|P|^e \prod_{k=1}^{q^{n}-1}|P|^{e-\min\{e,\mu(k)\}}&=|P|^{e+\sum_{k=1}^{q^{n}-1}(e-\min\{e,\lfloor\frac{\log_{q}k}{d}\rfloor\})}&=|P|^{eq^{n}-\sum_{k=1}^{q^{n}-1}\min\{e,\lfloor\frac{\log_{q}k}{d}\rfloor\}}&=|P|^{eq^{n}-(q-1)\Sigma_{k=1}^{n-1}q^{k}\min\{e,\lfloor\frac{k}{d}\rfloor\}},
Q = Q(y,z) &=\min\left\{\frac{\sigma L - kz - \left(\alpha(\sigma y \pm kz) + \widehat{\alpha}z\right)}{\sigma},\frac{\sigma\alpha y \pm \alpha kz + \widehat{\alpha}z - 2kz + \sqrt{(2\sigma L)^2 - 3(\sigma\alpha y \pm \alpha kz - \widehat{\alpha}z)^2}}{2\sigma}\right\}.
(STSTST)_\mathfrak{U}= \,|SG^{-1}_0|_\mathfrak{U}\;|G_0 TG^{-1}_1|_\mathfrak{U}\;|G_1 S\,{\overline{G_1 S}\,}^{-1}|_\mathfrak{U}\;|\overline{G_1 S}\, T\,{\overline{G_1 ST}\,}^{-1}|_\mathfrak{U}&\cdot|\overline{G_1 ST}\,S\,{\overline{G_1 STS}\,}^{-1}|_\mathfrak{U}\;|\overline{G_1 STS}\,T\,{\overline{G_1 STST}\,}^{-1}|_\mathfrak{U}.
\vec{A_{1}}= \left( \begin{array}{c}{x}\\{0}\\{0}\\ \end{array}\right) \vec{A_{2}}= \left( \begin{array}{c}{0}\\{x}\\{0}\\ \end{array}\right) \vec{\phi}= \left( \begin{array}{c}{- x}\\{x}\\{0}\\ \end{array}\right) \dot{\vec{\phi}}= \frac{1}{2}\left( \begin{array}{c}{x}\\{x}\\{0}\\ \end{array}\right) .
\overrightarrow{L}_{n,r_n}=(p_{1,1},\ldots ,p_{t,1}, m_{1,2},\ldots ,m_{q,2},\ldots, m_{1,n},\ldots ,m_{q,n}, p_{1,n},\ldots ,p_{t,n}, \lambda_n) = (L_1,\ldots,L_{nt+nq+1})
\bar{z}_1z_2 + \bar{z}_1^2+g(z_2,\bar{z}_1z_2 + \bar{z}_1^2)=\bar{z}_1 \bigl(z_2 +f(z_2,\bar{z}_1z_2 +\bar{z}_1^2)\bigr) + \bar{z}_1^2 + r(\bar{z}_1) .
\langle f,g\rangle = \mathrm{mes}(\Omega)\int_{\Omega}{fgdr}-\int_{\Omega}{fdr}\int_{\Omega}{gdr}= \mathrm{mes}(\Omega)\int_{\Omega}{gfdr}-\int_{\Omega}{gdr}\int_{\Omega}{fdr}= \langle g,f\rangle.~\Box
\begin{aligned}E_+(H) \coloneqq \int_0^1 \max_{p \in T^*M}H_s(p) ds, E_-(H)\coloneqq -\int_0^1 \min_{p \in T^*M}H_s(p) ds, \| H \|_{\mathrm{osc}}\coloneqq E_+(H)+E_-(H)=\int_0^1 \left(\max_{p \in T^*M}H_s(p) - \min_{p \in T^*M}H_s(p) \right) ds.\end{aligned}
\varepsilon^{2+k-\theta}-\frac{2^{q+1}}{T}\log\frac{3}{\delta}\varepsilon^{1+k-\theta}-\frac{2C_{\theta}}{T}\log\frac{3}{\delta}\varepsilon^k-\frac{2^{q+1}C_kq^k(2+\|f_q\|_\infty)^{k(q-1)}R^k}{T}\varepsilon^{1-\theta}&-\frac{2C_{\theta}C_kq^k(2+\|f_q\|_\infty)^{k(q-1)}R^k}{T}=0.
0 = \left ( -c_{\ell_0}+ 1 \right )\frac{r_{kc_{\ell_0}}}{r_{0c_{\ell_0}}}+ 1 + \sum_{i\in I}\frac{r_{ic_{\ell_0}}}{r_{0c_{\ell_0}}}- \frac{r_{k c_{\ell_0}}}{r_{0c_{\ell_0}}}= -c_{\ell_0}\frac{r_{kc_{\ell_0}}}{r_{0c_{\ell_0}}}+ 1 + \frac{|I|}{c_{\ell_0}- |I|}< \frac{-c_{\ell_0}}{c_{\ell_0}- |I|}+ 1 + \frac{|I|}{c_{\ell_0}- |I|}= 0,
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})\},
\P\big( \sum_{j=1}^m \rho_j Z_j(q_j) > A\big) \geq \P\big( \xi >A \big) - C_0 \frac{\sum_{j=1}^m \rho_j^3}{(\sum_{j=1}^m \rho_j^2)^{3/2}}&\geq 1- \Phi\Big(\frac{6}{\sqrt{\Phi(-2)(1-\Phi(1))}}\Big)- C_0 \frac{\sum_{j=1}^m \rho_j^3}{(\sum_{j=1}^m \rho_j^2)^{3/2}},
|b(n) - g(n)| = \left|\left(n^2 - z_1n + b(z_1) \right) - \left(n^2 - r(n)\cdot n + n \right)\right| &\le n|z_1 - r(n)| + |b(z_1) - n| &< 1.985n + (z_1 - 3) &< 1.985n + \sqrt{2n}&< 2n,
F = \left( \begin{array}{c c c c}{{0}}&{{- B_{z}}}&{{B_{y}}}&{{E_{x}}}\\{{B_{z}}}&{{0}}&{{- B_{x}}}&{{E_{y}}}\\{{- B_{y}}}&{{B_{x}}}&{{0}}&{{E_{z}}}\\{{- E_{x}}}&{{- E_{y}}}&{{- E_{z}}}&{{0}}\\ \end{array}\right) .
E_{l}^{k}=E_{m}^{(0)}+\Delta\epsilon_{l}+\frac{1}{N_{l}^{k}}\langle\Psi_{m}^{(0)}|(V_{l}^{k})^{+}[\hat{H}_{v},V_{l}^{k}]|\Psi_{m}^{(0)}\rangle
\begin{array}{lll}y_i = \prod_{j=1}^s\overline y_j^{b_{ij}}(\overline y_{s+\overline m}+\overline \beta)^{b_{i,s+1}}\mbox{for}1\le i\le s\mbox{and}y_{s+\overline m}= \prod_{j=1}^s\overline y_i^{b_{s+1,j}}(\overline y_{s+\overline m}+\overline \beta)^{b_{s+1,s+1}}\end{array}
\lambda_{1,p}(\beta) \le \frac{\int_0^1 \frac{| v' |^p F_\beta}{|\beta'|_g^{p-1}}\,dt}{\int_0^1 | v |^p F_\beta |\beta'|_g \,dt}= \frac{\int_0^1 \frac{| w' |^p F_\alpha}{|\alpha'|_g^{p-1}}\,dt}{\int_0^1 | w |^p F_\alpha |\alpha'|_g \,dt}
0 =-1-[z^1]g(q,z/q)+[z^1]g(q,z)g(q,z/q) &=-1-q^{-1}[z^1]g(q,z)+(1+q^{-1})[z^0]g(q,z)[z^1]g(q,z) &=-1+[z^1]g(q,z) &=\tilde{C}_0(q)-1,
{}_N\!\langle \varrho'_{n},\bar{\varrho}'_{n}| \varrho_{n},\bar{\varrho}_{n}\rangle_N = \prod_{\mu=0}^{p}\delta(\mathrm{Re}\varrho'_{n\mu}-\mathrm{Re}\varrho_{n\mu}) \delta(\mathrm{Im}\varrho'_{n\mu}-\mathrm{Im}\varrho_{n\mu})~.
\lvert R(n)\lvert <\frac{0.01}{2(\log n)^2}+\frac{4}{3}\cdot \frac{0.01}{(\log n)^3}&=\frac{0.01}{(\log n)^2}\left(\frac{1}{2}+\frac{4}{3\log n}\right)<\frac{0.005586}{(\log n)^2}.
\int_{B_{\alpha}(0)}\frac{r^{2}}{\lambda_{i}}(\frac{\lambda_{i}}{1+\lambda_{i}^{2}r^{2}})^{\frac{n-2}{2}(\frac{n}{n-2}+1)}=\lambda_{i}^{-4}O(\lambda_{i}, \ln \lambda_{i},1)=o(\frac{1}{\lambda_{i}^{n-2}}).
\{\chi_{\mu \nu}\}= \left( \begin{array}{cccc}0 0 0 0 0 \chi_{11}\chi_{12}\chi_{13}0 \chi_{12}\chi_{22}\chi_{23}0 \chi_{13}\chi_{23}\chi_{33}\end{array}\right);~~~ \chi_{11}+ \chi_{22}+ \chi_{33}= 0.
f_1 = x^9 - 3 x^7 - 12 x^6 + 3 x^5 + 6 x^4 - 61 x^3 - 12 x^2 + 24 x - 62, f_2 = x^9 - 3 x^7 - 18 x^6 + 3 x^5 + 18 x^4 - 109 x^3 - 18 x^2 - 214
\epsilon_{1}^{i}= e^{{\frac{1}{2}}A}\tilde{\eta}_{+}^{i}\ , \qquad \epsilon_{2}^{i}= \left( e^{-{\frac{1}{2}}A}- 2 \kappa^{D - 2}W ( \phi_{0}) e^{{\frac{1}{2}}A}x^{\mu}\Gamma_{\underline{\mu}}\right) \tilde{\eta}_{-}^{i}\ .
\sum_{m\geq 0}X^m \cdot\frac{\alpha^{m+1}-\beta^{m+1}}{\alpha-\beta}= \frac{1}{\alpha-\beta}\sum_{m\geq 0}\left( \alpha \cdot (X\alpha)^m -\beta (X\beta)^m\right) = \frac{1}{\alpha-\beta}\left( \frac{\alpha}{1-\alpha X}- \frac{\beta}{1-\beta X}\right) &=\frac{1}{(1-\alpha X)(1-\beta X)}.
I_0(s)=\int_a^{+\infty}e^{-su}(u^2-1)^{\frac{d-2}{2}}\, d u=\int_a^{+\infty}e^{-su}u^{d-2}\Bigl(1-\frac{1}{u^{2}}\Bigr)^{\frac{d-2}{2}}\, d u.
{}= \Bigg \{\sum_{\sigma = 0}^{N - 1}{\frac{w ( x_{3}, x_{1 3}, x_{1}| \sigma + e - h ) w ( x_{4}, x_{2 4}, x_{2}| \sigma + f - b ) s ( \sigma , b - g )}{w ( x_{4}, x_{1 4}, x_{1}| \sigma + e - h ) w ( x_{3}, x_{2 3}, \omega x_{2}| \sigma + f - b )}}\Bigg \}_{0},
\displaystyle E_{H\vert\xi}\left[Q\left(\sqrt{g\frac{\chi_{1}+\chi_2}{N_{0}}}\right)\right]= \int_{x_1=0}^{\tau}\int_{x_2=0}^{\infty}Q\left(\sqrt{g\frac{x_{1}+x_2}{N_{0}}}\right) \dfrac{f_{\chi_1}(x_1)f_{\chi_2}(x_2)}{P(\xi)}dx_2 dx_1.
\sum_{\mathbf{y}\in \mathbb{F}_{p}^{m}}\omega^{\mathbf{y}^{T}(\mathbf{C}+ \mathbf{C}^{T}) \mathbf{z}}=\begin{cases}p^m, (\mathbf{C}+\mathbf{C}^{T})\mathbf{z}=\mathbf{0}; 0, .\end{cases}
s = \langle \mathrm{\boldmath n}\cdot \mathrm{\boldmath j}_{0}\rangle = - \frac{1}{2}\left. \frac{\partial \mathrm{\cal L}_{\mathrm{e f f}}}{\partial V_{0}^{3}}\right|_{V_{\mu}^{3}= 0}= \frac{e B}{4 \pi}.
J =(x_{1},x_{2})\cap (x_{1},x_{4},x_{5})\cap(x_{1},x_{4},x_{7})\cap(x_{1},x_{6},x_{7})\cap(x_{3},x_{4},x_{5}) &\phantom{=..}\cap(x_{3},x_{4},x_{7})\cap(x_{3},x_{6},x_{7})\cap(x_{5},x_{6},x_{7}).
S(x-y)=\sum_{k=-\infty}^{\infty}S^{(k)}(x-y)=-i\frac{1}{Z^{(0)}[0,0,0]}\sum_{k=-\infty}^{\infty}e^{ik\theta}\left.\frac{\delta^2 Z^{(k)}[\eta,\overline{\eta},J]}{\delta\overline{\eta}(x)\delta\eta(y)}\right|_{J,\eta,\overline{\eta}=0}\; .
H^-(\mathbf{u})=\{(x_1, x_2, \dots, x_d)\in \mathbb{R}^d; u_1x_1 + u_2x_2+ \dots +u_d x_d < u_0 \}, & H^+(\mathbf{u})=\{(x_1, x_2, \dots, x_d)\in \mathbb{R}^d; u_1x_1 + u_2x_2+ \dots +u_d x_d > u_0 \}.
\oint_C \frac{d\zeta}{\zeta^j}\hat{f}=\sum_\alpha\oint_{\Gamma_\alpha}\frac{d\zeta}{\zeta^j}\ 2\log\left(\eta(\zeta)-P_{\alpha}(\zeta)\right)+4\mu\oint_0 \frac{d\zeta}{\zeta^{j+1}}\eta.
\int_{-\pi}^{\pi}p(\theta)^2d\theta =\frac{2\pi}{2n+1}\sum_{j=0}^{2n}p\left(\frac{2j}{2n+1}\pi\right)^2\leq\frac{2\pi}{2n+1}\left(\sum_{j=0}^{2n}p\left(\frac{2j}{2n+1}\pi\right)\right)^2= &=\frac{2n+1}{2\pi}\left(\frac{2\pi}{2n+1}\sum_{j=0}^{2n}p\left(\frac{2j}{2n+1}\pi\right)\right)^2=\frac{2n+1}{2\pi}\left(\int_{-\pi}^{\pi}p(\theta)d\theta\right)^2.
3(a) \ \mathrm{for}\k+1 = z(3(a) \ \mathrm{for}\k)+a_{k+1}(3(b) \ \mathrm{for}\k)+a_{k+1}^2(3(a) \ \mathrm{for}\k-1) 3(b) \ \mathrm{for}\k+1 = (2z+1)(3(a) \ \mathrm{for}\k)+a_{k+1}(3(b) \ \mathrm{for}\k)+a_{k+1}(3(c) \ \mathrm{for}\k) 3(c) \ \mathrm{for}\k+1 = -(3(a) \ \mathrm{for}\k)-a_{k+1}(3(c) \ \mathrm{for}\k)
\sum_{\substack{q\leq\Omega (q,Mp_0)=1}}\frac{1}{\varphi(qM)}\sum_{\substack{\psi\pmod{q}\xi\pmod{M}}}\sum_{\substack{n\leq X (n,q)>1}}\Lambda(n)\ll \frac{X^{1/2}}{M^{6/5}}\exp\left(-c^\sharp\sqrt{\log X}\right)(\log X)^2,
\frac{1}{2}\frac{d}{dt}|u_+(t)|^2+\mu_+|u_+|^2 =-(E_{\mu_+}B(u(t),u(t)), u_+(t)) &=-(B(u,u),u_+) &=-(B(u_++u_-+\eta g, u_++u_-+\eta g), u_+) &=-\eta (B(u_-, g), u_+)-\eta(B(g,u_-),u_+),
g(z)= \ \ \ \ 1+ z^{2c_1-1}+ z^{2(2c_1-1)}+ \dots + z^{(k-2)(2c_1-1)}+ z^{c_1+c_2}\left( 1+ z^{2c_1-1}+ z^{2(2c_1-1)}+ \dots + z^{(k-2)(2c_1-1)}\right) + \frac{z^{(k-1)(2c_1-1)}+ z^{c_2+1}}{z^{c_1}+ 1}.
\begin{aligned}k\sum_{j=1}^{n-1}\sum_{l=1}^k \frac{\binom{M_n}l(k)_l(k(j-1))_{M_n-l}}{(kn)_{M_n}}\frac 1{k(n-j+1)-l}= &k\sum_{l=1}^k\binom{c_nkn}l(k)_l\sum_{j=1}^{n-N}\frac{(k(j-1))_{c_nkn-l}}{(kn)_{c_nkn}}\frac 1{k(n-j+1)-l}+ &k\sum_{l=1}^k\binom{c_nkn}l(k)_l\sum_{j=n-N+1}^{n-1}\frac{(k(j-1))_{c_nkn-l}}{(kn)_{c_nkn}}\frac 1{k(n-j+1)-l}.\end{aligned}
u^*_{ij}(x_1,\ldots,x_k;y_1,\ldots,y_k) &= \sum_{i'=1}^k\sum_{j'=1}^k\frac{\pi_{i'j'}x_{i'}y_{j'}}{z}\left[u^*_{ij}(x_1 - \delta_{1i'},\ldots,x_k - \delta_{ki'};y_1 - \delta_{1j'},\ldots,y_k - \delta_{kj'}) + \delta_{ii'}\delta_{jj'}\right].
\begin{array}{llll}L_1 L_2 [z_3,z_1]=z_1, [z_3,z_2]=z_2 L_{3,a}[z_3,z_1]=z_2, [z_3,z_2]=az_1+z_2 L_{4}[z_3,z_1]=z_2, [z_3,z_2]=z_1 L_{5}[z_3,z_1]=z_2\end{array}
\left\{( 1 - x^{2}) \frac{d^{2}}{d x^{2}}+ \left[ \frac m{| m |}\sigma_{z}- ( 2 | m | + 2 ) \, x \right] \frac d{d x}- m \, ( m + 1 ) + \left( \lambda^{2}- \frac 1 4 \right) \right\}\left( \begin{array}{c}{{\xi_{\lambda m}}}\\{{\eta_{\lambda m}}}\\ \end{array}\right) = 0 .
\prod_{i=1}^r\left(\frac{m}{n_i}+1\right) \leq\sum_{i=0}^r\binom{r}{i}\left(\frac{m}{\sqrt[r]{n_1n_2\cdots n_r}}\right)^{r-i}\left(\frac{n}{r}\right)^{\frac{(r-1)i}{r}}& =\left(\frac{m}{\sqrt[r]{n_1n_2\cdots n_r}}+\left(\frac{n}{r}\right)^{1-\frac{1}{r}}\right)^r.
\mu_m(P_{n+\frac{k-2}{2}})=i^{\frac{2-k}{2}-m}C(n)\binom{k-2}{n+\frac{k-2}{2}}^{-1}=i^{\frac{2-k}{2}-m}\sum_{j}\frac{(-1)^{n+j+m}\binom{\frac{k-2}{2}-m}{j}\binom{\frac{k-2}{2}+m}{m-n+j}}{\binom{k-2}{n+\frac{k-2}{2}}}.
f(z) =(z-\alpha_1)^{n_1}\cdots(z-\alpha_s)^{n_s}, f'(z) =N(z-\alpha_1)^{n_1-1}\cdots(z-\alpha_s)^{n_s-1}(z-\beta_1)^{m_1}\cdots(z-\beta_p)^{m_p},
\begin{aligned}{\|Ju_2(t)\|_{L^2}}\le{}\|Ju_2(1)\|_{L^2}{}+ C \int^t_1 (\|u_1\|^2_{L^{\infty}}\|Ju_2\|_{L^2}+ \|u_1\|_{L^{\infty}}\|u_2\|_{L^{\infty}}\|Ju_1\|_{L^2})(\tau) \d\tau. \end{aligned}
W_4(u,z_1)=z_1K_\mu(uz_1)\sum_{s=0}^{N-1}\frac{A_s(z_1)}{u^{2s}}-\frac{z_1}{u}K_{\mu+1}(uz_1)\sum_{s=0}^{N-1}\frac{B_s(z_1)}{u^{2s}}+f(u,z),
l(e_n)^*\Omega =0 l(e_n)^* e_{i_1}\otimes\cdots\otimes e_{i_d}= \sum_{k=1}^d \delta_{n=i_k}q_{n i_1}\cdots q_{n i_{k-1}}e_{i_1}\otimes\cdots \otimes \hat{e}_{i_k}\otimes\cdots\otimes e_{i_d},
W_{n}^{4}= \sum_{i = 1}^{N}\left[ 2 x_{i}^{n + 2}\partial_{i}^{3}+ 3 ( n + 2 ) x_{i}^{n + 1}\partial_{i}^{2}+ ( n + 1 ) ( n + 2 ) x_{i}^{n}\partial_{i}- ( n + 2 ) x_{i}^{n + 1}( 1 - \theta_{i}\Pi_{i}) \partial_{i}^{2}\right]
\int_{\Omega_\ell^+\setminus \Omega_{\ell'+1}}|\tilde u_\ell^+|^2\leq \tilde\alpha \int_{\Omega_\ell^+\setminus \Omega_{\ell'}}\!\!\!|\tilde u_\ell^+|^2~~\tilde\alpha:=\frac{C_A}{\tilde\beta+C_A}\,.
\dot{L}=\sum_{i=1}^{2k+1}L_1L_2\ldots\dot{L}_i\ldots L_{2k+1}&=\sum_{i=1}^{2k+1}L_1L_2\ldots L_iU_{i+1}\ldots L_{2k+1}-\sum_{i=1}^{2k+1}L_1L_2\ldots U_iL_i\ldots L_{2k+1}&=LU_{2k+2}-U_1L=[L,U_1]\, ,
\begin{cases}\displaystyle{\partial_t u_k - i k^3 u_k + \gamma u_k+ \frac{ik}2 \sum_{k_1 + k_2 = k}v_{k_1}v_{k_2}= f_k}\displaystyle{\partial_t v_k - i \alpha k^3 v_k + \gamma v_k + ik \sum_{k_1 + k_2 = k}u_{k_1}v_{k_2}= g_k}.\end{cases}
D(P(X|Y)\|Q(X|Y)) =\int \log \left(\frac{dP_{X|Y}}{dQ_{X|Y}}(x,y)\right)P(d(x,y)) &=\int \left(\int \log \left(\frac{dP_{X|Y}}{dQ_{X|Y}}(x,y)\right)P(dx|Y=y)\right)P(dy)
\int_{t_1}^{t_2}\int_{|x|<t}\,\partial^{\alpha}\rho\,\partial_{\alpha}u\,\left(x^{\beta}\partial_{\beta}u+\left(\frac{d}{2}-1\right)u\right)\,dxdt=O\left(\epsilon^{-1}\left(\log{\frac{t_2}{t_1}}\right)^{\frac{1}{d}}\right).
\sum_{k=0}^{n}{n\choose k}2^k F_{jk}(\sqrt{5}F_j)^{n-k}\big ( 2^{1-(n-k)}- 1 \big ) B_{n-k}= n F_j 2^{1-n}\Big ( (L_j + 2\alpha^j)^{n-1}+ (L_j + 2\beta^j )^{n-1}\Big ) & = n F_j 2^{1-n}\sum_{m=0}^{n-1}{n-1\choose m}2^m L_{jm}L_j^{n-1-m}.
\Phi_\alpha (x_k (\beta)) - \Phi_\alpha (x_k) = R_{\Phi_\alpha}(x_k(\beta), x_k) + \langle \nabla \Phi_\alpha (x_k), x_k(\beta) - x_k \rangle & = R_{\Phi_\alpha}(x_k(\beta), x_k) - \frac{1}{\beta}\langle J_p (x_k(\beta)) - J_p (x_k), x_k(\beta) - x_k \rangle.
\left\{\frac{{\zeta_d}^{kb}T_0}{z^{\frac{b}{d}+1}}+\frac{{\zeta_d}^{k(b-1)}T_1}{z^{\frac{b-1}{d}+1}}+\cdots+\frac{{\zeta_d}^{k}T_{b-1}}{z^{\frac{1}{d}+1}}+\frac{\Theta}{z}\, \Bigg|\,k=0,\ldots,d-1 \right\},
\begin{array}{c}{2 \pi{\cal E}= \int_{M_{3}}d ( \mathrm{t r}( \Phi_{A}H^{A}) ) + \int_{M_{3}}d ( \mathrm{t r}( \Phi_{B}K^{B}) )}{= \int_{S^{2}}\mathrm{t r}( \Phi_{A}H^{A}) + \int_{S^{2}}\mathrm{t r}( \Phi_{B}K^{B})}\end{array}
\lim_{N \to \infty}N^{-1}S_j(\tau_N) = \frac{\nu q}{\lambda}\int_0^1 (1-py)^{-1}(1-y) y^{j-1}dy = \frac{\nu q}{\lambda}\sum_{k=0}^\infty \frac{p^k}{(j+k)(j+k+1)}, j\geq 1.
+ \left.{1 \over 60}F_{\mu\rho}~^{||\rho}F^{\mu\lambda}~_{||\lambda}-{1 \over 72}(F^{\mu\nu}F_{\mu\nu})\bar{R}-{1 \over 180}\bar{R}^{\lambda\nu\mu\rho}F_{\lambda\nu}F_{\mu\rho}-{1 \over 90}\bar{R}_{\mu\nu}F^{\rho\mu}F_{\rho}~^\nu \right)~~,
F(\mathbf{v}) = \begin{bmatrix}D 0 \ell\sin\Psi -\cos\Psi 0 D -\ell\cos\Psi -\sin\Psi 0 -\kappa\ell D - \kappa U D^0_{-1}0 0 0 D^0_{1}0 0 0 0 0 D^0_{-1}0 0 0 D^0_{1}0 \end{bmatrix}\mathbf{v}
\lim_{\Lambda}\omega_{\beta,\mu,\Lambda}^{0}(\frac{b^{*}_{k}b_{k}}{V}) =\lim_{\Lambda}\frac{1}{V}\left\{e^{\beta(\varepsilon_{k}-{\mu_{\Lambda}(\beta,\rho)})}-1 \right\}^{-1}= 0 \ , \ \forall k \in \Lambda^{*}\ ,
U_I(t,t_0)=\alpha(\tau) -\frac{i\omega_0}2(t-\tau)\left((A-\frac14)(aa^{\dag}+a^{\dag}a) + q (e^{i\omega_0 t_0}a^2 +e^{-i\omega_0 t_0}(a^{\dag})^2)\right)\alpha(\tau)+{\rm r.~t.~}\,.
1-e^{-\lambda I}=\lambda\int^{\zeta}_{0}dt\,e^{\xi_{t}}\exp\left\{-\lambda\int^{\zeta}_{t}ds\,e^{\xi_{s}}\right\}=\lambda\int^{\zeta}_{0}dt\,e^{\xi_{t}}\exp\left\{-\lambda e^{\xi_{t}}\widetilde{I}\right\},
\varphi(\rho_1) \varphi(\rho_n)^q \varphi(\rho_{i}) =x y^{-q}y^q x^i y^{-qi-B(1,n-i+1)}=x^{i+1}y^{-qi-B(1,n-i+1)}=x^{i+1}y^{-qi-B(1,n-i)+D(i)}=x^{i+1}y^{-q(i+1)-B(1,n-i)}y^{q+D(i)}=\varphi(\rho_{{i+1}})\varphi( \rho_n)^{q+D(i)}.
\AA^X := \sum_{k = 1}^{n - 1}\sum_{i = 1}^{\min\{k, p\}}\Big(2 x^k_i \partial^2_{x^k_i}+ 2 (p - k + 1)\partial_{x^k_i}\Big) \AA^Z := \AA^X + \sum_{i = 1}^{\min\{n, p\}}\Big(2 y_i \partial_{y_i}^2 + 2 (p - n + 1) \partial_{y_i}\Big)
\alpha_n(i) = \sum_{j \in W_n}\hat{\beta}_{n-1}(j) \lambda(j,i) &= \sum_{j \in W_n}\hat{\beta}_{n-1}(j) \lambda(j,i) + \sum_{j \in T'}\hat{\beta}_{n-1}(j) \lambda(j,i);\intertext{$T' \subset T_{n-1}^c$ and \eqref{e:prophatbeta}imply that the last sum is $0$, therefore}= \sum_{j \in W_n}\hat{\beta}_{n-1}(j) \lambda(j,i).
F(t;x,y,0,u,z,v) =\frac{y^2xz^2vt^2}{(1-ytu)(y-yzr+z)}+\frac{z(ytux-ytu+1)}{y-yzr+z}F(t;x,y,1,u,z,v) &\quad+\frac{y^2u^2vt^2z^2(1-v)(ytux-ytu+1)}{(1-ytu)(y-yzr+z)}F(t;x,y,1,u,1,v).
\|v-c(v)\|_{L^2({\cal H}_{d})}^2\leq C\mathcal{E}^{\partial\Gamma,d}(v,v),\ \{\rm where}\ \c(v)=\dfrac{1}{\mathcal{H}_d(\partial\Gamma)}\int_{\partial\Gamma}v(\xi)d\mathcal{H}_d(\xi).
R_{ijkl}=\sum_{p}\left(h^{p^{\ast}}_{ik}h^{p^{\ast}}_{jl}-h^{p^{\ast}}_{il}h^{p^{\ast}}_{jk}\right),\ \R_{ik}=\sum_{p}H^{p^{\ast}}h^{p^{\ast}}_{ik}-\sum_{p,j}h^{p^{\ast}}_{ij}h^{p^{\ast}}_{jk}.
2m^2(\rho_1^2\partial^2_{\rho_1}\hat{\phi}+\rho_1\partial_{\rho_1}\hat{\phi}+\rho_2^2\partial^2_{\rho_2}\hat{\phi}+ \rho_2\partial_{\rho_2}\hat{\phi})+ 2(k_{1,0}k_{2,0}-{\bf k_1}.{\bf k_2})\rho_1\rho_2\partial_{\rho_1}\partial_{\rho_2}\hat{\phi}+m^2\hat{\phi}+\lambda\hat{\phi}^3=0.
\left({\frac{d \tilde{x}}{d \tilde{\lambda}}}\right)^{2}+ \left[ \tilde{H}_{p + 1}^{-{\frac{4}{\Delta_{p + 1}}}}-{\frac{\tilde{\cal J}^{2}}{\tilde{H}_{p + 1}^{\frac{8 ( p + 2 )}{( D - 2 ) \Delta_{p + 1}}}\tilde{x}^{2}}}\right] \tilde{f}- \tilde{E}^{2}\tilde{H}_{p + 1}^{\frac{4 ( D - 2 p - 6 )}{( D - 2 ) \Delta_{p + 1}}}= 0 ,
\partial_i \Phi^u(x) = (\partial_i + u_i(x)\partial_z)|_{(x, u(x))}, \ \ \ \forall 1\leq i\leq n\,; \nu^{u,f}(x, u(x)) = \frac{1}{\sqrt{(1+f)(1+g_u^{ij}u_iu_j)}}\cdot (\partial_z - g_u^ij u_i(x)\partial_j)\,.
e^A =e^{2mt_2}\biggl(1+O\Bigl({1\over \log{n}}\Bigr)\biggr)={m^2n^2\over\log^2{n}}\biggl(1+O\Bigl({\log\log{n}\over\log{n}}\Bigr)\biggr), e^B =e^{2mt_1}\biggl(1+O\Bigl({1\over \log{n}}\Bigr)\biggr)={m^2n^2\over\log^2{n}}\biggl(1+O\Bigl({\log\log{n}\over\log{n}}\Bigr)\biggr).
\frac{1}{N^{2r}}\sum_{k_1=1}^N ... \sum_{k_l =1}^N |g_{k_1}|^{d_1}... |g_{k_l}|^{d_l}\leq \left( \frac{1}{N \eta}\right)^{2r - l}\frac{1}{N^l}\sum_{k_1=1}^N ... \sum_{k_l =1}^N |g_{k_1}|... |g_{k_l}| &\leq \left( \frac{1}{N \eta}\right)^{2r - l}\log (N)^{l}\leq \varphi^{\xi}(\Phi )^{4r - 2 l}.
\textbf{Y}_{1}= \sum_{k=1}^{K}\textbf{h}_{1,k}\big(\textbf{p}_{1,k}+\textbf{s}_{1,k}\big) + \sum_{j \neq 1}^{L}\sum_{k=1}^{K}\textbf{h}_{j,k}\big(\textbf{p}_{j,k}+ \textbf{s}_{j,k}\big) + \textbf{N}_{1}.
\frac{\prod_{j=1}^n (1-1/p^j)}{\prod_{j=1}^{\lfloor (n-r)/2 \rfloor}(1-1/p^{2j})}\frac{1}{p^{n(\lambda)+|\lambda|}\prod_{i \geq 1}\prod_{j=1}^{\lfloor m_i(\lambda)/2 \rfloor}(1-1/p^{2j})}.
G(x) = G(r) - \int_x^r \exp(\alpha t + \beta) g(t) \, dt &\geq G(r) - \int_x^r \exp(\alpha t + \beta) \bar g(t) \, dt = G(r) - \frac{1-e^{-\alpha(r-x)}\{\alpha(r-x)+1\}}{1-e^{-\alpha(r-a)}\{\alpha(r-a)+1\}}\{G(r) - G(a)\}.
\begin{aligned}\dot{x}= - y + a_{20}x^2 - 2 b_{20}x y + a_{20}b_{20}x^3 + r_{11}x^2 y, \dot{y}= x + b_{20}x^2 + a_{20}x y - b_{20}y^2 + a_{20}b_{20}x^2 y + r_{11}x y^2,\end{aligned}
\log(p+1) = A_p+b_p-\epsilon_p A_p =\int_{p}^{p+1}\log x \, dx b_p = \frac{1}{2}\big( \log(p+1)-\log(p)\big) \epsilon_p =\int_{p}^{p+1}\log x \, dx -\log p - \frac{1}{2}\big( \log(p+1)-\log(p)\big) &=\cdots &= \frac{2p+1}2\log \frac{p+1}p -1
\frac{e^{-rx}}{1-e^{-\beta x}}w_{n}\left( \frac{e^{-\beta x}}{1-e^{-\beta x}};0,\beta,r\right) =\frac{\beta^{n}}{2\pi i}{\displaystyle\int\limits_{a-i\infty}^{a+i\infty}}\left( \beta x\right)^{-t}\zeta\left( t-n,\frac{r}{\beta}\right)\Gamma\left( t\right) dt.
(\nabla_xY_{t_{k+1}}^{\theta}(t_k,x))^{-1}\partial_{\theta}Y_{t_{k+1}}^{\theta}(t_k,x)=\int_{t_k}^{t_{k+1}}(\nabla_xY_{s}^{\theta}(t_k,x))^{-1}\partial_{\theta}b(\theta,Y_{s}^{\theta}(t_k,x))ds.
= 1-\frac{1}{\lambda\beta^{\frac{2}{\alpha}}\kappa_{\alpha}\eta_i s^2}\exp\left\{-\lambda \beta^{\frac{2}{\alpha}}\kappa_{\alpha}(\inf(\mathcal{L}_i))^2\right\}\cdot &\left(1-\exp\left\{-\lambda \beta^{\frac{2}{\alpha}}\kappa_{\alpha}s^2\eta^2_i\right\}\right)
\big\||V|^{1/2}H_0^{-1/2}\big\|^2 = \big\||V|^{1/2}H_0^{-1}|V|^{1/2}\big\| \leq \big\||V|^{1/2}H_0^{-1}|V|^{1/2}\big\|_\mathrm{HS}= \frac{\|V\|_R}{4\pi}\,,
\left| \frac{1}{(\varphi(4))^k}\mathrm{Pf}\Big[ K_{N}\Big(4N+\frac{u_i}{\varphi(4)}, 4N+\frac{u_j}{\varphi(4)}\Big) \Big]_{i,j = 1}^{k}\right| \leq (2k)^{\frac{k}{2}}C^{k}\prod_{j=1}^{k}e^{-(a-b)u_{j}}.
\Big\|\sum_{|j|>\ell}\sum_{Q\in Q^j}w(Q) \psi_{j}(x,x_{Q})q_{j}h(x_{Q})\Big\|_{\dot{\mathcal F}^{\alpha,q}_{p,{\rm D}}}&=C\Big\|\Big\{\sum_{k\in \mathbb Z}\sum_{Q\in Q^k}\Big({\mathfrak R}^{k\alpha}\Big|q_k\Big(\sum_{|j|>\ell}\sum_{Q\in Q^j}w(Q) \psi_{j}(\cdot,x_{Q})q_{j}h(x_{Q})\Big)(x_Q)\Big|\Big)^q\chi_Q\Big\}^{1/q}\Big\|_{L^p_\omega},
\mathbb{E}[N_w^{\hat{x}}(G'[+]) - N_w^{\hat{x}}(H[+])\mid \hat{x}] = \frac{a-b}{n}\left((n^{++}-n_C^{++}) - (n^{+-}- n_C^{+-}) \right)(n_C^{++}- n_C^{+-})
R_s=R_1+R_2+R_3=\begin{cases}\mathcal{C}\left(\tfrac{aP_1}{1+bP_2}\right) + \mathcal{C}(P_2),\textrm{if}0\leq b< 1 \min\{\mathcal{C}(aP_1+bP_2),\mathcal{C}(aP_1)+\mathcal{C}(P_2)\}, \textrm{if}b\geq 1 \end{cases}