harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sup_{B_{R}(O)}G\leq G\phi\leq \max\left\{\begin{aligned}\frac{nKp}{2(p-1)}; & \min_{C_3\in(0,\frac{2(2-p)}{np})}\left\{\frac{((A+2K)R^{2}C_{3}+C_{1}^{2})np}{(2(2-p)-C_{3}np)R^{2}C_{3}}\right\}\end{aligned}\right\}=\tilde{C_{2}},
h(x,y)=-\frac{\sqrt{H_{0}(x)-U(y)}}{U'(y)}+\frac{\sqrt{H_{0}(x)-U(x)}}{U'(x)}=\frac{\sqrt{H_{0}(x)-U(y)}}{F(y)}-\frac{v(x)\sqrt{m(x)}}{\sqrt{2}F(x)}
W^*\pi(\lambda)g_{qpt}= W^*\left(\pi(\lambda)g + e^{\frac{-2\pi i t}{3}}e^{\frac{2\pi i kp}{M}}\pi(\lambda')g \right) = W^*\pi(\lambda)g + e^{\frac{-2\pi i t}{3}}e^{\frac{2\pi i kp}{M}}W^*\pi(\lambda')g.
\left[t^{N-j}\right]\frac{t^{N+1}P(1/t)}{1-yt}=\left[t^{N-j}\right] \left((1-t)^{N}- \frac{qt^{N}}{1-yt}\right)=\begin{cases}(-1)^{N-j}\binom{N}{N-j}j>0 (-1)^N-q j=0\end{cases}.
\frac{\partial}{\partial u}g(z,u)|_{u=1}=\frac{z+z^{2}+z^{3}+z^{6}}{1-z}exp(uz-z+u^{2}\frac{z^{2}}{2}-\frac{z^{2}}{2}+u^{3}\frac{z^{3}}{3}-\frac{z^{3}}{3}+u^{6}\frac{z^{6}}{6}-\frac{z^{6}}{6})|_{u=1}
|\eta(P_A)-\eta(P'_A)| \leq \lim_{n \rightarrow \infty}\log\left(\max_a |\mathcal{V}_a|\right)\cdot\|T_{\overline{a}^n}-T_{\overline{a}'^n}\|_1 = \log\left(\max_a |\mathcal{V}_a|\right)\cdot\|P_A - P'_A\|_1.
\begin{array}{r c l}{{d \Pi_{( 2 )}^{2}}}&{{=}}&{{\mathrm{c o s h}^{2}\chi \, d \phi^{2}- d \chi^{2}\, ,}}\\{{}}&{{}}&{{}}\\{{d \Omega_{( 2 )}^{2}}}&{{=}}&{{d \theta^{2}+ \sin^{2}{\theta}\, d \varphi^{2}\, ,}}\\ \end{array}
A=(\begin{matrix}L(\varphi)&L(\gamma)\\ \varphi&\gamma\\ \frac{\partial\varphi}{\partial n}&\frac{\partial\gamma}{\partial n}\\ \gamma&0\end{matrix}),b=(\begin{matrix}f\\g\\h\\ 0\end{matrix}),\varphi=\varphi(x_{i},x_{j}),\gamma=\gamma_{k}(X_{i}).(7)
\left(\mathcal F\left(L(\ell+1, 0) \otimes M_{1, r+1}\right), \mathcal F\left(L(\ell+1, 0) \otimes M_{1, r'+1}\right)\right) = &\qquad\qquad\qquad\qquad\qquad\qquad= \left(L(\ell, r\omega) \otimes L(1, \overline{r}\omega), L(\ell, r'\omega) \otimes L(1, \overline{r}'\omega)\right)
\sum_{i=1}^n[f_i(x_{kn})-f_i(x_{kn+i})] =\sum_{i=1}^n\sum_{j=1}^{i}[f_i(x_{kn+j-1})-f_i(x_{kn+j})] &\leq \sum_{i=1}^n\sum_{j=1}^{i}Ld(x_{kn+j},x_{kn+j-1}) &\leq \sum_{i=1}^n2\lambda_k L^2i=\lambda_k L^2n(n+1).
\tilde M_{n,\mathbf{p}_{k,1}}\cdot{\tilde M}_{n,\mathbf{p}_{k,2}}=\frac{4}{n^2}\sum_{\mathbf{i}\in \mathcal{J}}\prod_{\ell=1}^{k}\tilde I^{(p_{\ell,1})}_{i_1,i_2}\tilde I^{(p_{\ell,2})}_{i_3,i_4},
=\frac{1}{6}\sqrt[4]{27}F[2arctan(\sqrt[4]{3});\frac{1}{4}(\sqrt{6}+\sqrt{2})]=\frac{2}{9}\sqrt[4]{27}K[\frac{1}{4}(\sqrt{6}+\sqrt{2})]=\frac{2}{3}\sqrt[4]{3}K[\frac{1}{4}(\sqrt{6}-\sqrt{2})]
\tilde{S}_t= e^{-\alpha t}S_t ,\tilde{f}(t,x,\omega)=e^{-\alpha t}f(t,e^{\alpha t}x,\omega) , \tilde{k}(t,\xi,x,\omega)=e^{-\alpha t}k(t,\xi,e^{\alpha t}x,\omega).
\beta^{(1)}_2=\lim_m\frac{\sum_{n=3}^m\frac{1-c_{1n}}{1+c_{2n}}}{\sum_{n=3}^m\frac{1-c_{2n}}{1+c_{2n}}},\quad\beta^{(3)}_2=\lim_m\frac{\sum_{n=3}^m\frac{(1-c_{1n})(1-c_{2n})}{1+c_{2n}}}{\sum_{n=3}^m\frac{1-c_{2n}}{1+c_{2n}}}.
R^{\omega,\nu}f_k(rz) =\int_{\mathbb{D}}f_k(\sqrt r\xi)\overline{B_{\sqrt r z}^\nu (\xi)}\omega(\xi)dA(\xi) &\to \int_{\mathbb{D}}f(\sqrt r\xi)\overline{B_{\sqrt r z}^\nu (\xi)}\omega(\xi)dA(\xi) &=R^{\omega,\nu}f(rz)
s^{2}=\frac{1}{3N}\{\sum_{n=1}^{N}(x_{n,1}-\overline{x})^{2}+\sum_{n=1}^{N}(x_{n,2}-\overline{x})^{2}+\sum_{n=1}^{N}(x_{n,3}-\overline{x})^{2}\}.
g_q (y, M)={M \over 18\pi}x_a x_b \ [f^{(a)}_q (x_a,M^2)f^{(b)}_{\bar q}(x_b, M^2) + f^{(a)}_{\bar q}(x_a, M^2) f^{(b)}_q (x_b,M^2)]\ ,
\frac{d}{dt}h_{\min}\ge -(\sqrt{2\pi})^{n+1}e^{\frac{h_{\min}^2}{2}}\frac{1}{\det(\nabla^2h_{\min}+h_{\min}I)}h_{\min}^pf+h_{\min}&\geq -(\sqrt{2\pi})^{n+1}e^{\frac{h_{\min}^2}{2}}h_{\min}^{p-n}f+h_{\min}&=-((\sqrt{2\pi})^{n+1}e^{\frac{h_{\min}^2}{2}}h_{\min}^{p-n-1}f-1)h_{\min}.
\rho(v_B^{-1}) \circ (F \circ S)(\varphi) = \mu^l(v)^{-1}\mu^r\!\left(gv^{-1}v'?\right) \varphi\!\left( v b_i g v'' a_i \right) = \mu^l(v)^{-1}\mu^r\!\left(g\overline{Y}_j?\right) \varphi\!\left( v^2 b_i g \overline{X}_j a_i \right) &= \mu^l(v)^{-1}\mu^r\!\left(g S(a_j)S^{-1}(b_k)?\right) \varphi\!\left( v^2 g S^{-2}(b_i) b_j a_k a_i \right) = (\star)
T^*_{\widetilde B_1}(\widetilde{\mathcal B}^*V_0)= T^*_{\widetilde B_1}(\bar\Phi'\widetilde{\mathcal B}^*V_0)=\bar\Phi' T_{-B_2^*}^*(\tilde{\mathcal B}\tilde U_{0})= T_{-B_2^*}^*(\bar\Phi'\tilde{\mathcal B}\tilde U_{0})=0\mbox{on}\,\,\tilde \Gamma
\hat{\Gamma}_{\alpha_1\alpha_2}^\alpha\hat{\Gamma}_{\alpha_3\alpha_4}^{\alpha_1}\hat{\Gamma}_{\alpha_3\alpha_4}^{\alpha_2}=\hat{\Gamma}_{\alpha_1\alpha_2}^\alpha\hat{\Gamma}_{\alpha_3\alpha_4}^{\alpha_1}\hat{\Gamma}_{\alpha_2\alpha_4}^{\alpha_3},
P_{e}=P_{e|H_{0}}\cdot P_{H_{0}}+P_{e|H_{1}}\cdot P_{H_{1}}+\cdot\cdot\cdot+P_{e|H_{L-1}}\cdot P_{H_{L-1}}=\sum_{k=0}^{L-1}P_{e|H_{k}}\cdot P_{H_{k}}
(\mathcal{K}^{*}f)(t)=\begin{cases}C_{H}\cdot t^{\frac{1}{2}-H}\cdot\left(I_{1^{-}}^{H-\frac{1}{2}}\left(s^{H-\frac{1}{2}}f(s)\right)\right)(t), H>\frac{1}{2}; C_{H}\cdot t^{\frac{1}{2}-H}\cdot\left(D_{1^{-}}^{\frac{1}{2}-H}\left(s^{H-\frac{1}{2}}f(s)\right)\right)(t), H\leq\frac{1}{2}.\end{cases}
{\cal L}= \frac{1}{\kappa^{d - 2}}\sqrt{g}\left[ \ - R + \Lambda + \lambda \kappa^{2}~ \widehat{\mathrm{G}}+ \sum_{n = 1}^{\infty}\lambda_{n}\kappa^{2 n + 2}\Im_{n}[ \widehat{R}, \nabla , \Lambda ] \right]
k^{-q}\|u-v\|_{L^{q,\infty}(B)}^q = \frac{\Pi k^{\vartheta}}{\lambda}\ \Leftrightarrow \k = \left(\lambda \Pi^{-1}\|u-v\|_{L^{q,\infty}(B)}^q\right)^{\frac{1}{q+\vartheta}},
\mathbb{E}_{f_0}\biggl(\sum_{j \in J_1}\frac{N_j}{n}\log \frac{N_j}{np_j}\biggr) \leq \sum_{j=1}^k \mathbb{E}_{f_0}\biggl\{\frac{N_j}{n}\biggl(\frac{N_j}{np_j}- 1\biggr)\biggr\}- \mathbb{E}_{f_0}\biggl(\sum_{j \in J_2}\frac{N_j}{n}\log \frac{N_j}{np_j}\biggr)\leq \frac{k}{n}+ \frac{k}{n}\log n.
\sum_{1/\theta < j \le m}\frac{\cos(2\pi j \theta)}j = \Re \sum_{1/\theta < j \le m}\frac{e^{2\pi i j \theta}}j =\Re \sum_{1/\theta < j \le m}\frac{S_j-S_{j-1}}{j}=\Re \sum_{1/\theta < j \le m-1}\frac{S_j}{j(j+1)}+ \frac{S_m}{m}-\frac{S_{\lceil 1/\theta \rceil-1}}{\lceil 1/\theta \rceil}.
\theta_0\otimes v = \Bigl(\sum_{\mu < \nu}(-1)^{\mu + \nu}[x_{\mu},x_{\nu}] \otimes x_1 \wedge \ldots \wedge \hat{x}_{\mu}\wedge \ldots \wedge \hat{x}_{\nu}\wedge \ldots \wedge x_p\Bigr) \otimes v \theta_1(v) = \sum_{\nu=1}^p \Bigl( (-1)^{\nu+1}x_1 \wedge \ldots \wedge \hat{x}_{\nu}\wedge \ldots \wedge x_p\Bigr) \otimes x_{\nu}(v) \ .
\psi_{\vec{k}}^{(+)}(\vec{r})=\frac{4\pi}{\rho}\sum_{l=0}^{\infty}\sum_{m=-l}^{l}i^{l}e^{i\sigma_{l}}F_{l}(\eta,\rho)Y_{l}^{m}(\hat{r})Y_{l}^{m*}(\hat{k}),
\bar{D}^2 e^{-2 V}D^2 e^{2V}\rightarrow \bar{D}^2 e^{i \Lambda}e^{-2V}e^{-i \bar{\Lambda}}D^2 e^{i \bar{\Lambda}}e^{2V}e^{-i \Lambda}= e^{i \Lambda}\left(\bar{D}^2 e^{-2V}D^2 e^{2V}\right)e^{-i \Lambda}.
y_{c+d}^{(1)}(1+y_{a+c})(1+y_{b+d}) = -[P_{a+b+c}, P_{2b+c}, P_{b+c+d}, P_{b+2d}, P_{a+b+d}, P_{2a+d}, P_{a+c+d}, P_{a+2c}] &y_{c+d}^{(1)}\frac{1}{(1+y_{a+d}^{-1})(1+y_{b+c}^{-1})}= -[P_{a+b+c}, P_{b+2c}, P_{b+c+d}, P_{2b+d}, P_{a+b+d}, P_{a+2d}, P_{a+c+d}, P_{2a+c}]
\Gamma_{\nu\lambda}^\mu (x) \ = \ \frac 1 2 \g^{\mu\sigma}(x) \left( \frac{\partial g_{\sigma\lambda}(x)}{\partial x^\nu}\ + \ \frac{\partial g_{\nu\sigma}(x)}{\partial x^\lambda}\ - \ \frac{\partial g_{\nu\lambda}(x)}{\partial x^\sigma}\right)
\begin{array}{ll}L_1f(x)=\int_{\partial M}f(y)\hat{E}(x,y)d\sigma(y), L_2f(x)=\int_{\partial M}f(y)\partial_{\nu,y}\hat{E}(x,y)d\sigma(y), L_3f(x)=\int_{\partial M}f(y)\triangle_y\hat{E}(x,y)d\sigma(y), L_4f(x)=\int_{\partial M}f(y)\partial_{\nu,y}\triangle_y\hat{E}(x,y)d\sigma(y).\end{array}
{\cal H}( p ) = - 2 i \varepsilon^{\mu \nu \rho}\varepsilon^{\alpha \beta \lambda}g_{\nu \alpha}\frac{\mu^{\epsilon}}{( 2 \pi )^{d}}\int{\cal D}q \frac{1}{( p + q )^{2}q^{2}}I_{\beta \mu \lambda \rho}( q ) \; ,
c_2(F(\mathbf{x}_0),F(\mathbf{y}_0),\delta)=c_1(g_\delta(F(\mathbf{x}_0)),g_\delta(F(\mathbf{y}_0))) \neq\, c_2(F(\mathbf{x}_1),F(\mathbf{y}_1),\delta)=c_1(g_\delta(F(\mathbf{x}_1)),g_\delta(F(\mathbf{y}_1)))
{\bf P}^{\pi}(y^t, x^t) = \prod_{i=0}^tq_i(y_i|y_{i-M}^{i-1}, x_i)\pi_i(x_i|y_{i-J}^{i-1}), \nu_t^{\pi}(y_t|y_{t-J}^{t-1}) = \sum_{x_t\in{\cal X}_t}q_t(y_t|y_{t-M}^{t-1}, x_t)\pi_t(x_t|y_{t-J}^{t-1}),~t\in\mathbb{N}_0^{n}.
\langle x \rangle_{\sigma,N}= \frac{1}{|L_{\sigma,N}|}\sum_{k \in L_{\sigma,N}}x(k), \langle x \rangle_{\sigma}^+ = \limsup_{N \to \infty}\ \langle x \rangle_{\sigma,N}, \langle x \rangle_{\sigma}^- = \liminf_{N \to \infty}\ \langle x \rangle_{\sigma,N}.
\frac{\varphi_{n+1}-2\varphi_n+\varphi_{n-1}}{T^2}+\left( \mu^2-\nabla^2\right) \frac{\left( \varphi_{n+1}+4\varphi_n+\varphi_{n-1}\right)}6 ={}_{{}_{\!\!\!\!\!\!\!c}}\;\, j_n.
\mu^{L_n\cdots L_1}_{c_1,c_2}= L_n(\mu^{L_{n-1}\cdots L_{1}}_{L_{n-1}\cdots L_1(c_1),L_{n-1}\cdots L_1(c_2)}) \iota^{L_n\cdots L_1}= L_n(\iota^{L_{n-1}\cdots L_{1}})
\Bigg(\sum_{i=1}^kX_i\Bigg)\Bigg(\sum_{i=1}^k X_{i+k}\Bigg)=\sum_{i=1}^{2k}X_i+\sum_{p\geq 2}\sum_{(i_1,\dots,i_p)\in \mathbb{I}_{2k}^p}\alpha_{(i_1,\dots,i_p)}[X_{i_1},\dots,X_{i_p}].
g^{- 1}={\frac{4}{\xi_{i}^{n}\xi_{j}^{m}\lambda_{, \bar{z}}^{i}\lambda_{, z}^{j}\{\sigma_{m}, \sigma_{n}\}}}[ d \bar{z}\otimes d \bar{z}+ d z \otimes d z + \xi_{k}^{s}({\frac{\partial \sigma_{s}}{\partial p}}d p -{\frac{\partial \sigma_{s}}{\partial q}}d q ) \otimes d \lambda^{k}] .
R_f = \int\limits_{0}^{R}\underset{t>0}\int\left(\underset{j \in \psi}\prod\frac{1}{1+\max\{e^t-1,T+\Delta\}r^\alpha d_j^{-\alpha}}+\frac{1}{3}\underset{i \in \phi}\prod\frac{P[\eta(r)<T+\Delta]}{1+\max\{e^t-1,T\}r^\alpha d_i^{-\alpha}}\right)\mathrm{d}tf_R(r)\mathrm{d}r.
K =\sup_{1\leq t<\infty}\left\{-\int_{1}^{t}\frac{\left(R_{g(s)}-s^{2}\overline{R}\right)_{*}}{4}\exp\left(\int_{1}^{s}\frac{\tau(|M|^{*})^{2}}{2}d\tau\right)ds\right\}<\infty.
w= \langle u_n,\dots,u_2,u_1;\langle v_m,\dots,v_2,v_1;x\rangle\rangle = \sum \langle U_{2m},\langle U_{2m-1};v_m\rangle,\dots,U_4,\langle U_3;v_2\rangle, U_2,\langle U_1;v_1\rangle,U_0;x\rangle,
p(z) = 5 \cdot z^8 - 6 \cdot z^6 + 2 \cdot z^4 + 86 \cdot z^4 - 26 \cdot z^2 + 3 = z^4\cdot(5 \cdot (z^2)^2 - 6 \cdot (z^2) + 2) + 86 \cdot (z^2)^2 - 26 \cdot z^2 + 3.
D(H(p)) = \{\psi:\: H(p)\psi\in L^2(\Sigma):\, \psi(x,0)=0,\;\, x\in\mathbb{R},\;\; \psi(x,d)=0\;\;\;\; |x|\ge a \;\; \&\;\; \partial_z\psi(x,d)=0\;\;\;\; |x|<a, \;\; x\in\mathbb{R}\},
\left[ - \frac{g_{\mathrm{Y M}}^{2}N \Gamma ( \omega - 1 )}{8 \pi^{\omega}( 2 - \omega ) ( 2 \omega - 3 )}\right] \cdot [ x^{2}]^{( 2 - \omega )}2^{- J}\Delta^{J}( x ) \frac{J^{2}}{2}\langle \mathrm{T r}Z^{J - 1}\bar{Z}^{J - 1}- \frac{1}{N}\mathrm{T r}Z^{J - 1}\mathrm{T r}\bar{Z}^{J - 1}\rangle
\begin{array}{l}\nabla f(x^*)+\left[\displaystyle \sum_{i=1}^p[y^I]^*_j\nabla^2 c_j(x^*)-A^TA-{\cal J}c_{\gamma}(x^*)^T{\cal J}c_{\gamma}(x^*)-{\cal J}c_{\beta}(x^*)^T{\rm Diag}(\eta_{\beta}){\cal J}c_{\beta}(x^*)\right]\lambda^*=0 \end{array}
I_D^{+,i_1}(z):= \frac{1}{i_1!j_1!k_1!}\int_D\prod_{l=1}^{i_1}\frac{du_l}{1-u_l}\prod_{m=1}^{j_1}\frac{-dv_m}{1+v_m}\prod_{n=1}^{k_1}\frac{dw_n}{w_n}\frac{dt}{1-t}, I_D^{-,i_1}(z)= \frac{1}{i_1!j_1!k_1!}\int_D\prod_{l=1}^{i_1}\frac{du_l}{1-u_l}\prod_{m=1}^{j_1}\frac{-dv_m}{1+v_m}\prod_{n=1}^{k_1}\frac{dw_n}{w_n}\frac{-dt}{1+t}
\mathop{\max}_{\substack{S\subseteq \{\epsilon N\pm N^{3/4}\}1\leq|S|\leq 2}}\Big\{\frac{1}{\binom{N}{S}}\sum_{j\notin B}\Pr_{v\sim\mu_{t}}\big[|v H|=j\big] K_S(j)^2 \Big\}.
a_2(K_N) =a_2(K)+ lk(K',K'')N v_3(K_N) = v_3(K) - \frac{1}{4}(a_2(K')+a_2(K''))N + \frac{1}{8}\left(2 a_2(K) +lk(K',K'')^{2}\right)N & \quad+ \frac{1}{8}lk(K',K'')^{2}N^{2}a_4(K_N) =a_4(K) + a_3(L) N
\mathcal{S}=2\sum\limits_{\mu =\nu +\frac{1}{2}}^{+\infty}\mu e^{-\mu^{2}t}\sum\limits_{p=0}^{n-1}\gamma_{p}^{(\nu ,n)}\mu^{2p}-2\sum\limits_{\mu =\nu +\frac{1}{2}}^{\nu +\frac{n}{2}-1}\mu e^{-\mu^{2}t}\sum\limits_{p=0}^{n-1}\gamma_{p}^{(\nu ,n)}\mu^{2p}
G \bar{\psi}\gamma^{\mu}\psi \bar{\psi}\gamma_{\mu}\psi ={\frac{\pi ( \pi - g_{\mathrm{e f f}}^{2})}{g_{\mathrm{e f f}}^{2}}}\Bigl ( \bigl ( j^{0}\bigr )^{2}-{\frac{g_{\mathrm{e f f}}^{2}}{\pi}}\bigl ( j^{1}\bigr )^{2}\Bigr ) .
G(z,z')\lesssim \begin{cases}r'^{-n}r^{-s}(r/r')^{s-\frac n2+\nu'_0},r'>2r; r^{-1}d(z,z')^{-(n-1+s)},r\sim r'; r^{-n-s}(r'/r)^{1-\frac n2+\nu'_0},r'<\frac{r}{2};\end{cases}
\varepsilon=\varepsilon^aX^*e_a=\tilde{\varepsilon}^aX^*\tilde{e}_a=\tilde{\varepsilon}^aK(\sigma)^b{}_{a}X^*e_b,\quad\Longrightarrow \tilde{\varepsilon}^a=(K^{-1})^a{}_b\varepsilon^b.
+\infty > \int_{\mathbb B(a,r)}dd^c w \geq \int_{\mathbb B(a,r) \cap \Omega}d \mu \geq \sum_{j\geq j_0}\int_{\overline{\mathbb B}(a_j,r_j) \cap \{\varphi_j \geq -1\}}dd^c u_{j,k_j}= \sum_{j\geq j_0}\int_{\Omega '}dd^c u_{j,k_j}\geq \sum_{j\geq j_0}\frac{1}{2}=+\infty.
\lim\limits_{k\rightarrow\infty}\frac{\ln\ell_{k+1}}{\ln\ell_k}=\lim\limits_{k\rightarrow\infty}\frac{\ln(1+(pq)^{-k/2})}{\ln(1+(pq)^{(-k+1)/2})}=\lim\limits_{k\rightarrow\infty}\frac{(1+(pq)^{(-k+1)/2})(pq)^{-k/2}}{(1+(pq)^{-k})(pq)^{(-k+1)/2}}=(pq)^{-1/2}<1.
\begin{array}{l l l}{{f_{0}}}&{{=}}&{{x_{1}x_{2}x_{3}x_{4}\; ,}}\\{{f_{1}}}&{{=}}&{{x_{1}+ x_{4}+ x_{1}x_{4}\left( Q^{1 / 2}+ x_{2}+ x_{3}\right) \; .}}\\ \end{array}
(1-t^2)^{n-k}=-\int_t^1(\rho^2-t^2)^{n-k-2}\cdot(n-k)(n-k-1)\cdot 4t^2\omega_n(\rho)\rho d\rho begin{align*}5pt] \,+k\int_t^1(\rho^2-t^2)^{n-k-1}\cdot(n-k)\cdot 4\omega_n(\rho)\rho d\rho, n=2,3,4,\ldots, k=0,1,2,\ldots,n-2.
\widetilde{T}_{\widehat{D}}=\sqrt{\widetilde{N}}\max_{1\leq j\leq2q_0}\left(\frac{1}{n_1}\sum^{r_1^{(1)}}_{i=1}\widehat{D}_{ij}^{(1)}-\frac{1}{n_2}\sum^{r_1^{(2)}}_{i=1}\widehat{D}_{ij}^{(2)}\right).
\psi (X^1,\cdots, X^i+2\pi r_i',\cdots, X^n)=\psi (X^1,\cdots, X^i,\cdots, X^n) \ (i=1,\cdots, s), &\psi (X^1,\cdots, X^i+2\pi ,\cdots, X^n)=\psi (X^1,\cdots, X^i,\cdots, X^n) \ (i=s+1,\cdots, n).
\mathcal{L}=\sum_{i=1}^{N}{\sigma_i^2(1-\phi_{i}) + \sigma_i^2 \phi_{i}2^{-\frac{2c_i}{\phi_{i}}}}+\lambda \sum_{i=1}^{N}\phi_{i}\epsilon_s + \frac{2^{\frac{2c_i}{\theta_{i}}}-1}{h_i}- E & -\sum_{i=1}^N \beta_i c_i +\sum_{i=1}^{N}\mu_i (c_i-B_{max}) -\sum_{i=1}^{N}\eta_i \phi_{i}+ \sum_{i=1}^N \omega_i (\phi_{i}-1),
\bigoplus_{i=0}^{x-1}T_x(i) \bigoplus_{i=0}^{x-1}F_n(T_x(i)) \bigoplus_{i=0}^{x-1}F_n(T_x(\phi(i))) =\bigoplus_{i=0}^{x-1}T_x(i) \bigoplus_{i=0}^{x-1}F_n(T_x(i)) \bigoplus_{i=0}^{x-1}F_n(T_x(i)) &=\bigoplus_{i=0}^{x-1}T_x(i)
{\hat Q}_{\pm}^{BC}=\int dx^1 \sum_{a,b}\left[f^B\right]^{ab}\left[ (\partial_0 \phi^a \mp\partial_1 \phi^a) \psi_{\pm}^b \pm \sum_c \left[f^C\right]^{bc}\frac{\partial W^C}{\partial\phi^c}\psi^a_{\mp}\right]
\tfrac{1}{\sqrt{c^2+d^2}}\tfrac{d\det g}{\sqrt{c^2+d^2}}+(\tfrac{bd+ac}{\sqrt{c^2+d^2}})\tfrac{c}{\sqrt{c^2+d^2}}= \tfrac{d\det g}{\sqrt{c^2+d^2}}+ \tfrac{cbd+ac^2}{c^2+d^2}&= \tfrac{dad-dbc+cbd+ac^2}{c^2+d^2}=a;
\let\veqno\eqno \forall m \in \omega_M^{\vee}\cap M, \ \forall \gamma_1, \gamma_2 \in \Gamma, \ \ \ \ \ \ \ \h_{\gamma_1}(m){\sigma_Y}_{\gamma_1}^{\sharp}\left(h_{\gamma_2}\left(\tilde{\tau}^{-1}_{\gamma_1}(m)\right)\right) = h_{\gamma_1 \gamma_2}(m).
\tilde{E}\big[e^{\sigma B_T^H}|\mathcal{F}_{r}^{H}\big]=\sum\limits_{l=0}^{+\infty}\sum\limits_{k=0}^{l}\frac{\left( -1\right)^{l+k}(T-r)^{\left( l-k\right) H}}{\left( l-k\right) !}h_{l-k}\left( \frac{B_{T}^{H}-B_{r}^{H}}{(T-r)^{H}}\right)\psi_{k}^{(r,T)}(D_{T}^{H,l}e^{\sigma B_T^H}).
\sum_{\overset{{i+j+k=s}}{i,j,k\geq 0}}[T_i(u),T_j(v),T_k(w)] &=\sum_{\overset{{i+j+k=s}}{i,j,k\geq 0}}T_i\Big(D(T_j(u),T_k(v))w+\theta(T_j(v),T_k(w))u-\theta(T_j(u),T_k(w))v\Big),
\lim_{n\to\infty}\alpha_{n, k}= \sum_{j=0}^{m-k}\frac{(-1)^j}{(m+1)^{k+1+j}}{m+1 \choose m-k - j}{j + k +1 \choose j}&={m+1 \choose k+1}\sum_{j=0}^{m-k}\frac{(-1)^j}{(m+1)^{k+1+j}}{m-k \choose j}.
f(t_{\mathtt{D}_{i}},\beta_{\mathtt{D}_{i}})\leq F(t_{\mathtt{D}_{i}},\beta_{\mathtt{D}_{i}},\psi_{\mathtt{D}_{i}}^{(n)})=\frac{1}{2\psi_{\mathtt{D}_{i}}^{(n)}}t_{\mathtt{D}_{i}}^{2}+\frac{\psi_{\mathtt{D}_{i}}^{(n)}}{2}\beta_{\mathtt{D}_{i}}^{2}
\varepsilon_3=\varepsilon^b_1\varepsilon^c_2C^a{}_{bc}X^*e_a=\tilde{\varepsilon}^b_1\tilde{\varepsilon}^c_2\widetilde{C}^a{}_{bc}X^*\tilde{e}_a=\varepsilon^y_1\varepsilon^z_2(K^{-1})^b{}_{y}(K^{-1})^c{}_{z}\widetilde{C}^x_{yz}K^a{}_xX^*e_a,
\delta_{L}\psi ( x ) = -{\frac{i}{2}}\epsilon_{a b}\sigma^{a b}\psi , \quad \delta_{L}{e^{a}}_{\mu}( x ) = \epsilon{^a}_{b}e{^b}_{\mu}+{\frac{\kappa^{4}}{4}}\varepsilon^{a b c d}\bar{\psi}{^j}\gamma_{5}\gamma_{d}\psi{^j}( \partial_{\mu}\epsilon_{b c}) .
\begin{array}{rcl}\displaystyle\frac{\partial^{k}\lambda_{i}^{t}}{\partial t^{k}}= u_{k,1}c^{\frac{k}{2}}\|T\|^{k}\lambda_{i}^{t}+u_{k,3}c^{\frac{k-2}{2}}\|T\|^{k-2}(\lambda_{i}^{t})^{3}+u_{k,5}c^{\frac{k-4}{2}}\|T\|^{k-4}(\lambda_{i}^{t})^{5}+\ldots & +u_{k,k+1}(\lambda_{i}^{t})^{k+1},\end{array}
\begin{aligned}\bold{p}_{j}=[\psi_{j,1},\psi_{j,2},...,\psi_{j,j}, \rho^2\widetilde{t}_{1,1}^2 \psi_{j,1}+\Theta_{j,1}, & \rho^2\widetilde{t}_{2,2}^2 \psi_{j,2}+\Theta_{j,2},..., \rho^2\widetilde{t}_{j,j}^2 \psi_{j,j}+\Theta_{j,j}]^{T}, \bold{q}_{j}=M[\Gamma_{j,1},\Gamma_{j,2},...,\Gamma_{j,j},\Pi_{j,1},\Pi_{j,2},...,\Pi_{j,j}]^{T}.\end{aligned}
P(i\rightarrow f)=\sum_{n_1,\cdots,n_\ell=0}^\infty\,e^{-im(n_1r_1+\cdots +n_\ell r_\ell+D_G/2)(t_f-t_i)}\,<\psi_f|n_1,\cdots,n_\ell><n_1,\cdots,n_\ell|\psi_i>\ \ \ .
\begin{array}{ll}\begin{array}{ll}\frac{d}{d r}\sigma(r) = 0, r=0, \sigma(r) = 0, r \rightarrow \infty, \end{array}\begin{array}{ll}A(r) \neq 0, r \rightarrow\infty, A(r) = 1, r = 0.\end{array}\end{array}
\int_{\Omega_{bl}^+}S(A+ D u_{bl}^i) : D \varphi \: + \: \int_{\Omega_{bl}^-}S(D u_{bl}^i) : D \varphi = -\int_{y_2 = 0}S(A)n \cdot \varphi{\rm\,d}S, i=1,2
T_{sat}'(p_*;T_*)=T_*\cdot \frac{\frac{C_V(\gamma_V-1)}{p_*}-\frac{C_L(\gamma_L-1)}{p_*+\pi_L}}{C_V\gamma_V -C_L\gamma_L+\frac{q_V-q_L}{T_*}}< T_*\cdot \frac{\frac{C_V(\gamma_V-1)}{p_*}}{C_V\gamma_V}\,.
\prod_{p\nmid hk}\sum_{\substack{0\le q,\ell < \infty \min(\ell,q)=0}}p^{-qw}\phi^{\star}(p^q) \frac{\tau_A(p^{\ell})\tau_B(p^{\ell})}{p^{\ell(1+s_1+s_2)}}= \prod_{p\nmid hk}\left( 1 + \sum_{q=1}^{\infty}p^{-qw}\phi^{\star}(p^q)+ \sum_{\ell=1}^{\infty}\frac{\tau_A(p^{\ell})\tau_B(p^{\ell})}{p^{\ell(1+s_1+s_2)}}\right)
N \ge \lim_j \bigg| \frac{\partial_{x_i}u_{k_j}(s_j x)}{s_j x_n}\bigg| = \lim_j \bigg| \frac{\partial_{x_i}\tilde u_j(x)}{x_n}\bigg| =\bigg|\frac{\partial_{x_i}u_0(x)}{x_n}\bigg|
([\delta_{\varepsilon_1},\delta_{\varepsilon_2}]-\delta_{[\varepsilon_1,\varepsilon_2]_Q})A^a=0\Longleftrightarrow \nabla^\pm_\mu (T_{\nabla^\pm})^a{}_{bc}=2\rho^\nu_{[b|}(R_{\nabla^\pm})^a{}_{\nu\mu|c]}.
\frac{\sqrt{2}}{2 \pi}T_{2}K |_{M^{1 0}}= \frac{1}{( \beta m )^{\frac{1}{3}}}\frac{1}{1 6 \pi^{2}}\hat{I}_{4}= \frac{1}{\sqrt{m n}}\frac{1}{1 6 \pi^{2}}\hat{I}_{4}
R(\mathbf{h}_m,\mathbf{a}_m) = R(\mathbf{h}_m,\mathbf{a}_m,\alpha_{,m}) = \log^+\left(\left(\|\mathbf{a}_m\|^2-\frac{P|\mathbf{h}_m^*\mathbf{a}_m|^2}{1+P\|\mathbf{h}_m\|^2}\right)^{-1}\right).
X \equiv < \! 0 | \chi^{2}| 0 \! > = \int d^{3}x d^{3}y{\frac{F ( \mathbf{x}) F ( \mathbf{y}) + \mathbf{\nabla}G ( \mathbf{x}) \! \cdot \! \mathbf{\nabla}G ( \mathbf{y})}{4 \pi^{2}| \mathbf{x}- \mathbf{y}|^{2}}}
E^*_{i+k}A_1E^*_{i+k-1}A_1E^*_{i+k-2}\cdots E^*_{i+1}A_1E^*_{i}=\big((\frac{k}{2})!\big)^2 M^{\frac{2m-k}{2},\frac{i-1}{2}}_{\frac{i+k-1}{2},\frac{i-1}{2}}.
B(s,f) = \frac{1}{2\pi i}\int\limits_{3-i\infty}^{3+i\infty}\Gamma(w)B(w+s,f)X^w dw - \frac{1}{2\pi i}\int\limits_{-\delta_0+\delta_1-i\infty}^{-\delta_0+\delta_1+i\infty}\Gamma(w)B(w+s,f)X^w dw =: T_1(s,f) - T_2(s,f).
E_2 (r, a) = \frac{(\varepsilon - 1)^2}{16 \pi^2 a^4}\, \frac{23}{60}\, \frac{(d^4 - 10 d^2 - 15)}{(1+d)^4 (1-d)^4}\, , d=\frac{r}{a}\, , r < a.
|H_{2,1}(F_{f}/2)| \leq\frac{1}{96}\tau_{1}(1-\tau^2_{1})\left(|A|+|B|+|C|\right) &=\frac{1}{2304}\left(16+4\tau^2_{1}-17\tau^4_{1}\right) &\leq\frac{23}{3264}.
[\alpha(y),\alpha(z)] \cdot \alpha(x \cdot t) + [[x , y] , \alpha(z)] \cdot \alpha^{2}(t) &+ \alpha^{2}(y) \cdot ([x , z] \cdot \alpha (t)) - \alpha^{2}(x) \cdot (\alpha (y) \cdot (z \cdot t)) + \alpha^{2}(z) \cdot (\alpha (x) \cdot (y \cdot t))=0.
\begin{array}{ccccccc}{\cal I}^{hor}_2 = SO(2,3) \, \times \, SO(8) ,{\cal SC}_2 = Osp(8\vert 4), \null \null \null \null \null \null \null{\cal I}^{hor}_5 = SO(2,6) \, \times \, SO(5) ,{\cal SC}_5 = Osp(2,6\vert 4), \end{array}
\zeta_{v}(\varphi_{v},\chi_{v},s)=\int_{GL_{n}(F_{v})}{\xi^{\lambda_{v}}_{\varphi_{v}}\left(\begin{pmatrix}g 0 0 1_n\end{pmatrix}\right)}\chi_{v}(\det(g))\left|\det(g)\right|_{v}^{s-1/2}d^{\ast}g.
\sum_{p\in \P_j}\frac{1}{p}= \sum_{p\leq \lambda_j: p\nmid d}\frac{1}{p}- \sum_{p\leq \lambda_{j-1}:p\nmid d}\frac{1}{p}\geq j-\frac{1}{\lambda_j}- (j-1) = 1-\frac{1}{\lambda_j}.
H[f+g] =\exp\Big(\sum_{n=1}^{\infty}\frac{p_{n}}{n}\Big)[f+g] =\exp\Big(\sum_{n=1}^{\infty}\frac{p_{n}}{n}[f+g]\Big) =\exp\Big(\sum_{n=1}^{\infty}\frac{1}{n}(p_{n}[f]+p_{n}[g])\Big) =\exp\Big(\sum_{n=1}^{\infty}\frac{p_{n}}{n}\Big)[f]\cdot\exp\Big(\sum_{n=1}^{\infty}\frac{p_{n}}{n}\Big)[g] =H[f]H[g]
\begin{array}{r c l}{{}}&{{}}&{{t t^{- 1}= 1 = t^{- 1}t ~ ,}}\\{{}}&{{}}&{{t e t^{- 1}= q^{2}e ~ , ~ t f t^{- 1}= q^{- 2}f ~ ,}}\\{{}}&{{}}&{{[ e , f ] = \displaystyle \frac{t - t^{- 1}}{q - q^{- 1}}~ .}}\\ \end{array}
\mathrm{var}[S_{m}(y)-S_{k}(x)] =\mathrm{var}[\varphi^{D_{N},\Delta^{k}(x)}(y)-\varphi^{D_{N},\Delta^{k}(x)}(x)+\varphi^{\Delta^{k}(x),\Delta^{m}(x)}(y)] &\geq\mathrm{var}[\varphi^{\Delta^{k}(x),\Delta^{m}(x)}(y)]\geq\mathrm{var}[\varphi^{\Delta^{m+1}(x),\Delta^{m}(x)}(y)]\geq\gamma+o(1).
B ( x , y , z ) = \sum_{r > 0}{\frac{1}{r}}\left({\frac{y}{x}}\right)^{r}d + d \sum_{r , r^{\prime}> 0}B_{r r^{\prime}}\left({\frac{z}{x}}\right)^{r}\left({\frac{y}{z}}\right)^{r^{\prime}}.
\mathcal{W}^{(k)}:= \biggl\{w \in \mathbb{R}^k : \sum_{j=1}^k w_j \frac{\Gamma(j+2 \ell/d)}{\Gamma(j)}=0 \, \, \ell=1, \ldots, \lfloor d/4 \rfloor &\sum_{j=1}^k w_j= 1\, \, \, w_j=0 \, \, \, j \notin \{\lfloor k/d \rfloor, \lfloor 2k/d \rfloor, \ldots, k\}\biggr\}.
[h_\chi(z), H^{\beta}(w^2)] = [h_\chi(z), V^+(w)^{-1}\beta (w^2)w^{-2h_0}V^-(w)^{-1}] & =\Big(\sum_{n>0}\frac{z^{2n-2}}{w^{2n}}-i_{z, w}\frac{1}{z^2-w^2}-i_{w, z}\frac{1}{w^2-z^2}+\sum_{n>0}\frac{w^{2n}}{z^{2n+2}}\Big)H^{\beta}(w^2)=-\frac{1}{z^2}H^{\beta}(w^2).
Q\bigl(\boldsymbol{\theta},\boldsymbol{\theta}^{t}\bigr) =\mathsf{E}\Bigl\{\ln p\bigl(\mathbf{\mathbf{x}},\mathbf{y}\bigr)|\mathbf{y};\boldsymbol{\theta}^{t}\Bigr\}, \boldsymbol{\theta}^{t+1}=\arg\underset{\boldsymbol{\theta}}{\max}Q\bigl(\boldsymbol{\theta},\boldsymbol{\theta}^{t}\bigr),
N \sum_{i j k}\int_{x}\Biggl [ \tilde{A}_{i j}^{\mu}( x ) \tilde{A}_{\mu k l}( x ) \tilde{A}_{j k}^{\nu}( x ) \tilde{A}_{\nu l i}( x ) W^{i j k l}+ \tilde{A}_{i j}^{\mu}( x ) \tilde{A}_{\mu j k}( x ) \tilde{A}_{k l}^{\nu}( x ) \tilde{A}_{\nu l i}( x ) L^{i j k l}\Biggr ] .
\Big(\sum_{k=1}^{T}\eta_{k}\Big)^{-1}=\frac{1}{\min\Big(\frac{T}{18L\kappa}, T\gamma\Big)}=\max\Big(\frac{18L\kappa}{T},\frac{1}{T\gamma}\Big)\leq \frac{18L\kappa}{T}+\frac{1}{T\gamma}.
[y_1, y_1]=\alpha_2y_5, [y_1, y_2]=y_4, [y_2, y_1]=\frac{\alpha_5}{\alpha_3}y_4+\theta y_5, [y_2, y_2]=\beta_2y_5, [y_2, y_3]=\gamma_2y_5=-[y_3, y_2].
\begin{aligned}u_n(x) = p(x) u_j(x) \leq p(x) u(x) \leq a(x) j \in \{0, \ldots, n \}, p^L = u_{0}(x) = a_0(x) \leq \cdots \leq a_n(x) = p^U, \deg(u_j) \leq c_1 \deg(a_j) \leq c_2 j \in \{0, \ldots, n \}. \end{aligned}
\max_{\kappa,\lambda_{1},\lambda_{2}}\mathrm{R}\left(\frac{c_0}{\Phi_0(\nu;x)+H(\kappa,\lambda_1,\lambda_2;x)}-U(x)\frac{c_0}{\Phi_0(\nu;x+1)+H(\kappa,\lambda_1,\lambda_2;x+1)}-V(x)\right),
D_{{\cal F}}^{(N, \alpha)}(\Delta) =\frac{-i \pi^{1-N/2}\Gamma(2-N/2)}{2^{N-2}\alpha}\sum_{n=-\infty}^{\infty}e^{i 2\pi n \Delta/\alpha}\int_{0}^{\infty}dp\p^{N-3}J^2_{2\pi |n|/\alpha}(p r) .
V = V_{h}+ V_{v}a^{4}+{\frac{k^{4}a^{4}}{1 6 \pi^{2}}}\int_{0}^{\infty}d t t^{3}\ln \left[ 1 -{\frac{k_{\nu}( t ) i_{\nu}( a t )}{k_{\nu}( a t ) i_{\nu}( t )}}\right] ,
\Phi^{-1}|_{t=0}=\begin{pmatrix}\frac{1}{n-1}0 0 \dots 0 0 0 1 0 \dots 0 0 \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots \dots -\frac{12}{n-1}0 \dots \dots 0 1 \end{pmatrix}.
\left. I_{\mathrm{\scriptsize ~ g r a v}}\right|_{\mathrm{\scriptsize ~ s i n g}}= \frac{2^{1 0}}{3 5 \pi^{6}}\frac{1}{x_{1 3}^{8}x_{2 4}^{8}}\left[ s ( 7 t^{2}+ 6 t^{4}) + s^{2}( - 7 + 3 t^{2}) - 8 s^{3}\right] .
\sum_{i+j = n+1}^{}\sum_{\sigma}^{}(-1)^\sigma \epsilon (\sigma) ~ (-1)^{i (j-1)}~ l_j \big( l_i (x_{\sigma (1)}, \ldots, x_{\sigma (i)}), x_{\sigma (i+1)}, \ldots, x_{\sigma (n)}\big) = 0,
\mathfrak{s}(\left(\begin{array}{cc}(-1)^{k-1}1\end{array}\right))\mathfrak{s}(\left(\begin{array}{cc}1 (-1)^{k-1}z^{-1}&1\end{array}\right))\mathfrak{s}(\left(\begin{array}{cc}1 (-1)^{k-1}\end{array}\right))=\mathfrak{s}(\left(\begin{array}{cc}1 z^{-1}1\end{array}\right)).