harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
R e \tilde{\Pi}_{F}^{( q ) \nu \mu}( Q ) = - \frac{g^{2}}{2}N_{f}R e \int \frac{d^{4}K}{( 2 \pi )^{4}}t r \left[ \gamma^{\mu}S^{( 1 1 )}( K - Q ) \gamma^{\nu}S^{( 1 1 )}( K ) \right] .
x_{\beta_i}u \otimes \widetilde{C}^{m(u)}=x_{\beta_i}u \otimes \widetilde C^{m(x_{\beta_i}u)}\mbox{and}u \otimes x_{\beta_i}\widetilde{C}^{m(u)}\subset u \otimes \widetilde C^{m(u)}.
\limsup{N \to \infty}{\frac{A([0.1,0.2); N; (x_n)_n)}{N}}= \lim_{N \to \infty}{\frac{A([0.1,0.2); N; (x_n)_n)}{N}}= \frac{0.2 - 0.1}{1 - 0.1}= \frac{1}{9}< \frac{5}{9}.
A_{1}= A_{2}= \cdots = A_{k - 1}= \left( \begin{array}{c c}{{0}}&{{0}}\\{{1}}&{{0}}\\ \end{array}\right) \, , \quad A_{k}= \left( \begin{array}{c c}{{1}}&{{0}}\\{{0}}&{{0}}\\ \end{array}\right) \, .
H(x) = \langle\Pi_{X}(x+\Phi(x)) - x, \Phi(x) \rangle - \tfrac{1}{2}\|\Pi_{X}(x+\Phi(x))-x\|^2 & = \tfrac{1}{2}\|\Phi(x)\|^2 - \tfrac{1}{2}\|\Pi_{X}(x+\Phi(x))-(x+\Phi(x))\|^2 & = \tfrac{1}{2}\|(x+\Phi(x))-x\|^2 - \tfrac{1}{2}\|(x+\Phi(x))-\Pi_{X}(x+\Phi(x))\|^2 \geq 0.
|sA\Delta A^{c}| = |G\setminus (sA\Delta A) | & = |G| - |sA\Delta A| & \leqslant |G| - (2-\frac{2}{\epsilon}\beta)|A| & \leqslant \Big(\beta+ \frac{d\beta}{\epsilon}+\frac{2}{\epsilon}\beta\Big)|A|.
(\eta_{\mathrm{R}}- \eta) \Big( \rho,{m}| \bar \rho, \bar m \Big) = -h(\rho) + h(\bar \rho) + h'(\bar \rho)( \rho-\bar \rho) = - \int_{\bar \rho}^{\rho}( \rho- s) h''(s)\operatorname{d}s\, .
P_{s,f}^n \varphi(v) = \kappa(s)^n \frac{\nu_s(\langle f, \cdot \rangle^s \varphi)}{\nu_s(r_s)}\frac{1}{\langle f, v \rangle^s}r_s(v) + \frac{1}{\langle f, v \rangle^s}L_s^n \big(\langle f, \cdot \rangle^s \varphi \big)(v) & = \kappa(s)^n \nu_{s,f}(\varphi) r_{s,f}(v) + L_{s,f}^n \varphi(v),
R_j + \frac{1}{R_j}= 2 + \frac{\left(f_0(0) \alpha_j \frac{2k+1}{n}\right)^2}{2 + \sqrt{4 + \left(f_0(0) \alpha_j \frac{2k+1}{n}\right)^2}}& = 2 + \left(f_0(0) \alpha_j \frac{2k+1}{2n}\right)^2 + O\left(\frac{k^4\alpha_j^4}{n^4}\right),
E_{b_0}\Big[ \int_0^T \nabla L^{-1}[f_h](X_t).dW_t \Big]^2 = E_{b_0}\int_0^T \|\nabla L^{-1}[f_h](X_t)\|^2 dt \lesssim T\|L^{-1}[f_h]\|_{C^1}^2
-\frac{b^{1 - s}}{(q^{1 - s})_{s - 1}}\sum_{i = 0}^{\min\left\{m, s \right\}}\frac{(q^{-s})_{i}}{(q)_{i}}q^{i}\sum_{i_{1}= 0}^{n_{1}- i}\dotsm \sum_{i_{r}= 0}^{n_{r}- i}\sum_{j = 0}^{s - 1 - I_{r}}H (i, i_{1}, \dotsc, i_{r}, j).
\omega_1 = \left[\begin{matrix}\cos(\theta_1) \sin(\theta_1) \cos(\theta_2) \vdots \sin(\theta_1) \sin(\theta_2) \ldots \sin(\theta_{m-2}) \cos(\theta_{m-1}) \sin(\theta_1) \sin(\theta_2) \ldots\sin(\theta_{m-2}) \sin(\theta_{m-1})\end{matrix}\right]
(d_{\omega}F)(\gamma) = -2\,\mbox{Re}\,\langle\langle u,\,\bar\partial^{\star}\omega\rangle\rangle_\omega + 2\,\mbox{Re}\,\int\limits_X u\wedge\rho_\omega^{2,\,0}\wedge\overline{\rho_\omega^{2,\,0}}\wedge\bar\partial\omega_{n-3}
P\{T_1 = T_{\min}\}= & (-1)^{g-1}\sum_{k_g=1}^{M_g}\cdots \sum_{k_2=1}^{M_2}\sum_{k_1=0}^{M_1}(-1)^{k_1 + \cdots + k_g}\binom{M_g}{k_g}\cdots \binom{M_1}{k_1}\frac{k_2 p_2 + \cdots + k_g p_g}{k_1 p_1 + k_2 p_2 + \cdots + k_g p_g},
\begin{array}{lll}\Pi = \Pi_{\rm rd}\sqcup \Pi_1 \sqcup \Pi_0 \;, \hline\Pi_{\rm rd}= \{\alpha\in\Pi: \lambda_\alpha\geq 2\}\Pi_{1}= \{\alpha\in\Pi: \lambda_\alpha=1\}\Pi_{0}= \{\alpha\in\Pi: \lambda_\alpha=0\}\end{array}
\mathcal{W}_j=G(\mathcal{E}(u);\Upsilon_j)=G((0,E_1(u)) ; ( \varphi_{\iota+j},-2^{-1}m\varphi_{\iota+j}) )=-m^{-1}\int_\Sigma E_1(u)\varphi_{\iota+j}\, d\mu,
A^{\rm vp}_i(x_j)= -\frac{\Phi}{2\pi}\epsilon^{ij}\frac{x^j}{\rho^2}\Biggl[\frac{\alpha}{4\sqrt{\pi}}(m\rho)^5\,G^{30}_{13}\Biggl( (m\rho)^2\Bigg| \begin{array}{rrr}0 -\frac{1}{2}-\frac{1}{2}-\frac{5}{2}\end{array}\Biggr)\Biggr].
- \frac{\hat{y}_{23}}{\hat{y}_{12}\, \hat{y}_{34}}\,(\sum_{i = 0}^{4}c_n(s,t) v^n = (1+v)(s + v(1+s-t) + v^2) O(\frac{1}{(\hat{y}_{12}\, \hat{y}_{34})^2})
\big(\mathbf{H}(\mathbf{u})\big)(t, \cdot) = \exp(-t/\tau) \mathbf{H}_{0}+ \int_{0}^{t}\exp\big(-(t - s)/\tau\big) \mathbf{F}\big(\nabla_{\sigma}\mathbf{u}(s, \cdot)\big) \mathrm{d}s t \in [0, T].
u_{p,n,e}(k) = e\cdot\nu_p\left(\binom{n}{k}\right) + k = e\cdot\nu_p\left(\binom{n}{n-k}\right) + k > e\cdot\nu_p\left(\binom{n}{n-k}\right) + n-k = u_{p,n,e}(n-k)\mbox{\ \if\}k<n.
\Pr_{\rho\sim P_\epsilon}[ \rho\notin D_{k,l}(H\rho^\intercal)] \leq \Pr_{\rho\sim P_\epsilon}[\rho\in B]+\Pr_{\rho\sim P_\epsilon}[ \rho\notin D_{k,l}(H\rho^\intercal)\big|\rho\notin B] \leq 2e^{-\frac{l^2}{3\epsilon N}}+\max_{w\in\{\epsilon N\pm l\}}\Pr_{\rho\sim P_\epsilon}[ \rho\notin D_{k,l}(H\rho^\intercal)\big||\rho|=w].
\mathcal{F}_M(\lambda)=\mathrm{span}\left\{\prod_{i=1}^{q-1}\prod_{j=1}^{k_i}a^i_{-n_j^{(i)}}|\lambda\rangle_M\ |\k_i\geq0,\ 0<n_1^{(i)}\leq\dots\leq n_{k_i}^{(i)}\right\}\;,
[p.v.(\frac{1}{x})](u)=lim_{\epsilon\rightarrow0^{+}}\int_{\mathbb{R}\backslash[-\epsilon,\epsilon]}\frac{u(x)}{x}dx=\int_{0}^{+\infty}\frac{u(x)-u(-x)}{x}dxforu\in C_{c}^{\infty}(\mathbb{R})
u=v=\begin{pmatrix}x \lambda\end{pmatrix},u'=v'=\begin{pmatrix}x' \lambda\end{pmatrix},u^*=v^*=\begin{pmatrix}x^* \lambda^*\end{pmatrix}, F(u)=\begin{pmatrix}-A^T\lambda Ax-b\end{pmatrix}.
\phi(t) = \frac{\beta(t)}{\psi(t)}= \frac{a t^2 (1 + a t) (1 + a + a t)}{(1 - a t) (1 - t - a t^2) (1 - b t - a b t^2)}\left (1 + \frac{a t (1+ a t)}{(n - 1) (1 - b t - a b t^2)}\right )^{n - 1},
\partial_1 = \begin{pmatrix}1-t -1 -t -1 \cr 0 1 1 t \end{pmatrix};\partial_2 = \begin{pmatrix}-1 t t^4 -t^7 \cr -1 t^9 -t t^8 \cr1 -t^5 0 0 \cr 0 0 1 -t^5 \end{pmatrix}
\sum_{\substack{X^{\eta}\le p_1\le \dots\le p_\ell X^{\theta_1}\le\prod_{i\in\mathcal{I}}p_i\le X^{\theta_2}p_1\cdots p_\ell\le X/p_j}}^*S_{p_1\cdots p_\ell}(p_j)=o_{\mathcal{L},\eta}\Bigl(\frac{\#\mathcal{A}}{\log{X}}\Bigr),
\int_{0}^{T}- \psi'(t) (\tilde{\textbf{u}}^{\star}, \phi)_{0}~ dt + \sum_{j=1}^{d}\int_{0}^{T}\psi(t) (\textbf{P}A_{j}(\tilde{\textbf{u}}^{\star}+\bar{\textbf{u}})\partial_{x_{j}}\tilde{\textbf{u}}^{\star}, \phi)_{0}~ dt = 0.
\sum_{\substack{q<x^{1/2-\epsilon}(q,B)=1}}\sup_{(a,q)=1}\Bigl|\pi(x;q,a)-\frac{\pi(x)}{\phi(q)}\Bigr|\ll x\exp(-c_1\sqrt{\log{x}})+\log{x}\sum_{\substack{q<\exp(2c_1\sqrt{\log{x}}) (q,B)=1}}\sideset{}{^*}\sum_\chi\frac{|\psi'(x,\chi)|}{\phi(q)}.
\begin{array}{l}{\delta\over\delta C_{\mu}(\sigma)}W[C]=p^{\mu}(\sigma)=(p^0,p^1,p^2,p^3)= c(\sigma) \lambda^{\mu}(x(\sigma))= begin{align*}5pt]=c(\sigma)\left({1+x_-^2+x_1^2+x_2^2\over 2},\;\;-x_2+x_-x_1,\;\;x_1+x_-x_2,\;\;-{1+x_-^2-x_1^2-x_2^2\over 2}\right)\end{array}
\mathcal{I}_{2,1}={\mathrm{E}}\left[ A_{1}A_{2}1\left \{A_{1}A_{2}>\mathcal{C}\left( nR_{n}+\sqrt{n\log^{r+1}n}\right) \right \}\right] , & \mathcal{I}_{2,2}=\mathcal{C}\left( nR_{n}+\sqrt{n\log^{r+1}n}\right) \Pr \left(\bar c \Vert \widehat{\beta}-\beta \Vert_{2}>\Vert \widehat{\beta}-\beta \Vert_{2,n}\right) ,
u_{2k-1}^{(t)}=\frac{H_k^{(1,t)}H_{k-1}^{(0,t+1)}}{H_k^{(0,t)}H_{k-1}^{(1,t+1)}},k=1,2,\dots,m, &u_{2k}^{(t)}=\frac{1}{\delta^{(t)}}\frac{H_{k+1}^{(0,t)}H_{k-1}^{(1,t+1)}}{H_k^{(1,t)}H_{k}^{(0,t+1)}},k=0,1,\dots,m.
- 16 \left[ 9 \hat{a}^{\dagger}_e \hat{a}^{\dagger}_c \hat{a}^{\dagger}_d W^e_{\;\;fgb}V^{abcd}\hat{a}^f \hat{a}^g + 18 \hat{a}^{\dagger}_e\hat{a}^{\dagger}_f W^e_{\;\; fbc}V^{abcd}\hat{a}^f + 6\hat{a}^{\dagger}_e W^{e}_{\;\; bcd}V^{abcd}\right]
\| \gamma(t) v(t) - v(0) \| = \|v(t) - \gamma(t)^{-1}. v_0 \| & \le \| v(t) - v(t_n) \| + \| v(t_n) - g(t_n)^{-1}. v_0 \| + \| g(t_n)^{-1}. v_0 - \gamma(t)^{-1}. v_0 \| & \le \frac{1}{t_n}+ o_n(1) + \| \gamma(t)g(t_n)^{-1}. v_0 - v_0 \| & \le \frac{1}{t_n}+ o_n(1) + \frac{1}{m(n)}\| v_0 \|.
T= \alpha\begin{pmatrix}0 1 0 \ldots 0 0 0 1 0 \ldots 0 \vdots \vdots \ddots \ddots \ldots 0 \vdots \vdots . 0 \ddots 0 . . . . 0 1 0 0 . . \ldots 0 \end{pmatrix},
\begin{array}{r c l}{{\chi_{1 , k ( r + 1 ) + 1}^{( p , k p + 1 )}( q )}}&{{=}}&{{\displaystyle \sum_{n_{1}\geq \cdots \geq n_{k}\geq 0}\frac{q^{n_{1}^{2}+ \cdots + n_{k - 1}^{2}+ 2 n_{k}^{2}+ r ( n_{1}+ \cdots + n_{k - 1}+ 2 n_{k})}}{( q )_{n_{1}- n_{2}}\cdots ( q )_{n_{k - 1}- n_{k}}( q )_{2 n_{k}+ r}}\overline{{{F}}}_{p - 1 ; r}^{( 2 n_{k}+ r )}( q ) .}}\\ \end{array}
\overline{\psi}^{(1)}(x) P_+ \psi^{(1)}(x) \; \longleftrightarrow \;\frac{1}{2\pi}c^{(1)}:e^{-ia \Phi^{(1)}(x)}:_{M^{(1)}}:e^{-ib \Phi^{(2)}(x)}:_{M^{(2)}}\; e^{i\frac{\theta}{2}}\; ,
20\cdot15\cdot12/(\begin{matrix}47\\ 3\end{matrix})=20\cdot15\cdot12\cdot\frac{3!44!}{47!}=20\cdot15\cdot12\cdot\frac{3!}{45\cdot46\cdot47}=\frac{20}{47}\cdot\frac{15}{46}\cdot\frac{12}{45}\cdot3!
0 \geq (\nu_2 M - 1) \ln\left(1 - x^{\lambda}\right) \geq (\nu_2 M - 1) \ln\left(1 - \frac{1}{M^{1 + \lambda \varepsilon}}\right)= - \frac{\nu_2}{M^{\lambda \varepsilon}}+ O\left(\frac{1}{M^{1 + \lambda \varepsilon}}\right)
S_{S D}[ \vec{B}_{m}, V_{m n}] = \int d^{5}x \left\{-{\textstyle \frac{1}{4}}m^{2}\vec{B}_{m}^{T}\vec{B}_{m}-{\textstyle \frac{1}{4}}{\cal F}_{m n}{\cal F}_{m n}-{\textstyle \frac{m}{8}}\epsilon \vec{B}_{m}^{T}\eta \mathcal{D}\vec{B}_{m}\right\},
\widehat{\alpha}^\dagger_N =\alpha^\dagger\Big(-e^{-\pi\sqrt{N/3}}\Big),\intertext{and}\widehat{C}_N(q) = \frac{1}{a^2(-q)a^2(-q^N)}\,\operatorname{det}\!\left(\begin{matrix}a(-q) a(-q^N) q\dfrac{da(-q)}{dq}q\dfrac{da(-q^N)}{dq}\end{matrix}\right).
Z[A]=\exp \left[ - VT{m^{2}\over 2g^{2}}+{\rm Tr}\ln \Bigl[ \gamma_{\mu}\bigl( \partial_{\mu}- iA_{\mu}\bigr) + m \Bigr] \right] \Bigl( 1 + O(\mbox{$2$-loop}) \Bigr) \equiv{\mathrm e}^{-VT{\mbox{\Large$v$}}_{\rm T}}\Bigl( 1 + O(\mbox{$2$-loop}) \Bigr) \ ,
\acute{o}^A\bar{o}^{A'}+ o^A\acute{\bar{o}}{}^{A'}=o^A\dot{\bar{\iota}}{}^{A'}+ \dot{o}^A\bar{\iota}{}^{A'}+\iota^A\dot{\bar{o}}{}^{A'}+ \dot{\iota}^A\bar{o}{}^{A'}.
\left(\frac{(e^{-u}\phi)_t^{''}}{(e^{-u}\phi)_t^{'}})_t-2(e^{-u}\phi)\right)^{'}_t=\frac{2|h_z|^4}{(h+\bar{h})^2}\left(1-\frac{(e^{-u}\phi)^2}{(e^{-u}\phi)_t^{'}}\right)=2Q^2\left(1-\frac{(e^{-u}\phi)^2}{(e^{-u}\phi)_t^{'}}\right)
\begin{aligned}|N_{G}(h)\backslash V(T'\cup B)| \geq d_{G}(h)-|N_{G}(h)\cap V(B)| &-|N_{G}(h)\cap(V(T')\backslash\{u_{l+1},\cdots,u_{j-1}\}| &\geq (t+3)-2-(t-\lfloor\frac{j-l-1}{2}\rfloor) &\geq1+\lfloor\frac{j-l-1}{2}\rfloor.\end{aligned}
\Big|\sqrt{c^2k^2 +m^2c^4}-mc^2-\frac{k^2}{2m}\Big| = \Big|-\frac{k^4}{2mc^2}\frac{1}{(m+\sqrt{c^{-2}k^2+m^2})^2}\Big| \leq \frac{k^4}{4m^2 c^2}
\mathsf{Var}(F_\alpha(P_n)) = \sum_{i=1}^S \mathsf{Var}(P_n(i)^\alpha) + 2\sum_{1\le i<j\le S}\mathsf{Cov}(P_n(i)^\alpha,P_n(j)^\alpha) \le \sum_{i=1}^S \mathsf{Var}(P_n(i)^\alpha),
G_n^{(d)}(a,b) = - \nabla_{(a,b)}L_n (a,b,\tilde{\delta}_n, \tilde{\alpha}_n) = - (\partial_a L_n (a,b,\tilde{\delta}_n, \tilde{\alpha}_n),\partial_b L_n (a,b,\tilde{\delta}_n, \tilde{\alpha}_n))^T.
\int_0^{\infty}\exp \left(\epsilon i \omega(4\pi y_2^{1/2}\sqrt{1+u^{-2}},3\rho )- \epsilon i \omega(4\pi y_1^{1/2}\sqrt{1+u^{2}}, 3\rho )\right) u^{3\mu_2}&\times \psi(u) f^{\epsilon}_M\left(4\pi y_2^{1/2}\sqrt{1+u^{-2}}, 3\rho\right) f^{-\epsilon}_M\left(4\pi y_1^{1/2}\sqrt{1+u^{2}}, 3\rho\right) \frac{du}{u}
\left(\frac{d^{n+1}}{dt^{n+1}}B(t_0)h_1,h_2\right)=-i\alpha\left[\Big(\frac{d^n}{dt^n}(u|1)\Big)(t_0)(h_1|u(t_0))(1|h_2)-\Big(\frac{d^n}{dt^n}(1|u)\Big)(t_0)(u(t_0)|h_2)(h_1|1)\right]\ ,
d_p(\Phi(\xi_s^{\delta_{x_1},\delta_{x_2}}),\delta_{x_2})=d_p(\Phi(\xi_s^{\delta_{x_1},\delta_{x_2}}),\Phi(\delta_{x_2}))=d_p(\xi_s^{\delta_{x_1},\delta_{x_2}},\delta_{x_2})\leq\sqrt[p]{1-s}d_p(\delta_{x_1},\delta_{x_2}).
\sum_{i \in I_X}\left|e_B(X_i,Y)-\frac 12 |X_i||Y|\right|=\sum_{i \in I_X}\left|d_B(X_i,Y)-\frac 12\right| |X_i||Y| \leq \sum_{i \in I_X}\varepsilon |X_i||Y| \leq \varepsilon |X||Y| \leq \varepsilon N^2.
\begin{aligned}\nabla H_{3, \le 1}(u)= -3\,c_1\partial_x(v_x^2)-6\,c_1\partial_x\Pi_S[v_x\,z_x]-c_2\partial_{xx}(v^2)-2\,c_2\partial_{xx}\Pi_S[v\,z]+ &+c_2\pi_0[v_x^2]+2\,c_2\Pi_S[v_x\,z_x]+3\,c_3\pi_0[v^2]+6\,c_3\Pi_S[v\,z].\end{aligned}
\mathcal{I}_1 \ge \frac{c_2^2}{p\,2^{2\,p+1}}\,\left\|\frac{|u|^\frac{2}{p+1}}{\left(\displaystyle\int_\Omega |u|^\frac{2\,p}{p+1}\, dx\right)^\frac{1}{p}}-\frac{1}{U}\right\|^{p+1}_{L^{p}(\Omega)}&-\frac{2}{p\,(p+1)}\,\left(\frac{2}{(p+1)\, c_2^2}\right)^p\, (\mathcal{E}_f(U)-\mathcal{E}_f(U_0))^{p+1}.
\sum_{t=0}^{k-1}\binom{k-1}{t}(\frac{-p}{q})^{t}U_{mk+t}^{(k)}=U_{m}\sum_{t=0}^{k-1}\binom{k-1}{t}(-q)^{-t}p^{t}U_{m}^{k-1-t}U_{m+1}^{t}&=(-q)^{1-k}U_{m}\sum_{t=0}^{k-1}\binom{k-1}{t}(-qU_{m})^{k-1-t}(pU_{m+1})^{t}.
D_{\beta}I(p\|q)=\underbrace{\frac{1}{\beta(\beta-1)}}_{T}\left[\underbrace{\sum_{i}p^{\beta}_{i}}_{A}-\underbrace{\left(\sum_{i}p_{i}q^{\beta-1}_{i}\right)^{\beta}}_{X}\times\underbrace{\left(\sum_{i}q^{\beta}_{i}\right)^{1-\beta}}_{Y}\right]
\Delta_g u (x) = \left(\frac{1 - |x|^2}{2}\right)^2 \Delta u + (N - 2) \left(\frac{1 - |x|^2}{2}\right) \langle x, \nabla u (x)\rangle = \left(\frac{1 - |x|^2}{2}\right)^2 f(x),
\mbox{}\int_{\Bbb R^3}( | \nabla u_n \big|^{2}+ ( \lambda a(x) + 1 ) | u_n |^{2}) \eta_R dx+ \int_{\mathbb{R}^{3}}\phi_{u_n}u_n^{2}\eta_Rdx = \phi_\lambda'(u_n) \left( u_n \eta_R \right) - \int_{\Bbb R^3}u_n \nabla u_n \nabla \eta_R dx+ \int_{\Bbb R^3 \setminus \Omega'_\Upsilon}\tilde{f}(u_n) u_n \eta_Rdx.
\lambda\cdot f(x):=\lambda\cdot \sum_{j=0}^n a_j x^{[n-j]}= \sum_{j=0}^n a_j \sum_{h=0}^{n-j}(-1)^h \lambda^{[h]}x^{[n-j-h]}&= \sum_{k=0}^n \left(\sum_{j+h=k}(-1)^h a_j \lambda^{[h]}\right) x^{[n-k]}.
H_c ={\displaystyle \int d^3x ~{\cal H}_c}={\displaystyle \int d^3x \left ( \frac{1}{2}{\pi_i}{\pi_i}+{\frac{1}{4}}F_{ij}F_{ij}- \frac{m^2}{2}(A_0^2 - A_i^2) + A_0({\partial_i}\pi_i)\right ),}
|f_L(z)| \leq \sum_{|\alpha|\leq C\delta L}|a_{\alpha}|\left(\frac{\Gamma(|\alpha|+L)}{\alpha!\Gamma(L)}\right)^{1/2}|z^{\alpha}|+\sum_{|\alpha|>C\delta L}|a_{\alpha}|\left(\frac{\Gamma(|\alpha|+L)}{\alpha!\Gamma(L)}\right)^{1/2}|z^{\alpha}| &=:(I)+(II).
\rho^{(1)}_{\Lambda_{M'}}:=\frac{1}{\sqrt 2}(\rho_{\Lambda_{M'}}+\rho_{\Lambda_{M'\cup\Psi'}}),R^{(1)}_{\Lambda_{M'}}:=\frac{1}{\sqrt 2}(R_{\Lambda_{M'}}+R_{\Lambda_{M'\cup\Psi'}}),
\mathcal{H}(\lambda)\setminus \mathcal{H}(\lambda^+)\ =\{y_1,y_1-1,y_2,y_2-1,\cdots, y_m,y_m-1 \}\setminus \{1,x_1,x_1-1,x_2,x_2-1,\cdots, x_m,x_m-1 \},
S(p)=\left[-\frac{\sum_{i}p^{\alpha t}_{i}-\sum_{i}p^{\alpha t^{-1}}_{i}}{\alpha t-\alpha t^{-1}}\right]_{\alpha=1}=-\frac{\sum_{i}p^{t}_{i}-\sum_{i}p^{t^{-1}}_{i}}{t- t^{-1}}
J^{( 2 )}( 3 ; 1 , 1 ) = \frac{\mathrm{i}\pi^{2}}{\sqrt{k_{1 2}^{2}}}\ln \left( \frac{m_{1}+ m_{2}+ \sqrt{k_{1 2}^{2}}}{m_{1}+ m_{2}- \sqrt{k_{1 2}^{2}}}\right) .
p\cdot\phi_x'(0,0)= \sum_{k=0}^a\frac{p\cdot (-a)_k(\delta-\beta)_k(\delta-\gamma)_k}{(\delta)_k(-f-a)_k(1-f-a-\epsilon)_k}\cdot\frac{d\big((-f-x)_k\big)}{d x}\bigg|_{x=0}&+\sum_{k=0}^a\frac{(-f)_k(-a)_k(\delta-\beta)_k(\delta-\gamma)_k}{(\delta)_k(-f-a)_k(1-f-a-\epsilon)_k}\sum_{j=0}^{k-1}\bigg(\frac{p}{j-f-a}+\frac{p}{j+1-f-a-\epsilon}\bigg)
x_{0}:= \psi(\beta_{0}) = c_{4}=b_2 x_{1}:= \psi(\beta_{1}) = b_{1}b_{2}a_{2}a_{2}x_{2}:= \psi(\beta_{2}) = b_{1}b_{2}a_{2}{b_2}^{-1}{a_1}^{-1}a_{2}a_{1}x_{c}:= \psi(c) ={a_1}^{-1}{a_2}^{-1}a_{1}b_{2}a_{2}{b_2}^{-1}a:= C_5^{-1}C_4^{-1}(\beta_{2}) = a_{1}b_{2}b:= C_5^{-1}C_4^{-1}(c_{3}) = b_{1}b_{2}{b_1}^{-1}{a_1}^{-1}
T_3\equiv \frac{n!}{2p}\sum\limits_{\begin{smallmatrix}a=2 a \,\, \textrm{odd}\end{smallmatrix}}^{n-2}{\frac{B_{p-a}B_{p-n+a}}{a(n-a)}p}\equiv \frac{n!}{2}\sum\limits_{\begin{smallmatrix}a=2 a \,\,\textrm{odd}\end{smallmatrix}}^{n-2}{\frac{B_{p-a}B_{p-n+a}}{a(n-a)}}\pmod{p}.
\sum_{\substack{d\le H (d,N)=1}}\mu^2(d)\left|\pi(N;d,N)-\frac{\pi(N)}{\varphi(d)}\right| \leq 0.65 H\left(c_{\pi}(X_2)+\frac{1}{16\pi}\right) \sqrt{N}\log N \leq\frac{p_G(X_2)N}{\log^AN},
\frac{d}{dt}Q_{A}=\frac{d}{dt}\int_{x}e^{-\frac{x^{2}}{2A^{2}}}J^{0}(x)=-\int_{x}e^{-\frac{x^{2}}{2A^{2}}}\nabla\cdot J=\int_{x}\nabla(e^{-\frac{x^{2}}{2A^{2}}})\cdot J,
E\left( S_{2^{k}}\bar{S}_{2^{k}}|\eta_{0}\right) =E\left( S_{2^{k}}E(\bar{S}_{2^{k}}|\eta_{2^{k}})|\eta_{0}\right) =E\left( E(S_{2^{k}}E(\bar{S}_{2^{k}}|\eta_{2^{k}})|\eta_{0},\eta_{2^{k}})|\eta_{0}\right) & =E\left( E(S_{2^{k}}|\eta_{0},\eta_{2^{k}})E(\bar{S}_{2^{k}}|\eta_{2^{k}})|\eta_{0}\right) .
0 \leq X(a,a+b) - X(\lceil a \rceil , \lfloor a \rfloor + \lfloor b \rfloor ) = X(a, \lceil a \rceil) + X ( \lfloor a \rfloor + \lfloor b \rfloor , a+b) &\leq X (a , \lceil a \rceil) + X (\lfloor a \rfloor + \lfloor b \rfloor , \lfloor a \rfloor + 2 + \lfloor b \rfloor ),
\mathcal{R}_3=\Bigl\{(u,v,w,t):\, \theta_2-\theta_1<t<w<v<u<\theta_1,\,u+2v<1-\theta_1,u+v+2w<1, &u+v+w+2t<1,\,\theta_2<u+v<1-\theta_2, \{u+v,u+w,u+t,v+w,v+t,w+t\}\cap[\theta_1,\theta_2]=\emptyset\,\Bigr\}.
\int_{B_{r\lambda_n/2}(y_n)}\Phi(|\nabla u_n|) dx= \int_{A_{\lambda_n R,\lambda_n r}\cap B_{{r\lambda_n/2}}(y_n)}\Phi(|\nabla u_n|) dx=\int_{\Theta_n}\Phi(|\nabla u_n|) dx.
\|K(f\otimes e_j)\|_{L^{p'}(Y,\mathcal H)}= \|K_j f\|_{L^{p'}(Y)}\leq \|K_j\|_{L^p(X)\to L^{p'}(Y)}\|f\|_{L^{p}(X)}& = \|K_j\|_{L^p(X)\to L^{p'}(Y)}\|f\otimes e_j\|_{L^{p}(Y,\mathcal H)}\,,
\sum{\gamma_i(\gamma_i + \gamma_i^*)\theta_i^2}= -\frac{4c}{d}- 3c^2 - (j^2 + jk + l^2 + lm + n^2 + np + q^2 + qr) &= -4c\sqrt{c^2 + 1}+ 4c^2 - 3c^2 - (j^2 + jk + l^2 + lm + n^2 + np + q^2 + qr)
\sum_{m=0}^{\infty}\sum_{l=1}^{h(m,n)}\sum_{j=0}^\infty \|\vec a_m^l(j)\|<\infty,\sum_{m=0}^{\infty}\sum_{l=1}^{h(m,n)}\sum_{j=0}^\infty (j+1)\|\vec a_m^l(j)\|^2<\infty,
Z_{0}( 0 ) = - \int_{0}^{\infty}d y \left[ \ln ( 2 y I_{0}( y ) K_{0}( y ) ) - \frac{t^{2}}{8}( 1 - 6 t^{2}+ 5 t^{4}) \right] = \pi \left( 0 . 0 2 8 1 5 - \frac{1}{1 2 8}\right) .
h^{\prime\prime}(t) = 4\,f^{\prime\prime}(\log^2t)\cdot\frac{\log^2t}{t^2}- 2\,f^{\prime}(\log^2t)\cdot \frac{\log t}{t^2}+ 2\,f^{\prime}(\log^2t)\cdot \frac{1}{t^2}&= \frac{2}{t^2}\,\Big(2\,(\log^2t)\, f^{\prime\prime}\big(\log^2t\big) + (1-\log t)\, f^{\prime}\big(\log^2t\big)\Big)\,.
m m^{\dagger}= \rho^{2}\left( \begin{array}{c c c}{{x^{2}}}&{{x y}}&{{x z}}\\{{y x}}&{{y^{2}}}&{{y z}}\\{{z x}}&{{z y}}&{{z^{2}}}\\ \end{array}\right) = \rho^{2}\left( \begin{array}{c}{{x}}\\{{y}}\\{{z}}\\ \end{array}\right) ( x , y , z ) ,
e_x\alpha\beta e_y=(\alpha\beta)_{xy}[x,y] =\left(\sum_{x\le z\le y}\alpha_{xz}\beta_{zy}\right)[x,y]=\left(\sum_{z\in Z}\alpha_{xz}\beta_{zy}\right)[x,y] =\sum_{z\in Z}(\alpha_{xz}[x,z]\cdot\beta_{zy}[z,y])=\sum_{e_z\in E(e_x,\alpha,\beta,e_y)}e_x\alpha e_z\beta e_y,
\begin{array}{c c c}{{\partial^{\mu}C_{n \mu}^{F}= 0 ,}}&{{\quad \partial_{\mu}b_{n}^{F \mu}= 0 ,}}&{{\quad \partial_{\mu}\bar{C}_{n}^{F \mu}= 0 ,}}\\{{\partial^{\mu}C_{n \mu}^{B}= 0 ,}}&{{\quad \partial_{\mu}b_{n}^{B \mu}= 0 ,}}&{{\quad \partial_{\mu}\bar{C}_{n}^{B \mu}= 0 .}}\\ \end{array}
\|f_{\vec{S}_{\ell}}\|_{L^2(B^{n-1})}^2 \lesssim_{\varepsilon}r_{\ell}^{\ell/2}\Big(\prod_{i = 1}^{\ell}r_i^{-1/2}D_i^{\delta}\Big) R^{O(\varepsilon_{\circ})}\|f_{\vec{S}_{\ell}}^{\#}\|_{2}^2
E[g(W,\gamma,\theta_{0})] =E[E[DY|X]\{\gamma(X)-\gamma_{0}(X)\}]=E[\alpha_{0}(X)\gamma_{0}(X)^{-1}\{\gamma(X)-\gamma_{0}(X)\}] & =E[\alpha_{0}(X)\{\gamma(X)P_{0}(X)-1\}]=E[\alpha_{0}(X)\{\gamma(X)D-1\}]=-E[\alpha_{0}(X)\lambda(W,\gamma)].
E[|\widetilde{Z}_n^*(s)-\widetilde{Z}_n^*(t)|^2|\mathcal{F}^X] =\frac{2}{3}\sum_{i=1}^n\left\{\frac{K_h(t_{i-1}-s)}{\mathfrak{s}_n(s)}-\frac{K_h(t_{i-1}-t)}{\mathfrak{s}_n(t)}\right\}^2(X_{t_{i}}-X_{t_{i-1}})^4 &\leq c_2 r_n^2\frac{nh}{h^2}\cdot(nh)^2\cdot\frac{(s-t)^2}{nh^3}=c_2r_n^2\frac{n^2}{h^2}(s-t)^2
\begin{aligned}|\tilde Y_{s,t}(x,v)| \lesssim \frac{\delta (t-s)}{\langle d_{t,x}\rangle^2 + \langle x^\perp\rangle^2}, |\nabla_v \tilde Y_{s,t}(x,v)| \lesssim \frac{\delta (t-s)}{\langle d_{t,x}\rangle + \langle x^\perp\rangle}.\end{aligned}
\left[i.\rho +\frac{3}{2}\right].\left[i.\rho +\frac{5}{2}\right]\ldots \left[i.\rho +2.k+\frac{1}{2}\right]- &\left[i.\rho -\frac{3}{2}\right].\left[i.\rho -\frac{5}{2}\right]\ldots \left[i.\rho -2.k-\frac{1}{2}\right]=0
C_{i}^{I I A}= h_{1 0 i}^{1 1}, \qquad h_{0 i}^{I I A}= h_{0 i}^{1 1}, \qquad B_{i j}^{I I A}= A_{1 0 i j}^{1 1}, \qquad C_{0 i j}^{I I A}= A_{0 i j}^{1 1}
P_1 :=\prod_{i=r+1}^{t}(q-d_i-2i+1)\left(\prod_{i=t+1}^{m}(q-2i)+2(m-t)\prod_{i=t+1}^{m-1}(q-2i)+(m-t)(m-t-1)\prod_{i=t+1}^{m-2}(q-2i)\right), P_2 :=\left(\sum_{i=r+1}^{t}\prod_{j=1}^{i-1}(q-d_j-2j+1)\prod_{j=i+1}^{t}(q-d_j-2j+3)\right)\left(\prod_{i=t+1}^{m}(q-2i+2)+(m-t)\prod_{i=t+1}^{m-1}(q-2i+2)\right)
\sum\limits_{k=1}^n \sum\limits_{p=n+1}^{2n}d(p-k) = \sigma^2 n^2 +\int\limits_{\mathbb{R}\setminus \{0\}}e^{(n-1)u}\Bigl[\frac{e^u-e^{(n+1)u}}{1-e^u}\Bigr]\Bigl[\frac{e^{-u}-e^{-u(n+1)}}{1-e^{-u}}\Bigr](e^u-1)^2\,d\mathcal{M}(u).
\Gamma_{A^{\mu}A^{\nu}}( p , - p ) \equiv \Gamma_{\mu \nu}( p , - p ) = ( \eta_{\mu \nu}- \frac{p_{\mu}p_{\nu}}{p^{2}}) \Gamma^{T}( p^{2}) + \frac{p_{\mu}p_{\nu}}{p^{2}}\Gamma^{L}( p^{2}) \; .
T_{n}= ( 1 + \frac{1}{3}) - ( \frac{1}{3}+ \frac{1}{5}) + ( \frac{1}{5}+ \frac{1}{7}) + \cdots + ( - 1 )^{n - 1}( \frac{1}{2 n - 1}+ \frac{1}{2 n + 1}) .
L_{c}^{- 1}[\psi_{1}\otimes \cdots \otimes \psi_{m}]_{x_{1}^{0},...,x_{l}^{0}}(\underline{x};\underline{n},\underline{T})) =\bigl( \prod_{i = 1}^{l}L_{c}^{- 1}[\psi_{i}]_{x_{i}^{0}}(x_{i};n_{i},T_{i})\bigr)\bigl( \prod_{i = l + 1}^{m}\psi_{i}(x_{i})\bigr)
\widetilde{(g_{e}.1)^{2s}}(4q)=(ab)^{2s}c^{-s}\sum_{k>2q}{-s \choose k}{2k \choose k-2q}\frac{c^{-k}d^{k}}{2^{2k-1}}=c^{-s}\sum_{j=0}^{\infty}{2j+4q \choose j}\frac{d^{j+2q}\Gamma(j+2q-s)}{2^{2q-1}c^{j+2q}\Gamma(-s)}(1/2)^{j}.
a = h^{-M}= \begin{cases}h^{-\frac{\sigma}{k}- T\eta \lambda}= h^{-\frac{\sigma}{k}}(\log(h^{-1}))^T \ge h^{-\frac{1}{3}}0 < \delta \le 1, h^{-1- T\eta \lambda - t\eta}= e^{t}h^{-1}(\log(h^{-1}))^T \ge h^{-1}\delta = 0. \end{cases}
v_\kappa (z)\cdot z =\rho_\kappa^{-1}(m_\kappa \circ \, \phi_\kappa^{-1}(z) -z)\cdot z = \rho_\kappa^{-1}(m_\kappa \circ \, \phi_\kappa^{-1}(z)\cdot z -1 ) = -\frac{1}{2\rho_\kappa}| m_\kappa \circ \, \phi_\kappa^{-1}(z)- z|^2.
F_{6}\big(3^{-2/3}t\big) = \frac{1-q_2}{2}\exp\left(\frac{1}{3}\int_{t}^{\infty}\omega(s)ds +\frac{2}{3}\int_{t}^{\infty}\alpha(s)ds -\frac{1}{3}\int_{t}^{\infty}\frac{u_s(s)}{u(s)}(1+q_2(s))ds\right),
\begin{aligned}z_t = \mathcal{F}_\psi[z, u_\psi] \doteqdot z z_{\psi \psi}+ z_\psi \left( \frac{q-1}{\psi}- \frac{z}{\psi}\right) - \frac{1}{2}z_\psi^2 + \frac{2(q-1)}{\psi^2}z(1-z) - 2pz^2 u_\psi^2 u_t = \mathcal{E}_\psi[z, u] \doteqdot z u_{\psi \psi}+ u_\psi \frac{z + q-1}{\psi}- (p-1) e^{-2u}\end{aligned}
\sum_{j = 0}^k{( - 1)^j ( - 1)^{rj}\binom kj\left({\frac{{F_{n + r}}}{{F_n}}}\right)^j G_{m - (n + r)k + rj}}= ( - 1)^k ( - 1)^{rk}\left({\frac{{F_r}}{{F_n}}}\right)^k G_m ,n\ne 0\,.
\sum_{m=1}^\infty \Bigl|(L_tZ)_{{\nu_0}+m}(t,s;\tau,\xi)\Bigr| \leq C_8 (1+|v|_N)^{\nu_0+\frac{N}{1-\mu}}e^{C_8 (T\vee 1)(1+|v|_N)^{\frac{N}{1-\mu}}},
\times \exp\left\{\frac{1}{(1-x)^3(1-y)^5}\sum_{n=1}^{\infty}(-n^3x^{n+2}y^{n+1}+2n^3x^{n+2}y^{n+2}-n^3x^{n+2}y^{n+3}) \frac{z^n}{n^4}\right\}
\tilde{\Phi}^{1}= c \partial c , ~ \tilde{\Phi}^{2}= \frac{1}{3}c \partial c T_{B}, ~ \tilde{\Phi}^{3}= \frac 1 2 \partial c \partial^{2}c , ~ \tilde{\Phi}^{4}= \frac 1 6 \partial^{3}c c .
\| u \|^2_{\Sigma_1}:= \| u \|^2_{L^2(0,1)}+ \sum_{n=1}^{m_d}\| (\cdot)^{n-1}u^{(n)}\|^2_{L^2(0,1)}, &\| u \|^2_{\Sigma_2}:= \sum_{n=0}^{m_d-1}\| (\cdot)^{n}u^{(n)}\|^2_{L^2(0,1)}.
\begin{array}{ll}\int\limits_0^t \int\limits_{\mathbb{R}^3}\phi\partial_t\varphi_3 - k_1 v\cdot\nabla\phi \varphi_3\,\mathrm{d}x\mathrm{d}s = -\int\limits_{\mathbb{R}^3}\phi_0(x,\tau)\varphi_3(\cdot,0)\,\mathrm{d}x.\end{array}
E[\widetilde{V}_{0,n}^\ell] \le 2^\ell \sum_{\underline{x},\underline{y}}E\Big[ \prod_{i=1}^\ell L_n(x_i) \Big] E \Big[\prod_{i=1}^\ell L_\infty(y_i)\Big]\prod_{i=1}^\ell G(x_i,y_i) \,,
0\leq c_{\mathcal{A}}\left( n\right) \leq\left( n+q\right)^{\frac{k}{2}}c_{\mathcal{A}}\left( n\right) \asymp n^{k/2}\lim_{n\rightarrow\infty}\frac{c_{\mathcal{A}}\left( n\right)}{n^{k/2}}=1).
\mathrm{H}_{3 1}^{F}= - \frac{1}{4}{\cal F}_{3 1}\left( 9 \alpha_{3 1}+ 6 \beta_{3 1}\right) \int d^{3}{\vec{x}}\; i \pi ({\vec{x}}) \tau_{a}\partial_{a}{\xi}( \vec{x}) \left( \frac{\ell_{P}}{{\cal L}}\right)^{2 \left( 1 + \Upsilon \right)}
\int_{B_\rho(\zeta_i)}w_i^5\,U_j\,dx=\mu_i^{\frac{1}{2}}\mu_j^{\frac{1}{2}}\int_{B_{\mu_i}}U^5(z)\,\mu_j^{-\frac{1}{2}}\,U_j(\zeta_i+\mu_i z)\,dz.
\frac{1}{\kappa(s)^{n}r_{s}(v)}\mathbb{E}\Big[ r_{s}(G_n \!\cdot\! v) \| G_n v\|^s h\big( G_1 \!\cdot\! v, \log \|G_1 v\|, \ldots, G_n \!\cdot\! v, \log \|G_n v\| \big) \Big] &\qquad\qquad=\mathbb{E}_{\mathbb{Q}_{s}^{v}}\Big[ h\big( G_1 \!\cdot\! v, \log \|G_1 v\|, \ldots, G_n \!\cdot\! v, \log \|G_n v\| \big) \Big].
c'_{\alpha_1}= c_{\alpha_1}+ c_{\alpha_1+\alpha_2}+ c_{\alpha_1+2\alpha_2}-\pi_1, c'_{\alpha_1+\alpha_2}= 2\pi_1 - \pi_2, c'_{\alpha_1+2\alpha_2}= \pi_2 - \pi_1, c'_{\alpha_2}= c_{\alpha_1+\alpha_2}+ 2 c_{\alpha_1+2\alpha_2}+ c_{\alpha_2}- \pi_2,