harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
{\cal L}_{e f f}= I m \: [ \: \int d^{2}\theta \: d^{2}\bar{\theta}\frac{1}{2}K ( \Phi^{a}, \bar{\Phi}^{a}, V^{a}) + \int d^{2}\theta f_{a b}( \Phi ) W^{a}W^{b}+ c \: c \: ] ,
\mathcal R |_{M''}=U+\sum_{i\ne 1}^n V_i{\ss}_{1i}\,,\mathcal R^+ |_{M''}=U^++\sum_{i\ne 1}^n V_i^+{\ss}_{1i}^+\,, \quad U=\prod_{l\ne 1}^n a_{1l}\,,V_i=b_{1i}\prod_{l\ne 1, i}a_{il}\,,U^+=\prod_{l\ne 1}^n a_{1l}^+\,,V_i^+=b_{1i}^+\prod_{l\ne 1, i}a_{li}\,.
\mathbf{b}_T(\psi_{2,k,p},\theta_{2,k,p}) = \frac{1}{\sqrt{N}}\left[ e^{j2\pi n_1 \frac{d_R}{\lambda}\cos(\theta_{2,k,p})\sin(\psi_{2,k,p})}\right]^T_{n_1\in \mathcal{I}(N_1)}\otimes \frac{1}{\sqrt{N}}\left[ e^{j2\pi n_2 \frac{d_R}{\lambda}\sin(\theta_{2,k,p})}\right]^T_{n_2\in \mathcal{I}(N_2)}.
S_{\mathrm{g f}}\left[ A \right] = - \sum_{c_{0}= 1}^{N - 1}\int d^{4}x \frac{1}{2 \xi}\left( \frac{1}{L}\int_{0}^{L}d x_{3}\partial^{\mu}A_{\mu}^{c_{0}}\right)^{2}.
\Pi(\|\theta-\theta_{0}\|^2 / \varepsilon^2 \leq 2T) \geq F(\overline{k})\mathrm{S}_{M}( \|\theta-\theta_{0}\|^2 / \varepsilon^2 \leq 2T \mid \alpha= \overline{k}) &\geq F(\overline{k})M(T)\mathrm{S}( \|\theta-\theta_{0}\|^2 / \varepsilon^2 \leq 2T \mid \alpha=\overline{k}, d=T) &\geq \exp\{-1 -\gamma \overline{k}- (c_{3}+2+\eta) T \},
2(J_{n+1}-a^2a_{n+1}^2I)-(J_n-a^2a_n^2I) =(a^2b_n^2-3a^2a_{n+1}^2+a^2a_n^2)I &=\frac{a^2}{2}(a_n^2-b_n^2)I &=2a^2c_{n+1}^2I.
\hat{\pi}(\tau)-\hat{\pi}(\tau-1)=\binom{\tau-1}{r-1}p^r(1-p)^{\tau-r}\leq \binom{\tau}{r-1}p^r(1-p)^{\tau-r}\stackrel{p=o(1)}{\leq}\binom{\tau}{r-1}p^{r-1}(1-p)^{\tau-r+1}\leq \sum_{i=0}^{r-1}\binom{\tau}{i}p^i(1-p)^{\tau-i}=1-\hat{\pi}(\tau)
{\cal P}^{a\mu}= \alpha_2 \sqrt{-\gamma}\left[ \left( K_i K^i \gamma^{ab}- 2 K_i K^{ab\,i}- 2{\cal G}^{ab}\right) e^\mu{}_b + 2 (\widetilde\nabla^a K^{i}) n^\mu{}_i \right]\,.
\frac{n^{2d}}{m^2}\sum_{l = 2}^m \mathbb{E}_0[(D_{nl}^T)^2] = \frac{(m-1)n^{2d}}{2m}\mathrm{Var}_0\left[W^{(12)}_h\right] \frac{n^d}{m^2}\sum_{l = 2}^m \mathbb{E}_0\left[(D^Z_{nl})^2\right] = \frac{(m-1)n^{d}}{2m}\mathrm{Var}_0\left[U^{(12)}_h\right].
(-a-p)_k(p-a)_k=\frac{\Gamma_p(-a+k-p)}{\Gamma_p(-a-p)}\cdot\frac{\Gamma_p(-a+k+p)}{\Gamma_p(-a+p)}\equiv\frac{\Gamma_p(-a+k)^2}{\Gamma_p(-a)^2}=(-a)_k^2\pmod{p^2}.
C^n_{}( \varphi, \psi) (f) (\lambda_0, \ldots, \lambda_{n-1}) = \psi \left( f \left( \varphi(\lambda_0), \ldots, \varphi(\lambda_{n-1}) \right) \right) C^n_{\operatorname{cub}}( \varphi, \psi) (g) (\lambda) = \psi \left( g \left( \varphi(\lambda) \right) \right)
G^{(1)}_{m,n}=C(\dot{a_1-1},a_2,\ldots,a_d), A_{m,n}=G^{(1)}_{m,n}\vee C(\dot1,1,\dot1)\vee G^{(1)}_{m,n}, B_{m,n}=G^{(1)}_{m,n}\vee C(\ddot4)\vee G^{(1)}_{m,n}, C_{m,n}=G^{(1)}_{m,n}\vee C(\dot2,\dot2)\vee G^{(1)}_{m,n}.
\sup_{\theta \in A \times W_n^{(\eta)}}\ln(n)^q \left| \frac{1}{n}\sum_{i=1}^n f(X_{\frac{i-1}{n}}, \delta_0, \alpha_0) \left( h(z^n_i(\theta), \alpha) - h(n^{1/ \alpha_0}\Delta_i^n L,\alpha) \right) \right| \to 0,
Q_1 := \left\{q: q < \frac{\alpha}{2}(u_1 + u_2)\right\}, Q_i := \left\{q: \frac{\alpha}{2}(u_{i-1}+ u_i) < q < \frac{\alpha}{2}(u_i + u_{i+1}) \right\}, 1 < i < d, Q_d := \left\{q: q > \frac{\alpha}{2}(u_{d-1}+ u_d) \right\}, Q_{i,i+1}:= \left\{q: q = \frac{\alpha}{2}(u_i + u_{i+1}) \right\}.
\mathbf b = p(t,S_0)V(t,S_0)\bar{\mathbf L}_2[:,1] + w(t,S_0)V(t,S_0)\bar{\mathbf L}_1[:,1] + p(t,S_{M+1})V(t,S_{M+1})\bar{\mathbf L}_2[:,M+2] + w(t,S_{M+1})V(t,S_{M+1})\bar{\mathbf L}_1[:,M+2] + g(t,\mathbf S),
\frac1X \int_X^{2X}\Big(P_k \Big(x+\frac{x^{1-\frac1k}}{L};\theta \Big)-P_k(x;\theta)\Big)^2 \, dx= \mathcal M_k(X^{\theta},L)\Big(1+O(X^{-\varepsilon_2}) \Big) &+O\Big( \frac{X^{1-\frac1k-\varepsilon_2}}{L}\Big).
Q = \frac{6\cdot 16}{4!(2\pi)^4}\left( \frac{\zeta\pi}{2}\right)^2 \left( \frac{\zeta'\pi}{2}\right)^2 \sum \biggl\{\frac{-2\zeta^2}{\delta (\delta +\zeta /2)^2(\delta +\zeta)}\biggr\}\biggl\{\frac{-2\zeta^{\prime 2}}{\delta' (\delta' +\zeta^{\prime}/2)^2(\delta' +\zeta^{\prime})}\biggr\}=1\,,
\mathbb{E}[\breve{Z}_p^r\breve{Z}_p^t] =\frac{\tilde{E}_p^{r,t}}{\sigma_p^r \sigma_p^t},\mathbb{E}[\tilde{Z}_p^r\tilde{Z}_p^t] =\frac{\breve{E}_p^{r,t}}{\tau_p^r\tau_p^t},
\Delta_n(f,\widetilde{f}) = n[P_n(T\widetilde{f}||f^{\star})- P_n(f||f^{\star})] + n\tau^{-1}[\tau_1^{-1}D^{\prime}_n(Tf,f^{\star}) - D^{\prime}_n(T\widetilde{f},f^{\star})].
\|f_0\|_{L^\infty}+\kappa - C\delta \leq \iint_{\Omega\times\mathbb{R}^3}F(t_*,x,v)\psi(t_*,x,v) dxdv = \iint_{\Omega\times\mathbb{R}^3}f_0(x,v)\psi(0,x,v) dxdv begin{align*}3pt] \leq \|f_0\|_{L^\infty}.
\left( \begin{array}{c}{{m^{\prime}}}\\{{N^{\prime}}}\\ \end{array}\right) = \left( \begin{array}{c c}{{a}}&{{b}}\\{{c}}&{{d}}\\ \end{array}\right) \left( \begin{array}{c}{{m}}\\{{N}}\\ \end{array}\right) , \qquad \Theta^{\prime}= \frac{c + d \Theta}{a + b \Theta},
f_{0,n}(x) := f_n^*(x) , \f_{1,n}(x) := f_n^*(x)+ R h_n^\beta K\big(\tfrac{x-x_0}{h_n}\big) , \ \h_n = c_0 \left(\tfrac{f_n^*(x_0)}{n}\right)^{1/(2\beta+1)},
\int \! \! d^{d}p \, f ( p^{2}) = \Omega_{d}\int_{0}^{\infty}\! \! d p \, p^{d - 1}\, f ( p^{2}) \; ; \qquad \Omega_{d}= \frac{2 \pi^{d / 2}}{\Gamma ( d / 2 )}\quad ,
{\bf C}^{\bullet}_{{\bf b}, J}: ~ 0 \to B \to \prod_{i=1}^s B_{b_i, J}\to \prod_{i<j}\big(B_{b_i, J}\big)_{b_j, J}\to \cdots \to \big( \cdots \big(B_{b_, J}\big) \cdots \big)_{b_s, J}\to 0
\langle [\xi,\eta]_{\ast},[X,Y]\rangle=\langle \xi\otimes \eta,\gamma_{(lr-r)}[X,Y]\rangle &=\langle \xi\otimes \eta,(\mathrm{ad}_{X}^{(l)}\otimes 1)\gamma_{(lr-r)}(Y)\rangle +\langle \xi\otimes \eta,( \mathrm{ad}_{Y}^{(r)}\otimes 1)\gamma_{(lr-r)}(X)\rangle.
\|y_{1,k}^1\|^2+\|y_{2,k}^1\|^2=\left\|\begin{bmatrix}y_{1,k}^1 y_{2,k}^1\end{bmatrix}\right\|^2\leq \left\| \begin{bmatrix}u_{1,k}^1 u_{2,k}^1\end{bmatrix}\right\|^2=\|u_{1,k}^1\|^2+\|u_{2,k}^1\|^2.
\frac{d}{ds}A_{\alpha'}[\gamma_s]\Big\vert_{s=0}=\frac{d}{ds}\Big(\int_{\gamma_s}\alpha+\beta\Big)\Big\vert_{s=0}&=\int_{\gamma_s}df\Big\vert_{s=0}=\int_a^b df(\gamma(t))=f(\gamma(b))-f(\gamma(a)),
\mathcal{F}_{\varepsilon}^{\xi}(v,A):=\frac{1}{\sigma_{n-1}\varepsilon}\int_{A}W(v(t))~dt+\frac{\varepsilon}{2}\int_{A}\int_{A}J_{\varepsilon}^{\xi}(s-t)(v^{\prime}(s)-v^{\prime}(t))^{2}dsdt
f\star_{1}^{n}g \ (\mathbf{n}) \ = \iint_{S^{2}\times S^{2}}f(\mathbf{n}_{1})g(\mathbf{n}_{2})\mathbb{L}^{j}_{1}(\mathbf{n}_{1},\mathbf{n}_{2},\mathbf{n})d\mathbf{n}_{1}d\mathbf{n}_{2}\ ,
\sum_{n=1}^{\infty}\frac{p^{2}+\lambda_{n}^{2}}{p\left(p+\frac{1}{\pi}\right)+\lambda_{n}^{2}}\,\frac{1}{\left(\lambda_{n}^{2}+x^{2}\right)^{s}}=\frac{2\sin(\pi s)}{x^{2s-1}}\,\intop_{1}^{\infty}\left\{\frac{1}{\sigma(xt)e^{2\pi xt}-1}+\frac{1}{2}-\frac{1}{2\pi xt}\cdot\frac{1}{1+\frac{1}{\pi p}}\right\}\,\frac{dt}{(t^{2}-1)^{s}}.
\lambda_{\pm}=\frac{x^2-t}{2}\pm \mu,\mu=\sqrt{\frac{x^4}{4}+r_2x^2+r_1x+r_0}=\frac{x^2}{2}+r_2+\frac{r_1}{x}++\frac{r_0-r_2^2}{x^2}+\cdots,\!\!\!
\sum_j\frac{\Re(\eta_j)}{1+|\eta_j|^2}= \,\sum_j\frac{\Re(\theta_j)}{1+|\theta_j|^2}=f\sum_j\frac{\Re(\theta_j)}{f}\frac1{1+|\theta_j|^2}\ge \,\frac{f}{1+f^{-1}\sum_j\Re(\theta_j)|\theta_j|^2}\ge\,\frac{f}{1+f^{-1}\sum_j|\theta_j|^3}\ge \frac{f}{1+1/\Delta}\ge \frac{f}2\min\{1,\Delta\}.
\P_x\left\{(x_i): \frac{1}{n}\sum_{k=0}^{n-1}\xi(x_k,X(x_{k-1})\ldots X(x_0)c) \in [\ell^-_\xi -\epsilon,\ell^+_\xi+\epsilon]\right\}\geq 1-C_0 e^{-n\alpha_0},
\int_{\Sigma}\frac{1-\langle \nu(x), \nu (p) \rangle}{|p-x|^{N+s}}dx =\sqrt{1+\alpha^2}A_{m-2}\int_0^\infty( J_+(r) + J_-(r)) r^{N-2}dr &=\sqrt{1+\alpha^2}A_{m-2}\int_0^1(r^{N-2}+ r^s)( J_+(r) + J_-(r)) dr ,
u^1_0(t,y,z) = e^{\nu t \Delta}\left(u^1_{in \; 0}- t u_{in \; 0}^2\right) u^2_0(t,y,z) = e^{\nu t \Delta}u^2_{in \; 0}u^3_0(t,y,z) = e^{\nu t \Delta}u^3_{in \; 0}.
W\left(0,\exp(i t H)\right) = \exp(-it\lambda(H)) ,\; W\left(tu,I\right) = \exp(-itp(u)), q = p(iu) ,\; a(u)=\frac{1}{2}\left(q(u)+ip(u)\right) ,\; a^{\dagger}(u)=\frac{1}{2}\left(q(u)-ip(u)\right).
\mathcal{M}_{|\Delta_n|^2}(s) = \frac{\left({\alpha_n}/{b_{0,n}}\right)^{\alpha_n}}{2\Gamma(\alpha_n)}\int_0^{\infty}b^{\alpha_n-2}\exp\left(-\frac{\alpha_n b}{b_{0,n}}\right) &\left[\int_0^{\infty}\exp\left(-sx-\frac{x+\tilde{A}_n^2}{2b}\right)I_0\left(\frac{\tilde{A}_n\sqrt{x}}{b}\right)\mathrm{d}x\right] \mathrm{d}b.
l_\sigma(v^{2k}; 0,0) = l_\sigma(v^{-2k};0,0)^{-1}= ((uv)^{2k};0,-2k)^{-1}= ( ( \theta(0,2k)((uv)^{2k}))^{-1};0,2k ) & = ( ( (\theta(0,2k)(u) \theta(0,2k)(v))^{2k})^{-1}; 0,2k) = ((uv)^{-2k};0,2k),
28 = 1^2 + 3^2 + 3^2 + 3^2 * = 2^2 + 2^2 + 2^2 + 4^2 20 = 1^2 + 1^2 + 1^2 + 1^2 + 4^2 * = 2^2 + 2^2 + 2^2 + 2^2 + 2^2 40 = 1^2 + 1^2 + 1^2 + 1^2 + (2\cdot3)^2 * = 2^2 + 3^2 + 3^2 + 3^2 + 3^2,
\sum_{n=1}^{\infty}\frac{p^{2}+\lambda_{n}^{2}}{p\left(p+\frac{1}{\pi}\right)+\lambda_{n}^{2}}\,\frac{1}{x^{2}+\lambda_{n}^{2}}=\frac{\pi}{2x}-\frac{1}{1+\frac{1}{\pi p}}\cdot\frac{1}{2x^{2}}+\frac{\pi}{x}\,\frac{1}{\sigma(x)\,e^{2\pi x}-1},
\|u(t)\|_{\dot H^s}\ge\begin{cases}c(T-t)^{-(2s-1)/4}s\in(1/2,5/2),\s\neq 3/2, c\|u_0\|_{L^2}^{(5-2s)/5}(T-t)^{-s/5}s>5/2.\end{cases}
\sum_{i\le m}\langle \bar{u}, X_i\rangle^2= \sum_{i\le m}\bigl\langle \bar{u}, \frac{X_i}{\vert b_i\vert}\bigr\rangle^2 \vert b_i\vert^2 = \sum_{i\le m}\d(0, E)^2\cdot \vert b_i\vert^2= \d(0, E)^2.
e_{ij}=\begin{cases}\delta_{i,j}\;j\neq k, -1 \;i=j=k, \max(0,-b_{ik}) \;i\neq j=k,\end{cases}f_{ij}=\begin{cases}\delta_{i,j}\;i\neq k, -1 \;i=j=k, \max(0,b_{kj}) \;i=k\neq j.\end{cases}
(f_d^{-1})^{(1)}(x) = \frac{(e^d-1)e^x}{(e^x-1)(e^d-e^x)}, (f_d^{-1})^{(2)}(x) = \frac{(e^d-1)e^x(e^{2x}-e^d)}{(e^x-1)^2(e^d-e^x)^2}, (f_d^{-1})^{(3)}(x) = \frac{(e^d-1)e^x[e^{4x}+(e^d+1)e^{3x}-6e^de^{2x}+(e^{2d}+e^d)e^x+e^{2d}]}{(e^x-1)^3(e^d-e^x)^3}.
V=\left( \begin{array}{cccc}1 X_{1}X_{1}^2 \dots X_{1}^{m}1 X_{2}X_{2}^2 \dots X_{2}^{m}1 X_{3}X_{3}^2 \dots X_{3}^{m}\vdots \vdots \vdots \vdots 1 X_{m+1}X_{m+1}^2 \dots X_{m+1}^{m}\end{array}\right )\,,
\left( \left[ \mathbb{L}+i\mu\right] T_{\mu}u\right) \left( x\right)=\int_{\mathbb{R}}\left[ \mathbb{L}_{x}+i\mu\right] \left( \Gamma\left(x,\xi\right) e^{ix\xi}\right) \hat{u}(\xi)d\xi,
\int_\Omega K(x)\delta_i^{\frac{n+2}{n-2}}\lambda_i \frac{\partial\delta_i}{\partial\lambda_i}\,dx=\int_{B_{y_{j_i}}}\,\bigl[K(x)-K(a_i)\bigr]\delta_i^{\frac{n+2}{n-2}}\lambda_i \frac{\partial \delta_i}{\partial\lambda_i}\,dx+o\bigl(\frac{1}{\lambda_i^{n-2}}\bigr).
\begin{aligned}\lim_{k\rightarrow\infty}\big(h_{11j}(p_k)+h_{22j}(p_k)\big)=0, &\lim_{k\rightarrow\infty}\big(\lambda_1(p_k)h_{11j}(p_k)+\lambda_2(p_k)h_{22j}(p_k)\big)=0.\end{aligned}
\mathrm{P}^o_n = \mathrm{P}\left( \rho x_n \left(1- \frac{\epsilon_{1,k}\left( x_k +\frac{1}{\rho}\right)}{x_k (1+\epsilon_{1,k})}\right)<\epsilon_{1,n}, x_k> \frac{\epsilon_{1,k}}{\rho}\right) +\mathrm{P}\left(x_k\leq \frac{\epsilon_{1,k}}{\rho}\right),
\sum_{\mathbf{y}\in{\cal Y}^{n}}\Psi_{s,1}(\mathbf{y}) =\sum_{\mathbf{y}\in{\cal Y}^{n}}\prod_{i=1}^{n}\left(\sum_{x\in{\cal X}}P_{X}(x)W^{1-s}(y_{i}|x)V^{s}(y_{i}|x)\right) =\left[\sum_{x\in{\cal X}}P_{X}(x)\sum_{y\in{\cal Y}}W^{1-s}(y|x)V^{s}(y|x)\right]^{n},
\mathfrak{B}^r(\xi,t)=\begin{pmatrix}0 i\eta\beta^r(\xi)e^{-t\varphi(\xi,0)}\tau^{-i\nu-\frac{1}{2}}-i\eta\gamma^r(\xi)e^{t\varphi(\xi,0)}\tau^{i\nu-\frac{1}{2}}0 \end{pmatrix},
\Phi(x, \theta) = A(x) + \theta^{\mu}V_{\mu}(x) +{\textstyle \frac 12}\theta^{\mu}\theta^{\nu}F_{\mu \nu}(x) +{\textstyle \frac 16}\theta^{\mu}\theta^{\nu}\theta^{\rho}A_{\mu \nu \rho}(x)+{\textstyle \frac{1}{24}}\theta^{\mu}\theta^{\nu}\theta^{\rho}\theta^{\sigma}B_{\mu \nu \rho \sigma}(x)
\ \ \ +\sum_{i=1}^{m}h(Y_i,Y_{i+1},\ldots,Y_{N-m+i-1}|A) &\ \ \ +\sum_{i=m+1}^{N+1}h(\Psi_i^{N+1-m}(\Gamma_{N+1})|A) = \sum_{i=1}^{m}h(\Psi_i^{N+1-m}(\Gamma_{N+1})|A)+\sum_{i=m+1}^{N+1}h(\Psi_i^{N+1-m}(\Gamma_{N+1})|A) = \sum_{i=1}^{N+1}h(\Psi_i^{N+1-m}(\Gamma_{N+1})|A)
2\int d^2 x\, \partial_+ X^{\mu}\cdot \partial_- X^{\nu}\cdot\partial_{[\mu}B'_{\nu]}(X)= \int d^2 x\, \left[ \partial_+\left(B'_\mu \partial_- X^\mu \right) -\partial_-\left( B'_\mu\partial_+X^\mu\right) \right]\ .
1 = \beta(0,e_1) = \beta(e_1\cdot(e_1+e_2),e_1) = \beta(e_1,(e_1+e_2)\cdot e_1)\beta(e_1+e_2,e_1) = \beta(e_1,e_2)\beta(e_1+e_2,e_1)=a^2.
|h(T_1',0,\eta)| \leq \left( \prod_{j=1}^{k_0}j |\hat{W}(j)| (T_j'+\delta)\right) \int_{T_{j}\leq \tau_j \leq T_{j}'}|\psi|(\eta-\tau_1) |\psi|(\tau_1-2 \tau_2) \dots |\psi|((k_0-1)\tau_{k_0-1}- k_0\tau_{k_0}) |h_0|( k_0, k_0 \tau_{k_0}) d\tau_1 \dots d\tau_{k}.
\int_P (f\circ\pi_P)L_u^{\ast}(V\circ\pi_P)\,d{\rm vol}_P =\int_P g(L_u(f\circ\pi_P),V\circ\pi_P)\,d{\rm vol}_P &+\sum_{\alpha}\int_P g({\rm div}_E T_{\alpha_{\flat}(u)},V_{\alpha}{\rm grad}f\circ\pi_P)\,d{\rm vol}_P &-\sum_{\alpha}\int_P g({\rm div}_E T_{\alpha_{\flat}(u)},(f\circ\pi_P){\rm grad}_E\, V_{\alpha})\,d{\rm vol}_P.
\left\{\rho+ \mathrm{P}_0\xi_1 + i\eta \mathrm{P}_0\xi_1\left[ L -i\eta\mathrm{P}_1\xi_1-i\eta \rho \right]^{-1}\mathrm{P}_1 \xi_1 \right\}(\mathrm{P}_0 e)=0.
|h'(p^k)| < H^k + \sum_{j=1}^{\min(k,H)}(2H)^{k-j}\binom{H}{j}<(2H)^k \bigg(2^{-k}+ \sum_{j=1}^{\min(k,H)}\frac{1}{j! 2^j}\bigg)<(2H)^k,
D_i(G_k) =\begin{cases}\langle [y,x,\overset{i-1}{\ldots},x] \rangle D_{i+1}(G_k) \langle [y,x,\overset{i-3}{\ldots},x,y], [y,x,\overset{i-1}\ldots,x] \rangle D_{i+1}(G_k) \end{cases}
\sum_{n\ge0}\mathbf{E}^{(k)}_n(x;a,b,c)\frac{t^n}{n!}=\frac{2Li_k(1-(ab)^{-t})}{a^{-t}((ab)^{-t}+1)}c^{xt}=2a^{-t}\left(\sum_{n\ge0}(-1)^n(ab)^{-nt}\right)\left(\sum_{n\ge0}\frac{\left(1-(ab)^{-t}\right)^m}{m^k}\right)c^{xt}.
\sum_{\mu}\Delta_\mu(ab/t(pqt)^{1/2}|;q,t;p)\hat{R}^*_{\mu}\big(\hat{x};a,(pqt)^{1/2}/b;q,t,1;p\big)\hat{R}^*_{\mu}\big(\hat{y};b,(pqt)^{1/2}/a;q,t,1;p\big){}= \prod_{i,j}\frac{\big((pqt)^{1/2}x_iy_j;p,q\big)}{\big((pq/t)^{1/2}x_iy_j;p,q\big)}.
A_{n-1}=\min\{\dfrac{R_{n}-S_{n-1}}{2}+e,R_{n}-S_{n-1}+d[-a_{1,n}b_{1,n-2}], R_{n}-S_{n-1}+R_{n+1}-S_{n-2}+d[a_{1,n+1}b_{1,n-3}]\}\ge \min\{\dfrac{1-2e}{2}+e,(1-2e)+2e, (1-2e)+(2e+1)\}=\dfrac{1}{2}>d[a_{1,n-1}b_{1,n-1}].
z_1 = \frac{e^{i\psi/2}}{\sqrt{2}}\left(\cos \left(\frac{\theta_1+\theta_2}{2}\right)\cos \left(\frac{\phi_1 + \phi_2}{2}\right) + i \cos \left(\frac{\theta_1-\theta_2}{2}\right)\sin \left(\frac{\phi_1 + \phi_2}{2}\right)\right)
\sum_{S\in \mathcal{R}(G,r)}|E(G \setminus S)| = \sum_{j=2}^r |E(V_1V_j)|(n_1-1)(n_2-1)n_2^{r-2}+ \sum_{i,j\neq 1}|E(V_iV_j)|(n_2-1)^2n_1n_2^{r-3}.
\begin{array}{rcl}\langle (x,y), f^*(\alpha,\beta)\rangle_X = F^* t-t-\partial^*s + y^*(f^*(\beta)) &= F^* t-t-\partial^*s + (f\circ y)^*\beta &= \langle (f\circ x, f\circ y),(\alpha,\beta)\rangle_Y,\end{array}
\langle A^{-1}_{\varepsilon, \delta}\alpha, \alpha \rangle_{\omega_{\varepsilon}}\leq \sum_{j = 1}^{k}\frac{|\alpha_{j}|^{2}}{\varepsilon + \lambda_{j}}\frac{\varepsilon}{n-k}+ \sum_{l = k +1}^{n}\frac{|\alpha_{l}|^{2}}{\varepsilon}\frac{\varepsilon}{n-k-1}\leq (n-k-1)^{-1}|\alpha|_{i\partial \overline{\partial}\psi}^{2}.
U_{0}= \left( \begin{array}{c c}{Z_{1}+ Z_{2}}&{0}\\{0}&{Z_{1}+ Z_{2}+ \zeta}\\{- \sqrt{\zeta + \rho^{2}}\bar{z}_{2}}&{\rho z_{1}}\\{- \sqrt{\zeta + \rho^{2}}\bar{z}_{1}}&{- \rho z_{2}}\\ \end{array}\right)
b\cdot (u\dot{w}B, y) = b\cdot[ u\cdot w, p] = [b u\cdot w, p] = [ u'u\cdot w, u^{-1}b'u p] = (u'u\dot{w}B, \dot{w}u^{-1}b'upB) = (u'u\dot{w}B, b''\dot{w}pB) = (u' u\cdot w, b''\cdot y),
\begin{aligned}H(t) = H^{(0)}+tH^{(1)}&\psi_{n}(t) = \psi^{(0)}_{n}+t\psi^{(1)}_{n}+t^{2}\psi_{n}^{(2)}+... &\lambda_{n}(t) = \lambda^{(0)}_{n}+t\lambda^{(1)}_{n}+t^{2}\lambda_{n}^{(2)}+...\end{aligned}
\|f^{=d}\|_2^2 \le \left(\frac{e \cdot 12 r}{\theta}\right)^{d}\gamma_1\cdot \gamma \leq \left(\frac{33r}{d}\right)^d \gamma_1 \cdot \gamma \cdot \log^{d}\left(\frac{\gamma_2}{\gamma_1}\right),
\varkappa(m,p_1,p_2):=\sum_{s=0}^n\sum_{\nu\in I_s}k^{\frac{sr}{2}(\frac{1}{p_2}-\frac{1}{p_1})}(\prod_{\nu_j>k}\nu_j )^{\frac{r}{2}(\frac{1}{p_2}-\frac{1}{p_1})}|m(\nu)|^r<\infty,
(T_p h_\beta|_{S_p})^{-1}(X(h_\beta(p)) =(T_p h_\beta|_{S_p})^{-1}(X(h_\beta|_{S_p}\circ h_\alpha|_{S_p}^{-1}(h_\alpha(p)) &=(T_p h_\beta|_{S_p})^{-1}T_{h_\alpha(p)}(h_\beta|_{S_p}\circ h_\alpha|_{S_p}^{-1})(X(h_\alpha(p)) &=(T_p h_\alpha|_{S_p})^{-1}(X(h_\alpha(p))
\boldsymbol{Q}_{h}=-\frac{\hbar^{2}}{4}\left( \left\vert \dot{x}\right\vert^{2}\dot{\Delta}-\frac{1}{3}\right) -\frac{i\hbar}{\left\vert \dot{x}\right\vert}\left\langle \dot{x},\frac{\partial}{\partial \dot{x}}\right\rangle ,
\frac{1}{i}{\cal G}( x , x ; \chi ) = \frac{1}{( 2 \pi )^{2}}\int \frac{d^{2}k_{E}}{( k_{E}^{2}+ \chi )}\equiv \frac{1}{( 2 \pi )}\int \frac{d k}{2 \sqrt{k^{2}+ \chi}}
\|u\|_{H_\sigma(\Omega)}^2 =\int_\Omega u_{xx}^2+u_{yy}^2+2(1-\sigma)u_{xy}^2+2\sigma u_{xx}u_{yy}&\geq\int_\Omega u_{xx}^2+u_{yy}^2+2(1-\sigma)u_{xy}^2-|\sigma|(u_{xx}^2+u_{yy}^2)\geq(1-|\sigma|)\||\nabla^2u|\|_2^2.
A = \left( \begin{array}{ccccc}0 0 \ldots 0 a_{_{n-1}}a_{_0}0 \ldots 0 0 0 a_{_1}\ddots \vdots \vdots \vdots \ddots \ddots 0 \vdots 0 \ldots 0 a_{_{n-2}}0\end{array}\right)
\big|\langle v \rangle_h (t_1)-\langle v \rangle_h(t_2)\big|^{\frac{p}{2}}\leq c \Big||\langle v \rangle_h|^{\frac{p-2}{2}}\langle v \rangle_h(t_1)-|\langle v \rangle_h|^{\frac{p-2}{2}}\langle v \rangle_h(t_2)\Big|=c\big|[w]_h(t_1)-[w]_h(t_2)\big|
c_{+}^{*}(\chi_1,\mu_1,\lambda_1,\chi_2,\mu_2,\lambda_2):=2\sqrt{a}+\lim_{\varepsilon\to 0^+}(L_{\varepsilon}+L_{\varepsilon,2})=2\sqrt{a}+\frac{\sqrt{a}(D\sqrt{N a}+\chi_2\mu_2)}{b+\chi_2\mu_2-\chi_1\mu_1-M},
\mu^n: = \left\langle G(\phi^{n}),\phi^n\right\rangle=\int_{\mathbb{R}^d}\left(\frac12|\nabla\phi^n|^2+V|\phi^n|^2+\beta|\phi^n|^4-\Omega\overline{\phi^n}L_z\phi^n\right)\mathrm{d}\mathbf{x}=E(\phi^n)+\frac{\beta}{2}\int_{\mathbb{R}^d}|\phi^n|^4\mathrm{d}\mathbf{x}.
\Upsilon_{n}(w) = \frac{1}{2\pi i}\oint_{\Sigma_n}\frac{\prod_{j=0}^{n-1}(z + \theta j +a )}{\prod_{k=0}^{n}(z-j)}\int_0^1 x^{z+a+\theta w}dx dz & = \frac{-1}{2\pi i}\oint_{\Sigma_n}\frac{\prod_{j=0}^{n-1}(z + \theta j +a )}{\prod_{j=0}^{n}(z-j)}\frac{dz}{z+a+\theta w +1}.
R(t) =n!^{A-15}2^{-3}(2t+n)\frac{\Gamma(t)^{A+3}\Gamma(2t+4n+1)^3}{\Gamma(t+n+1)^{A+3}\Gamma(2t-2n)^3}&=n!^{A-15}2^{-3}(2t+n)\frac{\Gamma(t)^{A+3}\Gamma(2t+4n+1)^3\Gamma(2n-2t+1)^3}{\Gamma(t+n+1)^{A+3}}\cdot\frac{\sin(2\pi t)^3}{\pi^3}
\operatorname{P}\left(\left|\frac{1}{n}\sum_{i=1}^{n}\psi_{\tau}(y_{i})z_{i}-\mathbb{E}\psi_{\tau}(y_{i})z_{i}\right|\leq C\sqrt{\frac{\tau^{1-\delta}M_{\delta}^{1 / k}t}{n}}+C\frac{\tau t}{n}\right) \geq 1-2e^{-t}.
\Delta f=\frac{1}{r}\frac{\partial^{2}}{\partial r^{2}}(rf)+\frac{1}{r^{2}sin\theta}\frac{\partial}{\partial\theta}(sin\theta\frac{\partial f}{\partial\theta})+\frac{1}{r^{2}sin^{2}\theta}\frac{\partial^{2}f}{\partial\varphi^{2}},
\left[\begin{smallmatrix}C S &C S &-S C -S C\end{smallmatrix}\right]\left[\begin{smallmatrix}U &I_n &^tU^{-1}& I_n\end{smallmatrix}\right]=p\left[\begin{smallmatrix}I_n &I_n &^tZ I_n Z I_n\end{smallmatrix}\right]\left[\begin{smallmatrix}C S &C S &-S C -S C\end{smallmatrix}\right],
P ( A ) = \frac{D^{2}( k^{2}- 1 )}{2 ( 1 + q ) Q}\left[ 1 - \frac{2}{\Omega^{1 + q}}\frac{k}{\sqrt{k^{2}- 1}}Q ( - \mu / 2 )^{- 1 / 2}+ \frac{Q^{2}( - \mu / 2 )}{\Omega^{2 ( 1 + q )}}\right] .
\delta\left(U^{\theta}(t_k,x) \partial_{\theta}Y_{t_{k+1}}^{\theta}(t_k,x)\right)=\Delta_n(\sigma^{-1}(Y_{t_k}^{\theta})\partial_{\theta}b(\theta,Y_{t_k}^{\theta}))^{\ast}(W_{t_{k+1}}-W_{t_{k}})-R_{1}^{\theta,k}+R_{2}^{\theta,k}+R_{3}^{\theta,k}.
\begin{cases}\partial_{t}z - \sum_{i,j=1}^n\partial_{x_i}(A_{ij}\partial_{x_j}z) + bz=\chi_{\omega}g, \mbox{in}\Q, z=0 \ \mbox{or}\A\nabla z \cdot \nu =0, \mbox{on}\ \Sigma, z(0)= z_{0}, \mbox{in}\ \Omega. \end{cases}
\begin{array}{r c l c c c c}\mbox{\underline{sector}}{}\mbox{\underline{state}}{}{}\mbox{\underline{$R_1$}}\mbox{\underline{$R_2$}}\mbox{NS}{}|2s_3,2s_4\rangle, s_3=s_4{}+ + \mbox{R}{}|-,2s_1,2s_2\rangle, s_1=s_2{}+ + \end{array}
[\begin{matrix}R^{\prime}\\G^{\prime}\\B^{\prime}\end{matrix}]=[\begin{matrix}1&0&2-2\cdot K_{R}\\ 1&-\frac{K_{B}}{K_{G}}\cdot(2-2\cdot K_{B})&-\frac{K_{R}}{K_{G}}\cdot(2-2\cdot K_{R})\\ 1&2-2\cdot K_{B}&0\end{matrix}][\begin{matrix}Y^{\prime}\\P_{B}\\P_{R}\end{matrix}]
f(\eta):= Ae^{2\mu_j\eta}-Be^{2\mu_j\eta}(1-\mu_j^2e^{-\eta})^2= (A-B)e^{2\mu_j\eta}+\mu_j^2B\Big[2e^{(2\mu_j-1)h}-\mu_j^2e^{-2(1-\mu_j)h}\Big].
V_{7}=\frac{\varepsilon}{2}\sum_{i=1}^{d}\int_{\mathbb{T}^{d}}(\mu^{1}-\mu^{2})^{2}\frac{\partial}{\partial x_{i}}\left((\mu^{2})\Theta_{p_{i}q}(t,x,\mu^{1},Dw^{1})\right)\dx,
\begin{array}{r l}{{\mathrm{B e k e n s t e i n - V e r l i n d e ~ b o u n d}:}}&{{S_{\mathrm{B V}}= \frac{2 \pi}{n}E R ,}}\\{{\mathrm{B e k e n s t e i n - H a w k i n g ~ b o u n d}:}}&{{S_{\mathrm{B H}}= ( n - 1 ) \frac{V}{4 G_{n}R},}}\\{{\mathrm{H u b b l e ~ b o u n d}:}}&{{S_{\mathrm{H}}= ( n - 1 ) \frac{H V}{4 G_{n}}.}}\\ \end{array}
(-e^{\prime}\circ h)^{\tilde{c}}(x)=\inf_{y\in\overline{\Omega}\backslash L}\frac{| x-y|^{2}}{2\tau}1_{(\partial\Omega\times\partial\Omega)^{c}}+\Psi(y)1_{\Omega\times\partial\Omega}(x,y)-\Psi(x)1_{\partial\Omega\times\Omega}(x,y)+e^{\prime}\circ h(y)1_{\Omega}(y).
\left\lfloor\frac{(km+\ell)r}{mr+1}\right\rfloor-\left\lfloor\frac{(k'm+1)r}{mr+1}\right\rfloor = \ell-1 = \left\lfloor\frac{((k-k')m+\ell-1)r}{mr+1}\right\rfloor \, .
\sum_{k=1}^{\infty}|x_{k}y_{k}|\le(\sum_{k=1}^{\infty}|x_{k}|^{p})^{\frac{1}{p}}(\sum_{k=1}^{\infty}|y_{k}|^{q})^{\frac{1}{q}}forall(x_{k})_{k\in\mathbb{N}},(y_{k})_{k\in\mathbb{N}}\in\mathbb{R}^{\mathbb{N}}or\mathbb{C}^{\mathbb{N}}.
|K_n|\lesssim C^n(\|h_1\|_{L^{\infty}}+\|h_2\|_{L^{\infty}})^n|k|^{-2}\int_{z_1+\cdots +z_{m+1}=v-w}\prod_{j=1}^m(|k|e^{-|k||z_j|})\mathrm{d}z_1\cdots\mathrm{d}z_m\lesssim (C\varepsilon_0)^n|k|^{-1}e^{-|k||v-w|/2},
\xi^{i}= -{\frac{\rho d x_{0}^{i}}{D^{3 / 2}}}-{\frac{( x - x_{0})^{i}[ 2 \rho^{2}+ ( x - x_{0})^{2}] d \rho}{\rho^{2}D^{3 / 2}}}, \quad \quad \omega_{( 0 , 1 )}^{u}= q^{t}I_{u}\xi = -{\frac{( x - x_{0})^{t}I^{u}d x_{0}}{D^{2}}}.
L ={\frac{1}{4}}\int d^{2}\theta \left({\frac{1}{g^{2}( M )}}+{\frac{b_{0}}{8 \pi^{2}}}\log{\frac{M}{\mu}}\right) W^{a}W^{a}+ h . c . + \int d^{4}\theta \sum_{i}Z_{i}( \mu , M ) \Phi_{i}^{\dagger}e^{2 V_{i}}\Phi_{i}\, ,
\mathcal{P}_2^{\mathrm{ULA}}\left({x}\right)=\exp \left\{- 2\pi{\lambda_e}\int_0^\infty \int_0^{2\pi}{\rm{\mathbf{1}}}\left({\max \{{r_e},d\}}< \big(\frac{{P_t}{G_e(\varphi_{t_{e,o}})}\beta}{x \sigma_e^2}\big)^{\frac{1}{{\alpha_{\mathrm{NLoS}}}}}\right) \frac{1-{{f_{\Pr}}({r_e})}}{2\pi}{r_e}d{\varphi_{t_{e,o}}}d{r_e}\right\}
\frac{1}{X}\int_{\mathbb R}\Big|I_{k}(x;\theta_k, \delta)\Big|^2 w\Big( \frac{x}{X}\Big) \, dx &\le\frac{X^{1-2\delta}}{\pi^2}\int_Y^X \int_Y^X \frac{\zeta^k(\tfrac12-\delta+it)\zeta^k(\tfrac12-\delta-iv)}{(\tfrac12-\delta+it)(\tfrac12-\delta-iv)}X^{i(t-v)}\mathcal I(t-v) \, dv \, dt,
K_{M N}^{( 0 )}\equiv \left( \begin{array}{c c c}{g_{\mu}^{\rho}g_{\nu}^{\sigma}}&{0}&{- g_{\mu \nu}}\\{0}&{- g_{\mu}^{\rho}g_{\nu}^{\sigma}}&{0}\\{0}&{0}&{8}\\ \end{array}\right)
v_{j}=\frac{1}{\sqrt{|G|}}(\begin{matrix}1&e^{2\pi ij/|G|}&e^{2\cdot2\pi ij/|G|}&e^{3\cdot2\pi ij/|G|}&\cdot\cdot\cdot&e^{(|G|-1)2\pi ij/|G|}\end{matrix}).
0 =[K(\lambda,\varepsilon)-K(\lambda,0)]\left( Z^{\perp}(\lambda,\varepsilon)+\mathbf{I}\right) \left( a_{1}r_{1}+a_{2}r_{2}\right) & \ \ +\left( \mathbf{I}+K(\lambda,0)\right) \left( a_{1}r_{1}+a_{2}r_{2}\right) +\left( \mathbf{I}+K(\lambda,0)\right) \left( \left(\mathbf{I}-\mathbf{P}\right) r\right) .
V_{e f f}( b o s o n ) = \frac{1}{2}t r \ \int \frac{d \omega}{2 \pi}\int_{0}^{\infty}\frac{d s}{s}e^{i \omega \tau}e x p [ - s ( - \partial_{\tau}^{2}+ W ( \tau ) ) ] e^{- i \omega \tau}\,
\frac{\delta \rho}{\rho}\simeq A_{ei}\left( T_0 \lambda_0 \sqrt{2 \omega + 1}\right)^{\frac{4}{2 \omega - 1}}(2 \omega + 1)^{\frac{3}{2}}t_e^{- \frac{2 \omega + 1}{2 \omega - 1}}
\frac{m_{i}}{m_{j}}= \frac{M_{c}^{i}}{M_{c}^{j}}= \sqrt{\frac{( \vec{\alpha}_{i}\cdot \Lambda_{+}) ( \vec{\alpha}_{i}\cdot \Lambda_{-})}{( \vec{\alpha}_{j}\cdot \Lambda_{-}) ( \vec{\alpha}_{j}\cdot \Lambda_{+})}}.
\omega^{\mathrm{zm},-}\big|_{\partial Z_{1,0}}= \sum_{j=1}^m \phi_j, e^{4iR\lambda_j}C_{12}(\lambda_j)\varphi_j = \varphi_j, \big\lVert \varphi_j - \phi_j \big\rVert_Y <e^{-aR}\big\lVert \omega^{\mathrm{zm},-}\big\rVert_Y.
\Box \tilde \phi_{k+1}= \left(|\tilde \phi_{k+1, t}|^2- |\tilde \phi_{k+1, x}|^2\right) \tilde \phi_{k+1}+ \mathbf{1}_{\omega}f_{k}^{\tilde \phi^{\perp}_{k+1}}+ e_k, \; \tilde \phi_{k+ 1}[0]= (a, b)
\operatorname{Sp}(\mathcal{L}_{1,\alpha}\otimes I_{H^k(\mathbb{R}^{d-1})}+I_{H^k_{\alpha}(\mathbb{R})}\otimes\Delta_y)=\operatorname{Sp}(\mathcal{L}_{1,\alpha})+\operatorname{Sp}(\Delta_y).