harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\tilde f(x) = \big( \sqrt{(p+1)^2+m^2}+\sqrt{(p-1)^2+m^2}\big) \tilde h(x), \tilde u(x) = \tilde f(x) \sin x, \tilde V(x) = \lambda - \frac{1}{\tilde u(x)}(\sqrt{p^2+m^2}-m)\tilde f(x), \tilde H = \sqrt{p^2+m^2}-m + \tilde V(x)
[y^{p^r}_{12}, y^{p^s}_{1122}]=[x^{p^r}_{12}, x^{p^s}_{1122}]=x^{p^r}_{12}(1- x^{-p^r}_{12}x^{p^s}_{1122}x^{p^r}_{12}x^{-p^s}_{1122})x^{p^s}_{1122}.
P_A \left( \sigma(x^1) = y^1,\ldots,\sigma(x^p) = y^p \right) = \prod_{i = 1}^k w_A(x^i,y^i) \sum_{\theta}\mu_{A,\theta}(-1)^{\theta_1 g_1 + \theta_2 g_2}\det \limits_{i,j = 1}^k \left( K^{-1}_{A,\theta}( y^i,x^j) \right),
v(t) = e^{-itH}P_c u_0 -i\int_0^t e^{-i(t-s)H}P_c F(v(s)+a(s)\phi_0)\,ds, a(t) = e^{i\frac{1}{2}q^2t}a(0)-i\int_0^t e^{i\frac12 q^2(t-s)}\langle\phi_0, F(v(s)+a(s)\phi_0)\rangle\,ds.
\nabla u = r^{\mu-1}(\mu \phi \partial_r + \nabla_Y \phi), \langle \nabla_{\partial_r}\nabla u, \nabla u\rangle = \frac{1}{2}\partial_r|\nabla u|^2 = (\mu - 1)r^{2\mu-3}(\mu^2 \phi^2 + |\nabla_Y \phi|^2).
\begin{aligned}\mathbb{E}\Big(\int_{\Delta}^{T}|X_{t(\Delta)}^{\delta,\epsilon,h^{\delta}}-X_{t-\Delta}^{\delta,\epsilon,h^{\delta}}|^{2}dt\Big)\leq C_{T,H,a,M,|x|,|y|,\delta,\beta_{2}}(\Delta^{2}\vee\Delta^{2H}).\end{aligned}
G_\mathrm{\tilde{\rho}}(\tilde{\rho}) =1-\Pr\left\lbrace \frac{P_T|h_\xi|^2d^{-\alpha}}{\sigma^2_N+P_T\mathcal{I}}\geqslant \tilde{\rho}\right\rbrace &=1-\exp\left(-P_T^{-1}d^\alpha\tilde{\rho}\sigma^2_N \right)\mathrm{E}\left[\exp\left(-d^\alpha \tilde{\rho}\mathcal{I}\right)\right].
\mu_{V X^{(n)}}(A) := \mathbb{P}(VX^{(n)}\in A) = \mathbb{P}\Big(\sum_{j=1}^n n^{\kappa/p}\frac{Z_j}{(\lVert Z^{(n)}\rVert_p^p + W_n)^{1/p}}V_{\bullet,j}\in A\Big)
\int \partial_j \varphi \ \partial_j f \d \mu \geq \varrho_j \int | \partial_j \varphi |^2 d\mu + \int \sum_{k=1, \k \neq j}^N \partial_j \varphi \ \partial_j \partial_k H \ \partial_k \varphi \d \mu \geq \varrho_j \int | \partial_j \varphi |^2 d \mu - \sum_{k=1, \k \neq j}^N \kappa_{jk}\int \partial_j \varphi \ \partial_k \varphi \d \mu.
\partial_{\eta}^{j-1}L^{(g-\frac12)}_n(\eta)=(-1)^{j-1}L^{(g+j-\frac32)}_{n+1-j}(\eta), \partial_{\eta}^{j-1}P^{(g-\frac12,h-\frac12)}_n(\eta)=\frac{1}{2^{j-1}}(n+g+h)_{j-1}P^{(g+j-\frac32,h+j-\frac32)}_{n+1-j}(\eta),
\dfrac{\psi(x,z)}{\varphi(x,z)}=\dfrac{\left(z^2-\frac 1{x-1}\right)\sin(xz)+\left(\frac{zx}{x-1}\right)\cos(xz)}{\left(\frac{1-z^2x}{x-1}\right)\sin(xz)+\left(-\frac{zx}{x-1}+z^3\right)\cos (xz)}.
\begin{array}{| l | c | l | c |}\hline{{\mathrm{G e n u s}}}&{{\mathrm{C u r v a t u r e}}}&{{\mathrm{G r o u p}}}&{{\Lambda}}\\ \hline{{0}}&{{1}}&{{S O ( 1 , 2 )}}&{{- 1}}\\{{1}}&{{0}}&{{I S O ( 1 , 1 )}}&{{\, \, \, \, 0}}\\{{\geq 2}}&{{- 1}}&{{S O ( 2 , 1 )}}&{{\, \, \, \, 1}}\\ \hline{{}}\\ \end{array}
\Vert \kappa_{m}^{-1}h_{\mathcal{T}}^{-{m}}(\psi-\phi)\Vert_{L^2(\Omega)}^2 =\Vert \kappa_{m}^{-1}h_{\mathcal{T}}^{-{m}}\delta\phi\Vert^2_{L^2(\Omega)}= \lambda_h^2\kappa_{m}^2\Vert h_{\mathcal{T}}^{m}(1+ \delta)\phi\Vert^2_{L^2(\Omega)}.
\begin{array}{lll}\alpha_{0},\alpha_{1},\alpha_{2},\cdots,\alpha_{m}&= \overbrace{1,1,\cdots,1}^{k_{1}+1},\overbrace{0,0,\cdots,0}^{k_{2}-k_{1}}\overbrace{1,1,\cdots,1}^{k_{3}-k_{2}},\cdots,\overbrace{0,0,\cdots,0}^{k_{2q}-k_{2q-1}},\overbrace{1,1,\cdots,1}^{m-k_{2q}}\end{array}
\vert \zeta_0 \rangle = D_D(\zeta_0)\vert 0 \rangle = (1-\vert \zeta_0\vert^2)^{\lambda}\sum_{n=0}^{\infty}(2\lambda_0)^{n/2}\frac{\zeta_0^n}{\sqrt{n!}}f(n)!\vert n \rangle
\begin{matrix}1&2&3&4\\ 2&1&4&3\\ 3&4&1&2\\ 4&3&2&1\\ &L_{1}&&\end{matrix}\begin{matrix}1&2&3&4\\ 4&3&2&1\\ 2&1&4&3\\ 3&4&1&2\\ &L_{2}&&\end{matrix}\begin{matrix}1&2&3&4\\ 3&4&1&2\\ 4&3&2&1\\ 2&1&4&3\\ &L_{3}&&\end{matrix}
E[\phi(W,\gamma_{0},\alpha,\theta)] =\int\phi(w,\gamma_{0},\alpha,\theta)F_{0}(dw)=d\int g(w,\gamma(F_{\tau}^{\alpha}),\theta)F_{\alpha}(dw)/d\tau & =d\int g(w,\gamma_{0},\theta)F_{\alpha}(dw)/d\tau=0.Q.E.D.
\psi_A(x)=\,\frac{1}{\sqrt{N_A}}\,\sum^{N_A}_{k=1}\,\int_{\varphi^{-1}_{k*}(T_xM_4)}\,d^4z_k\, e^{\imath\,\vartheta_{k}(\varphi^{-1}_{k}(x),z_k)}\,n_{k}(\varphi^{-1}_{k}(x),z_k)\,|\varphi^{-1}_{k}(x),z_k\rangle
\nabla\times\left((\epsilon^{-1}-\epsilon_{0}^{-1})\nabla\times H\right)+\nabla\times\left(\epsilon_{0}^{-1}\nabla\times\widetilde{H_{\eta,\ell}}\right)-k^{2}\mu\widetilde{H_{\eta,\ell}}=0\mbox{in}\Omega.
\sqrt{-{\frac{1}{2}}( \nabla_{i}\zeta_{j}) ( \nabla^{i}\zeta^{j})}\vline{\mathrm{\raisebox{- 2 . 5 e x}{\scriptsize R = R_h}}}={\frac{b}{2 a^{2}}}\left( R_{h}{}^{2}- R_{d S}{}^{2}\right) .
\det \begin{pmatrix}b_{1,\varphi}(x,\xi',\xi_d=\pi_{1,2},\tau) b_{2,\varphi}(x,\xi',\xi_d=\pi_{1,2},\tau) begin{align*}4pt] b_{1,\varphi}(x,\xi',\xi_d=\pi_{2,2},\tau) b_{2,\varphi}(x,\xi',\xi_d=\pi_{2,2},\tau)\end{pmatrix}\neq 0.
\begin{aligned}\mathfrak{S}_{\delta, j}^{\Lambda*\mathbb{1}}\ll \big((x^{171}D^{624})^{1/1026}+ D^{9/10}+ (x^{-1}D^3)^{1/2}& \hskip 3,5mm+ (x^{19}D^{68}H^{19})^{1/114}+ (x^5D^{17}H^5)^{1/30}\big)x^{\varepsilon}\end{aligned}
\sum_{r=0}^k (-1)^r \frac{q^{k+r}}{(k+r)!}\frac{q^{k-r}}{(k-r)!}S(k+r,r) = \frac{q^{2k}}{(2k)!}\sum_{r=0}^k (-1)^r \binom{2k}{k+r}S(k+r,r) .
{\cal H}^{( 2 )}= \frac{2}{l}\gamma_{\mu \nu}\frac{\delta I^{( 2 )}}{\delta \gamma_{\mu \nu}}- \frac{\sqrt{- \gamma}}{2 \kappa}R + \frac{1}{2}\sqrt{- \gamma}\gamma^{\mu \nu}\partial_{\nu}\phi \partial_{\mu}\phi = 0 .
\left(B_{t+T-S}:0\leq t\leq S;\,\mathbb{P}_{0\to x}^{0,T}\middle|g_T=T-S\right)\stackrel{\mathcal{L}}{=}\left(B_t:0\leq t\leq S;\,\mathbb{P}_{0\to x}^{4,S}\right).
0 \le \psi_r(\lambda)\leq \psi_r\left(r^{-2}\left(y_{\frac{N-4}{2}}^{(1)}\right)^2\right)=-\frac{\pi}{\Gamma(\frac{N-2}{2})}\left({\textstyle\frac12}y_{\frac{N-4}{2}}^{(1)}\right)^{\frac{N-2}{2}}Y_{\frac{N-2}{2}}\left(y_{\frac{N-4}{2}}^{(1)}\right)=\gamma_N
g'''(z) +c_je^{z/3}g''(z)+\left(\frac{c_j^2}{3}e^{2z/3}+\frac{c_j}{3}e^{z/3}-K\right)g'(z) &+\frac{c_j}{3}e^{z/3}\left(\frac{c_j}{3}e^{z/3}+\frac{c_j}{9}-K\right)g(z)=0.
\sum_{i\neq j\neq k\neq l}^n\eta_1(i)\eta_1(j)\eta_2(k)\eta_2(l) \cr =\sum_{j\neq k\neq l}\Big(0-\eta_1(j)^2\eta_2(k)\eta_2(l)-\eta_1(j)\eta_1(k)\eta_2(k)\eta_2(l)-\eta_1(j)\eta_2(k)\eta_1(l)\eta_2(l)\Big)
\begin{array}{r c l}{{{\cal H}}}&{{=}}&{{( \Lambda^{T})^{- 1}\left( \begin{array}{c}{{P_{( p_{1}, p_{2})}}}\\{{0}}\\ \end{array}\right) \epsilon_{3}}}\\{{}}&{{=}}&{{\left( \begin{array}{c}{{P^{( 1 )}}}\\{{P^{( 2 )}}}\\ \end{array}\right) \epsilon_{3},}}\\ \end{array}
F_{l}( x ) = - \frac{1}{4}\left( \frac{d}{d x}( e_{l}s_{l}) \right)^{2}- x^{2}\Bigl ( s^{\prime \, 2}- s s^{\prime \prime}\Bigr ) \Bigl ( e^{\prime \, 2}- e e^{\prime \prime}\Bigr ) ,
R^{(\rho,\ell)}\cdot (\cdots c(m)_{p}\, c(m+1)_{p}\cdots ) = R^{(\rho,\ell)}\cdot (\cdots c(m)_{p}\, c(m+1)_{p}\cdots ) + b_1 \cdot R^{(\rho,\ell)}\cdot (\cdots c(m)_{p-1}\,c(m+1)_{p}\cdots ),
{\cal A}( \omega ) = 1 - \left|{\frac{1 + i{\frac{D}{C}}}{1 - i{\frac{D}{C}}}}\right|^{2}={\frac{3 6 \omega^{4}}{9 + 9 \omega^{2}+ 1 8 \omega^{4}+ \omega^{6}+ 4 \omega^{8}}}.
{R}_\theta(\mathbf{v})=\frac{1}{4}F_{H_1}(v_1)F_{H_2}(v_2)\left(1+\theta e^{-H_1 |v_1|-H_2|v_2|}\sinh \left(H_1 v_1\right)\sinh \left(H_2 v_2\right)\right),
\Gamma^{( 0 )}[ \phi ] = \int d^{4}x \Big ( \frac 1 2 \big ( \partial_{\mu}\phi ( x ) \partial_{\mu}\phi ( x ) + m^{2}\phi^{2}( x ) \big ) + \frac{g^{2}}{4 !}\phi \star \phi \star \phi \star \phi ( x ) \Big ) .
u_i^{\mathrm{a}}(\hat{\boldsymbol{\tau}}_i) \!=\! \underbrace{\frac{1}{1-\delta}\sum_{t \in \mathcal{T}}\! \left\{\!\!-p^{[t]}\!\!\left(\boldsymbol{d}^{[t]*}\right) \cdot d_i^{[t]*}\!-\! \frac{\epsilon_i}{2}d_i^{[t]*2}\!\right\}}_{\pi_\mathrm{total}=}\!-u_i^{\mathrm{s}}(\hat{\boldsymbol{\tau}}_i).
I_0[u_0,u_1; D] := \int_{D}\left[ ( |u_1(x)|^2 + |\nabla u_0(x)|^2 + |u_0(x)|^{p+1}) \langle x \rangle^{\alpha}+ |u_0(x)|^2 \langle x \rangle^{-\alpha}\right] \langle x \rangle^{\lambda(2-\alpha)}\,dx
\|g\|_{L^r(\mathbb F_q^d, d\textbf{m})}^r = \sum_{\textbf{m}\in \mathbb F_q^d}|g(\textbf{m})|^r, & \|f\|_{L^r(\mathbb F_q^d, d\textbf{x})}^r = q^{-d}\,\sum_{\textbf{x}\in \mathbb F_q^d}|f(\textbf{x})|^r,
u(t,x)-u(0,x) =\int_0^\infty e^{-\lambda s}P^\Omega_sP^\beta_s(g-g(0))(t,x)\,ds &=\int_0^\infty e^{-\lambda s}P^\Omega_sP^{\beta,}_s(g-g(0))(t,x)\,ds\in (\mathcal L_{\beta,\Omega}^{}),
\dot{X}(t) = F(t, X(t)), X(0) = X_0, X(t) \in \mathbb{R}^{n\times m}, t\in [0, T], \Vert F(t, X_1) - F(t, X_2)\Vert \leq L, \forall X_1, X_2 \in \mathbb{R}^{n\times m}, \forall t\in [0, T], \Vert F(t, X) \Vert \leq B, \forall X \in \mathbb{R}^{n\times m}, \forall t\in [0, T].
\mathbb E_x\left[e^{-p\int_{0}^{e(q)}\rm{\bf{1}}_{\{X^1_s \leq -b\}}ds}\rm{\bf{1}}_{\{X^1_{e(q)}>-y\}}\right]=\mathbb E_{-x}\left[e^{-p\int_{0}^{e(q)}\rm{\bf{1}}_{\{X_s \geq b\}}ds}\rm{\bf{1}}_{\{X_{e(q)}< y\}}\right],
Z = \int{\cal D}h_{\mu \nu}{\cal D}\theta_{\mu}{\cal D}\varphi{\cal D}b_{\mu}{\cal D}b{\cal D}c_{\mu}{\cal D}\bar{c}_{\mu}{\cal D}c{\cal D}\bar{c}\exp i \int d^{4}x \, L_{\mathrm{T}, a = 1}^{\prime},
\inf_{(\mathbf{h},\mathbf{g}) \in E^m}\mathcal{F}_N(\mathbf{h},\mathbf{g})=\liminf_{j \to \infty}\mathcal{F}_N(\mathbf{h}_j,\mathbf{g}_j)\ge \mathcal{F}_N(\mathbf{h}_*,\mathbf{g}_*).
\sum_{m=0}^\infty [ p^\#_m(o, o) - \pi (o)] \ge \sum_{m=0}^{\delta^2}[p^\#_m(o,o) - \frac1n] \ge \left(\sum_{m=0}^{\delta^2}p^\#_m(o,o) \right) - \frac1{f(n)},
B = - \frac{\vert e \vert H}{(2 \pi)^2}\frac{2 \sqrt{\pi}\vert e \vert H}{\Gamma (1 + \delta)}\int_{0^+}^{\infty}d \eta \; \eta^{- 3/2 + 1 + \delta}{\rm e}^{- ( 4 + \beta^2 m^2) \eta}{\rm csch}^2(\beta^2 \vert e \vert H \eta )
\frac{d^{l-m}}{dx^{l-m}}(x^{2}-1)^{l}=\sum_{j=0}^{l-m}(\begin{matrix}l-m\\j\end{matrix})l(l-1)...(1-l+m+j)\frac{(x+1)^{l}}{(x+1)^{l-m-j}}
\mathbb{E}_{\mu_{n}}S(\sigma)=\big(\,\mathbb{E}_{\mu_{n}}S^{1}(\sigma),\,\mathbb{E}_{\mu_{n}}S^{2}(\sigma),\,\mathbb{E}_{\mu_{n}}S^{3}(\sigma)\,\big) = \Big(\,\frac{1}{3},\,\frac{1}{3},\,\frac{1}{3}\,\Big) \mathbb{V}ar_{\mu_{n}}S(\sigma) = \mathbb{V}ar_{\mu_{n}}S^{1}(\sigma)+\mathbb{V}ar_{\mu_{n}}S^{2}(\sigma)+\mathbb{V}ar_{\mu_{n}}S^{3}(\sigma) =\frac{2}{3n}.
[ g_{I J}] = \frac{R_{2}}{\mathrm{I m}\, \tau}\left( \begin{matrix}{{1}}&{{\mathrm{R e}\, \tau}}\\{{\mathrm{R e}\, \tau}}&{{| \tau |^{2}}}\\ \end{matrix}\right) , [ \beta_{I J}] = \left( \begin{matrix}{{0}}&{{R_{1}}}\\{{- R_{1}}}&{{0}}\\ \end{matrix}\right)
n R_1 \geq H(A^n|LT_B')_\theta + f(\epsilon) &= \sum_{i=1}^n H(A_i |LT_B' A_{<i})_\theta + f(\epsilon) &= \sum_{i=1}^n H(A_i | U_i)_\theta + f(\epsilon) &= n H(A_T | U_TT)+ f(\epsilon) &= n H(A | C)+ f(\epsilon),
\eta_a(C(\mathsf{P}_i)) = \chi_a (\mathsf{P}_i) = q^{(\alpha_a - 2 \rho, \omega_i)}[d_a^{-1}(\alpha_a, \omega_i)]_{q_a}= \delta_{i a}q^{(\alpha_a - 2 \rho, \omega_a)}.
r_{5}:= b_{1}b_{3}- b_{2}^{2}+ 3b_{2}d_{2}- 3d_{2}^{2},r_{6}:= b_{1}^{5}- 3b_{1}^{3}b_{2}- 4b_{1}^{3}d_{2}+ 6b_{1}^{2}b_{3}+ 9b_{1}b_{2}d_{2}+ 3b_{1}d_{2}^{2}+ 3b_{2}b_{3}- 21b_{3}d_{2}.
h_i(u) b_{\lambda}&= \prod_{(x,y) \in A_{\lambda,i}}\dfrac{u - (\varepsilon_1(x - 1) + \varepsilon_2(y - 1))}{u - (\varepsilon_1 x + \varepsilon_2 y)}\prod_{(x,y) \in R_{\lambda,i}}\dfrac{u - (\varepsilon_1 (x + 1) + \varepsilon_2 (y + 1))}{u - (\varepsilon_1 x + \varepsilon_2 y)}b_{\lambda}.
D(A_{Y,U}) = \{u\in D(A_0^*);\; u(0-) = u_1(0+) = u_2(0+)=:u(0), \qquad\qquad-\alpha_- u''(0-) + \alpha_{+,1}u_1''(0+) + \alpha_{+,2}u_2''(0+) = \frac{\beta_- - \beta_{+,1}-\beta_{+,2}}{2}u(0), \qquad\qquad(u_1'(0+),u_2(0+))^\top = u'(0-) U\}, A_{Y,U}u = -A_0^*u.
\sigma_{1/w}=\begin{bmatrix}1 0 4\alpha 1 \end{bmatrix}\tau_{1/2}\begin{bmatrix}1 1/2 0 1 \end{bmatrix}\begin{bmatrix}\sqrt{N^{\prime\prime}_{1/w}}0 0 1/\sqrt{N^{\prime\prime}_{1/w}}\end{bmatrix}.
\frac{d}{d\theta}tan\theta=\frac{d}{d\theta}\frac{sin\theta}{cos\theta}=\frac{(sin\theta)^{\prime}\cdot cos\theta-sin\theta\cdot(cos\theta)^{\prime}}{cos^{2}\theta}=\frac{cos^{2}\theta+sin^{2}\theta}{cos^{2}\theta}
{\delta_{\rho}}{\mathcal{F}_{D}^{1}}={\delta_{\rho}}{\mathcal{F}_{D}^{0}}- \alpha{Q}\oint{d}\hat{s}( \xi ) \rho ( \xi ){\partial_{\hat{n}}}{\hat{G}_{D}}( \xi , \xi^{\prime}) + ({\alpha^{2}}- \alpha{Q}) \rho ( \xi^{\prime}) .
g_{2}(\beta_2) =\sum_{j=3}^{n}l_j\Big[q_j \frac{\psi(\beta_2 q_1)}{q_1}\Big(\frac{\psi'(\beta_1 q_2)\psi'(\beta_2 q_j)+\psi'(\beta_1 q_j)\psi'(\beta_2 q_2)}{\Delta}\Big) - \frac{q_j \psi(\beta_2 q_2)}{q_2}\Big(\frac{\psi'(\beta_2 q_1)\psi'(\beta_1 q_j)+\psi'(\beta_1 q_1)\psi'(\beta_2 q_j)}{\Delta}\Big) +\psi(\beta_2 q_j)\Big]
[x^{2n}]\left(\frac{x(\xi-1)}{1-x}\right)^{\ell}=[x^{2n-\ell}]\left(\frac{\xi-1}{1-x}\right)^{\ell}=(\xi-1)^{\ell}\binom{-\ell}{2n-\ell}(-1)^{2n-\ell},
\begin{array}{r l r l r l}{{\hat{O}}}&{{}}&{{\stackrel{\cal W}{\longleftrightarrow}}}&{{~}}&{{\tilde{o}( z ) \nonumber}}\\{{}}&{{\searrow \nwarrow}}&{{}}&{{~ ~ W \swarrow \nearrow}}&{{\nonumber}}\\{{}}&{{}}&{{~ o ( y , x )}}&{{}}&{{}}\\ \end{array}
\left( \begin{matrix}{a^{\prime}}\\{b^{\prime}}\\ \end{matrix}\right) = M \left( \begin{matrix}{a}\\{b}\\ \end{matrix}\right) = \left( \begin{matrix}{D}&{C}\\{B}&{A}\\ \end{matrix}\right) \left( \begin{matrix}{a}\\{b}\\ \end{matrix}\right) ,
a_1^{56}=z_4^{16}=:x_1, a_2^{57}=a_4^{56}=:y_2, z_3^{26}=z_4^{27}, &a_2^{57}-z_4^{27}= c^{25}=-a_4^{56}+z_3^{26},z_4^{35}=a_1^{56}-a_4^{56},
\frac{\overline{RL}}{\overline{LQ}}=\frac{\overline{BA}}{\overline{AC}},\frac{\overline{QN}}{\overline{NP}}=\frac{\overline{A^{\prime}C^{\prime}}}{\overline{C^{\prime}B}},\frac{\overline{PM}}{\overline{MR}}=\frac{\overline{CB^{\prime}}}{\overline{B^{\prime}A^{\prime}}}.
[\mathit{q}_{0}]-\left(\frac{\Gamma(1-2/{{c_{}}})}{\Gamma^2(1-1/{{c_{}}})}-1 \right) \left(\mathbb{E}[\mathit{q}_{0}]\right)^2=0,{{b_{}}}=\frac{\mathbb{E}[\mathit{q}_{0}]}{\Gamma(1-1/{{c_{}}})}.
B(e_{x},\phi)=I+\phi\frac{\partial B}{\partial\phi}|_{\phi=0}=[\begin{matrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{matrix}]-\phi[\begin{matrix}0&1&0&0\\ 1&0&0&0\\ 0&0&0&0\\ 0&0&0&0\end{matrix}]
\begin{array}{cc}\delta_1 v^i = \alpha \dot v^i, \delta_1 g = (\alpha g)\dot{},\end{array}\begin{array}{cc}\delta_2 v^i = \alpha \dot v^i, \delta_2 g = (\alpha g)\dot{}, \delta_2 \varphi = \alpha \dot \varphi - \dot \alpha \varphi.\end{array}
\eta \log x = D_2^{-1/(1+d_2)}(\log x)^{-\frac{\kappa(1+\delta)^2}{1+d_2}}\cdot (\log x) & \geq D_2^{-1/(1+d_2)}(\log x)^{\frac{(1-\kappa+\delta)(1+\delta)^2}{1+d_2}}\succ (1+\delta)\log\log x + \log 2,
\rho = \frac{\Pi_{1}( B^{2}E_{2}^{2}- E_{1}^{2})}{E_{1}( E_{2}^{2}- 1 )}- \frac{E_{1}T_{2}^{2}}{\Pi_{1}( E_{2}^{2}- 1 )}\frac{1}{1 + ( 1 - u^{2}) \sinh^{2}( x / \zeta_{B})}.
m(t) = 1 + \int_0^{t-1}q(u) \: du &= 1 + \int_0^{\tau_{k^*+1}/a_N}q(u) \: du + \sum_{j=k^*+1}^{j(t) - 1}\int_{\tau_j/a_N}^{\tau_{j+1}/a_N}q(u) \: du + \int_{\tau_{j(t)}/a_N}^{t-1}q(u) \: du.
\lVert (\widetilde{F}_{\eta}(v,\omega_k)-\xi)^{-1}-(\widetilde{F}_{\eta}(v,\omega)-\xi)^{-1}\lVert &=\lVert (\widetilde{F}_{\eta}(Uv,f_k(U\omega U^*))-\xi)^{-1}-(\widetilde{F}_{\eta}(Uv,U\omega U^*)-\xi)^{-1}\lVert
\triangle|\mathring{Ric}|^2=2|\nabla \mathring{Ric}|^2+2\mathring{R}_{ij}\mathring{R}_{ij,kk}=2|\nabla \mathring{Ric}|^2+2\mathring{R}_{ij}\mathring{R}_{li}R_{lkjk}+2\mathring{R}_{ij}\mathring{R}_{kl}R_{lijk}.
\int_{[-n^m,-1]}\frac{ns^{2}\,d\sigma_{\mu}^{-}(s)}{(\widetilde{x}_n-s)^{2}}=\frac{nq}{p}\sum_{k=1}^n\frac{k^{2m}\sigma_{\mu}^{+}([k^m,(k+1)^m))}{(\widetilde{x}_n+k^m)^2}&\leq\frac{4qc}{m\cdot n^{2m-3}}\sum_{k=1}^nk^{(2-a)m-1}&\leq\frac{4qc}{m\cdot n^{2m-3}}\int_1^{n+1}s^{(2-a)m-1}\;d\lambda(s) &\leq\frac{2^{(2-a)m+2}qc}{(2-a)m^2}\cdot\frac{1}{n^{am-3}}.
\sharp T_1(\varepsilon_1,\varepsilon_2,d,B)=\frac{\varepsilon_1-\varepsilon_2}{2(\alpha^\prime)^2 d^2}B^{2-\frac{1}{r}}+O_{\varepsilon_i}\left(\frac{B^{2-\frac{1}{r}}}{N d^2}\right)+O_{\sigma,\varepsilon_i}\left(\frac{B^\sigma N}{d^\sigma}\right),
G''(0) = \frac{n(n-1)}{r_n^2}I_0(f^*)V_n(X^n)-2\frac{n-1}{r_n}I_1(f^*)V_{n-1}(X^n)+2I_2(f^*)V_{n-2}(X^n) & = 2I_2(f^*)V_{n-2}(X^n)-\frac{n-1}{r_n}I_1(f^*)V_{n-1}(X^n) & = \frac{n2^n}{(n-2)!}\left (\frac{\sqrt{n-1}}{2}\arccos \left (1-\frac{2}{n}\right )1.062097-1\right ).
{\cal P}_{n}\approx \frac{\left[ 2 \sqrt{T^{2}- 4 d}+ 4 d - 1 \right]^{n + \frac 1 2}+ ( - 1 )^{n}\left[ 2 \sqrt{T^{2}- 4 d}- 4 d + 1 \right]^{n + \frac 1 2}}{\sqrt{\pi n}( 1 + 2 T + 4 d )^{n + \frac 1 2}\left( T^{2}- 4 d \right)^{\frac 1 4}}.
|A+ A'| \geq |A^* + A'| = |A^* + \{a_1, \dots, a_c\}| + \sum_{i=m+1}^d |A^* + \{b_i\}| & \geq (1 - 100^{-d^2})|A^{*}+ A^{*}| + (d-m)|A^*| & \geq (1- 100^{-d^2})(m+1 + 1/10)|A^*| + (d-m)|A^*| - m(m+1)/2 & \geq (d + 1 + 1/15) |A^*| - m(m+1)/2 &\geq (d+1+2/31)|A| - d(d+1),
q_{1}=\max\limits_{i=1,\ldots,m+1}\left\{\big\vert e_{i1}\big\vert \right\}q_{j}=\max\limits_{i=1,\ldots,m+1}\left\{\big\vert \widehat{e}_{ij}+f_{ij}\big\vert, \ \big\vert \widehat{e}_{ij}-f_{ij}\big\vert \right\}, \ \j=2,\ldots,m+1,
\mu^*=\frac{P_T+\textstyle\sum_{k=1}^{r_s}{\sigma^2}\left(\left(1-\rho^*\right)[\boldsymbol{\Lambda}]_{k,k}^2\right)^{-1}}{r_s\textstyle\sum_{k=1}^{r_s}\left(\left(\nu^*-\rho^*[\boldsymbol{\Lambda}]_{k,k}^2\right)\ln2\right)^{-1}}.
\mathfrak{S}_{\varpi}(1):= \mathfrak{S}_{\varpi}(x_1=1,x_2=1,\ldots) \hphantom{\mathfrak{S}_{\varpi}(1)}{}=\operatorname{DET}\left({(n-i+1)p+b-i+j+k+(i-1)r \choose k+ (i-1)r}\right)_{1 \le i,j \le n+1}.
\int_\Omega (\Phi'', \widehat{\Phi})dx + \int_\Omega \langle \nabla\Phi, \nabla \widehat{\Phi}\rangle dx+\int_{\Gamma}(\Phi, B \widehat{\Phi})d\Gamma +\int_\Omega (\Phi, A\widehat{\Phi})dx =0
F(x^{(\ell-1)}) - \eta^{k(\ell)-1}= \Big[F(x^{(\ell-1)}) - \tilde{F}^{k(\ell)-1}(x^{(\ell)})\Big]- \frac{1}{2}\big\|x^{(\ell)}- x^{(\ell-1)}\big\|_D^2 \leq \frac{1}{\beta}\Big[F(x^{(\ell-1)}) - F(x^{(\ell)})\Big] - \frac{1}{2}\big\|x^{(\ell)}- x^{(\ell-1)}\big\|_D^2.
SO_{\alpha}(G^*)-SO_{\alpha}(G^{*'})= (d_{G^*}(v_{1})-1)\left( \left(1+d_{G^*}^2(v_{1})\right)^{\alpha}-\left(1+(d_{G^*}(v_{1}+1))\right)^{\alpha}\right)+18^{\alpha}&-\left((d_{G^*}(v_{1}+1))^2+4\right)^{\alpha}\leq 18^{\alpha}-20^{\alpha}<0.
\left(\frac{1}{2\pi i}\right)^2 \iint\limits_{(\sigma)(\gamma)}L^\nu(s-w)\zeta(w)\frac{\Gamma(w) \Gamma(s - w)}{\Gamma(s)}V(s) X^s dw ds = \left(\frac{1}{2\pi i}\right)^2 \iint\limits_{(\epsilon)(\gamma)}L^\nu(s)\zeta(w)\frac{\Gamma(w) \Gamma(s )}{\Gamma(s+w)}V(s+w) X^{s+w}dw ds.
\psi (h ) =~ h, ~ h \in (\mathfrak{g}_{com})^0 = \mathfrak{g}^0, \psi ((h_1, \ldots, h_{n+1})) =~ h_1 + \cdots + h_{n+1},~ (h_1, \ldots, h_{n+1}) \in (\mathfrak{g}_{com})^n
i G_{a b}( x - y ) = \frac 1{i^{2}}\frac{\delta^{2}\ln Z [ \phi_{1}, \phi_{2}]}{\delta \phi_{a}( x ) \delta \phi_{b}( y )}= i \left( \begin{array}{c c}{G_{1 1}}&{- G_{1 2}}\\{- G_{2 1}}&{G_{2 2}}\\ \end{array}\right) \ .
\cos\pi\nu +1 = \frac{\pi e^{4h}}{(8h)^{q/2}}\bigg[\frac{1+\frac{3(q^2+1)}{64h}}{\Gamma[\frac{3}{4}-\frac{q}{4}]\Gamma[\frac{1}{4}-\frac{q}{4}]}+O(\frac{1}{h^2})\bigg]
\|g\|_{C(\Delta)}=\sup_{y\in\Delta}|\lambda \gamma(X,x^*,y)|=\|\lambda x^*\|_\infty=\|\lambda i_X(x^*)\|_{X^*}&=\sup_{x\in B_{X}}|\lambda\langle i_X(x^*),x\rangle|=\sup_{x\in B_{X}}|\lambda\alpha(X,x,x^*)| &=\sup_{x\in B_{X}}\langle g,x\rangle_X=\|\langle g,\cdot\rangle_X\|_{X^*}.
f(x_1, y_1,x_2,y_2)=|\varphi_{z_1}(z_2)|=\frac{\sqrt{(x_1 -x_2)^2+(y_1 -y_2)^2}}{\sqrt{(1- x_1 x_2 -y_1 y_2)^2+(x_1 y_2 -x_2 y_1)^2}}.
K_{2 0 0}( k^{2}) = \int_{- \pi / 2 + 3 \epsilon / 2}^{\pi / 2 - \epsilon / 2}d \theta_{2}\int_{- \pi / 2 + \epsilon / 2}^{\theta_{2}- \epsilon}d \theta_{1}~{\cal F}( \theta_{1})^{k^{2}- 1}{\cal F}( \theta_{2})^{k^{2}- 1}\left| 3 \sin \frac{\theta_{2}- \theta_{1}}{3}\right|^{- 2 k^{2}}.
\sum_{l\in P_1 \cup P_3}\frac{1}{4\tan \frac{\pi lb}{p}\tan\frac{\pi l}{p}}=\sum_{l\in P_1}\left(\frac{1}{4\tan \frac{\pi lb}{p}\tan\frac{\pi l}{p}}+\frac{1}{4\tan \frac{\pi lb^{-1}}{p}\tan\frac{\pi l}{p}}\right).
P\left(\frac{n-A_n^*}{f_1(n)}>\varepsilon\right) =1-P(A_n(t)>t,\quad\forall t=a_n,\ldots,\lfloor n-\varepsilon f_1(n)\rfloor) &=P\left(\bigcup_{t=a_n}^{\lfloor n-\varepsilon f_1(n)\rfloor}\{A_n(t)-t\leq 0\}\right) &=P\left(\bigcup_{t=a_n}^{\lfloor n-\varepsilon f_1(n)\rfloor}\{S_n(t)+a_n-t\leq 0\}\right),
\lim_{k\to +\infty}\|\sum_{j = 1}^2\alpha^k_j U_{y_k^j, \lambda^k_j}- \sum_{j = 1}^2\tilde{\alpha}^k_j U_{\tilde{y}_k^j, \tilde{\lambda}^k_j}\|_{\dot{H}^\gamma(\mathbb{R}^N)}= 0.
B^{( N )}( L , r , l | l ) = q^{( - e_{N - l + 1}+ e_{1}- 2 e_{l}) C^{- 1}m + g_{\frac{N}{2}}( l - 1 ) - g_{[ 1 , l - 1 ]}( N - l + 1 )}( 1 - q^{m_{l}})
I = ( x_6^2, x_5x_6, x_5^2, x_4x_5, x_3x_5, x_2x_5, x_1x_5, x_4^2x_6, x_3x_4x_6, x_2x_4x_6, x_1x_4x_6, x_3^2x_6,x_2x_3x_6, x_1x_3x_6, x_2^3x_6, x_1x_2^2x_6, x_1^2x_2x_6, x_4^4, x_3x_4^3, x_2x_4^3, x_1^4x_6, x_3^3x_4^2, x_3^4x_4, x_3^5) \subset K[x_0,\ldots,x_6],
\begin{array}{rcl}\hat{P}_4(x)\!\! \!\!\!=\!\!\! \!\!\left(16\pi^2 -160 \right) x^4+ \left(304\pi-32\pi^3 \right) x^3+\left(24\pi^4 - 208\pi^2 \right) x^2 begin{align*}1.0 ex]\!\! \!\!\! \!\!\! \!\!+\left(56\pi^3 -8\pi^5 \right) x +\pi^6 -2\pi^4,\end{array}
- \frac{1}{4}\exp \! \Big ( \sqrt{\frac{2 \kappa}{3}}\phi \Big ) [ V_{\mu \nu}{}^{j}V^{\mu \nu}{}_{j}+ \overline{{{\Theta}}}_{\mu \nu}\overline{{{\Theta}}}^{\mu \nu}+ 2 \sqrt{2 \kappa}( V_{\mu \nu}{}^{j}A_{j}+ \overline{{{\Theta}}}_{\mu \nu}\Theta ) f^{\mu \nu}+ 2 \kappa ( A_{j}A^{j}+ \Theta^{2}) f_{\mu \nu}f^{\mu \nu}]
\sum_{k=1}^{n}k^{\ell-1}S(k) =\sum_{k=1}^{n}k^{\ell-1}\sum_{\substack{a\in S 1\leqslant a\leqslant k}}1=\sum_{a\in S}\sum_{a\leqslant k\leqslant n}k^{\ell-1}.
{u}(\cdot,t)= e^{-A (t-t_0)}u_0 -\chi\int_{t_0}^t e^{-A(t-s)}\nabla u(\cdot,s) \cdot \nabla A^{-1}u(\cdot,s)ds +\chi\int_{t_0}^t e^{-A(t-s)}u(\cdot,s)\big(u(\cdot,s)-A^{-1}u(\cdot,s)\big)ds &+\int_{t_0}^t e^{-A(t-s)}u(\cdot,s)\Big(1+ a_0(s,\cdot)-a_1(s,\cdot)u(\cdot,s)-a_2(s,\cdot)\int_{\Omega}u(\cdot,s)\Big)ds.
P^{(\infty)}_{+}(z) = P^{(\infty)}_{-}(z)\begin{pmatrix}0 s_{j}-s_{j}^{-1}0\end{pmatrix}, z \in (-x_{j},-x_{j-1}), \, j = 2,...,m+1, & P^{(\infty)}_{+}(z) = P^{(\infty)}_{-}(z)e^{\pi i \alpha \sigma_{3}}, z \in (-x_{1},0).
[Y_{c(t)}^*]^{\perp}= \left\{v \in [X_{c(t)}^*]^{\perp}: \langle \eta_* e^{iy}, v \rangle_{L^2(\mathbb{R}\times \mathbb{T})}= \langle \eta_* e^{-iy}, v \rangle_{L^2(\mathbb{R}\times \mathbb{T})}= 0\right\}.
\frac{dN}{dt}=(N_r-N)D-I_{NR}\frac{N}{\kappa_{NR}+N}R, \frac{dR}{dt}=\bigg(\mu_{NR}\frac{N}{\kappa_{NR}+N}-(D+m_R(c_R))\bigg)R-I_{RP}\frac{R}{\kappa_{RP}+R}P, \frac{dP}{dt}=\bigg(\mu_{RP}\frac{R}{\kappa_{RP}+R}-(D+m_{P0})\bigg)P, \frac{dc_{T}}{dt}=(c_{r}-c_{T})D,
u_N(a + iy_0) \leq \int_{iy = 0}u_N(x) d\mu_{a + iy_0}(x) + \int_{iy = i\rho/2}u_N(x + iy) d\mu_{a + i y_0}(x) &= \int_{iy = 0}u_N(x + a) d\mu_{iy_0}(x) + \int_{iy = i\rho/2}u_N(x + a + iy) d\mu_{iy_0}(x).
A_j \ast A_k = 0 (j \neq k) , \sum_{k=1}^n A_k \in \{-1,1\}^{n \times n}, A_k A_k^T = I_{(n)}, A_j A_k^T + \lambda_{j,k}A_k A_j^T = 0 (j \neq k), B_j B_k^T - \lambda_{j,k}B_k B_j^T = 0 (j \neq k), \lambda_{j,k}\in \{-1,1\}, \sum_{k=1}^n B_k B_k^T = n b I_{(b)},
S^{B H}={\frac{\pi}{6}}\sum_{i = 1}^{N}\int_{\Sigma}d \sigma \left[ 2 \langle \hat{\phi}_{i}^{2}\rangle +{\frac{1}{m_{d , i}}}\langle \hat{\bar{\psi}}_{i}\hat{\psi}_{i}\rangle - 2 \langle \hat{V}_{i}^{2}\rangle \right] .
~s_1^*=\mathcal{F}\Big(r_3^*-\dfrac{\partial z_{g_3}}{\partial t},l_3^*,l_3^{**}\Big),~s_2^*=\mathcal{F}\Big(r_4^*-\dfrac{\partial z_{g_4}}{\partial t},l_4^*,l_4^{**}\Big).~~a.s.
\mathrm{(P2)}:\begin{cases}\underset{\{\textbf{u}_m\}_{m=1}^p}{\min}\ \rho \ \underset{m=1,\cdots,p}{\min}\|\mathbf w_k - \mathbf u_m\|\leq \rho, \ \forall k\in \mathcal{K},\end{cases}
H=\begin{pmatrix}1 0 0 0 ... 1 0 1 0 0 ... 1 0 0 1 0 ... 1 \cdot \cdot \cdot \cdot \cdot 1 \cdot \cdot \cdot \cdot \cdot 1 0 0 0 ... 1 1 0 0 0 ... 0 1 \end{pmatrix},
f(v_{\ell,m,n})=\sqrt{q}^{3\ell+m-n}\sum_{\substack{(i,j)\in\{1,3\}^2 i\neq j}}\{F_{iij}z_i^{\ell+m}z_j^n+F_{iji}z_i^{\ell+n}z_j^m+F_{ijj}z_i^\ell z_j^{m+n}\},
~~ s_x^2\langle y, \overline{\nabla f(x)}\rangle =(s_x-1)\frac{\langle x,\overline{\widetilde \nabla f(x)}\rangle}{\|x\|^2}\langle y,x\rangle -s_x\langle y, \overline{\widetilde \nabla f(x)}\rangle .
\begin{aligned}\frac{1}{2}\frac{d(b^2(t))}{dt}=b(g_0+g_1N-b)\leq C|b|-(1-g_1)b^2+g_1|b|\sqrt{c}, \frac{1}{2}\frac{d(c^2(t))}{dt}=c(2a_0+2a_1N-2c)\leq C|c|+2a_1|bc|+2a_1c\sqrt{c}-2c^2.\end{aligned}
\sum_{n=p}^{\infty}|\nabla^{k}I_{n,q}^{r}|=|\sum_{n=p}^{\infty}(\nabla^{k-1}I_{n,q}^{r}-\nabla^{k-1}I_{n-1,q}^{r})|=|\nabla^{k-1}I_{p-1,q}^{r}|
d(a-a^n) = \nu(n)c_a \mbox{\ \for\all\}a\in A c_{a+b}-c_a-c_b = d\frac{a^n + b^n - (a+b)^n}{\nu(n)}\mbox{\ \for\all\}a,b\in A ac_b + b^nc_a = c_{ab}\mbox{\ \for\all\}a,b\in A .