harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{2 \pi i}\int_{\Gamma'}\frac{z^{m-1}}{\prod (1-zy_j)^{1/\alpha}}F(x/z){\rm d}z = \frac{1}{2 \pi i}\int_{\Gamma''}\frac{1}{s^{m+1}}\frac{F(xs)}{\prod (1-y_j/s)^{1/\alpha}}{\rm d}s, = \frac{1}{2 \pi i}\int_\Gamma \frac{F(xs)}{\prod (s-y_j)^{1/\alpha}}{\rm d}s,
\mathcal{U}\left(\underline{u},\overline{v}\right)= e^{\beta z}\left[\beta\sigma\left(c-\beta\right)-f(1-\sigma e^{\beta z})q(1)e^{(\lambda_1-\beta)z}\right]+f(1-\sigma e^{\beta z})p(1-\sigma e^{\beta z}) begin{align*}0.2cm] \geq e^{\beta z}\left[\beta\sigma\left(c-\beta\right)-f(1)q(1)\right]>0
3U(u X^i) =\sum_{t=1}^3 u_t x_t^i=\sum_{t=1}^3 u_t (\sigma_1 x_t^{i-1}-\sigma_2 x_t^{i-2}+\sigma_3 x_t^{i-3}) &=\sigma_1\sum_{t=1}^3 u_t x_t^{i-1}-\sigma_2\sum_{t=1}^3 u_t x_t^{i-2}+\sigma_3\sum_{t=1}^3 u_t x_t^{i-3}&= 3\left(\sigma_1 U(u X^{i-1})- \sigma_2 U(u X^{i-2}) + \sigma_3U(u X^{i-3})\right).
\tilde{\Gamma}^{(4)}_{t}({\bf{q}}_1 ,{\bf{q}}_2 ,{\bf{q}}_3 ,{\bf{q}}_4) = \tilde{\Gamma}^{(4)}_{t, s = 0}({\bf{q}}_1 ,{\bf{q}}_2 ,{\bf{q}}_3 ,{\bf{q}}_4) \; ,
E_{st}(Z^{(2)}) = \mathbb L^5+2\mathbb L^4+\mathbb L^3, E_{st}(Z^{(3)}) = \mathbb L^7+3\mathbb L^6+5\mathbb L^5+2\mathbb L^4, E_{st}(Z^{(4)}) = \mathbb L^9+4\mathbb L^8+11\mathbb L^7+14\mathbb L^6+4\mathbb L^5, E_{st}(Z^{(5)}) = \mathbb L^{11}+5\mathbb L^{10}+17\mathbb L^9+35\mathbb L^8+30\mathbb L^7+6\mathbb L^6.
|2c_{b}(\lambda_{0}(t)+e_{2}(t))\int_{0}^{\infty}d\xi \frac{\cos(t\xi)}{t^{3}}\left(\partial_{\xi}\psi_{v_{2}}(\xi,\lambda_{1}(t))-\partial_{\xi}\psi_{v_{2}}(\xi,\lambda_{2}(t))\right)| &\leq \frac{C ||e_{1}-e_{2}||_{X}}{t^{3}\sqrt{\log(\log(t))}}
[F^0,K_1](\theta,r) = (0,\ast)^T, [F^0,K_0](\theta,r) = (\ast,0)^T, [F^1,K_1](\theta,r) = (0,\ast)^T, [F^1,K_0](\theta,r) = (\ast, -g'(\theta) k(\theta,r))^T,
\begin{cases}D^{1-\epsilon}_0y(t) = w(t), y(0) = \frac{\gamma_1-b_1s}{a_1}, D^{\epsilon}_0w(t) = z(t), w(0) = 0, D^{\alpha_2-1}_0z(t) = f(t,y(t),w(t)), z(0) = s.\end{cases}
F_\ell = \begin{pmatrix}\frac{a_\ell \beta_\ell - \Delta_\ell}{2\beta_\ell}\frac{\Delta_\ell (\beta_\ell^2 - \Delta_\ell)}{4\beta_\ell^3}\beta_\ell \frac{a_\ell \beta_\ell + \Delta_\ell}{2\beta_\ell}\end{pmatrix}.
\int \log\frac{1}{|x-t|}\,d\omega_{\Omega}(i,t) + \frac{1}{2}\log(1+x^2) = \int \log\frac{1}{|x-t|}\,d\omega_{\Omega}(i,t) - \int \log\frac{1}{|x-t|}\,d\delta_i(t) = \int g_\Omega(t,\infty)\,d\delta_i(t) = g_\Omega(i,\infty).
\| \sum_{n= 1}^\infty n^\alpha f_n(u) H_n(s) \|_{L^2(0, 1 ; L^r(S_2))}\leq\sum_{n= 1}^{N-1}n^\alpha \| f_n \|_{L^r(S_2)}+ C\| \sum_{n= N}^\infty n^\alpha f_n(u) H_n(s) \|_{L^2(0, 1 ; L^2(S_2))}& \leq C(N, \alpha) + C \left( \sum_{n= N}^{\infty}n^{2 \alpha}\| f_n \|_{L^2(S_2)}^2\right)^{1/2}< + \infty
B = P(x_1x_3x_2) + P(x_1dx_3dx_2) + P(x_1[x_2,x_3]) = P(x_3x_1x_2) + P(dx_3dx_1x_2) + P([x_1,x_3]x_2) + P(x_1dx_3dx_2) + P(x_1[x_2,x_3]) = P(x_3x_2x_1) + P(x_3dx_2dx_1) + P(x_3[x_1,x_2]) + P(dx_3dx_1x_2) + P([x_1,x_3]x_2) + P(x_1dx_3dx_2) + P(x_1[x_2,x_3]).
\left( \begin{array}{l}{\theta^{1}}\\{\theta^{2}}\\ \end{array}\right) \rightarrow \left( \begin{array}{l l}{\cos \frac{\alpha}{2}}&{- \sin \frac{\alpha}{2}}\\{\sin \frac{\alpha}{2}}&{\cos \frac{\alpha}{2}}\\ \end{array}\right) \left( \begin{array}{l}{\theta^{1}}\\{\theta^{2}}\\ \end{array}\right) .
g^{2}\, \Phi_{i}^{2}={\frac{m ( z = h_{i}; \mu )}{a^{\prime}( z = h_{i})}}={\frac{\displaystyle \prod_{j = 1}^{N - 1}( h_{i}- \mu_{j})}{\displaystyle \prod_{j = 1 , j \neq i}^{N}( h_{i}- h_{j})}}\, \cdotp
\begin{array}{c}{{P^{s}( b , c ) \, \vec{\hat{U}}_{\upsilon}^{( s ) \, a}( b , c ) = \hat{P}^{( s ) \, a}( b , c ) \, \vec{\hat{U}}_{\upsilon}^{( s ) \, a}( b , c ) = \vec{\hat{U}}_{\upsilon}^{( s ) \, a}( b , c ) \, ;}}\\{{\hat{P}^{( s ) \, f}( b , c ) \, \vec{\hat{U}}_{\upsilon}^{( s ) \, a}( b , c ) = \vec{0}, \qquad \quad a \neq f .}}\\ \end{array}
h^0(\mathcal{Y}_s,g_s^*(m\epsilon \mathcal{H}_s+mf_s^*\mathcal{O}(1)))=\frac{m^{d}\epsilon^{d-1}}{(d-1)!}((\mathcal{H}_s^{d-1}\cdot f_s^*\mathcal{O}(1))+O(\epsilon)+O((m\epsilon)^{-1})).
\theta(s,\sigma)\,=\, \frac{\pi}{2}(1+\sigma) -\left[\frac\pi2\,(1+\sigma)\,\log\left(\cos\frac{\pi\sigma}{2}\right)-\Xi\left(\frac{\pi}{2}\,(1+\sigma)\right)+(1+\sigma)\,\Xi\left(\frac{\pi}{2}\right)\right]\, s +o(s)\,,
y_{ij}:= T_{ij}^{(1)}T_{ji}^{(1)}-T_{jj}^{(1)}=y_{ji},z_{i}:=\beta T_{ii}^{(2)}-\frac{\beta}{2}T_{ii}^{(1)}\big (T_{ii}^{(1)}-1\big), p_{ij}:= \frac{\beta q_j}{q_i -q_j}, i \not= j,
\begin{array}{l}{\rm E}\left[ \left( \left({s_{l,i}+ s_{m,i}}\right) x_{(l,m),i\beta + k}x_{(\mathcal{A},\mathcal{B}),i\beta + k}\right)^2 \right] = 2{\rm E}[ ( x_{(l,m),i\beta + k}x_{(\mathcal{A},\mathcal{B}),i\beta + k})^2].\end{array}
\sum_{\substack{1 \leq i,j \leq M d(v_i,v_j) \geq \log m}}\left|\hat{\mu}(\xi_i -\xi_j)\right|^N = \sum_{\substack{1 \leq i,j \leq M d(v_i,v_j) \geq \log m}}e^{2c}m^{-2}\left( 1 + O\left( \frac{\log \log m}{\log m}\right) \right) \leq M^2 |\hat{\mu}(\xi)|^{2N}\left( 1 + O\left( \frac{\log \log m}{\log m}\right) \right).
d\bar{x}_i(s) = -(\Pi^Ta)_{ij}\frac{\partial V}{\partial x_j}\, ds + \frac{1}{\beta}\frac{\partial (\Pi^Ta)_{ij}}{\partial x_j}\, ds + \sqrt{2\beta^{-1}}(\Pi^T\sigma)_{ij}\, dw_j(s)\,, 1 \le i \le n\,.
\frac{2\sigma}{q_{\theta}}+\frac{d}{r_{\theta}}=\frac{d}{2}-s,\frac{2\sigma}{q_{\theta}}+\frac{d}{\tilde{r}_{\theta}}+\frac{2\sigma}{\tilde{q}_{\theta}}+\frac{d}{r_{\theta}}=d, \textmd{(Strichartz exponent relations)}
\Gamma_{s}^{N,\phi}(\pi^N) = \frac{2\kappa}{N^2}\sum_{(x,\sigma) \in V}\eta^N_s(x,\sigma) (\partial_x\phi(\tfrac{x}{N},\sigma))^2 +\frac{1}{N^2}\sum_{(x,\sigma)\in V}\sum_{\sigma'\in S}c(\sigma,\sigma')\eta^N_s(x,\sigma)(\phi(\tfrac{x}{N},\sigma')-\phi(\tfrac{x}{N},\sigma))^2 \ \ \ \ \ \ \ \ + R_4(\phi,s,N,\sigma),
= \int \mathrm{d}^{4}x \, \left( W_{e}^{2}( x , y ) - \frac{C^{2}}{( x - y )^{4}}\right) \phi ( x ) + \int \mathrm{d}^{4}x \, \frac{C^{2}\phi ( x )}{( x - y )^{4}}
\lambda^{\lambda_t}= (\underbrace{\lambda_t,\ldots,\lambda_t}_t, \gamma_{t+1},\ldots,\gamma_k,0,\ldots), \kappa^{\lambda_t}= (\underbrace{\lambda_t,\ldots,\lambda_t}_s, \kappa_{s+1},\ldots,\kappa_t, \gamma_{t+1},\ldots,\gamma_k,0,\ldots),
\left(A+R\right)^{T}_{i,k,j,l,n,m}\begin{pmatrix}\frac{\dot \alpha}{\alpha}\dot \beta^{m}\dot\xi^{l,j}\end{pmatrix}=\begin{pmatrix}\sigma \sigma_{k,i}\sigma_{n}\end{pmatrix}+O(\Vert v \Vert^{2}+\vert \delta J(u)\vert^{2})_{k,i,n},
\cos \pi \nu + 1 = \frac{\pi e^{4 h}}{( 8 h )^{q / 2}}\bigg [ \frac{1 + \frac{3 ( q^{2}+ 1 )}{6 4 h}}{\Gamma [ \frac{3}{4}- \frac{q}{4}] \Gamma [ \frac{1}{4}- \frac{q}{4}]}+ O ( \frac{1}{h^{2}}) \bigg ]
\Pi(m,n,\lambda,d)= &\sup\Bigg\{\bigg|\pi(A,B,\frac{\lambda}{2d},d)-\Big(1-\frac{\lambda\gamma-(1+\delta+\gamma)}{\lambda(\gamma+1)}\Big)^m\Big(\frac{1+\delta+\gamma}{\lambda\gamma}\Big)^n\bigg|: &~A,B\subseteq\mathbb{Z}^d, |A|=m, |B|=n, A\bigcap B=\emptyset\Bigg\},
{\rm A}(s,\tau,\tilde{w}_0){\rm A}(-s,\tilde{w}_0(\tau),\tilde{w}_0^{-1}) = \prod_i{\rm A}(s,\tau_i,\tilde{w}_i){\rm A}(-s,\tilde{w}_i(\tau_i),\tilde{w}_i^{-1}) \times \prod_{j<i}{\rm A}(s,\tau_{i,j},\tilde{w}_{i,j}){\rm A}(-s,\tilde{w}_{i,j}(\tau_{i,j}),\tilde{w}_{i,j}^{-1}) \, .
\begin{aligned}\mu_{2,1}^{T}(u)\leq c(1+T)(\|u_0\|_{H^{s}}+\|\,(|x|^{r_1}+|y|^{r_2})u_0\|_{L^2_{xy}}) &+c(1+T)T^\frac12(\mu_{1,1}^{T}(u))^2+cT^\frac12\mu_{1,1}^{T}(u)\mu_{2,1}^{T}(u).\end{aligned}
\max_{\substack{p_i\ge0,~R_i\ge0, i=1,\dots,K}}~ R ~ \rho_i\exp\left(\frac{(2^{R_i}-1)\sigma_i^2}{p_iQ_{ii}}\right) &~\times\prod_{k{\ne}i}\left(1+\frac{(2^{R_i}-1)p_kQ_{ki}}{p_iQ_{ii}}\right)\le1, &p_i\le{P}_i,~i=1,\dots,K,
\begin{aligned}u''+\frac{1}{8}(u^3+t^3v^3)+ 2=0,\,\,\,\,v''=\frac{1}{8}(\sqrt{tu}+13v^2) ,\t \in (0,1), \Delta u|_{t=1/5}=I_1(u(1/5)), \;\,\Delta u'|_{t=1/5}=N_1(u(1/5)) , \Delta v|_{t=2/5}=I_2(v(2/5)), \;\,\Delta v'|_{t=2/5}=N_2(v(2/5)) , u(0)=H_{1}(u(1/4)),\u(1)=L_{1}(v(3/4)), v(0)=H_{2}(v(1/3)),\v^{\prime}(1) =L_{2}(u(2/3)). \end{aligned}
P_f\left(\|\widehat{f}_n-\mathrm{E}_f\widehat{f}_n\|_\infty\geq\mathrm{E}_f\|\widehat{f}_n-\mathrm{E}_f\widehat{f}_n\|_\infty+\sqrt{2Q_2n^{-1}2^{\sum_{l=1}^dJ_{n,l}}x}\right)\leq e^{-x}.
\hat{h}_{0}( y ) = [ 2 y_{0}^{2}( \alpha_{1}e^{6 n}+ \alpha_{2}) / e_{l}]^{- 1 / 2}\sqrt{| y | + y_{0}}\left[ -{\frac{2}{s_{1}}}+ \ln \left({\frac{| y |}{y_{0}}}+ 1 \right) \right] \, ,
\mathrm{rank}\H_{\mathrm{prim}}^n(X_n(\mathbf{d}))=\begin{cases}\frac{2^{n+2}+2}{3}, \mbox{if}\ \mathbf{d}=3,\n\ \mbox{is even}; \frac{2^{n+2}-2}{3}, \mbox{if}\ \mathbf{d}=3,\n\ \mbox{is odd}; n+3, \mbox{if}\ \mathbf{d}=(2,2),\n\ \mbox{is even}; n+1, \mbox{if}\ \mathbf{d}=(2,2),\n\ \mbox{is odd}.\end{cases}
i{\cal G}_{2}( p ) = -{\frac{2 i e^{2}B^{2}p_{\bot}^{2}}{( p^{2}- m^{2})^{4}}}D + 2 e^{3}B^{3}\left[{\frac{1}{( p^{2}- m^{2})^{4}}}+{\frac{4 p_{\bot}^{2}}{( p^{2}- m^{2})^{5}}}\right] \bar{D},
S o l v \left( \frac{E_{7 ( 7 )}}{S U ( 8 )}\right) = S o l v \left( \frac{S L ( 2 , R )}{O ( 2 )}\right) \, \oplus \, S o l v \left( \frac{S O ( 6 , 6 )}{S O ( 6 ) \times S O ( 6 )}\right) \, \oplus \,{\cal W}_{3 2}
F_q(J_1,J_2,\gamma_1,\gamma_2) := \sum_{n_1,n_2}\frac{J_1^{n_1}J_2^{n_2}e^{i\gamma_1q^{2 [n_1]_q}+ i\gamma_2q^{2 [n_2]_q}}}{[n_1]_q![n_2]_q!}.
A_{k}(t,r,\theta,\varphi)=\sum_{l=1}^{\infty}\sum_{m=-l}^{l}\Bigr[c_{l}(r)\partial_{k}Y_{lm}(\theta,\varphi)+T_{l}(r)\varepsilon_{kp}\partial^{p}Y_{lm}(\theta,\varphi)\Bigr]e^{i \omega t}.
A(t_1:t_0)=\prod_{k=t_0}^{t_1-1}(1-\bar a_k)\stackrel{(a)}{\geq}\prod_{k=t_0}^{t_1-1}e^{-M(\gamma)\bar a_k}=e^{-M(\gamma)\sum_{k=t_0}^{t_1-1}\bar a_k}\geq e^{-M(\gamma)\sum_{k=0}^{\infty}\bar a_k}\geq e^{-M(\gamma)\Delta}>0,
b_n(t) =\sum_{i=1}^nK_h(t_{i-1}-t)\int_{t_{i-1}}^{t_i}\{\sigma^2(s)-\sigma^2(t)\}ds+\sigma^2(t)\left\{\frac{1}{n}\sum_{i=1}^nK_h(t_{i-1}-t)-\int_{-\infty}^\infty K_h(s-t)ds\right\}&=:\mathbb{I}_n(t)+\mathbb{II}_n(t).
\mbox{div}[(f^r)I = (\mathfrak{q}^{(D)})^r \mathfrak{q}^{(s)}] \stackrel{(*)}{=}\mbox{div}[(f^r)R ] + \mbox{div}[I] &= \mbox{div}[\mathfrak{q}^{(Dr)}] + \mbox{div}[\mathfrak{q}^{(s)}] = \mbox{div}[\mathfrak{q}^{(Dr + s)}].
\left|f\left(z\right)\right|=\left|\exp\left(\frac{\pi i}{2}e^{2\left|F\left(z\right)\right|}+e^{-2\left|F\left(z\right)\right|}\right)\right|\leq\exp\left(\pi e^{2\left|F\left(z\right)\right|}\right)=S_{f}
|{\hat \xi}|^2\leq \frac{32 e \sqrt{12\pi}}{3}\sum_{j=1}^2\int_{B_{2\hat s}(\xi_j)}\rho_{(\xi_j,t_j)}(\cdot,{\hat t}){\rm dist}(\cdot,J_{\xi_j})^2\, d\|V_{\hat t}\|.
x_1^2 + x_2^2 + x_3^2 + x_4^2 + x_5^2 = 0 \, , x_1^2 + i x_2^2 - x_3^2 - i x_4^2 = 0 \, , (i^2=-1.) x_1^2 - x_2^2 + x_3^3 - x_4^2 = 0 \, .
0 \rightarrow{\cal O}_{\pi^{*}{\cal S}}\rightarrow{\cal O}_{\pi^{*}{\cal S}}( \sigma |_{\pi^{*}{\cal S}}) \rightarrow{\cal O}_{\pi^{*}{\cal S}}( \sigma |_{\pi^{*}{\cal S}}) |_{\sigma |_{\pi^{*}{\cal S}}}\rightarrow 0 .
\begin{aligned}\alpha_1:\mbox{Hom}_{\tiny\mbox{[$n$]}_{\mathbb{K}}}(l,s) \to\mbox{Hom}_{\tiny\mbox{[$m$]}_{\mathbb{K}}}(\alpha(l),\alpha(s)) j_{ls}\mapsto \alpha_1(j_{ls})=\begin{cases}0, \mbox{if $l>s$}1, \mbox{if $l=s$}j_{\alpha(l)\alpha(s)}, \mbox{if $l<s$}\end{cases}.\end{aligned}
Z_f(s, \chi, D)=v(\bar{f}, D, \chi) +\sigma(\bar{f}, D, \chi)\dfrac{(1-q^{-1})q^{-s}}{1-q^{-1-s}}&+\sum_{P\in S(f, D)}{q^{-n}\int_{\mathcal{O}_K^n}\chi(ac(f_P(x)))|f_P(x)|^s|dx|},
\sup \left\{|\psi^{\zeta,(i)}_{x}(x)| + |\psi^{\zeta,(i)}_{xx}(x)| \, \mid \, x \leq 0 \right\}\leq C' \psi^{\zeta,(i)}(0) = u_{0}(0) \psi^{\zeta,(i)}_{x}(0) = u_{0,x_{i}}(0) \psi^{\zeta,(i)}_{xx}(0) = b_{i}\sup \left\{|\psi^{\zeta,(i)}(x) - u_{0}(x)| \, \mid \, x \in \overline{I_{i}}\right\}\leq C' \zeta^{2}
\sum_{i=1}^{n-1}[c(x_{i+1},y_i)-c(x_i,y_i)]+\c(x,y_n)-c(x_n,y_n) \leq \sum_{i=1}^{n-1}[h(x_{i+1})-h(x_i)]+h(x)-h(x_n) = \h(x)-h(x_1)=h(x)-h(s)=h(x),
f(\rho) = \sinh(\rho)-I_{r+1,r}(\rho)-d_r & = \sinh(\rho)\biggl(1+\frac{1}{r-2}\tanh^{r-2}(\rho)+\frac{r-1}{(r-2)(r-4)}\tanh^{r-4}(\rho) & +\dots+\frac{(r-1)(r-3)\cdots 5}{(r-2)(r-4)\cdots 2}\tanh^2(\rho)-\frac{(r-1)!!}{(r-2)!!}\biggr) & +\frac{(r-1)!!}{(r-2)!!}\arctan(\sinh(\rho))-d_r.
L^S(\tfrac{1}{2},\Pi\times \Pi') \sim_{E(\Pi)E(\Pi')}(2\pi i)^{dnn'/2}\prod\limits_{\imath \in \Sigma}\left(\prod\limits_{i=0}^{n}P^{(i)}(\Pi,\imath)^{sp(i,\Pi;\Pi',\imath)}\prod\limits_{j=0}^{n'}P^{(j)}(\Pi',\imath)^{sp(j,\Pi';\Pi,\imath)}\right).
U^+_{K}= \Big( U \setminus \{u_1\}\Big) \cup \{q_i^{-1}u_1, q_{s}u_1, q_s^{-1}q_i^{K-1}u_1, q_i q_i^{K-1}u_1 \}W^+_{K}= W \cup \{q_i^{-1}q_{s}^{-1}u_1, q_i^{K-1}u_1, q_i q_s q_i^{K-1}u_1 \},
F_\varepsilon(u)=\int_0^te^{-t/\varepsilon}\bigg(\int_{\mathbb{R}^n}\frac{\varepsilon^2|u''(t,x)|}{2}\,dx+\mathcal{W}(u(t,\cdot))-\int_{\mathbb{R}^n}f_\varepsilon(t,x)u(t,x)\,dx\bigg)\,dt,
\mathcal{D}_{\varepsilon}^{t}\left( k\right) :=\left( \left(z_{k}+\mathcal{C}_{\varepsilon}\left( t\right) \right) \cap\left(z_{k+1}-\mathcal{C}_{\varepsilon}\left( t\right) \right) \right)\ ,k=1,..,\mathfrak{k}_{N}-1\ .
\begin{array}{rl}I_5 = \displaystyle \sum_{i \in \mathbb{Z}_{n/3-1}}(X_i-X_{i+n/3})^2 +(X_i-X_{i+2n/3})^2 + (X_{i+n/3}-X_{i+2n/3})^2 &= \displaystyle \sigma\Big( 0; 2 , 0 , \dots, 0 ,-2 , 0, \dots, 0, -2, 0, \dots, 0 \Big) . \end{array}
\forall\;0\le k<n,\quad\prod_{0\le j<m}y_{k}^{(j)}=\mbox{Prod}_{\mathbf{P}_{k}}\left(\left(\mathbf{x}^{(m-1)}\right)^{\top^{\left(m-1\right)}},\cdots,\left(\mathbf{x}^{(j)}\right)^{\top^{j}},\cdots,\left(\mathbf{x}^{(0)}\right)^{\top^{0}}\right),
v_i^\prime=u_{n-i+1}-\frac{\left<u_{n-i+1},v_1^\prime\right>}{\left|\left|v_1^\prime\right|\right|^2}v_1^\prime-\sum_{\ell=2}^{i-1}\frac{\left<u_{n-i+1},v_\ell^\prime\right>}{\left|\left|v_\ell^\prime\right|\right|^2}v_\ell^\prime.
- \Delta_{g}\bigl( h_{2,\lambda}(r) \bigr) = -\left\{\frac{\partial^2}{\partial r^2}+ (n-1) S(r) \frac{\partial}{\partial r}\right\}h_{2,\lambda}(r) = \left( \frac{(n-1)^2}{4}{K_0}+ \lambda \right) h_{2,\lambda}(r)
\sum_{i=2}^N \sum_{l=1}^i c_{i,l}({\sf U}_\infty^{-1}\theta_0)^{i-2}(\theta_0^{-1}\theta_1)^{l-1}= \sum_{i=0}^{N'}c_i |x|^\frac{2(p-q)i}{1-q}, \sum_{i=3}^N \sum_{l=1}^i d_{i,l}({\sf U}_\infty^{-1}\theta_0)^{i-3}(\theta_0^{-1}\theta_1)^{l-1}= \sum_{i=0}^{N'}d_i |x|^\frac{2(p-q)i}{1-q}.
\sum_{r=D+1}^{\infty}\frac{1}{r^4}\leq \int_D^{\infty}\frac{1}{s^4}\,\mathrm{d}s = \frac{1}{3D^3}, \sum_{r=D+1}^{\infty}\frac{1}{r^2}\geq \int_{D+1}^{\infty}\frac{1}{s^2}\,\mathrm{d}s = \frac{1}{D+1}
I_{n o n - p l a n a r}^{T}= \frac{g}{8 \pi^{2}}\int p d p k d k \frac{J_{0}( \tilde{p}k )}{( e^{\beta ( \frac{k^{2}}{2}- \mu )}- 1 ) ( e^{\beta ( \frac{k^{2}}{2}- \mu )}- 1 )}.
V(G(\mathcal{F})) =\{S \in \binom{[n]}{\frac{k}{2}}:S \subseteq AA \in \mathcal{F}\}E(G(\mathcal{F})) =\{\{S_1,S_2\}: S_1 \cap S_2 = \emptyset, S_1,S_2 \in V(G(\mathcal{F}))\}.
\frac{d}{dt}\int \eta^2 u_k^p\;dx + \frac{4(p-1)}{p}\int (a\nabla (\eta u_k^{p/2}),\nabla (\eta u_k^{p/2}))\;dx = \frac{4(p-2)}{p}\int u_k^{p/2}(a\nabla (\eta u_k^{p/2}),\nabla \eta)\;dx+\frac{4}{p}\int u_k^{p}(a\nabla \eta,\nabla \eta)\;dx \;\;\;\;+ p\int (u\nabla a,\nabla (\eta^2u_k^{p-1}) )\;dx.
\frac{\left| u'_{R,r_1 \ldots r_n}(t) \right|}{\left| u'_{R,r_1 \ldots r_n}(r) \right|}=\left(\frac{q_{R,n}+(r-1) q_{R,n-1}}{q_{R,n}+(t-1) q_{R,n-1}}\right)^2 \leq\left(\frac{q_{R,n}}{q_{R,n}- q_{R,n-1}}\right)^2 \leq \left(\frac{N}{N-1}\right)^2.
{\hat{\cal T}}_J^R = \sum_{J^1, J^2}\left \langle \begin{array}{ll}R^1 R^2 J^1 J^2 \end{array}\right. \left | \begin{array}{l}R J \end{array}\right \rangle{\hat{\cal T}}_{J^1}^{R^1}(1){\hat{\cal T}}_{J^2}^{R^2}(2),
\displaystyle{\lambda v(x) - \frac{1}{2}\;\mbox{\rm Tr}\;[QD^2v(x)]- \left\langle A_Nx,Dv(x) \right\rangle- \inf_{{\eta}\in \Lambda}\left\{\left\langle L_N^{\delta, \varepsilon}{\eta},D^{G_N^{\delta,\varepsilon}}v(x) \right\rangle +l(x,{\eta}) \right\}=0.}
\sum_{n=1}^{\infty}\frac{h_k(n)}{n^s}= \prod_p (1+p^{-s}+ \cdots + p^{-(k-1)s}) &= \prod_p \frac{1-p^{-ks}}{1-p^{-s}}&= \frac{\zeta(s)}{\zeta(ks)}.
\mathcal{H}^{2}(B(z,R) \cap \Omega) = \sum_{k=1}^{i}\mathcal{H}^{2}(S_k), &= \sum_{k=1}^{i}\pi 4{r_{S_k}}^{2}\mbox{, by}\eqref{spherelocally2unifeq}&= \pi \left(4 \sum_{k=1}^{i}r^{2}\right), =4 \pi ir^{2}&= \pi R^{2}.
\int_0^z f(x)\,dx=\sum_{n=0}^N\int_0^zf_n(x)x^n\,dx =\sum_{n=0}^N\frac{\Theta(I_n)}{n+1}z^{n+1}&=\sum_{n=1}^N\frac{\Theta(I_{n-1})}{n}z^n+\Theta([0,1])\frac{\Theta(I_N)}{N+1}z^N.
\ll e^{O(p)}K \prod_{\substack{i = 1 i \ne i_0}}^{2 p}\left( \frac{K}{e^{h_i}}\right) \exp \bigg( 6 \pi \sum_{\substack{1 \le i < j \le 2 p h_i, h_j \ne \log K}}\frac{1}{\Vert t_{\sigma(j)}- t_{\sigma(i)}\Vert e^{\max(h_i,h_j)}}\bigg).
\Omega \wedge \overline{\Omega}= ( d\overline{Z}_{\gamma_e}\wedge d Z_{\gamma_m}+ d Z_{\gamma_e}\wedge d \overline{Z}_{\gamma_m}) \wedge d\theta_{\gamma_e}\wedge d\theta_{\gamma_m}& = \frac{2}{\pi}\cdot (S_1 - \ln |c|) dc_1 \wedge dc_2 \wedge d\theta_{\gamma_m}\wedge d\theta_{\gamma_e}
|f_n(t) - f(t)|\le \begin{cases}\exp(-n^2) + 2 - 2 \cos(2\pi i n^{-1}) 0 \le t \le n^{-1}; \exp(-n^2) + 2 - 2 \cos(2\pi i 2n^{-1}) n^{-1}\le t \le 2n^{-1}; 0 2n^{-1}\le t \le 1- n^{-1}; \exp(-n^2) + 2 - 2 \cos(2\pi i (1 - n^{-1})) 1- n^{-1}\le t \le 1.\end{cases}
R\left(\frac{sL_{1}^2}{4};\mu_{s}^{2}L_{1}^{2},\mu_{u}^{2}L_{1}^{2},\mu_{t}^{2}L_{1}^{2}; m_{0}L_{1},w \right)\equiv 16 \pi^{2}\frac{\sigma^{(\infty)}(s) - \sigma^{(0)}(s)}{g\sigma^{(0)}(s)}.
v_{\Lambda \Omega}= \frac{1}{f_{\Lambda}}\mathrm{{s g n}}( \sigma_{R_{i j}^{( \ell )}\Lambda}^{a}) \, \frac{v_{\Lambda_{i j}\left( R_{i j}^{( \ell )}\Lambda \right)_{i j}}}{f_{\left( R_{i j}^{( \ell )}\Lambda \right)_{i j}}}f_{\Omega}\, \# ( \Omega , \Lambda ) \, .
T(s,t,r) = \sum_{i = 0}^{s - 1}{\binom{t+i-1}iT(s - i,0,t + r + i)}+ \sum_{i = 0}^{t - 1}{\binom{s+i-1}iT(t - i,0,s + r + i)}&\quad= \sum_{i = 1}^s{\binom{s+t-i-1}{t-1}T(i,0,s + r + t - i)}& + \sum_{i = 1}^t{\binom{s+t-i-1}{s-1}T(i,0,s + r + t - i)}\,,
I_{ek}=\ \sum_{1\le m,n\le d}\Big\{\bigl((\sigma_{ek})_{mn}\bigr)_{x_m}\rho_{x_n}w_{ek}+(\sigma_{ek})_{mn}\rho_{x_mx_n}w_{ek}+(\sigma_{ek})_{mn}\rho_{x_n}(w_{ek})_{x_m}+(\sigma_{ek})_{mn}\rho_{x_m}(w_{ek})_{x_n}\Big\}
e^{-\mathcal{A}_{N,\varepsilon}t}-e^{-\mathcal{A}_N^0t}=-\frac{1}{2\pi i}\int_{\widetilde{\gamma}}e^{-\zeta t}\left( (\mathcal{A}_{N,\varepsilon}-\zeta I)^{-1}-(\mathcal{A}_N^0-\zeta I)^{-1}\right)\,d\zeta .
Q_{\perp}^{-}= \left( \frac{i g 2^{- 1 / 4}\sqrt{L}}{\pi}\right) \sum_{n , n_{\perp}}\frac{n_{\perp}}{\sqrt{n}}\left( A^{\dagger}( n , n_{\perp}) B ( n , n_{\perp}) - B^{\dagger}( n , n_{\perp}) A ( n , n_{\perp}) \right) \, .
E[X_j^{[r_1, r_2]}(t^* \wedge \tau)] \leq \int_{r_1}^{r_2}\mu e^{sj(t^* - u)}\: E[X_{j-1}^{\tau}(u)] \: du &\leq \frac{N \mu^j e^{sjt^*}}{s^{j-1}(j-1)!}\int_{r_1}^{r_2}e^{-sju}(e^{su}- 1)^{j-1}\: du &= \frac{N \mu^j e^{sjt^*}}{s^j j!}\bigg( (1 - e^{-s r_2})^j - (1 - e^{-sr_1})^j \bigg).
\Big|\sum_{m=2}^{n+1}\int_{\Gamma_+}(\alpha_{m,1}-\alpha_{m,2})(x,v,x'',v'')\phi(x,v)d \xi(x,v)\Big|\le \left\lbrace\begin{array}{l}C\|\phi\|_\infty \eta(1+|\ln(\eta)|),\textrm{}n=3, C\|\phi\|_\infty \eta\textrm{when}n\ge 4,\end{array}\right.
\log{(1 + \tau + t)}^2 - \log{(1 + \tau)}^2 = \log{\Bigr( 1 + \frac{t}{1+ (\chi t)^2}\Bigl )}\Bigl( \log{(1 + t + (\chi t)^2)}+ \log{(1 + (\chi t)^2)}\Bigr) \leq C \frac{t \log{(1 + (\chi t)^2)}}{1 + (\chi t)^2}\leq \frac{C}{\chi}.
(x_{\alpha\dot\alpha})=\left(\begin{array}{cc}z_2 z_1 -\bar{z}_1 \bar{z}_2 \end{array}\right), \qquad(\bar{x}^{\dot\alpha\alpha})=\left(\begin{array}{cc}\bar{z}_2 - z_1 \bar{z}_1 z_2 \end{array}\right).
\mathcal{O}_{\mathbb{P}(E)}(2km) \otimes \pi^{*}A^{-1}=K_{\mathbb{P}(E)}\otimes \mathcal{O}_{\mathbb{P}(E)}(2km) \otimes \pi^{*}A^{-2m}\otimes\mathcal{O}_{\mathbb{P}(E)}(r) \otimes \pi^{*}( A^{-1}\otimes K_{X}^{-1}\otimes \det E^{*}\otimes A^m) \otimes \pi^{*}A^{m}&= K_{\mathbb{P}(E)}\otimes \widetilde{L}
\mbox{m}\left(D_d\left(x,R_1\right),D_d\left(y,R_2\right)\right) = 2\left(R_1^2 + R_2^2 + R_1 + R_2 - 2r_0^2 - 4r_0 + 1\right), \mbox{where}, r_0 = \frac{R_1 - 1 + R_2 + (\gamma - \alpha)}{2}.
\begin{cases}u_{t}= \Delta u-\chi_1 \nabla \cdot (u \nabla v_1)+\chi_2 \nabla \cdot (u \nabla v_2) + u(a-bu),x\in\mathbb{R}^N 0=\Delta v_1-\lambda_1v_1+\mu_1u, x\in\mathbb{R}^N, 0=\Delta v_2-\lambda_2v_2+\mu_2u, x\in\mathbb{R}^N,\end{cases}
\int_{\mathbb{R}^N}|x|^l u^{\frac{l+N}{k+N-1}}\, dx = \frac{N-1}{k+N-1}\int_{\mathbb{R}^N}|y|^{\frac{l(N-1)-kN}{k+N-1}}v^{\frac{l+N}{k+N-1}}\, dy .
(1 -\frac{27}{64L^3})(x^2+y^2)^3+\frac{27}{64L^3}(x^3-3xy^2)^2 &-\frac{9}{4L}(x^2+y^2)(x^3-3xy^2)+(\frac{27}{16L^2}-L)(x^2+y^2)^2 &+(2-\frac{27}{64L^3})(x^3-3xy^2)-\frac{9}{8L}(x^2+y^2)+\frac{27}{256L^3}=0.
p\left( x|\theta\right) ={\displaystyle\prod\limits_{k=1}^{2l}}p\left( x_{k}|\mu_{k}\sigma_{k}\right) p\left( x_{k}|\mu_{k}\sigma_{k}\right) \overset{}{=}\frac{1}{\sqrt{2\pi\sigma_{k}^{2}}}\exp\left[ -\frac{\left( x_{k}-\mu_{k}\right)^{2}}{2\sigma_{k}^{2}}\right]
o_n(1) = \lambda \int_{\Omega}hu_n + \int_{\Omega}[f(u_n)-f'(u_n)u_n]u_n = -\mu \int_{\Omega}(p+2u_n^2)|u_n|^{p+2}e^{u_n^2}+ \lambda \int_{\Omega}hu_n,
\upsilon(\{\omega\in\Omega; \int\xi_{1}d\upsilon\leq\mathop{\lim\inf}_{n}\frac{\sum_{i=1}^{n}\xi_{i}}{n}\leq\mathop{\lim\sup}_{n}\frac{\sum_{i=1}^{n}\xi_{i}}{n}\leq-\int-\xi_{1}d\upsilon\})=1.
\int_{\Omega}f \dx < \frac{1}{t}\int_{0}^t \int_{\Omega}f \exp\Big(\frac{\partial u}{\partial t}\Big) \dx \d\tau = \frac{1}{t}\int_{0}^t \int_{\Omega}f^*(Yu)\det DYu \dx \d\tau = \frac{1}{t}\int_{0}^t \int_{\Omega^*}f^* \dy \d\tau = \int_{\Omega}f(x) \dx .
\frac{\partial}{\partial t}\frac{\delta\bar\ell^\mathcal{A}}{\delta\sigma^1}+\operatorname{div}^{\mathcal{A}_M}\frac{\delta\bar\ell^\mathcal{A}}{\delta\sigma^2}=-\operatorname{ad}^*_{\sigma^1}\frac{\delta\bar\ell^\mathcal{A}}{\delta\sigma^1}-\operatorname{Tr}\left(\operatorname{ad}^*_{\sigma^2}\frac{\delta\bar\ell^\mathcal{A}}{\delta\sigma^2}\right).
\chi(G) \leq \chi(A) + \chi(B) \leq 2 \sqrt{3t/y}+ (6 a)^{1/3}+ O(\sqrt{\frac{n}{\log n}}+ \frac{b^{1/3}\log \log((b^2/y^3)}{\log^{2/3}(b^2/y^3)}) = (6 a)^{1/3}+ O \Bigl( \sqrt{\frac{n}{\log n}}+ \frac{b^{1/3}\log \log(\frac{b^2}{t \log^6 n})}{\log^{2/3}(\frac{b^2}{t \log^6 n})}+ \frac{t^{1/3}}{\log n}\Bigr)
\frac{y^k}{2^{2k}}{}_2F_1\left(s+k+1, k+\frac{1}{2}, 2k+1, y\right) = \frac{y^k}{2^{2k}}\left[4\frac{\alpha(y)}{y}\right]^{s+k+1}{}_2F_1\left(s+k+1, s+1, k+1 , (\alpha(y))^2\right) & = [\alpha(y)]^k\left[4\frac{\alpha(y)}{y}\right]^{s+1}{}_2F_1\left(s+k+1, s+1, k+1 , (\alpha(y))^2\right).
\mathcal{N}(T) =||\mathbf{F}||^2_{s,\ast,T}+(||\dot{{\mathbf U}}||^2_{W^{2,\infty}_{\ast}(\Omega_T)}+||\varphi||^2_{W^{2,\infty}(\Gamma_T)}+||\mathbf{F}||^2_{W^{1,\infty}_{\ast}(\Omega_T)})||\hat{W}||^2_{s+4,\ast,T}.
\sum_{\mathcal{S}\in \Gamma(G)}P_{W^*}(\mathcal{S}) = \; \sum_{\mathcal{S}\in \Gamma(G)}\;\;\prod_{a' \in \mathcal{A}}P_{W^*_{a'}}(\mathcal{S}_{a'}) = \; \sum_{(\mathcal{S}_{a'})_{a' \in \mathcal{A}}\in \prod_{a' \in \mathcal{A}}\Gamma(G_{a'})}\;\; \prod_{a' \in \mathcal{A}}P_{W^*_{a'}}(\mathcal{S}_{a'}) = \; 1,
\begin{array}{lrlcccl}w_*:= \sup \{w\geq 0, \lim_{t\to +\infty}\sup_{0 \leq x\leq wt}|u(t,x)- 1| = 0 \}, w^*:= \inf \{w\geq 0, \lim_{t\to +\infty}\sup_{x\geq wt}|u(t,x)| = 0 \}. \end{array}
\mathrm{Pf}\begin{bmatrix}K_{N}(x_{j}, x_{k}) \end{bmatrix}= \sqrt{\det\begin{bmatrix}K_{N}(x_{j}, x_{k}) \end{bmatrix}}\propto \det(\Phi) \, \mathrm{Pf}\!\begin{bmatrix}E H -H^{t}\alpha \end{bmatrix}.
\psi_n(u)=\psi\left(\frac{E_{\sigma}(u)-(\beta(\sigma)-\epsilon(\sigma_n-\sigma))}{\epsilon(\sigma_n-\sigma)}\right)\psi\bigg(\frac{4}{\beta(0)}\frac{e^{-\frac{4}{\beta(0)}}}{\sigma}\left(\mathfrak{L}(u)-\frac{\beta(0)}{2}\right)\bigg).
\int_{\mathbb{R}^3}\frac{|\widehat{U}(t,\xi)|}{|\xi|}d\xi+\nu\int_0^t \int_{\mathbb{R}^3}|\xi||\widehat{U}(\tau,\xi)|d\xi d\tau + \int_{\mathbb{R}^3}\frac{|\widehat{H}(t,\xi)|}{|\xi|}d\xi+\mu \int_0^t \int_{\mathbb{R}^3}|\xi||\widehat{H}(\tau,\xi)|d\xi d\tau \leq C_{*}\delta_0,
\Pi=\begin{pmatrix}P P P \cdots P 1 P L_{1,1}L_{1,2}\cdots L_{1,b}1 \vdots \vdots \vdots \ddots \vdots \vdots P L_{a,1}L_{a,2}\cdots L_{a,b}1 1 1 1 \cdots 1 1\end{pmatrix}.
(\zeta'')^\tau_{\alpha_1,\dots,\alpha_s\beta_1,\dots,\beta_r}=\frac{\partial}{\partial(\partial^\lambda)}\big(\zeta^\tau_{\alpha_1,\dots,\alpha_s}\big) \cdot(\zeta')^\lambda_{\beta_1,\dots,\beta_r}-\frac{\partial}{\partial(\partial^\lambda)}\big((\zeta')^\lambda_{\beta_1,\dots,\beta_r}\big) \cdot\zeta^\tau_{\alpha_1,\dots,\alpha_s}.
w_0 = - [2(n-1)p_5 + q_4] \sum_i z_i^3 - 2p_5 \sum_{i\ne j}z_i^2 z_j - [2(n-1)p_4 + q_3] \sum_i z_i^2{}- 2p_4 \sum_{i<j}z_iz_j - [2(n-1)p_3 + q_2] \sum_i z_i - n(n-1)p_2 - nq_1.