harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{cases}w_1^{\prime\prime}+ A w_1 = 0 \,, w_i^{\prime\prime}+ A w_i+ C_{i i-1}w_{i-1}= 0 \,, 2 \le i \le n \,, \displaystyle{w_{i}^{\prime\prime}+ A w_{i}+ \sum_{k=n-1}^{i-1}C_{i k}\,w_k =0 \,, n+1 \le i \le n+p \,,}W_{|t=T}=W^T \,,\end{cases}
\frac{1}{\sqrt{- g}}{\cal L}_{f}= \overline{{\psi}}e^{\mu}{}_{a}\gamma^{a}( i \partial_{\mu}+ i \omega_{\mu}+ q V_{\mu}) \psi - M \exp \! \Big ( - \sqrt{\frac{\kappa}{6}}\phi \Big ) \overline{{\psi}}\psi .
\xi =P\left(Y_{j}>(1-\theta+c')^{1/2}\sqrt{2\log(2^{n})}\right) \geq\frac{(1-\theta+c')^{1/2}\sqrt{2\log(2^{n})}}{1+2(1-\theta+c')\log(2^{n})}\exp\left(-(1-\theta+c')\log(2^{n})\right) \geq C2^{-n(1-\theta+c')}
\frac{1}{2 \pi i}\int_{C_1}\zeta^k(s_1+1) \mathcal{D}_{k,\ell}(s_1,s_2) t^{s_1}ds_1 = \sum_{\substack{i_1+i_2 +i_3 = k-1 i_1,i_2,i_3 \ge 0}}\frac{\alpha_{i_1,k}\mathcal{D}_{k,\ell}^{(i_2,0)}(0,s_2) (\log t)^{i_3}}{i_{2}! i_{3}!}.
\limsup_{n \to \infty}T_{W_{nJ}, \psi}(\mu_n(Q^n)) \le \limsup_{n \to \infty}T_{W_{nJ},\psi_m}(\mu_n(Q^n)) \le \limsup_{n \to \infty}T_{W,\psi_m}(\mu_n(Q^n)) = T_{W,\psi_m}(\mu_\infty),
\chi(x) = \exp (2\pi i\{x\}) &= \exp \left( 2\pi i \sum_{k=\gamma}^{-1}a_k e^{\psi(k)}\right) &= \exp \left( \frac{2\pi i}{e^{-\psi(\gamma)}}\big( a_{\gamma}+ a_{\gamma+1}e^{\psi(\gamma+1) - \psi(\gamma)}+ \cdots + a_{-1}e^{\psi(-1) -\psi(\gamma)}\big) \right) &= \exp \left( \frac{2\pi i\ell}{e^{-\psi(\gamma)}}\right),
f(a(i) e(i)) = \sum_{j\in I'}{\tilde f}_{i}^j \, \left( a'(j) e'(j)\right) \tilde f_{i}^j = \frac{a(i)}{a'(j)}f_{i}^j f( \sqrt{a(i)}\, e(i)) = \sum_{j\in I'}{\tilde \tilde f}_{i}^j \left(\sqrt{a'(j)}\, e'(j)\right){\Tilde \Tilde f}_{i}^j = \frac{\sqrt{a(i)}}{\sqrt{a'(j)}}f_{i}^j.
C_1(\lambda) = \left[\begin{array}{cccc}\lambda P_d+P_{d-1}P_{d-2}\cdots P_0 \hline-I_n \lambda I_n & \ddots \ddots & -I_n \lambda I_n \end{array}\right] = \left[ \begin{array}{c}M_1(\lambda) \hline L_{d-1}(\lambda)\otimes I_n \end{array}\right]
\int_0^\infty(CL_2^u(y)+DL_2^v(y))\mathrm{e}^{-\lambda y^2}\mathrm{d}y=\frac{x^{1+\sqrt{mn}+c}}{(1+\frac{4t^\alpha}{\alpha}\lambda)^{\frac{3+\sqrt{mn}-c}{2}}}\mathrm{e}^{-\frac{\lambda x^2}{(1+\frac{4t^\alpha}{\alpha}\lambda)}}.
S_{m}^{( 1 )}= S_{m}^{( 1 )}( \Psi_{1}, g_{\mu \nu}^{B ( 1 )}) \mathrm{a n d}S_{m}^{( 2 )}= S_{m}^{( 2 )}( \Psi_{2}, g_{\mu \nu}^{B ( 2 )}) ,
p(D-2) \left(\prod_{i \in I_3}(D-i+1)\right) \partial_t v^{(2)}= \left(\prod_{i \in (I_3-1) \setminus I_2}(D-i)\right) p(D-2) \underbrace{\left(\prod_{i \in I_2}(D-i)\right) v^{(2)}}_{= w^{(2)}}&= \left(\prod_{i \in (I_3-1) \setminus I_2}(D-i)\right) v^{(3)}
{\cal H}_3 = |\nabla \psi_L|^2 + |\nabla \psi_R|^2 |+ |\nabla\psi_L||\nabla \psi_R|+ m_0^2 ( |\psi_L|^2 + |\psi_R|^2) + g_1 (|\psi_L|^4 + |\psi_R|^4) + g_2 |\psi_L|^2|\psi_R|^2 \,,
I_{\chi}(u_1, u_2) =\int_{\{u_1<u_2\}}\min\{|u_1-u_2|, 1\}(\theta_{u_1}^n-\theta_{u_2}^n)+\int_{\{u_2<u_1\}}\min\{|u_1-u_2|, 1\}(\theta_{u_2}^n-\theta_{u_1}^n) &=\int_X\max\{\min\{u_2-u_1, 1\}, -1\}(\theta_{u_1}^n-\theta_{u_2}^n).
k_{v^u_h}:=\begin{cases}i_{v^u_{h+1}}-i_{v^u_h}, 1\le h\le l_u-1\,,1\le u\le m+1\,, i_{v^u_1}-i_{v^u_{l_u}}, h=l_u\,,1\le u\le m+1\,.\end{cases}
z^{*}\Gamma_{\Phi_{1}(x;h)}z \geq C_{H}\|z^{*}D\Phi_{1}(x;h)\|_{L^{2}([0,1])}^{2}=C_{H}\int_{0}^{1}z^{*}J_{1}J_{t}^{-1}V(\Phi_{t})V^{*}(\Phi_{t})(J_{t}^{-1})^{*}J_{1}^{*}zdt.
\sum_{k=m-l+1}^{m+1}\binom{2m-k+1}{m}\binom{k-1}{m-l+1}h^{l-1}(1+h)^{m-l+1}=\binom{2m+1}{l-1}h^{l-1}(1+h)^{m-l+1}&=\left(\binom{2m}{l-1}+\binom{2m}{l-2}\right)h^{l-1}(1+h)^{m-l+1}.
0 = -\frac{(c-j+1)\gamma}{\xi}k_{1}^{j-1}(1) -j\left( 1+\frac{\mu}{\xi}\right) \sum_{n=0}^{j}p_{j,n}k_{2}(1) + \frac{\mu}{\xi}\sum_{n=1}^{j}n p_{j,n}k_{3}(1) -\frac{\mu}{\xi}\sum_{n=1}^{j+1}n p_{j+1,n}k_{4}(1)
W( 10^l | 1^k0 ) = \left[ A(01^k0...) - A(0^{\alpha+1}1 ...) \right] + \sum_{j=2}^l \left[ A(0^{j}1^k0) - A(0^{j+\alpha}1...) \right] + &+ \left[ A(10^l1^k0...) - A(10^{l+\alpha}1...) \right] &= b_k - a_{\alpha+1}+ \sum_{j=2}^l (a_j - a_{j+\alpha}) + d_l - d_{l+\alpha}\ .
\mathcal{E}_{q}(z)=E_{q}^{(1/2)}(z)={}_{1}\phi_{1}\left(0;-q^{1/2};q^{1/2},-q^{1/4}z\right)=\sum_{n=0}^{\infty}\frac{q^{n^{2}/4}}{(q;q)_{n}}z^{n}, z\in\mathbb{C}.
P(B_2^{(n)}) \leq P(S_n(n-f_1(n))\leq\lfloor n-\varepsilon' f_2(n)\rfloor-a_n) &\leq P(S_n'(n-f_1(n))\leq\lfloor n-\varepsilon' f_2(n)\rfloor) &=P(\mathrm{Bin}(n,1-\pi_n(n-f_1(n)))\geq n-\lfloor n-\varepsilon' f_2(n)\rfloor) &=P(\mathrm{Bin}(n,1-\pi_n(n-f_1(n)))\geq\lceil\varepsilon' f_2(n)\rceil),
{f_i}+{\sum\limits_{j = 0}^i{{w_{i,j}}\sum\limits_{k = j}^n{{\left(\lambda + q_k\right) f_k}{v_{j,k}}}}-{{\left({\frac{i}{n}}\right)}^\alpha}\sum\limits_{j = 0}^n{{w_{n,j}}\sum\limits_{k = j}^n{{\left(\lambda + q_k\right) f_k}{v_{j,k}}}}}& ={\left({\frac{i}{n}}\right)^\alpha}L .
\left( x\in (0,1)\mapsto k_{xx}(x,\cdot)\in L^2(0,1)\right) \in C^0([0,1];L^2(0,1)), \left( y\in (0,1)\mapsto k_{yy}(\cdot,y)\in L^2(0,1)\right) \in C^0([0,1];L^2(0,1)), k_{yy}(\cdot,0)=k_{yy}(\cdot,1)=0 \text{in}L^2(0,1).
\lim\limits_{R\rightarrow \infty}\lim\limits_{\epsilon\rightarrow 0}RHS=-\frac{N-2}{2}\Big(\mu \int_{\Omega}\frac{|u|^{2^{*}(t)}}{|y|^{t}}+\int_{\Omega}\frac{|u|^{2^{*}(s)}}{|y|^{s}}\Big)-\frac{N}{2}\int_{\Omega}a(x)u^2-\frac{1}{2}\int_{\Omega}u^2(x\cdot\nabla a).
\Psi^* \circ L \circ (\Psi^*)^{-1}= - a^{ij}(s,x)\partial_{x_i}\partial_{x_j}+ b^{j}(s,x) \partial_{x_j}+c(s,x), &\textup{where}\Lambda^{-1}\| \xi\|^2 \leq a^{ij}(s,x) \xi_i \xi_j \leq \Lambda \| \xi\|^2, &\textup{and}\| b(s,x)\| \leq \Lambda^{-1}, \ 0 \leq c(s,x) \leq \Lambda^{-1},
f_{X11}\left(x_{11}|\mathbb{A_{UE}}\right)=\frac{2\pi\lambda_{b}x_{11}\mathrm{e^{-\pi\lambda_{b}\max^2\left(x_{11},x_{11}^{\mu/2}\left(\frac{P}{N}\right)^{\frac{2-\mu}{2\alpha}}\left(\frac{1}{M}\right)^{\frac{1}{2\alpha}}\right)}}}{\mathbb{P\left[A_{UE}\right]}},
0=\left[\chi_{r^i\omega_i-\rho}\left(-\frac{2\pi\rho}{pP_\rho}\frac{\partial}{\partial\sigma}\right)\chi_{s^i\omega_i-\rho}\left(\frac{2\pi\rho}{p'P_\rho}\frac{\partial}{\partial\sigma}\right)-d_{\lambda}\right]\langle\mathcal{T}_{\rho}\rangle_{\rho}(\sigma)\ ,\lambda\in\mathcal{B}^{(q)}_{p,p'}\ ,
g_{ij}(x)=\delta_{ij}+\frac{1}{3}R_{iklj}x^kx^l + \frac{1}{6}R_{iklj,a}x^kx^lx^a+\left(\frac{1}{20}R_{iklj,ab}+ \frac{2}{45}R_{iklc}R_{jabc}\right)x^kx^lx^ax^b + O(r^{5}),
\alpha_j(x,\xi',\tau,\sigma)^2 = r(x, \xi' + i \tau d_{x'}\varphi) + (-1)^j \sigma^2 =r(x,\xi') -\tau^2 r(x,d_{x'}\varphi) +2i\tau\tilde{r}(x,\xi',d_{x'}\varphi) + (-1)^j \sigma^2,
b_k := \psi(p_k)\;\;\;\;\;\; (1\leq k\leq m), c_k := \psi(q_k)\;\;\;\;\;\; (1\leq k\leq m+1), a_k := \sum_{i=0}^{\ell-1}(g_{k+im}- h_{k+im})\;\;\;\;\;\; (1\leq k\leq m).
\begin{array}{ll}\widehat{P_{\leq N}f}(\xi) := \phi \left( \frac{\xi}{N}\right) \hat{f}(\xi) \widehat{P_{N}f}(\xi) := \psi \left( \frac{\xi}{N}\right) \hat{f}(\xi) \widehat{P_{>N}f}(\xi) := \hat{f}(\xi) - \widehat{P_{\leq N}f}(\xi)\end{array}
\sigma_{\mathfrak{X}(X)}(D) \leq \inf{\{\sigma \in \mathbb{R}\colon \mbox{$\sup_{N}\big\| \sum_{n=1}^{N}{\frac{a_{n}}{n^{\sigma}}n^{-s}}\big\|_{\mathfrak{X}(X)}< \infty$}\}}\,,
g_{2 1}= \pm \cos ( \sqrt{- H}t ) \quad , \quad g_{2 2}= \sqrt{- H}^{- 1}\sin ( \sqrt{- H}t ) \quad ; \qquad - \frac{\pi}{2}- \delta ( \epsilon ) < \sqrt{- H}t < \frac{\pi}{2}+ \delta ( \epsilon ) ,
E(r) \leq C\left(1 + \frac{1}{4r^2}\|v\|^3_{L^3(Q_{2r}(z_0))}+ \frac{1}{4r^2}\|\pi|^\frac{3}{2}_{L^\frac{3}{2}(Q_{2r}(z_0))}\right) =: C(1 + A(2r) + B(2r)),
\tau(\alpha) =\int_y^{x^\alpha}\frac{dS(v)}{v\log v}=\bigg[\frac{S(v)}{v\log v}\bigg]_y^{x^\alpha}+\int_y^{x^\alpha}\frac{S(v)(\log v+1)}{(v\log v)^2}\,dv &=\int_y^{x^\alpha}\frac{S(v)}{v^2\log v}\,dv+O\big((\log y)^{-1}\big),
p_{kl}^{i}=\sum_{r\leq s}\left(\frac{\partial^{2}L_{0}}{\partial y_{rs,j}\partial y_{kl,i}}-\frac{\partial (L_{EH})_{kl}^{ij}}{\partial y_{rs}}-\frac{\partial (L_{EH})_{rs}^{ij}}{\partial y_{kl}}\right) y_{rs,j}& =\sum_{r\leq s}Y_{kl}^{i;rs,j}y_{rs,j},
b^+=\frac{m\nu_m}{(2mh_1^+(0)+osc_\Omega\bold u)\mu_{m-1}},\; -h=h^+=\max\left\{\frac{\max\{1-\frac{\nu}{\mu_m};\frac{1}{2}\}}{mh_1^+(0)}h_1^+(t);h_2^+(t)\right\},
{}_2F_1(\frac{1}{2},\ell+1;\frac{3}{2};-x^2)|_{x\rightarrow \pm \infty}= \frac{(-1)^\ell\, \pi}{\Gamma(\frac{1}{2}-\ell)}\left[ \frac{\pi^{1/2}}{2\, \Gamma(\ell+1)|x|}- \frac{(-x^2)^{-(\ell+1)}}{2\, \Gamma(\frac{3}{2}+\ell)}\right] + O[\frac{1}{x^2}].
\mu(s\widetilde{\alpha},\sigma)=\gamma(G/P)^2q_F^{n(\omega) +n(\sigma \times \Pi(\sigma))-n(\sigma)}\dfrac{(1-\omega(\varpi)q_F^{-2s})(1-\omega^{-1}(\varpi)q_F^{2s})}{(1-\omega^{-1}(\varpi)q_F^{-1+2s})(1-\omega(\varpi)q_F^{-1-2s})}
\frac{1}{n}\sum\limits_{i=1}^n{2n \choose 2i}B_{2n-2i}\gamma_{2i+1}= \frac{1}{n}\sum\limits_{i=1}^n{2n \choose 2i}B_{2n-2i}\sum\limits_{j=0}^{i}{i\brack j}\gamma_{2j}&= \frac{1}{n}\sum\limits_{j=0}^{n}\gamma_{2j}\sum\limits_{i=j}^n{2n \choose 2i}B_{2n-2i}{i\brack j}.
\pi(x)=\left|\{p\in\mathbb P\mid p\le x\}\right|=\sum_{p\le x}1\theta(x)=\sum_{p\le x}\log p \Lambda(n)=\begin{cases}\log p \exists p\in\mathbb P,k>0:n=p^k 0 o.w.\end{cases}\psi(x)=\sum_{n\le x}\Lambda(n)
\mu_1(\rho_1) \ge \nu\left(\partial (\rho_1B)\cap \left({1\over \sqrt{3}}B+{\bf v}_1\right)\right) &=2\pi \sqrt{{1\over 3}+{4\over{57}}\sqrt{{2033}\over 3}}\left( \sqrt{{1\over 3}+{4\over{57}}\sqrt{{2033}\over 3}}-{{\sqrt{2033}}\over{57}}-{1\over \sqrt{3}}\right).
\begin{array}{l}(\frac{\partial f}{\partial x})^T(x)\Pi(x) + \Pi(x)\frac{\partial f}{\partial x}(x) begin{align*}3mm]+ (\frac{\partial h}{\partial x})^T(x)\frac{\partial h}{\partial x}(x) + \frac{\partial \Pi}{\partial x}(x)f(x)=0 \, ,\end{array}
[C_2 C_1^i R C_2 v]_a = \sum_{b,c,d=1}^k [C_2 C_1^i]_{ab}R_{bc}[C_2]_{cd}v_d = \sum_{b=1}^k (C_2 C_1^i) R_{ab}(C_2 v_b) & = \sum_{n_b =1}(C_2 C_1^i) R_{ab}(C_2 v_b) = \sum_{n_b =1}R_{a(i+1),b1}v_{b1},
\mbox{Problem 1:}~~\max_{\boldsymbol{P}}~~ \mathbb{E}\left[\sum_{j=1}^K \ln\left(1+\frac{h_{j}P_j}{\sigma^2+\sum_{i=1,i\neq j}^K h_{i}P_i}\right)\right], \mbox{s.t.}~~ P_i\ge 0, ~\forall i &\mathbb{E}\left[P_i\right] \le P_i^{av},~\forall i &\mathbb{E}\left[\sum_{i=1}^K g_iP_i\right] \le I.
\mathbb A\bigl((\pi_t)_{t\in[0,T]}\bigr) :=\frac 12\bigl[\mathcal F_\alpha^V(\rho_T) - \mathcal F_\alpha^V(\rho_0) +\mathcal E\bigl((\rho_t)_{t\in[0,T]}\bigr) +\mathcal E^\star\bigl((\rho_t)_{t\in[0,T]}\bigr)\bigr].
\overline{\mathcal{C}^I}=\mathcal{C}^{I}\cup \mathcal{C}^{III}\cup \mathcal{C}^{IV}\cup\mathcal{C}^{V}, ~\overline{\mathcal{C}^{II}}=\mathcal{C}^{II}\cup \mathcal{C}^{III}\cup \mathcal{C}^{IV}\cup\mathcal{C}^{V}.
||{\bf v}[{\bf u}^a+{\bf u}_{i}]||_{s,\ast,T}\le C\delta^2\begin{cases}\theta_i^{(s+2-\alpha)_+}s+2\not=\alpha, \theta_i s+2=\alpha,\end{cases}s\in\{6,\dots,\tilde{\alpha}-2\},
f=\lim_{t_m\to\infty}e^{it_mH}\Phi_a(x/t_m)e^{-it_mH}f\in S_a^1(B), &f=\lim_{t_m\to\infty}e^{it_mH}P_a^{{\varepsilon}_a}(t_m)e^{-it_mH}f\in S_a^1(B).
p(\boldsymbol{s}^{r}|\boldsymbol{s})=\prod_{m}\mathop{p(s_{m}^{r}|s_{m})}=\prod_{m}(1-s_{m})\delta(s_{m}^{r})+s_{m}\rho_{r}^{s_{m}^{r}}(1-\rho_{r})^{1-s_{m}^{r}},
\begin{aligned}\sum_{m_1,m_2=1}^\infty\sum_{m_3=0}^\infty \frac{z^{m_1+m_2+m_3}}{(m_1+m_2+m_3+a)^{s}}=\sum_{m_1,m_2,m_3=1}^\infty\frac{z^{m_1+m_2+m_3}}{(m_1+m_2+m_3+a)^{s}}&+\sum_{m_1,m_2=1}^\infty\frac{z^{m_1+m_2}}{(m_1+m_2+a)^{s}}\end{aligned}
\frac{d}{ds}|_{s=0}P_i(\exp_p(v + sw)) = \frac{d}{ds}|_{s = 0}\left( e_i \cdot (v + sw) \frac{g(|v + sw|)}{|v + sw|}\right) &= e_i \cdot w \frac{g(|v|)}{|v|}+ \frac{(e_i \cdot v)(v \cdot w)}{|v|}{\frac{d}{dr}}|_{r = |v|}\frac{g(r)}{r}.
\frac{1}{6}\frac{2 M}{\eta}K_{1}( 2 \eta M ) - 2 M \sum_{k = 1}^{\infty}\frac{1}{( k \pi )^{2}}\frac{K_{1}( 2 M \sqrt{\eta^{2}+ k^{2}\pi^{2}S^{2}})}{\sqrt{\eta^{2}+ k^{2}\pi^{2}S^{2}}}.
W_{2l+2}=xW_{2l+1}-W_{2l}=x^2W_{2l}-(xW_{2l-1}-W_{2l-2})-W_{2l-2}-W_{2l}= &=x^2W_{2l}-W_{2l}-W_{2l-2}-W_{2l}=(x^2-2)W_{2l}-W_{2l-2}.
\varphi_j(p) := \frac{\int_0^\infty e^{-\lambda s}p_j(s) ds}{\int_0^\infty e^{-\lambda s}\big(\sum_{k=j}^\infty p_k(s)\big) ds}= \begin{cases}\dfrac{1}{j+1}, p=0, 1-\dfrac{\int_0^{1}(1-p y)^{-1}y^{j}dy}{\int_0^1 (1-py)^{-1}y^{j-1}dy}, 0<p<1, \end{cases}
h \Phi_1^{\Sigma}(x)\theta_h(x)|v^{\Sigma}_{\Lambda}|^2v^{\Sigma}_{\Lambda}+ h Op_h^w(\Gamma)\left[\Phi_3^{\Sigma}(x)\theta_h(x)(v^{\Sigma}_{\Lambda})^3 + \Phi_{-1}^{\Sigma}(x)\theta_h(x) |v^{\Sigma}_{\Lambda}|^2\overline{v^{\Sigma}_{\Lambda}}+ \Phi_{-3}^{\Sigma}(x)\theta_h(x) (\overline{v^{\Sigma}_{\Lambda}})^3\right] & + hR(v) \,,
\mathrm{A r e a}= \sqrt{1 + 4 | \partial y_{3}/ \partial y |^{2}}d y \wedge d \bar{y}={\frac{\sqrt{W + \overline{{{W}}}+ | W^{\prime}|^{2}- y W^{\prime}- \bar{y}\overline{{{W^{\prime}}}}}}{y_{3}}}d y \wedge d \bar{y}
\frac{\varepsilon}{2}\frac{d}{dt}\sum_{i=1}^{N}\left\Vert v_{i}^{\varepsilon}-v_{i}^{0}\right\Vert_{L^{2}\left(\Gamma^{\varepsilon}\right)}^{2}\le C\max\left\{\varepsilon,\varepsilon^{\gamma}\right\}+C\varepsilon\sum_{i=1}^{N}\left\Vert v_{i}^{\varepsilon}-v_{i}^{0}\right\Vert_{L^{2}\left(\Gamma^{\varepsilon}\right)}^{2}.
g_{\Delta}(\omega) := \sum_{\zeta_{\Lambda}}\sum_{\eta_{\Lambda}}\mathbf{1}_{\zeta_{\Lambda}}(\omega) h_{\Delta}(\eta_{\Lambda}\omega_{\Lambda^c}) \frac{\gamma_{\Lambda}(\eta_{\Lambda}| \omega_{\Lambda^c})}{\gamma_{\Lambda}(\zeta_{\Lambda}|\omega_{\Lambda^c})}, \omega \in \Omega,
G_{k i_1 \dots i_n}= \sum_{\lambda = 1, \pi \in S_{n}}^{n!}a\delta_{k \pi(i_1}D_{i_2 \cdots i_{n})}+ \sum_{\lambda = 1, \pi \in S_{n}}^{n!}b \delta_{\pi(i_1 i_2}D_{i_3 \cdots i_n) k}+ \sum_{\lambda = 1, \pi \in S_{n}}^{n!}d \epsilon_{k s \pi(i_1}D_{i_2 \cdots i_n) s}+ f D_{k i_1 \dots i_n}
\partial^v_{n-1}\left[Q(u,v)(v_{n-1}-u_n)^2 (v_n-u_n)u_n^{k-1}\right] = Q(u,v)(v_{n-1}-u_n)(v_n-u_n)u_n^{k-1}\partial_{n-1}^v (v_{n-1}-u_n) = Q(u,v)(v_{n-1}-u_n)(v_n-u_n)u_n^{k-1}.
b_{\underline{{n}}\sigma}^{\dagger}= \frac{\pi / L_{\perp}}{\sqrt{8 \pi^{3}n}}b_{\underline{{p}}\sigma}^{\dagger}\, , \; \; a_{\underline{{m}}}^{\dagger}= \frac{\pi / L_{\perp}}{\sqrt{8 \pi^{3}m}}a_{\underline{{q}}}^{\dagger}\, , \; \;
\Phi \, = \, ( \, 1 - G^{R}K \, )^{- 1}\, \Psi \, = \, ( \, 1 + G^{R}K^{T}\, ) \, \Psi \, = \, ( \, 1 + G^{0}K^{T}- G^{\, \delta}K^{T}\, ) \, \Psi \, = \, G^{0}K^{T}\, \Psi
u_{0}=u_{0}({\bf i})=\begin{pmatrix}0{\bf i}{\bf i}0\end{pmatrix},~u_{\bf i}=u_{\bf i}({\bf i})=\begin{pmatrix}{\bf i}0 0 -{\bf i}\end{pmatrix}, &u_{\bf j}=u_{\bf j}({\bf i})=\begin{pmatrix}{\bf j}{\bf k}{\bf k}-{\bf j}\end{pmatrix}, ~u_{\bf k}=u_{\bf k}({\bf i})=\begin{pmatrix}{\bf k}-{\bf j}-{\bf j}-{\bf k}\end{pmatrix}.
\operatorname{l n}\left( \frac{\operatorname{c o s}\theta ( 0 , \varphi )}{\operatorname{c o s}\theta ( 1 , \varphi )}\right) ={\textstyle{\frac{1}{2}}}\operatorname{l n}\left( 1 + \frac{\operatorname{t a n}^{2}\eta_{1 2}}{\operatorname{c o s}^{2}\varphi}\right) .
\lim_{N\to\infty}\Big\|a_N^{-1}\big(f_MU_N\psi_N^{\operatorname{MF}}\big)_{jk}-\Phi^{\operatorname{gs}}_{jk}\Big\|_{L^2(\mathbb{R}^3)^{\otimes_{\operatorname{sym}}j}\otimes L^2(\mathbb{R}^3)^{\otimes_{\operatorname{sym}}k}}=0.
M_{k}^{\#}( \Gamma_{0}( N) ) :=\{f\in M_{k}^{!}( \Gamma_{0}( N) ) \mid \}, &S_{k}^{\# ,0}( \Gamma_{0}( N) ) :=\{f\in M_{k}^{\#}( \Gamma_{0}( N) ) \mid \}.
w_{n1}\big|_{\Gamma_0'\times(t_1+\delta,t_2-\delta)}=w_{n3}\big|_{\Gamma_0'\times(t_1+\delta,t_2-\delta)}, \frac{\partial w_{n1}}{\partial \nu}\Big|_{\Gamma_0'\times(t_1+\delta,t_2-\delta)}= \frac{\partial w_{n3}}{\partial \nu}\Big|_{\Gamma_0'\times(t_1+\delta,t_2-\delta)},
\Gamma_{+}^{(2)}= \left(\begin{array}{ccc}0 0 0 q^{-2}0 0 0 1 0\end{array}\right), \Gamma_{0}^{(2)}= \left(\begin{array}{ccc}-1 0 0 0 -q (q - q^{-1}) 0 0 0 q^{-2}\end{array}\right), \Gamma_{-}^{(2)}= \left(\begin{array}{ccc}0 -1 0 0 0 -1 0 0 0\end{array}\right).
I^{( 6 )}= \int d^{6}x \sqrt{- g^{( 6 )}}( e^{2 \psi - 2 \phi}( R^{( 6 )}+ 3 ( D \psi )^{2}- 8 D_{\alpha}\psi D^{\alpha}\phi + 4 ( D \phi )^{2}) - \frac{1}{1 2}H^{( 6 ) 2}) ,
\frac{e_1(L, F_1)}{|L|}= \frac{1}{\ell}\left ( \binom{r}{2}- 1 - \binom{r-\ell}{2}+1 \right ) & = \frac{1}{2\ell}\left ( r^2 - r - r^2 + 2r\ell - \ell^2 + r - \ell \right ) & = \frac{2r - \ell - 1}{2}& \geq \frac{r+2}{2}
\frac{(1-\eta)^2f_X((1-\eta)^2t)-(1+\eta)^2g_X((1+\eta)^2t)}{u_X((1+\eta)^2t)-v_X((1-\eta)^2t)}\le \frac{f_{U}(t)-g_{U}(t)}{u_{U}(t)-v_{U}(t)}= \rho_{U,n}(t).
Q_t\phi-Q_t\psi =\frac{\phi-\psi}{1+tA'_{\phi}}\circ \Phi^{-1}_{t,\phi}(x)+ &\quad+\left[\frac{\psi}{1+tA'_{\phi}}-\frac{\psi}{1+tA'_{\psi}}\right]\circ \Phi^{-1}_{t,\phi}(x)+ &\quad+\frac{\psi}{1+tA'_{\psi}}\circ \Phi^{-1}_{t,\phi}(x)-\frac{\psi}{1+tA'_{\psi}}\circ \Phi^{-1}_{t,\psi}(x).
\mathcal{Z}_{\zeta}^{({\tau_1},{\tau_2})}(c):=\begin{cases}-g_{{\zeta},{\tau_1},{\tau_2},c}+\chi{}(\cdot{},{\zeta})\cdot{}c B_{\zeta}({\tau_1})\cap{}C_{\zeta}({\tau_2}) -g_{{\zeta},{\tau_1},{\tau_2},c}B_{\zeta}({\tau_1})\setminus{}C_{\zeta}({\tau_2})\end{cases}
- \hat{C}^{( 1 )}( 0 ) \hat{C}^{( 2 )}( 0 ) = - C^{( 1 )}( - T / 2 ) C^{( 2 )}( - T / 2 ) = - C^{( 1 )}( \rho_{0}^{( 1 )}) C^{( 2 )}( \rho_{0}^{( 1 )}) ,
\begin{aligned}C(X-U_2^nX U_1^{-n})^\ast C=C\left(\sum_{m=0}^{n-1}U_2^m K U_1^{-m-1}\right)^\ast C = \sum_{m=0}^{n-1}CU_1^{m+1}K^\ast U_2^{-m}C=\sum_{m=0}^{n-1}U_2^{-m-1}CK^\ast CU_1^m = -\sum_{m=0}^{n-1}U_2^{-m-1}KU_1^m=X-U_2^{-n}X U_1^n,\end{aligned}
t^{|\lambda|}\frac{P_{\lambda}\big(x/t, t^{N-2}, \dots, t, 1; q, t\big)}{P_{\lambda}\big(t^{N-1}, \dots, t, 1; q, t\big)}= \frac{P_{\lambda}\big(x, t^{N-1}, t^{N-2}, \dots, t; q, t\big)}{P_{\lambda}\big(t^{N-1}, t^{N-2}, \dots, 1; q, t\big)}= \frac{P_{\lambda}\big(x, t, t^2, \dots, t^{N-1}; q, t\big)}{P_{\lambda}\big(1, t, t^2, \dots, t^{N-1}; q, t\big)},
\sum_{b=0}^a p^{(a-b)b}= 2 \bigg( p^{c(c+1)}+ \sum_{b=0}^{c-1}p^{(2c+1-b)b}\bigg) &= 2 \bigg( p^{c(c+1)}+ O\bigg( \sum_{b=0}^{c-1}p^{(c+\frac12)^2 - (c+\frac12-b)}\bigg) \bigg) = 2p^{c(c+1)}+ O\big( p^{(c+\frac12)^2-\frac12}\big).
\epsilon_{R P A}({\bf{q}},{\tilde{\omega}}) = 1 + \frac{v_{{\bf{q}}}}{V}\sum_{{\bf{k}}}\frac{n_{F}({\bf{k}}+{\bf{q}}/ 2 ) - n_{F}({\bf{k}}-{\bf{q}}/ 2 )}{{\tilde{\omega}}- \frac{{\bf{k . q}}}{m}}
p\left({d_{pF}^{}\leq{\delta^{}}}\right) = p\left({d_{pF}^2 \leq{\delta^2}}\right) = \int{\left({n - 1}\right){{\left({d_{pF}^2}\right)}^{n - 2}}d\left({d_{pF}^2}\right)}={\left( \delta \right)^{2n - 2}}.
\begin{aligned}u_k(x,t)\leq 1+ \frac{((|x|-\lambda^{-4/\alpha})_+^{\epsilon}-1)_+}{\lambda^{2(k_0+1)}}\leq 1+((|x|-\lambda^{-4/\alpha})_+^{\alpha/4}-1)_+=1+\psi_\lambda(x). \end{aligned}
\frac{c'_{1(r-1)}p^{m_{rr}-m_{1(r-1)}-m_{1r}}}{\prod_{k=2}^{r-1}p^{m_{k(r-1)}}}= \frac{c_{1(r-1)}p^{m_{rr}-m_{1(r-1)}-m_{1r}}}{\prod_{k=2}^{r-1}p^{m_{k(r-1)}}}+ \sum_{j=2}^r u_{jr}R_{2j}.
\exp\,\left( d_{1}+d_{2}+d_{3}\right) X.\exp\,\left( d_{1}+d_{2}+d_{3}\right) Y & =\exp\,\left( d_{1}+d_{2}+d_{3}\right) \left( X+Y\right) +\frac{1}{2}\left( d_{1}+d_{2}+d_{3}\right)^{2}\left[ X,Y\right] + & \frac{1}{12}\left( d_{1}+d_{2}+d_{3}\right)^{3}\left[ X-Y,\left[X,Y\right] \right]
\int_{\Omega}\frac{\partial\varphi}{\partial t}d\Omega=-\int_{\Omega}\nabla\cdot(\varphi u)d\Omega-\int_{\Omega}sd\Omega\Rightarrow\int_{\Omega}(\frac{\partial\varphi}{\partial t}+\nabla\cdot(\varphi u)+s)d\Omega=0.
d s^{2}={\frac{\Lambda^{2}}{z^{\prime 2}}}\Big \lbrack{\frac{\delta^{2}}{( z^{\prime}- \delta )^{2}}}d z^{\prime 2}+{\frac{( z^{\prime}- \delta )^{2}}{\delta^{2}}}\Big ( ( d \vec{x})^{2}- d t^{2}\Big ) \Big \rbrack
m\left(\left\{r>0 :\liminf\limits_{n \rightarrow \infty}(\inf\limits_{|z|=r}|z-z_n|^\frac{1}{n}) \leq (1-\epsilon)\right\}\right) = m\left(\limsup\limits_{n \rightarrow \infty}A_n\right) & \leq \lim\limits_{k \rightarrow \infty}m\left(\mathop{\cup}_{n \geq k}A_n\right)
|\alpha_3(t)| \leq C_1\int_0^1\lambda^{\beta}|\varphi(t\lambda)|d\lambda+C_2\int_1^{\infty}\lambda^{-1}|\varphi(t\lambda)|d\lambda &=C_1\int_0^t\lambda^{\beta}|\varphi(\lambda)|d\lambda\cdot t^{-(\beta+1)}+C_2\int_t^{\infty}\lambda^{-1}|\varphi(\lambda)|d\lambda &=o(t^{-(\beta+1)})+O(|\log(t)|) &=o(t^{-(\beta+1)}).
|k^3 - k_1^3 -\alpha(n-k_1)^3 + \alpha (n-k)^3 | & = |k-k_1|\left|\mu k^2[(1-\alpha)(1+ \mu) + 3 \alpha c_1] + 3\alpha \delta[1 + \mu - 2 c_1]k - 3 \alpha \delta^2\right| & \geq |k-k_1| \left[ (3\alpha c_1 + (1 - \alpha)(1 - \epsilon))|k| - 3 \alpha \epsilon(2c_1 + \epsilon - 1)|k| - 3 \alpha \epsilon^2 \right] \gtrsim |k_1 - k| |k|.
w_0 = \begin{cases}( \overline{2}, \overline{3}, \ldots , \overline{n+1}, 1) \textrm{if}k < n+1 \/; ( 1, \overline{2}, \overline{3}, \ldots , \overline{n+1}) \textrm{if}k = n+1 \/; \end{cases}
P_{jkl}^i = \nabla^{i}C_{kjl}-\nabla_{j}C^{i}_{kl}+ C^{i}_{kr}\nabla_{0}C^{r}_{jl}- C^{r}_{kj}\nabla_{0}C^{i}_{rl}, & Q^i_{jkl}= C^i_{lr}C_{jk}^r - C_{rk}^iC_{jl}^r,
b_i(I_m^{(k)})=\int_0^1I_m^{(k)}(x){\overline{\psi_i}}(x)dx=\begin{cases}0\ ,\ \hbox{if}\i=0\ \hbox{and}\i\geq a^k\ , \cr -\mathcal{A}\ , hbox{if}\ 1\leq i< a^k\,\end{cases}
\tau_{k+1, \ell+1}^{( \alpha ,\beta)}\tau_{k+1, \ell}^{( \alpha-1,\beta)}= \tau_{k+1, \ell}^{(\alpha,\beta)}\tau_{k+1, \ell+1}^{(\alpha-1,\beta)}+\tau_{k + 2, \ell+1}^{(\alpha-1,\beta)}\tau_{k, \ell}^{(\alpha,\beta)},
\sigma_{1}= \left( \begin{matrix}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{matrix}\right) , \quad \sigma_{2}= \left( \begin{matrix}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{matrix}\right) , \quad \sigma_{3}= \left( \begin{matrix}{{0}}&{{- i}}\\{{i}}&{{0}}\\ \end{matrix}\right) .
( \nabla_{a}E_{\alpha q}^{\underline{{\alpha}}}) E_{\underline{{\alpha}}, \beta r}={\frac{1}{4}}\hat{\Omega}_{a , b}^{~ ~ i}m_{b}^{~ c}\gamma_{c \alpha \beta}( \gamma_{i})_{q r}+{\cal D}_{a}h_{\alpha \beta}C_{q r}.
H_{0}=\frac{1}{2}\int d^{3}x\left(B_{a}^{ij}F_{ij}^{a}-A_{i}^{a}A_{a}^{i}+\left( \partial^{i}B_{0i}^{a}\right) \left( \partial_{j}B_{a}^{0j}\right) \right) \equiv\int d^{3}x\,h.
\lim_{\theta \to \infty}H_{1+\theta}^W(X)= \lim_{\theta \to \infty}\lim_{n\to\infty}\frac{1}{n}H_{1+\theta}(X^n) = \lim_{n \to \infty}\lim_{\theta \to \infty}\frac{1}{n}H_{1+\theta}(X^n) = \lim_{n \to \infty}\frac{1}{n}H_\infty(X^n) = H_\infty^W(X).
{\rm ch}[W(\Lambda_0)]=\sum_{r_1,r_2,r_3 \geq 0}\frac{q^{r_1^2+(r_2+r_3)^2+r_3^2-r_1(r_2+2 r_3)}}{(q)_{r_1}(q)_{r_2}(q)_{r_3}}.
{\displaystyle\sum\limits_{\mu_{1},\mu_{2}}}D_{\mu_{1},m_{1}}^{j_{1}}D_{\mu_{2},m_{2}}^{j_{2}}\left\vert j_{1}\mu_{1}j_{2}\mu_{2}\right\rangle =\sum_{j}C_{m_{1},m_{2},m}^{j_{1},j_{2},j}{\displaystyle\sum\limits_{\mu}}D_{\mu,m}^{j}\left\vert (j_{1}j_{2})j\mu\right\rangle
\mathcal{G}{\rm lct}_{d,p,\Lambda}\coloneqq\{{\rm glct}(K_X+B+M\mid P+N) \mid B+M\in \mathcal{G}\mathcal{B}(X)_{p,\Lambda}, P+N \in \mathcal{G}\mathcal{B}(X)_{p,\Lambda}, \dim X =d\}
T_{l}= \frac{1}{8}\left( \begin{array}{c c c c c}{0}&{- 1}&{1}&{- 2}&{2}\\{1}&{- 1}&{0}&{1}&{- 1}\\{2}&{- 3}&{1}&{0}&{0}\\ \end{array}\! \right)^{\! \! \! \sim},
\inf_{\Vert \tilde{D}\Sigma^{1/2}\Vert_F=1}\: Q_{2\xi}(\tilde{D}) \ge 1-\frac{4 \xi}{\sqrt{\pi}}\left(\frac{e}{2}\right)^{\frac{1}4}\sigma_{\max}\left(\Sigma^{1/2}\right)\: \sqrt{t}.
m_\alpha(x|\mu) = \mu \int_0^\infty f_\alpha(x|y) \, e^{-y/2}\,I_\mu(y/2) \frac{dy}{y}\rho_\alpha(x) = x \int_0^\infty f_\alpha(x|y) \, e^{-y/2}\,I_0(y/2) \frac{dy}{y}
\mathcal G_{1,(1;3)}^\ast(\tau)=- \frac{{\rm i}\pi}{3}-2\pi{\rm i}\sum_{n\ge 1}\frac{q^{n}-q^{2n}}{1-q^{3n}}, \mathcal G_{2,(1;3)}^\ast(\tau)=- \frac{\pi}{3\Im(\tau)}-(2\pi)^2\sum_{n\ge 1}n\frac{q^{n}+q^{2n}}{1-q^{3n}},
\left({\frac{1}{2}}p^{2}+{\frac{1}{2}}\sum_{\rho \in \Delta_{+}}g_{| \rho |}( g_{| \rho |}- 1 ) | \rho |^{2}V ( \rho \cdot q ) + ({\frac{\omega^{2}}{2}}q^{2}) \right) e^{W}={\cal E}_{0}e^{W}.
\frac{\partial^2 \lambda}{\partial t^2}- \nabla \cdot (a \nabla \lambda) = - (u - \tilde{u}) z_{\delta}, ~ x \in S_T, \lambda(\cdot, T) = \frac{\partial \lambda}{\partial t}(\cdot, T) = 0, \partial_{n}\lambda =0,~\mbox{on}~ \Omega_T\setminus S_T.
\lambda_l(t) = \gamma_l t^2 + \mu_l t^3 + \dots, \gamma_l \ge 0, \; \mu_l \in \mathbb{R}, l = 1, \dots, n, \varphi_l (t) = \omega_l + t \psi_l^{(1)}+ \dots, l = 1, \dots, n.
\left|z\right|^{K}\prod_{n=0}^{\infty}\left(1+r_{n}\right)^{\left(1-K\right)2^{1-n}}\leq\left|f\left(z\right)-f\left(0\right)\right|\leq\left|z\right|^{\frac{1}{K}}\prod_{n=0}^{\infty}\left(1+r_{n}\right)^{\left(1-\frac{1}{K}\right)2^{1-n}}
\varphi^*_{N, \sigma, v}=\varphi_{N, v}+ \sqrt{\frac{\|\sigma\|_2^2}{\log N}}\varphi'_{N, v}\mbox{for}v\in V_N\,, \mbox{and}M^*_{N, \sigma}= \max_{v\in V_N}\varphi^*_{N, \sigma, v}\,.
\ll\lambda^2_{max}\sum_{\substack{q<W^\prime R^2 (q,MP_f)=1}}\mu(q)^2\tau_{3k}(q)E(X,qM)+\lambda^2_{max}\sum_{\substack{q<W^\prime R^2 (q,MP_f)=1}}\mu(q)^2\tau_{3k}(q).
I_{1,1}^d=\iota(u_{xy}^{m,n})=\frac{u_{m+1,n+1}\, u_{m,n}-u_{m+1,n}\, u_{m,n+1}}{u_{m,n}\, u_{m+1,n}\, u_{m,n+1}\,\Delta x_m\,\delta y_n}= \frac{u_{m,n}\, u_{xy}^{m,n}- u_x^{m,n}\, u_y^{m,n}}{u_{m,n}\, u_{m+1,n}\, u_{m,n+1}}.