harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\bar{H}(\bar{g},\hat{g},S,y,\zeta ,w)=\frac 1{|S|}\sum_{(x,\kappa )\in S}(\bar{g}_{y,w}(\kappa )\zeta )^{|S|}\prod_{(x,\kappa )\in\bar{\eta}_{y,w},(x,\kappa )\notin S}((1-\zeta )\bar{g}(x,\kappa )).
\log \frac{1-\eta}{1-s}= \log \left( 1 + \frac{s(1-p)}{(1-s)(1-ps)}\right) =\left(1+o(1)\right) s (1-p) \frac{s(1-\eta)}{\eta(1-s)}= \frac{(1-\eta) (1-ps)}{p (1-s)^2}=\frac{1+o(1)}{p}.
\sum_{k=0}^{\lfloor n/2 \rfloor}\frac{2^k}{k!(n-2k)!}|a_n|^n \prod_{i<j}|z_i-z_j| \prod_{k=0}^{n}g(a_i) d\ell_E(a_n)d\ell_{n,k}(z_1,\dots,z_n) = \sum_{k=0}^{\lfloor n/2 \rfloor}\frac{2^k}{k!(n-2k)!}p_{n,k}(z_1,\dots,z_n,a_n)d\ell_E(a_n)d\ell_{n,k}(z_1,\dots,z_n).
\biggl<e^{S_4(\phi_s,\phi_f)}\biggr>_{0f}=\exp\biggl[\biggl<S_4(\phi_s,\phi_f)\biggr>_{0f}+\frac{1}{2}\biggl(\biggl<S_4^2(\phi_s,\phi_f)\biggr>_{0f}-\biggl<S_4(\phi_s,\phi_f)\biggr>^2_{0f}\biggr)+\cdot\cdot\cdot\biggr].
\chi^2(P,Q) = \sum_{a \in \mathcal{A}}\frac{P(a)^2}{Q(a)}- 1 & = \sum_{a \in \mathcal{A}}P(a) \, \exp\left( \log \frac{P(a)}{Q(a)}\right) - 1 & \geq \exp\left(\sum_{a \in \mathcal{A}}P(a) \, \log \frac{P(a)}{Q(a)}\right) - 1 begin{align*}0.1cm] = \exp\bigl( D(P \| Q) \bigr) - 1.
\left[ J^{\alpha \beta}( x ) , S_{-}^{\alpha^{\prime}\beta^{\prime}}( y ) \right] = - \delta ( x - y ) \left( \delta^{\alpha \alpha^{\prime}}S_{-}^{\beta \beta^{\prime}}( x ) + \delta^{\alpha \beta^{\prime}}S_{+}^{\beta \alpha^{\prime}}( x ) \right) ~ ~ ,
f(\eta) = C_Q\, \bigg(\frac{1-\cos\alpha_0}{1-\cos\eta}\bigg)^{(d-1)/2}\, \sqrt{\frac{1-\cos\alpha_0}{\cos\alpha_0-\cos\eta}}\,{}_2 F_1\bigg (1, \frac{d-1}{2};\frac{1}{2}; \frac{\cos\alpha_0-\cos\eta}{1-\cos\eta}\bigg) + F(\eta), \alpha_0 \leq \eta \leq \pi.
\sigma_{1}^{j - 1}\sum_{k= 0}^{s - 1}\lambda_{a_{1, 1}, k}^{j - 1}c^{1}_{1, b(a_{1, 1}, k)}+ \sum_{q_{i}\in Q}\sigma_{q_i}^{j - 1}\sum_{k= 0}^{s - 1}\lambda_{a_{1, 1}, k}^{j - 1}c^{q_i}_{1, b(a_{1, 1}, k)}+ \sum_{i = 1}^{|V|}\sigma_{v_i}^{j - 1}\lambda_{a_{v_{i, 1}}, b_{\gamma_{i}}}^{j - 1}\mu_{v_i, 1, 1}^{(b)}= 0 j \in [r].
|u'(c_n)| = \Big | l(c_n) \int_{c_n}^{\infty}f(s) ds \Big| = (b_n +1) \Big | \int_{x_n -a_n}^{\infty}f(s) ds \Big| = n^{2 \gamma}\Big | \sum_{k \ge n}(-1)^{k+1}a_k\Big| \ge n^{2 \gamma}\, \tfrac{a_{n}}{2}= n^{2 \gamma}\, \tfrac{1}{2 n^{\gamma}}\to \infty,
[-] \circ ( L\otimes [-] ) +[-] \circ ( L\otimes [-] ) \circ (L\otimes \overline{\gamma}_{L,L}) \overline{a}_{L,L,L}(\overline{\gamma}_{L,L}\otimes L) \overline{a}_{L,L,L}^{-1}\qquad{}+[-] \circ ( L\otimes [-] ) \overline{a}_{L,L,L}( \overline{\gamma}_{L,L}\otimes L) \overline{a}_{L,L,L}^{-1}( L\otimes \overline{\gamma}_{L,L}) =0,
\gamma_{0 1}( Z ) = \Phi_{0}( Z ) \Phi_{1}( Z )^{- 1}= - \delta^{- 1}\alpha \beta \gamma^{- 1}= - b_{+}^{- 1}b_{-}= - ( ( b_{3}- 1 ) b_{3})^{- 1}b_{-}^{2}
{}^{\Psi}\Phi_{p}^{(\alpha,\beta)}(b;c;z) =\sum_{n=0}^{\infty}\frac{{}^{\Psi}B_{p}^{(\alpha,\beta)}(b+n,c-b)}{B(b,c-b)}\frac{z^{n}}{n!}&=\frac{1}{B(b,c-b)}\int_{0}^{1}t^{b-1}(1-t)^{c-b-1}\exp(zt){}_{0}\Psi_{1}\left( \alpha,\beta;-\frac{p}{t(1-t)}\right)dt.
\begin{array}{c}{G_{\rho \sigma \nu}^{a b c d}= \beta_{1}( G_{\rho \sigma \nu}^{( 1 )}+ G_{\rho \sigma \nu}^{( 2 )}) - \beta_{2}( G_{\rho \sigma \nu}^{( 1 )}+ G_{\rho \sigma \nu}^{( 3 )})}\\{+ \beta_{3}( G_{\rho \sigma \nu}^{( 3 )}- G_{\rho \sigma \nu}^{( 2 )}) = 0}\\ \end{array}
r_{\varepsilon,\sigma}^2(\mathcal{E}_a,\Psi_{D,M},\beta) \leq \tilde{C}(\alpha,\beta) \varepsilon^2 \sqrt{\sum_{j=1}^{D \wedge M_1}b_j^{-4}}+ (7+4\sqrt{x_{\alpha/2}})\left[\sigma^2 \ln^{3/2}(1/\sigma) \vee a^{-2}_{D \wedge M_0}\right],
\hat{T}( w ) \hat{T}( z ) = \frac{\hat{c}/ 2}{( z - w )^{4}}+ \frac{2 \hat{T}( z )}{( z - w )^{2}}+ \frac{2 \partial_{z}\hat{T}( z ) - F ( z )}{w - z}+ O ( ( z - w )^{0}) \hspace{. 3 i n}.
\begin{array}{lcl}\{p^{n}, f\}_{\theta}= \sum_{k=0}{\theta}^{2k}f^{(2k+1)}p^{n-2k-1}\{p^{-n}, f\}_{\theta}= -\sum_{k=0}{\theta}^{2k}c^{2k+1}_{2k+n}f^{(2k+1)}p^{-2k-n-1}\end{array}
\Phi_{t+1}^{\pm}(x_1,x_2, \dots, x_k) - \Phi_{t+1}^{\pm}(x_2,\dots,x_k) &= x_1 \big( \Phi_t^{\pm}(x_1,x_2, \dots, x_k) - q^{\mp 2}\Phi_t^{\pm}(x_2,\dots,x_k) \big).
V(r)=\frac{1}{16}p_1^4r^2+\left((\alpha+1)(\alpha-1)+\frac{3}{4}-\frac{(D-1)(D-3)}{4}\right)\frac{1}{r^2}+\frac{4p_1^2}{(p_1^2r^2+4\alpha)}-\frac{32p_1^2\alpha}{(p_1^2r^2+4\alpha)^2}
\Gamma_{+}^{(3)}= \left(\begin{array}{ccc}0 0 q^2\end{array}\right)^T,\Gamma_{0}^{(3)}= \left(\begin{array}{ccc}0 - q^2 0\end{array}\right)^T,\Gamma_{-}^{(3)}= \left(\begin{array}{ccc}q^4 0 0\end{array}\right)^T.
\int_{0}^{1}\frac{\vert \phi^{\prime}\cos\phi(t)\vert}{\sin\phi(t)}\mathrm{d}t =\int_{0}^{t_{0}}\frac{\phi^{\prime}\cos\phi(t)}{\sin\phi(t)}\mathrm{d}t-\int_{t_{0}}^{1}\frac{\phi^{\prime}\cos\phi(t)}{\sin\phi(t)}\mathrm{d}t =-\log\sin\phi_{0}-\log\sin\phi_{1},
f(x) := a(x) - \pi_{l}(g,h_{1},h_{2})(x) & = a(x) - (h_{2}T_{2}^{*}(h_{1},g)(x) - gT(h_{1},h_{2})(x)) &= \underbrace{a(x) \Big( \frac{T_{2}^{*}(h_{1},g)(x_{0}) - T_{2}^{*}(h_{1},g)(x)}{T_{2}^{*}(h_{1},g)(x_{0})}\Big)}_{W_{1}(x)}- \underbrace{g(x) T(h_{1},h_{2})(x)}_{W_{2}(x)},
\mu (\overset{h_{N}-1}{\underset{i=0}{\cup}}(T^{q})^{i}(A^{*}_{N})) >(1-\frac{(q-1)(q-2)}{2})\frac{1}{b^{N-1}}\mu(\overset{h_{N}-1}{\underset{i=0}{\cup}}T^{i}(B^{*}_{N})) =(1-\frac{(q-1)(q-2)}{2}\frac{1}{b^{N-1}})h_{N}\mu (B^{*}_{N})
\|\dot r^*_{i,N}(t)\|\le \sqrt{d}L_T\left(2\|r^*_{i,N}(t)\|+\frac{1}{N}\sum_{j=1}^N\|r^*_{j,N}(t)\|+\sum_{k=1}^m\|q^*_{k,N}(t)\|+1+L_T\right)
\bar{g}_{\mu \nu}= \mathrm{d i a g . \}( - 1 , e^{2 \sqrt{\bar{\Lambda}}t}, e^{2 \sqrt{\bar{\Lambda}}t}, e^{2 \sqrt{\bar{\Lambda}}t}) , \mathrm{( d S_4 b a c k g r o u n d , \bar{\Lambda}> 0 )}
\nu_{2j+1}^{k}= \frac{1}{\Gamma(2k-2j)}\sum_{p=1}^{j+1}\frac{\Gamma(2k-2p+1)}{\Gamma(2j-2p+3)}\frac{B_{2j-2p+2}}{4^{p}}, 0 \leq j \leq k-2,
y - \frac{a}{2}z^{r}= ( s_{N_{f}- 2}t_{N_{f}- 2}z_{N_{f}- 2})^{r - 1}( s_{N_{f}- 2}t_{N_{f}- 2}- \frac{a}{2}s_{N_{f}- 2}t_{N_{f}- 2}z_{N_{f}- 2}) .
\int_0^\infty (\alpha-\alpha^2)^n \:\Box^n V|_{\alpha p}\: d\alpha = \int_0^\infty \alpha^n \:\Box^n V|_{\alpha p}\: d\alpha + \O \Big( \frac{1}{\varepsilon \underline{\omega}}\Big) &= \frac{1}{\underline{\omega}^{n+1}}\int_0^\infty \alpha^n \:\Box^n V|_{\alpha \hat{p}}\: d\alpha + \O \Big( \frac{1}{\varepsilon \underline{\omega}}\Big) \:,
\mu_b(w_2^{\ell_2}|w_1,u^{\ell_1})=\P(Z_{\ell_1+g+2}^{\ell_1+g+\ell_2}=w_2^{\ell_2}|Z_{\ell_1+g+1}=w_1,Z^{\ell_1}=u^{\ell_1}),
\mathcal{A}( Y , \Delta \theta_{j}) = \frac{V_{p}L^{4 - n}F ( \Delta \theta_{j})}{2^{p - 2}( 2 \pi^{2}\alpha^{\prime})^{( p + 1 - n ) / 2}}\sum_{\omega_{k}\in Z}\int_{0}^{\infty}t^{- \frac{p + n - 5}{2}}\exp^{- \frac{t \sum_{k}( Y_{k}+ 2 \pi \omega_{k}R )^{2}}{2 \pi^{2}\alpha^{\prime}}}d t \, ,
\sup_{t>0}|K_1(t,x,y)| \lesssim\begin{cases}e^{-|x|^2}, \mbox{if $\langle x,y\rangle \le 0$},{|x+y|}^{n}\exp{\left( \frac{|y|^2-|x|^2}{2}-\frac{|x-y||x+y|}{2}\right)}, \mbox{if $\langle x,y\rangle>0$}.\end{cases}
\mathbb{E}^{\mathbb{P}}\left[\Pi\left(t_{i},T_{j}\right)\left|\mathcal{F}_{0}\right.\right]=\mathbb{E}^{\mathbb{P}}\left[A\left(t_{i},T_{j}\right)\left|\mathcal{F}_{0}\right.\right]+M\left(t_{0},T_{j}\right).
|\xi+\eta-\rho|^{3/2}-|\xi-\rho|^{3/2}-|\eta|^{3/2}- (|\xi|^{3/2}-|\xi-\rho|^{3/2}-|\rho|^{3/2})= O \Big( \frac{|\xi-\rho|^2 + |\rho-\eta|^2}{|\eta|^{1/2}+ |\xi|^{1/2}}\Big) .
S_{, i j}^{G}= -{\cal G}_{i j}^{- 1}+ \left\{\begin{array}{c}{k}\\{i \j}\\ \end{array}\right\}{\cal E}_{k}+ \frac{\delta \chi^{\alpha}}{\delta \Phi^{i}}c_{\alpha \beta}\frac{\delta \chi^{\beta}}{\delta \Phi^{j}}.
\exp(-\sqrt{2}\varepsilon^{-1}|d_{\ell+1}(y)|) = \exp(-\sqrt{2}\varepsilon^{-1}f_{\ell,\ell+1}(y)) + O(\varepsilon^{\tfrac{1}{3}}) \exp(-\sqrt{2}\varepsilon^{-1}D_{\ell}(y)).
\begin{array}{l}{{{[}U_{a}, U^{b}{]}= S_{a}^{b}\in g^{0}}}\\{{\}}\\{{{[}U_{a}, U_{b}{]}= K_{a b}\in g^{+ 2}}}\\{{\}}\\{{{[}U^{a}, U^{b}{]}= K^{a b}\in g^{- 2}}}\\{{\}}\\{{{[}S_{a}^{b}, U_{c}{]}= U_{( a b c )}\in g^{+ 1}}}\\ \end{array}
I_\pm(\epsilon,R) =\frac{1}{2}{\rm Tr}\; SD_a(P_\pm-D_aP_\pm D_a)(P_\pm-D_aP_\pm D_a)^{2n+1+2\epsilon}\chi_R^a =\frac{1}{2}{\rm Tr}\; SD_aP_\pm(P_\pm-D_aP_\pm D_a)^{2n+1+2\epsilon}\chi_R^a -\frac{1}{2}{\rm Tr}\; SP_\pm D_a(P_\pm-D_aP_\pm D_a)^{2n+1+2\epsilon}\chi_R^a,
\sum_{d\lvert n}\mu(d)\lfloor \frac{n}{2ad}\rfloor =\sum_{d\lvert n}\mu(d)(\frac{n}{2ad}-\frac{1}{2a})=\sum_{d\lvert n}\mu(d)(\frac{n}{2ad})-\sum_{d\lvert n}\mu(d)(\frac{1}{2a}) &=\frac{1}{2a}\varphi(n) -\frac{1}{2a}\sum_{d\lvert n}\mu(d)=\frac{1}{2a}\varphi(n),
u^{n+2}_j = u^n_j - 2 a \Delta t (u_x)^n_j + a^2 \theta_o \Delta t^2 (u_{xx})^n_j - a \Delta t^2 (u_{xt})^n_j &+ a^2 \theta_e \Delta t^2 (u_{xx})^n_j + O(\Delta t^3) +O(\Delta x^3).
R_{W}^{2}={\frac{\int d^{3}x < \! \psi | T_{0 0}| \psi \! > r^{2}}{\int d^{3}x < \! \psi | T_{0 0}| \psi \! > r^{2}}}= R_{2}^{2}+ \Re \left({\frac{2 C}{E_{2}}}{\frac{\alpha}{\beta}}\right) .
P\left(\mbox{there exists}\,s\in [t,t+\Delta t]\cap S(\mathbf{X}): a_s> t+\Delta t, ~b_s> T-t \right) = P\left(\mbox{there exists}\,s\in S(g): a_s> t+\Delta t, ~b_s> T-t, ~s-U\in [t,t+\Delta t]\right).
\int^{1/K_p}_{-1/K_p}dy\,u^2_0(y)w(y)\sim\left.(1-K_p|y|)^{{8(q+1)(n-1)+4(p+1)(n+1)+(p+n)^2a^2_p}\over{(p+n)^2a^2_p+4(p+1)n}}\right|^{1/K_p}_{-1/K_p}
\mu_3(x) = \frac{-1}{2\sqrt{-1}}(([a,b]-[a^{\dagger},b^{\dagger}+ij+j^{\dagger}i^{\dagger}+fg+g^{\dagger}f^{\dagger}]), & ([a',b']-[a^{\dagger},b^{\dagger}]-gf-f^{\dagger}g^{\dagger})).
\mathbb{P}_0\left(\widetilde{\tau}^{(n)}_{M_n(b)}>(1-\epsilon)t_n\right) =\int_0^\infty\lambda_n(b)e^{-\lambda_n(b)s}\mathbb{P}_0(S_n(b)>(1-\epsilon)t_n-s)ds &\le (1-e^{-\lambda_n(b)t})\mathbb{P}_0(S_n(b)>(1-\epsilon)t_n-t)+e^{-\lambda_n(b)t},
\partial_{\mu,f,x}\Psi=\frac{\bar{V}^{f(x)+\partial_{\mu,x}fd^{\mu}x}_{f(x),x}- \bar{V}^{f(x)}_{x}}{d^{\mu}x}=\frac{\bar{V}^{f(x)+\partial_{\mu,x}fd^{\mu}x-f(x)}_{f(x),x}}{d^{\mu}x}=\frac{\bar{V}^{\partial_{\mu,x}fd^{\mu}x}_{f(x),x}}{d^{\mu}x}.
\varpi\bigl(z^{\left(t\right)}\bigr)\leq\max_{i,j,p,q}\left\{\gamma_{ij}\widetilde{\gamma}_{pq}\right\}\Bigl(\sum_{i=1}^{n}x_{i}^{\:\left(t-2\right)}\Bigr)^{2}\Bigl(\sum_{j=1}^{\nu}y_{j}^{\;\left(t-2\right)}\Bigr)^{2}
\liminf_{N\rightarrow+\infty}\frac{N}{\gamma_N^2}\log P(\frac{N}{\gamma_N}(\tau_c^N-\tau_c)>x) \geq -(J_{ini}(W^{\tau_c+\delta, b}_0)+J_{dyn, \tau_c+\delta}(W^{\tau_c+\delta, b})) &=-b^2J_{contra, \tau_c+\delta}(1)
\left\Vert h_{n,t}\right\Vert =O(n^{-1}),\sup_{x\in\lbrack0,1],t\in(-\pi,\pi]}\mid h_{n,t}(x)\mid=O\left( \frac{ln\left\vert n\right\vert}{n}\right) ,(h_{n,t},e^{i(2\pi k+t)x})=0
\begin{cases}\max \left( F(\lfloor \alpha_{m,t}(3)\rfloor ), G(\lfloor \frac{t-1}{2}\rfloor ) \right), \ \ \{\alpha_{m,t}(3)\}\leq 1/2; \max \left( F(\lfloor \alpha_{m,t}(3)\rfloor +1), G(\lfloor \frac{t-1}{2}\rfloor ) \right), \ \ \{\alpha_{m,t}(3)\}> 1/2.\end{cases}
{\mathsf{A}_k}^k =\tfrac{1}{n-2}\left[g^{jk}R_{kj}-\tfrac{1}{2(n-1)}Rg^{jk}g_{kj}\right] & =\tfrac{1}{n-2}\left[R-\tfrac{n}{2(n-1)}R\right] & =\tfrac{1}{2(n-1)}R
w_{\sigma,x_0}(x) := \frac{w(\sigma (x-x_0))}{\sigma^{-\frac{n+1}{2}}\left\| w\right\|_{L^2(B_{\sigma}^{+}(x_0))}}\rightarrow w_{x_0}(x) \mbox{in}L^2(B_{1}^+) \mbox{as}\sigma \rightarrow 0.
\mbox{whether}a)\mbox{}u(t,x,0)=u(t,x,L)=0, \mbox{or}b)\mbox{}u_y(t,x,0)=u_y(t,x,L)=0, \mbox{or}c)\mbox{}u(t,x,0)=u_y(t,x,L)=0, \mbox{or}d)\mbox{}u \mbox{is an $L$-periodic function with respect to $y$.}
f_{T_n, \underline{\chi}_n}(e_{\psi_n}g_{n_1,\cdots, n_s}(c_1,\cdots,c_s)e_{\psi_n})=J_{\pi_{\underline{\chi}_n}^{S}}(g_{n_1,\cdots, n_s}(c_1,\cdots,c_s)).
A'(x,\xi)\tilde{\tilde{\theta}}_h(x) \sharp \Big( \tilde{\theta}_h(\xi - \sum_{l=1}^3 i_l d\varphi(x))\Big) = A'(x,\xi) \tilde{\theta}_h(x)(\xi - \sum_{l=1}^3 i_l d\varphi(x)) + r'_0 \, ,
P_F(q)_{{\bf s}'{\bf s}}^{{\hat \mu}{\hat \nu}}= \sum_{N_X=0}^\infty d(N_X) \cdot{\bar \psi}_{{\bf s}'}q_{\hat \rho}\Gamma^{\hat \rho}\psi_{\bf s}{{\eta^{{\hat \mu}{\hat \nu}}}\over{q^2 + M_{N_X}^2}}.
8(s-1)^{2}{\large (}L(h){\large )}^{2}\left( x-h\right) +8(s-1)^{2}h^{2}{\large (}y-L(h){\large )}\dfrac{dy}{dx}-4(s-1){\large (}2\left( t-1\right) h^{2}+\left( s-t+2\right) h-s{\large )}\left( x-h\right) \dfrac{dy}{dx}-4(s-1){\large (}2\left( t-1\right) h^{2}+\left( s-t+2\right) h-s{\large )(}y-L(h){\large )}=0
{}[K, K]^{FN}_p = 2 \Big( (\imath_{e_i}\Psi) \wedge (\imath_{e_j}\nabla_{e_i}\Psi) + (\imath_{e_j}\imath_{e_i}\Psi) \wedge e^k \wedge \imath_{e_i}\nabla_{e_k}\Psi \Big) \otimes (e^j)^\#.
\begin{aligned}\frac{\partial}{\partial t}f +\frac{\partial}{\partial w}\mathcal{H}[f](w)f(w)+ \frac{\partial}{\partial w}\mathcal{K}[f](w)f(w)~dv=\frac{\varsigma}{2}\frac{\partial^2}{\partial w^2}(D(w)^2f(w)),\end{aligned}
\begin{pmatrix}0 -1 -1 0\end{pmatrix}\begin{pmatrix}\zeta^h 0 0 (-1)^h\zeta^{-h}\end{pmatrix}\begin{pmatrix}0 -1 -1 0\end{pmatrix}=\begin{pmatrix}(-1)^h\zeta^{-h}0 0 \zeta^{h}\end{pmatrix}=\begin{pmatrix}\zeta^{n-h}0 0 \zeta^{-n+h}\end{pmatrix}.
\mathfrak{L}_{0}= (\mathfrak{K}, a);\ \mathfrak{L}_{1}= (\mathfrak{L}_{0}, a_{1});\ \dots;\ \mathfrak{L}_{\nu}= (\mathfrak{L}_{\nu - 1}, a_{\nu});\ \dots; \mathfrak{N}_{0}= (\mathfrak{K}, b);\ \mathfrak{N}_{1}= (\mathfrak{K}, b^{2});\ \dots;\ \mathfrak{N}_{\nu}= (\mathfrak{K}, b^{2^{\nu}});\ \dots,
[0]:=(0,\,0,\,\dots,\,0) \in D_{n}^{\ast}, [1]:=(1/2,\,1/2,\,\dots,\,1/2) \in D_{n}^{\ast}, & [2]:=(0,\,\dots,\,0,\,1) \in D_{n}^{\ast}, [3]:=(1/2,\,\dots,\,1/2,\,-1/2) \in D_{n}^{\ast}.
d_{n}^i(g_n,\cdots , g_1,x) = \begin{cases}(g_n,\ldots, g_2,g_1x) \mbox{if}i=0 (g_n,\ldots, g_{i+1}g_i, \ldots, g_1,x) \mbox{if}0<i<n (g_{n-1},\ldots, g_1,x) \mbox{if}i=n \end{cases}
\underbrace{\sum_{(u, v) \in [r]\times [s]}\lambda^p_{(u-1)s + v}\cdot c\big((x_1, x_2,\ldots, x_t); (u, v)\big)}_{}&+ \underbrace{\sum_{v \in [s]}\rho \cdot c\big(\overline{(x_1, x_2,\ldots, x_t)}^{v, p}; (x_{\overline{v}^{\{t\}}}, v)\big)}_{}= 0.
\begin{array}{ll}f\cdot b = (e\cdot b)\cdot a(e\cdot b,f\cdot b) = (f\cdot a(e,f)^{-1}b)\cdot a(e\cdot b,f\cdot b) = (f\cdot b)\cdot (a(e,f)^{-1}a(e\cdot b,f\cdot b))\end{array}
\lim_{u\rightarrow0}\Bigr(u^{-n+\frac{\left\vert \alpha\right\vert}{2}}[b_{n,\alpha}]_{2n}+u^{-n+1+\frac{\left\vert \alpha\right\vert}{2}}[b_{n-1,\alpha}]_{2n}+\cdots\Bigr)=\hat g_\alpha.
h=(1-b^2)^{\frac{3}{2}}(x^{1}\dot{x}^{2}-x^{2}\dot{x}^{1})+\lambda\left( \sqrt{(\dot{x}^{1})^2+(\dot{x}^{2})^2}+b\cos(t+c) \dot{x}^1 + \sin (t+c) \dot{x}^2\right).
|-\lambda(t) \langle \left(\left(\frac{\cos(2Q_{\frac{1}{\lambda(t)}})-1}{r^{2}}\right)\left(v_{1}+v_{2}+v_{3}\right)+F_{0,2}\right)\chi_{\geq 1}(\frac{2r}{\log^{N}(t)}) \vert_{r=R\lambda(t)},\phi_{0}\rangle| &\leq C\lambda(t) \int_{0}^{\infty}|v_{4,c}(t,R\lambda(t))| \phi_{0}(R) R dR \leq \frac{C}{t^{2}\log^{3b+2N}(t)}
n_{\pi_1}=\frac{1}{2}\left([5]_q!-[2]_q![3]_q!(2+2(-1)^i+2(-1)^j+4(-1)^{i+j}) \right) &=\frac{1}{2}(q+1)^2(q^2+q+1)\left((q^2+1)(q^4+q^3+q^2+q+1)-(2+2(-1)^i+2(-1)^j+4(-1)^{i+j}) \right) &=\frac{1}{2}(q+1)^2(q^2+q+1)\left(\{q^6+q^5+2(q^4+q^3+q^2)\}+\{q-1\}-\{2(-1)^i+2(-1)^j)+4(-1)^{i+j}\}\right).
\frac{P_\nu(2\alpha-1)}{P_\nu(1-2\alpha)}f_{\nu}(\alpha)-g_\nu(\alpha)=-\frac{\pi}{2\sin(\nu\pi)}\frac{P_\nu(1-2\alpha)}{P_\nu(2\alpha-1)}-\frac{\sin^{2}(\nu\pi)\mathfrak Q_{\nu}({\alpha)}}{2\pi^2[P_\nu(1-2\alpha)]^{2}P_\nu(2\alpha-1)}.
\begin{aligned}|H_k(x,y)-H_k(x',y)| = \bigg|\int_{\Bbb R^N}\big[D_k(x,z)-D_k(x',z)\big]S_{k-M-1}(z,y)d\omega(z)\bigg| \leqslant C \int_{\{\|x-z\|\leqslant r^{2-k}\ \ \|x'-z\|\leqslant r^{2-k}\}}\frac{r^k\|x-x'\|}{V_k(x)+V_k(z)}(V_{k-M-1}(y))^{-1}d\omega(z) \leqslant Cr^k\|x-x'\|(V_{k-M-1}(y))^{-1}. \end{aligned}
\begin{array}{rclcc}\ell(r) = r^3 \, \frac{2 \, \sqrt{a^2+1}}{a^2-1}\, \frac{d}{dr}\, \left[H(r)\right]^{\frac{1-a^2}{1+a^2}}; \mbox{electric $0$-brane}p = - \frac{4}{\sqrt{a^2+1}}\, k ; \mbox{magnetic solitonic $0$-brane}\end{array}
\left( \begin{array}{c c}{A_{1 2}}&{A_{2 2}}\\{- A_{1 1}}&{- A_{1 2}}\\ \end{array}\right) = \left( \begin{array}{c c}{\partial_{\tau}a}&{\partial_{\tau}b}\\{\partial_{\tau}c}&{\partial_{\tau}d}\\ \end{array}\right) \left( \begin{array}{c c}{a}&{b}\\{c}&{d}\\ \end{array}\right)^{- 1},
m_1m_1 = 0 , m_1m_2 = m_2(m_1 \otimes 1 + 1 \otimes m_1) , m_1m_3 = m_2(1 \otimes m_2 - m_2 \otimes 1) - m_3(m_1 \otimes 1^{\otimes 2}+ 1 \otimes m_1 \otimes 1 + 1^{\otimes 2}\otimes m_1) .
{\rm Var}\left( \mathcal{N}^{c}(f_{\ell};{\cal V}(\xi_{\varepsilon ,i}))\right) =\iint_{{\cal V}(\xi_{\varepsilon ,i})\times{\cal V}(\xi_{\varepsilon ,i})}K_{2,\ell}(x,y) dxdy +\mathbb{E}\left[ \mathcal{N}^{c}(f_{\ell};{\cal V}(\xi_{\varepsilon ,i}))\right] -(\mathbb{E}\left[ \mathcal{N}^{c}(f_{\ell};{\cal V}(\xi_{\varepsilon,i}))\right])^{2}.
\mu_{\omega,\mathbf{c}}=S_{\omega,\mathbf{c}}(\Phi) =L(\Phi)+N(\Phi)+\omega Q(\Phi)+\mathbf{c}\cdot \mathbf{P}(\Phi)=\frac{1}{4-d}\left(2\omega Q(\Phi)+\mathbf{c}\cdot \mathbf{P}(\Phi)\right).
\int_{0}^{T}(u_{n}(X_{t}) -h^{*}(X_{t}))^{+}\,dA_{t}^{\nu_{n}}&= n \int_{0}^{T}(u_{n}(X_{t})-h^{*}(X_{t}))^{+}(u_{n}(X_{t})-h(X_{t}))^{-}\,dt \le 0,
M_{1}(\mathcal{NC}(A(n, p))) =(p^{3n}-p^{n})(p^{3n}-p^{n}-1)^{2}-4(p^{3n}-p^{n}-1)\dfrac{p^{n}(p^{2n}-1)(p^{2n}-p^{n}-1)}{2}~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+p^{n}(p^{2n}-1)(p^{2n}-p^{n}-1)^{2}= p^{9n}-2p^{8n}-p^{5n}+2p^{6n}= p^{8n}(p^{n}-2)+p^{5n}(2p^{n}-1)
\int_0^t \frac{1}{2}( u_r^2 , \nabla( u_r m_r)) dr = - \int_0^t \frac{1}{3}(u^3_r, \nabla m_r) dr \lesssim \| \nabla m \|_{\infty}\int_0^t \|u_r\|_{L^3}^3 dr .
r_x = \frac{x(x^2+y^2-1)}{16D},r_y=\frac{y(x^2+y^2-1)}{16D}, w =\frac{(x^2+y^2)^2}{64D}-(x^2+y^2)\left(\frac{\lambda^{-1}t^2}{4}+\frac{1}{32D}\right)+\frac{1}{4}\lambda^{-1}t^2+\frac{1}{64D},
R_{l}( r ) \stackrel{( r \rightarrow 0 )}{\sim}\frac{u_{l}( r )}{r^{({D}- 1 ) / 2}}= \mathrm{\boldmath \large \left\{\right.}\! \! \! r^{l}\mathrm{\boldmath \large ,}\! r^{- ( l + 2 \nu )}\mathrm{\boldmath \large \left. \right\}}\! \! ,
g = e^{x L_{- 1}}e^{t ( G_{- 1 / 2}^{+}+ G_{- 1 / 2}^{-})}\left( \prod_{n \geq 3}e^{\xi_{n - 1 / 2}^{\pm}G_{n - 1 / 2}^{\pm}}e^{u_{n}L_{n}}e^{\phi_{n - 1}U_{n - 1}}\right) e^{\phi_{1}U_{1}}e^{\xi^{+}G_{3 / 2}^{+}+ \xi^{-}G_{3 / 2}^{-}}e^{u_{2}L_{2}}.
\sum_{\mathcal{A}\in\mathcal{F}}\left(\frac{q}{q+1}\right)^{|\mathcal{A}|}\leqslant\sum_{\mathcal{A}\in\mathcal{F}}2^{-|\mathcal{A}|/q}\leqslant 2^{n-\binom{n}{k}/q}<1,
a_1^{35}=a_2^{25}=u_1^{26}=-u_2^{36}=-u_1^{37}=-a_4^{35}=:x_1, &a_4^{25}=a_3^{25}=-u_2^{37}=u_1^{27}=u_2^{26}=-a_1^{25}=:x_2, &u_2^{27}=-a_3^{25}=:x_3,u_1^{36}=a_2^{35}=:x_4.
e^{-\lambda_{N,s}t}\frac{\Delta_N(y)}{\Delta_N(x)}\det\left(p^{(N),s}_t(x_i,y_j)\right)^N_{i,j=1}>0 \ ,\ \forall \left(t,x,y\right)\in (0,\infty) \times \mathring{W}^N \times \mathring{W}^N,
\tilde{E}_{i}[ \eta | t ] = - \frac{2}{\bar{N}}\epsilon_{i j \rho \sigma}\dot{\eta}^{j}( t ) \int \delta \xi d s \, \omega ( \xi ( s ) ) E^{\rho}[ \xi | s ] \omega^{- 1}( \xi ( s ) ) \dot{\xi}^{\sigma}( s ) \dot{\xi}^{- 2}( s ) \delta ( \xi ( s ) - \eta ( t ) ) ,
\|g\|_{H^{-1/2}(\mathbb R^2)}^2\leq \frac{1}{4\pi}\int\limits_{\mathbb S^1\times \mathbb R}\Phi(\boldsymbol \omega, \tau)^2\,d\boldsymbol \omega\, d\tau, \Phi(\boldsymbol \omega, \tau)=\int\limits_{\boldsymbol \omega \cdot x=\tau}g(x)\, dl.
\begin{aligned}d \sqrt{\lambda_{1}}\geqslant \int_{L}\frac{| \nabla t |}{\sqrt{z(t)}}dl = \int_{- \frac{\pi}{2}}^{\frac{\pi}{2}}\frac{1}{\sqrt{z(t)}}dt \geqslant \frac{\left( \int_{- \pi / 2}^{\pi / 2}dt \right)^{3 / 2}}{\left( \int_{- \pi / 2}^{\pi / 2}z(t) dt \right)^{1 / 2}}\geqslant \left( \frac{\pi^{3}}{\int_{- \pi / 2}^{\pi / 2}z(t) dt}\right)^{1 / 2}.\end{aligned}
{1_{D(t_*^j) > \mu_4^{- 4}}}u(t) ={S_{1,\alpha}}(t - t_*^j){1_{D(t_*^j) > \mu_4^{- 3}}}u(t_*^j) +{S_{2,\alpha}}(t - t_*^j){1_{D(t_*^j) > \mu_4^{- 3}}}{\partial_t}u(t_*^j) &+ \int_{t_*^j}^t{{e^{- \alpha (t - s)}}}S_{2,\alpha}(t-s){1_{D(t_*^j) > \mu_4^{- 3}}}{\left|{u(s)}\right|^{p - 1}}u(s)ds.
\left( \tilde{M}_{\xi}^{2}( X ) \right)_{\mathrm{\scriptsize{I}}}+ \left( \tilde{M}_{\xi}^{2}( X ) \right)_{\mathrm{\scriptsize{I I}}}= M_{\xi}^{2}( X ) - \mu^{2}( \xi = \pi , \, \sigma ) .
|F_1|_{h_1,\delta_1}\leq \tilde M_1^2\varepsilon= b^2((b_1\frac{|\ln \varepsilon|}{4\pi h_1})^\tau e^{b_2e^4}R_0)^{2m^2(m+2)}\varepsilon<\varepsilon^{\frac{1}{2}}=:\epsilon_1.
\beta_{\epsilon,\alpha,C}^{\max}=\min(\tfrac{1}{2},\beta_{\epsilon,\alpha,C}^{\max})\le \beta_\epsilon < \frac{(1+\epsilon_g^{-1})\epsilon}{c_g^2\ln \tfrac{2\epsilon_g}{\epsilon}}, \forall\, 0<\epsilon<\min(1,\epsilon_g).
\frac{(q)^2_\infty}{[1/b]_\infty}\sum_{k=-\infty}^\infty\frac{(-b)^kq^{\frac{k(k+1)}{2}}}{1-bq^k}=- \sum_{n=0}^\infty\left(\frac{q^nb}{(1-bq^n)^2}+\frac{q^{n+1}/b}{(1-q^{n+1}/b)^2}\right)+ \sum_{\substack{n=-\infty n \neq 0}}^{\infty}\frac{(-1)^nnq^{n(n+1)/2}}{1-q^n}.
\sup_{D_{\bar{p}R}}\frac{f_1^{\bar{p}R/2,\bar{q}R}}{f_2^{\bar{p}R/2,\bar{q}R}}-m_{\bar{q}R}=\sup_{D_{\bar{p}R}}\frac{g}{f_2^{\bar{p}R/2,\bar{q}R}}\le c^4 \inf_{D_{\bar{p}R}}\frac{g}{f_2^{\bar{p}R/2,\bar{q}R}}&=c^4\left(\inf_{D_{\bar{p}R}}\frac{f_1^{\bar{p}R/2,\bar{q}R}}{f_2^{\bar{p}R/2,\bar{q}R}}-m_{\bar{q}R}\right),
p(N,\vec\ell;2)=\frac{(N-\ell)!\prod_j\ell_j!}{(N+t)(N-1)!}\left[\frac{(-1)^{N+\ell}\binom{N-1}{t-2}}{\binom{N+\ell}{\ell-t}}+\sum_{j=0}^{\ell-t}\frac{(-1)^j\binom{\ell-t}{j}\binom{N+j+1}{\ell}}{\binom{N+t+j}{j}}\right],
\widetilde{\psi}_{k}^{( 1^{2})}\big(x;\omega ,N,- \Vert \psi_{1}\Vert^{2}\big) = ( 1-k ) \psi_{k}(x;\omega,N) -z^{( 3N+5/2) /2}\exp ( -z/2) \frac{L_{1,k}^{N}(z) L_{1}^{N}(z)}{Q_{1}^{N}(z)},
\frac{\partial |\mathring{A}|^2}{\partial t}=2\nabla^j\left(\frac{\nabla_iH}{H^2}\right)\mathring{h}^i_j-\frac{4}{n-1}|\mathring{A}|^2-2\frac{\mathring{h}^k_i\mathring{h}^j_k\mathring{h}^i_j}{H},
\begin{cases}{\bf D}^{(z)}(t,x)~=~{\bf D}_1^{(z)}(t,x)+{\bf D}_2^{(z)}(t,x)+{\bf D}_3^{(z)}(t,x) begin{align*}2mm]{\bf D}_1^{(z)}(t,x)~=~I^{(z)}_{1}(t,x)\cdot \phi_0'(x-t),{\bf D}_2^{(z)}(t,x)~=~I^{(z)}_2(t,x)\cdot \phi_0'(x)\,, begin{align*}2mm]{\bf D}_3^{(z)}(t,x)~=~I^{(z)}_{31}(t)\cdot g_{|t|}(t-x)+I^{(z)}_{32}(t)\cdot\phi_0'(t).\end{cases}
\Phi(\mathrm{y},\tau; \mathrm{v},\mathrm{s}):=\ \begin{cases}\dfrac{\alpha}{4\pi(\tau-\mathrm{s})}\textbf{exp}\Big(-\dfrac{\alpha|\mathrm{y}-\mathrm{v}|^2}{4(\tau-\mathrm{s})}\Big) , \tau > \mathrm{s}0 ,\end{cases},
\mu\left(t,\left(\sum_{i=1}^n |x_i|^p\right)^{\frac1p}\right) \leq \left(\sum_{i=1}^n \mu\left(\frac tn, |x_i|^p\right)\right)^{\frac1p}=\left(\sum_{i=1}^n \mu^p\left(\frac tn, |x_i|\right)\right)^{\frac1p}.
{\cal C}_{k}^{\prime \prime}+ \biggl [ \frac{k^{2}}{1 + r^{2}}\biggl ( 1 + r^{2}\frac{a^{2}}{b^{2}}\biggr ) - \frac{{\cal G}^{\prime \prime}}{{\cal G}}\biggr ]{\cal C}_{k}= 0
\frac{d}{d u}\chi_{s + h}(u + h,\omega,x_0) &= J\Big(s + h,u + h,\omega, \chi_{s + h}\big(u + h,\omega,x_0\big)\Big)\, b\Big(\psi\big(s + h,u + h,\omega,\chi_{s + h}(u + h,\omega,x_0)\big)\Big) = J\Big(s ,u , \theta_h \omega, \chi_{s + h}\big(u + h,\omega,x_0\big)\Big)\, b\Big(\psi\big(s ,u ,\theta_h \omega,\chi_{s + h}(u + h,\omega,x_0)\big)\Big),
\sum_{\pi\in \mathcal{A}_B}e^{-\mu \sum_{j=1}^{n}|x_{b_j}-x_{\pi(2j)}|}\leq C_{d,\mu}^n e^{-\frac{\mu}{2\sqrt{d}}D_s(X\setminus Y_B, Y_B)}\leq C_{d,\mu}^n e^{-\frac{\mu}{2\sqrt{d}}D_s(X)}.
\Gamma = \Big\langle a_1,b_1,\ldots, a_g,b_g, e_1,\ldots, e_r, \, p_1,\ldots, p_s, h_1,\ldots,h_t \ \Big\vert\e_1^{m_1}= \ldots = e_r^{m_r}= \big(\prod_{j=1}^g [a_j,b_j]\big) e_1\cdots e_r p_1\cdots p_s h_1\cdots h_t = 1 \Big\rangle,
A_{L , i i^{\prime}}= \frac{1}{\sqrt{n n^{\prime}}}\int \frac{d z}{2 \pi}\frac{d z^{\prime}}{2 \pi}\frac{1}{z^{n + 1}( z^{\prime})^{n^{\prime}+ 1}}\left( 1 - z \right)^{\frac{n}{( e{\cal P}_{L})}( e Q )}\times
H_i(t,y,z^0,z^1,z^2,I_1,I_2;p_i)=\Big(r(t)y(t)+\sum_{k=0}^2b^k(t)^\tau z^k(t)^\tau+I_1(t)+I_2(t)\Big)p_i(t)+L_ie^{-\beta t}I_i^2(t),
x_1^{p^n}=x_2^{p^2}=x_3^{p^2}=x_4^p=1, & [x_1,x_2]=x_2^p, \; [x_1,x_3]=[x_1,x_4]=x_3^p, & [x_2,x_3]=x_1^{p^{n-1}}, \; [x_2,x_4]=x_2^p,\; [x_3,x_4]=1.
\overline{I}_{2}={(2ie\pi)^{2}\over 2}\int{d\overline{k}\over (2\pi)^{2}}\,\epsilon(\overline{k})\epsilon(\overline{k}+\overline{p})\left[n(|\overline{k}|) + n(|\overline{k}+\overline{p}|) - 2 n(|\overline{k}|)n(|\overline{k}+\overline{p}|)\right]
\partial_t \langle hw_\eta^q \rangle_{x,\eta}+ \frac{4(q-1)}{q}\langle h |\nabla w^{\frac{q}{2}}_\eta|^2\rangle_{x,\eta}& - 4\langle b\cdot \nabla w_\eta^{\frac{q}{2}},h w_\eta^{\frac{q}{2}}\rangle_{x,\eta}+ 4\theta\sum_{i=1}^d \langle b_i \nabla w_\eta^{\frac{q}{2}}, \frac{\kappa \eta_i \eta}{1+\kappa |\eta|^2}h w_\eta^{\frac{q}{2}}\rangle_{x,\eta}=0.
J_{+}^{(m)}=\sum_{\mathbf{k}\in \mathcal{S}_m}\binom{m}{\mathbf{k}}\frac{w_1^{k_1}w_2^{k_2}w_3^{k_3}}{2^{k_1+k_2+k_3-1}}\mathcal{J}_{+}^{(m,\mathbf{k})},