harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\frac{1}{G}= \frac{2^{[\frac{D+1}{2}]}N}{(4\pi)^{D/2}}\int^\infty_{1/\Lambda^2}\frac{ds}{s^{D/2-[(D-1)/2]}}e^{-s\rho^2}\prod_{k=1}^{[(D-1)/2]}eH_k \coth(eH_ks)~.
I = \int d x \left( 4 \Omega^{\prime}\rho^{\prime}+ 2 \Lambda e^{2 \rho}- 2 \Omega \psi^{' 2}- \frac{1}{2}\Omega \psi_{1}^{' 2}+ e^{- \psi_{1}- 2 \rho}\Omega^{2}A_{2}^{' 2}+ e^{\psi_{1}- 2 \rho}\Omega^{2}A^{' 2}\right)
TBV_{i}^{W}=|q^{W}_{i-1}(x_{i-\frac{1}{2}})-q^{W}_{i}(x_{i-\frac{1}{2}})|+|q^{W}_{i}(x_{i+\frac{1}{2}})-q^{W}_{i+1}(x_{i+\frac{1}{2}})|
\| P_k [A_{j,k_1}, \partial_0 A_{j,k_2}]\|_{L^2 \dot H^{-\frac12}}\lesssim \ 2^{-\frac16 (k_{max}- k_{min})}\| |D_x|^\frac16 A_{j,k_1}\|_{L^2 L^6}\| \partial_0 A_{j,k_2}\|_{L^\infty L^2}\lesssim \ 2^{-\frac16 (k_{max}- k_{min})}\|A_{j,k_1}\|_{S^1}\| A_{j,k_2}\|_{S^1}
W[X] = ig\int_0^1 d\tau A_b(X)\partial_{\tau}X^b+ \frac{(ig)^2}{2}\int_0^1 d\tau_1 \int_0^{\tau_1}d\tau_2[A_b(X(\tau_1)),A_c(X(\tau_2))] \partial_{\tau_1}X^b\partial_{\tau_2}X^c
\left| \sigma_{\infty}- \sigma_N \right| \leq C \left|A'(\sigma_{\infty}) - A'(\sigma_N ) \right| = C \left| A_N ' (\sigma_N ) - A' (\sigma_N ) \right| \overset{\eqref{e_quantitative_conv_an_der}}{\leq}C \frac{(|\sigma_N| +1)}{N}\overset{\eqref{h1}}{\leq}C \frac{|m| +1}{N}.
\lim_{\varepsilon\rightarrow 0}(\beta\varepsilon)^{n/2}\int_{\mathbb{R}^n}e^{i2\pi( x,(\alpha/\beta)^{\frac{1}{2}}(\beta\varepsilon)^{\frac{1}{2}}\eta+m)}e^{-i2\pi kx}a( x,(\alpha\varepsilon)^{\frac{1}{2}}\eta+m) e^{-\pi\varepsilon\beta|x|^2}dx &=\int_{\mathbb{T}^n}e^{i2\pi( x,m)-i2\pi kx}a(x,m)dx.
A_1 = \{(n-2)p(1-p)(1-\varepsilon_1) < |X_1|,|X_2|< (n-2)p(1-p)(1+\varepsilon_1)\}, A_2 = \{|Y| < (n-2)p^2(1+\varepsilon_2)\}, A_3 = \{C_1,C_2 < sp(1+\varepsilon_3)\},
\left|\begin{array}{rl}L^*u = 0 + \alpha_1 z^1{1}_{\mathcal{O}_{1,d}}+ \alpha_2 z^2{1}_{\mathcal{O}_{2,d}}\ \Q, &Lz^i = 0 - \dfrac{1}{\mu_i}u{1}_{\mathcal{O}_i}\ \Q, &u(0,t)=u(L,t) = 0, \ \z^i(0,t)=z^i(L,t) =0, \ \ \ (0,T), &u(T) = u^{T},\ \z^i(0)=0, \ \ \I.\end{array}\right.
\begin{aligned}\|\nabla^2 d\|_{L^2}^2 \le C_1 (\|\Delta d\|_{L^2}^2+\|d\|_{H^1}^2) &\le C_1 (\|\nabla d\|_{L^4}^4+\| \Delta d+|\nabla d|^2 d\|_{L^2}^2+\|d\|_{H^1}^2) &\le C_1 C_2 \|d-e_2\|_{L^\infty}^2\|\nabla^2 d\|_{L^2}^2 +C_1 \left(C_2\|d-e_2\|_{L^\infty}^4+\|\Delta d+|\nabla d|^2 d\|_{L^2}^2+\|d\|_{H^1}^2\right),\end{aligned}
V_{\mu, \xi}= p\sum_{j=1}^k\left((u^*_{\mu, \xi})^{p-1}- \left[\mu_j^{-\frac{n-2s}{2}}U\left(\frac{x-\xi_j}{\mu_j}\right)\right]^{p-1}\right)\eta_{j,R}+ p(1-\sum_{j=1}^k\eta_{j, R})(u^*_{\mu, \xi})^{p-1}.
\#\{\ell \in G_{p,m-w}\setminus \{0\}: e(\ell)=k\}= \#\{\ell\in G_{p,m-w}\setminus \{0\}:x^k \mid \ell\}-\#\{\ell\in G_{p,m-w}\setminus \{0\}:x^{k+1}\mid \ell\}= p^{m-w-k}-1-(p^{m-w-k-1}-1)=p^{m-w-k-1}(p-1),
\left\lbrace\begin{array}{lll}W''-{\Delta}W+\overline A_pW=0 \hbox{in}(0,+\infty) \times \Omega, \partial_\nu W + \overline B_pW = \overline D_pH \hskip1.0cm \hbox{on}(0,+\infty) \times \Gamma\end{array}\right.
I_{C_3\{1,2, 3\}}= \{\theta^1_p,\, \theta^2_p, \, \theta^1_z,\, \theta^2_z, \, \theta^3_z, \, \theta^4_z \}&= \{dy^1 - p^1dx, dy^2 - p^2dx,\, dz^1 - p^1 p^2 dx,\, dz^2 - xp^2 dx, \, &dz^3 - (y^1p^2 + xp^1p^2)\, dx, \,dz^4 - y^1p^1p^2 dx \}
\tilde{U}= \begin{pmatrix}U 0 0 1 \end{pmatrix}, \tilde{V}^\top = \begin{pmatrix}V^\top 0 0 1 \end{pmatrix}, \tilde{\Sigma}= \frac{1}{\sqrt{2\varepsilon}}\begin{pmatrix}\Sigma 0 0 \sqrt{\frac{p''}{2\varepsilon}}\end{pmatrix}.
-\frac{A(v)}{3}v^2 =-\frac{\rho_0(v)}{3}v+\mathcal{O}(v^2) &=-\frac{v}{6}(y(v)+1)-\frac{v}{3}\Big(\tfrac{1}{2}(y(v)+1)-\rho_0(v)\Big)+\mathcal{O}(v^2) &=-\frac{v}{6}(y(v)+1)+\mathcal{O}(v^2),
\aleph_{\alpha,L}\simeq \begin{dcases}\alpha L, 1< L \leq \frac{16}{3}, \frac{16}{3}\alpha^{\eta}, L = \frac{16}{3}\alpha^{-1+\eta}, \; \eta \in (0,1), \frac{1}{6\pi}\|(h_1-e_1)^{-1/2}x u_1\|^2, L \geq \frac{16}{3}\alpha^{-1}. \end{dcases}
\sqrt{\alpha}\frac{\sqrt{R-b}-\sqrt{R-a}}{\sqrt{R-b}+\sqrt{R-a}}\frac{\sqrt{b}+\nu\sqrt{a}}{\sqrt{b+Q}-\sqrt{a+Q}}=1, \frac{\nu\sqrt{ab}+\sqrt{(a+Q)(b+Q)}-Q}{2}+\sqrt{(R-a)(R-b)}=1,
\frac{u_{l,j,m}^{k+1}- u_{l,j,m}^k}{\tau}+ \frac{u_{l+1,j,m}^{k+1}- u_{l+1,j,m}^k}{\tau}- \frac{u_{l+1,j,m}^{k}- u_{l,j,m}^k}{\delta x_1}- \frac{u_{l+1,j,m}^{k+1}- u_{l,j,m}^{k+1}}{\delta x_1}= 0,
\begin{aligned}t H(L_t) = \frac{t_0^2}t \big\langle \Lambda_{t_0}, L_{t_0}\big\rangle+ 2 \frac{t_0 (t-t_0)}t \big\langle \Lambda_{t_0}, L_{t_0,t}\big\rangle+ \frac{(t-t_0)^2}t H(L_{t_0,t}) &\leq t_0 \|\Lambda_{t_0}\|_\infty + 2t_0 \|\Lambda_{t_0}\|_\infty +\frac{(t-t_0)^2}t H(L_{t_0,t})\end{aligned}
f(g(x))^{(n)}= \sum_{k_1+2k_2+\cdots+nk_n=n}\frac{n!}{k_1!1!^{k_1}\cdots k_n!n!^{k_n}}f^{(k_1+\cdots+k_n)}(g(x))\prod_{i=1}^n g^{(i)}(x)^{k_i},
\frac{B^{3}}{| B |}\left[ \prod_{a = 1}^{2}\delta \left( \frac{F^{a}}{| F |}+ \frac{B^{a}}{| B |}\right) + \prod_{a = 1}^{2}\delta \left( \frac{F^{a}}{| F |}- \frac{B^{a}}{| B |}\right) \right] ,
F_{2}=x^{2}-wx+\frac{1}{2}w^{2}+\frac{1}{2}c+\frac{1}{2}\cdot\frac{w^{5}}{d}+\frac{cw^{3}}{d}-2\frac{ew}{d}+\frac{1}{2}\cdot\frac{c^{2}w}{d}
I_1 =\int_{\lambda_2}^{\lambda^*}Q^{\frac{1}{2}}(\eta,x)d\eta=\frac{1}{x}\int_{\sqrt{4a b+i}}^{T}\sqrt{t^2-(4a b+i)^2}\; (1+o(1))d t &=\frac{T^2}{2x}-\frac{4a b+i}{4x}-\frac{4a b+i}{2x}\ln(2T)+\frac{4a b+i}{4x}\ln(4a b+i)+o\left (\frac 1 x\right ).
M_2 = -\frac{P_3(1)}{L}\sum_{w \mid q}\varphi(w) \mu(q/w) \mathop{\sum \sum}_{\substack{b \leq y_2 b \equiv \pm n (w) (bn,q)=1}}\frac{\Lambda(b) P_2[b]}{(bn)^{\frac{1}{2}}}F(n).
\begin{array}{c}{{- \frac{1}{2}\alpha^{\prime}D^{2}\tilde{T}- \frac{1}{1 2}\left( \frac{3 k}{k + 2}- 1 \right) \tilde{T}= 0}}\\{{\mathrm{w i t h ~ t h e ~ s o l u t i o n :}\qquad \tilde{T}( t , \theta ) = e^{\frac{1}{\sqrt{\alpha^{\prime}}}\tilde{p}_{0}( t - t_{0})}\frac{\sin [ ( n + 1 ) \theta ]}{( n + 1 ) \sin \theta}}}\\ \end{array}
\iota_S(y) - \iota_S(x) - \big<d,y - x\big> = -\big<d,h_\eta(\alpha_y) - h_\eta(\alpha_x)\big> = -\big<d,h_\eta(\alpha_y) - h_\eta(\alpha_x) - J_{h_\eta}(\alpha_y - \alpha_x)\big> \geq -\|d\| \cdot \gamma\|\alpha_y - \alpha_x\|^2 \geq -M \gamma C^2 \|x - y\|^2,
\sum_{j=1}^d\sum_{h=1}^{\infty}\sum_{\substack{p> \exp{Q}p^h\equiv b\,(kq)}}\frac{f_j(p)^h}{hp^{h(\sigma+it_p)}}=Y_b^\varphi(\sigma,q),b=1,\ldots,kq,\, (b,kq)=1,
I_h=-\int_{\mathbb R^2\times[0,T]}\zeta\varrho\mathbf u\cdot\nabla\, \,\Delta^{-1}\big(\, \big[\zeta\psi(\varrho)\big]_h\mathbf u- \big[\zeta\psi(\varrho)\mathbf u\big]_h\,\big)\, dxdt.
x-x_n =x-x_{n-1}-\frac{2}{a+b}S_{f, \tau}(x-x_{n-1}) &=\left(I_\mathcal{X}-\frac{2}{b+a}S_{f,\tau}\right)(x-x_{n-1}) &=\cdots=\left(I_\mathcal{X}-\frac{2}{b+a}S_{f, \tau}\right)^nx, \quad\forall x \in \mathcal{X}, \forall n \geq 1.
Q_m : \;\mathbb{P}_\mathbb{C}^{n}\setminus [\widetilde{\Lambda}_0] \rightarrow \mathbb{P}_\mathbb{C}^{m}z=[z_1,z_2,...,z_{n+1}] \mapsto Q_m(z)=[z_1,\cdots,z_m,z_{n+1}].
\mathrm{Re}((z-\xi)^2) = (\mathrm{Re}(z)-\xi)^2 - \Im(z)^2 \ge (|\mathrm{Re}(z)| - L_2)^2 - (L_1-|\mathrm{Re}(z)|)^2 &\ge (L_2-L_1)(L_1 + L_2 - 2 |\mathrm{Re}(z)|) & \ge (L_2 - L_1)^2 >0
\psi_{\tau}= (\psi^{a}_{\mu}\, , \, \psi , \, \psi^{a}) \quad , \quad \gamma_{\tau}= ( \gamma^{a}_{\mu}\, , \, \gamma , \, \gamma^{a}) \, , \, \, \xi = ( \gamma_{\tau}\, ,\, \omega^{a}) .
\frac{\pi^{2}R^{6}c ( R^{2})}{2 4}\int \mathrm{d}^{4}x A \raisebox{- 4 p t}{\stackrel{\displaystyle{( x )}}{\scriptstyle{\mu a}}}\left( \partial^{2}A \raisebox{- 4 p t}{\stackrel{\displaystyle{( x )}}{\scriptstyle{\mu a}}}- \partial_{\mu}\partial_{\nu}A \raisebox{- 4 p t}{\stackrel{\displaystyle{( x )}}{\scriptstyle{\nu a}}}\right)
{\cal Z}= 1 + \sum_{N = 1}^{\infty}\frac{\zeta^{N}}{N !}\left( \prod_{a = 1}^{N}\int d^{3}z_{a}\sum_{\alpha_{a}= \pm 1 , \pm 2 , \pm 3}^{}\right) \exp \left[ - \frac{g_{m}^{2}}{4 \pi}\sum_{a < b}^{}\frac{\vec{q}_{\alpha_{a}}\vec{q}_{\alpha_{b}}}{\left| \vec{z}_{a}- \vec{z}_{b}\right|}\right] .
\left. \frac{\partial^{2}}{\partial g_{k}\partial a_{\ell}}\left\langle \frac 1 N \, \mathrm{t r}\, \frac{1}{z - \phi}\right\rangle \right|_{g_{j}= 0}= 0 \ \ \mathrm{~ i f ~}\ \ \ell + 2 > k .
\beta = \beta^{\prime}/ ( g_{Y M}^{2}N )^{1 / 3}\quad R = R^{\prime}( g_{Y M}^{2}N )^{1 / 3}\quad \Delta_{l}^{2}= ( g_{Y M}^{2}N )^{1 / 3}\Delta_{l}^{\prime}{}^{2}/ N .
( q^{2 t_{j}}- 1 ) \left\langle \lambda_{j}^{\vee}, ( \sigma_{q})^{x}\gamma_{i}\right\rangle = ( q^{2 t_{i}}- 1 ) \, \left\langle \lambda_{i}^{\vee}, ( \sigma_{q})^{x}\gamma_{j}\right\rangle \, \, .
\Big\Vert \nabla \mathrm{H}\Big\Vert_{\mathbb{L}^4(\Omega)}= \delta^{-\frac{1}{2}}\Big\Vert \nabla \Tilde{\mathrm{H}}\Big\Vert_{\mathbb{L}^4(\mathrm{B})}= \mathcal{O}\Big(\delta^{-\frac{1}{2}}\Big).
F(x^k) - \eta^{k+1}\le F(x^k) - \eta^k - \frac{1-\beta}{2(N-1)}\bar{\mu}_k\big(F(x^k) - \eta^k\big) = \biggl(1-\frac{1-\beta}{2(N-1)}\bar{\mu}_k\biggr)\big(F(x^k) - \eta^k\big).
y^{2}~ t^{2}~ + ~ t ~ ( y^{2}- a_{1}^{2}) ~ ( y^{2}- a_{2}^{2}) ~ \ldots ~ ( y^{2}- a_{n}^{2}) ~ + ~ ~ \Lambda_{S U}^{2 b_{0}^{S U}}~ \Lambda_{S O}^{b_{0}^{S O}}~ y^{2}~ = ~ 0 ~ .
\sum_{n = 0}^{\infty}\frac{1}{y^{2}+ n^{2}}= \frac{\pi}{2 y}\left( \coth \pi y + \frac{1}{\pi y}\right) \simeq \frac{\pi}{2 y}+ \frac{1}{2 y^{2}}\equiv \bar{S}_{1}( y ) \,{.}
\left[{\frac{d n ( \omega | \mu )}{d \omega}}\right] ={\frac{1}{( 4 \pi )^{2}\kappa}}\int_{\Sigma}\left[ 2 b + a \left({\frac{\omega^{2}}{\kappa^{2}}}{\cal P}+ 2 \left( \frac 1 6 - \xi \right) R \right) \right] .
T_2(t,c) = \begin{cases}(1-c) \textup{if}\frac{1}{2}\leq t \leq 1, 0 \leq c < \frac{1}{2}t \textup{otherwise}\end{cases}C_2(t,c) = \begin{cases}(1-t) \textup{if}0 \leq t \leq \frac{1}{2}, \frac{1}{2}\leq c \leq 1 c \textup{otherwise}\end{cases},
\phi(g) = \sum_\Lambda \ \phi^{\Lambda}_{\alpha\beta}\D^{(\Lambda)}_{\alpha\beta}(g) = \int\limits_{SO(3)}dh\ \phi(gh) = \sum_\Lambda \ \int\limits_{SO(3)}dh\ \phi^{\Lambda}_{\alpha\beta}\D^{(\Lambda)}_{\alpha\gamma}(g)\D^{(\Lambda)}_{\gamma\beta}(h).
\mathsf{n}(r) r^j - \mathsf{n}(r) = \mathsf{n}(r) (r-1) \sum_{i=0}^{j-1}r^i = r \big(\mathsf{n}(r) - \mathsf{d}(r)\big) \sum_{i=0}^{j-1}r^i = \big(\mathsf{n}(r) - \mathsf{d}(r)\big) \sum_{i=0}^{j-1}r^{i+1}\in M_r.
V =\frac{1}{\frac{\pi^2}{3}-2}\left\{\int_0^1\frac{6 B^2(t)}{t(1-t)}\mathrm{d}t- \left[\int_0^1 6 B(t) \mathrm{d}t\right]^2\right\}&\ -\left[\frac{1}{\frac{\pi^2}{3}-2}\int_0^1 6 B(t) \ln \left(\frac{t}{1-t}\right)\mathrm{d}t\right]^2,
G_{23}(q) = q^{-23}-q + 263 q^{25}-3400 q^{49}+ 23374 q^{73}+ \cdots G_{47}(q) = q^{-47}+2q-2781q^{25}+ 102060q^{49}+ \cdots G_{71}(q) = q^{-71}-3q+ 17960q^{25}+\cdots
\frac{(q^d;q^d)_{n-1}^d}{(1-q)^{dn-d}}\sum_{k=0}^{n-1}\frac{(q^{d+1};q^d)_k^{d-2}(q,q^{1-d};q^d)_kq^{dk}}{(q^d;q^d)_k^d}\equiv 0\pmod{[n]^2}.
\left\|F_1 - G_1\right\|_{L_1}= \left\|\frac{1}{M_F}F - \frac{1}{M_G}G\right\|_{L_1}= \left\|\frac{M_G - M_F}{M_G}\frac{1}{M_F}\, F + \frac{1}{M_G}\, (F - G)\right\|_{L_1}\le \left\|M_G - M_F\right\|_{L_1}+ \left\|F - G\right\|_{L_1}\le 2 \left\|F - G\right\|_{L_1}
\begin{aligned}\biggr{|}\int_{\partial\Omega\cap B_{\eta}}\nabla c\cdot\nu c\psi^{2}\biggr{|}=\biggr{|}\int_{\partial\Omega\cap B_{\eta}}(\gamma-c) c\psi^{2}\biggr{|}&\le \gamma^{2}\int_{\partial\Omega\cap B_{\eta}}\psi^{2}&\le \gamma^{2}\int_{\partial\Omega}\psi^{2}.\end{aligned}
B' =(\I(X_1;Y_1|U_1),\ \min\{I(U_1;Y_2|U_2),\I(U_1,U_2;Y_2)- I(U_2;Y_3)\},\I(X_2;Y_3) \ ), C' =(\I(X_1;Y_1|U_1)+ \min\{I(U_1;Y_2|U_2),\I(U_1,U_2;Y_2)-I(U_2;Y_3)\},\ 0, \ \I(X_2;Y_3)\ ).
L_{\alpha}(x)=\frac{x^{1/2-\alpha}}{(1-2\alpha)\zeta(1/2)}+\sum_{|\gamma|<T_{v}}\frac{\zeta(2\rho)}{\zeta^{\prime}(\rho)}\cdot\frac{x^{\rho-\alpha}}{(\rho-\alpha)}+E_{\alpha}(x)+R_{\alpha}(x,T_{v})+I_{\alpha}(x),
(\varphi T_w^{R*})((T^R_{\lambda_R^+})^{-n}) =\varphi(T_w^{R*}(T^R_{\lambda_R^+})^{-n}) & =\varphi((T^R_{\lambda_R^+})^{-n}T_w^{R*}) & =\varphi((T^R_{\lambda_R^+})^{-n})e_G(\sigma)(j_R^+(T_w^{R*})) & =\varphi((T^R_{\lambda_R^+})^{-n})e_G(\sigma)(T_w^{*}) & =\varphi((T^R_{\lambda_R^+})^{-n}).
\zeta_{P{\cal K}(c)}(s)=\frac{1}{\Gamma (s)}\int_0^\infty\d\beta\beta^{s-1}{\rm Tr}e^{-\beta P{\cal K}(c)}\qquad , \qquad\zeta_{{\cal V}_1}(s)=\frac{1}{\Gamma (s)}\int_0^\infty\d\beta\beta^{s-1}{\rm Tr}e^{-\beta{\cal V}_1}\quad .
\frac{\left(-qz^{2};q\right)_{\infty}}{(q;q)_{\infty}}z^{\nu}=\sum_{m=0}^{\infty}\frac{q^{m^{2}}}{\left(q;q\right)_{m}}I_{\nu+m}^{(3)}(2zq^{m/2};q)\left(\frac{q^{\nu}}{z}\right)^{m},
i/(p-1)-v_p(i!)=\sum_{k=1}^r\{(i/p^k)-\lfloor i/p^k\rfloor\}+\sum_{k=r+1}^{\infty}i/p^k\le \sum_{k=1}^r((p-1)+\dots+(p-1)p^{k-1})/p^k+\sum_{k=r+1}^{\infty}i/p^k = r-(p^r-1)/p^r(p-1)+i/p^r(p-1)\le r-(p^r-1)/p^r(p-1)+(p^{r+1}-1)/p^r(p-1)=r+1.
\det\left(\partial_{p_j}f_i(m_j)\right) = \det\left( v_{p_j}(m_jp_j)(g(m_jp_j))^if(m_jp_j) \right) = \det\left( (g(m_jp_j))^{i-k}\right)\prod_j \partial_{p_j}f(m_j)(g(m_jp_j))^k
I_{+}= \frac{2 \pi i}{( 2 \pi )^{6}}\frac{z^{4}}{( 1 - z^{2})^{4}}\frac{\Theta ( \Delta x^{+})}{2}\int_{0}^{\infty}d k_{-}k_{-}^{3}e^{- i k_{-}Y^{2}}e^{i \frac{m^{2}\Delta x^{+}}{k_{-}}}
\delta^{(1)}(A_j)_{\overline{m'_j-1},{\bf m}_\perp,n}=ig{1\over2}[\delta\omega_{m_j,{\bf m}_\perp,n}+\delta\omega_{m_j-1,{\bf m}_\perp,n},(A_j)_{\overline{m'_j-1},{\bf m}_\perp,n}],
\langle \hat{\phi}(x)\hat{\phi}(x')\rangle_\beta=\int_{0}^{\infty}d\omega\int d^2k\left[n_\omega(\beta)~ U^{*}_{\omega,k}(x)U_{\omega,k}(x')+(n_\omega(\beta)+1)U_{\omega,k}(x)U^{*}_{\omega,k}(x')\right]~,
\alpha = A u_0{{2^\nu \Gamma(\nu+1) \Gamma(\nu)\Gamma(c) u_0^{-\nu}}\over{\Gamma(c-a) \Gamma(c-b)}},\beta = A u_0{{\Gamma(-\nu+1)\Gamma(-\nu)\Gamma(c) u_0^{\nu}}\over{2^\nu \Gamma(a) \Gamma(b)}}.
\mathbf{A}^\varepsilon(\lambda)\coloneqq \mathbf{Q}^\varepsilon-\lambda\mathbf{I}_{N}=\frac{1}{\varepsilon^\alpha h}\left(\mathbf{P}^\varepsilon-\mathbf{I}_{N}\right)-\lambda\mathbf{I}_{N},\lambda\in\mathbb{C},
\hat H^{2\! \times \! 2}_{\mathcal M}= - \frac{g^2}{2 r^3}\, \frac{\partial}{\partial r}\, r^3 \, \frac{\partial}{\partial r}+ \frac{2g^2}{r^2}\, \hat L^2 + \frac{r^4}{8\,g^2}\,\sin^2\!\vartheta\:\sin^2\!\psi \:.
\begin{array}{l}{\displaystyle{- Q ( z ) \frac{d^{2}\Psi ( z )}{d z^{2}}+ \left( ( j - \frac{1}{2}) Q^{\prime}( z ) + 2 P ( z ) \right) \frac{d \Psi ( z )}{d z}}-}\\{- \displaystyle{\left( \frac{1}{3}j ( j - \frac{1}{2}) Q^{\prime \prime}( z ) + 2 j P^{\prime}( z ) \right) \Psi ( z ) = E \Psi ( z )},}\\ \end{array}
( \Omega_{b})^{2}= \left( \begin{array}{c c c c c}{r^{2}}&{- 2 i v}&{0}&{\cdots}&{0}\\{2 i v}&{r^{2}}&{0}&{\ddots}&{0}\\{0}&{0}&{r^{2}}&{\ddots}&{\vdots}\\{\vdots}&{\ddots}&{\ddots}&{\ddots}&{0}\\{0}&{0}&{\cdots}&{0}&{r^{2}}\\ \end{array}\right)
G_{1}\equiv \sum_{m = 0}^{\infty}m^{2 k + 4}\alpha_{m}^{- 2 k - 9}, G_{2}\equiv \sum_{m = 0}^{\infty}m^{2 k + 3}\alpha_{m}^{- 2 k - 9}, k = 0 , 1 , 2 . . . ,
J_{L,k}^{\beta,\gamma}(x)=\sum_{i=0}^{k}(-1)^{k-i}\frac{\Gamma(k+\gamma+1)\Gamma(k+i+\beta+\gamma+1)}{\Gamma(i+\gamma+1)\Gamma(k+\beta+\gamma+1)(k-i)!i!L^i}x^i,
\sum_{i=1}^{n-1}\frac{1}{a_i}+ \frac{1}{a_n -1}= \sum_{i=1}^{n-2}\frac{1}{a_i}+ \frac{1}{a_{n -1}}+ \frac{1}{a_n -1}\leq \sum_{i=1}^{n-2}\frac{1}{a_i}+ \frac{1}{a_{n -1}-1}& \leq \cdots \leq \sum_{i=1}^{k-1}\frac{1}{a_i}+ \frac{1}{a_k-1}
P_q(k_0,k_1,k_2,\cdots, k_n)=\sum_{\tau_1,\tau_2,\cdots,\tau_n}P_q(\tau_0,\tau_1,\cdots,\tau_n)P_q(k_0-\tau_0,k_1-\tau_1,\cdots,k_n-\tau_n)q^{\sum_{i=0}^{n-1}\tau_i(k_{i+1}-\tau_{i+1})}.
P(\tau,\mu|X)=NormalGamma(\frac{n_{\mu}\mu_{0}+n\mu}{n_{\mu}+n},n_{\mu}+n,\frac{1}{2}(n_{\tau}+n),\frac{1}{2}(\frac{n_{\tau}}{\tau_{0}}+ns+\frac{n_{\mu}n(\mu-\mu_{0})^{2}}{n_{\mu}+n}))
\|K(t)\|_{\dot{H}^{s}}^{2}= \int_{\mathbb{R}^{2}}{|\xi|^{2s}|\widehat{K}(\xi,t)|^{2}}\,d\xi = \int_{\mathbb{R}^{2}}{|\xi|^{2s}e^{-2t|\xi|^{2}g(\xi)}}\,d\xi \leq Ct^{-(s+1)}g(A_{t})^{-(s+1)}.
\begin{array}{r l}{{}}&{{\displaystyle \int \frac{d^{4}k_{1}}{( 2 \pi )^{4}}\int \frac{d^{4}k_{2}}{( 2 \pi )^{4}}D ( k_{1}) D ( k_{1}+ k_{2}) \overline{{{D ( k_{2}+ p )}}}K_{1}}}\\{{\approx}}&{{\displaystyle 2^{- 1}( 4 \pi )^{- 4}p^{2}\ln ( p^{2}\tilde{p}^{2}) .}}\\ \end{array}
\theta(x) = \lim_{r\rightarrow 0^+}\frac{\| T \|(B_{r}(x))}{\mathcal{H}^k(S\bigcap B_r(x))}= \lim_{r\rightarrow 0^+}\frac{\| T \|(B_{r}(x))}{\Omega(k)r^k}\frac{\Omega(k)r^k}{\mathcal{H}^k(S\bigcap B_r(x))}&\leq 2^k\lim_{r\rightarrow 0^+}\frac{\| T \|(B_{r}(x))}{\Omega(k)r^k}= 2^k\Theta^k(||T|| , x)
u\triangleq[\begin{matrix}u^{(1)}\\ \vdots\\u^{(n)}\end{matrix}],f\triangleq[\begin{matrix}f^{(1)}\\ \vdots\\f^{(n)}\end{matrix}],g\triangleq[\begin{matrix}g^{(1)}\\ \vdots\\g^{(n)}\end{matrix}]
\sim{\frac{1}{( 4 \pi )^{d / 2}}}\sum_{n = 0}^{\infty}\left[ a_{n - \alpha}{\frac{\lambda^{d / 2 - n + \alpha - 1}}{\Gamma \left( \frac d 2 - n + \alpha \right)}}+ b_{n - \alpha}{\frac{\lambda^{( d - 1 ) / 2 - n + \alpha - 1}}{\Gamma \left({\frac{d - 1}{2}}- n + \alpha \right)}}\right] ,
\bigg\|\Big(\sum_j |T(f_1^j,\dots, f_m^j)|^s\Big)^\frac1s\bigg\|_{L^{q}(v)}\lesssim\prod_{i=1}^m\bigg\|\Big(\sum_j |f_i^j|^{s_i}\Big)^\frac1{s_i}\bigg\|_{L^{q_i}(v_i)},
\tau^{1}= \left( \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right) , \tau^{2}= \left( \begin{array}{c c}{0}&{- i}\\{i}&{0}\\ \end{array}\right) , \tau^{3}= \left( \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right) ,
\sum^{i^{*}-1}_{i=1}P_{\hat{Y}}(y_i) = \sum^{i^{*}-1}_{i=1}\Pr \{X \in A_{D}(y_i) \}+ \epsilon< 1 - \delta, &\sum^{i^{*}}_{i=1}P_{\hat{Y}}(y_i) =\sum^{i^{*}}_{i=1}\Pr \{X \in A_{D}(y_i) \}+ \epsilon\geq 1 - \delta, &P_{\hat{Y}}(y_1) \geq P_{\hat{Y}}(y_2) \geq \cdots \geq P_{\hat{Y}}(y_{k^*}),
({}^Q\Omega^-)^a{}_{bc}=N^a{}_x\rho^\mu_b\partial_\mu(N^{-1})^x{}_c,\quad({}^Q\Omega^+)^a{}_{bc}=((KN^{-1})^{-1})^a{}_x\partial^\mu_b(KN^{-1})^x{}_c,
\text{$\sigma$ is a non-negative, continuous function on$[-\overline{\tau},\infty)$}, \lim_{t\to\infty}\int_0^t \frac{1}{\sigma(s)}\,ds =\infty,\quad\lim_{t\to\infty}\sigma(t)=\infty, \lim_{t\to\infty}\int_{t-\tau(t)}^t \frac{1}{\sigma(s)}\,ds = 1, \lim_{t\to\infty}\frac{\sigma(t)}{t}=\infty.
\frac{\tilde{\cal E}_q[\phi(x)]}{L}= -\frac{1}{4\pi}\left(\frac{1}{2}\sum_j \omega_j^2 \log \frac{\tilde \omega_j^2}{m^2}+ \int_0^\infty \frac{dk}{2\pi}\tilde \omega(k)^2 \log\frac{\tilde \omega(k)^2}{m^2}\frac{d}{dk}\left(\tilde\delta(k) - \tilde\delta_1(k) \right) \right) \,.
\widehat{D}_{0}\pi_{a}+ i \varsigma{\frac{\partial E_{a}}{\partial \phi_{i}}}\tau_{i}- i \overline{{p}}^{\prime}{\frac{\partial \overline{{J}}^{a}}{\partial \overline{{\phi}}_{i}}}\overline{{\tau}}_{i}+ i E_{a}\beta - i \overline{{J}}^{a}\overline{{\gamma}}+{\cal O}( 1 / e^{2}) = 0
\bar{R}_{\;\;\;(\sigma )}^{\alpha \beta \;\;\;\delta (\rho )}=\varepsilon^{\alpha \beta \gamma \delta}\left( \left( D_{\gamma}\right)_{\;\;\sigma}^{\rho}+g\delta_{\sigma}^{\rho}\left( \partial_{\tau}B_{\gamma}^{\;(\tau )}+B_{\gamma}^{\;(\tau )}\partial_{\tau}\right) \right) .
\begin{cases}\overline{k}_{yyy}+\overline{k}_y+\overline{k}_{xxx}+\overline{k}_x-\lambda \overline{k}= -\lambda \delta (x-y) , \overline{k}(x,0)=\overline{k}(x,L)=0, , \overline{k}_y(x,0)=\overline{k}_y(x,L)=0, , \overline{k}(0,y)=\overline{k}(L,y)=0, .\end{cases}
\left[ q^{d}\right] (-1)^{(N-1)d}\prod_{i=1}^{N}\bigg( \frac{\alpha_i-z_{N+1}}{\alpha_i}\bigg)^{b_i}\frac{z_i^{d+1}}{P'(z_i)}\cdot\prod_{1\le i\ne j\le N}(z_i-z_j)\bigg{|}_{\epsilon=0}.
\begin{cases}du_n(t) = -\left[P_n \mathrm{A}u_n(t) + P_nB(u_n(t)) - |\nabla u_n(t)|^2_{L^2}u_n(t) \right]\,dt + \sum_{j=1}^m P_n C_j u_n(t) \circ dW_j(t), t \in [0,T], u_n(0) = \dfrac{P_n u_0}{|P_n u_0|}\end{cases}
\begin{aligned}Au(z) =\sum_{i,j=1}^n \sqrt{x_ix_j}a_{ij}(z) u_{x_ix_j}+\sum_{i=1}^n b_i(z)u_{x_i}+\sum_{i=1}^n \sum_{l=1}^m \sqrt{x_i}c_{il}(z)u_{x_iy_l}&+ \sum_{l,k=1}^m d_{lk}(z) u_{y_ly_k}+\sum_{l=1}^m e_l(z) u_{y_l},\quad\forall\, u\in C^2(B^{\infty}_R),\end{aligned}
S \> = \>\sum_{i=1}^{N}\left [ \xi_{i,\mu}\left ( \frac{\xi_{i+1}^\mu-\xi_{i-1}^\mu}{2}\right) -{\Delta t \over 2}\>\rho_{i,\mu}\left(\xi_{i+1}^\mu+2\xi_i^\mu+\xi_{i-1}^\mu \right) \right ] \quad,
\int_0^t F_2(u,y)\d y = \frac{1}{2}\int_0^t w^2 |\Gamma(u+iw)|\d w\geq \frac{1}{2}\int_0^{1}w^2 |\Gamma(u+iw)|\d w &\geq \frac{1}{6}\min_{\substack{u\in[-3/4,-1/4] w\in\times[0,1]}}|\Gamma(u+iw)|.
\int_{\omega_+}U_{\delta,e_n}^\frac{8n}{(n+2)(n-2)}=\begin{cases}O(\delta^\frac{n(n-2)}{n+2}) 4\leq n<6, O(\delta^3|\log \delta|) n=6, O(\delta^\frac{4n}{n+2}) n>6.\end{cases}
I_{\nu}(x)=\sum_{n\geq0}\frac{{\left(\frac{x}{2}\right)}^{2n+\nu}}{n!\Gamma(\nu+n+1)}=\frac{\left(\frac{x}{2}\right)^{\nu}}{\Gamma(\nu+1)}\cdot{}_0F_1\left(\nu+1,\frac{x^2}{4}\right)
\begin{aligned}\Delta(I(n_1,n_2)) = I(n_1,n_2)\otimes 1+\sum_{\substack{k_1+k_2=n_1+n_2 k_1,k_2\ge1}}\big( \delta_{n_1,k_1}+{\tt b}_{n_1,n_2}^{k_1}\big) I(k_1)\otimes I(k_2) + 1\otimes I(n_1,n_2). \end{aligned}
S = - \frac{T}{2}\int d^{3}\sigma \left[ \sqrt{- \gamma}\left( \gamma^{\alpha \beta}\Pi_{\alpha}^{a}\Pi_{\beta}^{b}\eta_{a b}- 1 \right) + \epsilon^{\alpha \beta \gamma}\Pi_{\alpha}^{A}\Pi_{\beta}^{B}\Pi_{\gamma}^{C}B_{A B C}\right]
\mathcal{F}\left( \rho\right) \overset{}{=}\frac{1}{2^{\frac{5}{2}}}\sqrt{\frac{4\left( 4-\rho^{2}\right)}{\left( 2-2\rho^{2}\right)^{2}}}\left( \frac{2+\rho}{4\left( 1-\rho^{2}\right)}\right)^{-\frac{3}{2}}.
[ W , e^{\hat{K}}] = \left[ W , \sum_{N = 0}^{\infty}\frac{1}{N !}\hat{K}^{N}\right] = \sum_{N = 0}^{\infty}\frac{1}{N !}\hat{K}^{N}\left( [ W , \hat{K}] \right) = e^{\hat{K}}( [ W , \hat{K}] ) \ .
z\frac{H_{k-1}^{(s+1,t)}(z)}{H_{k-1}^{(s+1,t)}}=\left(\frac{H_{k-1}^{(s,t)}H_{k}^{(s+1,t)}}{H_{k}^{(s,t)}H_{k-1}^{(s+1,t)}}\right)\frac{H_{k-1}^{(s,t)}(z)}{H_{k-1}^{(s,t)}}+\frac{H_{k}^{(s,t)}(z)}{H_{k}^{(s,t)}}.
a D_{1}^{\prime}e x p ( - i \frac{{\Gamma}_{2}}{|{\Gamma}_{2}|}{\pi}/ 4 ){\psi}_{j}^{\pm}= [ 1 + e x p ({\mp}\frac{\sin{\theta}_{j}}{|{\sin \theta}_{j}|}2 i{\theta}_{j}) ] ( \frac{{\Gamma}_{2}}{|{\Gamma}_{2}|}){\psi}_{j}^{\pm}.
\alpha_{N,t}:=\langle\psi_N,\widehat{m}\psi_N\rangle+|\mathcal{E}_N(\psi_N)-\mathcal{E}^{GP}(u)|-N(N-1)\langle\psi_N,g_\beta(x_1-x_2)\widehat{r}\psi_N\rangle.
\left\lbrace \begin{array}{ll}\frac{\partial v^{E,-}_i}{\partial t}-\sum_{j=1}^D v^{E,-}_j\frac{\partial v^{E,-}_i}{\partial x_j}=-\frac{\partial p^{E,-}}{\partial x_i}, \sum_{i=1}^D\frac{\partial v^{E,-}_i}{\partial x_i}=0, v^{E,-}_i(0,.)=h_i(.),~\mbox{for}~1\leq i\leq D,\end{array}\right.
\left(\begin{array}{c}0 0\end{array}\right)=\prod_{0\le i,j,k<2}\left(\mu_{0i}\mu_{0k}\nu_{0j}\nu_{0i}\omega_{0k}\omega_{0j}-\mu_{i1}\mu_{k1}\nu_{j1}\nu_{i1}\omega_{k1}\omega_{j1}\right)^{-2}\times
e^{-\gamma a}=\lim_{k\to\infty}\frac{\overline{\mu_c^{n*}}(\lambda_k+a)}{\overline{\mu_c^{n*}}(\lambda_k)}= \lim_{A \to \infty}n \int_{-\infty}^{A+}\lim_{k\to\infty}U_k(a-u)\mu_c^{(n-1)*}(du) =n \int_{-\infty}^{\infty}m_c(a-u;\{\lambda_k\})\mu_c^{(n-1)*}(du).
\operatorname{Re}\psi_3(i\rho,i\tau,\xi+i\eta)=-\alpha\tau^2-\beta\rho\eta\leqslant 0,\shortintertext{and}\operatorname{Re}\psi_4(i\rho,i\tau, \xi+i\eta)=-\alpha\tau^2-\beta\tau\eta \leqslant 0,
\left[ Ax - \dfrac{T(t)x-x}{t}\right]_j = [Ax]_j - \dfrac{[T(t)x]_j - [x]_j}{t}= \left (A_j[x]_j - \dfrac{T_j(t)[x]_j - [x]_j}{t}\right){\underset{t \to 0}{\overset{X_j}{\longrightarrow}}}[0]_j,
\left\{\begin{array}{c}\mathbf{L}_{a}\mathbf{R}_{a}\end{array}\right\}=\frac{k_{a}}{k}\frac{\partial}{\partial k}-\frac{1}{2}{\rm ctg}\frac{k}{2}\varepsilon_{ab}{}^{c}k^{b}\mathbf{D}_{c}\pm\frac{1}{2}\mathbf{D}_{a},
M(x ;z):=\begin{cases}e^{ic_+(x)\sigma_3}\left(\frac{\Psi_1^-(x ; z)}{a(z)},\Psi^+_2(x ;z)\right), \; z\in \mathbb{C}^+, begin{align*}5pt] e^{ic_+(x)\sigma_3}\left(\Psi_1^+(x ; z), \frac{\Psi^-_2(x;z)}{\bar{a}(z)}\right), \; z\in \mathbb{C}^-, \end{cases}
\sum_{j = 0}^k{3^j jG_j}= - \frac{9}{{121}}G_0 + \frac{{30}}{{121}}G_1 + \frac{{3^{k + 2}}}{{121}}(11k + 1)G_k &\qquad+ \frac{{3^{k + 1}}}{{121}}(11k - 10)G_{k + 1}\,.
g\left(R_{n}\right)=2\pi d_{k}^{2}p_{0}I\left(\frac{1}{R_{n}}\right)+2\pi d_{k}p_{0}I\left(\frac{R_{n}}{\varepsilon_{n}^{\alpha}}\right)+2\pi\left(p_{0}+\alpha_{k}\frac{R_{n}^{s_{k}}}{4}\right)I\left(\frac{1}{\varepsilon_{n}^{1-\alpha}}\right)+C_{4}.
\begin{matrix}a_{11}x_{1}+a_{12}x_{2}+\cdot\cdot\cdot+a_{1n}x_{n}&=&b_{1}\\a_{21}x_{1}+a_{22}x_{2}+\cdot\cdot\cdot+a_{2n}x_{n}&=&b_{2}\\ &\vdots&\\a_{n1}x_{1}+a_{n2}x_{2}+\cdot\cdot\cdot+a_{nn}x_{n}&=&b_{n}.\end{matrix}
Q(x_0+y_0,\dots,x_{\rho(X)}+y_{\rho(X)}) - Q(y_0,\dots,y_{\rho(X)}) = \log 2 \cdot x_0 + \frac{1}{2}\sum_{i=1}^{\rho(X)}\Lambda(q_i)x_i y_i + \sum_{i=1}^{\rho(X)}\Lambda(q_i)x_i^2 = \sum_{i = 0}^{\rho(X)}x_iQ_i(y_0, \ldots, y_{\rho(X)}) + \sum_{i=1}^{\rho(X)}\Lambda(q_i)x_i^2.
\mathbf{SP}_{4}(\lambda) = \left\{(x_1,\ldots,x_4) \in \mathbb{R}_{\geq 0}^4 \mid{\small\begin{array}{c}x_1\leq m_1, 2x_1+x_2+2x_3+2x_4\leq 2(m_1+m_2), x_4\leq m_2, x_1+x_2+x_3+2x_4\leq m_1+2m_2\end{array}}\right\}.