harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
(2 \sin(\tfrac{\phi}{2}))^{-\ell}= [(2 \sin(\tfrac{\pi}{6}+ \tfrac{\pi}{8(12n+j)}))^{-12n+j}]^{\frac{\ell}{12n+j}}\leq [(2 \sin(\tfrac{\pi}{6}+ \tfrac{\pi}{96}))^{-12}]^{\frac43}= 0.417\dots.
f^{+}_{\varepsilon}(z) =\frac{1}{2\sqrt{\pi\varepsilon}}\int\limits_{0}^{+\infty}e^{-\xi^2/4\varepsilon}\frac{1}{1-ze^{i\xi}}\,d\xi, \varepsilon>0\,,\,z\in\mathbb{D}\,, f^{-}_{\varepsilon}(z) =\frac{1}{2\sqrt{\pi\varepsilon}}\int\limits_{0}^{+\infty}e^{-\xi^2/4\varepsilon}\frac{1}{1-ze^{-i\xi}}\,d\xi, \varepsilon>0\,,\,z\in\mathbb{D}\,.
4 \alpha^{5/2}| \mathrm{Re}\langle u_{\alpha}\tilde{\Phi}_*^1\,|\, V_y \Phi_{\#}^y \rangle| \leq C \alpha^{5/2}\|\tilde{\Phi}_*^1\| \|V_y u_{\alpha}\| \|\Phi_{\#}^y\| = O(\alpha^4),
F_{n,x}^-(u) := \sum_{j=k}^{n-2}\binom{n-2}{j}p_{n,x,u}^j(1-p_{n,x,u})^{n-2-j}, \tilde{F}_{n,x}(u) := \sum_{j=k-1}^{n-2}\binom{n-2}{j}p_{n,x,u}^j(1-p_{n,x,u})^{n-2-j},
\alpha (u+v) =-\inf_{w\in K}\|u+v-w\|=-\inf_{w\in K}\|u+v-2w\| \ge-\inf_{w\in K}(\|u-w\|+\|v-w\|)=\alpha (u)+\alpha (v), \alpha (\lambda\,u) =-\inf_{w\in K}\|\lambda\,u-w\|=-\inf_{w\in K}\|\lambda\,u-\lambda\,w\|=-\inf_{w\in K}\lambda\|u-w\|=\lambda\alpha (u).
\frac{\partial}{\partial t_{p}}\left( \begin{array}{c c}{f}\\{{\overline{f}}}\\ \end{array}\right) = J_{1}\left( \begin{array}{c c}{{\delta}/{\delta f}}\\{{\delta}/{\delta{\overline{f}}}}\\ \end{array}\right) H_{p + 1}= J_{2}\left( \begin{array}{c c}{{\delta}/{\delta f}}\\{{\delta}/{\delta{\overline{f}}}}\\ \end{array}\right) H_{p},
\tilde{b}(t,\,x) =\left(b_{1}(t,\,x),\,b_{2}(t,\,x),\,\cdots,\,b_{m}(t,\,x)\right)^{\dagger}\tilde{\Gamma}(t,\,x) =\left\{\gamma_{c_{1},c_{2}}(t,\,x):1\le c_{1},\,c_{2}\le m\right\}
\sum_{k=0}^a\frac{(-a-p)_{p+k}(\frac12)_{p+k}}{(1)_{p+k}^2}\cdot z^k\equiv(1-z)^a\sum_{k=0}^a\frac{(-a-p)_{p+k}(\frac12)_{p+k}}{(1)_{p+k}^2}\cdot\frac{z^k}{(z-1)^k}\pmod{p^2}.
h_{\epsilon}'(t)=\chi'(t+\epsilon)\leq M\dfrac{\chi (t+\epsilon)}{t+\epsilon}\leq M\dfrac{\chi(t+\epsilon)-c\epsilon}{t+\epsilon}=M\dfrac{h_{\epsilon}(t)}{t+\epsilon}=\dfrac{M t}{t+\epsilon}\dfrac{h_{\epsilon}(t)}{t}\leq \dfrac{M}{1-\delta}\dfrac{h_{\epsilon}(t)}{t},
\tau \beta_t =(5\xi_x-\tau_t)\beta,\eta^1_x=2\xi_{xx},\eta^1_{xx}=\xi_{xxx},2\eta^1_{xxx}=\xi_{xxxx}, \eta^1_xu^2+(\eta^0_x +\eta^1_t+\eta^1_{xxxxx}\beta)u+\eta^0_t+\eta^0_{xxxxx}\beta=0, (\tau_t-\xi_x+n\eta^1)u+(5\eta^1_{xxxx}-\xi_{xxxxx})\beta-\xi_t+\eta^0=0.
[P,a^1]\cdots [P,a^{2k}]= (-1)^kPa^1(1-P)a^2P\cdots (1-P)a^{2k}P &\quad+(-1)^k(1-P)a^1Pa^2(1-P)\cdots Pa^{2k}(1-P) = (-1)^kPa_+^1(1-P)a_-^2P\cdots (1-P)a_-^{2k}P &\quad+(-1)^k(1-P)a_-^1Pa_+^2(1-P)\cdots Pa^{2k}_+(1-P).
-q_*\frac{N}{2}M_{0, 1}U(|x|)F_0(|x|)+\sum^N_{i=1}M_{1, i}U_1(|x|)Q_{1, i}\left( \frac{x}{|x|}\right) =\frac{q_*^2}{2^N\Gamma(N/2)}\biggr[-\frac{N}{c_*^2}M(\varphi)+\frac{1}{c_1^2}\biggr(\frac{x}{|x|}\cdot\Xi(\varphi)\biggr)U_1(|x|)\biggr].
\frac{\partial E}{\partial z}+\frac{\omega_{0}}{k_{0}c^{2}}\frac{\partial E}{\partial t}-\frac{1}{2k_{0}}i\nabla_{\perp}^{2}E=\frac{1}{\epsilon_{0}k_{0}c^{2}}\frac{\partial^{2}}{\partial t^{2}}P^{NL}
\widetilde{A}_\alpha^{DVD}:=\left(\begin{array}{c}\alpha\delta_n^+u_{m,n}+\delta_m^{(2)}\mu_nv_{m,n}+\mu_n(u_{m,n}^2+v_{m,n}^2)\mu_nv_{m,n}-\alpha\delta_n^+v_{m,n}+\delta_m^{(2)}\mu_nu_{m,n}+\mu_n(u_{m,n}^2+v_{m,n}^2)\mu_nu_{m,n}\end{array}\right)=\mathbf{0}.
\Bigl| \int u_{1}\overline{v}_{2}w_{3,\pm}dt dx \Bigr| & \lesssim (L_1L_2 L_3)^{\frac12-\delta}\Bigl(\frac{\min(N_1,N_2)}{\max(N_1,N_2)}\Bigr)^{\frac12}N_{\min}^{s-\frac12}\|u_{1}\|_{L_{t,x}^2}\|v_{2}\|_{L_{t,x}^2}\|w_{3,\pm}\|_{L_{t,x}^2},
\dfrac{d^{k}}{dx^{k}}\phi_{+\varepsilon ,\lambda}\left( x\right) =\beta_{2}\gamma^{-1}\dfrac{d^{k}}{dx^{k}}\left( \cos \sqrt{\lambda}\left(x-a\right) \right) +O\left( \left( \sqrt{\lambda}\right)^{k-1}e^{\left\vert t\right\vert \left( x-a\right)}\right) ,
\partial_{t}^{2}v_{2}(t,r) = \frac{-c_{b}}{2\pi}\int_{0}^{\pi}\sin^{2}(\theta)d\theta \int_{0}^{\infty}d\xi r \left(\frac{\sin(\xi t_{+})}{t_{+}^{4}}+\frac{\sin(\xi t_{-})}{t_{-}^{4}}\right) \partial_{\xi}^{4}\left(\frac{\chi_{\leq \frac{1}{4}}(\xi) \cdot \xi^{3}}{\log^{b-1}(\frac{1}{\xi})}\right)
\mathbf{E}\left[\|\mathcal{X}^{t}-\mathcal{X}^{\star}\|_{F(\mathcal{Q})}^{2}|\mathcal{X}^{0}\right]\leq\left(1-\mathop{\min}_{k=1,2,\cdots,l}\lambda_{\min}(\mathbf{E}[\widehat{\mathcal{Z}}_{(k)}])\right)^{t}\|\mathcal{X}^{0}-\mathcal{X}^{\star}\|_{F(\mathcal{Q})}^{2}.
\begin{cases}D^{\alpha_1}_ay_{ss}(t) = w_{ss}(t), y_{ss}(0) = 0, D^{1-\alpha_1}_aw_{ss}(t) = z_{ss}(t), w_{ss}(0) = 0, D^{\alpha_2-1}_az_{ss}(t) = f_{ss}(t,y(t),w(t)), z_{ss}(0) = 0.\end{cases}
\int_{0}^{1}f(x)G^2_{m_\gamma}(x)j_\gamma(dx) = \frac{-1}{u(\gamma')^2v(\gamma')}\int_{0}^{\gamma'}f(\gamma'^{-1}x)j(dx) \int_{0}^{x}dy\int_{y}^{\gamma'}G_m(z)dm(z) + \frac{m(\gamma')}{u(\gamma')^2v(\gamma')}\int_{0}^{\gamma'}f(\gamma'^{-1}x)j(dx)\int_{0}^{x}dy\int_{y}^{\gamma'}zdm(z) .
m(c,0) \leq \frac{I(\psi_n,c,0)}{K(\psi_n)^{\frac{2}{3}}}&=\frac{I(\psi_n,c,\gamma_n) -\gamma_n\int|\partial_x^{-1}\psi_n|^2dx}{K(\psi_n)^{\frac{2}{3}}}&< \frac{I(\psi_n,c,\gamma_n)}{K(\psi_n)^{\frac{2}{3}}}= m(c,\gamma_n).
((\iota\otimes\varphi)\Delta( x))\,\hat{}\,(\pi)=(\varphi\circ S^{-1})\,\hat{}\,(\pi)\hat{x}(\pi),((\varphi\otimes\iota)\Delta( x))\,\hat{}\,(\pi)=\hat{x}(\pi)(\varphi\circ S)\,\hat{}\,(\pi).
T_{\phantom{\mu}\nu}^{\mu}= 5 G_{( 5 )}^{\mu \alpha \beta \gamma \delta}G_{( 5 ) \nu \alpha \beta \gamma \delta}- \frac{1}{2}\delta_{\phantom{\mu}\nu}^{\mu}G_{( 5 )}^{2}\ , \qquad \mathrm{a n d}\quad G_{( 5 )}= d C_{( 4 )},
{\widetilde T}_{\mu \nu}= T_{\mu \nu}-{\widehat M}_{P}^{D - 4}\biggl [ - \partial^{\lambda}\partial_{\lambda}H_{\mu \nu}+ \partial_{\mu}\partial_{\nu}( \chi + g ) -{\frac{1}{2}}\eta_{\mu \nu}\partial^{\lambda}\partial_{\lambda}\chi \biggr ] ~ .
\exp \left\{- \beta{\cal V}V_{\mathrm{e f f}}( \varphi , \beta ) \right\}= \int [ D \phi^{\prime}] ~ [ D A_{\mu}] ~ [ D \bar{c}] ~ [ D c ] \exp \left( - I_{E}\left[ A_{\mu}, \phi^{\prime}, \bar{c}, c \right] \right) ,
2\omega_{12}+\theta_{12}=\frac{-B^1_{11,3}}{\mu}\omega_3,~~\omega_{13}=\frac{-B^1_{11,3}}{\mu}\omega_2,~~\omega_{23}=\frac{B^1_{11,3}}{\mu}\omega_1, &\omega_{1i}=0, i\geq 4,~~~~~~\omega_{2i}=0, i\geq 4,~~~~~~\omega_{3i}=0, i\geq 4.
V_{ef}\left( \phi_{c}\right) = V_{cl}\left( \phi_{c}\right) +\frac{1}{2}\frac{\hbar}{\Omega}\ln \det \left[ \frac{\delta^{2}S\left[ \phi_{c}\right]}{\delta \phi \left( x\right) \delta \phi \left( y\right)}\right] = V_{cl}\left( \phi_{c}\right) + V_{ef}^{\left( 1\right)}\left( \phi_{c}\right) ,
\{f, \{g,h\}\}+ \{g, \{h,f\}\}+ \{h, \{f,g\}\}= (X, \delta_X Df \wedge \delta_X( Dg, \wedge \delta_X Dh ))+ \circlearrowright + (X, \delta_X Df \wedge \delta_X( O_{X, Dg}Dh - O_{X, Dh}Dg))+ \circlearrowright + (X, \delta_X f \wedge \delta_X (D^2 g(\nu_{X}Df , \cdot) - D^2 f(\nu_{X}Dg , \cdot) ))+\circlearrowright,
F_j\left(\frac 1{w_1},\ldots, \frac 1{w_n},t\right)=\left(\frac 1{w_1}\right)^{m_{i1}}\cdot\ldots\cdot \left(\frac 1{w_n}\right)^{m_{in}}\cdot \widetilde{F_j}(w,t), j=1,\ldots, n.
III' =\sum_{i=0}^{s-2}(q^{s-2}-q^i)\sum_{k=1}^{p-1}\sum_{j=0}^{k-1}q^{j}+ \sum_{i=0}^{s-2}q^i \sum_{k=1}^{p-1}\sum_{j=0}^{k-1}(q^{j}-q^{j+p-1-k}) =(q-1)\sum_{i=0}^{s-3}\sum_{k=0}^{s-3-i}q^{i+k}\sum_{k=1}^{p-1}\sum_{j=0}^{k-1}q^{j}+ (1-q)\sum_{i=0}^{s-2}q^i \sum_{k=1}^{p-2}\sum_{j=0}^{k-1}q^j\sum_{i=0}^{p-2-k}q^{i}.
L_{n}^{( 0 )}= \frac{1}{2}\sum_{k}\alpha_{k}^{\mu}\alpha_{n - k}^{\nu}\eta_{\mu \nu}{,}\quad \bar{L}{}_{n}^{( 0 )}= \frac{1}{2}\sum_{k}\bar{\alpha}{}_{k}^{\mu}\bar{\alpha}{}_{n - k}^{\nu}\eta_{\mu \nu}
\bigl\langle \mathrm{d}( f \circ \pi ) , H_p v \bigr\rangle = \bigl\langle \mathrm{d}( f \circ \pi ) , T \mu (v) - V_p ( \nabla_v^\ast \mu ) \bigr\rangle = \bigl\langle \mathrm{d}( f \circ \pi \circ \mu ) , v \bigr\rangle - 0 , = \langle \mathrm{d}f, v \rangle .
\lim_{n\rightarrow +\infty}\mathbb{E}\left[ c[n] \right] = \frac{1}{\frac{\mu}{\tau}+ \frac{1}{\mu\tau}- \frac{1}{\mu\tau}}\cdot \left[ \frac{T}{\mu}- \left( 1-\mu^2-\frac{1}{\tau}\right) \frac{T}{\mu}- \left( \mu+\frac{1}{\mu\tau}\right)T \right] + T - \mu\frac{T}{\mu}= 0
\psi_{S}( x , t ; \alpha , \beta , \theta , X ) = \frac{\cos ( \alpha )}{\lambda}\frac{\exp \left[ i \theta + i m \sin ( \alpha ) ( \cosh ( \beta ) x - \sinh ( \beta ) t ) \right]}{\cosh \left[ m \cos ( \alpha ) ( \cosh ( \beta ) ( x - X ) - \sinh ( \beta ) t ) \right]}.
|||u-v_\zeta|||^2 \leq \mathcal{M}^\oplus(\alpha,\beta;\eta,\zeta,\boldsymbol{\tau},\boldsymbol{\rho},v_{\zeta}) := \mathcal{J}^\oplus(\alpha,\beta;\eta,\boldsymbol{\tau},v_\zeta) - \mathcal{J}^\ominus(\eta,{\zeta},\boldsymbol{\tau},\boldsymbol{\rho},v_\zeta)
\int_{\real^N}u(x,t)\eta(x,t)\,dx +\int_0^t\int_{\real^N}\left(-u(x,\tau)\eta_t(x,\tau)-u^m(x,\tau)\Delta\eta(x,\tau)\right)dx\,d\tau &=\int_0^t\int_{\real^N}(1+|x|)^{\sigma}u^p(x,\tau)\eta(x,\tau)dx\,d\tau.
(Y(u,z)v,w)_f =\langle fY(u,z)v,w\rangle=\langle Y(u,z)f(v),w\rangle =\langle f(v),Y(e^{zL_{1}}(-z^{-2})^{\deg}u,z^{-1})w\rangle =(v,Y(e^{zL_{1}}(-z^{-2})^{\deg}u,z^{-1})w)_f.
w_{R,\alpha}^+(t) = (1-t^2)^{-1/2}e^{-N \left(V \left( \frac{t}{\alpha}\right) + \varepsilon_R t \right)}, w_{R,\alpha}^-(t) = (1-\beta^{-2}t^2)^{1/2}e^{-N(V(\alpha t) + \varepsilon_R t)}\chi_{[-\beta, \beta]}(t),\beta=\alpha^{-2},
0= F_{(1)}(a)+F_{(3)}(a) = F_{(1)}(a)+3\frac{1}{(\sqrt{3})^{2}}F_{(3)}(a)~~~~~~~(t_{0}=1/\sqrt{3}) = \frac{\mathrm{d}}{\mathrm{d}t}(F_{(1)}(a)t+ F_{(3)}(a)t^{3})|_{t=t_{0}}= \frac{\mathrm{d}}{\mathrm{d}t}(F(ta))|_{t=t_{0}}= J(F)(t_{0}a)\cdot a
\mathcal{F}_{\nu_{1},...,\nu_{m},m,\mu}(z)=\Bigg[(m\ \mu+1)\int_{0}^{z}\prod_{j=1}^{m}\big(D_{\lambda}^{n,\gamma}\mathcal{I}_{\nu_{j}}(t)\big)^{\mu}dt\Bigg]^{1/(m\ \mu +1)},
\int\limits_{\mathbb{R}^{N}}\left\vert V_{0}\right\vert^{\frac{\left( N-s\right) p}{N-\mu-p}}\frac{dx}{\left\vert x\right\vert^{s}}=d^{1+\frac{\left( N-s\right) \left(p-1\right)}{N-\mu-p}}\int\limits_{\mathbb{R}^{N}}\left\vert U_{0}\right\vert^{\frac{\left( N-s\right) p}{N-\mu-p}}\frac{dx}{\left\vert x\right\vert^{\frac{s\left( N-p\right) -N\mu}{N-p-\mu}}}.
\tilde{G}_{ij}=\sum_{\beta=1}^{M \wedge N}\frac{\eta}{(E-\lambda_{\beta})^2+\eta^2}\xi_{\beta}(i) \xi_{\beta}(j),\ \tilde{G}_{\mu \nu}=\sum_{\beta=1}^{M \wedge N}\frac{\eta}{(E-\lambda_{\beta})^2+\eta^2}\zeta_{\beta}(\mu) \zeta_{\beta}(\nu),
S_{\mathrm{b o s o n i c}}= \frac{1}{2 \kappa_{1 1}^{2}}\int d^{1 1}x ( - g )^{1 / 2}\bigl ( R - \frac{1}{2}| F_{4}^{M}|^{2}\bigr ) - \frac{1}{6}\int A_{3}^{M}\wedge F_{4}^{M}\wedge F_{4}^{M},
\begin{array}{ccccc}\textrm{strength}: \!\!\! -1, \!\!\! +1, \!\!\! -1, \!\!\!\! \cdot \textrm{starting from x=0}: \!\!\! -y, \!\!\! y+2a_j, \!\!\! -y-2a_j, \!\!\!\! \cdot \textrm{starting from x=a}: \!\!\! -y+2a_j, \!\!\! y-2a_j, \!\!\! -y+4a_j, \!\!\!\! \cdot\end{array}
\hat V - V = \frac{6^{\frac{1}{3}}}{4}q_I ( \hat G^{IJ}- G^{IJ})q_J = - \frac{1}{8 \cdot 6^{\frac{1}{3}}}t^2 \:(q_I G^{I2})^2 \rightarrow 0\ , \qquad t^2 \rightarrow 0.
\sum_{n\geq0}U_{2k,2a-1}(n)q^n=U_{2k,2a-1}(1;q) &=\frac{(-q^2;q)_\infty(q^{2a},q^{4k-2a},q^{4k};q^{4k})_\infty}{(q^2;q^2)_\infty}+\frac{xq(-q^2;q)_\infty(q^{2a-2},q^{4k-2a+2},q^{4k};q^{4k})_\infty}{(q^2;q^2)_\infty}.
\bigoplus_{\begin{subarray}{c}\boldsymbol{b}=(b_1,\ldots,b_r)\in\mathbb N^r |\boldsymbol{b}|=b_1+\cdots+b_r=\delta\end{subarray}}S^{b_1}(V^{(1)})\otimes\cdots\otimes S^{b_r}(V^{(r)})\longrightarrow S^{\delta}(V),
\left( \frac{3}{2}- 2z^{-1}+ \frac{z^{-2}}{2}\right) \vec{U}(z) + k \sum\limits_{q=1}^{2}\vec{\omega}^{(q)}(z)\mathbf{B}_q\vec{U}(z) = k\vec{F}(z) - k\sum\limits_{q=1}^{2}\vec{\varpi}^{(q)}(z) \mathbf{B}_qU^0 + \frac{z^{-1}}{2}U^1 + \left( z^{-1}- \frac{z^{-2}}{2}\right)U^0,
\underbrace{\left(\sum_{i=1}^{M}\sqrt{\alpha_{n^{*},i}(k)}e^{j\theta_{n^{*},i}(k)}h^{i}_{s,n^{*}}(k)\right)}_{h^{eq}_{s,n^{*}}}x(k) + h_{ps}(k) x_{p}(k) + n(k)
\nabla^{2}-\frac{1}{c^{2}}\frac{\partial^{2}}{\partial t^{2}}=(A\partial_{x}+B\partial_{y}+C\partial_{z}+\frac{i}{c}D\partial_{t})(A\partial_{x}+B\partial_{y}+C\partial_{z}+\frac{i}{c}D\partial_{t}).
I(t) =\left( \int_0^T + \int_T^t\right)\omega(t-s,\lambda_1)p(s)G(\|u[\xi](\cdot,s-\rho(s))\|)d\tau & \le G(2\eta)\int_0^T \omega(t-s,\lambda_1)p(s)ds + \varepsilon \int_T^t \omega(t-s,\lambda_1)p(s)ds & \le G(2\eta)\omega(t-T,\lambda_1)\int_0^T p(s)ds + \varepsilon M & \le [G(2\eta)+M]\varepsilon,
A_{6,0}^{\#}= 988z^6 - 31122z^5 + 266760z^4 - 960336z^3 + 1662120z^2 - 1371249z + 433026 B_{6,0}^{\#}= 187z^6 - 11781z^5 + 141372z^4 - 636174z^3 + 1301265z^2 - 1226907z + 433026 K_{6,0}= 6539203 z^6 - 278018298 z^5 + 2790451188 z^4 - - 11179031292z^3 + 20974929426z^2 - 18462496536z + 6154165512
\sigma (Q(\hbar)(\xi)) = \left(1 + \hbar\sum_k\Psi_k\right)(Q_A(\xi) + \delta(\xi) + \mathcal{R}(\xi) + \hbar\delta_+(\xi) + \hbar\mathcal{R}_+(\xi)) = Q_A(\xi) + \delta(\xi) + \mathcal{R}(\xi) + \hbar\delta_+(\xi) + \hbar\mathcal{R}_+(\xi) + \hbar\sum_k\Psi_k(Q_A(\xi) + \delta(\xi) + \mathcal{R}(\xi)),
\left( \begin{array}{c c}{\left\{\bar{D}_{a}, D^{b}\right\}}&{\left\{\bar{D}_{a}, \bar{D}_{b}\right\}}\\{\left\{D^{a}, D^{b}\right\}}&{\left\{D^{a}, \bar{D}_{b}\right\}}\\ \end{array}\right) = 2 i m^{2}\Delta .
\Delta =\{x\in\mathbb{R}^2| \langle x,\pm(1,0)\rangle\leq4,\langle x,\pm(0,1)\rangle\leq4,\langle x,(1,1)\rangle\leq6,\langle x,(-1,1)\rangle\leq7, &\langle x,(-2,-3)\rangle\leq12\}.
\begin{aligned}(D''\eta_S)(\chi) =D_1''\cdot \eta_S(\chi)+\eta_S(D_2''\chi)=D_E''((1-P_S^S)(D_E''\cdot S(\chi)))+ +\eta_S(D_2''\chi )=(D_E'')^2(S(\chi))-D_E''\cdot P_S^S\cdot D_E''\cdot S(\chi)+\eta_S(D_2''\chi)= =-D_E''\cdot D_S''\cdot S(\chi)+P_1^S\cdot D_E''\cdot S(D_2''\chi)=-(D_S'')^2(S(\chi))=0,\end{aligned}
w^S_i \triangleq g^{\langle 1, S\rangle}_i - g^{\langle 0, S\rangle}_i = 1 + \beta \sum_{j \in N}p_{ij}g^S_j - \beta g^S_i = \begin{cases}(1-\beta) g^S_i i \in S 1 + \beta \sum_{j \in S}p_{ij}g^S_j \textrm{otherwise.}\end{cases}
F = \left( \begin{array}{c c}{{f_{1}( x )}}&{{0}}\\{{0}}&{{f_{2}( x )}}\\ \end{array}\right) ~ ~ , ~ ~{\tilde{F}}= \left( \begin{array}{c c}{{f_{2}( x )}}&{{0}}\\{{0}}&{{f_{1}( x )}}\\ \end{array}\right) , f_{1}( x ) , f_{2}( x ) \in C^{\infty}( C ,{\cal M}) .
\begin{array}{r@{}l}\delta \displaystyle=(-1)^{r(q+1)+1}* d *\sim (-1)^{r(q+1)+1}* s_2 *= - (-1)^{r(q+1)+1}{1\over 2}{C_{ij}}^k * c^i c^j \pi_k * begin{align*}0.3cm] \displaystyle= - (-1)^{r(q+1)+1 + (q-1)(r-q+1) + q(r-q)}{1\over 2}{C_{ij}}^k * c^i * * c^j * * \pi_k *={1\over 2}{C_{ij}}^k \pi^i \pi^j c_k = \overline{s}_2\; .\end{array}
1 + v^{- 4}= \sum_{l \in \bf{N}_{0}}\epsilon_{l}\frac{( ( l + 3 ) ! )^{2}}{( 2 l + 7 ) !}( v - 1 )^{l}\,_{2}F_{1}( l + 4 , l + 4 ; 2 l + 8 ; 1 - v )
\left( -\Delta + 1 + \frac{N-1}{r^2}- \left( p-1 \right) \left( |\cdot|^{N-\gamma}* Q^p \right)Q^{p-2}\right) \left( -\partial_r Q \right) = - \partial_r \left[ |\cdot|^{-(N-\gamma)}* Q^p \right] Q^{p-1}.
X_*(T)=X^*(\widehat{T}) \xrightarrow{res}X^*(Z(\widehat{M_S})^{\Gamma}) \to X^*(Z(\widehat{M_S})^{\Gamma})_{\mathbb{Q}}\cong \mathfrak{A}_{M_S ,\mathbb{Q}}\subset \mathfrak{A}_{\mathbb{Q}}
\mathbb{P}(z\leq Y_{m-1}\leq z+\alpha)\leq \begin{cases}\displaystyle \sqrt{2\alpha/\pi}, \: \mbox{if}m=2, \alpha/2, \: \mbox{if}m=3, \displaystyle \frac{\alpha}{2\sqrt{\pi(m-3)}}, \: \mbox{if}m\geq4. \end{cases}
\tilde{\Lambda}^{(m)}_{\alpha, 1, 2}(\beta) = \sum_{k>E(\lambda)}\partial_\nu\phi^{(\alpha)}_i\Big\vert_{\partial B_R}\frac{1}{(\lambda^{(\alpha)}_k - \beta)^{m + 1}}&\quad\times \Big(\int_{\partial B_R}\partial_\nu\phi^{(\alpha)}_k f{\rm d}s(y) + \int_{\partial B_R}\partial_\nu(\Delta\phi^{(\alpha)}_k)g{\rm d}s(y)\Big).
\frac{1}{\zeta(1-2\delta)\zeta(1-2\delta-2it)}\cdot \frac{1}{2\pi i}\int\limits_{(2A+1)/\log q -iq}^{(2A+1)/\log q +iq}\frac{\zeta(1+w_2+2\delta-2it)}{\zeta(1+w_2)}\hskip 125pt \cdot H_2(0,w_2,\delta,t) \Psi_{\Upsilon,M}(-2\delta-2it) \Psi_{\Upsilon,M}(w_2) dw_2.
w_0(x,\rho)=\sum_{m_1,\cdots,m_k \geq 0}{\Gamma(-\sum((m_k+\rho_k)l_0^{(k)}+1) \over \prod_{1\leq i \leq p}\Gamma(\sum(m_k+\rho_k)l_i^{(k)}+1)}x^{m+\rho}\;\;.
\begin{aligned}\sum_\rho\Big(\frac{\sin\frac12(\gamma-\tau)\log x}{\frac12(\gamma-\tau)\log x}\Big)^{\!2}= \frac{\log\frac{\tau}{2\pi}}{\log x}- \frac2{\log x}\sum_{n\le x}\frac{\Lambda(n)}{n^{1/2}}\Big(1-\frac{\log n}{\log x}\Big)\cos (\tau\log n) &+O\Big(\frac1{\tau\log x}\Big) +O\Big(\frac{x^{1/2}}{(\tau\log x)^2}\Big).\end{aligned}
\partial_{+}( \Gamma^{( 1 ) - 1}\, \partial_{-}\Gamma^{( 1 )}) = - \Gamma^{( 1 ) - 1}\, \Gamma^{( 2 )}, \qquad \partial_{+}( \Gamma^{( 2 ) - 1}\, \partial_{-}\Gamma^{( 2 )}) = \Gamma^{( 1 ) - 1}\, \Gamma^{( 2 )},
g = \alpha \cdot{(X - a_1)}^{e_1}\dots{(X - a_k)}^{e_k}{(X - b_1)}^{e'_1}\dots{(X - b_l)}^{e'_l}\rlap{,}h ={(X - c_1)}^{e''_1}\dots{(X - c_m)}^{e''_m}\rlap{.}
\begin{multlined}x_1^*+x_5^*+x_7^*+\Sigma_2^*+\Sigma_3^*+\Sigma_4^*> \shoveleft[3cm]{L+\frac{3M}{(4-3M)\alpha+3M}+\frac{L}{(1-L)\alpha+L}+\frac{M}{2}+\frac{1}{4}\approx 1.0047>1,}\end{multlined}
J(x,y,c\omega)=\ell_{1}J (a,b,ca)+\ell_{2}J(a,b,cb)+\ell_{4}J(a,b,c(ab))+\ell_{5}J(a,b,c(ac))+ &\ell_{6}J(a,b,c(bc))+\ell_{7}J\left(a,b,c((ab)c)\right)+\ell_{8}J\left(a,b,c(a(bc))\right)+\cdots
\frac{A(1, p)}{2 p^{\frac{1}{2}}}\sum_{m \geq 1}\frac{A(m, 1)}{m}H_{m, p}- \frac{1}{2 p^{\frac{3}{2}}}\sum_{m \geq 1}\frac{A(m, 1)}{m}H_{m p, p}&\qquad= \frac{L(1, f) \bigl(A(1, p) p - 1\bigr)}{p^{\frac{3}{2}}\pi}\int_0^{\infty}k(t) \tanh(\pi t) t \, d t + O(M T^{\frac{1}{7}+ \varepsilon}p^{\varepsilon})
\begin{array}{lcl}0 ={{\mathcal L}_{n}}(u)\star K[a]-K[a]\star{{\mathcal L}_{n}}(w) &= (u_{3}-w_{3}+2n\theta a'_{2}) p^{n-3}&+ (u_4-w_4+(n-3)\theta(w'_3-u'_3)+2n \theta a'_3 +4n \theta^2 a''_2)p^{n-4}& \vdots \end{array}
\big\{\Delta_{x,\S(\varphi,T)}\,:x \in \S(\varphi,T)|_{=s}\big\}=\big\{\S(\varphi,\Delta_{x,T})\,:x \in \S(\varphi,T)|_{=s}\big\}&=\big\{\S(\varphi,\Delta_{x,T})\,:x \in T|_{=s}\, \, \S(\varphi,\Delta_{x,T})\not= \phi \big\},
\mathbf{E}( \mathbf{x}) = \frac{1}{4 \pi}\int \rho_{e}( \mathbf{x}^{\prime}) \frac{\mathbf{r}}{r^{3}}d^{3}x^{\prime}+ \frac{1}{4 \pi}\int \mathbf{J}_{m}( \mathbf{x}^{\prime}){\bf \times}\frac{\mathbf{r}}{r^{3}}d^{3}x
W^*\bigl(1\otimes\pi_{\varphi}(x)\bigr)W =(\pi\otimes\pi_{\varphi})(\Delta x)W^*W=(\pi\otimes\pi_{\varphi})(\Delta x)(\pi\otimes\pi_{\varphi})(E) &=(\pi\otimes\pi_{\varphi})\bigl((\Delta x)E\bigr)=(\pi\otimes\pi_{\varphi})(\Delta x).
t_{d}=-\frac{d\phi}{d\omega}=t_{0}\frac{b_{0}+b_{1}\omega^{2}+b_{2}\omega^{4}+\cdot\cdot\cdot+b_{n-1}\omega^{2n-1}}{b_{0}+b_{1}\omega^{2}+b_{2}\omega^{4}+\cdot\cdot\cdot+b_{n-1}\omega^{2n-1}+b_{n}\omega^{2n}}
{\cal L}= \int dx \frac{dt}{2 \pi R'}\frac{1}{2g}[ \hbox{tr}\dot{Y}^{i}\dot{Y}^{i}+ \hbox{tr}\dot{A}^1 \dot{A}^1 +\frac{1}{2}\hbox{tr}[Y^{i},Y^{j}]^2- \hbox{tr}[\partial_1 Y^{i}- i[A^1,Y^{i}]]^2,
h=\mathcal F^{-1}\varphi, \tilde h=\mathcal F^{-1}\chi, &u_q:=\Delta_qu=\mathcal F^{-1}(\varphi(\lambda_q^{-1}\xi)\mathcal Fu)=\lambda_q^n\int h(\lambda_qy)u(x-y)dy, \mbox{for}q\geq 0, & u_{-1}=\mathcal F^{-1}(\chi(\xi)\mathcal Fu)=\int \tilde h(y)u(x-y)dy,
T_{n}(\theta) :=\mathrm{Tr}\left[\prod_{j=1}^pT_n(g_{j})T_n(h_{j})\right] &=\sum_{t_{1},\cdots,t_{2p}=1}^n\prod_{j=1}^p\widehat{g_{j}}(t_{2j-1}-t_{2j})\widehat{h_{j}}(t_{2j}-t_{2j+1}) &=\int_{\Pi^{2p}}\prod_{j=1}^{p}g_{j}(y_{2j-1})h_{j}(y_{2j})\cdot\prod_{j=1}^{2p}D_{n}(y_{j}-y_{j-1})\,\mathrm{d}y_{1}\cdots\,\mathrm{d}y_{2p},
\lim_{a \downarrow 0}\frac{c_{2}(d, a, m/(2da))}{a^{(d+1)/2}}= \left(\frac{d}{2 \pi^{2}}\right)^{(d-3)/2}\left[ \frac{d \{1+(d-1) \bar{\gamma}\}}{2 \pi (d-1) \bar{\gamma}}\right]^{2},
f(\Phi(A) ) =f((\pi(A))_{\mathcal{S}}) \le \frac{U(f(\pi(A))_{\mathcal{S}}U^* + V(f(\pi(A))_{\mathcal{S}}V^*}{2}&= \frac{U(\pi(f(A))_{\mathcal{S}}U^* + V(\pi (f(A))_{\mathcal{S}}V^*}{2}&= \frac{U\Phi(f(A))U^* + V\Phi(f(A))V^*}{2},
\Im(f(\zeta))-\Im(\zeta) \leq \Im(\beta) +\frac{1}{\Re\left(\zeta\right)\cdot\log\left(\Re\left(\zeta\right)\right)\cdots(\log^{\circ k}(\Re(\zeta)))^{1+\varepsilon}}& =\Im(\beta) + M_{\varepsilon,k}(\Re\left(\zeta\right)).
\left[ i g A_{i}( \xi ( s_{0}) ) \dot{\xi}^{i}( s_{0}) d s_{0}, i \tilde{g}\int_{\eta_{0}}^{\eta ( t_{0})}\delta \eta^{' j}( t_{0}) \tilde{E}_{j}[ \eta^{\prime}| t_{0}] \right] = 2 \pi i / N .
\int_0^{+\infty}h(t)^\lambda\, t^{\lambda-1}dt= \int_0^{+\infty}(h(t)t)^{\lambda-1}h(t) dt\leq \int_0^{+\infty}\left(\int_0^t h(s) ds\right)^{\lambda-1}h(t) dt\leq \left(\int_0^{+\infty}h(t) dt\right)^\lambda.
\mathcal{I}_2:=\int_{B_R}\int_{\mathbb{R}^N\setminus B_R}\frac{\Big(J_p(u_h(x)-u_h(y))-J_p(u(x)-u(y))\Big)}{|h|^{\gamma+t}}\, \delta_h u_h(x)\,\eta(x)^p\,d\mu(x,y),
\begin{cases}(\rho u)_x+(\rho v)_y = 0, (\rho u^2+p)_x+(\rho uv)_y = 0, (\rho uv)_x+(\rho v^2+p)_y = 0, \big((\rho E+p)u\big)_x+\big((\rho E+p)v\big)_y =0, (\rho uZ)_x+(\rho vZ)_y = -\rho\phi(T)Z.\end{cases}
F\left[ \begin{array}{c}\left\{\textbf{a},\textbf{m}\right\}\left\{\textbf{b},\textbf{n}\right\}\end{array}\;\middle|\;\{\textbf{x}\}\right] = \sum_{s_1=0}^{\infty}\dots\sum_{s_r=0}^{\infty}\frac{(\textbf{a})_{\textbf{m}\cdot \textbf{s}}}{(\textbf{b})_{\textbf{n}\cdot \textbf{s}}}\frac{\textbf{x}^\textbf{s}}{\textbf{s}!}
D^q_0(x)+\sum_{k=1}^{n^-}\sum_{j=1}^{n_k}D^q_{kj}(x)\left(\frac{\vartheta_k}{\vartheta_k+ s}\right)^j=\frac{1}{\psi_q^-(s)}\sum_{k=1}^{N}D^q_{k}\frac{e^{\gamma_{k,q}x}}{s+\gamma_{k,q}}, \x\leq 0,
\|T_{32}f\|_{\bf B}\le C \sum_{n=N_0+1}^\infty\| \tilde K_n\|_{{\mathcal S}_{\alpha_0, \beta_0}}\|f\|_{\bf B}\le C \Big(\sum_{n=N_0+1}^\infty n^{\alpha_0}(2r_0)^{n\delta}|z_0|^{-n}\Big)\|f\|_{\bf B}
\hat{h}_{\mathrm{r e d}}= \sum_{\vec{b}}\left\{- \frac{g^{2}L M}{4 \sqrt 2}\frac{1}{J}\frac{\delta}{\delta \varphi_{\vec{b}}}J \frac{\delta}{\delta \varphi_{\vec{b}}}+ \left( \frac{4 \sqrt 2}{g^{2}L M}\right) \sum_{\vec{\varepsilon}}( \varphi_{\vec{b}}- \varphi_{\vec{b}+ \vec{\varepsilon}})^{2}\right\}\, .
L_0 [a_n ,a_0 ] = -i \sum_{n>0}a_n \dot a_n^* - m^2 L a_0^2 -\sum_{n>0}\frac{m^2 L}{4 \pi n}a_n^* a_n \equiv -i \sum_{n>0}a_n \dot a_n^* - H \; ,
\sum\limits_{n\in\uppercase\expandafter{\romannumeral2}}\prod\limits_{k=1}^{\infty}|\left(\frac{1}{2}(1+\phi(k))+\frac{1}{2}(1-\phi(k))e(h b^n 2^{-k})\right)|\leq2^{(\frac{1}{2}+\rho(\varepsilon))r}:=2^{r(1-\eta_2)},
\xi(t;r,x) = S_{t-r}x + \int_r^t S_{t-u}\alpha(u,\xi(u;r,x)) du = S_{t-s}S_{s-r}x + S_{t-s}\int_r^s S_{s-u}\alpha(u,\xi(u;r,x)) du + \int_s^t S_{t-u}\alpha(u,\xi(u;r,x)) du = S_{t-s}\xi(s;r,x) + \int_s^t S_{t-u}\alpha(u,\xi(u;r,x)) du, t \in [s,\infty).
\begin{gathered}I^u_{k_1,\ldots ,k_d}a=\int^\infty_0 \bigg(\int^t_0 \ldots \int^{t_2}_0 a(t_1,\ldots,t_{d-1},t) dw_{k_1}(t_1)\ldots dw_{k_{d-1}}(t_{d-1})\bigg) dw_{k_d}(t)=\int^\infty_0 I^u_{k_1,\ldots ,k_{d-1}}(\tilde{a}(t)) dw_{k_d}(t)= =\mathcal{I}^u_{k_d}(I^u_{k_1,\ldots ,k_{d-1}}(\tilde{a}(\cdot))), \ \mu^u-\mbox{a.e.},\end{gathered}
F(x^*+h)-F(x_h^*) + \lambda \sum_{i\in I^c}\phi(|h_i|) &=h_{I^c}^T [\nabla F(x^*)]_{I^c}+ \lambda \sum_{i\in I^c}\phi(|h_i|)+ o(h_{I^c}) &\geq\sum_{i\in I^c}(\|[\nabla F(x^*)]_{I^c}\|_{\infty}- [\nabla F(x^*)]_i+1)|h_i| \geq 0.
\begin{array}{l}{{R ( \Delta ) = I ( \Delta ) = \Delta^{N - 1}( \Delta^{- 1}\Omega \Delta^{- 1})^{N - 1},}}\\{{R ( \Omega ) = \Delta^{N - 2}\Omega ( \Delta^{- 1}\Omega \Delta^{- 1})^{N - 1},}}\\{{I ( \Omega ) = \Delta^{N - 1}( \Delta^{- 1}\Omega \Delta^{- 1})^{N - 1}\Omega \Delta^{- 1}.}}\\ \end{array}
\Omega^1 =\left[ \begin{array}{rrrrrr}1 0 0 -1 0 0 0 1 -1 0 0 0 0 -1 1 0 0 0 -1 0 0 1 0 0 0 0 0 0 -2 2 0 0 0 0 2 -2 \end{array}\right].
\mathbb{P}[ | \Im \log \zeta (1/2 + i UT)| \geq V] \leq \sum_{r = 0}^{p-1}(1 +\log \log T)^r \mathbb{P}[ |I(UT, 2^{-r}H)| \geq (1-\varepsilon) 2^r V ] + \mathcal{O}_{\varepsilon,\varphi}(T^{-1/3}),
d_{K(s,t)}((x,y))=(s(e_{1})+t)e_{2}\wedge\cdot\cdot\cdot\wedge e_{k}+\sum_{i=2}^{k}(-1)^{i+1}s(e_{i})(e_{1}+\epsilon)\wedge e_{2}\wedge\cdot\cdot\cdot\hat{e_{i}}\cdot\cdot\cdot\wedge e_{k}
a_{ij}^{ll'}\left<Z_{i}, Z_{j}\right>=\displaystyle\sum_{k'=1}^{N}\left(\displaystyle\sum_{k=1}^{N}v^{lk'}_{ik}z_{ik}\right)\overline{\left(\displaystyle\sum_{k=1}^{N}v^{l'k'}_{jk}z_{jk}\right)}, \quad\quad \mbox{for all $l,l',i,j=1,2$.}
S = \int_{0}^{- 1}\frac{( 1 + u )^{- x}- 1}{x ( 1 - u )}\, d u = \frac{1}{2}\int_{0}^{1}\frac{1 - u^{- x}}{x ( 1 - u / 2 )}\, d u = \sum_{k \ge 1}2^{- k}\bigg ( \frac{1}{k}- \frac{1}{k - x}\bigg ) ,
G_k^0(x)-G_0^0(x) = \frac1{2\pi}\int_0^1\frac{e^{ix\xi}-1}{\xi}\frac{k}{\xi-k}~d\xi + \frac1{2\pi}\int_1^{\infty}\frac{e^{ix\xi}-\chi(\xi)}{\xi}\frac{k}{\xi-k}~d\xi.
\mathcal{R}^- = \sum_{m \geq 1}\sum_{n \geq 1}\frac{A(m,n)}{\sqrt{m^2 n}}g\Big(\frac{m^2 n}{N}\Big)\sum_{c > 0}\frac{S(n,-1;c)}{c}H_{m,n}^- \Big( \frac{4\pi \sqrt{n}}{c}\Big);