harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\varphi(b_0a_1b_1a_2b_2\cdots a_nb_n) =0+\varphi(b_0a_1a_2\cdots a_nb_n)\tau(b_1)\cdots \tau(b_{n-1}) &\varphi(b_0a_1a_2\cdots a_nb_n)\varphi(b_1)\cdots \varphi(b_{n-1}).
\mathbb E(\nu(N))\le \mathbb E(\nu(U_\varepsilon))\le\mathbb E(\liminf_{t\downarrow 0}\nu_t(U_\varepsilon))\le\liminf_{t\downarrow 0}\mathbb{E}\nu_t(U_\varepsilon)=\mu(U_\varepsilon)<\varepsilon\,.
32\left\{\widehat{L}(TM)\right\}^{(12)}=\left\{\widehat{A}(TM){\rm ch}[2240+309\widetilde{T_CM}+24(\wedge^2\widetilde{T_CM}+W_i+W_j)\right. &\left.+\widetilde{T_CM}\otimes \widetilde{T_CM}+\widetilde{T_CM}\otimes(W_i+W_j)+\wedge^3\widetilde{T_CM}]\right\}^{(12)}.
{\frac{4 i \pi z}{N \sqrt{2}}}= -{\frac{r - 1}{r}}+ \ln r +{\frac{( r - 1 ) ( \ln ( r - 1 ) - 1 )}{N r}}+{\frac{\bigl ( \ln ( r - 1 ) \bigr )^{2}}{2 N^{2}r}}+{\cal O}\bigl ( 1 / N^{3}\bigr ) .
\frac{\max\limits_{1\le j\le n}\sum\limits_{i=1}^j(X_j-\bar{X})-\min\limits_{1\le j\le n}\sum\limits_{i=1}^j(X_j-\bar{X})}{\Big\{\frac1{n}\sum\limits_{i=1}^n (X_i-\bar{X})^2\Big\}^{1/2}}
L_{i+1}+J^{(n-1)}_n = s_iL_is_i+\sigma_{i+1}+J^{(n-1)}_n &= s_i\left(\sum_{j=1}^{i-1}s_{j,i}\right)s_i+s_i+J^{(n-1)}_n &= \sum_{j=1}^{i-1}s_{j,i+1}+s_i+J^{(n-1)}_n &=\sum_{j=1}^is_{j,i+1}+J^{(n-1)}_n.
\delta ( \varepsilon ) \theta_{\alpha \beta \gamma}^{+ + \alpha^{\prime}\beta^{\prime}\gamma^{\prime}}= \left( P_{\theta}^{+}E P_{\theta}^{-}\right)_{\alpha \beta \gamma}^{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}}+ \left( P_{\theta}^{+}E P_{\theta}^{-}\right)_{\quad \alpha \beta \gamma}^{\alpha^{\prime}\beta^{\prime}\gamma^{\prime}}
\|f_v\|_{\varphi,2}^2 \asymp \|f_v|_{\Gamma^{\pm}}\|_{\varphi,2,\Gamma^{\pm}}^2 =\sum_n|v_n|^2e^{-2\varphi(\gamma_n)}(1+|\gamma_n|^2) =\|v\|_{\varphi,2,\Gamma^{\pm}}.
w_i (h) = \begin{cases}u_i^+ (h) - u_i^+ ((-1)^i \delta ) + d + \delta^2 \ \ \ h \in \bar J_i \setminus [-\delta, \delta ], d + (-1)^i h \delta \ \ \ h \in \bar J_i \cap [-\delta, \delta ], \end{cases}
\begin{array}{c c c}{{{\bf \mathrm{\large ~ \Sigma_{3 6 0}~}}}}&{{\mathrm{\normalsize ~{4 ~ ( \d e l t a_{i j})_{7 ~ \times ~ 7}}~}}}&{{\mathrm{\normalsize ~{6 ~ ( \d e l t a_{i j})_{7 ~ \times ~ 7}}~}}}\\{{}}&{{{\bf 1^{4}}}}&{{{\bf 1^{6}}}}\\ \end{array}
\left[\delta_{\zeta_1},\delta_{\zeta_2}\right]f=\{f,\{\zeta^\alpha_1\phi_\alpha,\zeta^\beta_2\phi_\beta\}\}=\{f,\zeta^\alpha_1\zeta^\beta_2{C_{\alpha\beta}}^\gamma\phi_\gamma\}\approx\zeta^\alpha_1\zeta^\beta_2{C_{\alpha\beta}}^\gamma\{f,\phi_\gamma\}\ .
z_{j0}= \sum_{i=1}^{n_j-1}\sigma_{ji}^+ + \sum_{i=1}^{n_j-1}\sigma_{ji}^- + \sigma_j + \eta \bar z_{j}= \bar s_j-\bar y_j =\sigma_{j}^+ - \sigma_{j}^- - \sigma_j\frac{\bar y_{j}}{\|\bar y_j\|}
\det( D\Phi_h(x)) =\frac{\det(I+\frac{h}{2}f'(\Phi_h(x)))}{\det(I-\frac{h}{2}f'(x))}=\frac{H_{\frac{h}{2}}\circ\Phi_h(x)}{H_{-\frac{h}{2}}(x)},
\partial (v_i) =0,1\leq i\leq 4; \partial (v_{ij}) =[v_i, v_j],1\leq i<j \leq 4; \partial (z) =0; \partial(v_{ijk}) =[v_i, v_{jk}]-[v_{ij}, v_k]-[v_j, v_{ik}]; \partial(w_{ijk}) =[v_i, v_{jk}]-[v_{ij}+z, v_k]-[v_j, v_{ik}].
\frac{P_{{E}^{\prime}/ X}( n )}{\alpha_{\dim ({E}^{\prime})}({E}^{\prime})}\le \frac{P_{{E}/ X}( n )}{\alpha_{\dim ({E})}({E})}\quad \quad \left( \frac{P_{{E}^{\prime}/ X}( n )}{\alpha_{\dim ({E}^{\prime})}({E}^{\prime})}< \frac{P_{{E}/ X}( n )}{\alpha_{\dim ({E})}({E})}\right) \, , \qquad ( n \gg 0 ) \, ,
\mathrm{E}[X^{n}|x_{p}<X<x_{q}] =\frac{\int_{\xi_{p}}^{\xi_{q}}(\sigma y+\mu)^{n}c_{1}g_{1}\left(\frac{1}{2}y^{2}\right)\mathrm{d}y}{F_{Y}(\xi_{p},\xi_{q})}=\frac{\sum_{i=0}^{n}\binom{n}{i}\mu^{n-i}\sigma^{i}\int_{\xi_{p}}^{\xi_{q}}y^{i}c_{1}g_{1}\left(\frac{1}{2}y^{2}\right)\mathrm{d}y}{F_{Y}(\xi_{p},\xi_{q})}.
\delta_t(T)\le \frac{\gamma{k\choose t}+ \Delta{k-1\choose t}}{{n-1\choose t}}=\frac{{k-1\choose t}}{{n-1\choose t}}\Big(\frac k{k-t}\gamma+\Delta\Big)\overset{\eqref{eq02}}{\le}\frac{{k-1\choose t}{n-1\choose k-1}}{{n-1\choose t}}={n-t-1\choose k-t-1}.
- i \overline{{p}}\Psi_{p}^{( 1 / 2 )}= \overline{{\varepsilon}}m \Psi_{p}^{( 1 / 2 )}, \overline{{\sigma}}_{p}^{( 1 / 2 )}\Psi_{p}^{( 1 / 2 )}= \overline{{s}}_{p}\Psi_{p}^{( 1 / 2 )},
\varphi(E_-(w)) = \varphi(E_-(w)E_{o_-}(\lambda_P^-))\sigma(E^P_{o_{-,P}}(\lambda_P^-)^{-1}) & = q(P,w)\varphi(E_-(w\lambda_P^-))\sigma(E^P_{o_{-,P}}(\lambda_P^-)^{-1}) & = q(P,w)\varphi(1)\sigma(E^P_-(w\lambda_P^-)E^P_{o_{-,P}}(\lambda_P^-)^{-1}) & = q(P,w)\varphi(1)\sigma(E^P_-(w))
\lim_{x \to 1^-}\sup_{v,w\in S(0,1)^2}\left(Bis_{(0,x)}(v,w)+1+\frac{\left\lvert \displaystyle \sum_{1\leq i,j\leq 2}g_{i\bar{j}}(0,x)v_i\overline{w_j}\right\rvert^2}{\left( \displaystyle \sum_{1\leq i,j\leq 2}g_{i\bar{j}}(0,x)v_i\overline{v_j}\right)\left( \displaystyle \sum_{1\leq i,j\leq 2}g_{i\bar{j}}(0,x)w_i\overline{w_j}\right)}\right)=0.
\tilde{y}_{t_{k,m}+2^{2-2k}}^{K}\circ \big(\tilde{y}_{t_{k,m}}^{K}\big)^{-1}\begin{aligned}[t] = \begin{cases}y_{2\tau_k}^{(i_k;L_k)}k = K\; (\mathrm{mod}\;2), \mathrm{Id}.\end{cases}& = : Y_{k,m}^K,\end{aligned}
\tilde t=T(t,x,u),\tilde x=X(t,x,u),\tilde u=U(t,x,u), \tilde u_{\tilde t}=U^t(t,x,u,u_t,u_x),\tilde u_{\tilde x}=U^x(t,x,u,u_t,u_x), \tilde f=F(t,x,u,u_t,u_x,f,g),\tilde g=G(t,x,u,u_t,u_x,f,g),
k \langle\langle T_{11}, \dotsc , T_{n(m+n)}, T \rangle \rangle / ( \operatorname{det}_J ) \longrightarrow k \langle \langle T_{11}, \dotsc , T_{nm}, T' \rangle \rangle T_{ij}\longmapsto T' T_{ij}\j = 1, \dotsc , m, T_{i m+j}\longmapsto 0 i \neq j, T_{i m+i}\longmapsto(1-T') T_{i m+i}.
\frac{\partial F}{\partial w^j_n}=-\frac{1}{2\pi i}\oint_c\frac{\partial H'}{\partial \alpha^j}\zeta^n\frac{d\zeta}{\zeta^2}-\frac{1}{2\pi i}\oint_0 \frac{\partial H}{\partial \alpha^j}\zeta^n\frac{d\zeta}{\zeta^2}
\frac{1}{n}\sum_{i=1}^n(y_i-f_m(x_i))^2 - \frac{1}{n}\sum_{i=1}^n(y_i-f(x_i))^2 + \frac{1}{n}\sum_{i=1}^n(f_m(x^{\prime}_i)-f(x^{\prime}_i))^2 \leq \frac{2v^2\epsilon^2_1(1+M_1/m_0)}{m_0}+ \frac{v^2M_1}{2m^2_0}.
\max \Biggl\{\sum_{i \in N}\sum_{j \in V}\sum_{t\in[n]}w^t_{ij}z^t_{ij}: \exists ~ (x, y) \geq 0 \eqref{eq:obm_lp_pol1}\eqref{eq:obm_lp_pol2}z^t_{ij}=\sum_{S\subseteq V\backslash j}x_{i,j}^{t,S}\Biggr\},
\int_v (1+v^2)[M_T(v)-M_{T(x)}(v)]^2M_T^{-1}dv= \int_v (1+v^2)\left[1- \frac{M_{T(x)}(v)}{M_T(v)}\right]^2M_T(v) \mathrm{d}v \sim \int_v (1+v^2) M_T(v) \left[ 1- \sqrt{\frac{T}{T(x)}}e^{-\frac{|v|^2}{2}(T(x)^{-1}-T^{-1})}\right]^2 \mathrm{d}v.
H^{d+3}_{}(Y,\mathbb{Q}/\mathbb{Z}(d+2)) = (H^{d+3}(Y,\mathbb{Q}/\mathbb{Z}(d+2))/(H^{d+3}(K_0,\mathbb{Q}/\mathbb{Z}(d+2))) \rightarrow \prod_{v \in X_0^{(1)}}H^{d+3}(Y_{K_{0,v}},\mathbb{Q}/\mathbb{Z}(d+2))/(H^{d+3}(K_{0,v},\mathbb{Q}/\mathbb{Z}(d+2)))
\lambda\left( t\right) =\dfrac{1}{{L}\left( t\right)}\int_{X\left(\cdot,t\right)}k^{\alpha}ds=\left({\displaystyle\int_{0}^{2\pi}}\frac{1}{k\left( \theta,t\right)}d\theta\right)^{-1}{\displaystyle\int_{0}^{2\pi}}k^{\alpha-1}\left( \theta,t\right) d\theta{\ \ \}
E_1^{-k,i+k}=\underset{{j\ge0\atop j\ge -k}}\bigoplus\underset{N\in\Theta_{2j+k+1}}\bigoplus E(N)\otimes_{K_0}H_{rig}^{i-2j-k}(N/K_0)\Longrightarrow H_{rig}^i(Y/{\mathfrak S}^{0},E)
\partial_t \rho_m = - \partial_x u(y_t) \rho_m = - \frac{w(y_t)}{c_\mu}\rho_m^{1-\alpha}- \frac{c_p}{c_\mu}\rho_m^{\gamma -\alpha+ 1}\geq - \frac{c_p}{c_\mu}\rho_m^{\gamma -\alpha+ 1}
F_1(x)=\frac{-1}{\sqrt{2}\sin(x)}\left(\frac{-\sin(x)}{1+\cos(x)}\right)^{-\frac{1}{\sqrt{2}}},F_2(x)=\frac{-1}{\sqrt{2}\sin(x)}\left(\frac{-\sin(x)}{1+\cos(x)}\right)^{\frac{1}{\sqrt{2}}}.
S_2(X,Y)= \frac{1}{2}|\tau(\psi)|^2\langle X,Y\rangle+\langle d\psi,\nabla\tau(\psi)\rangle\langle X,Y\rangle &-\langle d\psi(X),\nabla_Y\tau(\psi)\rangle-\langle d\psi(Y),\nabla_X\tau(\psi)\rangle
d^4(x'_4) =\phi^*(\bar{d}^4(x_4)) =\phi^*(y_1v_1v_2+y_2v_2v_1+2y_1u_1v_2 +2y_2u_2v_1+y_1v_1u_2 \;\;\;\;+y_2v_2u_1+2y_1u_1u_2 +2y_2u_2u_1+y_1u_2^2+y_2u_1^2) =2y'_2\gamma_1\gamma_2+2y'_1\gamma_1\gamma_2+y'_2\gamma_1^2+y'_1\gamma_2^2.
C_n(\kappa,{\bf P}_{Y_1})= \sup_{\frac{1}{n}{\bf E}\big\{\sum_{t=1}^{n}||X_t||_{{\mathbb R}^{n_x}}^2\big\}\leq \kappa}\sum_{t=1}^n I(X_t, V^{t-1};Y_t| Y^{t-1}) = \sup_{\frac{1}{n}{\bf E}\big\{\sum_{t=1}^{n}||X_t||_{{\mathbb R}^{n_x}}^2\big\}\leq \kappa}H(Y^n)-H(V^n) \in [0,\infty]
\begin{cases}\mathcal{Q}\,^{C}_{t}D_{T}^{q}\Theta(x,t)= A^{*}\mathcal{Q}\Theta(x,t) + C^{*}\mathcal{Q}z(t) \hbox{in}Q_{T}\Theta(\xi,t) = 0 \hbox{on}\Sigma_{T}\Theta(x,T) = 0 \hbox{in}\Omega.\end{cases}
C(n,k;q)\, =\sum_{\tau\not\equiv 0\,(\, p)}\bigl(a(\tau)-a_*(\tau)\bigr)+\sum_{\tau\not\equiv k\,(\, p)}a_*(\tau)\cr =n^2-\sum_{\tau\equiv 0\,(\, p)}\bigl(a(\tau)-a_*(\tau)\bigr)-\sum_{\tau\equiv k\,(\, p)}a_*(\tau),
T_{0}(z^{\prime})t(z)=\frac{\frac{c_1}{2}\omega(z)+\frac{c_2}{2}\Omega(z)}{(z^{\prime}-z)^{4}}+\frac{T_{0}(z)+2t(z)}{(z^{\prime}-z)^{2}}+\frac{\partial_{z}t(z)}{z^{\prime}-z}+\cdots\:\:\:.
C\!=\!\big\{(c_1\!+\!c_{12},\;c_2\!+\!c_{12}g)\,\big|\, c_1\!\in\! C_1, c_2\!\in\! C_2, c_{12}\!\in\! C_{12}\big\}\!=\!(C_1\times C_2)\oplus \widehat C_{12}.
0 =\nabla f(\bar x)+\sum\limits_{i\in I^g(\bar x)}\lambda_i\nabla g_i(\bar x)+\sum\limits_{j\in\mathcal P}\rho_j\nabla h_j(\bar x) &+\sum\limits_{l\in I^{0+}(\bar x)}\mu_l\nabla G_l(\bar x)+\sum\limits_{l\in I^{+0}(\bar x)}\nu_l\nabla H_l(\bar x)
\begin{array}{rl}(b, \lambda, p_{2}) \longleftrightarrow (c, \mu, p_{3}) ; P \leftrightarrow R , s \leftrightarrow u \, ; (b, \lambda, p_{2}) \longleftrightarrow (d, \nu, p_{4}) ; P \leftrightarrow Q , s \leftrightarrow t \, .\end{array}
\{\overline{a+I},\overline{b+I}\}_1 := \frac{1}{t-1}\Big((a+I)(b+I) - (b+I)(a+I)\Big) \mod t-1 &= \frac{1}{t-1}(ab-ba + I) \mod t-1 &= \frac{1}{t-1}(ab-ba) + (I+\langle t-1 \rangle) \mod t-1
V_\Delta + \sum_{F \in \Delta[1]}\ell^*(F) = \ell(\Delta) - 1 - \sum_{F \in \Delta[2]}\ell^*(F) V_{\Delta^\circ}+ \sum_{F^\circ \in \Delta^\circ[1]}\ell^*(F^\circ) = \ell(\Delta^\circ) - 1 - \sum_{F^\circ \in \Delta^\circ[2]}\ell^*(F^\circ).
\kappa_\alpha (F) =|\alpha|^{-\frac{1}{2}}\big |\log |\alpha| \big |\, \big ( |b_\Omega [f]| + \| F \|_{L^2 (\Omega)}+ \| f \|_{L^2 (\Omega_{6 R_0})}\big ) & + \big |\log |\alpha| \big | \, \| F_{12}-F_{21}\|_{L^1 (\Omega)}+ (\log |\alpha|)^2 \, \| F \|_{L^\infty_{2}(\Omega)}\,.
\Psi_{( h ; 1 )}( z ) \Psi_{( h^{\prime}; 1 )}( 0 ) = \ldots + \frac{C_{( h , h^{\prime}; 2 ) \iota}}{D_{\iota \iota^{*}}}z^{h_{\iota}- h - h^{\prime}}\chi_{\iota}( 0 ) + \ldots \, ,
P=\left(\begin{matrix}\alpha^{6}z + \alpha^{2}\alpha^{3}z + 1 0 0 \alpha^{6}z \alpha^{3}z \alpha^{6}0 z^{2}+ 1 \alpha^{4}z^{2}+\alpha^{6}z + \alpha \alpha z 0 \alpha^{4}z^{3}+ z^{2}+ \alpha^{6}z \alpha z^{3}+ \alpha^{6}z^{2}+\alpha^{3}z \alpha^{5}z^{2}+ \alpha z + \alpha^{6}1\end{matrix}\right).
a \mapsto \sum_{k,l=1}^m \langle x_k , \Phi( c_k^\ast a c_l) x_l \rangle = \sum_{k,l=1}^m \sum_{i,j=1}^n \langle T_ix_k , \phi((c_k a_i)^\ast a (c_l a_j)) T_j x_l \rangle
(vF_t(f_i))(1)=F_t(f_i)(v)=\sum_{u\in J}\sum_{u'\in J(N_{L_P,0}/tN_{L_P,0}t^{-1}H_{\Delta_P,0})}u'F_{t_1}(F_{t_2}(f_i(t^{-1}u'^{-1}vut)))
\mathbf{Reg}_T \leq \frac{1}{n}\sum_{i=1}^n\left\|H_i\right\|^2 \frac{\alpha^2 \sum_{i=1}^n\|H_i\|^2W_i}{1-\|Q\|}&~~~~~~~~~~~~~~~~~~+\frac{1}{T}\sum_{i=1}^n\left\|H_i\right\|^2\frac{C_T}{(1-\|Q\|)^2}.
\begin{array}{lll}L_{IJ}^{(-)}L_{JK}^{(-)}- (-1)^{P_J(P_I+P_K)}L_{JK}^{(-)}L_{IJ}^{(-)}= (q-q^{-1})(-1)^{P_KP_J}L_{JJ}^{(-)}L_{IK}^{(-)}, (I>J>K)\end{array}
\langle S_{\xi\chi}f,g\rangle =\langle \int_{\Theta}v(w)s(w)\pi_{G(w)}\xi_{w}^{\ast}\chi_{w}\pi_{F(w)}fd\mu(w),g\rangle &=\int_{\Theta}v(w)s(w)\langle f,\pi_{F(w)}\chi_{w}^{\ast}\xi_{w}\pi_{G(w)}g\rangle d\mu(w) &=\langle f,\int_{\Theta}v(w)s(w)\pi_{F(w)}\chi_{w}^{\ast}\xi_{w}\pi_{G(w)}gd\mu(w)\rangle &=\langle f,S_{\chi\xi}g\rangle.
( a \nabla_{\nu}b)^{2}- r_{0}^{2}(a-b)^{2}=a^{2}\nabla_{\nu}b^{2}-r_{0}(a-b)^{2}& \geq ( a \sharp_{\nu}b)^{2}+ \sum_{k=1}^{\infty}r_{k}\big[ a^{1-\frac{m_k}{2^k}}b^{\frac{m_k}{2^k}}- a^{1-\frac{m_k+1}{2^k}}b^{\frac{m_k+1}{2^k}}\big]^{2}
\zeta'(t)=||(\varepsilon\partial_x,\partial_y)u^\varepsilon_\psi(t')||_{B^\frac 12}+||(\varepsilon\partial_x,\partial_y)(Q_{11}^\varepsilon, \varepsilon Q_{12}^\varepsilon)_\psi(t')||_{B^\frac 12}+\sum_{m>0}m|c_m|e^{-m^2\pi^2t}+||u_{\phi}(t')||_{B^\frac 12}, \zeta(0)=0
g_{-1,-1}\textrm{is Egerv\'ary equivalent to}\begin{cases}g_{-1,1}\textrm{if and only if}n+m \textrm{is an even number}, g_{1,1}\textrm{if and only if}n \textrm{is an even number}, g_{1,-1}\textrm{if and only if}m \textrm{is an even number}, \end{cases}
\lVert u \rVert_{\dot{\mathrm{H}}^{\alpha,p}(\mathbb{R}_+,\mathrm{L}^p(\mathbb{R}^{n-1}))}:= \inf_{\substack{U_{|_{\mathbb{R}_+}}=u U\in \dot{\mathrm{H}}^{\alpha,p}(\mathbb{R},\mathrm{L}^p(\mathbb{R}^{n-1}))}}\lVert U\rVert_{\dot{\mathrm{H}}^{\alpha,p}(\mathbb{R},\mathrm{L}^p(\mathbb{R}^{n-1}))}
W_{r}(\rho_0,\rho_1) \le\left( \int_{\Omega\times\Omega}|x-x_0|^r\,d\gamma_{opt}\right)^{1/r}+\left( \int_{\Omega\times\Omega}|y-x_0|^r\,d\gamma_{opt}\right)^{1/r}&=\left( \int |x-x_0|^r\,d\rho_0\right)^{1/r}+\left( \int |y-x_0|^r\,d\rho_1\right)^{1/r},
\prod_{i\neq j}\frac{\Gamma(r_{j}-r_{i}-\nu_j))}{\Gamma(r_{i}-r_j-\nu_{i}+\nu_j)}=\prod_{i<j}\frac{r_j-r_i-\nu_j+\nu_i}{r_j-r_i}\frac{\Gamma(1+r_i-r_j)\Gamma(1+r_j-r_i)}{\Gamma(1+\nu_j+r_i-r_j)\Gamma(1+\nu_i+r_j-r_i)}.
\mathcal{D}= \mathfrak{B}\sqcup \bigsqcup_f \{\frac{bX^i}{f^j}:\ (j,p)=1,\j\geq 1, \ 0 \leq i< \deg(f),\b \in \mathcal{C}\}\sqcup \{b X^j,\ (j,p)=1,\j\geq 1, \b \in \mathcal{C}\},
\mathsf{H}_{univ}:= \tilde{\Lambda}[T_q, U_l, \langle d\rangle, q\nmid Np^n, l\mid Np\ell^{2r}, d\in \Delta_{Np^n\ell^{2r}}], \mathsf{H}_{univ}^B:= \tilde{\Lambda}[\tilde{U}_\ell, T_q, U_l, \langle d\rangle, q\nmid Np^n, l\mid Np, d\in \Delta_{Np^n\ell^{2r}}]
\widehat{\mathcal{P}^{\ast}}(\mathcal{U}_{(r)}^{e,\bullet},J_{(k-r)}T\mathcal{U}_{(r)}^{e,\bullet})\subset H_{inv}^{\ast}(\mathcal{U}_{(r)}^{e,\bullet},M)=H^{\ast}(J_{(r)}T,M)
\|f\|_{k,t}\le \sum_{|\alpha| \le k}\|\chi\|_{0,\tilde{p}}\|\partial^\alpha f\|_{0,p}&= |\mathrm{supp}(f)|^{1/\tilde{p}}\sum_{|\alpha| \le k}\|\partial^\alpha f\|_{0,p}&= |\mathrm{supp}(f)|^{1/t-1/p}\|f\|_{k,p}.
\mathrm{LC}_{s=1}^r[\mathcal{F}_{D,0}(z,s)]^{\mathrm{hol}}= \mathrm{LC}_{s=1}^r \biggl[ \frac{|D|^{\frac{s}{2}}L_D(s)}{2^{2s-1}\pi^{\frac{s-1}{2}}}\frac{\Gamma(s-1/2)}{\Gamma(s/2)}\frac{\zeta(2s-1)}{\zeta(2s)}\biggr] + \sum_{0 < d \equiv 0, 1 (4)}\frac{1}{\sqrt{d}}\mathrm{Tr}_{d,D}(F_{0,0,r}) q^d.
\det\begin{bmatrix}\partial_x\phi, \ \partial_a\phi, \ \partial_b \phi \partial_x\partial_{\theta}\phi, \ \partial_a\partial_{\theta}\phi, \ \partial_b\partial_{\theta}\phi \partial_x\partial^2_{\theta}\phi, \ \partial_a\partial^2_{\theta}\phi, \ \partial_b\partial^2_{\theta}\phi\end{bmatrix}\neq 0
E_{A_{j}}(u_{k}au_{l}(1-a^2)u_{l}^*au_{k}^*) = \tau ( u_{k}au_{l}(1-a^2)u_{l}^*au_{k}^* \omega ) & = \underbrace{\tau ( u_{k}au_{l}(1-a^2)u_{l}^*au_{k}^* \mathring{\omega})}_{= 0}+\underbrace{\tau ( u_{k}au_{l}(1-a^2)u_{l}^*au_{k}^* E_A(\omega ))}_{= 0}
\varphi(\hat{x},\hat{t})-\varphi(\hat{x},\hat{t}-h) \le\xi_1^{y,s,\varepsilon}(\hat{x},\hat{t})-\xi_1^{y,s,\varepsilon}(\hat{x},\hat{t}-h) &= -M_2(|\hat{t}-s|-|\hat{t}-h-s|) &\le M_2h
f_j^{(n)}(x_1,\ldots,x_{n-1},z,x;t) =\int_{\{0<t_1<\ldots<t_{j-1}<r<t_j<\ldots<t_{n-1}<t\}}g_{n-j}(t_j,x_j,\ldots,t_{n-1},x_{n-1},r,z,t,x) & f_{j-1}(t_1,x_1,\ldots,t_{j-1},x_{j-1},r,z)dt_1 \ldots dt_{n-1}dr
t_{uv}=(\kappa n)_{v}=(-\lambda\kappa\tau)t+(\kappa_{v})n+(\kappa\gamma)b, t_{vu}=(\lambda\tau n+\lambda_{u}b)_{u}=(-\lambda\kappa\tau)t+(\lambda\tau_{u}+2\lambda_{u}\tau)n+(\kappa\gamma)b.
D^{n+1}(a)= qD^n(ab^2) = q\sum_{k=0}^n\binom{n}{k}D^k(a)D^{n-k}(b^2) = qb^2D^n(a)+q\sum_{k=0}^{n-1}\binom{n}{k}D^k(a)2^{n-k}d^{2n-2k}c^2A_{n-k}\left(\frac{c^2}{d^2}\right).
D^{\frac{1}{2}}_h(s) = \frac{(2\pi)^{s - \frac{1}{4}}}{\Gamma(s - \frac{1}{4})}\langle P_h^{\frac{1}{2}}(\cdot, s), V\rangle = \sum_{m \in \mathbb{Z}}\frac{r_{3}(m^2 + h)}{(2m^2 + h)^{s - \frac{1}{4}}}- \mathfrak{E}_h^{\frac{1}{2}}(s).
{\widehat A}_tf(t):=\frac{\partial}{\partial t}\left\{t(1-t)\frac{\partial}{\partial t}\left[t(1-t)\frac{\partial f(t)}{\partial t}\right]+4\nu (\nu +1)t(1-t)f(t)\right\}+4\nu(\nu+1)\frac{2t-1}{2}f(t)=0
\Delta_n(x) =\frac{(-1)^{n}}{\pi x}\sum_{m\geq0}\frac{x^{2m+1}}{m!(m+1)!}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\frac{\sin^{2m+1}(t)\cos(2n t)}{\sin t}dt &=\sum_{m\geq0}\frac{x^{2m}}{m!(m+1)!}\frac{(-1)^{n}}{\pi}\int_{-\frac{\pi}{2}}^{\frac{\pi}{2}}\sin^{2m}(t)\cos(2n t)dt=\sum_{m\geq0}\frac{x^{2m}}{m!(m+1)!}T_n(m),
R[L^2](\langle t \rangle^{-(\alpha + \epsilon)}U(t):\:t \geq 1) \leq \sum_{n=0}^\infty 2^{-n\epsilon}R[L^2]( 2^{-n\alpha}e^{-2^{-n}t}U(t):\:t\in [2^n,2^{n+1}]) & \leq \sum_{n=0}^\infty 2^{-n\epsilon}\sup_{c < 0}R_c < \infty.
\mathcal{S}_{0,2}(V,V) =-\left(f\left(f+2\frac{\partial^2f}{\partial\theta_1^2}+2\frac{\partial^2f}{\partial\theta_2^2}\right)\right)(JX_1+JX_2) &-2\left(\frac{\partial f}{\partial\theta_2}\right)^2JX_1-2\left(\frac{\partial f}{\partial\theta_1}\right)^2JX_2.
(\int_{-\infty}^{\infty}e^{-x^{2}}dx)^{2}=\int_{-\infty}^{\infty}e^{-x^{2}}dx\int_{-\infty}^{\infty}e^{-y^{2}}dy=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}e^{-x^{2}-y^{2}}dxdy=
\qquad\begin{aligned}H^\textsc{cs}_\pm = -\frac{1}{2}\sum_{j=1}^N \partial_{x_j}^2 + \beta\,(\beta\mp1) \sum_{i < j}^N \frac{1}{4 \sin^2[(x_i - x_j)/2]}& = \frac{1}{2}\sum_{j=1}^N \, (z_j \, \partial_{z_j})^2 + \beta\,(\beta\mp1) \sum_{i < j}^N \frac{z_i \, z_j}{z_{ij}\,z_{ji}}\, .\end{aligned}
\ln\left(\left|z_i(t+1) - \bar{x}\right|\right)\leq \ln\left(\frac{2\|x(0)\|_1}{{y_{i}(t+1)}}\Lambda_{t,0}\right)\cr= \ln\left( 2\|x(0)\|_1 \right) + \ln\left(\frac{1}{{y_{i}(t+1)}}\right) \cr + \ln\left(\Lambda_{t,0}\right).
\begin{aligned}\left(\int_{\tilde f}\varphi \tau_1 + \bar{\partial}_X\varphi \wedge \tau_{01},\,\,\int_{\tilde f}\varphi \tau_{01}\right) =\left(\int_{D_1}\varphi \tau_1 + \bar{\partial}_X\varphi \wedge \tau_{01},\,\,\int_{D_1}\varphi \tau_{01}\right) &=\left(\int_{D_1}\tau_1 - \bar{\partial}_X((1 - \varphi)\tau_{01}),\,\,\int_{D_1}\varphi \tau_{01}\right).\end{aligned}
P_0^j (\Theta_k - \Theta_0)P_0^j = O^P({T}^{-2}) \mbox{for any}j, P_0^j (\Theta_k - \Theta_0)P_0^\ell = O^P({T}^{-1}) \mbox{for any}j\neq\ell, Q_0 (\Theta_k - \Theta_0) = (\Theta_k - \Theta_0) Q_0 = 0.
A^\bullet(X^m)_{\mathbb{Q}}= \bigoplus_{B \subset \{1,\ldots,m\}}Im(\Gamma^{\boxtimes \{1,\ldots,m\}\setminus B}\boxtimes (\Delta_2-\Gamma)^{\boxtimes B}) A^\bullet(X^n)_{\mathbb{Q}}= \bigoplus_{B \subset \{1,\ldots,n\}}Im(\Gamma^{\boxtimes \{1,\ldots,n\}\setminus B}\boxtimes (\Delta_2-\Gamma)^{\boxtimes B}).
P^{(-x_{1})}(z) = E_{-x_{1}}(z)\Phi_{\mathrm{Be}}(rf_{-x_{1}}(z);0)s_{2}^{-\frac{\sigma_{3}}{2}}e^{-\sqrt{r}g(z)\sigma_{3}}e^{\frac{\pi i \alpha}{2}\theta(z)\sigma_{3}},
\frac{\partial}{\partial k}[_{p}(x)_{n,k}]= -\frac{n}{k}\:_{p}(x)_{n,k}+ \frac{p}{k}\sum_{s=1}^{n-1}s\:_{p}(x)_{n,k}\:_{p}(x+(s+1)k)_{n-1-s,k}.
F_{\mu\nu}=iF_{\mu\nu}^{1}+F_{\mu\nu}^{5}\Gamma_{5}+F_{\mu\nu}^{a}\Gamma_{a}+F_{\mu\nu}^{a5}\Gamma_{a}\Gamma_{5}+\frac{1}{4}F_{\mu\nu}^{ab}\Gamma_{ab},
\Gamma_{2\hbar}(-d_{(0,0,0)}2\hbar) =\frac{(-1)^{d_{(0,0,0)}}}{d_{(0,0,0)}!}\cdot\Gamma_{2\hbar}(0)=\frac{(-1)^{d_{(0,0,0)}}}{d_{(0,0,0)}!}\cdot \frac{1}{e^{T_0}(1)},
\left(\frac{cf}{a^2}\right)^{\binom n2}q^{2\binom n3}\prod_{j=1}^n\frac{(q^{j-n}b/c,q^{n-j}bc/a,q^{j-n}e/f,q^{n-j}ef/a)_{j-1}}{(b,b/a,c,c/a,e,e/a,f,f/a)_{j-1}}.
\alpha_{t}({r}):=\begin{cases}\displaystyle{\int_{t}^\infty}e^{-\pi B({r},c(x))^2}dx \mbox{if}B({r},c)B\big({r},c^\perp\big)>0, begin{align*}2ex] -\displaystyle{\int_{-\infty}^{t}}e^{-\pi B({r},c(x))^2}dx \mbox{if}B({r},c)B\big({r},c^\perp \big)<0, 0 \mbox{otherwise,}\end{cases}
f - q|_g = \ - L^2(p|_{L^2(xy)}) z - p|_{L^2(xy)}L^2(z) + L^2(p|_{L^2(x)y}z) + L^2(p|_{xL^2(y)}z) \equiv \ - L^2(p|_{L^2(x)y}) z - L^2(p|_{xL^2(y)}) z - p|_{L^2(x)y}L^2(z) - p|_{xL^2(y)}L^2(z) &\ + L^2(p|_{L^2(x)y}) z + p|_{L^2(x)y}L^2(z) + L^2(p|_{xL^2(y)}) z + p|_{xL^2(y)}L^2(z) \equiv \ 0.
I :=\int_0^1\alpha(x(1-x))^2(D\phi)(D^2\phi)x^{3/2}(1-x){N/2-1}dx &=\int_0^1\frac{\alpha}{2}D(D\phi)^2x^{7/2}(1-x)^{N/2+1}dx &=-\int_0^1D\Big(\frac{\alpha}{2}x^{7/2}(1-x)^{N/2+1}\Big)(D\phi)^2dx.
{\displaystyle -2\mathrm{i}\Delta t(\partial \theta^{k}_{\Psi},\partial{\theta_{\Psi}^{k}})+\Delta t B\left(\overline{\mathbf{A}}^{k}_{h};\overline{\theta}_{\Psi}^{k},\partial{\theta_{\Psi}^{k}}\right) =J_1^{(k)}+J_2^{(k)}+J_3^{(k)}+J_4^{(k)},}
F ( s ) = \frac{1}{4 8}\int_{0}^{\infty}\! \! \! d \mu \, \tilde{c}( \mu ) \mu \sqrt{s}\left( ({\mu}^{3}{s}^{3 / 2}\! + \! 2 4 \mu \sqrt{s}) K_{0}( \mu \sqrt{s}) \! + \! ( 8 \mu^{2}s \! + \! 4 8 ) K_{1}( \mu \sqrt{s}) \right)
\delta G^{A B}\,\biggl( \partial_{A}\partial_{B}\varphi -\overline{\Gamma}_{A B}^{C}\, \partial_{C}\varphi \biggr) + \overline{G}^{A B}\,\biggl( \partial_{A}\partial_{B}\chi - \overline{\Gamma}_{A B}^{C}\partial_{C}\chi - \delta \Gamma^{C}_{A B}\partial\varphi\biggr) + \frac{\partial^2 V}{\partial \varphi^2}\chi=0,
\left(\frac{1}{2}f^{\prime}\,+\, e^f\,\frac{1}{12\sqrt{2}}\bar{g}\left(2e^{-\varphi}+e^{2\varphi}\right)\right)\gamma_0\,\varepsilon_a \,=\,-\frac{1}{6}\sqrt{\frac{1}{2}}e^fe^{\varphi}\frac{1}{2\bar{g}}\gamma^{\nu \rho}R_{\nu \rho}^{\;\;\;ij}\gamma_{ij}\gamma_0\,\varepsilon_a.
\int_0^{1}(1-py)^{-1}(1-y) y^{j-1}dy = \sum_{k=0}^\infty p^k \left( \int_0^{1}(1-y) y^{j+k-1}dy\right) = \sum_{k=0}^\infty \frac{p^k}{(j+k)(j+k+1)}.
\mathfrak{H}:=\left(\begin{array}{cccccccccc}H_3 H_{2}H_1 H_0 & H_3 H_2 H_1 H_0 & H_3 H_2 H_1 H_0 & H_3 H_2 H_1 H_0 & H_3 H_2 H_1 H_0 & H_3 H_2 H_1 H_0 & H_3 H_2 H_1 H_0 \end{array}\right) \in\mathbb F^{7\times 20}.
H_{p,m}(t)=h_{p,m}(t){t \choose p}{t+1-p \choose p}{t-(m+1)p \choose p -1}{t+mp \choose p -1}\prod\limits_{i=1}^{m^2p}\left((t+1-p)^2-\frac{i^2}{m^2}\right), \tilde{H}_{p,m}(t)=\tilde{h}_{p,m}(t){t \choose p}{t+1-p \choose p}{t-(m+1)p \choose p -1}{t+mp \choose p -1}\prod\limits_{i=1}^{m^2p}\left((t+1-p)^2-\frac{i^2}{m^2}\right),
P_{h}(\mathbf{t}_{\mathcal{N}}) = \sum_{\overline{nn'}\edge \of \ \Gamma^{orb}}P^{n,n'}_{h}(\mathbf{t}_{\mathcal{N}}) - \sum_{n\in \mathcal{N}}(\delta_{n,\mathcal{N}}-1)P_{h}^{n}(\mathbf{t}_{\mathcal{N}}),
\left|F(x,t,|y'|)y'- F(x,t,|y|)y\right|= |h(1)-h(0)| =\left|\int_0^1 h'(\tau) d\tau\right| \qquad=\left|\int_0^1 F(x,t,|\gamma(\tau)|)(y'-y) + F_z(x,t,|\gamma(\tau)|)\frac{\gamma(\tau)(y'-y)}{|\gamma(\tau)|}\gamma(\tau) d\tau \right| &\qquad\le |y'-y|\int_0^1 F(x,t,|\gamma(t)|) + F_z(x,t, |\gamma(\tau)|)|\gamma(\tau)| d\tau.
i \frac{\partial}{\partial t}\langle q',t\mid q,0 \rangle_l = -\frac{1}{2}\frac{\partial^2}{\partial q'^2}\langle q',t\mid q,0 \rangle_l + \left( \frac{l(l+1)}{2}\frac{1}{q'^2}+ V(q')\right) \langle q',t\mid q,0 \rangle_l,
(a_1, a_2, a_3) \leftarrow \begin{bmatrix}a_1 0 a_3 0 a_2 0 \end{bmatrix}\to \begin{bmatrix}a_1 a_2 a_2 0 \end{bmatrix}\leftarrow \begin{bmatrix}0 a_1 a_3 a_2 0 0 \end{bmatrix}\to (a_2, a_1, a_3),
\tilde{C}_{N}(z_{1},\overline{z_{2}}) = \frac{1}{(i(\tau_1-\tau_2+i(\eta_1+\eta_2)))^{2}}+ O_{L^1}(1/d_{N})+O_{L^1}\left(\log(d_{N})\sqrt{d_{N}^{3}/N}\right)
\left\langle\mbox{\boldmath$ \phi$}_{j},\mbox{\boldmath$ \psi$}_{j}\right\rangle_{\mathbf{X}_{j}}:=\left\langle \phi_{\operatorname*{D};j},\psi_{\operatorname*{N};j}\right\rangle_{\Gamma_{j}}+\left\langle\psi_{\operatorname*{D};j},\phi_{\operatorname*{N};j}\right\rangle_{\Gamma_{j}},
H_{b,c}(\lim_{n \to \infty}f_n)(x) = \sum_{y \sim x}b(x,y) \lim_{n \to \infty}(f_n(x) - f_n(y)) + c(x)\lim_{n \to \infty}f_n(x) = \lim_{n \to \infty}\left( \sum_{y \sim x}b(x,y)(f_n(x) - (f_n(y)) + c(x) f_n(x)\right) = \lim_{n \to \infty}H_{b,c}f_n(x) = 0,
\delta \,\sum_{k\in \mathbb{Z}}\,\sup_{|\gamma|\le \delta/2}\,g(\xi-k\delta-\gamma) \le \left(\delta+\sum_{k\in\mathbb{Z}\setminus \{0\}}\,\frac{\delta}{1+\delta^2\,(|k|-1)^2}\right) &=3\,\delta+ 2\,\sum_{n=1}^\infty\,\frac{\delta}{1+\delta^2\,n^2}\le3+2\,\int_0^\infty\,\frac{1}{1+x^2}\,dx=c<\infty.
\begin{bmatrix}\frac{\partial F_1}{\partial \lambda}\vdots \frac{\partial F_p}{\partial \lambda}\end{bmatrix}=\begin{bmatrix}a_{11}\ldots a_{1p}\vdots \ddots \vdots a_{p1}\ldots a_{pp}\end{bmatrix}\begin{bmatrix}F_1 \vdots F_p\end{bmatrix},
f ( x ; u , \gamma , \tau ) ~ \equiv ~{\frac{s i n h \big ({\frac{\pi i}{\tau}}( 1 - \gamma ) x \big ) ~ s i n h \big ({\frac{2 \pi i}{\tau}}\gamma u x \big ) ~ s i n h \big ({\frac{\pi i ( N - 1 )}{\tau}}\gamma x \big )}{x ~ s i n h \big ({\frac{\pi i}{\tau}}x \big ) ~ s i n h \big ({\frac{\pi i N}{\tau}}\gamma x \big )}}\ .
F_1(x) =-\frac{p((\tan^2(x)+1)/2)}{\sqrt{2}\sin(x)}=\frac{-1}{\sqrt{2}\sin(x)}\left(\frac{-\sin(x)}{1+\cos(x)}\right)^{\frac{-1}{\sqrt{2}}}, F_2(x) =-\frac{p((\tan^2(x)+1)/2)}{\sqrt{2}\sin(x)}=\frac{-1}{\sqrt{2}\sin(x)}\left(\frac{-\sin(x)}{1+\cos(x)}\right)^{\frac{1}{\sqrt{2}}}.