harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
d_{k+1}(x) =\sum_{i=0}^{(k+1)-3}(i+1)x^i +\sum_{i=k-1}^{2(k+1)-3}(2(k+1)+2-i)x^i &=\sum_{i=0}^{k-3}(i+1)x^i +\sum_{i=k-2}^{2k-3}(2k+2-i)x^i -x^{k-2}(1-2\sum_{i=1}^k x^i)+x^{2k-1}&=d_k(x)-x^{k-2}n_k(x)+x^{2k-1}.
f(x') - a_{n,k}\leq f(x') - f_{n-1}(x') = \sum_{j = n}^\infty 2^{- \gamma m j}g_j (x') \leq \frac{1}{2}2^{- \gamma m n}\sum_{j = 0}^\infty 2^{-\gamma m j}\le 2^{- \gamma m n}.
\left( \begin{array}{cc}\sqrt{\Delta}\:{\mathcal{D}}_+ imr - \lambda -imr - \lambda \sqrt{\Delta}\:{\mathcal{D}}_- \end{array}\right)\left( \begin{array}{c}X_+ X_- \end{array}\right) = 0 \left( \begin{array}{cc}{\mathcal{L}}_+ \lambda \lambda -{\mathcal{L}}_- \end{array}\right)\left( \begin{array}{c}Y_+ Y_- \end{array}\right) = 0
D_{pert, s\eta^i}(\psi+s\mathfrak{c}_{i+1}-s\mathfrak{h}_{i+1}) =sT^s_0(\mathfrak{c}_{i+1}-\mathfrak{h}_{i+1})\mbox{}mod(\ker(D|_{L^2_1})) &=s^2\mathcal{A}\mbox{}mod(\ker(D|_{L^2_1}))
\frac{1}{P(\alpha,\beta;(m+v_1)z+(hn+\mu+v_2))}=\left(\prod_{j=1}^g (h^{(j)})^{-\alpha_j-\beta_j}\right)\cdot P\left(-\alpha,-\beta;\left(\frac{m+v_1}{h}\right)z+n+\frac{(\mu+v_2)}{h}\right).
\forall s, t \in [1/2,1), \widetilde{z}_h^{(s)}= \widetilde{z}_h^{(t)}= \widetilde{u}_h \mbox{a.e. on}B_1 \setminus \left( \cup_{\mathcal{F}^{(s)}_h}B \bigcup \cup_{\mathcal{F}^{(t)}_h}B \right).
J^3\Big(\sum_{i+j+k=-s}\binom{-s}{i,j,k}x_1^{r_1+i}x_2^{r_2+j}x_3^{r_3+k}\Big) = \sum_{i+j+k=-s}\binom{-s}{i,j,k}(-1)^{r_1+r_2+r_3-s}\beta_{r_1+i+1}\beta_{r_2+j+1}\beta_{r_3+k+1}.
\begin{aligned}\gamma_1 = \left(\begin{array}{ccc}0 0 0 0 0 1 0 -1 0 \end{array}\right), \gamma_2 =\left(\begin{array}{ccc}0 0 1 0 0 0 -1 0 0 \end{array}\right), \gamma_3 =\left(\begin{array}{ccc}0 1 0 -1 0 0 0 0 0 \end{array}\right),\end{aligned}
u_\varepsilon= \gamma_\varepsilon-\frac{t_\varepsilon}{\gamma_\varepsilon}-\frac{t_\varepsilon}{\gamma_\varepsilon^3}-(A(\gamma_\varepsilon)-2\xi_\varepsilon)\frac{t_\varepsilon}{2\gamma_\varepsilon}+o\left(\frac{t_\varepsilon \tilde{\zeta}_\varepsilon}{\gamma_\varepsilon}\right)
p_{cov}^{c}= \,\, \frac{e^{-\frac{\pi^2 R^2 \lambda}{2}\sqrt{\frac{\gamma p_i}{p_c}}(1-e^{-k \pi \lambda \mu^2})}-1}{-\frac{\pi^2 R^2 \lambda}{2}\sqrt{\frac{\gamma p_i}{p_c}}(1-e^{-k \pi \lambda \mu^2})},
\left( f_{\omega}^{R *}\tilde{\beta}_{\omega^{\prime \prime}\omega}^{R *}+ f_{\omega}^{R}\tilde{\alpha}_{\omega^{\prime \prime}\omega}^{R}\right) \left( \tilde{\beta}_{\omega^{\prime \prime}\omega^{\prime}}^{R}f_{\omega^{\prime}}^{R}+ \tilde{\alpha}_{\omega^{\prime \prime}\omega^{\prime}}^{R *}f_{\omega^{\prime}}^{R *}\right) ,
\left\langle -\nabla \phi \left( x\right) ,y-x\right\rangle =-\phi \left(y\right) +\int_{0}^{1}\left\langle \nabla \phi \left( x+\lambda \left(y-x\right) \right) -\nabla \phi \left( x\right) ,y-x\right\rangle d\lambda & \leq M\left\vert y-x\right\vert^{2}=M\left\vert -\nabla \phi \left(x\right) \right\vert \left\vert y-x\right\vert^{2},
{\bar{\epsilon}}Q = - \mathrm{T r}\left\{\left({\bar{\epsilon}}\Gamma^{I}\Theta \right) \left( D_{\tau}X_{I}\right) + \left( 2 \Sigma^{0 I}D_{\tau}X_{I}- R \Sigma^{I J}[ X_{I}, X_{J}] \right) \epsilon_{\alpha}\Theta_{\alpha}^{\dag}\right\},
L = \pi^{i j}\dot{g}_{i j}- N^{\mu}T_{\mu}+ P_{\mu}\dot{N}^{\mu}+ \bar{\tilde{C}}_{\mu}\dot{\tilde{C}}^{\mu}+ \bar{C}_{\mu}\{\dot{C}^{\mu}- U_{\alpha \beta}^{\mu}N^{\alpha}C^{\beta}- \tilde{C}^{\mu}\}.
\pi={\cal P}+\frac{i}{W}\frac{\partial W}{\partial z}(\psi\overline{\psi}-\frac{3}{2}) ,\overline{\pi}=\overline{{\cal P}}-\frac{i}{W}\frac{\partial W}{\partial\overline{z}}(\psi\overline{\psi}-\frac{1}{2}) ,
\Phi(z)\left(\frac{z-a}{z}\right)^{\frac{c}{2}\sigma_3}\mathcal{P}(z)\left(\frac{z-a}{z}\right)^{-\frac{c}{2}\sigma_3}=\left(N^{c/2}\eta(z)\right)^{\sigma_3}H(z)\,{\cal F}(\zeta(z))\left(N^{c/2}\eta(z)\right)^{-\sigma_3}\Phi(z).
\sum_{k\in\mathbb{T}}\int\limits_{\gamma(t_{j},\varepsilon)}a_{k}(t)\Psi_{k,t}(x)dt=\int\limits_{(t_{j}-\varepsilon,t_{j}+\varepsilon)}\sum_{k\in\mathbb{T}}a_{k}(t)\Psi_{k,t}(x)dt
\left|\langle Pf, g \rangle_{L^2(\mathbb{T})}\right| = \left| \langle f, Pg \rangle_{L^2(\mathbb{T})}\right| \leq \|f\|_{L^{4/3}}\|Pg\|_{H^4}\leq \|f\|_{L^{4/3}}\|g\|_{L^\infty}.
\log \Upsilon ( x ) = \int_{0}^{\infty}\frac{d t}{t}\biggl [ \biggl ( \frac{Q}{2}- x \biggr )^{2}e^{- t}- \frac{\sinh^{2}\bigl ( \frac{Q}{2}- x \bigr ) \frac{t}{2}}{\sinh \frac{b t}{2}\sinh \frac{t}{2 b}}\biggr ]
c_M =\frac{bK+\sqrt{(1-b)b(K-K^2)}}{1-b-K}\intertext{and, hence, substituting \eqref{eq:cM}in \eqref{eq:Nullcline1}we obtain}h_M =h(c_M)=\frac{\Gamma}{\mu K_1}\frac{c_M(1+c_M)}{(K+c_M)(b+c_M)}.
J_{1,n}(x)=J_{1,n-1}(x)-e^{-\frac{x}{2}}e(\frac{i}{2},x)a_{n-1}(x)-\int_{x}^{\infty}\left( e(\frac{i}{2},t)e^{-\frac{t}{2}}\right)^{\prime}a_{n-1}(t)dt,
\mathcal{B}_\Theta=\displaystyle\left(\bigcup\limits_{i=1}^r\mathcal{B}_{0,i}\right)\cup\left(\bigcup\limits_{1\leq n<m\leq r}\mathcal{B}_{mn}\right)\cup\left(\bigcup\limits_{l_i>1}\mathcal{B}_{i}\right)
\vec{S}= \left[ \begin{array}{c}{\rm Re}\left({\mathcal{W}}\right){\rm Im}\left({\mathcal{W}}\right)\end{array}\right], \qquad\vec{a}=-\left[ \begin{array}{c}{\rm Im}(\phi\partial_x\bar{\phi}){\rm Im}(\phi\partial_t\bar{\phi})\end{array}\right],
\mathbf V(t) = \mathbf S(t-S) \mathbf U(S) + i \int_{S \wedge \tau_R}^{t \wedge \tau_R}\mathbf S(t-\sigma) \mathbf N(\mathbf u(s)) \, ds + i \int_{S}^t \mathbf S(t-s) \mathbf M(\mathbf V(s)) \, dW(s).
K(x_1,x_2,x_3)^{\langle \rho_1 \rangle}=K(y_1,y_2,y_3), y_1=x_1x_3, \y_2=x_2x_3, \y_3=x_3^2, K(x_1,x_2,x_3)^{\langle \rho_2, \rho_3 \rangle}=K(z_1,z_2,z_3), z_1=x_1x_2x_3, \z_2=x_2^2, \z_3=x_3^2
f_{d,\beta}(\bar{\rho}) =2\ln k + d\ln[1-c_\beta/k], f_{d,\beta}(\rho_s) =\frac{2k-s}{k}\ln k + \frac{d}{2}\ln\left[1-\frac{2}{k}c_\beta+\left(\frac{k-s}{k}+s\right)\frac{c_\beta^2}{k^2}\right].
\hat{q}(t)=\sqrt{\frac{\hbar}{2m\omega}}\left[a(t)+a^\dagger(t)\right]=\sqrt{\frac{\hbar}{2m\omega}}\left[a(t_0)\,e^{-i\omega(t-t_0)}+a^\dagger(t_0)\,e^{i\omega(t-t_0)}\right]\ ,
\frac{1}{2}\bigl|\Delta_j(x^{(j-1)}_0)\bigr|+O\left(e^{-Ax}\right)\leq \frac{1}{2}\bigl|\Delta_j(x^{(j)}_0)\bigr|+O\left(e^{-Ax}\right) \leq \bigl|\Delta_j(t)\bigr|\leq \bigl|\Delta_j(x^{(j)}_0)\bigr|+O\left(e^{-Ax}\right).
\begin{aligned}\int_{\mathbb{R}^{2d(N-s)}}\mathbf{1}_{Z_{(s+1):N}\in\mathcal{D}_{N-s}}\mathbf{1}_{|x_i-x_j|\leq \varepsilon}f_0^{\otimes (N-s)}(Z_{(s+1):N}) dZ_{(s+1):N}\leq & \leq \mathcal{Z}_{N-s-1}\varepsilon^d|B_1^d| \left\Vert f_0 \right\Vert_{L^\infty_x L^1_v}\end{aligned}
H_n:=\left|\begin{array}{cc}\Lambda_{n-1}(\theta_n^n)^\top \theta_n^n \boxed{m_{2n}}\end{array}\right|,\Lambda_{n-1}=(m_{i+j})_{i,j=0}^{n-1},\theta_i^j=(m_i,\cdots,m_{i+j-1}),
\frac{i}2 ( \sigma_{\mu}{}^{\rho})_{\alpha \beta}L_{\nu \rho}{}^{\alpha \beta}\ + \ \mathrm{h . c .}= 2 \mathrm{R e}\{b_{1}^{2}\}\sum_{k = 0}^{\infty}\frac{t_{\mathrm{a}i}( k ) \, [ \mathrm{a ~ i ~}]_{\mu \nu}{}( k ) + t_{\mathrm{b}j}( k ) \, [ \mathrm{b ~ j ~}]_{\mu \nu}{}( k )}{( k ! )^{2}}
| k_{\lambda}M |^{2}= k^{2}| M |^{2}= \frac{3 2 e^{2}g^{2}\pi^{2}}{k^{4}}U_{\mu \nu}U_{\alpha \beta}^{\dag}k_{\sigma}k_{\delta}\epsilon^{\mu \nu \lambda \sigma}\epsilon_{\lambda}^{\alpha \beta \delta}
K(h^{\frac{1}{2^{n-1}}},2)^{r_n}a^{2}\sharp_{\nu}b^{2}\leqslant a^{2}\nabla_{\nu}b^{2}- \sum_{k=0}^{n-1}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}\leqslant K(h^{\frac{1}{2^{n-1}}},2)^{R_n}a^{2}\sharp_{\nu}b^{2}
T_{+ 2 - 2}^{- 2}= 4 i \psi_{+ 2 \dot{q}}^{1 -}\psi_{- 2 \dot{q}}^{1 -}, \T_{\pm 2 - \dot{q}}^{- 2}= 4 i \chi \psi_{\pm 2 \dot{q}}^{1 -}, \T_{- \dot{q}- \dot{p}}^{- 2}= - 4 i \delta_{\dot{q}\dot{p}}( 1 + \chi^{2}) .
\left({\frac{\dot{a}_{0}}{a_{0}}}\right)^{2}={\frac{3 2 \pi^{2}G_{5}^{2}}{9 c^{6}}}( \varrho^{2}c^{4}+ 2 \sigma \varrho c^{2}) +{\frac{c^{2}a_{R \, 0}^{\prime \prime}}{a_{0}}}+{\frac{8 \pi G_{5}}{3 c^{2}}}\left( \Lambda +{\frac{4 \pi G_{5}}{3 c^{4}}}\sigma^{2}\right) -{\frac{k c^{2}}{a_{0}^{2}}},
\begin{cases}A(0, 0) = I, f(0)=0,\ \ \D' f(0)=0,\ \ \ ||f||_{C^{1,\alpha}(G \cap Q_{2}(0,0))}\le \delta, Du(0, 0)=e_n,\ \ \ \ ||u-x_n||_{H^{1+\alpha}(G \cap Q_{2}(0,0))}\le \delta, \max \bigg\{[A]_{\alpha, G \cap Q_{2}(0,0)}, ||g||_{H^{\alpha}(G \cap Q_{2}(0,0))}, ||b, c||_{L^{\infty}(G \cap Q_{2}(0,0))}\bigg\}\leq \delta.\end{cases}
-{\bf I}\partial\sb\tau\sp2+ge\sb0F\partial\sb\tau=\exp\left(\frac{ge\sb0}{2}F\tau\right)\left(-{\bf I}\partial\sb\tau\sp2+\frac{g\sp2e\sb0\sp2}{4}F\sp2\right)\exp\left(-\frac{ge\sb0}{2}F\tau\right)\,,
\kappa^2 =\lambda^2-||\nabla^S_{\dot\gamma_c^+}\dot\gamma_c^+||^2\geq (1-e^{-v(x_0)/2})^2-Me^{-v(x_0)/2}&=1-(2+M)e^{-v(x_0)/2}+e^{-v(x_0)}\geq 1-e^{-v(x_0)/4}\geq(1-e^{-v(x_0)/4})^2=\lambda_1^2=\kappa_1^2\,.
\sum_{n=0}^{2g_{u}}A_{u}^{*}(n) z^{n}= \sum_{n=0}^{2g_{u}}A_{u}^{*}(n) q^{g_{u}-n}z^{2g_{u}-n}= \sum_{n=0}^{2g_{u}}A_{u}^{*}(2g_{u}-n) q^{n-g_{u}}z^{n},
a \circledcirc b = (a|{}_* a\mathrm{hcod}\wedge_h b\mathrm{hdom}) \circ (a\mathrm{hcod}\wedge_h b\mathrm{hdom}{}_*| b) a \odot b = [a|{}_* a\mathrm{vcod}\wedge_v b\mathrm{vdom}] \bullet [a\mathrm{vcod}\wedge_v b\mathrm{vdom}{}_*| b]
\begin{aligned}{F_{\Phi_{(3)}}}_x (v)=\sum_{i=1}^{m}\big (6\langle B(v,e_{i}),B(v,e_{i})\rangle_{\mathbb R^q}-\langle B(v,v),B(e_{i},e_{i})\rangle_{\mathbb R^q}\big )<0\end{aligned}
\lim\inf_{k\to\infty}||x_k-q|| <\lim\inf_{k\to\infty}||x_k-p||=\lim\inf_{m\to\infty}||x_m-p|| <\lim\inf_{m\to\infty}||x_m-q||=\lim\inf_{k\to\infty}||x_k-q||.
q_{j}= q_{j}( a ) = \lim_{x \rightarrow a_{j}\atop x \in I}Q ( x ; a ) \ \ \mathrm{a n d}\ \p_{j}= p_{j}( a ) = \lim_{x \rightarrow a_{j}\atop x \in I}P ( x ; a ) , \ \j = 1 , \ldots , 2 m .
\widetilde{F}_{\eta,c,g}:=\eta W(2\omega^{-1}P_{c}v_g,-1)+d\Gamma(\omega)+\varphi(\overline{P}_c v_g)+\lVert \omega^{-1/2}P_{\omega}((c,\widetilde{m}])v_g\lVert^2 A_{\eta,c,g}:=\eta W(2\omega^{-1}P_{c}v_g,-1)+d\Gamma(\omega)+\lVert \omega^{-1/2}P_{\omega}((c,\widetilde{m}])v_g\lVert^2
\begin{array}{rcl}{{L}_{v}}={{C}_{V}}{{\left( V-{{V}^{des}}(t) \right)}^{2}}& +{{C}_{ICE}}{{\left(\frac{{{P}_{ICE}}}{{{\eta}_{ICE}}(\centerdot )}\right)}^{2}}+{{C}_{FR}}{{\left({{P}_{FR}}\right)}^{2}}\end{array}
\textstyle - r \sum_{u \in \mathbb F_p^\times}u (1 + u)^{r-1}= \textstyle - r \sum_{m=0}^{r-1}{r-1 \choose m}\sum_{u \in \mathbb F_p^\times}u^{m+1}= \textstyle - \sum_{m=1}^{r}m{r \choose m}\sum_{u \in \mathbb F_p^\times}u^{m}= L_r + r \delta_{s=p-1},
( G_{i j}^{E})_{i j = 1 , \cdots , M + 1}= \left( \begin{array}{c c}{{\displaystyle g_{r s}= ( \rho \cdot \cos \theta )^{\frac{2}{M}}{\bar{g}}_{r s}}}&{{\displaystyle 0}}\\{{}}&{{\vdots}}\\{{0 \cdots 0}}&{{{\dot{\vec{x}}}^{2}= \sin^{2}\theta}}\\ \end{array}\right) \qquad .
\breve Q_0^\mu(x)= \Big(\tau_0 -\frac 1{\Gamma(1-\mu)}\Big)I_-^\mu \big\{(1+x)^{1-\mu}\hat h_0(x)\big\}+ \frac 1{\Gamma(1-\mu)}I_-^\mu \big\{(1+x)^{-\mu}(\hat h_0(x)-1)\big\}.
0 = \frac{(L + i_j)!}{L!}\frac{1}{a_j^{i_j}}r_{L+i_j, j}= \sum_{\substack{h+n=L+i_j 0 \leq h \leq L}}\frac{(L + i_j)!}{h! n!}a_j^{n-i_j}c_h= \sum_{h=0}^L \binom{L + i_j}{h}a_j^{L-h}c_h
F(x,\lambda)= -\Big(E_\lambda(x)+ M_1(x,\lambda) +\frac{1}{2\lambda}E_\lambda(x) R Q(x) +O\Big(\frac{1}{\lambda^2}\Big)\Big) \int_0^x \Big(E_\lambda(-t)-M_1(t,\lambda) -\frac{1}{2\lambda}E_\lambda(-t) R Q(t) +O\Big(\frac{1}{\lambda^2}\Big)\Big) R f(t)\,dt
W_p\left\{\gamma\,:i = 1, 2, \ldots, n,\,\gamma(t_i) \in B_i\right\}= \left(D_{t_1}\left(\chi_{B_1}D_{t_2 - t_1}\left(\chi_{B_2}\cdots D_{t_n-t_{n- 1}}(\chi_{B_n})\cdots\right)\right)\right)(p)
\begin{aligned}\left|\sigma \int_0^T \int_{\Gamma(t)}e \, dx \int_{\Gamma(t)}v dx \, dt\right| \le\sigma \int_0^T |\Gamma(t)| \left(\int_{\Gamma(t)}e^2 dx\right)^{\frac12}\left(\int_{\Gamma(t)}v^2 dx\right)^{\frac12}\, dt \le \sigma \max\limits_{t\in[0,T]}|\Gamma(t)|\|e\|_0\|v\|_0.\end{aligned}
y(t_{n+1})=y(t_n) +\int_{t_n}^{t_{n+1}}[(1-\theta)f(t_n,y(t_n),y(qt_n))+\theta f(t_{n+1},y(t_{n+1}),y(qt_{n+1}))]{\rm d}t &+\int_{t_n}^{t_{n+1}}g(t_n,y(t_n),y(qt_n)){\rm d}B(t).
H_{L C}^{( A d S )}= \frac{1}{2 p_{-}}\int_{0}^{l}d \sigma \, \left( \pi_{i}\pi_{i}+ \frac{1}{( 2 \pi \alpha^{\prime})^{2}}\frac{R^{4}}{Y^{4}}Z^{i \, \prime}Z^{i \, \prime}\right) \ .
x^2 z y = x (q^2zx - q^2 + 1)y = q^2 (xz)(xy) + (1-q^2)xy = q^2 (1-q\nu_y)(1-q^{-1}\nu_z) + (1-q^2)(1-q^{-1}\nu_z) = 1 - q^3\nu_y - q^{-1}\nu_z +q^2\nu_y\nu_z.
|T^{\prime}| =t+(q+1)(t-2)[(m+1)(k-3)+m]+(2-q)(t-2)[((m-1)+1)(k-3)+ (m-1)] &=t+(t-2)[(q+1)(mk-2m+k-3)+(2-q)(mk-2m-1)] &=t+(t-2)[(q+1+2-q)(mk-2m)+(q+1)(k-3)-(2-q)] &=t+(k^2-k-5)(t-2)
\frac{1}{\sqrt{- g}}{\cal L}_{f}= \overline{{{\psi}}}e^{\mu}{}_{a}\gamma^{a}( i \partial_{\mu}+ i \omega_{\mu}+ q V_{\mu}) \psi - M \exp \! \Big ( - \sqrt{\frac{\kappa}{6}}\phi \Big ) \overline{{{\psi}}}\psi .
\left( \begin{matrix}{\lambda_{-}}\\{\lambda_{+}}\\ \end{matrix}\right) = \left( \begin{matrix}{\sin k_{2}x_{2}}\\{\cos k_{2}x_{2}}\\ \end{matrix}\right) e^{i ( \omega t - k_{1}x_{1})}\ \mathrm{w i t h}\k_{2}= \left( n + \mathrm{\frac{1}{2}}\right) \mathrm{\frac{\pi}{L}},
\eta_{S'}^{mic,Q}(\sigma ) = \sum_{\xi \in \Xi({^\vee}G)^{Q}}e(\xi)(-1)^{\dim S_{\xi}- \dim S'}\ \mathrm{tr}\left( \tau_{S'}^{mic}(P(\xi))(\sigma_{\xi})\right) \, (\pi(\xi), \mathcal{I}_{\xi}) ,
\widehat{h}_R(\omega) =e_I\otimes \widehat{h}(e_J)=\sum_{K}\langle \widehat{h}(e_J),e_K\rangle e_I\otimes e_K= &=\sum_{K}\frac{h^q}{q!}(e_J,e_K)e_I\otimes e_K=\sum_{K,L}\frac{h^q}{q!}(e_L,e_K)\langle e_L,e_J\rangle e_I\otimes e_K &= \sum_{K,L}\frac{h^q}{q!}(e_L,e_K) ( e_L\otimes e_K)\circ (e_I\otimes e_J) =\frac{h^q}{q!}\circ \omega.
j(\tau) :=\frac{\big(1+240\sum_{n=1}^{\infty}\sum_{d\mid n}d^3 q^n\big)^3}{q\prod_{n=1}^{\infty}(1-q^n)^{24}}&\hphantom{:}=\frac{\eta(\tau)^{24}}{\eta(2\tau)^{24}}+3\cdot2^{8}+3\cdot2^{16}\,\frac{\eta(2\tau)^{24}}{\eta(\tau)^{24}}+2^{24}\,\frac{\eta(2\tau)^{48}}{\eta(\tau)^{48}}begin{align*}2mm] \hphantom{:}=q^{-1}+744+196884q+\cdots,
I_{m}^{\alpha}:=\int_{0}^{\infty}\frac{\rho^{\alpha}}{(1+\rho^{2})^{m}}d\rho=\frac{2m}{\alpha+1}I_{m+1}^{\alpha+2}\alpha+1<2m I_{m}^{\alpha}=\frac{2m}{2m-\alpha-1}I_{m+1}^{\alpha}\alpha+1<2m I_{m}^{\alpha}=\frac{2m-\alpha-3}{\alpha+1}I_{m}^{\alpha+2}\alpha+3<2m.
\sigma_{\!\operatorname{ess}}(\widetilde{B}_{0}) = \emptyset \Leftrightarrow \lim_{\xi\to+\infty}\xi \int_{\xi}^{+\infty}\widetilde{w}(\tau) d\tau = 0, 0\not\in\sigma_{\!\operatorname{ess}}(\widetilde{B}_{0}) \Leftrightarrow \sup_{\xi\in\mathbb{R}_+}\xi \int_{\xi}^{+\infty}\widetilde{w}(\tau) d\tau < +\infty.
\frac{d}{dt}(\Im \int (x^2\chi(\frac{|x|}R))' u' \bar u dx) \geq \int_{|x|<R}|u'|^2dx+\frac{\alpha}{\alpha+2}\int_{|x|<R}|u|^{\alpha+2}dx-C\int_{|x|>R}(|u|^2+ |u'|^2+ |u|^{\alpha+2}) dx \geq \delta-C\int_{|x|>R}(|u|^2+ |u'|^2+ |u|^{\alpha+2}) dx,
\omega_{\pm}\equiv{\cal Q}_{\pm}(\mp \infty )=\pm 2\pi \displaystyle\frac{p-1}{p+1}(S-2S^{\pm})+2\pi \displaystyle\frac{p}{p+1}\left( \displaystyle\frac{\alpha}{\pi}-\delta +2k_{W}^{\pm}-2k_{\pm}\right) .
\mathcal{H}(w^n_{j-1}\vee k,w^n_j\vee k) \ge \mathcal{H}(w^n_{j-1},w^n_j)= \frac{w^{n+1}_j}{a_j}, \mathcal{H}(w^n_{j-1}\vee k,w^n_j\vee k) \ge \mathcal{H}(k,k)= \frac{k}{a_j},
{\rm Hilb}\big(\widetilde{\rm ANWC}_{W},t\big)=\prod_{i=1}^{l}{{1-t^{m_{i}+1}\over 1-t^{m_i}}}{\prod_{i=1}^{l}{1-t^{a_i}\over1-t}}=P_{\rm af\/f}(W,t) \prod_{i=1}^{l}(1-t^{a_i}),
\widehat{\mathbf{1}_{P_{\textbf{c}}}}(\pmb{\xi})=\frac{\exp{(2\pi i \langle \pmb{\xi}, \mathbf{c}\rangle)}}{p^{2kn}}\sum_{\mathbf{v}\in P_{\mathbf{c}}}\exp{\Big(2\pi i \frac{\langle \pmb{\xi}, \mathbf{v}-\mathbf{c}\rangle}{p^{2k}}\Big)}.
P_Z(t)=|Z+tB_2^n|= |B_2^2|\int_{tB_2^{n-2}}\left(1+\sqrt{t^2-|x|^2}\right)^{2}dx=t^{n-2}|B_2^{2}|\int_{B_2^{n-2}}\left(1+t\sqrt{1-|x|^2}\right)^{2}dx.
g_{\ell,j}(z)\equiv \sum_{n=1}^{\infty}a(\ell^{m_\ell}n)q^n-\sum_{n=1}^{\infty}a(\ell^{m_\ell+1}n)q^n\equiv\sum_{\substack{n=1 \ell\nmid n}}^{\infty}a(\ell^{m_\ell}n)q^n\pmod{\ell^j}.
(n+1)S_{n,l}S_{n,l}^{*}=t_{1}(S_{n,l}^{*}S_{n+1,l}+S_{n,l}S_{n-1,l}^{*})(1-S_{n,l}S_{n,l}^{*})+l(1-S_{n,l}S_{n,l}^{*}).
\psi_{a_{1}\cdots a_{2 k - 2}}= F_{\beta}^{- 1}\: ( N^{\dagger}\: N )^{- 1}\: N_{a}^{( \beta ) \: \dagger}\: D_{b}^{( \beta )}\: F_{\beta}\: \psi_{a_{1}\cdots a_{2 k - 2}a b}\: + \: \psi_{a_{1}\cdots a_{2 k - 2}}^{[ h ]}
\int_{0}^{2\pi}\int_{0}^{\pi}\int_{0}^{2\pi}D^{l}(\alpha\beta\gamma)_{nm}^{*}D^{l^{\prime}}(\alpha\beta\gamma)_{n^{\prime}m^{\prime}}sin\beta d\alpha d\beta d\gamma=\delta_{ll^{\prime}}\delta_{nn^{\prime}}\delta_{mm^{\prime}}\frac{8\pi^{2}}{2l+1}.
\left\vert{\xi}\right\rangle ={\frac{1}{\left({1 +{\vert \xi \vert}^{2}}\right)^{J}}}\sum_{m = 0}^{2 J}\xi^{m}{\binom{2 J}{m}}^{1 / 2}\left\vert{J , \- J + m}\right\rangle \quad .
\int_{B_r}\beta_{n+1}(1-2s)y^{-2s}\langle A\nabla U,\nabla U\rangle dz =\int_{B_r}(y+O(|z|)y)(1-2s)y^{-2s}\langle A\nabla U,\nabla U\rangle dz &=(1-2s)D(r)+O(r)D(r).
\sum_{\nu\in\mathbf{P}(\mu)}{}^{t}\Theta_{x_1^*}{}^{t}\Theta_{\mu,\tau}\Theta_{x_1}(v_\lambda)=\sum_{\nu\in\mathbf{P}(\mu)}\Theta_{x_1}\Theta_{e_{-\nu}}\phi_{\tau,\nu}\Theta_{x_1}(v_\lambda)=0
\sigma(0)\tau^{+}{=}\tau^{+}(0)\overline{\tau}^{-}(0) \tau^{+}{+}0 &{=}\tau^{+}(0)\left(\tau^{+}(0)\overline{\tau}^{-}{-}T(\tau^{+}(1)\overline{\tau}^{-})\right) &{=}0{-}\tau^{+}(0)Th &{=}(Th)(0)\tau^{+}{-}T((Th)(1)\tau^{+})+0 &{=}0{+}T\tau^{+},
G_{p}= \{x\in X\ |\ \sum_{i=0}^{N-1}\chi_{C_1}(x_i)=p\{\rm mod}\n\}, H_p= \{x\in X\ |\ \sum_{i=-N}^{-1}\chi_{C_1}(x_i)=p\{\rm mod}\n\}, L_p= \{x\in X\ |\ \sum_{i=N}^{N+n-1}\chi_{C_1}(x_i)=p\{\rm mod}\n\}
\Pr \left(N_{X,Y}\notin (1\pm\lambda) e_{D}(X,Y)p_{in}\mbox{for some pair}X \mbox{and}Y \right) \leq \binom{m}{k}^2 e^{-\frac{\lambda^2}3(1- \lambda)|X||Y|p\cdot p_{in}}\leq n^{2k}e^{-\frac{\lambda^2k^2pp_{in}}{6}}= (ne^{-\frac{\lambda^2 k p p_{in}}{12}})^{2k}= o(1/n).
\begin{array}{lll}\displaystyle{\left.\frac{d}{ds}\right|_{s=0}\exp (tX) \exp (t+s)Y \exp(tZ)}=\displaystyle{\left.\frac{d}{ds}\right|_{s=0}\alpha(t) \exp (s\operatorname{Ad}(\exp (-tZ))Y)}\ & \ =(\operatorname{Ad}(\exp (-tZ)Y)^L)_{\alpha(t)}. \end{array}
[ \mathbf{u}^{(3)}]^\varpi_j = ( u_0, u_{I^\varpi_{j,1}}, u_{I^\varpi_{j,2}}, \ldots, u_{I^\varpi_{j,q-1}}, u^{(1)}_i + \lambda u^{(2)}_i, u_{I^\varpi_{j,q+1}}, u_{I^\varpi_{j,q+2}}, \ldots, u_{I^\varpi_{j,\#_{I^\varpi_j}}}) .
p'_k(x) = \frac{-\beta_k e^{\alpha_k h_k+\beta_kx+\gamma_k}}{(1+e^{\alpha_k h_k+\beta_kx+\gamma_k})^2}=-\beta_k \frac{1}{(1+e^{\alpha_k h_k+\beta_kx+\gamma_k})}\frac{e^{\alpha_k h_k+\beta_kx+\gamma_k}}{(1+e^{\alpha_k h_k+\beta_kx+\gamma_k})}=-\beta_k p_k(x) (1-p_k(x)).
\dfrac{k^{\psi}}{\left| \omega(\phi) \right|}= \dfrac{k^{\psi}}{\left( \phi -t_1 \right) \left( t_2 - \phi \right)}= \dfrac{k^{\psi}}{\varphi k^{\psi}\left( t_2 - \phi \right)}\le \dfrac{1}{\xi_1 \, \xi_2'}.
\alpha_2(\mathbf h_2,\breve{\mathbf g}_2,\pmb{\phi}_2) = \alpha_2(\tilde{\pi}_2(t(2))\mathbf h_2, \tau_2(t(2))\breve{\mathbf g}_2,\omega_2(t(2))\pmb{\phi}_2) = \alpha_2(\tilde{\pi}_2(t(2))\mathbf h_2, \mathbf g_2,\omega_2(t(2))\pmb{\phi}_2),
Q_{\cal M}= \left( \begin{array}{c c c c c c c c}{1}&{1}&{- 3}&{1}&{0}&{0}&{0}\\{0}&{0}&{1}&{- 2}&{1}&{0}&{0}\\{0}&{0}&{0}&{1}&{- 2}&{1}&{0}\\{0}&{0}&{0}&{0}&{1}&{- 2}&{1}\\ \end{array}\right)
\sum_{s \geq 0}\big( E_{1,1}(s+1)h_1(-s+m-1) + h_1(s+m)E_{2,2}(-s) \big){}=\sum_{s \geq 0}\big( S_{1,1}(s+1)-S_{2,2}(s+m) \big) - \sum_{s=0}^{m-2}E_{1,1}(s+1)E_{2,2}(-s+m-1),
( \Phi , \Psi ) = N \int_{\Lambda}d A^{\mathrm{T r}}\prod_{x , i}\psi ( A_{x , i}^{\mathrm{T r}}) \det M^{\prime}( A^{\mathrm{T r}}) \Phi^{*}( A^{\mathrm{T r}}) \Psi ( A^{\mathrm{T r}}) ,
V ( L , \Delta \Theta ) ~ = ~ - \frac{2}{\pi}~ \frac{( 2 g_{\mathrm{\tiny ~ Y M}}^{2}N )^{\frac{1}{2}}}{L}( 1 - l^{2})^{\frac{3}{2}}\left( \int_{1}^{\infty}\frac{d y}{y^{2}\sqrt{( y^{2}- 1 ) ( y^{2}+ 1 - l^{2})}}\right)^{2}~ .
Z(\overline\delta,\overline\mu,\beta)=\frac{\beta}{\sqrt{\overline\mu - \frac{1}{2}(1-\beta)}}\frac{_3\big( \frac{1}{2}(\sqrt{\beta}+\frac{1}{\sqrt{\beta}})+\overline\delta \overline\mu\,|\, \frac{1}{\sqrt{\beta}}, \sqrt{\beta},\overline\delta \big)}{_3\big( \sqrt{\beta}\,|\, \frac{1}{\sqrt{\beta}}, \sqrt{\beta},\overline\delta \big)},
\Sigma = - \lambda_{1}\frac{2 + N_{1}}{9 6 \pi^{2}}M^{2}\ln \left( \frac{M^{2}}{4 \pi \mu^{2}}\right) - \lambda_{1}\frac{2 + N_{1}}{3}T^{2}h \left( \frac{M}{T}\right) + \sigma^{2}~ ~ .
\mathbb{E}\|\check{X}_{k+1}-Y_{k+1}\| = \left(1-\frac{2h}{(k+1)^{\gamma}}\right)\mathbb{E}\|\check{X}_k-Y_k\| +\frac{2\sigma h}{\sqrt{k+1}}= O\left(\frac{1}{k^{1/2-\gamma}}\right)
{\bf S}_{B H}={\frac{\bf A}{4 G_{N}}}; \qquad{\bf A}= 2 \pi^{2}( r^{2}e^{2 U})_{r = 0}^{\frac{3}{2}}= 2 \pi^{2}\Big ({\frac{Z_{h o r}}{3}}\Big )^{\frac{3}{2}}
p_{2Dc}\left( xy|\mu_{x}\sigma;\rho\right) \overset{}{=}\dfrac{1}{2\pi\Sigma^{2}\sqrt{1-\rho^{2}}}\exp\left\{\frac{-1}{2\left( 1-\rho^{2}\right)}\left[ \frac{\left( x-\mu_{x}\right)^{2}}{\sigma^{2}}+\frac{y^{2}\sigma^{2}}{\Sigma^{4}}-\frac{2\rho\left(x-\mu_{x}\right) y}{\Sigma^{2}}\right] \right\}
{\hat{M}}= - \frac{1}{2 \pi g P}\frac{1}{R^{\prime}}\frac{d}{d z}\frac{R{R^{\prime}}^{2}( 1 + D^{2}{R^{\prime}}^{2})}{( 1 +{R^{\prime}}^{2})^{3 / 2}}\frac{d}{d z}\frac{1}{R^{\prime}}
\frac{dP}{d\Omega_8}= 2\kappa^2 T_{0i}\hat{k}_i ={2\kappa^2 T_0\over 105^2\, \Omega^2_8}\omega^8\,{1 \over 3!}\, \left[ J^{ijk}J^{*}_{ijk}- 3 \, \hat{k}_l \hat{k^m}J^{lij}J^*_{mij}\right]
\begin{aligned}\int_0^T \int_0^1\Big( s \Theta a (w_x)^2 + s^3 \Theta^3\frac{(x-x_0)^2}{a}\w^2 \Big) e^{2s \varphi}\, dx dt \le C\int_0^T \int_{\omega}w^2 dxdt & + C\left(\int_0^T \int_0^1f^2e^{2s\varphi}dxdt +\int_0^T \int_0^{\beta_1}f^2 e^{2s \Phi_1(t,-x)}dxdt+ \int_0^T \int_{\lambda_2}^1 f^2e^{2s \Phi_2(t,x)}dxdt\right),\end{aligned}
df(\check{X}_t) = -2\check{X}_t^T \mathbf{L}\check{X}_tdt-2\alpha(t)\left(\nabla U_v(X(t))-\mathbf{1}\otimes\frac{1}{m}\sum_{i=1}^m \nabla u_i(X^{(i)}(t))\right)^T \check{X}_t dt+\sigma\alpha(t) dt+\sqrt{2\sigma\alpha(t)}\check{X}_t^T d\check{B}_t.
\frac{1}{N(\lambda)}\smallint\limits_Gd_\xi\textnormal{Tr}(\sigma_{\textnormal{glob}}(x,\xi))dx=\frac{1}{(2\pi)^n}\left( \smallint\limits_{T^*\mathbb{S}(G)}\sigma_{\textnormal{loc}}(x,\xi')d\mu_L(x,\xi')\right)\frac{\lambda^n}{N(\lambda)}+O(\lambda^{-1}).
\Gamma [ \phi ] ={\frac{1}{2}}\int_{0}^{\infty}{\frac{d T}{T}}\, \mathrm{e}^{- m^{2}T}\int{\cal D}x \, \mathrm{e x p}\left[ - \int_{0}^{T}\left({\frac{{\dot{x}}^{2}}{4}}+ U^{\prime \prime}\left( \phi ( x ) \right) \right) \right]
P(X>Y,X\geq T) =P(X>Y,X\geq T, Y\geq T)+P(X>Y,X\geq T,Y< T) &=\frac{1}{2}P(X\geq T, Y\geq T)+P(Y< T\leq X) &= \frac{1}{2}\left\{P(X\geq T, Y\geq T)+ P(Y< T\leq X)\right\}+\frac{1}{2}P(Y< T\leq X) & =\frac{1}{2}P(X\geq T) +\frac{1}{2}P(Y< T\leq X).
0 \geq F^{ij}\nabla_i\nabla_j (\log Q) = \ \frac{F^{ij}(2H\nabla_i\nabla_j H-4\nabla_i\nabla_j K)}{H^2-4K}+\alpha\frac{F^{ij}\nabla_i\nabla_j K}{K}+\frac{2F^{ij}\nabla_iH\nabla_j H}{H^2-4K}-(\alpha^2+\alpha)\frac{F^{ij}\nabla_iK\nabla_j K}{K^2}.
a_{j-1}w =(a_{n-1}a_1)^{i_1}\dotsm(a_ja_{n-j})^{i_{n-j}}a_{j-1}(a_{j-2}a_{n-j+2})^{i_{n-j+2}}\dotsm(a_{s+1}a_{s-1})^{i_{s-1}}a_s^{i_s}v\intertext{and}a_jw =(a_{n-1}a_1)^{i_1}\dotsm(a_ja_{n-j})^{i_{n-j}}a_j(a_{j-2}a_{n-j+2})^{i_{n-j+2}}\dotsm(a_{s+1}a_{s-1})^{i_{s-1}}a_s^{i_s}v.
Z ({\cal A}, G , q ) = \mathrm{e}^{F ({\cal A}, G , q )}= \sum_{N = 1}^{\infty}Z_{N}({\cal A}, G ) q^{N}= \prod_{n = - \infty}^{+ \infty}Z^{( \mathrm{c o v )}}( G , \mathrm{e}^{-{\cal A}v ( n )}q ) .
\begin{array}{ll}det_{q}L(u) = a (u + i \kappa /2) d (u - i \kappa /2) - b (u + i \kappa /2)c (u - i \kappa /2) \, = Q_{1}u^{2}+ Q_{2}u - Q_{3}+ \kappa^{2}/4 Q_{1}\quad ,\end{array}
\|I_1\|_{L^1(\mathcal G)}\leq |r_oB|^{1-{1\over p}}\left(\int_{r_oB}\big|[b,R_j](a)(g)dg\big|^pdg\right)^{1\over p}&\leq C|B|^{1-{1\over p}}\|a\|_{L^p(\mathcal G)}\leq C|B|^{1-{1\over p}}|B|^{{1\over p}-1}&=C,
K(s, t)=\begin{cases}w(s)-\left(w(a)+\psi(\lambda)\frac{w(b)-w(a)}{2}\right), ~~~~s\in[a, t), w(s)-\left(w(a)+(1+\psi(1-\lambda))\frac{w(b)-w(a)}{2}\right), ~~~~s\in[t, b].\end{cases}
\|\ \sum_{j=1}^{q}\frac{1}{\deg Q_j}m_{f}(r;Q_j) \le (n+1+\epsilon)T_{f}(r, s) &+c\left(m_0(\mathcal{L};r,s)+{\rm Ric}_{p}(r,s)+\kappa \log^{+}\mathcal Y\left(r^{2}\right)+\kappa \log^+r\right),
P_+(z)P_-^{-1}(z) = \mathbb{I}+\mathcal{O}(n^{-1}) n\to\infty, z\in \partial D\left(0,r_n\right), P(z)N^{-1}(z) = \mathbb{I}+\mathcal{O}(n^{-1}) n\to\infty, z\in \partial D\left(0,R\right).
\hat{\Phi}( x^{0}, x^{1}) = \frac{1}{\sqrt{2 \pi}}\int_{0^{+}}^{\infty}d k^{1}\frac{1}{\sqrt{k^{1}}}\left[ \hat{\Lambda}( k^{1}) \, e^{- i k^{1}( x^{0}+ x^{1})}\, \, + \, \, \hat{\Lambda}^{\dagger}( k^{1}) \, e^{i k^{1}( x^{0}+ x^{1})}\right] ,
\exp\left(\sum_{i, k \geq 0}I_{i, k}\hbar^{i - 1}\right) = \sum_{\{n_{i, k}\}}\prod_{i, k \geq 0}\frac{\hbar^{(i - 1)n_{i, k}}}{n_{i, k}!}I_{i, k}^{n_{i, k}}.