harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\phi(t)-\phi(p_2) =\sum_{i=2}^{n-1}\int_{p_i}^{p_{i+1}}\phi'(s)ds+\int_{p_{n}}^t \phi'(s)ds & \ge \sum_{i=2}^{n-1}\int_{p_i}^{p_{i+1}}-\dfrac{b_i}{s}ds+\int_{p_{n}}^t -\dfrac{b_n}{t}ds & = -\sum_{i=2}^{n-1}b_i\log(p_{i+1}/p_i)-b_n\log(t)+b_n\log(p_n) & \ge -b_n\log(p_{n+1})-b_n\log(t) & \ge -2b_n\log(t).
\frac{p_{1}}{p_0}=\frac{(\gamma+1)\rho_{1}-(\gamma-1)\rho_0-2\tilde{b}{\rho_{1}}}{(\gamma+1)\rho_0-(\gamma-1)\rho_{1}-2\tilde{b}{\rho_{1}}},~~~~~~~~~\rho_1>\rho_0 &u_1=\left(\frac{(p_1-p_0)(\rho_1-\rho_0)}{{\rho_0}{\rho}_{1}}\right)^{1/2},~~~\zeta=a_0\sec{\theta},~~~~v_1=0.
\begin{array}{ll}\mathfrak{g}(S,S')_{(0,0)}= \mathfrak{tri}(S)\oplus\mathfrak{tri}(S'), \mathfrak{g}(S,S')_{(1,1)}= \iota_0(S\otimes S'), \mathfrak{g}(S,S')_{(1,0)}=\iota_1(S\otimes S') , \mathfrak{g}(S,S')_{(0,1)}=\iota_2(S\otimes S').\end{array}
M_{X_{E}}(\rho_{1}(t))\equiv \int\limits_{\Gamma_{1}}d\mathbf{x}_{1}\rho_{1}(\mathbf{x}_{1},t)X_{E}(\mathbf{x}_{1},t)=-\int\limits_{\Gamma_{1}}d\mathbf{x}_{1}\rho_{1}(\mathbf{x}_{1},t)\ln \frac{\rho_{1}(\mathbf{x}_{1},t)}{A_{1}}\equiv S(\rho_{1}(t)).
e^{- W ( J , \xi )}= \int{\cal D}\phi_{i}{\cal D}\eta{\cal D}\overline{{{\eta}}}e^{- S - \frac 1{2 \xi}\int d^{d}x F^{2}+ \int d^{d}x \overline{{{\eta}}}( x ) \frac{\delta F}{\delta \phi_{i}( x )}\Delta_{i}\eta ( x ) + \int d^{d}x J_{i}( x ) \phi_{i}( x )},
T(X,Y) =\widetilde{\nabla}_XY-(-1)^{(k-1)(l-1)}\widetilde{\nabla}_YX-[X,Y] &=\sum_{i=1}^{k}\sum_{j=1}^l (-1)^{i+j}(\widetilde{\nabla}_{X_i}Y_j-\widetilde{\nabla}_{Y_j}X_i-[X_i,Y_j])\wedge X[i]\wedge Y[j] &=0.
X\times X^\ell\supset D_I:=\{(x,x_1,\dots,x_\ell)\mid x=x_a\,\forall\, a\in I\}\cong X\times X^{\ell-i}\,, X^{n+\ell}\supset \Delta_J:=\{(y_1,\dots,y_{n+\ell})\mid y_a=y_b\,\forall\, a,b\in J\}\cong X\times X^{\ell-i}\,,
\int_{-\infty}^{\infty}q^{\beta^{2}}\left(q^{-\beta+n+1};q\right)_{\infty}L_{n}^{(-\beta)}\left(-1;q\right)A_{q}\left(-q^{\beta+n}\right)d\beta=\frac{1}{\log q^{-1}\left(q;q\right)_{n}}\int_{-\infty}^{\infty}\exp\left(\frac{y^{2}}{\log q}\right)dy,
P m_{W^{( 1 )}}^{2}= m_{W^{( 2 )}}^{2}, \hspace{0 . 2 c m}P m_{Z^{( 1 )}}^{2}= m_{Z^{( 2 )}}^{2}, \hspace{0 . 2 c m}P W_{\mu}^{( 1 ) \pm}= ( - 1 )^{\delta_{\mu}}W_{\mu}^{( 2 ) \pm}, \hspace{0 . 2 c m}P Z_{\mu}^{( 1 ) \pm}= ( - 1 )^{\delta_{\mu}}Z_{\mu}^{( 2 ) \pm},
\unitlength .5cm\begin{picture}(11,7)\put(4,1){\line(1,0){5}}\put(4.5,3){\line(1,0){2}}\put(5.5,4){\line(1,0){2}}\put(9.5,4){\line(1,0){2}}\put(5,3.5){\line(0,-1){3}}\put(6,4.5){\line(0,-1){2}}\put(7,5.5){\line(0,-1){2}}\put(8.5,5.5){\line(0,-1){2}}\put(10,5.5){\line(0,-1){2}}\end{picture}
\partial_x\phi_N = \frac54 a x^{1/4}\phi_N\le c\,a_0\langle x\rangle^{1/4}\phi_N, \partial_x^2\phi_N \le c\,(a_0^2+a_0)\,\langle x\rangle^{1/2}\phi_N, \cdot \hskip10pt \hskip40pt \cdot \cdot \hskip10pt \hskip40pt \cdot \partial_x^5\phi_N \le c\,(a_0^5+a_0)\,\langle x\rangle^{5/4}\phi_N.
\hat{P}_{n}(x^{2})=\frac{4^{n}}{n!\,(\theta)_{2n}}\, W_{n}\!\left(\frac{x^{2}}{4};-\frac{1}{4}+\frac{\theta}{2},\frac{1}{4},\frac{1}{4}+\frac{\theta}{2},\frac{3}{4}\right)\!.
\oint \mathrm{d}u_{1}\oint \mathrm{d}u_{2}\oint \mathrm{d}u_{3}\frac{( u_{1}- u_{2})^{2}}{( u_{1}- u_{3})^{2}( u_{2}- u_{3})^{4}}\prod_{i = 1}^{4}\frac{( z_{i}- u_{1})^{1 / 2}( z_{i}- u_{2})^{1}}{( z_{i}- u_{3})^{1}}\, .
\iota_{\mathcal{L}_{\Pi\eta_1}\eta_2}X_1= \big(\mathcal{L}_{\Pi\eta_1}\eta_2 \big)(X_1) = a(\Pi\eta_1)(\eta_2(X_1))- \eta_2([\Pi\eta_1, X_1]_{A}) = a(\Pi\eta_1)(\eta_2(X_1))- \iota_{\eta_2}([\Pi\eta_1, X_1]_{A})
W_{\Lambda}\left( \phi ,\phi^{*},\overline{\phi},\pi\right) \equiv \overline{W}\left( \phi,\phi^{*},\overline{\phi},\pi,\varphi_{q}\left(\phi,\phi^{*},\overline{\phi},\pi\right),\varphi^{*}=0,\overline{\varphi}=0,\pi_{\varphi}=0\right)
I(G) = \ (x_1x_5,x_1x_6,x_1x_8,x_1x_{10},x_2x_5,x_2x_6,x_2x_9,x_2x_{11},x_3x_7,x_3x_8,x_3x_9,x_3x_{11},x_4x_7, \ \x_4x_8,x_4x_{10},x_4x_{11}, x_5x_8,x_5x_9,x_6x_{10},x_6x_{11},x_7x_9,x_7x_{10},x_8x_{11})
(N-M+\frac 12)_M(N-M+1)_{M}= \sum_{l=0}^M(-1)^{l}{M\choose l}(j-l+1)_l(N-M-j+l+\frac 12)_{M-l}\times \times (N-M-j+l+1)_{M-l}(j-l-2N+M+\frac{1}{2})_{l}
E[Y({\bar \rho}_j)|{\cal F}_{\tau_j}] = E \bigg[ \int_{\tau_j}^{{\bar \rho}_j \wedge \xi_j}\mu X_{j-1}(u) e^{-\int_{\tau_j}^u G_j(v) \: dv}\: du \bigg|{\cal F}_{\tau_j}\bigg] &\leq (1 + 3 \delta) s E \bigg[ \int_{\tau_j}^{{\bar \rho}_j \wedge \xi_j}e^{-s(u - \tau_j)}\: du \bigg|{\cal F}_{\tau_j}\bigg] &\leq 1 + 3 \delta.
\lim_{r\to 0}\{\Phi_{1},\Phi_{2},\Phi_{3},\Phi_{4},\Phi_{5}\}= \{0,3,3,\frac{3}{2},\frac{15}{2}\}\lim_{r\to \infty}\{\Phi_{1},\Phi_{2},\Phi_{3},\Phi_{4},\Phi_{5}\}= \{0,\frac{3}{2},\frac{9}{2},6,9\}
\breve{\Sigma}\cong N'_I(\Phi) \Sigma_0 \cong{\rm res}'_\tau(\breve{\Sigma}) \widetilde{\Sigma}_0 \cong{\rm res}_\tau(\breve{\Sigma}) \Sigma_1 ={\rm res}'_\tau(\breve{\Sigma}) \cup{\rm res}_\tau(\breve{\Sigma}).
A(x+\Phi_{m''}(X)(s-)+\tfrac{1}{2}\phi_{m''}(z)) \cdot \phi_{m''}(z) - A(x+X(s-)+\tfrac{1}{2}z) \cdot z & =\Big[A(x+\Phi_{m''}(X)(s-)+\tfrac{1}{2}\phi_{m''}(z)) - A(x+X(s-)+\tfrac{1}{2}z)\Big] \cdot \phi_{m''}(z) & + A(x+X(s-)+\tfrac{1}{2}z) \cdot (\phi_{m''}(z)-z).
\f_{2}= - e^{2}\int \frac{d^{4}k}{( 2 \pi )^{4}}\frac{d^{4}q}{( 2 \pi )^{4}}\frac{( k + p + q )^{2}+ q^{2}- k^{2}- p^{2}}{k^{2}\Big ( 1 + ( - 1 )^{n}k^{2 n}/ \Lambda^{2 n}\Big ) ( k + q )^{2}( k + p + q )^{2}q^{2}( q + p )^{2}}
e_{\theta}\cdot e^{t A}f =e^{t L_0}f^{\theta}, e_{\theta}\cdot \textrm{curl}\e^{tA}g =e^{tL_{0}'}\big(e_{\theta}\cdot \textrm{curl}\g \big), r e^{tL_0}\gamma =e^{tL_1}(r\gamma),
\langle J_{\lambda}^{'}(u),\varphi\rangle=\int_{\Omega}\nabla u\nabla\varphi dx-\int_{\Omega}\int_{\Omega}\frac{|u(x)|^{2_{\mu}^{\ast}}|u(y)|^{2_{\mu}^{\ast}-2}u(y)\varphi(y)}{|x-y|^{\mu}}dxdy-\lambda\int_{\Omega}u\varphi dx
P^{k_2+1}(z_2)|z_2|p'(|z_2|)\Big(\mathrm{Re}\big(b_{1k_2}(\alpha i-1)^{k_2}+g_2(z_2)\big)\Big) &+P^{k_1}(z_2)\mathrm{Re}\Big(a_{0k_1}(\alpha i-1)^{k_1}+g_1(z_2)\Big)\equiv 0
\begin{aligned}(z (t)-z(t_3))^{-1}(z(t_3)-z(t_2))= =\frac{t_3-t_2}{t-t_3}\Bigl(1+(t_2-t)\bigl(\frac12(z')^{-1}z''+\frac16(z')^{-1}z'''(t+t_2+t_3)-\frac14((z')^{-1}z'')^2(t+t_3)\bigr)\Bigr)+...\end{aligned}
L(b'') =L(1,b'')=L\big(U(b',b),b''\big)=LU\big(L(b',b''),L(b,b'')\big)=LU\big(0,L(b,b'')\big)= =LUL(b,b'')=L(b,b'')=L(b'',b)=LUL(b'',b)=LU\big(0,L(b'',b)\big)= =LU\big(L(b',b),L(b'',b)\big)=L\big(U(b',b''),b\big)=L(1,b)=L(b)
Q(M,P_i,\tau)= \left\{e^{\frac{1}{24}E_2(\tau)A_1}\widehat{A}(TX){\rm exp}(\frac{c}{2}){\rm ch}\left[\bigotimes_{n=1}^{\infty}S_{q^n}(\widetilde{T_CM})\otimes\bigotimes_{m=1}^{\infty}\wedge_{-q^m}(\widetilde{L_C})\right]\right. &\left.\cdot\varphi(\tau)^{8}{\rm ch}(\mathcal{V}_i)\right\}^{(14)}.
\biggl{(}{{\kappa_2-\kappa_1}\over{\kappa_2+\kappa_1}}\biggr{)}^2=e^{2\bigl{(}\log(1-\kappa_1/\kappa_2)-\log(1+\kappa_1/\kappa_2)\bigr{)}}=e^{-4\sum_{k=0}^{\infty}{1\over{2k+1}}\bigl{(}{\kappa_1\over{\kappa_2}}\bigr{)}^{2k+1}}.
\partial_{+}g_{L}g_{L}^{- 1}= \left( \begin{array}{c c}{{v_{L}\partial_{+}u_{L}+ b_{L}\partial_{+}a_{L}}}&{{u_{L}\partial_{+}a_{L}- a_{L}\partial_{+}u_{L}}}\\{{b_{L}\partial_{+}v_{L}- v_{L}\partial_{+}b_{L}}}&{{u_{L}\partial_{+}v_{L}+ a_{L}\partial_{+}b_{L}}}\\ \end{array}\right) ,
g_{\mathrm{hyp},s}(\sigma_{p}z,\sigma_{q}w)= \frac{4\pi\Im(z)^{1-s}}{2s-1}\mathcal{E}_{\mathrm{par},p}(\sigma_{q}w,s) +\delta_{p,q}\sum_{\substack{n\not = 0}}\frac{1}{|n|}W_{s}(nz)\overline{V_{\overline{s}}(nw)}+ O\big(e^{-2\pi(\Im(z)-\Im(w))}\big).
{\cal L}_\omega ={\cal L}_\omega^{<}+{\cal R}_\omega +{\cal R}_\omega^\bot \,,{\cal L}_\omega^{<}:={\cal V}_n{\mathfrak L}_n^{<}{\cal V}_n^{- 1}\,,{\cal R}_\omega :={\cal V}_n{\cal R}_n{\cal V}_n^{- 1}\,,{\cal R}_\omega^\bot :={\cal V}_n{\cal R}_n^\bot{\cal V}_n^{- 1}
\int_{Z(\eta)}\nu_{f \sqrt{d\mu}}\, d\mu= \nu_{f \sqrt{d \mu}}(Z(\eta) ) = \langle({\chi_{Z(\eta)}}f)\,\sqrt{d\mu}, f \sqrt{d \mu}\rangle =\int \chi_{Z(\eta)}\cdot |f|^2 d\mu \; = \int_{Z(\eta)}|f|^2 \, d\mu.
\sum_{k \in \mathcal{P}_H}\mathcal T_{\rm{com}}(k) = - \rho_z \sum_{p \in \mathcal P_L}\big( \widehat{g \omega}(0) + \widehat{g \omega}(p) \big) a_p^\dagger a_p + \mathcal E''+ \mathcal E' + \mathcal E,
\theta^0_1=\tilde{\theta}^0-\varepsilon^{-1}, \theta^0_2=\varepsilon^{-1}, t=-\varepsilon\tilde{t}, H=-\varepsilon^{-1}\tilde{H}, q_1=1-\tilde{p}_1-\tilde{p}_2, p_1=\tilde{q}_2, q_2=1-\tilde{p}_1, p_2=\tilde{q}_1-\tilde{q}_2.
\alpha_1 =\frac{1}{2}\left(8 n^2+8 n \alpha+3 \alpha^2+8 n \beta +4 \alpha \beta +4\beta^2\right), \beta_1 =-\frac{i}{2}\alpha (2n+\alpha) (2n+\alpha +2 \beta), \gamma_1 =-\frac{1}{16}(\alpha -2 \beta )(\alpha +2 \beta ) (4n+\alpha +2\beta) (4 n +3 \alpha +2\beta).
[\alpha_{\mathcal{L}}(a),\alpha_{\mathcal{L}}(b),[x,y,z]_{\mathcal{L}}]_{\mathcal{L}}= [[a,b,x]_{\mathcal{L}}, \alpha_{\mathcal{L}}(y), \alpha_{\mathcal{L}}(z)]_{\mathcal{L}}+ [\alpha_{\mathcal{L}}(x),[a,b,y]_{\mathcal{L}},\alpha_{\mathcal{L}}(z)]_{\mathcal{L}}+[\alpha_{\mathcal{L}}(x),\alpha_{\mathcal{L}}(y),[a,b,z]_{\mathcal{L}}]_{\mathcal{L}},
\sum_{\ell=0}^L \mathcal C_\ell \leq c_2 h_0^{-\gamma}\sum_{\ell=0}^L s^{\gamma\ell}\leq (2c_1h_0^{-\alpha})^{\frac{\gamma}{\alpha}}\varepsilon^{-\frac{\gamma}{\alpha}}<(2c_1h_0^{-\alpha})^{\frac{\gamma}{\alpha}}\varepsilon^{-\frac{1}{\mu_2}},
N \geq \frac{T}{(q-1)! \; T_{w_1,w_2,n}^{(q-1)}(w_1,\ldots,w_1)}= \frac{1}{(q-1)!}{n \choose w_1}{n-w_1 \choose w_1}\cdots{n-(q-2)w_1 \choose w_1}.
Pr(x|\sum_{i=1}^{m}x_{i}=n,x_{0},\alpha_{0},\alpha)=\frac{n!\Gamma(\sum_{i=1}^{m}\alpha_{i})}{\Gamma(n+\sum_{i=1}^{m}\alpha_{i})}\prod_{i=1}^{m}\frac{\Gamma(x_{i}+\alpha_{i})}{x_{i}!\Gamma(\alpha_{i})}
\mathcal{G}(O_n) =\mathcal{G}(h_0) \oplus \mathcal{G}(h_1) \oplus \dots \oplus \mathcal{G}(h_d) &= (k_0 \oplus k_1 \oplus \dots \oplus k_d)s + [ \mathcal{G}(r_0) \oplus \mathcal{G}(r_1) \oplus \dots \oplus \mathcal{G}(r_d) ] \mathcal{G}(r_i) < s =2^t &= (k_0 \oplus k_1 \oplus \dots \oplus k_d)s + \mathcal{G}((r_0,r_1,\dots,r_d))
N2_2(3n+1) = M2_2(3n+1) \pmod{3}, M2_2(5n) \equiv 0 \pmod{5}, N2_4(5n) + 4N2_2(5n) \equiv M2_4(5n) + 4M2_2(5n) \pmod{5}, N2_2(5n+1) \equiv 0\pmod{5}, M2_2(5n+1) \equiv 0\pmod{5}, N2_4(5n+1) \equiv M2_4(5n+1) \pmod{5}, N2_2(5n+3) \equiv M2_2(5n+3) \pmod{5}, N2_4(5n+3) \equiv M2_4(5n+3) \pmod{5}.
(r_h^1-p_h^0, \nabla\cdot \widetilde{\boldsymbol{e}}_u^1)=-(\nabla(r_h^1-r_h^0), \widetilde{\boldsymbol{e}}_u^1) \leq \frac{1}{8 \Delta t}\|\widetilde{\boldsymbol{e}}_u^1\|_0^2+C \Delta t\|\delta_t \nabla r_h^1\|_0^2.
\sup_{\gamma \in \mathcal{P}(\mathcal{V})}\left\{\int_{\mathcal{V}}g d \gamma - \lambda_1 \boldsymbol{K}_1(\mu_1, \gamma_1) - \lambda_2 \boldsymbol{K}_2(\mu_2, \gamma_2): \boldsymbol{K}_\ell(\mu_\ell, \gamma_\ell) < \infty \ell = 1, 2\right\}
L(x_{i}|t_{j})_{\varepsilon_{1}\varepsilon_{2}}^{\varepsilon_{1}^{\prime}\varepsilon_{2}^{\prime}}=L^{\prime}(x_{i}|\ominus t_{j})_{\varepsilon_{1}^{\prime}(1-\varepsilon_{2})}^{\varepsilon_{1}(1-\varepsilon_{2}^{\prime})}
\; - \frac{1}{N}\sum_{(b_1,\vec{b}_\ast):\;\mathcal{U}_{b_1,\vec{b}_\ast}= 0, \; \mathcal{U}_{b_1-1,\vec{b}_\ast}\neq 0}\mathcal{U}_{b_1-1,\vec{b}_\ast}\delta_{b_1,\vec{b}_\ast}.
[{\cal Z}_{A}^{I}, \bar{\cal Z}_{B}^{J}]_{P . B .}= i \delta^{I J}\delta_{A B}, \quad [{\cal Z}_{A}^{I},{\cal Z}_{B}^{J}]_{P . B .}= [ \bar{\cal Z}_{A}^{I}, \bar{\cal Z}_{B}^{J}]_{P . B .}= 0 ,
A(t)=\lambda_{1}(3E_{0}t+1)^{1/3}, B(t)=\lambda_{2}(3E_{0}t+1)^{-1/3}(3F_{0}t+1)^{1/3}, &C(t)=\lambda_{3}(3F_{0}t+1)^{-1/3}, D(t)=\lambda_{4}(3E_{0}t+1)^{1/3}(3F_{0}t+1)^{1/3},
P^{n}= | [ n ]_{q^{- 1}}! T ( P^{n}) = | [ n ]_{q^{- 1}}! \int_{\infty}^{z}d t_{1}\int_{\infty}^{t_{1}}d t_{2}\cdots \int_{\infty}^{t_{n - 1}}d t_{n}J ( t_{n}) \cdots J ( t_{1}) ~ .
\int \nabla \left( \psi^{2}u_{+}\right) \mathcal{A}\nabla u_{+}d\mu =\int\nabla \left( \psi^{2}u_{+}\right) \mathcal{A}\nabla ud\mu \leq -\int\left( \psi^{2}u_{+}\right) \phi d\mu \leq \left\Vert \phi \right\Vert_{X}\int \left\vert \nabla_{\mathcal{A}}\left( \psi^{2}u_{+}\right)\right\vert d\mu \ .
\omega_0^{inst}= d\left[ \frac{1}{2\pi}\cdot Re(c -c \ln c) + \frac{1}{2\pi}\cdot S \right] \wedge d\left( \frac{2\pi \cdot R \sigma}{S_1 - \ln|c|}\right) + dc_2 \wedge d\left( - \frac{S_2 + \arg c}{S_1 - \ln|c|}\cdot R \sigma \right)
\begin{array}{c}\medskip\partial_b T_b^a = 0 \,, 2 \pi T_b^a = 2 \rho^a \rho^c \partial_c x^{\mu}\partial_b x_{\mu}+ i\rho^a \psi^{\mu}\partial_b \psi_{\mu}- \delta_b^a \left (\rho^c \rho^d\partial_c x^{\mu}\partial_d x_{\mu}+ i \rho^c \psi^{\mu}\partial_c\psi_{\mu}\right) \,.\end{array}
<\psi_{qG}|\psi_{qG}>= \frac{\hbar\sqrt{2(q+1)}}{(q-1)|p|}\int\limits_{\frac{|\Im(p)|}{|p|}}^{\infty}\left\{z^2+\frac{[\Re(p)]^2}{|p|^2}\right\}^{\frac{2}{1-q}}\;dz.
{\cal A}= < \int_{S_{\rho}^{3}}\psi^{3}> = \int_{\cal M}9 6 \pi^{2}{\frac{a^{5}\rho^{2}d a}{( \rho^{2}+ a^{2})^{4}}}\cos \beta d \alpha d \beta = 3 2 \pi^{3}.
\Gamma^{\mu_{1}. . . \mu_{n}}\Gamma_{1 1}= ( - )^{n}\Gamma_{1 1}\Gamma^{\mu_{1}. . . \mu_{n}}={\frac{\epsilon^{\mu_{1}. . . \mu_{1 0}}}{( 1 0 - n ) !}}\Gamma_{\mu_{1 0}. . . \mu_{n + 1}}
\mathcal{B}_v(\lambda) \otimes \mathcal{B}_w(\mu) = \bigsqcup_{\pi \in \mathcal{B}_w(\mu)^\lambda}\left( \bigcup_{a_1, \ldots, a_l \geq 0}\tilde{f}_{i_1}^{a_1}\cdots \tilde{f}_{i_l}^{a_l}( C(\pi, e) ) \setminus \{0 \}\right) &\subsetneq \bigsqcup_{\pi \in \mathcal{B}_w(\mu)^\lambda}C(\pi, v) &= \mathcal{B}_v(\lambda) \otimes \mathcal{B}_w(\mu),
N(\psi,\bar\psi) = \sum_{l=1}^{r}\hat N_{l}(x,\psi,\bar\psi) &+ N(\psi,\bar\psi) - \sum_{l=1}^{r}\hat N_{l}(x,\psi,\bar\psi) &=: N^{(1)}(\psi,\bar\psi) + N^{(1,r)}(\psi,\bar\psi),
P = \int_{0}^{t}\bigg\{\int_{0}^{1}\chi_{\{x+z\in[0,1]\}}\bigg|\int_{0}^{1}\chi_{\{|x-y|\leq |z|\}}\bigg(\frac{\partial G}{\partial y}(t-s,x+z,y)-\frac{\partial G}{\partial y}(t-s,x,y)\bigg) & \times v^{\delta+1}(s,y)\d y\bigg|^p\d x\bigg\}^{\frac{1}p}\d s,
\frac{d}{dx}aslh(x)=\frac{d}{dx}atlh(\frac{x}{\sqrt[4]{x^{4}+1}})=(\frac{d}{dx}\frac{x}{\sqrt[4]{x^{4}+1}})[1-(\frac{x}{\sqrt[4]{x^{4}+1}})^{4}]^{-3/4}=
\log K_2(p,h) =-\frac{h+1}{3}\log N+2kp \log (4kp) \leq -\frac{\alpha}{3}\log N \log \log N +6 k^2 \log k \log N \log \log N =\left(-\frac{\alpha}{3}+6 k^2 \log k\right)\log N \log \log N \leq -\log N \log \log N.
w_{2p}'(t) \,\le\, -\frac{a}{p}\min_{\beta = 2,4}\biggl\{\Bigl( \frac{w_{2p}(t)}{w_p(t)}\Bigr)^{\beta p}\biggr\}w_{2p}(t) + pb\alpha^2 w_{2p}(t)~, t > 0~,
t_4 = -a_4 a_4 = -t_4 t_3 = -a_3 a_3 = -t_3 t_2 = -a_2 + \frac{3}{2}a_4^2 a_2 = -t_2 + \frac{3}{2}t_4^2 t_1 = -a_1 + 2 a_3 a_4 a_1 = -t_1 + 2 t_3 t_4 t_0 = -a_0 + \frac{1}{2}a_3^2 +a_2 a_4 -\frac{7}{6}a_4^3 a_0 = -t_0 + \frac{1}{2}t_3^2 + t_2 t_4 - \frac{1}{3}t_4^3
\left[ \widehat{q}_1,\widehat{p}_1 \right]=\left[ \widehat{q}_2,\widehat{p}_2 \right]=i, \left[ \widehat{q}_1,\widehat{q}_2 \right]= i \theta, \left[ \widehat{p}_1,\widehat{p}_2 \right]= i \eta,
\mbox{$T:\bigcup_{k\in\mathbb{N}}S_k\to \bigcup_{k\in\mathbb{N}}\Pi(k)$},\sigma\mapsto T(\sigma)=(T_j(\sigma))_{j=1}^{r(\sigma)}=(|x_j^{\langle\sigma\rangle}|)_{j=1}^{r(\sigma)},
B_{3 / 2}= \frac{5 \, \alpha \, \sqrt \pi}{6 4 \, R}, \quad B_{2}= \frac{4 \, \alpha}{4 5 \, R^{2}}, \quad B_{5 / 2}= \frac{2 6 9 \, \alpha \, \sqrt \pi}{2^{1 3}}\,{.}
X^+_{\mu,\nu}= A^+_{K+m}\cdots A^+_{K+2}A^+_{K+1}B^+_{\nu+n-1}\cdots B^+_{\nu+1}B^+_{\nu}, X^-_{\mu,\nu}= A^-_{K-m+1}\cdots A^-_{K-1}A^-_K B^-_{\nu-n}\cdots B^-_{\nu-2}B^-_{\nu-1}.
I^{\unlhd\lambda}I^{\unlhd\mu}=\sum_{\nu\unlhd\lambda,\,\pi\unlhd\mu}I^{\le\nu}I^{\le\pi}\subset\sum_{\nu\unlhd\lambda,\,\pi\unlhd\mu}\sum_{\rho\le\nu,\pi}I^{\le\rho}\subset\sum_{\nu\unlhd\lambda,\,\pi\unlhd\mu}\sum_{\rho\unlhd\nu,\pi}I^{\unlhd\rho}=\sum_{\rho\unlhd\lambda,\mu}I^{\unlhd\rho}.
\mbox{var}[S(u+du)-S(u)] =\mbox{var}[S(u+du)]+\mbox{var}[S(u)]-2\mbox{cov}\big(S(u+du), S(u)\big) &=\mbox{var}[S(u+du)]-\mbox{var}[S(u)]=\beta(\Lambda_{u+du}-\Lambda_{u})=\beta\frac{\lambda_i}{a_i}du.
A = \frac{1}{u_3\|u'\|_J}\begin{pmatrix}\det\begin{pmatrix}u_3 u_1 u_3' u_1'\end{pmatrix}\det\begin{pmatrix}u_3 u_2 u_3' u_2'\end{pmatrix}\det\begin{pmatrix}u_2 u_3 u_2' u_3'\end{pmatrix}\det\begin{pmatrix}u_3 u_1 u_3' u_1'\end{pmatrix}\end{pmatrix}={(a_{ij})}.
\frac{2-x}{(1-x)^2}\sum_{j=0}^\infty f_j \left(2-\frac{1}{x}, 2-\frac{1}{x}\right) = \frac{1}{2}\frac{3}{1-x}+ \frac{1-x}{2(1-2x)^2}\sum_{j=0}^\infty f_j \left(3-\frac{1}{x}, 3-\frac{1}{x}\right).
\mathcal{F}\big[R_nf\big](\xi)=\frac{1}{\pi(2\pi)^{n+1}}\int_{\mathbb{R}^{n+1}}\widetilde{M}_{n+1}(\xi;\eta_1,\ldots,\eta_{n+1})\widehat{f}(\xi-\eta_1-\ldots-\eta_{n+1}) &\times\widehat{h}(\eta_1)\cdot\ldots\cdot\widehat{h}(\eta_{n+1})\, d\eta_1\ldots d\eta_{n+1},
a_n = \sum_{j=0}^{n}\frac{n!}{j!}n^j = \sum_{j=0}^{n}n^{n-j}\prod_{i=0}^{j-1}(n-i) \equiv \equiv \sum_{j=0}^{d-1}n^{n-j}\prod_{i=0}^{j-1}(n-i) = n^{n-d+2}\sum_{j=0}^{d-1}n^{d-j-2}\prod_{i=0}^{j-1}(n-i) \pmod{d},
\begin{array}{r c l}{N = \sum_{a = 1}^{3}{\alpha^{a}}^{\dagger}\alpha^{a}\}&{,}&{\N^{\dagger}= N \ ,}\\{B^{\dagger}= \sum_{a = 1}^{3}{\alpha^{a}}^{\dagger}{\alpha^{a}}^{\dagger}\}&{,}&{\B = \sum_{a = 1}^{3}\alpha^{a}\alpha^{a}\ ,}\\ \end{array}
\begin{cases}\mathcal{Q}\;^{C}_{t}D_{T}^{q}\Psi(x,t)= A^{*}\mathcal{Q}\Psi(x,t) + \left\langle \mathcal{Q}\varphi(t), f \right\rangle_{L^{2}(D)}\chi_{_{D}}f(x) \hbox{in}Q_{T}\Psi(\xi,t) = 0 \hbox{on}\Sigma_{T}\Psi(x,T) = 0 \hbox{in}\Omega.\end{cases}
\begin{aligned}\mathcal{J}_{i}\big(v_{1}(\cdot), v_{2}(\cdot)\big)=\frac{1}{2}\mathbb{E}\left\{\int_{0}^{T}\left[l_{i}(t)\big(y^{v_{1}, v_{2}}(t)-k_{i}(t)\big)^{2}+m_{i}(t)\big(v_{i}(t)-n_{i}(t)\big)^{2}\right]dt\right. \left.+r_{i}\big(y^{v_{1}, v_{2}}(0)-h_{i}\big)^{2}\vphantom{\int_{0}^{T}}\right\}(i=1, 2).\end{aligned}
\begin{array}{l}z^3(\tau)={\displaystyle \frac{p_1}{p_3}}\tau + z^3(0) \quad , \, z^2(\tau)=(z^2(0)-{\displaystyle \frac{p_4(0)}{p_3}}\tau)\exp (i \gamma p_1 \tau ) = (z^4( \tau))^* \quad.\end{array}
{\cal L}_{b}( \frac{1}{2}, \alpha , g ) = \bar{\psi}[ \sigma_{\mu}( \partial_{\mu}+ i a_{\mu}+ i C_{\mu}) + M + i b ] \psi - \frac{i}{8 \pi \alpha}\epsilon_{\mu \nu \lambda}a_{\mu}\partial_{\nu}a_{\lambda}+ \frac{1}{g}b^{2}\; .
\boldsymbol{\Phi}_{\mathbf{s},c}^{\mathbb{\epsilon}}(x) = \tilde{\Phi}_{\frac{n_l}{2}-c}^{\epsilon_1}\left(\frac{xb_1}{a^{(1)}b^{(1)}N_0N_1}\right)\overline{\tilde{\Phi}_{\frac{n_l}{2}-c}^{\epsilon_2}\left(\frac{xb_1}{a^{(2)}b^{(2)}N_0N_1}\right)}
\int_{0}^{1}g_{\mathrm{hyp},s}(z,\sigma_{q}w)du=\frac{4\pi v^{1-s}}{2s-1}\mathcal{E}_{\mathrm{par},q}(z,s)+\frac{4\pi\delta_{p,q}}{2s-1}\big(v^{s}y_{p}^{1-s}-v^{1-s}y_{p}^{s}\big).
C_{BPS}^P(N,Y;\mathbf{l}^\ast,\alpha) &=\inf \{K>0 ; \forall H\in C^\infty(N)H|_Y\geq K, &\exists\mu\in\mathfrak{M}(N,X_H)|\langle \mathbf{l}^\ast,\rho(\mu,X_H) \rangle|\geq\mathbf{l}^\ast(\alpha)\}.
\left|a^{(n)}_1(t)-a^{(n)}_2(t)-{1\over 2}\cdot\left(\sigma^{(n)}_1(t)+\sigma^{(n)}_2(t)\right)\right|~=~\left|w^{(n)}(t,t+)-w^{(n)}(t,0-)\right|~\leq~M_0\sqrt{|t|},
\begin{aligned}f(x) \,\to\,\min g_i(x) \,\leq\,0 \qquad i\in\mathcal M h_j(x) \,=\,0 \qquad j\in\mathcal P G_l(x) \,\leq\,0 \qquad l\in I^{0+}(\bar x)\cup I H_l(x) \,\leq\,0 \qquad l\in I^{+0}(\bar x)\cup(I^{00}(\bar x)\setminus I)\end{aligned}
u_\varphi(t) = \left(\begin{array}{c c c}1 c_1 \, t^{p^{e_1}}\frac{1}{2}\, \lambda \, c_1^2 \, t^{2 \, p^{e_1}}0 1 \lambda \, c_1 \, t^{p^{e_1}}0 0 1\end{array}\right) (\, c_1 \in k \backslash \{0\}, \lambda \in k \backslash \{0 \}, e_1 \geq 0 \,) ,
\|b\|_{{\rm BMO}_\nu(\mathcal G)}\leq C \sum_{t=1}^\mathcal T \|b\|_{{\rm BMO}_{\nu,\mathcal D^t(\mathcal G)}(\mathcal G)}\leq C\sum_{t=1}^\mathcal T \sup_{S\in \mathcal D^t(\mathcal G)}w_\lambda(b;S){|S|\over \nu(S)}, 0<\lambda\leq{1\over 2^{Q+2}},
(\operatorname{id}\otimes\omega_{\Lambda_{\tilde{\varphi}}(a\sigma^{\tilde{\varphi}}_{-i}(b^*)),\Lambda_{\varphi}(p)})(W)=(\operatorname{id}\otimes\omega_{\Lambda_{\tilde{\varphi}}(a),\Lambda_{\tilde{\varphi}}(b)})(\Delta(p^*)),
\begin{array}{r c l}{\exp ({\cal K}_{0}( Q - 1 ) ) - 1}&{=}&{( f_{0}+ Q^{- 1 / 2}f_{1}) ( Q - 1 ) ,}\\{\exp ({\cal K}_{1}( Q - 2 ) ) - 1}&{=}&{Q^{- 1 / 2}f_{1}( Q - 2 ) .}\\ \end{array}
\tilde{Y}_O (P, G) := \sup \{\| \phi_1 * \phi_2 * \cdots * \phi_N \|_{q (P)}\mid ( \phi_1 , \phi_2 , \cdots , \phi_N ) \in \tilde{\mathcal{B}}\}, \tilde{Y}_R (P, G) := \inf \{\| \phi_1 * \phi_2 * \cdots * \phi_N \|_{q (P)}\mid ( \phi_1 , \phi_2 , \cdots , \phi_N ) \in \tilde{\mathcal{B}}\}.
S_{x}=\frac{\hbar}{2}[\begin{matrix}0&1\\ 1&0\end{matrix}],S_{y}=\frac{\hbar}{2}[\begin{matrix}0&-i\\i&0\end{matrix}],S_{z}=\frac{\hbar}{2}[\begin{matrix}1&0\\ 0&-1\end{matrix}]
R_{\mu k}\overline{[(RV)^3R]}_{\nu k}= R_{\mu k}(\sum (\sigma_i)^{3/2}(X^G_{i \mu_1})^3\overline{(z)^{3/2}R_{\nu a_1}R_{b_1 a_2}R_{b_2 a_3}R_{b_3 k}}.
p_{123}p_{145}-p_{124}p_{135}+p_{125}p_{134}=0\, , p_{123}p_{456}-p_{124}p_{356}+p_{125}p_{346}-p_{126}p_{345}=0 p_{234}A_1-p_{134}A_2+p_{124}A_3-p_{123}A_4=0 p_{123}^2p_{456}+\det{\begin{pmatrix}p_{234}p_{235}p_{236}p_{134}p_{135}p_{136}p_{124}p_{125}p_{126}\end{pmatrix}}=0\, .
\tilde{b}_{q,m}=\begin{cases}\lfloor b'_{q,m}\rfloor b'_{q,m}-\lfloor b'_{q,m}\rfloor\leq\beta \lceil b'_{q,m}\rceil b'_{q,m}-\lfloor b'_{q,m}\rfloor>\beta\end{cases},\,q\in\mathcal{Q}'_m,
\overline{MG}=\sum_{i}\overline{MG}_{i}=\sum_{i}\overline{p}^{\alpha}_{i}\overline{q}^{1-\alpha}_{i}&\overline{MH}=\sum_{i}\overline{MH}_{i}=\sum_{i}\frac{\overline{p}_{i}\overline{q}_{i}}{\left(1-\alpha\right)\overline{p}_{i}+\alpha\overline{q}_{i}}
\begin{aligned}\sin(\bar{\theta}_{i+1,j}- \bar{\theta}_{i,j}) + \sin(\bar{\theta}_{i-1,j}- \bar{\theta}_{i,j}) = \sin\bigg(\frac{2\pi}{N_1}\bigg) + \sin\bigg(-\frac{2\pi}{N_1}\bigg) = 0, \end{aligned}
\mu_h(B(a,r)) = \frac{\mu_n(I_n)}{\nu(I_n)}\,\nu(B(a,r)) \leq c_1\,g(\rho_n)^{-\eta}\,\rho_{n}^{\gamma h_{n}-1}\cdot g(r)^{-\beta}r^{\gamma h_{n+1}-1+s-\epsilon}\rho_n^{-\gamma h_{n+1}+1}\leq c_1\,g(r)^{-\eta-\beta}\,r^{\gamma h_{n+1}-1+s + \gamma(h_n-h_{n+1})-\epsilon}.
\Delta=I_{-\infty}^{+\infty}\frac{Re[f(x)]}{Im[f(x)]}=I_{-\infty}^{+\infty}\frac{b_{0}\omega^{n-1}-b_{1}\omega^{n-3}+...}{a_{0}\omega^{n}-a_{1}\omega^{n-2}+...}(26)
\frac{2 c_{b}}{\lambda(t)}\int_{0}^{1/2}d\xi \frac{\sin(t \xi)}{t^{2}}\left(\frac{b-1}{\xi \log^{b}(\frac{1}{\xi})}+\frac{b(b-1)}{\xi\log^{b+1}(\frac{1}{\xi})}\right)= \frac{2 c_{b}}{\lambda(t)}\int_{0}^{t/2}du \frac{\sin(u)}{t^{2}}\left(\frac{b-1}{u \log^{b}(\frac{t}{u})}+\frac{b(b-1)}{u\log^{b+1}(\frac{t}{u})}\right)
\gamma_d^{-k\binom d2}\gtrsim \prod_{\ell = k_3}^{k - 1}f_\ell = \prod_{j = 1}^{d - 1}\prod_{\ell = k_3}^{k - 1}\frac{1}{t_{\ell - j}^{d - j}}= \prod_{j = 1}^{d - 1}\left(\frac{A_{k_3 - j}}{A_{k - j}}\right)^{d - j},
F([f*g]) =[F\circ (f*g)]= [F \circ ((f*g)_-)_{\infty}] = [((F \circ f_-) * (F \circ g_-))_{\infty}] &= [(F \circ f_-)_{\infty}* (F \circ g_-)_{\infty}] = [(F\circ f) * (F \circ g)].
0\le \sum_{j=0}^{2N-1}\zeta_{j,N}\le{\frac 1 N}\sum_{j=0}^{2N-1}\sum_{{\ell=1}}^{\infty}\frac{d^{2\ell+1}}{(\ell!)^2}{(1-x_j^2 )}^\ell &= \sum_{\ell=1}^{\infty}\frac{d^{2\ell+1}}{(\ell!)^2}\left\{\frac 1N \sum_{j=0}^{2N-1}{(1-x_j^2 )}^\ell \right\}
\frac{d \phi}{d r^{2}}= \frac{r_{+}r_{-}}{2}\left( \frac{1}{r^{2}}- \frac{1}{r_{\beta^{\prime}}^{2}- r^{2}}\right) \frac{1}{\sqrt{( r_{+}^{2}- r^{2}) ( r^{2}- r_{-}^{2})}}
H=-\frac{1}{2}J\sum_{i,j}S_{i}^{z}S_{j}^{z}-g\mu_{B}H\sum_{i}S_{i}^{z}-\frac{1}{4}J\sum_{i,j}(S_{i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+})
S\,=\,-M_0^2\,\mathop{\int\!\!\!\int}_{\Sigma\hspace*{0.25cm}}d\Sigma\,\sqrt{-g}\left[\,1\,-\,\frac{\alpha}{2}\,r_s^2\,\Delta x^{\mu}\,\Delta x_{\mu}\right]\,-\,\sum_{a=1}^{2}\,m_a\,\int\limits_{C_a}^{}\!\!ds_a\,{,}
g(z_1, z_2, z_3, z_4) = [z_1, z_2][z_3, z_4] + [z_1, z_3][z_2, z_4] = [z_4, z_3] [z_2, z_1] + \bigl[ [z_1, z_2], [z_3, z_4] \bigr] + [z_4, z_2] [z_3, z_1] & + \bigl[ [z_1, z_3] , [z_2, z_4] \bigr] = g(z_4, z_2, z_3, z_1) + \bigl[ [z_1, z_2], [z_3, z_4] \bigr] + \bigl[ [z_1, z_3] , [z_2, z_4] \bigr]
\lim_{\epsilon\rightarrow 0}\frac{1}{\epsilon^4}B_4\!\left(\frac{i}{2}+\epsilon+c\,\epsilon^4\right)B_4\!\left(-\frac{i}{2}+\epsilon\right)|0\rangle_4=(0,0,0,2,0,0,-2,0,0,ic,0,0,2,0,0,0)^t.
F_w \left( \begin{array}{c}z x \end{array}\right) = F\left( \begin{array}{c}z x \end{array}\right) + \left(\begin{array}{c}\frac{\pi^2}{4}w 0 \end{array}\right)= \left( \begin{array}{c}z + q_1(z+ \delta x) \delta x - q_1(z+ \delta x)\end{array}\right) + \left( \begin{array}{c}\frac{\pi^2}{4}w 0 \end{array}\right),
X(t) =\int_{-\infty}^{kT+s}e^{{\bf A}(t-u)}{\bf e}dS(u)=\sum_{n=1}^{\infty}\int_{(k-n)T+s}^{(k-n+1)T+s}e^{{\bf A}(t-u)}{\bf e}dS(u) &=\lim_{d\rightarrow\infty}\sum_{n=0}^{d}\sum_{i=0}^{r-1}e^{{\bf A}t}\int_{(k-n)T+s_{j-2-i}}^{(k-n)T+s_{j-1-i}}e^{-{\bf A}u}{\bf e}dS(u)+e^{{\bf A}t}{\bf e}\int_{kT+s_{j-1}}^{kT+s}e^{-{\bf A}u}dS(u).