harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\begin{array}{l l}{D_{\alpha}= \partial_{\alpha}+ i \sigma_{\alpha \dot{\alpha}}^{\mu}\bar{\theta}^{\dot{\alpha}}\partial_{\mu},}\\{D_{\dot{\alpha}}= - \bar{\partial}_{\alpha}- i \theta^{\alpha}\sigma_{\alpha \dot{\alpha}}^{\mu}\partial_{\mu}.}\\ \end{array}
\Phi^{\left( c h i r a l ) \right)}= \exp \left( \frac 1 2 \theta^{\alpha}\sigma_{\alpha \dot{\beta}}^{\mu}\bar{\theta}^{\dot{\beta}}\partial_{\mu}\tilde{\partial}_{+}\right) \left( \phi + \theta^{\alpha}\chi_{\alpha}+ \theta^{\alpha}\theta_{\alpha}\tilde{\partial}_{+}f \right) ( x , y ) .
\left[ f \right]_{\mu, \mathcal{S}}= \left\{g \in \mathcal{M}( \mathcal{F}, \mathcal{S}) \colon ( \exists \, A \in \mathcal{F}\colon \mu(A) = 0 \{\omega \in \Omega \colon f(\omega) \neq g(\omega) \}\subseteq A ) \right\}.
\int_0^t ds \int_0^1 dz \left[G(s, x,z) -G(s,y,z)\right]^2 &= C\int_0^t ds \sum_{k\ge 1}\exp\left(-(2\pi k)^2 s\right)\big|1-\exp\left[i(2\pi k) d(x,y)\right]\big|^2 &= C\sum_{k \ge 1}\frac{\left[1- \exp(-(2\pi k)^2 t)\right]}{k^2}\big|1-\exp\left[i(2\pi k) d(x,y)\right] \big|^2.
\|uu_h^{\kappa}\|_{p^*}\leq c_\Omega \|uu_h^{\kappa}\|_{1,p}\leq M_{27}\left((\kappa+1)^{M_{28}}\right) \left[\|uu_h^{\kappa}\|_{\tilde{q}_1}^p+1\right]^{\frac{1}{p}}\leq M_{29}\left((\kappa+1)^{M_{28}}\right) \left[\|u^{\kappa+1}\|_{\tilde{q}_1}^p+1\right]^{\frac{1}{p}}<\infty.
(\mathbb E\|M_{N+1}\|^p)^{\frac 1p}\geq (\mathbb E\|f_{N}\|^p)^{\frac 1p}- \sum^N_{n=1}\Bigl(\mathbb E \Bigl\| (M^n_1-\sigma_n) \phi_{n}(\sigma_1,\ldots,\sigma_{n-1})\Bigr\|^p\Bigr)^{\frac 1p}\geq 2-\frac{\delta}{KL}\cdot N \cdot\max\{\|\phi_1\|, \|\phi_2\|_{\infty},\ldots, \|\phi_N\|_{\infty}\}>1.
\lim_{\eta \rightarrow 0}\lim_{\Lambda}\omega_{\beta,{\mu}_{\Lambda}(\beta,{\rho}, \eta),\Lambda,\eta}^{0}({b^{*}_{k}b_{k}}/{V}) =\lim_{\eta \rightarrow 0}\lim_{\Lambda}\frac{1}{V}\frac{1}{e^{\beta(\varepsilon_{{k}}-{\mu}_{\Lambda}(\beta,{\rho}, \eta)))}-1}= 0 \ ,
\left( \begin{array}{c}{u ( x , \tau )}\\{v ( x , \tau )}\\ \end{array}\right) = \left( \begin{array}{c}{a u_{0}( x ) \cos \Omega \tau + a^{3}u_{2}( x , \tau )}\\{a^{2}v_{1}( x , \tau ) + a^{3}v_{2}( x , \tau )}\\ \end{array}\right) .
\|\varphi\|_{p,q,\mu_{\beta}}^{2}:=\sum_{n=0}^{\infty}(n!)^{2}2^{nq}|\varphi^{(n)}|_{p}^{2},p,q\in\mathbb{N}_{0},\;\varphi\in\mathcal{P}\big(S_d'\big).
\begin{aligned}(E, v) :0\rightarrow E^{0}\overset{v}\rightarrow E^{1}\overset{v}\rightarrow \cdots \overset{v}\rightarrow E^{l}\rightarrow 0, (E', u) :0\rightarrow E^{l}\overset{u}\rightarrow E^{l+1}\overset{u}\rightarrow \cdots \overset{u}\rightarrow E^{l+m'}\rightarrow 0,\end{aligned}
\frac{\sigma_{m_n}- n}{m_n^{1/2}}\stackrel{d.}{\longrightarrow}Z_{2, B_1}, \mbox{as $n \rightarrow \infty$.}& \frac{\sum_{i=1}^{m}Q_i - m \mu_Q}{d(m)}\stackrel{d.}{\longrightarrow}Z_{\delta/2, B_2}, \mbox{as $m \rightarrow \infty$.}
\bar M_t^{\bar g}:= \bar g(Y_t)-\bar g(Y_0)-\sum_{i=1}^m\frac{1}{2}\int_0^t \Big(\bar \nabla^2 \bar g(Y_s)\big(Z_s^i,Z_s^i\big)+\left\langle \bar \nabla \bar g(Y_s), \bar A(Y_s)(Z_s,Z_s)\right\rangle\Big)ds &+\int_0^t \langle \bar \nabla \bar g(Y_s), \bar f(Y_s,Z_s)\rangle ds
S=\int\limits_0^1\left[-\frac{1}{2e}\left(\dot x^\mu-i\psi^\mu\chi+i\varepsilon^{\mu\nu\rho\zeta}b_\nu\psi_\rho\psi_\zeta+{\alpha\over2}b^\mu\right)^2-i\psi_\mu\dot\psi^\mu\right]d\tau\;,
J^{r+2}\Xi_V\rfloor p_1dp_1 \mathfrak{R}(d\rho) = - p_0d\big( J^{r+1}\Xi_V \rfloor p_1 \mathfrak{R}(d\rho) \big) = - d_H\big( J^{r+1}\Xi_V \rfloor p_1 \mathfrak{R}(d\rho) \big),
\langle f , \star \partial g \rangle_2=\langle \star f , \partial g \rangle_\mathcal{H}=\langle f \omega , \partial g \rangle_\mathcal{H}=\langle \omega , f \partial g \rangle_\mathcal{H}=-\langle \omega , g \partial f \rangle_\mathcal{H}=-\langle \star \partial f, g\rangle_2,
\frac{e^{-2\rho}}{\sqrt{\kappa-2}}+\frac{\kappa}{2 \sqrt{\kappa-2}}\rho=-\frac{\lambda^{2}x^{+}x^{-}}{\sqrt{\kappa-2}}-\frac{\kappa}{4\sqrt{\kappa-2}}\ln(-\lambda^{2}x^{+}x^{-})-\frac{m(x^{+}-x_{0}^{+})}{\lambda x_{0}^{+}\sqrt{\kappa-2}}\theta(x^{+}-x_{0}^{+})
{\cal E}= \frac{1}{2}\mid D_{0}\Phi \mid^{2}+ \frac{1}{2}\mid D_{i}\Phi \mid^{2}+ \frac{1}{4}F_{i j}^{2}+ \frac{1}{2}F_{0 i}^{2}+ \frac{\lambda}{8}( \mid \Phi \mid^{2}- 1 )^{2},
\frac{\delta C ( \kappa )}{\delta \sigma ( x )}|_{\sigma = 0}= + 2 \kappa^{2}\sqrt{g ( x )}\Psi_{\kappa}( x ) - 2 \int d \mu C ( \mu ) \Psi_{\kappa}( x ) \Psi_{\mu}( x ) \, \frac{\kappa^{2}}{\kappa^{2}- \mu^{2}}\sqrt{g ( x )}
\begin{cases}\eta(0,t)=0,\,\,\eta(L,t)=0,\,\,\eta_{x}(0,t)=f(t), t \in (0,\infty), w(0,t)=0,\,\,w(L,t)=0,\,\,w_{x}(L,t)=0, t \in (0,\infty).\end{cases}
\int_0^{\frac{\pi}{2}}\cos(2kx) \log\left( \tan x\right)\d x = 2(2n+1) \int_0^{\frac{\pi}{2}}\sin\left(2(2n+1)x\right) \left[ \int_0^x \log\left( \tan u \right) \d u \right] \d x = -(2n+1) \sum_{m=0}^{\infty}\frac{1}{(2m+1)^2}\int_0^{\frac{\pi}{2}}\left[ \cos\left( 4(m-n) x\right) - \cos \left(4(m+n)x \right) \right]\d x = - \frac{\pi}{2}\frac{1}{2n+1};
\big(\pi^\vee_Z(x)\xi\big)_i=\sum_{j,s}\sqrt{\frac{d_i}{d_j}}(\iota_i \otimes \bar{R}_s^*)(x^s_{ij}\otimes\iota_{\bar s})(\iota_s\otimes\xi_j\otimes\iota_{\bar s})(c^*_s\otimes\iota_{\bar s})(\iota_Z\otimes \bar R_s).
D_{s}^{H}F = \sum\limits_{i=1}^{n}\frac{\partial f}{\partial x_{i}}\left(\int_0^T \xi_{1}(t) dB^{H}(t),...,\int_0^T \xi_{n}(t) dB^{H}(t) \right)\xi_{i}(s), \ \s\in [0,T].
\varphi(q^{a_1})\varphi(q^{a_2})\varphi(q^{a_3})\varphi(q^{a_4}) = x_1( L(q) -2L(q^2))+x_2 ( L(q) -4L(q^4)) + x_3( L(q) -5L(q^5)) + x_4 ( L(q) -8L(q^8)) & + x_5 ( L(q) -10L(q^{10})) + x_6 ( L(q) -20L(q^{20})) & +x_7 ( L(q) -40L(q^{40})) +y_1 A_1(q) + y_2A_2(q) + y_3 A_3(q).
P \Bigg(\bigcap_{i=1}^M \bigg\{X_i \in \bigg((\sigma_{\rm w}^2 + \sigma_{\rm j,i}^2) \left (1 - \frac{\delta}{\sigma_{\rm w}^2 +c}\right ), &\qquad\qquad(\sigma_{\rm w}^2 + \sigma_{\rm j,i}^2) \left (1 + \frac{\delta}{\sigma_{\rm w}^2 +c}\right)\bigg ) \bigg\}\Bigg) > 1- \frac{\epsilon}{2}.
\begin{array}{c}{{C_{2}= \frac 1 2 L_{A B}L^{A B}= E_{o}( E_{o}+ 3 ) + s ( s + 1 )}}\\{{C_{4}= W_{A}W^{A}= - s ( s + 1 ) ( E_{o}- 1 ) ( E_{o}- 2 )}}\\{{W_{A}= \frac 1 8 \epsilon_{A B C D E}L^{B C}L^{D E}}}\\ \end{array}
\varepsilon_1 = \sum_{k\geq 1}k^{-4m/n}\|\partial_\nu\phi^{(1)}_{k + E}- \partial_\nu\phi^{(2)}_{k + E}\|_{L^2(\partial B_R)}, \varepsilon_2 = \sum_{k\geq 1}k^{-4m/n}\|\partial_\nu(\Delta\phi^{(1)}_{k + E}) - \partial_\nu(\Delta\phi^{(2)}_{k + E})\|_{L^2(\partial B_R)}.
\left[\begin{pmatrix}1 \frac{t^2-1}{t^2 v}0 1 \end{pmatrix}, \begin{pmatrix}\frac{1}{t}0 -\frac{t v}{(t^2-1)}t \end{pmatrix}\right] =\begin{pmatrix}t^2 0 v \frac{1}{t^2}\end{pmatrix}
Z^{\pm}_{\delta}=\int [dx]\ \theta \left(\pm\int^{\beta /2}_{-\beta /2}x dt\right)\exp\left( -\int^{\beta /2}_{-\beta /2}dt \left\{{1\over 2}\left[\dot{x}^2+(m^2+2\lambda_{\pm})x^2\right]+\delta (igx^3-\lambda_{\pm}x^2)\right\}\right).
P(1)T(1)=\sum_{i=0}^m p_i \sum_{i=0}^m t_i=\sum_{i=0}^{\frac{m}{2}-1}2(2\sum_{k=0}^{\frac{m}{2}-1}t_{k}+t_{\frac{m}{2}}) p_{i}+ t_{\frac{m}{2}}(2 \sum_{k=0}^{\frac{m}{2}-1}t_{k}+t_{\frac{m}{2}}) < 0.
0 = v_{dn}^{p^{\epsilon^{\prime}}}x = \left( \frac{p}{\pi^d}(v_n^A)^{\frac{p^{dn}-1}{p^n-1}}\right)^{p^{\epsilon^{\prime}}}x + \sum_{j=1}^{n-1}\left( v_j^Ax_j \right)^{p^{\epsilon^{\prime}}}x = \left(\frac{p}{\pi^d}\right)^{p^{\epsilon^{\prime}}}(v_n^A)^{p^{\epsilon^{\prime}}\frac{p^{dn}-1}{p^n-1}}x,
P_{i j}[ \alpha ( t ) , \alpha ( \mu ) ] \equiv \left( \begin{array}{l l}{{p}}&{{r}}\\{{r}}&{{q}}\\ \end{array}\right) , \qquad F_{k l}[ \alpha ( t ) , s ] \equiv \left( \begin{array}{l l}{{k}}&{{h}}\\{{h}}&{{g}}\\ \end{array}\right)
m(0)=\lim_{r\downarrow 0}\lim_{k\rightarrow\infty}\nu_{k,+}(B_r^+)+\nu_{k,-}(B_r^+)\geq 4\pi, &\lim_{k\rightarrow\infty}\nu_{k,+}(B_1^+)+\nu_{k,-}(B_1^+)\leq m(0)+\varepsilon/4,
P_K(V;{\bf a}_1, \ldots,{\bf a}_K, \rho_1, \ldots, \rho_K) =\frac{1}{Z(V)}\, \frac{C^K}{K!}\prod_{j=1}^{K}\rho_j^{\alpha}\ \Theta(\{{\bf a}_j\},\{\rho_j\}) \,.
\phantom{{}={}}\sum_{k=1}^{2^n-1}(\psi_{n,k}+\psi_{n,-k})(F)(\psi_{n,k}-\psi_{n,-k})(F) f(2^{-n}k)\varphi(a) &= \sum_{k=1}^{2^n-1}(\psi_{n,k}^2-\psi_{n,-k}^2)(F) f(2^{-n}k) \varphi(a) &= \sum_{k\in I_n}\psi_{n,k}^2(F) f(2^{-n}k) \varphi(a) = \Psi_n^A(f)(F)\cdot \varphi(a),
t = E{\int}{dr\, \Bigl(1 +{4M\over r}-{4q^2\over r^2}\Bigr)\left({E^2\Bigl(1 +{4M\over r}-{4q^2\over r^2}\Bigr) -{L^2\over r^2}- m^2\sqrt{1 +{4M\over r}-{4q^2\over r^2}}}\right)^{-1/2}}\ .
\sum_{i=1}^{n}l_t(x_{i,t}) =\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}l_{j,t}(x_{i,t})=\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}l_{j,t}(x_{j,t})+\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}(l_{j,t}(x_{i,t})-l_{j,t}(x_{j,t})) &\le\sum_{i=1}^{n}l_{i,t}(x_{i,t})+\frac{1}{n}\sum_{i=1}^{n}\sum_{j=1}^{n}G_1\|x_{i,t}-x_{j,t}\|,
I_{a}(t) = \int_{1}^{\infty}Ct^{\beta - 1}x^{- 1/\beta}w(x^{-1/\beta}, \beta, 1)dx \le t^{\beta - 1}\int_{1}^{\infty}x^{-1/\beta}Cdx \le C1^{1-1/\beta}t^{\beta - 1}= Ct^{\beta - 1}.
\widetilde L_4\left[ u^2 (1-u) (9-u) (25-u) \det\begin{pmatrix}D^{0}f_{1}(u) D^{0}f_2(u) D^0f_3(u) D^{1}f_{1}(u) D^{1}f_2(u) D^1f_3(u) D^{2}f_{1}(u) D^{2}f_2(u) D^2f_3(u) \end{pmatrix}\right]=0
C_2 f_N^{(2)}(t,x_1,v_1) =\int_{\mathbb{R}^d \times \mathbb{S}^{d-1}}\omega \cdot (v_2 - v_1) f_N^{(2)}(t,x_1,v_1,x_1 + \varepsilon \omega,v_2)d\omega dv_2
\begin{aligned}F(s_1,s_2):= \ \mu^2\lambda(s_1^2+\nu+1)^2+ \nu(s_2^2+\mu(1+\lambda))^2 -(\mu^2\lambda+\nu)(s_1^2+\nu+1)(s_2^2+\mu(1+\lambda))+(\mu^2\lambda-\nu)^2.\end{aligned}
{\bf a}= \left\| \begin{array}{c}{{a^{1}}}\\{{a^{2}}}\\{{a^{3}}}\\ \end{array}\right\| \quad{\bf b}^{T}= \left\| \begin{array}{c c c}{{b_{1}}}&{{b_{2}}}&{{b_{3}}}\\ \end{array}\right\| \quad r \in S O ( 3 )
{}_4F_3\biggl(\begin{matrix}r, \, 1-r, \, t, \, 1-t 1, \, 1, \, 1 \end{matrix}\biggm|1\biggr)=\sum_{k=0}^\infty\frac{(r)_k(1-r)_k(t)_k(1-t)_k}{k!^4}.
\partial_{\xi_1}V^{(i,m,j)}_2\bigl(\xi_1, x_2, t \bigr)\big|_{\xi_1=b_{i,m}\pm \tfrac{h_{i,m}}{2}}=\Big(\mp \delta_{\alpha_i,1}\kappa_i\big(v^{(i,m)}_0(x,t)\big) \pm \delta_{\beta_i,1}g^{(i)}_0\bigl(x,t\bigr)\Big)\Big|_{x_1=\varepsilon(j+b_{i,m})},
\begin{array}{lcl}\lambda^{(0)}= \gamma_{1,6}\lambda^{(0)}\gamma_{1,6}^{-1}, \lambda^{(0)}= -\gamma_{\Omega R,6}\lambda^{(0)^{T}}\gamma_{\Omega R,6}^{-1}\lambda^{(i)}= e^{2\pi iv_i}\gamma_{1,6}\lambda^{(i)}\gamma_{1,6}^{-1}, \lambda^{(i)}= -\gamma_{\Omega R,6}\lambda^{(i)^T}\gamma_{\Omega R,6}^{-1}\end{array}
a_{k+1}=a_k\,\frac{(1+4z_k)^2}{(1+2z_k)^2}+b_k\,\frac{4z_k(1+4z_k)^2}{(1+2z_k)^3(1-8z_k)}, b_{k+1}=2b_k\,\frac{(1+4z_k)^3(1-z_k)}{(1+2z_k)^3(1-8z_k)}, z_{k+1}=\frac{2z_k^2}{1+6z_k+(1+2z_k)\sqrt{1+8z_k}}.
g^{(2)}(y)=\frac{\hbar c \xi^2}{4\pi}\sum_{n=-\infty}^{\infty}\,y^2 \partial_y\left[ y{{\partial}_y}(I_n(a y)K_n(a y)) \right]^2=\frac{2 \hbar c \xi^2}{\pi^2}\int_{0}^{1}du\, \frac{u^2}{\sqrt{1-u^2}}y^2{{\partial}_y}(a y K_1(2a y u))^2 ,
h(z,w) =\frac{\eta_{-}^{2}}{4\pi}\frac{e^{-\eta_{-}u(w+1)}}{\sqrt{w^{2}-1}\sqrt{z^{2}-1}}\frac{z-w}{1-zw}\big(zw\big)^{M-N}\prod_{k = 1}^{N}\frac{\sigma_{k}z-\tau}{\tau z-\sigma_{k}}\frac{\sigma_{k}w-\tau}{\tau w-\sigma_{k}}.
-\frac{2\sin(\pi s)}{\pi}(-1)^j \int_1^\infty (t^2-1)^s t^{-2j-1}dt = \frac{i}\pi \int_{-\infty+i0}^{\infty+i0}(1+z^2)^s z^{-2j-1}dz = \binom{s}{j}.
\omega^{(1)}=\int_{S^3}\hat Q={3\over 4}{\rho^3 [\rho dx_0^4-x_0^4 d\rho]\over [\rho^2+(x_0^4)^2]^{5\over 2}},\quad\quad{\omega^{(1)}}^\prime=\int_{{S^3}^\prime}\hat Q={3\over 4}{\rho^3 [\rho dx_0^3-(x_0^3) d\rho]\over [\rho^2+(x_0^3-\bar x)^2]^{5\over 2}}.
\begin{aligned}{C1}: A({\mathbf{c}})(\tau)+A({\mathbf{d}})(\tau)=0 |\tau| \in \mathcal{U}_{1}\cup \mathcal{U}_{2}; &{C2}: C({\mathbf{c}, \mathbf{d}})(\tau)+C({\mathbf{d}, \mathbf{c}})(\tau)=0, |\tau| \in \mathcal{U}_{2}.\end{aligned}
\overline{H}(p)=\min_{q\in\mathbb{R}^d}\overline{H}(q),\;\;\textrm{or,}\;\; p\in\partial\left(\left\{\;\overline{H}(q)\leq\overline{H}(p)\;\right\}\right)\cap\partial\left(\textrm{Conv}\left(\left\{\;\overline{H}(q)\leq\overline{H}(p)\;\right\}\right)\right),
R_-^{-1}(z) R_+(z) = P(z) S_-^{-1}(z) S_+(z) P^{-1}(z) & = \mathbb I + z^{-\beta}e^{2n \varphi(z)}P(z) \begin{pmatrix}0 0 0 1 0 0 0 0 0 \end{pmatrix}P^{-1}(z).
p_{n}^{(\alpha,\beta)}(x) =\frac{1}{S_n(\alpha,\beta)}\sum_{k=0}^n\binom{n+\alpha}{n-k}\binom{n+\beta}{k}x^{n-k}(x-1)^k, S_n(\alpha,\beta) =\binom{2n+\beta+\alpha}{n}.
|k|x\left|\int_0^{\frac12}\frac{e^{i\xi}-1}{\xi}\frac{1}{\xi-kx}~d\xi\right| \le C|k|x\int_0^{\frac12}\frac{1}{|k|x-\xi}~d\xi &= C|k|x\log\left(\frac{|k|x}{|k|x-\frac12}\right) &=C|k|x\log\left(1+\frac{1}{2|k|x-1}\right) &\le C|k|x\log\left(1+\frac{1}{|k|x}\right)\le C|k|^{\epsilon}|x|^{\epsilon}.
B(0)^{\ell_1}\cdot B_1 \cdot B(0)^{\ell_2}\cdot B_1 \cdots B(0)^{\ell_k}\cdot B_1 \cdot B(0)^{\ell_{k+1}}= B(0)^{\alpha-\beta}B_1 =B(0)^{2 \alpha+k}B_1 & =B(0)^{2 \alpha+k-1}B_2(0)\,,
+\sum_{k=0}^{k=4}\frac{1}{\alpha+k}+\sum_{k=0}^{k=4}\frac{1}{\beta+k}-\frac{1}{1}-\frac{1}{2}-\frac{1}{3}-\frac{1}{4}-\frac{1}{5}-\frac{1}{1}-\frac{1}{2}].
G_{k_{1}k_{2}}^{NN}=\sqrt{(2k_{1}+1)(2k_{2}+1)}\cdot e^{-iN\omega_{L}t}\times\sum_{M}(\begin{matrix}I&I&k_{1}\\M^{\prime}&-M&N\\ \end{matrix})(\begin{matrix}I&I&k_{2}\\M^{\prime}&-M&N\\ \end{matrix})
\hat{\cal A}_{2}= - g{\frac{( - 1 )^{n + 1}}{2^{2 n - 1}n !}}({\frac{g}{\pi}})^{n}\epsilon_{\mu_{1}\nu_{1}. . . \mu_{n}\nu_{n}}\theta^{\alpha \beta}\partial_{\beta}( \hat{A}_{\alpha}\hat{F}_{\mu_{1}\nu_{1}}. . . \hat{F}_{\mu_{n}\nu_{n}}) .
\frac{A-B}{2}r^2(a_2e^{u_1}+a_1e^{u_2})\le J(ru_1'(r)+2, ru_2'(r)+2)+(A-B)r^2 &\quad\times\bigg[a_2e^{u_1}+a_1e^{u_2}-\frac{a_2(1+a_1)}{2}e^{2u_1}-\frac{a_1(1+a_2)}{2}e^{2u_2}+a_1a_2e^{u_1+u_2}\bigg] &=J(R_1u_1'(R_1)+2, R_1u_2'(R_1)+2)+o(1)\le C
\bigg\Vert \sum_{i=1}^n c_i (f_i-g_i) \bigg\Vert \leq \lambda_1 \bigg\Vert \sum_{i=1}^n c_i f_i \bigg\Vert +\lambda_2 \bigg\Vert \sum_{i=1}^n c_i g_i \bigg\Vert + \mu \bigg[ \sum_{i=1}^n |c_i|^2 \bigg]^{1/2}
\psi(C_1,C_2,\ldots,C_n) =\langle \psi(C_{1}),\psi(C_{2}),\ldots,\psi(C_{n})\rangle &=\langle \phi(C_{\sigma^{-1}(1)}),\phi(C_{\sigma^{-1}(2)}),\ldots,\phi(C_{\sigma^{-1}(n)})\rangle\in Z_{\phi}.
\begin{aligned}\left\{F,G\right\}_{\pi_1}=dG(\pi_1 dF)=dG \left(\pi_1 \alpha_{\frac{\delta f}{\delta k}}\right) &=\int \left[\left(\frac{\delta f}{\delta k}\right)''\left(\frac{\delta g}{\delta k}\right)'+D^{-1}_{s}\left(k\left(\frac{\delta f}{\delta k}\right)'\right)k \left(\frac{\delta g}{\delta k}\right)'\right] ds.\end{aligned}
[x+\gamma_j,B_u]\phi_j = -\frac2i\partial_x\phi_j + 2C_+(u\phi_j) -2[xC_+(u\phi_j)_x-C_+(x(u\phi_j)_x)] &=-\frac2i\partial_x\phi_j + 2C_+(u\phi_j) &=-2L_u\phi_j=-2\lambda_j\phi_j.
S [ X , g ] \ = \ \int_{0}^{1}d^{2}\sigma \frac{1}{2 \pi}\left\{\sqrt{g}g^{\alpha \beta}\partial_{\alpha}X^{I}\partial_{\beta}X^{I}\ - \i B^{I J}\varepsilon^{\alpha \beta}\partial_{\alpha}X^{I}\partial_{\beta}X^{J}\ \right\}\ ,
\widetilde{\rho}[\gamma]\equiv{\rm e}^{-{7+d\over 16}(\log \gamma'(0) + \log \gamma'(1) )}\;\;\;,\;\;\;\widetilde{\rho}[\gamma \circ \delta] = \widetilde{\rho}[\gamma]\widetilde{\rho}[\delta]\;\;\;;\;\;\;\gamma,\delta \in{\cal D}_L \;\;\;.
\left(\begin{array}{c}\varphi^{*}X_{1}\varphi^{*}X_{2}\end{array}\right) = A_{\varphi}\left( \begin{array}{c}X_{1}X_{2}\end{array}\right) + \left(\begin{array}{c}\sum_{k=1}^{n-3}\gamma_{1,k}Y_{k}\sum_{k=1}^{n-3}\gamma_{2,k}Y_{k}\end{array}\right)
F_{i}^{r}=H_{i+1}(A_{i}-P_{i}C_{i}^{\prime}(C_{i}P_{i}C_{i}^{\prime}+W_{i})^{-1}C_{i}-B_{i}(B_{i}^{\prime}S_{i+1}B_{i}+R_{i})^{-1}B_{i}^{\prime}S_{i+1})G_{i}^{\prime},
\hbox{\large $\frac{1}{2}$}( W^{(0)}_m, W^{(0)}_m)^a - V_m^a W^{(0)}_m = 0,\qquad\hbox{\large $\frac{1}{2}$}\{W^{(0)}_m, W^{(0)}_m \}_A - V_A W^{(0)}_m = 0,
\{d_{m}^{\mu , j}, d_{n}^{\nu , j^{\prime}}\}= - \eta^{\mu \nu}\delta^{j , j^{\prime}}\delta_{m , - n}; ~ ~ \{d_{m}^{\mu , k}, d_{n}^{\nu , k^{\prime}}\}= - \eta^{\mu \nu}\delta^{k , k^{\prime}}\delta_{m , - n}~ : R
{\cal L_{BCS}}= \psi^\dagger_\uparrow (i \partial_t - \epsilon (p))\psi_\uparrow + \psi^\dagger_\downarrow (i \partial_t - \epsilon (p))\psi_\downarrow - \lambda \psi^\dagger_\uparrow \psi^\dagger_\downarrow\psi_\downarrow \psi_\uparrow \, \, \, \, ,
k_{R}= \frac{1}{N n}\sum_{\alpha = 1}^{n}\sum_{\bf i}\frac{\partial^{2}V}{\partial ( \varphi_{\bf i}^{\alpha})^{2}}= 2 J d - m^{2}+ \lambda \frac{n + 2}{N n}\sum_{\alpha = 1}^{n}\sum_{\bf i}( \varphi_{\bf i}^{\alpha})^{2}~ .
\mathcal{M}_{{\lambda}}\left[\left(\mathrm{F}_1\ast\mathrm{F}_2\right);\alpha_{\mathrm{\Re}}\right]= \mathcal{M}_{{\lambda}}\left[\mathrm{F}_1;\alpha_{\mathrm{\Re}}\right]\mathcal{M}_{{\lambda}}\left[ \mathrm{F}_2;\alpha_{\mathrm{\Re}}\right]
\begin{aligned}e^{\frac{\pi(1-C\eta)}{b_{j}}}\leq s_{j}-s_{j,0}+e^{\frac{\pi(1-C\eta)}{b_{j,0}}}\leq s_{j}+e^{\frac{\pi(1-C\eta)}{b_{j,0}}}\leq s_{j}^{\frac{4}{3}}.\end{aligned}
K_{u v}^{x}\, k_{\mathbf{I}}^{u}\, k_{\mathbf{J}}^{v}\, - \,{\frac{\lambda}{2}}\, \varepsilon^{x y z}\,{\cal P}_{\mathbf{I}}^{y}\,{\cal P}_{\mathbf{J}}^{z}\, = \,{\frac{1}{2}}\, f_{\phantom{\mathbf{K}}\mathbf{I}\mathbf{J}}^{\mathbf{K}}\,{\cal P}_{\mathbf{K}}^{x}
\Gamma_{a,s}(l_{1s})=\tilde\Gamma_s^{-1}\Gamma_{a ,s}^{(o)}(l_{1s})\tilde\Gamma_s, \qquad \Gamma_{b,s}(l_{2s})=\tilde\Gamma_s^{-1}\Gamma_{b ,s}^{(o)}(l_{2s})\tilde\Gamma_s
\left\Vert U^{M}\left( t,s\right) \right\Vert_{\beta\rightarrow\beta}\leq\exp\left( K\left( \beta,d\right) \sum_{k=1}^{d}\left( M+1\right)^{\left( k/2-1\right)_{+}}\int_{J_{st}}\left\vert P_{k}\left( \tau,\theta,\theta^{\ast}\right) \right\vert d\tau\right) .
d_0 = -4\alpha^2\beta\nu(2\alpha^2 \nu+\sigma) & d_c = \sigma^2+4\alpha^2\nu \sigma + 8 \alpha^4\nu^2 & d_s = -\sigma^2-8\alpha^2\nu \sigma - 8 \alpha^4\nu^2=-d_c-4\alpha^2\nu \sigma & d_\tau = -\frac{\alpha}{\rho}
I = \ \sum_{2^\ell = 1}^{\alpha^{-2}t^{2}}\min \{2^{(k+1) \ell}|t|^{-1}(\|(w,r)\|_{H^{-1}}+ \|(g,k)\|_{H^{-1}}),2^{-(k+\frac32) \ell}\|(w,r)\|_{\dot H^{2k+2}}\}\lesssim \ |t|^{-\frac12}|t|^{-\frac{1}{8k+10}}(\|(w,r)\|_{H^{-1}}+ \|(g,k)\|_{H^{-1}}+ \|(w,r)\|_{\dot H^{2k+2}})
T_{U}^{\mu \nu , \, \alpha \beta}= \left[ - \frac 1 2 T_{1}( u ) + \frac 1 2 T_{2}( u ) - \frac 2 3 T_{3}( u ) + \frac 7{1 2}T_{4}( u ) - \frac 1 3 T_{5}( u ) \right]^{\mu \nu , \, \alpha \beta}
C_{BPS}^P(M\times I_R^n\times T^n,X\times T^n;\mathbf{l}^\ast,(0_M,e)) &=C_{BPS}^P(M\times I_R^n\times T^n,X\times T^n;K\mathbf{l}^\ast,(0_M,e))\leq\sum_{i=1}^nR_i\cdot|e_i|.
\mathcal{L}=\frac{1}{2}\int_0^T \int_{\Omega}(y-y_d)^2d\Omega dt+\frac{\nu}{2}\int_0^T\int_{\Omega}+\int_0^T\int_{\Omega}\left(-\frac{\partial y}{\partial t}+\Delta y-u\right) p_\Omega d\Omega dt+\int_0^T\int_{\partial\Omega}(y-h)p_{\partial\Omega}dsdt
+ \frac{C^{2}\, e^{2 \, \alpha + 2 \, \alpha^{*}}\, n^{4}\, \Theta ( y_{0}- x_{0})}{\left( \left( x_{0}- y_{0}- i \, \epsilon \right)^{2}- \left( x_{s}- y_{s}\right)^{2}\right)^{2}}+ \frac{C^{2}\, n^{4}\, \Theta ( y_{0}- x_{0})}{\left( \left( y_{0}- x_{0}- i \, \epsilon \right)^{2}- \left( x_{s}- y_{s}\right)^{2}\right)^{2}}
\left\{\frac{2^{3^{k+1}+k+2+\sum\nolimits_{i=0}^{k+1}2b_{i}3^{i}}-\sum\nolimits_{j=0}^{k}3^{k-j}2^{j+1+\sum\nolimits_{i=0}^{j}2b_{i}3^{i}}-3^{k+1}}{3^{k+2}}\right\}, &\left\{\frac{2^{2\cdot3^{k+1}-2+\sum\nolimits_{i=0}^{k+1}2b_{i}3^{i}}-\sum\nolimits_{j=0}^{k}3^{k-j}2^{3^{j+1}-1+\sum\nolimits_{i=0}^{j}2b_{i}3^{i}}-3^{k+1}}{3^{k+2}}\right\}=A.
i\hbar\frac{\partial}{\partial t}(\begin{matrix}\sqrt{n_{A}}e^{i\phi_{A}}\\ \sqrt{n_{B}}e^{i\phi_{B}}\end{matrix})=(\begin{matrix}eV&K\\K&-eV\end{matrix})(\begin{matrix}\sqrt{n_{A}}e^{i\phi_{A}}\\ \sqrt{n_{B}}e^{i\phi_{B}}\end{matrix}),
\alpha'_n=\frac{1+a}{1+aq^{2n}}q^n\alpha_n(a^2,q^2)\quad\mbox{and}\quad\beta'_n=\sum_{j\leq n}\frac{(-a)_{2j}}{(q^2;q^2)_{n-j}}q^{j}\beta_j(a^2,q^2).
L_{G}^{(i)}= \sup \left\{\frac{|G_{i}(p_{1},p_{2}) - G_{i}(q_{1},q_{2})|}{|p_{1}- q_{1}| + |p_{2}- q_{2}|}\, \mid \, (p_{1},p_{2}),(q_{1},q_{2}) \in \mathbb{R}^{2}\right\}, L_{G}= \max\{L_{G}^{(1)},\dots,L_{G}^{(K)}\}.
\mathcal{L}_{F}^{\alpha}[(D_{F}^{\alpha})^{n}f(S_{F}^{\alpha}(x))] & = (S_{F}^{\alpha}(s))^{n\alpha}\mathcal{F}_{F}^{\alpha}(s)-(S_{F}^{\alpha}(s))^{n\alpha-1}f(S_{F}^{\alpha}(0)) &-(S_{F}^{\alpha}(s))^{n\alpha-2}D_{F}^{\alpha}f(x)|_{x=S_{F}^{\alpha}(0)}-\ldots &-(D_{F}^{\alpha})^{n-1}f(x)|_{x=S_{F}^{\alpha}(0)}.
I\left( f\right) ={\displaystyle\sum_{k=-\infty}^{\infty}}\int_{0}^{R}2\pi f_{k}\left( r\right) w_{-k}\left( r\right) rdr={\displaystyle\sum_{k=-\infty}^{\infty}}\int_{0}^{R}\pi f_{k}\left( \sqrt{\rho}\right) w_{-k}\left( \sqrt{\rho}\right) d\rho.
z_2 \beta^{j+1}+ z_1 (\beta^j \nu_1 + \beta \nu_j) = [ x_1 \beta^j + x_2 \nu_j, y_1 \nu_1 + y_2 \beta ] = [x_1 \alpha^j, y_2 \alpha] \varphi = (z_2 \alpha^{j+1}) \varphi = z_2 \beta^{j+1}+ z_1 \nu_{j+1}
W'(x)= pA^{\tfrac{p}{2}-1}(x)\big(2h'(x)h''(x)+\frac{p-2}{2}h'^2(x)A^{-1}(x)A'(x)-\frac{1}{2}A'(x)\big) = pA^{\frac{p}{2}-1}(x)\big(h'(x)h''(x)+\frac{p-2}{2}h'^2(x)A^{-1}(x)A'(x)+\frac{m-1}{2}\sin 2h(x)h'(x)\big) = p(m-p)A^{\tfrac{p}{2}-1}(x)\tanh x\,h'^2(x),
III \le 2 c_6\int_0^{R/2}\left( \int_0^u t^{\frac{\alpha}{2}-1}dt \right){u^{-\alpha-1+\eta}}du =\frac{4 c_6}{\alpha}\int_0^{R/2}{u^{-\frac{\alpha}{2}-1+\eta}}du <\infty.
\check{\mathfrak{a}}[u,u]=\int_{\mathbb{R}^d}\left(\langle \check{g}(\mathbf{x})\mathbf{D}u,\mathbf{D}u\rangle + \check{v}(\mathbf{x})\vert u\vert^2\right)\,d\mathbf{x},u\in H^1(\mathbb{R}^d).
\eta_{a}(C(\mathsf{Q}))=\sum_{i,j}c_{j}^{i}\pi(E_{a}K_{2\rho}K_{a}^{-1}F_{a})_{i}^{j}-\sum_{i,j}c_{j}^{i}\pi(K_{a}F_{a}E_{a}K_{2\rho}K_{a}^{-2})_{i}^{j}.
\begin{array}{ll}G_i^0(X(T)) = \int_{0}^{T}f_i^0(s,x(s),u(s))ds+g_i^0(x(T)) \null \ge G_i^0(X_0(T))=\int_{0}^{T}f_i^0(s,x_0(s),u_0(s))ds+g_i^0(x_0(T))\end{array}
\delta_2\psi'(t) \leqslant \dfrac{\eta\delta_2^2}{2}\int_\Omega|\nabla u|^2dx+\dfrac{\eta}{2}\int_\Omega|\nabla \theta|^2dx+a_1\delta_2\int_\Omega|\theta|^2dx-\delta_2\int_\Omega f(t,u)(\Delta^{-1}\theta)dx &-\delta_2\kappa\int_\Omega u_t\theta dx-\delta_2a_0\int_\Omega|u_t|^2dx.
\quad\;[u,v,w]_{T_t}=\displaystyle\sum_{k=0}^{+\infty}\sum_{i+j=k}\Big(D(T_i(u),T_j(v))w+\theta(T_i(v),T_j(w))u-\theta(T_i(u),T_j(w))v\Big)t^{k},
\alpha \int_{\mathbb{R}^3}\mathrm{d}k \; f_y(k) \, \left\langle x u_{\alpha}\,\bigg|\, \frac{(h_{\alpha}-e_{\alpha})|k|}{3(h_{\alpha}-e_{\alpha}+|k|)}\,\bigg|\, x u_{\alpha}\right\rangle =\frac{\alpha^2}{L^3}- \aleph_{\alpha,L}\frac{\alpha}{L^4}+ O\Big( \frac{\alpha^3}{L^2}\log(\alpha^{-1})\Big),
\left(\delta_tV^{\frac{1}{2}}, V^{1}\right)\geq \frac{\left\|V^{1}\right\|^2- \left\|V^{0}\right\|^2}{2k_1}, \left(\delta_tV^{n-\frac{1}{2}}, V^{n-\frac{1}{2}}\right) =\frac{\left\|V^{n}\right\|^2- \left\|V^{n-1}\right\|^2}{2k_n}.
p^*(x_2, x_4 \mid x_1, x_3) = \sum_{x_0}p(x_0) \cdot p(x_2 \mid x_1, x_0) \cdot p(x_4 \mid x_3, x_0) = \alpha \cdot q(x_2 \mid x_1) \cdot q(x_4 \mid x_3) + (1-\alpha) \cdot r(x_2 \mid x_1) \cdot r(x_4 \mid x_3)
Q_{\alpha A}= T_{A}^{\, \, \, \, a}Q_{\alpha a}, \quad Q_{\dot{\alpha}\bar{A}}= T_{\bar{A}}^{\, \, \, \, \dot{a}}Q_{\dot{\alpha}\dot{a}}, \quad T_{\bar{A}}^{\, \, \, \, \dot{a}}= \left( T_{A}^{\, \, \, \, a}\right)^{*}
\theta_{S,\mu}\left(\gamma(\tau,z)\right)(c \tau + d)^{-\frac{g}{2}}e\left(-c S[z] (c \tau+d)^{-1}\right)=\sum_{\eta\in(2S)^{-1}\mathbb Z^g/\mathbb Z^g}\varepsilon_{S}(\eta, \mu;\gamma)\theta_{S,\eta}(\tau, z),
\sum_{k=2n}^\infty (1+k)^{-s}h_{k}(x)h_{k}(y) \leq \Big(\sum_{k=2n}^\infty (1+k)^{-s}|h_{k}(x)|^2 \sum_{k=2n}^\infty (1+k)^{-s}|h_{k}(y)|^2\Big)^{1/2}&\leq (1+2n)^{-s+1-\frac{1}{\alpha}}\, .
\begin{aligned}\mathcal{W}\,(f|D|^{-n})= 2^{m}v_{n-1}\int_{M}f~vol_{g}, \mathcal{W}\,(f |D|^{-n+2})= -2^{m}\frac{n-2}{24}v_{n-1}\int_{M}f R(g)~vol_{g}.\end{aligned}
E_2^{p,q}:=\mathcal{E}xt^p(R^{-q}a_{2,*}(a_1^*P^{\vee}),\mathcal{O}_{A_2})\Rightarrow \mathcal{E}xt^{p+q}(Ra_{2,*}(a_1^*P^{\vee}),\mathcal{O}_{A_2}).
J ( a ) ={\frac{1}{\Gamma ( 3 + \epsilon / 2 )}}\int_{0}^{\infty}d y y^{4 + \epsilon}{\frac{d}{d y}}\ln \left( 1 -{\frac{K_{\mu}( y ) I_{\mu}( a y )}{K_{\mu}( a y ) I_{\mu}( y )}}\right) ,
{\underbrace{\X_{\sigma^\prime ij}\}}_T= -{1\over E_j - E_i}\left({d\,\over d\sigma}f_{\sigma ij}\right)\int_\sigma^\infty d\sigma^\prime X_{\sigma^\prime ij}+{1\over E_j - E_i}\left(1-f_{\sigma ij}\right)X_{\sigma ij}.
S_1 := \{(t,w_2,\dots,w_n) \in U \,;\, a(t)=\kappa_\gamma(t)=0 \}, S_2 := \{(t,w_2,\dots,w_n) \in U \,;\, \hat\rho(t,w_2,\dots,w_n)=0, \kappa_\gamma(t)\neq0 \},
\lefteqn{\beta\,E^\dagger(x)-\Omega(x)+\Omega^\dagger}&\qquad=\inf_{r\geq 0}\bigl(\beta\,E^\dagger(x)-\Omega(x)+\Omega^\dagger-r\,\|F(x)-y^\dagger\|+r\,\|F(x)-y^\dagger\|\bigr) &\qquad\leq\inf_{r\geq 0}\bigl(D_\beta(r)+r\,\|F(x)-y^\dagger\|\bigr)
\sum_{n \geq 1}A(n, m) S(-n, p; c) \psi(n) = \sum_{d \bar d \equiv 1 \pmod*{c}}e\Bigl( \frac{-p \bar d}{c}\Bigr) \sum_{n \geq 1}A(n, m) e\Bigl( \frac{n d}{c}\Bigr) \psi(n)