harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
Q_{A,i}=\begin{cases}R_{A,i}\setminus{\displaystyle \bigcup_{(m,k) \in{M}_{A,i}'}}(\mathcal{R}_{A,m,i}\times\mathcal{T}_{k}^A) \ \textnormal{for}\A \geq 2^{101}, \R_{2^{100},i}\, \textnormal{for}\A = 2^{100},\end{cases}
\begin{aligned}\mbox{either}\;\; \limsup_{n \rightarrow\infty}\left(\frac{h^j_n}{h^k_n}+\frac{h^k_n}{h^j_n}\right) = \infty, &\mbox{or}\;\; (h^j_n) = (h^k_n)\;\;\mbox{and}\;\;\limsup_{n\rightarrow \infty}\frac{|t^j_n - t^k_n|}{(h^j_n)^\alpha}= \infty,\end{aligned}
\frac{\cosh \left( \nu \pi \right)}{\pi}\left| K_{_{\frac{1}{2}+ i \nu_{1}}}\left( x \right) \right|^{2}- \frac{e^{- \Lambda x}}{2 x}\ , \ \frac{\cosh \left( \nu \pi \right)}{\pi}\left| K_{_{\frac{1}{2}+ i \nu_{2}}}\left( x \right) \right|^{2}- \frac{e^{- \Lambda x}}{2 x}\ ,
({\frac{\rho}{4}}+ 2 \sin^{2}{F}) F^{\prime \prime}+{\frac{1}{2}}\rho F^{\prime}+ \sin{2 f}F^{2}-{\frac{1}{4}}\sin{2 F}-{\frac{\sin^{2}{F}\sin{2 F}}{\rho^{2}}}= 0
\sum_{j=1}^{\infty}2^{(-j+1)p(1-s_i)}\le 2\int_0^{2}t^{p(1-s_i)-1}\,dt=\frac{2^{1+p(1-s_i)}}{p(1-s_i)}\le \frac{2^{1+p}}{p(1-s_i)},
\frac{1}{2}[ \frac{1}{x + 2}- \frac{1}{x}+ \frac{1}{x + 3}- \frac{1}{x + 1}+ \frac{1}{x + 4}- \frac{1}{x + 3}\cdots - \frac{1}{x + 1 0}+ \frac{1}{x + 8}] = 0
P_{p}^{(0)}(z,w)=P_{p,s}^{(0)}(z,w)e^{p(\varphi(z)-\varphi(w))}s^{\otimes p}(z)\otimes (s^{\otimes p})^*(w) \in L^p_z\otimes (L^p_w)^*\,,\:\:z,w\in D.
U_0(t)-U_0(\lambda_a) \le U_0(D^{C_M}_1)+C_M(t-D^{C_M}_1)-U_0(\lambda_a) &\le C_M(D^{C_M}_1-\lambda_a)+C_M(t-D^{C_M}_1)=C_M\vert t-\lambda_a\vert
\left|\Gamma^{-1}(t)[K(\Gamma(t)f)\cdot \nabla](\Gamma(t)f)\right|_q\leq \mathcal{B}^2e^{\sum_{i=1}^N\left[\beta_i(t)\theta_i-\frac{t}{2}\theta_i^2\right]}|f|_{p}|\nabla f|_{p},\ \forall f\in W^{1,p}(\mathbb{R}^2).
{\psi}_{odd}^{-}= e^{-\frac{1}{2}z_{-}^2}\left( \begin{array}{c}z_{-}F(1-\frac{\nu}{2};\frac{3}{2};z_{-}^2) + \bar{G}\sqrt{\frac{\Delta}{2}}e^{i\frac{\pi}{4}}F(\frac{1-\nu}{2};\frac{1}{2};z_{-}^2) - \sqrt{\frac{\Delta}{2}}e^{i\frac{\pi}{4}}F(-\frac{\nu}{2};\frac{1}{2};z_{-}^2) + \bar{G}z_{-}F(\frac{1-\nu}{2};\frac{3}{2};z_{-}^2)\end{array}\right)
|x|^{-n}\, \left( \int_{B (\hat x, |x|/(8\breve c) )}| \nabla \bar e_{h,m}|^2 \, \, d \hat y \right)^{1/2}= o \left( |x|^{\frac{2-n-p}{p-1}}\right) \mbox{as}|x| \to \infty.
ds^2 = \Omega^2 ( e^{2 A}dx_{//}^2 + dr^2) +{L^2 \Omega^2 \over \rho^2}({d \theta^2 \over c}+ \rho^6 \cos^2 \theta ({\sigma_1^2 \over c X_2}+{\sigma_1^2 + \sigma_2^2 \over X_1})+{\sin^2 \theta\over X_2}d \phi^2 )
\|\eta\|^\ell_{\dot{B}_{p,1}^{\gamma_2}}\le C\|\eta\|^\ell_{\dot{B}_{2,1}^{\gamma_2+\frac n2-\frac np}}\le C\big(\|\eta\|^\ell_{\dot{B}_{2,\infty}^{\sigma-1}}\big)^{\theta_{3}}\big(\|\eta\|^\ell_{\dot{B}_{2,1}^{\frac n2-2}}\big)^{1-\theta_{3}},\theta_{3}=\frac{\frac np -\gamma_2-2}{\frac n2-1-\sigma}\in (0,1),
\frac{\partial^2u_i}{\partial x_j\partial x_k}=\frac{\partial \epsilon_{i,k}(\mathbf{u})}{\partial x_j}+\frac{\partial \epsilon_{i,j}(\mathbf{u})}{\partial x_k}-\frac{\partial \epsilon_{j,k}(\mathbf{u})}{\partial x_i}.
\underset{x_1,\cdots,x_N \to x \mathbf{1}}{\lim}e^{-\lambda_N t}\frac{\det\left(y_i^{j-1}\right)^N_{i,j=1}}{\det\left(x_i^{j-1}\right)^N_{i,j=1}}\det \left(p^{(N),s}_t(x_i,y_j)\right)^N_{i,j=1}dy=e^{-\lambda_N t}\Delta_N(y) \det \left(\partial^{(i-1)}_xp^{(N),s}_t(x,y_j)\right)_{i,j=1}^N dy.
\begin{aligned}\sum_{\alpha_1<n\leq \beta_1}f(a+nw_1+mw_2) =\int_{\alpha_1}^{\beta_1}f(a+xw_1+yw_2) dx &\quad+\int_{\alpha_1}^{\beta_1}\frac{\partial f(a+xw_1+yw_2)}{\partial x}P_1(x) dx &\quad+P_1(\alpha_1)f(a+\alpha_1w_1+yw_2) &\quad-P_1(\beta_1)f(a+\beta_1w_1+yw_2).\end{aligned}
|y^2-1| =|(y+1)(y-1)|\leq C|y+1|\leq CYv\leq Cv^{\frac{1}{2}}, |\hat{f}(v)-\hat{f}(0)| \leq \sup_{v'\in[0,v]}\bigg|\frac{d\hat{f}}{dv}(v')\bigg|v\leq CYv+C(\delta_2)v\leq Cv^{\frac{1}{2}},
\frac{2^{n_1}(1+o(1))}{\binom{n_1/2+\sqrt{n_1\ln n_1}+t}{t}}2^{n_2}\left(1- \sum\limits_{i=1}^t\frac{\binom{n_2}{i}(1+o(1))}{n_2^{2i}}\right) =\frac{t!2^{n+t}}{n_1^t}(1+o(1)).
W_0' = W_0 + \frac{1}{2}W_0^2 + N(4 + 8\alpha)W_2^2 + \mathcal{O}(|W_0|^3 + |W_2|^3) W_2' = 0 + W_0W_2 + (4 + 4\alpha)W_2^2+ \mathcal{O}(|W_0|^3 + |W_2|^3).
\begin{array}{c c c}{\it S}\\{\downarrow}\\{{\it S}^{*}}\\ \end{array}\begin{array}{c c c}{}&{\longrightarrow}&{}\\{}&{}&{}\\{}&{\longrightarrow}&{}\\ \end{array}\begin{array}{c c c}{{\bar{\it S}}}\\{\downarrow}\\{{\bar{\it S}}^{*}}\\ \end{array}\begin{array}{c c c}{}&{}&{}\\{}&{}&{}\\{.}\\ \end{array}
\lambda_1(U_2,a) \leq \lambda_1(TU_1,a) &\leq T^{-2}\lambda_1(U_1,a)+ \sigma T^{-2}|U_1|^{-\frac{2}{d}}&=T^{-2}\lambda_1(U_1,a)+ \sigma|U_2|^{-\frac{2}{d}}
\left[ -x \sum_{k=0}^{n_{c}-1}(-1)^{k}s_{k}(\phi^{2}) x^{n_{c}-1-k}+ m_{1}\cdots m_{n_{f}}\right]^{2}- \prod_{i=1}^{n_{f}}(x+m_{i}^{2}) = -\frac{\alpha_{n_{c}}^{2}}{\Lambda^{2}}x \prod_{a=1}^{n_{c}-1}(x-\alpha_{a})^{2}.
y(x,\l) = \int_0^x \Phi(x,\l) \Bigl(I_n - (C+D\Phi(1,\l))^{-1}D \Phi(1,\l)\Bigr)\Phi^{-1}(t,\l) i B f(t)dt + \int_x^1 -\Phi(x,\l) (C+D\Phi(1,\l))^{-1}D \Phi(1,\l)\Phi^{-1}(t,\l) i B f(t)dt = \int_0^1 G(x,t;\l)f(t)dt,
p\left(\gamma -\frac23\right)-q =C_gD_g\left(\frac{r_i}{r_i+\lambda_g}-\frac53\right)+C_iD_i\left(\frac{r_i}{r_i+\lambda_g}-\frac23\right) &=\frac{1}{3(r_i+\lambda_g)}\left(C_gD_g(-2r_i-5\lambda_g)+C_iD_i(r_i-2\lambda_g) \right).
\langle T_{i}^{k}\rangle_{{\mathrm{sub}}}^{(R)}=-\frac{\delta_i^k \xi^{-d-1}}{2^{d-1}\pi^{d/2}\Gamma (d/2)}\int_{0}^{\infty}\frac{\omega^d g^{(i)}(\omega )d\omega}{e^{2\pi \omega}+(-1)^d}
\dot{R}^{2}+ k = \frac{\mu}{R^{2}}+ \Bigl ( \frac{\tilde{\rho}^{2}}{\ell^{2}\sigma^{2}}- q^{2}\Bigr ) \frac{1}{R^{4}}+ \frac{2 \tilde{\rho}}{\ell^{2}\sigma}\, \frac{1}{R}+ \frac{\ell^{2}( \Delta \mu )^{2}}{1 6 \, ( R^{3}+ \tilde{\rho}/ \sigma )^{2}}\, .
5\prod_{j=1}^4(5-x_j) =5(5^4-5^3 \sum_{j=1}^4x_j+5^2\sum_{\j_1, j_2}x_{j_1}x_{j_2}- &5\sum_{\j_1, j_2, j_3}x_{j_1}x_{j_2}x_{j_1}+x_1x_2 x_3x_4) .
\tau(xk_\gamma,yk_\delta)=\tau(x,y)q^{-(\gamma,\delta)}(x\in U^+, y\in U^-,\gamma, \delta\in Q), &\tau(U^+_\gamma, U^-_{-\delta})=\{0\}(\gamma, \delta\in Q^+, \gamma\ne\delta), &\tau|_{U^+_\gamma\times U^-_{-\gamma}}\; (\gamma\in Q^+), &\tau(Sx,Sy)=\tau(x,y) (x\in \tilde{U}^{\geqq0}, y\in\tilde{U}^{\leqq0}).
\begin{gathered}N_{n,r,\delta}\geq (j_{\delta}+2) \cdot j_{\delta}\cdot \frac{r-1}{2}+N_{n^{\prime},(j_{\delta}+2)\cdot (r-1) + 1}n^{\prime}<n, N_{n,r,\delta}\geq (j_{\delta}+2) \cdot j_{\delta}\cdot \frac{r-1}{2}+ N_{n,(j_{\delta}+2)\cdot (r-1) + 1,\delta^{\prime}}\delta<\delta^{\prime}\leq \dim GL_n.\end{gathered}
\lim_{R \rightarrow \infty}\| (1 - \chi(\frac{x}{R})) u_{0}\|_{B_{1,2}^{1/2}(\mathbf{R}^{3})}+ \| (1 - \chi(\frac{x}{R})) u_{1}\|_{B_{1,2}^{-1/2}(\mathbf{R}^{3})}= 0.
\left[\!I_{M_{\rm\bf z}}\!-\!\Phi^{T}\bar{G}^{[2]}(\lambda_{0}^{[k]})\!\right]x_{2}\!=\!\Theta\left[\!\Theta^{-1}\bar{Y}^{[k]}\!-\!U_{1}^{T}\bar{G}^{[2]}(\lambda_{0}^{[k]})\bar{Y}^{[k]}\!\right]\alpha
\theta:=\frac{1}{C}\int_{F_\nu(x-)-q(x)}^{F_\nu(x-)}(1-p(v))\delta_{F_\nu^{-1}(v)}dv+\int_{F_\nu(x)}^{F_\nu(x)+\mu(\{x\})-q(x)}(1-p(v))\delta_{F_\nu^{-1}(v)}dv \mbox{and}\vartheta:=\frac{1}{C}\int_0^{F_\nu(x-)-q(x)}p(v)\delta_{F_\nu^{-1}(v)}dv+\int_{F_\nu(x)+\mu(\{x\})-q(x)}^1p(v)\delta_{F_\nu^{-1}(v)}dv
{\cal{B}}( x_{j}, y_{j}) = \left< \mathrm{e x p}\left[ \sum_{j = 1}^{n}( s_{j}\Phi ( x_{j}, t ) + t_{j}\Phi ( y_{j}, t ) ) \right] ~ \mathrm{e x p}\left[ - i \sum_{j = 1}^{n}( \omega ( x_{j}, t ) - \omega ( y_{j}, t ) ) \right] \right>
\tilde{m}_{M,n}(x,\Theta_{1},...,\Theta_{M})=\frac{1}{\sum_{j=1}^{M}N_{n}(x,\Theta_{j})}\sum_{j=1}^{M}\sum_{i=1}^{n}Y_{i}1_{X_{i}\in A_{n}(x,\Theta_{j})},
m_{A}^{\phantom{A}B}= \frac{1}{2 !}({\cal P}_{+}\Gamma^{M N})_{A}^{\phantom{A}B}C_{M N}+ \frac{1}{6 !}({\cal P}_{+}\Gamma^{M_{1}\ldots M_{6}})_{A}^{\phantom{A}B}Z_{M_{1}\ldots M_{6}}^{+}\; .
\| I_{h,T}u-u\|_{0,\overline{T}^s}+ h| I_{h,T}u-u |_{1,\overline{T}^s}\leqslant Ch^2 (|u|_{1,T}+|u|_{2,T}), ~~ s=\pm, ~\forall~T\in\mathcal{T}^i_h.
| \Xi \star \Xi \rangle_{M}= | \Xi \rangle_{M}, \ \ \ \mu_{3}^{M}= \det{}^{\frac{d}{2}}\left({\frac{1 - T}{1 - T + T^{2}}}\right) , \ \mu_{I}= \det{}^{\frac{d}{2}}( 1 - T ) .
\Phi = \,{\frac{- 1}{2 i \pi}}\, G_{1}^{0}G_{2}^{0}G_{3}^{0}\, \beta_{1}\beta_{2}\beta_{3}\, \left[ \, V_{1 2}^{0}\, \psi_{1 2}\, + \, V_{2 3}^{0}\, \psi_{2 3}\, + \, V_{3 1}^{0}\, \psi_{3 1}\, \right]
S = \int_{M}d^{D}x \sqrt{| \tilde{g}|}\left[ \tilde{R}- \left( \omega +{\frac{D - 1}{D - 2}}\right) ( \tilde{\nabla}\phi )^{2}\right] + 2 \int_{\partial M}d^{D - 1}x \sqrt{| h |}\tilde{K}\ .
V(\mathbf{j}) = \sum_{k=1}^{n-1}[(n-k+1)j_k - s(j_k)] - c(j_n,-\alpha(n)-1) \leq \sum_{k=1}^{n-1}[(n-k+1)j_k - s(j_k)]] =v(\mathbf{j}) < v(\mathbf{j}') = V(\mathbf{j}').
\nonumber \left( C_{\gamma_{1}\gamma_{2}\gamma}^{N_{1}N_{2}N}w_{\gamma_{1}}w_{\gamma_{2}}w_{\gamma}\right)^{2}= \int d g D_{0 0}^{( N_{1})}D_{0 0}^{( N_{2})}\overline{{D_{0 0}^{( N )}}}.
R_{\mathcal{BS}(\alpha)}(M(\beta)) =\begin{dcases}\dfrac{1}{1+2(1-\alpha)( \beta-1)}, 1< \beta\leq 1+ \dfrac{1-\alpha}{8\alpha}, \dfrac{2\sqrt{\alpha}}{(1+\alpha)\sqrt{1+16\alpha ( \beta-1)^2}}, 1+ \dfrac{1-\alpha}{8\alpha}\leq \beta< \frac{4}{3}.\end{dcases}
{}_a \mathtt{B}_{2a-1, a\nu-a+1, 1}(a^{{a}/{2}}z) =(2 \pi)^{\tfrac{a-1}{a}}a^{\tfrac{1}{2}(a^2\nu-2a\nu-a^2+a-1)}\left(\frac{z}{2}\right)^{-\tfrac{1}{2}(a-1)}& \times \mathtt{J}_{(\nu-1)+1/a}(z) \mathtt{J}_{(\nu-1)+2/a}(z) \ldots \mathtt{J}_{\nu}(z).
(MR_q f)(z) = \Gamma(s)^{-1}(Mf)(I-z) \cdot (Me^{-q})(z), (MR^h_q f)(z) = \Gamma(s)^{-1}(M f)(I-z) \cdot (Me^{-q})(z) \cdot (Mh)(s), (M(\Pi_q f)')(z) = p_0^{s+1}\Gamma(s+2)^{-1}(Mf)(I-z) \cdot (Me^{-q})(z),
2\int_{0}^{+\infty}\Phi(x)\cosh\Big(\frac{x}{2}\Big)\d x = \frac{1}{T^{1/4}}\int_{0}^L\Phi^+(x)e^{x/2}\d x + T^{1/4}\int_{0}^L\Phi^+(x)e^{-x/2}\d x.
T_1T_2(z_1^{m_1}z_2^{m_2}\eta) =M_{z_1}^{\alpha_1}(\alpha_2 z_1^{m_1}z_2^{m_2+1}U^{m_1}\eta) =\alpha_1 \alpha_2 z_1^{m_1+1}z_2^{m_2+1}U^{m_1}\eta =(I_{H^2({\mathbb{D}})}\otimes U^*) (\alpha_1 \alpha_2 z_1^{m_1+1}z_2^{m_2+1}U^{m_1+1}\eta) =\widetilde{U}T_2T_1 (z_1^{m_1}z_2^{m_2}\eta)
r((a,b_1)\odot(a, b_2)) =r(a,b_1)\odot r(a,b_2)=(p(a)\cdot q(b_1))\odot(p(a)\cdot q(b_2)) &=p(a)\cdot(q(b_1)\odot q(b_2))=0=r(0)) r(a,b_1\oplus b_2) =p(a)\cdot q(b_1\oplus b_2) &=p(a)\cdot (q(b_1)\oplus q( b_2))=p(a)\cdot (q(b_1)+q( b_2)) &=(p(a)\cdot q(b_1))+ (p(a)\cdot q( b_2))=r(a,b_1)+r(a,b_2) &=r(a,b_1)\oplus r(a,b_2) &=r((a,b_1)\oplus(a,b_2)).
V_{\begin{picture}(.4,.7)\put(0,0){\line(1,0){.3}}\put(0,.3){\line(1,0){.3}}\multiput(0,0)(.3,0){2}{\line(0,1){.3}}\end{picture}}[L_m] ={(\lambda^{1\over 2}- \lambda^{-1 \over 2}) \over (q^{1 \over 2}- q^{-1\over 2})}P_{L_m}[l = i\lambda, m = i (q^{1/2}- q^{-1/2})]~.
\frac{u([([ns]+1)x],[([ns]+1)y])}{u([[ns]x],[[ns]y])}= \frac{u([([ns]+1)x],[([ns]+1)y])}{h([ns]+1)}\frac{h([ns]+1)}{h([ns])}\frac{h([ns])}{u([[ns]x],[[ns]y])}\stackrel{n\rightarrow\infty}{\rightarrow}\lambda(x,y)\cdot 1\cdot \frac{1}{\lambda(x,y)}=1,
{K}^{(3)}_i(Z)= \sum_{j \in J_p}\sum_{l \in J_p}K_{ijl}^{(3)}Z_{j}Z_{l}, \tilde{K}_k^{(3)}(Z)=\sum_{j \in J_p}\sum_{l \in J_p}\tilde K_{kjl}^{(3)}Z_{j}Z_{l}
E = \frac{4 \pi}{\kappa g}\int_{0}^{\infty}\left[ \frac{1}{2}\varphi_{\infty}^{2}+ e^{2 \varphi_{\infty}}\frac{n^{2}}{2 x^{4}}\right] x^{2}d x = \frac{4 \pi}{\kappa g}n \ ,
\begin{aligned}a^{kl}\tilde{w}_{kl}= - b^k a^{ij}v_{ijk}+ 2 a^{kl}a^{ij}R^m_{kil}v_m v_j + 2 a^{kl}a^{ij}v_{ik}v_{jl}&+ 4 a^{kl}a^{ij}_k v_{il}v_j + a^{kl}a^{ij}_{kl}v_i v_j,\end{aligned}
\lim_{i\to\infty}F_{\Omega}(u_i)= \lim_{i\to\infty}\int_{B_{\delta}}\frac{e^{\alpha v_i^2}-1}{|h(y)|^{\beta}}|h'(y)|^2 + \lim_{i\to\infty}\int_{B_1\backslash B_{\delta}}\frac{e^{\alpha v_i^2}-1}{|h(y)|^{\beta}}|h'(y)|^2 \smallskip = \lim_{i\to\infty}A_1^i(\delta)+\lim_{i\to\infty}A_2^i(\delta).
\int_{- \infty}^{\infty}\epsilon \left[ \phi_{1}, \phi_{2}\right] d z \geq \int_{- \infty}^{\infty}d z \frac{d}{d z}\left({\phi_{1}}- \frac{2}{3}\phi_{1}^{3}- 2 \phi_{2}^{2}\phi_{1}\right) = \frac{\sqrt 2}{3},
( \sigma_{\mu})_{\alpha \dot{\alpha}}J^{\mu \alpha}= J_{\alpha \dot{\alpha}}^{\alpha}= i \frac{T_{G}}{4 \pi^{2}}\mathrm{T r}\left[ \bar{G}_{\dot{\alpha}\dot{\beta}}\bar{\lambda}^{\dot{\beta}}\right] , \qquad \vartheta_{\mu}^{\mu}= \frac{T_{G}}{1 6 \pi^{2}}\mathrm{T r}\left[ G_{\rho \sigma}G^{\rho \sigma}\right] .
< \lambda , \mu \mid \hat{\Phi}( \zeta ) \bar{\hat{\Phi}}( \zeta ) \mid \lambda , \mu > ={}^{*}\delta_{\lambda \tau}{}^{*}\delta_{\mu \nu}\Phi_{r}^{( \tau , \nu , \alpha )}( \zeta ) \Phi_{( \tau , \nu , \alpha )}^{r}( \zeta ) ,
G^+_{t,s}(x) = \left\{Z^\alpha_{t,s}(x) = 1, Z^{\beta}_{t,s}(x) = Z^{\Lambda}_{t,s}(x-1) = Z^{\Lambda}_{t,s}(x+1) =0 \right\}, G^-_{t,s}(x) = \left\{Z^\beta_{t,s}(x) = 1, Z^{\alpha}_{t,s}(x) = Z^{\Lambda}_{t,s}(x-1) = Z^{\Lambda}_{t,s}(x+1) =0 \right\}, G^0_{t,s}(x) = \left\{Z^\alpha_{t,s}(x) = Z^{\beta}_{t,s}(x) =0 \right\},
\aligned \quad||\sigma(t,x)-\sigma(t,y)||^2+\langle x-y,b(t,x)-b(t,y)\rangle &\quad+\sum_{i=1}^2\int_U|f_i(t,x,u)-f_i(t,y,u)|^2\nu(du) &\quad+\int_U\langle x-y,f_2(t,x,u)-f_2(t,y,u)\rangle\nu(du) &\le g(t)\eta(|x-y|^2),\t\ge 0, a.s. \endaligned
- \frac{(\partial_x u)(s,z)}{u(s,z)}= - k - \int_0^1 V(s-t) e^{-2kt}\,dt + \frac 1k \int_0^1 \int_0^{t_1}e^{-2kt_1}(1 -e^{-2kt_2}) V(s-t_1) V(s-t_2)\,dt_2\,dt_1 + O(\lvert k\rvert^{-2})
Z_a=\prod_{k\neq 0}\left(\det \frac{N^2}{i}\Omega k^2\right)^{-\frac{1}{2}}=\det\left( \frac{N^2}{i}\Omega\right)^{-\zeta(0)}e^{(N-1)\zeta'(0)}=\sqrt{N}\left(\frac{2T}{e^2 L}\right)^{\frac{N-1}{2}}\ \ .
A(x) = \sum_{k \in n \mathbb{Z}}e^{-\pi \alpha k^2}\left( \sum_{j=1}^{n}e^{2 \pi i k \varepsilon_j}e^{-2 \pi i k \frac{j}{n}}\right)e^{2 \pi i k x}= \sum_{k \in n \mathbb{Z}}e^{-\pi \alpha k^2}\left( \sum_{j=1}^{n}e^{2 \pi i k \varepsilon_j}\right)e^{2 \pi i k x},
h(r) = 2\int_{\varphi_{0}(r)}^{1-\varphi_{0}(r)}\log{|r -e(\varphi)|}\, d\varphi = \int_{\varphi_{0}(r)}^{1-\varphi_{0}(r)}\log(1+r^{2}-2r\cos(2\pi\varphi))\, d\varphi
\int_Y \langle d_b \phi, d_b\psi\rangle = \int_Y (d_b^*d_b\phi)\psi = \int_Y (-\Delta_Y\phi + (J\partial_r)^2\phi)\psi, \int_Y \langle d^c_b\phi, d_b\psi\rangle = \int_Y (d_b^*d^c_b\phi)\psi =(2-2n)\int_Y (J\partial_r\phi)\psi.
\frac{X_{\ell-1}(\gamma_{k^* + K})}{X_{\ell}(\gamma_{k^* + K})}\leq \frac{(1 + \delta) \delta s}{3 (1 - \delta) \mu}e^{-14s a_N/13}= \frac{(1 + \delta) \delta}{3(1 - \delta)}\bigg( \frac{s}{\mu}\bigg)^{-1/13},
\sum_{\lambda\in M_j}a(j)_{\lambda_1-\lambda_2,\lambda_2-\lambda_3,\dots, \lambda_{m}}p^{\sum_i \lambda_i \mu_i}= \sum_{d_1,\dots,d_{m}\geq 0}a(j)_{d_1,\dots,d_m}(p^{\mu_1})^{d_1}(p^{\mu_1+\mu_2})^{d_2}\cdots (p^{\mu_1+\dots+\mu_m})^{d_m}&= H_{m,p,\nu}(p^{\mu_1}, p^{\mu_1+\mu_2}, \dots, p^{\mu_1+\dots+\mu_m}).
P_{k \ell}({\bf u},{\bf v},{\bf w}) \,\,\, := \,\,\,P \bigl( \,({\bf c}_k -{\bf c}_\ell) \cdot{\bf u},\,({\bf c}_k -{\bf c}_\ell) \cdot{\bf v},\,({\bf c}_k -{\bf c}_\ell) \cdot{\bf w}\bigr),
\begin{array}{ll}D_{\mu}^i = \nabla_{\mu}+ i \frac{q}{2\sqrt{\beta}}Q_i \tilde H_{\mu}, D_{\mu}^I = \nabla_{\mu}+ i \frac{e}{2}E_I \tilde A_{\mu}-i \frac{\alpha e}{2 \sqrt{\beta}}E_I \tilde H_{\mu},\end{array}
\sum_{\tau' \in \Sigma'}e^{\frac{\tau'}{M}x}=\sum_{A_1=0}^{N-1}e^{\frac{A_1}{N}x}+\sum_{A_2=0}^{M-N-1}e^{\frac{A_2}{M-N}x}&=\frac{e^x-1}{e^{\frac{x}{N}}-1}+\frac{e^x-1}{e^{\frac{x}{M-N}}-1}.
(\begin{matrix}H(x,\lambda)&-J(x)^{T}\\diag(\lambda)J(x)&diag(c(x))\end{matrix})(\begin{matrix}p_{x}\\p_{\lambda}\end{matrix})=(\begin{matrix}-\nabla f(x)+J(x)^{T}\lambda\\ \mu1-diag(c(x))\lambda\end{matrix}),
0 =\partial_n^h(\beta)=\sum_in_i\partial_n^h(q_i)=\sum_in_i\left\{\sum_{p\in I_{n-1}(h)\cup N_{n-1}(h)}\left(\sum_{\gamma\in\mathcal{M}_G(q_i,p)}n_h(\gamma)\right)p\right\}&=\sum_{p\in I_{n-1}(h)\cup N_{n-1}(h)}\left\{\sum_in_i\left(\sum_{\gamma\in\mathcal{M}_G(q_i,p)}n_h(\gamma)\right)\right\}p.
\sum_{y<p\le x^{1/3}}\frac{1}{p\log p}\omega\left(\frac{\log x}{\log p}-1\right) =\frac{1}{\log x}\int_{3^-}^{u}v\omega(v-1)\,dG(v) &=\frac{1}{\log x}\left(\int_{3}^{u}\omega(v-1)\,dv+\int_{3^-}^{u}v\omega(v-1)\,d\left(G(v)-\log\frac{v}{2}\right)\right).
a(S) =C_{S}\,\det(S)^{k/2}\prod\limits_{p|L\;\;p|(N_1/L)}B_p^{S'}\Big(h\Big(v_p(L),v_p\Big(\frac{N_1}{L}\Big)\Big)\Big) =C_{S}\,\det(S)^{k/2}\bigg(\prod\limits_{p|L}B_p^{S'}\Big(h\Big(v_p(L),v_p\Big(\frac{N_1}{L}\Big)\Big)\Big)\bigg)\bigg(\prod\limits_{\substack{p\nmid L p|N_1}}B_p^{S'}\big(h(0,v_p(N_1))\big)\bigg).
\|(\mathbf{x}_{k+1}, \mathbf{y}_{k+1})-\mathcal{P}(\mathbf{x}_{k}, \mathbf{y}_{k})\| \leq \varepsilon_k \min\{1, \|(\mathbf{x}_{k+1}, \mathbf{y}_{k+1})-(\mathbf{x}_{k}, \mathbf{y}_{k}) \|\}, \hbox{where}\sum_{k=0}^{+\infty}\varepsilon_k < \infty.
\begin{array}{l}\begin{array}{lcl}Z(\vartheta ) = l\sinh \vartheta +\chi (\vartheta -\vartheta_{1})+\chi (\vartheta -\vartheta_{2})-\chi (\vartheta -\rho -i\sigma )-\chi (\vartheta -\rho +i\sigma ), \end{array}Z(\vartheta_{1,2})=2\pi I_{1,2}\; , Z(\rho \pm i\sigma )=2\pi I^{\pm}_{c}\; .\end{array}
D^m \Theta^{}(\mathcal{M}) - \Theta^{[m+1]}(D^m \mathcal{M}) = \int_0^1 D \mathcal{G}(\mathcal{N}(s), D^m \mathcal{M}) s \begin{pmatrix}\Lambda - \mathcal{M}\vdots D^{m-1}\left( \Lambda - \mathcal{M}\right) 0 \end{pmatrix},
P_{}\left(\{\mathbf{w}^{}_{i}\},\{\mathbf{v}^{}_{i}\}\right) = \left( \sum^{N}_{n=1}\mathbf{1}_n\left(\{\mathbf{w}^{}_{i,n}\},\{\mathbf{v}^{}_{i,n}\}\right)P_{c,n}\right. \left.+\sum^{K}_{i=1}\|\mathbf{w}^{}_{i}\|^2\right)+\lambda\left(\sum^{K}_{i=1}p^{}_i\right)
Q(x,u,y,v;\alpha,\beta)= (\alpha^2-1)\beta(xu+yv)-(\beta^2-1)\alpha(xy+uv)+(\alpha^2-\beta^2)(xv+uy)\phantom{=0\,.}&+(\alpha^2-1)(\beta^2-1)(\alpha^2-\beta^2)(4\alpha\beta)^{-1}\,.
u_{t}+\epsilon(uu_{x}+wu_z) =-p_{x}, \delta^{2}(w_{t}+\epsilon(uw_{x}+ww_z)) =-p_z, u_{x}+w_z =0, w =\eta_{t}+\epsilon u\eta_{x}, \p=\eta \rho g z=1+\epsilon \eta, w =0 z=0.
\mathcal{H}(w,u) = \prod_P (1-w^{d(P)})^{10}\Big( 1+ \frac{(|P|-1)w^{d(P)}(10-5w^{d(P)}+4w^{2d(P)}-w^{3d(P)}}{|P|(1-u^{d(P)})(1-w^{d(P)})^4}\Big).
\varphi ( x ) = \epsilon^{1 / 2}\Big \lbrack \left( \frac{C_{1}}{2 C_{2}}\right)^{1 / 2}\tilde{\varphi}_{0}+ \epsilon ( - \frac{D_{0}}{D_{1}}\tilde{\varphi}_{0}+ \varphi_{1 p}) \Big \rbrack ,
\begin{array}{llllll}\displaystyle D = 3 : \quad Sp(4;R) \to OSp(N;1|R)\cr\displaystyle D = 4 : \quad SU(2,2) \to SU(2,2|N)\cr\displaystyle D = 6 : \quad U_{\alpha}(4;H) \to UU_{\alpha}(N;4|R)\end{array}
\delta{\cal L}_1=-\delta\left\{{\lambda_{--}\over{1-\lambda_{++}\lambda_{--}}}A_+^2 +{\lambda_{++}\over{1-\lambda_{++}\lambda_{--}}}A_-^2 +{{1+\lambda_{++}\lambda_{--}}\over{1-\lambda_{++}\lambda_{--}}}A_+A_-\right\}
P_{Y_3 | Y^2,{\bf S}}(1 | 00,{\bf s}) = \frac{B_3^{00}}{M - B_1 - B_2^0}, \P_{Y_3 | Y^2,{\bf S}}(1 | 01,{\bf s}) = \frac{B_3^{01}}{B_2^0}, & P_{Y_3 | Y^2,{\bf S}}(1 | 10,{\bf s}) = \frac{B_3^{10}}{B_1 - B_2^1},\P_{Y_3 | Y^2,{\bf S}}(1 | 11,{\bf s}) = \frac{B_3^{11}}{B_2^1}.
F_g(u,{\cal P},T) = \begin{cases}E(u,{\cal P},T) + \min_{v \in u + \nabla T \mathcal{A}(\mathcal{P})}\lambda\Vert v- g \Vert^2_{L^2(\Omega)}(u,{\cal P},T) \in{\cal C}_g, + \infty , \end{cases}
\boldsymbol{L}_{JK}u(x)=\left( A_{JK}(u;K+p^{N}\mathbb{Z}_{p})\right) \Omega\left( p^{N}\left\vert x-J\right\vert_{p}\right) -A_{JK}u(x)\Omega\left( p^{N}\left\vert x-J\right\vert_{p}\right).
E_0(x)< \frac{11}{480}\frac{1}{4(x-\frac 12)^4}-\frac{29}{336}\frac{1}{5 x^5}+\frac{9031}{44800}\frac{1}{6 (x-\frac 12)^6}= \frac{11}{1920 (x-5)^4}-\frac{\Psi_2(9;x)(x-12)+12561000435989768}{67200 (-1 + x)^4 x^5 (-1 + 2 x)^6}< \frac{11}{1920 (x-5)^4}.
[B]_{j+1,j}=[A]_{j+1,j}d_{j}(x,y)d_{j+1}(y,x) = b_j [B]_{j,j+1}=[A]_{j,j+1}d_{j}(y,x)d_{j+1}(x,y) = c_j [B]_{j,j}= [A]_{j,j}d_{j}(x,y)d_j(x,y)=\frac{a_j}{(xy)^{\lfloor j/2\rfloor}}.
\|(Sf-Sg)(t_j)-(Sf-Sg)(t_{j+1})\| =\|\alpha_k (f-g)(P_k^{-1}(t_j))-\alpha_k (f-g)(P_k^{-1}(t_{j+1}))\| &=|\alpha_k|\|(f-g)(P_k^{-1}(t_j))- (f-g)(P_k^{-1}(t_{j+1}))\| &\leq \alpha_{\max}\|(f-g)(P_k^{-1}(t_j))- (f-g)(P_k^{-1}(t_{j+1}))\|.
S (d=11) =-T \int d^{p+1}\hat{x}\hspace{4pt}|\hat{k}|^{\beta}|\hat{k}^\prime|^{\gamma}\sqrt{\mid \partial_i \hat{X}^{\hat{\mu}}\partial_j \hat{X}^{\hat{\nu}}\hat{\Pi}_{\hat{\mu}\hat{\nu}}\mid}\, .
\frac{1}{2}y_{s B}\frac{\partial}{\partial y_{s B}}\psi_{0}= - \frac{1}{2}y_{s B}y_{s B}\psi_{0}= - \frac{1}{2}\sum_{s B}( y_{s B}^{2}- \frac{1}{2}) \psi_{0}- \frac{1}{4}\cdot 2 ( d - 1 ) \psi_{0}\; ,
S = - \frac{1}{4}\, \int d^{4}x \, \left[ ( 1 - \frac{1}{2}\, \theta^{\alpha \beta}F_{\alpha \beta}) \, F_{\mu \nu}F^{\mu \nu}+ 2 \mathrm{T r}\, ( \theta F^{3}) - \partial_{\beta}\left( \theta^{\alpha \beta}A_{\alpha}F_{\mu \nu}F^{\mu \nu}\right) \right]
(Z^\Lambda)''-\left({(ZNq) \over (ZNZ)}+q^0 \right){((Z^\Lambda )')^2 \over Z^\Lambda q^0 - q^\Lambda}+{1 \over 2}\left(\ln{(ZNZ) (\bar Z Nq) (\bar Z NZ)\over (\bar Z N \bar Z) (ZNq)}\right)' (Z^\Lambda )'=0\ .
( \Delta S )_{r e g .}\, = \,{\frac{i}{4 \pi}}\int d^{2}x ( c - \tilde{c}) \Big ( ( 1 - a ) \partial_{\mu}( A^{\mu}- \tilde{A}^{\mu}) - \epsilon^{\mu \nu}\partial_{\mu}( A_{\nu}- \tilde{A}_{\nu}) \Big )
N_1(t) =\int_\Gamma \Big(\varphi_1^{\eta_5}(x,t)+\varphi_2^2(x,t)\Big)d\sigma, N_2(t) = N_1(t)+\int_\Gamma \Big(\varphi_3^{\eta_5}(x,t)+\varphi_4^2(x,t)\Big)d\sigma + \int_U \Big(f_1^{\eta_6}(x,t)+f_2^4(x,t)\Big)dx
\approx\frac{1}{2}[\begin{matrix}1\\ \delta x\\ \delta u\end{matrix}]^{T}[\begin{matrix}0&Q_{x}^{T}&Q_{u}^{T}\\Q_{x}&Q_{xx}&Q_{xu}\\Q_{u}&Q_{ux}&Q_{uu}\end{matrix}][\begin{matrix}1\\ \delta x\\ \delta u\end{matrix}]
\frac{d}{d \tau}\, = \, \sqrt{\dot{x}^{2}}\, \frac{d}{d s \,}{;}\qquad \frac{d^{2}x_{\mu}}{d s^{2}\,}\, = \, \frac{\dot{x}^{2}\ddot{x}_{\mu}\, - \, ( \dot{x}\ddot{x}) \, \dot{x}_{\mu}}{( \dot{x}^{2})^{2}}\,{;}\qquad k_{1}\, \frac{\partial k_{1}}{\partial \ddot{x}_{\mu}}\, = \, - \, \frac{1}{\dot{x}^{2}}\, \frac{d^{2}x_{\mu}}{d s^{2}\,}{;}
\frac{d}{dt}(\begin{matrix}x\\y\\v_{x}\\v_{y}\end{matrix})=(\begin{matrix}v_{x}\\v_{y}\\ -\mu v_{x}\sqrt{v_{x}^{2}+v_{y}^{2}}\\ -g-\mu v_{y}\sqrt{v_{x}^{2}+v_{y}^{2}}\end{matrix})
I_{\varepsilon}^{1}\left( a\right) =\sum\limits_{\nu_{1},\nu_{2}\in{\cal N}_{\varepsilon}^{+}}\frac{1}{\rho \left( \nu_{1}\right) \rho \left( \nu_{2}\right)}\left[{\LARGE i}\right] +\sum\limits_{\nu_{1},\nu_{2}\in{\cal N}_{\varepsilon}^{+}}\frac{1}{\rho \left( \nu_{1}\right) \rho \left(\nu_{2}\right)}\left[{\Large ii}\right] ,
\| f (t,\eta)\|^p_{L^p(\Omega; \mathbb{R}^d)}+\| g(t,\eta)\|^p_{L^p(\Omega; \mathbb{R}^{d\times n})}&+ \| h(t,\eta)\|^p_{L^p(\Omega,L^p(\nu) )}+ \| h(t,\eta)\|^p_{L^p(\Omega,L^2(\nu))}& \leq K(1+\| \eta \|^p_{S^p(\Omega;\mathcal{D})}).
\hat{G}_{4}( p_{2}, - p_{3}, - p_{4}, p_{1}) = i \hat{S}( p_{2}) \otimes i \hat{S}( - p_{3}) \hat{\Gamma}_{4}( p_{2}, - p_{3}, - p_{4}, p_{1}) i \hat{S}^{T}( - p_{4}) \otimes i \hat{S}^{T}( p_{1}) ,
\begin{array}{c}{\Bigl ( a^{+}( P^{0}- P_{2}) + a^{-}\left( i P_{1}- P_{3}\right) \Bigr ) | \psi \rangle = 0 ,}{\Bigl ( a^{-}( P^{0}+ P_{2}) - a^{+}\left( i P_{1}+ P_{3}\right) \Bigr ) | \psi \rangle = 0 .}\end{array}
\mathcal{M}^{H_{q^d,P^\alpha}}= \left( \left( \mathcal{M}^{H_{q^d,P^\alpha}^\alpha}\right)^{H_{q^d,P^\alpha}^{\alpha-1}/H_{q^d,P^\alpha}^\alpha}\cdots \right)^{H_{q^d,P^\alpha}^1/H_{q^d,P^\alpha}^2}.
| T_{I I}( \theta , E_{0}/ e , \omega , t ) |^{2}= \sum_{k , p = 0}^{\infty}t_{2 k}^{p}( \theta ) \frac{E_{0}^{2 k + p}}{\omega^{2 k}}\cos^{p}( \omega t ) \sin^{2 k}( \omega t ) .
\Psi(x,f,g) :=g(x)\left(a(x)+2b(x)f(x)+\int_{(0,+\infty)}\theta\left(1-e^{-f(x)\theta}\right)\Pi^{L}(x,{\rm d}\theta)\right) &-\pi(x,g)\left(c(x)+\int_{(0,+\infty)}\theta e^{-\theta\pi(x,f)}\Pi^{NL}(x,{\rm d}\theta)\right).
\tau_k(x)-\tau_k(0)=\frac{3}{(1+x^2)^2}(2{{d}_{k}}(x-x^3)-{\omega_{k}}x^2), &\tau'_k(x) = \frac{6}{(1+x^2)^3}({{d}_{k}}(1-6x^2+x^4)-{\omega_{k}}(x-x^3)), &\tau''_k(x) = \frac{6}{(1+x^2)^4}[2{{d}_{k}}(-9x+14x^3-x^5)-{\omega_{k}}(1-8x^2+3x^4)].
d_1 = -{q \over \pi}, d_2 = -{p^2 + q^2 \over 2 \pi^2}, d_{n+2}={1 \over \pi}{2n + 1 \over n + 2}q d_{n+1}+{1 \over \pi^2}{n - 1 \over n + 2}\Big ( p^2 -{n^2 \over 4}\Big ) d_n \: \: (n \ge 1).
\beta ( k , p ) = \frac{- i}{2 \pi}\int_{- \infty}^{\infty}d x \frac{e^{i ( p_{1}+ q_{1}) x_{1}}}{\sqrt{4 \omega_{p}\omega_{k}}}F ( k , p , x ) \delta ( p_{2}+ q_{2}) \delta ( p_{3}+ q_{3})
\tilde{Q}^\alpha_*(\theta(k_1))= \begin{pmatrix}\tilde{Q}^{\alpha,\{k_1,k_2\}}_{*,0}(\theta(k_1)) \tilde{Q}^{\alpha,\{k_1,k_2\}}_{*,1}(\theta(k_1)) \cr\tilde{Q}^{\alpha,\{k_1,k_2\}}_{*,-1}(\theta(k_1)) \tilde{Q}^{\alpha,\{k_1,k_2\}}_{*,0}(\theta(k_1)) \tilde{Q}^{\alpha,\{k_1,k_2\}}_{*,0}(\theta(k_1)) \cr \ddots \ddots \ddots \end{pmatrix},