harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\Phi^n (f) (a_1, \ldots, a_n ) = f \big( P(a_1), \ldots, P(a_n) \big) - \sum_{k=0}^{n-1}\lambda^{n-k-1}\sum_{1 \leq i_1 < \cdots < i_k \leq n}~ Q \circ f \big( a_1, \ldots, P (a_{i_1}), \ldots, P(a_{i_k}), \ldots, a_n \big).
\Phi_s := \sum_{k=1}^j \Phi^{(k)}(s-s_0)^{k-1},\Phi^{(k)}:= \rho^{-s_0+m}\sum_{\ell=0}^{k-1}\tfrac{(-\log \rho)^\ell}{\ell !}\varphi^{(k-\ell)}.
\int_{\mathcal{T}^k}\prod_{j=1}^{k}\gamma_j^{m_j}\,d \nu_\alpha (\underline{\gamma})= \mathbf{E}\left[ \mathbb{X}_\alpha(n_1)^{m_1}\cdots \mathbb{X}_\alpha(n_k)^{m_k}\right]= g_\alpha(\underline{m})
\begin{array}{l c l}{\mathrm{N S 5}}&{:}&{0 1 2 3 4 5}\\{\mathrm{N S^{\prime}5}}&{:}&{0 1 2 3 8 9}\\{\mathrm{D 6}}&{:}&{0 1 2 3 7 8 9}\\{\mathrm{D 4}}&{:}&{0 1 2 3 6}\\{\mathrm{O 4}}&{:}&{0 1 2 3 6}\\ \end{array}
\begin{array}{c}g_1=z_0 g(z_0, z_1, z_2, z_3)+(z_0-z_5)\alpha(z_0, z_1, z_2, z_3) g_2=z_0 g(z_5, z_4, z_2, z_3)+(z_1-z_4)\alpha(z_0, z_1, z_2, z_3),\end{array}
\frac{f_s^{(j)}(0)}{j!}\frac{g_K^{(j)}(z)}{v^j}= \frac{f_s^{(j)}(0)}{j!}\frac{2^j(K-1)^j}{v^j}\frac{g_K^{(j)}(z)}{2^j(K-1)^j}=\frac{f_s^{(j)}(0)}{j!}\frac{2^j(K-1)^j}{v^j}\left( e^{Kz}\sum_{\ell=0}^j \frac{1}{2^j}\binom{j}{\ell}\frac{y^{(\ell)}(z)}{(K-1)^{\ell}}- \frac{y^{(j)}(z)}{(K-1)^j}\right).
\int_{\mathcal{Y}^d}\big(q\widetilde{m}^{q-1}| (\nabla_y\widetilde{m})^TK |^2+\widetilde{m}^{q+1}|\nabla_y^2\widetilde{w}|^2+ (q+1)\widetilde{m}^q (\nabla_y \widetilde{m})^T\nabla_y^2\widetilde{w}(\Lambda+ \nabla_y \widetilde{w})\big )dy= 0.
{\cal S}\, = \,{\frac{1}{2}}\int \, d x \, d t \, \left[ \frac{\dot{\phi}^{a}\dot{\phi}^{a}\, + \, \phi^{a}\partial_{i}\partial_{i}\phi^{a}}{\phi^{a}\, \phi^{a}}\, - \, \frac{( \, \dot{\phi}^{a}\dot{\phi}^{a}\, )^{2}}{( \, \phi^{a}\, \phi^{a}\, )^{2}}\right] \; \; ,
\psi=\left(\begin{array}{c}\psi_{1}\psi_{2}\end{array}\right)\ ,\quad\overline\psi\equiv\psi^{\dag}\gamma_{3}\equiv\left(\begin{array}{cc}\overline\psi_{1}-\overline\psi_{2}\end{array}\right)\equiv\left(\begin{array}{cc}\psi_{1}^{\dag}\sigma_{3}-\psi_{2}^{\dag}\sigma_{3}\end{array}\right)\ ,
H^{I_A (\mu_1, \mu_2) J_B (\nu_2, \nu_1)}={\rm Tr}\eta'^{i_{A(\mu_1 + 1)}}\cdots \eta'^{i_{A(\mu_2 - 1)}}\eta'^{j_{B(\nu_2 + 1)}}\cdots \eta'^{j_{B(\nu_1 - 1)}}
|A| \leq\sum_{i=0}^{\min\{k,N-1\}}\sum_{z\in B_i(x,y)}|A(z,i)| &\leq \sum_{i=0}^{\min\{k,N-1\}}|B_i(x,y)| \tbinom{N-1+i}{N-1}&\leq \sum_{i=0}^{N-1}N^i \tbinom{N-1+i}{N-1},
\begin{aligned}(i)~ \sup_{0 \leq t < \infty}\max_{1\leq i,j\leq N}\|x_i(t)-x_j(t)\|<\infty, &(ii)~ \lim_{t \to \infty}\max_{1\leq i,j\leq N}\|v_j(t)-v_i(t)\|=0,\end{aligned}
\int_{\Omega_\ell}A\nabla v_\ell^\varepsilon.\nabla v_\ell^\varepsilon = \int_{\Omega_\ell}a_{11}(\partial_{x_1}v_\ell^\varepsilon)^2 + 2(A_{12}.\nabla_{X_2}v_\ell^\varepsilon) \partial_{x_1}v_\ell^\varepsilon + (A_{22}\nabla_{X_2}v_\ell^\varepsilon).\nabla_{X_2}v_\ell^\varepsilon &= I_1(\varepsilon) + I_2(\varepsilon) + I_3(\varepsilon)\,.
\begin{array}{rcl}I_2(\rho^2,m) = \displaystyle \mu^{4-n}\int\frac{d^nk}{(2\pi)^n}[(k+p)^2+m^2]^{-2}\cos(k\cdot \rho) &= \displaystyle I_1(\rho^2,m)\cos(p\cdot \rho).\end{array}
H_+(s) =( (C_{\Sigma}^+ \tilde{\mu}(\tilde{v}_\Sigma -I) + g))(s)=( (C_{\Sigma}^- \tilde{\mu}(\tilde{v}_\Sigma -I) + g))(s) + \tilde{\mu}(s)(\tilde{v}_\Sigma(s) - I)) + g(s) \hphantom{H_+(s)}{}= H_-(s) + m_+ - m_-.
\rho h_{ij}'' + (1-n)h_{ij}' - h^{kl}h_{kl}'h_{ij}- \rho h^{kl}h_{ik}'h_{jl}' +\frac 12 \rho h^{kl}h_{kl}' h_{ij}'- 2\rho Ric_{ij}[h_\rho] =F_{ij},
\int_{\mathbb{R}_{\geqslant0}}f(y)\rho_{t}(\mathrm{d}y)=\lambda_{t}^{-1}\int_{0}^{t}\int_{\lbrace z>1\rbrace}\left(\int_{\mathbb{R}_{\geqslant0}}f(y)m_{\alpha(z,s),\beta(z,s)}(\mathrm{d}y)\right)\nu(\mathrm{d}z)\mathrm{d}s.
\big(x_1^{a_1}\cdots x_n^{a_n}\big)^{\perp}\cap \left(( y_0^{b_0-1}y_1^{b_1}\cdots y_m^{b_m})^\perp +(Y_0)\right) + \big( X_1^{a_1}\cdots X_n^{a_n}\big) = \left((x_1^{a_1}\cdots x_n^{a_n})^{\perp}+ (X_1^{a_1}\cdots X_n^{a_n})\right) \cap \left(( y_0^{b_0-1}y_1^{b_1}\cdots y_m^{b_m})^\perp +(Y_0)\right).
\int_{-\infty}^{\infty}|G_{x}(\tau,\omega)|^{2}d\omega=\int_{-\infty}^{\infty}|x(t)|^{2}e^{-2\pi(t-\tau)^{2}}dt\approx\int_{\tau-1.9143}^{\tau+1.9143}|x(t)|^{2}e^{-2\pi(t-\tau)^{2}}dt
p^{\sum_{i=1}^s \left[ (n+m)a_i - a_i^2\right]}\cdot \frac{\prod_{i=0}^{a_s-1}(1-p^{-n+j}) (1-p^{-m+j})}{\prod_{i=0}^{s-1}\prod_{j=1}^{a_{i+1}-a_i}(1-p^{-j})}=: N .
3 D_{i \alpha}( W ( \theta ) \tilde{\gamma}^{A})^{[ \beta \gamma ]}= \delta_{\alpha}^{\beta}D_{i \delta}( W ( \theta ) \tilde{\gamma}^{A})^{[ \delta \gamma ]}- \delta_{\alpha}^{\gamma}D_{i \delta}( W ( \theta ) \tilde{\gamma}^{A})^{[ \delta \beta ]}
\delta w^2_h(x,y) = \frac{1}{|h|^\beta}[(1-\eta(x+y))u(x+y+he)-(1-\eta(x+y))u(x+y) +(1-\eta(x-y))u(x-y+he)-(1-\eta(x-y))u(x-y) +2(1-\eta(x))u(x)-2(1-\eta(x))u(x+he)] = \frac{1}{|h|^\beta}[(1-\eta(x+y))u(x+y+he)-(1-\eta(x+y))u(x+y) +(1-\eta(x-y))u(x-y+he)-(1-\eta(x-y))u(x-y)].
\sigma=\frac{2}{J}V\cdot[\frac{1}{2\lambda_{1}}\frac{\partial W}{\partial\lambda_{1}}n_{1}\otimes n_{1}+\frac{1}{2\lambda_{2}}\frac{\partial W}{\partial\lambda_{2}}n_{2}\otimes n_{2}+\frac{1}{2\lambda_{3}}\frac{\partial W}{\partial\lambda_{3}}n_{3}\otimes n_{3}]\cdot V
\Lambda(f \otimes \mathrm{Ad}(g),k) = 2^{k+1-\nu(M)}C(N,M,\chi) \frac{\langle f,f\rangle}{\langle h,h\rangle}\frac{|\langle (\mathrm{id}\otimes U_M) F_{\chi |\mathcal H \times \mathcal H}, g \times g\rangle|^2}{\langle g,g\rangle^2},
\frac{1}{\sqrt{1-x^{4}}}arcsl^{*}(x)-\frac{1-x^{2}}{\sqrt{1-x^{4}}}arcsl(x)=\int_{0}^{1}\frac{x^{3}(y^{2}+1)}{\sqrt{(1-x^{4})(1-x^{4}y^{4})}}dy
\textbf{BR}(t)=-t\cdot{}_4F_3\left[\left\{\frac{1}{5},\frac{2}{5},\frac{3}{5},\frac{4}{5}\right\};\left\{\frac{1}{2},\frac{3}{4},\frac{5}{4}\right\};-\frac{3125 t^4}{256}\right],
H_{c}^{r}\, = \,{\frac{1}{2}}\int \left[ \left( \pi_{1}\phi_{2}- \pi_{2}\phi_{1}\right)^{2}\, - \,{\frac{1}{\phi_{1}^{2}+ \phi_{2}^{2}}}\left( \phi_{1}\partial_{i}\partial_{i}\phi_{1}+ \phi_{2}\partial_{i}\partial_{i}\phi_{2}\right) \right] d x \, .
\left\| \sum_{h=1}^{H_L}p_L (h \cdot) \right\|_2^2 \leq H_L\sum_{j_1,j_2=1}^{G_{L}}\frac{|u_{j_1}u_{j_2}|}{2}\frac{\gcd(j_1,j_2)}{\sqrt{j_1 j_2}}.
f(x,y)=\frac{1}{2\pi\sigma_{X}\sigma_{Y}\sqrt{1-\rho^{2}}}exp(-\frac{1}{2[1-\rho^{2}]}[(\frac{x-\mu_{X}}{\sigma_{X}})^{2}-2\rho(\frac{x-\mu_{X}}{\sigma_{X}})(\frac{y-\mu_{Y}}{\sigma_{Y}})+(\frac{y-\mu_{Y}}{\sigma_{Y}})^{2}])
\widetilde{\Pi}_{\nu_{G},k\nu_{T}}\left(x,y\right)=\frac{d_{\nu_{G}}}{(2\pi)^{d_{T}}}\int_{G}\int_{T}\chi_{\nu_{G}}(g^{-1})t^{-k\nu_{T}}\Pi\left(\tilde{\mu}^{G}_{g^{-1}}\circ\tilde{\mu}^{T}_{t^{-1}}\left(x\right),y\right)dtdg,
\lambda ={\frac{i q_{1}A_{q}^{1 / 2}- q_{2}}{i q_{2}A_{q}^{1 / 2}+ q_{1}}}={\frac{q_{1}q_{2}( A_{q}- 1 ) + i ( q_{1}^{2}+ q_{2}^{2}) A_{q}^{1 / 2}}{q_{1}^{2}+ q_{2}^{2}A_{q}}}\ .
^g(x)= \frac{1}{n-1}(k-1) \cdot^g(x) - 2(n-2) \cdot \sum_{j=1}^{k-1}\underbrace{K^g(X_j,X'_j)}_{=0}&+ \frac{1}{2(n-1)}\cdot (n-2(k-1)) \cdot^g(x) -(n-2)\sum_{j=1}^l \epsilon_i \underbrace{K^g(E_i,E_i)}_{=0}= \frac{n}{2(n-1)}\cdot^g(x),
F{{1,m \!\!\!\!\!}\atop \bigg{|}}_{\left( \begin{matrix}a b c d \end{matrix}\right)}(\tau, z_1, z_2) = (c \tau + d)^{-1}\, e^{- \frac{2\pi i mc z_1 z_2}{c \tau + d}}\F \left( \frac{a \tau + b}{c \tau + d}, \ \frac{z_1}{c \tau + d}, \ \frac{z_2}{c \tau + d}\right).
{\tilde{\delta}}\biggl[{\dot\varphi}^{a}-\omega^{ab}{\partial H\over\partial\varphi^{b}}\biggr]=\int~{\cal D}\lambda_{a}~\exp~i\int\lambda_{a}\biggl[{\dot \varphi}^{a}-\omega^{ab}{\partial H\over\partial\varphi^{b}}\biggr]dt
\Psi_q(d-\varepsilon)=\frac{(d-\varepsilon)^q+(2(d-\varepsilon)-1)^{2+\sin q}}{2}\;\;\textrm{and}\;\;\Psi_q(d+\varepsilon)=\frac{(d+\varepsilon)^q+(2(d+\varepsilon)-1)^{2+\sin q}}{2}.
P_ka(J_k)=\left(\begin{array}{cc}1 0\cr Y_{k}P_{k-1}\end{array}\right)\left(\begin{array}{cc}0 0\cr \xi a(J_{k-1})\end{array}\right)=\left(\begin{array}{cc}0 0\cr P_{k-1}\xi P_{k-1}a(J_{k-1})\end{array}\right).
lip_0(M) :=\left\{f\in Lip_0(M): \lim\limits_{\varepsilon\rightarrow 0}\sup\limits_{0<d(x,y)<\varepsilon}\frac{|f(x)-f(y)|}{d(x,y)}=0\right\}, S_0(M) :=\left\{f\in lip_0(M) : \lim\limits_{r\rightarrow \infty}\sup_{\stackrel{x y\notin B(0,r)}{x\neq y}}\frac{| f(x)-f(y)|}{d(x,y)}=0\right\}.
f(x) = \frac{1}{2K}\sum_{j = L+1}^M j^{-\theta}= \frac{1}{2K}\sum_{x - K x^\theta < j \leq x + K x^\theta}j^{-\theta}+ O(x^{-\theta}+ x^{\theta -1}) &= \frac{1}{2K (1-\theta)}\big((x + K x^\theta)^{1-\theta}- (x - K x^\theta)^{1-\theta}\big) + O(x^{-\theta}+ x^{\theta -1}) &= 1+ O(x^{-\theta}+x^{\theta-1}),
1 =\sigma(D_{w\lambda,\lambda}D_{w\lambda,\lambda}^{-1}) =q^{-(w\lambda-\lambda,w\lambda-\lambda)}\sigma(D_{w\lambda,\lambda}^{-1})\sigma(D_{w\lambda,\lambda}) =q^{2(\lambda,w\lambda-\lambda)}\sigma(D_{w\lambda,\lambda}^{-1})D_{w\lambda,\lambda}
q_1 := (1-\partial_x^2)^{-1}(\phi \circ \eta^{-1}), q_2 := -(1-\partial_x^2)^{-1}[(\phi \circ \eta^{-1})(\rho \circ \eta^{-1})], q_3 := -\partial_x(1-\partial_x^2)^{-1}[(\phi\circ\eta^{-1})((\partial_x\rho) \circ \eta^{-1})\partial_x \eta^{-1}],
\frac{B_{\phi(p^{l})-1}(\frac{1}{e})}{\phi(p^{l})-1}\equiv e\sum_{j=0}^{p^{l}-1}(\lfloor\frac{1+je}{p^{l}}\rfloor+\frac{1-e}{2})(1+je)^{\phi(p^{l})-2}\pmod{p^{l}}&\equiv e\sum_{\substack{j=0 (p,1+je)=1}}^{p^{l}-1}(\lfloor\frac{1+je}{p^{l}}\rfloor+\frac{1-e}{2})(1+je)^{-2}\pmod{p^{l}}.
S_{r}(f \circ_{}g)= \sum_{\substack{0\leq s,r s+t\leq r}}\frac{r!}{s!t!(r-s-t)!}\alpha^{s/p}\beta^{t/p}\gamma^{(r-s-t)/p}S_{r-t}(f) \circ_{}S_{r-s}(g) \,,
F_{\Delta_{s}}^{(s)}(\{\Delta_{i}\}|\{z_{i}\})=\int_{i\mathbb{R}}dP_{t}F_{\Delta_{s},\Delta_{t}}[\begin{matrix}\Delta_{2}&\Delta_{3}\\ \Delta_{1}&\Delta_{4}\end{matrix}]F_{\Delta_{t}}^{(t)}(\{\Delta_{i}\}|\{z_{i}\}).
R_p(J_n^2, 4n) = \sum_{k = 0}^\infty \frac{J_n^2(p^k)}{p^{4nk}}= 1 + \sum_{k = 1}^\infty \frac{(p^{nk}- p^{n(k-1)})^2}{p^{4nk}}&= 1 + (1 - 2p^{-n}+ p^{-2n})\sum_{k = 1}^\infty \frac{1}{p^{2nk}}&= 1 + \frac{p^{2n}- 2p^{n}+ 1}{p^{4n}- p^{2n}}&= \frac{p^{4n}- 2p^n + 1}{p^{4n}- p^{2n}}
S^*_{\lambda}(\zeta_\mu)=\zeta_{\mu+\check{\phi}(\lambda)}=\mathrm{X}^{\mu+\check{\phi}(\lambda)}q^{\check{\varphi}(\mu+\check{\phi}(\lambda))}\theta=\mathrm{X}^\mu(\lambda)q^{A(\lambda)}\mathrm{X}^{\check{\phi}(\lambda)}\zeta_\mu.
\{Q_{A}, Q_{B}\}= Z_{A B}= \left( C \Gamma^{\widetilde{\mu}_{1}\widetilde{\mu}_{2}}\right)_{A B}Z_{[ \widetilde{\mu}_{1}\widetilde{\mu}_{2}]}+ \left( C \Gamma^{\widetilde{\mu}_{1}\ldots \widetilde{\mu}_{6}}\right)_{A B}Z_{[ \widetilde{\mu}_{1}\ldots \widetilde{\mu}_{6}]},
g_4(x) = a\left[\mu x^2 + \delta(1-\mu)x + (c_A - \mu)\right] + \left[2\mu x + \delta(1 - \mu)\right] &= a\mu x^2 +\left[\delta a (1 - \mu) + 2 \mu\right] x+ a (c_A - \mu) +\delta(1-\mu) ,
H(\nu,\mathcal{E}) = \sum_{F\in\mathcal{F}}\sum_{E\in\mathcal{E},E\subset F}\phi(\nu(F)\nu_F(E)) &= \left(\sum_{F\in\mathcal{F}}\nu(F) \sum_{E\in\mathcal{E}}\phi(\nu_F(E))\right) + \left( \sum_{F\in\mathcal{F}}\phi(\nu(F)) \sum_{E\in\mathcal{E}}\nu_F(E) \right) &= H(\nu,\mathcal{E}|\mathcal{F})+H(\nu,\mathcal{F}) .
\frac{(\prod_{i=1}^{n}x_{i})^{1/n}}{(\prod_{i=1}^{n}(1-x_{i}))^{1/n}}\le\frac{\frac{1}{n}\sum_{i=1}^{n}x_{i}}{\frac{1}{n}\sum_{i=1}^{n}(1-x_{i})}
S[\phi,\chi]=\int[\frac{1}{2}\partial^{\mu}\phi\partial_{\mu}\phi-V(\phi)+\chi^{\dagger}i\overline{\sigma}\cdot\partial\chi+\frac{i}{2}(m+g\phi)\chi^{T}\sigma^{2}\chi-\frac{i}{2}(m+g\phi)^{*}\chi^{\dagger}\sigma^{2}\chi^{*}]d^{n}x
\nu_{k,i}^{x_i,y_i,\bar x_{k,i}}(x_k) =\frac{\L\big(f(y_i,x_k,\bar x_{k,i}),f(x_i,x_k,\bar x_{k,i})\big) \mu_k^{x_i,\bar x_{k,i}}(x_k)}{\bar \nu_{k,i}^{x_i,y_i,\bar x_{k,i}}},
\overline{H}_n(\mathbf{s})=\overline{H}_n(s_1,s_2,\ldots,s_r):=\sum\limits_{0\leq k_1<k_2<\cdots<k_r\leq n-1}\prod_{j=1}^{r}\frac{1}{(2k_j+1)^{s_j}},
\begin{aligned}L_1 = 2{K_1}_{xy}-{K_2}_{xx}-3{K_0}_{yy}- 6K_0{K_3}_x- 3K_3{K_0}_x* &+ 3K_0{K_2}_y + 3K_2{K_0}_y +{K_1}{K_2}_x - 2{K_1}{K_1}_y L_2 = 2{K_2}_{xy}-{K_1}_{yy}- 3{K_3}_{xx}+ 6K_3{K_0}_y + 3{K_0}{K_3}_y* &- 3K_3{K_1}_x - 3K_1{K_3}_x -{K_2}{K_1}_y + 2{K_2}{K_2}_x*\end{aligned}
\mathcal{R}_c = \left\{(x,y)~|~\langle J_Ax-J_Ay,x-y\rangle\leq(\tfrac{\delta}{2}+\tfrac{(1-\alpha-\delta/2)^2-\alpha^2+(\delta/2)^2}{2(1-\alpha-\delta/2)})\|x-y\|^2\right\}.
\alpha_j = \lVert \mathbf 1_{2^{j}\leq \lvert \xi \rvert \leq 2^{j+1}}m (\xi )\rVert_{\infty}\textup{and}\beta_j = \lVert \mathbf 1_{2^{j}\leq \lvert \xi \rvert \leq 2^{j+1}}\nabla m (\xi ) \cdot \xi \rVert_{\infty}.
\Delta:= d_{1, 0}^2 d_{1, -1}^2 d_{-1, 1}^2 d_{-1, 0}^2 t^8(16t^4 d_{1, 0}^2 d_{-1, 0}^2-32 t^4 d_{1, 0}d_{-1, 0}d_{1, -1}d_{-1, 1}&+16t^4d_{1, -1}^2 d_{-1, 1}^2-8t^2 d_{1, 0}d_{-1, 0}-8t^2d_{1, -1}d_{-1, 1}+1)
\times \sum_{{\bf{q}}_{1}\neq{\bf{q}},{\bf{0}}}n_{{\bf{k}}-{\bf{q}}/ 2 +{\bf{q}}_{1}}e^{i \mathrm{~}\theta ({\bf{k}}+{\bf{q}}_{1}/ 2 , -{\bf{q}}+{\bf{q}}_{1})}e^{- i \mathrm{~}\theta ({\bf{k}}-{\bf{q}}/ 2 +{\bf{q}}_{1}/ 2 ,{\bf{q}}_{1})}
\begin{aligned}\underset{[0,T]}{}({\rho}_4^2(t) \|y_t(t)\|^2 ) + \iint_Q{\rho}_4^2 |y_{xt}|^2 dxdt \leq C \left( \|y_t(0)\|^2 + \iint_Q \rho_3^2 |G_t|^2 dxdt \right. &+ \iint_{\mathcal{O}\times (0,T)}{\rho}_1^2 |f|^2 dxdt + \iint_Q{\rho}_0^2 |y|^2 dxdt + \left.\sum_{i=1}^2 \iint_Q{\rho}_0^2 |p^i|^2 dxdt \right).\end{aligned}
\alpha_{0}=-2d^{2},\beta_{0}=a^{2}-J_{1}d,\alpha_{1}=-4de,\beta_{1}=\frac{de}{4}+2ab-J_{1}e, \alpha_{2}=-2e^{2},\beta_{2}=1+2ac+b^{2}+\frac{e^{2}}{4},\beta_{3}=2bc,\beta_{4}=c^{2}
\langle F \rangle = \int d x_{1}. . . d x_{N}\psi^{*}F \psi = C^{2}\int d x_{1}. . . d x_{N}\prod_{i < j}| x_{i}- x_{j}|^{\beta}e^{- \omega \sum_{i}x_{i}^{2}}F ( x_{1}, . . . , x_{N}) .
\begin{array}{lccc}\overline{P_0}^G = 0 \overline{Q_0}^G = -s_2^2 \overline{P_1}^G = s_1 \overline{Q_1}^G = -s_2^2 \overline{P_{2}}^G = -s_1 \overline{Q_{2}}^G = -s_2^2 \end{array}
\{\pi_{\mu}( \textbf{x}, t ) , \pi_{\nu}( \textbf{y}, t ) \}^{*}= \{A_{\mu}( \textbf{x}, t ) , A_{\nu}( \textbf{y}, t ) \}^{*}= 0 \hspace{0 . 3 i n}( \mu , \nu = 1 , 2 , 3 , 4 ) .
d_{n}^{\cal O}( R ) = \frac{1}{n !}\frac{1}{( 2 \pi )^{n}}\int d \theta_{1}\ldots d \theta_{n}F_{n , n}^{\cal O}( \theta_{n}, \ldots , \theta_{1}| \theta_{1}, \ldots , \theta_{n}) e^{- E_{n}R}\, \, \,
\sum_{i}\gamma_{i}^{\left( D \right)}\left( a , b , c \right) \, \alpha_{i l}\left( a , b , c \right) \alpha_{i l^{\prime}}\left( a , c , b \right) = \left( - 1 \right)^{\left( m - 2 \right) \left( l - 1 \right)}\gamma_{l}^{\left( D \right)}\left( b , a , c \right) \delta_{l l^{\prime}}.
\mu_{X,b}(F):=\mbox{sup}\,\{p\in\mathbb{R}:\,|T_{b}^{l}F(y)|\leq \mbox{const}\,|y-b|^{p}, y\in X,\,l=\min\{k\in\mathbb{N},k\geq p\}\}\nu_{X,b}(F):=\max\{l\in\mathbb{N}:\,F\in\widehat{m}_{b}^{l}+\mathcal{F}_{b}(X)\}.
|G(\xi,t')-G(\xi,t)| \leq \Big| \pi \int_{t}^{t'}G(\xi,s)\mathrm{D}[Q(e^{-isA}z)][\xi] ds \Big| , &\leq{2 \pi |t-t'| |\xi| M r \sup_{s \in [t,t']}\sum_{j=2}^{r}\int_{\mathcal Z}|z|^{2j-1}d\tilde{\mu}_{s}(z)}, &\leq C|t-t'||\xi|,
G_{\theta}^{i j}( \nu ) = \delta_{i j}\left( - \frac{\bar{R}_{i}}{\bar{\gamma}_{i}}+ \nu \bar{R}_{i}^{2}\cot ( \nu R_{p}) \right) - ( 1 - \delta_{i j}) \nu \bar{R}_{i}\bar{R}_{j}\csc ( \nu R_{p}) \, .
\begin{array}{lll}[v_a]_i=1, [v_b]_i=0, \mbox{if}i \in \alpha; begin{align*}5pt] [v_a]_i=0, [v_b]_i=1, \mbox{if}i \in \gamma; begin{align*}5pt] [v_a]_i=t, [v_b]_i=1-t, \mbox{for some}t\in [0,1] \mbox{if}i \in \beta. \end{array}
\sum_{i=1}^{n}(y_{i}-\overline{y})^{2}=\sum_{i=1}^{n}(y_{i}-\hat{y}_{i})^{2}+\sum_{i=1}^{n}(\hat{y}_{i}-\overline{y})^{2}+\sum_{i=1}^{n}2(\hat{y}_{i}-\overline{y})(y_{i}-\hat{y}_{i}).
A_0 \leq A\left(D_{t_j}\right)\leq A\left(T_j\right) + A\left(R_j\right) \leq \frac{2\pi}{Q_j}L_0 + \frac{1}{2}L_0^2\sin\beta_j \left(\cos \beta_j + \sin\beta_j\tan(\alpha_j + \beta_j)\right) \leq \frac{2\pi}{Q_j}L_0 + L_0^2\sin\beta_j.
\varepsilon({\psi_{\lambda}},\sigma_{n!/2}) =\prod_{k=1}^s\varepsilon(\psi_{\underline\lambda,k},\sigma_{n!/2}) &=\prod_{k=1}^s(-1)^{\frac{(p-1)n_k}{2}}&=(-1)^{\frac{p-1}4\sum_{k=1}^sn_k}.
l_{\varphi}:=\int_{0}^{\infty}|\varphi'(t)|dt=\int_{0}^{\infty}\left|\frac{1}{\varepsilon}g(u(x_{0},0),\varphi(t))+L_{2,x}(u(x_{0},0),\,v(x_{0},t)))\right|dt\leq
\begin{aligned}\|P_h u - u_h\|_0^2 \leq C h^2 (\|\div \sigma\|_1 \|v\|_1 + (\|A\|_{1,\infty}+ \|A\|_{2,\infty})(\|\sigma\|_1 + \|p\|_1)(\|\tau\|_1 + \|q\|_1)) \leq C h^2 \|A\|_{2,\infty}(\|\div \sigma\|_1 + \|\sigma\|_1 + \|p\|_1)(\|v\|_1 + \|\tau\|_1 + \|q\|_1). \end{aligned}
\bigg(\frac{\partial^2}{\partial r^2}+\frac{g^2}{4}e^{\mp2i\theta}(\lambda_2-\lambda_1)\bigg)\psi^{(1)}_{\pm}=0, \;\;\psi^{(2)}_{\pm}=\frac{2e^{\pm i\theta}}{g(\sqrt{\lambda_1}+\sqrt{\lambda_2})}\frac{\partial}{\partial r}\psi^{(1)}_{\pm}.
|F_m|_{S_{I,m}(\delta)}\leq \sup_{0\leq j\leq m-1}|(\frac1{a_p})^{j+1}U_p^{nsp}(U_p^{sp})^jf|_{S_{I,m}(\delta)}&= \sup_{0\leq j\leq m-1}|(\frac{U_p^{sp}}{a_p})^j F_1|_{S_{I,m}(\delta)}&\leq \sup_{0\leq j\leq m-1}|F_1|_{S_{I,m-j}(\delta)}&= |F_1|_{S_{I,1}(\delta)}.
T_{1}= -{\frac{i}{2}}\left( \begin{matrix}{{0}}&{{1}}\\{{1}}&{{0}}\\ \end{matrix}\right) ; T_{2}={\frac{1}{2}}\left( \begin{matrix}{{0}}&{{- 1}}\\{{1}}&{{0}}\\ \end{matrix}\right) ; T_{3}= -{\frac{i}{2}}\left( \begin{matrix}{{1}}&{{0}}\\{{0}}&{{- 1}}\\ \end{matrix}\right) \ .
\forall m\in \mathrm{Int}_{E+2r_1\sqrt{k}}W_0(x,n) \setminus \left(n+L\mathbb{Z}^k\right): \> \nu(x,m)=0, \forall \, m\in \left(n+L\mathbb{Z}^k\right) \cap \mathrm{Int}_{E+\sqrt{k}}W_0(x,n): \> \nu(x,m)=1.
X_\bot^{\mu\nu}X_\bot^{\rho\sigma}= \frac{1}{6}\left(\,\pi^{\mu\sigma}\pi^{\nu\rho}- \varepsilon^{\mu\sigma\lambda}e^{(0)}_\lambda\varepsilon^{\nu\rho\tau}e^{(0)}_\tau \, \right)\cdot C + \frac{i}{4}\left(\, \pi^{\mu\sigma}\varepsilon^{\nu\rho\lambda}+ \pi^{\nu\rho}\varepsilon^{\mu\sigma\lambda}\, \right) e^{(0)}_\lambda \cdot Q_0.
\tilde{\cal H}=\omega\, q\cdot\!{\partial\over{\partial q}}-{1\over2}\sum_{j=1}^r{\partial^2\over{\partial q_j^2}}-\sum_{\rho\in\Delta_+}{g_{|\rho|}\over{\rho\cdot q}}\rho\cdot\!{\partial\over{\partial q}}.
\Phi_3(b;\,c;\,x,\,x^2)=\exp\left(2x\right)\,{}_2F_2 \left[ \begin{array}{c}c-\frac{b}{2}, \, c - \frac{b}{2}-\frac{1}{2}\,c,\,2c-b-1\end{array}\,;\, -4x\,\right]
\bar{X}_{t}^{\varepsilon ,\varphi}[\phi ]=x_{0}[\phi ]+\int_{0}^{t}b(\bar{X}_{s}^{\varepsilon ,\varphi})[\phi ]ds+\varepsilon \int_{\mathbb{X}\times\lbrack 0,t]}G(\bar{X}_{s-}^{\varepsilon ,\varphi},y)[\phi ]N^{\varepsilon^{-1}\varphi}(dy,ds),\,\phi \in \Phi
\varphi_{p}(s,x):=\sum_{n=1}^{\infty}\frac{p^{2}+\lambda_{n}^{2}}{p\left(p+\frac{1}{\pi}\right)+\lambda_{n}^{2}}\,\frac{1}{\left(\lambda_{n}^{2}+x^{2}\right)^{s}},\,\,\,\,(s)>\frac{1}{2}.
\Phi_G(m) = \frac{\Phi_G(m_0+i)}{\Phi_G(m_0)}= \frac{\binom{y-2m_0}{2}}{\binom{y-2m_0-2i}{2}}\frac{m_0+i+1}{m_0+1}& = \left(1+\frac{2i}{y-2m_0-2i+O(1)}\right)^2 \left(1+\frac{i}{m_0+1}\right) =1+\frac{\Theta(i)}{\lambda_1}.
\begin{array}{rcl}({\bf 1}+\tilde{\bf K})({\bf 1}+{\bf K}) = \frac{4}{(q+q^{-1})^{2}}({\bf 1}+(q^{2}+q^{-2})(q+q^{-1})^{2}{\bf P}) ({\bf 1}+{\bf K})({\bf 1}+\tilde{\bf K}) = \frac{4}{(q+q^{-1})^{2}}({\bf 1}+(q^{2}+q^{-2})(q+q^{-1})^{2}{\bf P}^{'})\end{array}
\begin{array}{rl}\tilde{w}\in\mathcal{D}^{\perp}\iff \langle\tilde{w},v\rangle=0\mbox{for all}v\in\mathcal{D}\iff \langle\sum\lambda_{i}w_{i},v\rangle=0\mbox{for all}v\in X \iff \sum\lambda_{i}w_{i}=0.\end{array}
F_b =\pi\theta'^{\delta}\frac{\pi\delta}{\sin(\pi\delta)}\sum_{k=1}^\infty \binom bk p^k \frac{\Gamma(\delta)}{\Gamma(k)\Gamma(\delta-k+1)}=\pi \theta^\delta R^2 \frac{\pi\delta}{\sin(\pi\delta)}\sum_{k=1}^\infty \binom bk \binom{\delta-1}{k-1}p^k .
i^{3,3}(H^3(Y;\mathbb{Q})) = 1, i^{2,2}(H^3(Y;\mathbb{Q})) = ph(Y,w) - 3, i^{1,2}(H^3(Y;\mathbb{Q})) = i^{2,1}(H^3(Y;\mathbb{Q}))= h^{1,2}(Z).
\{b_{n}\}=1,-\frac{1}{2},-\frac{1}{4},\frac{1}{3},-\frac{1}{6},-\frac{1}{8},\frac{1}{5},-\frac{1}{10},-\frac{1}{12},\frac{1}{7},-\frac{1}{14},-\frac{1}{16},\cdot\cdot\cdot
\hat{\rho}\Big(\big(\varphi(\hat{\pi}_{1}(t_{1}),t_{1}),\psi(t_{1})\big),\big(\varphi(\hat{\pi}_{2}(t_{2}),t_{2}),\psi(t_{2})\big)\Big) &~=\big |\varphi(\hat{\pi}_{1}(t_{1}),t_{1})-\varphi(\hat{\pi}_{2}(t_{2}),t_{2})\big | \vee | \psi(t_{1})-\psi(t_{2}) | &~=\rho\big((\hat{\pi}_{1}(t_{1}),t_{1}),(\hat{\pi}_{2}(t_{2}),t_{2}) \big),
\Delta_{\nu}(x)=\frac{4}{\pi}\int_0^{\frac{\pi}{2}}I_{2\nu}(2x\cos\theta)\sin^2\theta d\theta=\frac{1}{\sqrt{\pi}}\sum_{m\geq0}\frac{\Gamma\left(\nu+m+\frac{1}{2}\right)x^{2\nu+2m}}{m!\Gamma(\nu+m+2)\Gamma(2\nu+m+1)}.
F_{5}(t,r) = N_{2}(v_{5}) + \frac{\sin(2(v_{1}+v_{2}+v_{3}+v_{4}))}{2r^{2}}\left(\cos(2Q_{\frac{1}{\lambda(t)}}+2v_{5})-\cos(2Q_{\frac{1}{\lambda(t)}})\right) &+\left(\frac{\cos(2(v_{1}+v_{2}+v_{3}+v_{4}))-1}{2r^{2}}\right)\left(\sin(2Q_{\frac{1}{\lambda(t)}}+2v_{5})-\sin(2Q_{\frac{1}{\lambda(t)}})\right)
[B(m),B(n)]=m\delta_{m+n,0}c\quad\textrm{for}m,n\in 2\mathbb{Z}+1; &[B(m),X(n)]=2X(m+n)\quad\textrm{for}m\in 2\mathbb{Z}+1,n\in\mathbb{Z}; &[X(m),X(n)]=(-1)^{m+1}2B(m+n)+(-1)^{m}m\delta_{m+n,0}c\quad\textrm{for}m,n\in\mathbb{Z},
\frac 1 3 \varepsilon^{i j}\left( \begin{array}{l}{{2 \bar{\varepsilon}\gamma_{\mu \rho}\partial_{i}\theta \bar{\theta}\gamma^{\mu \nu}\partial_{j}\theta}}\\{{- \bar{\varepsilon}\gamma_{\mu \rho}\theta \partial_{i}\bar{\theta}\gamma^{\mu \nu}\partial_{j}\theta}}\\ \end{array}\right) \lambda_{\nu}\lambda^{\rho},
\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]\rightarrow \int\limits_{0}^{\infty}\int\limits_{0}^{\infty}d\nu_{1}d\nu_{2}\left[{\Large i}\right] \ .
S_{B o l t}= \frac{4 \pi^{2}( 1 5 r_{b}^{6}- ( 6 5 N^{2}- 3 \ell^{2}) r_{b}^{4}+ 3 N^{2}( 5 5 N^{2}- 6 \ell^{2}) r_{b}^{2}+ 9 N^{4}( 5 N^{2}- \ell^{2}) )}{3 ( 5 r_{b}^{2}+ \ell^{2}- 5 N^{2})}
\Lambda_{1/2}=\pm \frac{1}{\sqrt{2}}(1+i)\sqrt[4]{-E_+},\Lambda_{3/4}=\pm \frac{1}{\sqrt{2}}(1-i)\sqrt[4]{-E_+}begin{align*}5pt] \Lambda_{5/6}=\pm \frac{1}{\sqrt{2}}(1+i)\sqrt[4]{-E_-},\Lambda_{7/8}=\pm \frac{1}{\sqrt{2}}(1-i)\sqrt[4]{-E_-}.
\zeta(\alpha,t)-z_j(t)= \zeta(\alpha,0)-z_j(0)+\int_0^t \partial_{\tau}(\zeta(\alpha,\tau)-z_j(\tau))d\tau = \zeta(\alpha,0)-z_j(0)+\int_0^t D_{\tau}\zeta(\alpha,\tau)d\tau-\int_0^t b(\alpha,\tau)\partial_{\alpha}\zeta(\alpha,\tau)d\tau-\int_0^t \dot{z}_j(\tau)d\tau
\xi'_{n,j}(a)=F_{n,j}(a)x'_{j}(a)+F_{n,j}(a)\left(\frac{F'_{n,j}(a)x_{j}(a)}{F_{n,j}(a)}+\frac{E'_{n,j}(a)+\mu'_{n,j}(a)}{F_{n,j}(a)}\right).
\mu_m\langle\phi_m,\psi_n\rangle =\langle L\phi_m, \psi_n\rangle =\langle \phi_m, L\psi_n\rangle &=a_n\langle \phi_m,\psi_{n-1}\rangle+b_n\langle \phi_m,\psi_{n}\rangle+a_{n+1}\langle \phi_m,\psi_{n+1}\rangle,
\inf_{F\in \mathcal{F}_+(G)}\frac{H_\mu(\alpha_{F})}{|F|}+\mu(f) \, = \, \inf_{F\in \mathcal{F}_+(G)}\frac{H_\mu(\alpha_{F})+\mu(\sum_{F}{f})}{|F|}&\leq \, \inf_{F\in \mathcal{F}_+(G)}\frac{\operatorname{P}_{\sum_{F}{f}}(\mathcal{U}_{F})}{|F|}+\epsilon &\leq \, \operatorname{p}_f^{(nv)}(\pi)+\epsilon.
\begin{aligned}\cos(x)+\cos(y)= 2\cos\left(\frac{x+y}{2}\right)\cos\left(\frac{x-y}{2}\right) \Rightarrow \cos\left(\pi u_{0}+\frac{2\pi}{N}\cdot j\right)+\cos\left(\pi u_{0}-\frac{2\pi}{N}\cdot j\right)=2\cos\left(\pi u_{0}\right)\cos\left(\frac{2\pi}{N}\cdot j\right)\end{aligned}
\frac{a_{0}}{1+}\frac{-a_{1}}{1+a_{1}}=\frac{a_{0}}{1+\frac{-a_{1}}{1+a_{1}}}=\frac{a_{0}(1+a_{1})}{1}=a_{0}+a_{0}a_{1}=\sum_{k=0}^{1}\prod_{j=0}^{k}a_{j}
(\mathcal{T}Q)_j :=\min \left\{\sum_{l=1}^j Q_l-\sum_{l=1}^{j-1}(\mathcal{T}Q)_l,{E}_j \right\}, (\mathcal{T}{E})_j :=Q_{j+1}+{E}_j-(\mathcal{T}Q)_j,
\bigcup_{i=2}^{n}\bigcup_{\substack{1\le j_1,j_2\le i j_1\ne j_2 \gcd(i,j_1,j_2)=1}}\left\{\left<E^{p\times q}_{kl}\otimes E^{i\times i}_{j_1j_2}\otimes I_{t/i}\right>\right\}.
f(p,k)={2k^3(4\alpha^3 p^3+4\alpha^2 p^4+\alpha^2 p^2 k^2-3p^4 k^2+\alpha^2 k^4+3\alpha p k^4 + 3p^2 k^4) \over(3 \alpha (\alpha p +p^2)^3)}\Theta(p-k),
J_{h}( x ) = \lim_{N \rightarrow + \infty}\int_{- \infty}^{+ \infty}\frac{d k}{2 \pi}\, e^{i k \left\{x - ( 1 - \beta^{2}) \varepsilon \ln \left[ \frac{A}{N}\sin \frac{\pi \beta^{2}}{2 ( 1 - \beta^{2})}\right] \right\}}\, \frac{N \varepsilon \pi k \, \sinh \left( \pi k \beta^{2}- \frac{\pi k}{2}- \varepsilon i \pi \beta^{2}\right)}{R \sin \pi \beta^{2}\, \cosh k \frac{\gamma}{2}}\, .
\xi_i(0)=\max_{-1<z_1<\ldots<z_{i-1}<1}\min_{\substack{p\in S p(z_1)=\ldots=p(z_{i-1})=0}}\frac{\int_{-1}^1tp(t)w(t)dt}{\int_{-1}^1p(t)w(t)dt},\quad1\le i\le n.
E_{z}^{a}= \frac{N ( z , t )}{f^{\prime}( z , t )}\left( \begin{array}{c}{f}\\{1}\\{f^{2}}\\ \end{array}\right) , E_{u}^{a}= \frac{n ( u , t )}{\dot{f}( u , t )}\left( \begin{array}{c}{f}\\{1}\\{f^{2}}\\ \end{array}\right)