harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sup_{|\alpha|=k}\big\|D^{\alpha}\dot{\Delta}_j f\big\|_{L^{q, 1}\left(\mathbf{R}^{n}\right)}\leq C 2^{j\left(k+n\left(\frac{1}{p}-\frac{1}{q}\right)\right)}\|\dot{\Delta}_jf\|_{L^{p, \infty}\left(\mathbf{R}^{n}\right)},
\begin{aligned}\Big(\prod_{q=a_j}^{a_j\oplus(s_i-1)}\lambda_{j,q}\Big)x_{a(j,a_j\oplus s_i)}=\Big(\prod_{q=s_ia_n}^{s_ia_n+s_i-1}\lambda_{n,q}\Big)x_{a(n,(s_ia_n\oplus s_i)/s_i)}\end{aligned}
AG \geq \frac{2(2-\lambda)}{\lambda n}\phi G^{2}-\bigg(2K+\frac{12\left\vert b\right\vert}{n}\bigg)\phi G-2\left\vert \nabla\phi \right\vert G^{\frac{3}{2}}& \geq \frac{2(2-\lambda)}{\lambda n}\phi G^{2}-\bigg(2K+\frac{12\left\vert b\right\vert}{n}\bigg)\phi G-2\frac{C_{1}}{R}\phi^{\frac{1}{2}}G^{\frac{3}{2}}.
D(0, \varepsilon) = \sigma_{\varepsilon}([iT\bullet \varphi^{-1}(iI),S]_{\ast})=\sigma_{\varepsilon}([\varphi(iT)\bullet iI,\varphi(S)]_{\ast}) &=\sigma_{\varepsilon}(2i(\varphi(iT)+\varphi(iT)^{\ast})).
[\bar{a}_1 \bar{a}_2, \bar{a}_3] = [\bar{a}_1, \bar{a}_3] [\bar{a}_2, \bar{a}_3], [\bar{a}_1, \bar{a}_2 \bar{a}_3] = [\bar{a}_1, \bar{a}_2] [\bar{a}_1, \bar{a}_3], [\bar{a}_1, \bar{a}_2^{-1}] = [\bar{a}_1, \bar{a}_2]^{-1}, [\bar{a}_1^{-1}, \bar{a}_2] = [\bar{a}_1, \bar{a}_2]^{-1}\ \ \ \ \ \forall a_1, a_2, a_3 \in A.
\Big\langle : e^{i\varphi(f)}:_M e^{i\varphi(g)}:_M \Big\rangle_{C_m}\; = \;\exp \left( -\frac{1}{2}(f,C_m f) - (f,C_m g) - \frac{1}{2}(g,C_m g) \; +\right.
2 4 \mathrm{C l}_{2}( \theta_{5 , 3}) - 1 2 \mathrm{C l}_{2}( 2 \theta_{5 , 3}) - 8 \mathrm{C l}_{2}( 3 \theta_{5 , 3}) + 6 \mathrm{C l}_{2}( 4 \theta_{5 , 3}) = Z_{1 5}= 1 5 \sum_{k = 1}^{7}\left( \frac{k}{1 5}\right) \mathrm{C l}_{2}\left( \frac{2 \pi k}{1 5}\right)
\mathsf{d_v}:{\mathsf{\Omega}}^{r,s}\left({\sf J}^kY\right)\rightarrow{\mathsf{\Omega}}^{r,s+1}\left({\sf J}^kY\right),\mathsf{d_v}F:=\sum_{0\leq|I|\leq k}\sum_{a=1}^m\frac{\partial F}{\partial u^a_I}\vartheta^a_I,
\int_{\mathcal{T}}\exp( \varphi(t))) \, dt = \sum_{0 \le m \le \frac{\log \log T}{\log 2}}\int_{\mathcal{T}\cap \mathcal{P}(m)}\exp( \varphi(t))) \, dt + \int_{\mathcal{T}\cap \left( \bigcap_{m}\mathcal{P}(m)^c \right)}\exp( \varphi(t))) \, dt.
[ \tau ( \lambda ) Q ( \lambda ) \psi ] ({\vec{x}}) = T r \left[ \int d{\vec{t}}d{\vec{y}}\left( \prod_{i = N}^{1}L ( \lambda | x_{i}, \partial_{x_{i}}) R_{\lambda + \beta_{i}- 1}( t_{i}, x_{i}| t_{i - 1}, y_{i}) \right) \psi ({\vec{y}}) \right] ~ .
L_{k}^* R_{k^{-1}}^* \pi^* \omega = \rho(k) \pi^* \omega (\mathrm{Ad}k)^* \pi^* \omega = \rho(k) \pi^* \omega \mathrm{Ad}^*(k) \circ \rho(k) \pi^* \omega|_e = (\mathrm{Ad}{k^{-1}})^* \rho(k) \pi^* \omega|_e = \pi^* \omega|_e.
\frac{1 + g}{1 - g}= \lambda_{1}^{2}c o t h \frac{1}{2}k ( r + i t ) + \lambda_{2}^{2}c o t h \frac{1}{2}k ( r + i t - i c k^{- 1}) + \lambda_{3}^{2}c o t h \frac{1}{2}k ( r + i t + i c k^{- 1})
S(I_1,I_2) = \left\lbrace p \in \mathbb{P}: p\nmid 2N_1N_2, B_1(p) \in I_1, B_2(p) \in I_2 \right\rbrace S(I_1,I_2)(x) = \left\lbrace p\leq x: p\in S(I_1,I_2) \right\rbrace.
\Vert u \Vert_{C^k_{g,-a,V}}:= \Vert u \Vert_{C^k_{g,(S-f+1)^{-a/2},V}}, \Vert u \Vert_{C^{k,\alpha}_{g,-a,V}}:= \Vert u \Vert_{C^{k,\alpha}_{g,(S-f+1)^{-a/2},V}}.
P_{\perp}\begin{pmatrix}z \bar{z}\end{pmatrix}= \begin{pmatrix}P_{11}P_{12}P_{21}P_{22}\end{pmatrix}\begin{pmatrix}x+iy x-iy \end{pmatrix}= \begin{pmatrix}P_{11}+P_{12}i(P_{11}-P_{12}) P_{21}+P_{22}i(P_{21}-P_{22}) \end{pmatrix}\begin{pmatrix}x y \end{pmatrix}
\frac{1}{\sqrt[3]{a}+\sqrt[3]{b}}=\frac{\sqrt[3]{a^{2}}-\sqrt[3]{ab}+\sqrt[3]{b^{2}}}{(\sqrt[3]{a}+\sqrt[3]{b})(\sqrt[3]{a^{2}}-\sqrt[3]{ab}+\sqrt[3]{b^{2}})}=\frac{\sqrt[3]{a^{2}}-\sqrt[3]{ab}+\sqrt[3]{b^{2}}}{a+b}.
\begin{aligned}\Vert (v^1_e,g^1_e)\Vert_{L^\infty}+\Vert \langle Y\rangle^n(v^1_e,g^1_e)\Vert_{H^2}\leq C_0, &\Vert \langle Y\rangle^n(v^1_e,g^1_e)\Vert_{H^3}\leq C_0\epsilon^{-1/2}, &\Vert \langle Y\rangle^n(v^1_e,g^1_e)\Vert_{H^4}\leq C_0\epsilon^{-3/2},\end{aligned}
\begin{aligned}|X_t-X_s|^{p}= \int_s^t p\cdot |X_u-X_s|^{p-2}\cdot \bigl( (X_u-X_s)\cdot \mu(X_u) + \frac{1}{2}(p-1) \cdot \sigma^2(X_u)\bigr)\, du & + \int_s^t p \cdot (X_u-X_s)|X_u-X_s|^{p-2}\sigma(X_u)\, dW_u.\end{aligned}
I_{2}(y,a,b,d)=\frac{1}{a}h(d)\sum_{n=1}^{\infty}\int dp\,p^{d-3}\frac{1}{\sqrt{\vec{p}^{\,2}+m^{2}+(\frac{n\pi}{a})^{2}}}\frac{\cosh ((b-2y)\sqrt{\vec{p}^{\,2}+m^{2}+(\frac{n\pi}{a})^{2}})}{\sinh (b\sqrt{\vec{p}^{\,2}+m^{2}+(\frac{n\pi}{a})^{2}})}
b = 3(4\Delta)^{\frac{1}{3}}- 6(4\Delta)^{\frac{2}{3}}+ 9(4\Delta) - 10(4\Delta)^{\frac{4}{3}}+ 7(4\Delta)^{\frac{5}{3}}- \frac{22}{3}(4\Delta)^{\frac{7}{3}}+{\cal{O}}(\Delta^{\frac{8}{3}}) .
\lim_{\lambda\to0^+}\lambda\left\|\left[\int_{\mathbb{R}^n}\mathbf{1}_{E_{\lambda,\frac{\gamma}{q}}[f]}(\cdot,y)\left|\cdot-y\right|^{\gamma-n}\,dy\right]^\frac{1}{q}\right\|_{M_r^\alpha(\mathbb{R}^n)}=\left[-\frac{\kappa(q,n)}{\gamma}\right]^\frac{1}{q}\left\|\,\left|\nabla f\right|\,\right\|_{M_r^\alpha(\mathbb{R}^n)}.
X_3 = \left\{t\in \mathcal{L}\ :\ ||t||_{\infty}^2 \geq \frac{8 \pi^2 2^{\nu}\nu^2}{\omega^{\nu}\ln\left(1+\frac{2}{5}\pi^2r\right)}\left(\frac{1}{2}\nu^2 k\ln(k) + \frac{1}{2}\nu k \ln(R) \right)k^{-\nu+1}\right\}.
\mathcal{I}_{n,m}= \int_{T}^{\infty}t^{-\frac 3 2 -c}d \mathcal{J}_{n,m}(t) &\lesssim \int_{T}^{\infty}|\mathcal{J}_{n,m}(t)| \,t^{-\frac 5 2 -c}dt.
\ln \mathrm{d e t}^{\prime}( A_{a b}) = \ln [ \mathrm{d e t}^{\prime}( A_{+ +}) \mathrm{d e t}^{\prime}( A_{--}) ] = \int^{'}\frac{d^{3}q}{( 2 \pi )^{3}}\ln ( h_{+}({\bf q}, T ) h_{-}({\bf q}, T ) ) .
T^2_{r,j}(m_1,m_2) \leq \tau_{r,j}+ \sum_{\tau_{r,j}> x > y\geq 0}\frac{1}{\lambda_j^y \lambda_j^{x-y-1}(1-\lambda_j^{y-x})}&< \tau_{r,j}+\sum_{y,z=0}^{\infty}\frac{1}{\lambda_j^y \lambda_j^z (1-\lambda_j^{-1})}&= \tau_{r,j}+\Big(\frac{\lambda_j}{\lambda_j-1}\Big)^3 &<4\tau_{r,j}\Big(\frac{\lambda_j}{\lambda_j-1}\Big)^3.
\mathcal{I}_+(\psi_-,\psi_+) = -\tau_{+(T+T_0)}^{-1}(\Pi_+^{\psi_+,T}D_{+,T})^{-1}\Pi_+^{\psi_+,T}(e_{+,T}+ \mathcal L_+(\psi_{-,T},\psi_{+,T}) + \mathcal{R}_+(\psi_{+,T})).
\|\overline X(0,n)\|_2^2+\|\overline X(0,m)\|_2^2\le \|\overline X(0,n+m)\|_2^2+4\|\overline X(0,n+m)\|_2\|Z(n,m)\|_2 &\qquad+ 4\|Z(n,m)\|_2^2.
m =|A_1|+|A_2|+\sum_{\ell\geq 3}{|A_\ell|}\leq |A_1|+|A_2|+\sum_{\ell\geq 3}{\ell}{|A_\ell|}/3 &=(2|A_1|+|A_2|)/{3}+\sum_{\ell\geq 1}\ell|A_\ell|/3=(2|A_1|+|A_2|)/{3}+{2q}/{3}\leq{5q}/{3},
\begin{aligned}{_aD_x^{-\mu}}{_aD_x^{-\sigma}}u(x) ={_aD_x^{-\mu - \sigma}}u(x), x> a{_xD_b^{-\mu}}{_xD_b^{-\sigma}}u(x) ={_xD_b^{-\mu - \sigma}}u(x), b< x.\end{aligned}
{3 \over 2}{\sigma'' \over \sigma}+{3 \over 4}{\sigma'\gamma' \over\sigma \gamma}-{1 \over 4}\left({\gamma' \over \gamma}\right)^2+{1\over 2}{\gamma'' \over \gamma}=-{\Lambda_0 \over 4M^4}-{V^i_0 \over 4M^4}\delta(\rho-\rho_i)
H_{KL}(k_{x},k_{y},k_{z})=-\frac{\hbar^{2}}{2m}[(\gamma_{1}+\frac{5}{2}\gamma_{2})k^{2}-2\gamma_{2}(J_{x}^{2}k_{x}^{2}+J_{y}^{2}k_{y}^{2}+J_{z}^{2}k_{z}^{2})-2\gamma_{3}\sum_{m\ne n}J_{m}J_{n}k_{m}k_{n}]
Z [ T / e^{2}; x_{1}, \ldots , x_{K}] \; = \; \int [ d A ] [ d g ] ~ \left< A \right| e^{- H / T}\left| A^{g}\right> \mathrm{T r}~ g^{R_{1}}( x_{1}) \ldots \mathrm{T r}~ g^{R_{K}}( x_{K}) ~ ~ ,
\begin{pmatrix}x^{k+1}y^{k+1}\end{pmatrix}=\begin{pmatrix}x^{k}y^{k}\end{pmatrix}-\begin{pmatrix}I_n 0 -(1-\alpha)\frac{1}{s}A I_m\end{pmatrix}\begin{pmatrix}x^{k}-\widetilde{x}^k y^{k}-\widetilde{y}^k\end{pmatrix}.
\mathrm{I}= \left(\prod_{k=1}^{m-2}b_k(a_k)\right)|D|^{n_1}, \mathrm{II}= |D|^{-n_1}[F,\delta(a_{m-1})]|D|^{1+n_1}, \mathrm{III}= |D|^{-n_1-1}\left(\prod_{k=m}^{p-1}b_{k+1}(a_k)\right)|D|^{1+n_1+n_2}.
\delta_j^1 =\min\bigl\{1,\frac{d(a_j, c_j)}{C_{n+1}\alpha_{j}}\bigr\}, \delta_j^2 =\min\bigl\{1, \frac{d(c_j, d_j)}{C_{n+1}(\alpha_{m+j}+\beta_k)}\bigr\}, \delta_j^3 =\min\bigl\{1, \frac{d(d_j, b_j)}{C_{n+1}\alpha_{2m+j}}\bigr\}.
\frac{1}{|E^{(klm)}|}\int_{E^{(klm)}}(u^{(k)})^{(k)}\,ds = \frac{1}{|E^{(klm)}|}\int_{E^{(klm)}}(u^{(l)})^{(k)}\,ds, \frac{1}{|E^{(klm)}|}\int_{E^{(klm)}}(u^{(k)})^{(k)}\,ds = \frac{1}{|E^{(klm)}|}\int_{E^{(klm)}}(u^{(m)})^{(k)}\,ds
\psi_t =d_1\Delta \psi-\chi_1\nabla \cdot(\psi\nabla w)-\chi_1\nabla \cdot(u^{**}\nabla(w-w^{**})) &+ \psi\Big(a_0(t,x)-a_1(t,x)(u+u^{**})-a_2(t,x)v\Big)-a_2(t,x)u^{**}\phi,
\sum_{k,l\geq 0}c_e(k,l;s) x^{2k+2l}=\sum_{k,l\geq 0}{(qa^2/c^2,q^{2l}s^2;q^2)_k\over (q^2,q^{2l}q^3s^2/a^2c^2;q^2)_k}(q^2x^2/a^2)^k &\quad\quad\times{(c^2/q,s;q)_l (q^2s^2/a^4;q^2)_l\over (q,qs/a^2;q)_l (q^3 s^2/a^2c^2;q^2)_l}(q^2x^2/c^2)^l.
R^*_{r_i}= \frac{1}{2}\log_2[\frac{f_i^*\gamma_i^*}{2\beta_i^*\ln 2}g_{r_i}]^+, i=1, \cdots, M P^*_{r_i}=[\frac{f_i^*\gamma_i^*}{2\beta_i^*\ln 2}-\frac{1}{g_{r_i}}]^+ i = 1,\cdots, M
p_{}(x,y|\mu_{x},\mu_{y},\sigma)=\frac{\exp\left\{-\frac{1}{2\sigma^{2}(1-\rho^{2})}\left[ (x-\mu_{x})^{2}-2\rho(x-\mu_{x})(y-\mu_{y})+(y-\mu_{y})^{2}\right] \right\}}{2\pi\sigma^{2}\sqrt{1-\rho^{2}}}
P_{\mathcal{L}}(n) = \frac{3e(Y_{n}^{\mathcal{L}}) - K_{Y_{n}^{\mathcal{L}}}^{2}}{n^{d-3}}= n^{2}(f_{0}- d) + 2n(d-f_{1}+f_{0}) + 2f_{1}+ f_{0}- d - 4t_{2}
2\sum_{\mu = 1}^\infty{\frac{{H_{\mu ,3}}}{{\mu^7}}}+ 7\sum_{\mu = 1}^\infty{\frac{{H_{\mu ,2}}}{{\mu^8}}}= - 15\zeta (4)\zeta(6) + 14\zeta (3)\zeta (7) + 8\zeta (5)^2\,.
K^CF(t,x) = \int (O^{-1}(t,x,\xi') O(s,y,\xi')) F(s,y) (O^{-1}(t,x,\xi') O(s,y,\xi'))^{-1}e^{i R \xi_1}e^{i T \xi'^2}\tilde a_C(\xi) d\xi
\mu_{\omega,\mathbf{c}}=S_{\omega,\mathbf{c}}(\Psi_{\omega})=\omega^{2-\frac{d}{2}}S_{1,\frac{\mathbf{c}}{\sqrt{\omega}}}(\Psi)=\omega^{2-\frac{d}{2}}\mu_{1,\frac{\mathbf{c}}{\sqrt{\omega}}}.
\int_{{\mathbb{R}}^n}I(x,y)f(y)\,{\rm d}y = \int_{{\mathbb{R}}^n}k(t,s,x,z)\Big(\int_{{\mathbb{R}}^n}k(s,r,z,y)f(y)\,{\rm d}y\Big)\,{\rm d}z &= \int_{{\mathbb{R}}^n}k(t,s,x,z)\Gamma(s,r)f(z)\,{\rm d}z= \Gamma(t,s)\Gamma(s,r)f(x) &=\Gamma(t,s)f(x) = \int_{{\mathbb{R}}^n}k(t,s,x,y)f(y)\,{\rm d}y.
\mathsf{S}_\ell^k:=\sum_{i=\ell}^k\alpha_i,\widehat x^k:=\frac{\sum_{i=0}^k\alpha_i x^i}{\mathsf{S}_0^k},\widehat x^k_\ell:=\frac{\sum_{i=\ell}^k\alpha_i x^i}{\mathsf{S}_\ell^k},
\left(z_{4}sz_{1}sz_{2}^{i}\right)^{-1}z_{2}\left(z_{4}sz_{1}sz_{2}^{i-1}\right) =\left[\left(z_{4}sz_{1}sz_{2}^{i}\right)^{-1},z_{2}\right] \mbox{or}z_{2}\left(z_{4}sz_{1}sz_{2}^{i}\right)^{-1}\left(z_{4}sz_{1}sz_{2}^{i-1}\right) =e.
(x-1) T_n(\{q_{ij}\},x,y){}={\rm Coef\/f}_{[t_1 \cdots t_n]}\! \left[\Bigg( \sum_{\ell_1,\ldots,\ell_n\atop \ell_i \in \{0,1 \}, \, \forall\, i}\prod_{1 \le i < j \le n}(q_{ij}(y-1)+1)^{\ell_i \ell_j}(y-1)^{- (\sum_{j}\ell_{j})}t_1^{\ell_{1}}\cdots t_n^{\ell_{n}}\!\Bigg)^{\!(x-1)(y-1)}\right] .
\frac{d\lambda}{d\tau}=\frac{\lambda[-R(\lambda)\bar{R'}(\lambda)e^{\lambda \tau}e^{\bar{\lambda}\tau}+Q(\lambda)\bar{Q'}(\lambda)-\tau |Q(\lambda)|^{2}]}{|R'(\lambda)e^{\lambda \tau}+ Q'(\lambda)-\tau Q(\lambda)|^{2}}.
g(t) =\frac{1}{\sqrt{2 \pi (\sigma^2+\sigma_x^2)}}\exp(-\frac{t^2}{2 (\sigma^2+ \sigma_x^2)}) w(t) =\frac{\mathcal{Q}(\frac{-A-t\frac{\tilde{\sigma}^2}{\sigma^2}}{\tilde{\sigma}}) - \mathcal{Q}(\frac{A-t\frac{\tilde{\sigma}^2}{\sigma^2}}{\tilde{\sigma}})}{D}.
d(x',P^{x,y}_\gamma(x')) \leq d(nx',x')+d\big(nx',P^{y,y}_\gamma\big((nu^{-1}n^{-1})\cdot nx'\big)\big) &\leq d(x,y)+cd(y,nx')^\kappa=d(x,y)+cd(nu^{-1}x',nx')^\kappa &\leq d(x,y)+cCe^{C_1d(y,x)}d(x,x')^\kappa.
\widetilde{K}_n(x,y) = \left( \frac{2}{k A_k}\right)^{\frac{\alpha+1}{k}}K_n\left( \left( \frac{2}{k A_k}\right)^{\frac{1}{k}}x, \left( \frac{2}{k A_k}\right)^{\frac{1}{k}}y \right).
\beta(k,p)=\frac{-i}{2\pi}\delta(p_{2}+k_{2})\delta(p_{3}+k_{3})\int\! dx_1\frac{e^{ik_1x_1}}{\sqrt{4\omega_p\omega_k}}\left\{i\omega_k f_p^{+\ast}(t'(0,x))e^{ip_1x'_1(0,x)}-\partial_t\left\{f_p^{+ \ast}(t'(t,x))e^{ip_1x'_1(t,x)}\right\}\left|_{t=0}\right.\right\}
\sum_{w\in V(G)\setminus V}|\psi_w(K)| \leq \sum_{w\in V(K)\setminus V}\frac{6n^2r}{k_{[r]}}+\sum_{w\in V(G)\setminus V:V(K)\cap V_w\neq \emptyset}\frac{12n^2r^2}{|V|k_{[r]}}&\leq r\cdot \frac{6n^2r}{k_{[r]}}+n\cdot |V(K)\cap V|\cdot \frac{12n^2r^2}{|V|k_{[r]}}&= \frac{6n^2r^2}{k_{[r]}}\left(1+\frac{2n|V(K)\cap V|}{|V|}\right),
t^{1-2H}\frac{\partial u_{2/k}}{\partial t}= 2^{1/k}H \sum_{n=0}^{\infty}\frac{(-1)^n 2^{n/k}x^n t^{-\frac{2}{k}Hn-\frac{2}{k}H-2H}}{n!\Gamma(-\frac{n}{k}-\frac{1}{k})}
\begin{array}{ll}V^{\pm}_{n+\frac 12, 2}= \partial_{x}(\Psi^{a,\pm}+S_{\theta_n}\Phi^{\pm}_n)V^{\pm}_{n+\frac 12,1}+\partial_{z}(\Psi^{a,\pm}+S_{\theta_n}\Phi^{\pm}_n)V^{\pm}_{n+\frac 12,3}begin{align*}2mm] +u^{a,\pm}\partial_{x}(S_{\theta_n}\Phi^{\pm}_n)+w^{a,\pm}\partial_{z}(S_{\theta_n}\Phi^{\pm}_n)\end{array}
Z_{q}(\beta) = \frac{1}{\left(\hbar\omega\right)^{DN}}\left(\frac{\Gamma\left(\frac{1}{1-q}+1 \right)}{(1-q)^{DN}\; \Gamma\left(\frac{1}{1-q}+ DN + 1\right)}\right) \frac{1}{\beta^{DN}}.
S=\frac{1}{2}E_{\theta}H_{\phi}^{*}=\frac{1}{2}\frac{\zeta_{0}I_{h}^{2}l^{2}k^{2}}{(4\pi r)^{2}}sin^{2}(\theta)=\frac{\zeta_{0}}{32}I_{0}^{2}(\frac{l}{\lambda})^{2}\frac{1}{r^{2}}sin^{2}(\theta).
|\mu(\tau_{n}=v | \tau_{m}=u)- \mu(\tau_{n}=v)| = |\mu(\tau_{n}=v | \tau_{m}=u)- \sum_{u'\in I_{\mu}}\mu(\tau_{n}=v | \tau_{m}=u') \mu(\tau_{m}=u')| = |\sum_{u'\in I_{\mu}}\left ( \mu(\tau_{n}=v | \tau_{m}=u)-(\tau_{n}=v | \tau_{m}=u') \right ) \mu(\tau_{m}=u')| \leq \sum_{u'\in I_{\mu}}\zeta^{n-m}\mu(\tau_{m}=u')= \zeta^{n-m},
(e\sigma e)((e-y(s))^*(e-y(s))) =\sigma(e(1-x(s))^*e(1-x(s))e)\le\sigma((1-x(s))^*(1-s(s))), (e\rho e)(y(s)y(s)^*) =\sigma(ex(s)ex(s)^*e)\le\sigma(x(s)x(s)^*),
\mathcal{A}(x,g)(u,u_m,\xi_g)=\operatorname{Ad}_{g^{-1}}\left(\overline{\mathcal{A}}^1(x)u+\overline{\mathcal{A}}^2(x)(u_m)+TR_{g^{-1}}\xi_g\right),x=(t,m)\in U=I\times V.
a(x) + 1 = \big|\{m<0:f(m,x)\ge 0\}\big| - \big|\{n\ge 0:f(n,x)<0\}\big| + 1 =\big|\{m<1:f(m,x)\ge 0\}\big| - \big|\{n\ge 1:f(n,x)<0\}\big| =\big|\{m<1:f(m-1,\sigma(x))\ge - n(x)\}\big| - \big|\{n\ge 1:f(n-1,\sigma(x))<- n(x)\}\big| =\big|\{m<0:f(m,\sigma(x))\ge - n(x)\}\big| - \big|\{n\ge 0:f(n,\sigma(x))<- n(x)\}\big|.
p(y|x^{v-1},y^{v-1}) =\sum_{y_v}p(y_v|x^{v-1},y^{v-1})p(y|x^{v-1},y^{v}) p(y_{v}|x^{v-1},y^{v-1},y) =\frac{p(y_v|x^{v-1},y^{v-1})p(y|x^{v-1},y^{v})}{\sum_{y_v}p(y_v|x^{v-1},y^{v-1})p(y|x^{v-1},y^{v})}.
\Delta|\nabla u|^2 =<\nabla \Delta u, \nabla u>+|\nabla^2u|^2 =\varepsilon^{-2}(e^u+r^2e^{-u})|\nabla u|^2-\varepsilon^{-2}e^{-u}<\nabla r^2,\nabla u>+|\nabla^2 u|^2.
\int_{B_d(x)}\frac{1}{|x-y|^{n-2}(1+|y|)^{1+\sigma}}\, dy\leq \frac{C}{d^{1+\sigma}}\int_{B_d(0)}|y|^{2-n}\, dy= \frac{C}{d^{1+\sigma}}d=\frac{C}{d^\sigma}.
F_{g , ( 0 , \cdots , 0 , 1 , 0 , \cdots , 0 )}( \lambda ) ={\frac{( 1 - 2^{1 - 2 g}) | B_{2 g}|}{( 2 g ) !}}d^{2 g - 2}( \lambda^{\frac{d}{2}}- \lambda^{-{\frac{d}{2}}}) ,
\sigma=(\begin{matrix}1&2&3&4&5\\ 3&4&5&2&1\end{matrix})=(\begin{matrix}1&3&5\end{matrix})(\begin{matrix}2&4\end{matrix})=(\begin{matrix}1&3\end{matrix})(\begin{matrix}3&5\end{matrix})(\begin{matrix}2&4\end{matrix}).
{\cal F}^{(N_{F})}(\Psi)=i\frac{(4-N_{F})}{8\pi}\Psi^{2}\log{\left(\frac{C\,\Psi^{2}}{\Lambda^{2}_{N_{F}}}\right)}-\frac{i}{\pi}\sum_{k=1}^{\infty}{\cal F}^{(N_{F})}_{2k}\left(\frac{\Lambda_{N_{F}}}{\Psi}\right)^{2k(4-N_{F})}\Psi^{2}
\frac{1}{2\pi\mathrm{i}n}\oint_{|z|=1}\frac{f'_{n}(z)}{f_{n}(z)}\mathrm{d}z\leq\frac{1}{2\pi n\epsilon}\int_{0}^{2\pi}\bigg(\log|f_{n}(\mathrm{e}^{\mathrm{i}(x-\mathrm{i}\epsilon)})|-\log|f_{n}(\mathrm{e}^{\mathrm{i}x})|\bigg)\mathrm{d}x.
\frac{d}{dt}\langle N,T\rangle = \langle N_{t},T\rangle+\langle N,T_{t}\rangle=0 \langle N_{t},T\rangle =-\langle N,T_{t}\rangle=\tau\kappa \langle N_{t},B\rangle =-\langle N,B_{t}\rangle = -F(s,t).
\max\{A(\phi(x_0), \nabla\phi(x_0), \nabla^2 \phi(x_0)), \nabla F (\nabla \phi(x_0) )\cdot\nu\}\ge 0 \u(x_0)>0 \max\{B(\phi(x_0), \nabla\phi(x_0), \nabla^2 \phi(x_0)), \nabla F (\nabla \phi(x_0) )\cdot\nu\}\ge 0 \u(x_0)<0 \max\{-\mathcal{Q}_\infty \phi(x_0), \nabla F (\nabla \phi(x_0) )\cdot\nu \}\ge0 \u(x_0)=0
Z_{jk}|T \sim N (\mu_{n,m,k,T},\sigma^2_{n,m,k,T}),\qquad \mu_{n,m,k,T}=\frac{\eta}{n}\sum_{i\in A_{jk}}\sum_{\ell=1}^{n/m}\psi_{jk}(T^{(i)}_{\ell})f_{0}(T^{(i)}_{\ell})-f_{0,jk}\lesssim 2^{j/2}, \sigma^2_{n,m,k,T}= (\frac{\eta}{n})^2\sum_{i\in A_{jk}}\sum_{\ell=1}^{n/m}\psi_{jk}^2(T^{(i)}_{\ell})\lesssim 2^j\eta/n.
{\mathcal{B}}_{\varepsilon}(u,v):=\varepsilon^2 \left \langle \nabla u,\nabla v\right\rangle_\Omega + \left\langle b u,v\right\rangle_{\Omega}=F(v):= \left\langle f,v\right\rangle_{\Omega}\forall v\in H_{0}^{1}\left(\Omega \right),
\lambda_1 \left(\frac{x+s}{2}\right)= \frac{1}{2}\lambda_1(x)+\frac{1}{2}\lambda_1(s), & \lambda_4 \left(\frac{x+s}{2}\right)\leq \frac{1}{2}\lambda_4(x)+\frac{1}{2}\lambda_4(s), \lambda_2 \left(\frac{x+s}{2}\right)\geq \frac{1}{2}\lambda_2(x)+\frac{1}{2}\lambda_2(s).
\mathbf{W}_{\mathrm{a},i}= \frac{1}{\sqrt{\zeta_{\mathrm{a},i}}}\cdot \left\{\hat{\mathbf{H}}_{i,\mathrm{E}}^{\mathrm{H}}\left( \hat{\mathbf{H}}_{i,\mathrm{E}}\hat{\mathbf{H}}_{i,\mathrm{E}}^{\mathrm{H}}\right)^{-1}\right\}_{[1:K]},
C_q(2N)-C_q(N) = B_q(2N)+B_q\left(\left\lfloor\frac{N}{2}\right\rfloor\right) + 2B_q\left(\left\lfloor\frac{N}{4}\right\rfloor\right) + + B_q\left(\left\lfloor\frac{2N}{|q|}\right\rfloor\right) +B_q\left(\left\lfloor\frac{N}{2|q|}\right\rfloor\right) +2B_q\left(\left\lfloor\frac{N}{4|q|}\right\rfloor\right).
\prod_{\substack{n\geq 1 n\not\equiv 0,\pm (k_0 +1)}}(1-q^n)^{-1}= \sum_{n_1,n_2,\ldots,n_{k/2}\geq 0}\frac{q^{N_{1}^{2}+N_{2}^{2}+...+ N^{2}_{k/2}+N_{i}+N_{i+1}+\ldots+N_{k/2}}}{(q)_{n_1}(q)_{n_2}\cdots (q)_{n_{k/2-1}}(q^2)_{n_{k/2}}}.
H_f(x) =\limsup_{r\to 0}H_f(x,r)=\limsup_{r\to 0}\frac{L_f(x,r)}{l_f(x,r)}&\leq \limsup_{r\to 0}\frac{\big(1+O(r)\big)^2L_f'(x,r)}{l_f(x,r)}+\limsup_{r\to 0}\frac{o(r)}{l_f(x,r)}&\leq \limsup_{r\to 0}\big(1+O(r)\big)^2H_f'(x,r)+\limsup_{r\to 0}\frac{o(r)}{\big(1-O(r)\big)^2\|Df(x)\|_sr-o(r)}&=H_f'(x).
- \frac{1}{2}\sum_{n_{1}}\, \log \frac{\cos [ \pi ( \varphi_{2}+ \tau_{1}( n_{1}+ \varphi_{1}) - i \tau_{2}\mid n_{1}+ \varphi_{1}\mid ) ]}{\cos [ \pi ( \tau_{1}n_{1}- i \tau_{2}\mid n_{1}\mid ) ]}\, .
\sum_{0<s\leq t}G'(X_{s-})\Delta X(s) =\sum_{0<s\leq t}G'(X_{s-})\rho(X_{s-},\Delta L_s)\mathbf{1}_D(s) &=\int\limits_0^t\int\limits_{\mathbb{R}}G'(X_{s-})\rho(X_{s-},y)\nu (dy,ds)
\sigma ( 1 - \sigma ) T_{n}^{\prime \prime}+ ( 1 / 2 - 3 / 2 \sigma ) T_{n}^{\prime}+ \biggl [ \frac{k^{2}}{4}\frac{1 - \sigma}{\sigma}- \frac{n^{2}}{1 - \sigma}\biggr ] T_{n}= 0 ,
\det(A) = \frac{1}{(n_1+n_2)!}\sum_{\sigma \in S_{n_1+n_2}}sign(\sigma) \det \left( [p_1(a_{\sigma(1)}), \ \cdots, \p_1(a_{\sigma(n_1)})] \right) \ \cdot \det \left( [p_2(a_{\sigma(n_1+1)}), \ \cdots, \p_2(a_{\sigma(n_1+n_2)})] \right)
U\equiv\langle E\rangle=\sum_{r}P_{r}E_{r}=\sum_{r}\frac{e^{-\beta E_{r}}E_{r}}{Z}=\sum_{r}\frac{-\frac{\partial}{\partial\beta}e^{-\beta E_{r}}}{Z}=\frac{-\frac{\partial}{\partial\beta}\sum_{r}e^{-\beta E_{r}}}{Z}=-\frac{\partial logZ}{\partial\beta}.
\left|\mathbb{E}\left(\widehat{\alpha}_{t,\varepsilon}\right)^n\right|\leq c\,\int_{E^n}\int_{\mathbb{R}^{n}}\left|\mathbb{E}e^{\iota \sum_{i=1}^{2n-1}u_i(X^H_{\ell_{i+1}}-X^H_{\ell_i})}\right|dpd\ell,
\frac{\delta^{2}W}{\delta x_{\mu}( s ) \delta x_{\mu}( s^{\prime})}\mathop{=}_{s \to s^{\prime}}\frac{1}{| s - s^{\prime}|^{4}}( C_{1}+ C_{2}) ( 1 - \dot{\phi}^{2}) + \frac{1}{| s - s^{\prime}|^{2}}\left( \frac{( C_{1}+ C_{2})}{1 2}\dot{\phi}\stackrel{{\, . . .}}{\phi}+ \frac{C_{1}}{4}\ddot{\phi}^{2}\right) + O ( \phi^{4}) \; .
\lbrack ( 2 , 3 , + ) + ( 2 , 3 , - ) ] \times [ ( 2 , 3 , + ) + ( 2 , 3 , - ) ] = 2 \left( ( 1 , 1 , + ) + ( 1 , 1 , - ) + ( 1 , 3 , + ) +{( 1 , 3 , - )}\right)
|x_1-x_2|+|x_2-x_3|+|x_3-x_1| \geq (|x_1-x_2|^2+|x_2-x_3|^2+|x_3-x_1|^2)^{1/2}&=\sqrt{3}(|x_1-\bar{x}|^2+|x_2-\bar{x}|^2+|x_3-\bar{x}|^2)^{1/2}.
S^{\mathrm{t o t a l}}= - \int d^{p + 1}\xi ( - \sigma \cdot \sigma )^{\frac{1}{2}}+ g \int d^{p + 1}\xi \sigma^{\mu_{0}\mu_{1}\ldots \mu_{p}}\psi_{\mu_{0}\mu_{1}\ldots \mu_{p}}+ \int d^{D}X{\cal L}.
x=x_1x_2x_3=\left(\begin{array}{cc}1-z_1z_2 &1\end{array}\right)\left(\begin{array}{cc}1 -z_2(1-z_1z_2)^{-1}&1\end{array}\right)\left(\begin{array}{cc}1 0 -z_1 1\end{array}\right).
S_{(\lambda,\mu),(\lambda',\mu')}= \frac{\sum_{w\in W}(-1)^{l(w)}\xi^{\langle 2\rho+\lambda,(w\bullet(\lambda',\mu'))_2\rangle + \langle\mu,(w\bullet(\lambda',\mu'))_1+2\rho\rangle}}{\sum_{w\in W}(-1)^{l(w)}\xi^{\langle 2\rho,w\bullet 0\rangle}},
\Xi X = \left( \begin{array}{cc}m b \bar{b}n \end{array}\right) \left( \begin{array}{cc}M B \bar{B}N \end{array}\right) = \left( \begin{array}{cc}m M + b \bar{B}m B + b N \bar{b}M + n \bar{B}\bar{b}B + n N \end{array}\right),
\lambda_1= -\widehat A+\sqrt{\widehat A^2 -\gamma \widehat Q}=-0.027158, \lambda_2= -\widehat A-\sqrt{\widehat A^2 -\gamma \widehat Q}=-1.472842, &\alpha= \sqrt{\widehat A^2-\gamma \widehat Q}=0.722842.
B_{n+1}(t,q^{\frac{1}{2}})=B_{n}(t,q^{\frac{1}{2}})\cdot (1-x y^{n})^{-1}, \ \D_{n+1}(t,q^{\frac{1}{2}})=D_{n}(t,q^{\frac{1}{2}}) \cdot \prod_{k=0}^{n-1}(1- (q^{\frac{1}{2}})^{1-n+2k}x y^{n})^{-1}.
p_D(t, x, y)\ge c_1^{-1}\left(1\wedge \frac{\delta_D(x)}{1\wedge t^{1/\alpha}}\right)^{\alpha/2}\left(1\wedge \frac{\delta_D(y)}{1\wedge t^{1/\alpha}}\right)^{\alpha/2}\cdot\begin{cases}\Psi^1_{c_2,\gamma, T}(t, 5|x-y|/4) \hbox{if}t\in(0,T], \Psi^2_{c_2, T}(t, |x-y|) \hbox{if}t\in[T,\infty)\end{cases}
\partial_{x^{2}}f^{p}\left( x_{0}\right) =\pm{\displaystyle\sum\limits_{j=0}^{p-1}}\frac{\widetilde{F}_{x^{2}}^{j}-2\widetilde{F}_{xy}^{j}\widetilde{\nu}_{j}+\widetilde{F}_{y^{2}}^{j}\widetilde{\nu}_{j}^{2}}{\widetilde{F}_{x}^{j}}\partial_{x}f^{j}.
C^{2}.f({h})= \frac{1}{2}(x-y) (f({h}-2)-f({h}+2)){B}= \frac{1}{2}(f(h-4)xB-f(h)xB-f(h)yB+f(h+4)yB) \equiv \frac{1}{4}\big((-{h}^{2}+2{h}+\mu)f({h}-4)-2(\mu-{h}^{2})f({h}) +(-{h}^{2}-2{h}+\mu)f({h}+4)\big)
f_j^{(n)}( \pmb{t}_{n-1}, \pmb{x}_{n-1},r,z,t,x) = f_n(t_1,\ldots,t_{j-1},r,t_{j},\dots, t_{n-1},x_1,\dots, x_{j-1}, z, x_{j},\ldots,x_{n-1},t,x) = G_{t-t_{n-1}}(x-x_{n-1})\times \dots \times G_{t_j-r}(x_j-z)G_{r-t_{j-1}}(z-x_{j-1}) & \times \dots \times G_{t_2-t_1}(x_2-x_1) \mathbf{1}_{T_{n-1}^{j}(t,r)}(\pmb{t}_{n-1}),
- i \bar{\kappa}\Gamma^{a}\kappa ={\textstyle \frac{3 2}{R_{5}}}\tilde{\mathcal{C}}[ \Gamma_{s}( \hat{M}^{\hat{a}\hat{b}}) k_{( \hat{a}\hat{b})}+ \Gamma_{s}( \tilde{M}^{\tilde{a}\tilde{b}}) k_{( \tilde{a}\tilde{b})}] .
(I^{Q^{-}}_j,k_1^{Q^{-}}(j),k_2^{Q^{-}}(j))=\begin{cases}(I^{Q}_j,k_1^{Q}(j),k_2^{Q}(j)),~~ (I^{Q}_{j+1},k_1^{Q}(j+1),k_2^{Q}(j+1)),~~.\end{cases}
P\left(D_k(T_{i,k'})\geq D_\ast,T_{i,k'}\leq t\right) \leq \sum_{j=\lfloor 8tn^2/D_\ast \rfloor}^{\lceil 16tn^2/D_\ast\rceil}P\left(D_k(T_{i,k'}) \geq D_\ast ,D_{k-j}(T_{i,k'})\leq \frac{D_\ast}{2}\right).
\liminf_{R\rightarrow+\infty}\left\{R^{-\frac{d-1}{2}}\left\Vert\mathcal{D}\left( R\Omega-x\right) \right\Vert_{L^{p}\left( \mathbb{T}^{d}\right)}\right\}\leqslant\liminf_{R\rightarrow+\infty}\left\{R^{-\frac{d-1}{2}}\left\{\sum_{n\neq0}\left\vert R^{d}\widehat{\chi_{\Omega}}\left( Rn\right)\right\vert^{q}\right\}^{1/q}\right\}=0.
\varepsilon(E_{a}\triangleright(\mathsf{M}_{m}^{n})_{j}^{i}) =-\pi(E_{a}K_{a}^{-1})_{m}^{i}\pi(K_{a})_{j}^{n}+\pi(1)_{m}^{i}\pi(E_{a})_{j}^{n}=-\pi(E_{a})_{m}^{i}\pi(1)_{j}^{n}+\pi(1)_{m}^{i}\pi(E_{a})_{j}^{n},
U^{a b}= \left( \delta^{a b}\partial + f^{a b c}J_{1}^{c}\right) \sigma_{1}= \left( \begin{array}{c c}{0}&{\delta^{a b}\partial + f^{a b c}J_{1}^{c}}\\{\noalign{\vspace{6 p t}}\delta^{a b}\partial + f^{a b c}J_{1}^{c}}&{0}\\ \end{array}\right) .
\{A_{0}^{a}( x ) , A_{j}^{b}( y ) \}_{D B}= \frac{1}{m^{2}}\partial_{j}^{x}\delta^{a b}\delta ( x - y ) - \frac{g}{m^{2}}f^{a b c}A_{j}^{c}( x ) \delta ( x - y ) \nonumber
T \lrcorner D\omega = \frac{2\omega_{2n-2}}{\omega_{2n-3}}\Big( ((2n-3)c_1 -2(n-2)c_0) \beta_{2,0}+ (n-2)(3c_0-2c_1) \gamma_{2,0}+ (n-1)c_1 \gamma_{2,1}\Big)
{\cal M}= \left( \begin{matrix}{{0}}&{{m}}&{{2 m}}&{{3 m}}&{{\cdots}}\\{{m}}&{{1 / R}}&{{0}}&{{0}}&{{\cdots}}\\{{2 m}}&{{0}}&{{2 / R}}&{{0}}&{{\cdots}}\\{{\vdots}}\\ \end{matrix}\right) ,
\|C_{\mathrm{ext}}(t) - I \|^2 = \ \sup_{\|(x_n)\|=1}\|C_{\mathrm{ext}}(t) (x_n) - (x_n)\|^2 = \ \sup_{\|(x_n)\|=1}\sum_{n \in{\mathbb N}}\|C(nt) x_n- x_n\|^2 \leq \sup_{\|(x_n)\|=1}\sum_{n \in{\mathbb N}}r^2 \|x_n\|^2 = r^2,
{}_3 \langle \tilde I | V_3 \rangle_{123}={}_3 \langle 0| c^3_{-1}c^3_0 \exp\left[c^1 C b^2 + c^2 C b^1+ (c^1 - c^2) \Omega (b^1_0 - b^2_0) + (c^1 - c^2) \Gamma b^3_0 \right] (c_0 c_1 |0 \rangle)_{123}
W_{f i}[ K ] = \frac{1}{\sqrt{2 \pi \hbar}}\exp \left\{- \frac{i}{\hbar}\left( p_{f}q_{i}+ \sum_{j = 1}^{N}\epsilon \, H ( p_{f}- \epsilon K_{j - 1}- \sum_{l = 1}^{j}\epsilon \, J_{l}) \right) \right\}\, .