harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\bigl\Vert T_\psi^{A_0,A_1,A_2}\colon \mathcal S^2_n\times\mathcal S^2_n\to \mathcal S^1_n \bigr\Vert\,=\,\bigl\Vert \{\psi(\lambda_i^{(0)}, \lambda_k^{(1)},\lambda_j^{(2)})\}_{i,j,k=1}^n\colon \mathcal S^2_n\times\mathcal S^2_n\to \mathcal S^1_n \bigr\Vert.
V^{ab}{}_3=\left[\begin{array}{ccc}-\partial_1\partial_2f \frac12\big((\partial_1)^2-(\partial_2)^2\big)f 0 \frac12\big((\partial_1)^2-(\partial_2)^2\big)f \partial_1\partial_2f 0 0 0 0\end{array}\right]
{\bf \Phi}_{1}^{( n_{\rho}, m )}={\bf \Phi}_{1}^{( n_{\rho}, m )}( \rho , k ) = \left( \begin{array}{c}{\phi_{1}^{( n_{\rho}, m )}( \rho , k )}{\phi_{2}^{( n_{\rho}, m )}( \rho , k )}\end{array}\right)
\int\limits_{\psi_0}^{\pi/2}Cf(0,e_n,\psi)g(\psi_0,\psi)d\psi &= \int\limits_{\psi_0}^{\pi/2}\int\limits_0^{\infty}\int\limits_{S^{n-2}}f(\rho (\sin\psi) \omega, \rho \cos \psi)(\rho \sin\psi)^{n-2}g(\psi_0,\psi)d\omega d\rho d\psi.
\langle ( g_{1}\circ A_{1}) \cdots ( g_{n - 1}\circ A_{n - 1}) ( g_{n}\circ \Phi ) \rangle = \langle ( T \circ g_{1}\circ A_{1}) \cdots ( T \circ g_{n - 1}\circ A_{n - 1}) ( T \circ g_{n}\circ \Phi ) \rangle ,
\left[u+v'\right](f_1+g'_1,\dots,f_n+g'_n) = u(f_1+g'_1,\dots,f_n+g'_n) + v'(f_1+g'_1,\dots,f_n+g'_n) = u(f_1,\dots,f_n) + [u(g_1,\dots,g_n)]'+ [v(f_1,\dots,f_n)]'
\int_{\mathbf{B}_m}\int_{\mathbf{B}_m}\Phi_{0}(x, y)\,dx\,dy=\int_{\mathbf{B}_m}N(1)(y)\,dy=\int_{\mathbf{B}_m}\, \left( \frac{d^{2}}{8}- \frac{\left\vert y \right\vert^{2}}{6}\right) \, dy=\frac{\pi d^5}{60}.
c_b^{-2}\Vert u_{\mathrm{nc}}\Vert_{L^2(T)}^2 \le \Vert u_{\mathrm{nc}}\Vert_{b_T}^2 = (u_{\mathrm{nc}},u)_{b_T}-( u_{\mathrm{nc}},u-u_{\mathrm{nc}})_{b_T}.
E =_{\mathcal{H}^{n}}\left( F_{\ell}\cap \{z <a \}\right) \cup \left[ \widetilde{E}\cap \left( \{z< b \}\backslash \{z <a \}\right)\right] \cup \left[ \lambda ( r_{\ell}(b) - r_{\ell}(a) )e + \left( F_{\ell}\backslash \{z<b \}\right)\right].
S_{I,\varepsilon,\ell}(T):=\sum_{(n,\alpha)\in \Delta\cap \mathbb N^{r+1}}\sum_{\begin{smallmatrix}\beta\in\mathbb N^m \theta'(\beta,\alpha)\end{smallmatrix}}\sum_{\begin{smallmatrix}(k_j)_{j\in I}\in \mathbb N_{>0}^I (\ref{eq3.12})_{n,\beta}\end{smallmatrix}}\L^{-\sum_{j\in I}k_j\nu_j}\L^{-\varepsilon(n,\alpha)}T^{\ell(n,\alpha)}.
I_0(2nx) =\frac{1}{\pi}\int^{1}_{-1}{e^{-2ntx}\cdot \frac{1}{\sqrt{1-t^2}}dt}e^{-2nx}\cdot I_0(2nx) =\frac{1}{\pi}\int^{1}_{-1}{e^{-2nx(1+t)}\cdot \frac{1}{\sqrt{1-t^2}}dt},
-\otimes\mathfrak{S}_n:\mathbf{Mod}^{\phi,1}_{/\mathfrak{S}}\to \mathbf{Mod}^{\phi,1}_{/\mathfrak{S}_n}\mathfrak{M}\mapsto \mathfrak{M}\otimes_{\mathfrak{S}}\mathfrak{S}_n=:\mathfrak{M}_{\mathfrak{S}_n}.
W_a = \sum_{1\in \{i,j\}}U_R(x_i-x_j) \prod_{k\ne i,j}\theta_{2R}(x_j-x_k), W_b = \sum_{i,j \ge 2}U_R(x_i-x_j) \prod_{k\ne i,j}\theta_{2R}(x_j-x_k).
T_{+ p \pm 2}^{- \dot{q}}= - \frac{\chi}{2}\tilde{\gamma}_{\dot{q}p}^{i}\Omega_{\pm 2}^{- 2 i}+ \frac 1 2 \tilde{\gamma}_{\dot{q}q}^{i}\Omega_{+ p}^{- 2 i}\psi_{\pm 2 q}^{2 +},
{\cal E}= \frac 1{g_{Y M}^{2}}\int d^{2}x \ \left[ \e^{- a | T |^{2}}| D_{\alpha}T |^{2}+ \frac 1 4 e^{- b | T |^{2}}( 1 + c | T |^{2}) F_{\alpha \beta}F_{\alpha \beta}+ V ( T ) \ \right] ,
\begin{array}{l}\overset{\centerdot}{\mathop{SOC}}\,(t)=\frac{{{d}_{3,v}}}{W_{bat}^{MAX}}{{\left( P_{bat,nom}^{v}\right)}^{2}}\ \ \ -\left[ \ln \left({{d}_{2,v}}+{{d}_{1,v}}SOC(t) \right)+2{{d}_{3,v}}P_{bat,nom}^{v}\right. \ \ \ \left. +{{d}_{4,v}}\right]\frac{{{P}_{bat}}}{W_{bat}^{MAX}}\end{array}
|\hat{\mathbb{E}}_{\tau+}[X]-\hat{\mathbb{E}}_{\sigma+}[X]| \leq \sum_{i=1}^{n}\sum_{j=1}^{m}C(\sup_{(u_1,u_2)\in \Lambda_{|t_i-s_j|,T}}(|B_{u_2}-B_{u_1}|\wedge 1)+\sqrt{|t_i-s_j|})I_{\{\tau=t_i\}\cap \{\sigma=s_j\}}&\leq C(\sup_{(u_1,u_2)\in \Lambda_{\delta,T}}(|B_{u_2}-B_{u_1}|\wedge 1)+\sqrt{\delta}).
{\cal W}= \sum_{I = 1}^{N - R}\left( \sum_{b = 1}^{N^{\prime}/ 2}q_{I}^{b}T^{I}H_{b}\tilde{H}_{b}+ \frac{1}{2}\sum_{c = N^{\prime}+ 1}^{N^{\prime}+ N^{\prime \prime}}q_{I}^{c}T^{I}H_{c}H_{c}\right) ,
Q_{\mathrm{g r a v}}^{2}= R^{2}+{\frac{1}{1 2}}\left( F_{\mu \nu}F^{\mu \nu}\right)_{T^{2}}+{\frac{5}{3 6 N_{V}}}\left( F_{\mu \nu}F^{\mu \nu}\right)_{G_{( N_{V})}}~ .
\widetilde{\delta}(t) =1-e^{-t}, \sigma^2(t) =2\int_{0}^{t}\int_{0}^{x}e^{-x-y}(x^{2H}+y^{2H}-(x-y)^{2H})dydx &=\Gamma(2H+1,t)(1-2e^{-t})+e^{-2t}\Gamma^*(2H+1,t),
\langle \hat{\phi}^{2}\rangle ={\frac{b ( m )}{1 6 \pi^{2}}}, \langle \hat{\bar{\psi}}\hat{\psi}\rangle = 4 m \langle \hat{\phi}^{2}\rangle , \langle \hat{V}_{\mu}\hat{V}^{\mu}\rangle = 3 \langle \hat{\phi}^{2}\rangle .
\displaystyle \int_{D}d^{3}\xi \frac{\partial}{\partial \xi_{j}}\Bigl ( \beta ( \xi_{i}) \alpha_{j}( \xi_{i}) \Bigr ) = \int_{\partial D}d S_{j}\Bigl ( \beta ( \xi_{i}) \alpha_{j}( \xi_{i}) \Bigr ) = 0 ,
\|v_n - c_n (g) v_n u_{g^{-1}}\|^2_2 =\sum_h \| v^n_{hg^{-1}}- c_n (g) v^n_h \alpha(h,g^{-1})\|^2_2 \geq \sum_h \left (\| v^n_{hg^{-1}}\|_2 - \|c_n (g) v^n_h \alpha(h,g^{-1})\|_2\right )^2 = \sum_h \left (\| v^n_{hg^{-1}}\|_2 - \|v^n_h \|_2\right )^2.
\left[\frac{E(xb,x^{60})E(x^{11}b,x^{60})E(x^{31}/b,x^{60})E(x^{41}/b,x^{60})}{E(x/b,x^{60})E(x^{11}/b,x^{60})E(x^{31}b,x^{60})E(x^{41}b,x^{60})}\right]^N=b^{2N}.
\int_{\alpha}^{\zeta}{\left\vert{\tau^{2}-\alpha^{2}}\right\vert^{1/2}d\tau}=\int_{\sigma}^{x}{\left({\frac{\left\vert{t^{2}-\sigma^{2}}\right\vert}{1-t^{2}}}\right)^{1/2}dt},
\mathbb{P}\big( \forall x\in\mathbb{X}, \, \#(x,\mathbb{X}) < \infty \big) = 1 \; \Longrightarrow \; \mathbb{P}\big( \forall x\in\mathbb{X}, \, \#(x,\mathbb{X}) < \infty \big) = 1 ~.
\delta S_{lin}= \int d^6x\,\left(\frac1{12}H^*_{\mu\nu\rho}-\frac23 h^{+}_{\mu\nu\rho}\right)\delta H^{\mu\nu\rho}+\frac1{2\sqrt{u^2}}\epsilon^{\mu\nu\rho\sigma\tau\phi}H^-_{\mu\nu}H^-_{\rho\sigma}v_\tau\partial_\phi\delta a \ .
\|\tilde a\|_{l_2^d}^2 =\sum\limits_{i=1}^d\vert\tilde a_i\vert^2=4\sum\limits_{i=1}^d\vert g'(\xi_{h,i})a_i\vert^2 &\geq 4\sum\limits_{i=1}^d\left( g'(1) -c_1(h+h^2/4)\right)^2a_i^2 &=4\left(g'(1) -c_1(h+h^2/4)\right)^2.
\theta_{\Lambda, P}(\tau) = \sum_{X \in \Lambda}P(X) e^{\pi i \tau (X \cdot X)}&= \sum_{x\in \Lambda_1, y\in \Lambda_2}P_1(x)P_2(y) e^{\pi i \tau(x\cdot x)}e^{\pi i \tau (y\cdot y)}=\theta_{\Lambda_1, P_1}(\tau) \theta_{\Lambda_2, P_2}(\tau),
\frac{E_{\mathrm{c a s}}}{A}= \int d k_{x}d k_{y}\left\{\sum_{n}\sqrt{k_{x}^{2}+ k_{y}^{2}+ \left( \frac{n \pi}{a}\right)^{2}}- \frac{a}{\pi}\int d k_{z}\sqrt{k_{x}^{2}+ k_{y}^{2}+ k_{z}^{2}}\right\}
\mathcal{A}z:=(Au,0)~{\rm for}~z=(u,w)\in D(\mathcal{A}):=D(A)\times B^m, &F:z\in Z\mapsto(f(z),g(z))\in Z, &S:t\in[0,\infty)\mapsto(s(t),0)=\left(s_i(t)-A_i(A_i+A_e)^{-1}(s_i(t)+s_e(t)),0\right)\in Z, &\mathcal{F}(t,z)=S(t)-F(z).
A_t\cdot r\nabla \theta =\frac{1}{r}\Phi_t, (B\times u)\cdot r\nabla \theta =\frac{1}{r}u\cdot \nabla \Phi, \nabla \times B\cdot r\nabla \theta =-\frac{1}{r}\left(\Delta-\frac{2}{r}\partial_r \right)\Phi.
\prod_{i=2}^n \psi_i^{d_i}\left[ D_{1,n+1,n+2}\pi^*\psi_1^{d_1}(\psi_0+\psi_1+\psi_{n+1}+\psi_{n+2})\right. & \left. +D_{1,n+1,n+2}(D_{1,n+1}+D_{1,n+2}+D_{n+1,n+2})\pi^*\psi_1^{d_1}\right].
\int_{\mathbb{R}^{d}}u(t, x) \psi(x) d x= \int_{\mathbb{R}^{d}}\phi(x) \psi(x) d x+\frac{1}{2}\int_{t}^{T}\int_{\mathbb{R}^{d}}u(s, x) \Delta \psi(x) d x d s &+\int_{t}^{T}\int_{\mathbb{R}^{d}}u(s, x) \psi(x){W}(ds, x)dx,
\Delta_n^{1-\frac m2}\langle SAMPV_t^n(m_1,...,m_k), \bigotimes_{i=1}^k\bigotimes_{j=1}^{m_j}e_{p_{i,j}}\rangle_{\mathcal H^m}&\stackrel{u.c.p.}{\longrightarrow}\langle \int_0^t \rho_{\Sigma_s}^{\otimes k}(m_1,...,m_k)ds,\bigotimes_{i=1}^k\bigotimes_{j=1}^{m_j}e_{p_{i,j}}\rangle_{\mathcal H^m},
\frac{\mathrm{d}}{\mathrm{d}t}\int_{\Gamma}1 \, \mathrm{d}\mathcal{H}^d= - \int_{\Gamma}HV \, \mathrm{d}\mathcal{H}^d= - \int_{\Gamma}g(c)H^2 \, \mathrm{d}\mathcal{H}^d.
t^2 =\frac{\displaystyle\sum_{i=1}^{d}\|v_i\|_{\lambda_i}^2}{\displaystyle \sum_{{\substack{i,j=1 j\neq i}}}^{d}|v_iv_j|_2^2}= \frac{\displaystyle \sum_{i=1}^{d}\|u_i\|_{\lambda_i}^2}{\displaystyle \left(\frac{d-2}{d-1}-\frac{d-3}{d-2}\right)|u|_4^4 + \sum_{{\substack{i,j=1 j\neq i}}}^{d}|u_iu_j|_2^2}
\int_s^t \int_s^u \Phi_{n-1}(u-r) f(r) dr du = \int_s^{t}\int_s^{v_n}\int_s^{v_{n-1}}\cdots \int_s^{v_2}\prod_{i=2}^{n}\Phi(v_{i}-v_{i-1}) f({v_1}) dv_1 \cdots dv_{n}.
Q_s^n\varphi(v)= \frac{1}{\kappa(s)^n r_{s}(v)}P_s^n(\varphi r_{s})(v)= \int \varphi(G_n \!\cdot\! v) q_{n}^{s}(v, G_n) \mu(dg_1)\ldots \mu(dg_n),
\frac{\left| z\left(\left(\frac{1}{2}-\sigma\right) + j\omega\right)\right|}{\left|z\left(\left(\frac{1}{2}+\sigma\right) - j\omega \right)\right|}= \frac{\left|\left(\frac{3}{2}-\sigma\right) + j\omega\right|}{\left|\left(\frac{3}{2}+\sigma\right) - j\omega\right|}
f_{n-6}= f_{n-7}+A_{2-n}A_{4-n}A_{6-n}\psi_{n-6}^{(0)}+K_{6-n}\psi_{n-6}^{(0)}= O(r^{n-5})+\Big(K_{6-n}\beta_{n-6}^{(0)}\Big) \log r := b_{n-5}+O(r^{n-4}) \log r+O(r^{n-4}).
\Sigma_{T}= 3 2 g^{2}\int{\frac{d^{3}k}{( 2 \pi )^{3}}}{\frac{\sin^{2}{\frac{\tilde{p}\cdot k}{2}}}{k}}\left( n_{B}( k ) + n_{F}( k ) \right) + 4 g^{2}P^{2}\bar{\Sigma}\, ,
\sigma=-pI+\frac{1}{J}[\frac{2}{J^{2/3}}(C_{1}+\overline{I}_{1}C_{2})B-\frac{2}{J^{4/3}}C_{2}B\cdot B-\frac{2}{3}(C_{1}\overline{I}_{1}+2C_{2}\overline{I}_{2})I].
|F(y)-F(x)| = \left|\sum_{j,i}a_{j,i}(\phi_{j,i}(y)-\phi_{j,i}(x))\right| = \left|\sum_{j,i}(a_{j,i}-a_{\ell,m})(\phi_{j,i}(y)-\phi_{j,i}(x))\right| \leq \sum_{\substack{j,i\, \textrm{s.t.}2B_{j,i}\cap B_{\ell,m}\ne\emptyset}}|a_{j,i}-a_{\ell,m}|\frac{C}{r_{j,i}}d(y,x).
(\theta\otimes\rho)\bigl((R\otimes R)\Delta(S(a))\bigr) =\left(\rho RS\,\otimes\,\theta RS\right)(\Delta a)=\bigl(\rho\circ\tau_{-\frac{i}{2}}\,\otimes\,\theta\circ\tau_{-\frac{i}{2}}\bigr)(\Delta a) &=(\rho\otimes\theta)\Delta\bigl(\tau_{-\frac{i}{2}}(a)\bigr) =(\rho\otimes\theta)\bigl(\Delta(R(S(a)))\bigr).
\begin{array}[c]{ll}T_{k}^{\operatorname*{low}}:=T_{k}L_{\Gamma}, T_{k}^{\operatorname*{high}}:=T_{k}H_{\Gamma}, b_{k}^{\operatorname*{low}}\left( \cdot,\cdot\right) :=b_{k}\left(\cdot,L_{\Gamma}\cdot\right) , b_{k}^{\operatorname*{high}}\left(\cdot,\cdot\right) :=b_{k}\left( \cdot,H_{\Gamma}\cdot\right) .\end{array}
(\mathsf{N}_{m}^{n})_{j}^{i}\triangleleft X =(u_{m}^{i}\triangleleft X_{(1)})(u_{n}^{j*}\triangleleft X_{(2)})= \sum_{k,\ell}\pi(X_{(1)})_{k}^{i}u_{m}^{k}\pi(S(X_{(2)}))_{j}^{\ell}u_{n}^{\ell*}=\sum_{k,\ell}\pi(X_{(1)})_{k}^{i}(\mathsf{N}_{m}^{n})_{\ell}^{k}\pi(S(X_{(2)}))_{j}^{\ell}.
\aligned|\nabla u (\phi) | = |\int_D u \nabla \phi dvol|= |\lim_{s\to\infty}\int_D \alpha_s(u) \nabla\phi dvol| = |\lim_{s\to\infty}\int_D \alpha_s'(u) \nabla u\phi dvol| & \leq \limsup_{s\to \infty}\|\alpha'_s(u) \nabla u\|_{L^{p}(D)}\|\phi\|_{L^\lambda}\leq C \|\phi\|_{L^\lambda}.\endaligned
{\cal M}\displaystyle \int_{- \infty}^{\infty}\displaystyle \frac{d x}{2 \pi}\displaystyle \frac{1}{2}e^{- x}\,{\cal Q}( x ) \doteq \displaystyle \frac{1}{L}\displaystyle \int_{- \infty}^{\infty}\displaystyle \frac{d x}{2 \pi}e^{- x}\,{\cal Q}_{-}( x ) \, .
(\lambda+\partial_t-\Delta)^{-\frac{\alpha}{2}}h(t,x):=\int_{-\infty}^t\int_{\mathbb R^d}e^{-\lambda(t-s)}\frac{1}{(4\pi (t-s))^{\frac{d}{2}}}\frac{1}{(t-s)^{\frac{2-\alpha}{2}}}e^{-\frac{|x-y|^2}{4(t-s)}}h(s,y)dsdy,
j^{+}= \left( \begin{array}{c c c}{0}&{- 1}&{0}\\{0}&{0}&{0}\\{- 1}&{0}&{0}\\ \end{array}\right) j^{-}= \left( \begin{array}{c c c}{0}&{0}&{- 1}\\{1}&{0}&{0}\\{0}&{0}&{0}\\ \end{array}\right)
\bigg[ \prod_{I=1}^N: e^{- i 2 \sqrt{\pi}\omega{(I)}U_{I a}\Phi^{(I)}(x)}:_{M^{(I)}}\; e^{+ i \frac{\theta}{N}}\; + \; \prod_{I=1}^N: e^{+ i 2 \sqrt{\pi}\omega{(I)}U_{I a}\Phi^{(I)}(x)}:_{M^{(I)}}\; e^{- i \frac{\theta}{N}}\bigg] \Bigg\rangle_{K^\mu}\; ,
{p_{B,j,i}^{(1)}}= \frac{{x_{n,j}^{(\alpha )}+ 1}}{2}\sum\limits_{k = 0}^N{\;{\varpi_{N,k}^{(0.5)}}\,{\cal L}_{B,n,i}^{(\alpha )}\left({{x_{N,k}^{(0.5)}}; - 1,x_{n,j}^{(\alpha )}}\right)},i,j = 0, \ldots ,n.
(z-w)^{-4}-{1\over 2}\sum_{n=0}^\infty\left({-{1\over 2}\atop n+1}\right){n+1\over n+2}w^{-n-2}(z-w)^{n-2}={1\over 2}{z+w\over\sqrt{zw}}(z-w)^{-4}\,,
r\theta=r\frac{sin\theta}{cos\theta}-\frac{r}{3}\frac{sin^{3}\theta}{cos^{3}\theta}+\frac{r}{5}\frac{sin^{5}\theta}{cos^{5}\theta}-\frac{r}{7}\frac{sin^{7}\theta}{cos^{7}\theta}+\cdot\cdot\cdot
\mu^{2}= - \frac{1}{2}\left( \frac{\alpha^{2}- A^{2}}{l^{4}}+ \frac{1}{L^{2}}- \sqrt{{\left( \frac{\alpha^{2}+ A^{2}}{l^{4}}+ \frac{1}{L^{2}}\right)^{2}- \frac{4 A^{2}\alpha^{2}}{l^{4}}}}\right)
\begin{aligned}\lim_{n\rightarrow \infty}\Psi(n) \leq \|\varphi'\|_\infty^2\lim_{n\rightarrow \infty}\int_W \left(1_{G_n}(y)-1_G(y)\right)^2 d\ss(y) =\|\varphi'\|_\infty^2 \int_W\lim_{n\rightarrow \infty}\left(1_{G_n}(y)-1_G(y)\right)^2 d\ss(y) &=0. \end{aligned}
\mathbb{E}\left(\exp\left(- \gamma(\phi) \int_{\epsilon t}^{(1-\epsilon)t}\frac{\sqrt{t}\tilde{Z}_s^{\epsilon}}{(t-s)^{\frac{3}{2}}}ds \right)\right)=\mathbb{E}\left(\exp\left(-\gamma(\phi)\int_{\epsilon}^{1-\epsilon}\frac{\tilde{Z}^{\epsilon}_{\lambda t}}{(1-\lambda)^{\frac{3}{2}}}d\lambda\right) \right).
w_k(\sigma) = \frac{|X_\sigma(k-t-1)|}{{k+1 \choose t+1}\cdot |X(k)|}\leq \frac{{d-t \choose k-t}\cdot Q\cdot |X_\sigma(d-t-1)|}{{k+1 \choose t+1}\cdot |X(d)|}={d+1 \choose k+1}\cdot Q \cdot w(\sigma),
\sum_{m_1=0}^{rp-1}\sum_{m_2=0}^{sp-1}\sum_{m_3=0}^{tp-1}\binom{m_1+m_2+m_3}{m_1,m_2,m_3}\equiv_{p^3}\sum_{m_1=0}^{r-1}\sum_{m_2=0}^{s-1}\sum_{m_3=0}^{t-1}\binom{m_1+m_2+m_3}{m_1,m_2,m_3}.
M(t') = \int_0^{t'}\big[(\nabla \omega_\eta)^T\sigma\big](x(s))\, dw(s)\,, \mbox{and}\,\overline{M}(t') = \int_0^{t'}\big[(\nabla \omega_\eta)^T\sigma\big](y(s))\, d\bar{w}(s)\,.
\|y^{k+1}-z\|^2 \leq\|w^k-z\|^2-\|w^k-y^{k+1}\|^2 &=\|x^k-\lambda_k g(y^k)-z\|^2-\|x^k-\lambda_k g(y^k)-y^{k+1}\|^2 &=\|x^k-z\|^2-\|x^k-y^{k+1}\|^2+2\lambda_k\langle z-y^{k+1}, g(y^k)\rangle &=\|x^k-z\|^2-\|x^k-y^{k+1}\|^2+2\lambda_k\langle g(y^k), y^k-y^{k+1}\rangle-2\lambda_k\langle g(y^k), y^k-z\rangle.
V(r) = \frac{L(L+1)}{r^2}- \frac{Q}{r}f + \kappa B_1 \frac{r}{f}+ \kappa B_2 \frac{1}{f^2}+ \kappa B_3 \frac{r}{f^3}+ \kappa B_4 \frac{1}{f^4}+ \kappa B_5 \frac{r}{f^5}+ \kappa B_6 \frac{1}{f^6}.
S = \frac{1}{8 \pi G_{N}}\int_{\partial M}\Bigl [ ( K - K_{0}) +{\textstyle{\frac{1}{4}}}n_{\mu}\Bigl ( \sqrt{- g}e^{- \gamma \Phi}A_{\nu}^{\alpha}( L M L )_{\alpha \beta}F^{\beta \mu \nu}\Bigr ) \Bigr ] \quad .
\operatorname*{E}(X) =\frac{1}{\ln(2)}{\displaystyle\int\limits_{0}^{1}}\frac{x}{1+x}dx=\frac{1}{\ln(2)}-1=0.4426950408... & =\frac{1}{\ln(2)}{\displaystyle\int\limits_{0}^{1}}\frac{\{1/x\}}{1+x}dx=\operatorname*{E}\left\{\frac{1}{X}\right\}.
\begin{aligned}-2\varepsilon \mbox{Im}[(r\partial_ru)\overline{u}]+2 \varepsilon \lambda^{\frac12}r|u|^2+\varepsilon \lambda^{-\frac12}(r|\nabla u|^2-\lambda r|u|^2)= \varepsilon \lambda^{-\frac12}r|\nabla v_\lambda|^2.\end{aligned}
g(x)=f(x)\chi_{\Omega^c}+\sum_j\frac{1}{\mu(Q_{j})}\int_{Q_{j}}f(x_1,y_2)\,d\mu(y_2)\,\chi_{S_j}(x), &b(x)=\sum_{j}b_{j}(x)=\sum_{j}\left(f(x)-\frac{1}{\mu(Q_{j})}\int_{Q_{j}}f(x_1,y_2)\,d\mu(y_2)\right)\chi_{S_j}(x),
\ln \mathrm{d e t}_{{\mathrm{\scriptsize{Q E D}}_{3}}}= \frac{L}{2 \pi}\int_{m^{2}}^{\infty}\frac{d M^{2}}{\sqrt{M^{2}- m^{2}}}\ \ln \mathrm{d e t}_{{\mathrm{\scriptsize{Q E D}}_{2}}}( M^{2}) ,
a= -1+\alpha_N^0\beta_N^0 +a_1-\beta_N^0 a_3, c= -1+ \alpha_N^0\beta_N^0 +a_4-\alpha_N^0a_2 b= 1+ (1-a_1)(1-a_4) -\alpha_N^0\beta_N^0- (\alpha_N^0-a_3)(\beta_N^0 -a_2)\, .
S(AA')_{\hat{\gamma}_{AA'}^{(n)}}= 2\,g\left(n-\frac{1}{2}\right) = \ln n^2 + 2 + \mathcal{O}\left(\frac{1}{n^2}\right)\;, S(A')_{\hat{\gamma}_{AA'}^{(n)}}= g\left(\frac{n^2}{a}-\frac{1}{2}\right) = \ln\frac{n^2}{a}+ 1 + \mathcal{O}\left(\frac{1}{n^4}\right)\;,
T_{L}\ = \ - 3 \, \omega \, e^{- b_{L}d}\coth \left[ \omega ( x_{s}\! - \! L ) \right] \left( \frac{\tanh \left[ \, \frac{\omega}{2}( x_{s}\! - \! L ) \, \right]}{\tanh \left( \frac{\omega x_{s}}{2}\right)}\right)^{\cosh^{- 1}\left[ \omega ( x_{s}\! - \! L ) \right]}\ .
J[\psi^{(2)}]_{T,t}=\hbox{\vtop{\offinterlineskip\halign{\hfil#\hfil\cr{\rm l.i.m.}\cr$\stackrel{}{{}_{p_1,p_2\to \infty}}$\cr}}}\sum_{j_1=0}^{p_1}\sum_{j_2=0}^{p_2}C_{j_2j_1}\Biggl(\zeta_{j_1}^{(i_1)}\zeta_{j_2}^{(i_2)}-{\bf 1}_{\{i_1=i_2\ne 0\}}{\bf 1}_{\{j_1=j_2\}}\Biggr),
\zeta_{p,\infty}(s,c):=\sum_{m,n\neq0}\frac{p^{2}+\lambda_{n}^{2}}{p(p+\frac{1}{\pi})+\lambda_{n}^{2}}\cdot\frac{1}{\left(\lambda_{m}^{2}+c\,n^{2}\right)^{s}},\,\,\,\,(s)>1.
P\{\tilde{T}_1 < \cdots < \tilde{T}_g\}= \int_0^{\infty}\cdots \int_0^{t_3}\int_0^{t_2}f_g(t_g) \cdots f_2(t_2) f_1(t_1) \, dt_1 dt_2 \cdots dt_g & = \int_0^{\infty}\cdots \int_0^{t_3}f_g(t_g) \cdots f_2(t_2) F_1(t_2) \, dt_2 \cdots dt_g
0 = \left\| p/q \right\| - \left\| p/q' \right\| =\sum_{j=1}^{m}w_j ( ( q b_j - p c_j ) - ( q' b_j - p c_j ) )=\sum_{j=1}^{m}w_j ( q - q' ) b_j &=(q-q') \sum_{j=1}^{m}w_j b_j =(q-q') \sum_{j=1}^{m}w_j|1 \cdot b_j- 0 \cdot c_j| & = (q-q') \left\| 0/1 \right\|.
M=\begin{pmatrix}c_{0,0}c_{0,1}c_{0,2}c_{0,3}\dots c_{1,0}c_{1,1}c_{1,2}c_{1,3}\dots 0 c_{2,1}c_{2,2}c_{2,3}\dots 0 0 c_{3,2}c_{3,3}\dots \vdots \vdots \vdots \ddots \ddots \end{pmatrix}\ .
|B| \geq |N(u)\cap N(v)|-h &\geq |N(u)|+| N(v)|-|V|-h &\geq 2\left(\frac{1}{2}-\frac{1}{10h}\right)n -\left(\frac{1}{2}+\frac{1}{10h}\right)n -h &= \left(\frac{1}{2}-\frac{3}{10h}\right)n -h.
(\partial_t + \nabla_{A_{\infty}}^* \nabla_{A_{\infty}}) (\eta v) = - \nabla_{\eta v}(J L_{u_{\infty}}\mu(u_{\infty}) - \nabla_{A_{\infty}}^* L_{u_{\infty}}(\eta b) + \nabla (\eta b) \times d_{\infty}u_{\infty}& + \eta \partial_t q(a_t, \hat{u}_t) [b, v] + \eta' v
r_{\lambda}^{(1)}(t) = - \frac{1}{\lambda^{n-1}}\cdot g_{\lambda}(t) \theta(t)^{T}g_{\lambda}(t) - \sum_{j=1}^{n}\frac{1}{\lambda^{j-1}}\bigg( A_{j}(t) \theta(t)^{T}g_{\lambda}(t) + g_{\lambda}(t) \theta(t)^{T}A_{j}(t) \bigg),
R_{l}( r ) \stackrel{( r \rightarrow 0 )}{\sim}\frac{u_{l}( r )}{r^{({D}- 1 ) / 2}}= \mathrm{\boldmath \large ~ \left\{~ \right. ~}\! \! \! r^{l}\mathrm{\boldmath \large ~ , ~}\! r^{- ( l + 2 \nu )}\mathrm{\boldmath \large ~ \left. ~ \right\}~}\! \! \; ,
\int_{|1+z| \leq |x|^{-\delta}}\left| \frac{f(x(1+z)) - f(x)}{x f'(x)}- z I(|z| \leq 1)\right| \nu_U(\d z) & \leq (\alpha^{-1}\log |x| + 1) \nu_U ((-1-|x|^{-\delta}, -1+|x|^{-\delta})) \to 0.
H_{A}\ = \ \frac{\hbar^{2}}{2 m}\D_{A}^{2}\; + \; v \ = \, \frac{\hbar^{2}}{2 m}\; \left( - \triangle_{A}^{S}\; + \; \frac r 4 \; + \; c \left( F_{A}\right) \right) \; + \; v \ ,
g^{1}= 0 , \quad g_{a b c}^{2}= - g_{a b c}^{5}, \quad g^{3}= 0 , \quad g_{a b c}^{4}= \frac{1}{2}f_{a b c}^{1}, \quad g_{a b c}^{6}= - f_{a b c}^{1}, \quad g_{a b c}^{7}= f_{a b c}^{1}- g_{a b c}^{5}.
\int_{T-\tau}^T |\nabla_p H(\overline{x}(s),\overline{p}(s))-\dot{\overline{x}}(s)|^{2}\ds \leq c_4 \int_{T-\tau}^T |\overline{x}(s)-x(s)|^2\ds + c_5 |\overline{x}(T)-x(T)|^2 &\leq(c_4 \tau +c_5)\parallel x - \overline{x}\parallel_{\infty}^2.
\epsilon_{D}( \lambda , a ) = F ( D ) \sum_{n = 1}^{\infty}\int_{0}^{\infty}r^{D - 3}\left( r^{2}+ \left( \frac{n \pi}{a}\right)^{2}\right)^{\frac{1}{2}}e x p \left[ - \lambda \left( r^{2}+ \left( \frac{n \pi}{a}\right)^{2}\right)^{\frac{1}{2}}\right] .
\begin{cases}\|P(\psi)\|_{L^{2}(\mathbb{R})}\le C \|\psi\|_{L^{2}(\mathbb{R})}, \|P(\psi)\|_{H^{1}(\mathbb{R})}\le C \|\psi\|_{H^{1}(\mathbb{R})}, \|P(\psi)\|_{H^{2}(\mathbb{R})}\le C \|\psi\|_{H^{2}(\mathbb{R})},\end{cases}
\binom{p}{r}\binom{q}{s}\frac{s! r! (p+q-r-s)!}{(p+q)!}\left[ N^{\prime}_{R,p-r+2,q-s-2}(u) + N^{\prime}_{R,p+1-r,q-s-1}(u)\right] & \qquad\leq K_{1}^{p-r+2}K_{2}^{q-s+2}M_{p,q}\left[ K_{1}^{2}K_{2}^{-2}+ K_{1}K_{2}^{-1}\right] .
\frac{1}{2}\Delta|\nabla^3 W|^{2}= |\nabla^{4}W|^{2}+ \langle \nabla^3 W, \nabla \Delta \nabla^2 W \rangle +\frac{R}{4}|\nabla^3 W|^{2}+ 8 W_{ijkl,trs}W_{pjkl,tru}R_{pisu}+2W_{ijkl,pru}W_{ijkl,trs}R_{ptsu}+2W_{ijkl,tpu}W_{ijkl,trs}R_{prsu}\,,
\begin{bmatrix}\phi(\Delta_n^*\Delta_n) -\phi(\Delta_n)^*\phi(\Delta_n) \phi(\Delta_n^* a^*)-\phi(\Delta_n)^*\phi(a)^* \phi(a\Delta_n)-\phi(a)\phi(\Delta_n) \phi(aa^*)- \phi(a)\phi(a)^*\end{bmatrix}\geq 0.
\frac{dA}{dt}=\frac{(B+C)^{2}-A^{2}}{BC}+\frac{-A^{2}+B^{2}}{BD}a_{3}^{2}, \frac{dB}{dt}= \frac{(A+C)^{2}-B^{2}}{AC}+\frac{A^{2}-B^{2}}{AD}a_{3}^{2}, &\frac{dC}{dt}=\frac{(A-B)^{2}-C^{2}}{AB}, \frac{dD}{dt}= \frac{(A+B)^{2}}{AB}a_{3}^{2}.
\eta_l^+ := \prod_{\alpha \in \mathcal{O}_s^+}\left(\frac{e^\alpha + e^{- \alpha}}{2}\right)^{2|l|},\eta_l^- := \prod_{\alpha \in \mathcal{O}_s^+}\left(\frac{e^\alpha + e^{- \alpha}}{2}\right)^{-2|l|}\, .
P(V_n =k) \geq P(V_n=0) \frac{\beta_n^k}{k!}\left(1 - \frac{\frac{k^2\beta}{\lambda_n(1-\beta)}}{k}\right)^k \geq P(V_n=0) \frac{\beta_n^k}{k!}\left(1 - \frac{k^2\beta}{\lambda_n(1-\beta)}\right).
K_{C}=\sqrt{\frac{2}{3}}\cdot[\begin{matrix}1&-\frac{1}{2}&-\frac{1}{2}\\ 0&\frac{\sqrt{3}}{2}&-\frac{\sqrt{3}}{2}\\ \frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}&\frac{1}{\sqrt{2}}\end{matrix}]
Q=\frac{1}{2\pi}\left\{\lim_{r{\rightarrow}{\infty}}\left[\overline{\cal{N}}(r)-{\sin}\left(\overline{\cal{N}}(r)\,\right)\,\right]+2\pi\ell-\overline{\cal{N}}(0)+{\sin}\left(\overline{\cal{N}}(0)\,\right)\right\}\,.
d\boldsymbol{r}^{\prime}\boldsymbol{I}_{12}^{+}=d\boldsymbol{x}^{1}\boldsymbol{I}_{12}^{+}+d\boldsymbol{x}^{2}\boldsymbol{I}_{12}^{+}=2d\boldsymbol{x}^{1}\boldsymbol{I}_{12}^{+}.
\int_{0}^{t_2}ae^{b\tau}P_{new}(\tau)d\tau = \int_0^{t_1}ae^{b\tau}P_{new}(\tau)d\tau + \int_{t_1}^{t_2}ae^{b\tau}P_{new}(\tau)d\tau \leq \int_0^{t_1}ae^{b\tau}P(\tau)d\tau + \int_{t_1}^{t_2}ae^{b\tau}P(\tau)d\tau \leq T_{\delta}e^{bt_2}
\Gamma(\overline{\eta}^{\gamma \to \gamma'}) = \Gamma(\overline{\eta}) \cup \{\gamma \cup \gamma'\}\setminus \{\gamma,\gamma'\}\;\mbox{and}\; \tilde{\Gamma}(\overline{\eta}^{\gamma \to \gamma'}) = \tilde{\Gamma}(\overline{\eta}) \cup\{\overline{\eta}^{\mbox{\tiny $J$}}_{\gamma,I}\}.
\left|\frac{w(t) - w(x_n)}{t-x_n}- \frac{w_n (t) - w_n (x_n)}{t-x_n}\right| = \left| \frac{w(t) - w_n (t)}{t-x_n}\right| \leq \frac{2^{-i}|t- x_n|}{|t-x_n|}\leq2^{-m}.
l({\bf v}),\quad -i v_0\frac{\partial}{\partial{\bf v}},\quad -i [\frac{\partial}{\partial{\bf v}}+{\bf v}({\bf v}\frac{\partial}{\partial{\bf v}})+\frac{3}{2}{\bf v}],\quad -i v_0({\bf v}\frac{\partial}{\partial{\bf v}}+\frac{3}{2})
\begin{aligned}\lim_{|k| \rightarrow \infty}(z\Psi^-)_{21}(x ;k)=-\frac{1}{2i}\lim_{|z| \rightarrow \infty}\nu(x;z) &=-\frac{1}{4}\left(2 i \bar{u}_{xx}(x)+|u_x(x)|^{2}\bar{u}_x(x)\right) e^{ic_\pm}(x)=\widehat{\Psi}^-_{21}(x),\end{aligned}
{\not \! \nabla}^2 = \nabla^{\mu}\nabla_{\mu}- e^{\mu}_a e^{\nu}_b e^{\lambda}_c J^{ab}T^c_{\mu \nu}\nabla_{\lambda}+ \frac{1}{2}e^{\mu}_a e^{\nu}_b J^{ab}J^{cd}R_{cd \mu \nu},
\widetilde{[\chi_k, \left. \Delta \right|_{\mathit{D}}] u}=- \widetilde{(\Delta \chi_k) u}- 2 \widetilde{\nabla \chi_k\cdot \nabla u}=\widetilde{(\Delta \chi_k) u}- 2\nabla \cdot (\widetilde{(\nabla \chi_k )u}),
Z ( s ) = \int{\cal D}\psi{\cal D}\bar{\psi}{\cal D}b_{\mu}F ( \partial b ) \delta ( \varepsilon^{\mu \nu \rho}f_{\nu \rho}( b ) ) e^{- \int d^{3}x \bar{\psi}( \partial \! \! \! / + i ( s \! \! \! / + b \! \! \! / ) + m ) \psi}.
\phi_{3}( x , t ) \pm 2 \pi \approx 4 \operatorname{a r c t a n}\left( \epsilon_{1}\exp \left[ - x + \frac{a_{2}^{\prime}}{2}+ 2 \ln \left( \frac{1 - k_{3}}{1 + k_{3}}\right) \right] \right) \, .
\Bigg [ : e^{- i 2 \sqrt{\pi}\sqrt{\frac{\pi}{\pi \! + \! g N}}U_{1 b}\Phi^{( 1 )}( x )}:_{M^{( 1 )}}\; \prod_{I = 2}^{N}: e^{- i 2 \sqrt{\pi}U_{I b}\Phi^{( I )}( x )}:_{M^{( I )}}e^{+ i \frac{\theta}{N}}
g = \left( \begin{array}{c c}{1}&{0}\\{b}&{1}\\ \end{array}\right) \left( \begin{array}{c c}{e^{a / 2}}&{0}\\{0}&{e^{- a / 2}}\\ \end{array}\right) \left( \begin{array}{r r}{\cos \theta}&{\sin \theta}\\{- \sin \theta}&{\cos \theta}\\ \end{array}\right) ,
\int_{B(x,2r)}(-\det D^2u ) u^2\phi^4\,dx+ \int_{B(x,2r)}|Du|^4\phi^4\,dx &\quad= -\int_{B(x,2r)}|Du|^2 (Du\cdot D\phi^4) u \,dx +\frac12\int_{B(x,2r)}u^2\big[ |Du|^2\Delta\phi^4 -D^2\phi^4 Du\cdot Du\big] \,dx.
\mathfrak{v}_{\mathfrak{R},K}=\sum_{k\neq 0}\sum_{|l|\leq K}e^{ilt}\left( \begin{array}{c}e^{i(k-\frac{1}{2})\theta}V^+_{k,l}e^{i(k+\frac{1}{2})\theta}V^-_{k,l}\end{array}\right)-\sum_{l\neq 0; l\leq K}e^{ilt}\left( \begin{array}{c}v^+_{l}\sqrt{z}v^-_l\sqrt{\bar{z}}\end{array}\right).
\mathbf{L}_n = \mathbf{L}_n^0+ \frac{\mu}{4\pi}\int_0^{2\pi}e^{\gamma \varphi(\theta)}e^{in\theta}d\theta, \widetilde{\mathbf{L}}_n = \widetilde{\mathbf{L}}_n^0+ \frac{\mu}{4\pi}\int_0^{2\pi}e^{\gamma \varphi(\theta)}e^{-in\theta}d\theta.
u_{\epsilon_j}(t) = u_0 -i \int_0^tH_{\epsilon_j}(\theta)u_{\epsilon_j}(\theta)d\theta - i \int_0^tf(\theta)d\theta & = u_0 -i \int_0^tH_{\epsilon_j}(\theta)u(\theta)d\theta -i \int_0^tH_{\epsilon_j}(\theta)\bigl\{u_{\epsilon_j}(\theta)- u(\theta)\bigr\}d\theta - i \int_0^tf(\theta)d\theta,