harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\alpha_{\nabla}^E =(J \otimes 1)(Q_L(D^{L,E})+(D^{L,E})^2) = (\partial_A^{L^\vee}\otimes 1) \circ (J \otimes 1)(D^{L,E}) + [(j^\vee \otimes 1)D^{A,E}, (J\otimes 1)(D^{L,E})] =(Q_A + D^{L^\vee})(J\otimes 1)(D^{L,E})+[D^{A,E},(J\otimes 1)(D^{L,E})] =(Q_A + D^{L^\vee}+ [D^{A,E},-])(\beta) = \alpha,
\Phi_m(x,y) = \frac1{(4\pi)^{m/2}}\exp\left( \frac{-|x|^2-|y|^2}{4}\right) &=\frac1{(4\pi)^{(m-1)/2}}\exp\left( \frac{-|x|}{4}\right) \frac1{(4\pi)^{1/2}}\exp\left( \frac{-|y|^2}{4}\right) &=\Phi_{m-1}(x)\Phi_1(y).
\pi_{\psi}(g_1)\pi_{\psi}(g_2)f(w) =\epsilon_{g_1}\epsilon_{g_2}f(wg_1g_2) =\widetilde{c}_{X^{\ast}}(g_1,g_2)\pi_{\psi}(g_1g_2)f(w) = \widetilde{c}_{X^{\ast}}(g_1,g_2)\epsilon_{g_1g_2}f(wg_1g_2).
\begin{aligned}\mathbb F^f(c_{i-1}, c_{j+1})= \mathbb F^f(a-\epsilon', b+\epsilon'') \mathbb F^f(c_i, c_j)= \mathbb F^f(a+\epsilon', b-\epsilon'') \mathbb F^f(c_i, c_{j+1})= \mathbb F^f(a+\epsilon', b+\epsilon'') \mathbb F^f(c_{i-1}, c_j)= \mathbb F^f(a-\epsilon', b-\epsilon'').\end{aligned}
\langle DJ_1(f;v^1,v^2),w^2 \rangle = \alpha_1 \iint_{{\mathcal{O}}_{1,d}\times (0,T)}(y-y_{1,d})z\dxdt + \mu_1 \iint_{\mathcal{O}_1 \times (0,T)}v^1 w^2 dxdt
S = - \int d^{4}\sigma \sqrt{- \mathrm{d e t}\, ( G_{\mu \nu}+ e^{- \frac{\phi}{2}}F_{\mu \nu}- b_{\mu \nu})}+ \int \Big ( C_{4}+ C_{2}\wedge ( e^{- \frac{\phi}{2}}F - b_{2}) \Big ) .
\frac{p(x+\bar{h})\cdot q - p(x)\cdot q}{h}\ge \frac{\left(\frac{1}{1+\partial_{x_{1}}\varphi(x,p(x))h+o(h)}-1\right)p(x)\cdot q}{h}&=\frac{1}{1-\partial_{x_{1}}\varphi(x,p(x))h+o(h)}\left(-\partial_{x_{1}}\varphi(x,p(x))+\frac{o(h)}{h}\right) p(x)\cdot q.
\begin{aligned}A=\mathbb{E}\int_0^T\big[H_1^{v_1}(t)-H_1(t)\big]dt-\mathbb{E}\int_0^T\bigg[\big(b^{v_1}(t)-b(t)\big)q_1(t)+\big(\sigma^{v_1}(t)-\sigma(t)\big)k_1(t) + \sum_{j=1}^2\big(\sigma_j^{v_1}(t)-\sigma_j(t)\big)k_{j1}(t)-\big[f^{v_1}(t)-f(t)-\sum_{j=1}^2\big(z_j^{v_1}(t)-z_j(t)\big)h_j(t)\big]p_1(t)\bigg]dt.\end{aligned}
{\cal C}={\cal C}^{- 1}= \left( \begin{matrix}{{- 1}}&{{0}}&{{0}}&{{0}}\\{{0}}&{{1}}&{{0}}&{{0}}\\{{0}}&{{0}}&{{1}}&{{0}}\\{{0}}&{{0}}&{{0}}&{{1}}\\ \end{matrix}\right) \quad , \quad{\cal C}\alpha^{\mu}{\cal C}^{- 1}= \overline{{{\alpha}}}^{\mu \, \, \ast}\quad .
\begin{aligned}(f^n)^\#(z) \geq \exp\!\left((1+\delta(x_{n-1}))x_n - n\log (96\pi)- x_n-4\pi \log(2|z|) \right) = \exp\!\left(\delta(x_{n-1})x_n - n\log (96\pi)-4\pi \log(2|z|) \right) \geq \exp\!\left(\frac{x_n}{\log x_n}- n\log (96\pi)-4\pi \log(2|z|) \right).\end{aligned}
G_{r-1}^C (z | \underline \tau) = \exp \left \{\int_{C_1}\left [ (-1)^r \frac{e^{2\pi i z t}}{t ( 1 - e^{2\pi i t )}}\sum_{m\in C\cap\mathbb{Z}^r}e^{2\pi i t (m\cdot \underline \tau)}- \frac{e^{-2\pi i z t}}{t ( 1 - e^{2\pi i t})}\sum_{m\in C^\circ \cap\mathbb{Z}^r}e^{2\pi i t (m\cdot \underline \tau)}\right ] dt \right \},
\underset{y_{i_t}\in \mathbb{R}^{m\times 1}}{}\; ||e_{i_t}- A^T y_{i_t}||_{\infty}\; = \underset{x}{}\big\{e_{i_t}^Tx\; : \; Ax=0, \; ||x||_1 \leq 1 \big\}=\alpha_{1,\{i_t\}}.
\frac{\Delta f^{*}}{f_{f}}=\frac{-m_{F}}{\pi Z_{q}}(1-\frac{Z_{Liq}^{2}}{Z_{F}^{2}})=\frac{-m_{F}}{\pi Z_{q}}(1-J_{F}\frac{Z_{Liq}^{2}}{\rho_{F}})
D(p,l) =H^{l+1}(\Omega)\times (H^{l+2}(\Omega))^3 R(p,l) =H^{l}(\Omega)\times (H^{l+2}(\Omega))^3\times H^{l+\frac{5}{2}}(\Omega)\times H^{l+\frac{5}{2}}(\Omega)\times H^{l+\frac{3}{2}}(\Omega).
E(M)=\left(\begin{array}{ccccc}E(m_{11}) 0 0 \cdots 0 0 E(m_{22}) 0 \cdots 0 \vdots \ddots \vdots 0 \cdots 0 E(m_{n-1n-1}) 0 0 \cdots 0 0 E(m_{nn})\end{array}\right),
\mathcal{H}^{\ast}=\int d^{2}x\left[ \frac{1}{2}\left( \xi_{5}^{\ast2}+\xi_{6}^{\ast 2}\right) +\frac{1}{2}\left( \partial_{1}\xi_{3}^{\ast}-\partial_{2}\xi_{2}^{\ast}\right)^{2}+\frac{1}{2}m^{2}\left( \xi_{2}^{\ast 2}+\xi_{3}^{\ast 2}\right) +m\left( \xi_{2}^{\ast}\xi_{6}^{\ast}-\xi_{3}^{\ast}\xi_{5}^{\ast}\right) \right] .
f^n(u_h^n)-f(u^n) =(u_h^n)^3-(u_{Ih}^n)^3+ (u_{Ih}^n)^3-(u^n)^3-\Delta t \partial_t^+u_h^n-(u_h^{n-1}-u^{n-1}) &+\Delta t \partial_t^+u^n.
g_{\mathrm{sK}}(\dot q, \dot q) =\frac{1}{8}\int_{S_q}|\tau_{\dot q}|^2\, dA = \frac 18 \int_{S_q}\frac{|\dot q|^2}{|q|}\, dA = \frac 14 \int_{X}\frac{|\dot q|^2}{|q|}\, dA
|E_{4}| \leq C r \sup_{x \geq t}|\lambda''(x)| \int_{6r}^{\infty}dw \frac{r^{2}w}{w^{2}}\left(\frac{1}{1+w^{2}}+\frac{1}{\lambda(t+w)^{2-2\alpha}+ w^{2}}\right) &\leq C r \sup_{x \geq t}|\lambda''(x)|
K_{\bf a}^{( 1 )}( x , x^{\prime}, t ) ={\frac{\bf a}{4}}e^{{\frac{{\bf a}^{2}}{4}}t}e^{-{\frac{\bf a}{2}}( | x | + | x^{\prime}| )}\left( 1 - \Phi ({\frac{| x | + | x^{\prime}|}{2 \sqrt{t}}}-{\frac{\bf a}{2 \sqrt{t}}}) \right) ~ ~ ,
\rho(E)\equiv \sum_{\vec{n}}\delta(E-E_{\vec{n}})=\frac{1}{2\pi\mathrm{i}}\int_{c-\mathrm{i}\infty}^{c+\mathrm{i}\infty}\mathrm{e}^{\beta\, E}Z(\beta)\mathrm{d}\beta \underset{N\to \infty}{\longrightarrow}\frac{1}{2\pi\mathrm{i}}\int_{c-\mathrm{i}\infty}^{c+\mathrm{i}\infty}\mathrm{e}^{NS(\beta\, e)}\mathrm{d}\beta\ ,
f\nabla_X\omega =\nabla_X\omega(f) &:=(-1)^{(k-1)(k-2)}L_Xi_f\omega-\omega(\nabla_Xf) &=L_Xf\omega-\omega([X,f]) &=fL_X\omega+i_{[X,f]}\omega-\omega([X,f]) &=fL_X\omega.
|\Lambda, \Xi\rangle_{orb}=const \sum_{\nu, \nu', \nu_i}(-1)^\nu (-1)^{\frac{s_0^2}{2}}|\lambda, \Xi + \nu \beta_0 +\nu' \beta_{\gamma}+ \nu_1\beta^{(1)}+\dots +\nu_r\beta^{(r)}\rangle.
{d K_{2m, \infty}\over d\alpha}\!=\!{(2m)! \over \pi(2m+1)^2 2^{2(m-1)}}\!\int_0^{\pi/2}\!\!\!\big ( |\cos(\alpha-\varphi)|-|\cos(\alpha+\varphi)|\big )\Lambda_m(\varphi)d\varphi
=M_{iklmnp}^{(6)}+10r^{2}\langle\langle M_{iklm}^{(4)}\delta_{np}\rangle\rangle+72r^{4}\langle\langle M_{ik}^{(2)}\delta_{lm}\delta_{np}\rangle\rangle+240r^{6}\langle\langle\delta_{ik}\delta_{lm}\delta_{np}\rangle\rangle
S ={\frac{1}{\pi}}\int d^{2}x \left( \partial_{+}\left( \kappa \rho + 2 e^{- 2 \Phi}\right) \partial_{-}\left( \Phi - \rho \right) + \lambda^{2}e^{2 \left( \rho - \Phi \right)}+{\frac{1}{2}}\, \partial_{+}f_{i}\, \partial_{-}f_{i}\right) \; .
\sum_{k=0}^a\frac{(-a)_{k}(\beta)_{k}}{(1)_{k}^2}\cdot z^k\cdot\bigg(\sum_{j=0}^{k-1}\frac{1}{j+\beta}-2H_k\bigg) = (1-z)^a\sum_{k=0}^a\frac{(-a)_{k}(1-\beta)_{k}}{(1)_{k}^2}\cdot\frac{z^k}{(z-1)^k}\cdot\bigg(\sum_{j=0}^{k-1}\frac{1}{j+1-\beta}-2H_k\bigg).
\frac{i}{2}\mathrm{T r}\, \left( \hat{T}_{x}^{( 1 )}\frac{1}{\partial^{2}+ m^{2}}\sigma \frac{1}{\partial^{2}+ m^{2}}\right) - \frac{i}{4}\mathrm{T r}\, \left( \hat{T}_{x}^{( 0 )}\frac{1}{\partial^{2}+ m^{2}}\sigma \frac{1}{\partial^{2}+ m^{2}}\sigma \frac{1}{\partial^{2}+ m^{2}}\right) \, .
\lim_{n\rightarrow\infty}(|\nabla \tilde{u}_n|^2_2+|\nabla \tilde{v}_n|^2_2)=0,\ \ \lim_{n\rightarrow\infty}(|\nabla \tilde{u}_n|^2_2+|\nabla \tilde{v}_n|^2_2)\geq S^{\frac{2^*_\mu}{2^*_\mu-1}}_{H,L}.
\L^{(de)}_{(X/Y,E/a)}:={\iota_{Y^{de}}}_*\L^{(de)}_{(U/Y,E|_U/a)}\textup{, and}\phi^{(de)}_{(X/Y,E/a)}:={\iota_{Y^{de}}}_*(\phi^{(de)}_{(U/Y,E|_U/a)}):{F^{(de)}_{X/Y}}_*\L^{(de)}_{(X/Y,E/a)}\to\O_{X_{Y^{de}}},
\psi(F_{R})-\psi(F_{L}) =\int_{F_{L}}^{F_{R}}\eta'(y)A'(t,y)dy &=-\int_{F_{L}}^{F_{R}}\eta''(y)(A(t,y)-A(t,F_{L}(t)))dy+\left.\eta'(\cdot)(A(t,\cdot)-A(t,F_{L}))\right|_{F_{L}}^{F_{R}}&=-\int_{F_{L}}^{F_{R}}\eta''(y)(A(t,y)-A(t,F_{L}))dy+\eta'(F_{R}(t))(A(t,F_{R}(t))-A(t,F_{L}(t))),
\beta_i (F_1) = \begin{cases}\beta_{0, 2}(F_1) = d i = 0 \beta_{i, i+3}(F_1) ={\displaystyle \left [-d + \sum_{k \ge 2}k \, L_k \right ] \binom{d-1}{i+1}- \sum_{k \ge 2}k \, L_k \binom{d - k}{i+1}}i \ge 1. \end{cases}
\int_{0}^{\ell}d y \; \{\xi_{\alpha}( x ) , \xi_{\beta}( y ) \}\; \; \Omega^{L}\Big |_{\big ({\frac{\delta}{\delta \xi_{\beta}( y )}},{\frac{\delta}{\delta \xi_{\gamma}( z )}}\big )}= P_{\alpha \gamma}( x , z ) \quad .
\beta_k - \tilde{\beta}_k \le \sum_{i = m+1}\sum_{j > 0}j X_{i,j}(\beta_k - \tilde{\beta}_k)^2 \le \left( \sum_{i \ge m+1}\sum_{j > 0}j X_{i,j}\right)^2 \left[ (\beta_k - \tilde{\beta}_k)^2 \right] \le 2 \left( \left( \sum_{i= m+1}^S \sum_{j >0}j X_{i,j}\right)^2 + \left( \sum_{i = S+1}^{n}\sum_{j>0}j X_{i,j}\right)^2 \right)
u=\left(\begin{pmatrix}0 a_{12}0 a_{22}\end{pmatrix},\begin{pmatrix}b_{11}b_{12}0 b_{22}\end{pmatrix}, \begin{pmatrix}i_1 i_2\end{pmatrix}, 0, 0, b_{11}+f_2, \begin{pmatrix}f_1 f_2\end{pmatrix}\right)
D(P_{Y|X}\|Q_{Y|X}|P_{X}) :=D(P_{X}P_{Y|X}\|P_{X}Q_{Y|X}) *D_{1+s}(P_{Y|X}\|Q_{Y|X}|P_{X}) :=D_{1+s}(P_{X}P_{Y|X}\|P_{X}Q_{Y|X}),
\sigma\coloneqq \frac{2pt}{2(p+t)+pt}\wedge \frac{t}{4}=\begin{cases}\frac{2pt}{2(p+t)+pt}, \textnormal{if}\frac{6p}{2+p}\le t; \frac{t}{4}, \textnormal{if}\frac{6p}{2+p}\ge t. \end{cases}
\beta^{mat}:=\left(\begin{array}{cc}0 \beta^{(\eta)}_{12}\beta^{(\eta)}_{21}0 \end{array}\right) =\frac{i}{2}\left[\sigma_3, m^{PC}_{\eta,1}\right]=\left(\begin{array}{cc}0 i[m^{PC}_{\eta,1}]_{12}-i[m^{PC}_{\eta,1}]_{21}0 \end{array}\right).
\langle a,b\rangle_{l^2_n}=\frac{\pi}{n}\sum_{i=1}^{n-1}a_ib_i,\qquad\|a\|_{l^p_n}=\begin{cases}\Big(\frac{\pi}{n}\sum\limits_{i=1}^{n-1}|a_i|^p\Big)^{\frac{1}{p}},\quad 1\le p<\infty, \sup\limits_{1\le j\le n-1}|a_i|,\quad p=\infty,\end{cases}
\frac{-(-t\eta)^N}{1-t+t\eta}\cdot\frac{\prod_{\ell}(1-\eta^{\ell})^{\nu_{\ell}}}{\eta^N(\eta-1)}=\frac{-(-t)^N}{1-t+t\eta}\cdot\frac{\prod_{\ell}(1-\eta^{\ell})^{\nu_{\ell}}}{\eta-1}
d S^{2}={\frac{1}{I_{0}}}\{{\frac{1}{f_{0}}}[ e^{2 k_{0}}d z \, d{\bar{z}}+ \rho^{2}\, d \phi^{2}] - f_{0}\, d t^{2}\}+ I_{0}^{2}( A_{0 3}\, d \phi + d x^{5})^{2},
\frac{d^{p}H_{\lambda}( z )}{d z^{p}}= ( - 1 )^{p}p ! \frac{i}{2 \pi}\oint_{{\Gamma}_{1}}\oint_{{\Gamma}_{2}}d k_{1}d k_{2}\frac{k_{1}^{\Lambda}F ( k_{1}) k_{2}^{\Lambda}G ( k_{2})}{( z - k_{1}- k_{2})^{p}}
\begin{pmatrix}\xi' x' \eta' \end{pmatrix}=\begin{pmatrix}a \alpha b \gamma e \beta c \delta d \end{pmatrix}\begin{pmatrix}\xi x \eta \end{pmatrix}\,\, \mbox{and}\,\, (\xi'', x'', \eta'') = (\xi, x, \eta) \begin{pmatrix}a \alpha b \gamma e \beta c \delta d \end{pmatrix}.
A_{m}(t,q,y,u):= \left(\begin{matrix}q_{n-m+1,n-m+1}\begin{matrix}a_{n-m+1}(t,y,u) 0 \cdots 0 \end{matrix}\begin{matrix}q_{n-m+2,n-m+1}\vdots q_{n,n-m+1}\end{matrix}\boxed{A_{m-1}(t,q,y,u)}\end{matrix}\right); &A_{1}(t,q,y,u):=q_{n,n}
E_x\Big [\int_{0}^{\infty}\mathbf{1}_{[0,m]}(X_t) dt\Big ]= \int_{0}^{m}dy \int_{0}^{\infty}p(t,x,y) dt \ge \int_{\epsilon}^{m}dy \int_{R}^{\infty}\lambda f(t,x,y)dt=\infty.
\frac{\partial^{2}\tau}{\partial x^{2}}(x_*,y_*)=2+O(\varepsilon),\ \ \ \frac{\partial^{2}\tau}{\partial y^{2}}(x_*,y_*)=2+O(\varepsilon),\ \ \ \frac{\partial^{2}\tau}{\partial x\partial y}(x_*,y_*)=O(\varepsilon).
\Leftrightarrow\begin{cases}\frac{\tilde{\gamma}}{N}<\hat{c}<\frac{1}{N}| 2\tilde{\alpha}z_1z_2+\tilde{\omega}^2 | < | \tilde{\alpha}z_1^2+3\tilde{\beta}z_2^2-\tilde{\gamma}+\hat{c}N |\end{cases}
\int_{(-\infty,T)\times \Omega}( -u\phi_t+\sum_{|\alpha|\le m, |\beta|\le m}(A^{\alpha\beta}D^\beta uD^\alpha \phi)+\lambda u\phi)\,dx\,dt &=\sum_{|\alpha|\le m}(-1)^{|\alpha|}\int_{(-\infty,T)\times \Omega}f_\alpha D^\alpha \phi \,dx\,dt
|{\bar{B}}^n_k(t)-{\bar{B}}^n_k(t-)| \le \frac{1}{n}, \;|{\bar{\eta}}^n(t)-{\bar{\eta}}^n(t-)| \le \frac{2}{n}, \;|{\bar{Y}}^n_k(t)-{\bar{Y}}^n_k(t-)| \le \frac{K}{n}+ \sum_{j=K+1}^\infty \frac{jn_j}{n}
||\Delta_hf||_{L^p(\mathcal{D}_0)}^p = ||\tau_h f - f||_{L^p(\mathcal{D}_0)}^p\leq |h|^p ||\nabla f ||_{L^p(\mathcal{D}_1)}^p = |h|^p \sum_{j=1}^d||\partial_{x_j}f ||_{L^p(\mathcal{D}_1)}^p
F_{p}\left( \phi_{0}, \phi_{1}, \cdots , \phi_{p}\right) = \sum_{m_{0}\cdots m_{p}}\alpha_{m_{0}\cdots m_{p}}a_{m_{0}}\left( \phi_{0}\right) a_{m_{1}}\left( \phi_{1}\right) \cdots a_{m_{p}}\left( \phi_{p}\right)
\langle\!\langle (U'(t), -U (t)),(\Phi(t), \Phi'(t))\rangle \!\rangle & =\langle\!\langle (\widehat{U}_1,-\widehat{U}_0),( \widehat{\Phi}_0, \widehat{\Phi}_1)\rangle \!\rangle+\int_0^t \int_{\Gamma}(DH(\tau), \Phi(\tau)) dxdt,
\sigma_{i}= 2^{1 / 2}\left( \begin{array}{c c}{{e_{i}^{3}}}&{{e_{i}^{2}}}\\{{e_{i}^{1}}}&{{- e_{i}^{4}}}\\ \end{array}\right) ; \qquad \bar{\sigma}_{i}= 2^{1 / 2}\left( \begin{array}{c c}{{- e_{i}^{4}}}&{{- e_{i}^{2}}}\\{{- e_{i}^{1}}}&{{e_{i}^{3}}}\\ \end{array}\right) .
\int_{\alpha_0}^\eta \frac{g(\zeta) \, \sin\zeta \, d\zeta}{\sqrt{\cos\zeta - \cos\eta}}= - \frac{q\, \sqrt{\pi}\, (d-2) \, \Gamma((d-1)/2)}{2^{(d-2)/2}\, \Gamma(d/2)}\, \int_{\alpha_0}^\eta \frac{\sin\zeta \, d\zeta}{\sqrt{1-\cos\zeta}\, \sqrt{\cos\zeta - \cos\eta}}.
\int e^{-ix\cdot\xi}e^{i (x-y)\cdot\eta}e^{iy\cdot\zeta}a( x - h(x-y) , \eta) \widehat{u}(\zeta) \, d\zeta dy d\eta dx & = \int e^{i\widetilde{y}\cdot \varphi_1(\xi,\zeta,\eta)}\, e^{i\widetilde{x}\cdot \varphi_2(\xi,\zeta,\eta)}\, a( \widetilde{x}, \eta) \widehat{u}(\zeta) \, d\zeta d\widetilde{y}d\eta d\widetilde{x}
\hat{\delta}(x+y)= t(\mathcal M)-{\lim_{n \to \infty}\delta((x+y)e_n)}= (t(\mathcal M)-{\lim_{n \to \infty}\delta(xe_n)})+(t(\mathcal M)-{\lim_{n \to \infty}\delta(ye_n)}) = \hat{\delta}(x)+\hat{\delta}(y).
\lim_{t\rightarrow 0}\left|\phi_{t}(x-y))\right| = \lim_{t\rightarrow 0}\frac{1}{t^n}\left|\phi\Big(\frac{x-y}{t}\Big)\right|\lesssim \lim_{t\rightarrow 0}\frac{1}{t^n}\Big(\frac{|x-y|}{t}\Big)^{-n-1}\lesssim \lim_{t\rightarrow 0}t(|x-y|)^{-n-1}=0.
\omega_{0}^{2}- k_{n}^{2}\left[{\frac{1}{g}}+{\frac{1}{2}}k_{n}^{D - 4}\int^{\Lambda / k_{n}}{\frac{d^{D}q}{( 2 \pi )^{D}}}{\frac{1}{q^{2}( q + 1 )^{2}}}\right] = 0 .
K_\varepsilon(t) = h + \int_0^t \Big[ \int_0^1 \nabla_x b\big(s, \omega, X_{x}(s)+ \xi[X_{x+\varepsilon h}(s) - X_{x}(s)]\big)d\xi \Big] K_\varepsilon(s) ds &+ \int_0^t \Big[ \int_0^1 \nabla_x \sigma\big(s, \omega, X_{x}(s)+ \xi[X_{x+\varepsilon h}(s) - X_{x}(s)]\big) d\xi \Big] K_\varepsilon(s) dW(s),
\mathbf{I}_{h,\ell}\leq\left(h \sum_{m = 1}^N \left\|(u_m)_+ - (u_m)_+^{(\ell)}\right\|_{L^\gamma (K)}^\gamma \right)^{\frac{1}{\gamma}}+\left( h \sum_{m = 1}^N \left\| (u_{m - 1})_+ - (u_{m - 1})_+^{(\ell)}\right\|_{L^\gamma (K)}^\gamma\right)^{\frac{1}{\gamma}}begin{align*}2mm] =:I_1+I_2.
\bar\P_{i,a,b,c}:=\frac{1}{|\Theta_{i,a,b,c}|}\sum\limits_{\theta\in\Theta_{(i,a,b,c)}}\P_\theta, &\Theta_{(i,a,b,c)}:=\{\theta\in\Theta:\gamma_i(\theta)=a,\gamma_{-i}(\theta)=b, \lambda_{-i}(\theta)=c\},
( \psi^{-}, \phi^{+}) = \sum_{j}\int_{m_{0}^{2}}^{\infty}d{\mathsf s}\sum_{j_{3}}\int \frac{d^{3}\hat{\bf p}}{2 \hat{E}}\langle \psi^{-}| [ j ,{\mathsf s}] , j_{3}, \hat{\bf p}^{-}\rangle S_{j}({\mathsf s}) \langle^{+}j_{3}, \hat{\bf p}, [ j ,{\mathsf s}] | \phi^{+}\rangle .
f_{\tau_\mathrm{cb}, \rightarrow t}= \theta^{-t}\circ f_{\tau_\mathrm{cb}}\circ \theta^t f_{\tau_\mathrm{cb}, t \leftarrow}= \theta^{t}\circ f_{\tau_\mathrm{cb}}\circ \theta^{-t}f_{\tau_\mathrm{cb}, t \leftrightarrow t}= f_{\tau_\mathrm{cb}, \rightarrow t}\circ f_{\tau_\mathrm{cb}, t \leftarrow}
\phi_{m_{k}}^{2}( 0 ) = \frac{1}{{\cal A}_{n}R_{0}}\frac{\pi c}{8}\frac{( n + 1 )^{2}}{\Gamma^{2}[ ( n + 3 ) / 2 ]}\left( \frac{m_{k}}{c}\right)^{n}e^{- \frac{c}{2}\rho_{\mathrm{m a x}}}~ ,
A_1(k) := \int_{0}^{t}iske^{-isk^2}\iiint u^{\#}(\ell)\overline{u^{\#}}(n)u^{\#}(m) \,\mu^{\#,(1)}_{R,1}\left(k,\ell,n,m\right) \,d\ell dmdn\,ds, A_2(k) := \int_{0}^{t}e^{-isk^2}\iiint u^{\#}(\ell)\overline{u^{\#}}(n)u^{\#}(m) \, \partial_{k}\mu^{\#,(1)}_{R,1}\left(k,\ell,n,m\right)\,d\ell dmdn\,ds,
\begin{aligned}q^{-9}L_{4}(q^{32}) = -q^{-9}+ \sum_{\substack{n \geq 0 -n \leq j \leq n}}q^{(16n+3)^2 - 2(8j+3)^2}+ q^{(16n+13)^2 - 2(8j+3)^2}& + \sum_{\substack{n \geq 0 -n-1 \leq j \leq n}}q^{(16n+21)^2 - 2(8j+1)^2}+ q^{(16n+11)^2 - 2(8j+1)^2}.\end{aligned}
S_{e f f}= - 8 \frac{g^{2}}{( 4 \pi )^{2}}\int d^{4}x \epsilon^{\mu \nu \rho \alpha}a_{\mu}\partial_{\nu}b_{\rho \alpha}= - 8 \frac{g^{2}}{( 4 \pi )^{2}}{\cal S}_{B F}({\cal A}) .
\oint_{C}f(z)dz=\oint_{C}\frac{\frac{1}{(z+i)^{2}}}{(z-i)^{2}}dz=2\pi i\frac{d}{dz}\frac{1}{(z+i)^{2}}|_{z=i}=2\pi i[\frac{-2}{(z+i)^{3}}]_{z=i}=\frac{\pi}{2}
[\tilde{f}_{ik}(a),\tilde{g}_{kj}(b)] =[[\tilde{f}_{il}(1),\tilde{u}_{lk}(a)],\tilde{g}_{kj}(b)] &=[\tilde{f}_{il}(1),[\tilde{u}_{lk}(a),\tilde{g}_{kj}(b)]]-(-1)^{|a|}[\tilde{u}_{lk}(a),[\tilde{f}_{il}(1),\tilde{g}_{kj}(b)]] &=[\tilde{f}_{il}(1),\tilde{g}_{lj}(ab)]]
\beta_{(B^\rho_{1\infty})^*}(\nu_T, \mathcal N_{b_0}) \le \beta_{V^{\otimes d}_\kappa}(\nu_{T,\kappa}, \mathcal N_{b_0,\kappa}) + \sum_{i=1}^2 E\|\tilde Z_i - P_{(\kappa)}(\tilde Z_i)\|_{(B^\rho_{1\infty})^*}
=\int Dh_{\mu\nu}\exp\left\{-\int d^3x\left(g^2h_{\mu\nu}^2+V_{\rm total}\left[\varepsilon_{\mu\nu\lambda}\partial_\mu h_{\nu\lambda}\right]\right)+2\pi i\int\limits_{\Sigma}^{}d\sigma_{\mu\nu}h_{\mu\nu}\right\},
Z = C \, \left( \mu^{n_{c}- n_{f}}\Lambda_{1}^{3 n_{c}- n_{f}}\right)^{1 / ( 2 n_{c}- n_{f})}\omega^{k}, \qquad ( k = 1 , 2 , \ldots 2 n_{c}- n_{f}; \, \omega = e^{2 \pi i / ( 2 n_{c}- n_{f})}) ,
\Delta ( q_{1}, q_{2}; t ) ={\frac 1{2 \pi}}\sqrt{{\frac \pi{i t}}}\exp{\left({\frac{i ( q_{1}- q_{2})^{2}}{4 t}}-{\frac{i ( q_{1}+ q_{2}) t}2}-{\frac{i t^{3}}{1 2}}\right)}.
\dot{V}\left( \frac{\partial}{\partial L}\right) \langle W ( L ) \rangle - \int_{0}^{L}d J \langle W ( J ) W ( L - J ) \rangle \sim f ( z ) \exp \left\{- \frac{2}{5}\sqrt{6}z^{\frac{5}{2}}\right\}( 1 + \cdots )
|\psi_{I}(t)\rangle=[1-\frac{i\lambda}{\hbar}\sum_{m}\sum_{n}\int_{t_{0}}^{t}dt_{1}\langle m|V(t_{1})|n\rangle e^{-\frac{i}{\hbar}(E_{n}-E_{m})(t_{1}-t_{0})}|m\rangle\langle n|+...]|\psi(t_{0})\rangle.
\begin{aligned}Y_{\lambda}(x,p) =\sum_{n=1}^{\infty}\sum_{k=1}^{n}(k)_{p,\lambda}\frac{x^{n}}{n!}&=\sum_{n=1}^{\infty}\Big((1)_{p,\lambda}+(2)_{p,\lambda}+\cdots+(n)_{p,\lambda}\Big)\frac{x^{n}}{n!}\end{aligned}
[i_1, \cdots, i_{|\mathbf{a}|}]: =[n-a_{n+1-m}^{T}+1, n-a_{n+1-m}^T+2, \cdots, n; \quad\,\,\, (n-1)-a_{n-m}^T+1, (n-1)-a_{n-m}^T+2,\cdots, n-1; \quad\,\,\, \cdots; \quad\,\,\, m-a_1^T+1, m-a_1^T+2,\cdots, m].
1) f(\underbar{X}) = \sum\limits_{i=1}^n f_i(X_i), \mbox{or}2) f(\underbar{X}) = \prod_{i=1}^n f_i(X_i) \mbox{and}\ \prod_{i=1}^n \mathbb{E}[f_i(X_i)]. var[f_i(X_i)] \neq 0,
\begin{array}{c}\displaystyle(I-{}^tY_{22}(\tau))^{-1}\displaystyle=I+{}^tY_{22}(\tau)+({}^tY_{22}(\tau))^2(I-{}^tY_{22}(\tau))^{-1}\displaystyle=I+{}^tY_{22}(\tau)+{}^tY_{22}(\tau)M(\tau).\end{array}
\delta^\alpha_\gamma \delta^\beta_\delta = -\frac14 \hat C_{\gamma\delta}(\hat C^{-1})^{\alpha\beta}-\frac14 (\hat C \hat\gamma_A)_{\gamma\delta}(\hat\gamma^A\hat C^{-1})^{\alpha\beta}-\frac12 (\hat C \hat S_{AB})_{\gamma\delta}(\hat S^{AB}\hat C^{-1})^{\alpha\beta},
( J_{1}\cdot J_{1})_{2}= ( J_{1}^{2})_{2}= \{\, ( J_{1}^{2})_{2 , m}= \sum_{m_{1}}\sum_{m_{2}}\: \langle 1 , m_{1}; 1 , m_{2}| 2 , m \rangle \; J_{1 , m_{1}}J_{1 , m_{2}}\: | \, - 2 \le m \le + 2 \, \},
\begin{array}{lcl}\sum_{i,j=1}^{2}\langle[R(X_i,X_j),A]X_i,AX_j\rangle = \sum_{i,j=1}^{2}\langle[R(\widetilde X_i,\widetilde X_j),A]\widetilde X_i,A\widetilde X_j\rangle &= -\langle R(\widetilde X_1,\widetilde X_2)\widetilde X_1,\widetilde X_2\rangle(\lambda_1-\lambda_2)^2 &= 2K(|A|^2-2f^2),\end{array}
M_n= 4\pi \int_0^\infty dr~r^2\{\frac{F_\pi^2}{8}[ F'^2 +2\frac{sin^2nF}{n^2 r^2}] +\frac{1}{2e^2}~\frac{sin^2nF}{r^2}~[\frac{sin^2nF}{n^2 r^2}+ 2F'^2]\}
\frac{P'_{k}(x_0)}{P_k(x_0)}-\frac{P'_{k-1}(x_0)}{P_{k-1}(x_0)}= \frac{1}{x_0-r_k}+ \sum_{j=1}^{k-1}\frac{r_j-s_j}{(x_0-r_j)(x_0-s_j)},
|-\Phi(x-y)+\Phi(x)| = A\left|\log \frac{|x-y|}{|x|}\right|\le A \log 4 \frac{|x-y|}{|x|}&\le A \frac{|y|^{n-2}}{|x|^{n-2}}\left(\frac{|x|}{|y|}\right)^{n-2}\log 4\left(1 + \frac{|y|}{|x|}\right) &\le A \frac{|y|^{n-2}}{|x|^{n-2}}\max_{r\ge 1/2}r^{2-n}\log 4(1+r).
\mathbb{E}[ b({\bf x}_i) ] = \mathbb{E}\left[ \mathbb{E}[ b({\bf x}_i) | Y^{i-1}] \right] & = \sum_{\bf s}\sum_{y^{i-1}}P_{Y^{i-1},{\bf S}}(y^{i-1},{\bf s}) \sum_{{\bf x}_i}P_{{\bf X}_i | Y^{i-1}}b({\bf x}_i),
\sigma_{j}^{B W}( E ) \sim \left| a_{j}^{B W}( E ) \right|^{2}= \left| \frac{r_{\eta}}{E - ( E_{R}- i \frac{\Gamma}{2})}\right|^{2}\sim \frac{1}{( E - E_{R})^{2}+ ( \frac{\Gamma}{2})^{2}}\, ,
a\star_{1}^{n}b =\pi_{n}\cdot(ab)+\frac{\varepsilon_{abc}i}{\sqrt{n(n+2)}}c a\star_{1}^{n}a =\pi_{n}\cdot(a^{2})+\frac{1-\pi_{n}}{3}a\star_{1}^{n}a+b\star_{1}^{n}b+c\star_{1}^{n}c =1
\xi''(x) +(p-m)\tanh x \xi'(x)+(m-1)\cos 2h(x)\xi(x) +\xi'(x)\frac{p-2}{2}\frac{d}{dx}\log\big(h'^2(x)+(m-1)\cos^2r(x)\big) +h'(x)(p-2)\frac{d}{dx}\big(\frac{h'(x)\xi'(x)-\frac{m-1}{2}\sin 2h(x)\xi(x)}{h'^2(x)+(m-1)\cos^2h(x)}\big) &+\frac{\lambda}{\big(h'^2(x)+(m-1)\cos^2h(x)\big)^{\frac{p}{2}-1}}\frac{1}{\cosh^px}=0.
\frac{(m+n)!e^{(\gamma\cdot \mathbf 1)w\bar{\eta}}}{\pi^{m+n+1}}\Big(\frac{\gamma\cdot \mathbf 1}{\rho^{m+n+1}}+\frac{(m+n+1)\sum_{j=1}^{n}\gamma_je^{\gamma_jw\bar \eta}z_j\bar\zeta_j}{\rho^{m+n+2}}\Big).
\vert[\tilde{\bf T}^3u](z)-[\tilde{\bf T}^3v](z)\vert \le{2\over \vert 2\Lambda-1\vert}\int_{z_0}^z\vert g(t)\vert\vert [\tilde{\bf T}^2u](t)-[\tilde{\bf T}^2v](t)\vert\vert dt\vert \le \left\vert{2(z-z_0)]^3\over 3!(2\Lambda-1)^3}\right\vert\,\vert\vert g\vert\vert_\infty^3\,\vert\vert u-v\vert\vert_\infty.
\sum_{n=1}^{\infty}\frac{n(-q)_{n-1}q^{n(n+1)/2}}{(q)_n^{2}}=\sum_{n=1}^{\infty}\frac{nq^n(-q)_{n-1}}{(q)_n}-(-q)_{\infty}\sum_{j=1}^{\infty}\frac{q^{j^2}}{(q)_{j}^{2}}\sum_{n=1}^{j}\frac{q^n}{(1-q^{2n})}.
S^{(\rho)}=\sum_{\substack{x\leq n<2x n\equiv\nu_0\pmod{W^\prime}}}\biggl(\sum_{i=1}^k\chi_\mathbb{P}(nM+h_i)-\rho\biggr)\biggl(\sum_{d_i\mid nM+h_i}\lambda_{d_1,\ldots,d_k}\biggr)^2,
\zeta_{2}= \frac{2^{2 s - 2}\mu^{- 1}\Gamma ( s + \frac{1}{2}) \Gamma ( - s )}{\pi \Gamma ( s ) \Gamma ( \frac 1 2 - s )}\int d k \ \tilde{S}( k ) \tilde{S}( - k ) \left( \frac{k^{2}}{\mu^{2}}\right)^{- s - \frac{1}{2}}.
\left|p_1 \exp(i \eta x_1) Z_1 + p_{-1}\exp(-i \eta x_1 ) Z_{-1}\right|^2 = p_1^2 |Z_1|^2 + p_{-1}^2 |Z_{-1}|^2 + p_1 p_{-1}(Z_1 \overline{Z_{-1}}\exp(i2\eta x_1)+ \overline{Z_1}Z_{-1}\exp(-i2\eta x_1)) &= p_1^2 |Z_1|^2 + p_{-1}^2 |Z_{-1}|^2 + 2 p_1 p_{-1}|Z_1| |Z_{-1}| \cos(2 \phi_1 + 2 x_1 \eta) .
\tfrac{L_X^{(\alpha)}(s,t,z+z_0)}{L_X^{(\alpha)}(s,t,z_0)}=\tfrac{U\left(a+s,a+s-b-t+1,\tfrac{c+z_0+z}{\alpha}\right)}{U\left(a+s,a+s-b-t+1,\tfrac{c+z_0}{\alpha}\right)},z\ge 0,
p_1 \phi_k^R(1/n,\ldots 1/n) = \frac{p_1}{n}\left( \sum_i G_{k,i}\frac{1}{n}- \sum_{i,j}\frac{1}{n}G_{i,j}\frac{1}{n}\right) & = \frac{p_1}{n}(\hat G_{k,*}- \hat G)
\| \tau_{-\mathbf y}(f*\phi)(\mathbf x) (1+ d(\mathbf x,\mathbf y))^{\delta'}\|_{L^1 dw(\mathbf x)}&\leq C \|f(\cdot)(1+\|\cdot\|)^{\mathbf{N}/2+\delta}\|_{L^1(dw)}\|\phi(\cdot)(1+\|\cdot\|)^{\mathbf{N}+\delta}\|_{L^{\infty}}.
H_{S}= \sum_{m \geq 2}\frac{1}{m}P_{S}(R_{S}^{m})= \sum_{m \geq 2}\frac{1}{m}\sum_{n=1}^{\infty}\pi_{S}(n) R_{S}^{mn}= \sum_{m \geq 2}\frac{1}{m}\sum_{n=1}^{\infty}\pi_{X}(n) R_{S}^{(k+1)mn}= \sum_{m \geq 2}\frac{1}{m}\sum_{n=1}^{\infty}\pi_{X}(n) R_{X}^{mn}= \sum_{m \geq 2}\frac{1}{m}P_{X}(R_{X}^{m})= H_{X}.
\int_{\Psi_j}x^k e^{x^r / \hbar}dx = \int_0^\infty t^k e^{( \pi i + \arg \hbar i + 2 \pi i j ) (k+1) / r}e^{t^r\hbar^{-1}e^{\pi i + \arg \hbar i}}dt &= \exp\left((\pi i + \arg \hbar i + 2\pi i j) \frac{k+1}{r}\right) \int_0^\infty t^k e^{-t^r / |\hbar|}dt
{\Delta_{\gamma ,j}}= \frac{1}{k}\sum\limits_{i = 1}^k{\left({\frac{1}{{w\left({\left|{{\mu_i}}\right|}\right)}}{{\left({\frac{{{{\bar \mu}_i}}}{{\left|{{\mu_i}}\right|}}}\right)}^j}{w_j}}\right)\Delta_\gamma, \;\;\; j = 0, 1, \dots , m}.
||\T_f(r) \le\dfrac{n+k+1}{q-(2n-k+2)+n+k+1)}\sum_{i=1}^qN^{[k]}_{(f,a_{i})}(r)+o(T_f(r))+O(\max_{1\le i \le q}T_{a_i}(r)) &=\dfrac{n+k+1}{q-n+2k-1}\sum_{i=1}^qN^{[k]}_{(f,a_{i})}(r)+o(T_f(r))+O(\max_{1\le i \le q}T_{a_i}(r))\r\in N_I.
\frac{(-4096)^{N-1}G(N,k)}{2N^2\binom{2N}{N}^2}=\frac{2^{4 k-8}N \binom{2 N}{N}(-1)^{k+1}\binom{k+N}{N-k}\binom{2 N-2 k}{N-k}\binom{2 k+2 N}{k+N}}{\binom{2 k}{k}(2 k+2 N-1)}\in \mathbb{Z}.
c(l_{0}+1,l_{1};l_{0},l_{1}) =c(l_{0},l_{1};l_{0}+1,l_{1})=c_{0}\sqrt{\frac{(k_{1}-l_{0}-1)(k_{1}+l_{0})}{(l_{1}-l_{0})(l_{1}-l_{0}-1) (l_{1}+l_{0}) (l_{1}+l_{0}+1)}},
\#\Big\{(\mu_i,\mu_j) = (0,1), (\nu_i,\nu_j) = (1,1) \Big\}+\#\Big\{(\mu_i,\mu_j) = (1,1), (\nu_i,\nu_j) = (1,1) \Big\}+ &\#\Big\{(\mu_i,\mu_j) = (2,1), (\nu_i,\nu_j) = (1,1) \Big\}+\#\Big\{(\mu_i,\mu_j) = (2,1), (\nu_i,\nu_j) = (2,1) \Big\},
H^{2}= \frac{1}{2}\frac{( c_{1}+ k )^{2}}{r_{0}^{2}}\frac{( 1 - x^{2})^{5 / 2}}{x^{\frac{5 + 3 k + c_{1}}{2}}}\left[ E^{2}- x^{2 k}- \frac{l^{2}}{r_{0}^{2}}\frac{( 1 - x^{2})^{1 / 2}}{2 x^{1 / 2}}\right]
(T_{-2,\beta}f,g)_{w_{-2,\beta}}=\int_{-1}^{1}\left( (1-t)^{-1}(1+t)^{\beta+1}f^{\prime}(t)\overline{g}^{\prime}(t)+(1-t)^{-2}(1+t)^{\beta}f(t)\overline{g}(t)\right) dt.
Q = \alpha^2 x \left[ \left( r +\frac{\pi}{2}\frac{1}{\sqrt{x}}\right)^2 +\frac{\bar{c}^2}{x}\right],\quad R = \frac{\alpha^2}{x}\,F\!\left( \frac{1}{\sqrt{x}}; r \right), \quad{\bf S}^2 = \frac{\alpha^4}{x}\, G\!\left( \frac{1}{\sqrt{x}}; r \right),