harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
g ( \epsilon ) = \sum_{p}\epsilon ( p ) g_{p}= \sum_{p}\sum_{\sigma}\epsilon ( p ) [ d_{p}^{( \sigma )}\bar{d}_{\tilde{p}}^{( - \sigma )}- d_{p}^{( \sigma ) \, \dagger}\bar{d}_{\tilde{p}}^{( - \sigma ) \, \dagger}] \,{.}
\widetilde{T}=\{\{k_{i,1}, k_{i,2}\}: 1 \le i \le m \}\mbox{such that}k_{i,1}< k_{i,2}, \mbox{for all}1\le i \le m, \mbox{and}k_{i,1}< k_{i+1,1}, \mbox{for all}1\le i <m.
\eta_{2}={{C^{2 (D-3)}}\over{(D-4)!}}\left(1-{{C^{\prime^{2}}}\over{B^{2}}}+{{A^{\prime}C^{\prime}C}\over{A B^{2}}}-{{A^{\prime\prime}C^{2}}\over{AB^{2}}}+{{A^{\prime}B^{\prime}C^{2}}\over{AB^{3}}}+{{CC^{\prime\prime}}\over{B^{2}}}-{{B^{\prime}C^{\prime}C}\over{B^{3}}}\right)
\lbrack w^{1-p^{\prime}}]_{A_{_{\frac{p^{\prime}}{s^{\prime}}}}\left(B\right)}^{\frac{1}{p^{\prime}}}=|B|^{\frac{1}{s}}[w^{1-p^{\prime}}]_{A_{p^{\prime}}\left( B\right)}^{\frac{1}{p^{\prime}}}\Vert w^{\frac{1}{p}}\Vert_{L_{^{p}}(B)}^{-1}\Vert w\Vert_{L_{^{\frac{s}{s-p}}}(B)}^{\frac{1}{p}}.
\begin{cases}\aligned (N\Theta_1)^{2/N}(d_{1,n}+d_{2,n})\leq N J(u_n,v_n) \leq N(c_1+c_2)\Theta_1+o(1), d_{1,n}^{p-1}+\nu\alpha d_{1,n}^{\alpha/2-1}d_{2,n}^{\beta/2}\geq (N\Theta_1)^{2/N}, d_{2,n}^{p-1}+\nu\beta d_{1,n}^{\alpha/2}d_{2,n}^{\beta/2-1}\geq (N\Theta_1)^{2/N}. \endaligned\end{cases}
\psi_{\nu_{1}}(z)=\frac{1}{2\beta}\int_{[0,+\infty]}\frac{t\eta_{\mu_{1}}(z)}{1-t\eta_{\mu_{1}}(z)}\,\left(\frac{1}{t}+t\right)\,d\sigma(1/t),z\in\mathbb{C}\backslash\mathbb{R}_{+},
\frac{1}{\alpha}( \exp ( \alpha ( x ( t ) - t ) ) - 1 ) = \sum_{f = 1}^{\infty}e^{f t}\sum_{\sigma_{f}\in P_{f}}( - 1 )^{l ( \sigma_{f})}( \alpha + f )^{l ( \sigma_{f}) - 1}S ( \sigma_{f}) \prod_{j = 1}^{l ( \sigma_{f})}\frac{b_{d_{j}}}{d_{j}}.
\sum_{k=0}^{\infty}\frac{\Gamma \left( k+\alpha \right)}{\Gamma \left(k+1\right)}{_{1}F_{1}}\left( \alpha +k,\beta ,x\right) z^{k}=\left(1-z\right)^{-\alpha}\Gamma \left( \alpha \right){_{1}F_{1}}\left( \alpha,\beta ,\frac{x}{1-z}\right) .
F_i(g)= c_{i,0}+ \sum_{u=1}^{p-1}c_{i, u\mathbf{e}_{a+1}}\zeta_p^{ub_{a+1}}+\cdots +\sum_{u=1}^{r-1}c_{i,u\mathbf{e}_{k+1}}\zeta_r^{ub_{k+1}}
\hat{V}^{C}_{\sigma}(R_1,R_2) \sqrt{2\pi}\sigma = \frac{1}{2\pi\sigma^{2}}\frac{1}{N}\sum\limits_{n=1}^N exp \left ( -\frac{(x_{n}- y_{n})^{2}+ (z_{n}- s_{n})^{2}}{2\sigma^2}\right ) \sqrt{2\pi}\sigma
\int_{\partial M}\varepsilon\gamma\left(W_{\delta,q}+\delta V_{\delta,q}\right)\Phi d\sigma \le C\varepsilon\|W_{\delta,q}+\delta V_{\delta,q}\|_{L^{\frac{2(n-1)}{n}}(\partial M)}\|\Phi\|_{L^{\frac{2(n-1)}{n-2}}(\partial M)}\le\varepsilon C\delta\|\Phi\|_{H}=o(\varepsilon^{2})
s^{\ast}\left( c,d;\chi_{1},\chi_{2}\right) +\chi_{1}\left( -1\right)\chi_{2}\left( -1\right) s^{\ast}\left( d,c;\chi_{2},\chi_{1}\right)=-\chi_{2}\left( -1\right) P_{1}^{\ast}\left( 0,\overline{\chi_{2}}\right)P_{1}^{\ast}\left( 0,\overline{\chi_{1}}\right)
\ell(\gamma) \le \frac14 c\delta^{n-1}+ 8\delta^{n-1}+ \frac14 c\delta^{n-1}\le 9\delta^{n-1}= \frac{36}{c\delta^2}\cdot \frac14c\delta^{n+1}\le \frac{36}{c\delta^2}d(x,y).
\langle G'(u),u\rangle =bp(p-1)M_u^{p-2}\|u\|_{E^{\alpha,p}}^{2p}+pM_u^{p-1}\|u\|_{E^{\alpha,p}}^p &\ \ \ \ -\int_{0}^{T}f'_2(t,u(t))u^2(t)dt-\int_{0}^{T}f(t,u(t))u(t)dt &<M_u^{p-2}\|u\|_{E^{\alpha,p}}^p(bp^2\|u\|_{E^{\alpha,p}}^p+ap)-p^2\int_{0}^{T}f(t,u(t))u(t)dt &=a(p-p^2)M_u^{p-2}\|u\|_{E^{\alpha,p}}^p\leq0,
\alpha_1^{n_T}=\frac{t_2-t_1}{2},~\alpha_j^{n_T}=\frac{t_j-t_{j-1}}{2}2\le j\le n_T-1,\alpha_{n_T}^{n_T}=\frac{t_{n_T}-t_{n_T-1}}{2}
B_1(a,b,\lambda) = \sum_{k=1}^N(b_k+\alpha a_{k-1})E_{kk}+\lambda\sum_{k=1}^N E_{k+1,k}, B_2(a,b,\lambda) = \sum_{k=1}^N(b_k+\alpha a_k)E_{kk}+\lambda\sum_{k=1}^N E_{k+1,k}
\varrho(t,\boldsymbol{q})= \Bigl(\frac{1}{2}-\frac{5}{2 \sqrt{36 \lambda\tilde{\kappa}+25}}\Bigr)t^{\frac{2}{3}- \frac{5+\sqrt{25+36\lambda\tilde{\kappa}}}{6}}g( \boldsymbol{q}) +\Bigl(\frac{5}{2 \sqrt{36 \lambda\tilde{\kappa}+25}}+\frac{1}{2}\Bigr)t^{\frac{2}{3}- \frac{5-\sqrt{25+36\lambda\tilde{\kappa}}}{6}}g( \boldsymbol{q}) ,
\int_{\Lambda}\left[ \prod_{i = 1}^{n_{a}}d^{2}x_{i}^{( a )}d^{2}y_{i}^{( a )}\prod_{j = 1}^{n_{b}}d^{2}x_{j}^{( b )}d^{2}y_{j}^{( b )}\right] \Big | B ( a , b ) \Big |^{\frac{g}{\pi + g N}q ( N - 1 )}
\|P_{\delta}(W_{n;\textrm{in}})e^{-iH_0t_n}\psi \|=\|P_{\delta}(W_{n;\textrm{in}})e^{-iH_0t_n}\chi_{S_R}\psi \| \leq \|\chi_{S_R}e^{iH_0t_n}P_{\delta}(W_{n;\textrm{in}})\|_\textrm{op}\|\psi\|\xrightarrow{n\rightarrow \infty}0
-{\frac{1}{2}}{\hat{\Lambda}}_{( 1 ) l m n}{\hat{\Lambda}}_{( 1 )}^{l m n}+{\hat{\Lambda}}_{( 2 )}^{l m n}({\hat{\Lambda}}_{( 1 ) l m n}-{\hat{\Lambda}}_{( 0 ) l m p}\partial^{p}a \partial_{n}a ) ] .
\lim_{M\rightarrow\infty}\lim_{n\rightarrow\infty}III \leq \lim_{M\rightarrow\infty}\Big[ C_M \lim_{n\rightarrow\infty}\mathbb{E}\int_0^{T}\Vert X_n(s) - X(s) \Vert_H^2 ds \Big] = 0 .
\left|\mathbb{E}_\eta\left(\mathbb{1}_{\{\mathcal{T}_t(0) > \frac{1-p}{4}\delta^d t\}}g(\eta_t)\right)\right| = \left|\mathbb{E}_\eta\left(\mathbb{1}_{\{\mathcal{T}_t(0) > \frac{1-p}{4}\delta^d t\}}\mathbb{E}_\eta\left(g(\eta_t)|\mathcal{F}_{t,\Lambda^c}\right)\right)\right|,
Z((G-e)\cup (G/f),q,w) =(Z((G-e)\cup (G-f),q,w)Z((G/e)\cup (G/f),q,w))^{1/2}\geq &\geq (Z(H-\{e,f\},q,w)Z(H/\{e,f\},q,w))^{1/2}= &= Z((H/e)-f,q,w).
\frac{d\tau}{d\overline{\varphi}}\ =\ 2\overline{\chi}^\ast(\mathsf{H}'-\lambda )\overline{\varphi}\overline{\chi}\, ,\ \frac{d\tau}{d\overline{\chi}}\ =\ 2\overline{\chi}\bigl(\langle\overline{\varphi}|(\mathsf{H}'-\lambda )\overline{\varphi}\overline{\chi}\rangle_{r}-\mu\bigr)\, .
\left.\begin{array}{l}\gamma=x(\partial\Sigma) \hspace{3mm}={\gamma_\mu}'\circ{\gamma_{\mu-1}}'\circ\cdots\circ{\gamma_3}'\circ{\gamma_2}'\circ\gamma_{g+1}\circ\gamma_g\circ\cdots\circ\gamma_2\circ\gamma_1 \end{array}\right.
:E_{0,\ldots,0,1,\ldots,1}(q^{-n+k}s_1^{-1}z)F_{0,\ldots,0}(q^{n-k}z):=:\Lambda'(q^{-n+k}s_1^{-1}z)\bar\Lambda'(q^{n-k}z)Y_{n-k+1}(z):\,,
\lim_{c\to +\infty}e^{-c\langle \xi ,\rho_L\rangle}\Delta_\emph{U}(\xi) \phi_\xi^{\emph{bc}}(x+c\rho_L; g^{(c)})= \Delta_\emph{U}(\xi) \phi^{\emph{cs}}_\xi (x;g) .
J^{( N )}\left( N ; 1 , \ldots , 1 \right) = \mathrm{i}^{1 - 2 N}\; \pi^{N / 2}\; \frac{\Gamma \left({\textstyle{\frac{N}{2}}}\right)}{N !}\; \; \frac{\Omega^{( N )}}{V^{( N )}}.
q_\lambda^{1/2}q_\mu^{1/2}q_s^{-1}\theta(\lambda)c_{s,k}\theta(\mu_{n_s^{-1}}(-k)\mu) =q_\lambda^{1/2}q_\mu^{1/2}q_s^{-1}q_{\lambda\mu_{n_s^{-1}}(-k)\mu}^{-1/2}c_{s,k}E(\lambda\mu_{n_s^{-1}}(-k)\mu)
J_n =-\frac{d^2}{dt^2}\ln p(Y_1,\cdots, Y_n\mid R_{t_1},\cdots ,\exp(t\xi)R_{t_n}) & = \sum_{k=1}^n \left( \frac{1}{\left( \sigma_k^0 \right)^2}{H_k^0}^T H_k^0 + \frac{1}{\left( \sigma_k^1 \right)^2}{H_k^1}^T H_k^1 \right)
(I+L)Y^{-1}N^{\dagger}X^{-1}(I+R) =Y^{-1}N^{\dagger}X^{-1}+Y^{-1}N^{\dagger}X^{-1}R+LY^{-1}N^{\dagger}X^{-1}+LY^{-1}N^{\dagger}X^{-1}R &=Y^{-1}N^{\dagger}X^{-1},
ds^2 ={\frac{1}{\rho}}\left\{dy_i dy^i -{\frac{1}{\rho}}y_i dy^i y_{\bar j}dy^{\bar j}\right\}+ \left\{\rho^n dw d{\bar w}+ n \rho^{n-1}\left({\bar w}dw y_{\bar i}dy^{\bar i}+ w d{\bar w}y_i dy^i\right) \right\}
A_{u^M}:=\mathbb{P}[\bigcup_{m=1}^M\{(u_m,V_1)\in\mathcal{F}\}]; \pi(u^M;v) :=1\left\{\bigcup_{m=1}^M\{(u_m,v)\in\mathcal{F}\}\right\}&\quad\cdot1\left\{\imath_{U^M;\tilde{V}}(u^M;v)\le\log L-\gamma\right\}.
\gamma_j := \frac{n-m-1}{2}\cdot \frac{1}{(n-j)(n-j-1)}\cdot \prod_{i=n-m}^{n-j}\frac{2i}{2i+1}\textrm{for $1 \leq j \leq m$,}\gamma_0 := 1 - \sum_{j=1}^{m}\gamma_j,
V_3= t^{1+\alpha}\partial_t+\alpha xt^\alpha\partial_x-\bigg(\bigg(\frac{\alpha(c+1)t^\alpha}{2}+\frac{\alpha^2x^2}{4}\bigg)u+\frac{m\alpha t^\alpha}{2}v\bigg)\partial_u- &\bigg(\bigg(\frac{\alpha(c+1)t^\alpha}{2}+\frac{\alpha^2x^2}{4}\bigg)v+\frac{n\alpha t^\alpha}{2}u\bigg)\partial_v,
t \int_{- t}^{t}\frac{d x}{( x^{2}+ \lambda^{2})^{\kappa + 1}}= \frac{\pi t}{2^{\kappa}\kappa !}\left( - \frac{1}{\lambda}\frac{d}{d \lambda}\right)^{\kappa}\frac{1}{\lambda}- 2 ( - 1 )^{\kappa}\left( \frac{d}{d ( \lambda )^{2}}\right)^{\kappa}\sum_{k = 0}^{\infty}\frac{( - 1 )^{k}}{2 k + 1}\left( \frac{\lambda}{t}\right)^{2 k}.
\sum_{k=N}^{\infty}\frac{B_{2k}}{2k(2k-1)z^{2k-1}}= \frac{1}{4\pi i}\int_{c^{'}-i\infty}^{c^{'}+i\infty}\frac{1}{(2\pi z)^{s^{'}}}\; \frac{\Gamma(s^{'}) \zeta(s^{'}+1)}{\sin((s^{'}+1) \pi/2)}\; ds^{'}& + \;\; \frac{1}{2\pi i}\int_{C}\frac{1}{(2\pi z)^{2s-1}}\; \frac{\Gamma(2s-1) \zeta(2s)}{\sin(s \pi)}\; ds \;\;,
\begin{cases}\partial_t H^{\psi}_0=\partial_s^2 H^{\psi}_0+h(\psi) \mbox{in}~D_{a}, H^{\psi}_0 (s,0)=0, \forall~s\in\mathbb{R}, h(\psi):=\lambda_1(t)\sin \psi(s,t)-\lambda_2(t)\cos \psi(s,t), \end{cases}
\limsup_{n\to\infty}\sup_{\Sigma_0\cap B_r(0)}(1+|x|)^\tau |\tilde \varphi_n -\varphi| = 0 & \limsup_{n\to\infty}\sup_{\Sigma_0\setminus B_r(0)}(1+|x|)^\tau |\tilde \varphi_n -\varphi| \leq C r^{\tau-\tau'},
\displaystyle \lvert \tau\rvert_E^{-\dim(N'_S)/2}O_s(\xi_{+},\varphi) =\int_{(F^\times)^n}f_\varphi(a_1,\ldots,a_n) \prod_{k=1}^n \eta_{E/F}(a_k)^k \lvert a_k\rvert^{ks-(k-1)}d^\times a_1\ldots d^\times a_n
\rho_{q}( t ) = \frac{1}{2}\dot{\phi}^{2}+ V ; \quad P_{q}( t ) = \frac{1}{2}\dot{\phi}^{2}- V ; \quad w_{q}= \frac{1 - ( 2 V / \dot{\phi}^{2})}{1 + ( 2 V / \dot{\phi}^{2})}
\dot{u}_t(x)+F(t,x,u_t(x)) = -\gamma'(t)+F(t,x,u_t(x)) &\leq -\gamma'(t)+F(t,x,0)-\lambda_F(\phi(x)-\gamma(t)) &\leq \lambda_F|\phi(x)|-C+F(t,x,0) &\leq c_-,
\overline{Q}(t) = (b-t)\det\begin{bmatrix}\tilde{s}_0' \tilde{s}_1' \ldots \tilde{s}_{m-1}' 1 \tilde{s}_1' \tilde{s}_2' \ldots \tilde{s}_{m}' t \vdots \vdots \vdots \vdots \vdots \tilde{s}_{m}' \tilde{s}_{m+1}' \ldots \tilde{s}_{2m-1}' t^{m}\end{bmatrix},
\delta_\alpha [f, g]_\alpha = ~ (-1)^{m+n}[ \mu, [f, g]_\alpha]_\alpha = ~ (-1)^{m + n}[ [\mu, f]_\alpha, g]_\alpha + (-1)^{n+1}[ f, [\mu, g]_\alpha ]_\alpha = ~ (-1)^{n+1}[ \delta_\alpha f, g ]_\alpha + [f, \delta_\alpha g]_\alpha.
\Xi^\lambda (K_a F_a \otimes 1 \otimes 1 \otimes E_a) = \sum_{i,j,m,n,o,p}(2c_{j}^{i}-\delta_{j}^{i})\pi(K_{a}F_{a})_{m}^{j}c_{n}^{m}\pi(1)_{o}^{n}c_{p}^{o}\pi(E_{a}K_{\lambda})_{i}^{p}.
\left| \int\limits_0^1 x f_{yyy}(x,y, \xi) J_0(\gamma_m x) dx \right| = \left| \int\limits_0^1 \sqrt{x}(\sqrt x \, f_{yyy}(x,y\xi)) J_0(\gamma_m x) dx \right| \leq \frac{C_{27}}{(\gamma_m)^{5/2}}\max(\sqrt{x}f_{yyyxx}) \leq \frac{C_{28}}{m^{5/2}}.
\begin{aligned}\widetilde{m}(x;z) =z\int_{-\infty}^{x}e^{2 iz(x-y)}w(y) d y+\frac{1}{2i}w(x) &-\frac{1}{2 i}\int_{-\infty}^xe^{2iz(x-y)}\left[\left(2i\bar{u}_{yy}(y)+\bar{u}_y(y)|u_y(y)|^{2}\right)\widehat{\Psi}^-_{11}(y)+|u_y(y)|^2 \widehat{\Psi}^-_{21}(y)\right] dy,\end{aligned}
2\left\{\zeta_i^{(k)},\varphi_q^{(p)}\right\}_{\infty}= \left\{\ell_i^{(k)}, \psi_q^{(p-1)}+ \mbox{``higher terms''}\right\}_{\mathfrak{k}^*}& = \sum_{j=1}^i \frac{\partial}{\partial \psi_j^{(k)}}\left(\psi_q^{(p-1)}+ \mbox{``higher terms''}\right).
- i V^{2}\mathrm{e x p}( - i \frac{\pi}{6}( a_{1}+ a_{2}) ) \sum_{n^{\prime}}\sum_{e^{\prime}\neq 0}\frac{\mathrm{e x p}\frac{2 i \pi}{k}[ a e^{\prime}( \widetilde{n^{\prime}}- 1 / 2 ) - e^{\prime}( \widetilde{n}- 1 / 2 ) - e ( \widetilde{n^{\prime}}- 1 / 2 ) ]}{2 \mathrm{s i n}\pi e^{\prime}/ k}
X_{j}= \big(H^{j+1}(\Omega) \cap H^{1}_{0}(\Omega)\big) \times H^{j}(\Omega) j \geq 1, Y^{j}= \big(H^{j+1}(\Omega) \cap H^{1}_{0}(\Omega)\big) \times \big(H^{j}(\Omega) \cap H^{1}_{0}(\Omega)\big) j \geq 1,Y_{0}= X_{0}.
C_f(w) \, f(w;\lambda) = \varGamma((r-p-q)w+1-a-b) \cdot \dfrac{\sin \pi(p w+a) \, \sin \pi(q w+b)}{\sin(\pi r w)}&\phantom{=}\times C_1 \cdot \tilde{d}^w \cdot \dfrac{\prod_{i=0}^{p-1}\varGamma\left(w+\frac{i+a}{p}\right)\prod_{i=0}^{q-1}\varGamma\left(w+\frac{i+b}{q}\right)}{\prod_{i=0}^{r-1}\varGamma(w+v_i)},
\hat{h}_{1}^{( 1 )}= \displaystyle \frac{1}{\hat{q}_{1}^{( 0 )}}, \hat{h}_{i}^{( 1 )}= \displaystyle \frac{\hat{e}_{i - 1}^{( 0 )}\hat{h}_{i - 1}^{( 1 )}+ 1}{\hat{q}_{i}^{( 0 )}}, i = 2, \dots , N.
\frac{d^{2}p(\tau)}{d\tau^{2}}=\frac{-1}{4\pi^{2}}\left( \tfrac{9}{8}\wp^{\prime}\left( \left. p(\tau)\right \vert \tau \right) +\sum_{k=1}^{3}\tfrac{1}{8}\wp^{\prime}\left( \left. p(\tau)+\tfrac{\omega_{k}}{2}\right \vert \tau \right) \right) ,
\begin{array}{l l l}{{[ ( s x u \wedge [ r ] ) \, m \, ( s z u \wedge [ r ] ) ] \, j \, [ ( s x u \wedge [ r ] ) \, m \, ( r z d \wedge [ s ] ) ]}}&{{=}}&{{}}\\{{( \{s \}\wedge [ r ] ) \, j \, ( s x u \wedge r z d ) = s x u \wedge [ r ] ,}}&{{}}&{{}}\\ \end{array}
f \circ g ([(p_{k,l})_{k,l},w]) = f \left([(p_{k,l})_{k,l}], \left[ \prod_{l=1}^{N_1}p_{1,l}, \prod_{l=1}^{N_2}p_{2,l},\dots,\prod_{l=1}^{N_r}p_{r,l}, w \right] \right) (~\eqref{eq_isomor_prop2.8_11}) &= ([(p_{k,l})_{k,l},w]).
|\Phi_{j,+}-\Phi_{+}| = |\Phi_{j,+}-\Phi_{+}|(1_{(0,\infty)}(\Phi_j)+1_{(0,\infty)}(\Phi)-1_{(0,\infty)}(\Phi_j)1_{(0,\infty)}(\Phi) ) &\leq |\Phi_{j,+}-\Phi_{+}|(1_{(0,\infty)}(\Phi_j)+1_{(0,\infty)}(\Phi) ).
i \gamma^{\mu}\partial_{\mu}\left( \begin{array}{c}{\psi_{L}}\\{\psi_{R}}\\{\psi_{\nu}}\\ \end{array}\right) = \left( \begin{array}{c c c}{0}&{\phi^{2}}&{0}\\{\overline{{\phi^{2}}}}&{0}&{\overline{{\phi^{1}}}}\\{0}&{\phi^{1}}&{0}\\ \end{array}\right) \left( \begin{array}{c}{\psi_{L}}\\{\psi_{R}}\\{\psi_{\nu}}\\ \end{array}\right) .
{\cal V}^{ghost}_{{\mu_1}{\nu_1}\,{\mu_2}{\mu_3}}(k_1,k_2,k_3)=\frac\kappa 2\left[\eta_{{\mu_2}{\mu_3}}({k_2}_{\mu_1}{k_3}_{\nu_1}+{k_3}_{\mu_1}{k_2}_{\nu_1})- (\eta_{{\mu_1}{\mu_2}}{k_1}_{\nu_1}+ \eta_{{\nu_1}{\mu_2}}{k_1}_{\mu_1}){k_2}_{\mu_3}\right].
\begin{array}{r c l}{{L_{3}u_{\lambda , l , m}}}&{{=}}&{{m u_{\lambda , l , m}, \nonumber}}\\{{L_{\pm}u_{\lambda , l , m}}}&{{=}}&{{\sqrt{l ( l + 1 ) - m ( m \pm 1 )}\u_{\lambda , l , m \pm 1},}}\\ \end{array}
\begin{array}{lll}\displaystyle{I_{1}:= \int_{\mathcal{C}^d}\big( u_{n}(t,X_{t}(\omega))-u_{n}(s,X_{t}(\omega)) \big)^{4}dm^{n}(\omega)}\displaystyle{I_{2}:= \int_{\mathcal{C}^d}\big( u_{n}(s,X_{t}(\omega))-u_{n}(s,X_{s}(\omega)) \big)^{4}dm^{n}(\omega) \ .}\end{array}
\tilde K=\min\Big\{\frac{|\chi_1\mu_1-\chi_2\mu_2|}{\sqrt{\lambda_2}}+\frac{\chi_1\mu_1|\sqrt{\lambda_1}-\sqrt{\lambda_2}|}{\sqrt{\lambda_1\lambda_2}}\ ,\ \frac{|\chi_1\mu_1-\chi_2\mu_2|}{\sqrt{\lambda_1}}+\frac{\chi_2\mu_2|\sqrt{\lambda_1}-\sqrt{\lambda_2}|}{\sqrt{\lambda_1\lambda_2}}\Big\},
\int\int\phi_a~\frac{\partial\phi_b}{\partial x}~\frac{\partial^2\phi_c}{\partial y\partial t}~dx~dy=-\int\int\frac{\partial\phi_c}{\partial t}~\left(\frac{\partial\phi_a}{\partial y}\frac{\partial\phi_b}{\partial x}+\phi_a~\frac{\partial^2\phi_b}{\partial x\partial y}\right)~dx~dy
\begin{array}{l c l}{{\mathcal L}_{w}}&{=}&{w_{- 1}\star p + w_{0}}\\{}&{=}&{\{u_{- 1}v_{- 1}^{\prime}- u_{- 1}^{\prime}v_{- 1}\}\star p + \{u_{- 1}v_{0}^{\prime}- u_{0}^{\prime}v_{- 1}\}.}\\ \end{array}
Z_{\scriptscriptstyle S D}= \frac{1}{N_{\scriptscriptstyle S D}}\int_{\Omega^{\scriptscriptstyle 1}( M )}\mathcal{D}f e^{- i \bigl \langle f , ( \mathrm{1 \m k e r n - 5 m u I \!}+ \ast d_{\scriptscriptstyle 1}) f \bigr \rangle}
d(\beta) =\left(a_1T_1+a_2T_2 \right)^2 + \left(b_1T_1+b_2T_2 \right)^2+\left(a_1T_1+a_2T_2 \right) \left(b_1T_1+b_2T_2 \right), d(\gamma) =\left(a_1T_1+a_2T_2 \right) \left( b_1T_1+b_2T_2 \right)^2 +\left(a_1T_1+a_2T_2 \right)^2 \left( b_1T_1+b_2T_2 \right).
I_{F_1}(y_1,y_2,z)=ze^{(P\log y_1+ H\log y_2)/z}\sum_{d_1,d_2\geq 0}y_1^{d_1}y_2^{d_2}\frac{\prod_{a=-\infty}^0 (H-P+az)}{\prod_{a=1}^{d_2}(H+az)\prod_{a=-\infty}^{d_2-d_1}(H-P+az) \prod_{a=1}^{d_1}(P+az)^2}.
\sum_{n=1}^{N+1}\frac{(q^2;q^2)_{n-1}(q^2;q^2)_N(zq;q^2)_{N-n+1}(zq)^{n-1}}{(zq^2;q^2)_{n}(q^2;q^2)_{N-n+1}(zq;q^2)_{N+1}}=\sum_{n=1}^{N+1}\frac{(q;q^2)_{n-1}(q^2;q^2)_{N}(zq^2;q^2)_{N-n+1}(zq^2)^{n-1}}{(zq;q^2)_{n}(q^2;q^2)_{N-n+1}(zq^2;q^2)_{N+1}},
\binom{n}{2}(-1)^{k+1}\sum_{0 \le s \le (k-1)d/n}\binom{d}{s}(-1)^s = \omega_n^{i\binom{k}{2}}\Big( \binom{k}{2}\binom{d}{dk/n}+ \omega_n^i{n \brack k}^{'}_{\omega_n^i}\Big).
\| S_T^{\infty}(\phi_n) - S_T^{\infty}(\phi_m)\|_{C([-1,1];H^{s,0})}\leqslant \| S_T^{\infty}(\phi_n) - S_T^{\infty}(\phi_n^K)\|_{C([-1,1]; H^{s,0})}&+ \| S_T^{\infty}(\phi_m) - S_T^{\infty}(\phi_m^K)\|_{C([-1,1];H^{s,0})}&+ \| S_T^{\infty}(\phi_n^K) - S_T^{\infty}(\phi_m^K)\|_{C([-1,1]; H^{s,0})}.
x_{i_1i_2}x_{i_3i_4}\ast_g x_{j_1j_2}x_{j_3j_4}= \tfrac{1}{2}( x_{g(i_1)g(i_2)j_1j_2}x_{g(i_3)g(i_4)j_3j_4}+ x_{g(i_1)g(i_2)j_3j_4}x_{g(i_3)g(i_4)j_1j_2}).
\frac{Q^{1}( Q^{2})^{*}}{| Q^{1}| | Q^{2}|}= \mathrm{e}^{- i \omega_{Q}}, \qquad \frac{( a_{1})^{*}a_{2}}{| a_{1}| | a_{2}|}= \mathrm{e}^{i \omega_{V}},
\int_{\lfloor R \rfloor}^R f(y)dy = \int_{\lfloor R\rfloor}^R(R-y)^{1/2}(R+y)^{1/2}\, dy \le (2R)^{1/2}\int_{\lfloor R\rfloor}^R(R-y)^{1/2}\, dy = \frac23 (R-\lfloor R \rfloor)^{3/2}(2R)^{1/2}.
(\gamma_1, \ldots, \gamma_r)=(\underbrace{-1,\ldots, -1}_{|T_0|+|T_1|-|\bar t_0|}, \underbrace{0,\ldots,0}_{|T_0|+2|T_1|},\underbrace{1,\ldots,1}_{|T_1|+|\bar t_0|})
c^{\nu}_{2r}(h^2)=c^{-\nu}_{-2r}(h^2)\;\;\; and \;\;\;\frac{c^{\nu}_{2r}(h^2)}{c^{\nu}_{0}(h^2)}=\frac{c^{-\nu}_{-2r}(h^2)}{c^{-\nu}_{0}(h^2)}
\frac{d}{d\lambda}\|K(g_\lambda-f^*)\|\bigg|_{\lambda=0}=\frac{1}{2}\frac{\frac{d}{d\lambda}\|K(g_\lambda-f^*)\|^2\bigg|_{\lambda=0}}{\|K(g^*-f^*)\|}=\frac{\langle K(g^*-f^*), K(g-g^*)\rangle}{\|K(g^*-f^*)\|}
\lambda_{1 , 2}^{*}= \pm \sqrt{\frac 1 2 I_{1}+ \frac 1 2 \sqrt{I_{1}^{2}+ I_{2}^{2}}}, \quad \lambda_{3 , 4}^{*}= \pm \sqrt{\frac 1 2 I_{1}- \frac 1 2 \sqrt{I_{1}^{2}+ I_{2}^{2}}}
\frac{1}{2 \pi}\Omega = \left( \begin{array}{c c c c c}{0}&{x_{1}}&{}&{}&{}\\{- x_{1}}&{0}&{}&{}&{}\\{}&{}&{0}&{x_{2}}&{}\\{}&{}&{- x_{2}}&{0}&{}\\{}&{}&{}&{}&{\ddots}\\ \end{array}\right) .
\partial_{k}g_{i \bar{\jmath}}^{( 1 )}- \partial_{i}g_{k \bar{\jmath}}^{( 1 )}= ( v - w ) \left( h^{i l}( \bar{z}_{l}- \bar{z}_{j}) M_{l k}h^{k j}M_{j i}- ( i \leftrightarrow k ) \right) .
\int_t^{t+1}\frac{1}{2(s-t)}\int_{\mathbb{R}^n}\rho_{y,s}(x,t) \, d\mu_s (y) ds \leq c_7 (x,t) + \int_t^{t+1}\frac{1}{2(s-t)}2^{n-1}e^{-\frac{3}{64}\frac{R^2}{s-t}}D_1 \, ds <\infty
\Upsilon(\xi,\zeta)= \int_\zeta^{+\infty}\int_{|\xi''|<\delta,\xi''<\xi-\zeta'}\Gamma(\xi''+\zeta',\zeta')|\xi''|\psi_\delta(\xi'') d\xi''d\zeta' \leq C\int_{\zeta}^{\xi+\delta}\max_{|\xi''|<\delta,\xi''<\xi-\zeta'}\Gamma(\xi''+\zeta',\zeta') d\zeta'\; \delta \leq C \int_{\zeta}^{\xi+\delta}(1+|\xi|^{p-1}+|\zeta'|^{p-1}) d\zeta'\; \delta,
F_{0,0}(x,y) = \widetilde{F}_{0,0}(x,y) = 1, &F_{1,0}(x,y)=0, F_{1,1}(x,y) = \frac{2}{3}\pi^2 \Bigl( x^4 + y^4 \Bigr), &\widetilde{F}_{1,0}(x,y)=0, \widetilde{F}_{1,1}(x,y) = \frac{2}{3}\pi^2 \Bigl( x^2 + y^2 \Bigr)^2.
k_{u,v}=\frac{u-3v}{2v}, \lambda_{i,j}=\frac{i-1}{2}-\frac{1+(-1)^{i+j}}{4}-\frac{u}{2v}j, s_{i,j}=\frac{i}{2}-\frac{u}{2v}j, q_{i,j}=\frac{(uj-vi)^2-4v^2}{8v^2},
y_nC(x_n) = y_nC(z_n) + y_n z_nC'(z_n) \delta_n + y_nz_n^2C''(z_n)\delta_n^2/2 + o(C''(z_n)\delta_n^2) =y_nC(z_n) + y_n z_nC'(z_n) \delta_n+o(y_nC'(z_n)\delta_n).
\begin{array}{rl}X^{++}X^{++}= X^{+-}X^{++}= X^{++}X^{-+}= 0 \cr X^{+-}X^{-+}= X^{-+}X^{+-}= 0 \cr X^{++}X^{+-}= -{r^2\over 2}X^{++}= - X^{-+}X^{++}\cr X^{+-}X^{+-}= -{r^2\over 2}X^{+-}\ ,\qquad X^{-+}X^{-+}={r^2\over 2}X^{-+}\cr X^{+-}- X^{-+}= -{r^2\over 2}I .\end{array}
\mathcal{E}_2 := \left\{\exists \, y^{\prime}\in \left( w+[-4\lambda H, 5\lambda H) \times \{H^\prime\}\right) \cap \mathbb{L}: X^{y^{\prime}}_{H-H^\prime}- \pi_1(y^\prime) \geq \left(v-v^{\prime}\right)H \right\}.
\left( -\frac{d^{2}}{d\varphi_1^{2}}+\frac{9\lambda_1}{2\sin^{2}(3\varphi_1 )}\right) \Phi_{n_1}^{(1)}(\varphi_1 )=B_{n_1}^{(1)}\Phi_{n_1}^{(1)}(\varphi_1 ),
\langle i \vert Q^* \vert \tau \rangle = \sum_{\mu_1}^{M_A}\sum_{\nu=1}^{M_B}Q^*_{i,(\mu\nu)}\left(\tilde{T}_{\rm exp}\right)_{\mu\nu}= (P_A\tilde{T}_{\rm \exp}P_B^* )_{ii}= (T_{\rm exp})_{ii}= \langle i \vert \upsilon_{\rm exp}\rangle \ .
\alpha(Q(j),Q(i)) =\alpha(1,Q(i))-\alpha(1,Q(j)) &= |[1,Q(i))\cap Q(J)|-|[1,Q(j))\cap Q(J)| &= Q(i)-1 - |[1,Q(i))\cap Q(I\sqcup I+1)|-|[1,Q(j))\cap Q(J)| &= Q(i)-1 - |[1,i)\cap (I\sqcup I+1)|-|[1,j)\cap J| &= Q(i)-i + |[1,i)\cap J|-|[1,j)\cap J|\,\cdot
\min_{\mathbf{x}}\;\frac{1}{2t}\sum_{n=1}^{N}\sum_{\tau=1}^{t}\beta^{t-\tau}((\mathbf{g}_{n}^{(\tau)})^{T}\mathbf{x}-y_{n}^{(\tau)})^{2}+\mu^{(t)}\left\Vert \mathbf{x}\right\Vert_{1},
\mu (\mathrm{Res}_{\tau_\alpha:L \hookrightarrow L_{\alpha}}(x_1))_{\alpha}=\mu( x_\alpha)_\alpha, (y_i)_i=\left(\mathrm{Res}_{LK_i/K_i}(\tilde{y}_i)\right)_i, \mu\mathrm{Res}_{LK_{\mathcal{I}}/L}( x_1)=\mu z.
\begin{cases}\begin{aligned}\dot y_k =w_k, \dot w_k =(\Phi\star\mu)(y_k,w_k)+\frac{1}{m}\sum_{j = 1}^m \phi(y_k-y_j) (w_j-w_k)+u_k, \end{aligned}k=1,\ldots, m \begin{aligned}\partial_t \mu =- v\cdot\nabla_x\mu-\nabla_v \cdot \left[\left(\Phi\star\mu+\frac{1}{m}\sum_{j = 1}^m \phi(x-y_j) (w_j-v)\right)\mu\right],\end{aligned}\end{cases}
\begin{aligned}2\bar{R}_{ij0k}\bar{U}^{ij}(\bar{\nabla}_0 -\bar{\Delta})\bar{U}^{0k}=\frac{1}{t}P_{ijk}\bar{U}^{ij}\bar{U}^{0k}2\bar{R}_{0i0j}(\bar{\nabla}_0 -\bar{\Delta})\bar{U}^{0i}\bar{U}^{0j}=\frac{1}{t}M_{ij}\bar{U}^{0i}\bar{U}^{0j}. \end{aligned}
\Delta_z^{(D)}(h_s) = 1 \otimes h_s + z^s h_s \otimes 1,\Delta_z^{(D)}(C_s) = 1 \otimes C_s + z^s C_s \otimes 1, &\Delta_z^{(D)}(E_n) = 1 \otimes E_n + \sum_{k\geq 0}z^{n+k}E_{n+k}\otimes \phi_{-k}^-,\Delta_z^{(D)}(F_n) = z^n F_n \otimes 1 + \sum_{k\geq 0}z^k \phi_k^+ \otimes F_{n-k}.
\left(\begin{array}{c}z_1(x,t) z_2(x,t)\end{array}\right)= \left(\begin{array}{cc}\cos(\Omega t) \sin(\Omega t) -\sin(\Omega t) \cos(\Omega t)\end{array}\right)\left(\begin{array}{c}(1 + R(x))\cos(x) (1 + R(x))\sin(x)\end{array}\right)
e^{-t\rho_{Q}(X)}\Big(R^{\vee}\big(\exp(tX)\big)D\mu-R^{\vee}\big(\exp(tX)\big)\delta_{\emptyset}(D)\mu_{\emptyset}\Big) &\qquad=e^{-t\rho_{Q}(X)-\epsilon t}R^{\vee}\big(\exp(tX)\big)R^{\vee}(u)\mu \qquad\qquad+e^{-t\rho_{Q}(X)}\delta_{\emptyset}(D) \Big(R^{\vee}\big(\exp(tX)\big)\mu-R^{\vee}\big(\exp(tX)\big)\mu_{\emptyset}\Big).
d_{11}= \begin{cases}\frac{6N}{7}, \frac{N}{M}\in\left(0,1\right] \frac{6M}{7}, \frac{N}{M}\in\left(1,\frac{9}{7}\right] \frac{2N}{3}, \frac{N}{M}\in\left(\frac{9}{7},3\right] 2M, \frac{N}{M}\in\left(3,\infty\right) \end{cases}
\begin{array}{l}u(x-1)^{r}+u^2(x-1)^{k_1}p_1(x)+u^3(x-1)^{k_2}p_2(x) -(x-1)^{r-r_1}(u(x-1)^{r_1}+u^2(x-1)^{k_4}p_4(x)+u^3(x-1)^{k_5}p_5(x)) =u^2(x-1)^{k_1}p_1(x)-u^2(x-1)^{r-r_1+k_4}p_4(x) +u^3(x-1)^{k_2}p_2(x)-u^3(x-1)^{r-r_1+k_5}p_5(x), =u^2(x-1)^{\min\{k_1, r-r_1+k_4\}+\hat e_3}h_3(x)+u^3(x-1)^{k_2}p_2(x)-u^3(x-1)^{r-r_1+k_5}p_5(x),\end{array}
\begin{cases}\delta^0 (1) = (0) \delta^1 (1) = (1) \sigma(0) = \sigma (1) =1 \sigma (01)=0 \gamma((0)\otimes x) = \gamma( x\otimes (0)) =x \gamma((1)\otimes (1)) = (1) \gamma((01)\otimes (1)) = \gamma((1)\otimes (01)) =0\ .\end{cases}
\eta(\tau_i, \tau_n^{q-\ell}\tau_n) = \eta(\tau_i ,\tau_n)\eta(\eta(\tau_n, \tau_i), \tau_n^{q-\ell})=\tau_n^q\tau_1\eta(\tau_n^{q-1+D(i)}\tau_{i+1}, \tau_n^{q-\ell})=\tau_n^q \tau_1 \eta(\tau_n^{\ell-1+D(i)}\tau_{i+1},1) =\tau_n^q\tau_1,
G^{(+)}_{{D}}({\bf r},{\bf r}';k)={\mathcal G}^{(+)}_{{D}}({\bf R};k) \stackrel{\scriptscriptstyle(r \rightarrow \infty)}{\sim}- \frac{1}{4\pi}\left( \frac{k}{2 \pi}\right)^{({D}-3)/2}\, e^{i\gamma_{{D}}}\,\frac{e^{ikr}}{r^{({D}-1)/2}}\, e^{-i{\bf k}' \cdot{\bf r}'}\; ,
- \nabla \cdot \left(2\mu \vec \varepsilon (\vec u) + \lambda \nabla \cdot \vec u \vec I - b\,p\vec I\right) = \rho_b \vec g\,, begin{align*}0ex] \partial_t\Big(\frac{1}{M}p + \nabla \cdot (b \vec u)\Big) + \nabla \cdot \vec q = f \,, \vec q = - \vec K \nabla p \,,
m_{\alpha}^{2}= V_{0}+ m_{3 / 2}^{2}+ m_{3 / 2}^{2}n_{\alpha}\left( 1 + \frac{1}{3}\tilde{\delta}_{G S}\frac{d P}{d y}\right)^{- 2}| 1 - \frac{1}{3}\tilde{\delta}_{G S}\rho |^{2}( T + \bar{T})^{2}\left| \hat{G}( T , \bar{T}) \right|^{2}
\langle T^\nu_\mu(x)\rangle_{Ren.}(\mu)-\langle T^\nu_\mu(x)\rangle_{Ren.}(\mu')=\frac1{16\pi^2}\frac1{\sqrt{g}}\frac{\delta}{\delta g^{\mu\nu}}\int d^4x\sqrt{g}a_2(x)\ln(\mu/\mu') \ ,
c ( p ) = \frac{R_{+}( p ) \, \sinh \left( \frac{1}{2}\gamma p + 2 i \pi \alpha \right)}{R_{-}( p ) \, \sinh \frac{1}{2}\gamma p}, d ( p ) = - \frac{i \sin 2 \pi \alpha}{\sinh \frac{1}{2}\gamma p}.
\Phi_n\left(\frac{t+1}{2}\right) =P_n(t)\chi_{[-1,1)}(t)=C^n_{-1}\Phi_n(t+1)+C^n_1\Phi_n(t) &=C^n_{-1}P_n(2t+1)\chi_{[-1,0)}(t)+C^n_1P_n(2t-1)\chi_{[0,1)}(t).
\begin{aligned}\int_{0}^{s}\Bigl(\int_{-\infty}^{\infty}P(xu)dF_1(x) \Bigr)du \ge \sum_{m=1}^{\infty}\int_{A_m}^{A_{m+1}}\frac{1}{x}\Bigl(\int_{0}^{s}P(v)dv\Bigr)dF_1(x) \ge \ge \sum_{m=1}^{\infty}\frac{1}{A_{m+1}}\int_{0}^{A_m s}P(v) dv \Bigl( F_1(A_{m+1}) - F_1(A_m)\Bigr)\ge \end{aligned}