harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\sum_{{\bf{k}}}\Lambda_{{\bf{k}}}( -{\bf{q}}) \frac{- i \langle T a_{{\bf{k}}}^{n}({\bf{q}}) a_{{\bf{k}}^{'}}^{\dagger}({\bf{q}}^{'}) \rangle}{\langle T 1 \rangle}= G_{1}({\bf{q}},{\bf{k}}^{'},{\bf{q}}^{'}; n )
\hat{G}_{1,\varphi}(x) = \hat{\Lambda}(x) + \hat{M}_{0,\varphi}(x) = \left[I + \hat{N}_{1,\varphi}^{-}(x) (\Lambda^{+}_{0}(x))^{-1}\right] \hat{\Lambda}(x) \left[I + \hat{N}_{1,\varphi}^{+}(x) (\Lambda^{-}_{0}(x))^{-1}\right].
\epsilon_{1}^{\mu}= ( 0 , \epsilon^{i j}\frac{{p}_{j}}{| \vec{p}|}) \qquad \qquad \epsilon_{2}^{\mu}= ( \frac{| \vec{p}|}{M}, \frac{w_{p}}{M}\frac{{p}^{i}}{| \vec{p}|}) .
A_n=\frac{1}{\sqrt{n}}\sum_{j=1}^{n_1}\left[d_1\frac{X_{1j}-\mu_1}{\sigma_1}+d_2\sqrt{\frac{n_1}{n_2}}\frac{X_{2j}-\mu_1}{\sigma_1}\right], &B_n= \frac{d_2}{\sqrt{n_2}}\sqrt{n_2-n_1}\frac{1}{\sqrt{n_2-n_1}}\sum_{j=n_1+1}^{n_2}\frac{X_{2j}-\mu_2}{\sigma_2}.
\psi_{w_0}y_1^{c_1}\cdots y_j^{c_j}y_{j+1}^{c_{j+1}}\cdots y_n^{c_n}\psi_{w_{0}}y_{\min}=-\psi_{w_0}y_1^{c_1}\cdots y_j^{c_{j+1}}y_{j+1}^{c_{j}}\cdots y_n^{c_n}\psi_{w_{0}}y_{\min}.
\mathcal{I}_{\mathrm{D}}(\delta) = \inf_{\lambda \in \mathbb{R}_{+}^2}\left[\langle \lambda, \delta \rangle + \sup_{\pi \in \Pi(\mu_1, \mu_2)}\int g_{\lambda}(s_1, s_2) \, d \pi(s_1, s_2) \right].
d s^{2}= H^{\frac{p - 3}{4}}\Big ( - f d t^{2}+ \sum_{i = 1}^{p}( d x^{i})^{2}\Big ) + H^{\frac{p + 1}{4}}\left( f^{- 1}d r^{2}+ r^{2}d \Omega_{4 - p}^{2}\right) .
\frac{\partial^{2}\psi}{\partial\phi^{2}}=r^{2}sin^{2}\phi\frac{\partial^{2}\psi}{\partial x^{2}}-2r^{2}sin\phi cos\phi\frac{\partial^{2}\psi}{\partial x\partial y}+r^{2}cos^{2}\phi\frac{\partial^{2}\psi}{\partial y^{2}}-rcos\phi\frac{\partial\psi}{\partial x}-rsin\phi\frac{\partial\psi}{\partial y}
h(x,t) =e^{-\pi i a t^2/T}\,\sum_k \sum_{(q,m)\in\Gamma}\big[b_{(q,m),k}\,e^{-\pi i a T(t/T - (q-k))^2}\hskip-.25inH g(t- (q-k)T)\,\Phi_{(q,m)}(t , \, x -(q-k)T) \, e^{2\pi i (q-k)at}\big].
\varphi_\pm:\begin{pmatrix}x^1 x^2 x^3 x^4 \end{pmatrix}\mapsto\begin{pmatrix}x^1 x^2 x^3 x^4 \end{pmatrix}+a\begin{pmatrix}\cos(x^3+x^4) \pm\sin(x^3+x^4) 0 0\end{pmatrix}
\left\{\eta_i=k,\ \tau_i=k-\ell\right\}=\left\{Z_k\ge Z_{k-1},\Z_{k-1}< Z_{k-2}<\dots<Z_{k-\ell}, \tau_i=k-\ell\right\}=\left\{U_k\ge \frac{L_{k-1}}{L_{k-1}+R_{k-1}},\U_{k-1}<\frac{L_{k-2}}{L_{k-2}+R_{k-2}}, \dots,U_{k-\ell+1}<\frac{L_{k-\ell}}{L_{k-\ell}+R_{k-\ell}},\ \tau_i=k-\ell\right\}.
\left\{- \sum_{i = 1}^{3}\frac{\partial^{2}}{\partial x_{i}^{2}}+ \sum_{\stackrel{i , j = 1}{i \neq j}}^{3}\left({\frac{\omega^{2}}{8}}\left( x_{i}- x_{j}\right)^{2}+{\frac{2 \alpha}{\left( x_{i}- x_{j}\right)^{2}}}+ \Omega \left( \frac{x_{i}- x_{j}}{\sqrt{2}}\right) \right) \right\}\psi = E \psi ~ ,
P(C_{s}<R_{s}| \gamma_{M, max}>\gamma_{W}) = Q_{T}Q_{R}\sum_{i=0}^{Q_{T}Q_{R}}\binom{Q_{T}Q_{R}-1}{i}\frac{(-1)^i}{(i+1)}\Bigg( \frac{1}{1+\frac{(i+1)\overline{\gamma}_{W}}{\overline{\gamma}_{M}}}- \frac{e^{\frac{-(2^{R_{s}}-1)(i+1)}{\overline{\gamma}_{M}}}}{1+\frac{(i+1) 2^{R_{s}}\overline{\gamma}_{W}}{\overline{\gamma}_{M}}}\Bigg).
I_{n}(\theta_{0})=\mathbb{E}[(\frac{\partial}{\partial\theta}log\mathbb{P}(X_{n}|\theta))^{2}|_{\theta=\theta_{0}}]=-\mathbb{E}[(\frac{\partial^{2}}{\partial\theta^{2}}log\mathbb{P}(X_{n}|\theta))|_{\theta=\theta_{0}}],
\mu_M^H =\frac{3}{2}+\cos^2\biggl(\frac{5\pi}{8}\biggr)+\cos\biggl(\frac{5\pi}{8}\biggr)\cdot\sin\biggl(\frac{5\pi}{8}\biggr) &=2-\frac{1}{\sqrt{2}}&\cong 1,29.
n \geq \dim\biggl((X_0+X_1)+\sum_{i\in I}X_{i}\biggr) &=\dim(X_0+X_1)+\sum_{i\in I}\dim(X_{i}) &=\dim(X_0)+\dim(X_1)-\dim(X_0\cap X_1)+\sum_{i\in I}\dim(X_{i}) &= (k+1)d-\dim(X_0\cap X_1).
\left( \gamma ;q\right)_{0}=1~,~~~\left( \gamma ;q\right)_{m}=\left(1-\gamma \right) ~\left( 1-\gamma q\right) ~\left( 1-\gamma q^{2}\right)\cdot \cdot \cdot \left( 1-\gamma q^{m-1}\right) ~~~~~~m=1,2,3,...~,
\sum_{\sigma\subset \mu_\ell}c_{\ell}x_{\ell,\omega(\mu_\ell)-\omega(\kappa)}+ c_{\kappa}=0 \sigma \subset \kappa \sum_{\sigma\subset \mu_\ell}c_{\ell}x_{\ell,\omega(\mu_\ell)-\omega(\kappa)}=0 \sigma \not \subset \kappa.
G_s(\gamma_1) = \frac{\sqrt{J_1}}{E_q(J_1,J_2)}\left( F_q(J_1,J_2,\gamma_1)- F_q(J_1,J_2,-\gamma_1) \right) = \frac{2i\sqrt{J_1}}{E_q(J_1,J_2)}\sum_{n_1,n_2 =0}^\infty \frac{J_1^{n_1}J_2^{n_2}}{[n_1]_q![n_2]_q!}\sin(\gamma_1q^{2[n_1]_q});
\overline{\dot w^0}(x)-\overline p(F_0^{-1}(x))=0\,, \overline{\dot w}(t)-\sigma\overline p(t)+\sigma\int_0^\infty\overline p(t+s)F'(s)\,ds=0\,, \overline{\dot k}(x,t)-\overline p(\frac{t}{\mu}+F^{-1}(x))=0\,.
I_{12}= \int_1^2 \frac{\partial L}{\partial \dot{x}^i}d x^i = \int_1^2 d \tau \bigg( \frac{\partial L}{\partial \dot{x}^i}\dot{x}^i \bigg) = \int_1^2 p_i d x^i, &L_{geod}= \frac{\partial L}{\partial \dot{x}^i}\dot{x}^i = p_i \dot{x}^i.
\langle H_y\rangle_{\kappa \Phi_{\#}^y}= |\kappa|^2 \big( \|\Phi_{\#}^y\|_{\#}^2 + (e_{\alpha}+ \alpha \|\lambda_y\|^2 )\|\Phi_{\#}^y\|^2 +\alpha \|A^-_y \Phi_{\#}^y\|^2 + \alpha \langle V_y\rangle_{\Phi_{\#}^y}\big) &= |\kappa|^2 \|\Phi_{\#}^y\|_{\#}^2 + O(\alpha^4\log(\alpha^{-1})),
\nabla k(x,\lambda ;\xi ,z) \cdot \xi = p{\frac{|z - x|^2 |\xi - x|^{p-2}}{\lambda^p}}|\xi^{x,\lambda}-z|^{p-2}\big(|\xi|^2 - z^{x,\lambda}\cdot \xi \big) & - p |\xi - z|^{p-2}(|\xi|^2 - z \cdot \xi ).
\gamma^{-4}\left|\overline{\mathcal{G}}_N\right|=\gamma^{-4}\max_{1\leq j \leq N}\left|G_j\Gamma \right|\leq \xi_{N,\gamma}, &\xi_{N,\gamma}= \max \left(\frac{\lambda_N\left| d_2-d_1\right|}{\gamma},\left(\frac{\lambda_N}{\gamma}\right)^2 \vert \kappa( d_1-d_3)\vert \right )
S ={\frac{k A_{H}}{4 \ell_{P}^{2}}}+ 4 \pi{\frac{k}{\hbar}}\int_{H}{\frac{\partial{\cal L}}{\partial R_{\mu \nu \lambda \rho}}}g_{\mu \lambda}^{\perp}g_{\nu \rho}^{\perp}\sqrt{{}_{2}g}d^{2}x .
4C_2\underline{\gamma}^2\log{n}+\sigma_0^2(2\log{2^{\sum_{l=1}^dj_l}}+\underline{\gamma}\log{n}) &\qquad+4\sigma_0\underline{\gamma}\sqrt{C_2\log{n}(2\log{2^{\sum_{l=1}^dj_l}}+\underline{\gamma}\log{n})}\leq\kappa(\underline{\gamma})\log{n}+2\sigma_0^2\log{2^{\sum_{l=1}^dj_l}},
T^{\,b_{N+1}-b_{N}+k-b_{n}}\,e_{b_{N}}=v_{N}\,\Bigl(\prod_{j=b_{N}+1}^{b_{N+1}-1}w_{j}\Bigr)\,T^{\,k-b_{n}}\,e_{b_{n}}-T^{\,k-b_{n}}\,e_{b_{N}}&=v_{N}\,\Bigl(\prod_{j=b_{N}+1}^{b_{N+1}-1}w_{j}\Bigr)\Bigl(\prod_{j=b_{n}+1}^{k}w_{j}\Bigr)\,e_{k}-\Bigl(\prod_{j=b_{N}+1}^{b_{N}+k-b_{n}}w_{j}\Bigr)\,e_{b_{N}+k-b_{n}}.
{\hat{x}}_{a}^{i}( \varepsilon ) = e^{i{\hat{H}}\varepsilon}{\hat{x}}_{a}^{i}( 0 ) e^{- i{\hat{H}}\varepsilon}\simeq{\hat{x}}_{a}^{i}( 0 ) +{\frac{{\hat{\pi}}_{a}^{i}( 0 )}{m}}\varepsilon +{\frac{e}{2 m^{2}}}\sum_{b = 1}^{N}\epsilon^{i j}[{\hat{B}}({\hat{\bf x}}_{b}) \delta_{a b}+{\frac{e}{\theta}}\delta ({\hat{\bf x}}_{a}-{\hat{\bf x}}_{b}) ]{\hat{\pi}}_{b}^{j}( 0 ) \varepsilon^{2}
\nu(\mathcal{E}_4)= \frac{[\ker(\Psi^0_U(\mathcal{F}))]\prod_{v\in S-S_{\infty}}[H^0(K_v,\mathcal{F}_v)_{tor}]\left[\left(\frac{\prod_{v\in S_{\infty}}H^0(K_v,\mathcal{F}_v)}{H^0_{et}(U,\mathcal{F})}\right)_{tor}\right]}{[\ker(\beta)]\left[\left(\frac{\prod_{v\in S}H^0(K_v,\mathcal{F}_v)}{H^0_{et}(U,\mathcal{F})}\right)_{tor}\right]}.
\int x^{m}(a+bx^{n})^{p}dx=\frac{x^{m-n+1}(a+bx^{n})^{p+1}}{b(m+np+1)}-\frac{a(m-n+1)}{b(m+np+1)}\int x^{m-n}(a+bx^{n})^{p}dx
n\tau_{1}^{(n)}= \frac{1}{4}\sum_{l=1}^{n+1}\partial_{2}^{2}\tau_{2}^{(n-l-1)}(b_{2}^{(l)}-\tilde{b}_{2}^{(l)}) - \frac{1}{4}\sum_{l=2}^{n}\partial_{2}\tau_{2}^{(n-l)}\partial_{2}(b_{2}^{(l-1)}-\tilde{b}_{2}^{(l-1)}) + \frac{1}{4}\sum_{l=2}^{n}\tau_{2}^{(n-l)}\partial_{2}^{2}(b_{2}^{(l-1)}-\tilde{b}_{2}^{(l-1)}).
\beta_j = \begin{cases}4(N+1) \mbox{if}z_j = 0 4 \mbox{if}z_j\neq0. \end{cases}\mbox{and}\displaystyle{\lim_{r\to 0}\lim_{k \to + \infty}\frac{1}{2\pi}\int_{B_r(z_j)}e^{av_k}=0, \forall \, j=1,...,m}.
\sup_{\eth \in{\cal B}_\delta({\cal A}_n)}\big| \sum^n_{i=1}\big[ g_{{\cal A}_n}(\varepsilon_i,\eth) -g_{{\cal A}_n}(\varepsilon_i,\eth^{(K_n)}) \big] \big| \leq \frac{C_1}{2}n^{1/2}|{\cal A}_n|^{1/2}d_n^{1/2}.
\begin{aligned}\sup\limits_{x \in [0,L]}E_{u.h}\leq \left(F_0+\int_0^x F\right)^{1/2}\left\{1-2C\vartheta_0^{-8}\left(F_0+\int_0^x F\right)^2x\right\}^{-1/4}.\end{aligned}
\mathrm{B}_{p}\equiv \exp \{\frac{1}{p}t_{p}( \overline{{{\rho}}}_{\mathrm{t o t , -}}( p ) \overline{{{\rho}}}_{\mathrm{t o t , +}}( - p ) - \overline{{{\rho}}}_{\mathrm{t o t , +}}( p ) \overline{{{\rho}}}_{\mathrm{t o t , -}}( - p ) ) \}.
\sum_{k=1}^{n-3}\sum_{\ell=k+1}^{n-2}(n-\ell-1)^2(p-1)^2p^{n-4}&= (p-1)^2p^{n-4}\sum_{k=1}^{n-3}\sum_{\ell=k+1}^{n-2}(n-\ell-1)^2 &= (p-1)^2p^{n-4}\sum_{k=1}^{n-3}\sum_{i=1}^{n-k-2}i^2 &= (p-1)^2p^{n-4}\sum_{k=1}^{n-3}\frac{1}{6}(n-k-2)(n-k-1)(2(n-k-2)+1) &= \frac{1}{6}(p-1)^2p^{n-4}\cdot \frac{1}{2}(n-2)^2 (n-3)(n-1).
e_1(s) =\gamma_{\xi}^{'}(s), \kappa_j(s) =\langle\frac{de_j(s)}{ds},e_{j+1}(s)\rangle, \ 1\leq j\leq n-1, \kappa_n(s) =\langle\frac{de_n(s)}{ds},e_{2n}(s)\rangle, \tau(s) =\langle\frac{d\gamma(s)}{ds}, T\rangle.
\psi^{\hat v^*_{2,\alpha}}_{\alpha, 1}(\theta, x) \ = \ \inf_{v_1 \in{\mathcal M}_1}E^{v_1, \hat v^*_{2, \alpha}}_x \Big[ e^{\theta \int^\infty_0 e^{-\alpha t}r_1(X(t), v_1(t, X(t)), \hat v^*_{2, \alpha}(\theta e^{-\alpha t}, X(t))) dt}\Big].
\Theta^{\mu\nu}(x) ={1 \over 2}\int_S d^2 S^{\mu\nu}(x(\sigma)) \delta^4(x-x(\sigma)) ,\quad d^2 S^{\mu\nu}:= d\sigma^1 d\sigma^2{\partial(x^\mu,x^\nu) \over \partial(\sigma^1,\sigma^2)}.
{\cal{B}}(x_j,y_j)=\left<{\rm exp}\left[\sum_{j=1}^n(s_j\Phi(x_j,t)+t_j\Phi(y_j,t))\right]~{\rm exp}\left[-i\sum_{j=1}^n(\omega(x_j,t)-\omega(y_j,t))\right]\right>
\frac{1}{s}\le \frac{a}{p}+\frac{1-a}{q}\textrm{if}a=0 \textrm{or}a=1 \textrm{or}\frac{1}{s}+\frac{\gamma_1}{n}=\frac{1}{p}+\frac{\gamma_2-1}{n}=\frac{1}{q}+\frac{\gamma_3}{n}.
\widetilde{L}_{k,1}^{( 1^{2})}(z) =\big[ kz^{3}-( k+1) z^{2}+2( k-1) z+2( k-1) \big] L_{k}^{1}(z) +z^{2}( 2-z)^{2}L_{k-1}^{2}(z) .
(M_r{V}^x)_S = 2^{r-|S|-1}\sum_a\binom{|S|}{2}\binom{|S|-2}{a-2}+2^{|S|}\sum_b \binom{r-|S|-1}{2}\binom{r-|S|-3}{b-2}&= 2^{r-4}[|S|^2-|S|+(r-|S|-1)(r-|S|-2)]
{\tilde{T}}^{\mu \nu}= \frac 1 2 \left[ \nabla^{\mu}\Psi^{\dag}, \nabla^{\nu}\Psi \right]_{+}+ \frac 1 2 \left[ \nabla^{\nu}\Psi^{\dag}, \nabla^{\mu}\Psi \right]_{+}- \frac 1 4 \nabla^{\mu}\nabla^{\nu}\left[ \Psi^{\dag}, \Psi \right]_{+},
\langle de^{i\theta}U(x_0)\,|\, de^{i\theta}V(x_0)\rangle_g=\sum_{j, k=1}^{n-1}a_j(0, 0)\overline b_k(0, 0)\langle de^{i\theta}Z_j(x_0)\,|\, de^{i\theta}Z_k(x_0)\rangle_g.
\frac{2 r_{n}+ t}{r_{n}+ t}f^{( n , 0 )}( r_{1}, \dots , r_{n - 1}, r_{n}+ t ) - \frac{2 r_{n - 1}+ t}{r_{n - 1}+ t}f^{( n , 0 )}( r_{1}, \dots , r_{n - 1}+ t , r_{n}) = 0 .
\frac{2}{3}\cdot \frac{1}{X+1}- \frac{1}{3}\cdot \frac{1}{2X+1}= \sum_{k=0}^\infty \left ( \frac{1}{3}\cdot (2(-1)^k + 2^k) \right ) X^k = \sum_{k=0}^\infty \ell_k X^k \enspace ,
{\cal L}= \bar{\psi}_{\alpha}( i \partial \! \! \! / - e A \! \! \! / ) \psi_{\alpha}+ \frac{1}{2}( \phi^{*}\bar{\psi}_{\alpha}^{c}\psi_{\alpha}- \phi \bar{\psi}_{\alpha}\psi_{\alpha}^{c}) - \lambda N \phi^{*}\phi \, \, \, \, .
X\omega= -\dfrac{(\phi\xi^\perp\omega)\eta^\perp(\phi X)+(d\omega\wedge\eta^\perp\phi)(X,\phi\xi^\perp)}{||\xi^\perp||^{2}(1-||\xi^\perp||^2)}=-\dfrac{(\phi\xi^\perp\omega)\eta^\perp(\phi X)}{||\xi^\perp||^{2}(1-||\xi^\perp||^2)}.
\overline{M'_i}=(\mathop{q-2}\limits_{m-1},\underbrace{q-1, \ldots, q-1,}_{m-2-j_1}\mathop{q-2}\limits_{j_1},\ldots,\mathop{q-2}_{j_{i-2}},\underbrace{q-1, \ldots, q-1}_{j_{i-2}-j_{i-1}-1},\mathop{q-2}_{j_{i-1}},\underbrace{q-1, \ldots, q-1}_{j_{i-1}}).
G^{B^+_1}(x, y)\ge \frac{c_1 x^p_d}{|x-y|^{d+\alpha+\beta_1+\beta_2}}\begin{cases}y_d^{2\alpha-p+\beta_1+\beta_2}\left(\log (|x-y|/y_d)\right)^{\beta_4}2\alpha-2p+\beta_1+\beta_2<0; y_d^p \left(\log (|x-y|/y_d)\right)^{\beta_4+1}2\alpha-2p+\beta_1+\beta_2=0. \end{cases}
H_n(s) =\sum\limits_{k=1}^n(-1)^{k-1}\binom{n}{k}{_{s+1}}F_s\left(\left\{1\right\}^{s},1-k;\left\{2\right\}^{s};1\right), H_n(\overline{s})= \sum\limits_{k=1}^n(-1)^{k-1}\binom{n}{k}{_{s+1}}F_s\left(\left\{1\right\}^{s},1-k;\left\{2\right\}^{s};-1\right),
\delta = 3\alpha_1 -\alpha_2 + \alpha_1\alpha_3 - \alpha_1\alpha_2 + \alpha_2\alpha_3 - 2\alpha_3 + 2 \kappa_1 = \frac{(1-\alpha_3)(1 - 0.5 \alpha_2)}{\delta}\kappa_2 = \frac{0.5\alpha_1(1 + \alpha_3)}{\delta}\kappa_3 = \frac{\alpha_1(1 - 0.5\alpha_2)}{\delta}.
T(x)^{-1}=\frac{Q(x)}{xQ'(x)}=\frac{1}{xQ'(x)}\int_0^{x}Q'(u)\, du &=\int_0^1\frac{Q'(xt)}{Q'(x)}\, dt\rightarrow \int_0^1 t^{\alpha-1}\, dt=\frac{1}{\alpha}.
\Phi_{w_4}(y) = \int_{\Re \mu = 0}h(\mu) K_{w_4}(y; \mu )\, (\mu) d \mu, & \Phi_{w_5}(y) = \int_{\Re \mu = 0}h(\mu) K_{w_4}(-y; -\mu )\, (\mu) d \mu , & \Phi_{w_6}(y_1, y_2) = \int_{\Re \mu = 0}h(\mu) K^{(y_1), (y_2)}_{w_6}((y_1, y_2) ; \mu )\, (\mu) d \mu.
0=\delta S=\int\delta(\sqrt{-g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}})d\lambda=\int\frac{\delta(-g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda})}{2\sqrt{-g_{\mu\nu}\frac{dx^{\mu}}{d\lambda}\frac{dx^{\nu}}{d\lambda}}}d\lambda
\Pi_{0 0}( 0 , k ) \approx \frac{e^{2}}{8 \beta}\left[ \beta k + \frac{8 \ln [ 2 + 2 \cos ( e_{N}\beta \tilde{A}_{0}) ]}{\pi}\exp \left\{{- \frac{\pi \beta k}{8 \ln [ 2 + 2 \cos ( e_{N}\beta \tilde{A}_{0}) ]}}\right\}\right]
\mathcal{W}^{(\ell)}= \left\{(w_1, w_2): w_1 \in \{0, 1, \cdots, |A^{\ast}| \}, \right. \left. w_2 \in \{0,1, \cdots, (1+ \delta_{\ell}) k_{\ell}\}, \right. \left. w_1 + w_2 >0, |A^{\ast}| + w_2 \leq (1+ \delta_{\ell}) k_{\ell}+ w_1 \right\}.
\bigg|\sum_{v\in U_1}(X_k^v-\ell_0)-\sum_{v\in U_2}(X_k^v-\ell_0)\bigg|\le 2~~\mbox{and}~~[k/2]-1\le \sum_{v\in U_i}(X_k^v-\ell_0)\le [k/2]+1, i=1,2,
\mathcal{L}({\bf P}_2, \lambda,\mu)={\log}_2\det\left({\bf I}_N+{\bf H}_{22}{\bf P}_2({\bf R}_{-2})^{-1}{\bf H}_{22}^H\right)- \mu\left(\mbox{tr}{({\bf P}_2)-P}\right)+\lambda\left(\mbox{tr}\left({\bf H}_{12}{\bf P}_2{\bf H}_{12}^H-(\bar E-E_{11})\right)\right),
\mathcal{E}= \mathcal{E}(x_1,\ldots,x_k;y_1,\ldots,y_k) = \{M\in\mathcal{M}^{k\times k}(\mathbb{N}\cup\{0\}): m_{i,\cdot}\le x_i\ \mbox{and}\m_{\cdot,j}\le y_j\}
R_{\ell,\ell'}^1 =R_{N_1}(u^{\ell},u^{\ell'})=\frac{1}{N_1}\sum_{i=1}^{N_1}u_i^\ell u_i^{\ell'}\,\,\mbox{and}\,\,R_{\ell,\ell'}^2=R_{N_2}(v^{\ell},v^{\ell'})=\frac{1}{N_2}\sum_{i=1}^{N_2}v_i^\ell v_i^{\ell'}.
\epsilon_{_{\rm m}}=\biggl[\frac{8k^2s B_1}{\ell_0^2}+\frac{128k^2B_2}{9\omega_1^2}\ln2\biggr]\biggl[1+\frac{3}{4}\omega_1\eta +{\cal O}(\eta^3)\biggr]+\biggl[\frac{9k^2s B_1\omega_1^2}{8\ell_0^2}+k^2B_2\left(1+2\ln 2\right)\biggr]\biggr[\eta^2 +{\cal O}(\eta^3)\biggr].
(u_h^m-u_h^{m-1},u_{h}^{m,1/2})= \frac{1}{2}(u_h^m-u_h^{m-1},u_h^m+u_h^{m-1}) =\frac{1}{2}(\|u_h^m\|_2^2-\|u_h^{m-1}\|_2^2).
\int [d\varphi^I]\,\epsilon^a \, \lbrace\,\,{\partial\over \partial \chi^a}\Big( S_1 \,+\,\hbar M_1 \Big)\,+\,J^r{\overline R}^r_a \,\rbrace\,exp{i\over \hbar}\Big(S_{_\Sigma}+ \hbar M_1 \,+\, J^A \,\phi_A \Big)\,=\,0
\Delta h(x) =\frac{\hat C}{\mu_c}\sum_{y\in G}\omega_0(x,y) [\mathbf{d}^{-\beta}(y) -\mathbf{d}^{-\beta}(x) ] & =-\beta \frac{\hat C}{\mu_c}\sum_{y\in G}\omega_0(x,y)\xi^{-\beta-1}[\mathbf{d}(y) -\mathbf{d}(x) ],
(cz+d)^{-s}G(Vz,s;A,B;r_{1},r_{2}) =\left( e(-s)\sum\limits_{\overset{m+R_{1}\leq0}{d(m+R_{1})>c(n+}R_{2})}+\sum\limits_{\substack{m,n}}\right) \frac{f^{\ast}(-mb)f(-nc)}{((m+R_{1})z+n+R_{2})^{s}}& =G(z,s;B_{-b},A_{-c};R_{1},R_{2})+\left( e(-s)-1\right) g\left(z,s;B_{-b},A_{-c};R_{1},R_{2}\right)
\left( \sum\limits_{j_1,\ldots,j_{s+1}=1}^{n}\left\vert T_{s+1}\left( e_{j_1},\ldots,e_{j_{s+1}}\right) \right\vert^{\frac{1}{1-\left|\frac{1}{\mathbf{p}}\right|_{\leq s+1}}}\right)^{1-\left|\frac{1}{\mathbf{p}}\right|_{\leq s+1}}& \leq 2^{(s-1)\left[1-\left|\frac{1}{\mathbf{p}}\right|_{\leq s}\right]}\|T_{s+1}\|,
d_M(x,z) \leqslant d_M(x,f(x))+ d_M(f(x), y) + d_M(y,f(z)) + d_M(f(z),z) = d_M(x,f(x))+ d_M(f(x), \eta(x)) + d_M(\eta(z),f(z)) + d_M(f(z),z) \leqslant 4\delta.
\{x^{i}, x^{j}\}_{1}= 0 , \quad \{x^{i}, \theta^{j}\}_{1}= - \{\theta^{j}, x^{i}\}_{1}= \omega^{i j}, \quad \{\theta^{i}, \theta^{j}\}_{1}= - \{\theta^{j}, \theta^{i}\}_{1}= \frac{\partial \omega^{i j}}{\partial x^{k}}\theta^{k}
q^{-1}c_{n-3}= \Big[\sum\limits_{i=1}^{\frac{n}{2}-2}b_i(q^{-(\frac{n}{2}-i)}+q^{\frac{n}{2}-2-i})+b_{\frac{n}{2}-1}q^{-1}\Big]2h^2 +(q^{-(\frac{n}{2}-1)}+q^{-(\frac{n}{2}-3)}+\ldots+q^{\frac{n}{2}-5}+q^{\frac{n}{2}-3})
X_*(T)=X^*(\widehat{T}) \xrightarrow{res}X^*(Z(\widehat{M_S})^{\Gamma}) \to X^*(Z(\widehat{M_S})^{\Gamma})_{\mathbb{Q}}\cong \mathfrak{A}_{M_S, \mathbb{Q}}\subset \mathfrak{A}_{\mathbb{Q}},
\Bigl| \Bigl( \frac{dG}{d\xi_2'}\Bigr) (\xi_2') \Bigr| = \Bigl| \frac{2 \xi_1'}{{\xi_2'}^3}\Bigl({\xi_2'}^4 - \frac{9{\xi_1'}{\eta_1'}^3}{16}\Bigr) \Bigr| \geq 2^{-100}N_1^2,
K_{n,m}^{(\alpha,\beta)}(f;x,p,q)=\sum_{k=0}^{n+m}\left[\begin{array}{c}n+m k \end{array}\right]_{p,q}\prod_{s=0}^{n+m-k-1}(p^s-q^s\dfrac{x}{b_n}) \left(\dfrac{x}{b_n}\right)^k \int_0^1 f \left(\dfrac{(1-t)[k]_{p,q}+[k+1]_{p,q}t+\alpha}{[n+1]_{p,q}+\beta}b_n\right)\mathop{d_{p,q}t},
\partial_{+ 2}\psi_{+}^{m}= ( e_{+ 2}+ f_{+ 2}) \psi_{+}^{m}+ d_{+}\lambda_{+}\sigma^{m}\lambda_{+}, \ \partial_{- 2}\psi_{-}^{m}= ( e_{- 2}+ f_{- 2}) \psi_{-}^{m}+ d_{-}\lambda_{-}\sigma^{m}\lambda_{-}.
\,\, \int_0^\infty \frac{1}{2\pi i}\int_C e^{-\lambda}b_0^{l_0}\otimes \cdots \otimes b_0^{l_n}d\lambda (r^{m-1}dr ) = \,\, \int_0^\infty \frac{1}{2\pi i}\int_Ce^{-\lambda}(k r^2-\lambda)^{-l_0}\otimes \cdots\otimes (k r^2-\lambda)^{-l_n}d\lambda (r^{m-1}dr ),
\begin{aligned}d\rho_0 =D_1\lambda_1 + D_2\lambda_2+ D_4\lambda_4+ D_5\lambda_5+ D_6\lambda_6+ D_7\lambda_7 d\widetilde{\rho}_0 =D_1\lambda_1 + D_2\lambda_2+ D_4\lambda_4+ D_5\lambda_5\end{aligned}
\begin{cases}A' = (\sigma_{n-2}\cdots\sigma_{2}\sigma_{1}^{2}) \in M_{A}B' = (\sigma_{2}\cdots \sigma_{n-2}\sigma_{n-1}^{2}) \in M_{B}\Delta_{B}= (\sigma_{2}\sigma_{3}\cdots \sigma_{n-1})(\sigma_{2}\sigma_{3}\cdots \sigma_{n-2})\cdots(\sigma_{2}\sigma_{3})(\sigma_{2}) \in M_{B}.\end{cases}
0\le\Phi_{I,\lambda,P}(f)-\Phi_{I,\lambda,Q}(f) =f(t_{1})-f(t_{0})+f(t_{1})-f(t_{2})-|f(t_{0})-f(t_{2})|-\lambda =2f(t_{1})-2\max\{f(t_{0}),f(t_{2})\}-\lambda \le2[f(t_{1})-f(t_{0})]-\lambda,
\int_{ax}^{x}| \theta(t+\delta t) - \theta(t) - \delta t|^2 dt > \int_I | \theta(t+\delta t) - \theta(t) - \delta t|^2 dt = \delta^2 \int_I t^2 dt &> \delta^2\int_{ax}^{x/2}t^2 dt = \frac{\delta^2 x^3}{3}\left( \frac{1}{8}- a^3 \right).
\int_{0}^{\infty}\! d t I m T_{1}( t ) = - \frac{m^{2}}{4}\int_{4 m^{2}}^{\infty}\! \! \frac{d t}{t^{2}}\left( 1 - \frac{4 m^{2}}{t}\right)^{\frac{1}{2}}= - \frac{1}{2 4}\enspace .
\displaystyle \int_{0}^{T}B_{t}dB_{t}=\displaystyle \frac{1}{2}B_{T}^2 -\frac{1}{2}T=\displaystyle \lim_{\substack{n\longrightarrow +\infty}}\sum_{\substack{j\in J\setminus J_{K(n)}}}B_{t_{j}}(B_{t_{j+1}}-B_{t_{j}}).
\exists !\,u_0\in L^1({\mathbb{R}}^n)\mbox{such that}u(t,\cdot)=\Gamma(t,0)u_0\mbox{in}L^1({\mathbb{R}}^n)\mbox{for all}t>0; &u \mbox{is a global weak solution of \eqref{eq1}in}L^\infty_{wcs}(L^1),
\int_{\left[ - \frac{1}{2}, \frac{1}{2}\right]^d \backslash [-h,h]^d}\left| f(x) - \left| tP \cap \mathbb{Z}^d \right| \right| F_N (x) \, \mathrm{d}x = O \left( t^{d-1}\frac{\log N}{hN}\right) .
w_\alpha(0)=1\quad\lim_{\alpha\to\infty}\left( \left\vert\frac{\nabla w_\alpha(0)}{w_\alpha(0)}\right\vert^n+\left\vert\frac{\nabla^2 w_\alpha(0)}{w_\alpha(0)}\right\vert^{\frac{n}{2}}\right) \leq 1
I_{0}+ 2 I_{2}= - \frac{\Gamma \left( 2 - \frac{D}{2}\right)}{( 4 \pi )^{\frac{D}{2}}}\int_{0}^{1}d x 2 ( 1 - x ) \sum_{n}\left[ \left( 1 - \frac{4}{D}\right) \frac{1}{[ k_{0}^{2}+ m^{2}]^{2 - \frac{D}{2}}}+ \frac{4}{D}\frac{2 - D / 2}{\Gamma ( 3 )}\frac{1}{[ k_{0}^{2}+ m^{2}]^{2 - \frac{D}{2}}}\right] .
I_3=\int_{\Omega_-^{\varepsilon}}G_1^-\varphi_{x_2}-G_2^-\varphi_{x_1}\, dx=\int_{\partial \Omega^{\varepsilon}_-}\varphi (G_1^- n_2^{\varepsilon}-G_2^- n_1^{\varepsilon})\, d\sigma+\int_{\Omega^{\varepsilon}_{-}}\varphi((G_2^-)_{x_1}-(G_1^-)_{x_2})\, dx.
\begin{aligned}\mathbb{M}(\xi_n,z_n ) = \mathbb{M}(\xi^0,z^0 )+\sum_{j=1}^{J-1}2^{-1}\| \lambda^j_n \|_{L^2}^2 + 2^{-1}\| \gamma^J_n \|_{L^2}^2 +o(1), & \mathbb{E}(\xi_n,z_n ) = \mathbb{E}(\xi^0, z^0 )+\sum_{j=1}^{J-1}2^{-1}\|\nabla \lambda^j_n \|_{L^2}^2 + 2^{-1}\|\nabla \gamma^J_n \|_{L^2}^2 +o(1).\end{aligned}
{^{\pm}}H_{\mu \nu}^{a b}= \partial_{\mu}{^{\pm}}A_{\nu}^{a b}- \partial_{\nu}{^{\pm}}A_{\mu}^{a b}+ \frac{1}{2}f_{c d e f}^{a b}{^{\pm}}A_{\mu}^{c d}{^{\pm}}A_{\nu}^{e f}.
\left\|\sum_{n=1}^\infty a_n \omega_n\right\|^p =\left\|\sum_{n=1}^\infty a_n V\tau_n\right\|^p=\left\|V\left(\sum_{n=1}^\infty a_n \tau_n\right)\right\|^p &=\left\|\sum_{n=1}^\infty a_n \tau_n\right\|^p=\sum_{n=1}^\infty|a_n|^p, \{a_n\}_n\in \ell^p(\mathbb{N}).
\frac{i}{2}\mathrm{T r}\, \log \widehat{\cal H}= \left. \frac{i}{2}\mathrm{T r}\, \log \left( \widehat{1}\Delta + 2 \widehat{E}^{\lambda}\nabla_{\lambda}+ \widehat{\Pi}\right) \right|_{d i v}= \frac{1}{2 \epsilon}\int d^{2}x \, \sqrt{g}\, \mathrm{T r}\, \left( \widehat{\Pi}+ \frac{R}{6}\widehat{1}- \widehat{E}^{\lambda}\widehat{E}_{\lambda}- \nabla_{\lambda}\widehat{E}^{\lambda}\right) ,
\mathcal{C}_i(p) :=\{(p_l,g) \in \{p_1,\dots,p_m\}\times H^{-s}(W) \subset \widetilde{H}^s(W) \times H^{-s}(W), \ (p_l,g)\in \mathcal{C}_i \}\subset \mathcal{C}_i.
f_{l}( z ) = \sqrt{| z |}e^{i \left({\frac{1}{2}}\mathrm{a r g}\, z + \pi l \right)}\sum_{n = 0}^{\infty}b_{n}\left( \sqrt{| z |}e^{i \left({\frac{1}{2}}\mathrm{a r g}\, z + \pi l \right)}\right)^{- 5 n}+ f_{l}^{n . p .}( z ) ,
\begin{array}{c}\mathbf{E}^{(k)}_n(x;a,b,c)= \sum\limits_{m=0}^n\sum\limits_{j=0}^m\sum\limits_{i=0}^j\frac{2(-1)^{m-j+i}}{j^k}\binom{j}{i}(x\ln c-(m-j+i+1)\ln a-(m-j+i+1)\ln b)^n.\end{array}.
\begin{array}{l}P^*\{u\ne \pi_nx \ \& \v\ne \pi_ny\}= P^*\{(\pi_{n}x, \pi_{n}y)\notin W^n\}\le P^*\{\pi_{n}x\notin Q\}+P^*\{\pi_{n}y\notin Q\}+P^*\{\|\pi_{n}x-\pi_{n}y\|_X\ge \rho\}\le 2P^*\{x\notin Q_0\}+\delta < 2\varepsilon +\delta < 3\varepsilon\ \ \ (n\ge n_0).\end{array}
I' \le \prod_{j > r}\left( \iint \frac{d\xi d\eta}{\left(\xi^2+\delta^2\right)^{\frac{3}{4}}}\left(1 + 4\zeta_j^2\right)^{-\frac{3}{4(3 - r)\alpha_j}}\exp\left[-\frac{\zeta_j^2}{1+ 4\zeta_j^2}\cdot\frac{2b_j^2}{\alpha_j}\right] \right)^{\alpha_j}=:\prod_{j > r}\big(I_{j}\big)^{\alpha_j}~.
\kappa_m\left(\sum_{i=1}^n c_i Z_{\gamma_1,\gamma_2}(t_i) \right)=\frac{1}{2}(m-1)!A(\gamma_1,\gamma_2)^m C_m(\gamma_1,\gamma_2;\mathbf{t},\mathbf{c})\sim c (-2\gamma_1-1)^{1/2}\rightarrow 0.
\sum_{s \in S}\delta_i(s) x^s = \mathrm{H}_S(x) \prod_{j = 1}^i (1 - x^{b_j})^{-f_i}= \mathrm{H}_S(x) \prod_{j = 1}^i \sum_{l = 0}^{+\infty}\binom{f_j+l-1}{l}x^{l b_j}.
U_{l_0}(x,0) =U_{l_0}(x_1-c\tau-\sigma^*,x'-\rho,-\tau) &=\cdots=U_{l_0}(x_1-cn\tau-n\sigma^*,x'-n\rho,-n\tau)\overset{n\rightarrow\infty}{\longrightarrow}p^+_{l_0}
C(u) =\left[(\varrho q(u)-sq(0)h(u))+v(sh(u)-\varrho q(u)g(u))\right]/\varrho q(u) begin{align*}0.2cm] <\left[(\varrho q(1)-sq(0)h(0))+v(sh(1)-\varrho q(0)g(u))\right]/\varrho q(u) begin{align*}0.2cm] =\left\{\left[\varrho (h(1)+\mu)-s\mu h(0)\right]+v(sh(1)-\varrho \mu g(u))\right\}/\varrho q(u).
A_{ij}= \int_{S^2}\frac{W_0}{2}[R_{ij}+2 \tilde \nabla \tilde{X}^i \cdot \tilde \nabla (X_j^{(3)}+ P_j)+\tilde{X^i}(S_j-\tilde{\Delta}(X_j^{(3)}+12P_j)] dS^2 -\frac{3}{4}\int_{S^2}W_0^2\tilde{X}^i\tilde{X}^j dS^2 -30 \int_{S^2}P_i P_j dS^2 -10 \int_{S^2}\tilde X^i W_0P_j dS^2.
P= Q\cup \{b_1,b_2\}\cup \Big\{\{b_i,c^j_e\}: e\in \hat{E_i}, i,j\in \{1,2\}\Big\}& \cup \Big\{\{f^j_e,c\},\{f^j_e,e^j\},\{d^j_e,c^j_e\},\{d^j_e,e^{(3-j)}\}: j\in \{1,2\}, e\in(\hat{E_1}\cup \hat{E_2})\Big\}.
\mathbb{P}^{x_0}_{p,BE}[\omega_{BE}\in\cdot|C_w]=\frac{\sum\limits_{k\geq 1}(1-p)^{k-1}\mathbb{P}^{x_0}[T_k<\infty]\mathbb{P}^{x_0}[LE[0,T_k]\in\cdot|T_k<\infty]}{{\sum\limits_{k\geq 1}(1-p)^{k-1}\mathbb{P}^{x_0}[T_k<\infty]}}.
\left( \begin{array}{cc}l_{11}l_{12}l_{21}l_{22}\end{array}\right)= \left( \begin{array}{cc}1 0 1 0 \end{array}\right),\hspace{3mm}\left( \begin{array}{cc}1 1 0 0 \end{array}\right) \hspace{3mm}\mbox{etc.}
k_{\mu}M^{\mu}(k;p_{1}\cdot\cdot\cdot p_{n};q_{1}\cdot\cdot\cdot q_{n})=e\sum_{i}[M_{0}(p_{1}\cdot\cdot\cdot p_{n};q_{1}\cdot\cdot\cdot(q_{i}-k)\cdot\cdot\cdot q_{n})
- \sum_{i=1}^{\ell}s(\ell - i)(\tau_{j(t) + i + 1}- \tau_{j(t) + i}) \leq - \frac{(1 - \eta/3) s a_N}{q(t-1) k_N}\sum_{i=1}^{\ell}(\ell - i) &= - \frac{(1 - \eta/3) \ell (\ell - 1)}{2 q(t-1) k_N}\log \bigg( \frac{s}{\mu}\bigg).
\partial_t ||\eta(t)||^2 = 2 \int_0^1\partial_s \langle *F_{B_{s,t}}+ \mu(v_{s,t}), \eta(t) \rangle \, ds = -2 \langle \Delta_{B_{s,t}}\eta + L_{v_{s,t}}^*L_{v_{s,t}}\eta(t) , \eta(t) \rangle = - \Delta (||\eta(t)||^2 ) - 2 \int_0^1 \left( ||L_{u_{s,t}}\eta(t)||^2 + ||d_{A_{s,t}}\eta(t)||^2 \right)\, ds