harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\left( I^{\alpha,\alpha',\beta,\beta',\gamma}_{0+}f\right)(x):= &\qquad=\frac{x^{-\alpha}}{\Gamma(\gamma)}\int_0^x (x-\theta)^{\gamma-1}\theta^{-\alpha'}F_3\left(\alpha,\alpha',\beta,\beta',\gamma, 1-\frac{\theta}{x}, 1-\frac{x}{\theta}\right)f(\theta)d\theta.
k_{1}= \frac{1}{(1+c^{2})(a+1)}+\frac{c^{2}}{a+1}, k_{2}= \left(\frac{1}{(a+1)(1+c^{2})}+\frac{c^{2}(1-a)}{2(a+1)}-\frac{1}{2}\right)k_{1}, k_{3}= \frac{k_{1}}{6(c-\alpha)}.
E\{C_{i_1}\cdots C_{i_k}\}= \frac{(\phi \theta)^k}{(1-m/n)^{1-\phi \theta}}\frac{1}{i_1 i_2\cdots i_k}\left(\prod_{j=1}^k\left(1 + O\left(\frac{1}{(\log i_j)^2}\right)\right) +o(1)\right).
\begin{aligned}w^* =\left(1-\alpha P(w,v)\right)w+\alpha P(w,v)v-\beta\left(m(t)-w_d\right) &\qquad-\alpha\frac{\beta}{2}((P(w,v)-P(v,w))(w-v))+\Theta_1 D(w), v^* =\left(1-\alpha P(v,w)\right)v+\alpha P(v,w)w-\beta\left(m(t)-w_d\right) &\qquad-\alpha\frac{\beta}{2}((P(v,w)-P(w,v))(v-w))+\Theta_2 D(w). \end{aligned}
\|u\|_{\alpha}:= \|u\|_{\infty}+ [u]'_{\alpha}, [u]'_{\alpha}:= \sup_{M^2_{T,\delta}}\left\{\dfrac{|u(p,t)-u(p',t')|}{d(p,p')^{\alpha}+|t-t'|^{\alpha/2}}\right\},
\left| < i_{l + 1}^{1}, k_{l + 1}^{1}, i_{l + 1}^{2}, k_{l + 1}^{2}| e^{- \frac{\varepsilon}{2}H_{1 , 2}}| i_{l}^{1}, k_{l}^{1}, i_{l}^{2}, k_{l}^{2}> \right| .
\left\{\begin{array}{ll}\frac{\partial^2u}{\partial t^2}-{\rm div}\,(a(t, z)|Du_t|^{p-2}Du_t) + \beta(z)u_t - \Delta u = f(t, z, u) + \gamma u_t\ \mbox{in}\T\times\Omega, u|_{T\times\partial\Omega}= 0,\u(0, z)=u_0(z),\u_t(0, z)=u_1(z),\end{array}\right\}
L_{- n}= \sum_{i = 1}^{N}\left( \alpha ( a_{i}^{\dagger})^{n + 1}a_{i}+ \beta a_{i}( a_{i}^{\dagger})^{n + 1}+ ( \lambda -{\frac{1}{2}}) ( n + 1 ) ( a_{i}^{\dagger})^{n}\right) \, ,
O_{F_{22}}\xi_n\equiv -\frac{d^2}{dz^2}\xi_n-\frac{2m^2\alpha}{\lambda^2}(2\alpha+\lambda) sech^2(mz)\xi_n+m^2( \frac{2\alpha}{\lambda^2}(2\alpha+\lambda)-\frac{2\alpha}{\lambda})\xi_n= \tilde{\omega}^2_n\xi_n.
\rightarrow\bigoplus_{j\in I_{i}}K_{n-1}(RH_{j})\oplus\bigoplus_{j\in I_{i}}K_{n-2}(RH_{j})\rightarrow\bigoplus_{j\in I_{i}}K_{n-1}(RH_{j})\oplus KR_{n-1}^{G}(X^{i-1})
E_{\small C a s i m i r}^{\small s u r f a c e}={\frac{1}{6 \pi^{2}}}\; S \; \hbar c \; K^{3}\; \left[{\frac{\xi_{\small i n}}{n_{\small i n}}}+{\frac{\xi_{\small o u t}}{n_{\small o u t}}}\right] .
\Lambda - \lambda_1(\ell) = \frac{8 \pi^2 |\partial_2 \psi_{\varkappa, 2}(0)|^4}{\zeta_1 ''(\varkappa)}\cdot \ell^4 + \mathcal O(\ell^5) , begin{align*}6pt]\Lambda - \lambda_2(\ell) = \frac{\pi^2 \varkappa^4 |\partial_2 \psi_{\varkappa, 2}(0)|^4}{8 \zeta_1 ''(\varkappa)}\cdot \ell^8 + \mathcal O(\ell^9) .
\delta^{4}( 1 2 ) \equiv \delta^{4}( \theta_{1}- \theta_{2}) = \delta^{2}( \theta_{1}- \theta_{2}) \, \delta^{2}( \bar{\theta}_{1}- \bar{\theta}_{2}) \equiv \delta^{2}( 1 2 ) \, \delta^{2}( \bar{1}\bar{2}) .
w(\sigma) = \frac{|X_\sigma(d-t-1)|}{{d+1 \choose t+1}\cdot |X(d)|}\leq \frac{Q \cdot{d+1 \choose k+1}\cdot |X_\sigma(k-t-1)|}{{d+1 \choose t+1}\cdot |X(k)|}={d-t \choose k-t}\cdot Q \cdot w_k(\sigma),
[\begin{matrix}b_{0,j}\\b_{1,j}\\b_{2,j}\\b_{3,j}\end{matrix}]=[\begin{matrix}2&3&1&1\\ 1&2&3&1\\ 1&1&2&3\\ 3&1&1&2\end{matrix}][\begin{matrix}a_{0,j}\\a_{1,j}\\a_{2,j}\\a_{3,j}\end{matrix}]0\le j\le3
\Lambda_0 =\{-p^e\}, \Lambda_z =\Big\{{\sum}_{i=e-z}^{e-1}a_i(p^e-p^{i})+p^e-2p^{e-z}\colon 0\le a_i<p\Big\}, 0<z<e; \Lambda_e =\Big\{{\sum}_{i=1}^{e-1}a_i(p^e-p^i)-p^e-2\colon 0\le a_i<p\Big\}.
\rho^h_2 = \frac{2-18\eta + (1+24\eta)\cos(\pi h) - 6 \eta \cos(2\pi h)}{(1 - 48\eta )(2+ \cos(\pi h)}, \rho_2 = \lim_{h \to 0}\rho^h_2 = \frac{1}{1-48\eta},
c_{0 0}= c_{4 4}, \qquad c_{0 2}= c_{4 2}, \qquad c_{0 4}= c_{4 0}, \qquad c_{2 0}= c_{2 4}, \qquad c_{2 2}, \qquad c_{1 1}= c_{3 3}, \qquad c_{1 3}= c_{3 1}.
\aligned\partial (\frak N^{i+1}_{32}\circ \mathcal H^i)= \frak N^{i+1}_{32}\circ \frak H_{(ab)}^{i+1}\circ \frak N^{i}_{1}\cup\frak N^{i+1}_{32}\circ \frak H_{(b,21)}^i &\cup\frak N^{i+1}_{32}\circ \frak N^{i}_{2}\circ \frak H_{(ab)}^{i}\cup\frak N^{i+1}_{32}\circ \frak H_{(a,21)}^i\endaligned
\bar{R}^\mathrm{J}=\log{\left|\mathbf{I}+\mathbf{\tilde{A}}(\mathbf{Q}^{\prime}_\mathrm{z}+\sigma^2{\mathbf{\Omega}_\mathrm{z}^{+}}^{-1}{\mathbf{\Omega}_\mathrm{z}^{+}}^{-\mathrm{H}})^{-1}\right|}.
p^{n_{s}^{\prime}-n_{1}^{\prime}}q^{n_{2}^{\prime}-n_{1}^{\prime}}a_{n_{2}}-p^{n_{2}^{\prime}+n_{s}^{\prime}-2n_{1}^{\prime}}a_{n_{1}}=p^{n_{2}^{\prime}-n_{1}^{\prime}}q^{n_{s}^{\prime}-n_{1}^{\prime}}a_{n_{s}}\frac{N_{n_{2}}}{N_{n_{s}}}.
\left | \frac{N(y)}{N(1)}\right | = \left | \frac{c_5}{b_3}\right| , \left | \frac{N(1)}{N(x)}\right | = \left | \frac{\widetilde{c_5}}{\widetilde{b_3}}\right|, \left | \frac{N(x)}{N(y)}\right | = \left | \frac{\widehat{c_5}}{\widehat{b_3}}\right|.
\sum_{m=3}^\infty \mu_m \delta_m^2 = \frac 12 \sum_{m=3}^\infty \frac{d^m}{m}\sum_{i,j=2}^q \lambda_i^m \lambda_j^m = \frac 12 \sum_{i,j=2}^q \sum_{m=3}^\infty \frac{(d\lambda_i \lambda_j)^m}{m}= \sum_{i,j=2}^q \log \psi(d\lambda_i \lambda_j),
T_\eta f (x) \gtrsim\begin{cases}(b-a) b^{-1}\eta \le 0, (b-a) b^{\eta - 1}(b-a+x)^{-\eta}\eta \in (0,1), \log \Big( 1 + \frac{b - a}{x}\Big) \eta = 1, (b-a) b^{\eta - 1}(b-a+x)^{-1}x^{-\eta + 1}\eta > 1,\end{cases}
\left\{{\frac{D e t^{\prime}\left[ - ( d^{2}/ d \tau^{2}) + V^{\prime \prime}[ x_{c}( \tau ) ] \right]}{D e t \left[ - ( d^{2}/ d \tau^{2}) + \omega^{2}\right]}}\right\}^{- 1 / 2}\, s q r t{{\frac{S_{e o}}{2 \pi}}}\d \tau_{c}
\mathcal{D}_{p}^{\mathcal{T}}(F,f,\kappa)=\min\{\kappa u^{p}+\frac{N^{p}-1}{N^{p}-N}(F-u^{p-1}f):f\leq u\leq\min((\frac{F}{f})^{1/(p-1)},\frac{f}{\kappa})\}
\frac{\sum_{j=1}^s a_j (2\left\lceil t^{1/b_j}\right\rceil/t^{1/b_j})^{b_j}}{\prod_{j=1}^s (\left\lceil t^{1/b_j}\right\rceil /t^{1/b_j}+t^{-1/b_j})^{p(s)}}\longrightarrow \sum_{j=1}^s a_j 2^{b_j}.
\begin{array}{ll}\nabla(\mathcal{D}^{\alpha}\xi_t)= -k_1 \varrho\mathcal{D}^{\alpha}v_t-\sum\limits_{|\alpha_1|>0}k_1\mathcal{D}^{\alpha_1}\zeta \mathcal{D}^{\alpha_2}v_t-\sum\limits_{|\alpha_1|\geq 0}k_1\mathcal{D}^{\alpha_1}\zeta_t \mathcal{D}^{\alpha_2}v.\end{array}
T^\vee_\pi T_i = X_1T_1\cdots T_{n-1}T_i = X_1 T_1\cdots T_i T_{i+1}T_i T_{i+2}\cdots T_{n-1}= X_1 T_1\cdots T_{i-1}T_{i+1}T_iT_{i+1}T_{i+2}\cdots T_{n-1}&= X_1 T_{i+1}T_1 \cdots T_{i-1}T_iT_{i+1}\cdots T_{n-1}= T_{i+1}X_1T_1\cdots T_{n-1}= T_{i+1}T^\vee_\pi.
\left. \begin{array}{c}{{\theta_{\beta \bar{\gamma}}= i u_{\beta \bar{\gamma}}^{\bar{\alpha}}}}\\{{\theta_{\alpha \delta}= i u_{\alpha \delta}^{\gamma}}}\\ \end{array}\right\}\Leftrightarrow \left\{\begin{array}{c}{{\theta_{\alpha \beta}= i u_{\alpha \beta}^{\gamma}}}\\{{\theta_{\beta}= \theta_{\delta}}}\\ \end{array}\right.
\mathcal{L}_{n,k}^{(\alpha )}(x) = \varpi_{n,k}^{(\alpha )}\sum\limits_{j = 0}^n{{{\left({\lambda_{j}^{(\alpha )}}\right)}^{- 1}}\,G_{j}^{(\alpha )}\left(x_{n,k}^{(\alpha )}\right)\,G_{j}^{(\alpha )}(x)},k = 0, \ldots ,n.
\begin{aligned}S_{e,\mathcal{YM}^0}(X,Y) =\big (\exp (\frac{||R^{\nabla}||^2}2) -1 \big ) g(X,Y)-\exp (\frac{||R^{\nabla}||^2}2)\langle i_XR^{\nabla},i_YR^{\nabla}\rangle \, , \end{aligned}
\mbox{\boldmath $\sigma$}_k=(\mbox{\boldmath $\sigma$}_{k-1}^{(2)}, \cdots, \mbox{\boldmath $\sigma$}_{k-1}^{(M)}, \mbox{\boldmath $0$})+\mbox{\boldmath $e$}_k(H_1^T, H_2^T, \cdots, H_M^T) .
l_{Z_*,n}'(0)={\lambda_{Z_*,n}^3}\prod_{i=0}^{n-1}{\xi_{k}^{'''}(0) \over \xi_{k+1}^{'''}(0)}={\lambda_{Z_*,n}^3}{\xi_{0}^{'''}(0) \over \xi_n^{'''}(0)}={\lambda_{Z_*,n}^3}.
( S_{n N}F ) ( z_{1}, \cdots , z_{N}) = \prod_{\mu = 1}^{n}\oint_{C}d x_{\mu}F ( x_{1}, \cdots , x_{n}| z_{1}, \cdots , z_{N}) \Psi ( x_{1}, \cdots , x_{n}| z_{1}, \cdots , z_{N}) .
|v|^{-s}|\partial^\alpha f(z+v)-\partial^\alpha f(z)| &\quad=|v|^{-s}\left|\int_0^1v\cdot\nabla\partial^\alpha f(z+t'v)\,dt' \right| &\quad\leq|v|^{1-s}\sup_{i\in\{1,\ldots,n\}}\sup_{t'\in(0,1)}|\partial^{\alpha+e_i}f(z+t'v)| &\quad\leq|y-z|^{1-s}\sup_{\beta\in\mathbb{Z}_+^n,\,|\beta|=N+1}\sup_{t\in(0,1)}\left|\partial^{\beta}f(z+t(y-z))\right|.
\bigl(v_1, p^V\bigl((X_\alpha+\theta X_\alpha)^2 v_2\bigr)\bigr) =\bigl(v_1, (X_\alpha+\theta X_\alpha)^2 v_2\bigr) &=\bigl((X_\alpha+\theta X_\alpha)^2 v_1, v_2\bigr) &=|\alpha|^2B(X_\alpha,\theta X_\alpha)\bigl((1-s_\alpha)v_1, v_2\bigr) &=0.
\|\varphi\|^p_{1,2, p, \lambda ; W^{2, p}(\mathbb{R}^d)}\ = \ \| \varphi \|^p_{p, ; W^{2,p, \lambda}(\mathbb{R}^d)}+ \| \frac{\partial \varphi}{\partial t}\|^p_{p; L^{p, \lambda}(\mathbb{R}^d)}, \ 1 \leq p < \infty.
v(t,x)=\cos\left(t\sqrt{-\Delta_g}\right)u_0+\frac{\sin(t\sqrt{-\Delta_g})}{\sqrt{-\Delta_g}}\,u_1+\int_0^t\frac{\sin\bigr((t-s)\sqrt{-\Delta_g}\bigr)}{\sqrt{-\Delta_g}}\,f(s)\,\mathrm{d}s.
S(W_{\bullet,\bullet}^{L_{xy}};p) =\frac{16n(4n+1)}{9(2n+1)}\int_0^\frac{3}{2}h_{p,L}(t)dt =\frac{4n+1}{18n(2n+1)}\left\lbrace\int_0^\frac{3}{4}\frac{1}{(4n+1)^2}(2(4n-1)t+3)^2dt+\int_\frac{3}{4}^\frac{3}{2}(3-2t)^2dt\right\rbrace =\frac{4n^2+3n+1}{4n(2n+1)(4n+1)}.
V(\alpha) =\Big\{{\sum}_{i=1}^m(1-p^{-d_i})\colon \alpha\le (d_1,\ldots,d_m), \d_1=z\Big\}, \Gamma_z =\Big\{{\sum}_{i=0}^{z-1}a_i(p^z-p^i)\colon 0\le a_i<p\Big\}.
g = \displaystyle \prod_{j = 1}^{n}g_{j}^{( 1 )}= \displaystyle \prod_{j = 1}^{n}\Biggl ({\frac{a_{0}^{( j )}+ e^{-{k_{j}}( r + i t )}}{\bar{a}_{0}^{( j )}+ e^{{k_{j}}( r + i t )}}}\Biggr )
|U_l| =|U_{l-1}\cup S_{j_l}|\geq|U_{l-1}|+|S_{j_l}|-|U_{l-1}\cap S_{j_l}| &\geq((l-1)\rho-\Delta_{l-2}t(\rho))|M|+\rho|M|-|\bigcup_{i=1}^{l-1}{S_{j_i}\cap S_{j_l}}| &>(l\rho-\Delta_{l-2}t(\rho))|M|-(l-1)t(\rho)|M|=(l\rho-\Delta_{l-1}t(\rho))|M|,
\widehat{U}_1(x, a,b,\theta) = \frac{W_q(x)}{W_q'(b+)}\widehat{U}_1^0(a,b,\theta) - U_1^0(a,x,\theta), \theta \geq 0, \widehat{U}_i(x, a,b) = \frac{W_q(x)}{W_q'(b+)}\widehat{U}_i^0(a,b) - U_i^0(a,x), i = 2,3, 4.
\begin{aligned}F^{ii}\nabla_{j}u \nabla_{iij}u \geq \, F^{ii}\nabla_{j}u (\nabla_{j}U_{ii}- \nabla_{j}A^{ii}) - C |\nabla u|^2 \sum F^{ii}\geq \, (\psi_{p_{l}}- F^{ii}A^{ii}_{p_l}) \nabla_{j}u \nabla_{jl}u + \nabla_j u \nabla_j (u_t) - C (1 + \sum F^{ii}).\end{aligned}
\phi_1 := \phi_{23AB}(\tau, z) = 2 \zeta^{\pm 1}- 3 + (-3 \zeta^{\pm 2}+ 10 \zeta^{\pm 1}- 14) q + (2 \zeta^{\pm 3}- 14 \zeta^{\pm 2}+ 32 \zeta^{\pm 1}- 40) q^2 + O(q^3)
\pm{\eta_1\over 2}\pm \cdots \pm{\eta_{2m}\over 2}= \sum_{j\in I_1}{\theta_j+{\varphi_1\over 2}\over 2}- \sum_{j\in \bar{I_1}}{\theta_j+{\varphi_1\over 2}\over 2}+ \sum_{j\in I_2}{\theta_j-{\varphi_1\over 2}\over 2}- \sum_{j\in \bar{I_2}}{\theta_j-{\varphi_1\over 2}\over 2},
C_{\texttt{a}}(\xi_1,\ldots ,\xi_n;q)=C_{\texttt{a}}(\boldsymbol{\xi};q) := \prod_{1\leq j<k \leq n}\left(\frac{1-q e^{-i(\xi_{j}-\xi_k)}}{1-e^{-i(\xi_{j}-\xi_k)}}\right) ,
{d^{C G}}_{H}{d^{D H}}_{I}f^{I E F}f^{B A C}= \int d^{2}\sigma \sqrt{w ( \sigma )}Y^{C}Y^{D}\{Y^{E}\, , \, Y^{F}\}\{Y^{B}\, , Y^{A}\}{}- f^{D E F}f^{B A C}.
\int_{r\le |x|}\theta(z,t) dz \le \frac{1}{r}\int_{r\le |x|}{|x|}\theta(z,t) dz\le \frac{C(d_{\theta_0},\|\theta_0\|_{L^\infty{(S)}})}{r}\mbox{for any}r>0 \mbox{for any}t\ge0.
+ \C^{(3)}_{a9z}I_h^{az}-\frac{1}{2}\Bigl( h_{99}-\phi +\frac{1}{2}h_{zz}\Bigr) I_h^{99}- C^{(1)}_9 I_h^{z9}- \frac{1}{2}\Bigl( \phi - \frac{1}{2}h_{zz}+ h_{99}\Bigr) I_\phi -C^{(3)}_{abz}I_s^{ab}- 2 C^{(1)}_a I_s^{az}
\begin{aligned}X_1=\frac{1}{\Delta}[-\nu_{R,I}\nu_{T_1,T_2}\eta_{S,S,V,I}/\delta_{U}^2\nu_{T_1,T_2}+\nu_{R,I}\eta_{T_1,T_1,T_2}\nu_{S,V}\nu_{T,1,2}\nu_{S,I}/\delta_{R}^2 ].\end{aligned}
\Bigg(\sum_{l=1}^{j}Z_l+\sum_{l=j+1}^kB_l^{(j)}\Bigg)\Bigg(Z_{j+1}+\sum_{l=j+2}^kA_l\Bigg){}= \Bigg(\sum_{i=1}^kS_p\Bigg)\Bigg(\sum_{i=1}^{k}S_{p+i}\Bigg) =\sum_{i=1}^{2k}S_i+\sum_{q\geq 2}\sum_{(p_1,\dots,p_q)\in \mathbb{I}_{2k}^q}\alpha_{(p_1,\dots,p_q)}[S_{p_1},\dots,S_{p_q}].
A^2 = (E+M+M^*+F)(E+M+M^*+F)= = E+EM+EM^*+EF + ME+M^2+MM^*+MF + \quad\quad\ + M^*E+M^*M+M^{*2}+M^*F + FE+FM+FM^*+F,\quad\quad\quad\quad\quad
\kappa(X, K_X+\Delta+f^*L) =\kappa(F, K_F+\Delta|_F)+\kappa(A, \widehat{\det}f_*\mathcal{O}_X(lD)\otimes\mathcal{O}_A(L)) &=\kappa(F, K_F+\Delta|_F)+\dim V^0(A, f_*\mathcal{O}_X(lD)\otimes\mathcal{O}_A(L)).
\Omega^2_{\Gamma_N}\coloneqq \{x\in \Omega \colon \textrm{there exists}y\in \Omega \textrm{such that}b(y'')=y_{d-1}\textrm{and}t\ge 0, \textrm{satisfying}x=(y'', y_{d-1}- (y_d-a(y'))(2L)^{-1},y_d-t)\}.
\gamma_{\mathbf{e}_{1}\mathbf{e}_{2}\mathbf{e}_{3}}=\left( 1,e^{\int \tau (s)\,ds}+a\int \cosh \left( \int \tau (s)\,ds\right)\,ds,e^{\int \tau (s)\,\,ds}+a\int \sinh \left( \int \tau (s)\,ds\right)\,ds\right).
\nu_\lambda\big(\eta(O)=2\big) =\lim_{t\rightarrow+\infty}P_\lambda\Big(\eta_t^{(\mathbb{Z}^d,\emptyset)}(O)=2\Big) &\geq\limsup_{t\rightarrow+\infty}\frac{1}{E_\lambda\big(\zeta_t^2(O)\big)}\geq \frac{1}{\sup_{t\geq 0}E_\lambda\big(\zeta_t^2(O)\big)}>0
\sum_{k=0}^{n-1}\dfrac{u}{\sqrt{n\Delta_n}}\int_0^1\left(\partial_{\theta}b(\theta(\ell),X_{t_k})\right)^{\ast}(\sigma\sigma^{\ast})^{-1}(X_{t_k})M_{k,n,1}d\ell\overset{\P^{\theta_0}}{\longrightarrow}0.
( \theta_{i})^{\circ}= \eta_{i a}[ 4 ; - R^{\prime}]_{1 2 3 4}^{a b_{1}b_{2}b_{3}}\theta_{b_{1}}\theta_{b_{2}}\theta_{b_{3}}, \quad 1^{\circ}= \theta_{1}\theta_{2}\theta_{3}\theta_{4}.
\left(\begin{array}{c}u_E(T) u_{Ex}(T) u_{Exx}(T) \cdots\end{array}\right)-\left(\begin{array}{c}u_E(0) u_{Ex}(0) u_{Exx}(0) \cdots\end{array}\right)=T_E\left(\begin{array}{c}u_E(0) u_{Ex}(0) u_{Exx}(0) \cdots\end{array}\right),
T^{(n)}=T^{(n)}_{\geq 1}(y_n)=\sum_{k\geq 1}\frac{1}{\sqrt{\lambda_1^{(n)}+y_n-\lambda_k^{(n)}}}P_k^{(n)},y_n=\sqrt{\frac{\log n}{n}}\lambda_1^{(n)}.
\dim_H E_\varphi(\Phi) = \liminf_{\ell\to\infty}\frac{\sum_{j=1}^\ell j(1/c-1) \log\beta}{d\sum_{j=1}^{\ell+1}j/c \log\beta - (\ell+1)(1/c-1)\log \beta}= \frac{1-c}d.
c_{n,\sigma p}^{-1}K_{j}(x)= -\int_{B_{4}\setminus B_{3}}\frac{\phi^{p-1}(y)}{|x-y|^{n+\sigma p}}dy-\int_{B_{R}\setminus B_{4}}\frac{dy}{|x-y|^{n+\sigma p}}&+\int_{B_{R}^{c}}\frac{|\mathcal{A}_{\varphi}(y)|^{p-2}\mathcal{A}_{\varphi}(y)}{|x-y|^{n+\sigma p}}dy:=\sum^{3}_{i=1}J_{i},
K^{(2r+1,r_R)}_n (x)=\sum_{i=-r}^{r}E^{(ii)}-\frac{1-x^2}{h_n(b,d,x)}\Big(\sum_{r_R < i \leq r}E^{(-i,-i)}+t_nE^{(r+1-i,r+1-i)}{}-\sum_{r_R < i \leq r}E^{(r+1-i,-i)}+t_nE^{(-i,r+1-i)}\Big).
{3\over b^2}{a^{\prime}\over a}\left({a^{\prime}\over a}+{n^{\prime}\over n}\right)-{3\over{c^2n^2}}\left[{\dot{a}\over a}\left({\dot{a}\over a}-{\dot{n}\over n}\right)+{\ddot{a}\over a}\right]-{{3k}\over a^2}=-{{8\pi G_5}\over c^4}\Lambda.
- \vec{\sigma}\cdot \vec{p}\psi_{+ ( 2 )}- e \vec{\sigma}\cdot ( \vec{t}\wedge \hat{r}) A ( r ) \psi_{+ ( 2 )}+ i m \psi_{+ ( 2 )}- i 3 \, v \, \vec{t}\cdot \hat{r}\psi_{+ ( 2 )}\phi ( r ) = 0 ,
\Vert M_n g\Vert_{L_s}^s =\frac{1}{n}\left[ \sum_{\substack{l=0 l\equiv_2 0}}^{n-1}\int_0^1 \vert g(w) \vert^s \mathrm{d}w+\sum_{\substack{l=1 l\equiv_2 1}}^{n-1}\int_1^2 \vert \tilde g(z) \vert^s \mathrm{d}z\right].
|\begin{matrix}e_{1}&e_{2}&e_{3}\\ &&\\ \sum_{i}h_{1i}\frac{\partial q^{i}}{\partial s}&\sum_{i}h_{2i}\frac{\partial q^{i}}{\partial s}&\sum_{i}h_{3i}\frac{\partial q^{i}}{\partial s}\\ &&\\ \sum_{j}h_{1j}\frac{\partial q^{j}}{\partial t}&\sum_{j}h_{2j}\frac{\partial q^{j}}{\partial t}&\sum_{j}h_{3j}\frac{\partial q^{j}}{\partial t}\end{matrix}|
\begin{cases}2a_1-2\lambda a_5-2a_2\beta=0, 2\lambda a_4-2\alpha a_5-2\beta=0, 2\lambda a_2-2\lambda a_3+2a_5-2\alpha=0, 2\beta a_4 -(a_2+2a_4+1)\eta>0, 2\lambda-2a_4-a_2 \eta>0, 2\alpha a_3-2\lambda-\eta>0,\end{cases}
c_0 + b_0 = \frac{1}{\theta_0^2}c_0 = \sum_{l=0}^{K-1}\frac{\theta_l}{1-\theta_l}\frac{\theta_{-1}^2}{\theta_{l-1}^2}\prod_{j=1}^{l}\frac{1}{1-\sigma^K_{j}}b_0
c_p =\begin{cases}C_{12}(p'/2-1)^{-1/p'}, 1<p<2 , C_{12}\max \{\sqrt{2}; (\ln 2 + c_*^{-1})^{1/2}\}, p=2 , C_{12}(c_* p'/2)^{1/2-1/p'}(\Gamma (1-p'/2))^{1/p'}, 2<p< \infty, C_{12}(2\pi)^{1/2}c_*^{-1/2}, p = \infty.\end{cases}
f(\ell)=\frac{12}{\eta(\tau)\eta(\ell\tau)\vartheta_3^2(q)\vartheta_3^2(q^\ell)}\,\operatorname{det}\!\begin{pmatrix}\eta(\tau) \eta(\ell\tau) q\dfrac{d\eta(\tau)}{dq}q\dfrac{d\eta(\ell\tau)}{dq}\end{pmatrix}.
\widetilde X^{j,N}_{t+1}= \Psi\left(t, \widetilde X^{j,N}_t, \widetilde\mu^{j,N}_t,\Phi_j(t,\widetilde X^{j,N}_t),\xi^j_{t+1}\right), \widetilde X^{1,N}_{t+1}= \Psi\left(t,\widetilde X^{1,N}_t, \widetilde \mu^{1,N}_t, \widetilde \Upsilon^{1,N}(t,\widetilde X^{1,N}, \widetilde \mu^{1,N}),\xi^1_{t+1}\right), \mathbb{P}_N,
\begin{array}{l}\int_{M}(R-\frac{1}{2}Tor)\left( \gamma ,\gamma \right) d\mu+\int_{M}\left\vert \gamma_{1,1}\right\vert^{2}d\mu -\frac{1}{2}\int_{M}Tor^{\prime}\left( \gamma ,\gamma \right) d\mu +\frac{1}{4}\int_{M}(3Q+2Pu)ud\mu =0.\end{array}
U_{n2}= u_n + \tfrac{3}{4}\Delta t \varphi_{1}( \tfrac{3}{4}\Delta t J_n)F(u_n), u_{n+1}= u_n + \Delta t \varphi_{1}(\Delta t J_n)F(u_n) + \Delta t \tfrac{32}{9}\varphi_3 (\Delta t J_n) (N_n (U_{n2})- N_n(u_n )) .
T_{\alpha,\beta,m}[Q] :=(1-x^2)Q''+(\beta-\alpha-(\beta+\alpha+2)x-2(1-x^2)\left(\log(P_m^{(-\alpha-1,\beta-1)})\right)'Q' &+(\alpha-\beta-m+1)m-2\beta(1-x)\left(\log(P_m^{(-\alpha-1,\beta-1)})\right)'Q
\begin{array}{l}{{f_{l k}^{( + )}f_{c d}^{( + )}- f_{l d}^{( + )}f_{c k}^{( + )}= \| f^{( + )}\| \varepsilon_{l c m}\varepsilon_{k d n}f_{n m}^{( - )},}}\\{{f_{k l}^{( - )}f_{d c}^{( - )}- f_{d l}^{( - )}f_{k c}^{( - )}= \| f^{( - )}\| \varepsilon_{l c m}\varepsilon_{k d n}f_{m n}^{( + )},}}\\ \end{array}
\frac{\partial h}{\partial z_1}= \bar{z_1}g^1+ \delta \frac{\partial g^1}{\partial z_1}-iz_2g^2+D \frac{\partial g^2}{\partial z_1}=0 \frac{\partial h}{\partial z_2}= -\bar{z_2}g^1+ \delta \frac{\partial g^1}{\partial z_2}-iz_1g^2+D\frac{\partial g^2}{\partial z_2}=0
\Delta_g = \frac{\partial^2}{\partial x_n^2}+ \Gamma^\alpha_{n\alpha}\frac{\partial}{\partial x_n}+ g^{\alpha\beta}\frac{\partial^2}{\partial x_\alpha\partial x_\beta}+ \biggl( g^{\alpha\beta}\Gamma^\gamma_{\gamma\alpha}+ \frac{\partial g^{\alpha\beta}}{\partial x_\alpha}\biggr) \frac{\partial}{\partial x_\beta}.
\frac{1+a_{\varepsilon}}{2\varepsilon}\left( s-1/2\right)^{2}=\frac{1+a_{\varepsilon}}{2\varepsilon}(s^{2}-s-1/4) & \geq\frac{1+a_{\varepsilon}}{2\varepsilon}((1-\delta)s^{2}-1/4-\frac{1}{8\delta})\forall\delta>0\,.
\left| z - \frac{p_n}{q_n}\right| \leq \sum_{k = 0}^\infty \frac{1}{| q_{n + k}q_{n + k + 1}|}\leq \frac{C_0}{| q_n |^2}, \textrm{where}C_0 = \frac{r_0 \theta^2}{\theta^2 - 1}.
\begin{aligned}\sum_{j=0}^{[(k-1)/2]}U_{0,0}(2ja+t)+\sum_{j=1}^{[k/2]}U_{0,0}(2ja-t)=\sum_{n=-\infty}^{\infty}u_{n}T_{n}\cos n\pi t=0,t\in (0,a), \sum_{j=0}^{[(k-1)/2]}V_{0,0}(2ja+t)-\sum_{j=1}^{[k/2]}V_{0,0}(2ja-t)=\sum_{n=-\infty}^{\infty}v_{n}T_{n}\sin n\pi t=0,t\in (0,a),\end{aligned}
\|\widehat{\psi}+g\| \geq|\widehat{\psi}(w_0)+g(w_0)|\geq2|g(w_0)|-|h(w_0)| &\geq2|g(y_0)|-2|g(y_0)-g(w_0)|-|h(w_0)| &>2-2\delta-\delta=2-3\delta,
\varphi^*\left(\begin{array}{c c}a b 0 d \end{array}\right) = \left(\begin{array}{c c c}a^{2 \, p^{e_1}}0 b^{2 \, p^{e_1}}0 1 b^{p^{e_1}}\, d^{p^{e_1}}0 0 d^{2 \, p^{e_1}}\end{array}\right) (\, e_1 \geq 0 \, ) .
| N_{\alpha, \lambda_j,\gamma}(P)-\gamma P|\leq |N_{\alpha, \lambda_j,\gamma}(n_{\ell_j,j})-\gamma n_{\ell_j,j}| + \sum_{r=\ell_j}^k 30^2 15 \Big(\frac{\lambda_j}{\lambda_j-1}\Big)^{3/2}\sqrt{\tau_{r,j}}(3+\ln \tau_{r,j})^2 \omega(r).
\mathcal{H}^{+}= \ell^{2}(V_{0}) = \ell^{2}(v_{0}) \oplus \ell^{2}(v_{2}) , \mathcal{H}^{-}= \ell^{2}(V_{1}) = \ell^{2}(v_{1}) \oplus \ell^{2}(v_{3})
\mathcal{P}(x;0)=\hat{\phi}_0(x) \sum_{n=0}^{\infty}c_n\hat{\phi}_n(x), \ \c_n=\bigl(\hat{\phi}_n(x),\hat{\phi}_0(x)^{-1}\mathcal{P}(x;0)\bigr) \ \ (\Rightarrow c_0=1).
A^{-1}=\begin{pmatrix}B^{-1}0 0 0 D^{-1}0 0 0 E^{-1}\end{pmatrix}{\sum_{k=0}^{\infty}}(-1)^{k}\begin{pmatrix}0 CD^{-1}0 CB^{-1}0 0 0 0 0\end{pmatrix}^{k}.
m\left[\begin{array}{cc}a_{11}(A) 0 0 0 \end{array}\right]\left[\begin{array}{cc}0 M 0 I_{\mathcal{B}}\end{array}\right]+n\left[\begin{array}{cc}0 M 0 I_{\mathcal{B}}\end{array}\right]\left[\begin{array}{cc}a_{11}(A) 0 0 0 \end{array}\right]=0.
V^\mathcal{D}_\mathcal{N}(\mathbf{a}, \mathcal{B}^\mathcal{D}) = V_\mathcal{N}(\mathcal{B}) \cdot{\rm diag}\left( \beta_1^{(1)}, \beta_2^{(1)}, \ldots , \beta_{M_\ell}^{(\ell)}\right) ,
\Big|\int_{B_{2\rho}\setminus \overline{B_\rho}}T_2(\nabla \varphi, \nabla \Delta \varphi)\Big|\leq C\int_{B_{2\rho}\setminus \overline{B_\rho}}|\nabla \varphi||\nabla \Delta \varphi|d\mu_g \leq C\int_{B_{2\rho}\setminus \overline{B_\rho}}|u_\epsilon'||(\Delta u_\epsilon)'|dx+O(\epsilon^{n-6})=O(\epsilon^{n-6}).
\sum_{a=1}^q\left(\frac{f(a)}{q}\right)=\prod_{j=1}^k \left(\sum_{a_j=1}^{p_j^{\alpha_j}}\left(\frac{f(a_j)}{p_j^{\alpha_j}}\right)\right)= \prod_{j=1}^k \left(\sum_{a_j=1}^{p_j^{\alpha_j}}\left(\frac{f(a_j)}{p_j}\right)^{\alpha_j}\right),
\begin{pmatrix}\bar{x}_{t,n}\bar{x}_{t,1}\end{pmatrix}= \int_0^t S(t-s)\begin{pmatrix}F(s) G(s) \end{pmatrix}ds - \int_0^t S(t-s)\begin{pmatrix}0 \sigma_1 dW_s^{(1)}\end{pmatrix}.
\left. \frac{\langle \varphi_{4}^{2}\rangle}{M_{p , e f f}^{2}}\right\vert_{t = t_{p}}\sim C_{0}^{2}\left( \frac{H^{2}}{M_{p}^{2}}\right) \left( \frac{p}{H}\right)^{\frac{3 \Delta + 8}{\Delta + 2}}.
\begin{array}{rcl}\displaystyle{dX_t}= \displaystyle{b(X_t,v_t)dt+\sigma(X_t) dW_t - \gamma(X_t)d\xi_t}\displaystyle{d\xi_t}= \displaystyle{I_{\{X_t\in \partial D_{lm}\}}d\xi_t}\displaystyle{\xi_0}= \displaystyle{0,\ \ \ \X_0 = x\in \overline{D}_{lm}\, .}\end{array}
\hat{A}_i= \begin{pmatrix}U 1 0 \hdotsfor{3}0 \theta_{11}U 2 0 \hdotsfor{2}0 0 \theta_{11}U 3 0 \cdots 0 \hdotsfor{7}0 \hdotsfor{2}0 \theta_{11}U M-i 0 \hdotsfor{3}0 \theta_{11}U \end{pmatrix}, i=3,\cdots,M,
H = \frac{1}{2m}\left(\sum_i p_i^2 + \lambda \left(\sum_i x_i p_i\right)^2\right) +{\cal V}(r) = \frac{1}{2m}\left((1+\lambda r^2) \sum_i p_i^2 - \lambda \sum_{i<j}J_{ij}^2\right) +{\cal V}(r),
\|\tilde{G}\|_{C^{0,\gamma}(\mathcal{N}_{0}\cap (B_{1}\setminus B_{1/4}))}\leq C \lambda^{-\frac{3}{2}-\alpha}\lambda^{2-\frac{n+1}{p}}=C\lambda^{\frac{1}{2}-\alpha-\frac{n+1}{p}}.
\frac{|\lambda_2|}{\lambda_1^{\frac12}}\int|x||\nabla u^-_{R}|^2 -\frac{(d-1)}{2}\frac{|\lambda_2|}{\lambda_1^{\frac12}}\int\frac{|u_{R}|^2}{|x|}= \frac{|\lambda_2|}{\lambda_1^{\frac12}}\int\left|\nabla\left(|x|^{\frac12}u_R^-\right)\right|^2 -\frac14\frac{|\lambda_2|}{\lambda_1^{\frac12}}\int\frac{\left|u_R^-\right|^2}{|x|}\,.
\sum_{k=0}^n{n\choose k}2^k F_{jk}(\sqrt{5}F_j)^{n-k}B_{n-k}(\alpha) n F_j 2^{1-n}\Big ( (\sqrt{5}F_j + L_{j+3})^{n-1}+ (-\sqrt{5}F_j + L_{j+3})^{n-1}\Big ),
\sum_{l=1}^M\sum\limits_{\substack{j=1 j\neq l}}^M\mathbb{E}\left[y_l^* y_j|\hat{\mathbf{H}}\right]= 2\sum_{l=1}^{M-1}\sum_{j=i+1}^{M}\sum_{q=1}^M a_r^2\Re\left[\hat{\phi}^{\left(l,q\right)^*}\hat{\phi}^{\left(j,q\right)}\right]+\sum_{l=1}^M\sum\limits_{\substack{j=1 j\neq l}}^M a_c^2\hat{\phi}^{\left(l,c\right)}\hat{\phi}^{\left(j,c\right)}.
\left( \Gamma^{\mu}( i \partial_{\mu}\, + \, e A_{\mu}) \, - \, d \left( \begin{array}{c c}{{0}}&{{\gamma^{\mu}\gamma^{\nu}\gamma_{5}}}\\{{\gamma^{\mu}\gamma^{\nu}\gamma_{5}}}&{{0}}\\ \end{array}\right) F_{\mu \nu}\, - \, m \right) \psi = 0
G_{1}= \rho_{1}F^{3}+ \lambda F^{1}, G_{2}= \rho_{2}F^{4}+ \lambda F^{2}, G_{3}= \rho_{3}F^{5}+ \gamma \mathrm{div}\, F^{2}+ \tfrac{\tau_{0}\kappa}{\lambda + \delta}\mathrm{div}\, F^{6}.
\exp \left\{\mathrm{~ \cal{F}~}_{A_{2 N - 1}}( t ; \hbar ) \right\}= \langle 0 | \exp \left\{\sum_{k \not \equiv 0 ~ m o d 2 N , ~ k \geq 1}\frac{t_{k}}{\hbar}J_{k}\right\}g_{0}| 0 \rangle ,
\theta^0_1=\tilde{\theta}^0-\varepsilon^{-1},\theta^0_2=\varepsilon^{-1},t=\varepsilon\tilde{t},H=\varepsilon^{-1}\tilde{H}, q_1=\tilde{q}_1,p_1=\tilde{p}_1+\frac{-\tilde{\theta}^0+\varepsilon^{-1}}{\tilde{q}_1}, x=\varepsilon^{-1}\tilde{x},Y=\begin{pmatrix}1 0 0 0 \varepsilon 0 0 0 \varepsilon q_1\end{pmatrix}\tilde{Y}.
\limsup_{n\to \infty}\left[\frac{\sum_{i=0}^n \sigma_i\xi_{i+1}}{\sqrt{\sum_{k=0}^{n}\sigma^2_k}}-\frac{\sum_{i=0}^n \sigma_i\mu_{i+1}}{\sqrt{\sum_{k=0}^{n}\sigma^2_k}}\right]=\infty.
\Delta_F\otimes I(Y\otimes I_{V\otimes E}+ 1 \otimes (c\otimes \sigma(Y))) &=Y\otimes I_{V\otimes E\otimes F}+ 1\otimes \pi(Y)\otimes I_{V\otimes E}+ 1 \otimes (c\otimes \sigma(Y)) &=Y\otimes I_{V\otimes E\otimes F}+ 1\otimes (\pi\otimes c\otimes \sigma (Y))
\frac{\partial T^{ab}}{\partial w_{ij}}=\frac{\partial T^{ab}}{\partial w_{ji}},\frac{\partial T^{ab}}{\partial w_{ijk}}=\frac{\partial T^{ab}}{\partial w_{jik}},\frac{\partial T^{ab}}{\partial w_{ijkl}}=\frac{\partial T^{ab}}{\partial w_{jikl}}=\frac{\partial T^{ab}}{\partial w_{ijlk}}=\frac{\partial T^{ab}}{\partial w_{jilk}}.
\sum_{n = 1}^N e^{2\pi i \xi n}=\psi(N+1)^{-1}\sum_{n = 1}^N \psi(n) e^{2\pi i \xi n}+ \sum_{n = 1}^N \big(\psi(n)^{-1}- \psi(n+1)^{-1}\big) \sum_{k = 1}^n \psi(k) e^{2\pi i \xi k}.