harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
u(k,w) =\inf_{\mathbb{Q}\in B_{k}(\mathbb{P})}\mathbb{E}_{\mathbb{Q}}\left[U(x_0+ x_0 \langle w, X \rangle)\right] &\leq \inf_{\mathbb{Q}\in B_{k}(\mathbb{P})}\mathbb{E}_{\mathbb{Q}}\left[U(x_0+ x_0 \langle w^u, X \rangle)\right]=u(k,w^u).
c_{-j}:= - \frac{1}{2}\int_{-1/2}^{1/2}\left\{\log\left( 1 + \left( \frac{y/2\pi + \theta_1/2 + j}{x/2 \pi}\right)^2 \right) - \log\left( 1 + \left(\frac{y/2\pi + \theta_1/2 + j + \phi}{x/ 2 \pi}\right)^2 \right) \right\}\mathrm{d}\phi,
S = \sum_{n=0}^{\infty}\int d^6x \ [\dot{A}_{ab}^{(n)}E^{(n)}_{ab}- \frac{(-1)^n}{4}(E_{ab}^{(n)}E_{ab}^{(n)}+ B_{ab}^{(n)}B_{ab}^{(n)}) +{\Lambda}_{ab}^{(n+1)}{\Psi}_{ab}^{(n+1)}- 2 A_{ta}^{(n)}{\cal G}_a^{(n)}] ,
\begin{array}{l}\qquad \partial_0 x^\mu=-(p^\mu+(pP_y)P^\mu_y)-i(\bar\theta\Gamma^{\mu\nu}\lambda_\theta)P_{y\nu}, \qquad \partial_0 p^\mu=-\partial_1\left [ \partial_1 x^\mu-i(\bar\theta\Gamma^{\mu\nu}\partial_1\theta)P_{y\nu}+i\bar\theta\Gamma^\mu\lambda_\theta\right ], \qquad \partial_0\theta^\alpha=-\lambda^\alpha_\theta,\end{array}
\lim_{\beta \rightarrow \infty}\; \lim_{m \rightarrow 0^{-}}\langle{\bar{\psi}}\psi \rangle_{\beta}= \lim_{\beta \rightarrow \infty}\; \lim_{m \rightarrow 0^{+}}\langle \bar{\psi}\psi \rangle_{\beta}= \frac{e B}{4 \pi}\; .
I_q\otimes \mathcal{J}_{m-1}^{(q)}=I_q\otimes J_q \otimes \mathcal{A}_{m-2}^{(q)}, Q\otimes \mathcal{A}_{m-1}^{(q)}=(A_1-A_2)\otimes \mathcal{A}_{m-1}^{(q)},
||y(t)-y(s)||_{L^2}= \left| \left| \int_s^t(\partial_ty)(\tau)d\tau \right| \right|_{L^2}\le \left| \int_s^t||(\partial_ty)(\tau)||_{L^2}d\tau \right| \le |t-s|||\partial_ty||_{L^{\infty}(0,\tau_1;L^2)}\le M_1|t-s|,
I^2(x) =I^{2,1}(x)+I^{2,2}(x) =(2-\sigma)\int_{|y|>3/4(R_2''-R_2')}u(x+y)\partial_{y,e}[(1-\eta(x+y))\frac{\langle A(x)y,y\rangle\rho(x,y)}{|y|^{n+\sigma+2}}]dy &+(2-\sigma)\int_{|y|>3/4(R_2''-R_2')}u(x-y)\partial_{y,e}[(1-\eta(x-y))\frac{\langle A(x)y,y\rangle\rho(x,y)}{|y|^{n+\sigma+2}}]dy.
e T_{\mu \nu}={\tilde{\overline{\Psi}}_{L_{0}}}{\gamma_{\mu}}i \left( D_{\nu}- \frac{1}{2}\Gamma_{\alpha \nu}^{\alpha}\right){\tilde{\Psi}}_{L_{0}}- \frac{i}{2}g_{\mu \nu}\partial_{\alpha}({\tilde{\overline{\Psi}}}_{L_{0}}\gamma^{\alpha}{\tilde{\Psi}_{L_{0}}}) . \nonumber \,
0\leq \int_{\Omega}\big(F(x,D v)-F(x,D u)\big) d x+ \int_{B_{r_1}}\int_{B_{r_1}}a(x,y)\big(|v(x)-v(y)|^q-|u(x)-u(y)|^q\big) d \mu &\quad+2 \int_{\mathbb{R}^{N}\backslash B_{r_1}}\int_{B_{r_1}}a(x,y)\big(|v(x)-v(y)|^q-|u(x)-u(y)|^q\big) d \mu+\int_{B_{r_1}}(v-u)f(x) d x &=:I_{1}+I_{2}+I_{3}+I_{4}.
F^1(f,0) = \left( \frac{t}{2t^3 + t^2 + 2 + r}, 1 \right), F^2(f,0) = \left( \frac{t^2}{2t^3 + t^2 + 2 + r}, 2 \right), F^3(f,0) = \left( \frac{t^2 + 2 + r}{t^3}, -3 \right), F^4(f,0) = \left( \frac{2 + r}{t^2}, -2 \right), F^5(f,0) = \left( \frac{2 + r}{t}, -1 \right), F^6(f,0) = \left( r, 0 \right).
t_{k+1}=\inf\{t>t_{k}~|~\tilde{f}(\tilde{\vartheta}_{2}(t),\tilde{\eta}(t),\tilde{e}(t),\tilde{\xi}_{1}(t))\geq \sigma^{2}|\vartheta_{2}(\check{\xi}_{2}(t))\check{\xi}_{2}(t)|+\delta^{2}\}
(c^{T}\Sigma_{XX}^{-1/2}\Sigma_{XY}\Sigma_{YY}^{-1/2})(d)\le(c^{T}\Sigma_{XX}^{-1/2}\Sigma_{XY}\Sigma_{YY}^{-1/2}\Sigma_{YY}^{-1/2}\Sigma_{YX}\Sigma_{XX}^{-1/2}c)^{1/2}(d^{T}d)^{1/2},
KL (P_{f_{0,n}}, P_{f_{1,n}}) = 2n \int \big( \sqrt{f_{0,n}(x)}- \sqrt{f_{1,n}(x)}\big)^2 dx & \leq 2n \int |f_{0,n}(x) - f_{1,n}(x)| dx = 2nRh_n^{\beta+1}\| K \|_1 = 2Rc_0^{\beta+1}\|K \|_1 \leq 1,
P_{\xi_{\tau},\check{\xi}_{\tilde{\tau}}}\left(\pi_{\nu}(g_\nu)v_{\pi_{\nu}}\right)=\sum_{1\leq i,j\leq r}\left(\pi_{\nu}(g_\nu)v_{\pi_{\nu}},\phi_j\right)\int\limits_{K_{n,\nu}}\left<\xi_\tau,\tilde{\tau}\left(x^*\right)\check{\xi}_{\tilde{\tau}}\right>\left(\pi_{\nu}(x^{-1})\phi_{i},\phi_j\right)dx\cdot \phi_i.
\mathcal{L}(v^{n+1}-v^n,q^{n+1}-q^n,\Theta^{n+1}-\Theta^n) =\mathbb{F}^{m+1}(u^n-u^{n-1},p^n-p^{n-1},\theta^n-\theta^{n-1}), v^{n+1}(0)-v^n(0)=0, q^{n+1}(0)-q^n(0)=0,\Theta^{n+1}(0)-\Theta^n(0)=0.
\sum_{n=1}^{\infty}\frac{q^n}{(-q^n;q)_{n+1}(-q^{2n+2};q^2)_{\infty}}=\sum_{j=0}^{\infty}(-1)^j q^{6j^2+4j+1}(1+q^{4j+2}), \sum_{n=0}^{\infty}q^n (q^{n+1};q)_n (q^{2n+2};q^2)_{\infty}=\sum_{j=0}^{\infty}(-1)^j q^{3j^2+2j}(1+q^{2j+1}).
\mathbb{P}(u)_{k}(x)=\frac{1}{(2\pi)^{n/2}}\int_{\mathbb{R}^{n}}(\delta_{kj}-\frac{\xi_{k}\xi_{j}}{|\xi|^{2}})\hat{u}_{j}(\xi)e^{i\xi\cdot x}d\xi,1\le k\le n
\varphi_{m,k}(\hat{x})\varphi_{m,k}(\hat{y}) =\big(\hat{x}\oplus \big(\oplus_{j=k+1}^m \rho_{j,k}(x)\big)\big)\big(\hat{y}\oplus \big(\oplus_{j=k+1}^m \rho_{j,k}(y)\big)\big) &=\hat{x}\hat{y}\oplus \big(\oplus_{j=k+1}^m \rho_{j,k}(x)\rho_{j,k}(y)\big), \varphi_{m,l}(1_{B_l}) =1_{B_l}\oplus \big(\oplus_{j=l+1}^m \rho_{j,l}(1_{F_l})\big),
P^1=\langle 1,2\rangle \langle 2,2\rangle \ldots \langle \lambda-1,2\rangle \langle \lambda-1,1\rangle. &P^2=\langle 1,2\rangle \langle 2,2\rangle \ldots \langle \lambda-1,2\rangle \langle \lambda,2\rangle. &P^3=P^2 \langle \lambda,1\rangle. &P^4=\langle 2,2\rangle \langle 2,1\rangle \langle 3,1\rangle\ldots \langle \lambda,1\rangle.
H(\eta, \overline{\eta}^{\diamond}) = g(\eta^{\mbox{\tiny \sc Time}}+ (\overline{\eta}^{\diamond})^{\mbox{\tiny \sc Time}}) H^{*}(\eta, \overline{\eta}^{*}) H^{\mbox{\tiny $\mathcal{D}$}}(\eta, \overline{\eta}^{\diamond})\;,
f_{\Omega_{m},\Omega_{m+1}}^{\#}\left( x\right) =\sup_{\substack{B\left(\overline{x},r\right) \ni x \overline{x}\in\Omega_{m},r\leq\varepsilon_{m}}}\frac{1}{\left\vert B\left( \overline{x},r\right) \right\vert}\int_{B\left( \overline{x},r\right)}\left\vert f\left( y\right)-f_{B\left( \overline{x},r\right)}\right\vert dy.
- \frac{4}{\Gamma ( \frac{1}{2}d ) \Gamma ( 1 - \frac{1}{2}d )}\frac{( v^{2})^{\frac{1}{2}d - 1}}{1 + v^{2}}{}_{2}F_{1}\left({\textstyle{1 , \frac{1}{2}d - \frac{1}{2}; \frac{1}{2}d ; \frac{v^{2}}{1 + v^{2}}}}\right) \,{\cal I}_{0}\ ,
\langle \Psi | \int_{- \infty}^{\infty}{\widehat T}_{0 0}( t , 0 , 0 , 0 ) b ( t ) \d t | \Psi \rangle / \langle \Psi | \Psi \rangle \geq -{\frac{3}{3 2 \pi^{2}}}{\frac{\hbar c}{( c t_{0})^{4}}},
b=a-u \in (\dot{\mathrm{B}}^{\frac{1}{p}}_{p,1}(\Omega)+\mathrm{B}^{s}_{p,q}(\Omega))\cap\dot{\mathrm{B}}^{1+\frac{1}{p}}_{p,1}(\Omega)\subset \dot{\mathrm{B}}^{\frac{1}{p}}_{p,1}(\Omega)\cap\dot{\mathrm{B}}^{1+\frac{1}{p}}_{p,1}(\Omega)
f^{(\alpha)}(|p|) = \Omega_{\alpha}^2 \frac{2^{\alpha}}{\sqrt{\pi}}\frac{1}{(p-\frac{1}{2})!}\int_0^1\xi^{\frac{\alpha}{2}}\frac{d^p}{d\xi^p}\left\{\xi(1-\xi)\right\}^{p-\frac{1}{2}}{\rm d}\xi
\begin{pmatrix}-27.4310 +50.9483i -4.0647 + 5.7543i \hphantom{-.}0.0000 + 0.0000i \hphantom{-.}2.7694 - 1.2220i \hphantom{-.}0.6875 + 0.9203i\end{pmatrix}+ \mathbb{C}\begin{pmatrix}1-3i 0 1 0 0\end{pmatrix}
\left[\begin{array}{cc}\Sigma_1 0 0 0 \end{array}\right]\left[\begin{array}{cc}\tilde \Lambda_{11}+ \tilde \Lambda_{11}^H \tilde \Lambda_{12}+ \tilde \Lambda_{21}^H \tilde \Lambda_{21}+ \tilde \Lambda_{12}^H \tilde \Lambda_{22}+ \tilde \Lambda_{22}^H \end{array}\right]\left[\begin{array}{cc}\Sigma_1 0 0 0 \end{array}\right] = 0
C_p = \sum_j \left [(X_j.\gamma_{12}^p) u_j-\sum_k \gamma_{12}^k \gamma_{kj}^p u_j \right ] = \left (\sum_j u_j X_j \right ).\gamma_{12}^p - \sum_k \gamma_{12}^k \sum_j \gamma_{kj}^p u_j
\mathbf{W}^{t}=\sum_{i=1}^{n}\lambda_{i}^{t}\mathbf{v}_{i}\mathbf{u}_{i}^{\top}=\mathbf{1}\mathbf{u}^{\top}+\sum_{i=2}^{n}\lambda_{i}^{t}\mathbf{v}_{i}\mathbf{u}_{i}^{\top}
y= \sum_j\lambda_j\sum_{i_1,\ldots,i_{p-1}\in I}(-1)^{{i_1}+\ldots+i_{p-1}}(P_{1,j}+i_1)\wedge\ldots\wedge(P_{p-1,j}+i_{p-1})\otimes(Q_j-i_1-\ldots-i_{p-1}).
\stackrel{\mathrm{G}}{\tau}_{\alpha \beta}{}^{\! i}= \frac{1}{4 m i}\left[ \left( D^{i}\overline{{\Psi}}\right) \sigma_{\alpha \beta}\Psi - \overline{{\Psi}}\sigma_{\alpha \beta}\left( D^{i}\Psi \right) \right] \, ,
\int_{\Omega\,\cap\,B_{H_0}(x,1)}F(|z(t)|)\,dy \le\int_{\Omega\,\cap\,B_{H_0}(x,2)}F(|z(0)|)\,dy +C\int_0^t\int_{\Omega\,\cap\,B_{H_0}(x,2)}s^{\sigma-1}|z|^{1+\delta}\,dy\,ds+C\int_0^t\int_{\Omega\,\cap\,B_{H_0}(x,2)}s^{-\sigma}F(z)\,dy\,ds
\frac{h}{r^{3}}\frac{d}{d r}( h r^{3}\frac{d R}{d r}) + ( 1 + \frac{r_{1}^{2}}{r^{2}}) ( 1 + \frac{r_{5}^{2}}{r^{2}}) [ \omega^{2}- m^{2}+ ( \omega \sinh \sigma_{K}- m \cosh \sigma_{K})^{2}\frac{r_{0}^{2}}{r^{2}}] R = 0
\Sigma_{j}^{(1)}=\sum_{d\leq D}\lambda(d)\sum_{0<|h|\leq H}a_{h}W\left(-\frac{h}{d}\right)\ll\sum_{d\leq D}\sum_{1\leq h\leq H}\frac{1}{h}\left|W\left(\frac{h}{d}\right)\right|.
I(z,0,1,0,R_{1},R_{2}) &=-\frac{\pi i}{z}B_{2}\left( \frac{v+\left\{R_{2}\right\}}{k}\right) -\pi izB_{2}\left( \frac{\mu+1-\left\{R_{1}\right\}}{k}\right) +2\pi iB_{1}\left( \frac{\mu+1-\left\{R_{1}\right\}}{k}\right) B_{1}\left( \frac{v+\left\{R_{2}\right\}}{k}\right) .
\sum_{z^0 \in A}\pi^Q (z^0) f_{(0,1,1)}(z^0) \ = \ \sum_{z^0\in{\cal L}}\pi^Q (z^0) f_{(0,1,1)}(z^0) - \sum_{z^0 \in B}\pi^Q (z^0) f_{(0,1,1)}(z^0).
\Sigma(\rho)=\Sigma_{\delta ,k}(\rho)=\Sigma_{\pi, k}(\rho)=\{f\in L^*: f(L^2)=0, \hbox{such that there is}& \hskip1.1cmx\in X,\hbox{}x\neq 0, \hbox{with the property:}\rho (h)x=f(h)x,\hbox{}\forall\hbox{}h\in H\}.
{\mathbb P}\left\{x_{K_1}\geq a \right\}={\mathbb P}\left\{x_{i}\geq u_l \right\}\geq{\mathbb P}\left\{\cap_{j=1}^{i}\Omega_j \right\}= \prod_{i=1}^{i}\lambda_j \ge \prod_{i=1}^{K_1}\lambda_j.
0= g((\nabla_WA)Y,AZ)+g((\nabla_WA)Z,AY)+g(A^2\nabla_WY,Z)+g(A^2Y,\nabla_WZ) +c(1+||\xi^\perp||)\{g(\nabla_WY,Z)+g(Y,\nabla_WZ)\}= g((\nabla_WA)Y,AZ)+g((\nabla_ZA)W,AY)
\sigma=\sigma_{h}:= \frac{16^{1/4m}C \rho}{\gamma}\ge 2^{1/4m}, 2 ( 1 + 32^{1/4m})^{4m}C^{4m}\rho^{4m}=2( 2^{1/4m}\sigma \gamma +C\rho )^{4m}\le \gamma_0^{4m}, 2 (\gamma +C\rho )^{4m}\le \gamma_1^{4m}.
D_{12}^{(l)}\stackrel{1}{\widehat{F}^{(l)}}\stackrel{2}{\widehat{F}^{(1)}}D_{12}^{(l) \; -1}= \left( \begin{array}{cc}\widehat{F}^{(l+1)}0 0 -\alpha \beta \widehat{F}^{(l-1)}\end{array}\right),
\sum_{i=1}^n q_i \ln \bar{x}=\sum_{i=1}^n q_i \ln x_i(0) \Leftrightarrow \ \ \ln \bar{x}=\frac{1}{\sum_{i=1}^n q_i}\sum_{i=1}^n q_i \ln x_i(0)=\ln \prod_{i=1}^n x_i(0)^{\hat{q}_i}.
\int\limits_t^T\int\limits_t^sd{\bf w}_{\tau}^{(i_1)}d{\bf w}_{s}^{(i_2)}=\hbox{\vtop{\offinterlineskip\halign{\hfil#\hfil\cr{\rm l.i.m.}\cr$\stackrel{}{{}_{q\to \infty}}$\cr}}}\int\limits_t^T\left({\bf w}_s^{(i_1)}-{\bf w}_t^{(i_1)}\right)^{(q)}d{\bf w}_s^{(i_2)}.
\rho = \beta (\delta_{(\varphi_+,m_+)}+\delta_{(\varphi_-,m_-)}+ \delta_{(\widehat \varphi_+,\widehat{m}_+)}+\delta_{(\widehat \varphi_-,\widehat{m}_-)})+\gamma(\delta_{(\varphi_0,m_+)}+ \delta_{(\varphi_0,m_-)}+ \delta_{(\varphi_0,\widehat{m}_+)}+\delta_{(\varphi_0,\widehat{m}_-)}),
\sum_{m \geq 1}\frac{A(1, m)}{m}H_{m, p}&\quad= \frac{2}{\pi}\int_0^\infty k(t) \tanh(\pi t) t \biggl ( \frac{1}{2 \pi i}\int_{(1000)}p^{-u}\Bigl ( \sum_{m \geq 1}\frac{A(1, m)}{m^{1 + 2 u}}\Bigr ) F(u) \frac{\gamma\Bigl(\dfrac{1}{2}+ u, t\Bigr)}{\gamma\Bigl(\dfrac{1}{2}, t\Bigr)}\frac{d u}{u}\biggr ) \, d t.
B(n) \setminus T(A(n))=B(n)\setminus(T(A_1(n))\cup T(A_2(n))\cup T(A_3(n))) = \left\{(\mu,(c^d))\in B(n)\ \Big |\ \begin{array}{l}c+d+1\not \in \mu, \mu\neq (c,c-2) \mu_1-\mu_2\leq 2(c+d+1) \mu_1=3(c+d+1)\end{array}\right\}, = \left\{(\mu,(c^d))\in B(n)\ \Big |\ \mu \in \mathcal B(n,c,d) \right\}.
1=-\frac{\partial z}{\partial t}\int \frac{dF_0(x)}{(x-z)^2}, \ \ \ 0=-\frac{\partial^2 z}{\partial t^2}\int \frac{dF_0(x)}{(x-z)^2}+2(\frac{\partial z}{\partial t})^2\int \frac{dF_0(x)}{(x-z)^3}.
Y_t^{b,n}=\xi^b-\int_{t}^{T}g^b(s,Y^{b,n}_s,Z^{b,n}_s) +\lambda_s^n \frac{1-e^{\alpha (\xi^a_s-Y_s^{b,n})}}{\alpha}ds-\int_{t}^{T}Z_s^{b,n}\cdot dW_s, ~t\in[0, T],
A_1(0,t)= 1+\frac{\lambda^2}{4\pi^2}\frac{x^4+5x^2y^2}{|y|(x^2+y^2)^3}+\frac{\lambda^2y(3x^2-y^2)}{\pi^2(x^2+y^2)^3}= 1+\frac{\lambda^2}{4\pi^2}\frac{x^4+5x^2y^2-12x^2y^2+4y^4}{|y|(x^2+y^2)^3}= 1+\frac{\lambda^2}{4\pi^2}\frac{x^4+4y^4-7x^2y^2}{|y|(x^2+y^2)^3}
M1_k(n) = \sum_{m=-\infty}^\infty m^k M1(m,n), M2_k(n) = \sum_{m=-\infty}^\infty m^k M2(m,n), &M4_k(n) = \sum_{m=-\infty}^\infty m^k M4(m,n), N2_k(n) = \sum_{m=-\infty}^\infty m^k N2(m,n)
\langle(\phi_{\mathcal{A},\mathcal{B}})^* (j_X)^*(e_S \otimes e_T) ,(j_C)^*( e_S\otimes e_T)\rangle_{\mathcal{A},\mathcal{B}}= \langle\phi^* ( e_S \otimes e_T), e_S\otimes e_T\rangle.
v(X) = a_1X^{d_1}+ \ldots +a_tX^{d_t}+b, w_i(X) = a_1^{r^{-i}}X^{d_1}+ \ldots +a_t^{r^{-i}}X^{d_t}+b^{r^{-i}}, \: i \geq 1.
2\sum_{n=0}^{\infty}\textup{PD}_\textup{t}(3n)q^n =\dfrac{f_3^3}{f_1^5f_2^2}a^2(q)-\dfrac{f_3^3}{f_1^{5}f_2^2}a(q^2) &\equiv\dfrac{1}{f_4^2}\cdot\dfrac{f_3^3}{f_1}-\dfrac{a(q^2)}{f_4^2}\cdot\dfrac{f_3^3}{f_1}&\equiv\left(\dfrac{1}{f_4^2}-\dfrac{a(q^2)}{f_4^2}\right)\left(\dfrac{f_4^3f_6^2}{f_2^2f_{12}}+q\dfrac{f_{12}^3}{f_4}\right)~(\textup{mod}~4),
d = \frac{\binom{N_R-1}{r}\binom{N_T}{t}\binom{N_R-r-1}{t-1}t}{\binom{N_R-1}{r}\binom{N_T}{t}\binom{N_R-r-1}{t-1}t+\binom{N_R-1}{r+1}\binom{N_R-r-2}{t-1}\binom{N_T}{t-1}}
\alpha(f*\Phi\otimes \Psi)[v,w] =\sum_{\gamma}(f*\Phi)(v\gamma)\otimes m(v)\gamma\Psi(\gamma^{-1}w) =\sum_{\gamma,\varepsilon}f(\varepsilon)\varepsilon\Phi(\varepsilon^{-1}v\gamma)\otimes m(v)\gamma\Psi(\gamma^{-1}w) =\sum_{\varepsilon}f(\varepsilon)\varepsilon\alpha(\Phi\otimes \Psi)[\varepsilon^{-1}v,w]=f*\alpha(\Phi\otimes \Psi)[v,w]
( i j ( n + 1 ) ) \circ ( i_{1}i_{2}\ldots i_{l - 1}i ) ( j_{1}j_{2}\ldots j_{l^{\prime}- 1}j ) ( n + 1 ) = ( i_{1}i_{2}\ldots i_{l - 1}i j_{1}j_{2}\ldots j_{l^{\prime}- 1}j ( n + 1 ) )
B ( \alpha ) : = \frac{1}{3 2 \pi^{3}\alpha^{2}}\int_{0}^{\infty}\! d \tau \frac{\alpha \sin ( \pi / \alpha ) \, [ \, \cos ( \pi / \alpha ) - \cosh ( \tau ) + \tau \sinh ( \tau ) \, ] - \pi \, [ \, \cos ( \pi / \alpha ) \cosh ( \tau ) - 1 \, ]}{[ \, \cosh ( \tau ) - \cos ( \pi / \alpha ) \, ]^{2}\cosh^{4}( \alpha \tau / 2 )}.
\int\limits_{\mathbb{R}^{N}}\left\vert v\right\vert^{r}\frac{dx}{\left\vert x\right\vert^{s}}=d^{1+\frac{p-1}{p}r}\int\limits_{\mathbb{R}^{N}}\frac{\left\vert u\right\vert^{r}}{\left\vert x\right\vert^{N+sd-Nd}}dx
\beta'_n = \sum_{k=0}^n\frac{(\rho_1)_k(\rho_2)_k(aq/\rho_1 \rho_2)_{n-k}(aq/\rho_1 \rho_2)^k}{(aq/\rho_1)_n(aq/\rho_2)_n(q)_{n-k}}\beta_k.
\sigma_{ab}= r^2\tilde \sigma_{ab}+r^4 \sigma_{ab}^{(4)}+r^5 \sigma_{ab}^{(5)}+O(r^6) |H| = \frac{2}{r}+ r h^{(1)}+ r^2 h^{(2)}+ r^3 h^{(3)}+O(r^4) \alpha_H = r^2 \alpha_H^{(2)}+ r^3 \alpha_H^{(3)}+ r^4 \alpha_H^{(4)}+ O(r^{5}).
\frac{d^2\delta_1}{dv^2}=\frac{\lambda}{3\kappa^2}v\left(\frac{dV}{dv}-\frac{\kappa}{2}(1+y)-\frac{\kappa v}{2}\frac{dy}{dv}\right)+\frac{\lambda}{3\kappa^2}\left(V-c_{+0}-\frac{\kappa}{2}(1+y)v\right) &\qquad+(V-c_{+0})\frac{d\phi}{dv}+\frac{dV}{dv}\phi.
\operatorname{Re}\psi_3(i\rho,i\tau,\xi+i\eta)=-\alpha\rho^2-\beta\rho\eta \leqslant 0,\shortintertext{and}\operatorname{Re}\psi_4(i\rho,i\tau, \xi+i\eta)=-\alpha\tau^2-\beta\tau\eta \leqslant 0,
\frac{\mathrm{d}v}{\mathrm{d}t}= - \int_{0}^{- \infty}\mathrm{e}^{n \xi}\, \, \beta [ \alpha ( t - \xi ) ] \frac{\partial u [ \alpha ( t - \xi ) ]}{\partial \ln \alpha ( t - \xi )}\, \, \mathrm{d}\xi = - \beta [ \alpha ( t ) ] \frac{\partial v}{\partial \ln \alpha ( t )}.
\begin{aligned}\sum_{k=1}^{n-1}\frac{1}{k^{\omega}(n^\gamma-k^\gamma)}= \left( \sum_{k=1}^{[\frac n2]}+ \sum_{k=[\frac n2]}^{n-1}\right) \frac{1}{k^{\omega}(n^\gamma-k^\gamma)}& \leq C \left( \frac{1}{n^\gamma}\sum_{k=1}^{n}\frac{1}{k^{\omega}}+ \frac{1}{n^\omega}\sum_{k=[\frac n2]}^{n-1}\frac{1}{n^\gamma-k^\gamma}\right)\end{aligned}
\mathfrak{M}_{i}=\bigcup\limits_{q_{i}\leq P_{i}}\bigcup\limits_{\substack{1\leq a_{i}\leq q_{i}(a_{i},q_{i})=1}}\mathfrak{M}_{i}(a_{i},q_{i}),\mathfrak{m}_{i}=\left [\frac{1}{Q_{i}},1+\frac{1}{Q_{i}}\right ]\backslash\mathfrak{M}_{i},
f(tx+\sum_{p=1}^{\mathcal{N}}\phi^{(p)}\theta^{(p)}) = f(tx+\sum_{p \neq q}\phi^{(p)}\theta^{(p)}) +\sum_{i=1}^{N}\phi^{(q)}\theta^{(q)}_i \partial_i f(tx+\sum_{p \neq q}\phi^{(p)}\theta^{(p)}).
< \tau \vert g_{0}^{- 1}M_{+}^{- 1}M_{-}\vert \tau >^{( 1 )}= < \tau \vert ( g_{0}^{- 1}N_{-}g_{0}) N_{+}^{- 1}\vert \tau >^{( 1 )}= < \tau \vert N_{+}^{- 1}\vert \tau >^{( 1 )},
{\bf{\hat{g}}}= \frac{1}{( N - 1 )^{2}}\Bigg [ ( 2 - N ) \sum_{i = 1}^{N}\Big ( \frac{d{\hat{u}}_{i}}{{\hat{u}}_{i}}\Big )^{2}+ \sum_{r \neq s}\frac{d{\hat{u}}_{r}}{{\hat{u}}_{r}}\, \frac{d{\hat{u}}_{s}}{{\hat{u}}_{s}}\Bigg ]
H(z)=\frac{(1-q_{1}z^{-1})(1-q_{2}z^{-1})\cdot\cdot\cdot(1-q_{M}z^{-1})}{(1-p_{1}z^{-1})(1-p_{2}z^{-1})\cdot\cdot\cdot(1-p_{N}z^{-1})},
\begin{cases}\ \phi_1(\mu_1,\mu_2)^{-1}= \left[5\lambda\left(2\rho_1+\mu_2\right)\right]^{-1}+ 1, \ \phi_2(\mu_1,\mu_2)^{-1}= \left[5\lambda\left(2\rho_2+\mu_2\right)\right]^{-1}+ 1.\end{cases}
\psi_p(x)^*\psi_p(y) =\Psi_{(p,e)}(x\otimes e)^*\Psi_{(p\otimes e)}(y\otimes e) =\Psi_{(e,e)}(\langle (x\otimes e), (y\otimes e)\rangle =\Psi_{(e,e)}(\langle x, y\rangle\otimes e) =\psi_{e}(\langle x, y\rangle)\ (x, y\in X_p)
\varphi'(x) = 3\mathrm{f}' \mathrm{f}^2- 3\mathrm{h}^2(\mathrm{h}')^2\mathrm{f}'- \mathrm{h}^3 \mathrm{f}'' \mathrm{h}'- \mathrm{h}^3\mathrm{f}' \mathrm{h}'' &> \mathrm{h}'\mathrm{h}^2\big(-\mathrm{f}''\mathrm{h}-3\mathrm{f}'\mathrm{h}'\big)
\mathcal{F}_{12}^{(1)}=\mathcal{F}_{12}^{(2)}=\ldots=\mathcal{F}_{12}^{(N)}, \quad \mathcal{F}_{34}^{(1)}=\mathcal{F}_{34}^{(2)}=\ldots=\mathcal{F}_{34}^{(N)}, \quad \mathcal{F}_{56}^{(1)}=\mathcal{F}_{56}^{(2)}=\ldots=\mathcal{F}_{56}^{(N)}.
\overline{{{A}}}^{( N - 1 , M )}= \Bigl ( A_{m n}^{( N - 1 , M )}\Bigr )_{N - 1 \ge m , n \ge M - 1}, \quad \underline{{{A}}}^{( N , M + 1 )}= \Bigl ( A_{m n}^{( N , M + 1 )}\Bigr )_{N + 1 \ge m , n \ge M + 1}.
\begin{cases}0=\partial_{i}\left(A_{ij}(x)\partial_{j}u\right) +B_{i}(x)\partial_{i}u \Omega^h, begin{align*}4pt]u(x_0,x') = 0, x_0=h begin{align*}4pt]u(x)=f(x) \,x\in\partial\Omega, \end{cases}
S_{\mathrm{off}}^{c\geq p^2}:=\sum_{d_1,\,d_2<D}\,\, \,\ \sum_{Id_1, \,Jd_2<D}x_{Id_1}\bar{x}_{Jd_2}\sqrt{IJ}\sum_{L, M\geq 1}\frac{G_k(LMd_1d_2\slash p)}{\sqrt{LM}}\times\left(2 \pi i^{2k}\sum_{\substack{p|c c \geq p^2}}\frac{\mathcal{S}(IL, JM, c )}{c}\,J_{2k-1}\left(\frac{4\pi \sqrt{ILJM}}{c}\right)\right).
h_{\overline{D}_{0,v},\dots,\overline{D}_{n-1,v}}(Z;s_0,\dots,s_{n-1})=h_{\overline{D}_{0,v},\dots,\overline{D}_{n-1,v},\overline{D}_{f,v}}\big(X_\Sigma;s_0,\dots,s_{n-1},\tilde{f}s_f\big).
T r e^{- i s H}={\frac{T 2 a \ell^{2}}{4 \pi^{2}i}}{\frac{e^{- i s m^{2}}}{s^{2}}}[ 1 + 2 \sum_{n = 1}^{\infty}( - 1 )^{n}e^{i ( a n )^{2}/ s}] [ 1 + i s e B L ( i s e B ) ] ,
\big( K\circ v_j \big)^{(q)}= \sum \frac{q!}{m_1!\dots m_q!}(K^{(M_q)}\circ v_j) \prod_{s=1}^q \left( \frac{v_j^{(s)}}{s!}\right)^{m_s}&= \sum c_{m_1, \ldots, m_q}\frac{K^{(M_q)}\circ v_j}{F^{M_q}}\prod_{s=1}^q \big(f_0^{(s-1)}\big)^{m_s},
\frac{d}{dr}\log_{m+1}\frac{e}{r}= \frac{1}{\prod_{k=1}^m \log_k \frac{e}{r}}\left ( -\frac{1}{r}\right ), &\frac{d}{dr}\log_{m+1}\frac{1}{Cr^\alpha}= \frac{1}{\prod_{k=1}^m \log_k \frac{1}{Cr^\alpha}}\left ( -\frac{\alpha}{r}\right ).
t\left( t^{2}-z^{2}\right) \partial_{t}^{3}S+\left( 2t^{4}-2t^{2}z^{2}+3t^{2}+z^{2}\right) \partial_{t}^{2}S +2t(5t^{2}-z^{2})\partial_{t}S+2(3t^{2}+z^{2})S=0.
(-32 \varepsilon L , 32 \varepsilon L) = [-4L+t, 4L+t] \setminus \left( [ - (1 - 4\varepsilon ) 8L, - 4\varepsilon \cdot 8L ] \cup [ 4\varepsilon \cdot 8L, (1 - 4\varepsilon) 8L ] \right) \; .
\Pr \left\{\xi_1 \wedge \xi_2 < \left(a_{_{J_n}}^\eta \wedge \gamma\right) b \ \left|\frac{}{}\right. \J_n \right\}\leq \Pr \left\{M_n^{(-J_n)}< a_{_{J_n}}^\eta b \ \left|\frac{}{}\right. \J_n \right\}= \prod_{i=1, i \neq J_n}^n F\left( \frac{a_{_{J_n}}^\eta b}{a_i}\right).
f=\sum_{|I|=p,|J|=q,k}f_{IJk}dz_I\wedge d\overline{z}_J\otimes e_k\otimes \widetilde{e}=\sum_k f_k \otimes e_k,f_k=\sum_{|I|=p,|J|=q}f_{IJk}dz_I\wedge d\overline{z}_J\otimes \widetilde{e}.
-1 +\sum_{m=0}^\infty2\left(\frac{q^{6m+2}}{1+q^{6m+2}}-\frac{q^{6m+4}}{1+q^{6m+4}}-\frac{q^{6m+1}}{1-q^{6lm+1}}+\frac{q^{6m+5}}{1-q^{6m+5}}\right) & =-\frac{[q^{2},-q^3,-q^3;q^6]_\infty(q^6;q^6)^2_\infty}{[q,q,-q^2,-q^4;q^6]_\infty},
\langle n_+^{H}\rangle_{\Psi}\leq C \begin{dcases}C_B N\, K_L^{-2}K_{\ell}^2 \widehat{g}(0) , d=2, C_B N\,K_L^{-2}K_{\ell}^2 \sqrt{\rho a^3}, d=3. \end{dcases}
\sup_{0\leq s\leq t\leq T}E\left[\left|J_{s,t}^m-J_{s,t}\right|^p\right]^{1/p}\leq O(m^{1/2})+\tilde C T\sup_{r\in [0,T]}E\left[ (1+\|q_r\|^{\tilde p}+\|q_r-\tilde q_r^m\|^{\tilde p})^{2p}\right]^{1/(2p)}E\left[\|q_r^m-q_r\|^{2p}\right]^{1/{2p}}= O(m^{1/2}).
f_{1}(x) = \frac{1}{3}x, \f_{2}(x) = \frac{1}{3}x + \dfrac{1}{3}\begin{bmatrix}0 2 \end{bmatrix},\f_{3}(x) = \frac{1}{3}x + \dfrac{1}{3}\begin{bmatrix}2 0 \end{bmatrix},\f_{4}(x) = \frac{1}{3}x + \dfrac{1}{3}\begin{bmatrix}2 2 \end{bmatrix}.
\lim_{\tau \rightarrow -T_1^+}\frac{b_i(\tau)}{T_1 +\tau}=\lim_{\tau \rightarrow - T_1^+}\left(\frac{b_i(\tau)}{a(\tau)}\cdot \frac{a(\tau)}{T_1 +\tau}\right)=\xi_i^{-1}E(\xi).
\varphi_{\xi^-(\omega)(u-\omega_1),\xi^-(\omega)(v-\omega_2)}''(1) =(p-q)\|(\xi^-(\omega)(u-\omega_1),\xi^-(\omega)(v-\omega_2))\|^p - 2((\alpha+\beta)-q)\int_\Omega |\xi^-(\omega)(u-\omega_1)|^{\alpha}|\xi^-(\omega)(v-\omega_2)|^\beta dx<0,
\lim_{s \to 0}\frac{\int_0^{L_m^*}\frac{|w'|^2 V_{\gamma_m}}{\sigma_s}+\frac{k^2|w|^2 \sigma_s}{V_{\gamma_m}}\,dt}{\int_0^{L_m^*}|w|^2 V_{\gamma_m}\sigma_s \,dt}=\frac{\int_0^{L_m^*}\frac{|w'|^2 V_{\gamma_m}}{\sigma_0}+\frac{k^2|w|^2 \sigma_0}{V_{\gamma_m}}\,dt}{\int_0^{L_m^*}|w|^2 V_{\gamma_m}\sigma_0 \,dt}
\begin{cases}z_0 + z_1 + \cdots + z_{N-1}= 0 z_0z_1 + z_1z_2 + \cdots + z_{N-1}z_0 = 0 z_0z_1z_2 + z_1z_2z_3 + \cdots + z_{N-1}z_0z_1 = 0 \cdots \cdots z_0z_1z_2\cdots z_{N-1}= 1\end{cases}.
\frac{\partial}{\partial t}\left( \begin{array}{c}{{a ({\bf k})}}\\{{a^{*}({\bf k})}}\\ \end{array}\right) = \mathrm{\cal ~ f i n i t e ~ t e r m s}+ \lambda^{2}\left( \begin{array}{c c}{{A ( 0 ; t )}}&{{A^{*}( 0 ; t )}}\\{{A ( 0 ; t )}}&{{A^{*}( 0 ; t )}}\\ \end{array}\right) \ \; \; \left( \begin{array}{c}{{a ({\bf k})}}\\{{a^{*}({\bf k})}}\\ \end{array}\right)
\sum_{\substack{m,n \in \mathcal{M}' m\not=n}}r(m)r(n)\Phi\left(T\log{\frac{m}{n}}\right) \le \sum_{\substack{j,l \in \mathcal{J}j\not=l}}r(m_j)r(n_l)\Phi\left(T(|j-l|-1)\log{(1+T^{-1})}\right) \ll \sum_{\substack{j,l \in \mathcal{J}j\not=l}}r(m_j)r(n_l)\Phi\left(|j-l|-1 \right)
\frac{\partial^2}{\partial p^j \partial p^k}T_a(p) \:\frac{q^j}{2}\frac{q^k}{2}\;=\; -\frac{\partial}{\partial p^j}\left( (pq) \:T_a^{(1)}(p) \right) \frac{q^j}{2}\;=\; (pq)^2 \:T_a^{(2)}(p) \:-\:\frac{q^2}{2}\:T_a^{(1)}(p) \;\;\; .
\Vert \eta- \hat{\mu}\Vert_{L^2(\partial K)}= \Vert \eta- \hat{\mu}\Vert_{L^2 \left( \cup_{j=1}^{J}\mathcal{I}_j \right)}\le \sum_{j=1}^{J}\Vert \eta- \hat{\mu}\Vert_{L^2 \left( \mathcal{I}_j \right)}
{\cal I}_{L , M , 0 ; L^{\prime}, M^{\prime}, 0}= - 2 C^{2}( k + 2 ) \sum_{l = 0}^{k}\sum_{l^{\prime}, m^{\prime}}^{R}{\frac{\sin ( L , l )_{k}\sin ( L^{\prime}, l )_{k}\sin ( l . l^{\prime})_{k}}{\sin ( l , 0 )_{k}}}\delta_{\mu}^{( k + 2 )}( - )^{\frac{\mu ( l + 1 )}{( k + 2 )}}I_{l^{\prime}}^{m^{\prime}}( q ) \quad ,
1\,=\,\|\epsilon_i\wedge\epsilon_j\|_x\,=\,\lambda_i\lambda_j\|f_\ast (e_i)\wedge f_\ast (e_j)\|_x\,\leq\,\lambda_i\lambda_j\epsilon\phi^{-4}\|e_i\wedge e_j\|_x\,=\,\lambda_i\lambda_j\epsilon\phi^{-4}\,,
\epsilon_{2e}=1+\frac{1}{2^{1}}+\frac{1}{4^{2^{1}}}+\frac{1}{8^{4^{2^{1}}}}+\frac{1}{16^{8^{4^{2^{1}}}}}+\frac{1}{32^{16^{8^{4^{2^{1}}}}}}+...
\varphi(t,x):=J_0(t,x)P_{u_1}+\left\langle\nabla J_0(t,x), M_{u_1}\right\rangle+H(t,x)P_{u_1}+J_1(t,x)P_{u_0}-\sigma^{-1}\Delta J_0(t,x)P_{\theta_0},
\tilde{\Phi}(\tilde{\boldsymbol{\gamma}}^{*};V_a+\widetilde{\mathcal{V}}(V_a,Z_{a}h)Z_{a}h)=\tilde{\Phi}(\boldsymbol{\gamma}^{*};V_a)+\widetilde{\mathcal{V}}(V_b,Z_{b}h)Z_{b}h,
\frac{\nu}{2}\left(\|u_{t}\|_{L^{2}(\mathbb{R}^{2})}^{2}+\|D_{q}u_{t}\|_{L^{2}(\mathbb{R}^{2})}^{2}+\|qu_{t}\|_{L^{2}(\mathbb{R}^{2})}^{2}\right)=\|\sqrt{\nu}(\frac{-\partial_q+q}{\sqrt{2}})e^{-t(K_{\nu,\frac{\pi}{2}}+\nu^{\frac{1}{3}})}u\|_{L^2(\mathbb{R}^{2})}^{2}\le \frac{c}{t^{3}}\|u\|_{L^{2}(\mathbb{R}^{2})}^{2}\,.
t_n(j)=(-1)^{\frac{n-j}{2}}\frac{n}{n+j}\binom{\frac{n+j}{2}}{\frac{n-j}{2}}\cdot 2^j u_n(j)=(-1)^{\frac{n-j}{2}}\binom{\frac{n+j}{2}}{\frac{n-j}{2}}\cdot 2^j.
p_{S^m}(s^m)=\prod\limits_{t=1}^mp_S(s(t)), &p_{Y^{\rho m}_1,Y^{\rho m}_2|X^{\rho m}}(y^{\rho m}_1,y^{\rho m}_2|x^{\rho m})=\prod\limits_{q=1}^{\rho m}(y_1(q),y_2(q)|x(q)).
S_{m}^{( 1 )}= S_{m}^{( 1 )}( \Psi_{1}, g_{\mu \nu}^{B ( 1 )}) \hspace{0 . 5 c m}\mathrm{a n d}\hspace{0 . 5 c m}S_{m}^{( 2 )}= S_{m}^{( 2 )}( \Psi_{2}, g_{\mu \nu}^{B ( 2 )}) ,
p^2\xi_{20}(p,q) - p.q\xi_{11}(p,q) ={1\over 2}\left\{-\left[{1\over 2}+ m^2 \xi_{00}(p,q)\right] +{q^2\over 2}\xi_{01}(p,q) +{3p^2\over 2}\xi_{10}(p,q)\right\}
\phi_{i-1}(f)(x)=\mathrm{diag}\Big(f\big(\pi_1^{(i)}(x)\big),\dots,f\big(\pi_{n_i}^{(i)}(x)\big),f\big(x_1^{(i-1)}\big),\dots,f\big(x_{i-1}^{(i-1)}\big)\Big).