harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
\psi\left( d_-^{\ell}L(0v, 0w) \right) = \psi\left( t^{-\ell}d_{-}^{\ell+1}L(v1, w1) + qt^{-\ell}d_-^{\ell}L(v0, w0) \right) &= t^{-\ell}\psi\left(d_{-}^{\ell+1}L(v1, w1) \right) + qt^{-\ell}\left( d_-^{\ell}L(v0, w0) \right) &= t^{-\ell}p(v1, w1) + qt^{-\ell}p(v0, w0) &= p(0v, 0w).
\|F_m\|^2_{2,s}= \sum_{n = m}^{m^{1/s}}|F_m(n)|^2 \left(\frac{1}{n^{2s}}-\frac{1}{(n+1)^{2s}}\right) + \sum_{n = m^{1/s}+1}^{\infty}|F_m(n)|^2 \left(\frac{1}{n^{2s}}-\frac{1}{(n+1)^{2s}}\right) & \leq \sum_{n = m}^{m^{1/s}}|F_m(n)|^2 \left(\frac{1}{n^{2s}}-\frac{1}{(n+1)^{2s}}\right) + 4 m^2 m^{-2}.
\{A_\gamma(t),^2 E_S[f]\}=\frac{\tau_k}{2}\,\dot{\gamma}^a(t)\int_{\mathbf{x}\in S}\mathrm{d}S^{bc}(\mathbf{x})f^k(\mathbf{x})\mathcal{E}^{(3)}_{\gamma(t)}(\mathbf{x})v_{abc}(\mathbf{x})\,.
P_{n}^{(\alpha,\beta)}(z)=\frac{(-1)^{n}}{2^{n}n!}(1-z)^{-\alpha}(1+z)^{-\beta}\frac{d^{n}}{dz^{n}}\{(1-z)^{\alpha}(1+z)^{\beta}(1-z^{2})^{n}\}.
\frac{E_{0}\left( \ell ,\mu ,d\right)}{L^{d-1}}\approx -C\left( d\right) \frac{\Gamma \left( \frac{d}{2}+1\right)}{2^{d+3}\pi^{d/2}\Gamma \left( \frac{d}{2}\right) \ell^{d}}\left[ \left( \frac{d}{2}+1\right) \mu^{\frac{d}{2}-2}+2\mu^{\frac{d}{2}-1}\right] e^{-2\mu}.
\int_{|x-y|\leq\delta/C_{0}}e^{i\lambda \psi_{(1)}(x,\beta)}f_{i_{1}...i_{m}}(x)C^{i_{m}}(x,\beta;\lambda)dx=\mathcal{O}(e^{-\lambda/C}) i_{j}\in\{1,\dots, n\}j = 1, \dots , (m-1)
f(A\mathbf{z}) =4\langle A\mathbf{x},A\mathbf{y}\rangle^2-4\langle A\mathbf{x},A\mathbf{x}\rangle \langle A\mathbf{y},A\mathbf{y}\rangle &=4\langle\mathbf{x},\mathbf{y}\rangle^2-4\langle\mathbf{x},\mathbf{x}\rangle \langle\mathbf{y},\mathbf{y}\rangle &=f(\mathbf{z}),
\| \Delta_q \cos(nx) \|_{L^2}= \frac\pi2 \left( \sum_{m=-\infty}^{\infty}\varphi_q^2(m) \left| \delta_{n}(m) + \delta_{-n}(m)\right|^2 \right)^{1/2}= \frac\pi{2}\left( 2 \varphi_q^2(n) \right)^{1/2},
A^1x^+_1-A^1x^-_1+A^2x^+_1-A^2x^-_1 \succeq_{\mathcal{L}^{3}}b x_1^+,x_1^-,x_2^+,x_2^- \geq 0 x_j=x_j^+-x_j^- \j \in\{1,2\}.
\mathcal{H}^{\rm LB}(D)=\big\{[W^{\rm LB}(D, C)] \in H^{\rm LB}_2(X) \ \big| \C \in{\rm Col}_{X^2}^{\rm LB}(D) \big\},\mbox{and}\Phi_{\theta}^{\rm LB}(D)=\big\{\theta\big(W^{\rm LB}(D, C)\big) \in A \ \big| \C \in{\rm Col}_{X^2}^{\rm LB}(D) \big\}
c_{Y_{\ell}}\left(y_{\ell}, y_{\ell}^{\prime}\right)=\inf_{x_{\ell}, x_{\ell}^{\prime}\in \mathcal{X}}c_{\ell}\left(\left(y_{\ell}, x_{\ell}\right),\left(y_{\ell}^{\prime}, x_{\ell}^{\prime}\right)\right).
R_{0}( u ) = \prod_{l = 1}^{\infty}\left[ \frac{\Gamma ( 4 l \lambda - \frac{2 \lambda u}{\pi}) \Gamma ( 4 \lambda ( l - 1 ) + 1 - \frac{2 \lambda u}{\pi})}{\Gamma ( ( 4 l - 3 ) \lambda - \frac{2 \lambda u}{\pi}) \Gamma ( ( 4 l - 1 ) \lambda + 1 - \frac{2 \lambda u}{\pi})}/ ( u \to - u ) \right]
\rho \big(E\otimes x^my^n \big)=[n]_q\xi^{m+n}\lambda_1\lambda_2^{-n}x^{m+1}y^{n-1}, \rho \big(F \otimes x^my^n \big)=[m]_q\xi^{m+n}\lambda_1^{-1}\lambda_2^{-n}x^{m-1}y^{n+1}, \rho\big(K^{\pm 1}\otimes P \big)=P\big(q^{\pm 1}\xi x,q^{\mp 1}\xi\lambda_2^{-1}y\big),
\frac1{2\pi}\,\int_0^{2\pi}\frac{(\cos\theta)^{2j}(\sin\theta)^{2m-2j}}{(\sigma_1\cos^2\theta + \sigma_2\sin^2\theta)^{m+1}}d\theta= \frac1{2^{2m}}\frac{(2j)! (2m-2j)!}{m! j! (m-j)!}\, \sigma_1^{-\frac12-j}\sigma_2^{-\frac12-(m-j)};
-\frac{\,4\,m_{F}^{2}\,}{(4\pi )^{2}}\left( \xi_{0}-\frac{1}{6}\right)\int_{0}^{t}U^{2}(t)dt\,+\,\frac{1}{3(4\pi )^{2}}\,\sum_{i}\,N_{i}\,\int_{o}^{t}m_{i}^{2}(t)\,dt\,.
x_3 x_1-x_1x_2 =0, x_4x_2+x_2x_4=0, x_3 x_2-x_2x_3 =0, x_3x_4+x_4x_3=0, x_4 x_1+x_1x_4 =0, x_1x_2+x_2x_1=x_3^2+x_4^2.
\int\limits_{0}^{x}\int\limits_{\chi(\tau)}^{g(\tau)}\rho \phi(T)Z dyd\tau-\int\limits_0^x (g(\tau)-\chi(\tau))\bar{\rho}\phi(\bar{T})\bar{Z}d\tau = \int\limits_{0}^{x}\int\limits_{\chi(\tau)}^{g(\tau)}\rho (\phi(T)-\phi(\bar{T}))Z dyd\tau+\int\limits_{0}^{x}\int\limits_{\chi(\tau)}^{g(\tau)}(\rho-\bar{\rho})\phi(\bar{T})(Z-\bar{Z})dyd\tau = O(1)\delta_*^2.
{\mathcal J}^0(x) \;=\; \frac{q}{2\pi}\;{\rm div}\, \underline{\varphi}(x)~,~~\underline{{\mathcal J}}(x) \;=\; -\,\frac{q}{2\pi}~\frac{\partial}{\partial t}~\underline{\varphi}(x)~,
\widetilde{d}_T(f)(u_1,\cdots,u_{p+1}) &=\sum_{1\leq i < j \leq p+1}(-1)^{i+j}f(\rho(Tu_i)u_j-\rho(Tu_j)u_i,u_1,\cdots,\widehat{u_i},\cdots,\widehat{u_j},\cdots,u_{p+1}) &+\sum_{i=1}^{p+1}(-1)^{i+1}\Big([Tu_i,f(u_1,\cdots,\widehat{u_i},\cdots,u_{p+1})]+T\rho(f(u_1,\cdots,\widehat{u_i},\cdots,u_{p+1}))(u_i) \Big).
< 0 | N | 0 >_{\mathrm{l o c}}= \int{\frac{d^{3}\vec{q}}{( 2 \pi )^{3}}}{\frac{1}{2 \omega ( \vec{q})}}\left[ | J_{\mu}^{\mathrm{c l}}( \vec{q}) |^{2}-{\frac{1}{M^{2}}}\left| J_{\mu}^{\mathrm{c l}}( \vec{q}) q^{\mu}\right|^{2}\right]
\mathcal{F}_{\boldsymbol{\eta}}(\boldsymbol{v}) = \int_{Q}\Big((\boldsymbol{u}- \sigma \partial_t \boldsymbol{\eta}- \textbf{curl}\, \boldsymbol{\tau}) \cdot \boldsymbol{v}+ (\boldsymbol{\tau}- \nu \, \textbf{curl}\, \boldsymbol{\eta}) \cdot \textbf{curl}\, \boldsymbol{v}\Big) \, d\boldsymbol{x}\, dt.
E^{(2)}_{in}(z) = \left( \mathbb{I}- \frac{A_n^{(1)}(z)-A_n^{(1)}(0)}{n^3 z}\right) E_{in}^{(1)}(z), z \in D(0,r_n) E^{(2)}_{out}(z) = \left(\mathbb{I}- \frac{A_n^{(1)}(0)}{n^3z}\right)^{-1}, z \in A(0; r_n, R),
M(\lambda, \vec \gamma) \leq \dfrac{\lambda^2}{2}p + \lambda^3 \cdot\dfrac{8C_{V,f}^3 n}{(\sqrt{nh^d})^3}= \dfrac{\lambda^2}{2}\left(p + 2 \lambda \dfrac{8C_{V,f}^3}{\sqrt{nh^{3d}}}\right) \leq\dfrac{\lambda^2}{2}\left(p + \dfrac{16 \mathfrak g C_{V,f}^3}{\sqrt{nh^{3d}}}\right)\enspace .
\partial_{r}\varphi ={\frac{1}{3 ( r - r_{H})}}[ \eta_{P}e^{3 \varphi}{\bf P}_{i j \, \infty}+ \eta_{Q}e^{-{3 \varphi}}{\bf Q}_{i k \, \infty}]^{2}{\frac{1}{\eta_{P}{\bf P}_{i j \, \infty}+ \eta_{Q}{\bf Q}_{i k \, \infty}}}.
\|B_{1,M_n,n}\|^2_{L^2(\Omega)}= \Big\|\Phi_{M,n}( \widetilde u_T(\cdot,t) - \Phi_{M,n}( u_T)(\cdot,t)\Big\|^2 &=\frac{1}{n^2}\Big[\sum_{k=1}^n \sigma_k \epsilon_k \Big]^2+\sum_{p=1}^{M_n}e^{2(T-t)p^{2\beta}}\Bigg[ \dfrac{\pi}{n}\sum_{k=1}^n \sigma_k \epsilon_k\phi_p(x_k) \Bigg]^2.
C_{\alpha, 1}\varepsilon^2\sqrt{\sum_{k=1}^{D_\varepsilon}b_k^{-4}}-\sum_{k=1}^{D_\varepsilon}\theta_k^2\geq \left(2\left(1+\sqrt{2C_{\min}(\alpha,\beta)}\right)\sqrt{\sum_{k=1}^{D_\varepsilon}b_k^{-4}}\right) \varepsilon^2\sqrt{-\log(1-\beta)}+ 2b_{D_{\varepsilon}}^{-2}\varepsilon^2 \sqrt{-\log(1-\beta)}.
c_{v,f}=\frac{R_{g}}{M}\{\frac{3}{2}+(\frac{T_{f,v}}{T})^{2}\frac{exp(T_{f,v,i}/T)}{[exp(T_{f,v,i}/T)-1]^{2}}+1+\frac{2}{15}(\frac{T_{f,v}}{T})^{2}\},
c_{\mu,\nu;\pi}= \frac{\alpha^{d(\mu,\nu;\pi)/2}}{z_{\tilde{\mu}}z_{\tilde{\nu}}}\sum_{0 \leq i \leq m_1(\pi)}g_{\tilde{\mu},\tilde{\nu};\tilde{\pi}1^i}\cdot z_{\tilde{\pi}}\cdot i! \cdot \binom{n-|\tilde{\pi}|}{i}.
\left< S_0 x, S_0 y \right> \ =\ \sum_{i,j}\left<\varrho'(a_i)e b_i, \varrho'(\alpha_j)e \beta_j \right> \ =\ \sum_{i,j}b_i^*\left<e, \varrho'(a_i^*\alpha_j)e \right>\beta_j \ =\ \sum_{i,j}b_i^* \varphi(a_i^*\alpha_j) \beta_j \ =\ [x,y].
b_R^{2,3}:= \left| \begin{array}{ccc}1 \mathbb{E}R \mathbb{E}R^2 \mathbb{E}R \mathbb{E}R^2 \mathbb{E}R^3 \mathbb{E}R^2 \mathbb{E}R^3 \mathbb{E}R^4 \end{array}\right| = \sigma_R^2\mathbb{E}R^4 + 2(\mathbb{E}R)(\mathbb{E}R^2)\mathbb{E}R^3 - (\mathbb{E}R^2)^3 - (\mathbb{E}R^3)^2,
\beta(r_k)-r_{k+1}= \sum_{j=1}^N \beta( a_j^{1/2}r_{j,k}a_j^{1/2}) - \sum_{j=1}^N a_j^{1/2}r_{j,k+1}a_j^{1/2}&=: I_1 + I_2 + I_3 + I_4,
\sum_{i_1i_2\ldots i_k\in \mathfrak{A}_{c_1c_2\ldots c_k}}\mathsf{E}(M_{i_1i_2}M_{i_2i_3}\ldots M_{i_{k-1}i_k}M_{i_ki_1})\leq N \prod_{\ell=1}^{p}B(s_\ell)(s_{\ell}-1)!!
0 \le \pi_{\psi}(ae_P)ee\pi_{\psi}(ae_P)^* &\le \pi_{\psi}(ae_P)e^{1/2}ee^{1/2}\pi_{\psi}(ae_P)^* &\le \pi_{\psi}(ae_P)e^{1/2}he^{1/2}\pi_{\psi}(ae_P)^* &\le \pi_{\psi}(ae_P)he\pi_{\psi}(ae_P)^*=0 ()
\sum_{\ell = 0}^{\infty}{\frac{( - 1 )^{\ell}}{\ell + a}}\left( \begin{array}{c}{- 1 / 2}\\{\ell}\\ \end{array}\right) ={\frac{\Gamma ({\frac{1}{2}}) \Gamma ( a )}{\Gamma ( a +{\frac{1}{2}})}}.
S^{(s)}_{N, N}f = \sum_{m = 2}^{N}\left(S^{(s+1)}_{N, m}f - S^{(s+1)}_{N, m-1}f \right) \log m &=(\log N) S^{(s+1)}_{N, N}f+\sum_{m = 2}^{N-1}\big(\log m - \log (m+1)\big) S^{(s+1)}_{N, m}f.
\frac{1}{r_2^2}\int_{\{2a_0^f(\Psi)\Psi\le2\log r_2\}}|f|^2 =\lim_{p\rightarrow2a_0^f(\Psi)+0}\frac{1}{r_2^2}\int_{\{p\Psi<2\log r_2\}}|f|^2 &\ge G(0;\Psi,I_+(2a_o^f(\Psi)\Psi)_o,f).
-\delta{A_j}{\bar x_i^{4-j}}= \delta{I'_4}f(\bar x_i) ={I''_4}f(\bar x_i)\,\delta \bar{x_i}\Rightarrow \delta{\bar x_i}= \frac{{-\delta{A_j}\,\bar x_i^{4-j}}}{{I''_4}f(\bar x_i)}.
\psi_x = \left(\begin{matrix}-iz u - \bar u iz \end{matrix}\right) \psi begin{align*}2mm]\psi_t = i\left(\begin{matrix}-[2z^2 - |u|^2] -2izu+ u_x 2iz\bar u + \bar u_x 2z^2 - |u|^2\end{matrix}\right)\psi
v'' + \frac{N-1}{r}v' - \frac{N-1}{r^2}v = \frac{(-1)^k}{k}\binom{N-1}{k-1}\frac{v^k}{r^{k-1}}+ \frac{\lambda}{r^{N-1}}\int_0^{r}f(s) s^{N-1}ds,
\int_\Omega f_0(V) \psi^h_{\mu_j,\xi_j}dx = \int_{\Omega\setminus B(\xi,\rho)}f_0(V) \psi^h_{\mu_j,\xi_j}dx +\sum_{l=1}^k\int_{\mathcal{A}_l}f_0(V) \psi^h_{\mu_j,\xi_j}dx.
\frac{X_{\ell-1}(\gamma_{k^* + K})}{X_{\ell}(\gamma_{k^* + K})}\leq \frac{(1 + \delta) X_{{\ell}-1}(t^*) e^{\int_{t^*}^{\gamma_{k^* + K}}G_{{\ell}-1}(v) \: dv}}{(1 - \delta)X_{\ell}(t^*) e^{\int_{t^*}^{\gamma_{k^* + K}}G_{\ell}(v) \: dv}}= \frac{(1 + \delta) X_{\ell-1}(t^*) e^{-s(\gamma_{k^* + K}- t^*)}}{(1 - \delta) X_{\ell}(t^*)}.
\left |(\rho-1)a\alpha^n-(d_1\cdot \rho^\ell-(d_1-d_2))\rho^{m+k}\right| =\left| -(\rho-1)\Pi(n)-(d_2-d_3)\cdot \rho^k-d_3 \right| &\leqslant (\rho-1)\cdot \alpha^{-n/2}+(\rho-1)\cdot \rho^k+\rho-1 &< 2\cdot \rho^{k+1}.
(a-1)(b-1)d_0 - (d'-1)\frac{p^L-1}{d'}\geq (a - 1)(b - 1)\frac{(p^9-1)(p^{L-9}-1)}{ab}- \frac{b - 2}{b}(p^L - 1) = \frac{a-b+1}{ab}p^L - \frac{ab - a - b + 1}{ab}(p^9 + p^{L-9}- 1) + \frac{b-2}{b}> \frac{2}{ab}p^L - p^9 - p^{L-9}> 0.
{\cal F}_{\rm vac}(p) \equiv \frac{d}{d{\alpha}_{-p}}{\cal A}_{{\rm vac};+}- \frac{d}{d{\alpha}_p}{\cal A}_{{\rm vac};-}=(1-{\rm N}^2) \frac{e_{+}^2{\rm L}^2}{4{\pi}^2}\frac{1}{p}.
\iint_{\mathcal{C}_{\Omega}}y^av^{2}\,d\gamma(x)\,dy =\int_{0}^{\infty}y^a\int_{\Omega}v^{2}\,d\gamma(x)\,dy\leq C_{\Omega}\int_{0}^{\infty}y^a\int_{\Omega}|\nabla_{x}v|^{2}d\gamma(x)\,dy & \leq C_{\Omega}\iint_{\mathcal{C}_{\Omega}}y^a|\nabla_{x,y}v|^{2}\,d\gamma(x)\,dy.
q\cdot N_{d}^{\prime}=q^2+q\phi(b)+\frac{q\phi(b)}{q-1}\sum_{n=0}^{q-2}\frac{G_{-2n}G_{nd}G_{-n(d-1)}}{G_{-n}}T^n\left(\frac{4b}{a^d}\right) &=q^2+q\phi(b)+\frac{q\phi(b)}{q-1}+\frac{q\phi(b)}{q-1}\sum_{n=1}^{q-2}\frac{G_{-2n}G_{nd}G_{-n(d-1)}}{G_{-n}}T^n\left(\frac{4b}{a^d}\right).
p_{20}+ p_{30}+ \ell_{30}p_{10}=0 \iff \lambda_2 q_{20}+ \lambda_3 q_{30}+ \lambda_1 \ell_{30}q_{10}=0 &\iff \lambda_2 a_{03}c_{20}+ \lambda_3 a_{01}c_{30}+ \lambda_1 b_{03}c_{10}=0
M \left\| \sum_{|k| \sim M}\frac{\omega_k}{\sqrt{\lambda_k}}e^{2\pi i k x}\right\|_{L^p}= M \left\| \sum_{|k| \sim M}\Phi_M(k) \frac{\omega_k}{\sqrt{\lambda_k}}e^{2\pi i k x}\right\|_{L^p}= \left\| \sum_{|k| \sim M}\Psi_M(k) \omega_k e^{2\pi i k x}\right\|_{L^p}&\lesssim \left\| \sum_{|k| \sim M}\omega_k e^{2\pi i k x}\right\|_{L^p}\,.
\begin{aligned}\delta(t) = -\frac{\beta}{m-1}\,(T-t)^{-\alpha-1}+ \frac{C^{m-1}m}{a^2 (m-1)^2}\,(T-t)^{-\alpha m +2 \beta}\geq \frac{C^{m-1}m}{2 a^2 (m-1)^2}\,(T-t)^{-\alpha m + 2\beta}\,;\end{aligned}
\frac{d^2\phi}{d\theta^2}(0)= [\sum_{x,x'}W(x|x') \overline{P}^2_{0}(x')g(x,x')^2+2 W(x|x') g(x,x') \frac{d \overline{P}^2_{\theta}(x')}{d \theta}\Bigr|_{\theta=0}]-\eta(0)^2 = \mathsf{V}_0 [g(X,X')]+2 \sum_{x,x'}W(x|x') g(x,x') \frac{d \overline{P}^2_{\theta}(x')}{d \theta}\Bigr|_{\theta=0}
(\tfrac{1}{\sqrt{c^2+d^2}})\tfrac{-c\det g}{\sqrt{c^2+d^2}}+(\tfrac{bd+ac}{\sqrt{c^2+d^2}})\tfrac{d}{\sqrt{c^2+d^2}}= \tfrac{-c\det g}{\sqrt{c^2+d^2}}- \tfrac{bd^2+acd}{c^2+d^2}&= \tfrac{-cad+bc^2+bd^2+acd}{c^2+d^2}=b.
\delta \alpha_{1}+ \gamma \left( - \frac 1 2 C_{b c}^{a}B_{a}^{\mu \nu}A_{\mu}^{b}A_{\nu}^{c}\right) = \partial_{\mu}\left( - \frac 1 2 C_{b c}^{a}\varepsilon^{\mu \nu \lambda \rho}A_{\nu}^{b}A_{\lambda}^{c}\eta_{a \rho}\right) .
\hat{P}:= x (x,v_1^{-\alpha_1}) v_1^{-\alpha_1}(v_1^{-\alpha_1}, v_1^{\alpha_1}) v_1^{\alpha_1}(v_1^{\alpha_1},v_{2}^{\beta_1}) v_2^{\beta_1}\dots \dots v_{n-1}^{\alpha_{n-1}}(v_{n-1}^{\alpha_{n-1}},v_{n}^{\beta_{n-1}}) v_n^{\beta_{n-1}}(v_n^{\beta_{n-1}}, v_n^{-\beta_{n-1}}) v_n^{-\beta_{n-1}}(v_{n}^{-\beta_{n-1}},y) y
{d\over dt}X = i\sum_{\rho\in\Delta_{+}}g_{|\rho|}\,(\rho^{\vee}\!\!\cdot\hat{H})\,y_{|\rho|}(\rho\cdot q,(\rho^{\vee}\!\!\cdot\hat{H})\xi)\, (\rho\cdot\dot{q})\, \hat{s}_{\rho}
x_{w}= d_{u}(w'u^{k-1}) - p(x_{w'u^{k-2}}) + 2p(x_{w'u^{k-1}}) - \dim(w'u^{k-2}) + 2\dim(w'u^{k-1}) - \dim(w'u^{k}) & = d_{u}(w'u^{k-1}) - p(x_{w'u^{k-2}}) + 2p(x_{w'u^{k-1}})
j_{3}( \alpha , m , L ) = \langle \Omega | \bar{\chi}_{\alpha}\left( x \right) \gamma_{3}\chi_{\alpha}\left( x \right) | \Omega \rangle = \frac{1}{L}\sum_{k_{3}}\int \frac{d^{2}k}{( 2 \pi )^{2}}\frac{2 k_{3}}{E_{k}}
\varepsilon(i,j)=(-1)^{ij}, & e_1\cdot e_2= e_2\cdot e_1=-2e_3, &\begin{array}[t]{l}\alpha(e_1)=\sqrt{2}e_1,\alpha(e_2)=e_3-e_2, \alpha(e_3)=e_3.\end{array}
\begin{aligned}G\left(u(x_0)-u(y)\right)- G\left(v(x_0)-v(y)\right) \leq0 ,\forall y\in\mathbb{R}^n, G\left(u(x_0)-u(y)\right)- G\left(v(x_0)-v(y)\right) <0 ,\forall y\in\mathbb{R}^n\setminus\Omega.\end{aligned}
x= \sum \limits_{j\in \mathbb{J}}\Psi_j^*A_jx \iff \sum\limits_{j\in\mathbb{J}}(2e-c_j)\langle A_jx, \Psi_jx\rangle = \langle x, x\rangle \iff \sum\limits_{j\in\mathbb{J}}c_j^2\langle A_jx, A_jx\rangle = \langle x, x\rangle.
\Delta_{f}\psi = \psi'' + \psi' \left\{\frac{m}{d+R}- \langle x, \nabla r \rangle \right\}\leq \psi'' + \psi' \left\{\frac{m}{d+R}- |x-y| + |y - z| + |z| \right\}= \psi'' +\psi' \left\{\frac{m}{d+R}- d + |z| \right\}.
[\begin{matrix}\dot{x}_{1}\\ \dot{x}_{2}\\ \dot{x}_{3}\end{matrix}]=A[\begin{matrix}x_{1}(t)\\x_{2}(t)\\x_{3}(t)\end{matrix}]+B[\begin{matrix}u_{1}(t)\\u_{2}(t)\\u_{3}(t)\end{matrix}],
\varphi_{0}(a_{1},a_{2},\ldots,a_{n},c_{0},c_{1},\ldots,c_{n})\cdot w=A(\sum_{i=0}^{n}\alpha_{i}v_{i})+S(C(\sum_{i=0}^{n}\alpha_{n+1+i}v_{i}))+S(A^{-T}(\sum_{i=0}^{n}\alpha_{n+1+i}v_{i})),
\Delta(a, b, i) = \frac{i^2}{m}+ |i| - (a+i)^2 \frac{m+2}{2m}+ \frac{(b+i)^2}{2}+ \frac{|b+i|}{2}= |i| + \frac{|b+i|}{2}+bi - ai \frac{m+2}{m}+\frac{b^2}{2}- a^2\frac{m+2}{2m}= |i| + \frac{|b+i|}{2}+ (b- a \frac{m+2}{m}) i +\frac{b^2}{2}- a^2\frac{m+2}{2m}
{\rho'}_{x'\sigma'}^{\sigma''}\cdot \rho_{x\sigma}^{\sigma'}={\rho'}_{g'\sigma'}^{\sigma''}\cdot \rho_{g\sigma}^{\sigma'}={\rho'}_{u''g'\sigma'}^{u''\sigma''}\cdot \rho_{u''g'u'g\sigma}^{u''g'u'\sigma'}={\rho'}_{u''g'\sigma'}^{\sigma''}\cdot \rho_{u''g'u'g\sigma}^{u''g'\sigma'}
Y_m(\mu = 0,y) = \frac{\sqrt{m}\, \Gamma(my)}{m! \Gamma (my +1-m)}e^{-m\tau_0}= \frac{\Gamma(1+my) \Gamma(1+m(1-y))}{2\,\Gamma(1+m)}\, e^{-m\tau_0}B_m
\hbox{$\frac{(n-1)}{2}$}\vert \mu\vert - \hbox{$\frac12$}\ell(u_\mu)-\hbox{$\frac12$}\ell(v_\mu) =\hbox{$\frac{(n-1)}2$}\vert \lambda\vert - \hbox{$\frac12$}\ell(t_\mu)=\hbox{$\frac{(n-1)}2$}\vert \lambda\vert - \hbox{$\frac12$}\ell(t_\lambda)= n(\lambda).
\frac{r_j-r_i-\nu_j+\nu_i}{\Gamma(1+\nu_j+r_i-r_j)\Gamma(1+\nu_i+r_j-r_i)}\qquad{}=\frac{\sin(\pi(r_j-r_i-\nu_j+\nu_i))}\pi\frac{\Gamma(1-r_j+r_i+\nu_j-\nu_i)}{\Gamma(1+\nu_j+r_i-r_j)}\frac{\Gamma(1+r_j-r_i-\nu_j+\nu_i)}{\Gamma(1+\nu_i+r_j-r_i)}.
\|(I-\Pi)u\|_{L^2}\leqslant \;C\langle\lambda-\Re z\rangle^{-1}(\|v\|_{L^2}+\|(P_\lambda-z)\Pi u\|_{L^2})+C\langle\lambda-\Re z\rangle^{-3/4}|v_0| \leqslant \;C\langle\lambda-\Re z\rangle^{-1}\|v\|_{L^2}+|v_+|+C\langle\lambda-\Re z\rangle^{-3/4}|v_0|.
t_1(1-\cos\alpha) +2\left(t_2+\left|\dfrac{a_{n-1}}{a_n}\right|\right)\cos\alpha+t_1\sin\alpha &\leq \dfrac{2t_2|a_{k+1}|+2|a_k|}{|a_n|t_1^{n-k-1}}\cos\alpha+2\sin\alpha\sum\limits_{\nu=0}^{n-2}\left|\dfrac{a_\nu}{a_n}\right|\dfrac{1}{t_1^{n-\nu-1}}
V = \frac{1}{2}\int_{M}d^{2}x \left( \Omega^{- 1}F_{1 2}^{2}+ D_{1}\phi \overline{{D_{1}\phi}}+ D_{2}\phi \overline{{D_{2}\phi}}+ \frac{\Omega}{4}( | \phi |^{2}- 1 )^{2}\right)
n^{\mu}= \pm \left({\frac{\dot{a}}{F ( a )}}, \sqrt{F ( a ) + \dot{a}^{2}}, 0 , 0 \right) ; \qquad n_{\mu}= \pm \left( - \dot{a},{\frac{\sqrt{F ( a ) + \dot{a}^{2}}}{F ( a )}}, 0 , 0 \right) .
\limsup_{z\to0^+}\left|z^{\mu-1}W_2(u,z)-\Gamma(\mu)2^{\mu-1}u^{-\mu}\left(1-2\mu\sum_{s=0}^{N-1}\frac{B_s'(0)}{u^{2s+2}}\right)\right| \qquad{}\le\limsup_{z\to0^+}\left|\Gamma(\mu)2^{\mu-1}u^{-\mu}g_2(u,z)- \Gamma(\mu+1)2^\mu u^{-\mu-2}h_2(u,z)\right| .
\sum_{i,j=1}^3\int_{\mathbb R^3}\partial_i[(a_{ij}*\mu)v_j]f^2dv =\sum_{i,j=1}^3\int_{\mathbb R^3}\partial_i\bigg(\int_{\mathbb R^3}a_{ij}(v-v')v_j'\mu(v')dv'\bigg)f^2dv =\sum_{i,j=1}^3\int_{\mathbb R^3}\partial_i[a_{ij}*(v_j\mu)]f^2dv.
\lim_{n\to\infty}\widetilde s_n=\infty\,,\lim_{n\to\infty}\widetilde t_n-\widetilde s_n=\infty\,, \lim_{n\to\infty}\frac{1}{\widetilde t_n-\widetilde s_n}\ln\frac{\|X(\widetilde t_n)\xi\|}{\|X(\widetilde s_n)\xi\|}=\lambda\,.
\big( K\circ v_j \big)^{(q)}= \sum c_{m_1, \ldots, m_q}(K^{(M_q)}\circ v_j) \prod_{s=1}^q \big(v_j^{(s)}\big)^{m_s}= \sum c_{m_1, \ldots, m_q}\frac{K^{(M_q)}\circ v_j}{F_j^{M_q}}\prod_{s=1}^q \big(f_0^{(s-1)}\big)^{m_s},
A'_{11}=2\zeta^4 + \zeta^2 qr - \varphi_t A'_{12}=2i\zeta^3q -\zeta u_x e^{2i\varphi}+ i\zeta q^2 r A'_{21}=2i\zeta^3r +\zeta v_x e^{-2i\varphi}+ i\zeta qr^2 \ .
\begin{array}{ll}\frac{\partial \delta w(t,x)}{\partial t}=\sum_{ij}\left(a^{w^2}_{ij}-a^{w^1}_{ij}\right)\frac{\partial w^1}{\partial x_i\partial x_j}+\sum_{ij}a^{w^2}_{ij}\frac{\partial \delta w}{\partial x_i\partial x_j}-b^{w^2}_i\frac{\partial \delta w}{\partial x_i}+\sum_i (b^{w^2}_i(x)-b^{w^1}_i(x))\frac{\partial w^1}{\partial x_i},\end{array}
(\cot \langle \alpha_{1}, H \rangle)^{2}= \begin{cases}\frac{-(m_{1}+2m_{2}+6m_{3}+n_{1}) \pm \sqrt{(m_{1}+2m_{2}+6m_{3}+n_{1})^{2}-4(m_{1}+2m_{2}+m_{3}) (n_{1}+m_{3})}}{2(m_{1}+2m_{2}+m_{3})}(m_{3}>0) 1 (m_{3}=0).\end{cases}
\pi \left( \frac{\partial}{\partial x_1},\frac{\partial}{\partial x_1}\right) = \pi \left( \frac{\partial}{\partial x_2},\frac{\partial}{\partial x_2}\right) \mbox{and}\pi \left( \frac{\partial}{\partial x_1},\frac{\partial}{\partial x_2}\right) =0,
S_{\alpha}(s)u_{n}=S_{\alpha}(s)\big(\frac{1}{\nu(d,\alpha)}s_{n}\chi_{n^{-\alpha q/d}s_{n}^{-q/d}}(x_{n})\big) &=\frac{s_{n}}{\nu(d,\alpha)}S_{\alpha}(s)\chi_{n^{-\alpha q/d}s_{n}^{-q/d}}(x_{n}) &\geq s_{n}\chi_{n^{-\alpha q/d}s_{n}^{-q/d}+\sqrt{s}}(x_{n}) &\geq s_{n}\chi_{n^{-\alpha q/d}s_{n}^{-q/d}}(x_{n}).
\nu_j(n, s) = ~ (j-\psi(n))^2/2 + (j-\psi(n))(s + \log_{1/p}(1 + (p/q)^s) + \psi(n) + 1) - \log_{1/p}n \log_{1/p}(1 + (p/q)^s) + \psi(n)^2/2 + o(\psi(n)^2).
D_{\sigma,k-1}(h) =r^2D_{\sigma,k-1}\left(\frac{(z-\lambda)(g(z)-g(r\lambda))}{z-r\lambda}\right) \leqslant \frac{4r^2}{(1+r)^2}D_{\sigma,k-1}(g(z)-g(r\lambda)) &=\frac{4r^2}{(1+r)^2}D_{\sigma,k-1}(g).
\widehat{s}(t)=\frac{1}{M}\sum_p\|\boldsymbol{\alpha}[p]\|_2^2 s(t)+\frac{1}{M}\sum_p\sum_{p\neq l}\boldsymbol{\alpha}^{\mathrm{H}}[p]\boldsymbol{\alpha}[l]s(t-\tau_l+\tau_p)+\sum_p w_p(t),
\xi_{\alpha \beta}( \mu_{1}^{2}, \mu_{2}^{2}, \mu_{3}^{2}, p^{2}, q^{2}) = \int_{0}^{1}d z \int_{0}^{1 - z}d y \frac{z^{\alpha}y^{\beta}}{Q ( y , z , \mu_{1}^{2}, \mu_{2}^{2}, \mu_{3}^{2}, p^{2}, q^{2})},
\begin{aligned}W^j_0(x)\coloneqq \, \frac{|b^j|}{8\pi}\frac{E}{1-\nu^2}\Big((1-\log R^2)-\frac{|x|^2}{R^2}+\log|x|^2\Big)\Big\langle\frac{\Pi(b^j)}{|b^j|},x-x^j\Big\rangle\,.\end{aligned}
\frac{(q^m;q^m)_\infty}{(q)_\infty}\cdot\prod_{i=1}^{4}\theta(q^i;q^m)\prod_{i=5}^{\frac{m}{2}}\theta(q^i;q^m)^2 &= \prod_{i=5}^{\frac{m}{2}}\theta(q^i;q^m).
\sum_{0\le m_1< \cdots<m_{n}<\infty}\frac{(\alpha)_{m_1}}{{m_1}!}\frac{{m_n}!}{(\alpha)_{m_n}}\left\{\prod_{i=1}^{n}\frac{1}{(m_i+\alpha)^{a_i}(m_i+\beta)^{b_i}}\right\},
\mathbf{x}^{k+1}:= \underset{\mathbf{x}}{}f(\mathbf{x}) + (\sqrt{\eta}\boldsymbol{\nu}^k)^T \mathbf{E}_{}\mathbf{x}+ \frac{\rho}{4}\|\mathbf{E}_{}\mathbf{x}\|^2, & \boldsymbol{\nu}^{k+1}:= \boldsymbol{\nu}^k + \sqrt{\eta}\frac{\rho}{2}\mathbf{E}_{}\mathbf{x}^{k+1}.
\Lambda_{1}^{3 ( N + M ) - 2 N}\tilde{\Lambda}_{2}^{3 ( N - M ) - 2 N}\propto \mu^{2 N}\propto \lambda_{1}^{M}\tilde{\Lambda}_{1}^{3 N - 2 ( N + M )}\lambda_{2}^{- M}\Lambda_{2}^{3 N - 2 ( N - M )}\ .
g(t,q)=g(t_0,q_0)+\int_0^1 \partial_s g(t_0+s(t-t_0),q_0+s(q-q_0)) (t-t_0) +\nabla_q g(t_0+s(t-t_0),q_0+s(q-q_0))\cdot (q-q_0) ds
\lim_{n \rightarrow \infty}\| u_{0,n}\|_{\dot H^1}^2 + \| u_{1,n}\|^2_{L^2}- \left ( \sum_{j = 1}^J \| U^j_{L,n}(0) \|^2_{\dot H^1}+ \| \partial_t U^j_{L,n}(0) \|^2_{L^2}\right ) - \left ( \| w^J_{n}(0) \|^2_{\dot H^1}+ \| \partial_t w^J_{n}(0) \|^2_{L^2}\right ) = 0,
\frac{\partial}{\partial x^{i}}\langle J^{a i}( x ) J^{b j}( y ) J^{c k}( z ) \rangle |_{a n o m a l y}= - 2 \alpha d^{a b c}\epsilon^{i j k l}\frac{\partial}{\partial x^{i}}\delta^{( d )}( x - y ) \frac{\partial}{\partial x^{l}}\delta^{( d )}( x - z ) .
d \Theta^{\underline{{\beta}}}v_{\underline{{\beta}}}^{~ \underline{{\alpha}}}= \left( d \Theta^{\underline{{\beta}}}v_{\underline{{\beta}}}^{~ - q}, d \Theta^{\underline{{\beta}}}v_{\underline{{\beta}}}^{~ + q^{\prime}}\right) .
\nabla \pi_R =\left[\begin{array}{cccc}0 0 \frac{2\tau}{c_0}\pi_1\partial_s\pi_{T\Psi}\hat{R}(u, v, s) \frac{2\tau}{c_0}\pi_2\partial_s\pi_{T\Psi}\hat{R}(u, v, s) 0 0 \frac{2}{c_0}\pi_1\pi_{T\Psi}\hat{R}(u, v, s) \frac{2}{c_0}\pi_2\pi_{T\Psi}\hat{R}(u, v, s) 1 0 * * 0 1 * *\end{array}\right]
\mathcal{G}^k_{\log}(\vec{a};\vec{b}) =2\pi \times (i)^{k+1}\frac{\log(\langle \vec{a},\vec{b}\rangle)}{||\vec{a}||^{k+1}+||\vec{b}||^{k+1}}\int \limits_{|a_n|}^{|b_n|}\int \limits_{|a_{n-1}|}^{|b_{n-1}|}\cdots \int \limits_{|a_1|}^{|b_1|}\sqrt[k]{\sum \limits_{j=1}^{n}x^k_j}dx_1dx_2\cdots dx_n.
U_{t}(x)={\mathcal C}\,t^\frac{2\,s-N}{2}\,\left(1+\left(\frac{|x-x_0|}{t}\right)^2\right)^{\frac{2\,s-N}{2}},\mathcal{C}\in\mathbb{R}\setminus\{0\},\t>0,\x_0\in\mathbb{R}^N,
R^{(\alpha)}{}^{l}_{\phantom{1}k\bar{i}j}=\partial_{\bar{i}}\Gamma^{(\alpha)}{}^{l}_{jk}=\partial_{\bar{i}}\Big(\Gamma^{(0)}{}^{l}_{jk}-\frac{\alpha}{2}g^{l\bar{m}}T_{jk,\bar{m}}\Big) &=R^{(0)}{}^{l}_{\phantom{1}k\bar{i}j}-\frac{\alpha}{2}\partial_{\bar{i}}\Big(g^{l\bar{m}}T_{jk,\bar{m}}\Big).
\ln \left( k^\alpha \left(\frac{h}{h+\sigma}\right)^p \right)\leq\ln \left( k^\alpha \left(\frac{\overline{h}}{\overline{h}+\sigma}\right)^{c_2 \log k}\right) = \underbrace{\ln k}_{\ge 1}\underbrace{\left[ \alpha + c_2 \ln \frac{\overline{h}}{\overline{h}+\sigma}\right]}_{\leq \ln \varepsilon \leq 0}\leq \ln \varepsilon.
\begin{array}{@{}l@{}}{\displaystyle |\sum_{k=1}^{M}J_2^{(k),1}| \leq C\big(h^{2r+2}+(\Delta t)^{4}\big) +C\|\theta_{\Psi}^{M}\|_{\mathcal{L}^2}^{2}+ C\Delta t \sum_{k=1}^{M-1}{\|\theta_{\Psi}^{k}\|_{\mathcal{L}^2}^{2}}.}\end{array}
{\Psi}^{(1)}_{\mu}({\alpha},{\bf n},t,{\bf x})={\frac{1}{{(2\pi)}^{3/2}}}\sum_{\mu^{'}=-1}^{1}\int\frac{d{\bf p}}{p_0}\,{\xi}^{(1)}_{\bf p\mu\mu^{'}}({\alpha},{\bf n})\,\exp[-i(p\cdot{x})]{\Psi}^{(1)}_{\mu^{'}}({\bf p}).
L_{\nu}=\frac{2k\nu^{2}}{c^{2}}((1+\frac{c^{2}K_{\nu}}{2h\nu^{3}})ln(1+\frac{c^{2}K_{\nu}}{2h\nu^{3}})-(\frac{c^{2}K_{\nu}}{2h\nu^{3}})ln(\frac{c^{2}K_{\nu}}{2h\nu^{3}}))
Z_C[J,\zeta,\bar\zeta]=\int{\cal D}A_\mu{\cal D}\psi{\cal D}\bar\psi\delta\left(\partial_\mu A^\mu-C(x)\right) det{\cal M}_Le^{iS+i\int dx (J^\mu A_\mu+\bar\psi\zeta+\bar\zeta\psi)}.
G (\Theta(x),y)=(1+\alpha)\sum_{k=0}^\infty(\alpha\beta)^k\Big[\log|x-{\bf 2k}-\Theta(y)|-\beta\log|x+{\bf 2k+1}+\overline{\Theta(y)}|\Big] &\quad-(1+\alpha)\sum_{k=0}^\infty(\alpha\beta)^k\Big[\log|x-{\bf 2k}|-\beta\log|x+{\bf 2k+1}|\Big]+\frac{(1+\alpha)(1-\beta)}{1-\alpha\beta}\log|y|;
\lim_{\epsilon\downarrow0}f(\epsilon)(I-\lambda C_p G^{\omega,\lambda}_{p,p,p}(E\pm\iota\epsilon))(I+\lambda C_p G^\omega_{p,p}(E\pm\iota\epsilon))=0 \Rightarrow \left(\lim_{\epsilon\downarrow0}f(\epsilon) C_p G^{\omega,\lambda}_{p,p,p}(E\pm\iota\epsilon)C_p\right)(C_p^{-1}+\lambda G^\omega_{p,p}(E\pm\iota0))=0,
2c_{2}x_{1}=3(4l+2)c_{2}x_{1}=3nc_{2}x_{1}= [(n-1)c_{1}^{2}+nc_{2}]x_{1}-[(n-1)c_{1}^{2}-2nc_{2}]x_{1}= [2nc_{2}-(n-1)c_{1}^{2}]x_{1}\in{{^U}E}_{4}^{3,4},
\tilde g_i^{jk}=\hat g_i^{+,-}g_{i0}^{jk},\ \ \ 1\leq j,k \leq n-1, &\tilde g_i^{-,k}=-\hat g_i^{+,-}\alpha_{ks}^{(i)}(s,\tau,\beta^{(i)}),\ \ \tilde g_i^{+,k}=0,\ \ 1\leq k\leq n-1,\ \i=1,2.
A' - \mathcal{Z}_{M'}\,A'\, \mathcal{Z}_{N'}^T = \;\; A' - (\mathcal{C}+ VW^T)A' \mathcal{Z}_{N'}^T = \;\; A' - \mathcal{C}\, A'\, \mathcal{Z}_{N'}^T \,-\, VW^T \! A' \mathcal{Z}_{N'}^T = \;\; V' W'^T \,-\, VW^T \! A\, \mathcal{Z}_{N'}^T = \;\; Y Z.