harryrobert/nav2tex-decoder-latex-pretrain huggingface dataset:
N_1(T,\chi) = \frac{T}{\pi}\log{\frac{qT}{2m\pi}}- \frac{T}{\pi}- A(t_q,\chi) - B(t_q,\chi) + A(T,\chi) + B(T,\chi) &\quad\quad+ A(-t_q,\chi) + B(-t_q,\chi) - A(-T,\chi) - B(-T,\chi) + O(m^{1/2}\log{q}).
\Pi_{g_0}\circ D\Phi^W_{(g,\pi)}\circ \rho_g (D\Phi^W_{(g,\pi)})^* (f_m, X_m) = \Pi_{g_0}\circ \Phi^W_{(g,\pi)}(g,\pi) + \Pi_{g_0}(\psi, V) - \Pi_{g_0}\circ \Phi^W_{(g,\pi)}(\gamma_m, \tau_m)
\begin{aligned}|\alpha(g_p)(f \circ \rho) - f\circ \rho|_{\infty}^- \le \left|\left( \sum_{t \in K^M}|\alpha(t)f - f|\right) \circ \rho \right|_{\infty}^- &\le \sum_{t \in K^M}|\alpha(t)f - f|_{\infty}^- &\le M|K|^M \max_{k \in K}|\alpha(k)f-f|_{\infty}^-.\end{aligned}
\mu(\mathbb{R}^n)=\mu\left(\bigcup_{k\in\mathbb{N}}kQ\right) =\lim_{k\to\infty}\mu(kQ)\leq\lim_{k\to\infty}c_\mu k^{n_\mu}\mu(Q)=\lim_{k\to\infty}c_\mu k^{n_\mu}\cdot 0=0,
{\cal L}_{f}=\sqrt{2}i(\alpha' p^{+})\left\{\theta_{+}^{\dagger}(\partial_{\tau}+\partial_{\sigma})\theta_{+}+\theta_{-}^{\dagger}(\partial_{\tau}-\partial_{\sigma})\theta_{-}+2\theta_{+}^{\dagger}\tilde{\Omega}_{-}\theta_{-}+2\theta_{-}^{\dagger}\tilde{\Omega}_{+}\theta_{+}\right\}
D^{(g,\Sigma,e)}:=e_1\cdot\nabla^{\mathcal{S}\otimes \mathcal{\mathcal{I}}}_{e_1}+e_2\cdot\nabla^{\mathcal{S}\otimes \mathcal{\mathcal{I}}}_{e_2}+e_3\cdot\nabla^{\mathcal{S}\otimes \mathcal{\mathcal{I}}}_{e_3}
\Delta u_1 = 4\left(k_{11}e^{u_1}+k_{12}e^{u_2}-1\right)+4\pi\sum\limits_{j=1}^{N_1}{\delta_{p_j}(x)}\Delta u_2 =4\left(k_{21}e^{u_1}+k_{22}e^{u_2}-1\right)+4\pi\sum\limits_{j=1}^{N_2}{\delta_{q_j}(x)},
d_{2}\le 4^{(2n^3)^{149n^8}}3n^2(2n^3)^{148.5n^6}18^{n^2}(2n^3)^{148.5n^8}= 3n^2 18^{n^2}4^{(2n^3)^{149n^8}}(2n^3)^{(148.5n^8+148.5n^6)},
\delta\|v\|^2\leq -\langle T_{22}v,v\rangle=\Big|s\big\langle \big(A_{22}-A_{21}(\tilde{A}_{11})^{-1}A_{12}\big)v,v\big\rangle\Big|\leq |s|\big\|A_{22}-A_{21}(\tilde{A}_{11})^{-1}A_{12}\big\|\|v\|^2\leq \frac{\delta}{2}\|v\|^2,
\begin{aligned}I_{b}\leq k^+ w(x) \int_0^t\bigg(\int_{|y| \ge \max \{\frac{|x|}{2},1 \}}+\int_{1 \le |y| \le \frac{|x|}{2}}\bigg) \frac{\exp\Big(-\frac{|x-y|^2}{4s}\Big)}{\big(4\pi s \big)^{n/2}|y|^{\ell/\delta}}dyds =:k^+ (I_b^-+I_b^+) \, . \end{aligned}
\sum_{n_1, n_2 \geq 0}\frac{q^{\frac{3}{2}n_1^2 + 4 n_1n_2 + 4 n_2^2 - \frac{1}{2}n_1}}{(q;q)_{n_1}(q^2;q^2)_{n_2}}x^{n_1 + 2 n_2}= \sum_{n \geq 0}\frac{q^{n^2}}{(q;q)_n}x^n
\theta_n^{\mathrm{causal}}=\left(\sum_{i=1}^n E\left[W_{n,i}^{XX}\right]\right)^{-1}\sum_{i=1}^nE\left[W_{n,i}^{XX}\varphi_{n,i}\right], \shortintertext{and, with probability approaching one,}\theta_n^{\mathrm{causal}, \mathrm{sample}}=\left(\sum_{i=1}^n R_{n,i}E\left[W_{n,i}^{XX}\right]\right)^{-1}\sum_{i=1}^n R_{n,i}E\left[W_{n,i}^{XX}\varphi_{n,i}\right],
0 \le2\alpha_k[(f+g)(x^{k+1})-(f+g)(x_*)]\le\|x^k-x_*\|^2-\|x^{k+1}-x_*\|^2 &=(\|x^k-x_*\|+\|x^{k+1}-x_*\|)\cdot(\|x^k-x_*\|-\|x^{k+1}-x_*\|) &\le2M(\|x^k-x_*\|-\|x^{k+1}-x_*\|) &\le 2M\|x^k-x^{k+1}\|,
\bigl\| \textbf{x}^{(2)}\bigr\|_{\ell_1}= \bigl\| \textbf{x}^{(1)}\bigr\|_{\ell_1}- 2 |\bigl(\textbf{x}^{(1)}\bigr)_{k_2}| &=\bigl\|\textbf{x}\bigr\|_{\ell_1}-2 \left(|x_{k_1}|+\bigl| \bigl(\textbf{x}^{(1)}\bigr)_{k_2}\bigr|\right)\,.
\gamma_{\scriptscriptstyle{l}}= \Phi / 2 \pi - l + \frac{a^{2 l}}{\int_{0}^{a}d r r^{2 l - 1}e^{2 \phi ( r )}}- k^{2}a \chi_{\scriptscriptstyle{l}}^{\prime}( a ) + \mathrm{O}( k^{4}) .
VII_{A}\leq \sum_{i=1}^{d}\varepsilon F(M_{n})\int_{t}^{T}\int_{\mathbb{T}^{d}}\frac{1}{2\sigma_{10}}|\partial^{\alpha}\partial_{x_{j}}w^{n+1}|^{2}+\frac{\sigma_{10}}{2}|\partial^{\alpha}\partial^{2}_{x_{i}x_{j}}w^{n}|\dxd\tau.
[ Z_{1}, Z_{2}] = \imath s_{3}I + \imath l_{i}F_{i}( Q , P ) \ , \ \w h e r e \ \s_{3}= x_{i}^{1}y_{i}^{2}- x_{i}^{2}y_{i}^{1}\quad l_{i}= t^{1}y_{i}^{2}- t^{2}y_{i}^{1}\ .
A=\int_{0}^{\theta}\int_{0}^{r}dS=\int_{0}^{\theta}\int_{0}^{r}\tilde{r}d\tilde{r}d\tilde{\theta}=\int_{0}^{\theta}\frac{1}{2}r^{2}d\tilde{\theta}=\frac{r^{2}\theta}{2}
N(t) = 1 - 2t^4 - 2t^5 - 2t^6 + 2t^7 + 3t^8 + 3t^9 + 2t^{10}- 2t^{11}- 2t^{12}- 2t^{13}+ t^{17}= 1 - (2t^4 + 2t^5 + 3t^6 + t^7 + t^8) + (t^6 + 3t^7 + 4t^8 + 4t^9 + 3t^{10}+ t^{11}) - \cdots
\cfrac{a_0}{\big(\pi \prod_{k=1}^\infty \frac{\mu_k - \mu_0}{k^2}\big)^2}-\cfrac{1}{\pi}+ \sum_{n=1}^\infty \left(\cfrac{a_n n^4}{\big( \pi[\mu_0-\mu_n] \prod_{k=1, k \neq n}^\infty \frac{\mu_k - \mu_n}{k^2}\big)^2}- \cfrac{2}{\pi}\right)=-\cot\beta.
\sigma_{1}= \left[ \begin{array}{c c}{0}&{1}\\{1}&{0}\\ \end{array}\right] , \ \sigma_{2}= \left[ \begin{array}{c c}{0}&{- i}\\{i}&{0}\\ \end{array}\right] , \sigma_{3}= \left[ \begin{array}{c c}{1}&{0}\\{0}&{- 1}\\ \end{array}\right]
u^{n+1}_{j}= -2\theta_{1}^{2}\theta_{2}R^{3}u^{n}_{j+3}+2\theta_{1}\theta_{2}R^{2}u^{n}_{j+2}+(6\theta_{1}^{2}\theta_{2}R^{3}-R) &u^{n}_{j+1}+(1-4\theta_{1}\theta_{2}R^{2})u^{n}_{j}+ (R-6\theta_{1}^{2}\theta_{2}R^{3})u^{n}_{j-1}&+2\theta_{1}\theta_{2}R^{2}u^{n}_{j-2}+2\theta_{1}^{2}\theta_{2}R^{3}u^{n}_{j-3}.
\begin{aligned}- \nabla_y \cdot \left( D_0 \big(\nabla_y w_i + e_i\big)\right) = 0 \mbox{in}(0,T)\times \Omega \times Y^{\ast}, - D_0 \big(\nabla_y w_i + e_i \big) \cdot \nu = 0 \mbox{on}(0,T)\times \Omega \times \Gamma, w_i \mbox{is}Y\mbox{-periodic,}\int_{Y^{\ast}}w_i(t,x,y) dy = 0 \mbox{f.a.e.}(t,x) \in (0,T)\times \Omega.\end{aligned}
\lim_{t\to\infty}\,t^{\frac{N+A}{2}}\left[e^{-tL_k}\phi^{k,i}\right]\left(t^{\frac{1}{2}}y,t\right)=0\quad\mbox{in}L^2({\bf R}^N,e^{|y|^2/4}\,dy)\,\cap\,L^\infty(K), \lim_{t\to\infty}t^{\frac{N+2A}{2}}\frac{\left[e^{-tL_k}\phi^{k,i}\right](x)}{U_k(|x|)}=0\quad\,\,\qquad\mbox{in}L^\infty(B(0,R)),
I_4= \int_0^\infty (f_u^{(1)},f_h^{(1)})\cdot(u_p^1,h_p^1) x^{-2\sigma_1}{\rm d}y \leq \Vert (f_u^{(1)},f_h^{(1)})x^{\frac{1}{2}-\sigma_1}\Vert_{L_y^2}\Vert (u_p^1,h_p^1) x^{-\frac{1}{2}-\sigma_1}\Vert_{L_y^2}\leq Cx^{-\frac{3}{2}}+\delta_0\Vert (u_p^1,h_p^1) x^{-\frac{1}{2}-\sigma_1}\Vert_{L_y^2}^2.
\lambda = \sum_{Q \leq N_1}\sum_{\substack{(a,q) = 1 q \sim Q}}\tau_{-a/q}\bigg( \sum_{Q \leq 2^s \leq N}\phi^{(s)}\bigg)= \sum_{Q \leq N_1}\sum_{\substack{(a,q) = 1 q \sim Q}}\tau_{-a/q}\kappa( Q N \,\cdot\,).
\Psi_\xi(\theta) = \frac{\Gamma(\frac{1}{2}(-{\rm i}\theta +\alpha ))}{\Gamma(-\frac{1}{2}{\rm i}\theta)}\times\frac{\Gamma(\frac{1}{2}({\rm i}\theta +d))}{\Gamma(\frac{1}{2}({\rm i}\theta +d-\alpha))}, \theta\in\mathbb{R}.
V_{1}=\left[\begin{array}{ccccc|ccccc|ccccc|ccccc|ccccc}1 1 1 1 1 2 2 2 2 2 3 3 3 3 3 4 4 4 4 4 5 5 5 5 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 1 2 3 4 5 2 3 4 5 1 3 4 5 1 2 4 5 1 2 3 5 1 2 3 4\end{array}\right].
\gamma^{\textbf{ext}}_{1}\mathrm{U}_{\mathrm{e}}= -\mathbb{H}_{\alpha_\mathrm{m}}\Big[\mathrm{U}_{\mathrm{e}}\Big](\mathrm{x},t) - \Big(\frac{1}{2}I_{d}- \mathcal{K}_{\alpha_\mathrm{m}}^{*}\Big)\mathcal{S}_{\alpha_\mathrm{m}}^{-1}\Big(\frac{1}{2}I_{d}- \mathcal{K}_{\alpha_\mathrm{m}}\Big)\Big[\gamma^{\textbf{ext}}_{0}\mathrm{U}_{\mathrm{e}}\Big].
\bar{g}_{\mu \nu}= \mathrm{d i a g . \}( - 1 , e^{2 \sqrt{\bar{\Lambda}}t}, e^{2 \sqrt{\bar{\Lambda}}t}, e^{2 \sqrt{\bar{\Lambda}}t}) , \ \ \mathrm{( ~ d S_{4}~ b a c k g r o u n d , ~ \bar{\Lambda}> 0 ~ )}
\begin{array}{c}{{\left[ A^{-}, A^{+}\right]_{q \left( x \right)}= - 2 \alpha \left( x \right) \sinh \frac{i}{2}\frac{d}{d x}\left[ \alpha \left( x \right) \sinh 2 \rho_{\frac{s}{2}}\left( x \right) \right] \rightarrow}}\\{{\rightarrow \frac{d^{2}\rho \left( x \right)}{d x^{2}}}}\\ \end{array}
4\sum_{j<k\neq i}M_{jk}=2\sum_{j\neq i}\sum_{k\neq i}M_{jk}&= \sum_{j\neq i}( R_{\{1,2,j\}}+R_{\{2,3,j\}}+...+ R_{\{i-1,i+1,j\}}+...+ R_{\{n,1,j\}}&>0.
\prod_{i=1}^n \Big(1+ (t_{i+1}-t_{i})^{\frac{\beta}{\alpha}}\Big)^{-1}=\bigg[\prod_{i=1}^n (t_{i+1}-t_{i})^{1-\frac{\beta}{\alpha}}\bigg]\int_{[0,\infty)^n}\prod_{i=1}^n e^{-(t_{i+1}-t_{i})l_i}E_{\beta/\alpha}\Big(-l_i^\frac{\beta}{\alpha}\Big)dl_i.
\Pr (V_{n-1}\in du,V_{n}\in dv) =\Pr (V_{n-1}\in du,W_{n}\in d(v-u)) & =\Pr (V_{n-1}\in du)\Pr (W_{n}\in d(v-u)) & =\frac{\lambda^{n}e^{-\lambda v}u^{n-2}}{\Gamma (n-1)}dudv0<u<v<\infty.
-1 =\Big(u+\tfrac{d-a}{2c}v\Big)^2-\tfrac{D}{4c^2}v^2 &=\Big(u+\tfrac{d-a}{2c}v\Big)^2-\Big(\tfrac{\sqrt{D}}{2c}v\Big)^2 &=\Big( u+\tfrac{d-a+\sqrt{D}}{2c}v\Big ) \Big(u+\tfrac{d-a-\sqrt{D}}{2c}v \Big) &=\tilde u \tilde v.
F(x^-,x^+)=\left|{L \over \pi}\langle 0 | T^{++}(k) |n \rangle \right|^2\left({x^+ \over x^-}\right)^2{M_n^4 \over 8 \pi^2 k^3}K_4\left(M_n\sqrt{2 x^+ x^-}\right)
S_n'(y' - \sigma_{k+1}'(y')) = S_n'\Big[ y' - \frac{1}{w}\Big( \sigma'_1 \circ \sigma'_k(y') - (1-w) \sigma_{k-1}'(y') \Big) \Big] &= S_n' \Big(y'- \sigma_{k-1}'(y') \Big) - \frac{1}{w}S_n' \Big( \sigma'_1 \circ \sigma'_k(y') - \sigma_{k-1}'(y') \Big).
u(z) \ge 1+2\left(\beta-\frac{1}{2\log 2}\right)\int \frac{\mathrm{Arg}(zw^{-1})\Im(zw^{-1})}{|1-zw^{-1}|^{2}}\alpha(dw)=1+\left(\beta-\frac{1}{2\log 2}\right)\frac{1}{2\pi}\int_{0}^{2\pi}\frac{t\cos(t/2)}{\sin(t/2)}dt =1+2\left(\beta-\frac{1}{2\log 2}\right)\frac{1}{2\pi}\int_{0}^{\pi}t\cot(t/2)dt=2\beta\log 2>0.
s = 2(k-\ell) (|A|+ |B\setminus B'|) - n +\ell =\begin{cases}\ell + 2(k-\ell)|B\setminus B'|, \mbox{if}\frac{n}{k-\ell}\mbox{is even}k+ 2(k-\ell)|B\setminus B'|, \mbox{if}\frac{n}{k-\ell}\mbox{is odd}, \end{cases}
| \vec{p}_{f}| = \frac{\rho}{2}\sqrt{1 + \left[ \frac{m_{1}^{2}- m_{2}^{2}}{\sqrt{s}\rho}\right]^{2}}= \frac{1}{2 \sqrt{s}}\sqrt{s^{2}- 2 s ( m_{1}^{2}+ m_{2}^{2}) + ( m_{1}^{2}- m_{2}^{2})^{2}}= \frac{\lambda^{\frac{1}{2}}( s , m_{1}^{2}, m_{2}^{2})}{2 \sqrt{s}}
\rho ={\bf \nabla}.{\bf E}= -\left({\partial^2 \phi \over{\partial x^2}}+{\partial^2 \phi \over{\partial y^2}}\right)\qquad;\qquad{\bf j}={\bf \nabla}\times{\bf B}= -\left({\partial^2 \phi \over{\partial x^2}}+{\partial^2 \phi \over{\partial y^2}}\right)\,{\hat{\bf z}},
C_{\Gamma}:=\sup_{\mathbf{v}\in\mathbf{X}\backslash\left\{0\right\}}\frac{\left\Vert \Pi_{T}\mathbf{v}\right\Vert_{-1/2,\operatorname*{curl}_{\Gamma}}}{\left\Vert \mathbf{v}\right\Vert_{\operatorname*{curl},\Omega,1}}.
M_{1}(\mathcal{NC}(A(n,\nu))) =(2^{2n}-2^{n})(2^{2n}-2^{n}-1)^{2}-4(2^{2n}-2^{n}-1)(2^{n-1}(2^{n}-1)^{2})+2^{n}(2^{n}-1)^{3}= 2^{6n}-5\cdot2^{5n}+8\cdot2^{4n}-4\cdot2^{3n}= 2^{5n}(2^{n}-5)+2^{3n+2}(2^{n+1}-1)
h(x)=\sum_{i=1}^k\sum_{j=1}^{d_i}h(\alpha_{ij})\frac{\prod_{(k,l)\neq (i,j)}(x-\alpha_{kl})}{\prod_{(k,l)\neq (i,j)}(\alpha_{ij}-\alpha_{kl})}\Rightarrow a=\sum_{i=1}^k\sum_{j=1}^{d_i}\frac{h(\alpha_{ij})}{G'(\alpha_{ij})}
\begin{cases}\begin{aligned}\sum_{i,j,k,p}(\bar h^{p^{\ast}}_{ijk})^2=0, \ \ \ \ \bar\lambda_1\bar\lambda_2\neq 0, & S<\sup H^2=\bar H^2 \leq 3S-2 \ \ \ \S<\sup H^2<\dfrac43S,\end{aligned}\end{cases}
(\tau^* \circ \phi^*)(y^j) = \widetilde{x}^j + \sum_{|P| \geq 2}\widetilde{f}^j_P(\widetilde{x})\,\widetilde{\xi}^P &(\tau^* \circ \phi^*)(\eta^s_\mu) = \widetilde{\xi}^s_\mu + \sum_{|Q| \geq 2}\widetilde{g}^j_{Q}(\widetilde{x})\,\widetilde{\xi}^Q
\ddot{\phi}+3 \frac{\dot{a}}{a}\dot{\phi}+V'(\phi)+V'''(\phi)\frac{\hbar}{2}\int \frac{d^3k}{(2\pi)^3}\frac{\mid \varphi_k(t) \mid^2}{2}\coth\left[\beta_o\hbar W_k(t_o)/2\right] = 0
\begin{array}{lcl}\star_t\tau_3(t) = -\frac{\sqrt{6}\,\big(5y(t)^2+z(t)^2\big)}{21\,y(t)^3 z(t)^4}f^{3456}+\frac{\sqrt{6}\,\big(3y(t)^2-5z(t)^2\big)}{42\, y(t)^3 z(t)^4}(f^{1234}+f^{1256}) & +\frac{\sqrt{6}\,\big(2y(t)^2-z(t)^2\big)}{21\, y(t)^3 z(t)^4}(-f^{1367}-f^{1457}+f^{2357}-f^{2467}),\end{array}
L_{2}^{\ast}=\frac{\Gamma\left((m+1)/2\right)m^{3/2}\left[\left(1+\frac{\xi_{p}^{2}}{m}\right)^{-(m-3)/2}-\left(1+\frac{\xi_{q}^{2}}{m}\right)^{-(m-3)/2}\right]}{\Gamma(m/2)(m-1)(m-3)\sqrt{\pi}F_{Y}(\xi_{p},\xi_{q})},
\begin{array}{ll}a^{b}=u^{a}\nabla_{a}u^{b}={1\over V}\,\xi^{a}\nabla_{a}\,{1\over V}\,\xi^{b}={1\over V^2}\,\xi^{a}\nabla_{a}\,\xi^{b}&=-{1\over V^2}\,\xi^{a}\nabla^{b}\xi_{a}={1\over2}\,{1\over V^2}\,\nabla^{b}\left(-\xi^{a}\xi_{a}\right) &={1\over2}\,{1\over V^2}\,\nabla^{b}\,V^2=\nabla^{b}\ln V\;,\end{array}
\frac{dF}{dt}= \, 2 C_1 \mathbf{E}\Big(f(t) \int_0^t f(s) ds\Big) + C_2 \mathbf{E}\big(f(t)\big)^2 \le \, C_1 \mathbf{E}\Big(\int_0^t f(s) ds\Big)^2 + (C_1 + C_2) \mathbf{E}\big(f(t)\big)^2 \le \, F(t) + (C_1 + C_2) \big(g(t) + F(t)\big) = \, (C_1 + C_2 + 1) F(t) + (C_1 + C_2) g(t)\,.
v(i,j)^2 =\sum_{k=i}^jv_k^2+v_iv_{i+1}+v_{j}v_{j-1}+\sum_{k=i+1}^{j-1}(v_kv_{k-1}+v_kv_{k+1}) &=-2(j-i+1)+2+2(j-i-1)=-2.
\phi (p_{r(j_0 \beta j_1)}) = \phi (p_{r(\beta j_1)}) / 2 = \phi (p_{r(j_1 \beta^{-1})}) /2 = \phi (p_{r(\beta^{-1})}) /2 = \phi (p_{r(\beta)}) /2 = \phi (p_{r(j_0 \beta)}) = \phi (p_{r(\beta^{-1}j_0)}) = \phi (p_{r(j_1 \beta^{-1}j_0)}).
\frac{\partial}{\partial t_{p}}\left( \begin{array}{c c}{{f}}\\{{{\overline{{f}}}}}\\ \end{array}\right) = J_{1}\left( \begin{array}{c c}{{{\delta}/{\delta f}}}\\{{{\delta}/{\delta{\overline{{f}}}}}}\\ \end{array}\right) H_{p + 1}= J_{2}\left( \begin{array}{c c}{{{\delta}/{\delta f}}}\\{{{\delta}/{\delta{\overline{{f}}}}}}\\ \end{array}\right) H_{p},
V_{vdW}=\frac{A_{H}}{6}[ln\frac{h(4R_{P}+h)}{(2R_{P}+h)^{2}}+\frac{(2R_{P}^{2})}{h(4R_{P}+h)}+\frac{(2R_{P}^{2})}{(2R_{P}+h)^{2}}]
H_{0}\langle x \rangle^{\mu}f = -i \mu ( \alpha \cdot x ) \langle x \rangle^{\mu -2}f+ \langle x \rangle^{\mu}H_{0}f &= -i \mu ( \alpha \cdot x ) \langle x \rangle^{\mu -2}f - \langle x \rangle^{\mu}Qf\in \mathcal{L}^{2}(\mathbb{R}^{3}).
\int_{B_d(0)}\frac{1}{|x-y|^{n-2}(1+|y|)^{1+\sigma}}\, dy\leq \frac{C}{d^{n-2}}\int_{B_d(0)}\frac{1}{(1+|y|)^{1+\sigma}}\, dy\leq \frac{C}{d^{n-2}}d^{n-2-\sigma}=\frac{C}{d^\sigma}
A=\pi^{2}C\rho_{1}\rho_{2}=\frac{6\pi^{2}k_{B}T\rho_{1}\rho_{2}}{(4\pi\epsilon_{0})^{2}}\sum_{n=0,1...}^{\infty}\frac{\alpha_{1}(i\nu_{n})\alpha_{2}(i\nu_{n})}{\epsilon_{3}^{2}}
\sum_{n\geq 1}\frac{1}{\beta_{\nu,n}^2-1}=-\frac{\nu(\nu-2)J_{\nu}(1)+(3-2\nu)J_{\nu}'(1)+J_{\nu}''(1)}{2(2-\nu)J_{\nu}(1)+2J_{\nu}'(1)}=\frac{1}{2}\cdot \frac{J_{\nu}(1)+2(1-\nu)J_{\nu+1}(1)}{2J_{\nu}(1)-J_{\nu+1}(1)}.
J^{\mu~j}=-N^{\mu \nu}_{jb}\bigl( \varepsilon_{\nu\lambda}\partial^\lambda \Phi^b-\frac i2 \varepsilon^{bcd}\overline{\chi}^c \gamma_\nu \chi^d \bigr)={\cal J}^{\mu j}+ N^{\mu\nu}_{jk}K^k_{\nu},
\left|\bigcup_{{F\in C_2}}F\cap M\right| \le\left(c_2-q^{d-1}\right)(\Delta+zq^{d^2-d}\theta_{d-1})+q^{d-1}(\Delta+z'q^{d^2-d}\theta_{d-1}) &=c_2(\Delta+zq^{d^2-d}\theta_{d-1})+\underbrace{(z'-z)}_{\le\theta_d}q^{d^2-1}\theta_{d-1}.
\tilde{A}_{D'}= (A\otimes_{k}A)\left[\left(\frac{1\otimes t_{1}}{t_{1}\otimes 1}\right)^{\pm 1}, \left(\frac{1\otimes t_{2}}{t_{2}\otimes 1}\right)^{\pm 1}, \ldots,\left(\frac{1\otimes t_{r'}}{t_{r'}\otimes 1}\right)^{\pm 1}\right].
A_{12}=A_{13}=\cdots=A_{1m}=0, ~~C^1_{2,2}=C^1_{2,3}=\cdots=C^1_{2,m}=0, &\omega_{12}=(\frac{C^1_1}{\mu}-\frac{C^1_{2,1}}{C^1_1})\omega_1,~~~\omega_{1i}=0,~~~\omega_{2i}=-\frac{C^1_1}{\mu}\omega_i, ~~i\geq3.
\ll \frac{T_k (\log\log{R})^2}{\log{R}}\frac{r}{\phi_L(r)}\sum_{(e_m,a_mWB r)=1}\frac{Y_{\mathbf{r}'}}{\phi_\omega(e_m)}&\ll \frac{T_k(\log\log{R})^2\phi(a_m W B)W^{k-1}B^{k-1}\mathfrak{S}_{WB}(\mathcal{L})}{a_m\phi(WB)^k}\int_0^\infty F_2 (t_1,\dots,t_k)dt_m.
{}_{12}\langle V_2| ={}_{12}\langle V_2^r|(c_0^{(1)}+c_0^{(2)}),\ \ |V_3\rangle_{123}=\exp\left(-\sum_{r,s=1}^3c^{\dagger(r)}X^r{}^s_0b^{(s)}_0\right)|V_3^r\rangle_{123}.
\cos(d\sqrt{k})-\cos(s\sqrt{k}) = 2\sin\bigg(\frac{(s+d)\sqrt{k}}{2}\bigg) \sin\bigg(\frac{(s-d)\sqrt{k}}{2}\bigg) , 1 - \cos(t\sqrt{k}) = 2\sin^2\bigg( \frac{t\sqrt{k}}{2}\bigg) , \sin(t\sqrt{k}) = 2\sin\bigg( \frac{t\sqrt{k}}{2}\bigg) \cos\bigg( \frac{t\sqrt{k}}{2}\bigg) .
d x^{\mu_{1}}\wedge d x^{\mu_{2}}\wedge \cdots \wedge d x^{\mu_{r}}= \sum_{\mathrm{{\scriptsize ~ p e r m u t a t i o n s}}\, P}\mathrm{s i g n}( P ) d x^{\mu_{P ( 1 )}}\otimes d x^{\mu_{P ( 2 )}}\otimes \cdots \otimes d x^{\mu_{P ( r )}},
\delta ( p + q ) = ( 2 \pi )^{\, 4}\, \tilde{C}_{\alpha}^{\, \alpha_{0}}( q ) \, d e l t a_{\Phi_{c}( p )}\delta_{\Phi_{c}( q )}\, \Gamma_{0}^{\, \alpha , \, \alpha_{0}}( \Phi_{c}) \qquad \mathrm{f o r ~}\r = 0 \quad ,
(I - R_{0}R_{0}^{*})f =f(z) - \frac{1}{z}[ z f(z) - S(z) \tau^{*}(f) + S(0) \tau^{*}(f) ] = \frac{S(z) - S(0)}{z}\tau^{*}(f) = \tau \tau^{*}(f).
A_{3 , \zeta}= Q \rho \tau_{, \zeta}\ \ , A_{3 , \bar{\zeta}}= - Q \rho \tau_{, \bar{\zeta}}\ \ , \kappa^{2}= I^{3}={\frac{h^{3}e^{\tau_{0}\tau}}{( 1 -{\frac{2 m}{r}}) g_{2 2}^{\beta}}}
((x+z)^n+c,0)(z,x+z)^l=((x+z)^n+c,0)(z(x+z)^{l-1},(x+z)^l) = (z(x+z)^{n+l-1},(x+z)^{n+l})+ c(z(x+z)^{l-1},(x+z)^l)) \in \textrm{Diag}(A)
\left| f_1\left( \frac{k}{n}\right) - \gamma_{n,k}\right|\le\left( \frac{k}{n}\right)^{s - 1}c_{\operatorname{max}}\ \sum_{l=s}^{m}\frac{m^2}{n}\le\left( \frac{k}{n}\right)^{s - 1}c_{\operatorname{max}}\ \frac{m^3}{n},
\begin{aligned}p(N,\vec\ell;2)= \frac{(-1)^{N+\ell}N\prod_j\ell_j!}{(N)_{\ell}}&\times [z^{N-\ell}] (1-z)^{-t+1}\int_0^1\frac{(1-u)^{N+1}u^{\ell-t}}{(1-u+zu)^{\ell+1}}\,du ,\end{aligned}
S_2(\phi,\phi)=-\int d^D X \sqrt{-G}\left\{(D_M\phi)^{\dagger}D^M \phi+\frac{1}{2}\phi \frac{\partial^2 U}{\partial \Phi^2}\phi+\frac{1}{2}\phi^* \frac{\partial^2 U}{\partial{\Phi^*}^2}\phi^*+\phi^*\frac{\partial^2 U}{\partial \Phi\partial \Phi^*}\phi\right\}~,
\begin{array}{l l}{{n \left( \frac{1 - k}{1 + k}\right) = 2 G \left( U - T - J^{2}- \nu S^{2}\right)}}\\{{B^{2}= 1 - 8 G \left( U + \frac{( J^{2}+ \nu S^{2})}{2}\right)}}\\{{m^{2}= 4 G ( J^{2}+ \nu S^{2}) .}}\\ \end{array}
\sum_{n=1}^{\infty}\frac{(-1)^{n-1}z^n q^{\frac{n(n+1)}{2}}}{(1-c q^n) (z q)_n}= \frac{z}{c}\sum_{n=1}^{\infty}\frac{(zq/c)_{n-1}}{(zq)_n}(cq)^n.
{\large (}1-G(h_{+}){\large )}^{2}= \dfrac{4{\large (}2wst-(t^{2}-s^{2})v{\large )}^{2}}{{\large (}\sqrt{s^{2}+t^{2}}\sqrt{p_{1}}+\left\vert 2wst-(t^{2}-s^{2})v\right\vert{\large )}^{2}}
\phi( \exp_U (t X_0)) \, \phi( \exp_U (s Y)) \, \phi( \exp_U (t X_0))^{-1}= \exp_{\widetilde{U}}({\rm Ad}(\exp_{\widetilde{U}}( \sum_{i=1}^p \lambda_i (t) \widetilde{X}_i)) \sum_{j=1}^q \mu_j (s) \widetilde{Y}_j).
A_L \le \C(1+\varepsilon)^L 2^L+C(1+\frac{1}{\varepsilon})\sum_{\ell=1}^L(1+\varepsilon)^{\ell-1}2^{\ell-1}\psi^2_d(2^{L-\ell}) \le \C \, L^2 \, 2^L \varepsilon=\frac{1}{L}.
\gamma\Delta \vec{k}_l=\gamma(\vec{k}_{l+1}-\vec{k}_l)=\frac{\Lambda_l}{4M^4}\vec{n}_l, \,\,\, (l=1,2,\dots, L), \,\,\,{\rm with}\,\,\,\vec{k}_{L+1}=\vec{k}_1,
\eta_1(x_1,t) := \frac{h_{1,1}\, \partial_{x_2}v^{(1,1)}_0(x_1,-l_1,t)}{h_{1,1}\, \partial_{x_2}v^{(1,1)}_0(x_1,-l_1,t) + h_{1,2}\,\partial_{x_2}v^{(1,2)}_0(x_1,-l_1,t)}
\zeta_{2}= \frac{2^{2 s - 2}\mu^{- 1}\Gamma ( s + \frac{1}{2}) \Gamma ( - s )}{\pi \Gamma ( s ) \Gamma ( \frac 1 2 - s )}\int d k \ \tilde{S}( k ) \tilde{S}( - k ) \left( \frac{k^{2}}{\mu^{2}}\right)^{- s - \frac{1}{2}}\, .
\Vert f\Vert_{B^s_{\infty,\infty}(G)}=\sup_{\ell\geq 0}2^{ls}\Vert \psi_{l}(\mathcal{R}^{\frac{1}{2}})\Vert_{L^\infty(G)}\asymp \sup_{\ell\geq 0}2^{ls/2}\Vert \psi_{l}(\mathcal{R})\Vert_{L^\infty(G)}\lesssim \Vert f\Vert_{\Lambda^s(G)}.
\lim_{n\to \infty}\frac{1}{n^{2m}}\sum_{k=0}^{m}\sum_{I_{2k+1}}X_{j_1}\ldots X_{j_{2m+1}}X_{i_1}\ldots X_{i_{2m+1}}&=\sum_{k=1}^{m}\frac{a_k}{2^{2(m-k)}}\sum_{\ell=0}^{2k+1}\binom{2k+1}{\ell}^2 \ell!(2k+1-\ell)! \h_{2k+1}(\ell).
\sum\limits_{i=1}^N w_i e^{-\frac{T \sigma^2 R^{2 \alpha_d}}{c_0 \overline{g_p}t_i}}e^{-\rho_I \frac{\frac{T R^{2 \alpha_d}\gamma_0}{c_0 t_i \overline{g_p}}ArcCot[\frac{\delta_0}{\sqrt{L^2+\frac{T R^{2 \alpha_d}\gamma_0}{c_0 t_i \overline{g_p}}}}]}{\sqrt{L^2+\frac{T R^{2 \alpha_d}\gamma_0}{c_0 t_i \overline{g_p}}}}}
\| \prod_{i=1}^k \mathcal E_\lambda^\gamma f_i \|_{L^{\frac qk}(d\mu)}\le \| \prod_{i=1}^k \mathcal E_\lambda^\gamma f_i \|_{L^{\frac qk}(\mathbb R^d)}\| | \phi_\lambda | \ast \mu\|^{\frac k q}_{\infty}\le C \langle \mu\rangle_\alpha^{\,\,\frac kq}\lambda^{\frac{k(d-\alpha)}q}\| \prod_{i=1}^k \mathcal E_\lambda^\gamma f_i \|_{L^{\frac q k}(\mathbb R^d)}.
\prod_{i < j}F_{m i n}( \theta_{i j}) \longrightarrow [ F_{m i n}( i \pi ) ]^{3}\left( \frac{\sinh^{2}\theta_{1 2}}{\sinh^{2}\theta_{1 2}+ \sin^{2}\pi \alpha}\right) \left( \frac{\sinh^{2}\theta_{1 3}}{\sinh^{2}\theta_{1 3}+ \sin^{2}\pi \alpha}\right) \left( \frac{\sinh^{2}\theta_{2 3}}{\sinh^{2}\theta_{2 3}+ \sin^{2}\pi \alpha}\right) \, \, \, .
{\tau}_{\mathrm{R i n d l e r}}^{p \to n}= \frac{\pi^{2}a e^{\pi \frac{\Delta m}{a}}}{G_{F}^{2}m_{e}}\left[ \int_{- \infty}^{+ \infty}d{\omega_{R}}\frac{K_{i \frac{{\omega_{R}}}{a}+ \frac{1}{2}}( m_{e}/ a ) K_{i \frac{{\omega_{R}}}{a}- \frac{1}{2}}( m_{e}/ a )}{\cosh [ \pi ({\omega_{R}}-{\Delta m}) / a ]}\right]^{- 1}\, .
\partial_t\partial_zT+v\cdot\nabla_H\partial_zT-\left(\int_{-h}^z\nabla_H\cdot vd\xi\right)\partial_z^2T+u\cdot\nabla_HT \quad-(\nabla_H\cdot v)\left(\partial_zT+\frac{1}{h}\right)-\Delta_H\partial_zT-\varepsilon\partial_z^3T=0.
U_{_W}(t_i, t_{i-1}) \cong \frac{1}{2\pi}\int d\triangle_id\xi_i \exp\{-i(\xi_ix_i+\triangle_ip_i)\}\textrm{Tr}[\{1-i\epsilon \hat{H}(\hat{x},\hat{p})\}\exp i(\xi_i\hat{x}+\triangle_i\hat{p})].
\sum_{t=1}^{s-1}q_t\leq 1.01\frac{q_R}{B^R}\sum_{t=1}^{s-1}B^t\overset{\eqref{eq:b*}}{\leq}\frac{b^*}{1.01}\frac{q_R}{B^{R}}B^s\leq \frac{b^*}{1.01}q_s\,.
\widetilde G_n(x,y) = \left(\frac{1}{2\pi}\right)^{d/2}\int_{\mathbb{R}^d}\mbox{det}\left( e^{-i\theta_i(y_{j}-x_{i})}\right)_{i, j = 1}^d \prod_{i=1}^d (\varphi(\theta_i))^n d\theta_i.
\begin{aligned}\psi_k(t-t\wedge\tau_{\partial^TP}) =1\quad\hbox{if}\tau_{\partial^TP}< t-1/k, \psi_k(t-t\wedge\tau_{\partial^TP}) =0\quad\hbox{if}t-2/k <\tau_{\partial^TP},\end{aligned}
Z(\beta\,;R(S_{\nabla}^{{\bf C}}),\langle\cdot\,,\cdot\rangle_H,\langle\cdot\,,\cdot\rangle)=\pi^{\zeta}\,e^{-\frac{i\pi}{2}((\frac{2\theta}{\pi}\mp1)\zeta\,\pm\,\eta)}\,|\beta|^{-\zeta}\tau(M,\chi,\langle\cdot\,,\cdot\rangle_H)^{-1}
\int_{[0,1]^2}|\partial_{y}\phi_2(x,y)-1|^p dx &= \int_{\{(x,y)\in [0,1]^2: |f_i(x)-y|>\delta \forall i\}}|\partial_{y}\phi_2(x,y)-1|^p \,dx &~~~+\int_{\{(x,y)\in [0,1]^2: |f_i(x)-y|\leq \delta i\}}|\partial_{y}\phi_2(x,y)-1|^p \,dx &\leq (C\tau)^{p}\sqrt{|K|}.
H_{\mathrm{rad}}= \sum_\mathbf{m}\omega_\mathbf{m}\sum_{i,l}\Pi(\mathbf{m})_{il}b^+_{\mathbf{m},i}b_{\mathbf{m},l}H_{\mathrm{int}}= \sum_{\mathbf{m},i,l}( (ea/3)\sqrt{2\pi\omega_0}L^{-3/2})c(\omega_\mathbf{m}-\omega_0)\Pi(\mathbf{m})_{i,l}(E_{1i,0}b_{\mathbf{m},l}+E_{0,1i}b^+_{\mathbf{m},l})
e = [l^{-1},u][l^{-1},z]^{-1}= [l^{-1},u][z,l^{-1}] = l^{-1}z^{-1}uz l u^{-1}z l^{-1}z^{-1}l = l^{-1}z^{-1}[uz, l] z^{-1}l = l^{-1}[uz, l] l
x_{i}(k+s)= ~\sum_{j\in \mathcal{N}_{i}(k+s-1)}a_{ij}(k+s-1) x_j(k+s-1) \leq ~a_{ii}(k+s-1)x_{i}(k+s-1) &+\!\!\!\!\!\!\!\!\sum_{j\in \mathcal{N}_{i}(k+s-1)\setminus\{i\}}\!\!\!\!\!\!|a_{ij}(k+s-1)||x_j(k+s-1)| \leq ~a_{ii}(k+s-1)x_{i}(k+s-1) &+(1-a_{ii}(k+s-1))\|x(k)\|_{\infty}.
D_i(G_k)/D_{i+1}(G_k) = G_k^{\, p^l}\gamma_i(G_k)/G_k^{\, p^l}\gamma_{i+1}(G_k) \cong \gamma_i(G_k) / (G_k^{\, p^l}\cap \gamma_i(G_k)) \gamma_{i+1}(G_k) \cong \gamma_i(G_l) / \gamma_{i+1}(G_l).
\mathcal{A}\Omega_{n}=a^{p}a^{\dag q}\Omega_{n}=a^{p}\sqrt{\prod_{i=1}^{q}\left( n+i\right)}\cdot\Omega_{n+q}& =\sqrt{\prod_{i=1}^{q}\left( n+i\right)}\cdot a^{p}\Omega_{n+q}=\sqrt{\prod_{i=1}^{q}\left( n+i\right)}\sqrt{\prod_{j=0}^{p-1}\left(n+q-j\right)}\Omega_{n+\ell}
\begin{aligned}\partial_th_{ij}~=~ \alpha H^{-\alpha-1}\Delta h_{ij}-\alpha(\alpha+1)H^{-\alpha-2}\nabla_iH\nabla_jH+\alpha H^{-\alpha-1}|A|^2h_{ij}+(1-\alpha)H^{-\alpha}h_i^kh_{kj}+(1-\alpha)H^{-\alpha}g_{ij}+2\alpha H^{-\alpha-1}h_{ij}, \end{aligned}
\left[ \pi_{l}^{\left( N\right)}\left( \bar{x}\right) ,\,\pi_{k}^{\left(N\right)}(\bar{y})\right]_{x^{0}=y^{0}}\,=\,-i\,\theta \,\left( -1\right)^{N-1}\left( b+a\nabla^{2}\right) \epsilon_{ik}\partial_{l}\partial^{i}\delta^{2}\left( \bar{x}-\bar{y}\right) ,
\omega_\mu(\emph p_S(x))(T_x\emph p_S(u), T_x\emph p_S(v)) = \omega_\mu(\emph p_{J^{-1}\mu}(x))(T_x\emph p_{J^{-1}\mu}(u), T_x\emph p_{J^{-1}\mu}(v)) &= ((\emph p_{J^{-1}\mu})^*\omega_\mu)(x)(u,v).
f*P^l(x_1,x_2) -f(x_1,x_2) &=\int_{G^2}{P^l(-y_1,-y_2)f(x_1+y_1,x_2+y_2){d^2}_{{\mu}_{G^2}}\left(y_1,y_2\right)}&-f(x_1,x_2)\int_{G^2}{P^l(y_1,y_2){d^2}_{{\mu}_{G^2}}\left(y_1,y_2\right)}&=\int_{G^2}{(L_(y_1,y_2)f(x_1,x_2)-f(x_1,x_2))P^l(y_1,y_2){d^2}_{{\mu}_{G^2}}\left(y_1,y_2\right)}.
\phi_{T+1}^{(i)}(\hat{k}^{(i)}) = \begin{cases}\frac{1}{\mu}\tilde{R}_T(k^{(i)},\hat{k}^{(i)}), \hat{k}^{(i)}\neq k^{(i)}, 1- \sum_{l\neq k^{(i)}}\frac{1}{\mu}\tilde{R}_T(k^{(i)},l), \hat{k}^{(i)}= k^{(i)}.\end{cases}
\int_{\mathbb{R}^N}\int_{\mathbb{R}^N}\frac{|u(x)-u(y)|^{p-2}\,(u(x)-u(y))}{|x-y|^{N+s\,p}}\,(\varphi(x)-\varphi(y))\,dx\,dy=\int_{\Omega}F\,\varphi\,dx,
\left\langle E, (\partial_{\Sigma_0, \epsilon}^{*}F)^{(1)}[\phi] \otimes (\partial_{\Sigma_0, \epsilon}^{*}G)^{(1)}[\phi] \right\rangle = \left\langle E_{\Sigma_0}, F^{(1)}[\partial_{\Sigma_0, \epsilon}\phi] \otimes G^{(1)}[\partial_{\Sigma_0, \epsilon}\phi] \right\rangle.