ToniDO/test_latexformula · Datasets at Hugging Face

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\begin{align}-a^2\frac{dg^2\left(\frac{1}{a^2}\right)}{da^2}=g^2\beta_{\rm QCD}\left(g^2\right).\end{align}
\begin{align}Lz:=\eta y\cdot\nabla z+\eta Nz+\Delta_p z.\end{align}
\begin{align}\widehat{f}(x)= \frac{1}{Nh}\sum_{n=1}^N K\left( \frac{x-X(n)}{h}\right), x\in \mathbb{R}.\end{align}
\begin{align}\epsilon_{c,N} \rightarrow e^{- \left( 4 N / e \lambda \right)^{1/2} - N / e}.\end{align}
\begin{align}\mathcal{L}_{\boldsymbol{\lambda}}(f\ast_{\boldsymbol{\lambda}} g)=\mathcal{L}_{\boldsymbol{\lambda}}(f)\mathcal{L}_{\boldsymbol{\lambda}}(g), \mathcal{B}_{\boldsymbol{\lambda}}(fg)=\mathcal{B}_{\boldsymbol{\lambda}}(f)\ast_{\boldsymbol{\lambda}}\mathcal{B}_{\boldsymbol{\lambda}}(g),\end{align}
\begin{align} \mathcal{L}_\beta = \Delta - \beta(\tau) z \cdot \nabla - \beta(\tau ) Id \end{align}
\begin{align}\begin{array}{l}\textbf{Km} = \textbf{BR}[4,\{ 2,1,3,2,2,1,3,2,2,1,3,2, -1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1\}]\\\textbf{ColouredJones}[\textbf{Km},1][q^4](q^4-q^{-4})/(q^2-q^{-2})\\-q^{-30}+q^{-6}+q^{-2}+q^2+q^6+q^{10}-q^{22}-q^{26}-q^{30}+q^{38}\end{array}\end{align}
\begin{align}\|I_7\|&=\left\|\left\langle (1-A)Bh\partial_y\tilde{h}\partial_y u, u\right\rangle_{H^{3, 0}}\right\|\\&\lesssim\left\\|(1-A)Bh)\right\\|_{L^\infty}\\|\partial_y\tilde{h}\partial_y u\\|_{H^{3, 0}}\\|u\\|_{H^{3, 0}}+\\|\partial_y\tilde{h}\partial_y u\\|_{L^\infty}\\|(1-A)Bh\\|_{H^{3, 0}}\\|u\\|_{H^{3, 0}}.\end{align}
\begin{align} \beta_2( V ) &= 1 \\ \beta_4( V ) &= 1 + d \\ \beta_3( V ) &= \beta_3( V_{t} ) - s + d,\end{align}
\begin{align}\tilde{\mathcal{X}} = S(\mathcal{U}) - \mathcal{K}_{K,\zeta^}.\end{align*}
\begin{align} \tilde{D}=\frac{1}{\sqrt{F}}D+\frac{3F^{'}}{4F^{\frac{3}{2}}w_{1}w_{2}w_{3}}\gamma^{0}, \end{align}
\begin{align}\overline{A}^{(N-1,M)}=\Bigl(A^{(N-1,M)}_{mn}\Bigr)_{N-1\ge m,n\ge M-1},\quad\underline{A}^{(N,M+1)}=\Bigl(A^{(N,M+1)}_{mn}\Bigr)_{N+1\ge m,n\ge M+1}.\end{align}
\begin{align}y+\mathbb{K}y=\psi, \end{align}
\begin{align}L \, \in \, \mathrm{SO}(m,n) \, \Longleftrightarrow \, L^T \, \eta L \, = \,\eta \end{align}
\begin{align} \begin{aligned} c'(h) = -\kappa\left(1+\kappa\right) h^3 + \left[3\kappa + \left(1+\kappa\right)^2\right] h^2 -4\left(1+\kappa\right) h + 4 \end{aligned} \end{align}
\begin{align}p_L=-p_R\equiv p \quad (\mbox{mod}\,\,\, n).\end{align}
\begin{align}\mho _{\mathrm{Br}}\mathbb{I}_{\mathrm{BrBialg}}^{s}\overline{p}^{s}=p\mathbb{I}_{\mathrm{BrLie}}^{s}\mho _{\mathrm{Br}}^{s}\overline{p}^{s}=p^{s} \end{align}
\begin{align}\begin{aligned}&R_{A_{i}}C_{i}=0,\ B_{i}L_{D_{i}}=0\ (i=\overline{1,4}).\end{aligned}\end{align}
\begin{align}f(x,t):=\|\mathcal{W}[y+d] - \mathcal{W}[y] - \omega\|(x,t)\ge 0.\end{align}
\begin{align}&\frac{\partial q}{\partial t}=-\frac{\sqrt{3}e^{-\omega}}{4}q \ \alpha_2\times\alpha_3,\\&\frac{\partial q}{\partial u}=\frac{e^{-\omega}}{8}(4e^{\omega}q \alpha_2-4 q \alpha_3+\omega_v q\ \alpha_2\times\alpha_3),\\&\frac{\partial q}{\partial v}=-\frac{e^{-\omega}}{8}(4 q \alpha_2-4e^{\omega}q \alpha_3+\omega_u q\ \alpha_2\times\alpha_3).\end{align}
\begin{align}\mathbf G^{[2]}(\kappa h;\alpha) = \int_{-\pi}^{\pi} \frac{\sin^2(\theta)\, \|\widehat{\alpha}(\theta)\|^2}{\sinh^2(2\kappa h) + \sin^2(\theta)}\,\frac{d\theta}{2\pi}.\end{align}
\begin{align}\begin{pmatrix}-x & 1 \\1 & 0\end{pmatrix}\end{align}
\begin{align}f(w_0,w_1,1,w_3)=\frac{1}{x_2^4}f(x_0,x_1,x_2,x_3).\end{align}
\begin{align}A=\gamma^{(2n+1)}(P_{\rm F}-D_aP_{\rm F}D_a). \end{align}
\begin{align} T(\Sigma_{i\#j}^\varepsilon) = T(\Sigma) \circ \Gamma_{i,j,\varepsilon} \ .\end{align}
\begin{align} \tilde c^{(L)}_{l,m} = \frac{L! (L+1)!} {(L-l-m)! (L+1-m)! (l+2m)!} \, d_{[l,m]} \;.\end{align}
\begin{align}f=X_n^{q-1}a_{q-1}(X_1,\dots,X_{n-1})+X_n^{q-2}a_{q-2}(X_1,\dots,X_{n-1})+\cdots+a_{0}(X_1,\dots,X_{n-1}).\end{align}
\begin{align}{D}_{j_1\dots j_n}^{(s)}=(0,\infty)^n.\end{align}
\begin{align}G=\begin{pmatrix}\mathbf{1}_{n-1} & 1\\ G_{n-1} & \mathbf{0}_{n-1}^T\end{pmatrix},\end{align}
\begin{align} \xi_{u(t)}(q):=\xi(q,u(t))=\sum_{i=1}^k u_i(t) X_i(q),\qquad\forall\, q\in M, t\in[0,1],\end{align}
\begin{align}\displaystyle{ \raisebox{0.9ex}{\scriptsize{$\diamond$}}} \mbox{\hspace{-0.33ex}} \rho_\tau (Q) = \frac{\epsilon (q_0)}{\pi} \, Im \, \frac{1}{\displaystyle{ \raisebox{1.1ex}{\scriptsize{$\diamond$}}} \mbox{\hspace{-0.33ex}} D_\tau (Q)} \, . \end{align}
\begin{align}M^{L,\varepsilon}_p := \left\|\{x \in \{p_0, \ldots, p_{\|p\|-1} \} \colon\, \xi(x) \le (1-\varepsilon) a_L\}\right\|\end{align}
\begin{align}f_{i:n}(x)=\frac{f(x)}{\operatorname{B}(i,n-i+1)}\sum_{l=0}^{n-i}\binom{n-i}{l}(-1)^lF(x)^{i+l-1}.\end{align}
\begin{align} \widehat{\mu^{}_1} (k) \, = \, \tfrac{1}{3} \bigl( 1 + 2 \cos (\pi k) \bigr) \, = \, \begin{cases} 1, & , \\ - \frac{1}{3}, & , \end{cases}\end{align}
\begin{align} \gamma_{jk} = \gamma_{jk} - i \sigma_j \Phi \Phi^* \sigma_k + i \sigma_k \Phi \Phi^* \sigma_j.\end{align*}
\begin{align}[L^{+}(u)^{-1}]_{ij}=(-1)^{j-i}(qdetL^{+}(u-(n-1)h))^{-1}L^{+}(u-(n-1)h)_{1\cdots\hat{i}\cdots n}^{1\cdots \hat{j}\cdots n},\end{align}
\begin{align}\mu(D^4(x_i),x_{j})) = \mu(x_{n-6+i},x_{j}) = 0\end{align}
\begin{align}f(x)=\frac{2\theta}{\operatorname{B}(a,b)\,x^3}\exp\left(-\frac{a\theta}{x^2}\right)\left[1-\exp\left(-\frac{\theta}{x^2}\right)\right]^{b-1}.\end{align}
\begin{align} A_k \ &= \ \sum_{j=0}^{k} \, j! \, F_{j+1} \begin{Bmatrix}k \\ j \end{Bmatrix}, \\ B_k \ &= \ \sum_{j=0}^{k} \, j! \, F_{j+2} \begin{Bmatrix}k \\ j \end{Bmatrix}. \end{align}
\begin{align}\beta(x,y)=\beta(x',y')\,.\end{align}
\begin{align}\left\Vert f\right\Vert _{C_{\gamma ;\psi }\left[ a,b\right] }=\left\Vert\left( \psi \left( t\right) -\psi \left( a\right) \right) ^{\gamma}f\left(t\right) \right\Vert _{C\left[ a,b\right] }=\underset{t\in \left[ a,b\right] }{\max }\left\vert \left( \psi \left( t\right) -\psi \left( a\right) \right)^{\gamma }f\left( t\right) \right\vert\end{align}
\begin{align}\limsup_{n \to \infty} T_{W_{nJ}, \psi}(\mu_n(Q^n)) &\le \limsup_{n \to \infty} T_{W_{nJ},\psi_m}(\mu_n(Q^n)) \le \limsup_{n \to \infty} T_{W,\psi_m}(\mu_n(Q^n)) = T_{W,\psi_m}(\mu_\infty), \end{align}
\begin{align}\varphi'(\delta) = \frac{1}{\sqrt{\delta}} - \frac{1}{(1+\delta) \sqrt{(1+\delta)^2-1}} = \frac{1}{\sqrt{\delta}} \left (1- \frac{1}{(1+\delta)\sqrt{2+\delta}} \right ),\end{align}
\begin{align}I(x) = \{i\in [2n-1]: x_i,x_{i+1}\}.\end{align}
\begin{align}R_{s,j}&=\left(1+2^{\mathfrak{c}\|l\|}\lambda^{\gamma+\epsilon_3-\epsilon_4}\right)\sum_{m_j\in\mathcal{M}_j}\sum_{k_j\in\mathbb{Z}^2}\left\|a_{j,m_j,k_j,s}\right\|^2\\&\lesssim2^{\mathfrak{c}\|l\|}\lambda^{\gamma+\epsilon_3-\epsilon_4}\sum_{m_j\in\mathcal{M}_j}F_{j,m_j}(s).\end{align}
\begin{align}B_{i,r}-B_{i,l} = \frac{[v]}{\beta^2\sigma_3^{-2}-v_r} B_{i,l}.\end{align}
\begin{align} \forall \varepsilon > 0, \lim_{n \to \infty} \sum_{m=1}^n \mathbb{E}_{n,m}' [ \\|\chi_{n,m}' \\|_{L^2(\mu)}^2 : \\|\chi_{n,m}' \\|_{L^2( \mu)} > \varepsilon ] = 0 \end{align}
\begin{align}\lefteqn{ \int_{\{h>0\}}\,\int_0^{h(x)} f(x,z)\,dz\,dx}\\ &\overset{\eqref{ms}}=\int_{\{h>0\}}\,\int_0^{h(x)} \big(f(x,z)-f(x,h(x))\big)\,dz\,dx+\int_{\{h>0\}} h\,\kappa\,dx,\end{align}
\begin{align}{1\over g_1^2(q^2)}-{1\over g_2^2(q^2)} \simeq{1\over 24\pi^2}\ln\left({\sqrt{q^2}\over m_1}\right).\end{align}
\begin{align}j^{bc}_a = 0\end{align}
\begin{align}v_{a,q}(p)=\bigl(-\textrm{log} \rho(p)\bigl)^a \rho(p)^{\alpha}\end{align}
\begin{align} \left\|B_{n+1,\mu,0}(1)\Lambda_p\right\|_p \leq \left\|\sum_{j=1}^k \lambda_jS_{n+1,\mu,j}(1)\right\|_p \end{align}
\begin{align}(s_0s_1)^4=(s_{d-1}s_d)^4=1\quad\quad\quad\mbox{ and }\quad\quad\quad(s_{i-1}s_i)^3=1\quad\mbox{ for }2\le i\le d-1\end{align}
\begin{align} \sqrt{T}\left(f^{\lambda}_{\hat{\b{z}}} - f^{\lambda}_{\b{z}}\right) &=: f_{-1}+f_0,& f_{-1} &= \sqrt{T} (T_{\hat{\b{x}}}+\lambda)^{-1}(g_{\hat{\b{z}}} - g_{\b{z}}),& f_{0} &= \sqrt{T} (T_{\hat{\b{x}}}+\lambda)^{-1}(T_{\b{x}}-T_{\hat{\b{x}}}) f^{\lambda}_{\b{z}}. \end{align}
\begin{align}\pi = L(r) - R(r) \mbox{and} \pi' = L(r) + R(r)\end{align}
\begin{align} x(t)=1-t+o(t)\quad z(t)=\frac{2}{3\pi^2}t+o(t)\quad\end{align}
\begin{align}\zeta=-\Delta\psi+V\psi\in\mathrm{L}^2(\Omega).\end{align}
\begin{align}V(M_1 ,M_2 ) = \frac{1}{N} \mbox{Tr} \, M_1^2 + \frac{1}{N} \mbox{Tr} \,M_2^2 +\frac{1}{N^2 } \mbox{Tr} \, M_1^2 \, \mbox{Tr} \, M_2^2 \; .\end{align}
\begin{align}L = \displaystyle\frac{1}{2} (u^2+v^2) \left( \dot{u} \, \dot{u} +\dot{v} \, \dot{v} \right)-\displaystyle\frac{1}{u^2+v^2} \left( f(u)+g(v) \right)\end{align}
\begin{align}P_{\alpha }^{S}=Q^{S}\circ P_{\alpha }^{N}=Q^{S}\circ P_{\beta }^{N}\circ\Phi _{\beta ,\alpha }^{1}=P_{\beta }^{S}\circ \Phi _{\beta ,\alpha }^{1}.\end{align}
\begin{align}V^{(-1,-1)}_\zeta = \int\!{\rm d}^2z\,\Big(\zeta_{\mu \nu}\, \Psi^\mu \tilde\Psi^\nu \,e^{-\phi-\tilde\phi}e^{ik\cdot X}\Big)\,.\end{align}
\begin{align}u_1 = {1 \over v_1}, \quad u_2 = {v_2 \over v_1}, \quad v_1 = {w_1 \over w_2}, \quad v_2 = {1 \over w_2}, \quad w_1 = {1 \over u_2}, \quad w_2 = {u_1 \over u_2} ,\end{align}
\begin{align}K_M := \bigg\{ f\in L^p([0,T],H) : \int_0^T \Vert f(t) \Vert_V^p dt \leq M \bigg\}.\end{align}
\begin{align}\Sigma_{j=1}^n n(\mu,2j) = \Sigma_{a=1}^r a \# C_a.\end{align}
\begin{align} b_n = \biggl[ \frac{ \varphi_K (b-a) s^2(x) } { 4 \psi_K^2\rho^2_\mu(x) f_e^2 (0)}\biggr]^{1/5} N^{-1/5}_n.\end{align}
\begin{align} {{\rm min}} \left\{c +a_{2}+x_{2}^{2}, c +b_{2}-\sqrt{-c} x_{2}+x_{2}^{2}\right\}=0 \end{align}
\begin{align}-\phi'' - \|\phi\|^{2\mu} \phi +\omega \phi =0 \omega >0\end{align}
\begin{align}\\|\bar{b}-g\\|_\infty=\\|b_\infty\bar{b}-b_\infty g\\|_\infty=\\|\bar{b}_{n+1}-b_\infty g\\|_\infty\geq1,g\in H^\infty_n,\end{align}
\begin{align}M\int_{T_3}^t \tilde{I}_3(s)\,ds &\leq K_1 \int_{T_3}^t \frac{x'(s)}{f^2(x(s))} \cdot \frac{Mf(x(s))}{x'(s)}\,ds \\&\leq 2K_1 \int_{T_3}^t \frac{x'(s)}{f^2(x(s))}\,ds=2K_1 \int_{x(T_3)}^{x(t)} \frac{1}{f^2(u)}\,du.\end{align}
\begin{align}x_1^\prime = \varepsilon^2 y_1 \, , y_1^\prime = - \varepsilon^4 x_2 \, , x_2^\prime = \varepsilon^3 y_2 \, , y_2^\prime = \varepsilon x_1 \, , \end{align}
\begin{align}r_k(T)=\bigcup_{m<k}T(m),\end{align}
\begin{align} X : \det(\lambda I - A(z)) = 0 \end{align}
\begin{align}W_N (Z_N) = W_N^{\left(\tilde{i}_N (Z_N),\tilde{j}_N (Z_N)\right)}(Z_N)\end{align}
\begin{align}\int_{\theta}^t f_1(s_1) \int_{\Delta_{\theta,s_1}^p} g_1(r_1) \dots g_p(r_p) & dr_p \dots dr_1 ds_1 \\ & = \int_{\Delta_{\theta,t}^{p+1}} f_1(w_1) g_1(w_2) \dots g_p(w_{p+1}) dw_{p+1} \dots dw_1,\end{align}
\begin{align} r_1 = \frac{1}{4} \cdot \Bigl( \, 1+ \sum_{n=1}^{\infty}C_n^2 ( 1/4 )^{2n} \, \Bigr) .\end{align}
\begin{align}M^k = \left( \begin{array}{cccc} {\rm e}^{km} & k {\rm e}^{(k-1)m} M_{12} & \left( \frac{1-{\rm e}^{km}}{1-{\rm e}^{m}}\right) M_{13} & \left(M^{k}\right)_{14} \\ 0 & {\rm e}^{km} & 0 & \left( \frac{1-{\rm e}^{km}}{1-{\rm e}^{m}}\right)M_{24} \\ 0 & 0 & 1& 0 \\ 0 & 0 & 0 & 1\end{array} \right)\, ,\end{align}
\begin{align}S = \int dx^{+}dx^{-} \, {\rm tr} \Bigg[ \partial_{+}\phi\partial_{-}\phi+i\psi\partial_{+}\psi +\frac{g^2}{2}J^{+}\frac{1}{\partial_{-}^{2}}J^{+}-\frac{1}{2}ig^2 [\phi,\psi]\frac{1}{\partial_{-}} [\phi,\psi] \Bigg].\end{align}
\begin{align} {d+1 \choose \|I\|} \cdot \\|A\\| = \sum_{\sigma \in A} \frac{\|\{F \in X(d)\,\|\,\sigma \in F\}\|}{\|X(d)\|}= \frac{\|A\| \cdot k_I^{[d]}}{\|X \cap \prod_{i \in I} V_i\|\cdot k_I^{[d]} } = \frac{\|A\| }{\|X \cap \prod_{i \in I} V_i\|}\end{align}
\begin{align}\nu_N = \frac{1}{N} \sum_{k=1}^N \delta_{v_k}.\end{align}
\begin{align}(1) F_L = d\lambda\, V, (2) d_L V = 0\end{align}
\begin{align}D_{e_i}e_j=0,h_{ij}=\delta_{ij}\kappa_i,\kappa_1\geq \cdots\geq \kappa_n.\end{align}
\begin{align}T_{N,d,p}^{-1}u=vu=T_{N,d,p}v.\end{align}
\begin{align}{[k+1+\ell]\choose [q]}_+ \to \bigvee\limits_{r=0}^q \left({[k]\choose[r]}\times{[\ell]\choose[q-r]}\right)_+\end{align}
\begin{align} E'_{\ell}(x_1,\ldots,x_{2\ell-2},y,q^2y)=0 \,.\end{align}
\begin{align} \lim_{h_n\to \infty} \rho^{(2^{-h_n}T)}_t (x):= u_t(x),t\in \mathcal T, x\le \max\{x^{},R_0+c^T\}=:c^{*}\end{align}
\begin{align} \psi(s) = \sum_{j=0}^{\infty} b_j s^j = \sum_{j=0}^{\infty} \left( \prod_{k=1}^r \frac{(j+\nu_k+ \mu_k)!}{(j+\nu_k)!} \right) s^j,\end{align}
\begin{align}d{\bar s}^2=\bar g_{MN}dz^Mdz^N=g_{ab}(y)dy^a dy^b + r^2(y)d\sigma_n^2;\end{align}
\begin{align}\ddot{g}_{pm}(t)+3H \dot{g}_{pm}(t)+ \Bigl(p^2 e^{-2Ht} + m^2 \Bigr)g_{pm}(t)=0\ ,\end{align}
\begin{align} b_n(\mathrm{d}x) = \frac{v_{nF}(\mathrm{d}x)}{\sqrt{f(x)}} +v_{nF}(A)\frac{1}{\sqrt{F(A)} - F(A)} \sqrt{f(x)} \,\mathrm{d}x ,x\notin A,\end{align}
\begin{align}R(a-1)&=\Big\{(x_1,x_2,...,x_{n+1}) \mid (x_n,x_{n+1})=(0,0),(1,0),(0,1);\\&\ \ \ 0\leq x_i\leq 2\ \ \ \ 1\leq i\leq n-1;\ \\ \ x_i=2,\ \ \ \ x_j=0\ \ \ \ j\leq i-1\Big\}\\&\ \ \ \bigcup \Big\{(0,0,...,0,x_{n},x_{n+1})\mid (x_n,x_{n+1})=(2,0),(1,1)\Big\}.\end{align}
\begin{align}d(\chi,G):=\inf\{\\|\chi-\psi\\|_{L^2(0,L)};\;\psi\in G\}.\end{align}
\begin{align}\\|x(t)\\|\le \sqrt{\frac{\beta_2}{\beta_1}}\\|x^0\\|e^{-\bar\lambda t} \;\; t=0,\varepsilon, 2\varepsilon, ... ,\end{align}
\begin{align} \alpha_{g g^{-1}} = u_g^ u_g, u_{g g^{-1}} = u_{g} u_{g}^,u_{g h} = u_g \overline{\alpha_g}(u_h)\end{align}
\begin{align} \eta_S=\phi^{-1}\alpha\phi'\qquad\eta_F=\psi^{-1}\beta\psi'\ . \end{align}
\begin{align}\mathbb{E}(t):=\Xi(t)+\beta\Big[\alpha\Xi_1^{(0)}(t)+\Xi_2^{(0)}(t)\Big]+\beta\Big[\Xi_3(t)+\beta\Xi_4(t)\Big]+\beta^3\Big[\alpha\Xi_1^{(1)}(t)+\Xi_2^{(1)}(t)\Big],\end{align}
\begin{align}CT^{-1}_j (\gamma_j + \epsilon_j \beta_j) \le CT^{-1}_j \rho\epsilon_j (1 + \rho^{-1} B_j) \le \rho \hat{c}^{-1} PT^{-1}_j \epsilon_j = \hat{c}^{-1} \rho \epsilon_{j+1},\end{align}
\begin{align}\|0,f\rangle = \int d\theta f(\theta)\int d\hat n \|g(\hat n,\theta)\rangle~,\end{align}
\begin{align}(p_{10}-h_1)\Psi=0, \qquad (p_{20}-h_2)\Psi=0, \end{align}
\begin{align}(\chi_1^{\mu\oplus 0})^{-1}_\nu=&(\nu_1 (\overline{\partial \chi_1^{\mu\oplus 0}})(d\bar z-\mu\frac{{\partial \chi_1^{\mu\oplus 0}}}{\overline{\partial \chi_1^{\mu\oplus 0}}}dz)\\ &+\nu_2 (\partial \chi_1^{\mu\oplus 0})(-\bar\mu\frac{\overline{\partial \chi_1^{\mu\oplus 0}}}{\partial \chi_1^{\mu\oplus 0}}d\bar z+dz) )\circ(\chi_1^{\mu\oplus 0})^{-1}.\end{align*}
\begin{align}b(x)=\frac{x}{\sqrt{x^2+1}}.\end{align}

\begin{align*}-a^2\frac{dg^2\left(\frac{1}{a^2}\right)}{da^2}=g^2\beta_{\rm QCD}\left(g^2\right).\end{align*}

\begin{align*}Lz:=\eta y\cdot\nabla z+\eta Nz+\Delta_p z.\end{align*}

\begin{align*}\widehat{f}(x)= \frac{1}{Nh}\sum_{n=1}^N K\left( \frac{x-X(n)}{h}\right), x\in \mathbb{R}.\end{align*}

\begin{align*}\epsilon_{c,N} \rightarrow e^{- \left( 4 N / e \lambda \right)^{1/2} - N / e}.\end{align*}

\begin{align*}\mathcal{L}_{\boldsymbol{\lambda}}(f\ast_{\boldsymbol{\lambda}} g)=\mathcal{L}_{\boldsymbol{\lambda}}(f)\mathcal{L}_{\boldsymbol{\lambda}}(g), \mathcal{B}_{\boldsymbol{\lambda}}(fg)=\mathcal{B}_{\boldsymbol{\lambda}}(f)\ast_{\boldsymbol{\lambda}}\mathcal{B}_{\boldsymbol{\lambda}}(g),\end{align*}

\begin{align*} \mathcal{L}_\beta = \Delta - \beta(\tau) z \cdot \nabla - \beta(\tau ) Id \end{align*}

\begin{align*}\begin{array}{l}\textbf{Km} = \textbf{BR}[4,\{ 2,1,3,2,2,1,3,2,2,1,3,2, -1,-1,-1,-1,-1,-1,-1,-1,-1,-1,-1\}]\\\textbf{ColouredJones}[\textbf{Km},1][q^4](q^4-q^{-4})/(q^2-q^{-2})\\-q^{-30}+q^{-6}+q^{-2}+q^2+q^6+q^{10}-q^{22}-q^{26}-q^{30}+q^{38}\end{array}\end{align*}

\begin{align*}|I_7|&=\left|\left\langle (1-A)Bh\partial_y\tilde{h}\partial_y u, u\right\rangle_{H^{3, 0}}\right|\\&\lesssim\left\|(1-A)Bh)\right\|_{L^\infty}\|\partial_y\tilde{h}\partial_y u\|_{H^{3, 0}}\|u\|_{H^{3, 0}}+\|\partial_y\tilde{h}\partial_y u\|_{L^\infty}\|(1-A)Bh\|_{H^{3, 0}}\|u\|_{H^{3, 0}}.\end{align*}

\begin{align*} \beta_2( V ) &= 1 \\ \beta_4( V ) &= 1 + d \\ \beta_3( V ) &= \beta_3( V_{t} ) - s + d,\end{align*}

\begin{align*}\tilde{\mathcal{X}} = S(\mathcal{U}) - \mathcal{K}_{K,\zeta^*}.\end{align*}

\begin{align*} \tilde{D}=\frac{1}{\sqrt{F}}D+\frac{3F^{'}}{4F^{\frac{3}{2}}w_{1}w_{2}w_{3}}\gamma^{0}, \end{align*}

\begin{align*}\overline{A}^{(N-1,M)}=\Bigl(A^{(N-1,M)}_{mn}\Bigr)_{N-1\ge m,n\ge M-1},\quad\underline{A}^{(N,M+1)}=\Bigl(A^{(N,M+1)}_{mn}\Bigr)_{N+1\ge m,n\ge M+1}.\end{align*}

\begin{align*}y+\mathbb{K}y=\psi, \end{align*}

\begin{align*}L \, \in \, \mathrm{SO}(m,n) \, \Longleftrightarrow \, L^T \, \eta L \, = \,\eta \end{align*}

\begin{align*} \begin{aligned} c'(h) = -\kappa\left(1+\kappa\right) h^3 + \left[3\kappa + \left(1+\kappa\right)^2\right] h^2 -4\left(1+\kappa\right) h + 4 \end{aligned} \end{align*}

\begin{align*}p_L=-p_R\equiv p \quad (\mbox{mod}\,\,\, n).\end{align*}

\begin{align*}\mho _{\mathrm{Br}}\mathbb{I}_{\mathrm{BrBialg}}^{s}\overline{p}^{s}=p\mathbb{I}_{\mathrm{BrLie}}^{s}\mho _{\mathrm{Br}}^{s}\overline{p}^{s}=p^{s} \end{align*}

\begin{align*}\begin{aligned}&R_{A_{i}}C_{i}=0,\ B_{i}L_{D_{i}}=0\ (i=\overline{1,4}).\end{aligned}\end{align*}

\begin{align*}f(x,t):=|\mathcal{W}[y+d] - \mathcal{W}[y] - \omega|(x,t)\ge 0.\end{align*}

\begin{align*}&\frac{\partial q}{\partial t}=-\frac{\sqrt{3}e^{-\omega}}{4}q \ \alpha_2\times\alpha_3,\\&\frac{\partial q}{\partial u}=\frac{e^{-\omega}}{8}(4e^{\omega}q \alpha_2-4 q \alpha_3+\omega_v q\ \alpha_2\times\alpha_3),\\&\frac{\partial q}{\partial v}=-\frac{e^{-\omega}}{8}(4 q \alpha_2-4e^{\omega}q \alpha_3+\omega_u q\ \alpha_2\times\alpha_3).\end{align*}

\begin{align*}\mathbf G^{[2]}(\kappa h;\alpha) = \int_{-\pi}^{\pi} \frac{\sin^2(\theta)\, |\widehat{\alpha}(\theta)|^2}{\sinh^2(2\kappa h) + \sin^2(\theta)}\,\frac{d\theta}{2\pi}.\end{align*}

\begin{align*}\begin{pmatrix}-x & 1 \\1 & 0\end{pmatrix}\end{align*}

\begin{align*}f(w_0,w_1,1,w_3)=\frac{1}{x_2^4}f(x_0,x_1,x_2,x_3).\end{align*}

\begin{align*}A=\gamma^{(2n+1)}(P_{\rm F}-D_aP_{\rm F}D_a). \end{align*}

\begin{align*} T(\Sigma_{i\#j}^\varepsilon) = T(\Sigma) \circ \Gamma_{i,j,\varepsilon} \ .\end{align*}

\begin{align*} \tilde c^{(L)}_{l,m} = \frac{L! (L+1)!} {(L-l-m)! (L+1-m)! (l+2m)!} \, d_{[l,m]} \;.\end{align*}

\begin{align*}f=X_n^{q-1}a_{q-1}(X_1,\dots,X_{n-1})+X_n^{q-2}a_{q-2}(X_1,\dots,X_{n-1})+\cdots+a_{0}(X_1,\dots,X_{n-1}).\end{align*}

\begin{align*}{D}_{j_1\dots j_n}^{(s)}=(0,\infty)^n.\end{align*}

\begin{align*}G=\begin{pmatrix}\mathbf{1}_{n-1} & 1\\ G_{n-1} & \mathbf{0}_{n-1}^T\end{pmatrix},\end{align*}

\begin{align*} \xi_{u(t)}(q):=\xi(q,u(t))=\sum_{i=1}^k u_i(t) X_i(q),\qquad\forall\, q\in M, t\in[0,1],\end{align*}

\begin{align*}\displaystyle{ \raisebox{0.9ex}{\scriptsize{$\diamond$}}} \mbox{\hspace{-0.33ex}} \rho_\tau (Q) = \frac{\epsilon (q_0)}{\pi} \, Im \, \frac{1}{\displaystyle{ \raisebox{1.1ex}{\scriptsize{$\diamond$}}} \mbox{\hspace{-0.33ex}} D_\tau (Q)} \, . \end{align*}

\begin{align*}M^{L,\varepsilon}_p := \left|\{x \in \{p_0, \ldots, p_{|p|-1} \} \colon\, \xi(x) \le (1-\varepsilon) a_L\}\right|\end{align*}

\begin{align*}f_{i:n}(x)=\frac{f(x)}{\operatorname{B}(i,n-i+1)}\sum_{l=0}^{n-i}\binom{n-i}{l}(-1)^lF(x)^{i+l-1}.\end{align*}

\begin{align*} \widehat{\mu^{}_1} (k) \, = \, \tfrac{1}{3} \bigl( 1 + 2 \cos (\pi k) \bigr) \, = \, \begin{cases} 1, & , \\ - \frac{1}{3}, & , \end{cases}\end{align*}

\begin{align*} \gamma_{jk} = \gamma_{*jk} - i \sigma_j \Phi \Phi^* \sigma_k + i \sigma_k \Phi \Phi^* \sigma_j.\end{align*}

\begin{align*}[L^{+}(u)^{-1}]_{ij}=(-1)^{j-i}(qdetL^{+}(u-(n-1)h))^{-1}L^{+}(u-(n-1)h)_{1\cdots\hat{i}\cdots n}^{1\cdots \hat{j}\cdots n},\end{align*}

\begin{align*}\mu(D^4(x_i),x_{j})) = \mu(x_{n-6+i},x_{j}) = 0\end{align*}

\begin{align*}f(x)=\frac{2\theta}{\operatorname{B}(a,b)\,x^3}\exp\left(-\frac{a\theta}{x^2}\right)\left[1-\exp\left(-\frac{\theta}{x^2}\right)\right]^{b-1}.\end{align*}

\begin{align*} A_k \ &= \ \sum_{j=0}^{k} \, j! \, F_{j+1} \begin{Bmatrix}k \\ j \end{Bmatrix}, \\ B_k \ &= \ \sum_{j=0}^{k} \, j! \, F_{j+2} \begin{Bmatrix}k \\ j \end{Bmatrix}. \end{align*}

\begin{align*}\beta(x,y)=\beta(x',y')\,.\end{align*}

\begin{align*}\left\Vert f\right\Vert _{C_{\gamma ;\psi }\left[ a,b\right] }=\left\Vert\left( \psi \left( t\right) -\psi \left( a\right) \right) ^{\gamma}f\left(t\right) \right\Vert _{C\left[ a,b\right] }=\underset{t\in \left[ a,b\right] }{\max }\left\vert \left( \psi \left( t\right) -\psi \left( a\right) \right)^{\gamma }f\left( t\right) \right\vert\end{align*}

\begin{align*}\limsup_{n \to \infty} T_{W_{nJ}, \psi}(\mu_n(Q^n)) &\le \limsup_{n \to \infty} T_{W_{nJ},\psi_m}(\mu_n(Q^n)) \le \limsup_{n \to \infty} T_{W,\psi_m}(\mu_n(Q^n)) = T_{W,\psi_m}(\mu_\infty), \end{align*}

\begin{align*}\varphi'(\delta) = \frac{1}{\sqrt{\delta}} - \frac{1}{(1+\delta) \sqrt{(1+\delta)^2-1}} = \frac{1}{\sqrt{\delta}} \left (1- \frac{1}{(1+\delta)\sqrt{2+\delta}} \right ),\end{align*}

\begin{align*}I(x) = \{i\in [2n-1]: x_i,x_{i+1}\}.\end{align*}

\begin{align*}R_{s,j}&=\left(1+2^{\mathfrak{c}|l|}\lambda^{\gamma+\epsilon_3-\epsilon_4}\right)\sum_{m_j\in\mathcal{M}_j}\sum_{k_j\in\mathbb{Z}^2}\left|a_{j,m_j,k_j,s}\right|^2\\&\lesssim2^{\mathfrak{c}|l|}\lambda^{\gamma+\epsilon_3-\epsilon_4}\sum_{m_j\in\mathcal{M}_j}F_{j,m_j}(s).\end{align*}

\begin{align*}B_{i,r}-B_{i,l} = \frac{[v]}{\beta^2\sigma_3^{-2}-v_r} B_{i,l}.\end{align*}

\begin{align*} \forall \varepsilon > 0, \lim_{n \to \infty} \sum_{m=1}^n \mathbb{E}_{n,m}' [ \|\chi_{n,m}' \|_{L^2(\mu)}^2 : \|\chi_{n,m}' \|_{L^2( \mu)} > \varepsilon ] = 0 \end{align*}

\begin{align*}\lefteqn{ \int_{\{h>0\}}\,\int_0^{h(x)} f(x,z)\,dz\,dx}\\ &\overset{\eqref{ms}}=\int_{\{h>0\}}\,\int_0^{h(x)} \big(f(x,z)-f(x,h(x))\big)\,dz\,dx+\int_{\{h>0\}} h\,\kappa\,dx,\end{align*}

\begin{align*}{1\over g_1^2(q^2)}-{1\over g_2^2(q^2)} \simeq{1\over 24\pi^2}\ln\left({\sqrt{q^2}\over m_1}\right).\end{align*}

\begin{align*}j^{bc}_a = 0\end{align*}

\begin{align*}v_{a,q}(p)=\bigl(-\textrm{log} \rho(p)\bigl)^a \rho(p)^{\alpha}\end{align*}

\begin{align*} \left|B_{n+1,\mu,0}(1)\Lambda_p\right|_p \leq \left|\sum_{j=1}^k \lambda_jS_{n+1,\mu,j}(1)\right|_p \end{align*}

\begin{align*}(s_0s_1)^4=(s_{d-1}s_d)^4=1\quad\quad\quad\mbox{ and }\quad\quad\quad(s_{i-1}s_i)^3=1\quad\mbox{ for }2\le i\le d-1\end{align*}

\begin{align*} \sqrt{T}\left(f^{\lambda}_{\hat{\b{z}}} - f^{\lambda}_{\b{z}}\right) &=: f_{-1}+f_0,& f_{-1} &= \sqrt{T} (T_{\hat{\b{x}}}+\lambda)^{-1}(g_{\hat{\b{z}}} - g_{\b{z}}),& f_{0} &= \sqrt{T} (T_{\hat{\b{x}}}+\lambda)^{-1}(T_{\b{x}}-T_{\hat{\b{x}}}) f^{\lambda}_{\b{z}}. \end{align*}

\begin{align*}\pi = L(r) - R(r) \mbox{and} \pi' = L(r) + R(r)\end{align*}

\begin{align*} x(t)=1-t+o(t)\quad z(t)=\frac{2}{3\pi^2}t+o(t)\quad\end{align*}

\begin{align*}\zeta=-\Delta\psi+V\psi\in\mathrm{L}^2(\Omega).\end{align*}

\begin{align*}V(M_1 ,M_2 ) = \frac{1}{N} \mbox{Tr} \, M_1^2 + \frac{1}{N} \mbox{Tr} \,M_2^2 +\frac{1}{N^2 } \mbox{Tr} \, M_1^2 \, \mbox{Tr} \, M_2^2 \; .\end{align*}

\begin{align*}L = \displaystyle\frac{1}{2} (u^2+v^2) \left( \dot{u} \, \dot{u} +\dot{v} \, \dot{v} \right)-\displaystyle\frac{1}{u^2+v^2} \left( f(u)+g(v) \right)\end{align*}

\begin{align*}P_{\alpha }^{S}=Q^{S}\circ P_{\alpha }^{N}=Q^{S}\circ P_{\beta }^{N}\circ\Phi _{\beta ,\alpha }^{1}=P_{\beta }^{S}\circ \Phi _{\beta ,\alpha }^{1}.\end{align*}

\begin{align*}V^{(-1,-1)}_\zeta = \int\!{\rm d}^2z\,\Big(\zeta_{\mu \nu}\, \Psi^\mu \tilde\Psi^\nu \,e^{-\phi-\tilde\phi}e^{ik\cdot X}\Big)\,.\end{align*}

\begin{align*}u_1 = {1 \over v_1}, \quad u_2 = {v_2 \over v_1}, \quad v_1 = {w_1 \over w_2}, \quad v_2 = {1 \over w_2}, \quad w_1 = {1 \over u_2}, \quad w_2 = {u_1 \over u_2} ,\end{align*}

\begin{align*}K_M := \bigg\{ f\in L^p([0,T],H) : \int_0^T \Vert f(t) \Vert_V^p dt \leq M \bigg\}.\end{align*}

\begin{align*}\Sigma_{j=1}^n n(\mu,2j) = \Sigma_{a=1}^r a \# C_a.\end{align*}

\begin{align*} b_n = \biggl[ \frac{ \varphi_K (b-a) s^2(x) } { 4 \psi_K^2\rho^2_\mu(x) f_e^2 (0)}\biggr]^{1/5} N^{-1/5}_n.\end{align*}

\begin{align*} {{\rm min}} \left\{c +a_{2}+x_{2}^{2}, c +b_{2}-\sqrt{-c} x_{2}+x_{2}^{2}\right\}=0 \end{align*}

\begin{align*}-\phi'' - |\phi|^{2\mu} \phi +\omega \phi =0 \omega >0\end{align*}

\begin{align*}\|\bar{b}-g\|_\infty=\|b_\infty\bar{b}-b_\infty g\|_\infty=\|\bar{b}_{n+1}-b_\infty g\|_\infty\geq1,g\in H^\infty_n,\end{align*}

\begin{align*}M\int_{T_3}^t \tilde{I}_3(s)\,ds &\leq K_1 \int_{T_3}^t \frac{x'(s)}{f^2(x(s))} \cdot \frac{Mf(x(s))}{x'(s)}\,ds \\&\leq 2K_1 \int_{T_3}^t \frac{x'(s)}{f^2(x(s))}\,ds=2K_1 \int_{x(T_3)}^{x(t)} \frac{1}{f^2(u)}\,du.\end{align*}

\begin{align*}x_1^\prime = \varepsilon^2 y_1 \, , y_1^\prime = - \varepsilon^4 x_2 \, , x_2^\prime = \varepsilon^3 y_2 \, , y_2^\prime = \varepsilon x_1 \, , \end{align*}

\begin{align*}r_k(T)=\bigcup_{m<k}T(m),\end{align*}

\begin{align*} X : \det(\lambda I - A(z)) = 0 \end{align*}

\begin{align*}W_N (Z_N) = W_N^{\left(\tilde{i}_N (Z_N),\tilde{j}_N (Z_N)\right)}(Z_N)\end{align*}

\begin{align*}\int_{\theta}^t f_1(s_1) \int_{\Delta_{\theta,s_1}^p} g_1(r_1) \dots g_p(r_p) & dr_p \dots dr_1 ds_1 \\ & = \int_{\Delta_{\theta,t}^{p+1}} f_1(w_1) g_1(w_2) \dots g_p(w_{p+1}) dw_{p+1} \dots dw_1,\end{align*}

\begin{align*} r_1 = \frac{1}{4} \cdot \Bigl( \, 1+ \sum_{n=1}^{\infty}C_n^2 ( 1/4 )^{2n} \, \Bigr) .\end{align*}

\begin{align*}M^k = \left( \begin{array}{cccc} {\rm e}^{km} & k {\rm e}^{(k-1)m} M_{12} & \left( \frac{1-{\rm e}^{km}}{1-{\rm e}^{m}}\right) M_{13} & \left(M^{k}\right)_{14} \\ 0 & {\rm e}^{km} & 0 & \left( \frac{1-{\rm e}^{km}}{1-{\rm e}^{m}}\right)M_{24} \\ 0 & 0 & 1& 0 \\ 0 & 0 & 0 & 1\end{array} \right)\, ,\end{align*}

\begin{align*}S = \int dx^{+}dx^{-} \, {\rm tr} \Bigg[ \partial_{+}\phi\partial_{-}\phi+i\psi\partial_{+}\psi +\frac{g^2}{2}J^{+}\frac{1}{\partial_{-}^{2}}J^{+}-\frac{1}{2}ig^2 [\phi,\psi]\frac{1}{\partial_{-}} [\phi,\psi] \Bigg].\end{align*}

\begin{align*} {d+1 \choose |I|} \cdot \|A\| = \sum_{\sigma \in A} \frac{|\{F \in X(d)\,|\,\sigma \in F\}|}{|X(d)|}= \frac{|A| \cdot k_I^{[d]}}{|X \cap \prod_{i \in I} V_i|\cdot k_I^{[d]} } = \frac{|A| }{|X \cap \prod_{i \in I} V_i|}\end{align*}

\begin{align*}\nu_N = \frac{1}{N} \sum_{k=1}^N \delta_{v_k}.\end{align*}

\begin{align*}(1) F_L = d\lambda\, V, (2) d_L V = 0\end{align*}

\begin{align*}D_{e_i}e_j=0,h_{ij}=\delta_{ij}\kappa_i,\kappa_1\geq \cdots\geq \kappa_n.\end{align*}

\begin{align*}T_{N,d,p}^{-1}u=vu=T_{N,d,p}v.\end{align*}

\begin{align*}{[k+1+\ell]\choose [q]}_+ \to \bigvee\limits_{r=0}^q \left({[k]\choose[r]}\times{[\ell]\choose[q-r]}\right)_+\end{align*}

\begin{align*} E'_{\ell}(x_1,\ldots,x_{2\ell-2},y,q^2y)=0 \,.\end{align*}

\begin{align*} \lim_{h_n\to \infty} \rho^{(2^{-h_n}T)}_t (x):= u_t(x),t\in \mathcal T, x\le \max\{x^{**},R_0+c^*T\}=:c^{**}\end{align*}

\begin{align*} \psi(s) = \sum_{j=0}^{\infty} b_j s^j = \sum_{j=0}^{\infty} \left( \prod_{k=1}^r \frac{(j+\nu_k+ \mu_k)!}{(j+\nu_k)!} \right) s^j,\end{align*}

\begin{align*}d{\bar s}^2=\bar g_{MN}dz^Mdz^N=g_{ab}(y)dy^a dy^b + r^2(y)d\sigma_n^2;\end{align*}

\begin{align*}\ddot{g}_{pm}(t)+3H \dot{g}_{pm}(t)+ \Bigl(p^2 e^{-2Ht} + m^2 \Bigr)g_{pm}(t)=0\ ,\end{align*}

\begin{align*} b_n(\mathrm{d}x) = \frac{v_{nF}(\mathrm{d}x)}{\sqrt{f(x)}} +v_{nF}(A)\frac{1}{\sqrt{F(A)} - F(A)} \sqrt{f(x)} \,\mathrm{d}x ,x\notin A,\end{align*}

\begin{align*}R(a-1)&=\Big\{(x_1,x_2,...,x_{n+1}) \mid (x_n,x_{n+1})=(0,0),(1,0),(0,1);\\&\ \ \ 0\leq x_i\leq 2\ \ \ \ 1\leq i\leq n-1;\ \\ \ x_i=2,\ \ \ \ x_j=0\ \ \ \ j\leq i-1\Big\}\\&\ \ \ \bigcup \Big\{(0,0,...,0,x_{n},x_{n+1})\mid (x_n,x_{n+1})=(2,0),(1,1)\Big\}.\end{align*}

\begin{align*}d(\chi,G):=\inf\{\|\chi-\psi\|_{L^2(0,L)};\;\psi\in G\}.\end{align*}

\begin{align*}\|x(t)\|\le \sqrt{\frac{\beta_2}{\beta_1}}\|x^0\|e^{-\bar\lambda t} \;\; t=0,\varepsilon, 2\varepsilon, ... ,\end{align*}

\begin{align*} \alpha_{g g^{-1}} = u_g^* u_g, u_{g g^{-1}} = u_{g} u_{g}^*,u_{g h} = u_g \overline{\alpha_g}(u_h)\end{align*}

\begin{align*} \eta_S=\phi^{-1}\alpha\phi'\qquad\eta_F=\psi^{-1}\beta\psi'\ . \end{align*}

\begin{align*}\mathbb{E}(t):=\Xi(t)+\beta\Big[\alpha\Xi_1^{(0)}(t)+\Xi_2^{(0)}(t)\Big]+\beta\Big[\Xi_3(t)+\beta\Xi_4(t)\Big]+\beta^3\Big[\alpha\Xi_1^{(1)}(t)+\Xi_2^{(1)}(t)\Big],\end{align*}

\begin{align*}CT^{-1}_j (\gamma_j + \epsilon_j \beta_j) \le CT^{-1}_j \rho\epsilon_j (1 + \rho^{-1} B_j) \le \rho \hat{c}^{-1} PT^{-1}_j \epsilon_j = \hat{c}^{-1} \rho \epsilon_{j+1},\end{align*}

\begin{align*}|0,f\rangle = \int d\theta f(\theta)\int d\hat n |g(\hat n,\theta)\rangle~,\end{align*}

\begin{align*}(p_{10}-h_1)\Psi=0, \qquad (p_{20}-h_2)\Psi=0, \end{align*}

\begin{align*}(\chi_1^{\mu\oplus 0})^{-1}_*\nu=&(\nu_1 (\overline{\partial \chi_1^{\mu\oplus 0}})(d\bar z-\mu\frac{{\partial \chi_1^{\mu\oplus 0}}}{\overline{\partial \chi_1^{\mu\oplus 0}}}dz)\\ &+\nu_2 (\partial \chi_1^{\mu\oplus 0})(-\bar\mu\frac{\overline{\partial \chi_1^{\mu\oplus 0}}}{\partial \chi_1^{\mu\oplus 0}}d\bar z+dz) )\circ(\chi_1^{\mu\oplus 0})^{-1}.\end{align*}

\begin{align*}b(x)=\frac{x}{\sqrt{x^2+1}}.\end{align*}