AI Networking Cookbook: Practical recipes for AI-assisted network automation and development
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Let $I_a(g)=Q_g+a\sigma_2(A_g)$, where $A_g$ is the Schouten tensor and $Q_g$ is Branson's $Q$-curvature. On a closed connected manifold of dimension $n\ge4$ with a positive Einstein metric $g_0$, we prove that every smooth metric conformal to $g_0$ with nonnegative scalar curvature and constant $I_a(g)$ is Einstein fo...
Using classical conformal divergence identities with a variable reference curvature, we prove four main results. First, we establish a general mean-curvature estimate for conformal metrics on the round hemisphere with $A_g\in\overline{\Gamma_2^+}$ and positive prescribed $H_2$ data. When $\sigma_2(A_g)=0$ and the bound...
Let $(v,p)$ be a smooth stationary Navier--Stokes flow in $\mathbb R^3$ that vanishes at infinity, and set $\omega :=\operatorname{curl}v$. We prove the endpoint implication \[ \omega\in L^{9/5,\infty}(\mathbb R^3) \quad\Longrightarrow\quad v\equiv0. \] This weak-Lorentz condition is invariant under the far-field resca...
We prove a Liouville theorem for nonnegative entire strictly spacelike solutions of \[ \operatorname{div}\left(\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)+u^p=0 \qquad\text{in }\mathbb R^n. \] If $n=2$ and $p\geqslant1$, or if $n\geqslant3$ and $1\leqslant p<\frac{n+2}{n-2}$, every nonnegative $C^2$ solution satisfyi...
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