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When installing Ubuntu on a PC with an AMD APU like Athlon 200GE / Ryzen 2200G / Ryzen 2400G processor, the boot process doesn’t complete and the screen remains black. The OS won’t boot after the live CD screen.Read more

In this article Hyttinen and Kangas claim that universal abstract elementary classes which are categorical are essentially classes of vector spaces as soon as the model reaches a critical cardinality. The proof is grounded on the fact that they can equip the class with a closure operator which is Urbanik. I am not practical withRead more

Below is a problem, from an old Silk Road olympiad. Define an infinite sequence, $ a(n)$ , such that, $ a(1)=a(2)=1$ ; $ $ a(n)=a(a(n-1))+a(n-a(n-1)),\forall n\geq 3. $ $ Show that, for every $ n\geq 1$ , $ a(2n)\leq 2a(n)$ .Read more

The following inequality is an elementary exercise in convexity: let $ x,y$ be non-zero vectors in a normed space with $ \|x\|, \|y\|\leqslant 1$ . Suppose that $ \|x-y\| \geqslant 1$ . Then $ $ \left\|\frac{x}{\|x\|} – \frac{y}{\|y\|} \right\| \geqslant \|x-y\|.$ $ It is stated in Lemma 6 of H. Martini, K. J. Swanepoel, andRead more

Let $ \mathcal{E}_{\lambda}$ be the set of all elementary embeddings $ j:V_{\lambda}\rightarrow V_{\lambda}$ . If $ A\subseteq V_{\lambda}$ , then let $ j^{+}(A)=\bigcup_{\alpha<\lambda}j(A\cap V_{\alpha})$ , and let $ \mathcal{E}_{\lambda}[A]=\{j\in\mathcal{E}_{\lambda}\mid j^{+}(A)=A\}.$ We say that a non-trivial subalgebra $ X\subseteq\mathcal{E}_{\lambda}$ has the simplicity property if whenever $ \gamma<\lambda$ and $ \gamma$ is a limit ordinal, and $Read more

Let $ G$ be an arbitrary graph on $ n$ vertices and $ \mathcal L$ be its Laplacian. I need to show that \begin{equation}\tag{$ *$ } \langle \mathcal Lx,\mathcal L(|x|^{p-2}x)\rangle_{\mathbb R^n}\ge 0\qquad \hbox{for all }x\in \mathbb R^n \end{equation} and small $ p\in (2,\infty)$ (my guess is that $ p\approx 3$ will do), where $ |x|^{p-2}x$Read more