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Q&A for professional mathematicians
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Let $A$ be a $C^*$-algebra and $(a_{ij}) \in M_n(A)$ be a positive matrix. Does there exist a constant $C \ge 0$ (not depending on the $a_{ij}$) such that $$\lVert(a_{ij})\rVert \le C \Bigl\lVert\s...
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K-flat complexes and unbounded derived tensor product over a non-commutative she...
$\def\R{\mathscr{R}}\def\O{\mathcal{O}}$Let $X$ be a topological space and let $\R$ be a sheaf of unital non-commutative rings over $X$.
When $\R$ is commutative, there is much literature on existence of K-flat resolutions and unbounded derived tensor product of $\R$-modules [S; L, 2.2.5; GW, 21.94; SP, 06Y7].... -
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Number field with rational real points
Consider a number field $K$. How to classify or caracterize those $K$ intersecting the real line only in the rationals: $K\cap \mathbb{R}=\mathbb{Q}$ ?
An example are quadratic imaginary fields.
In general, we can wright $K=\mathbb{Q}[\alpha]\cong \mathbb{Q}[X]/\langle P\rangle$, where $P$ is the minimal polynomial of $\alpha$. A necessary condition is that $P$ has only non-real roots (in particular its degree is even). But this condition is... -
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Is there a increasing, convex, superlinear $\phi$ with $c_1 \phi(x)y \leq \phi(x...
The following version of a de la Vallée Poussin - criterion would be very helpful to me if it would be true. Can you say something about the truth value or give a reference?
Given a positive random variable $X$ with $\mathbb{E}[X] < \infty$, is there a increasing, convex function $\phi$ with $\lim_{x \to \infty} \frac{\phi(x)}{x} = \infty$ such that $\mathbb{E}[\phi(X)] < \infty$ and it holds for constants $c_1,... -
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Non-linear dynamical system and explosion
I study the following non-linear dynamical system$$dX_t=\lambda_1 \left((-\beta_1 X_t +\beta_2 \sqrt{Y_t})^2 -X_t \right)$$$$dY_t=\lambda_2 \left((-\beta_1 X_t +\beta_2 \sqrt{Y_t})^2 -Y_t \right)$$$$\text{with }Y_0>0 \text{ and } -\beta_1 X_0 +\beta...
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