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Q&A for professional mathematicians
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Hottest Questions Today - MathOverflow
Skip to main content Stack Exchange Network Stack Exchange network consists of 183 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, ...
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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 $$\lVert(a_{ij})\rVert \le C \Bigl\lVert\s...
Mathoverflow.net news digest
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0 days
Looking for References: Treatment of Integration of universally measurable funct...
I d be interested in books or papers that treat universally measurable functions and in particular integration theory of universally measurable functions. As a bonus it would be nice if it would also treat regular conditional probabilities/Markov kernels...
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0 days
Does heat semigroup preserve log-smoothness of measure?
It is known that, if $\rho$ is a $\alpha-$strongly log-concave density on $R^d$, then $P_t \rho = \rho * N(0,t I)$ is $\frac{\alpha}{1+t \alpha}-$strongly log-concave, for $P_t$ the heat semigroup.
I wonder if such a property holds for the log-smoothness of the measure - that is, if $ \nabla \log \rho$ is $L-$Lipschitz, then what can we say about $\nabla \log (P_t \rho)$? Is it also Lipschitz, and if so what is the constant?... -
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Hessian geometry and hyperbolic plane
A Hessian manifold $(M,D,g)$ is a manifold with a flat symmetric connection $D$ and a Riemannian metric $g$ that is locally expressed, in flat charts of $D$, as the Hessian with respect to $D$ of some function (called a local potential).
For no reason I was trying to see if there is a simply connected Hessian manifold whose metric is complete and of constant negative curvature, i.e. isometric to the hyperbolic plane. Reversing the viewpoint, this is equivalent to the question: can the... -
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Sharp constant for a normalized Erdős–Mordell gap with angular defect
We define the angular defect function for a triangle with angles $A,B,C$ ($A+B+C=\pi$):
$$\Phi(A,B,C) = \frac{1 + \cos A \cos B \cos C}{\sqrt{3}\,\sin A \sin B \sin C}.$$
For a point $P$ inside triangle $\triangle ABC$, let $\mathcal{E}(P) = PA+PB+PC - 2(d_a+d_b+d_c)$ be the Erdős–Mordell gap....
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