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Consider the family of Besov spaces $ B_{p,q}^{s}(\mathbb{R})$ with $ 0<p,q \leq \infty$ and $ s \in \mathbb{R}$ . Is there a natural way to define spaces of generalized functions $ f(t,x) \in \mathcal{S}'(\mathbb{R}^2)$ such that, for any test function $ \varphi \in \mathcal{S}(\mathbb{R})$ , we have $ $ \langle f(t,\cdot), \varphi(t) \rangle \in B_{p_1,q_1}^{s_1}(\mathbb{R})Read more

I am trying to prove the graph-theoretic generalization of the following geometric observation: let $ \mathbb{P}$ be a finite set of distinct points in the Euclidean plane and $ card\left(CH\left(\mathbb{P}\right)\right)\ge4$ , where $ CH\left(\mathbb{P}\right)\subseteq\mathbb{P}$ denotes the set of points constituting to the convex hull of $ \mathbb{P}$ , then the following observation can be made:Read more

I noticed in many papers, it’s common to assume the Riemann Hypothesis or variants thereof in order to get their results. There will be an “optimal” bound where RH is true, and one that is easier to obtain but not conditional. And I’ve seen other conjectures of this kind: Riemann Hypothesis Lindelöf Hypothesis Generalized RiemannRead more

In [1], Lin Weng shows that the Riemann hypothesis (RH) holds for certain linear combinations of shifted Riemann zetas. I would like to know whether these are all examples for which the RH is known. To be precise, we say that a Dirichlet series $ $ D(s)=\sum_{n=1}^\infty \frac{a_n}{n^s} $ $ satisfies the generalized Riemann hyporthesisRead more

Strong no loop conjecture: Let $ A$ be an artin algebra and $ S$ be a simple module in $ mod A$ , where $ mod A$ denotes the right finitely generated module category. If $ Ext_{A}^{1}(S,S)\neq 0$ , then $ pd S=\infty$ , where $ pd S$ denotes the projective dimension of $ S$Read more

I encountered an expression which manual simplification is manageable, where the results are nicely in terms of generalized mean (power mean). To avoid making this post too long, I created a post on Math StackExchange to display the kind of algebra I’m talking about.$ {}^1$ How can I formulate perhaps ComplexityFunction or adopt some goodRead more