Convergence of a particular double sum [closed]

Closed. This question needs details or clarity. It is not currently accepting answers. Want to improve this question? Add details and clarify the problem by editing this post. Closed 6 years ago. Improve this question Consider the following double sum: Q(n)=1n2n∑i=1n∑j=1[∂ijlnf(x)]2 where ∂ij is the partial second order derivative (bounded for all indices), the function … Read more

Table of LCM’s vs. table of products

In 2004 Kevin Ford established sharp asymptotics on Erdős’ problem on the number of different products a⋅b, a,b∈{1,…,n}. (http://arxiv.org/abs/math/0401223, see also discussion here: Number of elements in the set {1,⋯,n}⋅{1,⋯,n}) My naive question is whether there are much less different numbers of the form lcm(a,b), where a,b∈{1,…,n}. Answer AttributionSource : Link , Question Author : … Read more

Adams Spectral sequence and Pontrjagin-Thom construction [Reference request]

I will be grateful for any reference for the following statements/claims. 1) Let’s consider the case of $p=2$ and the classic Adams spectral sequence with the $E_2$-term given by $\mathrm{Ext}_{A}(\mathbb{F}_2,\mathbb{F}_2)$. If $\alpha$ and $\beta$ are two permanent cycles in the Adams spectral sequence, converging to elements $f\in{_2\pi_i^s}$ and $g\in{_2\pi_j^s}$, then is it true that $\alpha\beta$ … Read more

Centers of Noetherian Algebras and K-theory

I’ll start off a little vauge: Let E be a noncommutative ring which is finitely generated over its noetherian center Z. Denote by modE the category of finitely generated left E-modules and similarly for modZ. We have a functor F:modZ→modE which takes M to E⊗ZM, hence an induced map on (Quillen) K1-groups K1(F):K1(modZ)→K1(modE). I’m interested … Read more

Normal fields of geodesic spheres

This question is related to this one (https://math.stackexchange.com/questions/1383511/normal-curvature-of-geodesic-spheres) I’ve asked at math.stackexchange. Let (M,g) be a compact Riemannian manifold with no conjugate points and (˜M,˜g) its universal covering. Let ˆg the Sasaki metric on TM−{0} and dˆg its associated distance function. Fix ˜p∈˜M and R=1. Let ˜H:=˜M−¯B1(p). For x∈˜H, consider the geodesic sphere centered at … Read more

Existence of a block design

Let $\ell$ be an integer parameter. I want to ask the existence of the following design: There is a universal constant $\beta < 1$ such that for all sufficiently large $\ell$, the following holds: The universe size $d = O(\ell)$. There are $\ell^2$ sets in the design, $S_1, \dots, S_{\ell^2} \subseteq [d]$. $(\forall i \in … Read more

Applications and main properties of hyperfunctions

I am trying to get familiar with hyperfunctions, and I do have some familiarity with the classical theory of distributions. I am wondering whether hyperfunctions have any advantages over distributions. Are there any applications of the former which cannot be obtained using the latter? Any important examples? I was told one general property of hyperfunctions … Read more

How to compute (co)limits of enriched categories?

I’ve asked this question on math.stackexchange some time ago (https://math.stackexchange.com/questions/1380176/how-to-compute-colimits-of-enriched-categories) and I received no complete answers, so I’m posting it here. Let $\mathscr{V}$ be a monoidal category. Let $\mathbf{Cat}_{\mathscr{V}}$ be the category of (small) categories enriched over $\mathscr{V}$. I would like to know how to compute (co)limits in $\mathbf{Cat}_{\mathscr{V}}$. This leads to some questions. 1)When … Read more

Natural transformations of A∞A_\infty-functors (between dg-categories) are “directed homotopies” (reference?)

Let A and B be dg-categories over a field, viewed as A∞-categories. The A∞-category (actually, dg-category) of strictly unital A∞-functors A→B will be denoted by Fun∞(A,B). It is described explicitly (in the non-unital case) for example in P. Seidel’s book (“Fukaya category and Picard-Lefschetz theory”). Let Δ1 be the 1-simplex category, namely, the linear category … Read more