Determinant of quotient of unbounded operators

I have been trying to prove this for a while but failed so far. Let $A$ and $B$ are two positive, self-adjoint operators with compact resolvent on a Hilbert space $H$ defined on the same dense domain $D \subset H$ and suppose that $A^{-1}B$ extends from $D$ to a bounded operator on all of $H$. … Read more

Are ultralimits the Gromov-Hausdorff limits of a subsequence?

Let (Mi,pi) be a sequence of n-dimensional Riemannian manifolds with lower Ricci curvature bound −1. Fix a non-orincipal ultrafilter and let X be the ultralimit of the sequence. Does there exists a p∈X and subsequence of (Mi,pi) converging to (X,p) in the pointed Gromov-Hausdorff sense? Answer AttributionSource : Link , Question Author : dg.jan , … Read more

An inequality from the “Interlacing-1” paper

This question is in reference to this paper, http://annals.math.princeton.edu/wp-content/uploads/annals-v182-n1-p07-p.pdf (or its arxiv version, http://arxiv.org/abs/1304.4132) For the argument to work one seems to need that for the symmetric ±1 signing adjacency matrix of the graph, As, it holds that, max−root(∑s∈{0,1}mdet(xI−As))≤max−root(Es∈{0,1}m[det(xI−As)]) But why should this inequality be true? and the argument works because the polynomial on the … Read more

largest subgroup of Out(^F2)Out(\hat{F_2}) which preserves the Nielsen invariant

Let x,y be generators for the free group F2. It’s known that Aut(F2), and hence Out(F2) preserves the conjugacy class of the subgroup ⟨[x,y]⟩ generated by [x,y] (This conjugacy class is in some contexts called the nielsen invariant) On the other hand, if we view x,y as topological generators of ^F2 (hence fixing an embedding … Read more

Unibranch partial normalization

In a paper I recently read something about the “unibranch partial normalization” of a curve. Say, $R$ is a local integral domain with maximal ideal $\mathfrak{m}$ and fraction field $K$. Is it possible to find a minimal ring $R^u$ between $R$ and its normalization $R’$ in $K$ such that all localizations of $R^u$ in the … Read more

Conjectured new primality test for Mersenne numbers

How to prove that this conjecture about a new primality test for Mersenne numbers is true ? Definition: Let M_{q}=2^{q}-1 , S_{0} = 3^{2} + 1/3^{2} , \ and: \ S_{i+1} = S_{i}^{2}-2 \pmod{M_{q}} Conjecture: M_{q}\text{ is a prime iff: } \ S_{q-1} \equiv S_{0} \pmod{M_{q}} \text{ and iff: } \prod_{0}^{q-2} S_i \equiv 1 \pmod{M_{q}} … Read more

Has unconditional convergence ever been proved other than by deducing it from absolute convergence?

Nobody’s answering this question so I’ll try it here. This is really a reference request: Has a certain kind of proof ever been used? A series ∑nan converges absolutely if ∑n|an|<∞. It converges unconditionally if it converges to a finite number and all of its rearrangements converge to that same number. For series of real … Read more

Undecidability of the existential theory

Do you know if I can find the proof that the existential theory of Z with the structure of addition , divisibility and the relation (∃s∈Z)m=nps is undecidable, besides the one of the paper of J. Denef “The diophantine problem for polynomial rings of positive characteristic” ? In the paper of Denef, the corollary is … Read more

Flat resolutions of DG-schemes

Recall that a DG-scheme is a pair (X,OX), where (X,O0X) is a scheme, OX is a sheaf of commutative DG-algebras over (X,O0X), and each OiX is a quasi-coherent O0X-module. I would like to know if in this generality (without assuming things like X being quasi-projective over a field of characteristic zero), is it known that … Read more

Modules over an Azumaya algebra and modules over the associated Brauer-Severi variety

Assume A is an Azumaya algebra of rank r2 on a smooth projective scheme Y over C. Let f:X→Y be the Brauer-Severi variety associated to A. I read here in a comment that the category of modules over A is equivalent to the categroy of modules on X which restrict to every fiber as a … Read more