Determinant twist and $Pin _{\pm}$ structure on $4k$-dimensional bundles [Reference request]

Consider the automorphism $\varphi$ of $O(2n)$ given by $\varphi (A)=det(A)\cdot A$. The induced map in cohomology $H^*(BO(2n))$ sends $w_2$ to $w_2+(n+1)w_1^2$ (a proof is given at the end of the post). In particular, if $n$ is even, it switches $w_2$ and $w_2+w_1^2$. As these classes are the obstructions for existence of $Pin_{+}$ and $Pin_{-}$ structures, … Read more

Hasse diagrams of G/P_1 and G/P_2

in the Paper at the end, we can see Hasse diagrams for several projective, homogeneous G-varieties for G being a exceptional linear algebraic group. Note that D4/P1 is isomorphic to a six dimensional quadric, that i will denote as Q6. In an unfinished book by Gille, Petrov,N. Semenov and Zainoulline, which can be found … Read more

Is the family of probabilities generated by a random walk on a finitely generated amenable group asymptotically invariant?

Is the family of probabilities μn (convolution) generated by a random walk μ on a finitely generated amenable group G asymptotically invariant (‖gμn−μ‖L1→0 for any g∈G)? I am not familiar with random walks on amenable groups. Please indicate reference on the subject. Answer AttributionSource : Link , Question Author : anom , Answer Author : … Read more

Existence of martingales given some constraint on laws

Let X=(X)0≤t≤1 be a continuous martingale starting at 0, then denote by μ and ν the probability laws of ∫10Xtdt and X1. Then it is easy to see that the couple (μ,ν) is increasing in convex order, i.e. ∫Rf(x)dμ(x)  ≤  ∫Rf(x)dν(x) holds for all convex functions f:R→R of linear growth, see also “Peacocks and Associated Martingales, with … Read more

Dimension reduction for low-order moments of Rademacher-weighted sums of vectors

Let $x_1,\dots,x_n$ be vectors in a Euclidean space $H$. Let $\varepsilon_1,\dots,\varepsilon_n$ be independent Rademacher random variables (r.v.’s), so that $P(\varepsilon_i=\pm1)=1/2$ for all $i$. Does the inequality $$(1)\qquad E f\Big(\Big\|\sum_{i=1}^n\varepsilon_i x_i\Big\|\Big) \le E f\Big(\sum_{i=1}^n\varepsilon_i \|x_i\|\Big) $$ always hold if $f(x)=f_p(x):=|x|^p$ for some real $p\ge2$ and all real $x$? For $f=f_p$ with $p\ge3$, this inequality is due … Read more

What does “control of a deformation problem” mean?

Is the expression “control of a deformation problem’ ever defined? There are of course many examples relating a dg-Lie or L-infty algebra to a deformation problem, and the phrase is evocative. Is it ever defined? Answer AttributionSource : Link , Question Author : Jim Stasheff , Answer Author : Community

Generation of cohomology of graded algebras

Let A be an unital, associative, graded algebra over a base ring k. I’m happy to assume that k is a field if need be, and will insist that A free and of finite rank in each degree (locally finite). Further, A is connected: it vanishes in negative degrees and is of rank 1 (generated … Read more

Catenarity of monoid algebras

Let R be a commutative ring, let M be a commutative monoid, and let R[M] denote the corresponding monoid algebra. Suppose further that R is universally catenary. One may ask for conditions on M such that the ring R[M] is catenary. I know of the following results: If M is of finite type then R[M] … Read more

How distributive are the bad Laver tables?

Suppose that $n\in\omega\setminus\{0\}$. Then define $(S_{n},*)$ to be the algebra where $S_{n}=\{1,…,n\}$ and $*$ is the unique operation on $S_{n}$ where $n*x=x$ $x*1=x+1\,\text{mod}\, n$ and if $y<n$, then $x*(y+1)=(x*y)*(x*1)$. The algebra $(S_{n},*)$ satisfies the self-distributivity law $x*(y*z)=(x*y)*(x*z)$ if and only if $n$ is a power of $2$, and if $n$ is a power of $2$, … Read more