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 Aug10 comment Are there n-th roots of differential operators? Yes, if you write $\sqrt[1/2]{D_y} \sqrt[1/2]{D_y} g(y)$ as a multiple integral, there will be a term like $\int e^{2\pi i y(k-k')} dy$ which is $\delta(k-k')$ and everything takes care of itself. This is what makes the pseudodifferential calculus work. Aug10 answered Are there n-th roots of differential operators? Aug10 comment Computation of determinant of a matrix with elements from an arbitrary commutative ring I don't know the answer to this particular question, but there is a general phenomenon in algebraic geometry that computational complexity of certain algebraic problems is (inversely) related to the "niceness" of an associated geometric object. The presence of zero divisors definitely complicates the geometry. So I wouldn't find it completely surprising if cofactor expansion really is the most efficient method for an arbitrary commutative ring, without any additional assumptions. Aug3 comment The boundedness of an integral After integrating by parts and playing around a little bit, you can easily get an estimate like this provided you make an assumption like $|a| > a_0$, $|b| > b_0$. If you want an estimate that makes no such assumption, then I'm not sure there is any reason to expect such an estimate to exist. Aug3 revised Asymptotic behavior of a sequence based on a subsequence. deleted 177 characters in body Aug3 comment Asymptotic behavior of a sequence based on a subsequence. I didn't realize that $m$ is supposed to be fixed. I'll update my answer when I get a chance. Aug3 answered Asymptotic behavior of a sequence based on a subsequence. Jun24 awarded Yearling May1 comment What is the Lie algebra of the indefinite orthogonal group''? Just to add--these are all non-compact real forms of the corresponding complex groups $O(n, \mathbb{C})$, etc. So from the point of view of their complex representations, the signature $(p,q)$ does not really play a role. But the signature does play a role in the real representation theory (see e.g. the classification of spinors, which exhibits mod 8 periodicity). Apr28 answered Approximating the Hessian Apr28 awarded Supporter Apr27 answered An isomorphism in relative De Rham cohomology Apr27 awarded Commentator Apr27 comment An isomorphism in relative De Rham cohomology I'm a bit confused. If $E \to M$ is a vector bundle, then $M$ is a retract of $E$ and they have the same cohomology. It seems to me that the relative cohomology $H^\ast(E, E^o)$ should be isomorphic to the (reduced?) cohomology of the Thom space of $E$, which is certainly not isomorphic to $H^\ast(M)$ in general. Apr26 answered Scalar product on manifold. Apr25 comment Why is $W(V)\simeq D(k[X_1,\dots,X_n])$? @Kally what is your working definition of polynomial differential operators, if not the algebra generated by $x_i, \partial_j$ subject to the above relations? There are several equivalent definitions, so the answer depends on which you take as fundamental. Apr25 answered Reference request for “Hodge Theorem” Apr15 answered Symplectic positive definite matrix. Apr14 comment Embedding/Submersion Properties of Cotangent Maps (Pullbacks) Yes, $d\phi$ (or $\phi_\ast$, or $T\phi$) is the differential of $\phi$. I think (4) should be true as well, since $\phi$ will be a diffeomorphism on the total spaces and it is linear on the fibers. Apr14 answered Embedding/Submersion Properties of Cotangent Maps (Pullbacks)