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 Aug24 comment Formula for trace of compact operators on $L^2(\mathbb{R})$ given by integral kernels? If $K$ is nice, then for each fixed $z$, $K(z,x)$ is a well-behaved function of $x$, hence by the 3rd last equation can express $K(z,x)$ as the sum of $n$ of the integral of $K(z,y) e_n(y)$ with respect to $y$. Now plug in $z = x$ to obtain the result. Aug23 comment Formula for trace of compact operators on $L^2(\mathbb{R})$ given by integral kernels? Yes, I am well aware and absolutely agree. I answered as above, assuming $K(x,y)$ to be as nice as necessary, since it didn't seem that Mike was asking for the weakest possible hypotheses under which this holds, but rather whether it can be made rigorous at all. Aug10 comment Are there n-th roots of differential operators? This technique depends sensitively on both the given differential operator, as well as the space of functions on which it acts. In the case of the Dirac operator, it is not the Laplacian on functions that admits an algebraic square root, but rather the Laplacian on spinors. This also works for vector-valued functions, since we can decompose vectors into products of spinors. These kinds of $n$th roots are completely different than what I described in my answer, but both can be useful depending on the situation. 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 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 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. 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). 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. 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. 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. Apr13 comment Why is $W(V)\simeq D(k[X_1,\dots,X_n])$? Updated to include a sketch of the proof. You can find the details in the lecture notes by Cannas da Silva. Apr12 comment Exactness Axiom of Homology Theory $C_k(\cdot)$ are the chains, with boundary map $\partial: C_k(\cdot) \to C_{k-1}(\cdot)$, so that $H_k = \ker \partial_k / \mathrm{im} \partial_{k+1}$. They depend on what homology theory you are considering, but they are always there in the background somewhere. For the singular homology of a space $X$ they are formal linear combinations of maps $\Delta^k \to X$ where $\Delta^k$ is the standard $k$-simplex. For simplicial and cellular homology there are analogous definitions. Whenever you encounter any kind of homology you should think of it as the homology of some underlying chain complex. Apr8 comment Center of Clifford Algebra depending on the parity of $\dim V$? The parity issue is discussed in detail in E. Meinrenken's Notes math.toronto.edu/mein/teaching/clif_main.pdf . See Proposition 2.6 in particular, which gives the result over $\mathbb{C}$. Another good reference is Chevalley's book. Feb9 comment Non-closed subgroups of Lie groups Irrational flows on the torus are well-studied (and famous) examples in noncommutative geometry. So it might be worth looking at the noncommutative geometry literature to see if there is anything like what you're asking. Feb4 comment Expressing Differential Form in Different Coordinates @Confused I edited it to include a few more details of the evaluation of the pushforward. Nov16 comment Solving this Fourier transform? I haven't thought about it too much, but a quick look at the wikipedia page suggests that these functions might be closely related to the Laguerre polynomials, and the Laguerre polynomials are indeed related to Bessel functions. In any case I'm sure these functions are listed in tables of Fourier transforms, so they probably have somebody's name attached to them. Nov16 comment Solving this Fourier transform? Is there any reason you don't consider residues to be an exact analytic method?