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1d
comment compact inclusion of domains of unbounded operators
How do we know that $\mathcal{D}(L^{1/2}) \subset H^{1/2}$? It isn't obvious to me.
1d
comment compact inclusion of domains of unbounded operators
Are $\mathcal{D}(L)$, $\mathcal{D}(L^{1/2})$ equipped with their graph norms, or something else?
1d
revised Cohen-Macaulay rings and Normal rings
typo in title
Apr
21
answered Existence of a measure-preserving bijection
Apr
21
comment Functional minimization problem
@RyanBudney: Maybe $k$ is supposed to be fixed?
Apr
21
comment Real and imaginary part of an holomorphic function
What is your definition of "holomorphic function"?
Apr
20
comment What is the mathematical structure called if we replace commutative group by commutative monoid in the definition of linear space?
Based on your comments, maybe the term you are looking for is simply "cone", which is a subset of a (real or complex) vector space that is closed under addition and under multiplication by nonnegative scalars.
Apr
16
comment Rate of convergence in the Law of Large Numbers
Related to your other questions: mathoverflow.net/questions/73647/…
Apr
16
comment Rate of convergence in the Law of Large Numbers
I don't know an exact reference either, but I suspect that with a little work you can show that a distribution satisfying $E|X|^\alpha < \infty$ will satisfy the K-G hypotheses.
Apr
16
comment Rate of convergence in the Law of Large Numbers
Right. In the infinite second moment case, by "CLT-type result" I'm talking about a Kolmogorov-Gnedenko style stable law limit theorem giving the weak convergence of $(S_n - \mu n)/n^{1/\alpha}$ (typically the weak limit is a stable law, not the normal distribution).
Apr
16
comment Rate of convergence in the Law of Large Numbers
Maybe I'm confused, but doesn't a CLT-type result give you exactly what you want? In the finite-variance case, the classical CLT says that $(S_n - \mu n)/n^{1/2}$ converges weakly, so as a result $E_n/n^{1/2 + \epsilon} \to 0$ weakly and hence in probability (standard result). So a CLT-type result telling you that $(S_n - \mu n)/ n^{1/\alpha}$ converges in distribution would imply that $E_n / n^{1/\alpha + \epsilon} \to 0$ in probability.
Apr
15
comment Area enclosed by Brownian motion (without winding number)
@CarloBeenakker: Hmm, and I actually voted on that question. Seems that my memory doesn't last more than 2 years. Well, it got only a very partial answer there, maybe we will learn more from the MO crowd.
Apr
15
comment Lower order perturbations of 2nd order differential operators
Thinking in terms of spectra, self-adjoint operators tend to behave like real numbers, and skew-adjoint operators are imaginary numbers. It's a little like asking "if I take a real number, and add a very small imaginary number, might the result turn out to be real?" No, it never will.
Apr
15
comment Lower order perturbations of 2nd order differential operators
The problem is more fundamental than that. In Kato-Rellich the perturbation is at least symmetric, and the problem is to sort out the domain of the adjoint. An example is something like a second order operator perturbed by a zero order term. Here your perturbation is not even symmetric: integration by parts suggests that formally $(\beta X)^* = -\beta X + V$ for some zero-order term (potential) $V$. It's more likely to be skew symmetric.
Apr
14
comment Lower order perturbations of 2nd order differential operators
As an analogy in the elliptic setting, $P = d^2/dx^2$ is self-adjoint on $L^2(\mathbb{R}, dx)$ but $Q = d^2/dx^2 + d/dx$ is not (its adjoint is $d^2/dx^2 - d/dx$). However, $Q$ is self-adjoint on $L^2(\mathbb{R}, e^{-x} dx)$.
Apr
14
comment Lower order perturbations of 2nd order differential operators
Typically, not unless you change the measure in just the right way. Almost anything you try will be a counterexample.
Apr
14
revised Area enclosed by Brownian motion (without winding number)
in distribution
Apr
14
asked Area enclosed by Brownian motion (without winding number)
Apr
13
answered Diffusion semigroup generated by Laplacian
Apr
13
revised SDEs: Bounding the variance of a solution
added 192 characters in body