9
votes
Accepted
Bull's-eye Riemann sum
Not necessarily. Take small $\delta>0$. Spend time $\frac 23(1-\delta)$ at $(0,1)$. Then, in time $\delta/4$ travel the route $(0,1)\to(0,2)\to(-1,2)\to(-1,-2)$ so that the integral of the vertical ...
- 59.8k
5
votes
Accepted
How to understand the integral?
If I understand your question correctly, the answer is affirmative for Ricci-flat (i.e. flat) surfaces (not necessarily embedded in a Euclidean space) according to equation (14) of
Reilly, Robert C., ...
- 109k
5
votes
Extending a convex function to a higher dimensional domain
Yes: Just take $u(x,y):=v(x)$, which will be assumed in what follows.
Indeed, one can use Green's formula to show this, as is done in Christian Remling's answer.
More generally, the result holds for ...
- 118k
4
votes
Extending a convex function to a higher dimensional domain
$u(x,y)=v(x)$ works. By Green's formula, we can rewrite the LHS of your inequality as
$$
\frac{1}{2\pi} \int\!\!\!\int \Delta u\, dA = \frac{1}{2\pi}\int_{-1}^1 dy\int_{-\sqrt{1-y^2}}^{\sqrt{1-y^2}} ...
2
votes
Mean value theorem for Dirichlet series - optimize?
The following is an approach inspired by Selberg's treatment of the large sieve, and more directly by a sketch in Granville-Harper-Soundararajan (on the case of prime support). There remains an ...
- 19.4k
1
vote
Accepted
Mean value theorem in terms of Wirtinger calculus?
The desired relation can be written in components as
$$f_n(\mathbf{z}^{(1)},\mathbf{\bar z}^{(1)})-f_n(\mathbf{z}^{(2)},\mathbf{\bar z}^{(2)})=\int_0^1 d\tau\,\sum_{m}\left(\frac{\partial f_n}{\...
- 178k
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