All Questions
7 questions
2
votes
0
answers
946
views
On a deceptively tricky calculus problem
Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality
Let $A$ be a non-constant operator acting on $C^...
- 90
1
vote
0
answers
121
views
Does this integral have a closed form solution? [closed]
Let $x,y\in \mathbb{R}^2$, $B_r(0)=\{x||x|\leq r\}$. Does the following function (denoted as $f(r)$) have a closed form expression?
$$f(r)=\int_{B_r(0)}\int_{B_r(0)}\ln|x-y|dxdy.$$
- 63.9k
2
votes
0
answers
44
views
Derivatives of $G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt$ when $h$ is positive-homogeneous
Let $h:\mathbb R \to \mathbb R$ be a continuous which is positive-homogeneous of order $p \ge 1$, and define $G_h:[-1,1] \to \mathbb R$ by
$$
G_h(u):=\int_0^{2\pi} h(\cos t)h(\cos(t - \arccos(u)))dt.
$...
- 6,853
0
votes
0
answers
81
views
What is the standard terminology for the quantity $\|\nabla f\|_{L^2(\mu)} := \sqrt{\int_{\mathbb R^d}\|\nabla f(x)\|^2d\mu(x)}$?
Let $f:\mathbb R^d \to \mathbb R$ be a continuously differentiable function and let $\mu$ be a probability measure on $\mathbb R^d$.
Question. What is the standard teminology for the quantity $\|\...
- 6,853
-1
votes
1
answer
102
views
Is it true that $\nabla_x \int_0^\infty f(t,0) dt = 0 \implies \nabla_x f(t,0) = 0 \ \forall t>0$? [closed]
Let $f:\mathbb R_+ \times \mathbb R^N \to \mathbb R$ and $$F(x) = \int_0^\infty f(t,x) dt.$$ If $\nabla_x F(0) = 0$ do we have that $\nabla_x f(t,0) = 0$ for all $t \in \mathbb R_+$? If not, which ...
- 188k
2
votes
2
answers
197
views
About preserving real-rootedness of multivariable polynomials
Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted?
If ...
- 28.6k
2
votes
1
answer
264
views
Monotonicity of the integral
Let $R(x)$ be the residual function associated to the normal probability density, i.e.
$$R(x)~=~\int_x^{+\infty}\frac{1}{\sqrt{2\pi}}e^{-\frac{y^2}{2}}dy, \mbox{ for all } x\in R.$$
Define
$$\phi(...
- 109k