All Questions
12,776 questions
-3
votes
1
answer
411
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Can Hartogs' extension theorem be used to prove there's no naked singularity?
Maybe for the first time my question doesn't deal with number theory. Tonight a friend of mine told me about Hartogs' extension theorem and said his work in analytic microlocal analysis was somehow ...
- 6,984
-3
votes
2
answers
768
views
Question on Linear Operators
Let $V$ be a normed infinite dimensional vector space. Let $L: V \longrightarrow V$ be a bounded linear operator. Moreover assume that $L$ is 'locally nilpotent' that is:
$$ \forall v \in V \quad \...
- 741
-3
votes
1
answer
158
views
Randomness about coefficients of series
$B\subset \mathbb{N}\bigcup \{0\}$ is finite and not empty, infinite series:$$f(x)=\sum_{i=1}^{\infty}a_i x^i,a_i \in B$$ Now $f(x)$ is rational or has a natural boundary.
Now,the question :if $f(x)$...
- 3,094
-3
votes
1
answer
634
views
compactly supported harmonic functions [closed]
Do a significant class of compactly supported smooth functions u on Ω⊂Rn such that Δu≥0 exist?
Thanks!
- 25
-3
votes
1
answer
194
views
Bounding a number-theoretic integral
Find a good upper bound on $$\int_1^T\frac{\zeta'(s)}{\zeta(s)\zeta(1-s)}X^sdt,$$ where $s=c+it$ for a constant $c>1$ and $X>0$ is a parameter. If needed, we can assume RH.
My attempt here is ...
- 107
-3
votes
1
answer
392
views
A generalization of Chebyshev's sum inequality
From some my previous questions here and here and well-known rearrangement inequality. I pose an inequality as follows and I am looking for the proof or a reference.
Inequality: Let $y=f(x,y)$ is ...
- 1,776
-3
votes
1
answer
76
views
Minimal norm problem with linear combination of translation operator to be estimated
Follow up question from this one
Suppose $X = L^2(G)$, where $G$ is some locally compact group. Let $x, y \in G$ I for fixed $n$ I am seeking for an operator $H \in B(X)$ of the form
$$
H = H(\alpha_1,...
- 115
-3
votes
1
answer
63
views
How to show $\lambda_i \in \sigma_A(x)$?
Let $\sigma_A(x)$ be the spectrum of $x$ in $A$, and linear functional $\phi$ satisfying $\phi(x)\in \sigma_A(x)$ for every $x \in A$, consider $p(\lambda)=\phi((\lambda e-x)^n)$, and denote its roots ...
- 33
-3
votes
0
answers
55
views
Is the real and imaginary part of the Dirichlet eta function closest to its partial sums when trigonometric function changes signs?
To grasp the question we are concerned with three Theorems 1,2, and 3 in bold font below. First let us consider the Dirichlet eta function $\eta: \mathbb{C}\rightarrow \mathbb{R}$
$$
\eta(s) = \sum_{n=...
-3
votes
0
answers
68
views
Exercise generalizing (related to) Hölder's inequality
I came across this exercise and feel absolutely stuck:
Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
- 1
-3
votes
1
answer
451
views
Exponential decay of kernel
Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by
\begin{equation}
(Au)(\alpha) = \sum_{\beta}A(\alpha,\beta)u(\beta)
\end{equation}
where $\left|A(\alpha,\beta) \right|\le Ce^{-|\...
- 11
-4
votes
2
answers
530
views
Inverse square-law as a positive definite kernel?
Newtons law for gravity states that:
$$F_{12} = \frac{G m_1 m_2} {|x_1-x_2|^2}$$
The function :
$$k(x,y):=\exp(-| x-y|^2)$$
is known to be a positive definite function, called the RBF-kernel.
It ...
- 3,064
-4
votes
1
answer
213
views
Is the proof in "On Hilbert’s 8th Problem" published on Brazilian Journal of Probability and Statistics correct? [closed]
The article can be freely accessed here. The proof is only five pages. I am quite in doubt.
A new version (2021) of that paper can be found here.
- 263
-4
votes
2
answers
286
views
Does the Laplacian commutes with the indicator function [closed]
We define the laplacian operator $\Delta$ with the Neumann boundary conditions on the space $H^2(\Omega)$, where $\Omega$ is an open set of $\mathbb{R}^n$ with a smooth boundary $\partial\Omega$, and ...
-4
votes
1
answer
514
views
Meaning of the Mobius transformations video [closed]
What is this video trying to tell us?
http://www.youtube.com/watch?v=JX3VmDgiFnY
The statement that fractional linear transformations correspond to rotations of the sphere under the stereographic ...
- 1,783
-4
votes
1
answer
200
views
How I can choose $(t_1,t_2,...,t_{r}) \in (0,1)^{r}$ such that $f^{(k)}\left(1-2\prod_{j=1}^{k}t_{j}\right)=0$?
Let $f:\mathbb{R} \to \mathbb{R}$ be a real analytic function. Assume that $f$ has simple trivial zeros at each nonpositive integer. Then, all the $k$-th derivatives $f^{(k)}$ of $f$ have necessarily ...
- 1,197
-4
votes
1
answer
370
views
Is delta function symmetric against real axis? [closed]
Is $\delta\left(a+bi\right)=\delta\left(a-bi\right)$?
I wonder whether Dirac Delta (as defined via Fourier transform) is symmetric against the real axis.
We can write Delta function as
$$\delta(z) = \...
- 10.1k
-4
votes
1
answer
2k
views
Open mapping theorem for Riemann surfaces
What restriction must one impose on a Riemann surface M in order for all biholomorphic $f:M\to\mathbb{C}$ to be open mappings, aka mappings of $M$ onto open subsets $f(M)\subset\mathbb{C}$?
- 41
-4
votes
1
answer
144
views
Coordinate free computation of the second derivative of a functional [closed]
Let $F(g(f))$
be a functional that sends functions of a vector variable (from $n$-dimensional vector space) to $\mathbb R$.
$g$
is some function of scalar valued functions $f$.
I'm interested in a ...
- 1
-5
votes
0
answers
53
views
Every smooth function contains a bijection [closed]
Let $f:\mathbb{D}\rightarrow \mathbb{R}$ be a continuous non-constant over $\mathbb{D}$. Is there always a subset $\mathbb{A}\subseteq \mathbb{D}$ such that $f:\mathbb{A}\rightarrow \mathbb{R}$ is a ...
-6
votes
1
answer
614
views
Proof of formula for $\pi$ [closed]
The number $\pi$ can be expressed as $\pi=\lim_{n\to\infty} \frac{n\sqrt[n]{-1}-n}{\sqrt{-1}}$ or more poetically $\pi=\frac{\infty\sqrt[\infty]{-1}-\infty}{\sqrt{-1}}$. Here we choose the principal ...
- 16.6k
-6
votes
1
answer
138
views
Is the real part of the Eta function bounded by $2 \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}} $ [closed]
Consider the series defined by
\begin{equation}
f(\alpha,\beta) := \sum_{n=1}^{\infty}{\dfrac{(-1)^{n-1}}{n^{\alpha}}\cos(\beta\ln(n))}
\end{equation}
is it true that $$f(\alpha,\beta) \le 2\sum_{n=1}...
-6
votes
1
answer
180
views
An analog of Anderson's result in C* algebra setting [closed]
Let $\mathcal{A}$ be a unital $C^{*}$-algebra and $S(\mathcal{A})$ denote the states space of $\mathcal{A}$.
For $a\in \mathcal{A}$ , define $W(a) =\{\phi(a):\phi\in S(\mathcal{A})\}$
It's known that $...
- 307
-6
votes
1
answer
442
views
On gaps between consecutive zeros of the Riemann zeta function
Let $\gamma$ denote the imaginary part of a non-trivial zero of the Riemann zeta function. Do there exist some function $f$ such that $\gamma_{n+1} - \gamma_n > f(n)>0$ for all large $n$? To be ...
- 1,019
-6
votes
1
answer
112
views
Sign of real part and imaginary part zeta function at 1/2-x+iy and 1/2+x+iy [closed]
I want to know what the sign of the real part and imaginary part of $\zeta(1/2+x+iy)$ and $\zeta(1/2-x+iy)$ are ,are they the same? for example in this case they are the same
zeta(0.25+I 10)=0.74513-0....
- 1
-8
votes
1
answer
309
views
Is the Klein group related to the Klein bottle? [closed]
Is the group of symmetries of the rectangle-not-square related to the Klein bottle mathematically?
The reason I am asking is because I want to put a Klein bottle coffee cup in a joke about V_4 and ...
