All Questions
4,449 questions with no upvoted or accepted answers
8
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512
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Proving a proposition which leads the irrationality of $\frac{\zeta(5)}{\zeta(2)\zeta(3)}$
Question : Is the following $(\star)$ true for $a,b,c\in\mathbb Z$ ?
$$\begin{align}\int_{0}^{\frac{\pi}{2}}(ax^4+b\pi x^3+c{\pi}^{2}x^2)\log(\sin x)dx=0\Rightarrow a=b=c=0\qquad(\star)\end{align}$$
...
- 4,757
8
votes
0
answers
221
views
Density of odd and even eigenstates of an integral operator
Consider an integral operator $(Kf)(x)=\int_{-1}^{1}K(x-y)f(y)dy$, where the kernel $K(-x)=K(x)$ is an even function.
Let $\lambda_n$ be the ordered eigenvalues of $K$ and $f_n(x)$ the ...
- 81
8
votes
0
answers
357
views
Ultrapowers of Banach spaces without the continuum hypothesis
Let $\mathcal{U}$ be a non-trivial ultrafilter on the set of integers $\mathbb{N}$, and let $C(K)$ denote the Banach space of continuous functions on a compact $K$. Under the continuum hypothesis CH, ...
- 4,461
8
votes
0
answers
952
views
About generator and isomorphism problems for free groups operator algebras
Let $H$ be an infinite dimensional separable Hilbert space.
The $C^{*}$-algebras and von Neumann here unital and subalgebras of $B(H)$.
Definition : Let $\mathcal{A}$ be $C^{*}$-algebra (resp. a von ...
8
votes
0
answers
298
views
Spaces that never separate the Hilbert cube
I am interested in topological spaces such that whenever the space embeds into the Hilbert cube, the image of the embedding has a path-connected complement.
Any finite dimensional space has this ...
- 29.1k
8
votes
0
answers
508
views
Is there in ZFC a topological space which is normal, ccc, countably compact, first countable and non-compact?
I am looking for a space as in the title and since many very similar spaces do exist in the literature, I wonder whether someone has a reference (different from the ones I cite below) or just some ...
- 1,855
8
votes
0
answers
247
views
Construct a topologically $\infty$-dimensional separable metric space.
But don't assume knowledge of any topological dimension theory. Here is a specific approach (an open problem):
Does there exist a separable metric space $X$ such that the following two conditions ...
- 7,500
8
votes
0
answers
403
views
Is the product of a discretely Lindelöf space with [0,1] discretely Lindelöf ?
A space $X$ is discretely Lindelöf iff given any discrete subset $D$ of $X$, its closure in $X$ is Lindelöf. Such spaces were introduced by Arkhangel'skii about 15 years ago (if I am not mistaken) ...
- 1,855
8
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0
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452
views
Preduals of $\ell_1$
The space $\ell_1$ has loads of (isomorphic) predulas. They can be as weird as possible but I am interested in Banach lattices.
Question: Let $X$ be a Banach lattice with dual isomorphic to $\ell_1$. ...
- 191
8
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0
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1k
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Strictly singular operators and their adjoints
This is a question I thought about a while back and figured I'd throw it out there to see if anyone has some insight that I am missing.
Let $X$ and $Y$ be infinite dimensional separable Banach ...
- 1,073
8
votes
0
answers
751
views
The log kernel and Bochner Theorem
I was wondering if it possible to find a measure $\eta$ on $\mathbb{R}$ such that
$$
L(x):=\log\frac{1}{|x|}=\int e^{itx}\;d\eta(t)
$$
for every $x\in [0,1/2]$.
On a structural ground, this ...
- 2,135
8
votes
0
answers
196
views
Parametrizing derivations from the algebra of smooth functions on a manifold to its dual
$\newcommand{\Der}{\operatorname{Der}}$
$\newcommand{\Real}{{\mathbb R}}$
(Disclaimer: I fear this question may be a bit too basic for MO, but in my defence I have essentially zero differential ...
- 25.8k
8
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349
views
Finding a dimension-free bound for a certain multiplier on Euclidean space
The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
- 141
8
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0
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833
views
Is there a generalization of Brouwer's fixed point theorem?
In essence, this is the same problem as in
“The generalization of Brouwer's fixed point theorem?”.
But now I am determined to be careful. The main question is
the following:
Is there any ...
- 6,891
8
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0
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302
views
In a locally contractible space can we find local bases of contractible sets whose closures are locally contractible?
In a locally contractible topological space $X$ is it possible at each point $x$ to find a local basis of contractible sets $U_i\ni x$ such that the closure of each set $\overline{U_i} \subset X$ is ...
8
votes
0
answers
605
views
convergence rate in Wiener's approximation theorem
Wiener has the following fantastic results about approximations using translation families:
Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in \mathbb{...
- 1,839
8
votes
1
answer
207
views
Subspaces of $L_p([0,1])$ whose unit ball is compact for the topology of convergence in measure
Any information about the following questions would be welcome.
I wonder whether there are (well-known or easy) closed and infinite dimensional subspaces of $L_p([0,1])$ ($1<p<\infty$) whose ...
- 81
7
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0
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269
views
Looking for the eigenfunctions of the operator $T$ on $L_2(\mathbb R^+)$ defined by $Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy$
I'm looking to find a basis of eigenfunctions (and the corresponding eigenvectors) for the operator $T$ on $L_2(\mathbb R^+)$ defined by:
$$
Tf(x)=\int_0^\infty e^{-(x+y)^2/2}f(y)\,dy
$$
This operator ...
- 109
7
votes
0
answers
250
views
Proving this function is convex
Let $C$ be a symmetric positive definite matrix such that $0\leq c_{ij} \leq 1$, $c_{ii}=1$, and define $f$ as $$f(x)=\sum_{i}x_{i}\log(\sum_{j}c_{ij}x_{j})$$ for positive vectors $x$ (in fact let's ...
- 4,049
7
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0
answers
349
views
An open set which is not the union of a closed set and a countable set
The following fact is probably a known result:
Fact. Let $X$ be an uncountable Polish space. Then there exists an open subset of $X$ which is not the union of a closed set and a countable set.
Proof:...
- 1,511
7
votes
0
answers
131
views
Approximation of a continuous curve on commuting matrices
I have a continuous curve $A:\mathbf{R}_+\rightarrow \text{M}_N(\mathbf{R})$ such that
$[A(t),A(s)] \operatorname*{\longrightarrow}_{t,s\rightarrow +\infty} 0$, where $[A(t),A(s)] = A(t)A(s)-A(s)A(t)$....
- 3,425
7
votes
0
answers
294
views
Applications of Banach space homology
There is a well-developed theory of Banach space homology. What are some of its useful applications to Banach space theory and which important questions can one answer using it? In other words, how ...
- 175
7
votes
0
answers
148
views
Is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis?
In short, the question is in the title: is there a Hausdorff space with a $\sigma$-locally finite basis but no $\sigma$-discrete basis?
A bit of context:
Given a topological space $X$, a family $\...
- 775
7
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0
answers
151
views
Stochastic analysis on nuclear Fréchet spaces
This is a reference request question, so to make it clear what I am after, I will give a quick outline of the area I am thinking in and some questions that arise.
A lot of the time in infinite-...
- 439
7
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0
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164
views
Nontrivial examples of locally compact quantum groups
What are some families of locally compact quantum groups that are neither groups, duals of groups, compact, nor discrete?
- 3,816
7
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0
answers
177
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What is the current status of research on the von Neumann's inequality for $n \ge 3$?
Problem
Let $(T_1, \ldots, T_n)$ be a tuple of commuting contractions in Hilbert space $H$.
Does a constant $C_n \ge 1$ exist, for which it would be true, that:
$$\forall_{p \in \mathbb{C}[x_1, \ldots,...
- 63
7
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0
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272
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Generalizing uniform structures as Grothendieck topologies
Recently, I was reading a classical book "Sheaves in Geometry and Logic" by S. MacLane and I. Moerdijk, and then it stroke me that, that the definition of Grothendieck Topology bears some ...
- 519
7
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150
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The space of analytic associative operations
This question is a follow-up to this old one of mine.
Let $\mathcal{A}$ be the set of functions $\star:\mathbb{R}^2\rightarrow\mathbb{R}$ which are associative and $C^\omega$ (real analytic entire) in ...
- 20.5k
7
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162
views
Relation between the additive Haar measure on $(K,+)$ and the multiplicative Haar measure on $K^{*}$ for a global field $K$
The following question comes from my studying of Alain Connes's paper Trace Formula in Noncommutative Geometry and the Zeros of the Riemann Zeta Function. In it, on p. 11, Connes notes that if $K$ is ...
- 1,805
7
votes
0
answers
123
views
Steklov eigenvalue for circle valued functions
Let $(M,g)$ be a compact Riemannian manifold with boundary. It is well known that the first positive Steklov eigenvalue $\sigma_1$ of $M$ has the following variational characterization:
$$\sigma_1(M,g)...
- 2,095
7
votes
0
answers
80
views
Given composition rules, determining whether a continuous map between smooth functions is a pseudodifferential operator
Let $M$ be a closed manifold, and let $P:C^{\infty}(M)\rightarrow C^{\infty}(M)$ be a continuous linear map in the smooth Fréchet topologies. In what is to come, if it helps, one can assume further ...
- 808
7
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0
answers
198
views
The spectrum of the Banach algebra of certain arithmetic functions under Dirichlet convolution
I was thinking about using the tools of functional analysis to study some subring of arithmetic functions under Dirichlet convolution. If I let $D_s$ be the ring of arithmetic functions with finite ...
- 190
7
votes
0
answers
120
views
What is the closed cone generated by constant and coordinate functions and closed under taking $f\mapsto\max(f,0)$?
Let $C$ be the smallest closed convex cone of functions from $\mathbb{R}^n$ to $\mathbb{R}$ that contains all constant functions, all coordinate functions, and such that $\max(f,0)\in C$ whenever $f\...
- 2,772
7
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0
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138
views
The smallest cardinality of a cover of a group by algebraic sets
$\DeclareMathOperator\cov{cov}$A subset $A$ of a semigroup $X$ is called algebraic if $$A=\{x\in X: a_0xa_1x...xa_n=b\}$$ for some $b\in X$ and $a_0,a_1,...,a_n \in X^1=X\cup \{1\}$. The smallest ...
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7
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0
answers
193
views
Reduced group C*-algebra $C^*_r(\mathbb{Z}/2*\mathbb{Z}/2)$: norm of specific elements
Consider the free product of $\mathbb{Z}/2$ with itself with generators
$$
\mathbb{Z}/2*\mathbb{Z}/2=\langle u,v\mid u^2=1=v^2\rangle
$$
and regard its group $C^*$-algebra
$$
C^*(\mathbb{Z}/2*\mathbb{...
- 598
7
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0
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2k
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Algebraizing topology and analysis via condensed mathematics
I asked this question on Mathematics Stackexchange, but one of the users suggested that I ask this question at MathOverflow.
I've just come across a Twitter thread by Laurent Fargues explaining a work ...
- 79
7
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0
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225
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A weak analogue of smooth manifolds (reformulated)
It is widely known that $C^1$ manifolds are topological spaces locally homeomorphic to Euclidean spaces and possessing $C^1$ chart-converters. They have a tangent space at every point, regarding as ...
- 1,543
7
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0
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317
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Multiple Fourier series
In the book by Elias M.Stein and Guido Weiss "Introduction to Fourier Analysis on Euclidean Spaces" one states in page 268 the following theorem:
Theorem 1: The trigonometric series
$$\...
7
votes
0
answers
493
views
A locally compact, complete metric space in which the closure of open balls coincide with the closed ball is Heine-Borel
I saw the following result stated without a proof in a paper about the isometry group of metric measure spaces:
Let $X$ be a locally compact, complete metric space such that for all $x \in X$ and $R &...
- 99
7
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0
answers
207
views
Is the derivative the unique operation on points in the plane that preserves convexity?
Let $C(n)$ be the space of multisets of size $n$ of points in the Euclidean plane, topologised appropriately, and consider a surjective continuous map: $$D:C(n)\rightarrow C(n-1)$$
Such that the ...
- 1,949
7
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0
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242
views
Has this Banach algebra been studied?
Given $\Omega$ as $[0,1]^n$ or the closed unit ball in $\mathbb{R}^n$, we can consider the algebra of complex valued polynomials with pointwise multiplication and its closure with respect to the norm
...
- 71
7
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0
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391
views
Do algebraic completion/topological completion of fields always terminate? If so, are they unique?
Take the field $\mathbb{Q}$, If we complete it topologically with respect to the Euclidean norm, we get $\mathbb{R}$, then if we complete it algebraically, we get $\mathbb{C}$.
On the other hand, the ...
- 171
7
votes
0
answers
132
views
Smoothing property of a certain singular integral operator of non-convolution type
For simplicity, suppose that the dimension $d=2$, and let $g_s(x)$ be the Coulomb or Riesz potential defined by
$$g_s(x) := \begin{cases} -\frac{1}{2\pi}\ln|x|, & {s=0} \\ c_s|x|^{-s},& {0<...
- 873
7
votes
0
answers
351
views
Fractional Laplacian and chain rule
For the classical Laplacian, we have
$$\Delta (h(u)) = h'\Delta u + h''(u)|\nabla u|^2$$
for smooth functions $h$ and $u$.
Does a similar chain rule hold (up to a reminder term) also for the ...
- 161
7
votes
0
answers
294
views
Weaker version of the Borel lemma for vector-valued functions
Borel's lemma for Frechét-spaces $V$ says:
(i) For every $(v_j)_{j \in \mathbb{N}} \in V^\mathbb{N}$ there exists a smooth $f: \mathbb{R} \to V$ such that
$$f^{(j)}(0) = v_j.$$
For general locally ...
- 1,474
7
votes
0
answers
158
views
$C^*$ algebras whose nontrivial projections form a non empty compact connected set
Apart from $M_2(\mathbb{C})$. what is an example of a $C^*$ algebra $A$ whose set of non trivial projections form a non empty compact connected set?
Is there an example of this situation such that ...
- 356
7
votes
0
answers
440
views
Bounded open sets with same boundaries
Let $U_1$, $U_2$ two bounded open subsets of the euclidean plane.
and denote by $\partial U_1$ and $\partial U_2$ their topological boundaries.
Does $\partial U_1 = \partial U_2$ implies $U_1 = U_2$?
...
- 18.7k
7
votes
0
answers
3k
views
Definition of homogeneous Sobolev spaces
As we know the inhomogeneous Sobolev space (we only consider $s>0$)
$${H}^{s}\left(\mathbb{R}^{n}\right)=\left\{f \in L^2(\mathbb{R}^n):\int_{\mathbb{R}^{n}}|\xi|^{2 s}|\hat{f}(\xi)|^{2} \mathrm{d} ...
- 633
7
votes
0
answers
239
views
Does there exist a complete metric space which is Rothberger (or Menger) but not Hurewicz?
A topological space $X$ is said to be a
Menger space if for each sequence $(\mathcal{U}_n)$ of open covers of $X$ there is a sequence $(\mathcal{V}_n)$ such that for each $n$ $\mathcal{V}_n$ is a ...
- 505
7
votes
0
answers
221
views
adding one point from the Stone-Cech compactification
Let $X$ be any non-compact Tychonoff space and $\beta X$ be its Stone-Čech compactification.
The following fact is known: any point $p$ from the reminder $\beta X \setminus X$ is not a $G_{\delta}$-...
- 71