All Questions
12,823 questions
12
votes
3
answers
2k
views
Relevance of the complex structure of a function algebra for capturing the topology on a space.
This question is the outcome of a few naive thoughts, without reading the proof of Gelfand-Neumark theorem.
Given a compact Hausdorff space $X$, the algebra of complex continuous functions on it is ...
- 3,699
15
votes
4
answers
2k
views
Naive questions about "matrices" representing endomorphisms of Hilbert spaces.
This is a very basic question and might be way too easy for MO. I am learning analysis in a very backwards way. This is a question about complex Hilbert spaces but here's how I came to it: I have in ...
- 41.4k
3
votes
0
answers
468
views
Asymptotics related to the Erdos--Moser diophantine equation
I share the authorship of this question with Pieter Moree.
In our recent joint work with Y. Gallot (arXiv:0907.1356 [math.NT]) we attack
the Erdős--Moser diophantine equation
$$
1^k+2^k+\dots+(m-...
- 13.5k
2
votes
1
answer
341
views
Is there an nontrivial function whose 'period paralellograms' are Gosper Islands?
The Gosper island tiles the plane, so I'm curious if a nontrivial elliptic? function exists which would have a 'period gosper-island' instead of a period parallelogram. In this case, I'm using '...
- 975
7
votes
3
answers
1k
views
Can Stein's maximal principle be strengthened?
Let $T$ be an operator on $S(G)$ where $G$ is the line $R$ or the circle $T$, and $S(G)$ denotes the Schwartz space of functions on $G$.
We can ask if the operator T is bounded (as an operator from $...
- 13k
22
votes
2
answers
2k
views
Examples of loss of regularity by "creation of topology"
I would like to have a list as general as possible of examples of situations where the density of smooth objects into some "natural class" (the meaning of "natural" depending on the problem considered)...
- 2,041
21
votes
5
answers
18k
views
When is Sobolev space a subset of the continuous functions?
If we let $\Omega\subset\mathbb{R}^d$ with $d=1,2,3$ and define $\mathcal{H}^1(\Omega)=(w\in L_2(\Omega): \frac{\partial w}{\partial x_i}\in L_2(\Omega), i=1,...,d)$. My tutor has repeated several ...
- 3,217
1
vote
0
answers
133
views
Square powers of hemicontinuous operators
Let H be an infinite dimensional real Hilbert space.
A [not necessarily linear] mapping of H into itself is said to be hemicontinuous if it is continuous from each line
segment of H to the weak ...
- 4,060
5
votes
1
answer
7k
views
Dual Spaces of Sobolev Spaces
I will consider Sobolev spaces with $p=2$, only, so that they are Hilbert spaces. Hence the Sobolev inner product identifies each Sobolev space with its dual. In other words, I have an isomorphism $H^...
- 807
0
votes
1
answer
437
views
Positive Derivative
Let $\;f : \; \stackrel{\circ}{D}\; \subset \mathbb{R} \to \mathbb{R}$ differentiable in $x_0 \in \; \stackrel{\circ}{D}\;$ and $f\;'(x_0) > 0$.
Does exists a neighborhood $A \subset \; \stackrel{\...
7
votes
4
answers
6k
views
The characteristic (indicator) function of a set is not in the Sobolev space H¹
Is it true that the characteristic
(indicator) function of a subset of
Euclidean space with finite positive
measure is never in the Sobolev space
$H^1 = W^{1,2}$? And if so, what is the best/easiest/...
- 1,771
2
votes
2
answers
768
views
Elementary vector measure question: what am I doing wrong?
This is an edited post of a post I made on sci.math (e.g. to fit MO markup) with
an elementary question on vector measures. Since it is almost a week and I have
received no answers, I am trying here. ...
- 1,848
3
votes
1
answer
556
views
"Radon-Nikodym theorem" for nonabsolute continuous measures
Recently, in a particular problem I was solving, I needed some kind of Radon-Nikodym theorem for measures where one of them is not necessarily absolutely continuous with respect to other.
My colleague ...
- 349
7
votes
1
answer
286
views
a.e. convergence of the powers of an operator built from rotations
Consider two numbers $a,b\in R/Z$ and some integer $p\geq 1$. Let $T:L^p(R/Z)\rightarrow L^p(R/Z)$ be the operator given by
$$T(f)(x)=1/2(f(x+a)+f(x+b))$$
For which values of $a,b$ do we have almost ...
- 18.7k
13
votes
0
answers
1k
views
Constructive aspects of Caratheodory's theorem in convex analysis
Let me paraphrase Caratheodory's theorem in a probabilistic setup:
Let $X$ be a real-valued random variable. For $k = 1, \ldots, m$, let $f_k: \mathbb{R} \to \mathbb{R}$ be a continuous function such ...
- 1,839
28
votes
6
answers
12k
views
Almost orthogonal vectors
This is to do with high dimensional geometry, which I'm always useless with. Suppose we have some large integer $n$ and some small $\epsilon>0$. Working in the unit sphere of $\mathbb R^n$ or $\...
- 18.7k
94
votes
1
answer
11k
views
The mathematical theory of Feynman integrals
It is well known that Feynman integrals are one of the tools that physicists have and mathematicians haven't, sadly.
Arguably, they are the most important such tool. Briefly, the question I'd like to ...
- 23.5k
10
votes
2
answers
1k
views
Are operators with trivial spectrum nilpotent in a sense?
Being far from analysis, I recently learned about the Invariant subspace problem and came up with the following (perhaps simple or well-known) question.
Let $H$ be a separable complex Hilbert space ...
- 32.4k
9
votes
1
answer
893
views
Perturbations of an operator that disconnect the spectrum
The following question came to me while working on a technical matter about transversality in infinite dimension, and I'm really curious to know whether it has an affirmative answer at least under ...
- 60.5k
7
votes
1
answer
1k
views
Banach spaces with a certain separability property
In Ledoux and Talagrand's "Probability in Banach Spaces", for technical reasons they frequently assume that a Banach space $B$ has the property that the unit ball of $B^*$ contains a countable subset $...
- 11.4k
1
vote
2
answers
3k
views
unit sphere is weak dense in the unit ball
As I remember the following is true:
Fact: for every infinite-dimensional normed space $X$
the unit sphere $S$ is weak-dense in the unit ball $B$.
Please help me find a reference.
Thanks in ...
- 51
11
votes
2
answers
2k
views
What's wrong with compact-open topology on the space of maps?
Given a smooth vector bundle $E$ with non-compact base, let
$\Gamma(E)$ be the space of $C^\infty$ sections equipped with compact-open $C^\infty$-topology.
I have heard that $\Gamma(E)$ is not ...
- 29.1k
9
votes
1
answer
3k
views
Sequence that converge if they have an accumulation point
I am looking for classes of sequence, that converge iff they contain a converging sub-sequence.
The basic example of such sequences are monotone sequences of real numbers.
A more interesting examples ...
- 777
6
votes
1
answer
306
views
Measurable subgroups.
Let $G$ be a compact connected topological group and let $H$ be a subgroup of $G$. Suppose that $H$ is measurable with respect to the normalised Haar measure $\mu$ on $G$. Do we necessarily have $\mu(...
- 4,015
8
votes
1
answer
1k
views
Borel(X) = \sigma(X') for X non-separable
Let $X$ be a Banach space, $X' = \mathcal{L}(X, \mathbb{K})$ its dual space. Denote by $\mathcal{B}(X)$ the $\sigma$-algebra of Borel sets and denote by $\sigma(X')$ the $\sigma$-algebra which is ...
- 1,338
6
votes
2
answers
1k
views
Quantitative questions about the size of a finite epsilon net
Let $X$ be a metric space, and let $U \subset X$ be any set. A finite set $N = N(\epsilon) \subset U$ is called a finite $\epsilon$-net of $U$ if every point of $U$ is at most a distance of $\epsilon$...
- 943
7
votes
9
answers
5k
views
Would Euler's proofs get published in a modern math Journal, especially considering his treatment of the Infinite?
I was wondering how mathematicians of today would treat, for example, Euler's proof of zeta(2).
In William Dunham's book 'Journey through Genius' ( http://www.amazon.com/Journey-through-Genius-...
- 4,829
7
votes
1
answer
2k
views
Current status of Bloch constant and Landau constant bounds
The Bloch constant B (based on a theorem introduced by André Bloch in 1925 on the maximum radius of a one-to-one disk in the image of a normalized analytic function of the unit disk, see for instance ...
- 948
1
vote
1
answer
524
views
Evaluation of the following Series
Hi there,
I was wondering if you guys could be able to find the sum of the following series:
$ S = 1/((1\cdot2)^2) + 1/((3\cdot4)^2) + 1/((5\cdot6)^2) + ... + 1/(((2n-1)\cdot2n)^2) $, in which $\{...
- 4,829
2
votes
5
answers
786
views
Statistical Data Analysis
For personal research, I'm doing some analysis on collected data and trying to develop relationships between two variables where the data is collected through a data logger. I'm hypothesising that a =...
- 23
4
votes
1
answer
985
views
weak convergence in infinite dimensional spaces
Weak convergence can be tricky when dealing with infinite dimensional spaces. For example, the usual Levy's continuity theorem does not extend readily to separable Banach spaces.
Consider a (...
- 2,133
19
votes
5
answers
16k
views
What does "kernel" mean in integral kernel?
In functional analysis, there is the term "integral kernel". Examples are Possion kernel, Dirichlet kernel etc.
In algebra, the term kernel of a homomorphism refers to the inverse image of the zero ...
user2529
9
votes
3
answers
4k
views
Are there sigma-algebras of cardinality $\kappa>2^{\aleph_0}$ with countable cofinality?
A standard homework in measure theory textbooks asks the student to prove that there are not countably infinite $\sigma$-algebras. The only proof that I know is via a contradiction argument which ...
user2529
6
votes
0
answers
2k
views
Weak lower semi-continuity
Which conditions assure the weak lower semicontinuity of, say, an integral functional of the type
$F(u):=\int_\Omega f(u(x),Du(x))dx$ on $W^{1,2}(\Omega,\mathbb{R}^N)$ for a bounded, if you will even ...
7
votes
1
answer
1k
views
Is the absolutely continuous image of a nowhere dense set is also nowhere dense?
Let $f: [a, b] \subseteq \mathbb{R} \to \mathbb{R}$ be an absolutely continuous map. Does $f$ map a nowhere dense subset of $[a, b]$ to a nowhere dense set?
Remarks:
The answer is "no" if $f$ is ...
- 5,339
44
votes
1
answer
1k
views
Microwaving Cubes
First a little background. Microwaves do not heat uniformly. To help overcome this, your food is rotated, however this is not usually sufficient to produce totally uniform heating. Informally, this is ...
- 3,165
2
votes
2
answers
3k
views
Multiple outliers for two variable linear regression
Problem
Visually, the "extreme" outliers in the following graph are somewhat obvious:
Question
Given:
T - Set of all temperatures
Y - Set of all years
ΣT - Sum of temperatures.
ΣY - Sum of years.
...
- 43
39
votes
8
answers
13k
views
Can Cantor set be the zero set of a continuous function?
More generally, can the zero set $V(f)$ of a continuous function $f : \mathbb{R} \to \mathbb{R}$ be nowhere dense and uncountable? What if $f$ is smooth?
Some days ago I discovered that in this proof ...
- 5,339
12
votes
3
answers
1k
views
Drawing conclusions by NOT using AC.
The existence of non-measurable subsets and functions on $\mathbb{R}$ require the use of the axiom of choice. That is, there exist models of ZF in which all subsets of (and hence all functions defined ...
- 4,790
7
votes
2
answers
2k
views
What is the "Krein-Milman theorem for cones"?
Update: The question is completely answered. I had overlooked a reduction to the self-adjoint case, and the latter can be proved using a Hahn-Banach separation theorem. Thanks to Matthew Daws for ...
- 7,329
1
vote
1
answer
613
views
What is the value of the regularized incomplete beta function at x=0.5?
What is $I_{0.5}(a,b)$ where I is the regularized incomplete beta function?
- 598
16
votes
4
answers
11k
views
Fourier transform of Analytic Functions
Forgive me if this question does not meet the bar for this forum. But i would really appreciated some help.
I'm trying to construct a function according to some conditions in the frequency domain of ...
- 317
7
votes
2
answers
808
views
Is a subspace with a certain property dense in the dual of a vector space?
Suppose we have a normed vector space $V$ and its dual $V^*$, and suppose that $X \subseteq V^*$ has the property that for every $v \in V$, there is some $\phi \in X$ with $\Vert \phi \Vert = 1$ such ...
- 73
8
votes
1
answer
1k
views
What is exactly the (singularity) confinement property ?
This property seems to be used both in the context of differential equations and several kinds of discrete equation systems or automata.
It seems to be related in certain case to the Painlevé ...
- 948
18
votes
3
answers
3k
views
How did Gauss discover the invariant density for the Gauss map?
The Gauss map is defined on $(0,1)$ by the formula
$$
f(x)=\frac1x-\Big\lfloor\frac1x\Big\rfloor
$$
Then the density
$$
\rho(x)=\frac{1}{\log2(1+x)}
$$
is $f$-invariant.
It appeared in Gauss' diary. ...
- 1,414
11
votes
1
answer
654
views
Nonseparable Hilbert spaces as quotients of spaces of bounded functions
Is the following result true: the Hilbert space $\ell^{2}\left(2^{\Gamma}\right)$ is a quotient of $\ell^{\infty}\left(\Gamma\right)$ for any
uncountable $\Gamma$ ? [I think it is, but cannot remember ...
- 4,060
7
votes
1
answer
347
views
Nonexistence of determinantal functional equation for $\arccos$
Suppose I have distinct real numbers $a_i \in [-1,1]$, $i \in [k]$. I want to choose real numbers $b_j, j\in [k]$ such that the matrix $(\arccos(a_i b_j))_{i,j \in [k]}$ is nonsingular.
Is this ...
- 1,021
0
votes
1
answer
635
views
Topological dual and the notions of "smaller" and "larger" than...
Hi,
I've read this sentence but I can not understand what it means
[...] $\Phi'$ is the topological dual of some dense space $\Phi$ of $H_{aux}$ [...] Notice that the choice of $\Phi$ is subject to ...
- 733
5
votes
1
answer
403
views
Local form of a real-analytic function taking values in a Banach space
Let $B$ be an infinite-dimensional Banach space, and let $M\subset\mathbb{R}^n$ be a neighborhood of the origin in $\mathbb{R}^n$.
Suppose that $I:M\to B$ is a real-analytic function with $I(0)=0$ ...
- 6,548
6
votes
2
answers
4k
views
Bounded and weakly bounded sets in top. vector spaces
Consider a locally convex topological vector space V over the complex numbers. Is it true that every weakly bounded subset of V is indeed bounded? If not, what additional requirements are needed for ...
- 61