All Questions
12,823 questions
1
vote
1
answer
832
views
Differentiate under integral sign for iterated integral?
This is a bit of a trivial question, but as I don't know the answer immediately I thought I'd just ask.
Given the integral $\int_{0}^{t} \int_{0}^{t} f(x,x') dx dx'$, what is $\frac{\partial}{\...
- 11
0
votes
1
answer
634
views
How to invert this series?
Hi guys,
I'm working on a problem where I ended up with the following series:
$z(Q) = \exp(-Q) [ 1 + \frac{a_1}{Q} + \frac{a_2}{Q^2} + \ldots]$ valid around $Q \to \infty$
Is there a systematic ...
4
votes
3
answers
1k
views
De Rham cohomology and antiderivatives
A couple of recent questions about antiderivatives reminded me of the following, which I can't recall seeing tackled explicitly anywhere to my satisfaction and that I sketched to an ambitious calculus ...
- 15.4k
12
votes
4
answers
1k
views
Topologizing free abelian groups
For any set $S$ one can consider the free abelian group $\mathbb{Z}[S]$ generated by this set. Now suppose, there is a topology on $S$ given. Is it possible to find a topology on $\mathbb{Z}[S]$ in ...
- 11.1k
4
votes
1
answer
1k
views
When can a partial isometry $u$ in $\mathcal B(H \otimes K)$ be extended to a unitary in $1 \otimes \mathcal B(K)$?
Let $H$ and $K$ be Hilbert spaces, and let $u$ be a partial isometry in $\mathcal{B}(H \otimes K)$ between projections $p_0 = u^\ast u$ and $p_1 = u u^\ast$ such that $p_0, p_1 \leq 1 \otimes (1-q)$ ...
- 1,199
14
votes
3
answers
3k
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The difference between $l^1(G)$ and the reduced group $C^*$ algebra $C_r^*(G)$
Let $G$ be a group and $l^2(G)$ the Hilbert space on $G$. The complex group algebra $CG$ can be imbedded in $B(l^2(G))$, the set of all bounded linear operators, by left translation. The reduced group ...
- 1,373
3
votes
5
answers
1k
views
Approximate solutions for trisecting the angle or squaring the circle
Hello all, it is well-known by transcendence results or Galois theory that famous geometric problems such as trisecting an angle or "squaring the circle" (i.e. given a disk of radius 1 construct a ...
- 3,595
6
votes
2
answers
497
views
Can I detect the point of impact without looking at it?
I'm going to postpone the motivation for this question because the question itself involves no complicated maths and may well have a very simple solution so I don't want to put anyone off with high ...
- 26.8k
27
votes
4
answers
2k
views
Do Abel summation and zeta summation always coincide?
This is a more focused version of Summation methods for divergent series.
Let $a_n$ be a sequence of real
numbers such that $\lim_{x \to 1^{-}}
> \sum a_n x^n$ and $\lim_{s \to 0^{+}}
> \...
- 156k
1
vote
1
answer
994
views
On the convolution of generalized functions
It is provable that $f_\lambda\to f\Rightarrow f_\lambda*g\to f*g$ if $g$ has a compact support (shown in my textbook). In my particular case, $g=u(t+\triangle t)-u(t-\triangle t)$. Does for that ...
- 1,524
7
votes
4
answers
9k
views
Good example of a non-continuous function all of whose partial derivatives exist
What's a good example to illustrate the fact that a function all of whose partial derivatives exist may not be continuous?
- 1,479
14
votes
6
answers
2k
views
Finding questions between functional analysis and set theory
Are there some good questions on functional analysis whose solution depends on tools in set theory? My major is mathematical logic, I think tools in set theory, especially infinity combinatorics and ...
- 315
22
votes
6
answers
12k
views
Smooth dependence of ODEs on initial conditions
The following is a theorem known to many, and is essential in elementary differential geometry. However, I have never seen its proof in Spivak or various other differential geometry books.
Let $t_0$ ...
- 221
-6
votes
1
answer
434
views
On the extension of a limit [closed]
We know that $\lim_{p\rightarrow\infty}\left\Vert \left(x_{1},\cdots,x_{n}\right)\right\Vert _{p}=\max\left\{ \left|x_{1}\right|,\cdots,\left|x_{n}\right|\right\} =:\left\Vert x\right\Vert _{\infty}$
...
- 65
32
votes
0
answers
2k
views
$f\circ f=g$ revisited
This may be related to solving $f(f(x))=g(x)$. Let
$C(\mathbb{R})$ be the linear space of all continuous functions from
$\mathbb{R}$ to $\mathbb{R}$, and let $\mathcal{S}:=\{g\in C(\mathbb{R}) ; \...
- 4,060
20
votes
7
answers
2k
views
Extensional theorems mostly used intensionally
Some theorems are stated and proved extensionally, but in practice are almost always used intensionally. Let me give an example to make this clear -- integration by parts:
$$ \int_a^b f(x)g'(x)ds = \...
- 11.8k
0
votes
3
answers
248
views
how slow can the dimension of a product set grow?
Let us define the following "dimension" of a Borel subet $B \subset \mathbb{R}^k$:
$\dim(B) = \min\{n \in \mathbb{N}: \exists K \subset \mathbb{R}^n, ~{\rm s.t.} ~ B \sim K\}$,
where $\sim$ denotes "...
- 1,839
14
votes
0
answers
2k
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Schwartz kernel theorem for A-linear operators
Let $X,Y \subset \mathbb{R}^n$ be open subsets. Denote by $C^\infty(X)$ the smooth functions on $X$, let $\mathcal{E}'(Y)$ be its dual space considered as a space of distributions. Let $L(C^\infty(X), ...
- 7,637
2
votes
2
answers
584
views
A proof about an unconditional basis theorem
Hello everyone. I'm in a little trouble trying to find the proof of a theorem stated by W. T. Gowers. It is the Lemma 1.6 in his article 'An infinite Ramsey theorem and some Banach space dichotomies' (...
- 105
11
votes
1
answer
2k
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Spectral theory for self-adjoint field operators on a symmetric Fock space
Background
Suppose we have a finite-dimensional Hilbert space $H = \mathbb{C}^s$ (for a natural number s) and we construct the symmetric (or bosonic) Fock space built from it: $$F(H):= \mathbb{C} \...
- 195
77
votes
0
answers
4k
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2, 3, and 4 (a possible fixed point result ?)
The question below is related to the classical Browder-Goehde-Kirk fixed point theorem.
Let $K$ be the closed unit ball of $\ell^{2}$, and let $T:K\rightarrow K$
be a mapping such that
$$\Vert Tx-Ty\...
- 4,060
12
votes
3
answers
2k
views
Growth of the "cube of square root" function
Hello all, this question is a variant (and probably a more difficult one)
of a (promptly answered ) question that I asked here, at Is it true that all the "irrational power" functions are ...
- 3,595
9
votes
5
answers
3k
views
Geometric group theory and analysis
Geometric group theory is mainly concerned with topological and geometric properties of groups, spaces on which they act etc., so the ideas employed in GGT are mainly algebraic/geometric/topological. ...
- 2,449
2
votes
3
answers
4k
views
Show a linear operator is not compact
For $f\in L^2(0,\infty),$ define $(Tf)(x)=x^{-1}\int_0^x f(s)ds,$ for $x\in(0,\infty),$ then from hardy's inequality, $T\in B(L^2),$ my question is how to show that $T$ is not compact?
- 187
8
votes
1
answer
381
views
Estimating flat norm distance from a planar disc
Let $D\subset\mathbb R^2\subset\mathbb R^n$ be a unit planar disc in $\mathbb R^n$. Let $S$ be an orientable two-dimensional surface in $\mathbb R^n$ such that $\partial S=\partial D$. Of course, we ...
- 32.4k
50
votes
7
answers
16k
views
Way to memorize relations between the Sobolev spaces?
Consider the Sobolev spaces $W^{k,p}(\Omega)$ with a bounded domain $\Omega$ in n-dimensional Euclidean space. When facing the different embedding theorems for the first time, one can certainly feel ...
- 2,935
25
votes
6
answers
15k
views
Does every distribution define a Radon measure?
On the one hand, Wikipedia suggests that every distribution defines a Radon measure:
http://en.wikipedia.org/wiki/Distribution_(mathematics)#Functions_as_distributions (revision from February 2010, ...
- 2,895
5
votes
2
answers
410
views
How to find a solution to a particular Bottcher equation
Functional equations of the form $f(g(x))=(f(x))^p$, where $g(x)$ is known, is called Bottcher equation. Generally, we have only asymptotic formula for the solution $f(x)$ under certain conditions. In ...
- 1,858
2
votes
1
answer
271
views
Can we find an l-2 sequence if we know all l-p norms?
I'm wondering if there is a way to approximate the first $M$ terms of a non-increasing $\ell^2$ sequence $\{c_n\}$ if we know
$|c|_p^p = \sum c_n^p$ for $p=2,3,4,\dots$?
I've tried truncating the ...
- 143
123
votes
12
answers
29k
views
How to solve $f(f(x)) = \cos(x)$?
I found the following equation on some web page I cannot remember, and found it interesting:
$$f(f(x))=\cos(x)$$
Out of curiosity I tried to solve it, but realized that I do not have a clue how to ...
- 1,571
5
votes
1
answer
495
views
On the failure of the infinite dimensional Brouwer Theorem
Let $K$ be the closed unit ball of some infinite dimensional Banach
space, and let $H$ be an autohomeomorphism of $K$, having fixed
points. Can $H/2$ be fixed point free ?
Also, let ${\mathcal{F}}$ :=...
- 4,060
12
votes
1
answer
1k
views
Path integrals, localisation
Physicists use the "Atiyah-Bott formula" for path "integrals" (for instance the supersymmetric proof of the Atiyah-Singer index theorem. Is there some way to make atleast some of these ideas rigorous? ...
- 3,383
5
votes
1
answer
1k
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Orthogonal complements in Hilbert bundles
It's a standard fact that for a finite-dimensional vector bundle with an inner product, the othogonal complement of any subbundle is itself a locally trivial vector bundle.
What is known about the ...
- 8,803
1
vote
2
answers
3k
views
What does Gibbs phenomenon shows the nature of Fourier Series
As the title shows,we know that there is some points the series not approaching to the function.
Now,take the convergence theorem into consideration.As there is some the first break-points，the series ...
- 57
0
votes
1
answer
1k
views
How to prove that rational functions satisfy a Lipschitz condition in the *chordal metric*?
How can one show that rational functions satisfy a Lipschitz condition in the chordal metric on the Riemann sphere?
user4324
1
vote
1
answer
359
views
the Cech-cohomology of the sheaf of germs of plurisubharmonic functions defined on a domain in C^n
we all know that if we consider the sheaf of germs of a holomorphic functions defined on a domain in C^n,we have too many beautiful theorems characterizing the geometry of the domain by consider the ...
- 399
13
votes
1
answer
2k
views
Hausdorff Dimension and Hölder Continuity
Suppose we have a curve γ : [0,1] -> ℝn. It is well known that if this curve is Hölder continuous for some exponent α then the Hausdorff dimension of γ[0,1] is bounded above ...
- 183
6
votes
1
answer
1k
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Reference needed for: every idempotent in a C*-algebra is similar to a hermitian one
The result stated in the title is thoroughly standard - or that's the impression I got.
I seem to remember seeing it stated somewhere in a book I was reading in the library, and then reverse-...
- 25.8k
12
votes
1
answer
3k
views
Lower bounds on (truncated) Fourier transform of functions of constant modulus and bounded derivative
Let $f(x)=e^{i\phi(x)}$ define a function from $[0,1]$ to the complex unit circle through the real smooth function $\phi(x)$. Also, this function is periodic: $\phi(0)=\phi(1)=0\text{ mod }2\pi$ and ...
- 731
6
votes
4
answers
970
views
An identity for the cosine function
Let $x = \pi/(2k+1)$, for $k>0$.
Prove that
$$
\cos(x)\cos(2x)\cos(3x)\dots\cos(kx) = \frac{1}{2^k}
$$
I've confirmed this numerically for $n$ from $1$ to $30$.
I'm finding it surprisingly ...
- 581
5
votes
1
answer
467
views
Info about Elton–Odell theorem
Hello everyone, could anyone please tell me where can I find information about the Elton–Odell theorem?
It states:
For any infinite dimensional Banach space $X$ there is a $q > 1$ so that $X$ ...
- 105
4
votes
2
answers
2k
views
Convergence of Gaussian measures
Let $X$ be a separable Banach space with its Borel $\sigma$-algebra $\mathcal F$. Let $x_n \to x$ in $X$. Fix a Gaussian covariance operator $K$, and let $\mathbb P_n$ and $\mathbb P$ be Gaussian ...
- 8,512
64
votes
16
answers
13k
views
How helpful is non-standard analysis?
So, I can understand how non-standard analysis is better than standard analysis in that some proofs become simplified, and infinitesimals are somehow more intuitive to grasp than epsilon-delta ...
- 32.1k
41
votes
6
answers
87k
views
Fourier vs Laplace transforms
In solving a linear system, when would I use a Fourier transform versus a Laplace transform? I am not a mathematician, so the little intuition I have tells me that it could be related to the boundary ...
- 411
17
votes
1
answer
2k
views
Which Fréchet manifolds have a smooth partition of unity?
A classical theorem is saying that every smooth, finite-dimensional manifold has a smooth partition of unity. My question is:
Which Fréchet manifolds have a smooth partition of unity?
How is the ...
- 4,519
1
vote
2
answers
819
views
what is summation in the sense of a principal value?
In one paper I saw this equality:
$$\sum_{\eta=-\infty}^{\infty}\frac{z}{(z+\eta)}=\pi z\cot(\pi z)$$
which is the same as
$$\sum_{\eta=-\infty}^{\infty}\frac{1}{(z+\eta)}=\pi \cot(\pi z)$$
where ...
- 267
3
votes
0
answers
404
views
Wolff's application of CS to analysis
In the foreword of Tom Wolff's "Lectures on Harmonic Analysis", C. Fefferman writes "[Wolff made] (as far as I know) the first serious application of theoretical computer science to analysis." What ...
- 13k
13
votes
1
answer
1k
views
Which functions are Wiener-integrable?
I'm looking for either a few precise mathematical statements about Wiener integrals, or a reference where I can find them.
Background
The Wiener integral is an analytic tool to define certain "...
- 54.6k
15
votes
3
answers
2k
views
Disintegrations are measurable measures - when are they continuous?
This is a sequel to another question I have asked.
The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probability spaces; ...
- 8,512
4
votes
2
answers
1k
views
Can we extract information about how fast a function decay from its Laplace transform?
My question is whether we can extract information about how fast an integrable function converges to zero by looking at the asymptotics of its Laplace transform.
More concrete case, let $f:\mathbb{R} ...
- 1,839