All Questions
12,823 questions
8
votes
2
answers
915
views
Group homomorphisms and maps between function spaces
Let G and H be locally compact groups, and let $\theta:G\rightarrow H$ be a continuous group homomorphism. This induces a *-homomorphism $\pi:C^b(H) \rightarrow C^b(G)$ between the spaces of bounded ...
- 18.7k
54
votes
13
answers
90k
views
Good differential equations text for undergraduates who want to become pure mathematicians
Alright, so I have been taking a while to soak in as much advanced mathematics as an undergraduate as possible, taking courses in algebra, topology, complex analysis (a less rigorous undergraduate ...
2
votes
1
answer
362
views
Are these two notions of Lipschitz hypersurface equivalent?
Let $S$ be a subset of $\mathbb{R}^n$. I would like to call $S$
a Lipschitz(1) hypersurface if for every $x\in S$ there is a hyperplane $H$ so that the orthogonal projection onto $H$ is a bi-...
- 9,366
25
votes
1
answer
8k
views
Convergence of Fourier Series of $L^1$ Functions
I recently learned of the result by Carleson and Hunt (1968) which states that if $f \in L^p$ for $p > 1$, then the Fourier series of $f$ converges to $f$ pointwise-a.e. Also, Wikipedia informs me ...
- 871
5
votes
3
answers
565
views
Reference/Introduction to partial difference(NOT differential!) equations
The title says it all. Despite heavy googling I have not been able to find anything. What I am interested in, is theory (maybe modelling), not for the moment finite difference methods as ...
- 2,600
19
votes
6
answers
8k
views
Unbounded operator bounded in a dense subset
Let $X, Y$ be normed vector spaces, where $X$ is infinite dimensional. Does there exist a linear map $T : X \rightarrow Y$ and a subset $D$ of $X$ such that $D$ is dense in $X$, $T$ is bounded in $D$ (...
- 783
2
votes
4
answers
1k
views
An inequality question
Let $M$ be a $3\times2$ matrix. Is it true that for any $x\in\mathbb{R}^{2}$
with $\left\Vert x\right\Vert _{3}=1$ there is some subspace $V$
with dimension $2$ of $\mathbb{R}^{3}$, such that $\left\...
- 59
6
votes
1
answer
354
views
Positivity of "harmonic" summation
The settings for the problem are as follows. Given
a real number $\alpha\in[0,1]$, consider
a sequence of real (positive, negative and zero) numbers
$a_1,a_2,\dots,a_n,\dots$ satisfying
(1) $a_1=1$,
...
- 13.5k
8
votes
1
answer
713
views
Factoring operators $L_\infty \longrightarrow L_2$ as the composition of $n$ strictly singular operators, $n\in \mathbb{N}$
Motivation and background This question is motivated by the problem of classifying the (two-sided) closed ideals of the Banach algebra $\mathcal{B}(L_\infty)$ of all (bounded, linear) operators on $L_\...
- 2,378
6
votes
1
answer
578
views
Differential operators preserving the space of harmonic functions (aka higher symmetries of the Laplacian)
The article http://arxiv.org/abs/hep-th/0206233 (published in Ann. of Math. (2) 161 (2005), no. 3) deals with linear differential operators $D$ for which there exists another linear differential ...
- 8,597
7
votes
2
answers
948
views
Uniform variant of Stirling's approximation
Stirling's formula is usually stated in the form $\log \Gamma(s) = (s-\frac12) \log{s} - s + \log\sqrt{2\pi} + E(s)$, where
$E(s) = c_1/s + c_2/s^2 + \dots + O(s^{-K})$ for certain absolute ...
- 4,671
0
votes
1
answer
357
views
Questions on supremum
Is it true that $$\sup_{x\in\mathbb{R}}\frac{\left|\left(1+s\right)+tx\right|+\left|\left(1+t\right)x+s\right|}{1+\left|x\right|+\left|s+tx\right|}\geq1$$
for all $s,t\in\mathbb{R}$? Is it also true ...
- 65
4
votes
1
answer
311
views
Continuous functions on the states of a C*-algebra and its elements
Let $\mathcal A$ be a C*-algebra and $s(\mathcal A)$ the set of states on $\mathcal A$, with the weak* topology, as a subspace of the dual space. Suppose $f: s(\mathcal A) \to \mathbb C$ is a ...
- 41
152
votes
18
answers
24k
views
Why do we care about $L^p$ spaces besides $p = 1$, $p = 2$, and $p = \infty$?
I was helping a student study for a functional analysis exam and the question came up as to when, in practice, one needs to consider the Banach space $L^p$ for some value of $p$ other than the obvious ...
5
votes
1
answer
681
views
Does the norm of a normed linear space determine the form of its dual spaces elements?
Hello everybody,
As an introductory example, suppose $U \subset R^n$ is open and bounded, let $p = 2$. Then there is a constant $c>0$ s.t. $\forall u \in W^{1,p}_0 : \Vert u \Vert _ {W^{1,p}_0} \...
- 5,327
2
votes
2
answers
332
views
roots of recursive polynomials
I have recursive polynomials
$$Q_{n}(t)=tQ_{n-1}(t)+\frac{1-t^{2}}{n+1}Q'_{n-1}(t)$$
and
$$Q_{0}(t)=1$$
Is there a theory for finding a factorisation of recursive polynomials?
It is possible to ...
- 267
-3
votes
2
answers
260
views
On \ell_3 norm in R^2
Let $v,w\in\mathbb{R}^{2}$ and $v\perp w$. Is it true that $\left\Vert v\right\Vert _{3}\leq\left\Vert v+w\right\Vert _{3}$,
in which $\left\Vert \left(x,y\right)\right\Vert _{3}:=\sqrt[3]{\left|x\...
- 65
21
votes
3
answers
3k
views
When is $n/\ln(n)$ close to an integer?
As usual I expect to be critisised for "duplicating"
this question. But I do not! As Gjergji immediately
notified, that question was from numerology. The one I ask you here
(after putting it in my ...
- 13.5k
8
votes
3
answers
2k
views
Finitely additive translation invariant measure on $\mathcal P(\mathbb R)$
We know that a countably additive translation invariant measure with $\mu([0,1]) = 1$ cannot be defined on the power set of $\mathbb R$. This is because $[0,1]$ can be partitioned into countably many ...
- 2,745
1
vote
2
answers
2k
views
Fourier series of B-spline
The Fourier series of a function (B-spline) is given by:
$$s(x)=\sum_{j=-\infty}^{\infty}\operatorname{sinc}\Bigl[\pi\frac{j}{K}\Bigr]^{p}\exp[2\pi ijx]$$
But the B-spline has only finite support. How ...
- 267
2
votes
2
answers
679
views
L^2 space of holomorphic functions with given weight
Hi folks, what is known about the $L^2$ space of holomorphic functions of 1 complex variable with the scalar product
$\langle f, g \rangle = \int dzd{\bar z} \frac{ {\bar f(z)} g(z) }{(1 + z{\bar z})^...
- 362
13
votes
5
answers
1k
views
Does this sequence span $L^2$?
Consider the following sequence of functions in $L^2[0,\infty)$:
$$f_n(x)=e^{-x/n}x^n,\;\;n\geq 1$$
Does this sequence span $L^2[0,\infty)$ (that is, is the set of finite linear combinations
of these ...
- 520
1
vote
3
answers
658
views
Weighted Hardy Inequality for bounded domains
Hi,
I need an inequality similar to this one for bounded domain [0,L].
http://img94.imageshack.us/img94/3166/screenshot1qy.png
My u(x) is not 0 on the boundary.
I will appreciate if you can help me ...
- 13
2
votes
1
answer
272
views
Contractions and spaces
Suppose $X$ is a closed subspace of an $L^{1}$-space and $X$ is isometric to another $L^{1}$-space. Then we know that $X$ is in the range of a contractive projection on the $L^{1}$-space. Is there any ...
- 95
3
votes
3
answers
1k
views
Harmonic Functions
Suppose $f: \mathbb{R}\times\mathbb{R}\to\mathbb{R}$ is has continuous partial derivatives and
$$4f(x,y)=f(x+\delta,y+\delta)+f(x-\delta,y+\delta)+f(x-\delta,y-\delta) + f(x+\delta,y-\delta)$$
for ...
- 189
25
votes
4
answers
5k
views
Why is $ \frac{\pi^2}{12}=\ln(2)$ not true?
This question may sound ridiculous at first sight, but let me please show you all how I arrived at the aforementioned 'identity'.
Let us begin with (one of the many) equalities established by Euler:
...
- 4,829
5
votes
1
answer
623
views
For which values of $N$ is known the Lieb-Simon Inequality for $Z_N$ Models ?
Background:
Let $\mathbb Z^d$ denote the $d$-dimensional integer lattice with norm $|x|=\sum_i|x_i|$.
For each $x\in\mathbb Z^d$ we associate a spin variable, $\sigma_x$ taking values on the set
$\{...
- 2,044
4
votes
1
answer
466
views
Injection between non-isomorphic irreducible Hilbert space reps?
I must be missing something trivial here.
Let $G$ be, say, a reductive Lie group (or more generally any locally compact Hausdorff unimodular topological group). A unitary Hilbert space representation ...
- 41.4k
8
votes
3
answers
1k
views
When does a unitary Hilbert space rep of a reductive Lie group decompose into a direct sum of irreps with finite multiplicities?
I'm giving some lectures on the trace formula. Here's something I proved in the last lecture. Let $G$ be a locally compact Hausdorff unimodular topological group (e.g. a reductive Lie group), let $\...
- 41.4k
3
votes
1
answer
2k
views
A formula for the Jacobian of a flow
Let $U : \mathbb R^d \to \mathbb R^d$ be a smooth vector field, and let $F_t : \mathbb R \times \mathbb R^d \to \mathbb R^d$ be the corresponding smooth flow, defined by the differential equation $$\...
- 8,512
81
votes
3
answers
9k
views
Norms of commutators
If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\...
- 31.5k
4
votes
1
answer
2k
views
Existence of weak limits
Background: Three months ago, I asked this question, which is a bit related to the following (if the answer to it was Yes, then answer to this one would be Yes too, but since that was a No, it still ...
27
votes
1
answer
4k
views
Polynomials with rational coefficients
Long time ago there was a question
on whether a polynomial bijection $\mathbb Q^2\to\mathbb Q$ exists. Only one attempt
of answering it has been given, highly downvoted by the way. But this answer isn'...
- 13.5k
27
votes
1
answer
4k
views
Criteria for boundedness of power series
Consider a power series $\sum_{n=0}^{\infty} a_n x^n$ that is convergent for all real
x, thus defining a function $f: \mathbb{R} \to \mathbb{R}$.
Can one give necessary and sufficient criteria the ...
- 4,014
6
votes
2
answers
1k
views
Is perfect play possible in continuous rock-paper-scissors? game "step size" vs. "acceleration"
The first part of my question is simple: Is every game continuous in time and strategy-space also a game of perfect information with a good equilibrium? For example, consider rock-paper-scissors. The ...
- 2,383
37
votes
15
answers
13k
views
Geometric imagination of differential forms
In order to explain to non-experts what a vector field is, one usually describes an assignment of an arrow to each point of space. And this works quite well also when moving to manifolds, where a ...
- 2,041
17
votes
12
answers
5k
views
Looking for an interesting problem/riddle involving triple integrals.
Does anyone know some good problem in real analysis, the solution of which involves triple integrals, and which is suitable for second semester Analysis students?
Thanks!
- 459
0
votes
1
answer
250
views
asymptotic value
I have a more or less stupid problem with an asymptotic value of an integral.
consider
$\int_{\infty}^{Q} dx$ $\exp[-4x-a/x] \int_{\infty}^{x} dy$ $\exp[-2y -a/y]$ in the limit $Q \rightarrow \infty.$...
- 19
9
votes
0
answers
412
views
min/max of degenerate critical points and Newton diagrams
Given a smooth function of several variables, whose first derivatives vanish at the origin. Suppose the matrix of second derivatives is degenerate at the origin. For example all the second derivatives ...
- 2,232
66
votes
7
answers
10k
views
Why is the Hahn-Banach theorem so important?
Every time I hear it mentioned it is praised in the highest possible terms, and I remember one of my old lecturers saying that it is one of the 3 most important theorems in analysis. Yet the only ...
- 4,351
1
vote
1
answer
1k
views
Besicovitch Covering Constant for R^1
In the case where $E\subset\mathbb{R}^1$, a Besicovitch cover of $E$ is a cover by open intervals such that each point of $E$ is the center of some interval in the cover.
The Besicovitch Covering ...
- 111
1
vote
1
answer
433
views
Intersection of ideals in C*-algebra or even rings in general
Dear all,
here is a question that has been bothering me. It goes without saying that I would appreciate any help in answering it.
Let {I_k} be a countable sequence of two sided closed ideals in a C*-...
7
votes
3
answers
495
views
Noninteger iterates of functions: How to get ODE from flow at a given time?
Suppose you have the autonomous ordinary differential equation $dx(t)/dt = f(x(t))$ with $x: \mathbb{R} \to \mathbb{R}$ and the initial condition $x(0)=x_0$. Then, assuming some regularity conditions, ...
- 4,014
4
votes
2
answers
4k
views
Proof of Young's convolutions inequality for a general measure on $\mathbb R^d$
Is Young's inequality true for an arbitrary measure on $\mathbb R^d$? If so, where can I find a proof of it? In particular, where can I find the proof of the discrete version (i.e the version for $\...
- 2,745
6
votes
1
answer
508
views
Estimating the flow when we know the vector field
Suppose we have a $C^k$ vector field $v$ and let $\phi_t$ be the corresponding flow. I have estimates on $v$ and its derivatives: $|v| < C_0$, $|Dv| < C_1$, $|D^2v| < C_2$, ... $|D^kv| < ...
- 303
5
votes
2
answers
952
views
Good references for analytic solutions to nonlinear ordinary differential equations?
I am faced with a non-autonomous initial value problem for a function $x:[0,\infty) \to \mathbb{R}^2$ of the form
$$ x'(t) = f(t,x(t)) $$
for $f: [0,\infty) \times \mathbb{R}^2 \to \mathbb{R}^2$ with ...
- 33.3k
15
votes
2
answers
1k
views
Asymptotic approximation of $x^\alpha$ by entire functions
Given a non-integral real $\alpha$, is there an entire (see http://en.wikipedia.org/wiki/Entire_function) function $h(x)$ such that $x^{-\alpha}h(x)\longrightarrow 1$
for $x\rightarrow+\infty$ (with $...
- 17.9k
4
votes
2
answers
734
views
Analyzing the solution to a second-order, non-linear ODE
Let $\psi : [0,\infty] \to \mathbb R$ be a strictly positive, continuously differentiable function, and consider the non-linear ODE $$\ddot x = - \frac{1}{4} \frac{\psi'(x)}{\psi(x)} \left( \dot x^2 - ...
- 8,512
13
votes
8
answers
3k
views
Evaluation of the following series... $S = 1/(2\times3) + 1/(5\times6) + 1/(7\times8) + 1/(10\times11) + ... $
EDIT, Will Jagy, December 8, 2010: to anyone considering working on this, please first see http://mathoverflow.tqft.net/discussion/817/could-a-few-moderators-please-remove-one-of-my-questions/#Item_9 ...
- 4,829
0
votes
1
answer
625
views
A derivative of sorts?
Suppose $f$ is a continuous function of infinitely many real variables, and that 0 is an "identity element" for $f$ in the sense that
$$ f(0,\alpha,\beta,\gamma,\dots) = f(\alpha,\beta,\gamma,\dots). ...
