All Questions
12,823 questions
6
votes
2
answers
1k
views
Definable collections of non measurable sets of reals
Is there a definable (in Zermelo Fraenkel set theory with choice) collection of non measurable sets of reals of size continuum? More verbosely: Is there a class A = {x: \phi(x)} such that ZFC proves "...
- 9,631
7
votes
1
answer
2k
views
Why is 3 a bad constant in the Vitali covering lemma?
Hi,
recently I had to do with the Hardy-Littlewood maximal function and we used there the Vitali covering lemma with constant 5. Then, given an advice, I proved it with constant k>3. But I cannot ...
- 93
99
votes
28
answers
14k
views
Probabilistic proofs of analytic facts
What are some interesting examples of probabilistic reasoning to establish results that would traditionally be considered analysis? What I mean by "probabilistic reasoning" is that the approach should ...
- 1,695
5
votes
1
answer
514
views
Request for reference: Banach-type spaces as algebraic theories.
Sparked by Yemon Choi's answer to Is the category of Banach spaces with contractions an algebraic theory? I've just spent a merry time reading and doing a bit of reference chasing. Imagine my delight ...
- 26.8k
10
votes
1
answer
635
views
What's the nearest algebraic theory to inner product spaces?
Following the references to the accepted answer to Is the category of Banach spaces with contractions an algebraic theory? one discovers that there is an algebraic theory (infinitary) which is closely ...
- 26.8k
12
votes
2
answers
812
views
Inequality in Gaussian space -- possibly provable by rearrangement?
The following problem arose for my collaborators and me when studying the computational complexity of the Maximum-Cut problem.
Let $f : \mathbb{R} \to \mathbb{R}$ be an odd function. Let $\rho \in [...
- 6,694
20
votes
3
answers
4k
views
What is the origin of the term "spectrum" in mathematics?
The use of the term "spectrum" to denote the prime ideals of a ring originates from the case that the ring is, say, $\mathbb{C}[T]$ where $T$ is a linear operator on a finite-dimensional vector space; ...
- 118k
10
votes
1
answer
835
views
what was Hilbert's geometric construction in his 17th problem?
Hilbert's 17th problem asked if a nonnegative real polynomial is the sum of squares of rational functions. It was answered affirmative by Artin in around 1920. However, in his speech, he also asked if ...
user2529
29
votes
12
answers
6k
views
When does 'positive' imply 'sum of squares'?
Does anyone have examples of when an object is positive, then it has (or does not have) a square root? Or more generally, can be written as a sum of squares?
Example. A positive integer does not ...
4
votes
1
answer
945
views
Interpolation by a function whose second derivative is bounded
I don't know if this is an easy question for specialists in the field. Consider
the following interpolation problem : let $\varepsilon >0$, $X$ be a finite
set of real numbers and $g$ be a real-...
- 3,595
9
votes
3
answers
763
views
Approximating with translated Gaussians and low-frequency trig functions
Defining the translated Gaussians by $f_t(x)=\exp(-(x-t)^2)$ for $t,x\in\Bbb{R}$, we showed that the linear span of $\{f_t \mid 0 \le t < \epsilon\}$ is dense in $L^2(\Bbb{R})$, for any $\epsilon&...
- 524
15
votes
5
answers
4k
views
Discrete harmonic function on a planar graph
Given a graph $G$ we will call a function $f:V(G)\to \mathbb{R}$ discrete harmonic if for all $v\in V(G)$ , the value of $f(v)$ is equal to the average of the values of $f$ at all the neighbors of $v$....
- 85.6k
4
votes
3
answers
2k
views
Most important domains, extension theorems, and functions in several complex variables
For a new learner of several complex variables, the many domains (eg holomorphically convex, pseduconvex, Stein) and the many extension theorems (eg Riemann) and the many functions (plurisubharmonic) ...
user2529
6
votes
1
answer
726
views
The "ultimate" indefinite inner product space
This can be considered as a relative of Splitting a space into positive and negative parts.
Is there a real (non-trivial) vector space $V$, endowed with a nondegenerate symmetric bilinear pairing $\...
- 4,060
4
votes
1
answer
321
views
What functorial topologies are there on the space of linear maps between LCTVS?
Setup: we consider the category of locally convex topological vector spaces with morphisms as continuous linear maps. This time, I'm explicitly allowing the axiom of choice (or at least the Hahn-...
- 26.8k
13
votes
4
answers
2k
views
Is the category of Banach spaces with contractions an algebraic theory?
Consider the category of Banach spaces with contractions as morphisms (weak, so $\|T\| \le 1$). Is this an algebraic theory?
I suspect that this is true. The "operations" will be weighted sums, ...
- 26.8k
71
votes
2
answers
6k
views
Barrelled, bornological, ultrabornological, semi-reflexive, ... how are these used?
I'm not a functional analyst (though I like to pretend that I am from time to time) but I use it and I think it's a great subject. But whenever I read about locally convex topological vector spaces, ...
- 26.8k
5
votes
2
answers
765
views
Can we distinguish the algebraic and continuous duals of a Banach space without choice (or HBT)?
The algebraic dual of a normed vector space is the space of all linear functionals to the ground field (either $\mathbb{R}$ or $\mathbb{C}$ for this question). The continuous dual is the subspace of ...
- 26.8k
3
votes
3
answers
2k
views
Conditional expectation of convolution product equals..
Let $X, Y$ be two $L^1$ random variables on the probablity space $(\Omega, \mathcal{F}, P)$. Let $\mathcal{G} \subset \mathcal{F}$ be a sub-$\sigma$-algebra.
Consider the conditional expectation ...
- 47
14
votes
2
answers
783
views
Are two probability distributions uniquely constrained by the sum of their p-norms?
Let A, B and C be finitely supported probability distributions with at most d nonzero probabilities each. Now consider the following simultaneous equations using p-norms, for each value of p≥1, ...
- 2,649
97
votes
17
answers
17k
views
What's a nice argument that shows the volume of the unit ball in $\mathbb R^n$ approaches 0?
Before you close for "homework problem", please note the tags.
Last week, I gave my calculus 1 class the assignment to calculate the $n$-volume of the $n$-ball. They had finished up talking about ...
7
votes
2
answers
659
views
Chebyshev-like polynomials with integral roots
Chebyshev polynomials have real roots and satisfy a recurrence relation. I was wondering if one can find a sequence of polynomials with integral or rational roots with similar properties. More ...
- 23.5k
9
votes
1
answer
708
views
Hilbert spaces are induced by a bilinear form. How about n-linear forms?
A Hilbert space is a complete vector space equipped with scalar product, i.e. a symmetric positive definite bilinear form.
What if we replace 'bilinear' by 'n-linear'? One might wonder, whether the $...
- 5,327
12
votes
5
answers
7k
views
Applied mathematics Books (graduate level)
What are some good graduate level books on applied mathematics which explain in-depth the general modern problem-solving methods of the real-world typical hard problems?
There is a lot of books on ...
2
votes
3
answers
946
views
How can I measure the Morse index in infinite dimensions?
Let $V$ be a vector space over $\mathbb R$, and $a: V\otimes V\to \mathbb R$ a symmetric bilinear pairing. Recall that the Morse index of $a$ is the maximal dimension of any subspace $V_- \subseteq V$...
- 54.6k
7
votes
1
answer
314
views
How to see meromorphicity of a function locally?
Given a germ of an analytic function on a (compact, for simplicity) Riemann surface, how can one see (locally) whether this is a "germ of meromorphic function"? I.e. if I do analytic continuation ...
- 5,052
2
votes
4
answers
3k
views
Splitting a space into positive and negative parts
Let $V$ be a vector space over $\mathbb R$. A symmetric bilinear pairing on $V$ is a linear map $a: V\otimes V \to \mathbb R$. Because $\mathbb R$ is characteristic not-two, I will freely confuse ...
- 54.6k
17
votes
7
answers
3k
views
Why is Riemann-Roch an Index Problem?
I was in a lecture not long ago given by C. Teleman and at some point he said "Well, since Riemann-Roch is an index problem we can do..."
Then right after that he argued in favour of such a sentence. ...
- 2,132
4
votes
2
answers
730
views
Decomposition of Hölder continuous functions
Let $\alpha\in(0,1)$ and $\eta\in\Lambda_0^\alpha(\mathbb{R})$ be a compactly supported Hölder continuous function of order $\alpha$. I would like to show that, for any $n\in\mathbb{N}$, it is ...
- 852
36
votes
6
answers
2k
views
When are some products of gamma functions algebraic numbers?
I want to know when certain expressions of the form
$ {\Gamma(r_1/m) \Gamma(r_2/m) \ldots \Gamma(r_j/m) \over \Gamma(s_1/m) \Gamma(s_2/m) \ldots \Gamma(s_j/m)} $
are algebraic numbers. These ...
- 14k
4
votes
3
answers
609
views
When is $A : C(X) \to C(Y)$ a composition operator?
A composition operator $C\_T : C(X) \to C(Y)$ with $T \in C(Y, X)$ is defined by $C\_T f := f \circ T, f \in C(X)$.
I read in the book about Composition Operators by Singh and others that a ...
- 1,338
22
votes
9
answers
3k
views
When does the zeta function take on integer values?
Here $\zeta(s)$ is the usual Riemann zeta function, defined as $\sum_{n=1}^\infty n^{-s}$ for $\Re(s)>1$.
Let $A_n=${$s\;:\;\zeta(s)=n$}. The behaviour of $A_0$ is basically just the Riemann ...
- 7,013
1
vote
2
answers
1k
views
What do we know about the space of finite order distributions ?
Hi,
(Question updated)
My question is about the space of distributions of finite order $\mathcal{D}'_F$ (say on $\mathbb{R}^n$). What do we know about it ?
From in the information I gathered, it ...
- 11
15
votes
2
answers
2k
views
What is a projective space?
Is there a "recognition principle" for projective spaces?
What categories are there with projective spaces for objects?
Background: Although the title is a nod to What is a metric space?, ...
- 26.8k
21
votes
1
answer
1k
views
Is Dependent Choice all we really need?
http://en.wikipedia.org/wiki/Axiom_of_dependent_choice
Is DC sufficient for the understanding of objects that are countable in some suitable sense?
For example, is DC sufficient for the full ...
- 1,199
4
votes
1
answer
1k
views
Interpolation of sequences by analytic functions
Given a sequence of complex numbers $a_n$ with $n\in\mathbb{N}$, is it possible to find an analytic (or meromorphic) function that interpolates this sequence in the sense that $f(n)=a_n$?
If this is ...
- 1,219
11
votes
2
answers
862
views
Monotone Lipschitz embedding ?
In 1974, Aharoni proved that every separable metric space (X, d) is Lipschitz isomorphic to a subset of the Banach space c_0.
Thus, for some constant L, there is a map K: X --> c_0 that satisfies the ...
- 4,060
4
votes
5
answers
3k
views
Generalize Fourier transform to other basis than trigonometric function
The Fourier transform of periodic function $f$ yields a $l^2$-series of the functions coefficients when represented as countable linear combination of $\sin$ and $\cos$ functions.
In how far can this ...
- 5,327
28
votes
2
answers
4k
views
Terminology in category theory
A lot of the terminology of category theory has obvious antecedents in analysis: limits, completeness, adjunctions, continuous (functors), to name but a few. However, analysis and category theory ...
- 26.8k
45
votes
7
answers
16k
views
What is an intuitive view of adjoints? (version 2: functional analysis)
After realising that I don't have an intuitive understanding of adjoint functors, I then realised that I don't have an intuitive understanding of adjoint linear transformations!
Again, I can use 'em, ...
- 26.8k
8
votes
3
answers
1k
views
What is the term analogous to "Wronskian" for difference equations?
I am currently following a course on differential equations and difference equations (recurrence relations).
The teacher tries to make parallels between the two concepts, because the methods for ...
4
votes
1
answer
2k
views
Lebesgue measure of boundary of Caccioppoli set
Can anything be said about the measure of the topological boundary of a Cacciopoli set in $R^n$? Of course, the reduced boundary has finite (n-1)-dimensional Hausdorff measure, but this does not say ...
- 320
18
votes
3
answers
2k
views
What are the right categories of finite-dimensional Banach spaces?
This is inspired partly by this question, especially Tom Leinster's answer.
Let me start with some background. I apologize that this will be rather long, since I'm hoping for input from people who ...
- 11.4k
9
votes
2
answers
1k
views
Explicitly describing extreme points of infinite dimensional convex sets
I am currently trying to apply some results from Choquet theory - i.e., the generalisation of results by Minkowski and Krein-Milman for representing points in a compact, convex set C by probability ...
- 325
6
votes
3
answers
324
views
Inverses in convolution algebras
Let $G$ be a locally compact totally disconnected group, and to make life easy let's suppose its Haar measure is bi-invariant. Let $C_c(G)$ be the space of locally constant complex functions on $G$ ...
- 2,713
191
votes
34
answers
81k
views
What is convolution intuitively?
If random variable $X$ has a probability distribution of $f(x)$ and random variable $Y$ has a probability distribution $g(x)$ then $(f*g)(x)$, the convolution of $f$ and $g$, is the probability ...
11
votes
7
answers
1k
views
What are some interesting ways of making new metrics out of old metrics?
If $d(x,y)$ and $e(x,y)$ are metrics then $d(x,y)+e(x,y)$ and $\frac{d(x,y)}{1+d(x,y)}$ are metrics.
If $d_i(x,y)$ for $i=1,\dots,n$ are metrics then so is $\sqrt{\sum_{i=1}^n{d_i^2(x,y)}}$
Are ...
- 3,613
5
votes
3
answers
230
views
Is the Fell-Doran problem trivial in a topological setting?
The Fell-Doran problem is a problem in functional analysis. It goes as follows: Let $A$ be a complex unital algebra, $X$ a locally convex space, and $L(X)$ the algebra of all continuous endomorphisms ...
- 1,821
7
votes
4
answers
2k
views
Simultaneous Equations Involving Power Sums
Let $\ell$ be a positive integer greater than 1. The problem is to find a set of $n$ real positive numbers $x_i$ and $n+1$ numbers $y_i$ such that
$$\sum_{i=1}^n x_i^k= \sum_{i=1}^{n+1} y_i^k$$
for $k=...
- 731
3
votes
4
answers
1k
views
Is there a use for a Hilbert space that uses a different norm than the one induced by the inner product?
$l_1$ minimization / compressed sensing comes to mind. Does anyone have any concrete examples? Or is such a construct completely useless?
- 49