All Questions
5,628 questions
10
votes
5
answers
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views
Extracting a common convergent indexing from an uncountable family of sequences
Let $\mathcal{A}$ be some uncountable index set and $X$ be some separable reflexive Banach space.
For each $\alpha \in \mathcal{A}$, let
\begin{equation}
\{ x_n^{\alpha} \}_{n=1}^\infty
\end{equation}
...
- 3,477
10
votes
2
answers
3k
views
Continuous function from $[0,1]$ to $[0,1]$
Does there exist a continuous function $f:[0,1]\rightarrow [0,1]$ such that $f$ takes every value in $[0,1]$ an infinite number of times?
- 1,001
10
votes
4
answers
1k
views
Adventure with infinite series, a curiosity
It is easily verifiable that
$$\sum_{k\geq0}\binom{2k}k\frac1{2^{3k}}=\sqrt{2}.$$
It is not that difficult to get
$$\sum_{k\geq0}\binom{4k}{2k}\frac1{2^{5k}}=\frac{\sqrt{2-\sqrt2}+\sqrt{2+\sqrt2}}2.$$
...
- 43.2k
10
votes
2
answers
886
views
An attempt to generalize the previous inequality
In my previous MO question, the inequality was about a specific series and nicely answered by Cherng-tiao Perng. After testing with a few more numerical infinite sums, I came to realize that perhaps ...
- 43.2k
10
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1
answer
3k
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Can the supremum of continuous functions be discontinuous at every point of an interval?
Pether Luthy gave an example of a sequence of continuous real valued functions whose supremum was discontinuous on a set of positive measure. But does it exist a sequence of continuous real valued ...
- 231
10
votes
4
answers
3k
views
Measure 0 sets on the line with Hausdorff dimension 1
I use $\dim_H(E)$ to denote the Hausdorff dimension of a set $E \subseteq \mathbb{R}$ and $|E|$ to denote its Lebesgue measure. It is easy to see from the definition of Hausdorff dimension that if $\...
- 505
10
votes
2
answers
826
views
Is there a measure on $[0,1]$ that is 0 on meagre sets and 1 on co-meagre sets
I'm curious if there is a finite measure on the $\sigma$-algebra of subsets of $[0,1]$ with the Property of Baire, whose null sets are exactly the meagre sets.
I'd also be interested how "nice" such ...
- 345
10
votes
1
answer
1k
views
Within ZFC, is $2^{\aleph_0}<2^{\aleph_1}$ provable/independent?
So, I ask whether from the ZFC axioms one can prove X that every uncountable set has strictly more than continuum many subsets, or whether X is independent of the ZFC axioms. Note that (within ZFC) ...
- 3,584
10
votes
2
answers
598
views
Taylor series with coefficients in $\mathbb{Q}$
Is there a sequence of rational numbers $a_0, a_1, \dotsc$ such that $\sum\limits_{i\geq 0}a_i x^i$ converges absolutely to $2^x$ for every $x\in \mathbb{Z}$?
- 141
10
votes
1
answer
755
views
The $9$th tetration of $-\sqrt2$
Let $^na$ denote the $n$th tetration of $a$, so that $^0a=1$ and
$$^{n+1}a=a^{^na}$$
for $n=0,1,\dots$. (For complex $x$ and $y$, here we use the definition $x^y:=e^{y\ln x}$, where $\ln$ is the ...
- 128k
10
votes
1
answer
961
views
Ruling out the existence of a strange polynomial II
This is a refinement of my question asked earlier, which is answered beautifully in the negative by Thomas Browning. The example he gave was geometrically reducible. Now I want to ask the same ...
- 26.9k
10
votes
2
answers
1k
views
On equibounded sequences in $L^\infty$
Let $f_n: [0, 1] \to \mathbb R$ be a sequence of positive functions in $L^\infty$ (hence a fortiori in $L^1$) that are equibounded in $L^\infty$ norm - that is $\sup_{n \in \mathbb N} \|f_n\|_{L_\...
- 6,155
10
votes
2
answers
3k
views
Absolute continuity on $R^{n}$
I know the definition of absolute continuity if there is a function $f:(a,b)\rightarrow R$.
I wonder what is an analogy of this concept if we have a function $f:A\rightarrow R$, where $A\subset R^{n}$ ...
- 399
10
votes
2
answers
1k
views
Does a conditionally convergent sum with random signs converge almost surely?
Let $\sum a_n$ be a conditionally convergent sum of real numbers, and $\epsilon_n$ a sequence of independent identically distributed Bernoulli random variables with $\epsilon_n = 1$ or $-1$ with ...
- 6,155
10
votes
1
answer
668
views
On Pareto functions
The Pareto principle says that the top 20% of wealthy people people hold over 80% of the wealth. Suppose we had a non-negative function on $\mathbb R^n$ that satisfied this principle on every open ...
- 6,155
10
votes
2
answers
597
views
How to determine the asymptotics of $\sum_{n=0}^{\infty} e^{-\frac{2^n}{x}}$
I'm generally interested in being able to find an asymptotic expansion of
$$ \sum_{n=0}^{\infty} \left[ e^{- \frac{f(n)}{x}} \right] $$
As $x \rightarrow \infty$ and $f(n)$ is a smooth monotonically ...
- 2,689
10
votes
2
answers
1k
views
Proof in constructive mathematics that the principal square root function exists in any Cauchy complete Archimedean ordered field
In classical mathematics, there exists only one Cauchy complete Archimedean ordered field, the Dedekind complete Archimedean ordered field. However, in constructive mathematics, there are multiple ...
- 2,624
10
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1
answer
1k
views
Is there a strictly increasing differentiable function maps positively measurable set to zero measure set?
Let $g(t)$ be a strictly increasing differentiable function. Can it map positively measurable set to zero measurable set?
It's obviously that $\{g'>0\}$ is dense. If I can prove that the Lebesgue ...
- 1,168
10
votes
2
answers
9k
views
When do maximum and expectation commute?
Hi, I'm looking for conditions on $G(t,x)$ such that
$$
\sup\limits_{t\in [0,1]}E[G(t,X)]=E[\sup\limits_{t\in [0,1]}G(t,X)]
$$
where $X$ is a random variable (it's easy to see that $\sup\limits_{t\in [...
- 111
10
votes
3
answers
414
views
Is an open subset of a rigid space rigid?
Let $X$ be a locally compact Hausdorff space. Call $X$ rigid if its only autohomeomorphism is the identity, $\operatorname{Homeo}(X)=\{1\}$.
Questions:
Let $X$ be rigid. Is it true that every open ...
- 1,959
10
votes
2
answers
3k
views
Gluing two diffeomorphisms together
A fundamental construction in a first course on manifolds is to build a smooth function $\psi\colon \mathbb{R} \to \mathbb{R}$ with the property that for some $0<\delta<\epsilon$ we have
$\psi(...
- 8,903
10
votes
1
answer
329
views
Is there a triangle which makes dense set of angles by drawing medians?
This problem is a restatement of this question, first announced in MathStackExchange.
We start with a triangle $T$ in the Euclidean plane and we define $A_n$ as the set of angles of the $6^n$ ...
- 125
10
votes
3
answers
913
views
Inequality for functions on [0,1]
Let $a\in (0,1), \;\;\psi_a(x):=\prod_{j=0}^\infty (1-a^{2j+1}x).$
Question. Is it true that, for all $x\in [0,1]$ and all $k\in\mathbb{N},$ the following inequality holds:
$$\frac{x^k}{(1-a)(1-a^3)\...
- 783
10
votes
2
answers
6k
views
Who was the first to formulate the inverse function theorem?
Let $U\subset \mathbb{R}^n$ and let $F:U\to \mathbb{R}^n$. The 'classical' inverse function theorem gives a sufficient condition for the existence and differentiability of the inverse function of $F$.
...
- 103
10
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1
answer
1k
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Real analytic function, injective, non surjective and preserving the rationals ?
I'd like to prove the non-existence of a real analytic function, injective, non-surjective
that sends rationals to rationals.
Is it a classical result ? If not, any hints on how to prove it ?
Thanks ...
- 103
10
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2
answers
1k
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Does Rolle's Theorem imply Dedekind completeness?
I think the answer to the title question is "yes", but Gerald Edgar, in his comment on Does antidifferentiability of continuous functions imply Dedekind completeness? , points out an article (actually ...
- 19.7k
10
votes
2
answers
612
views
Proving the simple form of a function from statistical mechanics
I have discovered a pertinent solution to my problem in the article On the Kinetic Theory of Rarefied Gases by Harold Grad and the book Thermodynamik und Statistik by Arnold Sommerfeld, both of which ...
10
votes
1
answer
936
views
Derivative without extrema is monotone
This is a cross-post from Math.SE.
The question was asked there 3 months ago but didn't receive much attention aside from one comment asking for clarification. I feel like it might be non-trivial and ...
- 103
10
votes
2
answers
513
views
Is there a purely constructive presentation of the HK integral?
Treating the Riemann integral in a constructive setting is easy and straightforward. Treating the closely related but much more powerful Henstock-Kurzweil integral constructively is almost easy, ...
- 1,947
10
votes
2
answers
666
views
Reference request: Extensions of Wiener's Tauberian Theorem
Wiener's Tauberian Theorem says that linear combinations of translations of a function $f$ are dense in $L^1(\mathbb{R})$ if and only if the zero set of the Fourier transform of $f$ is empty. This is ...
- 710
10
votes
2
answers
1k
views
Harmonic oscillator discrete spectrum
Let us act intentionally stupid and assume we do not know that we can solve for the spectrum of the harmonic oscillator
$$-\frac{d^2}{dx^2}+x^2$$
explicitly.
Is there an abstract argument why the ...
- 501
10
votes
1
answer
1k
views
Extension of measures from the ball sigma-algebra to the borel sigma-algebra
Let $X$ be a metric space, $\Sigma_{1}$ the borel sigma algebra and
$\Sigma_{2}$ the sigma algebra generated by balls (open and closed).
If $\mu$ is a probability measure on $\Sigma_{2}$ can it be ...
- 307
10
votes
1
answer
1k
views
A strange Lipschitz function
Let $n \geq 3$. Does there exist a Lipschitz function $f: \mathbb R^n \to \mathbb R$ such that the following conditions hold?
The origin is a weak Lebesgue point of $\nabla f$, in the sense that the ...
- 6,155
10
votes
1
answer
817
views
Can a nowhere locally Hölder function be differentiable almost everywhere?
Fix $0 < \alpha < 1$. Suppose $f$ is nowhere locally $\alpha$-Hölder continuous - that is, it is not $\alpha$-Hölder on any open subinterval of $\mathbb R$. Is it possible for $f$ to be ...
- 6,155
10
votes
3
answers
849
views
Progress in robustifying mathematics - i.e. making mathematical theorems robust to small changes in hypotheses
The idea of making a mathematical theorem robust to small changes in its hypotheses has been known for some time. In areas such as group theory reasonable progress has been made leading to the theory ...
- 4,862
10
votes
2
answers
1k
views
Why is $\sup f_- (n) \inf f_+ (m) = \frac{5}{4} $?
This question is an old question from mathstackexchange.
Let $f_- (n) = \Pi_{i=0}^n ( \sin(i) - \frac{5}{4}) $
And let
$ f_+(m) = \Pi_{i=0}^m ( \sin(i) + \frac{5}{4} ) $
It appears that we have
$$\sup ...
- 763
10
votes
2
answers
2k
views
Continuous functions with convex level sets
Assume that $f:\mathbb{R}^{2}\to \mathbb{R}$ is a continuous function such that each level set $f^{-1}(c)$ is a convex set.
To what extent such functions are studied?
In particular:
Is there a ...
- 356
10
votes
1
answer
594
views
Are the polynomials in $\{1/t\}$ dense in $L^2(0,1)$?
Added. My question in the title was solved (in the negative) by Nik Weaver (in the answer below) and Mateusz Kwaśnicki (in the comments). In both solutions, the reason is that the $L^2$ density fails ...
- 13.8k
10
votes
1
answer
496
views
Does this Osgood-like condition imply continuity?
Let us consider a bounded, Borel function $F\colon \mathbb R^d \to \mathbb R^d$. Assume it satisfies the following
Osgood-like condition:
$$\tag{O}
\boxed{\vert \langle F(x) - F(y), x-y \rangle\...
- 391
10
votes
2
answers
343
views
A moment problem
Suppose $X, Y$ are two positive random variables such that $\mathbb{E}[X^\alpha] = \mathbb{E}[Y^\alpha]$ for all $\alpha \in (0, 1/2)$.
It is also known that the first moment exists for each of them, ...
- 205
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votes
2
answers
371
views
Can a vector-function $v:\mathbb{R}^n\to \mathbb{R}^n$ be an eigenvector of its own Jacobian matrix?
Good morning,
I've came across this question, which has been puzzling me for some days. Suppose we are given a vector-valued function $v:\mathbb{R}^n\to \mathbb{R}^n$, $v(x)=\left( v_1(x),\dots, v_n(...
- 311
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2
answers
1k
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Can the integration of integrable sections of a measurable function of two variables ever result in a non-measurable function?
I spent some time searching MathOverflow for a problem that would resemble the one given below, but it turned out to be a rather futile endeavor. I was led to this problem in my attempts to construct ...
- 1,396
10
votes
1
answer
571
views
Are “most” bounded derivatives not Riemann integrable?
Given $a,b\in\mathbb R$ with $a<b$. Let
$$X=\{f\in C([a,b]): f \text{ is differentiable on } [a,b] \text{ with }f' \text{ bounded }\},$$
and
$$A=\{f\in X: f' \text{ is Riemann integrable}\}. $$
It ...
- 299
10
votes
2
answers
917
views
Expressions for the inverse function of $f(x) = \ln(x)e^x$
Can the inverse of $ \ln(x)e^x $ be finitely expressed in terms of the Lambert-W function or any other well-known transcendental functions? It is clear that a closed-form elementary function ...
- 101
10
votes
2
answers
835
views
Functions that are approximately differentiable a.e
The classical definition of an approximately differentiable function is as follows:
Definition.
Let $f:E\to\mathbb{R}$ be a measurable function defined on a measurable set $E\subset\mathbb{R}^n$. ...
- 28k
10
votes
1
answer
2k
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Counting norms on an infinite dimensional vector space
It is known that whenever E is a finite dimensional real vector space, there is only one norm on E up to equivalence (actually one non discrete vector space topology).
Is it known what happens when E ...
- 101
10
votes
3
answers
2k
views
The intersection of $n$ cylinders in $3$-dimensional space
A standard question in vector calculus is to calculate the volume of the shape carved out by the intersection of $2$ or $3$ perpendicular cylinders of radius $1$ in three dimensional space. Such ...
- 11.4k
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votes
1
answer
349
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On a variant of Carlson’s theorem
My question is on whether or not there exists some monotone strictly decreasing sequence of positive numbers $c_1>c_2>\ldots$ such that given any $f$ which is a uniformly bounded holomorphic ...
- 4,135
10
votes
1
answer
394
views
Maximally continuous extension of continuous functions from $\mathbb Q$ to $\mathbb R$
Let $f: \mathbb Q \to \mathbb R$ be a continuous function.
An extension of $f$ is a function $\tilde f: \mathbb R \to \mathbb R$ such that $\tilde f = f$ on $\mathbb Q$.
We say an extension $\tilde f$ ...
- 6,155
10
votes
1
answer
872
views
Current vs Varifold
I know the basic definitions concerning current and varifold, and they are generalization of submanifolds. What are their respective pros and cons? What are their crucial similarities and differences?
- 1,630