All Questions
5,873 questions
18
votes
3
answers
3k
views
A curious sin-integral
While contending with a certain Fourier series, I stumbled on an incredibly simple evaluation (numerically) of a slightly complicated-looking sin-integral.
So, I wish ask:
Question. Is this really ...
- 43.2k
18
votes
4
answers
3k
views
Why is there no Borel function mapping every countable set of reals outside itself?
A choice function maps every set (in its domain) to an element of itself. This question concerns existence of an anti-choice function defined on the family of countable sets of reals. In an answer to ...
- 30.1k
18
votes
5
answers
3k
views
Bernoulli sum meets golden number
Let $B_n$ denote the Bernoulli numbers and let $\phi=\frac{1+\sqrt{5}}2$ be the golden ratio.
I encountered the following infinite sum and would like to ask:
Question. Is this true? If so, any ...
- 43.2k
18
votes
2
answers
3k
views
Is there an analytic non-linear function that maps rational numbers to rational numbers and it maps irrational numbers to irrational numbers?
Consider a function $h$ defined on real numbers, which is not of the form $kx+b$ i.e. a linear function. If $h$ maps rational numbers to rational numbers and it maps irrational numbers to irrational ...
- 303
18
votes
6
answers
3k
views
What's the use of Malgrange preparation theorem?
The Malgrange preparation theorem,which is the $C^{\infty}$ version of the classical Weierstrass preparation theorem,says that if $f(t,x)$ is a $C^{\infty}$ function of $(t,x)\in\mathbb{R}^{n+1}$ near ...
- 1,644
18
votes
2
answers
2k
views
Generalization of Darboux's Theorem
Darboux's Theorem. If $f:[a,b]\to\mathbb R$ is differentiable and $f'(a)<\xi<f'(b)$, then there exists a $c\in (a,b)$, such that $\,f'(c)=\xi$.
Does any of the following generalizations
Let $U\...
- 2,933
18
votes
2
answers
630
views
Is the notion of fixed point property for topological spaces an absolute notion?
Recall that a topological space $X$ has the fixed point property (FPP) if any continuous function $f: X\to X$ has a fixed point.
Is the notion of FPP for topological spaces an absolute notion? More ...
- 32.1k
18
votes
4
answers
4k
views
Problems in advanced calculus
I have been teaching Advanced Calculus at the University of Pittsburgh for many years. The course is intended both for advanced undergraduate students and the first year graduate students who have to ...
18
votes
2
answers
574
views
Existence of an antiderivative function on an arbitrary subset of $\mathbb{R}$
Let $f:\mathbb{R}\to \mathbb{R}$ be continuous at $x$ for every $x\in I$ where $I\subset \mathbb R$ could be arbitrary. Does there always exist a function $F:\mathbb{R}\to \mathbb{R}$ differentiable ...
- 1,503
18
votes
2
answers
1k
views
Characterisation of bell-shaped functions
This is an open problem that I learned from Thomas Simon. I will completely understand if the question is judged as non-research level (and it is indeed not related to my research), but I believe a ...
- 17.2k
18
votes
1
answer
3k
views
How bad can the second derivative of a convex function be?
One can easily construct an example of a measurable function $f:(a,b)\to \mathbb{R}$ which satisfies the following property:
$$\label{p}\tag{P}
f\notin L^1(I),\ \mbox{for each interval}\ I\subset (a,...
- 409
18
votes
2
answers
1k
views
Comparing "axiomatized function spaces"
This was previously asked and bountied at math.stackexchange with no response. I've also tweaked the language for clarity; see the edit history for the broader context, and note that the existing ...
- 21.1k
18
votes
2
answers
1k
views
An Entropy Inequality (generalized)
Let $X,Y$ be probability measures on $\{1,2,\dots,n\}$. For $0\le \alpha \le 1$, set $K=\sum_i X(i)^\alpha Y(i)^{1-\alpha}$ so that $Z:=\frac{1}{K}X^\alpha Y^{1-\alpha}$ is also a probability measure ...
- 351
18
votes
0
answers
1k
views
Does there exist a continuous open map from the closed annulus to the closed disk?
(Originally from MSE, but crossposted here upon suggestion from the comments)
In this MSE post, user Moishe Kohan provides an example of a non-continuous open and closed ("clopen") function $...
- 831
18
votes
1
answer
2k
views
Function of two sets intersection
Let $U$ be the set of all nonempty subsets of $[0,1]$ that are a union of finitely many closed intervals (where an "interval" that is a single point does not count as an interval). Does ...
- 1,209
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
17
votes
2
answers
1k
views
"Insanely increasing" $C^\infty$ function with upper bound
Let $C^\infty$ denote the collection of functions $f:\mathbb{R}\to\mathbb{R}$ such that for every positive integer $n$, the $n$-th derivative of $f$ exists. For $f\in C^\infty$ we set
$f^{(0)} = f$, ...
- 48.9k
17
votes
3
answers
2k
views
Is every Schwartz function the product of two Schwartz functions?
A Schwartz function on $\mathbb R^d$ is a $C^\infty$ function, such that all differentials of order $k \ge 0$ decay faster than any polynomial. They include the class $C^\infty_c(\mathbb R^d)$ of ...
- 475
17
votes
2
answers
1k
views
Is it consistent with ZFC that the real line is approachable by sets with no accumulation points?
Let $P$ denote the following proposition:
There exists a set $S$ of subsets of $\mathbb{R}$ such that
$S$ is totally ordered by inclusion;
each member of $S$ has no accumulation points;
the union of ...
- 708
17
votes
2
answers
905
views
Intersection of compact sets in the unit interval
Let $\mathscr K$ be an uncountable set such that every $K\in\mathscr K$ is a compact subset of $[0,1]$ with positive Lebesgue measure. Does it then follow that there exists an uncountable $\mathscr A\...
- 3,584
17
votes
1
answer
1k
views
Continuous functions of three variables as superpositions of two variable functions
Could we always locally represent a continuous function $F(x,y,z)$ in the form of $g\left(f(x,y),z\right)$ for suitable continuous functions $f$, $g$ of two variables? I am aware of Vladimir Arnold's ...
- 3,599
17
votes
2
answers
750
views
Approximation of smooth diffeomorphisms by polynomial diffeomorphisms?
Is it possible to (locally) approximate an arbitrary smooth diffeomorphism by a polynomial diffeomorphism?
More precisely: Let $f:\mathbb{R}^d\rightarrow\mathbb{R}^d$ be a smooth diffeomorphism for $d&...
- 173
17
votes
3
answers
1k
views
Decoupling a double integral
I came across this question while making some calculations.
QUESTION. Can you find some transformation to "decouple" the double integral as follows?
$$\int_0^{\frac{\pi}2}\int_0^{\frac{\pi}...
- 43.2k
17
votes
2
answers
1k
views
Kolmogorov superposition for smooth functions
Kolmogorov superposition theorem states that a continuous function $f(x_1,\ldots,x_n)$ can be written as
$$f(x_1,\ldots,x_n)=\sum_{q=0}^{2n}\Phi_q\left(\sum_{p=1}^{n}\phi_{q,p}(x_p)\right)$$
for ...
- 807
17
votes
2
answers
2k
views
"Find $\lim_{n \to \infty}\frac{x_n}{\sqrt{n}}$ where $x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$" -where does this problem come from?
Recently, I encountered this problem:
"Given a sequence of positive number $(x_n)$ such that for all $n$,
$$x_{n+1}=x_n+\frac{n}{x_1+x_2+\cdots+x_n}$$
Find the limit $\lim_{n \rightarrow \infty} \...
- 327
17
votes
3
answers
2k
views
The minimum of a sum of absolute values of inner products in $\mathbb{R}^d$
Consider a collection of unit vectors $v_1, \ldots, v_n$ in $\mathbb{R}^d$ (we think of $n$ being much larger than $d$). I would like to minimize the sum:
$$\sum_{i\neq j}|\langle v_i,v_j\rangle|.$$
...
- 2,288
17
votes
3
answers
975
views
Evaluating the sum $f(x):=\sum_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n$ and estimating bounds
For real variable $x$, the function
\begin{equation}
f(x):=\sum_{n=1}^\infty \frac{1}{n! n^n}(-x^2)^n
\end{equation}
clearly has infinite radius of convergence and defines a $C^\infty$ function on $\...
- 3,477
17
votes
2
answers
3k
views
The Riemann hypothesis as a problem in analysis
The recent post("Long-standing conjectures in analysis ... often turn out to be false") prompted me to think about a question which I have not given much though before: to what extent the ...
- 6,891
17
votes
3
answers
905
views
Existence of translation-invariant basis on $C_c(\mathbb R)$
Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
- 2,071
17
votes
1
answer
794
views
Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ with injective restriction $f|\mathbb Q^\omega$?
Question. Is there a continuous function $f:\mathbb R^\omega\to\mathbb R$ whose restriction $f|\mathbb Q^\omega$ is injective?
- 41.8k
17
votes
1
answer
986
views
Can two-point sets be Borel?
Recall that a two-point set is a subset of the plane which meets every line in exactly two points. Such a set was first constructed by Mazurkiewicz in 1914.
I wonder if the following question of ...
- 32.1k
17
votes
1
answer
2k
views
Hlawka inequality for determinants of positive definite matrices
It is mentioned here that if $A, B, C\in M_{n}(\mathbb C)$ are positive semidefinite, then $$\det (A+B+C)+\det C\ge \det (A+C)+\det (B+C)$$ (quoted from this article) and the special case ($C=\bf 0$) $...
- 13.4k
17
votes
2
answers
2k
views
Explicit and fast error bounds for polynomial approximation
Main Question
This question is about finding explicit, calculable, and fast error bounds when approximating continuous functions with polynomials to a user-specified error tolerance.
EDIT (Apr. 23): ...
- 697
17
votes
1
answer
580
views
Aperiodic monotile in $\mathbb{R}$
Motivation. Recently a group of researchers found an aperiodic monotile in $\mathbb{R}^2$, answering a long-standing question. There are many results in higher dimensions, so let's explore the lower ...
- 48.9k
17
votes
4
answers
1k
views
In choiceless constructivism: If $f'=0$ then is $f$ constant?
Prove, without any Choice principles or Excluded Middle, that if a pointwise differentiable function has derivative $0$ everywhere, then it is constant. The function in this case maps $\mathbb R$ to $\...
- 4,943
17
votes
1
answer
3k
views
Integrals of pullbacks and the Inverse function theorem(s?)
The usual story goes like this:
Smooth picture (?):
For a smooth bijection $\phi: M \to N$ between $n$-manifolds the following
is true:
$\phi^{-1}$ is a local diffeomorphism a.e.
...
- 7,789
17
votes
2
answers
4k
views
Is this statement which relates the Fourier transform of a function to its singularities correct?
I am working on a problem, which would possibly relate the Fourier transform/series with the jump singularities of the function where the function itself or one of its derivatives jump. ((some kind of ...
- 698
16
votes
6
answers
2k
views
Alternative proofs sought after for a certain identity
Here is an identity for which I outlined two different arguments. I'm collecting further alternative proofs, so
QUESTION. can you provide another verification for the problem below?
Problem. Prove ...
- 43.2k
16
votes
3
answers
1k
views
Can integration spoil real-analyticity?
Is there an example of a function $f:(a,b)\times(c,d)\to\mathbb{R}$, which is real analytic in its domain, integrable in the second variable, and such that the function
$$ g:(a,b)\to\mathbb{R},\qquad ...
- 353
16
votes
2
answers
2k
views
An analogue of the exponential function by replacing infinite series with improper integral
For every positive real number $x$ we define $$E(x)= \int_0^{\infty} x^t/t!\,\mathrm dt$$
where $t!=\Gamma(t+1)$. This is motivated by classical exponential function.
Is this function well defined (...
- 356
16
votes
3
answers
4k
views
Which functions have all derivatives everywhere positive?
Consider the class of functions from $\mathbb R$ to $\mathbb R$, such that the function is positive everywhere and its $n$th derivative is positive everywhere for all $n$.
The only examples I can ...
- 148k
16
votes
7
answers
6k
views
Understanding Gibbs's inequality
Short version
Gibbs's inequality is a simple inequality for real numbers, usually
understood information-theoretically. In the jargon, it states that
for two probability measures on a finite set, ...
- 27.7k
16
votes
2
answers
1k
views
How to generalize the various vector calculus theorems to distributions?
Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate ...
- 415
16
votes
3
answers
1k
views
A kernel 'more analytic' than $\exp(-x^2)$
I am looking for an analytic function $F: \mathbb{R} \rightarrow (0,\infty)$ with $\int_{\mathbb{R}} F(x) \, dx = 1$ and the property, that $\sum\limits_{k=0}^{\infty} |c_k| \varepsilon^k (2k)! < \...
- 1,295
16
votes
4
answers
2k
views
Is the $W^{1, \infty}$ limit of differentiable functions also differentiable?
Let $f_n$ be a sequence of differentiable functions on $[0, 1]$ with
$f_n \to f$ uniformly for some (necessarily) continuous $f$.
$f'_n - g \to 0$ in $L^{\infty}$ for some measurable $g$.
Is it true ...
- 6,155
16
votes
1
answer
686
views
Fourier's proof of reality of all roots of Bessel function $J_0(x)$
In his "Théorie de chaleur" Fourier proves that the zeros of Bessel function $J_0(x)$ are all real.
I want to ask if there is a modern version of this proof exist in literature?
If someone ...
- 784
16
votes
3
answers
1k
views
A natural center of a convex weakly compact set in Banach space
Question: Let $S$ be a convex weakly compact set in Banach space $H$. Propose a natural way to define the unique center $O \in S$.
Motivation: A lot! For example, in game theory $S$ can be a set of ...
- 161
16
votes
4
answers
1k
views
Proof of complete monotonicity of a binomial function
By plotting the function and its derivatives, one can easily be convinced that the function
$$f(x):=\log\binom{x}{p x}=\log\Gamma(x+1)-\log\Gamma(px+1)-\log\Gamma((1-p)x+1),$$ defined for $x>0$ and ...
- 1,324
16
votes
1
answer
3k
views
Did Euler know (unconsciously) to integrate by differentiating?
Considering a method to find the anti-derivative of an (sufficiently smooth) real function by differentiating published some years ago (equation (48) in Kempf et al., New Dirac Delta function based ...
- 4,014
16
votes
1
answer
888
views
Kakeya crossed-needles problem
The Kakeya needle problem asks for the
minimum area planar region in which one can completely turn around a line segment through
a series of translations and rotations. There is no minimum: There are &...
- 151k