All Questions
5,702 questions
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
2k
views
The Lefschetz operator
Let $\omega=\sum_{i=1}^n dx_i\wedge dy_i\in\bigwedge^2(\mathbb{R}^{2n})^*$ be a standard symplectic form. The following result is due to Lefschetz:
For $k\leq n$, the Lefschetz operator
$L^{n-k}:\...
- 28k
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
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
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
16
votes
2
answers
2k
views
Challenge: Non-Gaussian quartic integral and path integral in Quantum field theory
(1) It is well-known that we can get a Gaussian integral of this type, where $x$ is in $\mathbb{R}$:
$$
\int_{-\infty}^{\infty} dx e^{-ax^2}=\sqrt{(2\pi)/a}. \tag{i}
$$
We can generalize this ...
- 2,147
16
votes
6
answers
3k
views
A normal distribution inequality
Let $n(x) := \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}$, and $N(x) := \int_{-\infty}^x n(t)dt$. I have plotted the curves of the both sides of the following inequality. The graph shows that the ...
- 2,239
16
votes
2
answers
1k
views
Is there always a way up?
I am trying to find a simple criterion for a real continuous function $f$ on a connected, open subset $U$ of $\mathbb R^n$ that would imply the following property (P)
For any $x, y \in U$ such that $f(...
- 1,608
16
votes
1
answer
487
views
Bull's-eye Riemann sum
Let $f:[a,b] \to \mathbb{R}^2$ be a continuous curve on the plane.
Question: Are there numbers $a \leq x \leq c \leq y \le b$ such that $$(c-a)f(x)+(b-c)f(y) = \int_a^b f(t) \, dt \ ?$$
In other ...
- 2,479
16
votes
1
answer
2k
views
Are continuous functions almost completely determined by their modulus of continuity?
Given a function $f: \mathbb{R}\to\mathbb{R}$, we define its left modulus of continuity, $L(f): \mathbb{R} \times (0, \infty)\to [0,\infty]$ by
$$L(f)(x, e) := \sup \{d \ge 0 \,:\, f((x, x+d)) \...
- 2,069
16
votes
2
answers
528
views
Lipschitz constant for map between triangles
Let $T_1$ and $T_2$ be any two euclidean triangles with labeled sides. The sides are labeled respectively $e_1^1,e_2^1,e_3^1$ and $e_1^2,e_2^2,e_3^2$. Call $A:T_1\rightarrow T_2$ the affine map which ...
user99087
16
votes
1
answer
661
views
Does every real function have this weak derivation property?
After this question : Does every real function have this weak continuity property?
Natrualy there are an other (more difficult) :
Is it true that for every real function $f:\mathbb{R}\to\mathbb{R}...
- 4,074
15
votes
4
answers
3k
views
No Tonelli or Fubini
Whenever we can interchange summation (perhaps due to Tonelli-Fubini), good things happen. Otherwise, one has to struggle evaluating double sums in just one way, because the alternative results in a ...
15
votes
3
answers
2k
views
Asymptotic expansion of $\sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$
I've been trying to find an asymptotic expansion of the following series
$$C(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{n!{\sqrt{n}} }$$
and
$$L(x) = \sum\limits_{n=1}^{\infty} \frac{x^{2n+1}}{...
- 153
15
votes
1
answer
1k
views
Characterizing $\mathbf{R}$ as an ordered group
A standard characterization of $\mathbf{R}$ uses the order and the field structure: any linearly ordered field that is archimedean and complete is isomorphic to $(\mathbf{R}, +, \times, <)$ as an ...
- 18.7k
15
votes
2
answers
1k
views
Is there a $C_c^{\infty}( \mathbb{R}^d)$ function whose Fourier transform we can explicitly write down?
I noticed that although $C_c^{\infty}$-functions are dense in some quite large spaces and well understood (especially their Fourier transform) I have never encountered an explicit example of a ...
- 181
15
votes
1
answer
2k
views
Real polynomials that go to infinity in all directions: how fast do they grow?
Let $f(x_1, \cdots, x_n) \in \mathbb{R}[x_1, \cdots, x_n]$ be a polynomial. Define property $\mathbf{P}$ to be the property that there exists a compact set $K \subset \mathbb{R}^n$ and a positive ...
- 26.9k
15
votes
3
answers
1k
views
Version of Banach-Steinhaus theorem
I am wondering about the following version of the Banach-Steinhaus theorem.
Let $A$ be a closed convex subset contained in the unit ball of a Banach space $X$ and consider bounded operators $T_n \in \...
- 536
15
votes
4
answers
1k
views
Is the sequence of Apéry numbers a Stieltjes moment sequence?
Consider the sequence of Apéry numbers
$$
A_n = \sum_{k=0}^n \binom{n}{k}\binom{n+k}{k}\sum_{j=0}^k \binom{k}{j}^3
= \sum_{k=0}^n \binom{n}{k}^2\binom{n+k}{k}^2 .
$$
In an email, physicist Alan Sokal ...
- 41.1k
15
votes
2
answers
680
views
Are Fourier transforms of L^p stable under diffeomorphisms?
Let $\xi$ be a compactly supported distribution on $\mathbb R^n$ and assume that its Fourier transform is in $L^p$. Let $\phi:\mathbb R^n\to\mathbb R^n$ be a diffeomorphism. Does the Fourier ...
- 2,639
15
votes
1
answer
595
views
Solving a non linear equation
I've been trying to prove that the following equation has a unique solution in interval 0 < x < 1 :
$$ x = \Big(\frac{1 - (1-x^2)^K}{1 - (1-x)^K}\Big)^2 $$
Where K is a number (integer, if it ...
- 159