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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 ...
user23078's user avatar
  • 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\...
smyrlis's user avatar
  • 2,933
18 votes
3 answers
2k views

Elementary $\mathrm{Ext}^1$ intuition

$\DeclareMathOperator{\Hom}{\operatorname{Hom}}\DeclareMathOperator{\Ext}{\operatorname{Ext}}$I am wondering what sort of basic basic intuitive meaning $\Ext^1(M,N)$ has. As a base case: if $M$ and $N$...
alekzander's user avatar
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 ...
Mohammad Golshani's user avatar
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 ...
Paul's user avatar
  • 1,503
18 votes
1 answer
635 views

Is Carlitz's paper correct about the number of similarity classes of commuting matrices?

L. Carlitz has a paper, Classes of pairs of commuting matrices over a finite field, that computes the number of simultaneous similarity classes of of pairs of commuting matrices in $\operatorname{Mat}...
Yifeng Huang's user avatar
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 ...
Mateusz Kwaśnicki's user avatar
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,...
Tomás's user avatar
  • 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 ...
Noah Schweber's user avatar
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 ...
Daniel Friedan's user avatar
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 $...
D.R.'s user avatar
  • 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 ...
pi66's user avatar
  • 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!
Pandora's user avatar
  • 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}:\...
Piotr Hajlasz's user avatar
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 $f^{(0)} = f$, ...
Dominic van der Zypen's user avatar
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 ...
Paul Pfeiffer's user avatar
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 ...
Julian Newman's user avatar
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\...
TaQ's user avatar
  • 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 ...
KhashF's user avatar
  • 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&...
qp10's user avatar
  • 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}...
T. Amdeberhan's user avatar
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 ...
O.R.'s user avatar
  • 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} \...
Paresseux Nguyen's user avatar
17 votes
2 answers
1k views

The GCD-matrix: generalizing a result of Smith?

Let $M$ be the $n\times n$ matrix, known as the GCD matrix, of entries $M_{ij}=\gcd(i,j)$. In the paper H J S Smith, On the value of a certain arithmetical determinant, Proc. London Math. Soc. 7:208-...
T. Amdeberhan's user avatar
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: $$\sum_{i\neq j}|\langle v_i,v_j\rangle|.$$ ...
TOM's user avatar
  • 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 $\...
Isaac's user avatar
  • 3,477
17 votes
3 answers
3k views

Linear algebra from the categorical point of view

Is there any book of Linear algebra in the modern language of Category theory? I refer to the (systematic, formalist) study of the category whose objects are vector spaces and whose morphisms are ...
M. Carmona's user avatar
17 votes
1 answer
668 views

A coincidence or a fact: determinants of two matrices

While playing around with the MO question Determinant with factorials is not 0? about a determinant of the Hankel matrix of entries $(i+j-2)!$, having the value $\prod_{k=0}^{n-1}k!^2$, I stumbled on ...
T. Amdeberhan's user avatar
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 ...
Alex Gavrilov's user avatar
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?
Taras Banakh's user avatar
  • 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 ...
Mohammad Golshani's user avatar
17 votes
4 answers
10k views

Prime/undecomposable matrices

Prime matrices as defined in the following paper Prime matrices P. F. RIVETT AND N. I. P. MACKINNON carry over many properties of factorization as in natural numbers to matrices over the field of ...
Unknown's user avatar
  • 2,855
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): ...
Peter O.'s user avatar
  • 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 ...
Dominic van der Zypen's user avatar
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 $\...
wlad's user avatar
  • 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. ...
Saal Hardali's user avatar
  • 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 ...
Rajesh D's user avatar
  • 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 ...
T. Amdeberhan's user avatar
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 ...
H. Berbeleque's user avatar
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 (...
Ali Taghavi's user avatar
16 votes
4 answers
1k views

Reference for a linear algebra result

I asked the following question (https://math.stackexchange.com/questions/1487961/reference-for-every-finite-subgroup-of-operatornamegl-n-mathbbq-is-con) on math.stackexchange.com and received no ...
Stanley Yao Xiao's user avatar
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 ...
Will Sawin's user avatar
  • 148k
16 votes
4 answers
3k views

How many minors I need to check to conclude all minors will vanish ?

Given a $m \times n$ matrix $n>m$, I was trying to check if all its $m \times m$ minor vanish. I remember hearing that one really does not need to check all possible minors in order to conclude ...
Vagabond's user avatar
  • 1,795
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, ...
Tom Leinster's user avatar
  • 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 ...
YuerWu's user avatar
  • 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)! < \...
Ben Deitmar's user avatar
  • 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 ...
Nate River's user avatar
  • 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 ...
TPC's user avatar
  • 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 ...
Bogdan's user avatar
  • 161

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