Questions tagged [open-problems]

If it turns out that a problem is equivalent to a known open problem, then the open-problem tag is added. After that, the question essentially becomes, "What is known about this problem? What are some possible ways to approach this problem? What are some ways that people have tried to attack it before, and with what results?"

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2answers
307 views

A possible dynamical approach to the “Invariant Subspace Problem”

In the literature, is there any paper or research investigating the invariant subspace problem with consideration of differential operators acting on an appropriate Sobolev space?In particular is ...
2
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0answers
78 views

Open problems concerning Araujo's biseparating maps

Araujo stated the following four open questions at the end of his paper, page $518$ and $519.$ Question $1:$ Assume that there exists a biseparating map $T:A^n(\Omega:E)\to A^m(\Omega',F)$ which is ...
12
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1answer
994 views

What is known about the relationship between Fermat's last theorem and Peano Arithmetic?

As far as I know, whether Fermat's Last Theorem is provable in Peano Arithmetic is an open problem. What is known about this problem? In particular, what is known about the arithmetic systems $PA + \...
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0answers
240 views

Open problems in the theory of manifolds relating to construction [closed]

A while ago I stumbled across a paper of Thurston: Some Simple Examples of Symplectic Manifolds, where Thurston constructs closed symplectic manifolds with no Kaehler structure. My question is: What ...
9
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4answers
1k views

What is the smallest sphere whose surface includes 100 integer points?

Let $S(r)$ be the surface of the origin-centered sphere in $\mathbb{R}^3$. A point is an integer point if all its coordinates are integers. What is the smallest radius $r_n$ such that $S(r_n)$ ...
7
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0answers
335 views

On a question of Coste & Roy from 1979

On page 44 of their 1979 paper, Topologies for real algebraic geometry, Coste & Roy define a structure sheaf on the real Zariski spectrum of a commutative ring (which can be regarded as the real ...
3
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0answers
263 views

If $a^3+b^3+c^3=N$, then $x^3+y^3+z^3+t^3 = N$ in infinitely many ways?

It is well-known that, $$a^3+b^3+c^3 = N\tag1$$ for $N=1,\,2$ is solvable in the integers in infinitely many ways . However, it is an open question (but is conjectured) that if for general $N$ it has ...
3
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1answer
208 views

Asymptotic form of pdf of Escape Time of arithmetic fBm

I am trying to apply the Girsanov formula and Doobs optional sampling theorem to obtain an asymptotic form of first passage density of an fbm process with drift, but the answer i am getting seems ...
1
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1answer
194 views

Portability of Thompson theorem about solvability to Moufang loops

Say we have a finite Moufang Loop $Q$, $|Q|<\infty$. There is a theorem proved by Thompson that states: Group $G$, $|G|<\infty$ is solvable $\iff$ $\forall a, b \in G \langle a, b\rangle$ is ...
11
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3answers
1k views

Undecidable easy arithmetical statement

Is there a basic arithmetic statement which is known to be undecidable ? By basic arithmetic statement I do mean an easy statement in the spirit of the Collatz conjecture . By the way is there some ...
7
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1answer
150 views

On statistical bases in Banach spaces

Let $K$ be a subset of the positive integers $\mathbb{N}$. For each $n\in \mathbb{N}$, $K_{n}$ denotes the set $\{k\in K: k\leq n\}$ and $|K_{n}|$ denotes the number of the elements in $K_{n}$. The ...
3
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1answer
266 views

Can one find a Jordan curve which has exactly one inscribed rectangle?

In On the number of inscribed squares of a simple closed curve in the plane it is shown that Theorem: For every positive integer $n$ there is a simple closed curve in the plane (which can be ...
13
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1answer
239 views

Equilaterally triangulated surfaces with prescribed boundary

There is a problem in Richard Kenyon's list which I would like to post here, because although I have thought about it from time to time, I have not been able to make the slightest progress on it: ...
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0answers
100 views

What is an umbilic point of a convex polyhedron?

An umbilic point of a smooth ($C^2$) convex surface in Euclidean 3-space is a point where the principal curvatures are equal. Is there some good way to generalize this notion to convex polyhedra? See ...
12
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2answers
585 views

Intrinsic vs Extrinsic geometry of convex surfaces

By Alexandrov's isometric embedding theorem, any locally convex metric prescribed on the sphere admits a realization as a convex surface in Euclidean 3-space, which, by Pogorelov's rigidity result, is ...
74
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0answers
2k views

Converse to Euclid's fifth postulate

There is a fascinating open problem in Riemannian Geometry which I would like to advertise here because I do not think that it is as well-known as it deserves to be. Euclid's famous fifth postulate, ...
8
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0answers
194 views

Plank invariant measures on convex bodies

Let $K\subset R^2$ be a convex body, i.e., a compact convex set with interior points. A plank $P$ is the region between a pair of parallel lines in $R^2$. Let us say that $P$ intersects $K$ properly ...
32
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0answers
996 views

Converse of the Archimedean property of the sphere

In his remarkable book On the Sphere and Cylinder, where he came tantalizingly close to discovering calculus, Archimedes showed that the area of the portion of the sphere contained between a pair of ...
16
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2answers
516 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 ...
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0answers
228 views

Does this idea give an algorithm for constructing Hadamard matrices?

Fedor Petrov's answer of my preceding question shows that my question reduces to the famous Hadamard conjecture about Hadamard matrices of order $4k$. So I decided to study this conjecture and I got ...
15
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1answer
644 views

Does the image of the exponential map generate the group?

Let $G$ be a connected Fréchet-Lie group and let $\mathfrak g$ be its Lie algebra. Does the image $\exp(\mathfrak g) \subset G$ of the exponential map generate $G$?
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4answers
4k views

Is there always one integer between these two rational numbers?

It appears that for each integer $k\geq2$, there is always one integer $c$ that satisfies the inequalities below. Can this be proved? $$\frac{3^k-2^k}{2^k-1}<c\leq \frac{3^k-1}{2^k}.$$ Note that ...
14
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1answer
649 views

Who conjectured the Cartan determinant conjecture

The Cartan determinant conjecture states that every finite dimensional algebra of finite global dimension has the property that the determinant of its Cartan matrix is equal to one. Who stated this ...
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0answers
78 views

Modular which is metrizing but does not satisfy the $\Delta_2$ condition

Let $\Phi$ be a nice Young function (N-function) and $(\Omega,\mathcal{F},P)$ a probability space such that either $P$ is diffuse on a set of non-zero probability or $P$ is purely atomic and there are ...
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0answers
55 views

Splitting of ordinals of oscillation ranks of a Baire $1$ function

Denny and Tang proved that Theorem $2.3$ Let $(f_n)$ be a sequence in $\mathfrak{B}_1(K)$ converging pointwise to a function $f.$ Suppose $\sup\{ \beta(f_n):n\in\mathbb{N} \} \leq \beta_0$ and $\...
2
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0answers
214 views

A cubic system with two nested limit cycles with opposite orientations(2)

The second part of Hilbert's 16th problem not only concerns "The number of limit cycles of a polynomial vector field", but also the position and configuration of of those limit cycles with respect to ...
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0answers
479 views

Limit cycles as closed geodesics(2)

Hilbert 16th problem asks for a uniform upper bound $H(n)$ for the number of limit cycles of a polynomial vector field of degree $n$ on the plane. Here is an updated proof of the ...
4
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0answers
394 views

Limit cycles of quadratic systems and closed geodesics(Finitness of $H(2)$)

This question is inspired by this answer to the question Finding a 1-form adapted to a smooth flow. Assume that $V$ is a polynomial vector field of degree $2$ as follows:$$\begin{cases} x'=P(...
13
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3answers
1k views

Current state of the Komlos conjecture on vector balancing

Komlos Conjecture: the exists an absolute constant $K>0$ such that for all $d$ and any collection of vectors $v_1,\ldots, v_n\in \mathbb{R}^d$ with $\left\lVert v_i\right\rVert _2=1$ we can find ...
35
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6answers
8k views

Open problems in mathematical physics

What are good, still unsolved problems in mathematical physics that are in vogue? I always get the same answers: reference to Millennium Problems by the Clay Institute, or "there's still a lot to do ...
13
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3answers
424 views

Random N-body problem

Suppose there are $N$ unit-mass particles whose initial positions are uniformly distributed in a unit-radius disk. Each particle is assigned a randomly oriented initial velocity vector $v_i$ of length ...
15
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1answer
564 views

Open bilinear maps that are not uniformly open

A map $f\colon X\to Y$ between metric spaces is uniformly open whenever for each $\varepsilon >0$ there is $\delta >0$ such that for any $x\in X$ one has $$B_Y\big(f(x),\delta\big)\subseteq f\...
12
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2answers
612 views

Is the fundamental group of any compact hyperbolic 3-manifold embeddable into a p-adic group?

Is it true that for every compact hyperbolic $3$-manifold $M$ there exists a prime $p$, a finite field extension $K/\mathbb{Q}_p$, and an injective group homomorphism $$\tau \colon \pi_1(M) \to \...
3
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1answer
237 views

Hausdorff's question on $\omega_1$-gap

I read here that the following problem of Hausdorff is apparently still open. Is there a maximal branch $C$ in the poset $\omega^\omega$ with the eventual domination order, such that $C$ has no $\...
4
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1answer
226 views

Surface bundles over surfaces with(out) flat structure

I vaguely remember that I once attended a seminar or conference talk in which it was mentioned that the following question is open. Is there a (smooth) surface bundle over a surface $\Sigma_h \to E \...
6
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1answer
228 views

Every PD group is $\pi_1$ of an aspherical manifold

It is conjectured that for a discrete, finitely presented group $G$ such that $BG$ satisfies Poincaré duality, there actually exists a closed manifold $M$ which is homotopy equivalent to $BG$. This ...
14
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3answers
981 views

Can there be a polymath project for mathematical physics?

My hunch is that it might be possible to create something like https://polymathprojects.org/ for mathematical physics and I'd like to know whether MathOverflow users can recommend some appropriate ...
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0answers
492 views

The derivative of an integral function with indicator and max function as integrand

I encounter the following type of problem: \begin{equation} F(x) = \int_a^b \mathbf{1}_{\{v+x-h(v)\geq 0\}}\max\{h(v)-y-x,0\}dv \end{equation} where $\mathbf{1}_{\{z\geq 0\}}=1$ if event $z\geq 0$ ...
15
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0answers
538 views

Decidable open problems

Are there any significant open problems in mathematics which are clearly decidable (in that it is easy to write a clearly corresponding program which will eventually output either Yes or No (or ...
1
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1answer
108 views

Expectation of changing the gift choice [closed]

Suppose we are given two boxes, with one of gift valued $n$ dollars and the other one valued twice as much. We can pick a box, and after open it we have the choice of switching to another box. Shall ...
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2answers
1k views

Current state of Straus's illumination problem

In George W. Tokarsky's Polygonal Rooms Not Illuminable from Every Point (1995) it is stated that the problem Is a polygonal region illuminable from at least one point in the region? was still ...
16
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2answers
2k views

Is the Gromov conjecture still open?

Today I read about Gromov's definition of minimal volume for smooth manifolds. $$\min {\rm Vol}(M):=\inf_{|K_g|\leq1}\{{\rm Vol}(M,g)\}.$$ Gromov's conjecture states that for every closed simply ...
3
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1answer
542 views

What arguments do exist against defining completeness in NP using injective Karp reductions?

It is crucial to use the right notion of reduction to define completeness inside NP. Different notions of completeness inside NP may have significant impact on the properties of complete languages. ...
4
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0answers
154 views

Remaining models conjectured to converge to SLE(6) or CLE(6)

I am wondering which models are conjectured (eg. numerically) to converge to SLE(6) (Schramm-Loewner evolution with $\kappa=6$) or CLE(6) (conformal loop ensemble). I am searching for a research topic ...
4
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0answers
457 views

Does every separable Banach space have a Markushevich–Auerbach basis?

Let $X$ be a separable Banach space and $X^*$ be its dual, let $\{x_i\}$ be a sequence in $X$ with dense linear span and such that there exists a sequence $\{x_i^*\}$ in $X^*$ satisfying $x_i^*(x_j)=\...
4
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1answer
215 views

Can we solve the FGF problem by finding an appropriate action?

If we can find an action of the free group $\mathbb{F}_2$ on a measure space $X$ such that the crossed product $M=L^∞(X)⋊\mathbb{F}_2$ is a ${\rm III}_1$ factor with core isomorphic to $L(\mathbb{F}_2)...
2
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0answers
76 views

Pro-V topology on a free group

Let $W=G_p*G_q$, where $G_p$ and $G_q$ are pseudovarieties of all finite $p$-groups and all finite $q$-groups respectively, with $p$ and $q$ fixed prime numbers. The pro-$W$ topology on a group $G$ is ...
2
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1answer
211 views

On attempting a proof for $r > 1$, if $M = {2^r}{b^2}$ is an even almost perfect number which is not a power of two

(Preamble: I first thought that this question might be more appropriate for MSE. However, I posted it here nonetheless in the hope that someone with that brilliant idea can help with answering my ...
31
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1answer
1k views

A long-lasting conjecture of Pólya & Szegő

There is a conjecture by Pólya & Szegő (~1950, stated in p. 159 of their book Isoperimetric Inequalties in Mathematical Physics) which is as follows: "Of all $n$-gons of a fixed area, the regular ...
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0answers
238 views

On even almost perfect numbers other than powers of two

(Note: This question is an improved version of and has been cross-posted from this MSE post.) Let $\sigma(x)$ denote the sum of the divisors of $x$. If $\sigma(x) = 2x - 1$, then we call $x$ an ...

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