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 ...

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3
votes
0answers
156 views

What is the status of the subadditivity problem for analytic capacity?

Hi, Here is another question that concerns analytic capacity. For a compact set $K$ in the plane, define the analytic capacity of $K$ by $$\gamma(K):=\sup|f'(\infty)|,$$ where the supremum is taken ...
15
votes
2answers
756 views

Minimal graphs with a prescribed number of spanning trees

As its long ago since Erdős died and mathoverflow is the second best alternative to him (for discussing personal problems), I'd like to start a fruitful discussion about the following problem ...
11
votes
2answers
647 views

Interesting result on the Euler-Maschroni constant - what is the background?

Today I entered the following expression in maple: $$a_i = H_{10^i} - ln(10^i) - \gamma$$ Here $H_j$ equals $\sum_{k=1}^{j} 1/k$ and $\gamma$ is the Euler-Mascheroni constant. When I computed $a_n$ ...
4
votes
1answer
425 views

When is the norm of all positive operators on an ordered Banach space determined by their values on the positive cone?

I'm trying to investigate the interplay between the norm and cone of positive elements in ordered Banach spaces. In particular, I would like a nice characterization of when the norm of a positive ...
24
votes
0answers
1k views

Is S^2 x S^4 a complex manifold?

As observed by Calabi a long time ago, the manifold $S^2\times S^4$ admits an almost-complex structure (obtained by embedding it in $\mathbb{R}^7$ and using the octonionic product), which however is ...
-1
votes
1answer
2k views

On Odd Perfect Numbers

Hi all. My question today will be regarding what I consider to be a "stumbling block" while trying to research odd perfect numbers. Let $N = {q^k}{n^2}$ be an odd perfect number with Euler prime ...
1
vote
0answers
390 views

Reference Request - Jakob Weisblat's “The Search for the Odd Perfect Number” [closed]

Hi All! I am currently trying to locate an online copy of Jakob Weisblat's paper titled "The Search for the Odd Perfect Number". I could only get hold of the abstract: "A perfect number is a number ...
29
votes
1answer
912 views

Example of a compact Kähler manifold with non-finitely generated canonical ring?

A celebrated recent theorem of Birkar-Cascini-Hacon-McKernan and Siu says that the canonical ring $R(X)=\oplus_{m\geq 0}H^0(X,mK_X)$ of any smooth algebraic variety $X$ over $\mathbb{C}$ is a finitely ...
8
votes
1answer
362 views

Die-rolling Hamiltonian cycles

Let $R$ be a rectangular region of the integer lattice $\mathbb{Z}^2$, each of whose unit squares is labeled with a number in $\lbrace 1, 2, 3, 4, 5, 6 \rbrace$. Say that such a labeled $R$ is ...
12
votes
2answers
586 views

Are acyclic subcomplexes of finite contractible 2-complexes contractible?

Let $Y$ be a contractible finite simplicial 2-complex. Let $X$ be an acyclic subcomplex of $Y$ (i.e. $X$ connected, $H_1(X)=0$, $H_2(X)=0$). Is $X$ contractible? (Equivalently, is $\pi_1(X)$ ...
-1
votes
1answer
599 views

Question Re: Arian Berdellima's Papers On Odd Perfect Numbers [closed]

Hi everyone. I'd like to refer you to two papers by Arian Berdellima on odd perfect numbers: More Properties About Odd Perfect Numbers http://mpra.ub.uni-muenchen.de/31587/1/MPRA_paper_31587.pdf ...
18
votes
0answers
670 views

Given a lattice L with n elements, are there finite groups H < G such that L $\cong$ the lattice of subgroups between H and G?

If there is no restriction on $n$, this is a famous open problem. I'm wondering if any recent work has been done for small $n>6$. I believe the question is answered (positively) for $n=6$ by ...
4
votes
0answers
726 views

Has n^2*|sin(n)| limit? [closed]

Is there any sub-sequence $n_k$ of natural numbers such as $\lim(n_k^2|\sin(n_k)|) = 0$ ? when $k$ tends to infinity. In other words, does $\lim(n^2|\sin(n)|)$ exist (equal to infinity)? or there is ...
7
votes
1answer
257 views

Algorithmic decidability of equality in the ring of periods

Suppose two elements of the ring of periods are given by their systems of polynomial inequalities with rational coefficients. Is there a known algorithm deciding their equality? Is it known if their ...
9
votes
0answers
734 views

3-piece dissection of square to equilateral triangle?

At a workshop it was suggested that it likely remains an open problem whether or not there is a 3- or 2 -piece dissection of a square to an equilateral triangle. Can anyone confirm that this is ...
2
votes
0answers
163 views

What is the maximal density vertex subset of the 8-connected grid with induced vertex degrees $\leq 4$?

Let $G$ be the infinite graph defined by 8-connection of $\mathbb{Z}^2$. What is the maximal density vertex subset whose induced subgraph has maximum degree $\leq 4$? More precisely, what is the ...
3
votes
0answers
351 views

Compact surfaces smoothly immerse in: $\mathbb{R}^4$ or $\mathbb{R}^5$?

I wonder if someone can clarify whether it is known that every closed, orientable surface (2-manifold) has a smooth isometric immersion in $\mathbb{R}^4$? This topic has been discussed rather ...
13
votes
2answers
535 views

What is known about Ulam's problem of metric spaces with isometric squares?

Background In the book Problems in Modern Mathematics, S. Ulam asks the following question: Suppose $A$ and $B$ are metric spaces, such that $A^2$ and $B^2$ equipped with the 2-metric $d((x_1, ...
5
votes
1answer
712 views

Convergence speed of Jacobi eigenvalue algorithm for parallel ordering(Brent-Luk) ?

Is there estimate for convergence of the Jacobi eigenvalue algorithm for Hermitian matrices for "parallel ordring" (Brent-Luk ordering (see comment below)) ? For example for 4 4 matrices parallel ...
8
votes
2answers
441 views

Exact consistency-strength of “all projective sets are Ramsey”

I wonder if the exact consistency strength of "All projective sets have the Ramsey property" is still open. In Solovay's model, all sets have the Ramsey property, so the consistency strength of this ...
4
votes
1answer
485 views

What is the degree of a symmetric boolean function?

(previous title " Zero sum of binomials coefficients - a stronger version ") This is a stronger version of another question. Is there an $N\in \mathbb N$ and a sequence of non-constant functions $ ...
16
votes
0answers
813 views

When is a compact topological 4-manifold a CW complex?

Freedman's $E_8$-manifold is nontriangulable, as proved on page (xvi) of the Akbulut-McCarthy 1990 Princeton Mathematical Notes "Casson's invariant for oriented homology 3-spheres". Kirby showed that ...
10
votes
1answer
1k views

Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? [closed]

Is there a smooth $4$-manifold homeomorphic but not diffemorphic to $CP^2$? Are there known non-smooth examples homeomorphic $CP^2$?
3
votes
0answers
1k views

Open problems in “Algebraic geometry by robin hartshorne”

Hi, Is there a list of which of the open problems in algebraic geometry by robin hartshorne are still open ? (I searched the internet and didn't find one) Thanks from advanced, Matan Fattal
10
votes
0answers
483 views

Once differentiable, piecewise degree three polynomials on triangulated planar domains

Here is an easily described, but very difficult, problem that I (and a number of other people) really would like to see solved during our life times. The basic problem is to compute the dimension of ...
14
votes
1answer
571 views

Totally rational polytopes

Define a convex polytope in $\mathbb{R}^d$ as totally rational (my terminology) if its vertex coordinates are rational, its edge lengths are rational, its two-dimensional face areas are rational, ...
15
votes
1answer
432 views

Convex bodies with constant maximal section function in odd dimensions

In 1970 or so, Klee asked if a convex body in $\mathbb R^n$ ($n\ge 3$) whose maximal sections by hyperplanes in all directions have the same volume must be a ball. The counterexample in $\mathbb R^4$ ...
28
votes
0answers
1k views

Shortest closed curve to inspect a sphere

Let $S$ be a sphere in $\mathbb{R}^3$. Let $C$ be a closed curve in $\mathbb{R}^3$ disjoint from and exterior to $S$ which has the property that every point $x$ on $S$ is visible to some point $y$ of ...
15
votes
1answer
889 views

Axiom of choice and bases of vector spaces over a fixed field

Let $k$ be a field. In 1984 Andreas Blass proved that the axiom "for every extension $K|k$, every vector space over $K$ has a basis" implies the axiom of choice. He also raised the question Does ...
7
votes
1answer
370 views

A toy model for the t-section problem

Let $S(x)$ be the area of the yellow curvilinear triangle. I'd like to find a graph for which $S(x)=H(x)$ where $H $ is some prescribed function (small, smooth, vanishing near the endpoints to any ...
47
votes
21answers
13k views

Open problems with monetary rewards

Since the old days, many mathematicians have been used to attaching monetary rewards to problems they admit are difficult. Their reasons could be to draw other mathematicians' attention, to express ...
28
votes
8answers
4k views

Series whose convergence is not known

For most of the mathematical concepts I learn, it has more or less always been possible to find (at least google and find) unsolved problems pertaining to that specific concept. Keeping a bag of ...
11
votes
4answers
8k views

The Ramanujan Problems.

I originally thought of asking this question at the Mathematics Stackexchange, but then I decided that I'd have a better chance of a good discussion here. In the Wikipedia page on Ramanujan, there is ...
20
votes
1answer
788 views

Majorization and Schur Polynomials

Let me first define the majorization order (or dominance order) on partitions as $\lambda \succeq \mu$ iff $$\sum _{i=1}^{k}\lambda_i \geq \sum_{i=1}^{k}\mu_i$$ for all $1\le k\le l-1$ and ...
34
votes
4answers
2k views

The maximum of a polynomial on the unit circle

Encouraged by the progress made in a recently posted MO problem, here is a "conceptually related" problem originating from a 2003 joint paper of Sergei Konyagin and myself. Suppose we are given $n$ ...
11
votes
3answers
2k views

Zero divisor conjecture for finite fields

I believe that the most attractive "zero-divisor" conjecture is the existence of non-trivial zero-divisors in a group ring $\mathbb{C}[G]$ for a torsion free group $G$. For the sake of knowledge let ...
13
votes
3answers
1k views

Meromorphic 1-form and Picard's theorem

Let $D$ be the open unit disk in the complex plane and $U_1,U_2,\,\ldots\,,U_n$ be an open cover of the puntured disk $D^*= D\setminus\{0\}$. Suppose on each open set $U_j$ there is an injective ...
2
votes
0answers
544 views

Fixed point theorem for convex, closed multivalued mapping

There is well-known fixed point theorem theorem for multivalued l.s.c. maps, based on Michael selection theorem: Suppose, that $X$ is compact, convex and metrizable in locally convex Hausdorff ...
7
votes
1answer
382 views

Symplectic structures on a homotopy complex projective space

For $n>2$, there are infinitely many differentiable structures on the homotopy type of $\mathbb{C}P^n$. I want to know which differentiable structures support a symplectic form. My question is as ...
4
votes
1answer
715 views

Unsolved problems concerning Artinian Rings and Artinian Modules

I am preparing a write-up on Topics on Artinian Rings and Modules for a project. I hope to mention some unsolved problems in the domain of these objects along the way. Till now, I have been able to ...
22
votes
4answers
1k views

Trees in groups of exponential growth

Question: Let $G$ be a finitely generated group with exponential growth. Is there a finite generating set $S \subset G$, such that the associated Cayley graph $Cay(G,S)$ contains a binary tree? ...
4
votes
1answer
353 views

A question concerning products of finite cyclic groups

Let $m_1,\ldots, m_n$ be pairwise coprime natural numbers $\geq 1$. We consider the product $$G(m_1,\ldots,m_n) := \prod_{i=1}^{n} \mathbb{Z} / m_i \mathbb{Z}.$$ We define $M(n)$ as the set $n$-tuple ...
23
votes
1answer
1k views

Can you flip the end of a large exotic $\mathbb{R}^4$

Can you flip the end of a large exotic $\mathbb{R}^4$ Background Definition (Exotic $\mathbb{R}^4$): An exotic $\mathbb{R}^4$ is a smooth manifold $R$ homeomorphic but not diffeomorphic to ...
3
votes
3answers
732 views

Bounds on number of conjugacy classes in terms of number of elements of a group ?

What are bounds on number of conjugacy classes in terms of number of elements of a group ? (I allowed myself to edit the question in spirit of remarkable answers given to it by Gerry Myerson and ...
20
votes
3answers
4k views

Are nontrivial integer solutions known for $x^3+y^3+z^3=3$?

The Diophantine equation $$x^3+y^3+z^3=3$$ has four easy integer solutions: $(1,1,1)$ and the three permutations of $(4,4,-5)$. Elsenhans and Jahnel wrote in 2007 that these were all the solutions ...
7
votes
3answers
1k views

Kronecker Approximation theorem and Fibonacci numbers

There is a famous old theorem by Kronecker that for every positive real $\alpha$ and $\epsilon>0$ there exists a positive integer n such that $\alpha n$ is within $\epsilon$ of an integer. ...
0
votes
2answers
515 views

A simple question regarding the sum-of-divisors function

A good day to everyone. Consider the following "Conjecture": If $a, b \in M \subset \mathbb{N}$, then $1 < a < b$ and ... [plus some more conditions on $a, b$ and $M$...] if and only if ...
0
votes
1answer
364 views

Existence of Solutions to an Equation Involving the Sum-of-Divisors Function [Reference Request]

Let $\sigma(x) = \sigma_1(x)$ denote the sum of all the positive divisors of $x$. If $n \in \mathbb{N}$ is odd and $\gcd(n, \sigma(n)) = 1$, then do there exist any solutions to the following ...
2
votes
1answer
300 views

A question on a special type of function

Suppose I have a function $f$, and positive integers $x$ and $y$ such that $x$ is a square, $x \ne y$ and $y \ne \sqrt{x}$. Now, assume that: $|\frac{f(x)}{y} - \frac{f(y)}{x}| > 2$ ...
12
votes
3answers
906 views

Card game / options pricing / Brownian bridge question

We play a game. I shuffle a deck of cards and start dealing them face up. After any card you can say "stop", at which point I pay you 1 dollar for every red card dealt and you pay me 1 for every black ...