**7**

votes

**0**answers

212 views

### Is $Q_n(x)=\sigma_{n+1}(x)/\sigma_n(x)$ logarithmically convex on $\mathbf{R}$?

In 1975 J. van de Lune considered the monotony properties of the canonical Riemann Upper and Lower sums for $\int_0^1 t^xdt$, with $x>0$.
Writing $\sigma_n(x) := 1^x+2^x+\cdots+n^x$ these sums are
...

**5**

votes

**2**answers

289 views

### Measures of full Hausdorff dimension for self-affine sets

Consider the iterated function system $T_{1}(x)=(\beta x,\tau y)$, $T_{2}(x,y)=(\beta x+(1-\beta),\tau y+ (1-\tau))$ for $\beta\in(1/2,1)$ and $\tau\in (0,1/2)$ with self affine set ...

**6**

votes

**3**answers

841 views

### Kronecker's Jugendtraum for real quadratic fields?

Kronecker's Jugendtraum (or Hilbert's 12'th problem) is to find abelian extensions of arbitrary number fields by adjoining `special' values of transcendental functions. The Kronecker-Weber theorem was ...

**10**

votes

**1**answer

705 views

### The Class Number One Problem for Real Quadratic Fields

An approach to the Gauß class number one problem for imaginary quadratic fields is to determine the integral points on the modular curve $Y_{nonsplit}(n)$ for a suitable $n$. Here follows a quick ...

**2**

votes

**1**answer

342 views

### S-matrix conjecture: status?

Is the $S$-matrix conjecture still open? I mean the one listed as Problem 7 in this survey.

**9**

votes

**1**answer

505 views

### Minimizing the excursion of a sum of unit vectors

I have $n$ unit-length vectors $v_i$ in $\mathbb{R}^3$, whose
sum is zero:
$$ v_1 + v_2 + \cdots + v_n = 0 \; .$$
Now I form the closed polygon $P$ in space by placing them head to tail.
So the ...

**1**

vote

**0**answers

221 views

### Sums of Strongly z-ideals

In the rings of continuous functions,i.e.$(C(X))$ an ideal $I$ is called strongly $z$-ideal if it is an intersection of some maximal ideals of $C(X)$. i.e. $$I=\cap_{\alpha \in A} ...

**1**

vote

**1**answer

527 views

### A question concerns prime numbers

I have arrived to this conjecture in my work, I am not sure that is true or false. So I would appreciate if someone give a counterexample or prove it.
My question: Let n be a non-prime such that n-1 ...

**2**

votes

**0**answers

308 views

### weak hopf conjecture

Hi:
I am thinking of the following problem which is related to weak Hopf conjecture:
Let $E$ be the total space of a vector bundle over a compact nonnegatively curved manifold $B$. Let k=the rank ...

**16**

votes

**0**answers

407 views

### Actions on ℍⁿ generated by torsion elements

Let $n$ be a large integer.
I am looking for a cocompact properly discontinuous isometric action on $n$-dimensional Lobachevky space which is generated by elements of finite order.
Or equivalently, ...

**7**

votes

**9**answers

2k views

### Not especially famous, long-open problems which higher mathematics beginners can understand

This is a pair to
Not especially famous, long-open problems which anyone can understand
So this time I'm asking for open questions so easy to state for students of subjects such as undergraduate ...

**41**

votes

**30**answers

5k views

### Fundamental problems whose solution seems completely out of reach [closed]

In many areas of mathematics there are fundamental problems that are embarrasingly natural or simple to state, but whose solution seem so out of reach that they are barely mentioned in the literature ...

**8**

votes

**1**answer

273 views

### Maximal class of simple graphs of order $n$ with mutually distinct numbers of spanning trees

This problem in some ways related to this post.
Let $A_n$ be the set of all integers $x$ such that there exist a connected simple graph of order $n$ having precisely $x$ spanning trees. Study the ...

**171**

votes

**93**answers

29k views

### Not especially famous, long-open problems which anyone can understand

Question: I'm asking for a big list of not especially famous, long open problems that anyone can understand. Community wiki, so one problem per answer, please.
Motivation: I plan to use this list ...

**24**

votes

**1**answer

1k views

### Important open questions in the field of Tropical geometry

What are some of the important unanswered questions in the field of tropical geometry?

**43**

votes

**1**answer

1k views

### Local structure of rational varieties

I've been asked this question by a colleague who's not an algebraic geometer; we both feel that the answer should be "no", but I don't have a clue how to prove it.
Here's the question:
let $X$ be a ...

**15**

votes

**3**answers

2k views

### How to Tackle the Smooth Poincare Conjecture

The last remaining problem in this whole "everything is a sphere" business, is the Smooth Poincare Conjecture in dimension 4: If $X\simeq_\text{homo.eq.} S^4$ then $X\approx_\text{diffeo} S^4$. ...

**17**

votes

**1**answer

820 views

### Does smooth and proper over $\mathbb Z$ imply rational?

Does smooth and proper over $\mathbb Z$ imply rational?
I think someone told me that this is a standard conjecture. Is it a widely held? held at all? Did someone in particular make this conjecture? ...

**45**

votes

**2**answers

2k views

### vector balancing problem

I believe the solution posted to the arXiv on June 17 by Marcus, Spielman, and Srivastava is correct.
This problem may be hard, so I don't expect an off-the-cuff solution. But can anyone suggest ...

**1**

vote

**0**answers

160 views

### Entropy of Bernoulli walks on semi-groups.

Consider the Fibonacci semi-group $<L,R|LRR=RLL>$ with a Bernoulli walk $P(R)=p, P(L)=1-p$. Is the entropy $H(p)$ an unimodal function with maximum at p=0.5? Is this true for all finitely ...

**22**

votes

**0**answers

2k views

### Open problems/questions in representation theory and around ?

What are open problems in representation theory ?
What are the sources (books/papers/sites) discussing this ?
Any kinds of problems/questions are welcome - big/small, vague/concrete.
Some estimation ...

**16**

votes

**0**answers

448 views

### Avoidable words

Let $u(x_1,...,x_n)$ be a word, $k\in \mathbb{N}$. We say that $u$ is $k$-avoidable if there exists an infinite word in $k$ letters $\{a_1,...,a_k\}$ which does not contain values of $u$ (i.e. words ...

**3**

votes

**0**answers

162 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

**2**answers

799 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

**2**answers

660 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

**1**answer

478 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 ...

**30**

votes

**1**answer

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**

vote

**0**answers

403 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 ...

**30**

votes

**2**answers

1k 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

**1**answer

369 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

**2**answers

626 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

**1**answer

627 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

**0**answers

731 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

**0**answers

776 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

**1**answer

266 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

**0**answers

846 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

**0**answers

164 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

**0**answers

375 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

**2**answers

550 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

**1**answer

777 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

**2**answers

450 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

**1**answer

493 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 $ ...

**19**

votes

**0**answers

965 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

**1**answer

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

**0**answers

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

**0**answers

496 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

**1**answer

580 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

**1**answer

444 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 $\mathbb R^4$ ...

**29**

votes

**0**answers

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

**1**answer

983 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 ...