**5**

votes

**1**answer

819 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

456 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

502 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 $ ...

**22**

votes

**1**answer

1k 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 ...

**11**

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 $CP^2$?

**4**

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

503 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

589 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

450 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 $\mathbb R^4$ ...

**31**

votes

**1**answer

2k 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

1k 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

**1**answer

390 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 ...

**58**

votes

**23**answers

22k 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 ...

**33**

votes

**8**answers

5k 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 ...

**12**

votes

**4**answers

10k 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 ...

**22**

votes

**2**answers

989 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

**4**answers

3k 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

**3**answers

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

**3**answers

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 ...

**3**

votes

**0**answers

583 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

**1**answer

431 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

**1**answer

954 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 ...

**24**

votes

**4**answers

2k 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

**1**answer

360 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 ...

**24**

votes

**1**answer

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

**3**answers

821 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 ...

**26**

votes

**4**answers

5k 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

**3**answers

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

**2**answers

540 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

**1**answer

401 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

**1**answer

314 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:
$|\frac{f(x)}{y} - \frac{f(y)}{x}| > 2$
...

**14**

votes

**3**answers

1k 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 ...

**17**

votes

**1**answer

864 views

### What's the Kirby Diagram of a universal $\mathbb{R}^4$?

What's the Kirby diagram of a universal $\mathbb{R}^4$?
Background
Define $\mathcal{R}$ as the set of smoothings of $\mathbb{R}^4$. For two oriented elements $R_1$, $R_2$ in $\mathcal{R}$ we can ...

**13**

votes

**0**answers

477 views

### Comparing the Kazhdan-Lusztig and Steinberg pre-orders

Both Kazhdan-Lusztig and Steinberg have defined pairs of preorders on $S_n$. Kazhdan and Lusztig's preorders come from their basis:
We write $x\leq_L y$ if any left ideal spanned by K-L basis ...

**0**

votes

**1**answer

359 views

### Something new in old question about sums of three polynomial cubes ?

An old problem asks whether or not the polynomial
$$
t \in \mathbb{Q}[t]
$$
is a sum of three cubes, (of polynomials in $\mathbb{Q}[t]$).
Question: Something new known now ?
Somebody has an idea of ...

**14**

votes

**2**answers

760 views

### Free subgroups vs law

Consider the following two conditions for a group $G$:
(1) $G$ does not satisfy a nontrivial law.
(2) $G$ contains a non-abelian free subgroup.
Obviously (2) implies (1) and it is easy to ...

**59**

votes

**1**answer

3k views

### Nontrivial finite group with trivial group homologies?

I stumbled across this question in a seminar-paper a long time ago:
Does there exist a positive integer $N$ such that if $G$ is a finite group with $\bigoplus_{i=1}^NH_i(G)=0$ then $G=\lbrace ...

**14**

votes

**1**answer

615 views

### Amenability of groups in terms of a perturbation condition

Let $G$ be a countable group and $\lambda \colon G \to U(\ell^2 G)$ its left-regular representation. Suppose that there exists a constant $C>0$ such that for all $T \in B(\ell^2 G)$
$$\inf ...

**4**

votes

**1**answer

448 views

### Implication of Polignac's conjecture on prime distribution in models of PA

Polignac's conjecture (PC) is that there exists infinitely many pairs of consecutive prime numbers that are a distance $d$ apart for some natural number $d$. The twin prime conjecture is the ...

**19**

votes

**2**answers

1k views

### unboundedness of number of integral points on elliptic curves?

If $E/\mathbf{Q}$ is an elliptic curve and we put it into minimal Weierstrass form, we can count how many integral points it has. A theorem of Siegel tells us that this number $n(E)$ is finite, and ...

**6**

votes

**0**answers

298 views

### Real approximation for homogeneous spaces of linear algebraic groups

Let $X$ be a smooth geometrically integral variety over $\mathbf{Q}$, having a $\mathbf{Q}$-point.
We say that $X$ has the real approximation property if $X(\mathbf{Q})$ is dense in $X(\mathbf{R})$.
...

**8**

votes

**3**answers

1k views

### Prepending strings to primes.

Hello,
we all know that 31,331,3331,33331,333331,3333331,33333331 all are primes, and that 333333331 is not. Here we prepend the digit 3 to 31, to get a list of 7 primes.This gives me the following ...

**8**

votes

**1**answer

2k views

### The Invariant Subspace Problem: examples

Question. Is there a concrete example of a bounded linear operator on a Hilbert space for which it is not known if it has a non-trivial closed invariant subspace?
[Added 24.01.2011: According to ...

**22**

votes

**1**answer

2k views

### Iwasawa main conjectures vs Bloch-Kato conjectures

Let $p$ be a prime, $K$ be a number field, $S$ a finite set of finite places of $K$ containing the set $S_p$ of places above $p$ and the places at infinity, $G:=G_{K,S}$ the Galois group of the ...

**0**

votes

**1**answer

802 views

### Perfect Numbers - On Mersenne and Euler Primes

Hi, I apologize if there is already an (obvious) answer to my question, but please bear with me for the moment as I find it hard to see a good way to answer this question:
In the same way that the ...

**19**

votes

**2**answers

5k views

### Is the Invariant Subspace Problem interesting?

There's an amusing comment in Peter Lax's Functional Analysis book. After a brief description of the Invariant Subspace Problem, he says (paraphrasing) "...this question is still open. It is also an ...

**1**

vote

**1**answer

483 views

### Infinitely many pairs of primes?

Hi. I want to know how many (infinitely many) pairs of primes are known.
For convinience, let me give two definitions.
For any nonconstant polynomial $f(x)\in \mathbb{Z}[x]$, define $A_{f}=\lbrace ...

**3**

votes

**3**answers

2k views

### Re: Mordell's Equation $y^2 = x^3 + k$ and Perfect Numbers

I have already tried a somewhat exhaustive search of the literature, but couldn't find anything close to the problem that I am working on.
My question is: When does Mordell's Equation
$$y^2 = x^3 ...

**2**

votes

**1**answer

536 views

### Reference Request - Sharp Estimates for a Logarithmic Sum

Can anybody suggest a good (e.g. "non-technical") introduction to estimating bounds for logarithmic sums of the form
$$\sum_{i=1}^{r}{{\alpha_i}{\log(q_i)}}$$
where the $$\alpha_i$$ are positive ...

**4**

votes

**2**answers

926 views

### On Sorli's Conjecture Re: OPNs (Circa 2003)

In the PhD dissertation titled "Algorithms in the Study of Multiperfect and Odd Perfect Numbers" (hyperlinked here) and completed in 2003, Ronald Sorli conjectured that the exponent $k$ on the Euler ...