**0**

votes

**0**answers

37 views

### Understanding Prof. Keevash's proof on the “Existence of Designs” [on hold]

I have tried to read Prof. Kevvash's paper on the "Existence of designs". I am finding it very tough to read it linearly. I am comfortable with the nibble ideas and probabilistic methods in general.
...

**2**

votes

**2**answers

197 views

### Gauss Codes that produce classical knots as opposed to virtual knots

I have been doing some research in Gauss codes and have been reading Kauffman's paper Virtual Knot Theory. In section 3.3, Theorem 2, he states that
If $K$ is a virtual knot whose underlying Gauss ...

**13**

votes

**0**answers

287 views

### Does $S^4$ have a “symplecto-homeomorphic” structure?

The 4-sphere cannot be a symplectic manifold. In particular, it does not admit an atlas whose transition maps are symplectomorphisms ...

**27**

votes

**3**answers

769 views

### Is the fixed point property for posets preserved by products?

Recall that a partially ordered set (poset) $P$ has the fixed point property (FPP) if any order preserving function $f:P\longrightarrow P$ has a fixed point.
Theorem. Suppose $P$ and $Q$ are posets ...

**3**

votes

**1**answer

71 views

### reduction to np hard ordering problem

I am trying to show a reduction from a problem of ordering problem to an np-hard problem that has approximation poly-time algorithm.
My problem is: I have M auctions and in each auction I have N ...

**1**

vote

**1**answer

51 views

### A possible minimal aperiodic set of corner Wang Tile

From one of my previous question Aperiodic set of corner Wang Tile (although it is put on hold), I realize there is a systematic way to construct aperiodic corner type of Wang tile from edge type ...

**2**

votes

**0**answers

146 views

### Both NP-hard but different [closed]

What's the fundamental difference between the Knapsack problem and the travelling salesman (TSP) problem both of which are NP-hard, while the reality is that TSP could be solved much much faster?

**6**

votes

**2**answers

265 views

### Update on list of open problems for Cherednik/Symplectic Reflection Algebras

Background:
There are two lists of open problems about Cherednik or Symplectic Reflection Algebras from 2007: Ian Gordon's Problems, Chapter 9 in Symplectic Reflection Algebras, and Ginzburg & ...

**5**

votes

**3**answers

483 views

### Is the Manickam-Miklós-Singhi Conjecture solved?

This arXiv paper is claimed to contain a proof for the MMS conjecture. But it seems that this manuscript is not yet peer reviewed by other mathematicians. I personally tried to follow the paper, but ...

**7**

votes

**1**answer

194 views

### Subspaces of $l_{1}$ are not Lipschitz complemented in $l_{1}$

I have thought about the following question for several years. This question may be stupid or not interesting. My question is: Is there a subspace $U$ of $l{_1}$ such that the quotient $l_{1}/U$ is ...

**10**

votes

**1**answer

234 views

### Is there a non-zero ghost map between finite suspension spectra?

A morphism $f\colon X\to Y$ of spectra such that for every integer $n$ the induced map $\pi_n(f)\colon\pi_n(X)\to\pi_n(Y)$ on stable homotopy groups is zero is called a ghost map.
Not every ghost map ...

**9**

votes

**1**answer

248 views

### A special case of the integer Hodge conjecture

Let $X$ be a projective complex manifold of dimension $n$.
Are torsion cohomology classes in $H^{2n-2}(X,\mathbb{Z})$ algebraic?
(We may assume, without loss of generality, that $n=3$, because of the ...

**4**

votes

**0**answers

157 views

### Packing space by cones: Translates best?

Let $C$ be a right circular cone, the convex hull of a unit-radius disk
and a point directly above the disk center at height $h$.
Is the ...

**0**

votes

**0**answers

147 views

### Is there a “natural” reformulation of Hodge conjecture in terms of L-functions?

I just glanced at the Wikipedia article about the Hodge conjecture, and a (probably very naive, due to my huge lack of knowledge of the subject) question just came to my mind: can one associate ...

**11**

votes

**0**answers

255 views

### Generating random finite groups

I would like a method to efficiently generate a random finite group of a given order $n$.
If there are $g(n)$ non-isomorphic groups of order $n$,
ideally each group would occur with probability ...

**1**

vote

**1**answer

90 views

### simple cycle analog in 2D (with application in tiling)

We know that any closed cycle of a graph could be decomposed into sum of simple cycles. To translate this theorem into tiling of 1D (Wang tile). We know that any 1D periodic tiling could be ...

**7**

votes

**1**answer

393 views

### Is there a precise notion of “almost all” such that almost all finite groups are Galois groups of extensions of the rationals?

I vainly tried to define a notion of "almost solvable group" such that every "almost solvable group" is the Galois group of a finite extension of the rationals, but I can't figure out the right way to ...

**1**

vote

**1**answer

157 views

### relationship between corner tile and edge tile of wang tile

It is clear that any corner type of Wang Tile could be converted to edge type of Wang Tile by defining the edge color according to the corner color.
However, could we convert edge type of Wang Tile ...

**35**

votes

**1**answer

1k views

### improving known bounds for Pierce expansions; cash prize

Here's a problem that I thought of back in 1978 or so, and only a little progress has been made on it since then. I still think about it from time to time, but probably not that many people have ...

**6**

votes

**1**answer

154 views

### Integral straight-line embeddings of planar graphs

Wikipedia says (in the article on FÃ¡ry's theorem),
"Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The ...

**17**

votes

**3**answers

974 views

### What is Chern-Simons theory expected to assign to a point?

Let $G$ be a compact, connected, (simply connected?) Lie group and let $k \in H^4(BG, \mathbb{Z})$ be a cohomology class. Witten showed, at a physical level of rigor, that this data determines a ...

**3**

votes

**2**answers

188 views

### Conjecture of a subset of Wang tile which might be decidable

From the two papers proving the undecidability of Wang tile in 1966 by Berger and in 1971 by RM Robinson, the tiles used in proving undecidability has a general common feature:
The left color and ...

**26**

votes

**0**answers

379 views

### Which region in the plane with a given area has the most domino tilings?

I just finished teaching a class in combinatorics in which I included a fairly easy upper bound on the number of domino tilings of a region in the plane as a function of its area. So this led to ...

**6**

votes

**1**answer

542 views

### What keeps asymptotic Goldbach's conjecture out of reach of current technology?

Despite the rather recent progress in prime number theory (see the proof of the ternary Goldbach conjecture by H.A. Helfgott, and the striking result of Yitang Zhang improved by Tao, Maynard and ...

**5**

votes

**6**answers

687 views

### practical algorithms for np complete problems

Inspired by:
Conjecture on NP-completeness of tesselation of Wang Tile up to finite size
And the practicality of this topic (solving tessellation on a lattice):
coloring in lattice
Computational ...

**15**

votes

**1**answer

585 views

### Paul ErdÅ‘s: Determine or estimate the number of maximal triangle-free graphs on n vertices

Among the collections of the open problems of Paul ErdÅ‘s on the website of
Professor Fan Chung, there is one called "number of triangle-free graphs".
...

**7**

votes

**3**answers

846 views

### Objections to and arguments for the simplicity of all Riemann zeros

It seems to be that the simplicity of all the zeros is quite widely accepted as a working hypotheses, and it is known that a positive proportion are as such.
Titchmarsh explains in the last chapter ...

**3**

votes

**2**answers

289 views

### An integral related to the Euler Gamma function

The question is from the paper http://arxiv.org/abs/1312.7115 (A curious formula related to the Euler Gamma function, by Bakir Farhi): is it possible to express the integral
$$\eta=2\int\limits_0^1 ...

**3**

votes

**1**answer

168 views

### Real analytic ergodic diffeomorphisms of the two sphere

Does there exists a real analytic area preserving ergodic diffeomorphism on $S^2$?
(Possibly by perturbing a rotation in the real-analytic topology?)

**7**

votes

**1**answer

479 views

### Ramanujan's problem 754 still open?

In addition to the MO question The Ramanujan Problems. , I would like to ask the following.
Problem 754 from the list of the Ramanujan's problems ( ...

**9**

votes

**0**answers

179 views

### Convergence in $L^2$ of iterated expectations

Take a probability space $(\Omega,\mathcal{F},\mathbf{P})$ and random variable $X \in L^2(\Omega,\mathcal{F},\mathbf{P})$.
Define the iterated expectations of X as follows: $X_0 = X$, and, ...

**13**

votes

**1**answer

314 views

### An infinite, amenable, finitely presentable group with no non-trivial finite quotients

My question is a simple one: is there a group with the properties in the title?
In the absence of the 'finitely presentable' hypothesis, an example is provided by Juschenko--Monod's construction of a ...

**25**

votes

**2**answers

887 views

### A group-theoretic perspective on Frankl's union closed problem

Here is a group theoretic phrasing of a special case of the union closed conjecture:
Question: Given a finite group $G$, is there an element of prime power order which is contained in at most half ...

**14**

votes

**0**answers

427 views

### An open problem in convex geometry

Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...

**17**

votes

**2**answers

727 views

### Floors of rationals to powers: Infinite number of primes?

Let $r=a/b$ be a rational number in lowest terms, larger than $1$,
and not an integer (so $b > 1$).
Q. Does the sequence
$$ \lfloor r \rfloor, \lfloor r^2 \rfloor, \lfloor r^3 \rfloor,
...

**4**

votes

**0**answers

199 views

### Stability of analytic Zariski structures

Noetherian Zariski structures are introduced by Hrushovski and Zilber(1996).
An analytic Zariski structure is a generalization of Noetherian Zariski structures, introduced by Zilber and Peatfield.
...

**6**

votes

**4**answers

1k views

### Interactions of number theoretic conjectures and other fields of mathematics

There are many interesting open conjectures in number theory. My question is not about partial results or possible ways to prove them. It is about their interactions with the other fields of ...

**1**

vote

**1**answer

256 views

### Any results towards the irrationality of the sum of reciprocals of perfect numbers? [closed]

This question is a follow up to my comment to Sum of the reciprocal of perfect numbers. I would like to know which results have been published about the possible irrationality of the sum of ...

**16**

votes

**1**answer

650 views

### A Question on 1, 2 ,3 Conjecture

1, 2, 3 conjecture is well-known:
If $G$ is a simple graph which is not $K_2$ then one can assign a number among $1, 2, 3$ to every edge such that if we label each vertex with the sum of the numbers ...

**7**

votes

**4**answers

453 views

### A Special Pair of Models for ZFC (New Version)

Are there two models $M$ and $N$ for $\text{ZFC}$ such that:
(1) $M\subseteq N$
(2) $\aleph_{1}^{N}=\aleph_{1}^{M}$
(3) $\aleph_{2}^{N}=\aleph_{\omega +1}^{M}$
Update: According to Peter's useful ...

**6**

votes

**1**answer

552 views

### A Hot Betting On HOD

Remark: This question is based on an open question at the end of a paper by Hamkins, Kirmayer, and Perlmutter: "Generalizations of the Kunen Inconistency".
$HOD$ as an inner model of $ZFC$ lies ...

**3**

votes

**0**answers

94 views

### A challenging non homogenous fractional inequality

I have posted this question on Stackexchange but it has received no answer so far. It is a challenging generalization of several difficult inequalities, where none of the usual methods used in ...

**19**

votes

**0**answers

345 views

### Decidability of equality of elementary expressions

In the following definition the term expression is to be understood as a finite tree built from formal symbols without any predefined meaning assigned to them.
Define the set $\mathcal{E}$ of ...

**9**

votes

**2**answers

628 views

### Is the Steiner ratio Gilbertâ€“Pollak conjecture still open?

Gilbert-Pollak conjecture on the Steiner ratio: Consider a set $P$ of $n$ points on the euclidean plane. A shortest
network interconnecting $P$ must be a tree, which is called a Steiner minimum ...

**11**

votes

**1**answer

988 views

### Is $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$ correct, where $n$ is an integer?

Is it true that $\varliminf_{n \rightarrow +\infty} |n \sin n| = 0$, where $n$
runs over the integers?
The existence of the limes inferior follows from Dirichlet's approximation theorem,
but the ...

**2**

votes

**4**answers

660 views

### Product of exponents of prime factorization

Let $p(n)$ be the product of the exponents of the prime factorization of $n$. For example,
$$p(5184) = p(2^6 3^4) = 24 \;,$$
$$p(65536) = p(2^{16}) = 16 \;.$$
Define $P(n)$ as the number of iterations ...

**2**

votes

**0**answers

142 views

### What are the divisors of $2n^2 - \sigma_{1}(n^2)$ for composite $n$?

What are the divisors of $2n^2 - \sigma_{1}(n^2)$ for composite $n$?
Here, $\sigma_{1}$ is the classical sum-of-divisors function. For example, $\sigma_{1}(3^2) = 1 + 3 + {3^2} = 13$.
(The function ...

**1**

vote

**0**answers

208 views

### Can six square numbers be simultaneously represented in a single sum of consecutive odd numbers? [closed]

I had some free time from my work to do a little exploration regarding the existence (or non existence) of perfect cuboids. A solution is represented by the set of Diophantine equations:
$a^2 + b^2 = ...

**6**

votes

**0**answers

322 views

### Graphs with graphic imbalance sequences

Let $G$ be simple undirected graph and $e=uv\in E(G)$.
The imbalance of the edge $e$ is the value $imb(e)=|d(u)-d(v)|$.
Let $M_{G}$ denotes the imbalance sequence (or more correctly, multiset of ...

**2**

votes

**1**answer

211 views

### Passing C through a slot

Question: Given a closed curve C, what will be the (bounds on) dimension of the interval it will pass through?
i.e. which are the necessary and sufficient conditions for a planar compact set C to ...