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?"
551
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Open problems in hyperplane/subspace arrangements?
What are some open problems in hyperplane/subspace arrangements, preferably of the combinatorial algebraic topology kind, and where can one read about them? That is, where are they discussed, and ...
4
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1
answer
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Open problems books [closed]
As the title might indicate , I would like to look for recommendations for mathematical book that present open problems in depth with commentary.
The only book of this type that I've come across is ...
6
votes
1
answer
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Open problems in Banach spaces, universality
I have gathered a list of universality problems in Banach spaces which have been solved:
1.The non existence of a separable reflexive space universal for the class of separable reflexive spaces.
2....
18
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8
answers
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Open problems in continued fractions theory
I propose to collect here open problems from the theory of continued fractions. Any types of continued fractions are welcome.
19
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2
answers
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*The* open problem in General Relativity?
Q. Is there a single, clear mathematical question that has emerged as
the open problem in General Relativity?
I ask this on the ~100th anniversary of Einstein's (4-page!) 1915 paper,
"Die ...
4
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0
answers
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What is the status on questions related to Bhargava's factorial function?
In Manjul Bhargava's The Factorial Function and Generalizations he motivates a new type of factorial $n!_S$ using by generalizing a few theorems like:
For $k, l \in \mathbb{Z}$, we have $k! \times l!$...
2
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0
answers
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On Descartes / spoof odd perfect numbers
Descartes, Frenicle, and subsequently Sorli, conjectured that $k = 1$, if $N = {q^k}{n^2}$ is an odd perfect number given in Eulerian form (i.e., $q$ is prime with $q \equiv k \equiv 1 \pmod 4$ and $\...
4
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0
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Are there any interesting open questions having to do with submodularity, specially in the intersection of theoretical machine learning? [closed]
I was interested in knowing about open research topics related with sub modularity, specially within its intersection with theoretical machine learning (and related topics).
It seems to me that much ...
2
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1
answer
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Reference request: Research done on whether the Euler prime can be the largest factor of an odd perfect number
(Note: This was cross-posted from MSE.) I posted the following reference request in MSE three (3) days ago, but was unable to elicit any responses. I am cross-posting it to MO, hoping that it is ...
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2
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Can any finite lattice be realized as an intermediate subgroups lattice?
Let $G$ be a finite group and $H$ a subgroup.
Let $\mathcal{L}(H \subset G )$ be the lattice of all the intermediate subgroups between $H$ and $G$.
Question: Can any finite lattice be realized as ...
19
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2
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Open problems in Berkovich geometry
I would like to know if there is a state of the art recent reference on non-archimedean analytic spaces mentioning/listing open problems, conjectures, unresolved questions in the theory (*). I have ...
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Collection of conjectures and open problems in graph theory
Is there something similar to the Kourovka Notebook for graph theory (or anyway an organized, possibly commented, collection of conjectures and open problems)?
5
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1
answer
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Ruth-Aaron triples, etc
A Ruth-Aaron pair is two numbers $(n,n+1)$ such that
their sum of prime factors is equal, counting repeated prime factors.
(The name refers to Hank Aaron's 715 homeruns surpassing Babe Ruth's 714!)
So
...
3
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1
answer
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Open problems in compressed sensing
What are the main open problems in compressed sensing?
I am interested in theoretical as well as in numerical point of view.
19
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1
answer
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Is every projective space curve a set-theoretic intersection of two surfaces? What is known about this question?
I am sorry if this question has been already asked, couldn't find anything similar myself. I have recently recalled this long standing open problem of whether every irreducible curve in $\mathbb{P}^3(\...
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0
answers
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Long paths in the supercritical percolation.
I have a question on the length of the longest path, denoted by $\ell_n$, in the supercritical percolation on $[0,n]^d$, denoted by $C_n$.
We know that $C_n$ has a giant component whose size is of ...
4
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0
answers
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Reduction argument from a general vertex set V(G) to a prime power in Prof. Keevash's proof on the Existence of Designs
The proof flow of the paper "On the Existence of Designs" by Prof. Keevash as I understand it is the following:
-- Reduction from the general case to $V = \mathbb{F}_{p^a}$ (Lemma 6.3)
-- Covering ...
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1
answer
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Inverted pair of complex analytic families
I read the following "problem" in an old set of notes of Morrow and Kodaira which focused on deformations of complex manifolds:
Find a pair of complex analytic families $\lbrace M_t\rbrace$ and $\...
9
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0
answers
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Randomly placing nonoverlapping unit cuboids
Suppose one places unit cuboids of dimension $d$ with min-corners
uniformly distributed to lie in $[0,n]^d$, but with cuboid (strict) overlap forbidden.
At some point, the region is "saturated," ...
4
votes
3
answers
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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 ...
16
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0
answers
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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 $(\mathbb{R}^4,\omega_\text{std})\to(\mathbb{R}^4,\omega_\text{std})...
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3
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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
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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
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1
answer
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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 ...
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2
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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 & ...
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4
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Is the Manickam-Miklós-Singhi Conjecture solved? [closed]
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 ...
10
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1
answer
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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 ...
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answer
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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 ...
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1
answer
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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 ...
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0
answers
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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
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1
answer
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Rost Correspondence and minimal Pfister-Neighbors
In http://www.math.uiuc.edu/K-theory/0357/ Karpenko utters the following:
Conjecture 1.6. If an anisotropic quadric $X = Q$ possesses a Rost correspondence,
then the quadratic form (defining $X$) ...
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2
answers
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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/g(n)...
1
vote
1
answer
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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 ...
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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 ...
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1
answer
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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 ...
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1
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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
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1
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Integral straight-line embeddings of planar graphs
Wikipedia says (in the article on Fáry'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 ...
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3
answers
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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 $3$-...
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2
answers
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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 ...
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1
answer
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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 ...
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1
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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 ...
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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 ...
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2
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Determine or estimate the number of maximal triangle-free graphs on $n$ vertices
Among the collections of the open problems of Paul Erdős on the website of
Professor Fan Chung, there is one called "number of triangle-free graphs".
http://www.math.ucsd.edu/~erdosproblems/erdos/...
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3
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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 ...
11
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3
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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
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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?)
10
votes
1
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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 ( http://www.imsc.res.in/~rao/ramanujan/...
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votes
0
answers
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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, ...
15
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1
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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 ...
37
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2
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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 ...