All Questions
1,585 questions
94
votes
5
answers
10k
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Is there a dense subset of the real plane with all pairwise distances rational?
I heard the following two questions recently from Carl Mummert, who encouraged me to spread them around. Part of his motivation for the questions was to give the subject of computable model theory ...
- 236k
81
votes
4
answers
8k
views
Did Gelfand's theory of commutative Banach algebras influence algebraic geometers?
Guillemin and Sternberg wrote the following in 1987 in a short article called "Some remarks on I.M. Gelfand's works" accompanying Gelfand's Collected Papers, Volume I:
The theory of commutative ...
- 7,329
81
votes
3
answers
9k
views
Norms of commutators
If an $n$ by $n$ complex matrix $A$ has trace zero, then it is a commutator, which means that there are $n$ by $n$ matrices $B$ and $C$ so that $A= BC-CB$. What is the order of the best constant $\...
- 31.5k
68
votes
2
answers
2k
views
Continuous maps which send intervals of $\mathbb{R}$ to convex subsets of $\mathbb{R}^2$
Let $f : \mathbb{R} \longrightarrow \mathbb{R}^2$ be a continuous map which sends any interval $I \subseteq \mathbb{R}$ to a convex subset $f(I)$ of $\mathbb{R}^2$. Is it true that there must be a ...
- 629
63
votes
5
answers
10k
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Jean Bourgain's relatively lesser known significant contributions
Jean Bourgain passed away on December 22, 2018.
A great mathematician is no longer with us.
Terry Tao has blogged about Bourgain's death and mentioned some of his more recent significant contributions,...
52
votes
3
answers
5k
views
Is the "Napkin conjecture" open? (origami)
The falsity of the following conjecture would be a nice counter-intuitive fact.
Given a square sheet of perimeter $P$, when folding it along origami moves, you end up with some polygonal flat figure ...
- 1,306
51
votes
2
answers
5k
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A strengthening of the Cauchy-Schwarz inequality
Suppose $\mathbf{v},\mathbf{w} \in \mathbb{R}^n$ (and if it helps, you can assume they each have non-negative entries), and let $\mathbf{v}^2,\mathbf{w}^2$ denote the vectors whose entries are the ...
- 6,351
49
votes
5
answers
3k
views
If a unitsquare is partitioned into 101 triangles, is the area of one at least 1%?
Update: The answer to the title question is no, as pointed out by Tapio and Willie. I would be more interested in lower bounds.
Monsky's famous theorem with amazingly tricky proof says that if we ...
- 19k
49
votes
5
answers
5k
views
Is Lebesgue's "universal covering" problem still open?
The following problem has been attributed to Lebesgue. Let "set" denote any subset of the Euclidean plane. What is the greatest lower bound of the diameter of any set which contains a subset congruent ...
- 6,215
48
votes
6
answers
7k
views
Is there an "elegant" non-recursive formula for these coefficients? Also, how can one get proofs of these patterns?
Not sure if this is a "good" question for this forum or if it'll get panned, but here goes anyway...
Consider this problem. I've been trying to find a formula to expand the "regular iteration" of "...
47
votes
7
answers
5k
views
Intuitive proof that the first $(n-2)$ coordinates on a sphere are uniform in a ball
It is a classical fact that if $(x_1,\ldots,x_n)$ is a random vector uniformly distributed on the sphere $S^{n-1} \subseteq \mathbb{R}^n$, then the random vector $(x_1,\ldots,x_{n-2})$ is uniformly ...
- 11.4k
47
votes
6
answers
6k
views
Can we actually find any fixed points with Brouwer's theorem?
Background
At the risk of greatly oversimplifying matters, let me state a heuristic from Granas and Dugundji's beautiful book: fixed point theorems fall into two broad categories. The first class is ...
- 15.5k
43
votes
1
answer
5k
views
Can $L^p(\mathbb{R})$ and $ L^q(\mathbb{R})$ be isomorphic?
Let $p,q \in (1,\infty)$ with $p\neq q$. Are the Banach spaces $L^p(\mathbb{R})$, $L^q(\mathbb{R})$ isomorphic?
- 559
42
votes
10
answers
5k
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Is there a mathematical axiomatization of time (other than, perhaps, entropy)?
Since Euclid's axiomatization of space, we have developed a sophisticated mathematical model of space. Given a category of structures (measures), local space is modeled the spectrum of measurements ...
40
votes
5
answers
5k
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"Entropy" proof of Brunn-Minkowski Inequality?
I read in an information theory textbook the Brunn-Minkowski inequality follows from the Entropy Power inequality.
The first one says that if $A,B$ are convex polygons in $\mathbb{R}^d$, then
$$ m(...
- 22.8k
39
votes
3
answers
14k
views
Is the Invariant Subspace Problem interesting?
There's an amusing comment in Peter Lax's Functional Analysis book. After a brief description of the Invariant Subspace Problem, he says (paraphrasing) "...this question is still open. It is also an ...
- 1,241
38
votes
2
answers
13k
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What, exactly, has Louis de Branges proved about the Riemann Hypothesis?
I know this is a dangerous topic which could attract many cranks and nutters, but:
According to Wikipedia [and probably his own website, but I have a hard time seeing exactly what he's claiming] Louis ...
- 1,990
37
votes
4
answers
4k
views
Which differential equations allow for a variational formulation?
Many ODE's and PDE's arising in nature have a variational formulation. An example of what I mean is the following. Classical motions are solutions $q(t)$ to Lagrange's equation
$$
\frac{d}{dt}\frac{\...
- 7,583
37
votes
3
answers
3k
views
"Softness" vs "rigidity" in Geometry
According to common wisdom, there are structures in Geometry that have a more "topological" flavor, others that are more "geometrical", and others that are halfway between. Usually,...
- 23.3k
36
votes
2
answers
2k
views
Bodies of constant width?
In two-dimensional case one can generalize figures of constant width as figures which can rotate in a convex polygon.
Here is one example which can be used to drill triangular holes:
I would like to ...
- 45k
35
votes
2
answers
2k
views
Is it consistent with ZF that $V \to V^{\ast \ast}$ is always an isomorphism?
Let $k$ be a field and $V$ a $k$-vector space. Then there is a map $V \to V^{\ast \ast}$, where $V^{\ast}$ is the dual vector space. If we are in ZFC and $\dim V$ is infinite, then this map is not ...
- 156k
35
votes
2
answers
9k
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tr(ab) = tr(ba)?
It is well known that given two Hilbert-Schmidt operators $a$ and $b$ on a Hilbert space $H$, their product is trace class and $tr(ab)=tr(ba)$. A similar result holds for $a$ bounded and $b$ trace ...
- 43.2k
34
votes
1
answer
3k
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tr(ab)=tr(ba), part 2.
This is a Banach space version of Andre Henriques' question
Trace Question
for Hilbert spaces. Let $a:X\to Y$ and $b:Y\to X$ be bounded linear operators between Banach spaces s.t. $ba$ and $ab$ ...
- 31.5k
34
votes
1
answer
4k
views
Theme of Isbell duality
Let $C$ be a small category. Isbell duality provides an adjunction $\widehat{C} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}}\widehat{C^{\mathrm{op}}}^{\mathrm{op}}$....
- 63.1k
34
votes
3
answers
3k
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What is the best way to peel fruit?
A mango made me wonder about this. (See also this question, which is in a similar spirit.)
Fix $L >0$ and a smooth body (possibly nonconvex—pears or bananas are fair game!) $B \subset \mathbb{R}^3$...
- 15.4k
32
votes
4
answers
4k
views
Largest hyperbolic disk embeddable in Euclidean 3-space?
Hilbert proved that there's no complete regular ($C^k$ for sufficiently large $k$) isometric embedding of the hyperbolic plane into $\mathbb{R}^3$. On the other hand, the pseudosphere is locally ...
- 13.6k
32
votes
0
answers
921
views
Isometric embeddings of finite subsets of $\ell_2$ into infinite-dimensional Banach spaces
Question: Does there exist a finite subset $F$ of $\ell_2$ and an infinite-dimensional Banach space $X$ such that $F$ does not admit an isometric embedding into $X$?
There are some results of the ...
- 4,878
32
votes
2
answers
4k
views
Are there non-reflexive vector spaces isomorphic to their bi-dual?
Let $V$ be an infinite dimensional topological vector space and consider the natural application $\iota\colon V\to V^{**}$. The space $V$ is said to be reflexive if $\iota$ is an isomorphism.
Are ...
- 7,902
32
votes
19
answers
23k
views
Good books on theory of distributions
Hi all.
I'm looking for english books with a good coverage of distribution theory.
I'm a fan of Folland's Real analysis, but it only gives elementary notions on distributions.
Thanks in advance.
30
votes
2
answers
1k
views
Shortest path through $\sqrt{n}$ points out of $n$
Say I sample $n$ points uniformly at random in the unit square, and then I look for the shortest path through $\sqrt{n}$ of those points (rounding up, say). What happens to the length of this path as ...
- 335
30
votes
3
answers
1k
views
Diameter of m-fold cover
Let $M$ be a closed Riemannian manifold.
Assume $\tilde M$ is a connected Riemannian $m$-fold cover of $M$.
Is it true that
$$\mathop{diam}\tilde M\le m\cdot \mathop{diam} M\ ?\ \ \ \ \ \ \ (*)$$
...
30
votes
0
answers
1k
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Curves on potatoes
On twitter recently, Robin Houston brought up this problem from a mathematical puzzle book of Peter Winkler:
The puzzle is attributed to the book "The mathemagician and pied puzzler", and ...
- 68.9k
29
votes
1
answer
812
views
Running most of the time in a connected set
Let $P$ be a compact connected set in the plane and $x,y\in P$.
Is it always possible to connect $x$ to $y$ by a path $\gamma$ such that the length of $\gamma\backslash P$ is arbitrary small?
...
- 45k
28
votes
7
answers
13k
views
Regular borel measures on metric spaces
When teaching Measure Theory last year, I convinced myself that a finite measure defined on the Borel subsets of a (compact; separable complete?) metric space was automatically regular. I used the ...
- 18.7k
28
votes
2
answers
3k
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Probing a manifold with geodesics
Supposed you stand at a point $p \in M$ on a smooth 2-manifold $M$
embedded in $\mathbb{R}^3$.
You do not know anything about $M$.
You shoot off a geodesic $\gamma$ in some direction $u$,
and learn ...
- 151k
28
votes
5
answers
2k
views
Visibility of vertices in polyhedra
Suppose $P$ is a closed polyhedron in space (i.e. a union of polygons which is homeomorphic to $S^2$) and $X$ is an interior point of $P$. Is it true that $X$ can see at least one vertex of $P$? More ...
- 4,484
27
votes
1
answer
2k
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The Eyeball Theorem generalized
I have not seen the 2D Eyeball Theorem—that tangents from the centers of two circles, each encompassing the other, intersect each circle in the same segment length—generalized to higher ...
- 151k
26
votes
1
answer
846
views
Disc bounded by a plane curve
Let $\Sigma$ be a sphere topologically embedded into $\mathbb{R}^3$.
Is it always possible to find a disc $\Delta\subset\Sigma$ which is bounded by a plane curve?
It is easy to find an open disc ...
- 45k
26
votes
3
answers
2k
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Universality of zeta- and L-functions
Voronin´s Universality Theorem (for the Riemann zeta-Function) according to Wikipedia: Let $U$ be a compact subset of the "critical half-strip" $\{s\in\mathbb{C}:\frac{1}{2}<Re(s)<1\}$ with ...
- 7,127
26
votes
3
answers
2k
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About the category of von neumann algebras
I am looking for one (or more) reference about properties of the category of von Neumann algebra.
More precisely, in an answer of a previous question, Dmitri Pavlov mentions
that the $W^*$ category ...
- 357
26
votes
2
answers
5k
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Does Arzelà-Ascoli require choice?
Inspired by a recent Math.SE question entitled Where do we need the axiom of choice in Riemannian geometry?, I was thinking of the Arzelà--Ascoli theorem. Let's state a very simple version:
...
- 29.8k
25
votes
3
answers
994
views
Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas?
Does every convex polyhedron have a combinatorially isomorphic counterpart whose faces all have rational areas?
Does every convex polyhedron have a combinatorially isomorphic counterpart whose edges ...
- 513
25
votes
6
answers
3k
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Quantum fields and infinite tensor products
As I understand it, a naive interpretation of the state space of a quantum field theory is an infinite tensor product
$$\otimes_{x\in M} H_x,$$
where $x$ runs over the points of space. This ...
- 13.6k
25
votes
3
answers
2k
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Angle of a regular simplex
I find the following question embarrassing, but I have not been able to either resolve it, or to find a reference.
What is the vertex angle of a regular $n$-simplex?
Background: For a vertex $v$ ...
- 7,836
25
votes
2
answers
2k
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$f^3,f^2$ are the cube and quadratic of f respectively and both infinite differentiable on $R$,how to show so is $f$
Let $f$ be a real function with domain R.
If $f^2$ and $f^3$ are both infinitely differentiable on R,
how to prove $f$ is infinitely differentiable on R?
I have been thinking about this problem for a ...
- 295
24
votes
2
answers
2k
views
Is the Invariant Subspace Problem arithmetic?
Invariant Subspace Conjecture: A bounded operator on a separable Hilbert space has a non-trivial closed invariant subspace.
Can this conjecture be reformulated as an arithmetic statement, that is, $\...
- 6,891
24
votes
8
answers
4k
views
When does a metric space have "infinite metric dimension"? (Definition of metric dimension)
Definition 1 A subset $B$ of a metric space $(M,d)$ is called a metric basis for $M$ if and only if $$[\forall b \in B,\,d(x,b)=d(y,b)] \implies x = y \,.$$
Definition 2 A metric space $(M,d)$ has &...
- 2,680
24
votes
4
answers
2k
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A reinterpretation of the $abc$ - conjecture in terms of metric spaces?
I hope it is appropriate to ask this question here:
One formulation of the abc-conjecture is
$$ c < \text{rad}(abc)^2$$
where $\gcd(a,b)=1$ and $c=a+b$. This is equivalent to ($a,b$ being ...
user6671
23
votes
4
answers
2k
views
Are almost commuting hermitian matrices close to commuting matrices (in the 2-norm)?
I consider on $M_n(\mathbb C)$ the normalized $2$-norm, i.e. the norm given by $\|A\|_2 = \sqrt{\mathrm{Tr}(A^* A)/n}$.
My question is whether a $k$-uple of hermitian matrices that are almost ...
- 9,486
23
votes
9
answers
2k
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Nonseparable counterexamples in analysis
When asking for uncountable counterexamples in algebra I noted that in functional analysis there are many examples of things that “go wrong” in the nonseparable setting. But most of the examples I'm ...