58
votes
On which regions can Green's theorem not be applied?
I think this is an interesting and sort of deep question, so I'm going to answer it in part with the hope that my answer attracts even better answers.
I'll start with my first thought: surely there's ...
- 28.8k
37
votes
Accepted
Non-isomorphic rings that are localizations of each other
Example. Let $k$ be a field, and let $K = k(x_1,x_2,\ldots)$ be the fraction field of $k[x_1,x_2,\ldots]$. Let
$$A = K[y_1,y_2,\ldots],$$
and
$$B = A[y_1^{-1}].$$
Then $B$ is a localisation of $A$. If ...
- 27.3k
33
votes
A counterexample for Sard's theorem in $C^1$ regularity
My favourite example is as follows. Let the simple curve $\kappa:[0,1]\to K\subset \mathbb{R}^2$ be a parametrization of (half of) the Koch curve, and let $\phi:K\to[0,1]$ be its inverse; it is a ...
- 56.6k
33
votes
Chebyshev polynomials of the first kind and primality testing
Wow. This deserves a separate answer.
As I mentioned in a comment, motivated by the question, in a previous comment, by Igor Rivin whether an efficient primality test can be made if the statement in ...
- 17.5k
31
votes
Counterexamples against all odds
The most famous example is the so-called Riemann-Hilbert problem, which has a long and complicated history which I don't explain in detail. As it happens Hilbert's own formulation was not very exact, ...
30
votes
A counterexample for Sard's theorem in $C^1$ regularity
If $f\in C^1(\mathbb{R}^2,\mathbb{R})$, then the set of critical
values may have positive measure.
The classical construction due to Whitney was mentioned in the answer by T. Amdeberhan. However, the ...
- 27.3k
27
votes
On which regions can Green's theorem not be applied?
As mentioned elsewhere on this site, Sauvigny's book Partial Differential Equations provides a proof of Green's theorem (or the more general Stokes's theorem) for oriented open sets in manifolds, as ...
- 25.6k
26
votes
Accepted
Intuition behind counterexample of Euler's sum of powers conjecture
The search strategy is laid out in a paper devoted to finding all "small" non-negative solutions of $$x_1^5+\cdots x_n^5=y^5\text{ with }n \leq 6$$
L. Lander & T. Parkin, "A counterexample to ...
- 30k
25
votes
Accepted
Is a scheme Noetherian if its topological space and its stalks are?
This is false. The easiest counterexample I could come up with is the following "affine line with embedded points at every closed [rational] point":
Example. Let $k$ be an infinite field, let $R = k[...
- 27.3k
25
votes
On which regions can Green's theorem not be applied?
There is a fun reverse definition that is used for so called "currents" in geometric measure theory, objects for which then in Green's theorem always ends up trivially being true. But then ...
- 1,976
25
votes
Accepted
Smallest known counterexamples to Hedetniemi’s conjecture
Yes, Xuding Zhu did this in Relatively small counterexamples to Hedetniemi's conjecture (J. Comb. Theory B 146 (2021) pp. 141-150, doi:10.1016/j.jctb.2020.09.005, arXiv:2004.09028) where the sizes of ...
- 85.1k
24
votes
Uncountable counterexamples in algebra
Let $G$ be an abelian group.
The statement
If every subgroup of $G$ of finite rank is $\mathbf{Z}$-free, then $G$ is $\mathbf{Z}$-free.
is a theorem for $G$ countable, but false in general ($\mathbf{...
23
votes
Accepted
Conjecture: Given any five points, we can always draw a pair of non-intersecting circles whose diameter endpoints are four of those points
A counterexample is given by the following five points:
$$(0,0),(1,0),
\Big(-\frac{64867}{77629},\frac{3389}{60094}\Big),
\Big(\frac{5981}{56176},\frac{32211}{34172}\Big),
\Big(\frac{5925}{117812},-\...
- 118k
21
votes
A counterexample for Sard's theorem in $C^1$ regularity
This has been known for some time, including the higher-dimensional problem, in $\mathbb{R}^n$, that if $f\in C^k$ where $k<n$ then the set of critical points need not be of zero measure.
H. ...
- 41.8k
20
votes
Intuition behind counterexample of Euler's sum of powers conjecture
Even simply generating all quadruples $(a, b, c, d)$ with $1 \le a \le b \le c \le d \le 133$ should work fine. There are only about 13 million such quadruples. For each, we need to add together the ...
- 24.9k
20
votes
Counterexamples against all odds
Let $S$ be a finite set of (reduced) points in the projective plane
and let $I$ be the (saturated) homogeneous ideal of $S$.
Recall that $I^{(m)}$ is the $m$th symbolic power of $I$,
consisting of ...
20
votes
Accepted
Does $\mathbf{Cat}$ have the Cantor–Schröder–Bernstein property?
One can take some of the standard violations of CSB with other kinds of mathematical structures and transfer them to categories.
For example, with linear orders, we have the two linear orders $$\...
- 225k
19
votes
Accepted
How quickly can the derivative of an everywhere differentiable function change sign?
Fact 1 (Goldowsky-Tonelli): Let $F:(a, b) \to \mathbb{R}$ be continuous and have finite derivative everywhere. Suppose $F' \geq 0$ almost everywhere. Then $F$ is monotonically increasing.
For a proof ...
- 9,781
17
votes
Accepted
The $n$-th derivative has $n$ zeros. Can such a function be unbounded?
As suggested by Mateusz Kwaśnicki, the function $f : x \mapsto (1+x^2)^{s}$ is bell-shaped and unbounded for any $s \in (0,\frac{1}{2})$.
It is easy to see that $f^{(n)}(x) = P_n(x) (1+x^2)^{s-n}$ ...
- 7,199
17
votes
Accepted
Can you give an example of two projective morphisms of schemes whose composition is not projective?
Here is a locally Noetherian separated counterexample. I also give some motivation for this construction afterwards.
Definition. Let $Z$ be an infinite chain of affine lines: $Z = Z_1 \amalg_{p_1} ...
- 27.3k
16
votes
Accepted
Cocontinuous product-preserving functor between Grothendieck toposes
For any small category $J$, the colimit functor $\mathsf{Set}^J \to \mathsf{Set}$ preserves colimits. It preserves finite limits if and only if $J$ is filtered and it preserves finite products if and ...
- 7,596
15
votes
Accepted
Can a category be enriched over abelian groups in more than one way?
You can easily find examples among categories with one element: a category with one element is a (multiplicative) monoid, and $Ab$-enrichment over it is a choice of an addition which turns it into a ...
- 27.4k
14
votes
Accepted
Example of a ring with non-finitely generated unit group?
Let $A$ be a ring that is finitely generated as an abelian group. Let $I$ be the subgroup of torsion elements. By finite generation, $I$ is finite. On the other hand, $I$ is a two-sided ideal of $A$, ...
- 4,933
14
votes
Counterexamples against all odds
George Andrews and Cristina Ballantine's 2019 Almost partition identities builds on classical results to prove that various pairs of integer partition statistics are equal asymptotically 100% of the ...
14
votes
Counterexamples against all odds
(More of a comment than an answer, I suspect, but anyway…)
There are a bunch of results in graph theory that have to make exceptions for the Petersen graph, how do you rank that kind of counterexample?...
14
votes
Uncountable counterexamples in algebra
Countable torsion abelian groups are better behaved than uncountable ones. For example, Kaplansky’s “test problems”
If $G$ and $H$ are isomorphic to direct summands of each other, is $G\cong H$?
If $...
14
votes
Uncountable counterexamples in algebra
In rings: Let $R$ be a ring where idempotents lift modulo the Jacobson radical $J(R)$. Any countable set of orthogonal idempotents in $R/J(R)$ lifts to an orthogonal set of idempotents; but this ...
13
votes
A counterexample for Sard's theorem in $C^1$ regularity
I decided to challenge myself to make pictures of Piotr Hajlasz's example, partly for fun and partly for the next time I teach this. Let $C_3$ be the standard Cantor middle thirds set:
$$C_3 = \left\{ ...
13
votes
Accepted
External tensor product of irreducible representations is not irreducible?
You can generate examples from standard counterexamples to (generalisations of) Schur's lemma.
Let E/F be a field extension. Let $G=H=E^\times$, acting on the F-vector space E. Then the external ...
- 8,726
13
votes
Counterexamples against all odds
The generic oracle hypothesis is false. In particular, $\mathsf{IP}^G \ne \mathsf{PSPACE}^G$ for a generic oracle $G$, but $\mathsf{IP} = \mathsf{PSPACE}$ in real life. Similarly, the random oracle ...
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