Recent MathOverflow Questions

Thomason-Trobaugh Theorem

Math Overflow Recent Questions - Sat, 06/23/2018 - 08:18

Let $X$ be a scheme and $U$ be an open subscheme. The proof of the Thomason-Trobaugh Theorem implies that under some mild assumptions, for any perfect complex $F$ on $U$, we have that $F\oplus F[1]$ can be extended to a perfect complex on $X$. I'm just wondering whether there exists examples where $F$ is a perfect complex on $U$ but $F$ itself cannot be extended to $X$? I've found an example when $X$ is the cone $xy-z^{2}=0$ and $U$ is the complement of the origin. Is there an example for smooth $X$?

Clopen subsets of a closed subspace of a spectral space

Math Overflow Recent Questions - Sat, 06/23/2018 - 02:08

Let $X$ be a topological space. Set $K(X) := \{ A\subseteq X\mid A$ is quasi-compact and open $\}.$ A topological space $X$ is called spectral, if it satisfies all of the following conditions:

1) $X$ is quasi-compact and $T_0$. 2) $K(X)$ is a basis of open subsets of $X$. 3) $K(X)$ is closed under finite intersections. 4) $X$ is sober, i.e. every nonempty irreducible closed subset of $X$ has a (necessarily unique) generic point.

Let $C$ be a closed subset of a spectral topological space $X$. I am looking for equivalent conditions on $C$ under which if $A$ is a clopen(=Closed+Open) subset of $C$, then there exists a clopen subset $B$ of $X$ such that $A=C\cap B$?

On the structure of a finite group of order $144$

Math Overflow Recent Questions - Sat, 06/23/2018 - 01:00

Let $G$ be a finite group of order $144$. Suppose there is an irreducible $\mathbb{C}$-character $\theta$ such that $\theta(1)=9$.

QUESTION: Prove $G\cong A_4\times A_4$.

By using Magma, we know there is only one group of order $144$ with an irreducible $\mathbb{C}$-character $\theta$ of degree $9$. Now I want to prove this result without using Magma.

Asymptotic behavior of an integral depending on an integer

Math Overflow Recent Questions - Fri, 06/22/2018 - 08:57

A friend of mine, obtained a lower bound for the trace norm of matrices described in this question (for the special case $a_{ij} = \pm 1$). That lower bound is $ \frac{f(n)}{2\pi}$ where $$ f(n) := \int_0^\infty \log\left( \frac{(1+t)^n +(1-t)^n}{2} +n(n-1) t(1+t)^{n-2}\right)t^{- 3/2} \ \mathrm{d}t $$ Numerical computaions suggest that $$ f(n) = 4 \pi n + o(n) $$ How to justify it? Moreover, is it possible to obtain a good rate of convergence?

Idea behind definition of classifying space over an orbifold

Math Overflow Recent Questions - Fri, 06/22/2018 - 07:36

Today I was explaining to some one the notion of $\mathcal{G}$ spaces, covering spaces over orbifolds from Orbifolds as Groupoids: an Introduction.

Definition : Let $X$ be a locally compact Hausdorff space. An orbifold structure on $X$ is given by an orbifold groupoid $\mathcal{G}$ and a homeomorphism $f:|\mathcal{G}|\rightarrow X$. If $\mathcal{H}\rightarrow\mathcal{G}$ is a equivalnece, then $|\phi|:|\mathcal{H}|\rightarrow |\mathcal{G}|$ is a homeomorphism and the composition $f\circ|\phi|:|\mathcal{H}|\rightarrow |\mathcal{G}|\rightarrow X$ is viewed as defining an equivalent orbifold structure on $X$. An orbifold $\underline{X}$ is a space $X$ equipped with an equivalnece class of orbifold structures.

I have given definition of a $\mathcal{G}$-space as a smooth manifold $E$ with smooth maps $\pi:E\rightarrow \mathcal{G}_0$ and $\mu:E\times_{\mathcal{G}_0}\mathcal{G}_1\rightarrow E$ behaving like a group action map.

Then, I said, given a morphism of Lie groupoids $\phi:\mathcal{H}\rightarrow \mathcal{G}$ there is a functor from category of $\mathcal{G}$ spaces to the category of $\mathcal{H}$ spaces $\phi^*:(\mathcal{G}-\text{spaces})\rightarrow (\mathcal{H}-\text{spaces})$.

Definition : A covering space over a groupoid $\mathcal{G}$ is a $\mathcal{G}$ such that the map $\pi:E\rightarrow \mathcal{G}_0$ is a covering projection.

Then, I said the equivalnece of categories $\phi^*:(\mathcal{G}-\text{spaces})\rightarrow (\mathcal{H}-\text{spaces})$ restircted to covering spaces is still an equivalence. So, I can talk about covering space over an orbifold.

Definition : A covering space over an orbifold $\underline{X}$ is a covering space over Lie groupoid $\mathcal{G}$ representing $\underline{X}$ (in the sense defined above).

This notion is well defined : Suppose there is another Lie groupoid representing $\underline{X}$ in the equivalnece class, say $\mathcal{H}$ then, this $\mathcal{H}$ has to be morita equivalent with $\mathcal{G}$. So, category of covering spaces over $\mathcal{G}$ would be same(equivalent) as that of category of covering spaces over $\mathcal{H}$. So, the notion makes sense.

Then, he asked

"why do you do this much just to define the notion of covering space over a space $X$?"...

I did not think about this before and said at that moment that, $X$ is not merely a topological space in which case you can define covering map to be just local homeomorphism plus something, the usual definition.

One standard example in the classical notion of orbifold is quotient space of a manifold by a Lie group $M/G$.

Suppose you want to define the notion of covering space over $M/G$, you can not treat $M/G$ as simply a topological space, in which case you can define covering space to be a map $\pi:E\rightarrow X=M/G$ such that given $x\in X$ there is an open set $U$ containg $x$ and $\pi^{-1}(U)$ is disjoint union $\bigsqcup V_\alpha$ where $V_\alpha$ is mapped homeomorphically onto $U$ under $\pi$. You can not even treat $M/G$ as a smooth manifold where you would define smooth covering map just by replacing homeomorphically in above definition by diffeomorphically. In general $M/G$ is not a smooth manifold. It is more than a topological space and less than a smooth manifold. It is not easy/obvious to define what is a covering space over $M/G$ would be.. So, to make sense of covering spaces, we have to go to orbifold groupoid setup where notion of covering space would be just simple as above.

To summarize I said the following :

Orbifolds are neither just topological space where you can define covering maps to be local homeomorphisms plus something nor smooth manifolds where you can define covering maps to be local diffeomorphisms plus something. Orbifolds are something more than just a topological space and less than a smooth manifold. So, you need a separate approach to define covering spaces and groupoid approach is useful/straight-forward.

I just want to know if what I have said is actually a reason or did I misunderstood the idea.

Any comments are welcome.

Edit: This is just to bump this question up so that it gets some attention and in turn some comments.

Why torsion is only defined for linear connection on TM?

Math Overflow Recent Questions - Fri, 06/22/2018 - 04:22

The concept of curvature is defined for any linear connection on any vector bundle $E \to M$, but the concept of torsion is only defined for connection on the tangent bundle $TM$ of a manifold $M^n$, or for a connection obtained as the pullback of a connection on a vector bundle $E \to M$ isomorphic to $TM$ via an isomorphism $\theta \colon TM \to E$ equivalent to a solder form.

Why is that so ? If torsion can be interpreted as the twist of a moving frame along a curve, the same phenomena should occur for a connection on any vector bundle.

Is there a way to define a notion of torsion for any vector bundle ?

Can the number of countable models of a complete first-order theory decrease after adding constants?

Math Overflow Recent Questions - Thu, 06/21/2018 - 18:43

If $T$ is a countable complete first-order theory with infinite models, the number of countable models it has, $I(T,\omega)$, must be an element of $N=\{1,3,4,5,6,7,\dots,\omega,\omega_1,2^\omega\}$ (although we don't know if $\omega_1$ can happen). For which pairs $n,m\in N$ does there exist a countable complete theory $T$ with $n$ countable models but $m$ countable models after adding finitely many constants to the theory? Countably many new constants? In particular can we have $m<n$? EDIT: By 'adding constants,' I mean adding constants whose type is completely specified, i.e. expanding by constants and then passing to a complete theory in the expanded language.

Let $n\rightarrow m$ denote the statement "There exists a complete countable theory $T$ and a finite tuple of constants $\overline{a}$ such that $I(T,\omega)=n$ and $I(T_\overline{a},\omega)=m$." And let $n\rightarrow_\omega m$ denote the statement "There exists a complete countable theory $T$ and a countable set of distinct constants $A$ such that $I(T,\omega)=n$ and $I(T_A,\omega)=m$." Some easy results and relevant observations:

  • If $n\rightarrow m$ (resp. $n\rightarrow_\omega m$) and $k \rightarrow \ell$ (resp. $k \rightarrow_\omega \ell$), then $nk \rightarrow m\ell$ (resp. $nk \rightarrow_\omega m\ell$). (Take the disjoint union of the relevant theories.)
  • $n \rightarrow n$ for every $n\in N-\{\omega_1\}$. (This is obvious for $n=1$. There are easy examples for $n=\omega,2^\omega$ and the standard examples for $n=3,4,5\dots$ all have constants which do not increase the number of countable models.)
  • $n^2+n\rightarrow (n+1)^2$ for any $1<n<\omega$. (DLO with $n-1$ colors and a countable set of constants of order type $\omega + \omega^\ast$. By itself this theory has $n^2 + 1$ countable models. Adding a constant in between $\omega$ and $\omega^\ast$ makes the theory have $(n+1)^2$ models.)
  • $1\not\rightarrow n$ and $n\not\rightarrow 1$ for any $n\in N - \{1\}$.
  • $1\rightarrow_\omega 2^\omega$ (For example: DLO.)
  • $1\rightarrow_\omega \omega$ (For example: A structureless set.)
  • $n\not\rightarrow_\omega 1$ for any $n\in N$.
  • $1 \rightarrow_\omega n$ for every $2<n<\omega$. (The standard examples of Ehrenfeucht theories are $\omega$-categorical theories with countably many constants added.)
  • If a theory is not small, then it will have $2^\omega$ countable models after adding any countable set of constants.

Covering inequality for sets of intervals

Math Overflow Recent Questions - Thu, 06/21/2018 - 07:50

Let $I$ and $J$ be finite sets of open intervals $(a,b)\subset\mathbb R$. For a finite set of points $P\subset \mathbb R$ we denote those subsets of intervals from $I$ and $J$ containing some point from $P$ by $I_P,J_P$. Now suppose that \begin{align}\tag{*}\label{IP JP ineq}\lvert I_P\rvert\le \lvert J_P\rvert+1\end{align} for all finite subsets $P$, and that the inequality \eqref{IP JP ineq} is optimal in the sense that there exists at least one finite $P$ for which equality is achieved. Here $\lvert\cdot\rvert$ is simply the counting measure.

I have the strong suspicion (supported by numerical experimentation with random sets) that there has to exist some specific interval $(a,b)\in I$ such that $(a,b)$ contains at least one point from each set $P$ for which equality is achieved in \eqref{IP JP ineq}.

It is clear that the statement cannot be true for more general subsets than intervals but I couldn't come up with any argument yet. It is also clear that the claim cannot hold true if the $1$ in \eqref{IP JP ineq} is replaced by $0$.

Cargo Cult Science in mathematics?

Math Overflow Recent Questions - Thu, 06/21/2018 - 05:51

What would be some examples in the mathematical sciences of what Feynman once colorfully described as Cargo Cult Science (CCS) that are certifiably bogus according to reliable sources? Feynman's original essay can be found here. The formation of the cargo cults in the New Guinea Mountains in the 1930s is analyzed in Tomasz Witkowski's Psychology Led Astray: Cargo Cult in Science, on pages 17-20.

In the natural sciences there are some classical examples like phlogiston, ether (these two may have been historically justifiable and not in the same category as the other examples), and Lysenkoism. A more recent example is the Bogdanov affair attested to by an internal report of the CNRS to the effect that the theory had "no scientific value".

But it seems that in mathematics the examples are more rare. There is a number of published articles about $\tau (=2\pi)$ but this is not so much a CCS as a triviality. I remember reading somewhere that around 1900 there was some work on evaluating improper integrals that was genuinely bogus (as opposed to the techniques of summation of divergent series which are of course a legitimate field, with applications to physics, as nicely described in Varadarajan's 2007 article on Euler). Another example is Sergeyev's "grossone" CCS (now officially attested to by the unanimous statement of the EMS Surveys editors; meanwhile, Zentralblatt recycles Sergeyev's claims on how the grossone outperforms $\infty,\aleph_0,\omega$), involving numerous publications in refereed journals, books, collaborators, "international" conferences featuring keynote dinners in "exclusive" restaurants, "international" prizes, etc.

Are there other examples? Here it seems reasonable to impose a criterion of at least one publication in a refereed venue so as to filter out any number of oddballs posting at arxiv or vixra.

The examples developed so far (that meet the criteria for inclusion as above) are the following, in alphabetical order:

(1) Bogdanov, Grichka (a thesis in mathematics; a CNRS opinion of "no value");

(2) Santilli, Ruggero (publications in refereed venues; detailed rebuttal in MathSciNet review by Magill;

(3) Sergeyev, Yaroslav (publications in refereed venues, including books; unanimous negative opinion by editorial board as above. Note that this refers to his work in the grossone whereas his work on optimisation is another matter).

(4) Smarandache, Florentin (some of the publications seem to be legitimate, particularly in elementary geometry).

There is a couple of examples mentioned in the comments that I haven't had a chance to examine in detail yet.

How to extract an indiscernible sequence from a set in a stable theory?

Math Overflow Recent Questions - Wed, 06/20/2018 - 18:58

I was reading Chernikov's notes about stable theories, and he mentions the following fact:

If $T$ is stable and $A$ is some set of parameters large enough then there is some indiscernible sequence $I \subseteq A$ such that $|I|=|A|$.

I have been trying to find a reference of the previous fact but I have not been lucky. Any comment, hint or reference is highly appreciated!

A question on the sine function

Math Overflow Recent Questions - Wed, 06/20/2018 - 17:46

The Fejer-Jackson-Gronwall inequality involving the sine function is as follows: $$\sum_{k=1}^n\frac{\sin kx}k>0\quad\text{for all}\ n=1,2,3,\ldots\ \text{and}\ 0<x<\pi.$$

Here I ask the following related question.

QUESTION: Do we have $$\sum_{k=1}^n(-1)^k\left(\frac{\sin kx}k\right)^m<0<\sum_{k=1}^n\left(\frac{\sin kx}k\right)^m$$ for all $m,n=1,2,3,\ldots$ and $0<x<\pi$ ?

Actually I formulated this question in 2013. My numerical computation suggests that the answer should be positive. How to prove this?

Is there a good, general description of morphisms right orthogonal to effective epimorphisms?

Math Overflow Recent Questions - Wed, 06/20/2018 - 17:39

Let $C$ be a locally presentable, locally cartesian closed $\infty$-category. Then I think it's not hard to show that the class of effective epimorphisms in $C$ is closed under colimits and cobase-change, and is accessible, so that it forms the left class of a factorization system. If $C$ is an $\infty$-topos, then the corresponding right class consists of the monomorphisms in $C$. How about in general?

I guess I'm particularly interested in the case where $C$ is the unstable motivic category over a scheme $S$.

Lower semi-continuity of integration

Math Overflow Recent Questions - Wed, 06/20/2018 - 16:32

I've found many papers characterizing the weak lower semi-continuity of $$ \Phi(u)\triangleq \int_{x \in \Omega} f(x,u(x))dx, $$ on $L^2(\Omega)$ where $\Omega$ is a bounded subset of $\mathbb{R}^d$. Are there any results with regards to the same statement with $\Omega$ being all of $\mathbb{R}^d$ itself?

Here $f$ is a non-negative, Borel measurable function, which is convex in its second variable for a.e. one of its first variagbles.

Expression for the lattice operations on subspaces in Plucker embedding

Math Overflow Recent Questions - Wed, 06/20/2018 - 16:22

Suppose that $V$ is a finite dimensional $\mathbb Q$-vector space. To each subspace $S$ of dimension $k$, we can associate the line from the origin of $\Lambda^k(V)$ through the point $s_1\wedge \ldots \wedge s_k$ where $\{s_1,\ldots,s_k\}$ is an arbitrary basis for $S$. This is essentially representing subspaces via the Plucker embedding.

Now, suppose that I have a pair of elements $\alpha \in \Lambda^k(V)$ and $\beta\in \Lambda^{k'}(V)$ corresponding to some subspaces $S$ and $S'$. These elements are given in the usual coordinates on the exterior algebra relative to some basis for $V$.

Is there a simple way to compute the element of the exterior algebra corresponding to the subspace $S+S'$?

If $S\cap S'=0$, then $S+S'$ is represented simply by $\alpha\wedge\beta$, but it's not clear how to do a similar calculation if $S$ and $S'$ intersect non-trivially

While there is a reasonably straightforwards way to do this - in particular, given $\alpha$ and $\beta$, we can find a basis for each of the $S$ and $S'$ and then, using the exterior algebra, we can relatively easily find a basis for $S+S'$ and then pass back into the exterior algebra. However, this seems rather inelegant, especially given the really nice formula that exists if the spaces intersect trivially.

It is also trivial to do this, using Hodge duality, if one were able to compute an element representing $S\cap S'$ from $\alpha$ and $\beta$. This problem seems analogous to finding a $\gcd$ of $\alpha$ and $\beta$ in $\Lambda(V)$ considered as a ring.

Mainly, I'm wondering about this since I'm trying to mechanistically do various computations on some arrangements of codimension $2$ subspaces and have found the exterior algebra to be an good tool for similar computations on arrangements of hyperplanes, but moving from codimension $1$ to codimension $2$ requires knowing how to carry out this computation in greater generality.

Jumping times on Borel sets away from zero are stopping times

Math Overflow Recent Questions - Wed, 06/20/2018 - 16:00

The following comes from some remarks of Philip Protter at page 26 of the book Stochastic integration and Differential equations that I have not been able to prove yet.

Let $X$ a Levy process, under a filtration satisfying the usual conditions. If $\Lambda$ is a Borel set in $\mathbb{R}$ bounded away from zero (that is $0 \notin \bar{\Lambda}$), then the jumping times

\begin{align} &T_{\Lambda}^{1} = \lbrace t \geq : \Delta X_{t} \in \Lambda \rbrace \\ &\vdots \\ &T_{\Lambda}^{n} = \lbrace t > T_{\Lambda}^{n-1} : \Delta X_{t} \in \Lambda \rbrace \end{align} are stopping times.

My attempt Since the filtration satisfies the usual conditions, we only need to prove that $\lbrace T_{\Lambda} < t \rbrace \in \mathcal{F}_{t}$.

Let $\epsilon := d(0, \Lambda) >0$ and $M:= ( - \infty, - \epsilon] \cup [\epsilon, \infty)$, I am trying to prove

\begin{align} \lbrace T_{\Lambda} < t \rbrace = \left( \bigcup_{r \in [0, t) \cap \mathbb{Q}} \lbrace \Delta X_{r} \in \Lambda \rbrace \right) \cap \lbrace T_{M} < t\rbrace \end{align} If we can prove this equation, we are done. This is due to the fact that $\lbrace T_{M} \leq t\rbrace \in \mathcal{F}_{t}$ since \begin{align} \lbrace T_{M} < t\rbrace = \bigcap_{n} \bigcup_{r,s \in [0, t+1/n)\\ \vert r-s \vert < 1/n} \lbrace \vert X_{s} - X_{r} \vert > \epsilon \rbrace \end{align}

We know that the "$\supset$" is the easy part, but the "$\subset$" part is the only part that I need to prove. I was trying to prove this by contradiction, and seems that it is the best way.

If $w \in \lbrace T_{M} < t\rbrace $ and $w \notin \lbrace T_{M} < t\rbrace $ is a contradiction. This can be done using lemmas of discontinuities and the fact that $d(0, \Lambda) >0$. However the part $w \in \lbrace T_{M} < t\rbrace $ and $w \notin \left( \bigcup_{r \in [0, t) \cap \mathbb{Q}} \lbrace \Delta X_{r} \in \Lambda \rbrace \right)$ is the difficult one.

Any hint will be welcome.

Why do we only care about finding the value of a function defined on a Lie group at the identity (orbital integral problem)?

Math Overflow Recent Questions - Wed, 06/20/2018 - 15:29

The so-called orbital integral problem "amounts to determining the value of a function on $G$ at $e$ in terms of its integrals over (generic) conjugacy classes" (Helgason).

I suppose my question is quite vague, but why do we only care about the value at the identity? A function is surely not explicitly given by its value at the identity alone (even a smooth, compactly supported one).

Definition of the local Artin conductor for a representation of the Weil group

Math Overflow Recent Questions - Wed, 06/20/2018 - 14:51

Let $\rho$ be a continuous, finite dimensional complex representation of the Galois group $\operatorname{Gal}(\overline{F}/F)$, for $F$ a $p$-adic field. Is there a general notion of an Artin conductor of $\rho$?

If $\rho$ is a character of $\operatorname{Gal}(\overline{F}/F)$, then the conductor of $\rho$ is defined as a sum involving the higher ramification groups of a finite Galois extension of $F$ through which $\rho$ factors.

Is there moreover a definition of the Artin conductor of $\rho$ when $\rho$ is a representation of the local Weil group $W_F$?

Reference request: Existence/regularity for viscous Hamilton-Jacobi equations

Math Overflow Recent Questions - Wed, 06/20/2018 - 13:53

A basic PDE I would like to understand much better is the viscous Hamilton-Jacobi equation, such as: \begin{equation*} u - \epsilon \Delta u + H(Du) = f(x) \end{equation*} or \begin{equation*} u_{t} - \epsilon \Delta u + H(Du) = f(x) \end{equation*} with or without boundary conditions in the stationary case, or the Cauchy problem in the time-dependent case. I'm interested in the case when $\epsilon > 0$.

Very general viscosity solutions theory implies these equations have continuous solutions under mild assumptions on $H$ and $f$. However, my understanding is the Laplacian term should give us much better regularity than just continuity.

This is a relatively basic example and quite well-motivated from the point of view of stochastic control theory, but nonetheless I'm having trouble finding a down-to-earth reference that shows how to establish regularity for these equations without throwing in the kitchen sink. (In other words, I'm looking for a reference at the level of lecture notes so that I can avoid a little longer wading through Gilbarg-Trudinger or the parabolic equivalent.). It would be particularly nice if the reference in question used a fixed point theorem argument to get existence and regularity simultaneously, but I'm open to an alternative approach.

Is anyone aware of lectures notes that explain how to establish regularity for these equations? Alternatively, are there papers where this is explained in a compact way? My complaint as a student here is this is touched on only very briefly in Evans (in the discussion of fixed point theorems) and the more advanced textbooks on this strike me as extremely dense and somewhat old-fashioned. I may as well start working my way into those books, but if I can get a head start with something more concrete it would be nice.

Example of convex n-gon that cannot be decomposed into k congruent convex polygons

Math Overflow Recent Questions - Wed, 06/20/2018 - 12:47

I asked a related question here on MO without any answers yet.

The question is in the title - give an example of a convex $n$-gon that cannot be subdivided into $k>1$ congruent convex polygons. Even better, give a family that solves this for all combinations of $(n,k)$.

Intuitively, any generic $n$-gon should work, but the crux is in the details - I am very curious about what methods one can use to rigorously prove that a subdivision is impossible.

There are of course many variations, e.g., drop the convex restriction, and remove the restriction that the pieces are polygons.

Is there a computable homeomorphism between two different Cartesian powers of the computable real numbers?

Math Overflow Recent Questions - Wed, 06/20/2018 - 12:30

It's well know that it is surprisingly difficult to prove that $\mathbb{R}^n$ and $\mathbb{R}^m$ are not homeomorphic for $n\neq m$. Commonly proofs go through Brouwer's fixed point theorem, which is 'computably false' for dimensions greater than one: let $K$ denote the computable real numbers. For $n>1$, there are computable functions from $\left( [0,1]\cap K \right)^n$ to itself with no computable fixed point.

That by itself may not seem very bad, since the function may still extend to a continuous function on $[0,1]^n$ and have an incomputable fixed point, but a corollary of this is that there is a computable retraction of $\left( [0,1]\cap K \right)^n$ onto its boundary. Such a function clearly can't be extended to a continuous function on $[0,1]^n$. So we can see that the topological behavior of $K^n$, even when restricted to computable functions, is very different from the topological behavior of $\mathbb{R}^n$.

On the other hand, $K$ is homeomorphic to $\mathbb{Q}$ (although not computably so) and $\mathbb{Q}^n$ is homeomorphic to $\mathbb{Q}^m$ for any $n$ and $m$, so $K^n$ is homeomorphic to $K^m$ for any $n$ and $m$.

So the question is: Is there a computable homeomorphism between $K^n$ and $K^m$ for some $n\neq m$? If there is one I would assume we need $n,m>1$.


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