18
votes
Accepted
Ultraproducts of Banach spaces versus model theoretic ultraproduct
The ultraproduct of Banach spaces is the ultraproduct in the sense of metric structures in continuous logic. For a nice survey on this topic, see Model theory for metric structures by Ben Yaacov, ...
- 3,850
16
votes
Ultraproducts of Banach spaces versus model theoretic ultraproduct
As a logician, I take the model-theoretic notion of ultraproduct as the primary one, so the following formal connection describes how to get the Banach-space ultraproduct from the model-theoretic one.
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- 70.3k
11
votes
Ultraproducts of Banach spaces versus model theoretic ultraproduct
The "Banach space ultraproduct" is also referred to as the nonstandard hull, precisely in order to distinguish it from the model theoretic ultraproduct (which I shall simply call "...
- 98.8k
10
votes
Kissing number lower bound vs. upper bound - precise meanings?
Lots of questions here, I'll see how many I can address. To be clear, $K_L$ and $K_U$ are summaries of our current knowledge. There is some true kissing number in each dimension, and hopefully we'll ...
- 145k
5
votes
Accepted
Pythagorean theorem in Riemann metrics of non constant curvature
I think (see (??) below) there is no connected complete Riemannian manifold $(M,g)$ which is not flat and such that there is an isometry $f:(M,g)\to(M,\lambda^2g)$, where $\lambda\neq0,1$. I will ...
- 7,164
5
votes
Accepted
Approximate isometric embeddings of surfaces
I think that the answer is 'yes' if $U$ is simply-connected, because there is a way to construct a candidate 'approximate surface' from 'approximate solutions' of Gauss and Codazzi, but a more useful ...
- 104k
4
votes
Accepted
Is there an L-system for aperiodic tilings of the plane with the "hat" monotile?
(a second answer because this one is an answer)
So, I misled myself staring at the H8 in Smith et al. The way to solve this is to look at the F-supertile. That tile has 5 edges, and 4 of them are F-...
- 196
2
votes
Leech lattice shortest vector vs other 23 cases and E8 cases
Leech lattice $Λ24$ has larger distance between centers of the two balls (namely $D=2$), in contrast with other 23 classes of 24-dimensional lattice which as $D=√2$, also in contrast with the E8 ...
- 126k
2
votes
One-step problems in geometry
A the boundary of a convex polyhedron is shellable.
1
vote
One-step problems in geometry
I learned this one from my advisor: the Borromean rings are not realized by round circles.
Really this is two steps, since one needs to know that the Borromean rings are nontrivial (not the unlink), ...
1
vote
How to interpret couplings in optimal transport?
The interpretation of couplings as randomized transport maps is not without problems. In general, extreme points of the space of couplings need not be supported on the graph of a function; see, for ...
1
vote
What does the extension theorem for tilings state?
I first read about the extension theorem for tilings in a simpler form: if a finite protoset can tile an arbitrarily large disk, then it can the whole plane. My informal way of proving it is the ...
1
vote
Creating high quality figures of surfaces
The three.js JavaScript library is nice. See my artwork.
Shaders are nice as well. See the shadertoy website.
1
vote
Accepted
One-step problems in geometry
Let $A$ be a set of intially labelled points in $\mathbb{R}^d$. We may take any line containing at least $k$ labelled points and label any point on this line. For which minimal size $|A|$ (as a ...
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