431,925 research outputs found
The number of unit distances is almost linear for most norms
We prove that there exists a norm in the plane under which no n-point set
determines more than O(n log n log log n) unit distances. Actually, most norms
have this property, in the sense that their complement is a meager set in the
metric space of all norms (with the metric given by the Hausdorff distance of
the unit balls)
Algebraic K-theory of quasi-smooth blow-ups and cdh descent
We construct a semi-orthogonal decomposition on the category of perfect
complexes on the blow-up of a derived Artin stack in a quasi-smooth centre.
This gives a generalization of Thomason's blow-up formula in algebraic K-theory
to derived stacks. We also provide a new criterion for descent in Voevodsky's
cdh topology, which we use to give a direct proof of Cisinski's theorem that
Weibel's homotopy invariant K-theory satisfies cdh descent.Comment: 24 pages; to appear in Annales Henri Lebesgu
Asymptotic topology
We establish some basic theorems in dimension theory and absolute extensor
theory in the coarse category of metric spaces. Some of the statements in this
category can be translated in general topology language by applying the Higson
corona functor. The relation of problems and results of this `Asymptotic
Topology' to Novikov and similar conjectures is discussed.Comment: 38 pages, AMSTe
New Local Duals in Eternal Inflation
Global-local duality is the equivalence of seemingly different regulators in
eternal inflation. For example, the light-cone time cutoff (a global measure,
which regulates time) makes the same predictions as the causal patch (a local
measure that cuts off space). We show that global-local duality is far more
general. It rests on a redundancy inherent in any global cutoff: at late times,
an attractor regime is reached, characterized by the unlimited exponential
self-reproduction of a certain fundamental region of spacetime. An equivalent
local cutoff can be obtained by restricting to this fundamental region.
We derive local duals to several global cutoffs of interest. The New Scale
Factor Cutoff is dual to the Short Fat Geodesic, a geodesic of fixed
infinitesimal proper width. Vilenkin's CAH Cutoff is equivalent to the
Hubbletube, whose width is proportional to the local Hubble volume. The famous
youngness problem of the Proper Time Cutoff can be readily understood by
considering its local dual, the Incredible Shrinking Geodesic.Comment: 30 pages, 3 figure
Almost Commuting Orthogonal Matrices
We show that almost commuting real orthogonal matrices are uniformly close to
exactly commuting real orthogonal matrices. We prove the same for symplectic
unitary matrices. This is in contrast to the general complex case, where not
all pairs of almost commuting unitaries are close to commuting pairs. Our
techniques also yield results about almost normal matrices over the reals and
the quaternions.Comment: 13 pages, 3 figure
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