16,529 research outputs found
Around -independence
In this article we study various forms of -independence (including the
case ) for the cohomology and fundamental groups of varieties over
finite fields and equicharacteristic local fields. Our first result is a strong
form of -independence for the unipotent fundamental group of smooth and
projective varieties over finite fields, by then proving a certain `spreading
out' result we are able to deduce a much weaker form of -independence for
unipotent fundamental groups over equicharacteristic local fields, at least in
the semistable case. In a similar vein, we can also use this to deduce
-independence results for the cohomology of semistable varieties from the
well-known results on -independence for smooth and proper varieties over
finite fields. As another consequence of this `spreading out' result we are
able to deduce the existence of a Clemens--Schmid exact sequence for formal
semistable families. Finally, by deforming to characteristic we show a
similar weak version of -independence for the unipotent fundamental group
of a semistable curve in mixed characteristic.Comment: 23 pages, comments welcom
No-gaps delocalization for general random matrices
We prove that with high probability, every eigenvector of a random matrix is
delocalized in the sense that any subset of its coordinates carries a
non-negligible portion of its norm. Our results pertain to a wide
class of random matrices, including matrices with independent entries,
symmetric and skew-symmetric matrices, as well as some other naturally arising
ensembles. The matrices can be real and complex; in the latter case we assume
that the real and imaginary parts of the entries are independent.Comment: 45 page
Logical Dreams
We discuss the past and future of set theory, axiom systems and independence
results. We deal in particular with cardinal arithmetic
Logical closure properties of propositional proof systems - (Extended abstract)
In this paper we define and investigate basic logical closure properties of propositional proof systems such as closure of arbitrary proof systems under modus ponens or substitutions. As our main result we obtain a purely logical characterization of the degrees of schematic extensions of EF in terms of a simple combination of these properties. This result underlines the empirical evidence that EF and its extensions admit a robust definition which rests on only a few central concepts from propositional logic
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