79 research outputs found
Bianchi's classification of 3-dimensional Lie algebras revisited
We present Bianchi's proof on the classification of real (and complex)
-dimensional Lie algebras in a coordinate free version from a strictly
representation theoretic point of view. Nearby we also compute the automorphism
groups and from this the orbit dimensions of the corresponding orbits in the
algebraic variety describing all Lie brackets
on a fixed vector space of dimension . Moreover we clarify which orbits
lie in the closure of a given orbit and therefore the topology on the orbit
space with
Primary and secondary branch growth in black spruce and balsam fir after Careful Logging around small Merchantable Stems (CLASS).
Careful logging around small merchantable stems (CLASS) is a partial cutting treatment that consists of the harvest of 70%β90% of the merchantable volume of an irregular coniferous stand. In this treatment, regeneration, saplings and small merchantable stems (DBH < 15 cm) are preserved and can continue to grow and develop into the dominant layer of the new stand. The aim of this project was to examine the effects of CLASS on the primary and secondary growth of branches, as well as on branch diameter in black spruce and balsam fir trees in the boreal forest of Quebec, Canada. Primary and secondary growth were measured on five branches per tree while branch diameter was analysed from 15 whorls distributed within the crown of the 48 black spruce and 48 balsam fir trees sampled. Branch primary and secondary growth significantly increased after CLASS in the lower part of the crown in both species, and both types of growth increased proportionally. These findings suggest that CLASS may delay crown recession as the lower branches tend to survive and grow for a longer period. However, although radial growth increased in the years post-CLASS, this did not significantly influence the final branch diameter and should not lead to lumber downgrade
Polygonic spectra and TR with coefficients
We introduce the notion of a polygonic spectrum which is designed to
axiomatize the structure on topological Hochschild homology
of an -ring with coefficients in an -bimodule . For
every polygonic spectrum , we define a spectrum as the
mapping spectrum from the polygonic version of the sphere spectrum
to . In particular if applied to this gives a
conceptual definition of .
Every cyclotomic spectrum gives rise to a polygonic spectrum and we prove
that TR agrees with the classical definition of TR in this case. We construct
Frobenius and Verschiebung maps on by exhibiting
as the -fixedpoints of a quasifinitely genuine
-spectrum. The notion of quasifinitely genuine -spectra
is a new notion that we introduce and discuss inspired by a similar notion over
introduced by Kaledin. Besides the usual coherences for genuine
spectra, this notion additionally encodes that admits certain
infinite sums of Verschiebung maps.Comment: 61 pages, comments are welcom
Prismatic cohomology relative to -rings
We develop prismatic and syntomic cohomology relative to a -ring.
This simultaneously generalizes Bhatt and Scholze's absolute and relative
prismatic cohomology and shows that the latter, which was defined relative to a
prism, is in fact independent of the prism structure and only depends on the
underlying -ring. We give several possible definitions of our new
version of prismatic cohomology: a site theoretic definition, one using
prismatic crystals, and a stack theoretic definition. These are equivalent
under mild syntomicity hypotheses. As an application, we note how the theory of
prismatic cohomology of filtered rings arises naturally in this context
A note on quadratic forms
For a field extension we consider maps that are quadratic over but
whose polarisation is only bilinear over . Our main result is that all such
are automatically quadratic forms over in the usual sense if and only if
is formally unramified. In particular, this shows that over finite and
number fields, one of the axioms in the standard definition of quadratic forms
is superfluous.Comment: 12 pages, comments very welcome (in particular regarding previously
known results!
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