4,027 research outputs found
Consequences of the existence of Auslander-Reiten triangles with applications to perfect complexes for self-injective algebras
In a k-linear triangulated category (where k is a field) we show that the
existence of Auslander-Reiten triangles implies that objects are determined, up
to shift, by knowing dimensions of homomorphisms between them. In most cases
the objects themselves are distinguished by this information, a conclusion
which was also reached under slightly different hypotheses in a theorem of
Jensen, Su and Zimmermann. The approach is to consider bilinear forms on
Grothendieck groups which are analogous to the Green ring of a finite group.
We specialize to the category of perfect complexes for a self-injective
algebra, for which the Auslander-Reiten quiver has a known shape. We
characterize the position in the quiver of many kinds of perfect complexes,
including those of lengths 1, 2 and 3, rigid complexes and truncated projective
resolutions. We describe completely the quiver components which contain
projective modules. We obtain relationships between the homology of complexes
at different places in the quiver, deducing that every self-injective algebra
of radical length at least 3 has indecomposable perfect complexes with
arbitrarily large homology in any given degree. We find also that homology
stabilizes away from the rim of the quiver. We show that when the algebra is
symmetric, one of the forms considered earlier is Hermitian, and this allows us
to compute its values knowing them only on objects on the rim of the quiver.Comment: 27 page
Impact of Rare Decays and on Searches for Top-Associated Physics
Searches for top quark-associated physics such as or
in final states with multiple leptons require a careful accounting of expected
backgrounds due to the lack of reconstructible resonances. We demonstrate that
the rare top quark decays and , when a soft lepton is not detected, can contribute a non-negligible
background to such searches. Simulations in the LHC experiments typically do
not account for such decays and as such backgrounds to such searches may be
underestimated.Comment: 13 pages, 6 figure
Extending the Coinvariant Theorems of Chevalley, Shephard--Todd, Mitchell and Springer
We extend in several directions invariant theory results of Chevalley,
Shephard and Todd, Mitchell and Springer. Their results compare the group
algebra for a finite reflection group with its coinvariant algebra, and compare
a group representation with its module of relative coinvariants. Our extensions
apply to arbitrary finite groups in any characteristic.Comment: The applications and Examples in section 4 have been extende
The Auslander-Reiten quiver of perfect complexes for a self-injective algebra
We consider the homotopy category of perfect complexes for a finite
dimensional self-injective algebra over a field, identifying many aspects of
perfect complexes according to their position in the Auslander-Reiten quiver.
Short complexes lie close to the rim. We characterize the position in the
quiver of complexes of lengths 1, 2 and 3, as well as rigid complexes and
truncated projective resolutions. We describe completely the quiver components
that contain projective modules (complexes of length 1). We obtain
relationships between the homology of complexes at different places in the
quiver, deducing that every self-injective algebra of radical length at least 3
has indecomposable perfect complexes with arbitrarily large homology in any
given degree. We show that homology stabilizes, in a certain sense, away from
the rim of the quiver.Comment: 20 pages. This paper replaces paper the second half of [1] below, and
[2] replaces the first half of [1]. [1] Peter Webb, Consequences of the
existence of Auslander-Reiten triangles with applications to perfect
complexes for self-injective algebras, arXiv:1301.4701 [2] Peter Webb,
Bilinear forms on Grothendieck groups of triangulated categories,
arXiv:1709.0388
Micro Channel Cooler Performance Improvement by Insonation
The motivation for this work is the need to remove waste heat from laser diodes and high speed transistors in processes which are exponentially increasing past 1 kW/cm2 as anticipated by Moore\u27s Law. The hypothesis guiding the work is that ultrasonic insonation of micro coolers employed to dissipate these heat loads can improve heat removal. It is thought that the mechanism promoting the benefit is enhancement of the ability of the coolant to remove latent heat in two-phase operation by managing entrained bubble size near the cooler\u27s exit so as to forestall flow reduction or blockage caused by large bubbles, wedges and slugs accumulating there. Insonation experiments to prove the hypothesis have been done on several micro channel coolers in the range 4-80 kHz to quantify improvement in heat flux removal. In order to understand how insonation would produce benefit in heat removal, a research effort was undertaken to study the affect of 5-30 Pa acoustic fields on air bubbles rising in small aquariums. This involved developing a Faraday cage shielded acoustic probe, along with a force-beam calibration tool, for measuring field levels near a strongly electromagnetic-radiating ultrasonic source. Experiments were conducted on columns of pseudo monodisperse, sub-millimeter diameter air bubbles in water, and other fluids using bubble generators optimized for this purpose. A numerical analysis model based on energy balance of the acoustic work done on a bubble resulted in predicting mass transfer flux, and in quantifying bubble shrinkage and growth when irradiated on either side of its resonance. The model, and experiments show that bubble populations can be predictably altered by ultrasound. The research was concluded by identifying and quantifying micro channel cooler performance change when insonated in the range 4-80 kHz. It was discovered that 28 and 58 kHz radiation of exchangers having hydraulic diameters spanning 0.02 to 0.6 mm could produce heat flux removal improvements of 5 W/cm2 in devices normally removing less than 30 W/cm2, a factor of 17%. Peak thermal resistance improvement approaching 60 % has been observed
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