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 t→ℓ′νbℓℓt \to \ell' \nu b \ell\ell and t→qq′bℓℓt \to q q' b \ell\ell on Searches for Top-Associated Physics

Searches for top quark-associated physics such as $t\bar t W$ or $t\bar t H$ 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 $t \to \ell' \nu b \ell\ell$ and $t \to q q' b \ell\ell$, 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