6 research outputs found
Polynomial Time Algorithms in Invariant Theory for Torus Actions
An action of a group on a vector space partitions the latter into a set of orbits. We consider three natural and useful algorithmic "isomorphism" or "classification" problems, namely, orbit equality, orbit closure intersection, and orbit closure containment. These capture and relate to a variety of problems within mathematics, physics and computer science, optimization and statistics. These orbit problems extend the more basic null cone problem, whose algorithmic complexity has seen significant progress in recent years.
In this paper, we initiate a study of these problems by focusing on the actions of commutative groups (namely, tori). We explain how this setting is motivated from questions in algebraic complexity, and is still rich enough to capture interesting combinatorial algorithmic problems. While the structural theory of commutative actions is well understood, no general efficient algorithms were known for the aforementioned problems. Our main results are polynomial time algorithms for all three problems. We also show how to efficiently find separating invariants for orbits, and how to compute systems of generating rational invariants for these actions (in contrast, for polynomial invariants the latter is known to be hard). Our techniques are based on a combination of fundamental results in invariant theory, linear programming, and algorithmic lattice theory
Reducing the environmental impact of surgery on a global scale: systematic review and co-prioritization with healthcare workers in 132 countries
Background
Healthcare cannot achieve net-zero carbon without addressing operating theatres. The aim of this study was to prioritize feasible interventions to reduce the environmental impact of operating theatres.
Methods
This study adopted a four-phase Delphi consensus co-prioritization methodology. In phase 1, a systematic review of published interventions and global consultation of perioperative healthcare professionals were used to longlist interventions. In phase 2, iterative thematic analysis consolidated comparable interventions into a shortlist. In phase 3, the shortlist was co-prioritized based on patient and clinician views on acceptability, feasibility, and safety. In phase 4, ranked lists of interventions were presented by their relevance to high-income countries and low–middle-income countries.
Results
In phase 1, 43 interventions were identified, which had low uptake in practice according to 3042 professionals globally. In phase 2, a shortlist of 15 intervention domains was generated. In phase 3, interventions were deemed acceptable for more than 90 per cent of patients except for reducing general anaesthesia (84 per cent) and re-sterilization of ‘single-use’ consumables (86 per cent). In phase 4, the top three shortlisted interventions for high-income countries were: introducing recycling; reducing use of anaesthetic gases; and appropriate clinical waste processing. In phase 4, the top three shortlisted interventions for low–middle-income countries were: introducing reusable surgical devices; reducing use of consumables; and reducing the use of general anaesthesia.
Conclusion
This is a step toward environmentally sustainable operating environments with actionable interventions applicable to both high– and low–middle–income countries
Polynomial Time Algorithms in Invariant Theory for Torus Actions
An action of a group on a vector space partitions the latter into a set of
orbits. We consider three natural and useful algorithmic "isomorphism" or
"classification" problems, namely, orbit equality, orbit closure intersection,
and orbit closure containment. These capture and relate to a variety of
problems within mathematics, physics and computer science, optimization and
statistics. These orbit problems extend the more basic null cone problem, whose
algorithmic complexity has seen significant progress in recent years.
In this paper, we initiate a study of these problems by focusing on the
actions of commutative groups (namely, tori). We explain how this setting is
motivated from questions in algebraic complexity, and is still rich enough to
capture interesting combinatorial algorithmic problems. While the structural
theory of commutative actions is well understood, no general efficient
algorithms were known for the aforementioned problems. Our main results are
polynomial time algorithms for all three problems. We also show how to
efficiently find separating invariants for orbits, and how to compute systems
of generating rational invariants for these actions (in contrast, for
polynomial invariants the latter is known to be hard). Our techniques are based
on a combination of fundamental results in invariant theory, linear
programming, and algorithmic lattice theory.Comment: 27 page