76,013 research outputs found
Invariants of solvable Lie algebras with triangular nilradicals and diagonal nilindependent elements
The invariants of solvable Lie algebras with nilradicals isomorphic to the
algebra of strongly upper triangular matrices and diagonal nilindependent
elements are studied exhaustively. Bases of the invariant sets of all such
algebras are constructed by an original purely algebraic algorithm based on
Cartan's method of moving frames.Comment: 21 pages, enhanced and extended version. Section 2 reviews the method
of finding invariants of Lie algebras that was proposed in
arXiv:math-ph/0602046 and arXiv:math-ph/0606045. The computation is based on
developing a specific technique given in arXiv:0704.0937. Results generalize
ones of arXiv:0705.2394 to the case of arbitrary relevant number of
nilindependent element
Invariants of Lie Algebras with Fixed Structure of Nilradicals
An algebraic algorithm is developed for computation of invariants
('generalized Casimir operators') of general Lie algebras over the real or
complex number field. Its main tools are the Cartan's method of moving frames
and the knowledge of the group of inner automorphisms of each Lie algebra.
Unlike the first application of the algorithm in [J. Phys. A: Math. Gen., 2006,
V.39, 5749; math-ph/0602046], which deals with low-dimensional Lie algebras,
here the effectiveness of the algorithm is demonstrated by its application to
computation of invariants of solvable Lie algebras of general dimension
restricted only by a required structure of the nilradical.
Specifically, invariants are calculated here for families of real/complex
solvable Lie algebras. These families contain, with only a few exceptions, all
the solvable Lie algebras of specific dimensions, for whom the invariants are
found in the literature.Comment: LaTeX2e, 19 page
FoCaLiZe: Inside an F-IDE
For years, Integrated Development Environments have demonstrated their
usefulness in order to ease the development of software. High-level security or
safety systems require proofs of compliance to standards, based on analyses
such as code review and, increasingly nowadays, formal proofs of conformance to
specifications. This implies mixing computational and logical aspects all along
the development, which naturally raises the need for a notion of Formal IDE.
This paper examines the FoCaLiZe environment and explores the implementation
issues raised by the decision to provide a single language to express
specification properties, source code and machine-checked proofs while allowing
incremental development and code reusability. Such features create strong
dependencies between functions, properties and proofs, and impose an particular
compilation scheme, which is described here. The compilation results are
runnable OCaml code and a checkable Coq term. All these points are illustrated
through a running example.Comment: In Proceedings F-IDE 2014, arXiv:1404.578
Online Planner Selection with Graph Neural Networks and Adaptive Scheduling
Automated planning is one of the foundational areas of AI. Since no single
planner can work well for all tasks and domains, portfolio-based techniques
have become increasingly popular in recent years. In particular, deep learning
emerges as a promising methodology for online planner selection. Owing to the
recent development of structural graph representations of planning tasks, we
propose a graph neural network (GNN) approach to selecting candidate planners.
GNNs are advantageous over a straightforward alternative, the convolutional
neural networks, in that they are invariant to node permutations and that they
incorporate node labels for better inference.
Additionally, for cost-optimal planning, we propose a two-stage adaptive
scheduling method to further improve the likelihood that a given task is solved
in time. The scheduler may switch at halftime to a different planner,
conditioned on the observed performance of the first one. Experimental results
validate the effectiveness of the proposed method against strong baselines,
both deep learning and non-deep learning based.
The code is available at \url{https://github.com/matenure/GNN_planner}.Comment: AAAI 2020. Code is released at
https://github.com/matenure/GNN_planner. Data set is released at
https://github.com/IBM/IPC-graph-dat
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