Computational understanding and manipulation of symmetries

Attila Egri-Nagy, Chrystopher L Nehaniv, "Computational Understanding and Manipulation of Symmetries", in Chalup S. K., Blair A. D., Randall M. (Eds) Artificial Life and Computational Intelligence ACALCI, First Australasian Conference, Newcastle, NSW, Australia, February 5-7 2015, Proceedings, Lecture Notes in Computer Science, Vol. 8955, 2015 © Springer International Publishing Switzerland 2015 Final, published version of this paper is available online via doi: 10.1007/978-3-319-14803-8_2For natural and artificial systems with some symmetry structure, computational understanding and manipulation can be achieved without learning by exploiting the algebraic structure. This algebraic coordinatization is based on a hierarchical (de)composition method. Here we describe this method and apply it to permutation puzzles. Coordinatization yields a structural understanding, not just solutions for the puzzles. In the case of the Rubik’s Cubes, different solving strategies correspond to different decompositions

Program of Research in Structures and Dynamics

The Structures and Dynamics Program was first initiated in 1972 with the following two major objectives: to provide a basic understanding and working knowledge of some key areas pertinent to structures, solid mechanics, and dynamics technology including computer aided design; and to provide a comprehensive educational and research program at the NASA Langley Research Center leading to advanced degrees in the structures and dynamics areas. During the operation of the program the research work was done in support of the activities of both the Structures and Dynamics Division and the Loads and Aeroelasticity Division. During the period of 1972 to 1986 the Program provided support for two full-time faculty members, one part-time faculty member, three postdoctoral fellows, one research engineer, eight programmers, and 28 graduate research assistants. The faculty and staff of the program have published 144 papers and reports, and made 70 presentations at national and international meetings, describing their research findings. In addition, they organized and helped in the organization of 10 workshops and national symposia in the structures and dynamics areas. The graduate research assistants and the students enrolled in the program have written 20 masters theses and 2 doctoral dissertations. The overall progress is summarized

Tensor network states and algorithms in the presence of a global U(1) symmetry

Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. In a recent paper [arXiv:0907.2994v1] we discussed how to incorporate a global internal symmetry, given by a compact, completely reducible group G, into tensor network decompositions and algorithms. Here we specialize to the case of Abelian groups and, for concreteness, to a U(1) symmetry, often associated with particle number conservation. We consider tensor networks made of tensors that are invariant (or covariant) under the symmetry, and explain how to decompose and manipulate such tensors in order to exploit their symmetry. In numerical calculations, the use of U(1) symmetric tensors allows selection of a specific number of particles, ensures the exact preservation of particle number, and significantly reduces computational costs. We illustrate all these points in the context of the multi-scale entanglement renormalization ansatz.Comment: 22 pages, 25 figures, RevTeX

The Tensor Networks Anthology: Simulation techniques for many-body quantum lattice systems

We present a compendium of numerical simulation techniques, based on tensor network methods, aiming to address problems of many-body quantum mechanics on a classical computer. The core setting of this anthology are lattice problems in low spatial dimension at finite size, a physical scenario where tensor network methods, both Density Matrix Renormalization Group and beyond, have long proven to be winning strategies. Here we explore in detail the numerical frameworks and methods employed to deal with low-dimension physical setups, from a computational physics perspective. We focus on symmetries and closed-system simulations in arbitrary boundary conditions, while discussing the numerical data structures and linear algebra manipulation routines involved, which form the core libraries of any tensor network code. At a higher level, we put the spotlight on loop-free network geometries, discussing their advantages, and presenting in detail algorithms to simulate low-energy equilibrium states. Accompanied by discussions of data structures, numerical techniques and performance, this anthology serves as a programmer's companion, as well as a self-contained introduction and review of the basic and selected advanced concepts in tensor networks, including examples of their applications.Comment: 115 pages, 56 figure

Global models: Robot sensing, control, and sensory-motor skills

Robotics research has begun to address the modeling and implementation of a wide variety of unstructured tasks. Examples include automated navigation, platform servicing, custom fabrication and repair, deployment and recovery, and science exploration. Such tasks are poorly described at onset; the workspace layout is partially unfamiliar, and the task control sequence is only qualitatively characterized. The robot must model the workspace, plan detailed physical actions from qualitative goals, and adapt its instantaneous control regimes to unpredicted events. Developing robust representations and computational approaches for these sensing, planning, and control functions is a major challenge. The underlying domain constraints are very general, and seem to offer little guidance for well-bounded approximation of object shape and motion, manipulation postures and trajectories, and the like. This generalized modeling problem is discussed, with an emphasis on the role of sensing. It is also discussed that unstructured tasks often have, in fact, a high degree of underlying physical symmetry, and such implicit knowledge should be drawn on to model task performance strategies in a methodological fashion. A group-theoretic decomposition of the workspace organization, task goals, and their admissible interactions are proposed. This group-mechanical approach to task representation helps to clarify the functional interplay of perception and control, in essence, describing what perception is specifically for, versus how it is generically modeled. One also gains insight how perception might logically evolve in response to needs of more complex motor skills. It is discussed why, of the many solutions that are often mathematically admissible to a given sensory motor-coordination problem, one may be preferred over others