Topic
Piecewise linear function
About: Piecewise linear function is a research topic. Over the lifetime, 8133 publications have been published within this topic receiving 161444 citations.
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TL;DR: In this paper, the authors define a lattice statistical model on a triangulated manifold in four dimensions associated to a group G and show that the partition function of the model is invariant under the Alexander moves of the simplicial complex, thus it depends only on the piecewise linear topology of the manifold.
Abstract: We define a lattice statistical model on a triangulated manifold in four dimensions associated to a group G. When G=SU(2), the statistical weight is constructed from the 15j-symbol as well as the 6j-symbol for recombination of angular momenta, and the model may be regarded as the four-dimensional version of the Ponzano-Regge model. We show that the partition function of the model is invariant under the Alexander moves of the simplicial complex, thus it depends only on the piecewise linear topology of the manifold. For an orientable manifold, the model is related to the so-called BF model. The q-analog of the model is also constructed, and it is argued that its partition function is invariant under the Alexander moves. It is discussed how to realize the 't Hooft operator in these models associated to a closed surface in four dimensions as well as the Wilson operator associated to a closed loop. Correlation functions of these operators in the q-deformed version of the model would define a new type of invariants of knots and links in four dimensions.
379 citations
TL;DR: It is shown that the class of these functions is universal in the sense that if the problem of ultimately bounding the state in an assigned convex set via state feedback control can be solved via a Lyapunov function and a continuous state-feedback compensator then it can be solve via a polyhedral Lyap unov functionand a (possibly different) continuous control.
369 citations
23 Mar 2000
TL;DR: An experimental system called d/dt for approximating reachable states for hybrid systems whose continuous dynamics is defined by linear differential equations using an approximation algorithm whose accumulation of errors during the continuous evolution is much smaller than in previously-used methods.
Abstract: In this paper we describe an experimental system called d/dt for approximating reachable states for hybrid systems whose continuous dynamics is defined by linear differential equations. We use an approximation algorithm whose accumulation of errors during the continuous evolution is much smaller than in previously-used methods. The d/dt system can, so far, treat non-trivial continuous systems, hybrid systems, convex differential inclusions and controller synthesis problems.
360 citations
TL;DR: A new technique for fast “scan-along” computation of piecewise linear approximations of digital curves in 2-space is described and the application to the boundaries of the images of a lung and a rib in chest radiographs is illustrated.
359 citations
TL;DR: The toolkit as mentioned in this paper adapts a first-order perturbation approach and applies it in a piecewise fashion to solve dynamic models with occasionally binding constraints, such as a real business cycle model with a constraint on the level of investment and a New Keynesian model subject to the zero lower bound on nominal interest rates.
355 citations