1,917 research outputs found

    Big Toy Models: Representing Physical Systems As Chu Spaces

    Full text link
    We pursue a model-oriented rather than axiomatic approach to the foundations of Quantum Mechanics, with the idea that new models can often suggest new axioms. This approach has often been fruitful in Logic and Theoretical Computer Science. Rather than seeking to construct a simplified toy model, we aim for a `big toy model', in which both quantum and classical systems can be faithfully represented - as well as, possibly, more exotic kinds of systems. To this end, we show how Chu spaces can be used to represent physical systems of various kinds. In particular, we show how quantum systems can be represented as Chu spaces over the unit interval in such a way that the Chu morphisms correspond exactly to the physically meaningful symmetries of the systems - the unitaries and antiunitaries. In this way we obtain a full and faithful functor from the groupoid of Hilbert spaces and their symmetries to Chu spaces. We also consider whether it is possible to use a finite value set rather than the unit interval; we show that three values suffice, while the two standard possibilistic reductions to two values both fail to preserve fullness.Comment: 24 pages. Accepted for Synthese 16th April 2010. Published online 20th April 201

    A mathematical proof of the existence of trends in financial time series

    Get PDF
    We are settling a longstanding quarrel in quantitative finance by proving the existence of trends in financial time series thanks to a theorem due to P. Cartier and Y. Perrin, which is expressed in the language of nonstandard analysis (Integration over finite sets, F. & M. Diener (Eds): Nonstandard Analysis in Practice, Springer, 1995, pp. 195--204). Those trends, which might coexist with some altered random walk paradigm and efficient market hypothesis, seem nevertheless difficult to reconcile with the celebrated Black-Scholes model. They are estimated via recent techniques stemming from control and signal theory. Several quite convincing computer simulations on the forecast of various financial quantities are depicted. We conclude by discussing the r\^ole of probability theory

    Hierarchical bases for non-hierarchic 3Dtriangular meshes

    Get PDF
    We describe a novel basis of hierarchical, multiscale functions that are linear combinations of standard Rao-Wilton- Glisson (RWG) functions. When the basis is used for discretizing the electric field integral equation (EFIE) for PEC objects it gives rise to a linear system immune from low-frequency breakdown, and well conditioned for dense meshes. The proposed scheme can be applied to any mesh with triangular facets, and therefore it can be used as if it were an algebraic preconditioner. The properties of the new system are confirmed by numerical results that show fast convergence rates of iterative solvers, significantly better than those for the loop-tree basis. As a byproduct of the basis generation, a generalization of the RWG functions to nonsimplex cells is introduced

    Changing a semantics: opportunism or courage?

    Full text link
    The generalized models for higher-order logics introduced by Leon Henkin, and their multiple offspring over the years, have become a standard tool in many areas of logic. Even so, discussion has persisted about their technical status, and perhaps even their conceptual legitimacy. This paper gives a systematic view of generalized model techniques, discusses what they mean in mathematical and philosophical terms, and presents a few technical themes and results about their role in algebraic representation, calibrating provability, lowering complexity, understanding fixed-point logics, and achieving set-theoretic absoluteness. We also show how thinking about Henkin's approach to semantics of logical systems in this generality can yield new results, dispelling the impression of adhocness. This paper is dedicated to Leon Henkin, a deep logician who has changed the way we all work, while also being an always open, modest, and encouraging colleague and friend.Comment: 27 pages. To appear in: The life and work of Leon Henkin: Essays on his contributions (Studies in Universal Logic) eds: Manzano, M., Sain, I. and Alonso, E., 201

    Robust Singular Smoothers For Tracking Using Low-Fidelity Data

    Full text link
    Tracking underwater autonomous platforms is often difficult because of noisy, biased, and discretized input data. Classic filters and smoothers based on standard assumptions of Gaussian white noise break down when presented with any of these challenges. Robust models (such as the Huber loss) and constraints (e.g. maximum velocity) are used to attenuate these issues. Here, we consider robust smoothing with singular covariance, which covers bias and correlated noise, as well as many specific model types, such as those used in navigation. In particular, we show how to combine singular covariance models with robust losses and state-space constraints in a unified framework that can handle very low-fidelity data. A noisy, biased, and discretized navigation dataset from a submerged, low-cost inertial measurement unit (IMU) package, with ultra short baseline (USBL) data for ground truth, provides an opportunity to stress-test the proposed framework with promising results. We show how robust modeling elements improve our ability to analyze the data, and present batch processing results for 10 minutes of data with three different frequencies of available USBL position fixes (gaps of 30 seconds, 1 minute, and 2 minutes). The results suggest that the framework can be extended to real-time tracking using robust windowed estimation.Comment: 9 pages, 9 figures, to be included in Robotics: Science and Systems 201

    Sculptures in Concurrency

    Full text link
    We give a formalization of Pratt's intuitive sculpting process for higher-dimensional automata (HDA). Based on this, we show that sculptures, Pratt's Chu spaces, and Johansen's ST-structures are in close correspondence. We also develop an algorithm to decide whether a HDA can be sculpted and use this to show that some natural acyclic HDA are not sculptures. We believe that our result shed new light on the intuitions behind sculpting as a method of modeling concurrent behavior, showing the precise reaches of its expressiveness. We also show that there are sculptures whose unfoldings cannot besculpted, and that sculptures are the same as Euclidean cubical complexes. This exposes a close connection between geometric and combinatorial models for concurrency which may be of use for both areas
    corecore