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Maciej Wygralak - One of the best experts on this subject based on the ideXlab platform.

  • An Axiomatic Approach to scalar cardinalities of fuzzy sets
    Fuzzy Sets and Systems, 2000
    Co-Authors: Maciej Wygralak
    Abstract:

    Abstract We present an Axiomatic Approach to scalar cardinalities of fuzzy sets which is based on a system of three simple postulates. A characterization theorem for those cardinalities is given. The infinite family of possible scalar cardinalities of a fuzzy set generated by the postulates contains as particular cases all standard concepts of scalar cardinality, e.g. the sigma count of a fuzzy set, the cardinality of its core or support, and the cardinality of its t -level set.

Vladimir Pestov - One of the best experts on this subject based on the ideXlab platform.

  • An Axiomatic Approach to intrinsic dimension of a dataset.
    Neural networks : the official journal of the International Neural Network Society, 2007
    Co-Authors: Vladimir Pestov
    Abstract:

    We perform a deeper analysis of an Axiomatic Approach to the concept of intrinsic dimension of a dataset proposed by us in the IJCNN'07 paper. The main features of our Approach are that a high intrinsic dimension of a dataset reflects the presence of the curse of dimensionality (in a certain mathematically precise sense), and that dimension of a discrete i.i.d. sample of a low-dimensional manifold is, with high probability, close to that of the manifold. At the same time, the intrinsic dimension of a sample is easily corrupted by moderate high-dimensional noise (of the same amplitude as the size of the manifold) and suffers from prohibitively high computational complexity (computing it is an NP-complete problem). We outline a possible way to overcome these difficulties.

  • 2008 Special Issue: An Axiomatic Approach to intrinsic dimension of a dataset
    Neural Networks, 2007
    Co-Authors: Vladimir Pestov
    Abstract:

    We perform a deeper analysis of an Axiomatic Approach to the concept of intrinsic dimension of a dataset proposed by us in the IJCNN'07 paper. The main features of our Approach are that a high intrinsic dimension of a dataset reflects the presence of the curse of dimensionality (in a certain mathematically precise sense), and that dimension of a discrete i.i.d. sample of a low-dimensional manifold is, with high probability, close to that of the manifold. At the same time, the intrinsic dimension of a sample is easily corrupted by moderate high-dimensional noise (of the same amplitude as the size of the manifold) and suffers from prohibitively high computational complexity (computing it is an NP-complete problem). We outline a possible way to overcome these difficulties.

  • An Axiomatic Approach to intrinsic dimension of a dataset
    arXiv: Information Retrieval, 2007
    Co-Authors: Vladimir Pestov
    Abstract:

    We perform a deeper analysis of an Axiomatic Approach to the concept of intrinsic dimension of a dataset proposed by us in the IJCNN'07 paper (arXiv:cs/0703125). The main features of our Approach are that a high intrinsic dimension of a dataset reflects the presence of the curse of dimensionality (in a certain mathematically precise sense), and that dimension of a discrete i.i.d. sample of a low-dimensional manifold is, with high probability, close to that of the manifold. At the same time, the intrinsic dimension of a sample is easily corrupted by moderate high-dimensional noise (of the same amplitude as the size of the manifold) and suffers from prohibitevely high computational complexity (computing it is an $NP$-complete problem). We outline a possible way to overcome these difficulties.

  • Foundations of supermanifold theory: the Axiomatic Approach
    Differential Geometry and its Applications, 1993
    Co-Authors: Claudio Bartocci, Ugo Bruzzo, D. Hernandez Ruiperez, Vladimir Pestov
    Abstract:

    Abstract We discuss an Axiomatic Approach to supermanifolds valid for arbitrary ground graded- commutative Banach algebras B. Rothstein's Axiomatics is revisited and completed by a further requirement which calls for the completeness of the rings of sections of the structure sheaves, and allows one to dispose of some undesirable features of Rothstein supermanifolds. The ensuing system of axioms determines a category of supermanifolds which coincides with graded manifolds when B  R , and with G-supermanifolds when B is a finite-dimensional exterior algebra. This category is studied in detail. The case of holomorphic supermanifolds is also outlined.

János Fodor - One of the best experts on this subject based on the ideXlab platform.

Zhenyu Xiu - One of the best experts on this subject based on the ideXlab platform.

  • an Axiomatic Approach to bases and subbases in l convex spaces and their applications
    Fuzzy Sets and Systems, 2019
    Co-Authors: Bin Pang, Zhenyu Xiu
    Abstract:

    Abstract Considering a continuous lattice L as the lattice background, an Axiomatic Approach to bases and subbases in the framework of L-convex spaces is provided. Firstly, the concepts of bases and subbases in L-convex spaces are introduced and then L-convexity base axioms and L-convexity subbase axioms are proposed by abstracting the properties of bases and subbases, respectively. Secondly, some applications of L-convexity base axioms and L-convexity subbase axioms are investigated. It is shown that L-convexity base axioms can be used to demonstrate some relationship between spatial structures with respect to L-convex structures and L-convexity subbase axioms can be applied to define the join space and the product space of L-convex spaces.

Mich Tvede - One of the best experts on this subject based on the ideXlab platform.

  • Benchmark selection: An Axiomatic Approach
    European Journal of Operational Research, 2002
    Co-Authors: Jens Leth Hougaard, Mich Tvede
    Abstract:

    Abstract Within a production theoretic framework, this paper considers an Axiomatic Approach to benchmark selection. It is shown that two simple and weak axioms; efficiency and comprehensive monotonicity characterize a natural family of benchmarks which typically becomes unique. Further axioms are added in order to obtain a unique selection.