Advances in Computational Intelligence: 14th International by Margaret F. Shipley, Gary L. Stading (auth.), Salvatore

By Margaret F. Shipley, Gary L. Stading (auth.), Salvatore Greco, Bernadette Bouchon-Meunier, Giulianella Coletti, Mario Fedrizzi, Benedetto Matarazzo, Ronald R. Yager (eds.)

These 4 volumes (CCIS 297, 298, 299, three hundred) represent the complaints of the 14th overseas convention on info Processing and administration of Uncertainty in Knowledge-Based platforms, IPMU 2012, held in Catania, Italy, in July 2012. The 258 revised complete papers offered including six invited talks have been conscientiously reviewed and chosen from a number of submissions. The papers are equipped in topical sections on fuzzy laptop studying and online modeling; computing with phrases and choice making; tender computing in machine imaginative and prescient; tough units and complicated facts research: thought and purposes; clever databases and data approach; details fusion platforms; philosophical and methodological elements of sentimental computing; uncomplicated concerns in tough units; fortieth anniversary of the measures of fuziness; SPS11 uncertainty in profiling structures and functions; dealing with uncertainty with copulas; formal how you can care for uncertainty of many-valued occasions; linguistic summarization and outline of knowledge; fuzzy implications: conception and purposes; sensing and knowledge mining for educating and studying; idea and functions of intuitionistic fuzzy units; approximate points of knowledge mining and database analytics; fuzzy numbers and their purposes; info processing and administration of uncertainty in knowledge-based structures; aggregation features; vague percentages; probabilistic graphical types with imprecision: conception and purposes; trust functionality conception: fundamentals and/or functions; fuzzy uncertainty in economics and company; new developments in De Finetti's process; fuzzy measures and integrals; multi standards choice making; uncertainty in privateness and protection; uncertainty within the spirit of Pietro Benvenuti; coopetition; online game concept; probabilistic approach.

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N . (8) Property (8) clearly follows from (1) and proves that the image set of A∗ through the logarithmic function is given by a homogeneous linear system (8) and it is 34 M. Fedrizzi therefore a linear subspace of Rn×n , say L = {ln(B); B ∈ A∗ } = ln(A∗ ). Linear spaces are simple structures and allow working with tools of linear algebra. In [19] it is proposed to compute a consistent approximation of an inconsistent PCM A by the orthogonal projection of the transformed matrix ln(A) = (ln(aij ))n×n onto the linear subspace L.

Applic. 34, 41–47 (1997) 20. : A new measure of consistency for positive reciprocal matrices. Computers and Mathematics with Applications 46, 1839–1845 (2003) 21. : Inconsistency of pair-wise comparison matrix with fuzzy elements on the geometric mean. Fuzzy Sets and Systems 161, 1604–1613 (2010) 22. : Highlights and critical points in the theory and application of the Analytic Hierarchy Process. European Journal of Operational Research 74, 426– 447 (1994) 23. : A scaling method for priorities in hierarchical structures.

Proposition 2. If f, g are convex functions in F such that f g, then g is asymptotically linear. If f, g are concave functions in F such that f g then f is asymptotically linear. Proof. We have that f (n + 1) − f (n) g(m + 1) − g(m) for all n, m ∈ Æ Æ Æ. Then we have for all m ∈ , g(m + 1) − g(m) f (1) − f (0) = f (1) and so we can prove that for all n ∈ , g(n) nf (1). Since the function g is convex the difference g(n+ 1)− g(n) is increasing with respect to n and so we can prove that g(n + 1) − g(n) g(1) − g(0) = g(1) and then for all n ∈ , g(n) ng(1).

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