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By Miguel A. Sainz, Joaquim Armengol, Remei Calm, Pau Herrero, Lambert Jorba, Josep Vehi
This publication provides an cutting edge new method of period research. Modal period research (MIA) is an try to transcend the constraints of vintage durations by way of their structural, algebraic and logical positive aspects. the place to begin of MIA is sort of basic: It is composed in defining a modal period that attaches a quantifier to a classical period and in introducing the elemental relation of inclusion among modal periods during the inclusion of the units of predicates they settle for. This modal strategy introduces period extensions of the true non-stop features, identifies equivalences among logical formulation and period inclusions, and gives the semantic theorems that justify those equivalences, besides directions for arriving at those inclusions. purposes of those equivalences in several components illustrate the acquired effects. The publication additionally offers a brand new period item: marks, which aspire to be a brand new type of numerical remedy of error in measurements and computations.
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Additional info for Modal Interval Analysis: New Tools for Numerical Information
Sample text
Its first and second coordinates . R/ is called a “point-wise interval” or “degenerated interval”, graphical representations in a (Inf,Sup)-diagram are in Fig. 2, where the diagonal contains the point-wise intervals, proper intervals are represented by points in the left half-plane, and improper intervals are in the right half-plane. A/, allows the definition of modal inclusion and modal equality in terms of the inclusion and equality of the sets of predicates. B/. x/ W x 2 A0 . 9x 2 A0 / x D a is true.
R/: 1. X 0 / Â A; 2. X 0 / Ã A. Proof 1. X 0 ; 8/. X 0 ; 8/: 2. R/, and to identify also the set-theoretical expressions on the left of these equivalences by the analytical and programmable relations on the right. 12, because of their failure to provide the expected equalities, obstructs any straight path from the predicate-theoretical semantics of modal intervals to the semantics of their inclusion-lattice. Its ultimate meaning is the fact, common to all non-cheating real-life information processing, that interval processing of digital numerical information necessarily implies a certain degree of information loss.
R, it is necessary to give some preliminary definitions which will make it possible to avoid the use of the settheoretical extension. Rk / for the set of k-dimensional modal intervals. R/ are generalized in a natural way. Rki //, where kp C ki D k, and the original indices are supposed maintained. 4 The definition of the vectors Ap and A i would actually imply the rigorous definition of vectors with void components with their corresponding operations. It should be noted that this notation does not imply any permutation of the components of A, the vector A is not modified and, consequently, each component preserves its original index on A.