Intervals for the resultants of interval forces with existentially and/or universally quantified formulae with the help of the method of quantifier elimination

• Κύριος συγγραφέας: Ioakimidis, Nikolaos
• Άλλοι συγγραφείς: Ιωακειμίδης, Νικόλαος
• Μορφή: Technical Report
• Γλώσσα: English
• Έκδοση: 2020
• Θέματα: Intervals; Interval analysis; Interval arithmetic; Interval variables; Subtraction; Uncertainty; Uncertain variables; Uncertain forces; Interval forces; Collinear forces; Resultant; Quantifiers; Universal quantifier; Existential quantifier; Quantified formulae; Quantified/free variables; Quantifier elimination; Quantifier-free formulae; Control problem; Tolerance problem; Overestimation-free intervals; Symbolic computations; Mathematica; Διαστήματα; Ανάλυση διαστημάτων; Αριθμητική διαστημάτων; Μεταβλητές διαστήματος; Αφαίρεση; Αβεβαιότητα; Αβέβαιες μεταβλητές; Αβέβαιες δυνάμεις; Δυνάμεις διαστήματος; Συγγραμμικές δυνάμεις; Συνισταμένη; Ποσοδείκτες; Καθολικός ποσοδείκτης; Υπαρξιακός ποσοδείκτης; Τύποι με ποσοδείκτες; Μεταβλητές με ποσοδείκτες/ελεύθερες μεταβλητές; Απαλοιφή ποσοδεικτών; Τύποι χωρίς ποσοδείκτες; Πρόβλημα ελέγχου; Πρόβλημα ανοχής; Διαστήματα χωρίς υπερεκτίμηση; Συμβολικοί υπολογισμοί

The problem of the computation of the interval of the resultant of collinear uncertain forces represented by intervals without overestimation has been recently studied in two papers (i) by Elishakoff, Gabriele and Wang (2016) and (ii) by Popova (2017). In the first paper, a modification of classical interval arithmetic is proposed whereas the methodology proposed in the second paper is based on the algebraic extension of classical interval arithmetic. Here the general case of the computation of the interval of this resultant is studied in detail on the basis of the use of quantified formulae including the existential and/or the universal quantifiers with respect to the interval forces. Many quantified formulae are possible in a resultant problem and the method of quantifier elimination in its implementation in the computer algebra system Mathematica is used for the derivation of the related quantifier-free formulae. After the illustration of the present approach in the elementary subtraction problem, which is well known for the overestimation phenomenon, the same approach is illustrated in problems (originally studied in the above papers) concerning the resultants of two, three and four collinear forces with different directions as well as in the problem of three collinear forces acting on a box. Symbolic intervals with parameters one or two of the forces are also computed. The case of the resultant of many collinear interval forces is also successfully studied. The conclusion drawn is that several overestimation-free, exact intervals can be computed for the resultant of interval forces (frequently including a degenerate interval: sharp resultant) and the derived interval (if it exists) strongly depends on the quantifiers used for the interval forces.