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    On the convergence of the direct quadrature method for Cauchy type singular integral equations of the first kind
    (1983-02-10) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    The convergence of the direct quadrature method (based on the Gauss–Chebyshev quadrature rule) for Cauchy type singular integral equations of the first kind is demonstrated under appropriate conditions. The rate of convergence is also established.
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    Open Access
    Application of the method of quantifier elimination to Ben-Haim's info-gap decision theory (IGDT) under the presence of both horizon-of-uncertainty-related and ordinary interval uncertain variables
    (2023-06-13) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    Problems under uncertainty conditions appear frequently in practice. The use of classical interval analysis constitutes an interesting tool for the study of such problems with the uncertain variables assumed to be interval variables. Ben-Haim's info-gap (or information-gap) decision theory (IGDT) constitutes an interesting method for the study of such problems. The determination of the maximum value of the uncertainty parameter (or horizon of uncertainty) for the uncertain variables so that the performance requirement(s) is (are) satisfied is of primary importance in the IGDT. On the other hand, the method of quantifier elimination in computer algebra permits the transformation of quantified formulae to equivalent formulae, but free from the quantified variables. Here the method of quantifier elimination is applied to the mixed case with two or three uncertain variables where one (or two) of these variables is (are) ordinary interval variable(s) whereas the remaining uncertain variable(s) satisfies (satisfy) the popular fractional-error model of uncertainty in the IGDT. Therefore, here the horizon of uncertainty concerns only the latter variable(s). The present method is illustrated in the following five simple applications: (i) the problem of the area of a rectangle, (ii) the problem of the volume of a rectangular cuboid, (iii) the problem of the buckling load of a fixed–free column, (iv) the problem of the equivalent spring constants of two elastic springs connected in series and in parallel and (v) the similar problem for the resistances of three resistors.
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    An application of Ben-Haim's info-gap decision theory (IGDT) to Todinov's method of algebraic inequalities by employing the method of quantifier elimination
    (Κανένας, 2022-10-10) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    Problems under uncertainty conditions can be studied by using the very interesting and popular Ben-Haim's info-gap (or information-gap) decision theory (IGDT). On the other hand, recently, Todinov proposed an interesting and efficient method based on algebraic inequalities for the reduction of risk and uncertainty as well as for the generation of new knowledge and the optimization of systems and processes. One of the main problems where Todinov applied his new method is the problem concerning the equivalent resistances of n resistors in an electrical circuit connected both in series and in parallel. Here we consider the same problem, but now with the related algebraic inequality used as the performance requirement in Ben-Haim's IGDT. The methodology used here is based on the computational method of quantifier elimination. This method constitutes a very interesting approach for the transformation of quantified formulae to logically equivalent formulae, but now free from the quantifiers and the quantified variables. The same method is implemented in some computer algebra systems including Mathematica, which is used here. The problems studied here and related to the equivalent resistances of two or three resistors concern (i) two resistors with one horizon of uncertainty including the cases of parametric nominal value(s) of one resistance or both resistances here by using a fractional-error uncertainty model in Ben-Haim's IGDT, (ii) two resistors again, but with two horizons of uncertainty, (iii) three resistors with one horizon of uncertainty and (iv) two resistors again, but with the use of an ellipsoidal uncertainty model. The use of negated existentially quantified formulae instead of universally quantified formulae is also studied.
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    Applications of quantifier elimination to the proofs of algebraic inequalities in engineering problems related to Todinov's method for risk reduction
    (Κανένας, ) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    An interesting and efficient method based on algebraic inequalities for the reduction of risk and uncertainty as well as for the generation of new knowledge and the optimization of systems and processes has been recently proposed by Todinov. This method requires the proof of the related inequality and several classical approaches were successfully used by Todinov for this task. Here the use of the well-known method of quantifier elimination for the same task, that is for the proof of algebraic inequalities related to Todinov's method, as an additional approach appropriate for use with respect to some algebraic inequalities is proposed and actually used in some problems already proposed and successfully solved by Todinov on the basis of his own method of algebraic inequalities. The present approach is applied to some algebraic inequalities of engineering interest already proved by Todinov, more explicitly (i) to the inequality for the equivalent resistances of two resistors in an electrical circuit, (ii) to the inequality for the equivalent spring constants of two, three or four elastic springs, (iii) to inequalities concerning the supply of high-reliability components, (iv) to inequalities concerning ranking systems, (v) to the construction of a system with superior reliability and (vi) to the accumulated strain energy in bars under tension or in cantilevers under bending. Naturally, because of the well-known doubly-exponential computational complexity of quantifier elimination the present approach is applicable only when the related universally quantified formula contains a small total number of variables (free and quantified). Yet, the method of quantifier elimination can also be combined with the method of proof by induction and this additional possibility is also illustrated in two inequalities related to Todinov's method for their proof in the general case.
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    Problems under uncertainty : quantifier elimination to universally–existentially (AE) quantified formulae related to two or more horizons of uncertainty
    (Κανένας, ) Ioakimidis, Nikolaos; Ιωακειμίδης, Νικόλαος
    Problems under uncertainty appear frequently in practical applications. Ben-Haim's IGDT (info-gap decision theory) constitutes a very efficient method for the study of such problems. The three components (or elements) of Ben-Haim's IGDT are (i) the system model, (ii) the info-gap uncertainty model and (iii) the performance requirement(s). Appropriate (mainly positivity) assumptions can also be made. Here we use the IGDT only partially by restricting our attention to its first component, the system model, and to its second component, the info-gap uncertainty model, but paying no attention to the performance requirement(s) also very important in the IGDT. Here an emphasis is put on the use of (mixed) universally–existentially (AE) quantified formulae assuring the validity of the system model (under the assumptions made) for all values of the universally quantified uncertain variable(s) and for at least one value (or a set of values) of the existentially quantified uncertain variable(s) of course provided that these quantified variables satisfy the adopted info-gap uncertainty model here the popular fractional-error model. On the other hand, here we also assume that each uncertain variable (either universally or existentially quantified) has its own uncertainty parameter (or horizon of uncertainty). Next, by using the method of quantifier elimination in its powerful implementation in the computer algebra system Mathematica we transform the quantified formula to an equivalent QFF (quantifier-free formula) free from the quantifiers and the quantified variables, but, evidently, including the horizons of uncertainty. Two simple applications concerning (i) a product/quotient and (ii) the buckling load of a fixed–free column illustrate the present approach with the derivation of the related QFFs, some of which can also be verified manually.