Lecture Notes on Data Engineering and Communications Technologies 43 D Jude Hemanth Utku Kose Editors Artificial Intelligence and Applied Mathematics in Engineering Problems Proceedings of the International Conference on Artificial Intelligence and Applied Mathematics in Engineering (ICAIAME 2019) Lecture Notes on Data Engineering and Communications Technologies Volume 43 Series Editor Fatos Xhafa, Technical University of Catalonia, Barcelona, Spain The aim of the book series is to present cutting edge engineering approaches to data technologies and communications It will publish latest advances on the engineering task of building and deploying distributed, scalable and reliable data infrastructures and communication systems The series will have a prominent applied focus on data technologies and communications with aim to promote the bridging from fundamental research on data science and networking to data engineering and communications that lead to industry products, business knowledge and standardisation ** Indexing: The books of this series are submitted to ISI Proceedings, MetaPress, Springerlink and DBLP ** More information about this series at http://www.springer.com/series/15362 D Jude Hemanth Utku Kose • Editors Artificial Intelligence and Applied Mathematics in Engineering Problems Proceedings of the International Conference on Artificial Intelligence and Applied Mathematics in Engineering (ICAIAME 2019) 123 Editors D Jude Hemanth Department of ECE Karunya University Coimbatore, Tamil Nadu, India Utku Kose Department of Computer Engineering, Faculty of Engineering Suleyman Demirel University Isparta, Isparta, Turkey ISSN 2367-4512 ISSN 2367-4520 (electronic) Lecture Notes on Data Engineering and Communications Technologies ISBN 978-3-030-36177-8 ISBN 978-3-030-36178-5 (eBook) https://doi.org/10.1007/978-3-030-36178-5 © Springer Nature Switzerland AG 2020 This work is subject to copyright All rights are reserved by the Publisher, whether the whole or 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been made The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland Preface On behalf of the proceedings editors and the organization committee, it is with deep honor that I write this Preface to the Proceedings of the International Conference on Artificial Intelligence and Applied Mathematics in Engineering (ICAIAME 2019) held in Antalya, Manavgat (Turkey) The objective of the conference was to promote in academia, industry, organizations and governments, progress and expansion of knowledge concerning artificial intelligence and mathematical modeling techniques for advancing day-to-day affairs to make better and smart living The conference provided the opportunity to exchange ideas on machine learning, deep learning, robotics, algorithm design for intelligent solutions, image processing, prediction and diagnosis applications, operations research, discrete mathematics and general engineering applications, to experience the state-of-the-art technologies, identify solutions and build collaborations for real-time implementations In this context, the event provided a three-day, enjoyable scientific environment for all authors and participants to share–discuss their research results and experiences with an international audience Based on reviews from the scientific committee and the external reviewers, a total of 197 papers have been accepted to be presented within around 40 parallel sessions The proceedings are published by Springer under the Springer Series: Lecture Notes on Data Engineering and Communications Technologies, and the extended versions of papers with post-processing review will be published under some reputable journals In terms of international scope, ICAIAME 2019 included contributions by 18 different countries such as Algeria, China, Cyprus, Denmark, England, France, India, Iraq, Jordan, Kuwait, Lebanon, Mexico, Pakistan, Palestine, Switzerland, Trinidad Tobago, Turkey and USA It is great to see the outcomes of the research by the authors have found their way to the literature, thanks to valuable efforts shown in that remarkable event In addition to the contributed papers, a total of six invited keynote presentations were delivered by top experts in artificial intelligence and applied mathematics Dr Çetin Elmas highlighted the importance of ‘Artificial Intelligence in Project Management,’ Dr Jude Hemanth covered technical aspects of ‘Innovative Artificial v vi Preface Intelligence Approaches for Medical Image Analysis,’ Dr Paniel Reyes Cárdenas discussed ‘Diagrammatic Reasoning, Topological Mathematics and Artificial Intelligence,’ Dr Ali Allahverdi enlightened the audience with regard to ‘How to Publish Your Paper in a Reputable Journal,’ Dr Ender Özcan discussed ‘Recent Progress in Selection Hyper-Heuristics for Intelligent Optimisation’ and finally Dr Ekrem Savaş elaborated the topic titled ‘Some Sequence Spaces defined by Invariant Mean.’ The success of ICAIAME 2019 depends completely on the effort, talent and energy of researchers in the field of computer-based systems who have written and submitted papers on a variety of topics Praise is also deserved for the organizing and scientific committee members, and external reviewers, who have invested significant time in analyzing and assessing multiple papers, as holding and maintaining a high standard of quality for this conference The ICAIAME will act as strong base for researchers and scientists in the form of that excellent reference book I would like to thank all authors and participants for their contributions Anand Nayyar Preface Artificial intelligence (AI) is an exciting field of knowledge that had an explosion of sophistication and technical nuance in the last few years Let us consider only how the state of AI was purely hypothetical in many ways only 50 years ago and now we have not only developments that were envisaged in the wildest imaginations, but developments that were not even expected No doubt that AI is a field that has incited us to question about the nature of what we define as intelligence and the limits of our concepts about it However, though the discipline of AI in itself is essentially transdisciplinary, there is an important connection with philosophy that has not always been underlined properly: on the one hand because we need to every now and again stop and think the meaning of the achievements we have gotten thus far On the other hand, philosophy becomes important to even question what we want to achieve We need philosophy of AI to relate the achievements and plans that we engineer with the highest purposes of humankind Indeed, no discipline of knowledge is alien to human ethical issues and AI is not the exception One of the important lessons we have learned in the last few decades is, in my opinion, the ability to acknowledge that AI does not need to be necessarily modeled in human intelligence, and that human minds have aspects that cannot be translated into modeling due to its own very nature of being self-conscious in ways that artificial systems are not But the illustration also works for us: There are advantages that AI has given us that make humans recognize that we can flourish by integrating to our life developments that are exclusive of AI systems and we could not by ourselves For example, the world of mass communications has indeed made people be easily connected and promoted an encounter of cultures that otherwise can have little or no dialogue at all In this way, AI has made us more human and we can so give flesh to people who were not visible to us before A prominent aspect of the discussions between mathematicians, engineers, designers and philosophers is acknowledging that AI has grown in such a way that illustrates us for having new ideas that are informing ethics, aesthetics, art, experimental sciences such as chemistry and metaphysics, medicine and even philosophy vii viii Preface Philosophy of AI is then an important activity within the disciplines of AI: Engineers need the motivation to strive for better and deeper understanding of the capacities of managing information A philosophical dose of thought helps the engineer to understand that her or his contribution is absolutely valuable and crucial to the growth of humanity, and that technical advances are always a step forward in developing our humanity However, the philosophical dose of the engineer also helps her or him to acknowledge that there are ethical responsibilities to humanity, to truth and to the advancement of AI The drive that has led us to where we are has been an unrestricted desire for knowledge much more than economical rewards, for example This Springer edited collection at hand that came from the International Conference on Artificial Intelligence and Applied Mathematics in Engineering (ICAIAME 2019) is a great example of a sincere desire to have a dialogue guided by truth and openness, and the human exchange of ideas has been a model to follow for other disciplines, since the mathematicians and engineers are less prone to be affected by other egoistic interests but by a thirst of knowledge and inquiry All the contributions connect in fascinating and innovative ways Paniel Reyes Cárdenas Organization International Conference on Artificial Intelligence and Applied Mathematics in Engineering 2019 Web: http://www.icaiame.com E-Mail: info@icaiame.com Briefly About International Conference on Artificial Intelligence and Applied Mathematics in Engineering (ICAIAME 2019) held within April 20-21-22, 2019, at the Antalya, Manavgat (Turkey), which is the pearl of the Mediterranean, heaven corner of Turkey and the fourth most visited city in the world The main theme of the conference, which was held at Bella Resort & Spa with international participations along a three-day period, is solutions of artificial intelligence and applied mathematics in engineering applications The languages of the ICAIAME 2019 are English and Turkish Scope/Topics Conference Scope/Topics (as not limited to): In Engineering Problems: • • • • • • • • Machine Learning Applications Deep Learning Applications Intelligent Optimization Solutions Robotics/Soft Robotics and Control Applications Hybrid System-Based Solutions Algorithm Design for Intelligent Solutions Image/Signal Processing Supported Intelligent Solutions Data Processing-Oriented Intelligent Solutions ix Parametrical Analysis of a New Design Outer-Rotor Line 1029 performance, parameter estimation and high efficiency standards [10] In another study shows that LSSM is not only under full load, but also the performance of it during acceleration is also required for future testing standards [11] In many studies, the most common methods described for inner rotor LSSMs are adding magnet to induction motor or a complete motor design Parameters of the motor designing with each method have been optimized by analytical and FE analysis in terms of efficiency, initial performance, synchronization capability, torque and load capacity [12–26] Although many studies have been conducted on inner rotor LSSMs in the literature, no study has been found for outer rotor LSSM (OR-LSSM) that can be used in-wheel motor systems of electric vehicles In this study, a new OR-LSSM motor is designed and the effect of Outer Rotor parameters on the motor characteristics have been investigated with regression analysis (as a predictive modeling technique) that investigates the relationship between dependent and independent variables General Rotor Topologies and Proposed Model In the literature, design of LSSMs are mostly started by asynchronous motor design and then permanent magnets are added to rotor structure appropriately In this paper, the design processes are divided into three sections: calculation of initial parameters, cage design and PM placement with parametric analysis by using FEM software as shown in Fig Fig Motor design flow chart 1030 M Tümbek et al There are constraints for the motor outer geometry because of in-wheel systems The outer diameter of the motor is 300 mm and the width is 40 mm, in this study for considered system Initial parameters of the motor are given in Table Also, proposed motor model is given in Fig Table Motor parameters Motor design parameters Stator Outer diameter Stack length Number of slots Conductor per slot Rotor Outer diameter Inner diameter Core length Inertia Magnet Thickness Width 248 mm 40 mm 24 44 300 mm 250 mm 40 mm 0.02807 kgm2 20 mm mm In this study, 3D geometry and motor structure of the inner rotor motor used for electric vehicle is shown in Fig The proposed rotor geometry contains short circuit bars between each magnet Also, 24 pieces of magnets are used in groups of so as to face each stator pole Fig Proposed model geometry (a) 3D outer model (b) Inner model (c) 2D geometry (d) Cross section of the motor Parametrical Analysis of a New Design Outer-Rotor Line 1031 Parametric Analysis In 2D analysis of the Ansys Ansoft Maxwell, the motor geometry parameters can be defined as the first design variables In this way, analysis can be made by assigning different possible values for each parameter Also, the software allows to perform regression with sensitivity analysis The purpose of regression is to understand the relationship between parameters In other words, regression analysis helps to understand how to change dependent variables according to independent variables In this paper, the variables taken into consideration are bar diameter, bar distance from airgap, airgap length, magnet dimensions and distance magnet from airgap as shown in Fig 3, including skew angle of the bar Fig Rotor structures and design parameters 3.1 Airgap In general, because of the strong magnetism in the PM motor, the air gap of a LSSM is slightly longer than a similar-power induction motor The small air gap length reduces vibration, noise, and stray losses, while increasing efficiency However, when the air gap is too long, the flux leakages increase and cause a lower loading capacity In a study ranging in air-gap length between 0.3–0.5 mm, it is presented that the length of the airgap is not have an effect on Back-EMF Also, starting performance for the motor with a 0.5 mm air gap length is found to be poor [27] In the analysis for the air gap length with the range between mm and mm, nonlinear relationship with efficiency is investigated, as shown in Fig 3.2 Cage Bars Cage bars in the rotor of the LSSM produce starting torque required for the acceleration of the motor The design of the rotor bars shape is closely related to the initial performance, efficiency and synchronization of the motor In an analytical study, the 1032 M Tümbek et al Fig Effect of air gap length on motor efficiency effects on starting performance are investigated using different bar design By increasing the bar width and material conductivity, the initial impedance is reduced, thereby increasing the starting current and losses together with reducing start-up torque [28] In another study on asynchronous starting torque, it has been investigated that the rotor cage bars have a great effect on the moment of acceleration from zero speed to synchronous speed [29] Also, rotor bars have an effect of magnetizing NdFeB When the dimensions of the rotor bars are reduced, the resistance and magnetic flux density are increased, and the eddy current decreases However, smaller rotor bar dimensions adversely affect the motor characteristics [30] In the study on slot width and length, it has been shown that wide and long slots cause high flux density in the air gap This is also required for high torque at synchronous speed However, the starting torque and the synchronous torque are differently affected by the bar dimensions, so there is a limitation on the slot dimensions [31] It is reported that a successful synchronization is achieved for higher inertia values of the load system when deep bar slots are selected [32] In addition, it has been investigated in irregular bar structures to reduce current harmonics, torque fluctuation and Back-EMF [33] In this work supported by an analytical solution, effect of bar diameter on motor efficiency and effect of bar distance from airgap on motor efficiency are given in Figs and 6, respectively Fig Effect of bar diameter on motor efficiency Parametrical Analysis of a New Design Outer-Rotor Line 1033 Fig Effect of bar distance from airgap on motor efficiency 3.3 Magnets In the literature, different rotor structures have been formed with different magnet shapes and placement for inner rotor LSSMs (Fig 7) The Ansys Maxwell Program has templates for conventional inner rotor LSSM motors With the Rmxprt module included in this software, it is possible to carry out analytical analysis in a short time Some material information for each part in the motor model can be defined In a study conducted in the literature, NdFeB magnets provide more performance than SmCo-25 magnets In addition, asymmetrically distributed NdFeB magnets improve performance [34] Fig Different rotor shapes for inner LSSMs 1034 M Tümbek et al In the FEM analysis using the motor geometry in Fig 3, the proper magnet dimensions and placement are investigated In terms of efficiency, it is concluded that the thickness of the magnet relative to the rotor and magnet width are more effective (Figs 8, and 10) On the other hand, in order to determine the dynamic characteristics of the motor, it must be analyzed in the time domain Magnet geometry and dimensions are known to have an effect on synchronization problems and braking torque [35] Fig Effect of magnet distance from airgap on motor efficiency Fig Effect of magnet thickness on motor efficiency Fig 10 Effect of magnet length on motor efficiency Parametrical Analysis of a New Design Outer-Rotor Line 3.4 1035 Skew Angle The cogging torque effect, which causes a mechanical stress and the motor torque ripple, can be reduced by skewing the rotor as in induction motors In literature, it is shown that the torque ripples in LSSMs also reduced with skewed rotor by using 2D analysis [36] In addition, it is seen that it provides an increase in synchronization performance and a 67% reduction in cogging torque [37] In this study based on skew angle parameter, the cogging torque is reduced as shown Fig 11 Also, although the torque ripples in the steady state is minimized, its behavior at acceleration time has slowed down, as shown in Figs 12 and 13 Fig 11 Effect of skew angle on cogging torque Fig 12 Torque at time domain 1036 M Tümbek et al Fig 13 Motor speed with different skew angle Conclusions In this paper, sensitivity analysis for the parameters of OR-LSSMs for in-wheel systems is investigated and the results are given It is ensured that thanks to this analysis, if the design parameters are selected appropriately, the effects of the parameters on the motor characteristics can be observed comprehensively And then, prototype implementation can be done easy and most truly Also, in this study, the cogging torque effect on the synchronization, which also leads to a reduction in the torque quality of the motor, is presented together with solution of the problem References Leitman, S., Brant, B.: Build Your Own Electric Vehicle, 2nd edn McGraw-Hill Education, London (2008) Chau, K.T.: Electric Vehicle Machines and Drives: Design, Analysis and Application, 1st edn Wiley, New York (2015) Ehsani, M., Gao, Y., Gay, S., Emadi, A.: Modern Electric, Modern Hybrid, and Fuel Cell Vehicles Taylor & Francis Group, New York (2005) Chau, K.T., Li, W.L.: Overview of electric machines for electric and hybrid vehicles Int J Veh Des.: J Veh Eng Automot Technol Compon 64(1), 46–71 (2014) Chan, C.C., Chau, K.T., Jiang, J.Z., Xia, W.A.X.W., 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Department of Statistics, Să uleyman Demirel University, 32260 Isparta, Turkey ulasyamanci@sdu.edu.tr Department of Mathematics, Să uleyman Demirel University, 32260 Isparta, Turkey gurdalmehmet@sdu.edu.tr Abstract In 1974, Krivonosov defined the concept of localized sequence that is defined as a generalization of Cauchy sequence in metric spaces In this work, by using the concept of ideal, the statistically localized sequences are defined and some basic properties of I-statistically localized sequences are given Also, it is shown that a sequence is Istatistically Cauchy iff its statistical barrier is equal to zero Keywords: Statistical convergence spaces · Localized sequence · Ideal convergence · 2-normed 2000 Mathematics Subject Classification Primary 40A35 Introduction and Preliminaries Statistical convergence introduced by Fast [2] and Steinhaus [17] has many applications in different areas Later on, this concept was reintroduced by Schoenberg in his own study [16] The concept of statistical convergence is defined depending upon the natural density of the set A ⊆ N and the natural density of A is given |An | , where An = {a ∈ A : a ≤ n} and |An | denotes the by δ (A) := limn→∞ n number of elements in An Utilizing above information, we say that a sequence (xk )k∈N is statistically convergent to x provided that δ ({k ∈ N : |xk − x| ≥ ε}) = for every ε > In this case, we denote it st-lim xk = x For more detail informations about statistical convergent, see, in [6,20,23] On the other hand, I-convergence in a metric space was introduced by Kostyrko et al [7] and its definition is depending upon the definition of an ideal I in N c Springer Nature Switzerland AG 2020 D J Hemanth and U Kose (Eds.): ICAIAME 2019, LNDECT 43, pp 10391046, 2020 https://doi.org/10.1007/978-3-030-36178-5_92 1040 U Yamancand M Gă urdal A family I ⊂ 2N is called an ideal if the following properties are held: (i) ∅ ∈ / I; (ii) P ∪ R ∈ I for every P, R ∈ I; (iii) R ∈ I for every P ∈ I and R ⊂ P A non-empty family of sets F ⊂ 2N is a filter iff ∅ ∈ / F, P ∩ R ∈ F for every P, R ∈ F, and R ∈ F for every P ∈ F and every R ⊃ P An ideal I is called a non-trivial if I = ∅ and X ∈ / I The I ⊂ 2X is a non-trivial ideal iff F = F (I) = {N\P : P ∈ I} is a filter on X A non-trivial ideal I ⊂ 2X is called an admissible iff I ⊃ {{x} : x ∈ X} Recall that an admissible ideal I ⊂ 2N holds the property (AP ) if there is a family {Rn }n∈N such that (Pk \Rn ) ∪ (Rk \Pk ) for all k ∈ N and a limit set R= ∞ k=1 Rk ∈ I for every family {Pn }n∈N with Pn ∩ Pk = ∅ (n = k) , Pn ∈ I (n ∈ N) [7] Definition ([7]) Let {xn }n∈N be a sequence of real numbers It is called an I-convergent to K if the set A (ε) = {n ∈ N : |xn − K| ≥ ε} ∈ I for each ε > For more information about I-convergent, see the references in [9–11] The definition of 2-normed space was given by Gahler [3] After this definition, many authors studied statistical convergence, ideal convergence, ideal Cauchy sequence, star ideal convergent and star ideal Cauchy sequence on this space (see [4,5,15]) Depending upon the I-convergence and statistical convergence, the ideal statistical convergence was introduced in [1] Later on, the concepts of ideal statistically convergent and ideal statistically Cauchy were given in 2-normed spaces and important consequences were obtained in [18,19] Let (xk ) be a sequence in 2-normed space (X, , ): • It is an ideal statistically convergent to μ, if the set n∈N: |{k ≤ n : xk − μ, z ≥ ε}| ≥ δ n ∈I for each ε > 0, δ > and nonzero z in X In such case we can write it I-st- lim xk − μ, z = or I-st- lim xk , z = μ, z (see [18]) k→∞ k→∞ • It is an ideal statistically Cauchy sequence in X if there is a number N such that = 0, n ∈ N : |{k ≤ n : xk − xN , z ≥ ε}| ≥ δ δI n for every ε > 0, δ > and every nonzero z ∈ X (see [18]) • It is an I ∗ -statistically convergent to μ ∈ X iff there is a set B ∈ F (I) such that st- lim xmk − μ, z = and B = {b1 < b2 < < bk < } ⊂ N k→∞ I-Statistically Localized Sequences • It is an I ∗ -statistically Cauchy sequence iff there is a set B = < < bk such that st- lim k,p→∞ 1041 b1 < b2 xmk − xmp , z = A sequence (xn ) in a metric space X is said to be localized in some subset M ⊂ X if the number sequence d (xn , x) converges for all x ∈ M (see [8]) This definition has been extended to statistical localized and I-localized in metric space [12,13] and 2-normed spaces [21,22], and they obtained interested results about this concept In this paper, by using the concept of ideal, the statistically localized sequences are defined and some basic properties of I-statistically localized sequences are given Also, it is shown that a sequence is ideal statistically Cauchy iff its statistical barrier is equal to zero Main Results Our main definitions and notations are as following: Definition Let (xn )n∈N be a sequence in 2-normed space (X, , ) (a) It is called as the I-statistically localized in the subset M ⊂ X iff n∈N: |{k ≤ n : xn − x, z ≥ ε}| ≥ δ n ∈I exists for every x, z ∈ M , that is, the real number sequence xn − x, z is I-statistically convergent (b) The maximal set on which it is I-statistically localized is said to be Istatistically localor of (xn ) and it is denoted by locIst (xn ) (c) It is said to be I-statistically localized everywhere if (xn ) is I-statistically localor of (xn ) coincides with X (d) It is called as the I-statistically localized in itself if / locIst (xn )} ⊂ I {n ∈ N : xn ∈ From above definition, if (xn ) is an ideal statistically Cauchy sequence, then it is ideal statistically localized everywhere Actually, owing to |{n ≤ k : | xn − x, z − xn0 − x, z | k ε}| |{n ≤ k : xn − xn0 , z k we have |{n ≤ k : | xn − x, z − xn0 − x, z | k k ∈ N : |{n ≤ k : xn − xn0 , z ε}| ≥ δ k k∈N: ⊂ ε}| ≥ δ ε}| 1042 U Yamancand M Gă urdal So, the sequence is ideal statistically localized if it is ideal statistically Cauchy sequence Also, we are able to say that each ideal statistically convergence sequence is ideal statistically localized We know that if I is an admissible ideal, then every statistically localized sequence in 2-normed space (X, , ) is ideal statistically localized sequence in (X, , ) Definition We say the sequence (xn ) to be I ∗ -statistically localized in (X, , ) iff the number sequence xn − x, z is I ∗ -statistically convergent for each x, z ∈ X From above definition, one sees that every I ∗ -statistically Cauchy sequence or I -statistically convergent in 2-normed space (X, , ) is I ∗ -statistically localized in (X, , ) Note that for admissible ideal, I ∗ -statistically convergence and I ∗ statistically Cauchy criteria imply I-statistically convergence and I-statistically Cauchy criteria, respectively ∗ Lemma Let (xn ) be a sequence in (X, , ) If it is I ∗ -statistically localized on the set M ⊂ X, then (xn ) is I-statistically localized on the set M and ∗ (xn ) ⊂ locI (xn ) locIst st Proof Suppose that (xn ) is I ∗ -statistically localized on M Then, there is a set P ∈ I such that ε}| lim |{j ∈ N : xj − x, z j→∞ j exists for each x, z ∈ M and P C = N\P = {p1 < p2 < < pj } Then, the sequence xn − x, z is an I ∗ -statistically Cauchy sequence, which means that n∈N: |{k ≤ n : xk − xN , z ≥ ε}| ≥ δ n ∈ I Therefore, the number sequence xn − x, z is I-statistically convergent, which gives that (xn ) is I-statistically localized on the set M Now, we give our basic consequences about I-statistically localized sequences Proposition Let (xn ) be an ideal statistically localized sequence in (X, , ) Then it is ideal statistically bounded Proof Suppose that (xn ) is ideal statistically localized Then, the number sequence xn − x, z is ideal statistically convergent for some x, z ∈ X This means that k ∈ N : k1 |{n ≤ k : xn − x, z > K}| > δ ∈ I for some K > and δ > As a result, the sequence (xn ) is ideal statistically bounded Proposition Let I be an admissible ideal satisfying the property (AP ) and M = locIst (xn ) Also, a point y ∈ X be such that there exists x ∈ M for any ε > 0, δ > and every nonzero z ∈ M such that (1) k∈N: Then y ∈ M |{n ≤ k : | x − xn , z − y − xn , z | k ε}| > δ ∈ I I-Statistically Localized Sequences 1043 Proof To show that the number sequence xn − y, z is an I-statistically Cauchy sequence is enough Let be ε > and x ∈ M = locIst (xn ) is a point satisfying the property (1) From the (AP ) property of I, we get |{n ≤ k : | x − xkn , z − y − xkn , z | k ε}| → and |{(n, m) : | xkn − x, z − xkm − x, z | ε, n, m ≤ k}| → k as m, n → ∞, where K = {k1 < k2 < < kn < } ∈ F (I) Therefore, there is n0 ∈ N for any ε > 0, δ > and every nonzero z ∈ M such that k (2) (3) k n ≤ k : | x − xkn , z − y − xkn , z | (n, m) : | x − xkn , z − x − xkm , z | ε < ε , n, m ≤ k δ < δ for all n ≥ n0 , m ≥ m0 Since |{(n, m) : | y − xkn , z − y − xkm , z | ε, n, m ≤ k}| k ≤ |{n ≤ k : | y − xkn , z − x − xkn , z | ε}| k + |{n ≤ k : | x − xkn , z − x − xkm , z | ε}| k + |{n ≤ k : | x − xkm , z − y − xkn , z | ε}| , k we obtain by using (2) and (3) together with above inequality |{(n, m) : | y − xkn , z − y − xkm , z | k ε, n, m ≤ k}| < δ for all n ≥ n0 , m ≥ n0 So, |{(n, m) : | y − xkn , z − y − xkm , z | k ε, n, m ≤ k}| → as m, n → ∞ for the K = (kn ) ⊂ N and K ∈ F (I) Hence y − xn , z is an I-statistically Cauchy sequence, which finishes the proof Definition ([14]) Let a is a point in (X, , ) It is called a limit point of a set M in X if for an arbitrary Σ = {(b1 , ε1 ) , , (bn , εn )}, there is a point aΣ ∈ M , aΣ = a such that aΣ ∈ WΣ (a) Moreover, a subset L ⊂ K is called a closed subset of K if L contains every its limit point If L0 is the set of all points of a subset L ⊂ K, then the set L = L ∪ L0 is called the closure of the set L ... Springerlink and DBLP ** More information about this series at http://www.springer.com/series/15362 D Jude Hemanth Utku Kose • Editors Artificial Intelligence and Applied Mathematics in Engineering. .. and inquiry All the contributions connect in fascinating and innovative ways Paniel Reyes Cárdenas Organization International Conference on Artificial Intelligence and Applied Mathematics in Engineering. .. Mathematics in Engineering Problems Proceedings of the International Conference on Artificial Intelligence and Applied Mathematics in Engineering (ICAIAME 2019) 123 Editors D Jude Hemanth Department of