Newton’s divided gH-difference interpolation formula with unequal spacing for interval-valued function using Hukuhara difference

2021 ◽  
pp. 2250093
Author(s):  
Bapin Mondal ◽  
Md Sadikur Rahman
Keyword(s):  
Significant Role ◽  
Unknown Function ◽  
Interpolation Formula ◽  
Interval Uncertainty ◽  
Hukuhara Difference ◽  
Numerical Examples ◽  
Generalized Hukuhara Difference ◽  
Interval Valued Function ◽  
Definition Of ◽  
Interval Valued

Interval interpolation formulae play a significant role to find the value of an unknown function at some points under interval uncertainty. The objective of this paper is to establish Newton’s divided interpolation formula for interval-valued functions using generalized Hukuhara difference of intervals. For this purpose, arithmetic of intervals, Hukuhara difference and its some properties and concept of interval-valued function have been discussed briefly. Using Hukuhara difference of intervals, the definition of Newton’s divided gH-difference for interval-valued function has been introduced. Then Newton’s divided gH-differences interpolation formula has been derived. Finally, with the help of some numerical examples, the proposed interpolation formula has been illustrated.

scholarly journals Some Inequalities for LR-$$\left({h}_{1}, {h}_{2}\right)$$-Convex Interval-Valued Functions by Means of Pseudo Order Relation

2021 ◽  
Vol 14 (1) ◽  
Author(s):  
Muhammad Bilal Khan ◽  
Muhammad Aslam Noor ◽  
Khalida Inayat Noor ◽  
Kottakkaran Sooppy Nisar ◽  
Khadiga Ahmed Ismail ◽  
...  
Keyword(s):  
Applied Mathematics ◽  
Critical Role ◽  
Order Relation ◽  
Strong Relationship ◽  
New Class ◽  
Starting Point ◽  
Interval Valued Function ◽  
Fejér Inequality ◽  
Definition Of ◽  
Interval Valued

AbstractIn both theoretical and applied mathematics fields, integral inequalities play a critical role. Due to the behavior of the definition of convexity, both concepts convexity and integral inequality depend on each other. Therefore, the relationship between convexity and symmetry is strong. Whichever one we work on, we introduced the new class of generalized convex function is known as LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -convex interval-valued function (LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -IVF) by means of pseudo order relation. Then, we established its strong relationship between Hermite–Hadamard inequality (HH-inequality)) and their variant forms. Besides, we derive the Hermite–Hadamard–Fejér inequality (HH–Fejér inequality)) for LR-$$\left({h}_{1}, {h}_{2}\right)$$ h 1 , h 2 -convex interval-valued functions. Several exceptional cases are also obtained which can be viewed as its applications of this new concept of convexity. Useful examples are given that verify the validity of the theory established in this research. This paper’s concepts and techniques may be the starting point for further research in this field.

scholarly journals Generalized Hukuhara-Clarke Derivative of Interval-valued Functions and its Properties

2021 ◽  
Author(s):  
Ram Surat Chauhan ◽  
Debdas Ghosh ◽  
Jaroslav Ramik ◽  
Amit Kumar Debnath
Keyword(s):  
Directional Derivative ◽  
Numerical Examples ◽  
Lipschitz Continuous ◽  
Interval Valued Function ◽  
Clarke Derivative ◽  
Interval Valued

Abstract This paper is devoted to the study of gH-Clarke derivative for interval-valued functions. To develop the properties of gH-Clarke derivative, the concepts of limit superior, limit inferior, and sublinear interval-valued functions are studied in the sequel. It is proved that the upper gH-Clarke derivative of a gH-Lipschitz continuous interval- valued function (IVF) always exists. Further, it is found that for a convex and gH-Lipschitz IVF, the upper gH-Clarke derivative coincides with the gH-directional derivative. It is observed that the upper gH-Clarke derivative is a sublinear IVF. Several numerical examples are provided to support the study.

A novel difference and derivative of linearly correlated fuzzy number-valued functions

2022 ◽  
pp. 1-17
Author(s):  
Yonghong Shen
Keyword(s):  
Fuzzy Number ◽  
Fuzzy Numbers ◽  
Hukuhara Difference ◽  
Interval Numbers ◽  
Hukuhara Derivative ◽  
Generalized Hukuhara Derivative ◽  
Generalized Hukuhara Difference ◽  
Interval Valued

In the present paper, the notion of the linearly correlated difference for linearly correlated fuzzy numbers is introduced. Especially, the linearly correlated difference and the generalized Hukuhara difference are coincident for interval numbers or even symmetric fuzzy numbers. Accordingly, an appropriate metric is induced by using the norm and the linearly correlated difference in the set of linearly correlated fuzzy numbers. Based on the symmetry of the basic fuzzy number, the linearly correlated derivative is proposed by the linearly correlated difference of linearly correlated fuzzy number-valued functions. In both non-symmetric and symmetric cases, the equivalent characterizations of the linearly correlated differentiability of a linearly correlated fuzzy number-valued function are established, respectively. Moreover, it is shown that the linearly correlated derivative is consistent with the generalized Hukuhara derivative for interval-valued functions.

Interval SVM-Based Classification Algorithm Using the Uncertainty Trick

2017 ◽  
Vol 26 (04) ◽  
pp. 1750014 ◽  
Author(s):  
Lev V. Utkin ◽  
Yulia A. Zhuk
Keyword(s):  
Binary Classification ◽  
Interval Uncertainty ◽  
Training Data ◽  
Real Interval ◽  
Class Label ◽  
Numerical Examples ◽  
Probabilistic Uncertainty ◽  
Imprecise Dirichlet Model ◽  
Dirichlet Model ◽  
Interval Valued

A new robust SVM-based algorithm of the binary classification is proposed. It is based on the so-called uncertainty trick when training data with the interval uncertainty are transformed to training data with the weight or probabilistic uncertainty. Every interval is replaced by a set of training points with the same class label such that every point inside the interval has an unknown weight from a predefined set of weights. The robust strategy dealing with the upper bound of the interval-valued expected risk produced by a set of weights is used in the SVM. An extension of the algorithm based on using the imprecise Dirichlet model is proposed for its additional robustification. Numerical examples with synthetic and real interval-valued training data illustrate the proposed algorithm and its extension.

Generalized interval-valued hesitant intuitionistic fuzzy soft sets

2021 ◽  
pp. 1-12
Author(s):  
Admi Nazra ◽  
Yudiantri Asdi ◽  
Sisri Wahyuni ◽  
Hafizah Ramadhani ◽  
Zulvera
Keyword(s):  
Hesitant Fuzzy Set ◽  
Fuzzy Soft Set ◽  
Soft Sets ◽  
Binary Operations ◽  
Intuitionistic Fuzzy ◽  
Fuzzy Soft Sets ◽  
Comparison Table ◽  
Definition Of ◽  
Interval Valued ◽  
Intuitionistic Fuzzy Soft Sets

This paper aims to extend the Interval-valued Intuitionistic Hesitant Fuzzy Set to a Generalized Interval-valued Hesitant Intuitionistic Fuzzy Soft Set (GIVHIFSS). Definition of a GIVHIFSS and some of their operations are defined, and some of their properties are studied. In these GIVHIFSSs, the authors have defined complement, null, and absolute. Soft binary operations like operations union, intersection, a subset are also defined. Here is also verified De Morgan’s laws and the algebraic structure of GIVHIFSSs. Finally, by using the comparison table, a different approach to GIVHIFSS based decision-making is presented.

scholarly journals Hermite–Hadamard-type inequalities for the interval-valued approximately h-convex functions via generalized fractional integrals

2020 ◽  
Vol 2020 (1) ◽  
Author(s):  
Dafang Zhao ◽  
Muhammad Aamir Ali ◽  
Artion Kashuri ◽  
Hüseyin Budak ◽  
Mehmet Zeki Sarikaya
Keyword(s):  
Convex Functions ◽  
Fractional Integrals ◽  
Special Cases ◽  
Definition Of ◽  
The Given ◽  
Class Of Functions ◽  
Interval Valued ◽  
New Inequalities

Abstract In this paper, we present a new definition of interval-valued convex functions depending on the given function which is called “interval-valued approximately h-convex functions”. We establish some inequalities of Hermite–Hadamard type for a newly defined class of functions by using generalized fractional integrals. Our new inequalities are the extensions of previously obtained results like (D.F. Zhao et al. in J. Inequal. Appl. 2018(1):302, 2018 and H. Budak et al. in Proc. Am. Math. Soc., 2019). We also discussed some special cases from our main results.

Edges of Regression and Limit Normal Point of Conjugate Surfaces1

1998 ◽  
Vol 122 (4) ◽  
pp. 419-425 ◽  
Author(s):  
Ningxin Chen
Keyword(s):  
Differential Geometry ◽  
Contact Boundary ◽  
Boundary Line ◽  
Tangent Point ◽  
Common Tangent ◽  
Numerical Examples ◽  
Envelope Surface ◽  
The Common ◽  
Conjugate Surfaces ◽  
Definition Of

The presented paper utilizes the basic theory of the envelope surface in differential geometry to investigate the undercutting line, the contact boundary line and the limit normal point of conjugate surfaces in gearing. It is proved that (1) the edges of regression of the envelope surfaces are the undercutting line and the contact boundary line in theory of gearing respectively, and (2) the limit normal point is the common tangent point of the two edges of regression of the conjugate surfaces. New equations for the undercutting line, the contact boundary line and the limit normal point of the conjugate surfaces are developed based on the definition of the edges of regression. Numerical examples are taken for illustration of the above-mentioned concepts and equations. [S1050-0472(00)00104-5]

scholarly journals Generalized Euler-Lagrange Equations for Fuzzy Fractional Variational Problems under gH-Atangana-Baleanu Differentiability

2018 ◽  
Vol 2018 ◽  
pp. 1-15 ◽  
Author(s):  
Jianke Zhang ◽  
Gaofeng Wang ◽  
Xiaobin Zhi ◽  
Chang Zhou
Keyword(s):  
Fractional Derivative ◽  
Lagrange Function ◽  
Variational Problems ◽  
Sufficient Optimality Conditions ◽  
Lagrange Equations ◽  
Hukuhara Difference ◽  
Generalized Hukuhara Difference ◽  
Fractional Variational Problems ◽  
Fractional Calculus Of Variations ◽  
Necessary And Sufficient

We study in this paper the Atangana-Baleanu fractional derivative of fuzzy functions based on the generalized Hukuhara difference. Under the condition of gH-Atangana-Baleanu fractional differentiability, we prove the generalized necessary and sufficient optimality conditions for problems of the fuzzy fractional calculus of variations with a Lagrange function. The new kernel of gH-Atangana-Baleanu fractional derivative has no singularity and no locality, which was not precisely illustrated in the previous definitions.

Approach Integrated of Science, Technology, Engineering, and Mathematics – STEM

2021 ◽  
Vol 4 (4) ◽  
pp. 99-136
Author(s):  
Ibrahiem Mohammed Abdullah ◽  
Keyword(s):  
Mathematics Education ◽  
Significant Role ◽  
Arab World ◽  
Research Paper ◽  
Educational Activities ◽  
Science Technology ◽  
Mathematics Curricula ◽  
Integrated Approaches ◽  
And Mathematics ◽  
Definition Of

The research paper aims to highlight the STEM approach as one of the modern integrated approaches in the field of mathematics education. STEM which means the integration of Science, Technology, Engineering, and Math has its significant role in the development of curricula in the Arab world generally and particularly in mathematics curricula. This paper addresses the definition of STEM, the justifications for its emergence and the causes for the attention it recently receives. Moreover, the paper sheds light on its objectives, content, related teaching strategies, educational activities, evaluation, characteristics, advantages and obstacles found in its application.

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