On geometry of fixed figures via φ−interpolative contractions and application of activation functions in neural networks and machine learning models
In this work, we introduce novel postulates to establish fixed figure theorems with a focus on their extension to the domain of mvb−metric spaces. Consequently, we define conditions ensuring the existence and uniqueness of fixed circles, fixed ellipses, fixed Apollonius circles, fixed Cassini curves...
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| Format: | Article |
| Language: | English |
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Elsevier
2025-01-01
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| Series: | Ain Shams Engineering Journal |
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| Online Access: | http://www.sciencedirect.com/science/article/pii/S209044792400563X |
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| _version_ | 1841526253226557440 |
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| author | Khairul Habib Alam Yumnam Rohen Anita Tomar Mohammad Sajid |
| author_facet | Khairul Habib Alam Yumnam Rohen Anita Tomar Mohammad Sajid |
| author_sort | Khairul Habib Alam |
| collection | DOAJ |
| description | In this work, we introduce novel postulates to establish fixed figure theorems with a focus on their extension to the domain of mvb−metric spaces. Consequently, we define conditions ensuring the existence and uniqueness of fixed circles, fixed ellipses, fixed Apollonius circles, fixed Cassini curves, fixed hyperbola, and so on for self mapping. We also partially address an open problem demonstrating that a JS−contraction possesses a fixed elliptic disc. This property extends to smaller discs and ellipses within a complete mvb−metric space. By challenging the conventional assumption of zero self-distance, we pave the way for more accurate mathematical models applicable to real-world scenarios. Consequently, our research contributes not only to a deeper comprehension of mathematical concepts but also to practical utility across various scientific domains. Our findings are supported by illustrative examples. Additionally, we explore the concept of fixed figures in the context of Rectified Linear Unit (ReLU), a widely-used activation function in neural networks and machine learning models. Our exploration of fixed figures in the context of Rectified Linear Units (ReLU) further deepens our understanding of nonlinear systems and their relationship to neural network behavior. |
| format | Article |
| id | doaj-art-211df58d1c7d4f1aaf601c83be3885b3 |
| institution | Kabale University |
| issn | 2090-4479 |
| language | English |
| publishDate | 2025-01-01 |
| publisher | Elsevier |
| record_format | Article |
| series | Ain Shams Engineering Journal |
| spelling | doaj-art-211df58d1c7d4f1aaf601c83be3885b32025-01-17T04:49:19ZengElsevierAin Shams Engineering Journal2090-44792025-01-01161103182On geometry of fixed figures via φ−interpolative contractions and application of activation functions in neural networks and machine learning modelsKhairul Habib Alam0Yumnam Rohen1Anita Tomar2Mohammad Sajid3Department of Mathematics, National Institute of Technology Manipur, Langol, Imphal, 795004, Manipur, IndiaDepartment of Mathematics, National Institute of Technology Manipur, Langol, Imphal, 795004, Manipur, India; Department of Mathematics, Manipur University, Canchipur, Imphal, 795003, Manipur, IndiaPt. L. M. S. Campus, Sridev Suman Uttarakhand University, Rishikesh, 246201, Uttarakhand, IndiaDepartment of Mechanical Engineering, College of Engineering, Qassim University, Saudi Arabia; Corresponding author.In this work, we introduce novel postulates to establish fixed figure theorems with a focus on their extension to the domain of mvb−metric spaces. Consequently, we define conditions ensuring the existence and uniqueness of fixed circles, fixed ellipses, fixed Apollonius circles, fixed Cassini curves, fixed hyperbola, and so on for self mapping. We also partially address an open problem demonstrating that a JS−contraction possesses a fixed elliptic disc. This property extends to smaller discs and ellipses within a complete mvb−metric space. By challenging the conventional assumption of zero self-distance, we pave the way for more accurate mathematical models applicable to real-world scenarios. Consequently, our research contributes not only to a deeper comprehension of mathematical concepts but also to practical utility across various scientific domains. Our findings are supported by illustrative examples. Additionally, we explore the concept of fixed figures in the context of Rectified Linear Unit (ReLU), a widely-used activation function in neural networks and machine learning models. Our exploration of fixed figures in the context of Rectified Linear Units (ReLU) further deepens our understanding of nonlinear systems and their relationship to neural network behavior.http://www.sciencedirect.com/science/article/pii/S209044792400563XMetric spaceFixed Apollonius circlesFixed Cassini curvesFixed circlesFixed ellipsesFixed hyperbola |
| spellingShingle | Khairul Habib Alam Yumnam Rohen Anita Tomar Mohammad Sajid On geometry of fixed figures via φ−interpolative contractions and application of activation functions in neural networks and machine learning models Ain Shams Engineering Journal Metric space Fixed Apollonius circles Fixed Cassini curves Fixed circles Fixed ellipses Fixed hyperbola |
| title | On geometry of fixed figures via φ−interpolative contractions and application of activation functions in neural networks and machine learning models |
| title_full | On geometry of fixed figures via φ−interpolative contractions and application of activation functions in neural networks and machine learning models |
| title_fullStr | On geometry of fixed figures via φ−interpolative contractions and application of activation functions in neural networks and machine learning models |
| title_full_unstemmed | On geometry of fixed figures via φ−interpolative contractions and application of activation functions in neural networks and machine learning models |
| title_short | On geometry of fixed figures via φ−interpolative contractions and application of activation functions in neural networks and machine learning models |
| title_sort | on geometry of fixed figures via φ interpolative contractions and application of activation functions in neural networks and machine learning models |
| topic | Metric space Fixed Apollonius circles Fixed Cassini curves Fixed circles Fixed ellipses Fixed hyperbola |
| url | http://www.sciencedirect.com/science/article/pii/S209044792400563X |
| work_keys_str_mv | AT khairulhabibalam ongeometryoffixedfiguresviaphinterpolativecontractionsandapplicationofactivationfunctionsinneuralnetworksandmachinelearningmodels AT yumnamrohen ongeometryoffixedfiguresviaphinterpolativecontractionsandapplicationofactivationfunctionsinneuralnetworksandmachinelearningmodels AT anitatomar ongeometryoffixedfiguresviaphinterpolativecontractionsandapplicationofactivationfunctionsinneuralnetworksandmachinelearningmodels AT mohammadsajid ongeometryoffixedfiguresviaphinterpolativecontractionsandapplicationofactivationfunctionsinneuralnetworksandmachinelearningmodels |