Applications of Complex Uncertain Sequences via Lacunary Almost Statistical Convergence
We explore the realm of uncertainty theory by investigating diverse notions of convergence and statistical convergence concerning complex uncertain sequences. Complex uncertain variables can be described as measurable functions mapping from an uncertainty space to the set of complex numbers. They ar...
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2025-07-01
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| author | Xiu-Liang Qiu Kuldip Raj Sanjeev Verma Samrati Gorka Shixiao Xiao Qing-Bo Cai |
| author_facet | Xiu-Liang Qiu Kuldip Raj Sanjeev Verma Samrati Gorka Shixiao Xiao Qing-Bo Cai |
| author_sort | Xiu-Liang Qiu |
| collection | DOAJ |
| description | We explore the realm of uncertainty theory by investigating diverse notions of convergence and statistical convergence concerning complex uncertain sequences. Complex uncertain variables can be described as measurable functions mapping from an uncertainty space to the set of complex numbers. They are employed to represent and model complex uncertain quantities. We introduce the concept of lacunary almost statistical convergence of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo><</mo><mi>α</mi><mo>≤</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> for complex uncertain sequences, examining various aspects of uncertainty such as distribution, mean, measure, uniformly almost sure convergence and almost sure convergence. Additionally, we establish connections between the constructed sequence spaces by providing illustrative instances. Importantly, lacunary almost statistical convergence provides a flexible framework for handling sequences with irregular behavior, which often arise in uncertain environments with imprecise data. This makes our approach particularly useful in practical fields such as engineering, data modeling and decision-making, where traditional deterministic methods are not always applicable. Our approach offers a more flexible and realistic framework for approximating functions in uncertain environments where classical convergence may not apply. Thus, this study contributes to approximation theory by extending its tools to settings involving imprecise or noisy data. |
| format | Article |
| id | doaj-art-59086ff87e3445b4977a5c88e22a0cb5 |
| institution | DOAJ |
| issn | 2075-1680 |
| language | English |
| publishDate | 2025-07-01 |
| publisher | MDPI AG |
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| series | Axioms |
| spelling | doaj-art-59086ff87e3445b4977a5c88e22a0cb52025-08-20T02:45:43ZengMDPI AGAxioms2075-16802025-07-0114752610.3390/axioms14070526Applications of Complex Uncertain Sequences via Lacunary Almost Statistical ConvergenceXiu-Liang Qiu0Kuldip Raj1Sanjeev Verma2Samrati Gorka3Shixiao Xiao4Qing-Bo Cai5Chengyi College, Jimei University, Xiamen 361021, ChinaSchool of Mathematics, Shri Mata Vaishno Devi University, Katra 182320, IndiaSchool of Mathematics, Shri Mata Vaishno Devi University, Katra 182320, IndiaSchool of Mathematics, Shri Mata Vaishno Devi University, Katra 182320, IndiaChengyi College, Jimei University, Xiamen 361021, ChinaFujian Provincial Key Laboratory of Data-Intensive Computing, Key Laboratory of Intelligent Computing and Information Processing, School of Mathematics and Computer Science, Quanzhou Normal University, Quanzhou 362000, ChinaWe explore the realm of uncertainty theory by investigating diverse notions of convergence and statistical convergence concerning complex uncertain sequences. Complex uncertain variables can be described as measurable functions mapping from an uncertainty space to the set of complex numbers. They are employed to represent and model complex uncertain quantities. We introduce the concept of lacunary almost statistical convergence of order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>α</mi></semantics></math></inline-formula><inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo><</mo><mi>α</mi><mo>≤</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></semantics></math></inline-formula> for complex uncertain sequences, examining various aspects of uncertainty such as distribution, mean, measure, uniformly almost sure convergence and almost sure convergence. Additionally, we establish connections between the constructed sequence spaces by providing illustrative instances. Importantly, lacunary almost statistical convergence provides a flexible framework for handling sequences with irregular behavior, which often arise in uncertain environments with imprecise data. This makes our approach particularly useful in practical fields such as engineering, data modeling and decision-making, where traditional deterministic methods are not always applicable. Our approach offers a more flexible and realistic framework for approximating functions in uncertain environments where classical convergence may not apply. Thus, this study contributes to approximation theory by extending its tools to settings involving imprecise or noisy data.https://www.mdpi.com/2075-1680/14/7/526lacunary sequencealmost convergencestatistical convergencecomplex uncertain sequence |
| spellingShingle | Xiu-Liang Qiu Kuldip Raj Sanjeev Verma Samrati Gorka Shixiao Xiao Qing-Bo Cai Applications of Complex Uncertain Sequences via Lacunary Almost Statistical Convergence Axioms lacunary sequence almost convergence statistical convergence complex uncertain sequence |
| title | Applications of Complex Uncertain Sequences via Lacunary Almost Statistical Convergence |
| title_full | Applications of Complex Uncertain Sequences via Lacunary Almost Statistical Convergence |
| title_fullStr | Applications of Complex Uncertain Sequences via Lacunary Almost Statistical Convergence |
| title_full_unstemmed | Applications of Complex Uncertain Sequences via Lacunary Almost Statistical Convergence |
| title_short | Applications of Complex Uncertain Sequences via Lacunary Almost Statistical Convergence |
| title_sort | applications of complex uncertain sequences via lacunary almost statistical convergence |
| topic | lacunary sequence almost convergence statistical convergence complex uncertain sequence |
| url | https://www.mdpi.com/2075-1680/14/7/526 |
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